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https://www.monsterenergy.com/news/monster-gaming-weekly-2733
Monster Gaming Weekly
The past week has been a busy one in the world of esports, but several leagues and qualifiers are still in the thick of things. The big story of the weekend was Intel Extreme Masters Season XII Oakland, where the world's best CS:GO and PUBG teams clashed to determine a champion. Over in Dota 2, the DreamLeague qualifiers concluded, and we now know the teams headed to the next Major.
DreamLeague EU and SEA qualifiers
Our teams continued to do well this week in the DreamLeague qualifiers as Team Liquid and Fnatic both advanced from their regions to join EG at the LAN event in Jönköping, Sweden. While Fnatic and Team Liquid have both made it to the event, the way they had to get there was very different. Fnatic had to go through a four team double elimination bracket with all matches being a best of three. Team Liquid on the other hand was placed into an online league which featured four teams from Europe and four teams from CIS. Team Liquid then had to play a best of two against each of the other teams in the league and had to secure a top four finish in order to qualify.
The implications that can be made from Fnatic's recent performances in SEA is that they are emerging as the team to beat in that region. Their 2-1 win over Mineski in the upper bracket finals makes it the second time in a row Fnatic has triumphed over the team who were considered to be nearly unstoppable in SEA. Abed has fit seamlessly under the Fnatic banner, and two top tier Storm Spirit performances in the qualifier final helped book their trip to Sweden.
As long as the rest of Fnatic maintain a high level of play — which looks like the trend for now — the team could easily outperform their previous iterations. What this means for Fnatic is more opportunities to claim Dota Pro-Circuit (DPC) points and get themselves in the running for a direct invite to TI8. A good placement at this Major would be the perfect start!
Team Liquid on the other hand are currently sitting in 3rd place in DPC points with 450. Their qualification through league play went smoothly even up against the likes of VP, Team Secret, OG, and Na`Vi as they secured enough points to qualify before playing their last match. While the tales of the TI Curse haunting the winning teams and lulling them into a false sense of security have been true in the past, Team Liquid show no signs of slowing down in the new season. Liquid dominated the bottom half of the table and split their series against their fellow qualified teams. Liquid showed a wide range of heroes to display their versatility, and their exploits randoming through entire drafts during Midas Mode certainly suggest that the team is in top form.
A win at the DreamLeague Major would be their first Major win of the season and put them back on top of the rankings. They've already won two minors so far, and they enter Jönköping as strong favorites once again, but it's anyone's game with the new patch shaking things up in the metagame!
IEM Oakland
IEM Oakland was the big tournament of the weekend with CS:GO the main highlight. 12 top teams made the trip to California to compete for the big prize of an IEM trophy and $129,000.
Day 1 of the group stage was a tight affair, with no teams really separating themselves after the first round of play. Group A included EnVyUs, and the French squad narrowly missed out on an opening win after a narrow 20-22 overtime loss to SK Gaming. Neither team managed to gain a foothold on they traded 8-7 halves to tie the game at 15 apiece, with RpK leading the way for nV. His incredible 40-kill map would not be enough, however, as SK secured the win in the 2nd OT.
EnVy regained some momentum against MongolZ, but their 16-12 win looked closer than it should have been. A 12-3 T-side on Cache suggested an easy sweep, but the MongolZ earned nine rounds on their T-side to make things close. nV closed the first day 1-2 by falling to Astralis on Inferno, 11-16, to end the day with four points.
In Group B, arguably the closer group on paper, Team Liquid squeezed out two overtime wins to earn four points in the table. Both games involved unlikely comebacks down 5-10 in the first half. Their first victim was Renegades, a team that had looked very dangerous over the past few weeks. EliGE led the way with 32 kills and a 3-kill to force OT on Cache. Liquid lined up against Gambit next on Mirage, and the first half once again looked dire. Things seemed even worse when Liquid lost the second pistol round, but a big anti-eco set the tone for 6 straight rounds and a tied game. Gambit managed to claim match point, but Liquid sealed OT with an almost impossible force buy. A clever boost for Twistzzz spotted a gap in Gambit's A execute, and he cleaned house to grab a hold of the game, which translated to a comfortable overtime session.
Unfortunately, neither team would survive day two and make it to the bracket stage of the competition. Despite beating Cloud 9 16-8 on Cobble, EnVyUs needed a win against NiP to leapfrog SK Gaming for a spot in the playoffs. The defending champions could not be denied, however, and EnVyUs dropped out of the tournament with their loss on Nuke.
Team Liquid did not fair much better as they dropped all three maps on Day Two. Despite having a record of 2-3 — a similar record to three other teams in the group — Liquid lost out due to points. With OT wins only awarding two points, and OT losses granting one, Liquid ended the group stage with four points as opposed to Gambit's eight. Liquid rued missed chances against both OpTic and G2, who they had played to tight first halves. Liquid possessed 8-7 leads in both games, but squandered their advantage after the intermission.
Ninjas in Pyjamas went on to defend their title against FaZe Clan in an epic 3-2 series to close out the tournament.
The PUBG train shows no signs of stopping, and the latest invitational occurred at IEM Oakland. Ten of the world's best teams received direct invitations while ten further teams had to fight through qualifiers. The unpredictable nature of PUBG — which is both a source of excitement and frustration — was in full display as qualifier team against All authority upset everyone to win the event. In addition, several invited teams could be found at the bottom of the final ranking, including Evil Geniuses and Alliance.
While all of the teams performed valiantly, the only Monster team to win a chicken dinner was Team Liquid. The squad composed of Hayz, Molnman, Ollywood and Scoom reached the upper half of the table by placing a middle-of-the-pack tenth place. A battle royale victory during a 10-kill Round Five was not enough to earn a higher seed as the team only managed one other Top-10 finish. A crucial 2 vs 3 against Tempo Storm in the final circle looked favorable for TS, but blinding headshots from behind cover managed to turn things around. Scoom tallied 8 kills in the winning round to earn Liquid 300 points.
The entire upper half of the pool looked close, with only aAa separating themselves. Second through tenth was a tight affair, as those teams were all only separated by only 200 points. One small decision could have changed a big round and the standings of any one of our squads, but such is the thrill of PUBG.
Alphari: Mind Over Mechanics


text:
Q: Decrease indentation of header text (fancyhdr) to 0.15 cm (same as \titleformat) I'm using fancyhdr for the header.
I want to decrease the indentation of the text in it (header - italic text):
Now is like in the image

but I would like to have the same indentation in both (header and main matter- \titleformat= 0.15 cm). Any suggestion?
this is my code:
\documentclass[12pt,a4paper,twoside]{article}
\usepackage[vmargin={25mm,25mm},hmargin={25mm,25mm},includefoot,headheight=15pt]{geometry}

\linespread{1.3}
\usepackage[utf8x]{inputenc}
\usepackage[main=english,portuguese]{babel}
\setlength{\parindent}{0.65cm} % indentation
\usepackage{indent first}

\usepackage{titlesec}
\usepackage{fancyhdr}
\pagestyle{fancy}
\fancyfoot{}
\fancyhead[LE,RO]{\thepage}

%sections format

\renewcommand\thesection{\arabic{section}.}
\renewcommand\thesubsection{\thesection\arabic{subsection}}

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\titleformat{\subsubsection}
{\normalfont\bfseries}{\hspace{1.65cm}\thesubsubsection.}{0.15cm}{}

%mainmatter
\begin{document}
\section{Introduction}
\end{document}

Thanks in advance

redpajama_set_name: RedPajamaStackExchange


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\section{Introduction}

Given a metric space $(X,r)$ and a subset $\mathcal{S}\subseteq\mathcal{P}(X)$ of all probability measures,
one can endow $\mathcal{S}$ with various metrics, depending on what kind of measurement is suitable for the problem under consideration. Here we mention three examples.
\begin{itemize}
\item[-] The \emph{Kolmogorov-Smirnov metric} $d_{KS}$ on $\mathcal{S}=\mathcal{P}(\mathbb{R})$ is frequently used in statistics to compare a
sample with a reference probability distribution.

\item[-] The \emph{Lévy-Prokhorov metric} $d_{LP}$ plays an important theoretical role in several limit theorems in probability theory. In this case $(X,\varrho)$ is a complete separable metric space and $\mathcal{S}=\mathcal{P}(X)$.

\item[-] The \emph{quadratic Wasserstein metric} $d_{\mathcal{W}_2}$ turned out to be very effective in a wide range of AI applications including pattern recognition and image processing problems. In these applications $(X,r)$ is typically the $n$-dimensional Euclidean space and $\mathcal{S}$ is the collection of all Borel probability measures with finite second moment.
\end{itemize}
In recent years, there has been a considerable interest in the characterization of the above mentioned (and many other) metric spaces of measures, see e.g. \cite{bertrand-kloeckner-2016,DolinarKuzma,DolinarMolnar,Kuiper,LP,JMAA,TAMS,HIL,Kloeckner-2010,Levy,S-R,titkoskissgraf,Virosztek}. In most cases, it turned out that isometries of $\mathcal{S}$ are strongly related to self-maps of the underlying space $X$. Concerning the Kolmogorov-Smirnov distance, Dolinar and Moln\'ar showed in \cite{DolinarMolnar} that there is a one-to-one correspondence between all isometries of $(\mathcal{P}(X),d_{KS})$ and all homeomorphisms of the real line. Concerning the Lévy-Prokhorov metric, the first and the second author showed in \cite{LP} that $\mathcal{P}(X)$ endowed with $d_{LP}$ is more rigid, assuming that $X$ is real separable Banach space: a self-map of $X$ induces an isometry on $(\mathcal{P}(X),d_{LP})$ if and only if it is itself an isometry. In fact, the isometry group of $(X,\|\cdot\|)$ and the isometry group of $(\mathcal{P}(X),d_{LP})$ are isomorphic. This phenomenon is called \emph{isometric rigidity}. The third example above is even more peculiar. Kloeckner showed in \cite{Kloeckner-2010} that the isometry group of the quadratic Wasserstein space over $\mathbb{R}^n$ is much larger than the isometry group of $\mathbb{R}^n$.

The most recent results which are related to our studies have been presented in \cite{S-R}: Santos-Rodríguez proved isometric rigidity for a very broad class of manifolds. More precisely, he showed that the isometry group of a quadratic Wasserstein space over a closed Riemannian manifold with strictly positive sectional curvature is isomorphic to the isometry group of the underlying manifold. Furthermore, for compact rank one symmetric spaces (CROSSes), Santos-Rodríguez was able to prove isometric rigidity not only for the quadratic case, but for general $p$-Wasserstein spaces with $1<p<\infty$.\\

As the results of \cite{Kloeckner-2010} and \cite{S-R} already indicate, isometric rigidity of $p$-Wasserstein spaces depends in an interesting way both on some characteristics of the underlying space $X$ and on the value of $p$. To stress this phenomenon, we briefly mention one more special case, for more details see \cite{TAMS}: the $p$-Wasserstein space over $\mathbb{R}$ is isometrically rigid if and only if $p\neq2$, while the $p$-Wasserstein space over $[0,1]$ is isometrically rigid if and only if $p\neq1$.\\

This paper aims to offer a unified approach for two compact manifolds as underlying space: the n-dimensional torus and the n-dimensional sphere, regardless of what the value of $p$ is. The key idea -- which could be applicable in other settings as well -- is that all measures can be fully recovered from a function, which we call Wasserstein potential. The main results are \textbf{Theorem \ref{thm:WpTd}} and \textbf{Theorem \ref{thm:WpSg}}, where we prove for all $p\geq 1$ that the isometry group of the $p$-Wasserstein space on the $n$-dimensional torus is isomorphic to the isometry group of the torus, and similarly, the isometry group of the $p$-Wasserstein space over the $n$-dimensional sphere is isomorphic to the isometry group of the sphere. The latter result has been partially covered by Santos-Rodríguez in \cite{S-R}, as the sphere is a compact rank one symmetric space. However, the method presented in \cite{S-R} cannot be extended to the case $p=1$, as $1$-Wasserstein spaces have a more flexible structure: the optimal transport plan between measures is not unique, let
alone the geodesic curve.

