And reads the books that never pall."
L. Cranmer-Byng. Map shows route taken by Xuanzang, starting from Xian, (ancient Ch'ang An). Click to enlarge.
Xuanzang [600--664CE] was born into a world that beheld the tree of Buddhism slowly dying from the top. He bore witness, if unconsciously, to a time of transition and a noble faith in decay, and the swift, silent growth of jungle mythology around the crumbling temples of Buddha. His record of these sixteen years of travel is a priceless one, for through it we are able to reconstruct the world and ways of Buddhist India of the centuries that have passed. Yet far more priceless is that record, read between the lines, of a human soul dauntless in disaster, unmoved in the hour of triumph, counting the perils of the bone-strewn plain and the unconquered hills as nothing to the ideal that lay before him, the life-work, the call of the Holy Himalayas and the long toil of his closing years. It is difficult to over-estimate his services to Buddhist literature. He returned to his own country with no less than 657 volumes of the sacred books, seventy-four of which he translated into Chinese, while 150 relics of the Buddha, borne by twenty horses, formed the spoil reverently gathered from the many lands we call India.
And so we leave him to rest upon Mount Sumeru, where once his venturous soul alighted in the dreams of youth, with the serpents coiled beneath its base, with its seven circling hills of gold and the seven seas between, and the great salt ocean encompassing them all. There... "he waits with Maitreya until in the fullness of time the latter comes into this world. With him Xuanzang hoped to come back to a new life here to do again the Buddha's work for the good of others." Till then we leave him to the long interval of bliss transcending all planes of human ecstasy."
The Chinese monk Xuanzang was one of the greatest travelers in history and, while practically unknown in the West, is a household name in China and known to most educated Indians. The actual journey to what he called "the West" took place from 629 to 645 CE, of which he left a detailed account, "Buddhist Records of the Western World," regarded as one of the great classics of Chinese literature, translated into English in the nineteenth century by a clergyman scholar named Samuel Beal. Xuanzang went on horseback, on camel-back, on elephant-back, or on foot from the ancient capital of Chang-an (today's Xian) all the way to southern India, a distance of roughly five thousand miles, and then back via a somewhat different route, crossing the harshest deserts and the tallest mountains in the world in both directions. His purpose was to search out what he called the Law, the original classics of Buddhist thought that would enable Chinese Buddhism, a doctrine borrowed from India in a language very foreign to China, to be put on an authentic footing.
When he returned to China he wrote about the countries he had visited on his journey. But while the monk performed that task for the emperor, his concern was with an India that for him stood as the source of supreme wisdom. He went there to achieve exalted understanding, what he saw as the Ultimate Truth, that alone permits us to achieve the purpose of Buddhism, which is the cessation of otherwise inevitable and inescapable suffering. (From Ultimate Journey, Richard Bernstein, who retraced the path of the ancient Buddhist monk, probably in considerably greater comfort.) To be continued.
The Royal family is extinct, and the kingdom is governed by deputies from Kapisa. The towns and villages are deserted, and there are few inhabitants. At one corner of the royal residence there are about 1000 families. The country is rich in cereals, and produces a variety of flowers and fruits; it abounds also in sugarcane, from the juice of which they prepare "the solid sugar." The climate is warm and moist, and in general without ice or snow. The disposition of the people is timid and soft: they love literature; most of them belong to heretical schools [Hinduism]; a few believe in the true law [Mahayana Buddhism]. From old time till now this borderland of India has produced many authors of sastras; for example, Narayanadeva, Asanga Bodhisattva, Vasubandhu Bodhisattva, Dharmatrata, Manorhita, Parsva the noble, and so on. There are about 1000 sangharamas, which are deserted and in ruins. They are filled with wild shrubs, and solitary to the last degree. The stupas are mostly decayed. The heretical temples, to the number of about 100, are occupied pell-mell by heretics.
Inside the royal city, towards the northeast, is an old foundation. Formerly this was the precious tower of the patra [begging bowl] of Buddha. After the nirvana of Buddha, his patra coming to this country, was worshipped during many centuries. In traversing different countries it has now come to Persia.
Outside the city, about 8 or 9 li to the southeast, there is a pipala tree about 100 feet or so in height. Its branches are thick and the shade beneath somber and deep. The four past Buddhas have sat beneath this tree, and at the present time there are four sitting figures of the Buddhas to be seen here. During the Bhadrakalpa, the 996 other Buddhas will all sit here. Secret spiritual influences guard the precincts of the tree and exert a protecting virtue in its continuance. Sakya Tathagata sat beneath this tree with his face to the south and addressed Ananda thus: -- "Four hundred years after my departure from the world, there will be a king who shall rule it called Kanishka; not far to the south of this spot he will raise a stupa which will contain many various relics of my bones and flesh."
To the south of the Pipala tree is a stupa built by King Kanishka; this king ascended the throne four hundred years after the Nirvana, and governed the whole of Jambudvipa. He had no faith either in wrong or right and he lightly esteemed the law of Buddha. One day when traversing a swampy grove he saw a white hare, which he followed as far as this spot, when suddenly it disappeared. He then saw a young shepherd-boy, who was building in the wood hard by a little stupa about three feet high. The king said, "What are you doing?" The shepherd-boy answered and said, "Formerly Sakya Buddha, by his divine wisdom, delivered this prophecy: "There shall be a king in this victorious land who shall erect a stupa, which shall contain a great portion of my bodily relics." The sacred merits of the great king (Kanishka) in former births, with his increasing fame, have made the present occasion a proper one for the fulfilment of the old prophecy relating to the divine merit and the religious superiority of the person concerned. And now I am engaged for the purpose of directing you to these former predictions." Having said these words he disappeared.
Early Buddhism was a religion that depended heavily on royal support, as can be seen from the above legend of King Kanishka and the shepherd-boy, as related by Xuanzang. Before Kanishka Ashoka had also lent support to Buddhism. But in the dependence on royal support lay the seed of its own unraveling, in Gandhara and India; for when this royal support was not forthcoming, the Buddhist kings themselves having been overthrown by foreign invaders, the monastic system crumbled into dust.
This country is about 4000 li in circuit. There are some ten desert cities in this country, wholly desolate and ruined. The capital is overthrown and in ruins. Its circuit cannot be accurately measured. The royal precincts within the city measure some 14 or 15 li round. They were all built of brick. The foundation walls are still strong and high. It has been long deserted. The peopled villages are few and waste.
There is no supreme ruler; each of the towns appoints its own ruler. The ground is rich and fertile, and is cultivated according to the regular season. The climate is uniform, the manners of the people soft and obliging. There are 1000 or more ruined sangharamas remaining; by the side of the royal precincts there is still a sangharama with about 3000 (read 30) followers in it, who study the Little Vehicle of the Sammatiya school.
There are a couple of Deva [Hindu] temples, in which various sectaries worship. Within the royal precincts are some ruined foundation walls; these are the remains of the proper palace of Suddhodana-raja; above is built a vihara in which is a statue of the king. Not far from this is a ruined foundation, which represents the sleeping palace of Mahamaya, the queen. Above this they have erected a vihara in which is a figure of the queen.
By the side of this is a vihara; this is where Bodhisattva descended spiritually into the womb of his mother. There is a representation of this scene drawn in the vihara. The Mahastharvira school say that Bodhisattva was conceived on the 30th night of the month U-ta-lo-an-sha-cha (Uttarashadha). This is the 15th day of the 5th month (with us). The other schools fix the event on the 23rd day of the same month. This would be the 8th day of the 5th month (with us).
To the northeast of the palace of the spiritual conception is a stupa; this is the place where Asita the Rishi prognosticated the fortune of the royal prince. On the day when the Bodhisattva was born there was a gathering of lucky indications. Then Suddhodana-raja summoned all the soothsayers, and addressing them said , "With respect to this child, what are the fortunate and what the evil (signs)? As it is right, so do you clearly answer me." In reply they said, "According to the record of the former saints the signs are especially fortunate. If he remains in secular life he will be a Chakravartin monarch; if he leaves his home he will become a Buddha.
At the south gate of the city is a stupa. This is where the royal prince, when contending with the Sakya princes, cast the elephant away. The royal prince having contended in the public competition (of arts and athletic exercises), was left entirely alone among them all. And now the Maharaja Suddhodana, after receiving congratulations, was about to go back to the city.
By the side of this is a vihara in which is a figure of the royal prince. By the side of this again is a vihara; this was the sleeping apartment of the queen and the prince; in it is a likeness of Yasodhara and (the child) Rahula. By the side of the queen's chamber is a vihara with a figure of a pupil receiving his lessons; this indicates the old foundation of the schoolhouse of the royal prince.
Xuanzang is aware that Buddha was supposed to have been born at Kapilavastu and of the legends associated with him there. However, that he actually located the correct site is problematic. From his tone he doesn't seem to be in particular awe of the fact that he is actually witnessing the site where his master supposedly lived for 29 years before venturing forth to seek enlightenment.
The frontier line of Kalinga cannot have extended beyond the Godavari river on the southwest, and the Gaoliva branch of the Indravati river on the northwest (Cunningham). The chief town was probably Rajamahendri, where the Chalukyas perhaps established their capital. Either this place or Koringa, on the seacoast, agrees with the bearing and distance given in the text. If, however, we accept Mr. Fergusson's hypothesis that the capital of Konyodha was near Katak, and calculating the li to be one-seventh of a mile, we shall have to seek for the capital of Kalinga near Vijayanagram. (Samuel Beal).
This country is 5000 li or so in circuit; its capital is 20 li or so round. It is regularly cultivated and is productive. Flowers and fruits are very abundant. The forests and jungle are continuous for many hundred li. It produces the great tawny elephant, which are much prized by neighboring provinces. The climate is burning; the disposition of the people vehement and impetuous. Though the men are mostly rough and uncivilized, they still keep their word and are trustworthy. The language is light and tripping, and their pronunciation distinct and correct. But in both particulars, that is, as to words and sounds, they are very different from Mid-India. There are a few who believe in the true law, but most of them are attached to heresy. There are ten sangharamas, with about 500 priests, who study the Great Vehicle according to the teaching of the Sthavira school. There ae some 100 Deva temples with very many unbelievers of different sorts, the most numerous being the Nirgranthas [Jains].
In old days the kingdom of Kalinga had a very dense population. Their shoulders rubbed one with the other, and the axles of their chariot wheels grinded together, and when they raised their arm-sleeves a perfect tent was formed. There was a Rishi possessed of the five supernatural powers, who lived on a high precipice, cherishing his pure thoughts. Being put to shame because he had gradually lost his magic powers, he cursed the people with a wicked imprecation, and caused all dwelling in the country, both young and old, to perish; wise and ignorant alike died, and the population disappeared. After many ages the country was gradually repeopled by emigrants, but yet it is not properly inhabited. This is why at the present time there are so few who dwell here.
Not far from the south of the capital there is a stupa about a hundred feet high; this was built by Ashoka-raja. By the side of it there are traces where the four past Buddhas sat down and walked.
Near the northern frontier of this country is a great mountain precipice, on the top of which is a stone stupa about a hundred feet high. Here, at the beginning of the kapla, when the years of men's lives were boundless, a Pratyeka Buddha reached Nirvana.


text:
\section{introduction}

A smooth one-parameter family $\{\Gamma_t\}_{t\in I}$ of connected, immersed, planar curves $\Gamma_t\subset \mathbb{R}^2$ \emph{evolves by the $\kappa^\alpha$ flow}, $\alpha>0$, if
\begin{equation}\label{ka-flow}
\partial_t\gamma(\theta,t)=-\kappa^\alpha(\theta,t)\nu(\theta,t)\;\;\text{for each}\;\; (\theta,t)\in \Theta\times I
\end{equation}
for some smooth family $\gamma:\Theta\times I\to\mathbb{R}^2$ of immersions of $\Gamma_t$, where $\kappa(\theta, t)$ and $\nu(\cdot,t)$ are the curvature and the unit normal vector of $\gamma(\cdot,t)$. Our sign convention is that $\vec \kappa=-\kappa\nu$ is the curvature vector.

