Carbohydrate-loading and exercise performance. An update.
This review suggests that there is little or no effect of elevating pre-exercise muscle glycogen contents above normal resting values on a single exhaustive bout of high-intensity exercise lasting less than 5 minutes. Nor is there any benefit of increasing starting muscle glycogen content on moderate-intensity running or cycling lasting 60 to 90 minutes. In such exercise substantial quantities of glycogen remain in the working muscles at the end of exercise. However, elevated starting muscle glycogen content will postpone fatigue by approximately equal to 20% in endurance events lasting more than 90 minutes. During this type of exercise, exhaustion usually coincides with critically low (25 mmol/kg wet weight) muscle glycogen contents, suggesting the supply of energy from glycogen utilisation cannot be replaced by an increased oxidation of blood glucose. Glycogen supercompensation may also improve endurance performance in which a set distance is covered as quickly as possible. In such exercise, high carbohydrate diets have been reported to improve performance by 2 to 3%.
11897899 - Dietary strategies to promote glycogen synthesis after exercise.
3698159 - Supercompensation of muscle glycogen in trained and untrained subjects.
9781319 - Hepatic glucose production during exercise.
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text:
Grange 2017
The original and most powerful expression of Penfolds multi-vineyard, multi-district blending philosophy, Grange is arguably Australia's most celebrated wine and is officially listed as a Heritage Icon of South Australia. Crafted utilising fully-ripe, intensely-flavoured and structured shiraz grapes, the result is a unique Australian style that is now recognised as one of the most consistent of the world's great wines. With an unbroken line of vintages from the experimental 1951, Grange clearly demonstrates the synergy between shiraz and the soils and climates of South Australia. This release in 2021 commemorates the 70th anniversary since Grange was first made.
20/20 Points - Richard Hemming MW
98 Points - Mary Ann Worobiec, Wine Spectator
98 Points - Tony Love
97 Points - Josh Raynolds, Vinous
97 Points - James Halliday
The numbers have it: In its 70th Anniversary Year, this is only the 7th Grange to proclaim 100% Shiraz - 1951, 1952, 1963, 1999, 2000, 2011 and now 2017. A 10% shiraz hit-rate.
– Peter Gago, Penfolds Chief Winemaker
Huon Hooke
" Very deep, dark, dense colour with plenty of purple in the rim. The bouquet is classic Grange, loaded with toasty-smoky-oaky scents of smoked-meats and toasted barrels, dried herbs adding an extra fragrance, and hints of five-spice, especially star anise. It's very full-bodied and concentrated, powerful and long-lasting, but the tannins are beautifully crafted and supple, harmonious, and avoid any suggestion of astringency. The core of the wine is lusciously fruit-sweet and delicious, the robe of tannin complementary rather than domineering. The finish is tremendously long and satisfying. I can enjoy drinking this now: not usual with a new Grange release. A quite magnificent wine and a great Grange. "
Robert Parker's Wine Advocate
" Strongly marked—as always—by its 100% American oak elevage, the 2017 Grange backs up the cedar and vanilla notes with ample blackberry and cassis fruit. Full-bodied, ripe and almost decadently creamy in the mouth, it's loaded with substance, concentrated and rich, yet—in the context of Grange—relatively light and elegant-seeming on the finish. Only the seventh-ever Grange to be exclusively Shiraz, it originates from Barossa Valley (86%) and McLaren Vale (14%); Shiraz from other growing regions in South Australia failed to make the grade this year." - Joe Czerwinski
" Memorable, complex, aromatic and explosively deep, with a mix of palo santo, dark chocolate, black olive, espresso and hazelnut butter. The pure fruit at the core is a mix of ripe huckleberry, boysenberry and wild blackberry, with dense but polished tannins. Savory notes of dried rosemary and sage, cigar box, dried apricot and sandalwood linger on the epic finish. Drink now through 2045. " - MaryAnn Worobiec
Dense and deep, bright and lively crimson-red
Rampant scents derived from kitchen (winery) and garden (vineyard) …
A wine-loving forensic sniff or two, rather than an imagined spectrophotometric analysis reveals:
Venison carpaccio with a sweet balsamic reduction and freshly-pressed olive oil. Or, uncured pastrami?
A salsa verde type-emulsion with an emphasis on the parsley and extra virgin olive oil. Add tomato, omit lemon.
Enough of the savoury – wafts of sticky rice pudding, apricot pip, vanillin pod and coconut husk.
Subtle sweet and spiced sandalwood, ground cumin and wintergreen notes remind of an eighteen-month sabbatical in oak.
Where to start?! An advancing wave of flavours, understated power and charm. The swell builds and a more persuasive wave follows …and then more, with each sip.
The flavours: Laden with shiraz fruits from the Barossa to the North and McLaren to the South - 'maritime' and 'continental' climatic/varietal diversity. Red and black liquorice, soy and malt, burnt vanilla marshmallow …
Dark chocolate, wrapped in a satiny copha/beeswax candle-like coating. Supple oak nuances - dexterously integrated, no doubt compliments of barrel ferment
The texture: Mouthfeel is creamy/velvety, tannins are tight, creating a dense coating, with a 'rusty' edge. Ebullient acidity, whilst lively and racy, is well-managed and protective. To borrow a recently read descriptor – "cherry stone minerality"
Vintage Conditions
South Australia's wine regions experienced a cool and relatively wet winter and spring. Growers rejoiced as rainfall records were broken across parts of the state, Barossa Valley and McLaren Vale were well above winter long-term averages (122mm and 81mm respectively). This set the vines up with plentiful soil moisture profiles for the growing season and mitigated the need for irrigation. October was breezy, which challenged fruit-set, however the wind held off any potential frost events in the Barossa Valley vineyards. Cool conditions extended the growing season with flowering and veraison both later than expected. No heatwaves were recorded during summer and only a handful of days surpassed 40°C. Summer rainfall was above the long-term average, which helped to delay harvest. Drier and warmer weather prevailed in March, optimal conditions for grapes to finish ripening, develop deep colour and present strong varietal character. Harvest for shiraz grapes commence mid-March, a month later than the previous year.
VINEYARD REGION
Barossa Valley, McLaren Vale
wine analysis
Alc/Vol: 14.5%, pH: 3.61, Acidity: 7g/L
18 months in American oak hogsheads (100% new)
The Story of Grange
The development of Max Schubert's Grange is a modern tale of imagination, a battle against the odds and redemption. It began with a side trip to Bordeaux in 1949, where a wine 'capable of staying alive for a minimum of twenty years', first entered Schubert's mind. His first experimental vintage in 1951 began a new way of thinking that would eventually lead to a signature wine style, but not before Grange was discredited and Max Schubert forced to make the wine in secrecy.

redpajama_set_name: RedPajamaCommonCrawl


Your pass starts when you start. Just turn up, let reception know and your pass will be activated. Your pass must be used within 7 weeks, from first day.
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ALBANY, N.Y., Jan. 4, 2018 /PRNewswire/ -- AMRI, a global pharmaceutical contract research, development and manufacturing organization, today announced that Michael Mulhern has joined the company as chief executive officer, succeeding William S. Marth. Mr. Mulhern was previously chief executive officer of Sterigenics International, a leading global provider of contract sterilization, gamma technologies and medical isotopes. Mr. Marth will remain with the company in an advisory role. AMRI is a portfolio company of The Carlyle Group and GTCR.
Stephen H. Wise, managing director and global head of healthcare for Carlyle, said, "Michael is a highly accomplished executive with a 30-year track record of value creation in the healthcare and distribution sectors. His role as a member of AMRI's board of directors, coupled with his experience leading three private equity-backed companies, will make this a smooth transition."
Dean Mihas, managing director and head of healthcare for GTCR, said, "We thank Bill for his years of successful leadership of AMRI. Since joining the board of directors in 2012 and becoming CEO in 2014, Bill has been instrumental in building AMRI into a leading contract research, development and manufacturing organization. We wish him success in his future endeavors."
Mr. Mulhern said, "I am excited to join AMRI and look forward to working with such a talented group of professionals around the world to make a great company even better. I believe a cohesive, integrated AMRI will better enable us to serve our global customers. Bill and his team have accomplished great things, and I look forward to leading AMRI through its next phase of growth."
Since 2002, Mr. Mulhern has been CEO of three companies backed by GTCR – Sterigenics International, Fairmont Food Group, and American Sanitary – where he established a track record of product development, operational excellence, quality customer service and global growth. He previously held roles at Baxter International and Alliant Foodservice. Mr. Mulhern earned his B.A. in economics from Pennsylvania State University.
AMRI, a global contract research and manufacturing organization, partners with the pharmaceutical and biotechnology industries to improve patient outcomes and quality of life. With locations in North America, Europe and Asia, AMRI's team combines scientific expertise and market-leading technology to provide a complete suite of solutions in Discovery, Development, Analytical and Solid State Services, API Manufacturing and Drug Product. For more information about AMRI, visit https://www.amriglobal.com.


