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More info on Lucerne, Kings County, California
Lucerne, Kings County, California: Wikis
Categories: Kings County, California > Unincorporated communities in California
— Unincorporated community —
Location in California
Coordinates: 36°22′51″N 119°39′51″W / 36.38083°N 119.66417°W / 36.38083; -119.66417Coordinates: 36°22′51″N 119°39′51″W / 36.38083°N 119.66417°W / 36.38083; -119.66417
Elevation [1] 259 ft (79 m)
Lucerne is an unincorporated community in Kings County, California.[1] It is located on the Atchison, Topeka and Santa Fe Railroad 3.5 miles (5.6 km) north of Hanford,[2] at an elevation of 259 feet (79 m).[1]
The name Lucerne was applied in 1887, after Lucerne, Switzerland.[2]
^ a b c U.S. Geological Survey Geographical Names Information System: Lucerne, Kings County, California
^ a b Durham, David L. (1998). California's Geographic Names: A Gazetteer of Historic and Modern Names of the State. Quill Driver Books. p. 1067. ISBN 9781884995149.
Municipalities and communities of
Kings County, California
County seat: Hanford
Avenal | Corcoran | Hanford | Lemoore
Armona | Home Garden | Kettleman City | Lemoore Station | Stratford
Acebedo | Cimarron | Fort Roosevelt | Gepford | Grangeville | Guernsey | Halls Corner | Hamblin | Hardwick | Helm Corner | Hub | Hydril | Kettleman Station | Lucerne | Murray | Pitco | Reef Station | Remnoy | Rossi | Shell | Shirley | South Corcoran | Vanguard
Banner | Bean | Cloud | Cornwell | Cuneo | Hasset | Joneso | Kimble | Kingston | Lethent | Marsala | Millham City | Odessa | Orion | Robinson | Watertown
Categories: Kings County, California | Unincorporated communities in California | San Joaquin Valley geography stubs
Hidden categories: Infobox Settlement US maintenance

redpajama_set_name: RedPajamaCommonCrawl


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Cartoon by DonkeyHotey
Is President Donald Trump Going Senile In Front of Us?
Time to talk—again, about his mental health.
Mental health issues are often discussed with great care, in part due to its sensitiveness and how devastating the diseases are for the people suffering them and their relatives. But when it comes to President Donald Trump the conversation must not be avoided, nor should be out of the equation either. As a public servant, people hold authority over him (theoretically) which means his work and actions should be scrutinized closely, this includes the public debate about his mental condition which has intensified recently.
At 70, Trump is one of the oldest presidents in US history that has ever taken office, when couple with the many early dementia signs he shows, the age-risk factor, and the fact that his father Fred developed Alzheimer's in his 80s (family history puts him at risk of developing the condition), the issue takes high new relevance.
In a powerful study, scientists at Arizona State University analyzed cognitive decline in Alzheimer's disease in Presidents by going through their press conferences' speeches while in office. Transcripts of Ronald Reagan (1981–1989) and George H.W. Bush (1989–1993) have been diligently examined.
The results showed, that while President Bush didn't show any relevant signs of dementia, Reagan — diagnosed in 1994 with Alzheimer's — however, showed a significant deterioration in the number of unique words over time, a noteworthy proliferation in conversational fillers (well, so, um, ah, basically, literally, actually) and non-specific nouns (thing, something, anything) rose over time, that suggested he was suffering from dementia during his tenure as president, and we never knew about its existence.
Based on the rapid deterioration of his cognitive functions — which he has been displaying in the last 2 years. Many of Trump's physical characteristics point toward dementia: forgetfulness (leaving an executive order blank without signing it); incoherence speech, irritability, anger, impulsivity, and paranoia; degraded syntax: reliance on placeholders ('great, great, great'), limited vocabulary and reduced concentration.
Some experts think he is incapable of being in office. Some of them have even published a book concerning the issue. Last April, a prominent group of psychiatrists also warned the public during a conference at Yale University stating that the president was unfit for presidency due to mental illness. All these mental health professionals coming forward to talk about the president's cognitive functions is unprecedented, according to the New Yorker since the Goldwater rule was added to the American Psychiatric Association's code of ethics back 1973, professionals in the field have been forbidden to diagnose a person without examining them in person. But since the arrival of Trump to the spectrum, some of them have been skipping the rule by using the "duty to warn" term— often used in law — a concept that arises when a subject may be an imminent of danger to society.
Professionals have also argued that the Goldwater rule serves only to protect the guild from lawsuits and conflict of interests. But to others, this represents a huge issue. Recently professionals from the fields of psychiatry, psychology, medicine, public policy, and social work — just to name a few — have come together to register the phrase "Duty To Warn" as an organization to advocate for Trump's removal under the 25th Amendment and warn people about the President's mental instability.
President's Trump mental health is the elephant in the room, and we should be talking openly about it. If you have heard his rants, read his tweets or seen his awkward encounters on TV, you can tell how often Trump forgets facts and people's names, how he wanders around press conferences seemingly not having an idea 'how to behave' or what he is 'there' for, how his spoken and written vocabulary limitations are blatantly noticeable — the latter being worthy of a fourth-grader according to the Boston Globe. All these behaviors are frequently linked to the early onset of dementia.
If Trump's signs lead to the develop of a more serious condition, not only his job, his position as geopolitical leader are at risk, but if it worsens, and if it's not addressed accordingly by the Congress or the people, a mentally-ill individual in the White House serving as President could bring calamitous consequences, not only for the US economy and democracy but also for the entire world.
Even North Korean "Supreme leader" has come at him in a silver-tongued statement calling him "dotard" and "mentally deranged" (ouch). In what world are we living now that the Dictator of NoKo — Trump's new hype term — has more eloquence and a better rhetoric than (what is supposed to be) the leader of the free world?
Now let's take a closer look at the three most relevant symptoms of early dementia stage that Trump has showed so far:
1. Difficulty finding the right words
Unless you've been living under a rock you've probably seen the poorly functional use of language, and the lack of coherence of Trump's rhetoric. The gradual loss of the ability to understand and use of speech is one of the crucial factors of dementia according to the Mayo Clinic.
It doesn't matter if he continues to shout from every corner that he has "the best words" — which is another symptom in itself. Trouble finding the right words is a huge — yuuge — red flag, and that, as a high-level politician leaves much to be desired.
Being the leader of one of the superpowers of the world, Trump has to be careful with the language that he uses and his discourse should show some coherence at least. Unfortunately, that hasn't been the case, although has he been always like this? Apparently not, in the video embedded below you can tell that his speech has vastly changed since the 90's. You can see him speaking very calm, lucid, almost articulate in an interview on Letterman back in 1988, his gestures and movements are quite different as they are now. Spoiler alert: he was still an arrogant back then. The linguistic decline is striking though.
2. Being repetitive
An important deficit. Trump may have a substantial vocabulary and might be not using it as a conscious strategy, instead using a simple repetitive rhetoric — one that is widely used in advertising. But it seems that sometimes he's not even aware of it, and no, chanting "Make America Great Again" does not qualify for this category. Below some examples.
3. Behavioral problems
There is no doubt of Mr. Trump's lack of social awareness. He constantly engages in irrational and hostile outbursts, shows lack of tact (telling President's Macron wife that she was in 'such a good shape'), and aggression. These actions are in fact, an indicative of the state of his mind and its lack of rationality. Judging from his behavior Donald Trump might be excelling in this category. Some of the signs include:
Becoming suspicious of other people and paranoia
Emotional impediments are a threat for national security, and when you have a dementia-stricken individual who clearly doesn't know how to behave, taunting nuclear power when confronted by an insane military dictator, one can only argue that Trump's mental deterioration could put us all at extreme risk, his actions are dangerous from a geopolitical point of view. Nobody needs an unpredictable 'leader' in a world full of perils and challenges.
The questions around President's Trump mental health should never go away, But, can we use the 25th amendment to invoke him on the basis of dementia? Are Donald Trump's actions carefully calculated or is he showing the world a side of him that even he is not aware of? It would be interesting to see the results of a mental status examination on him, and him being impeached for it.
America Has a Historic Choice in 2020
Right Smack in the Middle in The Bigger Picture
Yes, there is a bias against Kamala Harris. No, it is not what sunk her campaign.
George Evans-Jones
"Shut Up, Hamlet, and Drive": Appeal for an Essential Art
Carla Seaquist
United States Voting Integrity is at Risk with Smartmatic Voting Systems
Joanne Cipressi
Christians Follow Christ — Not Politicians
Eric Sentell in The Bigger Picture
Let's Talk About the Real Looting Going On in America.
Lauren Elizabeth in Dialogue & Discourse
The Ins & Outs of Being Ideologically Dishonest (Cognitive Distortions and all)
John-Pierre Maeli
Some in the conservative media were slow to embrace Trump. Will they rebuke him now?
The Millennial Source


text:
\section{Introduction}
	The emission of two protons is an intriguing and exotic decay process, energetically possible in only a few nuclei close to the proton stability line. The first theoretical studies in this field were done in the sixties by Goldansky \cite{Goldansky_1960}, using a semi-classical 2-body formalism, who proposed two extreme mechanisms for the emission, sequential and simultaneous. Since then, various attempts have been made at describing the two proton emission. The simpler models, using semi-classical tools, have various degrees of success \cite{Sreeja_2019,Liu_HM_2021,Liu_YB_2021}. They generally depend on multiple parameters, but yield a relatively good predictive power. Nevertheless, when applied systematically to all known emitters, they reveal interesting patterns and help shed light on this difficult theoretical problem \cite{Delion_2022}.

