text:
\section{Introduction}

Determining the response of a quantum mechanical system to a time-dependent perturbation is a frequent endeavor of both experiment and theory in a large number of subfields of physics. A number of phenomena have been found to occur in quantum optics, in electronic spin and nuclear magnetic resonance, and in ultracold atoms and molecules to name just a few. Many reviews of driven quantum systems can be found including \cite{2015PhyS...90h8013C}. From the theory side, several approaches to the calculation of the transition probability between the states of the system can be found in textbook literature with the various techniques having strengths and weaknesses depending upon the form of the perturbation. Even when we restrict our attention to harmonic perturbations, one may compute the transition probability (or transition rate) between states using Floquet theory \cite{1965PhRv..138..979S}, the Rotating Wave Approximation (RWA) \cite{1940PhRv...57..522B}, Fermi's Golden Rule and others. A comparison between these techniques for a two-level system can be found in Dion \& Hirschfelder \cite{1976acp....35..265D}.

Perhaps one of the more unusual cases of a driven quantum mechanical system is for the flavor evolution in space/time of a neutrino as it propagates through matter with periodic fluctuations. Due to the difference in neutrino masses, a $N_f$ flavor neutrino has $N_f$ distinct eigenstates. Both Floquet theory and the Rotating Wave Approximation approaches have been used to calculate the effect of density fluctuations described by a single Fourier mode (FM) upon a two flavor neutrino \cite{Ermilova,Akhmedov,1989PhLB..226..341K,1999NuPhB.538...25A,2001PAN....64..787A,Kneller:2012id}. More recently Patton, Kneller and McLaughlin \cite{2014PhRvD..89g3022P} considered the case of two flavor neutrino evolution through \emph{aperiodic} matter fluctuations as one would find in a turbulent medium. Patton, Kneller and McLaughlin based their theoretical description of the evolution upon the Rotating Wave Approximation (RWA) and found it gave predictions which were in remarkably good agreement with numerical calculations on a case -by-case basis. They named their model Stimulated Transitions.
The evolution of neutrinos through random matter density fluctuations has been previously considered by many others \cite{1987PhLB..185..417S,1990PhRvD..42.3908S,Loreti:1995ae,Fogli:2006xy,Friedland:2006ta,2010PhRvD..82l3004K,Reid,2013PhRvD..88b3008L,2003PhRvD..68c3005F,Choubey,1996NuPhB.472..495N,1998PhRvD..57.3140H,2009PhLB..675...69K,1996PhRvD..54.3941B,Burgess} using statistical approaches. Note an alternative analysis of the similar problem of a two-level atom interacting with a stochastic electromagnetic field is found in Cummings \cite{1982NCimB..70..102C}.

But the two level quantum system / two flavor neutrino is a restricted case. It is well known that one finds richer phenomenology when the system possesses three or more eigenstates. Perhaps the best known examples are in the field of quantum optics where one has the phenomenon of electromagnetic induced transparency \cite{1990PhRvL..64.1107H,1991PhRvL..66.2593B,2003RvMP...75..457L,2005RvMP...77..633F} and coherent population trapping into dark states \cite{2012Sci...338.1609D} which has also been seen in quantum dots and solid-state systems  \cite{2008NatPh...4..692X,2011Natur.478..497T}. Since neutrinos have (at least) three flavors, not two, one wonders if some of the phenomenology of the three eigenstate systems can be found for a three flavor neutrino.

The goal of this paper is to present the generalization of the Stimulated Transition model developed by Patton, Kneller and McLaughlin to the case of multi-flavor neutrinos and subject to an arbitrary, Fourier-decomposed perturbing Hamiltonian. In order to facilitate applications to other quantum systems, the theory is described in section \S\ref{sec:theory} in the most general terms possible without reference to any particular quantum system. We then apply the theory in section \S\ref{induced} to three-flavor neutrinos passing through density fluctuations composed of two and three anharmonic FMs. We find the neutrino equivalent of electromagnetic induced transparency for two FMs, and then a new effect we call Restored Opacity when we consider three FMs. Our conclusions are presented in section \S\ref{conclusions}.

\section{Stimulated Transformation}
\label{sec:theory}

We begin with the general problem of the evolution in time of a arbitrary $N$ level quantum system due to a time dependent perturbation. At some initial time $t_1$ we prepare the system in some arbitrary state - represented by a column vector - which we decompose in terms of the $N$ eigenstates of some basis $(X)$. The system then evolves to a time $t_2$ and at which we decompose the state in terms of the $N$ eigenstates of a possibly different basis $(Y)$. The evolution is described by a matrix $S^{(YX)}(t_2,t_1)$ and the transition probabilities are the set of probabilities that the system in a given initial state $x$ of $(X)$ at $t_1$ is detected in the state $y$ of $(Y)$ at $t_2$. These transition probabilities are denoted by $P^{(YX)}_{yx}(t_2,t_1)$ and are related to the elements of $S^{(YX)}$ by $P^{(YX)}_{yx} = |S^{(YX)}_{yx}|^2$. Since $S^{(YX)}$ must be unitary, one needs $N^2$ independent real parameters in order to describe the matrix $S^{(YX)}$ but note only $(N-1)^2$ of the elements of $P^{(YX)}$ are independent. Hereafter we shall work with the case where the bases $(X)$ and $(Y)$ are the same although there are certainly circumstances where knowing the evolution from one basis to a different basis is useful. Note also that throughout this paper we set $\hbar=c=1$.

In the generic basis $(X)$ the evolution matrix can be found by solving the Schr\"{o}dinger equation
\begin{equation}
\imath {\frac{dS}{dt}}^{(XX)} = H^{(X)}\,S^{(XX)} \label{eq:dSdt}
\end{equation}
where $H^{(X)}$ is the Hamiltonian in the basis $(X)$. The initial condition is $S^{(XX)}(t_1,t_1) = 1$. We make no assumption about the structure of $H^{(X)}$ except that it be possible to separate the Hamiltonian into an unperturbed piece $\breve{H}^{(X)}(t)$ and a position dependent perturbation $\delta H^{(X)}(t)$ i.e.\ $H^{(X)}(t) = \breve{H}^{(X)}(t) + \delta H^{(X)}(t)$.

If $\breve{H}^{(X)}(t)$ is not diagonal then we introduce an instantaneous unperturbed eigenbasis $(u)$ by finding the unitary matrix $\breve{U}(t)$ defined by $\breve{H}^{(X)} = \breve{U}\,\breve{K}\breve{U}^{\dagger}$ where $\breve{K}$ is the diagonal matrix of the eigenvalues of $\breve{H}$, that is $\breve{K} = \rm{diag}( \breve{k}_1,\breve{k}_2,\ldots)$. The evolution matrix in the instantaneous unperturbed eigenbasis is related to the evolution $S^{(XX)}$ by $S^{(uu)}(t_2,t_1) = \breve{U}^{\dagger}(t_2) S^{(XX)}(t_2,t_1)\breve{U}(t_1)$
In this unperturbed eigenbasis
\begin{equation}
H^{(u)} = \breve{K} - \imath \breve{U}^{\dagger}\,\frac{d\breve{U}}{dt} + \breve{U}^{\dagger}\delta H^{(X)}\breve{U}
\end{equation}
We now write the evolution matrix in the unperturbed eigenbasis as the product $S^{(uu)} = \breve{S}\,A$ where $\breve{S}$ is defined to be the solution of
\begin{equation}
\imath \frac{d\breve{S}}{dt} = \left[ \breve{K} - \imath \breve{U}^{\dagger}\,\frac{d\breve{U}}{dt} \right] \,\breve{S}.
\end{equation}
If we know the solution to the unperturbed problem, $\breve{S}$, we can solve for the effect of the perturbation by finding the solution to the differential equation for $A$:
\begin{equation}
\imath \frac{dA}{dt} = \breve{S}^{\dagger}\,\breve{U}^{\dagger}\delta H^{(X)}\breve{U}\,\breve{S} \,A. \label{dAdt}
\end{equation}
In general the term $\breve{U}^{\dagger}\delta H^{(X)}\breve{U}$ which appears in this equation possesses both diagonal and off-diagonal elements. The diagonal elements are easily removed by writing the matrix $A$ as $A=W\,B$ where $W=\exp(-\imath\Xi)$ and $\Xi$ a diagonal matrix $\Xi=\rm{diag}(\xi_{1},\xi_{2},\ldots)$. Substitution into (\ref{dAdt}) gives a differential equation for $B$
\begin{equation}\label{dBdt}
\imath \frac{dB}{dt} = W^{\dagger}\left[\breve{S}^{\dagger}\breve{U}^{\dagger}\delta H^{(X)}\breve{U}\,\breve{S} -\frac{d\,\Xi}{dt}\right]\,W\,B \equiv H^{(B)} B
\end{equation}
and $\Xi$ is chosen so that $d\,\Xi/dt$ removes the diagonal elements of $\breve{S}^{\dagger}\breve{U}^{\dagger}\delta H^{(X)}\breve{U}\,\breve{S}$. Once $\Xi$ has been found, determining transition probabilities is reduced to solving for the $B$ matrix.

\subsection{Fourier-decomposed Perturbations}

We now consider the specific case of a constant potential for $\breve{H}^{(X)}$. This form for $\breve{H}$ means $\breve{S}$ is a diagonal matrix $\breve{S}=\exp(-\imath \breve{K}\,t)$. The perturbation $\delta H$ is taken to be a Fourier-like series of the form
\begin{equation}\label{HM}
 \delta H^{(X)} = \sum_a (C_{a} e^{\imath q_{a} t} + C_{a}^{\dagger} e^{-\imath q_{a} t})
\end{equation}
where $C_a$ is an arbitrary complex matrix and $q_a$ the frequency of the $a^{th}$ FM of the perturbation. We make no restriction on the number of FMs, the frequencies $q_a$ nor the size or structure of the matrices $C_a$. This generalization to arbitrary structure for the $C_a$'s is where we depart from previous analyses by Patton, Kneller \& McLaughlin \cite{2014PhRvD..89g3022P}. We also refer the reader to Brown, Meath \& Tran \cite{Brown:Meath:Tran} and Avetissian, Avchyan \& Mkrtchian \cite{2012JPhB...45b5402A} who considered the related but simpler problem of the effect of two lasers of different colors, i.e.\ two FMs, upon a two-level dipolar molecule.

