Keyboard is primarily the enter gadget within the pc utilizing which you can sort the information that you just want to present as enter, play games and perform various different duties. A keyboard has several keys though it depends and differs from one to the opposite manufacturer. Now there can be two varieties of keyboard depending on the arrangement of keys, one is the traditional keyboard and the other is the ergonomic keyboard. There have been some casualties, too. With previous bit graphics, it was cheap and easy to tile a vast world map and allow you to run around on the map with a representative icon of your airship, chocobo, or character sprite. Now we misplaced the overhead world map and simply selected locations from a static map with a menu. However, on the bottom, we had a 3D character working in precise THREE-D, fully rendered environments. Digital camera angles had been pre-set and couldn't be modified, but changed in battle.
Decide up a laptop sleeve, laptop computer case or a laptop computer bookbag to carry around your pc. You can even consider making your personal out of a sleeping bag or pillow. Dropping your laptop a couple of occasions might lead to crashing it, and wiping out all of your necessary paperwork. Below I discuss six extra nice gaming monitor choices (Form of like bonus footage). Also, do not forget to vote to your favorite! This can be a primary USB OBD II cable for your laptop computer and is just like the one I have and use. It works wonders and I extremely advocate it! Daria, I might trust that last mechanic who you talked about. It's true that if the light would not come back on then you need to be tremendous. Yet, you'll be able to nonetheless expect rather a lot out of either one of these processors. They're implausible for gaming, will not bottleneck the plethora of excessive efficiency $200 to $300 graphics playing cards, and are fairly good at multitasking. Personally, I use them in builds from $500 to $800. Should you're a photo editor, I might prioritize getting an i5 over a graphics card; however; gamers, will want to do the opposite.
When you've got determined to tackle the duty of assembling a pc your self, it will be higher that we spend a while in taking some safety precautions. The safety precautions that I am going to record in this article are extraordinarily necessary and all of them have to be taken seriously. A typical suite contains software that cater for word processing, presentation, graphic design and spreadsheet: It may very well be less, more or completely different, however serving totally different needs. I did let them know that the Better Business Bureau will get a cellphone name Monday morning, My lawyer Monday morning, the state TV's buyer shopper line and that I will probably be writing a DELL Assessment online. They still say 2-3 weeks!!!!! If you'd like more choices and more builds, I extremely suggest you take a look at my construct a gaming LAPTOP sequence on YouTube. A few of it will get a bit outdated every now and then but I typically redo all of my builds every quarter.
Whether or not you're looking for a laptop , tablet or desktop computer , you may find a vary of computing technology to suit each budget. Add to your Apple assortment with an iPad or cutting-edge MacBook Not forgetting the equipment to match, you'll find keyboards , monitors, plus a wide range of USB networking devices. Ensure your own home or enterprise is fully geared up with routers, Wi-Fi, printers and extra. Hello, Virginia, a few of these subjects are just good common sense discussion, but some go very deep and past even human understanding. For example, we have now been discussing in a non secular research group whether souls will likely be keen to occupy cloned bodies. Highly unscientific, however it is perhaps a good paper for a theology class. Anyway, its a very interesting article and well presented. I might love to be a student at present and present papers on a few of these subjects.
CAD, simple! A easy $300 laptop just isn't enough for dealing with serious engineering works in software like AutoCAD and Catia. Software installation dimension is big, processor requirement is swift and the software literally eats the RAM. This can be a assortment of wires through which knowledge is transmitted from one a part of a computer to a different. Additionally it is a communication system that transfers information between parts inside a computer or between computers. To sum up, it's a must to resolve whether or not you like a barely greater power consumption and eighty Hz refresh charge, or normal graphics card operation and 78 Hz refresh rate. Since the difference is a mere 2 Hz, I extremely suggest the latter. I like to recommend upgrading the video card to a older pc if you're a gamer, and you utilize Windows XP, Vista. Only improve your video card if it uses a AGP or PCI-Express videocard slot as a result of PCI based mostly video cards are slow and not much faster then Onboard video playing cards. Plus, they can be costly considering how gradual they are.


The Gujarat government is now set to tap the traditional knowledge of tribal healers of this hill station to promote health tourism. In Saputara, known as the 'abode of serpents', in Gujarat's Dang district, the traditional healers are mainly from the Bhil tribe. They are known far and wide for their effective snake bite cures. The picturesque hill station is also home to a large array of medicinal herbs. "Around 520 out of the 1,100 medicinal herbs listed in the pharmacopia of AYUSH (the Indian government's traditional medicine department) are found in Saputara," according to S.K. Nanda, the district in charge officer who has spent around three decades in the area. Covered with a canopy of mist, the hill station, located in the Sahyadri hills, boasts of many essential herbs used in ayurveda concoctions. "Among the major listed herbs found here are amla, harde, bherda, satavari, aswagandha, sarpagandha and tubers and roots which are crucial for ailments of chronic nature," Nanda said. Many people from far off come to the traditional healers, known as Bhagats, for relief, said Nanda. "The knowledge of the Bhagats is being documented," Nanda said, adding that the state government was planning to tap the region's health tourism potential. He said the government is providing a platform to traditional healers by giving them the opportunity to examine patients at state-run centres and also prescribe herbal drugs. "They are also being encouraged to grow the rare medicinal herbs that are found deep in the jungle. All this would help in conservation and also popularising the traditional, cheap and easy drug delivery that has no repercussions on the health of patients," Nanda said. According to Nanda, in 1981 when he was the district development officer of Dang, he had started a pharmacy where the medicinal herbs found locally would be collected and processed into formulations. "It is a GMP (Good Manufacturing Practice)-compliant factory that is helping the local tribals earn money for collecting the herbs and also earn wages for employment in the factory. The Bhagats are also able to prescribe the herbal formulations for simple ailments," he added.


This picture is the public figure of Changan sword WeChat.
In the process, the General Secretary kept asking questions and interrupting, exchanged in-depth with private entrepreneurs on some issues, and asked relevant departments to seriously study and properly solve them.
Chang An Jun found that the 10 entrepreneurs interviewed by the General Secretary today are all pioneers of high-tech innovation.
Liu Jiren: Chairman of Neusoft group.
The first doctor of computer application specialty trained in China is also the founder of Dongsoft, the first software listed company in China, and one of the founders and witnesses of the development of Chinas computer industry.
Lu Weiding: Executive Director, President of Zhejiang Wanxiang Group, Vice President of Zhejiang Federation of Industry and Commerce, son of Lu Guanqiu, the first generation private entrepreneur in China.
Wanxiang Group, developed from a farm machinery repair plant, is now a multinational enterprise with automotive parts manufacturing and sales as its main business. It is engaged in the R&D and production of luxury new energy vehicles in Southern California, USA.
Wang Xiaolan: President of times group.
Time Company was founded in Beijing in 1984. It is the earliest modern high-tech manufacturing pioneer enterprise in Zhongguancun area and in Chinas electromechanical industry. It currently has 1 billion 500 million yuan assets.
Sun Piaoyang: Chairman of Jiangsu Heng Rui medicine Limited by Share Ltd and vice chairman of Jiangsu Provincial Federation of industry and commerce.
Over the past 20 years, Hengruis annual R&D expenditure accounts for about 10% of sales, which is unique among domestic enterprises.
Lu Yong: general manager of star arrow special glass Co., Ltd.
The anti-irradiation glass cover produced by Star and Arrow is applied to every satellite and spacecraft launched in recent years in China as the armor of solar panels.
Tang Xiaoou: founder and chairman of Shang Tang technology.
He led the Chinese AI team and was selected into the worlds top ten AI pioneer laboratories. The original face recognition algorithm has an accuracy rate of 98.52%, which surpasses the human eye recognition ability.
Liu Hanyuan: Chairman of Tongwei groups board of directors and chairman of Tongwei Limited by Share Ltd. He was named the most successful businessman under the age of 40 by Fortune magazine.
Tongwei Group, which mainly deals in fish feed and aquatic products, drives nearly one million domestic farm households, and directly or indirectly drives more than 30 million Chinese farmers to become rich and increase their income.
Tan Jianfeng: founder and chairman of public technology, a senior expert in the field of information security.
He presided over the planning and design of Public Internet Security Authentication Platform, which was applied to WiFi access authentication in 31 provinces of China Telecom, and was called gatekeeper in the information age.
Liu Yi: Chairman of Anhui eco green Limited by Share Ltd, vice chairman of Anhui Federation of industry and commerce.
Aikelan means love, sky can be more blue, specializing in the development and industrialization of aftertreatment products of engine exhaust, and the products sell well in Europe and America.
Geng Zhe: Shandong and Information Chairman and general manager, the company is the new three board listed company (871991).
The products with the same information cover three series of products: intelligent water meter, intelligent heat meter and intelligent gas meter. They provide solutions for energy measurement and control system based on the Internet of Things.
About the private economy, the general secretary has spoken 4 times in a month and has a clear attitude.
The first time: on September 27, the Liaoning survey stressed: We should unswervingly encourage, support and guide the development of non-public economy.
Second time: On October 20, in a letter to the privately-owned entrepreneurs who were commended in the Wancun campaign, he said: The historical contribution of the private economy is indelible, the status and role of the private economy is beyond doubt, and it is wrong to deny and weaken any opinions and practices of the private economy.
Fourth time: On November 1, the private enterprise Symposium said, Private enterprises and private entrepreneurs are our own people.
The general secretarys intensive speech shows that Chinas support for the private sector is consistent.
Who are the people who are talking with the general secretary today?
On the morning of November 1st, Xi Jinping hosted a symposium on private enterprises and interviewed 10 entrepreneurs. We have made speeches and put forward opinions and suggestions on supporting the development of private enterprises in the new situation.


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This largely qualitative and partly quantitative study is a critical analysis of the representation of San people of Namibia in selected stories published in The Namibian, New Era, Informanté, The Southern Times, The Villager, The Windhoek Observer and Namibian Sun newspapers over two years from January 2012 to December 2013. The aim was to evaluate the economic, socio-cultural and political situation of the San people as reported by the seven newspapers. A purposive sampling technique was used to select the stories. Content analysis and critical discourse analysis (CDA) were used to analyse data and point out binaries and dichotomies inherent in selected articles. The framing theory and CDA informed the study. Recurrent themes from the literature review were compared and contrasted with media stories. The study found that media coverage of San people and issues that affected them was grossly inadequate, and that San people were a peripheral ethnic group when it came to access to health, media rights, education, land, self-identity and dignity. It also found that the media glossed over San-related issues. It concludes that there was a disconnection between non-journalistic writers and the media on San issues. San people were heterogeneous yet the media treated them as homogenous. Balanced reportage of issues that affect San people was conspicuously absent. Much of the reportage was event-driven, lacking analysis and balance. The study recommends a revision of the intermediary role of the media, journalistic ethics and a shift from symptoms to causes and viable solutions. It further recommends new media discourses that shift from stereotyping to a discourse that restores and upholds the identity of San people.


