Digital marketing, SEO, reputation management and performing IT tools will help your business be one step ahead of the competition as users' behaviors and search engines' capabilities develop over time. Commerx, will help you build a robust online brand.
We focus on the management of your brand or product and services search results within the digital space by using a combination of brand reputation monitoring, potentially damaging content removal and customer feedback mining to build and shape the perception of your brand.
Commerx Digital solutions will not only help your business get on top of major search engines results pages using various digital strategies but also use advanced digital technologies to transform businesses core operations.
At Commerx, we help businesses with content marketing, SEO and digital marketing, so the content we publish on behalf of your business is designed to address your target customers inquiries and position you as a subject matter expert. Our contents are designed to attract and convert in a non-aggressive nor in a-sales-focused way.


text:
using Eventing;
using Eventing.Core.Persistence;
using System;
using System.Collections.Concurrent;
using System.Threading;
using System.Threading.Tasks;

namespace Agrobook.Domain.Archivos.Services
{
    public class ArchivosService : EventSourcedHandler
    {
        private readonly IFileWriter fileWriter;
        private readonly ConcurrentDictionary<string, SemaphoreSlim> locks = new ConcurrentDictionary<string, SemaphoreSlim>();

        public ArchivosService(IFileWriter fileWriter, IEventSourcedRepository repository) : base(repository)
        {
            Ensure.NotNull(fileWriter, nameof(fileWriter));

            this.fileWriter = fileWriter;
        }

        public async Task<ResultadoDelUpload> HandleAsync(AgregarArchivoAColeccion cmd)
        {
            var @lock = this.locks.GetOrAdd(cmd.idColeccion, new SemaphoreSlim(1, 1));
            await @lock.WaitAsync();
            try
            {
                var resultado = await HandleAsyncWithPesimisticConcurrencyLock(cmd);
                return resultado;
            }
            catch (Exception)
            {
                throw;
            }
            finally
            {
                // This is the optimization of the locking mechanism.
                // Requires more testing
                if (@lock.CurrentCount < 1)
                    this.locks.TryRemove(cmd.idColeccion, out @lock);
                // In high contentious scenarios (multiple concurrent uploads, it gets null) I do not know why....
                @lock?.Release();
            }
        }

        public async Task HandleAsync(EliminarArchivo cmd)
        {
            var coleccion = await this.repository.GetOrFailByIdAsync<ColeccionDeArchivos>(cmd.IdColeccion);

            if (!coleccion.YaTieneArchivo(cmd.NombreArchivo))
                throw new InvalidOperationException("El archivo ni siquiera existe");

            if (coleccion.EstaEliminado(cmd.NombreArchivo))
                throw new InvalidOperationException("El archivo ya esta eliminado");

            coleccion.Emit(new ArchivoEliminado(cmd.Firma, cmd.IdColeccion, cmd.NombreArchivo));

            await this.repository.SaveAsync(coleccion);
        }

        public async Task HandleAsync(RestaurarArchivo cmd)
        {
            var coleccion = await this.repository.GetOrFailByIdAsync<ColeccionDeArchivos>(cmd.IdColeccion);

            if (!coleccion.YaTieneArchivo(cmd.NombreArchivo))
                throw new InvalidOperationException("El archivo ni siquiera existe");

            if (!coleccion.EstaEliminado(cmd.NombreArchivo))
                throw new InvalidOperationException("El archivo no está eliminado, no necesita restaurarse");

            coleccion.Emit(new ArchivoRestaurado(cmd.Firma, cmd.IdColeccion, cmd.NombreArchivo));

            await this.repository.SaveAsync(coleccion);
        }

        public async Task HandleAsync(RegistrarDescargaExitosa cmd)
        {
            var coleccion = await this.repository.GetOrFailByIdAsync<ColeccionDeArchivos>(cmd.IdColeccion);
            coleccion.Emit(new ArchivoDescargadoExitosamente(cmd.Firma, cmd.IdColeccion, cmd.NombreArchivo, coleccion.GetSize(cmd.NombreArchivo)));

            await this.repository.SaveAsync(coleccion);
        }

        private async Task<ResultadoDelUpload> HandleAsyncWithPesimisticConcurrencyLock(AgregarArchivoAColeccion cmd)
        {
            var coleccion = await this.repository.GetByIdAsync<ColeccionDeArchivos>(cmd.idColeccion);
            if (coleccion == null)
            {
                coleccion = new ColeccionDeArchivos();
                ColeccionDeArchivosIdProvider.ValidarElIdDeColecionPropuesto(cmd.idColeccion);
                coleccion.Emit(new NuevaColeccionDeArchivosCreada(cmd.Firma, cmd.idColeccion));
            }
            else if (coleccion.YaTieneArchivo(cmd.Descriptor.Nombre))
                return ResultadoDelUpload.ResponderQueYaExiste();

            if (await this.fileWriter.TryWriteUnindexedIfNotExists(cmd.FileContent, cmd.idColeccion, cmd.Descriptor))
            {
                coleccion.Emit(new NuevoArchivoAgregadoALaColeccion(cmd.Firma, cmd.idColeccion, cmd.Descriptor));

                await this.repository.SaveAsync(coleccion);

                return ResultadoDelUpload.ResponderExitoso();
            }
            else
                return ResultadoDelUpload.ResponderQueYaExiste();
        }
    }
}

redpajama_set_name: RedPajamaGithub


text:
Brown University Takes a Multifaceted Approach to Safeguarding Private Information
Innovative tactics keep data safe.
Anne Rawland Gabriel
Anne Rawland Gabriel is a freelance journalist who specializes in business technology. She is a frequent contributor to the CDW family of technology magazines.
Thanks to a multifaceted approach to data security, CISO David Sherry maintains an increasingly secure networking landscape at Brown University in Providence, R.I. By combining the best of technology solutions and human intellect, Brown officials are able to safeguard sensitive information.
Brown is not alone in its mission: Colleges and universities everywhere are amplifying data-security efforts as massive amounts of data are generated — and grow exponentially — through learning-management systems and the proliferation of wireless devices. Combined with the burgeoning student data collected by administrative offices, an untold amount of private information is now at risk on U.S. campuses.
Strategies to combat the security challenge vary, but many include technology solutions, such as modern identity-management systems.
North American educational institutions planning to invest the same amount, or more, in data security in 2013.
SOURCE: PwC "Global State of Information Security® Survey 2013," September 2012
At Brown, which has a student body of 8,600, Sherry and his staff of two receive nontechnical assistance with developing security strategies from a pair of multidisciplinary advisory organizations: the Brown Security Round Table, which meets every other week; and the Data, Privacy, and Records Management Steering Committee, which gathers quarterly.
"Both groups are very informative to us," Sherry says. "Since we don't have a huge staff, this extends our reach to different areas of the university."
On the technology side, Brown IT officials began the process of replacing the university's homegrown IAM (identity and access management) solution in early 2012. The existing system was developed incrementally over time and had swelled to a complex system of solutions and manual processes.
After thoroughly evaluating the available technologies, Brown chose the Oracle Identity Manager to address current demands.
"It provided the best fit for our security needs and allowed us to leverage our existing Oracle infrastructure," Sherry says. "With the new system, our attestation and account security will be significantly improved. This provides numerous benefits, including the ability to reduce the number of affiliated accounts and boost compliance. Simultaneously, we'll reduce complexity and enable automation of multiple identity management process."
IAM: A Spectrum of Technologies
IDC research director Sally Hudson says today's IAM solutions include "a spectrum of technologies that tells your system who is accessing your system, why they are accessing it, where they are coming from and how they are getting in."
In academia especially, she says, "it's important to relieve access barriers for students, faculty and other users while ensuring that each of these identities is assigned the correct access profile."
Modern IAM solutions accomplish the task in myriad ways, Hudson says, including fine-grained entitlements, automated provisioning/deprovisioning, single sign-on, authentication and advanced auditing and reporting for meeting regulatory compliance.
Self-service enrollment functionality also is perfectly suited to the higher education setting, Hudson says. "It saves significant amounts of time for both end users and IT."
Hudson advises institutions to "look for a solution based on industry standards, from a reputable and proven provider, that best addresses your institution's situation."
Back at Brown, the next project on the security list is adoption of a solution for mobile device management (MDM).
"MDM is a huge initiative for us," Sherry says. "We're in the process of evaluating manufacturers and expect to begin deployment within the next year."
Four Features
Hudson says institutions now shopping for an IAM solution should consider four features:
Automation: Modern IAM solutions streamline and automate security processes, from provisioning end users through the lifecycle, all the way to deprovisioning, when appropriate.
Customization: Best-of-breed solutions provide ample opportunities for IT managers to tailor solutions (i.e., alerts, challenge/response and end-user controls) to meet specific situations.
Flexible Delivery: Many solutions offer a range of deployment choices, including cloud-based and onsite solutions that may include a purpose-built appliance or a software application.
Self-Service: IT managers can customize solutions to enable self-service password resets or even self-provisioning for specific user types. Such features empower users and free up time for IT staff to concentrate on mission-critical projects.

Image courtesy of David Castillo Dominici / FreeDigitalPhotos.net

University Cyber Ranges Immerse Students in Cybersecurity Education
Hackers Evolve Attacks Methods in Higher Education Breaches
Security, Privacy and Confidentiality: What's the Difference?

redpajama_set_name: RedPajamaCommonCrawl


text:
Yes Co-Founder Chris Squire Dies
Richard E. Aaron, Getty Images
Chris Squire, stalwart co-founder of Yes, has died, bandmate Geoff Downes confirms via Twitter. The 67 year old had announced a hiatus from the band in May, following a diagnosis of acute erythroid leukemia.
"Utterly devastated beyond words to have to report the sad news of the passing of my dear friend, bandmate and inspiration Chris Squire," says Downes, who appeared on three Yes studio albums with the bass-playing legend – 1980's Drama, 2011's Fly From Here and 2014's Heaven and Earth.
Squire was the only constant in a band that began evolving almost as soon as it was founded in advance of a self-titled 1969 debut. Yes' Aug. 7 concert with Toto will mark the first-ever without him.
Yes later released a statement that marks Squire's sad passing:
It's with the heaviest of hearts and unbearable sadness that we must inform you of the passing of our dear friend and Yes co-founder, Chris Squire. Chris peacefully passed away last night in Phoenix Arizona, in the arms of his loving wife Scotty.
For the entirety of Yes' existence, Chris was the band's linchpin and, in so many ways, the glue that held it together over all these years. Because of his phenomenal bass-playing prowess, Chris influenced countless bassists around the world, including many of today's well-known artists. Chris was also a fantastic songwriter, having written and co-written much of Yes' most endearing music, as well as his solo album, Fish Out of Water.
Outside of Yes, Chris was a loving h usband to Scotty and father to Carmen, Chandrika, Camille, Cameron, and Xilan. With his gentle, easy-going nature, Chris was a great friend of many … including each of us. But he wasn't merely our friend: he was also part of our family and we shall forever love and miss him.
Squire's final work on Heaven and Earth helped that project to a Top 20 finish on the UK charts. Yes' best-selling albums include 1983's triple-platinum 90125 and 1971's double-platinum Fragile.
Born at the right time, on March 4, 1948, Squire's quickly developing musical genius was matched stride for stride by technology. He'd use a distinctive Rickenbacker 4001 electric to help redefine the role of the bass, even as he became Yes' undisputed leader.
"Our second album Time and a Word was where we first started to work with [new Yes engineer] Eddy Offord," Squire once told Ultimate Classic Rock. "He liked the way my tone was coming out of my Marshall 100-watt that I'd used ever since the beginning really, and the thing that he added to that by [using] certain compression and certain settings that he liked in the studio, so we just developed it really, from Time And A Word onwards and I suppose you could say that by the time we got to Fragile and "Roundabout," he'd really finessed the production of my bass sound."
Chris Squire's focus, beyond a lone 1975 solo effort called Fish Out of Water, was always on his band, as Yes released a total of 21 studio albums. He remained steadfastly in place until today, while the group – which has some 40 members, former members and occasional studio sidemen in its ranks – went through a series of lineup changes.
"It's really more by default than design that that situation has occurred," Squire remembered. "I think that in a lot of those instances there has been a lot of members of the band who have decided that they wanted to leave mainly to pursue solo projects and in the case of Rick Wakeman, that happened three or four times where he'd leave and come back and with Jon Anderson, that's happened in the past that he has gone off to do solo things and then come back to the fold. We must remember also that, of course, Alan White has also been there since 1972 when he came in, so he's been a constant compadre of mine ever since then."
White is now Yes' longest-tenured member. Guitarist Steve Howe joined in 1970, beginning the first of what would become three separate stints. Billy Sherwood, a '90s-era contributor, has already been announced as Squire's replacement on the upcoming Yes tour. Squire is only the second member of Yes to have died, following fellow co-founder Peter Banks in 2013.
Remembering Other Rockers We've Lost in 2015
Next: Top 10 Chris Squire Yes Songs
Source: Yes Co-Founder Chris Squire Dies
Filed Under: yes


There is no easy way to deal with separation. When a relationship comes to an end, especially if you have a long history with that person, it can be the most horrible feeling in the world.
But there are things you can do to make it easier, and it starts by admitting the truth to yourself. This is the hardest part because how do just let go of somebody who has been the love of your life?
The reality is that you need to accept things are over. If you can't find some closure, you are going to get stuck in a past relationship. And when you take final decision to end this painful relationship, you need to know the Important of Divorce Laws If You're Thinking About Filing.
Talking to a good friend or even a close family member won't bring your loved one back, but it will give you a way to vent. At the moment, there are a lot of emotions rushing through your body you can't seem to control. And talking is a good way to steer those emotions in one direction or take therapy for divorce to recover emotionally and mentally.
If you know of a quiet place where nobody will hear you scream, maybe it's time to take a trip. However, you might want to take somebody with you just in case you don't feel safe alone. Once you get to that spot, start screaming your lungs out. You'll be surprised at how well it works.
Whether you like to paint or take pictures, build houses or snorkeling, use the pain as motivation. Do your best stay proactive and busy, especially on the mornings when you really don't want to get out of bed.
Whatever you do, don't fall into limbo and stay there. Of course, you are going to cry for hours on end, but it does pass, eventually. Get more consultation to dealing with separation, you can consult https://deanhineslawyer.com online.


text:
Ronien ist ein Wohnplatz von Rühstädt im Landkreis Prignitz in Brandenburg.

