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Carapuça é uma espécie de barrete ou capuz de forma cônica carapuça in Dicionário infopédia da Língua Portuguesa com Acordo Ortográfico [em linha]. Porto: Porto Editora, 2003-2018. [consult. 2018-01-07 01:01:50]. Disponível na Internet: https://www.infopedia.pt/dicionarios/lingua-portuguesa/carapuça  e remonta ao período da Inquisição, em que os condenados eram obrigados a vestir trajes ridículos ao comparecer aos julgamentos. Além de usar uma túnica com o formato de um poncho, eles precisavam colocar sobre a cabeça um chapéu longo e pontiagudo, conhecido como carapuça. Daí a expressão "vestir a carapuça" ter se incorporado ao português escrito e falado com o sentido de "assumir a culpa".

A palavra vem do italiano antigo, carapazza, e do espanhol caperuza. Atualmente, a carapuça é utilizada em aves de rapina, na prática da falcoaria.

O fabricante de carapuças chama-se carapuceiro.

É conhecido também como um nome de família.

Chapéus


because they can. If you encounter, but you can not beat , please hurry to invite your friends list into your game together to engage it, and to drop more than a goodly portion of luck, but also a way of getting money to the Friends of the base!
used to increase this figure a bit of a challenge and entertainment, to be honest, not farm their significance, and this strange basically impossible to be done FARM their fall list and blue The strange world drop, as limited by chapter, but the yellow loading a relatively high probability it wants.


The Good Luck Duck: Port, Eh?
Thank you for your responses to my question about change. I am grateful, and awed by your stories. If you feel inclined to add a reply, I hope you'll give in to the urge.
They're living full-time in their Honda Element with their two substantial dogs, Walter and George. Both are swim coaches (the humans; the dogs swim avocationally). Toni describes her work with young kids as helping them relate to people better, speak more effectively with adults - a set of life skills - and almost incidentally she helps them become better swimmers. Scott works with older kids and young adults, superior swimmers who are headed for watery greatness. His work is more focused on the mechanics of swimming, and helping kids reach their scholarship and Olympic goals.
They have given themselves a year to find their own Utopia. A place that feeds their soul and jobs that feed their food habits. They've been offered jobs, but not in places where they want to live. Toni says she'd settle in a heartbeat if there were only a swimming pool in the Escalante Wilderness.
When we met them, they had snuggled in tightly against the Duck to shelter from the punishing winds we'd been enduring on Bird Island Basin. Which was fine, except that their too-cute rig was also snuggling the exhaust on our generator, so we got a little depleted waiting for them to move. Annie eventually took our rig for a drink and potty break and pulled back in a little further away.
They don't have a toilet or any refrigeration. I asked them about stealthing, but their handy pop-up makes them conspicuous, and without it they're sharing the bed with George and Walter.
The Powerfilm line of solar chargers is especially rugged and remarkably thin, as you'll notice because I'm remarking on it. The 30 watt charger folded up to be about the size of a spiral notebook (that's an iPad you can rip pages out of). If a panel should become damaged, it doesn't take the array out of the game.
We rode our bikes to dinner.
Bring mine with a chainsaw and an artichoke.
We each enjoyed an adult frozen beverage. Annie asked for the butchest thing they served.
This was a bit too much alcohol for Annie, and she had to walk her bike home. I mocked her from the sidewalk where I was resting; I don't remember the details, but the officer was nice.
Where are we going next! The giant car wash! Where then! I don't know! It's a mystery to all of us! As soon as I know, I'll tell someone!
Ipad you tear pages out of! hahah!
Michael, in case someone under 30 reads this blog.
Lou, you make the story sound much more interesting! You should write these for me! It was only an officer, and the offer was something like you can go sleep at home tonight if you can get up and walk away.
T drink in the background looks like my kind of margarita - on the rocks, no salt. mmmm.....the only problem with them is you can drink them, as opposed to sip slowly to avoid brain freeze. They tend to go quickly and then misbehave. You have to keep an eye on 'em for sure.
What does stealthing as used here mean? Just curious.
I love your bikes btw!
Les, the frozen drink was rowdy, for sure. I don't even know what kind of trouble the umbrella was getting up to while I was subduing the lime.
Rubye Jack, "stealthing" is RVing overnight in places where it's not officially allowed. Stealth vehicles are usually pretty ordinary-looking, like a car, or maybe a cargo van, and they don't raise suspicions or hackles.
I know, Judy. She has a tall persona.
The butchest drink she could find. THAT was the butchest drink she could find??? Seriously??? SO FUNNY!
trentandteresa.blogspot.com is the blog of the couple that just stayed 4 days at Big Bend.
Andra, that would test the boundaries of his minimalism, wouldn't it? And, the other drinks were pink and wearing pearls.
Teri, thanks for that. I'm going to check them out. Great to meet you today!
Janie, it wasn't a pretty sight. The Hello Kitty Kitty Kitty sported a boa.
I was really enjoying the artisteryness of the "Behind" series of photographs.
This new direction of "Under" could be a new expressive medium of artsy-fartsy ....um....espression!
Saw your message on my blog -thanks for thinking about me. We are good here - should be moved back in the house this week. Next on the list of repairs are my travel plans.
Meanwhile I will enjoy your epic journey. Your blog gets even better with the miles.
OK I'm behind here in a park with whimpy wifi and you are out all over the place having adventures with stealthers and butch drinks and cops. We need to catch up to you and just lurk in the background. It must be even funnier in person and it's mighty funny right here on the non ipad-ipod-iphone page.
Ish, stay tuned for more behindedness. I hadn't considered a whole Under series *jots down idea and pretends it's original*.
Kim, I'm glad you're doing alright. Such a cluster of emotional and financial setbacks ... glad to hear you're going to repair your travel plans, as well.
DD, it's funny, that's what the officer asked me, too! Coincidence?
Sherry, I make myself ridiculous so you don't have to!
I always figured they put umbrellas in drinks to help you keep your balance afterwards...like tightrope walkers who use umbrellas, y'know? Never worked for me, though, and now I know they won't keep a bicycle upright,either.
Sue, you're a Mythbuster! I suppose the plastic sword didn't help me, either.
Good for you. Way to live your life. Carpe Diem.
Hey Martha! Glad to have you following. Did we drop a tailpipe in Pleasanton?
Thanks, Nomad! I needed to carpe the guard rail on my way home.


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- YAK 40
- YAK 40K
- YAK 42D
Yakovlev YAK 40K VIP
US$ 2.500.000
类型: 喷气式飞机
品牌: Yakovlev
型号 1: YAK 40K VIP
最后一年的: 1/2010
地点: 俄罗斯, Kazan, UWKD
时间总计: 11497 h
价格: US$ 2.500.000
Aircraft Tools Consulting Ltd.
Presnensky Val st., 27
电话: +7 (495) 769 4242
移动电话: +7 (916) 650 4707
乘客数量: 10
登陆: 6763
品牌/型号: Ivchenko AI-25 turbofans Ivchenko AI-25 turbofans Ivchenko AI-25 turbofans
功率: 14.7 kN 14.7 kN 14.7 kN
Powerplant: 3 x Ivchenko AI-25 turbofans, 14.7 kN (3,300 lbf) each
The Ivchenko AI-25 is a family of military and civilian twin-shaft medium bypass turbofan engines developed by Ivchenko OKB of the Soviet Union. It was the first bypass engine ever used on short haul aircraft in the USSR. The engine is still produced by Ukrainian based aircraft engine manufacturing company, Motor Sich.
The AI-25 was designed to power the Yakovlev Yak-40 tri-jet airliner, often called the first regional jet transport aircraft, and is the starting point for the Lotarev DV-2 turbofan engine. The project was launched in 1965, with the AI-25s first test flight in 1966, and finally cleared for production in 1967. In 1972, the AI-25 was selected for the Polish PZL M-15 Belphegor, the world's only jet-powered biplane.
Development of the AI-25 continued and the uprated AI-25TL was designed for use by the Czechoslovak Aero L-39 Albatros military trainer with the first flight occurring in 1968. The L-39, would go on to become one of the most popular, and widespread trainer aircraft in the world, with over 3,000 L-39s produced, and with 2,900 examples still in active service today. A smaller version of the AI-25TL, the AI-25TLK has equipped the People's Republic of Chinas Hongdu L-11 fighter-trainer.
The AI-25TLK is also licensed built in the People's Republic of China as the WS-11. Another variant of the AI-25, is the AI-25TLSh in the 1990s, which underwent flight testing by the Ukrainian Ministry of Defense in 2002. Ivchenko-Progress is currently marketing the AI-25TLSh as an upgrade to existing L-39 and JL-8 operators that would extend the service life of the aircraft and improve performance. The latest AI-25 is the AI-25TL Series 2 designed for the Mikoyan MiG-AT.
General characteristics
Type: Turbofan
Length: 3,358 mm (132.2 in)
Diameter: ~611.6 mm (24.08 in)(intake casing)
Compressor: Axial, 3 fan/low pressure compressor stages, 9 high pressure compressor stages
Combustors: Annular
Turbine: 1 high pressure turbine stage, 2 low pressure turbine stages
Fuel type: Aviation kerosene, JP-1 for example
Oil system: Pressure spray with return
Maximum thrust: 16.867 kN (3,792 lbf)
Overall pressure ratio: 9.5:1
Bypass ratio: 2.0:1
Fuel consumption: 468.1 kg/h (1,032 lb/h)
Specific fuel consumption: 0.58 lb/lbf/h (59 kg/kN/h)
Thrust-to-weight ratio: 4.914:1
Service-Logs
Description of the Yak-40 (1,36 MB)

