text stringlengths 8 5.77M |
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Q:
Paths in PHP, which is correct -- absolute or relative?
In my code, I have included some files for use. When I specify the absolute path, I get a "Forbidden" message.
However, when I use the relative path, the code works.
I wonder why the absolute path was not being accepted.
Can someone explain what is going on?
Thanks in advance!
A:
To save yourself from trouble, always use absolute one.
As for your case, it's easy. You're just using a wrong path. Most likely you messed up a web root with filesystem root.
|
Comments from Tokyo, Japan
Upgrading to a Western Digital WD20EFRX hard disk
All hard disks will die, sooner or later. They only way to avoid that is to retire a drive early enough. Often I upgrade drives because I run out of disk space, and migrate the data to a bigger drive. However, this times it looks like one of my drives is about to die.
Over the last couple of months, one of my PCs that is processing data 24/7 has been seizing up periodically, so I was starting to get suspicious about its hard drives (it has two of them). This week the Windows 7 event viewer reported that NTFS had encountered write errors on the secondary drive. It’s a Samsung SpinPoint F2 EG (Samsung HD154UI, 1.5 TB) which basically has been busy non stop for over three years.
“Reported_Uncorrect” are fatal errors and “Current_Pending_Sector” are bad sectors the drive wants to replace with spare sectors as soon as it can. Neither is a good sign. So I have ordered a new drive, started a backup to another machine and will replace the drive with a new disk that I have ordered from Amazon.
The new drive is a 2 TB Western Digital WD20EFRX, which is part of WD’s “Red” series. These drives are specifically designed for 24/7 operation (as opposed for 8/5 office computers). The drive is 0.5 GB bigger, which is just as well as the old drive was getting close to filling up. Gradually I will be moving my processing to an Ubuntu server, which I already use as my main archive machine with a RAID6 drive array. |
There is a specter haunting the radical left, the specter of degeneration. Okay, that might sound a bit dramatic but the left today really is scared a great deal of something called various things to various people. Whether we call it reformism, opportunism, or revisionism, it mainly comes down to the violation of what the group and/or individuals in question consider as the core principles of their sub-ideology within the radical left. This phenomenon is a veritable obsession, and not a day goes by without the left bickering over it, leaving bitter factionalism and splits in its wake. Such is our obsession that many of the groups within the radical left point to the struggle against opportunism as one of their main tasks. But is it healthy to focus so strongly on fighting it? Or should we find other tasks to engage with? Surely the fight against ideas that deviate too much from our ideology’s basic principles is warranted? In this article we will consider the value of and the problems associated with this collective obsession.
The logic behind it mainly comes from a shared narrative that is relatively universal among the radical left. It goes something like this: the key to success in transforming our current capitalist society into a socialist one is ideas, and mainly the correct application of a set of ideas derived from a specific sub-ideology of the radical left, generally related to a particular socialist thinker and revolutionary experience. Failure on the other hand is again derived from ideas, and mainly the use of wrong ideas or the lapsing of a certain organization away from the previously mentioned correct ideas. To back this up, each of the sub-groups of the radical left generally have their own (long) list of revolutionary failures which are explained based upon a simple schema. It predictably goes something like this: in event A there was a possibility of socialism being accomplished, but this was sabotaged thanks to the application of the wrong ideas and/or the backstabbing of a rival radical leftist sub-ideology, who naturally adhere to wrong ideas.
Hence fighting against what your group regards as faulty ideas becomes absolutely crucial for any successful transition to socialism. Making the struggle against opportunism central to any left-wing project. Of course the accounts presented by radical leftists tend to be more complex than this, with ideas generally not being regarded as the only determinant for success. Yet I regard the image I just provided to be largely correct and, more importantly, to be upheld by most groups within the radical left. What is problematic about it is that it is overbearing and makes for an often vicious internal culture within the movement.
This vicious internal culture is quite unsurprising seeing that most of the radical left traces its lineage back to historical groups that were often themselves engaged in bitter disputes for most of their existence. The squabbles of bakunin and Marx, our own Cain and Abel of sorts (which one is Cain and which one is Abel I leave in the middle here), allow us to trace it back to the first international. The entire Leninist left even derives most of its ideas from a group named after a split (the bolsheviks literally meaning “the majority” after a dispute resulted in them splitting from the mensheviks). Which doesn’t even begin to explain the bloody history of dissent and its suppression among the bolsheviks. Progressing from slanderous polemics through banishment unto outright execution. That these groups and their main actors are often glorified as heroes and single minded sources of revolutionary truth can at times be somewhat worrying, not to disregard the often excellent theoretical contributions some of them made. Today, sectarian polemic also holds great potential for small groups who command little resources, as few are needed for the struggle against “opportunism” (basically a computer at this point). Thus making it somewhat rational for smaller groups to increasingly dedicate themselves to critique of other left-wing groups.
Now I do not claim that ideas or critique aren’t important, because they are. Yet our obsession with ideas and degeneration is hurting the left. The degeneration/betrayal “one size fits all” analysis that is peddled by most of the left is intellectually dishonest and self-serving, and quite frankly boring. It is not even successful in its own right, seeing as how many (read all) of our own narratives on revolutionary situations end in degeneration and betrayal. The complete lack of capacity to prevent degeneration is perhaps the biggest argument against this line of thinking, as it hasn’t prevented degeneration from taking place over and over again. Yet in the meantime the left is still stuck with being irrelevant in many countries, and having a rather unpleasant internal culture. An internal culture which can hardly be called democratic when its main goal is correcting perceived deviations. Instead the cultivation of a culture of vibrant internal discussion should be our priority. Something which of course is irrelevant when correct and incorrect ideas are already known, and with them the reasons for left-wing failure.
Paradoxically the more the radical left dedicates itself to critique the harder it becomes to develop a truly innovative culture of discussion and critique within it. As single-minded focus upon critique requires the organisation in question to put unwavering confidence in its own ideas, which are thus themselves put outside of possible critique, both from within the own organisation as from without. The establishment of a radical left with a truly democratic internal culture, thus requires transcending the boundaries put in place by small-group criticism. First and foremost by overcoming our fear of degeneration and by pointing to new perspectives on past revolutionary episodes. These should be based upon the structural considerations of each individual case, with economic conditions, the international situation, and the specific political culture of the left in that era being put in the foreground. In other words: looking at the structural underlying conditions that are not easily changed, instead of the ideas and actions of bickering leftist groups.
The left should overcome their endless narrative of betrayal and degeneration, by reconsidering many aspects of its internal culture and by daring to reanalyze historical events through different lenses. One of the basic tools for doing this is analysis based upon structural conditions, in essence a more determinist analysis. Afterwards we can look at how ideas and betrayal fit into this new picture. I hold no hope for converting the hard core of the small group left to this cause, as their practices are too heavily ingrained in their ideas and action. Yet rethinking our political culture is more relevant than ever today as we are seeing the rise of a revitalized radical left in many European countries, which is not at all perfect, but will hopefully provide openings for new militants to reconsider older forms of thinking and reshape our political culture into something more…pleasant, and hopefully productive. |
This invention relates generally to the field of fluid-solid contacting. More specifically, this invention deals with the delivery of fluids to beds of particulate material.
Fluid-solid contacting methods have a wide variety of applications. Such methods find common application in processes for hydrocarbon conversion and adsorption columns for separation of fluid components. When the fluid-solid contacting takes place in an adsorption column, the particulate material will comprise an adsorbent through which the fluid passes. In the case of hydrocarbon conversion, the fluid-solid contacting typically takes place in a reactor containing catalyst. Typical hydrocarbon conversion reactions that may be carried out are hydrogenation, hydrotreating, hydrocracking, and hydrodealkylation.
Fluid-solid contacting devices to which the method of this invention apply are arranged as an elongated cylinder usually having a vertical orientation through which an essentially vertical flow of fluid is maintained. Particulate material contained in this vessel is arranged in one or more beds. Fluid enters the vessel through an inlet located at an upstream end of the vessel. It is also commonly known to add or withdraw fluid from between the particulate beds. This is commonly done in adsorption schemes where the composition of the fluid passing between particle beds is changing or in hydrocarbon conversion processes where a quench system is used to cool fluid as it passes between beds.
Changes in the composition or properties of the fluid passing through the particular zone present little problem provided these changes occur uniformly. In adsorption systems these changes are the result of retention or displacement of fluids within the adsorbent. For reaction systems changes in temperature as well as composition of the fluid are caused by the particulate catalyst material contained in the beds.
Nonuniform flow of fluid through these beds can be caused by poor initial mixing of the fluid entering the bed or variations in flow resistance across the particulate bed. Variations in the flow resistance across the bed can vary contact time of the fluid within the particles thereby resulting in uneven reactions or adsorption of the fluid stream passing through the bed. In extreme instances, this is referred to as channeling wherein fluid over a limited portion of the bed is allowed to move in a narrow open area with virtually no resistance to flow. When channeling occurs, a portion of the fluid passing through the bed will have minimal contact with the particulate matter of the bed. If the process is one of adsorption, the fluid passing through the channel area will not be absorbed, thereby altering the composition of this fluid with respect to fluid passing through other portions of the absorbent bed. For a catalytic reaction, the reduction in catalyst contact time will also alter the product composition of fluid as it leaves different portions of the catalyst bed.
In addition to problems of fluid composition, irregularities in the particulate bed can also affect the density and temperature of the fluid passing through the bed. For many separation processes retained and displaced components of the fluid have different densities which tend to disrupt the flow profile through the bed. Nonuniform contacting with the adsorbent particles will exacerbate the problem by introducing more variation in the density of the fluid across the bed thereby further disrupting the flow profile of the fluid as it passes through the particle bed.
In reaction zones, temperature variations are most often associated with nonuniform catalyst contact due to the endothermic or exothermic nature of such systems. Nonuniform contact with the catalyst will adversely affect the reaction taking place by overheating or overcooling the reactants. This problem is most severe in exothermic reactions where the higher temperature can cause further reaction of feedstock or other fluid components into undesirable products or can introduce local hot spots that will cause damage to the catalyst and/or mechanical components.
Fluid flow into a vessel can disrupt the top surface of the bed. The disruption results from transverse fluid flow across the surface of the bed at velocities sufficient to move the individual bed particles. For a confined bed, this disruption or movement of the particles will cause the particles to abrade against each other generating smaller particles which are referred to as fines. These fines may increase pressure drop within the bed or escape from the bed thereby reducing the overall quantity of particles in the bed and possibly interfering with downstream operations. In unconfined beds, transverse fluid flow may also shift large quantities of particles so that the bed surface is highly irregular.
These transverse currents are the result of charging fluid into a relatively large diameter vessel through a relatively small diameter nozzle. Charging fluid into the vessel through a small diameter nozzle produces a high velocity jet that extends from the nozzle into the vessel. Impact of this jet on or adjacent to the surface of a relatively closed catalyst bed flares the fluid outward thereby producing eddy currents and fluid velocities transverse to the bed surface. The inlet effects associated with the relatively small diameter nozzle are compounded by the usual presence of an elbow just upstream of the nozzle which introduces another transverse velocity component into the fluid flow entering the vessel. The overall result of these inlet effects is often the piling up of particles around the periphery of the particle bed or the shifting of particles from one side of the bed to the other.
These detrimental inlet effects are avoided by uniformly dispersing the fluid as it enters the vessel. Uniform dispersal can be obtained by providing a sufficient length between the nozzle and the catalyst bed surface such that the fluid jet and any transverse velocities are substantially dissipated upstream of the particle bed. However, in most cases, it is impractical to provide the length necessary for dissipation of the inlet effects due to the excessive vessel tangent length that would be required. In fact, the trend in many industries is to decrease the length between the inlet nozzle and the particle bed surface in order to increase the total volume of particles in the vessel and thereby obtain greater fluid throughput or greater particle bed service life.
For these reasons, inlet distributors are commonly used to break up the fluid jet and redistribute fluid flow over the top surface of a particle bed. One such device is shown in U.S. Pat. No. 2,925,331 issued to Kazmierczak et al. where a fluid stream is downwardly directed onto the upper surface of the catalyst bed and passes first through a distributor consisting of a series of annular plates having inner diameters that progressively decrease in the direction of fluid flow so that portions of the fluid stream are in effect peeled off and redirected radially outward over the surface of the particle bed. It is also known in the hydrocarbon processing industry to attach cylindrical rings extending in the direction of fluid flow to the inner edge of the annular plates. Another type of distributor used to redirect and remix fluid flow upstream of a particle bed is shown in U.S. Pat. No. 3,598,541 issued to Hennemuth et al. and U.S. Pat. No. 3,598,542 issued to Carson et al. The Hennemuth distributor uses a series of circumferentially spaced holes to redistribute fluid within a fluid mixing device that communicates with the upper surface of a particle bed. The distributor disclosed in Carson uses a series of circumferentially spaced holes to radially discharge fluid across the upper surface of a particle bed. Thus, the prior art is well acquainted with a number of distribution devices for use in fluid solid contacting vessels.
Despite the use of different inlet distributors, bed disruption remains a problem. Distributors that use the annular plates or baffles of the Kazmierczak device reduce the severity of bed disturbances but have not eliminated it. Therefore, large scale shifting of particle bed surfaces, especially where fluid inlet velocities are high, still occurs. Such disruption is still known to occur even in cases where straightening vanes and other flow distribution devices are added to the upstream elbow as a means of eliminating a resulting transverse flow component. It has now been discovered that despite the presence of the baffles and additional redistribution devices, such as straightening vanes, fluid flow entering the vessel still remembers the change of direction that took place upstream of the inlet nozzle. |
Jose Mourinho could barely raise a smile but you can bet Jurgen Klopp did - a smile as wide as the Mersey.
If Klopp needed a beer after that painful derby at Goodison Park, this was the perfect excuse to crack one or two open.
Never mind Manchester City having to wait to formalise the Premier League title - that is pretty much a footnote to the immediate implications of this remarkable defeat.
Lifting and inspiring his team ahead of Tuesday’s return Champions League leg against Klopp’s Liverpool will be one of the toughest challenges faced by Guardiola in his bejewelled coaching career.
Physically, they looked spent, not just the effect of a gruelling, occasionally spiteful, derby game but perhaps the culmination of their relentless efforts over the entire piece this season.
(Image: Offside)
And where they are psychologically is anyone’s guess.
Can, for example, Raheem Sterling recover from this in the short term?
He will be reflecting on his misses in this game - chances to bury United long before the astonishing second half comeback - for a long time.
He will watch re-runs of Pep’s head-in-hands anguish when those opportunities were lifted into the stands and agonise. Guardiola knew how costly they could be.
How can Vincent Kompany and Nicolas Otamendi face Liverpool with any degree of confidence when their fallibility was again exposed?
The fact that John Stones cannot even get into the squad, incidentally, is a worry for Gareth Southgate and the player himself.
For all their expenditure, City’s central defensive area is still soft by elite class standards.
Both Paul Pogba’s goals and Chris Smalling’s winner were testament to that.
Kevin de Bruyne couldn't save the day from the bench (Image: Manchester City FC)
When the dust settles on this fractious occasion and on the season itself, that is one area Guardiola must address … again.
It is not the only one.
Additions in the attacking category are inevitable. With Sergio Aguero and Gabriel Jesus on the bench, there was no-one to take up a central role, although Bernardo Silva, all five foot-nothing of him, gave it the occasional crack.
As defensively vulnerable as they were here and at Anfield, City - with Jesus struggling - badly missed a useful attacking focal point.
That will be addressed by Sheikh Mansour’s wealth.
Obviously, the Sheikh was not in attendance, even though it would have given him the chance to rub shoulders with Gianni Infantino, in the extremely unlikely event he has a clue who the president of FIFA is.
Infantino must have been wowed by the frenetic occasion, if not the officiating.
There will be no English referees at the World Cup and no wonder, if Martin Atkinson’s performance is a barometer of the overall standard.
He was shocking, although Ashley Young would disagree, having been allowed to handle in the area and top Aguero with a full set of studs, also in the penalty box. |
Q:
Why does Checkstyle tries to create a Check for my Listener
I have implemented a Checkstyle Listener. It worked before, (I think with a 5.0 beta release), but now (with 5.0), checkstyle fails with the following CallStack
Unable to create Checker: cannot initialize module de.xyz.toxicity.TeamcityListener - Unable to instantiate de.xyz.toxicity.TeamcityListener
com.puppycrawl.tools.checkstyle.api.CheckstyleException: cannot initialize module de.xyz.toxicity.TeamcityListener - Unable to instantiate de.xyz.toxicity.TeamcityListener
at com.puppycrawl.tools.checkstyle.Checker.setupChild(Checker.java:177)
at com.puppycrawl.tools.checkstyle.api.AutomaticBean.configure(AutomaticBean.java:207)
at com.puppycrawl.tools.checkstyle.Main.createChecker(Main.java:138)
at com.puppycrawl.tools.checkstyle.Main.main(Main.java:115)
Caused by: com.puppycrawl.tools.checkstyle.api.CheckstyleException: Unable to instantiate de.xyz.toxicity.TeamcityListener
at com.puppycrawl.tools.checkstyle.PackageObjectFactory.createModule(PackageObjectFactory.java:156)
at com.puppycrawl.tools.checkstyle.Checker.setupChild(Checker.java:152)
... 3 more
Caused by: com.puppycrawl.tools.checkstyle.api.CheckstyleException: Unable to instantiate de.xyz.toxicity.TeamcityListenerCheck
at com.puppycrawl.tools.checkstyle.PackageObjectFactory.doMakeObject(PackageObjectFactory.java:99)
at com.puppycrawl.tools.checkstyle.PackageObjectFactory.createModule(PackageObjectFactory.java:153)
... 4 more
My configuration File looks like this
<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE module PUBLIC "-//Puppy Crawl//DTD Check Configuration 1.2//EN" "http://www.puppycrawl.com/dtds/configuration_1_2.dtd">
<module name="Checker">
<property name="severity" value="warning"/>
<module name="de.xyz.toxicity.TeamcityListener" />
<module name="FileLength">
<property name="max" value="500"/>
</module>
<module name="TreeWalker">
<module name="FileContentsHolder"/>
<module name="AnonInnerLength">
<property name="max" value="35"/>
</module>
// ... more modules like this follow
</module>
</module>
Everything works fine when my own Listener is removed from the config.
What really confuses me is: Why is checkstyle looking for the TeamcityListenerCheck class?
Such a class does not exist. Do I need it? What should it look like?
A:
Stupid me ... it was a simple classpath issue.
|
There is an undercurrent in the frayed relationship between Conor McGregor and the UFC that highlights the reality of the mixed martial arts business.
Currently, fighters are hired as independent contractors yet appear to be treated like employees without the benefits of such an arrangement.
US congressman Markwayne Mullin, himself a former professional mixed martial artist, intends to sponsor a piece of legislation that aims to shift the landscape of MMA by providing fighters such as McGregor with increased safeguards and control of their careers.
Five years from now, Mullin envisions all combat sports athletes fighting while enjoying far greater prospects of financial stability.
“It’s not to focus on bringing down the UFC,” said Mullin, a conservative Republican from rural Oklahoma. “I just want to make sure the athletes are valued the same as the league or promotion, or whatever you want to call the UFC at this point. Right now I feel the UFC treats their fighters not as an asset, but as a commodity. That mindset has to be changed. It’s a professional sport with professional athletes in it. It takes both to be successful.”
Mullin’s office has the majority of the legislation written, and remains focused on crafting final details that will be released in the coming weeks. The language could be made into law by amending the Muhammad Ali Boxing Reform Act or as standalone legislation.
Enacted in May 2000, the Ali Act sought to end what it described as the widespread abuse of boxers.
The lack of a centralized organization or league and the failure of state regulators to enforce the protections of fighters unevenly were among several issues the Ali Act cited as problematic for boxers. Insufficient oversight of promoters, the lack of an independent ranking system or uniform contracts, and the need for enhanced financial disclosures also joined the list.
While the Ali Act opened the business of boxing, paving the way for more lucrative contractual terms and freedoms for top boxers than those experienced by their MMA counterparts, enforcement by the Department of Justice has been criticized as toothless. Nonetheless, proponents believe a similar application of law to MMA would produce necessary upheaval in the way fighters and promoters operate, marking a huge shift from the sport’s current hybrid boxing-pro wrestling model.
There is strong support on both sides of the issue.
For years the proprietors of the UFC have opposed federal regulation for MMA. Zuffa, the UFC’s parent company, employs lobbyists in Washington to advocate that its league-like status combined with state regulation means its mixed martial artists does not need an “Ali Act.”
Representatives for the UFC visited Mullin’s office on 15 April this year to express just that opinion.
The UFC, according to Mullin, explained that their fighters are hired as subcontractors and treated as such. The congressman countered that if that was the case, UFC should have no problem with the potential legislation. Zuffa did not respond to a request for comment for this article.
“If they’re really subcontractors then the Ali Act is a framework to protect fighters if we want the sport to be successful,” Mullin said. “If we want it to be sustainable and respectable we need to make sure and understand that we need to take care of the fighters and the organization. I want the organization to be successful too, but the fighters need to be taken care of.
“They make the decision who fights and who doesn’t fight. There’s nobody that’s controlling the ranking system except the inner workings of the UFC. I think a third body needs to be looking at that. Needs to control that.”
Mullin told UFC representatives that by limiting endorsement opportunities and securing fighters to long-term contracts that include extension options, they appear to be operating like a league instead of merely a fight promoter.
Employment status affects many issues such as benefits, tax implications, and liability. In team sports, athletes are generally defined as employees. In individual sports, like boxing, they’re paid as independent contractors.
The UFC-fighter arrangement is a unique concoction of contractor and employee.
Among the key distinctions: employees work regularly with one company while contractors can provide services for various entities. UFC contracts address this via exclusivity clauses that lock in fighters to the promotion. The UFC provides a form of health insurance outside of competition, a rarity for contractors, yet fighters are expected to cover their own expenses for training among other costs.
Bellator MMA, a property of Viacom and a competitor to the UFC, has said it welcomes an Ali Act to open the sport, and the promotion’s president, Scott Coker, reiterated his support last week.
Competitors to the UFC see an opportunity for the world’s best fighters, many of whom are exclusive to the octagon, to become accessible. The movement to establish this kind of MMA commerce has been explored in various forms, including opened-and-closed FTC investigations and ongoing antitrust lawsuits that are in the discovery phase.
Discussion in the media regarding fighter rights generally focuses on the athletes organizing and establishing their own protections. Yet this argument fails to recognize that contractors are unable to unionize according to federal law.
Rather than unionizing, contractors can form an association, and there has been an ongoing effort towards that goal for several years.
When representatives from the budding fighters’ association visited Mullin to explain what they hoped to accomplish by folding MMA under the Ali Act, he was overjoyed.
“I said, ‘Are you kidding? I’ve been waiting three years for you guys to come to my office now let’s go with it.’ I’m very familiar with it but we’ll have to tweak some things,” he said. “They had the same thought. They said let’s include all combatants. I said I agree with that. Let’s get it on the table all at once. They’re gung-ho.
“We’ve had a tremendous amount of support from the business community and other fighters out there. The only one that’s really been opposed to this has been the UFC itself. They came in and they said it wasn’t going to do anything and wasn’t going to make any difference. And I said, ‘Then why are you guys in my office? If you’re not concerned about it, if you think everything is perfect, you shouldn’t be too concerned about it. But the truth is they know what’s happening right now with the fighters.”
Mullin said from his experience, timing can be more important than policy when it comes to getting legislation across the finish line. It happens that “the timing and the policy are perfect on this,” he said, and support from all sides, including labor unions, is welcome. The teamsters and other organized labor groups, including the NFL Players Association, have voiced support for the fighters association.
“I don’t care where the support comes from,” he said. “We all have a job to do. We also get to make a choice if we want to be part of the union or not part of the union. So if we’re all working for the same goal, and that’s to make sure the league and the fighters are on an even playing field and the can benefit from something they’re dedicating their lives to, let’s move.”
Mullin will need a sponsor on the Senate side, and the former presidential candidate John McCain, architect of the Ali Act, said he would support amending the law to cover mixed martial artists. Despite political differences Mullin said he and McCain — notorious for calling MMA “human cockfighting” (although he has since become more amenable) — have worked on other pieces of legislation.
“To see where his position has come. To see where the sport has moved to — being recognized as a true sport —is great,” Mullin said. “We’ll get a meeting with Senator McCain in the next week or so. Sit down and see if he has any interest in it. If not we’ll move on to someone else. I had another senator reach out and express interest.”
Mullin declined to name the senator because they had not yet discussed the legislation in depth.
Due to Mullin’s experience as a professional fighter — sherdog.com lists him at 3-0, though he complains it’s missing some wins — his concern about these issues began prior to his election to congress in 2012. The 38-year-old said he wanted to find a way to help fighters caught in “this cycle” yet had to wait because his friends in UFC “didn’t want to rock the boat”.
“I’m anti-regulation. I come from a business background. I can’t stand it,” Mullin said. “But when regulation is needed, that is the roll of the federal government.”
McGregor has made the crux of his fight with the UFC about contractually-obligated media duties. At the heart of his quarrel is who truly controls his career. Mullins’ bill may not relieve McGregor from the distraction of media tours and commercials, but it could very well produce important changes that alter the power dynamic between MMA’s biggest stars and their promoters. |
Q:
Let $X_n$ be a uniform distribution on $(-1,1)$. Let$ Y_n$ ~ Cauchy(0,1). Everything independent.
Let $X_n$ be a uniform distribution on $(-1,1)$. Let$ Y_n$ ~ Cauchy(0,1). Everything independent.
Let $Z_n$ = $X_n$ + $Y_n$
I want to study the law convergence of the sample mean of $Z_n$. That is:
$$ \overline{Z_n} = \frac{\sum X_i + \sum Y_i}{n} $$
So, first of all there is an hint: I cannot use the Law of large numbers. Why is that?
Anyway, I am really out of tools to attack this problem. Does someone have a hint?
EDIT: Managed to prove that sample mean of Cauchy is still Cauchy! Still not sure about the solution.
A:
Hint: what does $\sum X_i / n$ converge to? As you note, the sum $\sum Y_i / n$ is equal in distribution to a standard Cauchy.
|
Q:
find_in_set with join in laravel
How to get the desire output using laravel query. Tried that way does not get success please guide thanks a lot in advance
Is there any way we can set it in model if possible please guide
User
id name b_id
1 Alax 1,3
2 Rex 2,4
3 Lex 2,3
Books
id book_name book_author
1 Javascript jim
2 PHP json
3 LARAVEL rax
4 SYMPHONY Alax
Output
id name b_id
1 Alax Javascript, LARAVEL
2 Rex PHP, SYMPHONY
3 Lex PHP, LARAVEL
And Query:
$res = DB::table('user')->leftJoin('book', function($join){
$join->on(DB::raw("find_in_set(book.id, user.b_id)",DB::raw(''),DB::raw('')));
});
A:
Try this.
$data = \DB::table("user")
->select("user.*",\DB::raw("GROUP_CONCAT(book.book_name) as book_name"))
->leftjoin("book",\DB::raw("FIND_IN_SET(book.id,user.b_id)"),">",\DB::raw("'0'"))
->get();
\DB::raw("GROUP_CONCAT(CONCAT(book.book_name, ' - ', book.book_author) SEPARATOR ', ') AS book_name_author")
Your output looks like.
Illuminate\Support\Collection Object
(
[items:protected] => Array
(
[0] => stdClass Object
(
[id] => 1
[name] => Alax
[b_id] => 1,3
[book_name] => Javascript,LARAVEL
)
[1] => stdClass Object
(
[id] => 2
[name] => Rex
[b_id] => 2,4
[book_name] => PHP,SYMPHONY
)
[2] => stdClass Object
(
[id] => 3
[name] => Lex
[b_id] => 2,3
[book_name] => PHP,LARAVEL
)
)
)
|
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Graveyard of Empires TP
Afghanistan US Marines face a never-ending onslaught of Taliban But even hell can get worse The dead are coming back to life in The Graveyard of Empires and only together can both sides of the today's conflict survive tomorrow's undead assault Writer Mark Sable (Unthinkable Two-Face Year One) reunites with his Grounded co-creator Paul Azaceta (Amazing Spider-Man) to tell this critically claimed controversial tale of terror
BKGN2790 |
How local journalists fight government secrecy — every day
These men aren’t — and wouldn’t claim to be — household names. They’re not powerful First Amendment lawyers. They haven’t been parties in precedent-shattering U.S. Supreme Court cases. They’re not high-profile constitutional scholars.
And yet, what they’re doing to advance and protect the freedom of the press is as important as any Supreme Court precedent. They’re journalists, manning the front lines in battles against government secrecy and fighting the daily fight to provide their readers the most thorough and accurate information possible.
Mayfield, Lough and Giuliani, of course, aren’t alone. Every day, publishers, editors and reporters across the country push back veils of secrecy, ask hard questions and refuse to take “no” for an answer. Mayfield, Lough and Giuliani wouldn’t claim to be the nation’s best journalists, but they exemplify the best journalism has to offer.
Admittedly, I’m biased. Mayfield is a good friend and the publisher of Sauk Valley Media, one of my favorite clients. SVM publishes the Dixon Telegraph and the Sterling Daily Gazette in rural northwest Illinois. Lough serves as the papers’ executive editor, and Giuliani aggressively covers several beats. My bias, however, doesn’t change the fact that these men have teamed to increase the accountability and enhance the openness of the area’s local government bodies.
In just the last three weeks, for example, SVM has obtained opinions from the Illinois attorney general that the Lee County Board violated Illinois’ open-meetings law and that the Rock Falls Township High School District board of education failed to comply with the state’s freedom of information act.
At first blush, of course, these issues hardly seem like blockbusters. What makes them significant, however, is that they arise every day in every corner of the country, threatening to deny important information to the public and challenging the news media to respond.
The issue with the Lee County Board arose when the board on April 19 voted on two items — to reduce its size from 28 members to 24 and to fill an animal-control vacancy — even though the items did not appear on the board’s agenda.
Giuliani, who covers the board, believed the actions violated the state’s Open Meetings Act and said so, ultimately complaining to the Illinois Attorney General about the alleged violation. The county maintained its actions were legal, arguing that the vote on the size reduction was sufficiently publicized because the issue was referenced in minutes of a board committee and that the vote on the animal-control position did not require notice because the person hired was a “minimal income employee.”
The attorney general disagreed.
“[T]he public body cannot take action or make any decision with regard to items or topics not on the agenda of the regular meeting,” Assistant Attorney General Amanda Lundeen wrote. “[T]he Board is required to include in its meeting agendas public notice of any matter on which it intends to take final action, including the hiring of any employee which requires Board action.”
The issue with the school board arose in February, when a special-education teacher and wrestling coach accused of sending inappropriate text messages to a female student resigned from both positions near the end of a five-hour closed session. Shortly after the closed session ended, the board approved a resignation agreement negotiated during the closed meeting.
The board, however, refused to give Giuliani a copy of the agreement or related records, claiming it was permitted to keep the documents secret because they related to the adjudication of an employee disciplinary matter for which there had been no final outcome.
As permitted under a new Illinois law, SVM appealed the denial to the public access counselor in the attorney general’s office. (This new process works so well that media lawyers usually are not involved in the appeals, and I was not involved in this one.) Not surprisingly, the attorney general concluded that this exemption did not apply, as the employee’s resignation constituted a final outcome.
The district then asserted a second ground for keeping the documents confidential — that they would inappropriately disclose the student’s identity. Noting that SVM already had agreed that the district could and should redact the student’s name from the records, the attorney general also rejected this argument.
“[M]uch of the information in these documents relates solely to the district’s investigation and to [the teacher’s] reactions and reveals nothing about the student,” Assistant Attorney General Sarah Kaplan wrote. “The remaining information is not highly personal, and disclosure would not be objectionable to a reasonable person.”
The only documents the district could withhold, Kaplan said, were preliminary drafts or recommendations in which opinions were expressed or actions formulated.
Predictably, the district latched onto this exemption and is refusing to release any documents until the Attorney General determines which few records, if any, can be withheld under it.
No matter which records finally are released, it’s clear that SVM will continue holding local officials to the standards the open-meetings and open-records laws require. Giuliani knows how the laws work and how to enforce them. Lough devotes many of his weekend columns to raising awareness about these issues and mentors Giuliani and SVM’s other young reporters. And Mayfield continually provides the support, through resources and the papers’ editorial pages, that Giuliani and Lough need to maintain their fight.
As a result of these efforts, readers throughout northwest Illinois indisputably are better informed. Armed with this information, they join citizens across the country who — thanks to journalists like Giuliani, Lough and Mayfield — can monitor and evaluate the local boards and councils that make the decisions that directly affect their daily lives.
The story, you see, isn’t that Giuliani, Lough and Mayfield are special or unique. It’s that they’re not, that countless journalists like them every day battle and sacrifice and endure the wrath of angry elected officials to bring us important local news.
What can we do in return? For starters, we can pay attention, get and stay engaged with local issues. We also can get more directly involved, attend a public meeting, maybe even express our opinion.
And, occasionally, we might even tell the journalists fighting the daily fight that we appreciate their efforts.
The First Amendment Center is an educational organization and cannot provide legal advice.
Ken Paulson is president of the First Amendment Center and dean of the College of Mass Communication at Middle Tennessee State University. He is also the former editor-in-chief of USA Today.
Gene Policinski, chief operating officer of the Newseum Institute, also is senior vice president of the First Amendment Center, a center of the institute. He is a veteran journalist whose career has included work in newspapers, radio, television and online.
John Seigenthaler founded the Newseum Institute’s First Amendment Center in 1991 with the mission of creating national discussion, dialogue and debate about First Amendment rights and values.
About The First Amendment Center
We support the First Amendment and build understanding of its core freedoms through education, information and entertainment.
The center serves as a forum for the study and exploration of free-expression issues, including freedom of speech, of the press and of religion, and the rights to assemble and to petition the government.
Founded by John Seigenthaler, the First Amendment Center is an operating program of the Freedom Forum and is associated with the Newseum and the Diversity Institute. The center has offices in the John Seigenthaler Center at Vanderbilt University in Nashville, Tenn., and at the Newseum in Washington, D.C.
The center’s website, www.firstamendmentcenter.org, is one of the most authoritative sources of news, information and commentary in the nation on First Amendment issues. It features daily updates on news about First Amendment-related developments, as well as detailed reports about U.S. Supreme Court cases involving the First Amendment, and commentary, analysis and special reports on free expression, press freedom and religious-liberty issues. Support the work of the First Amendment Center.
1 For All
1 for All is a national nonpartisan program designed to build understanding and support for First Amendment freedoms. 1 for All provides teaching materials to the nation’s schools, supports educational events on America’s campuses and reminds the public that the First Amendment serves everyone, regardless of faith, race, gender or political leanings. It is truly one amendment for all. Visit 1 for All at http://1forall.us/
Help tomorrow’s citizens find their voice: Teach the First Amendment
The most basic liberties guaranteed to Americans – embodied in the 45 words of the First Amendment to the U.S. Constitution – assure Americans a government that is responsible to its citizens and responsive to their wishes.
These 45 words are as alive and important today as they were more than 200 years ago. These liberties are neither liberal nor conservative, Democratic nor Republican – they are the basis for our representative democratic form of government.
We know from studies beginning in 1997 by the nonpartisan First Amendment Center, and from studies commissioned by the Knight Foundation and others, that few adult Americans or high school students can name the individual five freedoms that make up the First Amendment.
The lesson plans – drawn from materials prepared by the Newseum and the First Amendment Center – will draw young people into an exploration of how their freedoms began and how they operate in today’s world. Students will discuss just how far individual rights extend, examining rights in the school environment and public places. The lessons may be used in history and government, civics, language arts and journalism, art and debate classes. They may be used in sections or in their entirety. Many of these lesson plans indicate an overall goal, offer suggestions on how to teach the lesson and list additional resources and enrichment activities.
First Amendment Moot Court Competition
This site no longer is being updated … And the competition itself is moving to Washington, D.C., where the Newseum Institute’s First Amendment Center is co-sponsoring the “Seigenthaler-Sutherland Cup National First Amendment Moot Court Competition,” March 18-19, in partnership with the Columbus School of Law, of the Catholic University of America.
During the two-day competition in February, each team will participate in a minimum of four rounds, arguing a hypothetical based on a current First Amendment controversy before panels of accomplished jurists, legal scholars and attorneys.
FIRST AMENDMENT CENTER ARCHIVES
State of the First Amendment survey reports
The State of the First Amendment surveys, commissioned since 1997 by the First Amendment Center and Newseum, are a regular check on how Americans view their first freedoms of speech, press, assembly, religion and petition.
The periodic surveys examine public attitudes toward freedom of speech, press, religion and the rights of assembly and petition; and sample public opinion on contemporary issues involving those freedoms.
See the reports. |
The present invention relates to a method and an apparatus for carrying out zero point corrections of temperature dependent characteristics of output signal signals of sensors.
Various types of sensors have their varying temperature dependent characteristics due to environmental temperatures therearound and piece-to-piece variations. Such a sensor has complex temperature dependent factors so that the temperature dependent characteristic of the sensor is usually different from a simple linear characteristic.
In order to correct the temperature dependent characteristic of the sensor, the environmental temperature around the sensor is measured as a temperature parameter signal, and the absolute level of the sensor signal outputted from the sensor, which includes an error depending on the environmental temperature, is corrected based on the measured temperature parameter signal. This results in that the slope of the sensor signal with respect to the environmental temperature is simple linearly corrected.
This absolute correction, however, fails to accurately correct the zero point correction, in other words, offset correction, of the temperature dependent characteristic of the sensor.
Then, as an example of the zero point correction of a temperature dependent characteristic of a sensor, a signal processing circuit that carries out the zero point correction of a temperature dependent characteristic of an oscillation gyro (a yaw rate sensor) is disclosed in Japanese Patent Publication H6-160100.
FIG. 7 illustrates the schematic structure of the disclosed signal processing circuit. In the signal processing circuit, a yaw rate signal obtained by an oscillation gyro 1 is amplified by an amplifier 2, and the amplified signal is synchronously detected by a synchronous detection circuit 3. The synchronous detected signal is quantized (digitized) by an analog-to-digital (A/D) converter 4 as digital data, and the digital data is smoothed by a microcomputer 5.
On the other hand, an environmental temperature around the oscillation gyro 1 is detected by a temperature sensor 6, and a temperature parameter signal based on the detected environmental temperature is inputted to the microcomputer 5 as temperature data.
The microcomputer 5 corrects the smoothed digital data based on the temperature data to obtain temperature characteristic data, storing the obtained temperature characteristic data on an EEPROM (Electrically Erasable and Programmable Memory) 7.
When measuring a yaw rate of an object by the oscillation gyro 1, the smoothed digital data corresponding to the yaw rate of the object detected by the oscillation gyro 1 is inputted to the microcomputer 5 through the amplifier 2, the synchronous detection circuit 3, and the A/D converter 4.
The microcomputer 5 reads out the temperature characteristic data from the EEPROM 7 to carry out the zero point correction of the digital data based on the read-out temperature characteristic data. A digital-to-analog converter (D/A) converter 8 converts the zero-point corrected digital data into analog data to output the analog data as the zero-point corrected yaw rate of the object. The above signal processing allows the zero point correction to be effectively executed.
The disclosed signal processing circuit, however, requires the microcomputer 5 for performing the zero point correction, causing the cost of the signal processing circuit to rise.
In addition, the disclosed signal processing circuit requires the A/D converter 4 for converting the yaw rate signal into the digital data, and the D/A converter 5 for converting the zero-point corrected digital data into the analog data corresponding to the zero-point corrected yaw rate of the object.
The analog-to-digital conversion processing (quantization processing) of the yaw rate signal itself, and the digital-to-analog conversion processing of the zero-point corrected digital data may increase the total processing time of the zero point correction processing, and, especially, quantization errors may be included in the zero-point corrected yaw rate of the object. |
# - Try to find Folly
#
# The following variables are optionally searched for defaults
# FOLLY_ROOT_DIR: Base directory where all folly components are found
#
# The following are set after configuration is done:
# FOLLY_FOUND
# FOLLY_INCLUDE_DIRS
# FOLLY_LIBRARIES
# FOLLY_LIBRARY_DIRS
include(FindPackageHandleStandardArgs)
set(FOLLY_ROOT_DIR "" CACHE PATH "Folder contains Folly")
if (NOT "$ENV{Folly_DIR}" STREQUAL "")
set(FOLLY_ROOT_DIR $ENV{Folly_DIR})
endif()
# We are testing only a couple of files in the include directories
if(WIN32)
find_path(FOLLY_INCLUDE_DIR folly/FBVector.h
PATHS ${FOLLY_ROOT_DIR}/src/windows)
else()
find_path(FOLLY_INCLUDE_DIR folly/FBVector.h
PATHS ${FOLLY_ROOT_DIR}/include)
endif()
find_library(FOLLY_LIBRARY folly
PATHS ${FOLLY_ROOT_DIR}/lib)
find_library(FOLLY_BENCHMARK_LIBRARY follybenchmark
PATHS ${FOLLY_ROOT_DIR}/lib)
find_package_handle_standard_args(FOLLY DEFAULT_MSG
FOLLY_INCLUDE_DIR FOLLY_LIBRARY)
if(FOLLY_FOUND)
set(FOLLY_INCLUDE_DIRS ${FOLLY_INCLUDE_DIR})
set(FOLLY_LIBRARIES
${FOLLY_LIBRARY}
${FOLLY_BENCHMARK_LIBRARY})
endif()
|
Pressure effects in differential mobility spectrometry.
A microfabricated planar differential ion mobility spectrometer operating from 0.4 to 1.55 atm in a supporting atmosphere of purified air was used to characterize the effects of pressure and electric field strength on compensation voltage, ion transmission, peak width, and peak intensity in differential mobility spectra. Peak positions, in compensation voltage as a function of separating rf voltage, were found to vary with pressure in a way that can be simplified by expressing both compensation and separation fields in Townsend units for E/N. The separation voltage providing the greatest compensation voltage and the greatest resolution is ion-specific but often occurs at E/N values that are unreachable at elevated pressure because of electrical breakdown. The pressure dependence of air breakdown voltage near 1 atm is sublinear, allowing higher E/N values to be reached at reduced pressure, usually resulting in greater instrumental resolution. Lower voltage requirements at reduced pressure also reduce device power consumption. |
1. Field of the Invention
This invention relates to the recovery of oil from petroleum reservoirs, and relates particularly to the use of hydrogen peroxide and its aqueous solutions to recover viscous oil from geological reservoirs.
2. Brief Description of the Existing Art
In excess of 4 trillion barrels of viscous oil are estimated to exist in Canada, Venezuela, Calif. and various other worldwide locations. Viscous oil may be defined as oil having a viscosity greater than about 100 centipoises at reservoir conditions. The known reserves of viscous oil are estimated to be at least three times the known worldwide reserves of easily recovered low viscosity oil. With present technology, most of the world's viscous oil reserves cannot be produced economically. The incentive to recover these vast reserves, however, is enormous and many methods have been tried to do so. The existing art for recovery of viscous oil includes the following methods.
Most of the present recovery methods rely on thermal techniques to reduce the viscosity of the oil and increase its ability to flow. One method uses mining techniques to dig up the oil-containing sand and liberate the viscous oil from the sand by washing with hot water. Another method uses hot solvent to dissolve the tarry hydrocarbon from the mined solids.
The most commonly used non-mining thermal methods are hot water injection, steam injection, and in situ combustion.
(a) Hot Water Injection
The simplest thermal method to reduce oil viscosity in situ is by injection of hot water. The water is heated at the surface, and then pumped down a metal pipe and into a subterranean oil-bearing formation. The hot water warms the oil and thereby reduces its viscosity, and the less viscous oil is able to move more easily toward a production well. This method, however, is limited to shallow reservoirs, and heat loss to the nonproductive overburden limits the maximum temperature at which one can inject hot water.
(b) Steam Injection
Steam injection is generally preferred over hot water injection because, pound for pound, steam will typically have 3 to 4 times more heat available for reducing oil viscosity than will hot water. Typically, steam is generated at the surface and injected in much the same manner as hot water. Steam also loses heat to the nonproductive overburden (typically 10 to 30% of its heat content) but because of steam's higher initial heat content it can be used at greater depths to generate higher downhole temperatures than can hot water.
The problems associated with steam injection are many and are well known to those skilled in the art. For instance, water treatment costs are high, and insulated injection tubing is required for deep reservoirs. Expensive and non-conventional completion methods must be used in steam injection, such as special cementing techniques, special expansion joints, special casing and couplings, etc. In addition, steam tends to "finger" through the reservoir to the production well, leaving large quantities of oil in place in the reservoir.
A common method for well stimulation and more rapid production of viscous oil involves injection of steam into a well for a short period of time (2 to 4 weeks) followed by a soak period of a few days. The soak period is followed by production from that same well for a period of 8 to 12 weeks. This method of well stimulation is commonly called Huff and Puff. In this method, the reservoir sand around the well bore is heated by injecting steam and allowing time for the steam to condense. This allows the oil bearing zone to extract the considerable latent heat of vaporization of the steam. The flow is then reversed by converting the former injection well to a production well. The hot oil near the injection well flows relatively easily into the well bore. Cooler oil from farther out in the reservoir moves radially into the heated zone where the oil extracts heat from the hot reservoir sand. Production is continued until the formation sand is too cool to lower the oil viscosity appreciably. The process can sometimes be repeated as many as five times before the operation becomes uneconomic.
(c) In Situ Combustion
In order to reduce excessive heat losses to the nonproductive overburden during hot water or steam floods, techniques have been devised to generate the desired heat in the oil bearing zone itself. In situ combustion is one such method. Typically, air is compressed to some pressure higher than reservoir pressure and injected into the formation. Spontaneous ignition of the hydrocarbon with air can sometimes take place, but ways to initiate the combustion have also been suggested. For instance, L.S. Melik-Aslanov et al (Russian Patent Certificate No. 570700, Aug. 30, 1977) suggests use of chromic acid solution to catalyze the rapid decomposition of hydrogen peroxide at the bottom zone of a well bore. The rapid decomposition is theorized to cause a high temperature near the well bore which enhances recovery by initiating combustion of the resident oil during subsequent injection and ignition of air-water foam. Another method for causing a high temperature at the bottom of a well bore is suggested by J. C. McKimmell (U.S. Pat. No. 3,561,533). He proposed to mix foams of two highly reactive compounds in the well bore--hydrogen peroxide and hydrazine, a common rocket propellant mixture--to effect chemical heating in a well.
Oxygen in air can react with hydrocarbons to produce heat, water, and carbon dioxide. Only about 20% of air, however, is oxygen. The remaining 80% is substantially all nitrogen, and nitrogen is inert, has low solubility in oil or reservoir fluids, and causes fingers of gas to move rapidly toward the production well. The nitrogen fingers provide an easy path for the steam and combustion front to follow, leaving a large amount of the resident oil in place. Premature arrival of the combustion front at a production well frequently signals the termination of the fire-flood. To help alleviate this condition, water is sometimes injected with the air. Water tends to occupy part of the nitrogen fingers and slows down the passage of air toward the fingers. Water also tends to prevent overriding of the air because water-air mixtures are much more dense than air alone.
Other techniques to minimize the adverse effects of inert gas fingering have been used, such as injection of pure oxygen, or mixtures of oxygen with water, flue gas, or carbon dioxide. See, for example, W. R. Shu, U.S. Pat. Nos. 4,454,916 and 4,474,237, and G. Savard, U.S. Pat. No. 4,557,329. Manufacture and compression of pure oxygen in the oil field, however, is expensive and hazardous.
More complete descriptions of the existing art may be found in Development of Heavy Oil Reservoirs, Briggs et al, J. of Petroleum Technology, Feb. 1988, p. 206; and in the books Enhanced Oil Recovery of Residual and Heavy Oils, M. M. Schumacher, Second Ed., Noyes Data Corp., Park Ridge, N.J., ISBN 0-8155-0816-6, and Fundamentals of Enhanced Oil Recovery, H.K. van Poollen and Associates, PennWell Publishing, Tulsa 1980. |
# Copyright 2016 The Chromium Authors. All rights reserved.
# Use of this source code is governed by a BSD-style license that can be
# found in the LICENSE file.
import("//testing/test.gni")
component("ports") {
output_name = "mojo_edk_ports"
sources = [
"event.cc",
"event.h",
"message_filter.h",
"message_queue.cc",
"message_queue.h",
"name.cc",
"name.h",
"node.cc",
"node.h",
"node_delegate.h",
"port.cc",
"port.h",
"port_locker.cc",
"port_locker.h",
"port_ref.cc",
"port_ref.h",
"user_data.h",
"user_message.cc",
"user_message.h",
]
defines = [ "IS_MOJO_EDK_PORTS_IMPL" ]
public_deps = [
"//base",
]
if (!is_nacl) {
deps = [
"//crypto",
]
}
}
source_set("tests") {
testonly = true
sources = [
"name_unittest.cc",
"ports_unittest.cc",
]
deps = [
":ports",
"//base",
"//base/test:test_support",
"//testing/gtest",
]
}
|
Roberto Mancini has said Carlos Tevez will not be leaving Manchester City this summer, after the club failed to agree a swap deal with Internazionale for the Italian club's striker Samuel Eto'o. Mancini also said a deal for the Partizan Belgrade defender Stefan Savic has been done and hinted at an interest in Arsenal's Samir Nasri.
"Yes, of course Tevez will stay, he is a fantastic player," the former Inter manager Mancini said. "There was talk of an exchange with Eto'o but Inter did not want to sell. I would say a deal for Savic has been done, but Nasri? No, not yet."
City have been linked with a variety of players in the transfer window and Mancini did nothing to quell speculation, naming a selection from Serie A. "I like Ezequiel Lavezzi, Marek Hamsik and Edinson Cavani. I like Javier Pastore a little less. Then, when I think of Udinese, I can think of Pablo Armero and Mauricio Isla. And I must say Antonio Di Natale. It's a pity that he is now getting on in years."
Nasri himself was reported on Saturday to have bemoaned his lack of trophies since arriving at Arsenal in 2008, saying that money is not his motivating factor. Negotiations have stalled on a new deal at the Emirates and, with the Frenchman's contract running out next summer, Arsenal may choose to cash in on him now rather than wait and lose him on a free transfer. Nasri's situation has sparked reported interest from both Manchester clubs and Chelsea.
"We already earn huge wages," Nasri said. "The priority is to make a big career and to win titles … With no titles under your belt, you can't be in the list for the Ballon D'Or. I came to England to get trophies because I haven't won anything in my career. I am hungry for titles. I play football because I love this sport and want to feel the emotion of winning. Lifting a trophy all together, this is the beauty and sense of team sports." |
Hmm… It looks like this post might be a bit outdated. Check out our newer content about optimizing garbage collection to improve app performance – HERE.
The 4 Java Garbage Collectors – How the Wrong Choice Dramatically Impacts Performance
When Garbage collection is one of those things that are generally known to impact performance, but beyond that – in terms of how it actually works – it’s pretty much a mystery to most of us. So, I thought I’d take a whack at covering the basics of GC, especially since this is an area that has seen some major changes and improvements with Java 8, especially with the removal of the PermGen and some new and exciting optimizations (more on this towards the end).
When we speak about garbage collection, the vast majority of us know the concept and employ it in our everyday programming. Even so, there’s much about it we don’t understand, and that’s when things get painful. One of the biggest misconceptions about the JVM is that it has one garbage collector, where in fact it provides four different ones, each with its own unique advantages and disadvantages. The choice of which one to use isn’t automatic and lies on your shoulders and the differences in throughput and application pauses can be dramatic.
What’s common about these four garbage collection algorithms is that they are generational, which means they split the managed heap into different segments, using the age-old assumptions that most objects in the heap are short lived and should be recycled quickly. As this too is a well-covered area, I’m going to jump directly into the different algorithms, along with their pros and their cons.
Garbage Collectors - Serial vs. Parallel vs. CMS vs. G1 (and what’s new in Java 8) #garbagecollection #gcperformance Click to Tweet
Psst! Looking for a solution to improve application performance? OverOps helps companies identify not only when and where slowdowns occur, but why and how they occur. Watch a live demo to see how it works.
1. The Serial Collector
The serial collector is the simplest one, and the one you probably won’t be using, as it’s mainly designed for single-threaded environments (e.g. 32 bit or Windows) and for small heaps. This collector freezes all application threads whenever it’s working, which disqualifies it for all intents and purposes from being used in a server environment.
How to use it: You can use it by turning on the -XX:+UseSerialGC JVM argument,
2. The Parallel / Throughput collector
Next off is the Parallel collector. This is the JVM’s default collector. Much like its name, its biggest advantage is that is uses multiple threads to scan through and compact the heap. The downside to the parallel collector is that it will stop application threads when performing either a minor or full GC collection. The parallel collector is best suited for apps that can tolerate application pauses and are trying to optimize for lower CPU overhead caused by the collector.
3. The CMS Collector
Following up on the parallel collector is the CMS collector (“concurrent-mark-sweep”). This algorithm uses multiple threads (“concurrent”) to scan through the heap (“mark”) for unused objects that can be recycled (“sweep”). This algorithm will enter “stop the world” (STW) mode in two cases: when initializing the initial marking of roots (objects in the old generation that are reachable from thread entry points or static variables) and when the application has changed the state of the heap while the algorithm was running concurrently, forcing it to go back and do some final touches to make sure it has the right objects marked.
The biggest concern when using this collector is encountering promotion failures which are instances where a race condition occurs between collecting the young and old generations. If the collector needs to promote young objects to the old generation, but hasn’t had enough time to make space clear it, it will have to do so first which will result in a full STW collection – the very thing this CMS collector was meant to prevent. To make sure this doesn’t happen you would either increase the size of the old generation (or the entire heap for that matter) or allocate more background threads to the collector for him to compete with the rate of object allocation.
Another downside to this algorithm in comparison to the parallel collector is that it uses more CPU in order to provide the application with higher levels of continuous throughput, by using multiple threads to perform scanning and collection. For most long-running server applications which are adverse to application freezes, that’s usually a good trade off to make. Even so, this algorithm is not on by default. You have to specify XX:+USeParNewGC to actually enable it. If you’re willing to allocate more CPU resources to avoid application pauses this is the collector you’ll probably want to use, assuming that your heap is less than 4Gb in size. However, if it’s greater than 4GB, you’ll probably want to use the last algorithm – the G1 Collector.
4. The G1 Collector
The Garbage first collector (G1) introduced in JDK 7 update 4 was designed to better support heaps larger than 4GB. The G1 collector utilizes multiple background threads to scan through the heap that it divides into regions, spanning from 1MB to 32MB (depending on the size of your heap). G1 collector is geared towards scanning those regions that contain the most garbage objects first, giving it its name (Garbage first). This collector is turned on using the –XX:+UseG1GC flag.
This strategy reduced the chance of the heap being depleted before background threads have finished scanning for unused objects, in which case the collector will have to stop the application which will result in a STW collection. The G1 also has another advantage that is that it compacts the heap on-the-go, something the CMS collector only does during full STW collections.
Large heaps have been a fairly contentious area over the past few years with many developers moving away from the single JVM per machine model to more micro-service, componentized architectures with multiple JVMs per machine. This has been driven by many factors including the desire to isolate different application parts, simplifying deployment and avoiding the cost which would usually come with reloading application classes into memory (something which has actually been improved in Java 8).
Even so, one of the biggest drivers to do this when it comes to the JVM stems from the desire to avoid those long “stop the world” pauses (which can take many seconds in a large collection) that occur with large heaps. This has also been accelerated by container technologies like Docker that enable you to deploy multiple apps on the same physical machine with relative ease.
Java 8 and the G1 Collector
Another beautiful optimization which was just out with Java 8 update 20 for is the G1 Collector String deduplication. Since strings (and their internal char[] arrays) takes much of our heap, a new optimization has been made that enables the G1 collector to identify strings which are duplicated more than once across your heap and correct them to point into the same internal char[] array, to avoid multiple copies of the same string from residing inefficiently within the heap. You can use the -XX:+UseStringDeduplicationJVM argument to try this out.
Java 8 and PermGen
One of the biggest changes made in Java 8 was removing the permgen part of the heap that was traditionally allocated for class meta-data, interned strings and static variables. This would traditionally require developers with applications that would load significant amount of classes (something common with apps using enterprise containers) to optimize and tune for this portion of the heap specifically. This has over the years become the source of many OutOfMemory exceptions, so having the JVM (mostly) take care if it is a very nice addition. Even so, that in itself will probably not reduce the tide of developers decoupling their apps into multiple JVMs.
Each of these collectors is configured and tuned differently with a slew of toggles and switches, each with the potential to increase or decrease throughput, all based on the specific behavior of your app. We’ll delve into the key strategies of configuring each of these in our next posts.
In the meanwhile, what are the things you’re most interested in learning about regarding the differences between the different collectors? Hit me up in the comments section 🙂
Additional reading –
1. A really great in-depth review of the G1 Collector on InfoQ.
2. The Complete Guide to Java Performance Monitoring
3. More about String deduplication on the CodeCentric blog. |
For scientists, the price of progress is specialization. When the goal of any researcher is to lay claim to a tiny niche in a crowded discipline, it’s hard for laypeople to find answers to the really important interdisciplinary questions. Questions like, "Is it possible to build a jetpack using downward-firing machine guns?”
Fortunately, such people can turn to Randall Munroe, the author of the XKCD comic strip loved by fans of internet culture.
... |
T.C. Memo. 2007-129
UNITED STATES TAX COURT
DEBRA ANNE BANDERAS, Petitioner v.
COMMISSIONER OF INTERNAL REVENUE, Respondent
Docket No. 7733-05. Filed May 22, 2007.
P filed joint Federal income tax returns with her
husband H for the 1997 and 1999 taxable years. The
returns were signed subsequent to H’s filing for
bankruptcy protection and reported balances due that
were not paid upon submission. Following H’s death, P
sought relief from joint and several liability under
sec. 6015(f), I.R.C., with respect to the 1997 and 1999
liabilities.
Held: P is not entitled to relief from joint and
several liability, pursuant to sec. 6015(f), I.R.C.,
with respect to her 1997 and 1999 taxable years.
James R. Monroe, for petitioner.
Miriam C. Dillard, for respondent.
- 2 -
MEMORANDUM FINDINGS OF FACT AND OPINION
WHERRY, Judge: This case arises from a petition for
judicial review filed in response to a determination concerning
relief from joint and several liability under section 6015.1 The
issue for decision is whether denial by respondent of
petitioner’s request for relief from joint and several liability
for the taxable years 1997 and 1999 constitutes an abuse of
discretion.
FINDINGS OF FACT
Some of the facts have been stipulated and are so found.
The stipulations of the parties, with accompanying exhibits, are
incorporated herein by this reference.
Petitioner married Julio C. Banderas (Dr. Banderas) in March
of 1972. For most of their married life, the couple resided in
Georgia, where Dr. Banderas practiced medicine, specializing in
orthopedic surgery, and petitioner stayed at home caring for
their children and children from a previous marriage of Dr.
Banderas. Petitioner later returned to school and in 1993
completed a registered nursing degree. Throughout the relevant
period and at the time of trial, petitioner was employed in the
nursing field, often working multiple jobs. During their
1
Unless otherwise indicated, section references are to the
Internal Revenue Code of 1986, as amended, and Rule references
are to the Tax Court Rules of Practice and Procedure.
- 3 -
marriage, petitioner and Dr. Banderas maintained and had equal
access to a joint checking account, and both wrote checks on the
account. Both also opened household mail. Dr. Banderas,
however, assumed primary responsibility in handling the family’s
financial affairs.
In the mid-1990s, Dr. Banderas became involved in a contract
dispute with a business associate, Alexander Doman (Dr. Doman),
who was to purchase Dr. Banderas’s medical practice in
preparation for Dr. Banderas’s retirement. The matter proceeded
to litigation and resulted in a $832,447 civil judgment against
Dr. Banderas on June 25, 1997. To collect on the judgment, the
Banderases’ joint checking account was levied in 1997.
Petitioner then, in August of 1997, opened a separate checking
account into which Dr. Banderas’s Social Security checks and
petitioner’s income were deposited and out of which living
expenses were paid.
During the pendency of the foregoing controversy,
Dr. Banderas retired, and he and petitioner moved to Florida in
early 1996. Thereafter, on October 3, 1997, Dr. Banderas filed a
voluntary chapter 7 bankruptcy petition in the U.S. Bankruptcy
Court for the Middle District of Florida. The bankruptcy case
was closed by order of that court on July 21, 2005, after
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disbursement by the trustee of more than $2.37 million.2
Meanwhile, on October 15, 1998, petitioner and Dr. Banderas
signed and timely filed a joint Form 1040, U.S. Individual Income
Tax Return, for 1997. The return reflected total adjusted gross
income of $304,673, consisting primarily of pension, annuity, and
Social Security income of Dr. Banderas but also including $5,025
of wage income earned by petitioner.3 A reported $10,250 was
shown as Federal income tax withheld, $250 of which represented
withholding from petitioner’s wages. The stated balance due was
$64,767. Petitioner was aware of the balance due at the time she
signed the return. No payment was submitted with the return.
The Banderases’ joint Form 1040 for 1998 was filed in late 1999
reporting a loss and is not at issue here.
By 1999, financial pressures had apparently led Dr. Banderas
to return to work. Then, on September 16, 1999, Dr. Banderas was
diagnosed with cancer. The cancer was terminal, and Dr. Banderas
died of complications from the disease on November 16, 1999. On
October 15, 2000, petitioner signed and timely filed a joint Form
1040 for 1999 as a surviving spouse. The return reflected total
adjusted gross income of $121,326, which amount incorporated wage
2
This Court takes judicial notice of the July 21, 2005,
order. See Fed. R. Evid. 201; Estate of Reis v. Commissioner, 87
T.C. 1016, 1027 (1986).
3
The Banderases rounded the amounts reported on their tax
returns.
- 5 -
and Social Security income of Dr. Banderas totaling $84,089 and
petitioner’s wages of $37,068. After subtraction of $10,063 in
Federal income tax withheld, $853 of which was attributed to
petitioner’s wages, the return reported a balance due of $10,262.
No payment was submitted with the return.
On or about June 12, 2003, the Internal Revenue Service
(IRS) received from petitioner a signed Form 8857, Request for
Innocent Spouse Relief. Petitioner indicated on the Form 8857
that she was requesting equitable relief under section 6015(f)
for underpayments of tax for 1997 and 1999,4 and she attached the
following explanatory statement:
Most of my married life - which was my entire
adult life until my husband passed away - I was a
“stay-at-home” wife and mother, with my husband working
hard to support us and to build what we thought was our
retirement plan.
In 1997, as a result of an unjust judgement
against my husband, an illegal hold was placed on our
joint checking account, which caused our check to the
IRS for the balance of our 1996 taxes to bounce due to
“unavailable funds”. Shortly afterwards, my husband
was advised to file for bankruptcy.
Before filing, we were reassured that the IRS
would be the #1 creditor, and that all taxes would be
paid before any other creditors. Had we not been
assured of this, my husband would have gotten a
distribution from the pension plan to pay any taxes
before filing.
During the bankruptcy proceedings, we realized
that our retirement plan was, in reality, his
retirement plan - and it was taken away. When my
4
Petitioner also listed 1996 as a year for which she was
requesting relief, but that year was not further considered after
the IRS explained that no balance was due.
- 6 -
husband passed away, the proceeds of his life insurance
policy also went to the bankruptcy court.
So I found myself, at almost 50 years of age, not
only without my beloved husband, but also with the
urgent need to prepare financially for my future.
In an attempt to pay off our debts, to support
myself since he passed away, and to try to prepare for
my future/retirement, I have been working 2-3 jobs at a
time, 70-100+ hours/week, something that I cannot
continue much longer.
Finding myself now responsible for an additional
bill of over $100,000 to the IRS, I am overwhelmed even
by the idea of how to pay it off. Because almost all
the taxes were based on my husband’s income, etc., and
the ensuing financial problems were a result of his
petition of bankruptcy and the court’s actions, I
respectfully request that I be relieved of the
responsibility of these debts as they present an
unfathomable hardship to me.
Petitioner also submitted to the IRS a Form 12510,
Questionnaire for Requesting Spouse, dated July 22, 2003. In
response to a question asking her to explain when and how she
thought reported balances due would be paid, petitioner wrote:
“By proceeds from my husband’s pension plan - never dreaming that
the court would take it away”. A further question asking what
efforts were made to pay reported taxes after relevant returns
were filed elicited the following answer:
Everything (pension plan) was frozen by the court - we
were waiting for the discharge to have the funds freed
up - in 99 we found that was never going to happen -
accountant said with such a loss we would not owe
anything “ever again”. Then my husband passed away;
with $400,000 left in bankruptcy court they are telling
me there will be none left for me or our kids.
Petitioner also on the Form 12510 completed a statement of her
average income and expenses. She listed wage income of
- 7 -
approximately $80,000 and monthly expenses totaling approximately
$3,700.
Petitioner’s request for relief was initially denied by the
IRS Examination Division on April 29, 2004. Petitioner responded
with a statement of disagreement requesting that the IRS
reconsider the denial. Her reasons for the continued dispute
paralleled those alluded to in her Forms 8857 and 12510, to wit:
Contrary to your conclusion, when my husband and I
signed the tax returns for 1997, we had every reason to
believe that we would be able to pay those taxes.
First of all, my husband’s attorney had assured us that
the IRS was always the first creditor in bankruptcy
proceedings. Therefore, we had no doubt that the taxes
would be paid through the court. As stated in my
original request for relief, had we even suspected that
this would not be the case, the amount for the taxes
could have - and would have - been withdrawn from the
pension plan monies before the bankruptcy was ever
filed. Also, had that suspicion existed, we still
would have had the security of knowing that the taxes
could be paid with pension plan funds after the
bankruptcy was discharged. Never in our wildest dreams
- or worst nightmares - did it ever occur to us that
the pension plan could be lost to the court. I believe
that it was two years later when we found that this
travesty of justice could actually take place.
While I knew when I filed and signed the 1999 return
that we had lost our pension plan, I believed, again
without a doubt, that those taxes would be paid.
Shortly after my husband’s death, before the return was
filed, I was told by several sources, including the
bankruptcy trustee, that remaining monies would go to
me as his beneficiary. When my husband passed away,
the bankruptcy court received an additional $750,000.00
- ¾ of a million dollars - from the proceeds of my
husband’s life insurance policy (monies which I still
contend rightfully belong to our children and me).
Therefore, although I no longer had faith or trust in
our “judicial” system, logic alone told me that there
would be more than enough monies to pay all taxes and
- 8 -
debts owed. It was not until last year, 2003, that the
trustee intimated that, although there was over
$400,000 remaining with the court, with all creditors
having been paid, that there “may not” be anything left
for me and/or the family. Over a year later, I am
still unable to fathom either the ludicrousness or
horrific injustice of this outcome.
Petitioner received the requested reconsideration of her
claim by the IRS Office of Appeals. Appeals Officer Mark Pearce
(Mr. Pearce) was assigned petitioner’s case in July of 2004. His
case activity records show that Mr. Pearce communicated with
petitioner or her representative, James R. Monroe (Mr. Monroe),
on a number of occasions and logged at least 8.75 hours
specifically on her case. After his review of the case,
Mr. Pearce concluded in an Appeals Case Memo prepared on February
22, 2005, that a weighing of the factors prescribed in Rev. Proc.
