text:
About Laurita
The Laurita Spina Bifida Project celebrates 1-year anniversary & success of 'Project Tanzania' with Jewel Award at Hispanicize 2017
Activism, Disability, Projects, Spina Bifida / By Laurita
ORLANDO, FL- April 27, 2017- (HISPANICIZE WIRE) – The Laurita Spina Bifida Project (The LSB Project), a non-profit organization whose mission is to start a global conversation about spina bifida, while connecting individuals and families living with it around the world, celebrated the one-year anniversary of its official launch on April 6th during Hispanicize 2017.
Launched during Hispanicize 2016 and attended by journalists, notable social media influencers, TV personalities, and marketing executives in Miami, The LSB Project celebrated another milestone on its anniversary when the organization's founder and president, Laurita Tellado, was awarded The Jewel Award for Best Creator Doing Social Good during the 2017 Tecla Awards.
Additionally, Tellado was surprised onstage by Manny Ruiz, founder and CEO of Hispanicize Media Group, Cristy Clavijo-Kish, Chief Development Officer of Hispanicize Media Group and 2017 Tecla Awards Chair, Vanessa James, founder of Vanessa James Media, and Felipe Arango, partner at BSD Consulting and leader of the #ParteDeLaSolution campaign that unites the Calvert Foundation, Kiva and the Latino Community Credit Union to connect people with opportunities to support Latino communities in the U.S. and Latin America through saving, lending, and investing. Together, they announced that Tellado and her organization will be offered a platform to promote their work, and Tellado will additionally be collaborating with Arango to create a network for content creators doing social good.
'Hispanicize is proud of Laurita's work with The Laurita Spina Bifida Project, and we congratulate her on receiving this accolade which is very special to the Hispanicize team," said Cristy Clavijo-Kish, Hispanicize Tecla Awards Chair and the company's Chief Development Officer. 'We are more committed than ever to supporting and elevating social good initiatives and offering social good creators a platform, and I believe the Hispanicize Tecla Jewel Award symbolizes Jewel's legacy of philanthropic work, as well as the power in unity that helps propel causes like Laurita's forward. "
"It is truly a humbling privilege and an honor for me to be awarded the first-ever Jewel Award at Hispanicize 2017," Laurita Tellado, Founder and President of The Laurita Spina Bifida Project, said. "Not only is this award very meaningful to me because of my close bond with the entire Hispanicize community; it meant so much to me that it is named in memory of my friend Jewel Figueras. She and I were very close, and I miss her so much. But, I felt her spirit that night, cheering us on."
The Laurita Spina Bifida Project raised $1,000 over the 2016 holidays, which was sent to the Association for Spina Bifida and Hydrocephalus Tanzania (ASBAHT) for the purchase of wheelchairs and other orthopedic supplies for children with spina bifida in dire need.
In March, The LSB Project received an update from ASBAHT on how the funds are being used.
"So far, the LSBP Tanzania project has supplied two wheelchairs to Halima, a primary school pupil, and Nsia, who is out of school due to [lack of] mobility aid. Nsia has also received a walker. She [has] yet to receive orthopedic shoes, which will protect her wound from further infection," Sifa Sylvia, ASBAHT Administrative Officer, reported.
"Besides Halima and Nsia, the funds have also been used to buy medicine for a child that is in need. Angel is almost 2 years old. She has hydrocephalus and has been operated 7 times so far, but her health progress is deteriorating each day. She is supposed to receive medicines, but her mother can't afford the price because she was abandoned by her husband and family members. They claim they are tired of spending on a child that doesn't heal," Sylvia said.
Tellado says she hopes the next fundraising initiative The LSB Project spearheads will involve helping people living with spina bifida in Latin America.
About The Laurita Spina Bifida Project
The LSB Project is a 501(c)3 tax-exempt, non-profit organization established in 2014 with the mission of starting a global conversation about spina bifida, while connecting individuals and families living with it around the world, and to educate health professionals from all fields on the issues affecting this community, including mental health and adult-centered care as part of the action plan.
The Laurita Spina Bifida Project can be found on Facebook
(https://www.facebook.com/Laurita-SpinaBifidaProject/), Twitter (@TheLSBProject) and instagram (@TheLSBProject).
About Laurita Tellado
Laura "Laurita" Tellado is a freelance writer, editor, and blogger, and the founder of Holdin' Out for a Hero and Espresso con Leche blogs, as well as the Founder and President of The Laurita Spina Bifida Project, a non-profit whose mission it so promote positive awareness of spina bifida and unite spina bifida communities across the globe.
In addition, Tellado is Editor of Hispanic PR Blog, and she manages social media and influencer relations for several clients.
Tellado was born in San Juan, Puerto Rico. An only child, she was three years old when her parents decided to move the family to Orlando, Florida, where she has resided ever since.
Follow her on Twitter and Instagram @Laurita Tellado.
About The Hispanicize Event
Now in its 8th year, Hispanicize 2017 Week (www.HispanicizeEvent.com) (#Hispz17) is the iconic, largest annual event for Latino trendsetters and newsmakers in digital content creation, journalism, marketing, entertainment and tech entrepreneurship.
Hispanicize 2017 is expected to gather more than 3,000 of the nation's most influential Latino professionals from the industries of digital content creation, journalism, music, marketing, film and business over four days. The event will take place in downtown Miami's JW Marriott Marquis Hotel, April 3-6, 2017.
The Hispanicize event is a launch pad for creative endeavors, new products, technologies, marketing campaigns, films, books, music and more targeting Latinos in the U.S. and/or Puerto Rico.
The Hispanicize event is owned and operated by Hispanicize Media Group, LLC, the parent company of DiMe Media, Hispanic Kitchen and the Hispanic PR Blog.
The Hispanicize Event can be found on Facebook https://www.facebook.com/Hispanicize, Instagram (@HispanicizeEvent) and Twitter (@Hispanicize).
© The Laurita Spina Bifida Project. All rights reserved.
How We Use Your Donations | #GivingTuesday November 30, 2021
Laurita Writes about Accessible Travel for Wanderful September 1, 2021
Orange County Convention Center and Partners Team Up to Support #ProjectUganda June 7, 2020
An Update on Jackline, a Young Girl in Uganda June 7, 2020
Buffalo Family Scores Nothing But Net with Inclusive Wheelchair Basketball League October 25, 2019
Support our local and global programs!
Donate with PayPal or Credit Card
How We Use Your Donations | #GivingTuesday
Laurita Writes about Accessible Travel for Wanderful
Orange County Convention Center and Partners Team Up to Support #ProjectUganda
An Update on Jackline, a Young Girl in Uganda
Buffalo Family Scores Nothing But Net with Inclusive Wheelchair Basketball League
Copyright © 2021 The Laurita Spina Bifida Project | Powered by ICON Websites


text:
(function(){
    'use strict';

    angular
    .module('ktp')
    .controller('NotesController', NotesController);

    function NotesController(Notes) {
        var vm = this;

        construct();

        function construct() {
            Notes.get().then(function(data){
                vm.notes = data;
            });
        }
    }
})();

redpajama_set_name: RedPajamaGithub


text:
CNN and Jim Acosta Sue Trump Over White House Access
Published Nov. 13, 2018 10:02AM ET
Jonathan Ernst/Reuters
CNN and Jim Acosta have filed a lawsuit against President Trump and several members of his staff, seeking a reinstatement of Acosta's White House press pass that was revoked last week. Acosta was denied entry to the White House after he refused to relinquish a microphone during a press conference with Trump that same day. The suit alleges that Acosta and CNN's First and Fifth Amendment rights were violated with the ban. CNN also cautioned that other news organizations could be similarly targeted by the Trump administration, "While the suit is specific to CNN and Acosta, this could have happened to anyone," the network said. "If left unchallenged, the actions of the White House would create a dangerous chilling effect for any journalist who covers our elected officials." The suit was filed in U.S. District Court in Washington, D.C. on Tuesday morning.
Ted Boutros, a lawyer for CNN along with former Bush Solicitor General Ted Olson, went onto the news channel to explain the case's argument:
Read it at CNN

redpajama_set_name: RedPajamaCommonCrawl


Featuring best casinos online reviews. Find top online casinos for Slots, Craps, Blackjack, UK and US Players. Only the best casinos reviewed and rated.
Best Casino Online is your guide to the latest promotional news and bonus reviews from some of the web's best online casinos.
Gambling.com recommends the Best Online Casinos. Exclusive Sign Up Bonuses, casino game reviews, live jackpots and strategies to help you win more. Winners start at Gambling.com!
Play all of your favorite casino games at the most trusted online casinos. Enjoy the variety of bonus offers available every day!
Listing the best online casinos for December 2013. Find safe, secure & trusted USA online casinos to play real money casino games.
The best online casinos for South Africans are all gathered in this page. Get fantastic welcome bonuses and play the best casino games available online.
Best Casinos Online helps players find the best casino online for their specific needs. We rank the top casinos for bonuses, comps, games and much more!
At TopCasino.com we offer a listing of the best most trusted and reputable online casinos in which to select from and enjoy playing your favorite casino games at.
OnlineCasinos.com is the leading guide to the best online casinos of 2013. Rankings of the top US online casinos by deposit method, software & games.
Best Online Casino Sites. With so many online casinos to choose from and play it's difficult for even the most savvy players to know where to play these days.
All the Latest Casino News, Reviews of all the Best Casinos and Software Platforms, Gaming Tips and Free Games. Best in Online Casinos, your one stop source to Online Casinos.
How to Choose the Best USA Online Casinos? If you want to enjoy the thrill of casino, but don't have the luxury of traveling to Vegas then why not try out an online casino?
Trying to find top online casinos? We compare the best online casinos worldwide, saving you time & money bonus hunting.
Singles Free Sites Australia - We're a 100% free online dating site. Singles Free Sites Australia View photos of singles in your area. ee who's online now! Singles Free Sites Australia chat with singles here for free.
Singles Dating Websites Australia. Singles Dating Websites Australia The best thing about our background check is that it is online, which makes it easy for you to do it from the comfort of your home, or at any Internet capable computer.
Australia Dating Sites - We're a 100% free online dating site. Australia Dating Sites View photos of singles in your area. ee who's online now! Australia Dating Sites chat with singles here for free.
100% free dating site for singles Singles Dating Sites Australia, Dating sites can be invaluable for many individuals trying to discover pals, lovers, and in many cases a new soul mates.
We're a 100% free online dating site! Free Singles Sites Australia, Internet dating sites can prove to be invaluable for most people that want to get buddies, associates, and in some cases a love of his life.
Australia Singles Dating Site - We're a 100% free online dating site. Australia Singles Dating Site View photos of singles in your area. ee who's online now! Australia Singles Dating Site chat with singles here for free.
Free Singles Dating Australia - We're a 100% free online dating site. Free Singles Dating Australia View photos of singles in your area. ee who's online now! Free Singles Dating Australia chat with singles here for free.
Single Sites Australia - We're a 100% free online dating site. Single Sites Australia View photos of singles in your area. ee who's online now! Single Sites Australia chat with singles here for free.
Australia's best FREE dating site! 100% Free Online Dating and Personal Ads for Australia Singles at Mingle2.com. Stop wasting money at other dating sites.
Free Single Dating Sites Australia. Free Single Dating Sites Australia The best thing about our background check is that it is online, which makes it easy for you to do it from the comfort of your home, or at any Internet capable computer.
Free Dating Sites Australia. Free Dating Sites Australia The best thing about our background check is that it is online, which makes it easy for you to do it from the comfort of your home, or at any Internet capable computer.
Free Singles Dating Sites Australia. Free Singles Dating Sites Australia The best thing about our background check is that it is online, which makes it easy for you to do it from the comfort of your home, or at any Internet capable computer.
Welcome - To Free Dating Australia. Online Dating That Works! Absolutely 100% free online dating site. Premium friend finder website for Australian singles.


We've provided comprehensive network design & architecture for dozens of Festivals and Live Events over the years, providing us with a unique advantage when planning each new project.
Supporting thousands of attendees, hundreds of traders operating with RFID, dozens of top up stations, production and staff users, Social Media teams and live streaming dedicated bandwidth are just some of the things involved in successfully deploying a Festival WiFi network.
We work closely with our events partners constantly developing and improving services to meet the ever changing and demanding need of todays digital landscape.
To find out how we can offer support to you and transform your event call us now on 020 3432 1222 or get in touch here.


