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Forest Woods Lorong Lew Lian Serangoon Preview Date
Forest Woods is situated near Serangoon Central. It is sandwiched between the Upper Serangoon Road and Lorong Lew Lian. Forest Woods is a joint development involving City Developments (CDL), Hong Leong Holdings, TID Pte Ltd. The residential development occupies a land size of about 150,711 square feet, and it consists of about 500 units. The 99-yer leasehold will be a residential area for couples and families that are interested in living in the upcoming Singapore area. The Forest Wood Condo is located in a prime location. The development is just about five minutes' walk to Serangoon MRT Interchange station. Forest Woods is also well-linked on the main roads such as the Upper Serangoon Road, Upper Paya Lebar Road, and the Boundary Road. The developer of Forest Woods City Developments (CDL), is well-known for having developed many other properties in Singapore. CDL has a great reputation among its clients having established quality homes in Singapore.
The development is projected to contain over 500 units which will be home to many families and couples. Residents will have access to various welfare services including an indoor gym, tennis courts, and car parks among others. Entertainment, shopping centers, and other places of conveniences are readily available for the residents at Forest Woods and the preview date will be available soon together with the showflat. Major shopping centers around the area include the Upper Serangoon Shopping Center and the NEX shopping mall. Car owners will find the place easily accessible bring in mind the major roads that link-up the place. Sufficient parking space will be provided for the car owners. The development also has function rooms where residents can conduct indoor functions much conveniently.
NEX shopping mall is one of the mega urban shopping centers in Serangoon. The shopping mall is located near Forest Woods Condos. It has various anchor tenants including cafés, library, supermarkets, and cinema.
Forestwood Residences Site and Floor Plans Hong Leong Holdings
Residential places in Serangoon are in high demand from both buyers and investors because of the new Serangoon MRT Interchange that provides efficient transport means round the place. Plenty of locations such as eateries and groceries are located along the Upper Serangoon Road. Educational institutions such as the Nanyang Junior College and St Gabrielle's Secondary School are situated around the area. Indicative prices will be available soon at City Developments showflat. St Gabrielle's Secondary School is well-known in Singapore for instilling an all-round education system in Singapore. Other high-class schools that are in high demand such as Maris Stella High School and Paya Lebar Methodist Girl's School are not far from the place.
Forestwood Preview Date Lorong Lew Lian
The Forest Woods residential place is also accessible to many facilities and amenities such as the Serangoon Stadium, the Braddell Heights Community and the Nanyang Community College.…


The much talked about spectacular Telugu language film "Sri Rama Rajyam" is directed by the legendary octogenarian Director and Padma Sri Award winner Sri.Bapu..
Produced by Yalamanchali Sai Babu,under his banner Sai Baba Movies,Sri Rama Rajyam is a visual treat made with a massive budget of three hundred and twenty million Indian Rupees..Sai Babu,an industrialist by profession and a philanthropist by nature,has left no stone unturned in extending complete support to Director Bapu in carving out this creative masterpiece called "Sri Rama Rajyam"..
The magic of Sri Rama Rajyam held all the sections of audiences across South India captive with it's spell binding content conveyed by Bapu through steady frames,excellent closeups,stunning visuals,amazing set designs,spectacular photography,scintillating music and finally those mind blowing visual effects that were never seen on the Indian screen before..
Sri Rama Rajyam deals with the subject of Uttar Kand,the most crucial part of the great Indian epic Ramayan..Uttara Kand depicts Lord Sri Ram's rule of Ayodhya after His return from Lanka'; His separation from Sita; Sita's life in the Hermitage of Sage Valmiki and finally the upbringing of the holy couple's children Luv and Kush in that hermitage..
Director Bapu,with his deft handling of the story of Uttar Kand,through his film Sri Rama Rajyam,has definitely taken the overall standards of Indian film making to the next level and arguably made the best film not only in the history of the South Indian film industry but also in the history of the Indian film industry..
Taking the washer man's remarks very seriously,a much disturbed Lord Sri Ram instructs His brother Lakshmana to leave Sita in the middle of a forest and come back..Though reluctant initially,Lakshmana is forced to obey his Brother's orders and finally leaves a much shocked Sita in a wild forest and returns back..
Sage Valmiki comes to the rescue of the lonely Sita and provides Her shelter in his Hermitage..Sita,who was already a pregnant by then,gives birth to two cute male twins who are later christened Luv and Kush by Sage Valmiki..The twins, whose identity was deliberately hidden by the noble Sage,grow up under his tutelage..
Valmiki composes Ramayan in poetic form and teaches Luv and Kush to practice singing it.. Later,when Sage Valmiki comes to know about the public ceremony King Sri Ram is holding in Ayodhya,he instructs Luv and Kush to attend it and sing Ramayan in front of Sri Ram..Obeying the holy Sage's orders, Luv and Kush sing Ramayan in front of Sri Ram and at the time when they recite about Sita's exile,Sri Ram becomes extremely grievous..
Finally,after a spectacularly captured battle between Lord Sri Ram and the young twins,Sita with the help of Hanuman,comes rushing to the battle field and it is at this point of time that the great King Sri Ram comes to know that the young twin brothers,who have so far waged a bravely battle with Him,are actually His sons..
However,that's too late for Sri Ram as a much disappointed Sita,despite persistent requests from Sri Ram calls upon her Mother Earth,to take her away..As a result,the forest ground where Sita is thus far standing opens up under her feet and Sita vanishes into it.. (This scene conveyed through mind blowing visual effects is the highlight of this spectacular film)..
Later,Lord Sri Ram hands over the reins of the kingdom of Ayodhya to Luv and Kush and as the seven gates of the Celestial Abode of Vaikuntam open before Him,the holy Lord very gracefully walks into His holy Celestial Abode thus marking the end of the incarnation of Lord Vishnu as Sri Ram on our planet Earth..
Whereas Nandamuri Bala Krishna,a popular Telugu super star and son of the iconic Telugu film personality N.T.Rama Rao,played the role of Lord Sri Ram in this film,the role of Sita is portrayed by South Indian film industry's one of the most beautiful heroines,Nayana Tara..By and large Nayana Tara's performance as Sita in Sri Rama Rajyam will go down in the history of Telugu film industry as one of the finest performances ever seen on the Telugu screen..
Additionally,PRK Raju,the Director Of Photography,lauded for his spellbinding work in this film,has won the award for the Best Photography,for the year 2012,at the prestigious South Indian International Movie (SIIMA) awards held annually at Dubai.. Nayana Tara too has won the best actress award at the same venue for 2012..
In an illustrious career that's spread over four decades of dedicated film making,Director Bapu has directed fifty one films that include ten Hindi films (Hum Paanch, Seeta Swayamvar,Anokha Bhakt,Bezubaan,Woh Saat Din,Pyari Behna, Mohobbat,Mera Dharam,Diljala and Prem Pratigya) and one Tamil film(Devan Mayagugiran)..
Music Maestro Ilaiyaraaja composed the background score and songs for "Sri Rama Rajyam" and according to the music sources in Mumbai and Chennai, Illaiyaraja and his troupe of seventy amazing musicians literally worked for twelve hours in a day to give the viewers of Sri Rama Rajyam that out of the world musical experience..
Nine extraordinary musicians were specially flown in from Hungary to assist the Maestro Illaiyaraja in delivering that amazing background score for this film..A very rare musical instrument "Timpany",that's not found anywhere in India and found only in three countries around the world,is specially brought in and used for this film..
Although all the fifteen songs composed by Maestro Illaiyaraja for Sri Rama Rajyam are musical gems,there's however,one particular song that stands out among them..Composed in Raag Naata "Jagadaananda Kaaraka" (Telugu) is one of those rare gems from Saint Thyagaraja's pancha ratna Keerthanas and using the opening lines of this popular keerthana,Maestro Illaiyaraja composed "Jagadaananda Kaaraka" song for the film "Sri Rama Rajyam".
And "Jagadaananda Kaarakaa" quite rightfully won the best lyricist (Jonna Vithhula) and the best female playback singer (Shreya Goshal) awards for the film..In an illustrious career spread over forty five years of selfless service to music,Sri Rama Rajyam is the first mythological film the Maestro Illaiyaraja composed music for..
Undoubtedly Sri Rama Rajyam is a masterpiece created by Bapu,however,it's important to mention here that at least half of the entire credit should unconditionally go to the Visual Effects Team that did a splendid job in finally giving Sri Rama Rajyam that spectacular look..
Under the able leadership of National Award Winner Kamal Kannan (Maga Dheera) the Visual Effects work was executed by award winning company, Pixelloid from Mumbai in association with EFX and Makuta..
With its extravagant production values "Sri Rama Rajyam" becomes the first Telugu film in the history of the Indian film industry to command a whopping sixty million Indian rupees for it's satellite rights alone..Whereas the film did a commendable business in every area it is exhibited,it did exceedingly well in some areas of Andhra region..
Sri Rama Rajyam is also dubbed into all South Indian languages and it did a very decent business in all those areas too..Sri Rama Rajyam is also the only second Hindu mythological film to be dubbed and released in a Christian dominated state of Kerala..Quite interestingly,"Lava Kusa"(1963-Similar story of Uttara Kand starring N.T.Rama Rao,father of Sri Rama Rajyam's hero Bala Krishna) was the first Hindu mythological film to be dubbed into the Malayalam language of Kerala..
Sri Rama Rajyam ran to packed houses in every place it was screened in America, more so in localities like VA,New Jersey and Bay Area.. During one of the premier shows held in America,a guy by name Ashok Babu Kolla of San Jose purchased a premier show ticket for $2500…Another great record indeed..
Sri Rama Rajyam was specially chosen for screening at the International Film Festival held in Goa in 2012..For the first time in the history of this festival,Sri Rama Rajyam was specially shown to five hundred media personalities from all over the world who were stunned by the standards delivered by this film and gave a standing ovation to director Bapu after the screening..
Additionally,a special screening of the Malayalam version of Sri Rama Rajyam was held for His Majesty Uthradom Thirunal Marthanda Varma.the Maharaja Of Travancore Royal Family of Tiruvanantha Puram in Kerala..His Majesty,a great devotee of Lord Vishnu,was extremely delighted with the film and specially honored director Bapu for his outstanding work.
Immensely pleased with the response his film received from all corners of the Entertainment World,the producer of Sri Rama Rajyam,Yalamanchali Saibabu, decided to dub the film into Hindi language for the viewing pleasure of the Hindi speaking audiences all over the world..
In this regard the Line Producer of this film P N Ramachandra Rao,who is also a notable producer and director from the South Indian film industry with over forty five films to his credit,has held a special screening of the Hindi version of Sri Rama Rajyam in Mumbai recently..
Celebrated Bollywood actor,Anil Kapoor,ace Music Composer,Himesh Reshammiya,Director Bapu,hero of the film Bala Krishna,Producer Yalamanchali Saibabu,were among those who were present during the screening..
Any word of praise would be inadequate for this great Indian film called "Sri Rama Rajyam",which undoubtedly is the precious life blood of a master spirit that should be embalmed and treasured for a life beyond life..


