text:
\section{Introduction}

Duality is an important concept in different branches of physics. It relates physical quantities and physical behavior of two different systems with each other. Prominent examples are the particle-wave duality in quantum mechanics and the duality of Maxwell's electrostatic and magnetostatic theory obtained by a Lorentz transformation from a rest frame to a moving one. The particle-hole duality (see e.g.~\cite{1}) for models of electrons in solid state physics and the Kramers-Wannier duality \cite{2} in statistical physics are further examples. The latter relates the free  energy of an Ising model with nearest neighbor coupling $J$ on a $2$-dimensional lattice $\mathcal{L}$ to the free energy of an Ising model with coupling $J^*$ on the dual lattice  $\mathcal{L}^*$. In particular, it links the low temperature behavior of the lattice system $\mathcal{L}$ to the high temperature properties of the dual system. For a self-dual lattice, i.e.~$\mathcal{L} = \mathcal{L}^*$, like the square lattice, the duality allows to determine the critical point for the phase transition of the corresponding Ising model \cite{2}.

An interesting duality has been discovered recently \cite{4,4a} for the Moshinsky atom \cite{4b}, a two electron model,
in three dimensions described by the hamiltonian $\hat{H}=\frac{\hat{\vec{p}}_1^{\,2}}{2m}+\frac{\hat{\vec{p}}_2^{\,2}}{2m}+ \frac{1}{2}m\omega^2 (\hat{\vec{x}}_1^{\,2}+\hat{\vec{x}}_2^{\,2}\big) + \frac{1}{2} D\,(\hat{\vec{x}}_1- \hat{\vec{x}}_2)^2$. Here $m$ is the electron mass, $m\omega^2$ the force constant of an external harmonic potential and $D$ the coupling constant of the harmonic interactions.
By introducing the dimensionless coupling constant $\Lambda=D/(m\omega^2)$ a duality relation
\begin{equation}\label{eq1}
S_q(\Lambda) = S_q(\Lambda ')
\end{equation}
for the R$\acute{e}$nyi entropy $S_q(\Lambda)=(1-q)^{-1}\ln\big[�\mbox{Tr}[(\rho_{1}(\Lambda)^q]\big],  \ q \neq 1$, has been found \cite{4}
for the 1-particle reduced density operator ($1$-RDO) $ \rho_{1}(\Lambda)$ of the ground state.
Condition (\ref{eq1}) means that for any given $\Lambda \in (-\frac{1}{2},0)$  there exists a unique coupling constant $\Lambda '(\Lambda) \in (0,\infty)$ (see Ref.\cite{4}) such that $S_q(\Lambda) = S_q(\Lambda '(\Lambda))$.
For the ground state of a generalized\footnote{Since the Moshinsky atom is a system of $2$ electrons we call its generalization to $N$ electrons generalized Moshinsky atom.} Moshinsky atom of three electrons in one dimension a similar duality has been observed for the natural occupation numbers $\lambda _k^{(1)}$   \cite{5,8}
\begin{equation}\label{eq2}
\lambda _k^{(1)} (\ell _- /\ell _+) = \lambda _k ^{(1)} (\ell_{-}^{*} / \ell_{+}^{*}) ,\    k \in \mathbb{N}
\end{equation}
with $[\ell_{-}^{*} / \ell_{+}^{*}](\ell _- ,\ell _+)= \ell _+ /\ell _-$.
$\ell_+$ and $\ell _-$ are two natural length scales related to both coupling constants of the model.
 Numerical analysis \cite{6} has provided strong evidence that this kind of duality is not restricted to the ground state of the Moshinsky atoms with two or three electrons, but holds for any number $N$ of electrons, and for all its eigenstates.
In this communication we will answer three questions

\begin{enumerate}
\item What is the origin of that duality?
\item Does it hold for any harmonic model and for any $N$-particle eigenstate?
\item Is the duality also given for the eigenvalues of the $M$-particle reduced density operator ($M$-RDO) with $M>1$
      and what is the role of the exchange symmetry?
\end{enumerate}

These questions are of fundamental relevance since reduced density operators (RDO) and their eigenvalues play an important role in atomic physics and quantum chemistry \cite{9,10,11}. They also have attracted strong attention in quantum information theory where the so-called quantum marginal problem is studied (see  Refs.\cite{12,15,16,17,18,19,20}). It asks whether given density operators (marginals) for subsystems of a multipartite quantum system are compatible in the sense that they can arise from a common total state. One of the prime examples originating from quantum chemistry is the $M$-particle $N$-representability problem \cite{9,10}, the problem of describing the family of $M$-RDO which can arise from an antisymmetric $N$-particle state. Stimulated by the results of Borland and Dennis \cite{13} and Ruskai \cite{14}, Klyachko \cite{15,16} in a ground-breaking work solved recently the $1$-particle $N$-representability problem. His solution has revealed so-called generalized Pauli constraints, restrictions on the eigenvalues of the $1$-RDO of fermionic systems, which are significantly strengthening Pauli's famous exclusion principle. Accordingly, answers to the questions raised above will add a new facet to the properties of RDO and their eigenvalues and therefore to this field of research.

\section{Harmonic model}
We consider a model for  particles with mass $m$ interacting harmonically. The corresponding hamiltonian is given by
\begin{equation}\label{eq3}
\hat{H} = \frac {1} {2m} \sum \limits _{i=1} ^N \hat{p} _i ^2 + \frac 1 2 \sum \limits _{1 \leq i , j \leq N}  D_{ij} \hat{x}_i \hat{x}_j \, .
\end{equation}
$\hat{x}_i$ and $\hat{p}_i$  are the position and momentum operators, respectively. $N$ is the number of degrees of freedom.
The interaction matrix $\textbf{D} = (D_{ij})$ is assumed to be positive definite and real-symmetric.
Note that this hamiltonian holds in any dimension. For $d>1$ the components  $\hat{x}_{r,\alpha}$ and $\hat{p}_{r,\alpha}$, for $r=1,\ldots,N'$ and $\alpha =1,\ldots,d$ of the position and momentum operators, respectively, should be relabeled, the same should be done for the  matrix, $ D_{ij}=D_{r\alpha,s\beta}$, and $N$ in Eq.~(\ref{eq3}) is given by $N=d \cdot N'$. Of course,  possible translational and  rotational symmetry would impose additional constraints on $\textbf{D}$.

For the case of identical particles the potential energy, the second term in Eq.~(\ref{eq3}), is invariant under particle permutations.
This restricts $\{D_{r\alpha,s\beta}\}$  to two independent matrices
$\textbf{D}^{(1)}$ and $\textbf{D}^{(2)}$ and the potential energy takes the form $\frac{1}{2}\sum_{r=1}^{N'}\,\hat{\vec{x}}_r^{\,t} \textbf{D}^{(1)}\, \hat{\vec{x}}_r + \frac{1}{2}\sum_{1\leq r <s\leq N'}(\hat{\vec{x}}_r-\hat{\vec{x}}_s)^t \textbf{D}^{(2)} (\hat{\vec{x}}_r-\hat{\vec{x}}_s)$, where
$D^{(2)}_{\alpha,\beta} = -D_{1 \alpha,2 \beta}$ and $D^{(1)}_{\alpha,\beta} \equiv D_{1\alpha,1\beta}-(N'-1)D^{(2)}_{\alpha,\beta}$ and
$\hat{\vec{x}}_r\equiv(\hat{\vec{x}}_{r 1},\ldots,\hat{\vec{x}}_{r d})^t$.
Consequently,  model (\ref{eq3}) (with corresponding $\textbf{D} > 0$) can also arise for translationally invariant models (i.e.~$\textbf{D}^{(1)}=\textbf{0}$) after a separation of the relative degrees of freedom from the free center of mass motion. Prominent examples for such harmonic systems are the Moshinsky atom \cite{4b} and its generalization to an arbitrary number of fermions \cite{7}, which were already mentioned in the introduction.
Note, hamiltonian (\ref{eq3})  can also be used to describe a $d$-dimensional  harmonic lattice  with $N'$ sites,  $r= 1,...,N' $  and dynamical matrix $\{D_{r\alpha,s\beta}\}$.