On her latest contribution to IRISES, the Bulletin of The American Iris Society, Australian hybridizer Heather Pryor writes candidly about her work on creating the ultimate orange Louisiana Iris (LA). She calls her work, "Heather's 'orange brick road of hybridizing endeavors'."
Her dedication to detail is evident and admirable, as is her use of descriptive language: "It has similar coloration to a zucchini or pumpkin flower; not quite yellow and not quite orange."
Take the road to "orange" with Heather Pryor by reading this wonderful article, you will be lead into a path of creativity, where you yourself may find it fascinating to create that which you desire; and in the end you will be asked to make your own decision regarding the outcome. Has she made it? Is this the color of her obsession?
After reading this wonderful article and looking at Heather Pryor's irises, I'm fully motivated to admonish my creative inner Self and find a way to Iris Heaven. Hope you will too.
Members of The American Iris Society receive a copy of IRISES, which is published quarterly and available in print or digital formats. More information on membership, please check our website.
Other pertinent information on Louisiana irises can be found in the website of the Society for Louisiana Irises.
Fantastic colors! Oranges are one of my husbands favorite colors and he really liked tandoori sizzler, we will be on the lookout for this one.
Thank you Carla. Heather Pryor beautiful LA irises are distributed in the U.S. by Iris City Gardens, in case you'd like to Google them. Happy gardening.
What's Happening in Luminata and Plicata Irises?


text:
Who Are Ya!? Kwesi Appiah
All you need to know about the latest addition to the Notts County ranks - Crystal Palace striker Kwesi Appiah…
NAME | KWESI APPIAH
DOB | 12 Aug 1990
BIRTHPLACE | Peterborough
HEIGHT | 5ft 11ins
POSITION | Striker
SQUAD NUMBER | 15
TWITTER |@kwes1appiah
BIOGRAPHY | Appiah began his career in non-league with Ebbsfleet, prior to joining Weymouth on a free transfer in 2009.
Peterborough United took the striker on loan, followed by King's Lynn, before he signed for Kettering in January 2010.
Following spells at Brackley Town and Thurrock, Appiah really hit form in 2011, scoring 35 goals in 33 appearances for Margate.
Understandably, his performances alerted a number of Football League clubs and, despite plenty of competition, Crystal Palace secured his services in January 2012.
Loan stints with League 2 Aldershot Town and Yeovil Town followed to give the promising youngster experience of professional football, but he also excelled in the Eagles' development squad, where he netted nine goals in 13 matches.
Earlier this term, Appiah joined Skrill Premier League outfit Cambridge and made a name for himself with 10 finishes in only 13 league starts.
Now eager to prove himself in Sky Bet League 1, the 23-year-old has joined Notts County on a one-month emergency loan with the hope of helping the Magpies climb away from the bottom four with more goals.


text:
\section{Introduction}
\label{s:intro}

Reconstruction in cosmology considers the following problem: one assumes that
the present spatial distribution of masses (galaxies and clusters, including
their dark-matter components) is known from observations, and one wants to
reconstruct the dynamical history of the Universe all the way to the earliest
epoch, when matter and radiation decoupled (nearly 14 billion years
ago). Peebles \cite{peebles89} introduced the reconstruction problem, and
proposed a variational formulation for solving it on a relatively small spatial
scale, that of the Local Group (which includes our own galaxy and neighboring
ones). On scales much larger than that of the Local Group, which have been
mapped in recent years through various projects such as the Sloan Digital Sky
Survey \cite{sdss}, reconstruction may be posed in the simplest cases as
an instance of optimal mass transport. Indeed, Frisch et al. \cite{nature}
showed that when the \textit{Zeldovich approximation} \cite{zeldovich} or
a refinement thereof (cf.~below) are applied to the relevant cosmological fluid
equations, the correspondence between the positions of mass elements initially
(at decoupling) and finally (at the present epoch) is the solution to an optimal
mass transport problem with quadratic cost. This solution is uniquely
prescribed by the \textit{marginals}: the mass distribution at decoupling
(essentially uniform) and its highly non-uniform present
distribution. A striking feature is that the sole knowledge of the current
positions of galaxies, without knowledge of their (proper)
velocities, yields nevertheless a unique solution for this kind of
large-scale reconstruction.

It was then shown by Brenier et al. and by Loeper \cite{mnras,loeper} that,
with prescribed marginals, unique reconstruction, not only of the Lagrangian
map, but of the full dynamical history of matter elements, carries over to the
Euler--Poisson model, whose validity extends much beyond that of the Zeldovich
approximation. Its unique solution is again obtained from an optimal transport
problem with a convex cost function, expressible as a space-time integral of
a suitable action, a problem whose numerical resolution remains a challenge.

As is well known, the mass transport problem was introduced by Monge
\cite{monge1781} more than two hundred years ago, and the theory took its
modern shape after the 1942 work of Kantorovich \cite{kantorovich} (see, e.g.,
Villani \cite{villani2009} for review).

The Zeldovich approximation \cite{zeldovich} was introduced in 1970 as
a first formulation in terms of Lagrangian coordinates of the growth
of density perturbations. It replaces the full Euler--Poisson
equations by basically the three-dimensional inviscid Burgers
equation (written here in standard fluid dynamical notation)
\begin{equation}
\partial_t{\bf v}+{\bf v}\cdot\nabla{\bf v}=0, \quad {\bf v}=\nabla\varphi.
\label{3dburgers}
\end{equation}
The validity of the Zeldovich approximation is controlled by how close one is
to decoupling, but in a
scale-dependent way: at very large scales, the Zeldovich approximation remains
valid up to the present epoch; at very small scales, the formation of
multi-stream caustics quickly ruins not only the validity of the Zeldovich
approximation, but even that of the
Euler--Poisson model. An immediate consequence of \eqref{3dburgers}
is that the velocity of any fluid particle remains constant in the
course of time and that the trajectories are straight lines. We denote
by $\bm q$ the initial (Lagrangian) fluid particle positions and
by $\bm x$ their (Eulerian) positions at the current epoch $t=T$. The
Lagrangian map associated with the Zeldovich approximation is
\begin{equation}
\bm q\mapsto\bm x=\nabla_{\bm q}\left(\frac{|\bm q|^2}{2}+T\varphi_0(\bm q)\right),
\label{burglagmap}
\end{equation}
where $\varphi_0(\bm q)$ is the initial velocity potential. For sufficiently
small $T$ and a sufficiently smooth initial potential, the Lagrangian map is thus
the gradient of a convex function, a property shared by the next-order
approximation, which will be discussed in Sec.~\ref{s:cosmo}. This is why
reconstruction is linked to optimal transport; indeed,
a theorem of Brenier \cite{brenierlong} states that the solution to the Monge
optimal transport problem with quadratic cost is a gradient
of a convex function, which satisfies a Monge--Amp\`ere equation. The method
of cosmological reconstruction in which one assumes that the Lagrangian map
has a convex potential and then numerically solves a quadratic-cost optimal
transport problem (after suitable discretization) is called the
Monge--Amp\`ere--Kantorovich (MAK) method \cite{nature}.

The Zeldovich approximation gives us some insight into the full temporal history
of mass elements. An important consequence of \eqref{3dburgers} is that for any
\hbox{$0\le t_1<t_2\le T$} the mapping of particle positions at time $t_1$ to their
positions at time $t_2$ is also a gradient of a convex function. When the
flow-induced mapping between any two times is potential, the flow is here
called \textit{omni-potential}. As we shall see, the velocity field associated
with such flows has the property of being simultaneously potential in Eulerian
coordinates (in cosmology, this constraint stems from the potential character
of gravity and the expansion of the Universe), as well as in Lagrangian
coordinates (which allows reconstruction by solving an optimal transport problem).

We are of course led to ask whether there exist omni-potential flows other than
Zeldovich/Burgers ones or trivial variants thereof. Investigating this issue is
the central topic of our paper.

In Section~\ref{s:geometry} we show that omni-potentiality can be reexpressed
geometrically and algebraically in terms of Hessian matrices and recast as a
set of one or several partial differential equations (depending on the space
dimension). In Section~\ref{s:algebra} we use an algebraic method to construct
explicit non-trivial examples of omni-potential flows in two and three
dimensions. These are rather special and we are led to ask how general are
omni-potential flows. In Section~\ref{s:wkb} we construct a fairly general
class of two-dimensional omni-potential flows, leaving the problem open in
higher dimensions. In Section~\ref{s:cosmo} we return to questions of
cosmological relevance: To what extent are the full solutions to the
cosmological equations omni-potential? Why is the validity of MAK
reconstruction better than that of the Zeldovich approximation, as pointed out
by Mohayaee et al.~\cite{royaetal}? In Section~\ref{s:conclusion} we list
some open problems and make concluding remarks. Finally, in the Appendix we
characterize sets of commuting symmetric matrices by constructing
suitable invariants.

\section{Criteria for omni-potentiality of flows}
\label{s:geometry}

In the present paper we study the kinematics of omni-potential flows.
We start by recalling the basic definitions of the Lagrangian and Eulerian
description of a flow --- a~motion of fluid regarded as a continuum
of infinitesimal fluid particles (whose mathematical abstraction is
a {\em point} particle).

Denote by ${\bf v}(\bm x,t)$ the velocity of the fluid measured at point $\bm x$
in space and at time $t$. It is usually called the {\em Eulerian velocity}
(the velocity measured at a fixed position in the laboratory frame).
The motion of a fluid particle satisfies the ordinary differential equation
$$\dot\bm x={\bf v}(\bm x,t),$$
which has to be supplemented by the initial condition
$$\bm x|_{t=0}=\bm q.$$
If the velocity field ${\bf v}(\bm x,t)$ is prescribed and sufficiently regular,
one can solve this initial value problem, at least locally in time, and obtain
the mapping $\bm q\mapsto\bm x(\bm q,t)$ called the {\em Lagrangian map}.
It takes a particle at the Lagrangian position $\bm q$ and carries it to the
Eulerian position $\bm x$. For a fixed $\bm q$, the curve
$\bm x(\bm q,t)$, parameterized by varying time $t$, is the \textit{trajectory
of the particle}, whose Lagrangian position is $\bm q$.
When, in a field associated with the flow,
we perform the substitution $\bm x\to\bm x(\bm q,t)$, we obtain its Lagrangian
description, in which the field is now a function
of the Lagrangian coordinates $\bm q$. For instance,
${\bf v}(\bm x(\bm q,t),t)$ is called the {\em Lagrangian velocity}.

In this paper, we consider flows ${\bf v}(\bm x,t)$ defined, for simplicity, in
the entire space ${\mathbb R}^d$, but restricted to a finite time interval
$[0,T]$. The flow induces a set of mappings of space: given two arbitrary
times $t$ and $\tau$, such that $0\le t<\tau\le T$, the
mapping from fluid particle positions at time $t$ to their positions at time
$\tau$ is here called the $(t,\tau)$-mapping. The $(0,\tau)$-mapping is just
the standard Lagrangian map.

As stated in the Introduction, for any two times $t$ and $\tau$, such that
$0\le t<\tau\le T$, the $(t,\tau)$-mappings induced by {\em omni-potential}
flows are required to be the gradients of a convex potential $\Phi(\bm q,t;\tau)$:
\begin{equation}
\bm q\mapsto\bm x=\nabla_{\bm q}\Phi(\bm q,t;\tau).
\label{potmap}\end{equation}
Such a mapping is here called potential and $\Phi(\bm q,t;\tau)$ is
called the $(t,\tau)$-potential. Given any three times $t_0$, $t$ and $\tau$
such that $0\le t_0\le t\le\tau\le T$, the $(t_0,t)$-mapping composed with
the $(t,\tau)$-mapping obviously yields the $(t_0,\tau)$-mapping.
This semigroup associativity, combined with omni-potentiality, implies
\begin{equation}
\Phi(\bm q,t_0;\tau)=\Phi\left(\nabla_{\bm q}\Phi(\bm q,t_0;t),t;\tau\right),
\label{associative}
\end{equation}
which is illustrated in Fig.~\ref{f:fig1}.

Differentiation of \eqref{potmap} with respect to time $\tau$ shows that
the velocity, ${\bf v}(\bm q,t;\tau)$, is also potential:
$${\bf v}(\bm q,t;\tau)=\nabla_{\bm q}\,\varphi(\bm q,t;\tau),$$
where
$$\varphi(\bm q,t;\tau)\equiv\partial_{\tau}\Phi(\bm q,t;\tau).$$
In this notation, ${\bf v}(\bm q,0;t)=\nabla_{\bm q}\varphi(\bm q,0;t)$ is
the Lagrangian velocity of a fluid particle, while its Eulerian velocity is
${\bf v}(\bm x,t;t)=\nabla_{\bm x}\varphi(\bm x,t;t)$; thus, in omni-potential
flow, the velocity is potential in both Eulerian and Lagrangian
coordinates. In three dimensions, this is equivalent, locally, to the
statement that the vorticity (the curl of the velocity) vanishes in
both coordinate systems. We shall call this dual potentiality in both Eulerian
and Lagrangian coordinates \textit{bi-potentiality}.

\begin{figure}
\includegraphics[width=85mm]{omni1.eps}
\caption{Potential composition of two potential maps.}
\label{f:fig1}\end{figure}

We demand that all $(t,\tau)$-potentials of omni-potential flows be convex in
the space variables. This implies that the associated $(t,\tau)$-mappings have
inverses that are also potential, the potentials associated with the inverse
mappings being the Legendre transforms of those of the direct mappings (see
Ref.~\cite{mnras}, Appendix~C and references therein). Actually,
for a sufficiently smooth flow, the converse is true: invertibility implies
convexity of the maps, as now explained. Let us denote by ${\cal H}(f)$
the Hessian of a (twice differentiable) function $f(\bm q)$, i.e., the $d\times d$
matrix with the entries
$${\cal H}_{ij}(f)\equiv\partial^2_{q_iq_j}f.$$
We now assume that the potentials are twice differentiable in the space
variables and that their Hessians
${\cal H}(\Phi(\bm q,t;\tau))$ depend continuously on their time
arguments. We then observe that, for coinciding times, the $(t,t)$-mapping is
the identity mapping, clearly having the convex potential
$$\Phi(\bm q,t;t)=|\bm q|^2/2,$$
whose Hessian is the identity. As the two times separate, loss of convexity
would require one or several eigenvalues of the Hessian to change sign and thus
to go through zero; at such an instant, the Jacobian matrix (which
for a potential mapping coincides with the Hessian of the potential) becomes
degenerate; then, generically, the inverse
mapping ceases to exist, i.e., the property of the $(t,\tau)$-mapping to be
bijective gets lost (in the cosmological context this amounts to {\em shell
crossing} leading to {\em multi-streaming}).

