Do you know what your life purpose is? Are you taking steps to ensure that you fulfil your calling?
Amazing things happen when you start to set specific, focused goals. Especaially when, with God's help, you create a detailed step by step plan to make sure you stay on target and achieve your goals.
Setting goals ia a step of faith. When you set goals you are choosing to launch out and actually do what you believe the Lord is asking you to do. And when you step out in faith it sets things in motion and releases God to act on your behalf.
God is delighted when you respond to His promptings and set goals in faith. It blesses His heart when you are willing to take risks and use the talents He has given you.
What do you need to put in place to make sure you actually do it?
If you'd like help with getting clear on your answers to these questions I will be offering affordable group coaching in September, so watch this space.
If that's too long to wait, contact me and we can arrange personal coaching to help you get started and keep going…I have special rates available for readers of my blog.
Wow! I am so excited about what you are doing here as I am interested in doing something similar with a specific group of people. I would love to discuss this with you. God is calling all His children back to Him in these times and putting us into His purpose for our lives. This is such an exciting time to be living in. I look forward to hearing from you.
Thank you for posting this! I am a goal oriented lady, but sometimes fail myself. If I look at it in another light. Pray on it, and have the strength from God, I am going to succeed!! Have a wonderful day!
Melissa I will be offerimg a group coaching program in the next few weeks – How To Discover Your Life Purpose – it might be just the help you need.


text:
Choral Festival
Pacific Chorale Academy
Choral Camp
Intro & Passage to the Arts
Concert Previews
Choral Sketches
Contact and Staff
For release: February 6, 2019
Andrew Brown, President & CEO
Ryan McSweeney, Director of Marketing & Box Office
Pacific Chorale to Celebrate "UnSung Heroes"
Concert on March 30th will highlight the accomplishments of women composers, including a world premiere by guest curator Karen P. Thomas.
Costa Mesa, CA – On March 30, 2019 at 7:30 p.m., in celebration of Women's History Month, Orange County's Pacific Chorale will highlight the creative genius of female composers in a concert titled "UnSung Heroes" at Segerstrom Center for the Arts' Renée and Henry Segerstrom Concert Hall. The evening's program will feature music composed by women ranging from the medieval mystic Hildegard von Bingen to 2013 Pulitzer Prize winner Caroline Shaw and will include a world premiere commission by Seattle composer and conductor Karen P. Thomas, who also serves as the concert's guest curator. Artistic Director Robert Istad will conduct the 140-voice Chorale with accompanist David Clemensen and guest musicians.
"It's no secret that women composers have been vastly underrepresented in music," says Istad. "Until fairly recently in history, women were strongly discouraged from pursuing careers in composition, and what music they did write was rarely publicly performed or published." According to the website Bachtrack, which maintains listings of concert, opera, and dance performances worldwide, women were represented in only 16% of classical music programming in the United States last year.
The Chorale's aim is to introduce audience members to a selection of incredibly talented women — some of them historic, some of them iconic, and many of them active today—who have made and are making an impact in music. Among those featured are the multi-talented 19th century musicians Clara Wieck Schumann and Fanny Mendelssohn Hensel—both of whom were overshadowed as composers in their lifetimes by the men in their families; the tragically short-lived French prodigy Lili Boulanger; veteran choral composer and arranger Alice Parker; African American composers Ysaÿe Barnwell and Rosephanye Powell; Galina Grigorjeva, representing the rich Estonian choral music tradition; and Hyo-won Woo from South Korea.
During this program, Istad will also recognize the major contributions of women in popular music, with selections from well-known English and American singer-songwriters Kate Bush, Carole King, and Joni Mitchell included in the evening's performance, and will prominently feature a number of the Chorale's female members as soloists.
Istad concludes, "You may not yet be familiar with all the names on this concert, but we think you'll agree their music is amazing!"
Prior to the concert, Segerstrom Center's second annual ¡DAMAS! celebration will take place on the Julianne and George Argyros Plaza beginning at 5:00 p.m. ¡DAMAS! celebrates women creators and will feature performances, activities, and a panel discussion which will include Karen Thomas. As part of the plaza activities, Pacific Chorale will create a "gallery walk" of posters of great moments for women in music history and will provide free concert tickets to groups who primarily serve women's causes. scfta.org/events/2019/¡damas!-free
Date & Time: Saturday, March 30 at 7:30 p.m.
Conductor: Robert Istad, Artistic Director
Featuring: Pacific Chorale • David Clemensen, piano
Location: Renée and Henry Segerstrom Concert Hall • 600 Town Center Dr., Costa Mesa, CA 92626
Tickets: From $25 to $115
Call 714/662-2345 or visit pacificchorale.org/events/unsung-heroes/
I Thank You God — Gwyneth Walker
Sorida (A Zimbabwe Greeting) — Rosephanye Powell
Was weinst du, Blümlein, Op. 23, No. 1* — Clara Schumann
Nach Süden, Op. 10, No. 1* — Fanny Mendelssohn Hensel
Soir sur la plaine — Lili Boulanger
Le Stelle (World Premiere) — Karen P. Thomas
Commissioned by ACFEA Tour Consultants for Rob Istad and Pacific Chorale, in support of Chorus America
O ignee spiritus — Hildegard von Bingen
Her Beacon Hand Beckons, from To the Hands — Caroline Shaw
This Woman's Work* — Kate Bush
In Paradisum — Galina Grigorjeva
Hark, I Hear the Harps Eternal — arr. Alice Parker
From the Wingbone of a Swan — Edie Hill
To See the Sky — Jocelyn Hagen
Both Sides Now* — Joni Mitchell
Wanting Memories, from Crossings — Ysaÿe M. Barnwell
(You Make Me Feel Like) A Natural Woman* — Carole King
Pal-So-Seong (8 Laughing Voices) — Hyo-won Woo
* solo selection
This project is supported in part by an award from the National Endowment for the Arts.
Pacific Chorale gratefully acknowledges its generous season sponsors:
Platinum Season Sponsor:
Phillip N. and Mary A. Lyons
Silver Season Sponsor:
Charles and Ling Zhang
ABOUT ROBERT ISTAD
Dr. Robert Istad has served as Pacific Chorale's Artistic Director since July 1, 2017. He has prepared choruses for many of the world's leading orchestral conductors, and has been featured on a number of recordings including The John Williams Collection on Sony Classical and Nostos: The Homecoming of Music on Yarlung Records. He continues in his role as Professor of Music, Director of Choral Studies, California State University, Fullerton. He is also serving as the President of the California Choral Directors Association and was recently elected to the board of directors of the national choral industry's association, Chorus America.
ABOUT KAREN P. THOMAS
Karen P. Thomas, composer and conductor, is the Artistic Director and Conductor of the Seattle Pro Musica. Her compositions are performed and broadcast throughout the world, by groups such as The Hilliard Ensemble and the Vocal Consort of Brussels. Her works are published by Santa Barbara Music Publishing and Northwest Choral Publishers, and have been featured at numerous ACDA and AGO conferences. A prize-winning composer, she has received grants from the NEA, The American Academy and Institute of Arts and Letters, and Meet the Composer, among numerous others. As a conductor, Ms. Thomas has produced ten critically-acclaimed CDs, and received the Margaret Hillis Award for Choral Excellence and the ASCAP-Chorus America Award. She has appeared as guest conductor at international festivals in Europe and North America, and the NPR radio show Saint Paul Sunday. She received the 2012 Outstanding Choral Director Award for Washington State, and currently serves on the boards of Chorus America, the American Choral Directors Association (NW Division) and the Greater Seattle Choral Consortium, and is a Commissioner on the Seattle Music Commission.
ABOUT PACIFIC CHORALE
Pacific Chorale has delighted national and international audiences with concerts of great choral music performed at the highest musical standards since 1968. Under the artistic leadership of Robert Istad, the organization produces a series of concerts each year at the Segerstrom Center for the Arts, where it serves as the sole resident choir. The mission of Pacific Chorale is to inspire our community through artistry and innovation in choral music performances and education programs.
Pacific Chorale is recognized for exceptional artistic expression, preserving and performing classical choral music as well as stimulating American-focused programming. Pacific Chorale presents a substantial performance season of its own and is sought regularly to perform with the nation's leading symphonies. Pacific Chorale has infused an Old World art form with California's hallmark innovation and cultural independence, developing innovative new concepts in programming, and expanding the traditional concepts of choral repertoire and performance. The organization boasts over 30 world premieres and has released 14 self-produced recordings.
In addition to its long-standing partnership with Pacific Symphony, with whom Pacific Chorale made its Carnegie Hall debut in April 2018, the Chorale has performed with such renowned American ensembles as the Los Angeles Philharmonic, the Boston Symphony, the National Symphony, San Diego Symphony, Los Angeles Chamber Orchestra, Philharmonia Baroque Orchestra, and Musica Angelica.
Pacific Chorale has also toured extensively to more than 19 countries in Europe, South America and Asia, and has collaborated with the London Symphony, the Munich Symphony, L'Orchestre Lamoureux and L'Orchestre de St-Louis-en-l'Île of Paris, the National Orchestra of Belgium, the China National Symphony, the Hong Kong Sinfonietta, the Estonian National Symphony, and the Orquesta Sinfonica Nacional of Argentina.
Pacific Chorale also places a significant emphasis on choral music education, offering grades 1-6 after-school programs, a high school student summer camp, and a community-wide singing event at Segerstrom Center for the Arts.
Archives for: February 5th, 2019
Release: Pacific Chorale to Celebrate "UnSung Heroes"
For release: February 6, 2019 Contacts: Andrew Brown, President & CEO [email protected] Ryan McSweeney, Director of Marketing & Box Office [email protected] 714/662-2345 Pacific Chorale to Celebrate "UnSung Heroes" Concert on March 30th will highlight the accomplishments of women composers, including a world premiere by guest curator Karen P. Thomas. Costa Mesa, CA – On March
Written by Ryan McSweeney on February 5, 2019
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redpajama_set_name: RedPajamaCommonCrawl


Today we share with all of you news from the Bhakti Yoga Sanga - Gaudiya Vaisnava Ashram - founded by Sriman Valihara das prabhu, and his mission and preaching in Italy.
Valihara das is an experienced teacher of bhakti yoga, the yoga of love, and belongs to a line of authentic spiritual masters in the ancient Vedic tradition known as Gaudiya Vaishnavism. He is the founder of Bhakti Yoga Sanga and for many years taught the science of yoga.
Valihara das was born in Milan in 1955. In his youth searching for a spiritual view of reality led him to make contact with Srila AC Bhaktivedanta Swami Prabhupada, who in 1975 accepted him as his disciple and gave him the name of Valihara das. For many years, following the instructions of his master, he undertook to make known the principles of the philosophy of bhakti to a number of people actively participating in the mission founded by his master.
In 1993 Valihara das during a trip to Vrindavan, India, met Srila Puri Goswami Maharaja Bhaktivaibhava, great realized soul, the spiritual brother and friend of Srila Prabhupada. Valihara das was impressed by his spiritual depth and became his disciple, receiving from him deep and inspiring spiritual instruction and a strong spiritual determination to spread more and more the message of Sri Chaitanya.
At the Bhakti Yoga Sanga blog http://bhaktiyogasanga.blogspot.com/ there are several articles published by Bhakta das which show the preaching activities of Sriman Valihara das prabhu.
The ashram is a vital center of spirituality whose members open their hearts to the vision of the Absolute. And 'the place of Divine Love, in which we pray to be able to act in a feeling of neglect under the gentle and careful guidance of the Guru, the divine master.
All shares in the ashram like to be considered always active, possessing the heart of the vocation to be the most insignificant. No one is motivated by selfish interest and exploitation, with a growth of artificial and illusory inner life.
Bhakti Yoga Sanga is an association that aims to introduce the techniques of bhakti yoga according to the teachings of the Vaishnava spiritual tradition, Sri Brahma Madhva Gaudiya Sampradaya.
The members of the association, under the loving guidance of a teacher and expert, learn to perceive the spiritual freedom, collaborating and supporting each other in a simple and natural lifestyle.