redpajama_set_name: RedPajamaCommonCrawl


Bali! Got to see it to believe it !!!
Bali is much more than just another tropical beach destination. Its heart and soul is its culture. Known as the "island of the Gods", Bali's stunning beauty features varied landscape with hills and mountains, rugged coastlines, sandy beaches and lush green rice terraces.


The Osbornes recently suffered a tragedy in our menagerie (which is almost good enough to be the title of this blog, but not quite).
Regular readers will know that I'm a bit of a yokel and that I live in semi-rural idyll with a variety of strange creatures – of the furry, feathered and buzzing variety, as well as two teenagers.
But our bucolic bliss has been shattered by the loss of two members of said mengerie to the evil Mr Fox, who evidently managed to find a way into the high security wing of the garden where the chickens and ducks live – or, in the case of the ducks, lived.
Until Mr Fox's visit we had a pair of Indian Runner ducks, which admitedly were a complete waste of space in terms of egg production (the books say they lay pretty much 365 days a year but if ours managed a dozen eggs last year I'd be surprised) but were highly entertaining. If you're not familiar with Indian Runners, they stand very upright and, as the name suggests, run everywhere. Or at least they did. Sadly, not quickly enough, it would seem.
Now, we're well aware that one of the risks one faces when one keeps any kind of poultry is from foxes. Foxes like ducks and chickens too, although for their calorific rather than their entertainment value. So one has to be alert to this particular threat and take appropriate precautions. Having lost birds to foxes in the past, we decided to create a secure area for them.
On two sides of this area is a five foot high fence, fronted with steel mesh and with electric wires top and bottom. On the third side is a brick wall of a similar height and on the fourth a stock-wire fence with a really, really spiky, tangly hedge in front of it. The hedge contains hawthorn, wild roses and pyracantha – it's vicious and regularly tears chunks out of me when I'm trimming it. All that's missing is a couple of searchlight towers and some armed guards, as our neighbours hilariously point out to us on more than the odd occasion. The point is that there was no way that a fox could get through it.
Except that over a period of time, while our backs were turned, wily old Mr Fox somehow managed to create a way through the seemingly impregnable hedge and eventually put his cunning plan into action. He must have torn himself to pieces actually getting through but, give him credit, he did it. And once he was in poor old Mr and Mrs Duck didn't stand a chance. Next morning, the only evidence was a pile of feathers and part of wing. Yuck!
All of this is an object lesson in risk management. As I point out *** warning, blatant and shameless plug approaching *** in my book 'Risk Management Simplified' the risk management process involves five simple steps, as illustrated below.
As I'm a consummate risk management professional I obviously followed the process to the letter in my poultry-based risk management efforts. Except that I didn't – not quite anyway.
What I neglected to do was to carry out the ongoing, regular monitoring and reviewing, to ensure that the countermeasures that I'd put in place continued to provide the level of risk mitigation that they did when I originally implemented them. Which is an absolutely key point that is so often overlooked.
I've now addressed the gap in my risk management strategy, by installing some electric fencing along the border in question. And unless Mr Fox either can either pole-vault or teams up with a local kangaroo, of which there aren't many in Evesham to my knowledge, he's not going to get in again. I think I might just keep an eye on things though!


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Economy > Trade
WTO agreement vindicates India's stand on food security: Nirmala Sitharaman
Commerce and industry minister Nirmala Sitaraman on Friday expressed satisfaction at the conclusion of the World Trade Organization (WTO) negotiations that has taken on board India's concerns on the effects of the trade facilitation agreement on public holding of food stocks by developing and poor countries.
On 5 August 2014, India decided not to join the consensus in the WTO on the implementation of the trade facilitation agreement till its concerns relating to the implementation of other Bali ministerial decisions, in particular, the decision on public stock-holding for food security purposes, were addressed.
There was much criticism of this stand in the subsequent months but India stood firm. However, in the ensuing months, India has succeeded to win over support from other WTO members, including the US, to find a way forward, she pointed out.
On Thursday, the General Council of the WTO adopted unambiguous decisions on public stock-holding for food security purposes, trade facilitation agreement and on post-Bali work.
WTO has made it clear that under the new mechanism, WTO members will not challenge the public stock-holding programmes of developing country members for food security purposes and that this arrangement will be in place until a permanent solution regarding this issue has been agreed and adopted.
This effectively does away with the ambiguity of the limited period peace clause as well as guard against the possibility of lack of effective cover for public stock-holding for food security purposes after 2017.
For India, this would strengthens the safeguard available for continuing the minimum support price policy which is a lifeline for millions of low income, resource poor farmers. It is also critical for food security in India and in countries which have similar policies, the minister stated in Parliament.
Moreover, she said, the commitment to find a permanent solution on public stock-holding for food security purposes by 31 December 2015 on a best-endeavour basis, introduces a sense of urgency in the process and would encourage other developing countries also to join the effort in pushing for a permanent solution at the earliest.
Importantly, since the General Council has unequivocally agreed to delink the negotiations for a permanent solution on public stock-holding for food security purposes from the agriculture negotiations on other issues under the Doha Development Agenda, this would avert the danger that countries like India would have to make concessions in some other area of the agriculture negotiations, in order to achieve a permanent solution to the food security issue.
As per the relevant provisions of the WTO Agreement, a General Council decision on these elements has the same legal status as a ministerial decision, she noted.
As per the Bali ministerial decision, the trade facilitation protocol was to be open for acceptance by members until 31 July 2015. WTO members have now decided to leave this open-ended. This would enable them to complete their internal processes for its acceptance.


