In the May issue of Consumer Reports magazine, Northern California regional grocer Raley's ranked in the top 10 of grocery stores nationally, which makes the food chain's labor problems all the more poignant.
Raley's actions in pulling out of further talks portends a possible strike by the union, which has to either walk out or accept cutbacks in healthcare and other compensation. The UFCW accuses Raley's of playing a "reckless game of chicken," according to the Sac Bee, and says it's not buying the company's argument that it needs such deep union concessions to remain competitive.
While Wal-Mart ranks near the bottom of Consumer Reports' list of grocers, it ranks high in one category: grocery prices. Despite its poor ratings for service, cleanliness and selection – the other aspects of the magazine's ratings system – Wal-Mart has become the nation's leading grocery chain. In this day of troubling economic news, price trumps everything.
Raley's may well be depending on the poor economy to keep its UCFW members on the job.
"We do not expect the employees to vote for a strike," wrote Chief Executive Michael Teel in a memo to employees. He noted that the company is asking its union employees to accept the same deal currently offered to non-union workers.
The union is in a tough spot. Despite its posturing, it cannot help but see the handwriting on the wall as big box stores including Wal-Mart, Costco and Target put their marketing might into the grocery business. (Of this group, only Costco ranks as a top tier supermarket in the Consumer Reports ratings). Wal-Mart is on the brink of opening some grocery-only stores in small communities. These outlets will add convenience and service to Wal-Mart's already prodigious pricing prowess.
Among other high-ranking grocery chains is Trader Joe's, a non-union shop that features small stores, personal service and good prices. Such alternative shopping outlets have cut deeply into the business of Raley's and other major grocers such as Save-Mart and Safeway. Save-Mart is currently in negotiations with the UCFW, but has not reached the kind of stalemate that Raley's is experiencing.
As an alternative to an immediate strike, the union could temporarily accede to Raley's demands, but file a grievance with the National Labor Relations Board. But for the grocer, a strike presents some genuine dangers.
Suppliers and non-union workers might not cross the picket line; and customers may elect to shop at other grocery stores rather than put up with the hassle of striking workers. The risk for a chain store like Raley's that depends on customer loyalty is that its customers may stay away even after the strike is settled.


The media, bloggers in particular, love them some frozen yogurt. And you know what they (we) love even more? Frozen yogurt wars. Combine that with the current probiotic craze, and we have the makings of a media yo-gasm.
And DC will stand on the sidelines of this big city fad no longer, because it looks like we're poised for our very own fro-yo fight.
But what's that on the horizon? LA-export Pinkberry is supposedly on its way, and a Canadian fro-yo chain, Yogen Früz, plans to open 16 stores in the area.
TangySweet should have a hometown advantage since it's a locally owned, small business. And owner Aaron Gordon told DCist he may open a few more shops around town.
So what do you think? Bring it on? Or this trend's so already been broughten.
At least we'll have a new dinner party debate topic.
Mr. Yogato (same as Harvard Square's Berry Line) is coming next week too!
I live in NYC, where Pinkberry has fast become as painfully ubiquitous on every street corner as Starbucks, CVS, Citibank, etc. Watch out, DC!
Thanks for the tip on Mr. Yogato! Sounds like it's opening soon (June 28?), though I'm a facebook holdout so I can't read the info that's linked in the bit about the grand opening.


Horses have evolved to eat small frequent forage meals, spending over 14.5 hours grazing each day.
Many horse owners try to mimic a more natural feeding pattern by providing free access to hay. But this often results in obesity because the horses eat too much.
The University of Minnesota looked at the effects of hay net design on the rate and amount forage eaten by horses. They wanted to see if hay net design mimics more natural feeding without causing weight problems.
Horses had access to hay inside the nets for two 4-hour periods: 7:00 to 11:00 a.m. and 4:00 to 8:00 p.m. daily. Throughout the trial, grass hay was fed at 1 percent body weight twice daily.
A second study showed horses feeding from the medium-holed net took just over 5 hours to eat the hay meal. Horses feeding from the small-holed nets took 6.5 hours to eat the meal. Both the control and large net resulted in consumption times of 3.2 and 3.4 hours, respectively.
We can expect that using small or medium hay nets to feed in a box stall setting twice daily, would allow horses to forage for 10 to 13 hours daily. This better mimics more a horse's natural grazing behavior.
Small and medium hay nets represent a simple and affordable way to extend foraging time when meal feeding horses. The use of the small and medium hay nets isn't practical for all horses. It take time (usually 4 to 5 feedings) for horses to adjust to eating from the nets.


A Public Open House will be held from 4:00pm to 7:00pm on Tuesday, July 17, 2018 at James Memorial Art Center (621 1st Avenue W, Williston, ND 58802-1714).
Additional Open Houses are taking place in Dickinson, Bismarck-Mandan, Fargo, Jamestown, Minot, Devils Lake, and Grand Forks. The dates, times, and locations of these Open Houses are provided on the project website.


Sewerage service fees. Recovery of fees.
Juridical consultants to EAAB (Bogotá Waterworks and Sewerage Company) in designing a juridical strategy that allows collection of unpaid fees, for use of the sewerage services.


text:
"The Carpet Crawlers" (a veces también llamada "Carpet Crawl", "The Carpet Crawl" o "Carpet Crawlers") es una canción del grupo británico de rock progresivo Genesis, publicada en su álbum conceptual The Lamb Lies Down on Broadway de 1974.

Historia
"The Carpet Crawlers" fue seleccionada para el segundo single de "The Lamb Lies Down On Broadway", acompañada por la canción "Evil Jam" en el lado-B, y lanzado en abril de 1975. "Evil Jam" es una interpretación en vivo de "The Waiting Room", también del álbum The Lamb Lies Down on Broadway. Fue grabada el 24 de enero de 1975 en la ciudad de Los Ángeles. El resto del concierto fue sacado en 1998, en los dos primeros discos de la caja Genesis Archive 1967-75.

Las estrofas de la canción están cantados con el registro más bajo de la voz de Gabriel, desarrollándose suavemente con el doble hi-hat de Phil Collins y los arregios de Tony Banks, guardando ciertas similitudes con la canción Ripples de su siguiente álbum.

Genesis continuó interpretándola en vivo tras la partida de Peter Gabriel y fue sacada del repertorio en 1984, reapareciendo primero en su gira "The Way We Walk" correspondiente a We Can't Dance de 1992 el último álbum de estudio con Collins como líder y vuelve a reaparecer muchos años después en su gira "Turn It On Again" (2007). Normalmente se le quitaba la primera estrofa de la canción, y Collins comenzaba cantando desde la segunda estrofa de la misma, de esta forma aparece grabada en el álbum Seconds Out (1977).

En 1999 fue grabada una nueva versión de la canción, reuniendo a los cinco miembros originales del grupo en el estudio, aunque no se encontraron los cinco simultáneamente en el estudio ya que cada uno grabó sus partes por separado.

Apariciones
La primera versión grabada en vivo de la canción aparece en el álbum Seconds Out de 1977. En este álbum aparece con el título de "The Carpet Crawl" y Collins comienza cantando la canción en la segunda estrofa de la misma, por lo que la primera estrofa no aparece.

Una nueva grabación de la canción aparece en el álbum compilatorio Turn It On Again: The Hits del año 1999, donde se le da el nombre de "The Carpet Crawlers 1999". En esta nueva grabación tanto Peter Gabriel como Phil Collins hacen las partes vocales. Sin embargo, la última estrofa de la canción no es interpretada.

Esta misma versión de la canción vuelve a aparecer en la nueva edición del álbum, Turn It on Again: The Hits - The Tour Edition del 2007.

La versión 1999 aparece también en el sencillo que acompañó al lanzamiento de la canción en el álbum de ese mismo año. Viene acompañada de las canciones Turn It On Again y Follow You, Follow Me en el lado-B.

Una versión remasterizada por Nick Davis de la versión original del álbum, aparece en el álbum compilatorio Platinum Collection del año 2004.

También fue incluida en el repertorio de canciones interpretadas en los conciertos de la gira "Turn It On Again 2007", siendo publicada en el álbum Live Over Europe 2007.

Versiones
Fue grabada como un cover por el grupo francés Matis y aparece en su álbum "Encore", publicado en 2006.
Aparece grabada en el álbum tributo "the Fox Lies Down" de John Ford.
También fue interpretada por la banda de new wave llamada "Human Dramas".

Formación
Peter Gabriel: Voz
Steve Hackett: Guitarra eléctrica, E-Bow
Mike Rutherford: Guitarra de 12 cuerdas, pedaleras
Phil Collins: Batería, coros
Tony Banks: ARP Pro-Soloist, piano eléctrico RMI

Enlaces externos
Video de la canción en vivo (1976)
Video de la versión 1999
Video grabado en Toronto, 7/11/2007
Letra original y traducida
Análisis de las letras (En inglés)

Canciones de Genesis
Canciones de 1974
Canciones en inglés
Composiciones en mi mayor
Composiciones en re mayor

redpajama_set_name: RedPajamaWikipedia


text:
\section{introduction}

Gabor \cite{Gabor} introduced a method using a family of elementary functions for reconstructing functions (signals)  in $1946$.
The idea of frames originated in the $1952$ paper by Duffin and Schaeffer  \cite{Duf}  to address some deep questions in non-harmonic Fourier series.  After some decades, Daubechies, Grossmann and Meyer \cite{DGM}  announced formally the definition of frame in the abstract Hilbert spaces in $1986$.  After their work, frame theory began to be widely used, particularly in the more specialized context of wavelet frames and Gabor frames.

Continuous frame is a concept of generalization of frames  proposed by G. Kaiser \cite{KAI} and independently by Ali, Antoine and Gazeau \cite{GAZ} to a family indexed by some locally compact space endowed with a Radon measure.
These frames are called as frames associated with measurable spaces by Gabrado and Han in \cite{GHN}. Askari-Hemmat, Dehghan and Radjabalipour in \cite{AHDR} called them as generalized frames and in Mathematical Physics they are referred to as coherent states \cite{COH}.

