text:
Call for self-defence: ACF, Middle Belt back Masari, Ohanaeze flays gov
August 19, 2021 easitimes	134 Views 0 Comments	INSECURITY, SELF-DEFENCE
NORTHERN socio-political organization, the Arewa Consultative Forum and the Middle Belt Forum have put their weight behind the call by Katsina State Governor, Bello Masari, on the people of his state to arm and defend themselves against bandits.
Masari had on Tuesday advised the people of the state to arm and defend themselves against bandits.
The governor said, "It is morally wrong for people to submit meekly to the bandits without any attempt to defend themselves. Security is everybody's business.
"It is the people's meek submission that emboldens the bandits to continue with their heinous activities with murderous frequency. We must free our minds of the notion that security is the sole responsibility of the government."
But the ACF on Wednesday joined the Katsina governor to ask northerners to take up arms and defend themselves against the rampaging bandits in the region.
The forum's National Publicity Secretary, Emmanuel Yawe, told one of our correspondents in a chat on Wednesday that the organisation was in full support of Masari' s order, adding that "Governor Masari has said the right thing".
Explaining that Masari's call mandating Katsina people to arm themselves against bandits terrorising communities in the state was in order, Yawe argued that it had been proven that bandits dreaded local hunters more than the regular forces.
The ACF spokesman said, "Not all weapons are banned. In the rural areas, some people are armed with bows and arrows. Even guns are allowed where police permit is issued. The arms are legal when such permits are issued.
"Do not also forget that local hunters in Adamawa gave Boko Haram hell. Boko Haram dreaded them more than the combined team of army and police.
"The locals know the terrain more than the army and police, who are assembled from all over the country in line with the Federal Government policy on recruitments into government agencies. No wonder they (local hunters) were recruited to join the fight against Boko Haram. Governor Masari has said the right thing."
Buy weapons for people, MBF tells Masari
On its part, the Middle Belt Forum expressed the need for government to buy and equip the people with weapons to enable them to defend themselves against terrorists' attacks
The National President of the MBF, Dr Bitrus Pogu, told The PUNCH while reacting to Masari's call on the people of his state to arm and defend themselves against bandits.
Pogu, who described the call as a welcome development, said, "If the governor has come to the reality of the security challenge facing the people, then let him go a step further and help the people to defend themselves.
"This is necessary because there is a problem here. No doubt, the people are ready to defend themselves, but do they have the resources to acquire weapons? The answer is no. So, the whole thing reverts back to the government.
"If the governor is serious about what he has said, let him help the communities that are vulnerable to acquire weapons and license them in such a way that vigilantes in those communities would be adequately armed to enable them to face any security threat by the terrorists in their communities. If he can do that, I believe the people will be able to defend themselves."
We'll challenge Southern Kaduna arms ban in court, says SOKAPU
Also, the Southern Kaduna Peoples Union said self-defence remained the constitutional and universal rights of citizens.
The spokesperson for SOKAPU, Luka Binniyat, said but for the brutal clampdown on firearms on Southern Kaduna, the destruction of lives and property in the area would have been minimal.
He added that the umbrella body of the people of the southern part of Kaduna would soon approach the court on the matter.
Binniyat said, "The truth is that self-defence is a constitutional and universal right. So, if an attacker comes to you with a gun? You surely need a gun to defend yourself against him.
"SOKAPU believes that had a brutal clampdown on firearm use not been imposed on Southern Kaduna, the huge loss of lives and property that we have suffered in the past years would have been minimal.
"We, therefore, totally back the advice by Masari for communities to purchase arms and defend themselves.
"Very soon, we shall go to court over this matter on whether or not citizens can defend themselves and whether such right includes the freedom to purchase arms for the exercise of such right."
Call dangerous, Nigeria heading for doom – Ohanaeze Ndigbo
However, the Igbo apex socio-cultural organisation, Ohanaeze Ndigbo, described the call by the Katsina State governor as a dangerous trend, adding that if the instrument of violence was spread among the citizenry, the country would be heading for doom.
The spokesman for Ohanaeze, Chief Alex Ogbonnia, who spoke with one of our correspondents, said instead of calling for citizens to arm themselves, the Federal Government should sit up "before it is too late".
He said, "Ohanaeze Ndigbo sees the governor's call as a dangerous trend because when you talk about government, it is a kind of social contract between the government and the governed. To that extent, the essence of government is for security of life and property.
"On the other hand, the monopoly of violence supposed to be within the government, but when the government does not have control of instrument of violence, it is a dangerous trend. It means the government is leaving the job which is supposed to do; the services which they are supposed to render to the society – security of life and property.
"It obvious that there's a problem in the country and we call on the Federal Government to sit up because it is tending towards doom to the nation, where the instrument of violence spread among the citizenry, and is no longer within the control and monopoly of the government," Ohanaeze further warned. PUNCH
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NCAA sanctions are not stopping Bill O'Brien from continuing to bring in quality talent to State College.
The Penn State head coach's latest work of magic came Wednesday night in the form of a commitment from Troy Vincent Jr., who is verbal pledge No. 12 for the Nittany Lions' 2014 recruiting class.
The 5-9, 189-pound cornerback from Gilman School (Baltimore, MD), who committed to the Nittany Lions while on campus for a visit Wednesday, is the son of former Philadelphia Eagles defensive back Troy Vincent.
Rated as a four-star prospect by Rivals.com, Vincent Jr. is considered one of the better cornerbacks nationally. While he is one of the more athletic rising senior cornerbacks, he has also been lauded for his smarts and awareness.
Vincent Jr. had other offers from major programs, including Michigan, Michigan State and Wisconsin. He is the third defensive back commit for PSU, joining fellow cornerback Daquan Worley and safety Marcus Allen. The Nittany Lions have made the defensive backfield a priority this cycle.
The Maryland product will join fellow Nittany Lions commit De'Andre Thompkins at The Opening, a camp in Beaverton, OR, that runs from June 30 through July 3.


text:
What to do in Barcelona
Enjoy Barcelona in total comfort
Archictecture & Art
Attractions & Monuments
Nature & Panoramic
Sant Jordi's Day in Barcelona: a stroll that you can't miss
20 April, 2016 | Julia P | 2 Comments
Sant Jordi is the saint patron of Catalonia. The atmosphere is simply unique in Barcelona
My grandmother Joana truly loved the city where she was born, grew up and lived for 90 years. She used to say that Barcelona was especially beautiful on two certain days: in September 24th, the day that citizens celebrate 'La Mercè' to honour one of the saint patronesses of Barcelona, and Sant Jordi's Day, in April 23rd.
For us, this day is the equivalent of Saint Valentine's. A romantic day to give a red rose to our loved ones, a tradition that dates back to the 15th century. In this gift, three characteristics come together: a single flower, symbolizing the 'exclusivity' dedicated to the lover; its red colour, as a sign of passion, and an ear of wheat, a symbol of fertility. The present also comes with a ribbon with the colours of the 'Senyera', the Catalan flag.
Sant Jordi is a romantic day to give a rose to our loved ones
Sant Jordi (which is the Catalan name of George) is the saint patron of Catalonia, and definitely a very special day to celebrate in Barcelona. Stalls are packed with roses to sell…and books! The reason is that April 23rd is also a remarkable day for literature: they say that in 1616 that day, William Shakespeare and Miguel de Cervantes died. Since 1929, Sant Jordi has been linked to roses and also books, which are sold with special discounts.
According to the Librarian Guild, they sell, on Sant Jordi's Day alone, around two-millions books, the 15% of what they sell during the whole year. And almost six millions roses are given on April the 23rd. We should mention that Sant Jordi's day is not an official public holiday in Catalonia, so shops are open and people go to work.
And that's the good thing of it! For the locals it's a great chance to take a few minutes off, walk around the city and look for a rose and a book for their loved ones. The atmosphere is simply unique! And we have some good news for you: This year Sant Jordi falls on a Thursday!
The hotspots to a stroll that you can't miss: La Rambla and Passeig de Gràcia
Passeig de Gracia is at the broad elegant avenue of Passeig de Gràcia that you will find Casa Batlló, one of the most fascinating buildings designed by Gaudí and one of the seven that have been declared World Heritage Site by UNESCO. Explore this artistic legend and skip the line with our best seller The Gaudí Tour.
Start to focus on certain details on the façade and the magic begins: The ceramic tiles and the roof resemble the skin of the dragon. From it, a chimney rises on one side. It's like a tower crowned by a four-armed cross which symbolises the sword of Sant Jordi that killed the beast. And on top-centre of the façade, a flower-blossom-shaped balcony: a metaphor of a rose…
By the way. Here is yours,
@->->–
Hope to see you soon in Barcelona!
Culture, History & Heritage	Love, Sant Jordi, Spring
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Elizabeth Gilbert – Tickets – The Lincoln Theatre – Washington, DC – June 6th, 2019 – I.M.P.
Elizabeth Gilbert, New York Times bestselling author of Eat, Pray, Love, returns to fiction with a story about love, life, and desire. City of Girls is set in the theater world of the 1940s, told from the perspective of Vivian Morris as she looks back on her youth with both pleasure and regret. Throughout the book, Gilbert explores themes of female sexuality and promiscuity, as well as the idiosyncrasies of true love. Ultimately, the story leads Vivian to a new understanding of the kind of life she craves and the kind of freedom it takes to pursue it. It will also lead to the love of her life, a love that stands out from all the rest. Written with a powerful wisdom about human desire and connection, City of Girls is a love story like no other.


Dr. Sandra Stotsky, professor of education reform at the University of Arkansas, offered to any state or school district a set of model standards that they can use as a English Language Arts curriculum framework as option if they would rather not use the Common Core State Standards.
Dr. Stotsky is known for her involvement in drafting the ELA standards for Massachusetts that recognized as some of the best in the nation.
She is offering these standards free of charge and you can see them below.


Tableau Conference On Tour is more than just an opportunity to connect with hundreds of link-minded data driven individuals. It's your chance to interact face-to-face with a growing global audience including Tableau partners, customers, and employees.
The Information Lab's goal is to help you make sense of data. We've guided thousands on their journey through data analytics by enabling them to learn Alteryx and Tableau to perform data preparation, blending and visualisation, all without reliance on long-term consulting engagements. Want to join them?
At Tableau on Tour London we'll be bringing a super knowledgeable team including 10 accredited professionals, 3 Tableau Zen Masters and an Alteryx Ace. Grab any of them to geek out with data or even get some impromptu Tableau & Alteryx training.
InterWorks is the leading data consultancy with a global reach from our offices across the US and Europe. Through innovative solutions and proven expertise, we cover the full course of your data journey, from network architecture and data management to visual analytics and custom development. Blending twenty years of consulting experience and a relentless drive for client satisfaction, we've helped countless organizations across every industry move further, faster and smarter toward their goal of a sustainable data-driven culture. Learn more at www.interworks.com.
EXASOL develops the world's fastest in-memory analytic database that gives you fast access to all of your data. With a live connection to Tableau, users no longer need to put up with the limitations of Tableau data extracts (TDE) and slow performance. Combining the enormous processing power of EXASOL and Tableau's ad hoc visual data discovery now puts the power of analyzing and visualizing full data sets in just seconds into the hands of every user.
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VirtDB provides solutions to prepare and push SAP R/3 or S/4HANA data to Tableau Desktop or Server and databases that conforms to SAP licensing policy. We enable your business users to access operational data faster, at minimal cost without compromising existing data governance.


