Right about now some of you will be laughing and saying good luck, nothing is going to change. That is where you would be wrong.
I first learned about the power of a parent's voice three years ago, when our school board decided to cancel the school buses. This would have effectively denied our children's right to an education as in the rural areas of Maple Ridge and Pitt Meadows, children relied on the school bus for transportation to get to school. I worked with a small group of very dedicated parents and together created the "Save the Maple Ridge School Bus campaign. We made history! We spread the news across the province and rallied together to fight for our children's right to an education.
We not only got the school buses back, but the government changed the way they provide school districts with funding for transportation. The bad news, of course, is that B.C still has the longest walk limits in CANADA to qualify for the school bus and the requirement to provide school buses has not been added back into The School Act…still…our voices were heard, and we got the school buses back.
My hope is that by speaking out about some of the problems in our education system, more parents will want to get involved and speak up.
It is not the teachers, the education assistants, other staff or the administrators that will be the driving force for change, it will be parents. The government will listen to parents.
It is very likely that some of you have absolutely no idea how different school is today, then when you last attended. Many of the kids we see today are different then what you remember when you were in school. Some kids have difficult home situations, some have special medical needs and learning differences while others have behavioural and mental health concerns. Some of these concerns have been confirmed by a doctor while other children are on wait list many years long to see a doctor. If our public education system states that "all children have a right to an education", we had better make darn sure we have the right supports in place… which, at this time, we do not.
There is something very wrong with our public school system when we are not able to help every child that needs help.
So, here I am, embarking on a journey to help fix the education system.
After working over 7 years as an education assistant in the public-school system day after day helping kids, I am increasingly aware of the vast number of children in our school system who need help but that are not receiving it. Some might consider schools today to be more of a daycare or child-minding facility as we simply do not have the man power, space, expertise and training to ensure every child can reach their full potential. Some days are so challenging that simply getting through it becomes the goal. Every child is different, each one with their unique strengths and areas of need. Our currently system does not adequately provide for all children.
Let's take that scary first step of talking about "inclusion"……. what it really means and if we are actually practicing it. Let's take a hard look at funding, space, staffing levels and what kind of training our education assistants, staff, teachers and administrators need to help children of all needs.
After working in our school district as an education assistant for so long, I feel compelled to speak up. I am discouraged with our current system and simply can no longer be silent.
I am convinced that if parents really knew what was going on in our schools, they would stand up and fight for a better and safer education for not only their children, but for all children.
It is time for me to stand up and fight for what I believe in.
I, Pascale Shaw, am running for school board trustee.


A consistent, high quality compost with a nutrient enriched formula and wetting agent to ensure easier watering.
Green Lane Allotments have teamed up with Kings Seeds of Colchester to enable us to offer each plot holder a comprehensive range of seeds at very competitive prices.
Every autumn, plot holders will receive a catalogue from Kings. Ordering through the allotment will generate a small income to the society.
If you have any questions please enquire at the site shop.
John Innes No. 1 ideal for pricking out or potting-up young seedlings or rooted cuttings. This composts has a carefully balanced nutrient content to suit most young plants.


Your tax-deductible donation ensures the continued growth and stability of the Rochester Gay Men's Chorus.
You can help us to continue to sing wonderful arrangements of beautiful songs in two ways: Support the Dollars to Notes campaign and be added to the list in our programs or sponsor an individual song and have your name listed with that song!
Do you have a business you would like to have allied with the RGMC? Click below for more information on how to advertise in our newsletters, concert programs, and even here on our website.
The James R. Moran Endowment for the RGMC, the Smith Memorial Scholarship Fund, and our bequest & legacy gift program give you several ways to give a targeted gift to the RGMC.
If you would like to donate items for us to sell at our next big garage sale to help support the efforts of the RGMC, please click below for more information.


text:
Why Black Church Leaders Often Get Away With Sexual Misconduct, and the Woman Breaking the Silence
Church & Ministries
Church Resource
Edition Current Edition: English Español 中文
Trump proclaims National Sanctity of Human Life Day on 47th anniversary of Roe v. Wade
Christian Association of Nigeria calls for 3 days of prayer, fasting after Boko Haram executes pastor
Anti-Christian violence, attacks on churches in Europe at all-time high in 2019: report
Senate impeachment trial of President Trump Day 1: Over 12 hours of debate, 'pettifogging' and foul language
Pregnancy center: 90% of our clients chose life in 2019
What does Satan want?
Hit pause on gene editing
CP Current Page: Church & Ministries | Saturday, November 10, 2018
By Leonardo Blair, Christian Post Reporter | Saturday, November 10, 2018
Facebook Twitter Email Print Whatsapp
Whatsapp Google Reddit Digg Stumbleupon Linkedin
Andrea Garrison is a model in New York City. | (Photo: Courtesy of Andrea Garrison)
It was Halloween night at the 5,000-member Inspirational Bible Church in Cincinnati, Ohio, and Pastor Victor S. Couzens was wielding the "blood" of Jesus like a "weapon" against everything he believes is attacking him and his congregation.
Nattily dressed with well-coiffed dreadlocks, he exuded a self-assured confidence as he spoke in tongues while sauntering across the stage toward a sleek, cross-adorned pulpit that rose automatically from the ground.
"I plead the blood. I take the blood as our defense. I take the blood as our offense. I take the blood as our weapon. I take the blood as our shield. I take the blood as our response. I plead the blood of Christ over you," he declared forcefully as his affirming followers joined him with hands raised in a din of prayers.
It was the ninth day of special prayer at Couzens' church. On this night, he seemed confident he was covered under the blood of Jesus and forgiven of all his sins. No weapon formed against him would prosper while he is covered by the blood of Jesus.
It had also been nine days since Andrea Garrison, a 30-year-old New York City model, sparked calls for his resignation. Garrison announced on Facebook Live that Couzens had been involved in a long-term sexual relationship with her and several other women simultaneously, and she won't stop speaking her truth until the church sits him down.
In recent months, Garrison said, she thought her relationship with the preacher was exclusive. He even led her to believe that she could become his wife. And Couzens kept her believing until one cold day in October, Garrison discovered he had proposed to another woman and didn't bother to tell her.
Couzens initially denied Garrison's claim in a now deleted Facebook post. She quickly responded, however, by releasing a trove of evidence of the relationship they had — racy emails, text messages, photos and checks allegedly showing the use of money from the church's account.
Portions of this trove were also shared directly with The Christian Post.
"I didn't take that kindly," Garrison told CP in a recent interview about the preacher's decision to brand her as a liar. "I approached him and said can you please take that (Facebook post) down. Don't call me a liar because we both know that I can destroy you with what I have. Don't do that. And he became arrogant, so I said, 'OK. If that's what you want to do.' And so I made my [Facebook] Live [video] and when I made my Live it just blew up from there."
Now, after hearing from several other deceived women she says are too afraid to speak out publicly against Couzens, Garrison wants to use her voice as a wrecking ball against the culture of silence that has enabled rogue pastors like Couzens to exploit women in the black church without penalty. She is also pushing to mobilize other women who have also been mistreated to force churches that have tolerated their behavior to start renouncing their sin.
Bishop Victor Couzens is leader of the 5,000-member Inspirational Bible Church – City of Destiny in Cincinnati, Ohio. | (Photo: Facebook)
"I feel like, at this point, it's no longer what he did to me, what he did to the other women. It's no longer about that. It's about the fact that the Church allows this. That the Church is OK with this. The Church is OK with sending threats to people who are just trying to expose something that's bad happening," Garrison said. "I'm tired of this type of stuff being swept under the rug and I'm tired of being hushed when all this could have been avoided with him just keeping it real."
Prevalence of Misconduct
The Rev. Donald H. Matthews is an ordained minister who has taught at numerous universities and seminaries in the areas of black studies, religious ethics, sociology, and psychology. He has more than 40 years of experience in ministry.
In his 2012 book, Sexual Abuse of Power in the Black Church: Sexual Misconduct in the African American Churches, Matthews pointed to research suggesting that black females are likely to experience three times the amount of sexual abuse compared to their counterparts in white churches.
And the cultural secrecy with which the issue is treated, he explained, are among several reasons that have prevented black churches from properly addressing the issue.
"It is important to recognize the suffering of African Americans in this area and to try and remedy it as well. This cannot be done as long as the ... taboo of discussing sexual matters continues in the African-American community and churches," Matthews said.
"This kind of sexual abuse of power represents a deep betrayal and leaves a woundedness and pain that has long-lasting and lingering effects that may never be fully healed. The pastoral abuse of power does not only negatively affect the persons directly and intimately involved with the pastor in his misconduct, it also has a harmful effect on the entire church," Matthews wrote.
Failed Attempt to Discipline
For now though, Garrison is one of few women who has come forward and publicly called out a black pastor over his sexual behavior. Meanwhile, Couzens has continued to lead his congregants in prayer and the blood of Jesus.
"Thank you Lord that all of our sins are under the blood ... that where sin abounds grace much more abounds and all of our sins are under the blood," he said on Halloween.
"That the sins of the people are under the blood. The sins of the priest are under the blood. The sins of the naysayers are under the blood. The sins of the yaysayers are under the blood. I command us to accept the blood of Jesus Christ," he urged.
As chatter about Couzens' lifestyle went viral in late October, Paul S. Morton, founding presiding bishop of The Full Gospel Baptist Church Fellowship International, was forced to weigh in after Garrison begged him to do something.
"The only reason I truly, honestly believe that Paul Morton did something is because I called him out," she said. "I posted a tweet and I said Paul Morton, the church has no faith in you. And I sent him a screen shot of people saying he won't do anything."
Morton responded with a public statement confirming that Couzens, a former key leader in his organization and a "spiritual son," had indeed been involved with "multiple" women. He also noted in an earlier statement that Couzens repented and promised to step down as pastor of Inspirational Bible Church.
In a since deleted suicidal statement on Facebook shortly after Morton's soft rebuke of his behavior, however, Couzens denied promising to step down. He also downplayed his multiple relationships. He later claimed the statement was the result of his account being hacked.
"People put words in my mouth. I never said I was resigning from anything. I never said I was stepping down from my church. I never said I have issues that I have to deal with and that I am guilty of anything. Yes! I have enjoyed female companionship. But no, it never was about sex or leading anyone on! If it was about sex, I could buy sex! They have Apps for that! But people print it and they write it and say I said it because that's what they want me to do and say! And it makes them look big," the deleted post said in part.
Even though Couzens blamed the suicidal statement on unconfirmed hacking, Morton removed the portion of his statement about Couzens being "sat down" from his duties as pastor and declined to explain why it was done when his office was contacted by CP.
Elders at Inspirational Bible Church have also refused to publicly address their pastor's behavior even as he continues to speak with tongues, promising to defend the Gospel of truth and light.
Bishop Victor Couzens is leader of the 5,000-member Inspirational Bible Church – City of Destiny in Cincinnati, Ohio. | (Screenshot: YouTube)
In his prayer on Halloween night, Couzens, without specifically referencing Garrison's campaign, said his church was under attack from evil forces and rallied his congregation to not respond like the world after Garrison reported to police that someone had tried to blackmail her into silence.
"I plead the blood over this ministry. I cover this house in the blood. No schism and no ism shall be able to separate or divide us. We are people of God. We live by grace. We live by mercy. We live by authenticity. We live by confession," Couzens said. "We will not be conformed to this world, we will not respond like the world. We won't fight like the world. We won't act like the world."
NYPD Investigates as Megachurch Pastor's Ex Says She Was Threatened With Blackmail
Megachurch Pastor Victor Couzens Denies Suicide Note, Asks God to Help Him Keep Job After Sex Scandal
Ohio Megachurch Pastor Had 'Multiple' Women, Bishop Paul S. Morton Confirms
Potter's House Denver Pastor Chris Hill Resigns Over Alleged Affair, Separates From Wife
Pro-Life Fashion Brand Celebrates it's 1st Birthday - #celebrateCOL
Wear Scriptures: Christian Clothing Company Encourages Christians to Speak Love to the Culture
Spiritual Practices for Your Busy Life
Want to Read More in 2020? Here's How You Can Finish a Book in 12 Minutes
Nigerian pastor who praised God in ransom video beheaded after refusing to deny Christ
'The Walking Bible' televangelist Jack Van Impe dies at 88
Kansas City Chiefs CEO gives glory to God as team heads to first Super Bowl in 50 years
Kanye West shares how Jesus saved him from devil; hundreds dedicate lives to Christ at youth conference
Mike Pence called 'one of the most persecuted Christians in America' at church MLK celebration
More In Church & Ministries
Southeast Christian Church announces plans for 10th campus, fourth since popular pastor's retirement
This week in Christian history: Pope warns against 'Americanism', Robert Boyle, Bible translator dies
Pastor of 5,000-member church plant identifies key ministry trends leading millennials to Christ
Subscribe Now We want to hear from you!
Overall Navigation
Email Whitelisting
2020 The Christian Post, INC. All Rights Reserved.

redpajama_set_name: RedPajamaCommonCrawl


Our students will be accepted to Cardijn College for their secondary education although families need to be aware that enrolment forms and interviews are still necessary and these commence from Year 6 for the Year 8 Cardijn intake. (Please note that the cut-off date for Year 8 applications to Cardijn occurs in November of the year your child is in Year 6).
Cardijn is committed to providing a high-quality high school education which embraces the full human growth and development of its students. The school commits itself to establishing academic excellence in all endeavours and the staff endeavours not only to bring out the best in the students, but inspire them to bring out the best in themselves.
Further information regarding enrolment at Cardijn College is available by contacting the Registrar, Mrs Michelle Gaffney on 8392 9449 or emailing registrar@cardijn.catholic.edu.au.


