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There were no histrionics, no last-minute deals or walkouts. In fact, no anything really. The Rio+20 Earth Summit last week ended with barely a whimper. Many of the halls emptied by early afternoon on the final day. With the concluding text agreed three days earlier, the journalists had little to report, unplugged their laptops, and headed for the beaches.
And yet, behind the scenes, something really interesting just may have happened in Rio.
The summit was never going to be like its illustrious predecessor in Rio de Janeiro 20 years earlier. At the original event, virtually all the world's leaders (including a reluctant George H.W. Bush) signed two groundbreaking treaties, on climate change and protecting biodiversity. Neither has turned out to be as effective as hoped back then. But the ambition was clear: saving the planet and achieving "sustainable development" for all.
Two decades on, the planet's ecosystems are still degrading fast. Any green gains have been wiped out by soaring consumption. Humanity's annual requirement for natural resources is about double what it was then. The rate of species extinctions is undiminished. Carbon dioxide emissions are up 40 percent, and the concentration of the heat-trapping gas this year
The text of the Rio+20 Declaration is lame even by the standards of international diplomacy.for the first time hit 400 parts per million (ppm) in the Arctic air — up about 40 ppm from 1992.
But the former will among politicians to address the challenge has dissipated. Where in Rio this time, were Obama, Merkel, and Cameron? Even Hillary Clinton only showed in Rio for a few hours. Where were the new treaties to sign? Europeans in particular had hoped to get agreement on drawing up a convention to protect the high seas, which are currently not covered by either national jurisdiction or international environmental law. But an unlikely alliance of the U.S. and Venezuela, backed by Japan, Russia, and Canada, vetoed the plan at the last.
The text of the Rio+20 Outcomes Document, the conference's main formal declaration, is lame even by the standards of international diplomacy. Beside a high-seas commitment, other proposals omitted from the declaration included an outright condemnation of subsidies for fossil fuels, support for mandatory environmental and sustainability reporting by large corporations, and a green light for green taxes. The conference did agree to start talks on setting sustainable development goals to augment the world's existing millennium development goals, but could not agree on what topics they might cover. Even a planned upgrade of the UN Environment Programme fizzled.
The process for drafting the Rio+20 declaration was as flawed as the outcome. The Brazilian hosts decreed that all text had to be agreed before the arrival of government ministers for the final three days. As the deadline approached, they simply deleted those sections causing controversy and replaced them with weak compromise text.
At most such conferences, ministers do deals to improve on the work of their civil servants. That often produces the late-night drama. Here, they were presented with a lowest-common-denominator fait accompli. There wasn't even a slot on the conference agenda for them to discuss the text they were signing. Any semblance of democratic accountability was lost.
The diplomats professed to be happy. The UN official in charge of the conference, Sha Zukang, declared a "successful" outcome. But environmentalists, scientists and social activists saw only that nobody had been committed to anything. Even conference veterans well versed in how to extract a smidgeon of success in grievous compromise could find nothing
'Twenty years ago, we agreed what to do. Now we have the tools to do it,' said a World Bank official.to cheer. WWF International lambasted a "colossal failure of leadership and vision"; Care International called it a "charade."
The recriminations began even before the conference closed. British Deputy Prime Minister Nick Clegg blamed developing countries for being "antagonistic to our European ideas on the green economy." Brazilian delegate and senator Eduardo Braga said, "Europe is too absorbed by its economic problems."
Both were right. But so too was the director of the UN Environment Programme Achim Steiner, when he said in mid-conference that it revealed "a world at a loss what to do." On the final day, he concluded: "We can't legislate sustainable development in the current state of international relations."
If not legislation, then what?
In part, the absence of ambition derives from a new realism about what international agreements alone can achieve. The World Bank's vice president of sustainable development, Rachel Kyte, remembered how 20 years before, governments appeared to believe that the planet could be put on a green path with some environmental laws to curb Western profligacy and some overseas aid for the developing countries. Nobody believes that anymore.
But there is optimism that a new way can be found. "We have failed to turn things round in the past 20 years," said Steiner, "but underneath that failure there is an extraordinary array of activity and innovation." The heart of that was a beguiling phrase: green economics. Rather than fighting the power of capital, or trying to legislate away its environmental downsides, the idea is to harness market forces to turn economies onto a green track. More carrot and less stick.
Green economics, though only briefly mentioned in the final declaration, was on everyone's lips in the corridors at Rio. Kyte insisted it was the answer. "Twenty years ago, we agreed what to do; now we have the tools to do it." Steiner chastised environmentalists who feared it. "If we do not go
Faced with governments reluctant to engage in its grand plans, the UN is increasingly turning to corporations.into the heart of economic policy, we will meet here at Rio+40 even more culpable. Markets are social constructs. They are not a force like gravity. They can be governed."
Green economics, as represented in Rio, meant primarily introducing metrics about the use and abuse of nature and natural resources into corporate and national accounting. This would, as Steiner put it, bring the environment within the "visible spectrum" of economic activity.
The theory goes that once natural capital turns up on balance sheets in the same way as man-made capital, then CEOs and policy makers will adopt greener ways. Natural capital accounting should mean environmental protection is seen as an investment, rather than a cost, said economist Sam Fankhauser of the London School of Economics. Corporations will husband resources better, while governments will switch taxation onto resource use and pollution rather than economic activity itself.
Faced with national governments reluctant to engage in its grand plans, the UN is increasingly turning to corporations to make a difference. This too is seen as part of the brave new world of green economics. Tim Wirth, the former U.S. senator who now heads the UN Foundation, which fosters UN links with business, told journalists that "magic" public-private partnerships were more important that any conference declaration.
Leading the charge to grab some of the action is Chad Holliday, former president of DuPont and now chairman of the Bank of America. UN Secretary General Ban Ki-moon has recruited him to co-chair his ambitious program to bring electricity and clean cooking fuel to the billions still without them by 2030. Holliday called the energy scheme "the greatest public-private partnership of all time" and said it would become the prototype for how to implement the planned UN sustainable development goals.
Many other CEOs have picked up the scent that there is plenty of business to be done. The UN claimed there were 1,500 corporate leaders in town for the conference, compared with only a handful in 1992.
The public-private partnerships will involve public funding, from governments and bodies such as the World Bank, leveraging further
NGOs warned in Rio that if nature had a dollar sign attached, corporations would soon take over.investment from the profit-seeking private sector. Ban's energy plan, dubbed "sustainable energy for all," has brought a commitment of $2.4 billion from the Brazilian government to connect its remaining citizens to the national grid by 2014. With such inducements, corporations have pledged $50 billon in further investment round the world.
And that is just the start. In another sign of how public-private partnerships might deliver, the Asian Development Bank announced here that it had launched a consortium of banks and corporations aiming to spend $175 billion over the next decade on "sustainable" transportation systems across the developing world, including rapid-transit bus networks, railways, and fuel-efficient vehicles.
Is this the salvation of the world, or a headlong rush into an unsustainable future under a green logo? NGOs warned in Rio that if nature had a dollar sign attached, corporations would soon take it over. They see the trading of carbon rights to forests, proposed under future climate change agreements, as just the start.
Meanwhile, ecologists pointed out that ecosystems do not function like warehouse inventories and that equating natural and man-made capital could be both bad economics and bad ecology. Green economies "need to be embedded in ecological principles and not simply focused on economic growth based on new, greener production systems," wrote Georgina Mace of Imperial College London in a paper in PLoS Biology, published to coincide with the conference.
But whatever the drawbacks of commodifying nature, something has to be done. The state of the host country today, 20 years after the first Earth Summit, underlines the failed resolve of the intervening years.
MORE FROM YALE e360
Putting a Price on The Real Value of Nature
Most worryingly, when I came to the first Earth Summit 20 years ago, everyone agreed that the model for how the developing world could achieve urban sustainability was the nearby Brazilian city of Curitiba. Its innovative bus transit system, trash recycling and poverty alleviation programs — set up by mayor and urban planner Jaime Lerner — were a huge success, as I saw on a short visit.
This time, I asked a new generation of delegates what the new urban model might be. Oh, Curitiba, they said. Any others? They scratched their heads. In two decades, nobody has successfully repeated the Curitiba experiment. Until there are thousands of Curitibas — thanks to green economics or some other means — the world will still be hurtling towards disaster. And we may not have another 20 years.