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Q: How to open the video file in my vps? I have purchase a vpn service(serverip is a.b.c.d),when  login into   my sftp with putty,i downloaded a video file name movie1 in /root , i want to watch the video file(in server ) played in my pc(client ),input mplayer  movie1 (mplayer have benn installed in sftp server).
The error message is :
mplayer: could not connect to socket
mplayer: No such file or directory
Failed to open LIRC support. You will not be able to use your remote control.
Playing movie1.
vo: couldn't open the X11 display ()!
vo: couldn't open the X11 display ()!
vo: couldn't open the X11 display ()!
How can  in my client   to watch it  to be displayed in my sftp server with mplayer command in putty console ?

A: In Putty, activate X-Forwarding for your connection.
But don't draw too much hope from this, I doubt your internet connection to your VPS is fast enough to watch movies from remote.
Sorry, I forgot this: Since you're using Windows, you'll need a local X Window Manager. Have a look at MobaXTerm instead of PuTTy.

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Today I went for a rock chic look to celebrate one of my best friend's graduation! My favorite part of this outfit is all the different necklaces. Even though they are all very different, they all seem to go together very well don't they.
I used to never mix metals, but now I think it looks pretty cool!
Do you ever mix metals?


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Liner Notes Cat. No. 80645 Release Date: 2006-01-01
Adaskin String Trio: Emlyn Ngai, violin; Steve Larson, viola; Mark Fraser, cello; Katie Lansdale, violin; John McDonald, piano; Annie Trépanier, violin
Robert Carl (b 1954) possesses one of the widest and most varied expressive ranges of any composer of his generation. This isn't the disc to prove that, however. In the string music here we focus in on what I think of as the central core of Carl's aesthetic: the devoutly ruminative music with expression markings like "achingly" and "fragile." In his string music, Carl aligns himself with Beethoven and Ives as a composer who asks musical questions, who frankly frames his own spiritual journey in sounds with all its doubts, hungers, epiphanies, and acquiescences.
At the end of Open (1998) or Carl's Second String Quartet (2000-01), we know the composer's response to the existential questions of life as definitively as we can know them at the end of Ives's Fourth Symphony, or Beethoven's Ninth. Carl can be playful, satirical, intellectual, or picturesque in his other works, but something about the soulful nature of strings makes him express his innermost yearnings through them.
And these three works, written within a three-year period, are even more closely related than this implies. Their motivic material seems cut from the same cloth: the imperious dotted-rhythm motive at the beginning of Open, the thirds encircling each other in the "Angel-Skating" Sonata (Violin Sonata No. 2) (1999), the leaping lines at the beginning of the Second Quartet, are all based on pairs of interlocking intervals, whose contours Carl weaves into tapestries whose underlying unity is felt viscerally without being analytically contrived.
Adaskin String Trio
Robert Carl: Music For Strings
Open for String Trio
Robert Carl
Violin Sonata No. 2: I. Rondo: Intimations
Violin Sonata No. 2: II. Variations: Vision
Violin Sonata No. 2: III. Chaconne-Berceuse: State of Grace
String Quartet No. 2: I. Fear of Death
String Quartet No. 2: II. Hymn: In Growing Awareness of Beauty's Presence
String Quartet No. 2: III. Love of Life
Elizabeth Brown: Mirage
George Rochberg: String Quartets 3-6
George Tsontakis: String Quartets Nos. 3 & 4
Robert Carl: From Japan
Robert Carl: The Geography of Loss
Robert Erickson: Complete String Quartets


Nigerian fashion influencer and lady celebrity stylist, Toyin Lawani, has said that every man is entitled to about 10 women as part of solution to cheating in relationships because of their polygamous nature.
"Every man is entitled to 10 women. Even pastors cheat. Our world is corrupt. Just make it work for your kids' sake," she writes on her Instagram handle.
She condemned women who think they can play both the roles of a dad and a mum in their kids' lives, saying: "to all the idiot mums that think a father's space can ever be replaced by a mother's space in their kids' lives, get this ehn? You can never grow manhood.


For 100 years now J. Hornig has been bringing delicious coffee to Austria. It's a popular family-run business with traditional roots. Nevertheless there's no better time than now to "restart" the brand. To focus on the future and to stop following the fashions of the last few centuries. J. Hornig still knows how coffee tastes best: Black, natural and home-made. It's the essence that counts most. And the pleasure when roasting coffee.


PORT ORCHARD — Teachers fought hard to reduce class sizes in the South Kitsap School District, but that has caused a big scheduling headache for students and administrators the first week of school.
The contract agreement between teachers and the district means lowering class sizes at the middle and high schools by two students this semester and an additional three next semester.
"Unfortunately what that's meant is some of our 7th and 8th graders — in at least one of our junior highs — has seen a second and maybe third schedule revision because we've tried to lower their class sizes," superintendent Michelle Reid said.
Administrators have been shuffling kids around to balance out the numbers and in some cases, brought in substitutes to handle the overflow.
The district laid off 57 teachers earlier this summer but brought back 54 of them. Some certificated employees who had been working in non-teaching roles are going back into the classroom.
In spite of the hiccups, Reid is confident things will be straightened out by next week.
"I think we're getting really close to being there and parents can know we're doing the best we can for the kids in the community," she said.
Next year, classes at the elementary level will be reduced by one student.
The problem was not caused by the contract. The problem came with the tight staffing ratios which were causing larger than ever before. The contract was mutually agreed on as a way to cut the number of big classes to provide a better education for students in SK.
…….causing CLASSES larger than ever before.
Still waiting to see any improvements at the high school. My daughter's advanced algebra class is sitting at 36 students. I know Dr Reid authorized hiring an additional teacher at SKHS but the class size numbers seem to indicate that one isn't going to fix things.


text:
\section*{Abstract}

After reviewing the algebraic derivation of the Doppler factor in the Li\'{e}nard-Wiechert potentials
of an electrically charged point particle,
we conclude that the Dirac delta function used in electrodynamics must be the one obeying the weak
definition, non-zero in an infinitesimal neighborhood, and not the one obeying the strong definition,
non-zero in a point. This conclusion emerges from our analysis of
a) the derivation of an important Dirac delta function identity, which generates the Doppler factor,
b) the linear superposition principle implicitly used by the Green function method, and
c) the two equivalent formulations of the Schwarzschild-Tetrode-Fokker action.
As a consequence, in full agreement with our previous discussion of the geometrical origin of the Doppler factor,
we conclude that the electromagnetic interaction takes place not between points in Minkowski space,
but between corresponding infinitesimal segments along the worldlines of the particles.

\section{Introduction}

We have recently investigated the geometrical origin of the Doppler factor in the Li\'{e}nard-Wiechert (LW)
potentials \cite{Galeriu2021a,Galeriu2021b}.
For an extended charged particle,
the geometrical origin of the Doppler factor was well understood.
In this case, first we had to find
the intersection of the worldtube of the charged particle with the past lightcone with
vertex at the field point,
and then we had to project this intersection onto the 3D space of the observer.
For a point charged particle,
the geometrical origin of the Doppler factor was not as obvious.
In this case it seemed that the electromagnetic interaction was taking place not
between point particles in Minkowski space, but between length elements
along the worldlines of the charged particles, with the endpoints of these
corresponding infinitesimal segments being connected by light signals.
Since the material point particle model has failed to provide an intuitive explanation of the Doppler factor,
we have suggested to replace the four-force
acting on a point particle with a four-force linear density acting on a
length element along the worldline of that particle.

We would have expected to reach a similar conclusion when considering
the algebraic derivation of the Doppler factor in the LW potentials.
But no such a conclusion has been proposed yet.
How could a geometrical treatment of a physics problem arrive at a conclusion that
was not also reached by an algebraic treatment of the same problem?
In order to elucidate this apparent paradox, we had to address some conceptual issues
afflicting the widely used Dirac delta function.

\section{The Li\'{e}nard-Wiechert potentials}

Assuming that the Lorenz gauge condition applies,
the electromagnetic four-potential $A^\alpha = (\phi, {\bf A})$
of a charged particle in motion is given by \cite{Jackson1999}
\begin{equation}
A^\alpha (x) = \frac{4 \pi}{c} \int G(x, x') \, J^\alpha (x') \, d^4x',
\label{eq:Green_fn_solution}
\end{equation}
where $x = (c t, {\bf x})$, $x'= (c t', {\bf x'})$, and $G(x, x')$ is the Green function.
For a point particle with electric charge $q$, position four-vector $X^\alpha = (c T, {\bf X})$,
four-velocity $U^\alpha = (\gamma c, \gamma {\bf V})$, and proper time $\tau$, the four-current density is given by \cite{Jackson1999}
\begin{equation}
J^\alpha(x') = \int q \, U^\alpha(\tau) \, \delta^{4D}\big( x' - X(\tau) \big) c \, d\tau.
\label{eq:current_density}
\end{equation}

The Green function must satisfy the equation
\begin{equation}
\Box_x G(x, x') = \delta^{4D}(x - x'),
\label{eq:Green_fn_equation}
\end{equation}
which is most often solved using the method of the Fourier transform.
However, since the integrand is singular, \lq\lq Green functions that differ in their behavior are obtained by choosing
different contours of integration relative to the poles\rq\rq \cite{Jackson1999}.
Not only the retarded and the advanced Green functions thus found are solutions
of (\ref{eq:Green_fn_equation}), but also any
linear combination whose coefficients add up to one.
All these Green functions share the property that
\lq\lq the differences between two solutions of the inhomogeneous equation is a solution
of the homogeneous equation\rq\rq \cite{Thirring1958}.

We could also start by noticing that a possible solution of (\ref{eq:Green_fn_equation}) is \cite{Anderson1967, Ryder1974}
\begin{equation}
G(x, x') = \frac{1}{4 \pi} \delta\big( (x - x')^2 \big) = \frac{1}{4 \pi} \delta\big( c^2 (t - t')^2 - ({\bf x} - {\bf x'})^2 \big).
\label{eq:symmetric_Green_fn}
\end{equation}
Next we use the Dirac delta function identity
\begin{equation}
\delta(w^2 - a^2) = \frac{1}{2 |a|} \big( \delta(w - a) + \delta(w + a) \big),
\label{eq:Dirac_delta_property_1}
\end{equation}
and, with $r = |{\bf x} - {\bf x'}|$, we rewrite the Green function (\ref{eq:symmetric_Green_fn}) as
one half of the retarded Green function (non-zero when $t'= t - r/c$)
plus one half of the advanced Green function (non-zero when $t'= t + r/c$).
\begin{equation}
G(x, x') = \frac{1}{2} \frac{\delta(c t - c t' - r)}{4 \pi r} + \frac{1}{2}\frac{\delta(c t - c t' + r)}{4 \pi r}.
\label{eq:symmetric_Green_fn_2}
\end{equation}
According to our previous discussion, a possible solution could also be
\begin{equation}
G(x, x') = f \frac{\delta(c t - c t' - r)}{4 \pi r} + ( 1 - f ) \frac{\delta(c t - c t' + r)}{4 \pi r},
\label{eq:symmetric_Green_fn_3}
\end{equation}
where $f$ is a yet undetermined real number.