We at Tri-Art are beside ourselves with joy in reading the words of Kamarr Richée, who felt he had to write about his experience listening to Herbie Hancock's Future Shock on the Pebbles Turntable through Audeze LCD-2 Headphones at CES 2015.
We've been granted permission to share his 'ode' to music with our fans. Please read on!
"there are moments in life that have been far beyond my limited, though certainly expansive, vocabulary. sitting in downtown Guadalajara, staring at the full moon and contemplating the gravity of the birth of my firstborn is one of them. the stirring in my soul when i sneak a view of the side of my wife's head as she doses in the passenger seat while i drive on the freeway. that moment when the sun first peeks around the corner of the rooftops after a tedious, gloomy morning with work and car troubles all along the way, suddenly melting the frost which was creeping along the edges of my heart. today I felt a similar such moment, and it will leave a point in my mind for years to come. the moment? listening to Herbie Hancock's Future Shock on a superbly looked after vinyl, using literally the world's best headphones, on a meticulously manufactured, pulley-operated, mostly wooden turntable in the middle of CES, possibly the only place in the world right now where one could actually truly experience a feeling of future shock.
i was first drawn in to the booth by a gentleman representing Audeze, the proud purveyors of the aforementioned headphones. my friend and i had already grown jaded to the claims of "improved sound quality" that almost every headphone company touts at the show. if you take their word for it everybody has the best sound, everybody has won every award, everyone is groundbreaking; more often, though, everyone is just proving what any teenager or rock guitarist could tell you: if you turn up the volume, you can hear more sound. Audeze was different, as they were the first booth adequately equipped to back up their claims. see, it really doesn't matter so much what headphones or speakers you use if your listening to compressed formats like the ubiquitous mp3. there were booths that claimed they could "reclaim" the lost audio through some special software magic, but that's a pipedream at best, maybe even a bold-faced lie. no, you need quality to make quality, and Audeze was actually using FLAC, what is referred to as a lossless audio format, to substantiate its claims. i was impressed. we exchanged business cards, and talked about getting in touch in the future, with us both coming from Orange County and all. i was working my way out of the booth, picking up my free trial code for TIDAL, the lossless answer to the more pedestrian offerings of other streaming services, when i saw the turntable setup by Tri-Art Audio. the modern use of bamboo has resulted in an array of elegant takes on otherwise tried designs, but Tri-Art's purists will tell you that like most things in the Hi-Fi world, this has everything to do with the sound. i was invited to choose my own album to listen to, but after thumbing through, could only think of one name i wanted to hear but had not seen: Herbie Hancock. i had, in my enthusiasm, flipped right by him, and was invited to what would be the highlight of my day. the set-up in its entirety is not cheap; i'm talking more than $4,000, but if you have the money, and actually care about what you put in your ears, it is more than worth it.


He finished Day IV of the third Test at Trent Bridge with five wickets.
England resumed the day at 311-9, with James Anderson and Adil Rashid at the crease.
Lastly, Kohli was proud of the fitness levels of his pacers.
That changed, however, when Pant dropped Buttler on one. I know what I'm capable of doing. Had an idea of Buttler since both of us played for Mumbai Indians (in the IPL).
England coach Trevor Bayliss said the first innings collapse to 161 all out was the decisive aspect in the Test and urged his batsmen to be wiser with their shot selection. England had better control percentages than India in the first two Tests.
While there are questions marks surrounding some of the spots in England's top order, they can usually at least rely on Alastair Cook and captain Joe Root for runs, but both have failed to hit a hundred on home soil since the Edgbaston Test against the West Indies a year ago.
"You can't just blast batsmen out in Tests, as proper lines and lengths are important".
"We're working extremely hard on it".
"The victory was much needed at this stage of the series and to have a performance like that, which was so clinical in all three departments, is something that was pleasing to me as a captain", said Kohli at the presentation ceremony.
"He is striking the ball well in the nets, and I wouldn't say necessarily that he is out of form. It was about the batsmen, what could we do to give them that cushion". As a batting group we spoke that if we step up we will definitely be on top. He pointed to Buttler's maiden Test century in a second innings stand of 169 with Ben Stokes as the template, as well as his opposite number, Virat Kohli, making exactly 200 runs for the second time this series.
But the second new ball put paid to any illusions of a miracle from the hosts. So, we had to make flawless use of it. "We targeted taking one wicket at a time", he said. "People are going through a lot", Kohli said.
Vince, the Hampshire captain who plays his cricket at the Ageas Bowl, scored 147 against their win against Nottinghamshire. In fact, my first spell in the Ranji Trophy was 10 overs. They also agreed to refund those who had bought fifth-day tickets in advance, after admitting: 'We slept on the day-five pricing policy and, frankly, we got it wrong'. "We didn't want to get rolled over".
That said, the latter could make way for Moeen Ali, fresh from a double-century and eight wickets for Worcestershire at Scarborough, should the pitch call for two spinners.
"The hundred was really satisfying". "It has been a long time coming, and a few months ago it was a million miles away".


Our garnet bracelet is made with 10mm garnet beads and mixed with Karen Hill Tribe Silver cubes and modern geometric charm. It has a Karen Hill Tribe toggle closure, and fits a 6 1/2 – 7 1/2 inch wrist. This bracelet stands on its own or can be combined and stacked with other bracelets from our collection or worn with a special bracelet of your own. Its beautiful colors and tones can be mixed with many looks in your wardrobe from casual to dressy.
Since ancient times, Garnet has graced the crowns of many emperors. In both the Hindu and Buddhist religions it is seen as a holy stone, that enlightens and gives wisdom. It is said to arouse the chi or life force. Garnet is January's birthstone.
Note: Karen Hill Tribe is a high purity silver (aprox 96%-99%) and is one of the principal tribes in Thailand. The beads are handmade by families and village.


See below for PCB routing tips and tricks.
Screenshot during the routing of a track on a PCB in Altium.
Having schematics in-front of you when routing is a must. Either print them out or put them on a second monitor if you are lucky enough to have one.
Sometimes it is not obvious from the rats nest to where a component should go. A classic example is decoupling caps. A rats nest will only show one pin connected to the power rail, and the other to ground. However, the schematic show (hopefully!) that it is associated with the power input to an IC, and therefore should be placed next to the IC.
Having the schematic visible also lets you place the components in a logical manner.


UWP support and other improvements!
The new release is available on Nuget.
Please also note that this release is built with Visual Studio 2015, which means that there is no longer Windows 8 or Windows Phone 8 support. Visual Studio 2015 requires to retarget these projects to Windows 8.1 and Windows Phone 8.1.
If you need any help you can get in touch via the JabbR #mvvmcross room, or open a topic on StackOverflow and tag it with "MvvmCross".


Removes a member from the clan.
The target username is removed from the clan and receives S>C 0x76 SID_CLANQUITNOTIFY. All online members then receive a S>C 0x7E SID_CLANMEMBERREMOVED to update their clan lists. You must be the leader or an officer of your clan and target username must be a member of lower rank.
If the target username is your own, you leave the clan and your rank may be anything except "Initiate" 0x00 or "Leader".


The exterior leather has softened and shows creasing throughout. The glazing alogn the edges of the bag is cracking and creasing. The bottom metal feet have surface scratches. The handles have surface scratches and the glazing along the edges are cracking as well. The interior is clean but shows heavy creasing throughout.


text:
After Signing Suh, What Will Dolphins Do About The Offense?
Filed Under:Amari Cooper, Brandon Gibson, Brian Hartline, Charles Clay, David Dwork, DeVante Parker, Drew Rosenhaus, Jimmy Graham, Kevin White, Miami Dolphins, Mike Wallace, National Football League, Ndamukong Suh, NFL, Rob Gronkowski, Ryan Tannehill
Wide receiver Mike Wallace #11 of the Miami Dolphins scores a third quarter touchdown as free safety Harrison Smith #22 of the Minnesota Vikings defends during a game at Sun Life Stadium on December 21, 2014 in Miami Gardens, Florida. (Source: Mike Ehrmann/Getty Images)
DAVIE (CBSMiami) – The Miami Dolphins appear to be on the brink of making a huge splash in free agency by signing the most coveted defensive player on the market, defensive tackle Ndamukong Suh.
The addition of Suh will instantly make Miami's defense much better and fix one of its biggest issues from last season, the inability to stop the run.
While there are still some holes that will need to be filled on the defensive side of the ball, the focus will have to shift at some point to the offense.
The Dolphins cut ties with a pair of receivers last week in Brian Hartline and Brandon Gibson and also exposed tight end Charles Clay to being nabbed by another team by using the transition tag on him.
Clay will be paid $7.071 million in 2015 but could get offered a larger contract by another team, which the Dolphins will have the option to match. The $7 million already makes Clay the third-highest paid tight end in the NFL for next season behind Rob Gronkowski ($15 million) and Jimmy Graham ($8 million).
On top of the Clay situation, there is still a lot of uncertainty surrounding Miami's top wideout Mike Wallace. Will the team hang on to him despite the very public friction between he and the coaching staff at the end of last season?
Wallace will make nearly $10 million in base salary in 2015, making him an unlikely trade target for other teams. Additionally, even if the Dolphins did trade him they'd still take a cap hit of $6.6 million this season, though after that he would be completely off the books.
Releasing Wallace wouldn't be very beneficial either, as the team would save just $2.5 million in cap space. That is unless they designate him as a post-June 1st release, which would free up $6.9 million in cap space in 2015 while still costing Miami $5.2 million against the cap in 2016.
It still very much up in the air whether the Dolphins will keep Wallace or not, but for now it seems the safest play may be to hang on to him and let Miami's new offense continue to develop with a premiere, proven playmaker for Ryan Tannehill to throw to.
Circling back to Hartline for a moment, it's not completely out of the question for him to return to the Dolphins at a lower rate.
Hartline's agent, Drew Rosenhaus, told CBS Miami radio that it's still very possible he may choose to return to Miami, citing the Dolphins stability at quarterback and the veteran receiver's familiarity with the team and the town as factors in his decision.
The Dolphins could also choose to upgrade at receiver through the NFL Draft if they want to move on from either or all of the aforementioned players.
Miami holds the 14th overall pick in April's draft and could potentially target one of the top receivers projected to go in the 1st round, such as Louisville's DeVante Parker, West Virginia's Kevin White or Alabama's Amari Cooper. Each are projected as early-to-mid 1st round picks.
If the Dolphins really want to nab one of the top two receivers, Cooper or White, they may have to trade up to do so. It's very possible that both of them will be off the board by the time Miami picks at 14.
These are all just some of the options that the Dolphins have to consider as the new league year begins at 4 p.m. on Tuesday. The Suh signing will keep the focus on the defense but Miami's work will be far from finished as their needs on both sides of the ball will need to be addressed in the coming weeks.
Follow David on Twitter (@DavidDwork)


text:
\section{Introduction}
Let $(M,g)$ is a Riemannian manifold, i.e. a differentiable $n$-dimensional manifold with a function $g$, which defines for all $p\in M$ a positive definite symmetric bilinear form
\begin{equation}
g_p:T_pM\times T_pM\rightarrow {\mathbb R}
\end{equation}
such that for any given vector fields $X,Y\in X(M)$ the map
\begin{equation}
g(X,Y): M\rightarrow {\mathbb R},~p\rightarrow g(X,Y)(p):=g_p(X_p,Y_p)
\end{equation}
is differentiable. The Riemannian metric $g$ allows to define a metric $d_M$ on $M$ via the length of curves
\begin{equation}
d_M(x,y):=\inf_{\mbox{$\gamma$ diff.}}\left\lbrace L(\gamma)|\gamma :[0,1]\rightarrow M, \gamma(0)=x, \gamma(1)=y\right\rbrace,
\end{equation}
with
\begin{equation}
    L(\gamma)=\int_0^1 \sqrt{g_{\gamma(t)}(\dot \gamma(t),\dot \gamma(t))} \,\mathrm dt.
\end{equation}
With this definition any connected Riemannian manifold becomes a metric space, and
it is well known that for any compact Riemannian manifold any two points $x,y\in M$ can be connected by a geodesic whose length is $d_M(x,y)$.
If $\nabla$ denotes the Levi-Civita connection, then a geodesic $\gamma$ is characterized by the equation
\begin{equation}
\nabla_{\dot\gamma}\dot\gamma=0,
\end{equation}
which becomes (in terms of the coordinates of the values of the curve $\gamma$)
\begin{equation}\label{geo1}
    \frac{d^2x^\lambda }{dt^2} + \Gamma^{\lambda}_{~\mu \nu }\frac{dx^\mu }{dt}\frac{dx^\nu }{dt} = 0\ ,
\end{equation}
where the well-known Christoffel symbols are
\begin{equation}\label{geo2}
    \Gamma^\kappa_{\; \mu \nu}=\frac{1}{2}g^{\kappa \rho} \left( \partial_\mu g_{\nu \rho}+\partial_\nu g_{\mu \rho}-\partial_\rho g_{\mu \nu} \right).
\end{equation}
This is an $n$-dimensional nonlinear ordinary differential equation with values in ${\mathbb R}^n$ which is difficult to compute numerically in general (note the quadratic terms). For computing the length of a geodesic it is easier to compute the solution of a  eikonal equation of the form
\begin{equation}\label{eik}
d^2=\frac{1}{4}\sum_{ij}a_{ij}(x)d^2_{x_i}d^2_{x_j}
\end{equation}
(boundary conditions considered later), where $x\rightarrow a_{ij}(x)$ are functions such that at each $x\in{\mathbb R}^n$ the matrix $(a_{ij}(x))$ is the inverse of the positive matrix $(g_{ij}(x))$ at each point $x$. Here $f_{x_i}:=\frac{\partial f}{\partial x_i}$ denotes the derivative of $f$ with respect to the variable $x_i$.
In general we shall write $\partial^{\alpha}f$, $\partial^{\alpha}_xf$ or $\frac{\partial}{\partial x^{\alpha}}f$ for the multivariate derivative with multiindex $\alpha=(\alpha_1,\cdots ,\alpha_n)$.
The connection between the length of a geodesic which is given in local coordinates as in \eqref{geo1}, \eqref{geo2} and the length function $d^2$ defined by equation \eqref{eik} is considered in section 2.
This way the problem of finding the length of a geodesic is reduced to solving a nonlinear first-order partial differential equation in some domain of Euclidean space.

