text:
\section{Introduction}

It has been known that treating the gauge fields as compact angle variables would reveal
some non-trivial aspects of gauge field theories  \cite{polya1,polya2,thooft1,lattice}.
Among studies based on the compactness of gauge fields, the lattice formulation of gauge
theories has provided an explanation for the confinement mechanism
as well as a basis for numerical studies at strong coupling regime \cite{lattice,kogut}.

By now there are specific instances of affinities
between coordinates and gauge fields. The oldest example
is the one by the special theory of relativity, by which it is understood that both space-time
coordinates and gauge fields transform as 4-vectors under the Lorentz transformations.
As another example, based on the duality proposed in
\cite{xpsi}, one can formulate the coordinate/field correspondence in both Abelian and
non-Abelian gauge theories \cite{cfdual}.
The other instance of relation between coordinates and gauge fields
is provided by T-duality of string theory. Accordingly,
the transverse coordinates of Dp-branes in the dual theory
are represented by the gauge fields of open string states,
leading to the correspondence \cite{tasi,9510017}
\begin{align}\label{1}
A_i \longleftrightarrow X_i/l_s^2,
\end{align}
in which $l_s$ is the string theory length.
At weak coupling the dynamics of $X_i$'s is captured by the
theory resulted from dimensional reduction of the ordinary U(1) gauge theory
\cite{tasi,9510017}. In particular the reduction on all spatial components of the
gauge field yields the D0-brane dynamics, namely \cite{tasi}
\begin{align}\label{2}
S_\mathrm{D0}=\int \! dt~\frac{m_0}{2}\, \dot{x}_i^2,
\end{align}
in which $m_0\propto 1/g^2 $ ($g$: gauge coupling) \cite{tasi}.
The transverse coordinates of $N$ number of Dp-branes are
represented by $N$ dimensional hermitian matrices \cite{9510135}.

In \cite{0bfath} the dimensional reduction of pure U(1) lattice gauge theory
is considered to model the dynamics of 0-branes at strong coupling regime.
The model by \cite{0bfath}
might be considered as the result of combination of
two themes mentioned earlier, 1) treating gauge fields as compact angle variables \cite{polya1,polya2,thooft1,lattice},
2) assuming similar characters between coordinates and gauge fields
\cite{xpsi,cfdual,tasi,9510017}.
The explicit form of the action after the dimensional reduction of
U(1) lattice gauge theory is:
\begin{align}\label{3}
S_0=\frac{1}{g^2}\sum_{n}\left(\cos\frac{x_{n+1}-x_{n}}{R}-1\right)
\end{align}
in which the coordinates appear as the compact angle variables depending on
discrete imaginary time label $n$.
By this form the worldline theory takes the form of
the 1D plane-rotator model of spin lattice systems \cite{mattis}.
Based on the prescription for the original lattice gauge theory
\cite{lattice}, using the transfer-matrix method the quantization of the model is
formulated. The exact energy spectrum as the function of gauge coupling
is obtained, with a minimum at critical coupling $g_c=1.125$ in the
lowest energy. As the direct consequence of the minimum,
the model exhibits a first-order phase transition between coexistent phases
with small and large couplings \cite{0bfath}. Based on discontinuous nature of the first-order phase transition, for $g< g_c$ and $T\approx 0$ the effective zero mean-square velocity $\langle v^2 \rangle $ is zero.

The purpose of the present work is to provide a general theoretical
ground for the models based on compact angle coordinates.
It is clarified that the dependence on the group elements
rather than the algebra ones would lead to invariance of
the action under the total shifts in the compact domain, like that
is happening in (\ref{3}).
It is observed that the proper dependence is achieved most naturally in
a discrete-time formulation of the theory.
The worldline action by the formulation
resembles the 1d spin chain Hamiltonian of magnetic systems,
with coordinates appearing as the spin degrees of freedom.
The transfer-matrix method is used to define the quantum theory \cite{wipf}.
As here the 1d spin chain is used for a ``particle-like" dynamics
interpretation, there should be a square-root of the mass in the definition
of the transfer-matrix elements \cite{wipf}. As a direct consequence of the presence
of the square-root, opposite to the 1d chains of magnetic systems, here
the energy spectra develops minima.
In a path-integral representation of the formulation, it is emphasized that
the square-root pre-factor in the definition of the transfer-matrix elements,
in contrast to the case with infinite extent coordinates, can not
be absorbed by a change of the integration variables.

It is well known that the 1d spin systems with short range
interactions do not exhibit the second-order phase transition
expected for these systems. However the present model, as
the consequence of the minima in the spectrum, exhibits a first-order phase transition.
As mentioned earlier, although the worldline by the model looks like a 1d spin chain,
due to the square-root pre-factor in matrix elements the spectra is different.
The phase transition nature by the model will be discussed based on
the behavior of the Gibbs free energy. In particular, the plot of $G$ versus
the thermodynamical variable $M$, as the effective mean-square-velocity,
develops cusp below a critical temperature $T_c$.
At the cusp the derivative $\frac{\partial G}{\partial M}$
is discontinuous, as expected in a first-order phase transition.

As examples for the formulation, the models based on the U(1),
$\mathbb{Z}_N$ and SU(2) groups are explicitly constructed.
In all of the examples the exact energy
eigen-values are obtained, leading to the first-order phase transition.

The organization of the rest of the paper is as follows. In Sec.~2 the
basic assumptions and ingredients for the formulation based on compact
coordinates are presented.
In Secs. 3, 4 and 5 three examples based on the groups
U(1), $\mathbb{Z}_N$ and SU(2) are presented explicitly. For
all the three groups the energy eigen-values are obtained exactly,
together with the discussion on the nature of the phase transitions in them.
Sec.~6 is devoted to conclusion and discussion.

\section{Basics and Formulation}

The transition amplitude between positions $x_0$ and $x_{N}$
at times $t_0$ and $t_N$ is represented by the path-integral \cite{sakurai}
\begin{align}\label{4}
\langle x_N,t_N|x_0,t_0\rangle=\lim_{N\to \infty} \int_{-\infty}^{\infty} \prod_{n=1}^{N-1}
\sqrt{\frac{m}{2\pi \,\mathrm{i}\,\hbar\, \epsilon}}dx_n ~ e^{\mathrm{i}\, S[t_0,t_N]/\hbar}
\end{align}
in which $\epsilon =(t_N-t_0)/N$, tending to zero
in the limit $N\to \infty$. It is noticed that in above all the intermediate positions
$x_n$'s have infinite extents, $-\infty < x_n <\infty$.
From the considerations mentioned earlier in Introduction,
here the main concern is about the finite-extent coordinates.
The finite-extent coordinates
are quite known in physics. The most familiar ones are the angle variables of polar
coordinates in 2D and 3D problems. The other example happens when the system
is defined inside a finite volume, like a box or a sphere.
As a definite example, let us consider a free particle on
a circle of radius $R$. Defining the angle variable $\phi=x/R$
with $-\pi \leq \phi \leq \pi$,  we have
\begin{align}\label{5}
L=\frac{1}{2}mR^2\dot{\phi}^2 ~~\to~~ H=\frac{p_\phi^2}{2mR^2}
\end{align}
leading to the eigen-functions and eigen-values as follow
\begin{align}\label{6}
\psi_n(\phi)=\frac{1}{\sqrt{2\pi}} e^{\mathrm{i}\, n \phi},~~~~E_n=\frac{n^2\hbar^2}{2mR^2}
\end{align}
Then the transition amplitude between two different positions is known to be
\begin{align}
\langle \phi_N,t_N|\phi_0,t_0\rangle&=\sum_{n=-\infty}^\infty
\psi_n(\phi_N)\psi_n^*(\phi_0)\, e^{-\mathrm{i}\,  E_n (t_N-t_0)/\hbar}
\cr
\label{7}
&=\frac{1}{2\pi}\sum_{n=-\infty}^\infty
e^{\mathrm{i}\,n(\phi_N-\phi_0)} e^{-\mathrm{i}\,n^2 \hbar(t_N-t_0)/2mR^2}
\end{align}
The above can be expressed in terms of the 3rd Jacobi theta function
\begin{align}\label{8}
\vartheta_3(z,\tau )=\sum_{n=-\infty}^\infty e^{\mathrm{i}\, \pi \,\tau \, n^2+2\,
\mathrm{i}\, n\, z}
\end{align}
by which we have \cite{tome}
\begin{align}\label{9}
\langle \phi_N,t_N|\phi_0,t_0\rangle=\frac{1}{2\pi}\vartheta_3
\left(\frac{\phi_N-\phi_0}{2},\frac{-\hbar (t_N-t_0)}{2\pi mR^2}\right)
\end{align}
Using the modular property of $\vartheta_3$
\begin{align}\label{10}
\vartheta_3(z,\tau )=(-\mathrm{i}\, \tau )^{-1/2} e^{-\mathrm{i}\, z^2/\pi \tau }\,
\vartheta_3\left(\frac{z}{\tau },-\frac{1}{\tau }\right)
\end{align}
the transition amplitude recasts as \cite{tome}
\begin{align}\label{11}
\langle \phi_N,t_N|\phi_0,t_0\rangle=\sum_{n=-\infty}^\infty
\sqrt{\frac{m R^2}{2\pi\,\mathrm{i}\,\hbar (t_N-t_0)}}
\exp\left(\frac{\mathrm{i}}{\hbar} \frac{mR^2 (\phi_N-\phi_0-2\pi n)^2}{2(t_N-t_0)}\right)
\end{align}
in which the summand is easily recognized as the
transition amplitude of a free particle experiencing the position difference
$x_N-x_0=R(\phi_N-\phi_0-2\pi n)$
during time $t_N-t_0$ \cite{sakurai}.
There is a nice interpretation for the sum as well. As the particle moves on the
circle from $\phi_0$ to $\phi_N$, it matters how many times
it rounds the circle. The sum on $n$, the so-called winding number, is responsible
for taking into account the contributions from different rounds to the
amplitude. So, nevertheless
the coordinate $\phi$ has finite extent, practically the particle
may travel long distances $\Delta x = R|\phi_N-\phi_0-2\pi n|$ for
$n=0,\pm 1,\pm2,\cdots$.
In fact (\ref{11}) may come to a form similar to (\ref{4}) with only
an extra summation. First let us present a time-sliced form of (\ref{11})
\cite{tome}:
\begin{align}
\langle \phi_N,t_N|\phi_0,t_0\rangle=&\lim_{N\to \infty}  ~
\int_{-\pi}^{\pi} \prod_{j=1}^{N-1}
\sqrt{\frac{mR^2}{2\pi \,\mathrm{i}\,\hbar\, \epsilon}}d\phi_j
\cr
\label{12}
&~ \times  \prod_{l=0}^{N-1} \sum_{n_l=-\infty}^\infty
\exp\left[\frac{\mathrm{i}}{\hbar}  \frac{mR^2 (\phi_{l+1}-\phi_l-2\pi n_l )^2}
{2\,\epsilon}\right]
\end{align}
It is noticed that at $l$-th time-slice the winding number $n_l$ is introduced \cite{tome}.
Now, by the following change in the integral variables \cite{tome}:
\begin{align}\label{13}
\sum_{n_l=-\infty}^\infty \int_{(2n_l-1) \pi}^{(2n_l+1)\pi} d\phi_l \to
\int_{-\infty}^{\infty} d\Phi_l
\end{align}
the expression (\ref{12}) recasts to  \cite{tome}:
\begin{align}\label{14}
\langle \phi_N,t_N|\phi_0,t_0\rangle=&\lim_{N\to \infty} \sum_{n=-\infty}^\infty \int_{-\infty}^{\infty}
 \prod_{j=1}^{N-1} \sqrt{\frac{mR^2}{2\pi \,\mathrm{i}\,\hbar\, \epsilon}}d\Phi_j \cr
&~~~\times \exp\left[\frac{\mathrm{i}}{\hbar} \sum_{i=0}^{N-1}
\frac{mR^2 (\Phi_{i+1}-\Phi_i)^2}{2\,\epsilon}\right]
\end{align}
in which $\Phi_N=\phi_N$ and $\Phi_0=\phi_0+2\pi n$.
In above all the intermediate angles $\Phi_j$'s
are integrated over the whole real-line $\mathbb{R}$.
In fact by the change of variable
(\ref{13}), integrating the intermediate angle $\phi_l$ over $[-\pi,\pi]$ together with sum over infinite possible rounds $n_l$ is replaced by integration of $\Phi_l$ over $(-\infty,\infty)$.