We refer to a solution as \emph{compact} if $\Theta\cong S^1$ and \emph{convex} if each of the timeslices $\Gamma_t$ bounds a convex domain, in which case the immersions $\gamma(\cdot,t)$ are proper embeddings.
Both compactness and convexity are properties that are preserved under the flow and it is known that, if the initial curve $\Gamma_0$ is compact and convex, the family converges to a single point in finite time \cite{BA98}. Moreover, if the initial surface is convex then it will immediately become strictly convex and smooth \cite{BA98}.
The solution $\{\Gamma_t\}_{t\in I}$ is called \emph{ancient} if $I$ contains the interval $(-\infty,t_0)$ for some $t_0\in\mathbb{R}$. In the compact case, by a time translation, we will assume that $I=(-\infty, 0)$. The goal of this paper is to construct and study convex ancient solutions for $\alpha\in (1/2, 1]$ as well as provide certain classification results.

When $\alpha=1$, the flow is the famous \emph{curve shortening flow} and such a classification is already known. Daskalopoulos, Hamilton and \v Se\v sum showed that the shrinking circles and the Angenent ovals
 are the only \emph{compact} examples \cite{DHS}. Their arguments are based on the analysis of a certain Lyapunov functional. Recently, \cite{BLT3}, a new proof of this result was given which  removes the compactness hypothesis by adding two more solutions: the stationary line and the self-translating grim reaper.

 This proof uses in an essential way X.J. Wang's dichotomy, which states that a convex ancient solution $\{\Gamma_t\}_{t\in(-\infty,0)}$ must either be entire (i.e. sweep out the whole plane, in the sense that $\cup_{t<0}\Omega_t=\mathbb{R}^2$, where $\Omega_t$ is the convex body bounded by $\Gamma_t$) or else lie in a  strip/slab region (the region bounded by two parallel  lines) \cite[Corollary 2.1]{Wa11}. Wang also proved that the only entire examples are the shrinking circles~\cite[Theorem~1.1]{Wa11}, thus reducing the classification question among solutions that lie in a slab.

In this paper, we aim to construct ancient compact convex solutions to the $\kappa^\alpha$ flow, for all $\alpha\in (1/2, 1]$, that lie in strip regions. These solutions can be thought of as the analogue to the Angenent ovals for the curve shortening flow.
Moreover, we prove that for $\alpha\in (\frac23, 1)$ the  Daskalopoulos, Hamilton and \v Se\v sum classification result extends to the $\kappa^\alpha$ flow. Finally, we show that any ancient non-compact convex solution to the $\kappa^\alpha$ flow, for all $\alpha\in (1/2, 1]$, must be a translating solution, therefore it is unique modulo rigid motions and parabolic rescalings by \cite{Urbas98}. We remark that this in fact is a corollary of the more involved result \cite[Theorem 1.1]{CCD18}.

\begin{theorem}\label{main thm} For any $\alpha\in (1/2, 1]$ there exists a convex compact ancient solution to the $\kappa^\alpha$ flow that lies between two parallel lines.
\end{theorem}

\begin{theorem}\label{thm2}
For $\alpha\in(\frac23,1]$, any ancient compact convex solution to the $\kappa^\alpha$ flow must be the solution constructed in Theorem \ref{main thm} or the shrinking circle, modulo rigid motions and parabolic rescalings.

For $\alpha\in (1/2, 1]$ any ancient non-compact convex solutions to the $\kappa^\alpha$ flow must be a translating solution.
\end{theorem}

Using the classification of translators by Urbas \cite{Urbas98}, we obtain the following corollary, which combined with the first part of Theorem \ref{thm2} provides a complete classification of ancient convex solutions to the $\kappa^\alpha$ flow for $\alpha\in (\frac23, 1]$.

\begin{corollary}
For $\alpha\in(1/2,1]$, modulo rigid motions and parabolic rescalings, Urbas' translating solution over a strip \cite{Urbas98} and the straight line are the unique non-compact convex ancient solutions to the $\kappa^\alpha$ flow.
\end{corollary}

The dichotomy theorem of X.J. Wang has been extended to the $\kappa^\alpha$ flow, $\alpha\in (1/2, 1]$, by S. Chen \cite{Chen15}, who also showed that if the solution sweeps the whole $\mathbb{R}^2$ then it must be the shrinking circle. We use this result to reduce our analysis to solutions that lie in a slab.

For $\alpha\in (1/2, 1)$, it does not seem possible to find an explicit solution as the Angenent oval. But for this range of $\alpha$'s,  it is known that, apart from the stationary line, and modulo rigid motions and parabolic translations, there is a unique translating solution to the $\kappa^\alpha$ flow and, in fact, this solution lies in a slab \cite{Urbas98}. This translating solution is essential in our construction of the solution described in Theorem \ref{main thm}

Our proof follows the ideas in \cite{BLT3}. We construct an ancient solution that lies in a strip by doubling (via reflection) compact pieces of the translating solution flowing them by the $\kappa^\alpha$ flow and taking a limit as the compact pieces become larger and larger. By analyzing the asymptotics of this solution, we are able to use Alexandrov reflection principle to show first that any such solution is reflection symmetric with respect to the mid-plane of the slab and second, for $\alpha\in (\frac23,1]$, that the solution is unique. Many of the techniques used in the construction as well as in the proof of uniqueness of this solution, have been used in \cite{BLT1, BLT3}. A major difficulty encountered for $\alpha<1$ is that the derivative of the enclosed area is no longer constant, which was a crucial ingredient in the previous works.

The fact that we can prove uniqueness only for $\alpha\in (\frac23,1]$ comes from the better asymptotics we get in this range, which is due to the following reason.
Urbas' unique convex translating solution is a graph over some bounded interval $I$, which, after a rotation, can be taken to lie on the $x$-axis. If we consider a timeslice that is contained in $I\times (0, +\infty)$ and let $R$ be the non-convex region of $I\times(0, +\infty)$ delimited by this timeslice, then the area of $R$ is finite if an only if $\alpha\in (\frac23, 1]$.  This area estimate allows us to obtain good enough asymptotics of the constructed solution in order to apply Alexandrov reflection and the maximum principle to prove uniqueness. We remark here that the same issue appears in the recent work of B. Choi, K. Choi and Daskalopoulos \cite{CCD20}, where they construct ancient solutions to the Gauss curvature flow under an initial finiteness of volume assumption. \\

The paper is structured as follows: In Section \ref{sec:AncientSol}, we gather facts about convex ancient solutions to the $\kappa^\alpha$ flow; In Section \ref{sec:Construction}, we construct families of old but not ancient solutions and show they satisfy estimates similar to the ones in Section \ref{sec:AncientSol} so that a subsequence of them converges to an ancient solution; Finally, in Section \ref{sec:Uniqueness}, we prove the  classification Theorem \ref{thm2}.

\subsection*{Acknowledgements} This research originated at  the  workshop ``Women in Geometry 2" at the  Casa Matem\'atica Oaxaca (CMO)  from June 23 to June 28, 2019.  We would like to thank CMO-BIRS for creating the opportunity to start work on this problem through their support of the workshop. Theodora Bourni thanks Mat Langford and Toti Daskalopoulos for useful discussions on the subject.

\section{Ancient solutions of the $\kappa^\alpha$-flow}
\label{sec:AncientSol}

Let  $\{\Gamma_t\}_{t\in (-\infty, 0)}\subset \mathbb{R}^2$ be a convex ancient solution to the $\kappa^\alpha$-flow and consider its parametrization by its turning angle
\[
\gamma:\Theta\times(-\infty,0)\to \mathbb{R}^2, \quad \Theta\subset[0, 2\pi)\,.
\]
The turning angle $\theta$ of the solution is the angle made by the $x$-axis and its tangent vector with respect to a counterclockwise parametrization.
Using spacetime translation and a space rotation if necessary, we assume that $\lim_{t\to 0}\Gamma_t\subset \{y\ge 0\}$ and that, at time $0$, the solution vanishes in the compact case whereas in the non-compact case it passes through the origin.

Let  $\nu=\nu(\theta, t)$ and $\kappa(\theta, t)$ be the outward pointing unit normal and curvature of $\Gamma_t$ at $\gamma(\theta, t)$, respectively. Then
\[
\nu(\theta, t)=(\sin\theta, -\cos\theta)\,,
\]
and the evolution of $\kappa$ is given by
\begin{equation}\label{kt}
\kappa_t=\kappa^2(\kappa^\alpha)_{\theta\th}+\kappa^{\alpha+2}\,.
\end{equation}

A very useful feature of the $\kappa^\alpha$ flow is that it satisfies a differential Harnack inequality, a consequence of which is that the curvature on an ancient solution is non decreasing \cite{BA94, Chow91}:
\begin{equation}\label{Har}
\kappa_t(\theta, t)\ge 0\,.
\end{equation}
This holds for non-compact solutions as well, since the curvature is bounded at all timeslices. Moreover, it is know that the inequality is strict unless the solution moves by translation. As a corollary we obtain that the ``ends'' of an ancient solution are translators in the following sense. For any $\theta\in \Theta$ and any sequence of times $t_i\to -\infty$ the sequence of flows
$\Gamma_t^i=\Gamma_{t+t_i}-\gamma(\theta, t_i)$
converges, after passing to a subsequence,  locally uniformly in the smooth topology to a convex translating solution to the $\kappa^\alpha$ flow. For a proof of this standard compactness argument see for instance \cite[Lemma 5.2]{BLT1}. In fact, the curvature $\kappa^T(\theta, t)$ of the limiting translator satisfies
\[
\kappa^T(\theta, t)=\lim_{j\to \infty}\kappa(\theta, t_{i_j}).
\]

\subsection*{The convex translating solution}
According to the theorem of Urbas stated below \cite{Urbas98} (see also \cite[Proposition 2.4]{CCD18}),  modulo translations, there is a unique convex translating solution of the $\kappa^\alpha$-flow moving with speed $1$ along the $e_2$ direction, for $\alpha\in(1/2,1]$, other than the straight line. We thus have that the limit $\Gamma^s_t=\Gamma_{t+s}-\gamma(\theta, s)$, as $s\to -\infty$, exists.

