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Indigenous Law Association at McGill Blog
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First Nations Child and Family Caring Society of Canada v. Canada, 2019 CHRT 39
The Indigenous Law Association at McGill Blog and the McGill Journal of Law and Health are collaborating on our blogs this year, in order to connect our readers with more content at the crossroads of health law, and Indigenous law and legal traditions. You can find this article cross-posted on their website.
Contributed by Maya Gunnarsson
The Canadian Human Rights Tribunal released their latest decision in First Nations Child and Family Caring Society of Canada et al. v. Attorney General of Canada (representing the Minister of Indigenous and Northern Affairs Canada) in September 2019. This ruling was the 9th issued by the Tribunal since 2016, when they found the Caring Society's allegations that Canada was discriminating against First Nations children in their provision of child and family services were substantiated.
This ruling is dedicated to all the First Nations children, their families and communities who were harmed by the unnecessary removal of children from your homes and your communities. The Panel desires to acknowledge the great suffering that you have endured as victims/survivors of Canada's discriminatory practices.
Caring Society v. Canada, 2019 CHRT 39 at para 13.
The purpose of this ruling was to determine the compensation Canada would be ordered to pay due to their discriminatory practices. More broadly, however, this decision attempted to bring justice to an extremely vulnerable group of people who have historically been denied access to justice through Canadian institutions. Since the initial ruling in 2016, the Tribunal had issued multiple remedial orders against Canada, finding that they were continuing to discriminate, and giving clear orders on what the federal government needed to do to comply with its 2016 decision. While the federal government may have been dragging its heels, the Tribunal firmly took a step towards reconciliation in Canada with this decision.
In 2008, the First Nations Child and Family Caring Society brought forward a complaint to the Canadian Human Rights Commission. They alleged that the federal government was discriminating in their provision of child and family services to First Nations children living on reserve and in the Yukon by providing inequitable and insufficient funding for those services, contrary to s. 5 of the Canadian Human Rights Act (CHRA). Whereas the federal government provides child and family services for First Nations, the provinces provide those services for non-Indigenous Canadians. At issue in the original complaint were the funding formulas the federal government used to provide these services, as well as their definition of Jordan's Principle—a principle intended to ensure First Nations children have access to the same healthcare and social services as non-Indigenous children. For a more detailed background on Jordan's Principle and the Canadian Human Rights Tribunal's previous rulings on this case, please see our previous blog post on the subject here.
In 2016, the Tribunal found that the Caring Society's complaint was substantiated and the federal government's provision of child and family services "resulted in denials of services and created various adverse impacts for many First Nations children and families living on reserves (para 458)." The Tribunal ordered the federal government to cease its discriminatory practice and to take measures to redress and prevent it. At the time of the judgement, the Tribunal stated that it had outstanding questions regarding the determination of compensation and deferred its ruling on the matter until those questions were answered.
The 2019 Decision
Under s. 53 of the CHRA, if the complainant's claim of discrimination is substantiated, the Tribunal may order the person (or entity) found to be engaging in the discriminatory practice to financially compensate the victim. Specifically, under s. 53 (2)(e) the Tribunal can order the person to pay up to $20,000 for pain and suffering that the victim experienced as a result of the discriminatory practice. Additionally, under s. 53(3), the Tribunal may order the person to pay an additional $20,000 where the person is found to be "engaging … in the discriminatory practice wilfully or recklessly."
In order to calculate the appropriate amount of compensation under s. 53(2)(e), the Tribunal had to determine whether the federal government's discriminatory behaviour resulted in harm to First Nations children, and the extent of their pain and suffering. In determining the appropriate amount of compensation under s. 53(3), the Tribunal had to assess whether Canada was aware of its discriminatory practices. These two sections of the CHRA provided the Tribunal with the authority to award a maximum of $40,000 to each victim of discrimination.
The Tribunal stated that it had heard extensive evidence demonstrating that First Nations children were denied essential services after significant and detrimental delays, causing harm to the children and their parents or grandparents caring for them. For example, some children were required to stay in hospital longer than was medically required, often away from their home communities, while their families waited on funding approval for medical equipment that they needed in order to leave the hospital. In some cases that approval never came.
The Tribunal further stated that the evidence showed some children and their families experienced serious mental and physical pain as a result of the delays and denials in services. Due to the funding formula, many children were removed from their homes and communities in order to access required services. These children were not living in abusive or unsafe homes; however, the only way for them to access services that were available to other Canadian children was to be put in the care of the State. The removal of children from their communities and families was traumatic, and caused great pain and suffering to both the children and their families. As a result, the Tribunal awarded $20,000, the maximum amount allowed for pain and suffering, to each First Nation child that was removed from their home since 2006, and to each of their parents or grandparents where the removal was unnecessary.
Further, the Tribunal found that "Canada's conduct was devoid of caution with little to no regard to the consequences of its behavior towards First Nations children and their families both in regard to the child welfare program and Jordan's Principle (para 231)." Canada knew that its policies were harming children, however, the government chose to put their financial interests above the best interests of children. As a result of this wilful and reckless behaviour, the Tribunal awarded each First Nation child that was removed from their home since 2006 and, where the removal was unnecessary, each of their parents or grandparents an additional $20,000, the maximum allowable amount.
The Tribunal's decision to award the maximum amount allowed by statute was rare. The decision repeatedly emphasized that the maximum amount of $40,000 for victims was reserved for the worst cases. The Tribunal also categorized the unnecessary removal of children from their homes, families, and communities as "a worst-case scenario (para 234)."
In 1998, the maximum amount the Tribunal could award victims of discrimination was raised from $5,000 for "reckless or wilful conduct or hurt feelings or loss of self-respect" to $20,000 for pain and suffering and $20,000 for reckless or wilful behaviour (para 5). Since then, the Tribunal has only ever ordered compensation to the maximum amount in both categories once: in Alizadeh-Ebadi v. Manitoba Telecom Services in 2017. This case involved an employee facing sustained discrimination at his workplace for over twelve years. Clearly, the Tribunal does not order the maximum amount of compensation lightly.
The impact of awarding the maximum amount allowed by statute to the victims cannot be overstated. In its decision, the Tribunal addressed the victims of Canada's discriminatory actions directly, stating: "No amount of compensation can ever recover what you have lost, the scars that are left on your souls or the suffering that you have gone through as a result of racism, colonial practices and discrimination (para 13)."
While the compensation of $40,000 cannot be considered proportional to the pain suffered by First Nations children and their families, ordering the maximum amount allowed by statute to be paid to each victim of Canada's racial discrimination recognizes the severity of the discrimination the Tribunal adjudicated. Further, it attaches the highest price tag it can onto the government placing its own interests above the best interests of children over whom it has a responsibility.
According to the final report of the Truth and Reconciliation Commission,
Reconciliation is about establishing and maintaining a mutually respectful relationship between Aboriginal and non-Aboriginal peoples in this country. In order for that to happen, there has to be awareness of the past, acknowledgement of the harm that has been incited, atonement for the causes, and action to change behaviour.
pg. 6-7
In this decision, the Tribunal acknowledged the past and detailed the specific harms of the government's behaviour. It attempted, to the best of its ability given the authority it had as set out by statute, to ensure atonement for the harms caused, through public renouncement and financial compensation. This decision also provides an impetus for the federal government to change its behaviour, as a financial cost is now associated with every subsequent act the government undertakes that neglects the needs of a First Nation child.
This ruling does not fix the entire First Nations Child and Family Services Program in and of itself. However, it signals to the government of Canada, to First Nations communities, and to the Canadian public, that egregiously discriminatory behaviour by the Canadian government is not acceptable. It signals that the lives of those children and their families impacted matter. And it underscores the importance of community, family, and culture in a First Nation child's life. In short, it is a significant step towards reconciliation.
On October 4, the Attorney General of Canada filed an application for judicial review of this decision. It calls for an order to set aside the decision and dismiss the claims for monetary compensation. This post will be updated following the release of the Federal Court's review.
Maya Gunnarsson is the Executive Online Editor of the McGill Journal of Law and Health and a second-year B.C.L./ LL.B. student at McGill University's Faculty of Law. Maya holds an M.A. in Canadian Studies and Indigenous Studies from Trent University where she focused her research on the role of the media in the ongoing crisis of violence against Indigenous women and girls. Prior to law school, Maya worked as a researcher at the Higher Education Quality Council of Ontario and interned at the Office of the UN High Commissioner for Human Rights in the Indigenous Peoples and Minorities Section.
Interview with Professor Aaron Mills on his new Indigenous Constitutionalism Course
The ILADA Blog team is very excited about McGill's new Indigenous Constitutionalism course, designed and delivered by Professor Aaron Mills. This interactive course uses Indigenous pedagogies, including the format of a sharing circle, to explore Anishinaabe legality, constitutionalism, and law.
In Professor Mills' words, "Through elder's teachings, stories, material culture, and the works of myriad Indigenous writers and orators, we'll begin to develop an understanding of the logic, structure, and principles of one view of Anishinaabe constitutionalism, what kinds of legal processes and institutions it supports, what kind of law these generate, and how that body of law changes through time and across places. We'll practise and apply Anishinaabe legal reasoning today, and will reflect upon a number of contemporary indigenous law revitalization projects in Canada, all in the context of contemporary colonialism. In so doing, we shall critically consider how colonialism structures indigenous-settler relationships on Turtle Island, complicating prospects for indigenous constitutional and legal revitalization."
As the first offering of this valuable course draws near its end, Sejeong Park, one of our Associate Editors, asks Professor Mills to share his thoughts on the challenges and potential impacts of these teachings.
Sejeong: Would you mind describing your course, "Indigenous Constitutionalism"?
Aaron: I think the course is… an adventure. (laugh) I think it's quite challenging. It asks students to bring their whole selves into the classroom. And so, like all classes, I certainly want students to come in with an inquiring mind. But it also takes seriously various modes of inquiry and participation. And so, it is in a circle format. Not exclusively, but that's the base structure. And then, depending on how the class feels and what's working effectively, we sort of mix it up. But the circle engages and respects people as people, not just as inquiring minds. And so, people participate and learn from one another and in different ways. And some of the contributions offered up are sometimes more from the heart or the spirit, or more from learned experience and body, in that sense. And this method of knowledge generation is relatively novel for the vast majority of the students. Hence, a challenge. (laugh)
Sejeong: It must be a challenge for you, too, sometimes. Or I assume so, especially if you are teaching with a different methodology than most courses would – or, at least, a methodology that is different from what most students are probably used to. Did you find any challenges in engaging students?
Aaron: Yeah, it certainly is a challenge, in the sense that, unlike a lecture and, to a certain extent, even a seminar format, there is not a whole lot I can to do prepare in advance. I have to be very on, in the moment. And my task as facilitator is to ensure that we engage meaningfully with the texts. And also, that everyone is participating in a good way and just being respected. And, especially towards the start of the course, if they're not effectively engaging with one another's contribution, it is my task to draw out those linkages, right? Not just to the material for the day, but also between one another's contributions. And so, yeah, it's tiring in a sense, but also quite exhilarating to see students, in every case, rise to this new challenge and develop capabilities which, in some cases, they did not even know they had. It's also exhilarating for me because it's a way of taking Indigenous law very seriously, on its own terms.
Sejeong: Could you share what you were thinking or feeling while you were designing the syllabus and any particular goals you may have had in mind?
Aaron: So, a great deal of excitement. That was probably the greatest feeling, because it gave an opportunity for me to share, not just the content but, as I want to keep emphasising, the way of learning that my Elders, Anishinaabe legends, and the land itself have shown me is the way that learning Indigenous law really happens. And so it's an opportunity to bring a small slice of that into students' lives. And so, for me that's terribly exciting because I think it helps people to grow and to be their best selves, but it also opens a door of possibility to a kind of transformative change that many of us don't have access to. Having said that, I should make it very clear that the course is not a political project to get students to act in certain ways. But it does have as one of its purposes empowering them by presenting another way of living law. And so, to expand their field of possibility.
Yeah. So, a lot of excitement and a deep, deep sense of responsibility. Because, you know, I state at the outset who my teachers are, and I have to honour their teachings, and, in a certain respect, I am representing their knowledge. And mistakes are of course my own, but I have to be mindful of the fact that I am not simply acting free-license, right? I, my community ties, and my connection to my place are not just a political act. They are also deeply meaningful to me in a personal sense. So, I have responsibilities that I have to live up to as I run the class. So that's sort of an invisible piece that is behind the scenes, that students might come to appreciate throughout the course. But it isn't the sort of class that I simply come in with a great deal of intellectual excitement and can make happen whatever I can make happen. There is a field of responsibility that I can move within. And at this point in my life, I know just the tiniest little bit myself. And so, I also have to be careful to not supersede what has been gifted to me and my own limitations.
Sejeong: What do you think or hope would be the impact of your course on your students, as future jurists or just in their careers?
Aaron: Right. Well, let me answer the "as jurists" part, because it's an easier answer and I think that it is not actually the main answer. This course isn't, in a strict sense, preparing them to be jurists. I think it can have – and ought to have – a very significant impact on that sort of practice, right? By, for instance, creating legal actors who are less colonial and less racist, and who have some of the interpersonal capacity to effectively engage the sort of persons who are. That can only be a good thing. And I, of course, hope that my course would have that sort of impact on, what is from a Western standpoint, conventional legal practice. However, this is sort of to beg one of the critical points of the course, but law actually means something different from an Anishinaabe standpoint, and which is not properly the domain of elites of a particular profession. It is part of everyday practice. It is part of just being people. And in the context of this interview, it would be very difficult for me to explain all that; it would be a very significant aside. But just taking that statement as it is then. If the goal is to enable people to practice the mode of law that Indigenous law belongs to – and so, I call it "rooted law" – that is something that is applicable to their lives generally. All their relationships, every time they form a judgement. So, in that sense, I hope it has a broader impact on students' lives.
Sejeong: I believe it's a one-year course, right?
Aaron: Yes, that's right.
Sejeong: I assume that students may have not given you feedback yet on how they feel about the course or the course material. But would you like to share any responses or reactions from them?
Aaron: Sure. I have to be somewhat cautious because some of the students have had very powerful responses and, you know, have shared some quite powerful things in our circle. And it's not for me to disclose their experience.
Sejeong: Right.
Aaron: But I found those moments really inspiring. And it's the sort of thing that a professor only hopes might happen: that a student is really impacted by something. But yeah, I suppose that, you know, a number of students have expressed an enlarged capacity for thinking and acting about law, and not just in respect of Indigenous peoples. So, from my standpoint, that is quite an outstanding outcome.
Sejeong: Right, of course.
Aaron: And they have voiced that sense of challenge. And they should! The course is designed so that we produce meaning together. So, when they have little context of what we are talking about, participating effectively in it is quite overwhelming. And I'd say that a vast majority of the class felt that way about the first whole section of the course. And now, they have become quite masterful. (laugh)
Sejeong: (laugh) Are you hoping to be able to offer this course again in the future?
Aaron: Mhm, yeah! I still don't know all those technical terms, but – at a couple of faculty council meetings ago, it was made a standard course. So, I am very thrilled about that.
Sejeong: Oh, congratulations!
Aaron: Yeah. I am looking forward to offering it next year and in future years. Um… One little thought for the last question actually.
One of the real highlights for me was being able to bring two of the folks in my circle of Elders, Sherry Copenace and Harry Bone, into our class. And the interaction between the Elders and the students was outstanding. And they continued reflective engagement even out of the class. Not just on the part of the students, but also the Elders! And so it was really magical, and I am just so thrilled to be at an institution that supports that, to have students who took that seriously and truly made the most of this experience. And what an amazing way to learn the law! (laugh)
Sejeong: Mhm, thank you so much! I don't know whether there was anything additional you wanted to share or talk about?
Aaron: No, I could probably talk about this course forever. (laugh)
During our small conversation after we ended the recording, Professor Mills added an additional comment:
When he was talking about his feeling of responsibility when teaching the course, it is also the responsibility he feels towards his Grandmother. He told me that a lot of his knowledge comes from his Grandmother's teachings – the circle, for example. This also makes his teaching the course more special, because it allows him to "keep these [the teachings, what would be considered "the past"] alive and relevant today."
Submissions by students of John Borrows' Anishinaabe Law Outdoor Education Experience
Last August, McGill law students visited the Chippewas of Nawash First Nation on Georgian Bay to learn about Anishinaabe law in an intensive field course. The course was led by Professor John Borrows with the help of Elders and other community members, and took place on the land and water. For four days, students worked with different sources of Anishinaabe law found in traditional stories, the environment, treaties, declarations, and customs and learned about Indigenous legal traditions.
As part of the ILADA's blog series on Indigenous languages and legal traditions, we asked the students to share their thoughts on the interconnection they observed between language and law during the field course. The following are some of their reflections.
En août dernié, un groupe d'étudiants en droit à McGill ont visité la Première nation des Chippewas de Nawash sur la baie Georgienne pour apprendre davantage à propos du droit Anishinaabe lors d'un cours intensif sur terrain. Professeur John Borrows a mené ce cours avec l'aide des Ainés de la communauté et cela a pris lieu sur la terre et dans l'eau. Pendant quatre jours, les étudiants ont appris à propos des traditions juridiques autochtones en travaillant avec différentes sources du droit Anishinaabe qui se trouvent dans les histoires traditionnelles, dans l'environnement, dans les traités, les déclarations, et les coutumes.
Dans le cadre du premier thème du blog d'ILADA sur les langues et les traditions juridiques autochtones, nous avons invité ces étudiants à partager leurs pensées sur les connexions qu'ils ont remarquées entre la langue et le droit pendant ce cours intensif. Les paragraphes qui suivent font état de leurs réflexions.
André (3L)
The Anishinaabe law course at Nayaashiinigmiing pushes the boundaries of what law school offers students. A small group of us, from 2L to 4L, came together along the Western shores of Georgian Bay at summer's end to listen and learn from nature and a community that has inseparable connection with their environment. The Anishinaabe law course required us to use our senses, it opened our minds and hearts to new epistemologies, and allowed for new relationships to emerge with our classmates, instructors, and the community.
One of the main lessons that I took away from the 4-day course was that Anishinaabe law and language are inextricably linked. Anishinaabe linguistic structure frames and delivers messages on how a community perceives the world around them. Examples of Anishinaabemowin (Anishinaabe language) were showcased time and time again over the intensive field course. Language and linguistics were emphasized in nearly every lesson, whether when deriving law from Anishinaabe stories or using Anishinaabemowin to describe the environment around us.
Professor John Borrows highlighted how the Anishinaabemowin language structure differed greatly from English and other Western languages. He also explained how language leads to disparate worldviews between the Anishinaabe and Westerners. One example of this is that Western languages (e.g. English or French) view the world as inanimate, Indigenous languages view the world as animate. Both the noun-oriented language of English or French and the action-oriented Indigenous languages, such as Anishinaabemowin, breathe life into their respectful worldviews.
"Indigenous legal traditions are best accessed in the context of language, stories, methods of communication, and styles of performance and discourse because these are mediums that frame understanding and encode values."
– Sákéj Youngblood Henderson
The ways in which Western and Indigenous cultures form and value relationships with people, objects, and environment are fundamentally different. For Anishinaabe, the distinction between natural objects and humans are seldom made; therefore, Anishinaabemowin reflects the relationship that the Anishinaabe have with the world around them. These relationships, in turn, form the laws that govern Anishinaabe way of life. It is these relationships that shape concepts of law and governance.
Over the course of the four days, it became clear to me that the rhetoric of Anishinaabemowin implicitly manifests Anishinaabe law and that, in the end of it all, the two cannot truly exist without each other.
Yulia (2L)
To say that Indigenous languages and laws are intertwined would be an understatement. We had multiple occasions to understand this close relationship between the two while on the field course in Neyaashiinigmiing. Anishinabemowin is so rich in verbs, and even nouns tell stories that naturally translate into legal principles. For instance, I particularly enjoyed my process of coming to understand land, which in the Western worldview is seen as property and an indication of one's social status, as a source of law. Even with the shortcomings of translation, it was extremely helpful to know that the word "land" in Anishinabemowin, 'akinoomaagewin', is a combination of the words "Earth" and "to teach", which is interpreted as "earth as teacher."
I am grateful for the opportunity to see the application of legal principles but also for the realization that sources of law and guiding principles are everywhere individuals can identify them: in the water, land, plants, animals, songs, dances, everyday life situations. In short, it was challenging yet inspiring to view the world around us as resource for reasoning and a tool for teaching. This, I think, can only be achieved by being in harmony and a loving relationship with one's surroundings, and more fundamentally, with oneself, because, as our firekeeper Ken said, we are already given everything we need.
Heather (2L)
In my experience, language has often been an afterthought in legal education; a way of articulating law, but not an integral part of the process of law itself. Aside from the few occasions where McGill's bilingual environment had me translating legal terminology between French and English, I took for granted that the language I used to talk about law was commonly understood and distinct from the law itself.
It was surprising then, when within the first few hours of arriving at Neyaashiinigmiing, we were introduced to Anishinaabemowin, the language of the Anishinaabe nation. Lessons in language came up just as naturally and as often as teachings about legal actors and sources of law. Over the course of the few days I spent there, I realized that it's actually quite difficult to conceive of Law without seeing how it relates to language, and vice versa.
Understanding just the bare basics of Anishinaabemowin has helped me grasp some of the foundational elements of Anishinaabe law. Anishinaabemowin is a verb-based language (and therefore quite different from English, which is noun-based), which means that action and relationship are emphasized. Noun forms are generally conveyed by joining together verbs, adding prefixes and suffixes, and creating descriptive phrases. My favourite example is the word for blueberry pie, miini-baashkiminasigani-biitoosijigani-bakwezhigan, which roughly translates to "exploding blueberry sauce-layered between bread-old Frenchman-bending over oven." It's quite a vivid picture! The use of compound words like this produces a language that highlights processes and relationships between different elements – a pie is nothing without the process of bending over an oven – and paints a picture of an animated world in constant movement. Full of descriptive words and relational phrases, it reflects a particular pattern of living and of practicing law.
I came away from this experience with the realization that Anishinaabe law is not an inert system of rules that is imposed on a community, but rather an intellectual and personal process. "Law" might be better understood as a verb, something that people do in relationship with one another, rather than a noun.
Breanne (2L)
One thing that really struck me was the extent to which law and language are indivisible. Law exists within language, and language is both a source of law and a medium of sharing law. I noticed that this made for a lawful community- there seemed to be an overall sense of familiarity with law because it is so deeply embedded within language. For me, this was a sharp contrast to the common law and civil law, which consists of largely unfamiliar terminology and jargon.
The relationship between law and language inspired me to be mindful about more than what was being said. To learn Anishinaabe law, I felt that I had to think holistically about language. I thought about why someone would choose to say something, how they said it, who said it, etc.
Debbie (2L)
The importance of being humble was an underlying theme of the Anishinaabe law course. Humility was somehow woven into every story, song and ceremony, and even directly present in some of the translations of the word "Anishinaabe". We learned that depending on how the components of the word are broken down, one can extract different meanings. "Anishinaabe" can be translated to mean "the pitiful people" (and being called "pitiful" is definitely the kind of thing that would humble you a little bit). However another meaning we can pull from the word "Anishinaabe" is "the good people". Throughout the course I learned how those two translations, the pitiful people and the good people, could work together to form a beautiful picture of humility.


