Stare at the beautiful sight of African elephants at twilight, and witness the split second before a grizzly bear shuts his jaws on a sockeye salmon.
Or study a western lowland gorilla walk through a cloud of butterflies she has disturbed, and see a greater bulldog bat's reflection as it catches a fish.
These extraordinary images feature in a new book bringing together the work of ten of the world's greatest wildlife photographers for the first time.
The Masters of Nature Photography features the artistry of Jim Brandenburg to the underwater images by David Doubilet and Paul Nicklen.
The book also includes Pål Hermansen's use of light and dark, the viewpoints of Frans Lanting and Anup Shah's intimate portraits.
In addition there are the visual statements of Michael Nichols, Thomas Mangelsen's landscapes and Christian Ziegler's images of wild behaviour.
Each photographer has a biography along with a portfolio of the ten images that they consider is most representative of their work.
For every photographer there is a brief biography defining the essence of their art and what drives and inspires them.
The photographers are all past winners of the Wildlife Photographer of the Year competition, which is now in its 49th year.
The Masters of Nature Photography will be published by the Natural History Museum of South Kensington, central London, on September 19.
A collection of some of the best pictures from past years of the competition is featured in Wild Planet, also published by the museum.


CAN I EXPECT MORE FROM MY VOLUNTEERS?
Managers of volunteers must keep in mind that we are talking about the key practices that make up a high-performance work system, not the traditional carrot-and-stick approach.
As I look back on working with my youth group I can honestly say, "Those were amazing times!" I witnessed youth experiencing God in new ways for the first time in their lives and expressing wonder in sharing communion with our living God. They were times that included a variety of challenges that every now and then did not turn out as I thought they would. The importance of working closely with a group of youth sponsors was primary in successfully navigating youth ministry. I was blessed by this team of adults and young adults who freely gave of their time to serve as mentors and leaders within the youth ministry. These volunteers and others were incredibly valuable. Without them I would ultimately have been overwhelmed by the immense work of the ministry.
Today, as part of Mission Canada, I see the vast numbers of volunteers that allow ministry to be fulfilled in our Pentecostal churches from coast to coast. The importance of volunteers and our need to involve them on a weekly basis must cause us to look deep and discover even more ways to improve not only our retention of volunteers, but the quality of our volunteers.
Kevin H. Johnson is the lead pastor of SonLife Community Church in Dartmouth, NS. During his studies at Saint Mary's University, he wrote and submitted a research paper on "Performance Management of Volunteers in the Non-Profit Sector." Therein he conducted extensive research contrasting and comparing the effects of proven high-performance work practices that have enhanced the performance of paid workers. He then applied these same practices to the performance of volunteer workers. His findings have shown that applying these same principles to the management of volunteers would likely yield organizations that perform better.
First, the research demonstrates that effective high-performance work systems used for paid staff are also beneficial in managing volunteers. Managers of volunteers must keep in mind that we are talking about the key practices that make up a high-performance work system, not the traditional carrot-and-stick approach.
Second, research is focused on interpersonal and motivational strategies for working with volunteers. It is designed for the retention and satisfaction of the volunteer and does not necessarily improve performance. The distinction being made in this research is that high-performance work systems actually improve volunteer performance in the area of delivering skilled services. In order to deliver skilled services, volunteers need to work according to specific job requirements, which leaves less room for volunteer negligence.
Third, the research indicates there are some very specific practices that non-profit organizations can apply to improve their volunteer effectiveness. Because there are limitations and constraints within any organization's ability to adopt new systems and behaviours, we will discuss these six practices in light of some of the limitations often encountered by non-profits.
Many non-profit organizations do not have the luxury of recruiting from a pool of applicants. If your choices for volunteers are limited, then you should work harder to orient volunteers to the mission, values and beliefs of the organization during the selection process. This will assist volunteers to function in ways that are consistent with the organization's goals.
Training is essential if volunteers are going to carry out their responsibilities properly. Training sessions should explain performance expectations, build skills, and teach problem-solving techniques. Furthermore, managers and staff should be trained in how to work with volunteers. However, training can be expensive, and availability of trainers is often limited. But even on a limited budget, alternative solutions exist—such as training through job shadowing or finding a qualified volunteer whose role is to provide training for other volunteers.
In this study, recognition of volunteers was not shown to be a driver of volunteer effectiveness. While this research is not able to determine the reasons for this, it may be due to the idea that volunteers often have a high affinity with the organizations to which they give their time. Thus, while recognition is likely valued and appreciated, it may not necessarily be a factor that is responsible for improving volunteer performance. Further research is needed to shed light on this. Consequently, it would be wrong to suggest that recognizing and appreciating volunteers holds little value.
Measuring performance factors was directly correlated to high volunteer performance. Measuring ensures that important factors for success are under observation and evaluation. If the items being measured are showing signs of concern, managers can address them before performance begins to suffer in a severe manner. Having meaningful measurements also creates an expectation within the volunteer that their work is significant to the organization and important enough to be evaluated.
Volunteers are considered "involved" when they are encouraged to be part of decision-making, are treated like full members, receive information about the organization, and are involved in their own job design. Whereas volunteers expect to be treated differently and also have a greater freedom than paid workers, letting them contribute to how their work is carried out, treating them like members of the organization, and sharing information freely with them will improve performance. Volunteers tend to be more effective when they feel their contribution is as valued as paid staff.
Effective leadership is especially important for the organization facing the challenges of delivering services through volunteers. Choosing and implementing a high-performance work system is no small task, so having capable leadership is critical to seeing this accomplished effectively. Non-profit managers should study and apply the principle of transformational leadership to assist them in their work with volunteers. Research indicates that this style of leadership can be learned, so there are many benefits to the manager and the performance of the organization's volunteers if leadership skills are improved.


It is more likely to see a habu snake in a bottle of Okinawan rice liquor than to see one slithering by the wayside. Even so, habus are very common in the Ryukyu islands, of which Okinawa is a part. Okinawa is home to three species of habu; the Okinawan habu (Protobothrops flavoviridis), endemic to the Ryukyu islands, is highly venomous.
A collaborative team of scientists from many research institutes and universities across Japan has now mapped the entire genome of the Okinawan habu, including the genes involved in venom production. Professor Nori Satoh and his team of researchers from Okinawa Institute of Science and Technology Graduate University (OIST), were a part of the study, which was published in Scientific Reports. Snake venom has attracted the attention of many scientists worldwide as it has tremendous potential in the development of new drugs. The findings from this study might advance efforts to identify novel therapies for treating ailments such as cancer, cardiovascular diseases and many others.
For the study, the researchers caught Okinawan habus in the wild using a long metal pole with a hook at one end, and collected venom, blood and tissue samples. Then, using the cutting-edge DNA sequencing facilities at OIST, the researchers identified nearly 60 genes from 18 gene families that help the snake make its venom.
Nearly 50 people—most often farm workers tending crops in fields and orchards—are reportedly bitten by habus every year in Okinawa alone. The venom of the Okinawan habu is hemotoxic, destroying blood cells and tissues. Though antidotes are now available, bites can still cause severe pain and permanent tissue damage in humans. The scientists hope the results of this study will contribute to more effective antidotes and treatments for habu bites. "The findings of this study are very important, especially to the people of Okinawa and the surrounding islands where the Okinawan habu is found," says Prof. Satoh.
Snake venom is a cocktail of complex proteins. "Though the Okinawan habu is not as venomous as cobras or taipans, it can produce a large amount of venom—up to 1 ml," says Prof. Hiroki Shibata, a collaborator from the University of Kyushu. Its long, sharp fangs help discharge the venom deep inside the prey, he explains.
The genome of any organism is a treasure trove of information. Mapping venom genes of the habu not only allows scientists to explore its potential in drug development, but also gives scientists the tools to study how venom toxins have evolved, their composition, and the mechanism by which they do their damage.
The researchers explored this venom's evolutionary history by constructing a phylogenetic tree of venom genes, including those that they identified in the habu. A phylogenetic tree helps illustrate the relationship between two or more organism groups by identifying the similarities and dissimilarities between them. The distance between the groups in a tree translates to different degrees of relatedness between them. The groups that are closely placed are more related to one another than those placed further away.
Venomous snakes around the world belong to one of the two families; Viperidae or Elapidae. The members of the family Viperidae, including the habu, have hemotoxic venom, whereas the members of the family Elapidae, such as cobras, have venom that is neurotoxic. In this study, the researchers found many common components among venom genes of both families, thereby tracing back the origin of the present-day venom toxins to a common ancestor that existed millions of years ago. "The venom gene copies originated a long time ago, possibly at the early stage of vertebrate evolution," explains Prof. Shibata. The scientists believe that combined information gathered from venom research so far will help unravel the enigma behind the process of venom production in snakes.
Since they are DANGEROUS & UNNECESSARY Animals, Remove those Genes and Release them. Once a number of them are released, they will copulate and produce non-venomous animals. REPEAT THE SAME FOR EVERY VENOMOUS SNAKE.
Also, Remove Canine Teeth Genes of Tigers, DOGS and Lions and Transfer Incisor Teeth Genes of Cattle !
They won't die. They will just chew Grass instead !
I mean, manipulate their zygotes while still in womb.
Needs more than just teeth changed, their entire digestive system and the enzymes produced are made for the diets they are on now. Such 'designer' critters you are basing more on looks and superficial design rather than how the critter would actually survive the dietary changes when the whole musculature, nervous system, programmed instincts and movements, skeleton and the several million years long program that has adapted them to where they are now is a LOT more than just a superficial change of how their teeth are arranged.


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The Potbelleez are a three-member Irish-Australian electro-house and dance music group, which formed in 2003 as a duo by DJs Dave Goode (a.k.a. David Greene) and Jonny Sonic (Jonathon Murphy). In 2005 they were joined by rapper Blue MC (Marisa Lock) on vocals and in 2006 by iKid (Ilan Kidron) on vocals. In October 2007, the group issued their breakthrough single, "Don't Hold Back", which peaked at No. 5 on the ARIA Singles Chart and, in 2011, it was certified 3× Platinum by ARIA. On 22 November 2008 their debut self-titled album was released, which reached No. 17 on the ARIA Albums Chart. On 27 May 2011 they issued a second studio album, Destination Now, which peaked at the same position. It spawned Gold and Multi Platinum-accredited singles "Hello" (2010) and "From the Music" (2011).

Music career
The Potbelleez were formed in Sydney in 2003 as an electro-house and dance music group. Both founders David Greene (a.k.a. Dave Goode) and Jonathon Murphy (Jonny Sonic) were born in Dublin, Ireland. They worked as DJs and producers. By 2002 they had independently relocated to Sydney and met each other by late that year. In 2003 Brisbane-born rapper Blue MC (Marisa Lock) was working in London as one half of a beatbox duo Streetbox with MC Xander on drum machine and vocals. By 2005 The Potbelleez were working in Sydney and when Blue MC returned to Australia she joined the group. Soon after the trio begun playing live in 2006 Ilan Kidron (a.k.a. iKid) joined on vocals. During 2006 the quartet began recording tracks in a terrace house in Darlinghurst. At the end of that year The Potbelleez signed to the Vicious Records label. In April 2007 their debut extended play, In the Junkyard, was issued with "Junkyard" and "Duurty Dreemz" spawned as singles. "Junkyard" is co-written by Greene, Murphy and Lock; while "Duuurty Dreemz" was co-written by Greene, Murphy, Kidron and H Walker. The Potbelleez are managed by Myles Cooper and signed to his Phat Planet Music Media company.

In October 2007, the group issued their breakthrough single, "Don't Hold Back", which debuted at No. 36 in its first week on the ARIA Singles Chart. The track was produced by The Potbelleez with Sam Littlemore, and was cowritten by Greene, Murphy, Kidron and Littlemore. It reached No. 1 on both ARIA's Club and Dance Charts. In February 2008 it peaked at No. 5 on the Singles Chart. In 2011, it was certified 2× Platinum by ARIA, with shipment of over 140,000 copies. The single also charted in New Zealand, peaking in the top 20 on the RIANZ Singles Chart. It appeared in the top 60 on both The UK and the German Singles Chart. The music video was directed by two-time ARIA Award-winning film maker James Hackett and was shot on location in Kings Cross inside the Moulin Rouge nightclub. Also in 2008, The Potbelleez, performed in the Big Top at the Homebake festival.

