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Canada | Brussels, 20 September 2017
EU-Canada trade agreement enters into force
On Thursday, 21 September, the Comprehensive Economic and Trade Agreement (CETA) between the EU and Canada enters into force provisionally.
Welcoming this milestone in the EU's trade policy, President of the European Commission Jean-Claude Juncker said: "This agreement encapsulates what we want our trade policy to be - an instrument for growth that benefits European companies and citizens, but also a tool to project our values, harness globalisation and shape global trade rules. This trade deal has been subject to an in-depth parliamentary scrutiny which reflects the increased interest of citizens in trade policy. The intense exchanges on CETA throughout this process are testimony to the democratic nature of European decision making and I expect Member States to conduct an inclusive and thorough discussion in the context of the ongoing national ratification processes of the agreement. Now it's time for our companies and citizens to make the most out of this opportunity and for everyone to see how our trade policy can produce tangible benefits for everyone".
Commissioner for Trade Cecilia Malmström said: "Things are about to change for our exporters. The provisional entry into force allows EU companies and citizens to start reaping the benefits of this agreement right away. This is a positive signal for the global economy, with the potential to boost economic growth and create jobs. CETA is a modern and progressive agreement, underlining our commitment to free and fair trade based on values. It helps us shape globalisation and the rules that govern global commerce. Moreover, CETA underlines our strong commitment to sustainable development and protects the ability of our governments to regulate in the public interest. This agreement also vastly strengthens our relationship with Canada, a strategic partner and ally with whom we have deep historical and cultural ties."
The provisional application of CETA on 21 September follows its approval by EU Member States, expressed in the Council, and by the European Parliament.
It will only enter into force fully and definitively, however, when all EU Member States have ratified the Agreement. The Commission will work with EU Member States and Canada to ensure its smooth and effective implementation.
What will CETA do?
CETA offers new opportunities for EU businesses of all sizes to export to Canada. It will save EU businesses €590 million a year – the amount they pay in tariffs on goods exported to Canada. As of 21 September CETA removes duties on 98% of products (tariff lines) that the EU trades with Canada. It also gives EU companies the best access ever offered to companies from outside Canada to bid on the country's public procurement contracts - not just at the federal level but at provincial and municipal levels, too.
The agreement will especially benefit smaller companies who can least afford the cost of the red tape involved in exporting to Canada. Small businesses will save time and money, for example, by avoiding duplicative product testing requirements, lengthy customs procedures and costly legal fees. Member States' authorities dealing with export promotion stand ready to help businesses to start exporting overseas, boost existing trade, and attract investment.
CETA will create new opportunities for European farmers and food producers, while fully protecting the EU's sensitive sectors. The EU has further opened its market for certain competing Canadian products in a limited and calibrated way, while securing improved access to the Canadian market for important European export products. Those include cheese, wine and spirits, fruit and vegetables, and processed products. CETA will also protect 143 EU "geographical indications" in Canada, high quality regional food and drink products.
The EU's 500 million consumers will also benefit from CETA. The agreement offers greater choice while upholding European standards, as only products and services that fully respect all EU regulations will be able to enter the EU market. CETA will not change the way the EU regulates food safety, including genetically modified products or the ban on hormone-treated beef.
The agreement also offers better legal certainty in the service economy, greater mobility for company employees, and a framework to enable the mutual recognition of professional qualifications, from architects to crane operators.
Moreover, EU Member States can continue to organise public services as they wish. A Joint Interpretative Instrument, which will have legal force, has further clarified this and other issues. It clearly and unambiguously outlines what Canada and the EU have agreed in a number of CETA articles.
Procedure and next steps
The EU and Canada signed CETA on 30 October 2016, following the EU Member States' approval expressed in the Council. On 15 February the European Parliament gave also its consent. On 16 May 2017 the Canadian side ratified CETA. This paved the way for provisional application as soon as Canada adopted all the necessary implementing rules.
CETA will be fully implemented once all EU Member States ratify the deal according to their respective constitutional requirements. At the time CETA will take full effect, a new and improved Investment Court System will replace the current investor-state dispute settlement (ISDS) mechanism that exists in many bilateral trade agreements negotiated in the past by EU Member States' governments. The new mechanism will be transparent and not based on ad hoc tribunals.
The framework for the EU-Canada relationship is set out in the Strategic Partnership Agreement (SPA), which allows for reinforced cooperation in strategic areas of shared interest and responsibility such as climate, security and foreign and security policy. The Strategic Partnership Agreement has been provisionally applied since 1 April 2017 and, together with CETA, enables an even further deepening of EU-Canada relations.
The EU's free trade agreements have been proven to spur European growth and jobs. One example is the EU-South Korea trade deal. Since it entered into force in 2011, EU exports to South Korea have increased by more than 55%, exports of certain agricultural products have risen by 70%, EU car sales in South Korea have tripled and the trade deficit turned into a surplus. This agreement was also applied provisionally for several years following EU-level ratification, pending its ratification by all EU Member States.
31 million jobs in Europe depend on exports. On average, each additional €1 billion of exports supports 14 000 jobs in the EU.
Factsheet: CETA overview
Factsheet: Shaping globalisation
Factsheet: The EU's interests - 6 guarantees
Factsheet: Sustainable development
Brochure: the benefits of CETA
Guide to CETA for businesses
Factsheet: Benefits for agriculture
CETA impact on Austria, Belgium, Bulgaria, Croatia, Cyprus, Czech Republic, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Netherlands, Poland, Portugal, Romania, Slovakia, Slovenia, Spain, Sweden, United Kingdom
CETA in your town - interactive map with information about the trade with Canada in local communities all around Europe
Factsheet: In figures: EU-Canada trade
Factsheet: The negotiating process
Ratifications by Member States
The full text of CETA
Meet the CETA champions: Learn why these small EU businesses are taking advantage of CETA
Last update: 20 Sep 2017 | Top of page


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Restaurants concerned Peach Bowl will impede on New Year's Eve business
by: Karina Cheung
COLUMBUS, Ohio (WCMH) — Restaurants owners across Columbus are hoping New Year's Eve will bring in big business as the Ohio State Buckeyes face off against the Georgia Bulldogs in Saturday's Peach Bowl.
Co-owner of Wolf's Ridge Brewing, Bob Szuter, said OSU game day during the regular season can be a tough day. On Saturday, the Peach Bowl is starting at 8 p.m., tight in the middle of prime New Year's Eve dining hours.
"Every Saturday for every restaurant across the city, it's tough during Ohio State football seasons," Szuter said.
What Ohio's midnight sports betting launch will look like at Columbus casinos
Still, Szuter said the restaurant is booked until 9 p.m. on Saturday for reservations.
"Ohio State in general brings a lot to the central Ohio community, so we're not too upset about it, just wished they played at [1 p.m.] on New Year's Eve instead of the prime-time hour there for restaurants," he said.
The President and CEO of the Ohio Restaurant Association, John Barker, is predicting many will plan around the big game.
"I think what you're going to see is that people who are really into football may not be going out to fine dining, or if they do, they're going to be back home or somewhere to be around a TV about 8 o'clock," Barker said.
Barker said casual restaurants usually see light traffic on New Year's Eve. He expects to see business split between fine dining, sports bars or both.
Best restaurant in Ohio visited by Guy Fieri is in CLE: report
"Big Bowl game like that, lots of time outs and big halftime show and all that, that game will probably run past 11 o'clock," Barker said. "So, that could be good for business as well if people are out enjoying themselves, so we're hoping for the best obviously and we're optimistic."
The last few New Year's Eves have been tough with the COVID-19 pandemic, then the wave of omicron, according to Szuter, but he said this year is feeling more normal.
"At the end of the day, we're looking pretty good. We're pretty full — almost completely full — for New Year's Eve right now," Szuter said. "There still are a few spaces available, but you know, compared to where we were last year, this is positive."

redpajama_set_name: RedPajamaCommonCrawl


James Bay has revealed his plans to release an illustrated book for his fans.
The singer has been busy sketching whilst on the road and is in the process of putting together the drawings for a comic based on the songs from his debut LP 'The Chaos and the Calm'.
Speaking to the Daily Star newspaper, he revealed: "I do still draw, in terms of a book, it terms of a book it's something I would never rule out because I do still sketch.
"When I had the idea for the 'Pink Lemonade' video I put out recently I had no other way to get it across to the director other than to draw the scenes.
"It definitely takes it to a new place. I've got a new look, a new sound. I just fancied a new chapter. There are moments on the album that sort of 'bridge' from 'The Chaos And The Calm' to the new album.
This Spring, he's due to perform in London at the Brixton Electric before embarking on a North American tour. He also has a handful of festival dates coming up including Isle of Wight, Glasgow Green, Unaltrofestival and We The Fest.
The 27-year-old singer has ditched his traditional guitar sound on his second record 'Electric Light – which is released – and he says it's because he needed to challenge himself.
He explained: "On this new record I made an effort to avoid that guitar because then it got harder.
The 'Hold Back The River' singer previously admitted he was inspired by Ed Sheeran and Taylor Swift to reinvent himself.
James has not only gone in a more pop-oriented direction, but he also chopped his long hair short and lost his Fedora hat for his transformation.
It's been three years since he dropped his Grammy nominated debut album 'Chaos and the Calm' and won the Brits Critics' Choice Award. Now he's change his aesthetic into something a little more contemporary, determined to break the rules of the pop industry. Details of his next album are yet to be announced.
He said: "Ed and Taylor have almost treated each release like they're trying to break themselves again as an artist on the world's stage.


for you from initial sketches to the final construction.
CGR MINDS is a company specially formed for providing the structured education in TOC (Theory Of Constraints", under the special agreement signed with TOCICO (Theory Of Constraints Internationals Certification Organization) – USA.
Students of other graduate courses.
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text:
Home > India's Authoritarian Streak
Wednesday, May 30, 2018 - 12:00am
India's Authoritarian Streak
What Modi Risks With His Divisive Populism
Manjari Chatterjee Miller
MANJARI CHATTERJEE MILLER is Associate Professor of International Relations at the Frederick S. Pardee School of Global Studies, Boston University. She is the author of Wronged by Empire: Post-Imperial Ideology and Foreign Policy in India and China (Stanford University Press, 2013).
Since the election of Narendra Modi as prime minister in 2014, India has grown deeply divided along ideological and religious lines. Under his watch, Modi's ruling Hindu nationalist Bharatiya Janata Party (BJP) has issued populist policies that harm religious minorities and dissenters, and its officials have largely remained silent about violence against such groups. In some instances, they have even gone so far as to laud such behavior.
In August 2015, Malleshappa Kalburgi, an Indian scholar who criticized Hindu idolatry, was murdered by Hindu right-wing groups. His death was later celebrated on Twitter by activists from Bajrang Dal, another right-wing Hindu group whose members are strong supporters of Modi. Later that year, a 50-year-old Muslim man, Mohammad Akhlaq, was killed by a Hindu mob for allegedly committing the "sin" of eating beef [1]. After the incident, Mahesh Sharma, a minister in the Modi government, insisted that the murder of Akhlaq was merely "the result of a misunderstanding." In 2017, after the murder of the journalist Gauri Lankesh, a prominent critic of Hindu nationalist groups, the BJP government accused opposition leaders of trying to "make political capital" out of the situation. In all of these cases, the prime minister failed to issue a statement of condolence or condemnation, leading the well-known Indian actor Prakash Raj to condemn Modi's silence as "chilling." This year, after the horrific rape of an eight-year-old Muslim girl by Hindu men who were determined to drive her community from their village, Modi only issued a condemnation after facing mounting pressure from opposition groups.
What is more, the government and Hindu nationalist activists now routinely suppress dissent and label protesters as "anti-national." [2] As a prominent example, in 2016, Kanhaiya Kumar, the president of the student union at Jawaharlal Nehru University in New Delhi, was arrested for "sedition" and accused of being "anti-national" after he and his fellow students engaged in a peaceful (albeit controversial) protest against the government's 2013 execution of Afzal Guru, a terrorist trained in Pakistan who helped carry out
Source URL: https://www.foreignaffairs.com/articles/india/2018-05-30/indias-authoritarian-streak
[1] https://www.foreignaffairs.com/articles/india/2015-07-01/indias-beef-beef
[2] https://www.foreignaffairs.com/articles/india/2016-03-01/speak-no-evil