	The modern consensus is that regardless of the underlying mechanism, $2p$ emission is the three body process by nature and rigor dictates it must be treated in the hyperspherical harmonics framework \cite{Grigorenko_2001}. A number of interesting models have been developed in the last two decades, among others, coupled channels (CC)-like ones \cite{Grigorenko_2001,Grigorenko_2003,Grigorenko_2010,Wang_2018}, and R-matrix description \cite{Brown_2003}. While detailed and exhaustive, these models are also quite complex and various technical difficulties arise in contrast to 2-body processes. The spurious 2-body bound states in the nucleus-proton interaction have to be removed in an accurate fashion. The interaction between the emitted protons breaks the spherical symmetry and accurately finding 3-body resonances requires a large number of partial waves to be considered. However, such drawbacks are unavoidable in fully microscopic 3-body calculations.

	For the above reasons we propose in this paper an alternative to a fully microscopic theory. It has been shown by Grigorenko \cite{Grigorenko_2001} that neglecting all but point-like Coulomb interactions between the emitted fragments leads to a good order of magnitude estimate of the lifetime of the decaying nucleus. However, such a model does not provide much insight into various nuclear quantities. We propose that pairing correlations between the emitted protons can be investigated by allowing the protons to interact also via a nuclear potential. A great simplification can still be made by considering the protons interact with the nucleus only through Coulomb potentials. However, this clearly cannot hold when the protons are close to the daughter nucleus. Consequently, in this region we use the prescription of Delion et al \cite{Delion_2013}, with some modifications, to compute the wave function of paired protons in a resonant 2-body state. In this sense our model is \textit{semi-microscopic}. The total wave function cannot be rigorously determined together with its derivative. However, we will show that interesting studies can be made on the influence of the $pp$ paring strength on the partial life-time of the decaying nucleus.

	The paper is structured as follows: in Section~\ref{sec:Formalism} we elaborate on the procedure to obtain the 3-body wave function and the decay width. We give recipes on the building of external and internal region wave functions, then use the current formulation to extract the (total and partial) decay width(s); in Section~\ref{sec:Results} we analyze various aspects of the $2p$ emission problem. We discuss the nature of the potential matrix, revealing that from a certain radius, the problem becomes practically uncoupled. We then study the partial waves obtained in the external region, showing that the asymptotic behavior is reached relatively soon outside the barrier. Then, we study the dependence and stability of the decay width on the proton pairing strength and matching point between internal and external regions, respectively.

	\section{Formalism}
	\label{sec:Formalism}
	The simultaneous emission of two protons from a parent nucleus $P$ can be written schematically as
	\begin{align}
		_{Z}^{A}P\to _{Z-2}^{A-2}D(\mathbf{r}_3) + p(\mathbf{r}_1) + p(\mathbf{r}_2),
	\end{align}
	where $D$ is the daughter nucleus in ground state, and $\mathbf{r}_j$ denote the position of the three fragments in the laboratory frame. We will work in the approximation of infinitely heavy nuclei, hence $D$ is at rest and we set $\mathbf{r}_3 = \mathbf{0}$. Also, the total kinetic energy released in this process is called $Q$-value and is the sum of the kinetic energies of the two emitted protons
	\begin{equation}
		Q_{2p}=\epsilon_1 + \epsilon_2.
	\end{equation}
	This process is governed by the time-dependent Scr$\ddot{o}$dinger equation
	\begin{equation}
		i\hbar\frac{\partial \Psi(\mathbf{r},t)}{\partial t} = H\Psi(\mathbf{r},t),
		\label{eq:tdse}
	\end{equation}
	where $H$ is the hamiltonian of the system and $\mathbf{r}$ denotes collectively all the position vectors involved in the system.
	There are 2 equivalent ways of choosing the remaining coordinate frames. The $T$ system consists of the relative position vector of the emitted protons and the position of their center of mass w.r.t. the nucleus. The $Y$ system consists of the positions of the two protons relative to the nucleus. In this work we will employ the $Y$ system hence $\mathbf{r} \equiv \lbrace \mathbf{r}_1,\mathbf{r}_2 \rbrace$. This choice is more natural (as will become clear later) for the semi-microscopic description we propose.

	Since all known $2p$ emitters have a partial half life of the order of $10^{-3}$s or below, we can readily employ the Gamow approximation assuming the $2p$ emission state is a resonant one
	\begin{equation}
		\Psi(\mathbf{r}_1, \mathbf{r}_2, t) = e^{-\frac{i}{\hbar}(Q_{2p}-i\frac{\Gamma}{2})t}\psi(\mathbf{r}_1, \mathbf{r}_2),
		\label{eq:Gamow_approximation}
	\end{equation}
	where, as usual, $\Gamma$ bears the significance of decay width and $\Gamma \ll Q_{2p}$. $\Gamma$ is real and positive in this approximation. We now replace Eq.~(\ref{eq:Gamow_approximation}) in Eq.~(\ref{eq:tdse}) and neglect for the moment $\Gamma$ and obtain the time-independent Schr$\ddot{o}$dinger equation for $\psi(\mathbf{r})$
	\begin{equation}
		H \psi(\mathbf{r}_1, \mathbf{r}_2) = Q_{2p}\psi(\mathbf{r}_1, \mathbf{r}_2).
		\label{eq:TIE}
	\end{equation}
	The expression of the Hamiltonian is given by (recoil effects are neglected in the assumption of an infinitely heavy nucleus)
	\begin{equation}
		H = -\hbar^2 \sum_{j=1}^{2}\frac{1}{2m_{p}}\Delta_{j} + \sum_{j}^{2}V_{j}(\mathbf{r}_{j}) + v(\mathbf{r_1},\mathbf{r}_2),
		\label{eq:3BodyHamiltonian}
	\end{equation}
	where the sums runs over the 2 protons, $m_p$ is the proton mass, $\Delta_{j}$ denotes the usual 3-dimensional laplacian associated to the coordinate $\mathbf{r}_{j}$, $V_{j}(\mathbf{r}_j)$ is the interaction potential between the nucleus and proton $j$ and $v(\mathbf{r}_1, \mathbf{r}_2)$ is the interaction potential between the emitted protons.

	The hyperspherical harmonics (HH) formalism (\cite{Avery_2018}) makes it possible to factorize the 6-dimensional space $(\mathbf{r}_1, \mathbf{r}_2)$ in one hyper-radial variable ($\rho$) and 5 hyper-angles. The usual convention one follows is
	\begin{align}
		\begin{aligned}
			\rho &= \sqrt{r_1^2 + r_2^2}, \hspace{1cm} \rho\in [0,\infty) \\
			\phi &= \arctan(r_2/r_1), \hspace{1cm} \phi \in \left[0, \frac{\pi}{2}\right]\\
			\theta_{1,2}, \varphi_{1,2} &= \text{Spherical angles of $\mathbf{r}_{j}$}.
		\end{aligned}
		\label{eq:HSCoords}
	\end{align}
	where $r_{j}=\left|\mathbf{r}_j\right|$. For briefness we denote $\Omega = (\phi, \theta_{1,2}, \varphi_{1,2})$. Using this transformation, Eq.~(\ref{eq:3BodyHamiltonian}) becomes
	\begin{align}
		\begin{aligned}
			H =& -\frac{\hbar^2}{2m_p}\left(\frac{\partial^2}{\partial \rho^2} + \frac{5}{\rho}\frac{\partial}{\partial \rho}\right) - \frac{\mathcal{L}^2(\Omega)}{\rho^2} +  \\
			&V_{1}(\rho\sin\phi) + V_{2}(\rho\cos\phi) +v(\rho,\Omega),
		\end{aligned}
		\label{eq:3BodyHamiltonianExplicit}
	\end{align}
	with $\mathcal{L}$, the grand-angular momentum, given by
	\begin{equation}
		\mathcal{L}^2(\Omega) = -\hbar^2\left[\frac{\partial^2}{\partial \phi} + 4\cot{2\phi} \frac{\partial}{\partial \phi} - \frac{1}{\hbar^2}\left(\frac{l_1^2}{\sin^2\phi} + \frac{l_2^2}{\cos^2\phi}\right)\right],
	\end{equation}
	and $l_{j}, j=1,2$ being the usual angular momenta of the two protons. The eigenvalue equation for $\mathcal{L}^2$ is
	\begin{equation}
		\mathcal{L}^2 \mathcal{Y}_c = \lambda_c \mathcal{Y}_c,
	\end{equation}
	with a multi-index $c$, $\lambda_c = K(K+4)$, $K = 2n+l_1 + l_2$ an even integer, $n$ an integer and
	\begin{align}
		\begin{aligned}
			&\mathcal{Y}_c = \mathcal{N}_c (\sin\phi)^{l_1}(\cos\phi)^{l_2} P_{n}^{l_1+1/2,l_2+1/2}(\cos(2\phi)) \times \\
			&\left[\left(i^{l_1}Y_{l_1,m_{1}}\otimes i^{l_2}Y_{l_2,m_2}\right)_{J,M_J}\otimes \left(\chi_{1}\otimes\chi_{2}\right)_{S,M_S}\right]_{L,M},
		\end{aligned}
	\end{align}
	with $\chi_{j}, j=1,2$ being the spin and projection of the $j$-th proton and $\otimes$ denotes angular momentum coupling. Here, $L,M$ are the total recoupled angular momentum of the two protons and its projection respectively, $P^{\alpha,\beta}_n$ are the Jacobi polynomials and $N_{c}$ is a normalization constant. It is now clear that $c = \lbrace K,L,M,j,s,l_1,l_2\rbrace$. We also note here that we are working in the adiabatic approximation, hence no coupling appears between the inert daughter core and the emitted protons.