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thinkable precisava de um novo design de página da web e criou um concurso na 99designs.
Um vencedor foi selecionado entre 29 designs enviados por 11 designers freelancers.
We're in need of a Web 2.0 style template for a custom sticker submission website we'll be launching soon called "images on stickers". a solid Web 2.0 look is a must.
Images on Stickers
Desired Color Scheme:
Up to designer.
Desired Style:
Accepted File Formats:
NOTE ADDED Saturday 11/11/06: Please keep the width 675px or smaller. I apologize, but I've just realized I want something smaller than "normal" sites. I want the content to be very focused and won't need much space to do that.
Here are also some sites to gain more inspiration from:
http://www.imeem.com (LOVE the header and logo - not the layout)
The deliverables should include one PSD for both the content page and the homepage. Please feel free to make a new logo - we're not stuck on the one shown below.
Here's an example layout someone else tried for us - we want more Web 2.0 styling though! This one isn't quite up to par but is a good example of what the content page needs to have. Please pay NO attention to the layout itself, just the items within. You have full creative freedom to organize the items as you see fit. We want a very catchy layout as the final product. The only thing we're set on is the image previews have to be 400 pixels wide.
The header should stay consistent throughout the site.
-an area to accommodate 10 random thumbnails of submitted stickers
-the main feature of the homepage should accommodate an ad to buy/get a sticker and an area showing the current number of stickers in the database... please make it sizeable to hold a big number... we might even get to a million so we want to make sure it fits.
- Login form
and please throw in anything else you see fit.
If you need some inspiration please visit a few of the sites here:

subset: commoncrawl


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A Fool's Fortune
by: Justin A Pilla & Jason D Cannoncro
Game Type(s):
Card Games
Players: 2 +
Age: 13 +
Available Languages
Status: In-Print
Stock Number: 487
Come play a game of mischief, magic, mirth, and lore. Delve into the
Book of Fate to reveal mysterious fortunes: realms near and far, resources
wondrous & wild. Cast your lot with a crew of assorted (and sordid)
characters. If you pay your dues and play your hand right, you may just
win A Fool's Fortune.
A Fool's Fortune is a two player card game in the tradition of Rummy,
where players race to make sets… but with several wily twists. Found within
the deck of 77 unique cards are 5 suits of Fortunes & and a host of talented
Characters player's may employ. Be the fi rst player to complete 3 sets to win
A Fool's Fortune!
You may also be interested in...


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If you have box tops, we would love to have them! Please send all box tops to Mrs. Suzanne Gash. Thank you!
DeKalb County School District is committed to providing a website that is accessible to the widest possible audience, regardless of technology or ability. This website endeavors to comply with best practices and standards defined by Section 508 of the U.S. Rehabilitation Act. If you need any assistance with our website contact Melanie Fitzgerald with Dekalb County Schools at 615-597-7987 or email to email@example.com sites on our school website. These linked sites are not under the control of the DeKalb County Schools, and the DeKalb County Schools are not responsible for the content of any of these linked sites or any link contained to a linked site, or any changes or updates to such sites. DeKalb County Schools are providing these links to you as a convenience, and the inclusion of any link does not imply endorsement of the site by DeKalb County Schools. We have no control over which sites are blocked by your school's network filter.
Copyright © 2020 DeKalb Middle School


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- Maps
- Contact Us
In any organization, managing conflict is a necessary skill. We can provide several levels of training in this area. We offer a program called We-Care that helps individuals identify their behavioral styles and how they change in conflict, and how improve their ability to control their reaction to conflict.
We also offer training for front line supervisors, both experienced and those new to supervision. Beyond this we have manager level training that we can deliver.
In today's interconnected workplace, soft skills are every bit as important as technical skills. We offer training in the areas of work ethics, team work, and communication skills.
Hutchinson Community College can provide hands on training and consulting in the area of Lean Manufacturing. From facilitating Kazien Events to improve your processes, to 6S projects to help organize and streamline your work area, to value stream mapping to identify where to start your Lean journey. HCC trainers use the proven methods of the "Toyota Production System" that has been widely adopted throughout industry to improve efficiency and effectiveness.
Hutchinson Community College can work with your organization using Six Sigma tools to improve quality and reduce process variation. With two Six-Sigma Greenbelts on staff, we can customize an affordable Six Sigma program for your company.


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Trailer FrenzyA special place to find the newest trailers for movies and TV shows you're craving.
Alec Baldwin plays a tattooed Santa Claus, who must team up with other bad ass "guardians" of the world (including Hugh Jackman as the Easter Bunny) to fight the fear that's destroying the children of the world. It's the trailer for the epically ridiculous Rise of the Guardians CG animated movie. But it's kind of all worth it, when you see that Santa Claus has inked sleeves that read "Naughty" and "Nice." What do you think? Too cheesy — or the next big kiddie sensation?
When an evil spirit hatches a plot to take over the world, the immortal Guardians join forces to protect the hopes, beliefs, and imagination of children all over the world.
Guardians hits theaters on Nov. 21st.


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|Our cabins are modern and comfortable. Complete with carpet in bedrooms, tile bathrooms and they have heat and whole house ventilation for summer temperatures.
|There are 10 cabins, each houses 16 persons with a camp capacity of 160 persons. The sleeping area is broken down into 4 cubicles with 4 beds. Cabins are wheelchair accessible.


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Temperature Gauge Repair End Piece
This is a replacement piece for the temperature gauge sensor plug on the engine for the temperature gauge sensor harness. The OEM GM plastic end piece for this temperature sensor wire harness end on all medium to high mileage Fieros usually fails and falls apart from the engine heat.
The wires to the temperature gauge sensor in most cases are usually in good condition. There is no need to buy a complete repair harness end piece when all you usually need is the plastic part.
I have seen many Fiero owners buy the expensive complete repair end just to remove the wires to get the new plastic piece and not use the new wires. All you have to do is plug the original wires into this new connector and flip the lock onto them.
Why buy a complete repair end when all you usually need is the plastic piece?
I do not offer free shipping.
Because I use postal flat rate boxes for heavy orders there is no program that can add the proper amount of shipping to every order. So all shipping is added manually by myself after you place your order. That is explained here: Shipping added later explanation
When you place an order the shipping amount will show $0.00. Soon after you place an order you will receive an email confirmation that shows the amount of shipping that was added to your order.
Shipping is $5.50 (USA)
Final shipping cost is determined by weight of the item(s) and distance from Milwaukee, WI. Shipping prices quoted are for items shipped to the Midwest area in the USA. If you are ordering a single item or multiple items the total shipping price is determined after you place the order. I try to keep my shipping prices to a minimum and use Postal Flat Rate boxes whenever I can utilize them for an order.


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Buy Between Parent & Child
Dr. Haim Ginott
Dr. Alice Ginott
Dr. H. Wallace Goddard
Video Clips of Haim and Alice
Between Parent and Teenager
Welcome to BetweenParentAndChild.com
Parenting may be the hardest yet most important--and ultimately the most rewarding--work a person can do. But parenting know-how does not come automatically.
Between Parent and Child may be the best parenting book ever written. Since it was first published in 1965 it has sold millions of copies. It is on the short list of all-time great self-help books in The Authoritative Guide to Self-Help Books. What is it so popular?
In a time when many parent advisors recommended therapy for parents, Haim Ginott believed that most parents merely needed better information. So, based on his experience working with troubled children, he taught parents how to talk with their children.
Perhaps Haims genius was helping parents capture the meaning behind childrens words and deeds. There is nothing quite as
soothing for children as being understood. There is nothing quite as helpful for solving parenting problems as the feeling of parents and children working together.
Ginotts approach was unique because he joined great compassion with solid limits. Perhaps that is why his book has become a classic. Newly revised and updated (by Haims widow and Arkansas Wally Goddard), the book continues to be a superb guide for parents.