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To end prison gangs, it's time to break up the largest prisons
By DAVID SKARBEK and COURTNEY MICHALUK
San Quentin State Prison, California's oldest correctional institution, sits on a 432-acre compound overlooking the beautiful San Francisco Bay. Inside, in grim juxtaposition to the prison's waterfront view, 701 men currently sit in the antiquated concrete cells that make up the largest death row in the Western hemisphere. In addition, the prison's four cellblocks also hold minimum-, medium- and maximum-security inmates, supervised by a prison staff of more than a thousand.
Yet the real power structure inside the prison has less to do with guards, or even the threat of death row, than with their fellow prisoners. From 2001 to 2012, 162 Californian prisoners were killed at the hands of other inmates, many in murders orchestrated by prison gangs.
Tales of gang-driven murder can be grisly. In 2009, Edward Schaefer was convicted of second-degree murder and vehicular manslaughter of a 9-year-old girl. Schaefer, who had a dozen prior convictions, hit the young girl with his motorcycle as she crossed the street. His blood alcohol level was more than twice the legal limit. Schaefer was sentenced to prison for 24 years to life, but his sentence was short-lived. After only 10 days at San Quentin, Schaefer walked into the prison yard and was stabbed seven times in the neck and chest by an affiliate of the Aryan Brotherhood prison gang. His attacker, fellow inmate Frank Souza, had manufactured a 7-inch "bone crusher" from a piece of a metal bunk bed. When authorities asked him why he killed Schaefer, Souza responded, "All I got to say, 9-year-old girl."
Prison life in America brings to mind a violent, chaotic and often cruel environment. But Schaefer's murder, like many inside prison walls, was not motivated by personal, racial or religious reasons. Instead, his murder represented a calculated move by the Aryan Brotherhood prison gang—in this case, a way for the gang to signal that there are acceptable crimes, and unacceptable ones, and they're willing to punish perpetrators of the latter.
Today, America is experiencing significant momentum on criminal-justice reform for the first time in a generation. Appalled by the statistics that show America imprisons far more of its population than any other developed nation, odd political couples like George Soros and Charles Koch are building coalitions to reform a prison system that is overused and disproportionately affects minorities. Senators Rand Paul (R-Ky.) and Cory Booker (D-N.J.) are both publicly on the offensive with criminal justice reform, and it makes sense. California, an early adopter of three-strikes laws, recently passed Proposition 47, which reduced the penalty for most nonviolent drug and minor theft offenses from felonies to misdemeanors.
But as these changes take place, our research suggests they'll be missing something important if they don't account for the important influence of prison gangs. The gangs that have formed in American prisons since the early 1960s have come to wield power far beyond prison walls, stretching into cities and influencing the lives of inmates long after they're released. The Mexican Mafia prison gang taxes the drug sales of Hispanic street gangs in Los Angeles. Its members order murders and regulate the drug trade from behind bars. People who join gangs in prison are also more likely to reoffend and to do so more quickly, thereby undermining rehabilitation efforts.
It's easy to imagine that prison gangs are a natural outgrowth of the harsh conditions of prison life—that as long as we have prisons, we'll have gangs. But economics suggests that it's more complicated than that. More specifically, there's good reason to believe that gangs aren't just a response to prisons, they're a response to the size of prisons. And unless we understand that, we won't be able to solve one of the most intractable problems in the criminal-justice system.
Before the 1960s, we know from ethnographic studies and interviews with prisoners, staff members and former offenders, inmates relied on a system of norms called the "convict code. " The code outlined the principles of good conduct, such as: Don't rat on another convict, don't break your word, and don't steal. Adherence to the code reduced the risk of victimization and serious injury at the hands of other inmates. This system dominated the prison scene until the mid-1950s, but it began to break down in the 1960s.
What changed? Prisons did. Partly as a result of a growing population, between 1945 and 1970, the prison population in California increased fourfold—from 6,600 inmates to roughly 25,000. Since the early 1990s, the California prison system has always held more than 100,000 inmates.
Prison growth also brought more young inmates and a more ethnically diverse population. Each of these shook up the consensus about what was acceptable behavior in prison. California's prisons went from housing a small, well-known group of inmates to housing a large, diverse population of strangers. The norms that facilitated a broader social cooperation behind bars were no longer effective. Violence went unchecked, and prison personnel were overwhelmed.
With prisons becoming less stable and more chaotic, a new type of order arose. In the 1960s and '70s, California saw the rise of some of the most notorious prison gangs, including the Aryan Brotherhood, Mexican Mafia and the Nuestra Familia.
It's easy to see these as chaotic, terrifying influences—and, in many ways, they are. But economics and social science suggest that they are also a very rational response to prison size. A recent book by one of us documents how changes in inmate demographics created a need for prison gangs. In small prisons, inmates' desire to be in good social standing constrains disruptive behavior. But as prison populations grew, it became too difficult to keep track of each man's reputation. If reputations are not well-known, then an inmate can violate the convict code without losing his standing in the community.
Gangs have such substantial control over inmates now that, at times, gang leaders and correctional officers actually work together to maintain order.
David Skarbek and Courtney Michaluk
Because of this failure of the code, gangs created a system of governance that provided protection to inmates and access to illicit goods. Gangs have such substantial control over inmates now that, at times, gang leaders and correctional officers actually work together to maintain order, even discussing which inmates should share a cell.
As these gangs formed, they established reputations for using violence. Their ability to do so gave them an advantage in the underground economy. Prisoners pay their debts to avoid suffering the consequences.
Prison gangs are racially segregated, but according to important research by sociologist Rebecca Trammell and others, they actually often have a broader concern for cooperation across color lines. Gang members are not allowed to initiate violence against a member of another gang simply for personal reasons. Gang leaders decide whether a conflict is worth pursuing, or if the matter can be settled without violence. A member's own gang might punish him for causing trouble with other gang members. "Shot callers" in gangs are those who know the prison yards well and can resolve disputes. If violence becomes too high-profile and attracts the attention of guards, it might interrupt business. Gangs claim territory and regulate their own members' behavior to bolster illicit markets. The larger the prison, the larger the profit opportunity for the gangs that control its commerce.
This is why prison gangs are so central to the culture and economy of large prison systems like California and Texas, but don't arise—or play little role—in smaller prison systems in places like Vermont or Wyoming. Likewise, when California had a small prison population, gangs did not exist. This also helps explain why gangs have relatively little influence in many English and European prisons, which hold far smaller populations than in California.
As incarceration has increased, the power and influence of prison gangs has grown with it. Today, states with big prison populations, like California and Texas, have the most prison gang activity. These two state prison systems alone hold about 70 percent of all prison gang members today. In 2006, an estimated 75 percent of all California inmates were gang members. There's no systematic data on gang membership across state prison systems. However, qualitative studies by Trammell and others document the fact that prison gangs have disproportionally large influence in the biggest prison systems.
Prison gangs serve an important economic function, but—to put it mildly—they are far from an ideal model of governance. Anything from name-calling to murder can serve as punishment for an inmate who violates gang rules. For inmates with low social standing like Schaefer, gangs murder on principle.
And the protection and access to drugs that one gains by joining a gang have a high cost to society outside the prison as well: Prisoners are members for life. Once released, members are required to take part in gang activity. This might include trafficking contraband to inmates or settling debts with those on the outside who owe the gang.
Understanding not only how, but also why, prison gangs arise is crucial for making smart criminal-justice policy decisions. Past attempts to stop prison gangs have failed in part because they simply don't address the larger structural reasons that explain why gangs exist. A typical approach is to lock up gang leaders in secure units to isolate them from other inmates, assuming perhaps that you can decapitate the crew by taking the leader away. Not in prison. A Department of Justice survey found that this strategy was ineffective because it simply created a void in which new leaders quickly emerge.
If we see prison gangs a different way—as an economically rational response to the conditions of large prisons—then the solutions start to look different. Throwing a gang member into secure housing simply means there is a job opening in the prison gang network. When gangs are understood to promote markets, instead of existing because of charismatic leaders or members' "violent nature," these findings are not surprising.
Correctional departments across the country have focused only on suppressing gang activity instead of addressing the underlying causes that give rise to them. If officials designed policies that recognized the governance role that gangs play, implemented these policies in a rigorous, scientific way subject to empirical assessment, and made key data available on gang membership and activities, then we'd know a lot more about how to limit the influence of prison gangs. What we do know from applying economics and solid social-science research to the prison population is that we have good reason to believe that the size of prisons is a crucial issue to confront. Officials can more easily monitor and control inmates in small prisons. If inmates are safe, they won't turn to gangs for safety. Small prisons also limit the size of the illicit marketplace, leaving fewer profits for gangs. When officials do not govern inmate life, inmates will turn to gangs for a brutal but effective source of order.
So, how can this be accomplished? One option would be to build more prisons that hold fewer prisoners. Scandinavian prisons, for example, typically house fewer than 100 inmates. Compared with California prisons today—which hold an average of several thousand—these create a dramatically different environment. A second option is simply to incarcerate fewer people for shorter sentences. The data suggest that reducing the incarceration rate from its current level will bring little to no increase in the crime rate. Recent initiatives, like California's Proposition 47, are aimed at reducing the prison population and creating a system more in line with practices in other developed, high-income countries. And it brings an added benefit, one we should factor into our policy decisions: It will also reduce the power of prison gangs.
David Skarbek is a lecturer in the Department of Political Economy at King's College London, and author of the new book, The Social Order of the Underworld: How Prison Gangs Govern the American Penal System (Oxford University Press). Courtney Michaluk is a second-year master's degree student in the economics department at George Mason University.
When reform hits real life
Why North Carolina might be the most innovative health care state in America
The new science fossil fuel companies fear
Free college is the wrong answer
Welcome to the Cannabis issue
David Skarbek
Courtney Michaluk
This story tagged under:
America's government is getting old
By Danny Vinik
Innovation vs. the ants
When cutting-edge research labs get old, they face a new kind of challenge: Upkeep is expensive, and it's not sexy.
By Jenny Hopkinson
A 40-year 'conspiracy' at the VA
By Arthur Allen
The little-noticed bombshell in Trump's immigration order
The travel ban got all the headlines, but experts are realizing another provision could clamp down on normal tourism and even diplomats.
That time Obama negotiated his own Carrier-style deal
Obamacare, the secret jobs program
By Dan Diamond
Why Paul Ryan won't accept — or dismiss — Donald Trump
By Danny Vinik and Darren Samuelsohn
Is Washington creating more Flints?
The untold story behind this year's water scandal is that D.C. has been turning its back on America's crucial water systems, leaving cities and states with the bill.
How Vermont beat Big Food
A new Obama emissions play
By Michael Grunwald
Arne Duncan's Wars
Here's what a Biden campaign would look like
Virtual schools are booming. Who's paying attention?
By Darren Samuelsohn
Obama's effort to 'nudge' America
FEMA head Craig Fugate explains what keeps him up at night
Uber vs. Laws
Will the "sharing economy" help kill off government regulation?
Leaked: What's in Obama's trade deal
I became a wired wonk
The U.S. government's predatory-lending program
Billions over budget. Two years after deadline. What's gone wrong for the 'clean coal' project that's supposed to save an industry?
Inside the new battle against Google
The anti-innovation presidency
Trump's attack dog on trade
By Megan Cassella
The teeth in Trump's new immigration rules
The charter war
By Colin Wilhelm
Obamacare's sinking safety net
By Paul Demko
Why the GOP is trying to stop the Pentagon's climate plan
Obamacare's 'Wellness' gamble
By Marianne LeVine
Inside the war on coal
The fight over fixing Puerto Rico
Inside the NSA's hunt for hackers
Obama: Time to tax oil
NASA chief: We could be on Mars in the 2030s
How to kill an agency
Ben Carson wants to end the VA? Good luck. Here's why it's nearly impossible.
Wrongometer
Donald Trump's imaginary immigration haul
Bank of Asphalt
Inside the fight against Obama's climate plan
The Capitol is not connected to the Internet
It's true. Here's why.
The Obamanomics legacy
A Q&A with Jason Furman.
What does Hillary stand for? Check her staff Twitter
What Jared's office actually does
By Nancy Scola
The strange story of the first Cuban import in half a century
Congress tries to hide a submarine
How Trump's attack on Mexico could hurt border security
Mexico has stepped up its cooperation—a lot. What happens when the president says it's shirking?
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The wrong state to have an accident
Obamacare was supposed to improve health care equality. But for some people, state politics has made the problem much worse.
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The holes in the overtime rule
Despite Obama's sweeping new regulation, America's 1930s-era labor law is still leaving a strange assortment of workers in the cold.
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The Flint of California
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The two words that rewrote American water policy
The Panama Papers: Where are the Americans?
Our best frenemy
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Congress just outlawed changes to the $1 bill
The publicly funded reports you can't read
The Katrina disaster that hasn't ended
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The 7-year-old government loan program that has never made a loan
The Internet of What?
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Tax Harvard!
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By Kevin Hartnett
President Barack Obama wants to make college more affordable. Here's a way to do it.