Medway Lodge is in a very good location very close to the High Street. It's minutes away from Gillingham train station which has very good links to the London.
This is a large double bedroom in a very quiet and clean house.
Facilities include free Wifi, Smart TV and Parking. There's a kitchen for our guest's use with all the necessary appliances.
Away on business or just need a break, Medway Lodge is the ideal place for you! This large double bedroom with its comfortable bed will make you feel warm and welcomed from the moment you walk in.
For the business person, its the perfect location to stay whilst on business.
***Please note that a cleaning fee of £20 and a refundable security deposit of £150 will be collected with each booking.


text:
« Best-Selling Prints
Buoys and Line
Friendship, Maine
A lobsterman's buoys and rope is left on the dock on a summer morning in Friendship Harbor, Maine.
Available as a Fine Art Limited Edition of 100 prints, which includes a signed Certificate of Authenticity with serial number. COA will be shipped separately. Photo © copyright by Rick Berk.
16" x 24" Limited Edition Print : $275.00 20" x 30" Limited Edition Print : $375.00 24" x 36" Limited Edition Print : $495.00
16" x 24" Limited Edition Print : $530.00 20" x 30" Limited Edition Print : $850.00 24" x 36" Limited Edition Print : $1,095.00 30" x 45" Limited Edition Print : $1,500.00 40" x 60" Limited Edition Print : $2,350.00
16" x 24" Limited Edition Print : $950.00 20" x 30" Limited Edition Print : $1,500.00 24" x 36" Limited Edition Print : $1,925.00 30" x 45" Limited Edition Print : $2,750.00 40" x 60" Limited Edition Print : $3,500.00
Related Galleries: Seascapes & Coastal, Maine Scenes, Limited Edition Fine Art Landscape Photographs


text:
\section{Introduction}
The Stokes problem
seeks unknown functions ${\bf u}$ and $p$ satisfying
\begin{eqnarray}
-\nabla\cdot A\nabla{\bf u}+\nabla p&=& {\bf f}\quad
\mbox{in}\;\Omega,\label{moment}\\
\nabla\cdot{\bf u} &=&0\quad \mbox{in}\;\Omega,\label{cont}\\
{\bf u}&=&0\quad \mbox{on}\; \partial\Omega,\label{bc}
\end{eqnarray}
where $\Omega$ is a polygonal domain in
$\mathbb{R}^d$ with $d=2,3$ and $A$ is a symmetric and positive definite $d\times d$ matrix-valued
function in $\Omega$.
For the nonhomogeneous boundary condition
\[
{\bf u}={\bf g}\quad \mbox{on}\; \partial\Omega,
\]
one can use the standard procedure by letting ${\bf u}={\bf u}_0+{\bf u}_g$. ${\bf u}_g$ is a known function satisfying ${\bf u}_g={\bf g}$ on $\partial\Omega$ and ${\bf u}_0$ is zero at $\partial\Omega$ and satisfies (\ref{moment})-(\ref{cont}) with different right hand sides.

The weak form in the primary velocity-pressure formulation for the
Stokes problem (\ref{moment})--(\ref{bc}) seeks ${\bf u}\in
[H_0^1(\Omega)]^d$ and $p\in L_0^2(\Omega)$ satisfying
\begin{eqnarray}
(A\nabla{\bf u},\nabla{\bf v})-(\nabla\cdot{\bf v}, p)&=&({\bf f}, {\bf v}),\quad \forall{\bf v}\in [H_0^1(\Omega)]^d\label{w1}\\
(\nabla\cdot{\bf u}, q)&=&0,\quad\quad\forall  q\in L_0^2(\Omega).\label{w2}
\end{eqnarray}

In the standard finite element methods for the Stokes and the
Navier-Stokes equations, both pressure and velocity are approximated
simultaneously. The primitive system is a large saddle point
problem. Numerical solvers for such indefinite systems are usually
less effective and robust than solvers for definite systems. On the
other hand, the divergence-free finite element method, discrete or
exact, computes numerical solution of  velocity by solving a
symmetric positive definite system in a divergence-free subspace. It
eliminates the pressure from the coupled equations and hence
significantly reduces the size of the system. The divergence-free
method is particularly attractive in the cases where the velocity is
the primary variable of interest, for example, the groundwater flow
calculation. The main tasks in the implementation of the
divergence-free method are to understand divergence-free subspaces,
weakly or exactly, and to construct bases for them.

Many finite element methods, continuous \cite{cr,gr,gun} and
discontinuous \cite{ckss,grw,kj,ls, wy-hdiv}, have been developed
and analyzed for the Stokes and the Navier-Stokes equations.
Divergence-free basis for different finite element methods have been
constructed \cite{gri1, gri2, gri3,gh1,gh2,wwy,yh1,yh2}.

A weak Galerkin finite element method  was introduced in
\cite{wy-stokes} for the Stokes equations in the primal
velocity-pressure formulation. This method is designed by using
discontinuous piecewise polynomials on finite element partitions
with arbitrary shape of polygons/polyhedra. Weak Galerkin methods
were  first introduced  in \cite{wy, wy-m} for second order elliptic
equations. In general, weak Galerkin finite element formulations for
partial differential equations can be derived naturally by replacing
usual derivatives by weakly-defined derivatives in the corresponding
variational forms, with an option of adding a stabilization term to
enforce a weak continuity of the approximating functions.  Therefore
the weak Galerkin method developed in \cite{wy-stokes} for the
Stokes equations  naturally has the form: find ${\bf u}_h\in V_h$ and
$p_h\in W_h$  satisfying
\begin{eqnarray}
(A\nabla_w{\bf u}_h,\nabla_w{\bf v})+s({\bf u}_h,{\bf v})-(\nabla_w\cdot{\bf v},p_h)&=&({\bf f},{\bf v}),\label{w3}\\
(\nabla_w\cdot{\bf u}_h,q)&=&0\label{w4}
\end{eqnarray}
for all the test functions ${\bf v}\in V_h$ and $q\in W_h$  where $V_h$ and $W_h$ will be defined later. The stabilizer  $s({\bf u}_h,{\bf v})$ in (\ref{w3}) is parameter independent.

Let $D_h$ be a discrete divergence free subspace of $V_h$ such that $(\nabla_w\cdot{\bf v},q)=0$ for ant $q\in W_h$.
Then the discrete divergence free WG formulation  is to  find ${\bf u}_h\in D_h$   satisfying
\begin{eqnarray}
(A\nabla_w{\bf u}_h,\nabla_w{\bf v})+s({\bf u}_h,{\bf v})&=&({\bf f},{\bf v}),\quad\forall{\bf v}\in D_h.\label{dfw}
\end{eqnarray}
System (\ref{dfw}) is symmetric and positive definite with many fewer unknowns. The main purpose of this paper is to construct  bases for $D_h$ in two and three dimensional spaces. A unique feature of these divergence free basis functions is that they can be obtained on general meshes such as hybrid meshes or meshes with hanging nodes. Numerical examples in two dimensional space are provided to confirm the theory. Although the Stokes equations is considered, the divergence free basis can be use for solving the Navier-Stokes equations.

\section{A Weak Galerkin Finite Element Method}\label{Section:wg-fem}

In this section, we will review the WG method for the Stokes
equations introduced in \cite{wy-stokes} with $k=1$.

Let ${\cal T}_h$ be a partition of the domain $\Omega$ consisting
mix of polygons satisfying a set of conditions specified in
\cite{wy-m}. In addition, we assume that all the elements $T\in{\mathcal T}_h$
are convex. Denote by ${\cal F}_h$ the set of all edges in 2D or
faces in 3D in ${\cal T}_h$, and let ${\cal F}_h^0={\cal
F}_h\backslash\partial\Omega$ be the set of all interior edges or
faces.

We define a weak Galerkin finite element space for
the velocity as follows
\[
V_h=\left\{ {\bf v}=\{{\bf v}_0, {\bf v}_b\}:\ \{{\bf v}_0, {\bf v}_b\}|_{T}\in
[P_{1}(T)]^d\times [P_{0}(e)]^d,\ e\subset{\partial T},\ {\bf v}_b=0\ \mbox{on}\
\partial\Omega \right\}.
\]
We would like to emphasize that there is only a single value ${\bf v}_b$
defined on each edge in 2D and face in 3D. For the pressure variable, we have
the following finite element space
\[
W_h=\left\{q:\ q\in L_0^2(\Omega), \ q|_T\in P_{0}(T)\right\}.
\]

For a given ${\bf v}\in V_h$, a  weak gradient and a  weak divergence  are defined locally on each $T\in {\mathcal T}_h$ as follows.

\smallskip

\begin{defi}
A  weak gradient, denoted by
$\nabla_{w}$, is defined as the unique polynomial
$(\nabla_{w}{\bf v}) \in [P_{0}(T)]^{d\times d}$ for ${\bf v}\in V_h$ satisfying the
following equation,
\begin{equation}\label{d-g}
(\nabla_{w}{\bf v}, q)_T = -({\bf v}_0,\nabla\cdot q)_T+ \langle {\bf v}_b,
q\cdot{\bf n}\rangle_{\partial T},\qquad \forall q\in [P_{0}(T)]^{d\times d},
\end{equation}
\end{defi}

\begin{defi}
A  weak divergence, denoted by
($\nabla_{w}\cdot$), is defined as the unique polynomial
$(\nabla_{w}\cdot{\bf v}) \in P_{0}(T)$ for ${\bf v}\in V_h$ that satisfies the following
equation
\begin{equation}\label{d-d}
(\nabla_{w}\cdot{\bf v}, \varphi)_T = -({\bf v}_0, \nabla\varphi)_T + \langle
{\bf v}_b\cdot{\bf n}, \varphi\rangle_{\partial T},\qquad
\forall \varphi\in P_{0}(T).
\end{equation}
\end{defi}

Denote by $Q_{0}$ the $L^2$ projection operator from $[L^2(T)]^d$
onto $[P_1(T)]^d$ and denote by
$Q_{b}$ the $L^2$ projection from $[L^2(e)]^d$ onto $[P_{0}(e)]^d$.
Let $({\bf u};p)$ be the solution of (\ref{moment})-(\ref{bc}).
Define $Q_h{\bf u}=\{Q_0{\bf u}, Q_b{\bf u}\}\in V_h$.
let $\mathbb{Q}_h$ be the  local $L^2$
projections onto $P_{0}(T)$.

