text:
\section{Introduction}
In this paper, we consider the Benjamin-Ono equation on the real line,
\begin{equation}\label{eq:bo}
\partial_tu + H\partial_{x}^2 u = \partial_x(u^2), \quad (t,x)\in [0,T]\times\real,
\end{equation}
where $H$ denotes the Hilbert transform, defined through the Fourier transform as
$$
\widehat{Hf}(\xi)=-i\sgn(\xi)\hat{f}(\xi), \quad \xi\in \real.
$$
The Benjamin-Ono equation models the propagation of unidirectional deep water waves \cite{benjamin, ono}. As a nonlinear dispersive equation, it is nonlocal, completely integrable and the nonlinear term presents a loss of derivative. In fact, since the linear dispersion effects are quite weak, one cannot handle the nonlinearity perturbatively in order to prove local well-posedness results for initial data $u_0\in H^s(\real)$. This was proven rigorously by Molinet, Saut and Tzvetkov \cite{molinetsauttzevtkov} and improved by Koch and Tzvetkov \cite{kochtzevtkov1}: the flow map is shown not to be $C^2$ for $s\in \real$, or even uniformly continuous for $s>0$.

A first step in the local well-posedness theory was given by Iorio \cite{iorio} for $s>3/2$. Several refinements ensued: Ponce \cite{ponce} for $s=3/2$, Koch-Tzevtkov \cite{kochtzevtkov2} for $s>5/4$ and Kenig-Koenig \cite{kenigkoenig} for $s>9/8$. Later, using a variant of the Hopf-Cole transform (connected to the Burgers equation),
$$
w\sim\frac{1}{2i}ue^{-i\partial_{x}^{-1}u},
$$
Tao \cite{tao} noticed that the worst interactions in the nonlinearity disappear, allowing for a rather direct proof of  local well-posedness in $H^1(\real)$. The method was later improved by Burq-Planchon \cite{burqplanchon} for $s>1/4$ and by Ionescu-Kenig \cite{ionescukenig} for $s\ge 0$ (later revisited by Molinet-Pilod \cite{molinetpilod} and by Ifrim-Tataru \cite{ifrimtataru}). One of the main difficulties is the transfer of bounds from $u$ to $w$ (and vice-versa), which has been handled either through paralinearization or by decomposing the solution into low and high frequencies. It is worth mentioning that in \cite{ifrimtataru}, the authors prove the local well-posedness in the weighted space $L^2((1+x^2)dx)$ (see also \cite{fonsecalinaresponce}). Finally, in \cite{molinetpilod}, the solution is shown to be unique for $s>1/4$, while conditional uniqueness (that is, under the assumption that the solution belongs to some auxiliary space) holds in the class $L^\infty((0,T), H^s(\real))\cap L^4((0,T), W^{s,4}(\real))$ for $s>0$. In the periodic case, unconditional uniqueness has been proven in \cite{kishimoto1} for $s>1/6$.

The main problem we wish to study in the context of the Benjamin-Ono equation is the nonlinear smoothing phenomena: the difference between the nonlinear and the free evolutions starting from the same initial data is in fact smoother than the initial data. For generic dispersive equations of the form $u_t+iL(D)u=N(u)$, one aims to derive the estimate
$$
\| u(t) - e^{-itL(D)}u_0\|_{L^\infty(H^{s+\epsilon}(\real))}\le C(t, \|u\|_{L^\infty((0,t), H^s(\real))}).
$$
This feature has been initially discovered by Bona and Saut \cite{BS1} in the context of generalized KdV equations and extended to many other contexts (see \cite{babin, bourgain, tzirakis6, tzirakis3, keraani, linaresscialom}, among others), by using either maximal estimates, Bourgain spaces or integration by parts in the time variable. Recently, the author and Silva \cite{CorreiaSilva} derived a unifying strategy, based on the infinite normal form reduction (INFR), to prove this phenomena for general dispersive equations and applied it to several classical examples, such as the Korteweg-de Vries and the modified Zakharov-Kuznetsov equations. In the context of the Benjamin-Ono equation on the real line, there are no results concerning this property (see \cite{HKO} for a  smoothing effect outside of the origin under high regularity and decay assumptions).

The INFR has been introduced in \cite{guo, ko, koy}. The main idea is to consider the profile of a solution $u$, $\tilde{u}(t)=e^{-itL(D)}u(t)$, whose equation concentrates all the dispersive information in an oscillatory integral. By integrating by parts in time (an idea used in \cite{babin} and \cite{tzirakis3}), one expects to gain some regularity (coming from the phase function) at the expense of having time derivatives in the nonlinearity, which, upon replacement, yield higher-order terms. Afterwards, one may integrate by parts once again, for as many times as one so desires. The INFR is simply the formal construction of an infinite iteration of this argument.

Evidently, if the phase is stationary in time, the argument is not as direct, requiring a decoupling into resonant and nonresonant parts. This is reminiscent of the space-time resonances methodology presented in \cite{germain, masmoudi}. The main difference in their argument is the use of vector-fields to deal with time resonances, while the INFR uses the resonance condition as a restriction on the domain of the oscillatory integral. However, as one may observe in \cite{CorreiaSilva} and in Section 4, the INFR analysis ultimately hinges on the study of space-time resonances. One may even argue that the INFR is both a refinement and a simplification of the method of \cite{germain, masmoudi}.

The INFR method has also been used by Kishimoto in \cite{kishimoto1, kishimoto2} (see also \cite{gebalin}) to prove unconditional uniqueness at low regularity. Indeed, the advantage of working with an infinite expansion in arbitrarily high-order terms is that the analysis can be carried out without the use of any auxiliary space. Moreover, it is worth pointing out that in all nonlinear smoothing results for one-dimensional dispersive equations presented in \cite{CorreiaSilva} and in the present paper, the critical regularity for local well-posedness is reached. We believe this fact is not random and further research into this matter is necessary.

As expected, we are unable to prove nonlinear smoothing directly for \eqref{eq:bo}: a consequence of our methodology is a Lipschitz continuity estimate for the flow, which would contradict \cite{kochtzevtkov1}. We proceed as for the derivative Schrödinger equation (see \cite{CorreiaSilva, tzirakis6}) and prove the phenomenon for the gauged version of \eqref{eq:bo}, inspired in  \cite{kishimoto1} and \cite{tao}. Through the INFR, we are able to prove nonlinear smoothing and unconditional uniqueness for the Benjamin-Ono equation in weighted Sobolev spaces.
\begin{thm}\label{teo:principal}
	Fix $s>0$. Define
	$$
	\Sigma=\left\{u \in H^s(\real): xu\in L^2,\ \hat{u}(0)=0 \right\}.
	$$
	\begin{enumerate}
		\item 	Given $u_0\in \Sigma$, there exists $T>0$ and a unique local solution of \eqref{eq:bo}
		$$
		u\in C([0,T], H^s(\real))\cap L^\infty((0,T), \Sigma)
		$$ with $u(0)=u_0$.
		\item The gauge mapping $\mathcal{G}: \Sigma \to  H^{s+1}(\real)$
		$$
		u \mapsto \mathcal{G}u=e^{-i\partial_x^{-1}u/2}-1
		$$
		is well-defined and it is continuous.
		\item Given solutions $u, v\in C([0,T], \Sigma)$ of \eqref{eq:bo}, one has the local Lipschitz estimate
		$$
		\|\mathcal{G}(u) - \mathcal{G}(v)\|_{L^\infty((0,T), H^{s+1})}\lesssim C\left(T,\|\mathcal{G}(u(0))\|_{H^{s+1}}, \|\mathcal{G}(v(0))\|_{ H^{s+1}}\right)\|\mathcal{G}(u(0)) - \mathcal{G}(v(0))\|_{ H^{s+1}}
		$$
		Moreover, for $\epsilon<\max\{ s, 3/4 \}$,
		$$
		\|\mathcal{G}(u) - e^{-tH\partial_{x}^2}\mathcal{G}(u(0))\|_{L^\infty((0,T), H^{s+\epsilon+1})}\lesssim C(T, \|\mathcal{G}(u)\|_{L^\infty((0,T),H^{s+1})}).
		$$
	\end{enumerate}

\end{thm}

The use of a weighted space is necessary in order for the gauge mapping to lie in $H^{s+1}(\real)$. This is a crucial part of our analysis. To lift this restriction, one must find a way to treat the low-frequency terms, possibly as in \cite{ionescukenig}. On the other hand, the necessity of weighted spaces to prove nonlinear smoothing is not surprising, as it has been observed for the KdV equation in \cite{imt}.

The remainder of this work is organized as follows: in Section 2, we recall the gauge transformation and derive the gauged equation. In Section 3, we give a brief overview of the INFR method and point out the required adaptations for the Benjamin-Ono equation. Finally, in Section 4, we prove the two essential estimates that allow for the application of the INFR, thus concluding the proof of the main theorem.

\section{The gauge transformation}

Given $v\in C([0,T], \mathcal{S}'(\real))$, we define
$$
\tilde{v}(t):=\widehat{e^{tH\partial_{x}^2}v(t)}.
$$
Given an initial data $u_0\in \Sigma$, \cite[Proposition 7.2]{ifrimtataru} ensures that any solution $u\in C([0,T], H^s(\real))$ satisfies
$$
\|\partial_\xi \tilde{u}(t)\|_{L^2} =\|(x-2tH\partial_x)u(t)\|_{L^2}\lesssim C(t, \|u_0\|_{L^2}, \|xu_0\|_{L^2}).
$$
Moreover, taking into account that
$$
\tilde{u}_t(t,\xi) = i\xi e^{it|\xi|\xi} \widehat{u^2}(t,\xi),
$$
the application of Riemann-Lebesgue lemma shows that $\tilde{u}_t \in C([0,T]\times \real)$. Consequently, $\hat{u}(t,0)= \tilde{u}(t,0)=0$ for $t\in [0,T]$. These properties allow us to define the antiderivative of $u$,
$$
\hat{F}(t,\xi)=\frac{\hat{u}(t,\xi)}{i\xi}, \quad (t,\xi)\in [0,T]\times (\real\setminus\{0\}).
$$
By Hardy's inequality, $F\in L^\infty([0,T], H^{s+1}(\real))$. From \eqref{eq:bo}, we deduce
\begin{equation}\label{eq:eqF}
	F_t + H  F_{xx}= \frac{1}{2}(F_x)^2.
\end{equation}
Take $\phi=\mathbbm{1}_{[-1,1]}$ and define the projections
$$
\widehat{P_\pm f} = \mathbbm{1}_{\pm \xi>0}\hat{f},
$$
$$
\widehat{P_{lo} f} = \phi \hat{f},\quad \widehat{P_{hi} f} = (1-\phi) \hat{f}.
$$
We write $P_{\pm lo}=P_{\pm}P_{lo}$ and $P_{\pm hi}=P_{\pm}P_{hi}$ and let $\chi_\cdot$ be the Fourier multiplier corresponding to $P_\cdot$. These projections are well-defined whenever $\hat{f}\in L^1_{loc}$. We define the gauge transform
$$
V=e^{-iF/2}-1,\quad w=\partial_{x}V=ue^{-iF/2},
$$
and thus
\begin{equation}\label{eq:ucomow}
u=we^{iF/2}=(1+\bar{V})\partial_xV.
\end{equation}
Observe that
$$
\|V(t)\|_{H^{s+1}}\le \sum_{n\ge 1} \frac{\|F(t)\|_{H^{s+1}}^n}{2^nn!}\lesssim \|F(t)\|_{H^{s+1}}e^{\|F(t)\|_{H^{s+1}}/2}.
$$
Set
$$
V_\pm = P_{\pm hi}V,\quad V_{lo}=P_{lo}V, \quad w_\pm=P_{\pm hi}w, w_{lo}=P_{lo}w.
$$
It follows from \eqref{eq:eqF} that
\begin{align}
(V_+)_t + H(V_+)_{xx}& = (V_+)_t - i(V_+)_{xx} \\&= P_{+hi}\left[ -\frac{i}{2} e^{-iF/2}\left( F_t - i F_{xx} -  \frac{1}{2}( F_x)^2 \right) \right] \\&= P_{+hi}\left[ - e^{-iF/2}\left( P_- F_{xx} \right) \right]\label{eq:eqV}\\&= P_{+hi}\left[ - (e^{-iF/2}-1)\left( P_- F_{xx} \right) \right]
\\&= -P_{+hi}\left(V_+P_-\partial_{x}((1+\bar{V})\partial_xV)\right)=:Q_+(V)+C_+(V)
\end{align}
where $Q$ is quadratic and $C$ is cubic.
Analogously, the equation in the large negative frequency range is
\begin{equation}\label{eq:eqV2}
(V_-)_t + H(V_-)_{xx}
= -P_{-hi}\left(V_-P_+\partial_{x}((1+\bar{V})\partial_xV)\right)=:Q_-(V)+C_-(V)
\end{equation}
In the low spectrum range, we have
$$
(V_{lo})_t + H(V_{lo})_{xx} = -P_{+lo}\left(V P_-\partial_{x}u\right) -P_{-lo}\left(V P_+\partial_{x}u\right)
$$
Due to the restriction on the frequencies, the two terms are equivalent to $P_{lo}(wu)$. Since $w,u \in L^\infty((0,T), L^2)$, $\widehat{wu}\in L^\infty((0,T)\times\real)$ and $(V_{lo})_t\in L^\infty((0,T), H^\infty(\real))$.

