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text:
"Osinbajo backing South-West for 2023, wake-up call to Ndigbo"
The Director-General of Voice of Nigeria, Mr Osita Okechukwu, on Monday said Vice-President Yemi Osinbajo's comment that a victory for President Muhammadu Buhari in the 2019 presidential election would guarantee the South-West the 2023 presidency was a "wake-up call" to the South-East.
Osinbajo made the comment while on a house-to-house campaign in Oyo town, where he held a 45-minute closed-door meeting with the Alaafin, Oba Lamidi Adeyemi.
Before now, it was believed that the South-East would have the opportunity to produce the president in 2023, although the Secretary to the Government of the Federation, Boss Mustapha, had at a rally in Owerri, Imo State, said the development would depend on Igbo support for Buhari in 2019.
Osinbajo's comment was faulted by groups, including the pan-Yoruba socio-political group, Afenifere, and the apex socio-political body of the Igbo, Ohanaeze Ndigbo, as well as the Peoples Democratic Party.
However, reacting to the development in a statement on Monday, Okechukwu said Ndigbo should see the VP's comments as a wake-up call.
Okechukwu said, "The Vice-President's call for Yoruba to vote for President Muhammadu Buhari in 2019 so as to have a shot at the Presidency in 2023 is constitutional and mostly importantly a wake-up call to Ndigbo.
"A wake-up call to Ndigbo in the true sense of the word that equity, natural justice and good conscience are the advantages on our side – for the South-East is the only geopolitical zone in the southern belt that has not presided over Nigeria since 1999, when the rotation of President convention took off.
"We appeal to Ndigbo to hearken to this clarion call as all we need is to vote for President Buhari so as to reinforce the equity, natural justice and good conscience which is on our side.
"Our vote for Buhari will engender the support of his mass followers in the North, for Buhari has a vote-bank of 10 to 12 million voters, which is a big arsenal."
The All Progressives Congress chief added, "On the issue of four years, Buhari's remaining four years post 2019 is cast in constitutional stone, while Atiku Abubakar's own is at best a pledge.
"Atiku's manifesto is six- year plan.
"Therefore, Buhari's option is the quickest and surest route to a Nigerian President of Igbo extraction."

redpajama_set_name: RedPajamaCommonCrawl


It's time for Santa Photos in Del Sur! Our time slots are now all filled. No worries though, we will have our stand by line as always. The stand by line usually moves rather quickly. For faster wait times, come towards the end of the shoot. Happy Holidays!


text:
The Upcoming.org Archives
Lee Scratch Perry May 25, 2006
BB King Blues Club & Grill
cover: $24.00 | In Advance $30 At the door
From People Funny Boy to Fight To The Finish, Lee ?Scratch? Perry?s productions and collaborations are uncountable. Considered as a living legend in regards to Reggae music, the one we call ?The Mighty Upsetter? has not only created, with People Funny Boy, the first reggae song of all times, but has also given birth to some fundamental concepts of modern music such as Dub, Rap or Remix. In the early 1970s, Perry formed a studio, The Black Ark, to experiment in the dub style and produce tracks from notable musicians like Bob Marley & the Wailers and The Clash, the legendary British punk group who were perhaps the earliest white artists to combing rock'n'roll with reggae rhythms. In 1998, he collaborated with the Beastie Boys on the track "Dr. Lee, PHD" for the Hello Nasty album. In 2003, Perry won a Grammy for Best Reggae Album with the album Jamaican ET.
Added by Fumio on March 7, 2006
Interested 10
acemystery
baby.squirrel
Fumio
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Imageyenation
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Latice
mburns
tbmtt


text:
Traceable Dynamic Measurement of Mechanical Quantities
Short Name: Dynamic, Project Number: IND09
Thomas Bruns (PTB)
Advancing the infrastructure for dynamic measurements: Ensuring reliable measurements of dynamic force, torque and pressure
Measurements of force, pressure and torque are important to most manufacturing sectors. However, while the sensors used to make these measurements (transducers) are calibrated in static conditions, they are often used in dynamic conditions, where force, pressure and torque vary with time, often varying over large ranges very rapidly. For example, an increase in fuel injection pressure in vehicle engines increases fuel efficiency and enables the development of cleaner cars. On ignition, pressures within engines rise several hundred-fold within a fraction of a second, and accurate measurement of this fast-changing pressure is needed to develop innovative and competitive products. In the energy sector, accurate assessment of rapidly-changing force, pressure and torque is needed to improve monitoring of safety-critical systems.
This project developed methods for calibrating dynamic measurements of force, pressure and torque. The project developed and extended the range of validated calibration methods and devices, traceable to primary standards, for a variety of force, pressure and torque transducers operating in dynamic modes. As a result of the project:
Force transducers can now be calibrated against primary standards for periodic measurement variations with frequency ranges up to 1 kHz and force amplitudes up to 10 kN.
Methods to calibrate pressure transducers for dynamic measurements of shock pressures with amplitudes up to 500 MPa and transducers for periodic torque were investigated and new services will be available in the near future.
Methods to calibrate amplifiers used in the dynamic mode traceable to national standards were developed.
Mathematical and statistical models were developed that estimate measurement uncertainties of dynamic measurements throughout the calibration chain.
The project outputs are already in use in the industrial and standardisation communities. Transducer manufacturers are using the improved calibration capabilities at NMIs and a transfer standard developed in the project is expected to become a commercial product. The methodology for calibrating sensor amplifiers has been incorporated into a German DKD guideline and an ISO draft standard, and a supplement to the international Guide to the Expression of Uncertainty in Measurement is planned. A web-based best practice guide providing practical information for engineers and technicians in industry is also available. Wider uptake of project results will, ultimately, allow European industry to accurately measure and control forces, pressures and torques, to support product and process innovation.
Final Publishable Report Traceable Dynamic Measurement of Mechanical Quantities (IND09) 2.01 MB
Final Publishable Summary Traceable Dynamic Measurement of Mechanical Quantities (IND09) 0.21 MB
MIKES (Finland)


text:
United States angle-down
New Zealand geography & geology
Milford Sound, Fiordland
By Miles Holden
You'll find a variety of awesome landscapes in New Zealand, all within easy reach of each other.
Mount Aspiring, New Zealand
By Small World Productions
Beech forest at Lake Gunn, Fiordland
By William Patino
White Island, Bay of Plenty
By Tourism New Zealand
Waitomo Glowworm Caves, Hamilton - Waikato
By Corin Walker Bain
Spectacular glaciers, picturesque fiords, rugged mountains, vast plains, rolling hillsides, subtropical forest, volcanic plateau, miles of coastline with gorgeous sandy beaches - it's all here. No wonder New Zealand is becoming so popular as a location for movies.
Lying in the south-west Pacific, New Zealand consists of two main islands - the North Island and the South Island. Stewart Island and many smaller islands lie offshore.
The North Island of New Zealand has a 'spine' of mountain ranges running through the middle, with gentle rolling farmland on both sides. The central North Island is dominated by the Volcanic Plateau, an active volcanic and thermal area. The massive Southern Alps form the backbone of the South Island. To the east of the Southern Alps is the rolling farmland of Otago and Southland, and the vast, flat Canterbury Plains.
Piopio, Hamilton - Waikato
By Tourism NZ
The Mangaotaki Rocks, near Piopio, featured in The Hobbit: An Unexpected Journey as Trollshaws Forest
How it began
New Zealand's oldest rocks are over 500 million years old, and were once part of Gondwanaland. This massive super-continent started to split up about 160 million years ago, and New Zealand separated from it about 85 million years ago.
New Zealand sits on two tectonic plates - the Pacific and the Australian. Fifteen of these gigantic moving chunks of crust make up the Earth's surface. The North Island and some parts of the South Island sit on the Australian Plate, while the rest of the South Island sits on the Pacific. Because these plates are constantly shifting and grinding into each other, New Zealand gets a lot of geological action.
Pohutu Geyser in Te Puia, Rotorua
By Shaun Jeffers Photography
Visit Te Puia to experience the might of the iconic Pohutu Geyser that erupts up to 20 times a day, spurting hot water up to 30m skyward.
Geothermal areas and hot springs
This subterranean activity blesses New Zealand with some spectacular geothermal areas and relaxing hot springs, as well as providing electricity and heating in some areas. Rotorua is the main hub for geothermal attractions, with plenty of mud pools, geysers, and hot springs in its active thermal areas — not to mention its trademark 'Sulphur City' smell. First settled by Maori who used the hot springs for cooking and bathing, Rotorua soon attracted European residents. The reputed health benefits of its hot pools quickly earned the area the name of 'Cureland'.
Beyond Rotorua, you can enjoy hot springs and other thermal activity in most regions of the North Island north of Turangi, as well as in Hanmer Springs and the West Coast in the South Island.
Wharariki Beach, Nelson
By Nelson Tasman
At low tide, when you can walk the length of Wharariki Beach past enormous sand dunes, rugged rocks and beautiful scenery.
Gentle, sandy beaches to wild, rugged coastlines
New Zealand has over 15,000 kilometres of beautiful and varied coastline. In the Far North and on most of the East Coast of the North Island you'll find long sandy beaches perfect for swimming, surfing and sunbathing. The North Island's west coast has dark sandy beaches, with sand heavy in iron. The north of the South Island has some beautiful sandy beaches, while the coastline around the rest of the South Island tends to be wilder and more rugged.
Hot air ballooning over Canterbury, Christchurch - Canterbury
By Ballooning Canterbury
Hot air ballooning over Canterbury
Mountain ranges to fertile farmland
About a fifth of the North Island and two-thirds of the South Island are mountains. Stretching from the north of the North Island to the bottom of the South, these mountains are caused by the collision of the Australian and Pacific Plates.
Over millions of years, alluvial deposits (eroded from the mountains by rivers) formed the vast Canterbury Plains in the South Island and a number of smaller plains in the North. These alluvial plains contain some of New Zealand's most fertile and productive farmland.
Fox Glacier, New Zealand
By Fox Glacier Guiding
On the South Island's West Coast you can find ancient glaciers that extend almost to the sea.
Glaciers of grinding ice
New Zealand's Southern Alps have a number of glaciers, the largest being Tasman glacier, which you can view by taking a short walk from Mount Cook village. New Zealand's most famous glaciers are the Franz Josef and Fox on the South Island's West Coast. Gouged out by moving ice over thousands of years, these spectacular glaciers are easily accessible to mountaineers and hikers. You can walk up to the glaciers or do a heli-hike — fly up by helicopter and walk down.
Fiordland National Park, Fiordland
By Fiordland Tourism
One of the most dramatic and stunning parts of New Zealand Fiordland is a mix of fiords, waterfalls and snow-capped mountains.
Sunken Mountains
Over thousands of years, the process of subduction has seen parts of the New Zealand landscape become submerged. The Marlborough Sounds and Fiordland are examples of high mountain ranges that have 'sunk' into the sea, creating spectacular sounds and fiords. These areas provide some of New Zealand most picturesque scenery, with steep lush hills plunging down to the deep still bays below. Clear, deep still water surrounded by beautiful bush makes these areas ideal for boating and kayaking.
Choose another airline Air New Zealand Air Tahiti Nui Hawaiian Airlines Qantas United Air New Zealand
Scenery and landscapes long-arrow-right
Volcanic & Geothermal long-arrow-right
Fox Glacier & Franz Josef Glacier long-arrow-right
Top New Zealand trips long-arrow-right


text:
\section{Introduction}

The Navier-Stokes  equations in $\R^3$

\begin{eqnarray}
\label{eqn:navier-stokes}
\partial_t u - \nu \Delta u  +  (u \cdot \nabla) u +  \nabla p & = & 0, \nonumber \\ \nabla \cdot u & = & 0, \nonumber \\ u_0 (x) & = & u(x,0)
\end{eqnarray}
with $\nu > 0$, describe the time evolution of the velocity $u$ and the pressure $p$ of a homogeneous incompressible viscous fluid and are of great  importance in the study of turbulent flows. The analytical and numerical study of these equations is very difficult and as a result of this, many different kinds of approximations and regularizations  have been proposed in order to have a better understanding of  the behaviour of its solutions.