Geografie
Der Ort liegt einen Kilometer südöstlich von Bälow und zwei Kilometer nördlich von Rühstädt. Er verfügt über keine eigene Gemarkung, sondern liegt auf der von Bälow. Die Siedlung besteht aus einigen wenigen Einzelhäusern die eng beieinander stehen. Am Ostrand liegt ein Landwirtschaftsbetrieb und 200 Meter südsüdwestlich ein Wasserwerk. Die Nachbarorte sind Lanken im Nordosten, Rühstädt im Süden, Scharpenlohe im Westen sowie Bälow im Nordwesten.

Einzelnachweise

Geographie (Rühstädt)
Ort im Landkreis Prignitz

redpajama_set_name: RedPajamaWikipedia


text:
\section{#1}\label{#2}\setcounter{equation}{0}}
  \textwidth 16truecm \textheight 8.4in\oddsidemargin0.2truecm\evensidemargin0.7truecm\voffset.9truecm
  \def\nnewpage{}

   \def\blackdot{{\color{red}{\hskip-.0truecm\rule[-1mm]{4mm}{4mm}\hskip.2truecm}}\hskip-.3truecm
   \def{{\color{purple}{\hskip-.0truecm\rule[-1mm]{4mm}{4mm}\hskip.2truecm}}\hskip-.3truecm
 \def\cor{\color{red}
 \def\cog{\color{green}
 \def\coe{\color{blue}
 \def\cob{\color{black}

\begin{document}

\def\overline\partial{\overline\partial}
\defL{L}
\def\int\!\!\!\!\int{\int\!\!\!\!\int}
\def\int\!\!\!\int{\int\!\!\!\int}
\def\int\!\!\!\!\int\!\!\!\!\int{\int\!\!\!\!\int\!\!\!\!\int}
\def\int\!\!\!\!\int\!\!\!\!\int\!\!\!\!\int{\int\!\!\!\!\int\!\!\!\!\int\!\!\!\!\int}
\def{\cor \underline{??????}\cob}{{\cor \underline{??????}\cob}}
\def\nto#1{{\coC \footnote{\em \coC #1}}}
\def\fractext#1#2{{#1}/{#2}}
\def\fracsm#1#2{{\textstyle{\frac{#1}{#2}}}}
\defU{U}
\def{}
\defp_{\alpha}{p_{\alpha}}
\defv_{\alpha}{v_{\alpha}}
\defU_{\alpha}{U_{\alpha}}
\defq_{\alpha}{q_{\alpha}}
\defw_{\alpha}{w_{\alpha}}
\deff_{\alpha}{f_{\alpha}}
\defd_{\alpha}{d_{\alpha}}
\defg_{\alpha}{g_{\alpha}}
\defh_{\alpha}{h_{\alpha}}
\def{+\delta}
\def\psi_{\alpha}{\psi_{\alpha}}
\def\psi_{\beta}{\psi_{\beta}}
\def\beta_{\alpha}{\beta_{\alpha}}
\def\gamma_{\alpha}{\gamma_{\alpha}}
\defT{T}
\defT_{\alpha}{T_{\alpha}}
\defT_{\alpha,k}{T_{\alpha,k}}
\deff^{k}_{\alpha}{f^{k}_{\alpha}}

\def\overline v{\overline v}
\def\overline u{\overline u}
\def\overline p{\overline p}
\def\overline q{\overline q}
\def\overline w{\overline w}
\def\overline z{\overline z}

\newcommand{{u}}{{u}}
\newcommand{\mathbb{R}}{\mathbb{R}}
\newcommand{\mathbb{N}}{\mathbb{N}}
\newcommand{\mathbb{Z}}{\mathbb{Z}}

\newcommand{B_r(x_0)}{B_r(x_0)}
\newcommand{Q_r(x_0,t_0)}{Q_r(x_0,t_0)}

\newcommand{\hilight}[1]{\colorbox{yellow}{#1}}

\def{}
\def{}
\def\fbox{\fbox{\bf\tiny I'm here; \today \ \currenttime}}{\fbox{\fbox{\bf\tiny I'm here; \today \ \currenttime}}}

   \baselineskip=15pt

\def\nts#1{{\color{red}\hbox{\bf ~#1~}}}

\def{\overline M}{{\overline M}}
\def\widetilde{\widetilde}
\newtheorem{Theorem}{Theorem}[section]
\newtheorem{Corollary}[Theorem]{Corollary}
\newtheorem{Proposition}[Theorem]{Proposition}
\newtheorem{Lemma}[Theorem]{Lemma}
\newtheorem{Remark}[Theorem]{Remark}
\newtheorem{definition}{Definition}[section]
\def\thesection.\arabic{equation}{\thesection.\arabic{equation}}
\def\hfill$\Box$\\{\hfill$\Box$\\}
\def\hfill$\Box$\\{\hfill$\Box$\\}
\def\comma{ {\rm ,\qquad{}} }
\def\commaone{ {\rm ,\qquad{}} }
\def\dist{\mathop{\rm dist}\nolimits}
\def\sgn{\mathop{\rm sgn\,}\nolimits}
\def\Tr{\mathop{\rm Tr}\nolimits}
\def\div{\mathop{\rm div}\nolimits}
\def\curl{\mathop{\rm curl}\nolimits}
\def\supp{\mathop{\rm supp}\nolimits}
\def\divtwo{\mathop{{\rm div}_2\,}\nolimits}
\def\re{\mathop{\rm {\mathbb R}e}\nolimits}
\def\indeq{\qquad{}\!\!\!\!}
\def\period{.}
\def\semicolon{\,;}
\defS{S}

\title{An anisotropic partial regularity criterion for the Navier-Stokes equations}
\author{Igor Kukavica}
\address{Department of Mathematics, University of Southern California, Los Angeles, CA 90089}
\email{kukavica@usc.edu}

\author{Walter Rusin}
\address{Department of Mathematics, Oklahoma State University, Stillwater, OK 74078}
\email{walter.rusin@okstate.edu}

\author{Mohammed Ziane}
\address{Department of Mathematics, University of Southern California, Los Angeles, CA 90089}
\email{ziane@usc.edu}

\begin{abstract}
In this paper, we address the partial regularity of suitable weak solutions of the incompressible Navier--Stokes equations. We prove an interior regularity criterion involving only one component of the velocity. Namely, if $(u,p)$ is a suitable weak solution and a certain scale-invariant quantity involving only $u_3$ is small on a space-time cylinder $Q_r(x_0,t_0)$, then $u$ is regular at $(x_0,t_0)$.
\end{abstract}

\maketitle

\section{Introduction}\label{sec:intro}
\setcounter{equation}{0}

The goal of this paper is to address the partial regularity of solutions of the 3D Navier--Stokes equations
 \begin{align}\label{NSE}
	\partial_t u - \Delta u + \sum_{j=1}^3\partial_j (u_j u) + \nabla p
           & = 0 \indeq \nonumber \\
	\div u & = 0 \indeq
	  \end{align}
where $u(x,t)=(u_1(x,t),u_2(x,t),u_3(x,t))$ and $p(x,t)$ denote the unknown velocity and the pressure.

The theory of partial regularity for the NSE ,
whose aim is to estimate the
Hausdorff dimension of the singular set and development of interior
regularity criteria, was initiated by Scheffer in \cite{S1,S2}.
In a
classical paper~\cite{CKN}, Caffarelli, Kohn, and Nirenberg proved
that
for a suitable weak solution
the one-dimensional parabolic Hausdorff measure (parabolic
Hausdorff length) of the singular set equals zero.
Recall that a point is regular if there exists a neighborhood
in which $u$ is bounded (and thus H\"older continuous);
otherwise,
the point is called singular.
Their
interior regularity criterion reads as follows: There exist two constants
$\epsilon_{\text{CKN}}\in(0,1]$ and $\alpha\in(0,1)$ such that if
  \begin{align*}
	\int_{Q_1}(|u|^3 + |p|^{3/2})\;dx dt \leq \epsilon_{\text{CKN}}
  \end{align*}
then
  \begin{align*}
	\|u(x,t)\|_{C^\alpha(Q_{1/2})} <\infty
  \end{align*}
where $Q_r = \{(x,t): |x| < r, -r^2 \leq t \leq 0 \}$.
Alternative proofs were given by
Lin~\cite{Li},
Ladyzhenskaya and Ser\"egin~\cite{LS},
an author of the present paper \cite{K1,K2},
Vasseur~\cite{V},
and Wolf~\cite{W1,W2}.
The problem of partial regularity of the solutions of the
Navier--Stokes equations has since then been addressed in various
contexts \cite{KP, RS1, RS2, RS3, Se1, Se2} and a variety of interior regularity
criteria has been proposed. In particular Wolf
proved in \cite{W2} the following: There exists
$\epsilon_\text{W}>0$ such that if
  \begin{align*}
	\int_{Q_1}|u|^3\;dx dt
        \le \epsilon_{\text{W}}
  \end{align*}
then the solution $u(x,t)$ is regular at the point $(0,0)$.

In a recent paper \cite{WZ}, Wang and Zhang proved an
anisotropic interior regularity criterion, which states:
For every $M>0$ there exists
$\epsilon_{\text{WZ}}(M)>0$ such that if
  \begin{align*}
	\int_{Q_1} (|u|^3 + |p|^{3/2})\;dx dt \leq M
  \end{align*}
and
  \begin{align*}
	\int_{Q_1}|u_h|^3 \;dx dt \leq \epsilon_{\text{WZ}}(M)
  \end{align*}
where $u_h = (u_1,u_2)$, then the solution $u(x,t)$ is regular at the
point $(0,0)$. Their result can be viewed as a local version of the
component-reduction regularity. Regularity is obtained by imposing
conditions only on some components of the velocity, rather that of three.
For a comprehensive review of such results we refer the reader to
\cite{M,PP}
and references therein.

The purpose of this paper is to prove an interior regularity criterion
involving only one component of the velocity.
Using a different argument from \cite{WZ},
we prove
the following stronger
statement: For every $M>0$ there
exists a constant $\epsilon(M)>0$ such that if
  \begin{align}
	\int_{Q_1} (|u|^3 + |p|^{3/2})\;dx dt \leq M
   \label{EQ15}
  \end{align}
and
  \begin{align*}
           \int_{Q_1} |u_3|^3
            \;dx dt
           \leq \epsilon(M)
  \end{align*}
then $u(x,t)$ is regular at the point $(0,0)$.
For the statement, cf.~Theorem~\ref{thm1} below.
Note that every suitable weak solution
satisfies \eqref{EQ15} for $M$ sufficiently large.
The contradiction
argument used to prove Theorem~\ref{thm1} may be also used to prove a
new interior regularity criterion
based on the pressure.
Namely, in Theorem~\ref{thm4} we prove that
if \eqref{EQ15} holds and if
  \begin{equation}
    \int_{Q_1} |p|^{3/2}
            \;dx dt
    \leq \epsilon(M)
   \label{EQ16}
  \end{equation}
then the solution is regular at $(0,0)$.

Also, as a corollary of Theorem~\ref{thm1} we obtain a
stronger version of the Leray's regularity criterion concerning
weak solutions.
Namely, by \cite{G2,Le}, if
$T$ is an epoch of irregularity, then
for any $q>3$
there is a sufficiently small $\epsilon>0$ such that
$
 \|u(\cdot,t)\|_{L^q}
\geq \fractext{\epsilon}{(T-t)^{(1-3/q)/2}}
$ for $t<T$ sufficiently close to $T$.
Recall that $T$ is an epoch of irregularity if $T$ is a singular time
for $u$, while the times $t<T$ sufficiently close to $T$ are regular.
In Corollary~\ref{cor12} we obtain that if $T>0$ is the first
singular time,
then
for all $q\ge3$
  \begin{equation*}
    \Vert (u_1,u_2)(\cdot,t)\Vert_{L^q}
    \geq
    \frac{M}{(T-t)^{(1-3/q)/2}}
  \end{equation*}
or
  \begin{equation*}
    \Vert u_3(\cdot,t)\Vert_{L^q}
    \geq
    \frac{\epsilon(M)}{(T-t)^{(1-3/q)/2}},
  \end{equation*}
for $t<T$ sufficiently close to $T$. (A similar statement holds when
$T$ is an epoch of irregularity.)
Similarly, using Theorem~\ref{thm4}, we obtain Corollary~\ref{cor2} which states that
if $T$ is the first singular time,
then
  \begin{equation*}
   \Vert u(\cdot,t)\Vert_{L^q}
   \geq
   \frac{M}{(T-t)^{(1-3/q)/2}}
  \end{equation*}
or
  \begin{equation*}
    \Vert p(\cdot,t)\Vert_{L^{q/2}}
    \geq
    \frac{\epsilon(M)}{(T-t)^{1-3/q}}.
  \end{equation*}
for $t<T$ sufficiently close to $T$.