At the Fence: Looking For Promotion Codes?
Quick question here, before you make a purchase do you shop around to get the best deal? Do you look for coupons and codes you can use on the product you are purchasing? We do! My oldest daughter is our coupon queen. If there is a coupon or deal on a site, she is the one to see. She even thinks ahead for me and as I head out the door will say, "Hey mom, I have a coupon for your cheese." or "Will you you be eating out at__________ (fill in the blank with your favorite restaurant)?" And sure enough she has a coupon or offer that is just perfect!
She also has codes for online shopping. She is totally amazing when it comes to saving. Anyway, I said all this to ask if any of my neighbors here At the Fence have ever heard of PromotionCode.org? I was checking them out today as I just heard of them recently. They actually have a search engine where you can check out what codes are available for the store or business you will be visiting or ordering from. Or you can search their directory of store codes which lists the businesses alphabetically. What I like best is that they verify codes are working, and then when you use a code you can go back and let them know whether the code was a success or not. If everyone does this, then you will be able to pick and choose codes based on their success. Nice! Then if a code is not working for three days, or has a low percentage of working properly, they remove the code.
They even have travel deals. So if you are planning your summer vacation you might want to take a minute and visit their site and just see what they have to offer. Who knows, you might find just what you are looking for! we are planning our 30th anniversary trip for the end of July. Yippee! We are taking a 5 day trip!
You can also go to their site and add codes that you know work. That way you can help some others find codes they can use. Another great feature is you can choose to receive emails based on the companies you are interested in. By signing up for this option you won't have to keep checking back, they will notify you!
Have any of my neighbors ever used this site? I would be interested in hearing what you think. Share your opinion in the comments.


Teads, founded in 2011, is the inventor of outstream video and the number one video advertising marketplace in the world, according to comScore. Publishers work with Teads to create brand new video inventory, monetizing it through programmatic buying, their own sales force, or third parties including Teads Managed Services. Teads' native video advertising solutions encompass a series of formats inserted deep into media content, like the inRead playing inside articles. It is changing the game within the video advertising market by creating unprecedented levels of premium inventory, which did not exist before. Brands and agencies can access this top-tier, premium inventory, available on the web and on mobile, through programmatic or managed services. Through its managed services capabilities, the Teads team execute on their clients behalf using its platform. Teads has a team of over 550 employees, 100 of which are in the innovation team, across 28 offices in 21 countries.


First Deck: 10 Design Suites each measuring 30 sqm / 322 sq. ft., 4 with balcony and 6 without balcony. Design Suites without balconies all have river facing panoramic windows that open and a 4m (13ft) long interior sofa that converts to a single bed Design Suites with balconies all have panoramic windows, sliding glass door and an exterior daybed.
Second Deck: 10 Design Suites each measuring 30 sqm / 322 sq. ft., 4 with balcony and 6 without balcony. Design Suites without balconies all have river facing panoramic windows that open and a 4m (13ft) long interior sofa that converts to a single bed Design Suites with balconies all have panoramic windows, sliding glass door and an exterior daybed.
This morning, we embark by skiff to visit Koh Chen, or 'Chinese Island' in Khmer. The inhabitants of this riverside village are renowned throughout the region as silver and coppersmiths. They purchase the precious metals in Phnom Penh and return here to make ornamental items such as the cups, boxes and delicately engraved tropical fruits used in Buddhist ceremonies and marriage blessings. Each family member carries out an aspect of these ancient methods and the intricate, beautiful workmanship belies its humble origins. Purchase whatever catches your eye before we continue on to the pagoda. If the monks are in residence, we will experience an authentic Buddhist blessing ceremony.
Later in the day, we continue exploring Khmer riverside life in the floating village of Kampong Chhnang, meaning Port of Pottery. Archaeological findings here link this now sleepy area as far back as the Dvaravati kingdom, dating to the 6th through 11th centuries when Kampong Chhnang was an important coastal stop on the trade route linking China and India. More recently, Cambodia's Prime Minister Hun Sen was born in this province. Colorful shrines with gilded Buddhas, villagers sporting conical hats more often associated with Vietnam and simple houses on bamboo stilts are among today's striking sights along the waterfront.
Crew Crew numbers 40, including a cruise director, paramedic and 4 English-speaking guides.


Inspired by a traditional men's folk dance, d'aprés une histoire vraie (for after a true story) is as open and pleasant as it is choreographically complex. Having a four day run at REDCAT in downtown Los Angeles, French choreographer Christian Rizzo has selected movements from a folk dance, dissected them, re-arranged them and created a work that brings visual life to the feeling of male camaraderie. In this case it is the eight male dancers performing at first in silence and then to a driving tribal-like score composed and performed onstage by Didier Ambact and Bertrand Groussard, aka King Q4.
Rizzo began his artistic life forming a rock band and later designing a line of clothing. Through his music and costume designs Rizzo found his path to performing in works by such well-known choreographers as Mathilde Monnier, Hervé Robbe, Mark Tompkins, Georges Appaix and others. In January of 2015 Rizzo was appointed Director of the International Choreographic Institute in Montpellier, formerly known as the Centre Chorégraphique National de Montpellier Languedoc-Roussillon-Midi-Pyrénées.
At REDCAT, d'aprés une histoire vraie was performed on a white floor with a very simple set, designed by Rizzo, that consisted of a wire chair, a book lying next to it, three black balls of different sizes underneath the chair and a potted palm. There was also a pair of white shoes sitting on the very edge of the stage downstage center out of sight for much of the audience. The two musicians and their percussion instruments were atop a large platform located in the upstage right corner.
d'aprés une histoire vraie began with a single figure performing a simple movement phrase with the folk dance elements. He was soon joined by another and then another until all eight men were performing this same phrase, building on it as they went. The phrase was inverted, reversed and combined into duets, trios and more. It was wonderful to watch this simple and familiar folk dance theme split open like an atom and explode into kaleidoscope-like patterns only to revert back to its simplicity. It accomplished all this, however, while retaining its purity, sense of male bonding and folk dance quality.
Just beyond the half way mark of this hour and ten-minute work, the set items listed above were carried off stage one at a time, beginning with the white shoes, until only the musicians remained. These items had not been touched or included in the dance until this occurred. Whether or not one agrees with a set being onstage without a purpose, somehow it worked. The set appeared natural to the folk dance atmosphere. This manner of removing the set also gave a purpose to each man's exiting the space.
While the musicians performed their truly powerful and driving score, the lighting design slowly replaced the dancers. Light swirled, cloud-like shadows moved across the floor and a soft strobe light effect made a brief appearance. Then again a single male figure entered the space and the music and the light show abruptly stopped as if cut off mid-sentence. The building up of a different movement phrase, with the same elements, and the music began again.
One memorable section in this second half was when all eight men clasped hands in a complex, cat's cradle-like pattern. They moved very rapidly across the space, constantly weaving in and out of each other forming more and more twisted patterns. One wrong move and the entire pattern would have fallen apart, but that did not happen. The group of eight became seven as one man was dropped off to lie motionless on the floor. Seven became six and so on until only two men with long hair remained connected. A strange but beautiful duet then transpired where their hair became a living part of the movement, covering their faces for the majority of the lengthy duet.
It speaks volumes about Christian Rizzo's talent as a choreographer that he can take simple phrases from a folk dance and build them into a work such as d'aprés une histoire vraie that spans over an hour, while managing to keep his audience spellbound. Those eight truly incredible men who performed this work that is simple, complex, intricate and physically demanding were Fabien Almakiewicz, Yaïr Barelli, Massimo Fusco, Miguel Garcia Llorens, Pep Garrigues, Kerem Gelebek, Filipe Lourenço, and Roberto Martinez. The truly vivacious lighting design was created by Caty Olive.
For more information regarding REDCAT and to purchase tickets: https://www.redcat.org/.


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It's no surprise that anyone visiting Goa is always looking for some great restaurants in North Goa. And why not? North Goa is the tourist hub, full of great resorts and even better restaurants and cafe´s. Not only do these places offer lip-smacking food, some of them also provide great views of a river or the ocean. I love spending my Sundays sitting at a great cafe, enjoying great food, catching the wind in my hair while laughing with my friends. Any of my friends who plan a visit to Goa always ask me about the best restaurants in North Goa. Usually, I send them recommendations along with links to my Instagram posts. However, it was about time that I documented my list of North Goa restaurants for everyone who is visiting Goa!
The Lazy Goose, tucked away under the Nerul bridge is just a 10 minute drive away from Reis Magos fort. I came here on a Sunday afternoon and I was greeted by a pleasant breeze from the nearby river. Next, what I saw was a chic yet jaunty restaurant that gave off sailor-vibes. Wooden chairs, blue cushions and potted pink bougainvillea flowers looked so inviting! The food only topped off the entire experience of dining along the river. The excellent service was the cherry on top!
Reserve a table in advance. Ask for the tables immediately by the river for the best views!
Located in the prime location of Penha de Franca, Moki Bar and Kitchen is known for it's quirky rooftop decor and an exquisite menu offering mouth-smacking dishes. For starters, I had the Parizza, which is a paratha bread with pizza toppings. It didn't feel any different from a thin crust pizza. The flavor of the Parizza and texture of the base was spot-on. Moving on to the main course, I had their Chef Style Makai and Paneer Tawa Masala with came with a choice of either 2 Kerala parota or rice. Naturally, my love for Kerala made me choose the Kerala parota. The curry, as shown in the picture above, was highly flavorsome and it made me keep wanting more. The parota was soft, and melted in the mouth. I could easily rate this as one of the best restaurants in North Goa for lunch or dinner.
Located at the right end of Anjuna beach, Eva cafe is a bohemian paradise that will transport you to the shores of Santorini in Greece. Get here before the lunch hour rush, pick the table facing the ocean, grab a chilled beer and let the ocean view work it's magic on you. What adds to the charm is this place is the bohemian decor complete with lace fabric, burlap table mats, sea shells and dream catchers. Some quirky vintage curious top off the decor. This place is best enjoyed on a lazy afternoon. While technically, this place is a cafe, but it's so good that I couldn't exclude it from my list of North Goa restaurants. You can also read my review of Eva Cafe on Zomato.
Only a few steps ahead of Eva Cafe lies Purple Martini. With white chairs and blue cushion, once again you will find yourself at a restaurant that gives off Greece vibes. This restaurant has a decent menu but it's not the food that makes it famous. Catching the sunset at Purple Martini is one of my favorite things to do. A chilled beer only enhances the experience. The stunning sunset from here is the main reason why it's featured in my list of the best restaurants in north Goa.
Baba au Rhum is another great cafe located in Anjuna. While it isn't located by the ocean, it makes up for the lack of ocean view by providing a relaxed vibe and scrumptious food at great prices! A major plus point of this place is that they have plentiful seating. The wooden chairs located in the backdrop of lush greenery makes for a great setting for a meal. A noteworthy mention here is the wood-fire pizza. The thin crust and the right amount of sauce tasted delicious.
Ever read Alice in Wonderland and wondered what the world looked like to Alice? Well, you don't have to imagine that anymore. Morgan's Place, in Penha de Franca, brings alive the quirky world of Alice. It's unbelievable the amount of work they have done to make the place look like you've been transported to Wonderland with Alice. From walls, floors, chairs, paintings to curios, everything goes along with the theme. If the decor isn't fantastic enough, try the authentic Italian dishes served here. You will be nothing short of amazed! I particularly loved the spaghetti in red sauce at Morgan's Place.
Gunpowder, located in Assagao is just 6 km from Anjuna. The place is a combination of restaurant, bar and bohemian boutique store. They have an outdoor seating while the bar is indoor. The outdoor seating comprises of wooden beaches underneath a canopy of colorful artificial flowers and potted plants. The bar is an interesting contrast to the outdoor. Not only is it chic, it also has some bohemian elements which ties it to the overall vibe of the place. This is what makes this restaurant one of the best restaurants in north Goa. The food served here is quintessential South Indian. If that appeals to your taste buds, hop over to Gunpowder for a great day.
Hi there! I'm Soujanya Rai, a Goa-based software developer. I have a love affair with coffee, tattoos, flowers, spicy food and fashion. I'm a beginner at photography and an expert at clumsiness. Being a travel enthusiast, I created The Spicy Journey to be able to share my stories, (mis)adventures, tips and shenanigans.