Their music has been featured on TV shows including Blood, Sweat and Gears, Crunch, So You Think You Can Dance and Australia's Next Top Model. Their tracks "Junkyard" and "Duuurty Dreemz" became the unofficial theme tunes to Channel Ten's Big Brother Up-Late show. In 2007 The Potbelleez performed "Don't Hold Back" at the Big Brother Australia finale to an audience of 1.9 million people, In 2008 their single, "Are You with Me", was used to open the finale. The Potbelleez opened Channel Ten's live New Year's Eve performance from Dawes Point under the Sydney Harbour Bridge, which was broadcast to over two million viewers. In February 2008 for the A-League Grand Final they performed "Junkyard", "Are You with Me" and "Don't Hold Back" at the Sydney Football Stadium.

In March and April 2008, The Potbelleez supported the German band Scooter on their Jumping All Over the World Tour in the German cities of Leipzig, Dresden, Hamburg, Berlin and Dortmund. Most of the gigs were sold out. The group continued their Moulin Nights residency at the Kings Cross night club, Moulin Rouge. In 2008, The Potbelleez generated their first DJ mix CD for Ibiza super-club Pacha, which was released in Australia through Central Station Records. The Potbelleez later mixed for Ministry of Sound Australia's Sessions Five, which was released in June 2008 in Australia. This release was certified Platinum with sales over 100,000.

The Potbelleez' follow-up single to "Don't Hold Back" was "Are You with Me", which was released in Australia on 31 May 2008 and was the Most Added Song to Australian radio on it debut. They performed the song at many live events, including on TV show Rove Live. Their single "Trouble Trouble" was aired on the Hot 30 Countdown exclusively on 25 September 2008. When speaking to Cam St Clair of Sea FM, Kidron announced that the single would hit shelves mid-October. It hit digital stores at the same time as the album. A music video has been made for the single. The video is of The Potbelleez at a wild party. The video can be seen on YouTube and has appeared frequently on Channel V. On 1 November 2008, The Potbelleez released their self-titled debut album. Also that month, Lars Brandle of Billboard described them as one of three key Australian electronica-dance groups that were receiving international recognition – alongside Melbourne-based bands Cut Copy and Midnight Juggernauts. At the ARIA Music Awards of 2008, "Don't Hold Back" was nominated in three categories, 'Best Dance Release', 'Breakthrough Artist – Single' and 'Highest Selling Single'.

In January 2009, The Potbelleez performed songs from their album for the annual Maccabi Junior Carnival at Fox Studios, Sydney. Lead singer, Ilan Kidron, also judged the carnival's talent quest. In February that year the group performed at the Good Vibrations Festival. Also that year, their single "Junkyard" appeared as the soundtrack for the Kia Soul "Hamster" TV commercial which aired in the US and Canada.

in 2010 The Potbelleez "Don't Hold Back" was first used in TV commercial for Jeep Motor Cars Australia. Jeep "Don't Hold Back" has since become one of the most successful and longest running TV Synch campaigns in Australia currently running for over 8 years. Their song "Hello" was used as the GO! TV channel's identification theme song from late 2010 to early 2011.

In March 2011, The Potbelleez began a 14-date sold-out tour with American R&B singer Usher on his OMG Tour. The tour included sold-out shows at Burswood Dome Perth, Rod Laver Arena Melbourne, Acer Arena Sydney, Brisbane and Newcastle Entertainment Centres and was their biggest tour to date. On 27 May 2011 The Potbelleez issued their second album, Destination Now. The lead single, "Hello", hit Australian radio ahead of the album in July 2010, whilst the second single, "Shake It", appeared in October that year. The band performed both tracks live at the AFL Grand Final Footy Show, held at Rod Laver Arena, Melbourne, The NRL Footy Show Grand Final Show at the Opera House and at the Los Angeles Galaxy vs Newcastle Jets soccer match, held at EnergyAustralia Stadium, Newcastle. The singles "Hello" and "From The Music" both charted in the top 20 of the ARIA Singles Chart at No. 19 and No. 16 respectively, while the album peaked at No. 17 on the related albums chart. In 2011 the group performed at the Irish arts-and-music festival, Electric Picnic. In October that year, "Are You with Me" was used for the opening theme of The Celebrity Apprentice Australia.

In August 2012 it was announced on the band's Facebook and Twitter accounts that MC Blue had departed the band to pursue a solo project. "It's been an unforgettable experience for me in one the hardest working bands in Australia, we are all immensely proud of what we have achieved over the years and I'm now super excited to be embarking on my solo journey" Blue MC.

They released their new single "Here on Earth" via Ministry of Sound on 4 July 2014.

In September 2016, "Shout (Say That You Want Me)" was recorded and performed by The Potbelleez themselves, and was also used in ads for Kmart. Ilan explained how the track came together: "We were in Nashville at a song writing camp and we were playing around with some big vocal tracks and got talking about iconic Australian anthems and 'Shout'. It was the incredible Isley Brothers who originally wrote and recorded 'Shout', but it was Australia's first rock n roll super star Jonny O'Keefe and his version in the early '60s that went on to become an Australian anthem! We were so excited to put our own spin on this classic belter".

Side projects
The Potbelleez' lead singer, Ilan Kidron, also performs in a cinematic fuzz folk duo, Freaks in Love, with his wife and film maker, Samantha Rebillet. Kidron provided original music for the score of 2002 short film, Inja, which is directed by Steve Pasvolsky and produced by Joe Weatherstone – it was nominated for an Academy Award for Best Live Action Short in March 2003. In 2014, Kidron formed the side project Go Comet! (stylised as GO COMET!) and released the song "Worlds Apart", which was used to soundtrack Seven News stories. The song reached number 68 on the ARIA Singles Chart in August 2014.

Members
 Dave Goode (a.k.a. David Greene) – producer, engineer, programming, DJ
 Jonny Sonic (Jon Murphy) – producer, engineer, programming, DJ
 Ilan Kidron (iKid) – vocals, guitar
 (former member - Marisa Lock (Blue MC) - 2005-2012)

Discography

Studio albums

EPs

Singles

Notes

Remixes
 May 2007: "Running" on Evermore's extended play Never Let You Go.
 July 2008: "Twisted" by Brian McFadden. The Potbelleez also produced the track.

Awards and nominations

AIR Awards
The Australian Independent Record Awards (commonly known informally as AIR Awards) is an annual awards night to recognise, promote and celebrate the success of Australia's Independent Music sector.

|-
| AIR Awards of 2011
|Destination Now
| Best Independent Dance/Electronic Album
|
|-

APRA Awards
The APRA Awards are presented annually from 1982 by the Australasian Performing Right Association (APRA), "honouring composers and songwriters". The Potbelleez have won two awards from six nominations, both for Dance Work of the Year.

|-
| rowspan="3"| 2009 ||rowspan="2"| "Don't Hold Back" (David Greene, Ilan Kidron, Jonathan Murphy, Sam Littlemore) || Most Played Australian Work ||
|-
| Dance Work of the Year||
|-
| "Are You With Me" (David Greene, Ilan Kidron, Jonathan Murphy, Sam Littlemore) || Dance Work of the Year ||
|-
| 2011 || "Hello" (Ilan Kidron, Jonathan Murphy, David Greene, Justin Shave, Marisa Lock) || Dance Work of the Year ||
|-
| rowspan="2"| 2012 || rowspan="2"| "From the Music" (David Greene, Ilan Kidron, Marisa Lock, Jonathan Murphy, Justin Shave) || Dance Work of the Year ||
|-
| Most Played Australian Work ||
|-
| 2013
| "Feed Off Me"
| Dance Work of the Year
|
|-
| 2014
| "Saved in a Bottle"
| Dance Work of the Year
|
|
|-

ARIA Awards
The ARIA Music Awards are presented annually from 1987 by the Australian Recording Industry Association (ARIA). The Potbelleez have received five nominations.

|-
| rowspan="3"| 2008 || rowspan="3"| "Don't Hold Back" || Breakthrough Artist – Single ||
|-
| Best Dance Release ||
|-
| Highest Selling Single ||
|-
| 2011 || "From the Music" || Best Dance Release ||
|-
| 2013 || "Saved in a Bottle" || Best Dance Release ||
|-

Other awards

References

External links

 Collection of photographs of The Potbelleez

APRA Award winners
Australian electronic dance music groups
Irish house music groups
Irish dance musicians