The paper is organized as follows. In the next section we state the
main results and introduce the notation used throughout the rest of the
paper. The proof is based on a contradiction argument and
Section~\ref{sec3} contains a regularity result for the limit
system, which turns out to be the Navier-Stokes system with $u_3 \equiv 0$. We would like to note that in order to prove Corollary~\ref{cor12} and Corollary~\ref{cor2} we require explicit estimates on the solutions of the considered limit system. Therefore, we cannot directly apply the results of Neustupa, Novotn{\'y} and Penel from \cite{NP,NNP}. Consequently, we need to modify this strategy to suit our needs.
The proofs of Theorems~\ref{thm1} and~\ref{thm5} are presented in
Section~\ref{sec4}, while
Section~\ref{sec5} contains the proofs
of Theorems~\ref{thm4} and~\ref{thm6}.

\section{The notation and the main results}\label{sec2}
\setcounter{equation}{0}

Let $D$ be an open, bounded, and connected subset of
${\mathbb R}^{3}\times (0,\infty)$.
We assume that $(u,p)$ is a suitable weak solution in
$D$, which means\\
(i) $u\in L_{t}^{\infty}L_{x}^{2}(D) \cap L_{t}^{2}H_{x}^{1}(D)$ and
$p\in L^{3/2}(D)$,\\
(ii) the Navier-Stokes equations
\eqref{NSE}
are satisfied in the weak sense, and\\
(iii) the local energy inequality holds in $D$, i.e.,
  \begin{align}
   & \left. \int_{\mathbb{R}^3} |u|^2  \phi\;dx  \right|_{T}
   +
   2\int\!\!\!\!\int_{{\mathbb R}^{3}\times(-\infty,T]} |\nabla u|^2\phi\;dx dt
   \nonumber\\&\indeq\indeq
    \le
    \int\!\!\!\!\int_{{\mathbb R}^{3}\times(-\infty,T]}
      \biggl(
       |u|^2(\partial_t\phi+\Delta \phi)
       + (|u|^2+2p) u\cdot \nabla\phi
      \biggr)\;dx dt
  \end{align}
for all $\phi\in C_{0}^{\infty}(D)$ such that $\phi\ge0$ in $D$
and almost all $T\in {\mathbb R}$.

Recall the following scaling
property of the Navier-Stokes equation: If $(u(x,t),p(x,t))$
is a solution, then so is
$(\lambda u(\lambda x,\lambda^2 t), \lambda^2 p(\lambda x, \lambda^{2} t))$.

Let $(x_0,t_0) \in D$. Denote by $B_r(x_0)$ the  Euclidean ball in $\mathbb{R}^3$ with center at $x_0$ and radius~$r>0$; we abbreviate $B_r = B_r(0)$. By $Q_r(x_0,t_0) = B_r(x_0) \times [t_0-r^2,t_0]$ we denote the parabolic cylinder in $\mathbb{R}^4$ labeled by the top center point $(x_0,t_0) \in D$. The following is the main result of the paper.

\begin{Theorem}\label{thm1}
	Let $(u,p)$ be a suitable weak solution of \eqref{NSE} in a neighborhood of $\overline{Q_r(x_0,t_0)} \subset D$ which satisfies
	\begin{align}\label{bound1}
		\frac{1}{r^2}
		\int_{Q_r(x_0,t_0)}
                   \bigl(
                      |u|^3 + |p|^{3/2}
                   \bigr)
                \;dx dt\leq M
           .
	\end{align}
Then there exists $\epsilon >0$ depending on $M$ such that if
	\begin{align}
		{\frac{1}{r^2} }
		\int_{Q_r(x_0,t_0)} |u_3|^3 \;dx dt\leq \epsilon
	\end{align}
	then $u$ is regular at $(x_0,t_0)$.
\end{Theorem}

The above theorem follows from the following stronger statement.

\begin{Theorem}\label{thm5}
For all $M,\epsilon_0,r>0$
there exist constants $\epsilon(M,\epsilon_0)>0$ and
$\kappa(M,\epsilon_0)\in(0,1)$ with the
following
property:
If $(u,p)$ is a suitable
weak solution of \eqref{NSE} in
${Q_r(x_0,t_0)} $ which satisfies
  \begin{align}
		\frac{1}{r^2}
		\int_{Q_r(x_0,t_0)}
	               (     |u|^3
                             +  |p|^{3/2} )
                 	\;dx dt\leq M
   \label{EQ17}
  \end{align}
and
  \begin{align}
		\frac{1}{r^2}
		\int_{Q_r(x_0,t_0)}
                   |u_3|^{3}
                \; dx dt
            \le \epsilon
   \label{EQ18}
  \end{align}
then
  \begin{align}
		\frac{1}{(\kappa r)^2}
		\int_{Q_{\kappa r}(x_0,t_0)}
	               (     |u|^3
                             +  |p|^{3/2} )
                 	\;dx dt\leq \epsilon_0
   \period
   \label{EQ20}
  \end{align}
\end{Theorem}

As a consequence of Theorem~\ref{thm5} we may deduce the following
improvement of a Leray's result from~\cite{Le}.

\begin{Corollary}\label{cor12}
Let $(u,p)$ be Leray's weak solution defined in a neighborhood
of $[0,T]$ with $T$ as the first singularity.
Then for every $M\ge1$ there exists
$\epsilon(M)\in(0,1]$ such that
  \begin{equation}
    \Vert (u_1,u_2)(\cdot,t)\Vert_{L^q}
    \geq
    \frac{M}{(T-t)^{(1-3/q)/2}}
  \end{equation}
or
  \begin{equation}\label{2.4}
    \Vert u_3(\cdot,t)\Vert_{L^q}
    \geq
    \frac{\epsilon(M)}{(T-t)^{(1-3/q)/2}},
  \end{equation}
for all $t\in(T/2,T)$ and $q\ge3$.
\end{Corollary}

Note that for $q=3$ a stronger statement has been
established in \cite{ESS}.
Also, observe that the statement  extends to the case
when $T$ is an epoch of irregularity by translating and
rescaling the time variable.

We first prove the corollary, while the proofs of
the theorems are provided in Section~\ref{sec4}.

\begin{proof}[Proof of Corollary~\ref{cor12}]
Assume that $u$ is regular on $(0,T)$ and
  \begin{equation}
    \Vert (u_1,u_2)(\cdot,t)\Vert_{L^q}
     \leq
    \frac{M}{(T-t)^{(1-3/q)/2}}
   \comma t\in(T/2,T)
   \label{EQ13}
  \end{equation}
and
  \begin{equation}
    \Vert u_3(\cdot,t)\Vert_{L^q}
    \leq
    \frac{\epsilon}{(T-t)^{(1-3/q)/2}}
   \comma t\in(T/2,T)
   \label{EQ14}
  \end{equation}
hold for some $M\ge1$ and
$\epsilon\in(0,M]$. We claim that $T$ is regular
if $\epsilon$  is sufficiently small.
The assumptions on the velocity and
the Calder\'on-Zygmund theorem imply
  \begin{equation}
   \|p(\cdot,t)\|_{L^{q/2}}
    \leq
    \frac{C M^2}{(T-t)^{(1-3/q)}}
   \period
   \label{EQ21}
  \end{equation}
Let $x_0\in{\mathbb R}^{3}$ be arbitrary.
Using H\"older's inequality, we
get
  \begin{align}
   \Vert u_j\Vert_{L^3(Q_{\sqrt T/2}(x_0,T))}
   \le
   C (\sqrt T)^{5/3-5/q}
   \Vert u_j\Vert_{L^q(Q_{\sqrt T/2}(x_0,T))}
   \le
   C T^{1/3} M
   \comma j=1,2
   \label{EQ22}
  \end{align}
where we used \eqref{EQ13} in the last step.
Therefore,
  \begin{equation}
   \frac{1}{(\sqrt T/2)^{2/3}}
   \Vert u_j\Vert_{L^3(Q_{\sqrt T/2}(x_0,T))}
   \le
   C M
   \comma j=1,2
   \period
   \label{EQ23}
  \end{equation}
Similarly, \eqref{EQ14} implies
  \begin{equation}
   \frac{1}{(\sqrt T/2)^{2/3}}
   \Vert u_3\Vert_{L^3(Q_{\sqrt T/2}(x_0,T))}
   \le
   C \epsilon
   \label{EQ24}
  \end{equation}
while by \eqref{EQ21}
  \begin{equation}
   \frac{1}{(\sqrt T/2)^{2/3}}
   \Vert p\Vert_{L^{3/2}(Q_{\sqrt T/2}(x_0,T))}^{1/2}
   \le
   C M
   \period
   \label{EQ25}
  \end{equation}
By Theorem~\ref{thm5},
there exists $\kappa\in(0,1)$ so that
if $\epsilon>0$ is sufficiently small
  \begin{align*}
        \frac{1}{(\kappa \sqrt T/2)^2}
	\int_{Q_{\kappa \sqrt T/2}(x_0,T)}
            (|u|^3 + |p|^{3/2})\;dx dt \leq \epsilon_{\text{CKN}}
   \comma x_0\in{\mathbb R}^{3}
   \period
  \end{align*}
Using the CKN theory, this provides a uniform bound for $u$ for $t$ in a neighborhood
of $T$. By Leray's regularity criterion, this shows that the time
$T$ is regular, as claimed.
\end{proof}

The strategy used in the proofs of Theorems~\ref{thm1} and~\ref{thm5}
enables us to prove the following two theorems.

\begin{Theorem}\label{thm4}
For every $M>0$, there exists a constant $\epsilon(M)>0$ with the
following
property:
If $(u,p)$ is a suitable
weak solution of \eqref{NSE} in a neighborhood of
$\overline{Q_r(x_0,t_0)} \subset D$ which satisfies
  \begin{align}
		\frac{1}{r^2}
		\int_{Q_r(x_0,t_0)}
                 |u|^{3}
                 	\;dx dt\leq M
   \label{EQ02}
  \end{align}
and
  \begin{align}
		\frac{1}{r^2}
		\int_{Q_r(x_0,t_0)}
	                    |p|^{3/2}
                \; dx dt
            \le \epsilon,
   \label{EQ04}
  \end{align}
then $u$ is regular at $(x_0,t_0)$.
\end{Theorem}

Although certain regularity criteria involving the pressure are known
(cf.~\cite{BG} for instance), the condition for regularity \eqref{EQ04} appears to be new. Theorem~\ref{thm4} follows in fact from a stronger result stated in Theorem~\ref{thm6}.

\begin{Theorem}\label{thm6}
For all $M,\epsilon_0,r>0$
there exist constants $\epsilon(M,\epsilon_0)>0$ and $\kappa(M,\epsilon_0)\in(0,1)$ with the
following
property:
If $(u,p)$ is a suitable
weak solution of \eqref{NSE} in
${Q_r(x_0,t_0)} $ which satisfies
  \begin{align}
		\frac{1}{r^2}
		\int_{Q_r(x_0,t_0)}
                     |u|^{3}
                 	\;dx dt\leq M
   \label{EQ17a}
  \end{align}
and
  \begin{align}
		\frac{1}{r^2}
		\int_{Q_r(x_0,t_0)}
	                    |p|^{3/2}
                \; dx dt
            \le \epsilon,
   \label{EQ19}
  \end{align}
then
  \begin{align}
		\frac{1}{(\kappa r)^2}
		\int_{Q_{\kappa r}(x_0,t_0)}
	               (     |u|^3
                             +  |p|^{3/2} )
                 	\;dx dt\leq \epsilon_0
   \period
   \label{EQ20a}
  \end{align}
\end{Theorem}

As a consequence of Theorems~\ref{thm4} and~\ref{thm6} we deduce
the following.

\begin{Corollary}\label{cor2}
Let $(u,p)$ be Leray's weak solution defined in a neighborhood
of $[0,T]$ with $T$ as the first singularity.
Then for every $M \geq 1$ there exists $\epsilon(M) \in (0,1]$ such that
  \begin{equation*}
    \|u(\cdot,t)\|_{L^q}
    \geq
    \frac{M}{(T-t)^{(1-3/q)/2}}
  \end{equation*}
or
  \begin{equation*}
     \|p(\cdot,t)\|_{L^{q/2}}
    \geq
    \frac{\epsilon(M)}{(T-t)^{(1-3/q)/2}},
  \end{equation*}
for all $t \in (T/2,T)$ and $q \geq 3$.