Below, we establish criteria for omni-potentiality. In Sec.~\ref{sub:commu} we
prove that a flow is omni-potential, whenever the Hessians of the potentials
$\Phi(\bm q,t;\tau)$, calculated at any two points of a trajectory, commute.
In Sec.~\ref{sub:app1} we present an equivalent condition for the commutativity
of Hessians: along any trajectory, at any time $t$,
the Hessian ${\cal H}(t)={\cal H}(\Phi(\bm q; 0,t))$ and its time
derivative should commute; this is used to show that
omni-potentiality is equivalent to potentiality of both the Lagrangian
and the Eulerian flow velocities (plus the convexity constraints). In
Sec.~\ref{sub:linea} we discuss simple examples of omni-potentiality:
Zeldovich and Zeldovich-type flows. Finally,
in Sec.~\ref{sub:eqn2d} we derive a partial differential equation
for the potential of a two-dimensional omni-potential flow. It
states that a suitable expression constructed from second-order derivatives
depends only on the Lagrangian coordinates, but not on time.
In other words, the Hessian of the potential possesses an {\em invariant},
whose value depends only on the trajectory. We construct similar invariants
in higher-dimensional spaces in the Appendix.

\subsection{Commutation of Hessians of the potential}
\label{sub:commu}

The semigroup associativity \eqref{associative} involves the composition
of two potential maps. We shall show that, in general, such a composition is
potential if and only if the Hessians commute. Basically this stems from the
well-known theorem that the product of two symmetric matrices is symmetric only
when they commute. The problem we are now addressing is illustrated
in Fig.~\ref{f:fig1}, which sketches the action of three mappings along the
same trajectory. We observe that the $(t,\tau)$-mapping is the composition of
the inverse of the $(t_0,t)$-mapping with the $(t_0,\tau)$-mapping. We shall
now show that its potentiality is equivalent to the commutation of the Hessians
of the $(t_0,t)$-mapping and of the $(t_0,\tau)$-mapping.

Specifically, we assume that, for any times $t$ and $\tau$, such that $0\le t\le\tau\le T$,
the $(t,\tau)$-mapping \rf{potmap} of ${\mathbb R}^d$ into itself is a bijection
that, together with its inverse, is smooth (i.e., has as many continuous
derivatives, as we might need). We denote
$$\Phi_1(\bm q)\equiv\Phi(\bm q,t_0;t),\qquad\Phi_2(\bm q)\equiv\Phi(\bm q,t_0;\tau),$$
where $\Phi(\bm q,t_0;t)$ and $\Phi(\bm q,t_0;\tau)$ are the potentials
of the $(t_0,t)$-mapping and the $(t_0,\tau)$-mapping, respectively.
The required potentiality of the $(t,\tau)$-mapping implies that the Jacobian
matrix $\left\|{\partial{\bm\xi}\over\partial\bm x}\right\|$ is symmetric:
$$\partial_{x_j}\xi_i=\partial_{x_i}\xi_j$$
for any pair of indices $1\le i,j\le d$. The converse is also true, at least
locally in space. By the chain rule,
\begin{eqnarray*}
{\cal H}_{mn}(\Phi_2)&=&\partial_{q_n}\xi_m\\
&=&\sum_{k=1}^d\partial_{x_k}\xi_m\,\partial_{q_n}x_k\\
&=&\sum_{k=1}^d\partial_{x_k}\xi_m\,{\cal H}_{kn}(\Phi_1).
\end{eqnarray*}
Therefore,
\begin{equation}
\left\|{\partial{\bm\xi}\over\partial\bm x}\right\|={\cal H}(\Phi_2)
{\cal H}^{-1}(\Phi_1),
\label{commute}\end{equation}
where ${\cal H}^{-1}(\Phi_1)$ denotes the matrix inverse to
${\cal H}(\Phi_1)$. For the r.h.s.~of \eqref{commute} to be a symmetric
matrix, the matrices ${\cal H}(\Phi_2)$ and ${\cal H}^{-1}(\Phi_1)$ must
commute, which is equivalent to the commutation of the two Hessians
${\cal H}(\Phi_2)$ and ${\cal H}(\Phi_1)$.

Reversing the arguments, we establish that commutation of
Hessians $\Phi(\bm q,t_0;t)$ along a trajectory,
$${\cal H}(\Phi(\bm q,t_0;t)){\cal H}(\Phi(\bm q,t_0;\tau))\!=\!
{\cal H}(\Phi(\bm q,t_0;\tau)){\cal H}(\Phi(\bm q,t_0;t)),$$
at any times $t$ and $\tau$ is \textit{necessary} for the mapping
${\bm\xi}(\bm x)$ to be potential. In particular, commutation of Hessians
of the potential $\Phi(\bm q,0;t)$ of the Lagrangian map along each trajectory
(for each fixed $\bm q$), together with invertibility or convexity, is equivalent
to omni-potentiality.

By the theorem on codiagonalizability of commuting (real) symmetric matrices
(see, e.g., Ref.~\cite{hornjohnson}, pp.~50--51) the equivalent condition is
that the Hessians of the potential $\Phi(\bm q,t_0;t)$ calculated at different
times $t$ for the same coordinate $\bm q$ and the same $t_0$ are
{\em codiagonalizable}, i.e., can be transformed into the diagonal form using
the same unitary matrix. In other words, along any trajectory, only
the eigenvalues of the Hessian of the potential but not the eigendirections
are allowed to vary.

So far, we have shown the commutation --- along a given trajectory ---
of the Hessians of the potentials of the $(t_0,t)$-mapping and
the $(t_0,\tau)$-mapping for the same starting time $t_0$ (e.g., for the
$(t_0,\tau_0)$-mapping and the $(t_0,t_1)$-mapping of Fig.~\ref{f:fig2}).
A similar argument proves the commutation of the Hessians of the
potentials of two mappings, such that the ending time of one of them coincides
with the starting time of the second one (e.g., for $(t_0,t_1)$-mapping and
the $(t_1,\tau_1)$-mapping). Combining these two results and relying again
on the theorem on codiagonalizability of symmetric commuting matrices,
we find that the Hessians of the potentials of the $(t_0,\tau_0)$-mapping and
of the $(t_1,\tau_1)$-mapping commute. Thus we have established that, along any
given trajectory, the Hessians associated with arbitrary pairs of times commute.

\begin{figure}
\includegraphics[width=85mm]{omni2.eps}
\caption{Commutation of Hessians of the potential along a trajectory.
A sketch of a trajectory and flow-induced mappings
from times $t_0$ and $t_1$ to times $\tau_0$ and $\tau_1$.}
\label{f:fig2}
\end{figure}

\subsection{Commutation of Hessians and their time derivatives:
bi-potential velocities}
\label{sub:app1}

Here we give an alternative formulation of omni-potentiality in terms
of commutation of Hessians and their time derivatives. We need some
preparatory material regarding $d\times d$ symmetric matrices with
smooth time dependence. Suppose at any two times $t$ and $t'$ they commute:
\begin{equation}
H(t)H(t')=H(t')H(t).
\label{tcomm}\end{equation}
Differentiating this equation in $t'$ and letting $t'=t$, we find that
at any time $H(t)$ commutes with its time derivative $\dot{H}(t)$:
\begin{equation}
H(t)\dot{H}(t)=\dot{H}(t)H(t).
\label{dcomm}\end{equation}
We shall show now that the converse is also true: if at any time $t$ (i)
relation \rf{dcomm} is satisfied, and (ii) all eigenvalues $\lambda_i$
of the symmetric matrix $H(t)$ are distinct, then \rf{tcomm} holds true
for any times $t$ and $t_1$.

Since $H(t)$ is symmetric, it can be expressed as
\begin{equation}
H(t)=U^t(t)\Lambda(t)U(t),
\label{dnf}\end{equation}
where $U$ is a unitary $d\times d$ matrix, and $\Lambda$ is diagonal.
Consider the identity $U(t)U^t(t)=I$, where $I$ is
the identity matrix; differentiating it in time yields
$$U\dot{U}^t=-\dot{U}U^t=-\left(U\dot{U}^t\right)^t.$$
Thus $X\equiv U\dot{U}^t$ is an antisymmetric matrix.

We substitute \rf{dnf} into \rf{dcomm}, and multiply it by $U$
on the left and by $U^t$ on the right, obtaining
$$\Lambda X\Lambda-\Lambda^2X=X\Lambda^2-\Lambda X\Lambda,$$
i.e., the matrix $X\Lambda-\Lambda X$ commutes with $\Lambda$.
This matrix is symmetric due to the antisymmetry of $X$.
By the theorem on codiagonalizability of commuting symmetric matrices,
the matrices $X\Lambda-\Lambda X$ and $\Lambda$ are thus
simultaneously diagonalizable; but $\Lambda$ is already diagonal, and hence
so is $X\Lambda-\Lambda X$. The entries of the latter matrix are
$(\lambda_j-\lambda_i)X_{ij}$, and therefore $X_{ij}=0$ for all
$i\ne j$ (we use here the condition that all eigenvalues of $H$
are distinct). The antisymmetry of $X$ implies that all diagonal
entries of $X$ also vanish, and thus $X=0$. Therefore,
$$\dot{U}=-XU$$
vanishes. In other words, variation of $H$ in time consists
solely of variation of its eigenvalues $\lambda_i$. This implies \rf{tcomm}.

The restriction that all the eigenvalues be distinct does not significantly
affect the generality of the statement: if they are not distinct at some
isolated times, the relation $\dot{U}=0$ remains satisfied at these times
by continuity.

Returning to the problem of omni-potentiality, we now take
$$H(t)={\cal H}(\Phi(\bm q,0;t))\equiv{\cal H}.$$
This Hessian $\cal H$ is the Jacobian of the Lagrangian map $\nabla\Phi(\bm q,0;t)$.
By the above result, omni-potentiality is equivalent to the commutation,
at any time, of the Jacobian and its time derivative.

We have seen earlier that omni-potential flow has a velocity which
is bi-potential, i.e., potential in both Eulerian and Lagrangian coordinates.
The statement just derived allows to prove the converse, namely that
a flow with a bi-potential velocity (and some convexity requirements)
is omni-potential. Let us denote by ${\bf v}^{\rm L}(\bm q,t)$ and
${\bf v}^{\rm E}(\bm x,t)$ the Lagrangian and Eulerian velocity, respectively.
Since $\bm x(\bm q,t)$ is the Lagrangian map, we obviously have
\begin{equation}
{\bf v}^{\rm E}(\bm x,t)={\bf v}^{\rm L}(\bm q(\bm x,t),t),
\label{vlageul}\end{equation}
where $\bm q(\bm x,t)$ is the inverse Lagrangian map, whose Jacobian is
${\cal H}^{-1}$. We now calculate the Eulerian velocity gradient,
using \eqref{vlageul}. By the chain rule, for any $i$ and $j$, we have:
\begin{eqnarray*}
\partial_{x_i}v^{\rm E}_j(\bm x,t)&=&\sum_{m=1}^d
({\cal H}^{-1})_{im}\,\partial_{q_m}v^{\rm L}_j(\bm q,t)\\
&=&\sum_{m=1}^d({\cal H}^{-1})_{im}\,\partial_{q_mq_j}\dot{\Phi}(\bm q,0;t)\\
&=&\sum_{m=1}^d({\cal H}^{-1})_{im}\dot{\cal H}_{mj}.
\end{eqnarray*}

Thus, the Eulerian velocity gradient is the product of the matrices
${\cal H}^{-1}$ and $\dot{\cal H}$. For the Eulerian velocity to
be potential, it is necessary and (locally) sufficient that this
product be a symmetric matrix. The commutation of the symmetric
matrices ${\cal H}^{-1}$ and $\dot{\cal H}$ is equivalent to the commutation of
${\cal H}$ and $\dot{\cal H}$. Equivalence to omni-potentiality follows from
the statement derived above.

