text:
\section{Introduction}
The Bohr-Mottelson collective model of the nucleus has recently attracted
a significant interest because of the possibility to derive
many ``new'' solvable
cases \cite{Iac1,Iac2,Iac3,RoBa,Cap,FV,FV2,Pepe,Bona1,Bona2,BonZ5,Mark2,
Rowe-new,LF,Pietr,CapNew,
RoI,RoTuRe}.
Intense efforts did arise because of the availability of models, based on the
square well potential, that are related with the issue of critical point
symmetries at the shape phase transition (E(5), X(5) and Y(5)) \cite{Iac1,
Iac2,Iac3}.
This has given rise from one side to many applications aimed at the survey of
existing experimental spectroscopic data and at the identification of
signatures for the new models \cite{Cast,Kru,Capal,Clark,Tonev,Liu,Frank}
in various mass regions, especially in connection with the effort to
build new descriptions for transitional nuclei.
On the other side a number of mathematical solutions of the Schr\"odinger
equation associated with the Bohr hamiltonian with various model-potentials
have been proposed \cite{Cap,FV,FV2,Pepe,Bona1,Bona2,BonZ5,LF,Pietr,CapNew,
RoI,RoTuRe}.
For some of those potentials, that in general are functions of the quadrupole
deformation variables $\beta$ and $\gamma$, an exact separation of variables
is possible, while in other cases an approximate separation holds.
Usually the spectrum (and transition rates) is derived analytically and
compared with available experimental data. We like to mention that
for a number of these new solvable cases extensive comparisons with
experimental data are still not available.

Recently a solvable model was proposed for the soft triaxial rotor with
a minimum in the potential along the $\gamma$ direction located at $\pi/6$
\cite{LF}.
The aim of the present paper is to extend and complete that solution,
proposing a solvable model for which the Schr\"odinger equation is separable.
The $\gamma-$angular part is then solved for a harmonic potential centered around any $\gamma_0$ in the
interval $~]0,\pi/3[~$, while the $\beta$ part is solved exactly for the
Kratzer-like potential, for the Davidson potential
and for the infinite square well potential.

Another reason for the search of analytic solutions for the more general
case with non-irrotational moments of inertia comes also from a recent
study \cite{Wood} in which the rigid triaxial rotor model is improved
by relaxing the assumption of irrotational flow moments of inertia. The
fit of the three components of the moment of inertia to experimental data
improves the description of a number of nuclear properties and suggests
that the irrotational assumption is not correct. Our model incorporates
this idea and extends the rigid model to $\gamma$-soft models.

\begin{figure}[!t]
\begin{center}
\epsfig{file=map-soft.eps,width=8cm,clip=}
\end{center}
\caption{Contour plot of the potential $V(\beta,\gamma)$ with minimum
in $\gamma_0=10^o$ and $\beta_0=0.2$. The situation depicted here
corresponds to a very soft deformed triaxial rotor, where the $V_1(\beta)$
potential (see Eq. (\ref{pote})) has a Kratzer-like form, while the
$V_2(\gamma)$ is a displaced harmonic oscillator.
The coordinates are $X=\beta\cos{\gamma}$ and $Y=\beta\sin{\gamma}$.
In the point $\beta=0$ the potential goes to $+\infty$ (black area).}
\label{soft10}
\end{figure}

To give an idea of the typical problem that we are considering
we show as an illustration in Fig. (\ref{soft10}) a contour plot of the
potential surface with a Kratzer-like potential in $\beta$ (with the minimum
at $\beta_0=0.2$) and a displaced harmonic oscillator potential in $\gamma$
(with the minimum at $\gamma_0=10^o$). The solution of the two-dimensional
Schr\"odinger equation for the Bohr hamiltonian containing this potential
surface is given by a set of eigenvalues and their corresponding eigenstates.
To each eigenstate (and in particular to the ground state) an average
quadrupole shape is associated. In the present case (Fig. \ref{soft10})
the wave functions are concentrated around the region of the minimum and
thus we are dealing with a soft triaxial rotor.

The present study may also be considered as an attempt to solve a simple
model that may be compared with microscopic theories (like HF, HF+BCS or HFB)
\cite{GG,Hay,Ansari}.
The potential surfaces used throughout the paper, rather than calculated
with a variational procedure in a microscopic framework, are
instead {\it a priori} given in
analytical form. A number of early microscopic studies, like for example
Ref. \cite{GG}, concluded that $\gamma-$softness was not only unavoidable
in triaxial nuclei, but that the potential barrier in the $\gamma$ direction
was so small to cast doubts on the applicability of the rigid rotor model.
We therefore reanalyze a number of osmium isotopes, which were treated within
a HFB formalism, and compare our results with the results of Ref.
\cite{Ansari}.

In the present paper, we use a potential of the form $V_1(\beta)+V_2(\gamma)/
\beta^2$, which allows for a separation of the Bohr hamiltonian in $\beta$
and $(\gamma,\theta_i)$ part (Section II). The solution of the
$(\gamma,\theta_i)$ part is amply presented in Section III.
In Section IV we discuss the quantum numbers associated with the present model
and presents a few comments on quasidynamical symmetry. In Section V we
discuss in detail some examples that illustrate the method of solving Eq.
(\ref{coupled}).
In Section VI, we solve analytically the $\beta$ part of the problem in the
Coulomb/Kratzer, harmonic/Davidson and infinite square well cases.
In Sections VII and VIII, we discuss the fitting procedure and its
accuracy and we apply the method to the chain of osmium isotopes and
make a comparison with other calculations. Here, we also stress the fact
that, in doing so, in general we go beyond an irrotational approach
and make use of three different moments of inertia used as parameters.
The need to take into account this possibility has been emphasized before
by Wood {\it et al.} \cite{Wood}. We also compare with the situation of irrotational
flow in which a value $\gamma_0$ can be deduced and implies that fluctuations
of the moment of inertia in the $\gamma$ direction are neglected, but
the softness is included. It is interesting to notice that the values of
$\gamma_0$ so obtained correlate well with results extracted from different
methods used to analyze data in the Os nuclei (Section VIII).
We analyze these isotopes, particularly because they are thought
to be situated in the transitional region between $\gamma-$rigid and
$\gamma-$soft shapes. In Section IX, we formulate the main conclusions
of the present study, while in appendix A, we explicitly present the
matrix elements used in the calculations.

\section{Formulation of the model}
The Schr\"odinger equation for the Bohr hamiltonian reads
\begin{equation}
\hat H_B \Psi(\beta,\gamma, \theta_\iota)\=
E  \Psi(\beta,\gamma, \theta_\iota)\,,
\end{equation}
where the hamiltonian is given by
\begin{widetext}
\begin{equation}
\hat H_B\=  -  { \hbar^2\over 2B_m} {1\over \beta^4}{\partial \over
\partial \beta}\beta^4{\partial \over \partial \beta} -
{\hbar^2\over 2B_m} {1\over \beta^2}{1\over \sin{(3\gamma)}}
{\partial \over \partial \gamma}
\sin{(3\gamma)}{\partial \over \partial \gamma}+
{\hbar^2\over 8B_m \beta^2}\sum_{i=1,2,3} {{\hat L}_i^2 \over
\sin^2(\gamma-2\pi\,i/3)} +V(\beta,\gamma)\,.
\end{equation}
\end{widetext}
Here ${\hat L}_i$ are the projections of the angular momentum
($\hat L$) on the body-fixed axes and $B_m$ is the mass parameter.
An exact separation of the variables $\beta$ and $\gamma$ may be
achieved when the potential is chosen as
\begin{equation}
V(\beta, \gamma)\= V_1(\beta)+{V_2(\gamma)\over \beta^2}\,.
\label{pote}
\end{equation}
The resulting set of differential equations (one containing only the $\beta$
variable and the other containing the $\gamma$ variable and the Euler angles,
$\theta_\iota$ with $\iota=1,2,3$),
after multiplication by $2B_m/\hbar^2$, reads
\begin{equation}
\Bigl\{-{1\over \beta^4}{\partial \over
\partial \beta}\beta^4{\partial \over \partial \beta}+u_1(\beta)-
\epsilon+{\omega\over \beta^2}\Bigr\}\xi(\beta)= 0
\label{be}
\end{equation}
$$
\Bigl\{-{1\over \sin{(3\gamma)}}
{\partial \over \partial \gamma}
\sin{(3\gamma)}{\partial \over \partial \gamma}+\sum_{i=1,2,3}
{{\hat L}_i^2 \over 4\sin^2(\gamma-2\pi\,i/3)}$$
\begin{equation}
+u_2(\gamma)-\omega
\Bigr\}\psi(\gamma,\theta_\iota)\=0\,,
\label{g-a}
\end{equation}
where $\omega$ is the separation constant, $\epsilon=(2B_m/\hbar^2)E$
 and $u_t=(2B_m/\hbar^2)V_t$ with
$t=1,2$. The wavefunctions satisfy $ \Psi(\beta,\gamma, \theta_\iota)\=
\xi(\beta)\psi(\gamma,\theta_\iota)$.
In Ref. \cite{LF} the same derivation has been proposed, but the problem
was restricted to deriving the solution of the soft triaxial rotor around
$\gamma\sim\pi/6$. In that case the rotational kinetic term may be
written in a very simple way and the solution of $\gamma-$angular part
may be given in a straightforward way. The main result of Ref.
\cite{LF} was, (i) to extend the Meyer-ter-Vehn formula
\cite{MTV} (strictly valid for a rigid rotor at $\gamma=30^o$) proving
that the corresponding formula for the $\gamma-$soft rotor requires the
addition of a (trivial) harmonic term in the $\gamma$ degree of freedom
and (ii) to show that the solution of the full problem
does not necessarily imply a trivial extension with some additive term
to include the $\beta-$vibrations, but rather yields an expression
for the energy levels in which the quantum numbers are intertwined in a
more complicated way.

Here we extend the results of Ref. \cite{LF}, addressing the more general
problem of a soft triaxial rotor (not confined to $\gamma\sim30^o$),
solving approximatively the $\gamma-$angular equation with a potential
of the form
\begin{equation}
u_2(\gamma)=C(\gamma-\gamma_0)^2=Cx^2\,,
\label{u2}
\end{equation}
that has a minimum in the interval $0^o<\gamma_0< 60^o$.
For symmetry reasons we can
restrict ourselves to the sector $0^o<\gamma_0 \le 30^o$.
The obstacle to the further separation of variables in Eq. (\ref{g-a})
is represented by the rotational term (second term), that mixes the
variable $\gamma$ with the projections of the angular momentum.
This term can be rewritten as $\sum_i A_i \hat L_i^2$.
We will follow two strategies to deal with the coefficients $A_i$.
\begin{itemize}
\item{
The first strategy is to approximate these coefficient as follows
\begin{equation}
A_i={1\over 4 \sin^2(\gamma_0-2\pi\,i/3)}\,,
\label{a-moment}
\end{equation}
and to replace the corresponding terms in Eq. (\ref{g-a}).
The physical meaning of this approximation is that the
fluctuations of the moments of inertia are completely neglected, while
the softness is taken into account. Making use of the form (\ref{a-moment}),
we can parametrise the components of the moment of inertia by
one single parameter.}
\item{
It has been shown in an extension of the rigid rotor model \cite{Wood}
that the components of the moment of inertia fitted to experimental data
do not necessarily agree with the irrotational approximation.