The computation of $d^2$ is still far from trivial, however. Even if the data $g_{ij}$ are analytic functions, power series expansion typically lead to power series solutions for $d^2$ with small radius of convergence.
Hence, the question is how we can approximate the function $d^2$ globally. Moreover, for some applications such as the accurate computation of diffusions we need the approximation of $d^2$ in strong norms (Sobolev norms of form
$H^{s,p}$ for possibly any positive real $s$. For that matter recall that $H^{0,p}\left( {\mathbb R}^n\right) =L^p\left( {\mathbb R}^n\right)$ and that for any $s\in {\mathbb R}$ we may define $H^{s,p}$ to be the set of all tempered distributions $\phi\in {\cal S}'$ such that $I_{-s}\phi$ is a function in $L^p\left( {\mathbb R}^n\right)$, where
$I_s$ is the pseudo-differential operator with symbol $\sigma_s(\xi)=\left(1+|\xi|^2\right)^{-\frac{s}{2}}$, i.e.
\begin{equation}
 I_s\phi ={\cal F}^{-1}\sigma_s{\cal F} \phi,~~\phi\in {\cal S}',
\end{equation}
${\cal F}$ denoting the Fourier transform. The goal of the present paper can then be formulated as follows: find for each $\epsilon >0$ and each real $s,p$ $(p\geq 1)$ an approximative solution $q^2_{s,p}$ to \eqref{eik} such that
\begin{equation}
\|d^2-q_{s,p}^2\|_{s,p}\leq \epsilon .
\end{equation}
We shall call $q_{s,p}^2$ an $H^{s,p}$ approximation to $d^ 2$ for reasons which will become apparent later.
Let us motivate this ambitious task by looking at a specific application.
There are a lot of applications for computations of the length of a geodesic, where applications to computations in general relativity are only one domain. Another important example is the leading term of the expansion of the fundamental solution of linear parabolic solutions (with variable coefficients). Varadhan showed that
the fundamental solution of the diffusion equation
\begin{equation}\label{PPDE}
\begin{array}{l}
\frac{\partial u}{\partial t}=\frac{1}{2}\sum_{i,j}a_{ij}\frac{\partial^2 u}{\partial x_i\partial x_j}+
\sum_i b_i\frac{\partial u}{\partial x_i},
\end{array}
\end{equation}
(where the diffusion coefficients $a_{ij}$ and the first order coefficients $b_i$ in \eqref{PPDE}
depend on the spatial variable $x$ only) is connected to the length $d$ of the geodesic with respect to the line element $ds^2=\sum_{ij}a^{ij}dx_idx_j$ ($a^{ij}$ being the inverse of $a_{ij}$) via the relation
\begin{equation}
d^2(x,y)=\lim_{t\downarrow 0}t\ln p(t,x,y).
\end{equation}
\begin{rem}
Solving equation \eqref{eik} we can assume that the matrix-valued function $x\rightarrow (a_{ij}(x))$ is symmetric, i.e.
$a_{ij}(x)=a_{ji}(x)$ for all $1\leq i,j\leq n$.  This is because
\begin{equation}
\begin{array}{ll}
d^ 2(x,y)=&\frac{1}{4}\sum_{ij}a_{ij}d^ 2_{x_i}d^ 2_{x_j}=\frac{1}{4}\sum_{ij}\frac{1}{2}\left( a_{ij}+a_{ji}\right) d^ 2_{x_i}d^ 2_{x_j}\\
\\
&+\frac{1}{4}\sum_{ij}\frac{1}{2}\left( a_{ij}-a_{ji}\right) d^ 2_{x_i}d^ 2_{x_j}=\frac{1}{4}\sum_{ij}\frac{1}{2}\left( a_{ij}+a_{ji}\right) d^ 2_{x_i}d^ 2_{x_j},
\end{array}
\end{equation}
so we can always substitute the matrix $a_{ij}$ by its symmetrization $\frac{1}{2}\left( a_{ij}+a_{ji}\right)$ without affecting the solution $d^ 2$.
\end{rem}
In \cite{Ka} we have seen that for $C^{\infty}$ coefficient functions $x\rightarrow a_{ij}(x)$ and $x\rightarrow b_i(x)$ and if some boundedness conditions of the derivatives are satisfied the fundamental solution has the pointwise valid form
\begin{equation}\label{WKBrep}
p(t,x,y)=\frac{1}{\sqrt{2\pi t}^n}\exp\left(-\frac{d^2(x,y)}{2 t}+\sum_{k= 0}^{\infty}c_k(x,y)t^k\right),
\end{equation}
where the functions $x\rightarrow c_k(x,y),~k\geq 0$ are solutions of recursively defined linear first order equations for each $y$. These equations can be solved by methods of characteristics or approximated by regular polynomial interpolation methods outlined in \cite{Ka2}. In the computation of the WKB-coefficients $d^2$ and $c_k,~k\geq 0$ the recursive relations for $c_{k+1}$ involve second order derivatives of $c_k$, and therefore implicitly derivatives of order $2k$ of the squared metric $d^2$. Hence it is of great interest to compute not only $d^2$ but also its derivatives up to a given order with high accuracy. The present work shows how his can be accomplished. In Section 2 we recapture some facts about the connection of the geodesic equation \eqref{geo1}, \eqref{geo2} and equation \eqref{eik}, and prove global existence, regularity and uniqueness of the latter (family) of equation(s) leading us to theorem 2.3. Then in Section 3 we provide further analysis of the family of eikonal equations which lead us to local representations of the solution.
In Section 4 we construct first a weak approximation of the solution (in $L^p$ sense), and then extend this to a recursive construction of an $H^{s,p}$-approximation.  In Section 5 we provide error estimates by using geometric information. Section 6 points out how the method may be applied for accurate approximation of diffusions, and we finish with a conclusion in Section 7.

\section{Global existence and regularity of the squared Riemannian distance $d^2$}
We shall only sketch the connection between geodesics and the eikonal equation \eqref{eik}. It is almost standard,
and details can be found in \cite{Ka} and \cite{Jo}. Our interest here is that the eikonal equation together with careful chosen boundary conditions has a global and unique solution. We shall have two different arguments for uniqueness: one is via uniqueness of an associated diffusion and WKB-representations (or, alternatively, Varadhan's result, cf. \cite{V}), but we will have the same insight from an other point of view when we look at local representations of the solution in the next Section.
We consider Riemannian manifolds where any two points can be connected by a minimal geodesic. For our purposes it is sufficient to consider manifolds which are geodesically complete.
Recall that a Riemannian manifold $M$ is geodesically complete if for all
$p\in M$ the exponential map $\exp_p: T_pM\rightarrow M$ is defined globally on $T_pM$. Here, $T_pM$ denotes the tangential space of the manifold $M$ at $p\in M$. The Hopf-Rinow theorem provides conditions for Riemannian manifolds to be geodesically complete. Especially we have
\begin{thm}
For a Riemannian manifold $M$ the following statements are equivalent:
\begin{itemize}
\item $M$ is complete as a metric space.

\item The closed and bounded sets of $M$ are compact.

\item $M$ is geodesically complete.
\end{itemize}
Each of these equivalent statements implies that geodesics are curves of shortest length. Moreover,
if $M$ is geodesically complete, then any two points of $M$ can be joined by a minimal geodesic.
\end{thm}
The connection between the arclength and equation \eqref{eik} can be established as follows. First equations for minimal geodesics are obtained from variation of the length functional. Second Hamilton-Jacobi calculus shows that the length functional satisfies the eikonal equation \eqref{eik}. Since this is known we only sketch the main steps for convenience of the reader.
Setting the variation of the length functional to zero we get
\begin{equation}
L\frac{d}{dr}\left(\frac{1}{L}2g_{ij}\dot x^i \right)+g_{ij,k} \dot x^i \dot x^j=0
\end{equation}
with $L\equiv \sqrt{g_{ij}(x(r))\dot x^i\dot x^j}$ and where we use Einstein summation. Parameterizing by arclength, i.e. setting $L\equiv 1$ (or $r=s$) we get
\begin{equation}
2g_{ij}\ddot x^i +2g_{ij,l}\dot x^l\dot x^i +g_{ij,k} \dot x^i \dot x^j=0
\end{equation}
which, upon multiplcation by $g^{mj}$ (entries of inverse of $(g_{mj})$) and rearranging becomes the geodesic equation \eqref{geo1},\eqref{geo2}.
In order to show on the other hand that the squared length functional satisfies \eqref{eik} we may consider the length functional
\begin{equation}
l(r,x,s,y)=\int_r^s L\left( x(u),\dot x(u)\right) du
\end{equation}
and invoke Hamilton-Jacobi calculus. This is done by
introducing the variables $p_i=L_{\dot x^i}$, and the associated Hamiltonian defined by
\begin{equation}
H(x,p)=\dot x^ip_i-L(x,\dot x).
\end{equation}
(here and henceforth we use Einstein summation if convenient).
Then we may write
$$x(t)\equiv x(t;r,x,s,y) \mbox{ and } p(t)\equiv p(t;r,x,s,y),$$
where $x(r;r,x,s,y)=x$ and $x(s;r,x,s,y)=y$.
and compute
\begin{equation}\label{1ls}
l_s=-H(x(s),p(s)).
\end{equation}
Then we may connect $p$ to $l_{y^k}$ by computing
\begin{equation}\label{ly}
\begin{array}{ll}
l_{y^k}=\int_r^s \left( \frac{\partial \dot x^i}{\partial y^k}p_i+\dot x^i\frac{\partial p_i}{\partial y^k}
-H_{x^i}\frac{\partial x^i}{\partial y^k}-H_{p_i}\frac{\partial p_i}{\partial y^k}\right) dt\\
\\
\int_r^s \stackrel{{\bf \cdot}}{\left( \frac{\partial x_i}{\partial y^k}p_i\right)}dt=\frac{\partial x_i}{\partial y^k}p_i \Big|^s_r=p_k(s;r,x,s,y).
\end{array}
\end{equation}
by invoking the canonical system of equations. This leads to
\begin{equation}
\frac{\partial l}{\partial s}+\sum_{ij}g^{ij}\frac{\partial l}{\partial y_i}\frac{\partial l}{\partial y_j}=0,
\end{equation}
and a similar equation with respect to the variables $x$. Then we get the equations for $l^2$ and $d^2$, i. e. the equations \eqref{lg} and \eqref{lf} below.

Recall that a minimal geodesic is a global distance minimizing geodesic. This minimal geodesic which connects $x$ and $y$ characterizes the Riemannian distance $d(x,y)$ in an obvious way. Moreover smoothness of $(x,y)\rightarrow d(x,y)$ for smooth diffusion and drift coefficients $a_{ij},b_i$ follows from
the following fact about ordinary differential equations.
\begin{thm}Let $F:{\mathbb R}^n\times {\mathbb R}^n\rightarrow {\mathbb R}$ be a smooth map. Consider the differential system
\begin{equation}
\frac{d^2x}{dt^2}=F\left( x,\frac{dx}{dt}\right),
\end{equation}
where $x$ is a map $I\subset {\mathbb R}\rightarrow {\mathbb R}^n$. Then for each point $(x_0,y_0)$
there exists a neighborhood $U\times V$ of this point and $\epsilon >0$ such that for $(x,v)\in U\times V$ equation (2.69) has a unique solution $x_v:]-\epsilon,\epsilon[\rightarrow {\mathbb R}^n$
with initial conditions $x_v(0)=x$ and $x_v'(0)=v$. Moreover, the map $X:U\times V\times ]-\epsilon,\epsilon[\rightarrow {\mathbb R}^n$ defined by $(t,x,v)\rightarrow X(t,x,v):=x_v(t)$ is smooth.
\end{thm}
Finally we get
\begin{thm}
Let $\Omega \subseteq {\mathbb R}^n$ be some domain. The function $d^2:\Omega\times \Omega \subseteq {\mathbb R}^n\times {\mathbb R}^n\rightarrow {\mathbb R}_+$ (the leading order term of the WKB-expansion of a parabolic equation with diffusion coefficients $a_{ij}$) is the unique function which satisfies the equations
\begin{equation}\label{lf}
d^2=\frac{1}{4}\sum_{ij}d_{x_i}^2a_{ij}d_{x_j}^2,
\end{equation}
\begin{equation}\label{lg}
d^2=\frac{1}{4}\sum_{ij}d_{y_i}^2a_{ij}d_{y_j}^2
\end{equation}
for all $x,y\in {\mathbb R}^n$ and with the boundary condition
\begin{equation}\label{lh}
d(x,y)=0 \mbox{ iff  $x=y$ for all $x,y\in {\mathbb R}^n$.}
\end{equation}
Moreover, the squareroot $d$ is the Riemannian distance induced by
\begin{equation}
\begin{array}{ll}
d(x,y):=\inf{\Bigg\{}\int_a^b&\sqrt{a^{ij}(\gamma)\stackrel{.}{\gamma}^i\stackrel{.}{\gamma}^j}dt|\gamma:[a,b]\rightarrow {\mathbb R}^n \mbox{ is piecewise } \\
&\mbox{ smooth with $\gamma(a)=x$ and $\gamma(b)=y$}{\Bigg \}}.
\end{array}
\end{equation}
The function $d^2$ is a $C^{\infty}$-function with respect to both variables.
\end{thm}
\begin{proof}
The variation of the length functional leads to the geodesic equation. On the other hand, Hamilton-Jacobi calculus leads us to the fact that the squared length functional $d^2$ satisfies the equation \eqref{eik}. It is clear that the squared length functional satisfies both equations \eqref{lf} and \eqref{lg} below. Moreover, it is clear that the squared length functional satisfies the initial condition \eqref{lh}.
Uniqueness is a bit more subtle. In \cite{V} Varadhan showed that
\begin{equation}\label{V}
d^2(x,y)=\lim_{t\downarrow 0}2t\ln p(t,x,y),
\end{equation}
where $p$ is the fundamental solution of a scalar parabolic equation with diffusion coefficient function $x\rightarrow a_{ij}(x)$. Since $p$ is unique for a strictly parabolic equation $d^2$ is uniquely determined
by the equation \eqref{V}. On the other hand one knows that for small $t>0$ $\ln p$ has for $C^{\infty}$ coefficients a representation of type \eqref{WKBrep} is valid (cf. \cite{Gi,Ka}). Plugging this into the correspondend parabolic equation leads to the eikonal equation \eqref{eik} which is, hence, satisfied by $d^2$. Moreover we know by $V$ and the fact that the squareroot of $d^2$ is a metric. Hence $d(x,y)=0$ if and only if $x=y$, and the same holds for $d^2$. Hence, we conclude that the global solution $d^2$ of the system of equations \eqref{eik},\eqref{lh} and \eqref{lh} is unique. Moreover, from the preceding theorem we can conclude that the function $(x,y)\rightarrow d^2(x,y)$ is also smooth with respect to both variables.
\end{proof}