One such regularization that has been extensively studied in recent years, in the case of bounded or periodic domains, is given by the Navier-Stokes-Voigt  equations

\begin{eqnarray}
\label{eqn:navier-stokes-voigt}
\partial_t \left(u - \a ^2 \Delta  u \right) - \nu \Delta u  +  (u \cdot \nabla) u +  \nabla p & = & 0, \nonumber \\ \nabla \cdot u & = & 0, \nonumber \\ u_0 (x) & = & u(x,0),
\end{eqnarray}
supplemented with appropiate boundary conditions. These equations were originally used by Oskolkov \cite{MR0377311} to study the motion of viscoelastic flows with relaxation time, i.e. the time that the flow takes to respond to an external force, of order $\a^2 / \nu$ and have been proposed by Cao, Lunasin and Titi \cite{MR2264822} as a good model for numerical simulations of the Navier-Stokes equations, provided $\a > 0$ is small.

The introduction of the regularizing term $- \a ^2 \partial_t \Delta  u$ in (\ref{eqn:navier-stokes-voigt}) has many important consequences. We first note that, assuming $u$ is a regular enough solution, (\ref{eqn:navier-stokes-voigt}) has a natural energy equality

\be
\label{eqn:energy-equality-nsv}
\frac{1}{2} \frac{d}{dt} \left(  \Vert u(t) \Vert ^2 _{L^2 (\R^3)} + \a^2 \Vert \nabla u(t) \Vert ^2 _{L^2 (\R^3)} \right) = - \nu \Vert \nabla u(t) \Vert ^2 _{L^2 (\R^3)},
\ee
instead of the usual one for (\ref{eqn:navier-stokes}). Moreover, Navier-Stokes-Voigt equations do not have a parabolic character, as Navier-Stokes equations do, behaving instead as a damped hyperbolic system. This leads to wellposedness for equation (\ref{eqn:navier-stokes-voigt}) both forward and backwards in time (see Cao, Lunasin and Titi \cite{MR2264822}, Kalantarov and Titi \cite{MR2570790}) and to differences in the energy spectrum when compared to (\ref{eqn:navier-stokes}) (see Levant, Ramos and Titi \cite{MR2655910}, Ramos and Titi \cite{MR2629486}). However, solutions to Navier-Stokes-Voigt equations share many other statistical and asymptotic properties with the Navier-Stokes equations when $\a > 0$ is small (see  Kalantarov, Levant and Titi \cite{MR2495891}, Larios and Titi \cite{Larios2010603}, Levant, Ramos and Titi \cite{MR2655910}, Ramos and Titi \cite{MR2629486}), while presenting advantages for the implementation of numerical models for their study. This supports the idea of using (\ref{eqn:navier-stokes-voigt}) in direct numerical simulations of (\ref{eqn:navier-stokes}) (see Borges and Ramos \cite{MR3048413}, Kuberry, Larios, Rebholz and Wilson \cite{MR2970841}, Layton and Rebholz \cite{MR3171807}). For more results concerning the Navier-Stokes-Voigt equations, see Bersellli and Bisconti \cite{MR2846786}, Ebrahimi, Holst and Lunasin \cite{MR3116155}, Garc\'{\i}a-Luengo, Mar\'{\i}n-Rubio and Real \cite{MR2891552}, Gao and Sun \cite{MR2863948}, Tang \cite{MR3217168}, Yue and Zhong {\cite{MR2806333} and references therein.

Only recently, Zhao and Zhu \cite{MR3316059}  adressed questions concerning the Navier-Stokes-Voigt equations in the whole space $\R^3$. In their work, they proved  existence of weak solutions  to  (\ref{eqn:navier-stokes-voigt}), but their main result  concerns the decay rate for the $H^1$ norm of solutions, which, for  initial data $u_0$ in $L^1 (\R^3) \cap H^1 (\R^3)$, is proved to be

\be
\label{eqn:decay-zhao-zhu}
\Vert u(t) \Vert _{H^1 (\R^3)} ^2 \leq C (1 + t) ^{- \frac{3}{2}}.
\ee
The main tool in the proof of these estimates is a clever adaptation of the Fourier Splitting method of M.E. Schonbek \cite{MR571048}, \cite{MR775190}, \cite{MR837929}, which takes into account the extra linear term present in (\ref{eqn:navier-stokes-voigt}) with respect to (\ref{eqn:navier-stokes}).

The main goal of this article is to prove a decay estimate for the Navier-Stokes-Voigt equations for {\em any} initial datum in $H^1 _{\a} (\R^3)$, where

\begin{displaymath}
\Vert u \Vert _{H^1 _{\a} (\R^3)} ^2 = \Vert u \Vert _{L^2 (\R^3)} ^2 + \a^2 \Vert \nabla u \Vert _{L^2 (\R^3)} ^2
\end{displaymath}
is the natural norm associated to (\ref{eqn:energy-equality-nsv}). To achieve this, we associate to every $u_0$ in $L^2 (\R^3)$ a decay character $r^{\ast} = r^{\ast} (u_0)$, which measures the ``order'' of $\widehat{u_0} (\xi)$ at $\xi = 0$ in frequency space and then find sharp decay estimates for solutions to the linear part

\be
\label{eqn:linear-part}
\partial_t \left( u - \a^2 \Delta u \right) - \nu \Delta u  = 0
\ee
in terms of $r^{\ast}$. We then use the Fourier Splitting Method and these estimates for the linear part to characterize the decay of Navier-Stokes-Voigt equations. This same strategy was followed by Niche and M.E. Schonbek \cite{2015arXiv150102105N}, \cite{Niche01042015} to obtain analogous results for the Navier-Stokes equations and some parabolic compressible approximations and for the dissipative quasi-geostrophic equation.

We state now one of the main results of this article.

\bt
\label{main-theorem}
Let $u_0 \in H^1 _{\a} (\R^3)$, with $\nabla \cdot u_0 = 0$ and $r^{\ast} = r^{\ast} (u_0)$, $- \frac{3}{2} \leq r^{\ast} < \infty$. Then for $\nu > \a^2$, every weak solution to (\ref{eqn:navier-stokes-voigt}) is such that for $t > 0$

\begin{displaymath}
\Vert u(t) \Vert _{H^1 _{\a}} ^2 \leq C_1 (C_2 + t) ^{- \min \{\frac{3}{2} + r^{\ast}, \frac{5}{2} \}},
\end{displaymath}
for some positive constants $C_i = C_i (\Vert u_0 \Vert _{H^1 _{\a} (\R^3)}, \a, \nu)$, $i = 1, 2$.
\et

The decay in Theorem \ref{main-theorem} is similar to the one for the Navier-Stokes equations

\be
\label{eqn:characterization-ns}
\Vert u(t) \Vert _{L^2 (\R^3)} ^2 \leq C_1 (C_2 + t) ^{- \min \{ \frac{3}{2} + r^{\ast}, \frac{5}{2} \}}, \qquad - \frac{3}{2} \leq r^{\ast} < \infty,
\ee
obtained by Bjorland and M.E. Schonbek \cite{MR2493562} (see also Niche and M.E. Schonbek \cite{Niche01042015}).  In our work, using the Fourier Splitting Method we obtain in frequency space the differential inequality (\ref{eqn:previous-to-main-inequality}), in which the decay rate is bounded from above by the average of the solution in a shrinking, time-dependent ball containing the small frequencies. The leading terms in the right hand side of this expression are the same as in the Navier-Stokes equations and, as the other terms provide faster decay rates, we recover the same estimate.

The bound from Theorem \ref{main-theorem} allows us to determine the decay rate for the difference between the whole solution $u(t)$ and the solution $\overline{u} (t)$ to the linear part (\ref{eqn:linear-part}) with the same initial datum.

\bt
\label{decay-nonlinear-part}
Let $u_0 \in H^1 _{\a} (\R^3)$, with $\nabla \cdot u_0 = 0$ and $r^{\ast} = r^{\ast} (u_0)$, $- \frac{3}{2} \leq r^{\ast} < \infty$. Let $w = u - \overline{u}$. Then for $\nu > \a^2$,

\begin{displaymath}
\Vert w(t) \Vert ^2 _{H^1 _{\a} (\R^3)} \leq C_1 (C_2 + t) ^{- \min \{ \frac{9}{4} + \frac{3}{2} r^{\ast}, \frac{13}{4} + \frac{1}{2} r^{\ast}, \frac{11}{2}  \}}.
\end{displaymath}
for some positive constants $C_i = C_i (\Vert u_0 \Vert _{H^1 _{\a} (\R^3)}, \a, \nu)$, $i = 1, 2$.
\et

This result has quantitative and qualitative differences with the analogous one for the $L^2 (\R^3)$ norm of $w = u - \overline{u}$ in the Navier-Stokes equations, where

\begin{displaymath}
\Vert w(t) \Vert _{L^2 (\R^3)} ^2 \leq C_1 (C_2 + t) ^{- \min \{\frac{7}{4} + r^{\ast}, \frac{5}{2} \}},
\end{displaymath}
see Niche and M.E. Schonbek \cite{Niche01042015} \footnote{We note that in \cite{Niche01042015} this estimate was proved for a compressible approximation to the Navier-Stokes equations. However, the proof carries over with no modifications due to the very similar structure of these equations, regardless of compressibility. For a similar result for Navier-Stokes, see Wiegner \cite{MR881519}.} .The decay rate from Theorem \ref{decay-nonlinear-part} is faster for $r^{\ast} > -1$, but for $r^{\ast} = -\frac{3}{2}$, the difference $w$ in the Navier-Stokes-Voigt equations may have arbitrarily slow decay, while in the Navier-Stokes equations we always have  some uniform decay with speed at least $(1 + t) ^{- \frac{1}{4}}$. We address this interesting point in Remark \ref{remark-slow-decay-nsv}.

This article is organized as follows. In Section \ref{section-decay-character} we recall the definition of the decay character of a function in $H^s (\R^n)$, with $s \geq 0$ and we prove Theorem \ref{characterization-decay-linear-part}, in which we caracterize the decay of solutions to the linear system (\ref{eqn:linear-part}), for initial data in $H^1 _{\a} (\R^n)$. Then, in Section \ref{section-proofs}, we prove Theorems \ref{main-theorem} and \ref{decay-nonlinear-part}.

\subsection*{Acknowledgments}
The author thanks Fabio Ramos and Mar\'{\i}a E. Schonbek for their comments and remarks on parts of this article.

\section{Decay character and characterization of decay of linear part}
\label{section-decay-character}

\subsection{Definition and properties of decay character}

The Fourier Splitting Method, introduced by M.E. Schonbek \cite{MR571048}, \cite{MR775190}, \cite{MR837929} to study decay of viscous conservation laws and the Navier-Stokes equations, relies on the study of small frequencies of the solution  to estimate the decay rates. In order to prove sharp estimates for the decay of the heat equation in terms of the initial data, Bjorland and M.E. Schonbek \cite{MR2493562} introduced the idea of {\em decay character $r^{\ast} = r^{\ast} (u_0)$} of a function $u_0$ in $L^2 (\R^n)$.  Roughly speaking, the decay character is the ``order'' of $| \widehat{u_0} (\xi)|$ in frequency space at $\xi = 0$. With these estimates in hand, Bjorland and M.E. Schonbek characterized the decay of solutions to the Navier-Stokes equations in terms of $r^{\ast}$, see (\ref{eqn:characterization-ns}).

Recently, Niche and M.E. Schonbek \cite{Niche01042015} generalized this notion in order to use data in $H^s (\R^n)$, for $s  > 0$ and to obtain results for other linear systems. We recall now some of these definitions and results.

