text:
\section{Introduction}
An important class of diagrams in finite-temperature and many-body
calculations is the set of the particle-hole ring diagrams. In gauge theories
the ring diagrams play a crucial role since they incorporate screening
corrections to the (bare) infinite-range interactions and their inclusion
becomes necessary in order to achieve a well behaved perturbative expansion
at all \cite{lebellac}. In theories with short-range interactions the
particle-hole bubbles provide also important corrections to the effective
interaction. An example for this feature is the induced interaction in Landau's
theory for normal Fermi-liquids \cite{baym,babu}. Likewise, the
energies and lifetimes of collective excitations of a system (phonons,
plasmons etc.) are determined from the poles of the particle-hole
polarization propagator.

The particle-hole ring diagrams have also been considered repeatedly in
effective field theory approaches to (three-dimensional) many-fermion systems
at finite density
\cite{hammer,schaefer}. Based on a contact-interaction proportional to the
s-wave scattering length $a$, the particle-hole ring diagrams contribute at third
and higher orders to the systematic low-density expansion of the equation of state
\cite{hammer2,steele}. The respective coefficients occurring in the expansion in
powers of $ak_f$ are however known only with moderate numerical accuracy.
Another feature of the particle-hole ring diagrams with a contact-interaction is
that their spin-factor (a polynomial of degree $n$ in $g$) varies in a somewhat
complicated way with the order $n$ and the spin-degeneracy $g$. For large
spin-degeneracy $g$ the particle-hole ring diagrams are actually distinguished
as the leading ones in a $1/g$-expansion (with $ag=$\,constant) and their
resummation to all orders has been studied in detail in refs.\cite{hammer,schaefer}.
Moreover, the resummed particle-hole ring diagrams could be used to derive
in the Bose limit a non-perturbative result with a non-analytical dependence
on the scattering length $a$ and the particle density \cite{hammer}.

The purpose of the present paper is to study in the same way the particle-hole
ring diagrams for fermions in two spatial dimensions. The analytical evaluation of
fermionic in-medium ladder diagrams in two dimensions together with their resummation
to all orders has been presented recently in (the appendix of) ref.\cite{2dlad}.
The pertinent
complex-valued polarization function $\Pi(q_0 ,|\bf q|)$ for fermions in two
dimensions is rederived in section 2. One finds that both its real and imaginary
part can be expressed in terms of simple square-root functions
of two dimensionless variables. This represents an essential simplification in
comparison to the three-dimensional case where logarithmic functions occur in the
expression for the real part. Given the polarization function the perturbative
contributions to the energy per particle $\bar E(k_f)$ as they arise from particle-hole
ring diagrams with a contact-interaction are calculated in section 3. The dependence
on the Fermi momentum is $k_f^2/M$ at each order and, most remarkably, the pertinent
numerical coefficients can be given as exact numbers (involving rationals and $\ln 2$)
from third up to twelfth order. The resummation of the two-dimensional particle-hole ring
diagrams to all orders is then studied in section 4. One finds a dependence on the
dimensionless coupling parameter $\alpha$, which is much more pronounced than the
dependence on $a k_f$ in the three-dimensional case. Moreover, the complete ring-sum
with all exchange-type diagrams included differs substantially from the resummation
of the leading $n$-ring diagrams only.
In the appendix a derivation of the formula $S_n(g)= (g-1)^n+(-1)^n(g^2-1)$ for
the general spin-factor of $n$-th order particle-hole ring diagrams is presented.

It is hoped that the results of the present paper will find applications in
special condensed matter systems (such as two-dimensional graphene \cite{graphene} or
strongly interacting  two-dimensional Fermi gases \cite{froehlich,randeria}).

\section{Polarization function}
\begin{figure}
\begin{center}
\includegraphics[scale=0.45,clip]{polarbubble.eps}
\end{center}
\vspace{-.5cm}
\caption{Elementary particle-hole polarization bubble with energy $q_0$ and
momentum $\bf q$ flowing through it. The wiggly line symbolizes the contact-interaction.}
\end{figure}

The basic quantity for treating the particle-hole ring diagrams is the complex-valued
polarization function $\Pi(q_0 ,|\bf q|)$, depending on energy $q_0$ and momentum
$|\bf q|$. For non-relativistic fermions in three dimensions its explicit derivation
can be found e.g. in the textbook \cite{fetter}. The expression for the polarizability of
the two-dimensional electron gas has been given long ago by Stern in ref.\cite{stern}.
Nevertheless, it is instructive to rederive the result for the two-dimensional
polarization function by presenting some details of the calculation.
In this work we consider spin-1/2
fermions of mass $M$ in two dimensions and introduce a two-body contact-interaction
with coupling constant $C_0= 2\pi \alpha/M$, where $\alpha$ is a dimensionless
parameter. When including this coupling constant and a factor $-2$ from the sum
over spins and the closed fermion loop in Fig.\,1, the  two-dimensional polarization
function reads:
\begin{eqnarray} \Pi(q_0 ,|\bf q|) &\!\!\!=\!\!\!& -{4\pi i \alpha \over M}\! \int\!
{dl_0 \over 2\pi}\int\!{d^2{\bf l} \over (2\pi)^2} \bigg\{ {i \over l_0-{\bf l}^2
/2M +i \epsilon} -2\pi\, \delta(l_0-{\bf l}^2/2M)\, \theta(k_f-|\bf l|)
\bigg\} \nonumber \\ && \times \bigg\{ {i \over l_0+q_0-({\bf l}+{\bf q})^2/2M
+i \epsilon} -2\pi \,\delta(l_0+q_0-({\bf l}+{\bf q})^2/2M)\, \theta(k_f-
|\bf l+\bf q\,|) \bigg\}\,. \end{eqnarray}
Here, each factor in the curly brackets is a non-relativistic particle-hole propagator
which has been rewritten identically as the sum of a vacuum-propagator and
a medium-insertion. The Fermi momentum $k_f$ appearing in the step-functions
$\theta(\dots)$ is related to the particle density (per area) by $\rho_2 =
k_f^2/2\pi$.  By shifting the integration variable $l_0\to l_0-q_0/2$, one deduces
immediately that $\Pi(q_0 ,|\bf q|)$ is an even function of the energy $q_0$. Therefore,
one can assume  $q_0\geq 0$ in the following.  Next, the momentum-space integrals are
evaluated for all four terms in eq.(1). The term with no step-function gives zero upon
$l_0$-integration with residue calculus,
since both poles lie in the lower $l_0$ half-plane. The two contributions with
one medium-insertion lead to a real part of the form:
\begin{equation} {\rm Re}\, \Pi(q_0 ,|{\bf q}|) = 4\pi \alpha  -\hspace{-0.44cm}\int
{d^2{\bf l} \over (2\pi)^2}\bigg\{ { \theta(k_f-|{\bf l}|)\over {\bf l}\cdot {\bf q}
+ {\bf q}^2/2-Mq_0 } +(q_0\to -q_0)\bigg\}\,, \end{equation}
where we have substituted ${\bf l}\to -{\bf l}-{\bf q}$ in the second contribution.
It is convenient to introduce dimensionless variables $(x, \nu)$ by setting $|{\bf q}|
= 2x k_f$ and $q_0=2\nu x k_f^2/M$, where the extra factor $x$ in the parametrization of
$q_0$ simplifies all following formulas. The remaining principal-value integral is
readily solved by using planar polar coordinates:
\begin{eqnarray} && {\alpha \over 2\pi x}\int_0^1 d\lambda -\!\!\!\!\!\!\int_0^\pi
d\varphi \,
\Big(x-\nu + \sqrt{\lambda} \cos\varphi\Big)^{-1} = {\alpha \over 2 x}\,{\rm sign}
(x-\nu)\,{\rm Re}\int_0^1 d\lambda \,\Big[(x-\nu)^2- \lambda \Big]^{-1/2} \nonumber \\ && =
{\alpha \over x}\,{\rm sign}(x-\nu)\,{\rm Re}\Big\{|x-\nu|-\sqrt{(x-\nu)^2-1}\Big\}\,.
\end{eqnarray}
The second term in eq.(2) is included by adding the same expression with $\nu \to
-\nu$. Scaling out the parameter $\alpha$, one gets finally for the real part of the
two-dimensional polarization bubble $ {\rm Re}\,\Pi(q_0,|{\bf q}|)=\alpha\,R(\nu,x)$
with the function:
 \begin{equation} R(\nu,x) =2 -{1\over x} \,{\rm Re}\Big\{ {\rm sign}(x+\nu)
\sqrt{(x+\nu)^2-1}+  {\rm sign}(x-\nu)\sqrt{(x-\nu)^2-1}\,\Big\}\,. \end{equation}
The imaginary part of the two-dimensional polarization function has the representation:
\begin{eqnarray} {\rm Im}\, \Pi(q_0 ,|{\bf q}|) &\!\!\!\!\!\!\!=\!\!\!\!\!\!\!& \alpha
\int\!d^2{\bf l} \, \delta(Mq_0-{\bf l}\cdot{\bf q})\Big\{\theta(k_f-|{\bf l}-{\bf q}/2|)
+ \theta(k_f-|{\bf l}+{\bf q}/2|) \nonumber \\ && \qquad \qquad  \qquad \qquad  \quad
-2 \,\theta(k_f-| {\bf l}-{\bf q}/2|)\, \theta(k_f-|{\bf l}+{\bf q}/2|)\Big\}\,,
\end{eqnarray}
where the shift $ {\bf l}\to {\bf l}- {\bf q}/2$ has been made.
By analyzing for the first term in eq.(5) the intersection of a straight line (as imposed
by $\delta(Mq_0-{\bf l}\cdot{\bf q})$) and a shifted circular disc in the $\bf l$-plane,
one finds that it contributes to the imaginary part with the term: $\alpha\, x^{-1}
\sqrt{1-(x-\nu)^2}$ for $x-1<\nu<x+1$. Via the substitution $\bf l \to -\bf l$
the contribution of the second term in eq.(5) is obtained by altering the sign of $\nu$
and thus it becomes: $\alpha\, x^{-1} \sqrt{1-(x+\nu)^2}$ for $-x-1<\nu<1-x$. Since
only positive $\nu$ need to be considered the latter condition is equivalent to
$0<\nu<1-x$. In order that the third term in eq.(5) contributes at all, the two
shifted Fermi discs must overlap, which requires $0<x<1$. In this case the third term
subtracts twice the second contribution in the region  $0<\nu<1-x$, and thus
effectively reverses the sign of that second contribution. Putting all three pieces
together and scaling out $\alpha$ again, one obtains for the imaginary part of the
two-dimensional polarization bubble
${\rm Im}\, \Pi(q_0,|{\bf q}|) = \alpha \, I(\nu,x)$ with the function:
\begin{eqnarray}&& I(\nu,x) = {1\over x} \sqrt{1-(x-\nu)^2}\,, \quad  {\rm for}
\quad |x-1|<\nu <x+1\,, \nonumber \\ && I(\nu,x) = {1\over x}\bigg\{ \sqrt{1-
(x-\nu)^2}-\sqrt{1-(x+\nu)^2}\bigg\}\,, \quad  {\rm for} \quad 0<\nu <1-x\,.
\end{eqnarray}
It is interesting to note that the imaginary part of three-dimensional polarization
function is of similar structure \cite{hammer,schaefer,fetter}. The parameter $\alpha$
gets replaced by $a k_f/2$ and the individual terms in eq.(6) come without square-roots.
The real part of the three-dimensional polarization function is composed of logarithms:
$\ln[(1+x+\nu) / |1-x-\nu|]$ and $\ln[|1+x-\nu| / |1-x+\nu|]$. One should also note that
the Lindhard functions of a Fermi gas in one, two, and three dimensions have been studied
extensively and compared with each other in ref.\cite{lindhard}. After translating the
dimensionless variables $(\nu,x)$ the present results for $R(\nu,x)$ and $I(\nu,x)$
written in eqs.(4,6) agree exactly with the earlier calculations in
refs.\cite{stern,lindhard}.