For the following technical analysis, to avoid annoying issues related to the dimensionality of physical quantities, we set several physical quantities as e.g.~$\hbar$ and $m$ to $1$ and treat any variable as e.g.~positions $x_i$, momenta $p_i$ and couplings $D_{ij}$ as dimensionless real numbers.
This could also be achieved by introducing an arbitrary length scale $L$ and dimensionless position operators $\hat{x}_i/L$, momentum operators
$\hat{p}_i/(\hbar/L)$ and hamiltonian $\hat{H}/(\hbar \Omega)$ with $\Omega\equiv\hbar/(L^2 m)$.
In position space $\hat{H}$ then takes the form
\begin{equation}\label{eq4}
H^{(x)} = - \frac {1}{2} \Delta_{\vec{x}} + \frac{1}{2} \vec{x}^t \textbf{D} \vec{x}\,
\end{equation}
where $\vec{x}\equiv (x_1,\ldots,x_N)^t \in \mathbb{R}^N$ and $\Delta_{\vec{x}}$ is the corresponding Laplace operator.
$H^{(x)}$ is densely defined on the separable Hilbert space $L^2(\mathbb{R}^N)$ of square-integrable functions $\Psi: \mathbb{R}^N\rightarrow\mathbb{C}$.
The corresponding time-independent Schr\"odinger equation reads
\begin{equation}\label{eq5}
H^{(x)} \Psi  = E \Psi \, .
\end{equation} \,
To determine the eigenenergies $E$ we just need to diagonalize the symmetric coupling matrix $\textbf{D}$,
\begin{equation}\label{eq8}
{\textbf{RDR}^t = \textbf{d}} \,,
\end{equation}
where $\textbf{d}\equiv \mbox{diag}(d_1,\ldots,d_N)$ is a diagonal matrix and $\textbf{R}$ an orthogonal matrix. The coordinate transformation from $\vec{x}$ to the pseudo-positions $ {\vec{y}}  = \textbf{R}\vec{x}$ is represented as a unitary transformation $U(\textbf{R})$ on $L^2(\mathbb{R}^N)$,
\begin{equation}\label{unitaryR}
(U(\textbf{R})\Psi\big)(\vec{y}) \equiv \Psi(\textbf{R}^{-1}\vec{y})\qquad,\, \forall \vec{y}\in \mathbb{R}^N\,.
\end{equation}
The transformed hamiltonian $H^{(y)}\equiv  U(\textbf{R})H^{(x)}U(\textbf{R})^\dagger$ takes the diagonal form
\begin{equation}\label{eq11}
H^{(y)} = \sum \limits ^N _{\mu = 1} \frac{1}{2} \frac{1}{\ell_\mu} \,[- \frac {\partial ^2}{\partial(y_\mu/\ell_\mu)^2} + (y_\mu / \ell_\mu)^2] \, .
\end{equation}
where we introduced the dimensionless length scales $\ell_\mu = \frac{1}{\sqrt{d_\mu}}$.
The normalized eigensolutions of $H^{(y)}$ are products of $1$-particle states $\varphi_\ell^{(\nu)}(y)$
\begin{equation}\label{eq12}
\Phi_{\bd{\ell}}^{(\boldsymbol{\nu})}(\vec{y}) = \prod \limits^N _{\mu = 1} \varphi ^{(\nu_\mu)}_{\ell_\mu}(y_\mu)
\end{equation}
with $\boldsymbol{\nu} = (\nu_1,\ldots,\nu_N), \; \nu_\mu \in \mathbb{N}_0$, $\bd{\ell}\equiv(\ell_1,\ldots,\ell_N)$ and
\begin{equation}\label{eq13}
\varphi _\ell^{(\nu)} (y) = \pi ^{- \frac 1 4} \ell ^{- \frac 1 2} (2^\nu \nu ! )^{-\frac 1 2} H_\nu (\frac y \ell) \exp{\big(-\frac {y^2}{2 \ell^2}\big)}\,.
\end{equation}
$H_\nu(z)$ are the Hermite polynomials. Then, the eigenstates of $H^{(x)}$ follow as
\begin{equation}\label{eq14}
\Psi_{\bd{\ell}}^{(\bd{\nu})}(\vec{x})= \big(U(\textbf{R})^\dagger\Phi_{\bd{\ell}} ^{(\bd{\nu})}\big)(\vec{x}) = \Phi_{\bd{\ell}} ^{(\bd{\nu})}(\textbf{R}\vec{x})\qquad,\,\forall \vec{x}\in \mathbb{R}^N.
\end{equation}
For later purpose, we have made explicit the dependence of $\Psi_{\bd \ell}^{(\bd{\nu})}$ on $\bd{\ell}\equiv (\ell_1,\ldots,\ell_N)$.
Moreover, from Eq.~(\ref{eq12}) and the explicit form of the $1$-particle states (\ref{eq13}) we can infer a homogeneous structure,
\begin{equation}\label{homog}
\Psi_{\bd{\ell}}^{(\bd{\nu})}(\vec{x}) = \alpha^{-\frac{N}{2}}\Psi_{\frac{\bd{\ell}}{\alpha}}^{(\bd{\nu})}\big(\frac{\vec{x}}{\alpha}\big) \equiv \big(V_N(\alpha)\Psi_{\frac{\bd{\ell}}{\alpha}}^{\bd{(\nu)}}\big)(\vec{x})\qquad,\, \forall \alpha \in \mathbb{R}^{+}\,.
\end{equation}
Here we have introduced the unitary rescaling operator $V_N(\alpha)$, which is rescaling the length by a factor $\alpha$ and is local,
$V_N(\alpha) = V_1(\alpha)^{\otimes^N}$, where $V_1(\alpha)$ the rescaling operator on $L^2(\mathbb{R})$.

Due to the harmonic character of the hamiltonian (\ref{eq4}) it is worth studying the Schr\"odinger equation (\ref{eq5}) also in the Fourier space.
For this we introduce the Fourier transformation, a linear bounded operator on the space of Schwartz functions $\varphi:\mathbb{R}\rightarrow \mathbb{C}$,
\begin{equation}\label{fourier}
\big(\mathcal{F}_1\varphi\big)(p)  \equiv \!\int_{\mathbb{R}}\mathrm{d}x\,\varphi(x)\,e^{-i \,x\,p}\,.
\end{equation}
For all $k \in \mathbb{N}$, $\mathcal{F}_1$ gives rise to Fourier operator $\mathcal{F}_k$ acting on Schwartz functions $\Phi:\mathbb{R}^k\rightarrow \mathbb{C}$ which is extended uniquely to the space $L^2(\mathbb{R}^k)$ according to the Plancherel theorem \cite{21}.
By identifying $L^2(\mathbb{R}^k)\cong \big(L^2(\mathbb{R})\big)^{\otimes^k}$ the local structure $\mathcal{F}_k = \mathcal{F}_1^{\otimes^k}$ w.r.t~ the $1$-particle Hilbert spaces $L^2(\mathbb{R})$ is obvious.

Applying $\mathcal{F}_N$ to Eq.~(\ref{eq5}) leads to the Schr\"odinger equation in momentum space
\begin{equation}\label{Hp}
H^{(p)} \tilde{\Psi} = E \tilde{\Psi}\,
\end{equation}
with $\tilde{\Psi}\equiv \mathcal{F}_N \Psi$, the Fourier transformation of $\Psi$ and
\begin{equation}\label{eq6}
H^{(p)} \equiv \mathcal{F}_N H^{(x)} \mathcal{F}_N^{\dagger}= - \frac{1}{2} \nabla_{\vec{p}}^t\,\textbf{D}\,\nabla_{\vec{p}} + \frac{1}{2}\,\vec{p\,}^2\, .
\end{equation}
Due to the unitarity of $\mathcal{F}_N$, $H^{(x)}$ and $H^{(p)}$ have the same spectrum. Below we will see that this dual description, using either the position or the momentum space, is the origin of the duality of the $M$-RDO and of its eigenvalues.

The eigenvalue problem for $H^{(p)}$ can be solved similarly as that for $H^{(x)}$. With the pseudo-momenta $\vec{\pi}= \textbf{R}\vec{p}$ and Eqs.~(\ref{eq8}), (\ref{unitaryR}), $H^{(p)}$ is transformed to
\begin{equation}\label{eq17}
H^{(\pi)} \equiv U(\textbf{R}) H^{(p)} U(\textbf{R})^\dagger = \sum \limits ^N _{\mu =1} \frac{1}{2} \frac{1}{\tilde{\ell}_{\mu}} [- \frac {\partial ^2} {\partial(\pi _\mu /\tilde{\ell}_\mu)^2} + (\pi_\mu/\tilde{\ell}_\mu)^2]\, ,
\end{equation}
with the reciprocal (dimensionless) length scales
\begin{equation}\label{lengthduality}
\tilde{\ell}_{\mu} \equiv \frac{1}{\ell_\mu}\qquad, \forall \mu =1,\ldots,N\,.
\end{equation}
Notice that the hamiltonians (\ref{eq11}) and (\ref{eq17}) are identical up to a swapping of $\bd{\ell}$ and $\tilde{\bd{\ell}}$. Since $H^{(x)}= U(\textbf{R})^\dagger H^{(y)}  U(\textbf{R})$ and $H^{(p)}= U(\textbf{R})^\dagger H^{(\pi)}  U(\textbf{R})$ the same also holds for $H^{(x)}$ and $H^{(p)}$. Consequently, we find
\begin{equation}\label{Hpeigen}
H^{(p)} \Psi_{\tilde{\bd \ell}}^{(\bd{\nu})} = E_{\tilde{\bd\ell}}^{(\bd{\nu})} \Psi_{\tilde{\bd \ell}}^{(\bd{\nu})}\,.
\end{equation}
Relating this to Eq.~(\ref{Hp}) with $\tilde{\Psi}\rightarrow \tilde{\Psi}_{\bd{\ell}}^{(\bd{\nu})}$ finally yields the important relation
\begin{equation}\label{eq19b}
\mathcal{F}_N \Psi_{\bd{\ell}}^{(\bd{\nu})} = \Psi_{\tilde{\bd{\ell}}}^{(\bd{\nu})} \,.
\end{equation}
It states that applying the Fourier operator $\mathcal{F}_N$ to an eigenstate of the hamiltonian (\ref{eq4}) does only lead to a rescaling of the length scales, i.e.~a replacement of $\bd{\ell}$ by $\tilde{\bd{\ell}}$ according to Eq.~(\ref{lengthduality}).

\section{Reduced density operators}
Instead of elaborating on eigenfunctions we focus on reduced density operators. To keep the notation simple we restrict ourselves for most of this section to $N$-particle systems with fermionic or bosonic exchange symmetry. The generalization to arbitrary/no symmetries will be discussed at the end of this section.

First, for a state $\Psi \in L^2(\mathbb{R}^N)$ we define the corresponding density operator $\rho:L^2(\mathbb{R}^N)\rightarrow L^2(\mathbb{R}^N)$,
\begin{equation}\label{NRDO}
\rho\,\Phi = \langle \Psi,\Phi\rangle_N\,\Psi\qquad,\,\forall \Phi \in L^2(\mathbb{R}^N).
\end{equation}
Here, $\langle \cdot, \cdot\rangle_N$ is the inner product on $L^2(\mathbb{R}^N)$ . By making use of the tensor product structure $L^2(\mathbb{R}^N)\cong L^2(\mathbb{R})^{\otimes^N}$ we can define for $m=1,\ldots,N-1$ the $M$-particle reduced density operator ($M$-RDO) $\rho^{(M)}$ of $\rho$,
\begin{equation}\label{mRDO}
\rho^{(M)} \equiv \mbox{Tr}_{N-M}[\rho]\,,
\end{equation}
where we introduced the partial trace $\mbox{Tr}_{N-M}[\cdot]$ (see e.g.~\cite{23}). Due to the exchange symmetry $\mbox{Tr}_{N-M}[\rho]$ is independent of the choice of the $N-M$ particles that are traced out. Since $\rho^{(M)}:L^2(\mathbb{R}^M)\rightarrow L^2(\mathbb{R}^M)$ we can study its eigenvalue equation on $L^2(\mathbb{R}^M)$,
\begin{equation}\label{eigeneqk}
\rho^{(M)} \chi^{(M)} = \lambda^{(M)} \chi^{(M)}\,.
\end{equation}
In quantum chemistry \cite{9} $\rho^{(M)}$ is typically represented w.r.t.~a basis of $L^2(\mathbb{R}^M)$ which leads to an infinite-dimensional $M$-particle reduced density \textit{matrix}.