\begin{theorem}\label{Urbas}\cite{Urbas98}
Modulo translations, there exists a unique strictly convex curve $G_0$ that satisfies
\[
\kappa^\alpha=-\langle \nu, e_2\rangle\,,\,\,\alpha\in (1/2,1]\,,
\]
where $\kappa$ and $\nu$ are the curvature and downward pointing unit normal to $G_0$. Moreover $G_0$ is a graph over a strip of width
\[
w_\alpha:=\int_{-\infty}^{+\infty}\frac{1}{(1+y^2)^{\frac12(3-\frac1\alpha)}}\, dy \,.
\]
Modulo translations, $G_t= G_0 +te_2$ is then the unique convex translating solution that moves with speed 1 along the $e_2$ direction.
\end{theorem}

\subsection*{Estimates for compact convex ancient solutions}
From now on, $\alpha\in(1/2,1)$ and $\{\Gamma_t\}_{t \in (-\infty, 0)}$ will denote a compact convex ancient solution to the $\kappa^\alpha$ flow which lies in the strip $\left[-\frac{w_\alpha}{2}\frac{w_\alpha}{2}\right]\times\mathbb{R}$ and in no smaller strip, where $w_\alpha$ is as in Theorem \ref{Urbas}. Using the parametrization by turning angle, we have
\[
\gamma:(-\pi, \pi]\times (-\infty, 0)\to \left(-\frac{w_\alpha}{2}, \frac{w_\alpha}{2}\right)\times\mathbb{R}\,.
\]
Since the solution lies in no smaller strip, it is not difficult to see that $\cup_{t\in (-\infty, 0)} \Gamma_t=(-\frac{w_\alpha}{2}, \frac{w_\alpha}{2})\times \mathbb{R}$, see for instance \cite[Lemma~5.1]{BLT1}.
We will use the differential Harnack inequality, to show bounds on the curvature and width of our convex ancient solutions.

\begin{proposition}\label{lowerkbbound}
For any $\theta\in(-\pi, \pi]$ and $t\in (-\infty, 0)$
\[
\kappa^\alpha(\theta, t)\ge \pm \langle \nu(\theta, t), e_2\rangle.
\]
\end{proposition}
\begin{proof}
We note first that the function $\kappa^\alpha\mp\langle\nu, e_2\rangle$ satisfies the Jacobi equation for the $\kappa^\alpha$ flow:
\begin{equation}\label{Jaceqn}
u_t=\alpha\kappa^{\alpha+1}(u_{\theta\th}+u)\,.
\end{equation}
The inequality is trivially true for $\theta=\pm \frac{\pi}{2}$. Recall that the flows $\Gamma^s_t=\Gamma_{t+s}-\gamma(\theta, s)$ converge as $s\to -\infty$ to a convex translator which lies in a strip of width $w_\alpha$, which by Theorem \ref{Urbas}, satisfies $\kappa^\alpha\mp\langle\nu, e_2\rangle\ge 0$.
The result then follows by the maximum principle.
\end{proof}

\begin{proposition}\label{X-est}
For any $t\in (-\infty, 0)$ and  any $ \theta\in \pm[0, \pi]$

\[
\pm \left(\langle\gamma(\pm \tfrac\pi2, t), e_1\rangle-\langle\gamma(\theta, t),e_1\rangle\right)\le \pm \int_\theta^{\pm\frac\pi2}\frac{\cos u}{|\cos u|^\frac{1}{\alpha}}\,du \,.
\]

\end{proposition}
\begin{proof} Recall that a convex curve $\gamma(\theta)$ with curvature $\kappa(\theta)$
satisfies
\begin{equation}
\label{eq:DiffPosition}
  \gamma(\theta_1) - \gamma(\theta_0)=\left(\int_{\theta_0}^{\theta_1}\frac{\cos u}{\kappa(u)}du\,,\,\,\int_{\theta_0}^{\theta_1}\frac{\sin u}{\kappa(u)}du\right)\,.
\end{equation}
For any $\theta\in[0, \frac{\pi}{2}]\cap \Theta$ we compute, using Proposition \ref{lowerkbbound} and the fact that $-\langle\nu,e_2\rangle=\cos \theta$,
\[
\langle\gamma(\tfrac\pi2, t), e_1\rangle-\langle\gamma(\theta, t),e_1\rangle=\int_\theta^{\pi/2}\frac{\cos u}{\kappa(u,t)}du\le \int_\theta^{\pi/2}\frac{\cos u}{|\cos u|^{\frac1\alpha}}\,du\,.
\]
The other cases are proved similarly.
\end{proof}
As a consequence of this width estimate and our condition that the ancient solution $\Gamma_{t}$ lies in no strip smaller than $\left[-\frac{w_\alpha}{2}, \frac{w_\alpha}{2}\right]\times\mathbb{R}$,  we can determine scale of the limiting translating solution:
\begin{proposition}\label{edges}
For any $\theta\in (-\pi, \pi]\setminus\{\frac\pi2, -\frac{\pi}{2}\}$, the sequence of flows
$\Gamma^s_t=\Gamma_{t+s}-\gamma(\theta, s)$
converges to a translating solution, which after a time and space translation, as well as a reflection about the $x$-axis in case $\theta\in (-\pi, \pi]\setminus(-\tfrac{\pi}{2}, \tfrac{\pi}{2})$, is given by $\{G_t\}_{t\in(-\infty, \infty)}$, as defined in Theorem~\ref{Urbas}.
\end{proposition}
\begin{proof} Given $\theta\in (-\pi, \pi]\setminus\{-\frac\pi2, \frac\pi2\}$, we have already seen that the sequence of flows
$\Gamma^s_t=\Gamma_{t+s}-\gamma(\theta, s)$, as $s\to -\infty$, converges to a translator, which by Theorem \ref{Urbas}, and after a time and space translation and possibly a reflection about the $x$-axis, is given by $\{\lambda G_t\}_{t\in (-\infty, \infty)}$, with the scale satisfying $\lambda\le 1$ since the translator must be contained in a slab of width $w_\alpha$ (and it might be contained in a smaller one). We want to show here that the scale satisfies $\lambda =1$.

Let us assume that $\theta\in[0, \tfrac\pi2)$ (the other cases being treated similarly). Since the solution $\{\Gamma_t\}_{t\in (-\infty, 0)}$ is contained in a slab of width $w_\alpha$ and in no smaller slab, for any $\epsilon>0$ there exists $t_\epsilon<0$ such that
\begin{equation}\label{widthest}
 \tfrac{w_\alpha}{2}-\langle\gamma(\tfrac\pi2, t) , e_1\rangle \le \epsilon \,,\,\,\forall t\le t_\epsilon\,.
\end{equation}
Note that, with the change of variable $y = \tan \theta$,
\begin{equation}\label{wath}
\frac{w_\alpha}{2}=\int_{0}^{+\infty}\frac{1}{(1+y^2)^{\frac12(3-\frac1\alpha)}}\,dy =\int_0^\frac\pi2\cos\theta^{1-\frac1\alpha}d\theta\,,
\end{equation}
and thus we can choose $\theta\in (0, \tfrac\pi2)$ so that$\int_\theta^\frac\pi2\cos\theta^{1-\frac1\alpha}d\theta<\epsilon$. Then, by Proposition \ref{X-est} and \eqref{widthest}
\[
\frac{w_\alpha}{2}- \langle\gamma(\theta, t) , e_1\rangle\le  \langle\gamma(\tfrac{\pi}{2}, t) , e_1\rangle- \langle\gamma(\theta, t) , e_1\rangle+\epsilon\le 2\epsilon\,.
\]
Hence the limit translator $\{\lambda G_t\}_{t\in (-\infty, \infty)}$ cannot be contained in a slab smaller that $w_\alpha-4\epsilon$, and since $\epsilon$ was arbitrary we have $\lambda=1$ which yields the result.
\end{proof}

\begin{proposition}\label{area} Let $A(t)$ be the area of the region enclosed by $\Gamma_t$. Then
\[
A(t)\le 2w_\alpha(-t)\,.
\]
\end{proposition}
\begin{proof}
 Using Proposition \ref{lowerkbbound} and the fact that $\langle \nu, e_2\rangle=-\cos\theta$, we have
\[
\begin{split}
-\frac{dA(t)}{dt}&=\int_{-\pi}^{\pi} \kappa^{\alpha-1}(\theta)\,d\theta\le4\int_0^\frac\pi2\cos\theta^{1-\frac1\alpha}d\theta\,.
\end{split}
\]
Hence, recalling \eqref{wath}, we have
\[
\begin{split}
-\frac{dA(t)}{dt}&\le 2w_\alpha\\
\end{split}
\]
and  integrating from $t$ to $0$,  yields the result.
\end{proof}

\section{Constructing a compact ancient solution}
\label{sec:Construction}

We will use the translating solution $\{G_t\}_{t\in(-\infty, \infty)}$, as described in Theorem \ref{Urbas}, to construct a compact ancient solution. By the uniqueness of $G_0$,  after a space and time translation,  we can assume that $G_0$ passes through the origin and is reflection symmetric with respect to the $y$-axis, so $G_0\subset \{y\ge 0\}$. Let $\Gamma^R$ be the convex curve we obtain after taking the union of $G_{-R}\cap \{y\le0\}$ and its reflection along the $x$-axis, see Figure \ref{Figure}. This curve is not smooth, but it is convex and thus, a solution to the $\kappa^\alpha$ flow  ``flowing out'' of this curve exists, in the sense of the following theorem of Andrews.

\begin{theorem} \label{convexsln}\cite{BA98}
For any convex curve $\Gamma_0$ bounding an open convex region in $\mathbb{R}^2$,
there exists a family of embeddings $\gamma: S^1\times (0, T)$ unique up to time-independent
reparametrization, which satisfy equation \eqref{ka-flow} and such that the image curves $\Gamma_t$
 converge to $\Gamma_0$ in Hausdorff distance as $t\to 0$. Moreover $\gamma\in C^\infty(S^1\times(0,T))$ and $\Gamma_t$ is strictly convex for all $t>0$.
\end{theorem}

\begin{figure}[htbp]
\begin{tikzpicture}[line cap=round,line join=round, scale=1.3]
  \pgfmathsetmacro{\alpha}{1.50};
  \pgfmathsetmacro{\am}{-1*\alpha};
  \pgfmathsetmacro{\p}{3.1416/2};
  \draw [->] (-2, 0) -- (2, 0) node[right] {$x$};

    \draw [line width=2pt,smooth,variable=\xi,green] plot [samples=100, domain={\am-.06}:{\alpha+.06}] (\xi,{ln(cos(\xi r))-ln(cos((\alpha+.06) r))});

  \draw [line width=2pt, domain={\am+.09}:{\alpha-.09},smooth,variable=\xi,teal] plot (\xi,{ln(cos(\xi r))-ln(cos(\alpha r))});
  \draw [line width=2pt, smooth, teal] ({\am+.09}, {ln(cos((\am + .09) r))-ln(cos(\alpha r))}) .. controls (\am+.03,0.46) and (\am+.03, 0.3) .. (\am+.03, 0);
  \draw [line width=2pt, smooth, teal] ({\alpha-.09}, {ln(cos((\am + .09) r))-ln(cos(\alpha r))}) .. controls (\alpha-.03,0.46) and (\alpha-.03, 0.3).. (\alpha-.03, 0);

  \begin{scope}[yscale=-1,xscale=1]