Fig.\,2 shows the support of the function $I(\nu,x)$ in the (positive) $x\nu$-plane.
The corresponding region lies in between the two straight lines $\nu = x-1$ and
$\nu = x+1$. When crossing the (dashed) line $\nu = 1-x$, the function $I(\nu,x)$
changes its form and it vanishes along the circumference (with the exception of the
line-segment $x=0,\, 0<\nu<1$). We anticipate that
the real-valued integrand relevant for calculating the interaction energy density from
particle-hole
ring diagrams involves a proportionality factor $I(\nu,x)$. Exploiting this property,
the function $R(\nu,x)$ can be restricted to the support of $I(\nu,x)$. Inspection
of eq.(4) shows that the last term can be dropped, since its radicand
$[\nu-(x+1)][\nu-(x-1)] $ is negative. Furthermore, in the region $0<\nu <1-x$
the other radicand $[\nu+(x+1)][\nu-(1-x)]$ becomes also negative. Altogether, the
function $R(\nu,x)$ simplifies considerably to:
\begin{eqnarray}&& R(\nu,x) = 2-{1\over x} \sqrt{(x+\nu)^2-1}\,, \quad  {\rm for}
\quad |x-1|<\nu <x+1\,, \nonumber \\ && R(\nu,x) = 2 \,, \quad  {\rm for} \quad
0<\nu <1-x\,,\quad 0<x<1\,, \end{eqnarray}
inside the relevant integration region.
\begin{figure}
\begin{center}
\includegraphics[scale=0.8,clip]{norbert.ps}
\end{center}
\vspace{-.5cm}
\caption{Integration region in the $x\nu$ quarter-plane in between the two straight
lines $\nu=x-1$ and $\nu = x+1$. The functions $R(\nu,x)$ and $I(\nu,x)$
have different forms below and above the dashed line $\nu=1-x$.}
\end{figure}

\section{Energy per particle and perturbative results}
In this section we calculate the energy per particle  $\bar E(k_f)$ as it arises from
ring diagrams with a contact-interaction. For orientation we note that the free Fermi
gas energy in two dimensions is $\bar E(k_f)^{(0)}= k_f^2/4M$. Likewise, the first-order
Hartree-Fock contribution from a contact-coupling $C_0=2\pi \alpha/M$ is given by $\bar
E(k_f)^{(1)}= -\alpha\,k_f^2/4M$. Obviously, the energy density arising from the leading
$n$-ring diagram (obtained by connecting $n$ polarization bubbles in Fig.\,1 to a closed
ring) is determined by the $n$-th power of the polarization function
$\Pi(q_0,|{\bf q}|)$. With inclusion of the symmetry factor $1/2n$ the corresponding
energy density reads:
\begin{equation} {k_f^2 \over 2\pi} \, \bar E(k_f)^{n-{\rm ring}}_{\rm lead}  = {i \over 2n}
\int\!{dq_0 \over 2\pi}\int\!{d^2{\bf q} \over (2\pi)^2} \,\Big[ \Pi(q_0,|{\bf q}|)\Big]^n
\,. \end{equation}
After transforming to the dimensionless variables $(x, \nu)$ one obtains the following
double-integral representation for the contribution of $n$-ring diagrams to the energy
per particle:
\begin{equation}\bar E(k_f)^{n-{\rm ring}} = {4k_f^2 \over \pi M} {\alpha^n \over n}{S_n
\over 2^n} \int_0^\infty \!\!dx\, x^2 \int_{\nu_{\rm min}}^{x+1}\!\! d\nu\, {\rm Im}
\Big[R(\nu,x)-i I(\nu,x)\Big]^n  \,, \end{equation}
with $\nu_{\rm min} = {\rm max}(0,x-1)$. In addition to the leading $n$-ring diagram
with spin-factor $2^n$, there are various exchange-type ring-diagrams due to the
Pauli exclusion-principle. For a contact-interaction these can all be summed together
by means of the total spin-factor $S_n = 1+3(-1)^n$. The corresponding sequence is
$S_n=(-2,4,-2,4,\dots)$ \cite{schaefer}. In the appendix an explicit derivation of
the formula $S_n = 1+3(-1)^n$, using multiple spin-traces, is given. The generalization
to arbitrary spin-degeneracy $g\geq 2$ is also presented there.

Based on the representation in eq.(9) the respective double-integrals have been studied
in detail by analytical and numerical methods. From third up to twelfth order analytical
results in closed form could be obtained for the contributions to the energy per particle
$\bar E(k_f)$. We list them one after the other:

\noindent
$n=3$: the integrand $I(I^2-3R^2)$ leads to the 3-ring contribution:
\begin{equation}\bar E(k_f)^{3-{\rm ring}}={k_f^2\over 4 M}\alpha^3(2\ln2-1)\,,\end{equation}
$n=4$: the integrand $4 IR(I^2-R^2)$ leads to the 4-ring contribution:
\begin{equation}\bar E(k_f)^{4-{\rm ring}}={k_f^2\over M}\alpha^4(2-3\ln2)\,,\end{equation}
$n=5$: the integrand $I(10I^2R^2-5R^4-I^4)$ leads to the 5-ring contribution:
\begin{equation}\bar E(k_f)^{5-{\rm ring}}={k_f^2\over M}\alpha^5\bigg(4\ln2-{11\over 4}
\bigg)\,,\end{equation}
$n=6$: the integrand $2IR(3R^2-I^2)(3I^2-R^2)$ leads to the 6-ring contribution:
\begin{equation}\bar E(k_f)^{6-{\rm ring}}={k_f^2\over M}\alpha^6\bigg({83\over 6}-20\ln2
\bigg)\,,\end{equation}
$n=7$: the integrand $I^7+7I R^2(5I^2R^2-3I^4-R^4)$ leads to the 7-ring contribution:
\begin{equation}\bar E(k_f)^{7-{\rm ring}}={k_f^2\over M}\alpha^7\bigg(24\ln2-{133\over 8}
\bigg)\,,\end{equation}
$n=8$: the integrand $8IR(I^2-R^2)(I^4-6 I^2R^2+R^4)$ leads to the 8-ring contribution:
\begin{equation}\bar E(k_f)^{8-{\rm ring}}={k_f^2\over M}\alpha^8\bigg({4657\over 60}-112\ln2
\bigg)\,,\end{equation}
$n=9$: the integrand $I(I^2-3R^2)(33I^4R^2-I^6-27I^2R^4+3R^6)$ leads to the 9-ring
contribution:
\begin{equation}\bar E(k_f)^{9-{\rm ring}}={k_f^2\over M}\alpha^9\bigg(128\ln2-{5323\over 60}
\bigg)\,,\end{equation}
$n=10$: the integrand $2IR(10I^2R^2-5I^4-R^4)(I^4-10I^2R^2+5R^4)$ leads to the
10-ring contribution:
\begin{equation}\bar E(k_f)^{10-{\rm ring}}={k_f^2\over M}\alpha^{10}\bigg(
{27947\over 70}-576\ln2 \bigg)\,,\end{equation}
$n=11$: the integrand $I^{11}+11I R^2(30I^6R^2-5I^8-42I^4R^4+15I^2R^6-R^8)$ leads to
the 11-ring contribution:
\begin{equation}\bar E(k_f)^{11-{\rm ring}}={k_f^2\over M}\alpha^{11}\bigg(640\ln2
-{149053\over 336}\bigg)\,,\end{equation}
$n=12$: the integrand $4I R(I^2-R^2)(I^2-3R^2)(3I^2-R^2)(I^4-14I^2R^2+R^4)$ leads to
the 12-ring contribution:
\begin{equation}\bar E(k_f)^{12-{\rm ring}}={k_f^2\over M}\alpha^{12}\bigg({4918777\over 2520}
-2816\ln2\bigg)\,.\end{equation}
Following the pattern of numerical coefficients in eqs.(10-19) one conjectures for the
$n$-ring contribution:
\begin{equation}\bar E(k_f)^{n-{\rm ring}}={k_f^2\over M}(-\alpha)^n\Big\{\zeta_n -(n-1)2^{n-6}
[3+(-1)^n]\ln2\Big\}\,,\end{equation}
with $\zeta_n $ a positive rational number. In the course of the calculation the
following observations have been made. The contribution to the double-integral in eq.(9)
from the (triangular) region $0<\nu<1-x$ (see Fig.\,2) can be solved analytically,
with a result equal to $\pi$ times a rational linear combination of $1$ and $\ln 2$.
The integral over the (rectangular) region $|x-1|<\nu<x+1$ is numerically extremely close
to a rational multiple of $\pi$. The corresponding values in the case $n=3$ are $\pi(7-12
\ln 2)/8$ and $-\pi/8$. In the cases $n = 4, 5, 6, 7$ the numerical integral over the
rectangular region $|x-1|<\nu<x+1$ vanishes with an accuracy $10^{-10}$ or better,\footnote{
For the analytical and numerical treatment of the integrals is it occasionally useful to
introduce new coordinates by $x = (\xi+\eta)/2$,  $\nu = (\xi-\eta)/2$. The integration
region in the $\xi\eta$-plane consists then of a triangle $-\xi<\eta<\xi$, $0<\xi<1$ and a
semi-infinite strip $-1<\eta<1$, $\xi>1$.} while for $n=8$ it has the value $-256\pi/15$.
For $n\geq 9$ the integrals over both regions develop divergences which cancel
in the sum. It is then advantageous to map the entire integration region onto a unit square
$0<x,y<1$ via substitutions: $\nu = (1-x)y$, $\nu=1-x+2x y$, and $\nu=x-1+2 y,\,
x \to1/x$. In order to illustrate the available precision we note that the large
integers $4657,\,  5323,\, 27947,\,149053$ and $4918777$ in eqs.(15-19) are
reproduced with a relative accuracy of $10^{-13}$ and better. The list of numerical coefficients
$c_n$ for the $n$-ring contribution to the energy per particle
 $\bar E(k_f)^{n-{\rm ring}} = c_n \alpha^n k_f^2/M$ is continued up to $n=30$
in Table\,1. One observes that the sequences of positive and
negative coefficients $c_n$ decrease both slowly in magnitude.