The above example shows that it is insufficient to
merely use finite-extent coordinates to be granted the non-trivial aspects.
The key point about the above treatment of motion on circle is that
the time-sliced form of the action appearing in the path-integral
(\ref{12}) is not invariant under the multi-round shifts
\begin{align}\label{15}
\phi_l \to \phi_l + 2\pi \, k_l,~~~~k_l=0,\pm1,\pm2,\cdots
\end{align}
with $k_l\neq k_j$ for $l\neq j$. It is noticed that the above shift is directly related to
the different number of possible rounds on circle represented earlier by $n_l$ in (\ref{12}).
Reminding that the group space of U(1) is circle, we can express
the main idea in a group theoretical language. Defining the U(1) group
element by $U_\phi=\exp(\mathrm{i}\, \phi)$, the action by the Lagrangian (\ref{5}) can be
expressed as \cite{tome}
\begin{align}\label{16}
S=\frac{1}{2} mR^2 \int dt \left(U_\phi^\dagger \dot{U}_\phi\right)
\left(U_\phi^\dagger \dot{U}_\phi\right)^\dagger=\frac{1}{2} mR^2\int dt~ \dot{\phi}^2
\end{align}
In above, although the starting point is taken to be the group element
$U_\phi$, the Lagrangian depends in fact on the algebra
element $\phi$. This observation for the group U(1) is general and holds
for other groups as well \cite{tome}.
Obviously the situation changes if the action would
be invariant under the shift (\ref{15}), namely due to
dependence on group elements $\exp(\mathrm{i}\,\phi_l)$'s
instead of the algebra elements $\phi_l$'s. Interestingly, once
the time parameter is assumed to be discrete the desired dependence is obtained.
By taking time as $t_n=n\, a\,$ for some finite value
$a$ and integer $n$, the worldline looks like a chain or 1D lattice with spacing
parameter $a$. On the $n$-th site of this chain the angle $\phi_n$ is sitting.
So at time step $n$ we have $U_n=\exp(\mathrm{i}\,\phi_n)$, by which the
discrete-time version of action (\ref{16}) is
\begin{align}
S&=\frac{1}{2}mR^2 \, a^{-1}\sum_n \left(U_n^\dagger (U_{n+1}-U_n)\right)
\left(U_n^\dagger (U_{n+1}-U_n)\right)^\dagger
\cr
&=\frac{1}{2}mR^2 \, a^{-1}\sum_n (U_n^\dagger U_{n+1}-1)
(U_{n+1}^\dagger U_{n}-1)
\cr
\label{17}
&=-\frac{1}{2}mR^2 \, a^{-1}\sum_n (U_n^\dagger U_{n+1}+U_{n+1}^\dagger U_{n}-2)
\end{align}
which obviously keeps invariance under the shift (\ref{15}).
The phenomenon observed here is partially in reverse direction
of what has happened in lattice formulation of gauge theories. Namely,
once one tries to introduce gauge symmetry to the theory on lattice
the algebra elements $A_\mu$'s are to be replaced by
the group elements $\exp(\mathrm{i}\, a \,A_\mu)$'s
in the action \cite{lattice}. Here as we were going to keep
the invariance under the shift (\ref{15}) the natural solution
appears to be defining the action on discrete-time worldline.
In the next sections we will use this as a basis for model building.

Before to end this section it is helpful to discuss the prominent role of
the imaginary time in the quantization of models with compact domain support in the
sense described in above. In particular let us consider the matrix
element by (\ref{4}) \cite{sakurai}
\begin{align}\label{18}
\langle x |\widehat{U}| x' \rangle = \langle x | \exp(-\mathrm{i}\,\Delta t\,\widehat{H}/\hbar)\,  | x'
\rangle= \sqrt{\frac{m}{2\pi \,\mathrm{i}\,\hbar\, \Delta t}}\, e^{\mathrm{i}\, S[t,t+\Delta t]/\hbar}
\end{align}
in which $\widehat{U}$ is the unitary time evolution operator, $\widehat{H}$ is the
Hamiltonian, and $S[t,t+\Delta t]$ is the action between times $t$ and $t+\Delta t$
\cite{sakurai}.  The basic observation is that the above representation in terms of the
action is not possible when the dynamical variables are to take values inside a compact domain
in the sense mentioned earlier. The reason can be easily understood for a system with
one dynamical variable $\lambda$ (field or coordinate). The
generalization to systems with more variables is then straightforward.
The identity $\widehat{U}\widehat{U}^\dagger=\widehat{\mathbb{1}}$
for the unitary evolution operator $\widehat{U}$,
defining $U(\lambda,\lambda'')=\langle \lambda|\widehat{U}|\lambda''\rangle$,
in the $\lambda$-basis takes the form:
\begin{align}\label{19}
\int_\Lambda d\lambda'' ~ U(\lambda,\lambda'') ~ U^\star(\lambda',\lambda'') =
\delta(\lambda-\lambda')
\end{align}
in which $\Lambda$ is the compact domain in which $\lambda$ takes values.
Now, by the representation like (\ref{19}), as the integrand consists of only
the ordinary regular functions and not the distribution ones,
there is no way that the integral over a compact domain
can develop a $\delta$-function.
By lacking the representation (\ref{18}) for a unitary time evolution operator,
the alternative is to assume that time is imaginary ($t\to -\mathrm{i}\,t$).
In the models with discrete time ($\Delta t =a$), by this alternative option
the one-step unitary operator $\widehat{U}_1=\exp(-\mathrm{i}\,a\widehat{H}/\hbar)$
is replaced by the so-called transfer-matrix operator $\widehat{V}$ whose matrix element
between two adjacent times $n$ and $n+1$ is given by \cite{wipf}
\begin{align}\label{20}
\langle \lambda_{n+1} | \widehat{V} | \lambda_n \rangle \propto
\sqrt{m}\,\exp\big(S_E(n,n+1)/\hbar\big)
\end{align}
in which $S_E(n,n+1)$ is the Euclidean action. Then by common eigen-states for
$\widehat{H}$ and $\widehat{V}$, the eigen-values of $\widehat{H}$ are defined
by \cite{lattice,wipf}
\begin{align}\label{21}
E_s=-\hbar\, a^{-1}\, \ln v_s
\end{align}
where $v_s$ is the corresponding eigen-value of $\widehat{V}$.
Provided that $\widehat{V}$ does not have negative eigen-values, the
above would give a consistent description of the quantum
theory based on an action with discrete imaginary time \cite{lattice,wipf}.
This approach is exactly what is chosen in lattice formulation of gauge
theories \cite{lattice}, turning space-time to a Euclidean one,
and it will be used in the present work as well.

In the formulation presented in above the discrete worldline is inlaid by
numbers as spin variables. An interesting extension
is to consider the case in which the site $n$ on worldline
is equipped with the spin operator $\hat{S}_n$, promoting
the worldline to a quantum spin chain. As an example,
let us consider the Heisenberg XYZ model, defined by the
Hamiltonian operator for two arbitrary adjacent sites
\begin{align}
\widehat{H}_\mathrm{XYZ}=-\frac{1}{2}
\left(\kappa_x\, \hat{S}^x_n\hat{S}^x_{n+1}+\kappa_y\, \hat{S}^y_n\hat{S}^y_{n+1}+
\kappa_z\, \hat{S}^z_n\hat{S}^z_{n+1} \right)
\end{align}
The spin operators in above are not restricted to a specific representation,
and generally belong to the $2s+1$ dimensional representation for
$s=\frac{1}{2},1,\frac{3}{2},\cdots$.
The transfer-matrix $\widehat{V}$ of the model, in analogy with (\ref{20}), is then defined by
\begin{align}
\widehat{V}\propto \sqrt{\kappa_x\kappa_y\kappa_z}\, \exp\big(-a\,\hat{H}_\mathrm{XYZ}\big)
\end{align}
The matrix $\widehat{V}$ in above is hermitian by construction, as it should.
The energy spectrum by the model can be obtained by the prescription (\ref{21}).
The detailed nature of the spectrum and the phase structure
by the model based on the quantum spin chain
is not discussed in here, and is left for future studies.

In summary, by the considerations mentioned in above, the followings are
the basis for model building:
\begin{enumerate}
\item Time is assumed to be discrete and imaginary, taking values $t_n=n\,a$ for integer $n$.
\item The action with discrete time depends on the group elements
to enhance the features by the compact nature of group.
\end{enumerate}

\section{U(1) Group}
For the $i$-th direction with coordinate $-\pi R^i \leq x^i \leq \pi R^i$, the U(1)
group element at $n$-th time step is taken as $U^i_n=\exp(\mathrm{i}\, x^i_n/R)$. Here for simplicity we take all radii $R^i$'s equal to $R$.
Following (\ref{17}) the Euclidean action takes the form
\begin{align}
S_E&=\frac{\kappa}{2} \sum_{n,\, i} \left(U^{i\,\dagger}_{n}\, U^{i}_{n+1} +
U^{i\,\dagger}_{n+1} \, U^{i}_{n}  -2 \right)
\cr
\label{22}
&=\kappa  \sum_{n,i}\left(\cos\frac{x^i_{n+1}-x^i_{n}}{R}-1\right)
\end{align}
The dimensionless constant $\kappa$ is the defining parameter
of the model (we have set $\hbar=c=1$).
The above, as discussed in previous section, is invariant under the shift:
\begin{align}\label{23}
x^i_n\to x^i_n+2\pi\, k^i_n R,
\end{align}
with $k^i_n$'s as integer numbers. At the first place let us check the limits:
\begin{align}\label{24}
\begin{split}
x/R& \ll 1\cr
x_{n+1}-x_n&\to a\, \dot{x}\cr
\sum_n&\to a^{-1}\int\! dt
\end{split}
\end{align}
leading to
\begin{align}\label{25}
S_E\simeq -\,\frac{a\kappa}{2R^2}\int dt ~\dot{x}^{2}_i
\end{align}
which describes the dynamics of an ordinary free particle with mass
$m_0=a\kappa/R^2$ in the imaginary time formalism.
As mentioned in Introduction, the action (\ref{22}) is used
in \cite{0bfath} to model the dynamics of 0-branes at strong coupling limit.
The action (\ref{22}) is the result of dimensional reduction of
U(1) lattice gauge theory along spatial directions, by setting:
\begin{align}\label{26}
\begin{split}
\kappa&=1/g^2\cr
a\, A^i&\to  x^i/R
\end{split}
\end{align}
The expression (\ref{22}) for the action is also known as the 1D plane-rotator
model of magnetic systems \cite{mattis}, although here it is interpreted as a discrete
worldline equipped by the angle variable coordinates $x^i$'s. In this section we
review the construction by \cite{0bfath}.
It is useful to define the new variables
\begin{align}\label{27}
y^i= x^i/R
\end{align}
taking values in $[-\pi,\pi]$, by which the action (\ref{22}) takes the form
\begin{align}\label{28}
S_0=\kappa \sum_{n,i}\left(\cos(y^i_{n+1}-y^i_{n})-1\right)
\end{align}
As the action is fully separable for each direction, it is sufficient to consider
only one copy, dropping the index $i$ hereafter.
As mentioned in Sec.~2, the action with discrete
imaginary time can be used to define the quantum theory based on
the transfer-matrix $\widehat{V}$, defined by its matrix elements
\begin{align}\label{29}
\langle  y_{n+1} |\widehat{V} |  y_n\rangle= \sqrt{\frac{\kappa}{2\pi}}
\,\exp\left[\kappa \left(\cos( y_{n+1}- y_n)-1\right)\right]
\end{align}
in which, recalling $m_0\propto \kappa$,
the normalization prefactor has to be inserted to match the propagator
(\ref{18}) (see also (\ref{20})) \cite{wipf}
\begin{align}\label{30}
\langle x_2,t_2|x_1 ,t_1\rangle \propto \sqrt{\frac{m_0}{2\pi}}\exp
\left(\frac{-m_0(x_2-x_1)^2}{2\,(t_2-t_1)}\right)
\end{align}
Using the identity for the modified Bessel function of the first kind:
\begin{align}\label{31}
\exp[\kappa\cos( y'- y)]=\sum_{s=-\infty}^\infty I_s(\kappa) \, e^{\mathrm{i}\,
s\,( y'- y)}
\end{align}
we have for (\ref{29})
\begin{align}\label{32}
\langle  y_{n+1} |\widehat{V} |  y_n\rangle= \sum_{s=-\infty}^\infty
 \sqrt{\frac{\kappa}{2\pi}}e^{-\kappa}I_s(\kappa)\, e^{\mathrm{i}\,s\,
( y_{n+1}- y_{n})}
\end{align}
by which one reads the normalized plane-wave
\begin{align}\label{33}
\psi_s(x)=\frac{1}{\sqrt{2\pi}}\exp(\mathrm{i}\,s\, y),~~~~~~~
-\pi\leq y \leq \pi
\end{align}
as eigen-function with the eigen-value
\begin{align}\label{34}
v_s(\kappa)=\sqrt{2\pi\kappa} \, e^{-\kappa}I_s(\kappa)
\end{align}
By the known properties of $I_s$-functions we have
$v_s=\sqrt{2\pi \kappa}\,e^{-\kappa}I_s(\kappa)\geq 0$.
This guaranties that the transfer-matrix method defined by (\ref{20})
and (\ref{21}) would lead to a consistent quantum
theory. Also by $I_s(z)=I_{-s}(z)$ the spectrum is doubly
degenerate for $s\neq 0$. The energy eigenvalues are found by (\ref{21})
and (\ref{34})
\begin{align}\label{35}
E_s(\kappa)=-\frac{1}{a} \ln \left[\sqrt{2\pi\kappa} \, e^{-\kappa}I_s(\kappa)  \right]
\end{align}
The behavior of above at the limit $\kappa \to\infty$
can be checked by the saddle point approximation for Bessel functions
\begin{align}\label{36}
I_s(\kappa)=\lim_{\kappa\to\infty}\frac{1}{2\pi}\int_{-\pi}^\pi d y~ \exp(\kappa\cos y +\mathrm{i}\,s\, y)
\simeq \frac{e^\kappa}{\sqrt{2\pi \kappa}} \exp\left(-\frac{s^2}{2\kappa}\right)
\end{align}
by which for (\ref{35}) we obtain
\begin{align}\label{37}
E_s\simeq  \frac{s^2}{2a\kappa}
\end{align}
matching the energy $E=p^2/(2m_0)$ of a free particle with
momentum  $p=s/R$ along the compact direction,
and mass  $m_0=\kappa\, a/R^2$ by (\ref{25}).
So in the limit
$\kappa\to\infty$ the spectrum approaches to that of an ordinary particle. For the intermediate $\kappa$ the spectrum is discrete.
In the limit $\kappa\to 0$, using
\begin{align}\label{38}
I_s(z)\simeq \frac{1}{s!}\left(\frac{z}{2}\right)^s,~~~~z\ll 1
\end{align}
we have
\begin{align}\label{39}
E_s=-(s+\frac{1}{2})\,\frac{\ln \kappa}{a}+O(s\ln s) +O(\kappa)
\end{align}
in which the 2nd term is independent of $\kappa$. Also at $\kappa\to 0$
\begin{align}\label{40}
E_{s+1}-E_s\simeq -\frac{\ln \kappa}{a} \gg \frac{1}{a}
\end{align}

