text:
\section{Introduction}

For an $O(n),n\geq 1$ model of a $d$-dimensional system with a temperature $T$ and geometry $\infty^{d-1}\times L$ the thermodynamic Casimir force is defined by  \cite{E90book}, \cite{BDT2000}
\begin{equation}
F_{\rm Casimir}^{(\tau )}(T,L)=-\frac{\partial f_{\rm ex}^{(\tau )}(T,L)}{\partial L}%
\text{,}  \label{def}
\end{equation}
where $f_{\rm ex}^{(\tau)}(T,L)$ is the excess free energy
\begin{equation}
f_{\rm ex}^{(\tau )}(T,L)=f^{(\tau)}(T,L)-Lf_b(T)\text{,}  \label{fexd}
\end{equation}
and the superscript $\tau $ denotes the boundary conditions. Here $f^{(\tau )}(T,L)$ is the full free energy per unit area of such a system subjected to the boundary conditions $\tau $ and $f_b
$ is the bulk free energy density. Accumulated evidence
\cite{E90book,K94,BDT2000,KD92a,D98,BE95,K97,HSED98,K99,GC99,GC2002,ML2001,
UBMCR2003,HK92,ES94,
DKD2003,SHD2003,DK2004,ZRK2004,FYP2005,DDG2006,DGS2006,GSGC2006,ZSRKC2007,
DSD2007,VGMD2007,MGD2007,H2007,H2009,GD2008,HHGDB2008,B2007,KK2006,DG2009,
GMHNHBD2009,MMD2010,
AM2010,H2010,H2011} supports the conclusion that if the boundary conditions are
identical---or sufficiently similar---at both surfaces bounding the system,
$F_{\rm Casimir}^{(\tau )}$ will be negative. In the case of a fluid confined
between identical walls this implies that the net force between those walls due
to the fluid will be {\em attractive} for large separations. On the other hand,
if the fluid wets one of the walls while the other wall prefers the vapor phase,
then the Casimir force is repulsive. This implies that if the boundary
conditions differ sufficiently the Casimir force can be expected to be positive,
or \emph{repulsive}, for the entire range of  thermodynamic parameters. In the
intermediate case in which one of the surfaces of a system belonging to the
Ising universality class has a weak preference for one of the phases of the
fluid while the other one exhibits a strong preference, or for a given ratio
of the surface fields and/or of surface enhancements on both
surfaces, it has been recently demonstrated \cite{AM2010,MMD2010,H2011} that one
can observe much richer behavior, with the Casimir force changing its sign once,
or even twice \cite{MMD2010}, as the temperature is adjusted. In addition, in
\cite{HGS2011} it has been
shown via Monte Carlo simulations that in a system with a geometry
$L_\perp \times L_{\|}^2$ subject to periodic boundary conditions both the
magnitude and the sign of the Casimir force depend on the aspect ratio
$a_r=L_{\perp}/L_{\|}$. In this case general arguments have been advanced to  suggest that
at the bulk critical point the Casimir force vanishes for $a_r=1$
and becomes repulsive for $a_r>1$. These results are supported by exact
calculations for the two-dimensional Ising model. For further information regarding the
results currently available on the critical Casimir effect the interested reader is
referred to general reviews \cite{K94,BDT2000} as well as articles devoted to
specific aspects of the critical Casimir force \cite{K99,G2009,TD2010rev,GD2011}.

The critical Casimir effect and the corresponding Casimir force discussed
above are due to spatial restrictions imposed on the thermal average of the order parameter and on its long ranged fluctuations in a system undergoing a second order phase transition.
Based on analogy, some hints regarding the general behavior of the thermodynamic
Casimir force can also be extracted  from the information available on the
quantum Casimir effect \cite{C48} which is due to the spatial restrictions
imposed on the possible fluctuations of the electromagnetic field. A series of
reviews devoted to different aspect of this latter effect are available
\cite{MT97,M94,KG99,BMM2001,M2001,M2004,L2005,GLR2008,BKMM2009,KMM2011,
RCJ2011}. For the quantum Casimir effect it has been demonstrated that if the
boundary conditions $\tau$ are symmetric so that reflection positivity holds,
the Casimir force $F_{\rm Casimir}^{(\tau )}$ is attractive \cite{B2007,KK2006}.
According to the theory of Dzyaloshinskii, Lifshitz, and Pitaevskii \cite{DLP61}
in any system with a slab-like structure in which a material $B$ separates two
identical half-spaces $A\equiv C$ the force is attractive as well. When $B$ is a
vacuum, again according to \cite{DLP61}, this remains true even when the
half-spaces $A$ and $C$ are not identical. This prediction, up to now, has been
verified for all materials for which the Casimir force has been measured.
Theoretical predictions exist, however, suggesting that in the latter case a
repulsive Casimir force can be generated by special selection of the material
properties of  $A$ and $C$; see, e.g., \cite{KKMR2002}. However, such a
situation has not been experimentally realized. The omnipresence of attractive
quantum Casimir electromagnetic force for objects in vacuum or air affects the
work of micro and nano-machines \cite{BR2001,L2005,GLR2008,RCJ2011} and might
cause sticking of their working surfaces. The possibility of realizing and controlling
a repulsive critical Casimir force might be one of the ways of overcoming the
above-mentioned difficulties.

In an attempt to shed additional light on the influence of differing boundary conditions on
the critical Casimir force, we consider a film system with $\infty^{d-1}\times
L$ geometry  consisting of local dynamical variables, say magnetic moments,
possessing $O(2)$ symmetry and constrained to lie in the $x$-$y$ plane. The
moments in one of the bounding surfaces are constrained to point in the same
direction in that plane and to be oriented at an angle  $\alpha$ with respect to
the similarly aligned spins in the other bounding surface. Furthermore, in the
case in which the moments have variable amplitudes, those amplitudes are fixed
at a non-zero value. Alternatively, one might think of the studied system as
a lattice gas of elongated, say, rod like molecules embedded on a lattice.
We investigate both the equilibrium behavior of those moments (or
molecules)---and the Casimir force that arises as a result of that behavior---as
a function of $T$ and $\alpha$. As we will see, among the results of our
calculation are a Casimir force that depends in a continuous way on both the
parameter $\alpha$ and the temperature and that can be attractive or repulsive.
In particular, by varying $\alpha$ and/or $T$ one controls both the sign and the
magnitude of the Casimir force in a reversible way.

We will refer to the boundary conditions described above as ``twisted'' boundary
conditions.   Subject to them, the moments within the system settle into a state
in which they rotate with respect to each other as the region between the
boundaries is traversed, creating a diffuse interface within it. The normalized
excess free energy per unit area of the system, $f_{\rm ex}^{(\alpha)}$, can be
related to the corresponding quantity in a system with zero twist of the
moments, which we will term a system with $(+,+)$ boundary conditions. The
quantified version of this relationship is
\begin{equation}
\label{falpha}
f^{(\alpha)}_{\rm ex}(T,L)=f^{(+,+)}_{\rm ex}(T,L)+\frac{1}{2L}\;\alpha^2 \,\Upsilon^{(\alpha)}(T,L),
\end{equation}
where $\Upsilon^{(\alpha)}(T,L)$ is the finite-size helicity modulus \cite{DG2009,D93,FBJ73} that characterizes the energy of the  system related to the diffuse interface in it. The excess free energy $f^{(\alpha)}_{\rm ex}(T,L)$ can also be resolved into regular and singular parts:
\begin{equation}
\label{fexdecomp}
f^{(\alpha)}_{\rm ex}(T,L)=f^{(\alpha)}_{\rm ex, \;reg}(T,L)+f^{(\alpha)}_{\rm ex,\;sing}(T,L).
\end{equation}
In the case of the singular part of the excess free energy in the vicinity of the bulk critical point one has,  according to  finite-size scaling theory \cite{BDT2000},
\begin{equation}\label{fexsing}
f_{\rm ex, \;sing}^{(\alpha)}(T,L)=L^{-(d-1)}X_{\rm ex}^{(\alpha)}\left(x_t\right)
\end{equation}
As a consequence, neglecting the ``background''  contribution to the Casimir force, one obtains
\begin{equation}\label{CasGeneral}
F_{\rm Casimir}^{(\alpha)}(T,L)=L^{-d}X_{\rm Cas}^{(\alpha)}\left(x_t\right).
\end{equation}
Here $x_t=a_t tL^{1/\nu }$ is the temperature scaling variable, $t=(T-T_c)/T_c$
is the reduced temperature, $a_t$ is a nonuniversal scaling factor, while
$X_{\rm ex}^{\left( \alpha\right) }$  and $X_{\rm Cas}^{\left( \alpha\right) }$
are {\em universal} (albeit geometry-dependent) scaling functions and $\nu $ is
the corresponding (universal) scaling exponent that characterizes the
temperature divergence of the bulk two-point correlation length, $\xi$, when one
approaches the bulk critical temperature from above, i.e. $\xi(t\rightarrow
0^+)\simeq \xi_0^+ t^{-\nu}$ with $\xi_0^+$ being some system dependent metric
factor. For the behavior of $\Upsilon^{(\alpha)}(T,L)$ near $T_c$ from (\ref{falpha}) and
(\ref{fexsing}) one derives
\begin{equation}\label{helmodfss}
\beta \Upsilon^{(\alpha)}(T,L)  =L^{-(d-2)}X_\Upsilon^{(\alpha)}\left(x_t\right),
\end{equation}
where $X_\Upsilon^{(\alpha)}$ again is universal scaling function. Requiring a
$L$-independent behavior of $\Upsilon$ in the limit $L\to \infty$, one obtains
$\Upsilon(T)\equiv \lim_{L\to\infty}\Upsilon^{(\alpha)}(T,L)$, with $\Upsilon(T)\ge 0$,
$X_\Upsilon^{(\alpha)}\left(x_t\right)\sim |x_t|^{(d-2)\nu}$ and, thus, $\Upsilon(t)\sim
|t|^{(d-2)\nu}$, which is in a complete agreement with \cite{FBJ73}.

 When such a diffuse interface is present within the system and $T<T_c$ from Eq. (\ref{falpha}) it is easy to see that
\begin{equation}
F_{\rm Casimir}^{\left(\alpha\right) }(T< T_c)\simeq  \frac{1}{2} \alpha^{2} \Upsilon(T) L^{-2},\; L\to\infty. \label{fssclowTas}
\end{equation}
Since $\Upsilon(T)\ge 0$, Eq. (\ref{fssclowTas}) implies that the Casimir force will be {\it repulsive} and, for $d>2$, much {\it stronger}, of the order of $L^{-2}$, than in systems with a compact interface where it is either of the order of $L^{-d}$, or smaller.

The structure of the article is as follows. In the next section \ref{latticemodel} we define a lattice three-dimensional mean-field XY model and present numerical results for the behavior of the Casimir force within it. Section \ref{GLMFmodel} presents analytical results for the scaling function of the Casimir force within the Ginzburg-Landau mean-field theory of the three-dimensional XY model. In both sections \ref{latticemodel} and \ref{GLMFmodel} we find interesting behavior of the force at a temperature $T_{\rm kink}$ below the critical one of the bulk system when $\alpha$ approaches $\pi$. We study this special case in section \ref{alphaequalpicase}. We deduce the existence of an additional second-order phase transition that is specific to this finite system. The article closes with a discussion presented in section \ref{discussion}. Technical details of the derivations are presented in appendixes at the end of the article.