\begin{lem}\label{lem:derivadaslimitadas}
	For $s\ge 0$, one has
	$$
\| (\tilde{V}_\pm)_t\|_{L^\infty((0,T)\times \real)}\lesssim \|V\|_{L^\infty((0,T),H^1)}^2 + \|V\|_{L^\infty((0,T),H^1)}^3.
	$$
\end{lem}
\begin{proof}
It suffices to prove that $\widehat{C}_\pm, \widehat{Q}_\pm \in L^\infty((0,T)\times \real)$. For $|\xi|<1$, $C\equiv Q \equiv 0$. For $|\xi|>1$, let us write $\widehat{Q}_+$:
$$
\widehat{Q}_+(V)=\int_{\xi_1>\xi>1} (\xi-\xi_1)\hat{V}(\xi_1)\hat{w}(\xi-\xi_1)d\xi_1 = \int_{\xi_1>\xi>1} \frac{\xi-\xi_1}{\xi_1}\left(\xi_1\hat{V}(\xi_1)\right)\hat{w}(\xi-\xi_1)d\xi_1 .
$$
Due to the restriction in the frequency space, the multiplier $(\xi-\xi_1)/\xi_1$ is bounded and
$$
\|\widehat{Q}_+(V)\|_{L^\infty((0,T)\times\real) }\lesssim \int |\xi_1\hat{V}(\xi_1)\hat{w}(\xi-\xi_1)|d\xi_1\lesssim \|V\|_{H^1}^2.
$$
The same argument applies for $Q_-$ and $C_\pm$.
\end{proof}

\section{The infinite normal form reduction}

In this section, we briefly explain the INFR method for general dispersive equations and how it can be used to prove nonlinear smoothing and uniqueness. We then conclude by stating the necessary adaptations to the Benjamin-Ono case. The discussion follows closely \cite{CorreiaSilva} and \cite{koy}.

For the sake of simplicity, let us start with a nonlinear dispersive equation with a single polynomial nonlinearity of order $k$,
$$
u_t + iL(D)u = N(u).
$$
Written in terms of the profile $\tilde{u}(t)=e^{-itL(D)}u(t)$,
\begin{align}
\tilde{u}(t,\xi) &= \tilde{u}(0,\xi) + \int_0^t \int_{\xi_1+\dots+\xi_k=\xi} e^{is\Phi(\Xi)}m(\Xi)\tilde{u}(s,\xi_1)\dots \tilde{u}(s,\xi_k)d\Xi ds\nonumber\\&=: \tilde{u}_0 + \int_0^t \mathcal{N}^{(1)}(\tilde{u}(s))ds\label{eq:geral}
\end{align}
We abuse the notation a bit, by forgetting possible complex conjugates of $\tilde{u}$.

We introduce some operators which will be crucial for the subsequent analysis. Given $\sigma\in (0,1)$, define the \textit{phase-weighted operator} as
\begin{equation}
\mathcal{F}\left[T_\sigma(u_1,\dots,u_k)\right](\x)=\int_{\x_1+\dots + \x_k=\x} \frac{1}{\jap{\Phi(\Xi)}^\sigma}m(\Xi)\hat{u}_1(\x_1)\dots \hat{u}_k(\x_k) d\x_1\dots d\x_{k-1}.
\end{equation}
Moreover, for any $\alpha\in \real$ and $M>0$, define the \textit{frequency-restricted operator}
$$
\mathcal{F}[T^{\alpha,M}(u_1,\dots, u_k)](\xi)=\int_{\substack{\x_1+\dots + \x_k=\x\\|\Phi(\Xi)-\alpha|<M}}m(\Xi)\hat{u}_1(\xi_1)\dots \hat{u}_k(\xi_k)d\x_1\dots d\x_k,
$$
The main assumption is the existence of appropriate multilinear bounds for these operators.

\begin{assumpt}
	For $\epsilon>0$ fixed, there exists $\sigma\in (0,1)$ such that
	\begin{equation}\label{eq:basicaepsilon}\tag{Bound$_{\sigma,\epsilon}$}
	\|T_\sigma(u_1,\dots,u_k)\|_{H^{s+\epsilon}}\lesssim \prod_{j=1}^{k}\|u_j\|_{H^s}.
	\end{equation}
	Moreover, there exist $\gamma,\beta$ satisfying
	$$
	\beta>0,\quad  \theta:=1-\max\{\gamma+\beta, \sigma+\gamma\}>0.
	$$
	such that, for any $\alpha\in \real$ and $M>1$,
	\begin{equation}\label{eq:frequencyrestrictedgen}\tag{Bound$^{\alpha,M}$}
	\|T^{\alpha,M}(u_1,\dots,u_k)\|_{H^s}\lesssim  \sup\{\jap{\alpha}^{\gamma},M^{\gamma}\}M^{\beta}\prod_{j=1}^{k}\|u_k\|_{H^s}
	\end{equation}
\end{assumpt}

\begin{nb}\label{nota:alphaimplicasigma}
	Observe that we only require a gain of regularity for the phase-weighted operator, which is centered around $\Phi=0$. For the frequency-restricted one, the phase is centered around $\alpha$, which may create additional difficulties in achieving a higher regularity. This separation is often useful (see \cite{CorreiaSilva}). However, it may happen that even for the frequency-restricted operator, a gain is already possible:
	\begin{equation}\label{eq:frequencyrestrictedepsilon}\tag{Bound$^{\alpha,M}_\epsilon$}
	\|T^{\alpha,M}(u_1,\dots,u_k)\|_{H^{s+\epsilon}}\lesssim  \sup\{\jap{\alpha}^{\gamma},M^{\gamma}\}M^{\beta}\prod_{j=1}^{k}\|u_k\|_{H^s}
	\end{equation}
	In this case, one has freely (Bound$_{\sigma,\epsilon}$) for $\sigma>\gamma+\beta$. Indeed, decomposing dyadically in $\Phi$,
\begin{align*}
	\|T_\sigma(u_1,\dots,u_k)\|_{H^{s+\epsilon}}&\lesssim \sum_{M \mbox{ dyadic}} \frac{1}{M^{\sigma}}\|T^{0,M}(u_1,\dots,u_k)\|_{H^{s+\epsilon}} \\&\lesssim \sum_{M \mbox{ dyadic}} M^{\gamma+\beta-\sigma} \prod_{j=1}^{k}\|u_k\|_{H^s} \lesssim \prod_{j=1}^{k}\|u_k\|_{H^s}.
\end{align*}
If $2\gamma+\beta<1$, the conditions of Assumption 1 are directly verified.
\end{nb}

For a fixed $N>1$ (which will be determined later), let us split the frequency domain into the near-resonant and nonresonant regions, depending on whether $|\Phi|$ is smaller or greater than $N$. We use the subscripts $1$ for the near-resonant term and $2$ for the nonresonant one:
$$
\tilde{u}(t,\x)=\tilde{u}(0,\x) + \int_0^t \mathcal{N}_1^{(1)}(\tilde{u}(s)) +\mathcal{N}_2^{(1)}(\tilde{u}(s)) ds.
$$
Using \eqref{eq:basicaepsilon},

\begin{equation}\label{eq:primeirotermo}
\left\|\mathcal{F}^{-1}\left(\mathcal{N}_1^{(1)}(\tilde{u}(t))\right)\right\|_{H^{s+\epsilon}}\lesssim N^\sigma\|T_\sigma(\tilde{u},\dots,\tilde{u})(t)\|_{H^{s+\epsilon}}\lesssim N^\sigma\|u(t)\|_{H^s}^k.
\end{equation}
For $\mathcal{N}_2^{(1)}$, one integrates by parts in time, using the relation
$$
e^{is\Phi}=\partial_s\left(\frac{1}{i\Phi}e^{is\Phi}\right)
$$
so that
$$
\int_0^t \mathcal{N}_2^{(1)}(\tilde{u}(s)) ds = \left[\mathcal{N}_0^{(2)}(\tilde{u}(s)) \right]_{s=0}^{s=t} + \int_0^t \mathcal{N}^{(2)}(\tilde{u}(s)) ds.
$$
Due to the factor $1/\Phi$ and the restriction on the frequency domain, the boundary terms are easily bounded:
$$
\left\|\mathcal{F}^{-1}\left(\mathcal{N}_0^{(2)}(\tilde{u}(s))\right)\right\|_{H^{s+\epsilon}}\lesssim \frac{1}{N^{1-\sigma}}\|T_\sigma(\tilde{u},\dots,\tilde{u})(s)\|_{H^{s+\epsilon}}\lesssim N^{-1+\sigma}\|u\|_{H^s}.
$$
For the remainder $\mathcal{N}^{(2)}$, one uses \eqref{eq:geral} to replace $\tilde{u}_t$ and obtain a new oscillatory integral which is of order $2(k-1)+1$ in $\tilde{u}$.