2003-61, 2003-2 C.B. 296, for evaluating claims under section
6015(f) did not support petitioner’s request for equitable
relief.
On March 3, 2005, the IRS issued to petitioner a notice of
determination denying her request for section 6015(f) relief.
The grounds listed in the notice for the denial were that
petitioner had established neither a reasonable belief that the
tax would be paid nor economic hardship. Petitioner filed a
timely petition with this Court contesting the adverse
determination and reflecting an address in Cape Coral, Florida.
A trial was held in November of 2005. At the time of trial,
- 9 -
petitioner had, within the last month, moved to Las Vegas,
Nevada, and had just begun a new job.
After posttrial briefs were filed, two Courts of Appeals,
those for the Eighth and Ninth Circuits, ruled that the Tax Court
lacked jurisdiction to consider denials of relief under section
6015(f) in proceedings where no deficiency had been asserted.
See Bartman v. Commissioner, 446 F.3d 785, 787 (8th Cir. 2006),
affg. in part and vacating in part T.C. Memo. 2004-93;
Commissioner v. Ewing, 439 F.3d 1009, 1013-1014 (9th Cir. 2006),
revg. 118 T.C. 494 (2002), vacating 122 T.C. 32 (2004). This
Court subsequently reached the same conclusion in Billings v.
Commissioner, 127 T.C. 7, 17 (2006).
Given these developments, the Court on August 30, 2006,
issued an order directing the parties to show cause why this case
should not be dismissed for lack of jurisdiction. Following
responses from both parties, the Court on October 24, 2006,
issued T.C. Memo. 2006-228, holding that we were constrained to
dismiss this case. An order of dismissal for lack of
jurisdiction was entered on the same date.
Thereafter, the Tax Relief and Health Care Act of 2006, Pub.
L. 109-432, div. C, sec. 408, 120 Stat. 3061, amended section
6015(e)(1) to provide that this Court may review the
Commissioner’s denial of relief under section 6015(f) in cases
where no deficiency has been asserted. The amendment applies
- 10 -
with respect to liability for taxes arising or remaining unpaid
on or after the December 20, 2006, date of enactment. Id.
Once again, in light of the change in the law, the Court on
December 26, 2006, issued an order directing the parties to show
cause why the earlier dismissal for lack of jurisdiction should
not be vacated. Respondent filed a response confirming that
petitioner’s liabilities for the relevant years remain unpaid and
that the Court now has jurisdiction to review the underlying
determination. The Court issued an order vacating the
October 24, 2006, dismissal on January 11, 2007. This matter is
now in a posture to be addressed on the merits.
OPINION
As a general rule, section 6013(d)(3) provides that “if a
joint return is made, the tax shall be computed on the aggregate
income and the liability with respect to the tax shall be joint
and several.” An exception to such joint and several liability
exists, however, for spouses able to satisfy the statutory
requirements for relief set forth in section 6015.
Section 6015 authorizes three types of relief. Subsection
(b) provides a form of full or partial relief available to all
joint filers and similar to, but less restrictive than, that
previously afforded by former section 6013(e), which was repealed
for taxes remaining unpaid as of July 22, 1998, or arising after
that date. Subsection (c) permits a taxpayer who has divorced or
- 11 -
separated to elect to have his or her tax liability calculated
proportionately as if separate returns had been filed.
Subsection (f) confers discretion upon the Commissioner to grant
equitable relief, based on all facts and circumstances, as
follows:
SEC. 6015. RELIEF FROM JOINT AND SEVERAL LIABILITY ON
JOINT RETURN.
(f) Equitable Relief.--Under procedures prescribed
by the Secretary, if--
(1) taking into account all the facts and
circumstances, it is inequitable to hold the
individual liable for any unpaid tax or any
deficiency (or any portion of either); and
(2) relief is not available to such
individual under subsection (b) or (c),
the Secretary may relieve such individual of such
liability.
The Court reviews a denial of relief under section 6015(f)
for abuse of discretion; i.e., whether respondent’s determination
was arbitrary, capricious, or without sound basis in law or fact.
Jonson v. Commissioner, 118 T.C. 106, 125 (2002), affd. 353 F.3d
1181 (10th Cir. 2003); Butler v. Commissioner, 114 T.C. 276, 292
(2000). Except as otherwise provided in section 6015, the
taxpayer generally bears the burden of proving such an abuse of
discretion. Rule 142(a); Alt v. Commissioner, 119 T.C. 306, 311
(2002), affd. 101 Fed. Appx. 34 (6th Cir. 2004); Jonson v.
Commissioner, supra at 113.
- 12 -
Relief under section 6015(b) or (c) is premised on the
existence of an understatement or deficiency. Because the
liabilities at issue in this case derive from unpaid taxes
reported on the 1997 and 1999 returns, petitioner is not eligible
for relief under subsection (b) or (c). Accordingly, there is no
dispute that petitioner satisfies the criterion for equitable
relief codified in section 6015(f)(2). The instant dispute thus
centers on the facts and circumstances inquiry of section
6015(f)(1).
As directed by section 6015(f), the Secretary has
promulgated guidance to structure the relevant facts and
circumstances analysis, the applicable version of which is set
forth in Rev. Proc. 2003-61, 2003-2 C.B. 296. Rev. Proc. 2003-
61, sec. 4.01, 2003-2 C.B. at 297, first lists seven threshold
conditions that must be met before the IRS will consider a
request for relief under section 6015(f): (1) The requesting
spouse filed a joint return for the year for which relief is
sought; (2) relief is not available under section 6015(b) or (c);
(3) the application for relief is made no later than 2 years
after the date of the IRS’s first collection activity; (4) no
assets were transferred between the spouses as part of a
fraudulent scheme; (5) the nonrequesting spouse did not transfer
disqualified assets to the requesting spouse; (6) the requesting
spouse did not file or fail to file the return with fraudulent
- 13 -
intent; and (7) absent enumerated exceptions, the liability from
which relief is sought is attributable to an item of the
nonrequesting spouse. Respondent here concedes that petitioner
meets these seven conditions.
Rev. Proc. 2003-61, sec. 4.02, 2003-2 C.B. at 298, then
gives three circumstances under which, if all are satisfied, the
IRS “ordinarily will grant relief” with respect to underpayments
on joint returns. The first of these elements requires that, on
the date of the request for relief, the requesting spouse be no
longer married to, be legally separated from, or not have been a
member of the same household as the nonrequesting spouse at any
time during the preceding 12-month period. Id. sec. 4.02(1)(a),
2003-2 C.B. at 298. The death of Dr. Banderas in November of
1999 preceded petitioner’s request by more than 12 months, and
this element therefore raises no barrier. The remaining two
elements, however, which address knowledge and economic hardship,
are at the crux of the instant controversy.
The second, or knowledge, element requires that:
On the date the requesting spouse signed the joint
return, the requesting spouse had no knowledge or
reason to know that the nonrequesting spouse would not
pay the income tax liability. The requesting spouse
must establish that it was reasonable for the
requesting spouse to believe that the nonrequesting
spouse would pay the reported income tax liability. If
a requesting spouse would otherwise qualify for relief
under this section, except for the fact that the
requesting spouse’s lack of knowledge or reason to know
relates only to a portion of the unpaid income tax
liability, then the requesting spouse may receive
- 14 -
relief to the extent that the income tax liability is
attributable to that portion. [Id. sec. 4.02(1)(b),
2003-C.B. at 298.]
Petitioner argues that respondent erred in concluding that
she possessed knowledge or reason to know that the taxes would
not be paid. As to 1997, petitioner contends that she had a
reasonable belief that the taxes would be paid either out of the
bankruptcy proceeding, from Dr. Banderas’s pension plan, or from
future earnings obtained through Dr. Banderas’s returning to
work. As to 1999, petitioner alleges that she reasonably
believed that the taxes would be paid from funds remaining after
the close of the bankruptcy proceeding, particularly in light of
the additional $750,000 received by the bankruptcy estate from a
life insurance policy on Dr. Banderas.
Petitioner testified regarding payment of the balances due
on the 1997 and 1999 returns, of which she was concededly aware.
The following colloquy on direct examination dealt with 1997:
Q Did your husband attempt to pay the 1997
taxes?
A We didn’t--we had no access to the money at
that time. It was in the Bankruptcy Court, and we--
Q So the bankruptcy was filed in 1997?
A Right. Before we filed the return.
Q And the return was filed in 1998?
A Right.
Q And all the funds were tied up in bankruptcy?
- 15 -
A Right.
Q What source did your husband attempt to use
to pay the 1997 taxes that were due?
A Well, we were told that the IRS was the
number one creditor. And we expected the Court, the
Bankruptcy Court to pay the taxes.
Q Any other source?
A If we--well, the pension plan. Because we
thought the pension plan was protected, and we never
thought that was going to be taken away from us.
Q How much was in the pension plan?
* * * * * * *
A Over a million dollars. Plus my husband’s
life insurance ended up being in there too. So over $2
million in the end.
Q Did somebody tell you that the pension funds
were exempt?
A We have always been told that the pension
plan was exempt. Yes.
* * * * * * *
Q What would have happened if the pension plan
didn’t pay for the taxes and the bankruptcy hadn’t paid
for the taxes? Was there any other source of--
A My husband would have had to go back to work,
which he ended up having to do anyway.
With respect to 1999, counsel inquired: “what was your intent
regarding the balance due that was shown on the 1999 tax return”?
Petitioner replied: “I anticipated money coming from the
Bankruptcy Court to pay it off”, and she went on to cite the
pension plan and life insurance.
- 16 -
The overarching general question raised by petitioner’s
contentions is one of timing. The standard suggested by
petitioner on brief highlights this issue: “The test should be
whether the requesting spouse had a reasonable belief that the
taxes would eventually be paid by the non-requesting spouse.”
(Emphasis added.) Petitioner was clearly aware on the dates she
signed the Forms 1040 that no liquid funds were available and on
hand to be submitted with the returns. She knew that the bulk of
Dr. Banderas’s assets were tied up in the bankruptcy; there is no
suggestion that any concrete steps had been taken, or were ever
taken, to try to obtain a distribution from the pension plan; and
Dr. Banderas had not yet returned to gainful employment at the
time of filing the return for 1997 and was deceased at the time
of filing the return for 1999.
It would thus appear that petitioner had not so much a
reasonable belief that the taxes would be paid as an inchoate
hope for a favorable change in circumstances that would, at some
undefined future point in time, place moneys at Dr. Banderas’s or
her own disposal to pay the taxes. Petitioner has at no juncture
made any attempt to identify when, in terms of a particular time
period, she thought the taxes would be paid. Furthermore,
although petitioner has generally alluded to being “told” that
the IRS would be the number one creditor in the bankruptcy and
that the pension plan was exempt, she did not call any witnesses
- 17 -
to establish the specific information she was given. As much as
we sympathize with the unfortunate, even tragic, string of events
that befell the Banderas household, we cannot accept a test so
nebulous as that proposed by petitioner. To do so would
essentially eviscerate the reasonable belief standard.
“Eventually” is simply too open-ended to place any meaningful or
administratively workable limits on qualification for relief.
The Court concludes that a reasonable belief that taxes
would be paid must at minimum incorporate a belief that funds
would be on hand within a reasonably prompt period of time.5 As
just indicated, petitioner has failed to make such a showing
here. Furthermore, because petitioner has never identified any
timeframe, the Court need not probe the contours of the how soon
question, or even whether the standard should be limited to
5
Taxpayers are required to pay their taxes when due. A
delay in payment not authorized by statute renders the Government
an involuntary creditor without security or other assurance that
the tax and interest thereon will be paid. When tax or interest
due is not paid, the burdens of Government are transferred to
other taxpayers, including those of future generations. Where
all funds needed to pay the taxes for a year of bankruptcy will
be trapped in the bankruptcy estate or for any other reason, the
bankrupt taxpayer and spouse may, at the taxpayer’s or spouse’s
option, elect to close their taxable years effective on the day
before the bankruptcy case was commenced. Sec. 1398(d)(2). This
election creates 2 short taxable years, the tax liability for the
first of which is included as a claim against the debtor’s estate
and is payable from the estate. Id.; see also sec. 6161(c)
(regarding authorized extensions of time to pay).
- 18 -
situations where moneys were thought to be available at the time
the relevant return was signed and filed.6
The foregoing conclusion is consistent with the result
reached by this Court in other contexts involving factual
contingencies regarding source of payment. In Vuxta v.
Commissioner, T.C. Memo. 2004-84, the Court addressed a
taxpayer’s request for relief under section 6015(f) for 1989,
1990, and 1991. The underlying returns showing balances due had
been filed on September 17, 1990, September 23, 1991, and May 30,
1996, respectively. The taxpayer and her husband had filed for
6
Placement of the modifying prepositional phrase (“On the
date the requesting spouse signed the joint return”) in the text
of Rev. Proc. 2003-61, sec. 4.02(1)(b), 2003-2 C.B. 296, 298,
would not appear to be conclusive on this issue. While the
Court’s placement of the phrase has varied, certain paraphrases
of the standard are amenable to a reading that could require a
belief that actual payment of the taxes would occur with the
filing of the return. E.g., Merendino v. Commissioner, T.C.
Memo. 2006-2 (“The relevant knowledge in the case of a reported
but unpaid liability is that the tax would not be paid when the
return was signed.”). The legislative history of the provision
might be open to a similar reading, in that the example supplied
indicates Congress intended for equitable relief to be granted
where a requesting spouse “does not know, and had no reason to
know, that funds intended for the payment of tax were instead
taken by the other spouse for such other spouse’s benefit.” H.
Conf. Rept. 105-599, at 254 (1998), 1998-3 C.B. 1008. Again,
however, the Court need not decide here whether such usages or
examples operate as limitations.
Additionally, guidelines set forth in the Internal Revenue
Manual (IRM) could suggest a more liberal reading, stating: “A
belief the tax would be paid is not reasonable if the RS
[requesting spouse] knew or had reason to know the NRS
[nonrequesting spouse] was not in an economic position, and was
not expected to be in an economic position within the foreseeable
future, to pay those taxes.” IRM sec. 25.15.3.8.3.2(2).
- 19 -
bankruptcy on May 1, 1992, and received a discharge on May 23,
1997. This Court concluded that, while the taxpayer lacked
knowledge or reason to know that the taxes for 1989 would not be
paid, she knew or had reason to know that the 1990 and 1991
liabilities would be unpaid because the return for 1990 preceded
the bankruptcy petition by only a little over 7 months and the
return for 1991 was filed during pendency of the bankruptcy case.
In Morello v. Commissioner, T.C. Memo. 2004-181, the
taxpayer’s husband lost his job in 1992, and she returned to work
after more than 20 years as a homemaker, in order to help relieve
resultant financial difficulties. By the time tax returns for
1992 through 1995 were signed and filed in 1996, financial
pressures, including mortgage foreclosure proceedings and the
fact that her husband was contemplating and then did file for
bankruptcy, had continued to plague the family for a number of
years. Because the taxpayer was aware of these circumstances, we
held that she had not established that it was reasonable for her
to believe that the balances shown as due on the returns would be
paid.
Given the foregoing, the Court cannot conclude that the
facts of this case are consistent with a reasonable belief that
the taxes would be paid. Accordingly, petitioner has not shown
that, at the times she signed the 1997 and 1999 returns, she had
no knowledge or reason to know that the taxes would not be paid.
- 20 -
Her sincere hope for payment through eventual receipt of funds
from various contingencies is not sufficiently concrete to fall
within the ambit of a meaningful reasonableness standard.
The third and final element of section 4.02 of Rev. Proc.
2003-61 places a requirement that the requesting spouse will
suffer economic hardship if relief is not granted. Id. sec.
4.02(1)(c), 2003-2 C.B. at 298. For purposes of making the
hardship determination, Rev. Proc. 2003-61, supra, references the
rules in section 301.6343-1(b)(4), Proced. & Admin. Regs. The
cited provision defines economic hardship as an inability to pay
“reasonable basic living expenses” and directs the Commissioner
to consider factors such as, among other things, the taxpayer’s
age, employment status and history, ability to earn, number of
dependents, and amount reasonably necessary for food, clothing,
housing, utilities, medical expenses, insurance (including home-
owner & health), transportation, current tax payments, etc. Sec.
301.6343-1(b)(4), Proced. & Admin. Regs.
The parties dispute what information the Court should
consider in its analysis of this element. Respondent’s position
is that consideration should take into account only those
materials provided to the Appeals Office during the underlying
administrative process. Petitioner disagrees and has sought to
supplement the administrative record in several respects.
Petitioner suggests that a purportedly insufficient quantity of
- 21 -
time spent by the Appeals officer in reviewing her case and the
officer’s failure to request updated financial information
justify augmentation of the record.
The Court stated its position on this issue in Ewing v.
Commissioner, 122 T.C. at 44 (in exercising our jurisdiction
under section 6015(e)(1)(A) “to determine” whether a taxpayer is
entitled to relief under section 6015(f), it is appropriate for
this Court de novo to consider evidence beyond the administrative
record), vacated on other grounds 439 F.3d 1009 (9th Cir. 2006).
However, because here the outcome flowing from a limited review
of the administrative record alone and that obtaining after
taking into account all information proffered by petitioner are
identical, the Court finds it unnecessary to comment further on
the merits of the parties’ differing views of the proper record
for review.
During the administrative phase, petitioner submitted a Form
12510 dated July 22, 2003, showing an excess of income over
expenses of approximately $2,980 per month. This portrayal
clearly fails to reflect economic hardship, and we do not read
petitioner’s arguments to contend otherwise. Nor does petitioner
suggest that she provided updated financial information to the
Appeals officer at any time prior to his final recommendations on
her case in early 2005. To the extent that petitioner intimates
that the burden rested entirely on the Appeals officer to request
- 22 -
new data, her point, in this particular context, is not well
taken.
As a general matter, taxpayers are usually responsible for
establishing their positions before the IRS. Second, on a more
specific and practical level, petitioner, despite being
represented by counsel, apparently never alerted the Appeals
officer of any relevant change in circumstances. Furthermore,
the types of items reported on the Form 12510 were not inherently
of a nature to raise a suspicion of dramatic change over the
course of the ensuing period. The only likely difference hinted
at on the face of the Form 12510 would have been a reduction in
income, as petitioner appended a comment that she might be unable
to continue to sustain multiple jobs. However, third-party
payers reported to the IRS even higher wages of $87,925 for the
completed taxable year of 2003 (the most recent year for which
records would have been available during at least the majority of
the time petitioner’s case was before Mr. Pearce). This fact was
noted by the Appeals officer in his recommendations. Third-party
payers similarly reported still higher wages to petitioner of
$90,974 for 2004. In this scenario, the Court, despite its
admiration for petitioner’s efforts to help herself, is
unconvinced that petitioner may abdicate all responsibility for
raising some issue of pertinent change or, conversely, that the
- 23 -
Appeals officer acted arbitrarily or capriciously in relying on
the information previously provided.
Additionally, the Court is not persuaded that petitioner’s
remarks concerning the quantum of time devoted to her case should
have any bearing on our analysis. The recorded time is in no way
patently insufficient, and the Court is not in a position to
divine any type of arbitrary standard as to the time it might
take an individual to properly evaluate a given set of facts.
At trial, petitioner sought to introduce three 1-page sheets
of financial information, all dated November 14, 2005 (the date
of the calendar call): (1) A listing of assets and liabilities;
(2) a listing of purported actual current income and expenses;
and (3) a listing of current income and expenses incorporating
national, regional, and local IRS standards. The listings were
not accompanied by any supporting documentation. Counsel for
respondent objected on grounds, in part, that she was not
provided with the exhibits until the morning of trial. The
Court, in light of the Standing Pretrial Order and citing the
prejudice to respondent’s ability to prepare for cross-
examination, sustained the objection, and the documents were not
admitted. Instead, petitioner testified regarding her estimates
of various assets, liabilities, income amounts, and expenses.
Petitioner then filed a posttrial brief and attached thereto
three tables, purportedly summarizing petitioner’s testimony and
- 24 -
bearing striking similarity to the exhibits not admitted at trial
(with the income and expense listings now bearing two columns,
one labeled November 14, 2005, and the other labeled February
2005). However, the tables incorporate noticeably more detail
and precision than can reasonably be gleaned from petitioner’s
often generalized approximations from the stand.
It is the status of the record as just described that
precipitated the Court’s comment that the outcome of the case is
unaffected regardless of any proffered information considered in
augmentation of the administrative record. The complete dearth
of any documentation to corroborate or substantiate the figures
renders petitioner’s various estimates unpersuasive in any event.
Moreover, the Court would observe that only the computations
purporting to detail income and expenses as of February 2005 show
an excess of expenses over income, and it is decidedly
questionable whether all of those expenses shown are properly
taken into account for purposes of section 301.6343-1(b)(4),
Proced. & Admin. Regs. (for example, charitable contributions and
payments on various liabilities).
The Court is therefore constrained to conclude that
petitioner has failed to establish economic hardship within the
meaning of Rev. Proc. 2003-61, sec. 4.02, 2003-2 C.B. at 298.
Accordingly, on account of failure to satisfy two of the three
elements enumerated as delineating those circumstances in which
- 25 -
the Commissioner will ordinarily grant relief, petitioner does
not qualify for equitable relief under Rev. Proc. 2003-61, sec.
4.02, 2003-2 C.B. at 298.
Where relief is not available under section 4.02 of Rev.
Proc. 2003-61, section 4.03 of the revenue procedure sets forth a
list of nonexclusive factors that the Commissioner will consider
in determining whether it would be inequitable to hold the
requesting spouse liable for all or part of the unpaid income tax
liability. As emphasized in Rev. Proc. 2003-61, sec. 4.03(2),
2003-2 C.B. at 298, no single factor is to be determinative;
rather, all factors are to be considered and weighed
appropriately.
Eight factors are listed, the latter two of which will weigh
in favor of granting relief if present but will not weigh against
relief if absent. The factors are directed toward whether: (1)
The requesting spouse is separated or divorced from the
nonrequesting spouse; (2) the requesting spouse will suffer
economic hardship without relief; (3) the requesting spouse did
not know or have reason to know that the nonrequesting spouse
would not pay the liability; (4) the nonrequesting spouse had a
legal obligation to pay the outstanding liability; (5) the
requesting spouse received significant benefit (beyond normal
support) from the enhanced assets or income resulting from the
unpaid liability; (6) the requesting spouse has made a good faith
- 26 -
effort to comply with income tax laws in subsequent years; (7)
the requesting spouse was abused by the nonrequesting spouse; and
(8) the requesting spouse was in poor mental or physical health
when signing the return or requesting relief. Rev. Proc. 2003-
61, sec. 4.03(2)(a) and (b), 2003-2 C.B. at 298-299.
As just described in connection with analysis of section
4.02 of Rev. Proc. 2003-61, the marital status factor weighs in
favor of granting relief, while the economic hardship and
knowledge elements weigh against granting relief. The remaining
factors are considered below.
The legal obligation factor asks whether the nonrequesting
spouse has an obligation to pay the outstanding liability
pursuant to a divorce decree or separation agreement. Id. sec.
4.03(2)(a)(iv), 2003-2 C.B. at 298. Because petitioner and
Dr. Banderas were never divorced or separated, this factor is
neutral.
The significant benefit factor probes whether the requesting
spouse received, directly or indirectly, from the assets or
income resulting from the unpaid liability any benefit in excess
of normal support. Sec. 1.6015-2(d), Income Tax Regs.; Rev.
Proc. 2003-61, sec. 4.03(2)(a)(v), 2003-2 C.B. at 299
(referencing sec. 1.6015-2(d), Income Tax Regs.). Evidence of
such benefit “may consist of transfers of property or rights to
property, including transfers that may be received several years
- 27 -
after the year of the * * * [relevant return]”. Sec. 1.6015-
2(d), Income Tax Regs. Normal support is measured by the
circumstances of the particular parties. Estate of Krock v.
Commissioner, 93 T.C. 672, 678-679 (1989); Levy v. Commissioner,
T.C. Memo. 2005-92.
Respondent notes that from 1999 to 2005, the Banderases or
petitioner sold several parcels of real property and received
approximately $8,000 to $10,000 on each of the three
transactions. Petitioner testified that the proceeds were used
primarily to pay living and moving expenses, to make a
downpayment on a new residence, and to contribute to the cost of
a daughter’s wedding. In 2004, petitioner received a $60,000
distribution from her individual retirement account. Petitioner
also acknowledged making a loan of conservatively at least
$18,000 during 2004 to a friend whose whereabouts were unknown at
the time of trial.
In the unique circumstances of this case, particularly the
fact that the majority of the Banderases’ assets were tied up in
the bankruptcy litigation, the described uses of the relatively
modest proceeds from the property transactions would not
generally appear to rise to the level of excess benefit. The
wedding costs, however, give us pause. Likewise, the $18,000
loan and the failure to apply any of the retirement account
distribution to outstanding taxes could signal a more telling
- 28 -
selective avoidance of tax liabilities. Nonetheless, the Court
is also cognizant that, while the bankruptcy establishes reason
to know that the taxes would not be paid at the time of filing,
its pendency could have engendered a degree of confusion in
petitioner’s mind as to responsibility for the taxes and the
eventual source of funds for their payment. The Court therefore
is unconvinced that the substantial benefit prong weighs strongly
either for or against relief.
The compliance with income tax laws factor addresses the
good faith efforts of the requesting spouse in subsequent years.
Rev. Proc. 2003-61, sec. 4.03(2)(a)(vi), 2003-2 C.B. at 299. The
record indicates that petitioner’s 2000 return was not timely
filed and that amounts due remained outstanding until at least
February of 2004. The 2001 and 2002 returns were timely filed,
but the balance due for 2001 was not paid until January of 2003.
The 2002 taxes were timely paid. As of the time of trial, IRS
records did not reflect that petitioner had filed her 2003 or
2004 returns. Petitioner testified that she believed she timely
filed the 2003 return but refiled it, in response to IRS
inquiries, shortly before the trial in November of 2005, at which
time she also filed her initial return for 2004. Petitioner also
explicitly conceded at trial and on brief that various of her
returns and related payments were delinquent on account of
financial difficulties, moves, misplaced documents, etc.
- 29 -
Hence, petitioner’s compliance with tax laws subsequent to
the filing of the 1999 return reveals shortcomings.
Additionally, petitioner’s generalized allusions to financial
pressures and information misplaced during moves are insufficient
to establish good faith efforts with respect to the series of
delinquencies occurring over a period of years. This factor
weighs against relief.
The presence of abuse is a factor weighing in favor of
relief, but its absence carries no contrary weight. Rev. Proc.
2003-61, sec. 4.03(2)(b)(i), 2003-2 C.B. at 299. Because
petitioner has at no point alleged any abuse by Dr. Banderas,
this factor is neutral.
The final factor enumerated in the revenue procedure, and
again weighing only in favor of and not against relief, is
directed toward whether the requesting spouse was in poor mental
or physical health when signing the return or seeking relief.
Id. sec. 4.03(2)(b)(ii), 2003-2 C.B. at 299. The nature, extent,
and duration of illness are to be taken into account. Id.
Petitioner’s statements at trial and on brief incorporate general
assertions of high blood pressure, depression, and bad knees.
Nonetheless, the record is bereft of any corroborating medical
documentation or any specific contentions regarding how such
ailments might have impacted petitioner’s ability to meet her
Federal tax obligations. In fact, petitioner acknowledged that,
- 30 -
except for her weight, she was in good health during the 1997 and
1999 years at issue and until 2000 when her blood pressure began
rising. The Court observes that petitioner was able to work
multiple jobs throughout much of the relevant period. Thus,
while the Court does not doubt that petitioner bore substantial
stress on account of the difficult circumstances besetting her
family, the record does not enable us to find particular health
issues of a nature that would affect the balancing of her claim
for relief. This factor, too, is neutral.
Accordingly, of those factors identified in the pertinent
revenue procedure, only one weighs clearly in favor of granting
relief. The remainder are either negative or essentially
neutral. The Court is therefore unable to conclude that
respondent’s denial of equitable relief under section 6015(f) was
an abuse of discretion.
The Court has considered all other arguments made by the
parties and, to the extent not specifically addressed herein, has
concluded that they are without merit or are moot. To reflect
the foregoing,
Decision will be entered
for respondent.
|
Cables, such as telecommunication cables and electrical power distribution cables, are ubiquitous and used for distributing electrical power and all manner of data across vast networks. The majority of cables are electrically conductive cables (typically copper), although the use of optical fiber cables is growing rapidly in telecommunication systems as larger and larger amounts of data are transmitted. As cables are routed across power or data networks, it is necessary to periodically open the cable and splice or tap into the cable so that power or data may be distributed to other cables or “branches” of the network. The cable branches may be further distributed until the network reaches individual homes, businesses, offices, and so on. The cables of the distributed lines are often referred to as drop lines, branch lines, or distribution lines.
At each point where the cable is opened, it is necessary to provide some type of enclosure to protect the cable. Commonly, the enclosure has one or more entry ports through which cables enter the enclosure. Depending upon the number of entry ports in the enclosure, the sizes of the entry ports, the number of cables entering the enclosure, and the sizes of the cables, the number of cables passing through the each entry port will vary. Often, especially with smaller diameter cables as are typically used in distributed lines, multiple cables are bundled for placement into a single larger entry port. This is particularly common where multiple smaller cables are routed from a single point to multiple locations, such as individual homes, buildings, offices, etc.
At each entry port, no matter the number of cables passing therethrough, it is often desirable or necessary to provide a seal around the cables to prevent the ingress of moisture, dust, insects, and the like into the enclosure. Current methods of providing a seal around the cables typically involve bundling cables with mastic material or rubber tape. Such sealing methods are replete with disadvantages. The quality of the resulting seal is highly dependent upon the skill of the installer, and is therefore typically inconsistent from one installer to another installer. Further, as the number of cables increases, it becomes more and more difficult to form a reliable seal. The use of mastic or rubber tape to form a seal also presents difficulties when it is desired to re-enter the enclosure, so as to add or remove cables in the entry port. Specifically, the old sealing materials must be removed without damaging the cables, and a new seal must be constructed. The removal and reconstruction of the seal requires the use of additional time and materials, and therefore adds to the cost of maintaining or expanding the network.
To address the above-described disadvantages of mastic or rubber tape seals, there have been attempts to use pre-formed grommets to expand the capacity of entry ports in an enclosure. Typically, the grommets are sized to fit within an enclosure entry port, and have a predetermined number of holes for accepting smaller diameter cables. The cables are threaded through the holes in the grommet, and the grommet is in turn secured within the entry port. Such grommets have the disadvantage that the cables must have a free end to thread through the grommet openings. However, in many applications a free cable end is not available because the cable has already been connected, spliced or terminated. In such situations, to avoid disconnecting, re-splicing or re-terminating the cable, the installer typically cuts through the body of the grommet into the opening, using a utility knife or the like; such that the cable can be inserted lengthwise into the opening. Successfully cutting into the grommet in this manner is very dependent upon the skill of installer, and becomes increasingly difficult as the number of cables increases and as the size of the grommet and cables decreases. Further, the grommet is left with several cuts extending directly from the openings to the outer surface of the grommet. The cuts degrade the reliability of the grommet, and are a potential pathway for the ingress of moisture into the enclosure.
A need exists for a seal that enables an installer to consistently and reliably bundle multiple cables for placement into a single entry port of an enclosure, independent of the skill level of the installer. Furthermore, a need exists for a seal that can be easily and quickly installed, and that can be easily re-entered and re-used. Still further, a need exists for a such a seal that can also be installed on an existing cable, without disconnecting, re-splicing or re-terminating the cable. |
[Hereditary pseudovitamin-D-deficiency rickets with alopecia as a consequence of extreme resistance of target organs to 1.25-(OH)2-cholecalciferol. Therapeutic progress in another disorder of vitamin-D-metabolism (author's transl)].
We report about a pair of Turkish siblings suffering from alopecia and severe rickets. In the elder sister we could show the causal role of extreme resistance of target organs to 1.25-(OH)2-cholecalciferol. The birth of a brother who developed the same symptoms together with two other similar cases in siblings recently published by other authors, made it possible to recognize the disorder as a typical syndrome-like entity, based on a hereditary defect. By successful administration of excessive (and so-far not previously published) doses of Vitamin-D3 a way for therapeutic progress in this new type of vitamin-D-dependent rickets is shown. |
Next year, Monash University will become the first University in Australia to introduce ‘trigger warnings’ on their course guides. It’s a policy that many are pushing for in Universities all across the country. Reportedly, the Monash Student Association pushed for the inclusion of content warnings, with Association President Abigail Stapleton saying that it was “a simple request” that was about making university “a better experience for students.“
Ironically though, folks are getting pretty offended over the attempt to stop people getting offended. IPA Research Fellow Matthew Lesh has said that trigger warnings “defeat the purpose of higher education and encourage academics to stop teaching certain materials.”
Australia is a few years behind America on the Safe Space/Trigger Warning debate, which over there is a constant point of controversy. The University of Chicago has recently come out strongly against both of the above, but the overall liberal arts campus culture has increasingly moved in that direction.
It got to the point where last year South Park satirised it – a sure indicator something has reached cultural saturation – in an episode that featured a viciously unkind depiction of Steven Seagal.
Source: Herald Sun.
Photo: Instagram. |
TAK-242 selectively suppresses Toll-like receptor 4-signaling mediated by the intracellular domain.
TAK-242, a small-molecule antisepsis agent, has shown to suppress lipopolysaccharide (LPS)-induced inflammation. In this study, we demonstrate that TAK-242 is a selective inhibitor of Toll-like receptor (TLR)-4 signaling. TAK-242 almost completely suppressed production of nitric oxide (NO) or tumor necrosis factor (TNF)-alpha induced by a TLR4-specific ligand, ultra-pure LPS, in mouse RAW264.7, human U-937 and P31/FUJ cells, whereas this agent showed little effect on other TLR ligands, Pam(3)CSK(4) (TLR1/2), peptidoglycan (TLR2/6), double strand RNA (TLR3), R-848 (TLR7) and CpG oligonucleotide (TLR9). Furthermore, TAK-242 potently inhibited nuclear factor (NF)-kappaB activation induced by ultra-pure LPS in HEK293 cells transiently expressing TLR4 and co-receptors, myeloid differentiation protein-2 (MD2) and CD14, whereas this agent showed little effect on other TLRs, TLR1/2, TLR2/6, TLR3, TLR5, TLR7 and TLR9. TAK-242 also inhibited ligand-independent NF-kappaB activation resulting from over-expression of TLR4. Although chimera receptors, which are consist of the extracellular domain of CD4 and the intracellular domain of human or mouse TLR4, showed constitutive NF-kappaB activation, TAK-242 potently inhibited the signaling from CD4-TLR4 chimera receptors. In contrast, the NF-kappaB activation mediated by TLR4 adaptors, myeloid differentiation factor 88 (MyD88), TIR-associated protein (TIRAP), Toll/IL-1R homology (TIR)-domain-containing adaptor protein-inducing interferon-beta (TRIF) or TRIF-related adaptor molecule (TRAM) was not affected by TAK-242. TAK-242 is therefore a selective inhibitor of signaling from the intracellular domain of TLR4 and represents a novel therapeutic approach to the treatment of TLR4-mediated diseases. |
Erigeron ovinus
Erigeron ovinus is a rare North American species of flowering plant in the daisy family, called the sheep fleabane. It has been found only in the southeastern part of the US state of Nevada (Clark County + Lincoln County).
Erigeron ovinus is a perennial herb up to 15 centimeters (6 inches) tall, with a large taproot, forming clumps of many individuals close together. Leaves are pinnatifid with long narrow lobes. The plant generally produces one or two flower heads per stem, each head with numerous yellow disc florets but no ray florets. The species grows on ridges and in cracks in rocks in conifer woodlands.
The oldest name for this plant is Erigeron caespitosus subsp. anactis. Conquist in 1947 sought to raise this from subspecies to the level of species, but opted to forgo the common (but not mandatory) practice of using the subspecific epithet as a species epithet. He chose a new epithet instead, ovinus.
References
Category:Flora of Nevada
ovinus
Category:Plants described in 1922 |
Thomas Was Alone
I loved Thomas Was Alone, the minimalist platformer released last year by Mike Bithell. The hilarious narration, memorable characters and awesome music defied my expectations I had after first hearing about the game and I was glad to see Thomas find a home on Steam. I'm never going to stop wanting for more people to play it. So it's great news that Thomas Was Alone will becoming to PlayStation 3 and PlayStation Vita.
The jumping-block game will support CrossBuy, meaning that you'll have it for both devices after buying it on one. The PlayStation edition will also feature a new director's commentary mode, and a new DLC episode with a new mechanic, voiceover by Danny Wallace and music by David Housden. Given that the voicework and music were both excellent in the original release, those are fantastic additions.
Thomas Was Alone joins indie darlings Super Crate Box and Retro City Rampage on the Vita. And Hotline Miami is on its way there, too. Say what you want about Sony's portable but it's turning into a great little avenue for indie games to reach wider audiences.
I really loved Thomas Was Alone, a charming indie puzzle platformer from developer Mike Bithell. Apparently, the folks who make games for Wired's UK division liked it, too, because they cranked out a game that cribs heavily from Thomas Was Alone.
CFBDSIR2149 Was Alone doesn't have the endearing narration of Danny Wallace or the excellent soundtrack by David Housden. But the newsgame tied to an article about an orphan planet does parrot the tone, text boxes and get-to-the-exit level design of Thomas. Despite being an unmistakable—and inferior—clone of Bithell's game, no acknowledgement of that inspiration is in the text. The developer of the CFBDSIR2149 game does mention Thomas Was Alone in the comments.
As of this writing, there hasn't been any change in the article text. No game deserves to be ripped off but it's especially troubling when it happens to an indie release. Everyone go play Thomas Was Alone. It's better anyway.
Update: Mention of Thomas Was Alone—and a link to the game on Steam—as inspiration has been added to the article. All is well. |
Synthesis and quality control of [(18) F]T807 for tau PET imaging.
The detailed synthesis and quality control of [(18) F]T807, radiotracer for tau protein aggregate imaging, are described. The radiotracer synthesis was accomplished in an average of 48 min with an average specific activity at end-of-synthesis of over 4.4 TBq/µmole (120 Ci/µmole) and an average radiochemical yield of 32%. Compliance with all standard US Pharmacopeia Chapter <823> acceptance tests was observed. |
'Persecuted' Pakistani Hindus seek refuge in India
When India and Pakistan separated in 1947, a large number of Hindus chose to live in Muslim-majority Pakistan. But now they want to leave. In August, over 500 Hindu families crossed the border from Pakistan into India, ostensibly on a pilgrimage. Soon after crossing the border, many of them announced that they would not go back to Pakistan due to the rising number of atrocities against the Hindu minority there, including kidnappings and forced conversions to Islam. |
I enjoy movies about clashes of cultures, and last night we watched a cute film, Sam’s recommendation, called The Hundred Foot Journey, which was a food movie, and I wouldn’t have thought it something that would interest him at all, so I suspect maybe some girl at school recommended it to him, because that’s why boys do most things.
It was the story of an Indian family who moved to England after the mother died, but England didn’t really work for them and so they moved to France and then their brakes failed and they nearly crashed and wound up in this spot in the French countryside and that was the establishing scene, you knew they would stay in that village and you knew that the hero would wind up with the pretty girl who helped tow their car to a service station, it was all very predictable, but that was also the lovely scenery part.
Then the father found the perfect spot for their restaurant but it was a dump which led to a fix-it-up montage scene with Indian music, but the problem was that across the road was the big, snobby restaurant, whose owner took an instant dislike to them and this led to a restaurant war which was the comedy part of the film.
But, the son, which surprises nobody because it was clearly announce in the very first scene of the movie, turns out to be a total genius of a chef and that leads to the rags and riches part of the story where he moves to Paris and becomes an acclaimed chef, and a bit of an arrogant jerk, but he misses the little village and especially the girl, and comes back and there is peace between the two restaurants and everybody lives happily ever after.
It’s like Ratatouille, but with humans. Cute, but I’d never say it was a must see.
Then, tonight, Helena wanted to watch My Big, Fat Greek Wedding II, which I found to be totally unwatchable. Some movies just should not have sequels, and that was one. |
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‘Gaza flotilla: riding on a wave of narcissism’
By painting Israel as the scum of the Earth and the Palestinians as the salt of the Earth, the flotilla crew are not only narcissistically advertising themselves as the noble saviours of Palestine. They are also hoping to relegitimise and reinvigorate the West’s moral imperative to act as the heroic rescuer of the Middle East. Indeed, for all the hyping up of the flotilla’s non-violent resistance and the crew’s ostensible backing of the Palestinians’ right to self-determination, in fact the flotilla crew are a war-thirsty, interventionist bunch. |
In contemporary vehicle structures (passenger cars, buses, trains, etc.), acoustic damping compounds with damping/soundproofing functions are provided in various components of the vehicle structures, such as accessories, panels, roofing, and flooring, to reduce or prevent transmission of vibrations generated by the structures and noise created by these vibrations. Common acoustic damping compounds include injectable and extrudable compounds based on bitumen, rubber, epoxy, and water-based (acrylate) dispersions whose matrix conforms to the shape of the vehicle. These acoustic damping compounds are usually applied to the surface of the vehicle frame and various application sites in the vehicle.
In order to prevent vibrations of the outer panels, prevent knocking of various components in the vehicle, and ensure the proper distance between components in a vehicle structure, a so-called undercoating is preferably applied between the outer panels and the roof arch, and between protective components and reinforcing components. This undercoating can reinforce the vehicle structure and the undercoating material can simultaneously function as an adhesive or sealant.
A composition integrating damping/soundproofing functions with the functions of an undercoating is the thermally curable composition disclosed in Patent Document 1, which comprises (a) an olefinic double-bond-containing polymer or copolymer based on a diene- and/or aromatic-substituted olefin and (b) a vulcanization system. When a composition described in Patent Document 1 is used, it serves as an integrated acoustic damping compound and undercoating, which previously required applying a plurality of different materials. This simplifies the manufacturing process and manufacturing equipment, and thus reduces costs. Because this composition is “pumpable” that is, can be distributed (pumped) by a pump, it can be applied by a robot and used advantageously in a vehicle production process which is highly automated.
However, compositions used in vehicle structures such those of automobiles will presumably be used in low-temperature environments and must have good low-temperature characteristics. A composition (one-part bonding agent, sealant, or coating) suitable for use in vehicle production having high tensile shear strength and high impact peel strength at low temperatures was disclosed in Patent Document 2 in the form of a high-temperature thermally curable reaction composition based on a natural and/or synthetic olefinic double-bond containing elastomer and a vulcanizer and containing a liquid polyene and polybutadiene. |
Reina Hispanoamericana 2019
Reina Hispanoamericana 2019 was the 29th edition of the Reina Hispanoamericana pageant. It was held on February 8, 2020 in Santa Cruz, Bolivia. At the end of that event, Nariman Battikah of Venezuela crowned Regina Peredo of México has her successor.
Final Results
Δ voted via the internet
Contestants
As of , 29 titleholders have been crowned
Hispanic Beauty Gala
Amazonas Girl
Lincy Colman
Best Costume
Stefani Zeceña
Miss Sports
Lincy Colman
Miss Personality
Laura Claro
Best Hair
Lincy Colman
Best Smile
Monserrat Báez
Miss Elegant Face
Valeria Vadell
Miss Photogenic
Yuanilie Alvarado
Miss Silueta
Valeria Badell
Crossovers
These are the contestants who previously competed or will be competing at other international beauty pageants:
Miss Universe
2017: Casandra Cherry
2019: Ketlin Lottermann
Miss World
2015: Stefanía Alemán
2017: Gabrielle Vilela (Top°40)
Miss International
2018: Stefania Aleman
2018: Cassandra Cherry
2020: Gabriela Irías (TBA)
Miss Grand International
2017: Diana Sofía Silva
2018: Gabrielle Vilela (Top°21)
Miss Eco International
2017: Diana Sofía Silva
Miss Planet International
2017: Marianella Chaves (Top°10)
Miss World Latin
2018: Stefani Zeceña
World Top Model
2018: Lincy Colman (4th Runner-Up)
2018: Tiffany de Freitas Brás
Notes
References
Category:Reina Hispanoamericana
Category:2019 beauty pageants |
Measure C Bond Expenditures as of June 30, 2014
Annual Performance and Financial Audit
California Proposition 39 requires annual performance and financial audits on use of general obligation bond proceeds. The performance and financial audits for the fiscal year ending June 30, 2014, were performed by an independent certified public accounting firm in accordance with generally accepted auditing standards and governmental auditing standards issued by the comptroller general of the United States. The result of the audits is that the district expended the general obligation bond proceeds in accordance with the requirements of Proposition 39. |
2008 Northern Pride RLFC season
2008 was the first competitive season for the Cairns based CRGT Northern Pride Rugby League Football Club. They competed in the QRL state competition, which in 2008 was called the Wizard Queensland Cup. 11 Clubs played 20 matches (10 home and 10 away) over 26 weeks.
The Pride finished 3rd, just losing the Preliminary Final in golden point extra-time to the Souths Logan Magpies, who went on to win the 2008 Grand Final. Foundation coach was Andrew Dunemann, who had played for the Canberra Raiders, Leeds Rhinos, Halifax RLFC and South Sydney Rabbitohs, and had been Under-20s coach for the Canberra Raiders. Assistant coach was David Maiden. Foundation captain was Chris Sheppard, who had played for the North Queensland Cowboys and St. George Illawarra Dragons.
2008 Season - CRGT Northern Pride
Staff
Coach: Andrew Dunemann
Assistant Coaches: David Maiden & Troy Cummings
Captain: Chris Sheppard
Chief Executive: Denis Keeffe
Chairman: John O’Brien
Operations Manager: Chris Sheppard
Commercial Manager: Brad Tassell
Competition: Wizard Queensland Cup
2008 Player awards
Friday 26 September 2008, Pride Leagues Club, Irene Street, Mooroobool
Yalumba Wines Most Improved Player – Hezron Murgha
EK Kitchens Best Back – Chey Bird
Skytrans Airlines Best Forward – Mark Cantoni
CRGT Player of the Year – Chris Sheppard
John O’Brien Perpetual Club Person of the Year – Pride Manager, Rob White
Player gains
This was the first season for the Northern Pride, so all players were new signings.
Ryan Bartlett from QRL's North Queensland Young Guns
Greg Byrnes from QRL's North Queensland Young Guns
Brett Anderson from QRL's North Queensland Young Guns
Warren Jensen from QRL's Wynnum Manly Seagulls
Josh Vaughan from QRL's Tweed Heads Seagulls
Joel Riethmuller from QRL's Ipswich Jets
Gordon Rattler from QRL's Ipswich Jets
Mark Cantoni from QRL's Eastern Suburbs Tigers
Ben Laity from QRL's Eastern Suburbs Tigers
Hezron Murgha from CDRL's Yarrabah Seahawks
Noel Underwood from CDRL's Yarrabah Seahawks
Farran Willett from CDRL's Yarrabah Seahawks
Kahu Wehi from CDRL's Tully Tigers
Ritchie Marsters from CDRL's Tully Tigers
Joshua Wehi from CDRL's Tully Tigers
Alex Starmer from CDRL's Ivanhoes Knights
Drew Campbell from CDRL's Ivanhoes Knights
Jason Roos from CDRL's Mareeba Gladiators
Chris Sheppard from CDRL's Mareeba Gladiators
Adam Mills from CDRL's Atherton Roosters
Stephen Sheppard from TDRL's Burdekin and CDRL's Mareeba Gladiators
Chey Bird from TDRL's (Townsville) Brothers
Ben Kerr from CDRL's Innisfail
Eric Warria from Australian Secondary Schools Rugby League Northern Territory team
Steve McLean from Canberra Raiders juniors.
In Round xx the following player was signed to the Pride:
Quincy To’oto’o-ulugia from NRL's Cronulla Sharks Jersey Flegg
In Round 10 the following players were signed to the Pride:
Chris Afamasaga from NRL's Gold Coast Titans . He walked out of the club after Round 18.
Jamie Frizzo from NRL's North Queensland Cowboys
In Round 12 the following players was signed to the Pride:
Rod Griffin from NRL's Wests Tigers and PNG Kumuls
In Round 16 the following players was signed to the Pride:
Luke Harlen from NRL's Wests Tigers
At the start of the season attempts were made to sign players from Papua New Guinea, but problems with visas prevented them playing for the Pride:
Michael Mark from PNG Kumuls
Jessie Joe Parker from PNG Kumuls
2008 squad
Ryan Ghietti
Chey Bird
Drew Campbell
Hezron Murgha
Noel Underwood
Gordon Rattler
Farren Wilett
Steve Sheppard
Eric Warria
Brett Anderson
Chris Sheppard
Jackson Nicolau
Josh Vaughan
Jason Roos
Alex Starmer (Prop)
Greg Byrnes
Ben Laity
Kahu Wehi
Adam Mills
Warren Jensen
Richie Marsters
Ben Kerr
Ben Vaeau
Matthew Bartlett
Joel Riethmuller
Mark Cantoni
Warren Jensen
Aaron Payne
Ashley Graham
Ben Vaeau
Carl Webb
Jackson Nicolau
Jacob Lillyman
John Williams
Justin Smith
Mark Henry
Matthew Bartlett
Matt Bowen
Ray Cashmere
Scott Bolton
Steve Southern
Ty Williams
Ryan Ghietti
Jordan Kane
2008 Season Launch
Team Launch - 14 December 2007
Season Launch - 7 March 2008 at 11.00am, Stockland Cairns. (Originally scheduled as part of the Esplanade Lagoon's 5-year celebrations but moved to Stockland Cairns because of wet season flooding in Cairns.)
Pre-Season Boot Camp
Croco Dylus Village Camp, Daintree River - 19–20 January 2008
2008 Jerseys
Trial Matches
Wizard Queensland Cup matches
2008 Ladder
Finals Series
2008 Northern Pride players
North Queensland Cowboys who played for the Northern Pride in 2008
2008 Televised Games
In 2008 games were televised by ABC TV and shown live across Queensland through the ABC1 channel at 2.00pm (AEST) on Saturday afternoons. The commentary team was Gerry Collins, Warren Boland and David Wright.
1: Northern Pride won 34-24: Round 5, Saturday 12 April 2008 against Souths Logan Magpies from Meakin Park, Logan
2: Northern Pride lost 4-34: Round 8, Saturday 3 May 2008 against Tweed Heads Seagulls from Cudgen Park, Cudgen
3: Northern Pride won 30-16: Round 11, Saturday 24 May 2008 against Ipswich Jets from Briggs Road Sporting Complex, Ipswich
4: Northern Pride won 26-10: Round 17, Saturday 12 July 2008 against Easts Tigers from Langlands Park, Stones Corner, Brisbane
5: Northern Pride won 34-16: Qualifying Final, Saturday 30 August 2008 against Ipswich Jets from Briggs Road Sporting Complex, Ipswich
6: Northern Pride lost 12-16 : Preliminary Final, Saturday 6 September 2008 against Souths Logan Magpies from Langlands Park, Stones Corner, Brisbane
References
External links
Northern Pride Official site
Northern Pride Facebook Page
Northern Pride Twitter Page
Northern Pride YouTube Page
Cairns Post - Northern Pride 2008 photo gallery
Category:Northern Pride RLFC seasons
Category:2008 in Australian rugby league
Category:2008 in rugby league by club |
Portable tools such as chain saws or power cutters are used in many different handling positions, even up side down. They are therefore usually crankcase scavenged and lubricant, e.g., oil is supplied to the crankcase. This lubrication system works in every handling position.
However, oil tends to collect in the crankcase so there is a surplus in the crankcase and tends to be a shortage for some lubricating places. By adding more oil this can of course be compensated for, but this will increase oil consumption and increase emissions of oil smoke in the exhaust gases.
There are even lubricating places that are very difficult to lubricate at all, e.g., a bearing on the crankshaft supporting a centrifugal clutch normally used for portable tools. Some tools use a sealed bearing that is pre-filled with grease.
The seals will wear resulting in loss of grease and the shaft will corrode increasing the wear of the seals and the loss of grease and shortening the life of the bearing.
Other tools use a duct arranged in the crankshaft so that one end of the duct reaches the bearing area. The other end of the duct either ends in the crankcase to get oilmist there, or ends in the outer end of the crankshaft to be lubricated with grease occasionally. In both cases the efficiency is limited and also dirt easily fills the respective duct so that the lubrication will be decreased or stopped. |
Q:
\DeclareSymbolFont undesired effect
I would like to have math operator names in Sans Serif.
\DeclareSymbolFont{operators}{OT1}{cmss}{m}{n} does the job but has also an effect on Greek capitals.
How do I change font for operator names keeping the rest of the formatting intact?
A:
If your aim is to change the typesetting of operator names to use sans serif type, you should define a new symbol font:
\documentclass{article}
\usepackage{amsmath}
% a new symbol font for names of operators
\DeclareSymbolFont{sfoperators}{OT1}{cmss}{m}{n}
% don't waste a math group
\DeclareSymbolFontAlphabet{\mathsf}{sfoperators}
% tell LaTeX to use sfoperators for names of operators
\makeatletter
\renewcommand{\operator@font}{\mathgroup\symsfoperators}
\makeatother
\DeclareMathOperator{\foo}{foo}
\begin{document}
$\sin\Gamma+\log(x-\Phi)-\foo(y)$
\end{document}
|
Eric Camilli is ready for the 2016 FIA World Rally Championship to get underway, opening a new chapter in his career as he makes the step up to WRC with the M-Sport World Rally Team.
The Frenchman impressed many in 2015 with his WRC2 performances of Team ORECA as he took on the might of the factory Škoda Fabia’s as was given the call up by M-Sport owner Malcolm Wilson over the off-season.
Camilli has been testing the Ford Fiesta RS WRC with co-driver Nicolas Klinger in prepation for the opening Rallye Monte-Carlo next week.
“These tests went very well,” said Camilli. “We were really able to work on various set up and develop a real harmony with our new team. It’s really special to bring these tests to a successful conclusion as official driver, but everything went as we wanted and we are really looking forward to start the season!
“The Rallye Monte-Carlo is an event both atypical, long and complicated due to its extreme weather conditions. The first target will be to learn the Fiesta RS WRC and to progress step by step while realizing a smart race. This is our first world race and we will know the level of the WRC. We will give the best of ourselves, as we always do.”
Monday marks the official start of the season as competitors get three days of recce underway before the shakedown and four days of rally action. |
HIV Experts Create the Roadmap for Providing PrEP to Uninfected Individuals to Reduce the Risk of HIV Infection
WASHINGTON, Aug. 24, 2011 /PRNewswire-USNewswire/ -- To stem the
estimated 2.6 million new HIV infections that occur worldwide each
year, more than 200 representatives from the scientific and
HIV/AIDS communities took an important step in assessing the safety
and public health implications of providing antiretroviral drugs to
uninfected men and women exposed to HIV through sexual contact
– a strategy called pre-exposure prophylaxis, or PrEP.
Assembling August 19 at an open public meeting and interactive
webcast convened by the Forum for Collaborative HIV Research, these
researchers, HIV/AIDS advocates, members of industry and
representatives from National Institutes of Health, the Centers for
Disease Control and Prevention (CDC), Food and Drug Administration
(FDA) and state public health departments applied the findings from
a number of large trials to discuss a roadmap for FDA and CDC to
develop guidance on the safe use of PrEP in otherwise healthy
individuals at high risk of acquiring HIV. Held with the
encouragement of FDA, this meeting has important implications for
medical practice in the U.S. because recent data strongly support
the efficacy of antiretroviral intervention for this purpose.
Although FDA has not yet approved PrEP to reduce HIV acquisition
in uninfected individuals, one form of PrEP recently studied for
use in healthy men or in couples where one partner is HIV positive
–a daily pill containing tenofovir plus emtricitabine
(TDF/FTC) – is FDA-approved for the treatment of HIV
infection. In women, studies have also demonstrated the efficacy of
prophylactic treatment with tenofovir applied as a vaginal gel.
"We now have findings from large studies that support a
conclusion that PrEP is effective in gay and bisexual men, who
represent more than half of new HIV infections in the U.S., and
now, there is evidence that PrEP may reduce HIV infection in
heterosexual men and women, the population hardest hit by HIV
worldwide," said Jur Strobos, M.D., Deputy Director of the Forum.
"We must however, apply these promising data to develop workable
strategies that mitigate risk that may be associated with the
prophylactic use of antiretrovirals. These include both medical and
socio-behavioral risk. We must ensure that people at greatest
risk for acquiring HIV receive a comprehensive package of
prevention services, including regular HIV testing, condom
provision, risk reduction counseling and management of other
sexually transmitted infections. The purpose of our meeting was to
help identify what the components of a complete package should
be."
Among the data reviewed at the meeting were results from the
large-scale, multinational iPrEx trial, which found that a daily
dose of TDF/FTC provided at least 44 percent protection to men and
transgender women who have sex with men (MSM). Among those patients
who were most adherent (used TDF/FTC on 90 percent or more of the
days during the trial), HIV risk was reduced by 73 percent.
Reinforcing these findings, a new CDC trial called the TDF2 study,
along with Partners PrEP conducted by researchers at the University
of Washington, showed that PrEP reduced the risk of HIV in 63
percent of the uninfected heterosexual men and women in the study
population. While unsuccessful in demonstrating efficacy for as yet
unknown reasons, data from the FEMPrEP study presented at the
meeting also confirmed that there were minimal if any safety
concerns associated with daily use of the antiretroviral
drugs. As limited studies, none could assess safety problems
that may arise with broader use – the purpose of this
meeting.
Charting the future use of PrEP as a prevention strategy, the
meeting participants considered how FDA's broad authority to
require Risk Evaluation and Mitigation Strategies (REMS) could be
applied to mitigate the risks associated with the scale-up of
prophylactic use of antiretrovirals. The participants focused
on several medical complications associated with use of the TDF/FTC
combination including a potential impairment in renal function in
some patients with long-term use and an early decrease in bone
mineral density that could stabilize over time.
Due to the need for further study of the long-term side effects
of PrEP, the panel recommended that communication plans be
developed for use by clinicians and patients that stress regular
testing of renal function and the panel agreed that additional REMS
measures, such as a Medication Guide, would be helpful to ensure
adequate monitoring of renal complications – a well-known
phenomenon with the use of tenofovir.
Also assessing available data on how tenofovir impacts bone
mineral density and vitamin D levels, the panel concluded that
additional educational materials are required to address these
medical risks, but that the decision to evaluate bone mineral
density should be left to the healthcare provider on a case-by-case
basis.
Other medical issues included handling of depression, frequent
in this population especially during periods of their lives when
PrEP might be most important; risks to infants of women who wish to
get pregnant on PrEP or who are breast-feeding; and
sexually-transmitted infection – which is not treated or
prevented by PrEP.
Targeted physicians for education should be infectious disease
experts, primary care givers including gynecologists, and
physicians who manage sexually transmitted infections even though
the latter have not previously been involved in longitudinal
care.
At the same time, the panel considered the potential for
patients to develop HIV drug resistance to antiretroviral drugs
used for prevention, noting that drug resistance has been well
documented in HIV/AIDS patients on treatment. Considered by some to
be the thorniest issue associated with the scale-up of PrEP,
resistant HIV is more difficult to treat and frequently requires a
more complicated treatment regimen. Moreover, even though the
development of resistance has not been a concern in ongoing PrEP
clinical studies, the panel members agreed that less frequent
testing for HIV seroconversion that is likely upon scale-up may
exacerbate the problem.
Because FDA can require a restricted distribution plan under
REMS, the panelists debated the use of this system for PrEP,
including requiring physician qualification and registration,
pharmacist distribution limitations, and mandatory patient testing
before a refill can be ordered and dispensed. Although restricted
distribution systems have proven effective in reducing the risks
associated with potent teratogens, like thalidomide, or to assure
the appropriate use of narcotics, panel members concluded that a
restricted distribution plan for PrEP could not be successfully
introduced when the same drug combination is available for
treatment and would impose substantial burdens on healthcare
providers and patients. Further, such a system would impair
access to needed preventative therapy.
The panelists also considered the issue of risk compensation.
Risk compensation would be the engagement in higher risk encounters
or lower use of condoms based on the false belief that these
protections are no longer necessary. While existing clinical trial
data suggests that risk behavior reduces and condom use increases
with PrEP, this was a topic that the panelists agreed should be
addressed in demonstration projects.
The panel members recognized that there are a number of
questions that still need to be answered through post-market and
ongoing studies, including the recommended frequency of HIV
testing; continued vigilance in assessing the development of
resistance on scale-up; and the possibility of risk compensation.
Until these findings are available, however, the panel urged the
development of a communications plan that will emphasize the need
and importance of regular testing, counseling, and, for medical
risks, close monitoring.
With regard to testing patients on PreP therapy for HIV
infection, the panelists expressed caution about requiring
complicated testing that may not be available in public health
clinics. In almost all settings, panel members agreed that routine
rapid enzyme-linked immunoassay testing should be sufficient.
Addressing the costs of testing, payors on the panels provided
assurance that, in covered patients, the cost of recommended
quarterly testing will most likely be reimbursed or compensated,
thus reducing the financial barriers to regular testing and to
PrEP. However, panel members urged the development of
mechanisms that will ensure access to regular testing for PrEP
patients without financial resources.
Next Steps
As FDA considers the findings from this public meeting,
conference participants urged healthcare providers and the public
to await further guidance from the CDC and FDA before considering
using PrEP. However, if providers believe that initiating
PrEP is urgent for a specific patient, CDC recommends following the
cautions and procedures previously published for PrEP use in MSM
(http://www.cdc.gov/mmwr/preview/mmwrhtml/mm6003a1.htm?s_cid=mm6003a1_w).
At the same time, CDC recommends that any healthcare provider
considering PrEP be apprised of the following precautions:
PrEP should only be used among individuals who have been
confirmed to be HIV-negative. Initial and regular HIV testing
is critical for anyone considering using PrEP. All
individuals considering PrEP must also be evaluated for other
health conditions that may impact PrEP use.
PrEP should never be seen as the first line of defense against HIV.
It was only shown to be effective in clinical trials when provided
in combination with regular HIV testing, condoms, and other proven
prevention methods.
Taking PrEP daily is critical. No other dosing regimen was
evaluated in these studies.
PrEP must be obtained and used in close collaboration with health
care providers to ensure regular HIV testing, risk reduction and
adherence counseling, and careful safety monitoring. Anyone
considering using PrEP should speak with his or her doctor.
PrEP has only been shown in clinical trials to reduce HIV infection
among heterosexual men and women and among men who have sex with
men. At this time, there are no data on its benefits or risks
among injection drug users.
Because pregnant and breastfeeding women were excluded from
participation in PrEP trials, further evaluation of available data
will be needed before any recommendations can be made regarding the
use of PrEP for women considering conception or in those who are
breastfeeding infants.
As a next step, the Forum for Collaborative HIV Research will post
the webcast early next week and will publish the proceedings of
this public meeting to advance the regulatory agenda. Once
published, the report will be distributed widely to the Forum's
many constituencies –government, industry, patient advocates,
healthcare providers, foundations, health insurers and academia
–with the goal of advancing research on PrEP and driving
public policy.
About the Forum for Collaborative HIV Research
Now part of the University of California (UC), Berkeley School
of Public Health and based in Washington, D.C., the Forum was
founded in 1997 as the outgrowth of a White House initiative which
called for an ongoing collaboration among stakeholders to address
emerging issues in HIV/AIDS and set the research strategy.
Representing government, industry, patient advocates, healthcare
providers, foundations and academia, the Forum is a public/private
partnership that is guided by an Executive Committee that sets the
research agenda. The Forum organizes roundtables and issues reports
on a range of global HIV/AIDS issues, including treatment-related
toxicities, immune-based therapies, health services research,
co-infections, prevention, and the transference of research results
into care. Forum recommendations have changed how clinical trials
are conducted, accelerated the delivery of new classes of drugs,
heightened awareness of TB/HIV co-infection, and helped to spur
national momentum toward universal testing for HIV. http://www.hivforum.org |
Stainless Steel Fabrication
Stainless Steel Fabrication
We at Metaline manufacture an extensive range of stainless steel products with a comprehensive range of fittings, welding consumable, and flanges. All the stainless products are available in different grades, surface finished and dimensions.
Our stainless steel products are increasingly the material of choice for metal fabrication. We supply stainless steel products that are available in a wide variety of dimensions and grades and our metal fabricators are well trained and provide flawless stainless steel fabricated products. Our company will asssist you in meeting all your stainless steel requirements.
Why choose us
Our professionals are well experienced and trained to handle the metal fabrication jobs. We have the best machineries to meet the requirements of our customers and satisfy them completely. Customer satisfaction and production of flawless products is our prime goal and our engineers and professionals strive to achieve the same. At Metaline you will certainly have all your requirements met which will be beneficial for the growth of your business.
Our engineers will work with you directly in order to fully understand your requirements as well as provide options regarding materials, finishes and other specifics. It’s our goal to establish a long-lasting relationship with all of our customers and we recognize that this initial step is crucial in setting the trajectory for the entire project.
We will take the submitted specification and prepare pre-production drawings/models/blueprints which we will submit to you for approval. Any changes that need to be performed can be done so at this stage and we’ll ensure that they are made in accordance with the original specifications.
Optionally, we offer delivery services as well for projects of any size and can ship your fabricated materials via ground, sea or air to any major worldwide destination.
General Information
Stainless Steel products are available in plate, sheet, and various other shapes. This can be bought as a Mill finish or can be polished to bright mirror finish depending on the application. The steel is manufactured with a low carbon content that makes it an ideal corrosion resistant product.
The two main types of stainless steel usually used are the 400 series and 300 series. The 300 series is mainly used in different applications like marine aerospace and any other areas which is directly exposed to different elements throughout the year. The 400 series stainless steel has properties which are corrosion resistant and is mainly ideal for interior decorative, architectural and appliances.
Commissioning and Project Closeout
One of the most critical stages of the construction process is the commissioning and closeout of the project.
Service Brochure
An overview of our construction services from Construction Management, Design-build, General Contracting to Small Jobs and Service Work
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About Company
Metaline is able to handle different projects with varying complexity. We offer complete Metal Fabrication from laser cutting, bending, engineering and design, to fabricated finished parts.
Metaline is one of the most diverse and largest metal fabricators in Toronto. |
Quick Android project with on-going work by sambero
The exact nature or technical specification for the project will be supplied to the winning bidder, however to ensure you’re able to adequately bid for this project here is a brief overview (written from the perspective of a developer, so I know what is being asked)… (Budget: $250-$750 USD, Jobs: Android, Mobile Phone) |
88 N.J. 16 (1981)
438 A.2d 323
HUDSON CITY SAVINGS BANK, A BANKING CORPORATION OF THE STATE OF NEW JERSEY, PLAINTIFF-RESPONDENT,
v.
