text:
U.S. Patriot Act
In: Historical Events
Submitted By Tray55
U.S.A. Patriot ACT
(Traylyn Redman)
September the 11th 200l is a day that we all know well and remember for the rest of our lives. It is a day that will down in U.S. history as one of the most historic and tragic events in our history. One response enlighten of these events that was put in place was the USA PATRIOT ACT. The reason why the USA PARTIOT ACT was signed is mainly because of poor intelligence that the U.S. has had prior to the 9/11 terrorist attacks. In addition to have a plan and reanalyze our intelligence we can make sure this attack would never happen again on American soil.
The patriot act is a critical and historical event in itself. The act was put into place very quickly after 9/11. The government and President Bush had seen the U.S. as a whole was a strong country but still had weaknesses and were vulnerable to attacks without taking swift action. The act was signed on October 26, 2001, just one month after the 9/11 attacks. The patriot act was put in place for many reasons and has been a benefit and various ways. There has still been criticism and skepticism of the act in just the short term of the act being signed. The act of information sharing is one factor that the critics argue. The critics argument that this information would get into the wrong hands or be used by other agencies for unrelated or wrong reasons. The act would permit wiretaps on terrorist or even any potential terror threat over more than one jurisdiction. This would make sense considering it would provide information and even faster response to prevent future attacks. The critics claimed that this would eventually lead to abuse and over use of what the main goal was intended for.
The act would permit quick searches like sneak and peak warrants which would allow the suspected criminal of terror to a search of home and valuables without any notice. The critics...
...U.S. Patriot Act Damian Coles AJS/552 June 14, 2012 Drew Christensen U.S. Patriot Act A discussion will be made to consider the advantages, and disadvantages of making the USA Patriot Act otherwise known as the patriot act a mandatory statue. As of now some of the patriot act has expired while other parts have been voted on and extended. To fully grasp why or why not the patriot act should become a permanent statue in American law. In order to fully understand the patriot act the reason for its existence and definition of what it is must be understood. It came into place after the horrific actions of a terrorist group on September 11, 2001. On that day countless amounts of Americans lives would be changed forever. Terrorist apprehended and attempted to crash planes into high value targets all at the same time. They were successful with one of the attacks being thwarted by the inhabitants of the airplane. It was an attack spearheaded by the terrorist organization Al Qaeda. The leader at the time of the attacks was an Osama Bin Laden, and he ordered the attacks. After these attacks on American soil the people and the government wanted affirmative action to be taken. The perpetrators of this heinous crime needed to be apprehended and an attack like this needs to be avoided. So the patriot act was created by congress and signed by former President George W. Bush and made a law on October 26, 2001. The law was unanimously voted into action by the House of Representatives...
...The USA PATRIOT ACT, commonly known as the Patriot Act, is an acronym for Uniting and Strengthening America by Providing Appropriate tools required to Intercept and Obstruct Terrorism Act of 2001. On October 23, 2001, Republican Rep. Jim Sensenbrenner introduced H.R. 3162 incorporating provisions from a previously sponsored House bill and a Senate bill also introduced earlier in the month. The next day on October 24, 2001, the Act passed the House 357 to 66, with Democrats comprising the overwhelming portion of dissent. The following day on October 25, 2001, the Act passed the Senate by 98 to 1, with Russ Feingold the only opposition. This act was signed then into law by former president George W. Bush on October 26, 2011, just six weeks after the September 11th attacks to the World Trade Center and the Pentagon. This act reduced restrictions in law enforcement agencies' gathering of intelligence with the U.S. The act also expanded the definition of terrorism to include domestic terrorism. The Patriot Act made many changes to U.S. law, such as the Foreign Intelligence Surveillance Act of 1978 (FISA), the Electronic Communications Privacy Act of 1986 (ECPA), the Money Laundering Control Act of 1986 and Bank Secrecy Act (BSA), as well as the Immigration and Nationality Act. It was strongly opposed by Senator, Russ Feingold; he was the only Senator to vote against the bill. The final Act had a number of sunsets that was supposed to expire on December 15, 2005. On May 26, 2011,......
Effects of Usa Patriot Act on Banking Privacy
...Effects of USA PATRIOT Act on Banking Privacy Introduction On September 11, 2001, the United States was attacked by the terrorists and the attack has completely changed the way we live and work. Its impact is so immense that it covers almost every aspect of our life including the privacy protection policy in the banking industry. After the September 11 terrorist attack, the U.S. Congress passed a law, the USA PATRIOT Act that makes it easier for government law enforcement and intelligence agencies to gather and share information related to terror-related investigations and it has changed how the banking industry or financial institutions handle the privacy of their customers' personal information. The purpose of this research paper is to explore the effects of the USA PATRIOT Act on banking industry's handling customers' private personal information. Some Background Information and History of Banking Privacy The USA PATRIOT Act is not an official title of the law. It is the acronym of the very long title of the Act: Uniting and Strengthening America by Providing Appropriate Tools Required to Intercept and Obstruct Terrorism (USA PATRIOT) Act of 2001. It was signed into law by President George W. Bush on October 26, 2001. But, before 2001, do we have any law that provides guidelines for the privacy of banking industry in the United States of America? Surely, there are several laws that are related to the financial institutions and the privacy protection......
Information Technology Act
...Information Technology Act Greg Bennett BIS/220 January 15, 2013 Nikia Johnson Information Technology Act Information technology continues to forge ahead and has done so from day one. Technology, moving forward, and becoming more advanced and sophisticated, adds growing concern over the ethical integrity of that technology. In correlation with those concerns, the United States government implemented various acts to aid control in these concerns and combat unethical behavior. Given the number of implemented acts, this paper focuses on two specific sections: the Fair Credit Reporting Act of 1970, and the U.S. Patriot Act of 2001. The implementation of the Fair Credit Reporting Act of 1970 enabled each American to be on an even plane with credit opportunities. The government executed the U.S. Patriot Act in 2001, in part because of the atrocity of 9/11, to lessen terrorist activities by the use of the Internet and cyberspace. Fair Credit Reporting Act of 1970 The Fair Credit Reporting Act (FCRA) ordained to "protect consumers from the disclosure of inaccurate and arbitrary personal information held by consumer reporting agencies" (Consumer Privacy Guide (2001). The Fair Credit Reporting Act (1970). 2001). At the time when this act was established, American citizens were not viewed equally by credit facilities when attempting to purchase homes or vehicles. This act was a means of governing and guaranteeing equal opportunity and consideration for......
The Usa Patriot Act--Good or Evil?
...THE USA PATRIOT ACT--GOOD OR EVIL? Abstract Terrorism has been around in one form or another since long before September 11th 2001. It is our extreme reaction to the real threat of a terrorist attack began 9/11/2001. Although the odds of dying from a terrorist attack in America are extremely remote, after the attacks of 9/11/2001 and the overwhelming show of concern by American citizens, Congress hastily put together a bill outlining the workings of the U.S. Patriot Act. The United States Patriot Act of 2001 was signed into law by President George W. Bush in 2001. President Barack Obama signed the Patriot Sunsets Extension Act of 2011, which created a 4 year extension of 3 key parts of the Patriot Act. Warrantless searches of business records, roving wiretaps, & conducting surveillance of individuals suspected of terrorist-related activities not linked to any specific organized terrorist groups. The Patriot Act dramatically reduced restrictions on law enforcement agencies' ability to search all records, documents, histories, etc., without the restraints of the normal operations of judicial law which most citizens assume will always protect them. The National and State Governments and their agencies work together to implement the Patriot Act with as minimal an "obvious" intrusion as possible into ours, the American citizens lives. It is obvious the security must be there, but with a minimal amount of perceived disruption to constitutionally given freedoms and rights.......
...The Patriot act was created due to a world crisis that happened in September 11, 2001 when terrorist invaded the United States of America and hijacked a couple of planes for the purpose to crash them in important places where they would get as many people as possible. One being the Twin towers or also known as the World Trade Center and also the pentagon which was not achieved due that some heroic civilians in the plane that decided to fight for their lives and the lives of others. Four passenger airliners were hijacked by 19 al-Qaeda terrorists so they could be flown into buildings in suicide attacks. Two of those planes,American Airlines Flight 11 and United Airlines Flight 175, were crashed into the North and South towers, respectively, of the World Trade Center Complex in New York City. Within two hours, both towers collapsed with debris and the resulting fires causing partial or complete collapse of all other buildings in the WTC complex, as well as significant damage to ten other large surrounding structures. A third plane, American Airlines Flight 77, was crashed into the Pentagon leading to a partial collapse in its western side. The fourth plane, United Airlines Flight 93, was targeted at Washington, D.C. but crashed into a field near Shanksville, Pennsylvania, after its passengers tried to overcome the hijackers. In total, almost 3,000 people died in the attacks, including the 227 civilians and 19 hijackers aboard the four planes. It also was the deadliest incident......
...The terrorist attacks of September 11, 2001, were of the nature of not easily forgotten. To help counter any future recurrences of such events, the U.S. government enacted what is most commonly known as the "Patriot Act." The full name of this act is "Uniting and Strengthening America by Providing Appropriate Tools Required to Intercept and Obstruct Terrorism (USA Patriot Act) Act of 2001," and was passed only 45 days after the terrorist attacks. There are many arguments on whether or not the Patriot Act is designed to work for the better of the world, or to take away many of the civil liberties that belong to citizens of the United States. On the surface, it seems like the Patriot Act is working towards the right goal: "To deter and punish terrorist acts in the United States and around the world, to enhance law enforcement investigatory tools, and for other purposes." However, many would argue that the last two objectives of the act (the majority of the act) have been taken too far. A closer look at the Patriot Act reveals well intended items, some of which also having consequences. Section 101 of the first title, Enhancing Domestic Security Against Terrorism, of the Patriot Act (USA Patriot Act) creates what is known as the Counterterrorism Fund. This fund is set up to recover costs that are spent in association with terrorism. The only drawback to the fund is that there is no limit to the spending. Section 102 condemns discrimination against Americans of Arab, Muslim,......
...USA PATRIOT ACT OF 2001 The USA PATRIOT Act of 2001 is a 342-page, sprawling piece of legislation that contains more than 150 sections and amends more than 15 federal laws. The law's full name is the Uniting and Strengthening America by Providing Appropriate Tools Required to Intercept and Obstruct Terrorism Act of 2001, hence the acronym USA PATRIOT Act. It deals primarily with combating terrorism and gives the executive branch of the federal government more tools to fight suspected terrorist activity, but it also aroused the anger of civil libertarians. Critics of the act have charged that the government gained the power to investigate and detain persons with little oversight from the courts. In the aftermath of the September 11, 2001, terrorist attacks, U.S. political leaders sought to address terrorism with new vigor. President George w. Bush and Attorney General John Ashcroft presented Congress with proposed legislation on September 17, 2001, that focused on intelligence gathering, immigration, criminal justice, and money laundering. The administration sought new powers to conduct searches of people suspected of terrorism; to detain and deport persons suspected of terrorist involvement; and to remove statutes of limitations on terrorism. In addition, the administration wanted the justice department to have the power to place wiretaps on the phones and computers of anyone suspected of terrorism. This initial proposal became the framework for the USA PATRIOT Act, which...
...Spring 2012 1) The purpose of this paper is to assess the leading issues in the continuing debate over the Patriot Act 1, Patriot Act 2 and Patriot Act 3 and its alleged abuse of our bill of rights by comparing and contrasting the positions between the Government need for them and the people's rights, freedoms and liberties. By offering this assessment, you will learn a lot about what policy making was going on to achieve this type of law. 2) What are the arguments surrounding the use of these extreme policies in the U.S. or are they not extreme? And 3) What are the implications that these policies having or are not having on the lives of Americans? Furthermore, the purpose here is to enhance your researching, writing, and analytical skills. 4) Compare and contrast the leading issues in the current controversial USA Patriot Act (1,2 and 3) situation. Your paper should compare and contrast the leading issues in the current controversial USA Patriot Act situation and how it affects your bill of rights. This assessment should explain the similarities and or differences between the rights of the citizens of the United States, the U.S. Government and the State positions. In addition, you should acknowledge the entities (i.e. U.S. citizens, government civil servants, interest groups, corporate interests…) that are endorsing the controversial USA Patriot Act and how this endorsement correlates, complements or conflicts with the position you maintain as an......
...The Patriot Act and Individual Privacy English Composition II 25 November 2012 The Patriot Act and Individual Privacy Over the last ten years, the Gallup's poll has shown a trending decline in the tolerance for the USA PATRIOT Act (Uniting and Strengthening America by Providing Appropriate Tools Required to Intercept and Obstruct Terrorism Act). Citizens are becoming more protective of their civil liberties than they were immediately following the terrorist attacks on September 11, 2001. Many U.S. citizens feel that the USA PATRIOT Act gives federal authorities excessive power and violates the protection provided to U.S. citizens by the U.S. Constitution. In the Bill of Rights, the Fourth Amendment to the U.S. Constitution protects "the right of the people to be secure in their persons, houses, papers, and effects, against unreasonable searches and seizures…" (U.S. Constitution. Amend. IV). The USA PATRIOT Act is believed to infringe upon U.S. citizen's civil liberties by granting federal authorities liberal room to work around the Bill of Rights. John McKay, former United States Attorney for the Western District of Washington stated that "the Patriot Act simply permits use of this longstanding tool in cases involving terrorism" (2005). In fact, these same liberal laws that make up the USA PATRIOT Act have been used to combat organized crime and drug dealers for many years. Suspected terrorist should fall into this same category and should......
First Amendment Issues
...Due Date: June 28, 2011 Persuasive Speech Outline: The USA PATRIOT ACT I. Introduction A. What if your home was raided by the police, and when you asked to see a warrant, an officer handed you a letter signed by the desk sergeant? What if the same officer informed you that he's taking your cellular phone, because your calls had been monitored? He arrests you, never informs you of the charge, and if released, tells you that you can't discuss the incident with anyone? You being a well versed citizen of your constitutional rights under the law begin to inform the officer of the first ten amendments, called the Bill of Rights, and your rights have been violated. What if the officer told you that two weeks prior the city legislators got together, and decided that catching criminals has become too difficult of a task? So, they got rid of the first, fourth, fifth, and tenth amendments of the Constitution and now citizens only have six? Well, you would probably advise the officer to zealously avoid the narcotics section of the evidence room. However, what if I told you that this ridiculous scenario I just described is in fact current law at a Federal level, and if all of us as citizens continue to turn a blind eye to the impact this law is having on our way of life, we will watch the very rights I have previously stated slowly, but assuredly dissipate. B. The first article of the U.S Military Code of Conduct is "I am an American, ......
...U.S. Government after 9\11 Markel Limbacker Kaplan College Abstract The U.S. Government has made many changes to the way it handles everyday functions. The acts of September 11, know as 9\11 made the government look at things differently. When the two hijacked planes crashed into the twin towers located in the middle of New York City the government woke up from its sleep that day. The security in the U.S. did a complete turnaround and the USA Patriot Act was born. U.S. Government after 9\11 Since the attack on American soil the government issued the USA Patriot Act. This was created to deter and punish terrorist acts in and around the United States. It was also designed to enhanced law enforcement and the way it handles investigations. The USA Patriot Act gives the government more power in controlling attackers. It gives more awareness to those in charge in protecting the government and its people. USA Patriot Act: The act was made to unite and strengthening America by providing appropriate tools required to intercept and obstruct terrorism. This was the main reason the act was created on October 26, 2001. The act allows the government to monitor money laundering, financial crimes, and finances of terrorists groups. It allows the government to monitor banks across the United States. Laws have been created to monitor bank officials inputting data into the system that will alert the government of it's actively. Anything that looks......
911paper
...there is still the public inquiring about the prisoners where would we have to transfer them? This scares those who live near a prison, many are terrorist so what should we do? Clean up and make Alcatraz Island also known as the Rock and put them there? From what we have learned that place is no better. Until there is peace throughout all the lands and countries there will and must be safeguards not just for us but for those who are suicide bombers and other terrorist. When someone threatened U.S. interests and safety, the government started thinking up ways to protect the citizens of our beautiful country. This is where the USA Patriot Act came from. The USA Patriot Act was signed into law by former President George W. Bush on October 26, 2001, just a little over a month after the attacks. It is an acronym that stands for, Uniting (and) Strengthening America (by) Providing Appropriate Tools Required (to) Intercept (and) Obstruct Terrorism Act of 2001. The easiest way to put what this act did was to say that it reduced the restrictions in law enforcement agencies' gathering intelligence with the United States, expanded the Secretary of the Treasury's authority to regulate financial transactions, particularly those involving foreign individuals and entities, and broadened the discretion of...
The Patriot Act; Agree of Disagree
...The Patriot Act: Agree or Disagree Amanda Collier ENG/215 August 27, 2012 Bruce Massis The Patriot Act: Agree or Disagree President George W. Bush signed into law on October 26, 2001 The USA PATRIOT Act. The title of the Act is a letter acronym: Uniting and Strengthening America by Providing Appropriate Tools Required to Intercept and Obstruct Terrorism Act of 2001 (PUBLIC LAW 107–56—OCT. 26, 2001) This Act was written to prevent and punish terrorist acts here in the United States as well as around the world, to help law officers, and to strengthen U.S. measures. Three essays have been studied on this subject, (U.S. Patriot Act – Damian Coles, and The Patriot Act: Wise beyond Its Years – John Ashcroft, and Cassady Pitt - U.S. Patriot Act and Racial profiling: Are there consequences of discrimination?) to come up with this critical argument essay. There are advantages and disadvantages of the Patriot Act. Although The Bill of Rights gives us privacy that could be invaded, is effectual for our country because this helps law enforcement have more resources and more freedom to help. Here are two disadvantages of the Patriot Act to be discussed. First racial profiling and, second America's right to privacy under the Bill of Rights. When someone is considered a criminal or terrorist just because of how they dress, look, or the religion they practice, this is racial profiling. Racial profiling was a problem in the United States even before 9/11 and the Patriot......
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text:
What is Skeleton?
Skeleton, the supportive framework of an animal body. The skeleton of invertebrates, which may be either external or internal, is composed of a variety of hard nonbony substances. The more complex skeletal system of vertebrates is made up of connective tissues. This includes bone and the several fibrous substances that form the joints that help to connect bone to bone and to muscle, it also encloses muscle bundles, and provides support by attaching the internal organs. For a more detailed discussion about the skeleton, go through the article.
Animal Skeleton
In addition to its supportive function, the animal skeletal system may provide protection, facilitate movement, and aid in certain sensory functions. A simple stiff, translucent, and nonliving envelope called a pellicle provides support to the body in many protozoans. In non-moving (sessile) coelenterates, such as coral, whose colonies attain great size, it is achieved by dead structures, both internal and external, which form supporting axes.
In the many groups of animals that are able to move, the movement is achieved either by external structures that are called exoskeletons or by the internal structures that are called endoskeletons. Many of the animals remain in their normal resting positions or erect by means of a specialized skeleton called a hydrostatic skeleton.
The skeleton's protective function is provided by the structures that are situated on the body surface, for example, the shells present on the crabs. These structures carry no muscle and form part of a protective surface armor. The similar structures that help in protecting are scales of fish, spines of echinoderms, the minute needle-like structures of sponges, and the tubes of hydroids. The bones present in the skull of the vertebrate helps to protect the brain. In most vertebrates and invertebrates, many skeletal structures provide protection and a rigid base for the insertion of muscles.
Let us discuss about the skeletal system types.
Types of Skeleton
There are certain types of skeletons that can characterize particular animal phyla, but in an animal, there are a limited number of ways in which it can form its skeleton. The calcareous internal skeleton present in the echinoderms is simply constructed but is not far different from the more elaborate bones of vertebrates. Skeletal fibers of similar chemical composition occur in unrelated animal groups; for example, coiled shells of roughly similar chemical composition are present in gastropods (e.g., snails), brachiopods (e.g., lamp shells), and cephalopods (e.g., chambered nautilus). The mechanical properties of various skeletal types may vary according to the needs of animals of particular habits or size ranges (e.g., aquatic, terrestrial).
The Different Types of Skeletal Systems are:
Hard structures
Semi Rigid structures
Hydrostatic structures
Elastic structures
Buoyancy devices
Calcareous Skeleton
The external shells found in the gastropods and bivalve mollusks are calcareous, stiff, and can be detached from the body. The laminated, or layered, shell grows by marginal and surface additions on the inner side. Muscles are present on the part of the shell, and the body of the animal that helps in the protection of the shell. The cephalopods and protozoans of the order Foraminifera form chambered calcareous shells that are numerous and large where the broken remains of the shells may form into a type of sand that covers large areas of tropical beaches, and the pieces may also convert into the rock. Protozoans of the order Radiolaria from skeletons of silica in the form of very complicated bars.
Coral skeletons are also partly inside and partly outside the animal. Calcareous depositions below a young coral polyp are secreted by the ectoderm, fixed to the surface to which the animal is attached, and thrown up into ridges, which form a cup into which the polyp can contract. Most of the soft tissue is then external to an axial calcareous skeleton, but in rapidly growing corals the skeleton is perforated, and soft tissue lies both inside and outside it.
The starfishes, brittlestars, and crinoids have various types of calcareous ossicles that are present in the mesoderm. These form units that have jointed segments along both the arms, spines that project from the body covering and articulate with ossicles, and calcareous jaws. Less well-organized calcareous deposits stiffen the body wall between the arms of the starfish.
1. What is the study of skeletons called? How many divisions does the human body have?
Ans: Osteology provides detailed information about the study of skeletons.
There are 206 named bones in the human body, these are subdivided into two divisions:
Axial Skeleton: It is formed by the vertical axis that consists of 80 bones. It includes the head, vertebral column, and breastbone.
Appendicular Skeleton: It consists of 126 bones, free appendages, and their respective attachments axial skeleton.
2. What is skeleton? Which are the longest bones and smallest bones in the human body?
Ans: The skeleton definition is as follows, it is the basic structure that provides support to the internal organs present in living beings.
The femur is the longest bone in the human body and stapes the smallest bone in the human body.