Clive Douglas Investments, in an entrepreneurial partnership with Clive Douglas and Richard Willis, has established Conexus as a niche hedge-fund combining an absolute return public equity fund with micro-cap private equity opportunities, targeting high-net-worth individuals as its investors. Metier injected capital and support for a stake in Clive Douglas Investments, assisting with the establishment of Conexus and providing execution advice on its deal flow.
Clive Douglas has extended his outstanding record in private client asset management with sound capital raising, investment performance and differentiated transaction access in Conexus.
The company has also partnered Anglo Platinum in pioneering a venture capital fund for the beneficiation and development of new technologies and downstream applications utilising platinum group metals. This fund has completed its first transaction in the development of fuel cells.
Douglas Investments also offers private clients bespoke advice in local and international portfolio management, and is a registered financial services provider.


text:
SOCIAL REGISTRY (...) 21 Titles
viewing 1 To 21 of 21 items
SIAN ALICE GROUP
SOCIAL REGISTRY
SOCIAL 068LP
$8.50 LOW STOCK LEVEL
"Released as an extremely limited edition 12-inch, Remix features the reworking of three tracks from Sian Alice Group's debut 59.59 by Alexis Taylor of Hot Chip, Brian Degraw of Gang Gang Dance and Spring Heeled Jack."
Troubled, Shaken, Etc.
Gatefold 2LP version on 140 gram vinyl.
SOCIAL 074CD
"Sian Alice Group -- a collective of UK musicians Sian Ahern, Rupert Clervaux and Ben Crook, amongst other collaborators -- are back with their sophomore album, Troubled, Shaken Etc. The band hones in on a genre-busting combination of organic improvisation, process-driven experimentation and the focused strength earned from a year of extensive travelling and touring the world together. You get your minimalist trances, electronic experimentation, techno thump, jazz motifs and Eastern music tropes in a joyous and complete listening experience. It's a snapshot (or, more closely, a yearbook) from an eventful 2008 that is more raw, emotional and loud than anything they've released thus far."
I.U.D.
The Proper Sex
"The Social Registry is proud to break your bones with I.U.D. -- Lizzi Bougatsos of Gang Gang Dance's (vocals, percussion) new project. Teaming with Sadie Laska of Growing and Extreme Violence (the duo is known here as Mr. Egglesby and Lil' Pickle -- they have different names for each release), The Proper Sex is a downright brutal industrial clusterfuck that's sure to fog the glasses of experimental music nerds and provide the catalyst for many a record store argument. Put The Boredoms, Einstürzende Neubauten, Meat Beat Manifesto, Tribe 8, Front 242 and some black metal tropes in a blender and then serve chilled at a party hosted by hyper-feminist vampires and you're getting close to the sound on this one. Recorded at Vacation Island Studios by Matt Boyton AKA Professor Funkenstein, The Proper Sex is creepy, relentless and (most of all) extremely fascinating. It's a varied listen that reflects its influences and cover art -- which mimics Sparks -- to a stunning effect. Like Gang Gang Dance, these rhythms are already cemented in a listener's head. It's just that this journey might be a descent into the inferno rather than up to the heavens." Includes poster of a naked horse-woman.
"All The Way, the debut full-length from Growing on The Social Registry, concludes an epic trip. Through six tracks Kevin Doria and Joe DeNardo complete the conceptual arc initiated on their 2006 album Color Wheel. If we still bought into genres, we would say that this record finally achieves a perfect balance of noise and pop, yet what we can say is that you'll potentially hum melodies long after your headphones are out. Recorded by Barry London at his Ocropolis Studio during the 2007-2008 winter, All The Way sounds like some sort of frozen urban chaos taking place at a superclub on another planet."
The Dusk Line
LP version.
$8.50 NOT IN STOCK
"Lateral finds the band reflecting on their past two (may we add highly underrated) albums while gazing to the future with newfound rhythmic motifs and otherworldly heights to their sound sculptures. The textures are warm and inviting, tempting the listener to focus on stunning minimalist changes within Growing's dense aural fog."
Mini-LP version.
KID KISHORE
The Social Club No. 9
SOCIAL 057EP
"Copenhagen native of Indian descent Kid Kishore strikes a balance between old and new, blending the sounds of India with the electronic music that permeates Denmark. Self-described as anything from 'Danish ghetto-tech' to 'golden age Bollywood,' his beat-heavy sound is a scattershot mix of musical styles from all over the sonic landscape."
"The Metabolismus collective has been making music since the mid-1980s, self-releasing most of their early material on cassette and vinyl LPs. The band comes from the town of Stuttgart, Germany and creates music that is rooted in the rich tradition of German experimental bands such as Can, Amon Duul, Popul Voh and Kraftwerk with touchstones in rock, folk and electronic music. With this release they are joined by Samara Lubelski and provide two beautiful tracks of arboreal shimmer with glimpses of soft '60s psychedelic sheen."
BONES, MIKE
The Sky Behind The Sea
"Mike Bones is New Jersey-born songwriter Mike Strallow, who lives and works in New York City. Of his pseudonym, Strallow offers: 'Bones is a nickname a friend gave to me when I was a kid. Making records as Mike Bones removes my voice from the song, at least in my own mind and leaves a song's meaning and intention ambiguous. I don't have to show my cards.' Although Bones has been writing songs for years, worked as a guitar player for The Mighty Flashlight, Damon McMahon, and Lizzie Trulie, and is a member of cult favorites Soldiers of Fortune (a band featuring members of Oneida, Home, and Oakley Hall), The Sky Behind The Sea marks his debut solo recording."
Sinister Militia
"Telepathe is a fluid band that ebbs and flows through genres of music all the while creating their own brand of dancey mysteriousness. Led by Busy Ganges and Melissa Lavaudais the group takes their eerie textural music and interweaves this with a rhythmic pattern that pulls you into their sound tapestry. The Sinister Militia 12" brings the listener into their creation. Tyler of !!! remixed Sinister Militia and accentuates all of the elements that make Telepathe the unique band they are. The eeriness and the beats are present in a more raucous splashy display."
GANG GANG DANCE
Retina Riddim
"In advance of Gang Gang Dance's upcoming studio full-length, The Social Registry are proud to present the band's first-ever DVD release, Retina Riddim. This film was made by Gang Gang Dance member, and visual artist, Brian DeGraw. Retina Riddim is by no means a tour film, documentary, live video, or anything as remotely stringent. Included on the DVD is a fan film, GGDbyOP, shot and edited on the road by Oliver Payne, a friend of the band's. As an added bonus Retina Riddim also contains an audio CD cut together by DeGraw specifically for this release, and a fold-out poster designed by Mr. DeGraw. Retina Riddim is both a document of Gang Gang Dance's progress over the past few years, and a look forward to the direction they are moving in."
ELECTROPUTAS
"The third installment from NYC's noise-mavens Electroputas is upon us and it is a heavy affair. Hand woven to thick analog tape over six months at the Junkyard Studio compound with Sean Maffucci of Icewater Scandal behind the board, this record opens up the stride of this criminally under exposed outfit. The sprawling results at times brings to mind such notable luminaries as Can, Chrome, Lightning Bolt or the more musical ramblings of Ciccone Youth. With this record the trio has managed to take their unrelenting live sound into a wholly new direction."
LUBELSKI, SAMARA
The Fleeting Skies
"The first proper solo adventure from Samara Lubelksi. Though she is most often recognized as a member of Hall Of Fame Samara also done time in The Sonora Pine, Tower Recordings, Metabolismis, Jackie O-Motherfucker & Salmon Skin and has also played alongside Tony Conrad & Tara Jane O'Neil. On these recordings Samara is joined by an array of friends & compatriots who help bring these ten folk/pop songs to perfectly arranged heights."
ICEWATER SCANDAL
Double LP version.
ARTANKER CONVOY
"The Social Registry 12" series continues with this silk and flame infected barnstormer from Artanker Convoy. This four track EP is sure to get feet moving and hips gyrating with their catchy and irresistible flow (people who have tried resisting have ended up in traction or worse). But this EP is not only for the floor-cutting set. It is layered with a multitude of different influences and experience (soul jazz, krautrock, bossanova, bubblegum pop to name a few) that set this group apart from the typical club ravers, they compose pieces that allow them to really stretch out and continue to evolve the more they play them. You can sit back and just enjoy this, but your body might have different ideas. Organic Grooves remixes one of the pieces shooting AT into an even higher orbit."
GHOST EXITS
Cincinnati Riot Blues
"We are proud to present the debut offering from NYC two piece Ghost Exits; a four song 12" EP. Ghost Exits have seen incarnations as a free jazz combo & rock band but it was late in 2001 when the current incarnation began to take shape. Trading in their drums & guitars for a sequencer and some out-moded analog synths Christopher Exit & Ivan Sunshine began looking to the British post-punk aesthetic and early 80s hip-hop for inspiration to create a new starting point from which to deliver their sound. Fast forward two years and you find the fully realized sound of Ghost Exits taking shape as they created a new beast to be reckoned with. It didn't take long for the rest of the NYC population to catch up with these two; and it wasn't long before they began landing shows with the likes of !!!, ESG, James Chance & The Warlocks."
"Gathering moss is not for Icewater Scandal; they are constantly percolating with ideas. They take what they have accomplished before redefining it and then heading for new horizons. Their first album travels over new territory from their debut EP released under the AM Radio moniker. Though still filtered through their concept of taking 60's experimental music a la the Dream Syndicate et al, and merging that with the elan of 80's indie rock; No Handle is at times more muscular and other times more cerebral, without losing any flair. Part of this can be attributed to their producer Lee Ranaldo of Sonic Youth. Recorded and mixed at SY's studio, Echo Canyon, there has been great care in getting the sound just right. Recorded, mixed and mastered all on analog equipment was a decision that allowed no ones and zeros to come between what was played and how the band wanted it heard."
PAINTING SOLDIERS
Horizon Falls
"From The Social Registry's new 12" EP series (strictly limited to 600 copies and packaged in identical metallic blue jackets): Andrea Hansen, though a member of the angular, surging rock group Icewater Scandal, has so much she wants to do that it cannot be contained within just one group. From her fertile mind and nimble fingers comes Painting Soldiers. Most of her work was recorded in a small one room apartment, on a lonely block above a gun shop, with just her guitar and a four-track recorder (and the occasional helping hand from various friends.) Andrea has been able to unleash her talent and imagination that is clear and unadorned with post production by Lee Ranaldo. From the gentle rollicking of 'Heaven And Mars' to the stormy psychedelic 'Mighty Fumble' this EP stands by itself. This debut 12" is sure to transport the listener into an existence that transcends the common and everyday."