So far, we have rewritten \eqref{eq:geral} as
$$
\tilde{u}(t,\xi) = \tilde{u}(0,\xi) + \mbox{controllable terms} + \int_0^t \mathcal{N}^{(2)}(\tilde{u}(s))ds.
$$

This concludes the first step in the INFR. One may now apply a recursive algorithm to expand the remainder integral $\mathcal{N}^{(2)}$. Fix $0<\delta<\theta/\beta$ and the sequence
$$
c_j=(j+1)^{2/\theta},\quad j\in \mathbb{N}.
$$
\vskip10pt
\underline{At the $J$-th step,}
\vskip10pt
\noindent\textit{Step 1.} Split the frequency domain into
\begin{itemize}
	\item Near-resonant: $\mbox{|Phase|}<c_J\mbox{|Phase at step 1|}^{\delta}$;
	\item Nonresonant: $\mbox{|Phase|}>c_J\mbox{|Phase at step 1|}^{\delta}$;
\end{itemize}
and write $$\int_0^t\mathcal{N}^{(J+1)}(\tilde{u}(s))ds=\int_0^t\mathcal{N}_1^{(J+1)}(\tilde{u}(s))+\mathcal{N}_2^{(J+1)}(\tilde{u}(s))ds.
$$
\textit{Step 2.} Integrate by parts in time the nonresonant term:
$$
\int_0^t \mathcal{N}_2^{(J+1)}(\tilde{u}(s))ds = \left[\mathcal{N}_0^{(J+2)}(\tilde{u}(s))  \right]_{s=0}^{s=t} + \int_0^t \mathcal{R}^{(J+2)}(\tilde{u}(s))ds.
$$
\textit{Step 3.} Use \eqref{eq:geral} to replace all instances of $\tilde{u}_t$ in $\mathcal{R}$ in order to obtain an oscillatory integral $\mathcal{N}^{(J+2)}$ of order $(k-1)(J+2)-1$ in $\tilde{u}$.
\vskip5pt
\noindent\textit{Step 4.} Repeat the algorithm for $\mathcal{N}^{(J+2)}(\tilde{u})$.

\vskip10pt
After an infinite amount of steps, one formally obtains the normal form equation
\begin{equation}\label{eq:NFE}\tag{NFE}
\tilde{u}(t,\xi)=\tilde{u}(0,\xi) + \sum_{j\ge 2} \left[\mathcal{N}_0^{(j)}(\tilde{u}(s))\right]_{s=0}^{s=t} + \int_0^t \sum_{j\ge 1} \mathcal{N}_1^{(j)}(\tilde{u}(s))ds.
\end{equation}

The basic multilinear estimates \eqref{eq:basicaepsilon} and \eqref{eq:frequencyrestrictedgen} can be propagated throughout the algorithmic procedure.

\begin{lem}[$H^s$ bounds for the \eqref{eq:NFE}]\label{lem:HsboundsNFE}
	For any $J\ge 2$,
	$$
	\left\|\mathcal{F}^{-1}[\mathcal{N}_1^{(J)}(\tilde{u})]\right\|_{H^{s+\epsilon}} \lesssim N^{-\theta-\delta\theta(J-2)+\delta\beta}\|u\|_{H^s}^{J(k-1)+1},
	$$
	$$
	\left\|\mathcal{F}^{-1}[\mathcal{N}_0^{(J+1)}(\tilde{u})]\right\|_{H^{s+\epsilon}}\lesssim N^{-\theta-\delta\theta(J-2)+\delta(\beta-1)}\|u\|_{H^s}^{J(k-1)+1},
	$$
	$$
	\left\|\mathcal{F}^{-1}[\mathcal{N}_1^{(J)}(\tilde{u}) - \mathcal{N}_1^{(J)}(\tilde{v})]\right\|_{H^{s+\epsilon}} \lesssim N^{-\theta-\delta\theta(J-2)+\delta\beta}\left(\|u\|_{H^s}^{J(k-1)} + \|v\|_{H^s}^{J(k-1)}\right)\|u-v\|_{H^s}
	$$
	and
	$$
	\left\|\mathcal{F}^{-1}[\mathcal{N}_0^{(J+1)}(\tilde{u})- \mathcal{N}_0^{(J+1)}(\tilde{v})]\right\|_{H^{s+\epsilon}}\lesssim N^{-\theta-\delta\theta(J-2)+\delta(\beta-1)}\left(\|u\|_{H^s}^{J(k-1)} + \|v\|_{H^s}^{J(k-1)}\right)\|u-v\|_{H^s}.
	$$
\end{lem}

\begin{nb}
	Since the estimates of Lemma \ref{lem:HsboundsNFE} decay exponentially in $J$ for $N\gg M:=\|u\|_{L^\infty((0,T),H^s)}$, one obtains a formal bound on \eqref{eq:NFE},
	\begin{align*}
	\|\mathcal{F}^{-1}[\tilde{u}(t)-\tilde{u}(0)]\|_{H^{s+\epsilon}} &\lesssim \sum_{J\ge 2} \left\|\mathcal{F}^{-1}[\mathcal{N}_0^{(J)}(\tilde{u}(t)) ]\right\|_{H^{s+\epsilon}} + \sum_{J\ge 2} \left\|\mathcal{F}^{-1}[\mathcal{N}_0^{(J)}(\tilde{u}(0)) ]\right\|_{H^{s+\epsilon}}\\&\qquad + \sum_{J\ge 1} \int_0^t \left\|\mathcal{F}^{-1}[\mathcal{N}_1^{(J)}(\tilde{u}(s)) ]\right\|_{H^{s+\epsilon}}ds\\&\lesssim \sum_{J\ge 2} N^{-\theta-\delta\theta(J-2)+\delta\beta}M^{(k-1)J+1} + TN^{1-\sigma}\\&\qquad +T\sum_{J\ge 2}N^{-\theta-\delta\theta(J-2)+\delta(\beta-1)}M^{(k-1)(J+1)+1}\\&\lesssim C\left(T,\|u\|_{L^\infty((0,T),H^s)}\right),
	\end{align*}
	which is precisely the nonlinear smoothing property. This can be shown to be valid for smooth initial data (such that $\tilde{u}_t\in H^s$). Therefore, if a nice well-posedness theory is available, one may argue by density and obtain the nonlinear smoothing result without any further considerations (this was the approach taken in \cite{CorreiaSilva}). However, for the Benjamin-Ono equation, due to the lack of a proper well-posedness result at low regularity, this argument only proves that solutions which can be approximated by smooth ones satisfy the nonlinear smoothing property.
\end{nb}

To make use of the formal computations, we have to justify the INFR procedure, which uses three properties:
\begin{enumerate}[label=(\subscript{P}{{\arabic*}})]
	\item  The product rule for the time derivative can be applied for a.e. $\xi\in \real$;
	\item  One can switch time derivatives and integrals in $\xi$;
	\item The remainder tends to zero as $J\to \infty$ (in some weaker norm).
\end{enumerate}
The first property can be ensured if, for a.e. $\xi\in \real$, $\tilde{u}_t(\cdot,\xi)\in L^\infty(0,T)$. As a rule of thumb, this can  be seen if $k<2d/(d-2s)^+$: since $u^k\in C((0,T),L^1(\real^d))$, by Riemann-Lebesgue Lemma, $\mathcal{F}(|u|^k)\in C((0,T)\times \real^d)$. Thus, for $\xi$ fixed, $\mathcal{F}(|u|^k)\in C((0,T))$. Since the difference between $|u|^k$ and $N(u)$ are absolute values and possibly spatial derivatives, this should be enough to see that  $\mathcal{F}(N(u))(\cdot,\xi)\in C((0,T))$.

The second property can be justified at the $H^s$ level, for $s\ge 0$, following \cite[Remark 4.5]{koy}. To prove that the remainder tends to $0$, we make the following
\begin{assumpt}
	Take $\gamma, \beta$ and $\sigma$ as in Assumption 1. There exists $\mu\in \real$ such that
	\begin{equation}\label{eq:definormamu}
	\| \jap{\cdot}^{-\mu} \tilde{u}_t\|_{L^\infty}<\infty.
	\end{equation}
	Moreover, one has the following weak phase-weighted and frequency-restricted estimates:
	\begin{equation}\label{eq:basicaepsilonfraca}
	\| \jap{\cdot}^{-\mu} T_\sigma(u_1,\dots,u_k)\|_{L^\infty}\lesssim \min_j\left\{ \|\jap{\cdot}^{-\mu} u_j \|_{L^\infty}  \prod_{l\neq j}\|u_l\|_{H^s}\right\}
	\end{equation}
	\begin{equation}\label{eq:freqrestrfraca}
	\| \jap{\cdot}^{-\mu} T^{\alpha,M}(u_1,\dots,u_k)\|_{L^\infty}\lesssim (|\alpha|+M)^{\gamma}M^\beta \min_j\left\{ \|u_j\|_{L^\infty}  \prod_{l\neq j}\|u_l\|_{H^s}\right\}.
	\end{equation}
\end{assumpt}

Notice that, at each step of the INFR, the remainder term $\mathcal{R}$ is essentially the boundary term $\mathcal{N}_0$ with one $\tilde{u}$ replaced by $\tilde{u}_t$. Then, applying the weak phase-weighted and frequency-restricted estimates, we have
\begin{lem}
	For any $J\ge 2$,
	$$
	\| \jap{\cdot}^{-\mu} \mathcal{R}^{(J+2)}(\tilde{u})\|_{L^\infty}\lesssim N^{-\theta-\delta\theta(J-2)+\delta(\beta-1)}\|u\|_{H^s}^{J(k-1)}\| \jap{\cdot}^{-\mu}\tilde{u}_t\|_{L^\infty}.
	$$
\end{lem}

\begin{thm}\label{teo:geral}
	Fix $s\ge 0$.
	\begin{enumerate}
		\item Under Assumptions 1 and 2 (with $\epsilon=0$), if ($P_1$) holds, then equation \eqref{eq:geral} has a unique solution. Moreover, the data-to-solution map is Lipschitz continuous in $H^s$.
		\item Given $\epsilon>0$, under Assumptions 1 and 2, if ($P_1$) holds, then equation \eqref{eq:geral} satisfies the nonlinear smoothing property of order $\epsilon$.
	\end{enumerate}
\end{thm}
\begin{proof}
	We start with the uniqueness statement. Given two solutions $u,v \in C([0,T], H^s(\real^d))$, the hypothesis ensure that both solutions satisfy \eqref{eq:NFE}. Applying the Lipschitz estimates of Lemma \ref{lem:HsboundsNFE}, for $N\gg \|u\|_{L^\infty((0,T), H^s)} + \|v\|_{L^\infty((0,T), H^s)}$, one has
	$$
	\|u-v\|_{L^\infty((0,T), H^s)}\lesssim \|u_0-v_0\|_{H^s}
	$$
	and the first claim follows. Given $\epsilon>0$, by hypothesis, the formal bounds from Lemma \ref{lem:HsboundsNFE} can be rigorously justified. The nonlinear smoothing property now follows from Remark 4.
\end{proof}

\begin{nb}
	All of the above considerations remain valid for systems of several dispersive equations and several nonlinear terms: one requires phase-weighted and frequency-restricted estimates for each term separately, for a uniform choice of parameters $\gamma, \beta$ and $\sigma$.
\end{nb}

\begin{nb}
The equations for  $V_\pm$ involve the low-frequency part  $V_{lo}$. As time derivatives fall onto this factor, \textit{we do not replace it by its evolution equation}. Instead, we use the fact that it is bounded in $H^s$, which is enough for the application of the iterated estimates. For these terms, no further reduction is necessary. A similar strategy was taken in \cite{CorreiaSilva} for the derivative nonlinear Schrödinger equation.
\end{nb}