\bd Let  $u_0 \in L^2(\R^n)$ and $\Lambda = (- \Delta) ^{\frac{1}{2}}$. For $s \geq 0$ the {\em s-decay indicator}  $P^s _r (u_0)$ corresponding to $\Lambda ^s u_0$ is

\begin{displaymath}
P^s _r (u_0) = \lim _{\rho \to 0} \rho ^{-2r-n} \int _{B(\rho)} |\xi|^{2s}| \widehat{u}_0 (\xi)| ^2 \, d \xi
\end{displaymath}
for $r \in \left(- \frac{n}{2} + s, \infty \right)$, where $B(\rho)$ is the ball at the origin with radius $\rho$.
\ede

\br \label{heuristics} \, Setting $r = q + s$, we see that the $s$-decay indicator compares $|\widehat{\Lambda ^s u_0} (\xi)|^2$ to $f(\xi) = |\xi|^{2(q + s)}$ near $\xi = 0$. When $s = 0$ we recover the definition of Bjorland and M.E. Schonbek \cite{MR2493562}.
\er

\bd  \label{df-decay-character} The {\em decay character of $\Lambda^s u_0$}, denoted by $r_s ^{\ast} = r_s ^{\ast}( u_0)$ is the unique  $r \in \left( -\frac{n}{2} + s, \infty \right)$ such that $0 < P^s _r (u_0) < \infty$, provided that this number exists. If such  $P^s _r ( u_0)$ does not exist, we set $r_s ^{\ast} = - \frac{n}{2} + s$, when $P^s _r (u_0)  = \infty$ for all $r \in \left( - \frac{n}{2} + s, \infty \right)$  or $r_s ^{\ast} = \infty$, if $P^s _r (u_0)  = 0$ for all $r \in \left( -\frac{n}{2} + s, \infty \right)$.
\ede

\br \label{remark-lp-l2} \, Let $u_0 \in L^2 (\R^n)$ such that $\widehat{u}_0 (\xi) = 0$, for $|\xi| < \delta$, for some $\delta > 0$. Then, $P_r ^s (u_0) = 0$, for any $r \in \left( - \frac{n}{2} + s, \infty\right)$ and  $r^{\ast} _s (u_0) = \infty$.
\er

\br If $u_0 \in L^p (\R^n) \cap L^2 (\R^n)$, with $1  \leq p \leq 2$, then $r^{\ast} (u_0) = - n \left( 1 - \frac{1}{p} \right)$, so if $u_0 \in L^1 (\R^n) \cap L^2 (\R^n)$ we have that $r^*(u_0) =0$ and if $u_0 \in L^2 (\R^n)$ but $u_0 \notin L^p (\R^n)$, for any $1 \leq p < 2$, we have that $r^*(u_0) = -  \frac{n}{2}$. For details, see Niche and M.E. Schonbek \cite{Niche01042015}.
\er

\smallskip

Let $u_0 \in H^s (\R^n)$, with $s > 0$. As $\widehat{\Lambda^s (u_0)} (\xi) = |\xi|^s \widehat{u_0} (\xi)$, the heuristics given in Remark \ref{heuristics} lead us to expect that $r^{\ast} _s (u_0) = r^{\ast} (\Lambda ^s u_0) = s + r^{\ast} (u_0)$. This is the content of the following Theorem.

\bt \label{decay-character-hs} (Theorem 2.11, Niche and M.E. Schonbek \cite{Niche01042015}) Let $u_0 \in H^s (\R^n)$, with $s > 0$.
\begin{enumerate}
 \item If $-\frac{n}{2} < r^{\ast} (u_0) < \infty$ then  $- \frac{n}{2} +s< r_s^{\ast}(u_0) < \infty$ and  $r_s^{\ast}(u_0) = s + r^{\ast} (u_0)$.
\item $r_s^{\ast}  (u_0) = \infty$ if and only if $r^{\ast} (u_0) = \infty$.
\item  $r^{\ast} (u_0) =- \frac{n}{2}$ if and only if $r_s^{\ast}(u_0) = r^{\ast} (u_0) + s = - \frac{n}{2} + s$. \end{enumerate}
\et

\subsection{Characterization of decay of linear part}  We address now the system

\be
\label{eqn:linear-part-nsv}
\partial_t \left(v - \a ^2 \Delta  v \right) - \nu \Delta v= 0,
\ee
which is the linear part of the Navier-Stokes-Voigt equations (\ref{eqn:navier-stokes-voigt}). From it, we obtain the energy equality

\be
\label{eqn:energy-inequality-linear-part}
\frac{d}{dt} \left( \Vert v(t) \Vert ^2 _{L^2 (\R^n)} + \a ^2 \Vert \nabla v(t) \Vert ^2 _{L^2 (\R^n)} \right) + 2 \nu \Vert \nabla v(t) \Vert _{L^2 (\R^n)} ^2 = 0
\ee
which naturally leads us to study the behaviour of the $H^1 _{\a}$ norm

\begin{displaymath}
\Vert v \Vert _{H^1 _{\a} (\R^n)} ^2 = \Vert v \Vert _{L^2 (\R^n)} ^2 + \a^2 \Vert \nabla v \Vert _{L^2 (\R^n)} ^2.
\end{displaymath}
Then in Fourier space, the solution to (\ref{eqn:linear-part-nsv}) with initial datum $v_0$ is

\begin{displaymath}
\widehat{v} (\xi, t) = e^{t \mathcal{M} (\xi)} \widehat{v_0} (\xi)
\end{displaymath}
where
\be
\label{eqn:nsv-multiplier}
e^{t \mathcal{M} (\xi)} = e^{ - \frac{\nu |\xi|^2}{1 + \a^2 |\xi|^2} t}.
\ee

\br
\label{facts-linear-part-nsv}
Note that (\ref{eqn:nsv-multiplier}) is bounded but not in any $L^p$, for $1 \leq p < \infty$. Then, solutions to (\ref{eqn:linear-part-nsv}) with initial data in Sobolev space remain there but are not regularized, i.e. they do not belong to higher order Sobolev spaces, as the kernel associated to (\ref{eqn:nsv-multiplier}) does not ``absorb'' derivatives as the heat kernel does. However, for small frequencies (\ref{eqn:nsv-multiplier}) behaves like a Gaussian, which allows us to the Fourier Splitting Method.
\er

In the following Theorem, we characterize the decay of solutions to (\ref{eqn:linear-part-nsv}) in terms of the decay character $r^{\ast} = r^{\ast} (v_0)$.

\bt
\label{characterization-decay-linear-part}
Let $v_0 \in H^1 _{\alpha} (\R^n)$, with $r^{\ast} (v_0) = r^{\ast}$. If $\nu > \a^2$, then for the solution $v(t)$ to (\ref{eqn:linear-part-nsv}) with initial datum $v_0$ we have

\begin{enumerate}
\item if $- \frac{n}{2} < r^{\ast} < \infty$, then there exist constants $C_1, C_2, C_3 > 0$ such that

\begin{displaymath}
C_1 (1 + t) ^{- \left( \frac{n}{2} + r^{\ast} \right)} \leq \Vert v(t) \Vert ^2 _{H^1 _{\alpha} (\R^n)} \leq C_2 (C_3 + t) ^{- \left( \frac{n}{2} + r^{\ast} \right)};
\end{displaymath}
\item if $r^{\ast} = - \frac{n}{2}$ given any $\eps > 0$, there exists $C = C(\eps) > 0$ such that

\begin{displaymath}
\Vert v(t) \Vert ^2 _{H^1 _{\alpha} (\R^n)} \geq C (1 + t) ^{- \eps}, \quad \forall \, t > 0,
\end{displaymath}
i.e. the decay is slower than any algebraic rate;
\item if $r^{\ast} = \infty$, given any $m > 0$ then there exists $C = C(m) > 0$ such that

\begin{displaymath}
\Vert v(t) \Vert ^2 _{H^1 _{\alpha} (\R^n)} \leq C (1 + t) ^{-m}, \quad \forall \, t  > 0,
\end{displaymath}
i.e. the decay is faster than any algebraic rate.
\end{enumerate}
\et

\medskip

{\bf Proof:} (1) Recall first that we have, by Theorem \ref{decay-character-hs}, that $r^{\ast} (\nabla v_0) = 1 + r^{\ast} (v_0)$. As $P_r (v_0) > 0$ there exist $\rho_0 > 0$ and $C_1 > 0$ such that for $0 < \rho < \rho_0$

\be
\label{eqn:decay-indicator-below}
C_1 < \rho ^{-2r - n} \int _{B(\rho)} |\widehat{v_0} (\xi)|^2 \, d \xi, \qquad C_1 < \rho ^{-2(r + 1) - n} \int _{B(\rho)} |\xi|^2 |\widehat{v_0} (\xi)|^2 \, d \xi.
\ee
Consider $B(t) = \{ \xi \in \R^n: |\xi| \leq \rho(t) \}$, for some continuous, positive and decreasing $\rho$ to be determined later. Then

\begin{eqnarray}
\label{eqn:inequality-lower-bound}
\Vert v(t) \Vert _{H^1 _{\alpha}} ^2 & \geq & \int _{B(t)} \left( 1 + \alpha ^2 |\xi|^2 \right) |e^{t \mathcal{M} (\xi)} \widehat{v_0} (\xi)|^2 \, d \xi \nonumber \\ & \geq & \int _{B(t)} \left( 1 + \alpha^2  |\xi|^2 \right) e^{- 2\nu |\xi|^2 t} |\widehat{v_0} (\xi)|^2 \, d \xi \nonumber \\ & \geq & C \rho ^{2r + n} (t) e^{- 2\nu \rho ^2 (t)} \rho ^{- 2r - n} (t) \int _{B(t)} |\widehat{v_0} (\xi)|^2 \, d \xi \nonumber \\ & + & C \rho ^{2(r + 1) + n} (t) e^{- 2\nu \rho ^2 (t)} \rho ^{- 2(r + 1) - n} (t) \int _{B(t)} |\xi|^2 |\widehat{v_0} (\xi)|^2 \, d \xi \nonumber \\ & \geq & C \rho ^{2r + n} (t) e^{- 2\nu \rho ^2 (t)} + C \rho ^{2(r + 1) + n} (t) e^{- 2\nu \rho ^2 (t)}
\end{eqnarray}
where we used (\ref{eqn:decay-indicator-below}) in the last line. Taking $\rho (t) = \rho_0 (1 + t) ^{-\frac{1}{2}}$ we obtain

\begin{displaymath}
\Vert v(t) \Vert _{H^1 _{\alpha}} ^2  \geq  C (1 + t) ^{- \left( r + \frac{n}{2} \right)} + C (1 + t) ^{- \left( r + 1 + \frac{n}{2} \right)} \geq C (1 + t) ^{- \left( r + \frac{n}{2} \right)},
\end{displaymath}
from which the result follows. To obtain the upper bound we use the Fourier Splitting method. We  follow the ideas in Zhu and Zhao \cite{MR3316059}. From the energy inequality (\ref{eqn:energy-inequality-linear-part})  we obtain

\begin{displaymath}
\frac{d}{dt} \left( \int _{\R^n} \left( 1 + \a^2 |\xi|^2 \right) |\widehat{u} (\xi, t) |^2 \, d \xi \right) + 2 \nu \int _{\R^n}  |\xi|^2  |\widehat{u} (\xi, t) |^2 \, d \xi = 0.
\end{displaymath}
Let

\begin{displaymath}
B(t) = \left\{\xi \in \R^n: |\xi| \leq \rho (t) = \left( \frac{g'(t)}{2 \nu g(t) - \a^2 g'(t)} \right) ^{\frac{1}{2}} \right\}
\end{displaymath}
where $g \in C^1 (\R)$, $g(0) = 1$, $g'(t) > 0$ and $2 \nu g(t) - \a^2 g'(t) > 0$, for all $t > 0$. We then have

\begin{align}
\label{eqn:inequality-before-decay-indicator}
\frac{d}{dt}  \left( g(t) \int _{\R^n} \left( 1 + \a^2 |\xi|^2 \right)  |\widehat{v} (\xi, t) |^2 \, d \xi \right)  \leq   g'(t) \int _{B(t)} \left( 1 + \a^2 |\xi|^2 \right) |\widehat{v} (\xi, t) |^2 \, d \xi \nonumber \\  \leq  g'(t) \rho^{2r + n} (t) \rho^{- 2r - n} (t) \int _{B(t)} |\widehat{v_0} (\xi)|^2 \, d \xi \nonumber \\  +   \a^2 g'(t) \rho^{2(r + 1) + n} (t) \rho^{- 2(r + 1) - n} (t) \int _{B(t)} |\xi|^2 |\widehat{v_0} (\xi)|^2 \, d \xi
\end{align}
where the first inequality was proved in page 186 in Zhu and Zhao \cite{MR3316059}. As $P_r (u_0) < \infty$ and $r^{\ast} (\nabla u_0) = 1 + r^{\ast} (u_0)$, there exist $\rho_0 > 0$ and $C > 0$ such that for $0 < \rho < \rho _0$