A continuous $g$-frame  is an extension of $g$-frames and continuous frames which was firstly introduced by Abdollahpour and Faroughi in \cite{AFC}. In recent years there has been shown considerable interest by functional analysts in the
study of continuous $g$-frames in Hilbert spaces.
For more details, see \cite{CO, RM}	and the references therein.

We begin with a few preliminaries which are needed in the sequel. Let $H, L$ be separable Hilbert spaces and $(\Omega, \mu)$ be a positive measure space. Let $\{H_{w}\}_{w\in\Omega}$ be a family of closed subspaces of $L$.  Throughout we consider the index set as $\Omega$ and we denote simply $\{H_{w}\}_{w}$ for  $\{H_{w}\}_{w\in\Omega}$.  We denote the set of all bounded linear operators from $H$ into $H_{w}$ by $L(H,H_{w})$  and the set of all bounded linear operators on $H$ with bounded inverse by $GL(H)$.  The set of all positive operators in $GL(H)$ is denoted by $GL^{+}(H)$ and $I_H$ represents the identity operator on $H$.  We note that if $P,Q\in GL(H)$, then $P^{\ast}, P^{-1}$ and $PQ$ are also in $GL(H)$.

\noindent We consider the space
\begin{equation*}
	\ell^{2}(\{H_{w}\}_{w})=\Big\{\{f_{w}\}_{w} : f_{w}\in H_{w},w\in\Omega, \int_{\Omega}\|f_{w}\|^{2}\ d\mu(w)<\infty\Big\}
\end{equation*}
with the inner product given by
\begin{equation*}
	\langle \{f_{w}\}_{w}, \{g_{w}\}_{w}\rangle =\int_{\Omega}\langle f_{w},g_{w}\rangle \ d\mu(w).
\end{equation*}
It is clear that $\ell^{2}(\{H_{w}\}_{w})$ is a Hilbert space.

\begin{lemma}\cite{CO}\label{lemma_3.1}
	Let $T:H\rightarrow H$ be a linear operator. Then the following are equivalent:
	\begin{itemize}
		\item[(1)] There exist constants $0< A\leq B<\infty$, such that $AI_{H}\leq T\leq BI_{H}$.
		\item[(2)] $T$ is positive and there exist constants $0< A\leq B<\infty$ such that
		\begin{equation*}
			A\|f\|^{2}\leq \|T^{\frac{1}{2}}f\|^{2}\leq B\|f\|^{2}, \quad \forall f\in H.
		\end{equation*}
		\item[(3)] $T\in GL^{+}(H)$.
	\end{itemize}
\end{lemma}

\begin{definition}\cite{PDA}
	Let $(\Omega,\mu)$ be a measure space with a positive measure $\mu$ and $P\in GL(H)$. A $P$-controlled continuous frame is a map $F:\Omega\rightarrow H$ such that there exist constants $0<A\leq B<\infty$ such that
	\begin{equation*}
		A\|f\|^{2}\leq \int_{\Omega}\langle f,F(w)\rangle \langle PF(w),f\rangle  \ d\mu(w)\leq B\|f\|^{2},\quad \forall f\in H.
	\end{equation*}
\end{definition}
\begin{definition}\cite{AFC} For each $w\in \Omega$, let $\Lambda_{w}\in L(H, H_w)$.
	We say that $\Lambda=\{\Lambda_{w}\}_{w}$ is a continuous $g$-frame for $H$ with respect to $\{H_{w}\}_{w}$ if
	\begin{itemize}
		\item[(1)] for each $f\in H$, $\{\Lambda_{w}f\}_{w}$ is strongly measurable.
		\item[(2)] there exist constants $0<A\leq B<\infty$ such that
		\begin{equation}\label{eq1}
			A\|f\|^{2}\leq \int_{\Omega}\|\Lambda_{w}f\|^{2}\ d\mu(w)\leq B\|f\|^{2},\quad\forall f\in H.
		\end{equation}
	\end{itemize}
	The numbers $A, B$ are called lower and upper frame bounds for the continuous $g$-frame, respectively. If only the right-hand inequality of \eqref{eq1} is satisfied, we call $\{\Lambda_{w}\}_{w}$
	is a continuous $g$-Bessel family for $H$ with respect to $\{H_{w}\}_{w}$ with bound $B$. If
	$A =B = \lambda $, we call $\{\Lambda_{w}\}_{w}$ the $\lambda$-tight continuous $g$-frame. Moreover, if $\lambda = 1$, $\{\Lambda_{w}\}_{w}$ is called the Parseval continuous $g$-frame.

	For a given continuous $g$-frame $\Lambda=\{\Lambda_{w}\}_{w}$ for $H$ with respect to $\{H_{w}\}_{w}$, there exists a unique positive and invertible operator (called the frame operator) $S_{\Lambda}:H\to H$ such that for each $f,g\in H$ :
	$$\langle S_\Lambda f, g\rangle=	\int_{\Omega}\langle f,\Lambda_{w}^*\Lambda_{w}g\rangle \ d\mu(w)
	$$ and $AI_{H}\leq S_{\Lambda}\leq BI_{H}$.

\end{definition}
\begin{definition}\cite{YM}
	Let $P,Q\in GL^{+}(H)$ and $\Lambda_{w}\in L(H, H_w)$.  We say that $\{\Lambda_{w}\}_{w}$ is a $(P,Q)$-controlled continuous $g$-frame for $H$ with respect to $\{H_{w}\}_{w}$ if $\{\Lambda_{w}\}_{w}$ is a continuous $g$-Bessel family and there exist constants $0<A\leq B<\infty$ such that
	\begin{equation}\label{eq3}
		A\|f\|^{2}\leq \int_{\Omega}\langle \Lambda_{w}Pf,\Lambda_{w}Qf\rangle \ d\mu(w)\leq B\|f\|^{2},\quad\forall f\in H.
	\end{equation}
	The numbers $A$ and $B$ are called controlled continuous $g$-frame bounds. If the right hand inequality of \eqref{eq3} holds for all $f\in H$, then $\{\Lambda_{w}\}_{w}$ is a called a $(P,Q)$-controlled continuous $g$-Bessel family with bound $B$.

	If $Q=I_{H}$, we call $\{\Lambda_{w}\}_{w}$ is a $P$-controlled continuous $g$-frame for $H$ with respect to $\{H_{w}\}_{w}$. If $Q=P$, we call $\{\Lambda_{w}\}_{w}$ is a $(P,P)$-controlled continuous $g$-frame for $H$ with respect to $\{H_{w}\}$.
\end{definition}

For a $(P,Q)$-controlled continuous $g$-Bessel family $\{\Lambda_{w}\}_{w}$ with bound $B$, the operator $T_{P\Lambda Q}:\ell^{2}(\{H_{w}\}_{w})\rightarrow H$ given by
\begin{equation*}
	T_{P\Lambda Q}\{f_{w}\}_{w}=\int_{\Omega}(PQ)^{\frac{1}{2}}\Lambda_{w}f_{w}\ d\mu(w),\quad \forall \{f_{w}\}_{w}\in \ell^{2}(\{H_{w}\}_{w})
\end{equation*}
is well-defined and its adjoint is given by
\begin{equation*}
	T_{P\Lambda Q}^{\ast}:H\rightarrow \ell^{2}(\{H_{w}\}_{w}),\quad T_{P\Lambda Q}^{\ast}f=\{\Lambda_{w}(QP)^{\frac{1}{2}}f\}_{w},\quad \forall f\in H.
\end{equation*}
$T_{P\Lambda Q}$ is called the synthesis operator and $T_{P\Lambda Q}^{\ast}$ is called the analysis operator of $\{\Lambda_{w}\}_{w}$. For a $(P,Q)$-controlled continuous $g$-frame $\{\Lambda_{w}\}_{w}$ with bounds $A$ and $B$, the operator
\begin{equation*}
	S_{P\Lambda Q}:H\rightarrow H, \quad 	S_{P\Lambda Q}f=\int_{\Omega}Q\Lambda_{w}^{\ast}\Lambda_{w}Pf\ d\mu(w),\quad\forall f\in H.
\end{equation*}
is called the frame operator of $\{\Lambda_{w}\}_{w}$. It is a positive invertible operator and $S_{P\Lambda Q}=QS_{\Lambda}P$.

\begin{example}
	Let $P$ and $Q$ be any positive definite matrices of order $2$.  For each $w\in [0, 2\pi]$, we define $\Lambda_{w}=\begin{pmatrix}
		\cos w\\\sin w
	\end{pmatrix}$.  Then $\{P\Lambda_{w}Q\}$ is a $(P,Q)$-controlled continuous $g$-frame for $\mathbb R^2$.
\end{example}

\section{ controlled continuous $g$-frames in Hilbert spaces}
In this section, we present certain conditions under which continuous $g$-frames become $(P,Q)$-controlled continuous $g$-frames. Given any operators $P,Q\in GL^{+}(H)$, if we have a  $(P,Q)$-controlled continuous $g$-frame $\{\Lambda_{w}\}_{w}$ for $H$ with respect to $\{H_{w}\}_{w}$, we shall get frame bounds of the continuous $g$-frame $\{\Lambda_{w}\}_{w}$ in terms of controlled continuous $g$-frame bounds and operator norms of $(PQ)^{\frac{1}{2}}$ and $(PQ)^{-\frac{1}{2}}$.  It is very useful in estimating bounds of some continuous $g$-frame provided it is a  $(P,Q)$-controlled continuous $g$-frame for some operators $P,Q\in GL^{+}(H)$.