Given this form for the perturbation, equation (\ref{dBdt}) indicates we need to consider the combination $\breve{U}^{\dagger}C_{a}\breve{U}$. If we write the diagonal elements of $\breve{U}^{\dagger}C_{a}\breve{U}$ as
\begin{equation}
 {\rm diag}( \breve{U}^{\dagger}C_a\breve{U} ) = \frac{F_a}{2\imath}  \exp(\imath\,\Phi_a)
 \label{eq7}
\end{equation}
where $F_a$ is a diagonal matrix of amplitudes $F_a={\rm diag}(f_{a;1},f_{a;2},\ldots)$ and $\Phi_a$ the diagonal matrix of phases $\Phi_a={\rm diag}(\phi_{a;1},\phi_{a;2},\ldots)$, then the matrix $\Xi$ is found to be $\Xi(t) = \sum_a \Xi_a(t)$
with
\begin{equation}
\Xi_a(t) = \frac{F_a}{q_a} \big[ \cos\Phi_a - \cos(\Phi_a + q_a\,t) \big].
\end{equation}
We denote the diagonal elements of $\Xi_a$ as $\xi_{a;1}, \xi_{a;2},\ldots$. Next we rewrite the off-diagonal elements of $\breve{U}^{\dagger}C_a\breve{U}$ as a matrix $G_a$ i.e\ $G_a ={\rm offdiag}(\breve{U}^{\dagger}C_a\breve{U})$. Putting together the solution for $\Xi$ and $\breve{S}$ and inserting the new matrix $G_a$, we find the Hamiltonian for $B$ is
\begin{equation}
\begin{split}
 H^{(B)} =&\exp(\imath\,\Xi)\exp(\imath\,\breve{K}\,t)\,\Big(\sum_a \left[ G_a e^{\imath q_a\,t} + G^{\dagger}_a e^{-\imath q_a\,t}  \right] \Big)\, \\
& \times \exp(-\imath\,\breve{K}\,t) \exp(-\imath\,\Xi)
\end{split}
\end{equation}
Written explicitly the element $ij$ of the Hamiltonian is
\begin{equation}
 H_{ij}^{(B)} = e^{\imath (\delta\breve{k}_{ij}t+\delta\xi_{ij})} \sum_a\Big[ G_{a;ij}e^{\imath q_a\,t} + G^{\star}_{a;ji}e^{-\imath q_a\,t}\Big]
\end{equation}
where $\delta\breve{k}_{ij} = \breve{k}_{i}-\breve{k}_{j}$ and $\delta\xi_{ij} = \xi_{i}-\xi_{j}$.
\\
\\
If we define
\begin{eqnarray}
x_{a;ij} & = & \frac{f_{a;i}}{q_a}\cos\phi_{a;i} - \frac{f_{a;j}}{q_a}\cos\phi_{a;j} \label{eq11}\\
y_{a;ij} & = & \frac{f_{a;i}}{q_a}\sin\phi_{a;i} - \frac{f_{a;j}}{q_a}\sin\phi_{a;j}  \label{eq12}\\
z_{a;ij} & = & \sqrt{x_{a;ij}^{2}+y_{a;ij}^{2}}   \label{eq13}\\
\psi_{a;ij} & = & \arctan\left( {\frac{{{y_{a;ij}}}}{{{x_{a;ij}}}}} \right) \label{eq14}
\end{eqnarray}

then the term $\delta\xi_{ij}$ is equal to
\begin{equation}
 \delta\xi_{ij} = \sum_a\Big[ x_{a;ij} - z_{a;ij}\cos(q_a\,t+\psi_{a;ij} )\Big].
\end{equation}
The presence of $y_{a;ij}$ in these equations is a new feature of the more general perturbing Hamiltonian we are considering. We now make use of the Jacobi-Anger expansion for $e^{\imath\delta\xi_{ij}}$
\begin{widetext}
\begin{equation}
e^{\imath\delta\xi_{ij}} = \prod_a \left\{ e^{\imath x_{a;ij}} \sum_{m_a=-\infty}^{\infty} (-\imath)^{m_a} J_{m_a}(z_{a;ij})\exp\Big[\imath\,m_a\left(q_a\,t + \psi_{a;ij}\right)\Big] \right\}.
\end{equation}
If we substitute this expansion into the expression for the elements of $H^{(B)}$ and define $\mu_{a,m_a;ij}$ and $\lambda_{a,m_a;ij}$ to be
\begin{eqnarray}
\lambda_{a,m_a;ij} & = & (-\imath)^{m_a}\,e^{\imath x_{a;ij}}\,J_{m_a}(z_{a;ij})\,e^{\imath m_a\psi_{a;ij}}
\label{eq17}\\
\mu_{a,m_a;ij} & = & (-\imath)^{m_a}\,e^{\imath x_{a;ij}}\,\left[ G^{\star}_{a;ji}\,J_{m_a+1}(z_{a;ij})\,e^{\imath(m_a+1)\psi_{a;ij}}-G_{a;ij}\,J_{m_a-1}(z_{a;ij})\,e^{\imath(m_a-1)\psi_{a;ij}}\right]
\end{eqnarray}
then we find the element $ij$ of the Hamiltonian is given by
\begin{equation}\label{HB_ij}
 H_{ij}^{(B)} = \imath \sum_a\left\{ \sum_{m_a}\mu_{a,m_a;ij}\,e^{\imath ( m_a q_a + \delta\breve{k}_{ij}) t}\prod_{b\neq a}\left[ \sum_{m_b} \lambda_{b,m_b;ij}\,e^{\imath m_b q_b t} \right]. \right\}
\end{equation}
\end{widetext}

\subsection{Rotating Wave Approximation}

Even though we started with a very general perturbing Hamiltonian, we have found a form for $H^{(B)}$ which has the same structure as that found by Patton, Kneller \& McLaughlin. From here on, we follow the same procedure to solve for the matrix $B$. First we adopt the Rotating Wave Approximation. The RWA amounts to selecting a particular value for the integers $m_a$ and $m_b$ in Eq.(\ref{HB_ij}) and dropping all others.
We do not specify a procedure for selecting those integers though algorithms exist. We expect there is not one procedure that can be adopted universally for all situations. There are some restrictions to be placed on the selection of the integers. In order that the resulting Hamiltonian be solvable we cannot make choices for $m_a$ and $m_b$ for every element $ij$ independently. Only $N-1$ elements are to be regarded as independent and a suitable set could be either those on the sub/superdiagonal or the off-diagonal elements in a particular row or column. The values of  $m_a$ and $m_b$ we select will be different for each independent element. We denote these integers by $n_{a;ij}$ since they are specific both to the frequency $a$ and the element of the Hamiltonian $ij$, and define
\begin{equation}
 \kappa_{ij} = \sum_a\mu_{a,n_{a;ij};ij} \prod_{b\neq a} \lambda_{b,n_{b;ij};ij}
\end{equation}
then $H_{ij}^{(B)}$ is simplified to
\begin{equation}
H_{ij}^{(B)} = -\imath \kappa_{ij} \exp\left[\imath \left( \sum_a n_{a;ij}\, q_a + \delta\breve{k}_{ij}\right) t\right],
\end{equation}
or, in full matrix, form
\begin{widetext}
\begin{equation}
H^{(B)} = \left( \begin{array}{cccc}
0 & - \imath \kappa_{12}\,e^{\imath \left[ \delta\breve{k}_{12} + \sum\limits_a n_{a;12}\,q_a \right]t} & - \imath \kappa_{13}\,e^{\imath \left[ \delta\breve{k}_{13} + \sum\limits_a n_{a;13}\,q_a \right]t} & \ldots \\
\imath \kappa^{\star}_{12}\,e^{-\imath \left[ \delta\breve{k}_{12} + \sum\limits_a n_{a;12}\,q_a \right]t} & 0 & \imath \kappa_{12}\,e^{-\imath \left[ \delta\breve{k}_{23} + \sum\limits_a n_{a;23}\,q_a \right]t} & \ldots \\
\imath \kappa^{\star}_{23}\,e^{-\imath \left[ \delta\breve{k}_{13} + \sum\limits_a n_{a;13}\,q_a \right]t} & \imath \kappa^{\star}_{23}\,e^{-\imath \left[ \delta\breve{k}_{23} + \sum\limits_a n_{a;23}\,q_a \right]t} & \ldots \\
 \vdots & \vdots & \vdots & \ddots
\end{array} \right)
\label{eq:hb}
\end{equation}
\end{widetext}
Again, we remind the reader that, for example, only $n_{a;12}$, $n_{a;23}$, $n_{a;34}$ etc.\ are independent: in all other cases the integer $n_{a;ij} = n_{a;i\ell} + n_{a;\ell j}$. As shown by Patton, Kneller \& McLaughlin, with this simplified Hamiltonian, Eq.(\ref{dBdt}), can be solved for the evolution matrix $B$ and we reproduce their solution here for completeness. Since both $n_{a;ij} = n_{a;i\ell} + n_{a;\ell j}$ and $\delta\breve{k}_{ij} = \delta\breve{k}_{i\ell} + \delta\breve{k}_{\ell j}$, we can factorize $H^{(B)}(t)$ into the form $H^{(B)}(t) = \Upsilon(t)\,M\,\Upsilon^{\dagger}(t)$ where the matrix $M$ is a constant, i.e. it contains the couplings $\kappa_{ij}$ only. The matrix $\Upsilon$ is of the form $\Upsilon(t)=\exp(\imath\,\Lambda\,t)$, where $\Lambda$ is also a constant matrix that depends only on $\delta\breve{k}_{ij}$, the integer sets $\{n_{a;ij}\}$ and
the frequencies $q_a$. Explicitly we can write
\begin{equation}\label{eq for M}
M = \left( \begin{array}{cccc}
    0 & -\imath \kappa_{12} & -\imath \kappa_{13} & \ldots \\
    \imath \kappa^{\star}_{12} & 0 & -\imath \kappa_{23} & \ldots \\
    \imath \kappa^{\star}_{13} & \imath \kappa^{\star}_{23} & 0 & \ldots \\
    \vdots & \vdots & \vdots & \ddots
    \end{array}\right),
\end{equation}
and one possible choice for the matrix $\Lambda$ is
\begin{widetext}
\begin{equation}
\Lambda  = \left( {\begin{array}{cccc}
\breve{k}_{1} + \sum\limits_a n_{a;1}\,q_a & 0 & 0 & \ldots \\
0 & \breve{k}_{2} + \sum\limits_a n_{a;2}\,q_a & 0 & \ldots \\
0 & 0 & \breve{k}_{3} + \sum\limits_a n_{a;3}\,q_a & \ldots \\
 \vdots & \vdots & \vdots & \ddots
\end{array}} \right),
\end{equation}
\end{widetext}
where $n_{a;i}$ are integers chosen so that $n_{a;i} - n_{a;j} = n_{a;ij}$. Using this factorization of $H^{(B)}(t)$ we find equation (\ref{dBdt}) can be rewritten as
\begin{equation}
\imath\Upsilon^{\dagger}\frac{dB}{dt} =  M\Upsilon^{\dagger}\,B
\end{equation}
Instead of solving for $B$ we solve for the combination $\Omega=\Upsilon^{\dagger} B$. The differential equation for $\Omega$ is found to be
\begin{equation}
\imath\frac{d\Omega}{dt} = \left(M +\Lambda \right)\,\Omega = H^{(\Omega)}\,\Omega.
\end{equation}
Since the matrix both $M$ and $\Lambda$ are constant matrices, the matrix $H^{(\Omega)}$ is also independent of $t$ meaning $\Omega$ has the formal solution $\Omega(t) = \exp(-\imath H^{(\Omega)} t)\,\Omega(0)$. The solution for $B$ is thus
\begin{equation}
B(t) = \Upsilon(t)\,\exp(-\imath H^{(\Omega)} t)\,\Upsilon^{\dagger}(0) B(0). \label{eq:soln for B}
\end{equation}
Now that we have the solution for $B$, the full evolution matrix in the basis $(u)$ is $S=\breve{S}\,W\,B$ but given that both $\breve{S}$ and $W$ are diagonal matrices, the transition probability between the unperturbed eigenstates is simply the square magnitude of the off-diagonal element of $B$. Note that we have made no restrictions on the number of FMs, the frequencies nor the size and structure of the amplitude matrices so this procedure for obtaining the RWA solution applies to both periodic \emph{and} aperiodic Hamiltonians.

For the particular case of two eigenstates we can write out the solution succinctly (after dropping the $12$ subscripts on $n_{a;12}$ and $\kappa_{12}$) by introducing the detuning frequency $p$ via $2p = \delta\breve{k}_{12} + \sum\limits_a {n_a q_a}$ and the Rabi flopping frequency $Q$ by $Q^{2} = p^{2}+ \kappa^{2}$. Using these quantities, $B(t)$ for two flavors is found to be
\begin{widetext}
\begin{equation}
B = \left( \begin{array}{cc}
 e^{\imath p t} \left[ \cos\left(Q\,t\right) -\imath\frac{p}{Q}\sin\left(Q\,t\right) \right] & -e^{\imath p t}\,\frac{\kappa}{Q}\sin\left(Q\,t\right) \\
e^{-\imath p t}\frac{\kappa^{\star}}{Q}\sin\left(Q\,t\right) & e^{-\imath p t} \left[ \cos\left(Q\,t\right) +\imath\frac{p}{Q}\sin\left(Q\,t\right) \right].
 \end{array} \right)
\end{equation}
\end{widetext}
and so we see the transition probability between the matter states 1 and 2 is
\begin{equation}
P_{12} = |B_{12}|^{2}= \frac{\kappa^2}{Q^2}\,\sin^{2}\left(Q\,t\right).
\label{transition}
\end{equation}
This is the result found by Patton, Kneller and McLaughlin \cite{2014PhRvD..89g3022P}. A transition probability that varies as $\sin^2 (Q\,t)$ is a generic prediction of \emph{all} RWA solutions of a two-level quantum system when the perturbing Hamiltonian can be expressed as a Fourier-like series even if the ratios between the frequencies of different modes are not rational numbers, which makes the series aperiodic.