\be
\label{eqn:decay-indicator-above}
\rho ^{-2r - n} \int _{B(\rho)} |\widehat{v_0} (\xi)|^2 \, d \xi \leq C, \qquad \rho ^{-2(r + 1) - n} \int _{B(\rho)} |\xi|^2 |\widehat{v_0} (\xi)|^2 \, d \xi \leq C.
\ee
Then, from (\ref{eqn:inequality-before-decay-indicator}) and (\ref{eqn:decay-indicator-above}) we obtain

\be
\label{eqn:previous-to-estimate}
\frac{d}{dt}  \left( g(t) \int _{\R^n} \left( 1 + \a^2 |\xi|^2 \right) |\widehat{u} (\xi, t) |^2 \, d \xi \right) \leq C g'(t) \rho^{2r + n} (t) + C g'(t) \rho^{2(r + 1) + n} (t).
\ee
Taking $g(t) = \left( \frac{1}{2 \nu} t + b \right)^k$, where $k > 1 + r + \frac{n}{2}$ and $b > \frac{k \a^2}{2 \nu ^2}$, we have that $2 \nu g(t) - \a^2 g'(t) > 0$ holds (because $\nu > \a^2$) and that $\rho(t) = C (t + b) ^{- \frac{1}{2}}$. From (\ref{eqn:previous-to-estimate}) the estimate

\begin{displaymath}
\Vert v(t) \Vert _{H^1 _{\alpha}} ^2 \leq C (t + b) ^{-k} + C (t + b) ^{- \left( r + \frac{n}{2} \right)} + C (t + b) ^{- \left( r + 1 + \frac{n}{2} \right) } \leq  C (t + b) ^{- \left( r + \frac{n}{2} \right)}
\end{displaymath}
follows.

\medskip
\noindent (2) As $r^{\ast} = - \frac{n}{2}$, for any  $r \in (- \frac{n}{2}, \infty)$ we have that $P_r (v_0) = \infty$. Then for any constant $\widetilde{C} = \widetilde{C} (r)$ there exists $\rho_0 > 0$ such that for $0 < \rho_0 < \rho$

\begin{displaymath}
\widetilde{C} < \rho ^{-2r - n} \int _{B(\rho)} |\widehat{v_0} (\xi)|^2 \, d \xi, \qquad \widetilde{C} < \rho ^{-2 (r + 1) - n} \int _{B(\rho)} |\xi|^2 |\widehat{v_0} (\xi)|^2 \, d \xi.
\end{displaymath}
As in the proof of the lower bound in (1) with an inequality similar to (\ref{eqn:inequality-lower-bound}), we obtain

\begin{displaymath}
C (1 + t)^{-r - \frac{n}{2}} \leq \Vert v(t) \Vert _{H^1 _{\alpha}} ^2.
\end{displaymath}
As this holds for any $r \in (- \frac{n}{2}, \infty)$, the estimate is established.

\noindent (3) Since $r^{\ast} = \infty$, for any $r \in (- \frac{n}{2}, \infty)$ we have $P_r (v_0) = 0$. So, for any $\widetilde{C} = \widetilde{C} (r)$ there exists $\rho_0 > 0$ such that for $0 < \rho_0 < \rho$

\begin{displaymath}
\rho ^{-2r - n} \int _{B(\rho)} |\widehat{v_0} (\xi)|^2 \, d \xi < \widetilde{C}, \rho ^{-2(r + 1) - n} \int _{B(\rho)} |\xi|^2 |\widehat{v_0} (\xi)|^2 \, d \xi < \widetilde{C}
\end{displaymath}
Proceeding as in the proof of the upper bound in (1)   with an inequality similar to (\ref{eqn:previous-to-estimate}), we obtain

\begin{displaymath}
\Vert v(t) \Vert _{H^1 _{\alpha}} ^2 \leq C (1 + t)^{-r - \frac{n}{2}} .
\end{displaymath}
As this holds for any $r \in (- \frac{n}{2}, \infty)$, the estimate is established. $\Box$

\section{Proof of Theorems \ref{main-theorem} and \ref{decay-nonlinear-part}}
\label{section-proofs}

Before proceeding to the proof of  Theorem \ref{main-theorem} we prove some Lemmas. Suppose that the solution to the Navier-Stokes-Voigt equation is regular enough such that

\be
\label{eqn:solution-fourier-space}
\widehat{u} (\xi, t) = \widehat{u_0} (\xi) \, e^{t \mathcal{M} (\xi)} - \int _0 ^t e^{(t -\tau) \mathcal{M} (\xi)} G(\xi, \tau) \, d \tau,
\ee
where $G(\xi,t) =  \mathcal{F} \left( (u \cdot \nabla) u - \nabla p \right) (\xi, t)$ and $\mathcal{F}$ denotes the Fourier transform.

\begin{Lemma} \label{estimate-modulus-uhat} For $\widehat{u}$ as in (\ref{eqn:solution-fourier-space}) we have

\be
\label{eqn:estimate-modulus-solution-fourier}
|\widehat{u} (\xi, t)|^2 \leq C  e^{2t \mathcal{M} (\xi)} |\widehat{u_0} (\xi)|^2  + C |\xi|^2 \left( \int _0 ^t \Vert u(\tau) \Vert _{L^2 (\R^3)} ^2  \, d \tau \right)^2.
\ee
\end{Lemma}

{\bf Proof:} The estimate follows from

\begin{eqnarray*}
|\mathcal{F} ((u \cdot \nabla) u) (\xi,t)| & \leq & |\mathcal{F} (\nabla \cdot (u \otimes u)) (\xi, t)| \leq C |\xi| |\mathcal{F} (u \otimes u)) (\xi, t)| \\ & \leq & C |\xi| \Vert \mathcal{F} (u \otimes u) (t) \Vert _{L^{\infty} (\R^3)} \nonumber \\ & \leq & C |\xi| \Vert u \otimes u (t) \Vert _{L^1 (\R^3)} \leq C |\xi| \Vert u(t) \Vert _{L^2 (\R^3)} ^2
\end{eqnarray*}
and from $\Delta p = - \nabla (u \cdot \nabla) u$, as

\begin{displaymath}
|\mathcal{F} (\nabla p) (\xi, t)| \leq |\mathcal{F} ((u \cdot \nabla) u) (\xi, t)| \leq C |\xi| \Vert u(t) \Vert _{L^2 (\R^3)} ^2. \qquad \Box
\end{displaymath}

{\bf Proof of Theorem \ref{main-theorem}:} \,  We again use the Fourier Splitting Method. We recall that existence of weak solutions to (\ref{eqn:navier-stokes-voigt}) has been established by Zhu and Zhao, see Theorem 2.1 in  \cite{MR3316059}. As usual in this method, we prove the estimates for regular enough solutions and we then pass to the limit  to establish it for weak solutions. Full details for this argument can be found in pages 267-269 in Lemarie-Rieusset \cite{MR1938147} and in the Appendix in Wiegner \cite{MR881519}.

 We follow the ideas of Zhu and Zhao \cite{MR3316059}.  As before, let

\be
\label{eqn:ball}
B(t) = \left\{ \xi \in \R^3: |\xi| \leq \rho (t) = \left( \frac{g'(t)}{2 \nu g(t) - \a^2 g'(t)} \right) ^{\frac{1}{2}} \right\}
\ee
where $g \in C^1 (\R)$, $g(0) = 1$, $g'(t) > 0$ and $2 \nu g(t) - \a^2 g'(t) > 0$, for all $t > 0$. The incompressibility condition leads to the energy inequality (\ref{eqn:energy-equality-nsv}), which is the same one as for the linear part, see (\ref{eqn:energy-inequality-linear-part}). We then obtain the same inequality as in the first line in (\ref{eqn:inequality-before-decay-indicator}), which after integration and use of Lemma \ref{estimate-modulus-uhat}  becomes

\begin{align}
\label{eqn:previous-to-main-inequality}
 g(t) \int _{\R^3} \left( 1 + \a^2 |\xi|^2 \right) |\widehat{u} (\xi, t) |^2 \, d \xi   \leq   C + \int _0 ^t g'(s) \int _{B(s)} \left( 1 + \a^2 |\xi|^2 \right) |\widehat{u} (\xi, s) |^2 \, d \xi \nonumber \\ \leq   C + C \int _0 ^t g'(s) \left( \int _{B(s)} \left( 1 + \a^2 |\xi|^2 \right) e^{2 t \mathcal{M} (\xi)} |\widehat{u_0} (\xi)|^2  \, d \xi \right) \, ds \nonumber \\  +   C \int _0 ^t  g'(s)  \left( \int _{B(s)} \left(1  +  \a^2 |\xi|^2 \right)  |\xi|^2 \left( \int _0 ^t \Vert u(\tau) \Vert _{L^2 (\R^3)} ^2  \, d \tau \right)^2  \, d \xi \right) \, ds.
\end{align}
We estimate now the right hand side of this inequality. For the first term we have, using the estimates from Theorem \ref{characterization-decay-linear-part}

\begin{align}
\label{eqn:estimate-linear-part}
 \int _0 ^t g'(s) \left( \int _{B(s)} \left( 1 + \a^2 |\xi|^2 \right) e^{2 t \mathcal{M} (\xi)} |\widehat{u_0} (\xi)|^2  \, d \xi \right) \, ds  \leq  C  \int _0 ^t g'(s) \Vert \overline{u} (s) \Vert ^2 _{H^1 _{\a}  (\R^3)}  \, ds \nonumber \\  \leq   C_1  \int _0 ^t g'(s) (C_2 + s) ^{- \left( \frac{3}{2} + r^{\ast} \right)}\, ds,
\end{align}
where $\overline{u}$ is the solution to the linear part of (\ref{eqn:navier-stokes-voigt}). For the second term, after integrating in polar coordinates in $B(t)$, we obtain

\begin{align}
\label{eqn:estimate-nonlinear-part}
 \int _0 ^t  g'(s) & \left( \int _{B(s)} \left(1  +  \a^2 |\xi|^2 \right)  |\xi|^2 \left( \int _0 ^t \Vert u(\tau) \Vert _{L^2 (\R^3)} ^2  \, d \tau \right)^2  \, d \xi \right) \, ds \nonumber \\ & \leq C \left( \int _0 ^t g'(s) \left( \rho^5 (s) + \rho^7 (s) \right)  \, ds \right) \left( \int_0 ^t \Vert u (\tau) \Vert _{L^2 (\R^3)} ^2 \, d \tau \right) ^2.
\end{align}
For a fixed $r^{\ast}$, we choose $g(t) = (C + t)^m$, for some $m > \max \{ \frac{1}{2},\frac{3}{2} + r^{\ast} \}$. Then $\rho(t) = C_1 (C_2 + t) ^{- \frac{1}{2}}$ and from (\ref{eqn:previous-to-main-inequality}), (\ref{eqn:estimate-linear-part}) and (\ref{eqn:estimate-nonlinear-part}) and the apriori estimate $\Vert u(t) \Vert _{L^2 (\R^3)} \leq C$, we obtain