Kratom Safety Recall : The FDA has recalled Kratom supplements as they are associated with salmonella outbreak. Some patients believe these dietary supplements help treat opioid dependency but there is no reliable evidence to support kratom's effectiveness for this issue.
Study Drug Fact Sheet: This fact sheet was provided through Chippewa Valley Coalition for Youth and Families and Henry Ford Hospital to help adults and teachers talk to students about the negative impact of using "study drugs".
Drugs that People Abuse: Learn about drug facts from the National Institute on Drug Abuse.
Alcohol Fact Sheet: Learn about alcohol from the National Institute on Drug Abuse.
Marijuana Fact Sheet: Learn about marijuana/weed/pot from the National Institute on Drug Abuse.
Heroin Fact Sheet: Learn about heroin (smack, junk) from the National Institute on Drug Abuse.
Tobacco and Nicotine Fact Sheet: Learn about tobacco and nicotine from the National Institute on Drug Abuse.
Spice (K2) Fact Sheet: Learn about K2/spice from the National Institute on Drug Abuse.
Pain Medicine Fact Sheet: Learn about pain medicine from National Institute on Drug Abuse.


text:
U.S. Open Track and Field was the name of the first top level indoor track and field meet in 2012. The U.S. Open was the opening competition in the 2012 IAAF Indoor Permit Meeting series  and was also part of the Visa Championship Series.  Held in Madison Square Garden in New York City, the inaugural event was announced for January 28, 2012, with television coverage on January 29 at 7 p.m. Eastern Time on ESPN2.