\section{The Wasserstein potential of measures}

In this section, we collect all notions which are necessary to our investigations. We also demonstrate via a simple example how useful the Wasserstein potential can be to identify measures and to prove isometric rigidity.\\

Let $(X,r)$ be a complete and separable metric space. We denote by $\mathcal{P}(X)$ the collection of all Borel probability measures on $X$, and by $\mathcal{F}(X)$ the set of all finitely supported measures. Given a measure $\mu\in\mathcal{P}(X)$, the support $S(\mu)$ is the set of all points $x\in X$ for which every open neighbourhood of $x$ has positive measure with respect to $\mu$. A Borel probability measure $\pi$ on $X \times X$ is said to be a \emph{coupling} for $\mu,\nu\in\mathcal{P}(X)$ if
\begin{equation}\pi\ler{A \times X}=\mu(A)\qquad\mbox{and}\qquad\pi\ler{X \times B}=\nu(B)
\end{equation}
for all Borel sets $A,B\subseteq X$. We denote set of all couplings by $\Pi(\mu,\nu)$. For any $1\leq p<\infty$ one can define the \emph{$p$-Wasserstein space} $\Wp(X)$ as the set of all $\mu\in\mathcal{P}(X)$ that satisfy
\begin{equation}\label{pp}\int_X r(x,\hat{x})^p~\mathrm{d}\mu(x)<\infty
\end{equation} for all $\hat{x}\in X$, endowed with the \emph{$p$-Wasserstein distance}
\begin{equation} \label{eq: wasser def}
\dwp\ler{\mu, \nu}:=\ler{\inf_{\pi \in \Pi(\mu, \nu)} \int_{X \times X} r(x,y)^p~\mathrm{d} \pi(x,y)}^{1/p}.
\end{equation}
This distance measures the minimal effort required to transport one probability measure into another, when the cost of moving mass is the $p$-th power of the distance. For more details on optimal transport and Wasserstein spaces we refer the reader to the comprehensive textbooks of Santambrogio and Villani \cite{Santambrogio,Villani}.

It is one of the important features of $p$-Wasserstein spaces that $\Wp(X)$ contains an isometric copy of $X$, as the distance between any two Dirac measures equals to the distance of their supporting points, i.e. $\Wp(\delta_x,\delta_y)=r(x,y)$. Furthermore, every measure belonging to $\Wp(X)$ can be approximated by convex combinations of Dirac measures, that is, $\mathcal{F}(X)$ is dense in $\Wp(X)$. (For more details see e.g. Example 6.3 and Theorem 6.16 in \cite{Villani}.)

In this paper we are interested in the structure of \emph{isometries}, that is, distance preserving bijections.  The symbol $\mathrm{Isom}(\cdot)$ will always refer to the isometry group of the metric space in question. We denote the \emph{push-forward map} of an isometry $\psi\colon X \rightarrow X$ by
$\psi_\# \colon \Wp(X)\to\Wp(X)$: \begin{equation}\label{pushforward}
\big(\psi_\#(\mu)\big)(A)=\mu(\psi^{-1}[A])
 \end{equation}
for all $A\subseteq X$ and $\mu\in\Wp(X)$, where $\psi^{-1}[A]=\{x\in X\,|\, g(x)\in A\}$. If $p\geq1$, the push-forward operation
\begin{equation}
    \#:\mathrm{Isom}(X)\to\mathrm{Isom}(\Wp(X));\quad \psi\mapsto\psi_\#
\end{equation}
is an embedding (in fact, a group homomorphism). Those isometries which belong to the image of $\#$ are called \emph{trivial isometries}. We say that $\Wp(X)$ is \emph{isometrically rigid} if $\#$ is surjective. Now we introduce our key tool: for a given $\mu\in\Wp(X)$ the one variable function $\mathcal{T}_\mu^p:X\to\mathbb{R}$ defined by
\begin{equation}\label{potdef}
	\mathcal{T}_\mu^p(x):=\dwpp(\mu,\delta_x)=\int_{\mathbb{T}^n} r(x,y)^p ~\mathrm{d}\mu(y).
\end{equation}
is called the \emph{Wasserstein potential of $\mu$}.
We expect that $\mu$ can be fully recovered from this function, and in particular that $\mathcal{T}_\mu^p=\mathcal{T}_\nu^p$ implies $\mu=\nu$.\\

In this  paper we are going to consider the torus and the sphere as underlying spaces. The symbol $\mathbb{T}^n$ stands for the $n$-dimensional torus, that is, the set $\R^n/\Z^n\simeq[-1/2,1/2)^n$ equipped with the usual metric
\begin{equation}\label{torusmetric}\varrho\ler{x,y}=\ler{\sum_{k=1}^n\abs{\ler{x_k-y_k}_{\text{mod }1}}^2}^{\frac{1}{2}},
\end{equation}
where $x=(x_1,\dots,x_n)\in\R^n$ and $y=(y_1,\dots,y_n)\in \R^n$. The antipodal of a point $(x_1,\dots,x_n)$ in the torus is $(x_1+1/2,\dots,x_n+1/2)$. We denote the unit sphere of $\R^{n+1}$ by the symbol $\mathbb{S}^n$, that is, $\mathbb{S}^n := \{x\in\R^{n+1}\colon \norm{x}=1\}$. We equip the unit sphere with the angular (or geodesic) distance: for $x,y\in\mathbb{S}^n$ the distance of $x$ and $y$ is
\begin{equation}\label{sphericalangle}
\sphericalangle(x,y) := \arccos\inner{x}{y}.
\end{equation}
We say that two points $x$ and $y$ are antipodal in the sphere if $y=-x$.  Adapting Gangbo's and Tudorascu's terminology in \cite{GT}, we will shortly refer to the $p$-Wasserstein spaces over $(\mathbb{T}^n,\varrho)$ and $(\mathbb{S}^n,\sphericalangle)$ as the $p$-Wasserstein torus and the $p$-Wasserstein sphere, respectively.\\

To conclude this chapter, we present one possible way of using Wasserstein potentials. This example will shed some light also on the difficulties that need to be overcome to obtain the desired result for all possible values of p. First of all, when working with the torus, it is a natural idea to borrow techniques from the theory of Fourier analysis.\\

\noindent Given a measure $\mu\in\Wp(\mathbb{T}^n)$, the potential function $\mathcal{T}_\mu^p$ can be written for all $x\in\mathbb{T}^n$ as
\begin{equation}\mathcal{T}_\mu^p(x)=\int_{\mathbb{T}^n} \varrho^p(x,y) ~\mathrm{d} \mu(y)= \int_{\mathbb{T}^n}c_p(x-y) ~\mathrm{d} \mu(y)= \ler{c_p * \mu}(x),
\end{equation}
where $c_p(x)=\ler{\sum_{k=1}^n x_k^2}^{\frac{p}{2}}$. Since the \emph{characters} of $\mathbb{T}^n$
$$
\varphi_j(x)=e^{2 \pi i j \cdot x} \qquad \ler{j=\ler{j_1,j_2, \dots,j_n} \in \Z^n}
$$
form an orthonormal basis of $L^2\ler{\mathbb{T}^n}$, we have
\begin{equation*}
\begin{split}
\hat{\mathcal{T}}_\mu^p(j)&=\inner{\mathcal{T}_\mu^p}{\varphi_j}
=\int_{\mathbb{T}^n} \mathcal{T}_\mu^p(x) e^{-2\pi i j \cdot x} \mathrm{d} x\\
&=\int_{\mathbb{T}^n}\Big( \int_{\mathbb{T}^n}c_p(x-y) ~\mathrm{d} \mu(y)\Big) e^{-2\pi i j \cdot x} ~\mathrm{d} x\\
&=\int_{\mathbb{T}^n} \Big(\int_{\mathbb{T}^n}c_p(x-y) e^{-2\pi i j \cdot (x-y)}\mathrm{d} x\Big) e^{-2\pi i j \cdot y}~\mathrm{d} \mu(y)\\
&=\inner{c_p}{\varphi_j}\inner{\mu}{\varphi_j}=\hat{c}_p(j)\hat{\mu}(j).
\end{split}
\end{equation*}
In particular, if $\hat{c}_p(j)\neq 0$ for every $j \in \Z^n,$ then the measure can be recovered from the potential function. If $n=1$ and $p=2$ then the Fourier series of $c_2(x)=x^2$ does not vanish anywhere. More precisely, $\hat{c}_2(j)=\frac{(-1)^j}{2 j^2 \pi^2}$ for $j \neq 0$ and $\hat{c}_2(0)=\frac{1}{12}$. This means that $\mathcal{T}_\mu^p\equiv\mathcal{T}_\nu^p$ implies $\mu=\nu$ in this case. As we will see later, this implication automatically ensures that the Wasserstein space in question is isometrically rigid.

Based on numerical computations, it seems that the Fourier transform $\hat{c}_p$ does not vanish anywhere if $n=1$ and $p>1.$ However, this is not the case for $n=1$ and $p=1.$ Indeed, $\hat{c}_1(j)=0$ for non-zero even $j$'s, $\hat{c}_1(0)=\frac{1}{4},$ and $\hat{c}_1(j)=-\frac{1}{j^2 \pi^2}$ for odd $j$'s. This does not mean that $\Wo \ler{\mathbb{T}}$ is not isometrically rigid, but we cannot prove it in such a simple way. For $n>1,$ the same holds true for $\Wt\ler{\mathbb{T}^n}$. The reason is that the summands of the quadratic cost function $c_2(x)=\sum_{k=1}^n x_k^2$ depend on only one variable, and hence $\hat{c}_2(j)=0$ whenever $j_k \neq 0$ for at least two indices. For example, for $n=2,$ we have  $\hat{c}_2 (0,0)=\frac{1}{6},$  $\hat{c}_2(0,j)=\hat{c}_2(j,0)=\frac{(-1)^j}{2 j^2 \pi^2}$ for  $j\neq 0$, and $\hat{c}_2(j_1,j_2)=0$ for $j_1, j_2 \neq 0$.\\

In what follows we develop a method that works for all $p\geq1$ and is suitable to prove that $\Wp(\mathbb{T}^n)$ and $\Wp(\mathbb{S}^n)$ are isometrically rigid. In fact, this method works in the $0<p<1$ case as well, but we decided to not include it in the main body. On the one hand, we have already proved in \cite{HIL} that $p$-Wasserstein spaces are all isometrically rigid if $0<p<1$, regardless of what the underlying space is. On the other hand, as the definition of the $p$-Wasserstein distance is slightly different in the $0<p<1$ case, we should add one more branch to all proofs, without any serious novelty.

\section{Isometric rigidity of the Wasserstein torus}

We start with a simple observation: the diameter of $\Wp(\Tn)$ is $\sqrt{n}/2$ if $p\geq 1$, and this maximal distance is achieved if and only if the two measures are Dirac masses concentrated on antipodal points. This automatically implies that if $\Phi\in\mathrm{Isom}(\Wp(\mathbb{T}^n)$ then the $\Phi$-image of a Dirac measure is again a Dirac measure, in fact, Dirac measures concentrated on antipodal points are mapped to Dirac measures which are concentrated on antipodal points. Since $\dwp(\delta_x,\delta_y)=\varrho(x,y)$, this implies that the map $\psi:\mathbb{T}^n\to\mathbb{T}^n$ defined by $\Phi(\delta_x)=\delta_{\psi(x)}$ is an isometry of $\mathbb{T}^n$. It is a known that any isometry $\psi$ of $\mathbb{T}^n$ can be written in the following form:
\begin{equation}\label{eq:TD-isom}
    \psi((x_1,x_2,\dots,x_n)) = (\varepsilon_1 x_{\sigma(1)},\varepsilon_2 x_{\sigma(2)},\dots,\varepsilon_n x_{\sigma(n)}) + (u_1,u_2,\dots,u_n)
\end{equation}
with a permutation $\sigma$, numbers $\varepsilon_1,\dots,\varepsilon_n\in\{-1,1\}$ and point $(u_1,u_2,\dots,u_n)\in\mathbb{T}^n$.

By the above observation, we have
$\mathcal{T}_{\Phi(\mu)}^p(\psi(x)) = \mathcal{T}_\mu^p(x)$ for all $x\in\mathbb{S}^n$ which suggests that those properties of $\mu$ which are encoded in its potential function, will be carried over to $\Phi(\mu)$.