\begin{table}
\begin{center}
\begin{tabular}{|c|c c c c c c c|}\hline
$n$ & 3 & 4 & 5 & 6 & 7 & 8 &9 \\ \hline
$100\, c_n$ &  $9.657359$  & $-7.944154$ & $2.258872$ &
$-2.961028$ & $1.053233$ & $-1.581756$ & $0.617245$\\ \hline
$n$ & 10  & 11& 12 &13 & 14 & 15& 16\\ \hline
$100 \, c_n$ & $-0.991886$ & $0.407651$ & $-0.682554$ &
$0.290058$ & $-0.499344$ & $0.217243$ & $-0.381589$         \\ \hline
$n$ & 17  & 18& 19 &20 & 21 & 22& 23\\ \hline
$100 \, c_n$ & $0.168933$ & $-0.301301$ & $0.135201 $ &
$-0.244050 $ & $0.110695$ & $-0.201763 $ & $0.092323$         \\ \hline
$n$ & 24  & 25& 26 &27 & 28 & 29& 30\\ \hline
$100 \, c_n$ & $-0.169628$ & $0.078189$ & $-0.144627 $ &
$0.067079 $ & $-0.124789$ & $0.058186 $ & $-0.108781$         \\ \hline
\end{tabular}
\caption{Numerical coefficients $c_n$ of the energy per particle
$\bar E(k_f)^{n-{\rm ring}} = c_n \alpha^n k_f^2/M$ arising from the $n$-th
order particle-hole ring diagrams with a contact-interaction $2\pi \alpha/M$.}
\end{center}
\end{table}

Let us also give an alternative representation for the ring energy
$\bar E(k_f)^{n-{\rm ring}}$ at $n$-th order. When following the method described
in the text book by Gross, Runge and Heinonen \cite{grossrunge} and evaluating the
key quantity defined in eq.(22.15) in two dimensions, one obtains:
\begin{equation} \bar E(k_f)^{n-{\rm ring}}= -{4k_f^2 \over \pi M}
{S_n\alpha^n \over n} \int_0^\infty\!\! dx\, x^2 \int_0^\infty\!\!dy\,
[K(x,y)\big]^n \,, \end{equation}
with the (two-dimensional) euclidean polarization function:
\begin{equation} K(x,y) = 1-{1\over \sqrt{2} x} \sqrt{x^2-y^2-1+
\sqrt{(1+x^2+y^2)^2-4x^2}} = 1 - {1\over x }\Big|{\rm Re}
\sqrt{(x+i y)^2-1} \Big|\,.\end{equation}
Note that the absolute magnitude in the short form removes the sign-ambiguity
of the complex square-root. The function $K(x,y)$ takes on values in the range $0<K(x,y)<1$
and its asymptotic behavior in the $xy$ quarter plane is:
\begin{equation} K(r \cos\varphi, r \sin \varphi) = {1\over 2r^2}
+{1-4 \sin^2 \varphi \over 8 r^4} +{\cal O}(r^{-6})\,.  \end{equation}
Using the representation in eq.(21) all previous results
collected in eqs.(10-19) could be reproduced with very high numerical accuracy.
\section{Resummation to all orders}
After the perturbative analysis of the two-dimensional ring diagrams with a
contact-interaction we study now their resummation to all orders. Returning to eqs.(8,9)
one sees that the resummation to all orders is provided by an arctangent-function
with two subtractions. The subtraction terms (linear and quadratic in $\alpha$) introduce
divergences (quadratic and logarithmic in the upper limit of the $x$-integration) which
get canceled by the integral over the arctangent-function. In order to avoid such
numerically delicate cancellations one better represents the derivative of $\bar E(k_f)$
with respect to $\alpha$ in terms of a convergent integral over a rational expression and
integrates back in $\alpha$:
\begin{eqnarray} \bar E(k_f)_{\rm lead} &\!\!\!\!\!\!\!=\!\!\!\!\!\!\!& {4k_f^2 \over \pi M}
\int_0^\alpha\!\! d z\, z^2\int_0^\infty \!\!dx\, x^2 \int_{\nu_{\rm min}}^{x+1}\!\!
d\nu\, I(\nu,x)\nonumber \\ && \times { I(\nu,x)^2-3 R(\nu,x)^2 + 2z R(\nu,x)
[ R(\nu,x)^2+I(\nu,x)^2] \over [1-z  R(\nu,x)]^2 +z^2   I(\nu,x)^2} =
{k_f^2  \alpha^3\over 4M}\,  G(\alpha) \,.  \end{eqnarray}
Here, we are considering first the sum of the leading $n$-ring diagrams with
spin-factors $2^n$. The dimensionless function $G(\alpha)$ is defined by scaling out
the free Fermi gas energy $k_f^2/4M$ and the leading power $\alpha^3$. Its non-vanishing
value at $\alpha=0$ is: $ G(0) = 4-8\ln 2=-1.54518$.

\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.45,clip]{2drings.eps}
\end{center}
\vspace{-.5cm}
\caption{The function $G(\alpha)$ resulting from the resummed particle-hole
ring diagrams.}
\end{figure}

Fig.\,3 shows the behavior of the function $G(\alpha)$ in the range
$-10< \alpha<10$ of the coupling parameter $\alpha$. One observes a sharp spike in
the vicinity of $\alpha=1/2$. The function $G(\alpha)$ crosses zero twice, at
$\alpha \simeq -6.33$ and  $\alpha \simeq 1.11$, and in the region $|\alpha|>2$ its
variation becomes rather weak. A detailed numerical study at large positive and
negative parameter values reveals that the asymptotic behavior of $G(\alpha)$ is given by:
\begin{equation}  G(\alpha) \simeq {2 \over 3} +{2\ln|4\alpha|-5/2\over \alpha}
\,. \end{equation}
This functional form has been found through a fit to the numerical data at large
$|\alpha|$. The behavior of the function $G(\alpha)$ in the region $0<\alpha<1$ around
the spike is shown with
magnification in Fig.\,4. One makes the interesting numerical observation that $G(1/4)
= -2$ and $G(1/2) = -4$ with an accuracy of $10^{-9}$. Furthermore, one sees that
$G(\alpha)$ decreases further and reaches its minimum value of about $-4.40$ at the
parameter value $\alpha \simeq 0.523$. One also recognizes from the magnified
representation in Fig.\,4 that the behavior of $G(\alpha)$ around the ``spike''
is actually completely smooth. All these observed properties together with the results in
eqs.(10-19) lead to the suggestion that the function $G(\alpha)$ could possibly be
expressed through some decent analytical formula (involving presumably elliptic or
hypergeometric functions). We have however not succeeded in finding such an analytical
formula.

\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.45,clip]{2dringfine.eps}
\end{center}
\vspace{-.5cm}
\caption{The function $G(\alpha)$ magnified in the region $0<\alpha<1$. }
\end{figure}

Next, we consider the complete ring-sum with inclusion of all exchange-type diagrams.
Based on the spin-factor $S_n= 1+3(-1)^n$ at $n$-th order (see eq.(9)) one obtains the
following representation of the resummed energy per particle:
\begin{eqnarray} \bar E(k_f) &\!\!\!\!\!\!\!=\!\!\!\!\!\!\!& {4k_f^2 \over \pi M}
\int_0^\infty \!\!dx\, x^2 \int_{\nu_{\rm min}}^{x+1}\!\!
d\nu\, \bigg\{\alpha I(\nu,x)\big[\alpha R(\nu,x)-1\big] \nonumber \\ && + \arctan
{\alpha I(\nu,x)\over \alpha R(\nu,x)-2} +3 \arctan{\alpha I(\nu,x) \over \alpha
R(\nu,x)+2} \bigg\} = {k_f^2 \alpha^3 \over 4M}\,  F(\alpha) \,.  \end{eqnarray}
The function $F(\alpha)$ defined by scaling out the free Fermi gas energy $k_f^2/4M$
and the leading power $\alpha^3$ is not independent. The relation to the function
$G(\alpha)$ introduced before in eq.(24) reads:
\begin{equation} F(\alpha) = {1\over 8}\Big\{G(\alpha/2)-3\,G(-\alpha/2)\Big\}\,.
\end{equation}
Fig.\,5  shows the behavior of the function $F(\alpha)$ in the range $-10<
\alpha<10$ of the coupling parameter $\alpha$. One observes now two peaks located
at $\alpha\simeq \pm 1 $, which simply arise from the minima of $G(\pm \alpha/2)$
(see Figs.\,3,4). The value at $\alpha=0$ is $F(0)=2\ln2- 1=0.386294$ and
asymptotically $F(\pm \infty)=-1/6$ is reached. In order to reveal the strong parameter
dependence in the region $-4< \alpha<4$ it is instrumental to scale out the
trivial factor $\alpha^3$ in eq.(26). When scanning through from large
negative to large positive values of coupling constant $\alpha$, the energy per
particle $\bar E(k_f)$ changes its sign four times. This is a rather striking parameter
dependence. Altogether, the main finding of the present work is that  complete
ring-sum with all exchange-type diagrams included leads to a quantitatively and
qualitatively different result for the energy per particle $\bar E(k_f)$ in comparison
to the standard resummation of the leading $n$-ring diagrams.

\begin{figure}[h!]
\begin{center}
\includegraphics[scale=0.45,clip]{falpha.eps}
\end{center}
\vspace{-.5cm}
\caption{The function $F(\alpha)$ resulting from the resummed particle-hole
ring diagrams.}
\end{figure}

\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.45,clip]{3dimrings.eps}
\end{center}
\vspace{-.5cm}
\caption{The functions $G_3(a k_f)$ and $F_3(a k_f)$ resulting from the resummed
particle-hole ring diagrams with a contact-interaction $4\pi a/M$ in three dimensions.}
\end{figure}