\section{The Casimir force in the lattice three-dimensional mean-field XY model}
\label{latticemodel}

Consider a lattice of dimensions $\infty^{d-1}\times L$, with each site
populated by an $O(2)$ fixed-length magnetic moment of magnitude $m$. We split
up the lattice into $(d-1)$-dimensional planes labeled ${1,\ldots,N}$, where
$L=N a$, with $a$ being the lattice constant taken in the remainder to be equal
to one. By translational invariance and neglecting the fluctuation within the
planes, all moments in plane $i$ must take the same value, equal to their mean
value, and must point in the same direction. However, due to the anisotropy
along the finite dimension, the moments will vary between planes. Let the moment
in plane $i$ be ${\bf m}_i$. We take a nearest-neighbor coupling with strength
$J$ both in the plane and out of it, and so the energy of a moment in plane $i$
will be
\begin{equation}
U_i=-J{\bf m}_i\cdot(2(d-1){\bf m}_i+{\bf m}_{i-1}+{\bf m}_{i+1})
\end{equation}
or, defining an effective magnetic field ${\bf H}_i=J(2(d-1){\bf m}_i+{\bf m}_{i-1}+{\bf m}_{i+1})$ at that site, $U_i=-{\bf m}_i\cdot{\bf H}_i$.

Approximating the moment ${\bf m}_i$ as being isolated, in an external magnetic field ${\bf H}_i$, we can assign it the local partition function
\begin{equation}
Z_i=\int_0^{2\pi} d\theta\ e^{\beta mH_i\cos\theta}=2\pi I_0(\beta mH_i)
\end{equation}
with $\beta=(k_B T)^{-1}$ and $\theta$ the angle between ${\bf H}_i$ and ${\bf m}_i$. On average, the component of ${\bf m}_i$ along ${\bf H}_i$ is
\begin{equation}
\langle m\cos\theta\rangle=\frac{1}{\beta}\frac{d\ln Z}{dH_i}=m\frac{I_1(\beta
mH_i)}{I_0(\beta mH_i)},
\end{equation}
where $I_0$ and $I_1$ are  the corresponding modified Bessel functions of the
first kind,
while the component normal to ${\bf H}_i$ is
\begin{equation}
\langle m\sin\theta\rangle=\frac{1}{Z}\int_0^{2\pi} d\theta\ m\sin\theta\ e^{\beta mH_i\cos\theta}=0
\end{equation}
so that the averaged moment is entirely along ${\bf H}_i$. Inserting the definition of ${\bf H}_i$ in terms of neighboring moments, we see that ${\bf m}_i$ must satisfy the equation
\begin{multline}
{\bf m}_i = \frac{2(d-1){\bf m}_i+{\bf m}_{i-1}+{\bf m}_{i+1}}{|2(d-1){\bf m}_i+{\bf m}_{i-1}+{\bf m}_{i+1}|}\\
\times R\left(\beta mJ\left|2(d-1){\bf m}_i+{\bf m}_{i-1}+{\bf m}_{i+1}\right|\right)m \label{latticeeom}
\end{multline}
in the mean-field approximation (all quantities now implicitly averaged), with $R(u)=I_1(u)/I_0(u)$.

\begin{figure*}[ht]
\includegraphics[angle=0,width=\columnwidth]{fig1a-lattice_illustration_abovekink.eps}
\includegraphics[angle=0,width=\columnwidth]{fig1b-lattice_illustration_belowkink.eps}
\caption{(Color online) Renderings of the moments ($d=3$, $N=30$, $\alpha=\pi$) for temperatures above (left) and below (right) the temperature where a kink occurs in the Casimir force.}
\label{lattice_twist_illustration}
\end{figure*}

If we define ${\bf m}_0={\bf m}_{N+1}=0$ for notational convenience, then the function
\begin{equation}
f(\{{\bf m}_i\},N)=\sum_{i=1}^N\left[\frac{1}{2}{\bf m}_i\cdot{\bf H}_i-\frac{1}{\beta}\ln\left(I_0\left(\beta mH_i\right)\right)\right] \label{latticefreeenergy}
\end{equation}
may be regarded as the total free energy functional of the system, because
minimizing with respect to $\{{\bf m}_i\}$ yields the self-consistency
conditions (\ref{latticeeom}) (See Appendix \ref{latticefreeenergyappendix}). In
order to compute the Casimir force on the system, we must also find the free
energy per site in the bulk, i.e. when the system is very large. In that case,
the moments $\{{\bf m}_i\}$ will all be identical (at least near the center) and
Eq. (\ref{latticeeom}) tells us that the equation
$m_{i,\textrm{bulk}}=mR(2dm\beta Jm_{i,\textrm{bulk}})$ determines their common
magnitude $m_{i,\textrm{bulk}}$. For $T>dJm^2/k_B$, the only solution is
$m_{i,\textrm{bulk}}=0$ while for $T<dJm^2/k_B$, there is a non-zero solution.
Thus, the bulk system exhibits a transition at $T_{c,\textrm{bulk}}=dJm^2/k_B$
between an ordered ($T<T_{c,\textrm{bulk}}$) and a disordered phase.

Letting $a$ be the lattice spacing along the finite dimension of the system, the bulk free energy density is
\begin{equation}
f_b=\frac{1}{a}\left[dJm_{i,\textrm{bulk}}^2-\frac{1}{\beta}\ln\left(I_0\left(2dm\beta Jm_{i,\textrm{bulk}}\right)\right)\right]
\end{equation}
and the Casimir force will be computed as

\begin{figure}[hb]
\includegraphics[angle=0,width=\columnwidth]{fig2-lattice_casimir_force.eps}
\caption{(Color online) The Casimir force in the lattice model ($d=3$, $N=30$) as a function of reduced temperature $t=(T-T_{c,\textrm{bulk}})/T_{c,\textrm{bulk}}$ for various values of the twist angle, $\alpha$.}
\label{lattice_casimir_force}
\end{figure}

\begin{multline}
F_{\textrm{Casimir}}=-\frac{\partial}{\partial L}\left(f-Lf_b\right)\\
\approx f_b-\frac{f(N+1)-f(N)}{a}.
\end{multline}

We now restrict our attention to the case $d=3$ and investigate the system numerically. This amounts to solving the simultaneous Eqs. (\ref{latticeeom}) subject to particular boundary conditions on ${\bf m}_1$ and ${\bf m}_N$. We are interested in how the behavior of the system depends on the twist angle, $\alpha$.

When the twist angle is zero, the Casimir force is purely attractive (i.e, negative), as expected for matching boundary conditions. These features are illustrated in the plot of Casimir force versus reduced temperature (Fig. 2). As the twist is increased, a low temperature region of repulsive Casimir force emerges. At a twist angle of $\pi/2$, the Casimir force becomes purely repulsive. When the system nears anti-symmetric boundary conditions, $\alpha=\pi$, the Casimir force develops a kink at a temperature $T_{\textrm{kink}}<T_{c,\textrm{bulk}}$.

The nature of the kink will be discussed at greater length in a subsequent section, but we begin to understand it from the renderings (Fig. \ref{lattice_twist_illustration}). Below the kink temperature, the moments achieve the twist of $\pi$ by rotating about the axis while maintaining almost their full length $m$. Above the kink temperature, the moments all reside in a plane, and the twist is localized to the center of the system, where the magnetization has shrunk to zero. The nearest-neighbor interaction between moments imposes a free energy penalty for both rotating with respect to neighbors and varying in length. The transition indicates the point at which these penalties trade off in dominance.

\section{The Casimir force in the Ginzburg-Landau mean-field theory of the three-dimensional XY model}
\label{GLMFmodel}

We now consider the continuous analogue of this system in the Ginzburg-Landau mean-field theory. The order parameter of the system is the magnetization profile ${\bf m}(z)$, where $z$ is the finite dimension of the system. The behavior of the system is found by minimizing the free energy functional (per unit area),
\begin{multline}
{\cal F}\left[ {\bf m};t,L\right]=\int_{-L/2}^{L/2} dz\,\left[\frac{b}{2}\left|\frac{d{\bf m}}{dz}\right|^2+\frac{1}{2}at\left|\textbf{m}\right|^2\right.\\
\left.+\frac{1}{4}g\left|{\bf m}\right|^4\right],
\label{LGenergyfunctional}
\end{multline}
with respect to ${\bf m}$ and subject to certain boundary conditions. The quantity $t$ represents the reduced temperature.

Switching to polar coordinates,
\begin{equation}
{\bf m}(z)=\left(\Phi(z)\cos\varphi(z),\Phi(z)\sin\varphi(z)\right),
\end{equation}
the free energy functional is rewritten as
\begin{multline}
{\cal F}\left[\Phi,\varphi;t,L\right]=\int_{-L/2}^{L/2} dz\,\left[\frac{b}{2}\left(\frac{d\Phi}{dz}\right)^2+\frac{b}{2}\Phi^2\left(\frac{d\varphi}{dz}\right)^2\right.\\
\left.+\frac{1}{2}at\Phi^2+\frac{1}{4}g\Phi^4\right]. \label{LGfreeenergy}
\end{multline}

Minimization with respect to $\varphi(z)$ gives
\begin{equation}\label{varphiEq}
\frac{d}{dz}\left[\Phi^2{\varphi'}\right]=0
\end{equation}
which leads to
\begin{equation}
\Phi(z)^2 \left(\frac{d\varphi}{dz}\right)=P_\varphi
\label{Pvarphidef}
\end{equation}
with an integration constant $P_\varphi$, independent of $z$, which roughly indicates the degree of twisting in the system. The condition from minimizing with respect to $\Phi$ is similarly computed as
\begin{equation}
b\frac{d^2\Phi}{dz^2}=b\Phi\left(\frac{d\varphi}{dz}\right)^2+at\Phi+g\Phi^3
\label{PhiEq}
\end{equation}
or, with the identification (\ref{Pvarphidef}),
\begin{equation}
b\frac{d^2\Phi}{dz^2}=b\frac{P^2_\varphi}{\Phi^3}+at\Phi+g\Phi^3.
\label{Phieom}
\end{equation}
The problem is now that of solving Eq. (\ref{Phieom}) subject to twisted boundary conditions:
\begin{eqnarray}
&\varphi(\pm L/2)=\pm \alpha/2,\nonumber\\
&\Phi(\pm L/2) = \infty,
\label{boundaryconditions}
\end{eqnarray}
i.e. where the moments at the boundaries are twisted by an angle $\alpha$ relative to one another.

Note that, because of reflection symmetry in Eq. (\ref{Phieom}) and the boundary conditions imposed on $\Phi$, we have that $\Phi(z)=\Phi(-z)$ and, thus, $\Phi'(z)=-\Phi'(-z)$, whence $\Phi'(0)=0$. From the symmetry of Eq. (\ref{Pvarphidef}) one concludes $\varphi(z)=-\varphi(-z)$ which leads to $\varphi(0)=0$.

Multiplying (\ref{Phieom}) by $d\Phi/dz$ and integrating with respect to $z$, we find a first integral
\begin{equation}\label{solPhiprime}
{P}_\Phi=-\frac{1}{2}b \left[\frac{P_\varphi^2}{\Phi^2}+\left(\frac{d\Phi}{dz}\right)^2 \right]  +  \frac{1}{2} a\, t\, \Phi^2+\frac{1}{4} g\, \Phi^4,
\end{equation}
with $P_\Phi$ being another integration constant independent of $z$. Let $\Phi_0\equiv\Phi(z=0)$ be the amplitude of the order parameter at the center of the interval. Then, taking into account that $\Phi'(0)=0$ one can conveniently express $P_\Phi$ as
\begin{equation}\label{PhiPhinode}
{P}_\Phi=-\frac{1}{2}b\frac{P_\varphi^2}{\Phi_0^2}  +  \frac{1}{2} a\, t\, \Phi_0^2+\frac{1}{4} g\, \Phi_0^4,
\end{equation}
from which it follows that
\begin{multline}\label{Phifirstintegral}
\left(\frac{d\Phi}{dz}\right)^2=P^2_\varphi\left(\frac{1}{\Phi_0^2}-\frac{1}{\Phi^2}\right)+\hat{a}t\left(\Phi^2-\Phi_0^2\right)\\
+\frac{\hat{g}}{2}\left(\Phi^4-\Phi_0^4\right),
\end{multline}
where
\begin{equation}\label{hatvar}
\hat{a}=\frac{a}{b}, \qquad \hat{g}=\frac{g}{b}.
\end{equation}
The last result allows us to express the boundary conditions as
\begin{widetext}
\begin{eqnarray}
\frac{L}{2}=\int_0^{L/2} dz=\int_{\Phi_0}^{\infty}d\Phi\,\frac{dz}{d\Phi}
=\int_{\Phi_0}^{\infty}d\Phi\,\frac{1}{\sqrt{P^2_\varphi\left(\Phi_0^{-2}-\Phi^{-2}\right)+\hat{a}t\left(\Phi^2-\Phi_0^2\right)+\frac{\hat{g}}{2}\left(\Phi^4-\Phi_0^4\right)}} \label{lengthcondition} \\
{\rm and}\qquad\frac{\alpha}{2}=\int_0^{L/2} dz\,\frac{d\varphi}{dz}=P_\varphi\int_{\Phi_0}^\infty \frac{d\Phi}{\Phi^2}\frac{1}{\sqrt{P^2_\varphi\left(\Phi_0^{-2}-\Phi^{-2}\right)+\hat{a}t\left(\Phi^2-\Phi_0^2\right)+\frac{\hat{g}}{2}\left(\Phi^4-\Phi_0^4\right)}}. \label{twistcondition}
\end{eqnarray}
\end{widetext}
These equations relate the integration constants, $P_\varphi$ and $\Phi_0$, to the system's external parameters, $L$ and $\alpha$.