The works on this project comprise of the part demolition of existing classrooms, new extensions and internal remodelling to classrooms and administration areas.
The works were programmed to fit around school operations in that the demolition and refurbishment was carried out during the school holidays where there were no pupils or staff in close vicinity. The new build works commenced once the school was operational, but allowed us to clearly segregate our site from the live school environment.
Stakeholder engagement throughout the project was key to its success. During the pre-construction stage, we worked collaboratively with the school and design team to finalise the design and create significant savings as the project was initially over budget. We ensured the client got the desired quality on the scheme, within their tight budget.
Working closely with the school with regards to added value engagement was also integral to the schemes success. We attended parents evening, carried out school assemblies to educate the pupils on health and safety; had a viewing platform for the pupils to keep an eye on site progress; supported the school in raising the profile of the construction works.


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Quiz: How Much Do You Know About Jimmy Carter?: HowStuffWorks
How Much Do You Know About Jimmy Carter?
Which of these American states is Jimmy Carter's birthplace?
On October 1, 1924, Jimmy Carter was born at the same Plains, Georgia, medical center where his mother worked as a nurse. He was the first U.S. president in history to be born in a hospital.
Paul Souders/DigitalVision/Getty Images
Jimmy Carter served a seven-year term in which branch of the military?
After earning a bachelor of science degree, Jimmy Carter enlisted in the Navy as a commissioned officer and served for seven years. He was the fifth president in a row to be a member of the branch.
Wiki Commons by National Archives and Records Administration
Rosalynn and Jimmy Carter have how many children?
Married since 1946, Rosalynn and Jimmy Carter are the parents of four children, although only Amy lived with them in the White House. Now in their 90s, the Carters enjoy spending time with their grandchildren and great-grandchildren.
Wiki Commons by BruiserBrody10
Any idea what President Carter's middle name might be?
James Earl Carter Sr. and Bessie Lillian Gordy Carter decided to keep the name in the family when Jimmy Carter was born. Known to the world as Jimmy Carter, President Carter's full name is James Earl Carter Jr.
The former president is often seen lending a hand to which of these charities?
Susan G. Komen Foundation
Jimmy Carter is one of the most humble presidents to have served the United States, and his humanitarian work continues today. Over the years, it has not been uncommon to see both him and his wife Rosalynn building homes for Habitat for Humanity.
UVA Center for Politics
Did Jimmy Carter run as a Republican or a Democratic candidate?
Jimmy Carter has always been known as a progressive politician. During his 1976 run for the White House, he maintained the same party affiliation as he did when he served as governor — Democrat.
Wiki Commons by Clawson
In what year did Jimmy Carter become the 39th president of the United States?
With 297 electoral votes, Jimmy Carter was elected by the American people on November 2, 1976. He did not officially take office until after the presidential inauguration on January 20, 1977.
The Carter Center via YouTube
Before becoming president, which job did Jimmy Carter hold?
United States Secretary of State
Governor of Georgia
Navy Admiral
Even as a gubernatorial candidate, Jimmy Carter was known for his liberal beliefs. He became the 76th governor of Georgia in 1971 and served right up until the time that he ran for the presidential office.
Wiki Commons by Greta Kempton
Which former U.S. president does Jimmy Carter say he admires the most?
Over and over again, President Carter has spoken of his admiration for the 33rd president of the United States, Harry S. Truman. Carter tells interviewers that he has great admiration for Truman because he never attempted to profit from the presidency.
Do you know which sport President Carter played in high school?
Many United States presidents have been the athletic type, but few of them were as active as Jimmy Carter was in high school. During his adolescent years, he played basketball, football and tennis.
chuckcollier/E+/Getty Images
When Jimmy Carter was five, he stole something. What was it?
A piece of candy
A penny
President Carter attributes his sense of honesty to the time he stole a penny from the church offering plate during his childhood. Carter says his father swiftly punished him, and he vowed never to steal anything again.
Iptv/PBS
Jimmy Carter shocked the nation when he interviewed with which magazine during the '70s?
In an effort to upset the reelection of Gerald Ford, Jimmy Carter's campaign team turned to the aid of media. During a 1976 interview with Playboy, Carter admitted that he had committed lust in his mind. Perhaps, his honesty earned a little trust with voters.
qbanczyk/E+/Getty Images
While living in his hometown, before he entered politics, Carter ran into trouble when he refused to join a local group. What was it called?
Citizens' Agricultural Council
The Georgia Citizens' Ensemble
Citizens for Climate Change
When the White Citizens' Council came knocking at Jimmy Carter's door, the future humanitarian president turned them and their white nationalist philosophies away. He was then harassed with violent threats, but he didn't let that sway him. In fact, it motivated him to run for the Georgia senate.
gremlin/E+/Getty Images
Is it true or false that Jimmy Carter claims to have seen a UFO?
It's true! Prior to becoming involved in politics, Jimmy Carter swears that he saw a UFO about thirty miles from his home. He vowed to release all the government's extra-terrestrial records while he was in office, but enthusiasts are still waiting.
Which of the government's vehicles did President Carter sell when he took office?
Presidential yacht
Camp David's golf carts
With his hero Harry S. Truman in mind, Jimmy Carter wanted to show the country that he was not in office to make a profit. In fact, one of the first things he did was sell the presidential yacht. He said that it was too excessive.
vip2014/Moment Open/Getty Images
What crop did the Carter family grow on their farm in Plains, Georgia?
The Jimmy Carter boyhood farm has been in the Carter family since 1928. Now a historic visitation center, the farm can be toured. Although it no longer produces peanuts, they were the farm's main crop when Carter was growing up.
Wiki Commons by Adam Jones
Jimmy Carter was awarded the Nobel Peace Prize in which year?
Out of all the presidents who have received the Nobel Peace Prize, Jimmy Carter had to wait the longest. In 2002, he was honored for his efforts to broker international peace and for his humanitarian efforts.
Wiki Commons by White House photographer. Ronald Reagan Presidential Library
Which of these U.S. presidents defeated Jimmy Carter during his bid for reelection?
Unfortunately, Jimmy Carter did not win his bid for reelection in 1980. With the Iran Hostage Crisis looming over him, he was considered a little too controversial for some. He lost the race to Ronald Reagan.
Wiki Commons by Gray wolf
Can you name the foundation the Carter family started in 1982?
Muscular Dystrophy Association
The Carter Center
Two years after leaving office in 1980, President Carter and his wife Rosalynn founded a not-for-profit charity named the Carter Center. The Carter Center operates in more than 80 countries.
chinaface/E+/Getty Images
Former President Carter has been seen purchasing his clothes at what retail store?
When a Dollar General opened in his hometown of Plains, Georgia, Jimmy Carter attended the ceremony. He is frequently seen there, purchasing his clothing and other household goods.
Can you figure out which of these books was not authored by Jimmy Carter?
"Why Not the Best?"
"White House Diary"
"An Hour Before Daylight"
"Life in Georgia"
Since leaving office, President Carter has authored over 30 books. He's never written a book called "Life in Georgia," though. Never using his prestigious former title for profit, he lives on the profits from the book and the stipend he receives from serving as commander in chief.
Wiki Commons by White House photo
Which country did President Carter negotiate two treaties with, during his first year in office?
Known for his presidential negotiation skills and his humanitarian leanings, President Carter stepped into a sticky situation in Panama. He managed to broker two deals with the country before he passed the reins to President Reagan.
Emory University via YouTube
Where did Jimmy Carter once spend time teaching as a professor?
Along with starting the Carter Center in 1982, Jimmy Carter became a member of the staff at Emory University. He taught there for 37 years before finally receiving tenure in 2019.
When Jimmy Carter blew out the candles on his 2019 birthday cake, how old did he turn?
When he blew out the candles on his 2019 birthday cake, President Carter turned 95. He holds the record as being the oldest former President of the United States, and he is extending that record daily. The next longest-lived is George H.W. Bush, who passed away at the age of 94 years, 171 days.
In November 2019, what sort of operation did Carter undergo?
When President Carter was diagnosed with cancer, he flew through the treatments and into complete remission. A 2019 surgery to remove a subdural hematoma from his brain had everyone waiting on pins and needles, but he enjoyed a full recovery.
How much do you think the current Carter family home is worth?
Former presidents like Clinton and Obama live in large, opulent homes that cost millions — not President Carter. The current Carter home in Plains, Georgia, is one he built himself in 1961, and it is valued at $167,000.
shulz/E+/Getty Images
When the former president cashes his pension check, how much does he receive?
$1.1 million per year
$543,000 per year
Former presidents are paid an annual stipend of $210,700 per year, and it's only a drop in the bucket for many. Carter prefers to live modestly and takes in most of his income from the sale of his books.
How has Jimmy Carter spent his Sundays since leaving the White House?
Coaching a local team
Teaching Sunday school
Tending to crops
Volunteering at a soup kitchen
Frequently on Sunday — but not every week — the former president teaches Sunday school at the Plains, Georgia, Maranatha Baptist Church. If you want to attend, you will have to get there early. No advance reservations are allowed, and a line forms around the block.
What condition was President Carter diagnosed with in 2015?
In 2015, President Jimmy Carter was diagnosed with metastatic melanoma that spread to his liver and his brain. After receiving immunotherapy, chemotherapy and radiation, he announced that he was cancer-free in 2016.
Wiki Commons by Dsdugan
How many visitors come through the doors of the Jimmy Carter National Historic Site annually?
It's often been said that the whole town of Plains, Georgia, is a tribute to its favorite former president. Every year, more than 70,000 guests contribute $4 million in revenue by visiting his birthplace.
Wiki Commons by Michael A. Schwarz
What food do Jimmy and Rosalynn Carter insist on making themselves?
Ever the humble couple, the Carters refuse to buy yogurt at the supermarket — they make it themselves! They are not big spenders with a vast wine cellar, either. They prefer to bring home the budget wine and enjoy it with their neighbors.
Wiki Commons by Mark Harrell
In Jimmy Carter's hometown, there's a statue honoring him. What kind of statue is it?
A bronze bust
A smiling peanut
A large medal
An oversized hammer
Right in the middle of Plains, Georgia, there's a big statue that pays tribute to the former president. A giant peanut with Jimmy Carter's iconic smile greets visitors as they head to the Jimmy Carter National Historic Site.
President Carter's 1979 "crisis of confidence" speech was sometimes known as what?
The "nice guy" speech
The "malaise" speech
President Peanut speech
President Farmer speech
Following Carter's "crisis of confidence" speech, many opposing lawmakers dubbed it the "malaise" speech because of his laid back stance on the Iran Hostage Crisis. Contrary to popular opinion, he never uttered the word "malaise" during this speech on July 15, 1979.
Wiki Commons by White House Staff Photographers
Who served as vice president while Jimmy Carter was in office?
Walter Frederick "Fritz" Mondale was always President Carter's choice for a running mate. The former diplomat proudly served as vice president from 1977 to 1981, before the pair lost to Ronald Reagan and George H.W. Bush.
What is often cited as the reason for President Carter's loss at the polls in 1980?
Panamanian Crisis
Canadian Citizens' Crisis
Right during election time, President Carter had more on his plate than campaigning for votes. His standoff with Iranian officials caused 52 Americans to be held hostage for 444 days. Many claim that this situation cost him the bid for reelection.
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Image: ABC
James Earl Carter Jr. began serving as president in 1977, but his legacy has far outlasted his time in office. From his tenure at a university to the library of books he has authored, there's a lot more to the man than simply being an ex-president. Throughout this quiz, you'll get the chance to tell us all you know about him. Will you get all of the questions right, though?
Alongside his wife Rosalynn, Jimmy Carter prefers to live a simple life. He buys his clothes at discount stores, makes his own yogurt and spends much of his spare time building homes for the needy through Habitat for Humanity. A man of honor, Carter swore to be like his presidential idol, Harry S. Truman, and he's never sought to make a profit from serving the nation. In fact, he once sold a governmental vehicle because he thought it was too excessive.
He's hailed by some as the greatest president the United States has ever seen. He's certainly one of the most relatable and fascinating! As you go through the questions about him, you'll expand your presidential knowledge even more. You'll also walk away with a smile and a new-found admiration for him. Give it your best shot, and see how many facts about the 39th president you know. How well will you do?