	In this work we deal only with spin singlet states, i.e. $L=M_L=J=M_J=S=M_S=0$, hence $l_1 = l_2 \equiv l$. This greatly simplifies the formalism and the multi-index $c$ becomes now $c=\lbrace K,l \rbrace$ with $K=2n+2l$.

	Similarly to the 3D case, we know expand the total spatial wave function as
	\begin{equation}
		\psi(\rho,\Omega) =  \rho^{-5/2}\sum_{c}g_{c}(\rho)\mathcal{Y}_c(\Omega),
		\label{eq:wf_expansion}
	\end{equation}
	where we have included the factor before the sum to cancel the first derivative in the hamiltonian. Upon inserting this expansion, together with the factorized hamiltonian, into Eq.~(\ref{eq:TIE}) and projecting onto channel a specific channel $c$, we obtain a system of coupled equations
	\begin{equation}
		-\frac{\hbar^2}{2m_p}\left(\frac{d^2}{d\rho^2}-\frac{\lambda_c}{\rho^2}\right) g_{c}(\rho) +\sum_{c'}V_{c,c'}(\rho)g_{c}(\rho) = Eg_c(\rho),
		\label{eq:HHSystem}
	\end{equation}
	where
	\begin{widetext}
	\begin{equation}
		V_{c,c'}(\rho) = \int d\Omega \mathcal{Y}_c(\Omega)\left[V_1(\rho\sin\phi) + V_2(\rho\cos\phi) + v(\rho,\Omega)\right]\mathcal{Y}_{c'}(\Omega),
		\label{eq:PME_Full}
	\end{equation}
	\end{widetext}
	are the potential matrix elements (PME). As usual, instead of solving Eqs.~\ref{eq:HHSystem}, we will solve the system for the associated fundamental matrix, which has on columns linearly independent solutions of \ref{eq:HHSystem}. By straight-forward generalization this system is given by
	\begin{eqnarray}
		&&-\frac{\hbar^2}{2m_p}\left(\frac{d^2}{d\rho^2}-\frac{\lambda_c}{\rho^2}\right) g_{c,c'}(\rho)
\\ \nonumber &+&\sum_{c''}V_{c,c''}(\rho)g_{c'',c'}(\rho) = Q_{2p}g_{c,c'}(\rho).
		\label{eq:HHSystemFundamMat}
	\end{eqnarray}
	It is instructive to rewrite $\lambda_c$ as $\lambda_c = l_c(l_c+1)$, with $l_c = K + 3/2$. This reveals one of the ways in which the 3 body decay is fundamentally different from the any 2 body process. The centrifugal barrier is present even in the lowest channel (i.e., when $K=0$).

	In our model we keep all partial waves with $l\le 7$ and $K \le 30$. We found that increasing $K$ above this value, while keeping $l$ constant induces no change in the decay width.

	For the computation of the wave function, we define two regions: internal and external, meeting at a mathcing radius $R_m$ and discuss them separately.

	\subsection{External wave function ($\rho>R_m$)}
	In this section we apply the tools presented above to build a three body wave-function in the external region for the two proton emission process.

	In order to solve the system given in Eq.~(\ref{eq:HHSystemFundamMat}), we need to specify the potentials. It is well known employing 2 body potentials that allow bound 2 body states introduces spurious effects. Such bound states are usually eliminated either through projections \cite{Sparenberg_1997} or through supersymmetric transformations \cite{Sparenberg_1997}. In order to avoid this extra difficulty, we consider  the nucleus-proton potential to be that of point charge interacting with a charged sphere of radius equal to the nuclear radius. We considered the proton-proton potential to be given by a simple central gaussian. We denote $|\mathbf{r}_1 - \mathbf{r}_2|\equiv r_{12}$ and
	\begin{equation}
		\label{vpp}
		v(|\mathbf{r}_1-\mathbf{r}_2|) = v(r_{12})=v_0e^{-\left(r_{12}/r_0\right)^2} + \frac{e^2}{r_{12}},
	\end{equation}
	where $v_0$ is a negative constant and $r_0 = 2$fm is the proton-proton interaction radius. Even though this potential allows shallow bound states (for $v_{0} \leq -30$ MeV and $r_0 = 2.0$ fm), we found no influence of this potential on our calculations.

	The system (\ref{eq:HHSystemFundamMat}) can be solved by the usual Numerov method. However, we found that much better stability can be achieved by employing the renormalized Numerov algorithm \cite{Johnson_1978}. In both cases the solution can be found by specifying the boundary conditions (BC) at $\rho\to\infty$ for each $g_{c,c'}$. This in itself can be achieved only in some approximation, since the system is not asymptotically decoupled (in principle). Grigorenko et al \cite{Grigorenko_2001} have analyzed the effects on accuracy and precision of multiple boundary conditions and determined that a good precision can be obtained through "diagonalized Coulomb" potentials. We follow this approach and, for the farthest two radial points, diagonalize the potential matrix. The Sommerfeld parameter for each channel $\eta_c$ is then used to for building the BC as outgoing Coulomb-Hankel functions \cite{NIST:DLMF}
	\begin{equation}
		g_{c,c'}(\rho)|_{\rho\to\infty} = \mathcal{H}_{\l_{c}}(\eta_c, k\rho)\delta_{c,c'}, \hspace{0.5cm} k=\sqrt{2m_pE}/\hbar.
	\end{equation}

	\subsection{Internal wave function ($\rho<R_{m}$)}
	Here we present the procedure to obtain the internal 3-body wave function, when the two protons are emitted from a paired state. In order to avoid using the complicated 3-body framework presented above in the very complex internal region, we first compute the energy levels and wave functions of bound and resonant protons inside the parent nucleus in the 2-body framework. Details of the procedure can be found in \cite{Delion_2010} together with the Woods-Saxon(WS) parametrization including spin-orbit interaction. Here we briefly outline the procedure.

	The single particle (sp) state diagonalizing a spherical WS + spin-orbit + Coulomb mean field with eigenvalue $\epsilon$ is a superposition of harmonic oscillator (ho) orbitals
	\begin{equation}
		| \psi_{\epsilon, l, j, m}\rangle = c^{\dagger}_{\epsilon, l, j, m} |0\rangle = \sum_{n}d_{\epsilon, l, j}^{n} |\varphi^{\beta}_{n,j,j,m}\rangle,
	\end{equation}
	depending upon the radial quantum number $n$, angular momentum $l$, total spin $j$, spin projection $m$ and the ho parameter
	\begin{equation}
		\beta = \frac{m_p \omega}{\hbar}.
	\end{equation}
	The coordinate representation of the ho wave function depending on $x\equiv(\mathbf{r}, s)$ is given by
	\begin{align}
		\begin{aligned}
			\varphi^{\beta}_{n,l,j,m}(x) &= \langle x| \varphi^{\beta}_{n,l,j,m} \rangle= \left[\phi_{n,l}^{\beta}(\mathbf{r})\otimes \chi_{\frac{1}{2}}(s)\right]_{j,m}\\
			\phi_{n,l,m}^{\beta} &= \mathcal{R}_{n,l}^{\beta}i^{l}Y_{l,m}(\hat{r}),
		\end{aligned}
		\label{eq:SPWF}
	\end{align}
	in terms of the radial ho function $\mathcal{R}_{n,l}^{(\beta)}(r)$, given by the Laguerre polynomial and the Euler Gamma function
	\begin{equation}
		{\mathcal R}_{n,l}^{(\beta)}(r)=(-)^{n}\left[\frac{2 \beta^{3/2}n!}{\Gamma(n+l+3/2)}\right]^{1/2}r^{l+1}e^{-\beta r^2} L_{n}^{l+1/2}(\beta r^2).
	\end{equation}