\section{Proof of the main results}

	In the context of the Benjamin-Ono equation, property (\subscript{P}{1}) follows directly from Lemma \ref{lem:derivadaslimitadas}. Therefore, by Theorem \ref{teo:geral}, our work is reduced to the proof of bounds \eqref{eq:frequencyrestrictedgen} and \eqref{eq:basicaepsilon} (in both strong and weak forms). Following Remark \ref{nota:alphaimplicasigma}, we further reduce our problem by showing the stronger version \eqref{eq:frequencyrestrictedepsilon} for each nonlinear term.
\begin{lem}[Frequency-restricted estimate for the cubic nonlinearities]\label{lem:cubic}
For $s>0$ and $0\le\epsilon<\min\{s,3/4\}$, set $$\gamma(\epsilon)=\max\left\{\frac{1}{4}+\frac{\epsilon-s}{2}, \epsilon-\frac{1}{2}, 0\right\}.
$$
One has
\begin{equation}\label{eq:estimcubicforte}
	\left\|T^{\alpha,M}[C_\pm](V_1,V_2,V_3) \right\|_{H^{s+\epsilon+1}}\lesssim (|\alpha|+ M)^{\gamma(\epsilon)}M^{1/2}\|V_1\|_{H^{s+1}}\|V_2\|_{H^{s+1}}\|V_3\|_{H^{s+1}}
\end{equation}
\begin{align}
&\left\|T^{\alpha,M}[C_\pm](V_1,V_2,V_3) \right\|_{L^\infty}\lesssim (|\alpha|+ M)^{\gamma(0)}M^{1/2}\label{eq:estimcubicfraco} \min_j\left\{ \|\hat{V}_j\|_{L^\infty}\prod_{k\neq j}\|V_k\|_{H^{s+1}}\right\}
\end{align}
\end{lem}
\begin{proof}
	We begin with \eqref{eq:estimcubicforte}. By duality and Cauchy-Schwarz, the estimate is reduced to a bound on
	$$
	I^{\alpha,M}_\epsilon=\int_{\xi>\xi_1>1} \frac{\jap{\xi}^{2s+2\epsilon+2}|\xi_2+\xi_3|^2   }{ \xi_1^2\jap{\xi_1}^{2s}\jap{\xi_2}^{2s+2} \jap{\xi_3}^{2s}}\mathbbm{1}_{|\Phi-\alpha|<M}d\xi_1d\xi_2
	$$
	where
	$$
	\Phi=|\xi|\xi - |\xi_1|\xi_1 + |\xi_2|\xi_2 -|\xi_3|\xi_3.
	$$
	Since $\jap{\xi}\le \jap{\xi_1}\jap{\xi_2}\jap{\xi_3}$, we may suppose that $s$ is as close to $\epsilon$ as we desire.

	Region A: $|\xi_1|\ll |\xi_2|, |\xi_3|$. Then $|\xi_2|\sim |\xi_3|\gg |\xi|$ and, writing $\xi_1=\xi-\xi_2-\xi_3$,
	$$
	|\partial_{\xi_2}\Phi| = 2||\xi_1|+|\xi_2||\sim |\xi_2|.
	$$
	Hence
	\begin{align*}
	I^{\alpha,M}_\epsilon \lesssim \int \frac{|\xi_2|^{2+2\epsilon}}{\jap{\xi_2}\jap{\xi_3}^{4s+1}}\mathbbm{1}_{|\Phi-\alpha|<M} \frac{d\Phi}{|\xi_2|}d\xi_3 \lesssim M.
	\end{align*}

	Region B: $|\xi_2|\lesssim |\xi_1|\ll |\xi_3|$ or $|\xi_3|\lesssim |\xi_1|\ll |\xi_2|$. This implies that $|\xi|\gg |\xi_1|$, which is impossible.

	Region C: $|\xi_1|\gtrsim |\xi_2|, |\xi_3|$.  Then, writing $\xi_2=\xi-\xi_1-\xi_3$,
	$$
	|\partial_{\xi_1}\Phi|\gtrsim |\xi_1|.
	$$
	Case I. $|\xi_2|\gtrsim |\xi_3|$. If $|\xi_2|\gtrsim |\xi|$,
	$$
	I^{\alpha,M}_\epsilon\lesssim \int_{\xi>\xi_1>1} \frac{\jap{\xi}^{2s+2\epsilon+2}|\xi_2|^2   }{ \xi_1^2\jap{\xi_1}^{2s}\jap{\xi_2}^{2s+2} \jap{\xi_3}^{2s}}\mathbbm{1}_{|\Phi-\alpha|<M}\frac{d\Phi}{|\xi_1|}d\xi_3 \lesssim \int_{\xi>\xi_1>1}\frac{1}{\jap{\xi_3}^{2s+1}} \mathbbm{1}_{|\Phi-\alpha|<M}d\Phi d\xi_3\lesssim M
	$$
	If $|\xi_2|\ll |\xi|$, since $|\partial_{\xi_1}\Psi|\gtrsim |\xi_1|\sim |\xi|$,
	$$
	|\Psi|\gtrsim |\xi(\xi_2+\xi_3)|.
	$$
Hence, for $a=\max\{2\epsilon-1,0\}$ and since $\epsilon<\min\{3/4, s\}$,
\begin{align*}
	I^{\alpha,M}_\epsilon&\lesssim \int \frac{|\xi|^{2\epsilon}|\xi_2+\xi_3|^2}{\jap{\xi_2}^{2s+1}\jap{\xi_3}^{2s+1}}\mathbbm{1}_{|\Phi-\alpha|<M} \frac{d\Phi}{|\xi_1|}d\xi_3\\&\lesssim  \int \frac{|\xi|^{2\epsilon-a}|\xi_2+\xi_3|^{2-a}|\Phi|^a}{\jap{\xi_2}^{2s+1}\jap{\xi_3}^{2s+1}}\mathbbm{1}_{|\Phi-\alpha|<M} \frac{d\Phi}{|\xi_1|}d\xi_3\\&\lesssim (|\alpha|+M)^a\int \frac{1}{\jap{\xi_3}^{2s+1}}\mathbbm{1}_{|\Phi-\alpha|<M} d\Phi d\xi_3 \lesssim (|\alpha|+M)^aM,
\end{align*}
	Case II. $|\xi_2|\ll |\xi_3|$. Then $\xi_3<0$. If $|\xi|\lesssim |\xi_2|$, we proceed as in Case I.
	Otherwise, since
	$$
	\Phi = \xi^2 -(\xi-\xi_2-\xi_3)^2 \pm \xi_2^2 + \xi_3^2 \sim \xi\xi_3,
	$$
	we have $|\xi\xi_3|^{1/2}\lesssim (|\alpha|+M)^{1/2}$. Writing $\xi_3=\xi-\xi_1-\xi_2$,
	$$
	|\partial_{\xi_1}\Phi|\sim ||\xi_1|-|\xi_3||=|\xi_1+\xi_3|\sim |\xi|.
	$$
	Therefore, since $2\epsilon-1<2-2s$,
	\begin{align*}
	I^{\alpha,M}_\epsilon&\lesssim \int \frac{|\xi|^{2s+2\epsilon+2}|\xi_3|^2}{|\xi_1|^{2+2s}\jap{\xi_3}^{2s} \jap{\xi_2}^{2s+2} } \mathbbm{1}_{|\Phi-\alpha|<M}d\xi_1 d\xi_2 \\&\lesssim \int \frac{|\xi|^{2s+2\epsilon+1}|\xi_3|^2}{|\xi_1|^{2+2s}|\xi_3|^{2s} \jap{\xi_2}^{2s+2} } \mathbbm{1}_{|\Phi-\alpha|<M}d\Phi d\xi_2  \\&\lesssim \int \frac{|\xi|^{2\epsilon-1}|\xi_3|^{2-2s}}{\jap{\xi_2}^{2s+2} } \mathbbm{1}_{|\Phi-\alpha|<M}d\Phi d\xi_2  \\&\lesssim \int \frac{|\xi\xi_3|^{1/2+\epsilon-s}}{ \jap{\xi_2}^{2s+2} } \mathbbm{1}_{|\Phi-\alpha|<M}d\Phi d\xi_2 \lesssim (|\alpha|+M)^{1/2+\epsilon-s}M.
	\end{align*}
	For estimate \eqref{eq:estimcubicfraco}, we bound directly
\begin{align*}
	&\|T^{\alpha,M}[C_+](V_1,V_2,V_3)\|_{L^\infty}\lesssim \int_{\xi_1>\xi>1} \frac{|\xi_2+\xi_3|}{\xi_1}\mathbbm{1}_{|\Phi-\alpha|<M}\hat{w}_1(\xi_1)\hat{V}_2(\xi_2)\hat{w}_3(\xi_3)d\xi_1d\xi_2 \\&\lesssim \left(\int_{\xi_1>\xi>1}\frac{|\xi_2+\xi_3|^2\jap{\xi_1}^2}{|\xi_1|^2\jap{\xi_2}^{2s+2}\jap{\xi_3}^{2s} }\mathbbm{1}_{|\Phi-\alpha|<M} d\xi_1d\xi_2\right)^{1/2}\|V_1\|_{L^\infty}\|V_2\|_{H^{s+1}}\|V_3\|_{H^{s+1}}
\end{align*}
As the integral is quite similar to $I^{\alpha,M}_0$, the proof for \eqref{eq:estimcubicforte} easily extends to this case. The same argument can be applied for $C_-$ and for the other combinations of $L^\infty$ and $H^s$ norms.
\end{proof}
\begin{lem}[Frequency-restricted estimate for the quadratic nonlinearities]\label{lem:quad}
	For $s\ge 0$ and $0\le\epsilon<\min \{s,3/4\}$, set
	$$
	\gamma(\epsilon)=\max\left\{\frac{1}{2}+\epsilon-s, 0\right\}.
	$$
	Then
	\begin{equation}\label{eq:estimquadforte}
	\left\|T^{\alpha,M}[Q_\pm](V_1,V_2) \right\|_{H^{s+\epsilon+1}}\lesssim (|\alpha|+ M)^{\gamma(\epsilon)}M^{1/2}\|V_1\|_{H^{s+1}}\|V_2\|_{H^{s+1}}
	\end{equation}
	\begin{align}
	&\left\|T^{\alpha,M}[Q_\pm](V_1,V_2) \right\|_{L^\infty}\lesssim (|\alpha|+ M)^{\gamma(0)}M^{1/2}\label{eq:estimquadfraco} \min\left\{ \|\hat{V}_1\|_{L^\infty}\|V_2\|_{H^{s+1}}, \|V_1\|_{H^{s+1}}\|\hat{V}_2\|_{L^\infty} \right\}
	\end{align}
\end{lem}
\begin{proof}
Here, we are led to study the integral
$$
J^{\alpha,M}=\int_{\xi>\xi_1>1} \frac{\jap{\xi}^{2s+2\epsilon+2}\xi_2^2}{\xi_1^2\jap{\xi_1}^{2s}\jap{\xi_2}^{2s}} \mathbbm{1}_{|\Psi-\alpha|<M}d\xi_2,
$$
where $\Psi=\xi^2-\xi_1^2 +\xi_2^2=2\xi\xi_2$. Once again, we assume that $s$ is close to $\epsilon$. Without loss of generality, suppose that $|\xi|>|\xi_2|$. Since $|\partial_{\xi_2}\Psi|\sim |\xi|$,
\begin{align*}
J^{\alpha,M}&\lesssim \int_{\xi>\xi_1>1} \frac{\jap{\xi}^{2s+2\epsilon+1}\xi_2^2}{\xi_1^2\jap{\xi_1}^{2s}\jap{\xi_2}^{2s}} \mathbbm{1}_{|\Psi-\alpha|<M}d\Psi\\&\lesssim \int_{\xi>\xi_1>1} |\xi|^{2\epsilon-1}|\xi_2|^{2-2s} \mathbbm{1}_{|\Psi-\alpha|<M}d\Psi\\& \lesssim \int_{\xi>\xi_1>1} |\Psi|^{\frac{1}{2}+\epsilon-s} \mathbbm{1}_{|\Psi-\alpha|<M}d\Psi \lesssim (|\alpha|+M)^{\frac{1}{2}+\epsilon-s}M,
\end{align*}
where we use, in the penultimate estimate, the fact that $2\epsilon-1<2-2s$. The $L^\infty$ bound follows as for the cubic nonlinearity.
\end{proof}