\end{Corollary}

\begin{proof}[Proof of Corollary~\ref{cor2}]
Assume that $u$ is regular on $(0,T)$ and
  \begin{equation}
    \Vert u(\cdot,t)\Vert_{L^q}
     \leq
    \frac{M}{(T-t)^{(1-3/q)/2}}
   \comma t\in(T/2,T)
   \label{EQ13a}
  \end{equation}
and
  \begin{equation}
    \Vert p(\cdot,t)\Vert_{L^{q/2}}
    \leq
    \frac{\epsilon}{(T-t)^{(1-3/q)/2}}
   \comma t\in(T/2,T)
   \label{EQ14a}
  \end{equation}
hold for some $M\ge1$ and
$\epsilon\in(0,M]$. We claim that $T$ is regular
if $\epsilon$  is sufficiently small.
Let $x_0 \in \mathbb{R}^3$ be arbitrary. Using H\"older's inequality, we obtain
	\begin{align}
		\|u\|_{L^3(Q_{\sqrt{T}/2}(x_0,T))}
         	\leq
		C(\sqrt{T})^{5/3-5/q}
		\leq
		CT^{1/3}M
         \end{align}
where we used (\ref{EQ13a}) in the last step. Thus, we get
  \begin{equation}
   \frac{1}{(\sqrt T/2)^{2/3}}
   \Vert u\Vert_{L^3(Q_{\sqrt T/2}(x_0,T))}
   \le
   C M
   \period
   \label{EQ23a}
  \end{equation}
Similarly, by \eqref{EQ14a} we have
  \begin{equation}
   \frac{1}{(\sqrt T/2)^{2/3}}
   \Vert p\Vert_{L^{3/2}(Q_{\sqrt T/2}(x_0,T))}^{1/2}
   \le
   C \epsilon
   \period
  \end{equation}
By Theorem~\ref{thm6},
there exists $\kappa\in(0,1)$ so that
if $\epsilon>0$ is sufficiently small
  \begin{align*}
        \frac{1}{(\kappa \sqrt T/2)^2}
	\int_{Q_{\kappa \sqrt T/2}(x_0,T)}
            (|u|^3 + |p|^{3/2})\;dx dt \leq \epsilon_{\text{CKN}}
   \comma x_0\in{\mathbb R}^{3}
   \period
  \end{align*}
This provides a uniform bound for $u$ for $t$ in a neighborhood
of $T$. By Leray's regularity criterion, this shows that the time
$T$ is regular, as claimed.
\end{proof}

\section{The limit system}\label{sec3}
\setcounter{equation}{0}

Let $D \subset {\mathbb R}^{3} \times (0,\infty)$ be a domain.
Consider the system
\begin{align}\label{eqn_limit}
	 \partial_t u_i - \Delta u_i + \sum_{j=1}^2 u_j \partial_j u_i + \partial_i p & =0 \indeq \text{ in $D$}
     \comma i=1,2 \nonumber \\
	 \partial_3 p& =0 \indeq \text{ in $D$} \nonumber \\
	 \partial_1 u_1 + \partial_2 u_2& =0 \indeq \text{ in $D$}
\end{align}
where $u(x_1,x_2,x_3,t)$ and $p(x_1,x_2,x_3,t)$ are unknown.
Note that the system \eqref{eqn_limit} stems from the Navier-Stokes equations by setting $u_3 = 0$.

Denote by $\mathcal{S}(u)$ the set of points where the solution $u(x,t)$ of \eqref{eqn_limit} is singular.
(The definition for a regular/singular point is the same as the one for the Navier-Stokes system.)
Therefore, we may conclude that the set $\mathcal{S}(u)$ is closed in $D$ and the partial regularity results regarding the Navier-Stokes equations imply that its $1$-dimensional parabolic measure (and as a consequence its $1$-dimensional Hausdorff measure) is equal to zero.

The following theorem, addressing regularity of the limiting system \eqref{eqn_limit},
is the main result of this section.

\begin{Theorem}\label{thm2}
Let $(u,p)$ be a
weak
solution of (\ref{eqn_limit}). Then $u$ is regular.
\end{Theorem}

We note that the results of Neustupa, Novotn{\'y} and Penel from \cite{NP, NNP} are not directly applicable in the considered setting since the weak solutions do not a priori have enough regularity to justify this approach. Moreover, in order to prove Corollary~\ref{cor12} and Corollary~\ref{cor2} we require explicit estimates on the weak solution of the system \eqref{eqn_limit} which cannot be obtained using the strategy from \cite{NP,NNP}. In particular, the presented proofs do not take advantage of epochs of irregularity.

The first step toward the proof of Theorem~\ref{thm2}, namely establishing
the regularity of the third component of the
vorticity $\omega_3=\partial_{1}u_2-\partial_{2}u_1$ stems however from the work of Neustupa, Novotn{\'y} and Penel mentioned above.

\begin{Lemma}\label{omega3reg}
	Let $(x_0,t_0) \in D$ and let $r>0$ be such that $Q_r(x_0,t_0) \subset D$. Then
	\begin{align}
		|\omega_3|^{q/2} \in L^\infty((t_0 - \rho^2,t_0),L^2(B_{\rho}(x_0))) \cap L^2((t_0-\rho^2,t_0), H^1(B_\rho(x_0)))
	\end{align}
	for any $q \in [2,\infty)$ and $\rho \in (0,r/2)$.
\end{Lemma}

\begin{proof}
Applying the curl operator to the system \eqref{eqn_limit} we note that $\omega_3$ satisfies the equation
\begin{align}\label{eqn_curl}
	\partial_t \omega_3 - \Delta \omega_3 + \sum_{j=1}^2 u_j \partial_j \omega_3 =0.
\end{align}
Without loss of generality we may assume that $(x_0,t_0) = (0,0)$. We denote $B_r=B_r(x_0) $ and $Q_r=Q_r(x_0,t_0)$.
Let $\eta$ be a smooth non-negative cut-off function, supported on $Q_{r}$, $\eta \equiv 1$ on $Q_{r/2}$ and such that $\eta$ vanishes on the lateral boundary of $Q_{r}$, that is $\eta =0$ on $B_{r} \times \{-r^2\} \cup \partial B_{r} \times (-r^2,0)$.
Fix $q \geq 2$.
Multiplying the equation~\eqref{eqn_curl} by
$|\omega_3|^{q-2}\omega_3\eta^2$ and integrating over $Q_{r}$, we obtain the estimate
\begin{align}
   &\sup_{-r^2/4 \leq t \leq 0} \int_{B_{r/2}} |\omega_3|^q \eta^2 \;dx + \iint_{Q_{r/2}} |\nabla (|\omega_3|^{q/2})|^2\eta^2
   \nonumber\\&\indeq
    \leq C(q) \iint_{Q_{r}}|\omega_3|^q (\eta |\Delta \eta| + |\nabla \eta|^2 + \eta |\partial_t \eta|)\;dxdt
	  + C(q) \iint_{Q_{r}}|u||\omega_3|^q \eta |\nabla \eta|\;dxdt,
\label{loc_q1}
\end{align}
where the second term on the right has been obtained from
\begin{align*}
	& - \sum_{j=1}^3 \iint_{Q_{r}} u_j \partial_j \omega_3 |\omega_3|^{q-2}\omega_3 \eta^2 \;dxdt
     = -\frac{1}{q} \sum_{j=1}^3 \iint_{Q_{r}} u_j \partial_j (|\omega_3|^q)\eta^2 \;dxdt
     \nonumber\\&\indeq
         \indeq = \frac{2}{q} \sum_{j=1}^3 \iint_{Q_{r}} u_j |\omega_3|^q \eta \partial_j \eta \;dxdt
	 \leq C \iint_{Q_{r}} |u| |\omega_3|^q \eta |\nabla \eta|\;dxdt
\end{align*}
using integration by parts and the divergence-free condition in the
second step.
This can be formally justified using a suitable mollification and passage to the limit. The estimate (\ref{loc_q1}) yields
\begin{align}\label{loc_q2}
    &\||\omega_3|^{q/2}\|^2_{L^\infty_t L^2_x(Q_{r/2})} + \|\nabla(|\omega_3|^{q/2})\|^2_{L^2(Q_{r/2})}
    \nonumber\\&\indeq
   \leq C(q)
\bigl(
\|u\|_{L^{10/3}(Q_{r})}\||\omega_3|^{q/2}\|_{L^{20/7}(Q_{r})}^2 + \||\omega_3|^{q/2}\|_{L^2(Q_{r})}^2
\bigr)
.
\end{align}
By the Sobolev embedding and interpolation, we obtain from (\ref{loc_q2})
\begin{align}
	\||\omega_3|^{q/2}\|_{L^{10/3}(Q_{r/2})} \leq C(q)\|u\|_{L^{10/3}(Q_{r})}^{1/2}\||\omega_3|^{q/2}\|_{L^{20/7}(Q_{r})} + C(q) \||\omega_3|^{q/2}\|_{L^2(Q_{r})}.
\end{align}
Since $2 < 20/7 < 10/3$ we may bootstrap the estimate. Namely, from~(\ref{loc_q2}) we obtain
\begin{align}\label{loc_q3}
	\||\omega_3|^{(7/6)q/2}\|_{L^{20/7}(Q_{r/2})}^{7/6} \leq C(q)\|u\|_{L^{10/3}(Q_{r})}^{1/2}\||\omega_3|^{q/2}\|_{L^{20/7}(Q_{r})} + C(q) \||\omega_3|^{q/2}\|_{L^2(Q_{r})}.
\end{align}
For $j=1,2,\ldots$ we define the sequences $q_j$ and $r_j$
by the recursive relationships $q_{j+1} = (7/6)q_j$
and $r_{j+1} = r_j/2$.
Then,
from \eqref{loc_q3},
\begin{align}\label{loc_q4}
	\||\omega_3|^{q_{j+1}/2}\|_{L^{20/7}(Q_{r_{j+1}})}^{7/6} \leq C(q_j)\|u\|_{L^{10/3}(Q_{r_j})}^{1/2}\||\omega_3|^{q_j/2}\|_{L^{20/7}(Q_{r_j})} + C(q_j) \||\omega_3|^{q_j/2}\|_{L^2(Q_{r_j})}.
\end{align}
Starting with $q_0=2$, we get $q_j = 2(7/6)^j$ and we conclude that for any $q \in [2,\infty)$
\begin{align}
	|\omega_3|^{q/2} \in (L^\infty_t L^2_x \cap L^2_t H^1_x)(Q_{r/2})
\end{align}
for $r$ sufficiently small. Using a covering argument, we obtain
\begin{align}\label{eqn3.54}
	|\omega_3|^{q/2} \in (L^\infty_t L^2_x \cap L^2_t H^1_x)(Q_\rho)
\end{align}
for every $\rho\in(0,r/2)$ with an explicit estimate.
\end{proof}

In order to prove Theorem~\ref{thm2},
we also need the following auxiliary result.

\begin{Lemma}\label{lemma3}\cite{G1}
	Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^3$. Let further $r \in (1,\infty)$ and $m \in \{0\} \cup \mathbb{N}$. Then there exists a linear operator $\mathcal{R}\colon W^{m,r}_0(\Omega) \to W^{m+1,r}_0(\Omega)$ with the properties
	\begin{itemize}
		\item[1.] $\div \mathcal{R}f = f$ for all $f \in W^{m,r}_0(\Omega)$ with $\int_\Omega f \;dx =0$, and
		\item[2.] there exists $C>0$ such that $\|\nabla^{j+1}\mathcal{R}f\|_{L^r(\Omega)} \leq C \|\nabla^j f\|_{L^r(\Omega)}$ for $j=1,\ldots,m$ and for all $f \in W^{m,r}_0(\Omega)$.
	\end{itemize}
\end{Lemma}

\begin{proof}[Proof of Theorem~\ref{thm2}]
First, note that the system \eqref{eqn_limit} may be rewritten as
\begin{align}\label{eqn_stokes}
	 \partial_t u_1 - \omega_3 u_2 & = -\partial_1 \left(p + \frac{1}{2}|u|^2\right) + \Delta u_1 \nonumber \\
	 \partial_t u_2 + \omega_3 u_1 & = -\partial_2 \left(p + \frac{1}{2}|u|^2\right) + \Delta u_2 \\
	\partial_3 p & = 0 \nonumber \\
	\partial_1 u_1 + \partial_2 u_2 & = 0. \nonumber
\end{align}
We show that any point $(x_0,t_0)$ is a regular point.
By translation, we can assume without loss of generality that $(x_0,t_0)=(0,0)$. We denote $B_r=B_r(x_0)$ and $Q_r=Q_r(x_0,t_0)$.  Let $r>0$ be as in Lemma \ref{omega3reg}. Let $\eta$ be a smooth cut-off function supported on $B_{r/2}$ and such that $\eta \equiv 1$ on $B_{r/4}$. Let $v = \eta u - V$, where $V(\cdot,t) = \mathcal{R}(\nabla \eta \cdot u(\cdot,t))$, with $\mathcal{R}$ being the operator defined as in Lemma~\ref{lemma3} with $\Omega = B_{r/2}$.
Note that we have
\begin{align}
	\int_{B_{r/2}} \nabla \eta \cdot u\;dx = \int_{B_{r/2}} \div(\eta u)\;dx = \int_{\partial B_{r/2}} \eta u \cdot n \;dS =0,
\end{align}
where $n$ is the outer normal vector to $\partial B_{r/2}$ thus we can apply Lemma \ref{lemma3}. Moreover, $\div V = \nabla \eta \cdot u$ in~$Q_{r/2}$.
 Note also that
\begin{align}
	 V & \in L^2((-r^2,0), W^{2,2}_0(B_{r/4})) \nonumber \\
	 \partial_t V & \in L^2(B_{r/4} \times (-r^2/16,0)).
\end{align}
Moreover, the Sobolev embedding and the control over $\partial_t V$ yield that $V$ is essentially bounded on $Q_{r/4}$.
In turn, the above defined $v$ solves the Stokes system
\begin{align}\label{eqn_local_v}
	& \partial_t v  - \Delta v + \nabla_2(\eta p + \frac{1}{2}\eta|u|^2) = (p + \frac{1}{2}|u|^2)\nabla_2 \eta - \partial_t V + \Delta V - \omega_3 V^\perp  - \omega_3 v^\perp
  .
\end{align}
Since $u$ is a weak solution, we obtain by interpolation $u \in L^5_tL^{30/11}_x(Q_{r/4})$. Thus $p \in L^{5/2}_t L^{15/11}_x(Q_{r/4})$, whence the first term on the right of (\ref{eqn_local_v}) belongs to $L^{5/2}_tL^{15/11}_x(Q_{r/4})$. The second, third, and fourth term belong to $L^2_tL^2_x(Q_{r/4})$, where for the fourth term we used the fact that $V$ is essentially bounded and the fact that by Lemma~\ref{omega3reg} applied with $q=2$ we have $\omega_3 \in L^\infty((-r^2,0),L^2(B_{r/4})) \cap L^2((-r^2,0),H^1(B_{r/4}))$ thus by interpolation $\omega_3 \in L^{10/3}_tL^{10/3}_x(Q_{r/4})$. The last term on the right of \eqref{eqn_local_v} belongs to $L^{5/2}_t L^{15/11}_x(Q_{r/4})$ since by interpolation $u,\omega_3 \in L^5_tL^{30/11}_x(Q_{r/4})$ and $V$ is essentially bounded on~$Q_{r/4}$. Therefore, in summary, the right side of \eqref{eqn_local_v} belongs to  $L^{5/2}_t L^{15/11}_x(Q_{r/4})$. The Stokes estimate (see \cite{SW}) applied to \eqref{eqn_local_v} yields $v \in L^{5/2}_t W^{2,15/11}_x(Q_{r/4})$ and thus by Sobolev embedding we obtain $v \in L^{5/2}_tL^{15}_x(Q_{r/4})$ which is a critical Serrin's regularity class.
\color{black}
\end{proof}