	Thus, the WS wave function can be rewritten as
	\begin{equation}
		\psi_{\epsilon, l, j, m}(x) = \langle x|\psi_{\epsilon, l, j, m} \rangle = \mathcal{R}_{\epsilon, l,j}(r)\mathcal{Y}_{j,m}^{(l,\frac{1}{2})}(\hat{r}, s),
	\end{equation}
	in terms of the radial WS wave function and spin-orbit harmonics, respectively
	\begin{align}
		\begin{aligned}
			\mathcal{R}_{\epsilon,l,j}(r) &= \sum_{n}d_{\epsilon, l,j}^{n} \mathcal{R}_{nl}^{\beta}(r)\\
			\mathcal{Y}_{j,m}^{(l,\frac{1}{2})}(\hat{r}, s) &= \left[i^lY_{l}(\hat{r})\otimes\chi_{\frac{1}{2}}\right]_{j,m}.
		\end{aligned}
	\end{align}

	A normalized pair state with a given spin/parity, labeled by $J$, is defined as follows
	\begin{align}
		\begin{aligned}
			|ab;JM\rangle &= \mathcal{N}_{ab}(J)\left[c^{\dagger}_{a}\otimes c^{\dagger}_{b}\right]_{J,M}|0\rangle\\
			\mathcal{N}_{a,b}(J) &\equiv \frac{\sqrt{1-\delta_{a,b}(-)^{J}}}{1+\delta_{a,b}},
		\end{aligned}
	\end{align}
	where $a\equiv(\epsilon_a, l_a,j_a), b=(\epsilon_b, l_b,j_b)$. In the configuration space, the pair state is given by
	\begin{widetext}
	\begin{align}
		\begin{aligned}
			\Psi_{abJM}(x_1,x_2) &= \langle x_1x_2|ab;JM\rangle  \mathcal{N}_{ab}(J)\mathcal{A}\lbrace\left[\psi_a(x_1)\otimes\psi_b(x_2)\right]_{JM}\rbrace\\
			&=\mathcal{N}_{ab}(J)\frac{1}{\sqrt{2}}\lbrace \left[\psi_a(x_1)\otimes\psi_b(x_2)\right]_{JM} - \left[\psi_a(x_2)\otimes\psi_b(x_1)\right]_{JM} \rbrace.
		\end{aligned}
	\end{align}
The complete wave function is given
	\begin{align}
		\begin{aligned}
			X_{\epsilon,l,j}\Psi_{abJM}(x_1,x_2) &= X_{\epsilon,l,j}\sqrt{2}\mathcal{N}_{ab}(J)\sum_{n_a,n_b} d^{n_a}_a d^{n_b}_b\Phi_{abJM}(x_1,x_2)\\
			\Phi_{abJM}(x_1,x_2) &= \left[\left(\phi^{(\beta)}_{n_a,l_a}(\mathbf{r}_1)\otimes \chi_{\frac{1}{2}}(s_1)\right)_{j_a}\otimes\left(\phi^{(\beta)}_{n_b,l_b}(\mathbf{r}_2)\otimes \chi_{\frac{1}{2}}(s_2)\right)_{j_b}\right]_{J,M},
		\end{aligned}
	\end{align}
	\end{widetext}
in terms the pair formation amplitude
\begin{eqnarray}
X_{\epsilon,l,j}&=&\frac{1}{2}\langle BCS|[c^{\dagger}_{\epsilon,l,j}\otimes c^{\dagger}_{\epsilon,l,j}]_0|BCS\rangle
\\ \nonumber&=&
\frac{\sqrt{2j+1}}{2}u_{\epsilon, l, j}v_{\epsilon,l,j},
\end{eqnarray}
depending on standard BCS amplitudes.
By changing $\Phi$ from $jj$ to $LS$ coupling one considers the singlet component
\begin{eqnarray}
&&\Phi_{abJM}(x_1,x_2)\rightarrow
\\ \nonumber
&&\left[\phi_{n_a,l_a}^{\beta}(\mathbf{r_1}) \otimes \phi_{n_b,l_b}^{\beta}(\mathbf{r_2}) \right]_{J} \otimes \left[\chi_{\frac{1}{2}}(s_1)
\otimes\chi_{\frac{1}{2}}(s_2)\right]_0
\\ \nonumber
&\times& \left\langle (l_al_b)J\left(\frac{1}{2}\frac{1}{2}\right)0;J| \left(l_a, \frac{1}{2}\right) j_a
\left(l_b\frac{1}{2}\right)j_b;J\right\rangle.
\end{eqnarray}
	Notice that all pair phases $(i)^{l_a+l_b} = (-)^{(l_a+l_b)/2}$ of products between sp wave functions (\ref{eq:SPWF}) have the same sign, due to the common angular momenta parities and therefore the product of these terms in the matrix element is positive.

	Next, we change the radial part by using the Talmi-Moshinsky (TM) transformation
\begin{eqnarray}
&&\left[\phi^{\beta}_{n_a,l_a}(\mathbf{r}_1)\otimes\phi^{\beta}_{n_b,l_b}(\mathbf{r}_2)\right]_{J,M}
\\ \nonumber
&=& \sum_{nlNL}\left[\phi_{n,l}^{\beta/2}(\mathbf{r})\otimes\phi_{N,L}^{2\beta}(\mathbf{R})\right]_{JM}\langle nlNL;J|n_al_an_bl_b;J\rangle,
\end{eqnarray}
	from absolute to relative and center of mass (cm) coordinates
	\begin{align}
		\begin{aligned}
			\mathbf{r} &= \mathbf{r}_1-\mathbf{r_2}\\
			\mathbf{R} &= \frac{\mathbf{r}_1 + \mathbf{r}_2}{2},
		\end{aligned}
	\end{align}
	by using in summation the conserving energy conditions
	\begin{equation}
		2n_a+l_a+2n_b+l_b = 2n+l+2N+L.
	\end{equation}
	The paired state wave function now has to be expanded in hyperspherical harmonics. We do this by the usual Fourier decomposition
	\begin{align}
		\Psi_{abJM}(x_1,x_2) &= \rho^{-5/2}\sum_{c}f_{c}(\rho)\mathcal{Y}_{c}(\Omega)\\
		f_c(\rho) &= \rho^{5/2}\int_{\Omega} d\Omega \Psi_{abJM}(x_1,x_2)\mathcal{Y}_{c}(\Omega).
	\end{align}
	\subsection{Decay width computation}
	Suppose we have complete knowledge of the spatial component of the wave function in Eq.~(\ref{eq:Gamow_approximation}). Without neglecting $\Gamma$, we can replace Eq.~(\ref{eq:Gamow_approximation}) into Eq.~(\ref{eq:tdse}), making use of Eq. (\ref{eq:3BodyHamiltonianExplicit}) but with an arbitrary potential $V(r,\Omega)$. For brevity we use $\Delta = \Delta_1 + \Delta_2$, the 6D laplacian and obtain for the wave function and its conjugate:
	\begin{align}
		\left(Q-i\frac{\Gamma}{2}\right) \psi =& \left[-\frac{\hbar^2}{2m_p}\Delta + V(\rho,\Omega)\right]\psi\\
		\left(Q+i\frac{\Gamma}{2}\right) \psi^{\dagger} =& \left[-\frac{\hbar^2}{2m_p}\Delta + V(\rho,\Omega)\right]\psi^{\dagger}.
	\end{align}
	Multiplying to the left both equations, the first by $\psi^{\dagger}$ and the second by $\psi$ and subtracting the first from the second, we obtain
	\begin{equation}
		\Gamma |\psi|^{2} = \frac{\hbar^2}{2m_pi} \left(\psi\Delta\psi^{\dagger} - \psi^{\dagger}\Delta\psi\right).
	\end{equation}
	Next, we integrate over the volume of a hypersphere of radius $R$, large enough to contain most of the wave function ($\int dV |\psi|^2 = 1$). The definition becomes then
	\begin{equation}
		\Gamma = \frac{\hbar^2}{2m_pi}\int_{0}^{R}d\rho \rho^{5}\int_{\Omega} d\Omega (\psi\Delta\psi^{\dagger} - \psi^{\dagger}\Delta\psi).
	\end{equation}
	We now use the partial wave expansion of Eq.~(\ref{eq:wf_expansion}) and the orthonormality of the HH to write
	\begin{equation}
		\Gamma = \sum_{c}\frac{\hbar^2}{2m_pi}\left(g_c(R)\frac{dg_c^{\dagger}(\rho)}{d\rho}|_{R} - g_c^{\dagger}(R)\frac{dg_c(\rho)}{d\rho}|_{R}\right).
		\label{eq:GammaDefintion}
	\end{equation}
	From the expression above, it would appear that the decay width depends on the hyper-radius of computation. However, if this hyper-radius is large enough, $g_c$ are approximately the Coulomb-Hankel functions. Since the quantity in brackets is nothing else than the Wronskian, it follows that the decay width is independent of computation point, at large hyper-radii. We have observed that convergence is achieved at moderate distances (30fm) for interesting nuclei.