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Get the Buying Guide BEFORE you waste money on products that don't work!Get it NOW! »
Editor's Rating & Review
How we review products
Very Blendable, Looks Natural.I really love Nars foundations for 2 reasons: Blending is a breeze, and always looks very natural. The very sheer, water-like formula dries on skin quickly, and is very buildable. I have never had any problems with breakouts; nor does it melt off my face, even in the scorching heat of Summer, and doesn't settle into fine lines, or creases like others do. Wears beautifully, all day (and night!) long.AshleyMichelle | 4 reviews
This review is: Helpful | Not helpful | Inappropriate | 2 of 3 people said helpful
long time userI've been using this foundation for over 2 years... I've only gone through 2 bottles. It's very sheer and light. Doesn't provide heavy coverage, so not good for blemishes. Evens out my undereye discoloration and that's about it. But it does the job. I'm just a little NARS crazy and am looking forward to changing over to smashbox once this expensive bottle is done.NIKKI104 | 31 reviews
This review is: Helpful | Not helpful | Inappropriate | 1 of 2 people said helpful
WORTH the moneyMy friend gave me this foundation because it was too light for her, and when I tried it I was IN LOVE!! It looks AMAZING : great coverage, great feel, great luminous finish. A bit heavy, but it just means you don't have to use much A LITTLE GOES A LONG WAY. Makes my skin look even, healthy, and flawless. I think it's my favorite foundation thus far. ( I usually use MAC )MeggieMae | 28 reviews
This review is: Helpful | Not helpful | Inappropriate | 1 of 4 people said helpful
My Reasons for Reporting This


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|A rendering of the touring exhibition "Barbie the Dreamhouse Experience," which is planned for Berlin and Sunrise, Fla.
EMS Entertainment/Mattel
Leaving Behind Malibu in Search of a New Dream Home
By Stuart Elliott
ONE of the best-known residents of Malibu, Calif., alongside billionaires like David Geffen and Larry Ellison, is embarking on a national promotional effort to help sell her home there for the eye-popping price of $25 million.
Potential buyers may want to try bargaining down the price when they realize the house has only one bedroom and one bathroom — and three walls.
Those clues ought to point to the owner: Barbie, the doll who has, in the imagination of millions of girls, lived on the beach — in a home formally known as the Barbie Malibu Dreamhouse — since the introduction of Malibu Barbie in 1971. Mattel, the toy maker that has marketed Barbie since 1959, is to announce on Thursday that Barbie is posting a "for sale" sign on her house in preparation for a move.
The announcement will kick off a campaign that is to culminate with a disclosure of Barbie's new home, which will be brought to life in a playset that will come out in the fall, in time for the 2013 Christmas shopping season.
The campaign is meant to mix actual and imaginary elements. For example, a section of Trulia, the real estate Web site, will carry the for-sale listing for what is being called "the dreamiest of dreamhouses." Josh Altman, a cast member of "Million Dollar Listing: Los Angeles," a reality series on the Bravo cable channel, is being "hired" by Barbie to list the house.
And four designers are forming a dream team, in Mattel's parlance, to advise Barbie in her hunt for a new home. They are Jonathan Adler, who brought the Malibu playset to life in 2009 as part of a commemoration of Barbie's 50th anniversary; Lulu de Kwiatkowski; Celerie Kemble; and Trina Turk.
Although the campaign will include some traditional advertising like an ad in Us Weekly magazine, the bulk of it will involve digital media, on platforms like trulia.com and the Barbie Web site, barbie.com; social media like Facebook and Twitter; and public relations.
The campaign will also include so-called experiential marketing in the form of "Barbie the Dreamhouse Experience" — a touring exhibition, in a partnership between Mattel and EMS Entertainment — that is to begin this year in two markets, Berlin and Sunrise, Fla.
When Mattel reported its fourth-quarter results on Friday, executives said that Barbie sales fell about 4 percent, representing the third quarterly decline during 2012. That trend has set off efforts by Mattel to generate interest in Barbie and address challenges like the sluggish economy and the growing interest among children in electronic toys.
Recent examples of Mattel initiatives include a campaign, predominantly in social media, to reunite Barbie and her boyfriend, Ken, after a seven-year separation; a Web video series, "Barbie Life in the Dreamhouse," that can be watched on barbie.com/dreamhouse and YouTube; and cruises with Barbie themes on Royal Caribbean International ships.
"Over all, we're pleased with the position of Barbie in the marketplace," said Stephanie Cota, senior vice president for girls' brands worldwide at the Mattel Brands division of Mattel. "No. 1 is a good place to be."
Still, "it's our job that she remain top of mind," she added, and "we are at our best when we recognize the cultural relevance of Barbie."
The campaign is being introduced amid signs that the battered American housing market is healing, like an acceleration in groundbreaking for new homes and a spate of prominent ads by national realty companies that included a commercial for Century 21 during Super Bowl XLVII on Sunday. Those may be indications that consumers are ready for a lighthearted campaign about a doll's foray into the real estate market.
Barbie is "a fun brand, an iconic brand, to partner with," said Ken Shuman, head of communications at Trulia in San Francisco. The location of and high asking price for Barbie's home fit with the focus of a Trulia blog "on luxury celebrity properties," he added.
Besides, "we're a Silicon Valley company, technology-focused," Mr. Shuman said, "and Silicon Valley has embraced Barbie, voting that computer engineer be her next career" in 2010.
Mr. Altman, who works frequently with Trulia, is also getting into the spirit of the campaign. He said he would be glad to show Barbie new homes "anywhere on the planet" because "she speaks God knows how many languages."
"And Ken is fine with whatever she wants to do," he added, laughing.
After suggesting that he might advise Barbie to buy "a skyscraper in Manhattan," Mr. Altman ended the conversation by declaring: "This is Barbie on the other line; I've got to go."
All the nontraditional elements of the campaign make it difficult to determine spending. Ms. Cota said it would be comparable to Mattel's spending on Barbie in recent years.
According to the Kantar Media unit of WPP, Mattel spent $11.6 million to advertise all Barbie products in major media in the first nine months of last year, compared with $8.4 million in the same period of 2011. Full-year spending was $18.4 million in 2011, $17.2 million in 2010, $23.7 million in 2009 and $31.7 million in 2008, Kantar Media reported.
The campaign is being led by an internal agency at Mattel, which is working with Attention, a digital and social media agency, and, for public relations, the HL Group, part of MDC Partners.


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Central City Opera's "La Bohème" moves action, and worthy singing, to 1930's Paris
Puccini's ever-popular "La Bohème" is defined by music and characters that keep the ears alert and the emotions engaged. And in the Central City Opera's all-new production of the time-tested treasure, director Kevin Newbury — in his first summer here — takes pains to ensure that the action and atmosphere don't interfere with the melody and character-driven plot.
Sung in Italian, this "Bohème" is set in 1930s Paris when the international capitol was a hub for artists, writers and thinkers. With an intentionally meager, yet magical, set design by David Korins — and despite Jessica Jahn's drab and dowdy costumes — Newbury evokes the romance and hard reality of bohemian life in a rundown apartment in the Latin Quarter.
Tenor Eric Margiore makes a notable CCO debut as Rodolfo, a poet who falls for the fair and fragile Mimi, sung by soprano Elizabeth Caballero. Demonstrating ample squillo – a resonant, poignant vocal quality that easily carries above the thick orchestration of the score – Margiore brought lyricism and expressivity to his role, especially as his voice became freer in the third and fourth acts.