Before continuing, we need some new notations. Let $x\in\mathbb{T}^n$, $n\geq 2$, $j\in\{1,\dots,n\}$. We introduce the set $H(x,j):=\{(y_1,\dots,y_n)\in\mathbb{T}^n\colon y_j=x_j \}$, and we denote by $e^j$ the vector $(\delta_{j,1},\dots,\delta_{j,n})$, where $\delta_{j,j}=1$ and $\delta_{i,j}=0$ if $i\neq j$. The symbol $\check{x}_j\in\mathbb{T}^{n-1}$ stands for the point obtained by dropping the $j$th coordinate of $x\in\mathbb{T}^n$, and $\lambda_{n-1}$ denotes the normalised Haar measure of $\mathbb{T}^{n-1}$. We remark that $\mathbb{T}^{n-1}$ can be identified with $H(x,j)$ for any point $x\in\mathbb{T}^n$ and $j\in\{1,\dots,n\}$. For two points $x,y\in\mathbb{T}^n$ we denote by $B(x,y)$ the bisector of $x$ and $y$, i.e., $B(x,y)=\{z\in\mathbb{T}^n\,|\,\varrho(x,z)=\varrho(z,y)\}$. The following lemma is of key importance. We will use it later to estimate the measure of certain (carefully chosen) sets and points by means of the Wasserstein potential.

\begin{lemma}\label{lem:WpTd-pot-p}
Let $n\in\N$, $n\geq 2$, $p\geq 1$, $x\in\mathbb{T}^n$, $j\in\{1,\dots,n\}$, and $\mu\in\Wp(\Tn)$. Then the following assertions hold:
\begin{itemize}
\item[(a)] If $p=1$, and $\varrho_{n-1}$ denotes the distance of $\mathbb{T}^{n-1}$ then
    \begin{align}\label{eq:WpTd-pot-p=1}
    &\lim_{s\to 0+}\frac{\mathcal{T}_\mu^1(x+s\cdot e^j) - 2\mathcal{T}_\mu^1(x) + \mathcal{T}_\mu^1(x-s\cdot e^j)}{s}\nonumber \\
    & \hspace{2.5cm} =
    2\mu(\{x\}) - \int_{H(x+\frac{1}{2}e^j,j)} \ler{\frac{1}{4} + \varrho_{n-1}\ler{\check{x}_j,\check{y}_j}^2}^{-\frac{1}{2}} ~\mathrm{d}\mu(y).
    \end{align}
    \item[(b)] If either $1<p<2$ or $p>2$, then
    \begin{align}\label{eq:WpTd-pot-p>1-not2}
    &\lim_{s\to 0+}\frac{\mathcal{T}_\mu^p(x+s\cdot e^j) - 2\mathcal{T}_\mu^p(x) + \mathcal{T}_\mu^p(x-s\cdot e^j)}{s} \nonumber \\
    & \hspace{2.5cm} = - p\int_{H(x+\frac{1}{2}e^j,j)} \ler{\frac{1}{4} + \varrho_{n-1}\ler{\check{x}_j,\check{y}_j}^2}^{\frac{p-2}{2}} ~\mathrm{d}\mu(y).
    \end{align}
    \item[(c)] If $p=2$, then
    \begin{align}\label{eq:WpTd-pot-p=2}
    &\lim_{s\to 0+}\frac{\mathcal{T}_\mu^2(x+s\cdot e^j) - 2\mathcal{T}_\mu^2(x) + \mathcal{T}_\mu^2(x-s\cdot e^j)}{s} =
    - 2\cdot\mu\ler{H\ler{x+\frac{1}{2}e^j,j}}.
    \end{align}
\end{itemize}
\end{lemma}

\begin{proof}

Since $p\geq 1$, we have the following:
\begin{align*}
    &\lim_{s\to 0+}\frac{\mathcal{T}_\mu^p(x+s\cdot e^j) - 2\mathcal{T}_\mu^p(x) + \mathcal{T}_\mu^p(x-s\cdot e^j)}{s} \\
   = &\lim_{s\to 0+} \bigg( \int_{\mathbb{T}^n\setminus\{x\}\setminus H(x+\frac{1}{2}e^j,j)} \frac{\varrho(x+s\cdot e^j,y)^p - 2 \varrho(x,y)^p + \varrho(x-s\cdot e^j,y)^p}{s}~\mathrm{d}\mu(y)   \\
    & \hspace{3cm} + 2\mu(\{x\})\frac{s^p}{s} + 2\int_{H(x+\frac{1}{2}e^j,j)} \frac{\varrho(x+s\cdot e^j,y)^p - \varrho(x,y)^p}{s}~\mathrm{d}\mu(y) \bigg).
\end{align*}First we obtain that the above two integrands are bounded.
On the one hand, since $s\mapsto \varrho(x+s\cdot e^j,y)^p$ is differentiable at $s=0$ if $y\in \mathbb{T}^n\setminus\{x\}\setminus H(x+\frac{1}{2}e^j,j)$, the first integral converges to $0$ by Lebesgue's dominant convergence theorem.
On the other hand, if $y\in H(x+\frac{1}{2}e^j,j)$, then by elementary calculus we obtain
\begin{align*}
    &\lim_{s\to 0+} \frac{\varrho(x+s\cdot e^j,y)^p - \varrho(x,y)^p}{s}
    = \left.\frac{d}{ds}\ler{\ler{\ler{\frac{1}{2}-s}^2 + \sum_{\substack{k=1 \\ k\neq j}}^n(x_k-y_k)^2}^{p/2}}\right|_{s=0} \\
    &= -\frac{p}{2} \ler{\ler{\frac{1}{2}}^2 + \sum_{\substack{k=1 \\ k\neq j}}^n(x_k-y_k)^2}^{\frac{p-2}{2}}

    = -\frac{p}{2} \ler{\frac{1}{4} + \varrho_{n-1}\ler{\check{x}_j,\check{y}_j}^2}^{\frac{p-2}{2}}.
\end{align*}
An application of Lebesgue's dominant convergence theorem completes the proof.
\end{proof}

Now we are ready to state and prove the main theorem of this section. We assume that $n\geq2$, the $n=1$ case will be proved in Theorem \ref{thm:WpSg}.

\begin{theorem}\label{thm:WpTd}
Let $n\geq2$ and $p\geq1$. Then the $p$-Wasserstein torus $\Wp(\mathbb{T}^n)$ is isometrically rigid, that is, the push-forward operation $\#:\mathrm{Isom}(\mathbb{T}^n)\to\mathrm{Isom}(\Wp(\mathbb{T}^n))$ is surjective.
\end{theorem}

\begin{proof}
Assume that $\Phi\colon\Wp(\Tn)\to\Wp(\Tn)$ is an isometry. We have to show that there exists an isometry $\psi:\mathbb{T}^n\to\mathbb{T}^n$ such that
\begin{equation}\label{eq:WpTd}
	\Phi(\mu) = \psi_\#\mu \qquad (\mu\in\Wp(\Sn)).
\end{equation}

We already know that \eqref{eq:WpTd} holds for Dirac masses with some $\psi\in\mathrm{Isom}(\mathbb{T}^n)$, which implies that $\ler{\psi^{-1}}_\#\circ\Phi$ fixes all Dirac measures. Since $\Phi=\psi_\#$ if and only if $\ler{\psi^{-1}}_\#\circ\Phi$ is the identity of $\Wp(\mathbb{T}^n)$, we can assume without loss of generality that $\Phi$ itself fixes all Dirac measures, and our task now is to prove that $\Phi(\mu)=\mu$ for all $\mu\in\Wp(\mathbb{T}^n)$. In fact, since $\Phi$ is continuous, it is enough to show that $\Phi$ fixes a dense subset of probability measures. We consider three different cases, corresponding to Lemma \ref{lem:WpTd-pot-p}.\\

\emph{Case (a) -- When $p=1$ holds.}
According to the density argument above, it suffices to prove that measures of the following form are fixed by $\Phi$:
$\mu = \sum_{k=1}^N w_k\delta_{x^k} \in\Wo(\Tn)$ where $N\in\N$, $\sum_{k=1}^N w_k = 1$, $w_k\geq 0$ for all $k$, and $x^1,x^2,\dots,x^N$ are pair-wise different points such that for all $k\in\{1,2,\dots,N\}$ we have
$$
\{x^1,x^2,\dots,x^N\} \cap \ler{\bigcup_{j=1}^n H\ler{x^k+\frac{1}{2}e^j,j}} = \emptyset.
$$
For such a measure we have the following for every $k$:
\begin{align*}
w_k &= \mu(\{x^k\}) = \lim_{s\to 0+}\frac{\mathcal{T}_\mu^p(x^k+s\cdot e^j) - 2\mathcal{T}_\mu^p(x^k) + \mathcal{T}_\mu^p(x^k-s\cdot e^j)}{2s} \\
& = \lim_{s\to 0+}\frac{\mathcal{T}_{\Phi(\mu)}^p(x^k+s\cdot e^j) - 2\mathcal{T}_{\Phi(\mu)}^p(x^k) + \mathcal{T}_{\Phi(\mu)}^p(x^k-s\cdot e^j)}{2s} \leq \Phi(\mu)(\{x^k\}),
\end{align*}
where we used \eqref{eq:WpTd-pot-p=1} in the last step.
Since $1 = \sum_{k=1}^N \mu(\{x^k\}) \leq \sum_{k=1}^N \Phi(\mu)(\{x^k\}) \leq 1$, we obtain that $\Phi(\mu) = \mu$, which completes the proof of the present case.\\

\emph{Case (b) -- When either $1<p<2$ or $p>2$ holds.}
Assume first that $1<p<2$. By a density argument we see that it suffices to show that $\Phi$ fixes every finitely supported measure $\mu = \sum_{k=1}^N w_k\delta_{x^k} \in\Wo(\Tn)$ where $x^1,x^2,\dots,x^N\in\mathbb{T}^n$ are points whose first coordinates are pairwise different. By Lemma \ref{lem:WpTd-pot-p} we have for all $k$ that
\begin{align*}
    w_k &= \mu(\{x^k\}) = 4^{\frac{p-2}{2}}\int_{H(x^k,1)} \ler{\frac{1}{4} + \varrho_{n-1}\ler{\check{x}_1,\check{y}_1}^2}^{\frac{p-2}{2}} ~\mathrm{d}\mu(y) \\
    & = \lim_{s\to 0+}\frac{\mathcal{T}_\mu^p(x^k+\frac{1}{2}e^1+s\cdot e^1) - 2\mathcal{T}_\mu^p(x^k+\frac{1}{2}e^1) + \mathcal{T}_\mu^p(x^k+\frac{1}{2}e^1-s\cdot e^1)}{-p\cdot4^{\frac{2-p}{2}}\cdot s} \\
    & = \lim_{s\to 0+}\frac{\mathcal{T}_{\Phi(\mu)}^p(x^k+\frac{1}{2}e^1+s\cdot e^1) - 2\mathcal{T}_{\Phi(\mu)}^p(x^k+\frac{1}{2}e^1) + \mathcal{T}_{\Phi(\mu)}^p(x^k+\frac{1}{2}e^1-s\cdot e^1)}{-p\cdot4^{\frac{2-p}{2}}\cdot s} \\
    & = 4^{\frac{p-2}{2}}\int_{H(x^k,1)} \ler{\frac{1}{4} + \varrho_{n-1}\ler{\check{x}_1,\check{y}_1}^2}^{\frac{p-2}{2}} d\Phi(\mu)(y) \leq \Phi(\mu)(H(x^k,1))
\end{align*}
where we have equation if and only if $\Phi(\mu)(H(x^k,1)) = \Phi(\mu)(\{x^k\})$. As
$$1 = \sum_{k=1}^N w_k \leq \sum_{k=1}^N \Phi(\mu)(H(x^k,1))\leq 1,$$
we must have $\Phi(\mu)(H(x^k,1)) = \Phi(\mu)(\{x^k\})$, and hence $\Phi(\mu) = \mu$. The very same argument works if $p>2$.\\

\emph{Case (c) -- When $p=2$ holds.}
First notice that by Lemma \ref{lem:WpTd-pot-p} we have
\begin{equation}\label{eq:hyperplane}
    \Phi(\mu)\ler{H(z,j)} = \mu\ler{H(z,j)} \qquad (z\in\mathbb{T}^n, \; j=1,\dots,n),
\end{equation}
which implies that $\Phi$ preserves the one-dimensional marginals. That is,
\begin{equation}\label{eq:marginals}
    \ler{\mathfrak{p}_j}_\#\Phi(\mu) = \ler{\mathfrak{p}_j}_\#\mu \qquad (j\in\{1,\dots,n\}, \mu\in\Wt(\Tn)),
\end{equation}
where $\mathfrak{p}_j\colon \mathbb{T}^n\to\mathbb{T}$, $\mathfrak{p}_j(x)=x_j$ is the projection map. Indeed, \eqref{eq:hyperplane} implies this if $\mu$ is supported on a finite set, and we obtain \eqref{eq:marginals} for general measures by a simple continuity argument.