When Li\'{e}nard \cite{lienard} and Wiechert \cite{wiechert} have derived their electromagnetic potentials,
they have only considered the causal solution.
From a mathematical point of view, this corresponds to setting $f = 1$ in (\ref{eq:symmetric_Green_fn_3}).
Equivalently, this also corresponds to replacing the $( 1 - f )$ contribution of the
advanced Green function with a retarded Green function of the same weight.
Why are we allowed to replace an advanced electromagnetic potential contribution with a retarded one?
The advanced potential is exactly equal to the retarded potential when the source particle is
at rest, or in motion with constant velocity, or in motion with constant acceleration (hyperbolic motion).
On top of that, by design, classical electrodynamics ignores any effect that the rate of change
in the acceleration of the source, or of the test particle,
may have on the electromagnetic interaction (the Lorentz four-force).
However, as if in opposition to the above statements,
we also notice that a consistent treatment of a system of two particles in circular motion
requires a time symmetric interaction \cite{Schild1963}.

Substitution of the causal Green function
and of the four-current density
into (\ref{eq:Green_fn_solution}) gives
\begin{multline}
A_{(ret)}^\alpha (x) = \frac{4 \pi}{c} \int \frac{\delta(c t - c t' - r)}{4 \pi r} \,
\int q \, U^\alpha(\tau) \, \delta^{4D}\big( x' - X(\tau) \big) c \, d\tau \, d^4x' \\
= q \int \frac{\delta(c t - c T - R)}{R} \, U^\alpha(\tau) \, d\tau
= q \int \frac{\delta(T - t + R/c)}{R} \, (1, {\bf V}/c)^\alpha \, dT,
\label{eq:Green_fn_solution_2}
\end{multline}
where $R = |{\bf x} - {\bf X}|$, since $x' = X(\tau)$ after the integration over $d^4x'$.
In the last step we have used the Dirac delta function identity
\begin{equation}
\delta(- a w) = \delta(a w) = \frac{1}{|a|} \delta(w),
\label{eq:Dirac_delta_property_2}
\end{equation}
and in the next step we use another Dirac delta function identity
\begin{equation}
\delta\big( g(w) \big) = \sum_n \frac{\delta(w - w_n)}{|g'(w_n)|},
\label{eq:Dirac_delta_property_3}
\end{equation}
where $g(w_n) = 0$ and the derivatives $g'(w_n) \neq 0$.

It is precisely at this step that the Doppler factor in the retarded LW potentials emerges.
In our case the function $g$ is \cite{Zangwill2013}
\begin{equation}
g(T) = T - t + \frac{R(T)}{c} = T - t + \frac{\sqrt{\big({\bf x} - {\bf X}(T)\big)^2}}{c},
\label{eq:function}
\end{equation}
the retarded time $T_r$ is the unique solution for which $g(T_r) = 0$,
\begin{equation}
T_r = t - \frac{R(T_r)}{c},
\label{eq:ret_time}
\end{equation}
the derivative $g'$ of the function $g$ is
\begin{equation}
g'(T) = 1 - \frac{{\bf V}(T) \cdot {\bf R}(T)}{c \, R(T)},
\label{eq:derivative}
\end{equation}
where ${\bf R} = {\bf x} - {\bf X}$, and the reciprocal of the Doppler factor is
\begin{equation}
g'(T_r) = 1 - \frac{{\bf V}(T_r) \cdot {\bf R}(T_r)}{c \, R(T_r)} > 0.
\label{eq:reciprocal_1}
\end{equation}

After the integration over $dT$, the retarded LW potentials of a charged point particle are obtained
\begin{equation}
(\phi_{(ret)}, {\bf A}_{(ret)})
= \frac{q}{R(T_r)} \frac{1}{1 - \frac{{\bf V}(T_r) \cdot {\bf R}(T_r)}{c \, R(T_r)}} \big( 1, {\bf V}(T_r)/c \big).
\label{eq:LWpotentials}
\end{equation}

We also notice that, from (\ref{eq:ret_time}), we can express $t$ as a function of $T_r$.
\begin{equation}
t(T_r) = T_r + \frac{R(T_r)}{c}.
\label{eq:field_point_time}
\end{equation}
It follows that $t'(T_r)$, the derivative of $t$
with respect to $T_r$, is exactly equal to $g'(T_r)$, the reciprocal of the Doppler factor.

Substitution of the acausal Green function
and of the four-current density
into (\ref{eq:Green_fn_solution}) gives
\begin{multline}
A_{(adv)}^\alpha (x) = \frac{4 \pi}{c} \int \frac{\delta(c t - c t' + r)}{4 \pi r} \,
\int q \, U^\alpha(\tau) \, \delta^{4D}\big( x' - X(\tau) \big) c \, d\tau \, d^4x' \\
= q \int \frac{\delta(c t - c T + R)}{R} \, U^\alpha(\tau) \, d\tau
= q \int \frac{\delta(T - t - R/c)}{R} \, (1, {\bf V}/c)^\alpha \, dT,
\label{eq:Green_fn_solution_2_adv}
\end{multline}
and, after a similar calculation, the advanced LW potentials of a charged point particle are obtained
\begin{equation}
(\phi_{(adv)}, {\bf A}_{(adv)})
= \frac{q}{R(T_a)} \frac{1}{1 + \frac{{\bf V}(T_a) \cdot {\bf R}(T_a)}{c \, R(T_a)}} \big( 1, {\bf V}(T_a)/c \big),
\label{eq:LWpotentials_adv}
\end{equation}
where the advanced time $T_a$ is
\begin{equation}
T_a = t + \frac{R(T_a)}{c}.
\label{eq:adv_time}
\end{equation}
Again, it follows that the derivative of $t$
with respect to $T_a$ is exactly equal to the reciprocal of the Doppler factor.

\section{Point particle or length element?}

The material point particle model seems to be in agreement with the above derivations.
Consistent with (\ref{eq:current_density}), the electric charge density is given by
\begin{equation}
\rho({\bf x'}, t') = q \, \delta^{3D}\big( {\bf x'} - {\bf X}(t') \big),
\label{eq:charge_density}
\end{equation}
and the electric current density is given by
\begin{equation}
{\bf J}({\bf x'}, t') = q \, {\bf V}(t') \, \delta^{3D}\big( {\bf x'} - {\bf X}(t') \big).
\label{eq:3D_current_density}
\end{equation}
It appears that the charged particle is indeed a point particle,
since the Dirac delta function is not equal to zero in just one point.
By definition \cite{Dirac1927}
\begin{eqnarray}
\delta(w) = 0 \ \ \  \text{when} \ \ \ w \neq 0, \label{eq:Dirac_delta_a} \\
\int_{- \infty}^{\infty} \delta(w) \, dw = 1. \label{eq:Dirac_delta_b}
\end{eqnarray}

In order to gain more insight, we focus on the derivation step that has produced the Doppler factor,
which is the application of formula (\ref{eq:Dirac_delta_property_3}).
How can a Dirac delta function, which is non-zero in just one point, return the value of a derivative?
To calculate the derivative of a function in a point we need to know the value of that function in an
infinitesimal neighborhood of that point, the information from just one point is not enough.
And how was formula (\ref{eq:Dirac_delta_property_3}) derived in the first place?
The proof involves an integration by substitution. A closer look at this method of integration
reveals the fact that the function in the integrand must be continuous, a property that the
Dirac delta function (\ref{eq:Dirac_delta_a}) certainly does not have. In fact, the Dirac delta function must also
be a differentiable function, since from this property it follows that
the electric charge density (\ref{eq:charge_density})
and the electric current density (\ref{eq:3D_current_density})
satisfy the continuity equation \cite{Landau4ed}.
Dirac explicitly assumed that the delta function is differentiable, also also provided a second,
quite different description of his function. This time the Dirac delta function \lq\lq is equal to zero except when
$x$ is very small\rq\rq \ \cite{Dirac1927}. Equation (\ref{eq:Dirac_delta_a}) is replaced by
\begin{equation}
\delta(w) = 0 \ \ \  \text{when} \ \ \ |w| > \epsilon,
\label{eq:Dirac_delta_c}
\end{equation}
where $\epsilon$ is a positive infinitesimal number.

We are, in fact, given two very different definitions of the same mathematical object!
This confusing situation has been recognized by Amaku {\it et al.} \cite{Amaku2021RBEF},
who have also discussed the criterion that allows us to distinguish one type of Dirac delta function from the other.
The Dirac delta function that is non zero in just one point, corresponding to the strong definition (\ref{eq:Dirac_delta_a}),
has associated with it an integral from zero to infinity of value one
\begin{equation}
\int_{0}^{\infty} \delta(w) \, dw = 1.
\label{eq:Dirac_strong_def}
\end{equation}
The Dirac delta function that is non zero in the infinitesimal  neighborhood of a point,
corresponding to the weak definition (\ref{eq:Dirac_delta_c}),
has associated with it an integral from zero to infinity of value one half
\begin{equation}
\int_{0}^{\infty} \delta(w) \, dw = \frac{1}{2}.
\label{eq:Dirac_weak_def}
\end{equation}

Which one of these two definitions actually applies to the
Dirac delta function used in our calculations of the LW potentials?
It is the weak definition of the Dirac delta function, as it is revealed by the proof of the
Dirac delta function identity  (\ref{eq:Dirac_delta_property_3}).
In the simple case when the function $g(w)$ has only one root $w_o$, with $g(w_o) = 0$,
this proof consists of two steps.

Step I. For a given test function $f(w)$, the domain of integration of the integral $\int \delta(g(w)) \ f(w) \ dw$
is restricted to an infinitesimal domain centered on the root $w_o$, since everywhere else the delta function is zero.
Inside this infinitesimal domain the function $g(w)$ is equal to its first order Taylor series approximation.
\begin{equation}
g(w) = g(w_o) + (w - w_o) \frac{dg}{dw}\Big|_{w_o} = (w - w_o) \, g'(w_o).
\end{equation}

Step II. A change of variables gets the constant factor $|g'(w_o)|$ out of the Dirac delta function,
a step equivalent to using the identity
\begin{equation}
\delta\big(a (w - w_o)\big) = \frac{1}{|a|} \delta(w - w_o).
\end{equation}

In Step I we have assumed that the Dirac delta function in non-zero in an infinitesimal neighborhood of a point.
In Step II we have assumed that the Dirac delta function is continuous, as required by the integration by substitution.
Only the weak definition (\ref{eq:Dirac_delta_c}) is consistent with these two assumptions.

\section{An analysis of the Green function method}

The same conclusion regarding the true nature of the Dirac delta function
used in electrodynamics
can be reached from a different perspective,
from an analysis of the Green function method that we have
used in order to find the electromagnetic LW potentials.
For simplicity, we only discuss the one dimensional problem.
The same conclusion holds when we use the Green function method in higher dimensions.

Consider a linear differential operator $\mathcal{L}$ and the ordinary differential equation
\begin{equation}
\mathcal{L} y(x) = f(x).
\label{eq:ODE}
\end{equation}
The Green function satisfies the equation
\begin{equation}
\mathcal{L} G(x, x') = \delta(x - x').
\label{eq:Green_fn_1D}
\end{equation}
The solution of the ordinary differential equation is calculated as
\begin{equation}
y(x) = \int G(x, x') \, f(x') \, dx'.
\label{eq:ODE_solution}
\end{equation}
The solution (\ref{eq:ODE_solution}) is usually justified by showing that,
when the operator $\mathcal{L}$ is applied on both sides
of equation (\ref{eq:ODE_solution}), the original equation (\ref{eq:ODE}) is obtained.
This extremely short and abstract proof does not seem to be conducive to intuitive understanding.