redpajama_set_name: RedPajamaWikipedia


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News Vespa Scooters | Historic Models
Vespa Scooters | Historic Models
Vespa Model History
From the very first 98cc model in 1946 to the 2003 Granturismo and the Vespa S in 2007, Piaggio has produced over 144 models, versions and variants of the Vespa (marked by different chassis codes): 144 models that trace the technical evolution of the world's most famous scooter.
By the time the Vespa ET4 was launched in 1996, over 20,000 modifications had been made to the original 1946 product and over 1,500 parts replaced.
It is difficult to pick out the most representative Vespas in an evolution that has lasted over 60+ years. Some Vespas are sought after by collectors because they belong to a special series, or because they were rapidly replaced by subsequent versions, and are highly priced in the period scooter market, which is extremely active all over the world.
Others, which were produced in greater numbers or stayed on the market longer, are classic scooter models that have left their mark in the history of two-wheeled mobility.
There is no lack of authentic technical records in the Vespa's history, each of which renews the tradition of innovation that has marked the evolution of the world's best selling motor scooter.
Vespa 98, 1946 – The first Vespa. It was powered by a 98 cc engine that delivered 3.2 bhp at 4,500 rpm with a top speed of 60 km/h. It was in production for two years: in 1946 vehicles no. 1 to no. 2,464 were produced, and no. 2,465 to no. 18,079 in 1947.
Vespa 125, 1948 – The first Vespa 125 cc. It differed from the 98 not only in engine size, but also for the introduction of rear suspension; the front suspension was also modified.
Vespa 125, 1953 – This marked the first important change to the engine: bore, stroke and timing gear were modified. Power output increased to 5 bhp at 5,000 rpm, and top speed to 75 km/h. The design of the fairing at the rear was also new.
Vespa 125 U, 1953 – The "Utility" version with spartan styling, which sold at 20,000 lire less than the more modern 125. The headlamp appeared high up on the handlebar for the first time in Italy (it had already been introduced on a number of exported models).
Vespa 150 GS, 1955 – Experts called it "the most popular, imitated and remembered model". There were numerous innovations: the 150 cc engine, 4-speed gearbox, standard long saddle, "faired" handlebar-headlamp unit, wheels with 10" tyres. This Vespa could reach 100 km/h. The design also changed, with a much more aerodynamic body.
Vespa 160 GS, 1962 – This was born to continue the market success of the first GS, with a completely new design. The exhaust silencer, carburettor and suspension were also new. The power output was 8.2 bhp at 6,500 rpm.
Vespa 150 GL, 1963 – Another new design for what has been called "one of the best-looking Vespas produced by Piaggio designers". The handlebar, trapezoid headlamp, front mudguard and trimmed-down rear lids were all new.
Vespa 50, 1964 – The first Vespa 50 cc, created to exploit the new Italian Highway Code, which made a number plate obligatory on larger engines. Extremely versatile and reliable, the engine featured a new layout, with the cylinder inclined 45° instead of horizontal. It was the last design to leave Corradino D'Ascanio's drawing board.
Vespa 180 SS, 1965 – It marked a new milestone in the growth of the engine (181.14 cc), with 10 bhp for a top speed of 105 km/h. The 180 SS (Super Sport) replaced the glorious GS 150/160 cc. Piaggio modified the front cowling, making it more aerodynamic and significantly improving comfort, handling and roadholding.
Vespa 125, 1966 – Unofficially known as the "new 125", it featured radical innovations in the design, frame, engine (inclined 45°) and suspension.
Vespa Super Sprint 90, 1966 – A special series derived from the Vespa 50/90 cc and the "new" 125, the hold-all was positioned between the saddle and the handlebar for a more "laid-back" riding style. The handlebar was narrow and low, and the mudguard and cowling were streamlined. With an engine capacity of only 90 cc, it could do 93 km/h.
Vespa 125 Primavera, 1968 – Together with the subsequent PX version, it was the most durable version of the Vespa. It derived from the "new" 125, but with considerable differences in the engine, which raised the top speed by 10 km/h. Great attention was paid to details, which included the classic, practical bag hook.
Vespa 180 Rally, 1968 – With this new vehicle, Piaggio extended the rotary timing fuel feed system to its entire production. The engine was new, the front headlamp new and more powerful, the frame, derived from the Vespa 150 Sprint, narrower and more aerodynamic than that of the Super Sport.
Vespa 50 Elestart, 1970 – It featured the great novelty of electric ignition, but the design was also completely revised and embellished compared to the 50 Special.
Vespa 200 Rally, 1972 – The Vespa with the largest engine. This model, with 12.35 bhp at 5,700 rpm, could reach 116 km/h.
Vespa 125 Primavera ET3, 1976 – The name stood for "Electronic 3 intake ports", and included important changes to the engine, which had more power and sparkle. Even the styling was changed from the standard Primavera (which remained in the range).
Vespa P 125 X, 1978 – The "PX" marked a new step forward in styling (the bodywork was completely redesigned) and performance. The hold-all was positioned behind the cowling. The same year the P 200 E also appeared, which could be equipped with separate lubrication and direction indicators incorporated in the body. Three years later the PX 150 E was launched, with performance halfway between the two models.
Vespa PK 125, 1983 – This replaced the Vespa Primavera (standard and ET3), which remained in production with the "Classic" body for the Japanese market, where it was the best-selling Western two-wheeler vehicle. The styling was new, and the PK body was completely different from that of previous scooters, because the welds of the body no longer overlapped but were integral.
Vespa PK 50, 1983 – Substantially identical to the PK 125, it appeared in two models, PK 50 and PK 50 S, both with 4-speed gearbox and electronic ignition.
Vespa PK 125 Automatic, 1984 – An automatic transmission was introduced on the Vespa, probably the most radical change (at least for the driver) since 1946. The presence of the automatic transmission was emphasised by the absence of the brake pedal, which was replaced by a lever on the left handlebar (which did not have to control the clutch as that was automatic). It was also available with automatic oil-petrol mixer and electric ignition. The following year the Vespa PK 50 Automatic was launched.
Vespa T 5 Pole Position, 1985 – The T 5 was the "extra-sporty" version of the PX series. With a new engine, aluminium cylinder and 5 intake ports, but the design was also new, particularly at the rear and around the front headlamp which incorporated an aggressive dome with a small Plexiglas windscreen. A spoiler was added on the cowling.
Vespa 50 N, 1989 – The changes to the Italian Highway Code meant that 50 cc vehicles were no longer bound by the 1.5 bhp limit, and Piaggio presented a new small Vespa with improved performance (over 2 bhp at 5,000 rpm), and new, smoother styling. A "Speedmatic" automatic version was also launched.
Vespa ET4 125cc, 1996 – The "new generation Vespa" launched on the 50th anniversary. A completely new project, it is the first Vespa ever powered by a 4-stroke engine. The Vespa ET is equipped with a front disk brake and an automatic CVT gearbox.
Vespa ET2 50cc, 1997 – Same as the ET4 125, but with a 50cc 2-stroke catalysed engine.
Vespa ET4 150cc, 1999 – First Piaggio scooter equipped with the new generation 4-stroke Leader engine, now on the 125cc model too.
Vespa ET4 50cc, 2000 – The first small Vespa with a 4-stroke engine, combining lively performance that will make no one regret the 2-stroke with quiet running and the reduction of polluting emissions. Fuel economy is outstanding: the Vespa ET4 50 has the highest range in the 50 cc class, with approx. 500 km on a full tank.
Vespa PX, 2001 – Classic design and unique features such as a four-speed gearbox have made the Vespa PX a cult scooter, a symbol of Italian style everywhere in the world. The 2-stroke 125, 150 and 200cc engines (displacements vary according to markets) with forced air cooling have electronic CDI ignition and electric start with a kick starter. The new PX now sports a powerful stainless steel front disc brake, 200 mm in diameter, guaranteeing prompt, safe and efficient braking. A reliable 150 mm rear drum modulates braking.
Vespa Granturismo 200L and 125L, 2003 – In 2003, the Granturismo made its appearance as the most powerful Vespa ever produced. In 200L and 125L versions, it combines the Vespa's emotional appeal with state-of-the-art technology: this was the first-ever Vespa to have sparkling four-stroke, four-valve, liquid-cooled engines that meet the new Euro2 emissions standards, as well as 12-inch wheels on the 200L and a two-disk brake system. The steel body is a uniquely Vespa touch.
Vespa LX, 2005 – Launched in Rome on 10 May, 2005, the LX is the 139th Vespa in almost 60 years. It is the sublime heir of a truly unique legacy, a designer scooter for those who want a stylish, avant-garde scooter for town use. A "compact" Vespa, the LX replaces the glorious Vespa ET (over 460,000 units sold since 1996). It is available in four modern, environment-friendly displacements: 50cc two and four strokes as well as a 125 and a 150cc four stroke.
Vespa GTS 250 i.e. – Fifty years after the launch of the Vespa GS (Gran Sport), the first sport scooter in history and still a sought-after treasure for collectors and fans, Vespa GTS 250 i.e. – launched on 25 May 2005 in Portofino – renews the GS blend of speed and style to become the fastest, most powerful and most high-tech Vespa in history.
With an avant-grade, extremely powerful 250cc four-stroke, four-valve electronic injection engine and two disc brakes with an optional ABS and brake servo, the Vespa GTS 250 i.e. was one of the first two-wheelers and the first 250cc to already meet the strict upcoming Euro 3 emissions limits.
Vespa GTS 125 – As of 2007, the Vespa GTS is also available in a 125 cc version. A perfect combination of elegance and performance, the Vespa GTS 125 offers all the class, exclusivity and high technology of the bigger GT scooter, but with a 125 cc engine.
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In a bid to raise awareness and fundraise for EAZA's 2011 Ape campaign Dublin Zoo and 2wheels cycles have teamed up to lead a 'Charity Cycle for Apes'. Spread over the 1st & 2nd of October, participants will travel 190km from Belfast Zoo to Dublin Zoo using only their hind legs and a two wheeled machine!
Fota wishes to show its support for the Ape Campaign and raise funds for our own contribution to this important cause. We have nominated Barry to take part as he put up the least resistance and his gangly features make him the most similar in appearance to a gibbon. He had this to say, 'A lot of the staff have been saying I don't have the legs so I'm going to try my hardest to get that monkey off my back. I want to emulate my hero Lance Apestrong'. Good luck to Barry. Visitors can now contribute sponsorship through reception and our gift shop and over the next number of weeks.
Throughout 2011 EAZA have been running the ape campaign. The six species of great apes and the sixteen species of gibbons are under threat from hunting, deforestation and disease. EAZA, the European Association of Zoos and Aquaria, represents and links over 320 zoological institutions across 35 nations. The purpose of such a network facilitator is to harbour co-operation within the European wildlife conservation community, and to promote education, research and progressive conservation.
Since the turn of the millennium EAZA has been involved in annual conservation campaigns that have addressed issues such as the bush-meat trade, population losses in iconic animals such as the big cats with the last campaign focussing on carnivore loss Europe wide. All of the apes are threatened and almost all are either endangered or critically endangered which emphasizes the need to improve upon our efforts at preserving them. The funds raised by the ape campaign will prove vital in a number of specific conservation projects in the coming years.


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What occupational health services do we offer?
Contact us at Union Quay Medical Centre to discuss the range of occupational health services available to your organisation, and to arrange an appointment with one of our specialists.
Please note:with complex assesment if the time gose over an hour that an hourly consultant rate is then applied.
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text:
\section{Introduction}

Entanglement is a primary feature of quantum mechanics lacking in classical mechanics.
It can be used for building revolutionary new technologies (e.g., quantum computer) and information processing.
One way of measuring this quantity is by computing entanglement entropy (EE).
The study of quantum many-body systems (QMBSs) is the study of the quantum properties of matter and quantum resources.
In this regard quantum many-body systems described by local Hamiltonians gain importance.
Restricting ourselves to $1D$ systems we know that gapped systems obey the area law \cite{gap1, gap2}.
The area law says that EE between two subsystems of a system is proportional to the area of the boundary between them. This is expected given the local nature of the interactions.
It is widely believed that for ``physically reasonable" quantum systems, the area law can be violated by at most a logarithm $\log(N)$ factor\cite{Swingle}, with $N$ the size of the quantum system.
This belief has been seriously challenged by both quantum information and condensed matter theorists in recent years.
It is of interest to find local Hamiltonians that exhibit more quantum fluctuations with their EE scaling as either the logarithm or the volume of the system size \cite{ira, Bravyi2012}.

More recently, a spin-$1$ local, frustration-free Hamiltonian called the Motzkin spin chain \cite{Bravyi2012,shor,Dell'Anna} (Its half-integer analog is called the Fredkin spin chain \cite{fredkin}
was shown to have unique ground states.
The model is constructed based on random walks, called the Motzkin walks.
Each Motzkin walk (path) can be mapped to a wave function of spins.
The Hamiltonian of the Motzkin chain is nearest neighbors with projectors as interactions.
It has unique ground state that can be seen as a uniform superposition of Motzkin walks. As the volume of the system $N\rightarrow\infty$, the random Motzkin walks converge to Brownian excursions \cite{Durrett}.
The EE of the system is proportional to $\sqrt{N}$.
It is exponentially more than the logarithm.

Such a spin chain can in principle be experimentally realized. The large amount of entanglement may be used as a
resource for quantum technologies and computation.
The model has since then been deformed to show phase transitions \cite{i1, i2}, generalized to possess new features using symmetric inverse semigroups \cite{p1, p2} and simplified to preserve translational invariance in \cite{caha}.
The ground state of the Motzkin chain can be analytically expressed as a matrix product state of the Motzkin walks using techniques from enumerative combinatorics \cite{enum}.
While there is no analytical description of the excited states, the model is not exactly solvable.

The full Hamiltonian of the uncolored
Motzkin chain was described by Bravyi et al. \cite{Bravyi2012}.
It is a spin chain with a 3-dimensional local Hilbert space given by a basis $\ket{u}$, $\ket{f}$, $\ket{d}$. Here $u$, $f$, and $d$ are used to abbreviate ``up'', ``flat'', and ``down'' of the Motzkin walks respectively.
This identification comes from that the states in this system can be mapped to paths (Motzkin walks) in the ``x-y" plane as done in \cite{shor}.
The Hamiltonian of the Motzkin spin chain with periodic boundary conditions is given by (the coupling constant $g$ was equal to $1$ in \cite{shor}):
\begin{equation}
 H_{M} = \sum_{j} H_{free,j} + g H_{int,j}
\end{equation}

The local Hamiltonians $H_{free}$ and $H_{int}$ are given in terms of two-site projectors,
\begin{eqnarray}
&&H_{free,j}=(\ket{u_jf_{j+1}}-\ket{f_ju_{j+1}}) (\bra{u_jf_{j+1}}-\bra{f_ju_{j+1}})\nonumber\\
&&~~+(\ket{d_jf_{j+1}} -\ket{f_jd_{j+1}})(\bra{d_jf_{j+1}}- \bra{f_jd_{j+1}})
\end{eqnarray}
and
\begin{equation}
H_{int,j} =(\ket{u_jd_{j+1}} - \ket{f_jf_{j+1}}) (\bra{u_jd_{j+1}} - \bra{f_jf_{j+1}}).
\end{equation}
The term $H_{int}$ can be viewed as a ``kinetic part" of the Motzkin model.
By turning on $g$, the system loses its symmetries and starts to ``interact".
Only the ground state of the chain can be analytically described by the Motzkin paths. The model is not exactly solvable.
Thus we will call the case with $g=0$ the non-interacting Motzkin chain or the free Motzkin spin chain.
While it is still strongly interacting, it is indeed integrable in the sense of quantum inverse scattering method \cite{Faddeev95,Korepin93} (or the algebraic Bethe Ansatz method (ABA)), as expanded on below. This fact allows us to solve the spectrum of the excited states completely.

In this paper we will analyze the case of the free Motzkin spin chain ($H_{FM}=H_{M}|_{g=0}$).
We observe that the resulting system is integrable. Its structure can be described (generated) by Yang-Baxter equation (\ref{Yang-Baxter-equation}). Then we will calculate its spectrum for periodic boundary conditions.
The model can be seen as a generalized Temperley-Lieb system.
This is understood from the local Hamiltonian, which is a representation of the generator of Temperley-Lieb algebra.
Due to the lack of commutation relations among the monodromy-matrix operators(\ref{T-matrix}), it is hard to construct the Bethe state and further steps of ABA.
We generalize Sklyanin's functional Bethe Ansatz \cite{Sklyanin92,Sklyanin95} and some ideas of Off-diagonal Bethe Ansatz \cite{Wang15} to approach the spectrum problem of the model. The latter method developed some techniques in computing the relations of products of transfer matrices. It is very useful in our current work.