    \draw [line width=2pt,smooth,variable=\xi,green] plot [samples=50, domain={\am-.06}:{\alpha+.06}] (\xi,{ln(cos(\xi r))-ln(cos((\alpha+.06) r))});

  \draw [line width=2pt, domain={\am+.09}:{\alpha-.09},smooth,variable=\xi,teal] plot (\xi,{ln(cos(\xi r))-ln(cos(\alpha r))});
  \draw [line width=2pt, smooth, teal] ({\am+.09}, {ln(cos((\am + .09) r))-ln(cos(\alpha r))}) .. controls (\am+.03,0.46) and (\am+.03, 0.3) .. (\am+.03, 0);
  \draw [line width=2pt, smooth, teal] ({\alpha-.09}, {ln(cos((\am + .09) r))-ln(cos(\alpha r))}) .. controls (\alpha-.03,0.46) and (\alpha-.03, 0.3).. (\alpha-.03, 0);
    \end{scope}

            \draw [line width=1.5pt ,smooth,variable=\xi,loosely dashed, black] plot [samples=100, domain={-1*(\p-.00005)}:{.01}] (\xi,{-ln(cos(\xi r))+ln(cos((\alpha+.06) r))});
        \begin{scope}[yscale=1,xscale=-1]
        \draw [line width=1.5pt ,smooth,variable=\xi,loosely dashed, black] plot [samples=100, domain={-1*(\p-.00005)}:{.01}] (\xi,{-ln(cos(\xi r))+ln(cos((\alpha+.06) r))});
        \end{scope}

       \draw[{Stealth}-{Stealth}, line width=1pt] (-\p,5.5) -- (\p, 5.5) node [pos=.5, above] {$w_{\alpha}$};

  \draw[color=green] (-.7,3.7) node {$\Gamma^R$};
  \draw[color=teal] (-0.7,-1.6) node { $\Gamma^R_t$};
    \draw[color=black] (1.2,4.5) node {$G_{-R}$};
  \draw[{Stealth}-{Stealth}, line width=1pt] (0,0) -- (\alpha-.03, 0) node [pos=.5, above] {$h_R(t)$};
  \draw[{Stealth}-{Stealth}, line width=1pt] (0,0) -- (0, {-ln(cos(\alpha r))}) node [pos=.5, fill=white] {$\ell_R(t)$};
  \draw[{Stealth}-{Stealth}, line width=1pt] (0,0) -- (0, -{-ln(cos((\alpha+.06) r))}) node [pos=.7, fill=white] {$R$};

  \draw[color=black] (2.5, 4) node[anchor=west] {$\Gamma^R = \Gamma^R_{T_R}$ is the initial curve};
  \draw[color=black] (2.5, 3.5) node[anchor=west] {$\Gamma^R_t$ is $t$-slice of the solution};
  \draw[color=black] (2.5, 3.0) node[anchor=west] {$\ell_R(t) = \sup_\theta\langle\gamma^R(\theta, t), e_2\rangle$};
  \draw[color=black] (2.5, 2.5) node[anchor=west] {$h_R(t) =  \sup_\theta\langle\gamma^R(\theta, t), e_1\rangle$};
  \draw[color=black] (2.5, 2.0) node[anchor=west] {$\mathcal{A}^R$ is the area enclosed by $\Gamma^R$};
  \draw[color=black] (2.5, 1.5) node[anchor=west] {$A_R(t)$ is the area enclosed by $\Gamma^R(t)$};

\end{tikzpicture}
\caption{The approximate solutions $\{\Gamma^R_t\}_{t\in (T_R, 0)}$.}
\label{Figure}
\end{figure}

Recall that the solution shrinks to a point in finite time. We can therefore translate time so that the extinction time is $t=0$ for our solutions. With this convention in mind, we define $\{\Gamma^R_t\}_{t\in (T_R, 0)}$ with $T_R \in (-\infty, 0)$ to be a solution to \eqref{ka-flow} with initial data given by $\Gamma^R$ in the sense of Theorem \ref{convexsln} and let $\gamma^R:S^1\times (T_R, 0)\to \mathbb{R}^2$ be a parametrization  of $\{\Gamma^R_t\}_{t\in (T_R, 0)}$ by turning angle, see Figure \ref{Figure}. Moreover, by the uniqueness of the solution given in Theorem \ref{convexsln}, this solution is reflection symmetric with respect to both coordinate axes, which furthermore implies that the extinction point is the origin.

We will construct an ancient solution  by taking a limit of such solutions as $R\to \infty$. For this, we will first prove that the curvature of and area enclosed by the solutions $\{\Gamma^R_t\}_{t\in (T_R, 0)}$ satisfy bounds similar to the estimates given for convex ancient solutions in Section \ref{sec:AncientSol}.
\begin{proposition}\label{lowerkbboundR}
For any $R>0$, $t\in (T_R,0)$ and $\theta\in S^1$,
\begin{equation*}\label{lowerkbboundR1}
\kappa^\alpha(\theta,t)\ge \pm \langle\nu(\theta,t), e_2\rangle\, .
\end{equation*}
\end{proposition}
\begin{proof} Because of Theorem \ref{convexsln}, the proof follows as in Proposition \ref{lowerkbbound}. Indeed, since
\[
\liminf_{t\to T_R}(\kappa^\alpha(\cdot, t)\pm \langle\nu(\cdot,t), e_2\rangle)\ge 0
\]
and $\kappa^\alpha\mp\langle\nu, e_2\rangle$ satisfies \eqref{Jaceqn}, the result follows by the maximum principle.
\end{proof}
Next we show a certain monotonicity of the curvature throughout the flows. In particular, the result tells us that the maximum of $\kappa^{\alpha}$ occurs at $\theta=0$, so that we just need a bound on $\kappa(0, t)$ in order to get a uniform bound on the curvature $\kappa(\theta, t)$ for any $\theta$.
\begin{proposition}\label{k-monotone-t}
For any $R>0$ and $t\in (T_R,0)$, $\Gamma^R_t$ satisfies
\begin{equation}\label{k-monotone}
\begin{cases}
(\kappa^\alpha)_\theta\le 0\,,\text{ for }\theta\in (0, \pi/2)\cup(\pi, 3\pi/2)\\
(\kappa^\alpha)_\theta\ge 0\,,\text{ for }\theta\in (\pi/2, \pi)\cup(3\pi/2, 2\pi)\,.
\end{cases}
\end{equation}
\end{proposition}
\begin{proof}
Note first that on $\Gamma^R$ \eqref{k-monotone} is true by convexity and the translator equation.

Let $v=(\kappa^\alpha)_\theta$. Then, recalling \eqref{kt}, we have
\[
\begin{split}
v_t=(\alpha+1)\kappa v ((\kappa^\alpha)_{\theta\th} + \kappa^\alpha)+\alpha\kappa^{\alpha+1}(v_{\theta\th}+v)\,.
\end{split}
\]
The strong maximum principle and the Hopf lemma then, applied to each of the four $\theta$- intervals implies the result.
\end{proof}
Next we prove some displacement and area estimates. For this we define
\[
\ell_R(t)=-\langle \gamma^R(0, t), e_2\rangle=\langle\gamma^R(\pi, t), e_2\rangle=\max_{\theta\in [0, 2\pi)}\langle\gamma^R(\theta, t), e_2\rangle\,,
\]
\[
h_R(t)=\langle\gamma^R(\tfrac{\pi}{2}, t), e_1\rangle=-\langle\gamma^R(\tfrac{3\pi}{2}, t), e_1\rangle=\max_{\theta\in[0, 2\pi)}\langle\gamma^R(\theta, t), e_1\rangle\,,
\]
and $A_R(t)$ to be the area enclosed by $\Gamma^R_t$, see Figure \ref{Figure}.

Note that, by the construction of the initial surface, we have
$\ell_R(T_R)=R$ and $\lim_{R\to \infty} h_R(T_R)=\tfrac{w_{\alpha}}{2}$. We first show some more precise estimates on $h_R(T_R)$ as well as $A_R(T_R)$.

\begin{proposition}\label{initial data} There exist constants $R_\alpha>0$ and $C_\alpha>0$ such that for all $R\ge R_\alpha$
\begin{itemize}
\item[(i)] $h_R(T_R) \ge \frac{w_\alpha}{2}- C_\alpha R^\frac{1-2\alpha}{1-\alpha}$\,,
\item[(ii)] $A_R(T_R)\ge 2w_\alpha R- C_\alpha R^\frac{2-3\alpha}{1-\alpha}$\,.
\end{itemize}
\end{proposition}
\begin{proof}
For any $\theta_0\in(0, \tfrac{\pi}{2})$, using \eqref{wath}, we have
\begin{equation}\label{th0est}
\begin{split}
\int_{-\theta_0}^{\theta_0}\cos\theta^{1-\frac1\alpha}d\theta&=w_\alpha-2\int_{\tan\theta_0}^{+\infty}\frac{1}{(1+y^2)^{\frac12(3-\frac1\alpha)}}\,dy   \\
& \geq w_{\alpha} - 2\int_{\tan\theta_0}^{+\infty}y^{-3+\frac1\alpha}\,dy = w_\alpha-\frac{2\alpha}{2\alpha-1}(\tan\theta_0)^{\frac1\alpha-2}\,.
\end{split}
\end{equation}
Let $\pm \theta_R$ be the turning angles at the two points of intersection of $\Gamma_R$ with the $x$-axis. Then by equation \eqref{eq:DiffPosition},
 \begin{equation}\label{thr}
 R=\int_0^{\theta_R}\frac{\sin u}{\kappa(u,0)} du= \int_0^{\theta_R}\frac{\sin u}{\cos u^\frac{1}{\alpha}} du=\frac{\alpha}{1-\alpha}(\cos\theta_R^{1-\frac{1}{\alpha}}-1)\,.
 \end{equation}
 Therefore
 \[
 \tan^2\theta_R=\cos\theta_R^{-2}-1=\left(\frac{1-\alpha}{\alpha} R+1\right)^\frac{2\alpha}{1-\alpha}-1\ge \left(\frac{1-\alpha}{2\alpha} R\right)^\frac{2\alpha}{1-\alpha}\,,
  \]
with the last inequality being true for all $R\ge R_\alpha$ for a constant $R_\alpha>0$.
 Now working as in \eqref{thr}, and using \eqref{th0est}, we obtain for $h_R(T_R)$
 \[
 \begin{split}
 h_R(T_R)=\int_0^{\theta_R}\cos u^{1-\frac1\alpha} du&\ge \frac{w_\alpha}{2}-\frac{\alpha}{2\alpha-1}(\tan\theta_R)^\frac{1-2\alpha}{\alpha}\\
 &\ge \frac{w_\alpha}{2}-C_\alpha R^\frac{1-2\alpha}{1-\alpha}\,,

 \end{split}
 \]
 for some constant $C_\alpha$ and for all $R\ge R_\alpha$.