\begin{theorem} Let 	$P,Q\in GL^{+}(H)$. Then $\{\Lambda_{w}\}_{w\in \Omega}$ is a  $(P,Q)$-controlled continuous $g$-frame for $H$ with respect to $\{H_{w}\}_{w}$ if and only if $\{\Lambda_{w}\}_{w}$ is a continuous $g$-frame for $H$ with respect to $\{H_{w}\}_{w}$.
\end{theorem}
\begin{proof}
	Assume that $\{\Lambda_{w}\}_{w}$ is a  $(P,Q)$-controlled continuous $g$-frame for $H$ with respect to $\{H_{w}\}_{w}$ with bounds $A$ and $B$.  Then for each $f\in H$, we have
	\begin{align*}
		A\|f\|^{2}&=A\|(PQ)^{\frac{1}{2}}(PQ)^{-\frac{1}{2}}f\|^{2}\\&\leq A\|(PQ)^{\frac{1}{2}}\|^{2}\|(PQ)^{-\frac{1}{2}}f\|^{2}\\&\leq\|(PQ)^{\frac{1}{2}}\|^{2}\int_{\Omega}\langle \Lambda_{w}P(PQ)^{-\frac{1}{2}}f,\Lambda_{w}Q(PQ)^{-\frac{1}{2}}f\rangle \ d\mu(w) \\&=\|(PQ)^{\frac{1}{2}}\|^{2}\Big\langle \int_{\Omega}Q\Lambda_{w}^{\ast}\Lambda_{w}P(PQ)^{-\frac{1}{2}}f\ d\mu(w),(PQ)^{-\frac{1}{2}}f\Big\rangle\\&=\|(PQ)^{\frac{1}{2}}\|^{2}\Big\langle QS_{\Lambda}P(PQ)^{-\frac{1}{2}}f,(PQ)^{-\frac{1}{2}}f\Big\rangle\\&=\|(PQ)^{\frac{1}{2}}\|^{2}\langle S_{\Lambda}P^{\frac{1}{2}}Q^{-\frac{1}{2}}f,Q^{\frac{1}{2}}P^{-\frac{1}{2}}f\rangle\\&=\|(PQ)^{\frac{1}{2}}\|^{2}\langle Q^{\frac{1}{2}}P^{-\frac{1}{2}}S_{\Lambda}P^{\frac{1}{2}}Q^{-\frac{1}{2}}f,f\rangle\\&=\|(PQ)^{\frac{1}{2}}\|^{2}\langle S_{\Lambda}f,f\rangle,
	\end{align*}
	hence,
	\begin{equation*}
		\frac{A}{\|(PQ)^{\frac{1}{2}}\|^{2}}\|f\|^{2} \leq \int_{\Omega}\|\Lambda_{w}f\|^{2}\ d\mu(w).
	\end{equation*}
	On the other hand for each $f\in H$, we have
	\begin{align*}
		\int_{\Omega}\|\Lambda_{w}f\|^{2}\ d\mu(w)&=\langle S_{\Lambda}f,f\rangle \\&=\langle (PQ)^{-\frac{1}{2}}(PQ)^{\frac{1}{2}}S_{\Lambda}f,f\rangle \\&=\langle (PQ)^{\frac{1}{2}}S_{\Lambda}f,(PQ)^{-\frac{1}{2}}f\rangle \\&=\langle S_{\Lambda}(PQ)(PQ)^{-\frac{1}{2}}f,(PQ)^{-\frac{1}{2}}f\rangle \\&=\langle QS_{\Lambda}P(PQ)^{-\frac{1}{2}}f,(PQ)^{-\frac{1}{2}}f\rangle \\&=\langle S_{P\Lambda Q}(PQ)^{-\frac{1}{2}}f,(PQ)^{-\frac{1}{2}}f\rangle\\&\leq B\|(PQ)^{-\frac{1}{2}}\|^{2}\|f\|^{2}.
	\end{align*}
	Finally, we conclude that $\{\Lambda_{w}\}_{w}$ is a continuous $g$-frame for $H$ with respect to $\{H_{w}\}_{w}$ with bounds $\frac{A}{\|(PQ)^{\frac{1}{2}}\|^{2}}$ and $B\|(PQ)^{-\frac{1}{2}}\|^{2}$.

	Conversely, suppose that  $\{\Lambda_{w}\}_{w}$ is a continuous $g$-frame for $H$ with respect to $\{H_{w}\}_{w}$ with bounds $A$ and $B$. Then,
	\begin{equation*}
		A\langle f,f\rangle \leq \langle S_{\Lambda}f,f\rangle \leq B\langle f,f\rangle, \quad\forall f\in H.
	\end{equation*}
	Since $P,Q\in GL^{+}(H)$, by Lemma \ref{lemma_3.1}, there exist constants $\alpha_{1},\alpha_{2},\beta_{1},\beta_{2}>0$ such that
	\begin{equation*}
		\alpha_{1}I_{H}\leq P\leq \beta_{1}I_{H}, \quad\alpha_{2}I_{H}\leq Q\leq \beta_{2}I_{H}.
	\end{equation*}
	So,
	\begin{equation*}
		\alpha_{1}\alpha_{2}A\|f\|^{2}\leq \int_{\Omega}\langle \Lambda_{w}Pf,\Lambda_{w}Qf\rangle \ d\mu(w)\leq \beta_{1}\beta_{2}B\|f\|^{2},  \quad\forall f\in H.
	\end{equation*}
	Therefore  $\{\Lambda_{w}\}_{w}$ is a  $(P,Q)$-controlled continuous $g$-frame for $H$ with respect to $\{H_{w}\}_{w}$.
\end{proof}
\begin{theorem}\label{th1}
	Let $P,Q\in GL^{+}(H)$. Then $\{\Lambda_{w}\}_{w}$ is a  $(P,Q)$-controlled continuous $g$-frame for $H$ with respect to $\{H_{w}\}_{w}$ if and only if $\{\Lambda_{w}\}_{w}$ is a  $((PQ)^{\frac{1}{2}},(PQ)^{\frac{1}{2}})$-controlled continuous $g$-frame for $H$ with respect to $\{H_{w}\}_{w}$.
\end{theorem}
\begin{proof}
	The proof follows from the following facts :

	\noindent For any $f\in H$, we have
	\begin{align*}
		\int_{\Omega}\langle \Lambda_{w}Pf,\Lambda_{w}Qf\rangle \ d\mu(w)&=\Big\langle \int_{\Omega}Q\Lambda_{w}^{\ast}\Lambda_{w}Pf\ d\mu(w),f\Big\rangle\\&=\langle QS_{\Lambda}Pf,f\rangle\\&=\langle QPS_{\Lambda}f,f\rangle\\&=\langle(PQ)^{\frac{1}{2}}S_{\Lambda}(PQ)^{\frac{1}{2}}f,f\rangle \\&=\Big\langle \int_{\Omega}(PQ)^{\frac{1}{2}}\Lambda_{w}^{\ast}\Lambda_{w}(PQ)^{\frac{1}{2}}f\ d\mu(w),f\Big\rangle\\&=\int_{\Omega}\langle \Lambda_{w}(PQ)^{\frac{1}{2}}f,\Lambda_{w}(PQ)^{\frac{1}{2}}f\rangle \ d\mu(w).
	\end{align*}
	Suppose that $\{\Lambda_{w}\}_{w}$ is a  $(P,Q)$-controlled continuous $g$-frame for $H$ with respect to $\{H_{w}\}_{w}$.  Hence
	\begin{equation*}
		A\|f\|^{2}\leq \int_{\Omega}\langle \Lambda_{w}(PQ)^{\frac{1}{2}}f,\Lambda_{w}(PQ)^{\frac{1}{2}}f\rangle \ d\mu(w)\leq B\|f\|^{2},\quad \forall f\in H
	\end{equation*}
	where $A$ and $B$ are continuous frame bounds of $\{\Lambda_{w}\}_{w}$. Thus $\{\Lambda_{w}\}_{w}$ is a  $((PQ)^{\frac{1}{2}},(PQ)^{\frac{1}{2}})$-controlled continuous $g$-frame for $H$ with respect to. $\{H_{w}\}_{w}$.  In a similar way, the other implication can be proved.
\end{proof}
\begin{theorem}
	Let $P,Q\in GL^{+}(H)$. Then $\{\Lambda_{w}\}_{w}$ is a  $(P,Q)$-controlled continuous $g$-frame for $H$ with respect to $\{H_{w}\}_{w}$ if and only if $\{\Lambda_{w}\}_{w}$ is a  $PQ$-controlled continuous $g$-frame for $H$ with respect to $\{H_{w}\}_{w}$.
\end{theorem}
\begin{proof}
	The proof is similar to that of Theorem \ref{th1}.
\end{proof}