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Related articles
From Wikipedia, "a codec is a device or computer program capable of encoding and/or decoding a digital data stream or signal."
In general, codecs are utilized by multimedia applications to encode or decode audio or video streams. In order to play encoded streams, users must ensure an appropriate codec is installed.
This article deals only with codecs and application backends; see List of Applications for a list of media players (MPlayer, mpv and VLC are popular choices).
Playing multimedia content requires two components:
- A capable media player
- The appropriate codec
It is not always necessary to explicitly install codecs if you have installed a media player. For example, MPlayer pulls in a large number of codecs as dependencies, and also has codecs built in.
List of codecs
- ALAC — Data compression method which reduces the size of audio files with no loss of information.
- CELT — Compression algorithm for audio. Like MP3, Vorbis, and AAC it is suitable for transmitting music with high quality. Unlike these formats CELT imposes very little delay on the signal, even less than is typical for speech centric formats like Speex, GSM, or G.729.
- Daala — New video compression technology. The effort is a collaboration between Mozilla Foundation, Xiph.Org Foundation and other contributors. The goal of the project is to provide a free to implement, use and distribute digital media format and reference implementation with technical performance superior to h.265.
- FAAC — Proprietary AAC audio encoder.
- FAAD2 — ISO AAC audio decoder.
- FLAC — Free Lossless Audio Codec.
- Fraunhofer FDK AAC — Complete, high-quality audio solution to Android (and Linux) users.
- JasPer — Software-based implementation of the codec specified in the emerging JPEG-2000 Part-1 standard.
- LAME — MP3 encoder and graphical frame analyzer.
- liba52 — Free library for decoding ATSC A/52 streams.
- libdca — Free library for decoding DTS Coherent Acoustics streams.
- libde265 — Open source implementation of the h.265 video codec.
- libdv — The Quasar DV codec (libdv) is a software codec for DV video.
- libmpeg2 — Library for decoding MPEG-1 and MPEG-2 video streams.
- MAD — High-quality MPEG audio decoder.
- Musepack — Audio compression format with a strong emphasis on high quality. It's not lossless, but it is designed for transparency, so that you won't be able to hear differences between the original wave file and the much smaller MPC file. It is based on the MPEG-1 Layer-2 / MP2 algorithms, but since 1997 it has rapidly developed and vastly improved and is now at an advanced stage in which it contains heavily optimized and patentless code.
- Nero AAC — AAC audio codec (decode/encode/tag) all-in-one.
- opencore-amr — Open source implementation of the Adaptive Multi Rate (AMR) speech codec.
- Opus — Totally open, royalty-free, highly versatile audio codec. Opus is unmatched for interactive speech and music transmission over the Internet, but is also intended for storage and streaming applications. It is standardized by the Internet Engineering Task Force (IETF) as RFC 6716 which incorporated technology from Skype's SILK codec and Xiph.Org's CELT codec.
- Schrödinger — Advanced royalty-free video compression format designed for a wide range of uses, from delivering low-resolution web content to broadcasting HD and beyond, to near-lossless studio editing.
- Speex — Patent-free audio compression format designed for speech.
- Theora — Open video codec developed by the Xiph.org.
- Vorbis — Completely open, patent-free, professional audio encoding and streaming technology.
- VP8 — High-quality, open video format for the web that's freely available to everyone.
- WavPack — Audio compression format with lossless, lossy, and hybrid compression modes.
- x264 — Free library for encoding H264/AVC video streams.
- x265 — Open-source project and free application library for encoding video streams into the H.265/High Efficiency Video Coding (HEVC) format.
- http://x265.org/ || x265 x265-hgAUR
- Xvid — Open source MPEG-4 video codec.
From http://www.gstreamer.net/:
- GStreamer is a library for constructing graphs of media-handling components. The applications supports range from simple Ogg/Vorbis playback, audio/video streaming to complex audio (mixing) and video (non-linear editing) processing.
Simply, GStreamer is a backend or framework utilized by many media applications. See GStreamer article.
From http://www.xine-project.org/about:
- xine is a free (gpl-licensed) high-performance, portable and reusable multimedia playback engine. xine itself is a shared library with an easy to use, yet powerful API which is used by many applications for smooth video playback and video processing purposes.
As an alternative to GStreamer, many media players can be configured to utilize the xine backend provided by xine-lib.
Note that the xine project itself provides a capable video player, xine-ui.
libavcodec is part of the FFmpeg project. It includes a large number of video and audio codecs. The libavcodec codecs are included with media players such as MPlayer and VLC, so you may not need to install the ffmpeg package itself.
Tips and tricks
Install MPlayer binary codecs
As an ultimate solution, you can try to install MPlayer binary codecs.
If you are not able to play some files go to http://www.mplayerhq.hu/design7/dload.html, read the instructions and install the codec you need to play your files.
They can also be found in the codecsAUR and codecs64AUR packages.
No H264, mpg4 or Musepack (.mpc) in Totem Player
If you see the "The H264 plugin is missing" warning with Totem media player, just install the Gstreamer libav library to fix it install gst-libav.


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- Home
SiriusXM Radios
Satellite Radio Accessories
- Bluetooth Accessories
Open Box and Refurbished Gear
- Quantity Discounts
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This program will help send more books our way, the better for me to review and the better for my students to continue feeding their appetite for new books to read!
Our hero Lenore is even running a contest on our behalf to win a BEA swag pack which will include a ton of signed bookmarks, at least 2 hot ARCs/books, and a very cool bag.
If you would like to be entered in the contest, please send at least two books in good condition to:
Cape Eleuthera Island School-RSDTwin Air
3406 S.W. 9th Avenue
Ft. Lauderdale, FL 33315
Please include an invoice that says the books are used and are donations and are valued at $1 each. This will help us with our strict customs process!
Thanks a million, Lenore!


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Over the next two months, we're excited to be hosting our annual BlackBerry Innovation Forums. This series of events is due to take place throughout Europe, the Middle East, and Africa, bringing together business leaders and mobile innovators who have three goals in common:
- To connect with business leaders to learn how they can transform their customer experience and deliver outstanding business results
- To collaborate with mobile innovators in order to discover how to embrace technology and come away with ideas that can immediately be put to work in their organization
- To explore the future of mobile in business, helping to optimise mobile strategy
The series this year continues from 2010 with the theme of "Enterprising Minds". With guest speakers lined up from Deloitte®, The Royal Bank of Scotland® (RBS) and Vodacom®. The very best of enterprising companies will discuss and share what motivates and inspires them in their work and operations.
Our first event was held in Johannesburg, South Africa on Thursday October 6th. Check out a video of Rory O'Neill, VP, Marketing EMEA talking about this year's theme 'Enterprising Minds'. We'll be posting regular updates from the events – follow @BBEMEABusiness for updates and to join the conversation using the #BIF11 hashtag!
Dates for the 2011 BlackBerry Innovation Forum are:
- Johannesburg – 6th October
- London – 12th October
- Amsterdam – 20th October
- Madrid – 3rd November
- Paris – 9th November
- Dubai – 16th November
- Milan – 24th November
- Frankfurt – 29th November
Find more information at www.blackberryinnovationforum.com!
Did you attend last year's Innovation Forum events? What are you looking forward to from this year's? Share in the comments below!


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At Floral Abundance we offer same day flower delivery, next day delivery and delivery at any time of your choosing. We deliver in Westlake and also offer national and international flower delivery via our relay partners. If you need flower delivery in Westlake, support a genuine local florist and receive the best quality there is! Our fresh flowers arrive daily from local farmers as well as Multiflora in Johannesburg and we only supply the best varieties!
Take your time to browse our products. We offer bouquets and arrangements suitable for same day delivery in Westlake. Please order before midday for guaranteed flower delivery in Westlake. Buy fresh! Buy local! Support local business and order from Floral Abundance. Want to make your gift extra special? Why not add a balloon or a choclate. Be sure to add your special message when ordering and we will write it in a card for you. We do flowers for any occasion and consult with you for your wedding day. Weekly corporate delivery is one of our specialities.
Below is a selection of flowers that can be delivered the same day in West Lake if ordered by 2.00pm (depending on stock levels). We can also deliver flowers the same day in the following areas Scarborough, Misty Cliffs, Simons Town, Kommetjie, Ocean View, Capri Village, Faerie Knowe, Sun Valley, Sunnydale, Milkwood Park, Noordhoek, Fish Hoek, Clovelly, Kalk Bay, St James, Muizenberg, Lakeside, Tokai, Steenberg, Retreat, Grassy Park, Lotus River, Constantia, Wynberg, Mitchells Plain and many more. Please call us on +27 798812011 to see if we can deliver flowers in your area.