redpajama_set_name: RedPajamaCommonCrawl


Used as the pharmaceutical intermediates of the Halofuginone.
andD purpose. It may be toxic or hazardous, and should be handled only by qualified individuals trained in laboratory procedures. The buyer will be responsible to warn and inform any person that may be in contact with the product of the potential risks and hazards, as well as to ensure that purchase, use and disposal of the product comply with local laws and regulations.


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Boatshed.com – redefining yacht brokerage for 20 years!
September 2019 marks the 20th anniversary of the launch of Boatshed.com with 20,000 reasons to keep growing after its first 20 years of success.
Back in 1999, while many of us might have been strutting our stuff to Britney Spears, munching popcorn while watching Notting Hill – or simply going into meltdown in fear of the impending Millennium Bug – something quite extraordinary was happening in a quiet corner of Milford Haven Marina.
With the experience of a dramatic air-sea rescue from a Force 10 storm in the Bay of Biscay the previous year, a passion for sailing was turning into a lifestyle as Boatshed.com opened its first brokerage office.
But for husband and wife team Neil and Mandy Chapman, little did they know at the time just how much their business idea was going to revolutionize the yacht brokerage industry.
Twenty years on – and more than 20,000 boat sales later – the Boatshed story is one of remarkable success and achievement.
True entrepreneurship takes the helm
Co-founder, Mandy Chapman, vividly remembers the early days – for all the wrong reasons. 'We didn't sell a single boat in our first six months and I was using my skills as a former graphic designer to build the website and to work on boat upholstery in the office to make ends meet,' recalls Mandy.
'And as we were using a film camera to take photos of the boats we were trying to sell, I was forever back and forth to the local chemists to get the films developed. It was costing us a small fortune,' added Mandy.
'We economized by staying on board our Rebel 41 yacht 'Supertaff', but our hard work and perseverance was soon to pay off. In our second six months we sold 118 boats and we purchased our first digital camera!'
So what was behind this impressive turnaround in sales? Co Founder of Boatshed.com, Neil Chapman, believes there were two key factors that influenced how Boatshed.com would flourish.
'As a start-up, we were living on the cusp of the digital age as far as yacht brokerage was concerned,' explained Neil. 'We saw an opportunity to seize the developing technologies and integrate them into a business model that could be rolled out on a global basis.
'At the heart of this,' added Neil, 'was the need to transform how buyers and sellers of boats could come together in a process that was faster, easier and more fulfilling in every aspect of the purchase and sale of a boat.'
Going for global coverage with local presence
Totally convinced about the potential of Boatshed.com, Neil and Mandy set about building an international team to provide the local presence and knowledge that was to be paramount in supporting the intuitive digital technologies that set the business apart from others in the industry.
Boatshed offices in Ireland and Gibraltar formed the early part of the expansion plans – but it was in 2005 that the Boatshed brand started to gain media exposure.
Never one to miss an opportunity for publicity, 'Supertaff' and her crew led the Sail 8 flotilla to France and back to help create awareness for Bob Geldof's Live 8 bid to get world leaders to tackle poverty at the forthcoming G8 Summit in Gleneagles.
"Hey Neil. It's Bob. You guys have done a great job," Bob who, thought Neil, as he took the call while awaiting clearance for the flotilla to enter Portsmouth Harbour. The conversation resumed some nine hours later as the Boatshed crew was invited backstage at Murrayfield where the Live 8 concert was being staged.
The Boatshed name took to the water again in 2005 with the sponsorship of an Offshore 225hp powerboat, while on dry land more brokerage offices were opening their doors in Portsmouth, the Hamble, the Algarve and Spain.
It was also the year in which Boatshed purchased the Grade II listed Engine House at Royal Clarence Yard in Gosport. With refurbishment to follow, this was to become the international headquarters for the Boatshed.com organization.
Rolling out the formula for success
The next five years saw continued growth with a new office in Cowes, expansion into the USA and Asia, and the introduction of Boatshed Performance to cater for the niche market of racing yachts.
By September 2006, more than 100,000 registered users had signed up to Boatshed.com – progress recognized with a Top 10 Award for its industry website ranking and a Young Entrepreneur of the Year Award from the Boating Business exhibition in 2007.
'The first 10 years of Boatshed.com exceeded all our expectations,' says Neil Chapman.
From selling 118 boats in our first year to achieving 124 boat sales in a single month in 2009 is quite phenomenal. But none of this could have been achieved without the enthusiasm and work ethic of the amazing team of people who have made Boatshed.com what it is today.'
Fast-forward to 2019 – and Boatshed.com is now firmly established and respected as one of the world's largest international yacht brokers. With representation through more than 70 offices in 16 countries, award winning technologies and 880,000 registered users, Neil, Mandy and the HQ team are relishing the challenges of the next 20 years.
'We never forget that our customers are the most important part of our business,' added Mandy.
'Boatshed's high level of local customer service has to fit and interconnect with national and international customer and brokerage requirements. And it's this fusion of local and international, traditional and modern that makes the essence of what Boatshed is all about.'
The Mayday Call that saved our lives – and started a business…
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text:
Home / General News / News / Trend / Don't try to control Obaseki – Oba of Benin warns Wike, other PDP Governors
Don't try to control Obaseki – Oba of Benin warns Wike, other PDP Governors
The Oba of Benin, Oba Ewuare II, has warned the Peoples Democratic Party (PDP) about godfatherism, which the party campaigned against before the September 19 Edo State governorship election.
The monarch stated this on Tuesday, when five PDP governors, Ahmadu Fintiri of Adamawa, Nyesom Wike of Rivers, Aminu Tanbuwal of Sokoto, Seyi Makinde of Oyo and Godwin Obaseki of Edo paid him a courtesy call.
Oba Ewuare told them not to control Obaseki, despite playing a big role in helping him win re-election.
He said, "I want to on this occasion plead with the PDP to guide against godfatherism and not to allow unscrupulous members to kill your party.
"I will tell the PDP governors not to become hidden godfathers that will be controlling Governor Obaseki.
"All hands should be on deck to work for Edo State and Edo people everywhere they may be. This is where everybody will be proud to belong to. I said the whole world is watching Edo State and I thank God and our ancestors that the election came without any violence."


is based on a 39 months contract with £399.00 Customer Deposit, £2,087.61 Chambers Discount, £1,100.00 Customer Saving and Guaranteed Future Value/Optional Final Payment of £11,273.00. 1.2% APR Representative.
As with everything in this high-powered luxury model, top-spec technology is a key feature both inside and out of the cabin. The Ford Mondeo Vignale is kitted out with the latest FORD SYNC 2 infotainment system which includes a SONY DAB Navigation System and 12-speaker Sony sound system. Additionally, you've got Active Park Assist (and front and rear parking sensors), which lends a helping hand to get into even the trickiest of spaces. Meanwhile Active City Stop and Traffic Sign Recognition also features. As standard the Vignale also comes with a reversing camera, LED headlamps and an impressive eight-inch sat nav display unit.
The Ford Mondeo Vignale take the safety of you and your passengers incredibly seriously. Awarded a full five-star safety rating from the experts at Euro NCAP, the Vignale is packed full of safety protection. Passive safety features include, a suite of airbags and a handy seatbelt reminder. Electronic Stability Control tops the active safety feature list – this effective feature detects a loss of steering and automatically applies the brakes to correct it. Plus, if you've got little ones onboard you can rest assured of their safety thanks to the handy ISOFIX child seat anchor points.
Not just a car, purchasing a Vignale offers you more than vehicle ownership. A new benchmark in Ford quality, the Mondeo Vignale redefines the level of luxury you can expect from both your vehicle, and the services attached to it.
Bespoke services are available to those who have purchased a Vignale, including collection and delivery (either at home or a work location) for vehicle servicing. Plus, buyers can call upon knowledgeable Vignale OneCall advisors 24 hours a day for additional support and the Vignale Service App helps to make caring for your vehicle easier than ever before.
Introducing the Ford Mondeo Vignale – the Mondeo, but not as you know it. This range-topping trim is packed full of luxurious extras and is ready to enhance your drive, making every journey a premium experience for you and your passengers.
Top class begins in production. Every Vignale model is hand-finished by six master craftspeople in the Vignale centre, a state-of-the-art facility in Valencia. To complement the cars premium look and feel both inside and out, under the bonnet you'll find an engine that perfectly balances rewarding performance with impressive efficiency. Laden with intelligent technologies, you can be sure you are in absolute control of this refined engine.
From its striking styling, including cutting-edge design features such as unique dark accents, a distinctive grille and 19-inch dark tarnish alloy wheels, to a driving experience that is second to none, the Vignale is making no apologies for its high-end spec.
You can take your pick between 2.0-litre EcoBoost petrol and TDCi diesel units, as well as the 184bhp petrol-electric hybrid powertrain.
If you chose the 2.0-litre EcoBoost 237bhp variant, you'll get a six-speed automatic transmission. The 176bhp 2.0-litre TDCi diesel comes with six-speed manual and PowerShift automatic transmissions, along with Ford's intelligent all-wheel drive system. Plus, the new 207bhp 2.0-litre bi-turbo diesel which delivers 450Nm of torque.
Vehicle performance was also on Ford's list of refinements for the Vignale. Ford's innovative Active Noise Cancellation system enhances interior improvements. Three cabin microphones detect engine noise and improve cabin ambience by cancelling out the sound using advanced audio technology to direct opposing sound waves through the audio system.
Although from the outside, the iconic Mondeo shape is instantly recognisable, its quirky metallic-finish, hexagonally-designed upper grille is immediately striking, sitting proudly at the front of the Mondeo Vignale. Additionally, there's a high-gloss lower grille with chrome bars and chrome door details. The high-gloss finish continues on the windshield, central and rear window pillars. Riding on 18-inch alloy wheels you can take your pick from options including Vignale Black, Vignale Silver and premium four-coat Vignale White.
Step inside the cabin and the executive feeling continues. The seats, which are laser cut for that additional precision, are available in exclusive Vignale leather trim. Small details are key, with the material featuring hexagonal quilting and tuxedo stitching. The soft-touch elements also feature on the instrument panel, arm-rest, door top-rolls and centre console.
The Ford Mondeo Vignale has taken the famous Mondeo and added refinement and luxury to every detail. Well-appointed with design features, safety enhancements and rounded off with an engine that rivals competitors in its class, the Vignale has redefined luxury in the Ford marque.
So what are you waiting for? Come and experience the luxurious driving experience offered by the Mondeo Vignale for yourself. If you have any questions, or you would like to take the car for a test drive please get in touch. Our team of experts would be delighted to help you down at our showrooms.