The stress tensor operator for a system with the free energy functional
(\ref{LGenergyfunctional}) is \cite{EiS94}
\begin{multline}\label{STO}
T_{k,l} = b \frac{\partial \mathbf{m}}{\partial x_k}\frac{\partial \mathbf{m}}{\partial x_l}\\
-\delta_{k,l} \left\{ \frac{1}{2} b \left[ {\Phi'}^2+\Phi^2{\varphi'}^2\right]+\frac{1}{2}at\Phi^2+\frac{1}{4}g\Phi^4\right\}\\
-b\left[\frac{d-2}{4(d-1)}+O\left(g^3\right) \right] \left[\frac{\partial^2}{\partial x_k \partial x_l} -\delta_{k,l}\nabla^2\right] \Phi^2.
\end{multline}
Calculating the $\langle T_{z,z} \rangle$ component, one obtains
\begin{equation}\label{STOfilm}
\langle T_{z,z} \rangle =\frac{1}{2}b\left[\left(\frac{d\Phi}{dz}\right)^2 +
\frac{P^2_\varphi}{\Phi^2}\right] - \frac{1}{2}at\Phi^2 - \frac{1}{4}g\Phi^4
\end{equation}
which, according to the general theory, is a $z$-independent quantity
equal to the pressure $-\partial f^{(\tau)}(T,L)/\partial L$ between  the
plates confining a fluctuating medium
\cite{K97,EiS94}. In our case, we see that
\begin{equation}
\langle T_{z,z} \rangle\equiv -P_\Phi,
\end{equation}
by (\ref{solPhiprime}). From Eqs. (\ref{PhiPhinode}) and (\ref{STOfilm}) one
observes that the Casimir force (the excess pressure over the bulk one) in this
system is
\begin{multline}\label{Casimir}
F_{\rm Casimir}(t,L)=-\left[-\frac{1}{2}b\frac{P_\varphi^2}{\Phi_0^2}  +  \frac{1}{2} a\, t\, \Phi_0^2\right.\\
\left.+\frac{1}{4} g\, \Phi_0^4 +\frac{1}{4g}(at)^2 \theta(-t)\right]
\end{multline}
where $\theta(x)$ is the Heaviside step function. Here, we have taken into account that the bulk free energy density $f_b$ for the system is $f_b(t<0)=-(a t)^2/4g$ while $f_b(t>0)=0$. As expected, the Casimir force is sensitive to the boundary conditions, through the quantities $P_\varphi$ and $\Phi_0$.

It is easy to show that $F_{\rm Casimir}(t,L)$  obeys the expected scaling. Indeed, in terms of the variables
\begin{multline}
z=L \zeta,\qquad \Phi=\sqrt{\frac{2}{\hat{g}}}X_\Phi L^{-1},\qquad \Phi_0=\sqrt{\frac{2}{\hat{g}}}X_0 L^{-1},\\
P_\varphi=\frac{2}{\hat{g}} X_\varphi L^{-3},\qquad \hat{a}t= x_t L^{-2}
\label{scalingvariables}
\end{multline}
the Casimir force reads (note that $\zeta$, $x_t$, etc. are all dimensionless)
\begin{equation}\label{cas}
F_{\rm Casimir}(t,L)=\frac{b}{\hat{g}}L^{-4}X_{\rm Cas}^{(\alpha)}(x_t),
\end{equation}
where
\begin{equation}\label{casscalingfunction}
X_{\rm Cas}^{(\alpha)}(x_t)=\left\{ \begin{array}{cc}
                             X_\varphi^2 / X_0^2- X_0^2\left(x_t+X_0^2 \right),  & x_t \ge 0 \\
                              X_\varphi^2 / X_0^2- \left(\frac{1}{2} x_t+X_0^2\right)^2, & x_t \le 0
                            \end{array} \right. .
\end{equation}
Taking into account that mean-field theories for short-ranges systems are effective $d=4$ theories, one concludes that Eq. (\ref{cas}) is in full agreement with the expected scaling behavior (\ref{CasGeneral}) of the Casimir force. From Eqs. (\ref{cas}) and (\ref{fssclowTas}) one can derive the low temperature asymptotic behavior of the scaling function $X_{\rm Cas}^{(\alpha)}(x_t)$ of the Casimir force. We find
\begin{equation}\label{casscalingfunctionasymp}
X_{\rm Cas}^{(\alpha)}(x_t)\simeq \frac{1}{2}  \alpha^2 |x_t|X_{\rm C},\; x_t\to-\infty,
\end{equation}
with $X_{\rm C}$ a constant. Appendix \ref{asymptotic} contains a derivation of the expression in  (\ref{casscalingfunctionasymp}). In that Appendix, we obtain the asymptotic expression for $X_{\rm Cas}^{(\alpha)}(x_t)$
\begin{equation}\label{casscalingfunctionasympXY}
X_{\rm Cas}^{(\alpha)}(x_t)\simeq \frac{1}{2}  \alpha^2 \left[|x_t|+4
 \sqrt{2|x_t|}+\frac{1}{2} \left(48-3 \alpha ^2\right)\right],
\end{equation}
when $x_t\to-\infty$. Note that Eq. (\ref{casscalingfunctionasympXY}) implies
that $X_{\rm C}=1$ for the $XY$ mean-field model.

One can further simplify (\ref{casscalingfunction}) by introducing the convenient combinations of scaling variables
\begin{equation}\label{newvarscaling}
\tau=x_t/X_0^2, \qquad \mbox{and} \qquad p=X_\varphi/X_0^3.
\end{equation}
Then the scaling function of the Casimir force reads
\begin{equation}\label{casscalingfunctionpandtau}
X_{\rm Cas}^{(\alpha)}(\tau)=\left\{ \begin{array}{cc}
                             X_0^4[p^2- \left(1+\tau \right)],  & \tau \ge 0 \\
                              X_0^4[p^2- \left(1+\tau/2 \right)^2], & \tau \le 0
                            \end{array} \right. .
\end{equation}
Eqs. (\ref{lengthcondition}) and (\ref{twistcondition}) then become
\begin{equation}\label{xoonpt}
X_0=\int_1^\infty \frac{dx}{\sqrt{(x-1)[x^2+x(1+\tau)+p^2]}},
\end{equation}
and
\begin{equation}\label{alphafinal}
\alpha= 2p \;X_0^3\int_0^{1/2} \frac{d\zeta}{X_\Phi^2(\zeta)}.
\end{equation}
\begin{figure}[ht]
\includegraphics[angle=0,width=\columnwidth]{fig3-Casimir3.eps}
\caption{(Color online) A plot of the dimensionless scaling function for the Casimir force,
$X_\textrm{Cas}$, versus $x_t$ (also dimensionless) for several values of
$\alpha$. The dotted curve is the Casimir scaling function in the Ising-like
case of a critical fluid under $(+,-)$ boundary conditions. For $x_t$ above a
certain value, this curve coincides with the one for the model studied here when
twisted by an angle $\alpha\approx\pi$.}
\label{plot:Casimir}
\end{figure}
\begin{figure}[ht]
\includegraphics[angle=0,width=\columnwidth]{fig4-latticevslg_casimir_force.eps}
\caption{(Color online) The scaling functions for the Casimir force in the Ginzburg-Landau
model (solid curves), overlaid with those from the lattice model with $N=50$
(data points), for several values of $\alpha$ (from top to bottom:
$\alpha=0.98\pi$, $\alpha=2\pi/3$, $\alpha=\pi/2$, $\alpha=\pi/3$, $\alpha=0$).}
\label{latticevslg_casimir_force}
\end{figure}

\begin{figure}[hb]
\includegraphics[angle=0,width=\columnwidth]{fig5-asymptotes.eps}
\caption{(Color online) Casimir force curves in the Ginzburg-Landau model for several values of $\alpha$, overlayed with their respective asymptotic expressions (dotted) given by Eq. (\ref{casscalingfunctionasympXY}) as proven in Appendix C.}
\label{asymptotes}
\end{figure}

\begin{figure*}[ht]
\begin{center}
\includegraphics[angle=0,width=\columnwidth]{fig6a-zeros_Casimir_force.eps}
\includegraphics[angle=0,width=\columnwidth]{fig6b-Casimir_Amplitudes.eps}
\end{center}
\caption{(Color online) Left: The positions $x_{t,0}^{(\alpha)}$ of the zeros of the Casimir
force in the $(x_t,\alpha)$-plane. Right: The Casimir amplitude $\Delta_{\rm
Cas}^{(\alpha)}$ as a function of the twist angle $\alpha$. This curve has been
initially reported in \cite{K97} based on representation
(\ref{KrechAmplitudes}) derived there. The Casimir amplitude
changes sign at $\alpha=\pi/3$.}
\label{plot:Casimir_zeros_and_amplitudes}
\end{figure*}

In order to determine the Casimir force scaling function $X_{\rm Cas}^{(\alpha)}(x_t)$, all one needs to know is the behavior of $X_0=X_0(x_t|\alpha)$ and $X_\varphi(x_t|\alpha)=p X_0^3$ as functions of $x_t=\tau X_0^2$ at a given fixed value of the angle $\alpha$. For that, one has to solve Eqs. (\ref{xoonpt}) and (\ref{alphafinal}) after determining the function $X_\Phi(\zeta)$ from
\begin{multline}\label{zetaonxo}
\zeta=\frac{1}{2}\\
+\frac{1}{2 X_0}\int_{[X_\Phi(\zeta)/X_0]^2}^\infty\frac{dx}{\sqrt{(x-1)[x^2+x(1+\tau)+p^2]}}
\end{multline}
with $0<\zeta\le 1/2$, which directly follows from (\ref{Phifirstintegral}).
The detailed knowledge of the behavior of the phase angle profile
$\varphi(\zeta)$ is not needed. The analytical treatment of Eqs.
(\ref{xoonpt})-(\ref{zetaonxo}) is performed in Appendix \ref{GLMF}. The
numerical evaluation of the expressions derived there leads to the results for
the Casimir force presented in Fig. \ref{plot:Casimir}. The comparison shows
excellent agreement between the continuum and the lattice model results from the
previous section - see Fig. \ref{latticevslg_casimir_force}. In order to
demonstrate it, we scale the lattice results $(t, F_{\rm Casimir})$ to $(a_t N^2
t, a_F N^4 F_{\rm Casimir})$, where the scaling factors $a_t$ and $a_F$ are
determined by forcing the Casimir force with $\alpha=0$ to agree between the two
models. This was done numerically for $N=50$, where we find $a_t\approx 2.977$
and $a_F\approx 7.480\times 10^{-5}$. Fig. \ref{asymptotes} shows a comparison between the low temperature behavior of the Casimir force with the analytically derived asymptotic behavior reported in Eq. (\ref{casscalingfunctionasympXY}). We find that, for all $\alpha$, the asymptotic behavior is achieved for $x_t\lesssim -150$.