We will therefore include this observation in our model, as
a second strategy, taking the components of the moment of inertia as
parameters.}
\end{itemize}

In the spirit of finding a simple solution, we
introduce in Eq. (\ref{g-a}) the further simplification
$~\sin{3\gamma}\sim  \sin{3\gamma_0}~$, obtaining
\begin{equation}
\Bigl\{ \overbrace{-{\partial^2 \over \partial x^2}+Cx^2}^{\hat D_2}-\omega
+\sum_{i=1,2,3} A_i{\hat L}_i^2 \Bigr\}\psi(x,\theta)=0\,,
\end{equation}
where the variable $x$, introduced in (\ref{u2}), has been used and
where the second order differential operator $\hat D_2$ has been defined.
The wavefunction may be written in the unsymmetrized form
as an expansion in terms of rotational wavefunctions, namely
\begin{equation}
\psi_{L,M}(x,\theta)\=\sum_{K'} a_{K'}^L(x)
{\cal D}^L_{M,K'}(\theta) \,.
\label{expa}
\end{equation}
where $K'$ is introduced to distinguish the multiple occurrence of
states with the same $L$. One may notice that the rotational basis has an
SO(2) symmetry reflected in the good quantum number $K'$.
We now multiply to the left by ${\cal D}^L_{M,K}$ and integrate over the
Euler angles, exploiting the orthonormality property of the Wigner functions
where possible. The result is the set of equations (one for each allowed value
of $K$)
\begin{equation}
({\hat D_2}-\omega) a^L_K(x) +\sum_{i}A_i\sum_{K'}
\langle LMK \mid
\hat L_i^2\mid LMK'\rangle  a^L_{K'}(x)=0\,,
\label{coupled}
\end{equation}
each of which contains matrix elements of the squared components of the
angular momentum (see appendix). We may rewrite it also as
\begin{widetext}
$$\Bigl({\hat D_2}-\omega+\sum_i A_i\langle LMK \mid
\hat L_i^2\mid LMK\rangle \Bigr) a^L_K(x)+$$
\begin{equation}
{(A_1-A_2)\over 4}\langle LMK \mid
\hat L_+^2\mid LMK-2\rangle a^L_{K-2}(x)+{(A_1-A_2)\over 4}\langle LMK \mid
\hat L_-^2\mid LMK+2\rangle a^L_{K+2}(x)=0
\label{undici}
\end{equation}
\end{widetext}
to highlight that only three of the $a$ coefficients appear in every equation.

\section{Algebraic approach}
A convenient way to treat the $\gamma-$angular part is to consider the
dynamic symmetry associated with the $\hat D_2$ operator, that is essentially
a harmonic oscillator hamiltonian. The advantage here lies in the fact
that the spectrum generating algebra allows us to write explicit
expressions for the $\gamma-$vibrational spectrum.

The infinitesimal generators of the sp(2,$\mathbb{R}$) Lie algebra
may be written as
\begin{equation}
\hat Z_1=-{1\over k}{\partial^2\over \partial x^2} \qquad \hat Z_2=k x^2
\qquad \hat Z_3=-i\Bigl({1\over 2} +x{\partial\over \partial x}\Bigr)\,,
\label{op}
\end{equation}
where $k$ is a constant. Similar approaches
have been used in Ref. \cite{RoBa,FV}.
The operators above close under commutation, i.e.:
\begin{equation}
[\hat Z_1,\hat Z_2]=-4i \hat Z_3 \quad
[\hat Z_3,\hat Z_2]=-2i \hat Z_2 \quad
[\hat Z_3,\hat Z_1]=2i \hat Z_1 \,.
\end{equation}
With the linear transformation
\begin{equation}
\hat X_1\={1\over 4} \bigl( \hat Z_1-\hat Z_2 \bigr) \qquad ~
\hat X_2\={1\over 2} \hat Z_3 \qquad ~
\hat X_3\={1\over 4} \bigl( \hat Z_1+\hat Z_2 \bigr)\,,
\label{tra1}
\end{equation}
one may recognize the standard commutation relations of the four isomorphic
Lie algebrae su(1,1)$\sim$so(2,1)$\sim$sp(2,$\mathbb{R}$)$\sim$
sl(2,$\mathbb{R}$)
\begin{equation}
[\hat X_1,\hat X_2]=-i \hat X_3 \qquad ~
[\hat X_2,\hat X_3]=i \hat X_1 \qquad ~
[\hat X_3,\hat X_1]=i \hat X_2.
\end{equation}
It is also very useful to define raising, lowering and weight operators
for this algebra,
\begin{equation}
\hat X_{\pm} = \hat X_1 \pm i \hat X_2 \qquad
\hat X_0 = \hat X_3\,,
\label{tra2}
\end{equation}
that obey the following commutation relations
\begin{equation}
[\hat X_+,\hat X_-]=-2\hat X_0 \qquad [\hat X_0,\hat X_\pm]= \pm \hat X_\pm
\,.
\end{equation}
The action of the above operators on orthonormal basis states for the irreps
of su(1,1) ($\mid n,\lambda \rangle$ with $n=0,1,2,..$)
is given by the equations:
\begin{eqnarray}
&\hat X_+&\mid n\lambda\rangle =\sqrt{(\lambda+n)(n+1)}\mid n+1,\lambda\rangle
 \\
&\hat X_-&\mid n+1,\lambda\rangle =\sqrt{(\lambda+n)(n+1)}\mid n\lambda\rangle
\\
&\hat X_0& \mid n\lambda\rangle ={1\over 2}(\lambda+2n)\mid n\lambda\rangle \,.
~~~~~~~~~~~~~~~\label{diago}
\end{eqnarray}
The values of $\lambda$ are found by comparing the standard
eigenvalue
equation for the Casimir operator of su(1,1) with the eigenvalue equation
for the same operator, but explicitly realized in the terms of the operators
defined in (\ref{op}):
\begin{equation}
\hat \mathbb{C}_2 \mid n\lambda\rangle \=
\Bigl(\hat X_3^2-\hat X_1^2-\hat X_2^2 \Bigr)\mid n\lambda\rangle\=
{1\over 4}\lambda(\lambda-2)\mid n\lambda\rangle\,.
\end{equation}
The action of the Casimir operator on a given function $\phi(x)$ gives
\begin{equation}
\hat \mathbb{C}_2 \mid \phi\rangle=
\Biggl\{ {1\over 8}\Bigl( \hat Z_1\hat Z_2+
\hat Z_2\hat Z_1\Bigr)-
{\hat Z_3^2 \over 4} \Biggr\}\mid \phi\rangle =
-{3 \over 16}\mid \phi \rangle\,.
\end{equation}
Therefore we obtain $\lambda={1/2,3/2}$.
As a consequence of the definitions given in this and in the
previous section, we notice that
\begin{equation}
\hat D_2\=\sqrt{C}(\hat Z_1+\hat Z_2)\=\sqrt{C}4\hat X_0\,,
\end{equation}
where the constant $k$ is defined as $k=\sqrt{C}$. The action of the operator
$4\hat X_0$ is given, for each allowed value of $\lambda$, by $(1+4n)$
and $(3+4n)$ respectively. Combining the two results we may write
that the eigenvalue equation for the $\hat D_2$ operator is given by
\begin{equation}
\hat D_2 \mid n_\gamma \rangle \= \sqrt{C}(1+2n_\gamma)
\mid n_\gamma \rangle \,,
\end{equation}
where we dropped the index $\lambda$ to cover the whole spectrum, and as
$\hat D_2$ is the harmonic oscillator operator in the $\gamma-$degree of
freedom, we can associate $n$ with the number of phonons in the
$\gamma-$variable, $n=n_\gamma$.

\section{Solution of coupled equations,
quasidynamic symmetry, quantum numbers and band structure}
\label{s1}\label{rules}
We discuss here the quantum numbers that appear when solving the
problem presented in the previous section.
The coupled set of equations (\ref{coupled}) contains the rotational
and $\gamma-$vibrational structure through the functions
$a^L_{K'}$  and the operator $\hat D_2$ respectively.
In the absence of $\gamma-$vibrations ($n_\gamma=0$), we are left with the
rotational structure (see Fig. \ref{rig}), which can be analyzed as following.
For each $L$ there is a set of allowed $K$'s: $\{K_i,...,K_f\}$  where
$K_i=0$ and $K_f=L$ for $L$ even (thus we have $L/2+1$ coupled equations)
and $K_i=2$ and $K_f=L-1$ for $L$ odd (giving $(L-1)/2$ equations).

For $\gamma=0$ and $\gamma=\pi/3$ the
projection of the third component of the angular momentum on the
intrinsic axis 3 gives a good quantum number ($K$), while for $\gamma=\pi/6$
the eigenvalue of the projection on the intrinsic axis 1 is a good
quantum number ($R$). In the intermediate regions, none of them may be taken
as a good quantum number.
In the expansion (\ref{expa}) the introduction of $K'$ as a label to
distinguish between the multiple occurrence of states with the same
value of $L$ is justified because it is referring to the rotational states.

It has been observed that in the triaxial region, moving from $\gamma=30^o$
toward $\gamma=0^o$, different groups of states may well
be classified into bands: a first band ($0^+,2^+, 4^+,...$) tends to the
finite axial rotor values and corresponds to $K=0$; a second band
($2^+,3^+,4^+,...$) is identified by its behaviour when $\gamma\rightarrow 0$
(in Fig. \ref{rig} this group of states somewhat
cluster around $\gamma\sim 10^o-12^o$); the beginning of a third band
($4^+,5^+,....$) is seen to escape to infinity at a quicker pace (leaving
Fig. \ref{rig} at around $\gamma\sim 20^o$).

The experimental observation that a classification in $\beta$ and $\gamma$
bands seems an almost universal feature of nuclear spectra reinforces
this choice. The labeling with the $K$ quantum number is often encountered
in the literature, although for what we have said here it may not be
considered adequate. We will in the following use the notation $K^*$ to
regroup the various eigenstates into bands, which are the counterpart of the
bands with good $K$, that are present in the axial cases.

\begin{figure}[!t]
\epsfig{file=data_figura.eps,width=0.48\textwidth}
\caption{Eigenvalues of Eq. (\ref{coupled}) as a function of the asymmetry for
$n_\gamma=0$ (relative to the lowest eigenvalue). In this case the
solution of the rotational part for the $\gamma-$soft triaxial rotor
corresponds to the outcome of the Davydov model \cite{Davy}. In addition,
the soft model
has infinitely many other copies (one for each possible value of $n_\gamma$)
of these sets of bands at higher energies (depending, of course, on the
magnitude of $C$).}
\label{rig}
\end{figure}

For phonons with total angular momentum $2$, one usually takes into
consideration two projections $\mid K\mid =0,2$.
These projections correspond to the so-called
$\beta-$ and $\gamma-$phonons, that generate axial and non-axial
quadrupole oscillations, respectively. Because of the non-axial character of
the $\gamma-$variable, the $\gamma-$vibrations are naturally incorporated
into Eq. (\ref{coupled}). This is obtained through the operator $\hat D_2$,
which adds the $\gamma-$vibrations to the basic structure of rotations of
a rigid triaxial body. Consequently, one can construct $\gamma-$bands
that consist of copies of the structure of the rigid triaxial rotor (Fig.