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Q: SQL to function in Codeigniter returns row I'm trying to get an entire row values from a new SQL query so.
How can I make a function
Select * from table1 t1,table2 t2,table3 t3 where t1.t1_id=t2.t1_id and t3.id=t2.t3_id

to something like this :
In model page
public function getID_researcher($lastname){
$query = $this->db->get_where('researcher', array('lastname' => $lastname));
return $query->row_array();
}

I need to return one row result base on the lastname which is from table1
To clarify here, 'researcher' is the table where it gets data but what I want is the new SQL for me to get the data.
I tried this one but still error.
public function getID_researcher($lastname){
$sql = = $this->db->query('Select * from table1 t1,table2 t2,table3 t3 where t1.t1_id=t2.t1_id and t3.id=t2.t3_id');

$query = $this->db->get_where($sql, array('lastname' => $lastname));
return $query->row_array();
}

A: to use this query
Select * from table1 t1,table2 t2,table3 t3 where t1.t1_id=t2.t1_id and t3.id=t2.t3_id

you can try to use active record db 'join' like this:
$this->db->from('table1 t1');
$this->db->join('table2 t2', 't1.t1_id = t2.t1_id', 'left');
$this->db->join('table3 t3', 't3.id = t2.t3_id', 'left');
$this->db->where('lastname', $lastname);
$query = $this->db->get();
$c = 0;
foreach($query->result() as $q){
  $result[$c]['name'] = $q->lastname;
  // put move variable here
$c++;
}
return $result;

hope this help

redpajama_set_name: RedPajamaStackExchange


text:
Q: Convert NSString separated by comma to NSArray
Possible Duplicate:
Convert NSArray to NSString in Objective-C

I have an array of an NSArray myArray, here is the content:
 (
            "<MPConcreteMediaItem: 0x1b7dd0> 8671085923283003246",
            "<MPConcreteMediaItem: 0x1b7e50> 16275751483823231324",
            "<MPConcreteMediaItem: 0x1b7e70> 4976771456698615367"
 )

I used the code below to assign to an NSString myString:
NSString *myString = [myArray description];

And the out put is still as expected:
 (
            "<MPConcreteMediaItem: 0x1b7dd0> 8671085923283003246",
            "<MPConcreteMediaItem: 0x1b7e50> 16275751483823231324",
            "<MPConcreteMediaItem: 0x1b7e70> 4976771456698615367"
 )

Now, say I want to convert myString back to an array named newArray, I used this:
NSArray *newArray = [[NSArray alloc]init];
newArray = [myString componentsSeparatedByString:@","];

But the content of the newArray is now different:
(
"(
    \"<MPConcreteMediaItem: 0x1b7dd0> 8671085923283003246\"",
        "
    \"<MPConcreteMediaItem: 0x1b7e50> 16275751483823231324\"",
        "
    \"<MPConcreteMediaItem: 0x1b7e70> 4976771456698615367\""
)"
)

Any idea what I need to do to fix this?

A: There are two methods

*

*- (NSArray *)componentsSeparatedByString:(NSString *)separator

*- (NSString *)componentsJoinedByString:(NSString *)separator
Use
NSString *myString = [myArray componentsJoinedByString:@","]; //instead of [myArray description];

A: This is what the NSArray prints out.
"(
\"<MPConcreteMediaItem: 0x1b7dd0> 8671085923283003246\"",
    "
\"<MPConcreteMediaItem: 0x1b7e50> 16275751483823231324\"",
    "
\"<MPConcreteMediaItem: 0x1b7e70> 4976771456698615367\""
)"

To print out the individual elements in the array try this and see what happens
NSString *myString1 = [myArray objectAtIndex:1];
NSString *myString2 = [myArray objectAtIndex:2];
NSLog(myString1);
NSLog(myString2);

A: Maybe I'm missing something but you are doing operations on "C" strings not NSStrings. You need more "@"s.


The list of Huawei phones getting Android 9 pie update are Mate 10, 10 Pro, Mate RS Porsche Design, P20, P20 Pro, Honor 10 & V10, Honor play.
On 7 July, Google released the latest Android operating system, Android 9.0 Pie. However, only the Google Pixel mobile phones can be upgraded to Android 9 Pie. Other mobile phone manufacturers are still working on it but Huawei has been quick here. It has already opened the internal beta test for some mobiles. Huawei mobile phones might be second to get the Android 9.0 Pie update after Google.
On August 8th, He Gang, the head of Huawei smartphone unit shared a message in Weibo: Good news about Android P! A lot of small partners are asking about the progress of Huawei Android P. At present, Huawei Mate 10 Pro, P20, Honory 10, and Honor V10 have started the internal test simultaneously.
This shows that Huawei officially started the internal beta test of Android 9.0 Pie on August 7. In addition to Huawei Mate 10 Pro, Huawei P20, Honor 10 and Honor V10, it is confirmed that Huawei Mate 10, Huawei Mate RS Porsche Design, Huawei P20 Pro and Honor Play has also started the internal beta test.
The eight Huawei mobile phones listed above will be the first to get the Android 9.0 Pie update. Is your mobile phone listed?
Finally, An important information about Android 9.0 pie. With the launch of Android 9.0, Google has made the "rollback protection" mandatory, which means that you cannot downgrade your system firmware. For example, If you are on Android 9, you can't go back to Android 8.
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text:
Swan Valley Funeral Services
| 363 Kelsey Trail
| Swan River, MB R0L 1Z0
| joe@sneathstrilchuk.com
Our Family Care Team
Henry Robert Fenski
October 5, 1947 ~ April 30, 2019 (age 71)
Henry Robert Fenski was born on October 5, 1947 to Robert and Katie (Daudrich) Fenske. He passed away peacefully on April 30, 2019 at the age of 71.
The Fenske children were brought up on the farm on NE 4-37-25 and later in the house on Learners Street on the edge of Minitonas. Of all the animals the family kept in the farmyard, Henry took most delight in the chickens, especially the exotic breeds that laid colorful eggs. He also enjoyed collecting coins as a young boy and added to his collection at every opportunity. This became a lifelong hobby. The family attended First Baptist Church in Minitonas and Henry was baptized at the age of 12. Henry loved to sing, and he had a beautiful deep voice. As a young boy, he sang out loudly from the front pew, much to the embarrassment of his siblings. Sadly, when Henry was 14 years old, his mother passed away after a lengthy battle with cancer. Henry went on to attend Bible School in Edmonton in the 60's and returned to Minitonas to continue farming with his father. Robert suffered many years with ALS, until his death in 1978.
Henry and Carol (Hart) were married in 1970 and raised three daughters, Jen, Christine and Amber on the farm three miles north of Minitonas. Aside from grain farming, Henry also worked for Siewert Logging. He later supplemented the farm income by cutting firewood, guiding hunters and pushing snow.
In 2003, when Jen and Chris purchased the farm from Henry, he relocated to St. Laurent, MB. He returned to Swan River in 2004 where he lived until March of this year, at which time he moved to the Pioneer Lodge in Minitonas. His last days were spent at the Swan Valley Health Centre, where he lost his battle with cancer.
Henry had many interests and hobbies over the years. He was an avid hunter and fisherman, taking every opportunity to have fun outdoors with his buddies. He took great pride in his numerous master angler catches and in developing his 'secret sausage recipe'. This homemade delicacy was a family favorite that he loved to share with friends. He also enjoyed playing cards, especially crib. Elvis gospel music was a favorite; and if anyone could sing along to Elvis and pull it off – it was Henry! Henry liked to spend money on new technology and toys; those who knew Henry well weren't surprised when he brought home an argo, an ultra-lite plane, a school bus, a new camcorder or a unique vehicle. He was usually up for a race or a wager, challenging friends and neighbours every chance he got. In recent years he spent countless hours building his coin collection. He very carefully created coin sets for each of his eight grandchildren and loved gifting them with coins each Christmas and birthday. Henry always made a point of phoning his children and grandchildren on their birthdays, eager to set a date to go out for a meal together.
Margaret has been a special friend to Henry in recent years. They enjoyed going out for supper, visiting, picking fruit, making wine and watching curling and Jets hockey. They also shared in fellowship with the close-knit community at the First Baptist Church in Swan River.
Despite many challenges over the years, we hold on to the good times and memories that make us smile.
Henry leaves to mourn his three daughters: Jen (Chris) Zamzow and their children Bryn, Neva and Matthew of Minitonas, Chris (Clint) Buchholz and their children Konnor and Finley of Campbell River, B.C. and Amber (Todd) Friesen and their children Josie, Charlotte and Kate of Kenville; sisters Lil (Leo) Hart of Swan River, and Florence (Richard) Holland of Kelowna, B.C.; brother Bill (Lin) Galloway of Swan River and special friend Margaret Zell of Minitonas along with numerous nieces and nephews. Henry was predeceased by his parents Katie (1962), Robert (1978), his sister Betty Watkins (2008) and nephew Wade Galloway (2014).
Thank you to those that have cared for Henry over the years and in these last months of his journey, including Dr. Gendy, Dr. Wickert, Dr. Kotb and the CancerCare team, home care and the staff at the Primary Health Care Clinic and Swan Valley Health Centre. Thank you everyone at the Pioneer Baptist Lodge in Minitonas for your kindness during the short time that Henry lived there. Dr. Koetting, Dr. Burnside and the amazing nurses and health care aides showed incredible care and compassion to Henry and his family in his final days.
As Henry requested, cremation and a private family burial has taken place. Donations in Henry's memory can be made to The Robert and Katie Fenske Charitable Foundation, Box 8, Minitonas, MB, R0L 1G0.
The Robert & Katie Fenske Charitable Foundation
Box 8, Minitonas MB R0L 1G0
CONTACT US | local_floristSEND FLOWERS
© 2019 Swan Valley Funeral Services Ltd.. All Rights Reserved. Funeral Home website by CFS