Announced as continuation of a 99-year indoor track and field tradition at Madison Square Garden, the meet displaced the Millrose Games, which has been the longest standing annual event at the arena for the previous 98 years.  The Millrose Games will continue at the Fort Washington Avenue Armory.

The event lasted just one year, and was cancelled in 2013.

Meet Records

Men

Women

References

Recurring sporting events established in 2012
Track and field competitions in the United States
Sports competitions in New York City
Annual sporting events in the United States
IAAF Indoor Permit Meetings
Track and field in New York City

redpajama_set_name: RedPajamaWikipedia


where <FLAVOR> is an optional string. If present, must be one of the supported emulations. If absent, CEMU prints the current emulation setting.
This permanently changes the mode of the account in which it is executed. Changing the mode of the SYSPROG account is not allowed as this must always be run in CACHE emulation mode.
The CREATE.INDEX verb is currently limited to creating indices on files in the context of the account where the file resides.
Moreover, CREATE.INDEX does not support the "NO.NULLS" option. If this option is specified, a warning message is issued. The NO.NULLS clause actually means "don't index empty strings, not NULL values ($c(128) in UniVerse) which makes it confusing.
The NO.NULLS clause is in fact just an attempt to speed up index building where the majority of values are the same. This is not an issue for Caché indices, nor are many duplicates of any single key.
Application which depend on not having NULL as one of the values on an index will have to be modified to account for this situation.
ON - display dates in international format. This is the default.
INFORM - set @SYSTEM.RETURN.CODE to 0, if dates are to be displayed in U.S. format, or 1, if they are displayed in international format.
(I) - display dates in international format; same as "ON".
(D) - display dates in U.S. format; same as "OFF".
The SP.COPY command has been added to control the placement of spooler jobs in the queue.
MVBasic programs (with suffix ".MVB") are compiled into an intermediate form before execution. These intermediate files have the suffix, ".MVI". Errors reported at runtime identify their locations in terms relative to these MVI files.
The MVI command allows programmers and support personnel to cross-reference line numbers given in terms of the MVI file back to the originating source code, if the source is available.
00006  CRT "IT IS IN ERROR"
00007  CRT "IT IS IN ERROR"
The LIST.LOCKS command lists the system lock table. Locks created by the MVBasic LOCK command are shown as LOCK nnn. Locks created by opening a sequential file are shown as FILE fname. All other locks are shown in their native format, including MVBasic record locks, which are shown on the global that holds the file.
The CLEAR.LOCKS command allows the user to release locks that are held by this process as a result of an MVBasic LOCK command. With an argument nnn, it releases that single LOCK nnn. Without an argument, it release all LOCKS held by the process.
The TANDEM command allows one multi-value user to connect to the terminal of another multi-value user, sharing terminal input and output. It is usually used as a support tool so that support staff can connect to a remote user and view or run their application.
There are two types of TANDEM user, the master and the slave. The master is the user who initiates the request and chooses the user to connect with. The slave is the user who accepts the connection request and it is the slave's terminal input and output that is shared.
Before a TANDEM session can exist, the slave user must issue a TANDEM ON command. By running this command, the user is acknowledging that their session may be connected to by a TANDEM master. Because of the way TANDEM is used, it is anticipated the TANDEM ON command will usually be run automatically when the user logs on through a logon proc/paragraph.
For security purposes, the TANDEM master user can only be run from the %SYS namespace. In MultiValue terms, the %SYS namespace is usually the more familiar SYSPROG account.
Both the master and slave users should be running the same terminal type. TANDEM does not translate terminal specific escape sequences. Hence if the master terminal was a Terminal (VT220) and the slave terminal was a WYSE 50, then none of the escape control sequences would be translated, and the screen of the master might look garbled.
ON - This is run from the tandem slave to acknowledge they can become a slave user. Without running this, a user cannot be connected to.
OFF - This is run from the tandem slave to refuse tandem requests.
nnn - By specifying a port number, you are asking to start a TANDEM session, as the master user. You must be in the %SYS namespace (SYSPROG account).
For legacy compatibility, TANDEM (N) and TANDEM (F) are the equivalent of TANDEM ON and TANDEM OFF respectively.
TANDEM to port 23 in VIEW ONLY mode.
To exit TANDEM enter ESC + "X"
Once a master connects to a TANDEM slave, the connection is in VIEW ONLY mode. This means that the master will see the output of the screen of the TANDEM slave, but the keyboards are not connected.
This sequences enters the TANDEM master user into FEED mode. This means the keyboards are now connected and anything the master user types at the keyboard is passed to the TANDEM slave user. Hence, the master can directly control the terminal of the slave. While in this mode, all keyboard characters are sent to the slave except for the ESCAPE key sequences. If you want to send ESCAPE to the slave, press ESCAPE twice.
This sequence exits the TANDEM session.
This sequence enters the TANDEM master into VIEW ONLY mode. The keyboards are disconnected. All keyboard input from the master user is ignored, except for these escape sequences. In VIEW mode the master user will simply view the output of the slave user.
This sequences enters both the TANDEM master and the TANDEM slave into message mode. What each user types is sent as a message to the other. The following example shows an instance of this feature.
While in the FEED mode, if the master user wants to send an ESCAPE character to the slave you must enter the ESCAPE character twice.
While in the VIEW ONLY mode, pressing the Q key will quit the session in the same manner as ESCAPE+X.
Shows: "TANDEM to port 23 in VIEW ONLY mode.
Shows: "To exit TANDEM enter ESC+X.
When a TANDEM master starts a session, the master must wait until the slave presses RETURN on their keyboard to finish their current terminal input before the session fully takes effect.
The slave must have executed the TANDEM ON command prior to this. This has both security and performance implications. The requirement that the slave terminal must declare itself as such means that a master cannot connect to a terminal arbitrarily. Also, executing TANDEM ON will marginally slow all subsequent terminal input/output incurs an additional performance penalty since all i/o involving the terminal does additional checking for available connections.
In Caché, if a user issues a verb that contains dashes in it, and the verb is not found in the VOC, the shell will replace the dashes with periods and attempt to find the transformed name in the VOC. If it is found, it will be used.
All standard verbs are represented in the VOC with periods in the name. This conversion allows them to be accessed in either form: with periods, or with dashes in the name.


Contribute to audit readiness culture within the LOC by generating a schedule for management monitoring and conducting the management monitoring tracking and ensuring closure of CAPA.
Keep a database of all clinical research in the LOC.
Liaise with the area pharmacovigilance hub, on the basis of PV activities for of all marketed products and products in clinical trials in Nigeria. Responsible for other pharmacovigilance activities in Nigeria performed at the LOC.
Coordinate LOC Medical governance meetings by driving and acting as the secretariat, driving agenda development in consultation with the CMD, taking minutes of meetings, tracking and timely closure of action.
Technical capabilities in advanced Excel and PowerPoint software, and the ability to analyse and present complex data.


text:
\section{Introduction and preliminaries}

Recall that a net $(x_\alpha)$ in a Riesz space $E$ is \emph{order convergent} to $x\in E$ (write $x_\alpha\xrightarrow{o}x$) if there exists another net $(y_\alpha)$ such that $\abs{x_\alpha-x}\leq y_\alpha\downarrow0$. Research of \emph{unbounded order convergence} has a long history in functional analysis literature. It is considered firstly by Nakano in \cite{N:48}, the notion of \emph{unbounded order convergence} was firstly introduced in \cite{W:77}. Several recent papers investigated unbounded order (norm, absolute weak, absolute weak*) convergence. A net $(x_\alpha)$ in a Riesz space $E$ is unbounded order convergent to some $x$ if the net $(|x_\alpha-x|\wedge u)$ converges to zero in order for all $u\in E_+$, we write $x_\alpha\xrightarrow{uo}x$. In \cite{GX:14,G:14,GTX:16,GLX:18}, some properties of $uo$-convergence in Riesz spaces and Banach lattices be studied. A net $(x_\alpha)$ in a Banach lattice $E$ is unbounded norm convergent to some $x$ if the net $(|x_\alpha-x|\wedge u)$ converges to zero in norm for all $u\in E_+$, we write $x_\alpha\xrightarrow{un}x$. The properties of $un$-convergence and $un$-topology, we refer to \cite{DOT:16,KMT:16}. Recently, the unbounded absolute weak convergence has been introduced. A net $(x_\alpha)$ in a Banach lattice $E$ is unbounded absolute weak convergent to some $x$ if the net $(|x_\alpha-x|\wedge u)$ converges to zero in weak for all $u\in E_+$, we write $x_\alpha\xrightarrow{uaw}x$ (see \cite{Z:16} for details). In \cite{W:19}, the unbounded absolute weak* convergence has been considered. A net $(x_\alpha^\prime)$ in a dual Banach lattice $E^\prime$ is unbounded absolute weak* convergent to some $x^\prime$ if the net $(|x_\alpha^\prime-x^\prime|\wedge u^\prime)$ weak* converges to zero for all $u^\prime\in E_+^\prime$, we write $x_\alpha^\prime\xrightarrow{uaw^*}x^\prime$.

It can be easily verified that, in $c_0$ or $l_p (1\leq p<\infty)$, $uo$-convergence, $un$-convergence and $uaw$-convergence of nets is the same as coordinate-wise convergence. In $l_p (1\leq p<\infty)$, $uaw^*$-convergence of nets is the same as coordinate-wise convergence. In $L_p(\mu) (1\leq p<\infty)$ for finite measure $\mu$, $uo$-convergence of sequences is the same as almost everywhere convergence, $un$-convergence and $uaw$-convergence of sequences is the same as convergence in measure. In $L_p(\mu) (1<p<\infty)$, $uaw^*$-convergence of sequences is the same as convergence in measure.