To readers familiar with other similar models but not with the Bethe Ansatz method, the central building block is the $R$-matrix. It essentially describes the scattering between two quasiparticles in the spin chain. The $R$-matrix satisfies the Yang-Baxter equation, which states that many-body scattering can be reduced to two-body scattering. This fact ensures the integrability of the periodic free Motzkin chain.

The $R$-matrix $R_{12}(\lambda)=P_{12}[(\lambda+\eta)\mathbbold{1}-\lambda\hat{e}_{12}]$ for the free Motzkin spin chain can be written as
\begin{equation}
\label{R-matrix-Motzkin}
\left(
\begin{array}{ccc|ccc|ccc}
\lambda+\eta &    &    &    &    &    &    &    &   \\
      &\lambda&    &\eta&    &    &    &    &   \\
      &    &    &    &    &    &\lambda+\eta&    &   \\
\hline
      &\eta&    &\lambda&    &    &    &    &   \\
      &    &    &    &\lambda+\eta&    &    &    &   \\
      &    &    &    &    &\lambda&    &\eta&   \\
\hline
      &    &\lambda+\eta&    &    &    &    &    &    \\
      &    &    &    &    &\eta&    &\lambda&    \\
      &    &    &    &    &    &    &    &\lambda+\eta\\
\end{array}
\right).
\end{equation}
Here $\lambda$ is the spectral parameter. $\eta$ is the crossing parameter. $P_{12}$ is the permutation operator, $\hat{e}_{12}$ is the corresponding generator \cite{PeriodicMotzkin} of the Temperley-Lieb algebra, and ${1\over2}\hat{e}_{12}$ is a projection operator.
\begin{equation}
\hat{e}_{12}=\hat{U}_{12}+\hat{D}_{12}.
\end{equation}
Unless otherwise specified, we use the standard notation: for any matrix $A\in {\rm End}({\rm\bf V})$, $A_j$ is an embedding operator in the tensor space ${\rm\bf V}\otimes
{\rm\bf V}\otimes\cdots$. It acts as $A$ on the $j$-th space and as an identity on the other factor spaces. So $R_{ij}(\lambda)$ acts as an identity on the factor spaces except for the $i$-th and $j$-th ones.
We denote the basis states by $\{\left|{u}\right>, \ket{f}, \ket{d}\}$ where $u$, $f$ and $d$ are used to abbreviate ``up'', ``flat'' and ``down'' respectively,
\begin{equation}
\ket{u}=\left(\begin{array}{c}
    1 \\
    0 \\
    0 \\
  \end{array}\right),\quad
\ket{f}=\left(\begin{array}{c}
    0 \\
    1 \\
    0 \\
  \end{array}\right),\quad
\ket{d}=\left(\begin{array}{c}
    0 \\
    0 \\
    1 \\
  \end{array}\right).
\end{equation}
The operators $\hat{U}_{j,j+1}$ and $\hat{D}_{j,j+1}$ are projectors to the states
\begin{equation}
\ket{u_j,f_{j+1}}\hspace{-0.5mm}-\hspace{-0.5mm}\ket{f_j,u_{j+1}},\;\text{and}\;\,\ket{d_j,f_{j+1}}\hspace{-0.5mm}- \hspace{-0.5mm}\ket{f_j,d_{j+1}}
\end{equation}
respectively.
The $R$-matrix satisfies the quantum Yang-Baxter equation (QYBE)
\begin{equation}
R_{12}(\lambda)R_{13}(\lambda+\nu)R_{23}(\nu)=R_{23}(\nu)R_{13}(\lambda+\nu)R_{12}(\lambda).
\label{Yang-Baxter-equation}\end{equation}
and possesses the following properties:
\begin{eqnarray}
&\mbox{Initial condition:}~~~~~~ R_{12}(0)= \eta P_{12},~~~~~\label{Initial-R}\\
&\mbox{Unitarity:}~  R_{12}(\lambda)R_{21}(-\lambda)= (\eta+\lambda)(\eta-\lambda)\,\mathbbold{1},\label{Unitarity}\\
&\mbox{Projection:}~~~  R_{12}(-\eta)=\eta P_{12}\hat{e}_{12}=- \eta\hat{e}_{12}.\label{Projection}
\end{eqnarray}
The partial transpose of this $R$-matrix is degenerate
\begin{equation}
\det(R^{t_1}_{12}(\lambda))=0.
\label{partial-trans}
\end{equation}
Therefore it does not satisfy crossing unitarity.
The monodromy-matrix $T(\lambda)$ can be constructed as
\begin{equation}
T(\lambda)=R_{0N}(\lambda-\theta_N)R_{0\,N-1}(\lambda-\theta_{N-1})\cdots R_{01}(\lambda-\theta_1),
\label{Monodromy-T}\end{equation}
where the index $0$ indicates the auxiliary space and the indices $\{1, \cdots, N\}$ denote
the physical or quantum spaces. $N$ is the number of sites and $\{\theta_j|j=1, \cdots, N\}$ are the inhomogeneous parameters.

Then the monodromy-matrix can be written as
\begin{equation}
\label{T-matrix}
T_0(\lambda)=
\left(
\begin{array}{ccc}
A(\lambda) & B_1(\lambda) & B_2(\lambda)\\
C_1(\lambda)&D_{11}(\lambda)&D_{12}(\lambda)\\
C_2(\lambda)&D_{21}(\lambda)&D_{22}(\lambda)\\
\end{array}
\right).
\end{equation}
Here the subscripts of $B$, $C$, $D$ operators indicate their position in auxiliary space $0$.
The transfer matrix $t(\lambda)$ is defined as the partial trace of monodromy matrix in the auxiliary
space
\begin{equation}
t(\lambda) = tr_0\{T_0(\lambda)\}=A(\lambda)+D_{11}(\lambda)+D_{22}(\lambda).
\label{trans}\end{equation}
Let us further introduce the vacuum state $|0\rangle$,
\begin{equation}
|0\rangle=|u_1\rangle,\cdots,|u_N\rangle=\left(\begin{array}{c}
    1 \\
    0 \\
    0 \\
  \end{array}\right)_{1} \otimes\cdots\otimes\left(\begin{array}{c}
    1 \\
    0 \\
    0 \\
  \end{array}\right)_{N} .
\end{equation}
The operators acting on the vacuum state give rise to
\begin{eqnarray}
&&A(\lambda)|0\rangle=a(\lambda)|0\rangle=\prod^N_{j=1}(\lambda-\theta_j+\eta)|0\rangle,\nonumber\\ &&D_{11}(\lambda)|0\rangle=d(\lambda)|0\rangle=\prod^N_{j=1}(\lambda-\theta_j)|0\rangle,\nonumber\\ &&D_{22}(\lambda)|0\rangle=0\quad,C_k(\lambda)\,|0\rangle=0,\quad B_k(\lambda)\,|0\rangle\neq 0.
\label{A-D-eigen}\end{eqnarray}
The QYBE (\ref{Yang-Baxter-equation}) implies that the monodromy-matrix $T(\lambda)$ satisfies the Yang-Baxter relation
\begin{equation}
R_{12}(\lambda-\nu)T_{1}(\lambda)T_{2}(\nu)=T_{2}(\nu)T_{1}(\lambda)R_{12}(\lambda-\nu).
\end{equation}
Therefore, one can prove that the transfer matrices with different
spectral parameters commute with each other
\begin{eqnarray}
[t(\lambda),t(\nu)]=0.
\label{trans-com}
\end{eqnarray}
So $t(\lambda)$ serves as the generating functional of the conserved quantities of the system.
This ensures the integrability of the periodic free Motzkin spin chain.

The Hamiltonian of periodic free Motzkin spin chain can be recovered from the transfer matrix through a logarithmic trace identity, such that
\begin{eqnarray}
H_{FM}=-\eta\;\frac{\partial\ln t(\lambda)}{\partial \lambda}\bigg{|}_{\lambda=0,\{\theta_j=0\}}+N=\sum^{N}_{j=1}H_{j,j+1}.
\label{Hamiltonian}\end{eqnarray}
Here the local Hamiltonian is
\begin{eqnarray}
H_{j,j+1}=H_{free,j}=\hat{e}_{j,j+1}=\hat{U}_{j,j+1}+\hat{D}_{j,j+1}\;,
\end{eqnarray}
with the periodic boundary conditions
$H_{N,N+1}=H_{N,1}$.

For the current free Motzkin chain model, it is difficult to implement the standard ABA method. Let us go along with the concept of functional BA \cite{Sklyanin92,Sklyanin95} and consider the product of the transfer matrices at two special points.
To do this, we follow the method in \cite{Cao13} and take advantage of the symmetry and properties (\ref{Initial-R}), (\ref{Unitarity}) and (\ref{Projection}) of the $R$-matrix. After a long calculation, we
find that the transfer matrix (\ref{trans}) satisfies the following operator identities
\begin{eqnarray}
t(\theta_j)t(\theta_j-\eta)=\hat{O}a(\theta_j)d(\theta_j-\eta),\;\, j=1,\cdots,N.~
\label{Operator-identity}\end{eqnarray}
Other than the identity operator in (\ref{trans-id-XXX}) for the spin-$1\over2$ case, here the operator $\hat{O}$ is a special operator.
Thus we can not simply convert the operator identities to the functional ones like in the XXX spin-$1\over2$ and all other cases. The final results show that $\hat{O}$ is a global operator acting highly non-trivial on all quantum spaces. One can not derive the functional relation by projecting the transfer matrix to its subspace. The fusion techniques do not work here.

So for the current model, the key point is to understand the structure of the $\hat{O}$ operator.
The form of operator $\hat{O}$ consists of the leading terms of operators $A(\lambda)$, $D_{11}(\lambda)$ and $D_{22}(\lambda)$ as polynomials of variable $\lambda$. It can be expressed as follows
\begin{eqnarray}
\hat{O}&=&{\displaystyle \lim_{\lambda \to \infty}{1\over\lambda^{2N}}(A(\lambda)+D_{22}(\lambda))D_{11}(\lambda)}\nonumber\\
&=&{\displaystyle \lim_{\lambda \to \infty}{1\over \lambda^N}(A(\lambda)+D_{22}(\lambda))}.
\label{O-operator}\end{eqnarray}
The second step of the formula above results from the leading terms of operator $D_{11}(\lambda)$ is proportional to the identity operator.

Moreover, the asymptotic behavior of the transfer matrix is
\begin{eqnarray}
\lim_{\lambda\to\infty}t(\lambda)&=&\lim_{\lambda\to\infty}A(\lambda)+D_{11}(\lambda)+D_{22}(\lambda)\nonumber\\
&=&\lambda^N(\hat{O}+\mathbbold{1})+\cdots
\label{trans-asymptotic}\end{eqnarray}
The operator identities (\ref{Operator-identity}) of the transfer matrix together with the asymptotic behavior (\ref{trans-asymptotic}) completely determine the spectrum of the transfer matrix.

The detailed analysis of the structure of operator $\hat{O}$ (also operators $A(\lambda)$ and $D_{22}(\lambda)$) and its eigenvalues are given in Supplemental Material.