\smallskip

We  introduce three bilinear forms as follows
\begin{eqnarray*}
s({\bf v},\;{\bf w}) &= &\sum_{T\in {\cal T}_h}h_T^{-1}{\langle} Q_b{\bf v}_0-{\bf v}_b,\;\;Q_b{\bf w}_0-{\bf w}_b{\rangle}_{\partial T},\\
a({\bf v},\ {\bf w})&=&\sum_{T\in{\mathcal T}_h}(A\nabla_w{\bf v},\ \nabla_w{\bf w})_T+s({\bf v},{\bf w}),\\
b({\bf v},\ q)&=&\sum_{T\in{\mathcal T}_h}(\nabla_w\cdot{\bf v},\ q)_T.
\end{eqnarray*}

\begin{algorithm}
A numerical approximation for (\ref{moment})-(\ref{bc})
can be obtained by seeking ${\bf u}_h=\{{\bf u}_0,{\bf u}_b\}\in V_h$ and
$p_h\in W_h$ such that
\begin{eqnarray}
a({\bf u}_h,\ {\bf v})-b({\bf v},\;p_h)&=&(f,\;{\bf v}_0),\quad\forall{\bf v}\in V_h\label{wg1}\\
b({\bf u}_h,\;q)&=&0,\quad\forall q\in W_h.\label{wg2}
\end{eqnarray}
\end{algorithm}

Define
\begin{equation}\label{3barnorm}
{|\hspace{-.02in}|\hspace{-.02in}|}{\bf v}{|\hspace{-.02in}|\hspace{-.02in}|}^2=a({\bf v},{\bf v}).
\end{equation}

The following optimal error estimates have been derived in \cite{wy-stokes}.
\begin{theorem}\label{h1-bd}
Let $({\bf u}; p)\in  [H_0^1(\Omega)\cap H^{2}(\Omega)]^d\times
(L_0^2(\Omega)\cap H^{1}(\Omega))$  and $({\bf u}_h;p_h)\in
V_h\times W_h$ be the solution of (\ref{moment})-(\ref{bc}) and
(\ref{wg1})-(\ref{wg2}), respectively. Then, the following error
estimates hold true
\begin{eqnarray}
{|\hspace{-.02in}|\hspace{-.02in}|} Q_h{\bf u}-{\bf u}_h{|\hspace{-.02in}|\hspace{-.02in}|}+\|\mathbb{Q}_hp-p_h\|&\le& Ch(\|{\bf u}\|_{2}+\|p\|_{1}),\label{err1}\\
\|Q_0{\bf u}-{\bf u}_0\|&\le& Ch^{2}(\|{\bf u}\|_{2}+\|p\|_{1}).\label{l2-error}
\end{eqnarray}
\end{theorem}

Define a discrete  divergence free subspace $D_h$ of $V_h$ by
\begin{equation}\label{D_h}
D_h=\{{\bf v}\in V_h;\;b({\bf v},q) =0,\quad\forall q\in W_h\}.
\end{equation}

By taking the test functions from $D_h$,
the weak Galerkin formulation (\ref{wg1})-(\ref{wg2}) is equivalent to the following  divergence-free weak Galerkin finite element scheme.

\smallskip

\begin{algorithm}\label{algorithm2}
A discrete divergence free WG approximation for (\ref{moment})-(\ref{bc}) is to find
${\bf u}_h=\{{\bf u}_0,{\bf u}_b\}\in D_h$ such that
\begin{eqnarray}
a({\bf u}_h,\ {\bf v})&=&(f,\;{\bf v}_0),\quad \forall {\bf v}=\{{\bf v}_0,{\bf v}_b\}\in D_h.\label{df-wg}
\end{eqnarray}
\end{algorithm}

System (\ref{df-wg}) is symmetric and positive definite with many
fewer unknowns. It can be solved effectively by many existing
solvers.

The main task of this paper is to construct basis for $D_h$. In the
next two sections, discrete divergence free bases will be
constructed explicitly for two and three dimensional spaces.

\section{Construction of Discrete Divergence Free Basis for Two Dimensional Space}

For a given partition ${\mathcal T}_h$,  let  ${\cal V}_h^0$ be the set of all
interior vertices. Let $N_F=card ({\mathcal F}_h^0)$, $N_V=card ({\cal
V}_h^0)$ and $N_K=card ({\mathcal T}_h)$. It is known based on the Euler
formula that for a partition consisting of convex polygons, then
\begin{equation}\label{key}
N_F+1=N_V+N_K.
\end{equation}
For a mesh ${\mathcal T}_h$ with  hanging nodes, the relation in (\ref{key}) is still true if we treat the hanging nodes as vertices.

First we need to derive a basis  for $V_h$. For each $T\in{\mathcal T}_h$ and
any ${\bf v}=\{{\bf v}_0,{\bf v}_b\}\in V_h$, ${\bf v}_0$ is a vector function with
two components and each component is a linear function. Therefore
there are six linearly independent linear functions $\Phi_{j+1},
\Phi_{j+2},\cdots,\Phi_{j+6}$ in $V_h$ such that they are nonzero
only at the interior of element $T$. For each $e_i\in{\mathcal F}_h^0$,
${\bf v}_b$ is a constant vector function. Thus, there are two linearly
independent constant  functions $\Psi_{i,1}$ and $\Psi_{i,2}$ in
$V_h$ which take nonzero value only on $e_i$. Then it is easy to see
that
\begin{equation}\label{basis}
V_h={\rm span}\{\Phi_1,\cdots, \Phi_{6N_K},\Psi_{1,1},\Psi_{1,2},\cdots,\Psi_{N_F,1},\Psi_{N_F,2}\}.
\end{equation}
For a given function ${\bf v}=\{{\bf v}_0,{\bf v}_b\}\in V_h$, it is easy to see that ${\bf v}_0$ can be spanned by the basis functions $\Phi_i$ and ${\bf v}_b$ by the  basis functions $\Psi_{j,k}$.

Next, we will find the dimension of $D_h$. Since the dimension for pressure space $W_h$ is $N_K-1$, it follows from (\ref{key}) that
\begin{equation}\label{bbb}
\rm{dim} (D_h)=\rm{dim} (V_h)-\rm{dim}(W_h)=6N_K+2N_F-N_K+1=6N_K+N_F+N_V.
\end{equation}

\begin{lemma}
The basis functions $\Phi_1,\cdots, \Phi_{6N_K}$ of $V_h$ in (\ref{basis}) are in $D_h$ and linearly independent.
\end{lemma}

\smallskip

\begin{proof}
Let $\Phi_i=\{\Phi_{i,0},\Phi_{i,b}\}$. The definition of $\Phi_i$ implies $\Phi_{i,b}=0$.
 For any $q\in W_h$, it follows from (\ref{d-d}), $\nabla q=0$ and $\Phi_{i,b}=0$,
\begin{eqnarray*}
b(\Phi_i,q)&=&\sum_{T\in{\mathcal T}_h} (\nabla_w\cdot\Phi_i,q)_T\\
&=&\sum_{T\in{\mathcal T}_h}(-(\Phi_{i,0},\nabla q)_T+{\langle} \Phi_{i,b}\cdot{\bf n}, q{\rangle}_{\partial T})\\
&=&0,
\end{eqnarray*}
which proves the lemma since the linear independence of  $\Phi_1,\cdots, \Phi_{6N_K}$ is obvious.
\end{proof}

\smallskip

For any $e_i\in{\mathcal F}_h^0$,let $\psi_{i,1}$ and $\psi_{i,2}$ be two basis functions of $V_h$ associated with $e_i$. Let ${\bf n}_{e_i}$ and ${\bf t}_{e_i}$ be a normal vector and a tangential vector to $e_i$ respectively. Define  $\Upsilon_i=C_1\Psi_{i,1}+C_2\Psi_{i,2}$ such that $\Upsilon_i|_{e_i}={\bf t}_{e_i}$. Obviously $\Upsilon_i\in V_h$ is only nonzero on $e_i$.

\smallskip

\begin{lemma}
Functions  $\Upsilon_1,\cdots,\Upsilon_{N_F}\in V_h$ are in $D_h$ and linearly independent.
\end{lemma}

\smallskip

\begin{proof}
Let $\Upsilon_i=\{\Upsilon_{i,0},\Upsilon_{i,b}\}$. For any $q\in W_h$, it follows from (\ref{d-d}) and $\nabla q=0$,
\begin{eqnarray*}
b(\Upsilon_i,q)&=&\sum_{T\in{\mathcal T}_h} (\nabla_w\cdot\Upsilon_i,q)_T\\
&=&\sum_{T\in{\mathcal T}_h}(-(\Upsilon_{i,0},\nabla q)_T+{\langle} \Upsilon_{i,b}\cdot{\bf n}, q{\rangle}_{\partial T})\\
&=&\sum_{T\in{\mathcal T}_h}{\langle} \Upsilon_{i,b}\cdot{\bf n}, q{\rangle}_{\partial T}\\
&=&0,
\end{eqnarray*}
where we use the fact ${\bf t}_{e_i}\cdot{\bf n}=0$.
Since $\Upsilon_i$ is only nonzero on $e_i$, $\Upsilon_1,\cdots,\Upsilon_{N_F}$ are linearly independent. We completed the proof.
\end{proof}

\smallskip

For a given interior vertex $P_i\in {\cal V}_h^0$, assume that there are $r$ elements having $P_i$ as a vertex which form a hull ${\cal H}_{P_i}$ as shown in Figure \ref{fig1}. Then there are $r$ interior edges $e_j$ ($j=1,\cdots, r$) associated with ${\cal H}_{P_i}$. Let ${\bf n}_{{e_j}}$ be a normal vector on $e_j$ such that normal vectors ${\bf n}_{e_j}$ $j=1,\cdots,r$ are counterclockwise around vertex $P_i$ as shown in Figure \ref{fig1}.  For each $e_j$, let $\Psi_{j,1}$ and $\Psi_{j,2}$ be the two basis functions of $V_h$ which is only nonzero on $e_j$. Define $\Theta_j=C_1\Psi_{j,1}+C_2\Psi_{j,2}\in V_h$ such that $\Theta_j|_{e_j}={\bf n}_{e_j}$. Define $\Lambda_i=\sum_{j=1}^r \frac{1}{|e_j|}\Theta_j$.

\begin{figure}[!tb]
\centering
\includegraphics[width=0.4\textwidth]{rotate-new.eps}\qquad \qquad
\includegraphics[width=0.4\textwidth]{basis2-new.eps}\\
  (a)\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad   (b)\\
\caption{(a) A 2D hull ${\cal H}_{P_i}$; (b) A hull with hanging node.}
\label{fig1}
\end{figure}

\smallskip

\begin{lemma}\label{lemma3}
Functions $\Lambda_1,\cdots,\Lambda_{N_V}\in V_h$ are in $D_h$ and linearly independent.
\end{lemma}

\smallskip

\begin{proof}
 Suppose that there exist constants  $c_1,\cdots, c_{N_V}$ such that $\sum_{i=1}^{N_V}c_i\Lambda_i=0$. Let $\Lambda_l$ be associated with a hull ${\cal H}_{P_l}$ such  that there exists $e_m$ as one of the interior edges of ${\cal H}_{P_l}$  and edge $e_m$ has  a boundary node as one of its end points. Since $\Lambda_i=0$ on $e_m$ for $i\neq l$,  we have
\[
0=\sum_{i=1}^{N_V}\int_{e_m}c_i\Lambda_i=\int_{e_m}c_l\Lambda_l=\int_{e_m}c_l\Theta_m=c_l{\bf n}_{e_m},
\]
which implies $c_l=0$. By this way, we can prove that all $c_i=0$ and $\Lambda_1,\cdots,\Lambda_{N_V}$ are linearly independent.
Next, we will show that $b(\Lambda_i,q)=0$ for all $q\in W_h$. Let $q_j\in W_h$ such that $q_j=1$ on $T_j\in{\mathcal T}_h$ and $q_j=0$ otherwise. So we only need to show that
\begin{equation}\label{qi}
b(\Lambda_i,q_j)=0,\quad\forall j=1,\cdots, N_K.
\end{equation}
Let $\Lambda_i$ and $q_j$ be associated with hull ${\cal H}_{P_i}$ and element $T_j$ respectively.
If $T_j\in{\mathcal T}_h$ is not in ${\cal H}_{P_i}$, we easily have $b(\Lambda_i,q_j)=0$. If $T_j\in {\cal H}_{P_i}$, let $e_s$ and $e_{s+1}$ be its two edges in ${\cal H}_{P_i}$ shown in Figure \ref{fig1},
\begin{eqnarray*}
b(\Lambda_i,q_j)&=&\sum_{T\in{\mathcal T}_h} (\nabla_w\cdot\Lambda_i,q_j)_T\\
&=& (\nabla_w\cdot\Lambda_i,q_j)_{T_j}\\
&=&-(\Lambda_{i,0},\nabla q)_{T_j}+{\langle} \Lambda_{i,b}\cdot{\bf n}, q{\rangle}_{{\partial T}_j}\\
&=&\int_{e_s}\frac{1}{|e_s|}{\bf n}_{e_s}\cdot{\bf n}+\int_{e_{s+1}}\frac{1}{|e_{s+1}|}{\bf n}_{e_{s+1}}\cdot{\bf n}\\
&=&0.
\end{eqnarray*}
We proved $\Lambda_i\in D_h$ for $i=1,\cdots, N_V$.
\end{proof}