It is also interesting to compare with the analogous results for the resummed ring
diagrams in three dimensions \cite{hammer,schaefer}. Using the product $a k_f$ of
scattering length $a$ and Fermi momentum $k_f$ as a dimensionless coupling
strength\footnote{The sign-convention is chosen such that $a>0$ corresponds to
attraction.} and appropriate polarization functions $I_3(\nu,x)$ and $R_3(\nu,x)$,
the only difference to eqs.(24,26) is an additional factor $3x$. The functions
$G_3(ak_f)$ and $F_3(ak_f)=[G_3(ak_f/2)-3G_3(-ak_f/2)]/8$ are defined
in the same way by dividing the resummed energy per particle $\bar E(k_f)$ by
the three-dimensional free Fermi gas energy $3k_f^2/10M$ and the leading power
$(ak_f)^3$. These two functions are shown by the dashed and full line in Fig.\,6 for
$-10<ak_f<10$. One observes a behavior similar to the one in
Figs.\,3,5, however with a less pronounced dependence on the dimensionless
coupling strength $ak_f$. The
function $G_3(ak_f)$ reaches its minimum of $-0.732$ at $a k_f \simeq 1.0$ and it
crosses zero at  $a k_f \simeq 3.75$. The
asymptotic behavior of $G_3(ak_f)$ is not obvious from Fig.\,6 and also very difficult
to extract by numerical methods \cite{hammer}. One can conclude that the two-dimensional
particle-hole ring diagrams with their multitude of analytical results represent a rather
special case.
\section{Summary and conclusions}
In this work we have studied the particle-hole ring-diagrams for a many-fermion in two
dimensions. Based on a contact-interaction the perturbative contributions to the energy
per particle $\bar E(k_f)$ could be given in closed analytical from third up to twelfth
order. The resummation to all orders exhibits a pronounced dependence on the
dimensional coupling parameter $\alpha$. A particular finding of the present work has been
that the complete ring-sum with all exchange-type diagrams included (corresponding to
spin-factors $1+3(-1)^n$) leads to a significantly different result for the energy per
particle than the standard resummation of the leading $n$-ring diagrams
(with spin-factors $2^n$).

\section*{Appendix: Calculation of general spin-factor}
In this appendix we present a derivation of the general spin-factor associated to
the particle-hole ring diagrams at $n$-th order. We start with the simplest case of
spin-1/2 fermions, i.e. spin-degeneracy $g=2$. While the leading $n$-ring diagram has
the obvious spin-factor $g^n = 2^n$, the problem consists in determining this factor
for all the exchange-type diagrams. For a four-fermion contact-interaction the
exchange-type diagrams required by the Pauli exclusion-principle are automatically
taken into account if the antisymmetrized form of the contact-interaction is used:
\begin{equation} {\bf 1} - {1\over 2}( {\bf 1}+\vec \sigma_1\cdot \vec \sigma_2) =
{1\over 2}( {\bf 1}-\vec \sigma_1\cdot \vec \sigma_2)\,, \end{equation}
as obtained by subtracting the spin-exchange operator. The resulting operator
$( {\bf 1}-\vec \sigma_1\cdot \vec \sigma_2)/2$ acts in the product spin-space of particle $1$
and particle $2$. The total spin-factor $S_n$ of the $n$-ring diagrams is now given
by the trace of the product of $n$ such operators multiplied in cyclic order of their
particle indices. The starting expression reads:
\begin{eqnarray} S_n &\!\!\!=\!\!\!& {1\over 2^n} {\rm tr}_1 {\rm tr}_2
\dots  {\rm tr}_n\{( {\bf 1}-\vec \sigma_n\cdot \vec \sigma_1)( {\bf 1}-\vec \sigma_1\cdot
\vec \sigma_2)\dots ( {\bf 1}-\vec \sigma_{n-1}\cdot \vec \sigma_n)\}\nonumber \\ & \!\!\!=
\!\!\!&  {2\over 2^n} {\rm tr}_2\dots  {\rm tr}_n\{( {\bf 1}+\vec \sigma_n\cdot
\vec \sigma_2)\dots ( {\bf 1}-\vec \sigma_{n-1}\cdot \vec \sigma_n)\}\,, \end{eqnarray}
and we have performed in the second line the trace in the spin-space of particle $1$.
Note the sign change in front of the spin-spin operator $\vec \sigma_n\cdot
\vec \sigma_2$. After taking $n-1$ spin-traces one arrives at:
\begin{equation}S_n=
{2^{n-1}\over 2^n} {\rm tr}_n\{ {\bf 1}+(-1)^n\vec \sigma_n\cdot \vec \sigma_n\} =
1+3 (-1)^n\,, \end{equation}
where the factor $3$ stems from $\vec \sigma^{\,2}= {\bf 3} $. The same sequence of
numbers $S_n=(-2,4,-2,4,\dots)$ has been reported in section IV of ref.\cite{schaefer}
without a derivation.

Next, we consider fermions with $g$ internal degrees of freedom. These ``flavor'' degrees
of freedom are assumed to form the $g$-dimensional representation of the
special unitary group $SU(g)$. The antisymmetrized contact-interaction which incorporates
the Pauli exclusion-principle is now obtained by subtracting from the unit-operator
${\bf 1}$ in two-particle flavor-space its Fierz transform:
\begin{equation} {\bf 1} - \bigg({{\bf 1}\over g}+{1\over 2}\vec \lambda_1\cdot
\vec \lambda_2 \bigg)\,. \end{equation}
Here, $\vec \lambda$ denotes $g^2-1$ traceless generators that are normalized as
tr$(\lambda^a\lambda^b) = 2 \delta^{ab}$, with indices $a,b \in\{1,\dots, g^2-1\}$. The
operator ${\bf 1}/g+\vec \lambda_1\cdot \vec \lambda_2/2$ has indeed the properties of a
flavor-exchange operator. Its eigenvalue in the $g(g+1)/2$-dimensional representation of
symmetric tensors is $1/g+ (g+2)(g-1)/g- (g^2-1)/g =1$, and its eigenvalue  in the
$g(g-1)/2$-dimensional representation of antisymmetric tensors is $1/g+ (g-2)(g+1)/g-
(g^2-1)/g =-1$. The starting expression for the spin-factor $S_n(g)$ reads:
\begin{eqnarray} S_n(g) &\!\!\!=\!\!\!& {\rm tr}_1 {\rm tr}_2\dots  {\rm tr}_n\bigg\{
\bigg({\bf 1}-{{\bf 1}\over g} -{1\over 2}\vec \lambda_n\cdot \vec \lambda_1\bigg)
\bigg({\bf 1}-{{\bf 1}\over g}- {1\over 2}\vec \lambda_1\cdot \vec \lambda_2\bigg)\dots
\bigg({\bf 1}-{{\bf 1}\over g}- {1\over 2}\vec \lambda_{n-1}\cdot \vec \lambda_n\bigg)
\bigg\} \nonumber \\ &\!\!\!=\!\!\!& {\rm tr}_2\dots  {\rm tr}_n
\bigg\{\bigg[g\bigg({\bf 1}-{{\bf 1}\over g}\bigg)^2 + {1\over 2}\vec \lambda_n\cdot
\vec \lambda_2\bigg]\dots \bigg({\bf 1}-{{\bf 1}\over g}- {1\over 2}\vec \lambda_{n-1}
\cdot \vec \lambda_n\bigg)\bigg\} \,, \end{eqnarray}
and we have shown the intermediate result of a single flavor-trace, where tr$\,{\bf 1}=g$.
After taking $n-1$ flavor-traces one arrives at:
\begin{equation}S_n(g) = {\rm tr}_n\bigg\{g^{n-1}\bigg({\bf 1}-{{\bf 1}\over g}\bigg)^n
+(-1)^n {1\over 2}\vec \lambda_n\cdot \vec \lambda_n \bigg\}\,, \end{equation}
which leads with $\vec \lambda\cdot\vec \lambda=2(g-1/g){\bf 1}$ to the final result:
\begin{equation}S_n(g) =(g-1)^n +(-1)^n (g^2-1) \,. \end{equation}
One realizes that  $g(g-1)$ is a common divisor of the polynomials $S_n(g)$ of
degree $n$ in $g$. The
spin-factors of the 3-, 4- and 5-ring diagrams are:
\begin{eqnarray}&& S_3(g) =  g(g-1)(g-3)\,, \nonumber \\ && S_4(g) =  g(g-1)(g^2-3g+4)\,,
\nonumber \\ && S_5(g) =  g(g-1)(g^3-4g^2+6g-5) \,. \end{eqnarray}
Although our derivation of the formula for $S_n(g)$ has been elementary, it seems
to represent a novel result. We note as an aside that the same method applied to
the symmetrized contact-interaction gives for bosonic $n$-ring diagrams the flavor-factor
$B_n(g)= (g+1)^2+g^2-1 = (-1)^n S_n(-g)$.

Let us finally generalize the expressions
for the resummed interaction energy per particle in eqs.(24,26) to arbitrary
spin-degeneracy $g$. The sum of the leading $n$-ring diagrams with spin-factors
$g^n$ takes the form:
\begin{equation}\bar E(k_f)_{\rm lead} = {k_f^2  \alpha^3 \over 4M }  \bigg\{
{g^2 \over 4}\, G\Big({\alpha g \over 2}\Big) \bigg\}\,, \end{equation}
and the complete ring-sum with inclusion of all exchange-type diagrams is
given by:
\begin{equation}\bar E(k_f) = {k_f^2\alpha^3\over 4M} \bigg\{ {(g-1)^3 \over 4g}
\, G\Big({\alpha \over 2}(g-1)\Big)- {g^2 -1\over 4g}\, G\Big(-{\alpha \over 2}\Big)
\bigg\} \,, \end{equation}
with the function $G(\alpha)$ defined in eq.(24) and displayed with its
$\alpha$-dependence in Figs.\,3,4.