\begin{proof}[Proof of Theorem \ref{teo:principal}]
The existence of solution and the definition of the gauge mapping are a consequence of the considerations of Section 2. Applying the INFR procedure and the estimates from Lemmas \ref{lem:cubic} and \ref{lem:quad}, the Lipschitz continuity, the nonlinear smoothing property and the uniqueness for the gauge transform $V=\mathcal{G}(u)$ follows. Finally, the uniqueness for $u$ follows from that of $V$.
\end{proof}

\section{Acknowledgements}
The author would like to thank Jorge Silva and Felipe Linares for helpful suggestions and comments. The author was partially supported by Funda\c{c}\~ao para a Ci\^encia e Tecnologia, through the grant UID/MAT/04459/2019.

redpajama_set_name: RedPajamaArXiv


Vitamin C is a nutrient your body uses for many critical functions including: maintaining your bones, muscles, and blood vessels, absorbing iron, and making sure you stay strong and healthy during the winter season. It helps to fight off free radicals that can damage your healthy cells. Vitamin C is naturally found in oranges and other citrus fruits, but it can also be taken orally. Dr. Bennett recommends taking 1000 mg of Vitamin C, five times a day to help boost your immune system. Vitamin C is shown to help increase the number of white blood cells in your body. If you have a low amount of white blood cells, then you could be more susceptible to illnesses.
Elderberries are a great natural way to boost your immune system. They contain chemical compounds that are shown to boost your immune system naturally. Elderberries can be consumed in many ways including tea, juice, jams and jellies, lozenges, and syrup. The syrup is probably the most common and is Dr. Bennett's go to for consuming Elderberry.
When you have a misalignment or subluxation in your spine it causes stress on your nervous system. This stress can lower your immune system which could lead to an increased chance of a cold or sickness. Our adjustments reduce and remove the stress on your nervous system which results in a strong immune system. Dr. Bennett recommends receiving a weekly adjustments to make sure your spine and nervous system are in tip-top shape.
Oregano essential oil is a great way to not only boost your immune system, but help minimize symptoms in the event of the cold or flu. The oil has many properties that help your body fight off the sickness including anti-inflammatory, antibacterial, and antiseptic properties. You can utilize Oregano is your diffuser by placing 4-5 drops in the water and diffusing it 3-4 times throughout the day.
Your gut is the main factor in your overall health. If you have a leaky gut, you probably have a lot of health issues going on. Did you know that 80% of your immune system is in your gut? If you don't have good gut health then your chances of getting sick this winter will increase drastically. You should take Turmeric and a multi-strain probiotic with at least 25 billion units to help heal your gut.


Butts Park ArenaReplay for women's First World War football match in Coventry to kick off on Bank Holiday Monday Butts Park Arena to host game when players from Coventry City Ladies represent two factory sides from the city from 1917 – Rudge-Whitworth and Humber Ladies.
Coventry City Ladies FCSky Blues' Steve Waggott joins board of Coventry City LadiesSky Blues development director Steve Waggott has been added to the board of Coventry City Ladies in a bid to improve links between the two sides.
SkyWomen's Football: Dermody back on the scoresheetHELEN Dermody netted her first league goal since November as she set Coventry City Ladies on their way to a 2-0 victory over Cardiff City.
SkyWomen's Football: Tenth goal of the season for Helen Dermody as Coventry City Ladies beat Cardiff City 2-0HELEN Dermody netted her first league goal since November as she set Coventry City Ladies on their way to a 2-0 victory over Cardiff City.


Shaken, stirred, muddled, and mixed—and everything in between—Savor Dallas' popular Shaken + Stirred event returns for a festive soiree on December 2nd at The Statler in downtown Dallas. Seven of the city's best mixologists will sling their signature drinks and battle it out to create the official cocktail of Visit Dallas 2018. The event's official sponsors include Savor Dallas, The Statler, Jack Daniels, Sourced, Downtown Inc and Sprint. This event is the ultimate cocktail party with custom spirit tastings, a swinging live band and a sneak preview of the restaurants of The Statler, led by chef Graham Dodds and chef Angela Hernandez.
A portion of the proceeds will benefit the Press Club of Dallas, a dynamic organization of journalists and professional communicators in North Texas committed to promoting the discipline of journalism, communication, literacy, current affairs and First Amendment freedoms. Cheers to having fun for a good cause! Don't wait til it's sold out! Click here to get tickets to the hottest party in Dallas.
Shaken + Stirred from last year was a smashing success! Just in case you missed it, here are some top highlights from the event.
Some winning cocktails from the day prepared by Sourced Craft Cocktails.
What could be better than great drinks with great people at a great venue like The Happiest Hour?
Make sure you don't miss out this year and get your tickets before Shaken + Stirred sells out. To purchase tickets for Shaken + Stirred, click here.


text:
Q: Where is Ruby's string literal juxtaposition feature officially documented? I recently realized that if you juxtapose a sequence of Ruby string literals (e.g. 'a' "b" 'c'), it's equivalent to the concatenation of those string literals. However, I can't find this language feature documented anywhere. I've searched using the terms "juxtaposition" and "concatenation", but only found reference to it in a couple of StackOverflow responses. Can anyone point me to a definitive reference?

A: There is a reference in The Ruby Programming Language, page 47.
It looks like it is deliberately in the parser, for situations where you want to split up string literals in code, but don't want to pay the price of concatenating them (and creating 3 or more strings). Long strings with no line breaks, and without requiring line-length busting code, are a good example
text = "This is a long example message without line breaks. " \
    "If it were not for this handy syntax, " \
    "I would need to concatenate many strings, " \
    "or find some other work-around"

A: In addition to the pickaxe reference, there are some unit tests:
# compile time string concatenation
assert_equal("abcd", "ab" "cd")
assert_equal("22aacd44", "#{22}aa" "cd#{44}")
assert_equal("22aacd445566", "#{22}aa" "cd#{44}" "55" "#{66}")

A: UPDATE
This is now officially documented in the RDoc that ships with Ruby.
Changes will propagate to RubyDoc the next time they build the documentation.
The added documentation:
Adjacent string literals are automatically concatenated by the interpreter:

  "con" "cat" "en" "at" "ion" #=> "concatenation"
  "This string contains "\
  "no newlines."              #=> "This string contains no newlines."

Any combination of adjacent single-quote, double-quote, percent strings will
be concatenated as long as a percent-string is not last.

  %q{a} 'b' "c" #=> "abc"
  "a" 'b' %q{c} #=> NameError: uninitialized constant q

ORIGINAL
Right now, this isn't anywhere in the official ruby documentation, but I think it should be. As pointed out in a comment, the logical place for the docs to go would be: http://www.ruby-doc.org/core-2.0/doc/syntax/literals_rdoc.html#label-Strings
I've opened a pull request on ruby/ruby with the documentation added.
If this pull request is merged, it will automatically update http://www.ruby-doc.org. I'll update this post if/when that happens. ^_^
The only other mentions of this I've found online are:

*

*The Ruby Programming Language, page 47 (mentioned in another answer)

*Ruby Forum Post circa 2008

*Programming Ruby

A: If you want o break a long single-quoted String-literal across multiple lines without embedding new line in it.
Simply break it into multiple adjacent string literals, the ruby interpreter will concatenate them during the parsing process.
str = "hello" "all"

puts str #=> helloall

remember though, that you must escape the newlines between the literals so that ruby does not interpret the new line as a statement terminator.
str = "hello" \
      " all" \
      " how are you."

puts str #=> hello all how are you

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text:
Great Energy Challenge
Hydrofracturing: An Energy Revolution
But is it a good revolution or a bad revolution?
According to Steve Forbes, editor in chief of Forbes magazine and sometime presidential candidate, the "energy crisis [is] over!" (In case you're wondering, that exclamation point is Mr. Forbes's.)
What ended the crisis? According to Mr. Forbes, there's an energy "revolution" on that's due to the application of a drilling technique known as hydraulic fracturing or, in the parlance of those in the know, just plain "fracking." Methinks that Mr. Forbes's pronouncement of the demise of the energy crisis is premature.
All Hail the Shale
The seeds of Mr. Forbes's revolution can be found in pockets of natural gas trapped inside shale, sedimentary rock formations often located deep beneath the surface of the Earth. Geologists tell us there are huge amounts of this "shale gas" hanging out in shale formations in the United States (and elsewhere), but because the gas is unable to migrate through the rock [pdf], these deposits had up until recently been classified as unrecoverable and thus of no use economically. (Related: "The Great Shale Gas Rush")
But All That Has Changed with Fracking
Getting to the shale gas is a relatively simple process. It involves both the use of horizontal drilling and fracking to maximize gas extraction. Once a vertical well reaches the target formation, one drills horizontally through it, then fractures the rock by injecting water, sand and a cocktail of likely toxic chemicals into the rock under high pressure to fracture it and liberate the trapped gas. The gas is then pumped out and voilà! — you have a whole new source of natural gas.
Make no mistake — this is a game changer. With natural gas now available from shale and other tight formations, the amount of potentially recoverable natural gas in the United States has grown by leaps and bounds. All of a sudden there is enough of the stuff to run our power plants and power our automobiles for decades, perhaps a century into the future.
But There Is a Problem
In the 1939 musical The Wizard of Oz, the bad witch was done in by a bucket of water. Shale gas may face a similar fate.
Evidence is mounting that fracking does bad things to people's drinking water. (Examples here, here, and here.) It appears that somehow all that fracturing and pumping are causing gas and chemicals to migrate upward into well water. Increasingly, people with wells in the vicinity of fracking operations are complaining of drinking water contaminated by natural gas or worse.
In at least one location, alternate water supplies have had to be found for and delivered to folks whose water is laced with natural gas to the point of posing a safety hazard — like the pipes going kaboom.
What's to Be Done?
Some advise not to worry about that water contamination thing.
Industry experts claim that contamination is just not possible. The fracking occurs so far below the water table, they say, there is no way for any contamination to occur. (If that's the case, one has to wonder why Congress exempted fracking from the Safe Drinking Water Act in 2005 and why companies have been slow or unwilling to disclose the chemicals used in the fracking process.) While not exactly an energy expert, Steve Forbes has entered the fray, assuring us that "the technology is there to get at these reserves in an extremely safe way" and declaring that "the Earth is awash in energy."
I am impressed with his confidence. But I wonder what it is based on. Has he bothered to visit and drink the water of some of the people whose wells have been contaminated? And how does he rationalize the presence of natural gas in their well water? Is it a coincidence that the contamination showed up once the fracking began, or have these people been drinking contaminated, flammable water all this time and just never realized it?
Others argue that getting at that shale gas is a duty — contaminated water be damned. One Pennsylvania man receiving royalties for fracking on his land reasons that "God made everything … he put [minerals] there for a purpose. It's in our best interest to use the resources." I guess there's something appealing about that … but then again I wonder about that whole apple thing and the instructions not to take a bite.
The Stakes Are Huge
We have a lot riding on this fracking thing. Getting at the shale gas would be good, opening up a huge new domestic source of albeit a fossil fuel but a relatively clean one. But you can't live without drinking water. Maybe, just maybe it would make sense to take a step back, spend some time trying to figure out what's going on with these folks' drinking water and then maybe even figure out how to frack without the contamination, should fracking prove to be the contamination source. But perhaps that's just too sensible.