Recall that a bounded subset $A$ of a Banach lattice $E$ is said to be \emph{L-weakly compact} if $x_n\rightarrow0$ for every disjoint sequence $(x_n)$ in the solid hull of $A$. They are the classes of L- and M-weakly compact operators and were introduced by P.Meyer-Nieberg in \cite{MN:74}. A continuous operator $T:E\rightarrow X$ form a Banach lattice to a Banach space is said to be \emph{M-weakly compact} if $Tx_n\rightarrow0$ for every norm bounded disjoint sequence $(x_n)\subset E$. Similarly, a continuous operator $T:X\rightarrow E$ form a Banach space to a Banach lattice is said to be \emph{L-weakly compact} whenever $y_n\rightarrow0$ for every disjoint sequence $(y_n)$ in the solid hull of $T(B_X)$. In \cite{W:19}, unbounded convergences and classical operators are connected. We have the following results.
\begin{theorem}\label{L-weakly compact sets}
(\cite[Theorem~3.1]{W:19})Let $E$ be a Banach lattice, then the following hold.
	\begin{enumerate}
		\item For a non-empty bounded subset $A\subset E$, the following are equivalent.
		\begin{enumerate}
			\item $A$ is L-weakly compact.
			\item\label{L-weakly compact sets-11} $\sup_{x\in A}|x^\prime_n(x)|\rightarrow0$ for every norm bounded $uaw^*$-null sequence $(x^\prime_n)\subset E^\prime$.
			\item\label{L-weakly compact sets-22} $\sup_{x\in A}|x^\prime_n(x)|\rightarrow0$ for every norm bounded $uaw$-null sequence $(x^\prime_n)\subset E^\prime$.
			\item $\sup_{x\in A}|x^\prime_n(x)|\rightarrow0$ for every norm bounded $uo$-null sequence $(x^\prime_n)\subset E^\prime$.
		\end{enumerate}
		\item For a non-empty bounded subset $B\subset E^\prime$, the following are equivalent.
		\begin{enumerate}
			\item $B$ is L-weakly compact.
			\item $\sup_{x^\prime\in B}|x^\prime(x_n)|\rightarrow0$ for every norm bounded $uaw$-null sequence $(x_n)\subset E$.
			\item $\sup_{x^\prime\in B}|x^\prime(x_n)|\rightarrow0$ for every norm bounded $uo$-null sequence $(x_n)\subset E$.
		\end{enumerate}
	\end{enumerate}

\end{theorem}
\begin{theorem}\label{L-weakly compact operators}
(\cite[Theorem~4.2]{W:19}) Let $E$ be Banach space and $F$ be Banach lattice, the following hold.
	\begin{enumerate}
		\item For a continuous operator $T:E\rightarrow F$, the following are equivalent.
		\begin{enumerate}
			\item $T$ is L-weakly compact operator.
			\item $y_n^\prime(Tx_n)\rightarrow0$ for each sequence $(x_n)\subset B_E$ and every bounded $uaw^*$-null sequence $(y_n^\prime)$ in $F^\prime$.
			\item $y_n^\prime(Tx_n)\rightarrow0$ for each sequence $(x_n)\subset B_E$ and every bounded $uaw$-null sequence $(y_n^\prime)$ in $F^\prime$.
			\item $y_n^\prime(Tx_n)\rightarrow0$ for each sequence $(x_n)\subset B_E$ and every bounded $uo$-null sequence $(y_n^\prime)$ in $F^\prime$.
		\end{enumerate}
		\item For a continuous operator $S:F^\prime\rightarrow E^\prime$, the following are equivalent.
		\begin{enumerate}
			\item $S$ is L-weakly compact.
			\item\label{L-weakly compact operators-adjoint-1} $S y_n^\prime(x_n)\rightarrow0$ for each sequence $(y_n^\prime)\subset B_{F^\prime}$ and every bounded $uaw$-null sequence $(x_n)$ in $E$.
			\item $S y_n^\prime(x_n)\rightarrow0$ for each sequence $(y_n^\prime)\subset B_{F^\prime}$ and every bounded $uo$-null sequence $(x_n)$ in $E$.
		\end{enumerate}
	\end{enumerate}

\end{theorem}

\begin{theorem}\label{M-weakly compact operators}
(\cite[Theorem~4.5]{W:19}) Let $E$ and $F$ be Banach lattices, then the following hold.
\begin{enumerate}
\item For a continuous operator $T:E\rightarrow F$, the following are equivalent.
\begin{enumerate}
\item $T$ is a M-weakly compact operator.
\item $Tx_n\rightarrow0$ for every $uaw$-null sequence $(x_n)$ in $B_E$.
\item $Tx_n\rightarrow0$ for every $uo$-null sequence $(x_n)$ in $B_E$.
\end{enumerate}
\item For the continuous operator $S:F^\prime\rightarrow E^\prime$, the following are equivalent.
\begin{enumerate}
\item $S$ is a M-weakly compact operator.
\item $S y^\prime_n\rightarrow0$ for every $uaw^*$-null sequence $(y_n^\prime)$ in $B_{F^\prime}$.
\end{enumerate}
\end{enumerate}
\end{theorem}
We can see that the unbounded norm convergence is the special case. In this paper, we should specialize in such a class of sets and operators. In section 2, we studied the classes of M-weakly compact with the real value range space, that is, continuous dual of Banach lattice of unbounded convergence. We found the exactly form of these duals and proved some classical results by orther ways. In section 3, using the unbounded norm convergence, we introduced a new classes of sets and operators named by weak (weak*) L-weakly compact sets, weak (weak*) L-weakly compact operators and weak M-weakly compact operators. Also we studied the basic properties of these sets and operators. In section 4, we studied that when are compact operators weak L- and M-weakly compact and vice-versa.

To state our results, we need to recall some definitions. Recall that
a Riesz space $E$ is an ordered vector space in which $x\vee y=\sup\{x,y\}(x\wedge y=\inf\{x,y\})$ exists for every $x,y\in E$. The positive cone of $E$ is denoted by $E_+$, $i.e.,E_+=\{x\in E:x\geq0\}$. For any vector $x$ in $E$ define $x^+:=x\vee0,x^-:=(-x)\vee0, \abs{x}:=x\vee(-x)$. An operator $T:E\rightarrow F$ between two Riesz spaces is said to be \emph{positive} if $Tx\geq0$ for all $x\geq0$. A sequence $(x_n)$ in a Riesz space is called disjoint whenever $n\ne m$ implies $\abs{x_n}\wedge\abs{x_m}=0$ (denoted by $x_n\perp x_m$). A set $A$ in $E$ is said order bounded if there exsits some $u\in E_+$ such that $\abs{x}\leq u$ for all $x\in A$. The solid hull $Sol(A)$ of $A$ is the smallest solid set including $A$ and is exactly the set $Sol(A):=\{x\in E:\exists y\in A,\abs{x}\leq\abs{y}\}.$ An operator $T:E\rightarrow F$ is called order bounded if it maps order bounded subsets of $E$ to order bounded subsets of $F$. A Banach lattice $E$ is a Banach space $(E, \Vert\cdot\Vert)$ such that $E$ is a Riesz space and its norm satisfies the following
property: for each $x, y\in E$ with $\abs{x}\leq\abs{y}$, we have $\Vert x\Vert\leq\Vert y\Vert$. It follows from \cite[Theorem~4.1]{AB:06} that if $E$ is a Banach lattice, then its norm dual $E^\prime$ is also a Banach lattice.

For undefined terminology, notations and basic theory of Riesz spaces and Banach lattices, we refer to \cite{LZ:71,LZ:83,AB:06,MN:91}.

\section{weak L-weakly compact and weak M-weakly compact operators}
To state our results, we need following results to show that why a unbounded convergent sequence must be bounded, which is carried to a norm convergent sequence by a continuous operator.
\begin{theorem}\label{finite-atom-operator}
Let $E$ be a Banach lattice and $X$ be a Banach space, for a continuous operator $T:E\rightarrow X$. If $Tx_n\rightarrow0$ for all disjoint sequence $(x_n)\subset E$ and the modulus $\abs{T}$ of $T$ exsits. Then the carrier $C_T=\{x\in E:\abs{T}(\abs{x})=0\}^d$ of $T$ is generated by finitely many atoms.
\end{theorem}
\begin{proof}
Suppose that there exsits infinite positive disjoint sequence of nonzero vectors $(x_n)$ in $C_T$ (since $(\abs{x_n})$ is also disjoint, hence we assume it is positive). Clearly, $\abs{T}(x_n)>0$ for all $n\in N$. Hence there exsits $y_n\in [-x_n,x_n]$ such that $T(y_n)\ne 0$. Since $(y_n)$ is also disjoint sequence, so $\big(\dfrac{y_n}{\Vert T(y_n)\Vert}\big)$ is also disjoint. But $T\big(\dfrac{y_n}{\Vert T(y_n)\Vert}\big)=1\nrightarrow0$, it is absurd. This proves the claim.

	It is similar to the proof of Theorem \ref{finite-atom-functional}, we have the result.
\end{proof}
Using \cite[Corollary~3.6]{GTX:16}, \cite[Lemma~2]{Z:16}, \cite[Lemma~2.3]{W:19} and \cite[Proposition~3.5]{KMT:16}, we have the following result.
\begin{corollary}
Let $E$ and $F$ be Banach lattices, for a continuous operator $T:E\rightarrow F$, assume that the modulus $\abs{T}$ and $\abs{T^\prime}$ exsit, then $C_T$ and $C_{T^\prime}$ are generated by finitely many atoms, if one of the following holds.
\begin{enumerate}
\item $Tx_n\rightarrow0$ for every $uo$-null sequence $(x_n)\subset E$.
\item $Tx_n\rightarrow0$ for every $un$-null sequence $(x_n)\subset E$ and $E$ has order continuous norm.
\item $Tx_n\rightarrow0$ for every $uaw$-null sequence $(x_n)\subset E$.
\item $T^\prime y_n^\prime\rightarrow0$ for every $uaw^*$-null sequence $(y_n^\prime)\subset F^\prime$.
\end{enumerate}
\end{corollary}
By observing the Theorem \ref{L-weakly compact sets}, Theorem \ref{L-weakly compact operators} and Theorem \ref{M-weakly compact operators}, we find that the $un$-convergence is special. Firstly, we study the case in special space. A vector $e$ in a Riesz space $E$ is said to be \emph{strong order unit} if the ideal $I_e$ generated by $e$ equal to $E$. According to \cite[Theorem~4.21 and Theorem~4.29]{AB:06}, a Banach lattice $E$ is lattice and norm isomorphic to $C(K)$ for some compact Hausdorff space $K$ whenever $E$ has a strong order unit. It follows from \cite[Theorem~2.3]{KMT:16} that the $un$-convergence implies norm convergence in $E$ iff $E$ has strong order unit.

Since every bounded $un$-null sequence in $l_\infty$ is norm convergent to zero. It is clearly that every $un$-null sequence in $B_{l_\infty}$ converges uniformly to zero on $B_{l_1}$ and $B_{ba(2^N)}$. So for the identical operator $I:l_1\rightarrow l_1$, every $un$-null sequence in $B_{l_\infty}$ converges uniformly to zero on $I(B_{l_1})$ and $I^{\prime\prime}(B_{ba(2^N)})$. And $I^\prime x_n^\prime\rightarrow0$ for every $un$-null sequence $(x_n^\prime)\subset B_{l_\infty}$.

Clearly, $B_{l_1}$ and $B_{ba(2^N)}$ are not L-weakly compact sets, $I$ and $I^{\prime\prime}$ are not L-weakly compact operators and $I^\prime$ is not M-weakly compact. Therefore we consider the classes of sets and operators.
\begin{definition}
Let $E$ be a Banach lattice, a bounded subset $A\subset E (B\subset E^\prime)$ is called \emph{weak(weak*) L-weakly compact} set whenever $\sup_{x\in A}\abs{x^\prime_n(x)}\rightarrow0 (\sup_{x^\prime\in B}\abs{x^\prime(x_n)}\rightarrow0)$ for every bounded $un$-null sequence $\{x_n^\prime\}\big(\{x_n\}\big)$ in $E^\prime (E)$. Clearly, $A (B)$ is weak (weak*) L-weakly compact set iff for each sequence $\{x_n\}\big(\{x_n^\prime\}\big)$ in $A (B)$, $x_n^\prime(x_n)\rightarrow0$ for every bounded $un$-null sequence $\{x_n^\prime\}\big(\{x_n\}\big)$ in $E^\prime (E)$.