\begin{definition} Let $(\Omega, \mu)$ be a positive measure space. For each $w\in \Omega$, let  $\Omega_{w}$ be a measurable subset of $\Omega$. We say that the collection 	$\{e_{w,v}\}_{v\in\Omega_{w}}$ is a continuous orthonormal basis of $H_{w}$ if
	\begin{itemize}
		\item[(1)] $f=\int_{\Omega_{w}}\langle f,e_{w,v}\rangle e_{w,v}\ d\mu(w)$, for all $f\in H_{w}$ ;
		\item[(2)] $\langle e_{w,v},e_{w,v}\rangle=1$, $\forall v\in \Omega_{w}$ ;
		\item[(3)] $\langle e_{w,v_{1}},e_{w,v_{2}}\rangle =0$, $\forall v_{1},v_{2}\in\Omega_{w}$ and $v_{1}\neq v_{2}$.
	\end{itemize}
\end{definition}
\begin{theorem}\label{th2}
	Let $P,Q\in GL^{+}(H)$. Then $\{\Lambda_{w}\}_{w}$ is a  $(P,Q)$-controlled continuous $g$-frame for $H$ with respect to $\{H_{w}\}_{w}$ if and only if $\{u_{w,v}\}_{w\in\Omega,v\in\Omega_{w}}$ is a  $(P,Q)$-controlled continuous $g$-frame for $H$, where $\{u_{w,v}\}_{w\in\Omega,v\in\Omega_{w}}$ is the sequence induced by $\{\Lambda_{w}\}_{w}$ with respect to $\{e_{w,v}\}_{w\in\Omega,v\in\Omega_{w}}$ (i.e., $u_{w,v}=\Lambda_{w}^{\ast}e_{w,v}$).
\end{theorem}
\begin{proof}
	For each $w\in\Omega$,  let $\{e_{w,v}\}_{v\in\Omega_{w}}$ be an continuous orthonormal basis for $H_{w}$.  Then we have
	\begin{equation*}
		\Lambda_{w}Pf=\int_{\Omega_{w}}\langle \Lambda_{w}Pf,e_{w,v}\rangle  e_{w,v}\ d\mu(v)=\int_{\Omega_{w}}\langle f,P\Lambda_{w}^{\ast}e_{w,v}\rangle e_{w,v}\ d\mu(v)
	\end{equation*}
	and
	\begin{equation*}
		\Lambda_{w}Qf=\int_{\Omega_{w}}\langle \Lambda_{w}Qf,e_{w,v}\rangle e_{w,v}d\mu(v)=\int_{\Omega_{w}}\langle f,Q\Lambda_{w}^{\ast}e_{w,v}\rangle e_{w,v} \ d\mu(v).
	\end{equation*}
	It is easy to check that
	\begin{align*}
		\langle \Lambda_{w}Pf,\Lambda_{w}Qf\rangle&=\int_{\Omega_{w}}\langle f,Pu_{w,v}\rangle\langle Qu_{w,v},f\rangle \ d\mu(v)\\
		\int_{\Omega}	\langle \Lambda_{w}Pf,\Lambda_{w}Qf\rangle \ d\mu(w)&=\int_{\Omega}\int_{\Omega_{w}}\langle f,Pu_{w,v}\rangle\langle Qu_{w,v},f\rangle \ d\mu(v)\ d\mu(w).
	\end{align*}
	Hence
	\begin{equation*}
		A\|f\|^{2}\leq\int_{\Omega}\langle \Lambda_{w}Pf,\Lambda_{w}Qf\rangle \ d\mu(w)\leq B\|f\|^{2},\quad\forall f\in H.
	\end{equation*}
	Thus
	\begin{equation*}
		A\|f\|^{2}\leq \int_{\Omega}\int_{\Omega_{w}}\langle f,Pu_{w,v}\rangle\langle Qu_{w,v},f\rangle \ d\mu(v)\ d\mu(w)\leq B\|f\|^{2},\quad\forall f\in H.
	\end{equation*}
	The proof is completed.
\end{proof}
\begin{theorem}
	Let $P,Q\in GL^{+}(H)$. Then $\{\Lambda_{w}\}_{w}$ is a  $(P,Q)$-controlled continuous $g$-frame for $H$ with respect to $\{H_{w}\}_{w}$ if and only if $\{Pu_{w,v}\}_{w\in\Omega,v\in\Omega_{w}}$ is a  $QP^{-1}$-controlled continuous $g$-frame for $H$, where $\{u_{w,v}\}_{w\in\Omega,v\in\Omega_{w}}$ is the sequence induced by $\{\Lambda_{w}\}_{w}$ with respect to $\{e_{w,v}\}_{w\in\Omega,v\in\Omega_{w}}$ (i.e., $u_{w,v}=\Lambda_{w}^{\ast}e_{w,v}$).
\end{theorem}
\begin{proof}
	From the proof of Theorem \ref{th2}, we have
	\begin{equation*}
		\int_{\Omega}	\langle \Lambda_{w}Pf,\Lambda_{w}Qf\rangle \ d\mu(w)=\int_{\Omega}\int_{\Omega_{w}}\langle f,Pu_{w,v}\rangle\langle Qu_{w,v},f\rangle \ d\mu(v)\ d\mu(w).
	\end{equation*}
	Then
	\begin{equation*}
		A\|f\|^{2}\leq\int_{\Omega}\langle \Lambda_{w}Pf,\Lambda_{w}Qf\rangle \ d\mu(w)\leq B\|f\|^{2},\quad\forall f\in H,
	\end{equation*}
	is equivalent to
	\begin{equation*}
		A\|f\|^{2}\leq \int_{\Omega}\int_{\Omega_{w}}\langle f,Pu_{w,v}\rangle \langle QP^{-1}Pu_{w,v},f\rangle  d\mu(v)\ d\mu(w) \leq B\|f\|^{2}, \quad \forall f\in H.
	\end{equation*}
	\noindent Hence the proof is completed.
\end{proof}
Combining all results presented in the section, we get the following result which gives several interesting characterizations of  $(P,Q)$-controlled continuous $g$-frames for $H$ with respect to $\{H_{w}\}_{w}$.
\begin{theorem}
	Let $P,Q\in GL^{+}(H)$. Then the following are equivalent:
	\begin{enumerate}
		\item $\{\Lambda_{w}\}_{w}$ is a  $(P,Q)$-controlled continuous $g$-frame for $H$ with respect to $\{H_{w}\}_{w}$.
		\item $\{\Lambda_{w}\}_{w}$ is a  $((PQ)^{\frac{1}{2}},(PQ)^{\frac{1}{2}})$-controlled continuous $g$-frame for $H$ with respect to $\{H_{w}\}_{w}$.
		\item $\{\Lambda_{w}\}_{w}$ is a  $QP$-controlled  continuous $g$-frame for $H$ with respect to $\{H_{w}\}_{w}$.
		\item $\{u_{w,v}\}_{w\in\Omega,v\in\Omega_{w}}$ is a  $(P,Q)$-controlled continuous $g$-frame for $H$, where $\{u_{w,v}\}_{w\in\Omega,v\in\Omega_{w}}$ is the sequence induced by $\{\Lambda_{w}\}_{w}$ with respect to $\{e_{w,v}\}_{w\in\Omega,v\in\Omega_{w}}$.
		\item $\{Pu_{w,v}\}_{w\in\Omega,v\in\Omega_{w}}$ is a  $QP^{-1}$-controlled continuous $g$-frame for $H$, where $\{u_{w,v}\}_{w\in\Omega,v\in\Omega_{w}}$ is the sequence induced by $\{\Lambda_{w}\}_{w}$ with respect to $\{e_{w,v}\}_{w\in\Omega,v\in\Omega_{w}}$.
		\item $\{\Lambda_{w}\}_{w}$ is a continuous $g$-frame for $H$ with respect to$\{H_{w}\}_{w}$.
	\end{enumerate}