\subsection{Zeldovich and Zeldovich-type flows}
\label{sub:linea}

In the Zeldovich approximation each particle keeps its initial velocity unaltered
in the course of time, and hence particles move along straight lines (at least
before multi-streaming occurs). The Lagrangian map at time $t$ is
$$\bm q\mapsto\bm x=\nabla_{\bm q}\left({|\bm q|^2\over2}+t\varphi_0(\bm q)\right),$$
where $\varphi_0(\bm q)$ is the velocity potential, prescribed at $t=0$.
The Hessian of this map is $I+t{\cal H}(\varphi_0)$, where $I$ is the identity
matrix and the matrix ${\cal H}(\varphi_0)$ is the Hessian of the initial
potential. For a given $\bm q$, the eigendirections of the associated Hessian are
those of the Hessian of the initial velocity potential. Clearly, all these
Hessians commute and, by the results of Sec.~\ref{sub:commu},
such a flow is omni-potential.

More general examples of omni-potential flows can be constructed by performing
an arbitrary nonlinear transformation of the time and by time-dependent zooming
of space. In space of any dimension $d\ge 2$, consider the flows defined by
the potentials
\begin{equation}
\Phi(\bm q,0;t)=\mu(t){|\bm q|^2\over 2}+\eta(t)\varphi_0(\bm q),
\label{genZel}\end{equation}
where $\mu(t)$ and $\eta(t)$ are arbitrary functions of time. Clearly,
these are again omni-potential.

In general, the trajectories associated with \eqref{genZel} are not straight
lines. However, if we look at them with a time-dependent magnifying glass which
applies a zooming factor $1/\mu(t)$, they become
straight. Furthermore, if we introduce a new time variable
$t'=\eta(t)/\mu(t)$, particles move again with a constant velocity.
Hence, the flows defined by \eqref{genZel} are trivial generalizations
of Zeldovich flows, and will here be called \textit{Zeldovich-type} flows.

Our goal is to find omni-potential flows that are not of this type.

\subsection{A linear second-order PDE for two-dimensional omni-potential flow}
\label{sub:eqn2d}

We derive here a differential equation for the potential of a two-dimensional
omni-potential flow. It turns out to be a linear second-order PDE.

Consider a symmetric $2\times2$ matrix $H$.
Suppose its eigenvector associated with eigenvalue $\lambda$
makes angle $\theta$ with the cartesian axis $q_1$. Thus,
$$H_{11}\cos\theta+H_{12}\sin\theta=\lambda\cos\theta,$$
$$H_{12}\cos\theta+H_{22}\sin\theta=\lambda\sin\theta.$$
In order to eliminate the eigenvalue $\lambda$, we multiply the first of these
equations by $\sin\theta$ and the second one by $\cos\theta$. Subtracting
afterwards the second equation from the first one, we obtain
\begin{equation}
{H_{11}-H_{22}\over H_{12}}=\cot2\theta.
\label{hess2d}\end{equation}
Prescribing the r.h.s.~of \eqref{hess2d} uniquely defines the orthogonal
frame of the two eigendirections. (The values of $\cot2\theta$ define the angle
$\theta$ modulo $\pi/2$; however, changing $\theta\to\theta+\pi/2$ swaps the
eigendirections, but does not affect the set of eigendirections.)

In an omni-potential flow, the eigendirections of the Hessians
of the $(0,t)$-potentials should depend only on the Lagrangian position $\bm q$
and not on the time $t$. Let $\Phi(\bm q,t)$ be a two-dimensional omni-potential
flow and let us denote by $g(\bm q)$ the common value of $\cot2\theta$ along
the particle trajectory emanating from $\bm q$. It then follows from \eqref{hess2d} that
\begin{equation}
(\partial^2_{q_1q_1}-\partial^2_{q_2q_2})\Phi=g(\bm q)\,\partial^2_{q_1q_2}\Phi
\label{gen2d}.\end{equation}

The search for two-dimensional omni-potential flow has thus been reduced to
finding solutions to \eqref{gen2d} for suitably prescribed functions $g(\bm q)$.

\section{Examples of omni-potential flows in two and three dimensions}
\label{s:algebra}

The main question that we address in this paper is whether omni-potential
flows exist that are not of Zeldovich type. In this section we give
a positive reply to this question both in the two- and three-dimensional spaces
by providing explicit examples of polynomial potentials for mappings induced
by such flows.

An example of an omni-potential flow in a space of arbitrary dimension is provided
by spherically-symmetric potentials of the form $\Phi(|\bm q|,t)$: a simple
calculation reveals the commutation of Hessians of such potentials, calculated
at different times at different points of a trajectory. This example
shares with Zeldovich flows the property that the trajectories are
straight lines --- in this case, in the radial direction. We would like
to construct less symmetric examples.

\subsection{Particular examples of two-dimensional omni-potential flow}
\label{sub:ex2d}

In ${\mathbb R}^2$, the problem of finding omni-potential flows has been recast
into the form of the partial differential equation \rf{gen2d} with the
initial condition $|\bm q|^2/2$ (which generates the identity map). We can
therefore construct examples of two-dimensional omni-potential flows
by finding different solutions to \rf{gen2d} for a given function
$g(\bm q)$ in the r.h.s. By linearity, any linear combination of such
solutions with time-dependent coefficients is also a solution to
\rf{gen2d}. For example, if $\Phi_1(\bm q)$ and $\Phi_2(\bm q)$ are two
sufficiently smooth independent solutions that are also independent of
$|\bm q|^2/2$ (which is always a solution to \rf{gen2d}), then the flow
with the potential $|\bm q|^2/2+\alpha_1(t)\Phi_1(\bm q)+\alpha_2(t)\Phi_2(\bm q)$
is omni-potential, and is typically not of Zeldovich type; for this,
the functions of time $\alpha_1$ and $\alpha_2$ must be linearly independent
and sufficiently small, so as not to spoil the convexity stemming
from the $|\bm q|^2/2$ term.

When $g(\bm q)$ is a ratio of homogeneous polynomials of $\bm q$ (say, of the same
degree $m$), solutions to \rf{gen2d} can be obtained by a purely algebraic
method. A solution can be sought
in the form of a homogeneous polynomial, $p^{(2)}_n(\bm q)$, of degree $n\ge m+2$;
then \rf{gen2d} reduces to a system of $m+n-1$ equations for the coefficients of
$p^{(2)}_n(\bm q)$ and $g(\bm q)$. (In what follows, $p^{(d)}_n$ denotes certain
homogeneous polynomials of degree $n$ defined in ${\mathbb R}^d$.) The function $g(\bm q)$
involves $2m+1$ independent coefficients (since the numerator and denominator
can be multiplied by any constant without changing $g(\bm q)$), and the polynomial
$p^{(2)}_n(\bm q)$ involves $n$ independent coefficients (since a solution
to \rf{gen2d} can be multiplied by any constant without yielding a new
independent solution). Comparison of the number of equations, $m+n-1$, with
the total number of the unknown coefficients, $2m+n+1$, suggests that we can
construct a family of such solutions, parameterized by $m+2$ coefficients
of $g(\bm q)$. However, the system of equations for the coefficients is,
in general, nonlinear, and hence its solvability cannot be established just
by counting the numbers of the unknowns and equations. When $g(\bm q)$ is the ratio
of linear functions, the equations for the coefficients of $p^{(2)}_n(\bm q)$ are
linear, and can be solved for any prescribed coefficients of~$g(\bm q)$.

Since the potential $\Phi$ is required to be a convex function on the entire
plane ${\mathbb R}^2$, we start by seeking homogeneous polynomials $p^{(2)}_n(\bm q)$
involving only even powers of $q_1$ and $q_2$. An instance of a solvable linear
system of equations yielding the coefficients of such polynomials is obtained for
\begin{equation}
g(\bm q)={aq_1^2-bq_2^2\over q_1q_2},
\label{gexa}\end{equation}
where the coefficients $a$ and $b$ may take arbitrary preset values. For such
$g(\bm q)$, a homogeneous polynomial of degree $2k$, $k\ge2$, solving \rf{gen2d}, is
\begin{eqnarray}
&&\hspace*{-8mm}p^{(2)}_{2k}(q_1,q_2)=\sum_{i=0}^k\left(\,\prod_{j=0}^{i-1}(2k-1+2j(a-1))\right.\nonumber\\
&&\hspace*{-8mm}\times\left.\prod_{j=0}^{k-1-i}(2k-1+2j(b-1))\right){k!\ q_1^{2i}q_2^{2(k-i)}\over i!(k-i)!(2k-1)}.
\label{poly2k}\end{eqnarray}
In particular, the first low-degree polynomial solutions are:
\begin{eqnarray}
p^{(2)}_4(q_1,q_2)&=&(2a+1)q_1^4+6q_1^2q_2^2+(2b+1)q_2^4,\label{p4}\\
p^{(2)}_6(q_1,q_2)&=&(4a+1)(2a+3)q_1^6+15(2a+3)q_1^4q_2^2\nonumber\\
&&\hspace*{-1cm}+15(2b+3)q_1^2q_2^4+(4b+1)(2b+3)q_2^6.\label{p6}
\end{eqnarray}
As one can see, the polynomial \rf{poly2k} vanishes identically for integer
$j\ge1$ and $\widehat{j}\ge1$ such that $j+\widehat{j}\le k-1$, and
\begin{equation}
\widehat{a}=1-{2k-1\over2\widehat{j}},\qquad\widehat{b}=1-{2k-1\over2j}.
\label{param}\end{equation}
For these isolated values in the plane of parameters, there exist nevertheless
two independent solutions, namely
$${\partial\over\partial a}\left.p^{(2)}_{2k}\right|_{a=\widehat{a},\,b=\widehat{b}}
\qquad\hbox{and}\qquad
{\partial\over\partial b}\left.p^{(2)}_{2k}\right|_{a=\widehat{a},\,b=\widehat{b}}.$$
This can be easily seen by differentiating \rf{gen2d} in $a$ and $b$ and
substituting the parameter values \rf{param}.

Clearly, $p^{(2)}_{2k}(\bm q)$ is convex, if all coefficients in \rf{poly2k} are
positive, i.e., if
\begin{equation}
\min(a,b)\ge-1/(2k-2).
\label{minabe}\end{equation}
Thus, the potentials
\begin{eqnarray}
\Phi(\bm q,t)=\mu_2(t){|\bm q|^2\over 2}+\sum_{k\ge2}\mu_{2k}(t)p^{(2)}_{2k}(q_1,q_2)
\label{exa2d}\end{eqnarray}
are convex for $\min(a,b)\ge0$, if in addition all $\mu_{2k}(t)$ are non-negative
(this condition is sufficient, but not necessary) and tend to zero fast enough
to guarantee convergence of the series at any point $\bm q$ and termwise differentiability
of \rf{exa2d} in the spatial variables. If the sum \rf{exa2d} is finite
and the maximum degree of the polynomials involved is $2K$, then the potential
is convex for $\min(a,b)\ge-1/(2K-2)$. The initial condition is satisfied
provided $\mu_2(0)=1$ and $\mu_{2k}(0)=0$ for all $k>1$. The convex potentials
\rf{exa2d} satisfy all requirements for omni-potential flows in the plane, and
are not of Zeldovich type.

So far, we have considered only even-degree homogeneous polynomial solutions or
linear combinations thereof. Is an admixture of odd-degree homogeneous
polynomials permitted? If such an odd-degree addition has a degree higher than
that of the highest even-degree homogenous polynomial comprised in the
solution, then convexity in the whole plane is ruled out. However, in a finite
domain, convexity need not be lost, provided the odd polynomial is scaled by
a sufficiently small factor. This is precisely what happens when $g(\bm q)$ is given by
\rf{gexa}: a homogeneous polynomial of odd degree $2k+1$ can be a solution to
\rf{gen2d} only for $k\ge2$ and
\begin{equation}
a=b=-1/(k-1).
\label{minabo}\end{equation}
The solution for these parameter values is
$$p^{(2)}_{2k+1}(q_1,q_2)=c_1\mathfrak{p}_k(q_1,q_2)+c_2\mathfrak{p}_k(q_2,q_1),$$
where
\begin{eqnarray*}
\mathfrak{p}_k(q_1,q_2)\!=\!\sum_{i=0}^{k-1}&&\hspace*{-4mm}\left(\,
\prod_{j=0}^{i-1}{(2(k-j)+1)(k-1-j)\over(j+1)(2j-1)}\right)\\
&&\hspace*{-3mm}\times\,q_1^{2i}q_2^{2(k-i)+1},
\end{eqnarray*}
and $c_1$ and $c_2$ are arbitrary constants.
For instance, for $a=b=-1$ fifth-degree polynomial solutions are
$$p^{(2)}_5(q_1,q_2)=c_1q_1^5-5c_1q_1^3q_2^2-5c_2q_1^2q_2^3+c_2q_2^5.$$
Comparison of conditions \rf{minabe} and \rf{minabo} shows that
the degree of an odd-degree polynomial solution for $g(\bm q)$ defined by \rf{gexa}
is higher than the degree of any convex even-degree polynomials existing
for the chosen $g(\bm q)$.