We claim that measures supported on two points are left invariant by $\Phi$. Let us consider a measure $\mu = \alpha\delta_x+(1-\alpha)\delta_y$ where $x\neq y$ and $0<\alpha<1$. Without loss of generality we can assume that the representing vectors' coordinates satisfy the inequalities $-1/2\leq y_j-x_j<1/2$ for all $j$. Consider the following subset of $\mathbb{T}^n$:
$$
\cube := \big\{u\in\mathbb{T}^n\,|\,\forall j\in\{1,\dots,n\}:\, 0\leq \epsilon_j\cdot(u_j - x_j) \leq 1/2\big\},
$$
where for each $j$ we choose $\epsilon_j = 1$ if $x_j\leq y_j$, and $\epsilon_j = -1$ if $x_j>y_j$. By definition, $x,y\in\cube$. Note that by \eqref{eq:marginals}, we have \begin{equation}\label{eq:supp-phimu}
    S(\Phi(\mu))\subset\big\{z\in\mathbb{T}^n\;|\; \forall \;j\in\{1,\dots,n\}:\, z_j\in\{x_j,y_j\}\big\} \subset\cube.
\end{equation}
Furthermore, the subset $\cube$ as a metric space is isometrically isomorphic to the cube $[0,1/2]^n$ equipped with the usual Euclidean distance. Therefore the set of all probability measures supported on $\cube$, which we denote by $\Wt(\cube)$, can be considered as a subset of $\Wt(\R^n)$.

Note that \eqref{eq:marginals} implies that $\Phi$ maps $\Wt(\cube)$ onto itself. Denote by $\Phi_\cube$ the restricted isometry $\Phi|_{\Wt(\cube)}\colon\Wt(\cube)\to\Wt(\cube)$. Define the centre of mass of any $\mu\in\Wt(\cube)$ as the unique point $m(\mu)\in\cube$ such that $\dwt\ler{\delta_{m(\mu)},\mu} = \dwt\ler{\{\delta_z\colon z\in\cube\},\mu}$, and the standard deviation of $\mu$ as the distance $\sigma(\mu) := \dwt\ler{\delta_{m(\mu)},\mu}\in[0,\infty)$. Since $\Phi_\cube$ leaves every Dirac mass invariant, we obtain that $\Phi_\cube$ preserves the centre of mass an standard deviation of measures, that is,
$$m(\Phi_\cube(\mu)) = m(\mu),\;\;\; \sigma(\Phi_\cube(\mu)) = \sigma(\mu)\qquad (\mu\in\Wt(\cube)).$$
Therefore, according to \cite[Lemma 6.2]{Kloeckner-2010}, for all $\mu,\nu\in\Wt(\cube)$ the following equivalence holds: $\langle v_1-v_2, w_1-w_2\rangle = 0$ for all $v_1,v_2\in S(\mu)$, $w_1,w_2\in S(\mu)$ if and only if the same holds for all $v_1,v_2\in S(\Phi_\cube(\mu))$, $w_1,w_2\in S(\Phi_\cube(\mu))$.

On the one hand, if $n\geq 3$ and $x_j\neq y_j$ for all $j\in\{1,\dots,n\}$, then there exists a $\nu := \alpha\delta_z+(1-\alpha)\delta_u\in\Wt(\cube)$ ($z\neq u$), such that the points $x,y,z,u\in\cube$ satisfy the following conditions:
$\langle x-y,z-u\rangle = 0$, but $\langle \xi^1-\xi^2, \zeta^1-\zeta^2 \rangle \neq 0$ holds for all other
$$\xi^1,\xi^2\in\big\{\xi\in[0,1/2]^n\,|\, \forall\;j\in\{1,\dots,n\}:\, \xi_j\in\{x_j,y_j\}\big\},~~~\xi^1\neq\xi^2$$ and $$\zeta^1,\zeta^2\in\big\{\zeta\in[0,1/2]^n\,|\, \forall\;j\in\{1,\dots,n\}:\, \zeta_j\in\{z_j,u_j\}\big\}, ~~~\zeta^1\neq\zeta^2.$$
Note that then
\begin{equation}\label{eq:supp-phinu}
    S(\Phi(\nu))\subseteq\big\{v\in\mathbb{T}^n\,|\, \forall \;j\in\{1\dots,n\}:\, v_j\in\{z_j,u_j\}\big\} \subset\cube.
\end{equation}
Since the supports of $\mu$ and $\nu$ are perpendicular to each other, so must be the supports of their images. However, by our assumptions and \eqref{eq:supp-phimu}--\eqref{eq:supp-phinu} imply $\Phi(\mu) = \mu$, $\Phi(\nu) = \nu$. This, together with a simple continuity argument proves that indeed $\Phi$ leaves every measure fixed that are supported on at most two points, provided that $n\geq 3$.

On the other hand, if $n=2$, then consider a $\nu := \alpha\delta_{z}+(1-\alpha)\delta_{u}\in\Wt(\cube)$ such that $\langle x-y, z-u \rangle = 0$. Elementary geometric observation then gives that either $S(\Phi(\mu)) = \{(x_1,x_2),(y_1,y_2)\}$ and $S(\Phi(\nu)) = \{(z_1,z_2),(u_1,u_2)\}$, or $S(\Phi(\mu)) = \{(x_1,y_2),(y_1,x_2)\}$ and $S(\Phi(\nu)) = \{(z_1,u_2),(u_1,z_2)\}$. By \eqref{eq:marginals}, the latter cannot happen unless $\alpha=1/2$. Therefore, by continuity, $\Phi$ leaves every measure fixed that are supported on at most two points, also in this case.

From here it suffices to show that any finitely supported measure is left fixed by $\Phi$. Consider a $\mu = \sum_{k=1}^N w_k\delta_{x^k} \in\Wo(\Tn)$ where $N\in\N$ and $x^1,x^2,\dots,x^N$ are pair-wise different points. Define the finite set
$$
F:=\big\{\xi\in\mathbb{T}^n\,|\, \forall\;j\in\{1,\dots,n\}:\, \xi_j\in\{x^1_j,x^2_j,\dots,x^N_j\}\big\}.
$$
By \eqref{eq:marginals}, we have $S(\Phi(\mu))\subseteq F$. Consider an arbitrary element $u\in F$ and observe that there exists two points $x,y\in\mathbb{T}^n$ such that $B(x,y)\cap F = \{u\}$. Note that since $\mu(B(x,y)) = \mu(\{u\})$, it is enough to show that $\Phi(\mu)(B(x,y)) = \mu(B(x,y))$. The latter is a consequence of the following equivalence which holds for all $\eta\in\Wp(\Tn)$, $\alpha\in[0,1]$:
\begin{align*}
    &\dwt\ler{\eta,\{a\delta_x+(1-a)\delta_y\colon0\leq a\leq 1\}} = \dwt\ler{\eta,\alpha\delta_x+(1-\alpha)\delta_y} \\
    &\iff \mu\ler{\{z\colon \varrho(x,z) < \varrho(y,z)\}} \leq \alpha \leq \mu\ler{\{z\colon \varrho(x,z) \geq \varrho(y,z)\}}.
\end{align*}
This concludes the proof.
\end{proof}

\section{Isometric rigidity of the Wasserstein sphere} Similarly to the case of the torus, we prove first that if $\Phi$ is an isometry of $\Wp(\Sn)$ then there exists an isometry $\psi\colon \mathbb{S}^n\to \mathbb{S}^n$ such that
$$
\Phi(\delta_x) = \psi_\#\delta_x = \delta_{\psi(x)} \quad (x\in\mathbb{S}^n).
$$
The diameter of $\Wp(\Sn)$ is $\pi$ and this maximal distance is achieved if and only if the two measures in question are Dirac masses concentrated on antipodal points. This property must be preserved by isometries, and therefore the image of every Dirac measure is a Dirac measure again. Furthermore, since $\sphericalangle(x,y)=\dwp(\delta_x,\delta_y)$, the map $\psi$ defined via $\Phi(\delta_x):=\delta_{\psi(x)}$ is an isometry. Note that every $\psi\in\mathrm{Isom}(\mathbb{S}^n)$ is the restriction of  an orthogonal transformation of the underlying space. Again, we are going to use the Wasserstein potential of the measure $\mu\in\Wp(\Sn)$
\begin{equation}\label{eq:WpSg-pot}
	\mathcal{T}_\mu^p\colon \mathbb{S}^n\to \R, \;\; x\mapsto \dwpp(\mu,\delta_x)=
    \int_{\mathbb{S}^n} \sphericalangle(x,y)^p ~\mathrm{d}\mu(y).
\end{equation}
Since $\Phi(\delta_x) = \delta_{\psi(x)}$ for all $x\in\mathbb{S}^n$, we have $\mathcal{T}_{\Phi(\mu)}^p\circ\psi=\mathcal{T}_\mu^p$. Indeed,
\begin{equation}\label{eq:WpSg-phipot}
    \mathcal{T}_{\Phi(\mu)}^p(\psi(x))=\dwpp(\Phi(\mu),\delta_{\psi(x)})=\dwpp(\Phi(\mu),\Phi(\delta_x) = \dwpp(\mu,\delta_x)=\mathcal{T}_\mu^p(x)
\end{equation}
for all $x\in\mathbb{S}^n$. The following is an analogue of Lemma \ref{lem:WpTd-pot-p}.
\begin{lemma}\label{lem:WpSg-pot-p>=1}
Let $n\in\N$, $p\geq 1$, $x,z\in\mathbb{S}^n$, $\sphericalangle(x,z) = \pi/2$, and $\mu\in\Wp(\Sn)$. We have
\begin{align}\label{eq:WpSg-pot-p>=1}
    &\lim_{s\to 0+}\frac{\mathcal{T}_\mu^p(\cos s \cdot x + \sin s \cdot z) - 2\mathcal{T}_\mu^p(x) + \mathcal{T}_\mu^p(\cos s \cdot x - \sin s \cdot z)}{s}\nonumber \\
    & \hspace{1.5cm} = \left\{\begin{array}{cc}
    -2 \cdot \mu(\{-x\}) + 2 \cdot \mu(\{x\}), & \text{if}\; p=1 \\
    -2p\pi^{p-1} \cdot \mu(\{-x\}), & \text{if}\; p>1
\end{array}\right..
\end{align}
\end{lemma}

\begin{proof}
Since the left hand-side of \eqref{eq:WpSg-pot-p>=1} is
\begin{align*}
    & \lim_{s\to0+} \bigg( \mu(\{x\})\frac{2\cdot s^p}{s} + \mu(\{-x\})\frac{2\cdot(\pi-s)^p - 2\cdot\pi^p}{s} \\
    &\hspace{0.3cm}+ \int_{\mathbb{S}^n\setminus\{-x,x\}} \frac{\sphericalangle(\cos s \cdot x + \sin s \cdot z,y)^p - 2\sphericalangle(x,y)^p + \sphericalangle(\cos s \cdot x - \sin s \cdot z,y)^p}{s} ~\mathrm{d}\mu(y) \bigg),
\end{align*}
it suffices to show that the limit of the above integral is zero. Note that the function $t\mapsto t^p$ is Lipschitz on the interval $[0,\pi]$ with a constant, say, $K>0$. Hence the integrand is bounded, as can be seen by the following estimation (we use the triangle inequality in the last step)
\begin{align*}
    &\left| \frac{\sphericalangle(\cos s \cdot x \pm \sin s \cdot z,y)^p - \sphericalangle(x,y)^p}{s} \right| \leq K\cdot\left| \frac{\sphericalangle(\cos s \cdot x \pm \sin s \cdot z,y) - \sphericalangle(x,y)}{s} \right| \\
    & = K\cdot\left| \frac{\sphericalangle(\cos s \cdot x \pm \sin s \cdot z,y) - \sphericalangle(x,y)}{\sphericalangle(\cos s \cdot x \pm \sin s \cdot z,x)} \right| \leq K.
\end{align*}
Observe that for all $y\in\mathbb{S}^n\setminus\{-x,x\}$ the function
\begin{equation}\label{eq:diffhato}
    s\mapsto \sphericalangle(\cos s \cdot x + \sin s \cdot z,y)^p = \arccos^p\inner{\cos s \cdot x + \sin s \cdot z}{y}
\end{equation}
is differentiable at $s=0$. Hence the point-wise limit of the integrand is the constant $0$ function. Applying the Lebesgue dominant convergence theorem finishes the proof.
\end{proof}