\section*{Acknowledgement}
I thank H.W. Hammer, T. Sch\"afer and R. Schmidt for informative discussions.
This work has been supported in part by DFG and NSFC (CRC 110).

redpajama_set_name: RedPajamaArXiv


'Too many heavyweight speeches after a good dinner', was Samuel Brittan's verdict on last year's Lord Mayor's banquet. Amid the ornate splendour of the City of London's medieval Guildhall sat the annual assembly of overfed industrialists, satiated financiers and boozy bureaucrats. You can imagine them dozing off as the monotonous heavyweights of the Bank of England and the Treasury droned on. Such a spectacle speaks volumes about the current state of economic thinking. 'Too many speeches', concluded Sir Samuel, 'and not enough jokes' (Financial Times, 4 November 1991).
Samuel Brittan is supposed to be the epitome of level-headed common sense. As the leading Financial Times columnist he inhabits a world of speculation about the shifts in stock markets, exchange rates, trade imbalances and budget deficits. His conclusions often turn up later in the pages of Hansard or the speeches of UK chancellors. He is a one-man think-tank, a shining star in the dull firmament of capitalist economics. So it is worth looking at his view of the slump and the prospects for recovery.
The knack of sound economic management, then, is to ensure an even rate of monetary expansion in line with the potential growth of the economy. The 'nominal GDP' money rule was first formulated by Brittan in his 1981 pamphlet, How to End the Monetarist Controversy, and has since found its way into Bank of England orthodoxy.
So inflation results once potential output has been reached, while potential output is defined as a situation beyond which inflation is generated. Bravo! Inflation results from an inflationary situation. What is an inflationary situation? It is a situation which generates inflation. There you have it: pure genius!
Sam Brittan's 'golden rule of boom and slump' rests simply on his own impressions about capitalism's growth 'pot-ential'. And these are gloomy indeed. By 'potential output', Brittan does not mean the ultimate possible devel-opment of the productive abilities of society ('in some engineering sense'). He is not even talking about full employment under capitalism. For Brittan, 'potential out-put' merely includes the scanty sums of idle capacity or unutilised capital reported in annual CBI surveys.
Even so, the notion that this meagre 'potential capacity' can be mobilised by a growth of the money supply is a complete fiction. It is also a shabby lie. The logic of the capitalist system is not to work towards some elusive level of 'potential output'. It works at a far lower level of profitable output, regardless of whether this leaves masses of spare capacity unused and idle.
Brittan holds out the promise that this 'excess capacity' could be drawn into action by a future injection of money into the system. But for the present he reconciles his readers to the degradation, impoverishment and unemployment of the slump. Recessions are inexorable cycles of 'creative destruction'. Sir Samuel's illusory economics promise greater things for tomorrow, while celebrating the 'creativity' of the slump today.
'It is in fact extremely rare for market economies to spiral downwards into a tailspin' (20 June); 'What goes down usually comes up' (4 July); 'The vast majority of recessions do not turn into great depressions' (4 July); 'My main reason for believing recession will be succeeded by recovery is that this is what has nearly always happened in the past' (1 August); 'Belief that every recession is a prelude to great depression clouds judgement' (19 December); 'It is often darkest before the dawn' (16 January).
From blasé summer confidence, Sir Samuel descended into downbeat autumn gloom. By late November he finally announced 'the postpon-ement of the recovery in the UK'. But even then he held out hopes for recovery based on extrapolating grandiose conclusions from a few months' figures for output and exports and surveys of industrialists' 'expectations' - the same facile 'teenagers' method that he has criticised for so many years.
By December, Brittan had lost hope in the recovery. No longer was the recession a quick U-shaped downturn followed by a sharp revival. It was not even to be a W-shaped 'double-dip' slump with a subsequent resurgence. More likely was a drawn out L-shaped slump, with the right hand part of the L drooping downwards.
So surely now the time had come to unveil the spectacular Sam Brittan recovery plan? Now you would expect him to resort to the stimulus of the (fraudulent) monetary 'transmission mechanism'. But no. It never existed. So what did Brittan come up with in the doldrums of February 1992?
Instead Brittan spent February preaching restraint, austerity and religious faith in 'self-correcting economic forces'. Now disillusioned with the efficacy of UK monetary machinery, Brittan the 'good European' could only promise (pray?) that sometime in the future 'the heavy cavalry of the Bundesbank will come up from behind' to the rescue.
Aside from this, Brittan's one idea to tap into his illusory 'potential capacity' was a Yuletide strategy for retail-led recovery based on shops introducing 'gift-wrapping to stimulate trade', abandoning the 'horror' of 'loud pop music' and leading the way with new lines in beakers ('Nightmare on Oxford Street', Financial Times, 30 December 1991). Yes, beakers. In all seriousness: 'Coloured plastic beakers which one can use after cleaning one's teeth or taking a drink of water.' No doubt it's just what 'one' needs after a heavyweight dinner at the Guildhall.


text:
Five tales of horror are dished out in the unforgettable ' The Mortuary Collection'
In my mind, The Mortuary Collection is a spiritual sequel to the masterful 2007 film Trick 'r Treat. They're not at all related or share any kind of connection through creators but I'm choosing to see them as beautifully respectful step-siblings. A unique, fresh, and twisted horror anthology that knows how to tell individual stories without sacrificing the role of the main narrative. Since Trick 'r Treat, horror anthologies have been drifting in and out of the spotlight with minimal success but The Mortuary Collection beats the snot out of them all. Speaking as someone who had, admittedly, low expectations, this film was a remarkable surprise. More than a surprise actually, I f***king loved it.
Written and directed by Ryan Spindell, The Mortuary Collection opens with the mysterious mortician Montgomery Dark (Clancy Brown) who, after holding a funeral for a young boy, is approached by a young woman named Sam (Caitlin Custer) inquiring about a job opening. She appears to be impressed with the idea of working in a mortuary and is even more impressed with Mr. Dark's magnificent book collection. He tells her that each addition to his library tells the story of someone who has died. Every soul has a book that details the events of their final moments, and she demands to hear some.
I'd probably do the same so her request is not that surprising.
The film is divided, and perfectly balanced, between the stories Mr. Dark tells and what is going on between him and Sam in the present. There is something strange about the mortuary that grows in prominence as the stories grow in their devilry.
There are four stories followed by a surprise fifth, and not a single one is boring. They follow a pattern that goes from decade to decade, from sin to sin, amusingly connected through the reappearing character Dr. Harold Kubler who pops up as an oblivious observer. They follow a woman who puts her nose where it doesn't belong, a prickish frat boy that doesn't use protection (definitely the funniest one), a man who can't bear the burden of his sick wife any longer, and a true tale of terror titled "The Babysitter Murders" that involves a deranged serial killer. Each warns against a particular "sin." Giving a worst-case scenario for every bad decision the participants make.
This is typical for horror; stories warning people that no evil deed goes unpunished. Evildoers, either by fate or guilt, will be punished for their choices. However, in a meta take on the anthology, the film acknowledges what it's doing and even comments on it, "your stories are predicable." The Mortuary Collection has an odd sense of humor that banks on audience expectation. A final girl, a knife-wielding maniac, women getting punished for premarital sex, and an obvious villain. All horror tropes you regularly expect to see, but the film wears them as a mask. Dangling these safe assumptions over your head like bait until you go to grab one and find out it's actually something completely different.
Everything leads up to a fifth and final tale of horror starring Sam and Mr. Dark themselves. It's a phenomenal conclusion that shocks as well as satisfies, revealing the ultimate punishment for a special monster that has hovered in the background since the opening credits, going unseen for much of the film but whose presence can be felt throughout.
I have no complaints regarding this film. Structurally, it's a perfect anthology. Not as bloody as some or thrilling as others, but it's well balanced, clever, and exhilarating in its own way. Holding it up is the double-sided coin of horror and humor, surrounded by a mix of moral debate, social norms, and honest to God good storytelling. The Mortuary Collection keeps you on your toes, which is what I really love about it. It was a blast and if for some reason you don't believe me, just look at what Rotten Tomatoes gave it because as of right now The Mortuary Collection stands at 94%. For a horror movie, having such a good critic score is a major achievement.
The Mortuary Collection will premiere on Shudder October 15.
All photos are property of Shudder, Trapdoor Pictures and Glass Eye Pix
Spiral (2019) review: A gripping display of fear and the hate it breeds
"The Wolf House" is the creepiest animated film since "Coraline"
Movie review: What exactly is "The Empty Man" that we all keep hearing about?
Review: 'The Djinn'
Horror anthology, Horror Movies, movie review, Shudder, The Mortuary Collection
Rachel Roth
Rachel Roth is a writer who lives in South Florida. She has a degree in Writing Studies and a Certificate in Creative Writing, her work has appeared in several literary journals and anthologies. @WinterGreenRoth
The Simpsons Treehouse of Horror: Best of Installments IX, X and XI
Undead and Uncut Follow-Along Colouring Book!

redpajama_set_name: RedPajamaCommonCrawl


Punk / Rock / Metal band formed by Chris Reifert (Death, Abscess, Autopsy) on guitars and vocals. Featuring guest appearances from Danny Coralles and Eric Cutler (both guitar players in Autopsy). Autopsy / Abscess- wise, the production is raw and primitive, as punk as it can be. Maybe the perfect mixture between Autopsy and the Ramones! 18 tracks of energetic raw punk rock with Chris Reiferts charismatic vocals.


Watch and listen to Wayne Rooney, Steph Houghton, Jack Rutter and Nathaniel Chalobah, explain how both parents and children can raise concerns they have about adults working in football.
Visit Childline, phone 0800 1111 or their website.
Visit the CEOP website (Child Exploitation & Online Protection) to help decide what to do.
Perhaps you would prefer to call NSPCC Helpline, 0808 8005000.
Not sure visit their website.
If you are worried about a child, reporting your concerns can help youth football stay safe.


text:
Per Jörgen Brink (ur. 10 marca 1974 w Delsbo) – szwedzki biegacz narciarski i biathlonista, trzykrotny medalista mistrzostw świata w narciarstwie klasycznym.

Kariera
Po raz pierwszy na arenie międzynarodowej pojawił w styczniu 1994 roku, startując na mistrzostwach świata juniorów w Breitenwang. Zdobył tam brązowy medal w sztafecie, w biegu na 10 km techniką klasyczną zajął 27. miejsce, a na dystansie 30 km stylem dowolnym uplasował się jedną pozycję wyżej.

25. miejsce w biegu na 10 km techniką klasyczną oraz 32. miejsce w biegu na 30 km stylem dowolnym. Podczas rozgrywanych rok później mistrzostw świata juniorów w Harrachovie zwyciężył w sztafecie, był drugi na 30 km stylem dowolnym oraz czwarty w biegu na 10 km klasykiem.

Jego olimpijskim debiutem były igrzyska w Salt Lake City w 2002 roku, gdzie zajął 24. miejsce w sprincie techniką dowolną. Cztery lata później podczas igrzysk olimpijskich w Turynie zajął 30. miejsce w biegu łączonym na 30 km oraz 51. miejsce na dystansie 50 km stylem dowolnym.

W 2001 roku wystartował na mistrzostwach świata w Lahti zajmując 19. miejsce w sprincie techniką dowolną. Mistrzostwa świata w Val di Fiemme w 2003 roku były najbardziej udanymi w jego karierze. Indywidualnie wywalczył brązowe medale w biegu łączonym na 30 km oraz na dystansie 50 km techniką dowolną. Ponadto wspólnie z Andersem Södergrenem, Perem Elofssonem i Mathiasem Fredrikssonem wywalczył także brązowy medal w sztafecie 4x10 km. Dwa lata później, podczas mistrzostw świata w Oberstdorfie jego najlepszym wynikiem było zaledwie 27. miejsce w sprincie stylem klasycznym.