From Eq. (\ref{casscalingfunctionpandtau}) one can also infer some general
properties of the Casimir force. Taking into account that, at any fixed $x_t$
and $\alpha$, $p$ is a definite function of $x_t$ and $\alpha$, i.e. that
$p=p(\tau|\alpha)$ one can, e.g., determine the coordinates $x_{t,0}^\alpha$ of
the zeros for the Casimir force for a given angle $\alpha$. According to Eq.
(\ref{casscalingfunctionpandtau}) one has that $X_{\rm Cas}^{(\alpha)}=0$ for
$p(\tau|\alpha)=\sqrt{1+\tau}$, with $\tau\ge 0$ and for $p(\tau|\alpha)=
1+\tau/2$ when $-2\le\tau\le 0$. A plot of the positions of these zeros in the
$(x_t,\alpha)$-plane  is presented in Fig.
\ref{plot:Casimir_zeros_and_amplitudes}.

The figure demonstrates how, by changing, e.g., the twist angle $\alpha$, one
can, at a given temperature $t$, make the Casimir force either repulsive or
attractive. For $0<\alpha<\pi/2$, this can also be achieved by changing the
temperature, i.e. the scaling variable $x_t$, at a given fixed value of
$\alpha$. We also conclude that, when $\alpha \to 0$, the position of the zero
value of the Casimir force approaches $-\infty$. This implies that when
$\alpha=0$, the Casimir force will be attractive for all temperatures. Actually,
for $\alpha=0$, $X_{\rm Cas}^{(\alpha)}(x_t)$ coincides with the known result
for the Ising model system \cite{K97}. The behavior of $X_{\rm Cas}^{(+,+)}(x)$
is shown as a thick black line in Fig. \ref{plot:Casimir}. These results are
briefly re-derived in Appendix \ref{GLMF} for the convenience of the reader.

When $\alpha\to\pi/2$, we observe in Fig. \ref{plot:Casimir_zeros_and_amplitudes} that $x_{t,0}^{(\alpha)}\to \infty$. Thus $\alpha>\pi/2$ implies that the Casimir force will be repulsive for all values of $x_t$. As $\alpha$ increases, the repulsive force becomes stronger. We see from Fig. \ref{plot:Casimir} that, when $\alpha=\pi/2$, the force is practically zero for all temperatures above the critical temperature of the finite system, while, for $\alpha>\pi/2$, it is repulsive in the whole temperature region. The cases $\alpha=2\pi/3$ and $\alpha=0.98 \pi$ illustrate these features in the figure. One observes numerically that, for $x_t>-10$, the $\alpha=0.98 \pi$ curve agrees with that of the mean-field Ising model with $(+,-)$ boundary conditions. At lower temperatures, there is an abrupt departure from the Ising model which will be discussed in the section to follow. The analytical expressions for the Ising model are known from \cite{K97}. For completeness, these results are recalled in Eq. (\ref{CasimiralphaeqpiKrech}) of Appendix \ref{GLMF}.

The behavior of the critical Casimir force, $\Delta_{\rm Cas}^{(\alpha)}=X_{\rm
Cas}^{(\alpha)}(x_t=0)/3$, as a function of $\alpha$, is illustrated in Fig.
\ref{plot:Casimir_zeros_and_amplitudes}. Note that the Casimir amplitude becomes
zero at $\alpha=\pi/3$, so that the Casimir force for $\alpha=\pi/3$ changes its
sign at $x_t=0$. This was initially reported in \cite{K97} and may also be seen
in Fig. \ref{plot:Casimir_zeros_and_amplitudes}. Note that in \cite{K97}
different type of parametrization of the amplitude and phase profiles is used---they are parametrized via the Casimir amplitudes.  This led to a restriction
of the results presented to the critical temperature only. For example, for the
determination of the Casimir amplitudes in \cite{K97} one has to solve, in our
notations, the following system of equations (see Eq. (3.16) in \cite{K97})
\begin{subequations}
\label{KrechAmplitudes}
\begin{equation}
 X_0=\int_1^\infty[x^3-1+(x-1)X_0^{-4}\Delta_{\rm Cas}^{(\alpha)}]^{-1/2}dx
\end{equation}
and
\begin{eqnarray}
\alpha &=& \sqrt{1+X_0^{-4} \Delta_{\rm
Cas}^{(\alpha)}} \\
&&\int_1^\infty x^{-1}[x^3-1+(x-1)X_0^{-4}\Delta_{\rm
Cas}^{(\alpha)}]^{-1/2}dx. \nonumber
\end{eqnarray}
\end{subequations}

\section{The transition at $\alpha=\pi$}
\label{alphaequalpicase}

The case $\alpha=\pi$ warrants special investigation because it features behavior reminiscent of a  phase transition. As mentioned in Section 3, the high temperature behavior of the system at $\alpha\approx\pi$ tracks that of the Ising model. However, we find that a kink develops in all quantities in the system at a temperature $t_{\textrm{kink}}$ below the bulk critical temperature of the system, and the system changes its character at this temperature. The lattice model also featured such a kink. In Section 2, we illustrated how the lattice system switches from a ``rotational'' state below the kink temperature to a ``planar'' state above it. We note that this phase transition-like behavior exists only in the finite system under the given boundary conditions and not in a thermodynamic sense. Below the transition temperature, the two coexisting phases are the rotational states with rotation plus or minus $\pi$. There is spontaneous symmetry breaking when the system orders in one of them. Above the critical temperature, there is a single state -- the ``planar'' one.

The system incurs free energy penalties when adjacent moments vary in length or direction. We see how each type of state mentioned would extremize the energy. The moments in the planar state minimize rotation: they reside in a plane and shorten to a length of zero at the center of the interval, where an abrupt reversal of direction occurs. The moments in the rotational state minimize length variation while gradually rotating from one end of the interval to the other. The temperature at which the kink occurs is the point at which the two energy penalties trade off in dominance. According to this description, the planar state is characterized by $X_0=X_\varphi=0$. Indeed, this is what comes out of the Ginzburg-Landau model above the kink temperature (see Fig. \ref{quantitykinkplots}). While these quantities vanish at high temperature, their ratio $X_\varphi/X_0$ remains non-zero at all temperatures.

\begin{figure}[ht]
\includegraphics[angle=0,width=\columnwidth]{fig7-quantitykinkplots.eps}
\caption{(Color online) The dimensionless quantities $X_0$ (solid), $X_\varphi/10$ (long dashing; scale reduced for ease of plotting) and $X_\varphi/X_0$ (short dashing) as functions of $x_t$ when $\alpha\approx\pi$. The vertical dotted line indicates $x_t=x_{t,\rm kink}$.}
\label{quantitykinkplots}
\end{figure}

In order to determine the kink temperature, we solve Eqs. (\ref{lengthcondition}) and (\ref{twistcondition}) simultaneously in the vicinity of that transition point, i.e. with $X_0=0$ and $X_\varphi/X_0$ finite but unknown. We find numerically (see Appendix \ref{tkinkappendix})
\begin{equation}
x_{t,\textrm{kink}}\approx -28.1099. \label{kinktemp}
\end{equation}
Additionally, series expanding the conditions (\ref{lengthcondition}) and (\ref{twistcondition}), we can find a first approximation to $X_0(x_t)$ in the vicinity of the kink. The result is that
\begin{equation}
X_0(x_t)\propto(x_{t,\textrm{kink}}-x_t)^{1/2}
\end{equation}
which agrees with standard mean field results for, e.g. the magnetization of a ferromagnet.

We now present another, more transparent, analysis of this transition. We claim that, at high temperatures, the system's only energy extremum is the planar state, while, at low temperatures, the system has access to the planar state, as well as two energetically equivalent rotational states with a gradual turn of either $\pi$ or $-\pi$. As previously described, the system favors the rotational state at low temperatures and ignores the planar state. Such a situation is commonly found in Ginzburg-Landau models, where the free energy has terms quartic and quadratic in a variable or field of interest. Depending on the coefficient of the quadratic term, there will be one fourth-order minimum at the origin or else a maximum at the origin and minima elsewhere. The planar state can be either an energy minimum or an energy maximum, while the rotational states are the symmetry-breaking minima that appear at sufficiently low temperature. By employing an approximation, we will show that this description fits the present system.

It is useful to turn to the scaling variables (\ref{scalingvariables}), in which the free energy functional, Eq. (\ref{LGfreeenergy}), becomes
\begin{multline}
{\cal F}=\frac{b}{\hat{g}} \frac{1}{L^3}\int_{-1/2}^{1/2} d\zeta \,\left[\left(\frac{dX_\Phi}{d\zeta}\right)^2+\frac{X_\varphi^2}{X_\Phi^2}\right.\\
\left.+x_t  X_\Phi^2+ X_\Phi^4\right]
\end{multline}
while the amplitude equation (\ref{Phieom}) reads
\begin{equation}
\frac{d^2X_\Phi}{d \zeta^2}=\frac{X^2_\varphi}{X_\Phi^3}+x_t X_\Phi+2X_\Phi^3.
\label{PhiEqScaled}
\end{equation}
Due to the boundary conditions (\ref{boundaryconditions}) the amplitude profile near the edges, $X_\Phi(\zeta\approx\pm 1/2)$, is largely temperature independent, i.e. it is almost the same above and below the transition temperature. The behavior near $\zeta=0$ must therefore account for the physics of the transition. Near the transition temperature, we have $X_0\approx 0$ and thus $X_\Phi(\zeta)\ll 1$ for $\zeta\approx 0$. Then Eq. (\ref{PhiEqScaled}) reduces to
\begin{equation}
\frac{d^2 X_\Phi}{d\zeta^2}=\frac{X^2_\varphi}{X_\Phi^3}
\end{equation}
with solution
\begin{equation}
X_\Phi(\zeta)=\sqrt{X_0^2+\left(\frac{X_\varphi}{X_0}\right)^2\zeta^2}\equiv\sqrt{X_0^2+M^2\zeta^2}, \label{asymptamp}
\end{equation}
defining $M\equiv X_\varphi/X_0$.

The free energy integral can now be computed in closed form using the asymptotic expression (\ref{asymptamp}). The expression for $X_\Phi(\zeta)$ is only valid up to some cutoff $\zeta=Y<1/2$, but the free energy from the edges of the interval ($|\zeta|>Y$) will not contribute to the behavior of the system, provided that $Y$ is large enough. The result of the integral is
\begin{multline}
{\cal F}\approx\frac{2bY}{\hat{g}L^3}\left[X_0^4 + \left(x_t+\frac{2}{3} Y^2 M^2\right)X_0^2\right.\\
\left. + \left(\frac{1}{5} Y^4 M^4 + \frac{1}{3}x_t Y^2 M^2  + M^2\right)\right] \label{approxfreeenergy}
\end{multline}
which is quartic in $X_0$. Recall that $X_0$ is the (scaled) amplitude of the order parameter at the center of the interval. In the planar state, $X_0=0$, while rotational states have $X_0>0$. Therefore we expect to always see an extremum at $X_0=0$, corresponding to the planar state, which will be a free energy minimum at high temperature and a maximum at low temperature. When the planar state is a maximum, two minima (rotational states) with $|X_0|>0$ should emerge. Indeed, this behavior is clear from the form of (\ref{approxfreeenergy}). The position of the non-zero minimum is found to be
\begin{equation}
X_0=\pm\sqrt{-\frac{1}{2}\left(x_t+\frac{2}{3} Y^2 M^2\right)}, \label{B0minimum}
\end{equation}
which is only real for sufficiently low temperature: $x_t\le -2 Y^2 M^2/3<0$. The transition occurs when equality holds. This does not fully determine the temperature of the transition because both $Y$ and $M$ are functions of $x_t$. The additional constraints are afforded by matching the hyperbolic expression (\ref{asymptamp}) for $X_\Phi(\zeta)$ with another expression correct near the edge of the interval.