text:
Jason Trembath
Outlier Studio (www.outlierstudio.com)
Director, Producer, Screenwriter and Transmedia Producer
Jason's Wall (4)
Jason Trembath was born on December 5th 1985 in Melbourne Victoria, before moving to Weipa, (Cape York) Queensland and then later growing up in Central Queensland.
Jason fell in love with movies at an early age and progressed from movies and music videos in scroll to short films and other projects in University.
After working on the Channel V Music Bus in 2004/05, Jason changed his degree to Business Marketing, so that he could learn how better promote and manage his future film projects.
In 2004, Jason enlisted into the Part-Time Army Officer training with the Australian Army Reserves. He graduated from the Royal Military College Duntroon in February 2007 as an Infantry Second-Lieutenant.
Jason was subsequently promoted to First-Lieutenant and then Captain. He has had 10 years experience in Military Management, which has been an incredible experience base for effective film producing.
After completing University, Jason sourced and attained funding to produce his first low/no budget feature film "Sharkmen" as a 'one-man crew' (Writer/Director/Producer/Cinematography/Visual Effects). He then sourced additional private financing to complete Post-Production on the film.
Due to how he completely produced "Sharkmen" on his own, in 2009 Jason was invited by the Director of Photography on "Desperate Housewives" to shadow and learn from him on set of Season 6 at Universal Studios, Los Angeles CA.
In November 2013 Jason was attached to be Transmedia Producer and Interactive Designer/Developer on the Australian Army's documentary on the Afghanistan War. "The Longest War: The Australian Army in Afghanistan" was launched on 7 May 2015 by the Chief of Army Lieutenant General David Morrison AO and Minister for Defence Kevin Andrews at the Australian War Memorial. (www.army.gov.au/the-longest-war)
Jason is now completing a one of a kind, interactive package to distribute "Sharkmen" and raise funding for his second feature film "Aussie Rough".
• Writer, Director and Producer
Transmedia Producer
Interactive, Visual and UX Designer
Key Skill Sets:
• Writing and Directing
• Film Production
• Transmedia Producing and Story Design
Interactive Website and App Designer
• Bachelor of Business, Marketing
• Advanced Diploma of Military Management (Australian Army)
• Diploma of Government (Management) (Australian Army)
• Certificate IV in Business, Frontline Management (Australian Army)
• Australian Defence Force Medal (For 4+ years of minimum National Service)
• Continual Interactive and Film Production Mastery Courses
• Robert McKee Story Seminar (continual participant)
• Jeff Gomez Transmedia Masterclass (3 attended, continual participant)
• Various AFTRS courses
• Various NIDA acting Courses
The Longest War - The Australian Army in Afghanistan
Outlier Studios
The Longest War - The Australian Army in Afghanistan (2015)
Documentary Transmedia Producer, Interactive Designer/Developer, Original Concept by, Interactive Director
Central Queensland University
Jason's network
Wild West Comedy
kered becker aka kezza
Matt Hodgkinson
David Le May
Cinematographer and director of photography
Cyrus Hallam
Alex Sarris
Author and screenwriter


text:
Runaway MaizyArtist
Play Like a Dream
By Runaway Maizy
A fast paced selection with a driving chorus. The song is about having an experience with another person that is unbelievable, out of this world, or like a dream. The song's catchy verse riff carries the piece into a soaring pop chorus. The breakdown in the bridge leads up to an explosion in the final chorus and a halftime breakdown. Great for sporting clips. [Tempo: Medium, Fast, Very fast / 200 BPM]
Vocal version (04:26)
Play That Is
A mid-tempo rock track full of emotion. The song deals with the end of a relationship and even though time has passed one party is still clearly not okay. The chorus involves a call and response between a male and female vocal until finally building in the finally chorus. [Tempo: Medium / 107 BPM]
Play Can You See
A mid-tempo syncopated pop-rock track. Full of dynamics, the song is a great pop instrumental hitting all the right accents. [Tempo: Fast / 107 BPM]
Play So Long
A mid-tempo, summer sounding song. The mellow verse transitions nicely into an action packed chorus. The song is about coming to terms with the fact that you've been with someone who has been dishonest, unfaithful, someone who has fooled you for, well so long. Male and female vocals. [Tempo: Fast / 162 BPM]
Play Release Me
This track runs right from the gate. Fast paced and in your face the verses syncopated guitars along with a smooth bridge provide the perfect pacing for huge open chorus. [Tempo: Fast, Very fast / 150 BPM]
Play Honorable Mention
A fast tempo track that continually builds. The song deals with heart-break, coming close to having something, but never actually attaining. It's fast paced, upbeat intro is perfect for the start of any segment or transition. Male vocals. [Tempo: Fast / 154 BPM]


So I have had a few things going on lately. I recently went up to Auckland and passed the Shimano mechanics course so I am now qualified to work on bikes which is pretty cool. Also I have just moved into central Chch with my lovely partner, Shannon. For those of you who don't know her yet I'm sure you will meet her at some of the races this season.
Just over a week ago I was hit by a car while riding to work. I got hit on my left going through a roundabout. Luckily I came out with no major damage but my bike was destroyed. Snapped through the frame in 4 places! The driver's insurance is covering it so should be all good and I will have a new bike soon. I obviously had some time off the bike to recover and let the bruising go down. But today I went out to the short track racing. It was only my second time on the bike since the accident so I planned just to see how my injuries held up first, then decide to ride or watch. I felt good and had no pain so I started the race. The first few laps went well and I was up in the front group and finding the pace fast but comfortable. Until Phil made an attack. I stayed with the others up the hill on that lap but the lungs couldn't keep up with the legs. So I settled into my own pace and rode home in forth. I felt like my legs were strong but it was just the fact that I pretty much hadn't riden for over a week so the tank was empty. Also eating donuts and sticky steamed buns at the Riccarton markets that morning wouldn't have helped either.
I'm looking forward to getting back into the training mode and after a week of solid riding I should be up the front at next weeks short track.