	Moreover, $\{\Lambda_{w}\}_{w}$ is a continuous $g$-frame for $H$ with respect to $\{H_{w}\}_{w}$ with bounds $\frac{A}{\|(PQ)^{\frac{1}{2}}\|^{2}}$ and $B\|(PQ)^{-\frac{1}{2}}\|^{2}$.
\end{theorem}
\section{Controlled continuous dual $g$-frames in Hilbert spaces}
In this section, we define  controlled continuous dual $g$-frames and characterize them by operator theory.
\begin{definition}\label{def1}
	Let $P,Q\in GL^{+}(H)$ and let $\{\Lambda_{w}\}_{w}, \{\Gamma_{w}\}_{w}$ be  $(P,P)$-controlled continuous and  $(Q,Q)$-controlled continuous $g$-Bessel families for $H$ with respect to $\{H_{w}\}_{w}$, respectively.
	We say that  $\{\Gamma_{w}\}_{w}$ is  a  $(P,Q)$-controlled continuous dual $g$-frame of  $\{\Lambda_{w}\}_{w}$ if \begin{equation*}
		f=\int_{\Omega}P\Lambda_{w}^{\ast}\Gamma_{w}Qf\ d\mu(w), \quad \forall f\in H.
	\end{equation*}
	In particular, if $Q=I_{H}$,  $\{\Gamma_{w}\}_{w}$ is called a  $P$-controlled continuous dual $g$-frame of $\{\Lambda_{w}\}_{w}$.
\end{definition}
\begin{definition}
	Let  $P,Q\in GL^{+}(H)$ and let $\{\Lambda_{w}\}_{w}, \{\Gamma_{w}\}_{w}$ be  $(P,P)$-controlled continuous and  $(Q,Q)$-controlled continuous $g$-Bessel families for $H$ with respect to $\{H_{w}\}_{w}$ respectively.
	For the pair $\{\Lambda_{w}\}_{w}$ and $\{\Gamma_{w}\}_{w}$, we define a  $(P,Q)$-controlled continuous dual $g$-frame operator $S_{P\Lambda\Gamma Q}$ by	\begin{equation*}
		S_{P\Lambda\Gamma Q}f=\int_{\Omega}P\Lambda_{w}^{\ast}\Gamma_{w}Qf\ d\mu(w),\quad\forall f\in H.
	\end{equation*}
	It is easy to check that $S_{P\Lambda\Gamma Q}$ is a well-defined bounded operator, and
	\begin{equation*}
		S_{P\Lambda\Gamma Q}=T_{P\Lambda P}T_{Q\Gamma Q}^{\ast}=PT_{\Lambda}T_{\Gamma}^{\ast}Q=PS_{\Lambda\Gamma}Q,
	\end{equation*}
	where $S_{\Lambda\Gamma}f=\int_{\Omega}\Lambda_{w}^{\ast}\Gamma_{w}f\ d\mu(w)$. Note that  $\{\Gamma_{w}\}_{w}$ is a  $(P,Q)$-controlled continuous dual $g$-frame of $\{\Lambda_{w}\}_{w}$ if and only if $S_{P\Lambda\Gamma Q}=I_{H}$.
\end{definition}
\begin{theorem}\label{th3}
	Let $P,Q\in GL^{+}(H)$ and let $\{\Lambda_{w}\}_{w}, \{\Gamma_{w}\}_{w}$ be  $(P,P)$-controlled continuous and  $(Q,Q)$-controlled continuous $g$-Bessel families with bounds $B_{\Lambda}$ and $B_{\Gamma}$ respectively. If $S_{P\Lambda\Gamma Q}$ is bounded below, then  $\{\Lambda_{w}\}_{w}$ and  $\{\Gamma_{w}\}_{w}$ are  $(P,P)$-controlled continuous and  $(Q,Q)$-controlled continuous $g$-frames respectively.
\end{theorem}
\begin{proof}
	Suppose that there exists a constant $\lambda>0$ such that
	\begin{equation*}
		\|S_{P\Lambda\Gamma Q}f\|\geq \lambda \|f\|, \quad \forall f\in H.
	\end{equation*}
	Hence
	\begin{align*}
		\lambda\|f\|&\leq \|S_{P\Lambda\Gamma Q}f\|\\&=\sup_{\|g\|=1}\Big|\langle \int_{\Omega}P\Lambda_{w}^{\ast}\Gamma_{w}Qf\ d\mu(w),g\rangle\Big |\\&=\sup_{\|g\|=1}\Big|\int_{\Omega}\langle \Gamma_{w}Qf,\Lambda_{w}Pg\rangle \ d\mu(w) \Big|\\&\leq \sup_{\|g\|=1}\bigg(\int_{\Omega}\|\Gamma_{w}Qf\|^{2}\ d\mu(w))\bigg)^{\frac{1}{2}}\bigg(\int_{\Omega}\|\Lambda_{w}Pg\|^{2}\ d\mu(w)\bigg)^{\frac{1}{2}}\\&\leq \sqrt{B_{\Lambda}}\bigg(\int_{\Omega}\|\Gamma_{w}Qf\|^{2}\ d\mu(w)\bigg)^{\frac{1}{2}}.
	\end{align*}
	Thus
	\begin{equation*}
		\frac{\lambda^{2}}{B_{\Lambda}}\|f\|^{2}\leq \int_{\Omega}\|\Gamma_{w}Qf\|^{2}\ d\mu(w),\quad \forall f\in H.
	\end{equation*}
	On the other hand, Since
	\begin{equation*}
		S_{P\Lambda\Gamma Q}^{\ast}=(PS_{\Lambda\Gamma}Q)^{\ast}=QS_{\Gamma\Lambda}P=S_{Q\Gamma\Lambda P},
	\end{equation*}
	then $S_{Q\Gamma\Lambda P}$ is also bounded below. Similarly, we can prove that $\{\Lambda_{w}\}_{w}$ is a  $(P,P)$-controlled continuous $g$-frame. Hence the proof is completed.
\end{proof}
\begin{theorem}
	Let $P,Q\in GL^{+}(H)$, $\{\Lambda_{w}\}_{w}$ and let $\{\Gamma_{w}\}_{w}$ be  $(P,P)$-controlled continuous and  $(Q,Q)$-controlled continuous $g$-Bessel families for $H$ with respect to $\{H_{w}\}_{w}$ respectively. Then the following conditions are equivalent:
	\begin{itemize}
		\item[(1)] $f=\int_{\Omega}P\Lambda_{w}^{\ast}\Gamma_{w}Qf\ d\mu(w)$,$\ \forall f\in H$.
		\item[(2)] $f=\int_{\Omega}Q\Gamma_{w}^{\ast}\Lambda_{w}Pf\ d\mu(w)$,$\ \forall f\in H$.
		\item[(3)] $\langle f,g\rangle=\int_{\Omega}\langle \Lambda_{w}Pf,\Gamma_{w}Qg\rangle \ d\mu(w)=\int_{\Omega}\langle \Gamma_{w}Qf,\Lambda_{w}Pg\rangle \ d\mu(w), \  \forall f,g\in H$.
		\item[(4)] $\|f\|^{2}=\int_{\Omega}\langle \Lambda_{w}Pf,\Gamma_{w}Qf\rangle \ d\mu(w)=\int_{\Omega}\langle \Gamma_{w}Qf,\Lambda_{w}Pf\rangle \ d\mu(w),\ \forall f\in H$.
	\end{itemize}
	In case the equivalent conditions are satisfied, $\{\Lambda_{w}\}_{w}$ and $\{\Gamma_{w}\}_{w}$ are  $(P,P)$-controlled continuous and  $(Q,Q)$-controlled continuous $g$-frames respectively.
\end{theorem}
\begin{proof}
	$(1)\Longleftrightarrow (2)$. Let $T_{P\Lambda P}$ be the synthesis operator of the  $(P,P)$-controlled continuous $g$-Bessel family $\{\Lambda_{w}\}_{w}$ and $T_{Q\Gamma Q}$ be  the synthesis operator of the  $(P,P)$-controlled continuous $g$-Bessel family $\{\Gamma_{w}\}_{w}$. Condition $(1)$ means that $T_{P\Lambda P}T_{Q\Gamma Q}^{\ast}=I_{H}$, and it  is equivalent to $T_{Q\Gamma Q}T_{P\Lambda P}^{\ast}$, which is identical to the statement $(2)$. Conversely, $(2)\implies (1)$ similarly.

	$(2)\Longleftrightarrow (3)$. It is clear that $(2)\implies (3)$. For any $f,g\in H$, $\langle f,g\rangle=\int_{\Omega}\langle \Lambda_{w}Pf,\Gamma_{w}Qg\rangle \ d\mu(w)$ shows that
	\begin{equation*}
		\Big\langle f-\int_{\Omega}Q\Gamma_{w}^{\ast}\Lambda_{w}Pf\ d\mu(w),g\Big\rangle=0,\quad\forall g\in H.
	\end{equation*}
	Hence $(3)\Longrightarrow (2)$ is proved.