When $\zeta\approx \pm 1/2$, $X_\Phi\to\infty$ and, according to (\ref{PhiEqScaled}) the amplitude profile is determined by
\begin{equation}
\frac{d^2 X_\Phi}{d\zeta^2}=x_t X_\Phi + 2 X_\Phi^3
\end{equation}
which, with $x_t<0$, is solved by
\begin{equation}
X_\Phi(\zeta)=\sqrt{|x_t|}\,\csc\left[\sqrt{|x_t|}\left(\frac{1}{2}-|\zeta|\right)\right].
\label{largezeta}
\end{equation}
Now imposing the continuity of $X_\Phi(\zeta)$ and $X'_\Phi(\zeta)$ from (\ref{asymptamp}) and (\ref{largezeta}) at $\zeta=Y$ allows us to solve for the parameters at the transition point. Proceeding numerically, we find:
\begin{eqnarray}
&x_{t,\textrm{kink}}\approx-22.4587,\qquad M_{\textrm{kink}} \approx 19.4498,\nonumber\\
&X_{0,\textrm{kink}}=0,\qquad Y_{\textrm{kink}}\approx0.2984.
\end{eqnarray}
Compared to the exact numerical results obtained in Appendix
\ref{tkinkappendix}, these values are consistent as a first approximation, as
are the behaviors of $X_0(x_t)$ and $M(x_t)$. In particular, the leading order
contribution to $X_0(x_t)$ goes as $(x_{t,\textrm{kink}}-x_t)^{1/2}$, in
agreement with the power law previously found. Finally, it is easy to
show that at the kink temperature the rate of the change of the phase in the
middle of the system diverges. Indeed, from Eq. (\ref{Pvarphidef}) and using the
definitions (\ref{scalingvariables}) one obtains
\begin{equation}
 \left[\frac{X_\Phi(\zeta)}{X_0}\right]^2\frac{d\varphi(\zeta)}{d
\zeta}=\frac{X_\varphi}{X_0^2}=\frac{M}{X_0}.
\label{relMkink}
\end{equation}
Thus, in the limit $\zeta\to 0$ one derives at
$x_t=x_{t,\textrm{kink}}$ that $(d \varphi(\zeta)/ d\zeta)|_{\zeta=0}=M/X_0\to
\infty$, since $X_{0,\textrm{kink}}=0$. Therefore, at the kink temperature all
the change in the phase of the moments happens at the middle of the system,
where there length becomes zero. The phase of the moments jumps there from $0$
to $\pi$.

\section{Discussion and concluding remarks}
\label{discussion}

We have studied the $O(2)$ model in a three-dimensional film geometry, and find
identical predictions from lattice and continuum mean-field theories. Consistent
with systems of similar type, the Casimir force with symmetric boundary
conditions ($\alpha=0$) is attractive, while the Casimir force with
anti-symmetric boundary conditions ($\alpha=\pi$) is repulsive. In particular,
the critical Casimir force, i.e. $F_{\rm Casimir}$ at the bulk transition
temperature, changes from repulsive to attractive. This is a standard result,
but we also find intermediate scenarios when $0<\alpha<\pi$ which feature
critical Casimir forces, and scaling functions for the Casimir force
(illustrated in Fig. \ref{lattice_casimir_force}), different from those of the
symmetric and anti-symmetric cases. The
Casimir force may therefore be continuously adjusted at constant temperature by
varying the twist $\alpha$, or at constant twist by varying the temperature.

Additionally we find that, when the boundary conditions are perfectly anti-symmetric ($\alpha=\pi$), the system undergoes a phase transition at a temperature below the bulk critical temperature. We are able to understand this transition as a symmetry-breaking effect: at high temperatures the moments of the system are confined to a plane, while at low temperatures they rotate about the $z$-axis by either $\pi$ or $-\pi$ to satisfy the boundary conditions. The high temperature behavior tracks that of an Ising model whose order parameter is always simply up or down, but our system departs from that behavior at the transition point, once moments find it energetically favorable to rotate.

\acknowledgments{D. Dantchev acknowledges the partial financial support of
Bulgarian NSF, grant No. DO 171/08. J. Rudnick acknowledges partial support from the NSF through grant DMR-1006128.}


The city of Sheffield has long been synonymous with cutlery and steel, and most previous books have understandably concentrated on the momentous changes which industrialization wrought on the area over the last two hundred years. The figures are astonishing: as early as the seventeenth century three out of every five men in the town worked in one branch or another of the cutlery trades and, in all, Sheffield had a smithy to every 2.2 houses; a hundred years later there were as many as six watermills per mile on rivers such as the Don, Porter and Rivelin, driving a wide range of industrial machinery and processes; local innovations included Old Sheffield Plate, crucible steel and stainless steel; during the mid-nineteenth century 60 per cent of all British cutlers worked in the Sheffield area, and the region manufactured 90 per cent of British steel, and nearly half the entire European output; small, specialized workshops producing a wide range of goods such as edge-tools and cutlery existed side by side with enormous steel factories (it has been estimated that in 1871 Brown s and Cammell s alone exported to the United States about three times more than the whole American output). Yet, as David Hey shows, the city s history goes back way beyond this. Occupying a commanding position on Wincobank, high above the River Don, are the substantial remains of an Iron Age hillfort, built to defend the local population. Celts, Vikings and Anglo-Saxons came and left a legacy recalled in many local names. By the twelfth century William de Lovetot had built a castle at the confluence of the Don and the Sheaf, and it is likely that is was he who founded the town of Sheffield alongside his residence. A century later can be found the first reference to a Sheffield cutler, so industry in the area can be said to be at least 700 years old, and no doubt stretches back even further."


We have a product we would like to add a video to. Is this possible?
I would suggest that you paste the embed code into the long description field of the product in Store Manager. Please make sure the dimensions are not too big. You might also think about having the video pop up in a different window if the customer clicks on it.


GLIDR puts feedback, discovery and validation at the center of your product development process. GLIDR gives you tools to set up a robust product roadmap, test your Ideas through Research & Experiments, and connect validated Evidence.
As an Enterprise Team Member, you may be added to multiple Projects by the Account Owner, allowing you to work on multiple projects simultaneously.
You can find the list of Projects in your Organization in the menu on the left-hand side of the screen. Projects you have been added to will appear with a filled in square icon to the left of the Project name.
GLIDR offers a Business Canvas and Product Canvas for each Project, allowing you to distinguish the ideas that are directly related to your market vs. related to your product.
You can categorize your Ideas by Roadmap, Status and Theme. When you start your Project, you'll find that the Roadmap and Status categories are already filled out with suggested titles, but your Account Owner or Project Admin can change them to fit your needs.
Keep it brief! There's a 100 character limit, but you can click on your Idea to add more detail and expand on your thoughts.
Once you've started adding and validating or invalidating your Ideas, you may decide to prioritize them according to your needs, or progress them as they move through the Product Roadmap. This can be done by dragging and dropping the individual cards, which can also be done in List View.
Once you've added your Evidence, you can connect it to your Ideas, or vice-versa. Clicking on the Connections tab will show you any existing connections you've made, and you can also add or edit connections to your Ideas.


A beautifully decorated, smart and spacious lateral flat in a well maintained purpose build block offering comfortable living space with modern touch finished by interior designers. The flat compromises of a bright reception room, specious master bedroom with king size bed, and en-suite shower room, further double bedroom, bathroom and modern fully fitted kitchen. Located in the heart of Chelsea minutes from King's Road with excellent shopping and transport facilities. Viewings highly recommended.


text:
Epic Warrior Women (2018) - TV Show
Epic Warrior Women highlights the women who fought just as courageously, skillfully and fiercely as male warriors of their time. With stunning re-enactments, original costumes, historic locations and CG animation, the three-part series chronicles the military achievements of the original Amazon horse warriors of the Steppe, probes the highly-trained female gladiators of Ancient Rome and concludes with the feared all-female regiments of Dahomey that shaped African history.
The three-part series chronicles the military achievements of the original Amazon nomad warriors of the Central Asian grasslands in 6th century BC, examines the highly trained female gladiators of Ancient Rome in 2nd Century AD, and discovers the terrifying all-female slave raiding regiments of Dahomey, who shaped African history at the end of the 19th century.
Epic Warrior Women Airs: Monday
Epic Warrior Women Status: Break
EPIC WARRIOR WOMEN - General Information
Series Premiere: March 19, 2018
Network: Smithsonian Channel
Airs: Monday at 08:00 pm
EPIC WARRIOR WOMEN - Season 1 -- 3 Episodes
1x1 -- Mar 19, 2018 -- Amazons
1x2 -- Mar 26, 2018 -- Gladiatrix
1x3 -- Apr 02, 2018 -- Africa's Amazons
Additional EPIC WARRIOR WOMEN Information
Epic Warrior Women on IMDb
Epic Warrior Women on TVmaze
Epic Warrior Women Cast
Adrienne Mayor
........... as Herself 2 episodes, 2018
Olzhas Togymbet
.......... as Enemy Chief 1 episode, 2018
Aimee Akapo Toulan
....... as Sukeoun 1 episode, 2018
Rebecca Dimech
........... as Gladiatrix Ardala 1 episode, 2018
Moa Enqvist Stefansdotter
as Signe 1 episode, 2018
Gaziza Tleubay
........... as Alkaya 1 episode, 2018
Ceciline Abissi
.......... as Nansica 1 episode, 2018
Dinara Baktybaeva
........ as Aella 1 episode, 2018
Anthea Zammit
............ as Gladiatrix Zonena 1 episode, 2018
Khulan Chuluun
........... as Samsi 1 episode, 2018
Sandra Cabrera Lopez
..... as Gladiatrix Achillia 1 episode, 2018
Nathalie Hounvo Yekpe
.... as Kings wife Obaseki 1 episode, 2018
Nicolas Dravo Houenou
.... as King Behanzin 1 episode, 2018
Aida Kubeeva
............. as Little Alkaya 1 episode, 2018
Alisa Vanlint
............ as Gladiatrix Alicia 1 episode, 2018
Ondassyn Bessikbassov
.... as Scythian Chief 1 episode, 2018
Jean Michel Coulon
....... as Ambassador Bayol 1 episode, 2018
Henry Zammit Cordina
..... as School Owner 1 episode, 2018
Suzanne Blier
............ as Herself 1 episode, 2018
Sean Buhagiar
............ as Roman Sponsor 1 episode, 2018
Terri Ochiagha
........... as Herself 1 episode, 2018
Marilu Vella
............. as Sponsor's Wife 1 episode, 2018
Zhibek Zhappasbayeva
..... as Alkaya's Mother 1 episode, 2018
Susuji Behanzin
.......... as Himself 1 episode, 2018
Bryan Bonnici
............ as Roman Writer 1 episode, 2018
Martine D'Souza
.......... as Herself 1 episode, 2018
Michael Tabone
........... as Druid 1 episode, 2018
Hermann Parzinger
........ as Himself 1 episode, 2018
Yahara Samare
............ as Gladiator Instructor 1 episode, 2018
Neville Refalo
........... as Arena Referee 1 1 episode, 2018
Zeinolla Samashev
Ray Degabriele
Almas Ordabayev
Ron Briffa
............... as Condemned Prisoner 1 episode, 2018
Michael Carter
........... as Himself 1 episode, 2018
Marcus Junkelmann
........ as 1 episode, 2018
Margareta Pettersson
..... as Schaman woman 1 episode, 2018
Amina Yerlan
............. as Samsi's Daughter 1 episode, 2018
Jaap Hogendoorn
.......... as Gladiator - Threax
Epic Warrior Women Torrent Download
Epic Warrior Women S01E02 Gladiatrix 480p x264-mSD [eztv] 235.79 MB 1 mo 4
Epic Warrior Women S01E01 Amazons INTERNAL 480p x264-mSD [eztv] 228.00 MB 1 mo 7
Epic Warrior Women S01E03 Africas Amazons 720p WEB H264-UNDERBELLY [eztv] 1.04 GB 1 mo 1
Epic Warrior Women S01E03 Africas Amazons WEB H264-UNDERBELLY [eztv] 669.32 MB 1 mo 6
Epic Warrior Women S01E02 Gladiatrix 720p WEB H264-UNDERBELLY [eztv] 1.04 GB 1 mo 1
Epic Warrior Women S01E01 Amazons WEB H264-UNDERBELLY [eztv] 669.52 MB 1 mo 5

redpajama_set_name: RedPajamaCommonCrawl


The Rain Gods are delighted to support NACUE's annual Varsity Pitch programme for 2016. At the semi-finals on 19 October, Double Dutch Drinks were selected as the winner of the Creative and Design Category that we have sponsored.
Well done to Joyce and Raissa and good luck for the final!