In order to prove Corollary~\ref{cor12} and Corollary~\ref{cor2} we need the following estimates on solutions of \eqref{eqn_limit}.
\begin{Lemma}\label{lem35}
Let $(u,p)$ be a solution of (\ref{eqn_limit}) and
	\begin{align*}
		\frac{1}{r^2} \int_{Q_{r}(x_0,t_0)}( |u|^3 + |p|^{3/2})\;dxdt \leq M
	\end{align*}
for some $r>0$ and $(x_0,t_0)$.
Then for any $\epsilon_0\in(0,1)$ we have
	\begin{align}
	  \frac{1}{(\kappa r)^2}
           \int_{Q_{\kappa r}(x_0,t_0)}( |u|^3 + |p|^{3/2})\;dxdt \leq \epsilon_{0}.
        \label{EQ06}
	\end{align}
for a constant $\kappa\in(0,1]$ depending only on $M$
and $\epsilon_0$.
\end{Lemma}

Consequently, under the assumptions of the theorem,
there exist $\kappa_0\in(0,1)$
and $K>0$ depending only on $M$ such
that
  \begin{equation}
   \kappa_0 r
   \Vert u\Vert_{L^\infty(Q_{\kappa_0 r})}
   \le
   K
   \period
   \label{EQ08}
  \end{equation}
The inequality \eqref{EQ06} implies \eqref{EQ08}
with $\kappa_0=\kappa/2$
using the
standard CKN theory (cf.~\cite{CKN,K1}).

\begin{proof}
Without loss of generality, we may assume that $(x_0,t_0)=(0,0)$. We denote $B_r(x_0) = B_r$ and $Q_r(x_0,t_0)=Q_r$. First, we note that by Lemma~\ref{omega3reg} we obtain $\omega_3 \in L^\infty((-r^2/16,0),L^2(B_{r/4})) \cap L^2((-r^2/16,0),H^1(B_{r/4}))$.
Similarly as in the proof of Theorem~\ref{thm2} we consider the abstract Stokes system (\ref{eqn_stokes}). Let $\eta$ be a smooth cut-off function supported on $B_{r/4}$ such that $\eta\equiv 1$ on $B_{r/8}$. We define $v = \eta u - V$ on $Q_{r/8}$. In particular  we obtain
\begin{align}
	& V \in L^2((-r^2/64,0), W^{2,2}_0(B_{r/8}(x_0))) \nonumber \\
	& \partial_t V \in L^2(B_{r/8}(x_0) \times (-r^2/64,0)).
\end{align}
Appropriate estimates follow from Lemma \ref{lemma3}, the Sobolev embedding theorem and interpolation.
On the other hand, the above defined $v$ solves the  Stokes system
\begin{align}\label{eqn_local_v2}
	& \partial_t v - \Delta v +  \nabla_2\left(\eta p + \frac{1}{2}\eta|u|^2\right) = \left(p + \frac{1}{2}|u|^2\right)\nabla_2 \eta - \partial_t V - \omega_3 V^\perp  + \Delta V - \omega_3 v^\perp.
\end{align}
Proceeding as in the proof of Theorem~\ref{thm2} we obtain that $v$ is in the critical Serrin's regularity class. Therefore, in order to prove \eqref{EQ06} we repeat the Stokes estimate on a smaller cylinder $Q_{r/16}$ which yields $v$ in a subcritical Serrin's regularity class. This combined with regularity properties of $V$ and \eqref{eqn3.54} gives us \eqref{EQ06} on a sufficiently small cylinder, that is on $Q_{\kappa r}$ for $\kappa \in (0,r/16)$ sufficiently small.
\end{proof}

\section{Proofs of Theorems~\ref{thm1} and~\ref{thm5}}
\label{sec4}
\setcounter{equation}{0}

In this section we present the proofs of Theorems~\ref{thm1} and~\ref{thm5}. In our considerations we use sequences of suitable weak solutions.
In the process, we need the following compactness result.

\begin{Lemma}\label{lemma_conv}
	Let $(u^{(n)}, p^{(n)})$ be a sequence of suitable weak solutions such that
	\begin{align}\label{bound_l1}
		\frac{1}{r^2}
		\int_{Q_r(x_0,t_0)}
                   \bigl(
                    |u^{(n)}|^3 + |p^{(n)}|^{3/2}
                   \bigr)\;dx dt\leq M
          ,
	\end{align}
and let $0 < \rho < r$. Then there exists a subsequence $(u^{(n_k)}, p^{(n_k)})$ such that $u^{(n_k)} \to u$ strongly in $L^q(Q_\rho(x_0,t_0))$ for all $1 \leq q < 10/3$ and $p^{(n_k)} \rightharpoonup p$ weakly in $L^{3/2}(Q_\rho(x_0,t_0))$.
\end{Lemma}

\begin{proof}[Proof of Lemma~\ref{lemma_conv}] Let $\phi \in
C^\infty_0(D)$ be such that $\phi \geq 0$ in $D$, $\phi =1 $ on
$B_\rho(x_0) \times (t_0-\rho^2,t_0)$ and
$\supp(\phi) \subset Q_r(x_0,t_0)$.
The local energy inequality for suitable weak solution yields
  \begin{align}
	& \int_{B_\rho(x_0)}|u(\cdot,t)|^2 \;dx + 2 \int_{Q_\rho(x_0,t_0)} |\nabla u|^2 \;dx dt \nonumber \\
	& \indeq \leq \int_{Q_r(x_0,t_0)} |u|^2(\partial_t \phi + \Delta \phi)\;dx dt
         +  \int_{Q_r(x_0,t_0)}(|u|^2 + 2p)u \cdot \nabla \phi\;dx dt
    \comma -\rho^2 \le t\le 0
  \end{align}
H\"older's inequality and the bound \eqref{bound_l1} imply that
there exists a constant $E>0$ (where $E=E(M,\rho)$) such that
  \begin{align}
	& \int_{B_\rho(x_0)}|u(\cdot,t)|^2 \;dx + 2 \int_{Q_\rho(x_0,t_0)} |\nabla u|^2 \;dx dt \leq E
    \comma -\rho^2 \le t\le 0
   .
  \end{align}
Possibly passing to a subsequence, we may assume that $u^{(n)} \rightharpoonup u$ in $L^2((t_0-\rho^2,t_0),H^1(B_\rho(x_0)))$ and weak-$\ast$ in $L^\infty((t_0-\rho^2,t_0),L^2(B_\rho(x_0)))$. We may also assume that $p^{(n)} \rightharpoonup p$ in $L^{3/2}(Q_\rho(x_0,t_0))$. The equations
\begin{align}
	\partial_t u^{(n)} = \Delta u^{(n)} - u^{(n)} \cdot \nabla u^{(n)} - \nabla p^{(n)} \indeq \text{in $Q_\rho(x_0,t_0)$}
\end{align}
and the weak convergence $u^{(n)} \rightharpoonup u$ in $L^2((t_0-\rho^2,t_0),H^1(B_\rho(x_0)))$ and weak-$\ast$ in $L^\infty((t_0-\rho^2,t_0),L^2(B_\rho(x_0)))$ along with the $L^{3/2}$ bound on $p^{(n)}$ imply that $\partial_t u^{(n)} \in L^{3/2}((t_0-\rho^2,t_0),(H^2_0)^\ast(B_\rho(x_0)))$ with a uniform bound
\begin{align}
	\|\partial_t u^{(n)}\|_{L^{3/2}((t_0-r^2,t_0),(H^2_0)^\ast(B_\rho(x_0)))} \leq C,
\end{align}
where the constant $C$ may depend on $E$.
Therefore, by the Aubin-Lions compactness lemma we conclude that $u^{(n)} \to u$ strongly in $L^{3/2}(Q_\rho(x_0,t_0))$. Since $u^{(n)}$ is bounded uniformly in $L^{10/3}(Q_\rho(x_0,t_0))$, by interpolation we get that $u^{(n)} \to u$ strongly in $L^q(Q_\rho(x_0,t_0)))$ for all $1 \leq q < 10/3$.
\end{proof}

We first prove the stronger result, namely Theorem~\ref{thm5}.

\begin{proof}[Proof of Theorem~\ref{thm5}]

Without loss of generality, we may assume that
$(x_0,t_0)=(0,0)$. Denote $Q_r=Q_r(x_0,t_0)$.
Fix $r>0$ and assume that there exists a sequence of suitable weak solutions $(u^{(n)},p^{(n)})$ with
  \begin{align}\label{bound21}
	\frac{1}{r^2}
	\int_{Q_r} (|u^{(n)}|^3 + |p^{(n)}|^{3/2})\;dx dt \leq M
  \end{align}
and
  \begin{align}\label{bound31}
     \frac{1}{r^{2}}\int_{Q_r}
     |u_3^{(n)}|^3 \;dx dt \to 0
     ,
  \end{align}
  but
   \begin{align}
		\frac{1}{(\kappa r)^2}
		\int_{Q_{\kappa r}}
	               (     |u^{(n)}|^3
                             +  |p^{(n)}|^{3/2} )
                 	\;dx dt > \epsilon_0
   \label{EQ33}
  \end{align}
for every $\kappa\in(0,1)$.
By Lemma \ref{lemma_conv}, we may
divide $r$ by $2$ and assume that
$u^{(n)} \to u$ strongly in $L^3(Q_r)$
and $p^{(n)} \rightharpoonup p$ weakly in $L^{3/2}(Q_r)$.
Note that $(u,p)$ solves the system (\ref{eqn_limit}). Theorem~\ref{thm2} implies that
  \begin{align}
	r^{1/6}\|u\|_{L^6(Q_{r}(x_0,t_0))}\le M_0<\infty
  \end{align}
where $M_0$ depends only on $M$.
By rescaling we now assume that $r=1$.
For $\kappa_1 \in (0,1)$,
which is to be determined below
we obtain
  \begin{align}
     \|u\|_{L^3(Q_{\kappa_1 })}
       \leq C \kappa_1 ^{5/6}\|u\|_{L^6(Q_{\kappa_1})}
       \leq C \kappa_1^{5/6}M_0
  \end{align}
using H\"older's inequality,
from where
  \begin{align}\label{conc1aa}
    \frac{1}{\kappa_1 ^{2/3}}\|u\|_{L^3(Q_{\kappa_1})}
    \leq C \kappa_1^{1/6}
       \|u\|_{L^6(Q_{\kappa_1 })}   \leq C\kappa_1^{1/6}M_0.
  \end{align}
There exists $\kappa_0 >0$ such that for $\kappa_1 \in (0,\kappa_0]$ we have
\begin{equation}\label{EQ01zz}
	C \kappa_1 ^{1/6}M_0
	\le
	\frac{1}{2}\epsilon_0^{1/3} \kappa_1^{1/12}.
\end{equation}
In particular, the inequalities \eqref{conc1aa} and \eqref{EQ01zz} then imply
  \begin{align}\label{cont_limitazz}
    \frac{1}{\kappa_1^{2}} \int |u|^3\;dxdt \leq \frac{1}{4}\epsilon_0 \kappa_1^{1/4}
   .
  \end{align}
Since $u^{(n)} \to u$ strongly in
$L^3_{{\rm loc}}(Q_{1})$,
we may choose $n$ large enough (depending on $\kappa_1$) so that
  \begin{align}\label{EQ31}
     \frac{1}{\kappa_1^{2}} \int_{Q_{\kappa_1 }}|u^{(n)}|^3\;dx dt
     \le
	\epsilon_0 \kappa_1^{1/4}
  \end{align}
for $n$ sufficiently large.