More insight comes from a comment by Arfken {\it et al.} \cite{Arfken}, who mention that
\lq\lq The fact that we can determine $\psi$ everywhere by an integration is a consequence
of the fact that our differential equation is linear, so each element of the source contributes additively.\rq\rq

Inspired by this comment, we generate an infinite set of points $x_k = 2 k \epsilon$,
where $k$ spans the set of
integer numbers, and we divide the $x$-axis into an infinite set of
infinitesimal intervals $[x_k - \epsilon, x_k + \epsilon)$. The source term is now written as
\begin{equation}
f(x) = \sum_k f_k(x),
\label{eq:source_term_sum}
\end{equation}
where
\begin{equation}
f_k(x) =
\begin{cases}
0, &x < x_k - \epsilon \\
f(x), &x_k - \epsilon \leq x < x_k + \epsilon \\
0, &x_k + \epsilon \leq x \\
\end{cases}
\label{eq:source_term_def}
\end{equation}
The equation (\ref{eq:ODE}) splits into a set of equations
\begin{equation}
\mathcal{L} y_k(x) = f_k(x),
\label{eq:ODE_set}
\end{equation}
and, due to the linearity of $\mathcal{L}$, the total solution is obtained as a sum
\begin{equation}
y(x) = \sum_k y_k(x).
\label{eq:solution_sum}
\end{equation}

In order to show that the sum in (\ref{eq:solution_sum}) is equal to the integral in (\ref{eq:ODE_solution}),
we start by introducing the Dirac delta function
\begin{equation}
\delta(x - x_k) =
\begin{cases}
0, &x < x_k - \epsilon \\
\frac{1}{2 \epsilon}, &x_k - \epsilon \leq x < x_k + \epsilon \\
0, &x_k + \epsilon \leq x \\
\end{cases}
\label{eq:Dirac_delta_d}
\end{equation}

We notice that
\begin{equation}
f_k(x) = f(x) \, \delta(x - x_k) \, 2 \epsilon,
\label{eq:33}
\end{equation}
and also that, in the limit $\epsilon \to 0$,
\begin{equation}
f_k(x) = f(x_k) \, \delta(x - x_k) \, 2 \epsilon.
\label{eq:34}
\end{equation}

Substitution of (\ref{eq:34}) into (\ref{eq:ODE_set}) gives
\begin{equation}
\mathcal{L} \frac{y_k(x)}{ f(x_k) \, 2 \epsilon} = \delta(x - x_k).
\label{eq:35}
\end{equation}

Comparison of (\ref{eq:35}) with (\ref{eq:Green_fn_1D}) makes it clear that
\begin{equation}
G(x, x_k) = \frac{y_k(x)}{ f(x_k) \, 2 \epsilon},
\label{eq:36}
\end{equation}
and that the total solution
\begin{equation}
y(x) = \sum_k y_k(x) = \sum_k G(x, x_k) \, f(x_k) \, 2 \epsilon,
\label{eq:solution_integral}
\end{equation}
in the limit $\epsilon \to 0$, is equal to the integral in (\ref{eq:ODE_solution})

In conclusion, our analysis of the Green function method reveals the fact that we have used of a Dirac delta function
(\ref{eq:Dirac_delta_d}) that obeys the weak definition. Only in this way we were able to split the source term $f(x)$,
which is a continuous function, into a sum of terms, each of them proportional to a Dirac delta function,
as seen in (\ref{eq:34}).

\section{The Schwarzschild-Tetrode-Fokker action}

Another strong argument in favor of the weak definition of the Dirac delta function is provided by a
discussion of the Schwarzschild-Tetrode-Fokker (STF) action.

Schwartzschild \cite{Schwarzschild1903} has discovered that the Lorentz force can be obtained
from the Lagrangian \cite{Goldstein1980}
\begin{equation}
L = \frac{1}{2} m v^2 - q \phi + \frac{q}{c} {\bf A}\cdot{\bf v},
\label{eq:L_1}
\end{equation}
which takes the form \cite{Goldstein1980}
\begin{equation}
L = - m c^2 \sqrt{1 - \frac{v^2}{c^2}} - q \phi + \frac{q}{c} {\bf A}\cdot{\bf v},
\label{eq:L_2}
\end{equation}
in Special Relativity.
Since $U_\alpha A^\alpha = \gamma \phi c - \gamma {\bf A}\cdot{\bf v}$,
the product of the Lagrangian (\ref{eq:L_2}) with the Lorentz factor $\gamma$
is a Lorentz invariant \cite{Jackson1999}.
\begin{equation}
L \gamma = - m c^2  - \frac{q}{c} U_\alpha A^\alpha,
\label{eq:L_3}
\end{equation}

Introducing the proper time, and since $dt = \gamma \, d\tau$, the action integral takes the form
\begin{equation}
\int_{t_i}^{t_f} L \, dt = \int_{\tau_i}^{\tau_f} \Big( - m c^2  - \frac{q}{c} U_\alpha A^\alpha \Big) d\tau.
\label{eq:action1}
\end{equation}

Tetrode \cite{tetrode}, realizing the importance of time symmetric electrodynamics,
has replaced the retarded LW four-potential
with the arithmetic mean of the retarded and advanced potentials.
Using the Dirac delta function identity (\ref{eq:Dirac_delta_property_1}),
from (\ref{eq:Green_fn_solution_2}) and (\ref{eq:Green_fn_solution_2_adv}) we obtain
\begin{multline}
A^\alpha (x) = \frac{1}{2} A_{(ret)}^\alpha (x) + \frac{1}{2} A_{(adv)}^\alpha (x) \\
= \frac{1}{2} q \int \frac{\delta(c t - c T - R)}{R} \, U^\alpha(\tau) \, d\tau
+ \frac{1}{2} q \int \frac{\delta(c t - c T + R)}{R} \, U^\alpha(\tau) \, d\tau \\
= q \int \delta\big( c^2 (t - T)^2 - ({\bf x} - {\bf X})^2 \big) \, U^\alpha(\tau) \, d\tau
= q \int \delta\big( (x - X)^2 \big) \, U^\alpha(\tau) \, d\tau.
\label{eq:timesympot}
\end{multline}

Quite remarkable, working before Dirac introduced his Dirac delta function in 1927,
Tetrode introduced his own
definition of the Dirac delta function, which in his notation was written as
$\delta(\sigma^2) = f(\sigma^2)/\sigma^2$.
He also mentioned that only the infinitesimal neighborhood of the lightcone
(\lq\lq da{\ss} nur die infinitesimale Umgebung des Lichtkegels\rq\rq) with vertex at the field point
contributes to the electromagnetic potentials, in effect giving the weak definition of the Dirac delta function.

Since in (\ref{eq:action1}) $q$, $U_\alpha$, and $\tau$ refer to the test particle at $x$,
while in (\ref{eq:timesympot}) $q$, $U^\alpha$, and $\tau$ refer to the source particle at $X$,
in order to substitute (\ref{eq:timesympot}) into (\ref{eq:action1}) we need a change in notation.
Let us label with \lq\lq1\rq\rq\ the test particle, and with \lq\lq2\rq\rq\ the source particle.
The action integral becomes \cite{tetrode}
\begin{equation}
W_1 = - \int_{\tau_{1 i}}^{\tau_{1 f}}  m_1 c^2 \, d\tau_1
 - \frac{q_1 q_2}{c} \int_{\tau_{1 i}}^{\tau_{1 f}} \int
\delta\big( (x_1 - x_2)^2 \big) \, U_{1 \alpha}(\tau_1) \, U_2^\alpha(\tau_2) \, d\tau_2 \, d\tau_1.
\label{eq:action2}
\end{equation}
Tetrode writes down the negative of (\ref{eq:action2}), which he acknowledges,
but this has no effect on the equations of motion obtained from the extremum condition.
The interaction term in Tetrode's action integral is still negative, due to the fact that
he is using the $(i c t, x, y, z)$ Minkowski formalism, equivalent to a metric tensor of
signature $(-, +, +, +)$, while we, following Jackson \cite{Jackson1999},
are using a metric tensor of signature $(+, -, -, -)$.

Tetrode then notices that the interaction term in (\ref{eq:action2}) is symmetric in the indices of the two
particles. For an infinitesimal variation of $x_1$
that vanishes at $\tau_{1 i} = - \infty$ and at $\tau_{1 f} = \infty$,
the condition $\delta W_1 = 0$ gives the equations of
motion of particle 1 in the electromagnetic field of particle 2. The same holds true for the motion
of particle 2 in the field of particle 1, provided that the whole action integral is also symmetric
in the indices of the particles.
\begin{multline}
W_{12} = - \int  m_1 c^2 \, d\tau_1 - \int  m_2 c^2 \, d\tau_2 \\
 - \frac{q_1 q_2}{c} \int \int
\delta\big( (x_1 - x_2)^2 \big) \, U_{1 \alpha}(\tau_1) \, U_2^\alpha(\tau_2) \, d\tau_2 \, d\tau_1.
\label{eq:action3}
\end{multline}

In the final step Tetrode generalizes the action integral (\ref{eq:action3}) to any number of electrically charged interacting particles.
Since $c \, d\tau = \sqrt{dx_\alpha dx^\alpha}$ and $U_\alpha \, d\tau = dx_\alpha$, the action function
for the whole world becomes
\begin{multline}
W = - \sum_A \int  m_A c \sqrt{dx_{A \alpha} dx_A^\alpha} \\
 - \sum_A \sum_{B > A} \frac{q_A q_B}{c} \int \int
\delta\big( (x_{A \beta} - x_{B \beta}) (x_A^\beta - x_B^\beta) \big) \, dx_{A \alpha} \, dx_B^\alpha.
\label{eq:action4}
\end{multline}

This is the exact expression from which Wheeler and Feynman started their theory of
time symmetric action-at-a-distance electrodynamics \cite{Wheeler1949}. Because they have used
a metric tensor of signature $(-, +, +, +)$, the scalar products bring in sign changes, and they
warn the readers about this issue in a footnote. In the same footnote they also give the definition of the
Dirac delta function, but this time, unlike Tetrode, they give the strong definition.
What difference does it make? The strong definition implies that the interaction takes place between
point particles, points in the 4D Minkowski space. The weak definition implies that the interaction
takes place between length elements, infinitesimal segments along the worldlines of the particles.