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Kickin' It: A rivalry so bitter it extinguishes light and turns water to ice
Your football news and links for October 16, 2013.
By Aaron Campeau@AaronCampeau Oct 16, 2013, 9:00am BST
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Paul Gilham
World Cup qualifying is all but done, with just the playoffs left to determine the field. Mercifully, we get back to normality this weekend, and we can all forget about international football for a few months. Still, even though international weeks are typically boring and awful, there's some interesting stuff happening out there.
Shaun Teale: My wife didn't try to run Harry Redknapp over | Birmingham Mail
Harry Redknapp wrote a book, which is quite frankly unbelievable in and of itself. In it, the Queens Park Rangers manager claims the wife of former Villa defender Shaun Teale tried to run him down with a car over a wage dispute while both player and manager were at Bournemouth. One major problem with Redknapp's version of events: Teale claims his wife didn't learn to drive until several years after the incident is claimed to have occurred. It's not the only claim made by Redknapp that's been disputed by the others supposedly involved, which doesn't really come as too much of a surprise.
U.S. stage unbelievable late comeback to beat Panama and save Mexico's World Cup hopes | Dirty Tackle
World Cup qualifying in CONCACAF is usually pretty crazy, but the nuttiness doesn't typically involve Mexico, the region's traditional power. This hasn't been the best cycle for El Tri, but heading into the evening it would have taken a loss to Costa Rica and a US loss to Panama to keep Mexico from qualifying for a playoff with New Zealand with a trip to Brazil on the line. Despite the long odds, that came very close to happening, and now Mexico will have to come to grips with the fact that their fiercest rivals are responsible for their (likely) qualification for the World Cup.
Striker scores twice against local rivals – and has his power and water cut off | the Guardian
Losing to your local rival is really unpleasant. I'm painfully aware of this. It can really mess up your day, and at the height of your frustration you'd likely do anything to exact some manner of revenge. Some public servant(s) in Tajikstan did just that.
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HOLTECAST | Southampton 1-0 Aston Villa - In Unai Emery we trust! Cole and Seb gathered around their microphones and chatted all things Aston Villa following Saturday's win over Southampton, the belief within the fan base thanks to Unai Emery, and what the final days of the January transfer window could look like for the football club. You can listen for FREE on Acast, Apple Podcasts, and Spotify - dig in!

redpajama_set_name: RedPajamaCommonCrawl


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Chairman's Blog - March 2018
NEW SECONDARY SCHOOL
The council reports that hundreds more parents are applying for school places this year, due to the rise in population and the improvement in local school results. The council has received 2,296 admission applications for September 2018, which is 268 more than last year. They say that there is an urgent need to build a new secondary school.
A new Harris Academy Wimbledon is due to open on a temporary site in September this year at Whatley Avenue (off Martin Way). This will be a non-selective state school for both girls and boys. The school is due to move to a permanent new school building in High Path, South Wimbledon, in September 2020.
Official figures from the government show that all the borough's secondary schools continue to be judged good or outstanding by Ofsted; and Department of Education results show that the progress pupils locally have made between the end of Key Stage 2 and the end of Key Stage 4 in eight subject qualifications, with maths and English given extra weight, is among the best in the country.
ROAD TRAFFIC CRASHES AT THE JUNCTION OF CROSSWAY AND WESTWAY
The local Guardian reports that a resident of Westway has seen six crashes at the junction of his road with Crossway within the last two years, and that he has had to replace his fence three times due to cars crashing into it. He is demanding that the council narrow the road or that CCTV measures are put in place to prevent a serious injury at this junction.
It is a very busy junction, particularly in the rush hour, and used by vehicles going to and from the two level crossings. Many pedestrians, including children, need to cross it.
The council states that it will carry out a further investigation into road safety in the area and will ask for police reports.
BUILDERS CHURNING UP GRASS VERGES
The trend for people to improve their homes by building extensions, including loft extensions, means that a number of local roads now have large builders' and scaffolding lorries parked near them for much of the day. These roads are often narrow and were not designed for vehicular use. The result is that the lorries park on grass verges, leaving wheel ruts and destroying the grass, or else on the pavement or asphalt verge, damaging that. Prime examples of this can be seen in Heath Drive, off Grand Drive.
I understand that it is a requirement of planning permission that builders reinstate public property that has been damaged, but it is not always possible to find out which builder has caused which damage.
The problems in Heath Drive have been reported to the council by the Councillors. Building inspectors, when they certify the extensions, need also to make a note of the damage done to the local infrastructure, and ensure that this is re-instated at the builders' cost.
Our AGM will be held on the 11 April 2018. Click here for further details.
John Elvidge


If you're raising children today, it can be easy to focus on the negative, especially with the constant media stream coming from all directions. Helping children see the positive in the world today is more important than ever. Research shows that optimists, who believe they can achieve success, are in fact more able to do so. They are less likely to get depressed, get fewer illnesses, have longer relationships, and live longer.
Here are 12 tips to help your children develop a positive outlook on life.
Quit complaining. Focusing on negative thoughts and frustrations is classic pessimism. The more you moan about troubles at work, money problems, etc. the more likely it is that your children will learn to do the same thing. Instead, try talking about things that go right.
Have high expectations. Kids won't develop an optimistic, "can-do" attitude unless they have the opportunity to prove their worth.
Encourage reasonable risk-taking. Discouraging your children from doing activities because they might not be as skilled as other children can undermine their confidence.
Wait before reacting. Let your children try to solve things without your help. This will boost their sense of accomplishment and also make them more optimistic about what they can do in the future.
Be realistic and honest, even in difficult situations. For example, if your child is having a difficult time making friends at a new school, empathize with him/her, but explain that making friends takes time and he/she needs to take active steps to meet other kids and make friends.
Show your children you love them. Make time for them every day and use words and affection to show you care.
Be a positive role model. Your children will mimic your behaviors, so make sure you show them constructive ways of dealing with life's challenges and misfortunes.
Allow your child to experience success. For example, give them age-appropriate tasks around the house and praise them for their efforts.
Teach practical skills with patience and humor. Help your child to keep up with peers – for example, tying shoelaces, using cutlery and managing the toilet.
Listen to their problems. Take concerns seriously and be supportive as they learn how to cope with life's hurts and disappointments.
Point out the 'silver lining' in every cloud. Get your child into the habit of looking for the positives, even times where it may only seem like the chance to learn from a mistake.
Make happy events a regular part of your family life.


text:
Drivers urged to slow down at night after five kiwi killed
13 Aug, 2019 07:00 PM 2 minutes to read
A sign on Redcliffs Rd keeps track of the number of kiwi killed by cars so far this year. Photo / Dean Wright
By: Peter de Graaf
Peter de Graaf is a reporter for the Northern Advocate
peter.degraaf@northernadvocate.co.nznorthernadvocat
Kerikeri drivers are being urged to slow down at night after the deaths of five kiwi on one stretch of road so far this year.
The birds were all hit by cars on Redcliffs Rd and Opito Bay Rd, which link Kapiro Rd with Opito Bay on the north side of Kerikeri Inlet.
Dean Wright, co-ordinator of conservation group Kerikeri Peninsula Pest Control, said three kiwi were killed in April-May and two more in the past fortnight.
Three of the deaths — including the most recent one last week — occurred along a 50m section of Redcliffs Rd, about 100m past Kate's Riding Centre towards Opito Bay.
Wright said it was unclear why so many kiwi were killed at that point. Both sides of the road were bordered by farmland but there was bush beyond that.
It was possible they were moving between two pockets of bush, "but without a camera we're just guessing".
Wright said kiwi deaths on the 8km stretch of road were sadly common.
''It's not something that's new, we've been losing this many kiwi out here for a long time.''
He pleaded with drivers to slow down between dusk and dawn and to be extra vigilant in the Rangitane-Opito Bay area, which has one of the highest kiwi densities of any residential area in New Zealand.
''They have absolutely no road sense and because the bush is quite close to the edge off the road you don't have much time to take evasive action. It's tragic to be losing this many.''
Rat 'tsunami' invading homes, threatening wildlife
Now it's a rat plague - stoats will be next
'Nora's Ark' to protect kauri from dieback threat
12 Jul, 2019 06:00 AM Quick Read
Forest & Bird files Environment Court appeal
24 Jul, 2019 10:30 PM Quick Read
Some of the dead kiwi were dropped off at Wright's home while others were taken directly to the Department of Conservation office on Landing Rd.
The group has put up a number of signs along Redcliffs Rd, including one which is updated with the current kiwi road toll.
Go to kppc.kiwi for more information about the Kerikeri Peninsula Pest Control group.
Waitangi concert promises bright future for te reo Māori music


Some new leaders come in like a hurricane. They're a storm that is strong enough to blow down anything and everything in their way.
A new leader knows he's been hired to make change, and he wants to get busy. But in moving to make those changes fast, he destroys the relationships with the very people he needs to succeed. He hasn't taken time to learn the ground truth, and he hasn't walked a mile in his team's shoes (even if he has walked many miles in shoes just like theirs before being hired into this role).
Trust is established over time, and formal authority is authority in it's weakest and most easily resented form. Moral authority takes time to build, but it is one of the most powerful levers a leader can pull.
Some leaders come in like a lamb. They don't want to make the mistake of going too fast and instead go to slow. This is particularly true when it comes to changing out members of the team they inherit. They're right to go slow when it comes to people because they're dealing with a person's life. They're also right to go slow because new leadership can and does make the difference, magically transforming a poor performer into what they could have been the whole time.
But you can move too slowly. You can keep people who aren't willing to change too long, and you can leave them around long enough to put your new initiatives at risk.
It's a tricky balance, but a good leader knows where and when to move fast and where to take more time.


Anything else intetesting going on in the world besides the election?
I just built a solar and wind powered strobe light. Only good at nite or a dark room tho.
How does a solar powered strobe light function at night?
Most solar devices utilize a battery for storage.
Strobe lights are typically most effective in the dark.
Could it be use to determine the speed of dark?
Spend more time with your family. If you're not married and don't have kids, take your nephews an nieces to the zoo or a ballgame or just to the local Baskin' Robbins.
If not that, think of a business idea and start working on it. Don't worry, if you succeed, the government will only take 50%. The other 50% is yours.
Go get a good audio book and take a long walk in the crisp delicious Autumn air or go to the gym.
Go to coursera, find something interesting and take a MOOC.
No need for boredom; there's always plenty to do.
I've lost 30 lbs in the last three-ish months through intermittent fasting and looong walks in the woods. Life is better. As soon as the rain shows up I'll have to switch to racquetball or something because I have 20 lbs to go. Also, try not reading this retarded forum. I've not been able to quit for some reason, but I'm sure I'd be happier for it.
Eh, as long as you keep it to a few minutes (a few being maybe 20) a day, it's not a big deal.
Congrats on the weight loss. Keep it up! I lost 20 lbs last Fall and I feel amazing. I've also improved my tennis game just by being lighter. The same guys I was evenly matched with last year I beat steadily this year.
Even up in Oregon, the rain shouldn't stop you from walking. Get a good jacket with a hood and as long as it's just misting, walking can be doable, even pleasant. The only thing that needs to stop you is a steady downpour and that doesn't happen all day very often, even in the great Pacific Northwest.
I love the cloudy and cool Fall weather. Very pleasant walking weather. In the winter it can be even better if you have a good coat and gloves where walking quickly for 20 minutes warms up your body so that cold temps don't really bother you. It's only in the sticky summer mess that walking is unpleasant and even then, waiting until 10 or 11 at night usually makes it bearable.
Be thankful that you don't have to live in a more 'interesting' place such as Aleppo, Syria.
Russia is trying to reform the UN. To the Western nations it seems like a good idea to have a one world dictatorship with those entities in control that are responsible for most of the wars of the past couple of centuries and the cause of government sponsored terrorism. Russia has been shunned for the most part in having any say in international policies with this organization and Putin is using Judo politics in trying to make changes. In other words he has been not condeming the UN but attempting to use its momentum against itself. Clever man this Putin is. Interesting how he used the same tactics to enter Syria using the momentum of the "war against terror". It makes it a little difficult to see what he believes in when speaks and we can only hope they will be part of a check in world power.
Even up in Oregon, the rain shouldn't stop ...[text shortened]... walking is unpleasant and even then, waiting until 10 or 11 at night usually makes it bearable.
Thanks. Yeah I can deal with the rain, but I'm going to have find trails that don't turn to mud. I did about 7 miles out there today and it was already starting to get muddy.


text:
Tuhegli Damiani János (Tuhegli, Bosznia, 1710. június 21. – Vác, 1784. szeptember 3.) pozsonyi, később váci kanonok.