	$(3)\Longleftrightarrow (4)$. $(3)\implies (4)$ is obvious. To prove $(4)\implies (3)$, we apply condition $(4)$ and get
	\begin{align*}
		\|f+g\|^{2}&=\int_{\Omega}\langle \Lambda_{w}P(f+g),\Gamma_{w}Q(f+g)\rangle \ d\mu(w)\\&=\int_{\Omega}\langle \Lambda_{w}Pf,\Gamma_{w}Qf\rangle \ d\mu(w)+\int_{\Omega}\langle \Lambda_{w}Pf,\Gamma_{w}Qg\rangle \ d\mu(w)\\&\quad +\int_{\Omega}\langle \Lambda_{w}Pg,\Gamma_{w}Qf\rangle \ d\mu(w)+\int_{\Omega}\langle \Lambda_{w}Pg,\Gamma_{w}Qg\rangle \ d\mu(w).
	\end{align*}
	Similarly,
	\begin{align*}
		\|f-g\|^{2}&=\int_{\Omega}\langle \Lambda_{w}Pf,\Gamma_{w}Qf\rangle \ d\mu(w)-\int_{\Omega}\langle \Lambda_{w}Pf,\Gamma_{w}Qg\rangle \ d\mu(w)\\&\quad -\int_{\Omega}\langle \Lambda_{w}Pg,\Gamma_{w}Qf\rangle \ d\mu(w)+\int_{\Omega}\langle \Lambda_{w}Pg,\Gamma_{w}Qg\rangle \ d\mu(w).
	\end{align*}
	\begin{align*}
		\|f+ig\|^{2}&=\int_{\Omega}\langle \Lambda_{w}Pf,\Gamma_{w}Qf\rangle \ d\mu(w)-i\int_{\Omega}\langle \Lambda_{w}Pf,\Gamma_{w}Qg\rangle \ d\mu(w)\\&\quad +i\int_{\Omega}\langle \Lambda_{w}Pg,\Gamma_{w}Qf\rangle \ d\mu(w)+\int_{\Omega}\langle \Lambda_{w}Pg,\Gamma_{w}Qg\rangle \ d\mu(w).
	\end{align*}
	\begin{align*}
		\|f-ig\|^{2}&=\int_{\Omega}\langle \Lambda_{w}Pf,\Gamma_{w}Qf\rangle \ d\mu(w)+i\int_{\Omega}\langle \Lambda_{w}Pf,\Gamma_{w}Qg\rangle \ d\mu(w)\\&\quad -i\int_{\Omega}\langle \Lambda_{w}Pg,\Gamma_{w}Qf\rangle \ d\mu(w)+\int_{\Omega}\langle \Lambda_{w}Pg,\Gamma_{w}Qg\rangle \ d\mu(w).
	\end{align*}
	By polarization identity,
	\begin{align*}
		\langle f,g\rangle&=\frac{1}{4}\bigg(\|f+g\|^{2}-\|f-g\|^{2}+i\|f+ig\|^{2}-i\|f-ig\|^{2}\bigg)\\&=\int_{\Omega}\langle \Lambda_{w}Pf,\Gamma_{w}Qg\rangle \ d\mu(w).
	\end{align*}
	In case the equivalent conditions are satisfied, $S_{Q\Gamma\Lambda P}=I_{H}$ implies $\|S_{Q\Gamma\Lambda P}\|=1$, hence by Theorem \ref{th3}, $\{\Lambda_{w}\}_{w}$ and $\{\Gamma_{w}\}_{w}$ are  $(P,P)$-controlled continuous and  $(Q,Q)$-controlled continuous $g$-frames respectively.
\end{proof}
\begin{theorem}
	Let $P,Q\in GL^{+}(H)$. A sequence $\{\Lambda_{w}\}_{w}$ is a  $(P,Q)$-controlled continuous $g$-Bessel family for $H$ with respect to $\{H_{w}\}_{w}$ with bound $B$ if and only if the operator $T_{P\Lambda Q}:\ell^{2}(\{H_{w}\}_{w})\rightarrow H$ given by
	\begin{equation*}
		T_{P\Lambda Q}(\{f_{w}\}_{w})=\int_{\Omega}(PQ)^{\frac{1}{2}}\Lambda_{w}^{\ast}f_{w}\ d\mu(w)
	\end{equation*}
	is well-defined and bounded with $\|T_{P\Lambda Q}\|\leq \sqrt{B}$.
\end{theorem}
\begin{proof}
	The necessary condition follows from the definition of  $(P,Q)$-controlled continuous $g$-Bessel sequence. We only need to prove the sufficient condition. Suppose that $T_{P\Lambda Q}$ is well-defined and bound operator with $\|T_{P\Lambda Q}\|\leq \sqrt{B}$. For any $f\in H$, we have
	\begin{align*}
		\int_{\Omega}\langle \Lambda_{w}Pf,\Lambda_{w}Qf\rangle \ d\mu(w)&=\int_{\Omega}\langle Q\Lambda_{w}^{\ast}\Lambda_{w}Pf,f\rangle \ d\mu(w) \\&=\langle QS_{\Lambda}Pf,f\rangle \\&=\langle (QP)^{\frac{1}{2}}S_{\Lambda}(QP)^{\frac{1}{2}}f,f\rangle\\&=\Big\langle \int_{\Omega}(QP)^{\frac{1}{2}}\Lambda_{w}^{\ast}\Lambda_{w}(QP)^{\frac{1}{2}}f \ d\mu(w),f\Big\rangle \\&\leq \|T_{P\Lambda Q}\|\bigg(\int_{\Omega}\|\Lambda_{w}(QP)^{\frac{1}{2}}f\|^{2}\ d\mu(w)\bigg)^{\frac{1}{2}}\|f\|\\&=\|T_{P\Lambda Q}\|\bigg(\int_{\Omega}\langle \Lambda_{w}Pf,\Lambda_{w}Qf\rangle \ d\mu(w)\bigg)^{\frac{1}{2}}\|f\|.
	\end{align*}
	Hence we get
	\begin{equation*}
		\int_{\Omega}\langle  \Lambda_{w}Pf,\Lambda_{w}Qf\rangle \ d\mu(w)\leq  \|T_{P\Lambda Q}\|^{2}\|f\|^{2}\leq B\|f\|^{2}.
	\end{equation*}
	This shows that $\{\Lambda_{w}\}_{w}$ is a  $(P,Q)$-controlled continuous $g$-Bessel family for $H$ with respect to $\{H_{w}\}_{w}$ with bound $B$.
\end{proof}
\begin{theorem}
	Let $P,Q\in GL^{+}(H)$ and  $\{\Lambda_{w}\}_{w}$ be a $(P,P)$-controlled $g$-frame for $H$ with respect to $\{H_{w}\}_{w}$ with the synthesis operator $T_{P\Lambda P}$. Then a  $(Q,Q)$-controlled continuous $g$-frame $\{\Gamma_{w}\}_{w}$ is a  $(P,Q)$-controlled continuous dual $g$-frame of $\{\Lambda_{w}\}_{w}$ if and only if
	\begin{equation*}
		Q\Gamma_{w}^{\ast}e_{w,v}=U(e_{w,v}\delta_{w}),\quad w\in\Omega, v\in\Omega_{w}
	\end{equation*}
	where $U:\ell^{2}(\{H_{w}\}_{w})\rightarrow H$ is a bounded left-inverse of $T_{P\Lambda P}^{\ast}$.
\end{theorem}
\begin{proof}
	If $\{g_{w}\}_{w}\in \ell^{2}(\{H_{w}\}_{w})$, then
	\begin{equation*}
		\{g_{w}\}_{w}=\int_{\Omega}g_{w}\delta_{w}\ d\mu(w)=\int_{\Omega}\int_{\Omega_{w}}\langle g_{w},e_{w,v}\rangle  e_{w,v}\delta_{w} \ \ d\mu(v)\ d\mu(w).
	\end{equation*}
	Roughly speaking $\{e_{w,v}\delta_{w}\}_{w\in\Omega, v\in\Omega_{w}}$ is a continuous orthonormal basis of $ \ell^{2}(\{H_{w}\}_{w})$.

	If there exists $U:\ell^{2}(\{H_{w}\}_{w})\rightarrow H$ is a bounded left-inverse of $T_{P\Lambda P}^{\ast}$ such that
	\begin{equation*}
		Q\Gamma_{w}^{\ast}e_{w,v}=U(e_{w,v}\delta_{w}),\quad w\in\Omega, v\in\Omega_{w}.
	\end{equation*}
	For any $f\in H$, we have
	\begin{align*}
		f&=UT_{P\Lambda P}^{\ast}f\\&=U\bigg(\int_{\Omega}\int_{\Omega_{w}}\langle \Lambda_{w}Pf,e_{w,v}\rangle e_{w,v}\delta_{w}\ d\mu(v) \ d\mu(w)\bigg)\\&=\int_{\Omega}\int_{\Omega_{w}}\langle f,P\Lambda_{w}^{\ast}e_{w,v}\rangle U(e_{w,v}\delta_{w})\ d\mu(v)\ d\mu(w)\\&=\int_{\Omega}\int_{\Omega_{w}}\langle f,Pu_{w,v}\rangle Q\Gamma_{w}^{\ast}e_{w,v}\ d\mu(v)\ d\mu(w)\\&=\int_{\Omega}Q\Gamma_{w}^{\ast}\bigg(\int_{\Omega_{w}}\langle Pf,u_{w,v}\rangle e_{w,v}\ d\mu(v)\bigg)\ d\mu(w)\\&=\int_{\Omega}Q\Gamma_{w}^{\ast}\Lambda_{w}Pf\ d\mu(w),
	\end{align*}
	where $u_{w,v}=\Lambda_{w}^{\ast}e_{w,v}$. By the definition of  controlled continuous dual $g$-frame,  $\{\Gamma_{w}\}_{w}$ is a  $(P,Q)$-controlled continuous dual $g$-frame of $\{\Lambda_{w}\}_{w}$.

	On the other hand, suppose that a  $(Q,Q)$-controlled continuous $g$-frame $\{\Gamma_{w}\}_{w}$ is a  $(P,Q)$-controlled continuous dual $g$-frame of $\{\Lambda_{w}\}_{w}$. For any $f\in H$, we have
	\begin{equation*}
		f=\int_{\Omega}P\Lambda_{w}^{\ast}\Gamma_{w}^{\ast}Qf\ d\mu(w)=\int_{\Omega}Q\Gamma_{w}^{\ast}\Lambda_{w}Pf\ d\mu(w),
	\end{equation*}
	that is, $T_{Q\Lambda Q}T_{P\Lambda P}^{\ast}=I_{H}$. Let $U=T_{Q\Gamma Q}$. Then $U:\ell^{2}(\{H_{w}\}_{w})\rightarrow H$ is a bounded left-inverse of $T_{P\Lambda P}^{\ast}$. A calculation as above shows that
	\begin{equation*}
		\int_{\Omega}\int_{\Omega_{w}}\langle f,Pu_{w,v}\rangle Q\Gamma_{w}^{\ast}e_{w,v}\ d\mu(v)\ d\mu(w)=f=\int_{\Omega}\int_{\Omega_{w}}\langle f,Pu_{w,v}\rangle U(e_{w,v}\delta_{w})\ d\mu(v)\ d\mu(w), \forall f\in H.
	\end{equation*}
	Combining this with the fact $\{e_{w,v}\}_{v\in \Omega_{w}}$ is a continuous orthonormal basis of $H_{w}$, we have
	\begin{equation*}
		Q\Gamma_{w}^{\ast}e_{w,v}=U(e_{w,v}\delta_{w}),\quad w\in\Omega, v\in\Omega_{w}.
	\end{equation*}
\end{proof}
\begin{theorem}
	Let $P\in GL^{+}(H)$ and $\{\Lambda_{w}\}_{w}$ be a  $(P,P)$-controlled continuous $g$-frame for $H$ with respect to $\{H_{w}\}_{w}$ with the synthesis operator and frame operator $T_{P\Lambda P}$ and $S_{P\Lambda P}$, respectively. Then $\{\Gamma_{w}\}_{w}$ is a  $P$-controlled continuous dual $g$-frame of $\{\Lambda_{w}\}_{w}$ if and only if
	\begin{equation*}
		\Gamma_{w}f=(Tf)_{w}+\Lambda_{w}S_{P\Lambda P}^{-1}Pf, w\in \Omega,v\in H,
	\end{equation*}
	where $T:H\rightarrow \ell^{2}(\{H_{w}\}_{w})$ is a bounded linear operator satisfying $T_{P\Lambda P}T=0$.
\end{theorem}
\begin{proof}
	If  $T:H\rightarrow \ell^{2}(\{H_{w}\}_{w})$ is a bounded linear operator satisfying $T_{P\Lambda P}T=0$, then $\{\Gamma_{w}\}_{w}$ is a $g$-Bessel family for $H$ with respect to $\{H_{w}\}_{w}$. In fact, any $f\in H$ we have
	\begin{align*}
		\int_{\Omega}\|\Gamma_{w}f\|^{2}\ d\mu(w)&=\int_{\Omega}\|(Tf)_{w}+\Lambda_{w}S_{P\Lambda P}^{-1}Pf\|^{2}\ d\mu(w)\\&\leq 2\bigg(\int_{\Omega}\|\Lambda_{w}S_{P\Lambda P}^{-1}Pf\|^{2}\ d\mu(w)+\|Tf\|^{2}\bigg)\\&\leq 2(B\|S_{P\Lambda P}^{-1}P\|^{2}+\|T\|^{2})\|f\|^{2},
	\end{align*}
	where $B$ is the upper bound of $\{\Lambda_{w}\}_{w\in\Omega}$. Furthermore,
	\begin{align*}
		\int_{\Omega}P\Lambda_{w}^{\ast}\Gamma_{w}f\ d\mu(w)&=\int_{\Omega}P\Lambda_{w}^{\ast}((Tf)_{w}+\Lambda_{w}S_{P\Lambda P}^{-1}Pf)\ d\mu(w)\\&=T_{P\Lambda P}Tf+\int_{\Omega}P\Lambda_{w}^{\ast}\Lambda_{w}S_{P\Lambda P}^{-1}Pf\ d\mu(w)=f.
	\end{align*}
	Thus $\{\Gamma_{w}\}_{w\in\Omega}$ is a  $P$-controlled continuous dual $g$-frame of $\{\Lambda_{w}\}_{w\in \Omega}$.