Fokker \cite{fokker}, deriving his results without the help of the Dirac delta function, completely avoids
the integrals from (\ref{eq:Green_fn_solution_2}) and (\ref{eq:Green_fn_solution_2_adv}) by
directly inserting into (\ref{eq:action1}) the expression of the time symmetric four-potential,
written with the help of (\ref{eq:LWpotentials}) and (\ref{eq:LWpotentials_adv}) as
\begin{multline}
A^\alpha (x) = \frac{1}{2} A_{(ret)}^\alpha (x) + \frac{1}{2} A_{(adv)}^\alpha (x) \\
= \frac{q}{2 R(T_r)} \frac{\big( 1, {\bf V}(T_r)/c \big)}{1 - \frac{{\bf V}(T_r) \cdot {\bf R}(T_r)}{c \, R(T_r)}}
+ \frac{q}{2 R(T_a)} \frac{\big( 1, {\bf V}(T_a)/c \big)}{1 + \frac{{\bf V}(T_a) \cdot {\bf R}(T_a)}{c \, R(T_a)}} \\
= \frac{q}{2} \frac{U^\alpha(T_r)}{\big( x_\beta - X_\beta(T_r) \big) U^\beta(T_r)}
- \frac{q}{2} \frac{U^\alpha(T_a)}{\big( x_\beta - X_\beta(T_a) \big) U^\beta(T_a)},
\label{eq:timesympot2}
\end{multline}
where $x - X(T_r) = \big( c t - c T_r, {\bf x} - {\bf X}(T_r) \big) = \big( R(T_r), {\bf R}(T_r) \big)$
and $x - X(T_a) = \big( c t - c T_a, {\bf x} - {\bf X}(T_a) \big) = \big( - R(T_a), {\bf R}(T_a) \big)$.
Fokker has used a metric tensor with signature $(+, -, -, -)$ and reduced electrostatic units,
instead of Gaussian units. This brings an extra $4 \pi$ in the denominator.

After we label with \lq\lq1\rq\rq\ the test particle, and with \lq\lq2\rq\rq\ the source particle,
the action integral becomes
\begin{multline}
W_{12} = - \int  m_1 c^2 \, d\tau_1 - \int  m_2 c^2 \, d\tau_2 \\
 - \frac{q_1 q_2}{2 c} \int \bigg(
\frac{U_{1 \alpha}(\tau_1) \, U_2^\alpha(T_{2 r})}{\big( x_{1 \beta} - x_{2 \beta}(T_{2 r}) \big) U_2^\beta(T_{2 r})}
- \frac{U_{1 \alpha}(\tau_1) \, U_2^\alpha(T_{2 a})}{\big( x_{1 \beta} - x_{2 \beta}(T_{2 a}) \big) U_2^\beta(T_{2 a})}
\bigg) d\tau_1.
\label{eq:action5}
\end{multline}

Based on Tetrode's expression (\ref{eq:action3}), we expect Fokker's expression (\ref{eq:action5})
to also be symmetric in the indices of the two particles. A permutation of the two indices gives
\begin{multline}
W_{21} = - \int  m_1 c^2 \, d\tau_1 - \int  m_2 c^2 \, d\tau_2 \\
 - \frac{q_1 q_2}{2 c} \int \bigg(
\frac{U_{2 \alpha}(\tau_2) \, U_1^\alpha(T_{1 r})}{\big( x_{2 \beta} - x_{1 \beta}(T_{1 r}) \big) U_1^\beta(T_{1 r})}
- \frac{U_{2 \alpha}(\tau_2) \, U_1^\alpha(T_{1 a})}{\big( x_{2 \beta} - x_{1 \beta}(T_{1 a}) \big) U_1^\beta(T_{1 a})}
\bigg) d\tau_2.
\label{eq:action6}
\end{multline}

The two action integrals (\ref{eq:action5}) and (\ref{eq:action6}) are indeed the same when
the retarded part of (\ref{eq:action5}) is equal to the advanced part of (\ref{eq:action6}),
while at the same time the advanced part of (\ref{eq:action5}) is equal to the retarded part of (\ref{eq:action6}).
We must have
\begin{equation}
\frac{d\tau_1}{( x_{1 \beta} - x_{2 \beta} ) U_2^\beta}
= \frac{d\tau_2}{( x_{1 \beta} - x_{2 \beta} ) U_1^\beta},
\label{eq:segments}
\end{equation}
regardless of whether $t_1 > t_2$ or $t_1 < t_2$. Equivalently, we can write (\ref{eq:segments}) as
\begin{equation}
( x_{1 \beta} - x_{2 \beta} ) \, dx_2^\beta = ( x_{1 \beta} - x_{2 \beta} ) \, dx_1^\beta,
\label{eq:segments2}
\end{equation}
an equation that describes two infinitesimal segments $dx_1$ and $dx_2$ whose endpoints are connected
by light signals. Indeed, this is the equation that emerges when we write
\begin{eqnarray}
( x_{1 \beta} - x_{2 \beta} ) ( x_1^\beta - x_2^\beta ) = 0, \\
( x_{1 \beta} + dx_{1 \beta} - x_{2 \beta} - dx_{2 \beta} )
( x_1^\beta + dx_1^\beta - x_2^\beta - dx_2^\beta ) = 0,
\label{eq:lamont}
\end{eqnarray}
and we keep only the first order terms in the infinitesimals \cite{lamont}.
In Fokker's article \cite{fokker}, with his notation, equation (\ref{eq:segments2}) is written as
$( R \cdot dy ) = ( R \cdot dx )$, and the infinitesimal segments connected by light rays are called
corresponding effective elements (\lq\lq entsprechenden effektiven Elementen\rq\rq).

Multiplying, in the numerator and in the denominator,
the retarded part of (\ref{eq:action5}) by a retarded $d\tau_{2 r}$ satisfying (\ref{eq:segments}),
and multiplying the advanced part by an advanced $d\tau_{2 a}$ also satisfying (\ref{eq:segments}),
we are able to write down the action function as
\begin{multline}
W_{12} = - \int  m_1 c^2 \, d\tau_1 - \int  m_2 c^2 \, d\tau_2 \\
 - \frac{q_1 q_2}{2 c} \int \bigg(
\frac{dx_{1 \alpha} \, dx_{2 r}^\alpha}{\big( x_{1 \beta} - x_{2 r \beta} \big) dx_{2 r}^\beta}
- \frac{dx_{1 \alpha} \, dx_{2 a}^\alpha}{\big( x_{1 \beta} - x_{2 a \beta} \big) dx_{2 a}^\beta}
\bigg),
\label{eq:action7}
\end{multline}
in a way that stresses the fact that
the electromagnetic interaction takes place between corresponding infinitesimal segments.
In the final step Fokker generalizes the action integral (\ref{eq:action7})
to any number of electrically charged interacting particles.

Dirac himself recognized that the action integral admits two equivalent formulations,
writing it without the Dirac delta function \cite{dirac1938}, like Fokker,
and with the Dirac delta function \cite{dirac1942}, like Tetrode.
In this way the STF action,
describing the interaction between corresponding infinitesimal segments,
reveals the fact that the Dirac delta function used in electrodynamics is the one that obeys the weak definition.

As a side note, Dirac \cite{dirac1942} also recognized that, in this context, the use of a Dirac delta function
obeying the strong definition is problematic, and suggested replacing it,
in the STF action, with a different function $\delta^*$
that, just like the Dirac delta function obeying the weak definition, depends on an
infinitesimal parameter ${\boldsymbol \lambda}$ that has the limit ${\boldsymbol \lambda} \to 0$.

\section{Concluding remarks}

In summary, the conclusion that we have reached based on our
algebraic derivation of the Doppler factor in the LW potentials of a point charged particle
is in full agreement with the conclusion that we have reached based on our geometrical derivation \cite{Galeriu2021a}.
The point particles are points only in the 3D Euclidean space, while in the 4D Minkowski space the
electromagnetic interaction takes place between corresponding infinitesimal segments along the worldlines
of the particles, just as Fokker has described in his groundbreaking article \cite{fokker}.
We have discovered that the Dirac delta function used in electrodynamics does not
obey the strong definition, as it is generally assumed, but instead it obeys the weak definition.
This conclusion emerges from our analysis of the Dirac delta function identity
responsible for the apparition of the Doppler factor,
from our analysis of the Green function method used in the derivation of the LW potentials,
and from our analysis of the STF action.

This is not the first time when, in electrodynamics, the use
of the Dirac delta function (\ref{eq:Dirac_delta_a}) that is non-zero in just one point has been questioned.
As early as 1935 Louis de Broglie \cite{deBroglie1935} has suggested replacing this
infinitely narrow needle-like function (\lq\lq une fonction en aiguille infiniment fine\rq\rq)
by a needle-like function of very small but finite width (\lq\lq une fonction en aiguille d'\'{e}paisseur tr\`{e}s petite, mais finie\rq\rq),
like for example
\begin{equation}
\delta(w) = \frac{1}{\sigma \sqrt{\pi}} e^{- \frac{w^2}{\sigma^2}},
\label{eq:debroglie}
\end{equation}
where $\sigma$ is a very small and positive parameter. This function is continuous and differentiable,
and can be easily generalized to higher dimensions, producing a Lorentz invariant expression.
The same idea, as explained in detail by Feynman \cite{FeynmanII}, has been explored by Fritz Bopp.
Feynman also encourages us to study such theories, in order \lq\lq to see the struggles of the human mind\rq\rq.
We hope that at least this goal was achieved!

\section*{Acknowledgments}

The author is very much indebted to David~H.~Delphenich for
translating the articles written by Tetrode and Fokker, upon personal request.

redpajama_set_name: RedPajamaArXiv


Wireless remotes and audio streamers.
Hearing aid accessories are keeping up with tech. Today, the world is more connected and mobile than ever before. And as time goes on the people will connect and engage with technology even more. With that wireless hearing aid accessories are keeping up with the human demand to connect and engage. Dr. Emily McMahan's hearing center offers the latest in hearing aid accessories and tech allowing you to do so much with your hearing aids. Like stream audio direct from your mobile phone. Or stream audio from your TV or music player. You control what you listen to on your hearing aids. And with all the many mobile apps on the market as free downloads, you can control your hearing aid's volume & audio source with your smartphone. You can find hundreds of hearing aid accessories at Alaska Hearing & Tinnitus Center.