	The advantage of using a semi-microscopic theory is that finding the entire wave function is not mandatory. Instead, we integrate the system ~\eref{eq:HHSystemFundamMat} from far away and match a linear combination of the matrix $g$ to the internal wave function $f$ at $R_m$.
	\begin{equation}
		f_{c}(R_m) = \sum_{c'} g_{c,c'}(R_m)N_{c'}.
	\end{equation}
	$N_{c}$ are called scattering amplitudes and the decay width is directly related to them, as we will show promptly. The obvious drawback of such a method is that only the wave function will be continuous, while its derivative will not. However, this is important only if the evaluation of the probability current is needed close to the nucleus. Describing emission from narrow resonances does not carry such constraints. The internal wave-function is real and normalized to 1 up to the hyper-radius $R_{m} \le R$. In this case, Eq.~(\ref{eq:GammaDefintion}) holds, but with $g_{c}(R) = \sum_{c} g_{c,c'}(R)N_{c'}$. The usefulness of the scattering amplitudes becomes even more obvious in the limit $R\to\infty$, where $g_{c,c'}(R)\to N_c\mathcal{H}_{l_{c}}(\eta_c, kR)\delta_{c,c'}$, the Wronskian is $2ki$ and the decay width becomes
	\begin{equation}
		\Gamma = \frac{\hbar^2 k}{m_{p}}\sum_{c}|N_{c}|^2.
		\label{eq:GammaFromAmplitudes}
	\end{equation}
	Following this recipe, we verified that Eq.~(\ref{eq:GammaFromAmplitudes}) and Eq.~(\ref{eq:GammaDefintion}) give the same result within machine precision for $R\ge 30$ fm. However, in contrast with a fully microscopic theory, $\Gamma$ depends in the matching radius between the internal and external wave functions. This is a well-known drawback of semi-microscopic theories. Nevertheless, it avoids the need to remove 2-body bound states from the nucleus-proton potentials and allows us the study of proton pairing on the life-time. Anyway, we will show that the decay width weakly
depends upon the matching radius in a relative large interval beyond the nuclear radius.

	\section{Results and Discussions}
	\label{sec:Results}
	In this section we will analyze various aspects of the 3-body problem using the semi-microscopic model built above. To this purpose we consider 3 nuclei, with input parameters specified in Table I.

	\begin{table}
		\caption{\label{tab:Nuclei} Parameters of analyzed nuclei. The second column contains the atomic number of the daughter nucleus. The third column contains the $Q$-value of the $2p$ decay. The forth column gives the angular momentum of the state from which protons are emitted. The last column contains the logarithm of the experimental decay width.}
		\begin{tabular}{|c|c|c|c|c|c|}
			\hline
			Nucleus & $Z_D$ & $Q_{2p}$(MeV) & l & $\log_{10}\Gamma_{\mathrm{exp}}$(MeV)  & Ref. \\
			\hline
			$^{19}$Mg & 10 & 0.750 & 2 & -10.121 & \cite{Goncalves_2017}\\
			$^{45}$Fe & 24 & 1.210 & 3 & -18.941 & \cite{Miernik_2007}\\
			$^{54}$Zn & 28 & 1.480 & 1 & -18.911 & \cite{Blank_2005}\\
			\hline
		\end{tabular}
	\end{table}
We first diagonalized the WS mean field for protons by adjusting its real part in order to obtain at the Fermi level the positive experimental proton energy $\epsilon=Q_{2p}/2$ (the paired nucleons have equal energies).
Then we solved BCS equations by using the inter-proton force given by the nuclear gaussian interaction in Eq. (\ref{vpp}).
By changing the nuclear strength $v_0$, we obtained the pairing gap at the Fermi level equal to the experimental pairing gap
$\Delta_F=\Delta_{exp}=12/\sqrt{A}$, considered as an input parameter.

	\subsection{Potential Matrix}
	We discuss here the nature of the potential matrix given by Eq.~(\ref{eq:PME_Full}). One fundamental difference between 3-body scattering and the 2-body analog is the channel coupling even at large hyperradius. Indeed the exact solution at large distances should account for situations in which there can be residual 2-body interactions. It is our purpose in this paper, however, to establish a set of approximations that simplify the picture as much as possible while retaining most of the mathematical rigor. The first aspect we draw attention to is the diagonality of the potential matrix.

	One way of measuring \textit{how diagonal the matrix is} consists in the estimate of the Pearson's correlation coefficient between rows and columns \cite{Stigler_1989}. In Fig.~\ref{fig:PM_correlations} we plot this quantity of the matrix (\ref{eq:PME_Full}) for $^{45}$Fe. In case of a diagonal matrix, this coefficient is 1. In our case, after the monopole turning point ($\simeq$100fm) stability at about 0.8. This implies the matrix is diagonally dominant, which, to first order allows for a decoupled treatment at infinity.
	The above consideration is reinforced in Fig.~\ref{fig:PM_squaresSum} where we plot the ratio of
	\begin{equation*}
		S_d = \sqrt{\sum_{c}V_{cc}^2} \hspace{1cm} \text{and} \hspace{1cm}
		S = \sqrt{\sum_{c,c'}V_{cc'}^2},
	\end{equation*}
	showing that the diagonal accounts for $95\%$ of the Froebinius norm. Again this points towards the possibility of using decoupled solutions at large radii.
	\begin{figure}
		\includegraphics[width=\columnwidth]{potentialDiagonaility_correlations_v070}
		\caption{\label{fig:PM_correlations}Pearson's correlation coefficient for the potential matrix as function of the hyper-radius (for $^{45}$Fe with $v_0$=-35 MeV and $r_0$=2 fm)}
	\end{figure}

	\begin{figure}
		\includegraphics[width=\columnwidth]{potentialDiagonaility_squares_v070}
		\caption{\label{fig:PM_squaresSum} Potential matrix diagonal dominance as function of hyper-radius (for $^{45}$Fe with $v_0$=-35 MeV and $r_0$=2 fm). See text for definitions.}
	\end{figure}

	We now investigate the eigenvalues of the potential matrix. To this purpose, we diagonalize the matrix through $VD=D\lambda^{(v)}$ where $D$ is the matrix of eigenvectors and $\lambda^{(v)}$ is a diagonal matrix with eigenvalues as non-zero elements. Assuming, then, the potential to be of type $V(r)/Q\simeq2\eta_c/kr$ when $r\to \infty$, we can extract the sommerfeld parameters as
	\begin{equation}
		\eta_c = \frac{1}{2}kr\frac{V(r)}{Q_{2p}} = \frac{1}{2}k r \frac{\lambda^{(v)}_c}{Q_{2p}}.
		\label{eq:Sommerfeld_diag}
	\end{equation}
	The problem that remains is how to assign these eigenvalues to the $(n,l)$ channels. To this purpose, in Fig. \ref{fig:PM_eigenvectors} we show the squared amplitude (weight) of each channel in the eigenvectors corresponding to the minimum and maximum eigenvalue of the potential. While there is strong mixing in the sub-barrier region, at large distances each eigenvector has a dominant component in one channel. This allows us to assign each eigenvalue, hence $\eta_c$, to one channel as shown in Fig. \ref{fig:PM_eigenvalues}.
	\begin{figure}
		\includegraphics[width=\columnwidth]{eigenVectors_minLambda}
		\includegraphics[width=\columnwidth]{eigenVectors_maxLambda}
		\caption{\label{fig:PM_eigenvectors} Weight of each $(n,l)$ channel in the eigenvectors associated to the lowest (upper panel) and highest (lower panel) eigenvalues. The same parameters as in Fig.~\ref{fig:PM_correlations} are used.}
	\end{figure}
	\begin{figure}
		\includegraphics[width=\columnwidth]{eigenValues_100fm_300fm.png}
		\caption{\label{fig:PM_eigenvalues} Sommerfeld parameters as function of the hyper-radius. The legend specifies the $(n,l)$ channel to which the eigenvalue has been ascribed. The same parameters as in Fig.~\ref{fig:PM_correlations} are used.}
	\end{figure}
	We note here that the largest Sommerfeld parameters of Eq.~(\ref{eq:Sommerfeld_diag}) do not reach an asymptotic behavior. This is expected since the problem is essentially coupled even at large distances. Nevertheless, outside the turning point, the Sommerfeld parameter has little relevance since the Coulomb function modulus is of the order of unity.