HAMPTON GARDENS LTD., A LIMITED PARTNERSHIP, DEFENDANT-APPELLANT, AND GROSSMAN, BROWN, WEINBERG & LAWSON, A PROFESSIONAL CORPORATION AS AGENT FOR THE LIMITED PARTNERS OF PLYMOUTH ASSOCIATES, LTD.; SHORE NATIONAL BANK; HOWELL WOODWORK, INC.; THE LEVEL LINE, INC. AND STATE OF NEW JERSEY, DEFENDANTS. GROSSMAN, BROWN, WEINBERG & LAWSON, A PROFESSIONAL CORPORATION, AS AGENT FOR THE LIMITED PARTNERS OF PLYMOUTH ASSOCIATES, LTD., PLAINTIFF,
v.
HAMPTON GARDENS, LTD., A LIMITED PARTNERSHIP, DEFENDANT.
The Supreme Court of New Jersey.
Argued October 5, 1981.
Decided December 17, 1981.
*18 Leon J. Sokol argued the cause for appellant (Greenstone & Sokol, attorneys).
James G. Lepis argued the cause for respondent (Lepis, Lepis & Curley and Dieffenbach, Witt & Birchby, attorneys).
The opinion of the Court was delivered by SCHREIBER, J.
We are called upon to decide whether a mortgagee may recover on a supersedeas bond the interest accruing during an unsuccessful appeal by the mortgagor of a foreclosure judgment. This question is to be resolved under the circumstances of this case, where the fair market value of the property purchased by the mortgagee for a nominal amount at a sheriff's sale is probably greater than the mortgage debt owed to the mortgagee and the surplus would have satisfied, in whole or in part, the interest accrued subsequent to the foreclosure judgment.
The facts are not in dispute. In 1974, the defendant Hampton Gardens, Ltd. purchased an 80 unit, two-story garden apartment complex located in Toms River, New Jersey, either assuming or subject to an existing first mortgage owned by the plaintiff Hudson City Savings Bank. The plaintiff declared a default in July 1976, went into possession and started foreclosure proceedings. The Superior Court, Chancery Division, entered a foreclosure judgment in favor of the plaintiff on October 11, 1977. The judgment recited the mortgage indebtedness of $983,161.19. It ordered that the mortgaged property be sold and the proceeds be applied to satisfy the debt, costs, counsel fees and interest *19 from September 15, 1977. Any surplus from the sale was to be deposited with the clerk of the court.
The defendant appealed and applied for a stay of the sale pending appeal. The trial court granted the stay on condition that defendant post a supersedeas bond for $75,000 "in conformity with the Court Rules [R.2:9-5 and R.2:9-6] in such case made and provided to secure interest, appellant [sic] costs, counsel fees, and against any losses plaintiff may sustain from its possession of the subject premises." The defendant thereupon posted a bond for $75,000.[1]
The Appellate Division affirmed the judgment of foreclosure and the defendant's petition for certification was denied. 81 N.J. 41 (1979). As the sole bidder at the sheriff's sale on April 10, 1979, plaintiff purchased the property for the nominal amount of $100. After the sale, plaintiff moved to recover the $75,000 that had been posted as a supersedeas bond on the grounds that the taxed costs on appeal were $865 and that accrued interest on the amount of the original decree at the legal rate of 8% from the date of judgment in 1977 to May 8, 1979[2] was $129,293.11. The defendant argued that plaintiff was not entitled to recover on the bond because the value of the property, which the bank had allegedly contracted to sell for $1,125,000, exceeded the sum of the mortgage debt plus accumulated *20 interest. Therefore, the plaintiff had suffered no loss of interest as a result of the delay engendered by the appeal.
The trial court granted plaintiff's motion to collect $75,000 on the bond because the calculated interest exceeded that amount. The defendant appealed and the Appellate Division affirmed in an unreported opinion. We granted plaintiff's petition for certification, 85 N.J. 486 (1981), and now reverse.
The supersedeas bond serves as a device to protect a party who has been successful at trial but has been forestalled from proceeding during an appeal. The bond's necessary terms and conditions are provided by the Rules Governing the Courts. Two rules apply to a bond posted pending an appeal of a foreclosure judgment. R.2:9-5 provides in part that a judgment adjudicating "the rights or liabilities of parties in respect of property which is the subject of an appeal or certification proceedings shall be stayed only upon the posting of a bond pursuant to R.2:9-6 or a cash deposit" unless the court otherwise orders on good cause shown. R.2:9-6(a) states that "[w]hen the judgment determines the disposition of the property in controversy ... the amount of the supersedeas bond shall be fixed at such sum only as will secure the damages recovered for the use and detention of the property, trial and appellate costs, and interest."
These rules have remained substantially intact since their initial adoption in 1948. See Rules 1:2-11 and 1:2-13 (effective September 15, 1948); R.R. 1:4-7 and R.R. 1:4-8 (1953). Their previous history is instructive in determining their purpose.
The original rule was modeled after the then existing Fed.R. Civ.P. 73(d), which codified the practice that had been fashioned by prior case law. 9 J. Moore, Federal Practice ¶ 208.06(2) (2d ed. 1980). That case law was superimposed upon the federal statutory framework governing appeals. We therefore look to federal statutory case law for guidance in resolving the issue presented.
*21 The twenty-second and twenty-third sections of the Judiciary Act of 1789, 1 Stat. 84-85, provided that when a judge signed a writ of error permitting an appeal there was to be fixed "good and sufficient security" for "all damages and costs," and upon affirmance the court in its discretion would determine just damages for the delay and single or double costs.[3] The Supreme Court subsequently defined what constituted good and sufficient security. In Catlett v. Brodie, 22 U.S. (9 Wheat.) 553, 554, 6 L.Ed. 158, 159 (1824), the Court held that the security should be sufficient to cover the entire amount of a money judgment. The Court wrote: "Whatever losses [the respondent] may sustain by the judgment's not being satisfied and paid, after the affirmance, these are the damages which he has sustained, and for which the bond ought to give good and sufficient security." Thereafter, the Supreme Court in 1867 adopted Rule 32 which provided that when the judgment determined the disposition of the property in controversy, as in replevin and in suits on mortgages, then the security was to be fixed "in an amount sufficient to secure the sum recovered for the use or detention of the property, and the costs of the suit, and `just damages for delay', and costs and interest on the appeal." 73 U.S. (6 Wall.) v (1867).[4]
*22 Interest on the appeal did not refer to the accumulation of interest on the indebtedness, but to interest on the judgment. The Court explained in Jerome v. McCarter, 88 U.S. (21 Wall.) 17, 32, 22 L.Ed. 515, 517 (1874), that the security was to provide "indemnity for loss" of interest "consequent upon the appeal, not for the payment of the interest." It has been the established practice in New Jersey that after foreclosure, interest runs at the legal rate and not at the rate stated in the bond or mortgage. Hoover Steel Ball Co. v. Schaefer Ball Bearing Co., 90 N.J. Eq. 515 (Ch. 1919); Deshler v. Holmes, 44 N.J. Eq. 581 (E. & A. 1888). See also Wilson v. Marsh, 13 N.J. Eq. 289 (Ch. 1861).
The Supreme Court expressed the view that the underlying purpose of the supersedeas bond was to protect the mortgagee from loss due to an appeal. In Jerome v. McCarter, a second mortgagee obtained a judgment of foreclosure. The amount due was in excess of $1,000,000. However, the property was subject to a prior encumbrance of more than $1,500,000. The second mortgagee claimed that a $10,000 supersedeas bond fixed by the district court was inadequate. The Supreme Court rejected this contention. The Court stated that the mortgagee could not suffer a loss based on the circumstances existing when the amount of the bond was fixed. If the value of the property was less than the amount due under the first mortgage as the mortgagee claimed, there would be no equity to be applied toward the second mortgage. Similarly, if the property depreciated in value pending the appeal, the mortgagee would suffer no loss. The mortgagee contended it would lose the opportunity to bid the property in at a reduced price and speculate upon its rise. The Court refused to recognize such a loss as legitimate *23 damages for delay. On the other hand, if, as the mortgagor claimed, the mortgage security would be sufficient to pay all the indebtedness with accruing interest when the appeal was decided, the mortgagee also would not incur a loss. The Court noted, however, that the amount of the bond could be modified if conditions changed during the appeal and the security obtained was no longer adequate.
The concept that the unsuccessful mortgagor may be liable on the supersedeas bond only for the loss occasioned by the delay is illustrated in Board of Supervisors v. Kennicott, 103 U.S. 554, 26 L.Ed. 486 (1881). A $40,000 supersedeas bond was posted in connection with a mortgagor's appeal of a foreclosure action. After the judgment was affirmed, the successful mortgagee sued on the bond asserting damages including interest accruing during the appeal on the judgment, which exceeded the amount of the bond. Judgment on the bond was reversed because there was no showing that the mortgagee had suffered a loss. Chief Justice Waite pointed out that "[s]o far as appears, the lands may have increased in value to an amount larger than the accumulation of interest, and the taxes may have been paid." 103 U.S. at 558, 26 L.Ed. at 488.
The federal precedents establish that the purpose of the supersedeas bond posted in foreclosure proceedings is to indemnify the mortgagee from loss in the respect stated in the supersedeas bond due to the delay occasioned by the appeal. Our rules governing supersedeas bonds serve the same purpose. If the mortgagee does not suffer a loss by reason of the delay, then no damages may be recovered under the bond.
It is important to recognize, however, that the losses protected against are only those specified. R.2:9-6(a) limits the requirements of the bond to a sum that will secure only against losses caused by use and detention of property, trial and appellate costs and loss of interest. The draft of the first proposed rule, R.1:2-8, listed these elements and damages for delay. Rules Governing The Courts of New Jersey (Tent. Draft, March 27, *24 1948). This was identical to Fed.R.Civ.P. 73(d). However, when the rule was adopted, the Court eliminated damages for delay. R.1:2-13(a) (effective September 15, 1948). In limiting the items for which an appellant might be responsible, the Court was balancing the protection to be afforded the respondent and the desirability of encouraging an appellant to prosecute a meritorious appeal. The broader the protection afforded in the bond, the greater the likelihood of a chilling effect on the litigant who desired to appeal. The rule reflects that accommodation of competing interests.[5]
The trial court in the instant case invoked R.2:9-6(a) which requires, as a condition of staying the sale pending appeal, that the mortgagor post a supersedeas bond to secure interest, costs, counsel fees and losses due to possession of the premises. If the mortgagee did not lose the interest to which it was lawfully entitled subsequent to the judgment of foreclosure, then it should not be allowed to recover that interest from the mortgagor. The purpose of the bond is to secure the mortgagee against harm. If no harm has been done, no interest is due.
Consequently, if as a result of the price received from the foreclosure sale to a bona fide third person, the mortgagee receives an amount sufficient to satisfy the indebtedness (principal and interest) due plus the interest accrued subsequent to the foreclosure judgment while the appeal was pending, then the mortgagee should not prevail on a supersedeas bond conditioned on payment of the same interest. See Gruber v. Ewbanks, 199 N.C. 335, 154 S.E. 318 (1930) (if, notwithstanding the stay, the creditor collects the debt, interest and costs by the sale of the *25 property after dissolution of the stay, the creditor sustains no damages by reason of the stay and may not recover on the supersedeas bond). Contra, Fidelity and Deposit Co. of Maryland v. Atlantic National Bank of Jacksonville, 234 So.2d 736 (Fla.App.), cert. den. 238 So.2d 111 (Fla.Sup.Ct. 1970); Jacksonville v. Brentwood Golf Course, Inc., 338 So.2d 1105 (Fla.App. 1976). If the sale proceeds are insufficient to cover that interest, then the action on the supersedeas bond may be maintained. A partial recovery of the interest from the sale proceeds would permit a proportionately decreased recovery on the bond.
In this case the mortgagee purchased the property for $100, though it is claimed that its value was at least $1,125,000. The amount due, $983,161 plus interest of $129,293 or a total of $1,112,454, was conceivably less than the value of the property obtained by the mortgagee.[6] Under these circumstances, the parties should be permitted to prove fair market value and net proceeds from the hypothetical sale.[7] The trial court may then determine what loss in interest, if any, the mortgagee suffered. In making this determination, the trial court should also factor in the other items covered by the bond: appellate costs, counsel fees and losses the mortgagee sustained from its possession of the premises during the appeal.
The judgment is reversed and the cause remanded for further proceedings in accordance with this opinion.
For reversal and remandment Chief Justice WILENTZ and Justices PASHMAN, CLIFFORD, SCHREIBER, HANDLER, POLLOCK and O'HERN 7.
For affirmance None.
NOTES
[1] The bond that was filed did not conform to R.2:9-6(a). The bond was in the form of security for a money judgment. It provided for payment of a $75,000 judgment together with interest and costs. The rule requires that in a mortgage foreclosure the bond be fixed in an amount to secure damages as may be recovered for use and detention of the property, trial and appellate costs, and interest. The parties have not raised any issue with respect to the form of the bond and we have considered the bond as having not entirely departed from the rule, insofar as interest during the appeal, costs, and use and detention of the property pending appeal are concerned. See Omaha Hotel Co. v. Kountze, 107 U.S. 378, 2 S.Ct. 911, 27 L.Ed. 609 (1883); Martin v. Clarke, 105 F.2d 685 (7th Cir.1939).
[2] The bidder was permitted to make payment four weeks after the sale.
[3] For English predecessors see statute of 3 James I, c.i; 13 Car. 2, c.2; 16 & 17 Car. 2, c. 8. See also L.Elmer, A Digest of the Laws of New Jersey, 159-160 (1838).
For the subsequent history of § 22 of the 1789 Judiciary Act see 1 Stat. 404 (1794), 12 Stat. 657 (1863), 15 Stat. 226 (1868) and 28 U.S.C. § 869 (1946). For the subsequent history of § 23 of the 1789 Act see 17 Stat. 198 (1872) and 28 U.S.C. § 874 (1946). Both 28 U.S.C. § 869 and § 874 were repealed by the 1948 Judicial Code, 28 U.S.C. § 1 et seq. (1948).
[4] Rule 32 has been revised and renumbered by the Court from time to time. See Sup.Ct.R. 29, 108 U.S. 573 (1884); Sup.Ct.R. 33(2), 266 U.S. 653, 678 (1925); Sup.Ct.R. 36(2), 275 U.S. 595, 621 (1928); Sup.Ct.R. 18(1), 346 U.S. 943, 966 (1954), and current Sup.Ct.R. 44.2 (eff. June 30, 1980). The provisions of Sup.Ct.R. 44.2 are substantially similar to former Fed.R.Civ.P. 73(d), which was abrogated in 1968 when the Supreme Court promulgated the Federal Rules of Appellate Practice. 43 F.R.D. 61, 115-116 (1967). Although its provisions were not transferred to the Appellate Rules, the substance of Fed.R.Civ.P. 73(d) retains vitality inasmuch as it had simply codified judicial practice. Federal Prescription Service, Inc. v. American Pharmaceutical Ass'n, 636 F.2d 755, 759 (D.C. Cir.1980); Poplar Grove Planting and Refining Co., Inc. v. Bache Halsey Stuart, Inc., 600 F.2d 1189 (5th Cir.1979).
[5] Where the subject matter may be adversely affected due to an appeal, the courts may relax the rules and expedite the appeal. See, e.g., DeSimone v. Greater Englewood Housing Corp. No. 1, 56 N.J. 428, 434 (1970), in which this Court granted direct certification of one cause, directed a losing party to appeal, if it so desired, within five days on a second related matter, set a briefing schedule and ordered oral argument, all of which took place in less than thirty days.
[6] The property was conveyed in April 1981 for $1,280,000. The plaintiff bank had appraised the property in 1972 for $1,135,000.
[7] It may be that in such proceedings determination of a deficiency on a bond or note can also be ascertained. See N.J.S.A. 2A:50-2 and -3 (Supp. 1980). We are referring this matter to the Supreme Court's Civil Practice Committee for its consideration.
|
1. Introduction {#sec1}
===============
Radiotherapy is commonly employed as part of a management of a wide variety of malignancies. To achieve local control, either alone or in combination with other modalities chemotherapy and/or surgery is used. It is reported almost half of all cancer patients receive radiotherapy as part of their treatment \[[@bib1]\]. Hence, irradiation of non-cancerous normal tissues during therapeutic radiation results in side effects that include self-limited acute toxicities, mild chronic symptoms, and aspects of organ dys-function \[[@bib2]\]. Improvement of chemical modifiers that can control radiation induced injury to normal tissue is getting increased attention and becoming important for reducing toxicities associated with therapeutic radiation \[[@bib3]\]. The mechanism by which damage to normal tissues is prevented is called radio-protection and the compounds involved are known as radio-protectors \[[@bib4]\]. These are administered either prior to or shortly after radiation exposure to alter response of normal tissues towards radiation and is considered an essential component of radiotherapy \[[@bib5]\].
Although several applications of azo compounds related to biological systems are reported \[[@bib6], [@bib7], [@bib8], [@bib9], [@bib10], [@bib11], [@bib12]\] not many have investigated them for their radio-protective properties. Although there is a study showing radio-protection by an azo compound, details regarding the process were not shown \[[@bib13]\]. The study was however interesting since not only did it show another application of an azo compound, it highlighted an useful biological attribute that might be exploited in future \[[@bib13]\]. In this study aspects related to radio-protection were investigated using 2-(2-hydroxyphenylazo)-indole-3^∕^-acetic acid (HPIA), an azo compound, whose preparation and characterization was reported earlier \[[@bib7]\]. Since not much is known on radio-protection by azo compounds, and because the compound we worked with (HPIA) was not much toxic to normal cells as realized from the IC~50~ values on HEK 293T cells in a previous study \[[@bib7]\], unlike most reports on azo compounds \[[@bib14]\], we performed a detailed study with it to identify the mechanism by which compounds with azo bonds might provide protection against toxic and injurious effects of ionizing radiation \[[@bib15]\].
The major attribute that a compound must possess in order to provide protection against damage caused by ionizing radiation is an ability to quench free radicals generated as a consequence of the radiolysis of water \[[@bib16], [@bib17], [@bib18]\]. It is now established that interaction of ionizing radiation e.g. γ rays with water forms radical and molecular products \[H·, ·OH, e~aq~¯, H~2~, H~2~O~2~ and H~3~O^+^\] \[[@bib16], [@bib17], [@bib18], [@bib19]\]. In N~2~O saturated medium, e~aq~¯ is converted to an equivalent amount of ·OH, according to [Eq. (1)](#fd1){ref-type="disp-formula"} \[[@bib20]\].
The products interact with a biological target modifying it in a manner having severe consequences in biology. Obviously then, preventing this from happening becomes important particularly with regard to protecting normal cells during radiotherapy \[[@bib2], [@bib3], [@bib4], [@bib5]\]. Thymine and calf thymus DNA were used as biological targets in model studies where they were irradiated in the absence and presence of HPIA by ^60^Co γ-rays in de-aerated (Ar saturated) and N~2~O saturated medium. To establish the mechanism by which radio-protection by HPIA is possible, radical scavenging experiments were designed and performed in the presence of the stable radical DPPH \[[@bib21], [@bib22], [@bib23]\]. To further establish if radio-protection of a biological target was due to scavenging of free radicals within cells, 2‛,7‛-dichlorofluorescin diacetate (DCF-DA) ROS depletion assay was performed on WI-38 lung fibroblast cells \[[@bib24], [@bib25], [@bib26], [@bib27], [@bib28]\]. It is worth mentioning here that we have been exploring biological attributes of complex formation of azo compounds for quite some time now and most of that work has dealt with the aspect of modifying an azo compound through complex formation leading to decreased formation of substances responsible for toxicity of azo compounds \[[@bib7], [@bib9], [@bib10], [@bib11]\]. Some of these studies also show that such modified azo compounds (complexed to metal ions) were more efficient in killing cancer cells than normal cells. This study shows azo compounds can be useful in protecting normal cells during radiotherapy as well.
2. Experimental {#sec2}
===============
2.1. Materials {#sec2.1}
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2-amino phenol and indole-3-acetic acid (IAA) \[E. Merck, India\] were used for preparing 2-(2-hydroxyphenylazo)-indole-3^∕^-acetic acid (HPIA) by diazo-coupling and re-crystallized from an ethanol-water mixture \[[@bib7]\]. Sodium nitrate of analytical grade (E. Merck, India) was used to maintain the ionic strength of the medium. Stock solutions of HPIA were either prepared in ethanol or in DMSO. All solvents were procured either from E. Merck, India or E. Merck, Germany. Thymine (CAS No. 65-71-4) and calf thymus DNA (CAS No. 73049-39-5) were purchased from Sisco Research Laboratories (SRL), India. Sodium dihydrogen phosphate and disodium hydrogen phosphate \[E. Merck, India\] were used to prepare phosphate buffer solutions. Calf thymus DNA was dissolved in triple distilled water containing NaCl, KCl and MgCl~2~ (E. Merck, India). Absorbance of DNA solutions was noted at 260 and 280 nm respectively to calculate A~260~/A~280~. Ratio of absorbance at 260 and 280 nm provides an estimate of the purity of DNA. The ratio (1.8 \< A~260~/A~280~ \> 1.9) obtained indicates DNA was sufficiently free of protein. Concentration was measured in terms of nucleotide taking molar extinction coefficient at 260 nm to be 6,600 M^−1^cm^−1^. Ethidium bromide (EB) (CAS No. 1239-45-8) purchased from SRL, India was used as the fluorescent probe to determine the extent of damage caused to double stranded calf thymus DNA. DPPH used for radical scavenging studies was purchased from SRL, India. Pure N~2~O and Ar gases used for purging experimental solutions prior to irradiation with ^60^Co γ-rays were purchased from Indian Refrigeration Stores, Kolkata, India.
2.2. Equipments {#sec2.2}
---------------
Absorption spectra were recorded on JASCO V-630 Spectrophotometer, Japan. Fluorescence was measured on JOBIN YVON Fluoromax spectrophotometer. A pair of 10 × 10 mm quartz cuvette was used for absorption and fluorescence experiments. A pH meter \[Equiptronix, EQ-610\] was used for recording pH. During radiation chemical experiments, solutions were subjected to γ-irradiation from a ^60^Co source having dose rate of 3.1 KGy/min (monitored by Fricke dosimetry). Loss of thymine was determined by HPLC (Shimadzu Corporation, Japan). A C~18~ column (Phenomenex) was used as the stationary phase. Change was followed at 254 nm using 5% methanol and 95% water as mobile phase. Flow rate was 1 ml per minute. DPPH radical scavenging study was done with the help of UV-Vis and EPR spectroscopy. EPR measurements were made on Jeol JES-FA 200 ESR spectrometer equipped with a Jeol microwave bridge. Spectroscopic parameters were 9.44 GHz (frequency), 100 mT (field sweep), 0.998 mW (microwave power) and modulation amplitude 3000 mT. EPR of samples were recorded in Jeol Quartz pyrex EPR tube no. 193 5D.
2.3. Methods {#sec2.3}
------------
### 2.3.1. Radiation chemical experiments {#sec2.3.1}
Aqueous solutions of thymine with or without HPIA were prepared. Concentration of thymine in an experimental solution was 10^−4^ M while that of HPIA was 10^−5^M. Prior to irradiation aqueous solutions of samples were saturated with Ar or N~2~O by purging them with the gases for at least 30 min. Following irradiation of thymine at different dose in the absence or presence of HPIA, solutions were analyzed by HPLC. Concentration of radiolysed thymine was calculated from the peak area of the HPLC chromatograms.
Calf thymus DNA (50 μM), in phosphate buffer, was irradiated with ^60^Co γ-rays at different dose (8 Gy, 12 Gy, 16 Gy, 20 Gy, 24 Gy). EB was added to aliquots of such irradiated DNA samples and fluorescence was recorded in the range 540 nm--640 nm following an excitation at 510 nm. Emission maxima of the EB-DNA adduct was obtained at 590--600 nm and served as a measure of the DNA remaining intact \[[@bib29], [@bib30], [@bib31]\]. Since reports suggest fluorescence intensity of the EB-DNA adduct is dependent on the concentration of EB, hence for maximum binding/saturation, concentration of EB was maintained at 800 μM (\~16 folds higher than the concentration of DNA in the experiment).
### 2.3.2. Radical scavenging experiments {#sec2.3.2}
#### 2.3.2.1. DPPH radical scavenging activity by electronic absorption spectroscopy {#sec2.3.2.1}
DPPH radical scavenging activity of HPIA was measured using UV-Vis spectroscopy. Owing to the presence of an odd electron, DPPH shows a strong absorption band at 517 nm. This gradually decreases if the odd electron pairs up in the presence of a radical scavenger, which was followed during the course of the experiment \[[@bib21], [@bib22], [@bib23]\]. There are not many studies that indicate interaction of an azo compound with DPPH, not to mention any that aims to establish an application where an azo compound by virtue of its attribute of quenching DPPH might become biologically useful \[[@bib32]\]. For our experiments, strength of the stock solution of DPPH being 6 mM, amount of DPPH taken for each experiment was 10 μL to arrive at a final concentration of 60 μM. The total volume of an experimental solution was 1mL (1000 μL). A reduced volume cuvette was used for the purpose. HPIA was taken from a stock of 3 mM in ethanol and appropriate amounts were added to each experimental solution to obtain final concentrations in the range 0--30 μM.
#### 2.3.2.2. DPPH radical scavenging by EPR spectroscopy {#sec2.3.2.2}
Electron paramagnetic resonance (EPR) measurements were performed at room temperature (298 K) to check the stability of a freshly prepared solution of DPPH in methanol \[[@bib33], [@bib34], [@bib35], [@bib36]\] for 30 min. No significant loss in signal was detected. To a 6 mM DPPH solution, different concentrations (0--5 mM) of HPIA were added. Solutions were mixed thoroughly and EPR signals were recorded 2 min after mixing HPIA and DPPH under identical conditions. DPPH denotes the radical DPPH•.
#### 2.3.2.3. Estimation of ROS by the DCFDA assay {#sec2.3.2.3}
The cell permeant reagent DCF-DA, a fluorogenic dye measures the activity of hydroxyl, peroxyl and other reactive oxygen species (ROS) within a cell \[[@bib24], [@bib25], [@bib26], [@bib27], [@bib28]\]. After diffusion in to the cell, DCF-DA gets de-acetylated by cellular esterases to a non-fluorescent form, later oxidized by ROS to 2‛,7‛--dichlorofluorescein (DCF), another highly fluorescent compound showing green fluorescence. Fluorescence due to DCF was detected using a fluorescence spectrophotometer (Hitachi, Japan). Excitation was done at 504 nm and emission measured at 529 nm. Stock solutions of DCF-DA (10 mM) were prepared in methanol and diluted further with culture medium to 100 μM. Cells were then treated with different concentrations of HPIA (0 μM, 1 μM, 5 μM, 10 μM) and allowed to stand for 30 min. ROS was induced by irradiation of cells with the help of ^60^Co γ rays. Cells were washed with ice cold Hanks balanced salt solution (HBSS) and incubated with 100 μM DCF-DA for 30 min at 37 °C. Subsequently, cells were lysed with alkali and fluorescence was recorded.
3. Results and discussion {#sec3}
=========================
3.1. Radiation chemical experiments on thymine {#sec3.1}
----------------------------------------------
2 × 10^−4^ M thymine was used in experiments to enable the application of high dose. This was necessary to ascertain the changes caused on thymine accurately when its aqueous solution was irradiated with ^60^Co γ rays. Since high concentrations were used, response obtained for a damage of the target can be said to be devoid of error that might occur for low concentrations.
From the extent of damage caused to thymine under Ar and N~2~O saturated conditions it was realized that HPIA provides significant protection to thymine compared to situations when it was subjected to irradiation in absence of HPIA. In fact, radiation-induced damage of thymine decreased enormously showing almost no change in presence of HPIA both in Ar and N~2~O saturated conditions. [Figure 1](#fig1){ref-type="fig"}A is the HPLC profile for the degradation of thymine with dose in the absence of HPIA in an Ar saturated medium while [Figure 1](#fig1){ref-type="fig"}B was obtained in presence of HPIA under exactly identical conditions. Degradation of thymine due to γ rays was also followed under N~2~O saturated conditions at different dose. [Figure 2](#fig2){ref-type="fig"} is a representative HPLC profile for thymine under different conditions and helps us to realize that HPIA provides significant protection to thymine. HPLC profiles of [Figure 2](#fig2){ref-type="fig"}A (in absence of radiation) & [Figure 2](#fig2){ref-type="fig"}C (with radiation under similar conditions but in presence of HPIA) are similar while that of [Figure 2](#fig2){ref-type="fig"}B (with radiation under similar conditions but in absence of HPIA) is different.Figure 1HPLC chromatograms recorded at 254 nm when 1 × 10^−4^ mol dm^−3^ thymine solution was irradiated with ^60^Co γ rays at different dose in absence (A) and presence of 1 × 10^−5^ mol dm^−3^ HPIA (B) under Ar saturated conditions.Figure 1Figure 2HPLC chromatograms of 1 × 10^−4^ mol dm^−3^ thymine recorded at 254 nm when subjected to (A) no radiation; (B) radiation of 200 Gy in absence of HPIA and (C) radiation of 200 Gy in presence of 1 × 10^−5^ mol dm^−3^ HPIA. Radiation provided under N~2~O saturated conditions.Figure 2
G (-thymine) was calculated for Ar and N~2~O saturated conditions ([Table 1](#tbl1){ref-type="table"}). Extent of degradation of thymine following irradiation in the absence and presence of HPIA in a N~2~O saturated medium was analyzed using HPLC ([Figure 3](#fig3){ref-type="fig"}). [Table 1](#tbl1){ref-type="table"} clearly indicates HPIA is a radio-protector since it prevents degradation of thymine due to γ irradiation.Table 1G (-thymine) under different experimental conditions.Table 1AdditiveG (-thymine)Ar saturatedN~2~O saturated―2.173.87HPIA0.060.12Figure 3Amount of thymine remaining when subjected to γ-irradiation in the absence (●) and presence of HPIA (■) under N~2~O saturated conditions.Figure 3
Many possibilities exist how this might be possible. HPIA could be interacting with the products of the radiolysis of water (e^-^~aq~, ·OH and ·H) for an Ar saturated medium or with ·OH for an N~2~O saturated medium, thus preventing them from interacting with thymine. It could interact with the radical products formed on thymine, following an initial attack on it by the products of the radiolysis of water and then reverse changes that might have occurred. It could be a combination of both these processes. Hence, to identify the mechanism by which HPIA prevents radiation-induced damage of thymine we decided to perform experiments with DPPH making use of a scavenging assay \[[@bib33], [@bib34], [@bib35], [@bib36], [@bib37]\].
3.2. Radiation chemical experiments on calf thymus DNA {#sec3.2}
------------------------------------------------------
To realize the consequences of interaction of γ-radiation with DNA in the absence and presence of HPIA, experiments were performed in N~2~O saturated medium where formation of ·OH is maximum \[[@bib20]\]. It is well established ·OH is the main species responsible for modification of double stranded DNA, hence experiments related to DNA were performed only in N~2~O saturated medium \[[@bib38], [@bib39], [@bib40]\]. [Figure 4](#fig4){ref-type="fig"}A, shows a gradual decrease in fluorescence of calf thymus DNA treated with EB following irradiation with ^60^Co γ rays in the absence of HPIA. However, when DNA from the same stock was irradiated at different dose in presence of HPIA there was hardly any decrease in fluorescence following treatment with EB ([Figure 4](#fig4){ref-type="fig"}B) clearly indicating HPIA provides protection to DNA.Figure 4Fluorescence spectra of calf thymus DNA saturated with EB following irradiation by ^60^Co γ rays at different dose in the absence (A) and presence (B) of HPIA. \[Calf thymus DNA\] = 50 μM, \[HPIA\] = 10 μM, \[EB\] = 800 μM.Figure 4
[Figure 5](#fig5){ref-type="fig"} shows the dose-effect curves for the modification of double-stranded calf thymus DNA irradiated in the absence and presence of HPIA. Plots were exponential with dose. D~37~ was calculated from initial slopes of each plot. Percentage loss of double-stranded DNA due to irradiation was calculated using \[DNA\]~0~/D~37~. \[DNA\]~0~ indicates initial concentration of calf thymus DNA used for irradiation. The value \[DNA\]~0~/D~37~ is independent of DNA concentration in the base pair region in which our investigations were performed. Following irradiation in the presence of HPIA, double stranded DNA remained almost unaffected as realized from the dose-effect curve in [Figure 5](#fig5){ref-type="fig"}. Fluorescence intensity of DNA bound EB ([Figure 4](#fig4){ref-type="fig"}B) for all radiation dose, not only remained constant with respect to the control, rather at each individual dose fluorescence intensity of the DNA-EB adduct was higher than when DNA was irradiated in the absence of HPIA, indicating that at all individual irradiation, HPIA provides almost complete protection to DNA against radiation-induced damage. If this wasn\'t the case, at least at higher dose there should have been greater decrease in fluorescence of the DNA-EB adducts.Figure 5Amount of double stranded calf thymus DNA remaining following irradiation of a 50 μM solution in the absence (●) and presence of HPIA = 10 μM (o).Figure 5
Interaction of radiation with water produces reactive free radicals \[·OH, ·H., e~aq~^−^, H~2~O~2~\] that are toxic if allowed to interact with several macromolecules in a biological system. These radicals affect the three dimensional structure of bio-molecules preventing them from performing their assigned functions. Hence, irradiation that is not under control initiates the breakdown of several important bio-molecules, a phenomenon commonly referred to as adverse effects of ionizing radiation. Using HPIA, this study made an effort to quench free radicals produced by irradiation, before they could cause undesired changes on a biological target, which extrapolated, might help to decrease toxic side effects associated with any ionizing radiation during radiotherapy,. There are studies on flavonoids procured from different sources and on tea extracts (catechins) etc. in this regard but not with an azo compound \[[@bib39], [@bib40], [@bib41], [@bib42]\]. Since azo derivatives are already in use for different purpose in the pharmaceutical industry, this property of scavenging radicals generated during radiolysis of water could be another application of such compounds \[[@bib39], [@bib40], [@bib41], [@bib42]\]; to provide protection to cells against radiation. The findings mentioned above for HPIA are important since experimental evidence suggests it is able to protect double stranded calf thymus DNA from radiation-induced damage. Hence, findings of this study may be used to develop molecules that can provide protection against radiation-induced damage.
It is also important to realize how a molecule like HPIA could be active as a radio-protective agent. To be a good radio-protector, a molecule should scavenge free radicals generated in solution when exposed to $\gamma$-radiation. This is possible either through direct scavenging of radicals formed in solution or by reversing a damage caused to a target. To establish the mechanism operative in case of HPIA that prevents radiation-induced damage of thymine and modification of calf thymus DNA, a study based on free radical reactions became necessary.
3.3. Mechanism of protection of radiation-induced DNA damage {#sec3.3}
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As mentioned earlier, radiation-induced damage of DNA is a consequence of the radiolysis of water that generates ROS, most of which are free radicals. Scavenging them is important as they are harmful. It is therefore important to either remove them as soon as they are formed or render them inactive so that they cannot interact with possible targets. Since HPIA provides protection to DNA from radiation-induced damage, it was necessary to study different aspects pertaining to radical scavenging and suggest a mechanism by which HPIA prevents radiation induced damage. Radical scavenging activity of HPIA was examined by techniques like DPPH scavenging assay that was followed by UV-Vis spectroscopy, EPR spectroscopy and DCF-DA assay performed on WI 38 lung fibroblast cells.
### 3.3.1. Assessment of scavenging of DPPH by UV-Vis spectroscopy {#sec3.3.1}
Free radical scavenging of HPIA was investigated by allowing it to interact with DPPH, a stable free radical making use of electronic spectroscopy \[[@bib34], [@bib35], [@bib36]\]. Radical scavenging was followed at 517 nm for different concentrations of HPIA. The ability of HPIA to scavenge DPPH in solution was realized from [Figure 6](#fig6){ref-type="fig"}. The deep violet color of DPPH gradually disappeared and a pale yellow color developed in solution as free radicals got neutralized providing a visual monitoring of the reaction. Concentration of the DPPH radical present in solution could therefore be realized from the change in absorption at 517 nm.Figure 6Changes in the UV-vis spectrum of DPPH (60 μM) in presence of HPIA (0--30μM) in ethanol.Figure 6
In the second pathway, in a subsequent step two distinctly different entities would exist in solution following the quenching of DPPH•. Whatever the mechanism, the conclusion is that HPIA has the ability to quench free radicals in solution which was realized using DPPH•.
### 3.3.2. Assessment of scavenging of DPPH by EPR spectroscopy {#sec3.3.2}
The EPR signal of DPPH was used to monitor the free radical scavenging activity of HPIA \[[@bib21], [@bib22], [@bib23], [@bib42], [@bib43], [@bib44]\]. [Figure 7](#fig7){ref-type="fig"}, shows for the same initial concentration of DPPH, increasing concentrations of HPIA lead to gradual decrease in its EPR response suggesting that with an increase in concentration of HPIA, a substantial amount of DPPH radical was no more present. Conversion of DPPH• to reduced DPPH might occur in two ways. One through adduct formation whereby one N atom of the azo bond in HPIA forms a bond with the nitrogen atom of DPPH• or with either of the benzene rings of DPPH• at the para position containing a lone electron \[[@bib36]\]. Such adduct formation immediately destroys the radical character of DPPH•. The new compound formed from HPIA would have a lone electron localized on the other nitrogen atom of the azo bond. The other path for quenching DPPH• could be through the transfer of an electron from a nitrogen atom of the azo bond in HPIA to DPPH•. As a consequence the nitrogen atom on which the lone electron was present in DPPH• becomes N¯ in reduced DPPH converting HPIA to a radical-cation (HPIA·^+^).Figure 7EPR spectra of DPPH (6 mM) with different concentrations of HPIA (0--5mM) in ethanol.Figure 7
### 3.3.3. Measurement of ROS by the DCF-DA assay {#sec3.3.3}
Reactive oxygen species (ROS) are also responsible for damage caused to DNA within cells and these are generated in good amount following irradiation. Presence of ROS was estimated in WI-38 lung fibroblast cells with the help of the DCF-DA assay that measures fluorescence due to DCF \[[@bib24], [@bib25], [@bib26], [@bib27]\]. Cells were previously treated with HPIA using three different concentrations (1 μM, 5 μM, and 10 μM) for 30 min. The experiment was performed under two separate heads i) when ROS profile was estimated based only on ROS present in WI-38 lung fibroblast cells and ii) when ROS was estimated after it was irradiated with ^60^Co γ rays with a dose of 2 Gy. A dose of 2 Gy was selected to make the study relevant to biological systems with particular reference to aspects like radiotherapy \[[@bib45], [@bib46]\]. The ROS profile in each case was done for two different times, one immediately after treatment of cells referred to in the discussion as that performed at 0 h and another after 4 h.
When ROS present in WI-38 lung fibroblast cells were measured immediately i.e. 0 hour in the absence of radiation it was seen that for 1 μM HPIA, quenching was almost negligible while for the measurement made after 4 h it was higher. Results indicate 1 μM HPIA was not sufficient to quench the formation of extra ROS generated in the next four hours. However, with increase in concentration of HPIA to 5 μM and 10 μM respectively, values obtained for ROS decreased in a manner expected both for 0 h and 4 h indicating increased concentrations of HPIA (5 μM and 10 μM) were sufficient for ROS quenching. The experiment clearly showed the ROS quenching ability of HPIA ([Figure 8](#fig8){ref-type="fig"}).Figure 8Effect of HPIA on ROS generation induced by ^60^Co γ rays on a WI-38 lung fibroblast cell line that was followed by the DCFDA assay using fluorescence spectroscopy.Figure 8
ROS was induced in WI-38 lung fibroblast cells with the help of ^60^Co γ rays at a dose of 2 Gy and subsequently followed by the DCF-DA assay. Monitoring of fluorescence of DCF reveals when HPIA was not used, generation of ROS at 0 h far exceeds that at 4 h ([Figure 8](#fig8){ref-type="fig"}). This is because at 0 h i.e. immediately after irradiation there is a high concentration of ROS which after 4 h for irradiated cells decrease, as metabolic activity of irradiated cells decrease substantially. Moreover, no new ROS is generated by such cells themselves. However, the ROS quenching mechanism of cells remain active; rather they become more active to be able to decrease the increased presence of ROS (following irradiation) in such cells. Therefore, for irradiation provided in the absence of HPIA, the assay performed at 4 h indicates that the presence of ROS was significantly less. When irradiation was provided in presence of HPIA, decrease in ROS was much more systematic at all concentrations of HPIA used; values recorded at 4 h being significantly less than that at 0 h. The experiment showed that the ROS quenching ability of HPIA was significant at 5 μM and beyond it. Results obtained for cells irradiated in the presence of 1 μM HPIA was almost comparable to that when no compound was present during irradiation; the only difference being, amount of ROS present at 0 h was slightly less when HPIA was present which is clear from the above discussion.
4. Conclusion {#sec4}
=============
Free radical scavenging and antioxidant activities of HPIA were examined. The study shows HPIA protects radiation-induced damage of thymine and prevents modification of double stranded calf thymus DNA compared to situations when experiments were performed on the same target in absence of HPIA. To investigate the mechanism of action of HPIA, DPPH radical quenching and DCF-DA assay were performed. Results show HPIA scavenged DPPH radical and depleted ROS generation in WI 38 lung fibroblast cells. Antioxidant efficacy of HPIA could be concluded from evaluation of radio-protection on thymine and calf thymus DNA following gamma irradiation of such targets in the absence and presence of HPIA. Results obtained are comparable to those of flavonoids. The DCF-DA assay further indicates HPIA mitigates reactive oxygen species−induced damage to cells by scavenging them. The DPPH scavenging ability of HPIA increased in a dose-dependent manner. A positive correlation of radio-protection with antioxidant activity due to HPIA was observed following different studies undertaken. At a concentration as low as 1 μM efficient radio-protection was observed due to HPIA on thymine and calf thymus DNA which is encouraging. However, the same concentration of 1 μM was not sufficient for ROS depletion for which slightly higher concentrations were required. Results suggest HPIA may be tried as an antioxidant supplement during radiotherapy for protection of normal cells.
Declarations {#sec5}
============
Author contribution statement {#sec5.1}
-----------------------------
Durba Ganguly, Ramesh C Santra, Swagata Mazumdar: Performed the experiments; Analyzed and interpreted the data.
Abhijit Saha: Contributed reagents, materials, analysis tools or data.
Parimal Karmakar: Conceived and designed the experiments; Analyzed and interpreted the data.
Saurabh Das: Conceived and designed the experiments; Analyzed and interpreted the data; Wrote the paper.
Funding statement {#sec5.2}
-----------------
Saurabh Das was supported by 10.13039/501100010426UGC-DAE CSR, KC Collaborative Research Scheme (UGC-DAE-CSR-KC/CRS/13/RC03/0886RCS) Durba Ganguly was supported by a project fellowship from 10.13039/501100010426UGC-DAE CSR, KC Collaborative Research Scheme. Ramesh C Santra was supported by a Senior Research Fellowship from 10.13039/501100001501UGC. Saurabh Das was supported by the RUSA 2.0 program operating at 10.13039/100009589Jadavpur University (Ref. no. R-11/438/19 dated 30.05.2019) Saurabh Das was supported by 10.13039/501100001501UGC, New Delhi as part of UPE II, Jadavpur University and the UGC-CAS II program.
Competing interest statement {#sec5.3}
----------------------------
The authors declare no conflict of interest.
Additional information {#sec5.4}
----------------------
No additional information is available for this paper.
SD is grateful to UGC for a one-time grant to Department of Chemistry, 10.13039/100009589Jadavpur University during the "International year of Chemistry" (2011) with which the EPR spectrometer was purchased. He is grateful to Prof. Kalyan K. Mukherjea and his research scholars for help in recording EPR data.
|
Brady Grey
Brady Grey (born 20 July 1995) is a former professional Australian rules footballer who played for the Fremantle Football Club in the Australian Football League (AFL).
Drafted with the 58th selection in the 2013 AFL draft from the Burnie Dockers Football Club in the Tasmanian State League, he played for Peel Thunder in the West Australian Football League (WAFL), Fremantle's reserve team during the 2014 season before suffering a stress fracture in his back in July.
Grey continued to play for Peel throughout 2015, and made his AFL debut for Fremantle in the final round of the 2015 AFL season, when Fremantle sent a weakened team to play Port Adelaide at Adelaide Oval. Twelve changes were made to the team, and Grey was one of four players to make their AFL debuts. He was delisted at the conclusion of the 2016 season, before he was subsequently re-drafted by Fremantle in the 2017 rookie draft.
Currently playing for the West Coast Eagles in the WAFL competition.
References
External links
WAFL Player Profile and Statistics
Category:1995 births
Category:Living people
Category:Fremantle Football Club players
Category:Peel Thunder Football Club players
Category:Australian rules footballers from Tasmania
Category:Indigenous Australian players of Australian rules football
Category:Burnie Dockers Football Club players
Category:West Coast Eagles (WAFL) players |
Uber Driver Kidnapped Her And Took Her To A Motel - a159482a
http://www.nbclosangeles.com/investigations/Uber-Driver-Arrested-Kidnap-With-Sexual-Intent-Charge-261730151.html
======
a159482a
If Uber really wants to take it to the next level, they have to ensure drivers
feel safe from incidents like this.
~~~
a159482a
I agree, it's not funny. The fact that they are trying to expand further with
such financing "Bloomberg reports that mutual fund giant Fidelity is competing
to lead a round of financing for startup car service Uber Technologies"
coupled with this incident, should impel Uber to have the right compliance
measures in place, as they are scaling, to ensure all users feel safe when
using the service.
|
Thin Film Transistor Liquid Crystal Displays (TFT-LCDs) as a kind of flat panel display apparatus are more and more applied to the high-performance display field because they have traits of small volume, low power consumption, radiation-free, relatively low production cost, etc.
A TFT-LCD includes a color filter substrate and an array substrate disposed to be aligned with each other, with a liquid crystal layer provided therebetween. By means of controlling the deflection of liquid crystal molecules in the liquid crystal layer, control of light intensity is realized, so as to achieve an objective of displaying images.
Generally, the structure of an array substrate may be as shown in FIG. 1a, and it includes a plurality of gate lines 10 and data lines 11 that crisscross over each other. A plurality of pixel units 12 arranged in the form of a matrix are defined by the crossing of the gate lines 10 and the data lines 11, and a pixel electrode 104 is provided within each of the pixel units 12. The sectional view of the array substrate taken along the A-A′ direction is shown in FIG. 1b, and it includes multilayer thin film structures from bottom to top, such as, a gate electrode 101, a gate insulating layer 102, a semiconductor active layer 103, a pixel electrode 104, and a source/drain metal layer 105. For example, the above thin film structures may be fabricated in such a manner that a thin film layer and a photoresist are formed sequentially on a substrate, and then are subjected to masking, exposure, development, etching, stripping and other process.
However, during production and processing, due to the impact of external environment or production process, a thin film layer that should be etched away may be left over on the substrate. For example, the semiconductor active layer 103 or the pixel electrode 104 lying in a region between two adjacent pixel units 12 (in correspondence with a photoresist fully-removed region) should be fully etched away. However, during exposure and development, because a photoresist in the above photoresist fully-removed region is affected by the film-plating process of the former layer and its own process, the photoresist may not be fully exposed and a superfluous photoresist retained region is formed. On this basis, by a subsequent production process, it is possible that a residual portion (forming a conduction layer 20) of the pixel electrode 104 shown in FIG. 1c, or a residual portion (forming a conduction layer 20) of the semiconductor active layer 103 shown in FIG. 1d is formed between two adjacent pixel units 12. In this way, pixel electrodes 104 in two adjacent pixel units 12 are electrically connected, and when one of the pixel units 12 is controlled for display, a pixel unit 12 that is adjacent and electrically connected to it is lit up as well, resulting in occurrence of an uncontrolled bright pixel point (bright dot defect). This adversely affects the display effects and the product quality.
In order to solve the above problems, laser bonding or cutting process is generally adopted to repair a pixel point that suffers from a bright dot defect. For example, the array substrate is detected by an optical detection instrument, and when a bright dot defect is found, the above conduction layer 20 may be cut, so that two adjacent pixel units 12 are not electrically connected. However, because thickness of the pixel electrode 104 is relatively smaller, the degree of identification of the optical detection is reduced, and this leads to increasing of the miss probability of detection. And, when the above repairing process is carried out, other thin film structure that has already been formed, such as, a passivation layer, a common electrode layer or the like (not shown in the figure) located on a surface of the pixel electrode 104, may be damaged. Thus, repair effect of the bright dot defect is degraded, and quality of the product is affected. |
Mugabe says no to whites, gays
Zimbabwean President, Robert Mugabe accused his political rivals Saturday of wanting to bring back “white people” and took a swipe at gay rights as campaigning gears up ahead of the July 31 election.
Mugabe, 89, clad in a white church robe and holding a biblical staff, appealed to thousands of members of an indigenous church in eastern Marange to support his bid for re-election after 33 years in power.
“We made a mistake in 2008 to vote for the people who love the white people. Voting for people who want to bring back the white people and thinking that there won’t be any development without white people,” he said.
The veteran leader will go head to head at the ballot box with longtime rival Prime Minister Morgan Tsvangirai.
The vote will end the pair’s tense power-sharing government that was forced by the chaotic 2008 polls.
Speaking in the diamond-rich area about 200 kilometres east of the capital Harare, Mugabe pushed his message of indigenisation of the economy. “The rich resources that our country is endowed with are for the black people, this is our country. And those who must rule this country must be black people,” he said.
Mugabe also attacked gay marriage, saying it was alien to Africa and criticised US President Barack Obama for urging Africa to respect gay rights on a recent visit to the continent.
“You heard it when Obama came to Africa saying Africa must allow gay marriages even women to marry each other so they can wed if they want,” he said.
“God destroyed the earth because of these sins. Weddings are for a man and a woman, who when married they bear children,” he said.
Mugabe, who once said gays and lesbians are worse than pigs and dogs, said animals are better off because they know their sexual orientation. |
NOTES
Source: U.S. Bureau of Economic Analysis
Release: Gross Domestic Product
Units: Billions of Chained 2012 Dollars, Seasonally Adjusted Annual Rate
Frequency: Quarterly
Notes:
BEA Account Code: A191RX
Real gross domestic product is the inflation adjusted value of the goods and services produced by labor and property located in the United States.For more information see the Guide to the National Income and Product Accounts of the United States (NIPA). For more information, please visit the Bureau of Economic Analysis. |
Jaguars' coaching search ends with Gus Bradley
Friday
Jan 18, 2013 at 1:15 AM
Ryan O'Halloran
Searching for a coach who was equal parts energetic and intense, tactician and leader, the Jaguars hired Gus Bradley on Thursday hoping he and new general manager Dave Caldwell can provide stability and success to a franchise that has been adrift in both departments since its last playoff appearance five years ago.
Now it's up to the new coach to put together a staff, with offensive coordinator the key component.
Bradley, 46, served as the Seattle Seahawks' defensive coordinator the last three years and wowed Caldwell during an interview Wednesday. Bradley, who will be introduced to the media on Friday, is the Jaguars' third head coach in as many years.
The move completed owner Shad Khan's massive makeover that started a day after the Jaguars completed a 2-14 season, the worst in team history.
Bradley and Caldwell no doubt talked about offensive coordinators during the interview and that process continued Thursday when the Jaguars asked for permission to speak with New Orleans assistant Pete Carmichael. But he opted to remain with the Saints.
The Jaguars have three staff openings (all on offense) and are expected to employ new offensive and defensive coordinators.
Only once the coordinators are in place will the process of deciding on the quarterback situation, the preferred schemes and whom to pick with the No. 2 overall draft choice in late April will begin.
Just as he was decisive last week in closing the door on acquiring current Jets quarterback Tim Tebow, Caldwell acted similarly in finding a replacement for Mike Mularkey.
The first interview wasn't until Monday. Only four candidates were interviewed. And among the eight new head coaches, Bradley is the only one with a defensive background.
"It was just a matter of time before Gus Bradley became a head coach in the NFL," Caldwell said in a statement.
HIRING STAFF NEXT
Bradley is expected to attend at least one day of Senior Bowl practices next week in Mobile, Ala., and he could have some of his staff in place by then.
Early defensive coordinator candidates to emerge are Seattle defensive line coach Todd Wash and Chicago linebackers coach Bob Babich. Wash and Bradley were on the same staffs in Tampa Bay and Seattle and Bradley coached for Babich at North Dakota State.
Current Jaguars defensive coordinator Mel Tucker, who interviewed on Monday with Caldwell, remains under contract but is expected to pursue another opportunity, which could be in Cleveland.
The only current Jaguars assistant coach with a tie to Bradley is assistant head coach-quarterbacks coach Greg Olson, who was on the same Buccaneers staff in 2008. He has coordinator and play-calling experience.
Carmichael won't be available as offensive coordinator, nor will former Arizona coach Ken Whisenhunt, who was hired by San Diego to run the offense. Veteran play-callers available include former Baltimore coordinator Cam Cameron and Bengals receivers coach Hue Jackson.
Bradley's background is running a 4-3 defensive scheme, which would mean less of a transition for the Jaguars' personnel. Seattle's defense allowed the league's fewest points this regular season; the Jaguars' defense was tied for 29th in that category.
"Watching Seattle's defense all year was a pleasure," Jaguars linebacker Russell Allen said. "They were physical and fast and aggressive and fun to watch. I'm excited about that."
"He's got a brilliant football mind," Seahawks coach Pete Carroll told reporters last week. "He's got a way of reaching people and touching people and getting the best out of them, both coaches and players. He's got everything you're looking for."
Caldwell opted not to wait to interview any assistants whose teams are playing in Sunday's conference title games, including San Francisco offensive coordinator Greg Roman, who was also Caldwell's college roommate.
Bradley interviewed with the Philadelphia Eagles on Tuesday and was told by owner Jeffrey Lurie he was one of two finalists.
"We were all in on Gus Bradley," general manager Howie Roseman told WIP Radio in Philadelphia. "He was incredibly impressive."
The Eagles, though, hired Chip Kelly after he reconsidered and left the University of Oregon. Soon after, Roseman said he called Caldwell to recommend Bradley.
"Gus more than met every criteria we insisted on from our new head coach and his intangibles and leadership abilities are exceptional," Caldwell said. "Gus is who the Jaguars need now and in the future."
Bradley's path to the Jaguars was unconventional. Sure, he was a position coach for two years (linebackers in Tampa Bay, 2007-08) before his three years as a coordinator with Seattle. But he didn't get to the NFL until he was nearly 40 years old.
A native of southeast Minnesota, Bradley played at North Dakota State and spent the first 16 years of his coaching career at the Division II or Football Championship Subdivision (Division I-A level).
Seven years later, Bradley is an NFL head coach.
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Activation of 5-HT2C (but not 5-HT1A) receptors in the amygdala enhances fear-induced antinociception: Blockade with local 5-HT2C antagonist or systemic fluoxetine.
It is well-known that the exposure of rodents to threatening environments [e.g., the open arm of the elevated-plus maze (EPM)] elicits pain inhibition. Systemic and/or intracerebral [e.g., periaqueductal gray matter, amygdala) injections of antiaversive drugs [e.g., serotonin (5-HT) ligands, selective serotonin reuptake inhibitors (SSRIs)] have been used to change EPM-open arm confinement induced antinociception (OAA). Here, we investigated (i) the role of the 5-HT1A and 5-HT2C receptors located in the amygdaloid complex on OAA as well as (ii) the effects of systemic pretreatment with fluoxetine (an SSRI) on the effects of intra-amygdala injections of 8-OH-DPAT (a 5-HT1A agonist) or MK-212 (a 5-HT2C agonist) on nociception in mice confined to the open arm or enclosed arm of the EPM. Nociception was assessed by the writhing test. Intra-amygdala injections of 8-OH-DPAT (10 nmol) or MK-212 (0.63 nmol) produced a pronociceptive effect and intensified OAA, respectively. Fluoxetine (2.5 mg/kg, intraperitoneally) did not change 8-OH-DPAT effects on nociception but antagonized the enhancement of the OAA produced by MK-212. Interestingly, prior injection of SB 242084 (a selective 5-HT2C antagonist) into the amygdala also blocked the MK-212 effects on OAA. These results indicate that 5-HT may facilitate nociception and intensify OAA, respectively, at 5-HT1A and 5-HT2C receptors located in the amygdala of mice. The impairment produced by systemic fluoxetine on the OAA enhancement provoked by intra-amygdala MK-212 suggests that this type of fear-induced antinociception may be modulated by SSRIs. |
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I can no longer watch any YT videos in Safari on iOS 7 - iPad 3. Trying to play will result in "this video is unavailable" message. I can click on the video title and it will open and play fine in Youtube app. (However embedded videos, such as posted in these threads, do not have their "titles" shown and that option is not available.)
Anyone else having this problem? Is there a setting I missed somewhere? |
Dr. Hani Sarie-Eldin, managing partner of Sarie-Eldin Legal Advisors and former head of the Capital Market Authority (CMA)(Photo from Al-Borsa News)
By Mohamed Ayyad
Egypt is suffering from the politicization of criminal law in the aftermath of the revolution of January 25, 2011 through state interference in investment and administrative disputes
Dr. Hani Sarie-Eldin, managing partner of Sarie-Eldin Legal Advisors and former head of the Capital Market Authority (CMA) has described this as a catastrophe that puts fear in the hearts of local and foreign investors and pushes them to scale down investment and dealings with state agencies.
Sarie-Eldin added in an interview with Al-Borsa newspaper that continuing to work with the same legislation and mechanisms of the past would generate the same problems people are currently unable to solve, especially the investment disputes left over from a corrupt system that still exists and runs on their complexity.
The involvement of criminal law in investment and administrative disputes came from the failure to conduct political trials for politicians and state officials as well as for well-intentioned investors. The state’s confusion and hesitation to resolve investment disputes will lead Egypt out of the fierce competition for a piece of the direct foreign investment cake.
The country is currently going through disputes with approximately 20 investors, as well as final judicial rulings returning several privatized companies to the state. It is difficult to implement them to change the legal status of the company shareholders or liquidate their assets, which is what has plagued the investment climate and made the state vulnerable to heavy fines with investors resorting to international arbitration.
According to Sarie-Eldin, the government was trying to coerce investors into settling disputes amicably and according to the conditions it sets, away from legal mechanisms and in clear breach of their contracts. It hinted at resorting to criminal law to force investors to succumb to the Government’s demands, leading to an unstable legal environment and a climate repellent to investment.
“The Government formed committees to rebalance contracts that suffer from a fiscal deficit and infringement of the rights of the state,” he said. “But it abused these committees by threatening the investor with prison in case of an unresolved dispute or failure to bend to government pressure, which led to the state’s general pattern of adopting physical coercion to settle commercial disputes. The committees became futile shams.”
He described the insertion of criminal law into commercial and administrative disputes as a “disaster” that led to government investors scared to make any decision that might subject them to condemnation and criminal accountability, either by privatizing land or implementing contracts. It ultimately became the easier decision for investors to void the contracts, which produced a wretched climate for investment and the complete paralysis of all state agencies under the pretext of protecting public money. It was, however, doing serious damage to public money, as he put it.
Assessing the investment climate required taking into account the legislative framework and judicial system, the speed of resolving and adjudicating investment disputes, the procedural and bureaucratic dimension, availability of infrastructure and investment-ready land and finally currency problems. He described these as the least influential compared to the legislative and procedural framework.
Thus, the chairman of the Department of Commercial Law at Cairo University said that the state’s confusion in resolving investment disputes was a disaster that required quick action from all sides, beginning with the President and Government and the heads of the regulatory agencies because the issue could no longer bear delays. Its resolution would have a very fast positive impact and would return investor confidence in Egyptian government contracts.
“The investor may bear the slowness of some of the procedures and bureaucracy and is asking for amendments to the Companies Act but will not accept being thrown in gaol or conduct a feasibility study for a project with a regime – and then the regime is toppled and another one comes along, and is brought down on the pretext of the right of the state. There are a number of matters that need to be relooked at.”
Sarie-Eldin gave the example of the real estate regulations of the Urban Communities Authority which prevent real estate investors, for instance, from marketing their plans until after their complete implementation. This was at odds with the nature of real estate investment. It was unacceptable to stop a multi-billion dollar project without five years of marketing. It also penalized the investor who might be slightly behind in their payments because of a financial crisis with the confiscation of the land, even if they had paid most of the installments.
He asked the Government to quickly and urgently review its executive decisions, especially its unrealistic and crippling real estate regulations, to revitalize real estate investment and lay regional foundations for pricing by all government agencies.
“The general impression of Egypt, internally and externally, is that it is a country that does not abide by the rule of law or respect contracts, and does not have a clear vision for resolving disputes in its decisions. The integrity of the state’s legal system and the stability of legal centers are the most important elements of an assessment of the investment climate in any country. The administrative apparatus may have made a mistake, but as long as the investor is not paying bribes or committing a crime, it is not permissible for him to be punished by the state. The injustice of the legal system is what threatens current investments and threatens another wave.”
Sarie-Eldin called for adjusting the legislative and economic systems. This included liquidation and bankruptcy laws, a safe exit and quick access to the market.
Despite this, Sarie-Eldin felt that the fundamental problem was operational and procedural more than it was legislative. Officials thinking it was easy to not carry out contracts under the pretext of protecting public funds was a crime because it limited the investment cycle and increased the problems the country faces and led it to standing before international arbitration centers. This exhausted the country’s resources while also halting multi-million dollar projects and limiting the arrival of new investments.
“Legal and legislative uncertainty and instability are at the heart of the damage to public funds and not the reverse. There is no doubt that the previous regime’s contracts, marred by corruption, are not necessarily financial but administrative. The government often signed contracts with investors then asked them for the possibility of changing the price years later. Investors consented and paid the agreed upon amount. After some years, they were surprised by the demands for them to pay more taxes despite the fact that the state’s administrative apparatus had not implemented [plans] to restore land or deliver energy.”
The legal advisor pointed out that other problems included the lack of discipline in communications, complaints about past judicial decisions, skepticism about financial receivables and serious intervention on the part of state oversight agencies away from the adjustment of irregularities.
Sarie-Eldin said the average citizen looking for job opportunities was the most hurt by government slowness in resolving international disputes. Despite the economy needing security, the stability of the economic climate and improvement of the investment climate were working to control lawlessness and limit demonstrations and preserve Senate staff.
The legal advisor described the phase that followed the January 25, 2011 revolution as an absurdity and asked those in charge of state administration for quick decision-making according to the terms and truths of law so as to not lose a place among the fierce competition for direct foreign investment. There are countries with higher taxes than Egypt’s that attract larger investments due to their calm investment climates and the state’s respect for its contracts. However, Egypt was prepared to mobilize greater investments for efficiency and licensing of its labor, alongside taxes, simple customs and infrastructure unaffected by recent events.
He turned to the crisis of companies that were returned to state possession years after privatization. “The Government has no choice but to implement the final verdicts and reclaim the assets as friendly solutions are no longer available,” he said.
Sarie-Eldin described these judicial verdicts nullifying the privatization of public sector companies as stemming from public pressure that put all assets sold within the circle of corruption.
The ex-chairman of the CMA wondered about how to revoke commercial contracts for which criminal provisions proving themselves marred by corruption were not issued and, at the same time, how to invalidate investment contracts according to criminal provisions for which judgments were not handed down in commercial disputes.
The legal advisor also wondered about how to return the assets of a company that had been traded on the stock exchange and had different legal statuses and a company whose assets had been completely liquidated. It was physically impossible to implement the positions and these would have been better resolved amicably before court, he said, underscoring that he meant the judgments issued against the buyer.
“What is happening right now is weakening the ability of any government to solve these disputes and settle the clash between all executive, judicial, and oversight administrations. The matter has become a state issue and must move quickly to achieve success.”
He played down the difficulty of resolving investment disputes on whose behalf final verdicts were signed, which became “amicable”. At the same time, he stressed the necessity of the government enforcing the final verdicts to maintain stable legal status. There were companies that had been completely liquidated and their assets sold and the original investor was no longer living, which made a physical settlement impossible. The situation also required taking into account the labor force that these disputes will leave behind.
According to Dr. Sarie-Eldin, international national arbitration does not work against Egypt. However, the state was going to international arbitration with badly formulated contracts, unspecialized legal representatives and an exhausted administrative apparatus. In addition, representatives of the government have their defense focus their arguments on cosmetic rather than substantive reasons – all of which has produced the impression of Egypt as a place that does not respect its contracts.
Dr. Sarie-Eldin explained that revolutions made it more difficult when dealing with investment contracts, which were signed by the former regime and stirred up great controversy.
“The current government must try to reconcile with the past,” he said. “We are still talking about the existing procedural and administrative framework without the reasons for the problems coming out. The current administration is working to treat the problems of the past with the same mode of thinking and legislation that it created, although it is necessary to define the problems by sector. There is no magic solution that can be generalized to get us out of this dilemma.
Dr.Sarie-Eldin asked those leading the administration to work quickly to get them on the correct path economically, legislatively, politically, procedurally and organizationally. “We started the two revolutions of January 2011 and June 2013, but they were not finished. As for the first, the regime we rose up against rules across many wings. The second is the religious regime, still increasing the state of political tension and imposing an unstable security reality in the street.”
“The truth is that what happened was only the toppling of individuals. The success of the revolution will only come when its principles are implemented. What happened was not the fault of the revolutionaries but rather of the administration. They did not undertake radical reforms in all areas.”
According to Dr. Sarie-Eldin, the Government currently has few believers in change. The administration was certainly corrupt and they lacked the ability to change and the tools to implement that change. The rest were employees so the administration’s corruption did not bother them. They were a part of it and were neither believers in change nor did they have the ability to do it.
He stressed that Egypt needed a government and regulatory and administrative agencies to define the required change, not exacerbate the problems. They needed to spread hope by broadcasting positive messages, and reassure the business community domestically and abroad.
He said that Interim President Adly Mansour was not governing except by the powers in the last constitutional declaration and the supreme word in the Supreme Council of the Armed Forces’ (SCAF) security dossier. The Government had limited power and was constrained by the scope of security and SCAF, but SCAF was the chief actor in it. As a result here was no one leading the government explicitly, which weakened the decisions taken, despite the fact that the government had a larger framework than ever before.
He described the economic situation as sad and wretched, and said the government was unable to radically resuscitate it right now. It had nothing to do but work to create a stable, balanced and sustainable political climate by way of greater honesty with the people and even accept extreme measures like restructuring subsidies, in parallel with working to address the budget deficit and conduct more in-depth tax reforms.
“The current phase requires painkillers and sedatives but cannot handle surgery in terms of dealing with Gulf aid with the utmost caution and using it with maximum efficiency to create to create job opportunities and direct it to the poorest regions,” he said.
The government called for the necessity of dealing with the transitional phase with the utmost discipline and caution, including all governorates, especially the poorest politically and socially, so as to not create problems that the government was not now prepared to solve. |
The effectiveness of metacognitive training for patients with schizophrenia: a narrative systematic review of studies published between 2009 and 2015.