\begin{figure}[t]
	\begin{center}
		\includegraphics[scale=1]{Fig-u1-1.pdf}
	\end{center}
	\caption{The few lowest energies by (\ref{35}) versus $\kappa$
		($E$ unit: $a^{-1}$).}
\end{figure}

The interesting observation by the spectrum (\ref{35}) is
about the energy of ground-state, which has a minimum at $\kappa_c=0.790$; see Fig.~1. As expected the existence of minimum leads to a
first order phase transition. The one-particle partition function may be evaluated by the definition
\begin{align}\label{41}
Z_1(\beta,\kappa):=\sum_{s=-\infty}^\infty e^{-\beta\, E_s(\kappa)}
\end{align}
or by means of the transfer-matrix operator ($\beta$ in $a$ units) \cite{wipf}
\begin{align}\label{42}
Z_1(\beta,\kappa)=\mathrm{Tr} \,\widehat{V}^\beta=
\int_{-\pi}^\pi \prod_{m=0}^{\beta -1}\sqrt{\frac{\kappa}{2\pi}}\, d y_m
\exp\left[\kappa\sum_{n=0}^{\beta-1} \left(\cos( y_{n+1}- y_n)-1\right)\right]
\end{align}
supplemented by the periodic condition $y_0=y_\beta$.
In the present case the equivalence of (\ref{41}) and (\ref{42}) is
checked by numerical evaluations. The basic observation by the compact angle variable in above is, in contrast to the situation with infinite extent coordinates, the normalization factor can not be absorbed by a change of integration variable. As the minimum of $E_0$ is in variable $\kappa$, we need the thermodynamical conjugate variable $M$, defined by ($T=\beta^{-1}$)
\begin{align}\label{43}
{M}(\beta,\kappa):= T \,\frac{\partial \ln Z_1(\beta,\kappa)}{\partial\; \kappa}
\end{align}
which is also interpreted as the equation-of-state of the system.
The Gibbs free energy can represent the exact nature of the
phase transition,
\begin{align}\label{44}
G_1=A_1+\kappa\,M
\end{align}
in which $A_1=-T\ln Z_1$ is the free energy per particle.
\begin{figure}[t]
	\begin{center}
		\includegraphics[scale=1]{Fig-u1-2.pdf}
	\end{center}
	\caption{The $G$-$M$ plots at four temperatures. The dashed pieces are
		not followed by the system due to the minimization of $G$.}
\end{figure}
The isothermal $G$-$M$ plots are presented in Fig.~2. Evidently below the critical temperature $T_c=0.335~a^{-1}$ the plots develop cusps, at which by the minimization of $G$ at equilibrium, the system follows the path with lower $G$ (solid-lines in Fig.~2). As the consequence, for $T<T_c$ there is a jump in first derivative of $\partial G/\partial M$, indicating that the phase transition is a first order one.
It is evident by now that the above phase structure is quite similar to
the gas/liquid transition, for which $G$-$P$ plots show exactly
the same behavior. In a similar way the equation-of-state (\ref{43}) should be modified
by the so-called Maxwell construction for $P$-$V$ diagram,
by which during isothermal condensation the pressure (here $M$) is fixed.
The results of the Maxwell construction for the present model are plotted as
isothermal $M$-$\kappa$ curves in Fig.~3. The flat part at $T_c$ corresponds to
values:
\begin{align}\label{45}
T_c=0.335:~~\kappa^*=1.403,~~~M^*=0.064.
\end{align}

\begin{figure}[t
	\begin{center}
		\includegraphics[scale=1.1]{Fig-u1-3.pdf}
	\end{center}
	\caption{The isothermal $M$-$\kappa$ plots. The straight-lines are due to the Maxwell construction,
		replacing the dashed parts.}
\end{figure}

\noindent For isothermal curves below $T_c$, the straight horizontal parts describe the coexistent phases of lower and higher $\kappa$'s during the phase transition.
The interesting fact about the equation-of-state modified by Maxwell construction is that $M$ always remains non-negative, that is $M\geq 0$.  This is specially important by
expectations from the variable $M$ at the limit $\kappa\gg 1$, at which we expect the ordinary behavior for particles. At this limit, back to
(\ref{25}) and (\ref{37}), we have
\begin{align}\label{46}
M \simeq \frac{1}{2} \langle \dot{y}^2 \rangle \propto \frac{T}{m_0}
\end{align}
where the proportionality is by the properties of free ordinary particles.
In fact the asymptotic tails in Fig.~3 for large $m_0\propto \kappa$
are explained by (\ref{46}).
There are also asymptotes at $\kappa\to 0$, although with different slopes. In fact the main difference between the case with present model and that of ordinary particles is about the existence of a phase transition. In particular, by the present model and below the critical temperature $T_c$, the two asymptotes by large and small masses (large and small $\kappa$'s)
are connected with a first-order phase transition.

One may define the order parameter for the present
model as well. For the ordinary magnetic systems with 2nd
order phase transition the order parameter is the
magnetization as the derivative of $G$ \cite{huang}. The
nonzero magnetization is interpreted as the magnetic ordering phase. However, due to the different nature of the 1st order phase transitions, the derivative of $G$ is discontinuous at cusps in Fig.~2.
This situation is again quite similar to the case with gas/liquid system, in which the volume difference of coexisting
phases, as the jump in the derivative $\partial G/\partial P$,
is taken as the order parameter \cite{huang}.
Similarly, in the present case the jump in $\partial G/\partial M$ defines the
order parameter, being simply the difference of $\kappa$'s of coexisting phases ($\kappa$-difference at ends of fixed-$M$ line in Fig.~3).
Like the gas/liquid system, the order parameter tends to zero
at the critical point (\ref{45}), and larger values of
order parameter (larger $\kappa$-difference)
at lower temperatures corresponds to lower fixed-$M$ line in Fig.~3.
In terms of the magnetic ordering $y_n\simeq y_{n+1}$,
this is the expected behavior with $M\propto \langle \dot{y}^2\rangle \simeq 0$ by (\ref{46}).

\section{$\mathbb{Z}_N$ Group}
The coordinates with a compact domain may form a discrete group such as
$\mathbb{Z}_N$. In this case the worldline looks like a spin chain
with discrete spin degrees sitting on its sites. In general the worldline
resemble the spin chain of Potts model ($\mathbb{Z}_2$
as of the Ising model). The members of
group $\mathbb{Z}_N$ are presented by
\begin{align}\label{47}
\{1,\varrho,\varrho^2,\cdots,\varrho^{N-1}\}
\end{align}
in which
\begin{align}\label{48}
\varrho=\exp(\mathrm{i}\,2\pi/N),~~~~~\varrho^N=1
\end{align}
At time-step $n$ the position may be represented by $r_n$ as
\begin{align}\label{49}
U_n=\varrho^{r_n}=\exp(\mathrm{i}\,2\pi\,r_n/N),~~~~~r_n=0,1,2,\cdots,N-1
\end{align}
by which the action takes the form
\begin{align}
S_E=\frac{\kappa}{2}\sum_n \left(
U^{\dagger}_{n}\, U_{n+1} +
U^{\dagger}_{n+1} \, U_{n}  -2 \right) \cr
\label{50}
=\kappa \sum_n \left( \cos \frac{2\pi(r_{n+1}-r_n)}{N} -1
\right)
\end{align}
The action is invariant under the shifts by $k_n$ being any integer
\begin{align}\label{51}
r_n\to r_n+k_n\, N
\end{align}
It is convenient to define the new variable
\begin{align}\label{52}
\frac{2\pi\,r}{N} = w = 0,\frac{2\pi}{N},\frac{4\pi}{N},\cdots,\frac{(N-1)2\pi}{N}
\end{align}
by which the action comes to the form:
\begin{align}\label{63}
S_E=\kappa \sum_n \left( \cos(w_{n+1}-w_n) -1  \right)
\end{align}
The transfer-matrix element then easily reads
\begin{align}\label{54}
\langle  {n+1} |\widehat{V} |  n\rangle &=\frac{1}{N} \sqrt{2\pi\kappa}
\,\exp\left[\kappa  \left( \cos(w_{n+1}-w_n) -1  \right)\right]
\\
\label{55}
& =\sum_{s=0}^{N-1} e^{-a\, E_s} ~\frac{1}{N}e^{\mathrm{i}\,s(w_{n+1}- w_{n})}
\end{align}
in which the plane-waves
\begin{align}\label{56}
\psi_s(w)=\frac{1}{\sqrt{N}}\, e^{\mathrm{i}\,s\,w}
\end{align}
satisfy the ortho-normality condition
\begin{align}\label{57}
\sum_w \psi_s^\star (w) \psi_{s'}(w) = \delta_{ss'}
\end{align}
\begin{figure}[t
	\begin{center}
		\includegraphics[scale=0.9]{Fig-zn-1.pdf}
	\end{center}
	\caption{The energies by (\ref{59}) for $\mathbb{Z}_4$ group.  }
\end{figure}
By the identity (\ref{31}) for Bessel functions, and using the condition (\ref{57}),
the sum on Bessel functions can be partitioned into $N$ cyclic ones, leading to
\begin{align}\label{58}
e^{-a\, E_s}=\sqrt{2\pi\kappa} \, e^{-\kappa}
\sum_{q=-\infty}^\infty I_{s+qN}(\kappa)
\end{align}
in which the sum is converging due to properties by $I_s$-functions.
So the energy eigen-values are simply given by
\begin{align}\label{59}
E_s(\kappa)=-a^{-1} \,\ln \left[ \sqrt{2\pi\kappa} ~ e^{-\kappa}
\sum_{q=-\infty}^\infty I_{s+qN}(\kappa)
\right]
\end{align}
for $s=0,1,2,\cdots, N-1$. Due to $I_s=I_{-s}$ we have the following degeneracy:
\begin{align}\label{60}
E_s=E_{N-s}
\end{align}
In fact the lowest and highest eigen-values are as follow
\begin{align}\label{61}
\begin{split}
E_\mathrm{min}&=E_0,\\
E_\mathrm{max}&=E_{N/2},~~~~~~~~N:\mathrm{even} \\
E_\mathrm{max}&=E_{(N\pm 1)/2},~~~N:\mathrm{odd}
\end{split}
\end{align}
\begin{figure}[t
	\begin{center}
		\includegraphics[scale=1.1]{Fig-zn-2.pdf}
	\end{center}
	\caption{The $G$-$M$ plots for $\mathbb{Z}_4$ group.  }
\end{figure}
In the limit $N\to\infty$ we expect to recover the
spectrum (\ref{35}) by U(1) group. It is in fact the case
using
\begin{align}\label{62}
\frac{I_{\pm\infty}}{I_\mathrm{finite}} \to 0
\end{align}
It can easily be checked that for $N=2~ \&~ 3$ there is no extrema in $E_0$. For
$N\geq 4$ there are both minimum and maximum.
The energy eigen-values are plotted in Fig.~4 for the group $\mathbb{Z}_4$, in which
the extrema are at
\begin{align}\label{63}
\kappa_\mathrm{min}=0.815,~~~~~~~~\kappa_\mathrm{max}=1.87
\end{align}
Again the one-particle partition function (\ref{41}), together with the thermodynamical functions
(\ref{43}) and (\ref{44}), can be defined for the system.
Like the case with U(1) group, the appearance of minimum leads to a
phase transition. The isothermal $G$-$M$ plots for the $\mathbb{Z}_4$ are plotted in Fig.~5.
As expected below the critical temperature $T_c=0.44$ the plots develop cusps, exhibiting
a first order phase between two co-existing phases with low and high $\kappa$'s.
However, at $T\approx 0$ there is a difference between the $\mathbb{Z}_N$ and
U(1) groups, that is here there is a finite higher $\kappa$ at which
the systems follows the path with finite $M$'s. This behavior is
evident by $T\approx 0$ curve in Fig.~5, in contrast to
U(1) group with $M=0$ toward $\kappa\to\infty$ at $T\approx 0$.
The two critical $\kappa$'s of $T\approx 0$ for $\mathbb{Z}_4$ group
at which the cusp starts and ends are as follow
\begin{align}\label{64}
\kappa_{c1}=0.69,~~~~~~~\kappa_{c2}=8.8
\end{align}