text:
Q: Intuition behind the null distribution of the deviance statistic in survival models I am reading Tutz & Schmid "Modeling Discrete Time-to-Event Data" (2016) chapter 4 Evaluation and Model Choice section 4.2 Residuals and Goodness-of-Fit. A goodness-of-fit statistic called deviance is defined as
$$
D = 2 \sum_{i=1}^N n_i \sum_{t=1}^k p_{it} \log\left( \frac{p_{it}}{\hat\pi_{it}} \right)
$$
where $\hat\pi_{it}$ is the estimated probability for a person belonging to group $i$ (of size $n_i$, among $N$ groups in total) to experience an event in the time period $t$ (from 1 to $k$), and $p_{it}$ is the corresponding observed proportion of persons in group $i$ that indeed experienced the event in the time period $t$.
A well-fitting model will produce values $\hat\pi_{it}$ that are close the the observed proportions $p_{it}$, though there will be discrepancies due to sampling variability inherent in $p_{it}$. This will yield $\frac{p_{it}}{\hat\pi_{it}}\approx 1$ and thus $\log\left( \frac{p_{it}}{\hat\pi_{it}} \right)\approx 0$ and thus $D\approx 0$. (Correct me if I am wrong.)
On the other hand, there is a later statement on the same page that asymptotically $D\sim\chi^2(N(k-1)-p)$. Now, I know $\chi^2(N(k-1)-p)$ can be obtained as a sum of squares of $N(k-1)-p$ independent N(0,1) random variables or as a sum of squares of a different number of dependent ones. However, I do not get the intuition why $p_{it} \log\left( \frac{p_{it}}{\hat\pi_{it}} \right)$ should behave like a square of N(0,1). Perhaps it should not? More generally, how is the asymptotic $\chi^2(N(k-1)-p)$ distribution obtained? (Intuition is welcome.)
Moreover, it seems to me that $\log\left( \frac{p_{it}}{\hat\pi_{it}} \right)$ and thus $p_{it} \log\left( \frac{p_{it}}{\hat\pi_{it}} \right)$ can be both lesser and greater than zero, and so $D$ could end up being negative. That is incompatible with a $\chi^2(N(k-1)-p)$ distribution. Am I getting this wrong?

A: This section of the Tutz and Schmid book assumes that there is no censoring, so this result is just the application of maximum-likelihood analysis to a multinomial distribution of events over the $k$ time periods.
Following Section 1.5.1 of the second edition of Agresti's Categorical Data Analysis, the multinomial log-likelihood function for a model with a set of probabilities $\pi$ given $n_j$ observations in $j$ categories (total $N$ observations) is:
$$ L(\pi)=\sum_j n_j \log(\pi_j)$$
where $\pi_j$ is the modeled probability of category $j$.
The displayed deviance formula follows the standard definition (e.g., Section 4.5 of Agresti): twice the difference in log-likelihood between a saturated model with the maximum likelihood:
$$\sum_j n_j \log(n_j/N)= \sum_j n_j \log(p_j),$$
where $p_j$ is the observed probability of category $j$, and that of the model in question:
$$\sum_j n_j \log(\pi_j).$$
As the data have maximum likelihood under the saturated model, the likelihood under any other model cannot exceed that. So the sum of terms contributing to the displayed deviance formula is necessarily non-negative, even if individual terms might be negative.
The asymptotic chi-square distribution comes from Wilks's theorem. Agresti provides a proof specific to the multinomial likelihood-ratio statistic in Section 14.3.4. This page and its links provide some intuition about Wilks's theorem. For degrees of freedom, I suppose you can think about this as $(k-1)$ independent time periods (no censoring, so the results for $k-1$ periods determine that for the $k^{th}$ period) for $N$ groups, less the $p$ fitted parameters.
In terms of using the deviance residuals in survival analysis (in models that incorporate censored data), Therneau and Grambsch note in Section 4.3:

The deviance residual was designed to improve on the martingale residual for revealing individual outliers, particularly in plotting applications. In practice it has not been as useful as anticipated.

I can't say that I've found them very useful, unlike the scaled Schoenfeld residuals for evaluating proportional hazards assumptions, or martingale residuals for evaluating functional forms.

redpajama_set_name: RedPajamaStackExchange


Performer is an FTP/Web Scripting tool designed to automate your daily essential Web tasks: carry out scheduled downloads, regular back ups, search users' hard disks for files and upload them to different accounts, send e-mail upon retrieval of a file to specific users, e-mail alert upon success/failure of file transfer, compare file sizes, and password-protect files. Compile your scripts to standalone executables (.exe). The program features 110 sample scripts to view and customize.


This series looks promising. Check out Andromeda's Fall and author Abigail Owen's great excerpt and cover reveal!
Excerpt from Tears of War by A. D. Trosper!
Nothing makes me happier than showing off authors I have come across, especially when they are as talented and nice as A. D. Trosper.


text:
\section{Introduction}

A defining property of a geodesic is that it is a locally distance minimizing curve. It is clear that a nontrivial closed geodesic can never be a \emph{globally} distance minimizing curve. Indeed, a closed geodesic can never minimize past half its length: traversing the geodesic in the opposite direction will always provide a shorter path. It is therefore natural to consider the largest interval on which a given closed geodesic is distance minimizing. This led Sormani \cite{Sor} to consider the notion of a 1/k-geodesic.

\begin{definition} A \emph{1/k-geodesic} is a closed geodesic $\gamma \colon S^1 \to M$ which is minimizing on all subintervals of length $l(\gamma)/k$, i.e. $$d(\gamma(t),\gamma(t+2\pi/k))=l(\gamma)/k \hspace{4mm} \forall t \in S^1.$$
\end{definition}

In order to study the existence of 1/k-geodesics on compact Riemannian manifolds Sormani employed the uniform energy $E \colon M^k \to \mathbb{R}$, a function first introduced in the study of Morse theory (Cf.~\cite{Mil}).

\begin{definition}\label{ue} The \emph{uniform energy} $E \colon M^k \to \mathbb{R}$ is defined for an element $\bar{x} = (x_1, \ldots , x_k)  \in M^k $ as
\begin{equation*}E(\bar{x}) = \sum_{i=1}^k \frac{d(x_i , x_{i+1})^2}{1/k} \hspace{9pt} where \hspace{6pt} x_{k+1} = x_1
\end{equation*}
\end{definition}

Sormani focused on the \emph{openly} 1/k-geodesics: those 1/k-geodesics that do not contain cut points at distance $l(\gamma)/k$ and therefore always minimize on some open neighborhood of subintervals of length $l(\gamma)/k$. She established a one-to-one correspondence between the openly 1/k-geodesics on a compact Riemannian manifold and the smooth critical points of the uniform energy \cite[Theorem 10.2]{Sor}. She then asked if her ideas could be extended to achieve a similar relationship between the full set of 1/k-geodesics on a compact Riemannian manifold and some notion of non-smooth critical points of the uniform energy \cite[Remark 10.4]{Sor}. We provide a positive answer to her problem by introducing the balanced points of the uniform energy.

\begin{definition}\label{balanced_intro}
Let $\widehat{qp}$ denote the collection of initial velocity vectors in $T_qM$ of unit-speed minimizing geodesics joining $q$ to $p$. Let $\bar{x} = (x_1, \ldots , x_k)  \in M^k$. We say $\bar{x}$ is a \emph{balanced point} of the uniform energy if $d(x_{i-1},x_i) = d(x_i, x_{i+1})$ and if there exists a vector $\bar{\xi} = (\xi_1, \ldots \xi_k) \in T_{\bar{x}}M^k $ with $\xi_i \in \widehat{x_i x_{i+1}}$ and $-\xi_i \in \widehat{x_i x_{i-1} }$ for every $i =1, \ldots, k$.
\end{definition}

\begin{thm}\label{combined} The 1/k-geodesics in a compact Riemannian manifold have a one-to-one correspondence with classes of balanced points of the uniform energy in $M^k$ of nonzero energy.
\end{thm}

Note that closed geodesics may disappear under the Gromov-Hausdorff convergence of Riemannian manifolds (see Example~\ref{sphere2}). In contrast, Sormani demonstrated the persistence of 1/k-geodesics under Gromov-Hausdorff convergence (Theorem~\ref{conv2}).  Classes of smooth critical points of the uniform energy may disappear even under the smooth convergence of Riemannian manifolds (see Example~\ref{sphere}). We combine Sormani's Theorem~\ref{conv2} with our Theorem~\ref{combined} to show that classes of balanced points of the uniform energy persist under Gromov-Hausdorff convergence.

\begin{cor}\label{persist} Let $M_i$ be a sequence of compact Riemannian manifolds converging to a compact Riemannian manifold $M$ in the Gromov-Hausdorff sense. Then any sequence of classes of balanced points of the uniform energies $E \colon M^k_i \to \mathbb{R}$ admits a subsequence which converges to a class of balanced points of the uniform energy $E \colon M^k \to \mathbb{R}$ or to the trivial class.
\end{cor}

We next turn our attention to the half-geodesics: those closed geodesics that minimize on any subinterval of length $l(\gamma)/2$. We first note that every closed manifold with non-trivial fundamental group admits a half-geodesic. Indeed, the shortest non-contractible closed geodesic, or the systole of the manifold, will always be a half-geodesic \cite[Lemma 4.1]{Sor}. Using Clairaut's relation, Wing Kai Ho \cite{Ho} produced surfaces diffeomorphic to the $2$-sphere that do not admit half-geodesics. In \cite{Ade}, the author produced surfaces diffeomorphic to the $2$-sphere that admit exactly $n$ half-geodesics for every $n\geq 0$.