The commutativity of the transfer matrices (\ref{trans-com}) means that they have common eigenstates. Let us assume that $|\Psi\rangle$ is an eigenstate of $t(\lambda)$. $|\Psi\rangle$ does not depend upon $\lambda$, and its corresponding eigenvalue is $\Lambda(\lambda)$,
\begin{eqnarray}
t(\lambda)|\Psi\rangle=\Lambda(\lambda)|\Psi\rangle.
\label{Lambda}\end{eqnarray}
The operator identities (\ref{Operator-identity}) imply that the corresponding eigenvalue $\Lambda(\lambda)$ satisfies the relation
\begin{eqnarray}
&&\Lambda(\theta_j)\Lambda(\theta_j-\eta)|\Psi\rangle=\hat{O}a(\theta_j)d(\theta_j-\eta)|\Psi\rangle\nonumber\\
&&~~=\{\omega\} a(\theta_j)d(\theta_j-\eta)|\Psi\rangle,\quad j=1,\cdots,N.
\end{eqnarray}
Thus we have
\begin{eqnarray}
\Lambda(\theta_j)\Lambda(\theta_j-\eta)
\hspace{-0.5mm}=\hspace{-0.5mm}\{\omega\} a(\theta_j)d(\theta_j-\eta),\;\, j=1,\hspace{-0.5mm}\cdots\hspace{-0.5mm},N.~~
\label{Lambda-identity}\end{eqnarray}
Here $\{\omega\}$ represents the eigenvalues of operator $\hat{O}$, which have $3^N$ numbers, such as
\begin{eqnarray}
\{2,1,\cdots,-1,\cdots,i,-i,\exp({2\pi i\over3}),\exp({4\pi i\over3}),\cdots,\nonumber\\
\exp({2\pi i\over N}),\exp({4\pi i\over N}),\cdots\}.~~~~~
\label{eigen-O}\end{eqnarray}
The eigenvalues $\{\omega\}$ of operator $\hat{O}$ includes two parts: The diagonal elements result in one single $2$ (the center element $2\times E^{M,M}$) and several $1$ (with degeneracy of $2^{N+1}-2$); The non-diagonal elements give rise to roots of unity, ranging from square roots to $N$-th roots.

In general, the distribution of $k$-th roots of unity as eigenvalues of operator $\hat{O}$ for a lattice of length $N$ is given by:
\begin{eqnarray}
f(N,k) = \left( \sum_{d|k} 2^d \mu(\frac{k}{d}) \right) \left[ \sum_{m = 1}^{\floor{\frac{N}{k}}} C(N,km)
\right]
\label{Formula_Mobius}\end{eqnarray}
where $\mu(n)$ refers to the M\"obius function in number theory. For $k=1$ this formula gives the correct diagonal result if one excludes the eigenvalue $2$. The $1$ with degeneracy of $2^{N+1}-2$ from the diagonal elements can be regarded as the $1$-st roots of unity.
In every specific $k$-th roots case, each kind of roots $\exp({2\pi i\over k})$, $\exp({4\pi i\over k})$,$\cdots$, $1$ has the same degeneracy. See the Supplemental Material for details.

According to the asymptotic behavior of the transfer matrix (\ref{trans-asymptotic}), we have that the leading terms of $\Lambda(\lambda)$ have the following values,
\begin{eqnarray}
&&\lambda^N\hspace{-0.5mm}\times\hspace{-0.5mm}\{3,2,\cdots,0,\cdots,1+i,1-i,1+\exp({2\pi i\over3}),\nonumber\\
&&1+\exp({4\pi i\over3}),\hspace{-0.5mm}\cdots\hspace{-0.5mm},1+\exp({2\pi i\over N}),1+\exp({4\pi i\over N}),\hspace{-0.5mm}\cdots\hspace{-0.5mm}\}.~~~
\label{Lambda-asymptotic}\end{eqnarray}
They are $\lambda^N$ multiplied by each eigenvalue of operator $\hat{O}$ plus $1$, respectively. They have a one-to-one correspondence with the eigenvalues of operator $\hat{O}$ (\ref{eigen-O}).

$\Lambda(\lambda)$, as a function of $\lambda$, is a polynomial of degree $N$.
The functional identities (\ref{Lambda-identity}) together with this asymptotic behavior (\ref{Lambda-asymptotic}), these $N+1$ conditions are sufficient to construct the $T-Q$ relation for $\Lambda(\lambda)$.

To explain the situation in the free Motzkin chain, we present a quick review of how to apply the functional BA approach to the XXX spin-$1\over2$ chain in Supplemental Material.
We propose a generalized $T-Q$ relation to describe the eigenvalues of the transfer matrix(\ref{trans}). This $T-Q$ relation together with the newly appeared $\hat{O}$ operator (\ref{O-operator}) in the current model allows us to calculate all spectrum with their degeneracies of the Hamiltonian.

Based on the properties (\ref{Lambda-identity}), (\ref{Lambda-asymptotic}), and also the configuration(the coefficients are calculated in Supplemental Material (\ref{coefficient-solution}))
of the coefficient $\omega$, let us propose the following conjecture for $\Lambda(\lambda)$,
\begin{eqnarray}
\Lambda(\lambda)=\omega\;a(\lambda)\frac{Q(\lambda-\eta)}{Q(\lambda)}+d(\lambda)\frac{Q(\lambda+\eta)}{Q(\lambda)}.
\label{T-Q}\end{eqnarray}
Here $\omega$ will go through each value in the set $\{\omega\}$. The function $Q(\lambda)$ is parameterized by $N$ Bethe roots $\{\lambda_j|j=1,\cdots,N\}$ as follows:
\begin{eqnarray}
Q(\lambda)=\prod^{N}_{j=1}(\lambda-\lambda_j).
\end{eqnarray}
To ensure $\Lambda(\lambda)$ is a polynomial, the residues of $\Lambda(\lambda)$ at the
poles $\lambda_j$ must vanish. So the $N$ Bethe roots satisfy the Bethe equations
\begin{eqnarray}
\omega\prod^N_{k=1}\frac{\lambda_j-\theta_k+\eta}{\lambda_j-\theta_k}
=\prod^N_{l\neq j\atop{l=1}}\frac{\lambda_j-\lambda_l+\eta}{\lambda_j-\lambda_l-\eta},\; j=1,\hspace{-0.5mm}\cdots\hspace{-0.5mm},N.~~
\label{BAEs}\end{eqnarray}

Then the energy spectrum of the Hamiltonian (\ref{Hamiltonian}) can be given in terms of the associated Bethe roots by
\begin{eqnarray}
E&=&-\eta\frac{\partial\ln\Lambda(\lambda)}{\partial \lambda}\bigg{|}_{\lambda=0,\{\theta_j=0\}}+N\nonumber\\
&=&-\sum^N_{j=1}\frac{\eta^2}{\lambda_j(\lambda_j+\eta)}
\end{eqnarray}

\section{Conclusions}
We studied the free Motzkin chain with periodic boundary conditions. Based on  intrinsic properties of the $R$-matrix, we derive the functional relations (\ref{Lambda-identity}) of the transfer matrix. These relations, together with the asymptotic behavior (\ref{Lambda-asymptotic}), allow us to construct a generalized $T-Q$ relation (\ref{T-Q}) for the eigenvalues of the transfer matrix and the associated Bethe equations (\ref{BAEs}).
The new parameter $\{\omega\}$ in the $T-Q$ relation is related to roots of unity. Its distribution can be described by the M\"obius function of the  number theory. Some patterns in number theory can be observed from the distribution ((\ref{Formula_Mobius}), see the TABLE \ref{tab}).
For example, when the lattice site $N$ is a prime number, the total number of square roots and the total number of the highest $N$-th roots are both $2^N-2$.  The $2^N-2$ is divisible by $N$, for any prime $N$.
The results allow one to study the thermodynamic properties. While now the Bethe equations have new parameters $\{\omega\}$, they have different values in different sub-spaces. One can not directly apply the Yang-Yang thermodynamic to this model.
Our method may apply to other  Temperley-Lieb  models with crossing-unitarity breaking.

\section*{Acknowledgments}
KH would like to thank Profs. Yupeng Wang, Wen-Li Yang, Guang-Liang Li, and Junpeng Cao for the helpful discussions.
The work of KH was supported by the National Natural Science Foundation
of China (Grant Nos. 11805152, 12047502 and 11947301), the Natural Science Basic Research Program of Shaanxi Province Grant No. 2019JQ-107, and Shaanxi Key Laboratory for Theoretical Physics Frontiers in China.
VK was supported by the U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers, Co-design Center for Quantum Advantage (C2QA) under contract number DE-SC0012704, and the SUNY center for QIS at Long Island
project number CSP181035.
The authors would like to thank the Simons Center for Geometry and Physics for hospitality.
This work was started during the workshop
``Entanglement and Dynamical Systems: December 10-14, 2018" held at the center.

\section{Supplemental Material}
In this section we give some basic notations.
We briefly review the functional Bethe Ansatz for XXX spin-$1\over2$ chain to illustrate our method.

The $\hat{U}_{j,j+1}$ and $\hat{D}_{j,j+1}$ operators can be expressed in terms of the states,
\begin{eqnarray}
&&\hat{U}_{j,j+1} = \ket{u_j,~f_{j+1}}\bra{u_j,~f_{j+1}} - \ket{u_j,~f_{j+1}}\bra{f_j,~u_{j+1}} \nonumber\\
&&~~- \ket{f_j,~u_{j+1}}\bra{u_j,~f_{j+1}} + \ket{f_j,~u_{j+1}}\bra{f_j,~u_{j+1}},
\end{eqnarray}and
\begin{eqnarray}
&&\hat{D}_{j,j+1} =  \ket{d_j,~f_{j+1}}\bra{d_j,~f_{j+1}} - \ket{d_j,~f_{j+1}}\bra{f_j,~d_{j+1}} \nonumber\\
&&~~- \ket{f_j,~d_{j+1}}\bra{d_j,~f_{j+1}} + \ket{f_j,~d_{j+1}}\bra{f_j,~d_{j+1}}.
\end{eqnarray}
Alternatively, they also have a Pauli matrix like representation,
\begin{equation}
\hat{U}_{j,j+1} = \frac{1}{2}~\left[ I^u_jI^u_{j+1} - u^x_ju^x_{j+1}- u^y_ju^y_{j+1}- u^z_ju^z_{j+1}\right],
\end{equation}
with
\begin{eqnarray}
&&I^u = \left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0\end{array}\right),~ u^x = \left(\begin{array}{ccc} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0\end{array}\right), \nonumber\\
&&u^y = \left(\begin{array}{ccc} 0 & -i & 0 \\ i & 0 & 0 \\ 0 & 0 & 0\end{array}\right), ~ u^z = \left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 0\end{array}\right).
\label{def-u}\end{eqnarray}
In a similar manner, we have the expression for operator $\hat{D}_{j,j+1}$,
\begin{equation}
\hat{D}_{j,j+1} = \frac{1}{2}~\left[ I^d_jI^d_{j+1} - d^x_jd^x_{j+1}- d^y_jd^y_{j+1}- d^z_jd^z_{j+1}\right],
\end{equation}
with
\begin{eqnarray}
&&I^d = \left(\begin{array}{ccc} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right),~ d^x = \left(\begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0\end{array}\right), \nonumber\\
&&d^y = \left(\begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & -i \\ 0 & i & 0\end{array}\right), ~ d^z = \left(\begin{array}{ccc} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1\end{array}\right).
\label{def-d}\end{eqnarray}

\subsection{A brief review of XXX spin-$1\over2$ chain}\label{sec:review XXX spin-1/2}