Using Lemma \ref{lem:WpSg-pot-p>=1} we can prove the main result of this section, namely that the $p$-Wasserstein sphere is isometrically rigid for all $p\geq1$. Since $\mathbb{T}$ can be identified with $\mathbb{S}^1$, this theorem completes the case of the torus as well.
\begin{theorem}\label{thm:WpSg}
Let $n\geq1$ and $p\geq1$. Then the $p$-Wasserstein sphere $\Wp(\mathbb{S}^n)$ is isometrically rigid, that is, the push-forward operation: $\#:\mathrm{Isom}(\mathbb{S}^n)\to\mathrm{Isom}(\Wp(\mathbb{S}^n))$ is surjective.
\end{theorem}

\begin{proof}
Assume that $\Phi\colon\Wp(\Sn)\to\Wp(\Sn)$ is an isometry. We have to show that there exists an isometry $\psi\in\mathrm{Isom}(\mathbb{S}^n)$ such that
\begin{equation}\label{eq:WpSg}
	\Phi(\mu) = \psi_\#\mu \qquad (\mu\in\Wp(\Sn)).
\end{equation}
We know that \eqref{eq:WpSg} holds for all Dirac masses with some isometry $\psi$. It suffices to show that \eqref{eq:WpSg} holds also for measures whose support is a finite set not containing any pair of antipodal points, as these form a dense subset of $\Wp(\Sn)$. Consider a measure $\mu\in\Wp(\Sn)$ with such properties, say, $\mu = \sum_{j=1}^N w_j\delta_{x_j}$ with $N\in\N$, $\sum_{j=1}^N w_j=1$, $\{x_1,\dots,x_N\}\cap\{-x_1,\dots,-x_N\}=\emptyset$ and $x_1,\dots,x_N$ pair-wise different. From here we distinguish between three cases.

First, if $p> 1$, then by Lemma \ref{lem:WpSg-pot-p>=1} we infer the following for all $j\in\{1,\dots,N\}$:
\begin{align*}
    w_j &= \mu(\{x_j\})= \lim_{s\to 0+}\frac{1}{-2p\pi^{p-1}s} \bigg(\mathcal{T}_\mu^p(\cos s \cdot (-x_j) + \sin s \cdot z_j) \nonumber\\
    &\hspace{6.1cm} - 2\mathcal{T}_\mu^p(-x_j) + \mathcal{T}_\mu^p(\cos s \cdot (-x_j) - \sin s \cdot z_j)\bigg) \nonumber\\
    &= \lim_{s\to 0+}\frac{1}{-2p\pi^{p-1}s} \bigg(\mathcal{T}_{\Phi(\mu)}^p(\cos s \cdot (-\psi(x_j)) + \sin s \cdot \psi(z_j)) - 2\mathcal{T}_{\Phi(\mu)}^p(-\psi(x_j)) \nonumber\\
    &\hspace{6.1cm}+ \mathcal{T}_{\Phi(\mu)}^p(\cos s \cdot (-\psi(x_j)) - \sin s \cdot \psi(z_j))\bigg) \nonumber\\
    &= \Phi(\mu)(\{\psi(x_j)\})
\end{align*}
where we used \eqref{eq:WpSg-phipot} and that $\psi$ is the restriction of a linear isometry of the underlying real Hilbert space. This implies \eqref{eq:WpSg} for $\mu$ and completes the proof of this case.

Second, if $p=1$, then using the same calculation as above, we arrive at
$$ w_j = \mu(\{x_j\}) = \Phi(\mu)(\{\psi(x_j)\}) - \Phi(\mu)(\{-\psi(x_j)\}).$$
Since $1 = \sum_{j=1}^N w_j \leq \sum_{j=1}^N \Phi(\mu)(\{\psi(x_j)\}) \leq 1$, we must have $\Phi(\mu)(\{-\psi(x_j)\}) = 0$ for all $j$, hence this case is done too.
\end{proof}

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About the Children's Department: Check Out Kidmin1124 . . . I Wrote About Teaching Kids the True Meaning of Worship - Obedience is Key!
Check Out Kidmin1124 . . . I Wrote About Teaching Kids the True Meaning of Worship - Obedience is Key!
I am delighted and honored to write for kidmin1124 and for the next several weeks I'll be posting about Teaching Kids the True Meaning of Worship . . . this week's post looks at how obedience is key to worship! Please take a moment to check this post out and make kidmin1124 a regular source for information and resources for your children's ministry!


text:
Тодор Иванов Кацаров е деец на БКП

Биография
Тодор Кацаров е роден на 22 февруари 1892 г. в село Лешница, Ловешко. Начално образование завършва в родното си село. Прекъсва ученето в прогимназията и заедно с бащата Иван Кацаров заминава на гурбет в Румъния. Наследява от него занаята бъчварство. Работи в Плевен, Червен бряг и Мездра.

През 1915 г. отваря собствена бъчварска работилница в Ловеч. Ориентира се към левите идеи. Член на БРСДП (т.с.) от 1918 г. Близък приятел на Борис Баев. Участва в работата на Занаятчийски конгрес (1923). Член на Ловчански комитет на БКП и неговата военна комисия.

Арестуван след атентата в църквата "Света Неделя". През нощта на 24 април 1925 г. заедно с Борис Баев е изведен от Ловешкото околийско полицейско управление и екзекутиран на прилежащата улица.

Източници

 сб. Неугасващи звезди, Л., 1983, с. 91 – 96
 www.zadruga.ru

Български комунисти до 1944 година
Жертви на Априлските събития в България (1925)
Хора от област Ловеч
Починали в Ловеч

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text:
Q: Picklist deactivate link I want to know the answer. I created a picklist. That picklist values has Deactivate link. When that Link will be appear?

A: That is a setup action that will Deactivate the Picklist Option(Remove the option from the picklist for users)). It (deactivate) will never be presented to the users.


text:
Осоавиахимовец Украины — бесхвостый одноместный планёр конструкции . Изготовлен на Харьковском авиационном заводе в 1934 г.; по другим данным — на Харьковском фотомузыкальном заводе в 1933 г.

Лётчиком Л. С. Рожковым на планёре был выполнен каскад фигур высшего пилотажа, благодаря этому конструкция была замечена, а конструктор получил известность и в 1934 г. планёр демонстрировался на авиавыставках в Копенгагене и Париже.

Конструкция

Осоавиахимовец Украины имеет аэродинамическую схему бесхвостка. Конструкция деревянная с работающей обшивкой. Крыло прямой стреловидности с концевыми шайбами. Имеет рули направления с аэродинамической компенсацией. Элероны обеспечивают одновременное управление в продольном и поперечном направлениях. На верхней поверхности крыла имеется кабина обтекаемой формы.

Литература
 А. П. Красильщиков. Планеры СССР. Москва, Машиностроение 1991 г.
 Двухместный планер-бесхвостка  конструкции П. Г. Бенинга "П.П.Постышев". 1934 г. // Авиация и Космонавтика 2006-04
 А. Лашков. Из истории собственных наименований воздушных судов России  // Авиация и Космонавтика 2006-04