\ref{rig}), where every copy possesses a different number of
$\gamma-$vibrations, characterized by $n_\gamma$. Once the
rotational and vibrational structure is determined, the eigenvalue $\omega$
can be calculated (in the standard way) and plugged into Eq. \ref{be} to
solve for the $\beta-$part of the problem. An elaborate discussion on the
$\beta-$part and the chosen potentials will be addressed in section VI.

One may describe the present situation in terms of a quasidynamical symmetry
\cite{RoL,RoWi} of a peculiar character: at $\gamma=0^o$ the group
SO(2) is a symmetry of the system, associated with $K$, while at $\gamma=30^o$
another SO(2) group is a symmetry of the system, associated with $R$, being
the chain U(5)$\supset$SO(3) common to the whole sector $0\le\gamma\le \pi/6$.
In the intermediate region, $0<\gamma< \pi/6$, the SO(2) symmetry is broken,
but it must be noticed (see Fig. \ref{rig}) that the structure of the
rotational spectrum, present at $\gamma=0^o$, {\it persists} in the whole
sector without being altered in a dramatic way. Only a smooth and slight
change may be seen. On the other side the structure of the 'maximally'
triaxial rotor at $\gamma=30^o$ {\it persists} also in the region around
$\gamma\sim 20^o-30^o$. In the intermediate region these groups of states
escape to infinity. It must be further noticed that the region where
the various states, that originate from the axial rotor side, are
most affected is exactly the region where the states coming from the
$\gamma=30^o$ triaxial rotor diverge. The strange character of this
quasidynamical symmetry is that (at variance with the cases
discussed by Rowe and collaborators \cite{RoL,RoWi,RoI},
where a true phase transition was present between two exactly solvable
limits associated with different symmetries and different group structures)
here we are dealing with a smooth transition between two limits which
formally have the same underlying group structure, SO(2), and there is no
critical point in between.
Therefore we conclude that the proposed labeling, that retains
the formal division in $\beta$,$\gamma$ and $K$
bands typical of an axial rotor,
is not only justified by the empirical observation that non-axial nuclei
display the same classification in bands, but it is also justified in
view of arguments based on a group theoretical approach.
It is not clear, at present, if a quantization procedure around a tilted
axis may help to shed light on this aspect.

\section{Examples}
\label{exam}
Equation (\ref{coupled}) is solved here in a few cases. Applying the recipe
discussed in Sec. \ref{s1}, the set of differential equations is turned in a
single algebraic equation in $\omega$. When $L=0$, only the value
$K'=0$ is present and therefore Eq. (\ref{coupled}) reduces to
\begin{equation}
\Bigl(\hat D_2-\omega +\sum_i A_i \langle 0\,0\,0\mid
\hat L_i^2\mid 0\,0\,0 \rangle \Bigr) a_0^0(x)=0\,,
\end{equation}
where all the matrix elements are evaluated to be zero and the first
solution is thus $\omega_{L=0,K^*=0,n_\gamma=0} =\sqrt{C}$
and we may write, in
general, $\omega_{L=0,K^*=0,n_\gamma} =\sqrt{C}(1+2n_\gamma)$.

States with $L=1$ are not present in this model.

For $L=2$, the two possibilities are $K'=0,2$, corresponding to
 2 coupled equations,
\begin{eqnarray}
(\hat D_2-\omega+3A_1+3A_2)a^2_0+\sqrt{3}(A_1-A_2)a^2_2=&0 \nonumber\\
\label{ew1} \\
(\hat D_2-\omega+A_1+A_2+4A_3)a^2_2+\sqrt{3}
(A_1-A_2)a^2_0=&0 \nonumber\,,\\
\label{ew2}
\end{eqnarray}
whose solutions are, $ \omega_{L=2,K^*,n_\gamma} =$
\begin{equation}
\sqrt{C}(1+2n_\gamma)+2\sum_iA_i\pm
2\sqrt{\sum_i A_i^2-\sum_{i<j} A_iA_j} \,.
\end{equation}
The rotational parts of these expressions reduce to
the correct values, $3$ and $6$ respectively, when $\gamma=30^o$.

For $L=3$ the only possibility is $K'=2$
and the solution becomes
\begin{equation}
\omega_{L=3,K^*=2,n_\gamma} =\sqrt{C}(1+2n_\gamma)+4A_1+4A_2+4A_3\,.
\end{equation}
Notice that when $\gamma=30^o$ two of the three components of the
moment of inertia ($A_2$ and $A_3$)
are equal to $1$ and the remaining ($A_1$) is $1/4$ so the rotational part
of the energy becomes $9$ (as obtained in refs. \cite{LF} and \cite{MTV}).
In general we have
$\omega_{L=3,K^*=2,n_\gamma} =\sqrt{C}(1+2n_\gamma)+ 4\sum A_i$.

For $L=4$ we can write the determinant of the matrix mentioned in Sec.
\ref{s1} as a third degree equation in $\omega$.
Its three solutions, corresponding to the cases $K^*=0,2,4$, may be
found analytically, although their expressions are rather lengthy.

It is possible to write an algebraic equation in $\omega$ for every value
of $L$, but this equation may be solved analytically only for the lowest
values. We must therefore resort to numerical computation for high values
of the angular momentum.

In Fig. \ref{rig} we plot the value of $\omega$ for various states for
$C=0$ with irrotational moment of inertia. This case correspond to the
well-known rigid rotor solution, that is a particular case of our model
(see Ch. 9 of Ref. \cite{PB}, for example).

Finally we notice that, as a consequence of the definitions given above,
 the following relation holds:
\begin{equation}
\omega_{200}+\omega_{220}=\omega_{000}+\omega_{320}.
\end{equation}
This expression is a generalization of the well-known relation
$E(2^+)+E({2'}^+)=E(3^+)$ \cite{PB}.

\begin{figure}[!t]
\epsfig{file=triaxia.eps,width=0.48\textwidth,clip=}
\caption{(Color online).
Plot of $u(\beta,\gamma)$ as in formula (\ref{pote}) for
$0\le \gamma \le \pi/3$. The potential $u_1(\beta)$ has a Kratzer-like form,
while the potential $u_2(\gamma)$ has a displaced harmonic dependence centered
around $\pi/6$ (for purpose of illustration).}
\label{polare}
\begin{picture}(1,1)(0,0)
\psset{unit=1.pt}
\rput(-30,160){$\beta$}
\psline{->}(-20,155)(-10,150)
\psarc{->}(40,220){35}{0}{25}
\rput(75,210){$\gamma$}
\end{picture}
\end{figure}

\section{Solution of the $\beta-$part}
Once the $\gamma-$part is solved with a particular choice for the moments of
inertia, we can determine the solution of the $\beta-$part inserting
the appropriate values for $\omega$ in Eq. (\ref{be}). With the substitution
$\xi(\beta)=\chi(\beta)\beta^{-2}$, Eq. (\ref{be}) is simplified to its
standard form:
\begin{equation}
{\partial^2 \chi(\beta)\over \partial \beta^2}+\Bigl\{\epsilon-u_1(\beta)
-{(2+\omega)\over \beta^2} \Bigr\} \chi(\beta)=0\,.
\end{equation}
As shown in \cite{LF,FV,FV2} an interesting case is represented by
the Kratzer-like potential (which reduces to a Coulomb-type potential
when $B=0$)
\begin{equation}
u_1(\beta)=-A/\beta+B/\beta^2\,.
\label{u1}
\end{equation}
We depict in Fig. \ref{polare}, as an illustration, the surface
corresponding to the reduced potential $u(\beta,\gamma)=u_1(\beta)+
u_2(\gamma)/\beta^2$ where the potential $u_1(\beta)$ has a Kratzer-like form,
while the potential $u_2(\gamma)$ has a displaced harmonic dependence centered
(as an example) around $\pi/6$. The depth of the pocket centered around the
minimum in $(\beta_0,\gamma_0)$ may be adjusted with the parameters of the
two potentials, $A$, $B$ and $C$.

Following \cite{LF} we can write the solution to equation (\ref{be})
directly as:
\begin{equation}
\epsilon(n_\gamma,n_\beta,L,K^*)={A^2/4 \over \Bigl(\sqrt{9/4+B+
\omega_{L,K^*,n_\gamma}} +1/2+n_\beta \Bigr)^2}\,,
\label{spec1}
\end{equation}
where the values of $\omega_{L,K^*,n_\gamma}$ have been found analytically or
numerically as discussed in the previous section.
Energies are usually redefined fixing the ground state to zero
and using the energy of the first $2^+$ state as unit, namely
\begin{equation}
\bar \epsilon(n_\gamma,n_\beta,L,K^*) = {\epsilon(n_\gamma,n_\beta,L,K^*)-
\epsilon(0,0,0,0)\over \epsilon(0,0,2,0)-
\epsilon(0,0,0,0)} ~~.
\label{spettro}
\end{equation}
Another interesting case that also leads to a solvable differential equation
is the Davidson potential \cite{Elli,Elli2,RoBa}:
\begin{equation}
u_1(\beta)=A_D\beta^2+{B_D \over \beta^2}\,.
\end{equation}
This is an extension of the harmonic potential (that can be obtained when
$B_D=0$).
Inserting this potential in Eq. (\ref{be}) gives formally the Laguerre
differential equation and the spectrum may be written as
\begin{equation}
\epsilon_D(n_\gamma,n_\beta,L,K^*)=\sqrt{A_D} \Bigl( 2n_\beta +
\widetilde \tau_{L,K^*,n_\gamma}+5/2\Bigr) \,,
\label{spec2}
\end{equation}
where, dropping the indices for simplicity,
$\widetilde \tau$ is the solution of $\widetilde \tau
(\widetilde \tau+3)=B_D+\omega$. The redefinition in
reduced energy units takes in this case a very compact form:
\begin{equation}
\bar \epsilon_D(n_\gamma,n_\beta,L,K^*) = {2n_\beta+\widetilde
\tau_{L,K^*,n_\gamma}-\widetilde \tau_{0,0,0}\over \widetilde
\tau_{2,0,0}-\widetilde \tau_{0,0,0}  }.
\end{equation}
The combination of the results for the $\gamma-$part in Sec.
\ref{s1}-\ref{exam} and the results for the $\beta-$part yields
spectra with an ample range of different possible behaviours.

Another interesting case is inspired by the E(5) and X(5) solutions
\cite{Iac1,Iac2}.
Starting again from Eq. (\ref{be}) one can adopt a different substitution,
 namely $\phi(\beta)=\beta^{3/2}\xi(\beta)$, and a change of variables,
$z=\sqrt{\epsilon}\beta$. These transformations, together with the choice
of the
potential as an infinite square well, yield the Bessel differential equation
\begin{equation}
\phi''+{\phi'\over z} +\Bigl(1-{\omega+9/4\over z^2} \Bigr)\phi=0\,.
\end{equation}
The solution of the equation above is given in terms of Bessel J functions
of irrational order
\begin{equation}
\phi_{s,\omega}(\beta)=c_{s,\omega} J_{\sqrt{\omega+9/4}}\Bigl(x_{s,\omega}
{\beta\over\beta_w}\Bigr)\,,
\label{wfnsq}
\end{equation}
where $c_{s,\omega}$ are normalization constants (given analytically in
\cite{compe}).