The potential 'comet of the century' is racing towards its date with the Sun on November 28th. Find out where it will be during its lead up to this fiery meeting.
The 110 (plus) Years Old Lamp is Still Working!


text:
Joseph Ranson (1906-1964) was an Irish Catholic priest, archivist and historian, who served as the last rector of the Irish College at Salamanca.

Born in 1906, in Killmuckridge, Co. Wexford, he was educated at Gorey CBS, and St. Peter's College, Wexford, his brother Fr. Robert (Bob) Ranson DD also became a priest, their father ran an undertakers business which was founded by their grandfather. Joe Ranson, trained and was ordained a priest in 1930 in Salamanca and served in the diocese of ferns. In 1949 he was appointed rector and archivist of the Irish College in Salamanca, where he served for 4 years, organising the archives for the transfer to St. Patrick's College, Maynooth, and the transfer of the properties to the Spanish state, and money received went to the Irish College Rome. Following his posting in Salamanca, he returned to Ireland as a priest in the diocese of Ferns. He was a member of the Royal Society of Antiquaries of Ireland, and the Uí Cinsealaigh Historical Society in Wexford, contributing to their journal The Past. He founded the Wexford County Museum, in Enniscorthy Castle which opened in 1962. He died on 17 November 1964.

References

Christian clergy from County Wexford
Irish antiquarians
Irish archivists
People educated at St Peter's College, Wexford
1906 births
1964 deaths
20th-century Irish Roman Catholic priests
20th-century antiquarians

redpajama_set_name: RedPajamaWikipedia


Every time you get in the car it is a risk. You are risking your life to get to your destination. It is well known that traffic accidents are one of the leading causes of injury-related deaths in the US (second only to suicide). But what if there was a way to avoid the roads that have the highest rates of fatal crashes? Would you reroute your commute to avoid these roads?
Sure! But how do I know what roads to avoid?
I'm glad you asked. The website riskyroads.org has done all of the legwork for you. They have taken statistics about fatal crashes from the National Highway Traffic Safety Administration and collaborated them to rank each major freeway in the country. They have ranked the roadways on a scale of 1-8, with eight being the worst.
How do they rank the roads?
They look at where all of the fatal accidents happen. If there are several accidents on the same road within 1,000 feet of each other, the road will receive a higher rating. The map of the country is then color-coded so that it is very easy for you to determine where the dangerous areas are in your city. Basically, any red or purple areas you will want to avoid if possible.
Just because someone got in a crash there doesn't mean I will.
That is very true. While distracted driving or speeding can occur anywhere, there are often other factors in play. Does the road have a really sharp turn? Is the road located near a popular bar, resulting in a higher occurrence of drunk drivers? Is the speed limit too high for the street conditions? Is the area simply a magnet for terrible drivers? The high rates of fatal accidents on particular streets probably happen for a reason. If they were a mere coincidence they wouldn't keep happening in the same spot.
So if I avoid those roadways I will be safer?
Of course, you can never guarantee when or where you will or will not get in an accident. But if you can reroute your commute to avoid these roads, you probably have a better chance of reaching your destination. If it adds a lot of time to your commute, it is probably not worth it. You will just want to make sure to drive extra defensively and follow all of the laws to ensure your safety. However, if you can alter the route you take without adding too much time to your commute, it may be worth a change. Sure, you may take a few more minutes to get where you are going, but at least you will get there safely.


text:
Q: JavaScript code executing after return In the following example, JavaScript seems to be completely ignoring my return statement, and just carrying on executing code.
var x = 1;
(function() {
  x = 2;
  return;
  var x = 3;
})();
console.log(x); // Outputs 1 in both Chrome and FF

Surely the code should output 2? If I remove the var keyword from var x = 3, it outputs 2 as expected. Is there some strange compiler optimization at work here?

A: No, the code shouldn't output 2 because variable declarations are hoisted so your code is equivalent to
var x = 1;
(function() {
  var x;
  x = 2; // changes the internal x variable
  return;
  x = 3; // does nothing because it's not reached
})();
console.log(x); // Outputs the outside x, which is still 1

The line
x = 2;

only changes the internal x variable which shadows the outside one.
The scope of a non global variable is the entire function in which it is declared. From the start of this function to its end.

redpajama_set_name: RedPajamaStackExchange


Every day, designers are challenged to shape the future in the industries they serve. But, how exactly can design catalyze innovation in organizations? What is design's role in shaping both physical and digital user experiences? In what ways can design humanize automation? IA Collaborative is partnering with IxDA to host an event for visual designers, UX designers, researchers, and more to share real-world case studies that explore how the consultancy answers these questions alongside the clients they serve.


text:
\section{Introduction}
TAP equations are sets of generalized mean field equations for computing approximate marginal moments in large probabilistic models with a dense connectivity of couplings with random strengths. Originally developed to compute the magnetisations of the Sherrington Kirkpatrick (SK) model of a spin-glass \cite{TAP}, there are now various generalizations especially to models in  the area of statistical data science~\cite{OW0,Kabashima,Donoha,Tramel18}.

Hence, it is an important to design and study the performance of iterative algorithms which can generate solutions to TAP equations. Several results for the dynamical properties of such algorithms have been derived for models with specific probability distributions of the couplings \cite{Bolthausen,Bayati,bayati2015universality}. More recent research has attempted to extend such results to more general (and possibly more realistic) classes of probability distributions \cite{Opper16,VAMP,takeuchi2017rigorous,Burak17,fletcher2018inference}.

In this paper we will address the dynamics of algorithms for the original case of Ising spin models. The model in its simplest formulation (with constant external fields) discussed in this paper
is less interesting for data science.  But its generalizations have been applied e.g. to modeling the dependencies of spikes recorded from ensembles of neurons \cite{schneidman2006weak,roudi2009ising}, to protein structure determination \cite{weigt2009identification} and gene expression analysis \cite{lezon2006using}. For a review, see \cite{nguyen2017inverse}. In any case, we expect that a proper understanding of algorithms for the simplest setting will also be highly useful for treating more complex models. We will consider TAP equations for the large class of rotation invariant random matrices \cite{Parisi}, which allow for nontrivial dependencies between matrix elements. We will show in Section~\ref{modelpr} that simpler cases such as the independent Gaussian couplings of the SK model or those of the {\em Hopfield model}  and its generalizations \cite{mezard2017mean} are special cases of the rotation invariant ensembles studied in this work.

The analysis of algorithms for TAP equations can be described in terms of the nonlinear dynamics of a system of nodes coupled by a dense random matrix.
In the thermodynamic limit of large systems, it is possible to derive an exact decoupling of the degrees of freedom for the statistical
properties of the dynamics using the so-called {\em dynamical functional} (DF) method \cite{Martin}. Unfortunately, the effective stochastic process of single variables contains couplings between the same variables at different times. This usually precludes closed form analytical results for the dynamical properties. In a previous paper,
motivated by existing algorithms for the case of specific random matrices, we have shown that such temporal self couplings can be canceled for the entire class of rotation invariant random matrices, if one includes specific memory terms in the algorithm. In that case,
the stochastic fields which drive the effective dynamics reduce to a Gaussian process.

Our previous construction of the so-called {\em single-step memory} (SSM) algorithm for solving TAP equations presented in \cite{Opper16} comes with two major drawbacks. First, the memory terms in the algorithm are based on the coefficients of
the power series expansion of a specific function derived
from the random matrix ensemble. Of course, these can be obtained easily, when an analytical
expression of the function is known. But for any given large matrix which is
assumed to be drawn at random from a rotation invariant (but otherwise
unknown) ensemble it is not clear
how to obtain such coefficients numerically in a simple way. This problem will make the algorithm not a good candidate for practical applications.
Second, from a theoretical perspective our analytical analysis is still incomplete. We were able to show by a linear stability analysis that the well-known {\em de Almeida Thouless} (AT) \cite{AT}
criterion which separates an ergodic region of the spin-model from
a more complex phase, is necessary for convergence of the algorithm. Unfortunately, we were
not able to get general analytical results for the asymptotic convergence rates.

In this paper we present a new construction for an algorithm for the Ising system which was motivated by similar approaches in the field of compressed sensing \cite{VAMP,takeuchi2017rigorous}. Remarkably, the updates of the algorithm
avoid the use of explicit memory terms and
can be easily applied to an Ising model with a given matrix of couplings.
The analysis by the DF method again shows a vanishing of the self-couplings.
This leaves us with an effective dynamics that is driven by a Gaussian process
but with a much simpler covariance structure compared to our previous method.
We are able to prove convergence  from a random initialization
of the algorithm in the thermodynamic limit provided that the AT criterion is fulfilled.
Moreover, we also obtain explicit expressions for the rate of convergence.
To our knowledge this is the first time that analytical results for the convergence of
such algorithms have been obtained.

The paper is organized as follows: In Section~2 we introduce the model and present motivating
examples for studying rotation invariant ensembles of coupling matrices. Section~3 provides a brief presentation on the TAP equations. In Section~4 we present our new algorithm for solving the TAP equations and in Section~5 we study its properties in the thermodynamic limit. Section~6 provides convergence properties of the algorithm. In section~7 we show how to compute
parameters which are needed by the algorithm before the iteration starts. For comparison,
in Section~8 we
give analytical expressions for the convergence rates of
the SSM algorithm of \cite{Opper16} for the special cases of the SK and Hopfield models. Comparisons of the theory with simulations are given in Section~9. Section~10 presents a summary and outlook. Major derivations of our results are located at the Appendix.