\begin{eqnarray*}
\Vert u(t) \Vert ^2 _{H^1 _{\a} (\R^3)}  & \leq & C (1 + t) ^{-m} + C_1 (C_2 + t) ^{- \left( \frac{3}{2} + r^{\ast} \right)} + C_3 (C_4 + t) ^{- \frac{1}{2}} \\ & \leq & C_1 (C_2 + t) ^{- \min \{ \frac{3}{2} + r^{\ast}, \frac{1}{2}  \}}.
\end{eqnarray*}
We now use this first preliminary decay to bootstrap, trying to find sharper estimates for (\ref{eqn:estimate-nonlinear-part}). Suppose $ \min \{ \frac{3}{2} + r^{\ast}, \frac{1}{2}  \} = \frac{3}{2} + r^{\ast}$. Then, for $g(t) = (C + t)^m$ with $m > \max \{\frac{3}{2} + r^{\ast}, \frac{3}{2} \}$, we have that  $\rho(t) = C_1 (C_2 + t) ^{- \frac{1}{2}}$ and then

\begin{displaymath}
C \left( \int _0 ^t g'(s)  \left( \rho^5 (s) + \rho^7 (s) \right) ds \right) \left( \int_0 ^t \Vert u (\tau) \Vert _{L^2 (\R^3)} ^2 \, d \tau \right) ^2 \leq C_1 (C_2 + s) ^{- \left( \frac{7}{2} + 2 r^{\ast} \right)}.
\end{displaymath}
The decay in (\ref{eqn:estimate-linear-part}) is still the slower one, so there is no improvement. For $ \min \{ \frac{3}{2} + r^{\ast}, \frac{1}{2}  \} = \frac{1}{2}$, we obtain

\begin{displaymath}
 \left( \rho^5 (s) + \rho^7 (s) \right) \left( \int_0 ^s \Vert u (\tau) \Vert _{L^2 (\R^3)} ^2 \, d \tau \right) ^2 \leq C_1 (C_2 + s) ^{-  \frac{3}{2}}
\end{displaymath}
so going back to (\ref{eqn:estimate-nonlinear-part}) we have

\begin{eqnarray*}
\Vert u(t) \Vert _{H^1 _{\a} (\R^3)} ^2 & \leq & C (1 + t) ^{-m} + C_1 (C_2 + t) ^{- \left( \frac{3}{2} + r^{\ast} \right)} + C_3 (C_4 + t) ^{- \frac{3}{2}} \\ & \leq & C_1 (C_2 + t) ^{- \min \{ \frac{3}{2} + r^{\ast}, \frac{3}{2}  \}}.
\end{eqnarray*}
We boostrap once again. Let $ \min \{ \frac{3}{2} + r^{\ast}, \frac{3}{2}  \} = \frac{3}{2} + r^{\ast}$, and consider $r^{\ast} \neq - \frac{1}{2}$.  By the same computations as before, now with $m > \max \{\frac{3}{2} + r^{\ast}, \frac{5}{2} \}$, we show that decay is still given by  (\ref{eqn:estimate-linear-part}). If $r^{\ast} =  - \frac{1}{2}$, then the integral of the $L^2$ norm in (\ref{eqn:estimate-nonlinear-part}) leads to a logarithmic term which after choosing $g(t) = (C + t) ^m$, with $m > \frac{5}{2}$, leads to

\begin{eqnarray*}
\Vert u(t) \Vert _{H^1 _{\a} (\R^3)} ^2 & \leq & C (1 + t) ^{-m} + C_1 (C_2 + t) ^{- 1} + C_3 \ln ^2 (1 + t) \, (C_3 + t) ^{- \frac{5}{2}} \\ & \leq & C (1 + t) ^{- 1}.
\end{eqnarray*}
If $ \min \{ \frac{3}{2} + r^{\ast}, \frac{3}{2}  \} = \frac{3}{2}$, then

\begin{displaymath}
\int _0 ^t \Vert u(\tau) \Vert _{L^2 (\R^3)} ^2 \, d \tau \leq C
\end{displaymath}
from which we obtain, after choosing $g(t) = (C + t) ^m$, with $m > \frac{5}{2}$

\begin{eqnarray*}
\Vert u(t) \Vert _{H^1 _{\a} (\R^3)} ^2 & \leq & C (1 + t) ^{-m} + C_1 (C_2 + t) ^{- \left( \frac{3}{2} + r^{\ast} \right)} + C_3 (C_4 + t) ^{- \frac{5}{2}} \\ & \leq & C (1 + t) ^{- \min \{\frac{3}{2} + r^{\ast},  \frac{5}{2} \}}.
\end{eqnarray*}
This finishes the proof of the Theorem. $\Box$

\smallskip

{\bf Proof of Theorem \ref{decay-nonlinear-part}} As before, we assume solutions are regular enough and we obtain estimates for them. The estimates ofr weak solutions are obtained as in Theorem \ref{main-theorem}. Take $w = u - \overline{u}$, where $\overline{u}$ is the solution to the linear part (\ref{eqn:linear-part}) with the same initial datum. Then $w$ solves the equation

\begin{displaymath}
\partial_t (w - \a^2 \Delta w) + (u \cdot \nabla) u = \nu \Delta w - \nabla p, \quad w_0 (x) = 0.
\end{displaymath}
Multiplying by $w$ and integrating we obtain

\begin{eqnarray*}
\frac{1}{2} \frac{d}{dt} \left( \Vert w(t) \Vert _{L^2 (\R^3)} ^2 + \a ^2 \Vert \nabla w (t) \Vert _{L^2 (\R^3)} ^2  \right) + \nu \Vert \nabla w (t) \Vert _{L^2 (\R^3)} ^2 =  \int _{\R^3} \overline{u} (u \cdot \nabla) u \, dx.
\end{eqnarray*}
We now use interpolation for $L^3$ in terms of $L^2$ and $L^6$, the embedding $H^1 (\R^3) \subset L^6 (\R^3)$, the decay for $\overline{u}$ from Theorem \ref{characterization-decay-linear-part} and the decay for $u$ from Theorem \ref{main-theorem} to obtain

\begin{eqnarray}
\label{eqn:ineq-cross-term}
\int _{\R^3} \overline{u} (u \cdot \nabla) u \, dx & = & - \int _{\R^3} \nabla \overline{u} (u \otimes u) \, dx  \leq  \Vert \nabla \overline{u} (t) \Vert _{L^2 (\R^3)} \, \Vert u(t) \Vert _{L^3 (\R^3)} \Vert u (t) \Vert _{L^6 (\R^3)} \nonumber \\ & \leq &  \Vert \nabla \overline{u}(t) \Vert _{L^2 (\R^3)} \Vert u(t) \Vert ^{1/2} _{L^2 (\R^3)} \Vert u(t) \Vert ^{1/2} _{L^6 (\R^3)} \Vert  u (t) \Vert _{L^6 (\R^3)} \nonumber \\ & \leq & \Vert \overline{u} (t) \Vert _{H^1 _{\a} (\R^3)} \Vert u(t) \Vert ^{1/2} _{L^2 (\R^3)} \Vert u(t)  \Vert ^{3/2} _{H^1 _{\a} (\R^3)}  \nonumber \\ & \leq & \Vert  \overline{u} (t) \Vert _{H^1 _{\a} (\R^3)} \Vert u(t)  \Vert ^{2} _{H^1 _{\a} (\R^3)}  \nonumber \\ & \leq & C_1 (C_2 + t) ^{- \min \{ \frac{9}{4} + \frac{3}{2} r^{\ast}, \frac{13}{4} + \frac{1}{2} r^{\ast} \}}.
\end{eqnarray}

\br \label{remark-slow-decay-nsv} If we were working with the Navier-Stokes equations, we would have

\begin{displaymath}
\int _{\R^3} \overline{u} (u \cdot \nabla) u \, dx \leq  \Vert \nabla \overline{u} (t) \Vert _{L^{\infty} (\R^3)} \Vert u(t) \Vert ^2 _{L^2 (\R^3)} \leq C (1 + t) ^{- \frac{5}{4}} \Vert u(t) \Vert ^2 _{L^2 (\R^3)},
\end{displaymath}
instead of (\ref{eqn:ineq-cross-term}), see page 590 in Niche and M.E. Schonbek \cite{Niche01042015}. This is so because the heat kernel strongly regularizes initial data, unlike the linear part of the Navier-Stokes-Voigt equation (see Remark \ref{facts-linear-part-nsv}) and this allows us to estimate a higher order norm on $\nabla \overline{u}$ and to have some uniform decay.
\er

Taking $B(t)$ as in (\ref{eqn:ball}) and following the same arguments as in the proof of Theorem \ref{main-theorem}, we have

\begin{eqnarray}
\label{eqn:inequality-with-integral}
\frac{d}{dt} \left( g(t) \Vert w(t) \Vert ^2 _{H^1 _{\a} (\R^3)} \right) &\leq& C g'(t) \int _{B(t)} \left( 1 + \a^2 |\xi|^2 \right) |\widehat{w} (\xi, t)| ^2 \, d \xi \nonumber \\ & + &  C_1 g'(t) (C_2 + t) ^{- \min \{ \frac{9}{4} + \frac{3}{2} r^{\ast}, \frac{13}{4} + \frac{1}{2} r^{\ast} \}}.
\end{eqnarray}
As

\begin{displaymath}
\widehat{w} (\xi,t) = \int _0 ^t e^{(t - s) \mathcal{M} (\xi) } \mathcal{F} ((u \cdot \nabla) u - \nabla p) (\xi,t) \, ds
\end{displaymath}
we obtain

\begin{displaymath}
|\widehat{w} (\xi, t)|^2 \leq C |\xi| \int_0 ^t \Vert u(s) \Vert  _{L^2 (\R^3)} ^2 \, ds,
\end{displaymath}
where we used that

\begin{displaymath}
|\mathcal{F} \left( (u \cdot \nabla) u - \nabla p \right) (\xi, t)| \leq C |\xi| \Vert u(t) \Vert ^2 _{L^2 (\R^3)},
\end{displaymath}
which we proved in Lemma \ref{estimate-modulus-uhat}. Hence

\be
\label{eqn:estimate-ball}
\int _{B(t)} \left( 1 + \a^2 |\xi|^2 \right) |\widehat{w} (\xi, t)| ^2 \, d \xi \leq C \left(\rho^5 (t) + \rho ^7 (t)  \right) \left( \int_0 ^t \Vert u(s) \Vert  _{L^2 (\R^3)} ^2 \, ds \right) ^2.
\ee
We will choose $g$ so that $\rho(t) = C_1 (C_2+ t) ^{- \frac{1}{2}}$. Assuming we have done so, from (\ref{eqn:inequality-with-integral}) and (\ref{eqn:estimate-ball}) we obtain

\begin{eqnarray*}
\label{eqn:inequality-with-g}
\frac{d}{dt} \left( g(t) \Vert w(t) \Vert ^2 _{H^1 _{\a} (\R^3) }\right) &\leq& C_1 g'(t)  (C_2+ t) ^{- \frac{5}{2}} \left( \int_0 ^t \Vert u(s) \Vert  _{L^2 (\R^3)} ^2 \, ds \right) ^2 \nonumber \\ & + & C_1 g'(t) (C_2 + t) ^{- \min \{ \frac{9}{4} + \frac{3}{2} r^{\ast}, \frac{13}{4} + \frac{1}{2} r^{\ast} \}}.
\end{eqnarray*}
If $r^{\ast} \neq - \frac{1}{2}$, then from the decay in Theorem \ref{main-theorem} we obtain

\begin{displaymath}
\left( \int_0 ^t \Vert u(s) \Vert  _{L^2 (\R^3)} ^2 \, ds \right)^2 \leq C_1 (C_2 + t) ^{- \min \{1 + 2 r^{\ast}, 3 \}}.
\end{displaymath}
which leads to

\begin{displaymath}
\frac{d}{dt} \left( g(t) \Vert w(t) \Vert ^2 _{H^1 _{\a} (\R^3) }\right) \leq  C_1 g'(t) (C_2 + t) ^{- \min \{ \frac{9}{4} + \frac{3}{2} r^{\ast}, \frac{13}{4} + \frac{1}{2} r^{\ast}, \frac{11}{2}  \}}.
\end{displaymath}
Taking $g(t) = C_1 (C_2 + t) ^m$, with $m > \max \{ \frac{9}{4} + \frac{3}{2} r^{\ast}, \frac{13}{4} + \frac{1}{2} r^{\ast}, \frac{11}{2}  \}$ we obtain

\begin{displaymath}
\Vert w(t) \Vert ^2 _{H^1 _{\a} (\R^3)} \leq C_1 (C_2 + t) ^{- \min \{ \frac{9}{4} + \frac{3}{2} r^{\ast}, \frac{13}{4} + \frac{1}{2} r^{\ast}, \frac{11}{2}  \}}.
\end{displaymath}
If $r^{\ast} = - \frac{1}{2}$ we obtain a logarithmic term from the integral of the $L^2$ norm of u, which does not change the estimate obtained in the previous line. This finishes the proof. $\Box$