The final results show that the spectrum of the free Motzkin chain can be constructed based on the XXX spin-$1\over2$ chain. Here we take the XXX spin-$1\over2$ chain as an example to further explain the situation in the free Motzkin chain. Following the concept of functional Bethe Ansatz \cite{Sklyanin92} or SoV method \cite{Sklyanin95}, we have the following operator identities for transfer matrix of XXX spin-$1\over2$ chain
\begin{eqnarray}
t^{({1\over2})}\hspace{-0.5mm}(\theta_j)t^{({1\over2})}\hspace{-0.5mm}(\theta_j-\eta)\hspace{-1mm}=\hspace{-1mm}\mathbbold{1}\cdot a(\theta_j)d(\theta_j-\eta),\; j\hspace{-1mm}=\hspace{-1mm}1,\hspace{-0.5mm}\cdots\hspace{-0.5mm},\hspace{-0.5mm}N\hspace{-0.5mm}.~~
\label{trans-id-XXX}\end{eqnarray}
Next, we consider the asymptotic behaviors of fused transfer matrix
\begin{eqnarray}
\lim_{\lambda\to \infty} t^{({1\over2})}(\lambda)=\mathbbold{1}\cdot(2\lambda^N+\cdots).
\label{Asym-XXX}\end{eqnarray}
Then let us assume the eigenstate of $ t^{({1\over2})}(\lambda)$ is $|\psi\rangle$, with its eigenvalue $\Lambda^{({1\over2})}(\lambda)$. Acting the above operator identities (\ref{trans-id-XXX}) and the asymptotic relation (\ref{Asym-XXX}) on $|\psi\rangle$, they should satisfy the corresponding functional relations
\begin{eqnarray}
\Lambda\hspace{-0.5mm}^{({1\over2})}\hspace{-0.5mm}(\theta_j)\Lambda\hspace{-0.5mm}^{({1\over2})}\hspace{-0.5mm}(\theta_j-\eta)\hspace{-1mm}=\hspace{-1mm}a(\theta_j)d(\theta_j-\eta),\; j\hspace{-0.5mm}=\hspace{-0.5mm}1,\hspace{-0.5mm}\cdots\hspace{-0.5mm},\hspace{-0.5mm}N.~~
\end{eqnarray}\begin{eqnarray}
\lim_{\lambda\to \infty} \Lambda^{({1\over2})}(\lambda)=2\lambda^N+\cdots
\end{eqnarray}
As we know, the $T-Q$ relation of the XXX spin-$1\over2$ chain is a polynomial function of degree $N$. This can be seen from the construction of the monodromy matrix.
The above $N+1$ functional relations are sufficient to determine such a $T-Q$ relation with Bethe equations
\begin{eqnarray}
\Lambda^{({1\over2})}(\lambda)=a(\lambda)\frac{Q(\lambda-\eta)}{Q(\lambda)}+d(\lambda)\frac{Q(\lambda+\eta)}{Q(\lambda)}.
\label{T-Q-XXX}
\end{eqnarray}
The function $Q(\lambda)$ is parameterized by $N$ Bethe roots $\{\lambda_j|j=1,\cdots,N\}$ as follows:
\begin{eqnarray}
Q(\lambda)=\prod^{N}_{j=1}(\lambda-\lambda_j).
\end{eqnarray}
To ensure $\Lambda^{({1\over2})}(\lambda)$ to be a polynomial, the residues at the
poles $\lambda_j$ must vanish. This requires the $N$ Bethe roots to satisfy the Bethe equations
\begin{eqnarray}
\prod^N_{k=1}\frac{\lambda_j-\theta_k+\eta}{\lambda_j-\theta_k}
\hspace{-1mm}=\hspace{-1mm}\prod^N_{l\neq j\atop{l=1}}\frac{\lambda_j-\lambda_l+\eta}{\lambda_j-\lambda_l-\eta},\;\, j=1,\hspace{-0.5mm}\cdots\hspace{-0.5mm},\hspace{-0.5mm}N.~~
\end{eqnarray}

\subsection{The structure of operators $A(\lambda)$ and $D_{22}(\lambda)$}

The operators $A(\lambda)$ and $D_{22}(\lambda)$ are central symmetric to each other with respect to the element $E^{{(3^N+1)/2},{(3^N+1)/2}}$ (the center element of the matrix). Which means each element $E^{i,j}$ (we adopt standard basis) in operator $A(\lambda)$ can be mapped to (equal to) the element $E^{3^N+1-i,3^N+1-j}$ in operator $D_{22}(\lambda)$. Here $3^N$ is the dimension of $N$ quantum spaces.

The operators $A(\lambda)$ and $D_{22}(\lambda)$ have the following asymptotic behaviors as $\lambda$ goes to infinity.
The leading terms of $A(\lambda)$ operator is a degenerated operator acting on column vectors in the up half quantum space $\begin{bmatrix}x_{1},x_{2},\dots,x_{M},0,\dots,0\end{bmatrix}^{\rm {T}}$, and $M={3^N+1\over2}$.
This degenerated operator can be seen as a series of row transformations acting on this diagonal matrix
\begin{eqnarray}
\operatorname{diag}(1,\vspace{-0.5mm}\cdots\vspace{-0.5mm},1,0,\vspace{-0.5mm}\cdots\vspace{-0.5mm},0)\vspace{-0.5mm}=\vspace{-0.5mm}\sum^M_{j=1}1\vspace{-0.5mm}\times\vspace{-0.5mm}E^{j,j},\;\, M\vspace{-0.5mm}=\vspace{-0.5mm}{3^N\vspace{-0.5mm}+\vspace{-0.5mm}1\over2}.~~~~
\end{eqnarray}
Here altogether $M$ elements take the value $1$, from the first diagonal element $E^{1,1}$ to the $M-$th diagonal element $E^{M,M}$, and all the rest elements are $0$.

Considering the symmetry between $A(\lambda)$ and $D_{22}(\lambda)$, the leading terms of the composite operator $A(\lambda)+D_{22}(\lambda)$ is a non degenerate operator. It can be expressed as
a series of row transformations acting on the diagonal matrix
\begin{eqnarray}
\operatorname {diag} (1,\vspace{-0.5mm}\cdots\vspace{-0.5mm},1,2,1,\vspace{-0.5mm}\cdots\vspace{-0.5mm},1)\vspace{-0.5mm}=\vspace{-0.5mm}\sum^{3^N}_{j=1\atop j\neq M}1\vspace{-0.5mm}\times\vspace{-0.5mm}E^{j,j}\vspace{-0.5mm}+\vspace{-0.5mm}2\vspace{-0.5mm}\times\vspace{-0.5mm}E^{M,M}~~~~~
\end{eqnarray}
This is just an identity matrix with its center ($M$-th) diagonal element plus $1$.

\subsection{Structure of operator $\hat{O}$ and its eigenvalues}

Operator $\hat{O}$ is composed of the leading terms of operators $A(\lambda)$ and $D_{22}(\lambda)$ as $\lambda\rightarrow\infty$.
To study this, besides the local operators $I^u$ in (\ref{def-u}) and $I^d$ in (\ref{def-d}),we further introduce the following notations,
\begin{equation}
S^+ =\left|u\right>\left<d\right|=
\left(\begin{array}{ccc}
 0 & 0 & 1 \\
 0 & 0 & 0 \\
 0 & 0 & 0
\end{array}\right),\;
S^- =\left|d\right>\left<u\right|=
\left(\begin{array}{ccc}
 0 & 0 & 0 \\
 0 & 0 & 0 \\
 1 & 0 & 0
\end{array}\right).
\end{equation}
For total lattice sites number $N$, let us set
\begin{eqnarray}
A_{\langle N\rangle}\equiv\lim_{\lambda\rightarrow\infty}\frac{A(\lambda)}{\lambda^N},\;
B_{\langle N\rangle}\equiv\lim_{\lambda\rightarrow\infty}\frac{B_2(\lambda)}{\lambda^N},\;\nonumber\\
C_{\langle N\rangle}\equiv\lim_{\lambda\rightarrow\infty}\frac{C_2(\lambda)}{\lambda^N},\;
D_{\langle N\rangle}\equiv\lim_{\lambda\rightarrow\infty}\frac{D_{22}(\lambda)}{\lambda^N}.
\end{eqnarray}
Here for $B_2(\lambda)$, the subscript $2$ refers to the second $B$ type operator in (\ref{T-matrix}). While the subscript of $B_{\langle N\rangle}$ is the number of total system sites, and these operators act nontrivially on all the $N$ quantum spaces. Then for the system with $N$ lattice sites,
\begin{eqnarray}
\hat{O}_{\langle N\rangle}=A_{\langle N\rangle}+D_{\langle N\rangle}.
\end{eqnarray}
Because of the symmetry between $A(\lambda)$ and $D_{22}(\lambda)$, we only need to analyze the structure of operator $A(\lambda)$.
According to the structure of the monodromy matrix and matrix multiplication, we have the following recursive relations for these operators:
\begin{eqnarray}
&A_{\langle1\rangle}=I^u_1,\quad B_{\langle1\rangle}=S^-_1, \nonumber\\
&A_{\langle N\rangle}=A_{\langle N-1\rangle}\otimes I^u_N +B_{\langle N-1\rangle}\otimes S^+_N,\nonumber\\
&B_{\langle N\rangle}=A_{\langle N-1\rangle}\otimes S^-_N+B_{\langle N-1\rangle}\otimes I^d_N.
\end{eqnarray}
Subscripts of $I^u$, $I^d$, $S^+$ and $S^-$ refer to the quantum spaces on which these local operators act.
Based on the above recursive relations, we have the following rules for the structure of the operator $A_{\langle N\rangle}$:
\begin{itemize}
\item The operator $A_{\langle N\rangle}$ is composed of local operators $I^u$, $I^d$, $S^-$ and $S^+$.
\item $A_{\langle N\rangle}$ is the sum of products of $N$ local operators acting on each of the quantum spaces(from $1$ to $N$, respectively).
\item $A_{\langle N\rangle}$ is the sum of all possible operator products permutations(terms).
\item $S^-$ and $S^+$ operators appear in pairs, and $S^-$ always comes before $S^+$ in one permutation(term). That means the quantum subspace number of $S^-$ is smaller than that of $S^+$.
\item $I^u$ can appear on its own. It can also come before $S^-$ or after $S^+$. $I^d$ only appears between $S^-$ and $S^+$.
\item Multiple $S^-S^+$ pairs appear in turns. They can not be nested within each other.
\end{itemize}

In the following, we take $N=3$ as an example. $A_{\langle3\rangle}$ can be represented as
\begin{eqnarray}
A_{\langle3\rangle}=I^u_1I^u_2I^u_3+I^u_1S^-_2S^+_3+S^-_1S^+_2I^u_3+S^-_1I^d_2S^+_3.
\end{eqnarray}
Considering the symmetry between $A_{\langle3\rangle}$ and $D_{\langle3\rangle}$, the operator $\hat{O}_{\langle3\rangle}=A_{\langle3\rangle}+D_{\langle3\rangle}$ has the matrix form
\begin{widetext}
\begin{equation}
\left(
  \begin{array}{ccccccccc|ccccccccc|ccccccccc}
1&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\
0&1&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&\color{blue}{1}&0&0&0&0&0&0&0&0\\
0&0&0&1&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\
0&0&0&0&1&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&\color{red}{1}&0&0&0&0&0\\
0&0&\color{blue}{1}&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&\color{red}{1}&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&\color{blue}{1}&0&0&0&0&0&0\\
\hline
0&0&0&0&0&0&0&0&0&1&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&0&0&0&1&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&\color{red}{1}&0&0&0&0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&0&0&0&0&0&\textbf{1}&\textbf{0}&\textbf{0}&0&0&0&0&0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&0&0&0&0&0&\textbf{0}&\textbf{2}&\textbf{0}&0&0&0&0&0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&0&0&0&0&0&\textbf{0}&\textbf{0}&\textbf{1}&0&0&0&0&0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&0&0&0&0&\color{red}{1}&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&1&0&0&0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&1&0&0&0&0&0&0&0&0&0\\
\hline
0&0&0&0&0&0&\color{blue}{1}&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&\color{red}{1}&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&\color{blue}{1}&0&0\\
0&0&0&0&0&\color{red}{1}&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&1&0&0&0&0\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&1&0&0&0\\
0&0&0&0&0&0&0&0&\color{blue}{1}&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&1&0\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&1\\
  \end{array}
\right).
\end{equation}
\end{widetext}
In this matrix, $A_{\langle3\rangle}$ occupies the left $14$ columns, and $D_{\langle3\rangle}$ occupies the right $14$ columns. The number $2=1+1$ in the center (the entry in the $14$-th row and $14$-th column, $2\times E^{MM}$, $M=\frac{3^3+1}{2}=14$) is the sum of value $1$ from $A_{\langle3\rangle}$ and another $1$ from $D_{\langle3\rangle}$.

In this matrix form, we divide the first quantum space into $3\times3$ blocks by solid lines. The block in the center, $\ket{f_1}\bra{f_1}=E^{2,2}_1$ (we adopt the standard basis, subscript refers to quantum space) is the operator $\hat{O}_{\langle2\rangle}$. The boldface $3\times3$ diagonal matrix $\operatorname {diag}\{1,2,1\}$ in the middle is just the operator $\hat{O}_{\langle1\rangle}$. Then the $2$ in the center can also be represented as $2\times E^{MM}=2\times E^{2,2}_1E^{2,2}_2E^{2,2}_3$ in quantum subspaces.

We can analyze the distribution of eigenvalues of $\hat{O}_{\langle3\rangle}$ from operator $A_{\langle3\rangle}$. The term $I^u_1I^u_2I^u_3$ corresponds to all the diagonal elements. Considering the symmetry between $A_{\langle3\rangle}$ and $D_{\langle3\rangle}$, the operator $\hat{O}_{\langle3\rangle}$ has $2^4-1=15$ diagonal elements, including one single $2$, and $14$ elements equal to $1$.