\section{Ising models with random couplings}
We consider Ising models with pairwise interactions of the spins $\matr s=(s_1,\ldots,s_N)^\top\in\{-1,1\}^{N}$ described by the (conditional) Gibbs distribution
\begin{equation}
p(\matr s\vert \matr J,\matr h)\doteq \frac{1}{Z}\exp\left(\frac{1}{2}\matr s^\top\matr J\matr s+\matr s^\top\matr h\right)\label{Gibbs}
\end{equation}
where $Z$ stands for the normalization constant. Our concern is to compute the vector of magnetizations
\begin{equation}
\matr {m}=\mathbb E[\matr s]
\end{equation}
where the expectation is taken over the Gibbs distribution. For the sake of simplicity of the analysis we limit our attention to the case where all external fields are equal
\begin{equation}
h_{i}=h\neq 0,~\forall i.
\end{equation}

We assume that the coupling matrix $\matr J=\matr J^{\top}$ is drawn from a rotation invariant matrix ensemble, i.e. $\matr J$ and $\matr V\matr J\matr V^\top$ have the same probability distributions for any orthogonal matrix $\matr V$ independent of $\matr J$. Equivalently, $\matr J$ has the spectral decomposition \cite{Collins14}
\begin{equation}
\matr J=\matr O^
\top\matr D\matr O \label{decom}
\end{equation}
where $\matr O$ is a Haar (orthogonal) matrix that is independent of a real-diagonal matrix $\matr D$. Though our analysis does not make reference to any specific coupling matrix model,
it is interesting to note, that certain models derived from products of simpler matrices
which were originally introduced in the context of communication theory \cite{ralfa} and recently reappeared in the statistical physics context \cite{mezard2017mean}, belong to the rotation invariant family.
\subsection{Motivation: Models with a product of random matrices}\label{modelpr}
Consider the product of $M$ matrices
\begin{equation}
\matr X\doteq\matr X_M\matr X_{M-1}\cdots \matr X_1 \label{prod}
\end{equation}
where $\matr X_m\in\mathbb{R}^{N_{m}\times N_{m-1}}$ and $N_0\doteq N$. We will consider
coupling matrices of the form
\begin{equation}
\matr {J}=\beta\matr X^\top\matr X-\frac{\beta}{N}{\rm tr}(\matr X^\top\matr X){\bf I}\label{Mlayer}
\end{equation}
where $\beta$ stands for the inverse temperature parameter. With this modeling of the coupling matrix the Ising model becomes equivalent to an M-layer probabilistic feed-forward neural network
with linear transfer functions. Specifically, for a set of hidden node vectors  $\mathcal U\doteq\{\matr u_0,\matr u_1,\cdots,\matr u_M\}$ with $\matr u_m\in \mathbb{R} ^{N_m\times 1}$ we define an $M$-layer linear neural network by the distribution
\begin{align}
	&p(\matr s,\mathcal U\vert \mathcal X,\matr h)\doteq\frac{1}{Z}e^{-\frac{\tilde\beta}{2}+\matr s^\top(\matr u_0+\matr h)}\times \nonumber \\ &\quad \times \left(\prod_{m=1}^{M}\delta(\matr u_{m-1}-\matr X_m^\top\matr u_m)\right){N}\left(\matr u_M;\matr 0,\frac 1 \beta {\bf I}\right)
\end{align}
where $\mathcal X\doteq \{\matr X_1,\ldots,\matr X_M\}$, $\tilde\beta\doteq\beta{\rm tr}(\matr X^\top\matr X)$ and ${N} (\cdot; \matr \mu,\matr \Sigma)$ denotes the Gaussian density function with mean $\matr \mu$ and covariance $\matr \Sigma$. It is easy to see that by integrating out the Gaussian variables $\mathcal U$, we arrive at the Gibbs distribution (\ref{Gibbs})
\begin{equation}
p(\matr s\vert \matr J,\matr h)=\int {\rm d}\mathcal U\; p(\matr s,\mathcal U\vert \mathcal X, \matr h).
\end{equation}

We will next discuss some interesting cases for which this model leads to rotation invariant ensembles.
	\subsubsection{Multi-layer Hopfield model}
Let $\matr J$ be as in \eqref{Mlayer} and let all $N_m\times N_{m-1}$ matrices $\matr X_m$ in \eqref{prod} be independent where the entries of $\matr X_m$ are independent Gaussian
random variables with zero mean and variance $1/N_{m}$ \cite{ralfa}. The case $M=1$
is essentially the well-known {\em Hopfield model} \cite{Hopfield}. Here, we ignore the fact that in the present case, the stored ``patterns'' are Gaussian not Ising variables.
Since we are not interested in memory retrieval properties of the model, the difference is not relevant.

The case $M=2$ coincides with Mezard's {\em combinatorial disorder Hopfield model} \cite{Mezard}. In general, random matrices $\matr J$ constructed in such a way belong
to rotation invariant ensembles and we call the model {\em multi-layer Hopfield model}.
\subsubsection{Multi-layer random orthogonal model}\label{mrom}
Let $\matr J$ be as in \eqref{Mlayer} and the $N_m\times N_{m-1}$ matrices $\matr X_m$ in \eqref{prod} are defined as
\begin{equation}
\matr X_m=\frac{1}{\sqrt{\rho_{m}}}\matr P^L_{\rho_{m}}\matr O_m(\matr P^L_{\rho_{m-1}})^\top\label{rom}
\end{equation}
where all $\matr {O}_m$ are independent $L\times L$ Haar matrices, and $\matr P^{L}_{\rho}$ is a $\rho L\times L$ matrix with $\rho\leq 1$ and
\begin{equation}
(\matr P^{L}_{\rho})_{ij}=\delta_{ij},\quad \forall i,j.
\end{equation}
For $M=1$ and $\rho_0=1$ we arrive at the {\em random orthogonal model} of Parisi and Potters \cite{Parisi}. In general, matrices $\matr {J}$ are also rotation invariant and we call the model \emph{multi-layer random orthogonal model}.

\subsubsection{SK model as the universal small $\alpha$-limit}
We next consider the scaling properties of the coupling matrix model \eqref{Mlayer} when the aspect ratio $\alpha\doteq \frac{N_0}{N_M}$  of the $N_M\times N_0$ matrix $\matr X$ in \eqref{prod} becomes small. For the 1-layer Hopfield model it is well-known that by rescaling
$\matr J\to \frac{1}{\sqrt{\alpha}}\matr J$
of the coupling matrix the model becomes equivalent to the SK model \cite{mezard2017mean} in the limit $\alpha\to 0$. To be precise, the limiting spectral distribution of $\frac{1}{\sqrt{\alpha}}\matr J$ becomes a Wigner semicircle law in the limit $\alpha\to 0$. We next show that this small-$\alpha$ limit holds in a universal sense for both the multi-layer Hopfield and multi-layer random orthogonal models. Specifically, the coupling matrices for $\alpha\leq 1$ are in general of the form
\begin{equation}
\matr J=\matr P^{N_M}_\alpha\matr Y(\matr P^{N_M}_\alpha)^\top.\label{skform}
\end{equation}
Let $f_{\alpha}$ denote the density of the limiting spectral distribution of $\frac{1}{\sqrt{\alpha}}\matr J$ and let $\sigma_{\matr Y}^2\doteq\lim_{N\to \infty}\frac{1}{N}{\rm tr}(\matr Y^2)$. Then,
we show in Appendix~\ref{dlimitsk} that
\begin{equation}
\lim_{\alpha\to 0}{f}_\alpha(x)=\frac {{\sqrt{4\sigma_{\matr Y}^2-x^2}}}{2\pi \sigma_{\matr Y}^2} \;, \quad \vert x \vert \leq 2\sigma_{\matr Y}.\label{limitsk}
\end{equation}
Actually, this result holds for any coupling matrix of the form \eqref{skform} with rotation invariant $\matr Y$ which has a compactly supported limiting spectral distribution with $\lim_{N\to \infty}\frac{1}{N}{\rm tr}(\matr Y)=0$. We note that $\sigma_{\matr Y}^2$ is a model-dependent parameter, e.g. for the multi-layer Hopfield model we have
\begin{equation}
\sigma_{\matr Y}^2=\beta^2(1+\sum_{m<M}\alpha_m)
\end{equation}
with $\alpha_m\doteq N_0/N_m$.  For a convenient computation of the model parameter $\sigma_{\matr Y}^2$ in general we refer the interested reader to Appendix~\ref{addsigma}.
\section{TAP equations}
The TAP equations are a set of nonlinear equations for approximating the magnetizations $\matr m$ of the Ising model with random couplings.
They are assumed (under certain conditions) to give the exact results in the thermodynamic limit \cite{Mezard} and were first proposed for the SK model in \cite{SK}. A generalization to rotation invariant matrices were originally derived in \cite{Parisi} using
a free energy approach. An alternative derivation was given in \cite{Adatap} using the cavity method. We also refer  to \cite{CakmakOpper18} for a rigorous approach on the self averaging assumptions made for cavity field variances in this approach. The TAP equations are given by
\begin{subequations}
\label{tap}
\begin{align}
\matr m&={\rm Th}(\matr \gamma)\\
\matr \gamma&=\matr J\matr m-{\rm R}(\chi)\matr m\\
\chi &= \mathbb E_u[{\rm Th}'(\sqrt{(1-\chi){\rm R}'(\chi)} u)] \label{chi}.
\end{align}
\end{subequations}
Here, for convenience we define the function
$${\rm Th}(x)\doteq\tanh(h+x) .$$ The random variable $u$ is a standard (zero mean, unit variance) normal Gaussian.

The only dependency on the random matrix ensemble of $\matr J$ is through the
R-transform which is defined as \cite{mingo2017free}
\begin{equation}
{\rm R}(\omega)={\rm G}^{-1}(\omega)-\frac{1}{\omega},  \label{Rtrans}
\end{equation}
where ${\rm G}^{-1}$ is the functional inverse of the Green-function
\begin{equation}
{\rm G}(z)\doteq \lim_{N\to\infty}\frac{1}{N}{\rm tr}((z{\bf I}-\matr J)^{-1}) . \label{Greens}
\end{equation}
${\rm G}(z)$ is bijective for real $z$ which are not in the support of the limiting spectral distribution of~$\matr J$. Hence, ${\rm R}(\omega)$ is well-defined for ${\rm G}_{\min}<\omega< {\rm G}_{\max}$ where we have defined ${\rm G}_{\max}\doteq\lim_{z\to \lambda_{\max}}{\rm G}(z)$
(and similarly for ${\rm G}_{\min}$) with $\lambda_{\max,\min}$ denoting the supremum and infimum of the support, respectively. One can show that
${\rm R}(\omega)$ is strictly increasing \cite{Guionnet} (unless $\matr J=c{\bf I}$ for some $c\in \mathbb{R}$), i.e. ${\rm R}'(\omega)>0$  where ${\rm R}'$ stands for the derivative of ${\rm R}$.

As an example, for the (1-layer) Hopfield model with  $\alpha_1\doteq N_0/N_1$
we have \cite{Opper16}
\begin{equation}
{{\rm R}({\omega})}=\frac{\alpha_1\beta^2\omega}{1-\alpha_1\beta\omega}.\label{R1}
\end{equation}
The corresponding equations \eqref{tap} agree with the well-known TAP equations of the Hopfield model previously obtained e.g.
in \cite{Mezard}. {Moreover,} for the 2-layer Hopfield model with $\alpha_2\doteq N_0/N_2$ we have (see Appendix~\ref{drmodel2})
\begin{equation}
{\rm R}({\omega})=\frac{\frac{1}{\beta \omega}-\alpha_1-\alpha_2-\sqrt{(\frac{1}{\beta \omega}-\alpha_1-\alpha_2)^2-4\alpha_1\alpha_2}}{2\alpha_1\alpha_2\omega}-\beta\label{rmodel2}
\end{equation}
and the resulting TAP equations \eqref{tap} coincide with those derived in \cite{mezard2017mean} by means of a mean field message passing method.