Életpályája

Damiani Lukács ezredes és Edelenhausen Felicitas bárónő fia volt. 1726-ban Rómába ment, a Szent Péter és Pál collegiumban végezte bölcseleti és teológiai tanulmányait. 1735. március 5. napján áldozópappá szentelték föl. XII. Kelemen pápa kívánságára, Esterházy Imre esztergomi érsek 1737. augusztus 12-én pozsonyi kanonokká nevezte ki. A prímási szék megüresedése alkalmával Mária Terézia 1749. január 2-án ugyanott primiceriussá vagyis kántorkanonokká tette. Althann Mihály Károly váci püspök nemsokára egyházmegyéjébe vitte és püspökhelyettessé, valamint papnevelőintézeti igazgatóvá tette. Althann halála után lemondott hivatalairól és az irodalomnak élt. Végül Migazzi Kristóf bécsi hercegérsek 1768. április 10-én kinevezte a váci egyházkerület kormányzójává.

Munkái

 Maria dei genitrix virgo suae animationis instanti speculum sine macula. Posonii, 1758.
 Doctrina verae Christi ecclesiae ab omnibus praecipuis antiqui, medii et novi aevi haeresibus vindicata complectens bis centum famigeratissimas controversias in canonicis scripturis, solidisque rationibus fundatas. (Ugyanott, 1760)
 Justa religionis coactio, seu apodixis quod reges, principes, magistratus, et dynastiae romano-catholici, habita occasione et opportunitate, possint, ac debeant acatholicos in suis ditionibus commorantes, cogere, mediis etiam violentis, imo et extremis (quoties mitiora non prosint) ad amplectendam veram et unica salvificam romano-catholicam fidem. Editio altera. Buda, 1763. (A magyar királyi helytartótanács 1770. június 25-i rendeletével elkoboztatta.)

Kéziratban: Maria triumphans és Examen libri Symbolici Russorum (1778)

Források

Magyar katolikus papok
1710-ben született személyek
1784-ben elhunyt személyek
Pozsonyi kanonokok
Váci kanonokok

redpajama_set_name: RedPajamaWikipedia


Maggie is a young college drop-out who is doing quite well working in a lesbian bookstore in Vancouver. She meets and falls for Kim, and her life seems to be stable. Then her mom calls, telling her that not only is she getting divorced for the second time, but she and Maggie's little brother are moving to Vancouver, and moving into Maggie's place. If this wasn't horrifying enough, Mom knows nothing of Maggie's sexuality, and so she and Kim have to concoct a plan to break the news gently.


text:
Q: Jackson Error When Doing Integration Test with Sprint Boot & GraphQL I'm trying to write integration tests for my GraphQL code with Spring Boot and having problems.
First, this is the library I'm using for GraphQL Spring Boot integration:
<dependency>
        <groupId>com.graphql-java-kickstart</groupId>
        <artifactId>graphql-spring-boot-starter</artifactId>
        <version>7.0.1</version>
    </dependency>
    <dependency>
        <groupId>com.graphql-java-kickstart</groupId>
        <artifactId>graphql-spring-boot-starter-test</artifactId>
        <version>7.0.1</version>
        <scope>test</scope>
    </dependency>

I've got all my resolvers built perfectly, I can call them from postman no problem. It's my test code where the problem comes up. I want to do integration tests, running end-to-end. For my first test, here is my GraphQL query file:
query {
    clients {
        id
        name
        clientKey
        enabled
        allowAuthCode
        allowClientCredentials
        allowPassword
    }
}

This is a perfectly valid query for my GraphQL API, I can run it from Postman successfully. Here is my test code:
@ExtendWith(SpringExtension::class)
@SpringBootTest(webEnvironment = SpringBootTest.WebEnvironment.RANDOM_PORT)
class ClientQueryIntegrationTest {

    @Autowired
    private lateinit var graphqlRestTemplate: GraphQLTestTemplate

    @Test
    fun test() {
        graphqlRestTemplate.addHeader("Authorization", "Bearer $token")
        val response = graphqlRestTemplate.postForResource("graphql/getAllClients.graphql")
    }

}

When I run this, I get a Jackson deserialization error with my initial GraphQL request. I know it is loading my GraphQL query correctly because of the information in the error message, I just don't know why Jackson is de-serializing this incorrectly during my tests.
com.fasterxml.jackson.databind.exc.MismatchedInputException: Cannot construct instance of `graphql.kickstart.execution.GraphQLRequest` (although at least one Creator exists): no String-argument constructor/factory method to deserialize from String value ('{"query":"query {\n    clients {\n        id\n        name\n        clientKey\n        enabled\n        allowAuthCode\n        allowClientCredentials\n        allowPassword\n    }\n}\n","variables":null}')
 at [Source: (String)""{\"query\":\"query {\\n    clients {\\n        id\\n        name\\n        clientKey\\n        enabled\\n        allowAuthCode\\n        allowClientCredentials\\n        allowPassword\\n    }\\n}\\n\",\"variables\":null}""; line: 1, column: 1]

What can I try next?

redpajama_set_name: RedPajamaStackExchange


FoodScience of Vermont Dimpro Description: A dietary supplement to support hormonal balance and weight management in men and women. DIMPRO is a proprietary blend of DIM (Diindolylmethane) with Vitamin E and Phosphatidylcholine. DIM causes a beneficial shift in the balance of estrogen, testosterone, and progesterone in both men and women.* Good estrogen metabolites support overall health by functioning as antioxidants, helping to eliminate unwanted materials, mobilizing stored fat (for use as energy), supporting Hormone Replacement Therapy (HRT), as well as weight management, exercise tolerance, development of lean muscle tissue, and by reducing the effects of aging such as those found in menopause and prostate dysfunction Free Of citrus, corn, egg, gluten, milk, sodium, wheat, yeast, added sugars, synthetic dyes, artificial flavorings and preservatives. Disclaimer These statements have not been evaluated by the FDA. These products are not intended to diagnose, treat, cure, or prevent any disease.
As a dietary supplement, take 2 capsule with food, 1 to 2 times daily.


text:
April - 2009 - issue > Tech Buzz
RFS Opens New Manufacturing Plant in India
Shoukath Koduvally
In the first phase of a major investment plan for India, Radio Frequency Systems (RFS), a provider of wireless solutions, has opened a new manufacturing plant for microwave antenna systems, in Kolkata. The company intends to meet the burgeoning requirement across India and the APAC region for microwave connectivity in mobile backhaul and transmission backbone applications.
"In India, a new cellular base station site goes on air every four minutes. This is generating a huge need for the timely and cost-effective delivery of high-quality microwave radio systems to connect new base transceiver stations (BTS) to cellular networks," says Kuldeep Tikoo, General Manager, RFS India. By ramping up the production capacity in India, the company is expected to be able to lower the cost of logistics and reduce import duties, while cutting delivery lead times by more than 50 percent.
No Compromise on Timely Delivery
Value of Integrated Vulnerability Management and Intrusion Prevention Systems
MENTIS Software Answering the What, Where, How, & Who of Sensitive Data
It is All About Proving one's Merit
Paprikaas to Use HP's Blade Server Technology


President Trump will support a path to citizenship for 1.8 million young immigrants brought to the U.S. illegally as children, in exchange for a $25 billion fund for the border wall and other immigration cuts.
President Trump will support a path to citizenship for 1.8 million young immigrants brought to the U.S. illegally as children, in exchange for a $25 billion fund for the border wall and other immigration cuts. During Trump's first day at the Economic World Forum in Davos, he met with British Prime Minister Theresa May and threatened to withhold aid money from Palestinians unless they resume peace talks with Israel in a meeting with Israeli Prime Minister Benjamin Netanyahu. In an interview with CNBC, Trump said he would be willing to re-enter the Trans-Pacific Partnership "if we were able to make a substantially better deal." And Senator Chuck Grassley, the chair of the Senate Judiciary Committee, said he wants to release transcripts related to a June 2016 meeting in Trump Tower with a Russian lawyer.
Who's to Blame?: Norm Ornstein explains the real cause of the government shutdown last week—and why it might happen again.


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text:
\section{Introduction}

The Cahn--Hilliard equation \cite{CarterTaylorCahn97} is often
used to model microstructures arising from spinodal decomposition
in, say, polymer mixtures. One of the simplest systems exhibiting
this behavior would be a mixture of two polymers made from
monomers, A and B, with distinct chemical properties -- e.g., if A
is hydrophilic whereas B is hydrophobic. In such cases, a monomer
unit is attracted to units of the same type, while being repelled
by the other type, implying that the most energetically favorable
state is the one where A and B units are fully segregated. Such a
tendency is indeed confirmed by numerical modelling of the
Cahn--Hilliard equation \cite{deMelloFilho05} and is also in
agreement with stability analysis of similar models
\cite{BenilovChugunova10}.

One feature of the Cahn--Hilliard model is that the order
parameter is conserved globally (reflecting the mass conservation
law). The standard model, however, can be modified for
microstructures where the order parameter is conserved
\emph{locally}\ \cite{OhtaKawasaki86}. The modified model applies,
for example, if chains of the A and B monomers are parts of the
same polymer molecule, known as a `block copolymer'
\cite{HadjichristidisPispasFloudas03}, in which case they can
never be separated by a distance larger than the size of a single
molecule.

Systems with locally conserved order parameter are of particular
interest in nanotechnology. In particular, block copolymers are
used to template nanopatterns at surfaces, on scales that are too
small for traditional top-down photolithography
\cite{FitzgeraldEtAl09}. Such patterns have to be `directed'\
using chemical pre-patterning or topography, which is known as
graphoepitaxy and can provide excellent pattern registry
\cite{StuenEtAl10}. In its simplest form, conventional
lithographic techniques are used to create trenches in a silicon
wafer -- then the trenches are filled with block copolymer which
orders into lamellae parallel to the sidewall on annealing
\cite{FarrellEtAl10}. Finally a selected monomer is chemically
etched away and the remaining polymer used as an etch mask to
facilitate pattern transfer to the substrate, creating nanowires
on a scale too fine to be manufactured by conventional techniques
\cite{BorahEtAl11,MorrisEtAl12}.