The aim of this paper is to review results of studies on the effectiveness of metacognitive training (MCT) for patients with schizophrenia in reduction of psychotic symptoms and cognitive biases. Furthermore, other variables, like social functioning, insight and neurocognitive functions, are analyzed. Systematic search in databases PubMed, EBSCO, Google Scholar, EMBASE, Cochrane Central Register of Controlled Trials and PsycINFO regarding studies on the effectiveness of the MCT was made. The review included 14 studies published in years 2009-2015, in which design of the study made comparison between MCT group and control group possible. Combined number of patients in MCT group was 354 and 355 in control group. The largest effect size was obtained for severity of delusions (d < 0.23; 1 >), especially reduction of conviction and distress of delusional beliefs. An effect size regarding negative symptoms reduction was small. Large effect size was observed for insight improvement (d < 0.45; 1.32 >). Positive impact of MCT on cognitive biases severity (d < 0.21; 0.83 >, especially jumping to conclusions) and improvement in some aspects of neurocognitive functions was observed (d < 0.2; 0.63 >). There was no improvement in social functioning of patients in MCT group. Follow-up studies show sustainability in symptoms improvement lasting at least 6 months. MCT is an effective form of therapy in reduction of delusions, cognitive biases related to schizophrenia and improvement of insight. Relatively easy accessibility and sustainability of therapeutic effects indicates that MCT can by effectively used in therapy of schizophrenia. To enhance training efficacy, especially in patients' general functioning, combining it with others forms of therapy is to be considered. |
Q:
Question involving Legendre symbols
Let r,p,q be distinct odd primes. Let 4r divide p-q. Show that
(r/p) = (r/q)
Where (a/b) is the Legendre symbol.
I'm sure we are suppose to use the law of quadratic reciprocity. I don't think this question is suppose to be difficult, but I cannot figure it out!
A:
Hint:
$$4r\mid (p-q)\iff p\equiv q\bmod r\quad\text{and}\quad p\equiv q\bmod 4$$
|
David Hansemann
David Justus Ludwig Hansemann (12 July 1790 – 4 August 1864) was a Prussian politician and banker, serving as the Prussian Minister of Finance in 1848.
Life
Hansemann was born in Finkenwerder, Hamburg, the son of a Protestant minister. After studying commerce, he was a representative for a Monschau cloth manufacturer. From 1817, he created different enterprises in Aachen, under which the predecessor company the Aachen native of Munich, which is today part of the AMB Generali. Into the 1820s and 1830s, he came by a large sum of money. His concern for the well-being of his employees and his readiness for generous donations were considered unusual. In 1821 he married Fanny Fremery, who came from a French Huguenot family.
David Hansemann engaged himself for the building of railways in the Rhine Province. He wrote several memoranda about railway. The moreover one he was shareholder of the Rhenish Railway Company (German: Rheinische Eisenbahn-Gesellschaft, RhE). In accordance with a royal cabinet order of February 1837, he became Vice-President of the RhE. He was considerably involved in the establishment of further railway companies, including the Cologne-Minden Railway Company and the Bergisch-Märkische Railway Company.
During the 1830s, Hansemann became increasingly involved in politics, and in 1843, he became a member of the provincial parliament (Provinziallandtag) for Rhenish Prussia. In the year 1847, he became a member of the Prussian United Parliament (Vereinigter Landtag). Hansemann was considered as one of the prominent heads of German Liberalism. Within liberalism, he, along with the later president of the Frankfurt Parliament, Heinrich von Gagern, belonged to the so-called "half ones", i.e. ready to compromise.
During the short-lived Prussian March Ministry under Gottfried Ludolf Camphausen, Hansemann was made Minister of Finance. He retained this post in the next administration led by Rudolf von Auerswald, until his resignation on 8 September 1848.
For the noble elite of Prussia, Hansemann was considered far too liberal; his subtly critical book Prussia and France of 1833 and different memoranda from the 1840s lead him to be thought of as a dangerous radical. On the other hand, he was considered by radicals to be a reactionary serving the elite: Karl Marx scornfully called him a "liberal lick-spittle."
He is honoured by having his portrait appear on the German Reichsbanknote for 50 Reichsmarks dated 30 March 1933.
This double resistance, a principal reason for the defeat of the liberalism after the March revolution in Germany, finally led to Hansemann leaving politics. After leaving political life, Hansemann returned to commerce, and in 1851 he formed the Disconto-Gesellschaft, which merged with Deutsche Bank in 1929.
Hansemann died in Schlangenbad, during a cure stay in the Taunus. He is buried in the Hansemann mausoleum at the Matthaeus Kirchhof in the Schöneberg district of Berlin. His son Adolph von Hansemann became one of the richest and most important entrepreneurs of the German Reich, although in contrast to his father, he was not a liberal.
Quotations
(1847) before the Prussian united federal state parliament became famous: * "in financial matters the cosiness stops."
From a memorandum of 1840: * "many governing or interfering the public administration into too many articles became rule. They have the official itself the inclination unconsciously hang-give to decide or estimate rather after cheapness the most various articles, which will leave fueglich to the discretion of the private people and Korporationen could, in own opinion." * "so everyone is in principle unfreely and politically minor, and the large majority does not carry by any means an active for demand for formally secured [... ] this kind of satisfaction of the people pleases some official splendid and as proof is stated, like the Prussian conditions the safest and most satisfying in Europe nevertheless would be."
From a letter from 1839: * "with the conviction that, if I dedicated time and mental effort completely the business my fortune would amount to probably now the double, I work much in general affairs. I judge fortunes only as means, not purpose. This means brings independence, to calming for the life span and the ability to give and in addition useful expenditures to make be able to the children a good education."
External links
The Aachener a hard High School over Hans man "the High School donated of the Aachener and Munich insurance created by Hans man"
Marx über Hansemann
David-Hansemann-Schule in Aachen
Preußen-Chronik über Hansemann
Die Deutsche Bank über Hansemann
David Hansemann to Prussian Interior Minister Ernst von Bodelschwingh (March 1, 1848)
Category:1790 births
Category:1864 deaths
Category:People from Hamburg
Category:German bankers
Category:Prussian politicians
Category:German railway entrepreneurs
Category:Finance ministers of Prussia |
117 F.2d 892 (1941)
KIDDER OIL CO.
v.
FEDERAL TRADE COMMISSION.
No. 7140.
Circuit Court of Appeals, Seventh Circuit.
January 15, 1941.
Rehearing Denied February 28, 1941.
*893 Robert J. Weiss, of Chicago, Ill., for petitioner.
Wm. T. Kelley, Chief Counsel, and Martin A. Morrison, both of Washington, D. C., for respondent.
Before SPARKS, MAJOR, and KERNER, Circuit Judges.
MAJOR, Circuit Judge.
This is a petition to review a cease and desist order of the Federal Trade Commission, (hereinafter called the "Commission") issued September 19, 1939, in a proceeding had under Section 5 of the Federal Trade Commission Act, 15 U.S.C.A. § 45. The controversy before the Commission, as here, was, in a broad sense, whether colloidal graphite in petitioner's product, known as "Koatsal," when added to lubricating oil in internal combustion engines, has the effect of substantially diminishing friction and reducing wear of the engine parts which move upon or against one another.
The issues involved arise from the complaint filed by the Commission, and petitioner's answer thereto. By the latter's answer, certain charges of the complaint were admitted and petitioner consented that a cease and desist order be entered in conformity thereto. It will only be necessary, therefore, to refer to the charges of the complaint which were denied by petitioner's answer. This portion of the complaint is as follows:
"In the course and conduct of its said business, as hereinabove described, respondent, in soliciting the sale of, and in selling its product, `Koatsal,' by pamphlets, labels attached to containers of the product, letters, post cards, testimonials, advertisements inserted in newspapers, periodicals, and magazines, and otherwise, has made extravagant, deceptive, misleading, and false statements and representations regarding the value, efficacy, and effect of its said product, and the results that are achieved by using it, among which are the following:
* * * * * *
"(b) That `Koatsal' performs amazing feats of lubrication never before possible and utterly impossible by any other method, that it perfects lubrication and is more efficient than any other method because it is scientifically correct;
* * * * * *
"(d) That `Koatsal' penetrates and adheres to all metal surfaces it reaches, `actually becomes a part of the metal, permeating the pores * * * literally "soaking" into it,' that the metal becomes plated with it and that moving parts ride on this plating;
"(e) That `Koatsal' reduces friction as much as 50%, provides perfect protection against burned out bearings and makes metal self-lubricating;
* * * * * *"
The portion of the cease and desist order now in controversy precludes petitioner, in connection with the offering for sale, sale *894 and distribution of its product "Koatsal," whether sold under that name or under any other name, from representing:
* * * * * *
"(2) That Koatsal penetrates and adheres to all metal surfaces it reaches, permeates the pores of the metal, soaks into the metal, and that the metal becomes plated with Koatsal and moving parts ride on this plating;
"(3) That an automobile conditioned with Koatsal will run any greater distance without oil in the crankcase without damage to any part than will an automobile conditioned with ordinary lubricating oil of the same quality used in Koatsal;
"(4) That the lubricating qualities of Koatsal are any greater than the lubricating qualities of the oil which it contains;
* * * * * *
As is common in cases of this character, the primary question for decision is whether the findings of fact as made by the Commission, upon which the cease and desist order is predicated, are sustained by substantial evidence. Other incidental issues, perhaps not necessary for decision of the main issue, are (1) whether the report made to the Commission by its Trial Examiner, who heard the witnesses, may properly be considered, and (2) whether this court should make certain findings of fact which were proposed to the Commission by the petitioner and which it is claimed were not included in the findings as made. We shall first discuss these incidental issues.
The Trial Examiner made an original report February 1, 1937. Thereafter, additional evidence was taken and the Examiner filed a supplemental report September 16, 1937, neither of which was incorporated in the transcript, certified and filed by the Commission. Upon a supplemental petition filed by petitioner, this court directed the Commission to certify the reports made by the Trial Examiner without prejudice, however, to the right of the Commission to renew its objection to our action in this respect. The reports are here, and it is again contended by the Commission that, under the statute and rules of the Commission, they are no part of the record. The statute, section 5 (c) provides that the Commission "shall certify and file in the court a transcript of the entire record in the proceeding, including all the evidence taken and the report and order of the Commission." It further provides that the court "shall have power to make and enter upon the pleadings, evidence, and proceedings set forth in such transcript a decree * * *." Rule 13 of the Commission's rules, adopted May 21, 1938, provides: "The Trial Examiner's report upon the evidence is not a decision, finding, or ruling of the Commission. It is not a part of the formal record in the proceeding, and is not to be included in a transcript of the record." It is pointed out by the Commission that there is no provision in the statute for a report "upon the evidence" by the Trial Examiner, that such report is provided for only by the Commission rule which expressly states that such report is not a part of the record to be included in the transcript, and that, therefore, it is not required. Three cases are cited which it is claimed sustain this position. Algoma Lumber Co. et al. v. Federal Trade Commission, 9 Cir., 56 F.2d 774, Arrow-Hart & Hegeman Electric Co. v. Federal Trade Commission, 2 Cir., 63 F.2d 108, and Federal Trade Commission v. Hires Turner Glass Co., 3 Cir., 81 F.2d 362. In the Algoma case, the holding was to the effect that it was not incumbent upon the Commission to certify the Examiner's report unless such report was referred to in the findings of the Commission. That was not done there, nor is it done here, but the court further held that as to whether the Examiner's report should be subsequently certified, was a matter resting in the Court's discretion. In the Arrow-Hart case, it was held that the Commission was not required to certify the Examiner's report unless such report was referred to in the Commission's finding. In the Hires case, the court denied a request that the Commission be required to certify the report of the Trial Examiner.
The Examiner is an agent of the Commission, appointed, and with authority to conduct the hearing and make a report. It is the practice, as we understand at any rate it was done in the instant case when his report is filed with the Commission that a copy be served upon the interested party. Exceptions are, and in this case were, filed to such report. An argument was had before, and brief submitted to the Commission in support of the exceptions thus made. The statute requires a "transcript of the entire record in the proceeding" and we think that the report thus became a part of the record in the proceeding, which, by the statute, was required to be certified. If we are right in this construction of the statute, the provision of *895 the Commission rule "It is not a part of the formal record in the proceeding, and is not to be included in a transcript of the record" is in conflict with the statute and of no effect. Assuming, however, that the statute is not susceptible of such construction, we further are of the opinion that it is a discretionary matter with the court, properly exercised in the instant case. Such conclusion is not inconsistent with our statutory duty to accept the findings of the Commission as to the facts if supported by substantial evidence. In the instant case, the findings of the Commission, upon which its cease and desist order rest, are at variance with the facts as reported by its Examiner. The Commission undoubtedly had the right to make findings contrary to the facts as reported by its Examiner, but that does not preclude us from considering such report in connection with our determination as to whether such findings are substantially supported. In Staley Manufacturing Company v. National Labor Relations Board, 117 F.2d 868, decided by this court November 14, 1940, in considering a similar situation, we said:
"* * * In the same connection, while the report of the Examiner is not binding on the Board, yet where it reaches a conclusion opposite to that of the Examiner, we think the report of the latter has a bearing on the question of substantial support and materially detracts therefrom."
Before considering petitioner's request that this court approve certain findings of fact which it proposed to the Commission, it seems appropriate to make some further statement concerning petitioner's product "Koatsal" as well as the use for which it was intended. Colloidal graphite is a product of the high-temperature furnace, consisting of pure graphite, which is an allotropic form of carbon subdivided into particles so small and fine that they can be suspended in oil, water, or other liquid medium and remain for a long time in suspension therein in a manner close to and having many properties of a true solution. Petitioner's product, which is an oil containing such graphite, is sold for comingling as an auxiliary or adjunct lubrication to the mineral oil which is universally used to lubricate engines. It is the colloidal graphite in the product, which, according to petitioner, produces the beneficial effect. It is generally recognized that in operating an automobile engine, it is dangerous to permit the supply of oil to diminish beyond a certain point; that, if the supply of oil reaches a certain lower limit, the scoring and wearing of cylinders, piston rings and bearings are likely to occur; that the engine without sufficient oil will heat, causing it to "seize" and be subjected to various other injurious effects. It is also generally recognized that there are two conditions of lubrication in an internal combustion engine which are known as "full-film" and "boundary" conditions. While this distinction apparently has a material bearing upon the findings of the Commission upon which its cease and desist order was issued, it is not necessary to review or quote the testimony of numerous witnesses in this regard for the reason that both the Examiner, in his report, and the Commission, in its findings, recognize such distinction. The following statement is from the Commission's findings:
"The primary function of any lubricant in aeroplane and automobile motors is to reduce friction. Friction is waste work and is reduced by the forming of a film of lubricant between the stationary and moving parts of a bearing holding them apart. There is practically no contact of metal to metal when a full film of lubricant is maintained between the moving and stationary surfaces. However, within this lubricating film particles next to the moving surface are in motion and those next to the stationary surface are stationary so that there must be a constant shearing of the film, which action transforms work into heat. Twice in each revolution, each piston pauses momentarily as it reverses direction. During that pause the tension in the rings tends to force out any lubricant between rings and the cylinder walls. In any automobile or aeroplane motor, even when running under full film conditions, there is a momentary shearing of the film.
"When an automobile is given a fresh charge of lubrication, the film is as full as it is possible to maintain between the metal surfaces. This is called full film lubrication. As the lubricant is consumed and only a thin film exists, boundary conditions are approached. Boundary condition is that stage of lubrication when the film is negligible."
The Commission also recognized that "boundary" conditions of lubrication occur under a variety of circumstances. Regarding such conditions, the following statement is made in its findings:
"Boundary conditions are often brought about by lowering the viscosity of the *896 lubricant, the introduction of grit between the surfaces and (of) the fluctuating temperatures under the wide range of speed and load of a motor. There are many other factors which cause boundary conditions."
Some of the "other factors" referred to by the witnesses are "when a person starts his engine on a cold day in the winter time, boundary lubrication undoubtedly exists in the engine, since the oil is not able to get to the surfaces for some little time after the engine is started"; "excessive use of the choke would give a momentary low viscosity to the oil on the cylinder walls brought about by dilution of the oil with gasoline," and "Throughout the ordinary life of an automobile engine, in ordinary operation, there will be frequent conditions of boundary conditions of oil lubrication. I think it happens all of the time. The fact that you do get wear continuously in any engine shows that it is always present."
Petitioner submitted to the Commission eleven proposed findings covering many of the details involved in the controversy, which the Commission refused to accept or reject as presented. It is now urged by the petitioner that this court has the authority to make such findings and should do so in order that the controversy be ended. In this respect, Federal Trade Commission v. Curtis Co., 260 U.S. 568, 580, 43 S.Ct. 210, 213, 67 L.Ed. 408, is relied upon. The language used indicates that the court has the authority to make additional findings, but at the same time the court states: "* * * the matter may be and ordinarily, we think, should be remanded to the commission the primary fact-finding body with direction to make additional findings, * * *."
Assuming that the court has such authority, we are of the opinion that it should be exercised, if ever, only in cases where the findings, as made by the Commission, are insufficient to dispose of the issues presented. In the instant case, we do not think that this can be said. It is true that many details incidentally involved are not determined by the Commission, but we know of no law or rule which imposes such requirement.
We are convinced that the findings as made by the Commission effectively dispose of all material issues adversely to petitioner. In order to illustrate this statement, we will refer to proposed findings 2 and 6, which furnish the strongest support to petitioner's contention.[1]
The Commission found: "that the viscosity and other properties and qualities of a film of lubricant are unaffected by the presence of the colloidal graphite in `Koatsal' whether a motor is operated under full-film or boundary conditions. `Koatsal' has the same qualities, properties, and characteristics as the oil therein contained has and no more. Its effect upon the metal surfaces of a bearing is the same effect as is produced by the oil therein contained and no more.
"In tests made on bearings with plain oil and also with graphite oil, it has been determined that in the presence of an ample supply of oil, `Koatsal' has no measurable effect on the friction, power, or economy of a gasoline engine. No reduction of friction is accomplished by conditioning a motor with `Koatsal.'
"Tests were also made by running automobiles and bearings to destruction and comparisons made as to the effect of `Koatsal' on durability of parts. It is determined from these tests that a bearing will run without substantial damage for an indefinite period after oil is drained from the crankcase so long as a film of oil is maintained between the moving and stationary surfaces of metal, whether the bearing has been conditioned with `Koatsal' or not. It is also determined from tests that as soon as the film of oil is removed and boundary conditions exist that a bearing is quickly destroyed, whether previously conditioned with `Koatsal' or not."
Petitioner argues that this finding is merely a blanket condemnation of its product, *897 with neither an affirmative or a negative finding by the Commission as to any one or more of the proposed findings submitted. As stated, this is not a detailed finding, as requested, but there can be no doubt but that it effectively disposes of all material issues adversely to the petitioner. In other words, according to the findings as made, petitioner's product is without merit, has none of the advantages claimed for it, and the benefits produced by its use are nil. This is so whether the existing conditions of lubrication are "full-film" or "boundary" conditions.
It is upon this basis that we approach the primary issue as to whether such finding is supported by substantial evidence. In doing so, it should be kept in mind that petitioner, before the Commission, and here, relies chiefly upon the claim that its product is beneficial when a motor is operated under "boundary" conditions. The theory and claim is that under "full-film" conditions, there is a complete separation of the contacting surfaces by an oil film, and that under such circumstances, the colloidal graphite does not come into play. On the other hand, so it is claimed, the graphite forms a graphoid surface on the contacting parts so that when boundary conditions exist, the oil film is held in close contact with the graphite layer thus formed, thereby preventing friction and wear.
The Commission relies largely, if not entirely, upon its Exhibit 19, together with the testimony of two witnesses, Dill and Brooks, as furnishing competent, substantial evidence to support the findings as made. Exhibit 19 is a report on tests made by the United States Bureau of Standards of Pyr-Oil, manufactured by the Pyr-Oil Company of LaCrosse, Wisconsin.[2] That company had distributed through the mails advertising matter in which representations were made for Pyr-Oil similar to those made by petitioner for its product. The tests were made at the request of the Post Office Department with a view of ascertaining whether the representations made by the manufacturer of Pyr-Oil were violative of postal regulations. The Commission in its brief states that the conclusions drawn by the experts from this report, are:
"1. In the presence of an ample supply of oil, Pyr-Oil has no measurable effect on the friction, power, or economy of a gasoline engine.
"2. No reduction of friction whatever was found after the engine had operated the equivalent of more than 1,000 miles as directed."
It will be noted that such conclusions are predicated upon the "presence of an ample supply of oil," while the findings of the Commission are applicable to a motor operated under either "full-film" or "boundary" conditions. In either case, so the findings state, "its effect upon the metal surfaces of a bearing is the same effect as is produced by the oil therein contained and no more." While the Commission, in its findings, as stated heretofore, recognized the distinction between "full-film" and "boundary" conditions, it appears to have completely ignored such distinction in its findings and in its argument presented to this court as to the benefits claimed by petitioner for its product with a motor operated under "boundary" conditions.
Brooks arrived at the following conclusion from the tests made: "I would say it is quite obvious that the use of Pyr-Oil in our destruction tests did not result in the engine operating materially longer."
The Commission states in its brief:
"* * * No evidence has been obtained by the Bureau of Standards that the use of graphite in the oil will lower friction in engine bearings when the engine is operated under full-film conditions.
"A witness for the petitioner testified that graphite in the lubricant would have no effect at all if the condition remained full-film lubrication. * * *"
The tests, as made by the Bureau of Standards, were not made with a motor in use upon a highway, but upon a Ford engine set on four pedestals and connected to an electric dinamometer. The report is of such length that we would not be justified in discussing it in detail. The report plainly discloses that the tests were conducted under conditions with a normal supply of oil. The report contains this statement.
"* * * It would appear then, that for an engine operating normally, a liquid lubricant is necessary to dissipate the heat whether Pyr-Oil is used or not and that in the presence of the liquid lubricant, the Pyr-Oil is superfluous.
*898 "The fact that engine parts wear is evidence that contact of moving parts does occasionally occur. It is possible that at such times a graphitic film on the surfaces would to some extent at least, protect them. However, as a comparative test for wear in normally operated engines would cover a very long period of time, no such test has been attempted. * * *"
The Auto Mechanics Department of the North Dakota Agricultural College conducted a test of Pyr-Oil and reached a conclusion as to its benefits contrary to that reached by the Bureau of Standards. In the report of the latter, in discussing the former, it is said:
"* * * The graphite deposited from the Pyr-Oil probably lubricated the bearing and enabled it to run longer and with less friction after the supply of oil had been cut off than would the small amount of oil which remained in the bearing alone. Bearings can undoubtedly be designed to run with graphitic lubrication alone. However, such a bearing would be incapable of sustaining loads comparable to those which can safely be put on an equal bearing lubricated with oil. The bearing in which the test was made at the North Dakota State College would probably have done as well with any form of graphite lubricant. * * * The possible benefit of the graphite would occur if for any reason the oil supply should so diminish that metallic contact would occur in the absence of the graphite. * * *"
The witness Dill testified: "* * * I am not a lubricating expert. I am not an expert on matters of lubrication of internal combustion engines, so that I have no idea what `boundary lubricating conditions' in regard to internal combustion engines means. * * * At all times during our test, we had a full and ample supply of oil film, so that the oil film was there and not broken. * * *"
At one point in his testimony, the witness Brooks stated: "I formed the opinion as the result of my tests and calculations that it was unlikely Pyr-Oil would be strongly beneficial in an engine."
The witness Dill also stated: "I can tell you that the engine ran approximately 10 minutes after the run without Pyr-Oil and approximately 12 minutes on the run with Pyr-Oil. That is just brute memory without any notes to refresh my recollection. It is of some tests I made seven years ago. I have run quite many tests on various matters since 1931. * * * I was first approached with regard to my being a witness in this proceeding last week. Then I first started to think again about Pyr-Oil."
Again this same witness, when interrogated concerning Commission's Exhibit 19, answered "Yes" to the following question: "And you say, elsewhere in the report, that, `The fact that engine-parts wear is evidence that contact of moving parts does occasionally occur. It is possible that at such times a graphitic film on the surfaces would to some extent, at least, protect them.'"
As to one test conducted by the Bureau of Standards, the witness answered "Yes" to the following question: "And you found in this case that the difference favored the use of Pyr-Oil by about 40 per cent"?
There also appears questions and answers as follows:
"Q. You draw the conclusion, then, that with Pyr-Oil added to the oil, before you drain the crankcase and eliminate the oil, you would not expect that it would run ten times as long as if the Pyr-Oil had not been put in ? A. That is my conclusion, to the best of my belief.
"Q. And that is as far as you can go with this test? A. Entirely."
From a study of Commission's Exhibit 19, and the testimony of Brooks and Dill, we have little hesitancy in reaching the conclusion that they afford no substantial support for the Commission's conclusion that petitioner's product is without merit. Certainly they fail to furnish even a scintilla of support that the product is without merit when used under "boundary" conditions. In fact, neither the report of the Bureau of Standards nor the testimony of these witnesses purport to relate to a test made under such conditions. On the contrary, they expressly limit themselves to a motor operated with a normal supply of oil, or under "full-film" conditions. Even when the product was employed under "full-film" conditions, the result of their test, according to the report and testimony, is anything but certain. They do not claim that the product is without benefit, but dispute that it is "strongly" or "materially" beneficial.
It is also of some significance that the tests upon which the Commission relies were made more than seven years previous to the hearing. Much of the testimony of Brooks and Dill was from memory, which *899 perhaps accounts for their uncertainty and indecision in numerous respects. It is no reflection upon the Bureau of Standards for us to conclude that the report and testimony concerning such tests are far from convincing. In this connection, it is difficult to understand why the Commission would go to hearing on such an important controversy, relying upon tests so uncertain and dubious both in their nature and result. A study of the record is convincing that the overwhelming weight of the testimony is contrary to the Commission's contention, and under such circumstances, it occurs to us that the Commission would have discerned the importance, and perhaps the necessity, of making such tests and experiments as would demonstrate, at least to a reasonable certainty, the validity of the charge which it had the burden of sustaining.
We think it is also relevant to point out that whatever probative value may be attached to the Commission's Exhibit 19, and the testimony of Brooks and Dill, is materially weakened by the Commission's Exhibit 6. This was a circular letter issued by the Bureau of Standards revised to April 3, 1934, which was subsequent to the tests referred to in Exhibit 19, and those made by Brooks and Dill. This circular letter was issued, so it was stated, in response to numerous inquiries regarding various lubricants containing graphite. To our minds this letter completely illustrates the fog of uncertainty and indecision which prevails in the Bureau of Standards concerning the results attending the use of graphite in lubricants. They again reaffirm their negative conclusion: "No evidence has been obtained that the use of graphite in the oil will lower friction in engine bearings, when the engine is operated under full-film conditions." The letter goes on to state: "When graphite is added to the mineral oil * * * the viscosity of the oil after mixing is lower than the viscosity of the original oil." At a later point in the letter is this statement: "No appreciable increase in engine horse power or in maximum car speed would be expected as a result of using graphite in an engine in good mechanical condition, unless the blending with the graphited preparation produces a decrease in viscosity." Taking these two statements together, they seem to lead to the incongruous result that the addition of graphite decreases the viscosity of the oil, but that no benefit will result from its use unless it produces such decrease. At another point, the letter states: "It is known that graphite tends to fill up the pores of cast-iron surfaces and to adhere very tenaciously. This may result in a somewhat smoother surface and may produce a slight lowering in the friction. * * * The use of a lubricated gasoline may be of some benefit in tending to reduce the wear on the piston rings and cylinders." Thus, all through this letter, as in Exhibit 19, and the testimony of Brooks and Dill, the information (if it may be called such) is what graphite may or might do, rather than what it does or does not do. Information of such a speculative and uncertain nature can afford little, if any, support to findings based thereon.
The report of the Commission's Examiner is before us and contains an exhaustive and, we think, accurate review of the evidence in the case. We shall not refer to the conclusion reached by the Examiner, except to state that his findings of matters material to the instant order are at variance with the findings of the Commission. As we have already said, the Commission is not bound by the findings of its Examiner, but under the circumstances of this case, we think the Examiner's report materially detracts from the Commissions claim that its findings are substantially supported. We recognize that our province is not to weigh the testimony, but we think it is not inappropriate to briefly refer to some of the direct positive testimony which contradicts many of the uncertain statements made by the witnesses relied upon by the Commission, and the inferences indulged in by such witnesses. In fact, the testimony of numerous of the Commission's witnesses is at variance with its findings, and contradicts the inferences of the witnesses Brooks and Dill, relied upon by the Commission.
Professor Linsenmeyer, a Commission witness, is head of the Mechanical Engineering Department at the University of Detroit, and has been for fourteen years. He has made extensive studies of lubricants. In one test he took two new Ford automobiles directly from the factory and drained the lubricant from both cars. One car was then filled with a colloidal graphite treated oil, and the other with an ordinary high-grade motor oil. The cars were driven upon the highway in a test under the same or similar conditions. The car with ordinary motor oil was destroyed at a distance of 3,739 miles, and the car containing *900 the colloidal graphite was driven a distance of 5,058 miles, at which time the drain plug was removed and the car operated a further distance of 16 miles. Regarding petitioner's claim that its product reduces friction as much as 50%, the witness testified:
"Well, the fact that we did get this improved economy and the decrease in friction, of my value of 36%, I think would not invalidate the claims of 50, for, after all, our condition was at the one speed. Perhaps if we had selected another speed it would have been a little more than 36%, so it met these claims of 50%. I think that is merely relative. I do know it would improve the frictional properties to a substantial amount."
Raymond Szymanowitz, another Commission witness, is a technical director of the Acheson Colloids Corporation. He has a degree of Bachelor of Science from Cooper Union. His experience has been extensive in the industrial field, and he has made special studies in connection with colloidal graphite. He testified concerning comparative tests of plain oil and graphited oil. In each case the oil supply was shut off after the motor had operated for thirty minutes. The result of these tests was illustrated by Commission's Exhibit 1. Referring to this Exhibit, the witness said: "* * * You will see in the case of the plain oil, there was a minimum rise in friction to seizure at this point. However, when the graphited oil was used the supply cut off at the same time, it continued to run and seizure took place I believe in 26 hours."
At another point this witness was asked: "Do you have any data showing the effect of what protection against burned out bearings colloidal graphite can give"? and he answered: "Well, the graph which I have referred to as Commission's Exhibit No. 1, for identification, you will note the line showing the point at which the oil supply became first exhausted or seriously diminished. In that particular exhibit you will find a graph which shows that even in the absence of oil the graphite has permitted the bearing to function unimpaired for quite a few hours."
He further related that the cars used in the test were run a certain distance, and at the end of that time the oil was removed from the crankcase. After the oil was thus drained, the cars whose lubricant had been treated with colloidal graphite would run as far as 47 miles. Concerning petitioner's representation that the use of colloidal graphite results in a car running an amazing distance without oil, the witness said: "It depends on what you mean by the words `amazing distance.' Some people might think that was amazing as a distance. * * *"
Frank S. Spring, another witness for the Commission, is the automobile engineer for the Hudson Motor Car Company, and has had experience with lubricants and colloidal graphite in motor cars. He testified "We have made various tests with it, and we find it is rather difficult to come to any very definite conclusions, because the difference between ordinary lubricant and a lubricant when this colloidal graphite is added, is not as great as the difference in the ordinary run of automobile engines. I think it is beneficial in new engines during the run-in period. We have made perhaps 100 tests, and my conclusion is that it is beneficial on new cars or new engines until they are well broken in."
It will be noted that so far we have alluded only to the testimony of witnesses for the Commission, and certain of its Exhibits. To go further might subject us to the criticism of weighing the testimony, which we are endeavoring to avoid. We shall, therefore, only make brief reference to the voluminous testimony, oral and documentary, offered by the petitioner.
Among such witnesses is George A. Abbott, head of the Chemical Department of the University of North Dakota. As a chemist of wide experience, he has interested himself for many years in lubricating problems, including that of colloidal graphite. He testified as to its beneficial effect, and, in substance, that colloidal graphite actually penetrates the metal with which it comes in contact. He further gave it as his opinion that the word "adsorption" was preferable to the word "penetrate."
C. G. Williams, Director of Research of the Institution of Automobile Engineers, testified that tests made by him disclosed that the addition of colloidal graphite to fuel brought about a reduction in wear ranging from 12 to 37 per cent for the piston rings and ranging from 27 to 50 per cent for the cylinder wear.
Petitioner's Exhibit 3 is an article on "The Graphoid Layer on Bearing Surfaces" by Professors Finch and E. J. Whitmore, scientists of England. This article *901 discloses that experiments with bearing surfaces lubricated with oils containing colloidal graphite point to the formation of an extremely thin and adherent layer parallel to the surface, and that its effect persists long after the supply of liquid lubricant is removed.
Doctor Charles F. Mabery, Professor of Chemistry, Case School of Applied Science, Cleveland, Ohio, in a paper before the American Society of Mechanical Engineers, states: "* * * Natural graphite serves an excellent purpose on cast-iron bearings, acting as a surface evener of the porous metals. * * * "
He further states: "There is probably no variety of lubrication in which colloidal graphite shows its economic value to better advantage than in reducing the friction on automobile bearings. * * *"
T. C. Thomsen, a chemist of repute, with reference to tests conducted by him, said: "* * * The tests showed that with colloidal graphite the piston-ring wear was halved."
Professor Erwin H. Hamilton, Associate Professor of Automotive Engineering at the New York University, testified that as a result of a number of researches and tests for various companies "an automobile conditioned with colloidal graphite will run longer a greater distance without oil in the crankcase, without damage to the parts, than will an automobile conditioned with ordinary lubricating oil. * * *"
A number of other witnesses gave testimony to the same effect. In addition, numerous articles from engineering, automobile and mechanical publications were offered in evidence in support of the claim made by the petitioner as to the beneficial results attained from the use of its product.
The conclusion follows, from what we have said, that the Commission's findings that petitioner's product produces no beneficial result under "boundary" conditions of lubrication, is without substantial support. On the other hand, the record convincingly discloses to the contrary. There is a difference of opinion, however, as to the extent of such benefits. This difference of opinion exists largely among petitioner's witnesses, as the Exhibits and witnesses relied upon by the Commission dealt mostly, if not entirely, with the effect under "full-film" conditions of lubrication. Petitioner's representation that its product will enable a motor car to operate an "amazing distance" without oil, or that its product is a "perfect" lubrication, evidently is some exaggeration. To what extent, however, it is difficult to say. Such terms are largely a matter of personal opinion. What might be an "amazing distance" to one person might cause no surprise to another. So far as we know, there is nothing "perfect" in this world, but still it is a common term, which undoubtedly means nothing more than that the product is good or of high quality. We can conceive of situations where the use of such words might be deceptive and even fraudulent. As used by petitioner, however, we are of the opinion that they are nothing more than a form of "puffing" not calculated to deceive.
Paragraphs 1 and 5 of the Commission's cease and desist order are not in controversy in view of the admissions contained in petitioner's answer to the complaint. Paragraphs 2, 3, and 4 of the order, set forth heretofore in this opinion, can not be affirmed as written for the reason that the findings upon which they are predicated are not substantially supported. Such paragraphs, therefore, are modified so as to require petitioner to cease and desist in the manner provided only as to the representations concerning "Koatsal" when the same is used as a lubricant in a motor operated under "full-film" conditions.
The cease and desist order is modified as indicated, and as modified is affirmed.
On Petition for Rehearing.
MAJOR, Circuit Judge.
We have before us a petition for rehearing, as well as a form of enforcement decree proposed by both petitioner and respondent. We are of the opinion that the decree proposed by the respondent is in conformity with our opinion. We held that the cease and desist order, concerning petitioner's product, when used under "boundary" conditions, was without substantial support. To go further is to enter a controverted field where we are not permitted, and did not intend, to enter. In other words, the testimony as to the effect of the product in question under "full film" conditions is in dispute and we did not hold that the evidence was insufficient to sustain respondent's order in this respect. The decree proposed by the petitioner, as contended for in its petition for rehearing, would require a holding in its favor when its product is used under "full film" as well as under "boundary" conditions. This argument was given due consideration *902 in the opinion and decided adversely to petitioner.
Our attention is called to an apparent error on page 2 of the opinion. The controverted issues are there stated under subdivisions (b), (d) and (e), which should be (d), (e) and (f). (b) is therefore eliminated and there is inserted in its place:
"(f) That an automobile conditioned with `Koatsal' can run an amazing distance without oil in the crankcase without damage to any part."
The petition for rehearing is denied, the decree proposed by petitioner refused, and that proposed by respondent approved.
NOTES
[1] The colloidal graphite in Koatsal, when mixed in sufficient quantities in the lubricant of an internal combustion engine, becomes firmly affixed and adheres to the surfaces of the moving parts, forming on the surface thereof what is called a graphoid surface.
6. When colloidal graphite is mixed in sufficient quantities in the lubricant of an internal combustion engine or in its gasoline fuel, it brings about, under boundary conditions of lubrication, a substantial reduction in the friction developed between the surfaces in moving contact with each other to a degree which the tests and experiments made fix as in the neighborhood of 50%, the result of one of such experimental tests being a reduction of exactly 50% in wear of the moving parts.
[2] It was stipulated that the percentage of colloidal graphite in Pyr-Oil, as tested by the Bureau of Standards, is the same as that contained in "Koatsal" as sold and distributed by the petitioner.
|
Kevin James shows some skin in his newest flick, “Here Comes the Boom.” Photo:
When 47-year-old Kevin James climbs through a window in a waistline-reducing black ensemble in the trailer for “Here Comes the Boom,” out today, audiences may feel a familiar feeling trickle through their spines. One minute and 50 seconds later, as a shirtless James, shimmering with sweat, whips out a one-two punch, it was clear the moment had arrived:
Ladies and gentleman, Kevin James is sexy as hell, and I’m here to prove it:
Exhibit A: Confidence doth a stud make
James spends roughly half of his new movie, which is about wrestling, shirtless. It takes a whole lot of guts for a 5-foot-8, over-200-pound man to bounce around, sans shirt, for all the world to see. His nude torso wrestles with Adonises, jiggling with pride, and our boy works it. Consequently, as the film progresses, man boobs begin to look more like pecs, and a belly starts to look more like a six-pack — because James’ confidence convinces you.
Exhibit B: D.I.L.F.s have more fun
With the rise of mega-D.I.L.F.s like Brad Pitt, it’s no longer lame to hop on the papa train. A man who has spread his God-given seed (James has three kids, ages 1, 5 and 7) is experienced in all the right ways: A father in his late 40s has seen the world, and he’s gone through (at least one) midlife crisis. A healthy 401(k) is a nice bonus, too.
Exhibit C: ‘Fat’ is the new black
Get over it, size-ists: Fat is in. If young pop stars such as Christina Aguilera and Lady Gaga can embrace their curves, certainly we can cut a cuddly middle-aged dude some slack. Besides, “Mike & Molly” is a successful sitcom starring two big-boned individuals, America recently rallied behind luscious lady anchor Jennifer Livingston and Doritos are available in a lot of really great flavors.
Exhibit D: Long Island accents are hot!
James’ Stony Brook, LI, tough tawk makes hearts flutter. He’s exactly the type of Long Island boy who will beat the bejeezus out of a guy one minute and write a misspelled love letter the next. He loves his mama’s home cooking, and he just wants to find someone in life who can dote on him the way she does.
Exhibit E: Never underestimate the power of a pole
On an unforgettable Season 8 episode of his long-running sitcom, “The King of Queens,” James (as Doug) let us in on a little secret. Disappointed by his wife Carrie’s (Leah Remini) pole-dancing skills in the bedroom, he jumps on that bad boy and shows her how it’s done. And surprise! Turns out he’s a natural, flipping and sliding around like a pro. “Magic Mike 2: More Magical”? It could work, Kevin. Just please don’t make another “Paul Blart: Mall Cop.” |
Soluble and Insoluble Yeast β-Glucan Differentially Affect Upper Respiratory Tract Infection in Marathon Runners: A Double-Blind, Randomized Placebo-Controlled Trial.
In a previous study, consumption of a dairy beverage incorporating insoluble β-glucan decreased upper respiratory tract infection (URTI) symptomatic days and severity in marathon runners. In this report, we extended our previous findings by presenting data on a dairy beverage containing soluble β-glucan and URTI in marathon runners. Healthy adults running in the 2017 Austin Marathon consumed dairy beverages (250 mL/day) containing 250 mg of insoluble (n = 69) or soluble (n = 76) baker's yeast β-glucan (Wellmune®) or placebo (n = 133) for the 45 days before, day of, and 45 days after the marathon (91 days total). Participants completed a daily online survey assessing compliance and URTI symptoms, which were evaluated using the Jackson Index and confirmed by the study physician. Total severity of URTI was significantly lower in the insoluble yeast β-glucan group compared to the placebo group, but was not different between the soluble yeast β-glucan group and placebo group. Severity ratings for nasal discharge were significantly lower in both the insoluble and soluble yeast β-glucan groups compared to the placebo group. Additionally, severity rating for sore throat was lower in the insoluble, but not the soluble yeast β-glucan group compared to the placebo group. The insoluble yeast β-glucan group, but not the soluble yeast β-glucan group also reported fewer URTI symptomatic days compared to the placebo group. The results suggest that soluble and insoluble yeast β-glucan, incorporated into a food matrix, differentially affected exercise-induced URTI in marathon runners. |
Q:
Symfony 2.1 custom field for check boxes
I have used custom form field tutorial http://symfony.com/doc/current/cookbook/form/create_custom_field_type.html and it created a gender selection in radio buttons. So far so good.
I need to show checkboxes with multiple selection. How can I add thhem?
A:
This field may be rendered as one of several different HTML fields, depending on the expanded and multiple options as shown in this table http://symfony.com/doc/current/reference/forms/types/choice.html#select-tag-checkboxes-or-radio-buttons from Symfony book.
|
Lewis Baines
Lewis Baines (born 10 October 1998) is an English professional footballer who plays as a defender for Chorley.
Career
Baines turned professional with Fleetwood Town in the summer of 2017, and spent loan spells with Bamber Bridge and Ashton United before signing a new contract in July 2018. He made his senior debut on 13 November 2018 in the EFL Trophy. He moved on loan to Chorley in December 2018, and to Stockport County in January 2019.
On 25 July 2019, Baines completed a permanent deal to join newly-promoted National League side Chorley.
References
Category:1998 births
Category:Living people
Category:English footballers
Category:Fleetwood Town F.C. players
Category:Bamber Bridge F.C. players
Category:Ashton United F.C. players
Category:Chorley F.C. players
Category:Stockport County F.C. players
Category:Association football defenders |
Q:
Setting the MSVC runtime in CMake
I'm following the instructions in the CMake FAQ entry "How can I build my MSVC application with a static runtime?" to centralize selection of the MSVC runtime for a bunch of nested CMake projects (they are pulled in as Git submodules and added to the master project using CMake's find_package() directive).
So, I wrote this CMake macro:
macro(configure_msvc_runtime)
if(MSVC)
# Default to statically-linked runtime.
if("${MSVC_RUNTIME}" STREQUAL "")
set(MSVC_RUNTIME "static")
endif()
# Set compiler options.
set(variables
CMAKE_C_FLAGS_DEBUG
CMAKE_C_FLAGS_MINSIZEREL
CMAKE_C_FLAGS_RELEASE
CMAKE_C_FLAGS_RELWITHDEBINFO
CMAKE_CXX_FLAGS_DEBUG
CMAKE_CXX_FLAGS_MINSIZEREL
CMAKE_CXX_FLAGS_RELEASE
CMAKE_CXX_FLAGS_RELWITHDEBINFO
)
if(${MSVC_RUNTIME} STREQUAL "static")
message(STATUS
"MSVC -> forcing use of statically-linked runtime."
)
foreach(variable ${variables})
if(${variable} MATCHES "/MD")
string(REGEX REPLACE "/MD" "/MT" ${variable} "${${variable}}")
endif()
endforeach()
else()
message(STATUS
"MSVC -> forcing use of dynamically-linked runtime."
)
foreach(variable ${variables})
if(${variable} MATCHES "/MT")
string(REGEX REPLACE "/MT" "/MD" ${variable} "${${variable}}")
endif()
endforeach()
endif()
endif()
endmacro()
I call this macro at the beginning of my root CMakeLists.txt (before any add_library() or add_executable() call is made) and add a little bit of debugging prints:
configure_msvc_runtime()
set(variables
CMAKE_C_FLAGS_DEBUG
CMAKE_C_FLAGS_MINSIZEREL
CMAKE_C_FLAGS_RELEASE
CMAKE_C_FLAGS_RELWITHDEBINFO
CMAKE_CXX_FLAGS_DEBUG
CMAKE_CXX_FLAGS_MINSIZEREL
CMAKE_CXX_FLAGS_RELEASE
CMAKE_CXX_FLAGS_RELWITHDEBINFO
)
message(STATUS "Initial build flags:")
foreach(variable ${variables})
message(STATUS " '${variable}': ${${variable}}")
endforeach()
message(STATUS "")
Then, I run CMake to generate a Visual Studio solution like so:
cmake -G "Visual Studio 9 2008" ..\.. -DMSVC_RUNTIME=dynamic
and I get the following outputs:
-- MSVC -> forcing use of dynamically-linked runtime.
-- Initial build flags:
-- 'CMAKE_C_FLAGS_DEBUG': /D_DEBUG /MDd /Zi /Ob0 /Od /RTC1
-- 'CMAKE_C_FLAGS_MINSIZEREL': /MD /O1 /Ob1 /D NDEBUG
-- 'CMAKE_C_FLAGS_RELEASE': /MD /O2 /Ob2 /D NDEBUG
-- 'CMAKE_C_FLAGS_RELWITHDEBINFO': /MD /Zi /O2 /Ob1 /D NDEBUG
-- 'CMAKE_CXX_FLAGS_DEBUG': /D_DEBUG /MDd /Zi /Ob0 /Od /RTC1
-- 'CMAKE_CXX_FLAGS_MINSIZEREL': /MD /O1 /Ob1 /D NDEBUG
-- 'CMAKE_CXX_FLAGS_RELEASE': /MD /O2 /Ob2 /D NDEBUG
-- 'CMAKE_CXX_FLAGS_RELWITHDEBINFO': /MD /Zi /O2 /Ob1 /D NDEBUG
Now, the thing is that when I start Visual Studio and examine the project properties under "C/C++, Code Generation", I see that the "Runtime Library" setting is not consistent with the options printed in the shell. Under the "Release", "MinSizeRel" and "RelWithDebInfo" configurations, I get the expected results ("Multi-threaded DLL /MD", but the "Debug" configuration still displays "Multi-threaded /MT").
Also, when I force use of the statically-linked runtime, I get similar results. If I run
cmake -G "Visual Studio 9 2008" ..\.. -DMSVC_RUNTIME=static
I get the following outputs:
-- MSVC -> forcing use of statically-linked runtime.
-- Initial build flags:
-- 'CMAKE_C_FLAGS_DEBUG': /D_DEBUG /MTd /Zi /Ob0 /Od /RTC1
-- 'CMAKE_C_FLAGS_MINSIZEREL': /MT /O1 /Ob1 /D NDEBUG
-- 'CMAKE_C_FLAGS_RELEASE': /MT /O2 /Ob2 /D NDEBUG
-- 'CMAKE_C_FLAGS_RELWITHDEBINFO': /MT /Zi /O2 /Ob1 /D NDEBUG
-- 'CMAKE_CXX_FLAGS_DEBUG': /D_DEBUG /MTd /Zi /Ob0 /Od /RTC1
-- 'CMAKE_CXX_FLAGS_MINSIZEREL': /MT /O1 /Ob1 /D NDEBUG
-- 'CMAKE_CXX_FLAGS_RELEASE': /MT /O2 /Ob2 /D NDEBUG
-- 'CMAKE_CXX_FLAGS_RELWITHDEBINFO': /MT /Zi /O2 /Ob1 /D NDEBUG
And yet all configurations produce the "Multi-threaded /MT" value for the "Runtime Library" setting.
What am I doing wrong, or if this is a bug in CMake (2.8.7) or something?
For what it's worth, if I generate Visual Studio 2010 project files, I get a different value for the "Debug" configuration, but still not the one I selected.
In all cases, the setting appears in regular font for the "Debug" configuration whereas it appears in bolded font for the other configurations, hinting that those are overrides. Moreover,if I open the XML project files, I find that the "Debug" configuration has no setting for the "RuntimeLibrary" attribute of the "Tool" element with the "Name=VCCLCompilerTool" attribute. All other configurations have an explicit setting.
A:
It seems all the while I was working on this, I forgot to remove the bad CMake configuration I'm trying to replace.
Further down the build script, I had left this little bugger:
set(CMAKE_CXX_FLAGS_DEBUG
"/DWIN32 /D_WINDOWS /EHsc /WX /wd4355 /wd4251 /wd4250 /wd4996"
CACHE STRING "Debug compiler flags" FORCE
)
Basically, I was overriding the results of by configure_msvc_runtime() macro with build flags that did not set the MSVC runtime.
A:
I took your code and generalized it to work for every existing configuration and not just for Debug/Release/RelWithDebInfo/MinSizeRel.
Also I made it to work with gcc too - check it out here
A:
This functionality will be improved with the release of cmake-3.15.
CMAKE_MSVC_RUNTIME_LIBRARY
CMP0091
It should be a matter of setting CMAKE_MSVC_RUNTIME_LIBRARY, for example (from docs) to set "multi-threaded statically-linked runtime library with or without debug information depending on the configuration":
set(CMAKE_MSVC_RUNTIME_LIBRARY "MultiThreaded$<$<CONFIG:Debug>:Debug>")
|
New City Council Speaker Corey Johnson pledged at his inauguration Sunday to tackle the city’s “affordability crisis” and its impact on working-class residents, the homeless and local businesses.
Johnson, a Democrat representing Manhattan’s West Side from the southern tip of Central Park down to Canal Street, committed to working with state lawmakers within the coming months to extend rent protections for more than one million city apartments and close loopholes in the rent stabilization law that Johnson said has allowed landlords to deregulate units.
“The affordability crisis that grips our city threatens the very existence of our neighborhoods. New Yorkers who have lived in the same community their entire lives now find themselves priced out, unable to afford their rent or even their groceries,” Johnson told the crowd at the ceremony held in an auditorium of the Fashion Institute of Technology. “Many working families are living literally paycheck to paycheck, one missed shift or one medical expense away from eviction or bankruptcy.”
Johnson was sworn in by Senate Minority Leader Chuck Schumer and was joined by dozens of his colleagues in the Council and other high ranking officials, including Mayor Bill de Blasio, who also spoke at the event.
The new leader of the city’s legislative body related the city’s soaring rents to his own experience moving to New York City from his small Massachusetts hometown, shortly after he came out as a gay teenager in 1999. It set off a “chain reaction” that helped him deal with his mental struggles and launch his career in politics, he said.
“When I came out to my family; when I came out at school, I was pretty much before that literally suicidal,” said Johnson, who illegally lived in a New York University dorm when he moved to the city. “That experience of coming here in 2001 as a 19 year old on a wing and on a prayer and on a dream with two suitcases and knowing two people — that’s becoming harder and harder in 2018. I don’t know how many 19-year-olds can come here who don’t come from a wealthy family and have that experience.”
Johnson, who is also the only openly HIV-positive elected official in the state, will take the reins after a crowded and contentious race for the speakership, which included tense moments between otherwise like-minded Democrats. Many of those losing candidates were in attendance. Though Council members Ydanis Rodriguez and Inez Barron — who cast the only dissenting vote for the speakership, a vote for herself — were no shows.
A spokeswoman for Rodriguez said the councilman missed the inauguration because the ceremony conflicted with his youngest daughter’s birthday party. Barron’s office did not respond to request for comment.
Johnson described the race as “emotionally challenging” but said he was still able to call the candidates his friends. The speaker and de Blasio certainly seemed to be on good terms as well. The mayor praised Johnson as a politician with a “burning desire for justice.”
“We send a message to the whole country that an HIV-positive man is one of the great leaders of our city,” de Blasio said.
The speaker was complimented as someone who could keep the mayor in check, however.
Both Schumer and City Comptroller Scott Stringer referenced Johnson’s announcement earlier this month of the creation of a new Council investigations unit to probe city agencies.
“Corey Johnson has made it clear that this Council is going to be a Council of independence, of issue-focus,” Stringer said, “and when you talk to the Council members, his colleagues, there’s really a moment here that Corey Johnson has become the person we need to lead us for the next four years.” |
#!/usr/bin/perl -w
use strict;
use File::Spec::Functions qw(catdir updir);
use FindBin;
use lib catdir $FindBin::Bin, updir, 'lib';
use bric_upgrade qw(:all);
exit unless test_table 'story_container_tile';
exit if test_column 'story_container_tile', 'related_story__id';
do_sql
# Handle story.
q{DROP INDEX fkx_sc_tile__related_story},
q{
ALTER TABLE story_container_tile
DROP CONSTRAINT fk_sc_tile__related_story
},
q{
ALTER TABLE story_container_tile
RENAME related_instance__id TO related_story__id
},
q{
CREATE INDEX fkx_sc_tile__related_story
ON story_container_tile(related_story__id)
},
q{
ALTER TABLE story_container_tile
ADD CONSTRAINT fk_sc_tile__related_story
FOREIGN KEY (related_story__id)
REFERENCES story(id) ON DELETE CASCADE;
},
# Handle media--no preexisting constraint or index.
q{
ALTER TABLE media_container_tile
RENAME related_instance__id TO related_story__id
},
q{
CREATE INDEX fkx_mc_tile__related_story
ON media_container_tile(related_story__id)
},
q{
ALTER TABLE media_container_tile
ADD CONSTRAINT fk_mc_tile__related_story
FOREIGN KEY (related_story__id)
REFERENCES story(id) ON DELETE CASCADE
},
# Add FK constraint for mediacontainer_tile.related_media__id
q{
CREATE INDEX fkx_mc_tile__related_media
ON media_container_tile(related_media__id)
},
q{
ALTER TABLE media_container_tile
ADD CONSTRAINT fk_mc_tile__related_media
FOREIGN KEY (related_media__id)
REFERENCES media(id) ON DELETE CASCADE
},
;
|
---
abstract: 'In this thesis, we study singular pseudo-differential operators defined by groupoids satisfying the Lauter-Nistor condition, by a method parallel to that of manifolds with boundary and edge differential operators. The example of the Bruhat sphere is studied in detail. In particular, we construct an extension to the calculus of uniformly supported pseudo-differential operators that is analogous to the calculus with bounds defined on manifolds with boundary. We derive a Fredholmness criterion for operators on the Bruhat sphere, and prove that their parametrices up to compact operators lie inside the extended calculus; we construct the heat kernel of perturbed Laplacian operators; and prove an Atiyah-Singer type renormalized index formula for perturbed Dirac operators on the Bruhat sphere using the heat kernel method.'
author:
- Bing Kwan SO
title: 'Pseudo-differential operators, heat calculus and index theory of groupoids satisfying the Lauter-Nistor condition'
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[Pseudo-differential operators, heat calculus and index theory of groupoids satisfying the Lauter-Nistor condition ]{}
[Bing Kwan SO ]{}
A thesis submitted in partial fulfillment of the requirements for\
the degree of Doctor of Philosophy\
at The University of Warwick.\
January 2010.
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$ \; \\ $ I wish to express my sincere gratitude to my supervisor Prof. J. Rawnsley for his continual guidance throughout the period of postgraduate studies. I wish to thank The Croucher Foundation, whose scholarship makes this project possible. Last but not the least, my parents for their continual support through the telephone from the other side of the globe, during the last three very difficult years of my lifetime.
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$ \; \\ $ Unless otherwise specified, all material in this thesis is the result of my own, independent, and original work to the best of my knowledge. The research was done under the supervision of Prof. J. Rawnsley during the period 2006-2010 for the degree of Doctor of Philosophy at The University of Warwick. The work submitted has not been previously included in a thesis, dissertation or report submitted to any institution for a degree, diploma or other qualification.
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SO, Bing Kwan
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$ \; \\ $ In this thesis, we study singular pseudo-differential operators defined by groupoids satisfying the Lauter-Nistor condition, by a method parallel to that of manifolds with boundary and edge differential operators. The example of the Bruhat sphere is studied in detail. In particular, we construct an extension to the calculus of uniformly supported pseudo-differential operators that is analogous to the calculus with bounds defined on manifolds with boundary. We derive a Fredholmness criterion for operators on the Bruhat sphere, and prove that their parametrices up to compact operators lie inside the extended calculus; we construct the heat kernel of perturbed Laplacian operators; and prove an Atiyah-Singer type renormalized index formula for perturbed Dirac operators on the Bruhat sphere using the heat kernel method.
Introduction: From singular to groupoid pseudo-differential calculus
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Traditionally, the way to study singular pseudo-differential operators involves studying underlying manifolds with embedded boundaries or corners. These boundaries are always defined by the zero set of some functions (known as the boundary defining functions $\rho $), with non-vanishing differentials near the boundary. As a consequence, a neighborhood of the boundary $\partial \mathrm M$ is of the form $[0 , 1) \times \partial \mathrm M$ (with the closed interval $[0, 1) $ parameterized by $\rho $). Then, one would typically consider the calculus of pseudo-differential operators whose kernels have poly-homogeneous expansions in $\rho $ near the boundary (see [@Mazzeo;EdgeRev] and the reference there).
The use of groupoids for studying the geometry of manifolds with boundaries (or corners) was a much later development. Early use of groupoids in pseudo-differential analysis include the convolution algebra defined on the holonomy groupoid of a regular foliation by Connes et. al. (see [@Connes;Book] for a review). The notion of pseudo-differential operators on a groupoid was developed by Nistor, Weinstein and Xu [@NWX;GroupoidPdO]. Subsequently, Monthubert [@M'bert;CornerGroupoids] showed that the $b$-calculus is, indeed, the vector representation of pseudo-differential operators on some groupoids. The theory is further formalized by Ammann et. al. into so called Lie manifolds, or manifolds with Lie structure at infinity [@Nistor;Polyhedral; @Nistor;LieMfld; @Nistor;Polyhedral2].
Despite the development of the groupoid theory, most, if not all analysis was done on examples very similar to the manifold with boundary case.
In this thesis, we study the analysis of pseudo-differential operators in a systematic way parallel to that of singular pseudo-differential operators on manifolds with boundary (i.e. [@Melrose;Book] etc.). Our work is motivated by the study of the Poisson (co)-differential operator and its homology. These invariants are difficult to compute. The only attempt to develop any form of machinery seems to be [@So;MPhil], and the Laplacian defined there is not elliptic in the usual sense. Also it is clear that the singularities are not explicitly defined by any boundary defining function. Moreover, even if the homology is computed directly, the result is often infinite dimensional, and therefore not very meaningful. For this reason, we consider renormalized index theory, which gives finite results.
The approach in this thesis is based on the principle that all singular pseudo-differential operators are defined by vector representations of operators on the\
groupoids. Therefore instead of studying the calculus of singular pseudo-differential operators, one only needs to study non-singular pseudo-differential operators on the groupoid. The main part of this thesis, Sections 2-5, is an account of the technical details on how this principle is implemented, particularly to the example of the Bruhat Poisson structure on the sphere ${\mathbb C}{\mathrm P}(1)$.
Here, we shall give an overview of our approach. In Section 2, We collect together background material from several standard sources, which is needed for the thesis. We begin with reviewing the well known formalism of uniformly supported groupoid pseudo-differential operators by Nistor et. al. [@NWX;GroupoidPdO]. The uniformly supported calculus is comparable to the small calculus in manifolds with boundary. We shall also define the notion of a Dirac operator on a groupoid. Then we shall introduce some examples, most notably the symplectic groupoids of the Bruhat Poisson structure on flag manifolds, where the Bruhat sphere is the simplest case.
Section 3 focuses on two questions, which are exact counterparts of [@Melrose;Book Chapter 5]:
1. What (elliptic) pseudo-differential operator on a groupoid has Fredholm vector representation?
2. What does the parametrix of a Fredholm operator on a the groupoid defining the Bruhat sphere look like?
Lauter and Nistor’s [@Nistor;GeomOp] theory gives a quick answer for (1), namely, if an operator is invertible on all the singular leaves, then its vector representation is Fredholm. In the simple case of the Bruhat sphere, question (1) therefore immediately reduces to checking the invertibility of the operator over the only singular leaf. Due to some invariance properties, the natural way to proceed is by Fourier-Laplace transform. We remark that our approach is again parallel to the indicial family formalism for manifolds with boundary (recall that given a partial differential operator $\varPsi$, the indicial family is defined to be the family of differential operators $ (e ^{-i \xi \rho } \varPsi e^ {i \xi \rho } ) |_{\partial {\mathrm M}}, \quad \xi \in {\mathbb C}$, see [@Loya;Pert; @Melrose;Book] for detailed definitions). Indeed, it can be said that Fourier-Laplace transform [*is the right*]{} definition for indicial family. We then turn to describe the parametrix of Fredholm operators on the Bruhat sphere, using the fact that the inverse of a properly supported, invariant pseudo-differential operator is an invariant pseudo-differential operator with exponential decaying kernel. We then generalize the notion to groupoids with sub-exponential growth, and prove that the resulting calculus has a composition rule similar to that of calculus with bounds.
In Section 4, we turn to the heat calculus of Laplacian operators. The treatment here is very different from that of [@Albin;EdgeInd; @Melrose;Book], and much simpler. That is not surprising because the source fibers are just non-singular manifolds with bounded geometry, and the heat kernel is essentially the leaf-wise heat kernel. Therefore the classical construction suffices. Perhaps the only unexpected result is the proof of transverse smoothness, which requires additional growth conditions on the differential of the product map. We shall leave the technical details to Section 4.2.
Given a perturbed Dirac operator that is Fredholm (one satisfying the conditions in Section 3), it is natural to seek an Atiyah-Singer type formula for its Fredholm index. That is the theme of Section 5. Again the technique we use is parallel to that of [@Albin;EdgeInd; @Loya;Pert; @Melrose;Book], and is fairly standard. We use the stereographic coordinates on the open leaf of the Bruhat sphere to define a renormalized trace. The we derive the local index formula. We do have to fall back to the machinery of [@Albin;EdgeInd] to describe the long time behavior of the heat kernel. However, it can be said that the ‘cheating’ occurs already when we use the stereographic coordinates, which effectively serves as a boundary defining function. Nevertheless, our result is stronger than that of [@Albin;EdgeInd] in the sense that we are able to derive an explicit trace defect formula.
Lie groupoids and pseudo-differential operators
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The differential geometry of Lie groupoids
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We begin our technical discussion with the basic definition of a Lie groupoid. Our definition follows the convention of [@Mackenzie;Book2], but with the source and target maps denoted by ${\mathbf s}$ and ${\mathbf t}$ instead of $\alpha $ and $\beta $.
A [*Lie groupoid*]{} ${\mathcal G}\rightrightarrows {\mathrm M}$ consists of:
1. Manifolds ${\mathcal G}$ and ${\mathrm M}$;
2. A unit inclusion ${\mathbf u} : {\mathrm M}\rightarrow {\mathcal G}$;
3. Submersions ${\mathbf s}, {\mathbf t}: {\mathcal G}\rightarrow {\mathrm M}$, called the source and target map respectively, satisfying $${\mathbf s}\circ {\mathbf u} = \operatorname{i d}_{\mathrm M}= {\mathbf t}\circ {\mathbf u};$$
4. A multiplication map $\mathbf m : \{ (a, b) \in {\mathcal G}\times {\mathcal G}: {\mathbf s}(a) = {\mathbf t}(b) \} \rightarrow {\mathcal G},
(a , b) \mapsto a b$ that is associative and satisfies $${\mathbf s}(a b) = {\mathbf s}(b) , \quad {\mathbf t}(a b) = {\mathbf t}(a),
\quad a ( \mathbf u \circ {\mathbf s}(a)) = a = ( \mathbf u \circ {\mathbf t}(a)) a ;$$
5. An inverse diffeomorphism $\mathbf i : {\mathcal G}\rightarrow {\mathcal G}, a \mapsto a^{-1}$, such that ${\mathbf s}(a^{-1}) = {\mathbf t}(a), \\ {\mathbf t}(a^{-1}) = {\mathbf s}(a)$ and $$a a^{-1} = \mathbf u ({\mathbf t}(a)), a^{-1} a = \mathbf u ({\mathbf s}(a)).$$
In this thesis, we assume that the groupoid ${\mathcal G}$ is Hausdorff. This extra assumption is clearly satisfied in all of the examples we shall shortly see. Note that many important groupoids, like holonomy groupoids of foliations, are not Hausdorff.
For simplicity we shall denote a Lie groupoid ${\mathcal G}\rightrightarrows {\mathrm M}$ by ${\mathcal G}$ and call it a groupoid; Also, with an abuse in notation we consider ${\mathrm M}$ as a subset of ${\mathcal G}$ via the unit inclusion $\mathbf u$. For each $x \in {\mathrm M}$, we write $${\mathcal G}_x := {\mathbf s}^{-1} (x) .$$
We say that a groupoid ${\mathcal G}$ is ${\mathbf s}$[*-connected*]{} if ${\mathcal G}_x $ is connected for all $x \in {\mathrm M}$.
Let ${\mathcal G}$ be a Lie groupoid and $a \in {\mathcal G}$. The [*right translation*]{} is the diffeomorphism: $$R_a : {\mathbf s}^{-1} (a) \rightarrow {\mathbf t}^{-1} (a), b \mapsto b a, b \in {\mathcal G}.$$
A [*right-invariant function*]{} on ${\mathcal G}$ is a smooth function $f$ such that $$f (b a) = f (b), \quad \forall a \in {\mathcal G}, b \in {\mathbf s}^{-1} (a) ;$$ A [*right-invariant vector field*]{} on ${\mathcal G}$ is a vector field $X$ such that $d {\mathbf s}X = 0 $ (i.e., $X$ is a vector field along the ${\mathbf s}$-fibers) and $$d R_a (X (b)) = X (b a), \quad \forall a \in {\mathcal G}, b \in {\mathbf s}^{-1} (a) .$$
From the definition, one immediately observes that any right invariant function $f \in C^\infty ({\mathcal G})$ can be written in the form $$f = {\mathbf t}^{-1} \tilde f, \; \text {where} \; \tilde f := \mathbf u ^* f \in C^\infty ({\mathrm M}).$$
### **Lie algebroids and singular foliations**
A [*Lie algebroid*]{} ${\mathcal A}$ is a vector bundle over ${\mathrm M}$, together with a Lie algebra structure $[ \cdot , \cdot ]$ on the space of smooth sections $\Gamma ^\infty ({\mathcal A})$, and a bundle map $\nu : {\mathcal A}\rightarrow T {\mathrm M}$ satisfying $$\nu ([X, Y]) = [\nu (X) , \nu (Y)],
\; \text {and} \; [X , f Y] = f [X, Y] + ({\mathfrak L}_{\nu (X)} f) Y,$$ for any $ X, Y \in \Gamma ^\infty ({\mathcal A}), f \in C^\infty ({\mathrm M}).$
Let $({\mathrm M}, \varPi)$ be a Poisson manifold [@Vas;Book]. Denote the contraction with the Poisson bi-vector field $\varPi$ by $\tilde \varPi : T^* {\mathrm M}\to T {\mathrm M}$. Define the bracket $$[ \omega _ 1 , \omega _2 ]
:= d (\omega _1 \wedge \omega _2 (\varPi))
+ \iota _{\tilde \varPi (\omega _1)} d \omega _2 - \iota _{\tilde \varPi (\omega _2)} d \omega _1,$$ for any 1-forms $\omega _1 , \omega _2$. It is easy to check that $T ^* {\mathrm M}$ is a Lie algebroid using $\tilde \varPi$ as the anchor map.