The boundary condition at the wall of the potential well, $\phi(\beta_w)=0$,
implies that the spectrum is given by
\begin{equation}
\epsilon_{SQ}(n_\beta,n_\gamma,L,K^*)=\Bigl({x_{s,\omega}\over \beta_w}
\Bigr)^2\,.
\end{equation}
Here and in Eq. (\ref{wfnsq}), $x_{s,\omega}$ is the s-th zero of the
Bessel function with index that depends on $\omega_{L,K^*,n_\gamma}$.
Notice that $n_\beta=s$.
The reduced spectrum takes the following form
\begin{equation}
\bar \epsilon_{SQ}(n_\beta,n_\gamma,L,K^*)={x_{s,\omega_{L,K^*,n_\gamma}}^2-
x_{0,\omega_{000}}^2\over x_{0,\omega_{200}}^2-
x_{0,\omega_{000}}^2}\,.
\end{equation}
This solution is somewhat similar to the so-called Z(5) solution \cite{BonZ5}
where an infinite square well in $\beta$ was combined with a harmonic
oscillator centered around $\gamma=30^o$. Major differences are the choice of
exact separation of variables made here, the extension of the solution to
the whole sector $0^o<\gamma<60^o$ and the possibility to
relax the hypothesis of
irrotational motion in deriving for the components of the moments of inertia.

\section{Random walk fitting procedure}
We are now equipped with a procedure to calculate the eigenvalues of
the $\gamma-$part of the problem and we have given a few analytic solutions
(Coulomb and Kratzer, harmonic and Davidson, and infinite square well) for
the $\beta-$part.
The energy spectrum (\ref{spec1}) depends formally on six parameters,
$A,B,C,A_1,A_2,A_3$, three of which come from the potentials (\ref{u1})
and (\ref{u2}), and the other three are the components of the moment
of inertia.
A closer look to Eq. (\ref{spettro}) reveals that, once the eigenspectrum
is scaled in the standard way, it does not really
depend on $A$, while (\ref{spec1}) does. We can now follow the strategy
to keep the irrotational hypothesis and, since the various
components of the moments of inertia are connected to each other by means
of the relations in (\ref{a-moment}), one has to deal with just one component
(or alternatively with $\gamma_0$). The other two components may be
thus determined inverting Eq. (\ref{a-moment}):
\begin{equation}
\gamma_0=arcsin\Bigl({1\over 2\sqrt{A_3}} \Bigr)\,,
\label{gafi}
\end{equation}
and, substituting the value of $\gamma_0$ in the definitions of $A_1$
and $A_2$, we obtain the results
\begin{equation}
A_1\= {4A_3\over \Bigl(\sqrt{12A_3-3}+1\Bigr)^2}\,, \qquad
A_2\= {4A_3\over \Bigl(\sqrt{12A_3-3}-1\Bigr)^2}\,.
\end{equation}
Therefore our model, using the Kratzer potential, has only three parameters,
$B$, $C$ and $A_3$, that may be used to fit experimental energy spectra.

This reduction of parameters, though convenient, may be very restrictive
(see \cite{Wood}) and we prefer to implement also a procedure to keep
all the components as independent parameters, at the price of complicating
the fit to experimental data. We distinguish the fitting procedures
with the words irrotational (3 parameters) and non-irrotational fit (5
parameters).

Similar considerations apply to the energy spectrum of the harmonic/Davidson
potential. In Eq. (\ref{spec2}) six parameters are formally present,
but using the reduced spectrum and the relations among the components
of the moment of inertia, one is lead to a dependence on three of them only
($B_D$, $C$ and $A_3$). Alternatively the non-irrotational fit includes 5
parameters ($B_D$, $C$, $A_1$, $A_2$ and $A_3$).
Likewise, for the square well case, (4) 2 parameters are
present in the (non-)irrotational fit.

Due to the number of parameters we have preferred to use a numerical method
based on a random walk procedure in order to determine the energy spectrum.

As a first step we consider an isotope and a
subset of its experimental levels (usually quite small, 4 levels
typically).
Starting from some initial set of parameters (typically
$B=C=20$ and $A_3=1$), we minimize the deviation of the
calculated and experimental energy values by walking
randomly in a suitable part of the many-dimensional parameter space.
The random walk is initially constrained to take only irrotational
moment of inertia (i.e. $A_1$ and $A_2$ are calculated directly from
$A_3$) and consists of a coarse-grain and a fine-grain stage, each of
which takes a certain number of steps.
The coarse and the fine phases of the procedure differ in the
percentual amount of the initial parameter values which is allowed
to change randomly (50\% and 10\% typically).
We usually end up with a set of parameters that correspond to some
local minimum of our hypersurface (although it must be noticed that
in this way we are not sure to catch the absolute global minimum).
From this set one may calculate a spectrum, which can be compared
with the experimental one and a total error, $\sqrt{\sum_i (1-E_i^{th}/
E_i^{exp})^2}$,
that gives an idea of the accuracy of the description that one
may get. Usually, using as input values energies of the ground and
$\gamma-$bands, the typical accuracy may range from 1\% to 0.1\%.
At this point one is still allowed to identify a value of $\gamma_0$,
calculated from the irrotational moments of inertia (\ref{gafi}), with
the angular position of the minimum.
As a final step we start from the set of parameters resulting from the
irrotational fit, but now we relax the irrotational hypothesis, treating
all the components of the moment of inertia as individual parameters.
Often the higher freedom of this fit reduces the error by a factor
2-10, but sometimes this procedure does not improve substantially
the quality of the fits.
It has to be stressed that at this point the identification with a
single value for $\gamma_0$ does not make sense anymore.

Our goal is, from one side, to give a simple model that goes beyond
the estimate of ground state energies and furnishes a decent
description of ground and $\gamma-$bands, and, from the other side,
allows one to extract a possible behaviour of the potential energy surface
associated with a given isotope.

One must keep in mind that this model
will work only when the potential energy surface has just one minimum.
The fact that, after fitting, some important discrepancies result between
calculated and experimental values may point out that a geometrical
model may not be well applicable after all.

\begin{figure}[!t]
\epsfig{file=fig4.eps,clip=,width=0.48\textwidth}
\caption{Experimental spectrum of $^{188}Os$ (a) and irrotational fit (b)
obtained with $B_D=62.301, C=82.66, A_3=2.3917$. Experimental data
are taken from \cite{NDS}.}
\label{188Os}
\end{figure}

\begin{figure}[!t]
\epsfig{file=fig5.eps,clip=,width=0.48\textwidth}
\caption{Experimental spectrum of $^{192}Os$ (a) and irrotational fit (b)
obtained with $B_D=132.182, C=44.6504, A_3=1.39954$. Experimental data
are taken from \cite{NDS}.}
\label{192Os}
\end{figure}

\section{The osmium isotopes}
We applied the procedure explained in the previous section to the case
of the osmium isotopes with $A=172-180,184-192$.
The irrotational fit with the Davidson potential, starting from the
$4^+$ and $6^+$ levels of the ground state band and the $2^+$ level of the
$K^*=2,n_\gamma=0$ band and the $0^+$ level of the $K^*=0,n_\gamma=1$ band,
leads very quickly to a set
of parameters ${B_D,C,A_3}$ that gives always a rather good description
of the low-lying
positive parity energy spectrum, with an estimate on the total error,
as defined above, which is in the range $0.01-0.09$. In figs.
\ref{188Os},\ref{192Os}
we compare the experimental energy levels of $~^{188,192}$Os (a) with the
irrotational fit (b), obtained from 1600 runs of coarse grained random walk
plus 1600 runs of fine grained random walk. In this case the non-irrotational
fit does not improve in a sizable way the quality of the fit.
 The absolute error in the case of the
$10^+$ state is $\simeq 100~$keV, or less.
In the best cases the accuracy for the $8^+$ and $10^+$ members of
the ground state band may reach the value $1/1000$.
It is observed that the ground-state and excited bands are fairly well
reproduced (which is not surprising since we used the positions of two
levels of each band as an input to our calculations). A bit more surprising
is the qualitative behaviour of other bands: usually we find that the
$(K^*=0,n_\gamma=1,n_\beta=0)$ and $(K^*=4,n_\gamma=0,n_\beta=0)$ bands
lie lower (in a few cases dramatically lower) than the first
$\beta-$band $(K^*=0,n_\gamma=0,n_\beta=1)$. In Fig. (\ref{bh}) we report the
reduced energy of the bandheads of the lowest bands for the whole chain.
In the osmium nuclei with lower mass number,
we find comparable energies
for the $\beta-$band and for the $K^*=0$ band, while the $K^*=4$ lies
at considerably higher energies. This is reflected in the magnitudes
of the $B_D$
and $C$ coefficients: for $^{192}$Os they are respectively, $\sim 130$
and $\sim 44$, but as soon as we move to lower mass numbers the value of $B_D$
drops to less than $1$ for the lightest ones. Thus, the $\beta-$vibrations
are found at much lower energies.

\begin{figure}
\epsfig{file=bands.eps,width=0.48\textwidth,clip=}
\caption{Reduced energies of the bandheads of the $(K^*,n_\gamma,n_\beta)=$
$(2,0,0)$, $(0,1,0)$, $(4,0,0)$ and $(0,0,1)$ bands (indicated with circles,
squares, diamonds and triangles respectively)as obtained from
the irrotational fit procedure for the whole chain of osmium isotopes.}
\label{bh}
\end{figure}

The example displayed in Fig. \ref{188Os} is rather rigid and the
(dimensionless) values of the parameters
are $B_D=62.3$, $C=82.66$ and $A_3=2.3917$. The latter corresponds
to a value of $\gamma_0\simeq 18.86^o$. The energy difference
between the minimum of the harmonic oscillator and the values at one of the
borders of the $\Delta\gamma=60^o$ wedge is roughly
\begin{equation}
{\hbar^2\over 2 B_m} C {(\gamma-\gamma_0)^2 \over \beta_0^2}\sim
{0.44\over \beta_0^2} MeV\,,
\label{Ceq}
\end{equation}
which, for a reasonable value of $\beta_0$ (e.g. values
smaller than 0.3),
is higher than the excitation energy of the higher lying states shown in Fig.
\ref{188Os}: $E_{8^+}\sim 1.5 MeV$.
In order to calculate the value in Eq. (\ref{Ceq}) we used the Bohr-Mottelson
prescription \cite{BM} for the mass coefficient,
$
B_m={1\over \lambda}{3\over 4\pi}A M_N R_0^2\,,
$
where $\lambda=2$, $M_N$ is the nucleon mass and $R_0=r_0A^{1/3}$
is the nuclear radius. Therefore we obtain
\begin{equation}
{\hbar^2\over 2B_m}={4\pi\hbar^2c^2\over3M_Nc^2r_0^2A^{5/3}}\sim
{120.72 MeV\over A^{5/3}}\,.
\end{equation}
This expression, combined with the proper mass number, has been used in
Eq. (\ref{Ceq}).
The evaluation of the depth of the potential well gives an indication
that the wavefunctions of the states that we have fitted are
all well confined inside the well.