\subsection{Examples of omni-potential flow in dimension $d\ge 3$}
\label{sub:ex3d}

The approach that has been applied in the previous subsection for construction
of an example in dimension two cannot be immediately generalized
to higher-dimensional spaces: while in ${\mathbb R}^2$ a single invariant determines
whether two symmetric matrices are codiagonalizable and hence equation \rf{hess2d}
fixes the set of eigendirections of a symmetric matrix $H$, in ${\mathbb R}^3$ at least
three such invariants must be considered simultaneously (see the Appendix).
In dimension three, equations \rf{inv3d1}--\rf{inv3d2} applied for the entries
of the Hessian of an unknown potential give rise to three partial differential
equations in the potential,
\begin{eqnarray}
&&{\partial^2_{q_1,q_3}\Phi\over\partial^2_{q_1,q_2}\Phi}\!\left(
{(\partial^2_{q_1,q_1}-\partial^2_{q_2,q_2})\Phi\over\partial^2_{q_1,q_2}\Phi}
+{\partial^2_{q_2,q_3}\Phi\over\partial^2_{q_1,q_3}\Phi}
-{\partial^2_{q_1,q_3}\Phi\over\partial^2_{q_2,q_3}\Phi}\right)\nonumber\\
&=&\!\!\!\left(g_1(\bm q)+{(\partial^2_{q_1,q_1}-\partial^2_{q_2,q_2})\Phi\over\partial^2_{q_1,q_2}\Phi}\right)\nonumber\\
&\times&\!\!\!\left(
{(\partial^2_{q_2,q_2}-\partial^2_{q_3,q_3})\Phi\over\partial^2_{q_2,q_3}\Phi}
+{\partial^2_{q_1,q_3}\Phi\over\partial^2_{q_1,q_2}\Phi}
-{\partial^2_{q_1,q_2}\Phi\over\partial^2_{q_1,q_3}\Phi}\right),\label{pde3d1}\\
&&{\partial^2_{q_2,q_3}\Phi\over\partial^2_{q_1,q_3}\Phi}
\left(g_1(\bm q)+{(\partial^2_{q_1,q_1}-\partial^2_{q_2,q_2})\Phi\over\partial^2_{q_1,q_2}\Phi}\right)\nonumber\\
&=&\!\!g_2(\bm q)-g_3(\bm q){(\partial^2_{q_1,q_1}-\partial^2_{q_2,q_2})\Phi\over\partial^2_{q_1,q_2}\Phi},\label{pde3d2}\\
&&g_3(\bm q)\left(
{(\partial^2_{q_2,q_2}-\partial^2_{q_3,q_3})\Phi\over\partial^2_{q_2,q_3}\Phi}
+{\partial^2_{q_1,q_3}\Phi\over\partial^2_{q_1,q_2}\Phi}
-{\partial^2_{q_1,q_2}\Phi\over\partial^2_{q_1,q_3}\Phi}\right)\nonumber\\
&=&\!\!{(\partial^2_{q_3,q_3}-\partial^2_{q_1,q_1})\Phi\over\partial^2_{q_1,q_3}\Phi}
+{\partial^2_{q_1,q_2}\Phi\over\partial^2_{q_2,q_3}\Phi}
-{\partial^2_{q_2,q_3}\Phi\over\partial^2_{q_1,q_2}\Phi},
\label{pde3d3}\end{eqnarray}
that must be satisfied simultaneously. Here the time-independent
quantities $g_k(\bm q)$ are related to the invariants $\gamma^{(3,k)}_{21}$
introduced in the Appendix:
\begin{eqnarray*}
g_1(\bm q)&=&\gamma^{(3,1)}_{21}+\gamma^{(3,3)}_{21},\\
g_2(\bm q)&=&\gamma^{(3,2)}_{21}+1,\\
g_3(\bm q)&=&\gamma^{(3,3)}_{21}.
\end{eqnarray*}
The invariants are combinations of the entries of a symmetric matrix,
which depend only on the set of eigendirections and not on the eigenvalues.
The algebraic theory of the invariants presented in the Appendix does not
take into account the properties of the Hessian, stemming from the specific
structure of its entries (for instance, each row and a column of the Hessian is
a gradient, which implies certain differential relations between the entries).
It is unclear how to prescribe $g_k(\bm q)$, taking into account these additional
constraints, for the equations to have at least two distinct solutions.

Because of this difficulty, instead of considering the invariants and solving
equations \rf{pde3d1}--\rf{pde3d3}, we exploit the fact that omni-potentiality
of flows amounts to commutation of the Hessians of the various
$(t_1,t_2)$-mappings along any trajectory (see Sec.~\ref{sub:commu}). We shall
construct our examples in ${\mathbb R}^d$ using the following strategy. The potential
is sought in the form of a linear combination of ``building blocks'' with
time-dependent coefficients. One of the building blocks is prescribed; we
take it to be a homogeneous polynomial, $p^{(d)}_m(\bm q)$, of degree $m$. All
the other building blocks are then required to have their Hessians commuting
with that of the prescribed building block. The function $|\bm q|^2$, whose
Hessian is the identity matrix, constitutes a trivial solution. We can try
finding other building blocks in the form of homogeneous polynomials
$p^{(d)}_n(\bm q)$ of some degree $n$. If we succeed, it is easy to show that
any linear combination of such building blocks (with the convexity restriction)
will define an omni-potential flow. We seek such polynomials by requiring the
vanishing of the non-diagonal entries of the commutator of the two Hessians, viz.
\begin{equation}
C(p^{(d)}_m,p^{(d)}_n)\equiv{\cal H}(p^{(d)}_m){\cal H}(p^{(d)}_n)-{\cal H}(p^{(d)}_n){\cal H}(p^{(d)}_m).
\label{commr}\end{equation}

Unfortunately, in general this strategy does not work, as now explained.
A homogeneous polynomial of degree $n$ has
$$(n+d-1)!\over n!(d-1)!$$
coefficients. Non-diagonal entries of the commutator $C$ are homogeneous
polynomials of degree $m+n-4$. The commutator is antisymmetric (recall that
the Hessians are symmetric matrices), hence we have to consider the $d(d-1)/2$
non-diagonal entries of $C$. Thus, in general, we have to solve
$$d(m+n+d-5)!\over2(m+n-4)!(d-2)!$$
equations, a number which is easily seen to exceed the number of coefficients,
$${(m+d-1)!\over m!(d-1)!}+{(n+d-1)!\over n!(d-1)!}.$$
So, the problem is overdetermined.

Nevertheless, potentials having all the required properties can be constructed
in ${\mathbb R}^d$ ($d\ge2$), if all the building blocks are restricted to be
homogeneous polynomials symmetric in all their arguments (i.e., invariant
under any permutation of the spatial variables $q_i\leftrightarrow q_j$). Such
building blocks have the following significant advantage: it suffices to
consider any of the polynomial equations arising from non-diagonal entries
of the commutator \rf{commr} (referred to as ``commutator equations'') -- all
these equations are equivalent by virtue of the symmetry.
In what follows, we implement this ``symmetric building block strategy''
in two cases, in ${\mathbb R}^d$ for $d\ge 3$ with just one unknown building block,
and in ${\mathbb R}^3$ with infinitely many ones.

Now, we focus on the symmetric polynomials
\begin{eqnarray}
p^{(d)}_4(\bm q)&=&\sum_{i=1}^dq_i^4+\widetilde{c}\sum_{i=2}^d\sum_{j=1}^{i-1}q_i^2q_j^2,
\label{p4gen}\\
p^{(d)}_6(\bm q)&=&\sum_{i=1}^dq_i^6+\widetilde{a}\sum_{i=1}^d\sum_{j=1}^dq_i^4q_j^2\nonumber\\
&&+\,\widetilde{b}\sum_{1\le i<j<k\le d}q_i^2q_j^2q_k^2.
\label{p6gen}\end{eqnarray}
(When $d=3$, the last sum in $p^{(d)}_6(\bm q)$ reduces to a single term
$\widetilde{b}q_1^2q_2^2q_3^2$.) We consider polynomials involving only even
powers of the spatial variables $q_j$, because we seek solutions that are
convex functions.

To be specific, we consider the commutator equation
$C_{12}(p^{(d)}_6,p^{(d)}_4)=0$ in ${\mathbb R}^d$, $d\ge3$. The l.h.s.~is a polynomial
of degree 6. Since in $p^{(d)}_4$ and $p^{(d)}_6$ any power of $q_1$ and $q_2$
is even, $C_{12}$ is proportional to $q_1q_2$, and every variable enters
into the polynomial $C_{12}/(q_1q_2)$ only in an even power.
Since both potentials are symmetric in $q_1$ and $q_2$, $C_{12}=0$
for $q_1=q_2$, and hence $C_{12}$ is divisible by
$q_1^2-q_2^2$. The polynomial $C_{12}/(q_1q_2(q_1^2-q_2^2))$
is of the second degree; it is thus just a sum of $q_j^2$ with certain
coefficients. Because the potentials are symmetric in $q_j$, it has the form
$$\alpha_1(q_1^2+q_2^2)+\alpha_2\sum_{j=3}^dq_j^2.$$
Hence, we have three independent parameters, $\widetilde{a}$, $\widetilde{b}$ and
$\widetilde{c}$, and two equations to satisfy. Calculating the coefficients
$\alpha_1$ and $\alpha_2$ and letting them vanish, we find
that the Hessians of $p^{(d)}_4(\bm q)$ and $p^{(d)}_6(\bm q)$ commute for
\begin{eqnarray}
\widetilde{a}&=&15\,\widetilde{c}/(12-\widetilde{c}),\label{p6a}\\
\widetilde{b}&=&75\,\widetilde{c}^{\,2}/((12-\widetilde{c})(3+\widetilde{c})).
\label{p6b}\end{eqnarray}
Thus, the potential
\begin{equation}
\Phi(\bm q,t)=\mu_2(t){|\bm q|^2\over 2}+\mu_4(t)p^{(d)}_4(\bm q)+\mu_6(t)p^{(d)}_6(\bm q)
\label{polydd}\end{equation}
defines a non-Zeldovich-type omni-potential flow in ${\mathbb R}^d$ ($d\ge3$).
Polynomials $p^{(d)}_4(\bm q)$ and $p^{(d)}_6(\bm q)$ are convex provided
$0\le\widetilde{c}<12$; hence, if all $\mu_i(t)\ge 0$, potential
\rf{polydd} is convex for $\widetilde{c}$ from this interval.
For $\widetilde{c}\ne 2$, $p^{(d)}_4(\bm q)$ and $p^{(d)}_6$ do not possess
spherical symmetry, and hence the potential \rf{polydd} is not spherically
symmetric. Restrictions of $p^{(d)}_4(\bm q)$ and $p^{(d)}_6(\bm q)$ onto the plane
$q_3=...=q_d=0$ coincide with the polynomials $p^{(2)}_4(q_1,q_2)$ \rf{p4} and
$p^{(2)}_6(q_1,q_2)$ \rf{p6} for $a=b=(6-\widetilde{c})/(2\widetilde{c})$.

Henceforth, for the sake of simplicity, we assume that the problem is
three-dimensional. In the remainder of the subsection we shall implement
the building block strategy with unknown blocks that are arbitrary even-degree
polynomials $p^{(3)}_{2n}(\bm q)$ for $n>2$. The polynomial $p^{(d)}_4(\bm q)$
\rf{p4gen} for $d=3$ remains our prescribed building block. By the theorem
on codiagonalizability of symmetric matrices, commutation with the Hessian
of $p^{(3)}_4(\bm q)$ implies, that the Hessians of any two polynomials from this
family commute. (This is taking place generically, i.e., at those points
in ${\mathbb R}^3$, where the Hessian of $p^{(3)}_4(\bm q)$ does not possess equal
eigenvalues; at non-generic points the commutation follows from continuity
of the Hessians and the commutation at generic points, which are present
in any neighborhood of a non-generic point.)