For the area estimate, we let $\mathcal{A}^R= A_R(T_R)$ be the area enclosed by $\Gamma^R$, so thanks to the symmetry with respect to the $x$-axis, $\frac12 \mathcal{A}^R$ is the area enclosed by $G_{-R}$ and the $x$-axis. Then,  following the proof of Proposition \ref{area} and estimating as above, we obtain
  \[
\frac12 \frac{d\mathcal{A}^R}{dR}=\int_{-\theta_R}^{\theta_R}\kappa^{\alpha-1}(\theta)\,d\theta= \int_{-\theta_R}^{\theta_R}\cos\theta^{1-\frac1\alpha}d\theta\ge w_\alpha-2 C_\alpha R^\frac{1-2\alpha}{1-\alpha}\,,
 \]
 for all $R\ge R_\alpha$.
Integrating from $R_\alpha$ to $R$ we obtain (ii).
\end{proof}

\begin{proposition}\label{t-Restimates} There exist constants $C_\alpha$ and $R_\alpha$ such that for all $R>R_\alpha$ and all $t\in (T_R,0)$ the following estimates hold.
\begin{itemize}
\item[(i)]  $-2w_a t\ge A_R(t)\ge -2w_\alpha t- C_\alpha(-t)^{2-2\alpha}$\,,
\item[(ii)] $R\ge -T_R\ge R- C_\alpha(1+ R^\frac{2-3\alpha}{1-\alpha})$\,,
\item[(iii)]$\frac{w_\alpha}{2}\ge h_R(t)\ge \frac{w_\alpha}{2}- C_\alpha(-t)^{1-2\alpha}$\,,
\item[(iv)] $-t\le \ell_R(t)\le \min\{-t + C_\alpha\left(1+R^{\frac{2-3\alpha}{1-\alpha}}\right), -C_\alpha t\}$\,,
\item[(v)]$1\le \kappa_R^\alpha(0, t)\le C_\alpha\left(1+\frac{1}{-t}\right)$\,.
\end{itemize}
\end{proposition}
\begin{proof}
The proof of the first inequality in (i) follows the corresponding proof of Proposition \ref{area} for ancient solutions, using here  Proposition \ref{lowerkbboundR} instead of Proposition \ref{lowerkbbound}.

By Proposition \ref{lowerkbboundR}, we have
\[
-\ell_R'(t)=\kappa^\alpha_R(0, t)\ge -\langle \nu(0, t), e_2\rangle=1\,.
\]
Integrating from $t$ to $0$ yields the first inequality of (iv). As we shall use it later, we point out that integrating from $T_R$ to $t$ yields
\begin{equation}\label{elles}
\ell_R(t)\le R-(t-T_R)\,.
\end{equation}
Applying the first inequality of (iv) with $t=T_R$ yields the first inequality in (ii). The second inequality in (ii) is a consequence of the first inequality in (i) applied with $t=T_R$ and estimate (ii) of Proposition \ref{initial data}.

The first inequality in (iii)  follows from the fact that $\Gamma^R$ is in the strip $\left(-\frac{w_{\alpha}}{2}, \frac{w_{\alpha}}{2}\right) \times \mathbb{R}$. To prove the second inequality, we use a technique from \cite[Lemma 4.4]{BLT1}.  Consider the circle centered on the negative $x$-axis and passing through $\gamma^R(\tfrac\pi2,t)$ and $\gamma^R(0,t)$. By Proposition \ref{k-monotone-t} and a simple maximum principle argument, it must be tangent to the curve at $\gamma^R(\tfrac{\pi}{2},t)$ from the inside, see \cite[Lemma 4.4]{BLT1} for details on this argument. We therefore have
\[
\kappa_R(\tfrac{\pi}{2}, t)\le \frac{2h_R(t)}{\ell_R^2(t)+h_R^2(t)}\,.
\]
We use this to compute
\[
-h_R'(t)=\kappa_R^\alpha(\tfrac{\pi}{2}, t)\le \left(\frac{2h(t)}{\ell^2(t)+h^2(t)}\right)^\alpha\le  \frac{2^\alpha h^\alpha(t)}{(-t)^{2\alpha}}\,,
\]
where in the last inequality we used the first inequality of (iv). Hence
\[
-\frac{1}{1-\alpha}(h_R^{1-\alpha}(t))'\le \frac{2^\alpha}{2\alpha-1}((-t)^{1-2\alpha})'
\]
and integrating from $T_R$ to $t$ we obtain
\begin{equation*}\label{hest}
h_R^{1-\alpha}(t)\ge h_R^{1-\alpha}(T_R) +\frac{2^\alpha(1-\alpha)}{2\alpha-1}\left((-T_R)^{1-2\alpha}-(-t)^{1-2\alpha}\right)\,
\end{equation*}
and using estimate (i) of Proposition \ref{initial data} and the first inequality in (ii) here yields the result.

To prove the second inequality in (i), we compute using Proposition \ref{lowerkbboundR} and  \eqref{eq:DiffPosition}
\[
-\frac{1}{4}\frac{dA_R(t)}{dt}=\int_{0}^{\frac\pi2} \kappa^{\alpha -1}\, d\theta \ge \int_{0}^{\frac\pi2} - \frac{\langle \nu,  e_2 \rangle}{\kappa(\theta, t)} \, d\theta = \int_{0}^\frac\pi2 \frac{\cos \theta}{\kappa(\theta,t)} d\theta=h_R(t).
\]
Integrating from $t$ to $0$, and using the second inequality of (iii), yields the result.

To prove the second inequality in (iv), we first note that  \eqref{elles} along with the second inequality in (ii) here implies that
\[
\ell_R(t)\le -t + C_\alpha\left(1+R^{\frac{2-3\alpha}{1-\alpha}}\right)\,.
\]
Furthermore, the first inequality in (i) and the second in (ii), along with  convexity, yield the uniform estimate $\ell_R(t)\le -C_\alpha t$.

Finally, the first inequality in (v) is straightforward by Proposition~\ref{lowerkbboundR}. The upper bound on $\kappa$ is a result of the differential Harnack inequality \cite{Chow91}, according to which, the quantity
\[
\kappa(\theta, t)(-t)^{\frac{\alpha}{\alpha+1}}
\]
is non decreasing in $t$. We thus have
\[
\begin{split}
\ell_R(t)&=\int_t^0\kappa^\alpha(0, s)ds\ge \kappa^\alpha(0, t)(-t)^\frac{\alpha^2}{\alpha+1}\int_0^t (-s)^{-\frac{\alpha^2}{\alpha+1}} ds\\
&=\frac{\alpha+1}{\alpha+1-\alpha^2}\kappa^\alpha(0, t)(-t)\,.
\end{split}
\]
which, along with the uniform bound  $\ell_R(t)\le -C_\alpha t$,  yields the result.
\end{proof}

The following is a more precise version of Theorem \ref{main thm}.

\begin{theorem}\label{conv}
For any $\alpha\in (1/2,1]$, there exists a compact convex ancient solution to the $\kappa^\alpha$ flow, $\{\Gamma_t\}_{t\in (-\infty, 0)}$, that sweeps out $(-\tfrac {w_\alpha}{2}, \tfrac{w_\alpha}{2})\times\mathbb{R}$. This solution is symmetric with respect to both coordinate axes and satisfies
\begin{equation}\label{h-t}
h(t)\ge\frac{w_{\alpha}}{2}-\frac{2}{(-t)^{2\alpha-1}}\,,
\end{equation}
where $h(t)=\langle\gamma(\frac\pi2, t), e_1\rangle$.
Moreover, if $\alpha\in (\tfrac23,1)$ then
\begin{equation}\label{ell-t}
\ell(t)\le -t+O(1)\,,
\end{equation}
as $t\to-\infty$, where $\ell(t)=\langle\gamma(\pi, t), e_2\rangle$.
\end{theorem}
\begin{proof}
Note that, by Proposition \ref{t-Restimates}, the diameter and the curvature of the solutions $\{\Gamma^R_t\}_{t\in (T_R, 0)}$ are bounded uniformly in $R$, since by Proposition \ref{k-monotone-t} the maximum of the curvature is at $\theta=0$. Recall also that, by Proposition \ref{t-Restimates}, $T_{R}\to -\infty$ as $R\to \infty$. Therefore there exists a sequence $R_j\to \infty$ such that the sequence of flows $\{\Gamma^{R_j}_t\}_{t\in (T_{R_j}, 0)}$ converges locally uniformly to a family of compact convex curves $\{\Gamma_t\}_{t\in (-\infty, 0)}$. The uniform lower bound on $h_R(t)$ of Proposition \ref{t-Restimates} shows that this family sweeps out  $(-\tfrac {w_\alpha}{2}, \tfrac{w_\alpha}{2})\times\mathbb{R}$ and that it satisfies \eqref{h-t}. In case $\alpha\in(\tfrac23,1)$, the estimate \eqref{ell-t} is a consequence of the estimate on $\ell_R(t)$ in Proposition \ref{t-Restimates} (iv), since $\frac{2-3\alpha}{1-\alpha}<0$.

Finally, this family is a weak solution of the $\kappa^\alpha$ flow, in the sense that it satisfies the avoidance principle with respect to any other smooth solution. This is easy to see since it is a limit of smooth flows. Therefore, by Theorem \ref{convexsln}, it is indeed a smooth solution.
\end{proof}

\section {Classification of convex ancient solutions}
\label{sec:Uniqueness}
In this section, we will classify all convex ancient solution  $\{\Gamma_t\}_{t\in(-\infty, 0)}$ to the $\kappa^\alpha$ flow for $\alpha\in(\frac23,1]$ which are contained in the strip $\Pi:=\{(x,y)
:|x|<w_\alpha/2\}$ and in no smaller strip. Moreover we will show, for any $\alpha\in(1/2,1]$, that the only non-compact ones are translating solutions. The proof is very similar to the corresponding proof for the curve shortening flow case ($\alpha=1$) given in \cite[Theorem 3.3]{BLT3}.
We first show that any compact convex ancient solution that lies in a strip is reflection symmetric with respect to the mid-line of the strip, which is proven by using
the Alexandrov reflection principle \cite{Chow97,ChGu01}. The proof follows \cite[Lemma~2.4]{BLT3} (cf. \cite{BrLo}).