Startuje także w zawodach cyklu FIS Marathon Cup. W 2010 i 2011 roku wygrywał najstarszy i największy maraton narciarski na świecie, szwedzki Vasaloppet, a w 2007 roku był trzeci. W 2010 roku był drugi w estońskim Tartu Maraton, a w 2011 zajął drugie miejsce w największym norweskim maratonie Birkebeinerrennet. W klasyfikacji generalnej najlepiej wypadł w sezonie 2011/2012, kiedy był trzeci za Czechem Stanislavem Řezáčem i Norwegiem Andersem Auklandem. Ponadto w 2003 roku był wicemistrzem Szwecji w biegu na 10 km techniką dowolną, a na tym samym dystansie techniką klasyczną wywalczył brązowy medal. Rok później był wicemistrzem Szwecji w sprincie stylem klasycznym.

Od 2007 roku uprawiał także biathlon. Jego największym sukcesem w tej dyscyplinie jest piąte miejsce w sprincie rozegranym 13 marca 2009 roku w kanadyjskim Vancouver. W 2008 roku wystartował na mistrzostwach Europy w Novym Měscie zajmując między innymi 12. miejsce w sprincie oraz 14. miejsce w sztafecie. Nie startował na mistrzostwach świata w biathlonie.

Osiągnięcia w biegach narciarskich

Igrzyska olimpijskie

Mistrzostwa świata

Mistrzostwa świata juniorów

Puchar Świata

Miejsca w klasyfikacji generalnej
 sezon 1998/1999: 86.
 sezon 1999/2000: 61.
 sezon 2001/2002: 16.
 sezon 2002/2003: 3.
 sezon 2003/2004: 22.
 sezon 2004/2005: 30.
 sezon 2005/2006: 58.

Miejsca na podium chronologicznie

FIS Marathon Cup

Miejsca w klasyfikacji generalnej
 sezon 2006/2007: 24.
 sezon 2009/2010: 9.
 sezon 2010/2011: 6.
 sezon 2011/2012: 3.
 sezon 2012/2013: 14.

Miejsca na podium

Visma Ski Classics

Miejsca w klasyfikacji generalnej
 sezon 2011: 4.
 sezon 2012: 3.
 sezon 2013: 4.
 sezon 2014: 7.
 sezon 2014/2015: 30.
 sezon 2015/2016: 23.
 sezon 2016/2017: 50.
 sezon 2017/2018: 32.
 sezon 2018/2019: 53.
 sezon 2019/2020: 112.

Osiągnięcia w biathlonie

Mistrzostwa Europy

Puchar Świata

Miejsca w klasyfikacji generalnej

Miejsca na podium
Brink nie stawał na podium zawodów Pucharu Świata w biathlonie.

Przypisy

Linki zewnętrzne
 Profil na stronie FIS
 Profil na stronie IBU

Medaliści Mistrzostw Świata w Narciarstwie Klasycznym 2003
Szwedzcy biegacze narciarscy
Szwedzcy biathloniści
Uczestnicy Zimowych Igrzysk Olimpijskich 2002
Uczestnicy Zimowych Igrzysk Olimpijskich 2006
Urodzeni w 1974
Szwedzcy olimpijczycy

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Representatives of state institutions, local government, and civil society organizations completed the training on monitoring that was held in Presevo on the last day on November and the first day of December. These 12 colleagues are members of the informal network for monitoring and advocacy that has been created for a mutual goal - to improve the position of refugees and migrants in Presevo, and the quality of coexistence in this town.
Training the colleagues to monitor and, thus, improve the system they are working within with refugees and migrants is one of the goals of the project "Building Together". During the following year, Local network will create two monitoring reports that will be used to advocate for further improvements to the system.
All the photographs from the training in Presevo can be found in the gallery, or in the Facebook album.


Colombian singer Shakira reached the top of the pop charts for the first time in the UK on Sunday with her new single 'Hips Don't Lie', featuring Wyclef Jean. Nelly Furtado was knocked to #2 with 'Maneater', while Pussycat Dolls moved up to #3 with 'Buttons'. Other notables include Sandi Thom, who slipped one to #4, Pink, who dropped two to #12, and Lily Allen, who debuted at #13 with 'Smile'.
Miss Piggy has another huge flop, Say Something already dropped off the UK singles charts after just 3 weeks and peaked at #27. Her singles are flopping as bad as her adventures tour.
Sad Lambs just because Mooriah can't hit #1 anywhere but one country. Shakira and Madonna are successful worldwide, Mooriah is a has-been flop worldwide.


text:
Father fears fate of teen charged in death of Army Veteran
Criminal history of teen charged in veteran`s death
By: Allison Mechanic
NEWPORT NEWS - The father of a teenage girl who is charged in the death of an army veteran mowing his lawn fears his daughter will be charged as an adult.
We are not identifying the teenager because of her age, but Calvin Casey, the teen's father, told News 3 exclusively that she has had run-ins with the law before.
"She went on the wrong path," said Casey. "She was with social services in Richmond and got hooked up with the wrong people for that I am very sorry."
Police say the teen was listed as a runaway and wanted on robbery charges in Henrico County when she drove Shawn Collier's car after he was arrested for outstanding warrants on Friday. Sources say on April 24th the teen along with two other women used the dating website Plenty of Fish to lure a Surry County man to the Coventry Garden Apartments. According to sources, the ladies asked the man to come to the back of building 216, where he was robbed by three men.
William Gordan, a neighbor in the apartment complex says he knew something illegal was going on.
"Luring people here to rob them, it's real shocking and kind of scary," Gordan said.
Now in addition to robbery, the teen faces several other charges including felony homicide, hit and run, evading and eluding police and operating a motor vehicle with no drivers license. Currently, the case is in Juvenile Domestic Relations Court, but Casey says he believes and fears his daughter will be charged as an adult.
"She made a mistake and she has to learn from it," said Casey. "But at the same time, I have to ask for you all to show her some sympathy too."
The Commonwealth's Attorney's Office says the case is currently still in Juvenile Domestic Relations Court so they can not comment on the case.


One of our most versatile playground markings, the mega matrix can be used to promote physical activity and play custom made games on. It also doubles up as a Noughts and Crosses grid.
Encouraging children to use their coordination skills to jump, step and hop from square to square, our Mega Matrix helps to improve stability whilst turning and landing and keeps children active both mentally and physically.
Our Mega Matrix makes a great addition to our range of Multi-Skills Playground Markings and is fully supported by Nationally Accredited CPD.


Passenger-only ferries poised for a Puget Sound return?
There are plans to connect Seattle and Port Townsend with passenger ferry service within two years.
So with the state out of the foot-ferry business and one other new service trying to make a go of it, drugs one might think we're on the verge of forming a new "mosquito fleet" of the kind people traveled on decades ago.
"I think we are," said Jim Pivarnik, deputy director of Port Townsend's port authority, which has a $1.3 million federal grant in hand to buy a boat for a new Seattle-to-Port Townsend route.


For some time now, the NHS has been rolling out the Personalised Mental Health Care Programme. Since hearing about it a few months ago, I haven't been given a clear idea of what it's about. But I do know that it enables Mental Health System Users to access funding for education and training opportunities if a professional agrees that there is benefit for your mental health.
So far, I'm really happy with this as I've been able to enrol on a one-day-a-week course with the Prince's Drawing School. I'm really enjoying the course and would strongly urge any disabled artists currently using the mental health system to look into the possibilities of this programme. Don't be put off by the vagueness you may come across from professionals. It's quite a new programme, so insist that they find out about it if they don't seem quite sure.
Find out more at www.rethink.org.
Another brilliant service I have been using since leaving the Recovery Centre earlier this year is Employment Support at St James House in Camden. This programme gives you good professional support in finding sustainable employment opportunities if, like me, you've been closed for repair for a while. Well worth checking out, if you live in Camden, and also having a look for local programmes if you don't.


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text:
There is a debate brewing in the wake of Jacob Hacker and Paul Pierson's new book, Off Center: The Republican Revolution and the Erosion of American Democracy. On the one side we have George Lakoff arguing that the boundaries between left and right in American politics can be reshaped by skilled politicians capable of shaping public perception. On the other, we have Hacker and Pierson arguing that the Republicans have succeeded by protecting themselves from backlash from America's centrist majority. They have done this not so much through the use of framing to reshape political identities as Lakoff argues, but through clever organizational strategies and bureaucratic tactics that buffer them from public opinion. (You can read about their book here, here, here, and here.)
What strikes me about this debate is that it involves two very different kinds of social scientific explanations, and nobody seems to have a strong sense of how the two fit together. On the one hand, we have a cognitive approach which looks at voter behavior in terms of their perceptions of political reality. On the other hand, we have an institutional analysis of the mechanisms used by the Republican party to secure access to the means to shape public perception.
The founders of modern sociology, Weber and Durkheim, both spent a great deal of attention on the process of rationalization: the shift from traditional values to modern rational-legal institutions. They worried about the dangers posed by this process. But they also overlooked the important ways in which modern, rational-legal, institutions are themselves shaped by values.
Once we understand this, we can see that it isn't really a question of ideological framing vs. organizational power, but that the two are highly interdependent. Republican ideology may not reflect the true nature of America's centrist politics, but more and more it reflects at least the professed beliefs of our political elite. It is a powerful force that ties them together and unifies them. That is why it is so hard for them to accept when Bush elevates cronyism over ideology in his nomination for the Supreme Court.
Values shape perceptions, they are an important ingredient in defining our shared interests, but they are much more than that. They are also a means by which we define our membership within a group. We adopt values contrary to our own self-interests because we often see larger gains to be made by allying ourselves with those in power, and by defending the value commitments of those we seek to ally ourselves with we bind ourselves to them. This is true whether you are a middle-class revolutionary adopting working class values, or a member of the working-poor embracing corporate values.
This is something that the Italian marxist philosopher-politician Antonio Gramsci understood. He compared the French revolution with the Italian Risorgimento and found that while the Jacobins had been able to embrace and ideology which emphasized their shared interests with rural peasants, the Italian elite had not been able to do the same. His writings on the "Southern question" discuss the long term impact of this failure for Italian unity. He also saw this as explaining the failure of Italy to follow in the revolutionary footsteps of the Soviet Union, since the workers of the North could not see their shared interests with southern peasants.
This last point is particularly important for understanding the United States, since race plays very much the same role here that the rural-urban divide did in Italy. It is too simple to say that the Democrats lost the South over race. At the same time, error and ignorance are not enough to explain why poor whites embrace corporate values. They do so because it is in their own interests to do so. Not because the policies resulting from such values will help them (the opposite is the case), but because in doing so they ally themselves with those in power. In a very fundamental way I don't see it as much different from becoming a member of the Communist Party in Soviet Russia or pre-reform China.
Membership has its privileges.
{lakoff, Hacker and Pierson, off center}


text:
Q: Why is there a computation error when returning function with structure I am trying to find the number of elapsed days between two dates. However, with some dates the program prints the wrong number of days between 2 dates.
The formula for number of elapsed days is: N=1461 x f(year,month) / 4+ 153 x g(month)/5 +day
In my program, I have f and g defined as functions that return different values of years and months depending on what the user inputs.
The main problem probably lies with how I scanned the user input with the local variables in the date structure.
N1 is the number of elapsed days for the 1st date. N2 is for the second date.
Below is my written code:
#include <stdio.h>
#include <stdbool.h>
#include <stdlib.h>
struct date {
        int month;
        int day;
        int year;
    };

int f( int theyear,  int themonth){
    if (themonth <=2)
        theyear = theyear-1;
    else
        theyear =theyear;
    return theyear;
}

int g( int Month){
    if (Month<=2)
        Month=Month+13;
    else
        Month = Month+1;

    return Month;
}

int main()