In fact the true shape of the potential is thought to depend on
$\gamma$ periodically (terms like $\cos{3\gamma}$ are the most important),
therefore the harmonic oscillator shape can only represent a convenient
approximation which is valid around the minimum.

\begin{figure}[!t]
\epsfig{file=osmium.eps,width=0.48\textwidth,clip=}
\caption{Values of $\gamma_0$ from irrotational fit to various osmium
isotopes (line with circles, lower panel) and corresponding
values of the parameter $C$ (middle panel) and $B_D$ (upper panel).
Predictions obtained in \cite{Werner} with a different method
are also shown (dashed line, lower panel).}
\label{gzero}
\end{figure}

We show in Fig. \ref{gzero} the values of $\gamma_0$ (black dots)
obtained from the irrotational fit for the whole chain of osmium isotopes
that we have analyzed together with the
corresponding values of the parameter $C$ (red squares, middle part)
and $B_D$ (diamonds, upper part). We notice that,
while the value of $\gamma_0$ decreases from the left to the middle and
then increases, the value of the parameter $C$ does not show any regular
behaviour with the mass number. When one considers the extremes of the mass
chain, the rotor becomes softer.
By applying relation (\ref{Ceq}) to the softest case ($^{172}Os$)
one can check that the harmonic approximation is still valid.

In Fig. \ref{gzero}, we also display (dashed line with blue triangles)
predicted effective values of $\gamma$ using $K$ shape invariants
\cite{Werner}. Despite the different technique used to obtain those
values (which are extracted using quadrupole matrix elements and $B(E2)$
values) they agree fairly well with our results. For these values, the
error bars result from errors on the measurement of the transition
rates. In our case an estimate of the theoretical error is quite difficult:
in principle we can repeat the fitting procedure many times and we can extract
a distribution of values of parameters. To understand if our procedure
is reliable we have repeated the fit on $~^{178}$Os 20 times. We have
collected the various sets of parameters and we present in Fig. \ref{accu}
the extracted values of $\gamma_0$. It is therefore possible to obtain an
estimate of the distribution of $\gamma_0$ values and to give mean values and
corrected standard deviations as
a simple estimate for the theoretical error.

\begin{figure}[!t]
\epsfig{file=accu.eps,width=0.48\textwidth,clip=}
\caption{(Color online).
Values of $\gamma_0$ (black dots) and of the parameter C (red
squares) extracted from 20 independent fits to $~^{178}$Os.
From these data one obtains an estimate of the theoretical
error associated with the fitting procedure described in the text.
Mean values, $\langle \gamma_0 \rangle$,
and mean values $\pm$ the corrected standard deviation,
$(\langle \gamma_0 \rangle\pm \sigma_{\langle \gamma_0 \rangle})$,
are indicated with dashed and dotted lines for both quantities,
respectively.}
\label{accu}
\end{figure}

Hartree-Fock-Bogoliubov calculations have been
carried out (Ref. \cite{Ansari}) in the Os-Pt region with results pointing
toward a prolate structure with $\gamma_0$ very close to $0^o$ and with a
flat potential in $\gamma$ (for the Os nuclei  with $A=186-192$).
In such self-consistent calculations the potential energy surface is
calculated variationally, starting from a given effective interaction.
In most cases no dynamical calculations are done on top of that and
energies are often associated with the exact
value of the minimum of this surface. Likewise, a few excited states can
also be predicted.
Our approach starts from a given mathematical expression for the potential
$V(\beta,\gamma)$, which is then used to solve the collective model and can
predict full energy spectra.
However, the particular choice of $V(\beta,\gamma)$ made at present may
still deviate from actual, more realistic, potential energy surfaces.
Besides, indications exist for ground state hexadecapole deformation
in the Pt-Os region, as suggested by
strong E4 coupling arising in $(\alpha,\alpha')$ and $(p,p')$ reactions.

Very recently a description of Os isotopes (among others) has been proposed
within the IBM-1 \cite{Mac}. The authors fit the low-energy positive-parity
spectra obtaining a good overall description. Both their work and ours,
which rely on models based on different ingredients, describe
equally well a good fraction of the spectral properties of Os isotopes,
although a number of differences clearly leave room for more detailed studies.
Another analysis of quadrupole moments, transition and transfer rates
of (platinum and) osmium isotopes in terms of IBM-2 \cite{bij} shows
that a reasonably good agreement with experimental data can be obtained with a
smoothly varying set of parameters.

In our present study to construct solutions to the Bohr hamiltonian for soft
triaxial nuclei, the model parameters used determine (i) the moments of
inertia (through the parameter $A_3$, or $A_i$ for the more general fit),
(ii) the stiffness parameter $C$ for the $\gamma-$oscillatory motion and
(iii) $B_D$ and $A_D$ determining the Davidson potential. There appear rather
large variations in particular in $B_D$ and $C$; however, the deduced value
of the quantity $\gamma_0$ (see fig. \ref{gzero} - lower part) still
exhibits a very smooth variation and agrees very well with results using a
totally different approach \cite{Werner}. We stress that the above variations
do not at all contradict the fact that smooth variations in the parameters
for the IBM-2 hamiltonian result in describing the observed data in the Os
nuclei, because totally different models are used.

\begin{figure}[!t]
\epsfig{file=quadmom.eps,width=0.48\textwidth,clip=}
\caption{(Color online). Electric quadrupole moments for osmium isotopes in
$eb$. Data (red squares with error bars) are taken from Ref. \cite{Ragh}.
The calculated values (black dots) have been obtained by adjusting an
additional parameter to reproduce the excitation energy of the first $2^+$
state.}
\label{qmom}
\end{figure}

In order to test even further our model, we present in fig. \ref{qmom} a
comparison of calculated and experimental static quadrupole moments of the
 first excited $2^+$ state. The definition of the quadrupole moment adapted
to the present case is:
\begin{equation}
Q=\sqrt{16\pi\over 5} \biggl( {2 \atop 2} {2 \atop 0} {2\atop -2}\biggr)
{3ZR_0^2 \over 4\pi} \langle 2_1^+ \mid\mid \alpha_2^*\mid\mid2_1^+\rangle
\end{equation}
where the reduced matrix element, once we exploit the trigonometrical
simplification for the $\gamma$ part and we insert the values of the
Clebsch-Gordan coefficients, is calculated to be
$$
\langle 2_1^+ \mid\mid \alpha_2^*\mid\mid2_1^+\rangle=\langle \xi_0(\beta)
\mid\beta\mid \xi_0(\beta)\rangle {10\over \sqrt{70}} \bigl( 1-
{1\over 4C}\bigr) $$
\begin{equation}
\bigl[ \cos{\gamma_0} (a_2^2-a_0^2)+2a_0^2a_0^2\sin{\gamma_0}\bigr] \,.
\end{equation}
The last expression still contains the matrix element of $\beta$ that
depends on the Davidson ground state wavefunction, which in turn contains
the parameter $A_D$. This parameter can be fixed by requiring that
$\epsilon_D(2^+)-\epsilon_D(0^+)$ matches with the absolute energy difference
in MeV. The results collected in fig. \ref{qmom}, are correct both in sign
and trend, but they underestimate a bit the measured values \cite{Ragh}.
A more complete study, including $B(E2)$ values, is in progress.

\section{Conclusions}
In the present paper, we have presented a method in order to solve
the Bohr-Mottelson collective model for a general soft triaxial nucleus.
By choosing a potential of the form $V(\beta,\gamma)= V_1(\beta) +
V_2(\gamma)/\beta^2$,  the Bohr Hamiltonian separates exactly in the
$\beta$ and $\gamma$ variables. Using a displaced harmonic oscillator
potential in the $\gamma$ direction, representing the softness in
the $\gamma$ variable, allows us to solve approximately the $\gamma$-angular
equation. In solving that part of the problem, we discuss two
possibilities to further separate the $\gamma$ variable from the
projections of the angular momentum. One approximation neglects
fluctuations of the moments of inertia in the $\gamma$ direction, but
keeps the softness. Under this assumptions, the three components of the
moment of inertia are related and result in one parameter.
The other approximation allows a fitting of the three components of the
moment of inertia independently. This is in line with the observation
that the experimental data do not agree so well with the assumption of
irrotational flow. We then present an algebraic method in order to find
approximate solutions to the set of coupled equations that result for
the motion in the $\gamma$ variable and also discuss at some length the
state labeling problem in order to characterize the various collective
bands that result. We subsequently study the equation for the $\beta$
degree of freedom, which can be solved exactly for a number of interesting
potentials. We thus consider the Coulomb/Kratzer potential, the Davidson
potential (which is an extension of the harmonic oscillator potential
in $\beta$) as well as the infinite square-well potential in our calculations.
In each case, we are able to derive analytic expressions, which are then
used to determine the full energy spectra. These energy spectra, that
describe oscillatory behavior in both the $\beta$ and $\gamma$ variables,
as well as the rotational motion, in general depends on 6 parameters:
three that characterize the $V(\beta,\gamma)$ potential and three that
determine the moment of inertia. Using appropriate scaling in the energy
spectrum only 5 remain to be determined. Going back to irrotational motion,
the three components of the moment of inertia are related and this reduces
the full set to just 3 parameters. Due to the number of parameters and the
very involved energy eigenvalue expression, we have used a numerical method
based on a random walk procedure in order to obtain the optimal fitting
parameter set. We have used this method to study the Osmium isotopes in
the interval $172 \le A \le 192$. One of the results is the fact that the
equilibrium $\gamma_0$ value shows a rather smooth dependence on $A$,
but the stiffness of the potential in the $\gamma$ variable indicates
rigid cases for intermediate masses and softer cases for isotopes
sitting at the extremes of the considered mass chain.
To elucidate this point it would be very helpful to search for solutions
of Eq. (\ref{g-a}) with more involved periodic potentials \cite{Stij}.
Still, the calculated values of
$\gamma_0$ are in rather good agreement with an independent approach
to extract an effective $\gamma$ value by Werner {\it et al.}.
We have in mind to study the rare-earth region in a more systematic
way using the methods presented here.

Recently other works have appeared that treat similar problems, yet
with different spirit and aims. Caprio \cite{CapNew} analyzes the numerical
diagonalization of a $\beta-$soft, $\gamma-$stabilized problem, evidencing
how the approximate separation of variables of the X(5) model may be questioned
on the basis of a strong $\beta-\gamma$ coupling. Despite the fact that
our approach is complementary (rather than solving exactly the numerical
problem, we postulate analytically solvable potentials and we try to see
if they can be profitably used), we reach similar conclusions. For example
we also find that a considerable degree of dynamical $\gamma-$softness is
needed (for realistic cases of $\gamma$ stiffness) to account for the energies
of the $\gamma-$bands' levels. Furthermore our method allows
 to look for the optimal value of the actual moments of inertia. A point of
difference is, instead, the pattern for the staggering: in the examples below
we rather find a scheme of the type $(2^+,3^+),(4^+,5^+),...$ similar to that
of the rigid triaxial rotor. Another very important series of paper by Rowe
and collaborators \cite{Rowe-new,RoI,RoTuRe} furnishes a rapidly
converging method for exact numerical treatment of the problem by using a
basis defined in terms of ``deformed'' wavefunctions in $\beta$ and
five-dimensional spherical harmonics.

~\\
The authors are most grateful to J.Wood for extensive discussions and
L.F. wishes to thank A.Vitturi for profound comments.