The non-compact case can be in fact dealt with in a much simpler way by looking at the asymptotic translators at $+\infty$ and $-\infty$. This idea follows the proof of \cite[Theorem 1.2]{CCD20}, by using here Proposition \ref{edges}. We present this result first.

\begin{theorem}\label{thm2II}
 Any convex, non-compact ancient solution to the $\kappa^\alpha$ flow, $\alpha\in (1/2,1]$, must be a translating solution.
\end{theorem}
\begin{proof}
Since the solution is convex, it is a graph over some interval $I$ (bounded or unbounded) which, without loss of generality, is an interval of the $x$-axis. Moreover, by \cite{CDK19}, this solution must in fact be eternal and we denote it by $\{\Gamma_t\}_{t\in (-\infty,+\infty)}$.  As we have already discussed, $\Gamma_t- \gamma(e_2,t)$ converges, as $t\to-\infty$, to a translator $T$. $T$ is either a straight line or it is defined between two parallel lines \cite{Urbas98}.  B. Choi, K. Choi and Daskalopoulos in \cite{CCD18} observed that, by the result in \cite{CDK19}, a convex complete graph over an open interval $I$
(either bounded or unbounded), under the flow, remains a convex complete graph over the same interval $I$. Hence, the solution is either entire or lies between two parallel lines.  $\Gamma_t- \gamma(e_2,t)$ converges, as $t\to+\infty$ to a translator $T_0$. In the case the solution is between two parallel lines, $T_0=T$ by \cite[Theorem 1.1]{CCD18}.  In case the solution is entire, then so is $T_0$ (which can be easily verified, for example by using translating solutions as barriers) and thus again $T=T_0$. Hence in both cases $T=T_0$ and by the rigidity case of the Harnack inequality we obtain the result.
\end{proof}

\begin{lemma}\label{lem:refl}
Let $\{\Gamma_t\}_{t\in(-\infty, 0)}$ be a strictly convex ancient solution to the $\kappa^\alpha$ flow, $\alpha\in(1/2,1]$, which is contained in the strip $\Pi:=\{(x,y)
:|x|<w_\alpha/2\}$ and in no smaller strip. Then $\Gamma_t$ is reflection symmetric about the $y$-axis for all $t<0$.
\end{lemma}
\begin{proof}
Set
\[
H_\beta:=\{(x,y)\in \mathbb{R}^2:x<\beta\
\]
and denote by $R_\beta$ the reflection about $\partial H_\beta$. By Proposition \ref{edges},  the `tips' (or tip if the solution is non compact) of this solution converge to the unique translator as defined in Theorem \ref{Urbas}, which is reflection symmetric. Therefore, by the convexity of $\{\Gamma_t\}_{t\in(-\infty,0)}$, given any $\beta\in (0, \frac{w_\alpha}{2})$, there exists a time $t_\beta$ such that
\[
(R_\beta\cdot\Gamma_t)\cap (\Gamma_t\cap H_\beta)=\emptyset
\]
for all $t<t_\beta$ (for more details of this fact, see \cite[Claim 6.2.1]{BLT1}). By the strong maximum principle and the boundary point lemma for strictly parabolic equations  this is true for all $t<0$, see \cite[Theorem 2.2]{Chow97}. Taking $\beta\to 0$ implies that $R_0\cdot \Gamma_t$ lies to the left of $\Gamma_t\cap H_0$. The result now follows by repeating the argument with $H_\beta:=\{(x,y)\in \mathbb{R}^2:x>-\beta\}$.
\end{proof}

\begin{theorem}\label{unique}
Modulo translations, there exists only one strictly convex ancient solution $\{\Gamma_t\}_{t\in(-\infty, 0)}$ to the $\kappa^\alpha$ flow $\alpha\in (\tfrac23, 1]$ contained in the strip $\Pi:=\{(x,y)
:|x|<\tfrac{w_\alpha}{2}\}$ and in no smaller strip.
\end{theorem}
\begin{proof}
After a time and space translation, we can assume that the solution gets extinct at the origin at time $t=0$. Let
\[
L(t)=-\langle \gamma(0, t), e_2\rangle+\langle \gamma(\pi, t), e_2\rangle\,.\]
By  Proposition \ref{lowerkbbound} we have $\kappa^\alpha(0, t)\ge 1$ and $\kappa^\alpha(\pi,t)\ge 1$ thus
\[
\frac{d}{dt}\left(L(t)+2t\right)\le 0\,.
\]
Therefore the limit
\[
L:=\lim_{t\to-\infty}(L(t)+2t)
\]
exists in $[0,\infty]$.

This quantity, for the particular compact solution we constructed in Theorem \ref{conv}, satisfies, by  \eqref{ell-t},
\[
L_0:=\lim_{t\to -\infty}(L_0(t)+2t)<\infty
\]
where $L_0(t)=2\ell(t)$, with $\ell(t)$ as in Theorem \ref{conv}.
We next claim that for any solution,
\begin{equation}\label{claim}
L=L_0\,.
\end{equation}
Suppose, contrary to the claim, that $L>L_0$ (the case $L<L_0$ is ruled out similarly). Then we can find $t_0$ such that
\begin{equation}\label{eq:ellt0}
L(t)>L_0(t) \,\,\text{ for all }t<t_0\,.
\end{equation}
Let $\{\mathrm{\Gamma^0}_t\}_{t\in(-\infty,0)}$ be the particular solution constructed in Theorem~\ref{conv}.
Define the halfspaces $H_\beta$ and the reflection $R_\beta$ about $\partial H_\beta$ as in Lemma \ref{lem:refl}. Given any $\beta\in(0,\frac{w_\alpha}{2})$, set
\[
\widetilde{\mathrm{\Gamma^0}}_t=(R_\beta\cdot \mathrm{\Gamma^0}_t)\cap \{(x,y)\in \mathbb{R}^2:x<0\}\,
\]
 and
\[
\widetilde \Gamma_t=\Gamma_t\cap \{(x,y)\in \mathbb{R}^2:x<0\}\,
\]
and let $(-r^0_t, r^0_t)=\partial\widetilde{\mathrm{\Gamma^0}}_t$ and $(r^-_t, r^+_t)=\partial\widetilde\Gamma_t$.
Then, by \eqref{eq:ellt0} and since $L_0$ is finite, there exists $c>0$ such that for all $t<t_0$, there exists $c_t\in (-c, c)$, such that $(-r^0_t+c_t, r^0_t+c_t)\subset (r^-_t, r^+_t)$. Moreover, by convexity and the convergence of the tips given in Proposition \ref{edges}, there exists $t_\beta<t_0$ depending on $\beta$ such that $(\widetilde{\mathrm{\Gamma^0}}_t+c_t)\cap\widetilde\Gamma_t=\emptyset$ for all $t<t_\beta$, with $c_t$ as above. It then follows by the strong maximum principle that $(\widetilde{\mathrm{\Gamma^0}}_s+c_t)\cap\widetilde\Gamma_s=\emptyset$ for all $t<t_\beta$ and $t<s<t_0$. Letting $\beta\searrow 0$, we find that there exists a constant $c_0$ such that  $\widetilde \Gamma_t\cap(\mathrm{\Gamma^0}_t+c_0)=\emptyset$ for all $t<t_0$. By Theprem \ref{lem:refl} we thus obtain that $\Gamma_t$ contains $\Gamma^0_t+c_0$ for all $t<t_0$, contradicting the fact that both curves reach the origin at time $t=0$. This finishes the proof of \eqref{claim}.

Now consider, for any $\tau>0$, the solution $\{\Gamma_t^\tau\}_{t\in(-\infty,0)}$ defined by $\Gamma_t^\tau=\Gamma_{t+\tau}$. Since
\[
L_\tau:=\lim_{t\to-\infty}(\ell(t+\tau)+t)> L= L_0\,,
\]
we may argue as above to conclude that $\Gamma_t^\tau$ lies outside $\mathrm{\Gamma^0}_t +c_\tau$ for some constant $c_\tau$ and for all $t<0$. Taking $\tau\to 0$, we find that $\Gamma_t$ lies outside $\mathrm{\Gamma^0}_t$ for all $t<0$. Since the two curves reach the origin at time zero, they intersect for all $t<0$ by the avoidance principle. The strong maximum principle then implies that the two coincide for all $t$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm2}]
The non-compact case has been shown in Theorem \ref{thm2II}. For the compact case and $\alpha\in (\frac23,1]$,
we have shown uniqueness as long as our solution is restricted in a slab. The theorem now follows by a result of Chen \cite{Chen15}, which states that if the solution is  not defined in a slab, then it must be the shrinking circle.
\end{proof}

\bibliographystyle{acm}

redpajama_set_name: RedPajamaArXiv


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text:
Why scientists get it wrong	June 1, 2010
Tags: climate change, Dr David Archibald, global warming, The Past and Future of Climate
Why scientists get it wrong
by David Archibald
Quadrant Online, June 1, 2010
Edited extract: "Why did so many scientists get it wrong?" from David Archibald's book – The Past and Future of Climate:
Click to order book
If the data and forecasts in this book are correct, then the Intergovernmental Panel on Climate Change (IPCC), the National Academy of Sciences in the United States, the Royal Society in the United Kingdom, the Bureau of Meteorology and the CSIRO in Australia are all wrong. How can this be? Firstly, there aren't that many scientists involved in the IPCC deliberations. The inner core is possibly twenty souls. Secondly, they were untroubled by the necessity to concoct fraudulent data to get their desired results. The only unknown question regarding the IPCC scientists is "Did they actually believe in the global warming that they were promoting?"
It turns out that they did, and possibly still do. That is shown by the Climategate emails released on 20th November, 2009. The Climategate emails are a selection of emails amongst members of the inner core plus minor characters. The fact that the IPCC scientists believed in the global warming they were promoting means that their morality at that level was better than expected, but it also means that they are a lot more stupid than expected. Nevertheless, their behaviour in promoting the notion of global warming using fraudulent statistics is reprehensible and hopefully they will be duly punished in this world or the next.
1.	Melissa - June 1, 2010
Gut instinct and groupthink peer review got us to where we are today. Back in the 80's and 90's every man and his dog would comment that things were changing. Back in their day they reminisced summers a certain way and flowers and trees blossomed earlier, later etc etc.
Another words climate is an emotive issue for most people that demand a near constant year on year for climate to be "normal". Couple that with the general modernization in the latter part of the twenty century and it all led to the collective gut instinct that climate was changing and theat human modernization was causing it, instinctive groupthink. Unfortunately, a group within the scientific community adapted the same priniciples as us ordinary lay people and that groupthink took hold like Japanese knotweed.
Science needs to fly by instrument, not instinct. Instinctive science can sometimes bring you to a correct scientific conclusion, more often than not it has created a lot of false hope and skewed science.
2.	Oliver K. Manuel - June 1, 2010
The sad fact is that our tax funds paid scientists to misrepresent and hide experimental data.
Solar scientists have used this technique for many years to promote the obsolete Standard Solar Model of the Sun.
NAS President Ralph Cicerone was confronted by eight (8) empirical facts in his latest warning about global warming on Physics World:
1. The Sun controls Earth's climate.
2. The Sun is a plasma diffuser that sorts atoms by mass.
3. The Sun is mostly Fe, O, Ni, Si, S, Mg and Ca beneath a veneer of lightweight H and He.
4. [These same elements are the main constituents of the Earth and ordinary meteorites.]
5. The Sun is the remains of the supernova that exploded 5 Gyr ago to make the solar system.
6. The Sun is heated by neutron repulsion (60%), neutron decay (5%), and fusion of the decay product (35%). H that escapes the fusion process is called SW Hydrogen
7. Solar neutrinos do not oscillate at the whim of politicians who control grant funds, but
8. The Climategate scandal revealed this trait in the modern consensus science community.
Oliver K. Manuel

redpajama_set_name: RedPajamaCommonCrawl


text:
https://sputniknews.com/20150527/1022609321.html
Medvedev: Russia Reviews Relations With EU, US Amid Cooperation With China
Russia is reviewing its relations with the European Union and the United States amid successful cooperation with China, Russian Prime Minister Dmitry Medvedev... 27.05.2015, Sputnik International
https://cdnn1.img.sputniknews.com/img/102187/26/1021872659_0:153:3080:1894_1920x0_80_0_0_46f0a17ce5e22a9d0cf58584bb2594cf.jpg
https://cdnn1.img.sputniknews.com/img/102187/26/1021872659_0:55:3080:1992_1920x0_80_0_0_f5e0f33aa961d2497dc6fb39c312d314.jpg
news, russia, us, china, dmitry medvedev, european union
© Sputnik / Michael Klimentyev / Go to the photo bankPresident Vladimir Putin meets with Chinese President Xi Jinping
Go to the photo bank
Russia is reviewing its relations with the European Union and the United States amid successful cooperation with China, Russian Prime Minister Dmitry Medvedev said Wednesday.
ST. PETERSBURG (Sputnik) – Moscow aims to develop ties with Beijing and implement the initiative on the connection between the Silk Road project and the Eurasian Economic Integration, Medvedev said during an international law forum.
"This is a positive experience, and it makes us to review how our relations with the countries of the European Union, the United States and a few other countries are developing, since last year had been rich in an international legal outlook on life under so-called sanctions."
Russia, China Air Military Influence to Exceed US in Future
26 May 2015, 13:54 GMT
In light of the West's sanctions against Russia, Moscow stepped up its cooperation with Beijing, including energy and defense sectors, with the two countries signing large energy deals and conducting joint military exercises.
The extension or lifting of Russia's responsive measures to the Western sanctions will be proportionate to the same decisions made by the West, the Russian prime minister said.
"I can only say that in the near future we will have to make a decision on either prolonging these responsive measures or not prolonging them. It's clear that our actions, and I can say this straightforward, will be equivalent and be based on the actions of our partners from other countries."
The Kremlin has repeatedly denied its involvement in the Ukrainian internal conflict and responded to Western restrictive measures with a ban on certain food imports from the countries that had sanctioned Russia. The list of banned products includes meat, poultry, fish, seafood, milk, dairy products and fruit and vegetables.


text:
Q: Local filesystem access in browser extensions I'm doing some investigation into options for an app I am building. Do browser extensions for all browser platforms have filesystem access like Firefox does?