In many ways the Lie algebroid plays the role of tangent bundle in our study. For example we have:
[@Fern'd;HoloAndChar] Let ${\mathrm E}$ be a vector bundle over ${\mathrm M}$. An ${\mathcal A}$-connection on ${\mathrm E}$ is a differential operator $\nabla ^{\mathrm E}: \Gamma ^\infty ({\mathrm E}) \to \Gamma ^\infty ({\mathcal A}' \otimes {\mathrm E}) $ satisfying the relations $$\begin{aligned}
\nabla ^{\mathrm E}_{f X} u =& \: f \nabla ^{\mathrm E}_X u \\
\nabla ^{\mathrm E}_X (f u) =& \: f \nabla ^{\mathrm E}_X u + {\mathfrak L}_{\nu (X)} u,\end{aligned}$$ for any $X \in \Gamma ^\infty ({\mathcal A}) , f \in C^\infty ({\mathrm M}), u \in \Gamma ^\infty ({\mathrm E})$.
As in the case of Riemannian manifolds, given a metric $g _{\mathcal A}$, i.e., a positive definite symmetric bi-linear form on ${\mathcal A}$, one can define the [*Levi-Civita ${\mathcal A}$-connection*]{} $\nabla ^{g _{\mathcal A}} $ on ${\mathcal A}$ by the formula $$\begin{aligned}
2 g_{\mathcal A}(\nabla ^{g _ {\mathcal A}} _X Y, Z)
:=& g_{\mathcal A}([ X, Y ], Z) - g_{\mathcal A}([Y, Z ], X) + g_{\mathcal A}([Z, X ], Y) \\
&+ {\mathfrak L}_{\nu (X)} g_{\mathcal A}(Y, Z) + {\mathfrak L}_{\nu (Y)} g_{\mathcal A}(Z, X) - {\mathfrak L}_{\nu (Z)} g_{\mathcal A}(X, Y),\end{aligned}$$ for any $X, Y, Z \in \Gamma ^\infty ({\mathcal A})$.
It is well known that every Lie groupoid ${\mathcal G}$ determines a Lie algebroid: Define the vector bundle $${\mathcal A}:= \{ X \in T_x {\mathcal G}: x \in {\mathrm M}\subset {\mathcal G}, d {\mathbf s}( X ) = 0 \}.$$ It is clear that restriction gives a 1-1 correspondence between $\Gamma ^\infty ({\mathcal A})$ and the space of right invariant vector fields on ${\mathcal G}$. Define $[ \cdot , \cdot ]$ to be the Lie bracket between invariant vector fields, and define $$\nu := d {\mathbf t}|_{{\mathcal A}} : {\mathcal A}\rightarrow T {\mathrm M}.$$ It is straightforward to check that ${\mathcal A}$ is a Lie algebroid over ${\mathrm M}$.
A Lie algebroid defined by some Lie groupoid as above is said to be [*integrable*]{}.
Note that not all Lie algebroids are integrable. See [@Fern'd;IntAlgebroid] for details.
For any Lie algebroid ${\mathcal A}\rightarrow {\mathrm M}$, the family of vector fields $${\mathcal F}:= \{ \nu (X) : X \in \Gamma ^\infty ({\mathcal A}) \}.$$ defines a (singular) integrable foliation on ${\mathrm M}$ in the sense of Sussmann [@Sussmann;SingFol]. We denote the leaf space of ${\mathcal F}$ by ${\mathrm M}/ {\mathcal F}$. For each $x \in {\mathrm M}$, we denote the leaf of ${\mathcal F}$ through $x$ by ${\mathcal F}_x$. Note that the leaves may be non-embedded sub-manifolds of ${\mathrm M}$. Given a singular foliation ${\mathcal F}$ defined by an integrable Lie algebroid ${\mathcal A}$, the following propositions, both are direct consequences of the results in [@Fern'd;HoloAndChar] (in particular Theorem 1.1), describe the leaves of ${\mathcal F}$.
\[SubmersionProp\] Let ${\mathcal G}\rightrightarrows {\mathrm M}$ be a Lie groupoid. For each $x \in {\mathrm M}$, the map ${\mathbf t}|_{{\mathcal G}_x } : {\mathcal G}_x \rightarrow {\mathrm M}$ is a submersion onto its image.
\[btImage\] Let ${\mathcal G}$ be an ${\mathbf s}$-connected Lie groupoid. Then for each $x \in {\mathrm M}$, one has $${\mathbf t}( {\mathcal G}_x ) = {\mathcal F}_x; \quad \text{and } \quad {\mathcal F}_x \cong {\mathcal G}_x / {\mathcal G}^x_x,$$ where ${\mathcal G}^x_x$ is the Lie group ${\mathcal G}^x_x := \{ a \in {\mathcal G}: {\mathbf s}(a) = {\mathbf t}(a) = x \}$, known as the isotropy subgroup.
### **Riemannian geometry of the ${\mathbf s}$-fibers** {#RiemVert}
Let ${\mathcal G}\rightrightarrows {\mathrm M}$ be a Lie groupoid over a compact manifold ${\mathrm M}$. Let ${\mathcal A}\rightarrow {\mathrm M}$ be its Lie algebroid. Fix a metric $g _{\mathcal A}$ (i.e. a symmetric, positive definite bi-linear form) on ${\mathcal A}$. For each $x \in {\mathrm M}$, $g _{\mathcal A}$ defines a Riemannian metric on the ${\mathbf s}$-fiber ${\mathbf s}^{-1} (x)$ by $$g_{\mathbf s}(X , Y ) := g _{\mathcal A}({\mathbf t}(a) ) (d R_a (X) , d R_a (Y) ).$$ Observe that $g_{\mathbf s}$ is right invariant in the sense that the right translation $$R_a : {\mathcal G}_{{\mathbf t}(a)} \rightarrow {\mathcal G}_{{\mathbf s}(a)}, \quad \forall a \in {\mathcal G}, X, Y \in T_a {\mathcal G}_x$$ is an isometry for any $a \in {\mathcal G}$. As a direct consequence of the assumptions, one has
For each $x \in {\mathrm M}$, the Riemannian manifold $({\mathcal G}_x , g_{\mathbf s})$ is a manifold with bounded geometry (see Appendix \[BdGeomNonSense\]).
Consider the ${\mathcal A}$-Levi-Civita connection: $$\begin{aligned}
2 g_{\mathcal A}(\nabla ^{\mathcal A}_X Y, Z)
:=& g_{\mathcal A}([ X, Y ], Z) - g_{\mathcal A}([Y, Z ], X) + g_{\mathcal A}([Z, X ], Y) \\
&+ {\mathfrak L}_{\nu (X)} g_{\mathcal A}(Y, Z) + {\mathfrak L}_{\nu (Y)} g_{\mathcal A}(Z, X) - {\mathfrak L}_{\nu (Z)} g_{\mathcal A}(X, Y),\end{aligned}$$ where $ X, Y, Z \in \Gamma ^\infty ({\mathcal A}).$ Let $R^{\mathcal A}$ be the curvature of $\nabla ^{\mathcal A}$.
Consider $\nabla ^{{\mathcal G}_x} _ {\tilde X} \tilde Y$, where $\tilde X, \tilde Y$ are right invariant vector fields, and $\nabla ^{{\mathcal G}_x}$ is the Levi-Civita connection of $({\mathcal G}_x, g _{\mathbf s}) $ for each $x \in {\mathrm M}$. Write $X:= \tilde X|_{\mathrm M}, Y:= \tilde Y|_{\mathrm M}$, then $X, Y \in \Gamma ^\infty ({\mathcal A})$. Then for any right invariant vector field $\tilde Z, a \in {\mathcal G}$, one has $$2 g_{\mathbf s}(a) (\nabla ^{{\mathcal G}_{{\mathbf s}(a)}} _{\tilde X} \tilde Y , \tilde Z)
= 2 g_{\mathbf s}(a) ((d R _a) ((\nabla ^{\mathcal A}_X , Y) ({\mathbf t}(a))) , (d R _a) (Z ({\mathbf t}(a)))).$$ It follows that for any $\tilde X, \tilde Y$ right invariant, the vector field $ a \mapsto \nabla ^{{\mathcal G}_{{\mathbf s}(a)}} _{\tilde X} \tilde Y (a) $ is also right invariant. Furthermore, $\nabla ^{{\mathcal G}_ {{\mathbf s}(a)}} _{\tilde X} \tilde Y (x) = \nabla ^{\mathcal A}_X Y (x)$ for any $x \in {\mathrm M}$.
By similar arguments, for any $\tilde X, \tilde Y, \tilde Z$ right invariant, $R (\tilde X, \tilde Y) \tilde Z $ is right invariant and one has $$R (\tilde X (a) , \tilde Y (a) ) \tilde Z (a)) = R ^{\mathcal A}(X ({\mathbf t}(a), Y ({\mathbf t}(a))) Z ({\mathbf t}(a))$$ for any $a \in {\mathcal G}$. Clearly, the right hand side $R ^{\mathcal A}(X ({\mathbf t}(a), Y ({\mathbf t}(a))) Z ({\mathbf t}(a)) $ is bounded since ${\mathrm M}$ is compact. Formulas for higher covariant derivatives also follow from these arguments.
Finally, to prove that the ${\mathbf s}$-fibers have positive injectivity radius, observe that ${\mathrm M}$ is compact. It follows that there exists $r_0 > 0$ such that $\exp ^{\nabla ^{\mathcal A}} $ is a diffeomorphism form the set $${\mathcal A}_{r_0} := \{ X \in {\mathcal A}:g _{\mathcal A}(X, X) < r_0 ^2 \}$$ onto its image. In proof of boundedness of curvature above, we saw that the Levi-Civita connection is obtained by right translating $\nabla ^ {\mathcal A}$. Therefore, for any $X \in T _a {\mathcal G}_{{\mathbf s}(a)}, a \in {\mathcal G}$, $$\exp ^{\nabla ^{{\mathcal G}_{{\mathbf s}(a)}}} X = (d R _a ) \circ \exp ^{\nabla ^{\mathcal A}} \circ (d R ^{-1} _a ) X .$$ It follows that the injectivity radius of ${\mathcal G}_{{\mathbf s}(a)} \geq r_0 $.
The bounded geometry of the ${\mathbf s}$-fibers means that the notion from manifolds of bounded geometry applies. In particular, we say that
A function $ u \in C ^\infty ({\mathcal G})$ is said to have [*bounded (fiberwise) derivatives*]{} if for any $x \in {\mathrm M}$, $u |_{{\mathcal G}_x} $ has uniformly bounded covariant derivatives.
### **Examples of Lie groupoids**
We give some examples of Lie groupoids relevant to Poisson geometry.
Let ${\mathrm M}$ be a manifold. The pair groupoid over ${\mathrm M}$ is the manifold ${\mathcal G}:= {\mathrm M}\times {\mathrm M}$ together with the operations: $$\begin{aligned}
\text {source and target maps: } & {\mathbf s}(x, y) = y, {\mathbf t}(x, y) = x, \quad \forall (x, y) \in {\mathrm M}\times {\mathrm M}, \\
\text {multiplication: } & \mathbf m ((x, y), (y, z)) = (x, z), \quad \forall (x, y) , (y, z) \in {\mathrm M}\times {\mathrm M}, \\
\text {inverse: } & \mathbf i (x, y) = (y, x), \quad \forall (x, y) \in {\mathrm M}\times {\mathrm M}, \\
\text {unit: } & \mathbf u (x) = (x, x), \quad \forall x \in {\mathrm M}.\end{aligned}$$ The anchor map is the identity on $T {\mathrm M}$. If, in addition, $\omega $ is a symplectic 2-form on ${\mathrm M}$, then $({\mathrm M}\times {\mathrm M}, \wp_1 ^* \omega - \wp_2 ^* \omega )$ is the symplectic groupoid of $({\mathrm M}, \omega )$, where $\wp_1 , \wp_2: {\mathrm M}\times {\mathrm M}\rightarrow {\mathrm M}$ is the projection to the first and second factor respectively.
\[Iwasawa example\] (See Lu and Weinstein [@LuWein;DressingTran]) Let ${\mathfrak g}$ be a complex semi-simple Lie algebra, let ${\mathfrak k}$ be a compact real form of ${\mathfrak g}$. Let $\theta $ be the Cartan involution on ${\mathfrak g}$ fixing ${\mathfrak k}$. Let ${\mathfrak a}$ be a maximal Abelian subalgebra of $i {\mathfrak k}$. Then ${\mathfrak h}= {\mathfrak a}+ i {\mathfrak a}$ is a Cartan subalgebra of ${\mathfrak g}$. Let ${\mathfrak g}= {\mathfrak h}\oplus \sum_{\alpha \in \Delta} {\mathfrak g}_\alpha$ be the root space decomposition. Choose a set of positive roots $\Delta^+$ and let ${\mathfrak n}= \sum_{\alpha \in \Delta^+} {\mathfrak g}_\alpha$. Then ${\mathfrak g}= {\mathfrak k}\oplus {\mathfrak a}\oplus {\mathfrak n}$ is an Iwasawa decomposition of ${\mathfrak g}$ (see [@Knapp;Book1 Chapter IV.4]).
Let $ \langle \cdot , \cdot \rangle $ be the imaginary part of the Killing form. Then $({\mathfrak g}, {\mathfrak k}, {\mathfrak a}+ {\mathfrak n}, \langle \cdot , \cdot \rangle)$ is a Manin triple (see [@Vas;Book Chapter 10]). Its corresponding Poisson Lie group structure can be written as $$\varPi_{\mathrm K}(g) := \frac{1}{2} \sum_{\alpha \in \Delta ^+}
(d R_g) (X_\alpha \wedge Y_\alpha ) - (d L_g) (X_\alpha \wedge Y_\alpha ) , \quad g \in {\mathrm K},$$ where $$X_\alpha := E_\alpha + \theta E_\alpha,
\; {\mbox {\rm and}} \; Y_\alpha := i E_\alpha - i \theta (E_\alpha ) \in {\mathfrak k}, \alpha \in \Delta ^+ ,$$ and $L_g , R_g$ denotes the left and right translation by $g$ respectively.
We turn to construction of the symplectic groupoid. From the construction of Iwasawa decomposition of Lie algebra above, one gets the Iwasawa decomposition of Lie group: $${\mathrm G}= {\mathrm K}{\mathrm A}{\mathrm N}.$$ Take ${\mathcal G}:= {\mathrm G}$ as a manifold. Define: $$\begin{aligned}
\text {source and target maps: } & {\mathbf s}(g) := k , {\mathbf t}(g) := k',
\text {where } g = a n k = k' a' n' \\
& \text {is the (unique) Iwasawa decomposition;} \\
\text {multiplication: } & \mathbf m (g_1 , g_2) := g_1 ({\mathbf s}(g_1))^{-1} g_2 ; \\
\text {inverse: } & \mathbf i (g) := k (n ')^{-1} (a ')^{-1} = n ^{-1} a ^{-1} k' ; \\
\text {unit: } & \mathbf u (k) := k \in {\mathrm G}\supset {\mathrm K}.\end{aligned}$$
\[BruhatExam\] [@Lu;PoissonCohNotes] Let ${\mathrm G}= {\mathrm K}{\mathrm A}{\mathrm N}$ be the Iwasawa decomposition as above. Let ${\mathrm T}\subset {\mathrm K}$ be the maximal torus with ${\mathfrak t}= i {\mathfrak a}$. Then the Poisson bi-vector field $\varPi _{\mathrm K}$ on ${\mathrm K}$ is ${\mathrm T}$-invariant. Hence one has a well defined Poisson manifold $$({\mathrm T}\backslash {\mathrm K}, d \wp_ {\mathrm T}( \varPi _{\mathrm K})),$$ where $\wp_ {\mathrm T}: {\mathrm K}\rightarrow {\mathrm T}\backslash {\mathrm K}$ is the natural projection onto coset space. This Poisson structure is known as the Bruhat Poisson structure.
Define the left action of ${\mathrm T}$ on ${\mathrm K}\times {\mathrm N}$ by $$g \cdot (k , n) := (g k, g n g ^{-1}), \quad \forall (k, n) \in {\mathrm K}\times {\mathrm N}, g \in {\mathrm T}.$$ It is easy to see that the projection onto $${\mathrm T}\backslash ({\mathrm K}\times {\mathrm N})$$ is a submersion. Define the groupoid operations on ${\mathcal G}:= {\mathrm T}\backslash ({\mathrm K}\times {\mathrm N}) \rightrightarrows {\mathrm T}\backslash {\mathrm K}$: $$\begin{aligned}
\text {source and target maps: } & {\mathbf s}( {}_{\mathrm T}(k, n) ) = {}_ {\mathrm T}k , {\mathbf t}( {}_{\mathrm T}(k, n) ) := {}_{\mathrm T}k' , \\
& \text {where } n k = k' a' n' \text { is the (unique) Iwasawa decomposition;} \\
\text {multiplication: } & \mathbf m ({}_{\mathrm T}(k _1 , n _1) , {}_ {\mathrm T}(k _2 , n _2) ) := {}_{\mathrm T}(k _2 , n _1 n _2) , \\
& \text {provided one has Iwasawa decomposition } n _2 k _2 = k _1 a' n' ; \\
\text {inverse: } & \mathbf i ( {}_{\mathrm T}(k, n)) := {}_{\mathrm T}(k', n^{-1}) , \\
& \text {where } n k = k' a' n' \text { is the (unique) Iwasawa decomposition;} \\
\text {unit: } & \mathbf u ({}_ {\mathrm T}k ) := {}_ {\mathrm T}(k , e) , e \in {\mathrm N}.\end{aligned}$$
Uniformly supported pseudo-differential calculus on a Lie groupoid
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In this section, we review the standard theory of pseudo-differential calculus developed by Nistor, Weinstein and Xu [@NWX;GroupoidPdO]. We refer to Appendix \[PDONonSense\] for notations on pseudo-differential operators (on ordinary manifolds).
A pseudo-differential operator $\varPsi $ on a groupoid ${\mathcal G}$ of order $\leq m$ is a smooth family of pseudo-differential operators $\{ \varPsi _x \}_{x \in {\mathrm M}}$, where $\varPsi _x \in \Psi ^m ( {\mathcal G}_{x} )$, and satisfies the right invariance property $$\varPsi _{{\mathbf s}(a)} (R_a^* f) = R_g^* \varPsi _{{\mathbf t}(a)} (f),
\quad \forall a \in {\mathcal G}, f \in C^\infty_c ({\mathcal G}_{{\mathbf s}(a)}).$$ If, in addition, all $\varPsi _x $ are classical of order $m$, then we say that $\varPsi $ is classical of order $m$.
For a pseudo-differential operator $\varPsi = \{ \varPsi _x \}$ on ${\mathcal G}$. The support of $\varPsi $ is defined to be $$\operatorname{Supp}(\varPsi) = \overline {\bigcup_{x \in {\mathrm M}} \operatorname{Supp}(\varPsi _x)}.$$ The operator $\varPsi $ is called properly supported if the set $$({\mathrm K}\times {\mathcal G}) \bigcap \operatorname{Supp}(\varPsi)$$ is compact for every compact subset ${\mathrm K}\subseteq {\mathcal G}$; The operator $\varPsi $ is called uniformly supported if the set $$\{ a b^{-1} : (a, b) \in \operatorname{Supp}(\varPsi) \}$$ is a compact subset of ${\mathcal G}$.
We denote the space of uniformly supported pseudo-differential operators (resp. classical pseudo-differential operators) on ${\mathcal G}$, of order $\leq m$, by $\Psi _\mu ^m ({\mathcal G})$ (resp. $\Psi _\mu ^{[m]} ({\mathcal G})$).
The way to define the total symbol for $\varPsi \in \Psi ^\infty ({\mathcal G})$ is similar to that of an ordinary pseudo-differential operator. Fix an ${\mathcal A}$-connection $\nabla$ (say, $\nabla _X Y := \nabla ^{\mathcal A}_{\nu (X)} Y$ for some usual connection $\nabla ^{\mathcal A}$). Then there is a neighborhood of the zero section $\Omega \subset {\mathcal A}$ such that the exponential map $\exp _\nabla : \Omega \rightarrow {\mathcal G}$ is a diffeomorphism onto its image. Fix a smooth function $\chi (g)$ supported on the image of $\exp _\nabla$ and equal to 1 on a smaller neighborhood of ${\mathrm M}$. Define $\Theta (g, h) := \chi (g) \exp ^{-1}_\nabla (g)$.
\[GpoidTotalDfn\] [@NWX;GroupoidPdO Equation (16)] Given $\varPsi \in \Psi ^\infty ({\mathcal G})$. Define $\sigma \in C^\infty ({\mathcal A}^*)$ by $$\sigma (\zeta ) := \varPsi _x (e^{i \langle \zeta , \Theta (\cdot ) \rangle} \chi ( \cdot))(x),
\quad \forall x \in {\mathrm M}\subset {\mathcal G}, \zeta \in {\mathcal A}^*_x.$$ The function $\sigma $ is called the total symbol of $\varPsi $ with respect to $(\nabla, \chi )$.
As in the case of manifolds, if there exist homogeneous symbols $\sigma _m , \sigma _{m-1} , \cdots$, of orders $m, m-1, \cdots$ respectively, such that $$\sigma - \sum_{l=0}^{N - 1} \sigma _{m - l} \in S^{m - N} ({\mathrm M})$$ for $N = 1, 2, \cdots$, then we say that $\varPsi $ is a classical pseudo-differential operator on ${\mathcal G}$. In this case, we define the principal symbol of $\varPsi $ as $$\sigma _{\mathrm {top}} (\varPsi ) := \sigma _{m}.$$
As in the case of manifolds, we denote the space of uniformly supported classical pseudo-differential operator of order $m$ by $\Psi ^{[m]} _\mu ({\mathcal G})$.
A classical pseudo-differential operator $\varPsi \in \Psi ^{[m]} _\mu ({\mathcal G})$ is said to be elliptic if $$\sigma _{\mathrm {top}} (\varPsi ) (X) \neq 0$$ for any $X \neq 0 \in {\mathcal A}^*$.
A pseudo-differential operator $\varPsi \in \Psi ^\infty ({\mathcal G})$ acts on $C^\infty ({\mathcal G})$ by $$\varPsi (u) (a):= \varPsi _{{\mathbf s}(a)} (u |_{{\mathbf s}^{-1} ({\mathbf s}(a)})).$$ It is easy to see that the composition $ \varPhi \circ \varPsi $ is well defined as long as either $\varPhi $ or $ \varPsi $ is uniformly supported. Furthermore, the composition respects the grading:
Let $\varPsi \in \Psi ^{[m]} ({\mathcal G}), \varPhi \in \Psi ^{[m']} ({\mathcal G})$ be such that either $\varPsi $ or $\varPhi $ is properly supported. Then $\varPhi \circ \varPsi \in \Psi ^{[m+m']} ({\mathcal G})$.
### **Example: Dirac operators on a groupoid**
In this section, we briefly describe the Dirac type operators on a groupoid ${\mathcal G}$ [@Nistor;GeomOp Section 6].
We begin with recalling the notion of Clifford algebra, following [@BGV;Book Chapter 3]. Let ${\mathrm V}$ be a finite dimensional vector space over ${\mathbb R}$ or ${\mathbb C}$. Let $B ( \cdot , \cdot ) $ be a symmetric bi-linear form on ${\mathrm V}$. Then the [*Clifford algebra*]{} of $({\mathrm V}, B )$, denoted by ${\mathrm {Cl} }( {\mathrm V}, B ) $, is the algebra generated by ${\mathrm V}$ with the relation $$v w + w v = - 2 B (v , w).$$ The algebra ${\mathrm {Cl} }({\mathrm V}, B)$ is ${\mathbb Z}_2$-graded by $${\mathrm {Cl} }({\mathrm V}, B) =
\mathrm {span} \{ 1 , v_{i_1} \cdots v _{i_{2 j }} : j = 1, 2 \cdots \} \oplus
\mathrm {span} \{ v_{i_1} \cdots v _{i_{2 j + 1}} : j = 0, 1, 2 \cdots \},$$ where $\{ v _i \}$ is any basis of ${\mathrm V}$.
A [*Clifford module*]{} of ${\mathrm {Cl} }({\mathrm V})$ is a ${\mathbb Z}_2$-graded vector space ${\mathrm E}= {\mathrm E}^+ \oplus {\mathrm E}^- $ such that the Clifford action $\gamma : {\mathrm {Cl} }({\mathrm V}) \to \operatorname{End}({\mathrm E})$ satisfies $$\begin{aligned}
\gamma ({\mathrm {Cl} }^+ ({\mathrm V})) {\mathrm E}^\pm & \subseteq {\mathrm E}^ \pm \\
\gamma ({\mathrm {Cl} }^- ({\mathrm V})) {\mathrm E}^\pm & \subseteq {\mathrm E}^ \mp.\end{aligned}$$
\[FormModule\] Let $B$ be an inner product on ${\mathrm V}$. Then $\wedge ^\bullet {\mathrm V}= (\bigoplus _{i=0} \wedge ^{2 i} {\mathrm V}) \oplus (\bigoplus _{i=0} \wedge ^{2 i + 1} {\mathrm V}) $ is a natural ${\mathrm {Cl} }({\mathrm V}, B)$ module, with action defined by: $$\gamma _\wedge (v) \omega := v \wedge \omega - \iota _{B (v , \cdot)} \omega ,
\quad \forall v \in {\mathrm V}, \omega \in \wedge ^\bullet {\mathrm V},$$ where $\iota $ denotes the contraction. It is easy to verify that such an action of ${\mathrm V}$ extends to ${\mathrm {Cl} }({\mathrm V})$.
Example $\ref{FormModule}$ also provides a canonical bijective map between ${\mathrm {Cl} }({\mathrm V})$ and $\wedge ^\bullet {\mathrm V}$ as vector spaces, namely, $$v \mapsto \gamma _\wedge (v) 1 , \quad v \in {\mathrm {Cl} }({\mathrm V}),$$ where $1 $ is the identity in the exterior algebra $\wedge ^\bullet {\mathrm V}$. It is easy to see that the ${\mathbb Z}_2 $ splitting of $ \wedge ^\bullet {\mathrm V}$ into even and odd orders gives a ${\mathbb Z}_2$ grading of the Clifford algebra ${\mathrm {Cl} }({\mathrm V})$.
Let ${\mathrm V}$ be an even dimensional vector space with inner product $B$. Let $e _1 , e_2 , \cdots , e_{2n} $ be an orthonormal basis of ${\mathrm V}$. Define $${\mathrm P}:= \operatorname{Span}\{ e_{2 i - 1 } + i e _{2 i} : i = 1, \cdots , n \} \subset {\mathrm V}\otimes {\mathbb C}.$$ Then ${\mathrm P}\oplus \bar {\mathrm P}= {\mathrm V}\otimes {\mathbb C}$. Define the action of ${\mathrm V}\otimes {\mathbb C}$ on ${\mathrm S}:= \wedge ^\bullet {\mathrm P}$ by $$\gamma _{\mathrm S}( v ) \omega :=
\begin{cases}
\: v \wedge \omega , & \forall v \in {\mathrm P}\\
\: \iota _{B (v , \cdot ) } \omega , & \forall v \in \bar {\mathrm P}.
\end{cases}$$ The Clifford module ${\mathrm S}$ is known as the [*spin representation* ]{} of the Clifford algebra ${\mathrm {Cl} }({\mathrm V})$.
Here, we list some basic facts about Clifford modules. See [@BGV;Book Chapter 3] for details.
\[CliffLem\] Let ${\mathrm V}$ be an even dimensional vector space over ${\mathbb R}$.
1. The complexified Clifford algebra ${\mathrm {Cl} }({\mathrm V}) \otimes {\mathbb C}$ is isomorphic to the matrix algebra $ \operatorname{End}({\mathrm S})$, where ${\mathrm S}$ is the spinor module;
2. The spinor module ${\mathrm S}$ is the only irreducible representation of ${\mathrm {Cl} }({\mathrm V})$;
3. For any Clifford module ${\mathrm E}$, $ \operatorname{End}({\mathrm E}) \cong {\mathrm {Cl} }({\mathrm V}) \otimes \operatorname{Hom}_{{\mathrm {Cl} }({\mathrm V})} ({\mathrm {Cl} }({\mathrm V}) , {\mathrm E}),$ with isomorphism given by $ v \otimes T \mapsto T (v) .$
We turn to consider bundles of Clifford modules. Let ${\mathcal G}\rightrightarrows {\mathrm M}$ be a groupoid. Let ${\mathcal A}\rightarrow {\mathrm M}$ be the Lie algebroid of ${\mathcal G}$, equipped with a metric $g _{\mathcal A}$. Abusing notation we also use $g _{\mathcal A}$ to denote the inner product on ${\mathcal A}'$. Then we define the [*Clifford bundle*]{}, to be the vector bundle $${\mathrm {Cl} }({\mathcal A}') := \bigcup _{x \in {\mathrm M}} {\mathrm {Cl} }({\mathcal A}'_x , g _{\mathcal A}(x)).$$ Note that ${\mathrm {Cl} }({\mathcal A}')$ is also ${\mathbb Z}_2 $-graded and we write $${\mathrm {Cl} }({\mathcal A}') := {\mathrm {Cl} }({\mathcal A}')^+ \oplus {\mathrm {Cl} }({\mathcal A}')^- .$$ Analogous to the case of Clifford algebras, we define:
A (bundle of) Clifford module is a ${\mathbb Z}_2$-graded Hermitian vector bundle $ {\mathrm E}= {\mathrm E}^+ \oplus {\mathrm E}^- $ over ${\mathrm M}$, with an action map $\gamma \in \Gamma ^\infty ({\mathcal A}\otimes {\mathrm E}\otimes {\mathrm E}')$, such that
1. For any $ \xi \in {\mathcal A}' \subset {\mathrm {Cl} }({\mathcal A}')$, $ \gamma (\xi ) : {\mathrm E}\to {\mathrm E}$ is skew-symmetric;
2. Each ${\mathrm E}_x , x \in {\mathrm M}$ is a ${\mathrm {Cl} }({\mathcal A}'_x)$-module.
A Hermitian ${\mathcal A}$-connection $\nabla ^{\mathrm E}$ is called [*Clifford*]{} if for any $X \in \Gamma ^\infty ({\mathcal A}), \xi \in \Gamma ^\infty ({\mathcal A}'), u \in \Gamma ^\infty ({\mathrm E})$, $$\nabla ^{\mathrm E}_X (\gamma (\xi ) u) = \gamma (\xi ) \nabla ^{\mathrm E}_X u + \gamma (\nabla ^{g _{\mathcal A}} _X \xi ) u,$$ where $\nabla ^{g _{\mathcal A}} $ is the Levi-Civita connection. It can be shown that Clifford ${\mathcal A}$-connections always exist (see [@Nistor;GeomOp Section 6]).
Consider the pullback bundle ${\mathbf t}^{-1} {\mathrm E}$. Any ${\mathcal A}$-connection $\nabla ^{\mathrm E}$ on ${\mathrm E}$ uniquely determines a right-invariant family of connections, still denoted by $\nabla ^{{\mathrm E}}$ for simplicity, on the ${\mathbf s}$-fibers of ${\mathcal G}$ by requiring that $$\nabla ^{{\mathrm E}} _{\tilde X} ({\mathbf t}^{-1} u) = {\mathbf t}^{-1} (\nabla ^{\mathrm E}_X u),$$ for any right-invariant vector field $\tilde X$ with $\tilde X |_{\mathrm M}= X$, and $u \in \Gamma ^\infty ({\mathrm E})$. Furthermore, if ${\mathrm E}$ is a ${\mathrm {Cl} }({\mathcal A})$-module, then ${\mathbf t}^{-1} {\mathrm E}|_{{\mathcal G}_x} $ is a ${\mathrm {Cl} }(T ^* {\mathcal G}_x )$-module for each $x \in {\mathrm M}$, and $\nabla ^{{\mathrm E}} $ is a Clifford connection in the usual sense.
The curvature of any even rank Clifford ${\mathcal A}$-connection $\nabla ^{\mathrm E}$ decomposes under the isomorphism $\operatorname{End}({\mathrm E}) \cong {\mathrm {Cl} }({\mathcal A}') \otimes \operatorname{End}_{\operatorname{End}{\mathrm {Cl} }({\mathcal A}')} ({\mathrm E}) $ as $$\gamma (R) + F ^{{\mathrm E}/ {\mathrm S}} ,$$ where $R $ is the Riemannian curvature of ${\mathcal A}$, considered as a section in $ \Gamma ^\infty (\wedge ^2 {\mathrm A}' \otimes {\mathrm {Cl} }({\mathrm A}) )$ with $\gamma $ acting on the ${\mathrm {Cl} }({\mathrm A})$ factor, and $F ^{{\mathrm E}/ {\mathrm S}} \in \Gamma ^\infty (\wedge ^2 {\mathcal A}' \otimes \operatorname{End}_{{\mathrm {Cl} }({\mathrm A}') } ({\mathrm E}))$ is known as the [*twisting curvature*]{}.
A (groupoid) [*Dirac operator*]{} is a differential operator from ${\mathbf t}^{-1} {\mathrm E}$ to itself of the form $$\eth = ({\mathbf t}^{-1} \gamma ) \circ \nabla ^{\mathrm E},$$ where $\nabla ^{\mathrm E}$ is a right-invariant, Clifford connection on the ${\mathbf s}$-fibers of ${\mathcal G}$; A [*perturbed Dirac operator*]{} is an operator of the form $$\eth + \varPsi \in \Psi ^1 _\mu ({\mathcal G}, {\mathrm E}),$$ where $\eth $ is a Dirac operator, and $\varPsi $ is an odd degree operator in $ \Psi ^{- \infty } _\mu ({\mathcal G}, {\mathrm E}) $ satisfying $$\varPsi (a ^{-1} ) = (\varPsi (a ) )^* , \quad \forall a \in {\mathcal G}.$$
It is easy to see that all Dirac operators are symmetric, hence essentially self adjoint. From our definition, it is also clear that any perturbed Dirac operators are also essentially self-adjoint.
### **The reduced kernel and convolution product**
Let ${\mathcal G}\rightrightarrows {\mathrm M}$ be a groupoid with compact set of units ${\mathrm M}$. Recall that we fixed a fiberwise metric $g_{\mathcal A}$ on the Lie algebroid ${\mathcal A}$ and extended it to a Riemannian metric on each ${\mathbf s}$-fiber by right translation. Hence, one has a family of Riemannian volume densities $\mu _x$ on ${\mathbf s}^{-1} (x) $. We shall also regard $\mu \in \Gamma ^\infty (| \wedge ^{\mathrm {top}} \operatorname{Ker}(d {\mathbf s}) |)$.
\[ConvDfn\] For any pair of functions $f, g \in C^\infty ({\mathcal G})$, such that $f (b) g (a b^{-1}) \in {\mathbf L}^1 ({\mathcal G}_{{\mathbf s}(a)}, \mu_{ {\mathbf s}(a) }), \quad \forall a \in {\mathcal G}$, the convolution product $f \circ g$ is defined to be $$f \circ g (a) := \int _{b \in {\mathcal G}_{{\mathbf s}(a)}} f (a b^{-1} ) g ( b ) \: \mu _{{\mathbf s}(a)} (b).$$
In particular, the convolution product is well defined for any pair $f, g \in C^\infty _c ({\mathcal G}) $, and $f \circ g \in C^\infty _c ({\mathcal G})$. The resulting algebra $( C^\infty _c ({\mathcal G}), \circ )$ is known as the [*convolution algebra*]{} of ${\mathcal G}$.
The convolution product can also be defined for sections of vector bundles. Let ${\mathrm E}, {\mathrm F}$ be vector bundles over ${\mathrm M}$, $f \in \Gamma ^\infty ({\mathbf t}^{-1} {\mathrm E}\otimes {\mathbf s}^{-1} {\mathrm E}'),
g \in \Gamma ^\infty ({\mathbf t}^{-1} {\mathrm E}\otimes {\mathbf s}^{-1} {\mathrm F}) $. Since one has natural identifications $$({\mathbf t}^{-1} {\mathrm E}\otimes {\mathbf s}^{-1} {\mathrm E}')_{a b^{-1}} \cong {\mathbf s}^{-1} {\mathrm E}_{{\mathbf t}(a)} \otimes {\mathbf t}^{-1} {\mathrm E}' _{{\mathbf t}(b)},
\text { and } ({\mathbf t}^{-1} {\mathrm E}\otimes {\mathbf s}^{-1} {\mathrm F}) _b \cong {\mathrm E}_{{\mathbf t}(b)} \otimes {\mathrm F}_{{\mathbf s}(b)},$$ the point-wise multiplication $$f (a b^{-1}) g (b) \in ({\mathbf t}^{-1} {\mathrm E}\otimes {\mathbf s}^{-1} {\mathrm F})_a$$ is well defined for each $a \in {\mathcal G}, b \in {\mathcal G}_{{\mathbf s}(a)}$, using the pairing between ${\mathrm E}' _{{\mathbf t}(b)} $ and ${\mathrm E}_{{\mathbf t}(b)}$. Hence the convolution product can be defined as:
\[ConvDfn2\] For any $f \in \Gamma ^\infty ({\mathbf t}^{-1} {\mathrm E}\otimes {\mathbf s}^{-1} {\mathrm E}'),
g \in \Gamma ^\infty ({\mathbf t}^{-1} {\mathrm E}\otimes {\mathbf s}^{-1} {\mathrm F}) $, such that $f (b) g (a b^{-1}) $ is a ${\mathbf L}^1 ({\mathcal G}_{{\mathbf s}(a)}, \mu_{ {\mathbf s}(a) })$ section with values in ${\mathrm E}_{{\mathbf t}(a)} \otimes {\mathrm F}_{{\mathbf s}(a)} $ for all $a \in {\mathcal G}$, then the convolution product $f \circ g$ is defined to be $$f \circ g (a) := \int _{b \in {\mathcal G}_{{\mathbf s}(a)}} f (a b^{-1} ) g ( b ) \: \mu _{{\mathbf s}(a)} (b)
\in \Gamma ^\infty ({\mathbf t}^{-1} {\mathrm E}\otimes {\mathbf s}^{-1} {\mathrm F}).$$
Alternatively, consider the set $$\tilde {\mathcal G}:= \{ (a , b) \in {\mathcal G}\times {\mathcal G}: {\mathbf s}(a) = {\mathbf s}(b) \}.$$ On $\tilde {\mathcal G}$ one defines the natural maps $$\begin{aligned}
\tilde {\mathbf t}: \tilde {\mathcal G}\to {\mathcal G}, & \quad \tilde {\mathbf t}(a, b) := a \\
\tilde {\mathbf s}: \tilde {\mathcal G}\to {\mathcal G}, & \quad \tilde {\mathbf s}(a, b) := b \\
{\mathbf s}^{(2)} : \tilde {\mathcal G}\to {\mathrm M}, & \quad {\mathbf s}^{(2)} (a, b) := {\mathbf s}(a) = {\mathbf s}(b) \\
{\boldsymbol \pi }: \tilde {\mathcal G}\to {\mathcal G}, & \quad {\boldsymbol \pi }(a , b) = a b ^{-1} .\end{aligned}$$ Note that $\tilde {\mathcal G}$ is just the fibered product groupoid of ${\mathcal G}$, with source and target maps $\tilde {\mathbf s}, \tilde {\mathbf t}$. Using the relations $${\mathbf t}\circ \widetilde {\mathbf m}= {\mathbf t}\circ \tilde {\mathbf t}, \quad
{\mathbf s}\circ \widetilde {\mathbf m}= {\mathbf t}\circ \tilde {\mathbf s}, \quad
\text { and } \quad {\mathbf s}\circ \tilde {\mathbf t}= {\mathbf s}^{(2)} = {\mathbf s}\circ \tilde {\mathbf s},$$ one naturally identifies the bundles (over $\tilde {\mathcal G}$): $$\begin{aligned}
\widetilde {\mathbf m}^{-1} ({\mathbf t}^{-1} {\mathrm E}\otimes {\mathbf s}^{-1} {\mathrm E}')
\cong & \: \tilde {\mathbf t}^{-1} ({\mathbf t}^{-1} {\mathrm E}) \otimes \tilde {\mathbf s}^{-1} ({\mathbf t}^{-1} {\mathrm E}') \\
\tilde {\mathbf s}^{-1} \otimes ({\mathbf t}^{-1} {\mathrm E}\otimes {\mathbf s}^{-1} {\mathrm F})
\cong & \: \tilde {\mathbf s}^{-1} ({\mathbf t}^{-1} {\mathrm E}) \otimes \tilde {\mathbf t}^{-1} ({\mathbf s}^{-1} {\mathrm F}).\end{aligned}$$ Hence, one can rewrite Definition \[ConvDfn2\] using the language of Appendix \[DGNonsense\] as $$\label{ConvDef3}
f \circ g (a)
= \int _{ (b' , b) \in \tilde {\mathbf t}^{-1} (a)}
\big( \widetilde {\mathbf m}^{-1} f (b' , b) \big)
\big( \tilde {\mathbf s}^{-1} g (b' , b) \big) \tilde \mu (b' , b),$$ where $\tilde \mu \in \Gamma ^\infty ( | \wedge ^{\mathrm {top}} \ker (d \tilde {\mathbf s}) |)$ is defined by $\tilde \mu = \mu $ at $ \tilde {\mathbf s}^{-1} (b') \cong {\mathbf s}^{-1} ({\mathbf s}(b))$, regarded as a family of measures (densities) on the fibers.
\[RedKer\] For any $\varPsi = \{ \varPsi _x \}_{x \in {\mathrm M}} \in \Psi ^\infty ({\mathcal G})$. The [*reduced kernel*]{} of $\varPsi $ is defined to be the distribution $$K_\varPsi (f) := \int _{\mathrm M}\mathbf u^* (\varPsi (\mathbf i^* f)) (x) \: \mu_{\mathrm M}(x),
\quad f \in C^\infty _c ({\mathcal G}),$$ where $\mathbf i$ and $\mathbf u$ denote respectively the inversion and unit inclusion.
Observe that, if $\varPsi \in \Psi ^{- \infty} ({\mathcal G})$, then $K_\varPsi \in C^\infty ({\mathcal G})$, i.e., there exists $\kappa \in C^\infty ({\mathcal G}) $ such that $$K_\varPsi (f) =
\int _{x \in {\mathrm M}} \left( \int _{b \in {\mathcal G}_x} \kappa (b) f (b^{-1}) \: \mathbf i ^* \mu _{{\mathbf s}(b)} \right) \mu _{\mathrm M},
\quad \forall f \in C^\infty _c ({\mathcal G}),$$ and one can recover $\varPsi $ by the formula: $$\varPsi (f) (a) = \int _{{\mathcal G}_{{\mathbf s}(a)}} \kappa (a b ^{-1}) f (b) \: \mu_{{\mathbf s}(a) } (b).$$
In [@NWX;GroupoidPdO], the authors defined the reduced kernel canonically using 1-densities.
One particularly important property of the reduced kernel of a pseudo-differential operator on a groupoid is the following:
[@NWX;GroupoidPdO Corollary 1] For any $\varPsi \in \Psi ^\infty ({\mathcal G})$, the reduced kernel is co-normal at ${\mathrm M}$ and smooth elsewhere.
### **Some representations of $\Psi ^\infty ({\mathcal G})$**
In this section, we recall some homomorphisms from $\Psi ^\infty _\mu ({\mathcal G})$ to other spaces of operators. The materials in this section can be found in [@Nistor;GeomOp]. Let ${\mathcal G}\rightrightarrows {\mathrm M}$ be an ${\mathbf s}$-connected Lie groupoid. Let ${\mathcal A}$ be the Lie algebroid of ${\mathcal G}$.
\[OpNorm\] Given any $\varPsi \in \Psi _\mu ^{-n-1} ({\mathcal G})$, define the 1-norm of $\varPsi $ by (see [@Nistor;GeomOp Equation (16)]) $$\| \varPsi \|_1 := \sup _{x \in {\mathrm M}}
\left\{ \int _{ {\mathcal G}_{x} } | \kappa (a) | d \mu _x (a) ,
\int _{ {\mathcal G}_{x} } | \kappa (a^{-1}) | d \mu _x (a) \right\},$$ where $\kappa (a)$ is the reduced kernel of $\varPsi $. Note that $\kappa $ is continuous because $\varPsi \in \Psi _\mu ^{-n-1} ({\mathcal G})$.
Next, we define the full norm of any $\varPsi \in \Psi _\mu ^0 ({\mathcal G})$ by $$\| \varPsi \| := \sup _{\rho } \| \rho (\varPsi ) \| _{{\mathbf H}},$$ where $\| \cdot \|_{\mathbf H}$ is just the operator norm, and the supremum ranges through all bounded representation $\rho $ of $\Psi ^0 _\mu ({\mathcal G})$ on ${\mathbf H}$ satisfying $$\| \rho (\varPsi ) \| _{\mathbf H}\leq \| \varPsi \|_1 , \quad \forall \varPsi \in \Psi ^0 _\mu ({\mathcal G}).$$
We denote the closure of $\Psi ^0 _\mu ({\mathcal G})$ under $\| \cdot \|$ by $${\mathfrak U}({\mathcal G}),$$ and the closure of $\Psi ^{- \infty } _\mu ({\mathcal G}) $ under $\| \cdot \|$ by $${\mathfrak C}^* ({\mathcal G}) \subset {\mathfrak U}({\mathcal G}).$$
Another important homomorphism is the so called vector representation, which defines the class of (leafwise)-differential operators on a manifold that we are interested in:
The [*vector representation*]{} is the homomorphism defined by $\nu : \Psi _\nu ^\infty ({\mathcal G}) \rightarrow \operatorname{End}(C^\infty ({\mathrm M}))$, $$(\nu (\varPsi ) u)(x) := \varPsi _x ({\mathbf t}^{-1} u)(x).$$
Equivalently, one can define $(\nu (\varPsi ))u$ to be the (unique) function on ${\mathrm M}$ satisfying $ (\nu (\varPsi ))u \circ {\mathbf t}= \varPsi (u \circ {\mathbf t}) $
Observe that if $X \in \Gamma ^\infty ({\mathcal A})$ is regarded as a differential operator on ${\mathcal G}$, then the vector representation of $X$ is just $\nu (X)$, the image of $X$ under the anchor map (regarded as a differential operator on ${\mathrm M}$), so there is no confusion using the same notation for both.
$ \; $
Elliptic and Fredholm operators
===============================
Using the same arguments as in the construction of parametrices of elliptic pseudo-differential operators on a manifold, one has:
Let $\varPsi \in \Psi ^{[m]} _\mu ({\mathcal G}) $ be elliptic. Then there exists an operator $Q \in \Psi ^{[-m]} _\mu ({\mathcal G})$, known as the parametrix of $\varPsi $, such that $$\label{CrudePara}
R _1 = \varPsi \circ Q - \operatorname{i d}\text { and } R _2 = Q \circ \varPsi - \operatorname{i d}$$ are elements in $ \Psi ^{- \infty } _\mu ({\mathcal G})$.
If ${\mathcal G}$ is the pair groupoid over a compact manifold, then all elements in $\Psi ^{-\infty } ({\mathcal G})$ are compact. It follows from Equation (\[CrudePara\]) that all elliptic operators are Fredholm. Unfortunately, in general, elements in $\Psi ^{- \infty } _\mu ({\mathcal G})$ are not compact operators. In the following section we review a Fredholmness criterion given by Lauter and Nistor [@Nistor;GeomOp]. Here, we first recall the notion of an invariant sub-manifold.
Let ${\mathcal G}\rightrightarrows {\mathrm M}$ be a groupoid. A proper sub-manifold $ {\mathrm Z}\subset {\mathrm M}$ is called an [*invariant sub-manifold*]{} if ${\mathbf s}^{-1} ({\mathrm Z}) = {\mathbf t}^{-1} ({\mathrm Z}) $. For an invariant sub-manifold, we denote ${\mathcal G}_{{\mathrm Z}} := {\mathbf s}^{-1} ({\mathrm Z})$. It is clear that ${\mathcal G}_{\mathrm Z}$ is a groupoid over ${\mathrm Z}$ by restricting the groupoid structure on ${\mathcal G}$. Also, for any $\varPsi = \{ \varPsi _x \} _{x \in {\mathrm M}} \in \Psi ^{\infty } ( {\mathcal G})$, define the [*restriction*]{} of $ \varPsi $ to be the operator $$\varPsi |_ {\mathrm Z}:= \{ \varPsi _x \} _{x \in {\mathrm Z}} \in \Psi ^{\infty } ({\mathcal G}_{\mathrm Z}).$$
Lauter and Nistor’s Fredholmness criterion {#LauNis}
--------------------------------------------
Let ${\mathcal G}\rightrightarrows {\mathrm M}$ be a groupoid with compact units ${\mathrm M}$. Assume that the anchor map $\nu : {\mathcal A}\to T {\mathrm M}$ is an isomorphism when restricted to some open dense subset ${\mathrm M}_0 \subseteq {\mathrm M}$. Then one can also define the metric $$g _{{\mathrm M}_0} (X, Y) := g_{\mathcal A}(\nu^{-1} X , \nu ^{-1} Y) , \quad \forall X, Y \in T_x {\mathrm M}_0 , x \in {\mathrm M}_0.$$ By definition, it is clear that ${\mathbf t}|_{ {\mathcal G}_{x} } : {\mathcal G}_x \rightarrow {\mathrm M}_0$ is a local isometry.
Following [@Nistor;GeomOp], we shall make the following assumptions:
\[BdGpoid\] An ${\mathbf s}$-connected groupoid ${\mathcal G}\rightrightarrows {\mathrm M}$ is said to be a [*Lauter-Nistor groupoid*]{} if
1. The unit set ${\mathrm M}$ is compact;
2. The anchor map $\nu : {\mathcal A}\rightarrow T {\mathrm M}$ is bijective over an open dense subset ${\mathrm M}_0 \subseteq {\mathrm M}$;
3. The Riemannian manifold $({\mathrm M}_0 , g _{{\mathrm M}_0}) $ has positive injectivity radius and has finitely many connected components ${\mathrm M}_0 = \coprod _\alpha {\mathrm M}_\alpha $;
4. As a groupoid, ${\mathcal G}_{{\mathrm M}_0} \cong \coprod _\alpha {\mathrm M}_\alpha \times {\mathrm M}_\alpha $, the pair groupoid.
Note that condition (2) implies the Lie algebroid is integrable, using the following result from Debord [@Debord;IntAlgebroid].
Let ${\mathcal A}$ be a Lie algebroid over ${\mathrm M}$, with anchor map $\nu : {\mathcal A}\rightarrow T {\mathrm M}$. Suppose that there exists an open dense subset $U \subset {\mathrm M}$ such that $\nu $ is injective on ${\mathcal A}|_U$. Then ${\mathcal A}$ is integrable.
Indeed, we shall mainly be studying examples where the groupoid is explicitly given.
The following lemma is useful for verifying assumption (4).
If all connected components of ${\mathrm M}_0$ are simply connected, then ${\mathcal G}_{{\mathrm M}_0} \cong \coprod _\alpha {\mathrm M}_\alpha \times {\mathrm M}_\alpha $.
Observe that, for each $x \in {\mathrm M}_0$, ${\mathcal G}_x $ is a covering of ${\mathcal F}_x $, the connected component in ${\mathrm M}_0 $ containing $x $. If all connected components of ${\mathrm M}_0$ are simply connected, then ${\mathcal G}_x \cong {\mathcal F}_x $ for all $x \in {\mathrm M}_0$. It follows from Proposition \[btImage\] that the isotropy subgroups ${\mathcal G}^x _x $ are trivial for all $x \in {\mathrm M}_0 $. Hence the assertion.
Since the Riemannian curvature is a local object, it follows that $( {\mathrm M}_0 , g _{{\mathrm M}_0} ) $ is a manifold with bounded geometry. Also, it is easy to see that for any vector bundle ${\mathrm E}\to {\mathrm M}$, the restriction ${\mathrm E}| _{{\mathrm M}_0} \to {\mathrm M}_0 $ is a vector bundle of bounded geometry. Hence one can consider the Sobolev spaces ${\mathbf W}^l ({\mathrm M}_0 , {\mathrm E})$ for any $l \in {\mathbb R}$.
Let $\varPsi = \{ \varPsi _x \} \in \Psi ^{[m]} _\mu ({\mathcal G}, {\mathrm E})$. For any $x \in {\mathrm M}_0$, assumption (4) enables one to identify $${\mathcal G}_x \cong {\mathrm M}_\alpha ,$$ where $M _\alpha $ is the connected component of ${\mathrm M}_0 $ containing $x$. Hence one identifies $\Gamma ^\infty ({\mathrm M}_\alpha , {\mathrm E}) \cong \Gamma ^\infty ({\mathcal G}_x , {\mathbf s}^{-1} {\mathrm E}) $. Under such identification, one has $$\label{VecSobo}
\nu (\varPsi) (f) |_{{\mathrm M}_\alpha } = \varPsi _x (f |_{{\mathrm M}_\alpha }) ,$$ for any $f \in \Gamma ^\infty ({\mathrm M}, {\mathrm E}) $. Since $\varPsi _x$ is a pseudo-differential operator of order $\leq m$, Equation (\[VecSobo\]) enables one to extend the vector representation $\nu (\varPsi)$ to a bounded map $$\nu _l (\varPsi) : {\mathbf W}^l ({\mathrm M}_0 , {\mathrm E}) \to {\mathbf W}^{l - m} ({\mathrm M}_0 , {\mathrm E}) ,$$ for any $ l \geq m$. In particular, if $\varPsi \in \Psi ^{- \infty } _\mu ({\mathcal G}, {\mathrm E})$, then $\nu _0 (\varPsi)$ is just the smoothing map $$\label{LNOpen}
\nu _0 (\varPsi) f (x) = \int _{y \in {\mathrm M}_\alpha } \psi |_{{\mathcal G}_{{\mathrm M}_0}} (x , y) f (y ) \mu _{{\mathrm M}_0} (y),
\quad f \in {\mathbf L}^2 ({\mathrm M}, {\mathrm E}) ,$$ where $\psi \in \Gamma ^\infty _c ({\mathcal G}, {\mathbf t}^{-1} {\mathrm E}\otimes {\mathbf s}^{-1} {\mathrm E}')$ is the reduced kernel of $\varPsi$, and we have used the identification ${\mathcal G}_{{\mathrm M}_0} \cong \coprod _\alpha {\mathrm M}_\alpha \times {\mathrm M}_\alpha $.
Recall that we defined ${\mathfrak U}({\mathcal G})$ and ${\mathfrak C}^* ({\mathcal G})$ to be the closure of $\Psi ^0 _\mu ({\mathcal G})$ and $\Psi ^{-\infty} _\mu ({\mathcal G})$ under the full norm $\| \cdot \|$ respectively. We shall denote ${\mathfrak J}:= {\mathfrak C}^* ({\mathcal G}_{{\mathrm M}_0})$ (the closure of pseudo-differential operators of order $-\infty$ on the groupoid over ${\mathrm M}_0$). The importance of ${\mathfrak J}$ lies in
For any $\varPsi \in {\mathfrak J}$, the vector representation $\nu (\varPsi )$ is a compact operator on ${\mathbf L}^2 ({\mathrm M}_0)$.
If $\varPsi \in \Psi _\mu ^{- \infty } ({\mathcal G}_{{\mathrm M}_0})$, then Equation (\[LNOpen\]) says that $\nu (\varPsi )$ is just a properly supported, smoothing operator on ${\mathrm M}_0$, which is well known to be compact. The assertion follows by taking limits.
One remarkable fact about these spaces is the following lemma:
\[ExactC\^\*\] [@Nistor;Family Lemma 2] One has short exact sequences $$\begin{aligned}
0 & \rightarrow {\mathfrak C}^* ({\mathcal G}_{{\mathrm M}_0}) = {\mathfrak J}\rightarrow {\mathfrak C}^* ({\mathcal G})
\rightarrow {\mathfrak C}^* ({\mathcal G}_{{\mathrm M}\setminus {\mathrm M}_0}) \rightarrow 0 \\
0 & \rightarrow {\mathfrak U}({\mathcal G}_{{\mathrm M}_0}) \rightarrow {\mathfrak U}({\mathcal G})
\rightarrow {\mathfrak U}({\mathcal G}_{{\mathrm M}\setminus {\mathrm M}_0}) \rightarrow 0.\end{aligned}$$
Another useful fact about Lauter-Nistor groupoids is that their vector representation is faithful. In other words:
[*[@Nistor;Polyhedral3]*]{} The map $\nu : \Psi ^{\infty } _\mu ({\mathcal G}, {\mathrm E}) \to \operatorname{End}(\Gamma ^\infty _c ({\mathbf t}^{-1} {\mathrm E}) )$ is injective.
Let $\varPsi = \{ \varPsi _x \} _{x \in {\mathrm M}} \in \Psi ^{\infty } _\mu ({\mathcal G}, {\mathrm E}) $ be such that $\nu (\varPsi ) = 0 $. First consider $\varPsi _x $ for arbitrary $x \in {\mathrm M}_0 $. For any $u \in \Gamma ^\infty _c ({\mathbf t}^{-1} {\mathrm E}|_{{\mathcal G}_x} )$, let $\tilde u \in \Gamma ^\infty _c ({\mathrm E})$ be the extension of $u$ by $0$. Then $$\varPsi _x (u ) = \nu (\varPsi ) (\tilde u) |_{{\mathrm M}_0 } = 0.$$ Therefore $\varPsi _x = 0$ for any $x \in {\mathrm M}_0 $. Now consider $x \in {\mathrm M}\setminus {\mathrm M}_0 $. For any $u \in \Gamma ^\infty _c ({\mathbf t}^{-1} {\mathrm E}|_{{\mathcal G}_x} ) $, let $\hat u \in \Gamma ^\infty _c ({\mathbf t}^{-1} {\mathrm E})$ be any extension of $u$. Then one has $$\varPsi (\hat u) = 0$$ on ${\mathcal G}_{{\mathrm M}_0}$, because $\varPsi _x = 0 $ for any $x \in {\mathrm M}_0$. Since $ \varPsi (\hat u) $ is continuous and ${\mathcal G}_{{\mathrm M}_0 }$ is dense in ${\mathcal G}$, it follows that $\varPsi _x (\hat u) = 0$ everywhere, hence $\varPsi = 0$.
In the following theorem, let $A $ be any fixed elliptic (pseudo)-differential operator of order $k > 0 $. (One can take $A $ to be, say, a Laplacian operator in Definition \[LapDfn\]). Then $(\operatorname{i d}+ A ^* A ) ^{- \frac{1}{2 k}} $ is well defined by functional calculus. Moreover, by [@Nistor;GeomOp Theorem 4] and its corollaries, $\nu \big( (\operatorname{i d}+ A ^* A ) ^{\frac{m}{2 k}} \big) : {\mathbf W}^m ({\mathrm M}_0 , {\mathrm E}) \to {\mathbf W}^ 0 ({\mathrm M}_0 , {\mathrm E}) $ is bounded for all $m$. With these preliminaries, the main result of Lauter and Nistor can be stated as:
\[Nis;Thm7\] [@Nistor;GeomOp Theorem 7] For any $\varPsi \in \Psi ^0 ({\mathcal G})$, or $\varPsi \in \Psi ^{[m]} ({\mathcal G})$ elliptic self-adjoint, the spectrum and essential spectrum of $\nu (\varPsi )$ satisfy $$\label{nuSpec}
{\boldsymbol \sigma }(\nu (\varPsi )) \subseteq {\boldsymbol \sigma }_{{\mathfrak U}({\mathcal G})} (\varPsi)
\; \text { and } \; {\boldsymbol \sigma }^e (\nu (\varPsi )) \subseteq {\boldsymbol \sigma }_{{\mathfrak U}/ {\mathfrak J}} (\varPsi ).$$ In particular, for any $\varPsi \in \Psi ^{[m]}_\mu ({\mathcal G})$ such that $ \varPsi (\operatorname{i d}+ A^* A)^{- \frac{m}{2 k}} $ is invertible in then ${\mathfrak U}({\mathcal G}) / {\mathfrak J}$, $\nu (\varPsi ) $ extends to a Fredholm operator from ${\mathbf W}^m ({\mathrm M}_0 , {\mathrm E}) $ to $ {\mathbf L}^2 ({\mathrm M}_0 , {\mathrm E})$; if\
$ \varPsi (\operatorname{i d}+ A^* A)^{- \frac{m}{2 k}} \in {\mathfrak J}$, then $\nu (\varPsi ) : {\mathbf L}^2 ({\mathrm M}_0 , {\mathrm E}) \to {\mathbf L}^2 ({\mathrm M}_0 , {\mathrm E})$ is compact.
By definition, for each $\lambda \in {\mathbb C}\setminus {\boldsymbol \sigma }_{{\mathfrak U}({\mathcal G}) / {\mathfrak J}}$, there exists $ Q \in {\mathfrak U}({\mathcal G}) $ such that $$(\varPsi - \lambda ) Q - \operatorname{i d}_{{\mathfrak U}({\mathcal G})} , Q (\varPsi - \lambda ) - \operatorname{i d}_{{\mathfrak U}({\mathcal G})} \in {\mathfrak J}.$$ Since $\nu $ maps ${\mathfrak J}$ to compact operators, it follows that $\nu (\varPsi )$ is Fredholm, hence $\lambda \in {\mathbb C}\setminus {\boldsymbol \sigma }^{e} (\nu (\varPsi ))$. The second inclusion follows by contra-positivity. The first inclusion is similar (with ${\mathfrak J}$ replaced by $\{ 0 \} $).
To prove that $\nu (\varPsi )$ is Fredholm (resp. compact) from the hypothesis, observe that $\nu (\varPsi ) = \nu ( \varPsi (\operatorname{i d}+ A^* A)^{- \frac{m}{2 k}} ) \nu ((\operatorname{i d}+ A^* A)^{\frac{m}{2 k}}) $, and use the well known fact that the composition between a Fredholm (resp. compact) operator and a bounded invertible operator is Fredholm (resp. compact).
Using the injectivity of the vector representation, and the fact that injective homomorphisms of $C ^*$-algebra preserve the spectrum [@Averson;Book p.12], the inclusion in Equation (\[nuSpec\]) can be sharpen to an equality. In particular:
\[Nis;Thm8\] [@Nistor;GeomOp Theorem 8] Suppose the groupoid ${\mathcal G}$ is Hausdorff. Then, for any $\varPsi \in \Psi ^0 ({\mathcal G})$, or $\varPsi \in \Psi ^{[m]} ({\mathcal G})$ elliptic self-adjoint, the spectrum and essential spectrum of $\nu (\varPsi )$ satisfy $${\boldsymbol \sigma }(\nu (\varPsi )) = {\boldsymbol \sigma }_{{\mathfrak U}({\mathcal G})} (\varPsi)
\; \text { and } \; {\boldsymbol \sigma }^e (\nu (\varPsi )) = {\boldsymbol \sigma }_{{\mathfrak U}/ {\mathfrak J}} (\varPsi ).$$
Suppose that ${\mathrm M}\backslash {\mathrm M}_0$ is a disjoint union of closed immersed invariant sub-manifolds $${\mathrm M}\backslash {\mathrm M}_0 = \bigcup _{j=1}^k {\mathrm Z}_k .$$ Then the hypothesis of Theorem \[Nis;Thm7\] can be made more explicit by
\[Nis;Thm10\] [@Nistor;GeomOp Theorem 10] For any $\varPsi \in {\mathfrak U}({\mathcal G})$, the spectrum $\varPsi + {\mathfrak J}$ in ${\mathfrak U}({\mathcal G}) / {\mathfrak J}$ can be written as a union $${\boldsymbol \sigma }_{{\mathfrak U}({\mathcal G}) / {\mathfrak J}} (\varPsi + {\mathfrak J}) =
{\boldsymbol \sigma }_{S ({\mathcal A}^*)} (\sigma _{\mathrm {top}} (\varPsi ))
\bigcup \bigcup _{j=1}^k {\boldsymbol \sigma }_{{\mathfrak U}({\mathcal G}_{{\mathrm Z}_j})} (\varPsi |_{{\mathrm Z}_j}),$$ where $\sigma _{\mathrm {top}} (\varPsi )$ is the principal symbol of $\varPsi $.
It suffices to prove that the homomorphism $$\varPsi + {\mathfrak J}\mapsto \sigma _{\mathrm {top}} (\varPsi ) \oplus \varPsi \rvert_{{\mathrm Z}_1 } \oplus
\cdots \oplus \varPsi \rvert _{{\mathrm Z}_j}$$ is injective. That is true because $\sigma _{\mathrm {top}} (\varPsi ) = 0 $ implies $\varPsi \in C^* ({\mathcal G})$, and the first exact sequence of Lemma \[ExactC\^\*\] implies $\varPsi \in {\mathfrak J}$.
Combining Theorem \[Nis;Thm7\] and Theorem \[Nis;Thm10\], we get
\[NisLem\] [@Nistor;GeomOp Theorem 10] Given an elliptic operator $\varPsi \in \Psi ^{[m]} _\mu ({\mathcal G}) , m \geq 0$. Suppose for all invariant sub-manifolds ${\mathrm Z}_j$, there exist $\varPhi _j \in \Psi ^{-m} ({\mathcal G}_{{\mathrm Z}_j} , {\mathrm E}|_{{\mathrm Z}_j} ) \bigcap {\mathfrak U}({\mathcal G}_{{\mathrm Z}_j}) $ such that $$(\varPsi |_{{\mathrm Z}_j} ) \varPhi _j = \varPhi _j ( \varPsi |_{{\mathrm Z}_j } ) = \operatorname{i d},$$ then $\nu (\varPsi )$ is Fredholm.
Application: Fredholm operators on the Bruhat sphere
-----------------------------------------------------
In this section, we study the Bruhat sphere ${\mathbb C}{\mathrm P}(1)$ in greater detail.