	Now we prove the converse. Assume that $\{\Gamma_{w}\}_{w\in\Omega}$ is a  $P$-controlled continuous dual $g$-frame of $\{\Lambda_{w}\}_{w\in \Omega}$. Define the operator $T$ as follows:
	\begin{equation*}
		T:H\rightarrow \ell^{2}(\{H_{w}\}_{w\in\Omega}),\quad f\rightarrow (Tf)_{w}, \quad f\in H
	\end{equation*}
	satisfying
	\begin{equation*}
		\Gamma_{w}f=(Tf)_{w}+\Lambda_{w}S_{P\Lambda P}^{-1}Pf, w\in\Omega.
	\end{equation*}
	For any $f\in H$, we have
	\begin{align*}
		\|Tf\|^{2}&=\int_{\Omega}\|\Gamma_{w}f-\Lambda_{w}S^{-1}_{P\Lambda P}Pf \|^{2}\ d\mu(w)\\&\leq \int_{\Omega}\|\Gamma_{w}f\|^{2}\ d\mu(w)+\int_{\Omega}\|\Lambda_{w}S_{P\Lambda P}^{-1}Pf\|^{2}\ d\mu(w)\\&\quad+2\bigg(\int_{\Omega}\|\Gamma_{w}f\|^{2}\ d\mu(w)\bigg)^{\frac{1}{2}}\bigg(\int_{\Omega}\|\Lambda_{w}S_{P\Lambda P}^{-1}Pf\|^{2}\ d\mu(w)\bigg)^{\frac{1}{2}}\\&\leq (B_{1}+A^{-1}+2\sqrt{B_{1}A^{-1}})\|f\|^{2},
	\end{align*}
	where $B_{1}$ is the frame upper bound of $\{\Gamma_{w}\}_{w\in\Omega}$ and $A$ is the frame lower bound of $\{\Lambda_{w}\}_{w\in\Omega}$. Thus $T$ is a linear bounded operator. Moreover, for any $f,g\in H$, we have
	\begin{align*}
		\langle T_{P\Lambda P}Tf,g\rangle&=\int_{\Omega}\langle P\Lambda_{w}^{\ast}Tf,g\rangle \ d\mu(w) \\&=\int_{\Omega}\langle P\Lambda_{w}^{\ast}(\Gamma_{w}f-\Lambda_{w}S_{P\Lambda P}^{-1}Pf),g\rangle \ d\mu(w) \\&=\int_{\Omega}\langle P\Lambda_{w}^{\ast}\Gamma_{w}f,g\rangle \ d\mu(w)-\int_{\Omega}\langle P\Lambda_{w}^{\ast}\Lambda_{w}S_{P\Lambda P}^{-1}Pf,g\rangle \ d\mu(w) \\&=\langle f,g\rangle -\langle f,g\rangle=0.
	\end{align*}
	That is $T_{P\Lambda P}T=0$. The proof is completed.
\end{proof}
\medskip

\section*{Declarations}

\medskip

\noindent \textbf{Availablity of data and materials}\newline
\noindent Not applicable.

\medskip

\noindent \textbf{Human and animal rights}\newline
\noindent We would like to mention that this article does not contain any studies
with animals and does not involve any studies over human being.

\medskip

\noindent \textbf{Conflict of interest}\newline
\noindent The authors declare that they have no competing interests.

\medskip
\bibliographystyle{amsplain}

redpajama_set_name: RedPajamaArXiv


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It's an honor to share an essay from yours truly recently published on the debut edition of DRYLAND, a new publication for L.A. poets, writers, and artists all around!
I had the fortune to connect with DRYLAND earlier this year after meeting one of their people at a writing circle with the InsideOUT Writers, and now it's a special treat to see my work on their site. The thing is: in all the time I spend on J.T., it almost slips my mind to send my work out elsewhere. But neither VONA nor DRYLAND could have happened if I didn't step out of my comfort zone to apply myself in other places for a little bit.
Just as with everything else, then: when we get so focused in one part of our lives, we really lose sight of certain other parts of ourselves through it. But with that in mind, let's not get caught up in this post! Instead, let's take a walk down memory lane with my essay: "20 Years After the L.A. Riots".
And thank you to DRYLAND, and of course, to all the J.T. supporters for a moment of their time.


This page displays sold house prices for Manor Drive in Reading.
Manor Drive in Shurlock Row, Reading consists predominantly of detached houses. Properties on Manor Drive typically have values around £1,150,000 - £1,550,000, but larger detached houses can command upwards of £1,550,000.
Map showing Manor Drive in Reading.


text:
BOOST_FIXTURE_TEST_SUITE(subsidy_tests, TestingSetup)

BOOST_AUTO_TEST_CASE(block_subsidy_test)
{
    const Consensus::Params& consensusParams = Params(CBaseChainParams::MAIN).GetConsensus();

    uint32_t nPrevBits;
    int32_t nPrevHeight;
    CAmount nSubsidy;

    // details for block 4249 (subsidy returned will be for block 4250)
    nPrevBits = 0x1c4a47c4;
    nPrevHeight = 4249;
    nSubsidy = GetBlockSubsidy(nPrevBits, nPrevHeight, consensusParams, false);
    BOOST_CHECK_EQUAL(nSubsidy, 50000000000ULL);

    // details for block 4501 (subsidy returned will be for block 4502)
    nPrevBits = 0x1c4a47c4;
    nPrevHeight = 4501;
    nSubsidy = GetBlockSubsidy(nPrevBits, nPrevHeight, consensusParams, false);
    BOOST_CHECK_EQUAL(nSubsidy, 5600000000ULL);

    // details for block 5464 (subsidy returned will be for block 5465)
    nPrevBits = 0x1c29ec00;
    nPrevHeight = 5464;
    nSubsidy = GetBlockSubsidy(nPrevBits, nPrevHeight, consensusParams, false);
    BOOST_CHECK_EQUAL(nSubsidy, 2100000000ULL);

    // details for block 5465 (subsidy returned will be for block 5466)
    nPrevBits = 0x1c29ec00;
    nPrevHeight = 5465;
    nSubsidy = GetBlockSubsidy(nPrevBits, nPrevHeight, consensusParams, false);
    BOOST_CHECK_EQUAL(nSubsidy, 12200000000ULL);

    // details for block 17588 (subsidy returned will be for block 17589)
    nPrevBits = 0x1c08ba34;
    nPrevHeight = 17588;
    nSubsidy = GetBlockSubsidy(nPrevBits, nPrevHeight, consensusParams, false);
    BOOST_CHECK_EQUAL(nSubsidy, 6100000000ULL);

    // details for block 99999 (subsidy returned will be for block 100000)
    nPrevBits = 0x1b10cf42;
    nPrevHeight = 99999;
    nSubsidy = GetBlockSubsidy(nPrevBits, nPrevHeight, consensusParams, false);
    BOOST_CHECK_EQUAL(nSubsidy, 500000000ULL);

    // details for block 210239 (subsidy returned will be for block 210240)
    nPrevBits = 0x1b11548e;
    nPrevHeight = 210239;
    nSubsidy = GetBlockSubsidy(nPrevBits, nPrevHeight, consensusParams, false);
    BOOST_CHECK_EQUAL(nSubsidy, 500000000ULL);

    // 1st subsidy reduction happens here

    // details for block 210240 (subsidy returned will be for block 210241)
    nPrevBits = 0x1b10d50b;
    nPrevHeight = 210240;
    nSubsidy = GetBlockSubsidy(nPrevBits, nPrevHeight, consensusParams, false);
    BOOST_CHECK_EQUAL(nSubsidy, 464285715ULL);
}

BOOST_AUTO_TEST_SUITE_END()