We provide a relationship between the half-geodesics (1/2-geodesics) and the Grove-Shiohama critical points of the distance function (Definition~\ref{GSdef2} and Proposition~\ref{GS2}). This notion of a critical point for the distance function was introduced by Grove and Shiohama \cite{Gro} during their proof of the celebrated diameter sphere theorem (Theorem~\ref{GSthm2}). Further work on the Grove-Shiohama notion of critical points was conducted by Grove-Petersen \cite{GP}, Gromov \cite{Gromov}, and Abresch-Gromoll \cite{AG}. We employ techniques from this field to study the behavior of the half-geodesics and establish the following result.

\begin{thm}\label{behavior} Let $M$ be a complete Riemannian manifold with sectional curvature $K \geq H>0$. Assume $M$ admits a half-geodesic $\gamma \colon S^1 \to M$ with $l(\gamma) > \frac{\pi}{\sqrt{H}}$. Then any closed geodesic which intersects $\gamma$ must have length at least $l(\gamma)$. Any half-geodesic which intersects at $\gamma(t)$ must also intersect at $\gamma(t+\pi)$ and have length equal to $l(\gamma)$.
\end{thm}

The paper proceeds as follows. In Section~\ref{sec_diff} we give the proof of a folk theorem that completely characterizes the differentiability of the distance function (Theorem~\ref{diff_cut_locus}). This result is applied in Section~\ref{critical} to study  the differentiability of the uniform energy. We are able to strengthen the standard relationship from Morse theory between the critical points of the uniform energy and the derivatives of its component distance functions (Theorem~\ref{sormani}). In Section~\ref{geo} we introduce the balanced points of the uniform energy and provide the proof of Theorem~\ref{combined}. In Section~\ref{balance5} we examine the behavior of the uniform energy functional under Gromov-Hausdorff convergence and prove Corollary~\ref{persist}. In Section~\ref{half2} we detail the relationship between half-geodesics and the Grove-Shiohama critical points of the distance function. It is here that we prove Theorem~\ref{behavior} and show how it applies to the round sphere, to oblate ellipsoids, and to real projective space (Examples~\ref{round}, \ref{oblate}, \ref{proj}).

\section{Differentiability of the Distance Function}\label{sec_diff}

Let $M$ be a complete Riemannian manifold. It is well known that the distance function $d_p \colon M \to \mathbb{R}$ is a continuous function on $M$ and is of class $C^{\infty}$ (i.e.~smooth) away from the cut locus of $p$ and the point $p$ itself (Cf.~\cite{Sak}). Less well known, and apparently absent from the literature, is the fact that $d_p$ is indeed differentiable at a certain subset of the cut locus of $p$. The main result of this section is Theorem~\ref{diff_cut_locus} which provides a complete characterization of the differentiability of the distance function $d_p$. In addition to filling a gap in the literature, this result will allow us to explore the differentiability of the uniform energy in the following section.

Let $M$ be a complete Riemannian manifold and $p \in M$ with $\gamma \colon [0,\infty) \to M$ a normalized geodesic, $\gamma(0)=p$. By continuity, the set of numbers $t >0$ such that $t=d(p,\gamma(t))$ is of the form $[0, t_0]$ or $[0, \infty)$. In the first case we say $\gamma(t_0)$ is the \emph{cut point} of $p$ along $\gamma$, and in the second case we say such a cut point does not exist. We define the \emph{cut locus} of p, denoted $C(p)$, to be the collection of all such cut points. A basic fact about the cut locus is as follows.

\begin{prop}[Cf.~\cite{doC}] Let $\gamma(t_0)$ be the cut point of $p=\gamma(0)$ along $\gamma$. Then
\begin{description}
\item{a)} either $\gamma(t_0)$ is the first conjugate point of $p$ along $\gamma$,
\item{b)} or there exists a geodesic $\sigma \neq \gamma$ from $p$ to $\gamma(t_0)$ such that $l(\sigma)=l(\gamma)$.
\item{} Conversely if (a) or (b) is satisfied, then there exists $\tilde{t} \in (0,t_0]$ such that $\gamma(\tilde{t})$ is the cut point of p along $\gamma$.
\end{description}
\end{prop}

We partition the cut locus into two disjoint subsets. We say that a cut point $q$ of $p$ is \emph{singular} if there exists a unique minimizing geodesic $\gamma$ joining $q$ to $p$. This can only occur if the cut points $p$ and $q$ are conjugate along $\gamma$. However being conjugate does not guarantee the existence of a unique minimal geodesic. We say that a cut point $q$ of $p$ is \emph{ordinary} if there exist multiple minimizing geodesics joining $q$ to $p$. Bishop \cite{Bis} showed that the set of ordinary cut points is dense in the cut locus. We are now ready to give a complete characterization of the differentiability of the distance function.

\begin{thm}\label{diff_cut_locus}Let $d_p \colon M \to \mathbb{R}$ denote the distance function from $p$. Then
\begin{description}
\item{a)} if $q \not\in C(p) \cup \{p\}$, we have that $d_p$ is smooth at $q$ and $\nabla d_p (q) = \dot{\sigma} (l)$ where $\sigma$ is the unique minimizing geodesic joining $p$ to $q$ and $l=d_p(q)$.
\item{b)} if $q \in C(p)$ is an ordinary cut point of $p$, we have that $d_p$ is not differentiable at $q$.
\item{c)} if $q \in C(p)$ is a singular cut point of $p$, we have that $d_p$ is differentiable at $q$ and $\nabla d_p (q) = \dot{\sigma} (l)$ where $\sigma$ is the unique minimizing geodesic joining $p$ to $q$ and $l=d_p(q)$.  However $d_p$ is not $C^1$ at its singular cut points.
\end{description}
\end{thm}

Sakai \cite{Sak} provides a detailed proof of parts (a) and (b) of Theorem~\ref{diff_cut_locus}. Ivanov \cite{Iva2} makes a statement similar to part (c) in a MathOverflow response. He does not provide a proof and it appears as though one does not exist in the literature. We provide one here for completeness.

\begin{definition} \label{d_p_definition} The \emph{one-sided directional derivative} of the distance function at the point $q \in M$ in the direction $v \in T_qM$ is defined to be
\begin{equation*}\label{d_p_def} D^+_v d_p(q) = \lim_{t \to 0^+} \frac{d_p(\exp_q (t \cdot v)) - d_p(q)}{t}
\end{equation*}
provided that this limit exists.
\end{definition}

\begin{lemma}[Cf.~\cite{Ito, Pla}]\label{direction_deriv} Let $\widehat{qp}$ denote the collection of initial velocity vectors in $T_qM$ of unit-speed minimizing geodesics joining $q$ to $p$. Then for a vector $v \in T_qM$ the one-sided derivative in the direction $v$ exists and is given by
\begin{equation*}\label{direc_deriv} D^+_v d_p (q) = \min{ \{ - \langle v,\xi  \rangle   :  \xi \in \widehat{qp}   \}. }
\end{equation*}
In particular, if there exists a unique minimizing geodesic $\sigma$ joining $p$ to $q$ then the two-sided directional derivative exists and equals $\langle v, \dot{\sigma}(l) \rangle$ where $l=d_p(q)$.
\end{lemma}

\begin{proof}[Proof of Theorem~\ref{diff_cut_locus}(c).]

Let $q$ be a singular cut point of $p$. Let $\sigma$ be the unique minimizing geodesic joining $p$ to $q$ and $l=d_p(q)$. It suffices to show that for every sequence $h_i \in T_qM$ with $|h_i| \to 0$, there exists a subsequence $k_i$ such that
\begin{equation} \label{lim}
\lim_{i \to \infty} \frac{d_p(\exp_q(k_i)) - d_p(q) - \langle \dot{\sigma}(l) , k_i \rangle}{|k_i|} =0.
\end{equation}
Write $h_i = t_i u_i$ for $u_i \in U_qM$ the unit tangent sphere and $t_i \to 0$. By compactness of the unit sphere we can choose a subsequence $v_i$ of the $u_i$ which converge to some $v \in U_qM$. Setting $k_i = t_i v_i$ we will show that the limit in (\ref{lim}) holds.