	\subsection{Wave function}
	Some insight into the nature of our problem can be gained through the examination of the wave function behavior channel by channel. Moreover, the case of $^{45}$Fe allows the study of the channel mixing since the paired protons have $l$=3 at the Fermi level, hence they cannot be on the lowest hyper-spherical channel on the nuclear surface. In Fig.~\ref{fig:WF_channels}, we plot external wave function components after matching at $r=$7fm. We see that immediately after the nuclear surface, the components populated by the BCS function are dominant. However, after a few tens of fm, the entire wave function \textit{flows} essentially in the lowest 2 hyper-spherical channel.
	\begin{figure}
		\includegraphics[width=\columnwidth]{wf_channels_diproton}
		\caption{\label{fig:WF_channels} Wave function components squared for $l=0$ and $l=3$ for the $^{45}$Fe nucleus with $v_0=-35$~MeV and $r_0=2$~fm.}
	\end{figure}
	This effect can be understood in terms of both the centrifugal barrier, which is significantly smaller in the $(0,0)$ and $(2,0)$ channels and of the potential matrix which couples every channel to the monopole. This is again quite different from the axially deformed 2-body case, where only neighboring channels are coupled. An important consequence is the following: despite the apparent complexity of the 3-body problem, the strong coupling of various channels to the monopole allows for a great simplification by discarding all but a few low-lying channels.

	\subsection{Decay width}
	A final check for the stability of our method concerns the decay width. The variation with distance of the decay width is given by Eq.~(\ref{eq:GammaDefintion}). The term in brackets is the Wronskian $W(g_c\dagger,g_c)$. In the 2-body case, it is proportional to the Wronskian of outgoing Coulomb functions, which, in turn is constant and equal to $2i$. We show in Fig.~\ref{fig:Gamma_vs_R} that the same holds for the 3-body case, still under the barrier, but at greater distances than in the 2-body case. In other words, the 3-body decay resembles the 2-body decay after a certain hyper-radius, when all couplings can be ignored.
	\begin{figure}[ht!]
		\includegraphics[width=\columnwidth]{Gamma_vs_hyperR}
		\caption{\label{fig:Gamma_vs_R}Decay width as function of the hyper-sphere radius $R$ in Eq.~(\ref{eq:GammaDefintion}) computed using the same parameters as in Fig.~\ref{fig:WF_channels}.}
	\end{figure}

	\begin{figure}[ht]
		\centering
		\includegraphics[width=\columnwidth]{Gamma_vs_rMatch_diProton_noPP_19Mg}
		\includegraphics[width=\columnwidth]{Gamma_vs_rMatch_diProton_noPP_45Fe}
		\includegraphics[width=\columnwidth]{Gamma_vs_rMatch_diProton_noPP_54Zn}
		\caption{\label{fig:Gamma_vs_V0} Decay width dependence on matching radius, for multiple proton-proton interaction strengths for $^{19}$Mg (a), $^{45}$Fe (b) and $^{54}$Zn (c). The parameter $r_0$ is fixed at $2$~fm in all cases.}
	\end{figure}

	\begin{figure}
		\centering
		\includegraphics[width=\columnwidth]{GammaChannel_vs_rMatch_diProton_noPP_19Mg}
		\includegraphics[width=\columnwidth]{GammaChannel_vs_rMatch_diProton_noPP_45Fe}
		\includegraphics[width=\columnwidth]{GammaChannel_vs_rMatch_diProton_noPP_54Zn}
		\caption{\label{fig:PartailGammas} Partial decay widths of the first 6 dominant channels at $v_0 = -45$ MeV and $r_0=2$~fm as function of the matching radius for $^{19}$Mg (a), $^{45}$Fe (b) and $^{54}$Zn (c). The dashed line with open diamond symbols is the sum of the displayed partial widths.}
	\end{figure}

	We turn now to analyzing the proton pairing effects. Fig. \ref{fig:Gamma_vs_V0} shows the dependence of the decay width as function of the matching radius for $^{19}$Mg, $^{45}$Fe and $^{54}$Zn. We note that the sequential (i.e. no proton interaction) and diproton cases bound the microscopic estimates from below and above respectively. This has been remarked in the past by several authors. It is also important to underline that including only the Coulomb interaction between the protons leads to serious underestimations of the decay width. This is true for both low masses (where the $pN$ and $pp$ potentials are of the same order of magnitude) and at high masses (where the $pN$ interaction is dominant on most of the radial range).

	The dependence on the proton pairing strength ($v_0$) shows some remarkable features. First of all, for all three nuclei, the value of $\Gamma$ from our model is in agreement with the experiment for $v_0 \simeq -43 $ MeV, which is a reasonable value, considering the degree of approximation in our work (the "bare" value is $v_0\simeq -35 $ MeV). We also note that the variation of the decay width with $v_0$ decreases as the mass number increases. This effect is caused by 2 components. The first is that the proton pairing strength inside the nucleus decreases with increasing mass. This leads to a smaller gap and, consequently, to a more confined wave function on the surface. The second component is the relative importance of the proton-proton nuclear interaction w.r.t. the proton-nucleus Coulomb interaction. Since the $pN$ Coulomb potential is more than twice stronger than the $pp$ interaction, it is expected that small variations of the latter will not matter much. For the $^{54}$Zn case, the stability plateau is large (between 8 and 11.5fm) at $v_0 = -45$ MeV. Towards the end of this interval, a variation of even 15 MeV in $v_0$ influences the total decay width by a factor less than 5. Notice also that the effective value of the nuclear strength $v_0$ inside nucleus, given by solving BCS equations, is by almost one order of magnitude smaller than its "bare" value in the freee space.

	In Fig. \ref{fig:PartailGammas} we plot the partial widths corresponding to the first 6 most important channels, again as function of the matching radius and for the same 3 nuclei. First of all, we note that in all 3 cases there is a dominant channel which is either the monopole (for $^{19}$Mg and $^{45}$Fe) or the $n=0$,$l=1$ channel (for $^{54}$Zn). This is expected due to the monopole centrifugal barrier being lower than on the other channels. For the $^{54}Zn$ case, the internal wavefunction is already built on the $l=1$ channels. The flow from $l=1$ channels towards $l=0$ channels is hindered by the Raynal-Revay coefficients.

	\section{Conclusions}

	In this paper we built a semi-microscopic model in hyper-spherical coordinates for the 2 proton emission process. We assumed that the protons are emitted from a paired state and that the transition happens between the ground states of the parent and daughter nuclei.

	By splitting the radial domain in an external region and an internal region, we avoid several difficulties associated to other models in literature. We achieved the plateau condition for decay widths beyond the nuclear radius. More importantly, we show that our model is sensitive to pairing correlations between the emitted protons. We studied the effect of these correlations on 3 nuclei, of significantly different masses. We showed that the partial life-times of these nuclei can be well reproduced using reasonable values for the $pp$ potential ($v_0\approx -45$MeV and $r_0\approx 2$fm).

	As expected, the decay widths predicted with our model lie between the the two extreme mechanisms proposed by Goldansky. In turn, this implies that the 2 proton emission process is a valuable tool for investigating the proton-proton potential both inside the nuclear medium and far away from it.

	\begin{acknowledgments}
	\end{acknowledgments}

	This work was supported by the grant of the Romanian Ministry Education and Research No. PN-18090101/2019-2021.

redpajama_set_name: RedPajamaArXiv


text:
India's valuation relative to EMs still less pricey than before, show data
Indian stocks may still have room to run, if the historical price-earnings ratio against their emerging market (EM) peers is any guide.
While the S&P BSE Sensex Index is near the most expensive on record, its premium over the MSCI EM Index is only around 45 per cent, below its five-year average and down from 76 per cent in 2018.
Foreign investors have bought a net $6 billion of Indian shares this year — the only inflows to an emerging Asia market outside of China.
"The broader market is still fairly undervalued," said Sumeet Rohra, a fund manager at Smartsun Capital Pte in Singapore, adding that gains have been concentrated in a small number of stocks.
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Welcome to watersports Portugal and holidays.
You want to actively spend your holiday in Portugal? Then you should pay your attention to the kite surfing, as it is one of the generally available and relatively new types of water sports.
The basis of kitesurfing is the movement through the water on board, due to traction develop and manage kite.
Lessons happening in lagoon (Lagos town), where the water level does not exceed the safe (maximum level 1.40m) and is regulated by the tides. Certificate instructors choose the most a good time for learning.
The duration of each lesson is 3 hours a day, in the training cost includes all necessary equipment and transfer to the spot and with the spot.
Lagos is an ancient port and very pretty city with more than 2,000 years of history. As it looked before, we do not know, in 1755, Portugal was shaken by earthquake and tsunami, which destroyed much of the historical beauty on the coast.
-Education of kitesurf takes place in small groups of no more than 4 people per instructor.
-All instructors are certified and well trained by IKO System.
Corresponding methods of teaching modern requirements. At the end of the theoretical and practical courses we offer IKO certificates, allowing the kite to rent equipment at stations around the world.
For more experienced kiters with their own equipment, we offer a safety net and a shuttle to the kite-lagoon.