\section{SU(2) Group}
As the case for a non-Abelian group here we consider the SU(2) group in
one spatial direction. Then at time-step $n$ the group element is
represented by
\begin{align}\label{65}
U_n=\exp(\mathrm{i}\,\bm{x}_n\cdot \bm{\sigma}/2R)
\end{align}
in which $\bm{x}_n=(x^1_{n},x^2_{n},x^3_{n})$ represents the three components in the SU(2) sector, and
$\bm{\sigma}=(\sigma_1,\sigma_2,\sigma_3)$ as Pauli matrices. The action then simply comes to the form
\begin{align}\label{66}
S_E=\frac{\kappa}{2} \sum_{n} \mathrm{Tr} \left(U^\dagger_{n}\, U_{n+1} +
U^\dagger_{n+1} \, U_{n} \, -\,  2\, \mathbb{1}_2 \right)
\end{align}
in which $\mathrm{Tr}$ is trace over the matrix structure, with $\mathrm{Tr}(\sigma_\alpha\sigma_\beta)=2\,\delta_{\alpha\beta}$.
As the requirement mentioned earlier, the above action is invariant under the shift:
\begin{align}\label{67}
|\bm{x}_n|\to |\bm{x}_n|+4\pi k_n R,
\end{align}
with $k_n$'s as integer numbers. Using the identity
\begin{align}
\frac{1}{2}\,\mathrm{Tr}\,(e^{\mathrm{i}\,\bm{x}_{n+1}\cdot \bm{\sigma}/2R}\,
e^{-\mathrm{i}\,\bm{x}_n\cdot \bm{\sigma}/2R})&=\cos\frac{r_{n+1}}{2R}\cos\frac{r_n}{2R}+\,\bm{\hat{x}}_{n+1}\cdot\bm{\hat{x}}_n\,
\sin\frac{r_{n+1}}{2R}\sin\frac{r_n}{2R}
\cr
\label{68}
&=: \cos\frac{\gamma_{n+1,n}}{2}
\end{align}
in which $r_n=|\bm{x}_n|$ and $\bm{\hat{x}}_n=\bm{x}_n/r_n$, the actions is simplified as
\begin{align}\label{69}
S_E=2 \kappa \sum_{n} \left(
\cos\frac{\gamma_{n+1,n}}{2} -1 \right)
\end{align}
In the limits $r_n\, \& \, r_{n+1} \ll R$ we have
\begin{align}\label{70}
\gamma_{n+1,n}^2\simeq \frac{1}{R^2}(\bm{x}_{n+1}-\bm{x}_n)^2
+O\left(\frac{r}{R}\right)^4
\end{align}
by which we have in the continuum limit
\begin{align}\label{71}
S_E\simeq -\frac{a\kappa}{4R^2} \int dt~\dot{\bm{x}}^2
\end{align}
In above again the minus sign is due to use of imaginary time in the formalism.
The action (\ref{71}) represents the free motion of
a free particle with mass $m_0=a\kappa/(2R^2)$.