{

    struct date time1, time2;
    int N1; int N2;

    printf("Enter time1 (mm,dd,yyyy): ");
    scanf("%i%i%i", &time1.month, &time1.day, &time1.year);

    printf("Enter time2 (mm,dd,yyyy): ");
    scanf("%i%i%i", &time2.month, &time2.day, &time2.year);
    N1=1461 * f(time1.year,time1.month) / 4 + 153 * g(time2.month) / 5 +3;
    N2 = 1461 * f(time2.year,time2.month)/4 +153 * g(time2.month)/5 +21;

    printf("%d \n",abs(N1 -N2));

    return 0;

}

Thank you very much. :)
Sample input/output:
Enter time1 (mm,dd,yyyy): 04 10 1994
Enter time2 (mm,dd,yyyy): 03 10 1994
18
As you can see, it prints out 18. However, it is supposed to be the number of days in a month.

A: change
N1=1461 * f(time1.year,time1.month) / 4 + 153 * g(time2.month) / 5 +3;
N2 =1461 * f(time2.year,time2.month)/4 +153 * g(time2.month)/5 +21;

to
N1=1461 * f(time1.year,time1.month) / 4 + 153 * g(time1.month) / 5 + time1.day;
N2 =1461 * f(time2.year,time2.month)/4 +153 * g(time2.month)/5 + time2.day;

?

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The ML5590 narrow and wide carriage models feature advanced specifications that ensure excellent productivity and reliability every time. Their robustness and flexibility make them the ideal choice for demanding environments.
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text:
By Sara Nathan
Landing in Israel, with the wide expanse of desert set out in great big dusty trails beneath you, is intoxicating and at once an adventure beckons.
You may hear the siren cry of the call to prayer from a mosque, watch a family of Hasidic Jews stride to synagogue on Shabbat — or catch a glimpse of a dog surfing on the glistening waves of a Tel Aviv beach (yes, really).
You will certainly be greeted by the excitable shrieks and smells of this tiny country which is enjoying a rapid tourism boom.
Israel is a country of extremes: heat, religion and politics and people and during any visit you will certainly have to embrace the chaos.
Grabbing the opportunity to be part of the country's growth with both hands, Virgin boss, British billionaire Richard Branson recently launched a route from London to Tel Aviv in September, saying: "With its thriving start-up scene, pristine beaches and vibrant nightlife, Tel Aviv was a key destination for Virgin Atlantic to establish a new route."
The new route aims to capture a share of the some 2,700 passengers flying from Tel Aviv to the United States each day, with London positioned as an attractive mid-point stopover for both tourists and business travelers.
While Tel Aviv may be set alongside the azure blue of the Meditteranean sea, it's way more than just a beach resort.
But it does give you the chance to gently sink into Israeli culture.
Virgin Atlantic founder Richard Branson arriving in Tel Aviv, Israel.
Streets away from the beach lies Carmel market, where I started my trip, only to be hit immediately with the smell of ripe fruit and vegetables. Blood red pomegranates are piled high upon each other as lascivious yellow mangos call out to you while nestled alongside stalls of sharp and sour olives.
It can be overwhelming. You need to learn how to haggle — and have a price in your mind — before you commit to buying anything.
And certainly don't go anywhere without trying a juice to cure whatever ails you from the renowned "Etrog Man" at the top of the market. Uzi- Eli is Israel's ultimate health guru and you can sip a concoction of passionfruit and goat's milk alongside cold-pressed etrog with grapefruit juice, or even Yemenite arak — an alcoholic drink favored in the Middle East nicknamed "the milk of lions" — made from raisins.
The past meets the present throughout the country. And so many hoteliers have used this to their advantage as they launch their new properties.
Arguably, the new jewel in Tel Aviv's crown is The Jaffa, a former French hospital and one-time convent which features a cavernous cocktail bar set in a former chapel and a wall dating back to the 13th century which runs through the lobby.
Meanwhile, The Setai nearby is converted from an Ottoman-era prison, and The Normanhas become a trendy darling that has played host to Kanye West.
More newcomers are set to arrive on the scene. The Selina Hotel in Neve Tzedek — think Tel Aviv's own Soho-style neighborhood, complete with fancy ice cream stores (make a beeline for Anita), will open this December followed by Soho House's Tel Aviv outpost next year.
The beauty of the city is that you can walk it easily, saunter along on the seafront up to Jaffa to explore the old flea markets, thriving stores and cafes, and then turn inland towards Rothschild Boulevard and walk back down through the hot streets to the recently renovated port.
As the light fades, you're faced with an almost different city altogether as the nightlife explodes around you.
Amid this "new Middle East", Israel boasts medical advances, computer science and even cherry tomatoes among its many developments that are on show at the Peres Center for Peace, founded in 1996 by the former President of Israel Shimon Peres.
The building by Italian architect Massimiliano Fuksa is set on the Jaffa coast next to the Arab neighborhood of Ajami, and features an interactive exhibit with a host of both Israeli and Arab contributors and a virtual reality chamber.
The food — ah the food — is much more than Israeli salad, a national dish of finely diced tomatoes, cucumbers and onion. Israeli food has hit a zenith thanks to chefs like Yotam Ottolenghi and Michael Solomonov, whose Philly eatery Zahav was crowned the outstanding restaurant of 2019 by the James Beard Foundation — known colloquially as the Oscar of the culinary world.
The beauty of Israel is that nothing is too far away. It's just a train ride — a high-speed route will open later this year — or just over an hour drive, from Tel Aviv to Jerusalem.
You don't have to be religious to let the sheer magnitude of the holy city wash over you.
At its core is the Old City, a maze of narrow alleyways and historic architecture that characterizes its four quarters — Christian, Muslim, Jewish and Armenian. It is surrounded by a fortress-like stone wall and home to some of the holiest sites in the world, including the Western Wall, where you can join Jews at their daily prayer, and follow the last days of Jesus, from the Last Supper Room to the Via Dolorosa along which Jesus made his way to the crucifixion at the site of the Church of the Holy Sepulchre, where you can view his tomb.
With the help of a tour guide (we explored with the brilliant Fanya Shtrul) you can see most of Jerusalem's must-see sites in the span of one afternoon.
Visiting Temple Mount, however, requires additional planning, as it's only open for limited times (note that modest clothing is also required).
As the city shut down for Shabbat and the sky turned the color of mulberry, we enjoyed Friday night dinner on the rooftop of the Mamilla hotel, with the city sprawled out in front of us, besieged by history.
Watergen, Tyto Care, OrCam, ECOncrete, Theranica, Temi and Eviation products make main list; Lemon...


text:
Last edited by Faekora
10 edition of The Complete Idiot"s Guide to World Religions found in the catalog.
The Complete Idiot"s Guide to World Religions
by Brandon Toropov
Published 2004 by Alpha Books in New York, USA .
US/CAN/UK
Other titles World religions
Statement by Brandon Toropov and Luke Buckles.
Series Complete Idiot"s Guides
Contributions Buckles, Luke.
LC Classifications BL80.3 .T67 2004
Pagination xxvi, 405 p. :
OCLC/WorldCa 656366374, 937705215
I'm a complete idiot and this book provides me with a guide to world religions. Read more. One person found this helpful. Helpful. Comment Report abuse-Mrs Gillian M. Scott. out of 5 stars Easy to follow. Reviewed in the United Kingdom on 11 September Verified s: The Complete Idiot's Guide to Islam, 3rd Edition: An In-Depth Look at One of the World s Most Important Religions: Emerick, Yahiya: Books -
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Rome and Carthage, the Punic wars.
At home with the word, 1987
Before the summer rain
Bsava Manual of Small Animal Ophthalmology
Memorials of the dead
The Fables of Aesop
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Oeuvres choisies
Madison River country
Montmorency Park National Historic Site of Canada
The relationship of discovery methods in mathematics to creative thinking and attitudes toward mathematics
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Absolutely. There are certain things you just cant be without no matter where you are or why.
I travel often and I cannot agree more these are my must-have item. The travel pillow which I can use on the long journey bus and flights so that my neck will not hurt when I woke up. The lip gloss is also my must-have item as it gets to dry up in the sky.
You have great timing! We are planning a family trip and I was just saying that I need to make a list of all the things we need.
I will have to check out the My Charge; we are headed to Disneyland soon, and my phone always dies! Great list!
These are great tips on what you take for traveling! I'll make a note for the future.
What a great list! On my next travel, I will keep in mind some of the items mentioned in this article. Thanks.
i used to love traveling and i have a few of the items you've mentioned. well, i still love traveling, only that i have toddlers which makes it impossible for me to leave them.
I definitely agree that something like the My Charge is essential for travel. There is nothing worse than running low on phone battery without being able to charge!
These are great suggestions! Traveling does a number on my hair and skin and all of these items are so important. And, I have to have a noise machine at a noisy hotel.
These have really good suggestions. The charger is a big must have for me.
I hadn't heard of a MyCharge before. I always have a charger on hand when I travel.
Those were a great travel essentials. Whenever I have a trip, I always amke sure to bring my portable phone charger too. It is very useful for me.
When we travel, we always wanted to have a comfortable trip. Neck pillow is always on my bag.
These are definitely must-have travel necessities. Most of these things are what I also pack.
Wipes do come in handy — more so the facial ones. They're perfect for removing makeup.
These are all great suggestions! I don't travel often but when I do I travel long distances, being comfortable is a must!
I have to say I travel really light so the only things I need apart from my clothes are my chargers and camera.