text:
Conventional (Family: Design-Bid-Build)
Summary: B104™–2017, Standard Abbreviated Form of Agreement Between Owner and Architect
Changes from the previous edition
Dispute Resolution—Mediation and Arbitration
AIA Document B104–2017 is intended for use on construction projects of limited scope and complexity.
AIA Document B104–2017 is intended for use on construction projects of limited scope and complexity. B104-2017 replaces B104-2007, Standard Form of Agreement Between Owner and Architect for a Project of Limited Scope. B104-2017 is an abbreviated version of B101™–2017. B104 contains a compressed form of basic services with three phases: design, construction documents, and construction. This document may be used with a variety of compensation methods. B104–2017 is intended to be used in conjunction with A104™–2017, Standard Abbreviated Form of Agreement Between Owner and Contractor, which it incorporates by reference.
For more complex projects, parties should consider using one of the following AIA owner/architect agreements: B101–2017 or B103™–2017. For smaller and less complex projects, parties may wish to consider B105™–2017, Standard Short Form of Agreement Between Owner and Architect.
For use and execution of a document, see its instructions »
AIA Document B104–2017 is a standard abbreviated form of agreement between Owner and Architect for building design and construction contract administration. B104–2017 is an abbreviated version of AIA Document B101–2017, Standard Form of Agreement between Owner and Architect, and it is intended for use on construction projects of limited scope and complexity. B104 contains a compressed form of basic services with three phases: design, construction documents, and construction. This document may be used with a variety of compensation methods, including percentage of construction cost and stipulated sum.
B104–2017 assumes that the Architect will provide cost estimates and will design the Project to meet the Owner's budget for the Cost of the Work. If the Owner will retain a third party to provide cost estimates, or if the Project will implement fast track, phased or accelerated scheduling, the parties should consider using AIA Document B103–2017, Standard Form of Agreement Between Owner and Architect for a Complex Project.
B104–2017 is intended to be used in conjunction with AIA Document A104–2017, Standard Abbreviated Form of Agreement Between Owner and Contractor, which it incorporates by reference. B104–2017 also can be used with Architect-Consultant agreements such as AIA Document C401™–2017. Before transmitting Instruments of Service or other information in digital form, parties should establish protocols for that transmission and may use AIA Document E203™–2013, Building Information Modeling and Digital Data Exhibit, for that purpose.
Significant features.
Standard of Care. B104–2017 contains a provision at Section 2.1 stating the applicable standard of care under which the Architect shall perform its services.
Copyright and Licensing of the Instruments of Services. In B104–2017, the Architect and the Architect's consultants are deemed the authors and owners of their respective Instruments of Service, and they retain all common law and statutory rights, including copyright. The Owner receives a license to use the Instruments of Service solely and exclusively for constructing, using, maintaining, altering and adding to the Project. This license will only terminate if the Architect rightfully terminates the Agreement for cause due to the Owner's default. In the absence of such a termination by the Architect, the Owner retains the license to use the Instruments of Service after completion of the Project or the Owner's termination of the Agreement. If the Owner subsequently uses the Instruments of Services without retaining the author of the Instruments of Service, the Owner agrees to release and indemnify the Architect for such uses. If the Owner rightfully terminates the Agreement for cause, however, the Owner is not required to release and indemnify the Architect for its further use of the Instruments of Service. If the Owner terminates the Agreement for its convenience, or the Architect terminates the Agreement due to the Owner's suspension of the Project, B104 provides at Section9.7.2 for the Owner to pay a licensing fee to the Architect for the Owner's continued use of the Architect's Instruments of Service.
Binding Dispute Resolution. Mediation is a condition precedent to any form of binding dispute resolution, but binding arbitration is not mandatory for disputes that fail to settle in mediation. Instead, the parties are required to select at Section 8.2.3 from three choices of binding dispute resolution: (1) arbitration, (2) litigation or (3) another method that they must identify.
Changes from the previous edition.
Document B104–2017 contains a number of significant format and content changes from B104–2007. A key formatting change between B104–2017 and B104–2007 is that B104–2017 reformats the Additional Services article. B104–2007 included two types of Additional Services: 1) Additional Services that were identified and contracted for at the time the Agreement was executed, and 2) Additional Services that arose as the Project proceeded. To avoid the potential confusion caused by having two types of Additional Services, B104–2017 has re-categorized the first type of Additional Services as Supplemental Services. Accordingly, in B104–2017, Supplemental Services refers to services that are not included as Basic Services but that are identified as the Architect's responsibility at the time the Agreement is executed. Additional Services now refer only to those services that may arise as the Project proceeds.
There are many other changes to foster clarity in the B104 Owner-Architect agreement as well. Described below are highlights of major changes in B104–2017, Standard Abbreviated Form of Agreement Between Owner and Architect.
Article 1 – Initial Information
§ 1.3 Use of Digital Data. This section requires the development of protocols for the transmission of Instruments of Service or other data in digital form through the use of AIA Document E203™–2013, Building Information Modeling and Digital Data Exhibit.
§ 1.3.1 This new section requires the parties to develop protocols for the use of, and reliance on, a building information model or portion thereof. Use of, or reliance on, a building information model without established protocols will be at the using or relying parties' sole risk. This provision requires the use of AIA Document E203–2013 and AIA Document G202™–2013, Project Building Information Modeling Protocol Form, for the establishment of these protocols. For more information on using the AIA's Digital Practice Documents, including E203–2013 and G202–2013, please see AIA's Guide, Instructions and Commentary to the 2013 AIA Digital Practice Documents.
Article 2 – Architect's Responsibilities
§ 2.2 Architect's Insurance Requirements. Section 2.2 contains a new provision wherein the parties may set forth the types and limits of insurance the Architect is required to carry for the Project. Specifically, B104–2017 lists General Liability, Automobile Liability, Workers' Compensation, and Professional Liability.
Article 4 – Supplemental and Additional Services
As noted above, B104–2007 included two types of Additional Services: 1) Additional Services that were identified and contracted for at the time the Agreement was executed, and 2) Additional Services that arose as the Project proceeded. B104–2017 re-categorizes the first type of Additional Services as Supplemental Services. In B104–2017, Additional Services only refers to those services that may arise as the Project proceeds.
Article 5 – Owner's Responsibilities
§ 5.2 Owner's Budget. Section 5.2 has been edited to clarify the Owner's responsibility with regard to updating the Owner's budget for the Project.
Article 6 – Cost of the Work
§ 6.7 Architect's Redesign Obligation. B104–2017 required the Architect to provide redesign services without additional compensation if the lowest bona fide bid or negotiated proposal exceeded the Owner's budget for the Cost of the Work. This obligation remains in B104–2017, however, an exception has been added, stating that the redesign services shall be compensated as an Additional Service if the lowest bona fide bid or negotiated proposal exceeds the Owner's budget due to market conditions the Architect could not reasonably anticipate. This approach is more similar to that taken under the Federal Acquisition Regulations' design within funding limitations obligation, where an architect's redesign efforts would be compensated if the unfavorable bids or proposals exceeding the budget are the result of conditions beyond the architect's reasonable control. (Design Within Funding Limitations 48 CFR 52.236-22).
Article 9 – Termination or Suspension
§ 9.6 Costs Attributable to Termination. This provision has been revised to clarify that costs payable to the Architect under a termination for convenience include costs attributable to the Architect's termination of consultant agreements.
§ 9.7 Termination Fee. A fill point has been added for the parties to identify the termination fee that the Owner would pay to the Architect in the event of a termination for convenience or if the Architect terminates because of an extended suspension of the Project, as well as a Licensing Fee if the Owner intends to continue to use the Architect's Instruments of Service.
§ 9.8 One-year Termination. A provision was added that terminates the Agreement one year from the date of Substantial Completion, unless otherwise stated in the Agreement. There are a number of provisions throughout the Agreement, such as Article 7 Copyrights and Licenses, where the terms and conditions will survive termination of the Agreement.
Article 11 – Compensation
§ 11.1 Compensation for Basic Services. Fill points have been added for two of the more popular methods for compensating Architects for Basic Services: Stipulated Sum and a Percentage Basis. Section 11.1 also includes a space for users to insert another compensation method, such as hourly billing rates or a multiple of direct salary expense, if the parties so choose.
§ 11.6 Calculation of Progress Payments For Percentage Basis Compensation. Language has been added to clarify how progress payments to the Architect are calculated when the parties have chosen a percentage basis as the method of compensation for the Architect's Basic Services. Progress payments are calculated by applying the percentages set forth in Article 11 to the Owner's most recent budget for the Cost of the Work. Section 5.2 requires the Owner to update its budget for the Cost of the Work as part of its budget for the Project throughout the course of the Project. Section 11.6 also clarifies that once a progress payment is made, it shall not retroactively be adjusted based on subsequent increases or decreases to the Owner's budget for the Cost of the Work.
This document contains provisions for mediation and arbitration of claims and disputes. Mediation is a non-binding process, but is mandatory under the terms of this agreement. Arbitration may be mandatory under the terms of this agreement. Arbitration is binding in most states and under the Federal Arbitration Act. In a minority of states, arbitration provisions relating to future disputes are not enforceable but the parties may agree to arbitrate after the dispute arises. Even in those states, under certain circumstances (for example, in a transaction involving interstate commerce), arbitration provisions may be enforceable under the Federal Arbitration Act.
The AIA does not administer dispute resolution processes. To submit disputes to mediation or arbitration or to obtain copies of the applicable mediation or arbitration rules, contact the American Arbitration Association at (800) 778-7879 or visit the website at adr.org.
Commentary: A201®-2007, General Conditions
Commentary: B101™-2007
FAQs: A121™–2018, Standard Form of Master Agreement Between Owner and Contractor where Work is provided under multiple Work Orders
FAQs: A221™–2018, Work Order for use with Master Agreement Between Owner and Contractor
FAQs: A421™-2018 and A422™-2018, Work Order Agreements
FAQs: A701™-2018, Instructions to Bidders
FAQs: B106™ – 2020, Standard Form of Agreement Between Owner and Architect for Pro Bono Services
FAQs: B107™–2020, Standard Form of Agreement Between Developer-Builder and Architect for Prototype(s) for a One- or Two- Family Residential Project with Limited Architectural Services
FAQs: B109™–2020, Standard Form of Agreement Between Owner and Architect for a Multi-Family Residential or Mixed Use Residential Project


text:
\section{Introduction}\label{s:intro}

Differential equations (DEs) which include ordinary differential equation (ODE) and partial differential equation (PDE), are used to model a wide variety of physical phenomena.
ODEs describe the dynamics of continuously changing processes by relating a process and its rate of change in biomedical research and other scientific areas.
PDE models are commonly used to model complex dynamic system in applied science such as physics, biology and finance.
The forms of DE models are usually proposed by experts based on their prior knowledge and understanding of the dynamic system.
But the values of the parameters in DE models are often unknown and need to be estimated from noisy measurements of the dynamics system.
Because most DEs have no analytic solutions and can only be solved with numerical method.
Traditional methods for estimating DE parameters require repeatedly solving DEs numerically under thousands of candidate parameter values.

A lot of statistical methods have been developed to estimate the parameters in DE modes by repeatedly solving the DE models numerically, which could be time consuming.
Specifically, a series of method in the study of HIV dynamics \cite{ho1995rapid}\cite{wei1995viral}\cite{wu1999population}\cite{wu2005statistical}\cite{wu1998estimation} and some hierarchical Bayesian methods \cite{putter2002bayesian} \cite{huang2006hierarchical} are proposed with use of numerical ODE solver.
As for PDEs, Muller and Timmer \cite{muller2002fitting} solved the target least-squares type minimization problem using an extended multiple shooting method.
It requires careful parameterization of the initial condition and then numerically solves the PDEs.
This leads to a pretty high computational load and there is growing need in developing efficient estimation methods for DE models.

Another strategy to estimate parameters of DE is the two-stage method, which in the first stage estimates the solutions and its derivatives from noisy observations using data smoothing methods without considering differential equation models, and then in the second stage estimates of the DE parameters are obtained by least squares.
Liang and Wu \cite{liang2008parameter} developed a two-stage method for a general first-order ODE model, using local polynomial regression in the first stage, and estimated asymptotic properties of the estimator.
Similarly, Chen and Wu \cite{chen2008efficient} developed local estimation for time-varying coefficients.
Bar, Hegger, and Kantz \cite{bar1999fitting} modeled unknown PDEs using multivariate polynomials of sufficiently high order, and the best fit was chosen by minimizing the least squares error of the polynomial approximation.
Based on the estimated functions, the PDE parameters were estimated using least squares \cite{muller2004parameter}.
The issues of noise level and data resolution were addressed extensively in this approach. See also Parlitz and Merkwirth \cite{parlitz2000prediction} and Voss et al. \cite{voss1999amplitude} for more example.

As mentioned before, a straightforward two-stage strategy, though easy to implement, has difficulty in estimating derivatives of the dynamic process accurately, leading to biased estimates of the DE parameter.
There are some advanced methods\cite{ramsay2007parameter}\cite{cao2011robust}\cite{cao2012penalized}, which approximate the solution and its derivatives by using basis-function with both considering noisy observations and equation constraint.
Then parameters are estimated by optimizing certain criteria.
Especially, Xiaolei Xun\cite{xun2013parameter} propose two joint modeling schemes: (a) a parameter cascading or penalized profile likelihood approach and (b) a fully Bayesian treatment.
This type methods usually have some sensitive hyper-parameters, which needed be carefully design and may have the parameter estimates converge to a local minima, otherwise global optimization is computationally intensive. In addition, The ability of basis functions to fit complex solution function and its derivatives is also a challenge.

Here we propose using a new statistical approach which  is based on Gaussian process, called Gaussian process regression with constraint (GPRC) method \cite{wang2020explicit}, to
accurately and robustly predict the solution and  derivatives appearing in the differential equations.
GPRC can efficiently make the predictions with both considering the observations and DE constraint in a joint Bayesian framework.
Compared with basis-function expansion method, the Gaussian process (GP) models \cite{seeger2004gaussian} provide a class of flexible and efficient  non-parametric fits
both for fitting noisy data and solving noisy DEs.
The prior knowledge can also be easily encoded by the covariance of the GP.
To achieve parameter estimation, we construct a likelihood by employing a new potential term which contains the DE  and data residuals and the Metropolis-Hastings(MH) algorithm is used to infer the unknown parameters in Bayes' rule.

The structure of the paper is organized as follows:
We first review the problem setup in Section \ref{set:Problemsetup}.
The review of GPRC method for estimating the solution and its derivatives is presented in Section \ref{set:method}. Specificly, the Picard iterative linearization for nonlinear DE is carefully formulated in \ref{set:picard}, based on the last example in \cite{wang2020explicit}.
Bayesian inference framework is introduced in Section \ref{set:inference}, where we propose a new potential of the likelihood.
Numerical examples are presented in Section \ref{set:numerical}, to demonstrate the effectiveness of the proposed method compared with two-stage method. Finally, Section \ref{set:conclusion} offers some closing remarks.
\section{Problem setup}
\label{set:Problemsetup}

Inferring the unknown parameters in differential equation is a classical problem in the inverse problem. Differential equations refer to the  relationship between the solution function and its derivatives. We consider a multidimensional dynamic process, $u({\bm{x}})$ , where ${\bm{x}} = (x_1, \dots, x_D)^T\in \mathbb{R}^D$ is a multidimensional argument.
The general formula for differential equation can be expressed as
\begin{equation}
\label{eq:pde}
\mathcal{F}({\bm{x}}, u,\frac{\partial u}{\partial x_1},\dots, \frac{\partial u}{\partial x_p},\dots, \frac{\partial^2 u} {\partial x_i\partial x_j},\dots; {\bm{\theta}}) = 0,
\end{equation}
 where ${\bm{\theta}}=(\theta_1,\dots,\theta_q)$ is the  vector of  unknown parameters. In the following, we denote $\mathcal{F}({\bm{x}}, u({\bm{x}}), \dots; {\bm{\theta}})$ by the short-hand notation $\mathcal{F}(u(\bm{x}); {\bm{\theta}})$. The left-hand side of Eq. \eqref{eq:pde} consists of the solution $u({\bm{x}})$ and its derivative terms ($\frac{\partial u} {\partial x_i}, \frac{\partial^2 u} {\partial x_i\partial x_i}, \dots, $).
In practice , we may not   directly observe $u({\bm{x}})$, but we  can observe the noisy measurement  $y({\bm{x}})$. For simplicity, we assume that there is a measurement error between $u({\bm{x}})$ and $y({\bm{x}})$, the noisy measurements satisfy
\begin{equation}
\label{y}
y_i = u({\bm{x}}_i) + \epsilon_i,
\end{equation}
where $i = 1,\dots, n$, is the observation index and $\epsilon_i$ is the measurement error. We assume this measurement error follows an independent and identically distributed Gaussian distribution with zero mean and variance $\sigma_{obs}^2$.