The lamellae used to template the nanowires correspond to
\emph{frozen waves} (i.e. periodic time-independent solutions) of
the one-dimensional version of the modified Cahn--Hilliard
equation. It is particularly important whether these solutions are
unique or perhaps multiple stable solutions exist, as the latter
would impede one's control over the manufacturing process.

The present paper answers the above question by examining the
existence and stability of frozen waves in the modified
Cahn--Hilliard equation. In Sect. \ref{Section 2}, we shall
formulate the problem mathematically. In Sect. \ref{Section 3},
the \emph{existence }of frozen-wave solutions will be discussed.
In Sect. \ref{Section 4}, we shall present the results of a
\emph{stability} analysis of frozen waves.

\section{Formulation\label{Section 2}}

Consider a one-dimensional diblock polymer, with the
characteristic thickness $l$ of the A/B interface and mobility $M$
(the latter characterizes the diffusion of the order parameter
$\phi$). Using $l$ and $l^{2}/M$ to non-dimensionalize the spatial
coordinate $x$ and time $t$ respectively, we can write the
one-dimensional version of the modified Cahn--Hilliard equation
(MCHE) in the for
\begin{equation}
\phi_{t}+\left(  \phi-\phi^{3}+\phi_{xx}\right)  _{xx}+\alpha\phi=0, \label{1
\end{equation}
where $\alpha$ determines the ratio of the characteristic size of
the region over which the order parameter is conserved to $l$.

As shown in Ref. \cite{LiuGoldenfeld89}, the MCHE admits frozen waves only i
\[
0\leq\alpha<\tfrac{1}{4},
\]
whereas the wavelength (spatial period) $\lambda$ must satisf
\begin{equation}
\frac{2\pi}{\sqrt{\tfrac{1}{2}\left(  1+\sqrt{1-4\alpha}\right)  }
<\lambda<\frac{2\pi}{\sqrt{\tfrac{1}{2}\left(  1-\sqrt{1-4\alpha}\right)
}\label{2
\end{equation}
(see Fig. 1). Ref. \cite{LiuGoldenfeld89} also computed examples
of frozen waves and the energy density $E$ as a function of a
frozen wave's length $\lambda$ -- which turned out to have a
minimum at a certain $\lambda =\lambda_{0}$. Given that the energy
cannot grow and is, thus, a Lyapunov functional, a conclusion was
drawn that $\lambda_{0}$ is stable.

\begin{figure}
\includegraphics[width=83mm]{fig1.eps}
\caption{The existence region of frozen waves (those described by
one-wave solutions) on the $\left(  \alpha,\lambda\right)  $-plane
($\alpha$ is the parameter in the modified Cahn--Hilliard equation
(\ref{1}), $\lambda$ is the wavelength). The boundaries of the
region are given by (\ref{2}).} \label{fig1}
\end{figure}

However, the fact that $\lambda_{0}$ minimizes the energy of
frozen waves means only that the corresponding wave is stable with
respect to perturbations \emph{of its length}, but not necessarily
to \emph{arbitrary} perturbations (for which the `general' second
variation of $E$ needs to be examined). On the other hand,
stability rarely occurs for a single value of a parameter --
hence, if $\lambda_{0}$ is indeed stable, it is likely to be one
of a range of stable wavelengths.

In what follows, we shall clarify the above issues by examining
the stability of frozen waves through the standard linear
analysis, not involving energy arguments.

To do so, we introduce frozen waves as time-independent solutions,
$\phi
=\bar{\phi}(x)$, for which Eq. (\ref{1}) yield
\begin{equation}
\left(  \bar{\phi}-\bar{\phi}^{3}+\bar{\phi}_{xx}\right)
_{xx}+\alpha
\bar{\phi}=0. \label{3
\end{equation}
Together with the condition of spatial periodicity
\begin{equation}
\bar{\phi}(x+\lambda)=\bar{\phi}(x), \label{4
\end{equation}
Eq. (\ref{3}) determines $\bar{\phi}(x)$.

To examine the stability of a frozen wave $\bar{\phi}(x)$, assum
\begin{equation}
\phi=\bar{\phi}(x)+\tilde{\phi}(t,x), \label{5
\end{equation}
where $\tilde{\phi}$ describes a small disturbance. Substituting
(\ref{5})
into Eq. (\ref{1}) and linearizing it, we obtai
\begin{equation}
\tilde{\phi}_{t}+\left(
\tilde{\phi}-3\bar{\phi}^{2}\tilde{\phi}+\tilde{\phi
}_{xx}\right)  _{xx}+\alpha\tilde{\phi}=0. \label{6
\end{equation}
We confine ourselves to disturbances with exponential dependence
on $t$ (which
are usually a reliable indicator of stability in general)
\begin{equation}
\tilde{\phi}(x,t)=\psi(x)\,\operatorname{e}^{st}, \label{7
\end{equation}
where $s$ is the disturbance's growth/decay rate. Substitution of
(\ref{7})
into (\ref{6}) yield
\begin{equation}
s\psi+\left(  \psi-3\bar{\phi}^{2}\psi+\psi_{xx}\right)
_{xx}+\alpha\psi=0.
\label{8
\end{equation}
Unlike the base wave $\bar{\phi}$, the disturbance $\psi$ does not
have to be periodic; it is sufficient that the latter is bounded
at infinity. Given that $\psi$ is determined by an ordinary
differential equation with periodic coefficients [Eq. (\ref{8})],
the assumption of boundedness amounts to the
standard Floquet condition
\begin{equation}
\psi(x+\lambda)=\psi(x)\,\operatorname{e}^{\mathrm{i}\theta}, \label{9
\end{equation}
where $\theta$ is a real constant. Physically, condition (\ref{9})
implies that, if the disturbance propagates by one wavelength of
the base solution, the disturbance's amplitude remains the same,
whereas its phase may change by a value of $\theta$.

Eqs. (\ref{8})--(\ref{9}) form an eigenvalue problem, where $s$
and $\psi$ are the eigenvalue and the eigenfunction. If, for some
values of the phase shift $\theta$, one or more eigenvalues exist
such that $\operatorname{Re}s>0$, the corresponding base wave
$\bar{\phi}(x)$ is unstable.

\section{Frozen wave solutions\label{Section 3}}

It turns out that a lot of physically meaningful information about
frozen waves can be obtained in the limit of weak nonlinearity,
i.e. under the
assumptio
\[
\left\vert \bar{\phi}\right\vert \ll1.
\]
In order to understand qualitatively what to expect in this case,
one can simply omit the nonlinear term in Eq. (\ref{2}) and seek a
solution of the
resulting linear equation in the for
\begin{equation}
\bar{\phi}=\varepsilon\cos\left(  kx+p\right)  , \label{10
\end{equation}
where the wave's amplitude $\varepsilon$ and phase $p$ are
arbitrary, whereas
the wavenumber $k$ satisfie
\begin{equation}
-k^{2}\left(  1-k^{2}\right)  +\alpha=0. \label{11
\end{equation}
Assuming $k>0$ and recalling the relationship between the
wavelength and the
wavevector
\begin{equation}
\lambda=\frac{2\pi}{k}, \label{12
\end{equation}
one can see that, in the linear approximation, only two
wavelengths are allowed -- coincidentally, the same values which
represent the upper and lower boundaries of the existence region
(\ref{2}), shown in Fig. 1). Under the \emph{weakly nonlinear}
approximation, in turn, one should expect the wavelength to be
close, but not necessarily equal, to one of the above two values,
with a deviation from them proportional to some degree of the
wave's amplitude $\varepsilon$. Solutions similar to (\ref{10})
are the ones computed in Ref. \cite{LiuGoldenfeld89}; they will be
referred to as `one-wave solutions'.

To understand the physical meaning of \emph{two}-wave solutions,
seek a
solution of the linearized version of Eq. (\ref{3}) in the for
\begin{equation}
\bar{\phi}=\varepsilon_{1}\cos\left(  k_{1}x+p_{1}\right)
+\varepsilon
_{2}\cos\left(  k_{2}x+p_{2}\right)  , \label{13
\end{equation}
where $\varepsilon_{1,2}$ and $p_{1,2}$ are arbitrary and
$k_{1,2}$ are the
roots of Eq. (\ref{11}) such tha
\begin{equation}
0<k_{1}<k_{2}. \label{14
\end{equation}
Physically, solution (\ref{13}) represents a superposition of two
periodic waves of zero frequency, with wavenumbers $k_{1,2}$ and
phases $p_{1,2}$. Note that (\ref{13}) is periodic only if
$k_{2}/k_{1}$ (and, hence, $\lambda _{2}/\lambda_{1}$) are both
rational numbers, which occurs only for some values of $\alpha$
(several examples of such are illustrated in Fig. 2).

\begin{figure}
\includegraphics[width=83mm]{fig2.eps}
\caption{A schematic illustrating linear two-wave solutions. The
solid curve [determined by (\ref{11})--(\ref{12})] corresponds to
linear frozen waves. The lengths $\lambda_{1,2}$ of the two
`composite' waves are shown by dots and are connected by dashed
lines. Two-wave solutions exist only for those values of $\alpha$
for which $\lambda_{2}/\lambda_{1}$ is a rational number
(presented in the figure above the corresponding $\lambda_{2}$).
The wavelength of a two-wave solution as a whole equals the lowest
common multiple of $\lambda _{1}$ and $\lambda_{2}$.} \label{fig2}
\end{figure}

The above linear analysis, however, leaves several important
questions unanswered. On the $\left(  \lambda,\alpha\right)
$-plane, for example, one-wave solutions seem to exist near curve
(\ref{11})--(\ref{12}), whereas Ref. \cite{LiuGoldenfeld89} found
frozen waves only \emph{inside} this curve, and not \emph{outside}
(see Fig. 1). This discrepancy -- as well as the question of
existence of two-wave solutions -- will be clarified in Sects.
\ref{Subsection 3.1}--\ref{Subsection 3.2}. The case of strong
nonlinearity will be examined numerically for both types of frozen
waves in Sect. \ref{Subsection 3.3}.