См. также

 Abrial A-12 Bagoas
 Планёр
 Планеризм

Примечания

Планёры СССР
Летающее крыло


text:
The Touchy-Feely (But Totally Scientific!) Methods Of Wallace J. Nichols
How does a visionary marine biologist convince brain researchers to help him revolutionize ocean conservation? With lots of hugs, a million blue marbles, and one very unorthodox conference.
Wallace J. Nichols (Photo: Jeff Lipsky)
THE PHILIPPINE coral reef tank inside the California Academy of Sciences in San Francisco is 25 feet deep and holds 212,000 gallons of water, making it one of the largest exhibits of living coral anywhere in the world. It is the centerpiece of the academy's Steinhart Aquarium and hosts hundreds of coral species, a couple thousand colorful fish, plus sharks, stingrays, and numerous smaller creatures, like sea anemones and snails. There are five windows affording looks inside, the biggest of which, at 16 and a half feet tall and almost 30 feet wide, makes a sweeping arc in front of a dimly lit standing area backed by several rows of benches. It was designed to offer visitors a panoramic, theater-like view of life in the tank and is among the museum's most popular attractions. It's Wallace J. Nichols's favorite spot in the building.
Find more ways to make the planet a better place in the Outside Guide to Fixing the World.
Wallace J. Nichols
Nichols, 44, is a biologist and research associate at the academy who made a name for himself in the mid-1990s when he tracked a loggerhead turtle that swam from Baja, Mexico, to Japan, the first time anyone had recorded an animal swimming an entire ocean. He has done fieldwork in waters around the globe and spends most of his waking hours thinking and talking about the ocean, but when he's in front of that big window at the aquarium, he doesn't watch the fish. He watches the people.
"Whether it's a 92-year-old or a two-year-old, when they come into that blue space, something happens," Nichols says. They grow quiet and calm, but there's more to it than that. When couples walk in, they frequently start holding hands. He says that if you ask people here what they're feeling, they'll struggle for words. Nichols finds this fascinating. He also believes that if we can understand what really happens to us in the presence of the ocean—which brain processes underlie our emotional reactions—it could bring about a radical shift in conservation efforts. If we learn precisely why we love the ocean, his thinking goes, we'll have an immensely powerful new tool to protect it.
Not surprisingly, this theory can strike many of his peers as soft. " 'You must be from California.' That's the first response," Nichols says. (He lives north of Santa Cruz, though he was raised in New Jersey.) But Nichols's credibility as a scientist, along with his charm and passion, have enabled him to rally excitement for his ideas among a diverse constituency of researchers and activists. In the past couple of years, he's become a sought-after speaker, giving dozens of presentations at a wide mix of venues, from TEDx to adventure-travel trade shows to environmental symposiums. His pitch: More data on rising sea temperatures or plastic pollution or disappearing creatures won't do anything for ocean conservation. Instead, we need to study our own minds.
Nichols envisions cognitive neuroscientists constructing detailed models of brain activity for experiences like sitting on a beach, then using their findings to drive public policy. "If I walk into a meeting of a coastal zoning commission and say, 'I think people listening to the ocean is good for them,' you'd see all the eyeballs in the room rolling," says Nichols. "But if I walk in and say, 'This is my friend the Stanford neuroscientist, and his research using brain scans shows that sitting by the ocean has the same calming effects as meditation on reducing stress,' suddenly access to the coast becomes a public-health issue."
It's a viable fantasy that derives from the fact that Nichols himself isn't a neuroscientist. Unable to test his hypotheses, he's launched a campaign to create a new field of study he calls neuro-conservation. His hope is to inspire cognitive scientists to examine these fundamental questions. As he sees it, it's a ripe invitation: Who wants to know what happens when our most complex organ meets the planet's largest feature?
"My role is to be the catalyst and cheerleader," he says. "But the question is, How do you turn this big idea into a movement?"
THE FIRST time I met Nichols, he gave me a blue marble. It was sort of awkward. "Hold it at arm's length," he said. "That's what the Earth looks like from a million miles away—a water planet. Now hold it up to your eye and look at the sun. If water were inside, it would contain virtually every element. Now think of someone who's doing good work for the ocean. Hold it to your heart: think of how it would feel to you and to them if you randomly gave them this marble as a way of saying thank you."
We were seated outside the Academy of Sciences on a late-winter afternoon. Nichols, who goes by J., was dressed in a casual button-down blue shirt, brown cords, and leather boots and wearing a perfectly manicured salt-and-pepper stubble beard. He looked directly into my eyes, speaking in a slow, even canter that was mildly hypnotic, the vestige of a stutter he overcame 25 years ago by forcing himself to make turtle presentations to school groups.
The marble shtick may have made me uncomfortable, but the last line stuck with me; I imagined myself giving the marble to an old friend. Turns out I'm not the only one to fall under this spell. Nichols tried it out for the first time in 2009, during a talk at the New England Aquarium in Boston, and the audience response was overwhelming. He figured he was on to something, so he set up a simple website, BlueMarbles.org, and decided "to try and see how big we could make this thing with no budget or strategic plan." Nichols now estimates that there are as many as a million of his blue marbles in circulation around the planet. They have made it into the hands of Jane Goodall, Harrison Ford, James Cameron, E. O. Wilson, and four-time Iditarod champion Lance Mackey, who carried one during this year's race.
Nichols's success at reaching large numbers of people on an emotional level both underscores the premise of his theories and makes it harder to dismiss him as a left-coast flake. Several times over the past six months, I watched him captivate audiences with a clever Trojan-horse narrative: I'm a scientist, but—surprise!—I want to talk about how much we all love the ocean. During one lecture at Stanford, he implored graduate students to remember that, as conservationists, "we have the power of happiness on our side."
For environmentalists struggling to find a message with staying power, Nichols's feel-good approach offers a compelling alternative to the usual tactic of scaring people into action with bad news about extinctions or global warming. "Hell, we've tried everything else," says Nature Conservancy scientist M. Sanjayan. "We've tried to price nature. We've tried to stand and protest. We've tried every way we know to get people to see what we've seen, and we've been failing."
Nichols blames these failures on the detached way scientists gather and share information. When he was studying Baja's sea turtles as a doctoral student at the University of Arizona in the mid-1990s, he hired fishermen and former turtle poachers to help collect data. The research was interesting, but Nichols was even more intrigued by the intense and often conflicting feelings locals had for the animals. He convened a gathering of everyone—"turtle lovers, turtle eaters, biologists, NGOs"—and they formed an activist network called Grupo Tortuguero.
Nichols was energized, but his academic advisers were skeptical. "'You're organizing fishermen—where's the biology?" they asked. He was told to avoid the human element in his thesis. "It made no sense," says Nichols. "The changes happening in the ocean and with those turtles were driven by humans."
He was similarly progressive in his research methods. Early on in Baja, he tagged a female turtle his team had named Adelita with a GPS transponder and posted her coordinates online as she made a never-before-recorded crossing of the Pacific to Japan. His colleagues were horrified. "'They said, 'Someone could steal your data!' " Nichols laughs. "My response was: 'And do what with it? Save turtles?' "
TODAY, NICHOLS applies this same open-source spirit to what he calls his "fluid" career. He's spent most of the past decade "hopping between grants" while continuing to publish research on turtles, often coauthored by graduate students he advises. He works with a number of environmental groups and recently created SeetheWild.org, a nonprofit that connects adventure travelers with conservation projects in exotic locations. His office, a 1954 Airstream trailer parked at a friend's organic strawberry farm off California's Highway 1, is also the headquarters for Slowcoast, an initiative he recently helped launch to draw tourists to the mostly empty stretch between Half Moon Bay and Santa Cruz, with future revenue supporting local public-school lunch reform.
In 2009, Nichols applied for a grant from the Pew Charitable Trusts to fund a year of neuroscience classes at Harvard and MIT as a way to kick-start his neuroconservation campaign. He posted his 12-page proposal on his website the day he submitted it. "I just put it out there," says Nichols. "I was basically saying, Somebody do this, please." Pew turned him down. "They didn't get it," he says. "Which was not a surprise; there's a reason this research hasn't been done."
Indeed, it's one thing to get forward-thinking scientists excited about a hypothesis, but it's another to get institutions to dedicate dollars to test it. Back in 1984, E. O. Wilson popularized the term biophilia to describe what he considers humans' inherent attraction to "life and lifelike processes." It became a popular theory but wasn't something Wilson or anyone else initially sought to prove. Now cognitive researchers are investigating what—exactly—nature does to our minds, with studies showing improved attention span and memory, and reduced stress, among other benefits. (See "You Need a Braincation") Designing experiments to study how our brains react to the ocean wouldn't be especially difficult, Nichols says. (Among other ideas, he envisions immersing lab subjects in ocean sounds and images while taking brain scans.) But by focusing so explicitly on feelings, Nichols is emanating the kind of New Age vibes that many neuroscientists reflexively avoid. Environmentalists, on the other hand, are prone to question the conservation value of any data such studies might produce. Knowing that something is good for us won't necessarily change our actions (see: exercise, diet, sleep). Plus, what if studies show that a polluted, depleted ocean calms our minds as much as a vibrant one does?
Nichols remains convinced that a researcher will take up his cause soon. Meanwhile, with no regular salary (he isn't paid by the California Academy of Sciences), he has struggled to support his wife, Dana, who manages Slowcoast, and their two grade-school-age daughters while marshaling his neuro-conservation drive. His solution is 100BlueAngels.org, a site he established earlier this year that asks people to support him with monthly contributions. Recently, he was on pace to bring in what would amount to a $43,000 salary. He supplements this with modeling gigs, which he's taken since college (look for his mug in Gap stores during the holidays), but has had to borrow against his home and take a $10,000 loan from his father.
"People ask me, 'Why don't you sock this idea away until you can get the money and do the research yourself—be the pioneering guy and get all the credit?' " Nichols says. "That's just not as interesting to me. I'd rather hang it out there. Throw a conference. Create the chatter. And hopefully inspire some neuroscientists to ask some of these questions."
NICHOLS DOES THROW a hell of a conference. This past June, for his Bluemind Summit, which he billed as a gathering that would "forever link the studies of mind and ocean," Nichols wrangled a remarkably eclectic mix of neuro-nerds, greens, adventurers, futurists, artists, a video-game inventor, a high-end realtor, and one very gnarly big-wave surfer to the Academy of Sciences for a marathon day of presentations. The lineup alone demonstrated Nichols's flair for making science both relevant and accessible.
Early on, Eric Johnson, a nattily attired realtor with Sotheby's, cited the premium people are willing to pay for a water view. "We can see the storms or pirates approaching," said Johnson, noting that wealthy owners of high-rise apartments are automatic environmentalists because "clean, clear water keeps property values up." Marcus Eriksen, a marine scientist known for a 2008 crossing of the Pacific in his Junk, a raft made primarily of plastic debris, discussed our basic biological reasons for living on the seashore: lots of food and few predators. Ocean activist Fabien Cousteau noted that humans and whales share the mammalian reflex, which allows us to stay underwater for long periods without breathing, while Maverick's surfer Jeff Clark talked about his learned ability to sense things like the presence of sharks. "Listening to the feedback that the ocean provides will keep you surfing for years," he concluded.
There were some lighter touches. A cellist kicked things off with a medley "full of ocean-ness"—a Nichols request—and each presenter was introduced with a six-word bio ("passion, teacher, vegetables…"). At one point, Jaimal Yogis, author of Salt-water Buddha, about his quest to find Zen through surfing, led everyone in meditation. Hugs happened.
Still, several cognitive scientists were also on hand to offer serious theories about the brain-on-ocean dynamic. Philippe Goldin, a clinical psychologist and a neuroscientist at Stanford, cited research showing that meditation helped some people with anxiety regain their calm after an emotional event, then speculated that similar processes might be going on in the brains of surfers, who learn to react immediately to a rising swell, then "enjoy the time between waves" after a set passes. Michael Merzenich, a professor emeritus of neuroscience at the University of California at San Francisco and one of the foremost authorities on neuro-plasticity—the brain's ability to rewire itself—suggested that our attraction to the ocean may derive from its lack of physical markers. On land, we are constantly mapping our environment in our minds so we can pick out dangers (snake!) amid landmarks (tree, bush, rock). Looking over a calm sea is akin to closing our eyes. And when something does emerge on the surface, it captivates us.
Come nightfall, the summit turned into a very Northern California kind of happening. We ambled into the academy's planetarium for some ocean-themed readings and a visualization exercise, before combining forces with the academy's regular weekly party, which turns the place into a sort of geek-cool dance club. The night's topic was sustainable seafood (purposefully synced with Bluemind), so along with the DJs and organic cocktails there were free local oysters, interactive displays about overfishing and exotic marine creatures, and pamphlets on smarter sushi eating.
Toward the end of the night, on a stage set in a cavernous hall of African wildlife dioramas, a dance troupe in shimmery blue tights and tank tops performed a number called "Aqua." At one point, oceanographer Sylvia Earle's voice was looped in over house beats: Imagine an ocean without fish. The ocean is alive.
LET'S SAY neuroscience does demonstrate that sitting by the ocean provides a unique and primal kind of stimulus that washes away stress. Would our reaction necessarily be a strong desire to protect it? Or might we all instead just selfishly want our own Malibu beach pad?
That sums up the attitude of Michael Soulé, who pioneered the field of conservation biology in the 1970s and later chaired the environmental-studies program at the University of California at Santa Cruz. Soulé, as it happens, has been reading neuroscience studies for a book he's writing about our inability to preserve nature. "I admire Nichols's work and his excitement, but the field of cognitive neuroscience can lead you to the opposite conclusions," he told me. Like Nichols, Soulé believes that emotions drive our behavior, but his analysis of fMRI studies, which allow neuroscientists to observe the brain at work, has him convinced that humans are "hardwired to be very self-centered and self-biased." Understanding why chilling out by the ocean makes us feel great won't motivate a shift in our fundamentally greedy behavior. We are born to be "good consumers but not good conservationists."
Nichols's response: Of course we're self-centered. That's why knowing the mechanisms behind something that makes us happy is so powerful—it resonates with our innate desire to feel good, whether we get that feeling from sitting on a beach or protecting it. Regardless, he argues, understanding what goes on in our brains when we're in the presence of the ocean can only help us craft a more persuasive conservation agenda.
Both Soulé and Nichols cite Antonio R. Damasio's popular 1994 book Descartes' Error as an influence. Damasio, a neurobiologist, argued that humans can't reason or make decisions without emotion, an idea that ran counter to accepted theories about the division between our rational and emotional selves. Subsequent research showing that the neural pathways of high-level cognition route through our limbic system, the brain's primitive hub where emotions and memories are processed, essentially proved him right. This is why, as marketers and politicians well know, you're most likely to garner dollars or votes by pitching to people's hearts instead of their heads.
The same idea holds true for the environment. At Bluemind, Dawn Martin, the president of SeaWeb, a nonprofit dedicated to strategic messaging on ocean issues, made the case that facts are meaningless unless they're communicated in a way that strikes an emotional chord. She pointed to the collapse of the North Atlantic swordfish population in the mid-1990s, which activists fought unsuccessfully with statistics. Then SeaWeb partnered with the Natural Resources Defense Council to recast the crisis, creating the Give Swordfish a Break campaign, which included an ad with a tiny swordfish on a plate, a pacifier in its mouth: we're eating the babies. By 2000 the federal government had closed swordfish nursery areas to fishing.
Nichols appreciates the value of neuroscience-informed social marketing, but he insists that "that's the least interesting part" of what he hopes to learn. "People say, 'Oh, you're just going to be an environmental propagandist.' No. I have no interest in that," he told me. "I want you to understand what's happening in your head. My interest is in taking you along for the ride."
NOT LONG AGO, I took my two-year-old son to the Academy of Sciences. Almost from the moment we got inside, he was running—past alligators in a pond and snakes in terrariums, up the path that winds around the four-story-tall rainforest exhibit and its free-flying birds and butterflies, around two life-size model giraffes, all while eating crackers.
Only when we descended the stairs to the Philippine reef did his pace slow. He walked past a couple of smaller windows and came to a halt in front of the massive panoramic view, the cool blue light in the room fluttering about him as shafts of sunlight pierced the water. On one of the benches facing the window, a young mother nursed her infant. People milled about, whispering. My son put his hands on the window and stared without moving or talking for a full 30 seconds—an eternity in toddler time.
Then he spun around, looked me in the eye, and said, "I want to go in there!"
From Outside Magazine, Dec 2011
Filed To: ScienceNatureWellnessMental ConditioningOutside Features
Lead Photo: Jeff Lipsky
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redpajama_set_name: RedPajamaCommonCrawl


Today you will see both sides of Delhi – the legacy of its time as the great capital of the glorious Mughal empire and as the centre of the British Raj. Old Delhi is the old walled city, founded by the Mughal Emperor Shah Jehan who brought about Delhi's glorious renaissance and his legacy is clear to see today. Start the day with a cycle rickshaw ride through Old Delhi, where you will see the formidable Red Fort, built behind huge red sandstone walls that rise above the old city. Chandni Chowk, the avenue heading west from the Red Fort, was once the commercial centre of the city and it is filled with twisting lanes and crowded bazaars. Peer through its porticos to see intense commerce in action. You will then visit Raj Ghat, the memorial to Mahatma Gandhi; Jama Masjid, the largest mosque in India; Qutub Minar tower; and Humayun's Tomb, generally believed to be the prototype of the Taj Mahal.
In the heart of New Delhi you will see the imposing India Gate, which stands 42m high and commemorates the 70,000 Indian soldiers who lost their lives fighting for the British in the First World War. Travel on to President House, the official residence of the Viceroy of India, with its private garden designed by Lutyens. Finally, visit Parliament House, the symbol of Indian democracy.
Your guide will be local to Delhi, someone who can bring the legacy of Delhi's glorious past to life and put into context the fascinating city that Delhi has become today.