\smallskip

\begin{theorem}\label{thm1}
Let $D_h$ be defined in (\ref{D_h}). Then for two dimensional space, $D_h$ is spanned by the following basis functions,
\begin{equation}\label{main}
D_h={\rm Span}\{\Phi_1,\cdots, \Phi_{6N_K}, \Upsilon_1,\cdots, \Upsilon_{N_F},\Lambda_1,\cdots,\Lambda_{N_V}\}.
\end{equation}
\end{theorem}

\begin{proof}
The number of the functions in the right hand side of (\ref{main}) is $6N_K+N_F+N_V$ which is equal to ${\rm dim}(D_h)$ due to (\ref{bbb}).
Next, we prove that
$$\Phi_1,\cdots, \Phi_{6N_K}, \Upsilon_1,\cdots, \Upsilon_{N_F},\Lambda_1,\cdots,\Lambda_{N_V}$$
are linearly independent.
Since $\Phi_i$ take zero value on all $f\in{\mathcal F}_h$, $\Phi_i$ will be linearly independent to all $\Upsilon_l$ and $\Lambda_m$.
Suppose
\begin{equation}\label{t1}
C_1\Upsilon_1+\cdots+C_{N_F}\Upsilon_{N_F}+C_{N_F+1}\Lambda_1+\cdots+C_{N_F+N_V}\Lambda_{N_V}=0.
\end{equation}
Multiplying (\ref{t1}) by $\Upsilon_i$ and integrating over $e_i$, we have
\[
C_i|e_i|=0,
\]
where we use the fact ${\bf t}_{e_i}\cdot{\bf n}_e=0$. Thus we can obtain $C_i=0$ for $i=1,\cdots,N_F$. By Lemma \ref{lemma3}, we can prove $C_i=0$ for $i=N_F+1,\cdots,N_F+N_V$. The proof of the lemma is completed.
\end{proof}

\section{Construction of Discrete Divergence Free Basis for  Three Dimensional Space}

Let ${\mathcal T}_h$ be a partition of $\Omega\subset \mathbb{R}^3$ consisting polyhedrons without hanging nodes.  Recall $N_F=card ({\mathcal F}_h^0)$, $N_V=card ({\cal V}_h^0)$ and $N_K=card ({\mathcal T}_h)$.  Denote by ${\mathcal E}_h$ all the edges in ${\mathcal T}_h$ and let ${\mathcal E}_h^0={\cal E}_h\backslash\partial\Omega$. Let $N_E=card ({\mathcal E}_h^0)$.

It is known based on the Euler formula that for a  partition consisting of convex polyhedrons, then
\begin{equation}\label{key3}
N_V+N_F+1=N_E+N_K.
\end{equation}

For each $T\in{\mathcal T}_h$ and any ${\bf v}=\{{\bf v}_0,{\bf v}_b\}\in V_h$, ${\bf v}_0$ is a vector function with three components and each component is a linear function. Therefore
there are twelve linearly independent linear functions $\Phi_{j+1}, \Phi_{j+2},\cdots,\Phi_{j+12}$ in $V_h$ such that they are nonzero only at the interior of element $T$. For each face $f_i\in{\mathcal F}_h^0$, ${\bf v}_b$ is a constant vector function with three component. Thus there are three linearly independent constant vector functions $\Psi_{i,1}$, $\Psi_{i,2}$ and $\Psi_{i,3}$ in $V_h$ which take nonzero value only on the face $f_i$. Then it is easy to see that
\begin{equation}\label{basis3}
V_h={\rm span}\{\Phi_1,\cdots, \Phi_{12N_K},\Psi_{1,1},\Psi_{1,2},\Psi_{1,3}\cdots,\Psi_{N_F,1},\Psi_{N_F,2},\Psi_{N_F,3}\}.
\end{equation}

Since the dimension for pressure space $W_h$ is $N_K-1$, (\ref{key3}) implies
\begin{eqnarray}
\rm{dim} (D_h)&=&\rm{dim} (V_h)-\rm{dim}(W_h)\label{bbb3}\\
&=&12N_K+3N_F-N_K+1=12N_K+2N_F+N_E-N_V.\nonumber
\end{eqnarray}

\begin{lemma}
The  functions $\Phi_1,\cdots, \Phi_{12N_K}$  in (\ref{basis3}) are in $D_h$ and linearly independent.
\end{lemma}

\smallskip

\begin{proof}
Let $\Phi_i=\{\Phi_{i,0},\Phi_{i,b}\}$. The definition of $\Phi_i$ implies $\Phi_{i,b}=0$.
 For any $q\in W_h$, it follows from (\ref{d-d}), $\nabla q=0$ and $\Phi_{i,b}=0$ that for any $q\in W_h$,
\begin{eqnarray*}
b(\Phi_i,q)&=&\sum_{T\in{\mathcal T}_h} (\nabla_w\cdot\Phi_i,q)_T\\
&=&\sum_{T\in{\mathcal T}_h}(-(\Phi_{i,0},\nabla q)_T+{\langle} \Phi_{i,b}\cdot{\bf n}, q{\rangle}_{\partial T})\\
&=&0,
\end{eqnarray*}
which finished the proof of the lemma since the linear independence of  $\Phi_1,\cdots, \Phi_{12N_K}$ is obvious.
\end{proof}

\smallskip

For any face $f_i\in{\mathcal F}_h^0$, let ${\bf n}$  be a unit normal vector of $f_i$ and let ${\bf t}_{1}$ and ${\bf t}_{2}$ be two linearly independent unit tangential vectors to the face $f_i$. Define  $\Upsilon_{i,1}=C_{1,1}\Psi_{i,1}+C_{1,2}\Psi_{i,2}+C_{1,3}\Psi_{i,3}$ and $\Upsilon_{i,2}=C_{2,1}\Psi_{i,1}+C_{2,2}\Psi_{i,2}+C_{2,3}\Psi_{i,3}$ such that $\Upsilon_{i,1}|_{f_i}={\bf t}_{1}$ and $\Upsilon_{i,1}|_{f_i}={\bf t}_{2}$ respectively. Obviously $\Upsilon_{i,1}$ and $\Upsilon_{i,2}$ are in $V_h$ and only nonzero on $f_i$.

\smallskip

\begin{lemma}
Functions  $\Upsilon_{1,1},\Upsilon_{1,2}\cdots,\Upsilon_{N_F,1}, \Upsilon_{N_F,2}\in V_h$ are in $D_h$ and linearly independent.
\end{lemma}

\smallskip

\begin{proof}
Let $\Upsilon_{i,j}=\{\Upsilon_{i,j,0},\Upsilon_{i,j,b}\}$ with $j=1,2$. For any $q\in W_h$, it follows from (\ref{d-d}) and $\nabla q=0$ for $j=1,2$,
\begin{eqnarray*}
b(\Upsilon_{i,j},q)&=&\sum_{T\in{\mathcal T}_h} (\nabla_w\cdot\Upsilon_{i,j},q)_T\\
&=&\sum_{T\in{\mathcal T}_h}(-(\Upsilon_{i,j,0},\nabla q)_T+{\langle} \Upsilon_{i,j,b}\cdot{\bf n}, q{\rangle}_{\partial T})\\
&=&\sum_{T\in{\mathcal T}_h}{\langle} \Upsilon_{i,j,b}\cdot{\bf n}, q{\rangle}_{\partial T}\\
&=&0,
\end{eqnarray*}
where we use the fact ${\bf t}_{j}\cdot{\bf n}=0$ with $j=1,2$.
Since $\Upsilon_{i,j}$ is only nonzero on $f_i$, $\Upsilon_{1,1},\Upsilon_{1,2}\cdots,\Upsilon_{N_F,1}, \Upsilon_{N_F,2}$ are linearly independent. We completed the proof.
\end{proof}

\begin{figure}
\begin{center}
\includegraphics[width=6cm]{divfree3D.eps}
\caption{A 3D Hull ${\cal S}_{E_i}$ .}  \label{fig2}
\end{center}
\end{figure}

\smallskip

For a given interior edge $E_i\in {\cal E}_h$,  assume there are $r$ elements having $E_i$ as one of their edges which form a solid denoted by ${\cal S}_{E_i}$ shown in Figure \ref{fig2}. Then there are $r$ interior faces $f_j$ ($j=1,\cdots, r$) in ${\cal S}_{E_i}$. Let ${\bf n}_j$ be a unit normal vector on the face $f_j$ such that normal vectors ${\bf n}_{j}$ $j=1,\cdots,r$ form oriented loop around the interior edge $E_i$ shown in Figure \ref{fig2}.  For each $f_j$, let $\Psi_{j,1}$, $\Psi_{j,2}$ and $\Psi_{j,3}$ be the three basis functions of $V_h$ which are only nonzero on $f_j$. Define $\Theta_j=C_1\Psi_{j,1}+C_2\Psi_{j,2}+C_3\Psi_{j,3}\in V_h$ such that $\Theta_j|_{f_j}={\bf n}_{j}$. Define $\Lambda_i=\sum_{j=1}^r \frac{1}{|f_j|}\Theta_j$.

\smallskip

\begin{lemma}
Functions $\Lambda_1,\cdots,\Lambda_{N_E}\in V_h$ are in $D_h$ .
\end{lemma}

\smallskip

\begin{proof}
Let $\Lambda_i$ and $q_j$ be associated with hull ${\cal S}_{E_i}$ and element $T_j$ respectively.
If $T_j\in{\mathcal T}_h$ is not in ${\cal S}_{E_i}$, we easily have $b(\Lambda_i,q_j)=0$. If $T_j\in {\cal S}_{E_i}$, let faces $f_s$ and $f_{s+1}$ be its two faces in ${\cal S}_{E_i}$ shown in Figure \ref{fig2},
\begin{eqnarray*}
b(\Lambda_i,q_j)&=&\sum_{T\in{\mathcal T}_h} (\nabla_w\cdot\Lambda_i,q_j)_T\\
&=& (\nabla_w\cdot\Lambda_i,q_j)_{T_j}\\
&=&-(\Lambda_{i,0},\nabla q)_{T_j}+{\langle} \Lambda_{i,b}\cdot{\bf n}, q{\rangle}_{{\partial T}_j}\\
&=&\int_{f_s}\frac{1}{|f_s|}{\bf n}_{f_s}\cdot{\bf n}+\int_{f_{s+1}}\frac{1}{|f_{s+1}|}{\bf n}_{f_{s+1}}\cdot{\bf n}\\
&=&0
\end{eqnarray*}
We proved $\Lambda_i\in D_h$ for $i=1,\cdots, N_E$.
\end{proof}

\smallskip

Unfortunately, $\Lambda_1,\cdots,\Lambda_{N_E}$ are linearly dependent. Let $P_i$ be an interior vertex in ${\mathcal T}_h$ and ${\cal G}_{P_i}$ be a hull formed by the elements $T\in{\mathcal T}_h$ sharing $P_i$. Let $e_j\in{\mathcal E}_h^0,\;j=1,\cdots,t$ with $P_i$ as one of its end point and $\Lambda_j,\; j=1,\cdots, t$ be the discrete divergence free functions associated with $e_j$. With appropriate choosing ${\bf n}_j$ in defining $\Lambda_j$, one can prove that $\sum_{j=1}^t\Lambda_j=0$. However, if we eliminate one function from $\{\Lambda_1,\cdots,\Lambda_t\}$ randomly, say $\Lambda_1$, we will prove that $\{\Lambda_2,\cdots,\Lambda_t\}$ are linearly independent in the following lemma.

\begin{lemma}\label{lemma9}
Functions $\Lambda_2,\cdots,\Lambda_t$ are linearly independent.
\end{lemma}

\smallskip

\begin{proof}
Let $f\in{\mathcal F}_h^0$ be an interior face in ${\cal G}_{P_i}$ with $e_1$ and $e_2$ as its two edges in ${\cal G}_{P_i}$. The definition of $\Lambda_j$ implies that only $\Lambda_1$ and $\Lambda_2$ are nonzero on $f$. Suppose that there exist constants  $c_2,\cdots, c_{t}$ such that $\sum_{i=2}^{t}c_i\Lambda_i=0$.  Then we have
\[
0=\sum_{i=2}^{t}\int_{f}c_i\Lambda_i=\int_{f}c_2\Lambda_2=\int_{f}c_2\Theta_f=c_2{\bf n}_{f},
\]
which implies $c_2=0$. By this way, we can prove that all $c_i=0$ and $\Lambda_2,\cdots,\Lambda_{t}$ are linearly independent.
\end{proof}

We start with $\{\Lambda_1,\cdots,\Lambda_{N_E}\}$ and eliminate one function for each ${\cal G}_{P_i}$ for $i=1,\cdots, N_V$. With renumbering the functions, we end up with $N_E-N_V$ discrete divergence free functions: $\{\Lambda_1,\cdots,\Lambda_{N_E-N_V}\}$.