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: Made dynamic highstock with PHP data I made a dynamic highstock (highchart) like this example, but with data from PHP. I have tried my code, but the results did not appear. How to make dynamic highstock with PHP?
PHP code :
<?php

include("connection.php");

$data = array();
$count = 0;
$result = mysqli_query($koneksi,"SELECT * FROM data ORDER BY time ASC ") or die ("Connection error");
while($row = mysqli_fetch_array($result)) {
$x = strtotime($row['time']) * 1000;
$y =    (float)$row['temperature'];
$data[] = array($x, $y);
$count++;
}
echo json_encode($data);
mysqli_close($koneksi);
?>

JSON from PHP:
[[1535981121000,40],[1535981432000,32.9],[1535981492000,32.7],[1535981552000,32.6],[1535981618000,32.6],[1535981672000,32.6],[1535981732000,32.6],[1535981793000,32.6],[1535981854000,32.6],[1535981913000,32.5],[1535981993000,32.4],[1535982034000,32.4],[1535982215000,32.4],[1535982287000,32.3],[1535982335000,32.3],[1535982396000,32.5],[1536576266000,45.2],[1536579542000,62.2],[1536653234000,30.2]]

Javascript code:
<script type="text/javascript">
    var chart;
function requestData() {
    $.ajax({
        url: 'chartTemp.php',
        success: function(point) {
            var series = chart.series[0],
                shift = series.data.length > 20;

            // add the point
            chart.series[0].addPoint(point, true, shift);

            // call it again after one second
            setTimeout(requestData, 1000);
        },
        cache: false
    });
}
document.addEventListener('DOMContentLoaded', function() {
    chart = Highcharts.stockChart('container', {
    chart: {
        events: {
            load: requestData
        }
    },

    time: {
        useUTC: false
    },

    rangeSelector: {
        buttons: [{
            count: 1,
            type: 'minute',
            text: '1M'
        }, {
            count: 5,
            type: 'minute',
            text: '5M'
        }, {
            type: 'all',
            text: 'All'
        }],
        inputEnabled: true,
        selected: 0
    },

    title: {
        text: 'Live random data'
    },

    exporting: {
        enabled: true
    },

    series: [{
        name: 'Random data',
        data: data[]
    }]
});