Financial support from the "FWO Vlaanderen" (L.F. and K.H.) and the
University of Ghent (S.D.B. and K.H.), that made this research possible,
is acknowledged.

\section*{APPENDIX: Matrix elements}
List of non-null matrix elements of operators as discussed in Eq.
(\ref{undici}) of Sect. II.
\begin{equation}
\langle LMK\mid \hat L_3^2 \mid LMK'\rangle =K^2 ~~~\mbox{~~if~~} K=K'
\end{equation}

\begin{widetext}
\begin{equation}
\langle LMK\mid \hat L_1^2 \mid LMK'\rangle =
\left\{
\begin{array}{lr}
\displaystyle
{L(L+1)-K^2\over 2} &~~~\mbox{~~if~~} K'=K\\
\displaystyle
{\langle LMK\mid \hat L_+^2 \mid LMK'\rangle \over 4}&~~~\mbox{~~if~~} K'=K-2\\
\displaystyle
{\langle LMK\mid \hat L_-^2 \mid LMK'\rangle \over 4}&~~~\mbox{~~if~~} K'=K+2\\
\end{array}
\right.
\end{equation}

\begin{equation}
\langle LMK\mid \hat L_2^2 \mid LMK'\rangle =
\left\{
\begin{array}{lr}
\displaystyle
{L(L+1)-K^2\over 2} &~~~\mbox{~~if~~} K'=K\\
\displaystyle
-{\langle LMK\mid \hat L_+^2 \mid LMK'\rangle \over 4}&~~~\mbox{~~if~~}
K'=K-2\\
\displaystyle
-{\langle LMK\mid \hat L_-^2 \mid LMK'\rangle \over 4}&~~~\mbox{~~if~~}
K'=K+2\\
\end{array}
\right.
\end{equation}
where
\begin{equation}
\langle LMK\mid \hat L_+^2 \mid LM(K-2)\rangle=
\sqrt{(L+K)(L+K-1)(L-K+2)(L-K+1)}
\end{equation}
and
\begin{equation}
\langle LMK\mid \hat L_-^2 \mid LM(K+2)\rangle=
\sqrt{(L+K+2)(L+K+1)(L-K-1)(L-K)} ~\,.
\end{equation}
\end{widetext}

redpajama_set_name: RedPajamaArXiv


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Hearing loss can cause many difficulties and often hearing aids in isolation are not enough. We have extensive experience in helping patients to understand the nature of their hearing loss and how listening can be improved by utilising other tactics and strategies. We are also able to offer advice and support with other assistive technology which may be able to help in situations you struggle in. These may be in meetings, on the phone or hearing the television. Hearing aids are not always able to resolve all difficulties an individual may be experiencing and other options need to be investigated. In these instances, hearing therapy may be useful.


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WEST LAFAYETTE, Ind. — The University of Iowa baseball team won its 20th game of the season and third straight Big Ten Series after splitting a Saturday doubleheader with Purdue at Alexander Field.
The Hawkeyes (20-14, 7-5) dropped game one, 9-5, to the Boilermakers (11-23, 5-6), but rebounded with a 6-3 win in the series finale.
"I'm happy to get out of here with a series win and start getting ready for next week," head coach Rick Heller said.
Game one saw a 4-0 Purdue lead disappear in the sixth when the Hawkeyes took a 5-4 lead. However, the Boilermakers responded with a five-run seventh to down Iowa, 9-5.
In the first two innings, sophomore starter Cam Baumann allowed six hits and four runs. The lefty from Fairfield, Iowa, battled back, surrendering only two hits and zero runs in his final three innings.
Once Baumann found his groove, the Iowa offense started to piece together a comeback. In the fourth, senior Chris Whelan scored on a double play after leading the inning off with a walk. Iowa tied the game at four in the fifth with three runs, including a two-run home run from senior Tanner Wetrich.
Wetrich then had the go-ahead RBI single in the top of the sixth that knocked in junior Ben Norman. Iowa's 5-4 lead was short lived.
Relievers Duncan Davitt and Trace Hoffman struggled in the seventh, allowing five runs on four hits and three walks.
Davitt walked the first two batters he faced before giving up a two-run triple, which forced Heller to go to the bullpen with Hoffman. The Boilermakers followed with back-to-back singles, including a back-breaker to pinch-hitter Milo Beam that scored the fourth and fifth runs of the inning.
Iowa had two runners on with one out in the eighth, but Whelan grounded into a 5-3 double play to end any comeback threat.
Game three was the series' most even matchup with the game being knotted at two until the seventh inning. After the first two innings, both starting pitchers found a groove until Iowa scored four runs in the eighth inning.
Junior Grant Judkins (4-3) was the winning pitcher, while Ryan Beard (0-5), who exited the game with an apparent leg injury, was the tough-luck loser. Judkins retired 10 straight batters from the second to the fifth inning and left in the eighth after recording two outs.
Eight of Iowa's 10 hits and all of Iowa's six runs on the offensive end were recorded in two innings – the second and the eighth.
In the second, the Hawkeyes scored two runs with two outs. Senior Luke Farley fouled off a few pitches before hitting an opposite field single to right field that scored sophomore Izaya Fullard. Next, junior Matthew Sosa bunted down the third base line and beat out the throw, plating senior Tanner Wetrich.
Iowa's four-run eighth inning started with three straight singles with junior Austin Martin drove in two runs. It was Martin's first and only hit of the series.
Following a Fullard fly out, junior Ben Norman smoked his team-leading fifth home run over the right center field fence, extending Iowa's lead to 6-2.
Judkins gave way to closer Grant Leonard in the eighth, who allowed one inherited runner to score. Leonard then struck out the final three batters in the ninth to convert his eighth save in as many chances.
"It was good to see in game two, a tight ball game, until late. We score four in the eighth to take the lead, but Purdue never went away. We got a great start out of Grant Judkins in game two and then Grant Leonard came in and did a nice job closing it out.
Iowa improves to 6-6 in six doubleheaders this season.
Iowa committed only one error in three games this weekend.
In two starts this weekend, Luke Farley was 3-for-9 with two RBIs.
With six RBIs this weekend, Ben Norman extended his team-leading RBI total to 25.
Grant Leonard's 10 career saves are ninth most in Hawkeye history.
Grant Judkins' start in game two was only the second time in nine starts he has allowed three or more earned runs.
On Tuesday, Iowa hosts Northern Illinois at Duane Banks Field with a 6:05 p.m. (CT) first pitch.


text:
Gig Preview: Kate Tempest at Brighton Dome
Gig Review: Alice Cooper at Br..
Critically acclaimed British spoken word artist, rapper, poet, novelist and playwright Kate Tempest released her third studio album, The Book of Traps & Lessons, on June 14th via Fiction/American Recordings. The album was produced by legendary producer Rick Rubin and Tempest's longtime collaborator Dan Carey. This is the first time Kate and Dan have worked with the legendary US producer. What has transpired is an incredible piece of work that takes the listener on a 45:02 minute journey of highs, lows and core bearing emotions. The tracks comprise a narrative thread and are intended to be listened to in one sitting.
The first track from the album, entitled 'Firesmoke' was premiered on BBC 6Music. Of "Firesmoke", Kate says, "I think of the song like a cleansing moment in the wider context of the record, in the same way that a relationship can be a cleansing moment in the wider context of a life. A kind of a turning point. I wrote it for my woman. Inspired by her and everything that falling in love with her has taught me."
Since her emergence in 2011 the London-born artist has redefined what it means to be a wordsmith in the Modern Age. To date, she has published three poetry collections, one book-length poem, a novel, staged three plays, and released two solo studio albums—Everybody Down (shortlisted for the 2014 Mercury Prize), and Let Them Eat Chaos (shortlisted for the 2017 Mercury Prize). Along the way, she entranced audiences on The Tonight Show Starring Jimmy Fallon, NPR's "Tiny Desk," and more. Not to mention, she garnered widespread critical acclaim from The New York Times, The New Yorker, Forbes, and more, to name a few.
Kate's UK tour kicked off last night at Oxford's O2 Academy and includes a show at the Eventim Apollo in Hammersmith tonight which will be her biggest headline show to date. Kate also visits the South Coast this Saturday (19th) with a date at the Brighton Dome.
UK TOUR DATES
14th – O2 Academy, Oxford
15th – Eventim Apollo, London
16th– O2 Institute, Birmingham
18th– O2 Academy 2, Leicester
19th– Dome, Brighton
20th – O2 Academy, Bristol
8th – LCR, Norwich
9th– O2 Academy, Sheffield
10th – Albert Hall, Manchester
12th – Belgrave Music Hall, Leeds
13th – Tramshed, Cardiff
16th – The Old Fruitmarket, Glasgow
17th – Boiler Shop, Newcastle
Tickets are available via www.katetempest.co.uk/tour
Website: katetempest.co.uk
Facebook: facebook.com/katetempest
Twitter: @katetempest
Instagram: @katetempest
Spotify: Kate Tempest
Apple Music: Kate Tempest
Photos and words courtesy of Chuff Media
Brighton, Brighton Dome, kate tempest, Novelist, Playwright, Poet, Rapper, The Book of Traps & Lessons, UK Tour
Russ Leggatt

redpajama_set_name: RedPajamaCommonCrawl


text:
Home Football Joan Laporta hints at strong Barcelona interest in Premier League midfielder
Joan Laporta hints at strong Barcelona interest in Premier League midfielder
Barcelona president Joan Laporta has hinted at major interest in Wolves midfielders Ruben Neves, appearing to suggest the club will dip into the transfer market depending on what happens with captain Sergio Busquets at the end of the season.
Busquets is out of contract in the summer a couple of weeks before he turns 35 and has been heavily linked with a switch to Inter Miami. He has previously set a February deadline for deciding whether he will be staying at Camp Nou or moving on.
If Busquets does leave, Barça will need to replace a player who has anchored their midfield since 2009 – Franck Kessie is tipped to depart and the squad's other midfielder play different roles.
Neves has spent nearly five years with Wolves since making a surprise switch from Porto in 2017, but is widely regarded as one of the best midfielders in the Premier League.
"Ruben Neves is a top player but it's part of private conversations. Busquets has to decide his future, he has proposals from MLS and so we're exploring options for the future," Laporta told Cadena SER.
"Xavi will decide who's the right player for us."
The interest seems like it would be reciprocated by Neves.
"If you ask in general, all players want to play for Barca!" he told Canal 11 in October. "It's one of the best clubs in the world. It's a privilege for me to be linked to this team."
Listen now as Scott Saunders hosts Graeme Bailey and Toby Cudworth in the latest episode of Talking Transfers. This week they discuss Man Utd's interest in Mohammed Kudus, Enzo Fernandez and Chelsea, Jude Bellingham's decision and Mykhaylo Mudryk's preferred move. Available on all audio platforms.
If you can't see this embed, click here to listen to the podcast!
Having previously been Porto captain at a young age, Neves has been ready to leave Wolves for some time in a bid to explore a new challenge and play at a higher level.
90min reported in April of last season that, in addition to Barcelona, there was interest in Neves from Arsenal, Tottenham, Atletico Madrid and Manchester United, although the latter went on to sign Christian Eriksen and Casemiro during the summer.