We rewrite the pressure equation as in~\cite{K1} as
  \begin{align}
   \Delta(\eta p^{(n)})
   &
   =
   -\partial_{ij}(\eta u_i^{(n)} u_j^{(n)})
   -u_i^{(n)} u_j^{(n)}\partial_{ij}\eta
   +\partial_{j}(u_i^{(n)} u_j^{(n)}\partial_{i}\eta)
   \nonumber\\&\indeq
   +\partial_{i}(u_i^{(n)} u_j^{(n)}\partial_{j}\eta)
   -p^{(n)}\Delta\eta
   +2\partial_{j}((\partial_{j}\eta)p^{(n)})
   \label{EQ34}
  \end{align}
where $\eta$ is a smooth cut-off function supported in
$B_{\kappa_1}$ identically $1$ on $B_{\kappa'\kappa_1}$
where $\kappa'\in (0,1/2]$
is to be determined below.
With $N=-1/4\pi|x|$ denoting the Newtonian potential,
we obtain
  \begin{align}
   \eta p^{(n)}
   &=
   R_i R_j(\eta u_i^{(n)} u_j^{(n)})
   -N*(u_i^{(n)} u_j^{(n)}\partial_{ij}\eta)
   +\partial_{j}N*(u_i^{(n)} u_j^{(n)}\partial_{i}\eta)
   \nonumber\\&\indeq
   +\partial_{i}N*(u_i^{(n)} u_j^{(n)}\partial_{j}\eta)
   -N*(p^{(n)}\Delta\eta)
   +2\partial_{j}N*((\partial_{j}\eta)p^{(n)})
   \nonumber\\&
   =
   p_1+p_2+p_3+p_4+p_5+p_6
   .
   \label{EQ09}
  \end{align}
For $p_1$, we have by the Calder\'on-Zygmund
theorem
  \begin{equation}
   \Vert p_1\Vert_{L^{3/2}(Q_{\kappa'\kappa_1 })}
   \le C \Vert u^{(n)}\Vert_{L^3(Q_{\kappa_1 })}^2
   .
   \label{EQ10}
  \end{equation}
For the rest of the terms, we use the fact that they
all contain derivatives
of $\eta$. This makes all the convolutions nonsingular
when $|x|\le \kappa'\kappa_1 $ (cf.~\cite{K1} or \cite{L} for details).
Using this, we obtain the estimate for
$p_2$, $p_3$, and $p_4$ which is as in \eqref{EQ10}.
For $p_5$, we have, as in \cite{K1},
  \begin{align}
 &  \Vert p_5\Vert_{L^{3/2}(Q_{\kappa'\kappa_1 })}
   \le
  {C(\kappa'\kappa_1)^2} \|p^{(n)}\|_{L^{3/2}(Q_1)}
     \le
  {C(\kappa'\kappa_1)^2} { M^{2/3}}
   \label{EQ12}
  \end{align}
  The same bound holds for $p_6$.
Summarizing, we obtain
  \begin{equation}
   \frac{1}{(\kappa'\kappa_1 )^{2/3}}
   \Vert p^{(n)}\Vert_{L^{3/2}(Q_{\kappa'\kappa_1 })}^{1/2}
   \le
   \frac{C_0}{(\kappa' \kappa_1 )^{2/3}}\Vert u^{(n)}\Vert_{L^{3}(Q_{\kappa_1 })}
   + C_0  { (\kappa' \kappa_1 )^{1/3}} M^{1/3}
   \label{EQ11a}
  \end{equation}
where $C_0$ is a constant.
Using \eqref{EQ31} we bound the right side of \eqref{EQ11a} by
\begin{equation}
	  \frac{C_0 \epsilon_0}{(\kappa' )^{2/3}}
	  \kappa_1^{1/12}
   + C_0  { (\kappa' \kappa_1 )^{1/3}} M^{1/3}
   = \frac{C_0 \epsilon_0}{(\kappa')^{2/3}}\kappa_1^{1/12}
   + C_0 (\kappa'\kappa_1)^{1/3}M^{1/3}.
   \label{EQ_last1}
\end{equation}
We can choose $\kappa'$ and $\kappa_1$
small enough so that the right side of \eqref{EQ_last1}
is smaller than $\epsilon_0^{1/3}/2$.
By possibly making $\kappa_1$ smaller,
we also have from \eqref{EQ31} that
  \begin{align}\label{EQ31z}
      \frac{1}{(\kappa'\kappa_1 )^{2}} \int_{Q_{\kappa'\kappa_1 }}|u^{(n)}|^3\;dx dt
     \le
	\epsilon_0 (\kappa'\kappa_1)^{1/4} \le \frac{1}{2} \epsilon_0.
  \end{align}
Thus, setting $\kappa = \kappa' \kappa_1$ we get
\begin{equation}
	\frac{1}{\kappa^{2}} \int_{Q_{\kappa }} (|u^{(n)}|^3+|p^{(n)}|^{3/2})\;dx dt  \le \epsilon_0,
\end{equation}
which leads to a contradiction with \eqref{EQ33}.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm1}]
The statement follows from Theorem~\ref{thm5}
by setting $\epsilon_0=\epsilon_{CKN}$.
\end{proof}

\section{Proofs of Theorems~\ref{thm4} and~\ref{thm6}}
\label{sec5}
\setcounter{equation}{0}

Since Theorem~\ref{thm6} is more general, we start with it first.

\begin{proof}[Proof of Theorem~\ref{thm6}]
In order to prove Theorem~\ref{thm6},
assume that there exists a sequence of suitable weak solutions $(u^{(n)},p^{(n)})$ satisfying
  \begin{align}
		\frac{1}{r^2}
		\int_{Q_r(x_0,t_0)}
	               (     |u^{(n)}|^3
                             +  |p^{(n)}|^{3/2} )
                 	\;dx dt\leq M
   \label{EQ17b}
  \end{align}
and
  \begin{align}
		\frac{1}{r^2}
		\int_{Q_r(x_0,t_0)}
	                    |p^{(n)}|^{3/2}
                \; dx dt
            \to 0,
   \label{EQ19b}
  \end{align}
but
  \begin{align}
		\frac{1}{(\kappa r)^2}
		\int_{Q_{\kappa r}(x_0,t_0)}
	               (     |u^{(n)}|^3
                             +  |p^{(n)}|^{3/2} )
                 	\;dx dt > \epsilon_0
   \period
   \label{EQ20b}
  \end{align}
for a certain $\kappa \in (0,1)$
to be determined explicitly below.
By Lemma~\ref{lemma_conv}, we may divide $r$ by $2$ and assume that $u^{(n)} \to u$ strongly in $L^3(Q_r(x_0,t_0))$. Note that $u$ solves the Burgers system
 \begin{align}
	\partial_t u_i - \Delta u_i
            + \sum_{j=1}^3 u_i\partial_j u_j &= 0 \indeq \text{ in $D$}
     \comma i=1,2,3
   \label{EQ05}
   \end{align}
with
  \begin{align}
	 \partial_1 u_1 + \partial_2 u_2 + \partial_3 u_3& =0 \indeq \text{ in $D$}.
   \label{EQ07}
  \end{align}
It is well-known that solutions of \eqref{EQ05} are regular (see e.g.~\cite{C}); alternatively, we may use
local $L^{6}$ estimates combined with the divergence-free condition \eqref{EQ07}. Therefore, we get
 \begin{align}
	r^{1/6}\|u\|_{L^6(Q_{r}(x_0,t_0))}\le M_0<\infty
  \end{align}
where $M_0$ depends only on $M$. By rescaling we now assume that $r=1$.

We then proceed as in the proof of Theorem~\ref{thm1}.
Namely, for $\kappa \in (0,1)$, H\"older's inequality yields
  \begin{align}
     \|u\|_{L^3(Q_\kappa(x_0,t_0))}
       \leq C \kappa^{5/6}\|u\|_{L^6(Q_\kappa(x_0,t_0))}
       \leq C\kappa^{5/6}M_0
  \end{align}
which implies
  \begin{align}\label{conc1}
   \frac{1}{\kappa^{2/3}}\|u\|_{L^3(Q_\kappa(x_0,t_0))}
   \leq C \kappa^{5/6}
       \|u\|_{L^6(Q_\kappa(x_0,t_0))}   \leq C\kappa^{5/6}M_0
   .
  \end{align}
Let $\kappa$ be sufficiently small so that
  \begin{equation}
   C\kappa^{5/6}M_0
   \leq
\frac16 \epsilon_0^{1/3}
   .
   \label{EQ011aa}
  \end{equation}
The inequalities \eqref{conc1} and \eqref{EQ011aa} then imply
  \begin{align}\label{cont_limita}
    \frac{1}{\kappa^{2}} \int_{Q_\kappa(x_0,t_0)}|u|^3\;dxdt
    \leq
   \frac16 \epsilon_{0}
   .
  \end{align}
Since $u^{(n)} \to u$ strongly in
$L^3_{{\rm loc}}(Q_{1}(x_0,t_0))$,
we may choose $n$ large enough so that
  \begin{align}\label{contr1}
     \frac{1}{\kappa^{2}} \int_{Q_\kappa(x_0,t_0)}|u^{(n)}|^3\;dxdt
     \le
     \frac12 \epsilon_{0}
    .
  \end{align}
From (\ref{EQ19b}) and (\ref{contr1}) for sufficiently large $n$ we obtain
\begin{align}
  	\frac{1}{\kappa^2} \int_{Q_{\kappa}(x_0,t_0)} (|u^{(n)}|^3 + |p^{(n)}|^{3/2})\;dx dt \leq \epsilon_{0}.
  \end{align}
 which contradicts \eqref{EQ20b}.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm4}]
Theorem~\ref{thm4} follows from Theorem~\ref{thm6} using the CKN criterion for
regularity.
\end{proof}

\bigskip

\noindent \footnotesize{\bf Acknowledgments.} I.K.\ was
supported in part by the NSF grants DMS-1311943, W.R. was supported in part by the NSF grant DMS-1311964,
while M.Z.\ was supported in part by the NSF grant DMS-1109562.
\normalfont
\normalsize

redpajama_set_name: RedPajamaArXiv


1. The pressures to develop and market a first product require taking some expedient shortcuts, such as hiring the most capable, but not necessarily the most team-oriented individuals; placing all priority on getting a workable product out the door, rather than building the product for maintainability and growth; putting in the most features rather than the best-tested features.
2. The top management habitually focuses on the race between funds running out and product delivery, rather than on internal communications, employee satisfaction (except with the potential value of their options), and leadership style. The command-and-control management style is workable for the first few years, but typically fails to inspire the organization to build itself into a self-renewing structure.
3. Having a focus on delivering a product using already-developed technology, the company does not need to invest in longer-term development of underlying technologies, or in the people who will bring in a steady stream of new technology.
The short-term focus of a startup must change soon after the deliver of the first few products. Companies that fail to incorporate longer-term thinking around their third year find themselves living from crisis to crisis. This makes the company unattractive to good managers and good technologists who don't necessarily get their jolllies from living in a startup environment, where "startup" means short-term thinking.
What sort of changes does your organization need, now that the product has been delivered? A new CEO who actually allows the organization to function as if there were competence at all levels? A seasoned technology executive who knows what to do to make the organization attractive to innovative people? A shift in emphasis to listening to customer feedback and involving existing customers in product decisions? Addition of a Quality department that actually has the teeth to delay a product introduction?
(1) "What happened to my adrenaline rush?" — the people who need crises to keep up their energy should pursue another startup.
(2) "I didn't know that a company could actually plan and execute with the future (beyond 1 month) in mind!" — the people who are stressed by the company's failure to plan and execute for the long term grow into steady, reliable contributors.


SUGAR LAND, Texas, June 7, 2018 -- Systel, Inc., an industry leader in rugged computing will demonstrate the latest in rugged computing technology at the Eurosatory International Defense and Security Exposition in Paris, France from June 11 to 15 in Booth DC451.
Systel will be demonstrating its fully rugged, high performance, high-density servers and Strike™ mission computers, ideal for multi-sensor C5ISR mission-critical applications.
"Our systems are purpose-built to withstand the rigors of deployments in harsh environments while providing extreme performance and high reliability," says Aneesh Kothari, vice president of marketing for Systel. "We are proud to support US and NATO armed forces through thirty years of proven experience providing the highest-caliber custom and COTS rugged computing solutions."
Systel is demonstrating numerous rugged, high-density computing embedded and server solutions in Booth DC451, USA Pavilion at Eurosatory 2018. For more information or to set up a meeting, please visit this link.
highest-caliber custom and COTS rugged computing solutions."
Systel, Inc. is a leading manufacturer of rugged embedded, server, and display computer products and solutions. Systel is headquartered in Sugar Land, Texas, with 30 years of experience in providing complex and advanced computer technologies to military, oil and gas, and manufacturing enterprises worldwide. Systel's pedigree of capabilities spans numerous platforms and includes some of the most advanced military vehicles such as the US Navy's ISR/ASW aircraft, General Atomics Predator® and Reaper® UAV fleets, and Counter-UAS Stryker ground vehicles. For more information, visit http://www.systelusa.com or contact Systel via email sales@systelusa.com or phone 888.645.8400.


text:
BMW E38 — автомобіль з кузовом «седан» класу «люкс». Дебютував у червні 1994 року, змінивши 2-е покоління 7-ї серії BMW E32. Модель випускалася аж до 2001 року, коли їй на зміну прийшла нова «сімка» BMW E65/E66 2002 модельного року. Існують такі варіанти цієї моделі: 728i, 730i, 735i (не доступні в США та Канаді), 740i і 740iL (V8), флагманський 750i V12 і дизельні варіанти 725tds, 730d і 740d.

Модель зазнала значних змін як у зовнішньому вигляді, так і під капотом. Дизайн багато в чому визначено попереднім кузовом Е32 (зовні) і деякими деталями всередині. Дизайн салону згодом став прототипом для п'ятої серії в кузові Е39 (зменшена копія). Ексклюзивні варіанти моделі були так само представлені в подовжених кузовах: 740iL і 750iL (колісна база подовжені на 140 мм) І ще більш розтягнутий 750 XLA (колісна база подовжена на 390 мм). В кінці 1998 року кузов піддався незначному рестайлігу.

Моделі 740iL і 750iL також випускалися в легкоброньованому (клас B4) виконанні Protection. Також на базі 740iL або 750iL існує версія броньовик (виконання Security, клас B6 або B7). У технічному оснащенні модель пішла далеко вперед. Особливо це стосується електронних систем. Вперше були впроваджені такі системи як:
 адаптивна автоматична коробка передач (система AGS);
 електронна система управління двигуном (ELM);
 система динамічної стабілізації руху (DSC);
 система регулювання положення кузова (NR);
 система електронного керування жорсткістю амортизаторів (EDC);
 система автоматичної стабілізації кузова (ASC)

Кузов і обладнання

Незважаючи на великі габарити, новий автомобіль здавався легким і елегантним. Завдяки нижчому і широкому передку з повністю інтегрованими в капот «ніздрями», довгою колісною базою, низькому профілю даху і укороченій задній частині він здавався меншим, ніж насправді. Спрофільований для кращої обтічності зад і кришка багажника, приховані за бампером вихлопні труби дозволили знизити до 0,3 коефіцієнт аеродинамічного опору. Додатковим ефектом від цього було також зниження шуму і забруднення автомобіля.