    </script>

I changed the javascript code above from this example

A: You are missing either,
// Set the JSON header (in .php)
header("Content-type: text/json");

Or like,
$.ajax({
method: "GET",
  url: "test.js",
  dataType: "json"
});

Also you have some syntax errors, at the end of javascript code add below (to confirm check in browser's console window),
)};

Your ajax call is returning all of the points in a single json, but the function you are calling looks like don't accept, so check again,
// add the point
chart.series[0].addPoint(point, true, shift);

redpajama_set_name: RedPajamaStackExchange


text:
United Democratic Alliance
Revision as of 06:27, 9 February 2018 by Leesa (talk | contribs) (→Background)
Daniel Shumba
United Democratic Alliance (UDA) is a political party formed by former Zanu-PF and United People's Party former member and president Daniel Shumba
According to the President of this party UDA.
"We have been on the ground for a while now," he said. "We are launching the party within 14 days.""For the record and in fact, the UDA and it's President Daniel Shumba are not associated with nor are we a partner or in alliance with the said New Patriotic Front," he said."Vengeance, entitlement, and exclusivity in whatever package is not acceptable and will not drive the emancipation of our nation," he said."The nation requires a leadership that is ready to truly deal with the broader economic challenges, repeal of repressive and unconstitutional laws, effectively combat corruption in a non-partisan manner. People want equal access to opportunities and to thrive.""The UDA will work with all progressive Zimbabweans in reclaiming our country. Democracy, constitutionalism, rule of law, inclusiveness are fundamental in fostering a new culture, new order, and a new thinking that is underpinned in Godly principles.
"So, clearly, we are not part of the New Patriotic Front or any attempt to recreate an old order in any form. Soon, in this month of February, we shall be officially launching UDA."[1]
↑ Expelled Zanu PF member forms party, , Published: 5 February 2018 , Retrieved: 8 February 2018
Retrieved from "https://www.pindula.co.zw/index.php?title=United_Democratic_Alliance&oldid=62529"
View edit history of this page.
Contributors: Leesa


What is a European Year?
Since 1983, the EU announces one topic to be the core attention for that year and for 2015, we want the topic to be development, issues of development would benefit from increased exposure and a special budget line.
First proposed at CONCORD's 2011 General Assembly, designating the year 2015 as the European Year for Development would be the first European Year to focus on a topic that lies mainly outside of the borders of the EU. As the end of the Millennium Development Goals (MDGs) fast approaches in 2015, CONCORD feels the dedicated year would give a strong boost to the on-going global process of elaborating a successor framework to the current Millennium Development Goals.
On October 17 at the 2012 European Development Days CONCORD co-organised a panel about the iniative, with partipants from European institutions.
The event brought together a panel of high level and expert speakers who shared their opinions with the audience on why EY2015DEV matters. These experts consisted of MEP Charles Goerens representing the EP; Ms. Mercedes Bresso, the First Vice President of the Committee of the Regions; Ms. Brenda King, representing the EESC; Mr. Peteris Ustubs, representing the Cabinet of Commissioner Piebalgs, and Mr. Andris Gobins, representing the membership of CONCORD. Marius Wanders, Board Member of CONCORD, was moderator.
On 23 October, the European Parliament voted in a large majority to call for 2015 to be a European Year for Development Cooperation. The vote came as part of an important report on EU Development Policy by Member of the European Parmiament, Charles Goerens.
Get involved – NEW web portal launched!
Also follow and add your thoughts to the topic via the #EY2015DEV hashtag on Twitter!


text:
Find information on:1. DDT2. EDTA3. orotic acid4. сarboplatin2.Propose the approximate shift/splitting pattern/etc. for each type of hydrogen thatwould be seen in an 'H-NMR spectrumIdentify the # of each type of each carbon (i.e. how many peaks in the 13Cspectra)3.Identify infrared (IR) active bonds within each molecule4.
Asked Sep 12, 2019
Find information on: 1. DDT 2. EDTA 3. orotic acid 4. сarboplatin 2. Propose the approximate shift/splitting pattern/etc. for each type of hydrogen that would be seen in an 'H-NMR spectrum Identify the # of each type of each carbon (i.e. how many peaks in the 13C spectra) 3. Identify infrared (IR) active bonds within each molecule 4.
Here, we are providing answer only for 1st question. If you need further questions to be answered, please repost the question.
DDT:
Dichlorodiphenyltrichloroethane, commonly known as DDT, is a colorless, tasteless, and almost odorless crystalline chemical compound, an organochlorine. It was originally developed as an insecticide. It is highly hydrophobic and nearly insoluble in water but has good solubility in most organic solvents, fats and Oils.
IUPAC name: 1,1'-(2,2,2-Trichloroethane-1,1-diyl) bis(4-chlorobenzene).
Molecular formula: C14H9Cl5.
Molar mass: 354.49 g/mol.
Structure of DDT:
CI CI CI CI CI
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text:
\section{Introduction and main results}

In the present paper we consider a nonlocal version of the boundary controllability problem for the heat equation in the one dimensional case. The standard problem to find a boundary control for the heat equation is a well-known topic and has been studied by several authors. We refer for example to the pioneer works of MacCamy, Mize and  Seidman \cite{maccamy1968approximate, maccamy1969approximate} and the books of Zuazua \cite{Zua1} and Lions \cite{Lio1}, and the references therein, for a complete analysis and review on this topic.

Next, we describe our problem and state the main result. We consider  the fractional heat equation in the interval $(-1,1)$. That is,
\begin{align}\label{HE-EX}
\begin{cases}
\partial_t u + (-\partial_x^2)^{s} u = 0 & \mbox{ in }\; (-1,1)\times(0,T),\\
u=g\chi_{\mathcal{O}\times(0,T)} &\mbox{ in }\; (\R\setminus (-1,1))\times (0,T), \\
u(\cdot,0) = u_0&\mbox{ in }\; (-1,1).
\end{cases}
\end{align}
In \eqref{HE-EX}, $u=u(x,t)$ is the state to be controlled,  $T>0$ and $0<s<1$ are real numbers, $(-\partial_x^2)^s$ denotes the fractional Laplace operator (see \eqref{fl_def}) and $g=g(x,t)$ is the exterior control function which is localized in a nonempty open subset $\mathcal O$ of ($\R\setminus (-1,1))$. We mention that it has been shown in \cite{War-ACE} that a boundary control (the case where the control $g$ is localized in a nonempty subset of the boundary) does not make sense for the fractional Laplace operator. By \cite{War-ACE}, for the fractional Laplace operator, the classical boundary control problem must be replaced by an exterior control problem. That is, the control function must be localized outside the open set $(-1,1)$ as it is formulated in \eqref{HE-EX}.

We shall show that for every $u_0\in L^2(-1,1)$ and  $g\in L^2((0,T);H^s(\R\setminus (-1,1)))$, the system \eqref{HE-EX} has a unique weak solution $u\in C([0,1];L^2(-1,1))$ (see Section \ref{sec-3}). In that case the set of reachable states is given by
\begin{align*}
\mathcal R(u_0,T)=\left\{u(\cdot, T):\; g\in L^2((0,T);H^s(\R\setminus (-1,1)))\right\}.
\end{align*}
We say that the system \eqref{HE-EX} is null controllable at time $T>0$ if, $0\in \mathcal R(u_0,T)$. The system is said to be exact controllable at $T>0$, if $\mathcal R(u_0,T)=L^2(-1,1)$.
We say that the system \eqref{HE-EX} is exactly controllable to the trajectories at $T>0$, if for any trajectory $\tilde{u}$ solution of \eqref{HE-EX} with initial datum $\tilde{u}_0\in L^2(-1,1)$ and without control $(g\equiv 0)$, and for every initial datum $u_0\in L^2(-1,1)$, there exists a control function $g\in L^2((-1,1);H^s(\R\setminus (-1,1)))$ such that the associated unique weak solution $u$ of \eqref{HE-EX} satisfies
\begin{align*}
u(\cdot,T)=\tilde{u}(\cdot,T),\quad \text{a.e. in }  (-1,1).
\end{align*}
The system is said to be approximately controllable at time $T>0$, if $\mathcal R(u_0,T)$ is dense in $L^2(-1,1)$.
We refer to Section \ref{sec-2} for the definition of the function spaces involved.

We mention that as in the classical local case ($s=1$) discussed in \cite[Chapter 2]{Zua1}, we have the following situation for the nonlocal case.
Since the system is linear, without restriction one may assume that $u_0=0$. In that case, solutions of \eqref{HE-EX} (with $u_0=0$) are of class $C^\infty$ far from $(\R\setminus (-1,1))$ at time $t=T$. This shows that the elements of $\mathcal R(u_0,T)$ are $C^\infty$ functions in $(-1,1)$. Thus, exact controllability may not hold. For this reason we shall study the null controllability of the system. However, it is a well-known result that the null controllability for the classical linear heat equation is equivalent to the controllability to the trajectories. In the fractional case, the above two concepts of controllability are again equivalent as shows our main result.

The following theorem is the main result of the paper.

\begin{theorem}\label{T1}
Let $0<s<1$ and let $\mathcal O\subset(\R\setminus (-1,1))$ be an arbitrary nonempty open set.
Then the following assertions hold.
\begin{enumerate}
\item If $\frac 12<s<1$, then the system \eqref{HE-EX} is null controllable at any time $T>0$.

\item If $0<s\le \frac 12$, then the system \eqref{HE-EX} is not null controllable at time $T>0$.

\item If $\frac 12<s<1$, then the system \eqref{HE-EX} is exactly controllable to the trajectories at any time $T>0$.
\end{enumerate}
\end{theorem}

We mention that in the proof of Theorem \ref{T1}, we shall heavily exploit the fact that the eigenvalues $\{\lambda_n\}_{n\in{\mathbb{N}}}$ of the realization of $(-\partial_x^2)^s$ in $L^2(-1,1)$ with zero Dirichlet exterior condition (see Section \ref{sec-2}) satisfy the following asymptotics (see e.g. \cite{kwasnicki2012eigenvalues}):
\begin{align}\label{lam}
\lambda_n=\left(\frac{n\pi}{2}-\frac{(2-2s)\pi}{8}\right)^{2s}+O\left(\frac{1}{n}\right)\;\text{ as }\, n\to\infty.
\end{align}

Recall that by Theorem \ref{T1}(b), the system \eqref{T1} is not null controllable at time $T>0$, if $0<s\le \frac 12$. It has been recently shown in \cite{War-ACE} that the system is indeed approximately controllable at any time $T>0$. The result obtained in \cite{War-ACE} is more general since it includes the $N$-dimensional case and the fractional diffusion equation, that is, the case where $\partial_tu$ is replaced by the Caputo time fractional derivative of $\mathbb D_t^\alpha u$ or order $0<\alpha\le 1$. Of course, the case $\alpha=1$ includes \eqref{HE-EX}.

The  null controllability from the interior (that is, the case where the control function is localized in a nonempty subset $\omega$ of $(-1,1)$) of the one-dimensional fractional heat equation has been recently investigated in \cite{biccari2017controllability} where the authors have shown that the system is null controllable at any time $T>0$ if and only if $\frac 12<s<1$.  The interior null controllability of the Schr\"odinger and wave equations have been studied in \cite{Umb}. The approximate controllability from the exterior of the super-diffusive system (that is, the case where $u_{tt}$ is replaced by the Caputo time fractional derivative $\mathbb D_t^\alpha u$ of order $1<\alpha<2$, has been very recently considered in \cite{CLR-MW}. The case of the (possible) strong damping fractional wave equation has been investigated in \cite{WaZa}.