As before, the action (\ref{69}) can be used to define the quantum theory
based on the transfer-matrix method.
The group manifold of SU(2) is known to be the
3-sphere $S^3$, for which the initial parametrization $\bm{x}=(x_1,x_2,x_3)$
in the spherical coordinates $\bm{x}=(r,\theta,\phi)$ is defined
in the intervals
\begin{align}\label{72}
0 \leq r \leq 2\pi R,~~~~~~~0\leq \theta \leq \pi,~~~~~~
0\leq\phi\leq 2\pi
\end{align}
Also it is convenient to use the replacement
\begin{align}\label{73}
\chi=\frac{r}{2R},~~~~~~~~~ 0\leq \chi \leq \pi
\end{align}
By the above parametrization the SU(2) invariant measure takes the form \cite{su2haar}
\begin{align}\label{74}
d\Omega_3=\sin^2 \!\chi \, d\chi\, d\Omega,~~~\mathrm{with}
~~~~d\Omega=\sin\theta \,
d\theta\, d\phi
\end{align}
satisfying
\begin{align}\label{75}
\int_{S^3} d\Omega_3 = 2 \pi^2
\end{align}
As the case with U(1) group, by the complete orthonormal spherical harmonics on
$S^3$ as the eigen-functions, we can read the eigen-values of
the matrix $\widehat{V}$. The normalized spherical harmonics
on $S^3$ are known to be \cite{su2haar}
\begin{align}\label{76}
\mathbb{Y}_{s \ell m}(\chi,\Omega)=\sqrt{\frac{2^{2\ell +1}(s+1)\, (s -\ell )!\, \ell !^{\,2}}{\pi (s+\ell +1)!}}\sin^\ell  \!\chi~C^{(\ell +1)}_{s-\ell }\!(\cos\chi)~Y_{\ell m}(\Omega)
\end{align}
in which $C^{(\ell +1)}_{s-\ell }(x)$ are the Gegenbauer polynomials and
$Y_{\ell m}(\Omega)=Y_{\ell m}(\theta,\phi)$ are the ordinary spherical harmonics
on 2-sphere $S^2$. In above all indices are integers and obey the ordering \cite{su2haar}
\begin{align}\label{77}
|m|\leq \ell  \leq  s=0,1,2,\cdots
\end{align}
The above harmonics are the normalized ones \cite{su2haar}
\begin{align}\label{78}
\int_{S^3}  d\Omega_3~ \mathbb{Y}_{s\ell m}(\chi,\Omega)~
\mathbb{Y}_{s'\ell' m'}^\star(\chi,\Omega)=
\delta_{ss'}\,\delta_{\ell \ell '}\,\delta_{mm'}
\end{align}
Using (\ref{68}) and in the $(\chi,\theta,\phi)$ parametrization of $S^3$,
 the action (\ref{69}) between the adjacent times $n$ and $n+1$ takes the form
\begin{align}\label{79}
S_E(n,n+1)=2\kappa\,\Big(\cos\!\chi_{n+1}\,\cos\!\chi_{n}+\bm{\hat{x}}_{n+1}\cdot\bm{\hat{x}}_n\, \sin\!\chi_{n+1}\, \sin\!\chi_{n}\,-1\Big)
\end{align}
in which
\begin{align}\label{80}
\bm{\hat{x}}_{n+1}\cdot\bm{\hat{x}}_n=\cos \theta_{n+1} \cos\theta_n
+\cos(\phi_{n+1}-\phi_n) \sin\theta_{n+1}\sin \theta_n
\end{align}
Then the matrix elements of the transfer matrix is given by
\begin{align}\label{81}
\langle  \bm{x}_{n+1} |\widehat{V} |  \bm{x}_n\rangle=
\sqrt{\frac{\kappa}{4\pi}} \exp\!\Big[ S_E(n,n+1)\Big]
\end{align}
Using the identity for real $w$
\begin{align}\label{82}
e^{w\, \bm{\hat{x}}\,\cdot\,\bm{\hat{x}}'}=  \sqrt{\frac{\pi}{2w}} ~
4\pi\sum_{\ell =0}^\infty \sum_{m=-\ell }^\ell  I_{\ell +1/2}(w) \,\,
Y_{\ell m}(\Omega)\,\,Y_{\ell m}^\star(\Omega')
\end{align}
in which the direction of two unit vectors $\bm{\hat{x}}$ and
$\bm{\hat{x}}'$ are given by ordinary solid-angles $\Omega$ and $\Omega'$,
respectively, and $I_{\ell +1/2}$ as before is the modified Bessel function.
Taking $w=2\kappa\sin\!\chi_n\sin\!\chi_{n+1}$, by (\ref{82}) we have
\begin{align}
\sqrt{\frac{\kappa}{4\pi}}
\exp\!\Big[ S_E(n,n+1)\Big]=\sqrt{\frac{\kappa}{4\pi}} \,
e^{2\kappa(\cos\!\chi_n\cos\!\chi_{n+1}-1)}
\sqrt{\frac{\pi}{4\kappa\sin\!\chi_n\sin\!\chi_{n+1}}}
\cr
\label{83}
\times 4\pi\sum_{\ell =0}^\infty \sum_{m=-\ell }^\ell  I_{\ell +1/2}(2\kappa \sin\!\chi_n\sin\!\chi_{n+1}) \, Y_{\ell m}(\Omega_n)\,Y_{\ell m}^\star(\Omega_{n+1})
\end{align}
for which we also have an expansion based on the energy eigen-values
and the eigen-functions as
\begin{align}
\sqrt{\frac{\kappa}{4\pi}}
\exp\!\Big[ S_E(n,n+1)\Big]&= \sum_{s=0}^\infty \sum_{\ell =0}^s \sum_{m=-\ell }^\ell
e^{-aE_{s\ell m}}\,  \mathbb{Y}_{s\ell m}(\chi_n,\Omega_n)
\,\mathbb{Y}_{s\ell m}^\star(\chi_{n+1},\Omega_{n+1})
\cr
\label{84}
&=\sum_{\ell =0}^\infty \sum_{m=-\ell }^\ell   \sum_{s=\ell }^\infty
e^{-aE_{s\ell m}}\,  \mathbb{Y}_{s\ell m}(\chi_n,\Omega_n)
\,\mathbb{Y}_{s\ell m}^\star(\chi_{n+1},\Omega_{n+1})
\end{align}
Using the orthonormality relation of $Y_{\ell m}$'s, by (\ref{83}) and (\ref{84}),
after the changes $\chi_n\to\chi$ and $\chi_{n+1}\to\chi'$, explicit expression (76) gives us
\begin{align}
\sum_{s=\ell }^\infty
\frac{2^{2\ell +1}(s+1)\, (s-\ell )!\, \ell !^{\,2}}{\pi (s+\ell +1)!}
e^{-aE_{s\ell }} \sin^\ell  \!\chi \sin^\ell  \!\chi'\,
C^{(\ell +1)}_{s-\ell }\!(\cos\!\chi) C^{(\ell +1)}_{s-\ell }\!(\cos\!\chi')
\cr  \label{85}
=\sqrt{\frac{\kappa}{4\pi}}
\sqrt{\frac{\pi}{4\kappa\sin\!\chi\,\sin\!\chi'}}\, 4\pi\,
e^{2\kappa(\cos\!\chi \cos\!\chi'-1)}
I_{\ell +1/2}(2\kappa \sin\!\chi \sin\!\chi')
\end{align}
\begin{figure}[t
	\begin{center}
		\includegraphics[scale=0.9]{Fig-su2-1.pdf}
	\end{center}
	\caption{The few lowest energies by (\ref{90}) for SU(2) group. }
\end{figure}
in which we have dropped the index $m$ in $E_{s\ell }$, as it is now an irrelevant one.
This indicates that the energy eigen-values has
at least $(2\ell +1)$-level degeneracy. Using the orthogonality
of the Gegenbauer polynomials
\begin{align}\label{86}
\int_0^\pi C^{(\ell+1)}_{s-\ell}\!(\cos\alpha)\, C^{(\ell+1)}_{s'-\ell}\!(\cos\alpha)
\sin^{2\ell+2}\!\!\alpha
\,d\alpha = \frac{\pi\, (s+\ell+1)!}{2^{2\ell+1}(s+1)(s-\ell)!\, \ell !^{\,2}}\, \delta_{ss'}
\end{align}
by multiplication the l.h.s. of (\ref{85}) by
$\sin^{\ell+2}\!\chi' \, C^{(\ell+1)}_{s-\ell}(\cos\chi')$ and
integration over $\chi'$  we have
\begin{align}
e^{-aE_{s\ell }} \, \sin^\ell  \!\chi  \,
C^{(\ell +1)}_{s-\ell }\!(\cos\!\chi)
= \int_0^\pi\!\! d\chi'
\sqrt{\frac{\kappa}{4\pi}}
\sqrt{\frac{\pi}{4\kappa\sin\!\chi\,\sin\!\chi'}}\, 4\pi\,
e^{2\kappa(\cos\!\chi\cos\!\chi'-1)}
\cr   \label{87}
\times\sin^{\ell+2}\!\chi' \, C^{(\ell+1)}_{s-\ell}\!(\cos\chi')
I_{\ell +1/2}(2\kappa \sin\!\chi\sin\!\chi')
\end{align}
\begin{figure}[t
	\begin{center}
		\includegraphics[scale=1.1]{Fig-su2-2.pdf}
	\end{center}
	\caption{The isothermal $G$-$M$ plots for SU(2) group. }
\end{figure}
Using the identity \cite{gegen}:
\begin{align}
\int_0^\pi e^{z \cos\chi \cos \chi'} \sin^{\ell+3/2}\!\chi'\,
C^{(\ell+1)}_{s-\ell}\!(\cos\!\chi') \,I_{\ell+1/2}(z\sin\!\chi \sin\!\chi')\,d\chi'
\cr    \label{88}
=\sqrt{\frac{2\pi}{z}}~ \sin^{\ell+1/2}\!\chi ~
C^{(\ell+1)}_{s-\ell}\!(\cos\!\chi) ~ I_{s+1}(z)
\end{align}
the integration in r.h.s. of (\ref{87}) can be calculated, setting $z=2\kappa$,
leading to
\begin{align}\label{89}
e^{-a\,E_s} =\sqrt{\frac{\kappa}{4\pi}} \,
\frac{2\pi^2}{\kappa}\,
e^{-2\kappa} \, I_{s+1}(2\kappa)
\end{align}
in which we have dropped the index $l$ from $E_s$ as well, since there is no
$\ell$-dependence in spectrum. By (\ref{89}) we find the energy as below with
a $(s+1)^2$-level degeneracy:
\begin{align}\label{90}
E_s=-\frac{1}{a}\ln \left[\sqrt{\frac{\kappa}{4\pi}} \,
\frac{2\pi^2}{\kappa}\,
e^{-2\kappa} \, I_{s+1}(2\kappa)
\right],~~~~s=0,1,2,\cdots
\end{align}
\noindent In Fig.~6 the four lowest energies are plotted, all
having minimum. This is in contrast to
the cases with U(1) and $\mathbb{Z}_N$ groups
where only $E_0$ has a minimum.
The one-particle partition function is given by
\begin{align}
Z_1(\beta,\kappa)=\sum_{s=0}^\infty (s+1)^2~ e^{-\beta\,E_s}
\end{align}
Again the equation-of-state as well as the Gibbs free energy can be obtained by
(\ref{43}) and (\ref{44}). The $G$-$M$ plots given in Fig.~7 develop cusp, indicating that
the system exhibits a first-order phase transition below the critical temperature $T_c=0.67$.
At $T\approx 0$ the critical $\kappa_c=0.29$ is obtained above which $M$ gets non-zero values.

\section{Conclusion and Discussion}
The general theoretical ground as well as specific examples are presented
for models based on compact angle coordinates.
The present construction might be considered as a continuation of the
theme by which the gauge fields are treated as compact angle variables \cite{polya1,polya2,thooft1,lattice}. In the present formulation the action
depends on the group elements rather than the
algebra elements, leading to the invariance under the total shifts inside
the compact domain. It is observed that a discrete-time
formulation of the theory is the
natural way to obtain the desired dependence on group elements.
The way toward the present formulation is in some sense in reverse
direction of what has happened in lattice formulation of gauge theories.
In particular, in formulation of gauge theories on lattice the invariance
under the gauge transformations requires that the gauge fields are introduced
to the theory via the group elements. Here the demand of invariance under
the total shifts of compact coordinates requires to treat time parameter
as a discrete one. The worldline action by the formulation
resembles the spin chain Hamiltonian of magnetic systems, with coordinates appearing
as the spin degrees of freedom.

The quantization of the model is formulated based on the transfer-matrix method
\cite{lattice,wipf}. The pre-factor in the definition of
the elements of transfer-matrix, in contrast to the case with infinite extent coordinates, can not be absorbed by a change of the path-integral
integration variables.
This particularly causes that the energy eigen-values develop minima as
functions of the defining parameter of the theory.

As examples for the formulation, the models based on the U(1),
$\mathbb{Z}_N$ and SU(2) groups are explicitly constructed.
In all of the models based on the three groups the exact energy
eigen-values are obtained. As the consequence of the minima in the spectrum
all the models exhibit the first-order transition between coexistent phases.

As mentioned earlier, by setting $\kappa=1/g^2$ the model by U(1) group is in fact the result of the dimensional reduction of pure U(1) lattice gauge theory. As one possible application of the present construction here we mention the attempt in \cite{0bfath} to fit the model by U(1) group to the expectations from the monopole dynamics. In particular, the phase transition for the particles with mass $m_0\propto 1/g^2$ by (\ref{25}) may provide a better understanding of the role of monopoles in confinement mechanism based on the dual Meissner effect in superconductors \cite{nambu,mandelstam,thooft2}. Based on the proposed mechanism, at strong coupling limit, at which the monopoles have tiny masses, the motion of monopoles around the electric fluxes prevents the fluxes to spread, leading to the confinement of the electric charges.  Instead at small coupling limit, where the monopoles are highly massive, the electric fluxes originated from source charges are likely to spread over space, leading to the Coulomb's law. It is expected that there is a critical coupling $g_c$ at which the transition from confined phase to the Coulomb phase occurs.

According to the model with group U(1), the two regimes with weak and strong couplings constants are related by a first-order phase transition. The behavior of the system at low temperatures, where the main contribution to the partition function is from the ground-state, is of particular interest. In the limit $T\to 0$, due to the Maxwell construction, we have $M=0$ for $g<g_c=1.125$; Fig.~8.
\begin{figure}[t
	\begin{center}
		\includegraphics[scale=1.1]{Fig-u1-4.pdf}
	\end{center}
	\caption{The isothermal $M$-$g$ plots by the model based on the U(1) group.}
\end{figure}
So as the consequence of discontinuous nature of the first order transition, at low temperatures and below $g_c$ we have $M\!\propto\! \langle v^2\rangle\approx 0$. This behavior is to be compared with (\ref{46}) for ordinary particles, by which there is an asymptote reduce of $M$ by increasing the mass at constant $T$. According to the present model, at low temperatures and below $g_c$, the particles with mass $m_0\propto g^{-2}$ are hardly moving ($\langle v^2 \rangle \approx 0$), leading to an exact Coulomb phase. On the other hand, exhibiting a high-slope increase of $\langle v^2 \rangle$ at $g_c$, the confined phase finds an instant govern once $g$ exceeds $g_c$ at low temperatures. This picture and specially the value of critical coupling constant are in agreement with theoretical and numerical studies \cite{0bfath}.

\vspace{4mm}
\textbf{Acknowledgement}: The author is grateful to M.~Khorrami
for helpful discussions on the role of the imaginary time in formulation of
gauge theories on lattice.
This work is supported by the Research Council of the Alzahra University.

redpajama_set_name: RedPajamaArXiv


text:
Q: What is a non recursive solution for Fibonacci-like sequence in Java? Given this pseudo code of a function
f(0) = 1;
f(1) = 3;
f(n) = 3 * f(n - 1) - f(n - 2); // for n >= 2.

Is there a non recursive way of doing this?

A: It's simple, in Java the solution looks like this:
public int f(int n) {

      int tmp;
      int a = 3;
      int b = 1;

      for (int i = 0; i < n; i++) {
          tmp = a;
          a = 3 * a - b;
          b = tmp;
      }

      return b;

}

All recursive solutions can be transformed into iterative solutions (the opposite is also true, see this post), albeit it's easier if the recursive solution it's in tail-recursive form.
The above algorithm can be understood as a dynamic-programming solution to the original recursion, it's very efficient since it only needs to save the two previous values at each point in the iteration.

A: Yes, all recursive algorithms can be converted into iterative ones. The recursive solution to your problem is something like (pseudo-code):
def f(n):
    if n == 0: return 1
    if n == 1: return 3
    return 3 * f(n-1) - f(n-2)

Since you only have to remember the previous two terms to calculate the current one, you can use something like the following pseudo-code:
def f(n):
    if n == 0:
        return 1
    if n == 1:
        return 3
    grandparent = 1
    parent = 3
    for i = 2 to n:
        me = 3 * parent - grandparent
        grandparent = parent
        parent = me
    return me

This simply handles the "recursive" termination condition first then iterates where it would normally call itself. At each iteration, you calculate the current term, then rotate the terms through the grandparent and parent.
There is no need to keep the grandparent around once you've calculated the current iteration since it's no longer used.
In fact, it could be said that the iterative solution is better (from a performance viewpoint) since terms are not recalculated as they are in the recursive solution. The recursive solution does have a certain elegance about it though (recursive solutions generally do).

Of course, like the Fibonacci sequence, that value you calculate rises very quickly so, if you want what's possibly the fastest solution (you should check all performance claims, including mine), a pre-calculated lookup table may be the way to go.
Using the following Java code to create a table of long values (that while condition is just a sneaky trick to catch overflow, which is the point at which you can stop building the array):
class GenLookup {
    public static void main(String args[]) {
        long a = 1, b = 3, c;
        System.out.print ("long lookup[] = { " + a + "L, " + b + "L");
        c = 3 * b - a;
        while ((c + a) / 3 == b) {
            System.out.print (", " + c + "L");
            a = b; b = c; c = 3 * b - a;
        }
        System.out.println (" };");
    }
}

gives you an array definition that you can just plug in to a lookup function, as per the following example:
public static long fn (int n) {
    long lookup[] = { 1L, 3L, 8L, 21L, 55L, 144L, 377L, 987L, 2584L, 6765L,