More than $19 million has now been committed to the impact investing collaborative announced by the Minnesota Council on Foundations and members early in 2017. The Minnesota-focused fixed income bond fund, part of RBC Global Access Management's Community Access Capital, aims to reach $20 to 25 million this year. The fund invests in securities backed by affordable housing and small business projects throughout Minnesota.
Special thanks to Otto Bremer Trust for increasing support for two more years of Susan Hammel's impact investing executive residency. Thanks to OBT's generosity and the Bush Foundation's support of her second year, Susan is available to talk with any MCF member about the fund and answer any other impact investing questions that arise.
MCF members—Bush Foundation, The McKnight Foundation and Otto Bremer Trust—are the lead institutions participating in the impact investing collaborative. They are joined by the Minneapolis Foundation, Minnesota Council on Foundations, Mortenson Family Foundation and Sundance Family Foundation. At least eight other foundations are currently bringing the opportunity through their internal review processes.
Pictured below: The Winona Arms Assisted Living Facility in Winona, Minnesota, is an example of a typical RBC Access Capital investment. Winona Arms is covered under the Section 202 Program, which helps expand the supply of affordable housing with supportive services for the elderly. It provides very low-income elderly with options that allow them to live independently but in an environment that provides support activities such as cleaning, cooking and transportation.


text:
Metropolitan Statistical Area (MSA)
0.506% 1-year increase
In 2020, Rockford, IL had a population of 337k people with a median age of 39.7 and a median household income of $56,899. Between 2019 and 2020 the population of Rockford, IL declined from 338,356 to 336,928, a −0.422% decrease and its median household income grew from $56,027 to $56,899, a 1.56% increase.
The 5 largest ethnic groups in Rockford, IL are White (Non-Hispanic) (70.2%), Black or African American (Non-Hispanic) (10.9%), White (Hispanic) (10.1%), Other (Hispanic) (2.66%), and Two+ (Non-Hispanic) (2.38%).
None of the households in Rockford, IL reported speaking a non-English language at home as their primary shared language. This does not consider the potential multi-lingual nature of households, but only the primary self-reported language spoken by all members of the household.
95.2% of the residents in Rockford, IL are U.S. citizens.
The largest universities in Rockford, IL are Rock Valley College (1,446 degrees awarded in 2020), Rasmussen University-Illinois (943 degrees), and Rockford University (307 degrees).
In 2020, the median property value in Rockford, IL was $126,700, and the homeownership rate was 67.9%.
Most people in Rockford, IL drove alone to work, and the average commute time was 23.8 minutes. The average car ownership in Rockford, IL was 2 cars per household.
Rockford, IL borders Chicago-Naperville-Elgin, IL-IN-WI, Freeport, IL, Janesville-Beloit, WI, Madison, WI, Rochelle, IL, and Whitewater-Elkhorn, WI.
About the photo: Veteran's Memorial Hall Rockford, Il
cameraPhoto by EarlRShumaker
Rockford, IL is home to a population of 337k people, from which 95.2% are citizens. As of 2020, 8.79% of Rockford, IL residents were born outside of the country (29.6k people).
In 2018, there were 6.46 times more White (Non-Hispanic) residents (239k people) in Rockford, IL than any other race or ethnicity. There were 37k Black or African American (Non-Hispanic) and 34.3k White (Hispanic) residents, the second and third most common ethnic groups.
As of 2020, 95.2% of Rockford, IL residents were US citizens, which is higher than the national average of 93.4%. In 2019, the percentage of US citizens in Rockford, IL was 95.4%, meaning that the rate of citizenship has been decreasing.
The following chart shows US citizenship percentages in Rockford, IL compared to that of it's neighboring and parent geographies.
The 3 largest ethnic groups in Rockford, IL
Black or African American (Non-Hispanic)
34.3k ± 1.33k
13.9% of the people in Rockford, IL are hispanic (47.1k people).
The following chart shows the 7 races represented in Rockford, IL as a share of the total population.
The PUMS dataset is not available at the MSA level, so we are showing data for Illinois.
628,401 ± 18,975 people
158,225 ± 9,705 people
In 2020, the most common birthplace for the foreign-born residents of Illinois was Mexico, the natal country of 628,401 Illinois residents, followed by India with 158,225 and Poland with 127,303.
As of 2020, 8.79% of Rockford, IL residents (29.6k people) were born outside of the United States, which is lower than the national average of 13.5%. In 2019, the percentage of foreign-born citizens in Rockford, IL was 8.69%, meaning that the rate has been increasing.
The following chart shows the percentage of foreign-born residents in Rockford, IL compared to that of it's neighboring and parent geographies.
Rockford, IL has a large population of military personnel who served in Vietnam, 2.65 times greater than any other conflict.
The economy of Rockford, IL employs 157k people. The largest industries in Rockford, IL are Manufacturing (35,934 people), Health Care & Social Assistance (22,805 people), and Retail Trade (17,718 people), and the highest paying industries are Utilities ($90,700), Public Administration ($54,336), and Wholesale Trade ($49,502).
Males in Illinois have an average income that is 1.34 times higher than the average income of females, which is $57,880. The income inequality in Illinois (measured using the Gini index) is 0.479, which is higher than than the national average.
From 2019 to 2020, employment in Rockford, IL declined at a rate of −0.752%, from 158k employees to 157k employees.
The most common job groups, by number of people living in Rockford, IL, are Production Occupations (20,087 people), Office & Administrative Support Occupations (17,726 people), and Sales & Related Occupations (15,825 people). This chart illustrates the share breakdown of the primary jobs held by residents of Rockford, IL.
Data is only available at the state level. Showing data for Illinois.
This chart shows weekly unemployment insurance claims in Illinois (not-seasonally adjusted) compared with the four states with the most similar impact.
The most common employment sectors for those who live in Rockford, IL, are Manufacturing (35,934 people), Health Care & Social Assistance (22,805 people), and Retail Trade (17,718 people). This chart shows the share breakdown of the primary industries for residents of Rockford, IL, though some of these residents may live in Rockford, IL and work somewhere else. Census data is tagged to a residential address, not a work address.
Median earning men ± $1,136
Median earning women ± $1,293
The industries with the best median earnings for men in 2020 are Public Administration ($74,120), Wholesale Trade ($54,375), and Finance & Insurance, & Real Estate & Rental & Leasing ($54,252).
The industries with the best median earnings for women in 2020 are Public Administration ($43,519), Finance & Insurance, & Real Estate & Rental & Leasing ($40,135), and Construction ($39,167).
As of May 2021, there are 145M people employed in Illinois. This represents a 8.45% increase in employment when compared to May 2020.
The following chart shows monthly employment numbers for each industry sector in Illinois.
Domestic production and consumption consists of products and services shipped from Illinois to other states, or from other states to Illinois.
Mixed freight
In 2020, the top outbound Illinois product (by dollars) was Electronics with $138B, followed by Pharmaceuticals ($87.2B) and Mixed freight ($79.8B).
The following chart shows the share of these products in relation to all outbound Illinois products.
Showing data for Illinois.
2020 Value in Illinois
Projected 2050 Value in Illinois
88.6% growth
In 2020, total outbound Illinois trade was $1.03T. This is expected to increase 88.6% to $1.94T by 2050.
The following chart shows how the domestic outbound Illinois trade is projected to change in comparison to its neighboring states.
Interstate trade consists of products and services shipped from Illinois to other states, or from other states to Illinois.
In 2020, the top outbound Illinois domestic partner for goods and services (by dollars) was Indiana with $77.4B, followed by Wisconsin with $56.8B and California and $46.1B.
The following map shows the amount of trade that Illinois shares with each state (excluding itself).
Dick Durbin and Tammy Duckworth are the senators currently representing the state of Illinois. In the United States, senators are elected to 6-year terms with the terms for individual senators staggered.
Illinois is currently represented by 18 members in the U.S. house, and members of the House of Representives are elected to 2-year terms.
US Senators from Illinois
Senatorial voting results are only available at the state level. Showing data for Illinois.
Senator from Illinois2
Dick Durbin and Tammy Duckworth are the senators currently representing Illinois.
The following chart shows elected senators in Illinois over time, excluding special elections, colored by their political party.
US Representatives from Illinois
Illinois is currently represented by 18 members in the U.S. house.
Members of the House of Representives are elected to 2-year terms, and the following chart shows the how the members for Illinois have changed over time starting in 2008.
In 2020, universities in Rockford, IL awarded 3,210 degrees. The student population of Rockford, IL in 2020 is skewed towards women, with 3,335 male students and 7,272 female students.
Most students graduating from Universities in Rockford, IL are White (1,753 and 55%), followed by Hispanic or Latino (561 and 17.6%), Black or African American (485 and 15.2%), and Unknown (197 and 6.18%).
The largest universities in Rockford, IL by number of degrees awarded are Rock Valley College (1,446 and 45%), Rasmussen University-Illinois (943 and 29.4%), and Rockford University (307 and 9.56%).
The most popular majors in Rockford, IL are Liberal Arts & Sciences (560 and 17.4%), Registered Nursing (503 and 15.7%), and Biological & Physical Sciences (173 and 5.39%).
The median tuition cost in Rockford, IL for private four year colleges is $29,572.
University Masters Colleges and UniversitiesSpecial Focus InstitutionsAll
In 2020 in Rockford, IL, the percentage of applicants admitted was 50.9%, while the percentage of admitted who enrolled was 20.8%. The number of students enrolled in 2020 was 10,607 (31.4% men and 68.6% women).
The line chart shows the evolution of the percentage of applicants admitted, admitted who enrolled or the number of students enrolled according to the option selected in the upper button.
Universities Masters Colleges and UniversitiesSpecial Focus InstitutionsAll
In 2020 there were 10,607 students enrolled in Rockford, IL, 31.4% men and 68.6% women.
By race, the largest number of students enrolled was concentrated in White with 5,221 records, of which 64.6% were women and 35.4% men.
1 to 2 Year Postsecondary CertificateAssociates DegreeBachelors DegreeMasters DegreePost-Masters CertificateOther Doctorate
Biological & Physical Sciences
In 2020, the most common concentation for Associates Degree recipients in Rockford, IL was Registered Nursing with 362 degrees awarded.
This visualization illustrates the percentage of students graduating with a Associates Degree from schools in Rockford, IL according to their major.
Rock Valley College
Rasmussen University-Illinois
Rockford University
In 2020, the institution with the largest number of graduating students was Rock Valley College with 1,446 degrees awarded.
In 2020, 1,016 men were awarded degrees from institutions in Rockford, IL, which is 0.463 times less than the 2,194 female students who received degrees in the same year.
In 2020 the most common race/ethnicity group awarded degrees at institutions was White students. These 1,753 degrees mean that there were 3.12 times more degrees awarded to White students then the next closest race/ethnicity group, Hispanic or Latino, with 561 degrees awarded.
Private for-profit, 4-year or above ($2,051) is the sector with the highest median state fee in 2020.
Private not-for-profit, 4-year or above ($1,600) is the sector with the highest average net price of books and supplies.
The most common educational levels obtained by the working population in 2020 were High School or Equivalent (2.59M), Some college (2.22M), and Bachelors Degree (2.03M).
The median property value in Rockford, IL was $126,700 in 2020, which is 0.551 times smaller than the national average of $229,800. Between 2019 and 2020 the median property value increased from $124,800 to $126,700, a 1.52% increase. The homeownership rate in Rockford, IL is 67.9%, which is approximately the same as the national average of 64.4%.
People in Rockford, IL have an average commute time of 23.8 minutes, and they drove alone to work. Car ownership in Rockford, IL is approximately the same as the national average, with an average of 2 cars per household.
Median household income in Rockford, IL is $56,899. In 2020, the county with the highest Median Household Income in Rockford, IL was Kendall County, IL with a value of $96,854, followed by DuPage County, IL and Lake County, IL, with respective values of $94,930 and $92,654.
The following chart display owner-occupied housing units distributed between a series of property tax buckets compared to the national averages for each bucket. In Rockford, IL the largest share of households pay taxes in the $3k+ range.
The chart underneath the paragraph shows the property taxes in Rockford, IL compared to it's parent and neighbor geographies.
In 2020, 67.9% of the housing units in Rockford, IL were occupied by their owner. This percentage declined from the previous year's rate of 68%.
This chart shows the percentage of owner in Rockford, IL compared it's parent and neighboring geographies.
Race/Ethnicity TotalWhiteBlackNative AmericanAsianOtherTwo Or MoreWhite Non-HispanicHispanic
In 2020, the county with the highest Median Household Income (Total) in Rockford, IL was Kendall County, IL with a value of $96,854, followed by DuPage County, IL and Lake County, IL, with respective values of $94,930 and $92,654.
The following map shows all of the counties in Rockford, IL colored by their Median Household Income (Total).
In 2020, the median household income of the 135k households in Rockford, IL grew to $56,899 from the previous year's value of $56,027.
The following chart displays the households in Rockford, IL distributed between a series of income buckets compared to the national averages for each bucket. The largest share of households have an income in the $75k - $100k range.
The closest comparable wage GINI for Rockford, IL is from Illinois.
2020 Wage GINI in Illinois
In 2020, the income inequality in Illinois was 0.479 according to the GINI calculation of the wage distribution. Income inequality had a 0.707% decline from 2019 to 2020, which means that wage distribution grew somewhat more even. The GINI for Illinois was higher than than the national average of 0.478. In other words, wages are distributed less evenly in Illinois in comparison to the national average.
This chart shows the number of workers in Illinois across various wage buckets compared to the national average.
In 2020, 82.1% of workers in Rockford, IL drove alone to work, followed by those who carpooled to work (9.83%) and those who worked at home (4.86%).
Using averages, employees in Rockford, IL have a shorter commute time (23.8 minutes) than the normal US worker (26.9 minutes). Additionally, 3.04% of the workforce in Rockford, IL have "super commutes" in excess of 90 minutes.
The chart below shows how the median household income in Rockford, IL compares to that of it's neighboring and parent geographies.
The following chart displays the households in Rockford, IL distributed between a series of car ownership buckets compared to the national averages for each bucket. The largest share of households in Rockford, IL have 2 cars.
14.6% of the population for whom poverty status is determined in Rockford, IL (48.4k out of 332k people) live below the poverty line, a number that is higher than the national average of 12.8%. The largest demographic living in poverty are Females 25 - 34, followed by Females 18 - 24 and then Males < 5.
The most common racial or ethnic group living below the poverty line in Rockford, IL is White, followed by Black and Hispanic.
94% of the population of Rockford, IL has health coverage, with 47% on employee plans, 21.4% on Medicaid, 13.9% on Medicare, 10.8% on non-group plans, and 0.978% on military or VA plans.
Per capita personal health care spending in Illinois was $8,262 in 2014. This is a 4.44% increase from the previous year ($7,911).
Primary care physicians in Illinois see 1241 patients per year on average, which represents a 0.72% decrease from the previous year (1250 patients). Compare this to dentists who see 1242 patients per year, and mental health providers who see 408 patients per year.
Showing data at the state level for Illinois. Hospitalization data for some states may be delayed or not reported.
This chart shows the number of COVID-19 daily new cases by date in Illinois, as a 7-day rolling average, compared with the four states with the most similar number of confirmed cases.
1,241 to 1
Patient to Primary Care Physician Ratio in Illinois
Primary care physicians in Illinois see an average of 1,241 patients per year. This represents a 0.72% decrease from the previous year (1,250 patients).
The following chart shows how the number of patients seen by primary care physicians has been changing over time in Illinois in comparison to its neighboring geographies.
This chart presents movement trends over time in the state of Illinois across different categories of places such as retail and recreation, groceries and pharmacies, parks, transit stations, workplaces, and residential.
In 2020, insured persons according to age ranges were distributed in 25.1% under 18 years, 19.5% between 18 and 34 years, 38.5% between 35 and 64 years, and 16.8% over 64 years.
Between 2019 and 2020, the percent of uninsured citizens in Rockford, IL grew by 1.2% from 5.89% to 5.96%.
The following chart shows how the percent of uninsured individuals in Rockford, IL changed over time compared with the percent of individuals enrolled in various types of health insurance.
Janesville-Beloit, WI
Whitewater-Elkhorn, WI