The fractional heat equation \eqref{HE-EX} defined in all the real line arises from a probabilistic process in which a particle makes long jumps random walks with a small probability, see for instance \cite{bucur2016nonlocal,Val}. Besides, this type of process occurs in real life phenomena quite often, see for example the biological observations in \cite{viswanathan1996levy}, and the marine predators  in \cite{humphries2010environmental}.

Regarding the exterior control problem, in many real life applications a control is placed outside
(disjoint from) the observation domain $\Omega$ where the PDE is satisfied.
Some examples of control problems where this
situation may arise are but not limited to:
 Acoustic testing, when the loudspeakers are placed far from the
aerospace structures \cite{larkin1999direct};
 Magnetotellurics (MT), which is a technique to infer earth's subsurface
electrical conductivity from surface measurements \cite{unsworth2005new, CWeiss_BvBWaanders_HAntil_2018a};
 Magnetic drug targeting (MDT), where drugs with ferromagnetic particles
in suspension are injected into the body and the external magnetic field is
then used to steer the drug to relevant areas, for example, solid tumors \cite{HAntil_RHNochetto_PVenegas_2018a, HAntil_RHNochetto_PVenegas_2018b,lubbe1996clinical} and Electroencephalography (EEG) is used to record electrical activities in brain \cite{niedermeyer2005electroencephalography,williams1974electroencephalography},
in case one accounts for the neurons disjoint from the brain, one will obtain
an external control problem. Besides, we mention that some preliminaries results about numerical analysis have been obtained  in the recent work \cite{AnKW}.

The study of nonlocal operators and nonlocal PDEs is nowadays a topic with interest to the mathematics and scientist communities due to the numerous applications that nonlocal PDEs provide. A motivation for this
growing interest relies in the large number of possible applications in the modeling of several complex phenomena for
which a local approach turns out to be inappropriate or limiting. Indeed, there is an ample spectrum of situations in
which a nonlocal equation gives a significantly better description than a local PDE of the problem one wants to analyze.
Among others, we mention applications in turbulence, anomalous transport and diffusion, elasticity, image processing, porous media flow, wave propagation in heterogeneous high contrast media (see e.g. \cite{HAntil_SBartels_2017a, BBC,Val,SV2} and their references).
Also, it is well known that the fractional Laplace operator is the generator of the so-called $s$-stable L\'evy process, and it is often used in
stochastic models with applications, for instance, in mathematical finance
(see e.g. \cite{AnKW,QDu_MGunzburger_RBLehoucq_KZhou_2013a}).
One of the main differences between nonlocal models and classical PDEs is that the fulfillment of a nonlocal equation at a point involves the values of the function far away from that point. We refer to \cite{BBC,Caf1, Caf3} and their references for more applications and information on this topic.

The rest of the paper is structured as follows. In Section \ref{sec-2} we introduce the function spaces needed to study our problem and we give some intermediate known results that are needed in the proof of our main results. In Section \ref{sec-3} we show the well-posedness of the system \eqref{HE-EX} and its associated dual system and we give an explicit representation of the solutions in terms of series for both problems. Finally, in Section \ref{prof-ma-re} we give the proof of our main results.

\section{Preliminary results}\label{sec-2}
In this section we give some notations and recall some known results as they are needed in the proof of our main results.
We start with fractional order Sobolev spaces.

For $0<s<1$ and $\Omega\subset{\mathbb{R}}$ an arbitrary open set,  we let
\begin{align*}
H^{s}(\Omega):=\left\{u\in L^2(\Omega):\;\int_\Omega\int_\Omega\frac{|u(x)-u(y)|^2}{|x-y|^{1+2s}}\;dxdy<\infty\right\},
\end{align*}
and we endow it with the norm defined by
\begin{align*}
\|u\|_{H^{s}(\Omega)}:=\left(\int_\Omega|u(x)|^2\;dx+\int_\Omega\int_\Omega\frac{|u(x)-u(y)|^2}{|x-y|^{1+2s}}\;dxdy\right)^{\frac 12}.
\end{align*}
We set
\begin{align*}
H_0^{s}(\overline{\Omega}):=\Big\{u\in H^{s}(\mathbb R):\;u=0\;\mbox{ a.e. in }\;\mathbb R\setminus \Omega\Big\}
=\Big\{u\in H^{s}(\mathbb R),\;\;\operatorname{supp}[u]\subset\;\overline{\Omega}\Big\}.
\end{align*}
Of course, if $\Omega={\mathbb{R}}$ in the above definition, then we mean that $H_0^{s}(\overline{{\mathbb{R}}})=H^s({\mathbb{R}})$.

We shall denote by $H^{-s}(\overline{\Omega})$ the dual of $H_0^{s}(\overline{\Omega})$, that is, $H^{-s}(\overline{\Omega})=(H_0^{s}(\overline{\Omega}))^\star$.

For more information on fractional order Sobolev spaces, we refer to \cite{NPV,War} and their references.

Next, we give a rigorous definition of the fractional Laplace operator. Let
\begin{align*}
\mathcal L_s^{1}(\mathbb R):=\left\{u:\mathbb R\to\mathbb R\;\mbox{ measurable},\; \int_{\mathbb R}\frac{|u(x)|}{(1+|x|)^{1+2s}}\;dx<\infty\right\}.
\end{align*}
For $u\in \mathcal L_s^{1}(\mathbb R)$ and $\varepsilon>0$ we set
\begin{align*}
(-\partial_x^2)_\varepsilon^s u(x):= C_{s}\int_{\{y\in\mathbb R:\;|x-y|>\varepsilon\}}\frac{u(x)-u(y)}{|x-y|^{1+2s}}\;dy,\;\;x\in\mathbb R,
\end{align*}
where $C_{s}$ is a normalization constant given by
\begin{align}\label{CNs}
C_{s}:=\frac{s2^{2s}\Gamma\left(\frac{2s+1}{2}\right)}{\pi^{\frac
12}\Gamma(1-s)}.
\end{align}
The {\em fractional Laplacian}  $(-\partial_x^2)^s$ is defined for $u\in \mathcal L_s^{1}(\mathbb R)$ by the following singular integral:
\begin{align}\label{fl_def}
(-\partial_x^2)^su(x):=C_{s}\,\mbox{P.V.}\int_{\mathbb R}\frac{u(x)-u(y)}{|x-y|^{1+2s}}\;dy=\lim_{\varepsilon\downarrow 0}(-\partial_x^2)_\varepsilon^s u(x),\;\;x\in\mathbb R,
\end{align}
provided that the limit exists.
We notice that if $u\in \mathcal L_s^{1}(\mathbb R)$, then $ v:=(-\partial_x^2)_\varepsilon^s u$ exists for every $\varepsilon>0$, $v$ being also continuous at the continuity points of  $u$.
For more details on the fractional Laplace operator we refer to \cite{Caf3,NPV,GW-CPDE,War} and their references.

Next, we consider the realization of $(-\partial_x^2)^s$ in $L^2(-1,1)$ with the zero Dirichlet condition $u=0$ in $\R\setminus (-1,1)$. More precisely, we consider the closed and bilinear form $\mathcal F:H_0^{s}(\overline{\Om})\times H_0^{s}(\overline{\Om})\to{\mathbb{R}}$ given by
\begin{align*}
\mathcal F(u,v):=\frac{C_{s}}{2}\int_{\mathbb R}\int_{\mathbb R}\frac{(u(x)-u(y))(v(x)-v(y))}{|x-y|^{1+2s}}\;dxdy,\;\;u,v\in H_0^{s}(\overline{\Om}).
\end{align*}
Let $(-\partial_x^2)_D^s$ be the selfadjoint operator in $L^2(-1,1)$ associated with $\mathcal F$ in the sense that
\begin{equation*}
\begin{cases}
D((-\partial_x^2)_D^s)=\Big\{u\in H_0^{s}(\overline{\Om}),\;\exists\;f\in L^2(-1,1),\;\mathcal F(u,v)=(f,v)_{L^2(-1,1)}\;\forall\;v\in H_0^{s}(\overline{\Om})\Big\},\\
(-\partial_x^2)_D^su=f.
\end{cases}
\end{equation*}
It is easy to see that
\begin{equation}\label{DLO}
D((-\partial_x^2)_D^s)=\left\{u\in H_0^{s}(\overline{\Om}),\; (-\partial_x^2)^su\in L^2(-1,1)\right\},\;\;\;
(-\partial_x^2)_D^su=((-\partial_x^2)^su)|_{(-1,1)}.
\end{equation}
Then $(-\partial_x^2)_D^s$ is the realization of $(-\partial_x^2)^s$ in $L^2(-1,1)$ with the condition $u=0$ in $\R\setminus (-1,1)$. It has been shown in \cite{SV2} that $(-\partial_x^2)_D^s$ has a compact resolvent and its eigenvalues form a non-decreasing sequence of real numbers $0<\lambda_1\le\lambda_2\le\cdots\le\lambda_n\le\cdots$ satisfying $\lim_{n\to\infty}\lambda_n=\infty$.  In addition, if $\frac 12\le s<1$, then the eigenvalues are of finite multiplicity.
Let $\{\varphi_n\}_{n\in{\mathbb{N}}}$ be the orthonormal basis of eigenfunctions associated with the eigenvalues $\{\lambda_n\}_{n\in{\mathbb{N}}}$. Then $\varphi_n\in D((-\partial_x^2)_D^s)$ for every $n\in{\mathbb{N}}$,  $\{\varphi_n\}_{n\in{\mathbb{N}}}$ is total in $L^2(-1,1)$ and satisfies
\begin{equation}\label{ei-val-pro}
\begin{cases}
(-\partial_x^2)^s\varphi_n=\lambda_n\varphi_n\;\;&\mbox{ in }\;(-1,1),\\
\varphi_n=0\;&\mbox{ in }\;\R\setminus (-1,1).
\end{cases}
\end{equation}

Next, for $u\in H^{s}(\mathbb R)$ we introduce the {\em nonlocal normal derivative $\mathcal N_s$} given by
\begin{align}\label{NLND}
\mathcal N_su(x):=C_{s}\int_{-1}^1\frac{u(x)-u(y)}{|x-y|^{1+2s}}\;dy,\;\;\;x\in\mathbb R \setminus\overline{\Om},
\end{align}
where $C_{s}$ is the constant given in \eqref{CNs}.
We notice that since equality is to be understood a.e., then \eqref{NLND} is the same as for a.e. $x\in\R\setminus (-1,1)$.
By \cite[Lemma 3.2]{GSU},  for every $u\in H^{s}(\mathbb R)$, we have that $\mathcal N_su\in H_{\rm loc}^s(\R\setminus (-1,1))$.
We mention that the operator $\mathcal N_s$ has been called "interaction operator" in  \cite{AnKW,QDu_MGunzburger_RBLehoucq_KZhou_2013a}. Several properties of $\mathcal N_s$ have been studied in \cite{DRV}.

The following unique continuation property which shall play an important role in the proof of Theorem \ref{T1} has been recently obtained in \cite[Theorem 16]{War-ACE}.

\begin{lemma}\label{lem-UCD}
Let $\lambda>0$ be a real number  and $\mathcal O\subset(\R\setminus (-1,1))$ an arbitrary  nonempty open set.
If $\varphi\in D((-\partial_x^2)_D^s)$ satisfies
\begin{equation*}
(-\partial_x^2)_D^s\varphi=\lambda\varphi\;\mbox{ in }\;(-1,1)\;\mbox{ and }\; \mathcal N_s\varphi=0\;\mbox{ in }\;\mathcal O,
\end{equation*}
then $\varphi=0$ in $\mathbb R$.
\end{lemma}

For more details on the Dirichlet problem associated with the fractional Laplace operator we refer the interested reader to \cite{BWZ1,Grub,RS2-2,RS-DP,War-ACE} and their references.