Hall, M 2006, 'Academic freedom and the university: Fifty years of debate' , South African Journal of Higher Education, 20 (3) , pp. 8-16.
Contemporary debates about academic freedom and institutional autonomy in South Africa's "liberal" universities began in the 1950s, stimulated by the policies and legislation for racial segregation. At the University of Cape Town (UCT), these debates were shaped by the influential T B Davie, and since 1959, UCT has offered a (usually) annual T B Davie Memorial Lecture at which the symbolic torch of academic freedom (extinguished during the apartheid years, and re-ignited after 1994) is carried in procession. But despite this ceremonial and its endurance there is not, and has not been since the mid 1980s, a university-wide consensus on the nature of academic freedom and its relationship with institutional autonomy.


If you have to ask how much a Ferrari costs, it may not be the car for you. An investment in aesthetics and performance, a Ferrari is more than an automobile; it's a marker of excellence. Models like the Ferrari Portofino can start in the lower $200,000s while the 488GTB and GTC4Lusso start in the mid- and high-$200,000s, respectively. The cost of your Ferrari will depend on the personal touches you take to customize it for your comforts, too.
This was the case in 1939 when Enzo Ferrari broke from Alfa Romeo's racing division, in 1947 when the first Ferrari-badged automobile was produced, and today, in the Ferrari Lake Forest showroom.
Ferrari has made a name for itself — in fact, Brand Finance named it the most powerful brand in 2014 — but you will find something more when you first drive home from Ferrari Lake Forest. In 1923, Benjamin Mancuso started the family in the auto business, and in 1981, Rick Mancuso started what would become Ferrari Lake Forest in a small but charming storefront in downtown Lake Forest.
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text:
Mark Twain, Caiaphas and the Mystery of Satan
Laughing jester *47 X 37 cm *1500s
During one of his stand-up routines late in life, Mark Twain said he'd always wanted to meet Satan. Shake his hand. (I paraphrase from memory)…"He must have had executive experience. After all, he is spiritual head of half the world…..and the political head of the whole of it." (drum roll).
Twain had a peculiarly American humorous look at Old Clootie, and I always felt Satan never liked being thought of that way, to be considered no more than a bully on the strip-joint side of town, or a petty politician at Town Hall. Either would have been a great slight in the Devil's eyes.
With Twain's sense of humor in mind, I'd always wanted to write a "Devil's History of the United States" based largely on the cat-and-mouse game he always thought he was playing with the Creator, Who (possibly) never considered Satan as anything more than an irritant, an annoying gnat. For you see, in history's more notable events, such as the Resurrection and creation of the United States, both of which turned History entirely on it its head, Satan was nowhere to be seen and God always one step ahead.
Indeed, the only face-to-face confrontation with God's plans, besides that apple-encounter with Eve in the Garden, was when Clootie met Jesus in the Wilderness just before He began His ministry. It couldn't have been a chance meeting.
In the process, he proved, even double-downed on Twain's observation that Satan was indeed the political head of the whole world, for he laid them all out in front of Jesus, offering them to Him…if only He'd bow down and worship him instead.
While Twain, in 1900, would have been correct to state that Satan was the spiritual head of "only" half the world, in whatever year the 30-year old Jesus was in the Wilderness, Satan was truly the spiritual head of the whole world as well, but that tiny dot in Judaea. However much haughty Protestant preachers in South Carolina want to go on about God always having been in charge of Man's affairs, neither was He, or did He even pretend to be. For you see, if we could color the world in reds and blue the way we do today, it would have been all blue, controlled entirely by pagans and barbarians who worshiped idols, even trees, except that one small, lone red dot on the world map, Judaea, itself only a glimmer of its former self, and here in the time of Christ, a province of back-to-back-to-back Persian, Greek and Roman conquerors, running back over 400 years.
What's more interesting, since preachers who preach the Old Testament never really mention it, is that from the beginning, after the Flood, Yahweh's (God's) domain was scarcely little more than a "thin red line" running from Abraham through Jacob and Joseph to Egypt, and then, at the able hands of Moses, out of Egypt back to the land God had originally promised in his Covenant with Abraham, 365 years earlier, and then approximately 1600 years of highs and lows that brought them to this tiny red area of Judaea. (That "thin red line" is my contribution to David M Poff's recent book, A User's Guide to Our Imperfect Union", worth a read if we can ever find a publisher.)
So when Twain was glibly remarking about Christianity having gained "half the spiritual world" in 2000 years (give or take) he was speaking about one of the most remarkable (and under-appreciated) events in all history…for you see, it all began when 120 men (and women) departed Jerusalem just 50 days after the Resurrection, having just been blessed with the gift-of-tongues, armed only with their personal knowledge of this Man, what He taught and that He had been crucified, and on the third day arose.
So, how did Satan miss this?
Enter Agatha Christie with a plot, and Mark Twain with the side bar.
I've always wanted to write this tale as an Agatha Christie short story, for the absence of God's greatest antagonist at these very crucial junctures, the Resurrection, and Pentecost from which the Word went forward, and eventually found its way to the shores of North America without really carrying any king's flag is a mystery in itself.
How could Old Clootie have missed this? How could the self-named arch-demon of God, knowing His plan from the Beginning, by tempting Eve, messing that plan up, then recognizing His Son who would soon begin preaching, having the presence of mind to try to seduce Him in the Wilderness, knowing the threat He posed to the kingdoms (plural) he'd amassed since the first Temple was destroyed 500 years earlier… How could he have missed what happened between the Crucifixion and Pentecost, just 50 days?
Well, I spent a lot of time in the Balkans and enjoyed many discussions (over rakia) with scientists mostly, some of whom were well read in the medieval history of their Orthodox Church. After 45 years under communism, their taste for religion was one not quite the same as you would recognize it. But they said some churches in the east had a special day for Caiaphas, the chief priest of the Sadducees, and principal go-between the Roman prefect, Pontius Pilate, and the people, and who convinced Pilate to order the crucifixion of Jesus. Western Christians had never seen Caiaphas as a good guy.
So, why would Christians set aside a feast day for Jesus' principal tormentor and adversary?
After securing the execution order, It's suggested that it was Caiaphas who asked Pilate to station a guard (or guards) at the tomb, warning him that some of Jesus's followers might come and steal the body away on the third day, because it had been foretold that this would happen. The record (Matthew) isn't clear whether it was Roman legions or local troops who were ordered there, since Matthew got all this information second-hand himself, but indeed according the Matthew, and angel came and rolled away the stone, and the guard(s) passed out from fear.
While the women were on their way, some of the guards went into the city and reported to the chief priests everything that had happened. When the chief priests had met with the elders and devised a plan, they gave the soldiers a large sum of money, telling them, "You are to say, 'His disciples came during the night and stole him away while we were asleep.' If this report gets to the governor, we will satisfy him and keep you out of trouble." So the soldiers took the money and did as they were instructed.
Apparently nothing ever made it on the nightly news in Jerusalem, or made it back to Pilate. Rumors spun around Jerusalem, but there were no confirmed sightings. So Caiaphas believed his butt was covered. Jesus reportedly appeared in several places over the next 30 days before the Ascension, but not within hearing-range of the centers of government or high Synagogue.
So where was Satan? Now some people think devils can just float around like drones, watching events unseen, but apparently even Satan had to have a snitch network. Even Mark Twain would have agreed that Satan didn't hang out on street corners using pimps for listening posts or runners. Satan, in whatever guise he used, hung with management. At the top. So what Satan told Caiaphas, his principal agent, was to urge Pilate to crucify him, and once that was done, insure His stay in the tomb was permanent.
He never once talked to a guard.
So Pilate, having no knowledge that Christ was even missing from the tomb, had no cause to raise any alarm. And Caiaphas, with the heart of a Lindsey Graham, figured as long a Pilate didn't know about that incident at the Tomb, neither was Satan.
So for 50 days no one was even looking for Jesus. And then came God's hole card, when the Holy Spirit came down onto 120 of His followers of them gathered in Jerusalem, Pentecost it's called, 10 days after the Ascension, and gave them all the gift of tongues, to head in 120 different directions, each with just-a-little-different description of just what Jesus said and did, and began preaching the Good News.
Many or most were martyred, only nobody knows how long they'd been at it before Satan ever found out. But by then it was too late. It took him 800 years to regain even half the territory he had lost.
Just imagine how many different ways good storytellers could tell that tale, from 33 AD to 800 AD, and challenge virtually every priest and preacher who ever came down the pike since, yet never mess up a single hair on the true story or message of Jesus Christ, or the grand plan of God.
A great lesson about how to use our thinking skills, for the one main thing we know that Satan is in command of the 7 deadly sins, and he really has figured out the ways of the 7 virtues, and knows how to compete with them.
And vice versa.
Lesson: Do not let someone move you to anger if the Evil behind that person is bigger than that person could ever possibly be. Think Swalwell or AOC. Swat him or her aside, and focus on that next level of evil, someone who can at least think past their own tiny vanities, appetites or ambitions. Piss on them. Then go to even the next, if need be. You will eventually get to the Big Guy in the front office and you will know you're close because of the smell of sulfur.
Then back away, and retool, unless you are well armed with knowledge and strength of heart.
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For Christmas Eve: Oysters with Bubbly Vinaigrette paired with The Phoenix.
Finely dice the shallots. Add them to a bowl with the red wine vinegar, MCC and freshly ground black pepper. Mix well and leave in the fridge for at least an hour before serving.
Shuck the fresh oysters and preserve the liquid in the shell. Serve the oysters lightly dressed with the bubbly mignonette dressing.