Силова структура кузова була значно посилена, особливо в зоні центрального тунелю, що дозволило добитися його кутової жорсткості в 25 000 Нм/град. Це істотно знизило шум і вібрацію, особливо в передній частині салону. Нові методи комп'ютерного проектування і застосування високоміцних сталей дозволили знизити частку кузова в загальній вазі автомобіля до величини менш ніж 20%.

Елегантний і комфортабельний салон зберігав стилістичну єдність із зовнішніми рисами автомобіля. Горизонтальні вставки з дерева розташовувалися не тільки на передній панелі, але і на дверях, роз'єднуючи і об'єднуючи різні за кольором і фактурою зони зверху і знизу.

За допомогою кнопок на кермовому колесі можна було управляти радіо, круїз-контролем, включенням рециркуляції повітря і телефоном. Всі кнопочки підсвічувались, а обрані значення відображалися на приладовій панелі.

Пропонувалися три види передніх сидінь. Стандартне мало електрорегулювання положення (вперед-назад), висоти подушки і кута нахилу спинки і подушки. У комфортабельного сидіння (comfort seat) додатково регулювався поперековий підпір і бічні опори на спинці. Контурне сидіння (contour seat) в доповненні до всього мало електрорегулювання довжини подушки і її бічних опор. У задніх сидінь регулювався нахил спинки, поперековий підпір і висота підголівника.

У 1998 році всі моделі отримали невелике оновлення зовнішнього вигляду. В очі відразу кидаються нові фари, які тепер підкреслені «хвилею» знизу, як на автомобілях нової 3-ї серії. Так передок більше схожий на типовий для автомобілів BMW. Покажчики поворотів по краях стали трохи вужчими, «ніздрі» не змінилися, але решітка всередині них стала більш опуклою. Ззаду в якості опції з'явилася хромована накладка на кришку багажника над номерним знаком, а також змінилися задні ліхтарі. У дорестайлингу ліхтарі були з пластику, а в рестайлингу зі скла.

Цікавим нововведенням стали ручки дверей, що підсвічувались. Як тільки автомобіль відчинявся з брелока, мініатюрні лампочки спалахували в усіх чотирьох дверних ручках, додатково висвітлюючи простір під ногами. У салоні тепер використовувалися нові матеріали обробки, замість радіотелефону, можна було замовити стільниковий телефон, підключений до інформаційної системи автомобіля і керований прямо з керма. Автомобіль стандартно стали обладнувати десятьма подушками безпеки.

Всі моделі з бензиновими двигунами випускалися як в звичайному виконанні, так і у версії з подовженою на 140 міліметрів колісною базою.

Довгобазні версії з найпотужнішими двигунами можна було замовити в захищеному (Protection) виконанні. У певних місцях на кузова такого автомобіля були встановлені панелі з кевлара, дуже міцного, але легкого матеріалу. Вага захищеної моделі зростав всього на 145 кілограмів. Крім цього автомобіль оснащувався броньованими стеклами, які захищали пасажирів від легкого вогнепальної зброї та спеціальними шинами, які, навіть прострелені, дозволяли проїхати до 500 кілометрів на швидкості 80 км/год.

Вершиною модельного ряду був лімузин, салон якого був майже на 40 сантиметрів довшим. Створений для задоволення потреб знаменитостей і особливо важливих персон автомобіль пропонував винятковий комфорт і міг оснащуватися будь-якими доступними опціями. Такі моделі виготовлялися в невеликій кількості за індивідуальними замовленнями.

Двигуни і модифікації

Ходова частина
На автомобіль встановлювалася передня незалежна підвіска з стійками типу Макферсон з двома кульовими шарнірами. До переднього шарніру кріпилася поперечна алюмінієва розтяжка, до заднього - C-подібний хитний важіль. Пружини підвіски мали бочкоподібну форму і навивати з дроту змінного перерізу. Підвіска приєднувалася до жорсткого підрамника, звареного з труб.

Особлива геометрія задньої багатоважільної незалежної підвіски забезпечувала оптимальну передачу сил, що діють на колеса. Так, гальмівна сила сприймалася нижнім трапецеїдальним штампованим важелем спеціального профілю. Поздовжні і бічні сили передавалися за допомогою двох верхніх тяг, направляючої і силової. І, за допомогою невеликої додаткової тяги спереду нижнього важеля, поздовжні сили безпосередньо впливали на нього, без створення непотрібних обертових і розгойдуючих моментів. Підвіска кріпилася до підрамника, на який також встановлювалася головна передача. Весь цей задній міст назбирати і регулювався окремо, а потім цілком монтувався на автомобіль.

На модель 750iL стандартно, а на інші - на замовлення, встановлювалася електронна система управління демпфуванням (EDC, Electronic Damper Control), яка автоматично змінювала жорсткість задніх амортизаторів, забезпечуючи спортивну керованість на хорошій дорозі і високий комфорт при русі по поганій. Крім того, модель 750i і всі довгобазні версії (L) стандартно обладнувалися задньої гідропневматичною підвіскою з системою автоматичної підтримки постійного рівня підлоги. Зберігаючи незмінною, незалежно від завантаження автомобіля, геометрію підвіски, ця система забезпечувала підвищений комфорт для задніх пасажирів.

Автомобілі оснащувалися кермовим механізмом типу гвинт-кулькова гайка і гідропідсилювачем з автоматично змінним тиском. На модель 750i (750iL) стандартно, а на інші моделі на замовлення встановлювалася система електронного регулювання, в залежності від швидкості автомобіля, зусилля на кермі.

На всіх колесах були встановлені дискові гальма, спереду з вентильованими дисками, ззаду - із суцільними, крім моделі 750i, у якій позаду також застосовувалися вентильовані гальмівні диски.

Крім антиблокувальної (ABS) і протисковзаючої (ASC) систем, якими стандартно обладнувалися всі автомобілі, на моделі з мотором V12 встановлювалася електронна система контролю стійкості руху (DSC). Трохи пізніше всі автомобілі стали комплектуватися електронною системою розподілу гальмівних сил (DBC).

Посилання

E38 90-е


text:
Afro Fashion Pop Up Shop
AFRO POP UP SHOP MILAN FIRST EDITION: An opportunity created by Afro Fashion Association for emerging designers. A new way to penetrate new markets, do cultural exchange, expand networks into the future and beyond.
Afro Pop Up Shop Milan, is the first of many appointments with Afro Fashion Week Milano, the event format that has been used around the world and is now landing for the first time in the Italian fashion capital through Afro Fashion Association born in 2015 with the aim to promote the international event with a huge intercultural impact
Afro Fashion Week Milano, the first large event that celebrates Afro Fashion in Italy, calls all the designers around the world through Afro Pop Up Shop, a scouting project with the aim to promote and to present the trends of the moment in the Afro fashion industry and organize their commercialization between different groups of sellers and buyers, in few word as a new way to penetrate new markets, do cultural exchange, expand networks into the future and beyond.
From 21st – 23rd April 2017 From 10:00 AM to 20:00 PM , the city of Milan, will be able, to discover, meet and buy the creations of international and national afro designers (From UK, FRANCE, MOROCCO, ITALY, TANZANIA).
10:00 am – 12:00 Opening Ceremony
12:30 - 18:00 pm Pop Up Shop
18:00 pm -19:30 pm Talk: Afro Beauty in Italy
19:45- 20:30 Afro cocktail
10:00am- 16:00 pm Pop Up Shop
16:00 pm- 17:30pm Performance live of Richard Kofi (self-taught painter and illustrator from the Netherlands).
17:45pm – 20:00 pm Pop Up Shop
10:00 am- 14:00 pm Pop Up Shop
14:00- 17:00 "Photo shooting pop up shop" with some of the designer's creations.
17:00pm- 19:30 pm Pop Up Shop
20:00- End
An event in collaboration with AfroItalian souls, Salambò Association, Nappytalia, Re-eye wear,
Afrofashion Pop-up Shop
Via Circo 1


Piraeus, Greece (PortSEurope) February 11, 2019 – The CEO of Piraeus Port Authority Sa (PPA), Captain Fu Chengqiu, and the President of Autorita' di Sistema Portuale del Mare Adriatico Settentrionale (AdSP – Northern Adriatic Sea Port Authority), Pino Musolino, have signed a Memorandum of Understanding (MoU) aimed at strengthening relations and cargo flows between the two ports.
Piraeus Port Authority is operated by China's COSCO Shipping Ports Limited.
The two parties will seek to establish a stable and mutual cooperation, in order to develop their ports and port services, supporting infrastructural connectivity policies and port services aimed at fostering the commercial flows between Europe, the Mediterranean and the Far East via the two ports.
The MoU also aims at improving the exchange of best practices and know-how in port management, specifically in the fields of information technology, investment and communication. The common target is to improve the performance of the two ports, which serve as fundamental networks in the upcoming international maritime connections along the new Silk Road.
Captain Fu Chengqiu, CEO of Piraeus Port Authority S.A. said: "Τhe port of Piraeus, the largest port of Greece and the Eastern Mediterranean, is an ideal hub between Asia and Central & Eastern Europe. It is the first deep sea EU port after crossing Suez Canal and offers combined transport solutions with efficient value added services for cargoes, which can be re-distributed by road, rail and sea.
The establishment of our cooperation with AdSP del Mare Adriatico Settentrionale is targeting to strengthen the trade links between Asia and Mediterranean and to enhance the role of ports as engines for the European economy. Piraeus port with its ongoing investment plan aims within the next years to contribute to the growth of the local and regional economy by developing almost every aspect of the shipping and maritime industries including containers, shipyards, logistics, cruise and ferries".
Pino Musolino, president of AdSP del Mare Adriatico Settentrionale (AdSP – Northern Adriatic Sea Port Authority), which includes the ports of Venice and Chioggia), declared: "In an upcoming scenario which foresees a powerful increase of trades between Asia and Europe along the new maritime Silk Road, it is necessary to put all our efforts on the rationalization of the logistics chains, starting from the ports and from the road connections with the inner markets, in order to shorten distances, to lower transports costs, to remove trucks from roads and improve the environmental sustainability of trades.
Venice, the westernmost point of arrival by sea in the Chinese strategy of the Belt and Road Initiative, has long been sharing this goal, working day-in and day-out with private and public operators in order to give substance and reliability to the commercial flows between Europe, the Mediterranean and the Far East.
The press release from AdSP del Mare Adriatico Settentrionale can be accessed here.
The press release from Piraeus Port Authority can be accessed here.


text:
require 'spec_helper'
require 'padrino-helpers'
require 'foundation-navigation/menu_item'

#

#   Job Board
#

module FoundationNavigation

  describe MenuItem do
    describe '#new' do
      it 'renders the a tag' do
        expect(MenuItem.new('Foo', '/foo').to_s).to have_tag(
          'li a',
          text: 'Foo',
          with: {href: '/foo'}
        )
      end
    end
  end

end


text:
NGC 4608 is een lensvormig sterrenstelsel in het sterrenbeeld Maagd. Het hemelobject werd op 15 maart 1784 ontdekt door de Duits-Britse astronoom William Herschel.

Synoniemen
 UGC 7842
 MCG 2-32-177
 ZWG 70.214
 VCC 1869
 PGC 42545

Externe links
  NASA/IPAC Extragalactic Database
  SIMBAD Astronomical Database
  SEDS

Zie ook
 Lijst van NGC-objecten

NGC-object in Maagd
Sterrenstelsel in Maagd
Virgocluster


WVS is an international charity providing veterinary aid to charities and animals all over the world.
From disaster emergency response to training and education, our aim is to provide a fast action veterinary response to charities and animals in need. From operations and sterilisation to vaccinations and veterinary aid parcels, we are helping animals all over the world.
WVS at the forefront of innovation! Find out about our veterinary apps.
WVS assists its associated charities by helping to find volunteers on an ongoing basis to help them provide better care for their animals. Volunteer placements are projects on which WVS is helping to find volunteers on behalf of these animal welfare charities. Usually they are available on an ongoing basis throughout the year. Your application process is dealt with by the WVS team who will also assist you with flight bookings, travel insurance and obtaining veterinary supplies for the charity you will be visiting, but the actual project itself is run by the charity on the ground. They will deal with your arrival and look after you throughout the duration of your stay.
These are projects which are organised and funded by WVS. They are usually run on specific dates throughout the year which are advertised in advance and involve a specific campaign, for example spay/neuter, rabies vaccination or veterinary training. For some of these projects, all or part of your travel costs are usually covered as well as your accommodation and travel throughout the project. WVS works with the charity on the ground to implement the project and provides the necessary veterinary supplies and equipment. WVS Projects also include our Emergency Response teams for which we often require very experienced volunteers who are available to take part on short notice.