text:
De Grand Prix schansspringen 2018 ging op 20 juli 2018 van start in het Poolse Wisła en eindigde op 3 oktober 2018 in het Duitse Klingenthal. De Grand Prix voor mannen bestond dit seizoen uit elf individuele wedstrijden en een wedstrijd voor landenteams. De Rus Jevgeni Klimov wist de Grand Prix op zijn naam te schrijven. De Grand Prix voor vrouwen bestond dit seizoen uit zes individuele wedstrijden. De Japanse Sara Takanashi won de Grand Prix voor vrouwen.

Mannen

Kalender

Eindstand

Vrouwen

Kalender

Eindstand

Gemengd

Kalender

Externe links
Mannen

Vrouwen

Grand Prix schansspringen
Schansspringen in 2018

redpajama_set_name: RedPajamaWikipedia


text:
A Little House in Tijuana = A New Life for A Single Mom of Three
· Living in a real house has changed Mari's life.
· She has a job and is studying to be a nurse, two things she never thought she would be able to do.
· Her children sleep safe and warm in their own beds. Many of her old worries are a distant memory.
By: Jenna Montgomery, IRT Staff
When I first met Mari two years ago, she was living with her three children in a tiny two-room shelter with dirt floors and a tarp roof on the outskirts of Tijuana, Mexico.
Now, thanks to you, Mari is living in a house made of four sturdy walls, a roof with shingles, a concrete floor, and three separate living areas, including a loft. Twenty five volunteers built this house for Mari in one day back in 2016. I wanted to see how her life had changed since then, so I paid her a visit.
The last time I saw Mari, on the day she got her house, she was full of hope. Full of hope for her future and for her children's futures. All she wanted back then was a safe place for her kids to sleep at night and a clean place for them to study and eat. I am so happy to report that these dreams have come true, even beyond her own expectations.
Sitting in Mari's kitchen, with her kids and their cousins on the sofa watching TV, Mari and I talked about her new life. Mari explained that she didn't realize all the ways having a real house would change her life.
"When we lived in the old house, we did not have a door. We didn't have much, but I was so afraid everything would get stolen if we left," said Mari. "Now, we have a door with a lock. This little lock has given me my freedom. I am able to leave the house without worry."
Emmanuel, Kevin, and Dulce Abril stand in front of the house they love so much. Dulce chose the purple paint, her favorite color!
With Mari's new found freedom, she enrolled in nursing school and found a job at a health clinic as a receptionist. Before, all she could do for money was odd jobs for her neighbors because she couldn't stray far from her home. She struggled to pay for food and school uniforms for her children.
She doesn't worry about money like she used to, or worry about the rain pouring into the house at night while they sleep. Their little family of four is very content and comfortable in their new home.
Mari and her children stand in the doorway of the makeshift shelter they lived in before IRT built their new home.
Mari's old "house", which has now been torn down, was so small that the four of them slept together every night on the bottom bunk of a set of bunk beds. She used the top bunk to store their clothes. Also, sleeping on the bottom bunk protected them from the rain, which would come in through the roof and drench all of their belongings, and the dirt floor would turn to mud.
Mari says the kids are so happy having more space. They love having their friends and their cousins over to a home they feel proud of. Mari also enjoys inviting her friends and family over for meals. Her family from other parts of the country can visit, now that there is space for them to stay.
"My family is everything. I am so much happier now that I get to see them more."
"Thank you for my new life," Mari said when we said good bye. "This house ensures my children are safe and healthy. What more could a mom want?"
NEW HOMES, NEW BEGINNINGS
"I Never Thought My Children Would Have a Home"
Adriana's Dream Brings Tears of Joy
A Life-Changing Gift
"We Will Have Roots"
Welcome to Fuentes del Valle
A Home of their own
(VIDEO) Life in Fuentes del Valle
The Best Day of Mari's Life
IN PICTURES: How can water change this one community?
A Poor Family's Leap of Faith for a Brighter Future:
We Must Not Ignore Our Struggling Neighbors in Mexico
Rosario's House of Hope


text:
Los Angeles Teacher Ratings
See 2009 rating
Geoff E. Jamora
A 5th grade teacher at Madison Elementary in 2010
These graphs show a teacher's "value-added" rating based on his or her students' progress on the California Standards Tests in math and English. The Times' analysis used all valid student scores available for this teacher from the 2003-04 through 2009-10 academic years. The value-added scores reflect a teacher's effectiveness at raising standardized test scores and, as such, capture only one aspect of a teacher's work.
Math effectiveness
Least effective
English effectiveness
See how this teacher would change under different statistical models »
The red lines show The Times' value-added estimates for this teacher. Jamora falls within the "more effective than average" category of district teachers in math and within the "more effective than average" category in English. These ratings were calculated based on test scores from 161 students.
Because this is a statistical measure, each score has a degree of uncertainty. The shading represents the range of values within which Jamora's actual effectiveness score is most likely to fall. The score is most likely to be in the center of the shaded area, near the red line, and less likely in the lightly shaded area. Teachers with ratings based on a small number of student test scores will a have wider shaded range.
The beige area shows how the district's 11,500 elementary school teachers are distributed across the categories.
Jamora's LAUSD teaching history
Years used for value-added rating. See FAQ for details.
Madison Elementary, 2010 - 2006
Victoria Avenue Elementary, 2005 - 2004
The Times gave LAUSD elementary school teachers rated in this database the opportunity to preview their value-added evaluations and publicly respond. Some issues raised by teachers may be addressed in the FAQ. Teachers who have not commented may do so by contacting The Times.
Source: The Los Angeles Times, Los Angeles Unified School District, Richard Buddin
Credits: Jason Felch, Stephanie Ferrell, Megan Garvey, Thomas Suh Lauder, David Lauter, Julie Marquis, Anthony Pesce, Sandra Poindexter, Ken Schwencke, Beth Shuster, Jason Song, Doug Smith
Los Angeles Teacher Ratings, the Los Angeles Times' database of value-added scores for Los Angeles Unified elementary schools and teachers.
» FAQ & About
» Read series
» Methodology
» Compare the models
Find a teacher...
Or, find a school
Teacher responses
Joseph J. Matthews
"Both of these charts seem to display normal distribution, yet in both cases you put the average ranking left of center. Why is this? What...
Ada B. Okoro
"First of all, I was a Math Coach from 2001 to 2005. I came back to the classroom July 1,2005. In order to smear my reputation or get as...
Karina J. Trinidad
"Unfortunately, test scores are the only thing that the public has at hand. I personally feel that so many important factors cannot be...
See all responses »
Six-figure L.A. city pensions
I C U: L.A. Connections
Prop 19: Contributions


text:
Hot N' Happening
Every story must have a message: Filmmaker Vinod Kapri
National Award-winning filmmaker Vinod Kapri, whose forthcoming movie "Pihu" centres on a two-year-old child left alone by her parents, believes all stories should leave their audience with a message.
Published: Monday,Nov 12, 2018 16:49 PM GMT-07:00
"Pihu" was screened for politicians here on Friday.
"For me, every story should have a message to convey. Pihu is a story of every household and this was a very important story to tell. The most difficult part was to bring this story to the audience and I thank (producer) Siddharth Roy Kapur and Ronnie Screwvala for supporting me and my film. People who love their family should go a watch this film."
The film's narrative is about a 2-year-old girl trapped in an apartment alone. It has just this one character throughout the narrative.
Kapur, who has produced the movie, it is this uniqueness that drew him to get associated with the project.
"He (Kapri) has kept it so gripping, so edge of the seat... It's like a thriller but a very emotional film at the same time and I think our audience will find it a very unique experience. Ronnie Screwvala and I both watched it and decided that this is something we would definitely have to back."
The special screening of the film here saw Union Science and Technology Minister Harsh Vardhan Harsh Vardha, Congress leader Rajeev Shukla, BJP's Delhi chief Manoj Tiwari, Rajya Sabha MP Amar Singh and more in attendance.
According to a statement, Shukla said after the screening: "I was not expecting that this film will be such a shocker; it takes time for you to absorb the incidents happening in the film. The girl has done a fantastic job and it is worth showing on an international platform."
"Pihu", starring Myra Vishwakarma, will release on November 16.
Ronnie Screwvala Amar Singh Manoj Tiwari Siddharth Roy Kapur dg Siddharth
Siddharth Roy Kapur gets featured in 500 Most Influential Leaders List ...
Siddharth wishes rumoured girlfriend Aditi Rao Hydari on her birthday ...


To ensure scientific publications are of the highest quality, the Committee on Publication Ethics (COPE) have established a series of international ethical standards and guidelines. These guidelines provide advice to editors and publishers on all aspects of publication ethics and, in particular, how to handle cases of research and publication misconduct. COPE also acts as a forum for editors and publishers of peer-reviewed journals to discuss all aspects of publication ethics.
Dove Medical Press requires authors to confirm that all patients or participants (or that person's parent or legal guardian, if the study participant is under 18 years of age) of any research, clinical trials or case studies have provided written informed consent for their details to be used in the manuscript. Similarly, any manuscript containing data, images or videos relating to an individual or group of individuals must have written patient consent for the publication of these details. Authors must also ensure all identifying patient data is anonymized and any nonessential identifying information should be omitted from the article. This includes but is not limited to the patient's name, address, place of employment, date of birth, telephone number, email address, medical records and patient ID, account numbers, license number, IP address and full or comparable photographic image or video. Dove Medical Press instructs all authors to read and follow the ICMJE guidelines for the Protection of Research Participants.
Where informed consent has been obtained, it must be indicated in the published article and authors must be able to provide details of this if requested by the editor. The forms must contain all the elements generally expected to be included in a patient consent form. The WHO provides useful informed consent form templates which can be adapted by investigators to suit the needs of their specific studies. It is also the responsibility of the authors to ensure they have adequately archived patient consent forms and inform the journal of this. If any doubt exists about patient identification the Publisher at Dove Medical Press will be the final arbiter.
All planning conduct and reporting of human research must be in accordance with the Declaration of Helsinki. This includes any study involving human subjects, medical records and human tissues. Authors should also seek the approval to conduct research from an independent local, regional or national review board (e.g. Institutional Review Board (IRB) or ethics committee). Any considerations or approvals made by this committee must not reduce or eliminate the protections for research subjects set out by the Declaration of Helsinki. If doubt remains whether the research was conducted in accordance with the Declaration of Helsinki, the author must demonstrate the rationale for the independent local, regional or national review boards decision. A statement declaring review board approval must be included in the manuscript.
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In some instances, a case report or case series containing information on less than three patients may not require IRB approval. This is because a case series of less than three subjects may not be considered generalized knowledge or research. However, this requirement is dependant on the institution, country or region implementing it and authors must ensure they have followed the correct regulatory requirements of their respective institution or country. Outside this requirement, all case reports, case series or clinical datasets must have IRB approval and authors must be aware that IRB approval cannot be given retroactively. A statement declaring IRB approval must be included in the manuscript. Details of Journals' Best Practices for Ensuring Consent for Publishing Medical Case Reports can be found on COPEs website.
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A: For those investigators who do not have access to formal ethics review committees, the principles outlined in the Declaration of Helsinki should be followed. If the study is judged exempt, a statement from the ethics committee is required. A statement should be included in the Materials and Methods section of your manuscript, regarding the decision.