From Lemma~\ref{direction_deriv} we have that $$\lim_{i \to \infty} \frac{d_p(\exp_q(t_i v)) - d_p(q) - \langle \dot{\sigma}(l) , t_i v \rangle}{t_i} =0.$$
Letting $\gamma_i=\exp_q(t_i v)$ and $c_i = \exp_q(t_i v_i)$ and applying the triangle inequality twice we have $$d_p(\gamma_i)-d(c_i,\gamma_i) \leq d_p(c_i) \leq d_p(\gamma_i)+d(c_i,\gamma_i). $$
We note that $\lim_{i \to \infty} \frac {\langle \dot{\sigma}(l) , t_i v_i \rangle}{t_i} = \lim_{i \to \infty} \frac {\langle \dot{\sigma}(l) , t_i v \rangle}{t_i}$ and $\lim_{i \to \infty} \frac{|t_i v - t_i v_i |}{t_i}=0$ so that the limit in (\ref{lim}) will hold if we show that $$\lim_{i \to \infty} \frac{d(c_i , \gamma_i )}{t_i} \leq \lim_{i \to \infty} \frac{|t_i v - t_i v_i | + O(t_i^2)}{t_i} =0.$$

Let $\tilde{\sigma}_{t_i}(s) \colon [0,1] \to T_qM$ be the constant speed path in $T_qM$ which traverses the arc of the circle of radius $t_i$ between $t_i v$ and $t_i v_i$ and $\sigma_{t_i}(s) \colon [0,1] \to M$ the curve in $M$ given by $\sigma_{t_i}(s)= \exp_q\,\tilde{\sigma}_{t_i}(s)$ so that $\sigma_{t_i}(0)=\gamma_i$ and $\sigma_{t_i}(1)=c_i$. We choose $i$ large enough that we can work in normal coordinates around $q$ where we know that $g_{ij} (\sigma_{t_i}(s))= \delta_{ij} + O(t_i^2)$ so that $|\dot{\sigma}_{t_i}(s)|^2= |\dot{\tilde{\sigma}}_{t_i}(s)|^2+O(t_i^2).$ Now computing the length of $\sigma_{t_i}(s)$ we have
\begin{align*}
l(\sigma_{t_i}(s) ) &= \int_0^1 |\dot{\sigma}_{t_i}(s)| \, ds = \int_0^1 \sqrt{  |\dot{\tilde{\sigma}}_{t_i}(s)|^2 + O(t_i^2)}\, ds =\int_0^1  |\dot{\tilde{\sigma}}_{t_i}(s)| + O(t_i^2) \, ds
\\ &= l(\tilde{\sigma}_{t_i}(s) ) + O(t_i^2) =   |t_i v - t_i v_i | +O(t_i^2)
\end{align*}
Finally we note that $d(c_i, \gamma_i ) \leq l(\sigma_{t_i}(s)) = |t_i v - t_i v_i |   +O(t_i^2) $ so that indeed the limit in (\ref{lim}) holds and we have shown that $d_p$ is differentiable at $q$ with $\nabla d_p (q)=\dot{\sigma}(l)$.

It is left to show that $d_p$ is not $C^1$ at singular cut points. Bishop \cite{Bis} showed that the ordinary cut locus is dense in the cut locus. Therefore, there exists a sequence $\{q_n\}$ of ordinary cut points converging to the singular cut point $q$. We know from part (b) of the theorem that $\nabla d_p$ is not defined at $q_n$ and can therefore conclude that $d_p$ is not $C^1$ at the singular cut locus.
\end{proof}

\section{Critical Points of the Uniform Energy}\label{critical}

The uniform energy (Definition~\ref{ue}) is given as a sum of distance functions, hence its differentiability will depend on the differentiability of its component distance functions. A priori, it may be possible for $E \colon M^k \to \mathbb{R}$ to be differentiable at points $\bar{x} \in M^k$ for which the individual terms $d(x_i,x_{i+1})^2$ are not differentiable. In Lemma~\ref{no_ordinary} we show that this phenomenon does not occur at the critical points of the uniform energy (Definition~\ref{cp}). We apply this lemma in Theorem~\ref{sormani} to strengthen the standard result from Morse theory relating the critical points of the uniform energy and the derivatives of its component distance functions (Cf.~\cite{Mil}). In particular, we are able to remove the standard assumption that the distance between each pair $(x_i,x_{i+1})$ is less than the injectivity radius of $M$.

\begin{definition}\label{cp} Let $\bar{x} = (x_1, \ldots , x_k)  \in M^k $. We say $\bar{x}$ is a \emph{critical point} of the uniform energy if $E$ is differentiable at $\bar{x}$ and its gradient is zero, i.e.~$\nabla E(\bar{x})=\vec{0}$.
\end{definition}

\begin{lemma}\label{no_ordinary} If $\bar{x} \in M^k$ is a critical point of the uniform energy then none of the pairs $(x_i, x_{i+1})$ are ordinary cut points, i.e.~all of the component distance functions are differentiable.
\end{lemma}

\begin{proof} We know that $\nabla E(\bar{x})=\vec{0}$ because $\bar{x}$ is a critical point. Letting $\bar{v}_i=(0, \ldots, v_i, \ldots, 0) \in T_{\bar{x}}M^k$ we have that $D_{\bar{v}_i} E (\bar{x}) =  \langle \bar{v}_i, \nabla E (\bar{x})  \rangle =0$ for every $\bar{v}_i \in T_{\bar{x}}M^k$. We show how to calculate $D_{\bar{v}_i} E (\bar{x})$ using the one-sided directional derivative of the distance function from Lemma~\ref{direction_deriv}. We have
\begin{align*}
D_{\bar{v}_i} E (\bar{x}) &=  k \, D^+_{v_i} (d_{x_{i-1}}^2 + d_{x_{i+1}}^2 )x_i \\
&=  2k \, d(x_{i-1},x_i) \, D^+_{v_i} (d_{x_{i-1}} )x_i + 2k \, d(x_i , x_{i+1}) \, D^+_{v_i} (d_{x_{i+1}} )x_i \\
&= c_1 \, \min{ \{ - \langle v_i,\xi  \rangle   :  \xi \in \widehat{x_i x_{i-1}} } \} + c_2 \, \min{ \{ - \langle v_i,\eta  \rangle   :  \eta \in \widehat{x_i x_{i+1}} } \}
\end{align*}
where $c_1=2k \, d(x_{i-1},x_i) $ and $c_2= 2k \, d(x_i, x_{i+1})$ are both positive constants.

We will show that the sets $\widehat{x_i x_{i-1}}$ and $\widehat{x_i x_{i+1}}$ each consist of a single vector so that by Theorem~\ref{diff_cut_locus} we can conclude that each of the component distance functions is differentiable. Assume by contradiction that one (or both) of the sets contains multiple vectors; say there exist $\xi_1$ and $\xi_2 \in \widehat{x_i x_{i-1}}$ with $\xi_1 \neq \xi_2$. Then for any $\eta \in \widehat{x_i x_{i+1}}$ we must have that either $\xi_1 \neq - \eta$ or  $\xi_2 \neq - \eta$ because $\xi_1 \neq \xi_2$. Without loss of generality assume that $\xi_1 \neq - \eta$. Then there must exist  $v_i \in T_{x_i}M$ such that $- \langle v_i, \xi_1  \rangle < 0$ and $- \langle v_i, \eta \rangle < 0$. This yields $D_{\bar{v}_i} E(\bar{x}) < 0$ which is a contradiction when $\bar{x}$ is a critical point. We therefore conclude that none of the pairs $(x_i, x_{i+1})$ are ordinary cut points and that all of the component distance functions are differentiable.
\end{proof}

\begin{thm}\label{sormani} We have that $\bar{x}$ is a critical point of the uniform energy if and only if $ d(x_{i-1},x_i) = d(x_i, x_{i+1})$ and $\nabla d_{x_{i-1}} x_i = -\nabla d_{x_{i+1}} x_i$ for every $i \in \{1, \ldots, k \}$.
\end{thm}
\begin{proof} Let $\bar{x}$ be a critical point so that by Lemma~\ref{no_ordinary} none of the pairs $(x_i, x_{i+1})$ are ordinary cut points and $\nabla d_{x_{i \pm 1}} x_i $ is well defined for every $i$. Computing via the chain rule we have
\begin{align}\label{chain}
\nabla E(\bar{x}) = 2k \, &[d_{x_k}x_1 \, \nabla d_{x_k}x_1 +d_{x_2}x_1 \, \nabla d_{x_2}x_1 , \, d_{x_1}x_2 \, \nabla d_{x_1}x_2 +d_{x_3}x_2 \, \nabla d_{x_3}x_2, \notag  \\
&  \ldots ,d_{x_{k-1}}x_k \, \nabla d_{x_{k-1}}x_k +d_{x_1}x_k \, \nabla d_{x_1}x_k ]
\end{align}
so that the critical point condition $\nabla E(\bar{x}) = \vec{0}$ implies that $ d_{x_{i-1}}x_i \, \nabla d_{x_{i-1}} x_i = -d_{x_{i+1}}x_i \, \nabla d_{x_{i+1}} x_i$. We note that $|\nabla d_{x_{i \pm 1}} x_i |=1$ and therefore conclude that $ d(x_{i-1},x_i) = d(x_i, x_{i+1})$ and $\nabla d_{x_{i-1}} x_i = -\nabla d_{x_{i+1}} x_i$ for every $i \in \{1, \ldots, k \}$.

In the opposite direction assume $ d(x_{i-1},x_i) = d(x_i, x_{i+1})$ and $\nabla d_{x_{i-1}} x_i = -\nabla d_{x_{i+1}} x_i$ for every $i$. Then by (\ref{chain}) we have $\nabla E(\bar{x}) = \vec{0}$ so that indeed $\bar{x}$ is a critical point of the uniform energy.
\end{proof}

\section{1/k-Geodesics and Balanced Points of the Uniform Energy}\label{geo}

This section introduces the balanced points of the uniform energy and extends the Sormani result \cite[Thm 10.2]{Sor} to provide a one-to-one correspondence between the classes of balanced points of the uniform energy and all 1/k-geodesics on a Riemannian manifold. We begin with an exposition of Sormani's ideas.