\bibliographystyle{plain}

redpajama_set_name: RedPajamaArXiv


Japanese DMM Securities, the world's second largest retail forex broker by volumes posted its September forex volumes. During the month, total volumes were $491.1 billion. The figure was 15.8% below August's figures. The figures followed a similar decline that was reported by GMO Click earlier this week.
After topping $1 trillion in volume during June, activity at DMM Securities has fallen more than 50% from its year high. The decline is similar to results from other Japanese brokers. Among the largest declines was seen at SBI Trade FX. The multi-asset broker was a late entrant into the OTC FX sector. However, they quickly made a big impact by offering consistent extraordinary (0.1 ) tight spreads in the USDJPY to build up its forex business. The result was that out of nowhere, SBI FX Trade grew into one the top Japanese FX brokers, hitting over $500 billion in volumes during June. However, with the broker beginning to widen spreads in July, volumes in August fell to less than 20% of the June highs. The changes at SBI Trade FX also allowed competing brokers to begin to price their offerings wider, which is directly correlated to the Japanese industry-wide decline in volumes taking place.
When did they started …SBI FX?
I dunno if SBI's spread moves are really that related? Markets have just been quieter, less trending.
SBI has started narrowing their spreads again recently though I see. Fortunately I have an account, now just need movement in the market.
SBI FX started last year as I recall.


Help raise up a new generation of leaders in Southern and Eastern Africa!
Watch how the Foursquare Global Missions Fund (GMF) strategically fulfills the Great Commission in Southern and Eastern Africa and around the world. Then, join us as we reach the unreached, together.


text:
La rivière Changja est une rivière située en Corée du Nord, et un affluent droit du fleuve le Yalu.

Géographie
Elle traverse le nord du pays et traverse la ville de Kanggye.

Voir aussi

 le fleuve Yalu

Notes et références

Cours d'eau en Corée du Nord
1Changja

redpajama_set_name: RedPajamaWikipedia


QuitAlcohol.comDetoxHow Will I Feel During Detox?
How Will I Feel During Detox?
If you have a substance abuse problem and are interested in getting your life back on track, one of the first things you must do is go through detoxification. Not only will this put you on the path to success, but it will go a long way in helping you avoid a relapse down the road.
Even though you are well aware that detox is necessary, this does not change the fact that you are sure to feel poorly along the way. As you wean your body off of drug or alcohol use, you are likely to face a variety of withdrawal symptoms.
Note: if you truly want to make it through detoxification in a successful manner, it is a must that you avoid the use of drugs and/or alcohol during this time. You may experience cravings, however, giving in to them will only put you back at square one.
In most people, it does not take long after beginning detox to experience some or all of these symptoms. You can expect these symptoms to come about within 12 to 24 hours.
When you go through detoxification at a substance abuse treatment facility, you can be rest assured that a qualified and experienced medical staff will watch over you the entire time. Additionally, they are able to prescribe medication that will help you fight against withdrawal symptoms, making it easier for you to complete this stage of recovery without too much discomfort.
How you feel during detox will depend on many factors, including the finer details of your substance abuse as well as your body. You should expect some symptoms, but always remember that the end result is going to be a better life.


text:
"A Coarsening of Society"
A fascinating story was featured this month on the cover of Harper €™s Magazine. Entitled €œHappiness is a Worn Gun: My Concealed Weapon and Me, € the article was written by author Dan Baum, a €œfairly typical liberal Democrat € who recently obtained a permit to carry a concealed handgun in Colorado. Baum €™s article provides a balanced and insightful perspective on the culture that surrounds the gun rights community.
In the piece, Baum traces his 49-year love affair with firearms, which dates back to the summers he spent at camp firing a .22 caliber rifle as an overweight child. As an adult gun owner, Baum readily admits that, €œThe sensual pleasure of handling guns is a big part of the habit … They are deeply satisfying to manipulate, even without shooting. € After deciding that hunting and range-shooting was not allowing him to be close enough to his firearms (Baum wanted to €œlive the gun life €), he decided to apply for a permit to carry a concealed weapon in his home state of Colorado.
A Right to Instant Gratification
In Colorado, just as in 37 other €œShall-Issue € states, the state must issue a permit to carry a concealed handgun to any citizen that passes an instant computer background check and meets a basic set of requirements. One of those requirements is to complete just three hours of training through an approved handgun safety course. This course need be taken only once-no additional training is required when a permit holder renews his/her permit every five years.
Baum, to his credit, went beyond the required three hours of training and took two separate handgun safety courses over the course of five days. He hoped to receive serious instruction as he took on the weighty and dangerous responsibility of carrying a weapon in public. In practice however, Baum found that his two classes €œtaught [him] almost nothing about how to defend [himself] with a gun € and €œwere less about self-defense than about recruiting [applicants] into a culture animated by fear of violent crime. €
At his first training class in Boulder, Baum €™s instructor €œpacked about twenty minutes of useful instruction into four long evenings of platitudes, Obama jokes, and belligerent posturing. € He also openly admitted to breaking the law, saying he refused to get a carry permit because €œI don €™t think I have to get the government €™s permission to exercise my right to bear arms. € A police officer taught the class a €œlegal implications € segment and encouraged the applicants to lie to police if stopped while wearing their guns. He then told the class that even though it is illegal to shoot a fleeing criminal, €œIf your aim is good enough, you have time to get your story straight before I get there. € The class was shown €œlurid films of men in ski masks breaking into homes occupied by terrified women € and spent time examining pictures of a man that had been €œslashed open with a knife. € In the course of four evenings, the applicants only handled their firearms once-shooting about 50 rounds at targets approximately 15 feet away.
Baum €™s second class was at the Tanner gun show in Denver (the same place where Columbine killers Eric Harris and Dylan Klebold obtained their firearms). Baum described it as a €œfifteen-minute recruiting pitch for the National Rifle Association (NRA) and a long-winded, paranoid fantasy about home invasion … €˜They know where your bedroom is, and they €™re there to kill you. €™ € The irony in this, as Baum states, is that only 87 Americans were murdered during home burglaries in 2008: €œStatistically, you had a better chance of being killed by bees. €
While Baum asserts that American citizens should be able to carry concealed weapons in public, he was deeply disturbed by his experience with the permitting process in Colorado, stating that, €œIt €™s a scandal…that people can get a license to carry on the basis of a three hour €˜course €™ given at a gun show. State requirements vary, but some don €™t even ask students to fire a weapon before getting a carry permit. € Baum recommends that state governments €œenforce high standards for instruction, including extensive live firing, role-playing, and serious examination of the legal issues … States should [also] require a refresher course, the way Texas does, before renewing a carry permit. € Furthermore, €œSince people can carry guns state to state, standards should be uniform. € As Baum notes, €œThe Second Amendment confers a right to keep and bear arms. It does not confer a right to instant gratification. €
A Different Mentality
An obsession with violent crime ( €œAt class, it was hard to discern the line between preparing for something awful to happen and praying for something awful to happen €) is just one element of a gun culture that Baum found himself immersed in once he decided to carry a handgun in public. In Baum €™s words, €œAnyone who tells you he has no fantasy life constructed around his gun either has been packing it for as long as he €™s been watching television or is flat-out lying. €
Another element is paranoia that national €œgun confiscation is nigh. € Baum recalled a man selling Yugoslav AK-47s at the Tanner gun show and yelling, €œBuy it now! Tomorrow they may not let you! € €œYou don €™t think [Obama €™s] waiting for his second term to come and get them? € he asked Baum. €œYou €™re dreaming. € To Baum, such fears seem wildly exaggerated: €œFor as long as I €™ve been voting, I €™ve reflexively supported waiting periods, background checks, the assault-rifle ban, and other gun control measures. None interfered with my enjoyment of firearms, and none seemed to me to be the first step toward tyranny. €
Baum recalled another gun show dealer yelling to potential customers, €œLiberals want to take away your gun and your McDonald €™s both. € Baum discovered a €œclass-based resentment that permeates modern gun culture, € citing an editorial in the NRA €™s America €™s First Freedom magazine that characterized their opposition as €œthose who sip tea and nibble biscuits while musing about how to restrict the rest of us. €
Then there are the color-coded €œconditions of readiness € that concealed handgun permit holders govern themselves by. Condition White is €œtotal oblivion to one €™s surroundings-sleeping, being drunk or stoned, losing oneself in conversation while walking in city streets, texting while listening to an iPod. € Condition Yellow is €œbeing aware of, and taking an interest in, one €™s surroundings-essentially, the mental state we are encouraged to achieve when we are driving. € It requires €œbeing mentally prepared to kill. € Condition Orange is being aware of a possible threat and Condition Red is responding to danger.
Baum notes that €œcontempt for Condition White unifies the gun-carrying community almost as much as does fealty to the Second Amendment. € He was told by one of his Boulder instructors that €œwhen you €™re in Condition White you €™re a sheep. € The American Tactical Shooting Association notes that the only time you should be in Condition White is €œwhen in your own home, with the doors locked, the alarm system on, and your dog at your feet. € Gun carriers are instructed to be in Condition Yellow at all times.
After experiencing Condition Yellow for months, however, Baum found it to be €œkind of exhausting. € He missed Condition White. €œCondition White may make us sheep, but it €™s also where art happens, € Baum says. €œIt €™s where we daydream, reminisce and hear music in our heads. Hardcore gun carriers want no part of that. €
Drinking the Kool-Aid
Despite being critical about several aspects of the gun culture, Baum simultaneously seems to endorse some of its most well-worn talking points. In a recent radio interview, he argues that guns laws are €œnot going to keep guns out of the hands of the people that you don €™t want to have them € and therefore €œonly really apply to the law-abiding. € He also suggests that €œShall-Issue € laws have had no negative effect on public safety. In making these claims, Baum examines long-term trends in violent crime rates in the U.S., but curiously fails to comment on gun death rates or compare America to other industrialized democracies. Gun violence prevention organizations do not argue that guns cause crime, but rather that the presence of guns makes attempted crimes, attempted suicides, and arguments/confrontations of all kinds more lethal.
One recent study found that U.S. homicide rates were 6.9 times higher than rates in 23 other high-income countries, driven by firearm homicide rates that were 19.5 times higher. These 23 nations uniformly have fewer guns per capita than the United States and far tougher gun laws. An examination of gun death rates within the United States finds that states with tough gun laws (including €œMay-Issue € concealed carry laws) like Hawaii, New York and New Jersey have the lowest gun death rates. States with weak laws (and €œShall-Issue € concealed carry laws) like Louisiana, Alaska and Nevada have the highest. This data provides little support for the notion that €œan armed society is a polite society. €
Baum also erroneously states, €œYoung black urban men killing each other-that is the gun problem [in the U.S.] right now. € In reality, the claim that gun violence is a €œblack problem" hardly makes sense when one considers that out of the 31,446 gun deaths that occurred in America in 2005, 21,958 of the victims were whites, and from 1976 to 2005, 86% of white murder victims were killed by whites. Additionally, in 2008 the FBI reported 14,180 gun deaths, only 844 of which were gang related. The ten states with the highest rates of gun death per capita in the U.S. in 2007 were Louisiana, Mississippi, Alaska, Alabama, Nevada, Arkansas, Tennessee, New Mexico, Arizona and West Virginia-predominantly rural states.
Finally, while acknowledging that €œa lot of gun rights people are like the Taliban, if you don €™t agree with them on absolutely everything you €™re a friend of tyranny and a monster, € Baum does not fully grasp the political consequences of the gun rights movement €™s agenda. He fails to perceive the €œInsurrectionist Idea € that animates the movement-namely, the belief that the Second Amendment gives individuals the right to confront €œtyrannical € government with force of arms. Insurrectionists will naturally oppose any laws that allow government oversight of firearms ownership because they want to remain anonymous should they one day decide to wage war against our government. As Cato Institute analyst David Kopel puts it, €œThe tools of political dissent should be privately owned and unregistered. €
Insurrectionism is a far-right-wing ideology that opposes a strong, activist government in nearly all of its forms and, as such, presents a threat to the broader progressive agenda that Baum claims to support. Baum recently got the cold shoulder when his Harper's article was featured at "The Truth About Guns" blog. Author Robert Farago and commenters at the blog called Baum into question over "the obvious conflict of interest between his liberal upbringing and the consequences of his acceptance of gun ownership," with one reader comparing the current Democratic Leadership in Congress to Joseph Stalin, Mao Zedong and Heinrich Himmler.
Not a Prop
Baum has decided that he will probably stop carrying his handgun in public. €œIt €™s uncomfortable, distracting, and freaks out my friends; it €™s not worth it, € he says. Baum felt that carrying his gun had €œmilitarized [his] life € and brought out impulses in him that he disliked, including €œsocial pessimism € and €œirrational fear € ( €œYou don €™t want to contribute to a coarsening of society by preparing to kill at a moment €™s notice €). Ultimately, he had to remind himself that his gun €œis not a prop, a political statement, or a rhetorical device, but an instrument designed to blow a ragged channel through a human being. €