text:
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Moustache History
1) History of the Moustache
2) Interesting Facts about Moustache
3) Moustaches Today
Many cultures regard the moustache as a symbol of masculinity. In some religions, a moustache is a piece of evidence that young teenage boys have grown into adult men. The process of nurturing moustaches went through different stages throughout history.
If you want to have a moustache and have a little hair on your face, SHIFT Hair Transplant clinic can help you. The hair transplant in Turkey provides any hair correction on the face.
History of the Moustache
Throughout history, moustaches and cultural attitudes have been changing. At times they have been praised, while at other times they have been banned. The moustache has experienced different periods, from the moment the first prehistoric people began to remove some of the hair on the face, and to this day when the trend is beginning to return. In the Middle Ages, the Prince of Wales was the first to wear a moustache on his face. However, only later in England during the reign of Queen Elizabeth could it be said that the moustaches became a fashion symbol. In the 17th century, there was an aesthetic and cultural trend in the cultivation of twisted, spiky moustaches, while in the 18th century, many romantics such as Lord Byron continued this tradition. During that period, the moustache was wrapped and quite "weak".
The dark age of the moustache emerged in the 19th century when the moustache was cultivated only by individuals, most often soldiers. After the end of the First World War, many were forced to shave. Since 1920, moustaches have returned to fashion and become a symbol of the intellect and seriousness. In the 1960s, new styles in shaping moustaches appeared.
Interesting Facts about Moustache
According to some studies, the oldest recorded moustaches date back to 300BC.
There can be between 10 and 20,000 hairs on the face. An average moustache contains up to 600 hairs.
Albert Einstein had his trademark moustache for over fifty years.
Moustaches can also help to protect the skin from cancer.
In certain parts of the United States, it is illegal for a man with a moustache to kiss a girl. Furthermore, many police officers have to get rid of their moustaches because their bosses think this will make them look more respectable.
In Ancient Egypt, there were both real and false moustaches, while today a moustache can be transplanted in many areas of the world. SHIFT Hair Transplant clinic can make these modifications during a hair transplant in Turkey.
Moustaches Today
Today, there is a global movement known as "Movember". People around the world are growing moustaches in November to help raise awareness of cancer in men.
There are many styles of moustache today. Some of the most famous are:
English – slightly bent aside;
Dali – based on the famous artist;
Natural – stylizing without tools;
Horseshoe – moustaches that grow vertically down;
Pencil;
Narrow, real and thin moustache, and many others.
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text:
Q: Extract price - years and months in separate columns I am very new to SQL and creating queries. I am trying to extract data to create some charts (in wpdatatables) that show year on year prices based on some criteria that I have set in the query. I am using case to build the year columns; I seem to be able to get data for one year but not the subsequent years.
Hoping someone can help, thanks in advance
My query is as follows.
select grid_calc.`Price`,
    monthname(grid_header.`EntryDate`) as Month,
       case when year(EntryDate) = 2018 then price else 0 end as price_2018,
       case when year(EntryDate) = 2019 then price else 0 end as price_2019,
       case when year(EntryDate) = 2020 then price else 0 end as price_2020,
       case when year(EntryDate) = 2021 then price else 0 end as price_2021
FROM grid_calc
  INNER JOIN grid_header
     ON grid_header.`BuyerID` = '234'
     AND grid_calc.`GridID` = grid_header.`GridID`
WHERE 1=1
   AND grid_header.`EntryDate` > '2018-01-01'
   AND grid_calc.`Teeth` = '2'
   AND grid_calc.`FeedType` = '434'
   AND grid_calc.`SexCategory` = '438'
group by monthname(EntryDate)
order by max(month(EntryDate));

The data I am getting..
Price   Month       price_2018  price_2019  price_2020  price_2021
4.8     January     4.8         0           0           0
4.85    February    4.85        0           0           0
5.05    March       5.05        0           0           0
4.9     April       4.9         0           0           0
4.7     May         4.7         0           0           0
4.65    June        4.65        0           0           0
4.85    July        4.85        0           0           0
4.85    August      4.85        0           0           0
4.8     September   4.8         0           0           0
5.2     October     5.2         0           0           0
6.85    November    0           0           6.85        0

P.S I know there is data in other years below is what I get when i only search on one year 2020
select grid_calc.`Price`,
    monthname(grid_header.`EntryDate`) as Month,
       case when year(EntryDate) = 2020 then price else 0 end as price_2020
FROM grid_calc
  INNER JOIN grid_header
     ON grid_header.`BuyerID` = '234'
     AND grid_calc.`GridID` = grid_header.`GridID`
WHERE 1=1
   AND grid_header.`EntryDate` > '2020-01-01'
   AND grid_calc.`Teeth` = '2'
   AND grid_calc.`FeedType` = '434'
   AND grid_calc.`SexCategory` = '438'
group by monthname(EntryDate)
order by max(month(EntryDate));

Price   Month       price_2020
5.7         January         5.7
6       February    6
6.45    March       6.45
5.85    April       5.85
5.95    May         5.95
5.95    June        5.95
6.45    July        6.45
6.25    August      6.25
6.05    September   6.05
6.6     October     6.6
6.85    November    6.85

Cheers.


text:
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Kangal - the origin of the ultimate flock guardian
Thie Kangal is a Turkish dog breed that is a part of the early Mastiff family of dog breeds. It possesses an impressive physique with its large muscular body, a curled tail, and a black mask. This dog breed served Turkish shepherds for thousands of years and up to this day, the Kangal is regarded as one of the best guardians in the dog world.
These dogs are closely related to the Anatolian Shepherd and throughout history were treated as two separate dog breeds. Eventually, the Anatolian Shepherd was renamed and their name from then was the Kangal. Today there is only one breed by the FCI Standard.
Most people are not familiar with Turkish dog breeds even though the Turkish Kangal history is rich and colorful. First, we need to say a bit more about the Kangal name.
Origin of the Kangal name
These dogs originated in the Sivas province, in the region of Anatolia in Turkey. The original name of this breed was the Anatolian Shepherd who was named after the region of their origin. Some dogs were regarded as Kangals and the Turkish Cynology Association didn't do much to clarify this confusing distinction between the breeds.
The Kangal dog's full name is Kangal Çoban Köpeği in their native language of the region they originated from. These dogs were given the "Kangal" part of the name after the city of Kangal in the center of the Anatolian region. The team of scholars that are credited for unofficially naming the Kangal dogs is Dr. Charmian (Biernoff) Stelle, Messrs J. Lloyd, D. Lyth, and Dr. Withof- Keus. They were mostly archaeologists that were researching that area and found much historic evidence of these dogs.
In 2018 the Turkish Cynology Association issued a decree that all Anatolian Shepherds will be named Kangals and that the Kangal will take the Anatolian Shepherds place in the official FCI registry.
Early Kangal history
This impressive dog breed is regarded as one of the world's best shepherd dogs. Though they are not shepherds in the traditional sense that they herd livestock, but rather guardians of the human shepherds and livestock. Since they were huge and powerful, they were used to defend livestock against predators such as wolves, bears and later in Africa against lions and cheetahs.
It is believed that the ancestors of this breed came to Anatolia when Persians tried to occupy the region we know today as Turkey. they were accompanied by three species of animals - horses for carrying heavy loads, sheep for their wool and meat, and dogs that were used for protection of both humans and sheep.
Those dogs were from the Molosser family of dog breeds and were big, strong and fearless. There are many tribes and civilizations that originated in the Anatolian region that has a great historical value and certainly, the Kangals were not the first dogs from that area.
No evidence of these types of dogs was found in that area before the Persian invasion around 1400 years ago.
Two interesting legends are surrounding the Kangal dogs and the first one says that a dog was given to the Ottoman sultan called Murat the IV by an Indian Maharajah. When the dog arrived at the sultan's palace, they made the dog fight a lion and the dog won. After the attack on their city, the dog got lost and it is believed that the Kangals are descendants of that dog.
The second legend simply says that these dogs derived from lions and tigers that were petted during the Assyrian time and were later used to keep livestock safe, just like the modern Kangal does.
Working history
These dogs are known for being fearless and tireless workers and protectors. They have been kept by Turkish farmers and shepherds for centuries and their main job was to keep sheep safe from mostly wolves.
They have been carefully bred through these times to eliminate all aggression displayed towards either humans or the flock of sheep they were trusted to guard. This way, a reliable and predictable character in these dogs was achieved and shepherds have been extremely satisfied with the work these dogs did.
The Turkish shepherds also managed to create a dog breed that is always alert, extremely confident and fearless yet loyal and gentle towards their family, especially children.
In more recent times, Kangals have been trusted to guard bigger cattle. They have been mainly employed to guard cows and their main predators have also changed. They were trusted to guard them against bears, wolves, and jackals.
They have developed a 3-step strategy when facing a threat. When they sense a predator they will first try to scare them off by barking. If the predator doesn't flee on the sight of the Kangal and still they decide to approach, these dogs will start growling. If that doesn't work, they will launch their attack.
When attacking they use their main weapons - speed and strength. They can reach speed up to 50km/h and have a bite force of 700 PSI (trice as strong as Pitbulls). The Kangals have the same attack strategy throughout history. They will run-up to the predator and try and knock them down by hitting them with their shoulder. After that, they try to attack the weakest points - the throat and the hind legs.
The UK Kennel Club recognized this breed in 2013. They have also said that all dogs that were previously registered as Anatolian Shepherds will be registered as the Turkish Kangals when the time for that occurs.
The FCI officially recognized and changed all the Anatolian Shepherds as Kangals in 2018 when the Turkish Cynology Association (KIF) issued a document which says that all Anatolian Shepherds will officially become Turkish Kangals and the two have then become a single dog breed.
This dog breed is somewhat rare so if you are interested in getting one, make sure you know how to handle these dogs and be prepared to wait and spend a bit more money on one.
World Dog Finder team
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MPH continues to work with big names in the Defense industry with clients such as EADS, DCNS, Nexter and Honeywell. It has deployed thousands of industry experts that have worked in areas of producing artillery, ammunition or integrating technology devices in the defense industry.
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A new report on the global HIV epidemic among young people was released by five U.N. organizations, as well as the World Bank and the International Labor Organization, on Wednesday. In a statement given after the report's release, the World Bank's managing director, Mahmoud Mohieldin, noted that "Existing prevention strategies have had limited success, so we have to look for creative new approaches to reverse the HIV/AIDS epidemic. These must address people's very basic needs for education, economic security, inclusion, dignity, and human rights." We at Global Camps Africa agree with Mr. Mohieldin and his colleages at the United Nations. That's why we are using an innovative approach to HIV/AIDS prevention and education in our programs: residential camp experiences for AIDS-affected children and youth in South Africa. We are reaching young people with voluntary HIV testing and counseling services, life skills, and HIV education through activities that are informal and fun. We create a secure environment that provides a foundation for learning based on trust, love, and acceptance.
Please join us in our work to empower children for an AIDS-free tomorrow. Find us at www.globalcampsafrica.org.