\subsection{Degenerate RWA}
If the FMs are such that the ratio between any pair of frequencies, $q_a$ and $q_b$, is a rational fraction then there are multiple sets of the integers $\{n\}$ which \emph{all} have the same detuning frequency. One can account for these degenerate rotating waves by simply adding them together so that $\kappa_{ij}$ becomes

\begin{equation}
\kappa_{ij} = \sum_{\{n\}} \left[ \sum_a\mu_{ij,n_{a;ij}} \prod_{b\neq a} \lambda_{ij,n_{b;ij}} \right].
\end{equation}
If all ratios between the frequencies are rational then accounting for the degeneracy in the RWA becomes an exercise in combinatorics. Even so, typically one finds the sum is dominated by one set - the one where the sum of the absolute values of the integers $\{n\}$ is smallest.

\section{Application to Neutrino Propagation}
\label{induced}

With the theory complete we now move on to testing whether it gives predictions which agree with numerical solutions. The problem we choose is the case of a three-flavor neutrino with an energy of $5\;{\rm MeV}$ propagating through fluctuating matter. Note, as commonly found in the literature on neutrino flavor transformation, we switch the variable from time $t$ to position along the neutrino trajectory $r$ where $r=c\,t$ since the neutrino wavepacket is localized in space and typically the energy of neutrinos is much larger than their rest mass hence they move at a speed close to $c$. As a consequence we switch from using `frequencies' to `wavenumbers'.

The Hamiltonian, $H$, governing the neutrino flavor evolution through matter is the sum of a constant vacuum term $H_V$ and a term
coming from the effect of matter $H_{M}$ \cite{Wolfenstein1977,M&S1986}. The vacuum Hamiltonian in the `flavor basis' is
\begin{equation}
H^{(f)}_V = \frac{1}{2E} U_V \left( \begin{array}{*{20}{c}} m_1^2-m_2^2 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & m_3^2-m_2^2 \end{array} \right) U_V^{\dagger}
\end{equation}
\\
where $U_V$ is the vacuum mixing matrix and $m_i$ the three neutrino masses. We set the squared mass differences $m_1^2-m_2^2 = -7.5 \times 10^{-5}\;{\rm eV^2}$ and $m_3^2-m_2^2 =2.32\times 10^{-3}\;{\rm eV^2}$ which are compatible with the mass-squared differences as given by the Particle Data Group \cite{PDG}. $U_V$ is parameterized by three mixing angles $\theta_{12}$, $\theta_{13}$ and $\theta_{23}$ - we set all possible phases to zero \cite{2009PhRvD..80e3002K} - and given by
\begin{widetext}
\begin{eqnarray}
U & = &
\left(\begin{array}{lll} c_{12}c_{13} & s_{12}c_{13} & s_{13}
\\
-s_{12}c_{23}-c_{12}s_{13}s_{23}
& c_{12}c_{23}-s_{12}s_{13}s_{23}
& c_{13}s_{23} \\
s_{12}s_{23}-c_{12}s_{13}c_{23} &
-c_{12}s_{23}-s_{12}s_{13}c_{23}
& c_{13}c_{23} \end{array}\right)
\label{eq:U}
\end{eqnarray}
\end{widetext}
where the notation is that $c_{ij} = \cos\theta_{ij}$ and $s_{ij} = \sin\theta_{ij}$. We take the angles to be $\theta_{12} =34^{\circ}$, $\theta_{13} =9^{\circ}$ and $\theta_{23} =45^{\circ}$ \cite{PDG}.

\subsection{Two Fourier Modes}

We first consider the case where the matter Hamiltonian is taken to be a constant upon which are superposed two FMs with wavenumbers $q_1$ and $q_2$ which are not in a rational ratio. The matter is regarded as affecting only the electron flavor type, not the other two flavors. The form of the Hamiltonian in the flavor basis with the first row/column indicating the electron flavor is thus
\begin{equation}
H^{(f)}_M(r) = V_{\star} \left[1 + A_1 \cos(q_1 r) + A_2 \cos(q_2 r)\right] \left( \begin{array}{lll} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array} \right)
\end{equation}
with $V_{\star}$ the potential from the constant background, $A_1$ and $A_2$ the amplitudes of the fluctuations. In what follows we set $V_{\star}$ to $V_{\star}=6\times 10^{-25}\;{\rm erg}$.

The vacuum Hamiltonian and the constant potential $V_{\star}$ form the `unperturbed' Hamiltonian $\breve{H}$. In the flavor basis $\breve{H}$ is not diagonal. We can diagonalize $\breve{H}$ by first finding its eigenvalues, denoted by $\breve{K} ={\rm diag}\left( \breve{k}_1, \breve{k}_2, \breve{k}_3\right)$, and then the unitary matrix $\breve{U}$ which satisfies $\breve{H}^{f} = \breve{U}\,\breve{K}\,{\breve{U}}^{\dagger}$. Since this is standard textbook quantum mechanics, we leave this as an exercise for the reader. For reference, the differences between the eigenvalues are found to be $\breve{k}_3 - \breve{k}_1 = 3.835\times 10^{-22}\;{\rm erg}$ and $\breve{k}_3 - \breve{k}_2 = 3.715\times 10^{-22}\;{\rm erg}$. Note that since $\breve{H}$ is a function of $V_{\star}$, the eigenvalues and unperturbed mixing matrix, $\breve{U}$ are also functions of $V_{\star}$. The level scheme we end up with is shown in figure (\ref{fig:levels}) with the three eigenstates of the unperturbed system denoted as
$\ket{k_1}$, $\ket{k_2}$ and $\ket{k_3}$.

The two FMs in the matter Hamiltonian are the Fourier-decomposed perturbation. The $C$ matrices for the two FMs are
\begin{equation}
 C_1 = \frac{1}{2} V_{\star} A_1 \left( \begin{array}{lll} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array} \right) \label{eq:C1}
\end{equation}
and
\begin{equation}
 C_2 = \frac{1}{2} V_{\star} A_2 \left( \begin{array}{lll} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array} \right) \label{eq:C2}
\end{equation}
giving diagonal elements of the $F_a$ matrices of the form
\begin{equation}
f_{a;i} = \frac{1}{2}|\breve{U}_{e i}|^{2} A_a V_{\star} \label{eq:f_a;i}
\end{equation}
while all the elements in the $\Phi_a$ matrices are $\pi/2$. This has the consequence that for both FMs
\begin{equation}
y_{a;ij} = z_{a;ij} =\frac{ A_a V_{\star} \left( |U_{ei}|^2 - |U_{ej}|^2 \right) }{2\,q_a} \label{y_a;ij}
\end{equation}
and therefore all of the phases $\psi_{a;ij}$ are also equal to $\pi/2$. The elements of the $G_a$ matrices are
\begin{equation}
G_{a;ij} = \frac{1}{2}\,\breve{U}_{e i}^{\star}\breve{U}_{e j}\,A_a V_{\star} \label{G_a;ij}
\end{equation}
If we put all these pieces together we find the $\lambda$'s are simply
\begin{equation}
\lambda_{a,m_a;ij} = J_{m_a}\left(z_{a;ij} \right) \label{lambda_a,ma;ij}
\end{equation}
while the $\kappa$'s are of the form
\begin{equation}
 \kappa_{a,m_a;ij} = \imath\left[ {{G_{a;ji}}{J_{{m_a} + 1}}\left( z_{a;ij} \right) + {G_{a;ij}}{J_{{m_a} - 1}}\left( z_{a;ij} \right)} \right]. \label{kappa_a,ma;ij}
\end{equation}

\begin{figure}[t]
\includegraphics[clip,width=\linewidth]{fig1.eps}
\caption{The three unperturbed eigenstates and the transitions between them in the two and three FM problems. The modes $q_1$ and $q_2$ are the FMs that drive transitions between the indicates states. The mode $q_3$ is the ``switch mode'' which switches on and off the effect of transitions induced by modes $q_1$ and $q_2$.}
\label{fig:levels}
\end{figure}
\begin{figure}[b]
\includegraphics[clip,width=\linewidth]{fig2.eps}
\caption{The transition probabilities from unperturbed eigenstate 1 to unperturbed eigenstate 3. The blue dashed line is the RWA result for the case $A_1=0.1, A_2=0$ and the red dashed line for $A_1=0.1, A_2=0.5$. The symbols represent the corresponding numerical results.}
\label{fig:ITvsr}
\end{figure}
\begin{figure}[t]
\includegraphics[clip,angle=270,width=\linewidth]{fig3.eps}
\caption{The amplitude of $P_{13}$ as a function of $q_1$. The parameters used are $V_{\star}=6\times 10^{-25}\;{\rm erg}, A_1=0.1$, and $A_2=0.5/0.2/0.1/0$ for the black/blue/green/red dashed line and symbols, $A_2=0$ for the blue dashed line and symbols. The symbols represent numerical results, while the dashed lines are from RWA evaluation.}
\label{fig:ITvsq1}
\end{figure}
Let us now set the wavenumbers for the two modes so that $q_1 \approx \breve{k}_3 - \breve{k}_1$ and $q_2 \approx \breve{k}_3 - \breve{k}_2$ as shown in figure (\ref{fig:levels}). The RWA integers we select for the $1,3$ element are thus $\{ n_{1;13}, n_{2;13} \} = \{ +1 , 0 \}$ and for the $2,3$ element we pick $\{ n_{1;23}, n_{2;23} \} = \{ 0 , +1 \}$. The integer set for the $1,2$ element must therefore be $\{ n_{1;12}, n_{2;12} \} = \{ +1 , -1 \}$ in order that $n_{a;12} + n_{a;23} = n_{a;13}$. The Hamiltonian $H^{(B)}$ is thus
\begin{widetext}
\begin{equation}
H^{(B)} = \left( \begin{array}{ccc}
    0 & -\imath \kappa_{12}\, e^{\imath\left(\delta\breve{k}_{12}+q_1-q_2\right) r} & -\imath \kappa_{13}\, e^{\imath\left(\delta\breve{k}_{13} + q_1 \right) r} \\
    \imath \kappa^{\star}_{12}\, e^{-\imath\left(\delta\breve{k}_{12}+q_1-q_2\right) r} & 0 & -\imath \kappa_{23}\, e^{\imath\left(\delta\breve{k}_{23} + q_2\right) r} \\
    \imath \kappa^{\star}_{13}\, e^{-\imath\left(\delta\breve{k}_{13} + q_1\right) r} & \imath \kappa^{\star}_{23}\, e^{-\imath\left(\delta\breve{k}_{23} + q_2\right) r} & 0
    \end{array}\right)
\end{equation}
with
\begin{eqnarray}
\kappa_{12} & = & \mu_{1,n_{1,12};12}\,\lambda_{2,n_{2,12};12} + \lambda_{1,n_{1,12};12}\,\mu_{2,n_{2,12};12} \nonumber \\
     & = &  \imath \,\left[ G_{1;21}\,J_{2}\left(z_{1;12}\right) + G_{1;12}\,J_{0}\left(z_{1;12}\right) \right] \,J_{-1}\left(z_{2;12}\right)
        \;+\; \imath \,J_{1}\left(z_{1;12}\right) \,\left[ G_{2;21}\,J_{0}\left(z_{2;12}\right) + G_{2;12}\,J_{-2}\left(z_{2;12}\right) \right] \\
\kappa_{13} & = & \mu_{1,n_{1,13};13}\,\lambda_{2,n_{2,13};13} + \lambda_{1,n_{1,13};13}\,\mu_{2,n_{2,13};13} \nonumber \\
     & = &  \imath \,\left[ G_{1;31}\,J_{2}\left( z_{1;13} \right) + G_{1;13}\,J_{0}\left(z_{1;13}\right) \right] \,J_{0}\left(z_{2;13}\right)
        \;+\; \imath \,J_{1}\left(z_{1;13}\right) \,\left[ G_{2;31}\,J_{1}\left(z_{2;13}\right) + G_{2;13}\,J_{-1}\left(z_{2;13}\right) \right] \\
\kappa_{23} & = & \mu_{1,n_{1,23};23}\,\lambda_{2,n_{2,23};23} + \lambda_{1,n_{1,23};23}\,\mu_{2,n_{2,23};23} \nonumber \\
     & = &  \imath \,\left[ G_{1;32}\,J_{1}\left(z_{1;23}\right) + G_{1;23}\,J_{-1}\left(z_{1;23}\right) \right] \,J_{1}\left(z_{2;23}\right)
        \;+\; \imath \,J_{0}\left(z_{1;23}\right) \, \left[ G_{2;32}\,J_{2}\left(z_{2;23}\right) + G_{2;23}\,J_{0}\left(z_{2;23}\right) \right]
\end{eqnarray}
Since we chose the vacuum mixing matrix to be pure real, the unperturbed matter mixing matrix $\breve{U}$ is also pure real. This is a result of the Naumov \cite{1992IJMPD...1..379N} and Harrison \& Scott \cite{2002PhLB..535..229H} identities. In this case we see from equation (\ref{G_a;ij}) that $G_{a;ij} = G_{a;ji}$. We also recall the identities for Bessel functions that $J_{-n}(z)=(-1)^{n}J_{n}(z)$ and $J_{n-1}(z) + J_{n+1}(z) = 2\,n\, J_n(z) / z$ and when combined, these identities mean that the expressions for the $\kappa$'s are the much simpler
\begin{eqnarray}
\kappa_{12} & = & -\frac{2\,\imath \,G_{1;12}}{z_{1;12}} \,J_{1}\left(z_{1;12}\right) \,J_{1}\left(z_{2;12}\right)
             \;+\; \frac{2\,\imath \,G_{2;12}}{z_{2;12}} \,J_{1}\left(z_{1;12}\right) \,J_{1}\left(z_{2;12}\right) \label{eq:kappa12:2}\\
\kappa_{13} & = & \frac{2\,\imath \,G_{1;13}}{z_{1;13}} \,J_{1}\left(z_{1;13}\right) \,J_{0}\left(z_{2;13}\right) \label{eq:kappa13:2}\\
\kappa_{23} & = & \frac{2\,\imath \,G_{2;23}}{z_{2;23}} \,J_{0}\left(z_{1;23}\right) \,J_{1}\left(z_{2;23}\right) \label{eq:kappa23:2}
\end{eqnarray}
\end{widetext}
We notice that both terms in $\kappa_{12}$ are proportional to the product of two Bessel functions $J_1$ so once we recall that the Bessel function $J_{n}(z) \sim z^{|n|}$ for small $z$, we see that the element $\kappa_{12}$ is smaller in magnitude than $\kappa_{13}$ and $\kappa_{23}$ since the values of $z_{a:ij}$ are very smaller. That is confirmed when we compute the numerical values and find $\kappa_{12} = 6.419\times 10^{-32}\,\imath\;{\rm erg}$, $\kappa_{13} = -3.888\times 10^{-27}\,\imath\;{\rm erg}$ and $\kappa_{23} = -1.311\times 10^{-26}\,\imath\;{\rm erg}$.