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Meet our homage to the mother of indulgence. These decadent
chocolate recipes are sure to steal the show on any table, though
we doubt they'll linger there long. So mouthwatering, they'll
have you skipping directly to dessert.
Get ahead of holiday baking, or any big baking occasion, by preparing and baking ahead of time. These tip and tricks will help you to get started.


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October, 20 2019 | 8:00 pm - 10:00 pm
541 E. St. Louis St. , Springfield, MO 65806
Come on out for a fun night of Open Mic comedy and great drink specials at Blue Room Comedy Club – located right inside Billiards of Springfield!
Blue Room Comedy Club is looking for the funniest person in Springfield! Come see these very funny amateur comedians. Mark your calendars, come out and join us support the up and coming comics from Springfield.
Wanna perform on Open Mic?
Email firstname.lastname@example.com to sign up
List Starts @ 7:30pm
Comic Meeting @ 7:45pm
(anyone who arrives after 7:45pm will not be allowed to sign up)
Show Starts @ 8pm


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Hoping to have a strong holiday season following the release of its first big fall title Too Human, Microsoft has begun its post-summer marketing push. Thursday in downtown Toronto, Microsoft showcased its diverse collection of holiday Xbox 360 software and Zune wares for press and industry goers with X'08 Canada, the Canadian version of the long-standing "X" exhibition series.
The showcase, which took place at the posh CiRCA nightclub in the heart of the city, reinforced Microsoft's message that a strong season of big games is coming for Xbox 360 owners. This year's showcase was a significant step up from previous years, with dozens of kiosks across the multifloor lounge. Microsoft's first-party games led the show while a hearty number of third-party titles played supporting roles. Electronic Arts, 2K Sports, Activision, Harmonix, Disney, Sega, Ubisoft, Square-Enix, Bethesda, Eidos, and Warner Brothers were all represented on the show floor.
Headlining the event was Gears of War 2, sequel to the 2006 hit from Epic. On the third floor of the club, a ten-man local multiplayer setup with ten big-screen LCD TVs and a lounge area played host to the most popular game on the floor. For the entire event, yells of excitement and screams of frustration could be heard from gamers using chainsaws and chain-guns to take it to each other. Gears of War 2 is without question the marquee title for the company's holiday season, and the layout made that abundantly clear.
Gears of War 2 was the headlining title at the event.
On the other end of the spectrum, Lips, Microsoft's new casual-friendly karaoke game designed to be the direct answer to Sony's popular SingStar franchise, was also being pushed hard. A few choice tracks including what appears to be the game's "trademark" song, "Young Folks" by Peter, Bjorn & John, were playable. The combination of motion-sensitive, wireless microphones and the ability to import your own music will likely make this first-party title a hit for social gatherings.
Microsoft also made a point of providing demos of the new Xbox 360 interface which is scheduled to land this fall with the seasonal firmware update. The new interface was unveiled at E3 earlier this year, and Microsoft reps were on hand to take users through the new features. The interface itself is certainly attractive, but the new "party" mode—a grouping system which allows up to eight players to engage in a voice or video chat, share pictures and movies, and join games together—is still far and away the most exciting feature.
One unexpected centerpiece on display in full form at the show was the Zune. Upon entering the show floor, some press—myself included—were led directly to the Zune area. The Canada-specific "Zune Originals" program was the main pitch of the presentation. Canadians looking to get a personalized Zune can pick up a "Zune Original"—a special customized model with laser-engraved artwork from six Canadian artists. These originals are currently available at no extra charge until mid-September. Inquires about the forthcoming addition of games to the Zune went unanswered, but it's clear that Microsoft believes that Xbox 360 gamers are a choice target market for the Zune media player, as demos for how to use the Zune to connect wireless to the Xbox 360 and share music were ongoing.
A strong collection of first- and third-party games was on display.
Other popular floor attractions included first-party games Banjo-Kazooie: Nuts & Bolts, the latest iteration in the famed Rare franchise which began life on the Nintendo 64, and Fable II, another big-name sequel to one of the original Xbox's exclusives. Big third-party crowd-drawers included the likes of Rock Band 2, Guitar Hero: World Tour, EndWar, Call of Duty 5, Far Cry 2, NHL '09, and Pure.
With both a plethora of Xbox 360 games and the Zune on display, Microsoft was raring to show off its wares for the coming holiday season. Whether or not the company can produce a repeat of last year's strong holiday sales performance is hard to say considering the vast improvement of PlayStation 3 software slated for release this fall. But Microsoft's strong line-up of first-party titles and leading-platform versions of big, multiplatform titles showed that the big guns are ready for the impending holiday war.
For hands-on impressions of games from the show-floor and more pictures of the event, check out Ars' gaming journal, Opposable Thumbs.


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Back in November 2010, it was rumored that director Christopher Nolan might shoot all of The Dark Knight Rises in IMAX. In May 2011, Warner Bros. sent out a press release saying that while not all of the movie would be shot in IMAX, the technology would be used "even more extensively" than it was in The Dark Knight, which contained about 40 minutes of IMAX. Then in December, Steve spoke with producer Emma Thomas and learned that the movie could feature more than 40 – 50 minutes of IMAX footage. Today, The Wall Street Journal reports that there will be "more than one hour's worth" of IMAX footage in The Dark Knight Rises. Aw, yeah.
Hit the jump for more. The Dark Knight Rises opens July 20th.
Speaking to The Wall Street Journal, Nolan explained that blockbuster movies can handle the technical requirements of the noisy, 90-pound cameras:
"There was a huge irony that we were told it would be too difficult to shoot a Hollywood movie on IMAX when we had this gigantic camera department, grips, electric, hundreds of people working for us," says the director, whose agreement to direct "The Dark Knight" was contingent on Warner Bros. allowing him to shoot the film in IMAX. "These were cameras that had been to the top of Mount Everest, to the bottom of the ocean and into outer space, but people thought we couldn't make a feature film. It was absurd."
There will be plenty of 3D and IMAX post-converted blockbusters this summer, but movies shot in IMAX, not 3D, is where those blockbusters should be trying to go. An IMAX movie is absolutely worth the cost of a ticket because it gives the viewer something they'll never be able to get at home. The Dark Knight Rises won't be entirely in IMAX, but I think we can all live with getting over an hour's worth of the spectacular format.
Finally, if you missed it, the new trailer for The Dark Knight Rises will premiere in front of The Avengers next week. Read about it here.