redpajama_set_name: RedPajamaGithub


text:
The Erasmus Charter is a member of the University of Sussex, created in 1961. The University of Sussex has 12 schools of study, and more than a third of the university's students are enrolled in postgraduate courses. Sussex has a diverse community of nearly 20,000 students, with around one in three being international students and over 1,000 academics, representing over 140 different nationalities.
Situated in Brighton, East Sussex, United Kingdom, all the students of Sussex have easy access to the South Downs National Park, Brighton's vibrant nightlife, shops, and the beautiful seafront. The university's campus is considered to be one of the most beautiful campuses in Britain, Sussex, it combines green open spaces with award-winning architecture designed by Scottish architect Sir Basil Spence.
The University of Sussex has produced many notable alumni, including 4 Nobel Prize winners, 15 Fellows of the Royal Society, and 10 Fellows of the British Academy. The University has also produced 24 fellows of the Academy of Social Sciences and a winner of the Crafoord Prize among its faculty.
University of Sussex was ranked "#226" in "Top World University" by "QS Global World Rankings."
University of Sussex was ranked "#151" in "University Ranking" by "The Times Higher Education."
University of Sussex was also ranked "#163" in "Global Universities" by "US News & World Report."
Here's what University of Sussex looks for in a student during the selection process. Learn about University of Sussex admission criteria, rate, and requirements below.
Acceptance Rate83 %
80 (iBT)
5G Mobile Communications and Intelligent Embedded Systems
22,500 GBP
The one-year master's course is designed to keep students up-to-date with the latest knowledge and skills required in modern digital technologies. The program is for students who would like to expand their knowledge and practical skills in the areas of mobile communications and intelligent embedded systems. Students will have access to a dedicated MSc Laboratory equipped with industry-standard hardware and software. Career opportunities include in fields such as mobile & satellite communications, aerospace, automotive & transport, defence & security and healthcare.
Wireless Computing, Mobile Computing
8,976 GBP
University Transcripts
1st September 1st August
The PhD program in accounting is ranked #17 in the world for citations in Business and Economics (The Times Higher Education, 2022). The program will help students to create research that has an impact on the areas of accounting, accountability, governance and performance management. The program specialises in research in social, environmental and ethical accountability and sustainability, data and analytics in accounting, capital market-based accounting research and governance and executive compensation.
1st September 1st June
1st January 1st October
GRE, GMAT, IELTS
1st September 30th April
1st January 31st October
The cost is an important aspect of choosing your dream college. Get detailed information about University of Sussex's tuition fees, living expenses and financial aid opportunities below.
Find miscellaneous information about students, campus and faculty of University of Sussex below.
Designed by Basil Spence, the campus of the University of Sussex has been visited by the Queen.
https://www.sussex.ac.uk/
Most asked questions related to the university
What is University of Sussex known for?
The University of Sussex is a major UK university for international students, located on the south coast of England. The University, which is best known for its world-class research, has 17,000 students enrolled in over 500 undergraduate, postgraduate, and foundation courses. Sussex is famously known for its courses in the Social Science and Humanities department.
Why should I go to the University of Sussex?
Studying at the University of Sussex allows you to make a difference in the world. Since its inception in 1961, the University has been the epitome of change, pushing for progress and demanding more in order to improve the lives of individuals and communities all over the world.
Is the University of Sussex good in athletics?
Yes. The Falmer sports centre alone has over 40 acres of playing field. The university also has agreements with Freedom Leisure, granting its students access to sports centres across West Sussex.
Does the University of Sussex have a degree in Psychology?
Yes. The School of Psychology in University of Sussex has the most academically diverse and influential communities of psychologists in the United Kingdom and offers a degree in Psychology.
Is the University of Sussex good for international students?
Sussex welcomes around 5,000 foreign students from over 100 countries each year. As a result, the University's campus is really international and multicultural. All international students have access to housing, and living on campus is one of the finest ways to get a feel for student life at Sussex.
Most discussed topics related to this university.
Posted by Robert Ferrero
Sussex India Scholarship deadline types of scholarshipslist of scholarshipsacademic scholarshipsscholarship for studying abroad
The Sussex India Scholarship deadline is 31st Jan 2021
Global - Representative

redpajama_set_name: RedPajamaCommonCrawl


Do you know someone who is about to give birth to a baby, or otherwise acquire one? That's great news!
A sleep deprivation test, so that new parents can FINALLY figure out whether or not they are tired.
A baby milestone chart, so that you will never forget the first time your baby vomited in the carseat.
Parenting tips you won't find anywhere else, possibly because they are not very helpful.
It's the perfect baby shower gift for making everyone laugh while also filling the expectant parents with quiet dread.


text:
UCL PSYCHOLOGY AND LANGUAGE SCIENCES
Faculty of Brain Sciences
UCL | PALS | Resources in Speech, Hearing & Phonetics
FaroSon - The Auditory Lighthouse
FaroSon is a Windows PC program for the real-time conversion of sound into a coloured pattern representing loudness, pitch and timbre. It is useful when teaching concepts of sound sensation, particularly when the audience is not suited to an acoustical account involving spectral analysis. It is also rather a pretty way to look at speech.
FaroSon (from "pharos" - lighthouse, and "son" - sound) is a program that turns sound signals into a scrolling coloured display in real-time. However, rather than presenting a physical analysis of the sound, Faroson attempts to present a psychological picture of the sound. While the horizontal axis of the pattern represents time, the other attributes of the picture are related to our subjective senses of loudness, pitch and timbre. The loudness of the sound is reflected in the brightness and saturation of the colours you see. The timbre of the sound is reflected in the colours themselves: sounds with predominantly bass character have a red colour, while sounds with a predominantly treble character have a blue colour. The pitch of the sound is reflected in the horizontal banding patterns: when the pitch of the sound is low, then the bands are large and far apart, and when it is high, the bands are narrow and close together. If the pitch of the sound is falling you see the bands diverge; when it is rising, you see the bands converge.
The input waveform is displayed at the top of the picture on a fixed amplitude scale to help you check that the input signal is a good size but not overloaded. At the bottom of the screen is a time axis, with tick marks every 100ms.
The program is only available for Windows PCs from
https://www.phon.ucl.ac.uk/downloads/faroson/faroson100.exe
To download the file, right click on the link above and choose "Save Target As". Save the file to your desktop or to a folder on your computer. Then run the file to install the program and to add an entry to your Start Programs menu. Once installed you can delete the downloaded file.
The signal is divided into overlapping frames of about 30ms, where each frame represents one vertical line of pixels. Time advances by about 1ms per horizontal pixel. Each frame is Hamming-windowed and a spectrum is computed using a discrete fourier transform. The relative preponderance of red, green and blue used to represent the frame is calculated from the relative energy in the low, mid and high frequency regions of the spectrum. The amplitude of the signal in each window is logarithmically compressed to generate a measure of loudness: this is used to modulate the brightness and saturation of the colour. The log amplitude spectrum is input to a further fourier analysis to derive the cepstrum, from which an estimate of fundamental frequency is found by determining the delay of the first rahmonic. The autocorrelation of the window is computed and the size of the autocorrelation peak is used to provide an estimate of periodicity. The fundamental frequency value and the periodicity measure are used to modulate the intensity of the presented colour in the vertical dimension.
Since FaroSon performs a lot of calculations in real-time, it can fail to keep up with the input signal. To show when this fault occurs, the bottom-most pixel of the window is used to as an indicator. If this horizontal line of pixels shows gaps, or if the display is choppy or has repeating sections, then you may need to adjust some configuration options. Choose File|Configuration and adjust the options as follows:
Controls how frequently the screen is updated. Range: from 5 frames/sec to 25 frames/sec. Default: 10 frames/sec. A higher rate gives a smoother scrolling display but puts more load on the graphics engine.
Undersampling rate
Controls how many frames are calculated directly and how many are calculated by interpolation. An undersampling rate of 3 means that one frame is calculated and two interpolated every three frames. Range: 1 to 5. Default: 3. A smaller value produces a display with fewer artefacts, but at a higher computational load.
Display waveform
Option to turn waveform display at top of window on and off.
Display timescale
Option to turn timescale display at bottom of window on and off.
Please send suggestions for improvements and reports of program faults to sfs@pals.ucl.ac.uk.
Please note that we are unable to provide help with the use of this program.
FaroSon is not public domain software, its intellectual property is owned by Mark Huckvale, University College London. However FaroSon may be used and copied without charge as long as the program remains unmodified and continues to carry this copyright notice. Please contact the author for other licensing arrangements. FaroSon carries no warranty of any kind, you use it at your own risk.
Some other pages on our site you may enjoy:
BROWSE - Simple tool for browsing audio files
BROWSE is a program for browsing audio recordings. With BROWSE you can drag and drop audio files in a range of formats onto the display to see and hear their contents. BROWSE allows you to zoom, scroll and save the audio files to other formats. More information.
CLEAR - Centre for Law-Enforcement Audio Research
The CLEAR project aims to create a centre of excellence in tools and techniques for the cleaning of poor-quality audio recordings of speech. More information.
Word count: . Last modified: 06:03 09-May-2019. by Mark Huckvale
© 2019 Speech, Hearing & Phonetic Sciences, UCL


text:
About The NTDC
Meet The Director General
Embassy Locations
Apply for Yellow Card
The Nigerian Media Assets
Tour Nigeria
The Nigerian National Assets
Familiarize with Nigeria before your trip.
The Nigerian Flag
The Nigerian flag is a vertical bicolour triband of green, white and green. The two green stripes represent Nigeria's natural wealth, while the white band represents peace. It was designed in 1959 by Michael Taiwo Akinkunmi and was officially adopted on October 1, 1960.
The Nigerian Coat of Arms
Nigeria's Coat of Arms features two white horses supporting a black shield, with a White "Y" shape running across the middle of the shield. Above the shield is a red eagle sitting on a green and white band placed on the shield. Beneath them are green grass, yellow flowers and a banderole.
The white horses represent dignity, the shield represents Nigeria's fertile soil, the eagle represents strength, while the green and white bands represent Nigeria's rich soil. The White 'Y' shape on the black shield represents the Niger and the Benue rivers. The yellow flower at the base is Nigeria's national flower, the Costus Spectabilis, and the banderole is Nigeria's national motto since 1978, 'Unity and Faith, Peace and Progress' (formerly 'Peace, Unity, Freedom').
The Nigerian National Anthem
Composed by Pa Benedict Odiase, MON
The Nigerian National Pledge
I pledge to Nigeria my Country
To be faithful, loyal and honest
To serve Nigeria with all my strength
To defend her unity
And uphold her honour and glory
The TOUR NIGERIA brand is an ambitious, timely and visionary brand for promoting domestic tourism in Nigeria. The vision of the master brand is to build the premier online destination for authentic Nigerian content, using technology, creativity, arts and culture to push the new national agenda. It will showcase the true spirit of Nigeria and tell the visual story of the most populous black nation in the world.
Address: Old Federal Secretariat, Area 1, Garki,
Abuja, Nigeria.
Email: hello@tournigeria.gov.ng
© 2020 Nigerian Tourism Development Corporation. All Rights Reserved. | Terms Of Use |
The Tour Nigeria trademark, phrase, symbol and design are registered trademarks of the Nigerian Tourism Development Corporation in the Federal Republic of Nigeria. NTDC reserves the right to initiate legal action if the trademarks are used without prior written consent.