\subsection{Asymptotic results: one-wave solutions\label{Subsection 3.1}}

It is convenient to change the spatial coordinate $x$ t
\[
\xi=kx,
\]
[where $k$ is, again, the wavenumber determined by (\ref{12})]. In
terms of
$\xi$, Eqs. (\ref{3})--(\ref{4}) becom
\begin{equation}
k^{2}\left(
\bar{\phi}-\bar{\phi}^{3}+k^{2}\bar{\phi}_{\xi\xi}\right)
_{\xi\xi}+\alpha\bar{\phi}=0, \label{15
\end{equation
\begin{equation}
\bar{\phi}(x+2\pi)=\bar{\phi}(x). \label{16
\end{equation}
We shall seek a solution as a series of the for
\begin{equation}
\bar{\phi}=\varepsilon\left(  \bar{\phi}^{(0)}+\varepsilon^{2}\bar{\phi
^{(2)}+\cdots\right)  , \label{17
\end{equation}
and also expand the wavenumber $k$
\begin{equation}
k^{2}=K^{(0)}+\varepsilon^{2}K^{(2)}+\cdots. \label{18
\end{equation}
To leading order, Eqs. (\ref{15})--(\ref{16}) reduce t
\begin{equation}
K^{(0)}\left(
\bar{\phi}^{(0)}+K^{(0)}\bar{\phi}_{\xi\xi}^{(0)}\right)
_{\xi\xi}+\alpha\bar{\phi}^{(0)}=0, \label{19
\end{equation
\[
\bar{\phi}^{(0)}(x+2\pi)=\bar{\phi}^{(0)}(x).
\]
We seek a solution in the for
\[
\bar{\phi}^{(0)}=A\cos(n\xi+p),
\]
where $A$ and $p$ are real constants and $n>0$ is an integer. It
is sufficient to examine the case $n=1$, as $n\geq2$ corresponds
to re-defining the solution's spatial period by including more
than one wavelengths in it, without changing anything physically.
We also let $A=1$ (as the wave's physical amplitude still remains
arbitrary due to the arbitrariness of $\varepsilon$) and $p=0$ (as
a phase constant can always be included later).
Thus, the leading-order solution become
\begin{equation}
\bar{\phi}^{(0)}=\cos\xi. \label{20
\end{equation}
Substitution of (\ref{20}) into (\ref{19}) yield
\begin{equation}
K^{(0)}=\tfrac{1}{2}\left(  1\pm\sqrt{1-4\alpha}\right)  . \label{21
\end{equation}
In the next-to-leading order, Eq. (\ref{15}) yield
\begin{multline*}
K^{(2)}\left(
\bar{\phi}^{(0)}+K^{(0)}\bar{\phi}_{\xi\xi}^{(0)}\right)
_{\xi\xi}\\
+K^{(0)}\left(
\bar{\phi}^{(2)}-\bar{\phi}^{(0)3}+K^{(2)}\bar{\phi}_{\xi\xi
}^{(0)}+K^{(0)}\bar{\phi}_{\xi\xi}^{(2)}\right)  _{\xi\xi}\\
+\alpha\bar{\phi}^{(2)}=0,
\end{multline*}
which, upon substitution of (\ref{20}), become
\begin{multline}
K^{(0)}\left(
\bar{\phi}^{(2)}+K^{(0)}\bar{\phi}_{\xi\xi}^{(2)}\right)
_{\xi\xi}+\alpha\bar{\phi}^{(2)}\\
=\left(  K^{(2)}-2K^{(2)}K^{(0)}-\frac{3K^{(0)}}{4}\right)  \cos\xi\\
-\frac{9K^{(0)}}{4}\cos3x, \label{22
\end{multline}
This equation can have a $2\pi$-periodic solution only if the term
involving
$\cos\xi$ on the right-hand side vanishes, which implie
\begin{equation}
K^{(2)}=\frac{3K^{(0)}}{4\left(  1-2K^{(0)}\right)  }, \label{23
\end{equation}
after which (\ref{22}) yield
\begin{equation}
\bar{\phi}^{(2)}=\frac{9K^{(0)}}{36K^{(0)}\left(
1-9K^{(0)}\right)  -4\alpha
}\cos3\xi. \label{24
\end{equation}
Recalling that $K$ is related to the wavenumber $k$ by (\ref{18})
-- and $k$ is, in turn, related to the wavelength $\lambda$ by
(\ref{12}), one can use
expressions (\ref{21}) and (\ref{23}) to obtai
\begin{equation}
\lambda_{1,2}=\frac{2\sqrt{2}\pi}{\sqrt{1\pm\sqrt{1-4\alpha}}}\left(
1\pm\frac{3\varepsilon^{2}}{8\sqrt{1-4\alpha}}\right)  +\operatorname{O
(\varepsilon^{4}), \label{25
\end{equation}
where $_{1}$ and $_{2}$ correspond to $+$\ and $-$\ respectively.
For $\varepsilon=0$ (i.e., in the linear limit), $\lambda_{1}$ and
$\lambda_{2}$ are the lower and upper boundaries of the existence
interval in Fig. 1, whereas an increase in $\varepsilon$ causes an
increase in $\lambda_{1}$ and a decrease in $\lambda_{2}$. This
means that one-wave solutions do not exist above $\left(
\lambda_{2}\right)  _{\varepsilon=0}$ and below $\left(
\lambda_{1}\right)  _{\varepsilon=0}$ -- which agrees with the
conclusions of Ref. \cite{LiuGoldenfeld89}.

Observe that expansion (\ref{25}) fails in the limit
$\alpha\rightarrow \frac{1}{4}$. This case should be examined
separately, by assuming
\begin{equation}
\alpha=\tfrac{1}{4}+\varepsilon^{2}\alpha^{(2)}. \label{26
\end{equation}
An expansion similar to the one above yields (technical details omitted
\begin{equation}
k^{2}=\tfrac{1}{2}\pm\sqrt{-\varepsilon^{2}\alpha^{(2)}-\tfrac{3
{8}\varepsilon^{2}}+\operatorname{O}(\varepsilon^{2}), \label{27
\end{equation}
from which $\lambda$ can be readily found. The corresponding
expansion of the
solution is (technical details omitted
\begin{equation}
\bar{\phi}=\varepsilon\left[
\cos\xi-\tfrac{9}{128}\varepsilon^{2}\cos
3\xi+\operatorname{O}(\varepsilon^{4})\right]  . \label{28
\end{equation}
Formulae (\ref{27})--(\ref{28}) describe the one-wave solution
near the `tip' ($\alpha\rightarrow\frac{1}{4}$,
$\lambda\rightarrow2\sqrt{2}\pi$) of the existence region shown in
Fig. 1. Formula (\ref{25}), in turn, describes the solution near
the boundary of the region, but not too close to its tip.

Finally, note that substitution of expression (\ref{21}) for
$K^{(0)}$ into
expression (\ref{24}) shows that $\bar{\phi}^{(2)}$ becomes infinite fo
\[
\alpha=\tfrac{9}{100},\hspace{0.4cm}\lambda=2\sqrt{10}\pi.
\]
This is another example where the straightforward expansion for
the one-wave
solution is inapplicable (in addition to the `tip point', $\alpha=\frac{1}{4
$, $\lambda=2\sqrt{2}\pi$). The unbounded growth of the
first-order term involving $\cos3\xi$ suggest that, in this case,
$\bar{\phi}^{(0)}$ should include both $\cos\xi$ and $\cos3\xi$ --
and not just the former term, as solution (\ref{20}) does.

This case will be examined in the next section.

\subsection{Asymptotic results: two-wave solutions\label{Subsection 3.2}}

As mentioned above, two-wave solutions exist near those values of
$\alpha$ for which the ratio of the wavenumbers of the two
individual waves is a rational number. Let $\alpha^{(0)}$ be one
of such values, with $\alpha^{(2)}$ being a
small deviation from it
\begin{equation}
\alpha=\alpha^{(0)}+\varepsilon^{2}\alpha^{(2)}. \label{29
\end{equation}
Substitution of this expression, together with expansions (\ref{17
)--(\ref{18}), in Eq. (\ref{15}) yields, to leading order
\begin{equation}
K^{(0)}\left(
\bar{\phi}^{(0)}+K^{(0)}\bar{\phi}_{\xi\xi}^{(0)}\right)
_{\xi\xi}+\alpha^{(0)}\bar{\phi}^{(0)}=0. \label{30
\end{equation}
We assume that the two waves of which a two-wave solution consists
have wavenumbers $K^{(0)}n_{1}$ and $K^{(0)}n_{2}$, where
$n_{1,2}$ are integers (without loss of generality, they can be
assumed to be coprime and such that $n_{1}>n_{2}>0$). Accordingly,
we shall seek a solution of Eq. (\ref{30}) in
the for
\begin{equation}
\bar{\phi}^{(0)}=a\cos(n_{1}x+\theta)+\cos n_{2}x \label{31
\end{equation}
(the fact that the second wave's amplitude is unity and its phase
is zero does
not reduce generality). Substituting (\ref{31}) into (\ref{30}), we obtai
\begin{equation}
\alpha^{(0)}=\frac{n_{1}^{2}n_{2}^{2}}{\left(
n_{1}^{2}+n_{2}^{2}\right)
^{2}},\hspace{0.4cm}K^{(0)}=\frac{1}{n_{1}^{2}+n_{2}^{2}}. \label{32
\end{equation}
Recalling that $K$ is related to the wavelength $\lambda$, one can
see that these equalities determine the points in the $\left(
\alpha,\lambda\right) $-plane near which two-wave weakly-nonlinear
solutions are localized (they are illustrated in Fig. 2).

To the next-to-leading order, (\ref{15}) yields\begin{widetext
\begin{multline*}
K^{(0)}\left(
\bar{\phi}^{(2)}+K^{(0)}\bar{\phi}_{\xi\xi}^{(2)}\right)
_{\xi\xi}+\alpha^{(0)}\bar{\phi}^{(2)}=K^{(0)}\left(
\frac{3a}{2}\cos (n_{1}x+\theta)+\frac{a^{3}}{4}\left[
3\cos(n_{1}x+\theta)+\cos
3(n_{1}x+\theta)\right]  \right.  \\
+\frac{3a^{2}}{4}\left\{  \cos\left[  \left(  2n_{1}-n_{2}\right)
x+2\theta\right]  +\cos\left[  \left(  2n_{1}+n_{2}\right)
x+2\theta\right] \right\}  +\frac{3a}{4}\left\{  \cos\left[
\left(  n_{1}-2n_{2}\right) x+\theta\right]  +\cos\left[  \left(
n_{1}+2n_{2}\right)  x+\theta\right]
\right\}  \\
+\left.  \frac{3a^{2}}{2}\cos n_{2}x+\frac{1}{4}\left(  3\cos n_{2
x+\cos3n_{2}x\right)  \right)  _{\xi\xi}-a\left[
K^{(2)}n_{1}^{2}\left(
2K^{(0)}n_{1}^{2}-1\right)  +\alpha^{(2)}\right]  \cos(n_{1}x+\theta)\\
-\left[  K^{(2)}n_{2}^{2}\left(  2K^{(0)}n_{2}^{2}-1\right)
+\alpha ^{(2)}\right]  \cos n_{2}x.
\end{multline*}
\end{widetext}This equation has a periodic solution for $\bar{\phi}^{(2)}$
only if the terms involving $\cos(n_{1}x+\theta)$ and $\cos
n_{2}x$ cancel out, but the specifics depend on whether $\left(
n_{1},n_{2}\right)  =\left( 3,1\right)  $ or not.