\smallskip

\begin{lemma}\label{lemma1}
Functions $\{\Lambda_1,\cdots,\Lambda_{N_E-N_V}\}$ are linearly independent.
\end{lemma}

\begin{proof}
The proof of the lemma is similar to the proofs of Lemma \ref{lemma3} and Lemma \ref{lemma9}.
\end{proof}

\smallskip

\begin{theorem}
Let $D_h$ be defined in (\ref{D_h}). Then for  three dimensional space, $D_h$ is spanned by the following basis functions,
\begin{equation}\label{main3}
D_h={\rm Span}\{\Phi_1,\cdots, \Phi_{12N_K}, \Upsilon_{1,1},\Upsilon_{1,2},\cdots, \Upsilon_{N_F,1},\Upsilon_{N_F,2},\Lambda_1,\cdots,\Lambda_{N_E-N_V}\}.
\end{equation}
\end{theorem}
\begin{proof}
The number of the functions in the right hand side of (\ref{main3}) is $12N_K+2N_F+N_E-N_V$ which is equal to ${\rm dim}(D_h)$ due to (\ref{bbb3}).
Similar to the proof of Theorem \ref{thm1}, we can prove that  $\{\Phi_1,\cdots, \Phi_{6N_K}, \Upsilon_{1,1},\Upsilon_{1,2},\cdots, \Upsilon_{N_F,1},\Upsilon_{N_F,2},\Lambda_1,\cdots,\Lambda_{N_E-N_V}\}$ are linear independent.
 \end{proof}

\section{Numerical Experiments}
In this section, we shall report several results of numerical examples for  two dimensional Stokes equations. The divergence-free finite element scheme introduced in Algorithm \ref{algorithm2} is used. The main purpose if to numerically validate the accuracy and efficiency of the WG scheme.

Let ${\bf v}_h\in D_h$ and $q_h\in W_h$, the error for the WG-FEM solution is measured in three norms defined
as follows:
\begin{eqnarray*}
{|\hspace{-.02in}|\hspace{-.02in}|} {\bf v}_h {|\hspace{-.02in}|\hspace{-.02in}|}^2:&=&\sum_{T\in\mathcal{T}_h}\bigg(\int_T|\nabla_w {\bf v}_h|^2dT+
+h_T^{-1}\int_{\partial T}({\bf v}_0-{\bf v}_b)^2ds\bigg),\qquad(\mbox{A discrete $H^1$-norm}),\notag\\
\|{\bf v}_0\|^2:&=&\sum_{T\in\mathcal{T}_h}\int_T|{\bf v}_0|^2dx,\qquad\qquad\qquad\qquad  (\mbox{Element-based $L^2$-norm})
\end{eqnarray*}

\subsection{Test case 1}
\begin{figure}[!htb]
\centering
\begin{tabular}{c}
  \resizebox{2.2in}{2.2in}{\includegraphics{ex2_derectV2.eps}}
\end{tabular}
\caption{Example 1: Level 1 of mixed polygonal mesh.}\label{fig:ex1_1}
\end{figure}

\begin{table}[h]
\caption{Test Case 1: Numerical error and convergence rates for the
Stokes equation with homogeneous boundary
conditions on the uniform rectangular meshes.}\label{ex1_rect} \center
\begin{tabular}{|c|cc|cc|}
\hline
$h$ & ${|\hspace{-.02in}|\hspace{-.02in}|} {\bf u}_h-Q_h {\bf u}{|\hspace{-.02in}|\hspace{-.02in}|}$&order & $\|{\bf u}_0-Q_0 {\bf u}\|$ &order \\
\hline\hline
   1/4     &8.1050e-01 &            &2.9957e-01  &        \\ \hline
   1/8     &6.9698e-01 &2.1769e-01  &9.9634e-02  &1.5882  \\ \hline
   1/16    &4.4578e-01 &6.4479e-01  &3.1031e-02  &1.6829  \\ \hline
   1/32    &2.4452e-01 &8.6638e-01  &8.5507e-03  &1.8596  \\ \hline
   1/64    &1.2620e-01 &9.5424e-01  &2.2131e-03  &1.9500  \\ \hline
   1/128   &6.3751e-02 &9.8519e-01  &5.5968e-04  &1.9834  \\ \hline
\end{tabular}
\end{table}

\begin{table}[h]
\caption{Test Case 1: Numerical error and convergence rates for the
Stokes equation with homogeneous boundary
conditions on the mixed polygonal meshes.}\label{ex1_derect} \center
\begin{tabular}{|c|cc|cc|}
\hline
$h$ & ${|\hspace{-.02in}|\hspace{-.02in}|} {\bf u}_h-Q_h {\bf u}{|\hspace{-.02in}|\hspace{-.02in}|}$&order & $\|{\bf u}_0-Q_0 {\bf u}\|$ &order \\
\hline\hline
   4.1016e-01   &8.1917e-01 &            &3.0927e-01 &      \\ \hline
   2.0508e-01   &7.0386e-01 &2.1887e-01  &1.0421e-01 &1.5694\\ \hline
   1.0254e-01   &4.6002e-01 &6.1359e-01  &3.3478e-02 &1.6382\\ \hline
   5.1270e-02   &2.5560e-01 &8.4781e-01  &9.4392e-03 &1.8265\\ \hline
   2.5635e-02   &1.3230e-01 &9.5007e-01  &2.4560e-03 &1.9424\\ \hline
   1.2818e-02   &6.6890e-02 &9.8401e-01  &6.2217e-04 &1.9810\\ \hline
\end{tabular}
\end{table}

The domain is set as $\Omega=(0,1)\times(0,1)$. Let the exact solution ${\bf u}$ and $p$ as follows,
\begin{eqnarray*}
{\bf u}=\begin{pmatrix}
10x^2y(x-1)^2(2y-1)(y-1)\\
-10xy^2(2x-1)(x-1)(y-1)^2
\end{pmatrix}
\mbox{ and }p=10(2x-1)(2y-1).
\end{eqnarray*}
It is easy to check that homogeneous Dirichlet boundary condition is satisfied for this testing. The right hand side function $f$ is given to match the exact solutions.

The first test shall be performed on the uniform rectangular meshes and the mixed polygonal meshes. The uniform rectangular meshes are generated by partition the domain $\Omega$ into $n\times n$ sub-rectangles. The mesh size is denoted by $h=1/n.$ Moreover, the WG divergence free algorithm is also test on the mixed polygonal type meshes. We start with the initial mesh shown as the Figure \ref{fig:ex1_1}, which contains the mixture of triangles and quadrilaterals. The next level of mesh is to refine the previous level of mesh by connecting the mid-point on each edge. The mesh size in this case is also denoted by $h$.

The error profile is reported in Table \ref{ex1_rect}-\ref{ex1_derect} for the rectangular meshes and mixed polygonal meshes, respectively. Both of the tables show the same convergence rate as the theoretical conclusion, which is $O(h)$ in the $H^1-$norm and $O(h^2)$ in the $L^2-$norm.

\subsection{Test case 2} The domain is given by $\Omega=(0,1)\times(0,1)$. Let the exact solutions ${\bf u}$ and $p$ as follows,
\begin{eqnarray*}
{\bf u}=\begin{pmatrix}
x(1-x)(1-2y)\\
-y(1-y)(1-2x)
\end{pmatrix},
\mbox{ and }p=2(y-x).
\end{eqnarray*}
The Dirichlet boundary condition and the right hand side function is set to match the above exact solutions. It is easy to check that the exact solution ${\bf u}$ satisfies the non-homogeneous boundary condition.

For this testing, the WG divergence free algorithm is perform on the triangular grids. The uniform triangular girds are generated by: (1) partition the domain into
$n\times n$ sub-rectangles; (2) divide each square element into two
triangles by the diagonal line with a negative slope. The mesh size
is denoted by $h=1/n.$

For the calculation of the pressure $p_h$, we shall make use of the basis function ${\bf v}\in V_h\backslash D_h$. This basis function is corresponding to the velocity ${\bf v}_b$ related of the normal direction on each edge. Let ${\bf v}\in V_h\backslash D_h$, the pressure $p_h$ is computed as follows,
$$b({\bf v},p_h)=a({\bf u}_h,{\bf v})-(f,{\bf v}_0).$$

Beside testing two norms of the error in velocity, we also measure the $L^2-$error in pressure. The numerical results in Table \ref{ex2_tri} show an $O(h)$ convergence in the ${|\hspace{-.02in}|\hspace{-.02in}|}\cdot{|\hspace{-.02in}|\hspace{-.02in}|}$ norm for velocity, $O(h^2)$ convergence in the $L^2$-norm for velocity, and $O(h)$ convergence in the $L^2-$norm for pressure, which are confirmed by Theorem \ref{h1-bd}.

\begin{table}[h]
\caption{Test Case 2: Numerical error and convergence rates for the
Stokes equation with non-homogeneous boundary
conditions.}\label{ex2_tri} \center
\begin{tabular}{|c|ccc|}
\hline
$h$ & ${|\hspace{-.02in}|\hspace{-.02in}|} {\bf u}_h-Q_h {\bf u}{|\hspace{-.02in}|\hspace{-.02in}|}$ & $\|{\bf u}_0-Q_0 {\bf u}\|$ &$\|p_h-p\|$ \\
\hline\hline
   2.5000e-01   &2.8901e-01   &4.2990e-02   &2.2624e-01\\ \hline
   1.2500e-01   &1.4367e-01   &1.0896e-02   &1.2246e-01\\ \hline
   6.2500e-02   &7.1997e-02   &2.7432e-03   &6.4525e-02\\ \hline
   3.1250e-02   &3.6052e-02   &6.8773e-04   &3.3224e-02\\ \hline
   1.5625e-02   &1.8037e-02   &1.7210e-04   &1.6871e-02\\ \hline
   7.8125e-03   &9.0202e-03   &4.3038e-05   &8.5037e-03\\ \hline
Conv.Rate       &9.9966e-01   &1.9934       &9.4871e-01\\ \hline
\end{tabular}
\end{table}

\section*{Acknowledgement}
We offer our gratitude to professor Eric Lord and professor David
Singer for their help on obtaining Equation (\ref{key3}).