\section{Further analysis of the equation for the squared metric $d^2$}

Next we observe that the local representation of the solution of the equations \eqref{lf}, \eqref{lg} with the boundary condition \eqref{lh} has a local representation which starts with the quadratic terms. This will be used in the construction of a global approximation. The analysis presented here gives us two other insights. First, a powere series ansatz leads atmost to local and not to global solutions. Even if there is a local power series representation of the solution at each point of the domain, we do not know how a global solution can be constructed from this information, because we do not know the location of the geodesic the length of which we want to compute. If we knew, then computing the length would be a rather trivial task. Even the derivatives of the length functional would be better computed from the explicit geodesic. However, as we mentioned the nonlinear ordinary differential equation describing the geodesic is harder to solve in general than the eikonal equation. Second, we shall see from an different point of view why the boundary condition \eqref{lh} leads to uniqueness of solutions $(x,y)\rightarrow d^2(x,y)$ of the system \eqref{lf}, \eqref{lg}, and \eqref{lh}.
We have \newpage
\begin{cor}
The local representation $d^2$ satisfying the equations \eqref{lf}, \eqref{lg}, together with the boundary condition \eqref{lh} is of the form
\begin{equation}\label{loc}
\begin{array}{ll}
d^2(x,y)&=\sum_{ij}a^{ij}(y)\Delta x^i\Delta x^j + \sum_{|\alpha < M }\frac{d^2_{\alpha}(y)}{\alpha!}\Delta x^{\alpha}\\
\\
&+\sum_{|\gamma|=M}\int_0^1(1-\theta)^{M-1}\frac{\Delta x^{\gamma}}{\gamma !}\partial^{\gamma}d^2(y+\theta \Delta x,y)d\theta.
\end{array}
\end{equation}
The coefficients $d_{\alpha}(y)$ are uniquely determined by a recursion obtained from the equations \eqref{lf}, \eqref{lg}. In coordinates with second order normal form, i.e. where $d^2$ is $\sum_{ij}\lambda_i(y)\Delta x^i\Delta x^j$ with $\lambda_i(y) , 1\leq i \leq n$ is the spectrum of $(a^{ij}(y))$,
the multiindex recursion is
\begin{equation}\label{rec}
\begin{array}{ll}
 d^2_{\beta}(y)=&\frac{1}{\left(1-\sum_i\beta_i\right)}{\Bigg (} \sum_i  \left(  \lambda^i_0\right)^2\frac{\lambda_i^{\beta\dot -2_i}}{(\beta\dot -2_i)!}1_{\left\lbrace \beta_i\geq 2\right\rbrace }\\
\\
&+\sum_i \sum_{|\alpha|\geq 1,|\gamma | \geq 3, \alpha+\gamma =\beta}\frac{\lambda_i^{\alpha}}{\alpha!}  \lambda^i_0 d^2_{\gamma}(y)\gamma_i\\
\\
&+ \sum_i  \sum_{\alpha\geq 0,|\delta|\geq 3,|\gamma| \geq 3,\alpha+\gamma+\delta\dot-2_i=\beta}\frac{\lambda_i^{\alpha}}{\alpha!}  \delta_i \gamma_i d^2_{\delta}(y)d^2_{\gamma}(y){\Bigg )}.
\end{array}
\end{equation}
This confirms uniqueness. (Note that there is no loss of generality if we choose the normal coordinates for the second order terms). In general the solution is not globally analytic in the sense that $d^2$ is not representable by a globally converging power series.
\end{cor}

\begin{proof}
A smooth solution $d^2$ of the eikonal equation has the representation
\begin{equation}
\begin{array}{ll}
d^2(x,y)&=d(y,y) + \nabla d(y,y)\cdot (x-y)\\
\\
&+\sum_{|\gamma|=2}\int_0^1(1-\theta)^{1}\frac{\Delta x^{\gamma}}{\gamma !}\partial^{\gamma}d^2(x+\theta \Delta x,y)d\theta.
\end{array}
\end{equation}
We abbreviate  $R(x,y)=\sum_{|\gamma|=2}\int_0^1(1-\theta)^{1}\frac{\Delta x^{\gamma}}{\gamma !}\partial^{\gamma}d^2(x+\theta \Delta x,y)d\theta.$
Since $d(y,y)=0$ we have
\begin{equation}
\begin{array}{ll}
d^2(x,y)=\nabla d^2(y,y)\cdot (x-y)+R(x,y)
\end{array}
\end{equation}
The 'only if'-condition of the boundary condition leads to $\nabla d^2(y,y)=0$. To see this assume that
$\nabla d^2(y,y)\neq 0$. Since $R(x,y)\leq C\|\Delta x\|^2$ there is a small $\Delta x$ such that
$\nabla d^2(y,y)\cdot \mu\Delta x >C\|\Delta x\|^2$ and $\nabla d^2(y,y)\cdot (-\mu)\Delta x <-C\|\Delta x\|^2$ for some $\mu\in (0,1]$. Hence there exists some $\rho$ such that with $x':=y+(\rho\mu)\Delta x$
\begin{equation}
d^2(x',y)=\nabla d^2(y,y)\cdot (\rho\mu)\Delta x + R(x',y)=0,
\end{equation}
contradicting one part of the boundary condition $d^2(x,y)=0~\mbox{ iff }~x=y$.
Next one computes that $\sum_{ij}a^{ij}(y)\Delta x^i\Delta x^j$ satisfies the equation
\begin{equation}
d^2(x,y)=\frac{1}{4}\sum_{ij}a_{ij}(y)d^2_{x_i}d^2_{x_j},
\end{equation}
and the uniqueness of theorem 2.3. (which we established by arguing with uniqueness of related diffusions and Varadhan's result, in the Atiyah-Singer spirit of short-range analytic expansions) identifies the coefficients $a^{ij}(y)$ as the second order terms of local representations around $y$. Having obtained this the representation \eqref{loc} is just a multivariate version of Taylor's theorem. Note, however, that we do not need to invoke the uniqueness of theorem 2.3. but just consider a recursion obtained from a power series ansatz starting with second order terms. However, this would complicate the matter a bit so we take advantage that we know the second order terms of a local representation by the preceding argument. Finally we have to establish the recursion in \eqref{rec}. The recursion shows directly that the higher order coefficients $d^2_{\beta}(y)$ for $|\beta|\geq 3$ are uniquely determined. Moreover, it is clear from \eqref{rec} that in general the convergence radius of the full power series is small (if not zero). Hence in general there is no globally analytic solution
 the function $d^ 2:\Omega\times \Omega \rightarrow {\mathbb R}$ globally analytic if for each $y\in \mathbb{R}^n$
the Taylor expansion of $d^2$ at $y\in \mathbb{R}^n$ and $x\in \mathbb{R}^n$
equals $d^2$ globally, i.e.
\begin{equation}\label{geolgth}
d^ 2(x,y)=\sum_{\alpha}\frac{\partial_{\alpha}d^2(y)}{\alpha !}(x-y)^ {\alpha}~~\mbox{forall}~x,y\in \mathbb{R}^n.
\end{equation}
Invoking the implicit function theorem equation \eqref{eik} is equivalent to
\begin{equation}\label{eik2}
d^2=\frac{1}{4}\sum_{i}\lambda_{i}(x)d^2_{x_i}d^2_{x_i},
\end{equation}
where $\lambda_i(x), 1\leq i\leq n$ is the spectrum of the positive $(a_{ij}(x))$. Since $d^2_{x_i}=2d d_{x_i}$ this is equivalent to
\begin{equation}\label{eik3}
1=\sum_{i}\lambda_{i}(x)d_{x_i}d_{x_i}
\end{equation}
The latter equation is easier but there is no Taylor expansion around $y$ as can be seen in the case of constant coefficients (and hence constant eigenvalues $\lambda$), where the solution is
\begin{equation}
d(x,y)=\sqrt{\sum_{i=1}^n \frac{\Delta x_i^2}{\lambda_i}}
\end{equation}
\begin{rem}
We use equation \eqref{eik2} mainly for the theoretical purposes of this corollary. In general it cannot be in general used for numerical purposes since this would imply that we have an efficient procedure to compute the eigenvalue functions of a space dependent matrix. Since we are looking for high precision in this paper, this is not possible in general. An exception is the case of dimension $n=2$ where we have
\begin{equation}
\lambda_{1,2}(x)=\frac{\mbox{tr}(A)(x)}{2}\pm\sqrt{\left( \frac{\mbox{tr}(A)(x)}{2}\right) ^2-\mbox{det}(A)(x)}
\end{equation}
where $A(x)=(a_{ij}(x))$.
\end{rem}
Next we plug in the power series expansion
\begin{equation}
d^2(x,y)=\sum_{i=1}^n\lambda^i_0\Delta x_i^2 +\sum_{|\beta |\geq 3 } d^2_{\beta}(y) \Delta x^{\beta}
\end{equation}
We have
\begin{equation}
d^2_{x_i}=2\lambda_0^i(y)\Delta x_i +\sum_{|\beta |\geq 3 } d^2_{\beta}(y) \beta_i \Delta x^{\beta \dot -1_i},
\end{equation}
where for any multiindex $\beta$ we define
\begin{equation}
\beta\dot-1_i=(\beta_1,\cdots,\beta_i,\cdots \beta_n)\dot-1_i:=\left\lbrace \begin{array}{ll} (\beta_1,\cdots,\beta_i-1,\cdots \beta_n) \mbox{ if } \beta_i\geq 1\\ (\beta_1,\cdots,0,\cdots \beta_n)~~\mbox{ else }  \end{array}\right.
\end{equation}
The term $\beta -2_i$ is defined analogously.
Plugging in the power series ansatz and using the relation $\lambda^0_i\left(  \lambda^i_0\right)^2=\lambda^0_i$, this leads to