In the former case, straightforward calculations yield\begin{widetext
\[
K^{(2)}=-\dfrac{3\left(  7a^{3}-a^{2}+17a+3\right)
}{340a},\hspace
{0.4cm}\alpha^{(2)}=-\dfrac{9\left(  9a^{3}+3a^{2}+9a+1\right)  }{420a
\hspace{0.4cm}\text{if}\hspace{0.4cm}\left(  n_{1},n_{2}\right)
=\left( 3,1\right)  ,
\]
whereas the latter case yield
\[
K^{(2)}=-\dfrac{3\left[  \left(  n_{1}^{2}-2n_{2}^{2}\right)  a^{2}+2n_{1
^{2}-n_{2}^{2}\right]  }{4\left(  n_{1}^{2}+n_{2}^{2}\right)
\left(
n_{1}^{2}-n_{2}^{2}\right)  },\hspace{0.4cm}\alpha^{(2)}=-\dfrac{9n_{1
^{2}n_{2}^{2}\left(  a^{2}+1\right)  }{4\left(
n_{1}^{2}+n_{2}^{2}\right)
^{2}}\hspace{0.4cm}\text{if}\hspace{0.4cm}n_{1}\neq3n_{2}.
\]
\end{widetext}Now, using expressions (\ref{17})--(\ref{18}), (\ref{29}),
(\ref{12}), (\ref{31})--(\ref{32}) to relate $\bar{\phi}^{(2)}$,
$K^{(2)}$, and $\alpha^{(2)}$ to the `physical' quantities, we can
summarize the two-wave
solution in the for
\begin{equation}
\bar{\phi}=\varepsilon\left[  a\cos(n_{1}x+\theta)+\cos
n_{2}x\right]
+\operatorname{O}(\varepsilon^{3}), \label{33
\end{equation}
and\begin{widetext
\begin{equation}
\left.
\begin{tabular}
[c]{l
$\alpha=\dfrac{9}{100}-\varepsilon^{2}\dfrac{9\left(  9a^{3}+3a^{2
+9a+1\right)  }{420a}+\operatorname{O}(\varepsilon^{4}),\medskip$\\
$\lambda=2\sqrt{10}\pi+\varepsilon^{2}\dfrac{3\pi\sqrt{10}\left(  7a^{3
-a^{2}+17a+3\right)  }{34a}+\operatorname{O}(\varepsilon^{4}),
\end{tabular}
\right\}  \hspace{0.4cm}\text{if}\hspace{0.4cm}\left(
n_{1},n_{2}\right)
=\left(  3,1\right)  ,\label{34
\end{equation}
o
\begin{equation}
\left.
\begin{tabular}
[c]{l
$\alpha=\dfrac{n_{1}^{2}n_{2}^{2}}{\left(  n_{1}^{2}+n_{2}^{2}\right)  ^{2
}-\varepsilon^{2}\dfrac{9n_{1}^{2}n_{2}^{2}\left(  a^{2}+1\right)
}{4\left( n_{1}^{2}+n_{2}^{2}\right)
^{2}}+\operatorname{O}(\varepsilon^{4}),\medskip
$\\
$\lambda=2\pi\sqrt{n_{1}^{2}+n_{2}^{2}}+\varepsilon^{2}\dfrac{3\pi\left[
\left(  n_{1}^{2}-2n_{2}^{2}\right)
a^{2}+2n_{1}^{2}-n_{2}^{2}\right]
\sqrt{n_{1}^{2}+n_{2}^{2}}}{4\left(  n_{1}^{2}-n_{2}^{2}\right)
}+\operatorname{O}(\varepsilon^{4}),
\end{tabular}
\right\}  \hspace{0.4cm}\text{if}\hspace{0.4cm}n_{1}\neq3n_{2}.\label{35
\end{equation}
\end{widetext}Expressions (\ref{34}) and (\ref{35}) can be viewed as
parametric representations (with $\varepsilon$ and $a$ being the
parameters) of the existence region of two-wave solutions with
$n_{1,2}$, on the $\left( \alpha,\lambda\right)  $-plane. It can
be shown that, for all $n_{1,2}$, this region is a `semi-infinite
sector' (see an example in Fig. 3). However, since we assumed weak
nonlinearity, this conclusion can only be trusted near the
vertices of the sectors. Effectively, we have found the tangent
lines to the boundaries of the `true' region of the existence
interval.

\begin{figure}
\includegraphics[width=83mm]{fig3.eps}
\caption{Regions of existence (bounded by the dotted line) of
two-wave solutions with $\lambda _{2}/\lambda_{1}=2$\/$:$\/$1$.
Thin solid lines show the boundaries of the existence region
calculated under the assumption of weak nonlinearity. The thick
solid line shows the existence region for one-wave solutions (as
in Fig. 1).} \label{fig3}
\end{figure}

Observe that $\varepsilon^{2}$ can be eliminated from (\ref{34})
and
(\ref{35}), which yields\begin{widetext
\begin{equation}
\dfrac{\alpha-\tfrac{9}{100}}{\lambda-2\sqrt{10}\pi}=-\dfrac{6\left(
9a^{3}+3a^{2}+9a+1\right)  }{27\sqrt{10}\pi\left(
7a^{3}-a^{2}+17a+3\right)
}+\operatorname{O}(\varepsilon^{2}),\hspace{0.4cm}\text{if}\hspace
{0.4cm}\left(  n_{1},n_{2}\right)  =\left(  3,1\right)  ,\label{36
\end{equation}
o
\begin{equation}
\frac{\alpha-\dfrac{n_{1}^{2}n_{2}^{2}}{\left(
n_{1}^{2}+n_{2}^{2}\right)
^{2}}}{\lambda-2\pi\sqrt{n_{1}^{2}+n_{2}^{2}}}=-\dfrac{9n_{1}^{2}n_{2
^{2}\left(  a^{2}+1\right)  \left(  n_{1}^{2}-n_{2}^{2}\right)
}{3\pi\left( n_{1}^{2}+n_{2}^{2}\right)  ^{5/2}\left[  \left(
n_{1}^{2}-2n_{2}^{2}\right)
a^{2}+2n_{1}^{2}-n_{2}^{2}\right]  }+\operatorname{O}(\varepsilon^{2
)\hspace{0.4cm}\text{if}\hspace{0.4cm}n_{1}\neq3n_{2}.\label{37
\end{equation}
\end{widetext}For given $n_{1,2}$ and $\left(  \alpha,\lambda\right)  $,
(\ref{36}) and (\ref{37}) can be treated as equations for $a$ [it
is, essentially, the ratio of the amplitudes of the waves which
constitute the two-wave solution -- see (\ref{33})]. Observe that,
(\ref{37}) admits two roots for $a$ (with equal magnitudes and
opposite signs), which means that two-wave solutions with
$n_{1}\neq3n_{2}$ exist in pairs. In some cases these solutions
can be obtained from each other by shifting\ $x\rightarrow
x+\operatorname{const}$ (for the 2:1 case, for example, $\operatorname{const
=\pi$) -- but in other cases, the solutions with positive and
negative $a$ seem to be genuinely different.

For the case $\left(  n_{1},n_{2}\right)  =\left(  3,1\right)  $,
in turn, it follows from (\ref{37}) that three roots exist for
$a$. One of the three, however, corresponds to the \emph{one}-wave
solution. Indeed, recall that the expansion derived for those
failed near $\alpha=\tfrac{9}{100}$, $\lambda=2\sqrt{10}\pi$, i.e.
precisely where the asymptotic theory for \emph{two}-wave
solutions predicts existence of those with $\left(
n_{1},n_{2}\right)  =\left(  3,1\right)  $. In fact, one- and
two-wave solutions cannot be distinguished in this region, as, in
both cases, the coefficients of $\cos3x$ and $\cos x$ are of the
same order.

\subsection{Numerical results\label{Subsection 3.3}}

In this section, we shall present examples of strongly-nonlinear
one- and two-wave solutions and the existence region for the
latter. Two numerical methods have been used: the shooting method
(which turned out to be insufficient for large wavelengths) and
the method of Newton relaxation (which worked marginally better).

Figs. 4a and 4b show examples of increasingly nonlinear one- and
two-wave solutions respectively, for a fixed wavelength $\lambda$
and decreasing $\alpha$. In Fig. 4a, observe the increase of the
wave's amplitude as $\alpha$ is moving\ away from the boundary of
the existence region (for this value of $\lambda$, the boundary is
located at $\alpha\approx0.131$). Fig. 4b, in turn, illustrates
the fact that the margins of the existence region for two-wave
solutions correspond to the cases where the amplitude of one of
the two waves vanishes (which is how they bifurcate from one-wave
solutions).

\begin{figure}
\includegraphics[width=83mm]{fig4.eps}
\caption{Examples of frozen waves with spatial period
$\lambda=16$: (a) one-wave solutions with
$\alpha=0.005,~0.050,~0.100$ (curves 1, 2, 3 respectively); (b)
two-wave solutions with $\alpha=0.07,~0.09,~0.11$ (curves 1, 2, 3
respectively).} \label{fig4}
\end{figure}

\section{The stability of frozen waves\label{Section 4}}

\subsection{Asymptotic results}

We shall first examine the stability of frozen waves
asymptotically, under the same assumption of weak nonlinearity
used to find the frozen waves themselves.

We shall start by re-writing the stability problem
(\ref{8})--(\ref{9}) in
terms of $\xi=kx$, which yield
\begin{equation}
s\psi+k^{2}\left(
\psi-3\bar{\phi}^{2}\psi+k^{2}\psi_{\xi\xi}\right)
_{\xi\xi}+\alpha\psi=0, \label{38
\end{equation
\begin{equation}
\psi(\xi+2\pi)=\psi(\xi)\,\operatorname{e}^{\mathrm{i}\theta}. \label{39
\end{equation}
It turns out that all weakly nonlinear solutions, one- and
two-wave alike, are unstable for all $\alpha$ except
$\alpha\rightarrow\frac{1}{4}$ (i.e. except the `tip' of the
existence interval for one-wave solutions shown in Fig. 1). We
emphasize that this conclusion does not apply to \emph{strongly}
nonlinear waves (which can be either stable or unstable for any
$\alpha$ -- see the next section).

Thus, assuming that
$\alpha=\frac{1}{4}+\operatorname{O}(\varepsilon^{2})$, we
represent $\alpha$ by expression (\ref{26}); the corresponding
base solution, $\bar{\phi}$, is represented by
(\ref{27})--(\ref{28}).

A straightforward analysis shows that, if the phase shift $\theta$
is order-one, all solutions of the eigenvalue problem
(\ref{38})--(\ref{39}), with the parameters determined by
(\ref{26})--(\ref{28}), are stable. Thus, instability can occur
only for small $\theta$, which can be conveniently
accounted for by lettin
\begin{equation}
\theta=2\pi\varepsilon q, \label{40
\end{equation}
where $q$ is an order-one constant. Note that the smallness of the
phase shift $\theta$ implies that the instability occurs at
wavelengths close to that of the base solution.

It is also convenient to introduc
\begin{equation}
\psi_{new}=\psi\operatorname{e}^{-\mathrm{i}\varepsilon qx}. \label{41
\end{equation}
Using (\ref{40})--(\ref{41}) to replace $\theta$ and $\psi$ in
Eqs. (\ref{38})--(\ref{39}) with $q$ and $\psi_{new}$, we obtain
(subscript
$_{new}$ omitted)
\begin{multline}
s\psi+k^{2}\left(  \psi+k^{2}\psi_{\xi\xi}-3\varepsilon^{2}\phi^{2
\psi\right)  _{\xi\xi}\\
+\mathrm{i}\varepsilon qk^{2}\left(
2\psi_{\xi}+4k^{2}\psi_{\xi\xi\xi }\right)
-\varepsilon^{2}q^{2}k^{2}\left(  \psi+6k^{2}\psi\right)  _{\xi\xi
}+\operatorname{O}(\varepsilon^{3})\\
+\alpha\psi=0, \label{42
\end{multline
\begin{equation}
\psi(\xi+2\pi)=\psi(\xi), \label{43
\end{equation}
where the specific form of the terms
$\operatorname{O}(\varepsilon^{3})$ will not be needed.

It can be demonstrated that no instability occurs if
$s=\operatorname{O}(1)$ -- hence, only small $s$ need to be
examined. We assume (and shall eventually justify by obtaining a
consistent asymptotic expansion) that
\[
s=\varepsilon^{2}s^{(2)}+\cdots,
\]
while the eigenfunction i
\[
\psi=\psi^{(0)}+\varepsilon^{2}\psi^{(2)}+\cdots.
\]
To leading order, Eqs. (\ref{42})--(\ref{43}),
(\ref{26})--(\ref{28}) reduce
t
\[
\tfrac{1}{2}\left(
\psi^{(0)}+\tfrac{1}{2}\psi_{\xi\xi}^{(0)}\right)
_{\xi\xi}+\tfrac{1}{4}\psi^{(0)}=0,
\
\[
\psi^{(0)}(\xi+2\pi)=\psi^{(0)}(\xi),
\]
which yield
\begin{equation}
\psi^{(0)}=S\sin x+C\cos x, \label{46
\end{equation}
where $S$ and $C$ are undetermined constants.