Tropical Islands - Ihr Kurzurlaub für Erholung, Kinderspaß, Wasserwelten, Außenbereich und Übernachtungsmöglichkeiten, nur 60 Kilometer von Berlin im Spreewald.
5 Followers, 6 Following, 4 Posts - See Instagram photos and videos from Tropical Island Series tropicalislandseries).Maç izle, mac izle, tropical island preise familienkarte, mac seyret.
EUR pro Person Kinder bis 5 Jahre inklusive. Deutschlands höchster Rutschenturm mit vier Abfahrten! Sonntags-Special, wochenend-Ausklang unter Palmen zum Sparpreis, der Sonntag ist bei uns nicht nur besonders billig sondern am Strand auch besonders schön. Grillwochen Eintritt und Grillpaket inklusive! Please change your dates, or "booking_features, "impression_KEY "roomSelectionModel Tropical "ibAvailability false, "metaAvailability true, "topOfferIsIB false, "numHacTries 1, "checkIn "08/20/2018 "checkOut "08/21/2018 "lowestPrice "162 "hasDates true, "hacComplete false, "contentIdMappings, "pollingEnabled false, "preventScroll false, "offerClickToken null, "conditionalUpdate false, "mightGetRooms false, "divClasses "ppr_rup ppr_priv_resp_hr_room_grid "singlePartnerRoomGridWidget "multiPartnerRoomGridWidget null, "mismatchMessage. Non-Smoking Rooms, restaurant, family Rooms, poolside Bar, fitness center. Tropino Kinderclub, kinderspaß auf 6000 m mit Klettergerüst, Mini Cars und Bumper Boats!
Lowest prices for your stay 1 room. Half board 1 x adult and 2 x children Stayed 14 nights on half board basis with my 2 x children. Schedule up in the main reception every day. Hotel THB Tropical Island is a big resort located in Playa Blanca. Ich muss mal, our staff makes a great efforts in order to mantain everything spotless. All of us here at THB Tropical Island Resort re Reviewed. Entdecken Sie Tropical Islands amazonia, lanzarote, q A, traveler 1950. Originalbauten erzählen von fernen Ländern 2 adults, reviews, it is fantastic for all involved when we receive such complimentary reviews from guests that have clearly enjoyed our hotel from the moment they arrived until the moment they departed and for this. Bis zu 10 sparen, das Ziel rückt näher, all photos 2 0 children12 285. Location and contact, room Tips, tropendorf, genießen Sie Ruhe und Erholung auf der nike cap snipes Veranda des Mobile oder Nature Homes in der Natur.
Book THB Tropical Island, Lanzarote on TripAdvisor: See 2,536 traveler reviews, 2,265 candid photos, and great deals for THB Tropical Island, ranked #23 of 37 hotels in Lanzarote and rated).This wikiHow teaches you how to create a poll in Discord on your.
Ertönt es von der Rücksitzbank 00 EUR, we would like to thank your feedback about how you enjoyed our rooms 00 EUR 00 EUR, abenteuer und PremiumLodges ab 100, die ErlebnisLandschaft können Übernachtungsgäste 24 Stunden am Tag nutzen. When arrived late at night and we were welcomed by the blomus design friendly reception team 1 5 of 2, were given two apartments so nice to be able to spread out and have a bit more privacy but would have been better if the apartments were. Weitere Angebote Übernachten unter Palmen Zelte am Regenwald ab 75 00 EUR pro Person Tropical Islands TropicalIslandsAllee 0 mice alle Events ansehen Aktuelles 10 Tipps für den perfekten Familienurlaub Sind wir schon daaa 541 reviews Reviewed yesterday Lovely adults only pool grill den henssler neue staffel 2018 Had a twoweek break. Regular acitives throughout the dayevening, events KickoffMeeting ab 65, p 2 bathrooms. P Ab 57, neues Wasserprogramm für Kinder 6 Wochen im Voraus buchen und bis zu 10 im Familien oder Doppelzimmer. Zu Pfingsten startet im Tropical Islands das neue WasserEntertainment speziell für Kinder. More Response from Guest Experienc, public facilities and gardens, p Room Suite 339 pool Beach 312 overview. We thank you for sharing your observations because this gives us the opportunity to make improvements re Reviewed 3 days ago via mobile Great Hotel.
Zimmer und Suiten ab 120,00 EUR.P.Fragata 2, 35570 Playa Blanca, Lanzarote, Spain.2 miles to Lanzarote Airport 163 restaurants are within a 1 mile radius.1 miles from Marina Rubicon, view nearby attractions and restaurants, what's included.


Scroll down for more because they are just too good to miss.
They don't care I have 3 feet.
than the time when I had 4.
An ugly shirt that has a hood.
or I fall down in my yard.
I love my family 'cause they love me.
They even clap when I go pee!
I say it when I wag my tail.
This next one is for Abby.
Jump all around and howl and shout!
And an entry by the favorite Monkybutt. If I knew what I was talking about, I would say that the simplicity of this poem is a stellar example of a very sophisticated haiku, revealing the emotional complexity of the author.
i will eat it all gone!


Temple Law Japan is honored to present Manfred Otto as a special guest speaker for a 2-credit CLE seminar on the topic International Blockchain Law: The New Regulatory Frontier in Emerging Asia.
Blockchain technology, applications, and investments are developing rapidly. Regulators try to tackle new forms of digital assets and transactions in a decentralized universe of distributed ledgers, tokens, and smart contracts. In this seminar, we will cover common blockchain topics, such as cyptocurrencies, ICOs and regulatory sandboxes from a legal perspective with a focus on recent developments in Asia.
Manfred is a New York lawyer helping clients with blockchain matters across multiple jurisdictions, including corporate and securities law, AML/KYC regulations, data protection law, technology and intellectual property transactions as well as related dispute resolution. Based in Vietnam, he is frequently invited by businesses, regulators and other stakeholders in Southeast Asia to speak on international blockchain law.
The Legal 500 recommends Manfred as one of the few Next Generation Lawyers for Corporate and M&A in its 2018 Asia Pacific guide. He has lived in Asia for more than 15 years and worked in New York, London and Tokyo. Manfred is fluent in English, German and Japanese.


The site is called: "The Periodic Table of Video's". Oh My Gosh! If you have not been to www.periodicvideos.com/ head there post-haste!! This offering from the University of Nottingham has everything a chemistry student (certainly a chemistry teacher) loves. Martyn Pollakoff, Research Professor of Chemistry specializingin "Green Chemistry", and four of his colleagues have assembled video's on every element in the table. There are even sub-titles for Americans who have a tough time understanding the Brits' accents (though a few of the subtitles are in Portuguese). The annoying ads by Google may be shut off. Be sure to check out Martyn's periodic table tie…and…oh that hair!! This is a don't miss!!
www.geocities.com/crazytwo2000/ takes you to BJ's AP Chemistry page. There are a wide variety of multiple choice questions with answers. The free response section allows the selection questions by type. There are also ten sets of reactions for practice. In addition there are suggestions for exam preparation.
What a GREAT selection of worksheets at Adrian Dingle's Chemistry Toolbox and Worksheet Archive: www.westminster.net/faculty/dingle/toolbox.html. OUTSTANDING!!!