Mahale Mountains National Park is home to some of the Africa last remaining wild chimpanzees and estimated to have roughly more than 800 residing in this area. Some of them are habituated to human.
This area was traditionally inhabited by the Batongwe and Holoholo people. Tongwe is a small ethnic group of Bantu, therefore this park is situated at the center of Tongwe Land that includes south of the Kigoma Region and north of the Katavi Region. The people had been highly attuned to the natural environment, living with visually no impact on the ecology.
The park was officially declared a National Park in 1985.
Set deep in the heart of African interior, a road free park located about 128 km south of where Stanley uttered that immortal greeting '' Dr Livingstone, I presume''. It is a scene reminiscent of an Indian Ocean Island beach idyll. Silky white coves hem in the sapphire waters of Lake Tanganyika, overshadowed by a chain of wild, jungle draped peaks towering almost 2 km above the shore; the remote, pristine and mysterious Mahale Mountains.
1.By Air: Mahale can be reached by chartered flight from Kigoma.Dar es salaam,Mwanza, Arusha or any airport in Tanzania.
(8-10hrs) to Lagosa point then arrange a boat with the park to pick you to the park.
The park can be visited throughout a year; however, the best time is during the dry season (May – November) but can be visited also in wet season (December - April), however you will need additional gears such as rain coat and boot for trekking and walking safari.
This activity is conducted from 06.30 to 18.30HRS.
Walking safari – Short and long walking safari are normally conducted along the existing trails depending on tourist preference such as enjoying physical features, bathing in pools and waterfalls as well as hiking safari.
Kayaking: This is the activity that offers a visitor an experience to move across water by using kayak. The equipment is offered by tented camps for visitors interested to do kayaking and is always done during the day when the lake is calm.
This is a mode of underwater diving in which a scuber diver uses a self- contained underwater breathing apparatus (scuba) to breath underwater, and this activity is conducted in lake show to give opportunity for visitors to view colored fish under the rock crevices.
Hiking: The hiking towards different peaks within Mahale Mountains National Park is a bit challenging but provides unique and opportunity to experience a diverse forest fauna and flora.
Mpira public Campsite: Located north of Kasiha Tourism Centre and approximately 7.5 km from the park headquarter. It can accommodate up to fifteen (15) people at time.
Sisindiko public Campsite: Located south of Kasiha Tourism Centre approximately 3.5km from mango house bandas. It can accommodate 20 people at a time.
Filming fee: charged per person/day and covers conservation, camping and filming fees. This is applicable to all nationalities.
These are species of Ape which is the most humanity's closest living relatives. Research of wild chimpanzees teaches us ''man's place in nature'' and provides us with key information that is relevant to re-constructing the social life of early man. Long term field studies on chimpanzees revealed many important facts about them such as hunting, tool-using, medicinal use of plants, social intelligence and existence of culture. Mahale National Park has the highest remaining wild chimpanzee's population in Africa.
Keep to the authorized trails.
Do not disturb wild animals in any way. Do not make noise.
Be considerate to fellow visitors – do not disturb them or the animals they are watching.
Do not take any pets or guns into the park.
Do not uproot, pick, cut or damage any plant or be in possession of any part of a plant indigenous to the park.
Do not light any fire or discard any burning object.
Do not discard any litter.
Between 7:00p.m and 6:00a.m. remain in the immediate vicinity of designated accommodation facilities (tented camps, tourist bandas, rest house or campsites).
A permit is valid for single entry within 24 hours only.


text:
#pragma once
#include <aws/eks/EKS_EXPORTS.h>
#include <aws/core/utils/memory/stl/AWSString.h>
#include <utility>

namespace Aws
{
namespace Utils
{
namespace Json
{
  class JsonValue;
  class JsonView;
} // namespace Json
} // namespace Utils
namespace EKS
{
namespace Model
{

  /**
   *
An object representing a node group launch template specification. The launch
   * template cannot include
   * SubnetId ,
   * IamInstanceProfile ,
   * RequestSpotInstances ,
   * HibernationOptions , or
   * TerminateInstances , or the node group deployment or update
   * will fail. For more information about launch templates, see
   * CreateLaunchTemplate  in the Amazon EC2 API Reference. For more
   * information about using launch templates with Amazon EKS, see Launch
   * template support in the Amazon EKS User Guide.

Specify either
   * name or id, but not both.

See Also:

   * AWS
   * API Reference

   */
  class AWS_EKS_API LaunchTemplateSpecification
  {
  public:
    LaunchTemplateSpecification();
    LaunchTemplateSpecification(Aws::Utils::Json::JsonView jsonValue);
    LaunchTemplateSpecification& operator=(Aws::Utils::Json::JsonView jsonValue);
    Aws::Utils::Json::JsonValue Jsonize() const;

    /**
     *
The name of the launch template.

     */
    inline const Aws::String& GetName() const{ return m_name; }

    /**
     *
The name of the launch template.

     */
    inline bool NameHasBeenSet() const { return m_nameHasBeenSet; }

    /**
     *
The name of the launch template.

     */
    inline void SetName(const Aws::String& value) { m_nameHasBeenSet = true; m_name = value; }

    /**
     *
The name of the launch template.

     */
    inline void SetName(Aws::String&& value) { m_nameHasBeenSet = true; m_name = std::move(value); }

    /**
     *
The name of the launch template.

     */
    inline void SetName(const char* value) { m_nameHasBeenSet = true; m_name.assign(value); }

    /**
     *
The name of the launch template.

     */
    inline LaunchTemplateSpecification& WithName(const Aws::String& value) { SetName(value); return *this;}

    /**
     *
The name of the launch template.

     */
    inline LaunchTemplateSpecification& WithName(Aws::String&& value) { SetName(std::move(value)); return *this;}

    /**
     *
The name of the launch template.

     */
    inline LaunchTemplateSpecification& WithName(const char* value) { SetName(value); return *this;}

    /**
     *
The version of the launch template to use. If no version is specified, then
     * the template's default version is used.

     */
    inline const Aws::String& GetVersion() const{ return m_version; }

    /**
     *
The version of the launch template to use. If no version is specified, then
     * the template's default version is used.

     */
    inline bool VersionHasBeenSet() const { return m_versionHasBeenSet; }

    /**
     *
The version of the launch template to use. If no version is specified, then
     * the template's default version is used.

     */
    inline void SetVersion(const Aws::String& value) { m_versionHasBeenSet = true; m_version = value; }

    /**
     *
The version of the launch template to use. If no version is specified, then
     * the template's default version is used.

     */
    inline void SetVersion(Aws::String&& value) { m_versionHasBeenSet = true; m_version = std::move(value); }

    /**
     *
The version of the launch template to use. If no version is specified, then
     * the template's default version is used.

     */
    inline void SetVersion(const char* value) { m_versionHasBeenSet = true; m_version.assign(value); }

    /**
     *
The version of the launch template to use. If no version is specified, then
     * the template's default version is used.

     */
    inline LaunchTemplateSpecification& WithVersion(const Aws::String& value) { SetVersion(value); return *this;}

    /**
     *
The version of the launch template to use. If no version is specified, then
     * the template's default version is used.

     */
    inline LaunchTemplateSpecification& WithVersion(Aws::String&& value) { SetVersion(std::move(value)); return *this;}

    /**
     *
The version of the launch template to use. If no version is specified, then
     * the template's default version is used.

     */
    inline LaunchTemplateSpecification& WithVersion(const char* value) { SetVersion(value); return *this;}

    /**
     *
The ID of the launch template.

     */
    inline const Aws::String& GetId() const{ return m_id; }

    /**
     *
The ID of the launch template.

     */
    inline bool IdHasBeenSet() const { return m_idHasBeenSet; }

    /**
     *
The ID of the launch template.

     */
    inline void SetId(const Aws::String& value) { m_idHasBeenSet = true; m_id = value; }

    /**
     *
The ID of the launch template.

     */
    inline void SetId(Aws::String&& value) { m_idHasBeenSet = true; m_id = std::move(value); }

    /**
     *
The ID of the launch template.

     */
    inline void SetId(const char* value) { m_idHasBeenSet = true; m_id.assign(value); }

    /**
     *
The ID of the launch template.

     */
    inline LaunchTemplateSpecification& WithId(const Aws::String& value) { SetId(value); return *this;}

    /**
     *
The ID of the launch template.

     */
    inline LaunchTemplateSpecification& WithId(Aws::String&& value) { SetId(std::move(value)); return *this;}

    /**
     *
The ID of the launch template.

     */
    inline LaunchTemplateSpecification& WithId(const char* value) { SetId(value); return *this;}

  private:

    Aws::String m_name;
    bool m_nameHasBeenSet;

    Aws::String m_version;
    bool m_versionHasBeenSet;

    Aws::String m_id;
    bool m_idHasBeenSet;
  };

} // namespace Model
} // namespace EKS
} // namespace Aws

redpajama_set_name: RedPajamaGithub


text:
New homeless liaison connects kids with resources
Casey Driscoll outside the HEART Locker in Kalispell on Thursday, Jan. 14. (Casey Kreider/Daily Inter Lake)
by HILARY MATHESON
Daily Inter Lake | January 16, 2021 12:00 AM
Casey Driscoll is the new homeless education liaison for Kalispell Public Schools and Evergreen School District and director of the HEART Program.
Since July, Driscoll has worked alongside former liaison and director Nichole Heyer as an AmeriCorps Vista member.
Heyer took a position as the youth homelessness demonstration project navigator for the Montana Legal Services Association.
As the homeless education liaison, Driscoll works directly with students identified as homeless under the McKinney-Vento Homeless Assistance Act to connect them with resources and support such as free meals and transportation to and from school. From year to year, 300 to 400 students in the two school districts are identified as homeless under the act, which defines homelessness as lacking a fixed, regular or adequate nighttime residence.
"I make sure the students are aware of their educational rights," Driscoll said about her primary role as a liaison.
Students do not have to be sleeping outside, in their car or a shelter to be considered homeless under this definition. It includes students who live in hotels, motels, trailers and campgrounds for example, or they could be living with friends or family members who aren't legal guardians.
"They might be in a position where they're OK and have adequate shelter, but it isn't guaranteed. This would be an example of a child staying with one or both of their parents, but their nighttime residence is with family friends, an aunt or uncle. It's still considered homeless because the parents aren't on the lease and they could get kicked out at any time."
"We work with teachers, counselors and staff at all levels, to help us identify those kids," Driscoll said. "They have children in the classroom all day and get a sense of a child's home life. We also use enrollment forms to identify which children are living in temp or a temporary situation."
The HEART Program, housed at 514 E. Washington St., is where students get connected with other basic necessities such as clothing.
As the program director, Driscoll manages the HEART Locker, a place for students to shop for free clothing, bedding and hygiene products. The HEART Locker Learning Center, which offers students a safe place to study, hang out or find part-time jobs after school; and student food pantries located in schools.
"It's a really unique position," she noted.
DRISCOLL HOLDS a master's degree in psychology from Stony Brook University where she worked in the career center and became interested in a career helping students. After graduation, she moved to Florida where she worked as a college recruiter. Her job entailed recruiting non-traditional students and she got experience helping people from a variety of backgrounds.
"They could be from low socioeconomic backgrounds, have a disability, be the first in their family to attend college — basically students at-risk of not continuing their education," Driscoll said. "I think that's where I realized I enjoy what I'm currently doing."
When she took a job in sales and quality assurance back in New York, Driscoll quickly learned she wanted to go back to working with students in an educational or nonprofit setting. AmeriCorps provided the opportunity to transition back into her desired career path.
Driscoll said she was very intentional about her AmeriCorps placement and looked for a small nonprofit setting where she could "wear many hats," and become "well-rounded."
When the liaison and director position opened up, Driscoll was excited.
"I wanted a job I could get behind and support," she said.
Currently, Driscoll said she is working on a website of community resources, applications or how-tos that are specific to the needs of homeless students in the valley. She said it's not always clear to students what to search for if they don't know what's available.
"We can still provide here on a case management basis, but sometimes it is easier to jump online and find it," Driscoll said.
While the learning center is currently closed as a precaution due to COVID-19, the locker remains open by appointment.
For more information or to make an appointment call 751-3630, email driscollc@sd5.k12.mt.us, or visit https://www.kalispellheartprogram.org.
Reporter Hilary Matheson may be reached at 758-4431 or hmatheson@dailyinterlake.com.