The polynomial
\begin{equation}
p^{(3)}_{2n}(\bm q)=\sum_{\mbox{$\scriptsize i,j,k\ge0\atop i+j+k=n$}}
\widetilde{a}_{i,j,k}\,q_1^{2i}q_2^{2j}q_3^{2k}
\label{p2n}\end{equation}
is symmetric, whenever
\begin{equation}
\mbox{$\widetilde{a}_{i,j,k}$ does not depend on the order of subscripts $i,j,k$.}
\label{p2ncoe}\end{equation}
Straightforward algebra yields
\begin{eqnarray*}
&&\hspace*{-6mm}C_{12}(p^{(3)}_{2n},p^{(3)}_4)=8q_1q_2\sum_{\mbox{$\scriptsize i,j,k\ge0\atop i+j+k=n$}}
\widetilde{a}_{i,j,k}\,q_1^{2i-2}q_2^{2j-2}q_3^{2k}\\
&&\hspace*{4mm}\times\left(ij(\widetilde{c}-6)(q_1^2-q_2^2)\right.\\
&&\hspace*{4mm}\left.+\,\widetilde{c}\,(-j(2j-1+2k)q_1^2+i(2i-1+2k)q_2^2)\right).
\end{eqnarray*}
Collecting similar terms in this sum, we find that it vanishes as long as
\begin{equation}
\widetilde{a}_{i,j,k}=\widetilde{a}_{i+1,j-1,k}\,\chi_j/\chi_{i+1}
\label{recur}\end{equation}
for any $i,\,j$ and $k$ such that $i+j+k=n$, where we have denoted
$$\chi_m=(\widetilde{c}\,(2n+2-3m)+6(m-1))/m.$$
Relation \rf{recur} can be regarded as a recurrence for coefficients
$\widetilde{a}_{i,j,k}$ for a fixed $k$. For $k=0$, we start the recurrence
assuming $\widetilde{a}_{n,0,0}=1$. This yields
$$\widetilde{a}_{i,n-1,0}={\prod_{m=1}^{n-i}\chi_m\prod_{m=1}^i\chi_m\over\prod_{m=1}^n\chi_m}.$$
We obtain now the starting values for the recurrence \rf{recur} for $k>0$
setting, in view of \rf{p2ncoe},
$$\widetilde{a}_{n-k,0,k}=\widetilde{a}_{n-k,k,0},$$
and find
\begin{equation}
\widetilde{a}_{i,j,k}={\prod_{m=1}^i\chi_m\prod_{m=1}^j\chi_m\prod_{m=1}^k\chi_m\over\prod_{m=1}^n\chi_m}.
\label{coeff2n}\end{equation}
Evidently, coefficients \rf{coeff2n} satisfy the symmetry condition \rf{p2ncoe}.
Hence, \rf{p2n} with the coefficients \rf{coeff2n} is a symmetric homogeneous
polynomial of degree $2n$, whose Hessian commutes with the Hessian of $p^{(3)}_4(\bm q)$.
The potential
\begin{equation}
\widetilde{\Phi}(\bm q,t)=\mu_2(t){|\bm q|^2\over 2}+\sum_{n\ge 2}\mu_{2n}(t)p^{(3)}_{2n}(\bm q)
\label{xpoly}\end{equation}
defines an omni-potential flow of a non-Zeldovich type in ${\mathbb R}^3$ (provided
the coefficients $\mu_{2n}(t)$ tend to zero sufficiently fast to guarantee convergence
of the series \rf{xpoly} and to allow its termwise differentiation). Since
the polynomial $p^{(3)}_{2n}(\bm q)$ is convex for
$$0\le\widetilde{c}<6(n-1)/(n-2),$$
the potential \rf{xpoly} is convex if all $\mu_{2n}(t)$ are non-negative and
$0\le\widetilde{c}\le 6$ (or for $0\le\widetilde{c}<6(N-1)/(N-2)$, if all
$\mu_{2n}(t)$ vanish for $n>N$).

Although we have constructed our example without prescribing the invariants,
it might be of interest to calculate them for the solutions that have been
obtained. Straightforward calculations yield the values of the invariants
for the potential $\widetilde\Phi(\bm q,t)$, for instance,
$$\gamma^{(3,1)}_{21}={(6+3\widetilde{c})q_2^2-(6+\widetilde{c})q_1^2\over2\widetilde{c}q_1q_2}
+{2q_2(q_1^2-q_2^2)\over q_1(q_2^2-q_3^2)},$$
which is consistent with a non-trivial dependence of the eigendirections
on the trajectories (labeled by the Lagrangian coordinates). We note that,
although the solution is symmetric in the spatial coordinates, the symmetry
is lost in the invariant. This stems from the invariant under
consideration being a nonlinear function of the projections of the
eigendirections on the plane $(q_1,q_2)$, and also from the components of the
eigendirections not being invariant under all permutations of coordinates (an eigenvector is
invariant under a permutation of the spatial variables $q_i\leftrightarrow q_j$
provided its $i$th and $j$th components are swapped).

We have used two approaches for constructing examples of omni-potential flow.
In three or more dimensions, we used the building-block strategy in which
the field of eigendirections of the commuting Hessians is characterized by
prescribing one of the blocks (in our construction the polynomial \rf{p4gen}).
The problem of commutation of Hessians then reduces to three linear equations
in the unknown block. These equations must be satisfied simultaneously, and we
found that in general no solution exists for an arbitrary prescribed block.
In two dimensions we have followed another approach, whereby the field of
eigendirections of the commuting Hessians is characterized by prescribing the
set of invariants, from which the field of eigendirections of the commuting
Hessians can be uniquely determined. In ${\mathbb R}^2$ just one such invariant,
$g(\bm q)$ (see \rf{gen2d}), should be considered. In ${\mathbb R}^3$ one must consider
three invariants, for instance, \rf{inv3d1}--\rf{inv3d2} of the Appendix, giving rise to three
nonlinear equations \rf{pde3d1}--\rf{pde3d3}. As in the former approach, these
equations must be satisfied simultaneously, and hence a solution does not
exist for an arbitrary set of prescribed invariants. Thus, whichever approach
is used for construction of omni-potential flows in ${\mathbb R}^3$, the prescribed
data must be tuned for a solution to exist. In the former approach the
equations are linear and thus simpler, but one is left with just one function
which can be tuned to achieve consistency of the three equations under
consideration; in the latter approach the equations are nonlinear and thus
more involved, but one has the freedom of tuning three a priori independent
scalar functions to gain consistency of the equations. Of course, in three
dimensions, the three invariants of the Hessian of the potential cannot be
prescribed as arbitrary functions. In other words, some conditions
on the invariants must hold for the three equations \rf{pde3d1}--\rf{pde3d3}
to be compatible. Can such conditions on the invariants be expressed in more
explicit form remains an open mathematical problem.

\section{A WKB approach to two-dimensional omni-potentiality}
\label{s:wkb}

So far we have obtained special cases of non-Zeldovich-type omni-potential
flows. How general are they? Can we, for example, in the two-dimensional case
prescribe an arbitrary smooth initial velocity potential $\varphi_0(\bm q)$ or,
more precisely, the invariant of its Hessian ${\cal H}(\varphi_0(\bm q))$:
\begin{equation}
g(\bm q)\equiv\frac{(\partial^2_{11}-\partial^2_{22})\varphi_0(\bm q)}{\partial^2_{12}\varphi_0(\bm q)}
=\frac{{\cal H}_{11}-{\cal H}_{22}}{{\cal H}_{12}}
\label{gintermsofphi0}\end{equation}
that appears in the general equation \eqref{gen2d}? In this section,
where we use only Lagrangian coordinates, $\partial_1$ and $\partial_2$ are short
for $\partial_{q_1}$ and $\partial_{q_1}$; similarly, $\partial^2_{11}$,
$\partial^2_{12}$ and $\partial^2_{22}$ denote the second Lagrangian derivatives.

We shall now show how the construction of non-Zeldovich-type omni-potential flow
with arbitrary invariant function $g(\bm q)$ can be done, using an idea of Arnold
for solving the linear equation which controls the stability of solution to the
Euler equation \cite{arnold1972}. Rather than trying to find the most general
solution to \eqref{gen2d}, we construct a special short-wavelength
solution through the WKB ansatz
\begin{equation}
\Phi(\bm q)=\mathrm{e}^{\mathrm{i}\kappa S(\bm q)}\left[A_0(\bm q)+\frac{1}{\kappa}A_1(\bm q)
+\frac{1}{\kappa^2}A_2(\bm q)+\ldots\right]+{\rm cc},
\label{wkb}\end{equation}
where the wavenumber $\kappa$ is taken very large and where cc stands for complex
conjugate (needed because we want real solutions). In WKB parlance, $S(\bm q)$ is
called the eikonal function and the functions $A_0(\bm q)$,
$A_1(\bm q)$, \ldots~are the amplitudes.

To the leading order, O$\left(\kappa^2\right)$, the WKB ansatz turns the linear
second-order PDE \eqref{gen2d} into the following nonlinear first-order PDE:
\begin{equation}
\frac{\left(\partial_1 S\right)^2-\left(\partial_2 S\right)^2}{\left(\partial_1 S\right)
\left(\partial_2 S\right)}=g(\bm q).
\label{nonlinearpde}\end{equation}
It is easily checked that \eqref{nonlinearpde} is equivalent to the
statement that, in the leading order, $\nabla_{\bm q}S(\bm q)$ is an eigenvector
of the Hessian ${\cal H}(\varphi_0(\bm q))$. Actually, this can be seen
directly, by an argument which also applies in space dimensions $d$ higher than
two. Assume that the leading WKB term for the potential has a fast spatial
dependence involving the phase factor $\mathrm{e}^{\mathrm{i}\kappa S(\bm q)}$, then the Hessian
will involve in the leading order a matrix factor $-\kappa^2(\partial_i S)(\partial_j S)$.
This is a degenerate matrix with one eigenvector of non-vanishing eigenvalue
in the direction of $\nabla_{\bm q} S(\bm q)$; all perpendicular vectors are associated
with the eigenvalue zero, which has multiplicity $d-1$. Omni-potentiality
requires that this degenerate matrix commute with the Hessian of the initial
potential or, equivalently, that $\nabla_{\bm q}S(\bm q)$ be an eigenvector
of ${\cal H}(\varphi_0(\bm q))$.

Returning to the two-dimensional case, we now construct the eikonal
function $S(\bm q)$. This construction will be done only locally in a
neighborhood $\Omega$, in which the Hessian ${\cal H}(\varphi_0(\bm q))$ is
sufficiently smooth and its eigenvalues are everywhere distinct. (We
recall that a double eigenvalue is an event of codimension two, which
typically takes place at isolated locations.) Let ${\bm n}^{(1)}(\bm q)$
and ${\bm n}^{(2)}(\bm q)$ be two unit eigenvectors of ${\cal H}(\varphi_0(\bm q))$,
chosen to depend smoothly on $\bm q$ in $\Omega$. The condition that the gradient
of the eikonal function be parallel to an eigendirection can now be expressed as
\begin{equation}
{\bm n}^{(1)}(\bm q)\cdot\nabla_{\bm q}S(\bm q)=0\quad{\rm or}\quad{\bm n}^{(2)}(\bm q)\cdot\nabla_{\bm q}S(\bm q)=0.
\label{twoeikonal}
\end{equation}
In words, these equations state that either ${\bm n}^{(1)}(\bm q)$
or ${\bm n}^{(2)}(\bm q)$ is normal to the level lines of the eikonal function.
Equivalently, the level lines of $S$ are the integral curves defined
by either ${\bm n}^{(2)}(\bm q)$ or ${\bm n}^{(1)}(\bm q)$. These form a
set of orthogonal curves. We thus have two classes of solutions.
We can prescribe $S$ arbitrarily on one of these curves ${\cal C}$ and
extend it locally by demanding that it remains constant on all the
curves orthogonal to ${\cal C}$. Note that these orthogonal
curves play here the role of rays in geometrical optics and are thus
conveniently called ``rays''.