The $1$-RDO, $\rho^{(1)}$, plays a particular role, since its eigenfunctions $\{\chi_k^{(1)}\}$ provide a $1$-particle description of the $N$-particle state $\Psi^{(N)}$ with the corresponding eigenvalues $\{\lambda _k^{(1)}\}$ as occupation numbers, where $ k \in \mathbb{N}$. In case of a $2$-particle observable $\hat{A}^{(2)}$ for a system of $N$ identical particles, $\rho^{(2)}$ arising from $\Psi^{(N)}$ is sufficient in order to calculate the expectation value $\langle\hat{A}^{(2)}\rangle_{\Psi^{(N)}}$. One important concept based on that is the simplified calculation of fermionic ground state energies: For hamiltonians with  $2$-particle interactions, the ground state can be obtained by minimizing the energy expectation value just over $2$-RDO $\rho^{(2)}$, which arise from  $N$-fermion states $\Psi^{(N)}$. The underlying set of states $\rho^{(2)}$ is much smaller than that of $N$-fermion quantum states, but the problem is then to determine the set of possible $\rho^{(2)}$'s, which is known as the $2$-particle $N$-representability problem \cite{10}. This makes obvious why reduced density operators particular for $M=1$ and $M=2$ are intensively studied in atomic physics and quantum chemistry, as well as, in quantum information theory as explained in the last paragraph of Section 1.

Before applying the concept of RDO to the eigenstates of our harmonic model (\ref{eq3}) we comment on the relation of Fourier operators $\mathcal{F}_1^{\otimes^N}$ and partial traces. Since $\mathcal{F}_k \equiv \mathcal{F}_1^{\otimes^k}$ is a unitary operator
and partial traces are respecting the local tensor product structure of $L^2(\mathbb{R}^N) \cong L^2(\mathbb{R})^{\otimes^N}$
we have
\begin{equation}\label{mRDOFourier}
\mbox{Tr}_{N-M}[\mathcal{F}_1^{\otimes^N}\,\rho \,(\mathcal{F}_1^\dagger)^{\otimes^N}]
= \mathcal{F}_1^{\otimes^M}\mbox{Tr}_{N-M}[\rho] (\mathcal{F}_1^\dagger)^{\otimes^M}
=  \mathcal{F}_k  \rho^{(M)} \mathcal{F}_M^\dagger\,.
\end{equation}
This means that if $N$-particle density operators $\rho$ and $\tilde{\rho}$ are conjugate by $\mathcal{F}_N$, $\tilde{\rho}=\mathcal{F}_N\,\rho \,\mathcal{F}_N^\dagger$, their $M$-RDO $\tilde{\rho }^{(M)}$ and $\rho^{(M)}$ are conjugate by $\mathcal{F}_M$, $\tilde{\rho}^{(M)} = \mathcal{F}_M \rho^{(M)}  \mathcal{F}_M^\dagger$.

Let us return to the harmonic model. For any arbitrary eigenstate $\Psi_{\bd{\ell}}^{(\bd{\nu})}$ we define its corresponding density operator
$\rho_{\bd{\ell}}$ and its $M$-RDO $\rho_{\bd{\ell}}^{(M)}$ according to Eqs.~(\ref{NRDO}) and (\ref{mRDO}). The same should  also be done for the Fourier transformation $\mathcal{F}_N \Psi_{\bd{\ell}}^{(\bd{\nu})}$ of $\Psi_{\bd{\ell}}^{(\bd{\nu})}$ which yields $\tilde{\rho}_{\bd{\ell}}$ and its $M$-RDO $\tilde{\rho}_{\bd{\ell}}^{(M)}$. Note that we skip the superscript $\bd{\nu}$ to keep our notation simpler.

Since the Fourier transformation $\mathcal{F}_k$ is not only a unitary operator but even an isometry (Plancherel theorem \cite{21}) we find for any $\Phi \in L^2(\mathbb{R}^N)$ by recalling Eq.~(\ref{NRDO}) and $\tilde{\Psi}\equiv \mathcal{F}_N \Psi$
\begin{equation}
\tilde{\rho}_{\bd{\ell}}\,\Phi \equiv \langle \tilde{\Psi}_{\bd{\ell}}^{(\bd{\nu})} , \Phi\rangle_N \tilde{\Psi}_{\bd{\ell}}^{(\bd{\nu})}
= \langle \Psi_{\bd{\ell}}^{(\bd{\nu})} , \mathcal{F}_N^{-1}\Phi\rangle_N \tilde{\Psi}_{\bd{\ell}}^{(\bd{\nu})}
= \mathcal{F}_N  \rho_{\bd{\ell}} \mathcal{F}_N^{-1}\,\Phi \,.
\end{equation}
Hence, we have $\tilde{\rho}_{\bd{\ell}} = \mathcal{F}_N  \rho_{\bd{\ell}} \mathcal{F}_N^\dagger$ and according to the comment in the first two lines below Eq.~(\ref{mRDOFourier})
\begin{equation}\label{mRDOconj}
\tilde{\rho}_{\bd{\ell}}^{(M)} = \mathcal{F}_M  \rho_{\bd{\ell}}^{(M)}\, \mathcal{F}_M^\dagger\,.
\end{equation}
Eq.~(\ref{mRDOconj}) together with Eq.~(\ref{eq19b}) implies for the $M$-RDO of any eigenstate of hamiltonian (\ref{eq4})
\begin{equation}\label{eq22a}
\rho_{\tilde{\bd{\ell}}}^{(M)} = \mathcal{F}_M  \rho_{\bd{\ell}}^{(M)}\, \mathcal{F}_M^\dagger\,.
\end{equation}

This result for $N$-particle systems with fermionic or bosonic exchange symmetry can be generalized to
arbitrary $N$-particle states. The only subtle difference is that without exchange symmetry the $M$-RDO depends
on the choice of the $(N-M)$ particles which are integrated out in Eq.~(\ref{mRDO}). To take this into account, we obtain now
for every choice of particles $\textbf{m}\equiv(m_1,\ldots,m_M)$ a corresponding $M$-RDO $\rho_{\bd{\ell}}^{(\textbf{m})}$.
For each of those $\Big(\!\begin{array}{c}N\\M\end{array}\!\Big)$ $M$-RDO the result Eq.~(\ref{eq22a}) holds separately,
\begin{equation}\label{eq22b}
\rho_{\tilde{\bd{\ell}}}^{(\textbf{m})} = \mathcal{F}_M  \rho_{\bd{\ell}}^{(\textbf{m})}\, \mathcal{F}_M^\dagger\,.
\end{equation}
Equation (\ref{eq22b}) is the duality relation for the $M$-RDO corresponding to a general $N$-particle state of a system described by hamiltonian (\ref{eq4}).

We close this section by deriving a homogeneous structure for the $M$-RDO following from the homogeneity relation
(\ref{homog}) for the $N$-particle eigenstates $\Psi_{\bd{\ell}}^{(\bd{\nu})}$. Recalling the unitary rescaling
operator $V_N(\alpha)$ from Eq.~(\ref{homog}), Eq.~(\ref{NRDO}) implies
\begin{equation}
\rho_{\bd{\ell}}\Phi =  \langle V_N(\alpha)^\dagger \Psi_{\bd{\ell}} , V_N(\alpha)^\dagger \Phi\rangle_N \Psi_{\bd{\ell}}
=  \langle \Psi_{\frac{\bd{\ell}}{\alpha}} , V_N(\alpha)^\dagger \Phi\rangle_N V_N(\alpha) \Psi_{\frac{\bd{\ell}}{\alpha}} \,.
\end{equation}
This yields the homogeneous structure
\begin{equation}\label{homogNRDO}
\rho_{\bd{\ell}} =  V_N(\alpha) \rho_{\frac{\bd{\ell}}{\alpha}} V_N(\alpha)^\dagger\qquad,\,\forall \alpha \in \mathbb{R}^+\,.
\end{equation}
By using the local structure $V_N(\alpha) = V_1(\alpha)^{\otimes^N}$, Eq.~(\ref{mRDO}), the unitarity of $V_1(\alpha)$ and
the properties of the partial trace it is an elementary exercise to show that the $M$-RDO inherit the same homogeneous structure (\ref{homogNRDO}),
\begin{equation}\label{homogMRDO}
\rho_{\bd{\ell}}^{(\textbf{m})} =  V_M(\alpha) \rho_{\frac{\bd{\ell}}{\alpha}}^{(\textbf{m})} V_M(\alpha)^\dagger\qquad,\,\forall \alpha \in \mathbb{R}^+\,,
\end{equation}
where $V_M(\alpha) = V_1(\alpha)^{\otimes^M}$ and $M= |\textbf{m}|$.

\section{Duality for eigenvalues and interaction matrices}
In this section we work out the consequences of the duality condition (\ref{eq22b}) of $M$-RDO.

Since the Fourier operator $\mathcal{F}_k$ is unitary the spectral consequences of result (\ref{eq22b}) are obvious:
Both density operators $\rho_{\bd{\ell}}^{(\textbf{m})}$ and $\rho_{\tilde{\bd{\ell}}}^{(\textbf{m})}$ have the same eigenvalues.
Moreover, due to the homogeneous structure (\ref{homogMRDO}) they depend only on the projective point $[\bd{\ell}]$ of $\bd{\ell}$ which is the equivalence class of $\bd{\ell}$ w.r.t.~the relation $\bd{\ell}\sim \bd{\ell}' :\Leftrightarrow \bd{\ell} = \alpha \bd{\ell}'$, for some $\alpha \in \mathbb{R}$.
We denote these eigenvalues by $\lambda_k^{(\textbf{m})}([\bd{\ell}])$, $k\in \mathbb{N}$, and order them decreasingly, $\lambda_k^{(\textbf{m})}([\bd{\ell}])\geq \lambda_{k+1}^{(\textbf{m})}([\bd{\ell}])$. Then, we have
\begin{equation}\label{NONduality}
\lambda_k^{(\textbf{m})}([\bd{\ell}]) = \lambda_k^{(\textbf{m})}([\tilde{\bd{\ell}}])\qquad,\,\forall k \in \mathbb{N}\,,
\end{equation}
where $\tilde{\bd{\ell}}\equiv \tilde{\bd{\ell}}(\bd{\ell})$ is given by Eq.~(\ref{lengthduality}) and we recall that we have skipped the superscript
$\bd{\nu}$ in (\ref{NONduality}), which labels the underlying $N$-particle energy eigenstate.