Respectively, let $X$ be a Banach space. A continuous operator $T:X\rightarrow E (T:X\rightarrow E^\prime)$ is said to be \emph{weak (weak*) L-weakly compact } operator if $T(B_X)$ is weak (weak*) L-weakly compact set in $E (E^\prime)$. Clearly, $T$ is weak (weak*) L-weakly compact iff $f_n(Tx_n)\rightarrow0$ for every sequence $(x_n)\subset B_X$ and every bounded $un$-null sequence $(f_n)$ in $E^\prime (E)$.

A continuous operator $T:E\rightarrow Y$ is said to be \emph{weak M-weakly compact operator} whenever $Tx_n\rightarrow0$ for every $un$-null sequence $(x_n)\subset B_E$.
\end{definition}

Then we study the basic properties of these sets and operators.

\begin{proposition}\label{solid-convex hull}
For Banach lattice $E$, the following hold.
\begin{enumerate}
\item Every subset and finite union of weak L-weakly compact set in $E$ is weak L-weakly compact set in $E$.
\item Every subset and finite union of weak* L-weakly compact set in $E^\prime$ is weak* L-weakly compact set in $E^\prime$.
\item The solid convex hull of weak L-weakly compact set in $E$ is weak L-weakly compact set in $E$.
\item The solid convex hull of weak* L-weakly compact set in $E^\prime$ is weak* L-weakly compact set in $E^\prime$.
\end{enumerate}
\end{proposition}
\begin{proof}
$(1)$ and $(2)$ clearly.

$(3)$ For a weak L-weakly compact subset $A$ of Banach lattice $E$, let $Sol(A)$ denote the solid hull of $A$.  $\sup_{x\in Sol(A)}|x_n^\prime(x)|=\sup_{x\in A}\{|y_n^\prime(x)|:|y_n^\prime|\leq |x_n^\prime|\}$. Clearly, $y_n^\prime\xrightarrow{un}0$ since $x_n^\prime\xrightarrow{un}0$. So we have $\sup_{x\in Sol(A)}|x_n^\prime(x)|\rightarrow0$, therefore $Sol(A)$ is also weak L-weakly compact set.

let $B$ denote the solid convex hull of $A$ as
$$B:=\{\sum_{i=1}^{n}\lambda_i x_i:x_i\in Sol(A),\lambda_i\geq0,\sum_{i=1}^{n}\lambda_i=1\}.$$
Then $|x^\prime_n(x)|=|x^\prime_n(\sum_{i=1}^{n}\lambda_i x_i)|\leq \sum_{i=1}^{n}|\lambda_i x_n^\prime(x_i)|\rightarrow0$ for every bounded $un$-null sequence $(x_n^\prime)$ in $E^\prime$. Hence $\sup_{x\in B}|x^\prime_n(x)|\rightarrow0$, therefore $B$ is weak L-weakly compact set.

$(4)$ is similar to $(3)$.
\end{proof}
Weak L-weakly compact set is not weak* L-weakly compact in general. The closed unit ball $B_{l_1}$ of $l_1$ is weak L-weakly compact, but not weak* L-weakly compact. In fact, if a Banach lattice has property (b), then every weak L-weakly compact set in $E^\prime$ is weak* L-weakly compact. Clearly, $c_0$ does not have property (b).

A Banach lattice $E$ has property (b) if and only if for each net $(x_\alpha)$ in $E$ with $0\leq x_\alpha\uparrow \hat{x}$ for some $\hat{x}\in E^{\prime\prime}$, $(x_\alpha)$ is order bounded in $E$. Every $KB$-space and $C(K)$ for some compact Hausdorff space $K$ have property (b) (see \cite{AAT:03} for details).
\begin{proposition}\label{bidual}
For a Banach lattice $E$, the following hold.
\begin{enumerate}
\item If $E$ has property (b), then $x_\alpha\xrightarrow{un}0$ in $E$ iff $x_\alpha\xrightarrow{un}0$ in $E^{\prime\prime}$.
\item If $E$ has property (b), then $x_\alpha\xrightarrow{uaw}0$ in $E$ iff $x_\alpha\xrightarrow{uaw}0$ in $E^{\prime\prime}$.
\item If $E$ has property (b), every weak L-weakly compact set in $E^\prime$ is weak* L-weakly compact.
\end{enumerate}
\end{proposition}
\begin{proof}
$(1)$ Let $(x_\alpha)$ be a $un$-null net in $E$ and for any $\hat{u}\in E^{\prime\prime}$. It follows from \cite[Theorem~4.3(1)]{KMT:16} that $x_\alpha\xrightarrow{un}0$ in the ideal $I_E$ generated by $E$. Its easy to verify that  $|x_\alpha|\wedge\hat{u}=|x_\alpha|\wedge\bigvee_{\beta\leq\alpha}|x_\beta|\wedge\hat{u}$ and $\bigvee_{\beta\leq\alpha}|x_\beta|\wedge\hat{u}\in I_E$. Since $E$ has property (b) and $(\bigvee_{\beta\leq\alpha}|x_\beta|\wedge\hat{u})\uparrow\leq\hat{u}$, so these exsit a $u_0\in I_E^+$ such that $(\bigvee_{\beta\leq\alpha}|x_\beta|\wedge\hat{u})\leq u_0$. Since $x_\alpha\xrightarrow{un}0$ in $I_E$, then $|x_\alpha|\wedge u_0\xrightarrow{}0$, so $|x_\alpha|\wedge\hat{u}=|x_\alpha|\wedge\bigvee_{\beta\leq\alpha}|x_\beta|\wedge\hat{u}$ is convergent to zero.

$(2)$ and $(3)$ by $(1)$.
\end{proof}
Clearly, every M-weakly compact operator is weak M-weakly compact, but the converse does not hold in general. The identical operator $I:l_\infty\rightarrow l_\infty$ is weak M-weakly compact, but not M-weakly compact. In which condition, every weak M-weakly compact operator is M-weakly compact.
\begin{theorem}\label{strong-M-weakly}
Let $E$ be a Dedekind $\sigma$-complete Banach lattice, the following are equivalent.
\begin{enumerate}
\item $E$ has order continuous norm.
\item For each Banach space $F$, every weak M-weakly compact operator $T:E\rightarrow F$ is M-weakly compact.
\item Every positive weak M-weakly compact operator $T:E\rightarrow l_\infty$ is M-weakly compact.
\end{enumerate}
\end{theorem}
\begin{proof}
$(1)\Rightarrow(2)$ Let $T:E\rightarrow F$ be a weak M-weakly compact operator. For a bounded disjoint sequence $(x_n)\subset E$, we have $x_n\xrightarrow{un}0$ by \cite[Proposition~3.5]{KMT:16}. Hence $Tx_n\rightarrow0$, therefore $T$ is M-weakly compact.

$(2)\Rightarrow(3)$ Obvious.

$(3)\Rightarrow(1)$ Assume that $E$ is not order continuous, it follows from \cite[Theorem~4.51]{AB:06} that $E$ contain lattice copy of $l_\infty$. According to \cite[Proposition~1.5.10(1)]{MN:91}, the identical operator $I:l_\infty\rightarrow l_\infty$ can extension to all of $E$. Moreover, $I$ has a positive extension to all of $E$ by \cite[Exercise~1.5.E1]{MN:91}. Hence there exsits a positive projection $P:E\rightarrow l_\infty$.

Let $T=P:E\rightarrow l_\infty$. For every bounded $un$-null sequence $(x_n)\subset E$, it follows from \cite[Theorem~4.3]{KMT:16} that $Tx_n\xrightarrow{un}0$. Since $l_\infty$ has strong order unit, hence $Tx_n\rightarrow0$ by \cite[Theorem~2.3]{KMT:16}. So $T$ is a positive weak M-weakly compact operator. But for the unit vectors $(e_n)$ of $l_\infty$, $\Vert Te_n\Vert=1\nrightarrow0$, therefore $T$ is not M-weakly comopact, it is absurb. Therefore $E$ has order continuous norm.
\end{proof}
Clearly, every L-weakly compact set in Banach lattice $E$ is weak L-weakly compact and weak* L-weakly compact. But the converse does not hold in general. In which condition, every weak (weak*) L-weakly compact set is L-weakly compact.
\begin{theorem}\label{weak-L-L-weakly compact}
Let $E$ be a Banach lattice, then the following hold.
\begin{enumerate}
\item\label{weak-L-L-weakly compact-1} $E^\prime$ has order continuous norm if and only if every weak L-weakly compact set in $E$ is L-weakly compact.
\item\label{weak-L-L-weakly compact-2} If $E$ is Dedekind $\sigma$-complete, then $E$ has order continuous norm if and only if every weak* L-weakly compact set in $E^\prime$ is L-weakly compact.
\end{enumerate}
\end{theorem}
\begin{proof}
$(1)\Rightarrow$ If $E^\prime$ is order continuous, it follows from \cite[Proposition~3.5]{KMT:16} that  $x_n^\prime\xrightarrow{un}0$ in $E^\prime$ for every disjoint $(x_n^\prime)\subset B_{E^\prime}$, hence every weak L-weakly compact in $E$ is L-weakly compact.

In orther words, according to \cite[Theorem~2.4]{W:19}, every $uaw^*$-null net is $un$-null whenever $E^\prime$ is order continuous. Using Theorem \ref{L-weakly compact sets}, we have the result.

$(1)\Leftarrow$ Assume that $E^\prime$ is not order continuous. It follows from \cite[Theorem~2.3.11 and Theorem~2.4.14]{MN:91}, $E$ contain a lattice complementably copy of $l_1$ and according to \cite[Theorem~4.69]{AB:06} there exsits a projection $P:E^\prime\rightarrow l_\infty$. Since $(l_1)^\sim_n=(l_1)^\prime=l_\infty$, so $x^\prime(x)=Px^\prime(x)$ for all $x^\prime\in E^\prime,x\in l_1$ by \cite[Proposition~1.4.10]{MN:91}. Let $B_{l_1}$ be the closed unit ball of $l_1$, since $l_\infty$ has strong order unit, therefore every $un$-null sequence in $l_\infty$ is norm convergent to zero by \cite[Theorem~2.3]{KMT:16}, hence $B_{l_1}$ is a weak L-weakly compact set in $E$ . But $B_{l_1}$ is not L-weakly compact, it is absurb. So $E^\prime$ is order continuous.

$(2)\Rightarrow$ is similar to $(1)\Rightarrow$.

$(2)\Leftarrow$ Assume $E$ does not have order continuous norm, then $E$ contains a lattice complementably copy of $l_\infty$ by \cite[Theorem~4.51]{AB:06}. So $E^\prime$ contains a lattice complementably copy of $ba(2^N)=(l_\infty)^\prime$. Since $E\subset (E^\prime)^\sim_n$, using \cite[Theorem~1.4.10]{MN:91}, we have $x^\prime(x)=x^\prime(Px)$ for every $x^\prime\in ba(2^N),x\in E$. Hence $B_{ba(2^N)}$ is a weak* L-weakly compact set in $E^\prime$, but not L-weakly compact. It is contraction, so $E$ has order continuous norm.
\end{proof}
It is similar to the duality property of L-weakly compact and M-weakly compact operators. We have also:
\begin{proposition}\label{weak L-strong continuous dual}
Let $E$ and $F$ be Banach lattices, then the following hold.
\begin{enumerate}
\item A continuous operator $T:E\rightarrow F$ is weak L-weakly compact iff $T^\prime:F^\prime\rightarrow E^\prime$ is weak M-weakly compact.
\item A continuous operator $T:E\rightarrow F$ is weak M-weakly compact iff $T^\prime:F^\prime\rightarrow E^\prime$ is weak* L-weakly compact.
\end{enumerate}
\end{proposition}
\begin{proof}
$(1)$ Since $y^\prime(y)=y^\prime(Tx)=(T^\prime y^\prime)(x)$, hence $\Vert T^\prime y_n^\prime\Vert=\sup_{y\in T(B_E)}\abs{y_n^\prime(y)}$. Assume that $T$ is weak L-weakly compact, then for a $un$-null sequence $(y_n^\prime)\subset B_{F^\prime}$, we have $\Vert T^\prime y_n^\prime\Vert=\sup_{y\in T(B_E)}\abs{y_n^\prime(y)}\rightarrow0$, hence $T^\prime$ is weak M-weakly compact. The converse is similar.