Next we write the equations for subleading corrections obtained by substituting
\eqref{wkb} in \eqref{gen2d} and identifying the coefficients of the various
positive and negative powers of the large parameter $\kappa$. We shall
only write the equations appearing at orders $\kappa^1$ and $\kappa^0$
(the higher-order equations have a similar structure). For what follows it is
convenient to use the compact notation introduced by Monge in his theory
of surfaces: ${\hat p}$, ${\hat q}$, ${\hat r}$, ${\hat s}$ and ${\hat t}$ stand respectively for
$\partial_1 S$, $\partial_2 S$, $\partial^2_{11} S$, $\partial^2_{12} S$ and $\partial^2_{22} S$. (We added hats to avoid
possible confusions.) Furthermore, we write $g$ for $g(\bm q)$. The leading-order
equation, \eqref{nonlinearpde}, repeated for convenience, and the two first
subleading equations are:
\begin{eqnarray}
&&\!\!\!\!\!{\hat p}^2 -{\hat q}^2 -g{\hat p}{\hat q}=0, \label{orderm2}\\
&&\!\!\!\!\!({\hat r} -g{\hat s} -{\hat t})A_0 +2({\hat p}\partial_1 A_0 -{\hat q}\partial_2 A_0)
- g({\hat p}\partial_2 A_0+{\hat q}\partial_1 A_0) \nonumber\\
&&\!\!\!\!\!=0, \label{order1}\\
&&\!\!\!\!\!({\hat r}-g{\hat s}-{\hat t})A_1 +2({\hat p}\partial_1 A_1-{\hat q}\partial_2 A_1)
- g({\hat p}\partial_2 A_1+{\hat q}\partial_1 A_1) \nonumber\\
&&\!\!\!\!\!-\mathrm{i}(\partial^2_{11}-\partial^2_{22}-g\partial^2_{12})A_0=0.
\label{order0}
\end{eqnarray}

Using \eqref{orderm2} to eliminate the function $g$, we can
rewrite \eqref{order1} and \eqref{order0} as
\begin{eqnarray}
&&\hspace*{-1cm}{\hat q}\partial_1 A_0-{\hat p}\partial_2 A_0 +\frac{{\hat p}{\hat q}({\hat r}-{\hat t})-{\hat s}({\hat p}^2-{\hat q}^2)}{{\hat p}^2+{\hat q}^2}A_0=0,
\label{order1simpler}\\
&&\hspace*{-1cm}{\hat q}\partial_1 A_1-{\hat p}\partial_2 A_1 +\frac{{\hat p}{\hat q}({\hat r}-{\hat t})-{\hat s}({\hat p}^2-{\hat q}^2)}{{\hat p}^2+{\hat q}^2}A_1
\nonumber\\
&&\hspace*{-1cm}-\,\mathrm{i}\frac{{\hat p}{\hat q}}{{\hat p}^2+{\hat q}^2}(\partial^2_{11}-\partial^2_{22}-g\partial^2_{12})A_0=0.
\label{order0simpler}
\end{eqnarray}
Equation~\eqref{order1simpler} is a first-order linear homogeneous transport
equation for the amplitude $A_0$ along the rays. It can be integrated starting
from arbitrary non-zero data on any curve orthogonal to the rays.
Equation~\eqref{order0simpler} for the amplitude $A_1$ is of the same sort, except that it has
an inhomogeneous term involving $A_0$. We may thus take vanishing data for $A_1$ on
an arbitrary curve orthogonal to the rays. Higher-order amplitudes satisfy
similar inhomogeneous transport equations.

Now we construct locally in space and time an omni-potential flow having a given
invariant function $g(\bm q)$. We take the initial potential $\varphi_0(\bm q)$
arbitrary, but sufficiently smooth. Hence, by the WKB method described
above we can construct a smooth eikonal function $S(\bm q)$ and smooth amplitude
functions $A_0(\bm q)$, $A_1(\bm q)$, \ldots. This, in principle, yields a smooth
solution, $\Phi(\bm q)$, to \eqref{gen2d}. (We shall not address here the issue
of the convergence of the WKB series \eqref{wkb}.) Because of the imaginary
exponential dependence on $\kappa$, the potential $\Phi(\bm q)$ has very large
second spatial derivatives O$\left(\kappa^2\right)$ and
has no reason to be convex. However, the following
time-dependent potential defines an omni-potential Lagrangian map:
\begin{equation}
\Phi(\bm q,t)=\frac{|\bm q|^2}{2}+t\varphi_0(\bm q)+f(t)\frac{\epsilon}{\kappa^2}\Phi(\bm q).
\label{arbitrary2d-omnipot}
\end{equation}
Here $f(t)$ is an arbitrary smooth function of time that vanishes, together
with $\dot f$, at $t=0$. For example, we can take $f(t)=t^2$. For
sufficiently small $t$ and sufficiently small $\epsilon$, the last two terms in
the r.h.s.~of \eqref{arbitrary2d-omnipot} will not spoil the convexity of the
first term. We have thus constructed (locally) omni-potential flows for a
quite arbitrary initial velocity. For large $\kappa$, the trajectories
resulting from \eqref{arbitrary2d-omnipot} differ only minutely from the
straight Zeldovich trajectories associated with the two first terms. However,
these flows are not of Zeldovich type because of the third term in the r.h.s.

If we try to extend the above WKB procedure from two to three
dimensions, we encounter an obstacle already in constructing
the eikonal function $S$. As we have seen, its gradient with respect
to the Lagrangian position ${\bm q}$ should be everywhere parallel to
an eigendirection of the Hessian ${\cal H}(\varphi_0)$.
Denoting by ${\bm n}(\bm q)$ a unit eigenvector, taken with
a smooth ${\bm q}$-dependence, we should then have
\begin{equation}
\nabla_{\bm q}S(\bm q)=\mu(\bm q){\bm n}(\bm q),
\label{intfactor}
\end{equation}
where $\mu(\bm q)$ is a scalar function. In other words, the 1-form
${\bm n}(\bm q)\cdot d\bm q$ should have an integrating factor (a factor which
makes it an exact 1-form). This is in general possible (locally) in two,
but not in three dimensions.

\section{Cosmological implications}
\label{s:cosmo}

So far our point of view has been kinematical: we constructed omni-potential
flows without any underlying dynamical equations. In cosmology the dynamical
setting is rather well known and discussed for example in
Ref.~\cite{bernardeau}. Let us just recall a few salient points. The most
widely accepted explanation of the large-scale structure seen in galaxy
surveys is that it results from small primordial fluctuations that grew under
gravitational self-interaction of collisionless cold dark matter
particles in an expanding universe.
The evolution of the mostly collisionless matter in the Universe is
described by the Vlasov--Poisson system in the position-velocity phase
space. At early times, i.e., close to the epoch of matter-radiation
decoupling, the expansion of the Universe selects a
single velocity solution at each position rather than a distribution
of velocities. This feature persists until particle crossing
(``shell-crossing'' in the cosmological language), where multi-stream
solutions are developed. As long as multi-streaming is ruled out or is
confined to scales sufficiently small to be neglected, the Vlasov--Poisson
system may be replaced by the Euler-Poisson system. Following the notation
of Ref.~\cite{mnras} and using Eulerian comoving coordinates, $\bm x$, together
with a time variable $\tau$ based on the amplitude factor of the growing mode
of linear theory, we can write the Euler--Poisson system as
\begin{eqnarray}
\label{comoveuler}
{\partial_\tau}\v+(\v\cdot{\nabla_{\bm x}})\v&=&-\frac{3}{2\tau}(\v+{\nabla_{\bm x}}\varphi_{\mathrm{g}}),\\
\label{comovcontinuity}
{\partial_\tau}\rho+{\nabla_{\bm x}}\cdot(\rho\v) &=& 0,\\
\label{comovpoisson}
{\nabla^2_{\bm x}}\varphi_{\mathrm{g}} &=& \frac{\rho-1}{\tau}.
\end{eqnarray}
Here, $\v$ is the peculiar velocity, $\rho$ the density (suitably normalized)
and $\varphi_{\mathrm{g}}$ the gravitational potential. As
$\tau\to 0$, to avoid singularities, the density must approach unity
everywhere; thus the distribution of matter is in the leading order uniform as
$\tau\to 0$. Similarly, $\v\to-{\nabla_{\bm x}}\varphi_{\mathrm{g}}$ as $\tau\to 0$;
thus the initial velocity is potential, but
otherwise arbitrary. It then follows from \eqref{comoveuler} that the velocity
stays potential at any later time.

Reconstruction handles the Euler--Poisson system as a two-point
boundary-value problem in which the initial density is prescribed as
uniform and the final (current) density is given by astronomical
observations. This is a mass transport problem whose cost function is
the action associated with the Euler--Poisson equations (see
Refs.~\cite{mnras} and~\cite{loeper}). Unfortunately, because this
action is a rather complicated functional, we do not yet possess
efficient numerical algorithms allowing us to solve this mass transport problem.

The situation simplifies with the Zeldovich approximation, which
amounts to setting to zero the r.h.s.~of \eqref{comoveuler}. The
remaining equations are then (i) the three-dimensional inviscid
Burgers equation, which implies that particles are moving with
constant velocity (in the coordinates here chosen), and (ii) the
continuity equation expressing mass conservation. The action to be
minimized for reconstruction reduces then to its kinetic energy term.
As a consequence, the cost function is just
the mass-weighted integral of the squared displacement of fluid
particles from their initial (Lagrangian) positions $\bm q$ to their
current (Eulerian) positions $\bm x$, as required by a theorem of Brenier
\cite{brenier}. After discretization, this mass transport problem
becomes an assignment problem, which can be solved by efficient
algorithms (see Sec.~4 of Ref.~\cite{mnras} and
Ref.~\cite{bertsekas}). This is the essence of the
Monge--Amp\`ere--Kantorovich (MAK) reconstruction method.

Mohayaee et al.~\cite{royaetal} tested the quality of MAK reconstruction
by applying it to final states of standard N-body simulations, that were
performed for various random initial conditions of cosmological relevance.
The authors of \cite{royaetal} noted the unprecedented accuracy of
the reconstructions down to a few megaparsecs. In particular, they performed
comparisons between three different Lagrangian maps: (i) the map based on
the N-body integration, (ii) the map obtained by applying the MAK procedure to
the current density field, calculated by the N-body integration, (iii) the map
obtained by applying the Zeldovich approximation, starting from the same
initial condition as for the N-body simulation. The conclusion of their
comparisons is that the N-body map is approximated much better by the MAK-generated
map than by the Zeldovich map. This is particularly striking in their Fig.~7,
which shows the negative Lagrangian divergence of the displacement $\bm x-\bm q$,
obtained by the three methods mentioned above.

Can we understand this good performance of MAK reconstruction on
sufficiently large spatial scales? First, let us make the rather
obvious observation that any Lagrangian map that is the (Lagrangian)
gradient of a convex function (here called the ``Brenier property'')
will be reconstructed exactly (in a discretized version), if we solve
the associated quadratic-cost optimal transport problem, for example,
by using the MAK procedure. Here is a trivial example of this: if we let an
initially quasi-uniform mass distribution evolve by pure Zeldovich
dynamics to a final distribution and there is no shell crossing, then
the MAK reconstruction is exact. If the solutions to the
Euler--Poisson equations had the Brenier property, MAK would perform
an exact reconstruction, but they don't: flows which
solve the Euler--Poisson equations have no Eulerian vorticity but do
generate Lagrangian vorticity \cite{varenna}.

However there exists a
refinement of the Zeldovich approximation which possesses the Brenier
property. This is given by the second-order of the Lagrangian perturbation theory
\cite{moutarde91,buchert92,buchert94,bouchetal95,catelan95,sahni-shandarin1996}.
Here we shall not describe the Lagrangian perturbation theory in any
technical details since the reader can find them in the above
publications. Nevertheless, in order to discuss some of its conceptual
problems, we describe briefly a few key steps.
One rewrites the Euler--Poisson equations in Lagrangian coordinates, to
obtain a set of nonlinear equations for the displacement $\bm x-\bm q$ and
its space and time partial derivatives. Assuming then that in a
suitable sense (see below) the displacement is small, O$(\epsilon)$,
one expands the equations in powers of the small parameter $\epsilon$.
Here, the only perturbed quantities are the deviations of the particle
trajectory from the homogeneous Hubble flow, i.e., from a purely
expanding Universe. At the first order, O$(\epsilon)$, one has the
Zeldovich approximation, which, as discussed in
Sec.~\ref{sub:linea}, is omni-potential. In particular, the Lagrangian
map is potential (by the Brenier property) and the velocity is
potential in both Eulerian and Lagrangian coordinates. As we have
said, with the Lagrangian perturbation theory, one may refine the
Zeldovich approximation. The second-order Lagrangian perturbation theory
(usually denoted by L2) captures significant gravitational physics, for example
some tidal effects \cite{buchertehlers93}, whose importance in the large scale
structure formation has been widely recognized (see, e.g.,
\cite{peeblesgroth76}). L2 has the remarkable property that, for
standard cosmological initial conditions, the Lagrangian map is still
potential. As a consequence, the velocity is also potential in
Lagrangian coordinates. With L2, in Eulerian coordinates the velocity is
potential only up to second order. One would have to sum the whole series
to arbitrarily high orders of the Lagrangian perturbation theory (i.e.,
to arbitrary orders of $\epsilon$) to recover the Eulerian potentiality of the
velocity (but note that convergence of the asymptotic series is not guaranteed
\cite{sahni-shandarin1996,nadkarnigosh-chernoff2011}). L2 having the Brenier property,
the Lagrangian and inverse Lagrangian maps can be reconstructed exactly as
an optimal transport problem, for example, by the MAK technique. This is
probably the main reason why MAK performs well (at sufficiently large scales).
Beyond the second order of the Lagrangian
perturbation theory, scales below the non-linearity scale are
expected to play a decisive role and Lagrangian vorticity is
unavoidable \cite{buchert94}. Such scales cannot be handled accurately
by standard MAK reconstruction.