I remember being little and creating sweet homemade gifts to give to my friends and family. I want my children to learn all about this too. The idea of creating something using nothing but a few crafts supplies, lots of time, and loads of love. That is what gift giving is all about!
I started thinking about some sweet homemade gifts that kids can make all on their own. Perfect for Christmas, Mother's Day, Father's Day, Teacher Gifts, you name it.
I bring you 45 Homemade Gifts Kids Can Make!
Wind Spinner – My son worked hard to make this creation his own! What a great gift for someone's backyard.
Yarn-Wrapped Hearts – An engaging fine motor craft a child can give to show her affection.
Lavender Lotion Bars – these little lotion bars are so simple to make, and little ones can help each step of the way! They are so gentle, making them perfect for a baby or young child.
Wrapped Up in Lights Fingerprint Frames from Fun-A-Day – I love how the children were "decorated" for the photos included with these frames!
Hand Print Mason Jar from Fun-A-Day – Children's hand prints make some wonderful gifts for family. This jar could be used as a vase, to hold spare charge or pencils, etc.
Beaded Napkin Rings from Buggy and Buddy – Kids who enjoy beading will have a blast making these table decorations as gifts.
Watercolor Bookmarks from Let's Explore – What a great way to take kids' art to make a useful gift.
Washi Tape Wooden Bracelets from Mama Miss – These bracelets look beautiful, and I'm sure adults and kids alike would enjoy receiving them as a present.
Tissue Paper Votive Holders from Craftulate – A simple and fun craft kids can make and give as table decoration.
Homemade Hot Chocolate Mix from Fireflies and Mud Pies – Who doesn't like delicious hot chocolate? Kids can create the simple recipe to share with family and friends.
LEGO Photo Pen Holder from LalyMom – Children who love building with LEGO bricks will enjoy making this pen holder.
Sandpaper Printed Towels from Toddler Approved – Such a cool process to make a meaningful keepsake!
DIY Clay Bowls from Frogs and Snails and Puppy Dog Tails – These lovely bowls are easy to make, and I know mamas will love using them every day.
Polymer Clay Beads and Jewelry from Babble Dabble Do – These beautiful, colorful beads can be used to make some gorgeous jewelry to give as gifts!
Crayon Resist Photo Display from Twodaloo – Such a cool idea! Make a special gift for displaying photos.
Shrinky Bracelet from Rust & Sunshine – Kids can explore shrinky art while creating a wearable gift.
Painted Mosaic Styrofoam Tile Picture Frame from Pink Stripey Socks – This colorful frame looks like it would be such fun to make with the kids.
Recycled Art Magnets from Hand Made Kids Art – Such a useful way to recycle kids' old artwork!
Homemade Chocolates from Nurture Store – Teach kids how to make delicious chocolates to share with others.
Fabric Resist Art Pillows from Meri Cherry – What a gorgeous art technique for the kids to use as they make pillows to give to their family!
Tie-Dyed Bookmarks from Happy Hooligans – Look at those colors! Who wouldn't love receiving this homemade bookmark as a present?
Modern Hand Painted Trivets from Inner Child Fun – Such fun designs to spice up a useful kitchen gift.
Citrus Sugar Scrub from Happily Ever Mom – I can almost smell the fresh scent of this homemade gift.
Homemade Snow Globes from Our Best Bites – Something tells me the kids will have as much making these as giving them as presents!
Styrofoam Printed Cards from The Artful Parent – A set of handmade cards would be the perfect gift for the letter writers in a child's life.
Tin Can Lanterns from Kids Craft Room – These lanterns are beautifully thought out and decorated! I think they'd look nice on a back porch.
Painting with Trains on Ceramics from Play Trains – What an amazing idea – let a child's love for trains (or cars or trucks) guide the art process of the gift!
Fizzy Scented Bath Bombs from Playdough to Plato – Bath bombs are gifts kids can make while learning more about science!
Abstract Painted Scarf from Crafting Connections – I never would have thought of making a scarf this way!
Personalized Notebooks from Mum in the Mad House – Prolific writers and note takers would get a lot of use out of these homemade books.
Melted Bead Sun Catchers from The Artful Parent – Kids can create a colorful piece of art by melting beads, then give the resulting piece as a present. Perfect to decorate someone's garden.
DIY Computer Wrist Rest from Pink Stripey Socks – Such a cheerful wrist rest, and it's even more meaningful because it's made from a child's art!
Sharpie Art Coasters from Paging Fun Mums – These colorful coasters would surely look like works of art in someone's living room!
Paper Pulp Pendants from Babble Dabble Do – What family member wouldn't love to wear these heart pendants a child made?
Glitter Candles from Lessons Learnt Journal – A glittery twist on a traditional child-made gift. Lovely!
DIY Painted Mugs from Glued to My Crafts – Good tips for how to ensure these painted mugs last through the dishwasher. Tea and coffee drinkers would get a lot of use out of them!
Lavender Heart Bath Bombs from Red Ted Art – A homemade gift that encourages relaxation!
Potato Soup Mix in a Jar from Inner Child Fun – Kids measure and pour as they put together a delicious soup mix for family and friends.
Hand Print Sun Catchers from Homegrown Friends – Such a beautiful hand print gift that is sure to be treasured by any who receives it!
DIY Flower Bottles from Studio Sjoesjoe – I love how such simple materials can be used to create beautiful gifts like these flower bottles.
Hand-Shaped Ring Dish from Mama.Papa.Bubba – Kids' hand prints always seem to make the best gifts for family!
Pony Bead and Prism Sun Catchers from Rhythms of Play – Simple and pretty beaded sun catcher anyone would love to receive from a child.
Peppermint Lotion Bars from Ed Snapshots – I bet these lotion bars smell decadent!
Felt Embroidered Coffee Cup Cozies from CraftWhack – For kids who like to sew and embroider, these look fun to make!
Eco Friendly Printed Napkin from The Brashear Kids – Let kids turn a basic napkin into something with more pizzazz! A great gift for those who like to entertain!
What do you think? Loads of inspiration – right? Meaningful and thoughtful, while also being a lovely gift to receive – Win-Win!
Love all of them. Can't wait for my boys to get started on some of these as gifts for grandparents and mom and dad.
What an extraordinary list! Thanks for so many fabulous ideas!
So happy you like them Kate!
this is a great list. The second "ado" should actually be adieu, as in the German word for goodbye. but maybe you already googled that.
Thank you Megan! I feel more cultured already – I just love this little online home of mine, so many helpful people! Adieu new friend!
So glad you like them Karen!
I was looking for anniversary for my parents and was having struggles. This helped for sure. Have a blessed day.
Awesome list! Totally stole some ideas and will be ending up under the tree this year 🙂 Thanks for sharing!
Yay! So glad you found some ideas perfect for your little one Feriha.
The crafty gifts you've shared are lovely. Thanks for such fun ideas! The teacher in me won't let this one go- adieu is French for goodbye. It means, "to God" and is an alternative to au revoir. Au revoir implies hope of meeting again, while adieu is more final. Though Megan is correct in stating adieu means goodbye, the German phrase for goodbye is Auf Wiedersehen.
Well thank you so much! Now I know 🙂 And thank you for taking the time to share it so informatively. I appreciate it!


Summer is coming! The Friends of the Library Mini Book Sale starts May 31, so kick off your summer right!
Register for a card online! You can start reading eBooks immediately! Your online card is good for 30 days and you can come see us for an upgrade! For more info about what you can do with your card, click here.
Assistive Technology! Everyone can enjoy the library--even our visitors with vision impairments. Find out how when you click here.
Looking for upcoming events? Get the scoop from our calendar! or check out our online newsletter! Or sign up to get library news straight to your email!
Have you checked out a Museum Pass? For more info, click here. For Museum Pass availability, click here.
Do you have books to donate? Find out how!
Click here for wireless printing!
Search the wonders of the world!
Book a room at the library for your event or meeting!
Mon-Wed: 10 a.m. to 9 p.m.
Thurs-Sat:10 a.m. to 6 p.m. & Sun: 1 p.m. to 5 p.m.
Mon & Thurs: 10 a.m. to 9 p.m.
Tues, Weds, Fri & Sat: 10 a.m. to 6 p.m.
Mon, Wed & Sat: 10 a.m. to 6 p.m.
Tues: 10 a.m. to 9 p.m.


The City of Brighton's Eye for Art program is proud to announce there are two winners for the 2018 People's Choice Award for the Eye for Art Sculpture-on-Loan Sculpture Walk at Carmichael Park: Lawrence Starck of Loveland and Kathy Wardle of Brighton will each receive $500.
Starck received the award for his "The Farmer" bronze sculpture and Wardle received it for "The Warrior" bronze sculpture.
These two sculptures, in addition to three others, have been on display around Carmichael Park since May 2017. During this time, visitors were encouraged to vote online for their favorite piece. Paper ballots were also handed out at events held at Carmichael Park like the Annual City BBQ. All five sculptures were created by local and regional artists.
The sculpture exhibit will be on display until the end of March 2019. Our visitors have had the pleasure of strolling among these visually engaging sculptures that are for sale and on display.


We offer FOR SALE a selection of antique and new furnishings, millwork, architectural salvage, doors, and Objets d'art. We can reproduce anything except musical instruments.
Hand crafted veneer and carved bathroom vanity, in walnut. Bespoke design.
Profile of Nesting tables with curved steel supports and wood tops. The white hutch cabinets and steel facades help to reflect the product, and maximize sales.
A British Invasion that you'll love. Rocker!
Barn board banquette and heritage window.
Bespoke kitchen and interior designed and crafted in-house.
Pictured is our mock-up of a classic panelled kitchen we built.


text:
This gallery features a variety of natural scenes (some with human-made features) and landscapes from around south Florida.
🌴Pura Vida🌴
The forest sees all...
Early afternoon☀light at Myakka River State Park
Above the trees at Myakka
Sunset in the Florida Keys
Afternoon showers at Green Cay Wetlands
Fort Zachary Taylor Fortress
Construction for this fort in Key West began in 1845 and was named after USA's 12th president. More info can be found here: http://www.fortzacharytaylor.com/park-and-fort.html
Afternoon showers in the Florida Keys
Many people who haven't been to the Florida Keys think it's all palm trees and sandy beaches, but the natural shoreline looks more like this with limestone outcroppings with three different species of mangroves on and around them. The shallow waters here are teeming with life! This is way more interesting than a manicured sandy beach anyway ;)
Small lighthouse in the Florida Keys
Curry Hammock State Park
Curry Hammock State Park in Marathon in the Florida Keys has one of the largest wild populations of state-endangered thatch palms (Thrinax radiata) in the US. This palm is native throughout the Caribbean and is a popular landscaping plant. Its range extends into southern Florida, but it is rare to find except in small patches primarily in the Florida Keys.
Peace and relaxation
Relaxing scene in Islamorada
Everglades sunset
Everglades sunset 2


text:
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    <meta http-equiv="X-UA-Compatible" content="IE=edge">
    <meta name="viewport" content="width=device-width, initial-scale=1">

    <title>Grab Api</title>

    <link rel="stylesheet" href="https://maxcdn.bootstrapcdn.com/bootstrap/3.3.4/css/bootstrap.min.css">

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redpajama_set_name: RedPajamaGithub


text:
Tower Hamlets sixth busiest borough for London's Air Ambulance in 2017
PUBLISHED: 15:17 15 March 2018
Rhiannon Long
London's Air Ambulance Charity relies on donations to keep services running. Picture: Matthew Bell
London's Air Ambulance treated 82 patients in Tower Hamlets last year, making it the sixth busiest borough for call-outs.
The majority of patients treated in 2017 were victims of stabbings or shootings. Picture: London Air Ambulance
The charity delivers care via helicopter and rapid response vehicle to patients in life-threatening situations.
In 2017 they treated 1,797 patients across the city, with almost a third of these being stabbing or shooting victims.
Lead clinician Dr Gareth Grier said: "Sadly, through 2017 the number of missions where a patient has been stabbed or shot has risen. It is not unusual now for our teams to perform open chest surgery for stab wounds twice in a single day. This would have been unheard of a few years back. "But our world leading treatments means that we can give these and other patients we treat the best possible chance of survival.
"The challenge we have is to innovate further to help our patients even more."
The charity costs £10 million a year to run and survives mainly from donations.


text:
<?php

 	require_once(__CA_LIB_DIR__."/core/Error.php");
	require_once(__CA_MODELS_DIR__."/ca_users.php");

 	class ProfileController extends ActionController {
 		# -------------------------------------------------------
 		function Edit() {
 			$t_user = $this->request->user;

 			$this->view->setVar("fname", $t_user->htmlFormElement("fname","
"._t("First name")."
^ELEMENT
"));
 			$this->view->setVar("lname", $t_user->htmlFormElement("lname","
"._t("Last name")."
^ELEMENT
"));
 			$this->view->setVar("email", $t_user->htmlFormElement("email","
"._t("Email address")."
^ELEMENT
"));
 			$this->view->setVar("password", $t_user->htmlFormElement("password","
"._t("Password (Leave blank to keep existing password)")."
^ELEMENT
", array('value' => '')));
 			$this->view->setVar("password_confirm", '
Password confirm (Leave blank to keep existing password)
<input name="password_confirm" value="" size="60" maxlength="100" autocomplete="off" type="password">
');