We now proceed to solve for $B$. Following the steps given above, the matrix $M$ is
\begin{equation}
M = \left( \begin{array}{cccc}
    0 & -\imath \kappa_{12} & -\imath \kappa_{13} \\
    \imath \kappa^{\star}_{12} & 0 & -\imath \kappa_{23} \\
    \imath \kappa^{\star}_{13} & \imath \kappa^{\star}_{23} & 0
    \end{array}\right).
\end{equation}
Next we need the matrix $\Lambda$. In order to construct this matrix we need to find a set of six integers such that $n_{a;i} - n_{a;j} = n_{a;ij}$. The simplest solution is $n_{1;1} = 1$, $n_{1;2} = 0$, $n_{1;3} = 0 $ and $n_{2;1} = 0$, $n_{2;2} = 1$, $n_{2;3} = 0$. With this choice we have

\begin{equation}
\Lambda  = \left( {\begin{array}{ccc}
\breve{k}_{1} + q_{1} & 0 & 0 \\
0 & \breve{k}_{2} + q_{2} & 0 \\
0 & 0 & \breve{k}_{3}
\end{array}} \right).
\end{equation}
Thus $H^{(\Omega)} = M+\Lambda$ is
\begin{equation}
H^{(\Omega)} = \left( {\begin{array}{ccc}
\breve{k}_{1} + q_{1} & -\imath \kappa_{12} & -\imath \kappa_{13} \\
\imath \kappa^{\star}_{12} & \breve{k}_{2} + q_{2} &  -\imath \kappa_{23} \\
\imath \kappa^{\star}_{13} & \imath \kappa^{\star}_{23} & \breve{k}_{3}
\end{array}} \right).
\end{equation}
$B$ has the formal solution $B(r) = \Upsilon(r)\,\exp(-\imath H^{(\Omega)} r)\,\Upsilon^{\dagger}(0)$ where $\Upsilon(r)=\exp(\imath\,\Lambda\,r)$ which can be evaluated using standard procedures. If we make the approximation that $\kappa_{12}$ is negligibly small compared to $\kappa_{13}$ and $\kappa_{23}$ and that the two wavenumbers are exactly on resonance, $q_1 = \breve{k}_3 - \breve{k}_1$, $q_2 = \breve{k}_3 - \breve{k}_2$, then we find the analytical expression for the $B$ matrix is
\begin{widetext}
\begin{equation}
B = \exp\left(\imath\,\left[\Lambda-\breve{k}_3\mathbb{1}\right]\,r\right) \,\left( {\begin{array}{*{20}{c}}
{\frac{{{{\left| {{\kappa _{23}}} \right|}^2}}}{{{Q^2}}} + \frac{{{{\left| {{\kappa _{13}}} \right|}^2}}}{{{Q^2}}}\cos \left( {Qr} \right)}&{\frac{{\imath{\kappa _{13}}\kappa _{23}^*}}{{{Q^2}}}\left[ {\cos \left( {Qr} \right) - 1} \right]}&{ - \frac{{{\kappa _{13}}}}{Q}\sin \left( {Qr} \right)}\\
{\frac{{\imath{\kappa _{23}}\kappa _{13}^*}}{{{Q^2}}}\left[ {\cos \left( {Qr} \right) - 1} \right]}&{\frac{{{{\left| {{\kappa _{13}}} \right|}^2}}}{{{Q^2}}} + \frac{{{{\left| {{\kappa _{23}}} \right|}^2}}}{{{Q^2}}}\cos \left( {Qr} \right)}&{ - \frac{{{\kappa _{23}}}}{Q}\sin \left( {Qr} \right)}\\
{\frac{{\kappa _{13}^*}}{Q}\sin \left( {Qr} \right)}&{\frac{{\kappa _{23}^*}}{Q}\sin \left( {Qr} \right)}&{\cos \left( {Qr} \right)}
\end{array}} \right),
\end{equation}
\end{widetext}
where $Q^2 = \left|\kappa_{13}\right|^2 + \left|\kappa_{23}\right|^2$ and $\mathbb{1}$ is a 3x3 unit matrix.
From this result we can extract the transition probability from unperturbed eigenstate 1 to unperturbed eigenstate 3 by taking the squared magnitude of $B_{13}$
\begin{equation}\label{p13}
{P_{13}} = \frac{{{{\left| {{\kappa _{13}}} \right|}^2}}}{{{Q^2}}}{\sin ^2}\left( {Qr} \right) = \left( {1 - \frac{{{{\left| {{\kappa _{23}}} \right|}^2}}}{{{Q^2}}}} \right){\sin ^2}\left( {Qr} \right).
\end{equation}
This result is interesting because it indicates the transition probability $P_{13}$ depends upon the wavenumber $q_2$ which is driving transitions from unperturbed eigenstate 2 to unperturbed eigenstate 3. In the extreme case when $\kappa_{23}$ is significantly larger than $\kappa_{13}$, the transition from states 1 to 3 is strongly suppressed. This is an analog of the Electromagnetically Induced Transparency (EIT) - see, for example, \cite{1990PhRvL..64.1107H,1991PhRvL..66.2593B,2003RvMP...75..457L,2005RvMP...77..633F} - in atomic physics where the presence of a second possible transition between atomic levels 2 and level 3 will inhibit the primary transition from atomic level 1 to level 3 leading to little absorption, and thus transparency, for the light frequency corresponding to the energy splitting of level 1 and 3.

To illustrate this neutrino version of induced transparency, in Fig.(\ref{fig:ITvsr}) we plot the transition probability as a function of $r$ when the system is at perfect resonance, namely when $q_1 = \breve{k}_3 - \breve{k}_1$ and $q_2 = \breve{k}_3 - \breve{k}_2$. The reader will observe that indeed, even though the wavenumber $q_1$ is exactly on resonance with the transition between neutrino states 1 and 3, the probability of being in state 3 has a maximum of only $10\%$ when $A_2 \neq 0$. When we remove the second FM $q_2$ the transition probability $P_{13}$ increases to $100\%$. Note also a) that the solution is periodic even though the two wavenumbers $q_1$ and $q_2$ do not form rational ratio, and b) how well the numerical solution to the problem agrees with the RWA solution. The predicted amplitude and the wavenumber match the amplitude and wavenumber of the numerical solution to within a few percent.

To see the effect of induced transparency more clearly, we fix $q_2$ at the resonance between states 2 and 3 and scan in $q_1$. The solution for $B$ can be found by evaluating the formal solution and from the element $B_{13}$ we extract the transition probability $P_{13}$. In Fig.(\ref{fig:ITvsq1}) we plot the amplitude of the oscillations in $P_{13}$ as a function of $q_1$. We see that in the presence of mode $q_2$, the transition probability has a peculiar shape with peaks off-resonance and local minimum at the resonance. If we turn off the second perturbing mode by setting $A_2$ to zero we recover the expected shape for a resonance at $q_1$. Again, we find the RWA is able to reproduce the shape of $P_{13}$ versus $q_1$ very well at all the values of $A_2$ used.

\subsection{Three Fourier modes}
\label{opacity}

Now we add a third FM to the perturbing Hamiltonian which we give an amplitude $A_3$ and wavenumber $q_3$. Thus the perturbing Hamiltonian in the flavor basis becomes
\begin{equation}
\delta {H^{\left( f \right)}}\left( r \right) = V_{\star}\sum\limits_{j = 1,2,3} {{A_j}\cos \left( {{q_j}r} \right)} \left( {\begin{array}{*{20}{c}}
1&0&0\\
0&0&0\\
0&0&0
\end{array}} \right)
\end{equation}
We shall leave $V_{\star}$ unchanged so that the unperturbed Hamiltonian is the same as the previous case of two FMs with the same eigenvalues. The structure of the coefficient matrices $C_1$ and $C_2$ are also unchanged from equations (\ref{eq:C1}) and (\ref{eq:C2}) and new coefficient matrix we must introduce for the new mode is structured as
\begin{equation}
 C_3 = \frac{1}{2} V_{\star} A_3 \left( \begin{array}{lll} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array} \right)
\end{equation}
Thus the form of $f_{a;i}$, $y_{a;ij}$, $G_{a;ij}$, $\lambda_{a,m_a;ij}$ and $\kappa_{a,m_a;ij}$ are the same as given previous but with the substitution of $q_3$ for the wavenumber.

Let us now set the wavenumbers for the first two modes so that $q_1 = \breve{k}_3 - \breve{k}_1$ and $q_2 = \breve{k}_3 - \breve{k}_2$ for the level diagram shown in figure (\ref{fig:levels}). Neither $A_1$ nor $A_2$ are zero and $A_2 > A_1$. For a two FM case this choice for the wavenumbers $q_1$ and $q_2$ and ratio of amplitudes would put the system exactly at the midpoint of figure (\ref{fig:ITvsq1}) so the transition probability $P_{13}$ is suppressed even though the wavenumber $q_1$ is exactly on resonance.