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text:
House panel seeks social media data on FBI threats made after Trump search
In addition to information about the companies' response to the online threats, the House panel also asked for their plans to minimize users' ability to incite violence.
U.S. Democratic lawmakers on Friday asked social media platforms to turn over data regarding an increase in online threats aimed at law enforcement following a search of former President Donald Trump's Florida home and asked how they planned to respond.
House Oversight Committee Chairwoman Carolyn Maloney and Stephen Lynch, Chairman of the Subcommittee on National Security, urged chief executives to act fast in letters addressed to eight internet companies, including Facebook-parent Meta Platforms Inc., Twitter Inc. and TikTok.
The requests come more than a week after a Columbus man was shot dead in a standoff with police after he tried to breach an FBI building in Ohio. Last week the FBI and U.S Department of Homeland Security alerted law enforcement agencies of an increase in threats.
The Democrats want to know if the uptick in online threats was linked to what they described as "reckless statements" issued by Trump and Republicans condemning the search.
"We urge you to take immediate action to address any threats of violence against law enforcement that appear on your company's platforms," the lawmakers said in the letters.
Letters were also made out to the Trump-backed platform Truth Social, Rumble, Gettr, Telegram and Gab.
Lawmakers said they would consider proposing legislation to protect law officers and improve coordination with federal agencies.
The FBI has been the subject of online threats since its agents searched Trump's Mar-a-Lago estate last week as part of an investigation into documents removed from the White House when Trump left office in January 2021. Agents removed 11 sets of classified records from the resort in Palm Beach, including some labeled "top secret" for the most sensitive U.S. national security information.
subcommittee on national security
stephen lynch


text:
KU ScholarWorks
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The New Divisia Monetary Aggregates
The_New_Divisia_Monetary_Aggregates.pdf (732.3Kb)
Barnett, William A.
Offenbacher, Edward K.
Spindt, Paul A.
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Economics Scholarly Works [173]
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Barnett, WIlliam A., Offenbacher, Edward K., and Spindt, Paul A. "The New Divisia Monetary Aggregates," (co-authored with Paul Spindt and Edward Offenbacher), Journal of Political Economy, December 1984.
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UNC-Duke tickets approaching Super Bowl prices because of Zion Williamson
ByMyron Medcalf
Zion Williamson has turned Wednesday'sNorth Carolina-Dukegame into the Super Bowl.
Well, it'll cost you the price of a Super Bowl ticket to see the game, at least.
If you're interested in squeezing into Cameron Indoor Stadium to watch the first installment of the legendary rivalry in 2019, you'll have to spend $2,500, the price of the cheapest available ticket. That's just shy of the $2,674 you would have paid on Jan. 27 for the cheapest available ticket to the Feb. 3 Super Bowl, per Kyle Zorn, a marketing strategist for secondary ticket seller TickPick.
One ticket to this week's UNC-Duke game sold for $10,652, per Vivid Seats. Zorn said there is only one other athlete who is comparable to Williamson and his impact on ticket prices: LeBron James after he signed with the Los Angeles Lakers last summer.
Ticket prices for Lakers road games have increased by 125 percent, Zorn said.
"As far as Duke this year, I haven't seen anything like this," Zorn said.
The presence of Williamson, the likely No. 1 pick in June's NBA draft, and projected top-five picksRJ Barrettand Cam Reddish has added more pop to the premier rivalry game in college basketball.
"Zion is a different bird. No question about that," Tar Heels coach Roy Williams said Tuesday. "We tried to recruit him very, very hard. He's got a combination of skill set that I've never seen before. There is a lot of attention but he deserves it. He's backed it up.
"I said today, somebody took exception, in a lot of ways this is the most gifted Duke team in the 16 years I've been back [in North Carolina]."
Some tickets for last season's national title game betweenVillanovaandMichiganwere as low as $67. Per Vivid Seats, the average price to attend a North Carolina home game is $70 this season, but the average price for the March 9 rematch against Duke is already at $991.
Zorn said the comparable ticket prices for Wednesday's UNC-Duke game (9 p.m., ESPN) were set about a week before the Super Bowl at Mercedes-Benz Stadium in Atlanta, and the venue size for Wednesday's game -- Cameron Indoor Arena's capacity is 9,314 -- is a significant factor in the skyrocketing prices.
"Between Zion Williamson, a No. 1 ranking and this being one of the biggest rivalries in sports, there's intense demand," Zorn said. "There's also a limited supply of inventory, which is having a major effect on price. Cameron Indoor Stadium has 13 percent the seating capacity of Mercedes-Benz Stadium. When you factor in that the floor seating is reserved for students, you're left with a short supply -- and massive demand."
Duke, the No. 1 team in America, will face No. 8 North Carolina in the 46th matchup between the two programs with both ranked in the top 10. Duke has won 23 such matchups and North Carolina has won 22, according to ESPN Stats & Information.
The Blue Devils are a consensus nine-point favorite Wednesday night. It's the third-highest spread between the two rivals as top-10-ranked teams since the 1993-94 season. The last time there was a double-digit favorite was 2001, when Duke was an 11.5-point favorite and lost outright. The total of 166.5 is also the fourth-highest between the two teams in the past 15 seasons.
Information from ESPN's Ben Fawkes was used in this report.
Copyright © 2023 ESPN Internet Ventures. All rights reserved.
SPORTS ESPN CAM REDDISH COACH K NORTH CAROLINA-TAR-HEELS RJ BARRETT ZION WILLIAMSON DUKE DUKE BLUE-DEVILS ESPNW SPORTS UNC MENS COLLEGE-BASKETBALL
Cold, Steady Rain Tomorrow


This meeting is for families interested in enrolling their student(s) in Highland Academy for the upcoming 2019/2020 school year. General school and enrollment information will be presented and questions will be answered. Note: If you attended the meeting on February 12th, you do not need to attend this meeting also.


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Rhythm Method Band
Rhythm Method Videos
Andre Deleo Music
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Singers/Musicians
VIP Premier
Take your time, look around, and learn all there is to know about Andre Deleo. Album release scheduled in November.
Holiday Single & more music
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Deleo
We had the pleasure of meeting Deleo in 2016. 'N'Topshape added the classically trained violinist to the Premier 2016 NCSS Grand Ball and Cherry Blossom Princess Coronation, and Sushi Reception. We received wonderful feedback about his performance. He continues to captivate the audience no matter where he performs.
Deleo is a Master Improvisationist in the genres of Jazz Fusin, R&B, Neo Soul, Hip Hop and Gospel. Deleo has used his violin and voice to create a signature style resulting in him being a musical powerhouse. His music will resonate with you!
Deleo was classically trained at the age of 9 and became the youngest member of the all adult Lakeland Symphony Orchestra. He studied under Daniel Heifetz and Gerald Fischbach at the University of Maryland. He has a recording housed in the National Archives in Washington DC as the violin soloist in the musical Elmer Gantry.
Deleo's music journey includes sharing the stage with the late Barry White on his final tour and performing with Stevie Wonder at Constitution Hall in Washington DC. His biggest honor was the opportunity to perform with a ten piece band for the 44th President Barack Obama's 2nd Inaugural Ball. He continues to perform in the Washington - Baltimore Metropolitan Area for Private Events, Weddings, Social Events, Corporate Events, Premier Events and more. He will travel out of town to perform with advance notice.
Deleo released his 1st Holiday Single and recently collaborated with Chevalier Music Publishing. Deleo is working in the studio and his new album release is coming soon! We are excited for him and his upcoming appearances.
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THIBODAUX, La. (AP) — A south Louisiana sheriff's office is offering four-day self-defense courses for kids.
The Lafourche Parish Sheriff's Office says this is the 19th year it has offered radKIDS defense courses. The name stands for "resist aggression defensively."
A news release says each week will have morning sessions for children aged 5 to 7, and afternoon sessions for those aged 8 to 12.
Parents can drop kids off or stay for the two-hour programs.
Advance registration is required because space is limited. The classes cost $10 per child, for materials.
Registration deadlines are Friday, June 16 for sessions in Raceland from June 20-23.
There's a Friday, July 7 deadline for sessions in Cut Off from July 11-14; and a Friday, July 14 deadline to register for sessions in Thibodaux from July 18-21.