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Category Archives: E-Government
Congress, Congressperson, Court, Court System, E-Government, Education, Emergency Services, Government, Healthcare, Issues, Law, Projects
Brazil's congressional leaders tackle the serious cell phone problems in their country
October 1, 2011 EPR Government News	Leave a comment
We Don't Make the Smartphone…We Make it Smarter, Safer, Courteous and Compliant.
With a registered bill on the table, U.S. based Try Safety First (TSF) is deploying time, talent and resources into Brazil to advance its new mobile phone protocol technology. With its white paper being published in more than twenty-four countries across the globe, it is Brazil where TSF is looking to launch.
When asked why Brazil , TSF CEO John Fischer said, "TSF is eager to work with all governments and strategic partners to advance our patent-pending protocol technology. But to date, it is the congressional leaders of Brazil who have demonstrated the greatest desire and responsiveness to tackle the serious cell phone problems in their country."
According to the World Health Organization, environment specific (prisons, court rooms, classrooms, in flight and behind the wheel) cell phone problems are costing theworld's top industrialized countries between one and three percent of GDP. Fischer agrees societal cell phone problems are dynamic, devastating and very costly. But he also expands the statement by saying they can easily be eradicated with the implementation of TSF protocol technology.
TSF is the world leader for mobile device protocol development. According to Fischer, the advancement of TSF's protocol technology will provide an optional tool for parents, teachers, judges, wardens, pilots and employers to easily eliminate cell phone problems within their respective environments. TSF's primary objective is to establish a uniform global standard throughout the wireless industry to increase the functionality of mobile devices to provide simple solutions to remedy grand societal problems.
For more information on TSF technology or for investment opportunities, please visit http://trysafetyfirst.com.
Brazilcell phone addictioncell phone distracted drivingcell phone problemscell phones in the classroomcell phones in the court roomcontraband prison phonesforeign investmentinvesting in Brazilinvestment opportunitynew technology launchsmall businesstexting accidentstexting behind the wheel
Congress, Democrats, E-Government, Education, Election, Events & Programs, Federal Government, Foreign Policy, Government, Issues, Online, Political Analyst, Politics, Presidency, Republicans, Senate, Society, Voters
2012ElectionNews.org Changes Its Name To USElectionNews.org
March 11, 2011 EPR Government News	Leave a comment
The popular U.S. election news website, 2012 Election News, has officially changed its name. Jillian Curtin, Editor-in-Chief of 2012ElectionNews.com has announced that the website has changed its name and URL to USElectionNews.org, to provide an accurate reflection of its ongoing focus.
"The response has been overwhelming," said Curtin. "What started out as a simple 2012 election discussion has turned into a national conversation about our personal rights, needs and dreams."
The name change is effective immediately and the site's new URL is http://uselectionnews.org. What began as a forum for the U.S. presidential elections in August 2010 has evolved and taken on a life of its own. The site was originally created as a non-partisan informational venue for news about U.S. elections and candidates. The website also provided voters with a forum for the free exchange of ideas on issues of concern.
In less than a year, US Election News has grown from a hub for local, national and global election news to a go to website for information on the economy, the war on drugs, workers unions, employment, government spending, worker safety and natural disasters relateing to our government leaders. The site also addresses immigration, healthcare, terrorist activities, the emerging green energy industry, social issues and international relations as releated to United States and Presidentail elections.
The multi-media site features a wide range of audio, video and written content. The publication informs voters about topics of concern that affects them on a local and national level, along with issues in far reaching locations around the globe. US Election News disseminates information on a non-partisan basis from Republicans, Democrats, Independents and Tea Party members.
US Election News has its headquarters in Cartersville, Georgia. The publication is always seeking qualified journalists to provide news and information for a balanced view on local and global issues. The site recently launched its complimentary Facebook page that can be found at http://www.facebook.com/2012ElectionNews. The Facebook site welcomes visitors and invites them to engage in lively civil discourse and debate on topics both foreign and domestic.
US Election News is dedicated to providing voters with a balanced view of the issues and actions taking place on a daily basis that affects all facets of life in a global community. The site welcomes visitor interaction and communication for a better understanding of government and world events that shape the history of the world each day.
US Election News can be contacted by phone at 202-379-2821 or by email via the convenient online form.
2012-election-newsdemocratic-partyelection-newspresidential-electionrepublican-party
Defense, E-Government, Federal Government, Government, Government Agency, Online
Foreign Translations, Inc. Announces Award of GSA Five-Year Contract Extension
August 13, 2010 EPR Government News	Leave a comment
Foreign Translations, Inc. (www.foreigntranslations.com), leader in document translation, website localization, and International SEO, announces the successful extension of its contract with the General Services Administration (GSA) as an authorized vendor of translation services for the Federal Government.
The 5-year contract renewal for multilingual translation and interpreting services extends through May 2015. Foreign Translations, Inc. was first awarded a GSA schedule contract in May 2005. Foreign Translations, Inc. has provided services to several government agencies including: transcription work for the Bureau of Alcohol, Tobacco, Firearms and Explosives and the United States Department of Treasury; multilingual translation and brochure design for the Peace Corps; and translation services for the U.S. Department of Transportation, the U.S. Department of Energy, the U.S. Department of Justice, the FDIC, the U.S. Army Corps of Engineers, the U.S. Air Force, the U.S. Navy, and the U.S. Department of Commerce.
According to Foreign Translations, Inc.'s CEO Ken Zwerdling, "this GSA contract allows us to support all the different branches of government, including continuing support for the U.S. military in Iraq and Afghanistan."
foreign translationsgovernment translationslanguage translationstranslation servicestranslationstranslations for the government
E-Government, Education, Events & Programs, Government, Government Agency, Online, Projects, Society
New Mexico Commission for Deaf and Hard of Hearing Persons Awards Contract to WeInterpret.Net
June 23, 2009 EPR Government News
We Interpret.Net has signed a contract with the New Mexico Commission for Deaf and Hard of Hearing Persons. Their local office is located at 6100 Seagull St. NE Suite B105 (Seagull Office Plaza) in Albuquerque, NM. We Interpret.Net along with is partner company, We Interpret Linked, both headquartered in the state of Maryland has been providing Sign Language Interpreting services since 1996 primarily on the East Coast. This new contract affords the citizens of New Mexico with state wide quality interpreting service coordination, internship opportunities for new interpreters, education and outreach, and emergency hospital service coordination in Las Cruces.
WeInterpret.Net is also a sponsor for Continuing Education Credits through the Registry of Interpreters for the Deaf. In addition to the management staff, full time interpreters, both Deaf and Hearing, will be hired. Individuals who would like to sub contract to provide Interpreting services are encouraged to register at www.WeInterpret.Net.
Appropriate communication methods vary by individual and We Interpret.Net can provide interpreters skilled in American Sign Language, Signed English, The Rochester Method, Cued Speech, Oral Interpretation and Tactile Methods for the Deaf/Blind individuals. In specific areas of New Mexico interpreters are in short supply. In addition to the on-site interpreters who will be directed from this office, there is a call center in Owings Mills, Maryland to provide Video Interpreting in the rural parts of the state. "Office staff who are also certified interpreters, along with our Video Interpreting solution allow the agency to respond to every request with qualified interpreters even on the busiest of days or during the overnight and weekend hours." Stated Deb McKerrow, Director of New Mexico opperations.
About WeInterpret.Net
Maryland Interpreting Services, a Maryland corporation doing business as We Interpret.Net (www.WeInterpret.Net) and We Interpret Linked (www.WeInterpretLinked.com) was established in 1996. Owner and President, David Stephenson is a nationally certified interpreter. Since its establishment, it has provided some 6 million hours of interpreting services to some 900 customers through offices in Utah, Virginia, Maryland and New Mexico. Staff Interpreters and Independent Sub-Contractors numbering over 1000 provide on-site and Video Remote Interpreting.
About New Mexico Commission for Deaf and Hard of Hearing Persons
The State of New Mexico, Commission for Deaf and Hard of Hearing Persons provides effective leadership, education, advocacy and programs to reduce barriers to the social, economic, educational, cultural and intellectual well – being of Deaf and Hard of Hearing New Mexicans and their families, friends and colleagues. (www.cdhh.state.nm.us).
cdhh.state.nm.uscontractE-GovernmentEducationEvents & ProgramsGovernmentinterpreting serviceNew MexicoNew Mexico Commission for Deaf and Hard of Hearing PersonsOnlineovernment AgencyProjectsSign Language Interpreting serviceSocietyWeInterpret.Net
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Welcome to EPR Government News
August 29, 2008 EPR Government News
EPR Government News is a new blog, part of EPR Network, that is going to be focused on and will be covering the Government news and stories from press releases published on EPR Network.
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An exciting and rare opportunity to join an established company going through a high tempo modernisation process. Full autonomy as the site expert, there is no management responsibilities.
A well established and long-standing brand across the UK and globally with a strong regional presence. With significant investment, the site is undergoing a large modernisation process introducing increased automation and placing greater emphasis on electrical and PLC systems.
We're looking for a Lead Site Electrical and Controls Engineer for their Newcastle Facility. Fully autonomous and with excellent PLC programming knowledge, this role has no man management responsibilities.
The Location: Based just outside of Newcastle, this role has good commuting links from both sides of the water following either the A1 or A19.