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English Rugby
Grewcock backs Shaw for World Cup
Tom Hamilton
August 18, 2011
Grewcock won 69 Test caps for England during a 10-year international career © Scrum.com
Former England lock Danny Grewcock believes his former side are spoilt for choice when it comes to finding the right second row mix for next month's Rugby World Cup.
England boss Martin Johnson is due to name his final 30-man squad on Monday with one of the key decisions he faces being whether to take three or four locks. The likes of Tom Palmer, Louis Deacon and Courtney Lawes have all impressed of late while the evergreen Simon Shaw remains a significant presence. Added to that impressive list of talent is relative newcomer Mouritz Botha and the ultra-versatile flanker-come-second-row Tom Croft.
Whatever blend Johnson opts for, Grewcock believes he cannot lose due to England's embarrassment of riches in that department. "I think it is exciting times in the second-row for England," Grewcock told ESPNscrum. "Louis Deacon had a great Six Nations and he really hit some form. Tom Palmer is playing some of the best rugby he has played for England.
"Courtney Lawes is the pick of the guys and he has played brilliantly for Northampton all year and he also performs at international level, which is difficult to do. I think you also have the experience of Simon Shaw, which is key, so I think Johnno has a lot of very good second-rows there."
And Grewcock believes, if Johnson does take four locks, that Shaw should be the man to fill that berth. Shaw, who won his 69th Test cap against Wales earlier this month, will be 38-years-old by the time England kick off their World Cup campaign but Grewcock believes he could still feature. "I think they'd go for Shaw over anyone else as he is part of the tried and tested second-row combination," Grewcock said.
"He's done it 60 odd times already and he's performed at the top level for a while and I do not think it would be a difficult decision to put him in the team at all so I reckon he'll be the fourth lock."
Question marks remain over the fitness of captain Lewis Moody but Grewcock, who has played with both Moody and England's most recent skipper Mike Tindall at club level, believes the Bath flanker is the right man to lead the side into the 2011 showpiece.
"Lewis Moody is a fantastic captain for England," Grewcock said. "The way they play and their good attitude is all part of his Lewis' captaincy. At the moment Lewis has had more experience with captaincy and if he is fit I would stick with him. However, he has an arsenal of experienced players behind him so if he is struggling then you have the likes of Mike Tindall and Nick Easter ready to step in.
"They are a very talented and a young squad but at the same time there are many leaders in there but if Lewis is fit I'd want to see him leading England out."
SSE, a leading energy company and National Community Partner of the RFU, is launching a new grassroots club rugby campaign, backed by Danny Grewcock and John Bentley. Tell us what your rugby club needs at www.SSErugby.com to win 2012 England International tickets.
© ESPN Sports Media Ltd.
Tom Hamilton is the Assistant Editor of ESPNscrum.
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Well, to begin with, we live in the most beautiful region in the United States. Ithaca and Central New York with its HUNDREDS of named waterfalls, Wineries, distilleries and breweries as well as the fabulous Finger Lakes make it a favored vacation destination for all. IDENTIFY you, your organization, your business with that beauty and by association, transfer some of that beauty to you!
Choose one of our magnificent images to grace your walls, your web site, your publications and more to bring the "Finger Lakes Flavor" to your project.
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Are you looking for the customer service that you have wanted but never received from a stock company? If you don't see what you need, request a photo using our convenient form. We will contact our contributors – more than 70 carefully reviewed photographic artists—and find your image.
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What is "Stock Photography?
If you are here for the first time, you may wonder what all of the pretty pictures are all about, and what are they for. Directly put, these images are available to choose from to help you make yourself look absolutely wonderful to your clients.
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With a field of over 70 photographers in the Central New Yorkarea, and many more in lands as far as Hawaiiand the UK, our stock is sure to have the image that helps tell your story, or set any mood you want!
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We also are full service commercial photographers, creating beautiful group photos from your meetings or workshops, designing beautiful photos for advertising pieces, in-process and complete architecture photos and much, much more.


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Take the battle to Avernus and blast your way through two new game modes in the Heroes expansion for the critically acclaimed shoot 'em up RESOGUN.
Take on wave after wave of relentless enemies and survive the day and night in Survival Mode and use the all new wrecking balls to cause maximum destruction in Demolition Mode.
Set your best score in new and improved leaderboards, and stake your claim on new trophies.
Get the RESOGUN Season Pass today and download the Heroes expansion now. The Season Pass will also give you access to future add-ons once they're released.
Heroes Expansion
Take on wave after wave of relentless enemies in Survival Mode, and cause maximum destruction in Demolition Mode.
Future add-ons will provide. . .
From Housemarque OY, the creators of the Super Stardust™ series & Dead Nation™ comes RESOGUN™, the next generation of shoot 'em up, developed exclusively for the PlayStation®4 system.
Unleash a diverse array of devastating weapons including lasers, missiles, overdrive, and nova-bombs to blast hordes of enemies and rescue the last survivors from distant galactic colonies that are under ruthless attack from an evil alien invasion force.
©2013 Sony Computer Entertainment Europe. Resogun is a trademark of Sony Computer Entertainment America LLC.


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On Saturday May 6th the Sheffield Steelkings kick of there 2017/18 Para hockey league season v Kingston Kestrels(13.45pm UK Time). The Steelkings would welcome support for the coming season. If you are based within the UK then it would be great to see you at a game. Games are played at Ice Sheffield. If you are elsewhere in the International Ice Hockey Community then you can follow the games at the following places:
Twitter: https://twitter.com/SteelkingsSH
Other teams accounts for the league are :
Kingston Kestrels: https://twitter.com/KestrelsParaIce
Peterbrough Phantoms: https://twitter.com/PhanParaHockey
Cardiff Huskies - https://twitter.com/cardiffhuskies
Manchester Mayhem - https://twitter.com/MancMayhemPIHC


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Post Your Comment
Everything about entrepreneurship, the good, bad and the ugly of it, fascinates me. I take a keen interest on startups and venture capital firms and have written extensively on fundraises, M&As and business strategies. I can safely say changing tracks from engineering to journalism has been one of my best decisions. When not working, I indulge in almost every Indian's poison, cricket, playing or watching. I am a foodie and video game buff.
(This story appears in the 12 April, 2019 issue of Forbes India. You can buy our tablet version from Magzter.com. To visit our Archives, click here.)


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WASHINGTON — Biologists reached back to elementary school to discover a promising new way to fight nasty bacteria: Apply the rules of grammar.
The unusual method to try to defeat drug-resistant microbes and anthrax borrows a page from "The Da Vinci Code" and the TV show "CSI: Crime Scene Investigation." Studying potent bacteria-fighters found in nature called antimicrobial peptides, biologists found that they seemed to follow rules of order and placement that are similar to simple grammar laws.
Using the new grammar-like rules for how these peptides work, scientists created 40 new artificial bacteria-fighters. They found that nearly half of them vanquished a variety of bacteria and two of them beat anthrax, according to a paper being published Thursday in the journal Nature.
This potentially creates not just a new type of weapon against hard-to-fight germs, but a way to keep churning out new and different microbe-attackers. In theory, that means that when bacteria evolve new defenses against one drug, doctors won't be stymied.
Using grammar as their guide, scientists could easily produce tens of thousands of new bacteria-fighters and test them for use as future drugs, said study lead author Gregory Stephanopoulos, a chemical engineering professor at the Massachusetts Institute of Technology.
It would likely take several years to develop the new medicines, but the process conceivably could be speeded up for fighting the worst bacteria, Stephanopoulos said.
In man's war with microbes, bacteria keep mutating to develop resistance to nature-derived drugs. However, this new method could allow scientists to jump several steps ahead of microbes, said Robert Berwick, a computational linguistics and engineering professor at MIT who wasn't part of the team.
Peptides are small proteins that attack the membrane walls of microbes causing them to rupture, said Georgetown University surgery professor Michael Zasloff, who first discovered antimicrobial peptides 19 years ago.
The key turns out to be in the way the peptides are made: stringing together amino acid molecules, which scientists represent with letters. That's when researchers saw a pattern that would make an English teacher beam.
"You have a string of letters and that string of letters reminds you immediately of a sentence, a kind of incomprehensible sentence, and you wonder in that sentence, 'Is that meaning hidden?'" asked Stephanopoulos. He used the example of a sentence: "Dave asks a question." What Stephanopoulos did was the equivalent of substitute different names for Dave and found that the peptide often still beat the bacteria.
Harvard evolutionary biologist Marc Hauser said that using grammar rules to decode genetics and medicine is growing more popular. But he said he worries that too many people are calling grammar what is really just simple code, not nearly as complicated as human language.
Berwick said the bacteria-fighting grammar rules are equivalent to the extremely basic spelling rule, "i before e except after c." The grammar rules Stephanopoulos developed are about what 2-year-olds learn on their own by listening to adults speak, he said.
Copyright 2006 The Associated Press. All rights reserved. This material may not be published, broadcast, rewritten or redistributed.