A: Chrome extensions do not allow arbitrary filesystem access. You could, however, write an NPAPI plugin, but this comes with some heavy caveats:
You can also use HTML5 filesystem APIs to access a sandboxed filesystem:

A: Dolphin browser looks to have it. Look at their API (I didn't test it yet).
Hopefully it will help you.

redpajama_set_name: RedPajamaStackExchange


Thirty Seconds Over Tokyo is a 1944 MGM war film. It is based on the true story of the Doolittle Raid, America's first retaliatory air strike against Japan four months after the December 1941 Japanese attack on Pearl Harbor.
Mervyn LeRoy directed Thirty Seconds Over Tokyo and Sam Zimbalist produced the film. The screenplay by Dalton Trumbo was based on the 1943 book of the same name, written by Captain Ted W. Lawson, a pilot who participated in the raid. In both the book and the film, Lawson gives an eyewitness account of the training, the mission, and the aftermath as experienced by his crew and others who flew the mission on April 18, 1942. Lawson piloted "The Ruptured Duck", the seventh of 16 B-25s to take off from the aircraft carrier USS Hornet.
This webpage uses material from the Wikipedia article "Thirty_Seconds_Over_Tokyo" and is licensed under the GNU Free Documentation License. Reality TV World is not responsible for any errors or omissions the Wikipedia article may contain.


This week, the Somethings had the pleasure to hang out with Kevin Gootee, a stand-up who is doing something pretty original on Amazon Prime.
Every episode features one or two stand-ups doing a set while Kevin and his panel talk about what they did right and what they did wrong. From constructive criticism to out and out wrecking, for a fan of comedy, it is a unique take on the artform.
If you haven't seen it, the new season is going to be appearing any minute now… but until then, catch up with Seasons 1 – 4 and then go follow Kevin Gootee (@KevinGootee) or go checkout the site at Comics Watching Comics.
Last but not least – if you are a stand-up comic who wants to give it a shot.. first, go watch the show. Trust us, know what you are walking into. Then, submit your information to comicswatchingcomics@gmail.com – you must have been a working comic for the last three years, but if you think you got it… you go get it!
And again, thanks Kevin for hanging.
Posted in: Podcasts.	Tagged: Comedy, Funny, guest, Television.


text:
Positive reinforcement to encourage voter turnout
Would publicizing the names of people who vote in elections help increase voter turnout? I think so, and that's why I introduced a notice of motion at last Monday's council meeting asking council to support the idea.
I'll explain more about the idea later. First, some background. Only 17,961 of 82,839 eligible voters cast ballots in last fall's election in Coquitlam. That's a turnout rate of just 21.7% .
No one knows exactly why the turnout is so low. Some optimists think it's because people are generally satisfied with the way civic affairs are being managed -- either that, or they don't think civic affairs are important enough to warrant the effort necessary to inform themselves and then cast a ballot.
Most analysts and experts are concerned about the lack of engagement. Of late, some have suggested that the province allow municipalities to make voting easier by opening up the process to on-line balloting. I don't think this is a good idea, though.
First, it would open the system to abuse. We seem to hear about a new computer virus every week or so these days. I think hackers would have a field day if elections went on-line.
Traditional way of encouraging voter turnout.
Maclean's magazine reported in its April 16 edition that, after the University of Western Ontario's alma mater's student council set up a on-line voting system, a Western alumnus hacked into the action and briefly changed a candidate's name to "Justin Bieber's Haircut." And at UBC's Senate election in 2010, 731 votes came from a single IP address.
I also think there should be some effort involved in voting, to better encourage fuller engagement with candidates and issues. Otherwise, you end up with "drive-by voting." Let's not cater to "slack-tivisim."
And that brings me to my motion. The current issue of Atlantic magazine reports on a Michigan study that found that voter turnout increases if people think the names of people who voted will be publicized.
Currently, municipal governments make public a record of all voters who cast ballots for eight weeks after the election. But members of the public are only allowed to view the list in person, and are not allowed to make photocopies.
My motion calls on the provincial government to amend all relevant legislation to allow for the publication, in print or on-line, and distribution of such lists. I can see a municipal government keeping a permanent record on its website, and perhaps even buying a supplement in the local newspaper thanking everyone who voted.
That's positive reinforcement at its best! And, of course, many potential voters would not want to be seen as being off the list, and they'd work to ensure they're on it.
Would people read such a list? Of course. Just think of how often you've scanned the names of the winners of the latest hospital lottery or participants in the Sun Run.
The biggest objection I've heard to my idea is that it would infringe on voter privacy or the secret ballot. I don't agree. The list would not say how the person voted. And, as stated above, there's already no absolute privacy, because the list is made public for eight weeks.
We'll be discussing my motion (seconded by Councillor Linda Reimer) at the month's end council meeting.
Posted by Terry O'Neill at 11:00 AM
Converso November 27, 2012 at 12:24 AM
What an odd practice it is to publish the names of voters. Or so it would seem, for I've never heard of it; how has this escaped my attention? Could it be an assumption on my part that such vestigial practice had been largely eliminated? After all, we no longer pillory the peccant in the public square, at least not literally.
It's shaming the person who does not take the time to vote that you really have mind, is it not? The social control of the traditional community?
I think -- my educated guess is -- that the list was maintained as an aid to the policing of the vote, to allow concerned citizens the opportunity to identify the casting of ballots by persons known to be unqualified to do so. The availability of the list for a limited period after balloting seems to fit with this purpose.
That I have voted, or not, is personal information that, in my opinion, should not be disclosed without my explicit authorization. I'm going to guess that the F.I.P.P.A. commissioner would agree.
The hottest place in Hell


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Mobile gambling is among the most important sorts of mobile software company in today`s mobile market. Anyone who has played computer games probably knowledgeable about electronic Arts. It`s an augmented reality game similar to Ingress in which you walk round the actual world, grab Pokemon, combat for Gyms, and struck Pokestops to restock online items. The death of game cheats could also be that consumer attitudes have shifted within our game diet.
An extremely sophisticated Android program built to spy on users was detected by security researchers. For the own information, you can even download the Locate My Device program from Google Play Store to get a simpler use of the feature. For instance, you can now loan players with a future buy out clause and appeal work license apps. Tap on any hero and from the modest pop up screen, you can set it to auto by simply checking the box beside a tap or spell the throw button should you would like `t want.
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About The Swedish Securities Council
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Information on good practice in the stock market
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The principals of the Swedish Securities Council
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Petitions to the Council
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A ruling by the Chair or the Director General can be re-examined by the Council at the request of the petitioner. Any review of such a decision is made by the Council in normal composition, unless it is clear that the ruling is to be amended.
If the Council issues a ruling with the support of the Financial Supervisory Authority, the decision may be appealed to the Financial Supervisory Authority by the party to whom the decision relates if the ruling has gone against the party.
The appeal is to be submitted in writing. In the submission, the complainant is to indicate which decision is being appealed and what amendment the party requests. The written appeal is to have been received by the Swedish Securities Council within three weeks of the complainant being informed of the decision or, where applicable, within three weeks of the publication of the decision.


text:
Karl Lagerfeld Is Releasing A Collaboration With Vans
It's official! Karl Lagerfeld is set to collaborate with Vans just in time for fashion month.
Text: Cassidy Morrison
Vans will officially collaborate with legendary designer Karl Lagerfeld to give the classic checkerboard pattern a chic upgrade. According to WWD, Vans' trademark pattern will now feature camouflage, "using a cameo motif depicting Lagerfeld's iconic profile, with boucle fabric and a quilted letter K among other leitmotifs."
The California-based favorite among skaters and street style aficionados will work with Lagerfeld on the hallmark "Old Skool" sneakers as well as other interpretations of the Vans' sneakers, and apparel. The collection will drop worldwide on Sept. 7 within the price range of $40 to $300. While images remain secretive now, this collection will no doubt become a street wear favorite.
In typical Lagerfeld fashion, the predominant colors will be black and white, including a T-shirt with an image of Lagerfeld donning a checkerboard tie, a black-and-white raglan sleeve bomber, and a leather backpack adorned with a quilted letter K.
Credits: Photo courtesy of WWD
V Edit: Borrowed From The Boys