We conclude this section with the following integration by parts formula.

\begin{lemma}
Let $u\in H_0^s(\overline{\Om})$ be such that $(-\partial_x^2)^s u\in L^2(-1,1)$. Then for every $v\in H^s({\mathbb{R}})$, the following identity
\begin{align}\label{Int-Part}
\frac{C_{s}}{2}\int_{{\mathbb{R}}}\int_{{\mathbb{R}}}\frac{(u(x)-u(y))(v(x)-v(y))}{|x-y|^{1+2s}}\;dxdy=\int_{-1}^1v(x)(-\partial_x^2)^su(x)\;dx+\int_{\R\setminus (-1,1)}v(x)\mathcal N_su(x)\;dx,
\end{align}
holds.
\end{lemma}

We refer to \cite[Lemma 3.3]{DRV} (see also \cite[Proposition 3.7]{War-ACE}) for the proof and more details.

\section{Well-posedness of the parabolic problem}\label{sec-3}

This section is devoted to the well posedness and the explicit representation in terms of series for solutions to the system \eqref{HE-EX} and its associated dual system.

Throughout the remainder of the article, $\{\varphi_n\}_{n\in{\mathbb{N}}}$ denotes the orthonormal basis of eigenfunctions of the operator $(-\partial_x^2)_D^s$ associated with the eigenvalues $\{\lambda_n\}_{n\in{\mathbb{N}}}$. Furthermore, for a given $u\in L^2(-1,1)$ and $n\in{\mathbb{N}}$, we shall let $u_n:=(u,\varphi_n)_{L^2(-1,1)}$ and for a given set $E\subseteq {\mathbb{R}}^N$ ($N\ge 1)$, we shall denote by $(\cdot,\cdot)_{L^2(E)}$ the scalar product in $L^2(E)$. Moreover, we shall denote by $\mathcal{D}(E)$ the space of all continuously infinitely differentiable functions with compact support in $E$.

\subsection{Representation of solution to the system \eqref{HE-EX}}

We have the following existence and explicit representation of solutions in terms of series.

\begin{theorem}\label{expl-2}
Let $T>0$ be any real number. Then for every $u_0\in L^2(-1,1)$ and \MW{$g\in L^2((0,T);H^s(\R\setminus (-1,1)))$}, the system \eqref{HE-EX} has a unique weak solution $u\in C([0,T];L^2(-1,1))$ given by
\begin{align}\label{4}
u(x,t)=\sum_{n=1}^{\infty}u_{0,n}e^{-\lambda_n t}\varphi_n(x)+\sum_{n=1}^{\infty}\left(\int_0^t (g(\cdot,\tau), \mathcal{N}_s\varphi_n)_{L^2(\R\setminus (-1,1))}  e^{-\lambda_n((t-\tau)}d\tau\right)\varphi_n(x).
\end{align}
\end{theorem}

\begin{proof}
The representation \eqref{4} of solutions to the system \eqref{HE-EX} is contained  in  \cite{War-ACE} for the case where $\partial_t$ is replaced with the Caputo time-fractional derivative $\partial_t^\alpha$ ($0<\alpha<1$), and $g\in\mathcal D((0,T)\times \R\setminus (-1,1))$. For the sake of completeness, we include the full proof.\\

{\bf Step 1}:  Firstly, let  $g\in\mathcal D((0,T)\times \R\setminus (-1,1))$ and let $v$ be the unique weak solution of the Dirichlet problem
\begin{equation}\label{DP}
\begin{cases}
(-\partial_x^2)^{s}v=0\;\;&\mbox{ in }\;\;(-1,1),\\
v=g&\mbox{ in }\;\R\setminus (-1,1).
\end{cases}
\end{equation}
It has been shown in \cite{War-ACE} that the Dirichlet problem \eqref{DP} has a unique weak solution $v\in H^{s}({\mathbb{R}})$ and there is a constant $C>0$ such that
\begin{align}\label{DP-E}
\|v\|_{H^s({\mathbb{R}})}\le C\|g\|_{H^s((\R\setminus (-1,1)))}.
\end{align}
Since $g$ depends on $(x,t)$, then $v$ also depends on $(x,t)$.  If in \eqref{DP} one replaces $g$ by $\partial_tg$,  then the associated unique solution is given by $\partial_tv$. From this, we can deduce that $v\in C^1([0,T]; H^{s}({\mathbb{R}}))$.

Now let $u$ be a solution of \eqref{HE-EX} and set $w:=u-v$. Then a simple calculation gives
\begin{align*}
w_{t}+(-\partial_x^2)^sw=u_{t}-v_{t}+(-\partial_x^2)^su-(-\partial_x^2)^sv=-v_{t} \;\;\mbox{ in }\;(-1,1)\times(0,T).
\end{align*}
In addition
\begin{align*}
w=u-v=g-g=0 \mbox{ in } (\R\setminus (-1,1))\times (0,T)\;\mbox{ and } w(\cdot,0)=u(\cdot,0)-v(\cdot,0)=u_0-v(\cdot,0) \mbox{ in }\;(-1,1).
\end{align*}
Since $g\in H_0^1((0,T);H^s(\R\setminus (-1,1)))$, we have that $v(\cdot,0)=0$ in $\Omega$.  We have shown that a solution $u$ of \eqref{HE-EX} can be decomposed as $u=v+w$, where  $w$ solves the system
\begin{equation}\label{wD}
\begin{cases}
w_{t}+(-\partial_x^2)^sw=-v_{t}\;\;&\mbox{ in }\;(-1,1)\times (0,T),\\
w=0  &\mbox{ in }\;(\R\setminus (-1,1))\times (0,T),\\
w(\cdot,0)=u_0\;\;&\mbox{ in }\;(-1,1).
\end{cases}
\end{equation}
We notice that $v_{t}\in C([0,T];H^{s}({\mathbb{R}}))$. The problem \eqref{wD} can be rewritten as the following Cauchy problem
\begin{equation}\label{C-P}
\begin{cases}
w_{t}+ (-\partial_x^2)_D^{s} w=-v_t\;\;&\mbox{ in }\; (-1,1)\times (0,T),\\
w(\cdot,0)=u_0&\mbox{ in }\;(-1,1).
\end{cases}
\end{equation}
It follows from semigroups theory that the Cauchy problem \eqref{C-P} (hence, the system \eqref{wD}) has a unique weak  solution $w\in C([0,T], L^2(\Omega))$ given by
\begin{align}\label{E1}
w(x,t)=\sum_{n=1}^\infty u_{0,n}e^{-\lambda_nt}\varphi_n(x)-\sum_{n=1}^\infty\left(\int_0^t \left(v_\tau(\cdot,\tau),\varphi_n\right)_{L^2((-1,1)}e^{-\lambda_n(t-\tau)}\;d\tau\right)\varphi_n(x).
\end{align}
Integrating \eqref{E1} by part, we obtain that
\begin{align}\label{E2}
w(x,t)=\sum_{n=1}^\infty u_{0,n}e^{-\lambda_nt}\varphi_n(x)-v(x,t)+\sum_{n=1}^\infty\left(\int_0^t \left(v(\cdot,\tau),\lambda_n\varphi_n\right)_{L^2((-1,1)}e^{-\lambda_n(t-\tau)}\;d\tau\right)\varphi_n(x).
\end{align}
Using the integration by parts formula \eqref{Int-Part}, we get that
\begin{align*}
\left(v(\cdot,\tau),\lambda_n\varphi_n\right)_{L^2((-1,1)}=\left(v(\cdot,\tau), (-\partial_x^2)_D^{s}\varphi_n\right)_{L^2((-1,1)}=-\left(g,\mathcal N_s\varphi_n\right)_{L^2(\R\setminus (-1,1))}.
\end{align*}
Substituting this identity into \eqref{E2} gives
\begin{align*}
w(x,t)=-v(x,t)+\sum_{n=1}^\infty u_{0,n}e^{-\lambda_nt}\varphi_n(x)+\sum_{n=1}^\infty\left(\int_0^t \left(g(\cdot,\tau),\mathcal N_s\varphi_n\right)_{L^2(\R\setminus (-1,1)}e^{-\lambda_n(t-\tau)}\;d\tau\right)\varphi_n(x),
\end{align*}
and we have shown  \eqref{4}. It is straightforward to verify that the series in \eqref{4} converges in $L^2(-1,1)$ uniformly in $[0,T]$. In addition, using \eqref{E1} and \eqref{DP-E}, we get that there is a constant $C>0$ such that for every $t\in [0,T]$,
\begin{align*}
\|u(\cdot,t)\|_{L^2(-1,1)}^2\le  &C\left(\|u_{0}\|_{L^2(-1,1)}^2+t^2\|v_t\|_{L^\infty((0,T);L^2(-1,1))}\right)\\
\le &C\left(\|u_{0}\|_{L^2(-1,1)}^2+t^2\|v_t\|_{L^\infty((0,T);H^s({\mathbb{R}}))}\right)\\
\le &C\left(\|u_{0}\|_{L^2(-1,1)}^2+t^2\|g_t\|_{L^\infty((0,T);L^2(\R\setminus (-1,1)))}\right).
\end{align*}
We have shown that $u\in C([0,T];L^2(-1,1))$.

(b) Secondly, let $g\in L^2((0,T);H^s(\R\setminus (-1,1)))$. Let $\{g_k\}_{k\ge 1}\subseteq \mathcal D((0,T)\times \R\setminus (-1,1))$ be a sequence of smooth functions such that $g_k$ converges to $g$ in $ L^2((0,T);H^s(\R\setminus (-1,1)))$, as $k\to\infty$, and let $u_k$ be the associated weak solution of \eqref{HE-EX}. Then, we have that $u_k\in C^\infty([0,T];L^2(-1,1))$ and satisfies the representation \eqref{4} with $g$ replaced by $g_k$. Passing to the limit, one can show that the limit function $u\in C([0,T];L^2(-1,1))$, satisfies \eqref{4} and is the unique weak solution to the system \eqref{HE-EX}. For the sake of brevity, we will not go into details.
\end{proof}

Theorem \ref{expl-2} is the fractional version of the classical local  heat equation with inhomogeneous boundary data, and it is the so-called \emph{boundary control semigroup formula}. We refer for instance to the book of Lasiecka and Triggiani \cite{lasiecka2000control} and the paper of Fattorini \cite{fattorini1968boundary} for more details on the local case.

\subsection{Representation of solution to the dual system}

Using the classical integration by parts formula, we have that the following backward system
\begin{equation}\label{Dual}
\begin{cases}
-\partial_t \psi +(-\partial_x^2)^s\psi=0\;\;&\mbox{ in }\; (-1,1)\times (0,T),\\
\psi=0&\mbox{ in }\;(\R\setminus (-1,1))\times (0,T),\\
\psi(\cdot,T)=\psi_0&\mbox{ in }\;(-1,1),
\end{cases}
\end{equation}
can be viewed as the dual system associated with \eqref{HE-EX}.