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Anna Marinenko, a designer from the Ukraine, wanted to create a tribute to the people of Japan, and from her mind sprung this simple clock:
I created Japan clock as reminder, that life goes on. And time heals. I'm sure Japanese people would be strong and will have great future. I was inspired with Japanese flag, blood drops and circles on the water. Clock made of thin plastic plates. There is opportunity to write notes on the clock's surface.
Marinenko is known for her 3D modelling work, including the Ghost Urns, which we've featured before. [Pursuitist]


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Easter Sunday Dinner
Sunday, April 24th
Serving 12:00pm – 6:00pm
Easter Dinner Reservations: (540) 837-1375
Break away from the city chaos and join us for Sunday Easter Dinner at L'Auberge Provençale Country Inn & Fine Dining Restaurant in White Post, VA. Celebrate the season in our sunny Peach Dining Room, or enjoy the views of the rolling hills and rural countryside by a window in the Blue Dining Room. If the weather is nice enjoy dinner, a cocktail or a glass of fine Virginia wine on the Patio! Enjoy the cool breeze, fresh air and scent of blooming flowers from the gardens of our Virginia bed and breakfast. Reserve a table for two, or gather the whole family for the celebration. Children are more than welcome and we have prepared a special Easter Menu just for them!
This year's Easter Menu will highlight the benefits of local dining. We're featuring the best of the farms, orchards and gardens that the Shenandoah Valley has to offer. The Dinner Menu on Easter Sunday will have the traditional items you're looking for with some delightful twists!
The menu will highlight Spring Lamb, Local "Ham", Ramps, Fiddleheads, Morels, Local Foraged Greens, Local Cheeses, Spring Asparagus, Fresh Peas and Homemade Biscuits.
Each Couple will also receive an "Easter Basket" with Housemade Chocolates.
Three Course Dinner: $62 per person
Five Course Dinner: $88 per person
Tasting Menu: $115 per person
Five Course Wine Pairings: $58 per person
Wine Tasting: $68 per person
Please make Reservations in advance as space is limited.
Escape to the country and celebrate this special day with great food, family and friends!