CM = 1.5V, VO = 2.5, and RL > 1M unless otherwise specified.
can result in exceeding the maximum allowed junction temperature of 150˚C. Output currents in excess of ±30 mA over long term may adversely affect reliability.
Note 4: Limits are guaranteed by testing or correlation.
Note 5: V+ = 15V, VCM = 7.5V and RL connected to 7.5V. For Sourcing tests, 7.5V ≤ VO ≤ 11.5V. For Sinking tests, 2.5V ≤ VO ≤ 7.5V.
Note 6: V+ = 15V. Connected as Voltage Follower with 10V step input. Number specified is the slower of the positive and negative slew rates.
boldface limits in this column. The LPC660AMJ/883 may also be procured to a Standard Military Drawing specification.
θJA with PD = (TJ–TA)/θJA.
Note 10: All numbers apply for packages soldered directly into a PC board.
Note 11: Do not connect output to V+when V+ is greater than 13V or reliability may be adversely affected.


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Study identifies protein that helps prevent active tuberculosis in infected patients
Written by Ibrahim on August 27, 2014 . Posted in Latest News
UCLA-led study has identified a protein that appears to play a key role in protecting people infected with Mycobacterium tuberculosis — the bacterium that causes tuberculosis — from developing the active form of the disease. The protein, interleukin-32, was discovered to be one biomarker of adequate host defense against TB.
The discovery could help doctors identify people who are at the greatest risk for the highly contagious and potentially fatal lung disease, and it could point the way toward new treatment strategies for TB.
The study, conducted in partnership with researchers from Harvard University School of Public Health and the University of Michigan School of Medicine, was published in the Aug. 20 online edition of the journal Science Translational Medicine.
The findings underscore the importance of maintaining sufficient levels of vitamin D to effectively combat the pathogen that causes TB. The researchers found that the protective protein, interleukin-32, can induce the killing of the TB bacterium only in the presence of sufficient levels of vitamin D.
An estimated one-third of the world's population is infected with tuberculosis, but the disease is latent in 90 to 95 percent of infected people, meaning that they experience no symptoms and are not contagious. Interleukin-32 contributes to maintaining that latent state and preventing active infection. In 2012, nearly 9 million people worldwide became sick with TB and there were 1.3 million TB-related deaths, according to the U.S. Centers for Disease Control and Prevention.
A new urgency for developing new approaches to identify individuals at risk, maintain immunity and treat active disease has arisen in recent years as TB has re-emerged as a global health threat — thanks in part to the rise of extremely drug-resistant bacteria.
"Until now, there had been no way to predict, based on biological factors, why latently infected individuals do not develop active tuberculosis," said Dennis Montoya, a postdoctoral scholar in the division of dermatology at the David Geffen School of Medicine at UCLA and the study's lead author. "We were surprised to find many differences between people with latent TB and healthy people, suggesting that people with latent TB may have activated immune systems that are protecting them from developing active infection."
In people with active TB, the disease-causing bacteria have overcome the defenses of the immune system, causing symptoms to develop in the lungs. Previous studies have focused on finding biological markers of disease progression in those patients. The new study, however, was the first to look for markers of protection in the blood of people with latent TB — those who are infected but have not developed symptoms.
The researchers analyzed gene expression profiles of hundreds of TB patients in four countries, as well as evaluating genes from activated immune cells shown in the laboratory to have the ability to kill the TB bacteria. They discovered that in people who were infected with TB but had higher levels of interleukin-32 (IL-32), the disease was more likely to be latent, and that — in laboratory experiments — IL-32 was able to stimulate the immune system to kill TB-causing bacteria, but only in the presence of sufficient vitamin D levels.
Vitamin D is produced in the skin upon exposure to sunlight, but approximately one-third of American adults — particularly members of ethnic minorities with darker-pigmented skin and lighter-skinned people who receive minimal sun exposure — lack sufficient levels of vitamin D, and vitamin D deficiency has been found to be associated with a higher risk for active TB.
"When vitamin D levels were low, IL-32 was not able to kill the bacteria," said Dr. Robert Modlin, the Klein Professor of Dermatology at UCLA and the study's senior author. "However, if we simulated the effect of supplementing individuals by adding vitamin D to the culture of the activated immune cells that had low vitamin D levels, IL-32 regained its ability to kill. Our findings suggest that raising standards for daily intake of vitamin D could help to protect against a TB pandemic."
Further studies are needed to determine if vitamin D supplementation helps prevent TB. But Modlin, a dermatologist, favors oral supplementation rather than increased sun exposure, which can increase the risk of skin cancer.
Although TB remains a leading killer in many parts of the world, the incidence of the disease has been declining in the U.S. since a resurgence that peaked in 1992, according to the CDC. A total of 9,945 cases were reported in 2012, and 569 people died from the disease in 2010, the most recent years for which U.S. data are available. In the U.S., unlike in many poorer countries, patients known to have latent TB infections are typically treated with antibiotics to prevent progression to the active disease. But strains of the tuberculosis bacteria are becoming resistant to the antibiotic drugs, increasing the threat of a multidrug-resistant TB pandemic that could affect the U.S.
"Our current strategy of antibiotic therapy, as well as methods of identifying which patients will develop severe disease, have remained essentially unchanged for decades," said Harvard's Barry Bloom, the study's co-author. "Our findings of specific markers in the blood that can tell us which people are at risk can help countries focus limited resources on patients who have lower levels of this protein. In addition, this may point the way to new treatments for a disease that greatly needs them."
The research builds on the team's previous work that found that vitamin D is important for killing TB bacteria.
Source: University of California, Los Angeles


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OneAmerica appoints Steven Crowe to its Board of Directors
November 15, 2022 November 10, 2022 by Talent4Boards Team
– USA, IN – OneAmerica, the national financial services provider, today announced the appointment of Steven Crowe to the board of directors.
"Steven's technology expertise and business acumen can further empower and inform our company's growth trajectory and digital service enhancements. His experience in harnessing technological advancements to better serve our customers will allow us to continue to focus on providing stellar products, exceptional service, and relationships our customers can depend on for a lifetime," said Chairman and CEO, Scott Davison.
About Steven J. Crowe
Steven Crowe has recently retired from Dell Technologies.
Steven Crowe of Scottsdale, Arizona, is a high-energy, results-oriented business leader with over 30 years of experience driving and accelerating revenue growth and profitability, most recently with Dell Technologies, a multinational I.T. technologies, solutions, and services giant. Driving large-scale organization transformation programs and pursuing top-notch customer advocacy was a career passion, and he arrives at a time when the life insurance, retirement services, and employee benefits provider delivered its strongest financial year to date in 2021.
At enterprise storage specialist EMC Corp., Crowe rose to senior vice president of sales in 2016, when the company was acquired by Dell Inc. in the most significant technology merger in history. He was part of the integration team and then promoted to the rebranded Dell Technologies as president of Global Specialty Sales & Pre-Sales. In that position, Crowe led over 8,000 team members globally, assisting customers with their IT transformation journeys, all while optimizing operating expenses at scale.
"I am deeply humbled to join the board at OneAmerica," said Steven Crowe. "Their history of strong financial stability and performance is driven by a genuine commitment to policyholders when it matters most, with aspirational service provided at the highest levels."
Steven Crowe earned a Bachelor of Science in business from Eastern Illinois University and a master's in business administration (concentration in portfolio management) from Illinois State University.
A national provider in the insurance and financial services marketplace for more than 140 years, the companies of OneAmerica help customers build and protect their financial futures.
OneAmerica offers a variety of products and services to serve the financial needs of its policyholders and customers. These include retirement plan products and recordkeeping services, individual life insurance, annuities, asset-based long-term care solutions, and employee benefit plan products.
Products are issued and underwritten by the companies of OneAmerica and distributed through a nationwide network of employees, agents, brokers, and other sources that are committed to providing value to our customers.
For more information: https://oneamerica.com/
- Disclaimer - News, data, and statements included in this release are intended exclusively for general information purposes. Talent4Boards does not guarantee that news is accurate or about the correct person and accepts neither liability for the consequences of the reader's reliance, nor responsibility for the accuracy of the information. Nothing in this release should be considered for decisions about referred securities. Products and brand names may be trademarks or registered trademarks of their respective owners.
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Categories Board - USA / Can., Board Privately Held Companies Tags Board of Directors, Financial Services, Non-Executive Director, USA