In practice, parameters ${\bm{\theta}}$ are often unknown and need to be determined. Our objective is to estimate the parameters from the limited noisy data $\{{\bm{x}}_i, y({\bm{x}}_i)\}_{i=1, \dots, n}$.

\section{GPRC}
\label{set:method}
For estimating the unknown parameter ${\bm{\theta}}$, accurate estimations of the solution and its derivatives are necessary and helpful.
To infer a nonlinear function from its noisy measurements at a given set of inputs is a classical statistical learning problem and a vast of well-established methods, ranging from polynomial and spline to kernel method and neural network, are available for many important applications.
However, if the interest is the derivatives and there is no additional observations of derivatives, the problemof estimating derivatives is more challenging and subtle than estimating the function itself and less attentions have been given to this issue.
In this section, we briefly introduce an estimator GPRC \cite{wang2020explicit}, which has shown the ability of accurately estimating the derivatives in ordinary differential equation and partial differential equation problems and here is used to estimate the solution and its derivatives with a guess parameter ${\bm{\theta}}$.
\subsection{ GPRC in linear differential equation}
\label{subset:gprc}
Specifically, the solution $u({\bm{x}})$ is assumed as a Gaussian process with zero mean and the covariance function $k_{uu}({\bm{x}},{\bm{x}}',\bm{\gamma})$, which is denoted as
\begin{equation}
\label{eq:GPprior}
u({\bm{x}}) \sim \mathcal{GP}(0, k_{uu}({\bm{x}},{\bm{x}}';\bm{\gamma})),
\end{equation}
where, $\bm{\gamma}$ denotes the hyper-parameters of the kernel function. One property of the Gaussian process in our favor is that any linear transformation, such as differentiation and integration, of a Gaussian process is still a Gaussian process. With the assumption \eqref{eq:GPprior}, we consider the case when $\mathcal{F}(\cdot; {\bm{\theta}})$ is a linear operator or a combination of different linear operators, denoted as $\mathcal{L}^{\bm{\theta}}$ with given ${\bm{\theta}}$. We introduce a random function $r({\bm{x}})$ as the residual of the linear differential equation $\mathcal{L}^{\bm{\theta}} u = 0$ for convenience:
\begin{equation*}
r({\bm{x}}):=\mathcal{L}^{\bm{\theta}} u(x).
\end{equation*}
Then the function $r({\bm{x}})$ is also a mean-zero Gaussian process
\begin{equation*}
r({\bm{x}}) = \mathcal{GP}(0, k_{rr}({\bm{x}},{\bm{x}}';\bm{\gamma})),
\end{equation*}
where $k_{rr}({\bm{x}},{\bm{x}}') = \text{Cov}(\mathcal{L}^{\bm{\theta}} u({\bm{x}}), \mathcal{L}^{\bm{\theta}} u({\bm{x}}'))$ denotes the covariance function of $\mathcal{L}^{\bm{\theta}} u$ between ${\bm{x}}$ and ${\bm{x}}'$. The following fundamental relationship between the kernels $k_{uu}$ and $k_{rr}$, $k_{ur}$, $k_{ru}$ is well-known (see  e.g. \cite{seeger2004gaussian,theore2003Solving}),\
\begin{equation}
\label{eq:cov_ru}
\begin{split}
k_{rr}({\bm{x}},{\bm{x}}';\bm{\gamma}) &= \mathcal{L}_{{\bm{x}}}^{\bm{\theta}} \mathcal{L}_{{\bm{x}}'}^{\bm{\theta}} k_{uu}({\bm{x}},{\bm{x}}';\bm{\gamma}),\\
k_{ur}({\bm{x}},{\bm{x}}';\bm{\gamma}) &= \mathcal{L}_{{\bm{x}}'}^{\bm{\theta}} k_{uu}({\bm{x}},{\bm{x}}';\bm{\gamma}),\\
k_{ru}({\bm{x}},{\bm{x}}';\bm{\gamma}) &= \mathcal{L}^{\bm{\theta}}_{{\bm{x}}}k_{uu}({\bm{x}},{\bm{x}}';\bm{\gamma}),
\end{split}
\end{equation}
Based on the Gaussian assumption and the covariance expressions between $u$ and $r$ in \eqref{eq:cov_ru}, a joint inference framework of Gaussian process regression for the available observation data of $u$ and $r$ can be naturally constructed.
By interpreting the equation $r := \mathcal{L}^{\bm{\theta}} u = 0$ as the constraint of the function $r$, GPRC will significantly improve the accuracy of estimation of solution and its derivatives due to the additional equation information.

\textbf{Training a Gaussian Process}

The training process is to find the optimal parameters $\bm{\beta} = \{\bm \gamma, \sigma_{obs}^2\}$ by maximum likelihood estimation (MLE).
Given $n$  noisy observations of the state $u$
at  $n$ points  $\{{\bm{x}}_i: 1\leq i\leq n\}$,  we  denote  the state vector ${\bm{y}} \equiv [y_1, y_2,\dots,y_n]^T\in {\mathbb R}^{n\times 1}$, the residual vector $\bm{r} \equiv [r_1, r_2,\dots,r_n]^T\in {\mathbb R}^{n\times 1}$ and
the training point matrix $X \equiv [{\bm{x}}_1, {\bm{x}}_2,\dots,{\bm{x}}_n]^T\in {\mathbb R}^{n\times D}$.
Let   $Y = \begin{bmatrix} {\bm{y}}\\ \bm{r}\end{bmatrix} \in {\mathbb R}^{2n}$.
Then
 the   negative log marginal likelihood
of  $p(Y\vert \bm{\beta})$
 has the following expression
\begin{equation}
\label{eq:logmarginal}
\begin{split}
-\log p(Y| \bm{\beta}) = \frac{1}{2}\log (\det{K})+\frac{1}{2}Y^T K^{-1}Y + \frac{n}{2}\log 2\pi,
\end{split}
\end{equation}
where  the $2n \times 2n$ matrix $K$ is defined by
\begin{equation*}
K = \begin{bmatrix}
&K_{uu}+ \sigma^2_u I & K_{ur}\\
&K_{ru} & K_{rr} + \sigma^2_r I
\end{bmatrix},
\end{equation*}
 The matrices $K_{uu}, K_{ur}$,  $K_{ru}$ and $K_{rr}$
 correspond to the kernel functions $k_{uu}, k_{ur}, k_{ru}$ and $k_{rr}$ respectively in \eqref{eq:GPprior} and \eqref{eq:cov_ru}, evaluated at the $n$ points $X $.  \\
The minimisation of the negative log marginal likelihood  is a non-convex optimisation task. However gradients are easily obtained
\begin{equation}
\label{eq:partialderivative}
\begin{split}
\frac{\partial L}{\partial \beta_k}=\frac{1}{2}tr(K\frac{\partial K^{-1}}{\partial \beta_k})+\frac{1}{2}Y^T\frac{\partial K}{\partial \beta_k}K^{-1}\frac{\partial K}{\partial \beta_k} Y,
\end{split}
\end{equation}
where $\beta_k $  indicates the $i$th hyper-parameter of $\bm{\beta}$ and therefore standard gradient optimisers can be used to minimize the negative log marginal likelihood \eqref{eq:logmarginal} within the gradient \eqref{eq:partialderivative}.

\textbf{Prediction}
After the hyper-parameters in the GPRC are computed,
the prediction of the function $u({\bm{x}})$ or its derivatives of interest,
denoted as a function $l({\bm{x}})$ (e.g., $l({\bm{x}})=\partial_{{\bm{x}}} u({\bm{x}})$),
at a new  test point ${\bm{x}}_*$ is described  below.
The covariance function of the GP $l({\bm{x}})$ is denoted by $k_{ll}({\bm{x}},{\bm{x}}')$ by the convenience.

With a given   point ${\bm{x}}_*$,  the differential equation provides  the fact $r({\bm{x}}_*)=0$, which
is a useful observation to incorporate the Bayesian inference.
To enhance this condition locally, $m$ equally-spaced points around ${\bm{x}}_*$, which are referred as the extended set $\chi:= \{ {\bm{x}}_*^j: 1\le j\le m  \}$ are actually considered in the prediction.
The set $\chi$ is similar to the window/scale  concept in local polynomial regression and thus adaptive strategy is possible.
Therefore, the value of the equation constraints in the extended set should be zero, i.e., $\bm{r}_{\chi}=\bm 0$.
These points ${\bm{x}}_*^j$ are supposed to resolve a characteristic length for the residual process $r({\bm{x}})$.

This is the posterior distribution for a specific set of test case,
\begin{align}
\label{eq:poster_u}
&\bar{u}({\bm{x}}_*) = K_{u_*\bullet}\widehat{K}_{u\bm{r}_{\chi}}^{-1}\begin{bmatrix}{\bm{y}}\\\bm{r}_{\chi}\end{bmatrix},
S_u({\bm{x}}_*)=K_{uu}({\bm{x}}_*,{\bm{x}}_*) - K_{u_*\bullet} \widehat{K}_{u\bm{r}_{\chi}}^{-1}K_{u_*\bullet}^T.\\
\label{eq:poster_l}
&\bar{l}({\bm{x}}_*) = K_{l_*\bullet}\widehat{K}_{u\bm{r}_{\chi}}^{-1}\begin{bmatrix}{\bm{y}}\\\bm{r}_{\chi}\end{bmatrix},
S_l({\bm{x}}_*)=K_{ll}({\bm{x}}_*,{\bm{x}}_*) - K_{l_*\bullet}\widehat{K}_{u\bm{r}_{\chi}}^{-1}K_{l_*\bullet}^T,
\end{align}
where
$$\widehat{K}_{u\bm{r}_{\chi}}=\begin{bmatrix}
K_u&K_{u \bm{r}_{\chi}}\\
K_{u \bm{r}_{\chi}}^T  &K_{\bm{r}_{\chi}}
\end{bmatrix}\in {\mathbb R}^{(n+m)\times (n+m)},$$
$K_u = K_{uu} + \sigma^2_u \bm{I}_u\in {\mathbb R}^{n\times n}$,
 $ K_{\bm{r}_{\chi}} = K_{\bm{r}_{\chi}\bm{r}_{\chi}}+ \sigma^2_r \bm{I}_{r} \in {\mathbb R}^{m\times m}$,
  $K_{u_*} = k_{uu}({\bm{x}}_*,{\bm{x}}_*)$,
$K_{l_*}= k_{ll}({\bm{x}}_*,{\bm{x}}_*)$
and $K_{u_*\bullet} = [K_{u_*u},  K_{u_*\bm{r}_{\chi}}]\in {\mathbb R}^{1\times (n+m)}$ and $K_{l_*\bullet} = [K_{l_*u} ,K_{l_*\bm{r}_{\chi}}]\in {\mathbb R}^{1\times (n+m)}$.  The posterior variances $S_u({\bm{x}}_*)$ and $S_l({\bm{x}}_*)$ can be used as good indicators of how confident the predictions are.