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Pell's equation
|Number theory index
|History Topics Index
We will discuss below whether Pell's equation is properly named. By this we mean simply: did Pell contribute at all to the study of Pell's equation? There is no doubt that the equation had been studied in depth for hundreds of years before Pell was born. In fact the first contribution by Brahmagupta was made around 1000 years before Pell's time and it is with Brahmagupta's contribution that we begin our historical study.
First let us say what Pell's equation is. We are talking about the indeterminate quadratic equation
nx2 + 1 = y2which we can also write as
y2 - nx2 = 1where n is a given integer and we are looking for integer solutions (x, y).
Now, although it is fair to say that Brahmagupta was the first to study this equation, it is equally possible to see that earlier authors had studied problems related to Pell's equation. To mention some briefly: Diophantus examines problems related to Pell's equation and we can reduce Archimedes' "cattle problem" to solving Pell's equation although there is no evidence that Archimedes made this connection.
Let us first note that
(b2 - na2)(d2 - nc2) = (bd + nac)2 - n(bc + ad)2and
(b2 - na2)(d2 - nc2) = (bd - nac)2 - n(bc - ad)2From this we see that if
b2 - na2 = 1 and d2 - nc2 = 1then
(bd + nac)2 - n(bc + ad)2 = 1and
(bd - nac)2 - n(bc - ad)2 = 1.In other words, if (a, b) and (c, d) are solutions to Pell's equation then so are
(bc + ad, bd + nac) and (bc - ad, bd - nac).This fundamentally important fact generalises easily to give Brahmagupta's lemma, namely that if (a, b) and (c, d) are integer solutions of 'Pell type equations' of the form
na2 + k = b2 and nc2 + k' = d2then
(bc + ad, bd + nac) and (bc - ad, bd - nac)are both integer solutions of the 'Pell type equation'
nx2 + kk' = y2.The proof that we have given is due to European mathematicians in the 17th Century (and we shall comment further on it later in this article), but Brahmagupta's lemma was discovered by Brahmagupta in 628 AD. The method was called samasa by the Indian mathematicians but we shall call it the 'method of composition'. In fact this method of composition allowed Brahmagupta to make a number of fundamental discoveries regarding Pell's equation.
One property that he deduced was that if (a, b) satisfies Pell's equation so does (2ab, b2 + na2). This follows immediately applying the method of composition to (a, b) and (a, b). Now of course the method of composition can be applied again to (a, b) and (2ab, b2+ na2) to get another solution and Brahmagupta immediately saw that from one solution of Pell's equation he could generate many solutions.
This was not the only way that Brahmagupta used the method of composition. He also noted that, using a similar argument to what we have just given, if x = a, y = b is a solution of nx2 + k = y2 then applying the method of composition to (a, b) and (a, b) gave (2ab, b2 + na2) as a solution of nx2 + k2 = y2 and so, dividing through by k2, gives
x = 2ab/k, y = (b2 + na2)/kas a solution of Pell's equation nx2 + 1 = y2.
How does this help? These values of x and y do not look like integers. Well if k = 2 then, since (a, b) is a solution of nx2 + k = y2 we have na2 = b2 - 2. Thus
x = 2ab/2 = ab,and this is an integer solution to Pell's equation. If k = -2 then essentially the same argument works while if k = 4 or k = -4 then a more complicated method, still based on the method of composition, shows that integer solutions to Pell's equation can be found. So Brahmagupta was able to show that if he could find (a, b) which "nearly" satisfied Pell's equation in the sense that na2 + k = b2 where k = 1, -1, 2, -2, 4, or -4 then he could find one, and therefore many, integer solutions to Pell's equation. Often he could find trial solutions which worked for one of these values of k and so in many cases he was able to give solutions.
y = (b2 + na2)/2 = (2b2 - 2)/2 = b2 - 1
y = (b2 + na2)/2 = (2b2 - 2)/2 = b2 - 1
For example, if we attempt to solve 23x2 + 1 = y2 we see that a = 1, b = 5 satisfies 23a2 + 2 = b2 so, by the above argument, x = 5, y = 24 satisfies Pell's equation. Applying the method of composition to the pair (5, 24) gives
x = 2×5×24 = 240, y = 242 + 23×52 = 1151as another solution. Applying the method of composition again yields
x = 11515, y = 55224and yet again gives
x = 552480, y = 2649601and so on.
Among the examples Brahmagupta gives himself is a solution of Pell's equation
83x2 + 1 = y2where he notes that the pair (1, 9) satisfies
83×12 - 2 = 92and applies his method to find the solution
x = 9, y = 82.We can now generate a sequence of solutions (x,y):
(9, 82),and so on.
(1476, 13447),
(242055, 2205226),
(39695544, 361643617),
(6509827161, 59307347962),
(1067571958860, 9726043422151),
(175075291425879, 1595011813884802)
(1476, 13447),
(242055, 2205226),
(39695544, 361643617),
(6509827161, 59307347962),
(1067571958860, 9726043422151),
(175075291425879, 1595011813884802)
One can only marvel at this brilliant work done by Brahmagupta in 628 AD.
The next step forward was taken by Bhaskara II in 1150. He discovered the cyclic method, called chakravala by the Indians, which was an algorithm to produce a solution to Pell's equation nx2 + 1 = y2 starting off from any "close" pair (a, b) with na2 + k = b2. We can assume that a and b are coprime, for otherwise we could divide each by their gcd and get a "closer" solution with smaller k. Clearly a, k are then also coprime.
The method relies on a simple observation, namely that, for any m, (1, m) satisfies the 'Pell type equation'
n.12 + (m2 - n) = m2.Bhaskara II now applies the method of composition to the pairs (a, b) and (1, m) to get
n(am + b)2 + (m2 - n)k = (bm + na)2.Dividing by k we note that
x = (am + b)/k, y = (bm + na)/kis a solution to
nx2 + (m2 - n)/k = y2.Since a, k are coprime we can chose m such that am + b is divisible by k. Bhaskara II now knows (but he gives no proof) that when m is chosen so that am + b is divisible by k then m2 - n and bm + na are also divisible by k. With such a choice of m he therefore has integer solutions
x = (am + b)/k, y = (bm + na)/kto the 'Pell type equation' nx2 + (m2 - n)/k = y2 where (m2 - n)/k is also an integer.
Next Bhaskara II knows that there are infinitely many m such that am + b is divisible by k. He chooses the one which makes m2 - n as small as possible in absolute value. If (m2 - n)/k is one of 1, -1, 2, -2, 4, -4 then we can apply Brahmagupta's method to find a solution to Pell's equation nx2 + 1 = y2. If (m2 - n)/k is not one of these values then repeat the process starting this time with the solution x = (am + b)/k, y = (bm + na)/k to the 'Pell type equation' nx2 + (m2 - n)/k = y2 in exactly the same way as we applied the process to na2 + k = b2. Bhaskara II knows (almost certainly by experience rather than by having a proof) that the process will end after a finite number of steps. This happens when an equation of the form nx2 + t = y2 is reached where t is one of 1, -1, 2, -2, 4, -4.
Bhaskara II gives examples in Bijaganita and the first we look at is
61x2 + 1 = y2.Using the above method he chooses m so the (m + 8)/3 is an integer, making sure that m2 - 61 is as small as possible. Taking m = 7 he gets
x = 5, y = 39as a solution to the 'Pell type equation' nx2 - 4 = y2. But this is an equation which Brahmagupta's method solves giving
x = 226153980, y = 1766319049as the smallest solution to 61x2 + 1 = y2.
Why do we suspect that Bhaskara II had no proof of the method? Well there are at least two reasons. Firstly the proof is long and difficult and would appear to be well beyond 12th Century mathematics. Secondly the algorithm always reaches a solution of Pell's equation after a finite number of steps without stopping when an equation of the type nx2 + k = y2 where k = -1, 2, -2, 4, or -4 is reached and then applying Brahmagupta's method. If experience of the algorithm is only via examples then, knowing how to proceed when k = -1, 2, -2, 4, or -4 is reached, it is natural to switch to Brahmagupta's method at that point. However, when one writes down a proof it should become clear that the algorithm switching to Brahmagupta's method is never necessary (although can reach the solution more quickly).
The next contribution to Pell's equation was made by Narayana who, in the 14th Century, wrote a commentary on Bhaskara II's Bijaganita. Narayana gave some new examples of the cyclic method. Here are two of his examples:
103x2 + 1 = y2.Choosing a = 1, b = 10 Narayana obtains
103×12 - 3 = 102.Choose m so that m + 10 is divisible by -3 with m2 - 103 as small as possible leads to m = 11 and we obtain
103×72 - 6 = 712.Next we must choose m so that 7m + 71 is divisible by -6 and m2 - 103 as small as possible. Take m = 7 to get the equation
103×202 + 9 = 2032.Continuing, choose m so that 20m+203 is divisible by 9 and m2 - 103 as small as possible. Take m = 11 to get the equation
103×472 + 2 = 4772.Now Narayana applies Brahmagupta's method, in the form we gave above for equations with k = 2, to obtain the solutions
x = 22419, y = 227528.His next example is a solution of Pell's equation
97x2 + 1 = y2which leads successively, by applying the cyclic method, to the equations
97×12 + 3 = 102Finally Narayana applies Brahmagupta's method to this last equation to get the solution
97×72 + 8 = 692
97×202 + 9 = 1972
97×532 + 11 = 5222
97×862 - 3 = 8472
97×5692 - 1 = 56042
97×72 + 8 = 692
97×202 + 9 = 1972
97×532 + 11 = 5222
97×862 - 3 = 8472
97×5692 - 1 = 56042
x = 6377352, y = 62809633Now the brilliant ideas of Brahmagupta, Bhaskara II and Narayana were completely unknown to the European mathematicians in the 17th Century. The European interest began in 1657 when Fermat issued a challenge to the mathematicians of Europe and England. Fermat wrote:-
We await these solutions, which, if England or Belgic or Celtic Gaul do not produce, then Narbonese Gaul will.Narbonese Gaul, of course, was the area around Toulouse where Fermat lived! One of Fermat's challenge problems was the same example of Pell's equation which had been studied by Bhaskara II 500 years earlier, namely to find solutions to
61x2 + 1 = y2.Several mathematicians participated in Fermat's challenge, in particular Frenicle de Bessy, Brouncker and Wallis. There followed an exchange of letters between these mathematicians during 1657-58 which Wallis published in Commercium epistolicum in 1658. Brouncker discovered a method of solution which is essentially the same as the method of continued fractions which was later developed rigorously by Lagrange. Frenicle de Bessy tabulated the solutions of Pell's equation for all n up to 150, although this was never published and his efforts have been lost. He challenged Brouncker who was claiming to be able to solve any example of Pell's equation to solve
313x2 + 1 = y2.Brouncker found the smallest solutions, using his method, which is
x = 1819380158564160, y = 32188120829134849and he sent these to Frenicle de Bessy claiming that they had only taken him "an hour or two" to find.
In Commercium epistolicum Wallis gave two methods of proving Brahmagupta's lemma which are both essentially equivalent to the argument we gave at the beginning of this article based on the result
(b2 - na2)(d2 - nc2) = (bd + nac)2 - n(bc + ad)2.In 1658 Rahn published an algebra book which contained an example of Pell's equation. This book was written with help from Pell and it is the only known connection between Pell and the equation which has been named after him.
Wallis published Treatise on Algebra in 1685 and Chapter 98 of that work is devoted to giving methods to solve Pell's equation based on the exchange of letters he had published in Commercium epistolicum in 1658. However, in his algebra text Wallis put all the methods into a standard form.
We should note that by this time several mathematicians had claimed that Pell's equation nx2 + 1 = y2 had solutions for any n. Wallis, describing Brouncker's method, had made that claim, as had Fermat when commenting on the solutions proposed to his challenge. In fact Fermat claimed, correctly of course, that for any n Pell's equation had infinitely many solutions.
Euler gave Brahmagupta's lemma and its proof in a similar form to that which we have given above. He was, of course, aware of the work of Brouncker on Pell's equation as presented by Wallis, but he was totally unaware of the contributions of the Indian mathematicians. He gave the basis for the continued fractions approach to solving Pell's equation which was put into a polished form by Lagrange in 1766. The other major contribution of Euler was in naming the equation "Pell's equation" and it is generally believed that he gave it that name because he confused Brouncker and Pell, thinking that the major contributions which Wallis had reported on as due to Brouncker were in fact the work of Pell.
Lagrange published his Additions to Euler's Elements of algebra in 1771 and this contains his rigorous version of Euler's continued fraction approach to Pell's equation. This established rigorously the fact that for every n Pell's equation had infinitely many solutions. The solution depends on the continued fraction expansion of √n. In the continued fraction of the square root of an integer the same denominators recur periodically. Moreover, the pattern in most of the recurring sequence is "palindromic". i.e. up to the last element, the second half of the periodic sequence is the first half in reverse. The last number in the repeating sequence is double the integer part of the square root.
For example √19 has the continued fraction expansion
which stands for
which recurs with length 6. The convergent immediately before the point from which it repeats is 170/39 and Lagrange's theory says that
x = 39, y = 170will be the smallest solution to Pell's equation
19x2 + 1 = y2.To find the infinite series of solutions take the powers of 170 + 39√19. For example
(170 + 39√19)2 = 57799 + 13260√19and
x = 13260, y = 57799will give a second solution to the equation. Again
(170 + 39√19)3 = 19651490 + 4508361√19giving
x = 4508361, y = 19651490as the next solution. Here are the first few powers of (170 + 39√19), starting with its square, which gives the first few solutions to the equation 19x2 + 1 = y2
57799 + 13260√19Although the continued fraction approach to solving Pell's equation is a very nice one for small values on n, the difficulty of the method has been analysed to see if it is the most efficient for large n. A polynomial time method in the length of the input n would be an algorithm which took time bounded by a fixed power of log n (the length of the input). The continued fraction method is not a polynomial time algorithm, and indeed it is now known that no polynomial time algorithm exists for solving Pell's equation.
19651490 + 4508361√19
6681448801 + 1532829480√19
2271672940850 + 521157514839√19
772362118440199 + 177192022215780√19
262600848596726810 + 60244766395850361√19
89283516160768675201 + 20483043382566906960√19
19651490 + 4508361√19
6681448801 + 1532829480√19
2271672940850 + 521157514839√19
772362118440199 + 177192022215780√19
262600848596726810 + 60244766395850361√19
89283516160768675201 + 20483043382566906960√19
Article by: J J O'Connor and E F Robertson


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