### **The Bruhat sphere and its symplectic groupoid**
The Bruhat Poisson structure is obtained by taking ${\mathrm G}= {\mathrm S \mathrm L}(2 , {\mathbb C}) , {\mathrm K}= {\mathrm S \mathrm U}(2) $, and ${\mathrm A}{\mathrm N}= $ set of upper diagonal matrices in Example \[BruhatExam\]. It is well known that the Bruhat sphere has two ${\mathcal A}$-leaves: ${}_{\mathrm T}e$ and its complement. As we have seen in Example \[BruhatExam\], the symplectic groupoid over the Bruhat sphere is $ {\mathrm T}\backslash ({\mathrm S \mathrm U}(2) \times {\mathrm N}) . $ Here, we describe the groupoid structure in greater detail.
\[BruhatPair\] Let $\alpha , \beta , w \in {\mathbb C}, |\alpha |^2 + |\beta |^2 = 1 $. Then we write $$[\alpha , \beta ] _{\mathrm T}^ { w } :=
\left(
\left(
\begin{smallmatrix}
\alpha & \beta \\
- \bar \beta & \bar \alpha
\end{smallmatrix}
\right),
\left(
\begin{smallmatrix}
1 & w \\
0 & 1
\end{smallmatrix}
\right)
\right) {\mathrm T}\in {\mathcal G}= {\mathrm T}\backslash ({\mathrm S \mathrm U}(2) \times {\mathrm N}) .$$ Also, recall that one can define stereographic coordinates $$\begin{aligned}
z =& \: x + \imath y \mapsto [ z , 1 ] \in {\mathbb C}{\mathrm P}(1) - [1 , 0] , \quad x , y \in {\mathbb R}\\
\dot z =& \: \dot x + \imath \dot y \mapsto [ 1 , \dot z ] \in {\mathbb C}{\mathrm P}(1) - [0 , 1] ,
\quad \dot x , \dot y \in {\mathbb R}.\end{aligned}$$ Then the source submersion ${\mathbf s}$ can be trivialized as $$\begin{aligned}
{\mathbf x}(z , w)
:=& \: \left[ \frac{\bar w - z}{(1 + |\bar w - z|^2)^\frac{1}{2}} ,
\frac{1}{(1 + |\bar w - z|^2)^\frac{1}{2}} \right] _{\mathrm T}^{ w },
\quad z, w \in {\mathbb C}. \\
\dot {\mathbf x}( \dot z , \dot w)
:=& \: \left[ \frac{\dot z \bar {\dot w} - 1}{(|\dot z|^2 + |\dot z \bar {\dot w} - 1 |^2)^\frac{1}{2}} ,
\frac{\dot z}{(|\dot z | ^2 + |\dot z \bar {\dot w} - 1|^2)^\frac{1}{2}} \right] _{\mathrm T}^{ \dot w },
\quad \dot z, \dot w \in {\mathbb C}.\end{aligned}$$
For any $ k = \left(
\begin{smallmatrix}
\alpha & \beta \\
- \bar \beta & \bar \alpha
\end{smallmatrix}
\right) \in {\mathrm K}, n =
\left(
\begin{smallmatrix}
1 & w \\
0 & 1
\end{smallmatrix}
\right) \in {\mathrm N},$ one has the Iwasawa decomposition $n k = k' a' n' $, where $$k ' = \left(
\begin{array}{cc}
\alpha ' & \beta ' \\
- \bar \beta ' & \bar \alpha '
\end{array}
\right) \in {\mathrm K},
\alpha ' = \frac{\alpha - \bar w \beta }{(|\beta |^2 + |\alpha - \bar w \beta |^2)^{\frac{1}{2}}},
\beta ' = \frac{ \beta }{(|\beta |^2 + |\alpha - \bar w \beta |^2)^{\frac{1}{2}}}.$$ Hence, one can easily write down the source, target and inverse maps $$\begin{aligned}
\label{CP2Formula}
{\mathbf s}( [\alpha , \beta ] _ {\mathrm T}^{ w }) =& \: [\alpha , \beta ] \\ \nonumber
{\mathbf t}( [\alpha , \beta ] _ {\mathrm T}^{ w }) =& \: [\alpha - \bar w \beta , \beta ] \\ \nonumber
([\alpha , \beta ] _{\mathrm T}^{ w })^{-1} =& \:
\left[ \frac{\alpha - \bar w \beta }{(|\beta |^2 + |\alpha - \bar w \beta |^2)^{\frac{1}{2}}},
\frac{ \beta }{(|\beta |^2 + |\alpha - \bar w \beta |^2)^{\frac{1}{2}}} \right] ^{- w} _{\mathrm T}. \end{aligned}$$ It follows that in the ${\mathbf x}$ and $\dot {\mathbf x}$ coordinates ${\mathbf s}({\mathbf x}(z , w) ) = [z , 1] , {\mathbf s}(\dot {\mathbf x}(\dot z , \dot w)) = [ 1 , \dot z ]$. The inverse can also be written down: $$([1 , 0] ^ w _ {\mathrm T}) ^{-1} = ([1 , 0] ^{- w} _{\mathrm T}) , ({\mathbf x}(z , w) )^{-1} = {\mathbf x}(z + \bar w , - w) ,
\quad \forall z , w \in {\mathbb C}.$$
It is clear that the symplectic groupoid defining the Bruhat Poisson sphere is a Lauter-Nistor groupoid. Indeed, many Poisson homogeneous spaces constructed by Lu (see [@Lu;PoissonCohNotes]), with open symplectic leaves, have symplectic groupoids satisfying the Lauter-Nistor conditions. Finally, note that we shall not use the symplectic structure in this thesis.
The Poisson bi-vector field can be also be explicitly written down [@Ryot;NecklaceCP1]. On the stereographic coordinate patch excluding ${}_{\mathrm T}e$, one has $$\varPi = (1 + x^2 + y^2 ) \partial _x \wedge \partial _ y;$$ On the opposite coordinate patch one has $$\varPi = (\dot x^2 + \dot y^2)(1 + \dot x^2 + \dot y^2) \partial _{\dot x } \wedge \partial _{\dot y}.$$
As an illustration, we describe the metric on the open leaf induced by the Poisson bi-vector field. For simplicity, take the round metric on the sphere $$(1 + x^2 + y^2)^{-2} (d x^2 + d y^2),$$ and the dual metric on ${\mathcal A}= T^* {\mathbb C}{\mathrm P}(1)$: $$g_{\mathcal A}:= (1 + x^2 + y^2)^2 ((\partial _x )^2 + (\partial _y)^2).$$ Then the metric on the open leaf ${\mathbb C}{\mathrm P}(1) - \{ {} _{\mathrm T}e \} $ is defined by $$\begin{aligned}
g_{\mathcal A}(\nu ^{-1} \partial _x, \nu ^{-1} \partial _x)
= \: & g_{\mathcal A}((1+ x^2 +y^2)^{-1} d y, (1+ x^2 + y^2)^{-1} d y) = 1 \\
= \:& g_{\mathcal A}(\nu ^{-1} \partial _y, \nu ^{-1} \partial _y), \\
g_{\mathcal A}(\nu ^{-1} \partial _x, \nu ^{-1} \partial _y) = \: & 0,\end{aligned}$$ where $\nu (\omega ) := \iota _{\omega } \varPi, \: \forall \omega \in T^* {\mathbb C}{\mathrm P}(1) $ is the anchor map
Here, we observe that the metric we obtained is just the Euclidean metric on ${\mathbb R}^2$. On the polar coordinates $ (\dot r , \dot \vartheta ) \mapsto \dot {\mathbf x}( \dot r e ^{i \dot \vartheta })$, the metric $g _{{\mathrm M}_0 } $ is just $ \dot r ^{-1} d \dot r ^2 + d \vartheta ^ 2 $. A metric of this form is known as ‘scattering metric’ in the edge calculus literature (see [@Albin;EdgeInd]). We shall use this fact later in Section 5. However, it is important to note that the compactification to the Bruhat sphere is [*not*]{} the same as the standard compactification to the disk with boundary.
### **Inverse and the Laplace-Fourier transform**
Observe that, over ${}_{\mathrm T}e$, one has $${\mathbf s}^{-1} ({}_{\mathrm T}e) = {\mathbf t}^{-1} ({}_{\mathrm T}e) = {\mathrm N}\cong {\mathbb R}^2$$ as a Lie group. Therefore, given any pseudo-differential operator $\varPsi = \{ \varPsi _x \} _{x \in {\mathbb C}{\mathrm P}(2)}$, it follows that $\varPsi _{{}_{\mathrm T}e}$ is an operator on ${\mathbb R}^2$ that is invariant under translation. As we shall see in this section, the simple structure on ${\mathbb R}^n$ enables one to study inverses through the Laplace-Fourier transform, which in turn gives a simple Fredholmness criterion.
Set $\nabla$ be the usual flat, translation invariant connection on ${\mathbb R}^n$, $\chi = 1$ on ${\mathbb R}^n \times {\mathbb R}^n$. One can regard ${\mathbb R}^n$ as a groupoid over a one point space. Recall, from Definition \[GpoidTotalDfn\], the total symbol of any properly supported $\varPsi _{ {} _{\mathrm T}e} \in \Psi ^\infty _\varrho ({\mathbb R}^n)$ is defined by $$\label{FourierLaplace}
\sigma (\zeta ) := (\varPsi _{ {} _{\mathrm T}e} )_p (e^{-i \langle p , \zeta \rangle}).$$ By virtue of Lemma \[Kennedy\], one has $$\varPsi _{ {} _{\mathrm T}e} (f) (p)
= \int _{\zeta \in {\mathbb R}^n} \sigma (\zeta ) e^{i \langle p, \zeta \rangle} \hat f (\zeta ) \: d \zeta.$$ It would be useful to consider $\varPsi $ as convolution with a distribution. Define $$\psi (f) := \varPsi _{ {} _{\mathrm T}e} (f (- p)) (0)
= \int _{\zeta \in {\mathbb R}^n} \sigma (\zeta ) \int _{q \in {\mathbb R}^n} e^{i \langle q, \zeta \rangle} f (q) \: d q d \zeta,$$ so that one has $$\varPsi _{ {} _{\mathrm T}e} (f) (p) = \psi _q (f (p - q)).$$ Note that $\psi $ is just the reduced kernel in Definition \[RedKer\], regarding ${\mathbb R}^n$ as a groupoid over a point.
Assume that one has the estimate $$C (1 + |\zeta |)^m \geq |\sigma (\zeta )| \geq C' (1 + |\zeta |)^m > 0$$ for some constants $C, C' > 0 $ (which implies that $\varPsi $ is elliptic of order $m$). It is straightforward to check that $(\sigma (\zeta ))^{-1} $ is also a symbol. Since the symbol map is a homomorphism, it follows that the inverse of $\varPsi $ is given by $$\varPsi _{ {} _{\mathrm T}e} ^{-1} (f) (p)
= \int _{\zeta \in {\mathbb R}^n} (\sigma (\zeta ))^{-1} e^{i \langle q, \zeta \rangle} \hat f (\zeta ) \: d \zeta .$$
Next, we describe the kernel of $\varPsi ^{-1}$ in greater detail. Note that Equation (\[FourierLaplace\]) is still valid for $\zeta \in {\mathbb C}^n$. Such extension is known as the Laplace-Fourier transform and shall be denoted by $\tilde \sigma (\zeta )$ or ${\mathfrak F}(f) $ if $f \in C ^\infty _c ({\mathbb R}^n)$. Indeed, one has
For any properly supported, invariant pseudo-differential operator $\varPsi $ on ${\mathbb R}^n$, the Laplace-Fourier transform $\tilde \sigma (\zeta )$ is a holomorphic function on ${\mathbb C}^n$.
In the case when $\varPsi $ is a differential operator, it was shown in [@Shimakura;Book Chapter 4.2] that the reduced kernel of $\varPsi ^{-1}$ decays exponentially, depending on the zeros of $\tilde \sigma (\zeta )$, i.e., the poles of $\tilde \sigma (\zeta )^{-1}$. Here, we prove a similar result for general pseudo-differential operators.
\[DecayProp\] Let ${\mathfrak H}$ be a holomorphic function on the strip $$S_\theta := \{ (\zeta _1 , \cdots \zeta _n ) \in {\mathbb C}^n : | \operatorname{i m}(\zeta _i) | < \theta , \quad \forall i \},$$ and satisfies the estimate $$\label{HoloEst}
\left| \partial _I {\mathfrak H}(\zeta ) \right| \leq C _I (1 + |\zeta |)^{m - |I|} , \quad \zeta \in S_\theta ,$$ for each multi-index $I$ and some $C_I > 0$, $m \in {\mathbb R}$. Let $\kappa$ be the distribution $$\kappa (f) := \int _{\zeta \in {\mathbb R}^n} {\mathfrak H}(\zeta ) \hat f (\zeta ) \: d \zeta, \quad f \in C^\infty _c ({\mathbb R}^n).$$ Then $\kappa $ is $C^\infty$ on ${\mathbb R}^n \backslash \{ 0 \}$. Furthermore, for any $0 < \varepsilon < \theta $, one has $$\kappa |_{{\mathbb R}\backslash \{ 0 \}} = e^{- \varepsilon |p|} F , \quad \forall | p | > 1$$ for some smooth function function $F$ with bounded derivatives.
First of all, since $\varsigma (\zeta ), \zeta \in {\mathbb R}^n$ is a symbol, it is well known that $\kappa $ is $C^\infty$ on ${\mathbb R}^n \backslash \{ 0 \}$, and for any natural number $N$ and multi-index $I$, there exists $C_{I, N} > 0$ such that $$\label{PdOKerEst}
|\partial _I \kappa ( p )| \leq C_{I, N} (1 + | p |)^N , \quad \forall \zeta \in {\mathbb R}^n , | p | \geq 1.$$
By the well known Paley-Weiner theorem, ${\mathfrak F}(f)$ is holomorphic on ${\mathbb C}$ for any $f \in C^\infty _c ({\mathbb R}^n)$, and for any natural number $N$, there exists constants $C_N$ such that $$\left| {\mathfrak F}(u) (\zeta ) \right| \leq C_N (1 + |\zeta |)^{-N}$$ for any $\zeta \in S _\theta $. Using Equation (\[HoloEst\]) in the hypothesis, the integrand $${\mathfrak H}(i (\varepsilon , \varepsilon , \cdots , \varepsilon ) + \zeta )
\times {\mathfrak F}(f) (i (\varepsilon , \varepsilon , \cdots , \varepsilon ) + \zeta ), \quad \zeta \in {\mathbb R}^n$$ lies in $L^1 ({\mathbb R}^n)$ for any $0 < \varepsilon < \theta $. Therefore we can use Fubini’s theorem to compute the integral $$\int _{\zeta \in {\mathbb R}^n} {\mathfrak H}(\zeta ) {\mathfrak F}(f) \: d \zeta
= \int \cdots \int \left( \int {\mathfrak H}(\zeta ) {\mathfrak F}(f) (\zeta ) d \zeta _1 \right) d \zeta _2 \cdots d \zeta _n.$$ We then use the Cauchy integral formula to shift the contour of $\zeta _1$-integration to $$\xi _1 + i \varepsilon , \xi _1 \in (-\infty , \infty).$$ The integral becomes $$\begin{aligned}
& \int \cdots \int \left( \int {\mathfrak H}(\xi _1 + i r, \zeta _2 , \cdots , \zeta _n )
\int e^{- i \langle (i \varepsilon + \zeta _1 , \zeta _2 , \cdots \zeta _n ) , q \rangle } f (q )
\: d q d \xi _1 \right) d \zeta _2 \cdots d \zeta _n \\
=& \int \cdots \int \left( \int {\mathfrak H}(\xi _1 + i \varepsilon , \zeta _2 , \cdots , \zeta _n )
\int e^{-i \langle (\xi _1 , \zeta _2 \cdots , \zeta _n , q \rangle}
(e^{\varepsilon q_1} f (q )) d q d \zeta _2 \right) d \zeta _3 \cdots d \zeta _n d \xi _1 \\
=& \cdots = \int {\mathfrak H}(\xi _1 + i \varepsilon , \xi _2 + i \varepsilon , \cdots , \xi _n + i \varepsilon )
\int e^{- i \langle (\xi _1 , \xi _2 , \cdots \xi _n ) , q \rangle } e^{\varepsilon (q_1 + \cdots + q_n)} f (q )
\: d q d \xi \end{aligned}$$ by using Fubini’s theorem and Cauchy integral formula repeatedly.
Define the distribution $$\tilde \kappa _\varepsilon ( g ):=
\int {\mathfrak H}(\xi _1 + i \varepsilon , \xi _2 + i \varepsilon , \cdots , \xi _n + i \varepsilon )
\int e^{- i \langle (\xi _1 , \xi _2 , \cdots \xi _n ) , q \rangle } g (q ) \: d q d \xi .$$ Since ${\mathfrak H}(\xi _1 + i \varepsilon , \xi _2 + i \varepsilon , \cdots , \xi _n + i \varepsilon )$ is a symbol for $\xi \in {\mathbb R}^n$ by assumption, using Equation (\[PdOKerEst\]) again, one conclude that $\tilde \kappa _\varepsilon $ is $C ^\infty $ on ${\mathbb R}^n \backslash \{ 0 \}$, and for any natural number $N$ and multi-index $I$, there exists $C_{I, N} > 0$ such that $$|\partial _I \tilde \kappa _\varepsilon ( p )|
\leq C_{I, N} (1 + | p |)^N , \quad \forall \zeta \in {\mathbb R}^n , | p | \geq 1.$$ Furthermore, by uniqueness of kernel, it follows that $$\kappa (p) = e^{\varepsilon (p _1 + \cdots + p _n) } \tilde \kappa _\varepsilon (p)$$ on ${\mathbb R}^n \backslash \{ 0 \}$. Since $ p _1 + \cdots + p _n - (- | p |) $ is bounded above on the subset $\{ p _1 , p _2 , \cdots p _n < 1 \}$, one can write $$\kappa = e^{- \varepsilon | p |} \tilde F$$ for some smooth function $\tilde F$ satisfying Equation (\[PdOKerEst\]) on the subset $$\{ | p | > 1 \} \bigcap \{ p _1 , p _2 , \cdots , p _n < 1 \}.$$ Repeating the arguments by considering the contours $$(\xi _1 \pm i \varepsilon , \xi \pm _2 i \varepsilon , \cdots , \xi _n \pm i \varepsilon ),$$ one gets a similar estimate on each quadrant. The assertion follows by combining these estimates.
Remark that the assumption of Proposition \[DecayProp\] is very mild. For example, one has
Let $P$ be a polynomial of order $n$, $P_{\mathrm {top}}$ be its highest order part. Let $f$ be a compactly supported function on ${\mathbb R}^n$. Suppose that $P_{\mathrm {top}} |_{{\mathbb R}^n} $ is elliptic, and $P + {\mathfrak F}(f) \neq 0 $ on ${\mathbb R}^n$. Then $P + {\mathfrak F}(f) \neq 0$ on some strip $S_\theta , \theta > 0$, and $(P + {\mathfrak F}(f))^{-1}$ satisfies the assumption of Proposition \[DecayProp\]
Also, we recall the following well know fact about the obstruction to existence of invertible perturbations (see, for example, [@Connes;Book] for an overview of the subject):
For any properly supported, invariant, elliptic pseudo-differential operator $\varPsi _{ {}_{\mathrm T}e } \in \Psi ^{[\infty]} _\varrho ({\mathbb R}^n)$, there exists $K \in \Psi ^{- \infty } _\varrho ({\mathbb R}^n )$ such that $\varPsi _{ {} _{\mathrm T}e} + K $ is invertible if and only if the $\mathbb K$-theoretic analytic index $$\operatorname{i n d}_{\mathrm {Ana}} (\varPsi _x) \in \mathbb K ^0 (C ^\infty _c (\mathbb R ^n ) , \circ)$$ vanishes. Here, $\circ $ denotes the convolution product on $C ^\infty _c ({\mathbb R}^n )$.
Finally, we end up with:
\[FredThm\] Let $\varPsi = \{ \varPsi _x \} _{x \in {\mathbb C}{\mathrm P}(2) } \in \Psi ^{[m]} _\mu ({\mathrm S \mathrm U}(2) \times {\mathrm N}/ {\mathrm T})$ be elliptic. Let $\tilde \sigma (\zeta )$ be the Laplace-Fourier transform of $\varPsi _{{}_{\mathrm T}e}$. Suppose $\sigma (\zeta )$ satisfies the estimate $$C (1 + |\zeta |)^m \geq |\tilde \sigma (\zeta )| \geq C' (1 + |\zeta |)^m$$ for some $C, C' > 0 $, on some strip $ \zeta \in S_\theta $ for some $\theta > 0 $. Then $\varPsi $ is Fredholm.
Given any $\varPsi $ as in the hypothesis. Let $\tilde \sigma (\zeta )$ be the Laplace-Fourier transform of $\varPsi _{{}_{\mathrm T}e}$. Put $\varsigma := \tilde \sigma ^{-1}$. Then $\varsigma $ satisfies the hypothesis of Proposition \[DecayProp\]. Hence $\varPsi _{{}_{\mathrm T}e}^{-1}$ has a reduced kernel of the form $$\psi = e^{- \varepsilon |p|} F (p)$$ on ${\mathbb R}^n \backslash \{ 0 \}$, where $F (p) \in C^\infty ({\mathbb R}^n \backslash \{ 0 \})$ satisfies Equation (\[PdOKerEst\]).
We need to prove that $\varPsi _{{}_{\mathrm T}e} ^{-1} \in {\mathfrak U}({\mathcal G}_{{}_{\mathrm T}e})$. To do so, write $\psi := \psi _\mu + \psi _e $, where $\psi _\mu $ is compactly supported and $\psi _e \in C^\infty ({\mathbb R}^n)$, $\psi _e = 0$ on a neighborhood of $0$. Let $\varPsi _\mu $ and $\varPsi _e$ be the corresponding pseudo-differential operators. Then $\varPsi _\mu \in \Psi ^{[-m]} _\mu ({\mathcal G}_{{}_{\mathrm T}e})$. It remains to consider $\psi _e$. Since $\psi _e$ decays exponentially, it is clear that one can find a sequence $\{ \kappa_j \}, j = 1,2, \cdots $ in $C^\infty _c ({\mathbb R}^n)$ such that $$\| \psi _e - \kappa_j \|_{{\mathbf L}^1 ({\mathbb R}^n)} \rightarrow 0.$$ Let $K _j \in \Psi ^{- \infty} _\mu ({\mathcal G}_{{\mathrm T}})$ be the corresponding invariant pseudo-differential operators. Then by Definition \[OpNorm\], $$\| \varPsi _e - K _j \| _1 \rightarrow 0.$$ It follows from Definition \[OpNorm\] of the full norm that $K _j \rightarrow \varPsi _e $. Hence $\varPsi _e \in {\mathfrak U}({\mathcal G}_{{}_{\mathrm T}e})$ as well. The result follows from Lemma \[NisLem\].
Proposition \[DecayProp\] and Theorem \[FredThm\] not only give a criterion for an operator $\nu (\varPsi )$, where $\varPsi \in \Psi ^{[\infty]} _\mu ({\mathcal G}) $ to be Fredholm, they also give a more precise description for the parametrix of $\nu (\varPsi )$ modulo compact operators.
\[ParaThm\] Let $\varPsi \in \Psi ^{[m]} _\mu {\mathcal G}$ be an elliptic operator satisfying the hypothesis of Theorem \[FredThm\]. There exists operators $Q \in \Psi ^{- [m]} _\mu ({\mathcal G})$, and $S \in \Gamma ^\infty ({\mathbf t}^{-1} {\mathrm E}\otimes {\mathbf s}^{-1} {\mathrm E}')$ (regarded as a reduced kernel in $\Psi ^{- \infty } ({\mathcal G})$) of the form $$S (a) = e ^{- \varepsilon \tilde d (a , {\mathbf s}(a))} \tilde \kappa , \quad a \in {\mathcal G},$$ for some $\varepsilon > 0 $, where $\tilde d $ is a smooth function on ${\mathcal G}^2$ satisfying $\tilde d - d \leq 1$, and $\tilde \kappa \in \Gamma ^\infty _b ({\mathbf t}^{-1} {\mathrm E}\otimes {\mathbf s}^{-1} {\mathrm E}) $ such that $$\nu (\varPsi ) \nu (Q + S) - \operatorname{i d}$$ is a compact operator.
By standard arguments one can find $Q \in \Psi ^{- [m]} _\mu ({\mathcal G})$ such that $$\varPsi Q - \operatorname{i d}= R _1 , \quad R _1 \in \Psi ^{- \infty} _\mu ({\mathcal G}).$$ On the other hand, Proposition \[DecayProp\] implies that one has $$(\varPsi _{{}_{\mathrm T}e} )^{-1} = \tilde Q + e ^{- \varepsilon \tilde d (a , {\mathbf s}(a))} F$$ for some $F \in \Gamma ^\infty ({\mathbf t}^{-1} {\mathrm E}\otimes {\mathbf s}^{-1} {\mathrm E}|_{{\mathcal G}_{ {} _{\mathrm T}e}} ) $ with bounded derivatives. Since both $Q _{{}_{\mathrm T}e}, \tilde Q $ are properly supported parametrices of $\varPsi _{{}_{\mathrm T}e}$, it follows that $$(\varPsi _{{}_{\mathrm T}e} )^{-1} - Q _{{}_{\mathrm T}e} = S _{{}_{\mathrm T}e}$$ for some $S _{{}_{\mathrm T}e} \in \Gamma ^\infty ({\mathbf t}^{-1} {\mathrm E}\otimes {\mathbf s}^{-1} {\mathrm E}|_{{\mathbf s}^{-1} ({}_{\mathrm T}e)} ) $ of the form $$S _{{}_{\mathrm T}e } (a) = e ^{- \varepsilon \tilde d (a , {\mathbf s}(a))} \kappa .$$
Let ${\mathrm U}\subset {\mathrm M}$ be a local trivialization of ${\mathbf s}$ around ${} _{\mathrm T}e $, i.e., ${\mathrm U}\times {\mathcal G}_{ {}_{\mathrm T}e } \cong {\mathbf s}^{-1} {\mathrm U}$. Fix any function $\chi \in C^\infty _c ({\mathrm U}) $ such that $\chi = 1 $ on a smaller neighborhood of ${} _{\mathrm T}e $. Define a section $ \tilde S (a) \in \Gamma ^\infty ({\mathbf t}^{-1} {\mathrm E}\otimes {\mathbf s}^{-1} {\mathrm E}) $ as follows: If $a \in {\mathbf s}^{-1} ({\mathrm U})$, $a$ is identified with a point $(x, p) \in {\mathrm U}\times {\mathcal G}_{ {} _{\mathrm T}e } $, and we define $$\tilde S (a) := S (p) \chi (x) .$$ Otherwise, we define $\tilde S (a) := 0 $. By the computations in the proof of Lemma \[CP1Est\], $\tilde S $ satisfies the estimate $$\tilde S (a) = e ^{- \varepsilon \tilde d (a , {\mathbf s}(a))} \tilde \kappa ,$$ where $\kappa (a) $ and $\kappa (a ^{-1})$ are both sections of bounded derivatives. Moreover, it is obvious that $\tilde S |_{{\mathcal G}_{{}_{\mathrm T}e)} } = S _{{}_{\mathrm T}e} $. It follows that $$\tilde R := \varPsi (Q + \tilde S) - \operatorname{i d}\in \Phi ^{- \infty } ({\mathcal G})$$ satisfies $\tilde R _{{}_{\mathrm T}e} = 0$. By Corollary \[CptCor\] below, it follows that that $\nu (\tilde R) = \nu (\varPsi ) \nu (Q + \tilde S) - \operatorname{i d}$ is a compact operator.
Exponentially decaying kernels
-------------------------------
Inspired by the results of Proposition \[DecayProp\] and Theorem \[FredThm\], we construct the pseudo-differential calculus with bounds, in parallel with the theory of poly-homogeneous distributions for manifolds with corners (see [@Melrose;Book Chapter 5]).
First, let ${\mathcal G}\rightrightarrows {\mathrm M}$ be a Lauter-Nistor groupoid. We say that
[ The groupoid ${\mathcal G}$ is of [*sub-exponential growth* ]{} if for any $\varepsilon > 0$, $$\int _{a \in {\mathbf s}^{-1} (x)} e ^{- \varepsilon d (x , a)} \mu _x (a) \leq C$$ for some constant $C$ independent of $x \in {\mathrm M}$; it is of [*polynomial growth*]{} if for some integer $N $ and constant $C$, $$\int _{a \in B (x, r) } \mu _x (a) \leq C r ^N .$$ ]{}
Clearly, polynomial growth implies sub-exponential growth.
[ Since each ${\mathbf s}$-fiber of the symplectic groupoid over the Bruhat sphere is quasi-isometric to the Euclidean space ${\mathbb R}^2$, the groupoid ${\mathrm T}\backslash ({\mathrm S \mathrm U}(2) \times {\mathrm N})$ is of polynomial growth. ]{}
Recall that in Section 2, given a Hausdorff groupoid ${\mathcal G}$, we defined the groupoid $ \tilde {\mathcal G}:= \{ (a , b ) \in {\mathcal G}\times {\mathcal G}: {\mathbf s}(a ) = {\mathbf s}(b) \} $. Also recall that for any $X \in \Gamma ^\infty ({\mathcal A})$, $X$ determines a right invariant vector field $X ^{\mathcal R}\in \Gamma ^\infty (\operatorname{Ker}(d {\mathbf s}))$. Here, we furthermore define vector fields on $\tilde {\mathcal G}$ by $$\begin{aligned}
X ^{\tilde {\mathcal R}} (a , b) :=& ( X ^{\mathcal R}(a ) , 0 ) \in T _a {\mathcal G}\times T _b ({\mathcal G}) \subseteq \tilde T {\mathcal G}\\
X ^{\tilde {\mathcal L}} (a , b) :=& ( 0 , X ^{\mathcal R}(b ) ),\end{aligned}$$ for any $( a , b ) \in \tilde {\mathcal G}$. Similarly, given any vector bundle ${\mathrm E}\to {\mathrm M}$, and ${\mathcal A}$-connection ${}^{\mathcal A}\nabla ^{\mathrm E}$ on ${\mathrm E}$, right translation defines a connection $\hat \nabla ^{{\mathbf t}^{-1} {\mathrm E}}$ on ${\mathbf t}^{-1} {\mathrm E}\to {\mathcal G}_x$, for each $x \in {\mathrm M}$. We shall consider the family of pullback connections $\hat \nabla ^{\tilde {\mathbf s}^{-1} ({\mathbf t}^{-1} {\mathrm E}) \otimes \tilde {\mathbf t}^{-1} ({\mathbf t}^{-1} {\mathrm E}' ) }$ on $ \tilde {\mathbf s}^{-1} ({\mathbf t}^{-1} {\mathrm E}) \otimes \tilde {\mathbf t}^{-1} ({\mathbf t}^{-1} {\mathrm E}' )
\to {\mathcal G}_x \times {\mathcal G}_x \subseteq \tilde {\mathcal G}$.
Fix a Riemannian metric $g _{\mathcal A}$ on ${\mathcal A}$, which in turn determines a metric on each of ${\mathcal G}_x$. For each $( a , b ) \in \tilde {\mathcal G}$, define $d (a , b) $ to be the Riemannian distance on ${\mathcal G}_{{\mathbf s}(a)} = {\mathcal G}_{{\mathbf s}(b)}$ between $a$ and $b$.
For each $\varepsilon > 0$, the $\varepsilon $-calculus of order $- \infty , \Psi ^{- \infty} _\varepsilon ({\mathcal G}, {\mathrm E})$, is defined to be the space of sections $\psi \in \Gamma ^\infty ({\mathbf t}^{-1} {\mathrm E}\otimes {\mathbf s}^{-1} {\mathrm E}' )$, regarded as reduced kernels, with the property that there exists some $\varepsilon ' > \varepsilon $ such that for all $(a , b ) \in \tilde {\mathcal G}, m = 0, 1, 2 \cdots , $ $$e ^{\varepsilon ' d (a , b) }
(\hat \nabla ^{ \tilde {\mathbf s}^{-1} ({\mathbf t}^{-1} {\mathrm E}) \otimes \tilde {\mathbf t}^{-1} ({\mathbf t}^{-1} {\mathrm E}' )} )^k
(\widetilde {\mathbf m}^{-1} \psi )(a , b) \leq C _k$$ for some constants $C _k > 0 $.
For each $m \in {\mathbb Z}, \varepsilon > 0$, the [*(classical) $\varepsilon $-calculus of order $m$*]{} is defined to be the space $$\Psi ^{[m]} _\varepsilon ({\mathcal G}, {\mathrm E})
:= \Psi ^{[m]} _\mu ({\mathcal G}, {\mathrm E}) + \Psi ^{- \infty } _\varepsilon ({\mathcal G}, {\mathrm E}).$$
As in the case of manifolds with boundary [@Mazzeo;EdgeRev; @Melrose;Book], we need to compute the composition rule of the calculus.
For any $\varepsilon _1 , \varepsilon _2 \geq 0 $ $$\Psi ^{- \infty } _{\varepsilon _1} \circ \Psi ^{- \infty } _{\varepsilon _2}
\subseteq \Psi ^{- \infty } _{\min \{ \varepsilon _1 , \varepsilon _2 \}}.$$
For simplicity we only consider the scalar case. It suffices to consider the convolution product $ u _1 \circ u _2 $ for any $u _1 \in \Psi ^{- \infty } _{\varepsilon _1} ({\mathcal G}),
u_2 \in \Psi ^{- \infty } _{\varepsilon _2} ({\mathcal G})$. In view of the formula $$u _1 \circ u _2 (a) = \int _{b \in {\mathcal G}_{{\mathbf s}(a)}} u _1 (a b ^{-1}) u _2 (b) \mu _{{\mathbf s}(a)} (b)
= \int _{c \in {\mathbf s}^{-1} ({\mathbf t}(a))} u _1 (c ^{-1}) u _2 (c a) \mu _{{\mathbf t}(a)} (c),$$ one can without loss of generality assume $\varepsilon _1 \leq \varepsilon _2$. Then by definition one has the estimates $ u _1 (a) \leq e ^{- \varepsilon ' _1 d (a, {\mathbf s}(a))} C ,
u _2 (a) \leq e ^{- \varepsilon ' _2 d (a, {\mathbf s}(a))} C' $ for some $\varepsilon '_1 > \varepsilon _1 , \varepsilon ' _2 > \varepsilon _2 $. One may further assume that $\varepsilon ' _1 < \varepsilon ' _2 $.
The hypothesis implies for any $a \in {\mathcal G}$ $$\begin{aligned}
| u _1 \circ u _2 (a) |
\leq & C _1 \int _{b \in {\mathcal G}_{{\mathbf s}(a)}}
e ^{- \varepsilon ' _1 d (a , b) } e ^{ - \varepsilon ' _2 d (b , {\mathbf s}(b))} \mu _{{\mathbf s}(a)} (b) \\
\leq & C _1 \int _{b \in {\mathcal G}_{{\mathbf s}(a)}}
e ^{- \varepsilon ' _1 |d (a , {\mathbf s}(a)) - d (b , {\mathbf s}(b))| - \varepsilon ' _2 d (b , {\mathbf s}(b))} \mu _{{\mathbf s}(a)} (b) \\
= & C _1 \int _{b \in B _a }
e ^{- \varepsilon ' _1 d (a , {\mathbf s}(a)) } e ^{- (\varepsilon ' _2 - \varepsilon ' _1) d (b , {\mathbf s}(b))}
\mu _{{\mathbf s}(a)} (b) \\
&+ C _1 \int _{b \not \in B _a}
e ^{ \varepsilon ' _1 d (a , {\mathbf s}(a)) } e ^{- (\varepsilon ' _2 + \varepsilon ' _1) d (b , {\mathbf s}(b))} \mu _{{\mathbf s}(a)} (b),\end{aligned}$$ where $B _a$ denotes the set $\{ b \in {\mathcal G}_{{\mathbf s}(a)} : d (b , {\mathbf s}(b)) < d (a , {\mathbf s}(a)) \} $ for each $a$. Hence for the first integral, one has $$\begin{aligned}
\int _{b \in B _a }
e ^{- \varepsilon ' _1 d (a , {\mathbf s}(a)) } & e ^{- (\varepsilon ' _2 - \varepsilon ' _1) d (b , {\mathbf s}(b))}
\mu _{{\mathbf s}(a)} (b)e ^{- \varepsilon ' _1 d (a , {\mathbf s}(a)) } \\
= & \: e ^{- \varepsilon ' _1 d (a , {\mathbf s}(a)) }
\int _{b \in B _a } e ^{- (\varepsilon ' _2 - \varepsilon ' _1) d (b , {\mathbf s}(b))} \mu _{{\mathbf s}(a)} (b) \\
\leq & \: e ^{- \varepsilon ' _1 d (a , {\mathbf s}(a)) }
\int _{b \in {\mathcal G}_{{\mathbf s}(a)}} e ^{- (\varepsilon ' _2 - \varepsilon ' _1) d (b , {\mathbf s}(b))} \mu _{{\mathbf s}(a)} (b),\end{aligned}$$ and the last integral is finite and only depends on ${\mathbf s}(a)$. As for the second integral, write $$\begin{aligned}
\varepsilon ' _1 d (a , {\mathbf s}(a)) - &(\varepsilon ' _2 + \varepsilon ' _1) d (b , {\mathbf s}(b)) \\
= &- \varepsilon ' _1 d (a , {\mathbf s}(a))
+ 2 \varepsilon ' _1 ( d (a , {\mathbf s}(a)) - d (b, {\mathbf s}(b)))
- (\varepsilon ' _2 - \varepsilon ' _1) d (b , {\mathbf s}(b)).\end{aligned}$$ Since $d (b, {\mathbf s}(b)) \geq d (a , {\mathbf s}(a)) $ for any $b \not \in B _a$. It follows that the second integral is again bounded by $$e ^{- \varepsilon ' _1 d (a , {\mathbf s}(a))}
\int _{b \in {\mathcal G}_{{\mathbf s}(a)}} e ^{- (\varepsilon ' _2 - \varepsilon ' _1) d (b , {\mathbf s}(b))} \mu _{{\mathbf s}(a)} (b).$$ Adding the two together and rearranging, one gets $ e ^{\varepsilon ' _1 d _{\mathbf s}(a)} (u _1 \circ u _2 ) (a) $ is a bounded function, as asserted.
To prove the assertion for derivatives, observe that by right invariance of $\mu$, $$\widetilde {\mathbf m}^* (u _1 \circ u _2 )(a, b) = \int u _1 (a c ^{-1} ) u _2 (c b ^{-1}) \mu _{{\mathbf s}(a)} (c) ,$$ for any $(a, b) \in \tilde {\mathcal G}$. It follows that for any $X , Y \in \Gamma ^\infty ({\mathcal A})$, $$\begin{aligned}
{\mathfrak L}_{X ^{\tilde {\mathcal R}}} \widetilde {\mathbf m}^* (u _1 \circ u _2 )(a, b)
=& \int {\mathfrak L}_{X ^{\tilde {\mathcal R}}} (\widetilde {\mathbf m}^* u _1 )(a, c ) (\widetilde {\mathbf m}^* u _2 )(c, b) \mu _{{\mathbf s}(a)} (c) \\
{\mathfrak L}_{Y ^{\tilde {\mathcal L}}} \widetilde {\mathbf m}^* (u _1 \circ u _2 )(a, b)
=& \int (\widetilde {\mathbf m}^* u _1 )(a, c ) {\mathfrak L}_{X ^{\tilde {\mathcal R}}} (\widetilde {\mathbf m}^* u _2 )(c, b) \mu _{{\mathbf s}(a)} (c)\end{aligned}$$ (here, note that ${\mathfrak L}_{X ^{\tilde {\mathcal R}}} {\mathbf m}^* u _1 (a, c)$ only differentiates in the $a $-direction), and so on for higher derivatives.
Note that the vector representation $\nu $ is well defined on $\Psi ^{ -\infty } _\varepsilon ({\mathcal G}, {\mathrm E})$ because of the sub-exponential growth assumption.
To apply the results of Section \[LauNis\], we first verify that
\[C\*SubsetProp\] For any $\varepsilon > 0$, $$\Psi ^{- \infty } _\varepsilon \subseteq {\mathfrak C}^* ({\mathcal G}, {\mathrm E}),$$ where ${\mathfrak C}^* ({\mathcal G}, {\mathrm E})$ is defined in Definition \[OpNorm\].
Let $\{\phi _n \} \in C ^\infty ({\mathbb R})$ be a series such that $0 \leq \phi _n \leq 1$, $\phi _n = 1 $ on $[0 , n) $, and $\phi _n = 0 $ on $[n +1 , \infty)$, and define $\chi _n (a) := \phi _n (d _{\mathbf s}(a)) \in C ^\infty _c ({\mathcal G})$. Given any $\kappa \in \Psi ^{- \infty } _\varepsilon ({\mathcal G}, {\mathrm E})$, Write $\kappa (a) = e ^{- \varepsilon d _{\mathbf s}(a)} u (a)$, where $u (a)$ is bounded.
Consider $\kappa _n := \chi _n \kappa \in \Gamma ^\infty _c ({\mathcal G}, {\mathrm E}) \cong \Psi ^\infty _\mu ({\mathcal G}, {\mathrm E})$. For any $x \in {\mathrm M}, n \in {\mathbb N}$, one has $$\begin{aligned}
\int _{a \in {\mathbf s}^{-1} (x)} | \kappa - \kappa _n | (a) \mu _x (a)
=& \: \int _{a \in {\mathcal G}_x \backslash B ({\mathrm M}, n) } e ^{- \varepsilon d _{\mathbf s}(a)}
(1 - \chi _n ) | u | (a) \mu _x (a) \\
\leq & \: e ^{- \frac{\varepsilon n }{2}}
\int _{a \in {\mathcal G}_x \backslash B ({\mathrm M}, n) } e ^{- \frac{\varepsilon}{2} d _{\mathbf s}(a)}
(1 - \chi _n ) | u | (a) \mu _x (a), \end{aligned}$$ where $B ({\mathrm M}, n) := \{ a \in {\mathcal G}: d (a , {\mathbf s}(a)) < n \}$. By the sub-exponential growth assumption, the integral is bounded by some constant $C$, independent of $x$. It follows that $\sup _{x \in {\mathrm M}} \| \kappa - \kappa _n \| _{{\mathbf L}^1 ({\mathbf s}^{-1} (x))} \to 0 $ as $n \to \infty $. Also, observe that $\kappa (a ^{-1}) = e ^{- \varepsilon d _{\mathbf s}(a)} u (a ^{-1})$ (since $d _{{\mathbf s}} (a ^{-1}) = d _{\mathbf s}(a)$). Applying exactly the same arguments to $\kappa (a ^{-1})$ one arrives at $$\| \kappa - \kappa _n \|_1 \to 0.$$ Hence $\kappa \in {\mathfrak C}^* ({\mathcal G}, {\mathrm E})$.
Combining the above Proposition \[C\*SubsetProp\] with Lemma \[ExactC\^\*\], one has:
\[CptCor\] For any $\varepsilon > 0$, $\varPsi \in \Psi ^{- \infty} _\varepsilon ({\mathcal G}, {\mathrm E})$ such that $\varPsi |_{{\mathrm Z}_j} = 0$, for any invariant sub-manifolds ${\mathrm Z}_j$, then $\nu _0 ( \varPsi ) : {\mathbf L}^2 ({\mathrm M}_0) \to {\mathbf L}^2 ({\mathrm M}_0)$ is a compact operator.
As a simple application of the calculus with bound, we can rewrite Theorem \[ParaThm\] as
For any $\varPsi \in \Psi ^{[m]} _\mu ({\mathcal G}, {\mathrm E})$ satisfying the hypothesis of Theorem \[ParaThm\], there exist $\tilde Q \in \Psi ^{[- m]} _\varepsilon ({\mathcal G}, {\mathrm E})$ such that $$\nu (\varPsi) \tilde Q - \operatorname{i d}\in \Psi ^{[- \infty]} _\varepsilon ({\mathcal G}, {\mathrm E})$$ is compact.
$ \; $
The heat calculus
=================
The heat kernel of perturbed Laplacian operators
-------------------------------------------------
In this section, we construct the heat kernel of some second order pseudo-differential operators on a groupoid ${\mathcal G}\rightrightarrows {\mathrm M}$ with ${\mathrm M}$ compact.
Given a vector bundle ${\mathrm E}$ over ${\mathrm M}$, fix an ${\mathcal A}$-connection $\nabla ^{\mathrm E}$ on ${\mathrm E}$. Then the pull-back defines a (family of ${\mathbf s}$-fiberwise) connection on the bundle ${\mathbf s}^{-1} {\mathrm E}\rightarrow {\mathbf s}^{-1} (x), x \in {\mathrm M}$, which we shall still denote by $\nabla ^{\mathrm E}$. Also, recall that we fixed a metric on ${\mathcal A}$, hence a Riemannian metric on the fibers ${\mathbf s}^{-1} (x), x \in {\mathrm M}$, which we shall still denote by $g _{\mathcal A}$. We define the Laplacian by taking the trace of the square of $\nabla ^{\mathrm E}$. More precisely:
\[LapDfn\] The [*Laplacian*]{} $\Delta ^{\mathrm E}\in \Psi ^2 _\mu ({\mathcal G})$ is the family of operators $\{ \Delta ^{\mathrm E}_x \}_{x \in {\mathrm M}}, $ where $$\Delta ^{\mathrm E}_x := \sum _{i=1}^n (\nabla ^{\mathrm E}_{X_i } \nabla ^{\mathrm E}_{X_i } - \nabla ^{\mathrm E}_{\nabla ^{\mathrm E}_{X_i} X_i}),$$ and $X_i$ is any local orthonormal basis of $T {\mathcal G}_x $.
Note that $\Delta ^{\mathrm E}$ is elliptic, and its principal symbol does not depend on the chosen connection $\nabla ^{\mathrm E}$.
We consider an operator of the form $$\label{GenLap}
\Delta ^{\mathrm E}+ F + K,$$ where $F \in \Gamma ^\infty ({\mathbf t}^{-1} {\mathrm E}\otimes {\mathbf t}^{-1} {\mathrm E})$, considered as a differential operator of order 0; and $K \in \Psi ^{- \infty }_\mu ({\mathcal G}, {\mathrm E})$. We shall denote the reduced kernel of $K$ by $\kappa $.
Since the restriction of ${\mathbf t}^{-1} {\mathrm E}$ to each ${\mathbf s}$-fiber $ {\mathcal G}_x $ is a vector bundle with bounded geometry, we have the Sobolev norms $ \| \cdot \| _{\infty , l } $ defined by Equation (\[L-Infty\]). For $u \in \Gamma ^\infty ({\mathbf t}^{-1} {\mathrm E})$, we define $$\| u \| _l := \sup_{x \in {\mathrm M}} \{ \| u |_{{\mathbf s}^{-1} (x)} \|_{\infty , l} \}.$$
Denote by ${\mathbf t}^{-1} {\mathrm E}\otimes {\mathbf s}^{-1} {\mathrm E}' \ltimes (0, \infty ) $ the pullback of ${\mathbf t}^{-1} {\mathrm E}\otimes {\mathbf s}^{-1} {\mathrm E}' \to {\mathcal G}$ by the projection ${\mathcal G}\times (0, \infty) \to {\mathcal G}$.
A [*(groupoid) Heat kernel*]{} of $\Delta ^{\mathrm E}+ F + K $ is a continuous section $$Q \in \Gamma ^0 ({\mathbf t}^{-1} {\mathrm E}\otimes {\mathbf s}^{-1} {\mathrm E}' \ltimes (0 , \infty ) ) ,$$ such that $ Q (a , t) , Q (a ^{-1} , t) $ are smooth when restricted to all ${\mathcal G}_x \times (0 , \infty ) $, and satisfies:
1. The heat equation $$(\partial _t + \Delta ^{\mathrm E}+ F + K) Q (a, t) = 0.$$ Here, we use the fact that ${\mathbf s}^{-1} {\mathrm E}' |_{{\mathcal G}_x} \cong {\mathcal G}_x \times {\mathrm E}'_x $, and let $\Delta ^{\mathrm E}+ F + K$ to act on the ${\mathbf t}^{-1} {\mathrm E}$ factor of $ Q (a, t) |_{{\mathcal G}_x} \in \Gamma ^\infty ({\mathbf t}^{-1} {\mathrm E}\otimes {\mathbf s}^{-1} {\mathrm E}' )
\cong {\mathrm E}' _x \times \Gamma ^\infty ({\mathbf t}^{-1} {\mathrm E})$ for each $t$ fixed;
2. The initial condition $$\lim _{t \rightarrow 0^+} Q \circ u = u,
\quad \forall u \in \Gamma ^\infty _c ({\mathbf t}^{-1} {\mathrm E}),$$ where $\circ $ denotes the convolution product.
Let $Q $ be a groupoid heat kernel. Then it is clear that for any $x \in {\mathrm M}$, $( a , b ) \in {\mathcal G}_x \times {\mathcal G}_x \mapsto Q (a b ^{-1} , t) $ is a heat kernel of $( \Delta ^{\mathrm E}+ F + K )_x $ on the manifold with bounded geometry ${\mathcal G}_x$. Using the uniqueness of the heat kernel on manifolds with bounded geometry, it is clear that:
A groupoid heat kernel $Q $ of $\Delta ^{\mathrm E}+ F + K $, if it exists, is unique.
### **The formal solution**
Before we start, we need to define some notation.
Recall that there exists $r_0 > 0$ such that $\exp ^\nabla $ is a diffeomorphism from the set $${\mathcal A}_{r_0} := \{ X \in {\mathcal A}:g _{\mathcal A}(X, X) < r_0 ^2 \}$$ onto its image. For each $x \in {\mathrm M}$, we denote the polar coordinates on ${\mathcal A}_x$, the fiber of ${\mathcal A}$ over $x$, by $(r , \vartheta )$. The image of ${\mathcal A}_{r_0} $ under $\exp ^\nabla$ is denoted by $B ({\mathrm M}, r_0 )$. Note that since $$d (\exp ^\nabla (r, \vartheta ) , x ) = r,$$ therefore $B ({\mathrm M}, r_0 ) = \{ a \in {\mathcal G}: d (a , {\mathbf s}(a)) < r_0 \}$, as expected. The exponential map also defines a local trivialization of ${\mathbf t}^{-1} {\mathrm E}$: For each $a = \exp ^\nabla X \in B ({\mathrm M}, r_0), E \in {\mathrm E}_{{\mathbf s}(a)}$ where $X \in {\mathcal A}_{{\mathbf s}(a)}$, define $ T (a) (E) \in {\mathbf t}^{-1} {\mathrm E}_a $ to be the parallel transport of $ E $ to $ a $ along the curve $ \exp ^\nabla \tau X , \tau \in [0, 1] $. Hence $T$ is a map from the set $\{ (a , E) : a \in B ({\mathrm M}, r_0), E \in {\mathrm E}_{{\mathbf s}(a)} \} $ to ${\mathbf t}^{-1} {\mathrm E}|_{B ({\mathrm M}, r_0)}$, and we denote its inverse map by $T^{-1} $. When restricted to ${\mathbf t}^{-1} {\mathrm E}|_{{\mathcal G}_x} $ for some $x \in {\mathrm M}$, the image of $T ^{-1}$, lies in $E _x$. In that case we shall still denote the restricted map by $T ^{-1} : {\mathbf t}^{-1} {\mathrm E}|_{{\mathcal G}_x \bigcap B ({\mathrm M}, r_0)} \rightarrow {\mathrm E}_x $.
Lastly, we let $J := \det (d \exp ^\nabla) \circ (\exp ^\nabla)^{-1} $ to be the Jacobian, and $V := d (a , {\mathbf s}(a)) \times d \exp^\nabla (\partial _r )$ be the radial vector field.
Consider a kernel of the form $$q (a , t) \Phi (a, t) \in \Gamma ^\infty ({\mathbf t}^{-1} {\mathrm E}\otimes {\mathbf s}^{-1} {\mathrm E}' \ltimes (0, \infty)),$$ where $q : B ({\mathrm M}, r_0 ) \times (0 , \infty ) \rightarrow {\mathbb R}$ is the Gaussian function $$q (a , t) := (4 \pi t)^{- \frac{n}{2}} \: e^{- \frac{d (a , {\mathbf s}(a))^2 }{4 t}}.$$ A straightforward calculation shows that:
One has $$(\partial _t + \Delta ^{\mathrm E}+ F) q (a , t) \Phi (a, t)
= q (a , t) (\partial _t + \Delta ^{\mathrm E}+ F + t ^{-1} \nabla ^{\mathrm E}_V + \frac{{\mathfrak L}_V J}{2 t J} ) \Phi (a, t).$$
\[HeatPowerSeries\] There exists a formal power series $$\Phi (a, t) = \sum _{i=1} ^\infty t^i \Phi _i (a), \quad \Phi _i \in \Gamma ^\infty$$ satisfying the equation $$(\partial _t + \Delta ^{\mathrm E}+ F + t ^{-1} \nabla ^{\mathrm E}_V + \frac{{\mathfrak L}_V J}{2 t J} ) \Phi (a, t) = 0.$$
Equating coefficients one gets $$\begin{aligned}
\nabla ^{\mathrm E}_V ( J ^\frac{1}{2} \Phi _0 ) = & \: 0 \\
\nabla ^{\mathrm E}_V ( J ^\frac{1}{2} \Phi _i ) + i \Phi _i =& - (\partial _t + \Delta ^{\mathrm E}+ F ) \Phi _{i-1} ,
\quad i = 1, 2 \cdots \end{aligned}$$ These are simple ordinary differential equations, with explicit solutions $$\begin{aligned}
\Phi _0 (\exp X) =& J ^{- \frac{1}{2}} T (\exp X) \\
\Phi _i (\exp X) =&
- J ^{ - \frac{1}{2}} T \int _0 ^1 J ^{\frac{1}{2}} T^{-1} ((\partial _t + \Delta ^{\mathrm E}+ F )
\Phi _{i-1} (\exp \tau X )) \tau ^{i-1} \: d \tau .\end{aligned}$$
Fix a cutoff function $\chi $ supported on $B ({\mathrm M}, r_0)$ such that $\chi = 1 $ on the smaller set $B ({\mathrm M}, \frac{r_0}{2}) := \{ a \in {\mathcal G}: d (a , {\mathbf s}(a)) \leq \frac{r_0}{2} \}$. Write $$G _N (a, t) := \chi (a) q (a, t) \sum _{i=1}^N t^i \Phi _i (a),
\quad t \in (0, \infty ).$$ Then one has
\[ParaProof\] For any $N > \frac{n}{2}$,
1. For any $k, l \in {\mathbb N}$, there exists a constant $C_{k, l}$ such that $$\| \partial _t ^k ((\partial _t + \Delta ^{\mathrm E}+ F) G _N) \|_l \leq
C_{k, l} t ^{N - \frac{n}{2} - k - \frac{l}{2}} ;$$
2. For any $t_0 > 0$, the map $$u \mapsto G _N (\cdot , t) \circ u, 0 \leq t \leq t_0$$ is a uniformly bounded family of operators on $\Gamma ^l ({\mathbf t}^{-1} {\mathrm E})$, and for any $u \in \Gamma ^l ({\mathbf t}^{-1} {\mathrm E})$, $$\lim _{t \rightarrow 0^+} \| G _N \circ u - u \| _l = 0.$$
On $B ({\mathrm M}, \frac{r_0}{2})$, from the proof of Lemma \[HeatPowerSeries\], one has $$\begin{aligned}
(\partial _t + \Delta ^{\mathrm E}+ F) G _N (a) =& \: t ^N q (a, t) \Phi _N (a) \\
=& \: (4 \pi )^{- \frac{n}{2}} t ^{N - \frac{n}{2}} e ^{- \frac{d (a , {\mathbf s}(a))^2}{4 t} } \Phi _N (a).\end{aligned}$$ It is elementary that $ e ^{- \frac{d (a , {\mathbf s}(a))^2}{4 t} }$ is bounded for any $a, t$, and $\Phi _N $ is smooth and hence has bounded derivatives. On ${\mathcal G}\backslash B ({\mathrm M}, \frac{r_0}{2}) $ observe that $e ^{- \frac {d (a , {\mathbf s}(a))}{4 t}}$ and all its derivatives decay faster than any powers as $t \rightarrow 0$. That proves (1) in the case $l = k = 0$. Other cases follow from a similar argument, with the additional observation that $$\begin{aligned}
\partial _t e^{- \frac {y^2 }{t}} =& - t ^{-1} (\frac{y ^2}{t} ) e ^{ - \frac{y ^2}{t}} = O (t^{-1}) \\
\partial _y e^{- \frac {y^2 }{t}} =& - t ^{-\frac{1}{2}} (\frac{y ^2}{t} )^{\frac{1}{2}} e ^{ - \frac{y ^2}{t}}
= O (t^{- \frac{1}{2}}) .\end{aligned}$$
To prove (2), write for any $a \in {\mathcal G}$, $$\begin{aligned}
G _N \circ u (a) := & \: \int _{{\mathcal G}_{{\mathbf s}(a)}} G _N (a b^{-1} ) u (b ) \mu _{{\mathbf s}(a)} (b) \\
= & \: \int _{{\mathbf s}^{-1} ({\mathbf t}(a))} G _N (c ^{-1}) u ( c a ) \mu _{{\mathbf t}(a)} (c)
\quad \text {(using the right invariance of $\mu$)} \\
= & \: \int _{{\mathbf t}(a)}
(4 \pi t)^{- \frac{n}{2}} e^{\frac{d (c ^{-1} , {\mathbf t}(c))^2 }{4 t}}
\chi (c ^{-1}) \Big( \sum _{i=0}^N t ^i \Phi _i (c^{-1}) \Big) u (c a) \mu _{{\mathbf t}(a)} (c).\end{aligned}$$ By right invariance and symmetry of the distance function $d (\cdot , \cdot)$, one has $d (c ^{-1} , {\mathbf t}(c)) \\ = d (c , {\mathbf s}(c)) $. Hence $\chi ( c^ {-1}) = \chi (c) $, and $ e^{\frac{d (c ^{-1} , {\mathbf t}(c))^2 }{4 t}} = e^{\frac{d (c , {\mathbf s}(c))^2 }{4 t}}$. Therefore the integrand is supported on $B ({\mathrm M}, r _0)$ and the integral can be computed by a change of variable $c = \exp ^\nabla X , X \in {\mathcal A}_{{\mathbf t}(a)}, g_{\mathcal A}(X, X) \leq r_0 ^2 $: $$\begin{aligned}
\int _{{\mathbf s}^{-1} ({\mathbf t}(a))}
(4 \pi t)^{- \frac{n}{2}} & e^{- \frac{d (c ^{-1} , {\mathbf t}(c))^2 }{4 t}}
\chi (c ^{-1}) \Big( \sum _{i=0}^N t ^i \Phi _i (c^{-1}) \Big) u (c a) \mu _{{\mathbf t}(a)} (c) \\
=& \int _{{\mathbf s}^{-1} ({\mathbf t}(a))}
(4 \pi t)^{- \frac{n}{2}} e^{- \frac{d (c , {\mathbf s}(c))^2 }{4 t}}
\chi (c) \Big( \sum _{i=0}^N t ^i \Phi _i (c^{-1}) \Big) u (c a) \mu _{{\mathbf t}(a)} (c) \\
=& \int _{X \in {\mathcal A}_{{\mathbf t}(a)}} (4 \pi t)^{- \frac{n}{2}} e^{- \frac{g _{\mathcal A}(X, X) }{4 t}} \chi (\exp ^\nabla X) \\
& \times \Big( \sum _{i=0}^N t ^i \Phi _i ((\exp ^\nabla X)^{-1}) \Big)
u ((\exp ^\nabla X) a) \det (d \exp ^\nabla )(X) \: d X.\end{aligned}$$ It is clear that the last expression converges to $$\chi (\exp ^\nabla 0) (\sum _{i=0}^N t ^i \Phi _i ((\exp ^\nabla 0)^{-1}))
u ((\exp ^\nabla 0) a) (\det (d \exp ^\nabla )(0)) = u (a),$$ since $(4 \pi t)^{- \frac{n}{2}} e^{- \frac {g _{\mathcal A}(X, X) }{4 t}}$ is just the Gaussian heat kernel on the usual Euclidean space.
### **From parametrix to heat kernel** {#LeviSeries}
In the last section we constructed an approximate solution to the heat kernel. In this section we use the method of Levi parametrix to construct a heat kernel. We turn to operators of the form $$\Delta ^{\mathrm E}+ F + K.$$
For each $N > \frac{n}{2}$, define the sections $ R ^{(k)} _n \in \Gamma ^\infty ({\mathbf t}^{-1} {\mathrm E}\otimes {\mathbf s}^{-1} {\mathrm E}' _ {[0, \infty )})$: $$\begin{aligned}
R ^{(1)} _N :=& \: (\partial _t + \Delta ^{\mathrm E}+ F + K ) G _N \\
R ^{(k)} _N :=& \: \int _0 ^t R _N (\cdot , t - \tau ) \circ R ^{(k-1)} _N (\cdot , \tau ) d \tau \\
=& \: \int _0 ^t \int _{{\mathbf s}^{-1} (a)} R_n (a b^{-1} , t - \tau ) R^{(k-1)} _N (b , \tau ) \mu _{{\mathbf s}(a)} (b) d \tau \\
Q ^{(0)} _N :=& \: G _N \\
Q ^{(k)} _N :=& \: \int _0 ^t G _N (\cdot , t - \tau ) \circ R ^{(k)} _N (\cdot , \tau ) d \tau, \quad k \geq 1 .\end{aligned}$$ Then one has the estimates
\[RGrowth\] Let $N > \frac{n + l}{2} $. There exists constants $\tilde C _l, l \in {\mathbb N}$ such that $$\| R ^{(k)} (\cdot , t) \| _l
\leq \tilde C _l \tilde C_0 ^k M ^k (1 + t ^{N - \frac{n+l}{2}} )^k t ^{k-1} ((k-1)!)^{-1}.$$
Using the same arguments as in the proof of (2) Lemma \[ParaProof\], one has $K G _N = \kappa (\cdot ) \circ G _N (\cdot , t) \rightarrow \kappa $ in the $\| \cdot \| _l $-norm as $t \rightarrow 0$. Therefore $ K G _N $ is a continuous section over $ {\mathcal G}\times [0, \infty)$, and its $l$-partial derivatives extends continuously to $t \in [0, \infty)$. Combining with (1) of Lemma \[ParaProof\], it follows that the integrand is a continuous section on ${\mathcal G}\times [0, t] $, so the integral exists (and is finite).
Combining (1) of Lemma \[ParaProof\] and the boundedness of $K$ to obtain for each $l$, $$\| R ^{(1)} _N (\cdot , t ) \| _l
= \| (\partial _t + \Delta ^{\mathrm E}+ F + K ) G _N (\cdot , t) \|_l
\leq \tilde C _l (1 + t ^{N - \frac{n+l}{2}} )$$ for some $\tilde C_l > 0$. Expand $R ^{(k)} $ as a multiple integral: $$\begin{aligned}
R ^{(k)} _N (a , t) =
\int _{0 \leq t_{k-1} \leq \cdots \leq t _1 \leq t }
& \int _{ b_1 , b_2, \cdots b_{k -1} \in {\mathbf s}^{-1} (a) } \\
R _N ^{(1)} (a b _1 ^{-1} & , t - t_1 ) R _N ^{(1)} (b_1 b_2 ^{-1} , t_1 - t_2) \cdots \\
\times R _N ^{(1)} & (b _{k-1} b _k ^{-1}, t _{k-2} - t _{k-1} )
R _N ^{(1)} (b _{k-1} , t_{k-1} ) \mu (b _1) \cdots \mu (b _{k-1}).\end{aligned}$$
Next, consider the domain of integration. Since both $G _N $ and $\kappa $ have compact supports, $R _N ^{(1)}$ is compactly supported for each $t \geq 0$. In particular, there exists $\rho > 0$ such that $R _N ^{(1)} (c _1 c _2 ^{-1} , t ) = 0 $ for any $c _1 , c _2 \in {\mathcal G}$ such that ${\mathbf s}(c _1) = {\mathbf s}(c _2) $ and $d (c _1 , c _2) \geq \rho $. Using the bounded geometry property of the ${\mathbf s}$-fibers, we take $$M := \sup _{c \in {\mathcal G}} \int _{B (a, \rho )} \mu _{{\mathbf s}(c)} < \infty.$$ Then it follows that the volume of the domain of integration is bounded by $$\int _{b _1 \in B (a , \rho )} \int _{b _2 \in B (b_1 , \rho )} \cdots \int _{b _{k-1} \in B (b _{k-2} , \rho )}
\mu _{{\mathbf s}(a)} (b _1) \cdots \mu _{{\mathbf s}(a)} (b _{k-1} ) \leq M ^{k-1}.$$ By elementary calculation, one also gets $$\int _{0 \leq t_k \leq \cdots \leq t _1 \leq t } d t _1 d t _2 \cdots d t _{k - 1} = t ^{k-1} ((k - 1)! )^{-1}.$$ Finally, one has for any $a \in {\mathcal G}$, $$\begin{aligned}
| R ^{(k)} _N (a , t) |_l \leq
\int _{0 \leq t_{k-1} \leq \cdots \leq t _1 \leq t }
& \int _{ b_1 , b_2, \cdots b_{k -1} \in {\mathbf s}^{-1} (a) }
| R _N ^{(1)} (a b _1 ^{-1} , t - t_1 ) |_l \\
\times & | R _N ^{(1)} (b_1 b_2 ^{-1} , t_1 - t_2)| _0 \cdots
|R _N ^{(1)} (b _{k-1} b _k ^{-1} , t _{k-2} - t _{k-1} ) |_0 \\
& \times | R _N ^{(1)} (b _{k-1} , t_{k-1} ) |_0 \mu _{{\mathbf s}(a)} (b _1) \cdots \mu _{{\mathbf s}(a)} (b _{k-1}) \\
\leq \int _{0 \leq t_{k-1} \leq \cdots \leq t _1 \leq t }
& \int _{b _1 \in B (a , \rho )} \int _{b _2 \in B (b_1 , \rho )} \cdots \int _{b _{k-1} \in B (b _{k-2} , \rho )} \\
& \tilde C _l \tilde C _0 ^{k-1} (1 + t ^{N - \frac{n+l}{2}} )^k
\mu _{{\mathbf s}(a)} (b _1) \cdots \mu _{{\mathbf s}(a)} (b _{k-1} ) \\
\leq \tilde C _l \tilde C_0 ^k M ^k t ^{k-1} (1 & + t ^{N - \frac{n+l}{2}} )^k ((k-1)!)^{-1}.\end{aligned}$$ The assertion follows by taking supremum over $a \in {\mathcal G}$.
\[QEst\] Assume that $l > 1, 2N > n + l$.