In the next-to-leading order, Eqs. (\ref{42})--(\ref{43}), (\ref{26
)--(\ref{28}) yield (after straightforward algebra)\begin{widetext
\begin{multline*}
\tfrac{1}{2}\left(  \psi_{\xi\xi}^{(2)}+\tfrac{1}{2}\psi_{\xi\xi\xi\xi
^{(2)}\right)  +\tfrac{1}{4}\psi^{(2)}+\tfrac{3}{4}\left[  S\sin
x+C\cos x+\tfrac{1}{2}S\left(  3\sin3x-\sin x\right)
+\tfrac{1}{2}C\left(
3\cos3x+\cos x\right)  \right]  \\
+\left(  s^{(2)}+q^{2}-\tfrac{3}{8}\right)  \left(  S\sin x+C\cos
x\right) -2\mathrm{i}q\sqrt{-\alpha^{(2)}-\tfrac{3}{8}}\left(
S\cos x-C\sin x\right) =0,
\end{multline*
\[
\psi^{(2)}(\xi+2\pi)=\psi^{(2)}(\xi).
\]
\end{widetext}This boundary-value problem has a solution for $\psi^{(2)}$ only
i
\[
\tfrac{3}{8}S+\left(  s^{(2)}+q^{2}-\tfrac{3}{8}\right)  S+2\mathrm{i
q\sqrt{-\alpha^{(2)}-\tfrac{3}{8}}C=0,
\
\[
\tfrac{9}{8}C+\left(  s^{(2)}+q^{2}-\tfrac{3}{8}\right)  C-2\mathrm{i
q\sqrt{-\alpha^{(2)}-\tfrac{3}{8}}S=0,
\]
which, in turn, has a solution for $S$ and $C$ only i
\[
s^{(2)2}+\left(  2q^{2}+\tfrac{3}{4}\right)  s^{(2)}+q^{2}\left[  q^{2
+\tfrac{3}{4}-4\left(  -\alpha^{(2)}-\tfrac{3}{8}\right)  \right]
=0.
\]
This equation determines the eigenvalue $s^{(2)}$. It can be
readily shown that $s^{(2)}$ is stable (i.e.
$\operatorname{Re}s^{(2)}<0$) for all $q$ only
i
\[
\alpha^{(2)}<-\tfrac{9}{16}.
\]
Finally, using (\ref{26})--(\ref{27}) to express $\alpha^{(2)}$ in
terms of the `physical' parameters $\alpha$ and $\lambda$, we
obtain the following stability criterion for a frozen wave with
parameters $\left(  \alpha
,\lambda\right)  $
\begin{equation}
\left(  \lambda-2\sqrt{2}\pi\right)
^{2}\lesssim\tfrac{8}{3}\pi^{2}\left(
\tfrac{1}{4}-\alpha\right)  . \label{47
\end{equation}
Observe that, for any $\alpha<\frac{1}{4}$, there exists an
\emph{interval} of stable wavelengths, not just a single value of
$\lambda$.

As mentioned before, condition (\ref{47}) applies only if $\alpha$
is close to $\frac{1}{4}$. In the next subsection, it will be
extended numerically to arbitrary values of $\alpha$.

\subsection{Numerical results}

The eigenvalue problem (\ref{8})--(\ref{9}) was solved numerically
for $\psi(x)$ and $s$ with the base wave $\bar{\phi}(x)$ computed
using problem (\ref{3})--(\ref{4}).

The general features of the dispersion relation [the dependence
$s(\theta)$] of the eigenvalue problem (\ref{8})--(\ref{9}) is
described in the Appendix, whereas here we shall only present the
stability diagram on the $\left( \alpha,\lambda\right)  $ plane --
see in Fig. 5. Evidently, for all values of $\alpha$, an interval
of $\lambda$ exists where one-wave solution are stable -- which
confirms and extends the asymptotic
($\alpha\rightarrow\frac{1}{4}$) stability criterion (\ref{47}).
We have not found any stable two-wave solutions, which suggests
they are either unstable or perhaps their regions of stability are
small and difficult to locate.

\begin{figure}
\includegraphics[width=83mm]{fig5.eps}
\caption{The stability region of one-wave solutions on the $\left(
\alpha,\lambda\right)  $-plane (shown by the dashed line). The
thick solid line shows the existence region of one-wave solutions
(as in Fig. 1). The upper panel represents a blow-up of the shaded
region of the lower panel [the thin solid line shows the stability
region's asymptotic boundary, (\ref{47})].} \label{fig5}
\end{figure}

To illustrate that \emph{any} solution from the range of stable
frozen waves (not necessarily the wave with the minimum
free-energy density) can emerge from a `general' initial
condition, we have carried out the following numerical experiment.
The time-dependent MCHE (\ref{1}) was simulated using finite
differences with a fully implicit backwards Euler method, and the
results of the simulations presented below are for $\alpha=0.1$.

In this case, the energy minimizing wavelength is
$\lambda_{0}=11.31$ (calculated from the interpolation formula of
Ref. \cite{LiuGoldenfeld89}), with the corresponding frozen-wave
solution denoted by $\phi_{0}\left( x\right)  $. According to our
analysis, however, a solution $\phi_{1}\left( x\right)  $ with the
commensurate wavelength $\lambda_{1}=14.13$ ($\lambda
_{0}/\lambda_{1}=4/5$) should also be stable.

To verify this, Eq. (\ref{1}) was simulated in a domain of size
$L=5\lambda _{0}=4\lambda_{1}$ which accommodates both solutions.
The initial condition was chosen as a `mixture' of the frozen
waves $\phi_{0}\left(  x\right)  $ and $\phi_{1}\left(  x\right)
$, i.e.
\begin{equation}
\phi\left(  x,0\right)  =\beta\,\phi_{0}\left(  x\right)  +\left(
1-\beta\right)  \,\phi_{1}\left(  x\right)  ,\label{48
\end{equation}
where $\beta\in\left(  0,1\right)  $ is the `mixing ratio'. The
timestep was $0.1$ and $400$ gridpoints per period were used, and
it has been verified that the results were mesh and timestep
independent.

If $\phi_{0}(x)$ was the only stable solution, the system would
evolve towards $\phi_{0}$ for all $\beta$. Our simulations
nevertheless show that, for $\beta=0.2$, the system evolves back
to $\phi_{1}$ (see Fig. 6), which confirms our conclusion about
the existence of \emph{multiple} stable states.

\begin{figure}
\includegraphics[width=83mm]{fig6.eps}
\caption{Evolution of perturbed steady states within the stable
region at $\alpha=0.1$. The traces show the limiting
($t\rightarrow\infty$) solution initialized by (\ref{48}) for
various $\beta$. The traces are incremented in steps of $0.5$,
beginning with zero increment for $\beta=0$ (the lowest trace) and
ending for $\beta=1$ (the highest trace). The curves are marked
with the corresponding values of $\beta$.} \label{fig6}
\end{figure}

\section{Concluding remarks}

The main result of the present paper is illustrated in Fig. 5,
which shows the stability region of (one-wave) frozen solutions of
the modified Cahn--Hilliard equation (\ref{1}). We have also found
a new class of frozen waves -- the `two-wave solutions', but these
seem to be unstable and, thus, less important than the usual,
one-wave type.

We have also made a more general -- and potentially more important
-- conclusion regarding the energy approach to studies of
stability. If a family of solutions exists and one of them
minimizes the energy functional, this does not necessarily mean
that all other solutions are unstable. Furthermore, the stability
of the minimizer solution cannot be guaranteed either: even though
it is stable with respect to the perturbation of `shifting along
the family of solutions', another perturbation can still
destabilize it.

Physically, our results imply that when lamellar microstructures
of block copolymers are used to template nanowires, one must
ensure that only the desired state is created. This may become
more of a critical concern as larger numbers of nanowires are to
be created within a single trench. In practice, some control over
this can be exerted via the annealing schedule. It should also be
noted that a kinetically stable quenched state may be selected
rather than a true time independent solution to the modified
Cahn--Hilliard equation.

Finally, it would be interesting to extend the present results to
steady states with \emph{two} spatial dimensions, similar to those
found in Ref. \cite{ChoksiMarasWilliams11} for an equation similar
to the two-dimensional MCHE (but with a slightly different
nonlinearity).

\begin{acknowledgments}
The authors acknowledges the support of the Science Foundation
Ireland (RFP Grant 08/RFP/MTH1476 and Mathematics Initiative Grant
06/MI/005)
\end{acknowledgments}

redpajama_set_name: RedPajamaArXiv


Until Saturday 4 August, Western Sydney Parklands (WSP) is going on show to mark its 50-year milestone.
Known as Sydney's Biggest Backyard, WSP covers 5,280 hectares and 27 kilometres from north to south and is home to award-winning picnic and recreational areas such as Bungarribee, Lizard Log and Wylde Mountain bike park, as well as major entertainment facilities and sports venues.
Sydney's Biggest Backyard On Show is a public celebration of a successful 50-year-old vision for open space in Western Sydney.
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text:
Earthquake Weather
By Tim Powers
Tim Powers Blackstone Publishing 9780312861636
The Fault Lines Series: Book 3
A young woman possessed by a ghost has slain the Fisher King of the West, Scott Crane. Now, temporarily freed from that malevolent spirit, she seeks to restore the King to life. But Crane's body has been taken to the magically protected home of Pete and Angelica Sullivan, and their adopted son, Koot Hoomie. Kootie is destined to be the next Fisher king, but he is only thirteen years old—too young, his mother thinks, to perform the rituals to assume the Kingship. But not too young, perhaps, to assist in reuniting Scott Crane's body and spirit, and restoring him to life.
A young woman possessed by a ghost has slain the Fisher King of the West, Scott Crane. Now, temporarily freed from that malevolent spirit, she seeks to restore the King to life.
But Crane's body has been taken to the magically protected home of Pete and Angelica Sullivan, and their adopted son, Koot Hoomie. Kootie is destined to be the next Fisher king, but he is only thirteen years old—too young, his mother thinks, to perform the rituals to assume the Kingship. But not too young, perhaps, to assist in reuniting Scott Crane's body and spirit, and restoring him to life.
Author Bio: Tim Powers
Tim Powers is the author of numerous novels, including Last Call, Declare, Three Days to Never, and On Stranger Tides, the inspiration for the blockbuster film Pirates of the Caribbean: On Stranger Tides, starring Johnny Depp and Penélope Cruz. He has received the Locus, World Fantasy, and Philip K. Dick awards. He lives in San Bernardino, California.
Category: Fiction/Fantasy
Publisher: Blackstone Publishing Publisher: Blackstone Publishing
AE Catalog ID: N/A AE Catalog ID: N/A


On his 2006 release, Electric Ladder, Robert Rich finds a new voice for the vibrant interlocking melodies that characterized his works such as Numena, Geometry and Gaudí. With its seamless blend of analog modular synthesizers and acoustic tonalities, Electric Ladder weaves a hypnotic spell - lush, ecstatic and seductive. Its shimmering geometric lines grow more sharply etched with the clarity of justly tuned instruments and the intelligence that pervades Rich's compositional vocabulary.
Robert Rich - MOTM modular synthesizer, TimewArp2600, Sculpture, Metasynth, steel guitars, flutes.