\vfill\eject

redpajama_set_name: RedPajamaArXiv


Jessica Springsteen and Tiger Lily competing at the Longines Masters of Paris Riders Masters Cup.
Riders Europe prevailed at the first leg of the 2018/2019 Riders Masters Cup season at the Longines Masters of Paris.
The Longines Masters of Hong Kong has just finished and the EEM World teams are already busy with the next stage of the Series: the Longines Masters of New York, from April 25 to 28, 2019, at the NYCB LIVE, Home of the Nassau Veterans Coliseum. The event will feature the two flagship events specific to the series: the Longines Speed Challenge and the Longines Grand Prix, but New York, like Paris, will also host a very special event.
This event is the Riders Masters Cup, a new intercontinental championship that has pitted the United States (Riders USA) against Europe (Riders Europe) since December 2017. The Longines Masters of New York will host the fourth edition of the Riders Masters Cup, which has been dominated by the Riders Europe team since its inception. The Riders USA team is determined to add a win to their record on April 27th at the NYCB LIVE, Home of the Nassau Veterans Coliseum. They will have the home team advantage in front of an audience that is committed to their mission. The crowd are more than just spectators, they are enthusiastic supporters in cheering on Riders USA. With world No. 2, McLain Ward, another top-10 rider in Beezie Madden, and a total of five riders in the top 25 entered from the U.S., it is obvious that for this fourth round, the reigning World Champions will have some serious artillery to combat the Riders Europe team.
The latter team keeps an essential asset: their team captain. Even though Switzerland's Philippe Guerdat now serves as national chef d'équipe of Brazil instead of France, he remains captain of the Riders Europe team.
And it is true that the champion Olympic coach played an essential role in the Riders Europe team's first three victories, both in terms of his selections and the strategy deployed during the competition. This was particularly evident in New York last year, where the Riders Europe team dominated throughout the event, winning the match in the last duel in which Guerdat lined up Harrie Smolders and his fast Cas 2 against McLain Ward and HH Gigi's Girl. The Riders USA team was still leading 7-6, but after Smolders' lightning fast round, Riders Europe took over for the win with a final score of 8-7. Guerdat's smart choice had paid off!
Launched in Paris in December 2017, the Riders Masters Cup has taken on a new dimension as a major intercontinental sporting challenge. Ultimately, the Riders Masters Cup is destined to become one of the greatest team battles between the continents such as the America's Cup in sailing, the Ryder Cup in golf, or the Davis Cup and Fed Cup in tennis. Of course, for the time being, the Riders Masters Cup is still writing its history, but at the first three editions, the event fascinated audiences in Paris and New York.
Whether you're looking back at the past season or further back in this sport, Europe and the United States have won most of the major team titles: the Olympic Games, World Championships or Nations' Cup finals. While the United States won the World Equestrian Games™ Tryon (USA) last year, the Old Continent immediately took its revenge in Barcelona at the Longines FEI Jumping Nations Cup™ Final with a 100% European podium (Belgium, France and Ireland). The Riders Masters Cup, created in 2016 by EEM and EEF (the European Equestrian Federation), continues this relentless battle between the two equestrian powers.
Robert Ridland serves as the current chef d'équipe of the Riders USA team and also was chef d'équipe of the US team that claimed gold at the 2018 WEG. For Europe, the chef d'équipe role has been entrusted to the head of the French gold medal team at the 2016 Rio Olympic Games, who is now the Brazilian team coach, Guerdat. The World Champion coach against the Olympic Champion coach! Team captains announce their short lists about a month before the event, while the final team of five riders, including one under 25 years of age, will be announced the day before the event. But the role of team captain goes far beyond selection alone. They also plan strategy, choosing the starting order of their riders. Then, depending on the results of the first round, the winning coach gets first choice of which rider he will line up against which opponent, and can even change his riders' horses. They play a crucial role as strategist, which undoubtedly influences the result, as we saw in New York where the result depended on the very last match-up!


text:
class MainWindow : public QWidget
{
    typedef enum
    {
        MENU_MUSIC=0,
        MENU_MF,
        MENU_PTHONE,
        MENU_HOME,
        MENU_APPLIST,
        MENU_SETTING,
        MENU_MAX
    }MENU_APP_TYPE;

    Q_OBJECT
public:
    explicit MainWindow(AppListInterface * pList,QWidget *parent = 0);
    ~MainWindow();
    QWidget* CenterWidget();
    //void InsertChildApp(int index,int funcId,QString text,QString on,QString off);
    //void DeleteChildApp(int index);
//    void ExeRPC();

    void StartVideoStream(const char* url);
    void StopVideoStream();

    bool InVideoStream();

    void BackToVideoStream();

    // add by fanqiang
    void SetSDLStatus(bool bConnect);

protected:
    virtual void showEvent(QShowEvent * e);

signals:

public slots:
    void onUpdateTime();
    void onMenuSelected();
    //void onChildAppSelected(int funcId);

    void OnVideoStreamMenuBtnClicked();
private:
    //VideoStream* videoWidget;
    // add by fanqiang
    QLabel *m_pSDLStatus;
    QLabel *m_pIcon;
    QLabel *m_pTime;
    QLabel *m_pNetStatus;
    QWidget *m_pCenter;
    QWidget  *m_pMainMenu;
    MenuButton  *m_pMenuTab[MENU_MAX];
    QTimer  *m_pTimer;
    //QList<CAppButton*> m_pChildApps;

    AppListInterface *m_pList;
    int  m_AppWidth;
    int  m_AppHeight;

    bool m_bInVideoStream;
    bool m_bsdlStatus;
};

#endif // APPLINKMENU_H

redpajama_set_name: RedPajamaGithub


text:
Real EstateDevelopmentTransportationArchitecturePublic ArtOpinions
DevelopmentUrbanized
$1.4 billion Calgary Cancer Centre expected to be complete by 2023
DH Calgary Staff
Rendering of the Calgary Cancer Centre (screenshot from the New Calgary Cancer Centre YouTube video/AHSChannel)
Something big this way comes.
Alberta Premier Rachel Notley was in Calgary earlier this week to announce that construction of the Calgary Cancer Centre was on time and on budget.
Company selected to construct new Calgary Cancer Centre for 2023
Canada to witness Super Blood Wolf Moon Total Lunar Eclipse this weekend
These were the most popular baby names in Alberta in 2018
And quite the significant budget at that, as the centre holds the title of the largest government infrastructure project in Alberta at $1.4 billion.
"All Albertans deserve top-quality, innovative cancer care," Notley said in a release from the Government of Alberta.
"I'm so excited to see continued progress on the Calgary Cancer Centre that gives hope and increased access to care to patients and families in southern Alberta. We continue to fulfill our promise to provide life-saving health services in communities across the province."
Premier Notley provides an update on construction at the Calgary Cancer Centre (Government of Alberta)
The release also noted that 30% of the concrete needed for the project's construction had already been poured — roughly 125,000 cubic metres or enough to fill 15 Olympic-sized swimming pools.
Once completed, the centre will host a 12-room radiation therapy department, with all rooms — or "vaults" as they're described in the release — featuring 1.8-metre-thick walls that will protect visitors from radiation exposure.
The centre will also see over 100 patient exam rooms, 160 inpatient unit beds, over 100 chemotherapy chairs, research laboratories, onsite parking with 1,650 spots, and additional space for clinical trials.
"I am so proud of our government's investment in this project," said Minister of Infrastructure Sandra Jansen, in the release.
"Our commitment to building vital health care infrastructure means cancer care in Alberta is now among the best in the world."
A rendering of the centre (Alberta Health Services)
The new centre is much-needed in the province, as the release states that Calgary's Tom Baker Cancer Centre reached full capacity in 2003.
In the years since, cancer rates have risen at a rate of 3 to 5% per year, with 54 Albertans discovering that they have cancer per day.
"Staff, patient advisers, and partners have been instrumental in the continued planning and execution of making the Calgary Cancer Centre a world-class facility for Albertans," said Alberta Health Service's CancerControl Alberta senior medical director, Dr. Matthew Parliment, in the release.
"We are very proud of all of the work accomplished to date and we are excited about what is to come."
Tours of a life-sized replica of the treatment rooms will be made available in February, and the full centre is expected to open in 2023 after six years of construction.


text:
Home » Cricket » Lockdown forces Ajinkya Rahane to go back to Karate
Lockdown forces Ajinkya Rahane to go back to Karate
Cricket | April 12, 2020
With everyone confined to their respective homes due to the 21-day curfew in India, the country's Test vice-captain Ajinkya Rahane has been busy away from the field.
Others have found one or two things doing, making it their daily routine during the lockdown put in place to curtail the spread of the coronavirus pandemic.
With some people improving on their culinary skills, Rahane revealed that he has been practicing Karate again, having got a black belt from the sport initially.
He further revealed that the situation has made him lend helping hands to his wife, Radhika as well in the household chores by mostly assisting her in looking after his daughter Arya. Before Arya wakes up in the morning, Rahane would have done his work out, which usually last about 30 to 45 minutes.
In a video of Rahane shared by the Board of Control for Cricket in India's (BCCI) on their social media handles, the Test vice-captain revealed that he further try practicing Karate again three to four times in a week while the rest of the day is dedicated to looking after Arya.
Due to the lockdown, he now has time for his daughter, and he is usually the one that goes to attend to her whenever she wakes up, which reduces the burden on his wife.
So, Rahane has joined other Indian athletes that are excited over the break that has given them time for their family.
Cricketers usually travel from one town to the other while playing the game. Additionally, Rahane also helps his wife in the cleaning and cooking when Arya goes to sleep.
He further revealed that his culinary skills have improved after getting a lot of cooking tips from his wife. Although he likes listening to music and reading books during his spare time, he has more time in his hands now as well. He further revealed that he is reading Swami Parthasarthy's The Holocaust of Attachment at the moment, which usually takes the other part of his days.
The last time Rahane was in action for India was in their Test series against New Zealand. He recently moved to Delhi Capitals from Rajasthan Royals.
Written by Oladipupo Mojeed
Three-time IPL champions Chennai Super Kings have released veteran off-spinner Harbhajan Singh befor... Read More
Indian legend Sunil Gavaskar has heaped praise on the team's number three batsman Cheteshwar Pujara.... Read More
Former Australian captain Ricky Ponting is finding it difficult to comprehend that Australia were no... Read More


I recently designed this wrap-around book cover and some interior vignettes for the same book: "Besonders Wertlos", a collection of film reviews and other articles by Jörg Buttgereit.
The vignettes accompany the respective articles' titles and represent their subjects, and the idea was to have all the vignettes appear somewhere on the cover.
I made about 50 vignettes in total. Some are somewhat hidden on the cover or only represented indirectly, but it should be easy to spot the nine below.


text:
East Bay Express: Fatal System Error
Booker Williams III hadn't missed a day of work for two years. So when the 34-year-old San Quentin State Prison correctional officer got a call on the morning of August 20, 2014 that his father was going to the hospital, he decided not to break his streak. Plus, his 62-year-old father, Booker Williams II, a lifelong Richmond resident, had been relatively healthy in recent years.
But when Williams III started getting more calls from family members about his father, he realized it could be serious. His aunt sounded worried. And an ambulance had taken the elder Williams to a different hospital than usual. In emergencies, Williams II, like most Richmond residents, would typically go to Doctors Medical Center (DMC) in San Pablo, a public hospital that serves West Contra Costa County. But thirteen days earlier, because of an ongoing financial crisis at the hospital, county public health officials and DMC executives had agreed to stop sending local ambulances to the facility.
For Williams II, a retired assembly line worker for mattress manufacturer Sealy, the trouble began at a routine appointment that morning at his primary physician's office in Hercules, just north of DMC. His doctor told him his breathing was off and his heart rate was well over one hundred beats per minute. "The doctor said he could have a heart attack in his office," recalled Betty Crater, Williams II's sister. He needed to get to an emergency room immediately. So they called 911 and an ambulance soon arrived.
But the paramedics had bad news... Read more at the Easy Bay Express.


text:
Q: Git folder merge between branches I have 2 branches

*

*master
|
 _ folder1
|
 _ folder2

*releases
|
 _ folder1
|
 _ folder2
Both have the same folder structure.
How can I merge only 1 folder from master branch to release branch, say folder1

A: Since changes to both folders are independent, the git way would have been to keep the changes in two separate branches, instead of commiting them to the one master branch. That would allow you to merge in just one of the branches.
Now, how can you get that state of two branches?
You will need the commit from that on the changes were developed, i.e. the last commit that was merged into releases. Use git log to find the commit id. This should be a commit on the master branch. Then create a tag there:
git checkout <commit id>
git tag begin_of_new_development

Checkout the master branch and create a new, checked out branch f1:
git checkout master
git checkout -b f1

Then use interactive rebase on the new branch to remove the commits that are not relevant to folder_1:
git rebase -i begin_of_new_development

From the list, remove all commits that don't work on folder_1.
You should now be left with a branch f1 that you can merge into you release branch. If you do the same for folder_2, you can later merge that changes into release, too. Be warned, however, to never leave one commit in both f1 and f2, because rebasing gives them new commit ids, and thus git can not know that they contain the same changes. Also, you can't merge master into release any more.