The TAP equations \eqref{tap} are well-defined when
\begin{align}
\eta{\rm R}'(\chi)&< 1 \label{cond2}~~ ({\rm AT})\\
\chi&<{\rm G}_{\max} \label{cond1}
\end{align}
where we have set
\begin{equation}
\eta\doteq\mathbb E_u[({\rm Th}'(\sqrt{(1-\chi){\rm R}'(\chi)} u))^2].\label{eta}
\end{equation}
The condition (AT) is sufficient for $\chi$ to have a unique solution in \eqref{chi}. This result can be easily derived by following the arguments of the derivation of \cite[Lemma 2.2]{Bolthausen}. There, the result refers to the SK model, i.e. ${\rm R}(\omega)=\beta^2\omega$.
Equality in \eqref{cond2}, i.e. $\eta{\rm R}'(\chi)=1$, agrees with the well-known AT
line of stability \cite{AT} for the Replica-symmetric ansatz for spin models with rotation invariant coupling matrices, see \cite[Eq. (46)]{Enzooo}. In the region of stability, we can also expect that the solution of the TAP equations will give us the asymptotically correct magnetisations in the thermodynamic limit.
The second condition \eqref{cond1} is needed to ensure that the Green-function has a unique
inverse, such that the R-transform can be defined properly. For the SK-model, e.g.
it restricts the valid inverse temperature regions to $\beta<\frac{1}{\chi}$. Similarly, for the
(1-layer) Hopfield model, one needs  $\beta<\frac 1 \chi\frac{\alpha_1}{1+\sqrt{\alpha_1}}$.

\section{Iterative solution of the TAP equations}
In this section we define a new iterative algorithm for solving the TAP equations \eqref{tap}. First, we recall that the only dependency on the random matrix ensemble in the TAP equations \eqref{tap} is via the R-transform ${\rm R}\doteq{\rm R}(\chi)$ (for short) and its derivative ${\rm R}'\doteq {\rm R}'(\chi)$.
Hence, if the limiting spectral distribution of the random matrix ensemble is known, these quantities can be computed exactly from the
Green-function (\ref{Greens}) via (\ref{Rtrans}). We will show in {Section~\ref{algorc}} a practical computation
of these quantities for concrete realizations of matrices $\matr J$.

We are looking for a solution to the TAP equations in terms of an iteration of a vector of auxiliary variables
$\matr \gamma(t)$, where $t$ denotes the discrete time index of the iteration.
The initialization of this vector is given by $\matr \gamma(0)=\sqrt{(1-\chi){\rm R}'} \matr u$ where $\matr u$ is a vector of independent normal Gaussian random variables.
We then proceed by iterating
\begin{subequations}
\label{dynamics}
	\begin{align}
		\tilde{\matr \gamma}(t)&=\frac{1}{\chi}{\rm Th}(\matr \gamma(t-1))-\matr \gamma(t-1)\\
		\matr \gamma(t)&=\matr A\tilde{\matr \gamma}(t) \label{TAP_dyn_two}
	\end{align}
\end{subequations}
for $t=1,2, 3, \ldots,T$ with the \emph{time-independent} random matrix
\begin{equation}
\matr A\doteq\frac{1}{\chi}(\lambda{\bf I}-\matr J)^{-1} -\bf I.
\label{def_matrixA}
\end{equation}
The variable $\lambda$ is defined as the (unique) solution of the scalar equation
\begin{equation}
{\rm G}(\lambda) = \chi
\end{equation}
or equivalent (see (\ref{Rtrans}))
\begin{equation}
{\rm R}(\chi) = \lambda - \frac{1}{\chi} \label{lamdef} .
\end{equation}
It is easy to see that the fixed points of $\matr \gamma(t)$ coincide with
the solution of the TAP equations for $\matr\gamma$, \eqref{tap},
if we identify the corresponding magnetisations by
\begin{equation}
\matr m = \chi(\tilde{\matr \gamma} +\matr \gamma).\label{magnez}
\end{equation}
\section{Dynamics in the thermodynamic limit}
Our goal is to analyze the convergence properties of the sequence $\matr \gamma(t)$
in the thermodynamic limit by studying
the deviation between the dynamical variables at different times
\begin{align}
	\Delta(t,s)\doteq \lim_{N\to\infty}\frac{1}{N} \mathbb E[\Vert \matr \gamma(t)-\matr \gamma(s) \Vert^2]
	\label{measure}
\end{align}
where the expectation is taken over the random matrix ensemble of $\matr A$ and the random initialization $\matr \gamma(0)$.
In the thermodynamic limit, the normalized squared distance will become self averaging and represents the typical squared distance corresponding to a single realization of the dynamics \eqref{dynamics}. We will show
later, that as $t,s\to\infty$, the deviation $\Delta(t,s)$ will converge to $0$
showing that the sequence of iterates $\matr \gamma(t)$ will have a limit which
thus solves the TAP equations.

Dynamical properties of fully connected disordered systems can be computed by a discrete time version of the method of DF developed by Martin, Siggia and Rose \cite{Martin}
and also used extensively for studying spin-glass dynamics, see e.g. \cite{sompo82}, \cite{Eisfeller}. This method provides us with exact results for the thermodynamic properties of the marginal distribution of a trajectory of arbitrary length $T$, for {\em single variables} (in our case $\gamma_i(t)$ and $\tilde\gamma_i(t)$) as long as the large-system limit $N\to\infty$ is taken before the long-time limit $T\to\infty$. The method is based on averaging the moment \emph{generating functional} for the trajectory of $\gamma_i(t)$ and $\tilde\gamma_i(t)$
\begin{align}
Z\{l_i(t)\}\doteq\int \prod_{t=1}^T &{\rm d}\tilde{\matr\gamma}(t) {\rm d}\matr \gamma(t)\; \delta\left(\matr{\tilde\gamma}(t)-{f}(\matr \gamma(t-1))\right)\times\nonumber \\&\times\delta\left(\matr \gamma(t)-\matr A\matr{\tilde\gamma}(t)\right)e^{{\rm i}\gamma_i(t) l_i(t)}
\end{align}
over the random matrix $\matr A$ and the random initialization $\matr \gamma(0)$. Here, for short we introduce the function
\begin{equation}
f(x)\doteq \frac{1}{\chi}{\rm Th}(x)-x.
\end{equation}
The generating functional obviously contains the correct dynamics \eqref{dynamics} within the Dirac delta $\delta$ functions. Moreover, since the partition function is normalized, i.e. $Z\{l_i(t)=0\}=1$, the replica-method is \emph{not needed} to perform the expectations.

By construction, the distribution of $\matr A$ induced by the matrix $\matr J$ is also rotation invariant. Hence we can follow the steps of a previous paper \cite[Appendix C]{Opper16}, by
essentially replacing all averages over the matrix $\matr J$ in that paper by averages over $\matr A$
and read off the result.
In our case, $\tilde{\matr\gamma}(t)$ plays the role of $\matr m(t)$~in~\cite{Opper16}.
We then find that the averaged generating functional
$\mathbb E [Z(\{l_i(t)\})]$ becomes
\begin{align}
& \int{\rm d}\mathcal N(\{\phi (t)\};\matr 0,\mathcal C_\phi) \
{\rm d}\mathcal N(\gamma_i(0);0, (1-\chi){\rm R}')
\nonumber \times \\
&\times\;\prod_{t=1}^{T}{\rm d}\tilde\gamma_i(t){\rm d}\gamma_i(t)\;\delta( \tilde\gamma_i(t)- f(\gamma_i(t-1)))\nonumber \\
&\times\delta\left(\gamma_i(t)-\sum_{s< t}\mathcal{\hat G}(t,s) {\tilde \gamma_i}(s)-\phi(t)\right) e^{{\rm i}\gamma_i(t)l_i(t)+\epsilon} \label{dftresult}
\end{align}
where $\epsilon=O(1/N)$ is a constant term that, as indicated, vanishes as $N \to \infty$ and $\mathcal N({\cdot};\matr \mu, \matr \Sigma)$ denotes the Gaussian distribution function with mean $\matr \mu$ and covariance~$\matr \Sigma$. Hence, through averaging, we manage to \emph{decouple}
the component $\gamma_i(t)$  in \eqref{dynamics} from the remaining components
for $N\to\infty$ and we obtain an \emph{effective stochastic process} for the dynamics of single, arbitrary components $\gamma(t)$ and $\tilde\gamma(t)$ (where we omit the index $i$ in the following)
of the vectors $\matr\gamma(t)$ and $\tilde{\matr\gamma}(t)$, respectively. This contains all information about the marginal statistics of these variables
induced by the randomness of $\matr A$ and $\matr \gamma (0)$. Note that ${\rm d}\mathcal N(\{\phi (t)\};{ \matr 0},\mathcal C_\phi)$
represents an average over a discrete time Gaussian process $\phi(t)$ with zero mean
and covariance $\mathcal C_\phi$.

The resulting effective stochastic process of single variables is immediately read off
from (\ref{dftresult}) as
\begin{subequations}
	\label{esp}
	\begin{align}
	\tilde\gamma(t)&=f(\gamma(t-1))\label{gam1}\\
	\gamma(t)&=  \sum_{s < t}{\hat {\mathcal G}(t,s)}\tilde\gamma(s) + \phi(t) \label{gam2}
	\end{align}
\end{subequations}
for $t=1,2,3\ldots$ with the initial condition $\gamma(0) = \sqrt{(1-\chi){\rm R}'} u$
where $u$ is standard normal Gaussian. For a trajectory of length $T$, ${\hat {\mathcal G}(t,s)}$ are given by the $(t,s)$ indexed entries of the $T\times T$ matrix ${\hat {\mathcal G}}$
which is defined by the matrix function
\begin{equation}
\hat{\mathcal G}\doteq{\rm R}_{\matr A}(\mathcal G) \label{Ghat}
\end{equation}
where ${\rm R}_{\matr A}$ is the R-transform corresponding to the random matrix
${\matr A}$ (see the definitions in eq (\ref{Rtrans})).
$\mathcal G$ is a $T\times T$ order parameter matrix, which must be computed from the
entire ensemble of trajectories of  $\gamma(t)$. Its
$(t,s)$ indexed entries are defined by the response function
\begin{equation}
\mathcal G(t,s)\doteq\mathbb E\left[\frac{\partial\tilde\gamma(t)}{\partial \phi (s)}\right]  .
\label{resp}
\end{equation}
By causality $\mathcal G$ is an upper triangular matrix (i.e. its $(t,s)$ indexed entries are zero for $s \geq t$).
The R-transform of a matrix can be defined from the power series expansion of the R-transform \cite{mingo2017free}
\begin{equation}
{\rm R}_{\matr A}(\omega)\doteq\sum_{n=1}^{\infty}c_{\matr A,n} \omega^{n-1} \label{R_series}
\end{equation}
in terms of the so-called {\emph  free cumulants} $c_{\matr A,n}$
by inserting the powers of the matrix $\mathcal G^{n-1}$ for $\omega^{n-1}$.
It follows that $\hat{\mathcal G}$ is
also upper triangular.  From the definition of the matrix $\matr A$, (\ref{def_matrixA}) and standard properties of R-transform, one can show that
\begin{align}
c_{\matr A,1} &= \lim_{N\to\infty} \frac{1}{N} {\rm tr} (\matr A) = 0\\
c_{\matr A,2} & = \lim_{N\to\infty} \frac{1}{N} {\rm tr} (\matr A^2) - c^2_{\matr A,1}  \\
& = \lim_{N\to\infty}
\left(\frac{1}{\chi^{2}}\frac{1}{N} {\rm tr}((\lambda {\bf I}-\matr J)^{-2})-1\right) \label{sline}\\
& =
\frac{\chi ^2{\rm R}'}{1-\chi^2{\rm R}'}
\label{c_results}
\end{align}
where in \eqref{sline} we use ${\rm G}'(\lambda) =-\lim_{N\to\infty} \frac{1}{N} {\rm tr}((\lambda {\bf I}-\matr J)^{-2})$ and re-express the result in terms of the R-transform of the limiting spectral distribution of $\matr J$.