We shall not set the mode $q_3$ to a particular value yet but we shall only consider wavenumbers such that $q_3$ is much smaller than $\breve{k}_3 - \breve{k}_1$, $\breve{k}_3 - \breve{k}_2$ and $\breve{k}_2 - \breve{k}_1$. i.e.\ $q_3$ is not on resonance with any pair of eigenvalue splittings. Thus the sets of RWA integers are very similar to the sets for the two FM case: for the $1,3$ element they are $\{ n_{1;13}, n_{2;13}, n_{3;13} \} = \{ +1 , 0, 0 \}$ and for the $2,3$ element we pick $\{ n_{1;23}, n_{2;23}, n_{3;23} \} = \{ 0 , +1, 0\}$. Again, these choices mean the integer set for the $1,2$ element is determined and must therefore be $\{ n_{1;12}, n_{2;12}, n_{3;12} \} = \{ +1 , -1,0 \}$ in order that $n_{a;12} + n_{a;23} = n_{a;13}$. The structure of the Hamiltonian for $H^{(B)}$ is exactly the same as in equation (\ref{eq:hb}).
As before, we find the expressions for the $\kappa$'s simplify greatly once we use the identities given earlier for the case where $G_{a;ij} = G_{a;ji}$, the vacuum mixing matrix is pure real, and from the Bessel functions. The final expressions are:
\begin{widetext}
\begin{eqnarray}
\kappa_{12} & = &  -\frac{2\,\imath\,G_{1;12}}{z_{1;12}} \,J_{1}\left(z_{1;12}\right) \,J_{1}\left(z_{2;12}\right) \,J_{0}\left(z_{3;12}\right)
                    \;+\; \frac{2\,\imath \,G_{2;12}}{z_{2;12}} \,J_{1}\left(z_{1;12}\right) \,J_{1}\left(z_{2;12}\right) \,J_{0}\left(z_{3;12}\right)  \\
\kappa_{13} & = &  \frac{2\,\imath \,G_{1;13}}{z_{1;13}} \,J_{1}\left(z_{1;13}\right) \,J_{0}\left(z_{2;13}\right) \,J_{0}\left(z_{3;13}\right) \\
\kappa_{23} & = &  \frac{2\,\imath \,G_{2;23}}{z_{2;23}} \,J_{0}\left(z_{1;23}\right) \,J_{1}\left(z_{2;23}\right) \,J_{0}\left(z_{3;23}\right)
\end{eqnarray}
\end{widetext}
which again shows $\kappa_{12}$ is much smaller than $\kappa_{13}$ and $\kappa_{23}$ when $z_{1;ij}$ and $z_{2;ij}$ are small. These expressions look very similar to those given in equations (\ref{eq:kappa12:2}) - (\ref{eq:kappa23:2}) for the two FM case, in fact the only difference is the presence of $J_{0}\left(z_{3;ij}\right)$. But the presence of this new term permits new phenomena because as we vary the wavenumber $q_3$ and/or its amplitude $A_3$ it becomes possible for either $z_{3;13}$ or $z_{3;23}$ to become equal to a zero of the Bessel function $J_0$. The effect will be to either switch off $\kappa_{13}$ or $\kappa_{23}$. If we switch off $\kappa_{13}$ then no transitions between states $1$ and $3$ can occur thus $P_{13}=0$ \emph{even though mode $q_1$ is on resonance}. At this value of $q_3$ the induced transparency has become complete. If we switch off $\kappa_{23}$ then now the effect of the third FM is to switch off the induced transparency effect and so we restore the amplitude of the oscillations of $P_{13}$ to 100\%. We call this effect Restored Opacity. Thus by scanning in the non-resonant mode $q_3$ we can tune the opacity of the system from zero to 100\% even though this mode is nowhere close to being resonant.
\begin{figure}[t!]
\includegraphics[clip,angle=270,width=\linewidth]{fig4.eps}
\caption{The probability $P_{13}$ as a function of $r$ for a three flavor neutrino model. The wavenumbers $q_1$ and $q_2$ are set to $q_1 = \breve{k}_3 - \breve{k}_1$ and $q_2 = \breve{k}_3 - \breve{k}_2$ with amplitudes $A_1=0.02$ and $A_2=0.1$. The third wavenumber is $q_3=5.24\times10^{-10}{\rm cm^{-1}}$. The dashed lines are the numerical solutions, the triangle symbols are the RWA prediction. The blue curve is for the case $A_3 =0$ and produces an example of Induced Transparency. The red curve is for $A_3=0.2$ and produces an example of Restored Opacity. }
\label{fig:ROvsr}
\end{figure}
To test these predictions we solve the for the transition probability $P_{13}$ numerically making no approximation. As for the two FM case, we set the potential $V_{\star}$ to $V_{\star}=6\times 10^{-25}\;{\rm erg}$ and the wavenumbers $q_1$ and $q_2$ are set to $q_1 = \breve{k}_3 - \breve{k}_1$ and $q_2 = \breve{k}_3 - \breve{k}_2$ with amplitudes $A_1=0.02$ and $A_2=0.1$. The third wavenumber $q_3$ is set to $q_3=5.24\times10^{-10}{\rm cm^{-1}}$ and we consider two cases: $A_3 =0$ and $A_3 =0.2$. The comparison between the numerical and RWA solutions is shown in figure (\ref{fig:ROvsr}). In the $A_3 =0$ case we expect induced transparency and indeed the figures shows that is correct with very small amplitude oscillations in $P_{13}$ even though the wavenumber $q_1$ is exactly on resonance between those pair of states. When we switch on the third mode we find $z_{3;23}$ is equal to a root of $J_0$ which means $\kappa_{23}=0$. This should return the amplitude of the oscillations of the transition probability $P_{13}$ back to unity and the figure indicates that does indeed occur: the presence of the third FM with this amplitude and wavenumber leads to a restoration of the opacity.

To further illustrate the power of the third FM, in figure (\ref{fig:ROvsq3}) we fix the amplitudes at $A_1=0.002,A_2=0.01$ and $A_3=0.02$, and scan in the wavenumber $q_3$. The purpose of using smaller amplitudes for the FMs is to suppress the fluctuations of the transition probability seen in the numerical results which make it hard to determine the transition amplitude. Note this choice also makes the corresponding value of $q_3$ which cause the Bessel functions to hit their roots smaller than in the example shown in figure (\ref{fig:ROvsr}). From every numerical solution we fit two sinusoids with amplitudes that enclose the oscillations of $P_{13}$ as seen in figure (\ref{fig:ROvsr}). The spread in amplitudes forms the width of the band for the numerical results shown in figure (\ref{fig:ROvsq3}).
The comparison of the theory and numerical solutions in figure (\ref{fig:ROvsq3}) indicate the theory does a very good job of reproducing the numerical results. At $q_3=5.24\times10^{-11}{\rm cm^{-1}}$, $\kappa_{23}$ is zero and therefore opacity is restored. When $q_3=4.22\times10^{-11}{\rm cm^{-1}}$ we find $z_{3;13}$ is a root of $J_0$ which forces $\kappa_{13}$ to be zero and thus we have complete transparency.
\begin{figure}[t!]
\includegraphics[clip,angle=0,width=\linewidth]{fig5.eps}
\caption{The amplitude of $P_{13}$ as a function of the wavenumber $q_3$. The potential $V_{\star}=6\times 10^{-25}\;{\rm erg}$ and the wavenumbers $q_1$ and $q_2$ are set to $q_1 = \breve{k}_3 - \breve{k}_1$ and $q_2 = \breve{k}_3 - \breve{k}_2$ with amplitudes $A_1=0.002$ and $A_2=0.01$. The amplitude of the third FM $q_3$ is $A_3=0.02$. The black solid line is from RWA evaluation and the thick red line is from the numerical solutions with the thickness of the band indicating the width of the fluctuations, an example of which is shown in figure (\ref{fig:ROvsr}). The values of $q_3$ which give Complete Transparency and Restored Opacity are indicated.}
\label{fig:ROvsq3}
\end{figure}

\section{Discussion and Conclusion}
\label{conclusions}

Mathematical tools for calculating the effect of a perturbation upon a quantum system are a valuable resource for both theorist and experimentalist. In this paper we have presented one such tool that is able to take an arbitrary perturbing Hamiltonian that has been decomposed into FMs and predict the evolution upon a multi-flavor neutrino / multi-level quantum system by using the Rotating Wave Approximation. We have placed no restriction on the dimensions of the perturbing Hamiltonian, the number of FMs, the size and structure of the amplitudes nor the wavenumbers. These lack of restrictions make the method applicable to a wide range of problems including those where the Hamiltonian is aperiodic / turbulent (but Fourier-decomposable).

We illustrated how our method works in practice by using it to analyze how a three flavor neutrino evolves when subject to density fluctuations composed of two and three anharmonic FMs. We found the RWA was able to predict the amplitude and wavenumber of the neutrino transition probabilities between pairs of states to within a few percent and we also discovered the equivalent of electromagnetic induced transparency when the expected maximal oscillations between a given pair of neutrino states could be switched off by the presence of a second resonant FM between another pair of states. When we added a third FM we found the neutrino evolution could be further controlled. At one value for the wavenumber the third FM was able to complete the transparency induced by the second but, at another value, we found the opacity could be restored. The three-flavor effects of induced transparency and restored opacity have obvious application to neutrinos and turbulence and is a area we shall pursue in future work.

\acknowledgments
The authors would like to thanks John Thomas, Bob Golub, Laura Clarke and Jason Bochinski for their helpful comments on this paper. This work was supported at NC State by DOE grants DE-SC0006417 and DE-FG02-10ER41577.

redpajama_set_name: RedPajamaArXiv


text:
Meet Jean Silva
It takes an avid resident to create the park of the future.
The World's Park
Ozgur Gungor | March 18, 2015
Jean Silva, The World's Park Community Advisor
Photo: Sam Holleran
She's one of nearly two dozen residents we recruited as Community Advisors to initiate community-led transformation of Flushing Meadows Corona Park (FMCP) in Queens. The group is spending four months at our Community Design School to improve wayfinding and circulation in the Park, and its connectivity to the surrounding communities. Read more about these sessions.
Name: Jean Silva
Occupation: Executive Director of the Flushing Meadows Corona Park (FMCP) Conservancy, a nonprofit dedicated to preserving, restoring, and maintaining the natural, historic, and cultural integrity of FMCP in order to ensure the park's continued educational, environment, and recreational benefits for users. We would like to bring the park back to the former glory it enjoyed during the two World's Fairs held in the past.
Neighborhood: Forest Hills, Community Board 6
Neighborhood Involvement: Flushing Meadows Corona Park Conservancy
I visit FMCP: Frequently and regularly organize volunteer efforts around Meadow and Willow Lakes.
Jean is focusing on how to improve access to the Park. Together with the members of her Team Access, she's exploring ways to:
Make it easier for people to get to the park from surrounding neighborhoods;
Increase the visibility of park entrances/exits;
Communicate the best entry points for people coming to the Park in different ways, like by foot, public transportation, car or bike, or using it in different ways like a soccer player or parents coming with their children.
At a community forum on March 1st, she introduced her preliminary design concepts, alongside the other Advisors, for feedback from the wider community. At the second forum on April 12, they will present their final concepts. It will also mark the opening of the exhibition at the Queens Museum that features all of these community-driven design proposals, on view until May 3, 2015.
Jean, like the rest of the Community Advisors, is eager to present her ideas to the NYC Parks Department, participate in further planning and design of FMCP, and motivate neighbors to do so too.
Read about our other Community Advisors.
Public Presentations
The World's Park Exhibition & Forum: "You Are Here"
Sunday, April 12, 2015 @ 2:00 PM
Reconnecting a regional park with its neighbors