Identity (\ref{NONduality}) is our main result and it is the generalization of the duality condition (\ref{eq2}) found in Ref.\cite{5}. It does not depend on the use of dimensionless quantities and is valid for \textit{arbitrary} number $N$ of particles, for \textit{any} eigenstate state of the harmonic model (3) and for any choice of the subsystem $\textbf{m}\equiv(m_1,\ldots,m_M) \subset \{1,\ldots,N\}$ of particles $m_1,\ldots,m_M$. Since the $d$-dimensional harmonic model can also be represented by hamiltonian (3), the duality relation also holds for \textit{arbitrary} dimensions.

Since \{$\ell_{\mu}/\ell_{\mu+1}\}_{\mu=1,...,N-1}$ is in a  one-to-one relation  with the projective point $[\bd{\ell}]$,
we can follow Ref.\cite{5} and  introduce
$\delta _\mu =  \ln (\ell_\mu/\ell_{\mu+1}) \; , \mu =1,\ldots,N-1$. Then
Eq.~(\ref{NONduality}) becomes
\begin{equation}\label{eq26}
\lambda_k^{(\textbf{m})} (\delta_1,\ldots,\delta_{N-1}) \equiv \lambda_k^{(\textbf{m})}(-\delta_1,\ldots,-\delta_{N-1}),\ k \in \mathbb{N}
\end{equation}
where we used for the function on $\{\delta_\mu\}$ the same symbol $\lambda_k^{(\textbf{m})}$ as for the function on $[\bd{\ell}]$ in order to avoid further overloading  notation. The duality in form of Eq.~(\ref{eq26}) implies that the eigenvalues $\lambda_k^{(\textbf{m})}$ are even functions in $\{\delta_\mu\}$. Since $\delta_\mu =0$ for $\ell_\mu \equiv \ell_{\mu +1}$, which corresponds to vanishing interactions, the dimensionless parameters $\delta_\mu$ are a measure of the interaction. If the decreasingly ordered eigenvalue functions $\lambda_k(\delta_1,\ldots,\delta_{N-1})$ are also analytic at $\delta_1=0,\ldots ,\delta_{N-1}=0$ their Taylor series around that point contain only even order terms in each $\delta_\mu$ \footnote{Although the spectrum of any RDO $\rho_{\textbf{m}}$
is analytic in $\{\delta_\mu\}$ the ordering $\lambda_k \geq \lambda_{k+1}$ of the eigenvalues leads to non-analytic behavior at $\delta_1,\ldots,\delta_{N-1}=0$ whenever two hypersurfaces $\lambda_k(\delta_1,\ldots,\delta_{N-1})$ and $\lambda_{k+1}(\delta_1,\ldots,\delta_{N-1})$ do cross at that point.}.
This has been the case for the ground state of the generalized Moshinsky atom investigated  in Ref.~\cite{5} and has significantly simplified the perturbational analysis of the natural occupation numbers for weak couplings.

In the case of a $r$-fold degeneracy of the length scales $\ell_\mu$, i.e.~there exits $\mu$ such that $\ell_\mu = \ell_{\mu+1} = \cdots = \ell_{\mu +r-1}$,  it is $\ell_{\mu +\nu}/\ell_{\mu + \nu+1} = 1$ for $\nu = 0,1,\ldots,r-2$. Then, the number of variables in
(\ref{eq26}) is effectively reduced by $(r - 1)$. This holds for the harmonic model studied in Ref.~\cite{5} where the degeneracy has led to $\ell_1 \equiv \ell_{-}$ and $\ell_2 = \ell_3 = \cdots = \ell_N \equiv \ell_{+}$.  Consequently, duality condition (\ref{NONduality}) for fermionic or bosonic states and $M=1$ reduces to Eq.~(\ref{eq2}).

If hamiltonian (\ref{eq3}) arises as a \textit{periodic} harmonic lattice model with long range interactions nothing changes qualitatively, if $D_{r\alpha,s\beta} \to 0$ with the distance between sites $r$ and $s$ going to infinity. However, if  $D_{r\alpha,s\beta}=const.$  for all $r \neq s$ then, depending on the lattice symmetry and dimension $d$ there exist $\mu$ such that $\ell_{\mu}/\ell_{\mu+1}=1$ ,
leading to a reduction of the number of independent variables of $\lambda_k^{(\textbf{m})}$ .

\section{Discussion and conclusions}
We have proven analytically that the recently discovered duality of the R$\acute{e}$nyi entropy (Ref.~\cite{4}) and of the natural occupation numbers (Ref.~\cite{5}) of the ground state of Moshinsky-type atoms with two and three electrons, respectively, is \textit{generic} for arbitrary $N$-particle systems with harmonic interactions, in any dimension. It originates from the duality of wave mechanics in position and momentum space in combination with the harmonic interactions.
The dualities (\ref{eq22b}) and (\ref{NONduality}) are valid for all RDO of any arbitrary \textit{pure} $N$-particle eigenstate. A possible fermionic or bosonic exchange symmetry for the case of identical particles is irrelevant.
Moreover, duality also holds for arbitrary $N$-particle pure states of system (\ref{eq3}), i.e.~states of the form $\Psi_{\bd{\ell}} =\sum_{\bd{\nu}}\,c_{\bd{\nu}} \Psi^{(\bd{\nu})}_{\bd{\ell}}$, which even generalizes to mixed states. As a consequence, for spinful particles described by hamiltonian (\ref{eq3}), the duality (\ref{NONduality}) holds for the eigenvalues of any orbital RDO. Moreover, it also holds for any full (i.e.~spin including) RDO.

What is the physical implication of that duality?

The kind of duality found here is quite analogous to the Kramers-Wannier duality, because the duality relation (\ref{eq22b}) or (\ref{NONduality}) allows to relate a harmonic model (\ref{eq3}) with interaction matrix  $\textbf{D}$ with a \textit{dual} model with interaction  matrix ${\textbf{D}}^{*}$, i.e.~a model with \textit{different} coupling constants. From section $2$ it is clear how the dual model looks like. First notice that the  hamiltonian
\begin{equation}
\hat{H}_{\alpha,\beta}(\textbf{D}) = \alpha \hat{T} + \beta \frac{1}{2}\,\hat{\vec{x}}^t\,\textbf{D}\,\hat{\vec{x}}\,,
\end{equation}
with $\hat{T}$ the kinetic energy operator, $\alpha,\beta \in \mathbb{R}^+$ coincides with that in Eq.~(\ref{eq3}),
$\hat{H}_{1,1}(\textbf{D})$, up to a rescaling of the energy and length scale. Therefore, we call $\hat{H}_{\alpha,\beta}(\textbf{D})$ and $\hat{H}_{\alpha',\beta'}(\textbf{D})$  equivalent. The spectrum of any RDO $\rho^{(\textbf{m})}$ of the $\boldsymbol{\nu}$-th eigenstate of $\hat{H}_{\alpha,\beta}$ is independent of $\alpha$ and $\beta$ and depends only on the equivalence class $[\hat{H}(\textbf{D})]$. Due to this ambiguity the dual model $\hat{H}(\textbf{D}^*)$ is only uniquely defined on the level of those equivalence classes. Recalling Eqs.~(\ref{eq8}),(\ref{lengthduality}) and $l_\mu \equiv \frac{1}{\sqrt{d_\mu}}$ we find
\begin{equation}\label{dualD}
\textbf{D}^*(\textbf{D}) = \textbf{D}^{-1} \,.
\end{equation}
A model $\hat{H}({\textbf{D}})$ is self-dual if its dual model belongs to the same equivalence class, i.e. $\hat{H}({\textbf{D}}^{*}({\textbf{D}})) \sim \hat{H}({\textbf{D}})$. In that case it is straightforward to show that Eq.~(\ref{dualD}) implies that $D_{mm} \equiv const$  and
$D_{mn} = 0$ for all $m \neq n$, which corresponds to noninteracting particles in an isotropic harmonic trap.
Summarizing, the duality (\ref{NONduality}) relates  two physical harmonic models corresponding to two equivalence classes of hamiltonians, $[\hat{H}(\textbf{D})]$ and $[\hat{H}(\textbf{D}^*(\textbf{D}))]$,
\begin{equation}
[\hat{H}(\textbf{D})] \leftrightarrow [\hat{H}(\textbf{D}^*(\textbf{D}))]\,,
\end{equation}
with $\textbf{D}^*(\textbf{D})$ given by Eq.~(\ref{dualD}).

As an example, consider identical particles in one dimension. As explained in section 2 there are only two coupling parameters $D^{(1)}$ and $D^{(2)}$.
Like in Refs.~\cite{5,8} there are just two length scales and their ratio depends on $D^{(2)}/D^{(1)}$, only. Then Eq.~(\ref{dualD}) leads to
\begin{equation}\label{eq28}
{D^{(2)}}^{*}/{D^{(1)}}^{*} = -\frac{{D^{(2)}}/{D^{(1)}}}{1+N{D^{(2)}}/{D^{(1)}}}.
\end{equation}
Since the eigenvalues $D^{(1)}$ and $D^{(1)}+ND^{(2)}$ (with multiplicity $N-1$) of the interaction matrix $\textbf{D}$ are both positive
Eq.~(\ref{eq28}) relates two relative interaction strengths ${D^{(2)}}^{*}/{D^{(1)}}^{*}$ and ${D^{(2)}}/{D^{(1)}}$ with opposite sign.
If the original model with $\textbf{D}$ describes attractive particles the dual one with $\textbf{D}^*$ describes repulsive particles, and vice versa.