1. There exists constants $C' _l$ such that $$\| Q _N ^{(k)} (\cdot , t) \|_l \leq C' _l \tilde C_0 ^k M ^k (1 + t ^{N - \frac{n+l}{2}} )^k t ^k (k!)^{-1};$$
2. The kernel $Q _N ^{(k)} (a, t)$ is continuously differentiable with respect to $t$ and $$(\partial _t + \Delta ^{\mathrm E}+ F + K) Q ^{(k)} _N = R ^{(k+1)} _N + R ^{(k)} _N.$$
Define the section $$B (a, t, s) := (G _N (\cdot , t - s ) \circ R ^{(k)} _N (\cdot , s)) (a),
\quad \forall a \in {\mathcal G}, t \in [0, \infty ), s \in [0, t].$$ Since $G _N (\cdot , t - s) $ is $C^l$ by our construction, by (2) of Lemma \[ParaProof\], one has for any $0 \leq s \leq t$, $$\begin{aligned}
\| b (\cdot , t , s) \| _l \leq & \: C' \int _0 ^t \| R ^{(k)} _N (\cdot , s) \| _l d s \\
\leq & \: C' \tilde C_0 ^k M ^k (1 + t ^{N - \frac{n+l}{2}} )^k \int _0 ^t s ^{k-1} ((k-1)!)^{-1} d s \\
\leq & \: C' \tilde C_0 ^k M ^k (1 + t ^{N - \frac{n+l}{2}} )^k t ^{k} (k!)^{-1},\end{aligned}$$ from which (1) follows. To prove (2), one has $$\begin{aligned}
(\partial _t + \Delta ^{\mathrm E}+ F + K) & (\int _0 ^t B (a, t, s ) d s ) (a, t) \\
=& \: B (a, t, t)
+ \int _0 ^t (\partial _t + \Delta ^{\mathrm E}+ F + K) G _N (\cdot , t - s) \circ R ^{(k)} _N (\cdot , s) d s \\
=& \: R ^{(k)} _N (a, t)
+ \int _0 ^t R ^{(0)} (\cdot , t - s) \circ R ^{(k)} _N (\cdot , s) d s \\
=& \: R ^{(k)} _N (a, t) + R ^{(k+1)} _N (a, t).\end{aligned}$$
Finally, we can construct the heat kernel
\[HeatKer\] For any $l, N$ with $2 N > n + l + 1$, the series $$\sum _{k=0} ^\infty (-1)^k Q ^{(k)} _N (\cdot , t)$$ converges to a limit $Q (\cdot , t) \in \Gamma ^0 ({\mathbf t}^{-1} {\mathrm E}\otimes {\mathbf s}^{-1} {\mathrm E}' \times (0, \infty) ) $, independent of $N$, in the $\| \cdot \|_l$ norm. Furthermore,
1. The section $Q$ is the heat kernel of $\partial _t + \Delta ^{\mathrm E}+ F + K $;
2. $ G _N $ approximates $Q$ in the sense that $$\| Q - G _N \| _l = O (t)$$ as $t \rightarrow 0$.
From (1) of Lemma \[QEst\], one has $Q ^{(k)} _N < \frac{1}{2 ^k }$ for sufficient large $k$. Convergence of the series $\sum _{k=0} ^\infty (-1)^k Q _N ^{(k)}$ follows from the comparison test. Assertion (2) follows from $ Q - G _N = \sum _{k=1} ^\infty Q _N ^{(k)}$, and implies the initial condition of (1), i.e., $$\lim _{t \rightarrow 0^+} \| Q \circ u - u \| _l = 0,$$ since $$\| Q \circ u - u \| \leq \| (Q - G _N ) \circ u \| _l + \| G _N \circ u - u \| _l \rightarrow 0.$$ To show that $(\partial _t + \Delta ^{\mathrm E}+ F + K) Q = 0$, observe that $\| (\partial _t + \Delta ^{\mathrm E}+ F + K) Q _N ^{(k)} \| _l \leq 2 ^{-k} $ for sufficient large $k$. Therefore one has $$\begin{aligned}
(\partial _t + \Delta ^{\mathrm E}+ F + K) \sum _{k=1}^\infty (-1) ^k Q _N ^{(k)}
=& \: \sum _{k=1} ^\infty (-1)^k (\partial _t + \Delta ^{\mathrm E}+ F + K) Q _N ^{(k)} \\
=& \: R ^{(1)} _N + \sum _{k=2} ^\infty (-1)^k (R ^{(k)} _N + R ^{(k-1)} _N) \\
=& \: 0.\end{aligned}$$
We shall denote the heat kernel of the Laplacian $\Delta ^{\mathrm E}+ F + K$, as constructed above, by $$e ^{- t (\Delta ^{\mathrm E}+ F + K) } := Q (\cdot , t).$$
Alternatively, let $e ^{-t (\Delta ^{\mathrm E}+ F)} $ be the heat kernel of $\Delta ^{\mathrm E}+ F$ constructed using the same method as above. Then a heat kernel of $\Delta ^{\mathrm E}+ F + K$ is given by $$\label{HeatPert}
e ^{-t (\Delta ^{\mathrm E}+ F + K)} = e ^{-t (\Delta ^{\mathrm E}+ F)} + \sum_{i = 1} ^\infty t^i \tilde Q ^{(i)} ,$$ where $$\tilde Q ^{(i)}
:= \int _{0 < \tau _0 < \cdots < \tau _i < 1}
e ^{-t (\Delta ^{\mathrm E}+ F)} (\cdot , \tau _0 t) \circ \kappa
\circ e ^{-t (\Delta ^{\mathrm E}+ F)} (\cdot , \tau _1 t) \circ \kappa \circ \cdots
\circ \kappa \circ e ^{-t (\Delta ^{\mathrm E}+ F)} (\cdot , \tau _i t),$$ and the integration is over the Lebesgue measure.
As in the case of manifolds with bounded geometry, the heat kernel of Laplacian on groupoids satisfies the following ‘off diagonal’ estimate:
\[HeatDecay\] Fix $\varepsilon > 0$ such that for any $a \in {\mathcal G}$, $\kappa (a b ^{-1}) = 0$ and $ G _N (a b ^{-1}, t) \\ = 0 $ for any $t$, whenever $b \in {\mathcal G}_{{\mathbf s}(a)} \backslash B (a, \varepsilon )$. Let $t > 0$ be fixed. For any $\lambda > 0$, there exists $C > 0 $ such that $$\label{QDecay}
| e ^{- t (\Delta ^{\mathrm E}+ F + K) }(a, t) |
\leq C e ^{- \lambda d (a, {\mathbf s}(a))}, \quad \forall a \in {\mathcal G}, d (a , {\mathbf s}(a) ) > 2 \varepsilon ,$$ and $Q (a ^{-1} , t) \in {\mathbf L}^1 ({\mathcal G}_{{\mathbf s}(a)})$
Let $I \in {\mathbb N}$ be such that $I \varepsilon \leq d (a , {\mathbf s}(a)) \leq (I + 1) \varepsilon $. Then $Q _N ^{(k)} (a, t) = 0 $ for any $k < I$. Therefore one has $$\begin{aligned}
| Q (a, t) | e ^{\lambda d (a, {\mathbf s}(a))}
\leq & \: \sum _{k = I} ^\infty e ^{\lambda (I + 1) \varepsilon }
C' _0 \tilde C_0 ^k M ^k (1 + t ^{N - \frac{n}{2}} )^k t ^k (k!)^{-1} \\
=& \: e ^{\lambda (I + 1) \varepsilon }
\frac{ C' _0 \tilde C_0 ^I M ^I (1 + t ^{N - \frac{n}{2}} )^I t ^I }{I!}
\times \sum _{k = 0} ^\infty \frac{ \tilde C_0 ^k M ^k (1 + t ^{N - \frac{n}{2}} )^k t ^k I! }{(k + I)!}.\end{aligned}$$ It is clear that the last expression goes to $0$ as $I \rightarrow \infty$, so Equation (\[QDecay\]) is proved. From Equation (\[QDecay\]), one has $$Q (a ^{-1} , t) \leq C e ^{- \lambda d (a ^{-1}, {\mathbf t}(a))} = C e ^{- \lambda d (a, {\mathbf s}(a)) }.$$ It follows that $Q (a^{-1} , t) \in {\mathbf L}^1 ({\mathcal G}_{{\mathbf s}(a)})$ because ${\mathcal G}_{{\mathbf s}(a)}$ has at most exponential volume growth.
### **The heat kernel of the vector representation**
We turn to study the heat kernel of $\nu (\partial _t + \nabla ^{\mathrm E}+ F + K)$, where $\nu$ is the vector representation. The construction becomes very simple, once we know the heat kernel of $(\partial _t + \nabla ^{\mathrm E}+ F + K)$.
\[VectorHeat\] If $Q $ is a heat kernel of $\partial _t + \nabla ^{\mathrm E}+ F + K$, then $\nu (Q)$ is a heat kernel of $\nu (\partial _t + \nabla ^{\mathrm E}+ F + K)$ in the sense that $$\begin{aligned}
\nu (\partial _t + \nabla ^{\mathrm E}+ F + K) \nu (Q) f =& \: 0, \quad \forall t > 0 \\ \nonumber
\lim _{t \rightarrow 0^+} \| \nu (Q) f - f \| =& \: 0\end{aligned}$$ for any $f \in \Gamma ^\infty ({\mathrm E})$.
By Proposition \[HeatDecay\], $\nu Q$ is well defined for each $t \leq 0$. By definition one has $${\mathbf t}^{-1} ( \nu (\partial _t + \nabla ^{\mathrm E}+ F + K) \nu (Q) f)
= (\partial _t + \nabla ^{\mathrm E}+ F + K) Q ({\mathbf t}^{-1} f) = 0.$$ The second equality follows by a similar argument.
One important observation from Theorem \[VectorHeat\] is that the heat kernel of the vector representation is not a smoothing operator. However, if ${\mathcal G}\rightrightarrows {\mathrm M}$ is a Lauter-Nistor groupoid in the sense of Definition \[BdGpoid\], then for any $f \in \Gamma ^\infty _c ({\mathrm E}| _{{\mathrm M}_0})$, one has $$\nu (K) (f) (x) = \int _{a \in {\mathcal G}_x } \kappa (a ^{-1} ) f ({\mathbf t}(a)) \mu _{x} (a)
= \int _{y \in {\mathrm M}_\alpha } \kappa |_{{\mathcal G}_{{\mathrm M}_0}} (x , y ) f (y) \mu _{{\mathrm M}_0},$$ where ${\mathrm M}_\alpha $ is the connected component of ${\mathrm M}_0$ containing $x$ and we have used the identification ${\mathcal G}_{{\mathrm M}_0} \cong \coprod _\alpha {\mathrm M}_\alpha \times {\mathrm M}_\alpha $.
### **Application: Heat kernel in edge calculus**
As an application of our construction, we give a simple proof to Albin’s conjecture on generalization of [@Albin;EdgeInd Theorem 4.3]. We refer to the same paper for details.
\[AlbinConj\] A Laplacian operator on any manifolds ${\mathrm M}$ with iterated complete edge has a heat kernel.
By [@Nistor;LieMfld], the pseudo-differential calculus is defined by a groupoid ${\mathcal G}$ over the compactification ${\mathrm M}$ of ${\mathrm M}_0 $. In particular, any Laplacian on ${\mathrm M}_0$ is the vector representation of a Laplacian operator on ${\mathcal G}$. The lemma follows from our constructions above.
Transverse regularity of the heat kernel
-----------------------------------------
In the last Section, we proved that the series $ \sum _{k=0} ^\infty (-1)^k Q ^{(k)} _N (\cdot , t) $ converges to the heat kernel $Q (\cdot , t) $ in the $\| \cdot \| _l $ norms. It follows that $Q $ is smooth on each ${\mathbf s}$-fiber. In this section, we consider the problem of regularity of the heat kernel $Q$.
### **Riemannian metrics and connections on the groupoid ${\mathcal G}$**
Let ${\mathcal G}$ be a groupoid with compact units ${\mathrm M}$, let ${\mathbf s}$ be the source map. As in the beginning of this section, we have already fixed an invariant metric $g _{\mathcal A}$ on the foliation $\ker (d {\mathbf s}) \subset T {\mathcal G}$. We shall extend $g _{\mathcal A}$ to $T {\mathcal G}$. Fix a distribution ${\mathcal H}\subset T {\mathcal G}$ complementary to $\ker (d {\mathbf s})$. Then the differential $d {\mathbf s}$ identifies ${\mathcal H}\cong {\mathbf s}^{-1} T {\mathrm M}$. It follows that any metric on ${\mathrm M}$ defines a metric $g _{\mathcal H}$ n ${\mathcal H}$. We define the metric $g _{\mathcal G}$ on ${\mathcal G}$ by taking the orthogonal sum of ${\mathcal H}$ and $\ker (d {\mathbf s})$.
The distribution ${\mathcal H}$ canonically induces a splitting $$T \tilde {\mathcal G}= \ker (d \tilde {\mathbf s}) \oplus \ker (d \tilde {\mathbf t}) \oplus {\mathcal H}^{(2)},$$ such that ${\mathcal H}= \{ d \tilde {\mathbf s}(X ) : X \in {\mathcal H}^{(2)} \} = \{ d \tilde {\mathbf t}(X ) : X \in {\mathcal H}^{(2)} \}$ (see [@Heitsch;FoliHeat]). Indeed, one can write down ${\mathcal H}^{(2)}$ explicitly: $${\mathcal H}^{(2)} := ({\mathcal H}\times {\mathcal H}) \bigcap T \tilde {\mathcal G}.$$ Also, note that the relation ${\mathbf s}^{(2)} = {\mathbf s}\circ \tilde {\mathbf t}= {\mathbf s}\circ \tilde {\mathbf s}$ implies $$\ker (d {\mathbf s}^{(2)} ) = \ker (d \tilde {\mathbf s}) \oplus \ker (d \tilde {\mathbf t}).$$
Given any metric $g _{\mathcal G}$ as above, the splitting $T \tilde {\mathcal G}= \ker (d \tilde {\mathbf s}) \oplus \ker (d \tilde {\mathbf t}) \oplus {\mathcal H}^{(2)} $ defines a metric on $\tilde {\mathcal G}$, which shall be denoted by $\tilde g _{\mathcal G}$.
Next, we equip $T {\mathcal G}$ with a special connection, following [@Heitsch;FoliHeat]. Recall that one has identification $T {\mathcal G}= \operatorname{Ker}( d {\mathbf s}) \oplus {\mathbf s}^{-1} T ^* {\mathrm M}$. Denote the orthogonal projection onto $\operatorname{Ker}( d {\mathbf s})$ by $P ^{\mathcal V}$. Take the Levi-Civita connection $\nabla ^{T {\mathcal G}}$ on $({\mathcal G}, g _{\mathcal G}) $. Then $\nabla ^{T {\mathcal G}} $ induces a connection on ${\mathcal V}:= \operatorname{Ker}( d {\mathbf s})$ by $$\nabla ^{\mathcal V}_X Y := P ^{\mathcal V}\nabla ^{T {\mathcal G}} _X Y
\quad \forall X \in T {\mathcal G}, Y \in \Gamma ^\infty (\operatorname{Ker}(d {\mathbf s}) ) \subset \Gamma ^\infty (T {\mathcal G}).$$ We define the connection $\nabla ^{{\mathcal V}\oplus {\mathcal H}} $ on $ T {\mathcal G}$ by taking the direct sum of $\nabla ^{\mathcal V}$ and ${\mathbf s}^{-1 } \nabla ^{T {\mathrm M}}$.
### **The regularity theorem**
In this section, we state and prove our transverse regularity theorem. Let ${\mathcal G}\rightrightarrows {\mathrm M}$ be a groupoid with ${\mathrm M}$ compact. We shall assume that the Lie algebroid ${\mathcal A}$ is orientable. Let $\mu $ be the ${\mathbf s}$-fiber-wise invariant Riemannian volume form.
Recall that $\tilde {\mathcal G}:= \{ (a , b ) \in {\mathcal G}\times {\mathcal G}: {\mathbf s}(a ) = {\mathbf s}(b) \} $ and $\widetilde {\mathbf m}(a , b) = a b ^{-1} , \quad \forall (a , b ) \in \tilde {\mathcal G}$. Also, we write $\widetilde {\mathbf m}_* $ to denote the differential of $\widetilde {\mathbf m}$, regarded as a bundle map, i.e. $\widetilde {\mathbf m}\in \Gamma ^\infty (\operatorname{Hom}T (\tilde {\mathcal G}, \widetilde {\mathbf m}^{-1} T {\mathcal G}))$; and ${\mathfrak L}^{(m)} \mu $ to denote the $m$-th Lie derivative of $\mu$. (see Appendix \[DGNonsense\]).
\[EstReg\] Assume that
1. The source map ${\mathbf s}: {\mathcal G}\to {\mathrm M}$ is a fiber bundle;
2. For each $m \in {\mathbb N}$, there exist constants $C _m, \varepsilon _m > 0$ such that $$| (\nabla ^{\operatorname{Hom}T (\tilde {\mathcal G}, \widetilde {\mathbf m}^{-1} T {\mathcal G})}) ^m \widetilde {\mathbf m}_* | (b' , b)
\leq C _m e ^{\varepsilon _m ( d _{\mathbf s}(b' , {\mathbf s}(b')) + d _{\mathbf s}(b , {\mathbf s}(b)))};$$
3. The Lie derivatives of $\mu $ satisfy the estimate $$| {\mathfrak L}^{(m)} \mu (X _1 ^{\tilde {\mathcal H}} , \cdots , X _m ^{\tilde {\mathcal H}} ) (b' , b)|
\leq C _m e ^{\varepsilon _m ( d _{\mathbf s}(b' , {\mathbf s}(b')) + d _{\mathbf s}(b , {\mathbf s}(b)))} |X _1 | \cdots |X _m| .$$
Then for any $F \in \Gamma ^\infty ({\mathbf t}^{-1} {\mathrm E}\otimes {\mathbf s}^{-1} {\mathrm E}'), K \in \Psi ^{- \infty } _\mu ({\mathcal G}, {\mathrm E})$, the heat kernel $e ^{-t (\Delta ^{\mathrm E}+ F + K ) } \in \Gamma ^\infty _b .$
Recall from Lemma \[HeatKer\] that the heat kernel is defined to be the sum $$e ^{- t (\Delta ^{\mathrm E}+ F + K)} = \sum _{k = 0 } ^\infty ( -1 ) ^k Q ^{(k)},$$ where, using Equation (\[ConvDef3\]), the $Q ^{(k)} $ have the form: $$\begin{aligned}
Q ^{(0)} (a , t) =& \: G _N (a , t)\\
Q ^{(k)} (a , t) =& \: \int _0 ^t \int _{(b' , b) \in \tilde {\mathbf t}^{-1} ({\mathbf s}(a))}
\widetilde {\mathbf m}^{-1} G _N (b' , b, t - \tau ) \tilde {\mathbf s}^{-1} R ^{(k-1) } (b', b, \tau) \tilde \mu (\tilde b) d \tau,\end{aligned}$$ where $R _N ^{(k)} $ is defined by taking convolution product of $R _N ^{(1)} := (\partial _t + \Delta ^{\mathrm E}+ F + K ) G _N $ with itself $k$-times.
Fix a connection $\nabla ^{\mathrm E}$ on ${\mathrm E}\to {\mathrm M}$. We denote by $\nabla ^{{\mathbf t}^{-1} {\mathrm E}\otimes {\mathbf s}^{-1} {\mathrm E}'}$ to be the tensor of the pullbacks of $\nabla ^{\mathrm E}$ by ${\mathbf s}$ and ${\mathbf t}$. Hence $\nabla ^{{\mathbf t}^{-1} {\mathrm E}\otimes {\mathbf s}^{-1} {\mathrm E}' }$ is a connection on ${\mathcal G}$. Pulling-back again by $\tilde {\mathbf t}$, one has the bundle $\tilde {\mathbf t}^{-1} ({\mathbf t}^{-1} {\mathrm E}\otimes {\mathbf s}^{-1} {\mathrm E}' ) $ over $ \tilde {\mathcal G}$, and the corresponding connection $\nabla ^{\tilde {\mathbf t}^{-1} ({\mathbf t}^{-1} {\mathrm E}\otimes {\mathbf s}^{-1} {\mathrm E}' )}$.
We begin with estimating the covariant derivatives of $R _N ^{(1)}$. Taking covariant derivative throughout the proof of (2) of Lemma \[ParaProof\], one gets $$\nabla ^{{\mathbf t}^{-1} {\mathrm E}\otimes {\mathbf s}^{-1} {\mathrm E}' } (\kappa \circ G _N )
\to \nabla ^{{\mathbf t}^{-1} {\mathrm E}\otimes {\mathbf s}^{-1} {\mathrm E}' } G _N ,$$ as $t $ goes to $0$. Modifying the arguments of the proof of (1) of Lemma \[ParaProof\] in the same manner, one gets the estimate $$\| (\nabla ^{{\mathbf t}^{-1} {\mathrm E}\otimes {\mathbf s}^{-1} {\mathrm E}' } )^m ((\partial _t + \Delta ^{\mathrm E}+ F) G _N ) \|_0
\leq C _m ^{(1)} t ^{N - \frac{n}{2} - \frac{l}{2} - m}.$$ Combining the two, it follows that $$\| (\nabla ^{{\mathbf t}^{-1} {\mathrm E}\otimes {\mathbf s}^{-1} {\mathrm E}' } ) ^m R ^{(1)} _N (\cdot , t) \| _0
\leq C _m ^{(1)} (t ^{N - \frac{n + l}{2} - m} + 1)$$ for some constants $C _m ^{(1)}$ independent of $t$.
Next, we estimate the derivatives of $R _N ^{(k)}$. Write $$R _N ^{(k)} (a, t) = \int _0 ^t \int _{(b', b) \in \tilde {\mathbf s}^{-1} (a)}
\widetilde {\mathbf m}^{-1} R _N ^{(1)} (b' , b, t - \tau) \tilde {\mathbf s}^{-1} R _N ^{(k-1)} (b' , b, \tau)
\tilde \mu (b', b) d \tau .$$ Then the corollaries of Lemma \[DFiberInt\] imply for any (local) vector field $X$ on ${\mathcal G}$, $$\begin{aligned}
(\nabla &^{{\mathbf t}^{-1} {\mathrm E}\otimes {\mathbf s}^{-1} {\mathrm E}' } _X R _N ^{(k)}) (a, t) \\
=& \int _0 ^t \int _{(b', b) \in \tilde {\mathbf s}^{-1} (a)}
\nabla ^{\tilde {\mathbf t}({\mathbf t}^{-1} {\mathrm E}\otimes {\mathbf s}^{-1} {\mathrm E}')}
\big( \widetilde {\mathbf m}^{-1} R _N ^{(1)} (b' , b, t - \tau) \tilde {\mathbf s}^{-1} R _N ^{(k-1)} (b' , b, \tau) \big)
(X ^{\tilde {\mathcal H}}) \tilde \mu \, d \tau \\
&+ \int _0 ^t \int _{(b', b) \in \tilde {\mathbf s}^{-1} (a)}
\widetilde {\mathbf m}^{-1} R _N ^{(1)} (b' , b, t - \tau) \tilde {\mathbf s}^{-1} R _N ^{(k-1)} (b' , b, \tau)
\big( {\mathfrak L}^{(1)} \tilde \mu (X ^{\tilde {\mathcal H}}) \big) \, d \tau ,\end{aligned}$$ where $X ^{\tilde {\mathcal H}} \in \Gamma (\tilde {\mathcal H}) \subset \Gamma (T \tilde {\mathcal G})$ is the horizontal lift of $X$. Observe that for all $t > 0$, $ R _N ^{(1)} (\cdot , t )$ is supported on a set of the form $\{ a \in {\mathcal G}: d _{\mathbf s}(a , {\mathbf s}(a) ) \leq \rho \} $ for some $\rho > 0 $. It follows that $ R _N ^{(k-1)} $ is supported on the set $\{ a \in {\mathcal G}: d _{\mathbf s}(a , {\mathbf s}(a) ) \leq (k - 1 ) \rho \} $; and $ \widetilde {\mathbf m}^{-1} R _ N^{(1)} (b', b, t - \tau)$ is supported on the compact set $ \{ d _{\mathbf s}( b' , b ) \leq \rho \}. $ Hence, for each $a \in {\mathcal G}$ fixed, the domain of integration can be re-written as $$B (a , \rho ) := \{ b \in {\mathcal G}: {\mathbf s}(b ) = {\mathbf s}(a) , d _{\mathbf s}(a , b ) \leq \rho \},$$ and whose volume is bounded by some constant $M$ independent of $a$.
Expanding the first integrand using Leibniz rule, one gets: $$\begin{aligned}
\nabla ^{\tilde {\mathbf t}({\mathbf t}^{-1} {\mathrm E}\otimes {\mathbf s}^{-1} {\mathrm E}')} &
\big( \widetilde {\mathbf m}^{-1} R _N ^{(1)} (b' , b, t - \tau) \tilde {\mathbf s}^{-1} R _N ^{(k-1)} (b' , b, \tau) \big)
(X ^{\tilde {\mathcal H}}) \\
=& \big( \nabla ^{\widetilde {\mathbf m}^{-1} ({\mathbf t}^{-1} {\mathrm E}\otimes {\mathbf s}^{-1} {\mathrm E}' )}
\widetilde {\mathbf m}^{-1} R _N ^{(1)} (b' , b, t - \tau) (X ^{\tilde {\mathcal H}}) \big)
\tilde {\mathbf s}^{-1} R _N ^{(k-1)} (b' , b, \tau) \\
&+ \widetilde {\mathbf m}^{-1} R _N ^{(1)} (b' , b, t - \tau)
\big( \nabla ^{\tilde {\mathbf s}^{-1} ({\mathbf t}^{-1} {\mathrm E}\otimes {\mathbf s}^{-1} {\mathrm E}' ) }
\tilde {\mathbf s}^{-1} R _N ^{(k-1)} (b' , b, \tau) (X ^{\tilde {\mathcal H}}) \big) \\
=& \big( (\widetilde {\mathbf m}^{-1} \nabla ^{{\mathbf t}^{-1} {\mathrm E}\otimes {\mathbf s}^{-1} {\mathrm E}' }
R _N ^{(1)} (b' , b, t - \tau ) ) (\widetilde {\mathbf m}_* X ^{\tilde {\mathcal H}} ) \big)
\tilde {\mathbf s}^{-1} R _N ^{(k-1)} (b' , b, \tau) \\
&+ \widetilde {\mathbf m}^{-1} R _N ^{(1)} (b' , b, t - \tau )
\big( \tilde {\mathbf s}^{-1} ( \nabla ^{ {\mathbf t}^{-1} {\mathrm E}\otimes {\mathbf s}^{-1} {\mathrm E}' }
R _N ^{(k-1)} (b' , b, \tau ) (X ) ) \big),\end{aligned}$$ where the last line follows from of Equation (\[PullbackDer\]) and the observation that $d \tilde s (X ^{\tilde {\mathcal H}} ) = X $. Using hypothesis (2), one has for any $(b' , b) \in \tilde {\mathcal G}$, $$\begin{aligned}
\big| ( \widetilde {\mathbf m}^{-1} \nabla ^{{\mathbf t}^{-1} {\mathrm E}\otimes {\mathbf s}^{-1} {\mathrm E}' }
R _N ^{(1)} ) & (\widetilde {\mathbf m}_* X ^{\tilde {\mathcal H}} ) \big| (b', b, \tau) \\
\leq & \| \nabla ^{{\mathbf t}^{-1} {\mathrm E}\otimes {\mathbf s}^{-1} {\mathrm E}' } R _N ^{(1)} (\cdot , \tau ) \| _0
| X (b ' b ^{-1} ) | C _0 e ^{\varepsilon _0 ( d _{\mathbf s}(b' , {\mathbf s}(b')) + d _{\mathbf s}(b , {\mathbf s}(b)))} \\
\leq & C _m ^{(1)} (t ^{N - \frac{n + l}{2} - 1} + 1) | X (b ' b ^{-1} )|
C _0 e ^{\varepsilon _0 ( d _{\mathbf s}(b' , {\mathbf s}(b')) + d _{\mathbf s}(b , {\mathbf s}(b)))} .\end{aligned}$$
Now, one can estimate $ \| \nabla ^{{\mathbf t}^{-1} {\mathrm E}\otimes {\mathbf s}^{-1} {\mathrm E}' } R _N ^{(k)} \| _0$. For any $a \in {\mathcal G}$, $$\begin{aligned}
\big| \nabla ^{{\mathbf t}^{-1} {\mathrm E}\otimes {\mathbf s}^{-1} {\mathrm E}' } & R _N ^{(k)} (a , t ) \big| \\
\leq \int _0 ^t & \int _{ B (a , \rho ) }
C _0 e ^{\varepsilon _0 2 k \rho }
\| \nabla ^{{\mathbf t}^{-1} {\mathrm E}\otimes {\mathbf s}^{-1} {\mathrm E}' } R _N ^{(1)} (\cdot , t - \tau ) \| _0
\| R _N ^{(k - 1)} (\cdot , \tau ) \| _0 \mu (b ) \: d \tau \\
+& \int _0 ^t \int _{b \in B (a , \rho )} \| R _N ^{(1)} (\cdot , t - \tau ) \| _0
\| \nabla ^{{\mathbf t}^{-1} {\mathrm E}\otimes {\mathbf s}^{-1} {\mathrm E}' } R _N ^{(k - 1)} (\cdot , \tau ) \| _0 \mu (b ) \: d \tau\\
+& \int _0 ^t \int _{b \in B (a , \rho )} \| R _N ^{(1)} (\cdot , t - \tau ) \| _0
\| R _N ^{(k - 1)} (\cdot , \tau ) \| _0 C _0 e ^{\varepsilon _0 2 k \rho } \mu (b ) \: d \tau,
\intertext{where we used hypothesis (3) to estimate the last term, }
\leq \int _0 ^t & \int _{b \in B (a , \rho )} C _1 ^{(1)} (1 + (t - \tau )^{N - \frac{n + l}{2} - 1})
C _0 e ^{\varepsilon _0 2 k \rho } \\
& \times \tilde C_0 ^{k-2} M ^{k-2} (1 + \tau ^{N - \frac{n+l}{2}} )^{k-1} \tau ^{k-2} ((k-2)!)^{-1}
\mu (b) \: d \tau \\
+& \int _0 ^t \int _{b \in B (a , \rho )} \tilde C _0 (1 + (t - \tau )^{N - \frac{n+l}{2}} )
\| \nabla ^{{\mathbf t}^{-1} {\mathrm E}\otimes {\mathbf s}^{-1} {\mathrm E}' } R _N ^{(k - 1)} (\cdot , \tau ) \| _0 \mu (b ) \: d \tau \\
+& \int _0 ^t \int _{b \in B (a , \rho )} \tilde C _0 (1 + (t - \tau )^{N - \frac{n+l}{2}} ) \\
& \times \tilde C_0 ^{k-2} M ^{k-2} (1 + \tau ^{N - \frac{n+l}{2}} )^{k-1} \tau ^{k-2} ((k-2)!)^{-1}
C _0 e ^{\varepsilon _0 2 k \rho } \mu (b ) \: d \tau \\
\intertext{}
\leq \int _0 ^t & \tilde C _0 (2 + t ^{N - \frac{n+l}{2}} ) M
\| \nabla ^{{\mathbf t}^{-1} {\mathrm E}\otimes {\mathbf s}^{-1} {\mathrm E}' } R _N ^{(k - 1)} (\cdot , \tau ) \| _0 \: d \tau \\
&+ (C _1 ^{(1)} + \tilde C _0 ) \tilde C _0 ^{k - 2} M ^{k - 1} C _0 e ^{\varepsilon _0 2 k \rho }
(2 + t ^{N - \frac{n+l}{2}} ) ^k t ^{k-1} ((k-1)!)^{-1}\end{aligned}$$ Using an induction argument, it is straightforward (but tedious) to obtain the following estimate for any $N > \frac{n}{2} + m + 1$: $$\| \nabla ^{{\mathbf t}^{-1} {\mathrm E}\otimes {\mathbf s}^{-1} {\mathrm E}' } R _N ^{(k)} (\cdot , t) \| _0
\leq k \tilde C _1 ^k M ^k e ^{\varepsilon _0 2 k \rho } (t ^{N - \frac{n}{2} } + 2 ) ^k t ^{k-1} ((k - 1) !)^{-1},$$ for some constant $ \tilde C _1 $.
It is straightforward (if not tedious) to repeat the same arguments above to get estimates for higher derivatives: $$\| (\nabla ^{{\mathbf t}^{-1} {\mathrm E}\otimes {\mathbf s}^{-1} {\mathrm E}' })^m R _N ^{(k)} (\cdot , t) \| _0
\leq k^m \tilde C _m ^k M ^k e ^{\varepsilon' _m 2 k \rho } (t ^{N - \frac{n}{2} } + m ) ^k t ^{k-1} ((k - 1) !)^{-1},$$ for some constants $ \tilde C _m , \varepsilon ' _m$. Finally, arguments similar to the proof of (1) Lemma \[QEst\] gives the estimate $$\| \nabla ^{{\mathbf t}^{-1} {\mathrm E}\otimes {\mathbf s}^{-1} {\mathrm E}' } _X Q _N ^{(k)} (a , t) \| _0
\leq k C ' _1 e ^{2 \varepsilon _0 k R} t ^k ((k - 1 )! ) ^{-1} .$$ It follows that $ \sum _{k = 0 } ^\infty (-1 )^k Q ^{(k)} _N $ converges uniformly in all derivatives up to order $m$, provided $N > \frac{n}{2} + m $. Since $N$ is arbitrary, it follows that $ e ^{- t (\Delta ^{\mathrm E}+ F + K)} \in \Gamma ^\infty _b ({\mathbf t}^{-1} {\mathrm E}\otimes {\mathbf s}^{-1} {\mathrm E}' )$.
### **Example: the Bruhat sphere**
We again look at the example of the Bruhat sphere. We shall explicitly define a metric on the groupoid ${\mathcal G}= {\mathrm T}\backslash ({\mathrm K}\times {\mathrm N})$. Observe that ${\mathcal G}$ is an associated bundle over ${\mathrm K}= {\mathbb C}{\mathrm P}(1)$. It is well known that one has identifications as vector bundles $$\begin{aligned}
T {\mathcal G}\cong & \: (T {\mathrm T}) \backslash (T {\mathrm K}\times T {\mathrm N}) \\
\operatorname{Ker}(d {\mathbf s}) \cong & \: {\mathrm T}\backslash ({\mathrm K}\times T {\mathrm N}).\end{aligned}$$
Observe that ${\mathcal G}= {\mathrm T}\backslash ({\mathrm K}\times {\mathrm N})$ is an associated bundle of the principal bundle ${\mathrm K}\to {\mathrm T}\backslash {\mathrm K}$, hence the arguments in [@KMS;Book Section 11] can be used to fix a complementary distribution to $\operatorname{Ker}(d {\mathbf s})$. Fix an $\operatorname{a d}{\mathrm K}$-invariant metric $g _{\mathfrak k}$ on ${\mathrm K}$, the Lie algebra of ${\mathrm K}$. Let ${\mathfrak t}^\bot $ be the orthogonal complement of ${\mathfrak t}\subset {\mathfrak k}$. Define ${\mathrm T}^\bot$ to be the distribution on ${\mathrm K}$ $${\mathrm T}^\bot := \{ d R ^{\mathrm K}_k (X) : k \in {\mathrm K}, X \in {\mathfrak t}^\bot \} \subseteq T {\mathrm K},$$ and the distribution on ${\mathcal G}$ $${\mathcal H}:= \{ d \wp _{\mathrm T}(X , 0) \in T ({\mathrm T}\backslash ({\mathrm K}\times {\mathrm N})) : (X , 0) \in {\mathrm T}^\bot \times T {\mathrm N}\},$$ where $\wp$ denotes the projection onto he coset space. It is easy to see that ${\mathrm H}$ is a distribution complementary to $\ker (d {\mathbf s}) = {\mathrm T}\backslash ({\mathrm K}\times T {\mathrm N})$. To define a metric on ${\mathrm H}$, one simply takes the pullback of the round metric $g _{\mathfrak k}$ on ${\mathrm T}\backslash {\mathrm K}$, more explicitly, $$g _{\mathcal H}({\wp}_{\mathrm T}(d R_k ^{\mathrm K}(X _1 ) , 0) , {\wp} _{\mathrm T}(d R ^{\mathrm K}_k ( X _2 ) , 0)) := g _{{\mathfrak k}} (X _1 , X _2).$$ Finally, we define a metric $g _{\mathcal G}$ on ${\mathcal G}= {\mathrm T}\backslash ({\mathrm K}\times {\mathrm N})$ by taking the orthogonal sum of $g_{\mathbf s}$ and $g _{\mathcal H}$.
In our special case ${\mathcal G}= {\mathrm T}\backslash ({\mathrm K}\times {\mathrm N})$, $\tilde {\mathcal G}^{(2)} $ is diffeomorphic to ${\mathrm T}\backslash ({\mathrm K}\times {\mathrm N}\times {\mathrm N})$, where ${\mathrm T}$ acts on ${\mathrm K}$ by right multiplication and on ${\mathrm N}\times {\mathrm N}$ by conjugation, and the diffeomorphism is given explicitly by $${}_{\mathrm T}(k , n_1 , n_2)
\mapsto ({}_{\mathrm T}(k , n_1) , {} _{\mathrm T}(k , n_2)) \in {\mathcal G}\times {\mathcal G}\supseteq \tilde {\mathcal G}^{(2)}.$$
Consider the map $\widetilde {\mathbf m}: \tilde {\mathcal G}^{(2)} \to {\mathcal G}, \widetilde {\mathbf m}(a, b) := a b ^{-1}$. One has the commutative digram $$\begin{CD}
{\mathrm K}\times {\mathrm N}\times {\mathrm N}@> \widehat {\mathbf m}>> {\mathrm K}\times {\mathrm N}\\
@VVV @VVV \\
\tilde {\mathcal G}^{(2)} \cong {\mathrm T}\backslash ({\mathrm K}\times {\mathrm N}\times {\mathrm N})
@>\widetilde {\mathbf m}>> {\mathcal G}= {\mathrm T}\backslash ({\mathrm K}\times {\mathrm N}),
\end{CD}$$ where $\widehat {\mathbf m}: {\mathrm K}\times {\mathrm N}\times {\mathrm N}\to {\mathrm K}\times {\mathrm N}$ is defined to be the function $$\widehat {\mathbf m}(k, n_1 , n_2) := (k', n_1 n_2 ^{-1} ).$$
We verify that the metric we constructed satisfies the assumptions of Theorem \[EstReg\]. Hence the heat kernel of a Laplacian operator on the Bruhat sphere is smooth.
\[CP1Est\] For each $m \in {\mathbb N}$, there exists a polynomial $p _m $ on ${\mathrm N}\times {\mathrm N}= {\mathbb R}^2 \times {\mathbb R}^2$ such that, for any ${} _{\mathrm T}(k, n_1 , n_2 ) \in \tilde {\mathcal G}^{(2)}, X \in T _{ {}_{\mathrm T}(k , n_1 , n_2) } \tilde {\mathcal G}^{(2)} $, $$\big| (\nabla ^{\operatorname{Hom}T (\tilde {\mathcal G}, \widetilde {\mathbf m}^{-1} T {\mathcal G})})^m \widetilde {\mathbf m}( X ) \big|_{g _{\mathcal G}}
\leq | p _m (n_1 , n_2) | | X | _{g ^{(2)} _{\mathcal G}}.$$
We prove this lemma by direct computation. First, one obtains a formula for $\widetilde {\mathbf m}$ using Equation (\[CP2Formula\]). Namely, for any $k =
\left(
\begin{smallmatrix}
\alpha & \beta \\
- \bar \beta & \bar \alpha
\end{smallmatrix}
\right) ,
n_1 =
\left(
\begin{smallmatrix}
1 & w' \\
0 & 1
\end{smallmatrix}
\right) ,
n_2 =
\left(
\begin{smallmatrix}
1 & w \\
0 & 1
\end{smallmatrix}
\right) ,$ one has $$\widetilde {\mathbf m}({}_{\mathrm T}(k , n_2 , n_2)) =
\left[ \frac{\alpha - \bar w \beta }{(|\beta |^2 + |\alpha - \bar w \beta |^2)^{\frac{1}{2}}},
\frac{ \beta }{(|\beta |^2 + |\alpha - \bar w \beta |^2)^{\frac{1}{2}}} \right] ^{w' - w} _{\mathrm T}.$$ First consider the $ {\mathcal H}$-component. Any vector $X \in {\mathrm T}^\bot $ can be written in the form $$X = \partial _t \Big| _{t = 0}
\left(
k e^{ t \left(
\begin{smallmatrix}
0 & v \\
- \bar v & 0
\end{smallmatrix}
\right) } , n_1 , n_2 \right) ,
\quad v \in {\mathbb C}.$$ Then, $d \widetilde {\mathbf m}(X) $ is by definition: $$\begin{aligned}
& d \widehat {\mathbf m}(X) :=
\partial _t \Big| _{t = 0}
\widetilde {\mathbf m}\left(
e^{ t \left(
\begin{smallmatrix}
0 & v \\
- \bar v & 0
\end{smallmatrix}
\right) } k , n_1 , n_2 \right) = \\
&
\left(
\left(
\begin{array}{cc}
\frac{(|\beta |^2 - |\alpha |^2 )(\bar w _2 v - w _2 \bar v)
+ |w _2|^2 (\bar \alpha \bar \beta v - \alpha \beta \bar v )}{2 Q }
& \frac{v}{Q } \\
- \frac{\bar v}{Q } & \frac{- (|\beta |^2 - |\alpha |^2 )(\bar w _2 v - w _2 \bar v)
- |w _2|^2 (\bar \alpha \bar \beta v - \alpha \beta \bar v )}{2 Q }
\end{array}
\right) k',
0 \right) ,\end{aligned}$$ where we denoted $Q := |\beta |^2 + |\alpha - \bar w _2 \beta | ^2 $. It follows that $$d \widetilde {\mathbf m}(d \wp _{\mathrm T}(X)) = \frac{1}{Q} d \wp _{\mathrm T}(X) .$$ Similarly, one has $$\begin{aligned}
\partial _t \big| _{t=0} \widehat {\mathbf m}& (k , w _1 + z _1 t , w _2 )
= (0 , z _1 , 0 ) \\
\partial _t \big| _{t=0} \widehat {\mathbf m}& (k , w _1 , w _2 + z _2 t ) \\
&=
\left(
\left(
\begin{array}{cc}
\frac{(\alpha - \bar w _2 \beta )\bar \beta z _2 - (\bar \alpha - w _2 \bar \beta ) \beta \bar z _2 }{2 Q } &
\frac{- |\beta | ^2 \bar z _2 }{Q} \\
\frac{ | \beta | ^2 z _2 }{Q} &
\frac{- (\alpha - \bar w _2 \beta )\bar \beta z _2 + (\bar \alpha - w _2 \bar \beta ) \beta \bar z _2 }{2 Q }
\end{array}
\right) k',
- z _2 \right) .\end{aligned}$$ We estimate a lower bound for $Q $ in terms of $w$. Since it is clear that $Q \neq 0$, we may without loss of generality assume that $|w _2 | > 1$. Suppose $| \beta | \leq \frac{1}{2 | w |} \leq \frac{1}{2}$. Then the relation $| \alpha | ^2 + |\beta |^2 = 1 $ implies $| \alpha | \geq \frac{3}{4}$. It follows that $| \alpha - \bar w \beta | ^2 \geq (\frac{3}{4} - \frac{1}{2} ) ^2 = \frac{1}{4} \geq \frac{1}{4 | w_2 |^2} $. Hence $|\beta |^2 + | \alpha - \bar w \beta | ^2 \geq \frac{1}{4 |w |^2 } $ in both cases.
Finally, from the above computations, we observe that the coefficients of the $m$-th covariant derivatives of $\widetilde {\mathbf m}$ are of the form $$Q ^{- m} p _I (\alpha , \bar \alpha , \beta , \bar \beta , \omega , \bar \omega ),$$ where $p _I$ are polynomials. The assertion follows.
It is easy to see that the ${\mathbf s}$-fiber-wise Riemannian volume form $\mu$ also satisfies similar estimates. Therefore we conclude that
For any vector bundle ${\mathrm E}\to {\mathbb C}{\mathrm P}(1) $, any Riemannian metric $g _{\mathcal A}$ on ${\mathcal A}$, $F \in \Gamma ({\mathrm E}\otimes {\mathrm E}') $ and $K \in \Psi ^{- \infty } _\mu ({\mathcal G}, {\mathrm E})$, the heat kernel $$e ^{- t (\Delta ^{\mathrm E}+ F + K ) }
\in \Gamma ^\infty _ b ({\mathbf t}^{-1} {\mathrm E}\otimes {\mathbf s}^{-1} {\mathrm E}' \ltimes (0, \infty) ) .$$
Short time asymptotic expansion of the heat kernel
---------------------------------------------------
Let ${\mathcal G}\rightrightarrows {\mathrm M}$ be a groupoid with ${\mathrm M}$ compact, and the Lie algebroid ${\mathcal A}\to {\mathrm M}$ of even rank $\varkappa$. Let $\nabla ^{\mathrm E}$ be a Clifford ${\mathcal A}$-connection and $\eth$ be the corresponding Dirac operator. Then a straightforward calculation shows that $$\eth ^2 = \Delta ^{\mathrm E}+ (\frac {1}{4} \tilde R + F ^{{\mathrm E}/ {\mathrm S}}),$$ where $\tilde R$ is the scalar curvature and $F ^{{\mathrm E}/ {\mathrm S}} $ is the twisting curvature. Therefore the construction of the heat kernel above applies.
Before stating our main result Lemma \[AsymLem\], we first need to define some notation. Let ${\mathbb C}^{k \times k} $ be the set of all matrices with coefficients in ${\mathbb C}$. Given any power series $h : {\mathbb C}^{k \times k} \to {\mathbb C}$ $$h ( Z _{i j} ) = h (0) + \sum _{I} h _I Z _{I} ,$$ where the sum is over all multi-indexes $I = \{i_1 j_1 , i_2 j_2 , \cdots , i_p j_p \} $, and $ Z _I := Z _{i_1 j_1} Z _{i_2 j_2} \\ \cdots Z _{i_p j_p}$. Let $( \omega _{i j } ) \in \wedge {\mathrm V}'$ be a matrix of 2-forms on some vector space ${\mathrm V}$. We [*define* ]{} $h (\omega _{i j} )$ to be the polynomial $$( \omega _{i j} ) \mapsto h (0 ) + \sum _I h _I \omega _I \in \bigoplus _{l \text { even} } \wedge ^l {\mathrm V}',$$ where $\omega _I := \omega _{i_1 j_1} \wedge \omega _{i_2 j_2} \wedge \cdots \wedge \omega _{i_p j_p} $.
In particular, take $h$ to be the Taylor series expansion of $$Z \mapsto
\sqrt {\det \Big( \frac{Z}{2} ( \sinh \frac{Z}{2} )^{-1} \Big)} : {\mathbb C}^{\varkappa \times \varkappa } \to {\mathbb C},$$ where $ Z \mapsto \frac{Z}{2} ( \sinh \frac{Z}{2} )^{-1}
: {\mathbb C}^{\varkappa \times \varkappa } \to {\mathbb C}^{\varkappa \times \varkappa } $ is defined by the power series of $ z \mapsto \frac{z}{2} (\sinh \frac{z}{2}) ^{-1} : {\mathbb C}\to {\mathbb C}$. Define the $\widehat {\mathbb A}$-genus by $$\widehat {\mathbb A}:= h (R).$$ It is straightforward to check that $\widehat {\mathbb A}\in \Gamma ^\infty (\wedge {\mathcal A}')$ is a well defined section.
\[AsymLem\] The heat kernel $e ^{- t \eth ^2} \in \Gamma ^\infty ({\mathbf s}^{-1} {\mathrm E}\otimes {\mathbf t}^{-1} {\mathrm E}' \ltimes (0 , \infty) )$ has an asymptotic expansion $$e ^{- t \eth ^2 } (x, t) \cong (4 \pi )^{- \frac{n}{2}} \sum _{i = 0} ^\infty t ^{i - \frac{n}{2}} Q _i (x) ,
\quad \forall x \in {\mathrm M}\subset {\mathcal G},$$ for some $Q _i \in \Gamma ^\infty ({\mathrm {Cl} }({\mathcal A}') \otimes \operatorname{End}_{{\mathrm {Cl} }({\mathcal A}')} ({\mathrm E}) ) $. Furthermore
1. The coefficient $Q _i \in {\mathrm {Cl} }_{2 i} ({\mathcal A}') \otimes \operatorname{End}_{{\mathrm {Cl} }({\mathcal A}')} ({\mathrm E}) $;
2. One has $$\big( \operatorname{s t r}Q _{\frac{\varkappa }{2}} \big) \mu _{\mathcal A}= \text{order $ \varkappa $ component of } \widehat {\mathbb A}\wedge \exp F ^{{\mathrm E}/ {\mathrm S}}.$$
Regarding all operators involved as families of operators along the ${\mathbf s}$-fibers, the heat kernel of $\eth |_{{\mathbf s}^{-1} (x)}$ is just $$Q _x (a, b) := Q (a b ^{-1}), {\mathbf s}(a ) = {\mathbf s}(b) = x.$$ The computations of the asymptotic expansion of $Q _x (a, a) = Q (x)$ is very standard. See, for example, [@BGV;Book Chapter 4].
For convenience, we denote the order $ \varkappa $ component of $\widehat {\mathbb A}\wedge \exp F ^{{\mathrm E}/ {\mathrm S}} $ by $\Omega _\varkappa (\widehat {\mathbb A}\wedge \exp F ^{{\mathrm E}/ {\mathrm S}} ) $.
It is easy to compute the asymptotic expansion of the heat kernel of the operator $\eth ^2 + K$. From Equation (\[HeatPert\]), the heat kernel of $\eth ^2 + K$ can be written as $$e ^{- t (\eth ^2 + K)} (a , t) = e ^{- t \eth ^2} (a , t)
+ \sum _{i= 1} ^\infty (-1) ^i t ^i \tilde Q ^{(i)} (a , t) ,$$ where $\tilde Q $ is the heat kernel of $\Delta ^{\mathrm E}$, and $\tilde Q ^{(i)} := \int _{0 \leq \tau _0 \leq \cdots \leq \tau _i \leq 1}
\tilde Q (\cdot , \tau _0 t) \circ \kappa \circ \tilde Q (\cdot , \tau _1 t) \circ \kappa \circ \cdots
\circ \kappa \circ \tilde Q (\cdot , \tau _i t).$ Since $\tilde Q ^{(i)} (\cdot , 0)$ are smooth, it follows immediately that
The heat kernel $e ^{- t (\eth ^2 + K) } \in \Gamma ^\infty ({\mathbf s}^{-1} {\mathrm E}\otimes {\mathbf t}^{-1} {\mathrm E}' \ltimes (0, \infty ) )$ of the Laplacian $ \eth ^2 + K $ has an asymptotic expansion $$e ^{- t (\eth ^2 + K) } \cong (4 \pi )^{- \frac{n}{2}} \sum _{i = 0} ^\infty t ^{i - \frac{n}{2}} Q _i (x) ,
\quad \forall x \in {\mathrm M}\subset {\mathcal G},$$ for some $Q _i \in \Gamma ^\infty ({\mathrm {Cl} }({\mathcal A}') \otimes \operatorname{End}_{{\mathrm {Cl} }({\mathcal A}')} ({\mathrm E}) ) $. Furthermore
1. The coefficient $Q _i \in {\mathrm {Cl} }_{2 i} ({\mathcal A}') \otimes \operatorname{End}_{{\mathrm {Cl} }({\mathcal A}')} ({\mathrm E}) $;
2. One has $$\big( \operatorname{s t r}Q _{\frac{\varkappa }{2}} \big) \mu _{\mathcal A}= \Omega _\varkappa (\widehat {\mathbb A}\wedge \exp F ^{{\mathrm E}/ {\mathrm S}} ).$$
The renormalized trace and index theorem
========================================
Consider a Fredholm operator on $\Gamma ^\infty ({\mathrm E}) $ of the form $\nu (\eth + \varPsi )$, where $\eth$ is a Dirac operator and $R \in \Psi ^{- \infty} _\mu ({\mathcal G}, {\mathrm E})$.
We saw that the heat kernel is not a smoothing operator in general, and the usual trace formula $$\int _{{\mathrm M}} \kappa (x , x) \mu (x)$$ cannot be applied. Instead, one need to consider an extension of the trace functional, known as the renormalized trace.
The renormalized integral
--------------------------
We shall only consider the case of the Bruhat sphere. In this case, one has the two stereographic projection coordinates $$r e ^{i \vartheta } \mapsto [r e ^{i \vartheta }, 1] \text{ and }
\dot r e ^{- i \vartheta } \mapsto [1 , \dot r e ^{- i \vartheta }] ,$$ and one can consider the cutoff integrals $$\int _{ r \leq r_0} \int _{0 \leq \vartheta \leq 2 \pi } f ([r e ^{i \vartheta }, 1]) r d \vartheta d r
= \int _{\dot r \geq \frac{1}{r_0}} \int _{0 \leq \vartheta \leq 2 \pi }
f ([1 , \dot r e ^{- i \vartheta }]) \frac{1}{\dot r ^3} d \vartheta d \dot r$$ for any $f \in C^\infty ({\mathbb C}{\mathrm P}(1))$ as $r _0 \to \infty$. Using standard arguments reviewed in [@Paycha;Renorm], one has:
\[0Renorm\] For any $k = 1, 2, \cdots,$ and $F \in C ^\infty _c ({\mathbb R})$, one has the expansion as $r _0 \to \infty$: $$\int _{\frac{1}{r _0}} ^\infty F \lambda ^{- k} d \lambda
= \sum _{j = 1}^{k-1} C_j r _0 ^j + R \log r_0 + C _0 + O (r _0 ^{-1})$$ for some constants $ C_0 , \cdots , C_j , R $. In particular, the constant term $C _0 $ is given by the formula $$C_0 = \partial _\lambda ^{k - 1} F (0) \sum _{j = 1} ^{k - 1}\frac{1}{j} +
\frac{1}{(k - 1)!} \int _0 ^\infty \partial _\lambda ^k F (\lambda ) \log \lambda d \lambda
:= \sideset{_{\mathrm R}}{_0 ^\infty} \int F (\lambda ) \lambda ^{- k} d \lambda .$$
We return to the case of the Bruhat sphere. Given any section $\omega \in \Gamma (\wedge ^2 {\mathcal A}')$, such that $\omega $ is two times differentiable on ${\mathbb C}{\mathrm P}(1) $ and three times differentiable on ${\mathbb C}{\mathrm P}(1) \setminus \{ {}_{\mathrm T}e \}$, we define:
\[CP1Renorm\] The [*renormalized integral*]{} of $\omega $ is the number $${\sideset{_{\mathrm R}}{}\int}\omega :=
\sideset{_{\mathrm R}}{_0 ^\infty} \int
\Big( \int _0 ^{2 \pi} ( f \circ \dot {\mathbf x}) (\dot r e ^{- i \vartheta }) d \vartheta \Big) \dot r ^{- 3} d \dot r ,$$ where $ \omega = f \mu _0 $, and $\mu _0 $ is the volume form defined by the round metric. Here, note that $(\dot r , \vartheta ) \mapsto f \circ \dot {\mathbf x}(\dot r e ^{- i \vartheta })$ is two times differentiable on the whole ${\mathbb R}^2$-space.
We may choose other volume forms instead of the round one, and the result depends on our trivialization. This discrepancy is well known. See [@Paycha;Renorm] for a review.
The renormalized trace and trace defect formula
------------------------------------------------
With the renormalized integral defined, it is natural to define the renormalized trace.
Let ${\mathrm E}$ be a ${\mathbb Z}_2$-graded vector bundle over ${\mathrm M}$. Let $\mu $ be a fixed ${\mathbf s}$-fiberwise volume on ${\mathcal G}$ identifying $\Psi ^{- \infty } ({\mathcal G}, {\mathrm E}) \cong \Gamma ^\infty ({\mathbf t}^{-1} {\mathrm E}\otimes {\mathbf s}^{-1} {\mathrm E}' )$. For any $K \in \Psi ^{- \infty } ({\mathcal G}, {\mathrm E}) $ with reduced kernel $\kappa \in \Gamma ^\infty ({\mathbf t}^{-1} {\mathrm E}\otimes {\mathbf s}^{-1} {\mathrm E}' )$, the [*renormalized (super)-trace*]{} of $K$ is defined to be $$\sideset{_{\mathrm R}}{} \operatorname{S t r}( \varPsi ) := {\sideset{_{\mathrm R}}{}\int}\operatorname{s t r}(\kappa |_{{\mathrm M}} ) \mu .$$
In this section, we compute explicitly $${\sideset{_{\mathrm R}}{}\int}\big( f \circ g (x) - g \circ f (x) \big) \mu _{{\mathrm M}_0} (x),$$ where for simplicity we assume $f, g \in C^\infty ({\mathcal G})$ and $g$ is compactly supported (hence the convolution products are well defined). In general, the expression is non zero. Hence proving that the ‘renormalized trace’ is not a trace.
One has the trace defect formula $$\begin{aligned}
{\sideset{_{\mathrm R}}{}\int}\big( & f \circ g (x) - g \circ f (x) \big) \mu _{{\mathrm M}_0} (x) \\ \nonumber
=& - \pi \int _{\mathbb C}\big( \operatorname{r e}( w' ) \partial _{\dot x} + \operatorname{i m}( w') \partial _{\dot y} \big)
\big( f ([1 , 0] ^{ \bar w' } _{\mathrm T}) g([1 , 0] ^{ - \bar w' } _{\mathrm T}) \big) | d w' |^2 . \end{aligned}$$
By Definition \[ConvDfn\], the convolution product $f \circ g$, written in Notation \[BruhatPair\], is given by the formula $$f \circ g (z)
= \int _{w \in {\mathbb R}^2 }
f ({\mathbf x}(z + \bar w , - w) ) g ({\mathbf x}(z, w)) | d w |^2 .$$ As in Lemma \[0Renorm\], we need to consider $$\int _{z \in B (0, r_0) } \int
f ({\mathbf x}(z + \bar w , - w) ) g ({\mathbf x}(z, w)) | d w |^2 | d z |^2$$ as $r _0 \to \infty $. Performing the $z$-integral first and changing variable $z' = z - \bar w , w' = - w$, the integral becomes $$\int \int _{z' \in B (- \bar w', r_0) }
f ({\mathbf x}(z ', w') ) g ({\mathbf x}(z' + \bar w', - w')) | d z' |^2 | d w' |^2.$$ On the other hand, one has $$\int _{z \in B (0, r_0)} g \circ f |d z |^2
= \int \int _{z \in B (0, r_0)} f ({\mathbf x}(z , w)) g (z + \bar w , - w) |d z|^2 |d w|^2 .$$ Combining the two integrals, one gets $$\begin{aligned}
\int _{z \in B (0, r_0)} f \circ g - g \circ f & \: \mu _{{\mathrm M}_0} (z) \\
= \: & \int \int _{z \in B (- \bar w , r_0) \backslash B (0, r_0)}
f ({\mathbf x}(z , w) ) g ({\mathbf x}(z + \bar w , - w) ) |d z |^2 |d w |^2 \\
&- \int \int _{z \in B (0 , r_0) \backslash B (- \bar w, r_0)}
f ({\mathbf x}(z , w) ) g ({\mathbf x}(z + \bar w , - w) ) |d z |^2 |d w |^2 \end{aligned}$$ after canceling the common domain. In order to compute the integral, one needs to parametrize the domains $B (- \bar w , r_0) \backslash B (0, r_0)$ and $ B (0 , r_0) \backslash B (- \bar w, r_0)$. For each $w \in {\mathbb C}$, consider the sets $$\begin{aligned}
S ^+ _w :=& \: \Big\{ - \frac{ r_0 e ^{i \varphi } \bar w }{| w |} - \lambda \bar w
: - \frac{\pi}{2} \leq \varphi \leq \frac{\pi}{2} , 0 \leq \lambda \leq 1 \Big\} \\
S ^- _w :=& \: \Big\{ - \frac{ r_0 e ^{i \varphi } \bar w }{| w |} - \lambda \bar w
: \frac{\pi}{2} \leq \varphi \leq \frac{3 \pi}{2} , 0 \leq \lambda \leq 1 \Big\}.\end{aligned}$$ It is elementary to see that $$S ^+ _w \backslash S ^- _w = B (- \bar w , r_0) \backslash B (0, r_0)
\text { and } S ^- _w \backslash S ^+ _w = B (0 , r_0) \backslash B (- \bar w , r_0)$$ modulo sets of measure 0. With these natural parameterizations, one has $$\begin{aligned}
\int _{z \in B (0 , r_0 )} (f \circ g - g \circ f) & \: | d z |^2 \\
= \int \int _0 ^{2 \pi} \int _0 ^1 & f ( {\mathbf x}(- \frac{ r_0 e ^{i \varphi } \bar w }{| w |} - \lambda \bar w , w ))
\\ \times &
g ({\mathbf x}(- \frac{ r_0 e ^{i \varphi } \bar w }{| w |} - (1 - \lambda ) \bar w , - w ))
r _0 |\bar w |\cos \varphi d \lambda d \varphi | d w |^2.\end{aligned}$$ Next, we approximate $f $ by its Taylor series at ${\mathbf s}^{-1} ([1 , 0]) $ as $r _0 \to \infty$. More precisely, define the trivialization $$\dot {\mathbf x}(\dot z , \dot w)
:= \left[ \frac{1}{(1 + |\dot z|^2 )^\frac{1}{2}} , \frac{\dot z}{(1 + |\dot z|^2 )^\frac{1}{2}} \right] ^{\dot w}_{\mathrm T}.$$ Write $\dot z = \dot x + i \dot y , \dot w = \dot u + i \dot v, \; \; \dot x , \dot y , \dot u , \dot v , \in {\mathbb R}$. Using the change in coordinate formula ${\mathbf x}(z , w) = \dot {\mathbf x}(\frac{1}{z} ,\frac{z ^2}{|z |^2} w)$ and the expansions $$\begin{aligned}
\frac{1}{- \frac{ r_0 e ^{i \varphi } \bar w }{| w |}- \lambda \bar w }
=& - \frac{|w|}{r_0 e ^{i \varphi } \bar w }
\left(1 - \frac{\lambda | \bar w |}{r_0 e ^{i \varphi }}
+ O (r _0 ^{-2}) \right) \\
\frac{ r_0 e ^{i \varphi } \bar w + \lambda \bar w |w|}{ r_0 e ^{- i \varphi } + \lambda |w| }
=& e ^{2 i \varphi } \bar w + 2 i e ^{2 i \varphi } \sin \varphi \frac{\lambda \bar w |w|}{r _0 } + O (r ^{-2}),
$$ one gets $$\begin{aligned}
f \Big( {\mathbf x}\Big( - & \frac{ r_0 e ^{i \varphi } \bar w }{| w |} - \lambda \bar w , w \Big) \Big) \\
=& f ([1 , 0] ^{ e ^{2 i \varphi } \bar w } _{\mathrm T})
- \Big( \frac{ \operatorname{r e}(e ^{- i \varphi } w )}{r_0 |w|} \partial _{\dot x}
+ \frac{ \operatorname{i m}( e ^{- i \varphi } w )}{r_0 |w|} \partial _{\dot y} \Big) f ([1 , 0] ^{ e ^{2 i \varphi } \bar w } _{\mathrm T})\\
&+ \frac{2 \lambda |\bar w| \sin \varphi }{r_0}
\big( \operatorname{i m}(e ^{2 i \varphi } \bar w) \partial _{\dot u}
- \operatorname{r e}(w ^{2 i \varphi } \bar w) \partial _{\dot v} \big) f([1 , 0] ^{ e ^{2 i \varphi } \bar w } _{\mathrm T})
+ O (r _0 ^{-2}).\end{aligned}$$ Combining with a similar expression for $g$, the integrand has an expansion: $$\begin{aligned}
\label{ExpandInt}
\nonumber
f \Big( {\mathbf x}\Big( - & \frac{ r_0 e ^{i \varphi } \bar w }{| w |} - \lambda \bar w , w \Big) \Big)
g \Big( {\mathbf x}\Big( - \frac{ r_0 e ^{i \varphi } \bar w }{| w |} - (1 - \lambda ) \bar w , - w \Big) \Big) \\ \nonumber
= \: & f ([1 , 0] ^{ e ^{2 i \varphi } w } _{\mathrm T}) g ([1 , 0] ^{ - e ^{2 i \varphi } w } _{\mathrm T}) \\
&- \Big(\frac{ \operatorname{r e}(e ^{i \varphi } w )}{r_0 |w|} \partial _{\dot x}
+ \frac{ \operatorname{i m}(e ^{i \varphi } w )}{r_0 |w|} \partial _{\dot y} \Big)
\Big( f ([1 , 0] ^{ e ^{2 i \varphi } \bar w } _{\mathrm T}) g([1 , 0] ^{ - e ^{2 i \varphi } \bar w } _{\mathrm T}) \Big) \\ \nonumber
&+ f ([1 , 0] ^{ e ^{2 i \varphi } \bar w } _{\mathrm T})\Big( \frac{2 (1 - \lambda ) |\bar w| \sin \varphi }{r_0} \Big)
\big( \operatorname{i m}(e ^{2 i \varphi } \bar w) \partial _{\dot u}
- \operatorname{r e}(w ^{2 i \varphi } \bar w) \partial _{\dot v} \big) g([1 , 0] ^{ e ^{2 i \varphi } \bar w } _{\mathrm T}) \\ \nonumber
&+ g ([1 , 0] ^{ - e ^{2 i \varphi } \bar w } _{\mathrm T}) \Big( \frac{2 \lambda |\bar w| \sin \varphi }{r_0} \Big)
\big( \operatorname{i m}(e ^{2 i \varphi } \bar w) \partial _{\dot u}
- \operatorname{r e}(w ^{2 i \varphi } \bar w) \partial _{\dot v} \big) f([1 , 0] ^{ e ^{2 i \varphi } \bar w } _{\mathrm T}) \\ \nonumber
&+ O (r _0 ^{-2}).\end{aligned}$$ We compute the (renormalized) integral of each terms in Equation (\[ExpandInt\]). First consider the last term: $$\begin{aligned}
\int _0 ^1 \int _0 ^{2 \pi} \int _{\mathbb C}&
g ([1 , 0] ^{ - e ^{2 i \varphi } \bar w } _{\mathrm T}) \Big( \frac{2 \lambda |\bar w| \sin \varphi }{r_0} \Big)
\big( \operatorname{i m}(e ^{2 i \varphi } w) \partial _{\dot u}
- \operatorname{r e}(e ^{2 i \varphi } w) \partial _{\dot v} \big) f([1 , 0] ^{ e ^{2 i \varphi } \bar w } _{\mathrm T}) \\
\times & r_0 | w | \cos \varphi | d w |^2 d \varphi d \lambda \\
=& \int _0 ^1 \int _0 ^{2 \pi} \int _{\mathbb C}g ([1 , 0] ^{ - \bar w' } _{\mathrm T}) \big( \operatorname{i m}(e ^{2 i \varphi } \bar w) \partial _{\dot u}
- \operatorname{r e}(e ^{2 i \varphi } \bar w) \partial _{\dot v} \big) f([1 , 0] ^{ e ^{2 i \varphi } \bar w } _{\mathrm T}) \\
& \times 2 | w' |^2 \lambda \sin \varphi \cos \varphi | d w' |^2 d \varphi d \lambda ,\end{aligned}$$ by changing variable $w' := e ^{ 2 i \varphi } w $. The integral vanishes since $\int _0 ^{2 \pi} \cos \varphi \sin \varphi d \varphi = 0$. Using the same arguments the integral of the first and the third term are both $0$. It remains to consider the second term. Again we change variable $w' := e ^{ 2 i \varphi } w $ to get: $$\begin{aligned}
- \int _0 ^1 \int _0 ^{2 \pi} \int _{\mathbb C}&
\Big( \frac{ \operatorname{r e}( e ^{- i \varphi } w ')}{|w'|} \partial _{\dot x}
+ \frac{ \operatorname{i m}(e ^{- i \varphi } w' )}{|w'|} \partial _{\dot y} \Big)
\Big( f ([1 , 0] ^{ \bar w' } _{\mathrm T}) g([1 , 0] ^{ - \bar w' } _{\mathrm T}) \Big) \\
& \times | w' | \cos \varphi | d w' |^2 d \varphi d \lambda.\end{aligned}$$ Applying the identities $\int _0 ^{2 \pi} \cos \varphi \sin \varphi d \varphi = 0, \int _0 ^{2 \pi} \cos ^2 \varphi d \varphi = \pi$, one finally obtains: $$\begin{aligned}
{\sideset{_{\mathrm R}}{}\int}\big( & f \circ g (x) - g \circ f (x) \big) \mu _{{\mathrm M}_0} (x) \\
=& - \pi \int _{\mathbb C}\big( \operatorname{r e}( w' ) \partial _{\dot x} + \operatorname{i m}( w') \partial _{\dot y} \big)
\big( f ([1 , 0] ^{ \bar w' } _{\mathrm T}) g([1 , 0] ^{ - \bar w' } _{\mathrm T}) \big) | d w' |^2 . \end{aligned}$$
The McKean-Singer formula and index formula
--------------------------------------------
We recall the derivation of index formulas using the McKean-Singer formula.