text:
Padel – sport rakietowy. W Stanach Zjednoczonych i Kanadzie sport jest znany jako paddle. Padel zawiera w sobie zarówno elementy tenisa jak i squasha.

Historia

Sport pochodzi z Acapulco w Meksyku. Stworzony został przez Enrique Corcuera w 1969. Głównym inicjatorem rozwoju padla był Alfonso de Hohenloh, który wybudował pierwszy kort w Club Marbella w Hiszpanii. W 1975 nowa dyscyplina zawędrowała do Argentyny, stając się w ciągu kilku lat najpopularniejszym sportem praktykowanym w tym kraju. W kolejnych latach padel zyskał popularność w Ameryce i na kontynencie europejskim. Popularność sportu wzdłuż wybrzeża Costa del Sol w południowej Hiszpanii i Algarve w południowej Portugalii ujawniła ją dużej liczbie turystów brytyjskich, co przyczyniło się do zwiększenia popularności tego sportu w Wielkiej Brytanii i założenia w 2011 federacji brytyjskiej.

Sport ten jest drugą dyscypliną po piłce nożnej pod względem popularności w Hiszpanii. W największym państwie na Półwyspie Iberyjskim w padla gra kilka milionów osób w wieku od 4 do ponad 80 lat.

W Polsce zrzesza się coraz więcej miast z aspiracjami do uprawiania tej dyscypliny sportu. Powstała także Polska Federacja Padla. W 2017 roku powołana została kadra narodowa, która swój debiut odbyła na Mistrzostwach Europy w Portugalii tego samego roku.

Nazwa "Paddle" pochodzi od słowa "wiosło". Drugą nazwą jest "Tenis Łopatkowy", wywodzi się ona od rakiety, ponieważ była ona wykonywana z drewna i miała kształt podobny do łopatki.

Zasady gry
Gra w padla odbywa się na korcie o sztucznym podłożu, o wymiarach 20x10 m. Aktualnie większość kortów do padla ma ogrodzenie wykonane z drucianej siatki z "oczkami". Tylna i część bocznych ścian wykonana jest z hartowanego szkła bądź bloczków betonowych. Na powierzchni klasycznego kortu do tenisa mieszczą się trzy korty do padla.

Piłki są podobne do tenisowych, ale mniej sprężyste, a co za tym idzie niżej się odbijają. Rakieta do padla nie posiada naciągu jak rakieta tenisowa, wykonana jest z bardzo lekkich, wytrzymałych materiałów, np. włókna węglowego. Musi posiadać nieelastyczny sznurek przymocowany do rączki, który podczas gry obligatoryjnie jest przekładany na nadgarstek jako zabezpieczenie. Wymiary rakiety nie mogą przekroczyć 45,5 cm długości, 26 cm szerokości i 38 mm grubości. W płaszczyźnie główki zwykle znajduje się około 60 otworów o średnicy 13 mm każdy. Wedle wytycznych Międzynarodowej Federacji Padla obowiązkowy jest przynajmniej jeden otwór.

Mecz padlowy dzieli się na sety, one na gemy, które składają się z punktów. Celem gry jest wygranie dwóch setów. Punkty liczy się jak podczas meczu tenisa ziemnego. Rozgrywkę każdego z nich rozpoczyna  serwis, który w padlu wykonuje się "od dołu". Gracz serwujący odbija piłkę o ziemię, a następnie przebija ją po przekątnej na karo serwisowe rywali. W momencie kontaktu z rakietą piłka musi znajdować się poniżej linii pasa serwującego. W czasie wymiany piłka może odbić się tylko raz o powierzchnię kortu, a następnie musi być przebita przez gracza na drugą stronę siatki – do przeciwnika. Gracz ma wolny wybór, czy zagrać piłkę od razu z powietrza, czy poczekać na uderzenie piłki od powierzchni kortu. Piłka musi trafić dokładnie w pole kortu, bez wcześniejszego uderzenia o metalową siatkę lub ścianę po stronie rywali (dozwolone jest odbicie piłki o ścianę po własnej stronie kortu). Po uderzeniu o powierzchnię kortu piłka może odbić się od ogrodzenia (drucianej siatki) i/lub ściany kortu. Jeśli piłka uderzy o drucianą siatkę, zanim zostanie zwrócona do przeciwnika – punkt zdobywa przeciwnik. Tak samo dzieje się, gdy piłka po uderzeniu o kort wróci na stronę rywali sama, bez kontaktu z rakietą oraz, gdy wyleci poza kort i dotknie podłoża. Gra odbywa się według opisanych zasad do momentu, gdy piłka odbije się więcej niż jeden raz od powierzchni którejś ze stron lub trafi w siatkę dzielącą kort bądź zostaną złamane inne reguły gry.

Przypisy

Linki zewnętrzne
  Strona Polskiej Federacji Padla
 World Padel Tour
 Official website of the International Padel Federation
 Kacper Bartosiak, Tenis uwięziony w klatce. Padel wkracza na salony

redpajama_set_name: RedPajamaWikipedia


The Zoning Board of Appeal (ZBA) on April 9 approved the proposal for a 21 unit residential structure at 50 Stedman St. This proposal has gone through a lengthy process with neighborhood associations, especially the Stonybrook Neighborhood Association (SNA), who voted not to oppose the project in February.
The building was before the ZBA for violations of insufficient off-street parking, forbidden use, and excessive FAR. Architect Luccio Trabucco said the project has gone through an 18 month process, through which they have been able to meet "most, if not all concerns of the abutters." Trabucco confirmed that they have met with the Stonybrook Neighborhood Association and the Stonybrook Neighborhood Association, who both expressed non-opposition to the project.
The proposed residential building is in an industrial subdistrict, and the current site of Wentworth Service Station. The SNA was originally concerned about what would happen to Bill Miceli, the owner of the auto repair shop, as well as the setbacks for the abutting property. These issues were addressed by the developer, leading to the SNA's vote of non-opposition.
Trabucco said the building would consist of 15 two bedroom units, six one bedroom units, and 21 parking spaces. Four of the units will be affordable, and the Floor Area Ratio is a little over 2.
Enrique Peen from the Mayor's Office of Neighborhood Services spoke in support of the project, and said that after "extensive community meetings," there are "no further concerns." Councilor O'Malley and the Carpenter's Union also went on record in support. No one spoke in opposition of the project.


text:
Category Archives: Sona NSS News
Sona bags University level NSS Awards
Anna University, Chennai appreciates all the affiliated engineering colleges in Tamilnadu for the NSS activities and gives award every year. The award function for this year was conducted on 28.09.2018 which was presided over by Prof. M. K. Surappa, Vice Chancellor, Anna University and Dr.R.Ramesh, NSS Coordinator, Anna University. Sona College of Technology received two awards for the year 2017-18. The award was given by Prof. M. K. Surappa, Vice Chancellor, Anna University. The best NSS Programme Officer award was received by Dr. P.Iyyanar AP/IT, and the best female NSS volunteer award was received by Ms.S.Soniya, Final year FT. The Management, Directors, the Principal and all the HOD's congratulate the award winners.
This entry was posted in Sona NSS News on October 1, 2018 by Sona College of Technology.
BEST NSS VOLUNTEER AWARD BY FINAL YEAR FT STUDENT
Final year Fashion Technology student Ms. S.Sonia Received best NSS volunteer award from Anna University on 28.09.2018.
This award is issued in the University level NSS Award function at Anna University, Chennai.
This entry was posted in Sona College News, Sona FT News, Sona NSS News on September 28, 2018 by Sona FT Department.
Sona's NSS wing bags two prestigious awards
Anna University, Chennai appreciates all the affiliated engineering colleges in Tamilnadu for the NSS activities and gives award every year. The award function for this year was conducted on 25.10.2017 which was presided over by Dr.T.V.Geetha Dean CEG & Member, Convener Committee, Anna University, Dr.Madan Mohan Goel Director, RGNIYD and Dr.S.Ganesan Registrar, Anna University. Sona College of Technology received two awards for the year 2016-17. The award was given by Dr.Madan Mohan Goel Director, RGNIYD. The best NSS Programme Officer award was received by Mr. P. Jaya Prakash AP/IT, and the best female NSS volunteer award was received by Ms. A.Lavina Anitha, Final Civil. The Management, Directors, the Principal and all the HOD's congratulate the award winners.
Programme officer award 1 among 12
Female volunteers award 1 among 10
This entry was posted in Sona College News, Sona NSS News on November 2, 2017 by Sona College of Technology.
World Blood Donor Day'16
On account of 'World Blood Donor Day ', our Sona NSS wing in association with SKS Hospital had a 'Blood Donation Awareness' Rally on 18th June 2016.
Rally was inaugurated by SKS Hospital Director Dr.Suresh K Viswanathan at Sona College of technology. The Rally took place from our college premises to SKS hospital with 70 NSS volunteers between 7.00 am and 8.45 am. Herewith attached few snapshots of the awareness rally.
Our Sona NSS wing has 'Blood Donor club' through which we are actively donating more than 300 units of blood every year in and around Salem, also we educate the freshers about blood donation to save the life.
This entry was posted in Sona NSS News and tagged Blood Donation Awareness, Blood Donor club, Sona NSS wing, World Blood Donor Day on June 20, 2016 by Sona College of Technology.