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Titan joins HP to make smartwatch
Titan's smartwatch is called Juxt and will also be sold globally - initially in the US, UK and Canada - through HP's distribution network.
Bengaluru: L eading watchmaker Titan has developed and launched smartwatches in collaboration with global technology giant HP. It's a segment that has caught the imagination of many, thanks to a variety of attractive products from the stables of Apple, Samsung, Motorola, Sony and Microsoft in recent times.
"In the next one year, we will launch it in five or six more countries," Bhaskar Bhat, managing director of Titan, said.
Asked how the watch was different from others in the market, Bhat said: "Others are calculators on your wrist. Ours is very well-designed with functionalities enhanced by technology, but without losing the category identity as a watch."
Titan, he said, worked on the analog display and design, and HP worked on the electronics, and connectivity and cloud technologies. The watches are priced between Rs 16,000 and Rs 20,000, which the company says is lower than for equivalent ones from global brands.


We had a great educational visit to Roskilly's farm today. Toby Roskilly gave us a fascinating tour of the ice cream factory and we learned about how the business has grown since it was established in the 70s.
We even went into the blast chiller where it was minus 25 degrees! It was so cold and we tried to imagine what it have been like for Shackleton's crew to have been stranded in sub zero temperatures for nearly a year.
To round off a great morning we were all treated to a delicious organic Roskilly's ice. This should really inspire us to design and make some terrific ice creams.


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Taking a look at Fair Green over the coming week and the average daytime maximum temperature will be around 16°C, with a high for the week of 19°C expected on the afternoon of Monday 22nd. The mean minimum temperature will be 8°C, dipping to its lowest on the morning of Saturday 27th at 5°C. Expect the week ahead to have some days seeing a little precipitation and some days with rain. The indicators are that Thursday 27th will have the most precipitation with an accumulation of around 10.0mm. On the whole winds are likely to be moderate.


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\section{Introduction}
\label{sec:intro}
In engineering and science, improving the accuracy and precision of measurement tools, such as microscopy, X-ray crystallography and MRI, is of great interest. However, due to the physical limitations in measurement tools, sometimes we can only indirectly or partially observe a signal of interest, e.g., obtaining only low-frequency information, only low-resolution image, or only the magnitude of a signal. The microscope is a good example of a measurement tool having such physical limitations ranging from low-frequency measurements to phaseless measurements \cite{brenner1959negative,zhang2006functional,benveniste2002mr,stolyarova2007high,huang2009super}.

To overcome the limitation of low-frequency measurements, researchers have investigated recovering a signal from only its low-frequency Fourier measurements, and referred to it as \textit{super-resolution}. The authors in \cite{candes2014towards} and \cite{tang2012csotg} proposed SDP based methods for the recovery of signals in the continuous domain under certain separation conditions, by employing \textit{Total Variation Norm Minimization (TVNM)} and \textit{Atomic Norm Minimization (ANM)} respectively.
Besides, to address the issue of phaseless measurements, people have studied the \textit{phase retrieval} to obtain phase information from the magnitude measurements of a signal \cite{fienup1982phase,gerchberg1972practical}. Recently, in \cite{candes2015phase}, the authors proposed a trace-norm minimization to solve the phase retrieval problem with the use of masks.

Super-resolving a signal from only magnitudes of low-frequency Fourier measurements is often ill-posed due to lack of both phase information and high-frequency information; and hence it is a challenging problem. The authors in \cite{chen2014algorithm,jaganathan2016phaseless} considered the \textit{phaseless super-resolution} aiming at recovering signals with only low-frequency magnitude measurements. In the noiseless setting, the authors in \cite{chen2014algorithm} proposed a combinatorial algorithm for signal recovery using only low-frequency Fourier magnitude measurements, but this algorithm requires additional distinguishing conditions on the signal impulses. In the noisy setting, this combinatorial algorithm suffers from error propagation. Instead of assuming the distinguishing conditions on signals, the authors in \cite{jaganathan2016phaseless} used masks to obtain different types of magnitude measurements. The authors provably showed that under appropriate choice of masks, an SDP formulation can be used to recover time-domain impulse signals on the discretized grid.

In this paper, we consider super-resolving time-domain impulse signals located off the grid from only low-frequency Fourier magnitude measurements. To tackle the continuous parameter domain, we propose a novel SDP formulation, employing ANM to recover signals from Fourier magnitude measurements. For example, our approach applies to the magnitude measurements used in \cite{candes2015phase,jaganathan2016phaseless}. In numerical experiments, we show the successful recovery of signals in the continuous domain from only low-frequency magnitude measurements. Furthermore, we compare our algorithm to a simple combining algorithm performing phase retrieval followed by ANM. Our method shows better recovery performance than the simple combining algorithm. In the future work, we will consider the noisy magnitude measurement case.

\textbf{Notations:} In this paper, we denote the set of complex numbers as $\mathbb{C}$. We reserve calligraphic uppercase letters for index sets, e.g., $\mathcal{N}$. We use $|\mathcal{N}|$ as the cardinality of the index set $\mathcal{N}$. We use the superscripts $*$, $T$, and $H$ to denote conjugate, transpose, and conjugate transpose respectively. We reserve $i$ for the imaginary number, i.e., $i^2 = -1$. We denote a time-domain signal as a lowercase letter, and its frequency-domain signal as its uppercase letter. To denote a ground true signal, we use the superscript $o$, e.g., $x^{o}$. For the index of a vector and a matrix, we start with the index $0$; hence, we denote the first element of the vector $X$ as $X_0$, and the top-left element of a matrix $Q$ as $Q_{0,0}$.

\section{Problem Formulation and Background}
Let $x^{o}(t)$ be a sum of Dirac functions expressed as follows:
\par\noindent
\small
\begin{align}
\label{eq:sigmodelstd}
    x^{o}(t) = \sum_{j=1}^{k} c^{o}_j \delta(t-t^{o}_j),
\end{align}
\normalsize
where $\delta(t)$ is the Dirac delta function, $c^{o}_j \neq 0 \in \mathbb{C}$, and $t^{o}_j \in [0,1)$. Its Fourier transform is given by:
\par\noindent
\small
\begin{align}
\label{eq:FourierSignalModel}
    X^{o}_f = \sum_{j=1}^{k} c^{o}_j e^{-i 2 \pi f t^{o}_j} = \sum_{j=1}^{k} |c^{o}_j| a(t^{o}_j,\phi^{o}_j)_f,\; f\in\mathcal{N},
\end{align}
\normalsize
where $f \in \mathcal{N} = \{0,1,...,n-1\}$, $a(t^{o}_j,\phi^{o}_j) \in \mathbb{C}^{|\mathcal{N}|}$ is an \textit{atom} vector, with the $f$-th element given by $a(t^{o}_j,\phi^{o}_j)_f = e^{-i (2\pi f t^{o}_j - \phi^{o}_j)}$. Simply, $X^{o} = V^{o}c^{o}$, where $X^{o} \in \mathbb{C}^{n}$, $V^{o} = [a(t^{o}_1,0),...,a(t^{o}_k,0)]$, and $c^{o} = [|c^{o}_1|e^{i\phi^{o}_1}, ..., |c^{o}_k|e^{i\phi^{o}_k}]^T$. We also define the minimum separation of $x^{o}(t)$, denoted by $\bigtriangleup_t$, as the closest distance between any two different time value $t^{o}_j$'s in cyclic manner \cite{candes2014towards,tang2012csotg}, i.e.,
\par\noindent
\small
\begin{align}
    \Delta_t = \underset{t^{o}_i,t^{o}_j \in [0,1),\;i\neq j}{\min}\;\;|t^{o}_i - t^{o}_j|.
\end{align}
\normalsize

The goal here is to find $x^{o}(t)$ from the low-frequency Fourier magnitude measurements. We state the phaseless super-resolution problem with masks as follows:
\par\noindent
\small
\begin{align}
\label{prob:orig}
    & \text{Find}\;\; x(t)  \nonumber  \\
    & \text{subject to} \;\; Z[r,l]

     = \bigg| \int_{0}^{1} D_r(t) \sum_{j=1}^{k} c^{o}_j \delta(t-t^{o}_j) e^{-i 2\pi l t} dt \bigg|,\\
    & \quad\quad\quad \text{for}\; {-R \leq r \leq R}\;\text{and}\;l \in \mathcal{N}, \nonumber
\end{align}
\normalsize
where $Z[r,l]$ is the $l$-th frequency magnitude obtained by using the $r$-th mask function $D_r(t)$. Depending on the mask function $D_r(t)$, one can have different types of magnitude information. For example, if we choose $1 + e^{-i 2\pi t}$ for $D_r(t)$, we have $|X^o_l + X^o_{l+1}|$, $l \in \mathcal{N}$.