Although the original hamiltonian $\hat{H}({\textbf{D}})$ and its duals  $\hat{H}^{*}\equiv\hat{H}({\textbf{D}^{*}})$ have different spectra and different eigenfunctions, the eigenvalues of $\rho^{(\textbf{m})}$ of corresponding  quantum states of $\hat{H}$ and  $\hat{H}^{*}$, e.g.~their ground states,   are identical.  The same is true for any function of these eigenvalues, like  entanglement entropies and correlation functions. For the  specific models investigated in  Refs.~\cite{4,5}, the duality condition implies that the interactions in the dual  model are \textit{repulsive} (\textit{attractive}), if those of the original one  are \textit{attractive} (\textit{repulsive}) (see also Eq.~(\ref{eq28})). For the general model (\ref{eq3}) the interactions can be a mixture of attractive and repulsive terms under the constraint of positive definiteness of ${\textbf{D}}$.
Duality   can  interchange attractive and repulsive terms.
Despite the different nature of their interactions  this duality relates a generalized Moshinsky atom or
a harmonic lattice (not necessarily periodic)  with a \textit{dual} one  leaving the physical properties as entanglement, correlation functions, etc.~unchanged.
Whereas the Kramers-Wannier duality \cite{2} links low and high temperature behaviour,  the duality found in the present contribution connects strongly localized states with $\ell_\mu$ ``small'' with weakly localized states of the dual model for which $\ell ^{*}_\mu$ is ``large'', or vice versa.

\section*{Acknowledgments}
CS acknowledges financial support from the Swiss National Science Foundation (Grant P2EZP2 152190).

\section*{References}

redpajama_set_name: RedPajamaArXiv


Posted in Personal on January 25, 2016, 7:31 PM by ViperGirl85.
Posted in Personal on January 4, 2016, 7:57 PM by ViperGirl85.
Third Row at WWE Tomorrow!
Posted in Personal on December 29, 2015, 7:19 PM by ViperGirl85.
Posted in Personal on December 1, 2015, 9:26 PM by ViperGirl85.
Posted in Personal on November 3, 2015, 8:54 PM by ViperGirl85.
Weekend, WWE and Writing - Oh My!
Posted in Personal on October 20, 2015, 9:32 PM by ViperGirl85.
Posted in Personal on October 6, 2015, 5:36 PM by ViperGirl85.
Posted in Personal on August 13, 2015, 8:31 PM by ViperGirl85.
Posted in Personal on July 18, 2015, 11:31 PM by ViperGirl85.
Posted in Sports on April 30, 2015, 8:37 PM by allen0187.
Posted in Sports on April 30, 2015, 8:11 PM by allen0187.
Posted in Sports on April 27, 2015, 10:41 PM by allen0187.
Ladies and Gentlemen, Your New INTERCONTINENTAL CHAMPION - W-A-D-E-B-A-R-R-E-T-T-!!!
Posted in Sports on April 25, 2015, 1:29 AM by allen0187.
Posted in Personal on April 1, 2015, 9:38 PM by ViperGirl85.
Posted in Sports on March 29, 2015, 12:17 AM by ViperGirl85.
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We Ordered Our WWE SummerSlam Tickets!
Posted in Personal on March 16, 2015, 9:56 PM by ViperGirl85.
Posted in Sports on March 13, 2015, 12:31 AM by allen0187.
I Met Shawn Michaels For The Third Time!
Posted in Personal on March 9, 2015, 10:30 PM by ViperGirl85.
I'm Meeting Shawn Michaels Again!
Posted in Personal on February 20, 2015, 10:52 PM by ViperGirl85.
I wrote a post a while back stating that I was going to WWE SummerSlam. You can catch that post right here: http://personapaper.com/article/14702-heading-to-wwe-summerslam-in-2015 (if I am not allowed to link it, please let me know and I'll check it.
Posted in Personal on February 18, 2015, 9:53 PM by ViperGirl85.
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Posted in Writing on October 26, 2014, 8:50 PM by allen0187.
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Posted in Sports on August 18, 2014, 1:04 AM by allen0187.
I'm liking the #ZebCoulter gimmick and of course, I'm loving the Jack Swagger-Ruzev rivalry that 's going on. This rivalry will culminate in a #FlagMatch at #SummerSlam. Of course, Zeb Coulter has proven himself to be a good mouthpiece for #JackSwagger .
Posted in Sports on August 10, 2014, 11:35 PM by allen0187.
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Posted in Sports on June 12, 2014, 1:14 PM by melody23.


Dr. Zakarian received his Doctoral Degree in Chemistry in 2001 in the field of natural products synthesis, working in the group of Professor Robert Holton. He subsequently pursued his postdoctoral studies at the University of California, Irvine in the laboratories of Professor Larry Overman, between 2002 and 2004. After first four years in Florida, Dr. Zakarian joined the faculty at UC Santa Barbara in June 2008. He is a recipient of the Amgen Young Investigator Award, Eli Lilly Grantee Award, and the NSF Faculty Early Career Development Award.
In our research, we are applying our core interest in the synthesis of complex natural products as a starting point to contributions in different fields of chemical science, including the development of organic synthesis on both strategic and methodological levels, asymmetric synthesis, development and applications of catalysis, and various aspects of bioorganic chemistry, especially the elucidation of the mode of action of unusual biologically active natural products.
As an example, in the past few years our efforts have focused on a subclass of marine toxins belonging to the cyclic imine group. We have achieved a scalable synthesis of one member in this class of complex natural products that enabled the elucidation of its mechanism of toxicity in collaboration with a research group in France. This research project also served as a starting point for our contributions to the development of new methodology based on enolate chemistry, which is an ongoing effort in our group.
Beaumont, S.; Ilardi, E. A.; Tappin, N. D. C., Zakarian, A. "Marine Toxins With Spiroimine Rings: Total Synthesis of Pinnatoxin A" Eur. J. Org. Chem. 2010, 5743–5765.
Gu, Z.; Zakarian, A. "Total Synthesis of Rhazinilam: Axial-to-Point Chirality Transfer in an Enantiospecific Pd-Catalyzed Transannular Cyclization" Org. Lett. 2010, 12, 4224–4227.
Herrmann, A. T.; Saito, T.; Stivala, C.; Tom, J.; Zakarian, A. "Regio- and Stereocontrol in the Rhenium-Catalyzed Transpositions of Allylic Alcohols" J. Am. Chem. Soc. 2010, 132, 5962–5963.
Lu, C.-D.; Zakarian, A. "Unified Total Synthesis of (±)-Trichodermamide B and the Putative Biosynthetic Precursor of (±)-Aspergillazine A Using the Oxaza-Cope Rearrangement" Angewandte Chemie, Int. Ed. 2008, 47, 6829–6831.
Stivala, C.; Zakarian, A. "Total Synthesis of (+)-Pinnatoxin A" J. Am. Chem. Soc. 2008, 130, 3774–3776.
Ilardi, E. A.; Stivala, C. E.; Zakarian, A. "Hexafluoroisopropanol as a Unique Solvent for Stereoselective Iododesilylation" Org. Lett. 2008, 10, 1727-1730.
Stivala, C.; Zakarian, A. "Total Synthesis of (+)-Pinnatoxin A" J. Am. Chem. Soc. 2008, 130, 3774-3776.
Qin, Y.; Stivala, C. E., Zakarian, A. "Acyclic Stereocontrol in the Ireland-Claisen Rearrangement of alpha-Branched Esters" Angewandte Chemie, Int. Ed. 2007, 46, 7466-7469.
Zakarian, A.; Lu, C.-D. "Development of the 1,2-Oxaza-Cope Rearrangement" J. Am. Chem. Soc. 2006, 128, 5356-5357.


Cards such as Credit and Debit cards are by far the most common non-cash payment method used by consumers today. These cards are used for in-store purchases, to withdraw cash from an Automatic Teller Machine (ATM), or to make purchases over the phone or Internet.
Newer payment systems such as chip and pin have assisted in improving customer efficiency and reducing some types of card fraud such as skimming, however it's also important to consider what further opportunities are created by the technology for fraud offenders, who are constantly seeking better ways to exploit weaknesses in the system.
Online frauds are most commonly carried out by the theft of genuine card details that are then used to make purchases over the Internet, by phone or mail order. The genuine cardholder may not be aware of this fraud until he or she receives a transaction SMS for the amount debited or check their statement.
Offline frauds or Counterfeit card fraud is the use of altered or illegally reproduced cards including the replication/alteration of the magnetic stripe and changes to the details on the face of the card with intent to defraud. Most of this fraud involves a technique called skimming.


This Verona farmers market is located at Market Dock - Augusta County Government Center Verona, VA 24482. Drop by to see for yourself. Call to learn more about its assortment of organic food, vegetables, fruits, crafts and local specialties. Begins at 12, on Wednesdays. Takes SFMNP. Click the Edit link if you frequent this market to provide information about what is offered.


http://www.wsj.com/articles/i-was-wrong-about-obamacare-1469997311 Large health systems deliver "personalized" care in the same way that GM can sell you a car with the desired options. Yet personal relationships of the kind often found in smaller practices are the key to the practice of medicine. They are the relationships that doctors want to forge with patients, and vice versa. It may sound old-fashioned, but what I have learned is that we do not need to sacrifice this unique feature of our health-care system as we move forward in adapting new value-based payment models and improving the health of patients.
Bloomberg News (6/27, Ramkumar, 2.07M) reports that "during the study period, deductibles grew 86 percent and coinsurance costs rose 33 percent." Although "co-payment costs fell…so did the number of insurance plans that charged them." The data indicated "out-of-pocket costs exceeded the annual growth of insurance premiums…which increased 5.1 percent" annually.


Summer..the most beautiful season, with fireflies emerging, seas warming and the sun shining brightly..
Create unforgettable memories with your loved ones during your summer holidays with our Demre – Kekova boat tour.
Let's take a look at this delightful tour route!
Our Kekova boat trip starts between 10:00 and 10:30 am as we depart from Demre Harbour and head towards our first stop.
Discover the fascinating secrets of Kekova's history on this unique adventure! The sunken city was flooded after an earthquake two thousand years ago and never fails to leave an impression. View the enchanting ruins of the Sunken City as we cruise alongside and capture unforgettable photos.
Our boat will anchor in Kalekoy (Simena) Port for 1.5 hours, giving you time to visit its authentic village and soak up the unique culture. Alternatively, you can spend your time here swimming in the beautiful blue waters.
We'll stop to swim and eat in Gokkaya Bay, giving you time to experience its beautiful natural setting and enjoy the various water sports.
Take time to admire the majestic natural structure of the Pirates Cave, which is renowned for its large size and softly curving lines.
There are eight types of turtles known around the world, five of which are found in the Mediterranean. In Karemlik Bay, you'll have the chance to witness these joyful creatures in what is a natural paradise.
After an exciting journey that lasts all day long, our tour ends at Demre Harbour.
Lettuce, Arugula, Parsley, Green onion, Cress, Mint, Carrot grater, Tomato slices, Olive oil with lemon sauce.