\subsection{Picard iterative linearization for nonlinear differential equation}
\label{set:picard}
For nonlinear differential equations, the equation constraint can't be formulated as a Gaussian process since the product of Gaussian processes is not a Gaussian process anymore.
Here an iterative linearization method is introduced for nonlinear equations, motivated by the Picard iteration method\cite{coddington1955theory} , which is an extension of method used in Van der Pol equation example in \cite{wang2020explicit}.

Assume there have an initial guess of the solution $\hat{u}_0$ and its  derivative $\hat{l}^i_0$ which can be fitted by any classical method, like Gaussian process regression and basis function expansion.
Here the hat notation $\hat{\cdot}$ represents an estimation and we are only interest in certain derivative terms which is useful for lineariztion.
So the $i$th derivative term $\hat{l}^i$ is used to represent all derivatives of interest for simplicity.
Then any nonlinear term of the nonlinear equation, which is composed of $K$ solution or its derivative terms, can be simply approximated by replacing $K-1$ terms with the corresponding given initial guess of the solution $\hat{u}_0$ and its derivatives $\hat{l}^i_0$, and only keep one term.
This is a quite straightforward linearlization strategy and can be easily applied.
There is a trick that replacing the solution or low order derivatives, would be preferred than the one who replacing the high order derivatives of the nonlinear term in the approximation.
That is because the instability of numerical differentiation of the fitted function solely from the data which is the measurement of solution with adding noise, will be more serious in estimating higher order derivative. The linearization of $\mathcal{F}(u(\bm{x}); {\bm{\theta}})$ is expressed as
\begin{equation}
\label{eq:linear}
\mathcal{F}(u, l^i,\dots;  \hat{u}_0, \hat{l}^i_0,\dots; {\bm{\theta}}) = 0.
\end{equation}
Based on linearization in the Eq. \eqref{eq:linear}, GPRC introduced in Section \ref{subset:gprc} can be used to estimate a new solution $\hat{u}$ and derivative $\hat{l}^i$.
Then an iterative scheme can be constructed with the new pair $\{\hat{u}, \hat{l}^i\}$.
Roughly speaking in the $n$th cycle, given the current guess of solution $\hat{u}_n$ and derivatives $\hat{l}^i_n$, we can compute a new (and possibly better) approximation $\{\hat{u}_{n+1}, \hat{l}^i_{n+1} \}$ using
\begin{equation}
\label{eq:picard_iteration}
\{\hat{u}_{n+1}, \hat{l}^i_{n+1} \}= \textbf{GPRC}(\mathcal{F}(u, l^i, \dots; \hat{u}_n, \hat{l}^i_n,\dots;{\bm{\theta}}), \-y),
\end{equation}
where $\textbf{GPRC}(\cdot)$ indicates the estimator of GPRC whose output is the posterior mean functions of the solution and its derivatives.
Our stopping criterion is the sum of RMSE values of solution and the RMSE values of all derivatives of interest, which is expressed by
\begin{equation*}
e = \int (||\hat{u}_{n+1} - \hat{u}_{n}||^2 + \sum_i ||\hat{l}^i_{n+1} - \hat{l}^i_{n}||^2)p({\bm{x}}) d{\bm{x}},
\end{equation*}
where $|| \cdot||^2$ denotes the L2 normalization. If $e$ is small than a give small positive threshold $\epsilon_{rmse}$,  then we think the iteration reaches stability.

\section{Inference}
\label{set:inference}
We are interest in the problem of estimating the unknown parameters $\bm{\theta}$ from the indirect observations ${\bm{y}}$  and the DE constraint. So for inferring the unknown parameters, both data equations \eqref{y} and DE constraint equation \eqref{eq:pde} should be satisfied.

In Bayesian setting, the prior belief about the parameter $\bm{\theta}$ is encoded in the probability distribution $\pi_{prior}({\bm{\theta}})$. Here we use a uniform distribution as prior for highlighting the action of likelihood function. Our aim is to infer the distribution of ${\bm{\theta}}$ conditioned on the given data $\bm{y}$, i.e., the posterior distribution $\pi({\bm{\theta}}|\-y)$. By the Bayes' rule, we have
\begin{equation}
\label{eq:posterior_mcmc}
\pi({\bm{\theta}}|\-y)\propto \exp(-\alpha*\eta({\bm{\theta}};\-y;\mathcal{F}(\cdot)))*\pi_{prior}({\bm{\theta}}),
\end{equation}
where $\alpha$ is a scale hyperparameter, which is used to adjust the likelihood value to a reasonable scale, and $\eta({\bm{\theta}};\-y;\mathcal{F}(\cdot))$ is called potential of the likelihood $\exp(-\alpha*\eta({\bm{\theta}};\-y;\mathcal{F}(\cdot)))$, specific
\begin{equation}
\label{eq:potential}
\eta({\bm{\theta}};\-y;\mathcal{F}(\cdot)) = \frac{1}{2}||\-y -  \hat{u}(X;{\bm{\theta}})||^2 +\frac{1}{2}\int [\mathcal{F}(\hat{u}(\-x;{\bm{\theta}});{\bm{\theta}})]^2 p(\bm{x})d\bm{x}.
\end{equation}
Fidelity to the PDE can be measured by $\int [\mathcal{F}(\hat{u}(\-x;{\bm{\theta}});{\bm{\theta}})]^2 p(\bm{x})d\bm{x}$, while fidelity to the data can be measured by
$||\-y -  \hat{u}(X;{\bm{\theta}})||^2$.  Equal weights for data and equation information are assumed in E.q \eqref{eq:potential}.

The unnormalized posterior \eqref{eq:posterior_mcmc} can be easily sampled using MCMC method such as  Metropolis-Hastings (MH) algorithm \cite{chib1995understanding}, Gibbs sampling \cite{gilks1992adaptive} and DRAM \cite{haario2006dram} et al.
Here we use MH sampling algorithm to draw samples from the unnormalized posterior \eqref{eq:posterior_mcmc}.
The Metropolis-Hastings (MH) algorithm is one of the most popular MCMC methods.
An MH step of invariant distribution $\pi({\bm{\theta}})$ and proposal distribution  $ \pi_q({\bm{\theta}}^{i+1} |{\bm{\theta}}^i) $ involves sampling a candidate value ${\bm{\theta}}^{i+1}$ given the current value ${\bm{\theta}}^i$  according to $ \pi_q({\bm{\theta}}^{i+1}| {\bm{\theta}}^i)$.
The Markov chain then moves towards ${\bm{\theta}}^{i+1}$ with acceptance probability $\mathcal{A}({\bm{\theta}}^{(i)},\bm{\theta}^{i+1})=\min\{1,\frac{\pi({\bm{\theta}}^{i+1}|\-y)q({\bm{\theta}}^{(i)}|{\bm{\theta}}^{i+1})}{\pi({\bm{\theta}}^{(i)}|\-y)q({\bm{\theta}}^{i+1}|{\bm{\theta}}^{(i)})}\}$, otherwise it remains at ${\bm{\theta}}^i$.
In our work, we draw from $\pi({\bm{\theta}}|\-y)$ using MH sampling:
\begin{enumerate}
\item Initialise $\bm{\theta}^{(0)}$
\item For $i$ = 0 to $N$ -1\\
(a) Sample ${\bm{\theta}}^{i+1}$ from the proposal distribution $\pi_q(\bm{\theta}^{i+1}|\bm{\theta}^{(i)})$.\\
(b) Compute $\mathcal{A}({\bm{\theta}}^{(i)},\bm{\theta}^{i+1})=\min\{1,\frac{\pi({\bm{\theta}}^{i+1}|\-y)\pi_q({\bm{\theta}}^{(i)}|{\bm{\theta}}^{i+1})}{\pi({\bm{\theta}}^{(i)}|\-y) \pi_q({\bm{\theta}}^{i+1}|{\bm{\theta}}^{(i)})}\}$ and sample $u$ from $\mathcal{U}[0,1]$.\\
(c) If $u < \mathcal{A}({\bm{\theta}}^{(i)},\bm{\theta}^{i+1})$ then accept ${\bm{\theta}}^{i+1}$.  Otherwise, set ${\bm{\theta}}^{(i+1)}={\bm{\theta}}^{(i)}$.
\end{enumerate}

\section{Numerical implementation}
The GPRC method can be used for estimating the solution and derivatives in either ODE or PDE problems \cite{wang2020explicit}.
Here we investigate the parameter estimation problem in three scenarios: linear ODE, nonlinear ODE and nonlinear PDE, to verify the effectiveness of our method and make a comparison with two-stage method.
It's worth noting that only one iteration is applied in Picard linearization procedure in Eq. \eqref{eq:picard_iteration} because we only replace the solution $u$ in nonlinear terms that is always estimated well directly from data.
In two-stage method, we also use Gaussian process regression as statistical model to approximate the state and derivatives from the observations \cite{rai2019gaussian}.
Same Gaussian process based regression model, used in two-stage method and our method can eliminate the error caused by the model difference.
In two-stage method, the data information in Eq. \eqref{eq:potential} is fixed due to the solution and its derivative estimations would not vary with different $\bm{\theta}$.
Then the unknown parameters can be detected by only minimizing the parametric equation constraint residual.
So the potential is reduced to
\begin{equation*}
\eta({\bm{\theta}};\mathcal{F}(\cdot)) = \int [\mathcal{F}(\hat{u}(\-x;{\bm{\theta}});{\bm{\theta}})]^2 p(\bm{x})d\bm{x}.
\end{equation*}
The same radial basis function (RBF) kernel are shared in both GPRC and GPR (used in two-stage method),
\begin{equation*}
k_{uu}({\bm{x}}, {\bm{x}}',\bm{\gamma}) = \gamma^2_\alpha \exp(-\frac{1}{2}\sum_{d=1}^D\gamma_d({\bm{x}}_d-{\bm{x}}_{d}')^2).
\end{equation*}
with hyper-parameter $\gamma^2_\alpha$ and $\gamma_d$.
The kernel $k_{lu}({\bm{x}}, {\bm{x}}',\bm{\gamma})$ can be computed based on \eqref{eq:cov_ru}. For instance the kernel of first order derivative term $\frac{du(t)}{d t}$ in ODE can be expressed as:
\begin{equation*}
k_{lu}(t, t',\bm{\gamma}) = \frac{k_{uu}(t, t',\bm{\gamma})}{d t}=-k_{uu}(t,t',\bm{\gamma})\gamma_d(t-t').
\end{equation*}
The expression of other kernel for a general differential operator can be computed similarly.
\label{set:numerical}
\subsection{$1$D toy example}
First we consider the following  $1$-dimension linear ordinary differential equation,
\begin{equation}
\label{eq:exam1ode}
u''(t)+\theta_1u'(t)+\theta_2u(t)=0.
\end{equation}
where $u$ is a function of time $t$, $'$ and $''$ denotes first  and second order derivative operators, respectively.
${\bm{\theta}} = [\theta_1, \theta_2]$ is the unknown parameters which need to refer.
Our true parameter values are taken as $\theta^*_1=1$,$\theta^*_2=3$. Then $21$ observations are evenly measured in the time interval $[0,10]$ with Gaussian noise $\mathcal{N}(0, \sigma^2_{obs}=0.1)$.  The observations and the true solution $u$ are shown in Fig. \ref{fig:linearode}.
\begin{figure}[htpb]
\centering
\includegraphics[width=0.48\linewidth]{figs/predict_u.eps}
\caption{Illustration of the data and the true solution}
\label{fig:linearode}
\end{figure}
We assume the prior of the sate function $u$ is a zero-mean Gaussian process expressed in \eqref{eq:GPprior}.
As discussed in Section \ref{set:method}, the equation constraint $r = u'' + \theta_1 u' + \theta_2u$ is also a Gaussian process, $
r\sim \mathcal{GP}(0,k_{rr}({\bm{x}},{\bm{x}}';\bm{\gamma})).$
With the property of covariance, the covariance between $r$ and $u$ can be expanded as
\begin{align*}
{\mathrm{Cov}}(r,u) &={\mathrm{Cov}}(u'' + \theta_1u' + \theta_2u,u)\\
&={\mathrm{Cov}}(u'',u) + \theta_1{\mathrm{Cov}}(u',u)+\theta_2{\mathrm{Cov}}(u,u).
\end{align*}
So,  $k_{ru} = k_{u''u} + \theta_1k_{u'u}+\theta_2k_{u,u}$.
 Similarly, the kernel functions
 corresponding to covariances $
   {\mathrm{Cov}}(r,u'), {\mathrm{Cov}}(r,u'')$ and ${\mathrm{Cov}}(r,r)$ are expressed as
 \begin{align*}
 k_{ru'} &= k_{u''u'} + \theta_1k_{u'u'}+\theta_2k_{u,u'},\\
 k_{ru''} &= k_{u''u''} + \theta_1k_{u'u''}+\theta_2k_{u,u''},\\
 k_{rr}& = k_{u'' u''} + {\theta_1}^2 k_{u'u'} + {\theta_2}^2  k_{uu} + 2{\theta_1} k_{u''u'}+2{\theta_2} k_{u''u}+ 2{\theta_1}{\theta_2} k_{u'u}.
\end{align*}