Finally, the zero-mean Gaussian process $\{\phi(t)\}$ has a
covariance matrix given by
\begin{equation}
\mathcal C_\phi = \sum_{n=1}^{\infty}c_{\matr A,n}\sum_{k=0}^{n-2} \mathcal {G}^k\mathcal {\tilde C}(\mathcal {G}^\top)^{n-2-k} \label{cov}
\end{equation}
where
\begin{equation}
\mathcal {\tilde C}(t,s)\doteq \mathbb E[\tilde\gamma(t)\tilde \gamma(s)]
\label{tilde_cov}.
\end{equation}
and the $c_{\matr A,n}$ are defined by the R-transform (\ref{R_series}).

It may seem that we have not gained much from this approach, because interactions between different dynamical variables are now replaced by couplings of single variables over time.
In principle, all matrix elements of  $\hat{\mathcal G}$, $\mathcal G$ and $\mathcal C$ could be (approximately) computed sequentially in time, e.g. by lengthy Monte Carlo simulations of the process
$\phi(t)$ \cite{Eisfeller}. On the other hand, the memory terms preclude an analytical treatment in general. Surprisingly, as we shall show in Appendix \ref{vanishmemory} the memory terms vanish, i.e. we get
\begin{equation}
\hat{\mathcal G}=\mathcal{G}= {\matr 0}.\label{memoryless}
\end{equation}
The memory-free property \eqref{memoryless} implies that
\begin{equation}
\gamma (t)=\phi(t) \label{phi_gamma}
\end{equation}
for $t=1,2,\ldots,T$ are Gaussian random variables with a much simplified covariance structure that is given by
\begin{align}
{\mathcal C}&= c_{\matr A,2}\tilde{\mathcal C}\\
&=\frac{\chi ^2{\rm R}'}{1-\chi^2{\rm R}'}\tilde{\mathcal C}.
\end{align}
Explicit results can be computed by the recursion (see Appendix~\ref{dcovfield})
\begin{subequations}
	\label{covfield}
\begin{align}
 \mathcal C(t,s)&=\frac{g(\mathcal C(t-1,s-1))}{1/{\rm R'}-\chi^2}  \label{covf} \\
 \mathcal C(t,t)&= (1-\chi){\rm R}'
\end{align}
\end{subequations}
where we have defined
\begin{equation}
g(x)\doteq {\mathbb E[{\rm Th}(\gamma_1){\rm Th}(\gamma_2)]} -\chi^2x\label{kfunc}
\end{equation}
such that the expectation is taken over the random variables $\gamma_1$ and $\gamma_2$ which are jointly Gaussian  with zero mean and equal variance $(1-\chi){\rm R}'$ and covariance
\begin{equation}
x\doteq \mathbb E[\gamma_1\gamma_2].
\end{equation}
To initialize the recursion (\ref{covf}), we note that by (\ref{phi_gamma}) we have
$\mathcal C(t,s) = \mathbb E[\gamma(t) \gamma(s)]$ and we can include the initialization $\gamma(0)$ in the
definition. It is easy to see that the random variable $\gamma(0)$ is independent
from $\gamma(t)$ for $t > 0$. Hence, we get
\begin{equation}
\mathcal C(t,0) = 0,\quad t>0\label{s1}.
\end{equation}

\section{Convergence of the single-variable dynamics}
The convergence properties of the dynamics can now be obtained from the covariance
using the equality
\begin{align}
\Delta(t,s) & = \mathcal C(t,t)+\mathcal C(s,s)-2\mathcal C(t,s)\label{deltaC} \\
& = 2(1-\chi){\rm R}'-2\mathcal C(t,s) \label{npt}.
\end{align}
From these results, we are able to show that
\begin{align}
	&\Delta(t+1,s+1)<\Delta(t,s),\quad \forall t,s.\label{res1}
	\end{align}
This means that with increasing time, the distance between pairs of iterates
separated by a fixed amount of time strictly decreases. We thus get
\begin{equation}
\lim_{t,s\to \infty} \Delta(t,s)=0\label{conver}
\end{equation}
and the dynamics is convergent.  Let us denote the convergence rate of $\Delta(t,s)$ by
\begin{align}
\mu_{mf}\doteq\lim_{t,s\to \infty}\frac{\Delta(t+1,s+1)}{\Delta(t,s)}
\end{align}
where for convenience we introduce the abbreviation ``\emph{mf}'' in the notation to emphasize the thermodynamic memory-free property \eqref{memoryless}. We obtain
\begin{equation}
\mu_{mf}=1-\frac{1-\eta{\rm R}'}{1-\chi^2{\rm R}'}.\label{res2}
\end{equation}
Moreover, it turns out that $\ln\mu_{mf}$ gives the exponential decay rate:
\begin{equation}
\lim_{t\to \infty}\frac{1}{t}\ln \Delta(t,\infty)=\ln \mu_{mf}\label{res3}
\end{equation}
where $\Delta(t,\infty)\doteq \lim_{s\to\infty}\Delta(t,s)$. Notice that when the model parameters approach the AT-line, i.e. $\eta{\rm R'}=1$, the convergence becomes arbitrarily slow. The derivations of \eqref{res1}, \eqref{res2} and \eqref{res3} are located at Appendix~\ref{dres13}.

\section{Algorithmic considerations}\label{algorc}
So far we have assumed that the limiting spectral distribution of $\matr J$ is analytically known beforehand for computing the quantities ${\rm R}$ and ${\rm R}'$. In the sequel, we present a practical approach that bypasses this need.

Specifically, we first compute the spectral decomposition
\begin{equation}
\matr J=\matr O^\top\matr D\matr O.\label{sdcm}
\end{equation}
The Green function and its derivative are then replaced by their finite-size approximations as
\begin{align}
{\rm G}(z)&= \frac{1}{N} {\rm tr}((z{\bf I}-\matr D)^{-1}) \\
{\rm G}'(z)&= - \frac{1}{N}{\rm tr}((z{\bf I}-\matr D)^{-2}).
\end{align}
The necessary R-transform and its derivative
are then obtained by solving the  simple fixed-point equations
\begin{subequations}
	\begin{align}
		\lambda &={\rm R}+\frac{1}{\chi}\\
		{\rm R}&=\lambda-\frac{1}{{\rm G}(\lambda)}\\
		{\rm R}'&=\frac{1}{{\rm G}(\lambda)^2}+\frac{1}{{\rm G}'(\lambda)}\\
		\chi &= \mathbb E_u[{\rm Th}'(\sqrt{(1-{\rm G}(\lambda)){\rm R}'} u)].
	\end{align}
\end{subequations}
The matrix $\matr A$ is then computed as
\begin{equation}
\matr A=\frac{1}{\chi}\matr O^\top(\lambda{\bf I}-\matr D)^{-1}\matr O-\bf I.\label{resolvent}
\end{equation}
These steps have to be performed before the main iterations and have $O(N^3)$, i.e.
\emph{cubic} computational complexities. On the other hand, the iterations are just based on matrix vector multiplications and evaluations
of scalar nonlinear functions. These have only \emph{quadratic} computational complexity per iteration.
\section{Comparison with a previous algorithm in the cases of SK and Hopfield Models}
In a  previous paper \cite{Opper16}, we have introduced the SSM iterative algorithm for solving TAP equations for models with general rotation invariant coupling matrices. This algorithm and its corresponding DF analysis is based on the
coefficients of power series expansions related to R-transforms of the random matrix ensemble. This practically limits the applicability of the method to cases where such coefficients
are known analytically. Nevertheless, both the algorithm and its analysis become simple for the SK and Hopfield models. Hence, it is interesting to derive the corresponding convergence rates of SSM algorithms for these models which is what we show in the sequel.

As for the SK model we first note that the necessary R-transform quantities are given by
\begin{equation}
{\rm R}=\beta^2\chi ~~\text{and}~~ {{\rm R}'}=\beta^2
\end{equation}
where $\chi$ is the solution of \eqref{chi}. The SSM dynamics for the SK model is nothing but
Bolthausen's celebrated convergent dynamics \cite{Bolthausen}:
\begin{subequations}
	\label{amp_sk}
	\begin{align}
		\matr m(t)&={\rm Th}(\matr \rho(t-1))\\
		\matr \rho(t)&=\matr J\matr m(t)-{\rm R}\matr m(t-1)\label{psi_sk}.
	\end{align}
\end{subequations}
Here, $\matr m(0)\doteq \matr 0$ and $\matr \rho(0)\doteq\sqrt{(1-\chi){\rm R}'}\matr u$ where $\matr u$ is a vector of independent normal Gaussian random variables.

Second, for the Hopfield model, the R-transform and its derivative are (see (\ref{R1}))
\begin{equation}
{\rm R}=\frac{\beta^2\alpha_1\chi}{1-\beta\alpha_1\chi} ~~\text{and}~~ {{\rm R}'}=\frac{\beta^2\alpha_1}{(1-\beta\alpha_1\chi)^2}\label{hopfieldR}
\end{equation}
where $\chi$ is the solution of \eqref{chi}. In the spirit of Bolthausen's dynamics \eqref{amp_sk}, we introduce a SSM construction for the Hopfield model as
\begin{subequations}
	\label{amp}
	\begin{align}
	\matr m(t)&={\rm Th}(\matr \rho(t-1))\\
	\matr \phi(t)&=\matr J\matr m(t)-{\rm R}\matr m(t-1)\\
	\matr\rho(t)&=(1+\frac{\rm R}{\beta})\matr \phi(t)-\frac{\rm R}{\beta}\matr\rho(t-1) \label{psi_hop}
	\end{align}
\end{subequations}
where $\matr m(0)\doteq \matr 0$ and $\matr \rho(0)\doteq\sqrt{(1-\chi){\rm R}'}\matr u$.

Note that the variable $\matr \gamma(t)$ in the dynamics \eqref{dynamics}
has the same fixed points as the variables $\matr \rho(t)$ in the dynamics \eqref{amp_sk} and \eqref{amp} for the SK and Hopfield models, respectively. In order to analyze the convergence properties of the SSM algorithms \eqref{amp_sk} and \eqref{amp} in the thermodynamic limit we define
\begin{equation}
\Delta_{ssm}(t,s)\doteq \lim_{N\to\infty}\frac{1}{N} \mathbb E[\Vert \matr \rho(t)-\matr \rho(s) \Vert^2]. \label{deltaphi}
\end{equation}
For both the SK and Hopfield models, we have
\begin{equation}
	\Delta_{ssm}(t+1,s+1)<\Delta_{ssm}(t,s),\quad \forall t,s\label{delta_ssm}.
\end{equation}
Thus, $\Delta_{ssm}(t,s)$ converges to $0$ as $t,s\to\infty$. In particular, the rate of convergence is obtained as
\begin{align}
	\mu_{ssm} & \doteq \lim_{t,s\to \infty}\frac{\Delta_{ssm}(t+1,s+1)}{\Delta_{ssm}(t,s)}=\eta{\rm R}'\label{rho_con}.
\end{align}
The derivations of \eqref{delta_ssm} and \eqref{rho_con} are given in Appendix~\ref{amp_con}.