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3-Hour Ceasefire in Aleppo which Opposition Says is Insufficient - ASHARQ AL-AWSAT English Archive
3-Hour Ceasefire in Aleppo which Opposition Says is Insufficient
by Taha Abed alWahed, Nazeer Rida and Said Abdul Razzak | Aug 11, 2016 | Middle East |
Moscow, Beirut, Ankara-Russia's Foreign Ministry has announced a daily three-hour ceasefire starting Thursday, to allow humanitarian convoys to enter the city of Aleppo safely.
"The daily ceasefires will be observed from 10 a.m. to 1 p.m. local time starting Thursday," General Sergei Rudskoi of the Russian military's General Staff said during a press conference in Moscow Wednesday.
The Russian general added that all military action, air and artillery strikes would be halted for that period.
Rodskoi said Russia, together with the Syrian authorities of Bashar Assad's regime, was ready to help all interested organizations safely deliver humanitarian aid to Aleppo's residents.
However, Rudskoi excluded opposition-controlled areas from receiving the humanitarian aid.
Syria's opposition had welcomed the Russian decision, but said the period of ceasefire was short and "insufficient."
Ahmad Ramadan of the opposition Syrian National Coalition told Asharq Al-Awsat: "We welcome any form of ceasefire, but it should be accompanied with a respect to U.N. Resolution 2254 and Articles 12, 13 and 14, which stipulate the immediate halt of attack on civilians, because we believe that the Russians are mostly targeting civilians."
Ramadan said that a three-hour ceasefire means there would be 21 hours of attacks on Syrian civilians, adding that previously announced ceasefires proved inapplicable.
Rudskoi said the question of joint control over delivery of humanitarian aid via the Castello Road was being discussed with the U.N. and the U.S.
On Wednesday, the U.N. said the three-hour daily ceasefire in Syria would not be sufficient to deliver all the needs of civilians currently trapped in the city, which has been under fierce attack.
AFP quoted U.N. chief Stephen O'Brien as saying Wednesday he was willing to consider the Russian plan, but that a 48-hour pause in fighting was needed to meet the entire humanitarian needs in the Syrian city.
In Ankara, Turkish Foreign Minister Mevlut Cavusoglu said his country was building a "strong mechanism" with Russia to try to find a solution in Syria, adding that a delegation, including foreign ministry, military and intelligence officials will go to Russia on Wednesday for talks.
Cavusoglu's announcement came one day after a meeting between Turkish President Recep Tayip Erdogan with his Russian counterpart Vladimir Putin in Moscow for the first time since the countries agreed to fix their deteriorating relations caused by Turkey's downing of a Russian plane last November.
"We share the same point of view with Russia on the ceasefire, delivering humanitarian aid to Syrians and the political solution in the country," he said, adding that National Intelligence Organization (MIT) chief Hakan Fidan and representatives from the Foreign Ministry and Turkish Armed Forces (TSK) will arrive to St. Petersburg on Wednesday night for talks with Russian officials.
"We will lift the level of our delegation in the future to include the foreign ministers of both countries," Cavusoglu said.
Despite the agreement between Russia and Turkey, Erdogan's spokesman Ibrahim Kalin said there were differences between both sides concerning the fate of the head of the Syrian regime and the mechanism for achieving the transitional political process, the ceasefire and the protection of unity of Syrian territories.
Former commander of Aleppo operations Colonel Abdul-Jabbar Akidi told Asharq Al-Awsat that the city of Aleppo is moving towards being liberated from the criminal regime. He said despite information that the regime had asked for the help of Iraqi forces and members of the so-called Hezbollah to support its forces fighting in Aleppo, these reinforcements would not be capable "to restore the collapsing morale of Assad's troops."
Taha Abed alWahed, Nazeer Rida and Said Abdul Razzak


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FYI: More FTC Actions Against Fraudulent Loan Mod Companies
The Federal Trade Commission announced several new developments in its crackdown on fraudulent loan modification and foreclosure relief companies. One settlement order bans 16 marketers from the mortgage modification or foreclosure relief business, another requires payment of $11.4 million for violating a previous court order the FTC obtained against the same scammer for fraudulent telemarketing practices, and a new FTC lawsuit charges another online marketing operation with masquerading as a government mortgage assistance program.
The FTC is focusing on companies that charge consumers up-front fees and made false promises that they could get their loans modified or prevent foreclosure.
- Impersonation of MakingHomeAffordable.gov
- "Federal Loan Modification Law Center," charging largely upfront fees of as much as $3k but not delivering services
- False claims of being able to obtain mortgage loan modifications for consumers in all or virtually all cases, false "money-back guarantees," and false statements of affiliation with HOPE NOW
- Misleading "official government agency seals" or logos and links to federal government Web sites
For more detailed information, and copies of the court filings, please see:
http://www.ftc.gov/opa/2010/06/loanmods.shtm
The United States Court of Appeals for the Seventh Circuit recently held that borrower inquiries to a loan servicer that either reque...
T he Secretary of the Treasury designate d July 21, 2011, as the "designated transfer date" p ursuant to Title X of the Dodd-Fr...
The Federal Reserve Board and Federal Trade Commission propose to: (1) revise and add to the content requirements for risk-based pricing no...
On June 23, 2016, New York Governor Cuomo signed S8159 (attached) into law, which among other things contains provisions relating to: - ...
As referenced in our prior update below, t he Illinois legislature passed an amendment to its foreclosure statute allowing a borrower to ...
As widely reported, the CFPB recently issued a proposed rule and request for comment to: (1) implement the Dodd-Frank Act's...
FYI: More FTC Actions Against Fraudulent Loan Mod...
Ralph is a member of the Governing Committee of the Conference on Consumer Finance Law. He is also the immediate past Chair of the Preemption and Federalism Subcommittee for the ABA's Consumer Financial Services Committee. He served on the Law Committee for the former National Home Equity Mortgage Association, and completed two terms as Co-Chair of the Consumer Credit Committee of the Chicago Bar Association.
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I can't believe June is over already! Only two months of summer left :( This month has been Vampire month for me, with the new True Blood novel release and of course, the Eclipse soundtrack :) But let's get down to the beauty products!
I got this a while ago with one of those Boots £5 off vouchers and started using it properly when the days became sunnier. I absolutely love it! Even though it's SPF 30 it's light and sinks into the skin quickly, acting as a moisturiser, primer and sun block in one. It has a slightly cream/beige colour rather than being the usual stark white that SPF creams can be, but does have a faint sun-block scent. It's my summer essential and I won't leave the house without it on!
This mineral foundation featured in my May favourites too, but I have to mention it again because I have practically used it every day. In the hot weather it's great for setting liquid foundation in place and I've also been using it alone in the day. With the right brush it gives good coverage - I like to use a pore-filling primer underneath for a smoothest finish. This powder really came through for me when I realised to my horror that it was the only foundation I'd packed for my mini cruise last week! (usually I bring a heavy coverage liquid like MAC Studio Fix as it needs to last from 6pm to 4/5am) However I packed on loads with my Inglot kabuki brush and my skin glowed and was almost flawless. One of the best things about mineral foundation is the lack of tide marks around your jawline/neck - a serious issue for me and my pale skin! The shade 'Fairly Light' is one shade darker than their lightest (Fair) which suits me better, but I got this in a set, and in the summer I can make it work for me.
This MSF was part of the MAC Pinkzapoppin! set released around Christmas time. I use it as a bronzer and it gives a very light and flattering colour. It's very iridescent and has a pinkish undertone, which suits pale skin much better than your typical orangey brown bronzers. I've used it a lot and barely made a dent in it - definitely good value!
When I got this foundation for the first time about five years ago I didn't get on with it at all. But after experimenting with different ways of application and the release of an even lighter shade this really works for me when I want to put on a fast, good coverage and oil absorbing foundation on rushed mornings. Of course it's not the greatest for tide marks, but it is easier to blend these away with the mousse formulation than with a thick liquid. Lately I've been applying it with a duo fibre brush and blending any problem areas with a makeup sponge and this method gives a lighter, smoother finish.
I've had this mascara longer than is probably hygienic, but it's still going strong and hasn't dried out. The brush looks pretty standard but it is a holy grail mascara. Pure black, separates lashes, lengthens and holds the curl well (although the waterproof version would do this better). I paid about £19.50 for it at the time, but it's worth it as I've bought countless lower end brand mascaras since which could never match up. This mascara makes me excited about my lashes!
I started using this last year following Lollipop26's recommendation. St. Tropez always seems too orange and expensive for me to bother, but I paid £2.99 for this rip-off on Amazon Marketplace, and it's been one of my discoveries of the year. I no longer have to look like the living dead with my pale, almost blue skin. I only apply it on special occasions or when I'm feeling particularly fat! I've had it on for the most of June. It's best applied with with a tan mitt for no streaking and non-freaky hands. It barely streaks, the colour is natural on me, and it fades quite evenly. However this bottle is about a year old (I did a massive bulk order when I saw how great it was) and I think it must have gone off a bit as it looks darker than normal - time for a new one! The only drawback is that it is drying, but I've found this an issue with all the fake tans I've tried.
I know it's not beauty related, but it's been a big favourite this month!
I got a bit obsessive and read all the previous novels, along with watching the TV series'. I'm halfway through the latest book and glad to see that Charlaine's writing skills have become slightly more sophisticated. Don't get me wrong, I love the fictional world she's created, set in the American South and I've been vampire mad ever since Buffy, but as a writer I find her repetitive. I wish someone more imaginative had the idea for this series. (And I want more Eric action!) I feel the same way about Twilight, again I love this series, but wish a less 'cringey' writer had that famous meadow dream.
What's your favourite June product?
You can get better coverage with minerals if you make a water/glycerin mix (10 part water 1 part glycerin) and spray a little on your kabuki before the ol' swirly-tap. And I love Hypnose too, I was really dissapointed in it at first but after a few weeks it dried down a tad and I love it now.
I'm popping to Westfield today with the old man's debit card! What primer should I get??
Also do me a favour.. where u have my button, can you change the code from .jpg to .png? That will make the corners transparent so it looks better!
Sorry, you need to change the name of the button too so where it says 'heels2.jpg' change it to 'heels2-1.png' or just copy the code again!
Just got back from Boots & Superdrug.. got me some goodies! I got the Loreal one you recommended.. gonna try it out today.
That mascara is absolutely AMAZING..!!
I use St. Moriz as well, you cant find it here in the US so i have to buy it on Amazon, but its been sold out lately....i LOVE that stuff!
@Angel: I hope you like it, I forgot to tell you that Avon do a great one too, the consistency is similar to the L'Oreal one but slightly lighter, which some people might prefer, and it's about 1/2 the price, you can order online or from the catalogue.
@allmylove: SPF25? That's fab, I used to use Palmers but got bored of the scent and found that it didn't sink in quick enough on rushed mornings!
I am most probably going to buy the St.Moriz self tanner so I can develop a gradual tan before vacation.
Lovely and informative post dear!
Hi Aimee, so did I, but then I noticed they brought out Champagne, and although the number is higher, when lined up in store Champagne is to the far left of Ivory and therefore lighter. Ivory is too yellow for me, although I did own that shade too before they copped on and brought out a lighter colour! Champagne is pretty pale with pink undertones.
Quick Tip & Hello Again!
Free Benefit Makeup with Glamour Mag!
MAC Venomous Villains Coming Autumn 2010!


Өчигдөрхөн л намын бүлгийн дарга хэмээн олон нийтэд мэдэгдэл хийж ЖДҮХС-аас зээл авсан гишүүдтэй хатуухан яриатай, бүрэн эрхээс нь түтгэлзүүлэх тэр байтугай Ерөнхий сайд ч ажлаа муу хийж байна хэмээн донгодож явсан Д.Хаянхярваа өнөө суудалдаа үлдэх эсэхээ шийдүүлэхээр болчихлоо.
Д.Хаянхярвааг бүлгийн доторх эв нэгдлийг хангаж чадаагүй, хариуцлагагүй зохион байгуулах чадваргүй гэж үзэн ажлаа хүлээлгэж өгөхийг сануулсан нь түүний яаран сандран мэдэгдэл хийсэнтэй ч холбож үзэж болмоор… Ёстой л түрүүлж дуугарсан хөхөөний ам хөлдөхийн үлгэр боллоо.
Д.Хаянхярваа ЖДҮХС-аас зээл авсан уу?
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Sorry for taking so long on this release. There aren't any big functional differences this round, but there are a lot of bug fixes. In preparation for some code completion, I've revamped the way the modeling works (which is reflected in the outline) and I've spent a lot of time writing, rewriting, and refining a lot of regexes. As such, there's a good chance I've introduced new bugs into the outline, so I'd appriciate posts in this thread of code samples along with the problem in the outline. This will be to further refine the modeling system.


text:
package org.wso2.carbon.mss.internal.router;

import io.netty.buffer.ByteBuf;
import org.wso2.carbon.mss.HttpResponder;

/**
 * HttpHandler would extend this abstract class and implement methods to stream the body directly.
 * chunk method would receive the http-chunks of the body and finished would be called
 * on receipt of the last chunk.
 */
public abstract class BodyConsumer {
    /**
     * Http request content will be streamed directly to this method.
     *
     * @param request
     * @param responder
     */
    public abstract void chunk(ByteBuf request, HttpResponder responder);

    /**
     * This is called on the receipt of the last HttpChunk.
     *
     * @param responder
     */
    public abstract void finished(HttpResponder responder);

    /**
     * When there is exception on netty while streaming, it will be propagated to handler
     * so the handler can do the cleanup.
     *
     * @param cause
     */
    public abstract void handleError(Throwable cause);
}

redpajama_set_name: RedPajamaGithub


It's officially the happiest time of the year! Why? Because the biggest Shopbop sale of them all is finally here! With so many gifts for him and for her, the possibilities are endless! Get that designer handbag you have been pining after or that watch you know she will love. All the savings will make this the best guilt free shopping you've done all year!
And if the budget is tight, don't fret. There is a huge selection of great gifts under $100 (everything you see above!) Another budget tip. Order with a friend to maximize the discount percentage!