In \cite{jaganathan2016phaseless}, Jaganathan \textit{et al.} consider the case when the signal $x^{o}(t)$ is located on the grid, i.e., $t^{o}_j \in \{0,1,2,....n-1\}$. By $n$-point DFT, (\ref{prob:orig}) is equivalent to
\par\noindent
\small
\begin{align}
\label{prob:PLS_discrete}
    & \text{Find}\;\; x  \nonumber  \\
    & \text{subject to} \;\; Z[r,l]
     = | \langle f_l, D_r x \rangle |,\\
    & \quad\quad\quad \text{for}\; {-R \leq r \leq R}\;\text{and}\;l \in \mathcal{N}, \nonumber
\end{align}
\normalsize
where $x \in \mathbb{C}^{n}$ is a complex valued $k$-sparse vector, $D_r \in \mathbb{C}^{n \times n}$ is a diagonal matrix, and $f_l$ is the conjugate of the $l$-th column of the $n$ point DFT matrix. The authors in \cite{jaganathan2016phaseless} proposed the following semidefinite relaxation-based program for the phaseless super-resolution in the discrete domain by denoting $Y=xx^H$ and relaxing the rank-1 constraint on $Y$:
\par\noindent
\small
\begin{align}
\label{prob:PLS_discrete2}
    & \underset{Y}{\text{minimize}}\;\; ||Y||_1 + \lambda \text{Tr}(Y)  \nonumber  \\
    & \text{subject to} \;\; Z[r,l]^2
     = \text{Tr}( D_r^H f_l f_l^H D_r Y ),\\
    & \quad\quad\quad \text{for}\; {-R \leq r \leq R},\;l \in \mathcal{N},\;\text{and}\;Y \succeq 0,\nonumber
\end{align}
\normalsize
for some $\lambda > 0$.

This paper makes no assumption of $t^{o}_j$ being on the grid. In the next section, we propose an ANM based semidefinite relaxation of (\ref{prob:orig}) to deal with impulse functions off the grid.

\section{Phaseless Super-Resolution in the Continuous Domain}
We define the atomic norm of a vector $X \in \mathbb{C}^{|\mathcal{N}|}$ as follows:
\par\noindent
\small
\begin{align}
\label{eq:atomicNorm}
    & ||X||_{\mathcal{A}} = \inf \{ \sum_j |c_j|: X_l = \sum_j |c_j| a(t_j, \phi_j)_l,\; \substack{{t_j \in [0,1),}\\{ \phi_j \in [0,2\pi)}}\}.
\end{align}
\normalsize
We have the following new proposition for the atomic norm:
\begin{proposition}
\label{pro:sqr_atomic}
For any $X \in \mathbb{C}^{|\mathcal{N}|}$, $\mathcal{N} =\{0,1,...,n-1\}$,
\par\noindent
\small
\begin{align}
\label{eq:sqr_atomic_norm}
    ||X||_{\mathcal{A}}^2 = \inf_{u,s} \bigg\{\frac{1}{|\mathcal{N}|}\text{$s$Tr(Toep($u$))}\;:\;\begin{bmatrix} \text{Toep($u$)} & X \\ X^H & s\end{bmatrix} \succeq 0 \bigg\},
\end{align}
\normalsize
where $\text{Tr}(\cdot)$ is the trace operator, and $\text{Toep($u$)}$ is the Toeplitz matrix whose first column is $u=[u_0, u_1,...,u_{n-1}]^T$. Moreover, suppose after the Vandermonde decomposition \cite{cara1911uber,cara1911uber2,toeplitz1911zur}, $\text{Toep($u$)} = VDV^H$, where $ V = [a(t_1,0),...,a(t_r,0)]$ and $D$ is a positive diagonal matrix. Then, there exists a vector $c$ such that $X=Vc$ and $\sum_j|c_j| = ||X||_{\mathcal{A}}$.
\normalsize
\end{proposition}
Proposition \ref{pro:sqr_atomic} is similar to Proposition II.1 in \cite{tang2012csotg}; however, Proposition \ref{pro:sqr_atomic} considers the trace of $s\text{Toep($u$)}$ instead of the sum of trace of $\text{Toep($u$)}$ and $s$. Proposition \ref{pro:sqr_atomic} is essential to derive our new SDP formulation handling phaseless measurements. For readability, we place the proof of Proposition \ref{pro:sqr_atomic} in Appendix.

Motivated by Proposition \ref{pro:sqr_atomic}, we propose the following squared atomic norm minimization for the phaseless super-resolution in the continuous domain, simply \textit{phaseless ANM}:
\par\noindent
\small
\begin{align}
\label{eq:ANMPhaseless}
	& \underset{X}{\text{minimize}}\;\;  ||X||_{\mathcal{A}}^2 \nonumber\\
	& \text{subject to}\;\; a_r(X)=b_r, \; r=1,2,...,q,
\end{align}
\normalsize
where $q$ is the total number of magnitude measurements, $a_r(X)$ is the magnitude mapping function, $|\langle a_r, X \rangle |$, $a_r \in \mathbb{C}^{|\mathcal{N}|}$, and $b_r$'s are magnitude measurement results.

From Proposition \ref{pro:sqr_atomic}, we can change (\ref{eq:ANMPhaseless}) to
\par\noindent
\small
\begin{align}
\label{eq:ANM_SDP_PL1}
	& \underset{u,X,s}{\text{minimize}}\;\;  \frac{1}{|\mathcal{N}|} \text{$s$Tr(Toep($u$))} \nonumber\\
	& \text{subject to}\;\; U \triangleq \begin{bmatrix} \text{Toep($u$)} & X \\ X^{H} & $s$ \end{bmatrix} \succeq 0, \nonumber \\
    & \quad\quad\quad\quad \;\;a_r(X)=b_r, \; r=1,2,...,q,
\end{align}
\normalsize
where $u,X \in \mathbb{C}^{|\mathcal{N}|}$ and $s \in \mathbb{C}$. From the positive semidefiniteness of $U$, $s \geq 0$, and $\text{Toep($u$)} \succeq 0$. Besides, if $X_j \neq 0$, $j \in \mathcal{N}$, then $s \neq 0$ from the non-negativeness of all principal minors of $U$ \cite{meyer2000matrix}. However, because of the magnitude constraints, (\ref{eq:ANM_SDP_PL1}) is a non-convex program.

By the Schur complement lemma \cite{boyd2004convex}, $U \succeq 0$ implies $\text{$s$Toep($u$)} - XX^{H} \succeq 0$. Since $s\text{Toep($u$)} = \text{Toep($su$)}$, by defining $Q=XX^{H}$ and $u'=su$, and getting rid of the rank constraint on $Q$, we have the following SDP relaxation for the phaseless ANM:
\par\noindent
\small
\begin{align}
\label{eq:ANM_SDP_sim}
	& \underset{Q\succeq 0, u'}{\text{minimize}}\;\; \frac{1}{|\mathcal{N}|}\text{Tr(Toep($u'$))}\nonumber\\
	& \text{subject to}\;\; \text{Toep($u'$)} - Q \succeq 0, \nonumber\\
    & \quad\quad\quad\quad\;\; A_r(Q) = b^2_r, \; r =1,2,...,q,
\end{align}
\normalsize
where $A_r(Q)$ is a mapping function, $\text{Tr}(A_r Q)$. Here, $A_r = a_r a_r^H$.