\begin{equation}
\begin{array}{ll}
&\left( \sum_{|\beta |\geq 3} d^2_{\beta}(y)\Delta x^{\beta}\right) \left(1-\sum_i\beta_i \lambda_0^i\lambda^0_i\right) \\
\\
=&\left( \sum_{|\beta |\geq 3} d^2_{\beta}(y)\Delta x^{\beta}\right) \left(1-\sum_i\beta_i\right)\\
\\
=&\left( \sum_i  \sum_{|\alpha|\geq 1}\frac{\lambda_i^{\alpha}}{\alpha!}\Delta x^{\alpha}\right)\left(  \lambda^i_0\right)^2\Delta x_i^2+\\
\\
&+\left(\sum_i \sum_{|\alpha|\geq 1}\frac{\lambda_i^{\alpha}}{\alpha!}\Delta x^{\alpha} \right)\left( \lambda^i_0\sum_{|\beta| \geq 3}d^2_{\beta}(y)\beta_i\Delta x^{\beta}\right) \\
\\
&+\left( \sum_i  \sum_{\alpha}\frac{\lambda_i^{\alpha}}{\alpha!}\Delta x^{\alpha}\right)\times\\
\\
&\left(  \sum_{|\beta|\geq 3,|\gamma|\geq 3}\beta_i \gamma_i d^2_{\beta}(y)d^2_{\gamma}(y)\Delta x^ {\beta\dot-1}\Delta x^ {\gamma\dot-1}\right).
\end{array}
\end{equation}
This leads to
\begin{equation}
\begin{array}{ll}
&\sum_{|\beta |\geq 3} d^2_{\beta}(y)\Delta x^{\beta}\\
\\
=&\frac{1}{\left(1-\sum_i\beta_i\right)}{\Bigg (}\left( \sum_i  \sum_{|\alpha|\geq 1}\frac{\lambda_i^{\alpha}}{\alpha!}\Delta x^{\alpha}\right)\left(  \lambda^i_0\right)^2\Delta x_i^2+\\
\\
&+\left(\sum_i \sum_{|\alpha|\geq 1}\frac{\lambda_i^{\alpha}}{\alpha!}\Delta x^{\alpha} \right)\left( \lambda^i_0\sum_{|\beta| \geq 3}d^2_{\beta}(y)\beta_i\Delta x^{\beta}\right) \\
\\
&+\left( \sum_i  \sum_{\alpha}\frac{\lambda_i^{\alpha}}{\alpha!}\Delta x^{\alpha}\right)\times\\
\\
&\left(  \sum_{|\beta|\geq 3,|\gamma|\geq 3}\beta_i \gamma_i d^2_{\beta}(y)d^2_{\gamma}(y)\Delta x^ {\beta\dot-1}\Delta x^ {\gamma\dot-1}\right){\Bigg )}.
\end{array}
\end{equation}
Simplifying and renaming multiindices in order to collect for multiindices of order $\beta$ we get
\begin{equation}
\begin{array}{ll}
&\sum_{|\beta |\geq 3} d^2_{\beta}(y)\Delta x^{\beta} \\
\\
=&\frac{1}{\left(1-\sum_i\beta_i\right)}{\Bigg (} \sum_i  \sum_{|\alpha|\geq 1}\left(  \lambda^i_0\right)^2\frac{\lambda_i^{\alpha}}{\alpha!}\Delta x^{\alpha+2_i}+\\
\\
&+\sum_i \sum_{|\alpha|\geq 1}\sum_{|\gamma| \geq 3}\frac{\lambda_i^{\alpha}}{\alpha!}  \lambda^i_0 d^2_{\gamma}(y)\gamma_i\Delta x^{\alpha +\gamma} \\
\\
&+ \sum_i  \sum_{\alpha}\sum_{|\delta|\geq 3,|\gamma|\geq 3}\frac{\lambda_i^{\alpha}}{\alpha!}  \delta_i \gamma_i d^2_{\delta}(y)d^2_{\gamma}(y)\Delta x^ {\alpha+\gamma+\delta\dot-2_i}{\Bigg )}.
\end{array}
\end{equation}
The latter equation leads directly to \eqref{rec}.
\end{proof}

Let us draw some consequences out of our theoretical considerations. There is neither an explicit solution nor leads a power series ansatz to a global solution in general. Neither does it help to have local solutions in terms of power series.  Such representations are not sufficient for our purposes, since we are interested in a global solution for $x\rightarrow d^2(x,y)$ and do not know the intermediate points on the corresponding geodesic in order to compute the global $d^2$ by means of local power series representations.
This motivates our later construction of regular polynomial interpolation of $d^2$ as seemingly unavoidable.

\section{Regular polynomial interpolation algorithm for the Riemannian metric and its derivatives}
For the moment let us denote again an interpolation polynomial which approximates the squared Riemannian distance $d^2$ in the $L^p$-sense on some bounded domain $\Omega$ by $q_{0,p}^2$ and one that approximates the squared Riemannian distance $d^2$ in the $H^{s,p}$-sense (again on $\Omega$) by $q_{s,p}^2$. How can we check that a given polynomial is an approximation in either sense?
The equation \eqref{eik} gives us itself a hint how an approximation $q_{s,p}^2$ of $d^2$ performs.
In order to obtain the $L^p$ error of an $L^p$ approximation $q^2_{0,p}$ of $d^2$ we may plug in the approximation
$q_{s,p}^2$ into the right side of equation \eqref{eik} and subtract the left side, i.e. we compute
\begin{equation}
\frac{1}{4}\sum_{ij}a_{ij}(x)\frac{\partial q^2_{0,p}}{\partial x_i}\frac{\partial q^2_{0,p}}{\partial x_j}-q^2_{0,p}=r_{0,p}(x),
\end{equation}
We shall see that $r_{0,p}\in O(h^3)$ locally (with $h$ the mesh size of the interpolation points) implies that
\begin{equation}
\|d(x,y)-q_{0,p}\|_{L^p(\Omega)}
\end{equation}
converges to zero as the number of interpolation points $N$ goes to infinity in such a way that the mesh size of the set of interpolation points $h$ goes to zero. Note that $q_{0,p}$ denotes the squareroot of $q_{0,p}^2$.
We call an approximation $q^2_{s,p}$ an $H^{s,p}$-approximation if it approximates not only $d^2$ in the $L^p$ sense but can be plugged in into all the derivatives of \eqref{eik} of order $m$ (i.e. multivariate derivatives $\alpha$ for $|\alpha|\leq  m$ of the eikonal equation) such that in
\begin{equation}
\frac{\partial^{\alpha}}{\partial x^{\alpha}}\left(\frac{1}{4}\sum_{ij}a_{ij}(x)\frac{\partial q^2_{0,p}}{\partial x_i}\frac{\partial q^2_{0,p}}{\partial x_j} \right)-\frac{\partial^{\alpha}}{\partial x^{\alpha}}d^2(x,y)=:r_{\alpha ,p}
\end{equation}
the right side  staisfies $r_{0,p}\in O(h^{3+m})$ locally  implies that
\begin{equation}
\|d(x,y)-q_{0,p}\|_{H^{s,p}(\Omega)}
\end{equation}
converges to zero as the number of interpolation points $N$ goes to infinity in such a way that the mesh size of the set of interpolation points $h$ goes to zero.
Accordingly, we call such $q_{0,p}^2$ ($q_{s,p}^2$) an $L^p$- ($H^{s,p}$) approximation of the boundary value problem \eqref{eik}.
In the next subsection we construct a $L^p$-approximation and refine the construction in the following subsection in order to construct $H^{s,p}$-approximations.
\subsection{Polynomial interpolation of eikonal equation in $L^ p$ sense}

We may write the eikonal equation \eqref{eik}
\begin{equation}\label{eik0}
d^2(x,y)=\frac{1}{4}\sum_{ij}a_{ij}d^ 2_{x_i}d^ 2_{x_j}=\frac{1}{4}\left\langle \nabla d^2, A\nabla d^2\right\rangle.
\end{equation}
Assume that $A=(a_{ij})$ is constant. The solution \eqref{eik} with the boundary condition $d^2(x,y)=0$ iff $x=y$ is
\begin{equation}
d^2(x,y)=\left\langle \Delta x, A^ {-1}\Delta x\right\rangle,
\end{equation}
where $\Delta x=(x-y)$, and  $A^ {-1}=:(a^ {ij})$ denotes the inverse of the matrix $A$.
This is easily verified by observing that
\begin{equation}
\nabla d^ 2= 2 A^ {-1}x.
\end{equation}
Define
\begin{equation}
d^ 2_{A^ {-1}(x^j)}(x,y)=\sum_{ml} a^ {lm}(x_j)(x^ l-y^l)(x^ m-y^m),~~j=0,\cdots,N
\end{equation}
we get the first recursively defined approximation algorithm for the Riemannian distance based on $N+1$ interpolation points $x^0=y, x^1, x^2, \cdots ,x^N$.
Note that the squared distance is a function
\begin{equation}
d^ 2:\Omega \times \Omega \subseteq {\mathbb R}^n\times {\mathbb R}^n \rightarrow {\mathbb R}_+,
\end{equation}
where we define ${\mathbb R}_+:=\left\lbrace x| x\geq 0\right\rbrace $.
There are several ways to approximate the function $d^2$.
In order to approximate this function we approximate first the function $x\rightarrow d^2(x,y)$, then the function
$x\rightarrow d^ 2(x, x^1)$ and so on up to $x\rightarrow d^ 2(x, x^N)$.

We start with the approximation of $x\rightarrow d^2(x,y)$.
First define
\begin{equation}
d_{00}^2(x,y)=d^ 2_{A^ {-1}(y)}(x,y)
\end{equation}
Next define
\begin{equation}
d_{10}^2(x,y)=d^ 2_{A^ {-1}(y)}(x,y)+c_{10}\Pi_{l=1}^n(x_l-y_l)^2d^ 2_{A^ {-1}(x_1)}(x,y),
\end{equation}
and determine a real number $c_{10}$ such that
\begin{equation}
d_1^2(x_1,y)=\frac{1}{4}\sum_{ij} a_{ij}(x_1) d^ 2_{1,x_i}(x_1,y)d^ 2_{1,x_j}(x_1,y),
\end{equation}
i.e. the eikonal equation \eqref{eik} with respect to $x$ and fixed parameter $y$ is satisfied at $x_1$.
Proceeding we get a series $d^2_{10}, d^2_{20}, \cdots, d^2_{k0},\cdots $ of approximations of the form
\begin{equation}
d_{k0}^2(x,y)=d^ 2_{A^ {-1}(y)}(x,y)+\sum_{j=1}^kc_{j0}\Pi_{r=0}^j\Pi_{l=1}^n(x_l-x^r_l)^2d^ 2_{A^ {-1}(x^j)}(x,y).
\end{equation}
Having determined the real numbers $c_{10},\cdots c_{(k-1)0}$ we obtain the real number $c_{k0}$ by solving
\begin{equation}\label{eikatk}
d_{k0}^2(x_k,y)=\frac{1}{4}\sum_{ij} a_{ij}(x_k) d^ 2_{k0,x_i}(x_k,y)d^ 2_{k0,x_j}(x_k,y).
\end{equation}
for $c_{k0}$.
Continuing this procedure for $N$ interpolation points we get a polynomial of the form
\begin{equation}\label{dN1}
d_{N0}^2(x,y)=d^ 2_{A^ {-1}(y)}(x,y)+\sum_{j=1}^Nc_{j0}\Pi_{r=0}^j\Pi_{l=1}^n(x_l-x^r_l)^2d^ 2_{A^ {-1}(x^j)}(x,y).
\end{equation}
with $N$ real numbers $c_{j0}$ obtained recursively by plugging in $d^2_{j0}$ with one degree of freedom $c_{j0}$ into \eqref{eikatk}.

Analogous constructions are done to approximate $x\rightarrow d^2(x,x^j)$ for $k=1,\cdots ,N$ with
\begin{equation}\label{dNk}
d_{Nk}^2(x,x^k)=d^ 2_{A^ {-1}(y)}(x,x^k)+\sum_{j=1}^Nc_{jk}\Pi_{r=0}^j\Pi_{l=1}^n(x_l-x^r_l)^2d^ 2_{A^ {-1}(x^j)}(x,x^k),
\end{equation}
with $c_{jk}$ computed analogously.
The construction of the functions $d^2_{N0},\cdots , d^2_{NN}$ suffices to approximate $d^2$ (we do not need to synthesize these functions into one function, for example by a Lagrangian polynomial interpolation). Note that for $j=0,\cdots N$ the function $d^2_{Nk}$ satisfies the equation
\begin{equation}\label{bdpk}
\begin{array}{ll}
d^2(x,x^k)=\frac{1}{4}\sum_{ij}a_{ij}(x)d^2_{x_i}(x,x^k)d^2_{x_j}(x,x^k)\\
\\
\mbox{ with boundary condition }\\
\\
d^2(x,x^k)=0~\mbox{ iff }~x=x^k.
\end{array}
\end{equation}
at all interpolation points $x^0,\cdots x^N$ by construction.
\begin{rem}
Note that in the preceding construction no restrictions on the choice of the interpolation points are made.
This does not mean that one may search for an optimal choice of interpolation points and improve efficiency and convergence. We are free to choose a certain set of interpolation points (for example Chebyshev nodes).
But these are purely computational aspects which will be exploited elsewhere.
\end{rem}

\begin{rem}
Note that we have constructed an approximation of the squared metric $d^2$. The metric $d$ is then approximated naturally by the squareroot of the approximation of the squared metric, i.e. we consider the function
\begin{equation}
x\rightarrow d_{Nk}(x,x^k):=\sqrt{d^2_{Nk}(x,x^k)}
\end{equation}
to be the approximation of the metric function $x\rightarrow d(x,x^k)$.
\end{rem}