$(2)$ is similar to $(1)$.
\end{proof}
Clearly, every L-weakly compact operator is weak L-weakly compact and weak* L-weakly compact. But the converse does not hold in general. In which condition, every weak (weak*) L-weakly compact operator is L-weakly compact.

\begin{theorem}\label{weak-L-weakly compact operator}
Let $E$ be a Banach lattice, then the following are equivalent.
\begin{enumerate}
\item $E^\prime$ has order continuous norm.
\item For each Banach space $X$, every continuous operator $T:X\rightarrow E$ is weak L-weakly compact iff $T$ is L-weakly compact.
\item Every positive operator $T:l_1\rightarrow E$ is weak L-weakly compact iff $T$ is L-weakly compact.
\end{enumerate}
\end{theorem}
\begin{proof}
$(1)\Rightarrow(2)$ According to \cite[Proposition~3.5]{KMT:16}, every disjoint sequence is $un$-null.

$(2)\Rightarrow(3)$ obvious.

$(3)\Rightarrow(1)$ Assume $E^\prime$ is not order continuous, it follows from \cite[Theorem~4.69]{AB:06} that $E$ contains a lattice complementably copy of $l_1$. Let $S:l_1\rightarrow E$ be the canonical injection. Since $E^\prime$ contains the lattice complementably copy of $l_\infty$ which has strong order unit, so $B_{l_1}$ is weak L-weakly compact set. Hence the operator $T=S:l_1\rightarrow E$ is positive weak L-weakly compact.  But $T$ is not L-weakly compact. It is absurb, therefore $E^\prime$ has order continuous norm.
\end{proof}
Dually, we have:
\begin{theorem}\label{weak*-L-weakly compact operator}
Let $E$ be a Dedekind $\sigma$-complete Banach lattice, then the following are equivalent.
\begin{enumerate}
\item $E$ has order continuous norm.
\item For each Banach space $X$, every continuous operator $T:X\rightarrow E^\prime$ is weak* L-weakly compact iff $T$ is L-weakly compact.
\item Every positive operator $T:ba(2^N)\rightarrow E^\prime$ is weak* L-weakly compact iff $T$ is L-weakly compact.
\end{enumerate}
\end{theorem}
\begin{proof}
We prove $(3)\Rightarrow(1)$, $(1)\Rightarrow(2)\Rightarrow(3)$ are obvious.

Assume $E$ does not have order continuous norm, then $E$ contains a lattice complementably copy of $l_\infty$. Hence $E^\prime$ contains a lattice complementably copy of $ba(2^N)$. Let $T:ba(2^N)\rightarrow E^\prime$ be the canonical injection. Since $E$ contains the lattice complementably copy of $l_\infty$ which has strong order unit, so $B_{ba(2^N)}$ is weak* L-weakly compact set. Therefore $T:ba(2^N)\rightarrow  E^\prime$ is positive weak* L-weakly compact since $E$ contains a lattice complementably copy of $l_\infty$. But $T$ is not L-weakly compact. It is contraction, hence $E$ has order continuous norm.
\end{proof}
The following results give some equivalent characterizations of weak (weak*) L-weakly compact sets and weak (weak*) L-weakly compact operators by disjoint sequence.
\begin{theorem}\label{disjoint-weak L}
Let $E$ be a Banach lattice, $A$ be a bounded solid subset of $E$ and $B$ be a bounded solid subset of $E^\prime$. Then the following statements hold.
\begin{enumerate}
\item If $E$ has order continuous norm, then the following are equivalent.
\begin{enumerate}
\item $A$ is weak L-weakly compact.
\item\label{disjoint-d-1} For every positive disjoint sequence $(x_n)$ in $A$ and each bounded $un$-null sequence $(x_n^\prime)$ in $E^\prime$, we have $x_n^\prime(x_n)\rightarrow0$.
\end{enumerate}
\item If $E^\prime$ has order continuous norm, then the following are equivalent.
\begin{enumerate}
\item $B$ is weak* L-weakly compact.
\item For every positive disjoint sequence $(x_n^\prime)$ in $B$ and each bounded $un$-null sequence $(x_n)$ in $E$, we have $x_n^\prime(x_n)\rightarrow0$.
\end{enumerate}
\end{enumerate}
\end{theorem}

\begin{proof}
$(1)(a)\Rightarrow(1)(b)$ obvious.

$(1)(b)\Rightarrow(1)(a)$ Let $(x_n^\prime)$ be an any bounded $un$-null sequence in $E^\prime$. we have to show that $\sup_{x\in A}|x_n^\prime(x)|\rightarrow0$. Assume by way of contradiction that $\sup_{x\in A}|x_n^\prime(x)|$ does not converge to zero as $n\rightarrow\infty$. Then there exsits some $\epsilon>0$ such that $\sup_{x\in A}|x_n^\prime(x)|>\epsilon$. Note that the equality $\sup_{x\in A}|x_n^\prime(x)|=\sup_{0\leq x\in A}|x_n^\prime|(x)$ holds, since $A$ is solid. \cite[Theorem~8.1]{KMT:16} and \cite[Proposition~5]{Z:16}, we have $|x_n^\prime|\xrightarrow{w^*}0$.
According to the proof of \cite[Theorem~2.5]{CCJ:14}, we can find a strictly increasing subsequence $(n_k)_{k=1}^\infty\subset N$ such that $|x_{n_{m+1}}^\prime|(4^m\sum_{k=1}^{m}x_{n_k})<\frac{1}{m}$. Let
$$x=\sum\limits_{k=1}^\infty2^{-k}x_{n_{k}}, y_m=(x_{n_{m+1}}-4^m\sum\limits_{k=1}^{m}x_{n_k}-2^{-m}x)^+.$$
Then by \cite[Lemma~4.35]{AB:06}, $(y_m)$ is positive disjoint sequence in $A$, since $A$ is solid. It is similar to \cite[Theorem~2.5]{CCJ:14}, we have $\abs{x^\prime_{n_{m+1}}}(y_m)\nrightarrow0$, it is contraction to our assumption,
we complete the proof

$(2)$ is similar to $(1)$.
\end{proof}
It is similar to \cite[Theorem~4.3]{W:19}, we have the following result.
\begin{theorem}\label{weak L-weakly compact operator-disjoint}
Let $E$ and $F$ be Banach lattices, the following statements hold.
\begin{enumerate}
\item For a positive operator $T:E\rightarrow F$, if $F$ has order continuous norm, then the following are equivalent.
\begin{enumerate}
\item $T$ is weak L-weakly compact.
\item $y_n^\prime(Tx_n)\rightarrow0$ for each positive disjoint sequence $(x_n)\subset B_E$ and every bounded $un$-null sequence $(y_n^\prime)$ in $F^\prime$.
\end{enumerate}
\item For a positive operator $S:F^\prime\rightarrow E^\prime$, if $E^\prime$ has order continuous norm, then the following are equivalent.
\begin{enumerate}
\item $S$ is weak* L-weakly compact.
\item $S y_n^\prime(x_n)\rightarrow0$ for each positive disjoint sequence $(y_n^\prime)\subset B_{F^\prime}$ and every bounded $un$-null sequence $(x_n)$ in $E$.
\end{enumerate}
\end{enumerate}
\end{theorem}

\section{compactness of weak L-weakly compact and weak M-weakly compact operators}\label{}
There is no direct relationship between relatively compact set and weak (weak*) L-weakly compact set. The close unit ball $B_{ba(2^N)}$ is weak and weak* L-weakly compact set, but not relatively compact. The singleton $\{x\}=(1,...1,....)\in l_\infty$ is relatively compact set, but not weak and weak* L-weakly compact. If a Banach lattice $E$ has order continuous norm, then every relatively compact set in $E$ is L-weakly compact, moreover it is weak L-weakly compact.

The following result shows that when the class of relatively compact sets and the class of weak (weak*) L-weakly compact sets coincide.
\begin{proposition}\label{compact=wl=w*l}
For a Dedeking $\sigma$-complete Banach lattice $E$, the following are equivalent.
\begin{enumerate}
\item $E$ is atomic and both $E$ and $E^\prime$ are order continuous ($E^\prime$ is atomic and both $E$ and $E^\prime$ are order continuous).
\item The class of relatively compact sets, the class of L-weakly compact sets and the class of weak L-weakly compact sets in $E$ coincide.
\item The class of relatively compact sets, the class of L-weakly compact sets and the class of weak* L-weakly compact sets in $E^\prime$ coincide.
\end{enumerate}
\end{proposition}
\begin{proof}
According to \cite[Theorem~2.3]{AE:10}, $E$ is atomic and both $E$ and $E^\prime$ are order continuous iff $E^\prime$ is atomic and both $E$ and $E^\prime$ are order continuous.

$(1)\Rightarrow(2)$ Since $E^\prime$ is order continuous, according to Theorem \ref{weak-L-L-weakly compact}\eqref{weak-L-L-weakly compact-1}, we only need to show the class of relatively compact sets and L-weakly compact sets coincide. Since $E$ has order continuous norm, therefore every relatively compact set in $E$ is L-weakly compact.

For a L-weakly compact set $A\subset E$. According to \cite[Proposition~3.6.2]{MN:91}, for every $\epsilon>0$, there exists some $x\in E^a_+$ such that $A\subset [-x,x]+\epsilon\cdot B_E$. Since $E$ is atomic, it follows from \cite[Theorem~6.1]{W:99} that $[-x,x]$ is compact. Using \cite[Theorem~3.1]{AB:06}, we have $A$ is relatively compact.

$(2)\Rightarrow(1)$ According to Theorem \ref{weak-L-L-weakly compact}\eqref{weak-L-L-weakly compact-1}, we have $E^\prime$ is order continuous.

Assume that $E$ is not order continuous, then $E$ contains a lattice copy of $l_\infty$. Clearly the singleton $\{x\}=(1,1...)\in l_\infty$ is compact, but not L-weakly compact. It is absurd, therefore $E$ is order continuous.

Suppose that $E$ is not atomic. According to \cite[Theorem~6.1]{W:99}, $[-x,x]$ is not relatively compact for all $x\in E_+$. But $[-x,x]$ is L-weakly compact, it is contraction, so $E$ is atomic.

$(1)\Leftrightarrow(3)$ is similar to $(1)\Leftrightarrow(2)$.
\end{proof}

Compact operator is not these operators in general. Considering the compact operator (rank is 1) $T:l_1\rightarrow l_\infty$ define as $T(x_n)=(\sum_{n=1}^{\infty}x_n,\sum_{n=1}^{\infty}x_n,...)\in l_\infty$ for each $(x_n)\in l_1$. It is clear that $T$ is not weak (weak*) L-weakly compact and weak M-weakly compact.

Weak (weak*) L-weakly compact and weak M-weakly compact operator is not compact in general. Considering the identical operator $I:l_1\rightarrow l_1$, $I$ is weak L-weakly compact, $I^\prime$ is weak M-weakly compact and $I^{\prime\prime}$ is weak* L-weakly compact. But $I$, $I^\prime$ and $I^{\prime\prime}$ are not compact.