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Baja's San Miguel Surf Break Protected as a State Park
September 23, 2021 by Serge Dedina Leave a Comment
On September 19, Baja California Governor Jaime Bonilla signed into law the establishment of the 169-acre Arroyo San Miguel State Park, just north of Ensenada. The new park, the first state park in the history of Baja California, includes the beach at the famed San Miguel river mouth, a cobblestone point, as well as the riparian watershed of the same name.
"The San Miguel break, which was the first place where surfing began in Mexico, is an iconic site for the surfing community, so we celebrate this decree," said Eduardo Echegaray, president of the Baja California Surfing Association.
Daniel Dedina enjoys an early winter session at San Miguel. Photo: WILDCOAST
The new state park helps to strengthen the Bahia de Todos Santos World Surfing Reserve that includes San Miguel, Todos Santos Island, and Salsipuedes among other surf breaks. The reserve proposal was supported by a coalition of organizations including Pronatura-Noroeste, Save the Waves, Northwest Environmental Law Center (DAN), Pro Esteros, Island Conservation Group (GECI)and WILDCOAST.
That's me surfing San Miguel during the annual contest there each January. It is a special place that the designation of the state park can help to safeguard.
"With this decree, Governor Bonilla has left a legacy for the people of Baja California, and in particular Ensenada," said Gustavo Danemann, executive director of Pronatura Noroeste. "The Arroyo San Miguel State Park is the first natural protected area to be established under state jurisdiction, and with this, it sets a precedent on the vision we have in Baja California for the protection and public and sustainable use of our landscapes and natural areas."
Similar to the San Mateo Creek watershed within San Onofre State Beach, Arroyo San Miguel State Park is a classic riparian oak watershed – a vibrant coastal and terrestrial ecosystem as well as a fabled surf spot.
"Arroyo San Miguel State Park is dominated by riparian habitats with oaks, willows and reeds as well as species of native vegetation, especially increasingly threatened coastal sage scrub," said WILDCOAST Mexico director Mónica Franco. "It is also home to resident and migratory birds such as quail and peregrine falcons."
Map of new Arroyo San Miguel State Park
The most important element for surfers is that it restricts development along the shoreline and along the western end of the watershed and is a big victory for the environmental and surfing community of Baja California.
"The designation of the Arroyo San Miguel State Park is a truly community initiative," said Danemann. "The Baja California government demonstrated fundamental support and commitment to make this possible. We are confident that the new state administration will also support the conservation of the Arroyo San Miguel State Park, and we reaffirm our commitment to continue collaborating in this project."
Community cleanup prior to a surf contest at Arroyo San Miguel State Park, Baja California, Mexico. The Ensenada surf and environmental community have been very involved in the development of the state park and the protection of the surf break, wetland, and watershed. Photo: WILDCOAST
Originally published in The Inertia.
Filed Under: Baja California, coastal protection, Mexico, Surfing Tagged With: Baja, Baja California, conservation, Pronatura-Noroeste, protected areas, San Miguel, Save the Waves, Serge Dedina, state parks, Surfing, WiLDCOAST


– this time with festival pic!
I missed a trick at Jisc DigiFest yesterday. All the other presenters at David Kernohan's splendid "Whatever happened to the MOOC?" session kicked off with a festival anecdote. David and Viv Rolfe, even had pictures of themselves playing at festivals! How cool is that?! I, however, launched straight into open education policy and practice :} Afterwards, James Clay rightly complained about my lack of festival-going anecdote. So here, as promised and by way of recompense, is a picture of me at Glastonbury in 1992. And once you've all stopped laughing, there's a copy of my presentation below.
"We've heard about some really inspiring open education developments, many of which have their roots in the UKOER programmes. We know that it's notoriously difficult to measure the impact of short term innovation funding, but two years after the end of UKOER, it's interesting to look back and see that the programme does seem to have had a hugely positive impact right across English higher education. One of the aims of UKOER was to embed open education practice across the sector and it's actually starting to look as if has done just that.
The situation is rather different north of the border. Scottish institutions were not eligible to participate in the UKOER and, arguably, this has resulted in lower awareness of the potential benefits of open education, and open education practice is less well embedded across the sector.
Although there have been no comparable large scale funding initiatives, we have seen a number of innovative open education developments within Scottish education, particularly in the area of open badges and MOOCs, and groups like the Open Knowledge Foundation and Wikimedia UK have also made real efforts to engage with the education community.
In an attempt to join up these initiatives and disseminate open education practice more widely, Cetis, SQA, Jisc RSC Scotland and ALT Scotland, launched Open Scotland, a voluntary cross sector initiative that aims to raise awareness of open education, encourage the sharing of OER, and promote the development of open policy and practice.
Open Scotland partially takes its inspiration from Nordic OER "a network of stakeholders to support uptake, adoption and collaboration around OER in the Nordic countries" and we've also been inspired by Higher Education institutions in Wales who came together to agree a statement of intent to adopt open educational principles. We see this statement as a positive development and are interested to see what impact it will have in practice.
Open Scotland has undertaken a number of awareness raising activities including the Open Scotland Summit, which brought together senior managers, policy makers and key thinkers to explore the development of open education policy and practice in Scotland. The Open Scotland blog was launched to disseminate news relating to all aspects of openness in education and to act as a focal point for discussion and debate. We have also just this week released the first draft of a Scottish Open Education Declaration. This is based on the UNESCO Paris OER Declaration, but extends its scope to focus on open education in general, rather than OER in particular. We invite all those with an interest in open education to contribute to shaping this draft declaration so we can reach a consensus on open education principles that will benefit all sectors of Scottish education.
There have also been some significant developments at Government level. In a speech earlier this year, the Cabinet Secretary for Education outlined the Scottish Government's vision of higher education and acknowledged the potential of MOOCs to form new pathways to learning, to widen participation and promote a culture of collaborative development and reuse. While it is hugely encouraging that the Scottish Government has started to acknowledge the potential of open education, there is some concern that the scope of this vision is insufficiently broad and may fail to encompass the wider benefits of open education to the Scottish sector as a whole. We all know that MOOCs are just one component of the wider open education landscape.
This entry was posted in higher education, MOOCs, oer, openscot, policy and tagged cetis, digifest14, moocs, oer, open, openscot. Bookmark the permalink.


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Open your PSD project and you'll see the easy-to-use interface, with controls and tools that simply make your life easier. Whether you're working on your design or getting the lay of the land, the tools make it easy. Things are as easy as using Microsoft Office, save for one detail: The layers are visible, and you can always go back and forth.
Our review prompted us to get two copies of the iPhone X, one for each of us, to play with, and show what we think of it. The iPhone X is sleek, and the colors are vivid. They're also reflective, so the colors look different when they hit your camera.
All the advanced features come to Photoshop for iPad. An intuitive user interface makes it easy to find, resize, rotate or crop photos. And there's just no other app that lets you bring all those special effects to your photos for a quick touch-up or to include in a collage.
Gets right to the point in our Apple Watch review, but even with a few shortcomings it does some things well. The heart rate monitor tracked my pulse and that seemed like a good feature to see if it could work when I'm wearing flannel and boots, and not just a t-shirt. The lack of connectivity is one of the areas where it failed, though, and where people who might want to use their Apple Watch as a fitness tracker might see it lacking.
Adobe Photoshop Lightroom 6 has several new features such as drastically improved performance. Lightroom now allows users to transfer multiple collections and locations into a single project. The new Marvelous viewer tool for images makes it easier to organize, edit and decorate websites. Also, the viewfinder now includes a floating "rotation tool."
Adobe Photoshop Elements is one of the popular applications for a digital photo editor. Along with other related applications, it's created by the company in 2000. It's a light version of the famous Adobe Photoshop. It uses the original format of the photo editing carried out by Adobe Photoshop. It's an edit-oriented program which lets users to edit and extract photos, modify content, convert the size, and other functions. With the ability to operate on both the Mac, Windows and Linux.
The basic version of Photoshop was referred to as Photoshop Lightroom. Now the software has evolved as Adobe Photoshop Lightroom and its functionality has increased across the latest versions. This software is a complete photo editing program. It's powerful, simple, and efficient photo editing tool that enables you to easily recover lost files and optimize your photos.
Photoshop has always been the most popular software for digital editing and photo manipulation. And it has effectively catered to users based on their needs and skills. Even if you are a hobbyist, a professional photographer, or a designer, Photoshop has been a great tool in your endeavours. With its incredible power, user-friendly interface, and advanced features it has helped users to achieve their creative goals.
The transition from Photoshop CS and CS2 to Photoshop CS3 was the opening of new graphics editing sessions. Canonical, a creator of the software was also happy to be supported by Canonical to create this great software. Canonical announced the release of the Mac and Windows version of Photoshop, and after the success of the initial launch of Photoshop CS5, Canonical introduced Photoshop CS6. So the popular software is very popular and very cross-platform which has won hearts of users all over the world.
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The next feature on the list is probably the most commonly used feature of photo editing. Image transforms allows you to easily modify the shape of any photo. You can retouch images without changing anything about the pixels of the image. This allows you change images in shapes you can easily retouch. Just take any picture and you'll see this feature in action.
2. Opacity mask: Unlike many other software products, Photoshop is pretty thorough with its opacity mask feature. You're able to quickly select which parts of an image you want to mask out, everything that's not on the mask is preserved. This is an extra type of photo editing that turns out extremely effective, and you can do so much with it.
If you're looking for a simple way to clean up your images and make them look better, you should try de-skewing photos. To do so, you just need to align the edges of your image to the edges of your photo. Next, you'll use a blur effect to smooth out the edges of the photo. Choose the quick blur effect if you'd only like a rough blur effect, while you can choose the regular blur or the angle blur tool for a much smoother look. There are tons of apps like this for free on iOS and Android.
Adobe is certainly known for its large selection of fonts. While no other software package has as many choices as Photoshop, the fonts in the package vary greatly in typeface. You can preview fonts, measure type size, enlarge, shrink, retouch, and smooth them. You can even duplicate, swap, or embed the font.
If you have an image that's 10,000 pixels wide, and you want to view it at 13,200 pixels wide, you can press the Hand–Click to Fit to fit them on the screen. You can also set up cropping to the right side to adjust the image's sizes. For example, the cropped image only takes up 5,000 pixels by 5,000 pixels.
Image skew tool version 2.0 lets you change the perspective of the image based on the position of the mouse. This feature lets you adjust the image to make it more attractive. You can adjust the appearance of the drawing and images with various colours.
For projects with a lot of text, you can edit it easily. Photoshop Elements' Text tool remembers the last place where you dragged the line cursor, including the pressure. This means that you'll avoid extra work when you have to change the space where you put the text.
It's important to know the difference between rel=nofollow and rel=noopener . While some websites use the former, the latter is a better alternative. As rel=nofollow doesn't add relevant content, the latter one saves the user's web browser's resources.
Photoshop CC features the Camera RAW engine, which allows designers to open RAW files without having to convert the files to other formats. However, the Photoshop CC has been totally restructured and revamped and is not backward-compatible with the Adobe Photoshop CS. Therefore, you should opt for the required version if you want to use the new features and functions. The free service of Adobe Photoshop Elements includes the basic features of the Photoshop family.
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Adobe Photoshop has been at the forefront of the image editing industry since its inception in 1984. Its basic editing tools are incorporated in the application. The Macintosh version of Photoshop was first introduced in 1991 and was later distributed on PCs. Since its introduction, the application has become more accurate. The latest version CC 2019, finally allows the Macintosh employees to be able to use the application. It is among the most popular image-editing tools in the world and the Macintosh version is
Although its closest rivals may be Pixelmator or Corel Painter, Adobe's Photoshop still has much to offer. For starters, there's a whole suite of tutorials available to help. You can also browse a massive gallery of free images, which has options for finding similar images and adding the top 20 best images of all time.
Adobe Photoshop CC 2017 is the latest version (significantly upgraded since 2016). It adds numerous new features, including Adobe Sensei (AI) and Adobe Link (beta), the new face-recognition feature called Adobe InDesign Link, and a series of improvements to individual editing features
Since much of the iOS experience is driven by iOS apps, it's understandable that a version of Photoshop Elements was left off the list, but downloading Photoshop Elements 2016 on an iOS device is a breeze, from Apple's App Store. A one tap download allows you to download the app, register it, and get to work on images. From there, it's just like using the app on a PC. It works, of course, but it's easy to beat expectations.
The new Tool Settings dialog now displays information dynamically when you hover over a tool or the image itself, making it easier to see how a tool affects the image and quickly disable and enable it as needed. 2019 is your last chance to activate features by year and 2019 for the Creative Cloud subscription. In 2020, they will no longer be available. Photoshop will no longer accept subscription renewals after December 31, 2020.
Developed specifically for large file sizes, XLR8 supports both RAW and JPEG formats and allows you to quickly manage a large number of files simultaneously. The new autodetect process makes the tool learn directly from your code points if you have a set of images to process, thus helping to prevent you from having to adjust the pipeline settings each time you open a new set of images.
If you've used the new "Paint" feature in Photoshop CC 2019 for editing in Layers, you'll be pleased to know that when you start painting, even in the Apply dialog you'll see how "Paint Everything" will affect the image as you move your tools. You can still paint in the traditional way, and the painting functionality also remains in place.
Photoshop CC 2015 is among some of the most advanced software programs available for those who are aspiring to become professional photographers. We have seen similar updates for the past few versions of Photoshop, adding capabilities and features that help make the software more user-friendly. Adobe has also positioned Photoshop CC 2015 to work across all platforms—Windows, Mac, iOS, and Android devices.
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After starting its journey almost 40 years ago, Adobe Photoshop has seen amazing evolution with each new release and version. The application is always loved by designers and photographers and anyone who is using Photoshop for image editing. From basic features to AI-powered tools and innovative features, there are few tools that are used by those designers and photographers to make correct adjustments and edit their photos. In this post, we will show you the top ten most awesome tools available in Photoshop CC. Let's take a look at the top ten tools that Adobe developed with time and remained highly used by the designers and photographers.
People told us there would be a Photoshop revolution when they adopted the first Digital Darkroom. In fact, something happened. The world of photography and digital imaging took off. Artists, hobbyists, professionals, and even businesses and governments embraced digital technology and put their creativity to work.
From a technical standpoint, digital photography and picture-editing software had improved greatly since the advent of Photoshop. The features that made it work for professionals were also accessible to photographers, plus new features like the ones we announced today. But the power of Photoshop evolved through new interfaces, a reimagined architecture, and improvements in the continuity between the desktop and mobile versions. This would be worth doing for a hobbyist or even a seasoned photographer.
But there was also a reason to make the change. Because the features that were essential for professionals weren't good enough for the occasional creative person, making the switch would mean a better image editing experience for people who'd used print tools to create art in the past. It also meant a less fragmented, more polished Photoshop experience when it asked the professional for action. And hey, even the gimpy iPhone user on the other end of the world could benefit from it.
The latest version of the graphics program, CS6 (Compatibility Level of 6), is on a whole new level. The new features and innovations in design of Photoshop 2015 bring back some of the most sought after features in its previous version, CS6. Such as; stitching a bunch of images to create the illusion of a panorama; warp distort a picture with rich brushes; drag and drop graphical elements and more.
Photoshop is the most loved and widely used image editing software in the world. At its heart it's a paint program that has many powerful features, such as the ability to mix colors and perform powerful fine-tuning tasks. Photoshop's real power lies in its nondestructive editing tools, such as layer masks and adjustment layers. With these tools, you can make adjustments to your image without deleting or altering any part of it; simply make changes through the layers of the image. You can make a brush tool with painting strokes that can be applied to the pixels of the layer to be edited. This gives you maximum control over the changes.
This year Adobe released Photoshop CS6. It has over 40 new features and enough new for people to seriously consider buying a copy; this release is the most anticipated new version of Photoshop since version 6. It's a lot like previous versions in functionality but at the same time its interface has been completely redesigned to be easier to use and more powerful. It's a powerful and cropeditor for both the professional and creative amateur. Those interested in the professional Photoshop will get some of the best new and redesigned features. Creative types will still cast their eyes on the latest version of Photoshop as Adobe's lineup of new tools has inspired and delighted many.
Photoshop CC – Photoshop CC is a new version of Photoshop. It is the updated version of the previous Photoshop for Windows. It is available as a standalone version. Advantages with this version is the inbuilt help system, which allows instant and visual help, viewing and understanding of the different button interaction. With the help of hired in-app templates, users can quickly create the layouts for different projects. and apps to save time and money.
Mask – Thanks to the new version of Adobe Photoshop, a new mask option has been introduced. This feature allows you to select a particular section of an image, which gets blurred automatically. It is a better way to create masks and includes the auto calibration options.
Lens Correction – During the previous version of Adobe Photoshop, it did not provide any view-based image correction. With this new version of Photoshop, the company has introduced a viewing-based image correction feature, which enables the user to define the boundaries of the existing objects in the image or provide these masks directly. The feature also allows the use of vector shapes.
Clarity – This is one of the most important Photoshop features in the world of digital photography. Photoshop Clarity allows the users to get an edge of the objects in the background, while retaining the clarity of the foreground object. This amazing tool prevents the over sharpening from taking place, while retaining the smoothness of the objects. It is an inbuilt feature with this version of Photoshop, which is not available in any other Photoshop software.