\subsection{Construction of $H^{s,p}$-approximations}

We refine the construction of the preceding section by construction of an approximation which solves not only \eqref{eik}, (or the set of equations \eqref{lf}, \eqref{lg} with boundary conditions \eqref{lh}), but also all multivariate derivatives of \eqref{eik} up to a given order $m$ at the interpolation points. It turns out then that these polynomials are $H^{s,p}$-approximations for $s\leq m$. The approximation is constructed recursively again.
For a multiindex $\beta$ of order $|\beta|=m\geq 3$ we denote the approximations of order $ d^2_{M(\beta_m)^{n,N}}$ or just $d^2_{M(\beta_m)}$ if we do not want to refer to the number of interpolation points $N$ and the dimension of the problem $n$ explicitly. The choice of the mesh is free again (in principle). We just assume that a set $\left\lbrace x_1,\cdots ,x_N \right\rbrace$ of interpolation points is given. Again we may construct functions
$x\rightarrow d^2_{M(\beta)0}(x,y)$, $x\rightarrow d^2_{M(\beta)0}(x,x^1)$,..., and $x\rightarrow d^2_{M(\beta)0}(x,x^N)$. We shall construct the first function $x\rightarrow d^2_{M(\beta)0}(x,y)$ for arbitrary multiindex $\beta$. The other functions can be constructed completely analogously.
We start with the $L^p$-approximation.
\begin{equation}
d_{N0}^2(x,y)=d^ 2_{A^ {-1}(y)}(x,y)+\sum_{j=1}^Nc_{j0}\Pi_{r=1}^j\Pi_{l=1}^n(x_l-x^r_l)^2d^ 2_{A^ {-1}(x_j)}(x,y),
\end{equation}
where the numbers $c_{j1}$ have been determined according to section 4.1..
Next we define $d^2_{M(\beta)0}(x,x^N)$ for multiindices of order $|\beta|=3$. Let $\beta^0,\cdots, \beta^k,\cdots ,\beta^{R}$ a list of multiindices of order $3$. The length $R$ of this list is dependent of the dimension $n$ of course. Start with $\beta^0=(\beta^0_1,\cdots,\beta^0_n)$ and let $\gamma^0$ be an multiindex with $|\gamma|=2$ such that $\beta^0-\gamma =1_i$ for some index $i$. Define (recall that $x^0=y$)
\begin{equation}
d^2_{\beta^0 0}(x,y)=d^2_{N0}(x,y)+\frac{1}{\beta^0!}c_{\beta^0}^0(x-y)^{\beta^0}.
\end{equation}
Then plug $d^2_{\beta^0 0}(x,y)$ into the equation
\begin{equation}\label{beta0}
\begin{array}{ll}
\partial^{(\beta^0-\gamma^0)}_xd^2(x,y)=\partial^{(\beta^0-\gamma^0)}_x\left(\frac{1}{4}\sum_{ij}a_{ij}(x)\frac{\partial d^2}{\partial x_i}\frac{\partial d^2}{\partial x_j} \right),
\end{array}
\end{equation}
evaluate at $x=x^0=y$ and solve for the real number $c_{\beta^0}^0$. Then proceed recursively: having defined the function $x\rightarrow d^2_{\beta^0 (k-1)}(x,y)$ define
\begin{equation}
d^2_{\beta^0 k}(x,y)=d^2_{\beta^0 (k-1)}(x,y)+c_{\beta^0}^k\Pi_{l=0}^{k-1}(x-x^l)^{\beta^0+{\bf 1}}\frac{1}{\beta^0!}(x-x^k)^{\beta^0},
\end{equation}
where ${\bf 1}=(1,1,\cdots,1)$.
Then plug $d^2_{\beta^0 k}(x,y)$ into the equation \eqref{beta0}, evaluate at $y$, and solve for $c_{\beta^0}^k$. When $k=N$ we have got the approximation
\begin{equation}
d^2_{\beta^0 N}(x,y)=d^2_{N0}(x,y)+\sum_{k=0}^Nc_{\beta^0}^k\Pi_{l=0}^{k-1}(x-x^l)^{\beta^0+{\bf 1}}\frac{1}{\beta^0!}(x-x^k)^{\beta^0}.
\end{equation}
with $N+1$ real numbers $c_{\beta^0}^k$ for $0\leq k\leq N$ determined recursively. Note that the function
$x\rightarrow d^2_{\beta^0 k}(x,y)$ satisfies the equations \eqref{eik} and \eqref{beta0} at all interpolation points $x^0,\cdots,x^N$. Then we take the next multiindex $\beta^1$ from the list of multiindices of order $3$ (i.e. $|\beta^1|=3$) where we may assume that $\beta^1-\gamma^1=1_k$ for some multiindex $\gamma^1$ with $|\gamma^1|=2$ and some index $k$. An analogous construction as in the case of $\beta^0$ can be done. The only difference is that we start with $d^2_{\beta^0 N}(x,y)$ instead of $d^2_{N0}(x,y)$. We get an approximation of the form
\begin{equation}
d^2_{\beta^1 N}(x,y)=d^2_{\beta^0 k}(x,y)+\sum_{k=0}^Nc_{\beta^1}^k\Pi_{l=0}^{k-1}(x-x^l)^{\beta^1+1}\frac{1}{\beta^1!}(x-x^k)^{\beta^1}.
\end{equation}
where the real numbers are computed recursively by plugging the current approximation into the equation
\begin{equation}\label{beta0}
\begin{array}{ll}
\partial^{(\beta^1-\gamma^1)}_xd^2=\partial^{(\beta^1-\gamma^1)}_x\left(\frac{1}{4}\sum_{ij}a_{ij}(x)\frac{\partial d^2}{\partial x_i}\frac{\partial d^2}{\partial x_j} \right),
\end{array}
\end{equation}
evaluating at the current interpolation point and solving for the currently undetermined real number $c_{\beta^1}^k$.
Doing this for all the multiindices of order $3$ in the list above we get the approximation
\begin{equation}
d^2_{M(\beta_3)}(x,y):=d^2_{\beta^N N}(x,y).
\end{equation}
Note that by construction the function $x\rightarrow d^2_{M(\beta_3)}(x,y)$ satisfies the equation \eqref{eik} and all its first order derivative equations
\begin{equation}
\partial^{i}_xd^2=\frac{1}{4}\partial^{i}_x\left( \sum_{lm}a_{lm}(x)d^2_{x_l}d^2_{x_m}\right) ,~~1\leq i\leq n,
\end{equation}
at all interpolation points $x^0=y,x^1,\cdots, x^N$. This completes the stage of construction for multiindices of order $3$. Next assume that the construction for the approximation
\begin{equation}
x\rightarrow d^2_{M(\beta_m)}(x,y)
\end{equation}
of order $m$ has been completed. Then we may list the multiindices of order $m+1$, i.e. consider a list of multiindices $\delta^0,\delta^1,\cdots, \delta^{R_{m+1}}$ such that $|\delta|=m+1$. The procedure is then quite similar as in the stage for multiindices of order $3$. Therefore we give a very short description. Starting with the multiindex $\delta^0$ there is a multiindex $\beta^k$ of order $m$ (i.e. $|\beta^k|=m$) such that $\delta^0-\beta^k=1_i$ for some index $i$. Then we get successive approximations
\begin{equation}
d^2_{\delta^0 k}(x,y)=d^2_{M(\beta_m)}(x,y)+\sum_{r=0}^{k}c_{\delta^0}^r\Pi_{l=0}^{r-1}(x-x^l)^{\delta^0+{\bf 1}}\frac{1}{\delta^0!}(x-x^r)^{\delta^0},
\end{equation}
where the real numbers $c_{\delta^0}^k$ are succesively determined by plugging in the function $x\rightarrow d^2_{\delta^0 k}(x,y)$ into the equation
\begin{equation}
\partial^ {\beta^k}d^2=\frac{1}{4}\partial^ {\beta^k}\left( \sum_{lm}a_{lm}(x)d^2_{x_l}d^2_{x_m}\right) ,
\end{equation}
evaluated at the interpolation point $x^k$ (Note that $\partial^{\beta^k}=\partial^ {\delta^0-1_i}$).
After $N+1$ steps we get the approximation function $x\rightarrow d^2_{\delta^0 N}(x,y)$. Having defined
$x\rightarrow d^2_{\delta^l N}(x,y)$ for $l=0,\cdots p-1$ the next multiindex $\delta^r$ may be such that there is an multiindex $\beta^h$ of order $m$ such that $\delta^{r}-\beta^h=1_i$ for some index $i$. We may then define $x\rightarrow d^2_{\delta^p k}(x,y)$
\begin{equation}
d^2_{\delta^p k}(x,y)=d^2_{M(\beta_m)}(x,y)+\sum_{r=0}^{k}c_{\delta^p}^r\Pi_{l=0}^{r-1}(x-x^l)^{\delta^p+{\bf 1}}\frac{1}{\delta^p!}(x-x^r)^{\delta^p},
\end{equation}
and determine the constants $c_{\delta^p}^r$ by plugging in the function $x\rightarrow d^2_{\delta^r k}(x,y)$ into the equation
\begin{equation}
\partial^ {\beta^h}d^2=\frac{1}{4}\partial^ {\beta^h}\left( \sum_{lm}a_{lm}(x)d^2_{x_l}d^2_{x_m}\right) ,
\end{equation}
and evaluate at $x^k$.
Finally, we get the approximation of order $m+1$, namely
\begin{equation}\label{betam}
d^2_{M(\beta_{m+1})}=d^2_{\delta^{R_{m+1}} N}(x,y).
\end{equation}
Note that this approximation satisfies the eikonal equation \eqref{eik} and all its derivatives
up to order $m+1$, i.e. all equations
\begin{equation}
\partial^{\alpha}_x d^2=\frac{1}{4}\partial^{\alpha}_x \left( \sum_{lm}a_{lm}(x)d^2_{x_l}d^2_{x_m}\right)
\end{equation}
with $|\alpha|\leq m+1$ at all interpolation points $x^1,\cdots x^N$.
\begin{rem}
Note that at some stage of the construction we may have a multiindex $\gamma$ such that $\gamma-\alpha=1_{i_0}$ for some $\alpha$ and some index $i_0$. Then  the terms in the $\alpha$th derivative of the eikonal equation evaluated at $x^k$ that do not annihilate a term of form $c_{\gamma}^k\Pi_{l=0}^{k-1}(x-x^l)^{\gamma+{\bf 1}}(x-x^k)^{\gamma}$ are quite easily computed. For this reason the constants of the form $c_{\gamma}^k$ are quite easily computed. You can see very easily this by writing the $\alpha$th derivative of the eikonal equation invoking symmetry $a_{ij}=a_{ji}$. We have
\begin{equation}\label{alpha}
\begin{array}{ll}
\partial^{\alpha}d^2(x,y)=\partial^{\alpha}\left(\frac{1}{4}\sum_{ij}a_{ij}(x)\frac{\partial d^2}{\partial x_i}\frac{\partial d^2}{\partial x_j} \right)\\
\\
=\frac{1}{2}\sum_{ij}a_{ij}(x)\left( \partial^{\alpha}\frac{\partial d^2}{\partial x_i}\right) \frac{\partial d^2}{\partial x_j} +\frac{1}{4}\sum_{ij}\left(\frac{\partial^{\alpha}}{\partial x^{\alpha}} a_{ij}(x)\right) \frac{\partial d^2}{\partial x_i} \frac{\partial d^2}{\partial x_j}\\
\\
 +\frac{1}{4}\sum_{ij}\sum_{\beta <\alpha}\sum_{\gamma \leq\beta}\binom{\alpha}{\beta}\binom{\beta}{\gamma}
\left( \partial^{\beta}a_{ij}(x)\right)
 \left( \partial^{\alpha -\beta-\gamma}\frac{\partial d^2}{\partial x_i}\right)  \partial^{\gamma} \frac{\partial d^2}{\partial x_j}
\end{array}
\end{equation}
If the indicated approximation is plugged into \eqref{alpha} and evaluated at $x^k$ only the terms $\frac{1}{2}\sum_{j}a_{i_0j}(x)\left( \partial^{\alpha}\frac{\partial d^2}{\partial x_{i_0}}\right) \frac{\partial d^2}{\partial x_j}$ (evaluated for approximations $d^2_{\gamma k}$ at interpolation point $x^k$) do not annihilate terms of  form $c_{\gamma}^k\Pi_{l=0}^{k-1}(x-x^l)^{\gamma+{\bf 1}}(x-x^k)^{\gamma}$.
\end{rem}

\section{Error estimates for the regular polynomial interpolation algorithm}
We first consider error estimates for $L^p$-approximations, and then extend our estimates to $H^{s,p}$-approximations. In the whole Section we consider a bounded domain $\Omega \subseteq {\mathbb R}^n$ and assume that the coefficient functions $a_{ij}$ are $C^{\infty}$.
\subsection{Error estimates for $L^p$ approximation}