(C)1997 SQUARE ENIX CO., LTD. All Rights Reserved.
(C)1998-2010 Copyright TechnoBrain CO.,LTD.(C)2010 Sonic Powered Co.,Ltd.
(C)2010 Sony Computer Entertainment Europe. Published by Sony Computer Entertainment Inc. Developed by Just Add Water( Developments) Ltd.
(C)SEGA /(C)Crypton Future Media, Inc.
(C)1998-2000GENEON UNIVERSAL ENTERTAINMENT JAPAN, LLC. ALL RIGHTS RESERVED.
(C)SUCCESS /(C)Tears Ichikawa Soft /(C)HAMSTER Co.
(C)2007 Sonic Powered Co., Ltd. (C)1998-2007 Copyright TechnoBrain CO., LTD.


Explore Havana from the writer's perspective!
Unique way to discover Havana following the footsteps of a great legend of literature. This Novel Prize winner fell in love with Cuba attracted initially by marlin fishing, subject he developed later in The Old Man and the Sea. Room 511, in Hotel Ambos Mundos housed him every time he came since 1928, until he bought a quiet villa, Finca La Vigia in 1940, places from where he wrote multiple novels and essays. The American author was a lover of cocktails and a regular to El Floridita and La Bodeguita del Medio bars.
Departing from Varadero: 07:30-18:00 (2 hrs drive to Havana).
Visit to Finca La Vigia Museum, Hemingway's former farm. Admission fee: 5.00 CUC per person.
Visit to Cojimar fishermen community, famous for harboring Hemingway's fishing boat and inspiration for the novel The Old Man and the Sea.
Panoramic tour along the seafront avenue, "Malecón", the Revolution Square, Central Park, Opera House, Capitol building, former Bacardi Building and Museum of the Revolution, former Presidential Palace. Stop at these places for pictures.
Visit to Saint Francis of Assisi Square, Arms Square, Old Square and Cathedral Square.
Visit to El Floridita bar, Sloppy Joe's bar, La Bodeguita del Medio bar, Hotel Ambos Mundos (Ernest Hemingway's regular bars and hotel in Old Havana).
This gem of Eclecticism opened in 1930 and became famous for excellent service and a long list of illustrious guests such as Nat King Cole, Winston Churchill, Frank Sinatra, Walt Disney, Marlon Brando, and several American Mafiosi as Meyer Lansky, Lucky Luciano. It stands where in the 18th Century the Santa Clara Battery was emplaced, part of the old city´s defense system, of which two cannons remain in the hotel´s garden. It was declared in 1998 as National Monument.
This construction was conceived in 1953 as Civic Square, on the 100th anniversary of the Cuban national hero´s birth and finished in 1959 when it was renamed as Revolution Square. It consists of a 142 meters tower representing a five pointed star and a huge statue of José Martí whose Memorial is in the interior of the base. Since 1959 it has been Cuba´s political and administrative center and the main venue for political rallies and official celebrations. Pope John Paul II in 1998 and Pope Benedicto XVI in 2012 celebrated masses together with thousands of worshipers.
Construction of this fortress was designed by Bautista Antonelli and built between 1589 and 1630. The original lighthouse was rebuilt many times, the one that still stands today is from 1845 and has its original lamp. Havana was a highly prized target for enemies and pirates for its extremely favorable strategic position in the Caribbean and became the most fortified city in any Spanish colony. It was declared a UNESCO site as part of the city´s defense system along with other fortresses as La Cabaña, La Real Fuerza, La Punta, etc.
It was designed by Belgian architect Paul Belau to host the social activities of Havana´s large and affluent Spanish community. The magnificent façade is decorated with four sculpture groups depicting Charity, Education, Music and Theater. It was built over the foundations of the Tacón Theater which opened on November 1837. The present building was inaugurated on 1915 and has witnessed world famous artist as Fanny Essler, Verdi, Sarah Bernhardt, Arthur Rubinstein, Anna Pavlova and Alicia Alonso, etc. Today, it houses the National Ballet of Cuba.
The Cuban Capitol, Inaugurated on May 20th, 1929 was the seat of the House of Representatives and the Senate until 1959. It marks the 0 kilometer of Cuba's highways system. Its almost 92 meters high dome, the Chamber of Deputies, the Hall of Lost Steps and the statue "the Republic", made by the Italian artist Angelo Zanelli, the third largest indoor statue in the world are among its many attractions. Today, it houses the Ministry of Science, Technology and Environment.
Ernest Hemingway's favorite place to enjoy his daiquiri. In 1953 it was considered one of the seven most famous bars in the world, and in 1992, the American Academy of Gastronomical Science presented the Floridita with the Best of the Best Five Star Diamond Award. Kwon as the "cradle of the Daiquiri", it still keeps its original Regency décor from 1950s.
The hustle and bustle at San Francisco Square bothered religious services at the Basilica. The Franciscan fathers requested the creation of a square for exclusive commercial use, named New Square. After the creation of other urban areas in the 19th century it lost its initial name becoming Old Square. Cristina Market was built in the center in honor of the Queen, being demolished in 1908. Administrative and business offices sprang up with the 20th Century and a small amphitheater with a parking lot beneath were built in 1952 which are no longer part of this square after its restoration.
A very well-known bar where Hemingway loved to drink mojitos and a one of the city's great tourist attractions, where countless visitors have left their imprint in a souvenir, a photo or on its graffiti covered walls. Here you will find signatures by Errol Flynn, Nat King Cole, Ernest Hemingway, Pablo Neruda, Gabriel García Márquez, Alejo Carpentier and Nicolás Guillen, among many others.
It was originally known as Swamp Square because of the marshy land and subterranean springs. A church built by the Jesuit order (1748-1776) was promoted to the status of cathedral 1789, thus the original plaza became Cathedral Square. Some aristocratic buildings appeared later as Lombillo, Marquis of Aguas Claras and Marquis of Arcos palaces. The cathedral´s Baroque façade is grandiose with two large asymmetrical towers and its neoclassical interior treasures frescoes by Italian Giovani Perovani and Jean Baptiste Vermay. Pope John Paul II visited it in 1998.
Its location made it the center of imports, export, sailors and troops disembarking. The construction of a humble Franciscan convent, later one of the most beautiful buildings in the area, gave its name to the open space. The Fountain of the Lions, donated by Villanueva Count in 1836 and sculpted in Carrara marble, supplied for many years the ships docked here with drinking water. The Stock Exchange (1909) and the Customs House (1914) were added to the physiognomy of the square.
Neoclassic monument inaugurated in 1828 as a remembrance to the foundation of Saint Christopher of Havana. It stands where there was a silk-cotton tree, under whose shadow the first Mass and city council were held on November 16th, 1519. Inside, there are three canvases by Jean Baptiste Vermay depicting these scenes. Every year on November 16th, locals would circle the tree three times, knock on its trunk three times and ask for their wishes as part of a town tradition.
This seafront promenade runs for 7 km alongside the city´s historic quarters, from the colonial center to the skyscrapers of El Vedado. This busy boulevard is lined with many attractive buildings whose pastel hues have faded in the sun and salty air. It is magical at sunset, popular among fishermen, and the perfect place for people to hang out, meet friends and socialize.
This fortress was built in 1558-1577 to protect the city from pirate attacks and is the oldest military construction in Havana. Despite its perfect Renaissance architecture, it couldn´t be efficient as it was built too far inside the bay, in a low area. It was used as warehouse, Governor´s residence, barracks, National Archive, Army Headquarters, National Library, and Arms Museum, etc. The Giraldilla, a sculpture cast in bronze by Havana artist Jerónimo Martínez Pinzón was placed as a weather vane on top of the Northwest tower in the 1630s and became a city´s symbol.
The Central Plaza was built in 1877 after the old city walls were demolished. Since February 1905, its large esplanade is presided by a statue of José Martí, which replaced a marble statue of Isabel II of Spain. The Central Plaza is surrounded by significant buildings, such as the Grand Theater of Havana, the Inglaterra, Telégrafo, Plaza and Parque Central Hotels, the old Gómez Mena mall, the Universal Art building of the Fine Arts National Museum and the Payret Movie Theater.
Cuba´s first motion picture theater, opened in 1897, a year after the Lumiere brothers presented their invention in Paris.
Few steps away from the village´s foundation site, it was built between the Parochial Church and the Real Fuerza Castle, spot that grouped the political, military, religious and civilian functions of an emerging population. Used for military exercises in the 1700s, it is surrounded by important public buildings of the time as the Royal Post Office (1772-1790), subsequently the Lieutenant Governor's Palace; the Governor´s Palace, the City´s Museum since 1968; the Count of Santovenia's Palace, today Santa Isabel Hotel, and El Templete.
At San Francisco de Paula on the outskirts of Havana, this villa was built in 1887 and bought by Hemingway in 1940 where he lived for 20 years and wrote some his novels. It was turned into a museum in 1962 after his death and displays over 1000 books library, hunting trophies, the author personal belongings, etc. Two curios features in the garden are the cat cemetery and the author´s fishing boat El Pilar which he also used to patrol the sea north of Cuba, on the outlook of Nazi submarines that were in the area to sink ships laden with sugar intended for the allied troops.
This small fishermen town, east of Havana, was the place that harbored Hemingway´s boat "El Pilar" and became the setting for his book "The Old Man and the Sea". There is a monument featuring a bust of the writer and nearby on the seafront, a small fort (1649) stands as the eastern most defense point of the city where this community emerged.
This charming hotel from 1923, eclectically decorated, is rich in literary memories. The American writer Ernest Hemingway stayed here for long periods during the 1930´s writing fishing chronicles and began to write his novel For Whom the Bell Tolls. Room 511, where he mostly stayed is now a museum that features personal belongings and pictures of the author.
Sloppy Joe's was a spectacular Havana bar in the 30's. Ernest Hemingway was a regular with his pal, Joe Russell who was the owner of another Sloppy Joe's Bar in Cayo Hueso, a name Hemingway had suggested in honor to Mr. Garcia, the owner, who ran his business very messy and careless at first, hence the name of Sloppy Joe. A sandwich served here, made of shredded beef "ropa vieja", was worldwide known as Sloppy Joe´s. It was also a very popular place for having the longest mahogany drinks cabinet of Cuba.