After solving (\ref{eq:ANM_SDP_sim}), we can find the optimal $\hat{Q}$ and optimal $\text{Toep($\hat{u}$)}$. Our analysis of (\ref{eq:ANM_SDP_sim}) in the following section shows that under certain conditions, $\hat{Q} = X^{o}{X^{o}}^H$. We can recover $X^{o}$ up to global phase by the eigenvalue decomposition of $\hat{Q}$. More importantly, because of the structure of $\text{Toep($\hat{u}$)} = V^{o} D {V^{o}}^H$ for some diagonal matrix $D$, we can apply any parameter estimation method such as Prony's method \cite{kay1988spectral,stoica2005spectral,blu2008sparse} or a matrix pencil method \cite{hua1990matrix,sarkar1995using} to find the time location $t^{o}_j$'s.

\section{Performance Analysis}
\label{sec:performance}
We first consider the analysis of (\ref{eq:ANM_SDP_sim}) given a rank-1 matrix $Q$. And then, we provide the analysis of (\ref{eq:ANM_SDP_sim}). Finally, we look at one scenario having magnitude measurements from a set of masks, in which (\ref{eq:ANM_SDP_sim}) provides the desired signal recovery.
\begin{theorem}
\label{thm:atomic_norm}
For a given rank-1 positive semidefinite matrix $Q=XX^H$, $X \in \mathbb{C}^{|\mathcal{N}|}$, the following optimization problem provides the squared atomic norm of $X$, i.e., $||X||_{\mathcal{A}}^2$:
\par\noindent
\small
\begin{align}
\label{eq:main}
	& \underset{u}{\text{minimize}}\;\; \frac{1}{|\mathcal{N}|}\text{Tr(Toep($u$))}\nonumber\\
	& \text{subject to}\;\; \text{Toep($u$)} - Q \succeq 0.
\end{align}
\normalsize
\end{theorem}
\begin{proof}
We can prove it by using Proposition \ref{pro:sqr_atomic}. Defining $u=u's$, where $s>0$ is a scalar. Then we can re-state the constraint as $Toep(u') - \frac{1}{s}XX^H \succeq 0$. By the Schur complement lemma, we have the optimization problem in Proposition \ref{pro:sqr_atomic}. Therefore, from Proposition \ref{pro:sqr_atomic}, the optimal value of (\ref{eq:main}) is the same as $||X||^2_{\mathcal{A}}$.
\end{proof}

\begin{corollary}
\label{thm:atomic_norm_subset}
If (\ref{eq:ANM_SDP_sim}) gives a rank-1 solution to $Q$, then (\ref{eq:ANM_SDP_sim}) minimizes the squared atomic norm of $X$ among all vectors $X$ satisfying the given constraints $a_r(X)=b_r$, $r=1,2,...,q$.
\end{corollary}
\begin{proof}
From Theorem \ref{thm:atomic_norm}, (\ref{eq:ANM_SDP_sim}) provides the minimum squared atomic norm of $X$ among all vectors $X$ satisfying the constraints $a_r(X)=b_r$, $r=1,2,...,q$.
\end{proof}

Let us consider the case when we have low-frequency Fourier magnitude measurements from a set of masks. The main difference between \cite{candes2014towards,tang2012csotg} and our setting is that we have only magnitude measurements, instead of measurements offering both phases and magnitudes.
\begin{theorem}
\label{thm:nonconvx}
Given the magnitude measurements $|X^{o}_j|$, $|X^{o}_j + X^{o}_{j+1}|$, and $|X^{o}_j - iX^{o}_{j+1}|$, $j \in \mathcal{N}=\{0,1,...,n-1\}$, (\ref{eq:ANM_SDP_sim}) provides the unique solution $Q=X^{o}{X^{o}}^H$, and $x^{o}(t)$ is uniquely obtained up to global phase if the following conditions hold: $X_j \neq 0$, $\forall j \in \mathcal{N}$, and $\Delta_t \geq 4/|\mathcal{N}|$.
\end{theorem}
\begin{proof}
Given magnitude data, $|X^{o}_j|^2$, $|X^{o}_{j+1}|^2$, $|X^{o}_j + X^{o}_{j+1}|^2$, and $|X^{o}_j - iX^{o}_{j+1}|^2$, we can find $Q_{j,j}$, $Q_{j+1,j+1}$, $Q_{j,j+1}$ and $Q_{j+1,j}$, which are the elements of the diagonal, sub-diagonal, and super-diagonal of the matrix $Q$, by simply solving linear equations on $Q$ together. From Lemma \ref{lemma:Q} in Appendix \ref{app:lemma:Q}, we can uniquely recover $Q=X^{o}{X^{o}}^H$ and $X^{o}$ up to global phase. According to Proposition \ref{pro:sqr_atomic} and Theorem \ref{thm:atomic_norm}, (\ref{eq:ANM_SDP_sim}) with $X^{o}$ is essentially the same as the optimization problem dealing with the standard ANM \cite{tang2012csotg} or TVNM \cite{candes2014towards}. Therefore,(\ref{eq:ANM_SDP_sim}) provides unique $x^{o}(t)$ up to global phase if the separation condition holds, i.e., $\Delta_t \geq 4/|\mathcal{N}|$.
\end{proof}

\section{Numerical Experiments}
\label{sec:experiment}
We compare our phaseless ANM against the standard ANM \cite{tang2012csotg} using measurements offering both phases and magnitudes, as well as against a simple algorithm which first performs the phase retrieval \cite{candes2015phase} and then applies the standard ANM \cite{tang2012csotg} to recover the impulse functions from the recovered signal using the phase retrieval. We use CVX \cite{cvx} to solve (\ref{eq:ANM_SDP_sim}).

Fig. \ref{fig:anm_sep_combine} (a) and (b) show the probability of successful recovery from the standard ANM and the phaseless ANM respectively. We conducted 50 trials for each parameter setting and measured the success rate. At each trial, we chose one time impulse $t^{o}_1$ uniformly at random in $[0,1)$, and another time impulse $t^{o}_2$ by adding the separation $\Delta_t$ to $t^{o}_1$ in the cyclic manner. We sampled the real part and imaginary part of time coefficients $c^{o}_j$'s uniformly at random in (0,1). We consider low frequencies, i.e., $\mathcal{M}=\{0,1,...,m-1\}$, where $m<n$, $\mathcal{M} \subseteq \mathcal{N}$. For a set of masks in the phaseless ANM, we use the same masks as those of Theorem \ref{thm:nonconvx} over the index set $\mathcal{M}$. The x-axis represents the separation condition $\Delta_t$ varied from $1/n$ to $11/n$, and y-axis is the number of low-frequency Fourier measurements $m$, varied from $2$ to $30$. In fact, for the phaseless ANM, the number of magnitude measurements is $3m-2$. We evaluated the recovery performance for the signal dimension $n = 32$.  We calculated the Euclidean distance between the estimated and true time locations. If the distance is less than $10^{-3}$, then we consider the estimation successful. Numerical experiments show that our phaseless ANM can find the exact time locations in the continuous domain with the same performance as the standard ANM. For large $k$, e.g., $k=10$, our method also provides the same performance as the standard ANM. We omit the simulation results in this paper due to the space limitation.
\begin{figure}[!t]
\centering
  \includegraphics[width=3.3in]{fig_anm_sep_combine.eps}
  \caption{\small The probability $P$ of successful recovery by varying the separation condition $\Delta_t$ and the number of measurements $m$ when $n = 32$. (a) Standard ANM. (b) Phaseless ANM}
\label{fig:anm_sep_combine}
\end{figure}
\begin{figure}[!t]
\centering
  \includegraphics[width=3.3in]{fig_PLANM_PRANM.eps}
  \caption{\small The probability $P$ of successful recovery by varying the number of magnitude measurements $q$ and sparsity $k$ when $n = 32$. (a) Phase retrieval and then standard ANM (b) Phaseless ANM}
\label{fig:PLANM_PRANM}
\end{figure}

One can think of a simple method conducting the phase retrieval first, and then doing the standard ANM. To compare our algorithm with this simple method, we further carried out numerical experiments by varying the number of magnitude measurements $q$ and the number of sparsity $k$ in (\ref{eq:sigmodelstd}). In this simulation, instead of using a set of masks used in Theorem \ref{thm:nonconvx}, we randomly chose a vector $a_r$ for each magnitude measurement in (\ref{eq:ANMPhaseless}). Fig. \ref{fig:PLANM_PRANM} (a) and (b) show the probability of successful recovery from the simple combining algorithm and the phaseless ANM respectively. The x-axis is the number of magnitude measurements $q$, and y-axis is the number of sparsity $k$. With randomly chosen magnitude measurements, our method outperforms the simple combining algorithm.