        17711L, 46368L, 121393L, 317811L, 832040L, 2178309L, 5702887L,
        14930352L, 39088169L, 102334155L, 267914296L, 701408733L,
        1836311903L, 4807526976L, 12586269025L, 32951280099L, 86267571272L,
        225851433717L, 591286729879L, 1548008755920L, 4052739537881L,
        10610209857723L, 27777890035288L, 72723460248141L, 190392490709135L,
        498454011879264L, 1304969544928657L, 3416454622906707L,
        8944394323791464L, 23416728348467685L, 61305790721611591L,
        160500643816367088L, 420196140727489673L, 1100087778366101931L,
        2880067194370816120L, 7540113804746346429L };

    if ((n < 1) || (n > lookup.length))
        return -1L;

    return lookup[n-1];
}

Interestingly enough, WolframAlpha comes up with a formulaic approach that doesn't even use iteration. If you go to their site and enter f(0)=1, f(1)=3, f(n)=3f(n-1)-f(n-2), you'll get back the formula:

Unfortunately, it may not be as fast as the iteration, given the limited number of input values that result in something that can fit in a Java long, since it uses floating point. It's almost certainly (but, again, you would need to check this) slower than a table lookup.
And, it's probably perfect in the world of maths where real-world limits like non-infinite storage don't come into play but, possibly due to the limits of IEEE precision, it breaks down at higher values of n.
The following functions are the equivalent of that expression and the lookup solution:
class CheckWolf {
    public static long fn2 (int n) {
        return (long)(
            (5.0 - 3.0 * Math.sqrt(5.0)) *
                Math.pow(((3.0 - Math.sqrt(5.0)) / 2.0), n-1) +
            (5.0 + 3.0 * Math.sqrt(5.0)) *
                Math.pow(((3.0 + Math.sqrt(5.0)) / 2.0), n-1)
            ) / 10;
    }

    public static long fn (int n) {
        long lookup[] = { 1L, 3L, 8L, 21L, 55L, 144L, 377L, 987L, 2584L, 6765L,
            17711L, 46368L, 121393L, 317811L, 832040L, 2178309L, 5702887L,
            14930352L, 39088169L, 102334155L, 267914296L, 701408733L,
            1836311903L, 4807526976L, 12586269025L, 32951280099L, 86267571272L,
            225851433717L, 591286729879L, 1548008755920L, 4052739537881L,
            10610209857723L, 27777890035288L, 72723460248141L, 190392490709135L,
            498454011879264L, 1304969544928657L, 3416454622906707L,
            8944394323791464L, 23416728348467685L, 61305790721611591L,
            160500643816367088L, 420196140727489673L, 1100087778366101931L,
            2880067194370816120L, 7540113804746346429L };
        if ((n < 1) || (n > lookup.length)) return -1L;
        return lookup[n-1];
    }

Now we need a mainline to compare them:
    public static void main(String args[]) {
        for (int i = 1; i < 50; i++)
            if (fn(i) != fn2(i))
                System.out.println ("BAD:  " + i + ": " + fn(i) + ", " + fn2(i)
                    + " (" + Math.abs(fn(i) - fn2(i)) + ")");
            else
                System.out.println ("GOOD: " + i + ": " + fn(i) + ", " + fn2(i));
        }
    }

This will output:
GOOD: 1: 1, 1
GOOD: 2: 3, 3
GOOD: 3: 8, 8
GOOD: 4: 21, 21
GOOD: 5: 55, 55
GOOD: 6: 144, 144
GOOD: 7: 377, 377
GOOD: 8: 987, 987
GOOD: 9: 2584, 2584
GOOD: 10: 6765, 6765
GOOD: 11: 17711, 17711
GOOD: 12: 46368, 46368
GOOD: 13: 121393, 121393
GOOD: 14: 317811, 317811
GOOD: 15: 832040, 832040
GOOD: 16: 2178309, 2178309
GOOD: 17: 5702887, 5702887
GOOD: 18: 14930352, 14930352
GOOD: 19: 39088169, 39088169
GOOD: 20: 102334155, 102334155
GOOD: 21: 267914296, 267914296
GOOD: 22: 701408733, 701408733
GOOD: 23: 1836311903, 1836311903
GOOD: 24: 4807526976, 4807526976
GOOD: 25: 12586269025, 12586269025

Looking good up to here, some more:
GOOD: 26: 32951280099, 32951280099
GOOD: 27: 86267571272, 86267571272
GOOD: 28: 225851433717, 225851433717
GOOD: 29: 591286729879, 591286729879
GOOD: 30: 1548008755920, 1548008755920
GOOD: 31: 4052739537881, 4052739537881
GOOD: 32: 10610209857723, 10610209857723
GOOD: 33: 27777890035288, 27777890035288
GOOD: 34: 72723460248141, 72723460248141
GOOD: 35: 190392490709135, 190392490709135
GOOD: 36: 498454011879264, 498454011879264

But then something starts going awry:
BAD:  37: 1304969544928657, 1304969544928658 (1)
BAD:  38: 3416454622906707, 3416454622906709 (2)
BAD:  39: 8944394323791464, 8944394323791472 (8)
BAD:  40: 23416728348467685, 23416728348467705 (20)
BAD:  41: 61305790721611591, 61305790721611648 (57)
BAD:  42: 160500643816367088, 160500643816367232 (144)
BAD:  43: 420196140727489673, 420196140727490048 (375)

The fact that the above are tantalisingly close, and that the number of digits in the error is proportional to the number of digits in the result, indicates it's probably a loss-of-precision problem.
After this point, the formulaic function just starts returning the maximum long value:
BAD:  44: 1100087778366101931, 922337203685477580 (177750574680624351)
BAD:  45: 2880067194370816120, 922337203685477580 (1957729990685338540)
BAD:  46: 7540113804746346429, 922337203685477580 (6617776601060868849)

And then our lookup function breaks down as well since the numbers are too big for a long:
BAD:  47: -1, 922337203685477580 (922337203685477581)
BAD:  48: -1, 922337203685477580 (922337203685477581)
BAD:  49: -1, 922337203685477580 (922337203685477581)

A: [ Oops, I thought this was a Perl question. Still, the code should be readable enough to a Java developer. ]
This is really just moving the recursion to userland, but you could use:
sub f {
    my ($n) = @_;
    my @f = (1,3);
    $f[$_] = 3 * $f[$_-1]- $f[$_-2] for 2 .. $n;
    return $f[$n];
}

Of course, this begs for caching. There's no need to recalculate values we already know.
my @f = (1,3);
sub f {
    my ($n) = @_;
    $f[$_] = 3 * $f[$_-1]- $f[$_-2] for @f .. $n;
    return $f[$n];
}

A: The function is defined in terms of itself, so in one sense any implementation is recursive, unless some mathematician comes and tells us f(n) can be evaluated without evaluating f(n-1) and f(n-2). As others have shown, there is a way to implement it in Java function that does not call itself.

A: Answers here are correct, but they work in O(n), while you can do it in O(log n), exponentially faster. Observe that
[f(n)  ] = [3 -1] [f(n-1)]
[f(n-1)]   [1  0] [f(n-2)]

Let vn be the vector [f(n), f(n-1)] and A the matrix as above, so you get vn = A vn-1, therefore vn = An-1 v1. Compute (n-1)-th power of matrix A using binary exponentiation and multiply it by v1. For more on linear recurrences see here.

A: If your question is about whether an equivalent non-recursive definition of the function can be found, you should search for properties of the Fibonacci sequence.
Your sequence can be found by writing the Fibonacci (without the first 2 numbers) and removing every 2nd number: 1, 3, 8, 21, 55, 144, ...
sqrt5 = sqrt(5)
phi = ( 1 + sqrt5 ) / 2
fibonacci(n) = round( power( phi, n ) / sqrt5 )
f(n) = fibonacci( 2*n + 2 )

A: def func(n):
    f= array(n+1)
    f[0]=1
    f[1]=3

    for i in 2:n :
        f[i] = 3*f[i-1]-f[i-2]
    return f[n]

A: The Fibonacci series sequence of numbers starts as: 0,1,1,2,3,5,8,13,21,34,55....
this can be defined by simple recurrence relation F(n)=F(n-1)+F(n-2) for n>1 and two initial conditions, F(0)=1 and F(1)=1
Algorithm Fibonacci
//Computes the nth Fibonacci number
//Input: A non-negative integer
//Output: The nth Fibonacci number
1. Begin Fibo
2. integer n, i;
3.    if n<=1 then
4.     return n;
5.  else
6.    F(0)<-0; F(1)<-1;
7.    for i<-2 to n do
8.     F(i)<-F(i-1)+F(i-2);
9.     F(i-2)=F(i-2);
10.    F(i-1)=F(i);
11. done
12. end if
13. end Fibo

A: Here is simply a function with minimum line of code and maximum flexibility.
You can add any "initial values" and any other recursive "function" you want simply.
def fib(n):
  fibs = [1, 3]       # <--  your initial values here
  if n == 1:
    return fibs[0]
  if n == 2:
    return fibs[:1]
  for i in range(2, n):
    fibs.append(3*fibs[-1] - fibs[-2])  # <-- your function here
  return fibs

And the result is:
n=10
print(fib(n))

[1, 3, 8, 21, 55, 144, 377, 987, 2584, 6765]

A: This is my non-recursive solution (javascript):

function fibonacci(n) {
  let f = 0;
  let g = 1;
  for (let i = 0; i <= n; i++) {
    console.log(f);
    f = f + g;
    g = f - g;
  }
}

fibonacci(10);

A: As requested by @paxdiablo I'm making this an answer. It's a recurrence relation and can be solved non-recursively, similar to the fibonacci sequence mentioned in another answer. It turns out to be (Python notation).
def f(n):
    return int((13**0.5-3)/(2*13**0.5)*((3-13**0.5)/2)**n + (0.5+3/(2*13**0.5))*((3+13**0.5)/2)**n)

However, this forumula does most probably not work for large n, because of limited float precision. The given python version fails for n = 30:
>>> print ([f(n) == 3 * f(n-1) + f(n-2) for n in range(2, 30)])
[True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, False]
>>> print([f(n) for n in range(30)])
[1, 3, 10, 33, 109, 360, 1189, 3927, 12970, 42837, 141481, 467280, 1543321, 5097243, 16835050, 55602393, 183642229, 606529080, 2003229469, 6616217487, 21851881930, 72171863277, 238367471761, 787274278560, 2600190307441, 8587845200883, 28363725910090, 93679022931153, 309400794703549, 1021881407041801]

Warning: I used a "+" instead of a "-", so the formula is wrong. See comments.

A: I don't know why nobody has proposed the following algorithm, which is way simpler and faster:
class Fibonacci {

    public void compute(int n) {
        long i=0;
        long a=0,b=1,d=0;

        while (i<n) {

            System.out.println(a);
            d=a;
            a=b;
            b=b+d;
            i+=1;
        }
    }

    public static void main(String[] args) {

        var fib = new Fibonacci();

        fib.compute(90);
    }

}

Output:
0
1
1
2
3
5
8
13
21
34
55
89
144
233
377
610
987
1597
2584
4181
6765
10946
17711
28657
46368
75025
121393
196418
317811
514229
832040
1346269
2178309
3524578
5702887
9227465
14930352
24157817
39088169
63245986
102334155
165580141
267914296
433494437
701408733
1134903170
1836311903
2971215073
4807526976
7778742049
12586269025
20365011074
32951280099
53316291173
86267571272
139583862445
225851433717
365435296162
591286729879
956722026041
1548008755920
2504730781961
4052739537881
6557470319842
10610209857723
17167680177565
27777890035288
44945570212853
72723460248141
117669030460994
190392490709135
308061521170129
498454011879264
806515533049393
1304969544928657
2111485077978050
3416454622906707
5527939700884757
8944394323791464
14472334024676221
23416728348467685
37889062373143906
61305790721611591
99194853094755497
160500643816367088
259695496911122585
420196140727489673
679891637638612258
1100087778366101931
1779979416004714189

redpajama_set_name: RedPajamaStackExchange


I put together a roundup of Media Query websites for the fine people over at The Inspiration Blog. Make sure to check out the full roundup on TIB.
Thanks guys for this excellent post.
Great resource.Nice stuff.thanks for sharing this post.


text:
< Return
Derek Boshier
Redbud Gallery, Houston, Texas
Review by Donna Tennant
Derek Boshier, "David Bowie (The Lodger)," 2016, acrylic on canvas, 36 x 18"
Continuing through April 30, 2019
British pop artist Derek Boshier has deep ties to Houston. After joining the faculty of University of Houston in 1980 and regularly exhibiting in various Houston venues for more than 20 years, he relocated to Los Angeles. This show spans the last 15 years, roughly Boshier's time in Los Angeles since leaving Houston.
"Kings on Road" is one of seven paintings from 2013 contrasting dual images that comment on popular culture or politically charged events. "Kings" refers to a major street in London associated with the mod culture of the 1960s and the punk culture of the 1970s. The image references fashion icons such as Vivienne Westwood and Mary Quant. In this series, the canvases are divided by a diagonal line; on the right, Boshier paints these photographs in white, black and gray, while on the left, he presents a bright caricature inspired by the subject matter. In "Lee Harvey Oswald," the photograph shows Jack Ruby shooting Oswald in the basement of Dallas police headquarters, and on the left, a tiny superhero figure punches Oswald violently in the face. In "Saigon," Boshier uses photojournalist Eddie Adams' memorable 1968 photograph of the execution of a Viet Cong prisoner at close range, while on the left he paints two hands holding revolvers against a bright red background. "Kent State" juxtaposes John Paul Filo's 1970 image of a young student crying next to a classmate's body with two sneering sadistic cops.
Boshier also revisits his famous cover design for David Bowie's album with the Clash, "The Lodger" (1979), for which he and Bowie collaborated with photographer Brian Duffy. Bowie was laid on a metal support and pressed beneath a plate of glass to create the illusion that he has a broken nose and appears to be falling. In Boshier's 2016 "David Bowie (The Lodger)" the familiar image seems to be electrified as it hovers above the skyline of Los Angeles.
The largest painting in the show, titled "Mexico," juxtaposes florescent reds, oranges, and blues against a grisaille background. The colorful images depict stereotypical images of Mexico — mariachis, masks, serapes and cactus — floating over a monochromatic background of gruesome images of the dead and dying. There are machine guns, firing squads and hangings. Comments on politics and our current state of affairs always seem to find a place in Boshier's work.
The world of celebrity continues to pique Boshier's interest, as the most recent series of "Ghosts of Houston" demonstrates. Consisting of ink drawings on paper of well-known deceased Houstonians — including artists Dick Wray and Casey Williams, writers Donald Barthelme and Molly Ivins, curator Walter Hopps and uber-collector Dominque de Menil — each portrait utilizes the same background photograph, a brick street in Freedman Town. Located near downtown, it is Houston's oldest African-American community. The figures, posed standing in the middle of the street, are painted in black ink, their faces rendered in a loose comic-book character style. The "Ghosts" of the cultural elite are provocatively joined to anonymous presence of the residents of Freedman Town.
Now in his early 80s, Boshier has spent his career blurring the distinction between high and low art ever since his student days at the Royal College of Art in London in the early 1960s with fellow students David Hockney, Allen Jones, R.B. Kitaj and others in who would become associated with the British Pop Art movement. Boshier stopped painting in the '70s and turned to other media, including drawing, printmaking, film, writing, sculpture, installation and photography. His aesthetic approach remains catholic, blending pop art, comic book style and film imagery. The current show conveys the effect of the thousands of images that bombard us daily and the impact of digital technology on the individual. What we get is the product of Boshier's acerbic wit, engaged content and distinctive style.
Redbud Gallery
View each day's exhibition openings and special events
Show me Openings & Events
All Exhibitions & Events

redpajama_set_name: RedPajamaCommonCrawl


text:
You are not logged in. [Log In] Science a GoGo's Home Page » Forums » General Discussion » Not-Quite-Science Forum » 19 yrs of BLM violations, enforcement decision
#51637 - 04/10/14 02:27 AM 19 yrs of BLM violations, enforcement decision