The allergen free market is quickly shifting from compromised copies of gluten-filled favorites to an area of experimental food and flavor, ripe for innovation.
The future of food is one of today's most hotly debated topics by everyone from scientists and celebrity chefs to Bill Gates. But, we believe that necessity remain the mother of invention. And there is one relatively new and breaking influence that has the power to endure and steer yet another radical change for the future direction of food: allergen free.
In the burgeoning allergen free market, necessity is creating totally new approaches to healthy eating. This market is quickly shifting from compromised copies of gluten-filled favorites to an area of experimental food and flavor, ripe for innovation.
But first, let's take a step back and look at the context. Allergen free (or Free From which is the name convention now adopted in the UK) brands are rapidly becoming mainstream and it is now such a recognized market that we think of them as a new way of eating for everyone, not just a restriction for the few.
We are seeing both existing and new brands respond to food allergies, sensitivities and the lifestyle changes of consumers by developing new languages and aesthetics that are changing the perception of this market from staid, limited, mundane and bland to a fresh, exciting, varied and easy way to eat. In this way, these brands are building (a new) desire amongst their established audience and – maybe more significantly – now drawing in a brand new food savvy audience.
New brands such as Ilumi are leading this change.
Ilumi is a range of 100% natural nut, milk and gluten free gourmet convenience meals, designed to reflect the exciting truth behind its food; that it's full of taste, variety and cooking expertise while still being allergen free. The brand philosophy focuses on food without sacrifice, creating tolerance rather than intolerance and bringing to life a forward thinking, food-focused approach and establishing a connected community around the brand.
It is the creation of this complete world of food – and food experience – that has already galvanized a foodie revolution and, with the growing influence of this market, is now taking this connection to food to a whole new level. Foodies are looking to get the most from their food emotionally, physically and intellectually and buying into the idea – and celebration – of 'being in the know' that is at the core of the new Free From approach.
Allergen free consumers are foodies not faddies and with this growing market high on consumers', retailers' and the nation's agenda, this presents a new and exciting opportunity for future thinking and innovation. Our food brands, experts and ideators need to adopt and adapt this approach mainstream, finding new and ever more creative product solutions that stimulate and challenge the assumptions of food culture as we know it.


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      Source:map-lazy-load.js

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redpajama_set_name: RedPajamaGithub


text:
Share this Story: ALDRICH: Waiting on justice — Wrongful conviction conference coming to Winnipeg
ALDRICH: Waiting on justice — Wrongful conviction conference coming to Winnipeg
Josh Aldrich
David Milgaard (right) speaks to the media along with his lawyer Greg Rodin (left) in Calgary, about wrongful convictions as he takes part in the launch of the first annual International Wrongful Conviction Day on Thursday October 2, 2014. Photo by Stuart Dryden /Stuart Dryden/Calgary Sun/QMI Ag
Justice, especially in the most heinous of cases, is something we all crave and strive for.
In most cases, the police and court system do a reasonable job in tracking down killers and putting them behind bars.
ALDRICH: Waiting on justice — Wrongful conviction conference coming to Winnipeg Back to video
Except when they do not. Not only does the search for real justice end, but an innocent bystander's life can be ruined in the process.
On Friday, five men wrongfully convicted of murder and three lawyers will host a conference at the Canadian Museum for Human Rights in Winnipeg to discuss these issues.
The museum is an apt backdrop considering what it stands for, but also for what this city has been through in the last year with 44 homicides in 2019 and the first two of 2020 this past weekend. The Winnipeg Police Service generally does an admirable job in chasing down those responsible in these crimes and finding justice for the loved ones of those killed, but there is a history of exceptions to this rule.
James Driskell smiles while speaking to media at a news conference after the release of the report of the 2006 inquiry into Driskell's wrongful conviction in 1991 for the murder of Perry Dean Harder. Photo by JASON HALSTEAD/SUN MEDIA /JASON HALSTEAD
Four of the five now free men on the panel should be familiar names to Winnipeggers. James Driskell was wrongly convicted and incarcerated for 13 years for the death of Perry Harder in 1990. David Milgaard was wrongly convicted and incarcerated for 23 years for killing Gail Miller in 1969. Frank Ostrowski was wrongly convicted and incarcerated for 23 years in the shooting death of Robert Nieman in 1986. Tom Sophonow was wrongly convicted and incarcerated for 4.5 years in the death of Barbara Stoppel in 1981.
The fifth is Brian Anderson who served 10 years for a murder he has spent more than 40 years trying to prove he did not commit and will spend the rest of his life on parole.
"Often times it includes tunnel vision by the police where they affix early on in the investigation on a suspect and ignore evidence that tends to exculpate the person and focus on evidence that tends to implicate the person, don't uncover other leads or look in other directions and the result of course is there is a case made where it is incomplete in terms of the investigation," said Greg Rodin, who represented Milgaard and Kyle Unger — another Manitoba man wrongfully convicted of murder and released.
These cases are not just at the feet of the police. According to Rodin, who is one of the conference organizers, there are often issues all along the chain of justice in these instances, this includes Crown prosecutors not turning over all evidence, intentionally or otherwise, to the defence, and defences that are ill-equipped to defend their client.
Frank Ostrowski. Winnipeg Sun files Photo by DEAN PRITCHARD/Winnipeg Sun /DEAN PRITCHARD/Winnipeg Sun
Manitoba Justice has been working to rectify these issues.
"There has been extensive training and education for justice system participants to significantly reduce the risk of wrongful convictions, including Crowns, police, defence counsel and forensic scientists," said a spokesman for Manitoba Justice in an email. "This training has led to improvements in areas such as disclosure, understanding tunnel vision, the limitations of eyewitness identification and testimony, understanding the risks associated with forensic and expert testimony and many more. Legal developments in disclosure and plea inquiry requirements have also reduced the potential for wrongful guilty pleas."
The Winnipeg Police Association did not respond to requests for an interview.
Prime Minister Justin Trudeau recently called for the creation of an independent Criminal Case Review Commission which would streamline the process for review of potential wrongful conviction cases. The current process requires an application to the Minister of Justice just to be reviewed and it can take years.
Thomas Sophonow. Postmedia files Photo by David Clark /PRV
According to Innocence Canada, there are currently 81 cases being reviewed for wrongful conviction, and all but three of them are for murder or manslaughter. Eight of the cases are in Manitoba.
It is still not known what the commission will look like or how it will operate or when it will come into being, and it will be one of the major topics discussed on Friday by the panel.
Also discussed will be the hell these men went through while their lives were put on hold in the name of closure.
Rodin is encouraging the public to come out to this free conference — it starts at 2 p.m. — and to form a greater understanding of the issues at hand.
"Participate in the process of making a better justice system that's less likely to fail, but also one where it does fail — as it inevitably will — it can be corrected, the fear can be remedied and in a timely fashion," said Rodin.
Jaldrich@postmedia.com
Twitter: @JoshAldrich03

redpajama_set_name: RedPajamaCommonCrawl


text:
New Census Data Show Household Incomes Are Rising Again, But Share Going to Middle Class Is at Record Low
Article Sep 12, 2017
Too many families are being left behind as a record-high share of income goes to the top.
Alex Rowell
David Madland
Advancing Racial Equity and Justice, Building an Economy for All, Restoring Social Trust in Democracy, Strengthening Health and Ending the Pandemic, Census, Economic Justice, Economy, Income Inequality, Jobs, Racial Equity and Justice, Trade, Trump Administration, Wages+10 More
Madia Coleman
Associate Director, Media Relations
mcoleman@americanprogress.org
Peter Gordon
pgordon@americanprogress.org
Jerry Parshall
Senior Director, Safety and Justice Campaign; Director, State and Local Government Affairs
jparshall@americanprogress.org
A couple walks past abandoned buildings in a poor, urban neighborhood in Cincinnati, Ohio, on April 11, 2009. (AP/Al Behrman)
The latest Census Bureau data show that for the second straight year, the typical U.S. household saw its income rise in inflation-adjusted terms in 2016, the last year of the Obama administration, and incomes have now recovered to approximately pre-Great Recession levels. The median U.S. household income was $59,039 in 2016, a 3.2 percent increase from real 2015 levels.
While the data contain some good news, the overall story is still quite bleak.
Median household income is now at roughly the same level it was in the late 1990s—meaning that household income has been effectively stagnant for two decades. Furthermore, 2016's higher level of income when compared to years prior to 2013 could be partially driven by the survey's redesign in 2013, which resulted in finding higher incomes than the previous survey design. It also represents a very small share of economic growth over this period.
The majority of gains continue to go to those at the very top, and most households—regardless of education level or race—continue to see little income growth. In 2016, the share of the nation's income going to the top 5 percent hit a record high, while the share going to the middle class—the middle 60 percent of households—fell to a record low. Since 1967, the first year that the Census Bureau began releasing income-share data, the middle class has never received a smaller share.
On the positive side, the data show that lower-income workers have made gains in recent years and that the poverty rate has fallen dramatically postrecession, suggesting that policies such as increasing the minimum wage can make a significant impact on people's income but that much more needs to be done. Unfortunately, the policies that the Trump administration is pursuing are unlikely to raise incomes for most Americans.
The data show that lower- and middle-income households are seeing slower income growth than those on the higher end of the income spectrum. The average household in the middle 60 percent saw its income grow by 2.7 percent from 2015 to 2016, with households in the group between the 20th and 40th percentiles seeing the fastest growth. Households in the lowest-earning 20 percent saw slower income growth than in 2015, but their income still grew by 2.6 percent after another year of job growth and successful campaigns to raise state and local minimum wages. But the highest-earning households saw the largest growth, with the income of the top 5 percent of households climbing by 5.6 percent and the income of the top 20 percent climbing by 4.4 percent.
This growth compounds on previous years of unequal growth. Since the end of the recession, the top 5 percent of households have seen their incomes grow more than twice as fast as the middle 60 percent, on average. Since the end of the recession, incomes of the top 20 percent have grown by 1.6 percent annually, on average, and incomes of the top 5 percent have grown by 2.3 percent on an annualized basis. Over the same time period, the incomes of the middle 60 percent grew by just 1.1 percent.
In light of this unequal growth, it is no surprise that the middle class is bringing home a record low share of income. The middle 60 percent of households earn just 45.4 percent of the total U.S. national income. The top 20 percent take home 51.5 percent, and just the top 5 percent of households earn 22.6 percent of the national income—more than the bottom 40 percent of households' earnings combined. The declining middle-class share of income has a real impact on middle-class families. The average middle-class family would have earned about $10,800 more in 2016 had the middle 60 percent of households earned the same share of national income that they earned in 1968.
These new data also show an economy that is not working properly for many households regardless of their educational attainment levels. Since 2010, households headed by individuals with less than a high school diploma have actually seen the largest income growth as a share of their income, even as their average income remained relatively low at just $40,424 in 2016. This is likely due in part to the fact that individuals with less than a high school education have seen the largest reduction in their unemployment rate over the past six years, as well as due to state- and local-level minimum wage increases. Households headed by individuals with a high school diploma, some college education, or a four-year degree have seen their incomes effectively stagnate, growing at or below 0.5 percent per year, on average, from 2010 to 2016.
Finally, the most recent data show that incomes of black and Hispanic households remain dramatically lower than those of non-Hispanic white households. The median Hispanic household earned $47,675, 73.3 percent of the median white household income in 2016. Meanwhile, the median black household earned even less: $39,490, just 60.7 percent of the median white household income. These racial income gaps have fallen since 2015. However, the black income gap is still dramatically larger than it was in 2000. Without concrete efforts to fight the structural biases that lead to such gaps, it is possible they will stagnate in upcoming years.
While middle-class families saw continued income gains in 2016, America's economic growth is still too concentrated within its wealthiest households. This unfair economy is not a force of nature, but instead was caused by policy choices. Policymakers must take a stand firmly on the side of workers and their families to help them share in the growth that is being captured by the wealthy.
There are clear ways that policymakers could help reverse this continued growth in inequality. American workers deserve a new system of labor law that gives them greater power in the workplace and beyond, making it easier for workers to raise wages and improve benefits across an industry. Policymakers should boost working families' economic security by improving access to child care, requiring companies to offer paid sick leave, and instituting fair scheduling requirements. The government should crack down on mergers and monopolies that hurt workers and consumers. And finally, the government should ensure that every American willing and able to work is able to find a good job, both by aiming for full employment through monetary policy and by providing a backstop of well-paying public jobs in communities across the country.
Without these and similar bold policy actions, working Americans will likely face continued years of stagnant incomes. Unfortunately, President Donald Trump's policy agenda moves in the opposite direction, with many of the policies he has pushed—from weakening overtime standards to reducing workers' rights—set to lower workers' pay.
Alex Rowell is a research associate for Economic Policy at the Center for American Progress. David Madland is a senior fellow at the Center.
Senior Fellow; Senior Adviser, American Worker Project
Explore The Series
The State of the U.S. Labor Market: Pre-August 2017 Jobs Release
The State of the U.S. Labor Market for Latinas: Pre-October 2017 Jobs Release
Revolutionizing the Workplace: Why Long COVID and the Increase of Disabled Workers Require a New Approach
Mia Ives-Rublee, Rose Khattar, Anona Neal
The Wage Gap Persists—But Recent Reforms Are Making a Difference
Rose Khattar, Lauren Hoffman
Opinion: Hispanic and Latino workers deserve equal economic opportunity, not a return to the pre-pandemic status quo
Latino Workers Continue To Experience a Shortage of Good Jobs
Rose Khattar, Jessica Vela, Lorena Roque