We have the following existence result.
\begin{theorem}\label{theo-48}
Let $T>0$ be a real number and  $\psi_0\in  L^2(-1,1)$. Then the system \eqref{Dual} has a unique weak solution $\psi\in C([0,T];L^2(-1,1))$ which is given by
\begin{align}\label{eq-25}
\psi(x,t)=\sum_{n=1}^{\infty}\psi_{0,n}e^{-\lambda_n(T-t)}\varphi_{n}(x).
\end{align}
In addition the following assertions hold.
\begin{enumerate}
\item There is a constant $C>0$ such that for all $t\in [0,T]$,
\begin{equation}\label{Dual-EST-1}
 \|\psi(\cdot,t)\|_{L^2(-1,1)}\le C\|\psi_0\|_{L^2(-1,1)}.
\end{equation}
\item For every $t\in[0,T)$ fixed, $\mathcal{N}_s \psi(\cdot,t)$ exists, belongs to $L^2(\R\setminus (-1,1))$ and is given by
\begin{align}\label{norm-der}
\mathcal{N}_s \psi(x,t)=\sum_{n=1}^{\infty}\psi_{0,n}e^{-\lambda_n(T-t)}\mathcal{N}_s\varphi_{n}(x).
\end{align}
\end{enumerate}
\end{theorem}

\begin{proof}
 Using the spectral theorem for selfadjoint operators with compact resolvent we are reduced to look for a solution $\psi$ of the form
\begin{align*}
\psi(x,t)=\sum_{n=1}^\infty(\psi(\cdot,t),\varphi_n)_{L^2(-1,1)}\varphi_n(x).
\end{align*}
Replacing this expression in \eqref{Dual} and letting $\psi_n(t):=(\psi(\cdot,t),\varphi_n)_{L^2(-1,1)}$, we get that $\psi_n(t)$ solves the following ordinary differential equation:
\begin{align*}
-\psi_n^{\prime}(t)+\lambda_n\psi_n(t)=0, \;t\in (0,T);\;\;\mbox{ and }\;\psi_n(T)=\psi_{0,n}.
\end{align*}
It is straightforward to show that $\psi$ is given by \eqref{eq-25}. The estimate \eqref{Dual-EST-1} and the identity \eqref{norm-der} can also be easily justified.
\end{proof}

\section{Proof of the main result}\label{prof-ma-re}

In this section we give the proof of the main results of this work, namely Theorem \ref{T1}. For this purpose, we need first to establish some auxiliaries results that will be used in the proof.

\begin{lemma}\label{lem-1}
The system \eqref{HE-EX} is null controllable at time $T>0$ if and only if for each initial datum $u_0\in  L^2(-1,1)$, there exists a control function $g\in L^2((0,T);H^s(\R\setminus (-1,1)))$ such that the solution $\psi$ of the dual system \eqref{Dual} satisfies
\begin{align}\label{NSC-NC}
\int_{-1}^1u_0(x)\psi(x,0)\;dx=\int_0^T\int_{\mathcal{O}} g(x,t)\mathcal{N}_s\psi(x,t)\;dxdt,
\end{align}
for each $\psi_0\in L^2(-1,1)$.
\end{lemma}

\begin{proof}
Let $g\in L^2((0,T);H^s(\R\setminus (-1,1)))$. Multiplying \eqref{HE-EX} by the solution $\psi$ of \eqref{Dual}, integrating by parts and using the integration by parts formula \eqref{Int-Part}, we obtain that
\begin{align*}
0=&\int_0^T\int_{-1}^1\Big(u_t(x,t)+(-\partial_x^2)^su(x,t)\Big)\psi(x,t)\;dxdt\\
=&\int_{-1}^1\Big(u(x,T)\psi(x,T)-u(x,0)\psi(x,0)\Big)\;dx-\int_0^T\int_{-1}^1u(x,t)\psi_t(x,t)\;dxdt\\
&+\int_0^T\int_{-1}^1u(x,t)(-\partial_x^2)^s\psi(x,t)\;dxdt+\int_0^T\int_{\R\setminus (-1,1)}\Big(u(x,t)\mathcal N_s\psi(x,t)-\psi(x,t)\mathcal N_su(x,t)\Big)\;dxdt\\
=&\int_{-1}^1\Big(u(x,T)\psi(x,T)-u(x,0)\psi(x,0)\Big)\;dx +\int_0^T\int_{\R\setminus (-1,1)}u(x,t)\mathcal N_s\psi(x,t)\;dxdt.
\end{align*}
The above identity implies that
\begin{align}\label{Ob-In}
\int_{-1}^1u(x,0)\psi(x,0)\;dx -\int_{-1}^1u(x,T)\psi(x,T)\;dx=\int_0^T\int_{\R\setminus (-1,1)}u(x,t)\mathcal N_s\psi(x,t)\;dxdt.
\end{align}

Now if \eqref{NSC-NC} holds, then it follows from \eqref{Ob-In} that $\int_{-1}^1u(x,T)\psi(x,T)\;dx=0$. Thus, we can deduce that $u(\cdot,T)=0$  in $(-1,1)$  and the system \eqref{HE-EX} is null controllable.

Conversely, if the system \eqref{HE-EX} is null controllable, that is, if $u(\cdot,T)=0$  in $(-1,1)$, then \eqref{NSC-NC} follows from \eqref{Ob-In} and the proof is finished.
\end{proof}

Finally, for the proof of Theorem \ref{T1} we also need the following technical result.

\begin{lemma}\label{uni-bound}
Let $\{\varphi_{k}\}_{k\in{\mathbb{N}}}$ be the orthogonal basis of normalized eigenfunctions associated with the eigenvalues $\{\lambda_k\}_{k\in{\mathbb{N}}}$. Then, for every nonempty open set $\mathcal{O}\subset (\R\setminus (-1,1))$, there exists a scalar $\eta>0$ such that for every $k\in{\mathbb{N}}$,
the function $\mathcal{N}_s\varphi_k$ is uniformly bounded  from below by $\eta$ in $L^2(\mathcal{O})$. Namely,
\begin{align}
\exists\; \eta>0,\; \forall \;k\in{\mathbb{N}},\; \|\mathcal{N}_s\varphi_k\|_{L^2(\mathcal{O})}\geq \eta.
\end{align}
\end{lemma}

\begin{proof}
We prove the result in several steps.

{\bf Step 1}:
Firstly,  since $\varphi_k=0$  in $\R\setminus (-1,1)$ for every $k\in{\mathbb{N}}$, it follows from the definition of the operators $(-\partial_x^2)^s$ and $\mathcal N_s$ that for almost every $x\in\mathcal O\subseteq(\R\setminus (-1,1))$, we have that
\begin{align}\label{eq-eg}
(-\partial_x^2)^s\varphi_k(x)=C_{s}\mbox{P.V.}\int_{{\mathbb{R}}}\frac{\varphi_k(x)-\varphi_k(y)}{|x-y|^{1+2s}}\;dy=C_{s}\int_{-1}^1\frac{\varphi_k(x)-\varphi_k(y)}{|x-y|^{1+2s}}\;dy=
\mathcal N_s\varphi_k(x).
\end{align}
We have shown that $(\mathcal N_s \varphi_k)|_{\mathcal O}=((-\partial_x^2)^s\varphi_k)|_{\mathcal O}$ for every $k\in{\mathbb{N}}$. We notice that \eqref{eq-eg} holds not only for $\varphi_k$, but for all functions in $H_0^s(\overline{\Om})$.

Secondly, let us introduce the auxiliary function $q:{\mathbb{R}}\to[0,\infty)$ defined by:
\begin{align}\label{q_def}
	q(x) := \begin{cases}
		0 & x\in \left(-\infty,-\frac 13\right),
		\\[7pt]
		\displaystyle \frac 92 \left(x+\frac 13\right)^2 & x\in \left(-\frac 13,0\right),
		\\[10pt]
		\displaystyle 1-\frac 92 \left(x-\frac 13\right)^2 & x\in \left(0,\frac 13\right),
		\\[7pt]
		1 & x\in \left(\frac 13,+\infty\right).
	\end{cases}
\end{align}
For any $\alpha>0$, we define the function $F_\alpha:{\mathbb{R}}\to{\mathbb{R}}$ as follows:
\begin{align*}
F_\alpha(x) = F(\alpha x):= \sin\left(\alpha x+\frac{(1-s)\pi}{4}\right)-G(\alpha x),
\end{align*}
where $G$ is the Laplace transform of the function
\begin{align*}
\gamma(y):=\frac{\sqrt{4s}\sin(s\pi)}{2\pi}\frac{y^{2s}}{1+y^{4s}-2y^{2s}\cos(s\pi)}\exp\left(\frac 1\pi\int_0^{+\infty}\frac{1}{1+r^2}\log\left(\frac{1-r^{2s}y^{2s}}{1-r^2y^2}\right)\,dr\right).
\end{align*}
Now, we define the sequence of real numbers
\begin{align*}
\mu_k:=\frac{k\pi}{2}-\frac{(1-s)\pi}{4},\quad k\geq 1.
\end{align*}
It has been shown in \cite[Example 6.1]{kwasnicki2010spectral} that $F_{\mu_k}$ is the solution of the system
\begin{align*}
\begin{cases}
\fl{s}{F_{\mu_k}(x)}=\mu_k F_{\mu_k}(x) & x> 0,
\\
F_{\mu_k}(x)=0 & x\le 0.
\end{cases}
\end{align*}
In other words, $\{F_{\mu_k}\}_{k\geq 1}$ are the eigenfunctions of  $(-\partial_x^2)^s$ on the interval $(0,\infty)$  with the zero Dirichlet exterior condition, and $\{\mu_k\}_{k\geq 1}$ are the corresponding eigenvalues.
Let us now define
\begin{align*}
	\varrho_k(x):=q(-x)F_{\mu_k}(1+x)+(-1)^k F_{\mu_k}(1-x),\quad x\in{\mathbb{R}},\quad k\geq 1.
\end{align*}
Notice that $F_{\mu_k}(1+x)=0$ for $x\leq -1$ and $F_{\mu_k}(1-x)=0$ for $x\geq 1$. This fact, together with the definition \eqref{q_def} of the function $q$ imply that, for all $k\geq 1$, $\varrho_k(x) = 0$ for $x\in{\mathbb{R}}\setminus (-1,1)$. In addition, it follows from \cite[Lemma 1]{kwasnicki2012eigenvalues} that $\{\varrho_k\}_{k\in{\mathbb{N}}}\subset D((-\partial_x^2)_D^s)$ and  there is a constant $C>0$ such that
\begin{align*}
\left| \fl{s}{\varrho_k}(x)-\mu_k^{2s}\varrho_k(x)\right|\leq \frac{C(1-s)}{\sqrt{2s}}\mu_k^{-1},\;\;\textrm{ for all }\;\; x\in (-1,1),\;k\geq 1.
\end{align*}
Furthermore, by \cite[Proposition 1]{kwasnicki2012eigenvalues}, there is a constant $C>0$ such that for every $k\ge 1$, we have
\begin{equation}
\|\varrho_k-\varphi_k\|_{L^2(-1,1)}\le
\begin{cases}
\frac{C(1-s)}{k}\;\qquad&\mbox{ when }\; \frac 12\le s< 1\\\
 \frac{C(1-s)}{k^{2s}}\;&\mbox{ when }\; 0<s<\frac 12.
 \end{cases}
\end{equation}

{\bf Step 2:}
Now, let $\mathcal{O}\subset \R\setminus (-1,1)$ be an arbitrary nonempty open set and assume that for every $\eta>0$ there exists $k\in{\mathbb{N}}$ such that
\begin{align}\label{contr}
\|\mathcal{N}_s\varphi_k\|_{L^2(\mathcal{O})}<\eta.
\end{align}
It follows from \eqref{contr} that there is subsequence $\{\varphi_{k_n}\}_{n\in{\mathbb{N}}}$ such that
\begin{align}\label{contr-2}
\|\mathcal{N}_s\varphi_{k_n}\|_{L^2(\mathcal{O})}<\frac 1n,
\end{align}
for $n$ large enough. Since $L^2(\mathcal O)\subset H^{-s}(\mathcal O)$, it follows from \eqref{contr-2} that there is a constant $C>0$ independent of $n$ such that for $n$ large enough, we have
\begin{align}\label{contr-3}
\|\mathcal{N}_s\varphi_{k_n}\|_{H^{-s}(\mathcal{O})}\le \frac Cn.
\end{align}