19 yrs of BLM violations, enforcement decision finally made !
1) 5 years is the Federal limitation.
2) the maximum fine provided is only $500.00 (from what I can find)
§4170.2-1 Penal provisions under the Taylor Grazing Act.
Under section 2 of the Act any person who willfully commits an act prohibited under §4140.1(b), or who willfully violates approved special rules and regulations is punishable by a fine of not more than $500.
http://www.ecfr.gov/cgi-bin/retrieveECFR...ART&ty=HTML
3) the Fed Govt has probably already spent several million.
why not just give this man free use of the land that he has already been using at no cost for 19 years and its free use was probably a large part of what made America what it once was for decades or centuries before the BLM folk started charging for its use.
his cattle provide fertilizer for the plants , ground aeration
for the plants , and unless each of his cows also has a calf
he cant be charged for just the cow.
the BLM page clearly states the following
Federal Grazing Fee
The Federal grazing fee, which applies to Federal lands in 16 Western states on public lands managed by the BLM and the U.S. Forest Service, is adjusted annually and is calculated by using a formula originally set by Congress in the Public Rangelands Improvement Act of 1978. Under this formula, as modified and extended by a presidential Executive Order issued in 1986, the grazing fee cannot fall below $1.35 per animal unit month (AUM); also, any fee increase or decrease cannot exceed 25 percent of the previous year's level. (An AUM is the amount of forage needed to sustain one cow and her calf, one horse, or five sheep or goats for a month.) The grazing fee for 2014 is $1.35 per AUM, the same level as it was in 2013.
http://www.blm.gov/wo/st/en/prog/grazing.html
this could actually lead to a skirmish between any enforcement
officials and the militia organizations who have vowed to
protect this man and his property.
3/4 inch of dust build up on the moon in 4.527 billion years,LOL and QM is fantasy science.
#51653 - 04/11/14 04:11 PM Re: 19 yrs of BLM violations, enforcement decision [Re: paul]
this man and his ancestors have been grazing their cattle on
this land since the 1870's !!!
Nevada didn't become a state until 1864 , Im going to
guess that this mans ancestors played a very important role
in the establishment of and the economic development of the
State of Nevada.
when the BLM gained access to these lands Im certain there were provisions made for the public who were currently using the lands.
in that all current permits , leases , agreements , contracts
that allow the public to use the land at the time of the
acquisition of the land by the BLM would remain in place and would not be dishonored by the BLM.
The BLM's multiple-use mission, set forth in the Federal Land Policy and Management Act of 1976, mandates that we manage public land resources for a variety of uses, such as energy development, livestock grazing, recreation, and timber harvesting
http://www.blm.gov/wo/st/en/info/About_BLM.html
43 U.S. Code § 1769 - Existing right-of-way or right-of-use unaffected; exceptions; rights-of-way for railroad and appurtenant communication facilities; applicability of existing terms and conditions
(a) Nothing in this subchapter shall have the effect of terminating any right-of-way or right-of-use heretofore issued, granted, or permitted. However, with the consent of the holder thereof, the Secretary concerned may cancel such a right-of-way or right-of-use and in its stead issue a right-of-way pursuant to the provisions of this subchapter.
from what I understand this man has not consented , he is
the last man standing who has not consented.
his right of way is not and never will be affected by the
Federal Land Policy and Management Act of 1976 unless he
consents.
he cannot be charged with a violation because he never
gave his consent therefore he does not fall under the
jurisdiction or laws or regulations set fourth in the bill.
others have consented and they are affected by the bill.
Title 43 U.S. Code.
http://www.law.cornell.edu/uscode/text/43/315
43 U.S. Code § 315 - Grazing districts; establishment; restrictions; prior rights; rights-of-way; hearing and notice; hunting or fishing rights
Whenever any grazing district is established pursuant to this subchapter, the Secretary shall grant to owners of land adjacent to such district, upon application of any such owner, such rights-of-way over the lands included in such district for stock-driving purposes as may be necessary for the convenient access by any such owner to marketing facilities or to lands not within such district owned by such person or upon which such person has stock-grazing rights.
grant = gift
I suppose the next thing that needs to be established is
whether the land in question is located within the jurisdiction boundaries of the U.S. Code.
is the State of Nevada located within the boundaries of
the United States of America?
OK , NOW IT MAKES SENCE.
I should have thought about it in the correct manner.
its not about the permits.
its not about anything that has anything to do with the
BLM and its mission.
its about forcing this man to sell his land so that the land
can be resold for many millions of dollars due to its underground water content!!!
the video says that a defense company wants to build on the
you should listen to this video first to set the mood and
to gain insight into the historic value of water
cool clear water ....water...
if the video is still there of course.
currently its not there or there are so many people watching it that its not responding.
AS NOT SEEN ON TV , you wont see this on TV.
because it shows the People getting results from protesting
against tyranny.
Im certain that if todays mainline news agencies were around
before 1776 during the first skirmishes and sabre rattling
and protest that led up to the revolt and first shots fired
in the American Revolution which resulted in the Liberties
that these People are protecting they also would not have covered those events as they did not cover this event.
#51703 - 04/15/14 12:51 AM Re: 19 yrs of BLM violations, enforcement decision [Re: paul]
I don't think this rule of anarchy is going to do any good for our country. If impromptu gatherings of armed people are going to support people who are not obeying the law, it will lead to the neutering of the federal government. That gathering of people was in contradiction to the laws of the United States, and it has set a dangerous precedent. If the laws cannot be enforced the government may as well not exist. What those people were doing was unlawful, and yet they are being made out to be heroes. Mr Bundy does not have the right to ignore the laws of the United States at his whim. The fact that he has been doing so for years only adds to the wrong he and his friends are doing. Usurping the laws of the land is sedition, if not treason. I predict that nothing good will come of this. If this continues, I fear for the union.
#51711 - 04/15/14 06:30 PM Re: 19 yrs of BLM violations, enforcement decision [Re: Amaranth Rose II]
this entire charade involving Mr Bundy is nothing more than
a illegal land grab that involves politicians doing favors for
industry , favors that already include millions of dollars of tax payers
money spent to attempt to remove Mr Bundy , and further contracts
between a agency of the United States and
industry that is to be paid for by industry , if you step back and look at
this charade and the agencies and people involved it resembles organized crime
more than government of the people of a nation.
if you or anyone else believe that this type of criminal acquisition of peoples property
and or attempts at the / any unconstitutional removal of peoples rights as U.S. Citizens
by or through any legislation , policy or bill written
within the laws of the United States of America is legal
then procedures to pursue Mr Bundy in a court of law for the
criminal act involved should be pursued , the only recourse that any entity pursuing Mr Bundy would have would be
the refusal of Mr Bundy to concede his rights as a land owner whos land is adjacent to the lands managed by the BLM.
it is not illegal for Mr Bundy to refuse to concede his rights as a U.S. Citizen to the BLM.
you speak of fear for the union , if this type of practice
continues there will be no union to fear for , there will
only be criminals having there way with the Citizens of the
United States and their property for personal monetary gain.


there is always a smile when your memories call.
while the last one is hidden behind a broken wall.
long ago misplaced and not a memory at all.
and soon you were just one step from a wife.
a chance you deserved before your life was done.
and still so young with those untouched ways.
and grew tall with them during brighter days.
all your dreams disappeared at twenty-one years.
and always that smile when your memories call.


text:
Q: Is there a way to sync my chats across sessions while using prosody? I am trying to setup my own prosody severer. I am using two clients (Pidgen/WinX and Xabber/Android). I am noticing that messages that I send to a contact appear only in that session. While replies from my contact appear on all sessions.
Is there a way around this? Is this a server issue?

A: I discovered that this feature is dependent on both the server and the client. The server and the client need to implement XEP-0280 protocol ("Message Carbons").

*

*Pidgen does not support this feature.

*The Xabber Beta android app supports it.

*Gajim supports this extension.

*On the server side, we had to include mod_carbons.

A: As you already wrote yourself, there's XEP-0280 (Message Carbons), but you would might want to have Message Archive Management (XEP-0313) as well, which is mod_mam for Prosody.


a productive little tobacco factory on the outskirts of town.
Tobacco was already introduced on Java as early as the 16th century. Smoking of cheroots quickly became very popular among all layers of the population, men and women alike. As tobacco use grew ever more popular in Europe as well, the colonial masters saw the potential for export. Up to date, a lot of Indonesian cigarette factories still churn out millions of cigarettes, mainly for the domestic market. The cigar story is different though. The most successful enterprise of tobacco plantation and processing, starts at the end of the 19th century, when Mr Nienhuys, saw the potential of the fertile volcanic soils in the surroundings of Medan, North Sumatra and persuaded the sultan of Deli to give permission for the cultivation of tobacco. Until this day, the cover leaves from Deli, are the best in the world and various cigar producers worldwide are using the leave to delicately wrap their cigars.
While Yogyakarta is the city famous for the Sultan's Palace and the Borobudur temple nearby, few people know about a productive little tobacco factory on the outskirts of town. Join us on a tour of smiling girls, sorting, drying, molding and wrapping exquisite tobacco into the sought after smokes, in and outside Indonesia.


text:
var map, filter, filterStrategy;
var animationTimer;
var currentDate;
var startDate = new Date(1272736800000); // lower bound of when values
var endDate = new Date(1272737100000); // upper value of when values
var step = 8; // sencods to advance each interval
var interval = 0.125; // seconds between each step in the animation

function startAnimation() {
    if (animationTimer) {
        stopAnimation(true);
    }
    if (!currentDate) {
        currentDate = startDate;
    }
    var spanEl = document.getElementById("span");
    var next = function() {
        var span = parseInt(spanEl.value, 10);
        if (currentDate < endDate) {
            filter.lowerBoundary = currentDate;
            filter.upperBoundary = new Date(currentDate.getTime() + (span * 1000));
            filterStrategy.setFilter(filter);
            currentDate = new Date(currentDate.getTime() + (step * 1000))
        } else {
            stopAnimation(true);
        }
    }
    animationTimer = window.setInterval(next, interval * 1000);
}

function stopAnimation(reset) {
    window.clearInterval(animationTimer);
    animationTimer = null;
    if (reset === true) {
        currentDate = null;
    }
}

// add behavior to elements
document.getElementById("start").onclick = startAnimation;
document.getElementById("stop").onclick = stopAnimation;
var spanEl = document.getElementById("span");

var mercator = new OpenLayers.Projection("EPSG:900913");
var geographic = new OpenLayers.Projection("EPSG:4326");
map = new OpenLayers.Map("map");

var osm = new OpenLayers.Layer.OSM();

filter = new OpenLayers.Filter.Comparison({
    type: OpenLayers.Filter.Comparison.BETWEEN,
    property: "when",
    lowerBoundary: startDate,
    upperBoundary: new Date(startDate.getTime() + (parseInt(spanEl.value, 10) * 1000))
});

filterStrategy = new OpenLayers.Strategy.Filter({filter: filter});

var flights = new OpenLayers.Layer.Vector("Aircraft Locations", {
    projection: geographic,
    strategies: [new OpenLayers.Strategy.Fixed(), filterStrategy],
    protocol: new OpenLayers.Protocol.HTTP({
        url: "kml-track.kml",
        format: new OpenLayers.Format.KML({
            extractTracks: true
            //,extractStyles: true // use style from KML instead of styleMap below
        })
    }),
    styleMap: new OpenLayers.StyleMap({
        "default": new OpenLayers.Style({
            graphicName: "circle",
            pointRadius: 3,
            fillOpacity: 0.25,
            fillColor: "#ffcc66",
            strokeColor: "#ff9933",
            strokeWidth: 1
        })
    }),
    renderers: ["Canvas", "SVG", "VML"]
});

map.addLayers([osm, flights]);
map.setCenter(new OpenLayers.LonLat(-93.2735, 44.8349).transform(geographic, mercator), 8);

redpajama_set_name: RedPajamaGithub


it is you need to do inside of js and still let the dialplan do its thing.
> Using bridge in the dialplan directly works.
> Putting the following instead doesn't work either.
> Using the following instead doesn't work either.
> Still don't understand why.
> comes in at line #548 from the provider side of what I assume is the A leg.
> Here's the log with siptrace.
> differently so that those 2 endpoints can talk to each other?
> "FreeSWITCH Log" for the syntax highlighting.
> I just verified that making an outbound call to my cell phone still works.
> Now back to the case I'm having problem with.
> cell phone can communicate with each other.
> > anything both ways. Any idea?


text:
Q: Maximum Likelihood estimator for $\alpha$. I'm trying to work this question and I need an explaination ( The starting point ).
How do I find the MLE for $\alpha$ and MSE for the MLE for $\alpha$ if we have the following:
$ X_{1}, ..., X_{n} $ are constants and $u_1, ...u_n$ are iid $ N(0, \sigma^2 )$ and $\sigma^2$ is assumed to be known.  Suppose $ Z_1, ..., Z_n$ are random sample that satisfy $$ Z_j = \alpha X_j + u_j $$

A: The density for obtaining $Z_j$, given $\alpha$, is that of $u_j=Z_j-\alpha X_j$, so the likelihood is proportional to
$$
\prod_{j=1}^n\exp\left(-\frac1{2\sigma^2}\left(Z_j-\alpha X_j\right)^2\right)=\exp\left(-\frac1{2\sigma^2}\sum_{j=1}^n\left(Z_j-\alpha X_j\right)^2\right)\;.
$$
Thus the maximum likelihood is attained at the minimum of the quadratic function of $\alpha$ in the exponent. Since you asked for a starting point, I'll leave it at that for now.