\begin{definition}[\cite{Sor}, Definition 3.1]\label{kk} A \emph{1/k-geodesic} is a closed geodesic $\gamma \colon S^1 \to M$ which is minimizing on all subintervals of length $l(\gamma)/k$, i.e. $$d(\gamma(t),\gamma(t+2\pi/k))=l(\gamma)/k \hspace{4mm} \forall t \in S^1.$$ The 1/k-geodesics come in two disjoint flavors:
\begin{description}
\item{a)} A \emph{strict} 1/k-geodesic is a 1/k-geodesic that contains a pair of cut points at distance $l(\gamma)/k$.
\item{b)} An \emph{openly} 1/k-geodesic does not contain cut points at distance $l(\gamma)/k$, and therefore will always minimize on some open neighborhood of subintervals of length $l(\gamma)/k$.
\end{description}
\end{definition}

\begin{prop}[\cite{Sor}, Theorem 3.1]\label{all} Any closed geodesic is a 1/k-geodesic for a sufficiently large number k.
\end{prop}
\begin{proof} This follows from compactness of the circle and the local minimizing property of geodesics.
\end{proof}

\begin{definition}[\cite{Sor}, Theorem 10.1] Let $\bar{x} \in M^k$ be a critical point of the uniform energy as in Definition~\ref{cp} (i.e.~$\nabla E(\bar{x})=\vec{0}$). We say $\bar{x}$ is a \emph{smooth} critical point of the uniform energy if none of the pairs $(x_i, x_{i+1})$ are (singular) cut points, i.e.~the component distance functions are all smooth functions.
\end{definition}

\begin{lemma}[\cite{Sor}, Lemma 10.2] Let $\bar{x} = (x_1, \ldots , x_k)  \in M^k $ be a smooth critical point of the uniform energy. Then there exists a unique associated closed geodesic $\gamma \colon S^1 \to M$ with $\gamma(2\pi i /k)=x_i$.
\end{lemma}

\begin{definition}[\cite{Sor}, Definition 10.2] A smooth critical point is called \emph{rotating} if the unique associated geodesic $\gamma \colon S^1 \to M$ is such that for every $t \in S^1$ the point $(\gamma(t), \gamma(t+2\pi/k), \ldots , \gamma(t + 2\pi(k-1)/k) ) \in M^k$ is also a smooth critical point of the uniform energy. In this case we call the set $\{(\gamma(t), \gamma(t+2\pi/k), \ldots , \gamma(t + 2\pi(k-1)/k)) \in M^k  :  t \in S^1 \}$ a \emph{smooth class of critical points} of the uniform energy in $M^k$.
\end{definition}

A smooth critical point need not be a rotating smooth critical point. Indeed, for $\gamma$ the unique closed geodesic associated to the smooth critical point, there may be a $t_0 \in S^1$ such that $(\gamma(t_0), \gamma(t_0+2\pi/k), \ldots , \gamma(t_0 + 2\pi(k-1)/k) )$ is not a smooth critical point. An example where this occurs is the over-under geodesic on the doubled square \cite[Ex 9.2]{Sor}. Any four-tuple of evenly spaced points on this geodesic is a smooth critical point, except for the four-tuple of midpoints. Both the over and the under geodesics minimize between pairs of adjacent midpoints, hence these pairs are cut points. This geodesic is therefore a strict 1/4-geodesic.

\begin{thm}[\cite{Sor}, Theorem 10.2]\label{sormani3} Openly 1/k-geodesics in a compact Riemannian manifold have a one-to-one correspondence with smooth classes of critical points of the uniform energy in $M^k$ of nonzero energy.
\end{thm}

Sormani then asks if her ideas can be extended to achieve a similar relationship between the strict 1/k-geodesics and some notion of non-smooth critical points of the uniform energy \cite[Remark 10.4]{Sor}. In order to extend her ideas it will be necessary to consider those points $\bar{x} \in M^k$ which contain pairs $(x_i, x_{i+1})$ of cut points. We note that Lemma~\ref{no_ordinary} concludes that the critical points of the uniform energy cannot contain pairs $(x_i, x_{i+1})$ of ordinary cut points. It is therefore necessary to consider a more general notion than critical point of the uniform energy. We introduce the balanced points of the uniform energy for this purpose. The remainder of this section is used to establish Theorem~\ref{combined} which gives a one-to-one correspondence between 1/k-geodesics and classes of balanced points of the uniform energy.

\begin{definition}\label{balanced} Let $\widehat{qp}$ denote the collection of initial velocity vectors (in $T_qM$) of unit-speed minimizing geodesics joining $q$ to $p$.  Let $\bar{x} = (x_1, \ldots , x_k)  \in M^k $
\begin{description}
\item{a)} we say $\bar{x}$ is a \emph{balanced point} of the uniform energy $E \colon M^k \to \mathbb{R}$ if $d(x_{i-1},x_i) = d(x_i, x_{i+1})$ and if there exists a vector $\bar{\xi} = (\xi_1, \ldots \xi_k) \in T_{\bar{x}}M^k $ with $\xi_i \in \widehat{x_i x_{i+1}}$ and $-\xi_i \in \widehat{x_i x_{i-1} }$ for every $i \in \{1, \ldots, k \}$.
\item{b)} such a balanced point is called \emph{uniquely balanced} if none of the pairs $(x_i, x_{i+1})$ are ordinary cut points, i.e. every pair is joined by a unique minimizing geodesic.
\item{c)} such a balanced point is called \emph{smooth} if none of the pairs $(x_i, x_{i+1})$ are cut points and is called \emph{non-smooth} if at least one of the pairs are cut points (ordinary or singular).
\end{description}
\end{definition}

\begin{remark}\normalfont It follows immediately from Theorem~\ref{sormani} that $\bar{x}$ is a critical point of the uniform energy if and only if it is a uniquely balanced point. Similarly, $\bar{x}$ is a smooth critical point of the uniform energy if and only if it is a smooth balanced point.
\end{remark}

\begin{definition} We say a closed geodesic $\gamma \colon S^1 \to M$ is \emph{associated} to the balanced point $\bar{x} \in M^k$ if there exists $t \in S^1$ with $\bar{x}=(\gamma(t), \gamma(t+2\pi/k), \ldots , \gamma(t + 2\pi(k-1)/k) ) \in M^k$ and $\dot{\gamma}(t+2\pi (i-1) /k) \in \widehat{x_i x_{i+1}}$ and $-\dot{\gamma}(t+2\pi (i-1) /k) \in \widehat{x_i x_{i-1}}$ for every $i \in \{1, \ldots, k \}$.
\end{definition}

We note that a balanced point need not have an associated closed geodesic. An example of such a balanced point is given by the five-tuple of midpoints on the doubled pentagon \cite[Remark 10.4]{Sor}.

\begin{definition} A balanced point is called \emph{rotating} if there exists an associated closed geodesic $\gamma \colon S^1 \to M$ such that for every $t \in S^1$ the point $(\gamma(t), \gamma(t+2\pi/k), \ldots , \gamma(t + 2\pi(k-1)/k) ) \in M^k$ is a balanced point with associated closed geodesic $\gamma$. In this case we call the set $\{(\gamma(t), \gamma(t+2\pi/k), \ldots , \gamma(t + 2\pi(k-1)/k)) \in M^k  :  t \in S^1 \}$ a \emph{class of balanced points} of the uniform energy in $M^k$.
\end{definition}

An example of a balanced point with associated closed geodesic which is not rotating is given by the corners on the doubled square \cite[Ex 10.1]{Sor}.

\begin{definition} A \emph{non-smooth class of balanced points} of the uniform energy in $M^k$ is a class of balanced points such that $(\gamma(t_0), \gamma(t_0 +2\pi/k), \ldots , \gamma(t_0 + 2\pi(k-1)/k) ) \in M^k$ is a non-smooth balanced point for some $t_0 \in S^1$.
\end{definition}

An individual non-smooth balanced point may be associated to multiple closed geodesics, and thus multiple non-smooth classes of balanced points. An example where this occurs is the four-tuple of midpoints on the doubled square which is associated to the two closed over-under geodesics \cite[Ex 9.2]{Sor}. The four-tuple $[(0,0),(.25,.5),(.5,1),(.75,.5)]$ on the standard flat two-torus provides another illustrative example.

\begin{lemma}\label{lem1} For every 1/k-geodesic $\gamma \colon S^1 \to M$ and every $t\in S^1$ the point $\bar{x}=(\gamma(t), \gamma(t+2\pi/k), \ldots , \gamma(t + 2\pi(k-1)/k) ) \in M^k$ is a rotating balanced point of the uniform energy. Therefore, to each 1/k-geodesic we can associate a unique class of balanced points in $M^k$.
\end{lemma}
\begin{proof} We have that $d(x_{i-1},x_i) = d(x_i, x_{i+1})$ because the geodesic $\gamma$ is parametrized by arc length. Then letting $\xi_i$ correspond to the velocity vector of $\gamma$ at $x_i$ we see that $\bar{x}$ is indeed a balanced point of the uniform energy. The rotating condition is satisfied because $\bar{x}$ was generated by the closed 1/k-geodesic $\gamma$.
\end{proof}

\begin{lemma}\label{strict_lemma} For every strict 1/k-geodesic there exists a $t_0 \in S^1$ such that $(\gamma(t_0), \gamma(t_0+2\pi/k), \ldots , \gamma(t_0 + 2\pi(k-1)/k) )$ is a non-smooth balanced point of the uniform energy. Therefore, to each strict 1/k-geodesic we can associate a unique non-smooth class of balanced points in $M^k$.
\end{lemma}
\begin{proof} By Lemma~\ref{lem1} we know that the k-tuple will be a rotating balanced point. The fact that $\gamma$ is a strict 1/k-geodesic means that it contains cut points at distance $l(\gamma)/k$.
\end{proof}

\begin{lemma}\label{relation} To any non-smooth class of balanced points in $M^k$ we can associate a unique strict 1/k-geodesic.
\end{lemma}
\begin{proof} By definition, any class of balanced points in $M^k$ comes with a unique associated 1/k-geodesic. The fact that the class of balanced points is non-smooth ensures that there will be a $t_0 \in S^1$ such that $(\gamma(t_0), \gamma(t_0 +2\pi/k), \ldots , \gamma(t_0 + 2\pi(k-1)/k) ) \in M^k$ is a non-smooth balanced point and therefore that the associated 1/k-geodesic will be strict.
\end{proof}

\begin{thm}\label{strict} Strict 1/k-geodesics in a compact Riemannian manifold have a one-to-one correspondence with non-smooth classes of balanced points of the uniform energy in $M^k$ of nonzero energy.
\end{thm}
\begin{proof} This fact follows directly from Lemma~\ref{strict_lemma} and Lemma~\ref{relation}.
\end{proof}

This theorem successfully extends the ideas of Sormani to the case where the 1/k-geodesics are not open and the balanced points are non-smooth. We can now combine Theorem~\ref{sormani3} and Theorem~\ref{strict} to obtain Theorem~\ref{combined} which gives a correspondence between classes of balanced points in $M^k$ (smooth or non-smooth) and 1/k-geodesics (open or strict).