By equation \eqref{eq:poster_u} and \eqref{eq:poster_l},
 the posterior distributions $p(u_*|{\bm{y}}, \bm{r}_{\chi}) = {\mathcal{N}}(m_u, \Sigma_u)$, $p(u_*'|{\bm{y}}, \bm{r}_{\chi}) = {\mathcal{N}}(m_{u'}, \Sigma_{u'})$ and $p(u_*''|{\bm{y}}, \bm{r}_{\chi}) = {\mathcal{N}}(m_{u''}, \Sigma_{u''}) $ are  given below
\begin{align*}
m_{u_*} &= K_{\bullet u_*}^T\widehat{K}_{u\bm{r}_{\chi}}^{-1}Y,
\quad
\Sigma_{u_*} =K_{u_*} - K_{\bullet u_*}^T\widehat{K}_{u\bm{r}_{\chi}}^{-1}K_{\bullet u_*}\\
m_{u'_*} &= K_{\bullet u_*'}^T\widehat{K}_{u\bm{r}_{\chi}}^{-1}Y,
\quad
\Sigma_{u'_*} = K_{u'_*} - K_{\bullet u_*'}^T\widehat{K}_{u\bm{r}_{\chi}}^{-1}K_{\bullet u_*'}\\
m_{u''_*} &= K_{\bullet u_*''}^T\widehat{K}_{u\bm{r}_{\chi}}^{-1}Y,
\quad
\Sigma_{u''_*} = K_{u''_*} - K_{\bullet u_*''}^T\widehat{K}_{u\bm{r}_{\chi}} ^{-1}K_{\bullet u_*''},
\end{align*}
where   $Y = [{\bm{y}}, \bm{r}_{\chi}]^T$.
This is  the posterior estimation of state and derivative functions.
In GPRC algorithm, we set $\sigma_r^2= 10$  and $\chi= [t_*]$, which means only thee predicted point is included in the extend set $\chi$. Although we would sacrifice some precision of prediction due to the small extend set, this setting can efficiently speed up the calculation.
Then we can obtain the posterior estimation of solution and its derivatives within any given ${\bm{\theta}}$.
A Gaussian proposal distribution $q({\bm{\theta}}^*|{\bm{\theta}}^{(i)})=\mathcal{N}({\bm{\theta}}^{(i)},0.6^2)$ is used as the proposal distribution in our MH sampler with $\alpha = 100$ in our likelihood function \eqref{eq:posterior_mcmc}.
$5000$ samples are drawn from the posterior.
As a comparison same number of samples are obtained by two-stage method and the both posterior results are shown in  Figure \ref{fig:linearode_posterior}.
Obviously from the figure on the left, we can find that the samples (red scattered points) computed by our method have a wider variance, which means a bigger uncertainty of the posterior is estimated.
In the marginal pdf figures, the first dimension marginal pdf value at $\theta_1^*$ computed by our method (red line) are larger than one computed by two-stage method (blue dash)  and the second dimension marginal pdf values computed by two method at $\theta_1^*$ are equal. These results indicate the posterior estimation computed by our method is more credible.
This point would be better clear in a small-noise case because the bias caused by the numerical error of derivatives estimation would be highlights when the measure noise is small.

We then consider the small-noise case where we use the same implementation configurations as in the large-noise case.
We change the noise level from $0.1$ to $0.05$ and collect the same number of observations at same locations.
The samples from posterior are shown in Figure \ref{fig:linearode_posterior} (right).
Similar to the large-noise case, the figure shows that the results of the two-stage are of low accuracy, while our method yield rather good results.

\begin{figure}[htpb]
\centering
\includegraphics[width=0.48\linewidth]{figs/ex1posterior_largenoise.pdf}
\includegraphics[width=0.48\linewidth]{figs/ex1posterior_smallnoise.pdf}
\caption{Posterior samples and marginal pdfs, computed by our method (red) and two-stage method (blue) in different noise levels (Left: $\sigma^2_{obs}=0.1$; and Right: $\sigma^2_{obs}=0.05$). Posterior samples are shown in the scatter diagram and true parameter (black line) and two marginal pdfs are drawn in subplots corresponding to different noise levels. White star is our true parameter location;}
\label{fig:linearode_posterior}
\end{figure}

\subsection{Van der Pol equation}
The Van der Pol model  is described by 1-dimensional nonlinear ODE equation,
\begin{equation}
\label{eq:vande}
u''- \mu(1-u^2)u' + u=0,
\end{equation}
where $\mu$ is the unknown parameter. For this nonlinear differential equation, we first use the GPR method to obtain initial guess of the  solution $ \hat{u}_0$ only from data.
Then we can obtain a linearizaiton form of \eqref{eq:vande}, by using the approximation $\hat{u}_0$ to replace the original $u$ of nonlinear term, expressed as
\begin{equation*}
u''+\mu(1-\hat{u}_0^2)u'+u=0.
\end{equation*}
In this example 41 observations are measured at every $0.5$ time units on the interval [0,20] with given true parameter $\mu^*=0.5$.
The clean data are corrupted with additive Gaussian noise with variance $\sigma_{obs}^2=0.01 $.
The figure on the left in Fig \ref{fig:vdp_equation} shows the true solution $u$ and the observations.
In GPRC,  $\sigma_r^2=0.1$ is set and $10$ points which are evenly selected in the domain $[t_*-1, t_*+1]$, form the extend set $\chi$ .  In the MH sampling procedure, the  design of the proposal distribution $q(\theta^*|\theta^{(i)})$ is $\mathcal{N}(\theta^{(i)},0.6^2)$.
The histogram of the posterior samples computed by both our method and two-stage method are shown in the Fig. \ref{fig:vdp_equation} (right).
In the figure, the true parameter (black line) are closer to the mean of red histogram of samples computed by our method than the one of blue histogram, computed by two-stage method.
\begin{figure}[htpb]
\centering
\includegraphics[width=0.45\linewidth]{figs/illustration.eps}
\includegraphics[width=0.5\linewidth]{figs/posterior.pdf}
\caption{Left: the solution $u$ (black line) with true parameter $\mu^*=0.5$ and our noisy observations (red star); Right: histograms of the posterior samples computed by our method (red) and two-stage method (blue), separately. Black line is our true parameter $\mu^*=0.5$; }
\label{fig:vdp_equation}
\end{figure}

\subsection{The KdV Equation}
The KdV equation is a nonlinear, dispersive partial differential equation for function $u$ of two real variables, space $x\in \mathbb{R}^{1\times1}$ and time $t$:
\begin{equation*}
\frac{\partial u}{\partial t} + \theta_1 u \frac{\partial u}{\partial x} + \theta_2 \frac{\partial^3 u}{\partial x^3} = 0,
\end{equation*}
It is an asymptotic simplification of Euler equations used to model waves in shallow water.
It can also be viewed as Burger's equation with added dispersive term.
In this equation two parameters need to be estimated from some noisy observations of function $u$. The true solution is created using a spectral method with $512$ spatial points and $200$ timesteps. More details about this model refer \cite{rudy2017data}.
This is a highly non-linear PDE. So $200$ solution values are randomly selected from the given discrete solution in the time domain $[15, 20]$ and spatial domain $[-21, 26]$, within our true parameters are $\theta^*_1=6$ and $\theta^*_2=1$.
A Gaussian noise is added on the selected solution values with zero mean and $0.1$ variance.
The true solution and the observations are shown in the Fig. \ref{fig:example3_post} (left).

In MH sampler procedure, $5000$ samples are obtained by the given proposal distribution $\mathcal{N}(0, 0.6I)$.  The posterior samples and marginal distributions are shown in right figure of Fig. \ref{fig:example3_post}.
As shown, the true parameter (white star) is closer to the center of the posterior samples computed by our method than the one of two-stage method. In addition, at the true parameters $\theta^*_1$ and $\theta^*_2$, the values of marginal PDF computed by our method are larger than the one computed by two-stage method, express that our method yields a better posterior estimation.
\begin{figure}[htpb]
\centering
\includegraphics[width=0.48\linewidth]{figs/kdv.eps}
\includegraphics[width=0.45\linewidth]{figs/ex2posterior.eps}
\caption{Left: the illustration of true solution and the observations (black scattered points); Right: posterior samples (scattered points) and two corresponding marginal pdfs computed by our method ( red) and two-stage method ( blue). White star is our true parameter;}
\label{fig:example3_post}
\end{figure}

\section{Conclusion}
\label{set:conclusion}
In conclusion, we make use of a powerful tool: the Gaussian Process Regression with Constraint method (GPRC) to the problem of parameter estimation.
We propose a new potential function which combines the data information and equation information, by using the accurate estimation of solution and its derivatives computed by GPRC.
Then posterior samples are drawn from the product of prior and likelihood with this type potential by MH sampler.
A two-stage method are also illustrated in each example for comparison.
With both linear and nonlinear examples, we illustrate that the proposed method can be a competitive tool to estimate the unknown differential equation parameters.
Finally we discuss some limitations of the proposed method as well as some issues that need to be address in the future.
Firstly for any given parameters, the GPRC would estimate the solution and derivatives by a training procedure which is time-costing.
Maybe an efficient surrogate can be constructed for the likelihood and  an adaptive procedure\cite{wang2018adaptive} can be employed for efficient sampling in future.
Secondly, the observations with large bias would lead to ineffectiveness of our method.
GPRC would reduce the numerical error of derivative operator by considering the DE information, but the biased observations probably let it fail.

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