DO YOU NEED A COLORING BOOK?
I've had the privilege of working with great people to create great coloring books. If you would like a coloring book for your city, your business, or for a special project, I'd love to make that happen.


text:
ag analytics
Job Theory: The theoretical foundation of predictable growth
Organizations around the world have devoted countless resources to the challenge of innovation through the years. Companies spend billions every year on compiling data, acquiring tools and hiring highly skilled personnel to succeed with innovation.
The innovation processes within companies are perfected down to every little detail — whether it is data collection, data analysis, requirements, measurements, risk or design. It might appear as though companies have mastered an awfully precise, scientific process. But in reality, innovation is still a painful hit-or-miss game for most companies and leaves innovation to chance.
Naturally, the question about how companies can successfully innovate with greater predictability arises. In Competing Against Luck, Clayton Christensen (2016) presents the answer in the form of a theory on customer choice, of which one of the powerful key components is that customers don't simply buy products and services — they hire them to do a job in their lives that resolves their struggles for progress.
Why is Jobs to Be Done important?
The fundamental problem with innovation in most companies is that they ask the wrong questions. In the age of Big Data, masses and masses of data are compiled and analyzed with advanced techniques and tools. But many companies tend to look for patterns and cross-references to describe customers, linking characteristics to purchase decisions — the "what" — but failing to uncover the true cause of customers purchasing products and services — the "why."
Many companies are simply working towards the wrong goal, getting better and better at the wrong things. Instead, innovators should start asking themselves the right question to get the right answers:
What causes a customer to purchase and use a particular product or service?
At its heart, Jobs Theory explains why customers pull certain products and services into their lives. It reveals the causal mechanisms behind purchase decisions and, furthermore, where to look for successful new product ventures. Consequently, innovation success becomes predictable, and you can compete against luck while others are still counting on it.
Recommended watch: Clayton Christensen on "The Milk Shake Dilemma" — an illustration of uncovering a job
Defining the job
What are innovators missing then? Christensen argues that the innovation process has been lacking a theory as an essential tool to solving the innovators' challenges — a theory that provides an answer to the question of what causes a customer to purchase and use a particular product or service. Jobs Theory provides this answer. Putting it simply, customers don't just buy products and services; they pull them into their lives to make progress in particular circumstances.
Christensen calls this progress a "Job to Be Done" — or just a "job" — and customers can either "hire" or "fire" products to fulfill or solve these jobs. More specifically, a job consists of three components: progress, circumstance and dimensional complexity.
The definition of a job is the progress that a person is trying to make in a particular circumstance. The choice of the word "progress" is deliberate, as it represents movement towards a goal or an aspiration. A job is thus always a process to make progress, and it entails all the struggles and barriers customers encounter when seeking to solve their jobs.
A certain desired progress is always strongly influenced by the circumstance in which it arises. The circumstance — meaning the specific context — is fundamental for defining a job, and thus for creating successful.
Several questions could be important to ask when probing the circumstance of the job, such as who, when, where, while doing what and what will you be doing next, but also extends to other contextual factors, such as life stage, family status or financial status. Managers usually constrain themselves to looking at product attributes, customer characteristics, trends or competitive response, but those are insufficient factors when predicting customer behavior. It is all about making the job and its specific context the unit of analysis rather than other factors that correlate with, but don't cause, a purchase decision.
The circumstance is this cause. It links a need to a context, which becomes a job, making it specific and a driver for choice. You might find yourself "being hungry in the morning on your way to work." That's a need, but it's too vague and general to provide directions for where to look for product innovations.
When linking a specific circumstance to that need, such as "I'm alone on a long, boring commute and want to entertain myself, while staying satiated until noon," it suddenly provides clear guidelines on what job and struggles are critical, and thus why people might "hire" a thick, long-lasting milk shake instead of a banana that's gone too quickly.
Simultaneously, circumstance also explains why a parent might "hire" the same product for its child to feel like a good parent for very different reasons. For those reasons, a circumstance is always fundamental for defining a job and innovating with success.
Functional, social and emotional dimensions
Finally, when defining a job, it is insufficient to focus entirely on the functional or practical need. A job also has social and emotional dimensions, which can be far more important than the functional. Think of childcare. One would of course think of the functional dimension when hiring a solution — safely taking care of one's children in a location and manner that works well in one's life — but the choice probably relies more on emotional and social dimensions, such as who one will trust with one's children.
It is important to note that we don't create jobs, we discover them. Jobs are persistent and remain the same over time, but the way we solve them changes. Think of technology and how it has changed the job of teenagers communicating without the intervention of parents. The underlying job has not changed, but the solution has changed dramatically; from passing notes in class to Snapchat. They are two very different solutions, and that is exactly why it is dangerous to fall in love with a solution instead of falling in love with solving the underlying job.
Understanding and focusing on circumstance-specific hiring criteria triggers a whole new set of important insights of which a very notable one is that the competitive field is likely very different than first assumed.
"We compete with everything you do to relax (…) We compete with video games. We compete with drinking a bottle of wine. That's a particularly tough one. We compete with other video networks, playing board games."
— Netflix CEO, Reed Hastings
Companies will have to consider and shape their offerings around the experiences that consumers will seek in solving their jobs and help them surpass any barrier that gets in their way of making the desired progress.
Competitive advantages will be determined by who understands and solves their customers' jobs the best. Because how could companies ever hope to create solutions that consumers will hire over all other potential products if they don't know what they are really competing with?
Premium prices
By successfully identifying and deeply understanding the customers' jobs and its competitors, companies will possess valuable information about how to design products and services that solve the job and the struggles it entails better than all existing solutions.
Such products and services are highly valued by customers, and they are consequently willing to pay premium prices for those solutions. It can thus be very profitable to invest in such market research.
But be aware that your findings are strongly influenced by the questions you ask. Identifying and deeply understanding the customers' jobs does not rely on the tools you use, but what you are looking for.
Job Hunting — where do we find the jobs?
Starting to look for the right things does not necessarily mean already gathered data and research have to be discarded. However, asking the right questions and piecing the data and research together in the right way is key. Think of it as equipping a job lens; capturing the story of customers in their moments of struggle or desire for progress, which in turn will allow for a different understanding of the competitive landscape and the priorities and tradeoffs customers are willing to make.
Although jobs are everywhere around us, they can be difficult to see without the right approach. In other words, you will need a job-hunting strategy. Five approaches are presented below.
1. Competing with nothing — nonconsumption
First, and where the biggest potential lies, is what Christensen calls "nonconsumption." Too often, companies consider only how they can gain market shares from competitors, but not where they can find overlooked demand. When there are no acceptable existing solutions that satisfy consumers' jobs, they choose to do nothing about the unresolved job. This is exactly where highly successful companies like Airbnb have made breakthrough innovations, and thus where the biggest growth opportunities lie. You won't see nonconsumption if you're not looking for it, but if you find it, you will compete with nothing.
2. Finding a job close to home
Looking for a job in your everyday life can be a very useful approach, and the reason is simple. If you encounter a struggle that prevents you from resolving a job, chances are that other people have as well. In fact, some of the world's greatest innovators have discovered poorly resolved jobs with inspiration from their own lives, such as when Sony co-founder Akio Morita introduced the Walkman.
3. Workarounds and compensating behaviors
Sometimes, customers are deeply unhappy with existing solutions to jobs they want to resolve very badly. They care about solving the job so badly that they bundle several solutions, creating their own to compensate for the poorly resolved job. These kinds of workarounds are golden opportunities for innovators just waiting to be seized. It requires deep understanding of the circumstance in which the customer struggles to utilize the huge potential that lies within workarounds and compensating behavior, but succeeding will in turn allow for charging premium prices.
4. Unusual uses
Companies can learn a lot by investigating unusual uses of long-established products. That is, when customers use them in ways that are different than the company had intended. By studying how customers use products for different purposes than initially intended, jobs that are hidden in plain sight can be identified and better products introduced for already-existing customers.
5. Look for what people don`t want to do
Finally, a great way of looking for customers struggling to make progress is looking for so called "negative jobs". In contrast to "positive jobs," negative jobs are those that customers don't want to do. Honing in on what people don't want to do is a great source of inspiration and provides plenty of innovation opportunities.
Hiring and firing solutions
Customers can't always articulate their requirements accurately or completely — their motivations are often complex and the decision-making process complicated, and it starts way before entering the store. Despite the complexity, there are still some useful concepts to learn. There are always two opposing forces battling for dominance in the moment of choice: "forces compelling change" and "forces opposing change."
The forces compelling change to a new solution
There are two forces pushing and pulling customers to change solutions. First, there has to be a significant frustration causing customers to want to take action. It might not be enough that the struggle is simply nagging or annoying the customer — it has to be substantial enough before the customer starts looking elsewhere.
The second force, the pull of an enticing new product or service, has to be present and has to solve the customer's struggle for progress better than the current solution. Although the push of the current solution and the pull of another solution is where companies often direct their efforts, the forces opposing changes are not to be neglected.
The forces opposing changes
There are two unseen, yet incredibly powerful, forces at play that many companies ignore completely: the forces holding a customer back. First, customers are often stuck in the "habits of the present." Even though their job is getting solved poorly, they're comfortable with the solution and therefore stick with it, or they're simply used to solving the job in a certain way. This has to do with the tendency of customers seeking to avoid losses, which is a stronger psychological driver than the allure of gains. Second is "anxiety of choosing something new." The anxiety of the unknown is powerful, because customers worry that it will not be a better solution, or the solution will not work at all — the "What if…?"
Innovators too often focus on the forces compelling for change by trying to make sure the new solution is sufficiently resolving the customers' struggles to cause them to switch, but completely ignoring the forces opposing that change. It is of course important to make a product compelling enough for customers to consider switching, but the opposing forces are important as well. There is almost always some kind of friction involved with switching from one solution to another. There is less friction for solutions with only functional dimensions, but when the decision involves "firing" a solution with emotional and social attachments, it is far harder to let go. That holds true even in the B2B market. A plant manager ordering supplies has be sure she can trust the supplier. If not, she might become anxious about looking bad or unprofessional in the eyes of her colleagues or even getting fired.
More on Job Theory with Clayton Christensen, Karen Dillon and Taddy Hall
This article is strongly inspired by "Competing against luck" by Clayton Christensen, Taddy Hall, Karen Dillon and David S. Duncan (2016). We believe the information relevant for any manager, and we strongly recommend reading the book itself to obtain a better understanding of customer choice. The following is a short description from the book:
After years of research, Harvard Business School professor Clayton Christensen and his coauthors Taddy Hall, Karen Dillon and David S. Duncan have come to one critical conclusion: Extensive data on customers is not the key to successful innovation and growth.
Customers don't simply buy products or services; they "hire" them to do a job. Understanding customer jobs drives innovation success, and the "Jobs to Be Done" approach is used by some of the world's most respected companies, including Amazon, Uber and Airbnb to name a few.
Christensen and his coauthors contend that by understanding what causes customers to "hire" a product or service, any manager can predict what innovations will succeed, creating products customers not only want to "hire," but that they will pay premium prices to bring into their lives.
Jobs To Be Done
More from ag analytics
We are ag analytics. A hybrid tech analytics consultancy combining the best in strategic thinking, data science and preparatory technology. www.aganalytics.dk
Simon Knapp
Paralysis Through Analysis — How To Balance Working On Projects vs. Learning Something New
Dr Mehmet Yildiz
Insights from My Research into Formal/Informal Learning for Technical Leaders
ISMsalesgroup
Why Your Staff Retention Sucks! (3 Essential Tips to Fix This)
Jennifer Friebely
ILLUMINATION'S MIRROR
Where is That B*tch?
Garry Mkrtchyan
productleague.com
On Being A Mentee
CareerKeeper
Getting Fired: What I learned from getting laid off from my first full-time job
Fred Carpenter
The tone of your voice says a lot
Abigail Willford
Leave Dates
Why Your Company Needs an Annual Leave Policy


Do I have to schedule events in advance?
Any events that require facilitators (team building challenge course, zipline, rock wall, archery, etc) must be scheduled at least 1 week in advance. Call us at (765)584-9622 or email Jackie Welch, Camp Yale Director, at jackierwelch@gmail.com to schedule an event.
What are Camp Yale's Hours of Operation?
Facilitated Hours are available from March-October as scheduled by groups. Walking trails and disc golf are open year round from dawn to dusk.
How many team building elements can my group complete in one day?
This depends on each individual group. Some groups are able to complete all elements during their team building event, but it is not unusual for groups to not complete every element. This gives a group something to look forward to for their next visit to Camp Yale!
Can I camp at Camp Yale?
You can tent camp at Camp Yale for $5/person/night. You would have access to our restrooms with showers, fire pit, and shelter house. You can also park campers in our parking lot for $30/night.


Ability to do Reverse Engineering on GWI units.
To send in your GWI unit for repair use the shipping address below.
If you have any technical questions or wish to offer technical information about any GWI units then fill out the form below.