text:
Ongoing art exhibits around Tuscany
Trying to make plans for your holidays? Check out a selection of some of the most interesting ongoing exhibits in Tuscany. If you're also going to be in Florence, make sure to check out the current exhibitons in the city and surroundings. Also check out our blog for news on interesting events.
Don't stop there: check out local events including festivals, sagras and special holidays throughout Tuscany, a selection of those that are easy to find and fun to participate.
- what's happening in Tuscany
- what's happening in Florence
Keith Haring and Street Art | Palazzo Blu | Pisa
One of the leading exhibitions in Tuscany is that of Keith Haring at Palazzo Blu, where more than 170 works by the father of street art are exhibited for the first time in Europe tracing his artistic production, all coming from the Nakamura Keith Haring Collection.
The exhibition, open until April 17, 2022, develos over seven different sections that help the visitor retrace the life of Keith Haring through his works. The themes dealt with are very broad and indicative of the artist's personality, from the work with children to the birth of the mural in Pisa, up to those that have characterized his artistic production: AIDS prevention, gay rights, racism, apartheid, war and many others.
Also at Palazzo Blu, "Attitude" comes to life, an exhibition that collects the various personal attitudes of artists who began to paint spontaneously in the urban context, combining the energy of the street with creativity. Four rooms to recount the various souls of street art, from the rebellious to the more institutional. The works will be visible until April 3, 2022.
HOURS: Monday / Friday (10am-7pm) Saturday / Sunday (10am-8pm)
WHEN: until April 3 and 17, 2022
INFO: Palazzo Blu
Salvador Dalì: from Galileo Galilei to Surrealism | Palazzo delle Papesse | Siena
A unique exhibition of its kind, the result of the encounter between Dali's interest in science and Galileo Galilei's discoveries. About 150 of the most important works of the Catalan master arrive in Siena, which tell of his great passion for mathematics, physics and psychoanalysis. The path unfolds through extremely complicated concepts, such as those of space and time, translated by Salvador Dalì into the language of art.
The location is not accidental either, the exhibition "Salvador Dalì in Siena, from Galileo Galilei to Surrealism" is in fact housed inside the Palazzo delle Papesse in Siena, where the father of modern science lived after his abjuration.
Given its current success, the event has been extended until April 30, 2022.
HOURS: every day from 10am to 7pm
WHEN: until April 30, 2022
INFO: Dalì a Siena
"Painters of Light" | Ex Cavallerizza | Lucca
One hundred works from Italian and non-Italian museums and private collections arrive on display at the Ex Cavallerizza in Lucca, in Piazzale Verdi, for "I Pittori della Luce / Painters of Light". The itinerary, curated by Vittorio Sgarbi, aims to show the importance of light in 17th century painting, from the masterpieces of Caravaggio - the first to use all the power of this element on the canvas - to Pietro Paolini, one of the most important artists in Lucca in the new naturalistic school.
Among the other works you'll find those of Pieter Paul Rubens, Jusepe de Ribera and many others.
The exhibition is open until October 2, 2022 and, until March, you can see "Il Cavadenti / The Toothe Extractor" by Caravaggio, on loan from the Uffizi Gallery.
TIMETABLE: every day from 10a. to 8pm
WHEN: until October 2, 2022
INFO: Contemplazioni
Andy Warhol – ICONS! | PALP Palazzo Pretorio | Pontedera
The exhibition Andy Warhol - ICONS! showcases the life of the most influential artist of the 1960s, narrator of modern society who transformed the fetishes of the American collective imagination into art. The event presents more than 140 works arranged in 5 sections that cover his works from the beginning to the end of the 1980s.
It starts with the colorful Liz (Taylor) of '64 and ends with the iconic Marilyn. And again, three splendid Cows and other creations that have become legend: Brillo Box, Flowers, Eletric Chair, Campbell's Soup. Andrew Warhol was one of the most influential artists of the twentieth century, who with his multifaceted production (works, photos, films, design, illustrations) told the evolution of the Stars and Stripes.
HOURS: from Tuesday to Sunday (10am-7pm)
WHEN: until March 20, 2022
INFO: PALP Pontedera
"Magazzini Generali" | Museo della Città | Livorno
The idea was born in the period in which the Livorno civic museums were closed due to the pandemic: the staff took the opportunity to review the inventories and from there the exhibition "Magazzini Generali: The Livorno Collections from Deposits to the Museum" was born. An exhibition path inside the Museum of the City of Livorno that brings to light more than 90 works, including paintings, sculptures, photographs and objects hitherto hidden from the public.
The setting is also original and worthy of a deposit: the works are placed on or near cardboard boxes on which you will find captions and focuses for further information.
HOURS: from Tuesday to Sunday 10am-7pm, Friday and Saturday until 8pm
WHEN: until February 27, 2022
INFO: Museo della Citta di Livorno
Hi Woman! | Palazzo Pretorio | Prato
The Museum of Palazzo Pretorio in the city of Prato welcomes "Hi Woman! The news of the future" a wide-ranging exhibition that welcomes the works of 22 international artists. The focus of the exhibition revolves around the concept of annunciation, a word that contains multiple meanings as well as a milestone of Western culture.
Thus, not surprisingly and yet extraordinary, in organizing this exhibition it was decided to exhibit the works of 22 women, emphasizing the centrality of the female subject in ancient and contemporary narration.
HOURS: every day from 10:30am to 6:30pm, except Tuesdays
INFO: Palazzo Pretorio di Prato
Spazio Radicale e L'Arte e la Città | Centro Pecci | Prato
La mostra Spazio Radicale è una nuova occasione per valorizzare il patrimonio di immagini, oggetti e idee che compongono le varie raccolte e si trovano oggi negli archivi del Centro Pecci. Il percorso presenta le perlustrazioni dello "spazio" proposte da architetti "radicali" e artisti contemporanei, muovendosi tra spazi reali e immaginati. "Spazio Radicale" è visitabile fino al 30 aprile 2022 dal martedì alla domenica.
L'Arte e la Città è invece una mostra che indaga i rapporti fra arte contemporanea e ambiente urbano utilizzando opere provenienti dalle collezioni museali e rari materiali d'archivio. Dipinti, fotografie, sculture che parlano di incontri, scenari e soggetti dell'ambito urbano, che ci svelano anche inserimenti artistici pensati e realizzati per Prato. È aperta fino al 12 giugno 2022.
The Spazio Radicale exhibition is a new opportunity to showcase the treasure of images, objects and ideas that make up the various collections found today in the archives of the Pecci Center. The itinerary presents the explorations of the "space" proposed by "radical" architects and contemporary artists, moving between real and imagined spaces. "Spazio Radicale" can be visited until April 30, 2022 from Tuesday to Sunday.
Art and the City, on the other hand, is an exhibition that investigates the relationship between contemporary art and our urban environment using works from museum collections and rare archive materials. Paintings, photographs, sculptures that speak of encounters, scenarios and urban subjects, which also reveal artistic insertions designed and created for Prato. It is open until June 12, 2022.
HOURS: from Tuesday to Sunday 10am-8pm
WHEN: until April 30 and June 12, 2022
INFO: Centro Pecci
Jenny Saville | 5 locations | Florence
The Florentine exhibition invites the public to discover the work of Jenny Saville through her pictorial production, highlighting the close relationship between the artist and the masters of the Italian Renaissance, in particular with some of Michelangelo's masterpieces.
The works are exhibited in 5 different locations. There is still time to visit the exhibition: it is open until February 27, 2022 at the Museo Novecento (where you'll find the largest number of paintings by the British artist), at the Museo degli Innocenti and at Casa Buonarroti. Instead, it closes on February 20 in the Salone dei Cinquecento in Palazzo Vecchio and at the Museo dell'Opera del Duomo in Florence.
The Museo del Novecento is holding several activities tied to the exhibition: guided tours every Sunday at 3pm and 4.30pm; painting workshop for children up to 12 years, every Sunday at 4.30 pm. Reservation is required.
Here the other exhibitions in progress in Florence.
HOURS (MUSEO DEL NOVECENTO): 10am-8pm every day except Thursday
WHEN: until February 20 and 27
INFO: Museo Novecento
What to See in Tuscany
Exhibits in Florence
3 Towers with a View in Florence
Abbey of Monte Oliveto Maggiore
The Abbey of San Galgano
Cerreto Guidi, the villa
Loro Ciuffenna in Valdarno
Author: Lourdes Flores
I'm from California but have called Florence my home for over a decade. I love to explore Italy; it is a lot of fun to try to see everything like I'm seeing it for the first time, keeping you, our readers, always in mind. I enjoy sharing what I know and helping others as they make their travel plans for Tuscany through our Forum. If you have itinerary-related questions, please post them there!
List of ongoing exhibits currently in Florence,...
Where to enjoy the best panoramic views of...
Explore the beautiful area of Crete Senesi and...
Visit the old Abbey of San Galgano and see the...
Useful information on how to best visit the...
Helpful Itinerary Ideas for the town Loro...
Questions? The right place to ask is our Forum*
* Questions posted on Comments above will no longer receive replies: please ask on our Forum!


text:
Сульфоксиды — органические соединения с общей формулой R1R2SO.

Номенклатура
Название ациклических сульфоксидов образуются от двух органических радикалов, связанных к атомом серы, к которым прибавляется слово «сульфоксид», например:
(CH3)2SO — диметилсульфоксид, (C2H5)2SO — диэтилсульфоксид. Название циклических сульфоксидов образуется из название соответствующего гетероцикла с окончанием «оксид», например, тиациклогексан-1-оксид.

Химическая структура молекул сульфоксидов
Структура молекул сульфоксидов представляет собой искажённую пирамиду. В случае диметилсульфоксида характеристики молекулы следующие:

ИК-спектры сульфоксидов характеризуются полосами поглощения в области 1050—1060 см−1, которые соответствуют валентным колебаниям группы S=O.