text:
12 Top Walk In Freezer Wiring Diagram Images
Walk In Freezer Wiring Diagram
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Fast, utterly reliable and delivered at a competitive price.
We know that busy landlords need a quick fix without compromising on quality - and that is exactly what we provide. From leaking pipes and faulty radiators, through to blocked toilets and boiler repairs, our experienced team will fix the problem and ensure that your tenants are kept happy and warm. We also carry out gas safety checks to help ensure that you fulfil your duty of care as a landlord.
All gas installations need to be regularly inspected to ensure that they are in safe working order. Boilers and fires should be checked annually to reduce risk of carbon monoxide poisoning. All of our experienced heating engineers are gas safe registered and qualified to carry out gas safety checks.
Servicing a boiler helps ensure that you get the very best out of it, both in terms of performance and longevity. Our expert engineers have the latest equipment and will be able to service any boiler and make any required adjustments to ensure that it remains in tip top condition.
When things go wrong, tenants don't like to be kept waiting. We offer a rapid response service to diagnose the problem and get it fixed quickly and with minimum fuss. We also provide a 24/7 emergency service for urgent issues that need immediate attention.
Our team of experienced plumbing, heating and gas engineers provide a responsive and reliable service. We have the knowledge and expertise to diagnose and fix any plumbing or heating issue, to get the job done 'right, first time'. To provide our landlords with complete peace of mind we offer a 90 day, no quibble, labour guarantee on all work that we undertake.


text:
<?php

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    public function setContext($value) {
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    public function __construct($code, $message, $file, $line, $context) {
        parent::__construct($message, $code);

        $this->file = $file;
        $this->line = $line;
        $this->setContext($context);
    }
}

class EHandler {
    private function __construct() {
    }

    public static function error_handler($code, $message, $file, $line, $context) {

        if (error_reporting() === 0) {
            return;
        }

        throw new CustomException($code, $message, $file, $line, $context);
    }

    public static function exception_handler($e) {
        try {
            ob_start();
            //$lines = explode("\n", $output);
            //var_dump($lines);
            echo $e->getMessage();
            echo '
';
            echo get_class($e);
            echo '
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            echo  $e->getFile();
            echo '
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            echo  $e->getLine();
            echo '
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            var_dump($e->getTrace());

            $dump = ob_get_clean();
            if (Config::$debug) {
                echo $dump;
            } else {
                echo file_get_contents(Config::$dataDir . '/custom/error.html');
            }
        } catch (Exception $e) {
            echo file_get_contents(Config::$dataDir . '/custom/error.html');
            echo 'ErrorHandler';
        }
        exit;
        /*echo '
oops! something went seriously wrong.

It would be pretty awesome if you could mail me (kobelipkens@gmail.com) the following info:
';
        var_dump($e);
        echo '
' . $e->message . '
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        $file = basename($e->file);
        echo $file . ': ' . $e->line . '
';
        exit;*/
    }
}


Here we come face to face on how far young ladies can go in engaging themselves in degrading and embarrassing actions just to acquire a man of their dreams. On the other hand is certain family that can only acquire their selfish interests through lies, intrigues and deceit. Interesting is how their dirty secrets and lies are exposed leading to a terrible disgrace and mockery by an entire community.


Last week we used Jan de Weille's ECG (electrocardiogram) equipment to do a quick experiment. Gaspard and Sandra Bébié Valérian of art-act were starting a new project on mycelium and asked Jan (a scientist, who had also worked on sonification of electric fish and motoneurones) some questions about sonification of champignons.
An image of our first experiment. Jan thinks we just mesured the apparatus' imperfections. (We had no idea about the response range nor about delays.) I was excited to see the potentials change when I talked to the mushroom, and Gaspard was excited about all. We had a nice range of reactions from scientific to almost chamanic.
Wondering what this tells about art-science projects.
27-28 09 Distant Feelings, Video Vortex, Spazju Kreattiv, Valletta, Malta.


text:
Q: Is it possible to define an unnamed trait and use it as a mixin in Scala? I've implemented the cake pattern using structural types instead of wrapper traits. I am now wiring up my dependencies like this:
trait GreeterDependency { def greeter = HelloGreeter }
val printer = new Printer with GreeterDependency

It would be nice if I could do something like this instead:
val printer = new Printer with trait { def greeter = HelloGreeter }

However, I get a syntax error. Is there a way to define an unnamed trait and use it as a mixin like this?
(For clarity, here is all my code: http://ideone.com/vMDFYD)

A: I'm not completely certain what you're going for, but I think what you're looking for is to instantiate a trait with its abstract members defined in an anonymous class. If that's the case, you can just do:
val printer = new Printer { def greeter = HelloGreeter }

It's a pattern I'm using a lot right now to make up for the fact that traits can't define constructor parameters.
Full refactoring based off of the Ideone link in the question:
trait Greeter {
  def apply(name: String): String
}

object HelloGreeter extends Greeter {
  def apply(name: String) = s"Hello, $name!"
}

trait Printer {
  def greeter: Greeter

  def apply(name: String) = println(greeter(name))
}

object Main extends App {
  val printer = new Printer { def greeter = HelloGreeter }
  printer("rightfold")
}

Running example here: http://ideone.com/mAumNY

redpajama_set_name: RedPajamaStackExchange


Dr. Buch (pronounced "Bush") is recognized for his expertise in male infertility analysis and treatment, throughout the Metroplex, across the nation, and globally. His expertise is held in the highest regard by peers, and his warm, down to earth manner wins the confidence of patients.
Bachelor's degree from University of Michigan.
Medical Doctorate at University of Michigan's medical school.
Training in surgery and urology at Albany Medical Center of Union College in Albany, New York, with a special interest in male infertility and vasectomy reversal.
Two-year Fellowship in male infertility and microsurgical vasectomy reversal at Baylor College of Medicine in Houston, Texas, where he was one of only five American Urological Association Research Scholars in that program.
Shortly after Baylor, Dr. Buch was recruited to create a new Division of Male Infertility and Microsurgical Vasectomy Reversal at the University of Nebraska Medical Center in Omaha. His contributions to his field were acknowledged by the American Fertility Society (now the American Society of Reproductive Medicine) with the Male Reproduction Urology Prize Paper Award.
Upon relocation to Hartford, Connecticut in 1990, Dr. Buch worked with the University of Connecticut Medical Center. He published additional research papers, and trained residents and research Fellows in male infertility and microsurgical vasectomy reversal. This became his second successful new training program, and it prompted his decision to seize an opportunity for private practice in the Dallas-Ft. Worth area.
Dr. Buch has held a number of academic positions, and currently serves as Clinical Associate Professor of Urology at University of Texas Southwestern Medical School in Dallas. He has hospital affiliations at Frisco Ambulatory Surgery Center and Baylor Medical Center in Frisco. He maintains active membership in a broad range of professional associations, and participates on many committees.
Sharon Vogel is a very important part of the NTMIC Sperm Lab team. Sharon is valued by staff, doctor clients, and patients for her skill and devotion to excellence, as well as experience in easing couples through the emotional aspects of infertility treatment.
She was certified as a Medical Technologist by the American Society of Clinical Pathologists in February 1985. Sharon earned a Bachelor of Science degree in Animal Science from Oklahoma State University in Stillwater, Oklahoma, and continued her education at Texas Tech University and Methodist Hospital in Lubbock, Texas.
During her career, Sharon has worked for several physician office laboratories as a generalist, with extensive experience in phlebotomy, hematology, chemistry, and microbiology. She became an andrology specialist in September 2002, at North Texas Male Infertility Center.