Members of Chippewa Moraine Chapter will gather on Monday evening, November 20, at 6:30 PM to conduct the Chapter Annual Membership Meeting, as prescribed by the Ice Age Trail Alliance Bylaws, Article II, Section 17. The meeting will be held at the Chippewa Falls Public Library, 105 W Central St, Chippewa Falls WI.
The Agenda includes IATA directed items and chapter-specific items.
Chapter officers include Chair, Vice Chair(s), Secretary, and Treasurer.
As part of the Ice Age Trail Alliance Bylaw Amendments of 2017, existing Chapter Bylaws are no longer in effect. However, chapters may create "Operational Guidelines" to address matters not currently contained in the Alliance Bylaws. We have drafted an Operational Guidelines document (OGs) that reflects matters in our prior chapter bylaws not in conflict with the Alliance Bylaws, as well as clarifying some additional questions created by the new Bylaw/OG scheme. Pending approval by the IATA Executive Director, the OGs will be adopted at the Chapter Annual Membership Meeting. Copies of the OG draft are available for download and review.


text:
License this image
To license this image or for more information please contact our Licensing team.
We can manage your request more quickly if you include the following information:
Image reference number: XXX
Title and author (if applicable)
Publisher/Producer/Broadcaster
Planned release date
Print run or number of units
Image size and position, example 1/4 page, full screen, front cover, inside
Languages/territorial distribution
Digital rights - format details and license period
By downloading this image you agree that you will not use it commercially, as defined in our full terms and conditions.
You can use the image for:
Non-commercial research and private study
Teaching and instruction
Non-commercial publishing (up to A5-size, and in print runs of up to 4000 copies)
Non-commercial online use, up to 768 pixels, and for up to 5 years
Please indicate that you accept all terms to proceed
I agree to the full terms and conditions
Other uses, including exhibition catalogue and display, broadcast, advertising, book jackets and commercial packaging, are covered by our commercial terms.
If you would like a larger file, or to use the image in other ways, contact our Licensing team.
Alternatively search more than 1.25 million objects from the V&A Collections Search the collections
Not currently on display at the V&A
1828-1830 (made)
Great Britain (made)
By the early 19th century the formal coat was completely cut away in front, offering little protection or warmth. A new style of coat - the frock-coat - was introduced between 1815 and 1820. It had skirts to the knees and buttoned in front. It was influenced by the style of the greatcoat (overcoat), but more fitted in shape. Worn first as informal daywear, the frock-coat soon became the formal day-coat of the 19th century. This example in brown silk and worsted has a shawl collar of velvet in the style of the late 1820s. The weight of the skirts of the new frock-coat made it difficult to cut the fronts in one piece, so a waist seam was introduced in the 1820s.
SCRAN
Europeana Fashion Project
Silk and worsted, faced with silk velvet, lined with silk, metal, hand-sewn
Frock coat made of worsted, silk and silk velvet, Great Britain, 1828-1830
Frock coat made of brown worsted and silk. With a wide curving dark brown silk velvet collar and shaped matching cuffs. The coat is just below knee length. It has a waist seam and slightly flared skirt with a lapped vent at the centre back flanked by two pockets set within a pleat fastening with self covered buttons and headed by a brown tassel. The shoulders are padded and the sleeves wide, gathered at the shoulder and tapering slightly to the wrist. Four pairs of brown tapes with frogs are stitched down the front lapping across to fasten, but the real securing is done by two black metal hooks and eyes. There are vertical slip pockets at the front, fastening with two worked brown silk buttons. The coat is lined with brown silk, the back is interlaced, the fronts are faced back, and with the brown silk of the collar.
Given by Mr John Morant
Textiles and Fashion Collection
T.136-1967
About this object record
Explore the Collections contains over a million catalogue records, and over half a million images. It is a working database that includes information compiled over the life of the museum. Some of our records may contain offensive and discriminatory language, or reflect outdated ideas, practice and analysis. We are committed to addressing these issues, and to review and update our records accordingly.
You can write to us to suggest improvements to the record.
Suggest Feedback
Record created December 15, 1999
Record URL
https://collections.vam.ac.uk/item/O13925/frock-coat-unknown/
Download as: JSONIIIF Manifest
Shuffle objects
The V&A's collections also include the National Art Library and the V&A Archives.
Hear about collections, exhibitions, courses and events from the V&A and ways you can support us.
You can change your preferences or opt out of hearing from us at any time using the unsubscribe link in our emails. Read our full privacy notice.
Daily except Tuesday: 10.00 – 17.00
Cambridge Heath Rd,
Bethnal Green, London E2 9PA
Queen Elizabeth Olympic Park, Stratford, London


However strong the perimeter security, in the vast majority of organisations there are far too many opportunities for hackers or malware attacks to slide in undetected, writes Mark Kedgley, Chief Technology Officer, New Net Technologies.
Forensic-level monitoring of system changes provides a means whereby subtle breach activity can be exposed, but just having the means to detect changes is only part of the solution.
In the same way that seemingly clear pond water is revealed to be teeming with life when placed under a microscope, the amount of noise created on a daily basis by critical upgrades, system patches and required updates once visible is overwhelming. And when it comes to breach detection, it is virtually impossible to distinguish between the expected file and registry changes prompted by these changes and nefarious activity.
Change control provides a procedural means to compartmentalise expected change activity and, in theory at least, isolate unexpected changes including breach activity. But despite pleas from security experts and the advice of best practice guidelines, including ITIL, COBIT and FISMA, to deploy effective change control, the vast majority of organisations simply cannot make it work. For most, the constraint of highly bureaucratic change requests creates delays in urgent system changes that add more risk – it is perceived – than a potential security breach.
Yet the business damage that occurs when hackers can hide in plain sight in this way is widely recognized.
From Target to Home Depot and most recently the Carabank APT, estimated to have stolen $1B from banks around the world, the fall out of a major breach is horrendous. The negative publicity is even worse when monitoring tools were in place, had identified a suspicious system change or two and were sending out alerts to raise the alarm. The fact that alerts simply got lost in the noise is no excuse. Did no one have the time to investigate or, more realistically, did no one actually believe a breach was likely?
This perception of risk remains a real problem. For all the high profile security disasters, the damage to both reputation and revenue, most people in IT and security still have an 'it won't happen to me' attitude. To be fair, this is not just head in the sand – it is a realistic response to the fact that achieving system lock down and any kind of actionable breach visibility are, for most companies, simply unattainable.
The problem? Noise. File Integrity Monitoring (FIM) is a great tool and an essential component of the security toolkit. It provides a complete view of every single change that occurs across the IT infrastructure, but unless it is used hand in hand with rigid, zero-tolerance change control, the amount of noise generated on a daily and weekly, let-alone monthly, basis is unmanageable.
Who has time go through the information deluge that occurs after each set of Microsoft patches is applied to check that nothing untoward has been buried unnoticed within the system changes? The alternative, a centrally imposed, bureaucratic change control process is equally unworkable because it places unrealistic constraints on individuals desperate to respond to new security threats or calls from the business for an urgent fix.
So what is the solution? Organisations simply cannot leave the door open and allow the ill-intentioned exploit of this continual state of change to wander in unchallenged. There has to be a way of cutting down the noise; of distinguishing the mass of justified change from the unexpected – and doing that in a time frame that makes investigation both possible and useful.
But how? In theory, a company could take a manual snapshot of an individual machine pre and post patch and identify the changes. But with hundreds of files added or updates made during patching, plus registry and security policy changes on each and every machine, this is simply not possible across any size of IT estate. Furthermore, not all of these patches will be applied at the same time, or even in a predictable manner due to re-boot requirements – there is no way of keeping track of this process manually.
However, it is now possible to use automated, intelligent FIM to provide a list of all the changes that occur as a result of that patching exercise on one machine and create a 'patch blueprint'. Ideally, patches will be pre-staged on a test or non-production machine prior to roll-out anyway to verify them as having no negative impact on application operation, but any one installation can do the job. With the blueprint in place, each time that same sequence of changes is spotted anywhere else across the IT estate, the activity observed can be confidently attributed to known, named patches, delivering true 'closed-loop' intelligent change control. This not only copes with the huge volume of change that occurs but also manages the time issues associated with the long tail of patching exercises. With each change analyzed automatically in real-time, all that is left for investigation are the truly exceptional events including potential breach activity.
By pre-staging patches as part of the regular patching process and extracting a template of changes, a company can roll out the patches across the entire organization, knowing that all changes will be automatically compared with that blueprint. While a company will generate a huge amount of noise when FIM is first turned on, by tracking the IT infrastructure through a few cycles of weekly patching, systems upgrades and updates, the organization can quickly categorise good behavior – leaving just the unexplained and unexpected for investigation or review. Malware and other breach activity – if present – will be present within these residual changes.
In practice, within a month or so, the information deluge has been reduced to a highly filtered, highly categorized and labeled view of the entire IT infrastructure. The other compelling aspect of this approach is that it removes or at least lessens the need for strict and bureaucratic change management. Because the volume of noise created by patches and upgrades is reduced, necessary changes can be allowed to be made immediately as required and then retrospectively assessed to confirm that the changes made on the system dovetail exactly with those planned. As a result, good change control is still employed – but without the constraints of time and bureaucracy that too often derail best intentions.
Once an organization is in this position of homeostatic equilibrium, it is pretty straightforward to head off breaches and malware infections – control has been attained without imposing unworkable constraints over regular day to day activity.
Enabling every IT team, of any size and maturity, to confidently embrace change control is a massive step forward in IT security. However, this is not just a question of making the process simpler and easier to manage; it is about a change in attitude. Organizations need to accept that an attack is not only possible but that it is likely – and that requires a mind-set shift.
Just consider this real-life situation: a company initially assumed their FIM solution was at fault when alarms were repeatedly raised. On investigation, the technology was doing its job, and in fact, an attack was in progress; despite investing in prevention and detection tools, there was no actual expectation that they would ever be required. Is it any wonder that hackers continue to find new ways in?


Can the Samsung Galaxy S8 Win Back the people?
Dual Camera is the new black but what these two lenses actually do on each phone seems to differ quite a bit. The iPhone 7 Plus and the LG G6 have quite different philosophies when it comes to what gives you the best camera phone experience. The G6 gives you a wide angle and the iPhone 7 Plus a 2x optical zoom.
The iPhone 7 Plus features a dual rear-facing camera, and both sensors are 12 megapixels. One sensor offers a f/1.8 aperture, while the other sports a f/2.8, so the camera should cover all your needs as far as lighting goes. Other features include optical image stabilization and phase detection autofocus.
The G6, at least on paper, seems to offer a better camera. The both of the two lenses on the camera sit at 13MP, with one offering a f/1.8 aperture and the other sitting at f/2.4. On top of that, the standard view camera offers optical image stabilization and laser autofocus.
While both phones use a dual camera system, the uses are different. Apple designed the phone with a focus on zooming in, giving users a 2X optical zoom. LG focused on delivering a wider view, which allows you to get more in your photo. In my day to day usage of both phones, I definitely have used the 2X zoom on the iPhone a lot, but I have to say that I find the wide angle lens a lot more useful. Shooting for Instagram's square format the wide angle lets me have more room to frame my photo better or to just get more in.
Since the G6 is clearly more versatile in capturing landscape we were curious how it did on the zoom feature, so while we were in Neuschwanstein Castle in Germany we thought we'd put it to a test.
Another big difference between the G6 and iPhone 7 Plus is that Apple shares information between the two lenses. This means that each lens can focus on a different part of the scene, this lets you change focus after the fact. Apple calls this "Depth," and gives a much shallower focal range than a regular smartphone, letting you create blurred "bokeh" effects. With a machine learning system, it can also separate a subject from its background for more DSLR-like images. Shooting portraits is great and something very useful in taking photos of your friends or yourself. The difference in photo quality between these two shots with backlighting is massive and clearly, pulls the iPhone into the lead.
However, I still side on the wide angle lens being more useful in my day to day life even if though portraits are so much better on the iPhone.


Grooming Mitts are used to remove loose hair from your dog's top coat.
The patented rubber bristles trap hair and debris when rubbed over your cat's coat.
Ideal to massage in shampoo when washing your dog.