We have
\begin{thm}
The approximations $d^2_{Nk}$ defined in \eqref{dNk} are $L^p$- approximations of the boundary value problems
of form \eqref{bdpk}, i.e. $L^p$- approximations for functions of form $x\rightarrow d^2(x,x^k)$ for $p>1$.
\end{thm}
\begin{proof}
Let $x$ and $y$ be two points connected by a geodesic curve $\gamma$ given in local coordinates with values in ${\mathbb R}^n$. Let us assume also that $x$ and $y$ are interpolation points. We have no solution for the curve $\gamma$ in general, but there are lets say $k$ points $z^0=x,z^1 \cdots z^k=y$ in the image of the curve $\gamma$ with Euclidean distance less than a certain mesh size $h$. Clearly,
\begin{equation}
d(x,y)=\sum_{i=0}^N d(z^i,z^{i+1})
\end{equation}
Next define an approximative distance along the geodesic of form
\begin{equation}
d_g(x,y)=\sum_{i=0}^n d_g(z^i,z^{i+1}),
\end{equation}
where $d_g$ is the squareroot of $d_g^2(z^{i},z^{i+1}):=\sum_{lm}a^{lm} (z^i_m-z^{i+1}_m)(z^i_l-z^{i+1}_l)$.
Since $y$ is fixed $d$ is approximated by $d_{N0}$ and we estimate
\begin{equation}\label{metest}
d(x,y)-d_{N0}(x,y)=d(x,y)-d_g(x,y)+d_g(x,y)-d_{N0}(x,y)
\end{equation}
Our analysis showed that the local approximation of $d^2$ by $d_g^2$ is of order $O(h^3)$ hence the approximation of $d$ by $d_g$ is of order $O\left( h^{\frac{3}{2}}\right)$, hence with generic constant $C$ we have for the first summand on the right hand side of \eqref{metest}
\begin{equation}
|d(x,y)-d_g(x,y)|=\sum_{i=0}^N |\left( d(z^i,z^{i+1})-d_g(z^i,z^{i+1})\right)|\leq C\sqrt{h}
\end{equation}
The modulus of the first summand on the right hand side can be estimated by
\begin{equation}
|d(x,y)-d^g(x,y)|\leq C\sqrt{h}
\end{equation}
Since $\Omega$ is a compact bounded domain, the $C^{\infty}$ coefficient functions  $a^{ij}$ are  Lipschitz Only locally Lipschitz is needed). Assuming a suitable choice of the points on the geodesic for the second summand we get by an elementary argument that
\begin{equation}
\|d_g(x,y)-d_{N0}(x,y)\|_{L^p}\leq \sum_{i=0}^N \|d(z^i,z^{i+1})-d_{g}(z^i,z^{i+1})\|_{L^p}
\leq Ch^{p-1}.
\end{equation}

\end{proof}

\subsection{Error estimates for $H^{s,p}$ approximation}

\begin{thm}
The approximations $d^2_{M(\beta_m)}$ defined in \eqref{betam} are $H^{s,p}$- approximations of the boundary value problems
of form \eqref{bdpk} for $s\leq m$, i.e. $H^{s,p}$- approximations for functions of form $x\rightarrow d^2(x,x^k)$ for $p>1$.
\end{thm}

\begin{proof}
For fixed $y$ the function $x\rightarrow d^2(x,y)$ and the function $x\rightarrow d^2_{M(\beta_m)}(x,y)$ both satisfy the eikonal equation and its derivatives at any interpolation point by construction. That means that for all interpolation points $x_j,~1\leq j\leq N$ and all derivatives $\gamma\leq m$ we have
\begin{equation}
\partial^{\gamma}_xd^2(x_j,y)=\partial^{\gamma}_x d^2_{M(\beta_m)}(x_j,y).
\end{equation}
Next recall a multivariate version of Taylor's theorem
\begin{thm}
If $f\in C^{\infty}$, then for all positive integers $M$ we have
\begin{equation}
\begin{array}{ll}
f(x+y)=\sum_{|\alpha|<M}\frac{(\partial{\alpha}f)(x)}{\alpha!}y^{\alpha}\\
\\
+M\sum_{|\gamma| =M}\frac{y^{\gamma}}{\gamma !}\int_0^1(1-\theta)^{M-1}(\partial^{\gamma}f)(x+\theta y)d\theta
\end{array}
\end{equation}
\end{thm}
Applying this formula, we see from our construction of $x\rightarrow d^2_{M(\beta_m)}(x,y)$ that the local order of approximation of $x\rightarrow d^2(x,y)$ is $O(h^{3+m})$. A similar reasoning as in the preceding Section leads to the result. Note here that the same reasoning holds when $y$ is replaced by another interpolation point $x^j$.
\end{proof}

\begin{rem}(Sharper error estimates)
A little analysis shows that the local order of approximation is
\begin{equation}
d^2(x,y)-d^2_{M(\beta_m)}(x,y)\leq C
P\frac{h^{m}}{m !},
\end{equation}
where $P$ is the number of multiindices of order $m$ and
\begin{equation}
C:=2\max\left\lbrace \sum_{|\gamma| =M}\sup_{x\in \Omega}\partial^{\gamma}d^2(x,y),\sum_{|\gamma| =M}\sup_{x\in \Omega}\partial^{\gamma}d^2_{M(\beta)}(x,y)\right\rbrace
\end{equation}
Similarly for $|\beta | \leq m $ we get
\begin{equation}
\partial^{\beta}d^2(x,y)-\partial^{\beta}d^2_{M(\beta_m}(x,y)\leq C
\frac{h^{m -|\beta|}}{(m -|\beta |) !},
\end{equation}
Since we are working on a bounded domain and $d^2$ is $C^{\infty}$ there is some bound $C$, but not a priori known. However bounds for $C$ may be obtained from a priori estimates by inspection of the eikonal equation.
It is clear that
\begin{equation}
 d^2=\frac{1}{4}\sum_{ij}a_{ij}d^2_{x_i}d^2_{x_j}.
\end{equation}
is equivalent to
\begin{equation}
 d^2=\frac{1}{4}\sum_{i}\lambda_{i}d^2_{x_i}d^2_{x_i},
\end{equation}
and, hence
\begin{equation}
|d^2_{x_i}|\leq \frac{4d^2}{\lambda_{\min}},
\end{equation}
where $\lambda_{\min}=\min_i \inf_{x\in \Omega}\lambda(x)$.
Further a priori estimates for the derivatives may be obtained from derivatives of the eikonal equation.
\end{rem}

\section{Analytic approximations of the fundamental solution of parabolic equations}

Consider the parabolic equation
\begin{equation}
\frac{\partial p}{\partial t}-\frac{1}{2}\sum_{ij}a_{ij}(x)\frac{\partial^2 p}{\partial x_i\partial x_j}-\sum_i b_i(x)\frac{\partial p}{\partial x_i}=0
\end{equation}
on some domain $(0,T)\times \Omega \subseteq (0,T)\times {\mathbb R}^n$, and where
$x\rightarrow (a_{ij}(x))$ is a matrix-valued $C^{\infty}$-function with symmetric positive matrix $(a_{ij}(x))$
for all $x\in \Omega$, and $x\rightarrow b(x)$ is also a $C^{\infty}$-function.
Plugging in the Ansatz
\begin{equation}\label{WKBrep}
p(t,x,y)=\frac{1}{\sqrt{2\pi t}^n}\exp\left(-\frac{d^2(x,y)}{2 t}+\sum_{k= 0}^{\infty}c_k(x,y)t^k\right),
\end{equation}
leads to the recursive equation \eqref{eik} for $d^2$. Given $d^2$ the first order recursive equation
\begin{equation}\label{c01e}
-\frac{n}{2}+\frac{1}{2}Ld^2+\frac{1}{2}\sum_{i} \left( \sum_j\left( a_{ij}(x)+a_{ji}(x)\right) \frac{d^2_{x_j}}{2}\right) \frac{\partial c_{0}}{\partial x_i}(x,y)=0,
\end{equation}
together with the boundary condition
\begin{equation}\label{c01b}
c_0(y,y)=-\frac{1}{2}\ln \sqrt{\mbox{det}\left(a^{ij}(y) \right) }
\end{equation}
determines $c_0$ uniquely for each $y\in {\mathbb R}^n$. Furthermore, having computed all WKB-coefficient functions $c_l$ up to order $k$, for $k+1\geq 1$ the coefficient function $c_{k+1}$  can be computed via the first order equation
\begin{equation}\label{1gaa}
\begin{array}{ll}
(k+1)c_{k+1}(x,y)+\frac{1}{2}\sum_{ij} a_{ij}(x)\Big(
\frac{d^2_{x_i}}{2}\frac{\partial c_{k+1}}{\partial x_j}
+\frac{d^2_{x_j}}{2} \frac{\partial c_{k+1}}{\partial x_i}\Big)\\
\\
=\frac{1}{2}\sum_{ij}a_{ij}(x)\sum_{l=0}^{k}\frac{\partial c_l}{\partial x_i} \frac{\partial c_{k-l}}{\partial x_j}
+\frac{1}{2}\sum_{ij}a_{ij}(x)\frac{\partial^2 c_k}{\partial x_i\partial x_j}
+\sum_i b_i(x)\frac{\partial c_{k}}{\partial x_i},
\end{array}
\end{equation}
with the boundary conditions
\begin{equation}\label{Rk}
c_{k+1}(x,y)=R_k(y,y) \mbox{ if }~~x=y,
\end{equation}
$R_k$ being the right side of \eqref{1gaa}.
We will show in a subsequent paper that equations \eqref{c01e}
\eqref{c01b}, and \eqref{1gaa},\eqref{Rk} can be solved or approximated to higher order if \eqref{eik} is solved or approximated to higher order. We have
\begin{prop}
In order to compute the WKB-approximation up to order $k$ a $H^{s,p}$ approximation of $d^2$ for $s\geq 2k$ is sufficient.
\end{prop}
\begin{proof}
In each recursion step \eqref{1gaa} an operator of order $2$ is applied to the previously computed WKB-coefficients.
\end{proof}

\section{Conclusion and final remarks on computational issues}

We have established a stable algorithm for efficient computation of the length of geodesics as a function of two arbitrary points on $C^k$- Riemannian manifold with minimal geodesic as well as of  partial derivatives of the length functional (of principally any order) accurately. We established error estimates in arbitrary Sobolev norms. We showed how the algorithm can be applied in order to compute fundamental solutions of irreducible linear parabolic equations. There are many obvious applications to mathematical physics and finance as well as  to statistics, e.g. to the maximum log-likelihood method, to option pricing, computing transition amplitudes etc.. Finally, we remark that the interpolation polynomials should not be evaluated in the way they are constructed. Here careful implementation of Horner schemes is needed. But this computational issues will be considered in a subsequent paper.

redpajama_set_name: RedPajamaArXiv


Samsung continually lead absolutely everyone with Galaxy word collection and now new range 9 is on its way with hell of appears which are past your mind that what they are going to do with all different manufacturers like huawei and lg. next in line samsung galaxy observe 9 leaked for an excessive amount of time. now its design is likewise discovered displaying that bezels on the perimeters are nearly gone at the same time as the mythical aspect on facets of samsung's galaxy word 9 is spell binding but still you'll get a few appears of be aware eight on this cellphone.
Disclaimer : Samsung Galaxy Note 9 price in Pakistan is updated on daily basis and rate/price for Samsung Galaxy Note 9 is valid for all cities of Pakistan included Islamabad, Rawalpindi, Lahore, Karachi, Faisalabad, Peshawar, Sialkot, Quetta etc. Samsung Galaxy Note 9 price depends on many factors thats include the price of dollors, manufacturar price and import cost and duties etc. Our team entered specs from diffrent resources and not responsible for any typo error in specs for Samsung Galaxy Note 9, please report any such error in bug report panel for Samsung Galaxy Note 9. Samsung Galaxy Note 9 price in Pakistan.


Holiday shopping season is almost here, and the web traffic is beginning to spike. With surveys showing that nearly two-fifths of holiday shopping could take place online, you really need to be prepared this year. Ramping up your holiday eCommerce email marketing strategy is a great place to start, and here are some ways to do that.
A whopping 56% of consumers already think you know their purchase history, so you might as well use it to cater offers to their preferences and tastes. This can be particularly useful in your holiday ecommerce email marketing campaigns.
For example, say you know a customer purchased something they marked as a gift around this time last year. You can use that information to remind them to pick up a gift again, with a specific featured item or coupon offer.
Perhaps you have a segment of people who purchased last minute last year and had to pay for expedited shipping. You could use that information now to remind them to shop early, hit the deals you're offering, and save on shipping (or qualify for free shipping).
When you send emails like this, it shows your audience that you're paying attention, catering to their needs, and puts you atop their nice list during a bustling and busy holiday shopping season.
Creating set-it-and-forget it campaigns like this account for about 12% of revenue generated for Harney & Sons. Wouldn't it be great to add 12% to your revenue without having to actively manage the initiatives? This example is just one way to set up automated emails.
Customer reviews are still a major purchase influencer, and it's easy to automate the process of requesting them from customers. Much like the above example, you can set emails to automatically generate about a week after a shipment goes out, depending on shipment method, requesting a product review and linking directly to the product. This keeps you top-of-mind, makes it easy for the customer to act, and can help garner business when other customers see reviews on products they view on your website.
Equally important is also making sure that these emails look good across devices, so your customers can read your message easily no matter where they are. This means incorporating large, tappable calls-to-action, eye-catching images, and short, snappy text.
These are just a couple examples of how you can begin ramping up your holiday eCommerce email marketing efforts as we dive headlong into the holiday season. For more tips to win the holidays, check out ShippingEasy's Holiday Hub, packed with all the resources you need to succeed this year.


text:
/**
 * @module ol/format/filter/Or
 */
import {inherits} from '../../util.js';
import LogicalNary from '../filter/LogicalNary.js';