\section{Appendix}
\subsection{Proof of Proposition \ref{pro:sqr_atomic}}
We follow the proof of \cite[Proposition II.1]{tang2012csotg}.
\begin{proof}
\label{prf:sqr_atomic}
Let us denote the optimal value of the right hand side of (\ref{eq:sqr_atomic_norm}) by SDP($X$). In order to show $||X||_{\mathcal{A}}^2 = \text{SDP($X$)}$, we will show that (1) $||X||_{\mathcal{A}}^2 \geq \text{SDP($X$)}$ and (2) $||X||_{\mathcal{A}}^2 \leq \text{SDP($X$)}$.

The proof of (1) is easily shown by considering a feasible solution of SDP($X$). For $X=\sum_j |c_j| a(t_j, \phi_j)$, by choosing a feasible solution, $\text{Toep($u$)} = \sum_j |c_j| a(t_j,\phi_j) a(t_j,\phi_j)^H$, and $s=\sum_j |c_j|$, we have
\par\noindent
\small
\begin{align*}
    \begin{bmatrix} \text{Toep($u$)} & X \\ X^H & s \end{bmatrix} = \sum_j |c_j| \begin{bmatrix} a(t_j,\phi_j) \\ 1 \end{bmatrix}\begin{bmatrix} a(t_j,\phi_j) \\ 1 \end{bmatrix}^H  \succeq 0.
\end{align*}
\normalsize
For this feasible solution, $\frac{1}{|\mathcal{N}|} s \text{Tr(Toep($u$))} = (\sum_j |c_j|)^2$, which is $||X||_{\mathcal{A}}^2$. Thus, $\text{SDP($X$)} \leq ||X||_{\mathcal{A}}^2$.

For the proof of (2), we will show that for any $u$, $s$, and $X$, $\frac{1}{|\mathcal{N}|}s \text{Tr(Toep($u$))} \geq ||X||_{\mathcal{A}}^2$. Suppose for some $u$, $s \neq 0$, and $X$, the matrix $U$ in (\ref{eq:ANM_SDP_PL1}) is positive semidefinite. From the positive semidefinite condition, we have $\text{Toep($u$)} \succeq 0$ and $s > 0$. From the Vandermonde decomposition \cite{cara1911uber,cara1911uber2,toeplitz1911zur}, for any positive semidefinite $\text{Toep($u$)}$, we have $\text{Toep($u$)} = VDV^H$, where $V = [a(t_1,0)\;a(t_2,0),...a(t_r,0)]$, and $D$ is a diagonal matrix having $d_j$ as its $j$-th diagonal element. Since $VDV^H = \sum_{j=1}^r d_j a(t_j,0) a(t_j,0)^H$ and $||a(t_j,0)||^2_2=|\mathcal{N}|$, we have $\frac{1}{|\mathcal{N}|} \text{Tr(Toep($u$))} = \text{Tr($D$)}$. Also, from the Vandermonde decomposition and $U \succeq 0$, $X$ is in the range space of $V$; in fact, if $X$ is not in the range of $V$, we can always find a vector $z$ such that $z^H Uz < 0$. Therefore, $X=Vw =\sum_{j=1}^{r} w_j a(t_j,0) $, where $w \in \mathbb{C}^r$. By the Schur complement lemma \cite{boyd2004convex},  $U$ in (\ref{eq:ANM_SDP_PL1}) is expressed as follows:
\par\noindent
\small
\begin{align}
\label{eq:schur}
    VDV^H -\frac{1}{s}Vww^HV^H \succeq 0.
\end{align}
\normalsize
It is noteworthy that we can always find a vector $q$ such that $V^Hq = sign(w)$, where $sign(w)^H w = \sum_{j=1}^{r} |w_j|$, by choosing $q = V(V^H V)^{-1} sign(w)$. This is because $V^H$ has full row rank.
By choosing $q$ such that $V^H q = sign(w)$,  we have
\par\noindent
\small
\begin{align*}
    \text{Tr($D$)} = q^H VDV^H q \geq \frac{1}{s} q^H Vww^H V^H q = \frac{1}{s} (\sum_j |w_j|)^2,
\end{align*}
\normalsize
where the inequality is from (\ref{eq:schur}). Therefore, we have
\par\noindent
\small
\begin{align*}
    \frac{1}{|\mathcal{N}|}s\text{Tr(Toep($u$))} = s\text{Tr($D$)} \geq (\sum_j |w_j|)^2 = ||X||_{\mathcal{A}}^2.
\end{align*}
\normalsize

If $s=0$, from the sufficient and necessary condition for the positive semidefiniteness of a Hermitian matrix, all of $U$'s principal minors need to be non-negative \cite{meyer2000matrix}. Thus, $X_j = 0$, $\forall j \in \mathcal{N}$. In this case, Proposition \ref{pro:sqr_atomic} still holds.
\end{proof}

\subsection{Lemma for the positive semidefinite matrix $Q$}
\label{app:lemma:Q}
\begin{lemma}
\label{lemma:Q}
Let $Q \in \mathbb{C}^{|\mathcal{N}| \times |\mathcal{N}|}$, and $X^{o} \in \mathbb{C}^{|\mathcal{N}|}$. Suppose (1) $Q \succeq 0$, (2) $Q_{j,j} = Q^{o}_{j,j}$, $Q_{j,j+1} = Q^{o}_{j,j+1}$, and $Q_{j+1,j}=Q^{o}_{j+1,j}$, $j \in \mathcal{N}=\{0,1,...,n-1\}$, where $Q^{o}=X^{o}{X^{o}}^H$, (3) $X^{o}_j \neq 0$, $\forall j \in \mathcal{N}$. Then, $Q$ is uniquely determined as $Q=X^{o}{X^{o}}^H$.
\end{lemma}
\begin{proof}
From the fact that a Hermitian matrix is positive semidefinite if and only if all of its principal minors are non-negative \cite{meyer2000matrix}, all of $Q$'s principal minors are required to be non-negative. Let us prove our lemma by induction. When $|\mathcal{N}|=3$, the determinant of $Q$ is $-|X^{o}_1 Q_{0,2} - X^{o}_0 X^{o}_1 {X_2^{o}}^{*}|^2$, where $Q_{0,2}$ is unknown. To be $-|X^{o}_1 Q_{0,2} - X^{o}_0 X^{o}_1 {X_2^{o}}^{*}|^2 \geq 0$, $X^{o}_1 Q_{0,2} - X^{o}_0 X^{o}_1 {X_2^{o}}^{*} = 0$. Since $X^{o}_1 \neq 0$, $Q_{0,2}$ is determined uniquely as $X^{o}_0 {X_2^{o}}^{*}$. When $|\mathcal{N}|=4$, we can consider the top-left $3 \times 3$ submatrix of $Q$ and the bottom-right $3 \times 3$ submatrix of $Q$ to determine $Q_{0,2}$ and $Q_{1,3}$ respectively. And then, we can deal with $3 \times 3$ principal submatrix of $Q$ having $Q_{0,4}$ to determine $Q_{0,4}$. In the similar way, when $|\mathcal{N}|=n$, we can uniquely determine every unknown variables in $Q$. We omit the detailed explanation due to the space limitation.
\end{proof}

\small
\bibliographystyle{IEEEbib}

redpajama_set_name: RedPajamaArXiv


Welcome to Bhubaneswar property portal, where you can find Property listings for your dream homes, flats, lands and other properties for sale in Bhubaneswar. You can also search real estate agents and developers in our agent and builder directory section and read the latest news and activities of the Bhubaneswar real estate market.
Now a days the city is chosen as the best popular location for both to stay and investment for its pollution free pre planned city and also it is one of the best tourist location of eastern India. The state Odisha attracts the investors in search of all types of attraction, starting from beautiful temples to beaches.
Investors walking up to Bhubaneswar Real Estate for its huge bargains which can be found. Prices of flats and houses are down 2 times what few years before. But rent increasing high in many localities like Patia and Khandagiri due to rapid development of engineering colleges and multinational companies. BhubaneswarProperty.in always tries to provide the bargain price for customer. All the properties in our website carefully chosen to offer the maximum benefit for our visitors.