text:
Coordinator's Blog
Mastering the COVID-19 quarantine
Various languages
Domestic Violence | High Impact | Awareness Raising
Risk Assessment | Statistics | Recommendations
Training Tools | First Responders | Police Coperation
Two Lessons from IMPRODOVA: Exploring the Frontline Response
How to better serve the victims of domestic violence? The IMPRODOVA research project has collected field studies with frontline responders (law enforcement, NGOs, health professionals) in eight European countries to provide an answer. There is no silver bullet, but in terms of concrete solutions improvements that are implementable in the short term, two key lessons stand out: the organizations who deal with victims should rely on specialized personnel and improve on inter-organizational collaboration.
Frontline responders who are specialists of domestic violence probably serve the needs of victims in a better way than frontline responders who are generalists. By "specialist", we mean police officers (or social workers, or medical professionals) whose job specialty consists in handling domestic violence cases. By "generalist", we refer to those police officers (or social workers, or medical professionals) who indifferently handle all the cases that they encounter in their work. The key variable therefore is whether victims make themselves known to specialists or generalists.
Generalists typically are less knowledgeable about domestic violence, less inclined to take non-physical violence seriously, more inclined to rely on personal discretion, and less likely to make informed and helpful referrals. Generalists' varying skills and zealousness mean that some investigations can be critically crippled by mistakes made in the early stages of the procedure, leaving at-risk women in a particularly dangerous situation of having reported a crime to the police, not having been taken seriously enough, and now facing retaliation from their partners.
Conversely, specialists are better trained, more knowledgeable about the different types of violence, abuse and control dynamics, and more likely to follow protocols and procedures designed to safeguard the victim's interests. In some countries, such as Hungary, Slovenia or Portugal, virtually all police officers on the frontline response to domestic violence are generalists. In other countries, such as Scotland and Finland, most frontline response is made of specialists. In France, Germany and Austria, it varies according to locations, with some places served with specialized units and others with only generalists.
IMPRODOVA teams find that specialists have received relevant trainings and have more experience. Their expertise ensures a prompt and suitable treatment of "normal" cases, and they can manage more complex and technical cases. Their organizational structure reflects an advanced process of division of labour (cf. the Scottish case and the task-sharing in the Scottish police between the local, divisional and national levels). Specialists are more capable of correctly interviewing the victim, of appropriately filling case documentation, of giving correct advice to the victim, of appropriately orienting the victim to support structures. This is manifest from the Finnish, German and Scottish cases. Being specialized on one type of crime, specialists do not face the problem of managing conflicting priorities—a typical problem for generalists. They usually have more reasons (and perhaps more time) to attend additional trainings, academic workshops, and to visit NGOs. They play the role of internal and external experts for DV prevention and investigation (towards non-specialized police officers, NGOs, public administrations). Moreover, they work to increase awareness of their non-specialized colleagues.
Most important, specialized units also are more likely to work in close partnerships with other types of professionals, for instance the embedded social worker at the police station, like in France. Victims have multifaceted needs, and a single profession cannot provide the kind of support that may alleviate the victims' burden. This is why being invested in a network of professionals from other sectors is so important. For instance, social workers who take care of the social-work needs of victims (housing, children, access to poor relief, etc.) allow police officers to focus on investigations and procedural aspects of the case.
Making different professions involved in meaningful partnerships is hard. The Scottish "Multi-Agency Risk Assessment Conferences" (MARAC) is an impressive example of how different worlds (from police to housing to social work to education) come to talk to each other to make sure that every angle of the victims' concerns are addressed. Three core issues are at stake to make partnerships work. First, there must be mechanisms of coordination between actors who sometimes has reservations about working with other professions. One partner needs to have the legitimacy to make people and organizations work together as team partners. Second, pragmatic solutions must be found regarding the management of information, in worlds where professional secrecy often is sacred. Third, trainings must be available so that newcomers can be brought up to speed.
Professional specialization and inter-organizational partnerships are not glamorous. But they make a difference, and should be pursued, because they benefit victims.
2019-09-30 by Francois Bonnet, Thierry Delpeuch and Margarita Vassileva
"Détection des violences au sein du couple"
March 3th, 2021, Online videoconference
February 12th, 2021, Online videoconference
January 11th, 2021, Online videoconference
Don't look away, medicine! - domestic violence in an international context from different perspectives
EUROCRIM - Online conference of the European Society of Criminology
10-11 September 2020 – Online
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Workshop by the Community of Users on Secure, Safe and Resilient Societies on the human and societal aspects of crime in the time of the pandemic
Prof. Michele Burman presented on 'The IMPRODOVA Project' at a workshop organised by the European Commission on the human and societal aspects of crime in the time of the pandemic.
Project Presentation at OSCE Meeting
On November 27, 2020 the Annual Meeting of the OSCE Police Academy Network took place online focussing on "Use of Force" as general theme. Reference point of most presentations were responses to what seems to trigger police violence and what can be done about the problem. IMPRODOVA Coordinator Prof. Joachim Kersten covered German police education and training including IMPRODOVA research findings.
Another successful elective course on domestic violence at the medical faculty in Münster
On 24 and 25 October 2020, the second compulsory elective course for medical students on the topic of "Domestic Violence in an International Context" was held - due to the Corona Pandemic - online again. The elective course was designed to be interactive and allowed the 16 students to engage in lively discussions.
More News ...
Significance of the Children's House and IMPRODOVA projects for effective high impact violence management in Slovenia
Considerations on domestic violence – Japanese and German perspectives
Women must not pay the price for COVID-19! Good practice examples to prevent further risk to victims of male violence
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East Wing Apartment. Sleeps 2, 1 bedroom, 1 large bathroom, kitchen.
South Wing Apartment. Sleeps 5, 1 lounge/principal bedroom, 1 twin bedroom, 1 bathroom, large kitchen.
Nelsons Suite. Sleeps 2, 1 bedroom, 1 large bathroom.
Breakfast facilities available in conservatory.
All bookings incur a £20 cleaning fee per visit.
Swafield Hall can provide accommodation for up to 12 guests. Please contact the owners to arrange.


Recently, our topics have examined some of the reasons that adults decide to wear braces. These excellent resources offer more information on the current state of orthodontic technology, how you can tell whether you are a viable candidate for braces, and more! To learn more, contact King Orthodontics at (919) 246-4433.
This brochure from the American Association of Orthodontists explains what an orthodontist is and what makes the profession distinct from dentistry.
The Washingtonian takes a look at Invisalign and how these clear, removable aligners have shaken up the world of orthodontics.
Visit this website to learn more about the Damon system of orthodontia and why it's both faster and more comfortable than other braces treatments.
Is now the right time for you to get braces? Find out by reading this informative article from Oral-B.


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Baptist Life
Latest Vatican City News
The Latest: China says COVID-19 hospitalizations above 1,000
Jan. 14, 2021 9:40 PM EST
BEIJING — China says it is now treating more than 1,000 people for COVID-19 as numbers of cases continue to surge in the country's north. The National Health Commission said Friday that 1,001 patients are under care for the disease, 26 of them in serious condition. It says 144 total new cases were...
Australia financial unit admits huge Vatican transfer error
VATICAN CITY (AP) — Australia's financial intelligence agency has admitted it vastly misreported the amount of money transferred from the Vatican to Australia over the past six years by nearly $1.5 billion. A statement Wednesday from the Vatican confirmed that the actual amount of money transferred...
Probe highlights Vatican legal system's limited protections
Jan. 12, 2021 2:16 AM EST
VATICAN CITY (AP) — A criminal investigation into a Vatican real estate investment is exposing weaknesses in the city state's judicial system and a lack of some basic protections for those accused — highlighting the incompatibility of the Holy See's procedures with European norms. The...
Pope prays for dead in Capitol rioting, appeals for calm
VATICAN CITY (AP) — Pope Francis said Sunday that he is praying for those who died in the U.S. Capitol rioting and has appealed for calm to prevail in the United States to help safeguard that nation's democratic values. During his traditional Sunday noon remarks at the Vatican, Francis noted that five...
The Latest: Mexico sees record 16,105 new virus cases
Jan. 9, 2021 10:22 PM EST
MEXICO CITY — Mexico has posted another daily record for newly confirmed coronavirus cases, with 16,105 new infections reported Saturday, and a near-record of 1,135 deaths related to COVID-19 in the latest 24-hour period. The country has now topped 1.5 million total infections and over 133,000 deaths so...
Pope Francis 'astonished' by mob attack on US Capitol
Jan. 9, 2021 2:22 PM EST
VATICAN CITY (AP) — Pope Francis has told an Italian broadcaster that he was "astonished" by the mob attack at the U.S. Capitol due to the democratic traditions of the United States. The pope said in an interview with Mediaset that the private broadcaster is set to air on Sunday that even...
Australian agency reviewing data on Vatican money transfers
Jan. 8, 2021 12:56 AM EST
ROME (AP) — The Australian government's financial intelligence agency said Thursday it is reviewing its data after questions were raised about its report that $1.8 billion were transferred from the Vatican to Australia over six years. The agency, Austrac, said it was working with the Vatican to get...
The Latest: Japan's prime minister weighs state of emergency
TOKYO — Japanese Prime Minister Yoshihide Suga said Monday vaccine approval was being speeded up and border controls beefed up to curb the spread of the coronavirus, and he promised to consider declaring a state of emergency. Tokyo Gov. Yuriko Koike and the governors of Saitama, Chiba and Kanagawa had...
Prelate blocked for months by Belarus resigns Minsk post
Jan. 3, 2021 9:03 AM EST
VATICAN CITY (AP) — Pope Francis on Sunday accepted the resignation of Minsk' archbishop who had been blocked for months by Belarusian authorities from returning to his homeland after criticizing a crackdown on anti-government protesters there. Monsignor Tadeusz Kondrusiewicz, in charge of the...
The Latest: Ariz. gov. rejects call for online learning
PHOENIX — Arizona's Gov. Doug Ducey has rejected the state's top education official's call for Ducey to order public schools to use only online instruction for the next two weeks unless they have waivers from health officials. Amid a coronavirus surge in the state, Superintendent of...
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Property Location In the heart of Davao, Marco Polo Davao is close to NCCC Mall of Davao and Monument of Peace and Unity. This 4-star hotel is within close proximity of People's Park and San Pedro Cathedral. Rooms Make yourself at home in one of the 245 air-conditioned rooms featuring minibars and LCD televisions. Wired and wireless Internet access is available (surcharge). Bathrooms have complimentary toiletries and hair dryers. Conveniences include phones, as well as safes and desks. Rec, Spa, Premium Amenities Pamper yourself with a visit to the spa, which offers massages, body treatments, and facials. If you're looking for recreational opportunities, you'll find a health club, an outdoor pool, and a spa tub. This hotel also features wireless Internet access (surcharge), babysitting/childcare, and gift shops/newsstands. Dining Take advantage of the hotel's 24-hour room service. Business, Other Amenities Featured amenities include high-speed (wired) Internet access (surcharge), a business center, and limo/town car service. Event facilities at this hotel consist of conference space and meeting rooms. A roundtrip airport shuttle is available for a surcharge.


Fire energy boosts your spirits and confidence. It's the perfect day to show off what you've been keeping under wraps.
If your charms won't work on one person who's to say they won't on another? Appeal to the second-in-command or the friend of a friend.
Maybe it's a song that you can't get out of your head or a word that keeps popping up. These are signals from your unconscious. Look up the title or word and you'll get the message.
It may look as if you haven't accomplished your objective, but you secretly know better. Communication through a third party confirms this.
Mentoring someone is more than just performing your good deed for the day. What's common to you is uncommonly valuable to someone else.
Watch it, because you're starting to sound patronizing. Don't turn your nose up at people; bring them up to speed instead.
Don't get drawn into a shady deal because you believe it will expedite matters. Do things by the book now and you won't be sorry later.
Could loved ones and friends be any more self-absorbed? Yes. But it's good because it forces you to be upfront about what you want.
Expect to be bombarded by unsolicited advice– but don't take it the wrong way. People are excited by what you're doing and want to weigh in with their opinions.
There's nothing wrong with showing a little vulnerability. If anything people will be more sensitive. You often act as if you don't need support.
Strong personalities mean strong opinions, so you have to find a way to marshal the energies without being overrun by them.
A friend's latest crisis du jour may be too much to handle right now. There's no shame in begging off.


Quick and easy honey and oat balls. A great option for those with little time in the morning or who don't like breakfast. The honey gives an initial boost of energy after waking up and the oats provide a slow releasing energy.
Method: Simply add all of the ingredients in a large bowl and mix well. Form into ten balls. Can refridgerate for one week.