Square roots of unity are caused by $S^-S^+$ pairs. Each term of $A_{\langle3\rangle}$ expression consists of $3$ local operators. In each term with a $S^-S^+$ pair, the center element $E^{2,2}$ (in its own subspace) of the rest operator $I^u$(or $I^d$) will lead to a square root.
In $A_{\langle3\rangle}$ operator, these elements are $E^{2,2}_1S^-_2S^+_3$, $S^-_1S^+_2E^{2,2}_3$ and $S^-_1E^{2,2}_2S^+_3$. Considering $D_{\langle3\rangle}$, there are $6$ elements related to square roots in $\hat{O}_{\langle3\rangle}$. This number can be calculated by combination
$C(3,2)\times2=6$. The $6$ elements are marked with red color in the above matrix. These elements will lead to $6$ square roots of unity: $-1$, $-1$, $-1$, $1$, $1$ and $1$.

The $S^-S^+$ pairs are also the basic components of higher order roots of unity.

Cubic roots come from terms with one $S^-S^+$ pair. Unlike the case of square roots, in terms with a $S^-S^+$ pair ($3$ local operators in each term), the rest one operator $I^u$(or $I^d$) should take $E^{1,1}$(or $E^{3,3}$). This $E^{1,1}$ (or $E^{3,3}$) makes the order plus one: $2+1=3$. Then in $A_{\langle3\rangle}$ operator, these elements are $E^{1,1}_1S^-_2S^+_3$, $S^-_1S^+_2E^{1,1}_3$ and $S^-_1E^{3,3}_2S^+_3$. Considering $D_{\langle3\rangle}$, there are $6$ elements in $\hat{O}_{\langle3\rangle}$. We calculate the number as $C(3,2)\times C(1,1)\times2=6$.
As marked in blue in the $\hat{O}_{\langle3\rangle}$ matrix form, the $6$ elements will lead to $6$ cubic roots: $\exp({2\pi i\over3})$, $\exp({2\pi i\over3})$, $\exp({4\pi i\over3})$, $\exp({4\pi i\over3})$, $1$ and $1$.

Next let us take operator $A_{\langle6\rangle}$ as an example to explain more rules of eigenvalues of $\hat{O}$. The operator $A_{\langle6\rangle}$ can be expressed as the sum of the products of $6$ local operators
\begin{widetext}
\begin{eqnarray}
&&A_{\langle6\rangle}=I^u_1 I^u_2 I^u_3 I^u_4 I^u_5 I^u_6
+I^u_1 I^u_2 I^u_3 I^u_4 S^{-}_5 S^{+}_6
+I^u_1 I^u_2 I^u_3 S^{-}_4 S^{+}_5 I^u_6
+I^u_1 I^u_2 S^{-}_3 S^{+}_4 I^u_5 I^u_6
\nonumber\\
&&+I^u_1 S^{-}_2 S^{+}_3 I^u_4 I^u_5 I^u_6
+S^{-}_1 S^{+}_2 I^u_3 I^u_4 I^u_5 I^u_6
+I^u_1 I^u_2 I^u_3 S^{-}_4 I^d_5 S^{+}_6
+I^u_1 I^u_2 S^{-}_3 I^d_4 S^{+}_5 I^u_6
\nonumber\\
&&+I^u_1 S^{-}_2 I^d_3 S^{+}_4 I^u_5 I^u_6
+S^{-}_1 I^d_2 S^{+}_3 I^u_4 I^u_5 I^u_6
+I^u_1 I^u_2 S^{-}_3 I^d_4 I^d_5 S^{+}_6
+I^u_1 S^{-}_2 I^d_3 I^d_4 S^{+}_5 I^u_6
\nonumber\\
&&+S^{-}_1 I^d_2 I^d_3 S^{+}_4 I^u_5 I^u_6
+I^u_1 S^{-}_2 I^d_3 I^d_4 I^d_5 S^{+}_6
+S^{-}_1 I^d_2 I^d_3 I^d_4 S^{+}_5 I^u_6
+S^{-}_1 I^d_2 I^d_3 I^d_4 I^d_5 S^{+}_6
\nonumber\\
&&\hspace{-0.5mm}+\hspace{-0.5mm}I^u_1 I^u_2 S^{-}_3 S^{+}_4 S^{-}_5 S^{+}_6\hspace{-0.5mm}
+\hspace{-0.5mm}S^{-}_1 S^{+}_2 I^u_3 I^u_4 S^{-}_5 S^{+}_6\hspace{-0.5mm}
+\hspace{-0.5mm}S^{-}_1 S^{+}_2 S^{-}_3 S^{+}_4 I^u_5 I^u_6\hspace{-0.5mm}
+\hspace{-0.5mm}I^u_1 S^{-}_2 S^{+}_3 I^u_4 S^{-}_5 S^{+}_6\hspace{-0.5mm}
+\hspace{-0.5mm}I^u_1 S^{-}_2 S^{+}_3 S^{-}_4 S^{+}_5 I^u_6\hspace{-0.5mm}
+\hspace{-0.5mm}S^{-}_1 S^{+}_2 I^u_3 S^{-}_4 S^{+}_5 I^u_6
\nonumber\\
&&+I^u_1 S^{-}_2 I^d_3 S^{+}_4 S^{-}_5 S^{+}_6
+I^u_1 S^{-}_2 S^{+}_3 S^{-}_4 I^d_5 S^{+}_6
+S^{-}_1 S^{+}_2 I^u_3 S^{-}_4 I^d_5 S^{+}_6
+S^{-}_1 I^d_2 S^{+}_3 I^u_4 S^{-}_5 S^{+}_6
\nonumber\\
&&\hspace{-0.5mm}+\hspace{-0.5mm}S^{-}_1 S^{+}_2 S^{-}_3 I^d_4 S^{+}_5 I^u_6\hspace{-0.5mm}
+\hspace{-0.5mm}S^{-}_1 I^d_2 S^{+}_3 S^{-}_4 S^{+}_5 I^u_6\hspace{-0.5mm}
+\hspace{-0.5mm}S^{-}_1 S^{+}_2 S^{-}_3 I^d_4 I^d_5 S^{+}_6\hspace{-0.5mm}
+\hspace{-0.5mm}S^{-}_1 I^d_2 S^{+}_3 S^{-}_4 I^d_5 S^{+}_6\hspace{-0.5mm}
+\hspace{-0.5mm}S^{-}_1 I^d_2 I^d_3 S^{+}_4 S^{-}_5 S^{+}_6\hspace{-0.5mm}
+\hspace{-0.5mm}S^{-}_1 S^{+}_2 S^{-}_3 S^{+}_4 S^{-}_5 S^{+}_6
\nonumber\\
\end{eqnarray}
\end{widetext}
Among them, the term $I^u_1I^u_2I^u_3I^u_4I^u_5I^u_6$ corresponds to all the diagonal elements. Similarly, together with $D_{\langle6\rangle}$, there are $2^7-1=127$ diagonal elements in $\hat{O}_{\langle6\rangle}$, and one of them is $2$. All the rest of the elements are $1$.

The number of square roots of unity is $(C(6,2)+C(6,4)+C(6,6))\times2=2^6-2=62$, and half ($62/2=31$) of them are $-1$, the other half are $1$. Single $S^-S^+$ pair in one term leads to square roots, when all the rest operators take $E^{2,2}$. Recurring $S^-S^+$ pairs ($2$ or $3$) in one term with the rest operators taking $E^{2,2}$, will also give rise to square roots. This means we should also take terms with $S^-S^+S^-S^+$ and $S^-S^+S^-S^+S^-S^+$ into account.

The number of cubic roots is $(C(6,2)\times C(4,1)+C(3,1))\times2=126$. The three kinds of cubic roots are $\exp({2\pi i\over3})$, $\exp({4\pi i\over3})$ and $1$, and each has degeneracy $126\over3$.
In the expression of operator $A_{\langle6\rangle}$, let us consider terms with one $S^-S^+$ pair and four $I^u$(or $I^d$) operators. When one $I^u$(or $I^d$) operator takes $E^{1,1}$(or $E^{3,3}$) in its subspace, and the other three $I^u$ take $E^{2,2}$, each such choice in one term will lead to a cubic root. And recurring $I^uS^-S^+$($I^u$ takes $E^{11}$), $S^-I^dS^+$($I^d$ takes $E^{33}$) and $S^-S^+I^u$($I^u$ takes $E^{11}$) patterns in one term also lead to cubic roots.

The number of $4$-th roots is $(C(6,2)\times C(4,2))\times2=180$. The four kinds of $4$-th roots are $i$, $-1$, $-i$ and $1$, and each kind has degeneracy $180\over4$.
$4$-th roots come from terms with a $S^-S^+$ pair and two $I^u$(or $I^d$) taking $E^{1,1}$(or $E^{3,3}$) in its subspace, and the other two $I^u$(or $I^d$) taking $E^{2,2}$. Two $S^-S^+$ pairs in one term with the rest two operators taking $E^{2,2}$, will reduce to square roots, as we mentioned above.

The number of $5$-th roots is $(C(6,2)\times C(4,3)+C(6,4)\times C(2,1))\times2=180$. The five kinds of $5$-th roots are $\exp({2\pi i\over5})$, $\exp({4\pi i\over5})$, $\exp({6\pi i\over5})$, $\exp({8\pi i\over5})$ and $1$, and each kind has degeneracy $180\over5$.
$5$-th roots come from terms with a $S^-S^+$ pair and three $I^u$(or $I^d$) taking $E^{1,1}$(or $E^{3,3}$) in its subspace, and the other one $I^u$(or $I^d$) taking $E^{2,2}$.
$5$-th roots also come from terms with two $S^-S^+$ pairs, and one of the rest two operators $I^u$(or $I^d$) taking $E^{1,1}$(or $E^{3,3}$) in its subspace, and the other taking $E^{2,2}$.

The number of $6$-th roots is $(C(6,2)\times C(4,4)+C(6,4)\times C(2,2)- C(3,1))\times2=54$.
The six kinds of $6$-th roots are $\exp({2\pi i\over6})$, $\exp({4\pi i\over6})$, $-1$, $\exp({8\pi i\over6})$, $\exp({10\pi i\over6})$ and $1$, each kind has degeneracy $54\over6$.
$6$-th roots come from terms with all operators not taking their center element $E^{2,2}$ in their subspace. This will contain terms such as $S^{-}_1S^{+}_2E^{1,1}_3E^{1,1}_4E^{1,1}_5E^{1,1}_6$ and $S^{-}_1S^{+}_2S^{-}_3S^{+}_4E^{1,1}_5E^{1,1}_6$.
While there are exceptions, recurring $I^uS^-S^+$, $S^-I^dS^+$ and $S^-S^+I^u$ pairs such as $E^{1,1}_1S^-_2S^+_3E^{1,1}_4S^-_5S^+_6$, $S^-_1E^{3,3}_2S^+_3S^-_4E^{3,3}_5S^+_6$ and $S^-_1S^+_2E^{1,1}_3S^-_4S^+_5E^{1,1}_6$ will reduce to cubic roots. $S_1^-S_2^+S_3^-S_4^+S_5^-S_6^+$ will reduce to square root (as mentioned above).

For the case of $\hat{O}_{\langle7\rangle}$, according to the above rules, we directly write down the distribution of different eigenvalues.

Eigenvalues from diagonal elements: a single $2$, multiple $1$ with degeneracy of $2^8-2=254$;

Number of square roots of unity: $(C(7,2)+C(7,4)+C(7,6))\times2=2^7-2=126$. $-1$ and $1$ are roots of degeneracy ${126\over2}=63$.

Number of cubic roots: $(C(7,2)\times C(5,1)+ C(3,1)\times7)\times2=252$. $\exp({2\pi i\over3})$, $\exp({4\pi i\over3})$ and $1$ are roots of degeneracy ${252\over3}=84$.

Number of $4$-th roots: $(C(7,2)\times C(5,2))\times2=420$. $i$, $-1$, $-i$ and $1$ are roots of degeneracy ${420\over4}=105$.

Number of $5$-th roots: $(C(7,2)\times C(5,3)+C(7,4)\times C(3,1))\times2=630$. $\exp({2\pi i\over5})$, $\exp({4\pi i\over5})$, $\exp({6\pi i\over5})$, $\exp({8\pi i\over5})$ and $1$ are roots of degeneracy ${630\over5}=126$.