The following conclusions can be obtained directly.
\begin{proposition}
Let $X$ be a Banach space and $E$ be a Banach lattice, then the following hold.
\begin{enumerate}
\item If $E$ has order continuous norm, then every compact operator from $X$ into $E$ is weak L-weakly compact.
\item If $E^\prime$ has order continuous norm, then every compact operator from $X$ into $E^\prime$ is weak* L-weakly compact.
\item If $E^\prime$ has order continuous norm, then every compact operator from $E$ into $X$ is weak M-weakly compact.
\end{enumerate}
\end{proposition}
The following result shows that when the weak L-weakly compact operator is compact.
\begin{theorem}\label{wl is compact}
Let $F$ be a Banach lattice, then the following are equivalent.
\begin{enumerate}
\item $F^a$ is atomic and $F^\prime$ has order continuous norm.
\item For each Banach space $X$, every weak L-weakly compact operator $T:X\rightarrow F$ is compact.
\item For each Banach lattice $E$ without order continuous dual, every positive weak L-weakly compact operator $T:E\rightarrow F$ is compact.
\end{enumerate}
\end{theorem}
\begin{proof}
$(1)\Rightarrow (2)$ Since $F^\prime$ has order continuous norm, so every weak L-weakly compact operator $T:X\rightarrow F$ is L-weakly compact by Theorem \ref{weak-L-weakly compact operator}.  According to \cite[Proposition~3.6.2]{MN:91}, for any $\epsilon>0$, there exists some $y\in F^a_+$ such that $$T(B_E)\subset [-y,y]+\epsilon\cdot B_E.$$ Since $F^a$ is atomic, it follows from \cite[Theorem~6.1(5)]{W:99} that the order interval $[-y,y]$ is norm compact. Using \cite[Theorem~3.1]{AB:06}, we have $T(B_E)$ is relatively compact set in $F$, so $T$ is also compact.

$(2)\Rightarrow (3)$ obvious.

$(3)\Rightarrow (1)$ We prove $F^\prime$ is order continuous first. Assume $F^\prime$ is not order continuous, by \cite[Theorem~4.69]{AB:06}, $F$ contains the lattice copy of $l_1$. Let $S:l_1\rightarrow F$ be the canonical injection of $l_1$ into $F$.  Let $E=l_1$, the positive operator $T=S:l_1\rightarrow F$ is positive weak L-weakly compact since $l_\infty=(l_1)^\prime$ is a $AM$-space with strong order unit, but not compact. Therefore $F^\prime$ has order continuous.

Then we prove that $F^a$ is atomic. Since $E^\prime$
is not order continuous then, by \cite[Theorem ~2.4.14]{MN:91}, there is a norm bounded disjoint
sequence $(u_n)$ of positive elements in $E$ which does not converge weakly to zero. Without loss of generality, we may assume that $\Vert u_n\Vert\leq 1$ for all $n$ and that there are $\phi\in E^\prime_+$ and $\epsilon>0$ such that $\phi (u_n) > \epsilon$ for all $n$. It follows from \cite[Theorem~116.3]{LZ:83} that the components $\phi_n$ of $\phi$, in the carriers $C_{u_n}$, form an order bounded
disjoint sequence in $E^\prime_+$ such that \text{$\phi_n (u_n) = \phi (u_n)$ for all $n$ and $\phi_n (u_m) = 0$ if $n\neq m$.} Note that $0 \leq \phi_n \leq \phi$ for all $n$.

Assume that $F^a$ is not atomic, it follows from \cite[Theorem~6.1]{W:99} that there
exists some $0 \leq y \in F^a$
such that $[0, y]$ is not norm compact. Now, fix a sequence $(y_n)$
in $[0, y]$ which has no norm convergent subsequence in $F^a$ and none in $F$.

Define an operator $T : E \rightarrow F$ by
$$T(x)=\sum_{n=1}^{\infty}(\frac{\phi_n(x)}{\phi(u_n)})y_n$$
for $x\in E$. Note that in view of the inequality
$$\sum_{n=1}^{\infty}\Vert (\frac{\phi_n(x)}{\phi(u_n)})y_n\Vert\leq \frac{1}{\epsilon}\Vert y\Vert\sum_{n=1}^{\infty}\phi_n(|x|)\leq \frac{1}{\epsilon}\Vert y\Vert\phi(|x|)$$
for each $x \in E,$ the series defining $T$ converges in norm for each $x \in E$ and $T (u_n) = y_n$ for all $n$. Hence the operator $T$ is well defined and it is also easy to see that $T$ is a
positive operator.

Since $(y_n)$ has no norm convergent subsequence in $F$, so $T$ is not compact by Grothendieck theorem (\cite[Theorem~5.3]{AB:06}). However, $T$ maps norm bounded subset in $E$ to a order bounded subset in $F$. To see this, note that for all $x \in B_E$, we have
$$|T(x)| \leq T(|x|) =\sum_{n=1}^{\infty}(\frac{\phi_n(|x|)}{\phi(u_n)})y_n\leq \frac{1}{\epsilon}\big(\sum_{n=1}^{\infty}\phi_n(|x|)\big)y\leq \frac{1}{\epsilon}\phi(|x|)y\leq \frac{1}{\epsilon}\Vert\phi\Vert y.$$
So $T$ is L-weakly compact operator, hence $T$ is weak L-weakly compact operator. It is contraction, so $F^a$ is atomic.
\end{proof}

The following results show that when an operator is both compact and weak L-weakly compact.
\begin{theorem}\label{compact=w L-weakly compact}
Let $F$ be a Banach lattice, then the following are equivalent.
\begin{enumerate}
\item $F$ is atomic and both $F$ and $F^\prime$ are order continuous.
\item For each Banach space $X$, every continuous operator $T:X\rightarrow F$ is compact operator iff $T$ is weak L-weakly compact.
\item For each Banach lattice $E$ without order continuous dual, every positive operator $T:E\rightarrow F$ is compact operator iff $T$ is weak L-weakly compact.
\end{enumerate}
\end{theorem}

\begin{proof}
$(1)\Rightarrow(2)$ Since $F$ is order continuous, so every compact operator $T:E\rightarrow F$ is weak L-weakly compact. Since $F$ is an atomic Banach lattice with order continuous dual, it follows form Theorem \ref{wl is compact} that every weak L-weakly compact operator is compact.

$(2)\Rightarrow(3)$ Obvious.

$(3)\Rightarrow(1)$
According to Theorem \ref{wl is compact}, we have $F^a$ is atomic and $F^\prime$ is order continuous. We claim that $F$ has order continuous norm.

Suppose that $F$ is not order continuous, by \cite[Corollary~2.4.3]{MN:91}, there exists a disjoint sequence $(x_n^\prime)\subset B_{F^\prime}$ such that $x_n^\prime\nrightarrow0$ in $\sigma(F^\prime,F)$. That is, there is some $y\in F_+$ with $x^\prime_n(y)\nrightarrow0$. As $E\ne\{0\}$, we may fix
$u\in E$ and pick a $\phi\in E^\prime$
such that $\phi(u)=\Vert u\Vert=1$ holds.

Now, we consider operator $T : E\rightarrow F$ defined by $$T(x)=\phi(x)\cdot y$$ for each $x\in E$. Clearly, $T$ is a positive compact operator (its rank is 1). But it is not an weak L-weakly compact operator.
If not, as the singleton $\{u\}$ and
$T(u) = \phi(u) · y = y$. Since $x^\prime_n(y)\nrightarrow0$, but $x_n^\prime\xrightarrow{un}0$ by \cite[Proposition~3.5]{KMT:16}, therefore $T$ is weak L-weakly compact. It is absurd, so $F$ is order continuous. Hence $F$ is atomic.
\end{proof}
The following result shows that when a weak M-weakly compact operator is compact.
\begin{theorem}\label{sc is compact}
Let $E$ be a Dedekind $\sigma$-complete Banach lattice, then the following are equivalent.
\begin{enumerate}
\item $(E^\prime)^a$ is atomic and $E$ has order continuous norm.
\item For each Banach space $Y$, every weak M-weakly compact operator $T:E\rightarrow Y$ is compact.
\item For each Banach lattice $F$ without order continuous norm, every positive weak M-weakly compact operator $T:E\rightarrow F$ is compact.
\end{enumerate}
\end{theorem}
\begin{proof}
$(1)\Rightarrow(2)$ For a strong continuous operator $T:E\rightarrow Y$. Since $E$ has order continuous norm, according to Theorem \ref{strong-M-weakly}, $T$ is M-weakly compact. It follow from \cite[Theorem~5.64]{AB:06} that $T^\prime:Y^\prime\rightarrow E^\prime$ is L-weakly compact, hence $T^\prime$ is weak* L-weakly compact. By Theorem \ref{weak-L-L-weakly compact}, $T^\prime$ is L-weakly compact.

According to \cite[Proposition~3.6.2]{MN:91}, for any $\epsilon>0$, there exists some $x^\prime\in (E^\prime)^a_+$ such that $$T^\prime(B_{Y^\prime})\subset [-x^\prime,x^\prime]+\epsilon\cdot B_{E^\prime}.$$ Since $(E^\prime)^a$ is atomic, it follows from \cite[Theorem~6.1(5)]{W:99} that the order interval $[-x^\prime,x^\prime]$ is norm compact. Using \cite[Theorem~3.1]{AB:06}, we have $T(B_{Y^\prime})$ is relatively compact set in $E^\prime$, so $T^\prime$ is also compact. It follows Schauder theorem (\cite[Theorem~5.2]{AB:06}) that $T:E\rightarrow Y$ is compact.

$(2)\Rightarrow(3)$ obvious.

$(3)\Rightarrow(1)$ At first, we prove that $E$ has order continuous norm. Assume that $E$ is not order continuous, then there exsits a positive projection $P:E\rightarrow l_\infty$. Let $F=l_\infty$. Considering the operator $T:P:E\rightarrow l_\infty$, $T$ is a positive weak M-weakly compact operator since $l_\infty$ has strong order unit. But $T$ is not compact. It is contraction, so $E$ has order continuous norm.

Then, we claim that $(E^\prime)^a$ is atomic. We assume that $(E^\prime)^a$ is not atomic and construct a
positive weak M-weakly compact operator from $E$ into $F$ which is not compact.

Since the norm of $F$ is not order continuous then, by \cite[Theorem~4.14]{AB:06}, there
exists a disjoint order bounded sequence $(y_n)$ in $F_+$ which does not converge to zero
in norm. We may assume that $0 \leq y_n \leq y$ and $\Vert y_n\Vert=1$ for all $n$ and some $y\in F_+$.

Since $(E^\prime)^a$ is not atomic, there exsits a $0\leq\psi\in (E^\prime)^a$ such that the order interval $[0,\psi]$ is not norm compact by \cite[Theorem~6.1]{W:99}. Choose a sequence $(\phi_n)$ in $[0,\psi]$ which
has no norm convergent subsequence in $E^\prime$. Also, since $\psi\in (E^\prime)^a$, by \cite[Theorem~2.4.2]{MN:91}, the order interval
$[0,\psi]$ is weakly compact. By the Eberlein-Smulian Theorem (\cite[Theorem~3.40]{AB:06}), we may assume, by extracting a subsequence if necessary, that $(\phi_n)$ converges
weakly to some $\phi\in [0,\psi]$. So $(\phi_n)$ converges weakly* to $\phi$.

Now, define two operators $S,T : E \rightarrow F$ by
$$S(x) = \phi (x) y+\sum_{n=1}^{\infty} (\phi_n-\phi)(x) y_n,\quad T(x)=\psi(x)y$$
for each $x\in E$. It follows from the proof of Wickstead in \cite[Theorem~1]{W:96} that $0\leq S\leq T$ and that $S$ is not compact.