/**
 * @classdesc
 * Represents a logical `<Or>` operator between two ore more filter conditions.
 *
 * @constructor
 * @param {...module:ol/format/filter/Filter} conditions Conditions.
 * @extends {module:ol/format/filter/LogicalNary}
 * @api
 */
const Or = function(conditions) {
  const params = ['Or'].concat(Array.prototype.slice.call(arguments));
  LogicalNary.apply(this, params);
};

inherits(Or, LogicalNary);
export default Or;

redpajama_set_name: RedPajamaGithub


text:
Q: Excel , Multi-lookup/Match formula I would like to use a formula in Sheet B to obtain the following transformation.
I want to vlookup stack, over, flow, super, user and the associated id's and put them into the Sheet B format. The formula would be copied horizontally across many 'Names' and then down.
Current, sheet A:
+-------------+-------+-------+
| Position_ID | Name  | Value |
+-------------+-------+-------+
|  5963650267 | stack |    10 |
|  5963650267 | over  |    20 |
|  5963650267 | flow  |    30 |
|  5963650267 | super |    40 |
|  5963650267 | user  |    50 |
|  5963650268 | stack |    90 |
|  5963650268 | over  |   110 |
|  5963650268 | flow  |    80 |
|  5963650268 | super |    70 |
|  5963650268 | user  |    20 |
+-------------+-------+-------+

Expected, Sheet B, headers and positions ids are already pre populated:
+-------------+-------+------+------+-------+------+
| Position_ID | stack | over | flow | super | user |
+-------------+-------+------+------+-------+------+
|  5963650267 |    10 |   20 |   30 |    40 |   50 |
|  5963650268 |    90 |  110 |   80 |    70 |   20 |
+-------------+-------+------+------+-------+------+

A: Assuming the data in Sheet A is located at A1:C11 (adjust as required), enter this Formula Array in Sheet B at B2 then copy to all required cells (i.e. C2:F2 and B3:F3)
=INDEX('Sheet A'!$C$1:$C$11,
MATCH(CONCATENATE($A2,"|",B$1),
CONCATENATE('Sheet A'!$A$1:$A$11,"|",'Sheet A'!$B$1:$B$11),0))

Formula Array must be entered by holding down CTRL + SHIFT + ENTER

A: Apologies for the formatting - but if you add the vlookups to the empty shell of position_ids by name on sheet b it should give you the grid you're looking for.
Sheeta! ID&Name Position_ID Name    Value
        =C2&D2      1       stack   10
        =C3&D3      1       over    20
        =C4&D4      1       flow    30
        =C5&D5      1       super   40
        =C6&D6      1       user    50
        =C7&D7      2       stack   90
        =C8&D8      2       over    110
        =C9&D9      2       flow    80
        =C10&D10    2       super   70
        =C11&D11    2       user    20

Sheetb! stack   over    flow    super   user
1   =VLOOKUP($A14&B$13,$B$2:$E$11,4,FALSE)  =VLOOKUP($A14&C$13,$B$2:$E$11,4,FALSE)  =VLOOKUP($A14&D$13,$B$2:$E$11,4,FALSE)  =VLOOKUP($A14&E$13,$B$2:$E$11,4,FALSE)  =VLOOKUP($A14&F$13,$B$2:$E$11,4,FALSE)
2   =VLOOKUP($A15&B$13,$B$2:$E$11,4,FALSE)  =VLOOKUP($A15&C$13,$B$2:$E$11,4,FALSE)  =VLOOKUP($A15&D$13,$B$2:$E$11,4,FALSE)  =VLOOKUP($A15&E$13,$B$2:$E$11,4,FALSE)  =VLOOKUP($A15&F$13,$B$2:$E$11,4,FALSE)

Sheetb! stack   over    flow    super   user
1   10  20  30  40  50
2   90  110 80  70  20

redpajama_set_name: RedPajamaStackExchange


Medisafe Project, is a cloud-synced app for your mobile device. Medisafe Project, is helping people implement healthier living habits, and preventing medication over-doses/under-dosing. Using everyday technology, the app helps to remind users when their medication needs to be taken. Also but just as importantly, it informs loved ones and caretakers if the user forgets.
HAIFA, ISRAEL (November 14, 2012) – MediSafe Project (www.MediSafeProject.com) launched today the first-ever cloud-synced mobile app helping families prevent emergencies caused by over- or under-dosing medications.
Simple to use, available for all Android devices, and compatible with iPhones and iPads as an alpha version, MediSafe Project is the first mobile health app not only reminding users when it's time to take their medication, but also using families', friends' and caretakers' intervention instincts by sending them alerts when a loved one misses a dose.
Inspired by their diabetic father's accidental insulin double-dose, brothers Rotem and Omri 'Bob' Shor view MediSafe Project as a way to lower hospitalization and mortality rates, promote sustainable behavior changes that prolong health, decrease long-term healthcare costs and help pharmaceutical companies understand patients' barriers to medication compliance.
Part of the first graduating class of the Microsoft Accelerator for Windows Azure, MediSafe Project also has vital, Big Data implications for pharmaceutical companies. From anonymous demographics, geolocation and patient behaviors to physician trends and other market aspects, MediSafe Project's broad analytics can help pharmaceutical companies better understand how people receive and take their medication. Says Shor: "The dashboard is able to show you which demographics have higher non-compliance for a drug compared to the general population, the doctors or areas that have the lowest prescription rates with a city, competitive drugs patients are switching to, and so on." Discussions with several Big Pharma companies are underway, and Shor expects partnership deals to begin closing when MediSafe Project exits Beta status in 2013.
Visual, Easy-to-use User Interface – The app's homescreen resembles an interactive pillbox. Users can customize different meds to look like different pills, and the center MediSafe logo turns into a mouth swallowing meds when users signify taking medication by dragging preconfigured pills into it.
FDA Drug Database Compatible – Users can type in medications' generic or brand names, or take pictures with their devices' cameras of the FDA's universal National Drug Code (NDC) number – included on all original pharmaceutical packaging – and the correct pharmaceutical name, manufacturer and medication strength will automatically be recorded, ensuring an error-free medication list in the event of a medical emergency.
Prescription Refill Alerts – Users receive alerts before medication courses are completed, allowing them to order prescription refills for medications taken chronically.
Privacy Protections – Syncing medication schedules with caretakers is optional. Aggregated usage statistics are anonymous. Viral sharing is limited to email.
Available now as a free download in Google Play and iTunes App Store (preview version), MediSafe Project is a complete ecosystem of patients, families, doctors, drug stores and pharmaceutical companies to solve the hazardous and costly problem of medication non-compliance.
MediSafe Project is the first cloud-synced mobile app helping families prevent emergencies caused by over- or under-dosing medications. In addition to reminding users when it's time to take their medication, MediSafe Project sends alerts to selected family members, friends and caretakers when a loved one misses a dose. Aggregated patient behavior data, physician trends and other market aspects are available to help pharmaceutical companies better understand how people receive and take their medications. Inspired by the accidental insulin overdose of the co-founders' father, MediSafe Project is first ecosystem of patients, families, doctors, drug stores and pharmaceutical companies to solve the hazardous and costly problem of medication non-compliance. Download the MediSafe application from the Google Play and iTunes App Store (preview version). For more information, visit www.MediSafeProject.com.


Volodymyr Zelenskiy played a history teacher who becomes President of Ukraine Vasyl Holoborodko in a Ukrainian satirical TV series "Servant of the People," first aired in 2015.
One morning, a history teacher from Kyiv, Vasyl Holoborodko, wakes up as the president of Ukraine. This is not the opening of a Franz Kafka novel, but a satirical Ukrainian television series "Servant of the People," which follows an honest man's attempts to change a corrupt system.
"Why do our politicians come to power and make the same mistakes? It's because they are only good at math! They only know how to subtract, divide, and multiply their wealth," Holoborodko rants on the show.
Created by one of most successful television producers of Ukraine, Volodymyr Zelenskiy, the first two seasons of "Servant of the People" have been a hit. Holoborodko will return to the country's television screens in March for the third season.
In the meantime, Zelenskiy, who plays Holoborodko, will try his chances in the March 31 presidential election.
His savvy populist campaign and TV fame already won him third place in the polls.
Zelenskiy appeals to voters dissatisfied with traditional ruling elites who will vote in protest, according to political expert Volodymyr Fesenko, head of the Kyiv-based Center for Applied Political Research.
This populism and his show business background have won Zelenskiy comparisons to the U. S. President Donald Trump.
As with Trump during the 2016 US presidential election, Zelenskiy has found fertile ground in the disillusionment of ordinary citizens with the powers that be.
In Ukraine, voters are so weary of the same cohort of politicians who have failed to deliver on their promises in the past that they are ready to elect someone new, even if that someone has neither political experience nor a political program.
Ukrainians aren't new to populism. But Zelenskiy's brand of populism differs from the populism of veteran politicians like ex-Prime Minister Yulia Tymoshenko or member of parliament Oleg Lyashko.
Nobody knows what policies Zelenskiy stands for as an aspiring leader of the nation.
While other contenders in the presidential race have rolled out elaborate election programs full of promises, Zelenskiy doesn't even have one yet. Yet he turns its absence into an advantage.
"I will do it differently. We will write my political program with you," he said in a video address on Facebook on Jan. 8, asking social media users to send him what they think are Ukraine's biggest problems. The video has been viewed over a million times and received hundreds of comments with suggestions for his program.
An entertainer by trade, Zelenskiy said he would turn his election campaign into a reality show. In an interview with Ukrainska Pravda on Jan. 21, he shared his plans to live broadcast the work of his campaign office.
"We have to have an open country with an open government," he said.
Zelenskiy's campaign indeed has been different from those of other candidates. He announced his presidential bid on TV on New Year's Eve, minutes before the end of the year. In that same time, other channels broadcast President Petro Poroshenko's annual speech.
He threw a pizza party to celebrate his official nomination by his political party, which is called Servant of the People, like his TV show. He communicates with his audience through videos on Instagram and Facebook.
The actor smartly planned the promotion of the new season of "Servant of the People" to go hand in hand with his election campaign, making Zelenskiy indistinguishable from his character, a role model President Holoborodko.
And this, experts say, is a problem.
"Zelenskiy, of course, has a right to partake in politics. But we don't know anything about his team, his political orientation, his weak and strong sides. It is voting for a pig in a poke," said Oleksiy Haran, a professor of political science of Kyiv Mohyla Academy and research dırector at the Democratic Initiative Foundation.
But Zelenskiy is not Holoborodko.
Unlike his character, he is not an Everyman. A 41-year-old native of the industrial city of Kryvyi Rih, with more than 630,000 people more than 400 kilometers southest of Kyiv, Zelenskiy is a wealthy and successful television mogul.
Over 15 years of its existence, his production studio Kvartal 95 has created dozens of top rated TV shows, musicals, sitcoms, and movies for Ukrainian viewers and Russian-speaking audiences abroad. Zelenskiy said his company sells their shows to 21 countries.
Zelenskiy was quick to react to his first crisis when journalists caught him lying about shutting down his filmmaking business in Russia that turned out to be active and receive funds from the Russian government.
Less than a week after the scandal broke on Jan. 17, he announced that he was leaving the ownership of a Cyprus-registered firm, Green Family Ltd, listed as a founder of three companies in Russia and Kvartal 95 in Ukraine.
Zelenskiy gained fame thanks to his scathing political satire. He has parodied numerous former and sitting politicians and oligarchs on stage, ridiculing their corruption and greed.
He scripts, directs, hosts, and acts in several prime-time comedy shows on 1+1, one of the most watched TV channels in Ukraine.
He is on the big screen too. A romantic comedy he directed starred, "Me, You, Him, Her," has set a box office record with Hr 61.9 million ($2.2 million) in revenues in three weeks since its release in cinemas on Dec. 27. The movie has become the biggest box office success in the history of independent Ukraine, according to B&H Film Distribution Company.
Unlike Holoborodko, Zelenskiy doesn't shy away from wealthy patrons. His presidential bid is believed to be backed by Ihor Kolomoisky, the self-exiled Ukrainian oligarch behind 1+1. In his recent interviews, the actor claimed they had only business relations with Kolomoisky and that 1+1 owed Kvartal 95 a lot of money.
Kolomoisky has been quite open about his dislike of President Poroshenko whom he accuses of illegally expropriating PrivatBank (it was nationalized in 2016 after revelations of large-scale fraud) and an attempt to seize control over 1+1 TV channel.
There are a few things we know about Zelenskiy's political views from his recent interviews. His manner of speech is informal and metaphorical; he appeals to emotions and tends to over-simplify the solutions he proposes to current issues of Ukraine's domestic and foreign policies.
"I want our people to get fatter," he said. "So that the concept of 'saving for a rainy day' went into the background. So that the subculture of poverty was removed," he said in a December interview with Ukrainian journalist Dmytro Gordon.
He believes (falsely, as analysts from VoxCheck platform proved) that the interest rates on loans from the International Monetary Fund are unreasonably high.
In his words, negotiating the end of the Donbas war with Russian president Vladimir Putin sounds like a piece of cake, and the current administration has not tried hard enough.
Fighting corruption and oligarchy is quite simple too for Zelenskiy.
"We will impose the same rules for everyone. No exceptions," he said in an interview with Ukrainska Pravda.
Iryna Bekeshkina, director of the Democratic Initiatives Foundation, believes Zelenskiy stands a low chance of winning the presidency: He is leading the polls among young people under 30, but his voter base might not vote at all, she said.
Young people are generally less active voters than the older population. In addition, in Ukraine, a voter has to go through a bureaucratic procedure if he wants to cast a ballot not at the polling station near his or her place of registered residency.
"A lot of young people moved to bigger cities or abroad. I doubt many will bother changing the polling station," she told the Kyiv Post.