Физические свойства
Сульфоксиды являются бесцветными малолетучими жидкими или кристаллическими веществами. Растворяются в большинстве органических растворителей, низшие сульфоксиды растворяются в воде.

Свойства некоторых сульфоксидов:

Методы получения
Сульфоксиды получают в основном окислением соответствующих сульфидов. Окислителями выступают пероксид водорода в кислой среде, оксиды азота, гипохлориты и некоторые другие окислители.

Химические свойства

Литература


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What's your favorite… hard to choose, right?!


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The London Plane Tree
October 29, 2012 in Trees | Tags: American, bark, beautiful, Brooklyn, huge, Hybrid, London, New York, plane, stately, storm, street, sycamore, tree | by Wayne | 5 comments
The tree in front of my house right now (you'll have to imagine the roaring)
Hurricane Sandy is nearly in Brooklyn: the sky looks like a sepulcher and dark winds are roaring down the street. The gale is howling in the huge London plane trees outside which are swaying and bending as though they were bamboo. This is nothing to sneer at since trees are nearly a meter (three feet) in diameter and twice as tall as the two and three story houses. The trees are probably as old as the neighborhood—which was built about a century ago. Hopefully the trees and I will all be standing tomorrow and not floating out in the Atlantic on our way towards Newfoundland.
A London plane tree (Platanus × acerifolia) at Vassar College
For purely academic reasons I looked up London plane trees and I was gratified to discover that they are "fairly wind resistant." I was also cheered to learn that the trees are a strange hybrid of Eurasian plane trees and American sycamores. Only in 17th century Europe were the new world trees planted in proximity to their old world relatives. The tree loving English realized what a beautiful and hardy tree this is and they began planting the hybrid plane trees along streets.
The trees really are beautiful. Like the magnificent rainbow eucalyptus, London Plane trees have mottled bark albeit in muted splotches of cream, gray, brow, and verdigris rather than in insanely colorful stripes. The trees can grow to 35 meters in height and can be up to 3 meters in circumference–in fact there is (hopefully still) one that big by a nearby church.
London plane trees at Union Square Park (NYC)
Resistant to pollution and able to survive with highly compacted roots, the London plane tree is a perfect ornamental city tree—so much so that the NYC Parks Department tries to limit its planting since the hybrid sycamore/plane makes up more than 10% of the trees in the city. Ironically the logo of the NYC Parks Department is a London plane tree leaf crossed with a Maple leaf.
Support NYC Parks!
The London plane tree is said to have beautiful wood which looks like freckled pink lace. The tree also grows ample crops of spiky seed balls which are eaten by squirrels and birds. The true worth of the tree is as a magnificent living specimen tree. I am devoutly wishing for the best for the plane trees on my street (and not only because they tower over the stone house I am inside).
Olympics Opening Ceremonies and the Spirit of the Times
July 26, 2012 in China, Opinion, Politics, Uncategorized | Tags: Beijing, Boyle, Ceremony, costume, dance, Danny, London, moral, Olympic, Olympics, Opening, politicians, ridiculous, spectacle, stars, Tempest | by Wayne | Leave a comment
The London Olympics Stadium
The 2012 Olympics are starting tomorrow. I'm looking forward to watching (and blogging about) some of the esoteric sports which only get their moment of glory every four years—especially the sailing, boating, and shooting sports which are my favorite. Before we get to the actual Olympics though, we have to get through the opening ceremonies, which are always a huge sloppy mess. Like costumed mascots, which fascinate and appall the viewer with a unique combination of human and inhuman elements (in fact the 2012 Summer Olympics already feature completely ludicrous mascots) there is something simultaneously evocative and revolting about such international mass spectacles. If you can tolerate the agonizing kitsch and the eye-wateringly lurid spectacle, there are always insights into the host nation and the larger zeitgeist of each era.
The Opening Ceremony of the 2004 Summer Olympics in Athens
Sydney 2000 Summer Olympics Opening Ceremony
The principle elements of opening ceremonies generally include pyrotechnics, has-been pop stars, dreadful dance routines, strange performance art, posturing politicians, and crazy costumes. There is also a moral lesson or story (which is meant to be an undercurrent but which is usually fairly overt) presented in a peculiar opera-like mash of dance, cameo celebrity appearances, and moveable sets.
Like all Americans, I boycotted the 1980 Moscow Olympics opening
Each host nation always manages to bring its own special horrible thing to the opening ceremony–for example the Beijing opening ceremony featured mass dance routines that would put North Korea to shame. Tens of thousands of majorettes all marched in place for hours in high heeled boots with big fake smiles that said "they have my family!"
2008 Beijing Olympics Opening Ceremony (featuring clapping marching performers)
The overarching message of the Beijing opening ceremony seems to have been that China had a very ancient and superior culture but then fell on hard times (through no fault of its own) before building a brighter & better homogenous society which is poised to take leadership of the world. During the bombastic (but compelling) performance, the cameras kept cutting to the grandstand filled with world leaders. Putin stared at the spectacle with icy hatred in his eyes and a hard frown. George Bush Junior kept slumping over in his seat with disinterest as Laura plucked at his elbow.
Crazy Costumes from the 2008 Beijing Olympics opening Ceremony
At least China still envisions a future in outer space (2008 Beijing Olympics opening Ceremony).
England, of course, is not lacking in dried-up rock stars and supernumerary VIPS, but preliminary reports indicate tomorrow's opening ceremony will also be a chronological morality tale put together by England's foremost director. The 2012 Olympics opening ceremony was designed by Danny Boyle, the director of Bollywoodwesque Slumdog Millionaire, zombie horror film 28 Days Later, and heroin-soaked black comedy Trainspotting. According to The Daily Mirror:
The whole ceremony is based on William Shakespeare's brilliant play, The Tempest. The title in particular is borrowed from a stirring speech made by the native Caliban to his master Prospero. "Be not afraid," says Caliban, "for the isle is full of noises. Sounds, and sweet airs, that give delight and hurt not."
In addition to the bard, Boyle apparently intends to pay homage to England's agrarian past with a comprehensive cavalcade of live farm animals. The second act will feature the hardships of the industrial revolution and the amorality of England's colonial ascendancy—which is meant to provide a dark and upsetting counterpoint to the initial bucolic splendor. Finally modern England will appear as a land of toleration and rock-and-roll! [Of course all of this could be wrong. Boyle has been trying to keep his program secret, and this information is based on leaks and speculation.]
2010 Spirit Bear? Does anyone else remember this?
This year is already looking exciting in terms of political drama. Putin will not attend since he is angry with Great Britain (as apparently is President Obama, though nobody has yet fathomed why). An awkward Mitt Romney will be there, trying [and failing] to fit in with actual world leaders. But the real excitement will focus on the central performance, a train wreck of public art featuring farm animals, Elton John, industrial grime, James Bond, the Spice girls, and medieval kings. What does that say about the zeitgeist? Find out tomorrow!
2012 Olympic Mascots
May 27, 2010 in Humor, Mascots, Opinion | Tags: 2012 Olympic Mascots, hoodie, lion, London, Mandeville, Mascots, Olympic, Olympics, pigeon, unicorn, Wenlock | by Wayne | 6 comments
In our continuing exploration of the uneasy world of mascots, it's time to meet Wenlock and Mandeville, mascots for the 2012 London Olympics. Hmm, oh dear… They each have a camera for an eye, which seems eerily appropriate given England's dystopian fascination with Orwellian surveillance equipment. They do not have mouths, probably so that they are unable to scream. Understanding their back story makes them no less disquieting: according to their creators, they are steel nuggets handicrafted by an eccentric grandfather and then given life by children's love for sports.
"The mascot will help us engage with children which is what I believe passionately in," London organising committee chairman Sebastian Coe told Reuters.
"The message we were getting was that children didn't want fluffy toys, they didn't want them to be human but they did want them rooted in an interesting story. "By linking young people to the values of sport, Wenlock and Mandeville will help inspire kids to strive to be the best they can be."
Um, what? Toy designers know how easily children can be (mis)lead during marketing research. You have to watch their hands and eyes in order to find out their real answers. Or maybe I'm wrong and English children really do like mouthless, handless, soulless one-eyed robot-monsters.
Come on English designers! Just slap a bearskin on a bulldog and head for the pub. Everyone would be happy and you would have an enduring winner instead of the travesties which Wenlock and Mandeville so clearly are. As an added bonus, here are some alternate ideas for 2012 London Olympics mascots:
Trafalgar the pigeon (by Danny Ihns)
The lion and the unicorn from the UK coat of arms (as re-envisioned by Woodrow Phoenix)
'Dodgee – the Olympic Hoodie' (created by Aaron Robinson)


Inauguration of the MSc Ayurvedic Hospital Management Programme of 2017 intake was held on July 02, 2017 at Centauria Hill Resort, Ratnapura. Guest of Honour, Senior Professor APG Amarasinghe of the Institute of Indigenous Medicine, University of Colombo delivered the keynote speech at the inauguration. Dr. Iraj Ratnayake, Dean, Faculty of Graduate Studies, Prof. Wasantha Rathnayake, Dean, Faculty of Management Studies, Dr. MSM Aslam, Chairperson of the Board of Study in Management, Dr. Athula Gnanapala, Coordinator of the MSc Ayurvedic Hospital Management Programme were also addressed the gathering. Academic staff of the Faculty of Management Studies, Administrative staff of the university and 2017 intake of 42 Ayurvedic Medical Practitioners who are attached to Department of Ayurveda at national and provincial levels were also among the delegates.


We don't typically think about planning for the unexpected in our businesses (because we're not expecting it!), but it's vital in order to keep things running when you can't work. Meg found herself suddenly needing to take unexpected time off because of some health issues, which taught her a lot about preparing her freelance business for a leave of absence. Learn more about it on her website!
We learned so much valuable information on this episode, and it doesn't just apply to independent contractors. It applies to every business! If your business can't run without you, even for a day, start now to implement these tips so you can keep things running smoothly no matter what life throws your way.
For more tips, especially for Pinterest VAs, check out Episode 6: How to Manage Multiple Pinterest Accounts with Google Profiles. This game-changing hack will help you streamline your workflow and stay organized!
On today's episode, Annette and guest host Meg Bateman of MegBateman.com talk about something most VAs and other independent contractors don't think about much—how to plan for expected (or unexpected!) personal time off. And stick around for today's peculiar pin—cooking chicken in milk.
Independent contractors don't get paid unless they work. If you need to take time off, the only way to keep getting paid is to keep the workflow going. Work ahead at least a week, but two is even better. This doesn't just give you a cushion for necessary time off, it will also reduce your stress!
This might be a virtual assistant (or even a spouse), but someone needs to be able to take over for you when you're "out of the office." It's also smart to consider a backup for your backup in case they also need to take some time off at the same time as you!
If you don't have a backup person, consider your professional network. Instead of seeing them as your competition, approach a trusted colleague with a win-win collaboration. If you can cover for each other in the event of a vacation, surgery, or family emergency, your business will have an extra layer of protection without adding an additional team member.
Keep all your information in one place with the capability to share access. Evernote, Google Docs, and Google Keep are all great options.
If you have to take PTO, a quick check-in email is helpful so they understand that service will continue uninterrupted, when you'll back, etc.
Going back to work can be harder than preparing for leave. When you're back in the office, you may need some "recovery time." Whether your leave is vacation or medical, it may take you a while to get back into the swing of work, so take this into account when you're working to get ahead.
It may take some time to implement all these action steps, but the results will be well worth the effort. Not only will your business run smoothly in case of planned or unplanned time off, but your stress load will decrease significantly knowing you're prepared!
Huge thanks to Meg for this incredibly helpful information! If you know someone who could benefit from this information, please share the podcast or this blog post.
To connect with Meg online, visit her website MegBateman.com or on Facebook or Instagram (you can follow her healing journey here).
Don't forget to check out today's Peculiar Pin How to Cook Chicken in Milk, and follow our peculiar pins board for future ones.