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/**
    File Name:          15.9.5.2-2.js
    ECMA Section:       15.9.5.2 Date.prototype.toString
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    var OBJ = new MyObject( new Date(0) );

    testcases[tc++] = new TestCase( SECTION,
                                    "var OBJ = new MyObject( new Date(0) ); OBJ.toString()",
                                    "error",
                                    OBJ.toString() );
    test();

function MyObject( value ) {
    this.value = value;
    this.valueOf = new Function( "return this.value" );
    this.toString = Date.prototype.toString;
    return this;
}
function test() {
    for ( tc=0; tc < testcases.length; tc++ ) {
        testcases[tc].passed = writeTestCaseResult(
                            testcases[tc].expect,
                            testcases[tc].actual,
                            testcases[tc].description +" = "+
                            testcases[tc].actual );

        testcases[tc].reason += ( testcases[tc].passed ) ? "" : "wrong value ";
    }
    stopTest();
    return ( testcases );
}

redpajama_set_name: RedPajamaGithub


Guest Lecture Series are lecture events that focus on their experience with math and science and what they did to be successful in that particular area. Each lecture will include two speakers, a break in between, and a question and answer session following each event.
The Lectures are designed to create a positive learning environment for students of all age groups. We strive to provide the most effective and motivated lectures by professors and highly achieved students. We hope that these events will not only motivate students but also expand their perspectives in Math and Science.
There will be about 4 - 5 total lectures during the school year. The specific date for the next lecture will be announced a day or two after the previous event ends. There will be more information sent through email about 2 weeks before each lecture. This event is free of charge, but there will be fundraising events at the lecture.
You will gain valuable lessons regarding Science and Math.
You will hear from famous professors, successful high school students, and experienced college students.
You will be able to interact with the professors and ask questions during the discussion period.
You will gain motivation by listening to the lectures.
Interested and motivated students who enjoy learning about Science and Math.


Features: �Keep-warm function(up to 12 hours)� �Non-stick inner pan� �Retractable power cord � �Spatula & 6 oz.
measurement cup �Made in Japan� Model : JNP�� 1800 Capacity(liters/oz)� 1.
9 Qty/Ctn(pcs) 4� Measurement(m3)� 0.
131 Electric Rating cook/warm(Watt)� 650/59 .
Hot Dog Steamer Model - CST-1412 Patented Steaming System Steams up to 12 Hot Dogs & 12 Buns at one time Cooking Timer Steams a variety of foods Includes a ""Hot Dogs of The World"" Recipe Booklet with 40 Hot Dog variations .
5CUPS/ Features: Non-stick inner pan� �15 cup large rice cooker� �Keep-warm function(up to 12 hours) � �Spatula & 6 oz.
measurement cup� Model : JAZA10U� Capacity(liters/oz)� 1.
00� Qty/Ctn(pcs) �� 4� Measurement(m3) � 0.
106� Electric Rating cook/warm(Watt) � 500/57 .
TIGER JNPS10U RICE COOKER 5.
5CUP HUY Main Features: Keep-warm function(up to 12 hours) Non-stick inner pan Retractable power cord Spatula & 6 oz.
measurement cup Warranty: Limited manufacturer warranty.
� Please contact manufacturer for warranty service Warranty applies to residents of United States only No warranty for physical damage .
3oz / 3cups Electric Rice Cooker Made in Japan Keep-warm function(up to 12 hours) Spatula & 6 oz.
Dinner is easier with the Black & Decker 6-Cup Rice Cooker. Cooking takes just 20-30 minutes and the unit keeps the delicious fluffy rice ready to serve long after cooking is complete with an automatic keep warm function. Use it as a steamer too. The included steaming basket lets you prepare healthy meals with vegetables fish and more. Cleanup is a cinch with the dishwasher-safe nonstick bowl and tempered glass lid. Plus you can use the cooker to prepare soups stews and even oatmeal!. The durable glass lid gives you a look in at rice during cooking with a cool-touch plastic handle for easy removal. The included steaming basket lets you make healthy meals with vegetables fish and more. The nonstick cooking bowl is completely removable for eash dishwasher cleanup. Different colored lights show when the unit is cooking rice or keeping it warm. The cooker lets off steam at the perfect pace with a small vent in the lid. 6 Cup Capacity Automatic Keep Warm Removable Nonstick Bowl Dishwasher-Safe Bowl an This item cannot be shipped to APO/FPO addresses. Please accept our apologies.
Dinner is easier with the Black & Decker 3-Cup Rice Cooker. Cooking takes just 20-30 minutes and the unit keeps the delicious fluffy rice ready to serve long after cooking is complete with an automatic keep warm function. Cleanup is a cinch with the dishwasher safe nonstick bowl and tempered glass lid. Plus you can use the cooker to prepare soups stews and even oatmeal! 3-Cup Capacity Automatic Keep Warm Removable Non Stick Bowl Dishwasher-Safe Bowl and Glass Lid Measuring Cup and Spoon Included Indicator Lights Tempered Glass Lid Built-In Lid Holder Steam Vent. This item cannot be shipped to APO/FPO addresses. Please accept our apologies.
The Canner 9.5-Quart Pressure Canner and Cooker with removable non-stick cooking chamber serves large meals and cleans up easily. Over 20 built-in safety features Separate programs for various cooking types at the push of a button. Pressure cook slow cook bake brown meat steam and cook rice or risotto. Separate water bath/steam functions for canning. Can four quarts six pints eight jelly jars or ten half-pint jars in one batch. This item cannot be shipped to APO/FPO addresses. Please accept our apologies.
TIGER JNP0720/RICE COOKER/4CUPS Product Description Features: Electronic control heating makes better rice.
Special keep-warm heating elements keep rice warm up to 12 hours.
Retractable power cord reduces counter top clutter and stays attached for storage.
Non-stick PTFE coated rice cooking and steamer pans make clean-up easy.
High quality double wall construction with locking top to seal in flavor.
Includes measuring cup, spatula, cooking and steamer pans.
Water level marks on inner surface of ricecooking bowl.
Built in carrying handle for safe and easy moving.
�Features: �Keep-warm function(up to 12 hours)� �Non-stick inner pan� �Retractable power cord � �Spatula & 6 oz.
measurement cup �Made in Japan� Model : JNP� 0550 Capacity(liters/oz) �0.
6 Qty/Ctn(pcs)��� �4 Measurement(m3) �0.
074� Electric Rating cook/warm(Watt) �305/23 .
Kitchen Kettle multi-cooker steams vegetables and rice cooks pasta. Roasts beef pork and poultry. Makes soups and casseroles. Heavy cast aluminum base for even heat distribution. Premium nonstick surface inside and out for stick-free cooking and easy cleaning. This item cannot be shipped to APO/FPO addresses. Please accept our apologies.
Big dinners just got easier thanks to the Black & Decker 16-Cup Rice Cooker. Cooking takes just 20-40 minutes and the unit keeps the delicious fluffy rice ready to serve long after cooking is complete with an automatic keep warm function. Use it as a steamer too! The included steaming basket lets you prepare healthy meals with vegetables fish and more. Cleanup is a cinch with the dishwasher-safe nonstick bowl and tempered glass lid. Plus you can use the cooker to prepare soups stews and even oatmeal The durable glass lid gives you a look in at rice during cooking with a cool-touch plastic handle for easy removal. The included steaming basket lets you make healthy meal with vegetables fish and more. The nonstick cooking bowl is completely removable for easy dishwasher cleanup. The built-in holder props the lid conveniently on either of the side handles. Different colored lights show when the unit is cooking rice or keeping it warm. The cooker lets off steam at the perfect pace with a small vent This item cannot be shipped to APO/FPO addresses. Please accept our apologies.
The Black & Decker 28-Cup Rice Cooker makes it easy to create rice and more for family dinners and large gatherings. The nonstick removable cooking bowl features an extra-large capacity for rice and grains soups and stews or even oatmeal and pasta. The included steaming basket increases your cooking options with healthy preparation of fish veggies and more. Plus the cooking bowl tempered glass lid and included accessories are all dishwasher-safe for easy cleanup. The durable glass lid allows you to see the rice during cooking and removal is easy with the cool-touch plastic handle. The included steaming basket allows you to make healthy meals with vegatables fish and more. The Removable nonstick cooking bowl is dishwasher-safe and completely removable for easy cleanup. The built-in holder props the lid conveniently on either of the side handles. Colored lights show when the unit is cooking rice or keeping it warm. The cooker lets off steam at the perfect pace with a small vent in the lid. 28-Cup Ca This item cannot be shipped to APO/FPO addresses. Please accept our apologies.
Make meal preparations easier with this Brentwood Rice Cooker. A Steamer is included with a Non-Stick Coated Inner Pot. Rice Cooker includes and an Auto Shut-Off for your safety protection. With a Tempered Glass Lid to top it off so you can prepare a diverse meal everytime. 400Watts White Colored Stay Cool Handles 5 cup-uncooked This item cannot be shipped to APO/FPO addresses. Please accept our apologies.
Cuisinart 4-Cup Rice Cooker/Steamer Brushed stainless housing makes up to 7 cups of rice. Includes stainlesss steaming basket recipe book. Brushed stainless steel housing with chrome-plated handles Makes 4-7 cups of cooked rice Traditional lever control with warm and cook settings Warm and cook indicator lights Automatically switches to "Warm" when rice is cooked Tempered glass cover with stainless rim and chrome-plated knob Stainless steel steaming pot Measuring cup and rice spoon Instruction/Recipe book Limited 3-year warranty BPA Free - See more at: http://www.cuisinart.com/products/slow_cookers_rice_cookers/crc-400.html#sthash.SQrg3XG5.dpuf This item cannot be shipped to APO/FPO addresses. Please accept our apologies.