From a computational point of view, for the cases of the SK model and the Hopfield model, the SSM algorithms  \eqref{amp_sk}  and \eqref{amp} are more convenient than our new algorithm \eqref{dynamics}, because
 the latter requires the computation of a matrix inverse \eqref{def_matrixA} (before the iteration starts). Nevertheless, it is interesting to note from \eqref{res2} that
\begin{equation}
\mu_{mf}-\mu_{ssm}=\frac{{\chi^2{\rm R}'}(\eta{\rm R}'-1)}{1-\chi^2{\rm R}'}<0\label{iden}.
\end{equation}
Thus, in the thermodynamic limit the SSM algorithms for SK and Hopfield models have both a slower convergence compare to the algorithm \eqref{dynamics}.

\section{Simulation Results}
In this section, we compare our analytical results with simulations of the algorithm
for large random matrices.
We note from \eqref{deltaC} that the dynamics \eqref{dynamics} in the thermodynamic limit
is described in terms of the limiting "covariance" expression
\begin{equation}
\mathcal C(t,s)=\lim_{N\to\infty}\frac{1}{N} \mathbb E[\matr \gamma(t)^\top \matr \gamma(s)]. \label{covdef}
\end{equation}
whose explicit analytical expression is given in \eqref{covfield}. In Figure~\ref{fig1}
\begin{figure}
	\epsfig{file=figure1,width=1.04\columnwidth}
	\caption{Discrepancy between theory and simulations for the
	covariance in case of a $2$-layer Hopfield model with $N/N_1=1$ and $N/N_2=2$, $N=10^{4}$ and $\beta=0.25$. }\label{fig1}
\end{figure}
we illustrate how well $\mathcal C(t, s)$ predicts $\frac{1}{N}\matr \gamma(t)^\top \matr \gamma(s)$ for a $2$-layer Hopfield model with a single realization of the dynamics. Note that we assume the large-system limit $N\to\infty$ is taken before the long-time limit $t\to\infty$. Thus, we get excellent agreement between theoretical predictions and simulations on single instances for finite-time properties of large systems. The discrepancy between theory and simulations for large times increases
as the model parameters approach the AT line. This can be seen in Figure~\ref{fig2},
\begin{figure}
	\epsfig{file=figure2,width=1.03\columnwidth}
	\caption{Asymptotics of the algorithm for $2$-layer Hopfield model with $N/N_1=1$ and $N/N_2=2$, $N=10^{4}$ and $h=1$. The inverse temperature $\beta=0.35$ gives the AT line. The flat lines around $10^{-30}$ are the consequence of the machine precision of the computer which was used. }\label{fig2}
\end{figure}
close to the AT line (i.e. $\beta=2.29$) where \eqref{res2} fails to predict the exponential decay for a single-realization of dynamics. In fact the dynamics has started to lose its self-averaging properties in this regime, i.e. the dynamics for different realisations of the random matrix and random
initial conditions yield remarkably different asymptotic convergence behavior, see also in Figure~\ref{fig4}. In Figure~\ref{fig1} and Figure \ref{fig2}, we compute the necessary quantities $\rm R$ and $\rm R'$ by the algorithmic approach in Section~\ref{algorc}.

Figure~\ref{fig3} refers to the SK and (1-layer) Hopfield models
\begin{figure}
	\epsfig{file=figure3,width=1.03\columnwidth}
	\caption{Comparison of the convergence rates with SSM algorithms for the SK and Hopfield models. $N=10^4$.}\label{fig3}
\end{figure}
where we compare the convergence rates of the dynamics \eqref{dynamics} with the SSM dynamics for a single-realization. The simulation results confirm the thermodynamic bound \eqref{iden}. However, as we illustrate in Figure~\ref{fig4} (for the Hopfield model)
\begin{figure}
	\epsfig{file=figure4,width=1.03\columnwidth}
	\caption{Asymptotics of the algorithm for Hopfield model with $h=1$, $\beta=0.75$ and $N=10^4$. We show the results of two different realizations (reds and blues) for the random
	matrix and initial conditions in the dynamics \eqref{dynamics} and \eqref{amp}. The inverse temperature $\beta=1.08$ gives the AT-line. }\label{fig4}
\end{figure}
that when the model parameters approach the AT line, after few iteration steps the dynamics \eqref{dynamics} shows strong fluctuations with respect to different realizations of
while the SSM dynamics shows negligible fluctuations.

So far we have shown examples of the performance of the algorithm on matrices
drawn at random from spherical invariant distributions. On the other hand, all necessary
parameters (R-transform and its derivative) of the algorithm can be computed from a given matrix without explicit knowledge of the distribution. The same parameters appear in the analytical
results of the theory and depend only on the (limiting) spectrum of the matrix.  What happens if we run the algorithm on a matrix which is not a typical realization from a spherical invariant random matrix ensemble? To get a first impression on the robustness of our results we  compare the theoretical results (obtained from the spectrum) with simulations based on a model
with \emph{randomly-signed Hadamard} matrices. To understand this model we first consider the
(spherical symmetric) random orthogonal model discussed by Parisi and Potters \cite{Parisi} (i.e. the 1-layer random-orthogonal model with $\rho_0=1$). The coupling matrix can be written in the form
\begin{equation}
\matr J=\beta\matr O^\top\matr D_{\rho}\matr O
\end{equation}
where $\matr O$ is a Haar matrix and $\matr D_{\rho}={\rm diag}(d_1,\cdots,d_N)$ has
random binary elements $d_i=\mp 1$ with $\vert\{d_i=1\}\vert=\rho N$.
Motivated by a recent study \cite{Greg} in random matrix theory we now introduce a non-rotation invariant coupling matrix ensemble ---  called \emph{randomly-signed Hadamard}
by replacing the random Haar matrix by a different random orthogonal matrix $\tilde{\matr O}$:
\begin{equation}
\matr J=\beta\tilde{\matr O}^\top\matr D_{\rho}\tilde{\matr O}~~ \text{with}~~\tilde{\matr O}\doteq \frac{1}{\sqrt{N}}\matr H_{N}\matr Z. \label{rsh}
\end{equation}
Here, $\matr Z$ is an $N\times N$ diagonal matrix whose diagonal entries are independent and composed of binary $\mp1$ random variables with equal probabilities and $\matr H_N$ is the $N\times N$ Hadamard matrix which is {\em deterministically} constructed \cite{hadamard1893resolution}
from the recursion
\begin{align}
\matr H_{2^k}=\left[\begin{array}{cc}
\matr H_{2^{k-1}}& \matr H_{2^{k-1}}\\
\matr H_{2^{k-1}}&-\matr H_{2^{k-1}}
\end{array}\right], ~~ k\geq 1
\end{align}
with $\matr H_{1}\doteq1$. Note that $\tilde{\matr O}$ is an orthogonal matrix with the {\em binary entries} $\tilde O_{ij}=\mp \frac{1}{\sqrt{N}}$. Both coupling matrices have the same spectra and thus the same
R-transform which can be easily obtained as
\begin{equation}
{\rm R}(\omega)=\frac{(\beta \omega-\rho)+\sqrt{(\beta \omega-\rho)^2+4\rho^2\beta\omega}}{2\rho\omega}-\beta.
\end{equation}
In Figure~\ref{fig5}~and Figure~\ref{fig6}
\begin{figure}
	\epsfig{file=figure5,width=1.03\columnwidth}
	\caption{Discrepancy between theory and simulations for the
	covariance in case of randomly-signed Hadamard model with $N=2^{13}$, $h=2$, $\alpha=1/2$ and $\beta=4$.}\label{fig5}
\end{figure}
\begin{figure}
\epsfig{file=figure6,width=1.03\columnwidth}
\caption{Asymptotics of the algorithm for randomly-signed Hadamard model with $N=2^{13}$, $h=2$, $\alpha=1/2$. The inverse temperature $\beta=6.8$ gives the AT line.}\label{fig6}
\end{figure}
we illustrate how well our theoretical results describe the dynamics \eqref{dynamics} when the coupling matrix $\matr J$ is modeled by a randomly-signed Hadamard ensemble. The simulation results suggest that our theoretical results can be valid for larger classes of coupling matrices than the rotation invariant matrix ensembles.
\section{Summary and Outlook}
In this paper we have introduced an iterative algorithm for solving the TAP equations of Ising models with arbitrary rotation invariant coupling matrices. We have shown that the algorithm is convergent in the thermodynamic limit when the AT-line criteria is fulfilled. We have also obtained a compact analytical expression for the rate of convergence. The analytical results are supported by numerical simulations of large systems on single instances of random matrices. Preliminary simulations for random matrices which are not generated from rotation invariant ensembles but show similar weak dependencies between matrix elements indicate that our theory may extend to larger classes of matrices. Preliminary analytical calculations suggest that some of our results on convergence can be derived under the weaker property of {\em asymptotic freeness} \cite{mingo2017free} (which is fulfilled  by rotation invariant matrix ensembles). Such extensions of our approach would also be important for applications to TAP-style mean field equations
for statistical inference in probabilistic models. A first step in this direction would be to consider Ising models with non-constant or random external fields. Such problem setting plays a role when  the probability distribution of the spins is conditioned on observed spin variables. This is a relevant problem in machine learning e.g. in the training  of restricted Boltzmann machines~\cite{Tramel18}. More complex extensions would be to the important case of models with continuous  random variables. Of course, the latter case
would require new and nontrivial generalizations of our basic algorithm. These generalizations
might be motivated e.g. by the so-called {\em expectation propagation algorithms} \cite{Minka1,OW5} which are
frequently used in machine learning.  A different direction of research is in the optimization of algorithms. It will be interesting to investigate which properties makes our new method converge faster than the previously defined SSM algorithms. On the other hand it will also be important to
get a better understanding of finite size fluctuations, because these will influence the robustness of convergence, when the dynamics is close to the AT instability.

\section*{Acknowledgment}
We are grateful to funding from the BMBF (German ministry of education and research) joint project {\bf01 IS 18037 A} : BZML- Berlin Center for Machine Learning.

redpajama_set_name: RedPajamaArXiv


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Q: Underline to specific text - FPDF I have a table created using FPDF. My problem is how to add an underline to specific text?
For example,
Name | Gender | Age
Dame | Female | 20
I want the "Name, Gender & Age" is underlined
This is my code so far
$p->Cell(20,5,'Name','LTR',0,'C',0);
$p->Cell(40,5,'Gender','TR',0,'C',0);
$p->Cell(40,5,'Age','TR',0,'C',0);

while(retrieve from DB)

Thanks in advance

A: Use SetFont for underlines. You can leave the font family blank to keep it the same.
$p->SetFont('','U');
$p->Cell(20,5,'Name','LTR',0,'C',0);