A: From the discussion in the comments I assume two things:

*

*Changes in folder1and folder2 are independent.

*There are only changes on the master branch (at least to folder1).

Subsequently, the following instructions will overwrite any changes made to folder1 on the release branch. They basically copy the files in folder1 over from master to release.
Check out the release branch.
git checkout release

Then, checkout all files in folder1 from the master branch. Doing so, leaves release the checked out branch. Also, all those file get added to the staging index.
git checkout master -- folder1

Commit the changes.
git commit -m "Copy over changed files in folder1 from master branch."

redpajama_set_name: RedPajamaStackExchange


text:
The Royal wedding 2018
Guests and wedding party
Charity Gift Fund
Coming to Windsor?
How to follow the wedding
On Monday 27 November 2017 The Prince of Wales announced the engagement of his son Prince Harry to Ms. Meghan Markle.
Prince Harry and Ms. Meghan Markle have chosen to release this official portrait photograph to mark their engagement. The photograph was taken by photographer @alexilubomirski earlier this week at Frogmore House, Windsor.
The wedding will take place at St George's Chapel, Windsor on Saturday 19 May.
The service will begin at St George's Chapel at midday and will be conducted by the Dean of Windsor, The Rt Revd. David Conner. The Most Revd. and Rt Hon. Justin Welby, Archbishop of Canterbury, will officiate as the couple make their marriage vows.
At 1pm, following the service, the couple will travel around Windsor in a horse-drawn carriage, providing an opportunity for members of the public to see them and join in with the celebrations.
The Prince of Wales announced the engagement of Prince Harry to Ms. Meghan Markle on 27 November 2017.
Prince Harry informed Her Majesty The Queen and other members of his family. He also sought and received the blessing of Ms. Markle's parents.
The couple will live in Nottingham Cottage at Kensington Palace.
On the afternoon that their engagement was announced, Prince Harry and Ms. Meghan Markle appeared for a photo call in the gardens of Kensington Palace.
The couple took part in a broadcast interview which was aired on the evening of the announcement
The Royal wedding will be a day of celebration in the UK and across the world. The Ministry of Defence alongside the Lord Chamberlain's Office (the department of the Royal Household responsible for organising ceremonial events) are planning a spectacular day of pageantry.
More than 250 members of the Armed Forces will perform ceremonial duties at the wedding.
The Ministry of Defence have released details of the regiments and units which hold a special relationship with Prince Harry and which will provide ceremonial support at the wedding.
Find out more about military involvement in the wedding.
Horses and carriages for the Royal wedding will be provided by the Royal Mews - part of the Royal Household - who provide carriages for other occasions, from Trooping the Colour to Royal Ascot.
Find out more about the horses and carriages which will be used.
Guests watching from inside the castle grounds
Prince Harry and Ms. Meghan Markle have said they want their Wedding Day to be shaped so as to allow members of the public to feel part of the celebrations too.
On 2 March 2018, Kensington Palace announced that 2,640 people will be invited into the grounds of Windsor Castle. Read the Kensington Palace press release in full.
Once invitations had been issued, details of selected individuals were released including young people who have shown strong leadership, and those who have served their communities. Read more about individuals who have been invited.
Around 600 people have been invited to the service at St George's Chapel. Find out more about the invitations.
Kensington Palace have announced that Prince Harry has chosen his brother The Duke of Cambridge to be his Best Man. Read the press release in full.
During a media briefing, the Communications Secretary to Prince Harry announced the involvement of the bride and groom's families in their wedding day, including the attendance of all three siblings of Diana, Princess of Wales, and Ms. Meghan Markle's parents' movements on the day. Read the Communications Secretary's briefing in full.
Prince Harry and Ms. Meghan Markle are incredibly grateful for the goodwill shown to them since the announcement of their engagement and are keen that as many people as possible benefit from this generosity of spirit. The couple have therefore asked that anyone who might wish to mark the occasion considers making a donation to one of seven organisations they would like to support, reflecting their shared values.
Find out more about these seven charities.
Kensington Palace have announced which suppliers have been chosen for some of the key elements of the wedding day:
Claire Ptak, an American pastry chef who is based in London has been chosen to create a lemon elderflower wedding cake for Prince Harry and Ms. Meghan Markle. Find out more.
The invitations
The wedding invitations have been produced by Barnard & Westwood. The company has held a Royal Warrant for Printing & Bookbinding by Appointment to Her Majesty The Queen since 1985. Find out more.
Prince Harry and Ms. Meghan Markle have chosen floral designer Philippa Craddock to create the church flowers for their wedding. Philippa will direct a team, including florists from St George's Chapel and Buckingham Palace, to create the displays at St George's Chapel and for St George's Hall. Find out more.
With the most recent first, here is a comprehensive list of the announcements which have been made regarding the Royal wedding:
4 May 2018 – Briefing from the Communications Secretary regarding involvement in the wedding day from both families.
1 May 2018 - The couple have selected an Ascot Landau carriage for their procession through Windsor Town.
26 April 2018 – The Duke of Cambridge will be Best Man.
10 April 2018 – Details of individuals invited to Windsor Castle for the wedding.
9 April 2018 - The seven organisations chosen by the couple for charitable donations announced.
31 March 2018 - Philippa Craddock to create the floral arrangements for St George's Chapel and St George's Hall.
23 March 2018 - 600 people have been invited to the service. The invitations have been produced by Barnard & Westwood.
20 March 2018 – Claire Ptak to make the Royal wedding cake.
15 March 2018 – Details of military involvement in the wedding day.
2 March 2018 - 2,640 people to be invited into the grounds of Windsor Castle.
12 February 2018 – Timings and overview of the wedding day.
28 November 2017 - The couple will marry in St George's Chapel, Windsor.
27 November 2017 – Prince Harry and Ms. Meghan Markle announce their engagement.
On the day of the wedding, members of the public will be able to watch live footage of the wedding and the carriage procession on big screens located on the Long Walk and in Alexandra Gardens. Food stalls and other facilities will be available.
If you are planning to come along and join in the celebrations, make sure you plan ahead!
Rail operators will be putting on extra trains and there will be park and walk and park and ride services to help people get into town. A 6,000 space pre-booked car parking area will be at the Review Ground, just off the Long Walk. Visitors are encouraged to book all travel arrangements early so they can make the most of their trip.
Visit the Royal Borough of Windsor and Maidenhead's website for the latest information on visiting Windsor for the Royal Wedding.
Visit the South Western Railway and Great Western Railway websites to plan your journey to Windsor.
As Prince Harry and Ms. Meghan Markle get married you'll be able to follow live in many ways.
As well as the live broadcast on the main channels in the UK – BBC One, ITV and Sky – people can follow the Royal Wedding on social media on Facebook, Twitter and Instagram.
You can also watch the wedding live online on our official Royal Family YouTube channel.
An Open Letter from The Duchess of Cambridge to Midwives
The Duchess of Cambridge has written an open letter to midwives across the country
The Duke of Sussex hosts The Rugby League World Cup Draw live from Buckingham Palace
Statement from Her Majesty The Queen
The Duchess of Rothesay presents an honorary degree to The Princess Royal at the University of Aberdeen
Prince Harry's visit to the Caribbean
A statement from Her Majesty The Queen
I offer my heartfelt condolences to the Omani people, and wish to send a personal message of friendship as Oman mourns the death of His Majesty.
A message from The Duke of Cambridge to the people of Oman
Message of condolence from The Queen on the passing of Sultan Qaboos bin Said al Said
A message of condolence from The Queen to the Governor-General of Canada
The Countess of Wessex and Lady Louise meet England Hockey
Statement on discussions with The Duke and Duchess of Sussex
The Countess of Wessex trains for her DofE Diamond Challenge
The Queen's message of condolence to all Australians
Members of the Royal Family remember the war dead.
The Queen and members of The Royal Family on Christmas Day 2019
Behind the scenes- Garden Parties
The Queen's Christmas Broadcast 2019
The Royal Family supports community initiative 'Together at Christmas'
The Queen's Gold Medal for Poetry 2019 awarded to Lorna Goodison
The Duke and Duchess of Cambridge visit India and Bhutan
State Visit by the Emperor and Empress of Japan
The Countess of Wessex takes salute at The Sovereign's Parade
A Royal Recipe: Cinnamon Stars
The Duke of Gloucester- Charities and Patronages
As you climb each step think of the thousands of officers who have gone before you; those who fought and survived, those who fought and died in the many campaigns since Sandhurst began in 1801.
The Countess of Wessex's speech at the Sovereign's Parade, Royal Military Academy Sandhurst
The Princess Royal, Patron of English Rural, opens a new affordable housing development in Kent.
The Queen hosts a reception for NATO leaders
The Queen- Interests
'The Not Forgotten' Christmas Party with The Princess Royal
A message from Her Majesty The Queen following the Terror Attacks at London Bridge
The Queen and the Armed Forces
The Countess of Wessex visits Oxfordshire
Her Majesty The Queen visits The Royal Philatelic Society
The Prince of Wales visits The Solomon Islands
The Queen and Law
The Duke of Gloucester visits Rutland
My generation and those following are acutely aware that we cannot simply carry on as we are.
A speech by The Duke of Cambridge at the Tusk Conservation Awards
Your ability to communicate the beauty and vulnerability of our natural environment remains unequalled
The Queen's speech on awarding Sir David Attenborough the Chatham House Prize
Royal Tour of the Western Balkans
New Zealand has faced up to the most painful periods of her past in a way that offers an example to the world. She has done so with courage, compassion and tolerance – qualities which, it seems to me, define the New Zealand character,
Remarks by HRH The Prince of Wales at Waitangi, New Zealand
The Prince of Wales and The Duchess of Cornwall visit New Zealand
The Duke and Duchess of Gloucester reopen The Princess Alice Garden
The Queen and Government
It's because of all of your efforts that 50,000 young people are getting access to such amazing experiences every single year.
A speech by The Duke of Sussex at the OnSide Awards at the Royal Albert Hall
A statement by His Royal Highness The Duke of York
The Duke of Cambridge visits Kuwait and Oman
Find out more about the programme for His Royal Highness's visit
St James's Palace


There's no right answer when someone asks you: "How old do you think I am?"
Faced with such a dilemma, most of us aim low—erring on the side of flattery rather than honesty. But the truth is, accurately guessing someone's age is a difficult task, perhaps best left to amusement park workers and street performers.
Why is it that some people just look older (or younger) than they really are? Scientists may have found the answer.
Chronological age is very different from biological age—the condition of chromosomes after each cellular division—according to Nilesh Samani of the University of Leicester, co-author of a February 7 report published in Nature Genetics. Biological age, Samani says, is related to the length of telomeres—stretches of DNA at the ends of chromosomes, which protect these precious packages of genes from daily wear and tear. We're born with telomeres of a certain length, and these get shorter as our cells divide, resulting in aging, scientists think.
But not all telomeres are born equal. Telomere length varies from person to person, but is similar in siblings, suggesting it might be under some genetic control. Samani and his colleagues analyzed more than 500,000 genetic variations (naturally occurring, single-nucleotide differences) spanning the genome in blood cells collected from almost 3,000 people. The researchers found that individuals carrying a particular genetic variant had shorter telomeres—meaning less padding for those fragile genes. The variant lies near a gene called telomerase RNA component, or TERC, and earlier studies in animals have shown that low TERC expression is associated with shorter telomeres, and faster biological aging.
The telomeres of people carrying one copy of the variant looked on average three to four years older, according to their lengths, than those of someone the same chronological age without the variant. There also was a gene dosage effect: Two copies of the variant (one inherited from each parent) resulted in an additive aging effect of six to eight years—meaning that a 50-year-old carrying two copies of the variant had the telomeres of someone aged 58. The results suggest that some people are genetically programmed to age at a faster rate, according to study co-leader Tim Spector, from King's College London.
The odds of carrying the variant are relatively high, Samani says. "In the populations we studied, about 7 percent of people carried two copies of the variant and about 38 percent of people [carried] one copy." But it's unknown whether shorter telomeres make their carriers physically appear older than they really are. "We haven't looked at that—it's an interesting question," he says; pondering how to measure the age someone looks in a follow-up study. Maybe we'll take pictures of the participants and a have a panel guess their age," he adds jokingly.
Samani, a cardiologist and professor of cardiology, is more interested in whether people carrying the variant are at a higher risk for developing age-associated illnesses like heart disease. "I see people in their eighties who have very normal arteries, and people in their forties with cardiovascular disease," he says. Previous work published by Samani has linked telomere shortening with heart disease, suggesting that biological age may be more relevant to age-related diseases than chronological age. "We've got a marker for biological aging—telomere length—so we're studying whether we can relate it to the increased risk of getting some of these age-associated diseases. That would certainly put the concept of biological aging on firm foot[ing]."
Samani now plans to investigate how the variant is making telomeres shorter. "It could be regulating TERC, but that's got to be shown in further studies," he says. People carrying the variant might further accelerate the biological aging process if they smoke, are obese or don't exercise—all of which are bad news for telomeres.