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A: No. For studies using client-owned animals, a high standard (best practice) of veterinary care and involves informed client consent statement should be included in the Materials and Methods section.
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A: Yes. You must specify in the Materials and Methods section of your manuscript, the ethical review committee approval, and the international, national, and/or institutional guidelines followed regarding the animals welfare.
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A: No. However, if cells have been gifted, or not sourced from an accredited commercial provider, we require confirmation that the research has been approved by a named institutional review board or ethics committee.


Treat your pooch to a late summertime dip at one of the City of Alexandria's Annual Doggie Day Swims. For the first time, dogs under 25 pounds can enjoy a pool to themselves at Itty Bitty Doggie Dive on Sunday, September 13, from 10 a.m. to 2 p.m. at Charles Houston Recreation Center's Memorial Pool, 901 Wythe St. Dogs 25 pounds and up can enjoy Big Dog Day Swim on Saturday, September 19 from 10 a.m. to 2 p.m. at Old Town Pool, 1609 Cameron St.
Following the end of the summer pool season, this annual event provides dogs with the opportunity to swim and play games in one of the City's pools. This year, due to the generous donation by Your Dogs Best Friend, admission is free! Register Online Now to reserve your dog (Itty Bitty Doggie Dive: activity number 134390 or Big Dog Day Swim: activity number 134391).
Dog handlers must be 16 years of age or older, and cannot swim or enter the pool with their dogs. Dogs must be sociable and remain leashed while on the pool deck. All dogs should wear current vaccination tags or must have a current rabies certificate or letter from their veterinary stating that they have received the shots in order to gain entry to the event. Space is limited to 100 dogs in the pool area at one time.
For additional information, contact Ralph Baird, Aquatics Supervisor, at 703.746.5435 and ralph.baird@alexandriava.gov or visit www.alexandriava.gov/Aquatics.


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'American Horror Story:Coven' 3.09 "Head" Review
The first two seasons of American Horror Story dealt almost exclusively with a contained setting be it the infamous murder house populated by both the living and dead, or Briarcliff Manor the torturous mental asylum at the center of last season. Part of the reason why Coven seemed so exciting at first had to do with the idea of several locations and of several factions of an impending witch war. It didn't take long however for the show to spend way too much time with a set group of characters at Miss Robichaux's Academy and short-change a number of other characters and stories introduced in the first couple of episodes. Throughout the winter finale, 'Head', a number of these backseat storylines were given the chance to shine but this late in the game is it enough to save what's been a fairly weak season of American Horror Story?
There are four episodes left when the series comes back in January and there still isn't a great deal to be excited about plot-wise this season, but perhaps that's because Coven is being treated more as a showcase for its characters than for its overall story. At this point, there's little reason to care about the Supreme mystery, or how the war on women witches will end when both plots seem to have fairly forgone conclusions. While there have been shocking acts this season, the conclusion of most storylines have actually been fairly predictable. What worked best about this episode was how it decided to just give us scenes that worked on their own to entertain, enlighten and build character. The exposition scenes involving The Corporation couldn't hold a candle to the fabulously macabre revenge Myrtle Snow put forth against former Council members Pembroke and Quentin. Now more than ever American Horror Story is committed to giving us more memorable moments than a traditional plotline. Frankly that's what made this episode one of the better ones of the season.
That said, if you do still care about the Supreme mystery storyline it seemed as though they subtly dropped one hell of a potential game-changing hint tonight. This is something I started to notice when the promotional photos for this episode came out that while Myrtle looked like Hell warmed over last week...this week her skin looked far better. Not only that but she is now capable of some extremely delicate magic when she gave Cordelia back her sight with the 'donated' eyeballs. Myrtle raved about the hippie swamp witch who knew nothing of what she could fully be capable of, and she claimed to Fiona that there were secrets in the flames the likes of which no one else could know of. It makes one wonder if the witches that have died and been resurrected are indeed on another level than they were in their first lives. It's similar to when Buffy died and was brought back that another slayer was called up in her place. I wonder if we're dealing with a scenario where because Misty's ability seems to shift the rules it's also creating a scenario where if witches are rejuvenated by her then they become something far more powerful. A potential spoiler for the entire coven structure of one Supreme ruling over all. It would be highly amusing if because this woman came into their lives that they were able to cast aside the old ways in favor of a modern coven where power is shared and very potent throughout its members. An entire coven of Supremes taking their line to the next level.
It's especially clear at this point that the women must join their considerable forces together
in order to survive. The metaphor of achieving ultimate sisterhood in a male-dominated and predatory world is pretty heavy-handed this year as evidenced by the cold open of young Hank and his father on their first hunting trip together. An exercise in showing that all kinds of hatred is learned, not naturally felt as we see later on with Hank's continued issues with his particular family business. In fact the entirety of The Corporation storyline exists to give the ladies the only kind of enemy they could truly put aside all their differences and their egos aside for--men who wish to quash the existence of women with power. If you remove the pretense of magic and other tenets of the occult from the story then this is basically a group of men who do not believe women should be allowed any inkling of stature that could put women above them--that is simply unthinkable. Due to the fact that The Corporation seems entirely male-driven it is then to Marie LaVeau's eventual peril what happened when she became involved in their misdeeds since the show considers her actions as having turned against her own kind--women not tribe or coven--an ultimate sin. She was put into the position of a woman that didn't consider the bigger picture and now that a man has destroyed her world embracing the idea of the sisterhood is the only thing she has left. When I mentioned above that a foregone conclusion is likely with this storyline I'm referring to the consequences of this team up. It likely will be that only together these women will be able to defeat their enemies...and then have plenty of time to return to back-stabbing one another.
Ultimately that's the thing about this show taking on too much of a preachy overtone with its thematic issues. They want to have it both ways and this was especially evident in the scene of Hank's slaying of Marie's tribe while elsewhere Delphine seemed to have a breakthrough in her bigotry. It always seemed as though both sides would suffer great casualty once the witch-hunters took control of the battle. It also was extremely uncomfortable to watch especially the attempted juxtaposition of this slaughter with Delphine's tears of regret. If you consider this as objectively as possible then Hank's violent rampage was a necessary evil in order to push story forward in the final episodes. Marie is a character who will be important to Coven and at this point in the war she needed to be joined with other people with power. Focusing on the visual of it however is a far harder thing to forgive. Whoever made that decision in the editing room made a mistake as this came off another mixed message for a major Coven theme. It was as though the white woman realizing she was wrong held some kind of profound triumph to be found within it while just downstairs mass murder occurred purely as a means of getting from point A to point B of the Coven story. I had a feeling that drawing the initial battle lines along sides of race would lead to something like this and I don't think that just because Marie seeks sanctuary with the 'white witches' (her words) that people will just let the massacre of so many African-American characters slide. Let's hope that Misty Day makes a house call to Cornrow City or there will be an even bigger issue here with this creative choice.
For the most part however, the winter finale held the kind of excitement to it that's been missing for most of the year. I can't wait until January to see what Ryan Murphy and his team have these amazing performers do next even if the season three plotlines themselves aren't much to anticipate on their own. Now that we're this far into the season it's for the best to try to sit back and enjoy the glamorous lunacy unless they get pushy with the themes again because part of what made season one a batshit wild ride was the fact that the themes of adultery, family, and forgiveness didn't overpower the stories they were given to play out in. So let's hope we get more over-the-top then soapbox in the coming episodes.
Posted by Keysha C. at 10:47 PM
Labels: American Horror Story, American Horror Story:Coven, FX, Reviews, TV
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While estimated shipping dates for most iPhone 7 and iPhone 7 Plus models have quickly slipped to 2-3 weeks or later, including November for most Jet Black models, customers who elected not to pre-order online or slept through their alarm may still have a chance to purchase the model they desire sooner than anticipated.
iPhone 7 and iPhone 7 Plus reservations will reopen on September 17 at 8:00 a.m. local time in the United States, and on September 17 at 6:00 a.m. local time in the United Kingdom, enabling customers to reserve an iPhone and pick it up at a local Apple retail store within a set 30-minute window.
In Canada, the "Reserve and Pick Up" program remains live at the time this article was published, with many iPhone 7 models still available for pick up on launch day. iPhone 7 Plus reservations have sold out in all of Ontario, but some 5.5-inch models remain available at select stores in other provinces.
"Reserve and Pick Up" remains live in Australia, Belgium, Germany, Hong Kong, the Netherlands, and Switzerland as well for select models, at the time this article was published. Reservations will reopen on September 17 at 6:00 a.m. local time in Italy. Apple is accepting September 17 reservations in the United Arab Emirates.
iPhone 7 and iPhone 7 Plus launch at Apple retail stores on September 16. Apple traditionally accepts walk-in customers on a first come, first served basis, with a separate line for Reserve and Pick Up customers. However, Apple anticipates demand to outstrip supply as usual, so in-store stock could be especially limited.
Authorized resellers such as Best Buy and stores of carriers such as AT&T, Verizon, Sprint, and T-Mobile also generally begin to receive extremely limited iPhone stock on launch day, but availability is not guaranteed due to tight supply. Prospective buyers are best to call their local stores ahead of time.
I thought nobody would buy it with the headphone jack missing. What does Apple need to do to make people not want it so that I can have it on launch day?
To the people gleefully celebrating these stories about pre-orders, just stop and ask yourself why Apple would preemptively announce they won't be releasing first weekends sales figures. Companies DO NOT do that if they believe the sales numbers are going to be great.
Pre-orders "selling out" means absolutely nothing in the big scheme of things. We all know they could artificially decide to limit the early numbers, to give the impression of high demand. It's not that complicated. If sales are big, they'll let us know. If they don't want to talk numbers, there is always a reason for that, and it's not a positive one.
I see a lot of people on Twitter who are part of the iPhone Upgrade Program that weren't able to reserve a phone, at least not the black models. One, Schiller should have said on stage that the black models would be in limited quantities at launch and two, people who are part of the iPhone Upgrade Program should get first dibs on stock.
The cult is alive and kicking like its June 2007.
Now I understand why there are always lines in front of their stores with new introductions. Only being able to pick it up at a store makes the impression its very popular. Pure marketing procedure as is the low in stock and first come, first served thing. I'm very curious about the numbers this year as competitors seem to have the same or better features and build quality at half the price (specifically interested in China numbers, that's the place all high tech is happening today).
I heard that sales of the Note 7 are on fire.
They need to manufacture more phones prior to the launch date.
If they could they would, you really think apple likes not selling phones.
I thought nobody would buy it with the headphone jack missing. What does Apple need to do to [...] so that I can have it on launch day?
So you think all the people seeing "sold out" today are just going to go buy Android phones? No, this just means more people will think about lining up at Apple Stores, or hovering around their computer hoping to get in on the next round of pre-orders when they open up. Saying "sold out" doesn't harm them right now. If this went on for a month or more, sure, but we're nowhere near that point right now.
Ever heard of an "impulse buy"? Checking site after site and seeing "sold out, available November" is a real buzz-kill and all those impulse buyers will say "Eh...my iPhone 6 or 6s is fine...I'll just keep it and save the money".
Are they going to buy Android, maybe...maybe not. Are they going to buy an iPhone 7 today? Definitely not. Business 101 -- no stock = no money.
This hasn't sold out, Apple just didn't order much for the launch.