We claim that $S$ is weak M-weakly compact operator. For this, note that $0 \leq S^\prime \leq T^\prime$ and $T^\prime(h)=h(y)\psi$ for all $h\in F^\prime$. Then for every $h\in B_{F^\prime}$, we have
$$|S^\prime(h)|\leq S^\prime(|h|)\leq T^\prime(|h|)\leq |h|y\psi\leq \Vert y\Vert\psi,$$
Since $[-\psi,\psi]$ is L-weakly compact set, so $S^\prime$ is L-weakly compact operator, hence $S^\prime$ is weak* L-weakly compact operator. According to Proposition \ref{weak L-strong continuous dual}, $S$ is weak M-weakly compact. It is contraction, therefore $(E^\prime)^a$ is atomic.
\end{proof}
\begin{corollary}\label{w*l is compact}
	Let $E$ be a Dedekind $\sigma$-complete Banach lattice, then the following are equivalent.
	\begin{enumerate}
		\item $(E^\prime)^a$ is atomic and $E$ has order continuous norm.
		\item For each Banach space $Y$, every weak* L-weakly compact operator $T^\prime:Y^\prime\rightarrow E^\prime$ is compact.
		\item For each Banach lattice $F$ without order continuous norm, every positive weak* L-weakly compact operator $T^\prime:F^\prime\rightarrow E^\prime$ is compact.
	\end{enumerate}
\end{corollary}

The following result shows that when an operator is both compact and weak M-weakly compact.
\begin{theorem}\label{sc=compact}
Let $E$ be a Dedekind $\sigma$-complete Banach lattice, then the following are equivalent.
\begin{enumerate}
\item $E^\prime$ is atomic and both $E$ and $E^\prime$ are order continuous.
\item For each Banach space $Y$, every continuous operator $T:E\rightarrow Y$ is weak M-weakly compact iff $T$ is compact.
\item For each Banach lattice $F$ without order continuous norm, every positive operator $T:E\rightarrow F$ is weak M-weakly compact iff $T$ is compact.
\end{enumerate}
\end{theorem}
\begin{proof}
$(1)\Rightarrow(2)$ For a strong continuous operator $T:E\rightarrow Y$. Since $E^\prime$ is atomic and $E$ is order continuous, it follows from Theorem \ref{sc is compact} that $T$ is compact. Using $E^\prime$ is order continuous and Schauder theorem (\cite[Theorem~5.2]{AB:06}), we can get the converse.

$(2)\Rightarrow(3)$ obvious.

$(3)\Rightarrow(1)$ According to Theorem \ref{sc is compact}, we have $(E^\prime)^a$ is atomic and $E$ is order continuous. We claim that $E^\prime$ is order continuous.

Assume that $E^\prime$ is not order continuous. It follows from \cite[Theorem~4.69]{MN:91} that there exsits a positive projection $P:E\rightarrow l_1$. Fix a vector $0<y\in F_+.$ Define the operator $S:l_1\rightarrow F$ as follows:
$$S\left(\lambda_{n}\right)=\left(\sum_{n=1}^{\infty} \lambda_{n}\right) y$$
for each $(\lambda_n)\in l_1$. Obviously, the operator $S$ is well defined. Let
$$T=S \circ P : E \rightarrow l_{1} \rightarrow F$$
then $T$ is a positive compact operator since $S$ is a finite rank operator (rank is 1). Let $\left(e_{n}\right)$ be the standard basis of $l_{1}$. Clearly, $e_n\xrightarrow{un}0$, according to \cite[Theorem~4.3]{KMT:16}, $e_{n}\xrightarrow{un}0$ in $E$, but $T(e_n)=y\nrightarrow0$. Hence, $T$ is not an weak M-weakly compact operator. It is absurd, so $E^\prime$ is order continuous, therefore $E^\prime$ is atomic.
\end{proof}

\begin{corollary}\label{w*l = compact}
	Let $E$ be a Dedekind $\sigma$-complete Banach lattice, then the following are equivalent.
\begin{enumerate}
\item $(E^\prime)^a$ is atomic and both $E$ and $E^\prime$ are order continuous.
\item For each Banach space $Y$, every continuous operator $T^\prime:Y^\prime\rightarrow E^\prime$ is weak* L-weakly compact iff $T$ is compact.
\item For each Banach lattice $F$ without order continuous norm, every positive operator $T^\prime:F^\prime\rightarrow E^\prime$ is weak* L-weakly compact iff $T$ is compact.
	\end{enumerate}
\end{corollary}

redpajama_set_name: RedPajamaArXiv


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At WSP, we constantly push the boundaries to meet the high expectations of our clients. Our expertise makes it possible for architects to realize their aesthetic visions, and for our clients to meet their commercial and operational goals. We have a long history of innovation and a reputation for blending imaginative thinking with common sense solutions. The buildings we engineer perform well throughout their lifetime and make a positive contribution to society and the environment.
Creating the Highest Value
We add value to our projects by listening carefully to our clients in order to ensure that we align with their objectives and the needs of building occupants, in addition to delivering structures that are safe, sustainable and cost-effective. From the earliest design stages, we plan for the diverse uses of a project through its whole life-cycle, focusing on efficiency and economy. We seek to deliver flexible solutions able to adapt to future needs, while optimizing the building structure to maximize property value.
1 UNDERSHAFT, LONDON
Delivering Technical Excellence for Any Type of Building
With world-class engineering skills, technical back up and in-depth knowledge of current design standards, we deliver the best, most resilient and efficient structures for any type of building, anywhere in the world and in the most challenging conditions.
Our work includes assuring perfect acoustic performance for the concert hall at the Guildhall School of Music and Drama in London, with a 'box-in- box' design to isolate the space from external sounds and vibrations. Other examples include overcoming challenging site conditions, heavy stone masonry and a heritage designation for a magnificent renovation of the Victoria Memorial Museum of Nature in Ottawa, Canada.
Overcoming Complex Structural Engineering Challenges
We regularly design structures on constrained inner-city sites with all of the associated challenges; for example, making use of existing basements and foundations, threading piles around underground services and infrastructure and working around underground rail tunnels. We provide commercial solutions to minimize the impact of wind on tall buildings through the innovative use of high strength concrete and steel and the intelligent use of acceleration damping. We also ensure the stability of buildings in challenging seismic conditions and, through our expertise in analyzing the potential impact of extreme events on structures such as bomb blasts, we develop strategies for effective mitigation measures.
VICTORIA MEMORIAL MUSEUM OF NATURE, OTTAWA, CANADA
Saving Time, Saving Money
We have driven advances in engineering and innovative techniques to streamline building construction, minimize costs and reduce commercial risks. We pioneered the method of top-down construction for The Shard in London, saving approximately four months on the program. The technique is now used in the majority of our high-rise projects. We helped accelerate the build time for Alder Hey Children's Hospital in Liverpool through the use of modular design. Many of the structural components were manufactured off-site, and by using precast concrete panels on the perimeter to distribute the load to the foundations, we also enabled maximum flexibility of room layouts inside the hospital.
Sustainable Structures
Sustainability credentials of buildings are becoming increasingly important to our commercial clients by allowing them to deliver properties that are attractive to tenants and allow the flexibility to extend the useful life of a building for maximum return on investment. Our holistic approach takes into account the whole life-cycle of the building, from the earliest design stages through construction and operation to demolition.
During the design process we routinely seek to identify opportunities to reuse existing materials or incorporate recycled and/or locally sourced materials into structures. We use carbon and energy assessment tools to calculate the embodied carbon for all structural options, and we advise on ways to reduce construction waste through material selection and construction methods. For Queen Elizabeth University Hospital in Glasgow, WSP implemented an innovative on-site waste sorting system which resulted in 96% of waste being diverted from landfill (exceeding the target of 80%), while also reducing traffic movements for removal of waste from the site.
ONE WORLD TRADE CENTER, NEW YORK
Structural Engineering for High-Rise Buildings
We are leaders in the structural design of high-rise buildings, with a portfolio of iconic towers including the World Trade Center and The Shard. We are now delivering a new generation of super-slender high rise residential developments with buildings such as 432 Park Avenue, the exceptionally slim, 426m high residential, hotel and commercial tower in New York, currently the tallest residential building gracing city skylines in the western hemisphere. These slender buildings help developers make the most of limited real estate in urban areas and to capitalize on small and constricted sites to create highly desirable residences with magnificent views and high market values.
Our Structural Engineering Consultancy Services
We deliver all kinds of building projects from the earliest planning stages through to operation, in roles ranging from lead consultant to specialist services provider. Our core structural services include structural design, engineering for high wind and seismic environments, peer review and value engineering, forensic engineering and structural investigations, building renovations and alterations, blast resistant design, construction inspection, 3D visualization and BIM and steel detailing.
Enduring Relationships with the World's Top Clients and Architects
WSP is proud to have built enduring relationships with some of the world's top developers and signature architects, for whom we are delivering exciting and complex schemes across the globe. We apply our depth of international experience obtained over many years to the thinking process that precedes construction, providing valuable contributions from the outset to achieve our clients' and architects' aspirations, while responding to the constraints of the project and the local market.
Thanks to our integrated and flexible global structure, our clients benefit from close, long-term relationships with local delivery teams who understand the challenges they face, backed by a global network of specialists. This allows us to deliver the most innovative solutions to every project, along with the highest levels of experience, technical expertise and in-depth knowledge of local cultures, practices, standards and regulations.
Visit The Possible
Commercial and Mixed Use
Mission Critical Facilities
Sports and Stadia
Building Information Modelling - BIM
Green Building Design
Mechanical, Electrical and Plumbing
Property Condition Assessment
Wind Engineering and Modelling
450 Elizabeth Street, Melbourne
450 Elizabeth Street is a 70-storey residential tower standing on a 1000m² street corner site in central Melbourne. Known as "The Lighthouse", it will contain 610 apartments with communal facilities including a pool and a gymnasium. It also has an automated car parking system located in the podium.
Optimizing Learning in Energy Efficient Welsh Schools
Thanks to WSP's sustainable designs, primary school students and teachers in Wales were treated to a new and improved learning environment. Two primary schools in Carmarthenshire have been designed and built according to Passivhaus, the world's most rigorous energy and comfort standard.
Torre Mayor, Mexico City
High-rise buildings in earthquake zones must meet the same standards of safety, comfort and buildability as anywhere else, but with a whole other set of challenges.
London, England, UK
Strata SE1, London
At 43 storeys, Strata SE1 was the tallest building in south London when it was completed. It is home to more than 1,000 people and is the first ever building with wind turbines integrated into the building form itself.
One57, New York
Rooms with views in one of Manhattan's tallest residential buildings
Johannesburg, Gauteng, South Africa
Montecasino Entertainment World and Teatro, Johannesburg
The largest in the southern hemisphere, the Teatro is a core component of the extension to one of Johannesburg's premier casino and entertainment centres. The theatre is designed for large-scale productions and can accommodate 1,900 people at full capacity.
Manhattanville, Columbia University
An academic mixed-use project spanning 17 acres of the old Manhattanville manufacturing zone in West Harlem. It will boast 6.8 million ft² of space for teaching, research, underground parking and support services, and create new facilities for civic, cultural, recreational and commercial activity.
Kerry Centre, Hong Kong
A 138m-high, 30-storey tower, with nearly 70,000m2 of construction floor area. This was the first time that WSP in Asia used Building Information Modelling (BIM) to cover all design items in a project.
Arthur Ashe Stadium, New York
Arthur Ashe Stadium is the largest outdoor tennis-only venue in the world by capacity, and it is the main stadium of the US Open.
3 Hudson Boulevard, New York
3 Hudson Boulevard is a new mixed-use commercial tower in the Hudson Yards district. The environmentally sustainable tower will include premium corporate offices and exclusive residences, and is designed to maximize sunlight and provide 360 degree views.

redpajama_set_name: RedPajamaCommonCrawl


text:
Cyornis ruficrissa — вид горобцеподібних птахів родини мухоловкових (Muscicapidae). Ендемік Калімантану.

Таксономія
Cyornis ruficrissa раніше вважався підвидом рудохвостої джунглівниці, однак був визнаний окремим видом в 2021 році.

Підвиди
Виділяють два підвиди:
 C. r. ruficrissa (Sharpe, 1887) — гора Кінабалу;
 C. r. isola (Hachisuka, 1932) — гори на півночі Калімантану.

Поширення і екологія
Cyornis ruficrissa живе в гірських тропічних лісах Калімантану.

Примітки

Мухоловкові
Птахи, описані 1887
Птахи Індонезії
Птахи Малайзії