text:
Brian L. Sykes
Brian is a partner with Vandeventer Black and concentrates his practice in Workers' Compensation Defense. He works with insurance carriers and self-insured businesses defending workers' compensation claims filed pursuant to the Virginia Workers' Compensation Act and the Longshore and Harbor Workers' Compensation Act. Brian also works with risk managers, human resource professionals and executives on preventative strategies.
Brian received his B.A. from The College of William & Mary, his M.Ed. from Virginia Commonwealth University, and his J.D. from Washington & Lee University School of Law. Before attending law school, Brian was a School Counselor at Charlottesville High School.
He is a member of the American, Virginia and Norfolk & Portsmouth Bar Associations (Workers' Compensation and Labor & Employment Law Sections) and the Virginia State Bar. Brian is an At-Large Member of the Board of the Virginia Association of Defense Attorney's (VADA). He previously served on the Board of Directors for the Norfolk Ronald McDonald House and on the Corporate Advisory Board for the Children's Hospital of The King's Daughters (CHKD). In addition, he served as a member of the Virginia Bar Association's Lifestyle Balance Committee, as the Co-Chair of the Eastern Virginia Pro Bono Legal Aid Hotline for the Young Lawyers' Division of the Virginia Bar Association, and on the Board of the Young Lawyers' Section of the Norfolk & Portsmouth Bar Association.
Maritime & Admiralty Law
B.A., Psychology, College of William and Mary
M.Ed., School Counseling, Virginia Commonwealth University
J.D., Washington & Lee University
Virginia Association of Defense Attorneys – Worker's Compensation Division, At-Large Member
Martindale-Hubbell® AV® Preeminent™ Peer Review Rating
Coastal Virginia Magazine – "Top Lawyers" Workers' Compensation (2016-present)
Listed in The Best Lawyers in America® – Workers' Compensation Law-Employers (2018-present)
Vandeventer Black Attorneys Listed as 'The Best Lawyers in America' of 2019
Twenty-two Vandeventer Black Attorneys Recognized as Top Lawyers
The Application of the Amorphous "Death Presumption" in Virginia Workers' Compensation


Sometimes parenting backwards makes the most sense.
The other day, my daughter forgot where I parked our car in the garage when we visited the library. Inside, I did a little cheer.
Not because I didn't have to climb yet another flight of stairs, arms overloaded with books (ok, a little bit because of that).
But because, after three and a half years together, this tough, resilient little girl lowered her guard for a second. She trusted me to get her the five miles from the library to our home.
For this moment, skipping in a parking garage, she was just a child.
Sometimes, you see, I feel like I am parenting backwards.
With children who are securely attached, parents spend the first eighteen years of their kid's lives helping them become more independent.
My daughter came to me at age four. She was more independent than any other preschooler I have known.
And I've spent every single day since trying to coax and prod her to let me do things for her—even things she can already do for herself.
Because I am her mom. And that's what moms do.
Securely attached children learn in their first years of life that the world is safe. That their needs will be met. That mamas can be trusted.
This is a much harder lesson to learn when that safety has been violated. When those needs have not always been met. When you've learned, of necessity, how to do things for yourself.
As a tiny child, my daughter learned that she must always have an escape route. She learned that she better know exactly where she is at all times and exactly how to get back to someplace familiar. She would never, ever fall asleep in the car. She was always looking, studying, making sure she knew where she was.
And so I turned my head as she insisted that we were on the wrong level of the parking garage. And I smile. Because it meant that she is learning to trust. This is childhood.
I turn my head as she picks out an outrageous outfit—complete with mismatched accessories. And I smile. Because it means she has better things to think about than what her friends at school will say. This is childhood.
I turn my head as this big girl, almost as tall as I am, crawls into my lap and snuggles close. And I smile. We missed four years of this physical affection. But we are making up for it now. I sniff her coconut-smelling hair and nestle my chin on top of her head. This is childhood.
I will keep on parenting backwards.
Independence is good. But today, for us, attachment is better. Trust is better. Childhood is better.


text:
Research states that prejudice comes from a basic human need and way of thinking
By Divya Menon, American Psychiatric Association
(Medical Xpress) -- Where does prejudice come from? Not from ideology, say the authors of a new paper. Instead, prejudice stems from a deeper psychological need, associated with a particular way of thinking. People who aren't comfortable with ambiguity and want to make quick and firm decisions are also prone to making generalizations about others.
In a new article published in Current Directions in Psychological Science, a journal of the Association for Psychological Science, Arne Roets and Alain Van Hiel of Ghent University in Belgium look at what psychological scientists have learned about prejudice since the 1954 publication of an influential book, The Nature of Prejudice by Gordon Allport.
People who are prejudiced feel a much stronger need to make quick and firm judgments and decisions in order to reduce ambiguity. "Of course, everyone has to make decisions, but some people really hate uncertainty and therefore quickly rely on the most obvious information, often the first information they come across, to reduce it" Roets says. That's also why they favor authorities and social norms which make it easier to make decisions. Then, once they've made up their mind, they stick to it. "If you provide information that contradicts their decision, they just ignore it."
Roets argues that this way of thinking is linked to people's need to categorize the world, often unconsciously. "When we meet someone, we immediately see that person as being male or female, young or old, black or white, without really being aware of this categorization," he says. "Social categories are useful to reduce complexity, but the problem is that we also assign some properties to these categories. This can lead to prejudice and stereotyping."
People who need to make quick judgments will judge a new person based on what they already believe about their category. "The easiest and fastest way to judge is to say, for example, ok, this person is a black man. If you just use your ideas about what black men are generally like, that's an easy way to have an opinion of that person," Roets says. "You say, 'he's part of this group, so he's probably like this.'"
It's virtually impossible to change the basic way that people think. Now for the good news: It's possible to actually also use this way of thinking to reduce people's prejudice. If people who need quick answers meet people from other groups and like them personally, they are likely to use this positive experience to form their views of the whole group. "This is very much about salient positive information taking away the aversion, anxiety, and fear of the unknown," Roets says.
Roets's conclusions suggest that the fundamental source of prejudice is not ideology, but rather a basic human need and way of thinking. "It really makes us think differently about how people become prejudiced or why people are prejudiced," Roets says. "To reduce prejudice, we first have to acknowledge that it often satisfies some basic need to have quick answers and stable knowledge people rely on to make sense of the world."
Ironic Effects of Anti-Prejudice Messages
Provided by American Psychiatric Association
Citation: Research states that prejudice comes from a basic human need and way of thinking (2011, December 20) retrieved 28 January 2022 from https://medicalxpress.com/news/2011-12-states-prejudice-basic-human.html
Mental simulations of social thought and action
Is there a hidden bias against creativity?
How your brain reacts to mistakes depends on your mindset
The first step to change: Focusing on the negative
Sexism and gender inequality
Many adults inaccurately perceive their own BMI and body size
Maternal consumption of soft drinks may be linked to ADHD symptoms in offspring
Psychiatric disease associated with increased risk of death from cardiovascular disease and diabetes
Blood markers can predict depression in pregnancy


text:
<?xml version="1.0" encoding="utf-8"?>
<resources>
    <string name="app_name">glass-lowfreq-mqtt-sensor</string>
    <string name="hello_world">Hello world!</string>
    <string name="action_settings">Settings</string>
    <string name="sensor_name">My MQTT Arduino</string>
    <string name="custom_keyword">mqtt sensor</string>
</resources>


Buy tickets to the event here!
On Sunday, October 28, 2018 DAWG will be hosting our fourth annual Wine for Wags fundraiser. We're excited to announce our new venue: The Santa Barbara Carriage Museum. Mingle with friends old and new, enjoy music, a delicious meal, libations and a silent auction to benefit our DAWGs.
DAWG is a non-profit 501(c) (3) corporation, which operates solely on the generosity of our community. We support our dogs through funds gathered from donations, grants, sponsors, and fundraising events. Without your support we cannot continue with our life-saving work. DAWG provides a safe and loving environment for dogs awaiting permanent adoption, and promotes their adoption through outreach programs in the Santa Barbara community.
2017 marked DAWG's 26th year of service to the dogs and families of Santa Barbara and the surrounding areas. Through the hard work, dedication, and generosity of wonderful volunteers and community members we have grown from literally nothing, to an amazing facility that has placed over 8,000 dogs into forever homes. We save lives every day. Without DAWG, hundreds, even thousands, of adoptable dogs would have died unnecessarily. Over the last 25 years we have built kennels, an office, a bathing station, isolation and quarantine areas, Little Dog Land, Happy Sutherlin's Meet & Greet Garden, exercise yards, surgical rooms and a veterinary facility to help us care for our dogs and give them love and shelter while we find them their forever homes. We are currently seeking sponsors, food, wine and beer donors and donations for our silent auction.
Interested in being a sponsor or gracious donor?


text:
Q: What stack does a vectored exception handler use (Windows)? I have a toy programming language that uses a separate stack for each coroutine/fiber/thread. The stack is initially allocated using MEM_RESERVE via VirtualAlloc on Windows, to reduce committed memory. I've added a vectored exception handler that catches page faults that occur when the stack grows, and calls VirtualAlloc again on these pages with MEM_COMMIT.
The problem I have is that my vectored exception handler (set up via AddVectoredExceptionHandler) is not running. I suspect this is due to a double stack fault. The theory is that the program runs out of MEM_COMMIT stack, and starts the exception handler. Since the exception handler uses the same stack (possibly?) it also immediately faults, and thus never runs.
So, does the handler use the same stack as the coroutine/fiber that faulted? Is there any way to make vectored exception handlers use a special stack? (On Unix-like systems, I use sigaltstack for this purpose).

A: Reading the VC2008 documentation:

Vectored Exception Handling
Vectored handlers are not frame-based, therefore, you can add a handler that will be called regardless of where you are in a call frame.

The "regardless of where you are in a call frame" makes me expect that the exception handler uses the current stack (sp register).
To make sure, reserve some space before "the program runs out of MEM_COMMIT stack" to allow the handler to use the stack, and use a debugger to check the  assembler sequence of instructions and the registers.


text:
Protomicroplitis glaphyra är en stekelart som först beskrevs av De Saeger 1944.  Protomicroplitis glaphyra ingår i släktet Protomicroplitis och familjen bracksteklar. Inga underarter finns listade i Catalogue of Life.

Källor

Bracksteklar
glaphyra

redpajama_set_name: RedPajamaWikipedia


We are delighted to introduce you to our school.
Lord Grey opened in 1973 as a community comprehensive school following the amalgamation off Bletchley Grammar and Wilton Secondary schools. New accommodation was provided to supplement the existing facilities. The school was incorporated as a Grant Maintained School on 1st April 1993 and moved to Foundation status on 1st September 1999, under the New Framework for Schools. Today the school is an Academy and has a strong Governing Board and a super reputation in the local community. It has enjoyed a period of calm and stability in the last few years and this has allowed it to focus on teaching and learning, quality staffing, excellent pastoral care and first rate safeguarding. The school is always striving to deliver its very best exam results in public exams for Year 11 and the Sixth Form.
£7.5 million capital improvement grant.
Achieving the Investor in People standard on our first attempt.
Specialist College status (MFL) gained.
This school is an exciting place to be, in which to learn and teach; the facilities have been improved and enhanced. As the school grows, so too will the opportunities for our students and staff.