Finally, let us discuss briefly the thorny issue of the validity of the
Lagrangian perturbation theory. As mentioned before, the
Euler--Poisson equations are a consequence of the Vlasov--Poisson
equations only as long as multi-streaming is absent. The problem is
that, with any cosmologically realistic initial condition at
decoupling, multi-streaming appears immediately or, anyway, well before the
present epoch. The situation is somewhat similar to what we would have
with a one-dimensional Burgers flow in which the initial velocity
would be spatially non-differentiable or nearly so, for example, a
velocity field whose space dependence is the Ornstein--Uhlenbeck
process: shocks would then appear after an arbitrarily short time. This, of
course, constitutes no reason to discard the Euler--Poisson equations for
cosmology. One way to keep them is to \textit{regularize} the initial
velocity field by applying a low-pass filter, i.e., by setting to zero all
Fourier amplitudes whose wavenumbers exceed a cutoff $K =1/L$, where
$L$ is the regularization scale. It may then be shown that no
shell-crossing and thus no multi-streaming will take place for at least
some finite time $T_\star(L)$. Roughly, $T_\star(L)$ is the inverse of
the largest shear (velocity gradient) of the regularized initial
velocity. Up to this time the Euler--Poisson equations are valid.
Furthermore, it may be shown that, up to time $T$, the small parameter which
controls the validity of the Lagrangian perturbation theory (including that
of the Zeldovich approximation) is $T/ T_\star(L)$.
Of course, the true problem is not regularized, or only barely so. Comparison
with N-body simulations suggests nevertheless that the Lagrangian
perturbation theory remains valid for $T/T_\star(L)\ll 1$ when looking only
at scales greater than $L$. Understanding this from a theoretical point of view
is a challenge, which has been partially addressed by Buchert \cite{varenna}.

\section{Concluding remarks}
\label{s:conclusion}

The main question that we have addressed in this paper concerns
omni-potentiality, the (convex) potential character of the mapping
from any time $t_1$ to any time $t_2 \ge t_1$ with $0\le t_1\le t_2\le
T$. First, we have considered a class of flows of ``Zeldovich type'',
comprised of pure Zeldovich/Burgers flows and those obtained from them
by application of arbitrary nonlinear transformations of the time
variable and arbitrary time-dependent scale factors. Such flows are
trivially omni-potential. So are spherically symmetric flows. We then
have investigated (i) the existence of non-trivial omni-potential
flows, (ii) their genericity: can we prescribe the initial velocity
potential in an arbitrary way?

The flows have been characterized by their Lagrangian maps $\bm q\mapsto\bm x(\bm q,t)=
\nabla_{\bm q}\Phi(\bm q,t)$ in terms of the scalar potential $\Phi$. As shown in
Sec.~\ref{sub:commu}, omni-potentiality implies that along any particle
trajectory the Hessians ${\cal H}(\Phi(\bm q,t))$ commute and thus have common
eigendirections. The field of such eigendirections is prescribed as a function
of the initial (Lagrangian) position $\bm q$, for example, by the eigendirections
of the Hessian of the initial velocity potential $\phi$. The set of
eigendirections of a real symmetric $d\times d$ matrix depends on $d(d-1)/2$
parameters and can be characterized, for example, by $d(d-1)/2$ of the
invariants discussed in the Appendix. As we try to determine a
single scalar function $\Phi$, the situation is rather different in two and
higher dimensions.

When $d=2$, we have a single invariant expressible as a ratio of
suitable combinations of spatial second derivatives of $\Phi$.
Thus, prescribing the field of the invariant values, $g(\bm q)$, we obtain
a linear second-order PDE, \rf{gen2d}, for $\Phi$.
For a suitable family of fields $g(\bm q)$, we have found in
Sec.~\ref{sub:ex2d} non-trivial omni-potential flows that are linear
combinations of homogeneous polynomials, thus ensuring the existence of such
non-trivial flows. Using a WKB method, in Sec.~\ref{s:wkb} we have then been able
to construct omni-potential flows, at least locally in space-time, for arbitrary
smooth $g(\bm q)$. These flows are actually close to Zeldovich flows with straight
trajectories (but are not of ``Zeldovich type''). Extending this
construction globally in space and avoiding the rapid spatial oscillations
inherent to a WKB method constitute interesting open problems.

When $d=3$, omni-potentiality can be expressed in either of two equivalent
ways. One is to demand the commutation of the Hessians of $\Phi$ with those
of a prescribed $\phi$; this gives $d(d-1)/2$ linear homogeneous second-order
PDEs. The other involves working with the invariants, introduced in the
Appendix, which are rational functions of the entries of the
Hessians of $\Phi$; this gives $d(d-1)/2$ nonlinear second-order PDEs. In both
approaches we have one unknown scalar function, $\Phi$, which has to satisfy
more than one equation. Hence, there is an issue of compatibility of these
equations. However, by restricting the potential $\Phi$ to possess a suitable
finite symmetry group, we have obtained a fairly large class of non-trivial
solutions that are even-degree homogeneous polynomials. Whether non-Zeldovich-type
omni-potential flows exist for an arbitrary smooth $\phi$ remains an open
problem. In dimensions $d>3$ the situation is basically the same.

We have shown in Sec.~\ref{sub:app1} that omni-potentiality of a flow is
equivalent to having at each time a velocity field that is potential in
both Eulerian and Lagrangian coordinates. Such double potentiality
was frequently considered in cosmology. It is of particular relevance
when performing reconstruction by convex optimization techniques such
as the Monge--Amp\`ere--Kantorovich (MAK) procedure. Note that the
Euler--Poisson flow is potential in Eulerian coordinates, but not in
general in Lagrangian coordinates. As discussed in Sec.~\ref{s:cosmo},
the approximate Euler--Poisson flow
obtained by the second-order Lagrangian perturbation theory (L2) is
exactly potential in Lagrangian coordinates --- and thus its inverse
Lagrangian map can be obtained exactly by the
Monge--Amp\`ere--Kantorovich (MAK) procedure --- however, it is only
approximately potential in Eulerian coordinates and thus must be
qualified as an approximately omni-potential flow. In particular,
it does not represent an example of an exactly omni-potential three-dimensional
flow for an arbitrary smooth initial velocity.

We finally wish to mention a concrete open problem of cosmological
interest: as mentioned, MAK gives the exact inverse Lagrangian map for
L2 (and this contributes to explaining why MAK works so well when
tested with N-body simulations). However, in L2, between the Eulerian
position $\bm x$ and its Lagrangian antecedent $\bm q$, the trajectory is
not given exactly by the Zeldovich approxima\-tion that has a constant
velocity $(\bm x-\bm q)/\tau$ (in the coordinates we used in
Sec.~\ref{s:cosmo}). The actual L2 trajectory is slightly curved and
its current (peculiar) velocity differs slightly from
$(\bm x-\bm q)/\tau$. It would be of interest to find~how to perturbatively
handle such discrepancies. Given that the full Euler--Poisson
reconstruction problem and the Zeldovich approximation to it both have
convex optimi\-zation formulations, the question arises, whether L2
and higher-order approximations possess such formulations.

\begin{acknowledgments}
We are particularly grateful to J.~Bec, Y.~Brenier, T.~Buchert, S.~Colombi and
A.~Sobolevskii for extensive fruitful discussions. Thanks are also due to
F.~Bouchet, K.~Khanin, R.~Mohayaee, A.~Nusser and E.B.~Vinberg. UF, OP and VZ were
supported by the grant \hbox{ANR-07-BLAN-0235} OTARIE from Agence Nationale de
la Recherche, France. OP and VZ were supported by the grant 11-05-00167-a from
the Russian foundation for basic research. Several visits of BV, OP and VZ to
Observatoire de la C\^ote d'Azur (France) were
supported by the French Ministry of Higher Education and Research.

\end{acknowledgments}

redpajama_set_name: RedPajamaArXiv


text:
'I Am Jazz' Episode 9 Recap & Spoilers: Ahmir's Mom Misgenders Jazz: 'You Are Dating A Man, I'll Never Accept That'
By Hannah Preston On 2/27/19 at 9:25 AM EST
Culture Movies & TV
Tuesday's episode of I Am Jazz started where it last left off, finding the location of Jazz. Even though her family was worried about her whereabouts, Jazz was safe and said she did not have her phone by her because she was too distracted by her romantic date with new boyfriend Ahmir.
Jazz Needs Independence–Now
While her parents were glad to know she was safe, they started to "grill" her on not having her phone nearby or keeping her sister updated on her whereabouts. This family discussion then led into Jazz asking for more independence after she turns 18.
"Jazz appears to be looking for her independence," her father Greg said to the camera. "We've always given her enough space to make her own decisions, but it seems like dating is piling onto this independence need."
The independence topic doesn't stop with her parents but continues on in a conversation with Ari. As the two are strolling through their neighborhood, Jazz proposed the idea to start apartment hunting and possibly have her sister move in with her.
"I know if I'm talking about moving out on my own, my parents might not be supportive," Jazz said to the camera. "But they would definitely be more inclined to allow me to move out if they knew I'd be living with Ari."
"So she's really a part of my master plan," she added.
Ari may not have seemed in no rush to move out of her parents' house, so the topic of apartment hunting hasn't quite reached a conclusion.
A Transphobic Mom?
Moving on from Jazz's need for independence to an even bigger bombshell: Ahmir's "transphobic" mother. During a conversation between the young couple, it was revealed that Ahmir's mother "wasn't as accepting as [he] thought" of his relationship with the transgender teen.
"I'm shocked," Jazz said to the camera. "I finally found someone that was fully accepting of me, and yet his mom who raised him doesn't accept me."
It was also revealed that not only was his mother not accepting of the relationship, but also not acknowledging who Jazz real was–a girl.
"The most hurtful thing my mom said was that Jazz was still a man and that I would be gay for being in a relationship with Jazz," Ahmir said to the camera. "As soon as my mom said that, I lost it. I'm not gay, I'm not attracted to men and Jazz is a female."
Not only was his mother's statements hurtful, but things were amped up after she threatened to kick him out of the house.
"It's really scary that Ahmir being in a relationship with me could cause him to not even have a home," Jazz said to the camera.
Following the unfortunate revelation, Jazz reports back to her parents to fill them in on the not-so-great news. Similar to Jazz's response, her parents were shocked at Ahmir's mother's reaction.
"I'm not fond of transphobic people," Jeanette said. "I really don't want any of my children to date anyone transphobic or with a family that's transphobic."
While it's not for certain how the transphobic elements of Ahmir's family will pan out, both Jeanette and Jazz expressed an interest in speaking with Ahmir's mother personally.
Checkup With Dr. Bowers
In the past weeks, the Jennings family has gone through turmoil after turmoil, and after the Ahmir bombshell, it was time for some good news. During last week's episode, Dr. Marci Bowers told Jazz's parents that she would travel to Florida to examine Jazz in person to see how things are progressing and if she is in need of a third surgical procedure.
As the Jennings family held their breath in the doctor's room, both Dr. Bowers and Dr. Jess Ting were "shocked in a good way" of her progression. It was then ruled that a third procedure was not necessary and Jazz could return to regular physical activity.
Apartment Hunting
Following some beachfront apartment hunting with Ari, Jazz has become very adiment about moving. While Ari isn't sure whether she's ready to move out of her parents home, Jazz decided to bring up the topic with Jeanette and Greg.
"I want to be able to exercise a little bit more freedom and a little bit more control in my life, so by the time I do go to college, I can be fully prepared for that lifestyle," the teen said to her parents.
The discussion took a tense turn when her parents, Jeanette and Greg Jennings, asked if they were "prohibiting" Jazz from exercising her independence.
"Last I checked, we are pretty lenient—pretty cool parents," Jeanette responded. "You do just about everything you want."
In addition to the shock of Jazz's sudden announcement, her parents then called her maturity level into question. Though she said she plans to pay everything herself, her father proposed that she may not have the full understanding of "what it takes to live on your own."
While it's not clear what Jazz's financial state is, both of her parents believe that it would be far too expensive for her to live on her own—that's when Greg decided to ask whether there was an outside source for the sudden need to become independent.
"Does this have anything to do with partners," Greg questioned.
Yes, it could be speculated that Ahmir may play a role in Jazz suddenly wanting to get a place of her own, but she stated that was her sole reason.
"I think Ahmir has a big role in [Jazz] wanting to move out," Jeanette said to the camera. "If Ahmir doesn't have a place to live, I feel really bad about that, but Jazz needs to realize that you don't live with someone you're in a relationship with unless you're very serious."
The episode ended on another cliffhanger with Jazz threatening to move out, even without her parents' approval.
Find out what happens next when I Am Jazz returns next Tuesday on TLC at 9 p.m. ET.
'I Am Jazz' Episode 9 Recap & Spoilers: Ahmir's Mom Misgenders Jazz: 'You Are Dating A Man, I'll Never Accept That' | Culture
'I Am Jazz' Episode 9 Sneak Peek: Is Jazz Moving Out?
'I Am Jazz' Episode 8 Recap & Spoilers