Twitter is a great marketing tool.
The global popularity of Twitter seems puzzling, what with its 140-characters limit. The micro-blogging site has more than 24 million unique visitors per month, with 500 million messages—called "tweets"—per day. As some people have discovered, while many media outlets choose to bombard users with extensive amounts of data and text, Twitter whittles it down to a few succinct sentences and gives users only what need.
[related|post]Twitter is the perfect tool for your business to quickly share information and build ties with customers. Its appeal and relevance to branding lies in its convenience: it requires very little effort to announce a sale or launch, to anticipate the release of new products, or to answer customer concerns.
• Get named and create a following. When creating a Twitter account, make sure that your username is short, easy to remember (and type), and captures the essence of your company or brand well. For many businesses, the company name is enough. After creating an account, you need to be "followed" by as many people as possible, meaning you have to get people to subscribe to your tweets. If you have friends who are already on Twitter, ask them to follow you. It may be a small number, but it's a start. Spread the word about your Twitter account through your other accounts with social media.
• Become a follower as well. Make an effort to subscribe to the Twitter pages of other companies, brands, or personalities that are linked or relevant to your brand because they're a good source of information for you and your followers. By doing this, you also expand your reach, opening you up to more potential followers.


The laundry is just an extra room, but its design is as important as the others.
Doing your laundry takes an average of 3 hours and 23 minutes a week, according to the Australian Bureau of Statistics. It's a short span of time, but a lot of people still hesitate to wash their own dirty clothes. They have lots of reasons, but they don't want to do it mainly because it's a painful chore.
Working in a laundry room with poor ergonomics strains your back and hip muscles. It gives you a terrible back pain sometimes.
We are taller on average today so the (old) standard 900mm bench height might not work for all of us. Consider raising your bench height up to 920mm to improve the ergonomics. The additional height will enhance people's posture and will help avoid back pain. Designers can extend the bench height up to 1000mm., in case they work for a homeowner who is long-legged and taller than average.
You don't need to crouch uncomfortably when loading and unloading the washers if you elevate it. From the floor level, raise the front loaders until the drum is 750-1000mm high. Consider integrating the appliances into a joinery for a more seamless look. If you're buying a new front loader, choose a model with drums positioned higher in the machine's body.
Having more than one laundry hamper will help you separate the white, bright, and dark pieces of clothing before your wash day. Provide two to three baskets and label each one. This system orders everyone in your family to sort their dirty clothes themselves, so you don't have to. It will save you time and you will avoid backaches due to bending too much.
We like this laundry hamper that's integrated into the cabinetry. The doors tilt out together with the removable bags. Other designs feature pull-outs with removable plastic baskets.
It's easier to open and access the contents of drawers compared to cupboards. Imagine yourself looking for the detergent and fabric conditioner in your laundry room. With more cabinets, you are forced to rummage uncomfortably up to the back of the cupboard. Drawers, on the other hand, will show you all of its contents in just one pull, therefore you can quickly find the items you need. Consider having drawers with various depths as it can help you organise your stuff.
If there's no room for custom drawers, consider shallow cabinets and open shelves for storage. Shallow cabinets are easier to access compared to the standard ones. Use pullable baskets and containers to organise the contents of the shelves.
Choose a large, deep, and wide sink for your laundry. You need something for heavy-duty chores such as hand washing your clothes, bathing your pets, washing your household cleaning tools, and rinsing your garden tools. Next, install a tap with flexible hose and nozzle. It will help you control and direct the water when you need it.
You need enough landing spaces near your workstations. Your front loaders need ample floor space for loading dirty clothes and bench space for folding the clean ones. The same law applies to the sink—you need space beside it to rest the newly hand washed clothes.
Include a door in your laundry which leads to your outdoor space so the sun and wind can dry your clothes for you.
Ironing boards that are integrated into the joinery are more stable compared to the thin and wobbly portable ones. It is effortless to store and offers a streamlined look when not in use.
It's a hassle to carry heavy laundry baskets from one floor to another. If your bedrooms, bathroom, and laundry aren't on the same floor, then a flimsy laundry chute is a convenient addition. Linen chutes are lightweight alternatives to heavy laundry baskets.
You can also connect multiple laundry shoots upstairs to the laundry downstairs through a system of tubes. This contraption will make your laundry a breeze. You save you a lot of time and effort as you don't have to crouch to get the dirty clothes and carry it all downstairs.
Make your laundry area more functional through adding dedicated storage for your household cleaning items. Have a good combination of deep and shallow drawers, tall and short cupboards, hooks for hanging, and parking space. Organise and store your mops, brooms, vacuum cleaners, gardening tools, and sporting gears here.
Find a Melbourne architect & designer for this project.


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Shetland Sheepdogs are known for their excitable, eager to please and hardworking personalities. Here's all you need to know about these fluffy friends!
The Shetland Sheepdog breed, also called the Sheltie, originated in the early 20th century when Collies and Shetland herding dogs were interbred. They used to be known for tending to the small sheep of the Shetland Islands, but today they are raised as farm dogs and family pets.
The Shetland Sheepdog breed is beloved for their affectionate, playful, and obedient disposition. These pups are the ultimate companion dogs—incredibly smart, loving, and eager to please. Shelties love intellectual stimulation and any activity that involves engaging with their humans!
Shetland Sheepdogs are some of the brightest dogs around. They can understand a new command with less than five repetitions and obey it the first time almost without fail!
Shelties are affectionate, loyal, and devoted to people they are familiar with. They thrive with guardians who give them lots of intellectual stimulation and exercise.
Shelties shed in clumps that can be pulled or brushed off of the main coat, rather than shedding individual hairs. This makes grooming much easier!
Shelties do best with a sensitive, attentive owner since they require lots of intellectual stimulation.
Some Shelties have a strong herding instinct and the entire breed likes to run in wide-open areas. Make sure they get lots of exercise and pay attention to how they're expending that energy. You don't want your Sheltie rescue trying to herd a bunch of kids on tricycles at the park!
Shetland Sheepdogs do well with children if they are around them from an early age; since this is a smaller breed, it's easy for a child to accidentally injure them, so be sure to supervise the socialization process between your Sheltie and your children.
If a Sheltie doesn't get an adequate amount of exercise and intellectual stimulation, they can resort to excessive barking and nervous behavior; however, this is easily remedied! Just an hour of exercise with your pup will set them back at ease.
Shetland Sheepdogs are lively, brilliant and playful dogs. Because of their popularity, Shelties are victims of overbreeding. Poor breeding practices like puppy mills can exacerbate health issues in the breed, such as hip dysplasia, eye problems, and hypothyroidism. Help Shelties avoid this sad fate. If you're looking for a smart, loving, and bubbly companion, consider opening your home to a Shetland Sheepdog rescue.


This summary discusses the various methods of name resolution used by Microsoft Windows clients. On a network using TCP/IP, it is necessary to resolve NetBIOS and host names to IP addresses in order to connect to network resources. Microsoft Windows clients will follow a certain sequence to resolve a name to an address and completing process when it successfully resolves that name to an IP address.
There are two main types of name resolution used in almost every situation: NetBIOS resolution and host name resolution. Clients (mainly legacy) connecting to resources on Microsoft servers, typically through Windows File Manager or Network Neighborhood, use NetBIOS name resolution. Host name resolution resolves the names of TCP/IP resources that do not connect through the NetBIOS interface. The are many examples of applications that rely on host name resolution such as web browsers, Ping, FTP, and Telnet.
Legacy clients (pre-Windows 2000) will use the NetBIOS name resolution process before attempting to use host name resolution processes. Windows 2000 and later clients will mainly perform host name resolution prior to attempting NetBIOS name resolution.
Verifies if it is the local machine by its registered name NetBIOS name.
When resolving NetBIOS and host name the client will skip the steps listed above for which it is not configured for. For example, if there is no DNS or WINS server configured in the TCP/IP settings, those steps will be skipped and the client will continue down the list of options.


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