 			$va_profile_prefs = $t_user->getValidPreferences('profile');
 			if (is_array($va_profile_prefs) && sizeof($va_profile_prefs)) {
 				$va_elements = array();
				foreach($va_profile_prefs as $vs_pref) {
					$va_pref_info = $t_user->getPreferenceInfo($vs_pref);
					$va_elements[$vs_pref] = array('element' => $t_user->preferenceHtmlFormElement($vs_pref), 'formatted_element' => $t_user->preferenceHtmlFormElement($vs_pref, "
".$va_pref_info['label']."
^ELEMENT
"), 'info' => $va_pref_info, 'label' => $va_pref_info['label']);
				}

				$this->view->setVar("profile_settings", $va_elements);
			}

 			$this->render('Profile/profile_html.php');
 		}
 		# -------------------------------------------------------
 		function Save() {
 			$t_user = $this->request->user;

			# --- process incoming registration attempt
			$ps_email = strip_tags($this->request->getParameter("email", pString));
			$ps_fname = strip_tags($this->request->getParameter("fname", pString));
			$ps_lname = strip_tags($this->request->getParameter("lname", pString));
			$ps_password = strip_tags($this->request->getParameter("password", pString));
			$ps_password_confirm = strip_tags($this->request->getParameter("password_confirm", pString));

			if (!caCheckEmailAddress($ps_email)) {
				$va_errors["email"] = _t("E-mail address is not valid.");
			}else{
				$t_user->set("email", $ps_email);
			}
			if (!$ps_fname) {
				$va_errors["fname"] = _t("Please enter your first name");
			}else{
				$t_user->set("fname", $ps_fname);
			}
			if (!$ps_lname) {
				$va_errors["lname"] = _t("Please enter your last name");
			}else{
				$t_user->set("lname", $ps_lname);
			}
			if($ps_password && ($ps_password != $ps_password_confirm)){
				$va_errors["password_confirm"] = _t("Please confirm your password");
				$va_errors["password"] = _t("Please re-enter your password");
			}

			// Check user profile responses
			$va_profile_prefs = $t_user->getValidPreferences('profile');
			if (is_array($va_profile_prefs) && sizeof($va_profile_prefs)) {
				foreach($va_profile_prefs as $vs_pref) {
					$vs_pref_value = $this->request->getParameter('pref_'.$vs_pref, pString);
					if (!$t_user->isValidPreferenceValue($vs_pref, $vs_pref_value, true)) {
						$va_errors[$vs_pref] = join("; ", $t_user->getErrors());

						$t_user->clearErrors();
					}
				}
			}

			# get names of form fields
			$va_fields = $t_user->getFormFields();

			# loop through fields
			foreach($va_fields as $vs_f => $va_attr) {
				switch($vs_f){
					case "user_name":
					case "active":
					case "userclass":
						# noop
						break;
					# -------------
					case "password":
						if(!$va_errors[$vs_f] && $_REQUEST[$vs_f]){
							$t_user->set($vs_f,$_REQUEST[$vs_f]); # set field values
							if ($t_user->numErrors() > 0) {
								$va_errors[$vs_f] = join("; ", $t_user->getErrors());
							}
						}
						break;
					# -------------
					default:
						if(!$va_errors[$vs_f]){
							$t_user->set($vs_f,$_REQUEST[$vs_f]); # set field values
							if ($t_user->numErrors() > 0) {
								$va_errors[$vs_f] = join("; ", $t_user->getErrors());
							}
						}
						break;
					# -------------
				}
			}

			// Save user profile responses
			if (is_array($va_profile_prefs) && sizeof($va_profile_prefs)) {
				foreach($va_profile_prefs as $vs_pref) {
					$t_user->setPreference($vs_pref, $this->request->getParameter('pref_'.$vs_pref, pString));
				}
			}

			if(sizeof($va_errors) == 0){
				$t_user->setMode(ACCESS_WRITE);
				$t_user->update();
				if($t_user->numErrors()) {
					$va_errors["register"] = join("; ", $t_user->getErrors());
					print $va_errors["register"];
				}else{
					$this->notification->addNotification(_t('Your profile has been updated.'), __NOTIFICATION_TYPE_INFO__);
				}
			}
			$this->view->setVar('errors', $va_errors);

			$this->Edit();
 		}
 		# -------------------------------------------------------
 	}
 ?>


Add some style to your weekend cook-out! Whether you're grilling hamburgers or preparing a five course meal, you'll be able to take on the mess. But, even more impressive, Saint James offers a specially designed garment, to protect your body against the harmful solar rays. This polyamid microfiber offers a skin protection twice as much superior to Cotton. You will also appreciate its easy care, drying and active breathing properties.
I bought this to wear at our lake house, it's perfect, nothing screams summer by the water better than navy and white stripes. It's a great apron, fits well, and has a well positioned pocket at the front. I look forward to many years of use.
This knit apron is large, so it will fit almost everyone. It has handy pockets in front and is nice and soft. I recommend it!


text:
Young South Sudanese leader: 'Let'..
Young South Sudanese leader: 'Let's stop being politically correct'
3AW Breakfast
A young South Sudanese man who himself became a victim of gang violence has told how he believes Melbourne's gang crime problem should be tackled.
In 2011 Nelly Yoa came to the aid of a stranger and was stabbed with a machete, almost bleeding to death.
He now volunteers with offenders of Sudanese descent, telling 3AW Breakfast the current approach to gangs clearly isn't working.
"I was sort of in disbelief when the police said gangs don't exist," Mr Yoa said.
"Let's stop being politically correct and call it for what it is."
Click PLAY below to hear the full details
Mr Yoa also penned a powerful piece in today's Age, calling for action.
"It discredits all the hard work the majority of the South Sudanese community has done," he said.
"This is getting out of control, the wider community is sick of it.
"Someone's going to get killed."


Each one of us has been created by God for a special purpose and is therefore uniquely special. Together we pray , grow and learn in the footsteps of Jesus.
Please hover over the tabs to view different policies.


Gas-powered Weed Eater push mowers contain two springs that control the performance of the carburetor. The longer, thicker spring is popularly known as the primary spring along with also the shorter, thinner spring is known as the secondary spring. Both springs work together in concert to control the speed of the mower's engine during idle and operation. Installing these springs correctly is crucial to make certain that your Weed Eater mower's engine operates as it needs to be.
Disconnect the spark plug wire from the spark plug. This prevents accidental motor ignition during the repair.
Remove the air filter cover. Remove the air filter. Unscrew the screws that hold the air filter foundation in place and pull the foundation in the machine to get the area where the springs should be installed atop the carburetor.
Attach the stretched loop at the conclusion of the main spring to the hook to the governor arm linkage. Insert the round, circular loop at the other end of the primary spring to the corresponding hole in the throttle control lever. It is simple to locate this hole by choosing the one that lines up with the end of the spring.
Attach either end of this small secondary spring to the hook to the governor arm linkage. This is the exact same one where you connected the stretched loop of the main spring. Hook the other end of the secondary spring to the empty hole to the throttle control lever.
Reattach the air filter base to the side of the mower by screwing it into place. Reinstall the air filter and the air filter cap.
String trimmers are a blessing for homeowners with obstacles or structures within their lawns that lawn (San Diego, CA) mowers can't get around. The Echo GT 200R is an air-cooled, two-stroke gas trimmer that comes with a nylon line head. If you're looking for a more secure and powerful head, the Maxi-Cut head is a three-blade head with permanent blades rather than trimmer line. Altering the nylon string head for the Maxi-Cut head isn't too difficult and can be achieved in a matter of 10 minutes with basic tools.
Align the hole in the top plate with the hole in the bearing housing (located on the shaft between the shield and head). Install the locking tool (a small, 3/16-inch diameter tool) into the hole.
Remove the 3/8-24 locknut on the bottom of the cotton mind employing a 9/16-inch wrench.
Remove the washer and slide the cotton line head off the shaft, taking good care to catch the huge washer that lies between the shaft assembly and the nylon head.
Tighten the nuts and bolts on each one of the 3 blades with a normal screwdriver and wrench until the blades have been tight enough.
Slide the threaded shaft into the hole on the Maxi-Cut head.
Put in the flat washer and locknut. Tighten the locknut until the head does not rotate on the beam.
Start the trimmer and ensure the blade is running smoothly; it shouldn't wobble on the shaft, nor should it sound as though the blades have been loose.
Cracks develop in driveways when climate extremes of cool and warm trigger growth and contraction of the concrete. In the summer months, when it rains or you clean water and the vehicle the Lawn Care, water erodes the soil beneath the concrete and sifts down through the cracks. Erosion causes the soil to to be in, which creates pressure on the on the concrete, causing it to crack. In the winter, when ice or snow melts through the day and incurs the cracks, it may freeze. The issue basically grows larger when the driveway isn't caulked instantly.
Scrape out the crack using a screwdriver to loosen rocks, dirt or remnants of caulking.
Scrub using a wire brush to complete breaking free materials aside.
The crack that is sweep clear using a whiskbroom of free materials.
Cracks which are under ¼ inch-wide from a gun full of concrete fix caulk sealant. Place the tip of the caulk tube to the crack and hold the trigger down, filling the crack as you pull the caulk tube over the crack to somewhat over-flowing.
Press the caulk using the the top of concrete with caulk blade that is plastic.
Allow the caulk to dry in line with the directions of the manufacturer's.
Repeat the procedure for every single crack which you need to caulk.
Seal cracks that are bigger by pushing a foam backer rod to the crack and along its size using a screwdriver. The backer rod should be ½ inch below the the top of concrete to permit space for the caulk to be used along with it.
Till it's overflowing fill the crack with fix caulk sealant.
To ensure it's level with all the the top of concrete spread over the crack.
Colorful torenias (Torenia fournieri) are usually called by their genus name or wishbone blossoms because of their wishbone-shaped stamen. They are annuals indigenous to tropical Asia and come in a huge array of colors. You can buy their seeds in many garden supply centres, but hybrids seeds yielding amethyst-colored flowers Flagstaff are patented. You sow and germinate seeds of patented plants exactly the identical way as conventional cultivars, but there are limitations on how you're permitted to propagate them.
Standard torenia flower cultivars bear rose, pink, purple, purple, blue or white blooms. You might buy seeds of 2 torenia amethyst hybrids on the marketplace, torenia amethyst (Torenia fournieri "Summer Wave Amethyst") or torenia big amethyst (Torenia fournieri "Summer Wave Large Amethyst").
Torenia amethyst cultivars are patented. Nurseries who develop patented plant Boise hybrids get a royalty for every Stump Removal sold. You cannot legally propagate a plant that is patented by putting a cutting. You can, however, save and sow their seeds.
Torenia seeds develop in tiny pods that ripen at the end of the growing season. Eliminate the completely mature seeds in the pods, let them dry thoroughly, store them in a cool, airtight container to sow following spring. Seeds saved from hybrid crops like the torenia amethyst cultivars might not resemble their parent in size or colour. The pollen they get can also impact their appearance.
Torenias will climb as summer annuals U.S. Department of Agriculture plant hardiness zones 2 through 11. They require temperatures over 40 Fahrenheit; frost will destroy them. Prior to plant their tiny seeds function 4 to 5 pounds of 10-10-10 or 5-10-10 fertilizer into the top 4 to 6 inches of 100 square foot of rich, well-drained dirt. If you live in a northern climate, plant them in sunlight. If you live in a warmer climate, plant them in a place with morning sun and afternoon shade Press the seeds into the dirt after the last spring frost has passed and the soil has warmed. Torenia will grow from 8 to 12 inches high, spreading approximately 6 inches broad, so plant them in 6 to 8 inches apart. At 70 F, they will germinate in 10 to 15 days. It's a good idea to plant them indoors 10 to 12 weeks before you expect the last spring frost in your area. They evolved in a humid, warm climate, so keep the dirt of germinating seeds and seedlings moist but not soggy.