Baton Rouge, LA, March 30, 2004 - Investigators from the U.S. Chemical Safety and Hazard Investigation Board (CSB) today provided a community update on the ongoing investigations into three accidents that occurred at Honeywell International in East Baton Rouge Parish over a 24-day period last summer.
The information was prepared for presentation at a CSB community briefing tonight at the Leo S. Butler Community Center in East Baton Rouge Parish, with CSB Board Member Dr. Gerald Poje presiding, joined by Board Member Rixio Medina. The meeting is scheduled for 7 p.m. at 950 East Washington Street, Baton Rouge, LA 70802.
Investigator Lisa Long reported that the chlorine release on July 20, 2003, which injured eight Honeywell employees with four of them hospitalized, was caused by a failure of a chlorine cooler, which allowed liquid chlorine to enter the refrigerant system.
Ms. Long said chlorine gas escaped through a hole in a chlorine cooler tube. Ms. Long reported that operators first detected the release at 3:05 a.m. Local authorities activated a siren system at 4 a.m. advising residents to remain in their homes for safety. Ms. Long said, "The company attempted to contain the release by spraying the chlorine cloud with water, but it was not until 7 a.m. that the all-clear signal was given."
Ms. Long reported that Honeywell's testing of chlorine levels, beginning at 5:30 a.m., found concentrations of 1.2 parts per million parts of air (ppm). Concentrations of 1 to 3 ppm can cause mucous membrane irritation; much higher concentrations cause chest pains and death. By 7 a.m. the company reported concentrations below 0.35 ppm, (zero-point-thirty-five ppm).
Investigator Mike Morris reported that a worker, Delvin S. Henry, was fatally injured after he was sprayed with spent, or used, antimony pentachloride (pronounced ANT-imony penta CHLOR-ide) on July 29, 2003, a highly corrosive chemical that can cause serious chemical burns and lung damage.
The operator was engaged in an operation to prepare large used refrigerant cylinders for off-site testing. Five years earlier, the cylinder, originally labeled as antimony pentachloride, had been relabeled as a refrigerant before shipping it to the Baton Rouge plant. The re-labeling was performed by a specialty company in Denver after consulting with Honeywell operators in California.
Though he had worked in the plant for about 3 years, the operator was new to this particular job, and evidently believed all of the cylinders he was working with were empty or contained only small amounts of refrigerant and were labeled as such.
Unable to properly purge, or clear, the contents of one of these cylinders by hooking up hoses to the cylinder's drain valves, the operator removed a plug from the back of the cylinder. The spent antimony pentachloride sprayed out under pressure, enveloping the operator, resulting in his death the next day.
CSB Investigator Johnnie Banks reported that the Honeywell plant had been rapidly shut down following the July 20, 2003, chlorine accident. After the July 29 incident the company ordered the facility to remain shut down until procedures and equipment could be checked to ensure the facility could be operated safely. On Aug. 13, liquid hydrofluoric acid, known in the industry as HF, splashed onto a worker.
The rapid shutdown after July 20 had left significant amounts of the HF, a colorless chemical that can quickly destroy human tissue, in an internal piping system. A decision was made to remove it. Operators began draining it into a sewer through a one-inch pipe where flowing water siphoned the HF into the wastewater sewer system.
On Aug. 13, the system was suspected of being clogged, and an operator opened and closed two valves in an attempt to clear the blockage. The operator then reopened the valves, resulting in a pressure surge that ran through the one-inch pipe, called a "venturi stick."
Pressurized by nitrogen, hydrofluric acid and water escaped from the sewer, splashing onto the worker. CSB investigators found the pipe had not been properly secured, and was suspended by a single rope.
The operator was aided by a maintenance supervisor, who experienced a coughing spell from the fumes. The operator was treated and released at the local hospital; the maintenance supervisor was released from the hospital the next day.
Ms. Long said her investigation team will continue to gather information and analyze several areas of interest, such as the company's hazard evaluation process, plant safety programs and community notification issues.
In prepared remarks for the community meeting, Dr. Poje said, "These incidents, three in a row in a period of less than four weeks, show the need for continuing safety vigilance at chemical plants and other facilities where potentially dangerous chemicals are processed. While this investigation is not complete, I emphasize the need for chemical companies to constantly review their processes, and I would urge chemical facilities, local authorities and neighborhood groups to work together to ensure that notification and emergency response systems are in order."
The Board does not issue citations or fines but does make safety recommendations to plants, industry organizations, labor groups, and regulatory agencies such as OSHA and EPA. The Board designates formal responses to its recommendations as acceptable or unacceptable, open or closed. Further information about the CSB is available from www.csb.gov. For more information, please contact Sandy Gilmour who will be in Baton Rouge March 30 at 202-251-5496 (cell). In Washington, contact Daniel Horowitz, 202-261-7613 / 202-441-6074 (cell).


text:
Tanqueray Rangpur Gin, 750 mL (82.6 Proof)
Tanqueray Rangpur Gin uses rare Rangpur limes during the production process to obtain zest and exotic citrus undertones, this flavor creates a unique, full-bodied spirit with a smooth finish. The perfect inspiration for your own ingenious cocktails, mix Rangpur Gin with a splash of premium tonic water and a wedge of lime for a classic gin and tonic. Includes one 82.6 proof 750 mL bottle of Rangpur Gin. Tanqueray was developed over 180 years ago by Charles Tanqueray, a clergyman's son who chose not to follow the family calling and instead opened a distillery in Bloomsbury, London. Not afraid to mix up brave ideas in his own unique style, he invented a gin which is still celebrated to this day. The perfect mix of ingenuity, heritage and hard work, Tanqueray is a dry, crisp gin with a rich juniper flavor that makes the most refreshing tasting gin and tonics and the driest of martini cocktails. Please drink responsibly. One 750 mL bottle of Tanqueray Rangpur Gin Distilled with rare Rangpur limes Balanced blend of juniper, coriander, angelica and licorice Exotic citrus undertones and a smooth finish 82.6 proof 2017 San Francisco World Spirits Competition Winner


Come find out why living in the Country is a COOL thing to do--this 3 Bedroom, 2 Bath Ranch home is just delightful offering over 2600sqft of finished living space. Her traditional front Living Room, Combo Dining and open Kitchen with Appliances speak Entertainment Ready, along with her spacious Family Room with a gas Mantled Fireplace. Looking for more Room to Roam, come check out her Finished Basement and her 3+ Acres of mature landscaping. 3 Tier Deck, Above-ground Pool, Security System, and a Pole Barn with concrete floors, the list is endless. Home Sweet Home, come you will see!!!
Dundee Elementary School Elementary School / Public 6 3.84 mi.
Dundee Middle School Middle School / Public 6 3.84 mi.
Dundee Community High School High School / Public 8 3.86 mi.
Clayton H. Symons Elementary School * Elementary School / Public 7 6.24 mi.
Paddock Elementary School * Elementary School / Public N/A 6.48 mi.
Riverside Academy * High School / Public N/A 4.29 mi.
13006 Oelke Road, Maybee, MI 48159 is a 3 bedroom, 2 bath single family home was sold for $225,500.


Burgundy sleeveless dress with a ruched center seam for a gathered effect down the front and back. Slim fit. Slightly flared skirt. Concealed back zip. High neckline. Twisted top seam. Seamed back yoke. Color: Burgundy. 70% Viscose, 30% Ramie.


text:
  La Chapelle est
       un village
       géographiquement positionné dans le département de Savoie en Rhône-Alpes. Elle comptait 311 habitants en 2008.

À coté de La Chapelle sont situées les communes de
Sonnaz localisée à 5 km, 1 257 habitants,
Drumettaz-Clarafond à 5 km, 2 180 habitants,
Mouxy localisée à 6 km, 1 842 habitants,
Pugny-Chatenod à 6 km, 833 habitants,
Voglans à 6 km, 1 639 habitants,
Saint-Jean-d'Arvey située à 5 km, 1 397 habitants,
 entre autres. De plus, La Chapelle est située à seulement neuf km de Chambéry.

Le nombre de logements, à La Chapelle, se décomposait en 2011 en dix appartements et 196 maisons soit
 un marché relativement équilibré.

La Chapelle est situé à seulement 10 Kilomètres de Jacob-Bellecombette, les élèves qui aurons besoin de se loger à moindre coût pourront envisager de prendre un appartement à La Chapelle. La Chapelle est également un bon investissement locatif a cause de sa proximité de Jacob-Bellecombette et de ses Universités. Il sera aisé de trouver un logement à vendre.

Si vous envisagez de emmenager à La Chapelle, vous pourrez aisément trouver une maison à vendre.


text:
HomeResearch
Study Shows Security Market Doubling By 2006
By Sharon Gaudin
The IT security market is expected to more than double between 2001 and 2006, according to a new report just released by industry analyst group International Data Corp.
The total market, including software, hardware and services, is expected to jump from $17 billion in revenues in 2001 to $45 billion three years from now, according to Brian Burke, senior research analyst for IDC's Security Products service.
Growth in security-related hardware is leading the charge with a compound annual growth rate of 25% between 2001 and 2006. Security services, according to IDC's report, falls in line next with a 24% growth rate and software follows that with 16%.
"Although many IT markets declined from 2001 to 2002, a recent IDC survey shows security spending remains a top priority for most organizations," says Burke. "Our survey results show that security was rated as a top priority in 2002. And security was the only IT budget to increase in 2001."
Burke says administrators are focusing their efforts on worm and virus attacks. Secure content management is the industry's biggest growth area right now, with antivirus software making up 85% of that spending, according to Burke.
"Antivirus companies had a huge, huge year," he adds. "They grew an average of 33% to 34%."
Burke also says administrators are turning to email content filtering, working to stop the deluge of spam, as well as to abide by an influx of privacy regulations in the health care and financial industries.
"They're starting to understand that firewalls aren't enough anymore," he adds. "They're adopting a layered approach, protecting mail servers and file servers, and installing antivirus protection at the gateway. They're looking at protection on each layer."
IDC's report, however, flies in the face of recent analyst observations that security and network administrators are backing off of security spending. Many industry watchers say spending peaked after the Sept. 11 terrorist attacks, amid fears that terrorists could target the Internet and corporate networks. After such a big focus — both in terms of time and budget — on security, a lot of IT leaders and CFOs are holding off on new spending right now.
Eric Hemmendinger, research director for the Security and Privacy Group at the Boston-based Aberdeen Group, says the economic slowdown has tightened budgets and greatly slowed security spending.
"We do have more awareness of the issues than we did a few years ago," says Hemmendinger. "But the economic slowdown has crippled spending and that has kept people from going off and solving the problems they might have otherwise."
Sharon Gaudin
Rich Content Drives Asia-Pacific Storage
HP Goes Green
Report: IT Recovery Still a Work in Progress
CIOs Finally Ready to Start Hiring Again
IT Spending Seen on the Rise Again in 2010


text:
Dosinia physema is een tweekleppigensoort uit de familie van de Veneridae. De wetenschappelijke naam van de soort is voor het eerst geldig gepubliceerd in 1870 door Römer.

Veneridae


The U.S.S. Maine, a second-class battleship commanded by Capt. Charles D. Sigsbee, was dispatched to Havana harbor in January 1898, ostensibly as a good will gesture, but in reality to protect American lives and property in the troubled colony. For three weeks it lay moored to a buoy 500 yards off the arsenal. The Spaniards clearly felt some resentment towards the American presence, but there was no overt display.
On February 15, 1898, at 9:40 p.m., the Maine exploded, killing 260 men aboard. Public opinion in the United States, urged on by the yellow press, assumed that Spain had plotted the treachery. Scant attention was given to the possibility that an accidental internal explosion had occurred. Cries of "Remember the Maine, To Hell with Spain" were heard throughout the country and war fever increased. Congress responded by appropriating $50 million for military preparedness.
President McKinley, despite his anger, still proceeded cautiously. A navy court of inquiry concluded on March 28 that the explosion had been caused by an external source, probably a mine.1 Spain was blamed by an increasing segment of the American population. McKinley explored means to maintain peace and prayed earnestly over the dilemma, but soon realized that he could not hold back the push toward war. Theodore Roosevelt, a leader of the war faction, opined, "Let the fight come if it must. I rather hope that the fight will come soon. The clamor of the peace faction has convinced me that this country needs a war."
President McKinley requested a declaration of war and Congress responded affirmatively on April 25, but had earlier disavowed any interest in annexing Cuba. The first military action followed within a week.
The wreck of the Maine remained in Havana harbor until 1911, when U.S. Army engineers built a coffer dam around the wreck, sealed the after hull of the ship and floated it out to sea. It was sunk with its flag flying on March 16, 1912. The remains of 66 crew members were removed during the raising and buried in Arlington National Cemetery.
1. The court of inquiry's judgment was upheld by another investigation of the Maine incident in 1911. However, during the 1970s, a third navy investigation revealed that the explosion had been internal and the likely cause was spontaneous combustion in the coal holds.
The War Lovers: Roosevelt, Lodge, Hearst, and the Rush to Empire, 1898 by Evan Thomas.
Fighting for American Manhood: How Gender Politics Provoked the Spanish-American and Philippine-American Wars by Kristin L. Hoganson.
Mr. Speaker!: The Life and Times of Thomas B. Reed The Man Who Broke the Filibuster by James Grant.


Belief in God answers two questions: what, if anything, is it that Jews, Christians, and Muslims are agreeing about when they join in claiming that there is a God; and what, if any, prospects are there for rationally defending or attacking this claim?
In the context of a sustained argument for particular answers to these questions, Tim Mawson tackles many of the most prominent topics in the philosophy of religion. He argues that those who believe that there is a God are best interpreted as believing that there is a being who is essentially personal, transcendent, immanent, omnipotent, omniscient, eternal, perfectly free, perfectly good, and necessary; and non-essentially creator of the world and value; revealer of Himself; and offerer of everlasting life. Having explored the meaning and consistency of this conception of God in the first half of the book, Mawson goes on to consider whether or not belief or the absence of belief in such a God might be the sort of thing that does not rationally require argument and, if not, what the criteria for a good argument for or against such a God's existence might be. Having established some criteria, he uses them to evaluate specific arguments for and against the existence of such a God. He looks at the Argument to Design; the Cosmological Argument; the Ontological Argument; the Argument from Religious Experience; the Argument from Apparent Miracles; the Problem of Evil; and Pascal's Wager. Finally, he explores the relation between faith and reason.
In the course of his argument, Mawson makes striking new claims and defends or attacks established positions in new ways. His conversational style, lively wit, and enlightening examples make Belief in God as pleasurable as it is instructive and thought-provoking. It makes an ideal text for beginning undergraduate courses and for anyone thinking about these most important of questions.