\section{Balanced Points Under Gromov-Hausdorff Convergence}\label{balance5}

In this section we study the behavior of the uniform energy functional under Gromov-Hausdorff convergence. In \cite[Example 9.3]{Sor} Sormani showed that classes of smooth critical points of the uniform energy can disappear even under the smooth convergence of Riemannian manifolds (see Example~\ref{sphere}). In contrast, we prove Corollary~\ref{persist} which shows that classes of balanced points of the uniform energy persist under Gromov-Hausdorff convergence. We begin with an important result by Sormani concerning the persistence of 1/k-geodesics under Gromov-Hausdorff convergence.

\begin{thm}[\cite{Sor}, Theorem 7.1]\label{conv2} Let $M_i \to M$ be a sequence of compact Riemannian manifolds converging in the Gromov-Hausdorff sense. Let $\gamma_i \colon S^1 \to M_i$ be a sequence of $1/k$-geodesics. Then a subsequence of the $\gamma_i$ converge point-wise to a continuous curve $\gamma \colon S^1 \to M$, and $\gamma$ is either a $1/k$-geodesic or trivial.
\end{thm}

\begin{ex}[\cite{Sor}, Example 7.2]\normalfont \label{sphere2} This example illustrates that unlike 1/k-geodesics, closed geodesics may disappear under Gromov-Hausdorff convergence. Consider the sequence of ellipsoids $M_i$ given by $x^2 + y^2 + (z/c_i)^2 =1$. We see that the equators $\gamma_i=(\cos{t},\sin{t},0)$ of these ellipsoids are all closed geodesics. As $c_i \to 0$ the sequence $M_i$ converges in the Gromov-Hausdorff sense to a doubled disk. The equator on the doubled disk is not a locally length minimizing curve, hence not a closed geodesic, and we have demonstrated a disappearing sequence of closed geodesics under Gromov-Hausdorff convergence.

Let us view this example in light of Theorem~\ref{conv2}. We note by Proposition~\ref{all} that each equator $\gamma_i$ is a 1/k-geodesic for some integer $k$. However, as $c_i \to 0$ this integer $k$ grows without bound. The sequence of equators $\gamma_i$ is therefore not a sequence of 1/k-geodesics for any fixed integer $k$, and the example does not contradict the statement of Theorem~\ref{conv2}.
\end{ex}

\begin{ex}[\cite{Sor}, Example 9.3]\normalfont \label{sphere} This example illustrates that classes of smooth critical points of the uniform energy can disappear even under the smooth convergence of Riemannian manifolds. Consider again the ellipsoids given by $x^2 + y^2 + (z/c)^2 =1$. We note that the distance between cut points along the equators $\gamma_c$ is a continuous function of the parameter $c$. There will therefore be some $c_0$ such that $\gamma_{c_0}$ is a strict $1/3$-geodesic. Then for all $c>c_0$ we have that $\gamma_c$ is an openly $1/3$-geodesic and by Theorem~\ref{sormani3} corresponds to a class of smooth critical points of the uniform energy. Choosing a sequence $c_i \to c_0$ with $c_i > c_0$ and considering the associated sequence $M_i$ of ellipsoids converging to $M_{c_0}$ we see that the classes of smooth critical points associated to the equators disappear in the limit space $M_{c_0}$.

We note by Theorem~\ref{combined} that each of the equator $1/3$-geodesics (even that in the limit $M_{c_0}$) is associated to a class of balanced points of the uniform energy, leading us to the statement of Corollary~\ref{persist} that classes of balanced points persist in the limit.
\end{ex}

\begin{proof}[Proof of Corollary~\ref{persist}] Let $M_i$ be a sequence of compact Riemannian manifolds converging to a compact Riemannian manifold $M$ in the Gromov-Hausdorff sense. Assume that each $M_i$ admits a class of balanced points of the uniform energy $E \colon M_i^k \to \mathbb{R}$. Then by Theorem~\ref{combined} we know that each class corresponds to a $1/k$-geodesic $\gamma_i \colon S^1 \to M_i$ and by Theorem~\ref{conv2} there exists a subsequence of the $\gamma_i$ converging to $\gamma\colon S^1 \to M$, where $\gamma$ is either a $1/k$-geodesic or trivial. When $\gamma$ is nontrivial Theorem~\ref{combined} tells us that there exists a corresponding class of balanced points of the uniform energy $E \colon M^k \to \mathbb{R}$.
\end{proof}

We note that individual balanced points of the uniform energy need not persist under Gromov-Hausdorff convergence. In \cite[Example 2.2]{Sor} we see that each of the handled spheres $M_j$ admits a balanced point of the uniform energy $E \colon M^2_j \to \mathbb{R}$ which disappears on the standard sphere in the limit.

\section{Half-geodesics and Grove-Shiohama Critical Points of Distance}\label{half2}

The half-geodesics are those closed geodesics which minimize on any subinterval of length $l(\gamma)/2$. It is clear that a half-geodesic will never be an openly half-geodesic and will always be a strict half-geodesic. In Section~\ref{geo} we generalized a result of Sormani to provide a relationship between the strict 1/k-geodesics and the non-smooth balanced points of the uniform energy. We will now apply this result to study the half-geodesics via critical point methods. For an element $\bar{x} = (x_1,x_2)  \in M^2 $ we calculate the uniform energy to be
\begin{equation*}E(\bar{x}) = \sum_{i=1}^2 \frac{d(x_i , x_{i+1})^2}{1/2} = 4 \cdot d(x_1,x_2)^2
\end{equation*}
We know that the distance function is not everywhere smooth, so we recall the definition of a Grove-Shiohama critical point of the distance function and relate this notion to the balanced points of the uniform energy $E \colon M^2 \to \mathbb{R}$.

\begin{definition}\label{GSdef2} A \emph{Grove-Shiohama critical point} of $d_p \colon M \to \mathbb{R}$ is a point $q \in M$ such that for any $v \in T_qM$ there exists $\xi \in \widehat{qp}$ such that $\measuredangle {(v, \xi)}  \leq \pi/2$.
\end{definition}

The original application of this critical point definition is the celebrated Grove-Shiohama \cite{Gro} diameter sphere theorem.

\begin{thm}[Grove-Shiohama]\label{GSthm2} Let $M$ be a complete Riemannian manifold with sectional curvature $K \geq H > 0$ and diameter $> \frac{\pi}{2\sqrt{H}}$. Then $M$ is homeomorphic to the sphere.
\end{thm}

\begin{prop}\label{GS2} If $(p,q) \in M^2$ is a balanced point of the uniform energy $E \colon M^2 \to \mathbb{R}$ then we have that $q$ is a Grove-Shiohama critical point of $d_p$ and that $p$ is a Grove-Shiohama critical point of $d_q$.
\end{prop}
\begin{proof} The balancing condition ensures that there are vectors $\xi_1, \xi_2 \in \widehat{qp}$ which meet at angle $\pi$ in $T_qM$. Thus $q$ is a Grove-Shiohama critical point of $d_p$. The proof that $p$ is a Grove-Shiohama critical point of $d_q$ is equivalent.
\end{proof}

The Grove-Shiohama definition proved fruitful and sparked a comprehensive study of the critical points of the distance function. The survey by Cheeger \cite{Che} addresses applications of the Grove-Shiohama notion of critical points. Proposition~\ref{GS2} allows us to apply this critical point theory to the study of half-geodesics. We follow established methods from critical point theory in employing the theorem of Toponogov to prove Theorem~\ref{behavior}.

\begin{proof}[Proof of Theorem~\ref{behavior}.] We note that $\text{diam}(M) \geq d(\gamma(t),\gamma(t+\pi) ) >  \frac{\pi}{2\sqrt{H}}$ so that by the diameter sphere theorem $M$ is homeomorphic to a sphere.

First assume by contradiction that $\sigma \colon S^1 \to M$ is a closed geodesic with $l(\sigma)<l(\gamma)$ and $\sigma(0)=p=\gamma(t)$. Then by Theorem~\ref{combined} and Proposition~\ref{GS2} we know that $p$ is a Grove-Shiohama critical point of $d_q$ where $q=\gamma(t+\pi)$. Therefore there exists a minimum geodesic $\tau$ from $p$ to $q$ with $\alpha = \measuredangle{(\dot{\sigma}(0),\dot{\tau}(0) )} \leq \pi/2$. Let $l(\sigma)= \frac{\pi}{\sqrt{H_1}} < l(\gamma) =2 \, l(\tau)= \frac{\pi}{\sqrt{H_0}}$ so that $H_0 < H_1$ and $H_0 < H$. Set $H_2 = \min{ \{H, H_1\} }$. Although $\sigma$ is not a segment we know $l(\sigma) \leq  \frac{\pi}{\sqrt{H_2}}$ so that we can apply Toponogov's theorem to the hinge $\{\sigma, \tau \}$. This yields a comparison triangle in the sphere $S_{H_2}$ with side lengths $\{ B= \frac{\pi}{2\sqrt{H_0}} , C= \frac{\pi}{\sqrt{H_1}}, D \}$ where $B \leq D \leq \frac{\pi}{\sqrt{H_2}}$. We note that $\frac{\pi}{2} < B \sqrt{H_2} \leq D \sqrt{H_2} \leq \pi$ so that by Cosine Law we have
\begin{align*}
0 &>\cos{B\sqrt{H_2}} \geq \cos{D\sqrt{H_2}} \\
&= \cos{B\sqrt{H_2}} \cdot \cos{ C\sqrt{H_2}} +  \sin{B\sqrt{H_2}} \cdot \sin{ C\sqrt{H_2}} \cdot \cos{\alpha} \\
&\geq  \cos{B\sqrt{H_2}} \cdot \cos{ C\sqrt{H_2}}
\end{align*}
a contradiction. Thus such a closed geodesic $\sigma$ must have $l(\sigma)\geq l(\gamma)$.