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Renovation Diaries: don't fight it
Hey diary.
We are now more than a year past our renovation completion timeline. A horrible combination of the pandemic, material shortages, rising costs, and a general lack of poor planning brought us here. It's easy to feel overwhelmed or want to point fingers at everyone involved, but we are trying to lean into the spirit of the season and choose peace.
It's easier said than done, but we take great comfort in knowing how much is complete. Our lobby is looking as sexy as ever. Our floors are complete. The new doors are in, and it's so quiet now in the rooms we know everyone is going to sleep well. The library is finally (finally!) painted and we personally pulled the staples out of that original floor.
So, yeah. Things are looking up. They are still a year behind schedule, but they're... up. Whatever that's supposed to mean.
Best wishes and positive thoughts are being sent to city leadership now, and we are praying that the final inspections are easy, that they are kind to small business, and reasonable. Fingers and toes crossed over here!
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New program is helping young kids in Louisville stay away from violence
Updated: 9:04 PM EDT May 20, 2021
Munashe Kwangwari
SCRUBS WITH THE STETHOSCOPE CHILDREN AS EARLY AS PRESCHOOL ARE BEING TAUGHT HAD A STEER CLEAR OF CRIME. IT'S A NEW INITIATIVE SPEARHEADED BY A LOCAL ACTIVIST AND HEALTH PROFESSIONALS GIVING KIDS THE INSPIRATION NEEDED TO FIGHT VIOLENCE IN THEIR COMMUNITY WHILE THE SOUND OF GUNFIRE SHATTERS A SENSE OF SECURITY IN SOME LOCAL NEIGHBORHOODS. IT ALSO SHATTERS THE INNOCENCE OF YOUNG CHILDREN. IT'S THE REASON WHY LOCAL ACTIVISTS CHRISTOPHER 2X AND U OF L HEALTH PROFESSIONALS ARE SHOWING CHILDREN HOW THEY CAN BRING OUT CHANGE I CAN BE A DOCTOR SO WE CAN HELP A LOT OF PEOPLE. IT'S THE NEW MINDSET COMING FROM KIDS IN THE FUTURE HEALERS PROGRAM. WE'RE SAYING WHY NOT INSPIRE THEM TO BE PART OF THE HEALTHCARE ARENA THE SAME WAY Y'ALL HEAR ABOUT SHOOTING THE DAMAGE OUT HERE BOY. I CAN BE THE HEALERS JUST LIKE THE PEOPLE IN THE HOSPITAL WITH THE HELP OF U OF L MEDICAL STUDENTS KIDS IN THE PROGRAM PARTICIPATE IN EDUCATIONAL HANDS-ON ACTIVITIES TO LEARN ABOUT GENERAL. OUT OF ME AND SURGERY IT WAS FOREIGN TO ME. I DON'T KNOW WHY IT'S ABOUT HELPING PEOPLE THAT THEY HURT THE SICK. OR BLEEDING. THAT'S WHY IT'S FOR KIDS. THE HOPE IS TO INSPIRE KIDS TO MAKE A DIFFERENCE SOMETHING DR. KEITH MILLER WITH U OF L HEALTH SAYS MAYBE THE ANSWER TO CURBING THE SPREAD OF VIOLENCE PEOPLE HAVE GENUINE INTEREST AND AND THEY FIND MOTIVATION AND SEEING THAT I CAN DO THIS TOO. OFFICIAL SAINT EARLY INTERVENTION IS IMPORTANT BECAUSE IT HELPS KIDS. DEVELOP A PLAN TO DO SOMETHING POSITIVE IN LIFE TO REACH A CERTAIN GOAL BY HAVING SEEN SOMEBODY WHO'S ALREADY THERE. I THINK IT MAKES A BIG DIFFERENCE AN IDEA THAT'S PROVING TO BE TRUE FOR NINE YEAR OLD SHAMAN RICHARDSON. I DID LIKE IT IF A DOCTOR FOR A DAY I WANT TO BE A GOOD PERSON
A new program is inspiring young kids in Louisville to create a better future for themselves and the community.It's called the Future Healers Program, the result of a collaboration between local activist Christopher 2X, UofL Health and University of Louisville medical students.Leaders of the program note while constant gun violence shatters a sense of security in some local neighborhoods, it also shatters the innocence of young children. They believe this program will be one of the leading efforts in preventing the younger generation from getting into the line of crime."We're saying, 'Why not inspire them to be a part of the health care arena,'" Christopher 2X said. "The same way they hear about shootings and damage out here, we are saying to them, they can be the healers like the people in the hospital."The program offers educational hands-on activities to learn about general anatomy and surgery."It was fun to me," said 9-year-old future healer, Shamun Richardson. "It's about helping people if they're hurt, or sick, or bleeding. That's why it's for kids."The hope is to inspire kids to make a difference, something UofL Health officials believe may be the answer to curbing the spread of violence. Transplant surgeon Dr. Christopher Jones said getting into the minds of the kids this early is important because it helps them develop a plan to do something positive in life."When you're trying to reach a certain goal, by having seen someone who is already there, I think it makes a big difference," Jones said.It's an idea that's proving to be true for 9-year-old Shamun."I did like being a doctor for a day," Shamun said. "I want to be a good person, not a bad person."In mid-February, leaders conducted a pilot session of the future healers program with about 50 preschool and middle school children at the Chestnut Street YMCA. The session was virtual to adhere to COVID-19 mitigations.Christopher 2x said their hope is to bring the sessions back this summer, offering in-person opportunities for those interested.
A new program is inspiring young kids in Louisville to create a better future for themselves and the community.
It's called the Future Healers Program, the result of a collaboration between local activist Christopher 2X, UofL Health and University of Louisville medical students.
Leaders of the program note while constant gun violence shatters a sense of security in some local neighborhoods, it also shatters the innocence of young children. They believe this program will be one of the leading efforts in preventing the younger generation from getting into the line of crime.
"We're saying, 'Why not inspire them to be a part of the health care arena,'" Christopher 2X said. "The same way they hear about shootings and damage out here, we are saying to them, they can be the healers like the people in the hospital."
The program offers educational hands-on activities to learn about general anatomy and surgery.
"It was fun to me," said 9-year-old future healer, Shamun Richardson. "It's about helping people if they're hurt, or sick, or bleeding. That's why it's for kids."
The hope is to inspire kids to make a difference, something UofL Health officials believe may be the answer to curbing the spread of violence. Transplant surgeon Dr. Christopher Jones said getting into the minds of the kids this early is important because it helps them develop a plan to do something positive in life.
"When you're trying to reach a certain goal, by having seen someone who is already there, I think it makes a big difference," Jones said.
It's an idea that's proving to be true for 9-year-old Shamun.
"I did like being a doctor for a day," Shamun said. "I want to be a good person, not a bad person."
In mid-February, leaders conducted a pilot session of the future healers program with about 50 preschool and middle school children at the Chestnut Street YMCA. The session was virtual to adhere to COVID-19 mitigations.
Christopher 2x said their hope is to bring the sessions back this summer, offering in-person opportunities for those interested.


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SPbU is opening a representative office in the Republic of Korea
On May 23, St. Petersburg State University will open its foreign representative office in Seoul. The project will be implemented at Hankuk University of Foreign Studies.
The VIP guests at the opening ceremony will be Rector of SPbU Nikolai Kropachev, SPbU Deputy Rector for Foreign Affairs Sergey Andryushin, Head of SPbU Rector's Office Maria Gusarova, and Director of the Directorate of the Russia — Republic of Korea Dialogue forum Inna Tsoi.
In Korea, the position of SPbU Representative Office Director (http://builder.hufs.ac.kr/user/spbupredstvo/) will be held by Kim Hyun Taek, Vice President for International Affairs at Hankuk University.
The main areas of work for the office will be organizing joint educational and research projects, holding academic conferences and other events for students and teachers from Korean universities, as well as promoting the Russian language, history and culture in Korea and the Korean language, history and culture in Russia.
Saint Petersburg State University is the largest center of Korean studies in Russia. The University has agreements with 18 universities and research organizations of the Republic of Korea. SPbU Institute of Interdisciplinary Research of Korea carries out a whole number of ambitious research projects jointly with the Korea Foundation and the Academy of Korean Studies. With every passing year, SPbU students show more and more interest in academic mobility programs with Korean partner universities of SPbU; at the same time, foreign internships at Korean universities now involve not only students specializing in Korean studies, but also those who study in the fields of sociology, international relations, management, journalism, history and economics.
Moreover, St. Petersburg State University provides organizational support for the Russia — Republic of Korea Dialogue civil society forum, which has been established as an international platform for discussing topical issues. The main objective of the Forum is to build a constructive dialogue between the representatives of the two countries and form a solid basis for Russian-Korean cooperation which would be independent of the current political situation.


My favorite go to recipe for these cookies is by my friend Jenny Keller. They are easy to make, no fail, and are super yummy.
After you have frosted them, simply turn upside down onto a plate filled with sprinkles and push down gently to coat. These are the sprinkles I used.
This is the exact cookie cutter I used.


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Arsenal star Pierre-Emerick Aubameyang to reveal new masked superhero celebration
Coral BarryMonday 11 Mar 2019 11:00 pm
Aubameyang could not find his mask after scoring against United (Picture: Getty)
Pierre-Emerick Aubameyang hopes to reveal his new masked superhero celebration on Thursday when Arsenal take on Rennes in the Europa League.
Arsenal welcome Rennes to the Emirates trailing 3-1 from the first leg and on the brink of an early exit from Europe.
Aubameyang, who has two goals in six Europa League appearances this season, was thwarted in his attempts to show off his new celebration against Manchester United on Sunday.
Aubameyang has a history of elaborate celebrations (Picture: Getty)
The striker scored a penalty to make it 2-0 to Arsenal, but could not find his prop behind the goal.
'I had a special celebration planned but I couldn't find my mask,' Aubameyang said after the game.
'I asked a member of the staff to put it in the back of the net but when I scored the penalty I couldn't find it.
'But no worries. Maybe next time.'
Aubameyang is Arsenal's main goal threat and has scored a remarkable 30 times in just 52 appearances for the club.
Arsenal will need the Gabon international firing on all cylinders against Rennes as Unai Emery aims to rescue the Gunners' European campaign.
Aubameyang played in Arsenal's 3-1 defeat in France which saw Sokratis sent off in the first half.
Alex Iwobi grabbed a crucial away goal for Arsenal, but the Gunners cannot afford to concede again at home.
Arsenal FCEuropa LeagueManchester United FCPierre-Emerick AubameyangUnai Emery