Number of $6$-th roots: $(C(7,2)\times C(5,4)+C(7,4)\times C(3,2)-C(3,1)\times7)\times2=378$. $\exp({2\pi i\over6})$, $\exp({4\pi i\over6})$, $-1$, $\exp({8\pi i\over6})$, $\exp({10\pi i\over6})$ and $1$ are roots of degeneracy ${378\over6}=63$.

Number of $7$-th roots: $(C(7,2)\times C(5,5)+C(7,4)\times C(3,3)+C(7,6)\times C(1,1))\times2=126$. $\exp({2\pi i\over7})$, $\exp({4\pi i\over7})$, $\exp({6\pi i\over7})$, $\exp({8\pi i\over7})$, $\exp({10\pi i\over7})$, $\exp({12\pi i\over7})$ and $1$ are roots of degeneracy ${126\over7}=18$.

Further, we summarize the combination rules for the eigenvalues of operator $\hat{O}_{\langle N\rangle}$:
\begin{itemize}
\item Number of diagonal elements is $2^{N+1}-1$, and one of them is $2$. All the rest are $1$.
\item Number of square roots is $C(N,2)+C(N,4)+\cdots=2^N-2$. Square roots come from $S^-S^+$ pair(s), and recurring $S^-S^+$ also give rise to square roots.
\item Higher order roots are based on $S^-S^+$ pair(s). In one term of operator products, each element $E^{1,1}$ in $I^u$(or $E^{3,3}$ in $I^d$) will increase the order by $1$, and each additional $S^-S^+$ pair will increase the order by $2$. The element $\ket{f}\bra{f}=E^{2,2}$ in the center of each subspace will maintain the order number. Briefly speaking, in a certain term, each element in local operator outside the center of its subspace

    makes the order of the root plus $1$. In contrast, elements in the center $E^{2,2}$ of local operators ($I^u$ and $I^d$) have no contribution to the order.
\item Terms with recurring higher order patterns (like $S^-S^+$ is related to square root and $S^-S^+E^{1,1}$ is related to cubic root) will reduce their order to a single pattern. When counting $n$-th order terms, if $n$ is a composite number, we should consider all kinds of recurring patterns (which means all divisors of $n$) that lead to reduced orders.
\item In every specific $m$-th roots case, each kind of roots $\exp({2\pi i\over m})$, $\exp({4\pi i\over m})$,$\cdots$, $1$ has the same degeneracy.
\end{itemize}

\subsection{The distribution formula for the eigenvalues of operator $\hat{O}$}
Let us recall the definition of $\hat{O}$. It is the leading term in the taylor expansion of the transfer matrix, i.e. $t(\lambda)= A(\lambda) + D_{11}(\lambda) + D_{22}(\lambda) = (\mathbbold{1} + \hat{O})\lambda^N +\cdots$ Equivalently, it may be defined as:

\begin{eqnarray}
\mathbbold{1} + \hat{O} = \text{tr} \{ \Tilde{R}_{0N} \Tilde{R}_{0N-1} ... \Tilde{R}_{02} \Tilde{R}_{01} \}
\end{eqnarray}
where $\Tilde{R} = \lim_{\lambda \rightarrow \infty} \frac{R(\lambda)}{\lambda} = \mathbbold{1}  - (\ket{ud} - \ket{du})(\bra{ud} - \bra{du})$ can be seen as a permutation matrix which permutes the states $\ket{u} \otimes \ket{d}$ and $\ket{d} \otimes \ket{u}$. One immediate consequence of this is that the leading term of $D_{11}$ is $\mathbbold{1}$ (since the auxiliary space is left invariant), which we chose to cancel out with the $\mathbbold{1}$ term in our definition. \\

The second consequence is that the operator maps basis states to basis states. Apart from the state $\ket{f} \otimes\ket{f} \otimes ... \otimes \ket{f} \otimes \ket{f}$ which is mapped to itself by both $A_{\langle{N}\rangle}$ and $D_{\langle{N}\rangle}$ and so is an eigenstate with eigenvalue 2, the operator acts as a permutation matrix on the basis states. It acts as a left cyclic shift of all sites not equal to $\ket{f}$. For example, on a chain of length $6$, the state $\ket{u} \otimes \ket{u} \otimes\ket{d} \otimes\ket{f} \otimes\ket{u} \otimes\ket{d}$ can be mapped to  $\ket{u} \otimes \ket{d} \otimes\ket{u} \otimes\ket{f} \otimes\ket{d} \otimes\ket{u}$. \\

As a result, finding the spectrum of $\Hat{O}$ is equivalent to finding the cycle decomposition of the corresponding permutation. Also based on the combination rules in the last section, we can write down the distribution of eigenvalues for arbitrary $\hat{O}_{\langle N\rangle}$.

Then the number of $k$-th roots for a lattice of length $N$ is given by:
\begin{eqnarray}
f(N,k) = \left( \sum_{d|k} 2^d \mu(\frac{k}{d}) \right) \left[ \sum_{m = 1}^{\floor{\frac{N}{k}}} C(N,km)
\right]
\end{eqnarray}
where $\mu(n)$ refers to the M\"obius function. For $k = 1$ this formula gives the correct diagonal result if one excludes the eigenvalue $2$.

Some results in number theory can be observed from the distribution. For example, when $N$ is a prime number, the total number of square roots is $2^N-2$. It is equal to the total number of the highest $N$-th roots in this case. As we mentioned above, the degeneracies of all kinds of $N$-th roots are equal. This will lead to a result in number theory: $2^N-2$ is divisible by $N$, for $N$ is prime.

We can tabulate the first few values, which gives in TABLE \ref{tab}:

\begin{widetext}
\begin{table*}
\begin{ruledtabular}
    \centering
    \begin{tabular}{ccccccccccc}
Length & Diag & sqrt & 3rd & 4th & 5th & 6th & 7th & 8th & 9th & 10th \\ \hline
 N = 1 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
 N = 2 & 6 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
 N = 3 & 14 & 6 & 6 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
 N = 4 & 30 & 14 & 24 & 12 & 0 & 0 & 0 & 0 & 0 & 0 \\
 N = 5 & 62 & 30 & 60 & 60 & 30 & 0 & 0 & 0 & 0 & 0 \\
 N = 6 & 126 & 62 & 126 & 180 & 180 & 54 & 0 & 0 & 0 & 0 \\
 N = 7 & 254 & 126 & 252 & 420 & 630 & 378 & 126 & 0 & 0 & 0 \\
 N = 8 & 510 & 254 & 504 & 852 & 1680 & 1512 & 1008 & 240 & 0 & 0 \\
 N = 9  & 1022 & 510 & 1014 & 1620 & 3780 & 4536 & 4536 & 2160 & 504 & 0 \\
 N = 10 &  2046 & 1022 & 2040 & 3060 & 7590 & 11340 & 15120 & 10800 & 5040 & 990 \\
\end{tabular}
\end{ruledtabular}
    \caption{The categories and quantities of the eigenvalues for the operator $\hat{O}$ from $N=1$ to $10$. The results from direct diagonalization are coincident with the numbers we count from the combination rules $f(N,k)$. We remark that in the diagonal elements, the eigenvalue $2$ is excluded, and $1$ can be regarded as the $1$-st root of unity.}
    \label{tab}
\end{table*}
\end{widetext}

This was computed with the following implementation in Mathematica:

\begin{verbatim}
  g[n_]:= DivisorSum[n,(2^#)*MoebiusMu[n/#]&]
  binomSum[n_,k_]:=Sum[Binomial[n, i],{i,k,n,k}]

  f[n_,k_]:=g[k]*binomSum[n,k]
\end{verbatim}

\subsection{Configuration of $\{\omega\}$ in the $T-Q$ relation}\label{sec:Configuration of omega}

Due to $\{\omega\}$, other than XXX spin-${1\over2}$ case, there must exist coefficients before the two terms of the $T-Q$ relation(\ref{T-Q}). Assuming $\omega$ is one eigenvalue of operator $\hat{O}$, then the corresponding coefficient of the leading term of $\Lambda(\lambda)$ as $\lambda\rightarrow\infty$ should be $\omega+1$, as shown in (\ref{Lambda-asymptotic}). Let $\tilde{a}$ and $\tilde{d}$ represent the coefficients before $a(\lambda)$ and $d(\lambda)$ terms, respectively.
Based on the identities (\ref{Lambda-identity}) and the asymptotic behavior (\ref{trans-asymptotic}) and (\ref{Lambda-asymptotic}), these coefficients should satisfy the following equations
\begin{eqnarray}
\left\{
\begin{aligned}
&\tilde{a}\cdot \tilde{d}=\omega,\\
&\tilde{a}+\tilde{d}=\omega+1.
\end{aligned}
\right.
\end{eqnarray}
The equations have two sets of solutions:
\begin{eqnarray}
\left\{
\begin{aligned}
&\tilde{a}=\omega,\\
&\tilde{d}=1;
\end{aligned}
\right.
\qquad\text{and}\qquad
\left\{
\begin{aligned}
&\tilde{a}=1,\\
&\tilde{d}=\omega.
\end{aligned}
\right.
\label{coefficient-solution}\end{eqnarray}
After carefully calculations, we find that only the first solution set is appropriate.
Basically speaking, this is because $\{\omega\}$ are eigenvalues of the operator $\hat{O}$, and the operator $\hat{O}$ can be related to $A(\lambda)+D_{22}(\lambda)$, which correspond to $a(\lambda)$ (\ref{A-D-eigen}). In contrast, the operator $D_{11}(\lambda)$ corresponds to $d(\lambda)$, and $D_{11}(\lambda)$ is proportional to identity operator under the limit $\lambda$ tends to $\infty$.

redpajama_set_name: RedPajamaArXiv


Founded by a childhood cancer survivor and her family, the free annual camp is celebrating its 24th year in South Jersey.
TABERNACLE - William "Tre" Walker remembers arriving at Camp No Worries as a sad young teen, half dreading another week at the annual sleep-away camp in the Pine Barrens.
By then, the Pennsauken resident had been coming every summer for a free week of camp, ever since he was diagnosed with acute lymphoblastic leukemia in kindergarten. He knew what to expect: time outside to play with kids like himself, who knew what it was like to be stressed out, cooped up and pitied.
"I didn't know if I could put myself in the mood to be happy, but the second I stepped foot on that campground, it's almost like my body just had this response of like, 'You're going to be happy now,'" Walker recalled. Joy flooded through him, and "everything that could be happy was happy," he said.
"And that's when I realized how important it was. You actually do forget the rest of the world. It is a place of no worries — just have fun, go be you."
This week, the 21-year-old college senior is among more than 175 volunteers who run the annual camp for kids affected by childhood cancer, both patients and siblings. Founded in 1995 by the family of Kasey Hall Massa, the program started with 14 campers. Today, it draws up to 100 kids.
Now 43, Massa survived a malignant brain tumor, diagnosed when she was 11 years old. She works full time as a social worker at MD Anderson Cancer Center at Cooper, and calls leading Camp No Worries a "hobby" that takes volunteers 11 months a year to organize, fund and run.
Once operated under the YMCA of Burlington and Camden Counties, Camp No Worries is an independent nonprofit that relies on donors to support the $1,200 cost to send each child for the week.
Like Walker, many of the counselors are former campers, able to teach kids what life after cancer can look like.
Patrick and Sheila Koreck of Sicklerville sent all four of their kids to Camp No Worries. Their 9-year-old daughter, Claire, was diagnosed with Ewing sarcoma five years ago. Claire and her twin brother, Declan, her 12-year-old sister, Maeve, and 7-year-old sister, Sarah, look forward to the week all year long, their mother said.
An oncology nurse who works with adult patients, Sheila Koreck said people don't realize how a cancer diagnosis affects every member of the family, especially little children who don't understand why their sibling is at the hospital, or why they miss so much time with their mom.
"Only people who go through it really understand what you've been through," Koreck said.
Walker, the former camper, said his job is to sit alongside the kids who are reluctant to participate, encouraging them to try one new thing each day. By Wednesday, he said, usually everyone is in "camp mood," slapping high-fives with everyone they encounter, freely acting silly.
When they're older, he said, the campers will realize how important this experience was. They will remember a time when they were free from stress and worries, a place where they could just be.