{\bf Step 3}: Using the triangle inequality, we get that there is a constant $C>0$ such that
\begin{align}\label{nnn}
\|\varrho_{k_n}-\varphi_{k_n}\|_{H_0^s(\overline{\Om})}^2\leq &C\|(-\partial_x^2)^s\varrho_{k_n}-(-\partial_x^2)^s\varphi_{k_n}\|_{L^2(-1,1)}^2\notag\\
\leq &C \Big(\|(-\partial_x^2)^s\varrho_{k_n}-\mu_{k_n}^{2s}\varrho_{k_n}\|_{L^2(-1,1)}^2+\|\varrho_{k_n}(\mu_{k_n}^{2s}-\lambda_{k_n})\|_{L^2(-1,1)}^2\notag \\
&+\|\lambda_{k_n}\varrho_{k_n}-(-\partial_x^2)^s\varphi_{k_n}\|_{L^2(-1,1)}^2\Big).
\end{align}
It follows from \eqref{nnn} and  Step 1 that there is a constant $C_{k_n}(s)>0$ which converges to zero as $n\to\infty$, such that
\begin{align*}
\|\varrho_{k_n}-\varphi_{k_n}\|_{H_0^s(\overline{\Om})}^2 \leq C_{k_n}(s).
\end{align*}
Let the operator $L$ be defined by
$$L:  H_0^{s}(\overline{\Om})\to H^{-s}(\mathcal O),\; v\mapsto Lv:=((-\partial_x^2)^sv)|_{\mathcal O}=(\mathcal N_sv)|_{\mathcal O},$$
where we recall that $H^{-s}(\mathcal O)=(H_0^s(\overline{\mathcal O}))^\star$.
 It has been shown in \cite[Lemma 2.2]{GRSU} that the operator $L$ is compact, injective with dense range. Let $B_1:=\overline{B}\left(\varrho_{k_n},C_{k_n}(s)\right)$ be the closed ball in $H_0^s(\overline{\Om})$ with center in $\varrho_{k_n}$ and radius $C_{k_n}(s)$. Since $L$ is a compact operator, we have that the image of $B_1$, namely $L(B_1)$, is totally bounded in $H^{-s}(\mathcal{O})$. Therefore, for every $\varepsilon>0$ there exists $N\in{\mathbb{N}}$ and $\{\psi_1,\ldots,\psi_N\}\subseteq B_1$ such that
\begin{align*}
L(B_1)\subseteq \bigcup_{j=1}^{N}\overline{B}_{H^{-s}(\mathcal O)}(L(\psi_j),\varepsilon).
\end{align*}
We observe that,  the eigenfunction $\varphi_{k_n}$ belongs to $B_1$. Thus, there exists $j\in\{1,\ldots,N\}$ such that
\begin{align*}
L(\varphi_{k_n})\in \overline{B}_{H^{-s}(\mathcal O)}(L(\psi_j),\varepsilon).
\end{align*}
We have shown that for $n$ large enough,
\begin{align*}
\|L(\varphi_{k_n})-L(\psi_j)\|_{H^{-s}(\mathcal{O})}\leq \varepsilon.
\end{align*}
Since $\psi_j\in B_1$, firstly we obtain that $\varphi_{k_n}\rightarrow \psi_j$, as $n\to\infty$ in $H_0^s(\overline{\Om})$  and secondly we have that $\psi_j$ is an element of the spectrum $\{(\varphi_k,\lambda_k)\}_{k\geq 1}$. That is, $\psi_j$ is a solution of \eqref{ei-val-pro}. Finally, as $L(\varphi_{k_n})$ converges to zero in $H^{-s}(\mathcal{O})$ (by \eqref{contr-3}), we can deduce that $L(\psi_j)=\mathcal N_s\psi_j=(-\partial_x^2)^s\psi_j=0$ a.e. in $\mathcal O$. It follows from the unique continuation property (Lemma \ref{lem-UCD})  that $\psi_j=0$  a.e. in ${\mathbb{R}}$, which is a contradiction. The proof of the lemma is finished.
\end{proof}

\begin{proof}[{\bf Proof of Theorem \ref{T1}}]
Let $u$ be the unique weak solution of \eqref{HE-EX} and $\psi$ the unique weak solution of the dual problem \eqref{Dual}.
Recall that by Lemma \ref{lem-1}, the system \eqref{HE-EX} is null controllable if and only if the identity \eqref{NSC-NC} holds.
It is a well known result that \eqref{NSC-NC}  is equivalent to the following observability inequality for the dual system:  there exists  a constant $C>0$ such that
\begin{align}\label{obs-ine}
\|\psi(\cdot,0)\|_{L^2(-1,1)}^2\leq C\int_0^T\int_{\mathcal{O}}\left|\mathcal{N}_s \psi(x,t)\right|^2\;dxdt.
\end{align}
From Section \ref{sec-3}, the solution $\psi$ of \eqref{Dual} is given by
\begin{align*}
\psi(\cdot,t)=\sum_{n=1}^{\infty}\psi_{0,n}e^{-\lambda_n(T-t)}\varphi_n(x),
\end{align*}
and the nonlocal normal derivative is given by
\begin{align*}
\mathcal{N}_s\psi(\cdot,t)=\sum_{n=1}^{\infty}\psi_{0,n}e^{-\lambda_n(T-t)}\mathcal{N}_s\varphi_n(x).
\end{align*}
Therefore, using the above representations of $\psi$ and $\mathcal{N}_s\psi$, and employing the orthonormality of the eigenfunctions in $L^2(-1,1)$, then the observability inequality \eqref{obs-ine} becomes
\begin{align}\label{obs-ine-1}
\sum_{n=1}^{\infty}|\psi_{0,n}|^2 e^{-2\lambda_n T}\leq C\int_0^T\int_{\mathcal{O}}\left|\sum_{n=1}^{\infty}\psi_{0,n}e^{-\lambda_n (T-t)} \mathcal{N}_s\varphi_n(x)\right|^2dxdt.
\end{align}
By Lemma \ref{uni-bound},  $\|\mathcal{N}_s\varphi_n\|_{L^2(\mathcal{O})}$ is uniformly bounded from below by $\eta>0$. Therefore, we can deduce that
\begin{align*}
\int_0^T\int_{\mathcal{O}}\left|\sum_{n=1}^{\infty}\psi_{0,n}e^{-\lambda_n (T-t)} \mathcal{N}_s\varphi_n(x)\right|^2dxdt\geq \eta^2\int_0^T \left|\sum_{n=1}^{\infty}\psi_{0,n}e^{-\lambda_n (T-t)}\right|^2 dt.
\end{align*}
Using the change of variable $T-t\mapsto t$, we obtain that
\begin{align*}
\int_0^T\int_{\mathcal{O}}\left|\sum_{n=1}^{\infty}\psi_{0,n}e^{-\lambda_n t} \mathcal{N}_s\varphi_n(x)\right|^2dxdt\geq \eta^2\int_0^T \left|\sum_{n=1}^{\infty}\psi_{0,n}e^{-\lambda_n t}\right|^2 dt=\eta^2\left\|\sum_{n=1}^{\infty}\psi_{0,n}e^{-\lambda_n t}\right\|_{L^2(0,T)}^2.
\end{align*}
Thus, the observability inequality \eqref{obs-ine} holds if the following estimate is proved:
\begin{align}\label{rusell and fatorini}
\left\|\sum_{n=1}^{\infty}\psi_{0,n}e^{-\lambda_n t}\right\|_{L^2(0,T)}^2\geq C\sum_{n=1}^{\infty}|\psi_{0,n}|^2 e^{-2\lambda_n T}.
\end{align}
It is a well known result for parabolic equations (see e.g. \cite{fattorini1971exact}), that an inequality of the type \eqref{rusell and fatorini} holds if and only if the eigenvalues $\{\lambda_n\}_{n\in{\mathbb{N}}}$ satisfy the following M\"{u}ntz condition. That is, the series
\begin{align}\label{serie}
\sum_{n=1}^{\infty}\frac{1}{\lambda_n}
\end{align}
is convergent. As we have mentioned in the introduction, the eigenvalues $\{\lambda_n\}_{n\ge 1}$ satisfy  \eqref{lam}. That is,
\begin{align}\label{lam1}
\lambda_n=\left(\frac{n\pi}{2}-\frac{(2-2s)\pi}{8}\right)^{2s}+O\left(\frac{1}{n}\right)\;\text{ as }\, n\to\infty.
\end{align}
Therefore, we have the following two situations.
\begin{itemize}
\item If $0<s\leq \frac 12$, then the series \eqref{serie} will have the behavior of the harmonic series, which implies that it is divergent.
\item On the other hand, if $\frac 12<s<1$, hence, $2s>1$, then using \eqref{lam1} we can deduce that the series \eqref{serie} is convergent.
\end{itemize}
The proof of Parts $(a)$ and $(b)$ is complete.

Now, we show Part (c). As a direct consequence of the previous controllability result, we obtain that the system \eqref{HE-EX} is exactly controllable to trajectories at any time $T>0$ if $\frac 12<s<1$. Since, this result is standard in control theory, then we  only give the main idea. Indeed, let $\tilde{u}_0\in L^2(-1,1)$ and let $\tilde{u}$ be the solution of \eqref{HE-EX} with zero exterior condition $(g\equiv 0)$.
Let $z$ be the unique weak solution of the system:
\begin{align}\label{CT}
\begin{cases}
\partial_t z + (-\partial_x^2)^{s} z = 0 & \mbox{ in }\; (-1,1)\times(0,T),\\
z=g\chi_{\mathcal{O}\times(0,T)} &\mbox{ in }\; (\R\setminus (-1,1))\times (0,T), \\
z(\cdot,0) = u_0-\tilde{u}_0&\mbox{ in }\; (-1,1).
\end{cases}
\end{align}
Next, we consider the function $u:=z+\tilde{u}$.  Since the system \eqref{CT} is linear, then using the same arguments as in the proof of Part $(a)$, we can easily deduce that there exists a control $g\in L^2((0,T);H^s(\R\setminus (-1,1)))$ such that  $z(\cdot,T)=0$ a.e. in $(-1,1)$.
Thus, $u(\cdot,T)=z(\cdot,T)+\tilde{u}(\cdot,T)=\tilde u(\cdot, T)$ a.e. in $(-1,1)$.
We have shown Part (c) and the proof is finished.
\end{proof}