Next assume by contradiction that $\sigma \colon S^1 \to M$ is a half-geodesic with  $\sigma(0)=p=\gamma(t)$ but which does not contain the point $\gamma(t+\pi)=q$. Then by Theorem~\ref{combined} and Proposition~\ref{GS2} we know that $\sigma(\pi)=x \neq q$ is a Grove-Shiohama critical point of $d_p$. Let $\gamma_2$ be a minimal geodesic from $q$ to $x$. We know there exists a minimal geodesic $\gamma_0$ from $x$ to $p$ such that $\measuredangle{(-\dot{\gamma}_2(d(q,x)),\dot{\gamma}_0(0))} \leq \pi/2$. Since $p$ and $q$ are mutually critical there exists minimal geodesics $\gamma_1$ and $\tilde{\gamma}_1$ from $p$ to $q$ such that $\measuredangle{(\dot{\gamma}_1(0),\dot{\gamma}_0(d(p,x)))} \leq \pi/2$ and $\measuredangle{(-\dot{\tilde{\gamma}}_1(d(p,q)),\dot{\gamma}_2(0))} \leq \pi/2$. Now apply the triangle version of Toponogov's theorem to both $\{\gamma_0, \gamma_1, \gamma_2 \}$ and $\{\gamma_0, \tilde{\gamma}_1, \gamma_2 \}$ yielding comparison triangles in the sphere $S_H$. Because triangles in the sphere are determined up to congruence by side lengths we get a unique comparison triangle, each of whose angles is $\leq \pi/2$. This implies that the comparison triangle is completely contained in an octant of the sphere, hence has side lengths $\leq \frac{\pi}{2\sqrt{H}}$, a contradiction to $l(\gamma_1)=l(\gamma)/2>\frac{\pi}{2\sqrt{H}}$. We conclude that the half-geodesic $\sigma$ must contain the point $\gamma(t+\pi)$ and therefore must have length $l(\gamma)$.
\end{proof}

\begin{remark}\normalfont We note that the techniques in the first part of the proof of Theorem~\ref{behavior} are inspired by Berger's proof of the minimal diameter theorem (Cf.~\cite{Pet}). The second part of the proof is similar to the Grove-Shiohama proof of the diameter sphere theorem (Cf.~\cite{Che}).
\end{remark}

\begin{ex}\normalfont \label{round} The round sphere of constant sectional curvature $H$ satisfies the assumptions of Theorem~\ref{behavior}. The only half-geodesics are the great circles which all have length $\frac{2\pi}{\sqrt{H}}$. Any half-geodesic through $p$ contains its antipode $-p$.
\end{ex}

The theorem of Bonnet-Myers states that a complete Riemannian manifold with sectional curvature $K \geq H>0$ has an upper diameter bound of $\frac{\pi}{\sqrt{H}}$ and therefore an upper bound of $\frac{2\pi}{\sqrt{H}}$ on the length of half-geodesics. Moreover, Cheng's rigidity theorem \cite[Theorem 3.1]{Cheng} states that the round sphere is the only such manifold to realize this upper diameter bound. The following example illustrates that Theorem~\ref{behavior} has applications beyond the round sphere by constructing complete Riemannian manifolds with sectional curvatures $K \geq H>0$ and admitting half-geodesics of lengths strictly between $\frac{\pi}{\sqrt{H}}$ and $\frac{2\pi}{\sqrt{H}}$. This example also demonstrates that there are non-round metrics on the sphere which admit half-geodesics of length twice the diameter.

\begin{ex}\normalfont \label{oblate} Starting with the round two-sphere of curvature $H_0 > H$ we construct an oblate ellipsoid by shrinking the sphere along the $z$-axis while keeping the equator fixed. We allow the construction to proceed until the curvature at the poles is exactly $H$. The resulting surface is a complete Riemannian manifold with sectional curvature $K \geq H>0$. In this construction the meridians remain half-geodesics and have length twice the new diameter. We note that the equator now fails to minimize between antipodal points and is no longer a half-geodesic. If $H_0$ is not too much greater than $H$ then the new meridians will have length strictly between $\frac{\pi}{\sqrt{H}}$ and $\frac{2\pi}{\sqrt{H}}$. The conclusions of Theorem~\ref{behavior} therefore apply to any closed geodesics that intersect a meridian.
\end{ex}

The next example demonstrates that the strict lower bound imposed on the length of the half-geodesics in Theorem~\ref{behavior} is sharp.

\begin{ex}\normalfont \label{proj} The standard real projective space of constant sectional curvature $H$ admits half-geodesics of length $\frac{\pi}{\sqrt{H}}$. Through any point $[p]$ on the identified equator there exists a half geodesic $\gamma$ with $\gamma(0)=[n]$ and $\gamma(\pi)=[p]$. Thus there exist multiple half-geodesics through $[n]$ which fail to meet again after time $\pi$.
\end{ex}

\begin{cor}\label{half_geo_cor} Let $M$ be a complete Riemannian manifold with sectional curvature $K \geq H >0$ and diameter $> \frac{\pi}{2\sqrt{H}}$. Let  $p,q \in M$ be points which realize the diameter. Then any half-geodesic through $p$ must contain $q$ and have length twice the diameter.
\end{cor}
\begin{proof} Berger's lemma (Cf.~\cite{CE}) says that diameter realizing points are always mutual Grove-Shiohama critical points. The proof of Theorem~\ref{behavior} shows that $q$ is the unique Grove-Shiohama critical point of $d_p$ and the corollary follows.
\end{proof}

\begin{lemma}[Gromov, Cf. \cite{Che}]\label{gromov2} Let $M$ be a complete non-compact Riemannian manifold with sectional curvature $K \geq 0$. Then for every $p \in M$ the distance function $d_p$ has no critical points outside of some ball $B(p,R_p)$. In particular, $M$ is homeomorphic to the interior of a compact manifold with boundary.
\end{lemma}

We note that in the compact setting we always have a universal upper bound on the length of the half-geodesics, namely twice the diameter. In the non-compact setting such a universal bound does not exist, but we offer the following pointwise result.

\begin{cor} Let $M$ be a complete non-compact Riemannian manifold with sectional curvature $K \geq 0$. Then for every $p \in M$ there exists a constant $R_p$ such that any half-geodesic through $p$ must have length $\leq 2 \, R_p$.
\end{cor}
\begin{proof} This follows from Proposition~\ref{GS2} and Lemma~\ref{gromov2}.
\end{proof}

As a final application of Theorem~\ref{combined} and Proposition~\ref{GS2} we give a positive result on the existence of half-geodesics on the flat two-torus.

\begin{ex}[half-geodesics on 2-torus]\label{torus}\normalfont We use our methods to show that any flat torus $T$ with fundamental domain a rectangle has exactly four half-geodesics through any point $p \in T$. Let $p$ be at the center of a fundamental domain for $T$. It is known that the midpoints of the sides together with the corners of the domain project to the three Grove-Shiohama critical points of $d_p$ \cite[Example 1.7]{Che}. By Proposition~\ref{GS2} we identify four candidate half-geodesics through $p$: the two lines through $p$ parallel to the sides and the two diagonals of the domain. By symmetry of the torus we have that each of these candidate half-geodesics corresponds to a class of balanced points of the uniform energy in $T^2$. We therefore apply Theorem~\ref{combined} to conclude that each of these lines is indeed a half-geodesic.
\end{ex}

\section{Acknowledgements}
The author would like to thank Carolyn Gordon and Craig Sutton for their guidance through all stages of the research process. The author would also like to thank Christina Sormani for suggesting the original problem and for helpful discussions and direction.

\nocite{*}
\bibliographystyle{abbrvnat}

redpajama_set_name: RedPajamaArXiv


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News	By Nick Welsh	| Wed Sep 12, 2018 | 9:52am
Sol Linver Named New County Undersheriff
He Succeeds Barney 'Yoda' Melekian as Sheriff Brown's Right-Hand Man
Sheriff Bill Brown (left) and new Undersheriff Sol Linver
After serving as Sheriff Bill Brown's unofficial Yoda for the past three-and-a-half years, Bernard Malekian has now been officially been replaced as undersheriff by Sol Linver, a 34-year veteran of the Santa Barbara County Sheriff's Office and until now the third-highest ranking member of that agency. Shortly after Brown won re-election outright in June, Malekian quietly announced he'd taken a job working for county administrator Mona Miyasto, for whom he will attempt to herd the cats of the county's often fractious public safety departments.
Linver was quick to note he has big shoes to fill. Malekian may not have walked on water with many elected officials who took a dim view of his former boss Brown, but he came close. Malekian — known simply as "Barney" in the world of county government or sometimes "Yoda" — could often make Brown's case more effectively than Brown could, and he took care never to upstage Brown in the process. Smart, understated, and a straight shooter, Malekian was enough cop even in his 60s to subdue an unruly motorist. His appointment, however, had not been without controversy.
Undersheriff Bernard Melekian
Paul Wellman (file)
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