text:
module Weaver where

import Control.Monad (zipWithM)
import Data.Functor ((<$>))
import Data.List (intercalate)

import Data.Document
import Data.Text
import Parser
import Utils

-- | 'msgLine' @title cntLoc styLoc@ produces a warning message for a
-- given content and style 'Srcloc'.
msgLine :: String -> Srcloc -> String -> String
msgLine title (n, cnt) sty =
    intercalate "\n" ["In line " ++ show n ++ ", " ++ title,
                      prefix "  " cnt,
                      "does not match style",
                      prefix "  " sty]

-- | 'msgLines' @title cntLoc1 cntLoc2 styLoc1 styLoc2@ produces a
-- warning message that spans multiple lines of content and style.
-- This warning message is adjusted to possibly equal ranges of
-- content and style lines.
msgLines :: String -> Srcloc -> Srcloc -> Srcloc -> Srcloc -> String
msgLines title (n1, cnt1) (n2, cnt2) (n1', sty1) (n2', sty2) =
    let
        pre | n1 == n2 = "In line " ++ show n1
            | otherwise = "In lines " ++ show n1 ++ "-" ++ show n2
        cnt | n1 == n2 = cnt1
            | otherwise = intercalate "\n" [cnt1, "...", cnt2]
        sty | n1' == n2' = sty1
            | otherwise = intercalate "\n" [sty1, "...", sty2]
    in
      intercalate "\n" [pre ++ ", " ++ title,
                        prefix "  " cnt,
                        "does not match style",
                        prefix "  " sty]

-- | 'msgHeading' @cntLoc styLoc@ produces a suitable warning message
-- for a 'Heading' element given the 'Srcloc' of the content and the
-- 'Srcloc' of the style.
msgHeading :: Srcloc -> Srcloc -> String
msgHeading loc (_, styStr) = msgLine "heading" loc styStr

-- | 'msgPararaph' @cntLoc styLoc@ produces a suitable warning message
-- for a 'Paragraph' element given the 'Srcloc' of the content and the
-- 'Srcloc' of the style.
msgParagraph :: Srcloc -> Srcloc -> String
msgParagraph loc (_, styStr) = msgLine "paragraph" loc styStr

-- | 'weaveText' @loc txt1 txt2@ weaves 'Text' @txt1@ with style of
-- 'Text' @txt2@ where @loc@ is the 'Srloc' of the produced 'Style'
-- elements.  'Nothing' is returned if the style cannot be applied.
weaveText :: Srcloc -> Text -> Text -> Maybe Document
weaveText loc (Emphasis cnt) (Emphasis sty) =
    Just $ Style loc (trim sty) [[Plain $ trim cnt]]

weaveText loc (Footnote cnt) (Footnote sty) =
    Just $ Style loc (trim sty) [[Plain $ trim cnt]]

weaveText loc (Plain cnt) (Plain sty) =
    Just $ Style loc (trim sty') [[Plain $ trim cnt]]
    where sty' | isParagraph sty = init sty
               | otherwise = sty

weaveText _ _ _ = Nothing

weaveExtraLine :: Srcloc -> [Text] -> Text -> Maybe Document
weaveExtraLine loc txts (Emphasis sty) = Just $ Style loc (trim sty) [txts]
weaveExtraLine loc txts (Footnote sty) = Just $ Style loc (trim sty) [txts]
weaveExtraLine loc txts (Plain sty) = Just $ Style loc (trim sty) [txts]

-- | 'weaveLine' @loc txts1 txts2@ weaves 'Text's @txts1@ with style
-- of @txts2@ where @loc@ is the 'Srcloc' of the produced 'Style'
-- elements.  'Nothing' is returned if the style cannot be applied.
weaveLine :: Srcloc -> [Text] -> [Text] -> Maybe [Document]
weaveLine _ txts1 txts2 | length txts1 < length txts2 = Nothing
weaveLine loc txts1 txts2 | length txts1 == length txts2 = zipWithM (weaveText loc) txts1 txts2
weaveLine loc txts1 txts2 =
     let (matTxts, unmatTxts) = splitAt (length txts2 - 1) txts1
     in do docs <- weaveLine loc matTxts $ init txts2
           doc <- weaveExtraLine loc unmatTxts $ last txts2
           return $ docs ++ [doc]

weaveExtraLines :: Srcloc -> [[Text]] -> [Text] -> Maybe Document
weaveExtraLines loc lns [Emphasis sty] = Just $ Style loc (trim sty) lns
weaveExtraLines loc lns [Footnote sty] = Just $ Style loc (trim sty) lns
weaveExtraLines loc lns [Plain sty] = Just $ Style loc (trim sty) lns
weaveExtraLines _ _ _ = Nothing

-- | 'weaveLines' @loc lns1 lns2@ weaves lines @lns1@ with style of
-- lines @lns2@ where @loc@ is the 'Srcloc' of the produced 'Style'
-- elements.  'Nothing' is returned if the style cannot be applied.
weaveLines :: Srcloc -> [[Text]] -> [[Text]] -> Maybe [[Document]]
weaveLines _ lns1 lns2 | length lns1 < length lns2 = Nothing
weaveLines loc lns1 lns2 | length lns1 == length lns2 = zipWithM (weaveLine loc) lns1 lns2
weaveLines loc lns1 lns2 =
    let (matLns, unmatLns) = splitAt (length lns2 - 1) lns1 in
    do docss <- weaveLines loc matLns $ init lns2
       doc <- weaveExtraLines loc unmatLns $ last lns2
       return $ docss ++ [[doc]]

-- | 'weaveStyle' @doc style@ combines content 'Document' @doc@ and
-- style 'Document' @style@ in a single styled 'Document'.
weave :: Document -> Document -> (Document, [String])
weave doc style =
    let (docs, errs) = weave' doc style in (ensureDocument docs, errs)
    where weave' cnt@(Heading loc1 lns1) (Heading loc2 lns2) =
              case weaveLines loc1 lns1 lns2 of
                Nothing -> ([cnt], [msgHeading loc1 loc2])
                Just docs -> (concat docs, [])

          weave' cnt@(Paragraph loc1 txts1) (Paragraph loc2 txts2) =
              case weaveLine loc1 txts1 txts2 of
                Nothing -> ([cnt], [msgParagraph loc1 loc2])
                Just docs -> (docs, [])

          weave' cnt@(Content docs1) sty@(Content docs2) | length docs1 < length docs2 =
              let
                  (cntLoc1, cntLoc2) = rangeloc cnt
                  (styLoc1, styLoc2) = rangeloc sty
              in
                ([cnt], [msgLines "content" cntLoc1 cntLoc2 styLoc1 styLoc2])

          weave' (Content docs1) (Content docs2) =
              let
                  (matDocs, unmatDocs) = splitAt (length docs2) docs1
                  (docss, errss) = unzip $ zipWith weave' matDocs docs2
              in ([Content (concat docss ++ unmatDocs)], concat errss)

          weave' (Section doc1) (Section doc2) =
              let (doc', errs) = weave' doc1 doc2 in
              ([Section $ ensureDocument doc'], errs)

          weave' cnt sty =
              let
                  (cntLoc1, cntLoc2) = rangeloc cnt
                  (styLoc1, styLoc2) = rangeloc sty
              in
                ([cnt], [msgLines "document" cntLoc1 cntLoc2 styLoc1 styLoc2])


Victor Oladipo credits solid defense for the Indiana Pacers' strong offensive production.
"I think defensively we were really good (and) I think that's why we were effective offensively," Oladipo said after scoring 25 points with seven rebounds and leading Indiana to a 132-112 victory over the Brooklyn Nets on Saturday night.
"When we play at a high level defensively, we give ourselves a great chance to win," Oladipo said. "We have to build on it. We just have to keep getting better. If we do that and continue to keep growing, continue to keep playing with confidence on good days and bad days, the sky's the limit for us."
Kyle O'Quinn came off the bench and scored 14 of his 16 points in the fourth quarter. Bojan Bogdanovic and Tyreke Evans each scored 15 points for the Pacers, who bounced back following a 118-101 loss at Milwaukee on Friday, the first of a back-to-back.
Indiana forced Brooklyn into 20 turnovers, committed 11 of their own and outscored the Nets 30-12 in points off turnovers.
"We got stops tonight," Pacers coach Nate McMillan said. "I thought our defense was good. They made some tough 3s in that first half. We made them work for those shots in the first half and we felt like if we continued to work like that, that eventually they would break. In the second half, we kind of wore them down and they started to miss those same shots that they were making. You have to force misses in order to get out and play your pace."
O'Quinn scored back-to-back baskets, Oladipo made a 3-pointer, and Evans scored twice, including a layup, to give the Pacers a 122-99 lead with 6:01 remaining.
"We made some mistakes defensively and we allowed them to start getting downhill," Harris said. "Kyle O'Quinn had a big fourth quarter. Oladipo hurt us most of the game, but had a good fourth quarter, as well. We have to clean up a lot of stuff on the defensive end, especially doing it consistently for four quarters."


Controliss Regency Neutral 240V AC mains RTS remote control electric roller blind: This fabric is a formal pattern of bold stripes. Contemporary, neutral and duotone, the limited palette of shades of grey and light brown effortlessly blends into any design scheme.
Chronis Smart RTS incorporates Somfy's RTS technology, which eliminates the need for wiring between the control switch and the blind. You are free to install the control point anywhere you want, with no disturbance to décor.
Chronis Smart RTS is a 7-day programmable timer giving one up and one down command per day to a blind or several motorized blinds simultaneously.
Cosmic mode to operate blinds in conjunction with sunrise and sunset times.
Security mode to simulate presence in the home by bringing forward or delaying the programmed times of operation by up to 15 minutes.
Date and time must be set whenever battery is replaced.