We'll all agree that some rules are an evil necessity. Little things like murder, treason or driving slow in the hammer lane are to be expected and respected. Some rules are very personal. I will shower everyday. I will wear big boy pants when the temp dips below 50º.
Then there are the rules that verge on common sense. I'm talking about vegetables in desserts. How can anyone with a shred of moral decency tell me that a main meal side dish belongs in the reason for cleaning your plate?
In an effort to cow our morale and force us to be healthy, my employer has loaded the snack food machines with baked, low sodium, air-popped sticks and twigs.
I'm here to tell you there is a saint in our midst. A rebel in the hen house! For if you look closely, you will find the the rules have been broken and the start of that slippery slope.
Next Who Says You Can't Go Back?
Carrot cake for breakfast … yum!


Terence Cheng is Senior Lecturer at the School of Economics, University of Adelaide. He is a health economist whose current research interests are in the areas of health care insurance, medical labour market, the economics of ageing, and applied microeconometrics.
Terence has published in both general and field journals in economics including the Economic Journal, Journal of Health Economics, Health Economics, Canadian Journal of Economics, and multidisciplinary journals such as Social Science and Medicine and Health Policy. His research has been featured in Australian and international media. He has also worked on consultancy and commission research by international organisation and governmental clients such as the World Bank, the Department of Health and Human Services (Victoria, Australia), and Health Workforce Australia.
In 2019, Terence was appointed a Senior Research Fellow of the Centre for Research on the Economics of Ageing (CREA), and Visiting Associate Professor of the School of Economics at the Singapore Management University. He was a recipient of an Australian Government Endeavour Research Fellowship in 2018 where he visited New Zealand to collaborate with researchers to establish a new research program on the economics of well-being and ageing.
Prior to joining Adelaide, Terence worked at the University of Melbourne where he was involved in the MABEL Longitudinal Survey of Australian Doctors. He remains an honorary fellow of the Melbourne Institute. He has a PhD from the Australian National University, and Master and Bachelor degrees from the National University of Singapore.
Click here for CV (pdf format).


text:
Q: Prestashop 1.7.x override javascript from module with my own module I am writing my own module that need to override ps_shoppingcart.js
I try to read in documentation:
https://devdocs.prestashop.com/1.7/themes/reference/overriding-modules/#overriding-templates-and-assets
└── modules
    ├── css
    │   ├── external-lib.css
    │   └── style.css
    ├── js
    │   └── app.js

I know that kind of override it is for themes, but I was hope that was working for modules too, and unfortunately I don't find nothing about module.
Another idea may be override Ps_Shoppingcart class and call my own javascript.
Do you any idea how to solve it or some doc link, tutorial etc.
UPDATE Solved
This is the solution that I found it, I am not sure if is the best way but it is work.
I override the ps_shoppingcart.php and then I try to load my own javascript.
That probably because path error, was not loaded, so I included it from my module and now it is work
ps_shoppingcart.php override I don't include any javascript
if (!defined('_PS_VERSION_')) {
    exit;
}
class Ps_ShoppingcartOverride extends Ps_Shoppingcart
{
    public function hookHeader()
    {
        return;
    }
}

If someone know better solution, I'd be happy to hear it
Thanks
Gabriele

A: You can use unregisterJavascript FrontController function in your module hookHeader to delete some js instance and then register your own js to override its functionality.
This line in your case:
$this->context->controller->unregisterJavascript('modules-shoppingcart');

Now, you can register your
$this->context->controller->registerJavascript('instance-name', 'js-resource-path);


text:
Драган Киряков Ца́нков (9 ноября 1828, Свищов, Османская империя — 24 марта 1911, София, Болгария) — болгарский политический деятель. Третий по счёту глава правительства автономного Болгарского княжества (апрель — декабрь 1880 года).

Биография
Учился в Одессе, Киеве и Вене. С 1857 года жил в Константинополе, был учителем во французском училище. Тогда же и там же основал в католическом монастыре болгарскую типографию, в которой печатал болгарские книги и журналы.

С 1859 по 1863 год издавал еженедельную газету «България», в обычном тоне подцензурных газет, издаваемых в Турции. Тем не менее газета старалась, насколько это было возможно, пробуждать в болгарах чувство национальной самостоятельности и за невозможностью отстаивать её против турок вела кампанию против греков, в особенности по вопросу о самостоятельности болгарской церкви. Увлечение этой борьбой в связи с отсутствием веры в Россию, разбитую в Крымской кампании, скоро привело Цанкова к пропаганде католицизма; «только слепые, — писал он в 1859 году, — не могут отличить истинно христианского дела римской пропаганды от истинно дьявольского дела панславизма и панэллинизма».

В 1860 году ездил в Рим, где принял католицизм; в декабре был одним из инициаторов унии константинопольских болгар с Римом. Спустя несколько лет он раскаялся в этом и перешёл обратно в православие.

В 1863 году занял в Свищове место драгомана и директора паспортов.

В 1864 году был приглашён Мидхадом-пашой, управлявшим тогда Дунайским вилайетом, в Рущук, где занимал различные должности на турецкой государственной службе (инспектора турецкого пароходства, члена суда и др.); в то же время основал и вёл болгарскую типографию в Рущуке. Позже переменил ряд должностей в различных городах; между прочим, был цензором болгарских книг в Константинополе. Во всех этих должностях он, оставаясь в глубине души болгарином, умел ладить с турецкими властями и быть исправным турецким чиновником. За это время он написал: «Кратка Българска История» (Царьград, 1868) и «La Bulgarie» (вместе с М. Балабановым, Лондон, 1876), перевёл на болгарский язык «Тьмниците ми от Силвио Пелико» и редактировал в разное время, каждый раз неподолгу, журналы: «Читалище», 1870, «Источно время», 1874, «Руководитель на основното учение», 1874 (педагогический журнал).

От революционной деятельности Цанков стоял в стороне, но тем не менее пользовался значительным уважением даже среди радикальных элементов. Во время русской оккупации он занял место тырновского вице-губернатора; в Тырнове был избран в первое (учредительное) народное собрание (1879), где занял место в рядах либералов. После вступления на престол Александра Баттенберга Цанков был недолго болгарским дипломатическим агентом в Константинополе.

Глава Правительства

В марте 1880 г. ему было поручено сформировать кабинет; кроме председательства он взял иностранные дела, Каравелову поручил финансы. Его кабинет не вызвал доверия России, вооружил против себя духовенство и пал в ноябре 1880 г. Новый кабинет составил Каравелов; Цанков получил в нём внутренние дела, но в декабре должен был выйти в отставку вследствие столкновения с князем. Цанков считался тогда либералом (Каравелов — радикалом) и не был ещё русофилом; к этому времени относится его сказанная в письме, но затем опубликованная фраза, что если русские будут продолжать вести себя по отношению к Болгарии так, как ведут себя теперь, то болгары заявят: «мы не хотим ни русского меда, ни русского жала».

После государственного переворота 1881 года Цанков был арестован и затем интернирован во Враце. После восстановления конституции в 1883 г. он вторично составил кабинет, в котором взял себе внутренние дела, а консерватору Стоилову поручил министерство юстиции. Этот кабинет вёл борьбу с радикальной оппозицией (Каравелов, Стефана Стамбулов).

В начале 1884 г. Цанков распустил народное собрание. Выборы происходили при небывалой до тех пор и после того свободе и законности; никаких насилий и фальсификаций не было. Выборы дали большинство Каравелову, и Цанков тотчас уступил ему место. Каравелов повёл политику, враждебную России (присоединение Восточной Румелии, война с Сербией, начало раздоров с которой восходит ко времени кабинета Цанков, захватившего кусок сербской территории вследствие изменения течения р. Тимока), а Цанков стал определённым и ярким русофилом. Личная роль его в государственном перевороте 9 августа 1886 года, свергшем князя Александра, не совсем ясна.

Занял место в рядах созданного им временного правительства (9—12 августа ст. ст. 1886 года), во главе которого стоял митрополит Тырновский Климент, и подписал прокламацию, оправдывавшую переворот той изменой, которую допустил князь по отношению к России.

Эмиграция
После торжества Стамбулова Цанков должен был покинуть Болгарию и поселиться в Санкт-Петербурге, где жил на пенсию от русского правительства. В это время он считался вождём партии «цанковистов», бывшей партией непримиримой оппозиции; в действительности, однако, престарелый и больной Цанков был неспособен к деятельности и был только ширмой, именем; за него действовали Бендерев, Груев и всего более его зять Людсканов.

От его имени время от времени выпускались манифесты, в которых говорилось о необходимости безусловно следовать указаниям России, великой освободительницы и покровительницы Болгарии, и о невозможности примириться с князем Фердинандом, как незаконным правителем, даже в том случае, если бы он дал отставку Стамбулову. После падения последнего Цанков вернулся в Софию на основании амнистии, был принят князем в аудиенции и немедленно примирился с ним. Он был избран в народное собрание, но не мог играть сколько-нибудь заметной роли, так что даже цанковистский кабинет Данева 1902 г. был составлен без него.

Цанков — человек, не имеющий серьёзного образования, плохо владеющий иностранными языками, даже русским, несмотря на многие годы, проведённые в России, но искусный дипломат, воспитавшийся в турецкой школе, умеющий превосходно лавировать между партиями и личностями и извлекать из них пользу.

В 1902 г. выработал законопроект о народном просвещении, направленный против начальных учителей, чем вызвал со стороны последних сильную неприязнь.

Кроме общих сочинений по истории Болгарии, из коих в цанковистском духе написаны: Drandar, «Cinq ans de règne» (П., 1884) и его же, «Les événements politiques en Bulgarie» (П., 1896), см. Ю. Иванов, «Българский периодически печатъ» (т. 1, София, 1893).

Примечания

Литература

Политики Болгарии
Премьер-министры Болгарии
Министры иностранных дел Болгарии
Министры внутренних дел Болгарии
Академики Болгарской академии наук

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