\bibliographystyle{plain}

redpajama_set_name: RedPajamaArXiv


Animal Lib: Report animal cruelty/abuse on free call 1800751770. Anonymity is assured. Also 0409144803.
WRES Statewide Wildife Emergency Rescue. 0427 301 401. Mainly covers Central Victoria.
Wildlife Victoria 1300094535 (You can also leave details on their website at http://www.wildlifevictoria.org.au/wildlife-victoria-rescue .). Wildlife Victoria have a poor record of responding to calls and leave people waiting on line. Plus they almost only refer to the other wildlife caring organisations anyway, so consider ringing one of the other organisations directly instead.
Australian Fauna Care. They can be reached by e-mailing info [AT] fauna.org.au .
Phyllis Facey: 08 96461469; 0437483175. Location: Beverley West Australia, about 130kms east of Perth. Registered wildlife carer and rescuer. Works mostly with marsupials but will take in some birds and hand on to other carers. Licenced to carry a sedative to sedate roadside trauma roos if necessary.
We hope to make this list more comprehensive for Australia. In particular we need contact details for New South Wales, South Australia, Western Australia, the Northern Territory and north Queensland. Please contact us or leave a comment if you know the contact details of any wildlife rescuers.
Almost certainly there is more information on the internet. We will add more links.
Take care not to endanger yourself or others by stopping your vehicle suddenly in a dangerous location.
Approach any wild animal with care. Kangaroos particularly, are capable of lashing out with their very powerful hind legs, even if severely injured. Most animals are capable of inflicting nasty bites and scratches especially when frightened or in pain.
IF THE ANIMAL IS NOT DEAD.
If the animal has not been killed, do not approach it . Call for assistance immediately. Give accurate directions and if possible stay with the animal and warn others of its presence. If you are able to stay until assistance arrives, this also ensures that the animal is located.
Check the underbelly for a pouch. Animals such as kangaroos, possums and wombats carry their young is a pouch. A bulge or movement inside a pouch may indicate live young. Gently remove the joey, taking care not to further injure the animal. An animal that has been dead for several hours, even overnight, may still have live young in its pouch.
A cold animal that is sick, injured or orphaned will stay cold unless it is heated by some external source. Wrapping the animal in a towel or jumper alone will not make it warm. Small animals can be placed under the front of a person's jumper. This is a safe and reliable heat source. Alternative emergency heat sources are hot water bottle or other plastic container filled with warm water, or an electric blanket on a low or mid setting with the animal wrapped in a towel. The animal must not be placed directly onto the heat source. Take care not to heat the animal too quickly and avoid extreme fluctuations in temperature. Aim for a constant temperature of 30-32 degrees C.
Native animals are easily stressed and this alone may be enough to kill the animal. Keep it as quiet as possible, away from loud noises, and do not allow children to play with it. A joey requires the same care as a premature human baby.
An animal will suffer in much the same way as a human. Seek immediate veterinary care if the animal is injured and even if you wish to care for the animal yourself, seek advice on the care and hand-rearing of orphans. Please see our contacts page, members of the Native Animal Network are available at all time to provide advice and assistance.
The following diets should only be used short term, when assistance is delayed in obtaining the proper milk formula. The correct diet in native animals is essential. Special milk formula such as Wombaroo have been specially developed for native animals and are recommended for rearing orphans long term.
DO NOT FEED the animal cow's milk that comes from a carton or bottle as this can cause diarrhoea. Do not feed baby formula. Use only tinned or lactose free powdered milk in an emergency.
50% tinned full cream evaporated milk and 50% boiled water. This should be fed using a long teat or soft piece of fine rubber tubing. An eyedropper or syringe may also be used.
50% Powdered full cream milk and 50% boiled warm water with some Baby Cereal added. This should be fed from a bottle or a short teat or by spoon.
On a number of occasions, James Sinnamon, the administrator of this site has been contacted on 0412 319669 by people seeking help for injured wildlife. Whilst James is more than happy to provide what help he is able to to help save the life of a possum, koala, kangaroo, echidna or other native creature, he is probably not the best person to contact first as he, himself, has little direct experience in providing first-aid for wildlife.
Update, 11 August 2013: As a result of an emergency, where, just now, we tried to save, from amputation, the leg of a kangaroo near Griffith, NSW, we have realised that this page needs more phone numbers (and other information) to increase the likelihood that the necessary help and information will be received in sufficient time to save the life of a native animal or to enable his/her injury to be fixed. If you can help, or know of someone else who is able to help, please contact James Sinnamon on 0412 319669.
Australian Wildlife Protection Council - is NOT a rescue group.
Please check to see if BARN is still answering their phone 24 hours. Have been told that they stopped operating after last years floods.
Campbell Newman has NO respect for wildlife. He has made a shallow promise to help the Sea Turtles and Dugongs. He has also made a promise to let anyone cruelly MURDER Flying Foxes. No one is making promises to stop the cruel MURDERING of Macropods. He also has NOT made any promises for Qld's koalas.
Sorry, I have removed the AWPC from the Wildlife Rescuers list. But they are incredibly helpful and in touch with many wildlife carers, so deserve all our support.
Thanks for the tip on Cambell Newman. It's really a choice between two clowns in Queensland, I'm afraid.
2 wildlife rescue and rehabilitation organisations in Melbourne covering the bay area from Brighton to Mornington Peninsula are Animalia 0435 822699 and AWARE 0412 433727.
A possum was attacked by a domestic dog in the northern Brisbane suburb of Lutwyche earlier this evening at roughly 7.30PM Queensland Time (8.30PM elsewhere on the Eastern Australian seaboard). It sadly died an hour later in spite of efforts by Lea-ann to contact the Brisbane Area Network Association animal rescuers listed above and the care she gave to the injured possum.
The phone number for BARN, 0405 056 066, was inactive when Lee-ann tried to ring.
It seems unlikely that BARN would have been able to save the possum even if Lea-ann had been able to reach them. Still, let's hope that the mobile service works better next time a possum's life is threatened with injury.
Another way to better assure that injured possums receive more timely treatment in future would be for more contact phone numbers to be listed here. If you know of any other numbers through which people able to help injured wildlife can be reached, please leave a comment here or 'phone me on 0412 319669.
Phone numbers of more people able to treat injured Australian possums & other wildlife needed http://candobetter.net/?q=node/2820#comment-33873 A possum died tonight.
Today a candobetter.net editor received a call from Evelyn, a person who was picniking in St Vincents Garden, Albert Park. She saw a dog off-leash attack a possum, then the human accompanying the owner throwing the possum about. I left a message about the location and nature of the problem on Wildlife Victoria's site (after waiting about 10 minutes on their phone). Animal Lib had initially said that they did domestic animals, but later phoned back, unasked, to say they would send someone out if the girl wished and were given her contact number. They said to count them in in future for this kind of problem. In the mean time Evelyn had found a towel and wrapped the possum and was going to take it to a local vet. it was a ringtail, by her description. A person who had done a possum care course in the area was also contacted and said they would ring Evelyn.
After that I looked at the above page and realised that people were ringing here first because our number was at the top. So I have placed that number at the base of the page and improved the display of more directly relevant resources for finding rescuers.
I tried to help Evelyn find somewhere to take the ring tail possum.The local vet which she thought was open 24 hours was closed. I rang the Lort Smith Animal hospital in North Melbourne who were open and said they would take the possum. They said that if there were any visible injuries that the possum would be put down as possums "do not respond well to antibiotics". They gave me 2 other choices north of the city in Collingwood and Kensington but Evelyn said they were all too far for her. I then drove to meet Evelyn intending to take the possum to a vet. I contacted her by phone on the way to get more directions. During this phone call l I learned that the possum had died. Just one more clash with burgeoning humanity and our wild life!


text:
Susan Point: Works on Paper
The River — Woven by Time (2002)
by Susan Point
Medium: serigraph
Size: 32.75 × 11.25 inches
Edition: 100
Reference Code: SP2002-01
Other available artwork you might like by Susan Point:
Butterflies II (1993)
People Together (2006)
Raven's Song (2008)
Susan Point
OC, RCA
Coast Salish (Musqueam)
Susan began making limited edition prints on her kitchen table in 1981 while working as a legal secretary. She received several early commissions, which established her reputation for innovative proposals and for completing projects on time, on budget and at the highest level. She took courses in silver, casting and carving, all of which led to monumental sculptures in mixed media, and she was the first Northwest Coast artist to work in glass. She continues to release a number of print editions each year, but her focus has been on commissioned sculpture.


text:
Bae faces military service stint
Planet Golf — 31 December 2014 by GW staff and news services
Although a dispute over potential mandatory military service hangs over his head, Sang-Moon Bae plans to compete in next week's Hyundai Tournament of Champions.
According to his manager, Bae, a two-time winner on the PGA Tour, has a valid green card and is contacting lawyers to "work through the military issues" in his native South Korea. Bae was granted U.S. residency two years ago.
"Sang-Moon intends to stay in the United States as long as it is lawful to do so and play on the PGA Tour this year, beginning with next week's Hyundai Tournament of Champions in Kapalua," his manager said in a statement. "As this is an ongoing legal matter, he will make no additional statements until the issue is resolved."
At issue is a South Korean requirement that states all men between the ages of 18-35 must complete two years of military service. Since a peace treaty wasn't signed after the 1950-53 Korean War, South Korea is still technically at war with North Korea.
Bae, who won the Frys.com Open in October, is ranked second in the FedExCup and 84th in the world. According to the Yonhap news agency, Bae's mother said earlier this week that the golfer had his request for the extension of his overseas travel permit  rejected by the Military Manpower Administration.
In the past, the Korean government has been known to waive the military service requirement for athletes winning gold at the Asian Games or any Olympic medal. Bae is the highest ranked golfer in Korea and a likely candidate to represent his country when golf returns to the Olympic in 2016 at Rio de Janeiro.


This smooth and even 3ply sport, is a warm, soft, and slightly fuzzy yarn perfect for colorwork projects or lighter weights hats and gloves.
164yds, 6.5sts/inch, US5 (3.75mm) needles.
Classic Silk is a blend of 50% cotton, 30% Silk, and 20% Nylon. It has a great nubbly look, a bit of sheen, and a great raw silk feel. 5 sts per inch on a US6 (4mm) needle.
We knit this shawl with Classic Elite's "Classic Silk" yarn. 3 balls is enough for the shawl pictured.


text:
Billboards in West Brandywine Township
See billboard prices and buy ads online. Want billboards beyond West Brandywine Township? Search all Philadelphia, Pennsylvania
Billboard Rates in West Brandywine Township
Find, compare and buy billboard ads in West Brandywine Township, Pennsylvania online! Before viewing available billboards and rates in West Brandywine Township, take a second to learn about the advertising audiences you can reach in the area. West Brandywine Township, Pennsylvania is home to 8,432 individuals across approximately 2,919 households. When buying billboards in West Brandywine Township, you might also consider the economic characteristics of the area. The median housing value in West Brandywine Township is $346,717, and the average household size is 2.99. The population of 8,432 is expected to grow by 6.4%. In terms of homeownership in West Brandywine Township, Pennsylvania: for every 12.33 homeowner(s) there is 1 renter. Workers in West Brandywine Township fall into the following categories: service industry (9% of workers), blue-collar jobs (21% of workers), and white-collar roles (70% of workers). The median household income (HHI) is $95,242 and the average HHI is $114,816. According to The U.S. Census, the male to female ratio in West Brandywine Township, Pennsylvania is 1.02 to 1. The Census also provides an ethnic breakdown of the residents you could reach with billboards in West Brandywine Township. The area includes 103.0% white people, 1.3% Asian people, 1.7% black people, and 1.2% Hispanic individuals. This area has a high school graduation rate of 81%. Of those that went on to pursue a degree, 29% dropped out of college. Of the 71% who completed college, 28% achieved a postgraduate degree.
On average, billboard advertising in West Brandywine Township, Pennsylvania costs $600-$3,611
About West Brandywine Township
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With its long white sandy beaches, inviting luxury hotels, crystal clear skies and turquoise waters, there are many reasons to visit Cancun this year. Seriously, it is one of the most popular tourist resorts in the world and people come back time and time again. Cancun's all-inclusive hotels will cater to your every need from relaxing rooms, enticing food and lively entertainment on a nightly basis.
There is so much to see and do in the area from ancient ruins to taking a day trip to Cozumel island where you can go snorkelling, diving and kitesurfing to your heart's content. So, if you are thinking of a Cancun holiday this year when should you go and what are the best attractions to see while you are there?
When should you holiday in Cancun?
There isn't really a bad time to visit this tourist focused Mexican city – it is hot and lively all year round! If you love holidays in the sun then Cancun is the place for you. The average temperature sits comfortably between 23°C – 28°C throughout the year and regularly rises above 30°C. Between April to November you will find the highest temperatures with scorching days and pleasant nights to get your holiday tan and even over the winter period you will rarely find the thermometer fall below 21°C.
What are the 5 best attractions to see around Cancun?
The name means 'secret well' and these natural wonders provide an enjoyable tourist excursion in Cancun. Essentially sinkholes filled with water, the Cenotes consist of natural caves and underground rivers that you can explore by snorkelling, diving or swimming.
A Unesco World Heritage site, Chichen Itza is a must see while you are visiting Cancun. Gasp at the imposing temples, pyramids and cosmic symbolism while overlooking the sports fields that were used for ancient games. The complex was inhabited from 600 AD to around the 1200s so there is plenty to explore whether on a guided tour on at your own pace.
One thing that Cancun is renowned for it its theme parks – in fact, the numbers rise into the double figures. You can spend a whole week exploring the different theme parks; some with exhilarating rides, others with water sports. Some even give you unlimited access to food and drink while you are there.
Experience the vibrant nightlife in Cancun by visiting Coco Bongo which is a club with a difference. Live shows start every 10-15 minutes and you can purchase a VIP ticket which gets your drinks all night long as well as your own personal waiter.
Love Mexican food and want to learn how to cook the Cancun way? Can cook is a local cookery school that teaches small groups how to conjure up authentic dishes. Learn from the experts and impress your friends and family with your new-found skills when you get home.
We can find a Cancun holiday deal that suits your dates or you can see our latest Cancun deal here. Cancun is a holiday destination that has ballooned in popularity and for good reason. It provides a great way to experience both the old and new Mexico from ancient ruins to lively nightlife and everything else in-between. The tourist spot offers something for everyone and there are plenty of holiday deals to visit on a budget. Whether you are after a relaxing time in the sun sipping a tequila or you desire a more active holiday exploring underground rivers or impressive Mayan structures, Cancun has it all and it is one of the most sought after destinations for Mexico holidays in 2018.
Let us plan your next holiday; we can do all the legwork so you don't have to AND we can get you a superb price!
Book your next adventure today and pay the balance in instalments. Get a quote or call us on 0207 254 0136.
What are the advantages of booking with a travel agent?