Fix a Riemannian metric $g _{\mathcal A}$ on ${\mathcal A}$. Denote the invariant ${\mathbf s}$-fiberwise Riemannian volume form by $\mu $. Let ${\mathrm E}$ be a ${\mathrm {Cl} }({\mathcal A}')$ module, and $(\eth + \varPsi )$ be a perturbed Dirac operator. Consider $$\lim _{t \to \infty } \sideset{_\mathrm R}{} \operatorname{S t r}(e ^{- t(\eth + \varPsi )^2})
- \lim _{t \to 0^+} \sideset{_\mathrm R}{} \operatorname{S t r}(e ^{- t (\eth + \varPsi)^2})
= \int _0 ^\infty \partial _t \sideset{_\mathrm R}{} \operatorname{S t r}(e ^{- t (\eth + \varPsi)^2} ) d t.$$ For the right hand side, one has $$\partial _t \sideset{_\mathrm R}{} \operatorname{S t r}(e ^{- t(\eth + \varPsi)^2} )
= \sideset{_\mathrm R}{} \operatorname{S t r}(\partial _t e ^{- t(\eth + \varPsi)^2} )
= \sideset{_\mathrm R}{} \operatorname{S t r}([\eth + \varPsi , (\eth + \varPsi ) e ^{- t(\eth + \varPsi)^2}] ).$$ One can then use the trace defect formula in the last section to compute $\sideset{_\mathrm R}{} \operatorname{S t r}([\eth + \varPsi , (\eth + \varPsi ) e ^{- t(\eth + \varPsi)^2}] )$. The actual calculation is very complicated. Nevertheless we denote the result by $$\boldsymbol \eta (\eth + \varPsi)
:= \int _0 ^\infty \sideset{_\mathrm R}{} \operatorname{S t r}([\eth + \varPsi , (\eth + \varPsi ) e ^{- t(\eth + \varPsi)^2}] ) \:
d t.$$
It remains to study the limits $\lim _{t \to 0 ^+ } \sideset{_\mathrm R}{} \operatorname{S t r}(e ^{- t(\eth + \varPsi )^2}) $ $ \lim _{t \to \infty } \sideset{_\mathrm R}{} \operatorname{S t r}(e ^{- t (\eth + \varPsi)^2}) $. Much work have already been done. We first consider the $t \to 0^+$-limit.
\[tto0Prop\] For the $t \to 0$ limit, one has $$\lim _{t \to 0 ^+ } \sideset{_\mathrm R}{} \operatorname{S t r}(e ^{- t(\eth + \varPsi )^2})
= \sideset{_\mathrm R}{} \int \Omega _\varkappa ( \widehat {\mathbb A}\wedge \exp ( - F ^{{\mathrm E}/ {\mathrm S}}) ) .$$
Recall that, by Lemma \[AsymLem\], one has the asymptotic expansion $$e ^{- t (\eth + \varPsi )^2} (x, t)
\cong (4 \pi )^{- \frac{n}{2}} \sum _{i = 0} ^\infty t ^{i - \frac{n}{2}} Q _i (x).$$ Since $\operatorname{s t r}Q _i = 0 $ for any $i < \frac{n}{2} = 1$, it follows that $$\begin{aligned}
\label{tto0}
\lim _{t \to 0^+} (4 \pi )^{- \frac{n}{2}} \sum _{i = 0} ^\infty t ^{i - \frac{n}{2}} \operatorname{s t r}(Q _i (x))
=& (4 \pi )^{-1} \operatorname{s t r}Q _{\frac{n}{2}} (x) \\ \nonumber
=& (4 \pi )^{-1} \frac{ \Omega _\varkappa ( \widehat {\mathbb A}\wedge \exp ( - F ^{{\mathrm E}/ {\mathrm S}}) ) }{ \mu },\end{aligned}$$ by (2) of Lemma \[AsymLem\]. Since ${\mathrm M}$ is compact, the convergence in Equation (\[tto0\]) is uniform in all derivatives. Since Definition \[CP1Renorm\] of the renormalized integral only involves integration and evaluation of the derivatives of the integrands, it follows that $$\lim _{t \to 0 ^+ } \sideset{_\mathrm R}{} \int \operatorname{s t r}e ^{- t (\eth + \varPsi )^2} (x, t) \mu =
(4 \pi ) ^{-1} \sideset{_\mathrm R}{} \int \Omega _\varkappa ( \widehat {\mathbb A}\wedge \exp ( - F ^{{\mathrm E}/ {\mathrm S}}) )$$ as well.
As a direct consequence of Proposition \[tto0Prop\], one has
For any perturbed Dirac operators $\eth + \varPsi $, not necessary Fredholm, one has $$\label{CheatEq}
\lim _{t \to \infty } \sideset{_\mathrm R}{} \operatorname{S t r}(e ^{- t(\eth + \varPsi )^2})
= (4 \pi )^{-1} \sideset{_\mathrm R}{} \int \Omega _\varkappa ( \widehat {\mathbb A}\wedge \exp ( - F ^{{\mathrm E}/ {\mathrm S}}) )
+ \boldsymbol \eta (\eth + \varPsi ),$$ provided the limits on both sides exist.
We turn to study the behavior as $t \to \infty $. Let $\eth + \varPsi $, be a perturbed Dirac operator. Note that $\eth + \varPsi$ is essentially self-adjoint. In addition, we assume that $\eth _{ {}_{\mathrm T}e} + R _{{}_{\mathrm T}e} $ is invertible. It follows from Corollary \[NisLem\] that $\nu (\eth + \varPsi ) $ is Fredholm. Since one has ${\mathcal G}_{{\mathrm M}_0} \cong {\mathrm M}_0 \times {\mathrm M}_0$, it follows that $0$ is (at most) an isolated point of $\boldsymbol \sigma (\eth _x + R _x ) $ for $x \neq {} _{\mathrm T}e $. Our last objective is to study the behavior of the renormalized integral $$\sideset{_\mathrm R}{} \int \operatorname{s t r}e ^{ - t ( \eth + \varPsi )^2 } \mu ,$$ as $t \to \infty$.
From our assumptions, it is clear that the null space $\operatorname{Ker}(\nu ((\eth + \varPsi )^2))$, is finite dimensional. Denote by $P ^0 $ the projections onto $\operatorname{Ker}(\nu ((\eth + \varPsi )^2))$. Let $u _1 , \cdots , u _N \in {\mathbf L}^2 ({\mathrm M}_0 , {\mathrm E}) $ be any orthonormal basis of $\operatorname{Ker}(\nu ((\eth + \varPsi )^2))$. Then $P ^0 _{x} $ has a kernel $$\sum _{i =1 } ^N u _i (y) u _i (y') ,
\quad (y , y') \in {\mathrm M}_0 \times {\mathrm M}_0 \cong {\mathcal G}_{{\mathrm M}_0}.$$ Consider the regularity of $ u _i $. Applying the parametrix formula $$\nu _0 (Q _1) \nu _0 (\varPsi) - \operatorname{i d}= \nu _0 (R _1)$$ to $u _i$, where $ Q _1 \in \Psi ^{[- m ]} _\mu ({\mathcal G}, {\mathrm E}) , R _1 \in \Psi ^ {- \infty} _\mu ({\mathcal G}, {\mathrm E})$, one has $$u _i = \nu _0 (R _1) u _i ,$$ for each $i $. Using Lemma \[SoboBdLem\], it follows that $u _i \in {\mathbf W}^\infty ({\mathrm M}_0 , {\mathrm E})$.
By the identification $\nu ((\eth + \varPsi )^2) \cong (\eth + \varPsi )^2 _x , x \neq {}_ {\mathrm T}e$, $\operatorname{Ker}((\eth + \varPsi )^2 _x )$ is finite dimensional and consists of elements in ${\mathbf W}^\infty ({\mathcal G}_x , {\mathbf t}^{-1} {\mathrm E})$. Denote the projection onto the kernel of $(\eth + \varPsi) ^2 _x $ by $P ^0 _x$ (note that $P ^0 _{ {} _{\mathrm T}e } = 0 $ since $(\eth + \varPsi )^2 _x $ is invertible). Then, using again the fact that $0$ is at most an isolated point of ${\boldsymbol \sigma }_{{\mathbf L}^2 }( (\eth + \varPsi )^2 _x ) $, one has the following well known variation of [@Simon;SpecHeatKer]:
\[SimonLongTime\] There exists some $\lambda > 0$ such that for each $x \in {\mathrm M}$, $$\lim _{t \to \infty } e ^{t \lambda } (e ^{- t (\eth - \varPsi )^2 _x } - P ^0 _x ) = 0$$ in all Sobolev norms.
Unfortunately, we do not know any direct way to prove that $$\sideset{_{\mathrm R}}{} \operatorname{S t r}( e ^ {- t (\eth + \varPsi )^2 }) \to \sideset{_\mathrm R}{} \operatorname{S t r}( P^0 )$$ as $t \to \infty $. Instead, we observe that $\nu ((\eth + \varPsi )^2 ) $ can be identified with an edge operator on ${\mathrm M}_0 = {\mathbb R}^2 $, studied in [@Albin;EdgeInd]. From Lemma \[AlbinConj\], the heat kernel $e ^ {- t (\eth + \varPsi )^2 }$ coincides with the heat calculus constructed in [@Albin;EdgeInd Section 4]. Furthermore, it is easy to see that Definition \[CP1Renorm\] coincides with [@Albin;EdgeInd Equation (6.1)], for the heat kernel. Therefore, by [@Albin;EdgeInd Lemma 6.1], one has $$\label{ttoinfty}
\lim _{t \to \infty } \sideset{_\mathrm R}{} \operatorname{S t r}(e ^{- t(\eth + \varPsi )^2})
= \sideset{_\mathrm R}{} \operatorname{S t r}(P ^0) = \operatorname{i n d}(\nu (\eth + \varPsi ) ).$$ Note that the last equality follows from the fact that $\operatorname{s t r}(P ^0 )$ is an integrable function on ${\mathrm M}_0$, hence the renormalized integral coincides with the usual integral, which turns out to be $\operatorname{i n d}(\nu (\eth + \varPsi ) ) $ because $P ^0$ is just the projection to the null space of $\nu ((\eth + \varPsi )^2) $.
Finally, combining Equations (\[CheatEq\]) and (\[ttoinfty\]), and results in Section 3, one gets
For any self adjoint perturbed Dirac operator $\eth + \varPsi \in \Psi ^1 _\mu ({\mathcal G}, {\mathrm E})$ on the symplectic groupoid ${\mathcal G}= {\mathrm T}\backslash ({\mathrm S \mathrm U}(2) \times {\mathrm N})$ of the Bruhat sphere, such that the Fourier-Laplace transform $ {\mathfrak F}((\eth + \varPsi )_{{}_{\mathrm T}e} ) $ is invertible on a tubular neighborhood of the real axis, $\nu _0 (\eth + \varPsi ) : {\mathbf W}^1 ({\mathrm E}) \to {\mathbf W}^0 ({\mathrm E})$ is Fredholm; and its Fredholm index is given by the Atiyah-Singer index formula: $$\operatorname{i n d}(\nu _0 (\eth + \varPsi ))
= (4 \pi ) ^{-1} \sideset{_\mathrm R}{} \int \Omega _\varkappa ( \widehat {\mathbb A}\wedge \exp ( - F ^{{\mathrm E}/ {\mathrm S}}) )
+ \boldsymbol \eta (\eth + \varPsi ).$$
$ \; $
Concluding remarks
==================
In this last section, we make some remarks and highlight some open problems.
Our first objective in generalizing the calculus on manifolds with boundary was to extend the uniformly supported pseudo-differential calculus to include the parametrix of Fredholm operators. We did so for the Bruhat sphere case in Section 3, where we used the exponentially decaying calculus. In the general case, one would derive an invertibility criterion on the ${\mathbf s}$-fibers over the invariant sub-manifolds. That involves understanding the representation theory of the isotropy subgroup ${\mathcal G}^x _x $ on sections over the ${\mathcal G}^x _x $-principle bundle ${\mathbf s}^{-1} (x)$. It is known that the kernel of inverse of an uniformly supported pseudo-differential operator on a manifold with bounded geometry has exponential decay [@Shubin;BdGeom]. The only remaining problem is whether one can use a tubular neighborhood theorem to extend the fiber-wise inverse to the whole groupoid. In the same vein, Medadze and Shubin [@Shubin;Lie2] proved that the space of pseudo-differential operators on an unimodular Lie group with exponentially decaying kernel is closed under functional calculus. It would be interesting to prove an analogue for Lie groupoids. More precisely:
Let ${\mathcal G}\rightrightarrows {\mathrm M}$ be a groupoid with compact units ${\mathrm M}$ and polynomial growth. Then the exponentially decaying calculus $$\bigcup _{\varepsilon > 0 } \Psi ^{[\infty ]} _\varepsilon ({\mathcal G})$$ is closed under holomorphic functional calculus.
The main difficulty in proving the conjecture lies in proving that the inverses of a smooth family of pseudo-differential operators is still a smooth family. Such a result would enable one to construct, say, complex powers of elliptic operators in a framework more concrete than the axiomatic approach of [@Nistor;CplxPwr].
The discussion on extended calculus cannot be complete without mentioning what is missing in our construction, compared with the case of edge manifolds. In the latter case, one can construct a ‘very residual’ calculus, consisting of functions (sections) on ${\mathrm M}_0 \times {\mathrm M}_0 $ with poly-homogeneous expansions near the singularities. The full calculus is formed by adding the residual calculus to the decaying calculus. Then it was shown that the full calculus contains the generalized inverses of (semi)-Fredholm operators. The proof of these results uses order-by-order cancellations of the boundary defining function near the singular leaves. It is not clear what analogue should be used for groupoids. However, the techniques used in [@Nistor;Polyhedral; @Nistor;Polyhedral2], and the occurrence of stereographic coordinates (which just measures the distance from the opposite of the singularity) in Section 5 might offer a strong hint.
Our next task was to construct the heat kernel of perturbed Laplacian operators on a groupoid in Section 4. The proof of existence is fairly classical. The mystery lies in the proof of transverse smoothness of the heat kernel, which requires considering a (rather arbitrary) transverse metric and bounding the derivatives of the multiplication operator. At this point, we conjecture that a transverse metric satisfying the hypothesis of Theorem \[EstReg\] exists for all Hausdorff groupoids, and can be constructed by gluing exponential coordinates (as in Nistor [@Nistor;IntAlg'oid]).
We went on to derive an Atiyah-Singer type index formula on the Bruhat sphere in Section 5. We cheated by using the stereographic coordinates on the Bruhat sphere to define the renormalized integral. Therefore the arguments cannot be easily generalized beyond the flag manifolds. We further cheated by using known results from edge calculus to show that the renormalized trace converges to the Fredholm index. We expect a direct proof of Equation (\[ttoinfty\]) would be possible by better understanding the resolvent and/or null space projection of the Laplacian operator, that would involve results in functional calculus or residue calculus, as described earlier. One immediate observation form the renormalized index theory is that the renormalized index, as well as the $\mathbb K$-theoretic index, of an elliptic (pseudo)-differential operator are well defined even for non-Fredholm operators. We have not studied the connection between the two, but the arguments involved should be straightforward (see, for example, [@Nistor;Family Proposition 3]).
On the side of generalizing the renormalized trace, we think one possible way to proceed is to use the $Q$-weighted trace machinery developed by Paycha et. al. (see [@Paycha;Renorm] for an introduction), but that is more speculation than educated guess...
And the thesis ends here. However, the work in this thesis is just the beginning of a vast subject concerning singular pseudo-differential calculus defined by groupoids. In the limited space and time we had, we were only able to achieve some success in the simplest case, namely the Bruhat sphere; but the potential of the techniques illustrated here, is unlimited.
Some preliminaries on differential geometry and pseudo-\
differential calculus
========================================================
Notes on submersions and pullback vector bundles {#DGNonsense}
-------------------------------------------------
In this section, we define some notations concerning pullback of vector bundles and recall some basic facts. Let ${\mathrm B}_1 , {\mathrm B}_2 $ be manifolds, $ \pi : {\mathrm B}_2 \to {\mathrm B}_1 $ be a smooth map, and ${\mathrm E}$ be a vector bundle over ${\mathrm B}_1$. Denote the bundle projection by $\wp : {\mathrm E}\to {\mathrm B}_1 $.
The [*pullback bundle*]{} is the vector bundle over ${\mathrm B}_2$: $$\pi ^{-1} {\mathrm E}:= \{ (x , e) \in {\mathrm B}_2 \times {\mathrm E}: \pi (x ) = \wp (e) \},$$ with bundle projection $\pi ^{-1} \wp (x , e) := x $ and the fiber-wise linear operations.
One has a natural map $\pi _{\mathrm E}: \pi ^{-1} {\mathrm E}\to {\mathrm E}$ determined by the commutative diagram $$\begin{CD}
\pi ^{-1} {\mathrm E}@> \pi _{\mathrm E}>> {\mathrm E}\\
@VVV @VVV \\
{\mathrm B}_2 @> \pi >> {\mathrm B}_1
\end{CD}
\quad .$$ Consider the particular case ${\mathrm E}= T {\mathrm B}_1$. One has $\pi _{T {\mathrm B}_1 } : \pi ^{-1} T {\mathrm B}_1 \to T {\mathrm B}_1 $. On other hand, one also has the differential $d \pi : T {\mathrm B}_2 \to T {\mathrm B}_1 $, These two maps determine a bundle map $ \pi _* \in \Gamma ^\infty (\operatorname{Hom}( T {\mathrm B}_2 , \pi ^{-1} T {\mathrm B}_1 ))$ by $$\pi _* ( X ) := (x , d \pi (X) ) , \quad \forall X \in T _x {\mathrm B}_2 .$$ Also recall that one can “pullback" a section to a section of the pullback bundle, i.e., one has the naturally defined map $\pi _{\mathrm E}^{-1} : \Gamma ^\infty ({\mathrm E}) \to \Gamma ^\infty (\pi ^{-1} {\mathrm E}) $, $$(\pi _{\mathrm E}^{-1} f) (x) := f (\pi (x) ) , \quad \forall f \in \Gamma ^\infty ({\mathrm E}), x \in {\mathrm B}_2 .$$
Given any connection $\nabla ^{\mathrm E}$ on ${\mathrm E}$, recall that the pullback connection $\nabla ^{\pi ^{-1} {\mathrm E}} $ is a connection on $\pi ^{-1} {\mathrm E}$ characterized by $$(\nabla ^{\pi ^{-1} {\mathrm E}} )_X (\pi ^{-1} f) (x) = ( x , \nabla ^{\mathrm E}_{d \pi (X)} f (\pi (x)) ),$$ for any $x \in {\mathrm B}_2 , X \in T _x {\mathrm B}_2$. It follows, by using the canonical identification $$\operatorname{Hom}(\pi ^{-1} T {\mathrm B}_1 , \pi ^{-1} {\mathrm E}) \cong \pi ^{-1} \operatorname{Hom}(T {\mathrm B}_1 , {\mathrm E}),$$ that one can write $$\label{PullbackDer}
\nabla ^{\pi ^{-1} {\mathrm E}} (\pi _{\mathrm E}^{-1} f) = (\pi ^{-1} _{\operatorname{Hom}(T {\mathrm B}_1 , {\mathrm E}) } (\nabla ^{\mathrm E}f )) \circ \pi _* ,$$ for any section $f \in \Gamma ^\infty ({\mathrm E})$. Moreover, applying covariant derivatives to Equation (\[PullbackDer\]) and using the Leibniz rule, one gets $$\begin{aligned}
(\nabla ^{\pi ^{-1} {\mathrm E}} )^2 (\pi _{\mathrm E}^{-1} f)
=& \nabla ^{\operatorname{Hom}(T {\mathrm B}_1 \otimes \pi ^{-1} {\mathrm E}) } \nabla ^{\pi ^{-1} {\mathrm E}} (\pi _{\mathrm E}^{-1} f) \\
=& \nabla ^{\operatorname{Hom}(T {\mathrm B}_1 \otimes \pi ^{-1} {\mathrm E}) }
((\pi ^{-1} _{\operatorname{Hom}(T {\mathrm B}_1 , {\mathrm E}) } (\nabla ^{\mathrm E}f )) \circ \pi _* ) \\
=& ((\pi ^{-1} _{\operatorname{Hom}(T {\mathrm B}_1 , \operatorname{Hom}T (T {\mathrm B}_1 , {\mathrm E})) } (\nabla ^{\operatorname{Hom}(T B_1 , {\mathrm E})} \nabla ^{\mathrm E}f )
\circ \pi _* ) \circ \pi _* \\
&+ (\pi ^{-1} _{\operatorname{Hom}(T {\mathrm B}_1 , {\mathrm E}) } (\nabla ^{\mathrm E}f )) \circ (\nabla ^{\operatorname{Hom}(T B_2 , \pi ^{-1} T B_1 )} \pi _* ) \\
=& (\pi ^{-1} _{\operatorname{Hom}(T {\mathrm B}_1 \otimes T {\mathrm B}_1 , {\mathrm E}) } ( \nabla ^{\mathrm E})^2 f ) \circ (\pi _* \otimes \pi _* ) \\
&+ (\pi ^{-1} _{\operatorname{Hom}(T {\mathrm B}_1 , {\mathrm E}) } (\nabla ^{\mathrm E}f )) \circ (\nabla ^{\operatorname{Hom}(T B_2 , \pi ^{-1} T B_1 )} \pi _* ),\end{aligned}$$ and so on for higher derivatives.
Suppose, furthermore, that one has a fiber bundle structure ${\mathrm Z}\to {\mathrm B}_2 \to {\mathrm B}_1$. Since $\pi $ is now a submersion, $ {\mathcal V}: = \ker (d \pi ) \subseteq T {\mathrm B}_2 $ defines a (regular) integrable foliation. We shall assume that ${\mathcal V}$ is orientable. Hence all fiber $\pi ^{-1} (p) \cong {\mathrm Z}$ are orientable. Fix a complementary distribution ${\mathcal H}$ to ${\mathcal V}$. For any (local) vector field $\tilde X \in \Gamma (T {\mathrm B}_1 )$, denote the horizontal lift of $\tilde X $ by $\tilde X ^{\mathcal H}$.
Given any $\omega \in \Gamma ^\infty (\wedge ^k {\mathcal V}' )$, the Lie differential (with respect to ${\mathcal H}$ ) is the section $ {\mathfrak L}^{{\mathcal H}} \omega \in \Gamma ^\infty (\operatorname{Hom}({\mathcal H}, \wedge {\mathcal V}' ))$, $$\begin{aligned}
{\mathfrak L}^{\mathcal H}\omega (X) (V_1 , V_2 , \cdots , V_k ) (p)
=& \: {\mathfrak L}_{\tilde X ^{\mathcal H}} (\omega (V _1 , V _2 , \cdots , V _k )) (p) \\ \nonumber
&- \sum _{i = 1 } ^k \omega (V _1 , \cdots , [ \tilde X ^{\mathcal H}, V _i ] , \cdots , V_k ) (p),\end{aligned}$$ for any $X \in T _p {\mathrm B}_2 $, where $\tilde X $ is any local extension of $d \pi (X) $.
Let $\varkappa $ be the rank of ${\mathcal V}$. For any $\mu \in \Gamma ^\infty _c (\wedge ^\varkappa {\mathcal V}) $, consider point-wise average $\langle \mu \rangle \in C ^\infty (B _1 ) $, defined by $$\langle \mu \rangle (p) := \int _{ x \in \pi ^{-1} (p) } \mu |_{\pi ^{-1} (p) } .$$
\[DFiberInt\] For any vector $X \in T _p {\mathrm B}_1 , p \in {\mathrm B}_1 $, one has the formula $${\mathfrak L}_X (\langle \mu \rangle )(p ) = \int _{x \in \pi ^{-1} (p) } {\mathfrak L}^{\mathcal H}\mu ( X ^{\mathcal H}) .$$
First consider the trivial case ${\mathrm B}_2 \cong {\mathrm U}\times {\mathrm Z}, {\mathrm U}\subseteq {\mathbb R}^n$ and ${\mathcal H}$ be the distribution along $ {\mathrm U}\times \{ z \}, z \in {\mathrm Z}.$ By linearity, one may assume that $ X = \partial _j $. Fix a volume form on ${\mathrm Z}$ and denote by $\mu _0$ its pullback to ${\mathrm U}\times {\mathrm Z}$ by the projection map onto ${\mathrm Z}$. Then one can write $\mu = f (p , z ) \mu _0 $ for some $f \in C ^\infty _c ({\mathrm B}_2)$. Differentiating under the integral sign, one gets $${\mathfrak L}_X \langle \mu \rangle = \int _{z \in {\mathrm Z}} (\partial _j f (p, z)) \mu _0 (z).$$ It is clear that ${\mathfrak L}^{\mathcal H}\mu _0 = 0 $. It follows that ${\mathfrak L}^{\mathcal H}\mu (\partial _j ) = (\partial _j f (p, z)) \mu _0 (z)$, and the assertion follows.
Let ${\mathcal H}'$ be any other complementary distribution. Then one has for any vector field $X$, $X ^{{\mathcal H}'} = X ^{{\mathcal H}} + X ^{\mathcal V}$ for some vector field $X ^{\mathcal V}\in \Gamma ^\infty ({\mathcal V})$. Using the definition, it is easy to check that $${\mathfrak L}^{{\mathcal H}'} \mu (X ^{{\mathcal H}'} ) - {\mathfrak L}^{\mathcal H}\mu (X ^{\mathcal H}) = {\mathfrak L}_{X ^{\mathcal V}} \mu ,$$ where the right hand side is just the Lie derivative on the integrable foliation ${\mathcal V}$. Integrating fiber-wisely, one gets $$\int _{x \in \pi ^{-1} (p) } {\mathfrak L}^{{\mathcal H}'} \mu ( X ^{{\mathcal H}'} )
= \int _{x \in \pi ^{-1} (p) } {\mathfrak L}^{\mathcal H}\mu ( X ^{\mathcal H}) + \int _{x \in \pi ^{-1} (p) } {\mathfrak L}_{X ^{\mathcal V}} \mu .$$ The second term on the right hand side vanishes by Stoke’s theorem. Therefore one still gets $$\int _{x \in \pi ^{-1} (p) } {\mathfrak L}^{{\mathcal H}'} \mu ( X ^{{\mathcal H}'} ) = {\mathfrak L}_X \langle \mu \rangle .$$
Finally, the general case follows because the assertion is local and one can always restrict to local trivializations.
We shall briefly describe several obvious generalizations to Lemma \[DFiberInt\]. Fix a connection $\nabla ^{T {\mathrm B}_1} $ on ${\mathrm B}_1 $. For any $\omega \in \Gamma ^\infty (\wedge ^k {\mathcal V})$, define ${\mathfrak L}^{(n)} \omega \in \Gamma ^\infty (\operatorname{Hom}(\otimes ^n {\mathcal H}, \wedge ^k {\mathcal V})) $ inductively by $$\begin{aligned}
{\mathfrak L}^{(1)} \omega := & \: {\mathfrak L}^{\mathcal H}\omega \\
{\mathfrak L}^{(m + 1)} \omega (X _0 , \cdots , X _m )
:= & \:
{\mathfrak L}^{\mathcal H}({\mathfrak L}^{(m)} \omega (\tilde X _1 ^{\mathcal H}, \tilde X _2 ^{\mathcal H}, \cdots , \tilde X _m ^{\mathcal H})) (X _0 ) (p) \\ \nonumber
&- \sum _{i = 1 } ^m ({\mathfrak L}^{(m)} \omega )
(\tilde X _1 ^{\mathcal H}, \cdots , (\nabla ^{T {\mathrm B}_1 } _{\tilde X _0 } \tilde X _i )^{\mathcal H}, \cdots , \tilde X _m ^{\mathcal H}) (p),\end{aligned}$$ for any $X _0 , \cdots X _m \in {\mathcal H}_p $, where $\tilde X _i $ is any local extension of $d \pi (X _i) $. Then a straightforward computation using the Lemma \[DFiberInt\] and the definitions gives
For any $\mu \in \Gamma ^\infty _c (\wedge ^\varkappa {\mathcal V})$, $X _1 , \cdots , X _m \in T _p {\mathrm B}_1 , p \in {\mathrm B}_1$, one has $$\nabla ^m (\langle \mu \rangle )(X _1 , \cdots , X _m ) (p)
= \int _{x \in \pi ^{-1} (p)} {\mathfrak L}^{(m)} \mu (\tilde X _1 ^{\mathcal H}, \cdots , \tilde X _m ^{\mathcal H}) (x),$$ where $\tilde X _i $ is any local extension of $X _i $.
Lemma \[DFiberInt\] can also be generalized in a different direction Let ${\mathrm E}$ be a vector bundle over ${\mathrm B}_1 $. For any $f \in \Gamma ^\infty _c (\pi ^{-1} {\mathrm E}) , \mu \in \Gamma ^\infty (\wedge ^\varkappa {\mathcal V})$, define $$\langle f \mu \rangle (p)
:= \sum _{i = 1 } ^l \langle f _i \mu \rangle (p) e _i (p) \quad \in \Gamma ^\infty ({\mathrm E}), \quad p \in {\mathrm B}_1,$$ where $e _1 , \cdots , e _l $ is any local basis around $p$ and $f = \sum _{i = 1 }^l f _i \pi ^{-1} (e _i ) $ on $\pi ^{-1} (p)$. The definition is independent of choice of a local basis. Let $\nabla ^{\mathrm E}$ be any fixed connection on ${\mathrm E}$. Then a simple application of Lemma \[DFiberInt\] leads to
Given any $f \in \Gamma ^\infty _c (\pi ^{-1} {\mathrm E}) , \mu \in \Gamma ^\infty (\wedge ^\varkappa {\mathcal V})$. Then for any vector field $X \in \Gamma ^\infty (T {\mathrm B}_1 ), p \in {\mathrm B}_1$, $$\nabla ^{\mathrm E}\langle f \mu \rangle (X) (p)
= \int _{x \in \pi ^{-1} (p) } (\pi ^{-1} (\nabla ^ {\mathrm E}) f) (X ^{\mathcal H}) \mu (x) + f ({\mathfrak L}^{\mathcal H}\mu (X ^{\mathcal H})) (x) .$$
Preliminaries on pseudo-differential calculus {#PDONonSense}
----------------------------------------------
In this section, we recall some basic definitions and results about pseudo-differential calculus. All materials in this section are classical and can be found in, say, Hormander [@Hormander;1].
### **Distributions and kernels**
Let $\Omega \subseteq {\mathbb R}$ be an open subset. We denote by $C^\infty _c (\Omega )$ the space of smooth compactly supported functions on $\Omega $. The space $C^\infty _c (\Omega ) $ is equipped with the $C^\infty $-topology: $$u _n \rightarrow u \; \text {if} \;
\sup _{x \in {\mathrm K}} | \partial _x ^I (u_n - u) | \rightarrow 0,$$ for any compact subset ${\mathrm K}$ and any multi-index $I$.
A distribution (on $\Omega $) is a continuous linear map $\phi : C^\infty _c (\Omega ) \rightarrow {\mathbb C}$. We shall denote the space of distributions by $$C^\infty _c (\Omega )'.$$ For any open subset ${\mathrm U}\subset \Omega $, the restriction of $\phi $ to ${\mathrm U}$ is defined to be the restriction of $\phi $ to $C^\infty _c ({\mathrm U})$ (extended to $C^\infty _c (\Omega )$ by 0). The support of $\phi $, denoted $\operatorname{Supp}(\phi )$ , is the collection of points $x \in \Omega $ such that the restriction of $\phi $ to any open neighborhood of $x$ is non-zero. We say that $\phi \in C^\infty (\Omega )$ if there exist $\kappa \in C^\infty (\Omega )$ such that $$\phi (u) = \int _\Omega \kappa (x) u (x) \: d x, \quad \forall u \in C^\infty _c (\Omega ).$$ Note that such $\kappa $, if it exists, is unique.
The most important result about distributions is the Schwartz distribution theorem:
For any continuous map $A : C ^\infty _c ({\mathrm M}) \to C ^\infty _c ({\mathrm M})' $, there exists a unique continuous linear functional $K : C ^\infty _c ({\mathrm M}\times {\mathrm M}) \to {\mathbb C}$ such that $$(A f ) (g) = K (f (x) g (y)), \quad \forall f , g \in C ^\infty _c ({\mathrm M}).$$
### **Pseudo-differential operators on a manifold**
Let $\Omega $ be an open subset on ${\mathbb R}^n$, and $m \in {\mathbb R}$. A symbol of order $\leq m$ is a smooth function $\sigma (x , \zeta )\in C^\infty (\Omega \times {\mathbb R}^n)$ such that for any compact ${\mathrm K}\subset \Omega $ and multi-index $I, J$, there is a constant $C^{\mathrm K}_{I,J}$ such that $$\left|
\partial ^I _x \partial ^J _\zeta \sigma (x, \zeta )
\right|
\leq C^{\mathrm K}_{I, J} (1 + |\zeta |^2)^\frac{m - |J|}{2} \quad \forall x \in {\mathrm K}.$$
The set of symbols on $\Omega $ of order $\leq m$ shall be denoted by $\mathbf S^m (\Omega )$; and define $$\mathbf S ^{-\infty} (\Omega ) := \bigcap_{m \in {\mathbb R}} \mathbf S^m (\Omega ),
\mathbf S^\infty (\Omega ) := \bigcup_{m \in {\mathbb R}} \mathbf S^m (\Omega ).$$
A symbol $\sigma_l \in \mathbf S ^l (\Omega ) $ is called homogeneous of order $l$, if $$\sigma_l (x, \lambda \zeta ) = \lambda ^l \sigma (x, \zeta ),
\quad \forall x \in \Omega , |\lambda | \geq 1, |\zeta | \geq 1 .$$ A symbol $\sigma \in \mathbf S ^m (\Omega )$ is said to be classical of order $m, m \in {\mathbb Z}$ if there are homogeneous symbols $\sigma _m , \sigma _{m-1} , \cdots$, of orders $m, m-1, \cdots$ respectively, such that $$\sigma - \sum_{l=0}^{N - 1} \sigma _{m - l} \in \mathbf S^{m - N} (\Omega )$$ for $N = 1, 2, \cdots$.
The set of classical symbols of order $m \in {\mathbb Z}$ is denoted by $\mathbf S^{[m]} (\Omega )$.
Let ${\mathrm M}$ be a manifold. A function $\sigma \in C^\infty (T^* {\mathrm M})$ is called a symbol of order $\leq m$ if for every coordinate patch $({\mathrm U}, {\mathbf x})$, $$\sigma \circ ({\mathbf x}^*) \in \mathbf S ^m ({\mathbf x}({\mathrm U})).$$ Here, we have identified $T^* ({\mathbf x}({\mathrm U})) \cong {\mathbf x}({\mathrm U}) \times {\mathbb R}^n$. The symbol $\sigma $ is said to be homogeneous (resp. classical) if $ \sigma \circ ({\mathbf x}^*) $ is homogeneous (resp. classical).
The set of symbols of order $\leq m$ (resp. classical symbols of order $m$) is denoted by $\mathbf S^m ({\mathrm M})$ (resp. $\mathbf S ^{[m]} ({\mathrm M})$).
A pseudo-differential operator on $\Omega \subseteq {\mathbb R}^n$ of order $\leq m$ is a linear operator $\varPsi : C^\infty _c ({\mathrm U}) \rightarrow C^\infty ({\mathrm U})$ of the form $$(\varPsi u)(x)
= (2 \pi)^{-n} \int_{\zeta \in {\mathbb R}^n} \int_{y \in \Omega}
\sigma (x, \zeta ) e^{i \langle \zeta , x-y \rangle} u(y) \; d y \; d \zeta,
\quad u \in C^\infty_c (\Omega ),$$ for some symbol $\sigma \in \mathbf S ^m (\Omega )$. If $\sigma $ is classical, i.e., $\sigma \in \mathbf S ^{[m]} (\Omega ), m \in {\mathbb Z}$, then we say that $\varPsi $ is a classical pseudo-differential operator of order $m$.
A pseudo-differential operator on a manifold ${\mathrm M}$ of order $\leq m$ is a linear operator $\varPsi : C^\infty _c ({\mathrm M}) \rightarrow C^\infty ({\mathrm M})$ such that for any coordinate patch $({\mathrm U}, \mathbf x)$, the induced map $$u \mapsto (\mathbf x^{-1} )^* (\varPsi (\mathbf x^* u)), \quad u \in C^\infty_c ({\mathbf x}({\mathrm U}))$$ is a pseudo-differential operator on ${\mathbf x}({\mathrm U}) \subseteq {\mathbb R}^n$ of order $\leq m$.
The set of pseudo-differential operators on ${\mathrm M}$, of order $\leq m$ (resp. classical pseudo-differential operators of order $m$), is denoted by $\Psi^m ({\mathrm M})$ (resp. $\Psi ^{[m]} ({\mathrm M})$). We also define $$\Psi ^{-\infty} ({\mathrm M}) := \bigcap_{m \in {\mathbb R}} \Psi ^m ({\mathrm M}),
\Psi ^\infty ({\mathrm M}) := \bigcup_{m \in {\mathbb R}} \Psi ^m ({\mathrm M}).$$ Note that $\Psi ^{- \infty} ({\mathrm M}) = \bigcap _{m \in {\mathbb Z}} \Psi ^{[m]} ({\mathrm M})$.
Let $\varPsi \in \Psi ^\infty ({\mathrm M})$ be a pseudo-differential operator with distributional kernel $\kappa (x, y)$. The support of $\varPsi $, denoted $\operatorname{Supp}\varPsi $, is defined to be the support of $\kappa $. The operator $\varPsi $ is said to be properly supported if for any compact subset ${\mathrm K}\subset {\mathrm M}$, the set $$({\mathrm K}\times {\mathrm M}) \bigcap \operatorname{Supp}(\varPsi)$$ is a compact subset of ${\mathrm M}\times {\mathrm M}$.
We denote the space of properly supported pseudo-differential operators of order $\leq m $ by $\Psi ^m_\varrho ({\mathrm M})$. It is clear that a properly supported $\varPsi \in \Psi^\infty ({\mathrm M})$ extends uniquely to a linear operator from $C^\infty ({\mathrm M}) $ to itself. It follows that the composition of two pseudo-differential operators $\varPsi \circ \varPhi $ is well defined whenever one of them is properly supported.
### **The symbol of a pseudo-differential operator**
Fix a connection $\nabla$ on ${\mathrm M}$. Then there is a neighborhood of the zero section $\Omega \subset T {\mathrm M}$ such that the exponential map $\exp _\nabla : \Omega \rightarrow {\mathrm M}\times {\mathrm M}$ is a diffeomorphism onto its image. Fix a smooth function $\chi (x, y)$ supported on the image of $\exp _\nabla$ and equal to 1 on a smaller neighborhood of the zero section. Define $\Theta (x, y) := \chi (x, y) \exp ^{-1}_\nabla (x, y)$.
\[TotalSym\] Given a $\varPsi \in \Psi ^m ({\mathrm M}), m \in {\mathbb R}$. Define $\sigma (\varPsi ) \in S^m ({\mathrm M})$ by $$\sigma (\varPsi) (\zeta ) := \varPsi (e^{i \langle \zeta , \Theta (x, \cdot) \rangle } \chi (x, \cdot))(x),
\quad \zeta \in T^*_x {\mathrm M}.$$ The function $\sigma (\varPsi) $ is called the total symbol of $\varPsi $ with respect to $(\nabla, \chi )$.
If the total symbol $\sigma (\varPsi )$ is classical, i.e., there exists homogeneous symbols $\sigma _m , \\ \sigma _{m-1} , \cdots$, of orders $m, m-1, \cdots$ respectively, such that $$\sigma - \sum_{l=0}^{N - 1} \sigma _{m - l} \in \mathbf S^{m - N} ({\mathrm M})$$ for $N = 1, 2, \cdots$, then we say that $\varPsi $ is a classical pseudo-differential operator on ${\mathrm M}$. In this case, we define the principal symbol of $\varPsi $ as $$\sigma _{\mathrm {top}} (\varPsi) := \sigma _{m}.$$
We denote the space of classical pseudo-differential operators on ${\mathrm M}$ by $\Psi ^{[m]} ({\mathrm M})$.
\[InvPrincipal\] It can be shown that if the total symbol with respect to some $(\nabla, \chi )$ is classical, then the total symbol with respect to any set of $(\nabla' , \chi ')$ is classical. Also, it is well known that the principal symbol is independent of $\nabla$ and $\chi $.
The following lemma asserts that a pseudo-differential operator $\varPsi $ can be recovered from its total symbol, up to a smoothing operator.
\[Kennedy\] [@Kennedy;Intrin Proposition 3.1] Any pseudo-differential operator $\varPsi $ on ${\mathrm M}$ can be written in the form $$\varPsi u (x) =
\int _{\zeta \in T^*_x {\mathrm M}} \int _{y \in {\mathrm M}}
\sigma (\zeta ) e^{-i \langle \zeta , \Theta (x, y) \rangle} \chi (x, y) u(y) d y d \zeta
+ \int_{y \in {\mathrm M}} \kappa (x, y) u (y) d y,$$ for some $\kappa (x, y) \in C^\infty ({\mathrm M}\times {\mathrm M})$.
### **Pseudo-differential operators between sections of vector bundles**
It is straightforward to generalize the notion of pseudo-differential operators to sections of a vector bundle: Let ${\mathrm E}\rightarrow {\mathrm M}$ be a vector bundle of rank $k$. Let $({\mathrm U}, {\mathbf x})$ be a trivial coordinate patch. Then any smooth section $s \in \Gamma ^\infty ({\mathrm E}|_{\mathrm U})$ can be regarded as a ${\mathbb C}^k $-valued smooth function on ${\mathbf x}({\mathrm U})$. We say that a linear map $\varPsi : \Gamma ^\infty _c ({\mathrm E}) \rightarrow \Gamma ^\infty ({\mathrm E})$, is a pseudo-differential operator if for any pair of standard basis vectors of ${\mathbb C}^k$, $\mathbf e_i $ and $ \mathbf e_j, i, j = 1, \cdots, k$, the induced map $$u \mapsto \langle \mathbf e_i , (\mathbf x^{-1} )^* (\varPsi (\mathbf x^* u \mathbf e_j) \rangle,
\quad u \in C^\infty_c ({\mathbf x}({\mathrm U})),$$ is a pseudo-differential operator on ${\mathbf x}({\mathrm U}) \subseteq {\mathbb R}^n$.
We denote the set of pseudo-differential operator, of order $\leq m$, on ${\mathrm E}\rightarrow {\mathrm M}$ by $ \Psi ^m ({\mathrm M}, {\mathrm E}), $ and so on.
It is clear that the notion of (total and principle) symbol of an element in $\Psi ({\mathrm M}, {\mathrm E})$ can be generalized in a similar manner. However, in this case, the symbol is an element in $$\Gamma ^\infty (\wp ^{-1} ({\mathrm E}\otimes {\mathrm E}')),$$ where $\wp : T^* {\mathrm M}\rightarrow {\mathrm M}$ is the natural projection. Likewise, an operator $\varPsi \in \Psi ^m ({\mathrm M}, {\mathrm E})$ is said to be [*elliptic* ]{} if its principal symbol $\sigma (\zeta )$ is invertible (as a matrix) whenever $\zeta \neq 0$.
Finally, note that a smoothing operator on $\Gamma ^\infty _c ({\mathrm E})$ is of the from $$u \mapsto \int _{y \in {\mathrm M}} \kappa (x, y) u (y ) d y,$$ where $\kappa (x, y) \in \Gamma ^\infty ({\mathrm E}_x \otimes {\mathrm E}'_y )$, and the integrand is considered as a map from ${\mathrm M}$ to ${\mathrm E}_x$, for each $x \in {\mathrm M}$.
Manifolds with bounded geometry {#BdGeomNonSense}
--------------------------------
In this section, we a study special class of manifolds, namely, manifolds of bounded geometry in the sense of Shubin [@Shubin;BdGeom]. Our objective is to define various Sobolev spaces, which would serve as the natural domain for the pseudo-differential operators. We shall refer the general theory to [@Nistor;GeomOp].
A Riemannian manifold ${\mathrm M}$ is said to have bounded geometry if
1. $ {\mathrm M}$ has positive injectivity radius;
2. The Riemannian curvature $R$ of ${\mathrm M}$ has bounded covariant derivatives.
### **Basic properties**
Here, we recall some basic results concerning manifolds of bounded geometry.
\[BallCover\] [@Shubin;BdGeom Lemma 1.2] There exists $\epsilon _0 > 0$ such that for any $0 < \varepsilon < \varepsilon _0 $, there is a countable set $\{ x_\alpha \} \subset {\mathrm M}$ such that the balls $B (x_\alpha , \varepsilon )$ is a cover of ${\mathrm M}$, and any $x \in {\mathrm M}$ belongs to at most $N$ balls $B (x_\alpha , 2 \varepsilon ) $, for some $N$ independent of $x$.
Recall that for every point $x$ in a Riemannian manifold ${\mathrm M}$, the exponential map is a homeomorphism from an open neighborhood of $0 \in T_x {\mathrm M}$ to an open neighborhood of $x$. Its inverse thus defines a local coordinate patch, known as the [*(geodesic) normal coordinates (around $x$)*]{}.
\[BddPartition\] Let $\{ (B ( x _\alpha , \varepsilon ) , {\mathbf x}_\alpha ) \} $ be a cover by normal coordinates patches, such that the conclusion of Lemma \[BallCover\] holds. Then there exists a partition on unity $\theta _\alpha $ subordinated to $\{ B (x_\alpha , \varepsilon ) \}$, such that for any $k \in {\mathbb N}$, all $k$-th order partial derivatives of $\theta _\alpha $ are bounded by some $C_k$, independent of $\alpha $.
Let ${\mathrm M}$ be a manifold with bounded geometry. A vector bundle ${\mathrm E}\rightarrow {\mathrm M}$ is said to have bounded geometry if for any $k \in {\mathbb N}$, there exist $C_k > 0$ such that for any trivial normal coordinate patches, the all $k$-th order partial derivatives of the transition function is bounded by $C_k$.
### **Sobolev spaces**
\[SoboDfn\] Let ${\mathrm E}$ be a vector bundle of bounded geometry. Fix a normal coordinates cover $\{ ({\mathrm U}_\alpha , {\mathbf x}_\alpha ) \}$ of ${\mathrm M}$ such that ${\mathrm E}| _{{\mathrm U}_\alpha }$ is trivial, and a locally finite partition of unity $\{ \theta _\alpha \} $ subordinated to $\{ {\mathrm U}_\alpha \}$, as in Lemma \[BddPartition\]. Regard $\theta _\alpha s$ as a smooth vector valued function on ${\mathbb R}^n$ through local coordinates.
On $\Gamma ^\infty ({\mathrm E})$, define the $\infty $-norms $$\| s \| _{\infty , l} := \sup _\alpha \{ |\partial ^I \theta _ \alpha s (x) | : x \in U _\alpha , |I| \leq l \}$$ for each $l \in {\mathbb N}$. We say that a section $s \in \Gamma ^\infty ({\mathrm E}) $ has bounded derivatives if $\| s \| _{\infty , l} < \infty $.
For each $m \in {\mathbb R}$, define the 2-norms $$\label{L-Infty}
\| s \| _{2, m} := \Big( \sum _{\alpha }
\| \theta _\alpha s \| _{{\mathbf W}^m (U _\alpha ) } ^2 \Big)^{\frac{1}{2}},$$ where $ {\mathbf W}^m (U _\alpha ) $ is the $m$-th Sobolev norm on $U _\alpha \subset {\mathbb R}^n$. We denote the completion of $\Gamma ^\infty _c ({\mathrm E}) $ with respect to $\| \cdot \| _{2 , m}$ by ${\mathbf W}^{m} ({\mathrm M}, {\mathrm E})$.
Observe that, since all transition functions are uniformly bounded, the equivalence classes of these norms are independent of the choices made.
For $m \in {\mathbb Z}$, ${\mathbf W}^m ({\mathrm M})$ can be equivalently defined by the collection of distribution $ u \in C _c ^\infty ({\mathrm M})' $ such that $ {\mathfrak L}_{X _1} {\mathfrak L}_{X _2} \cdots {\mathfrak L}_{X _m} u \in {\mathbf L}^2 ({\mathrm M}) $ for any collection of vector fields $ X _1 , \cdots , X _m $ with unit length.
As in the case of ${\mathbb R}^n$, one has the Sobolev embedding
\[SoboEm\] For any integer $m, l$ such that $m > l + \frac{n}{2}$, $${\mathbf W}^m ({\mathrm M}) \subseteq C ^l _b ({\mathrm M}).$$ Furthermore, there exists a constant $C$, depending only on $m, l, n$, such that $$\| u \| _{0, l} \leq C \| u \| _{2 , m}$$ for any $u \in {\mathbf W}^m ({\mathrm M})$.
\[SoboDecay\] Let $u \in {\mathbf W}^m ({\mathrm M})$, where $m > l + \frac{n}{2}$ for some integer $l$. Fix any point $x _0 \in {\mathrm M}$. For any $\varepsilon > 0$, there exist integer $N _0 $ such that for any integer $N > N _0$, $$\sup _{x \not \in B (x_0 , N)} | u (x) |_l \leq \varepsilon .$$
Fix smooth functions $\chi _j , j \in {\mathbb N}$ such that $0 \leq \chi _j \leq 1 $, $\chi _j = 0$ on $B (x _0 , j)$, and $\chi _j = 1 $ on ${\mathrm M}\backslash B (x _0 , j+1)$. Since $\chi _j \to 0$ as $j \to \infty$, it follows that $\| \chi _j u \| _{2, m} \to 0 $. By the previous Lemma, one has $$\sup _{x \not \in B (x_0 , j)} |\chi _j (x) u (x) |_l = \| \chi _j u \| _{0, l} \leq C \| \chi _j u \| _{2 , m} .$$ The assertion follows because $$\sup _{x \not \in B (x_0 , j + 1)} | u (x) |_l
= \sup _{x \not \in B (x_0 , j + 1)} |\chi _j (x) u (x) |_l
\leq \sup _{x \not \in B (x_0 , j)} |\chi _j (x) u (x) |_l ,$$ for all integer $j$.
On a manifold with bounded geometry, a class of ‘uniformly bounded’ pseudo-differential operators can also be defined. Fix any covering $\{ U _\alpha , {\mathbf x}_\alpha \}$ of ${\mathrm M}$ by normal coordinates. Let $\varPsi \in \psi ^m _\varrho ({\mathrm M})$. Recall that $( {\mathbf x}_\alpha ^{-1} )^* \psi {\mathbf x}_\alpha ^* $ is a pseudo-differential operator on $U _\alpha $. Let $\sigma _\alpha \in \mathbf S ^m (U _\alpha )$ be the total symbol of $( {\mathbf x}_\alpha ^{-1} )^* \psi {\mathbf x}_\alpha ^* $. Then we say that
The pseudo-differential operator $\varPsi $ is [*uniformly bounded*]{} if
1. The support of $\varPsi $ is contained in the set $$\{ (x , y ) \in {\mathrm M}\times {\mathrm M}: d (x , y ) < r \}$$ for some $r > 0$;
2. For any multi-indexes $I , J$, there exists a constant $C _{I J} $, independent of $\alpha $, such that $$| \partial _x ^I \partial _\zeta ^J \sigma _\alpha | \leq C _{I J} (1 + |\zeta |)^{m - |J|} .$$
We denote the set of all, uniformly bounded pseudo-differential operators of order $\leq m$ by $\Psi ^m _b ({\mathrm M})$.
Finally, we can state the main result on boundedness of pseudo-differential operators on Sobolev spaces.
\[SoboBdLem\] For any $\varPsi \in \Psi ^m _b ({\mathrm M}, {\mathrm E}), u \in {\mathbf W}^l ({\mathrm M}, {\mathrm E})$, $ \varPsi u \in {\mathbf W}^{l - m} ({\mathrm M}, {\mathrm E}) $. Furthermore, the map $u \mapsto \varPsi (u) $ is a bounded map from ${\mathbf W}^l ({\mathrm M}, {\mathrm E}) $ to ${\mathbf W}^{l - m} ({\mathrm M}, {\mathrm E})$.
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1307 Lethal Performance Shelby Project Lethal Performance Egr Delete
To keep intake temps as low as possible on the drag strip, and clean up the engine compartment a bit Jesse installed Lethal Performance’s EGR Delete kit (PN LP-EGRDELETE-2; $29.95). This cap blocks off the connection at the header. |
"Previously on Tru Calling:" "You're in medical school now, Miss Davies." "Are you committed to saving lives or not?" "Yes, I'm committed." "Don't you see?" "I gave you a gift." "I gave you the perfect excuse to walk away." "Actually, you gave me every reason to stay." "You missed quite a day." "Jack came back, and my father moved to town." "Are you offering me a job?" "What do you say?" "Wanna come work for the old man?" "He does love his sister, and that bond is stronger than you think." "So's the bond between father and son." "Good people have to die too, Tru." "You know that as well as I." "You say you're not a killer." "What do you call what you're doing?" "My job." "Every cadaver has a story." "Treat it with respect and you will be privileged to learn from it." "Break into groups." "Four to a cadaver." "Excuse me." "Are you the girl who answered Seidel's trick question... last week and made him look bad?" " I don't know if I made him look bad." " Guys, this is her." "We want you." "How awkward." "She means for our lab group." "We need a fourth." " And you want me?" " Absolutely." "You cut Seidel off at the knees, which was so hot." "Obviously, you're brilliant, and because you're auditing you're not gonna affect the grade curve." "Tyler says that you're the perfect woman." " Jensen." " Tru." "You got a really deft hand with that duodenum." "I suppose you hear that all the time though." "Thanks." "Sorry, I work nights." "It was a long shift." " Didn't get much sleep." " You must get jaded quickly if you're yawning in a room with dead bodies." " What is with this bag?" " What's wrong with the bag?" " It's a gift for Lexi." " Jensen finally looked up from his books long enough to get a girlfriend." "We were all very relieved." " "Essence of Roses. "" " Essence of Roses." "Avery, what...?" "What are you doing?" "I can't put it on myself." "What if I don't like it?" "...or that you smell bad." "I don't know, it depends on where she sprays it." "Nice." "Very nice." "That's no way to handle a gallbladder, Mr. Lee." " Oh." " Oh." "Lexi's perfume." "I'm sorry." "Hey, smells better than formaldehyde." "What is that smell?" "What smell?" "It smells like roses and formaldehyde." " So, Harry, what's with the camera?" " Oh, it's nothing." "Just my first solo gig at Dad's law firm." "The training wheels are off, baby." "Check it." "These two guys got in a fight on a golf course and one of them is filing an AB." " AB?" " Yeah, assault and battery charge." "He claims he was too hurt that he couldn't leave the house." "Well, I happen to know that he's a major hockey fan and he's got tickets to tonight's game." "So you're gonna be there taking shots of him jumping up and down and high-fiving his friends." "And when I lay those pictures down on Dad's desk, he's gonna love me." "Love me." "I'm glad you are getting along." "I know how much that means to you." "Yeah, it's great." "You know, you two are getting along, right?" "And..." "I don't know, maybe we were just too hard on him." "Maybe." " Hi, mind if I join you?" " Oh, Dr. Allen." " No, go ahead." " Oh, you can sit down." "Oh, damn." " I forgot to ask for sugar." " Oh, have one of mine." " Oh, shoot." " Is it all right?" "Yeah, it's just a little wool-polyester fiber." "Did you get the memo about the fire drill?" " No, I didn't." " Tomorrow afternoon." "I like to warn people because sometimes that alarm goes off at the most inconvenient times, when you're in the toilet or you're elbow deep in someone's chest cavity, and..." "Well, I suppose I should go back across the street." " Okay." " Bye." "Bye." "Thanks." "I'm just cursed." "I'm just..." "I'm sure it wasn't that bad." "When the saints were around my cradle, they said:" ""He will be able to analyze 60 kinds of bullet wounds but when it comes to conversational skills, none."" "I thought you didn't even like small talk." " I hate small talk." " So then why would it bother you if...?" "Ah." "What was that?" "What was that "ah"?" "It's just an "ah," Davis." "There's no agenda." "Okay." " Harry." " Hey, it's me." " I got those pictures." " Congratulations." "Yeah, it came at a price though." "The guy's got a mean right hook." " Ouch." "It's worth it." "I can't wait to show Dad." "I am so happy for you." "I'll call you later, okay?" " Yeah." " Bye." " Yeah, about the "ah"..." " Talk to her again." "Find a topic you can both get excited about." "I find that my interests don't overlap with the rest of humanity's." "Okay." "Thomas Burrell, M.D." "Looks like a note he scribbled to himself." " What is it, a grocery list?" " "Varden" and an arrow pointing to "DSM3."" "Must be a psychiatrist." " How do you figure?" " Well, you see..." "Please." "Help me." "That's no way to hold a gallbladder, Mr. Lee." "I am so sorry." "I'll thank you not to bring your feminine trinkets into the lab, Miss Davies." "No." "I'm sorry, I'll go tell him it was mine." "It's all right, but I have to go." "Is it me, or does Tru seem a little abrupt?" "Help me." " Hey." " Hi." " Is he in?" " Yes, go on in." "Thanks." "I rewound." "Got it." "Day two." "Cause of death?" "Shot in the back at Fourth Street Market." "What's your strategy?" "My daughter is expecting you and her record's been good since Luc, which by the way was a stretch of the rules on your part." "I told you." "Luc was meant as a lesson." "Not a lesson she seems to have learned from." "I am not losing this guy to Jack." " You did well against him last time." " That's the point." "There's an opportunity here to take things to another level." "Learn from the mistake you made with her mother." " Marrying her?" " Killing her." "We know that won't help." "The ability only goes to someone else." "No, Tru has to be rendered powerless." "And I have an idea on how we can do that." " Who's the victim?" " Thomas Burrell, M.D." "Must be a psychiatrist." "His office is at 87 Third Street." " I'm going straight there." "Maybe I can head him off before he goes to the market." "Hey." "I'm sorry to bother you but your receptionist seems to be away." "Oh." "Um..." "Well, I'm sure she'll be back in a minute." " Can I help you?" " I hope so." "I..." "I've been going through a lot of changes lately and I was thinking maybe it's time to talk to someone about them." " Could you be more specific, Mr...?" " Goodwin." " Paul Goodwin." " Dr. Burrell." "What do you hope to get out of therapy, Paul?" "Well, I have a very high-stress job and I wouldn't mind learning some coping skills." "May I ask what it is you do?" "You might say I'm a repo man." "People cheat, try to hold on to what's no longer theirs and when that happens, I come and take it back." "Well, I can see where that might be stressful." "People must get angry with you all the time." "They do." "They fail to realize that what I do isn't personal." "Laws are there for a reason." " Someone has to say no." " Mm-hm." "Well, Tuesday at 5 for a beginning?" " Great." " Good." "Excuse me." "I'll be right back." "I don't know where Jan's gone." "This is Dr. Burrell." "May I help you?" "I see." "And who made the referral?" "Thanks, I'm ready." "No." "That shouldn't be a problem at all." "I'll just lighten my caseload." "No, I tell you what." "Well, Jan's out of the office right now." "When she gets back I'll just get her to move some things around." "No." "No problem." "You're welcome." "Hi." "You know what, I gotta run." "Tuesday at 5, right?" " That'll be fine." " Great." "I should mention, I've got an ex-girlfriend who's just a little Fatal Attraction." "She cornered my last shrink with a lot of questions." "Claimed the doctor was in some sort of "danger. "" "She was trying to invade my privacy, you know?" "No, I promise you, I take doctor-patient confidentiality very seriously." "I figured that." "Thanks, doc." "See you Tuesday." "Dr. Burrell." "Oh, you're too late." "The good doctor just left." "How do you know these things?" "How do you get to a victim before I do?" "Does Apple tell Microsoft?" "We're not business rivals." "This is about people's lives, Jack." "Our business is people's lives, Tru." "Nice try, Jack, but I'm not gonna stand here arguing with you because I've got a job to do." "Love the perfume." "See you." " Dr. Burrell." " Yes?" "I need to talk to you." "It's an emergency." " Would you mind coming with me?" " Coming where?" "I can't talk about it here, but it'll only take a few minutes." "If you wanna talk, make an appointment with my office." "No, there isn't time." "Dr. Burrell." "Dr. Burrell!" " Police!" " My God!" "She killed that man!" "No, I..." "Hands in the air." "You're under arrest." "I didn't do it." "Someone shot him in the back, and they're getting away right now!" "They were arguing." "She shot him when he turned." " It did look like that." " I was only talking to him." " Ma'am, your name would be?" " Kelly Robson." "Tru, what's the deal?" "I thought you were supposed to be the responsible one." "Handcuffs?" "No problem." "I see this all the time." "If you're in this predicament again, a good penknife will get you out." "I don't usually carry a penknife." "Well, any sharp blade will do." "How many times have you been in these?" "You know what?" "Don't answer that." "What's that smell?" " Hi." " I'm in trouble." "I tried to stop the murder, but things didn't go well." " I need to see you." " I would advise against that." "Are there, by any chance, cops there?" "That would be correct." "I'm sorry that we won't be seeing each other today." "I got to him too late." "Jack was there." "He set me up." "It all happened so quickly." "I was standing there listening to him lie to the police so I got out of Dodge." "Sometimes situations like this can have a legal solution." " You want me to turn myself in?" " No." "I think he wants me to talk to Dad." "Well, he's got a point, Tru." "Dad's a criminal-defense lawyer." "I mean, this is what he does." "Okay, message received." "I'd better go." "Call you later." " Okay." " Who was that?" "That was my contractor." "He's working on my bathroom and he said he'd have it ready today, but..." "They never show up." " Dad, something happened." " I know." "The police were here." "Come sit with me." "Honey, you're gonna have to tell me everything." " Son, would you mind waiting outside?" " No, it's okay." "Harrison, please." "Sure." "Yeah." "Outside." "Dad, this is all a setup." "I didn't do it." "You should know that the police have already begun investigating you." "They found your name in his appointment book for therapy sessions." " So, they know you were his patient." " I wasn't his patient." "They also found a file with your name on it in his office." "I never even met this man before he was killed." "A number of witnesses say they saw you having an argument with this man." "What was that about?" "I never even had a gun when I was arrested." "Does that mean anything?" "Plenty of prosecutions go ahead without the murder weapon." "A place that crowded, hundreds of people tramping through the crime scene anybody could've kicked it, anyone could've picked it up." "The police will not see that as significant, especially because they have a witness." "This witness is the man that framed me." "His name is Jack Harper." "He's looking for a way to bring me down." "Why would he wanna do that?" "Because we have this past, Dad." "It's complicated." "Dad, why did you ask Harrison to step outside?" "Because I thought you might have something to tell me that you wouldn't want him to hear." "I have to go." "Tru." " He thinks I'm guilty." " No." "You just misunderstood him." "Harry, he believes that I killed someone, that I'm a murderer." "He must've had his mind made up before I even got there." "I was stupid to expect that just because he's my father..." "I'm on my own in this." "No." "Look, you are not on your own, okay?" "You got me." "I'm not some hotshot lawyer, okay?" "But I know people." "I know a guy who can get us some fake IDs, passports." "We can go to Mexico, Thailand." "Really." "I'll find a job." "You could look for an opening at a morgue." "There's dead people all over the world, right?" "I don't think it'll work, but thank you." "Come on, let's go." " Good job." " It is the humanitarian approach." "What harm could she do behind bars for the next 50 years?" "Okay, so she could save a few felons, I'll give her that." "Did you know that when the guillotine was invented the idea came from a physician?" "Dr. Guillotin." "He thought that was humanitarian too." "Well, it was quick." "You sure you can talk?" " Yeah, you okay?" " Usually, if the day were this bad I'd want do it over, except I'm doing it over already." "Burrell's body got here a half-hour ago." "Okay, you'll find a piece of paper on the body." "And there's a note scribbled." ""Varden" and "DSM3."" "Yesterday, you knew he was a psychiatrist from what you found on the body." " How's that possible?" " It must be the DSM3." "That's the Diagnostic and Statistical Manual of Disorders." "It's kind of the bible of psychiatric diagnosis." "It's weird, though, because the third edition was replaced by number four back in the '90s." "Is any of this helping you, or...?" "I wish I could say it was." "Because I have this crazy idea, I guess." "Right now, I'm so open." "Well, if you can get in contact with Burrell's body he might ask for help again." "And that way your day could restart, and you can get yourself out of this mess." "It's a long shot, but it may be the only shot we have." "The problem is that there's still way too many police here." "If I can't get to the body, maybe you could bring it to me." "Well, I find that cadavers tend to get noticed in public." "I know one place where they're old news." "The anatomy lab on campus." "It should be empty this time of day." " I'll meet you there." " Okay." "Yeah, and fleeing to Thailand was a crazy idea." "Look, I can do this on my own, but they'll be searching for my car." "Lend me yours?" "I'm driving." "Hey." "So some alone time with Mr. Gorenstein?" "Yeah, I'm just reviewing this morning's work." "What about you?" " Me?" "Same thing." " Well, pull up a stool." "Wouldn't you rather pull up a stool at the coffee shop, say?" " Are you asking me out for coffee?" " No, I'm just confused is all." "I mean, you seem like the last person in our class who'd need to review." "I mean, you're brilliant." "You graduated college early, masters in chemistry." "Teachers are already impressed with you, so take a break." "Did Avery tell you that in the two minutes it took me to get sponges this morning?" " Maybe." " Yeah." "It's not the professors I really need to convince." "Heard of the Bettendorf Clinic?" "Right up there with Mayo and Johns Hopkins." "My dad's kind of their poster child." "Some people say he's the top diagnostician in the country." " Hard to live up to." " Yeah." "No, besides from the inhumanly high standards he's kind of like Martha Stewart, you know?" "And I just keep folding the napkins wrong." "I know what you mean." "Yeah, I never know what my dad's gonna be." "Which is why there's no point in being here on a nice day trying to impress these guys." "Um..." "Tru?" "Look, I'm in trouble, and I can't explain why but I need you to trust me." "Don't tell anyone I'm here." "What are you doing?" "Tru?" "Tru, is there something I should know here?" "Tru?" "I'm Officer Gomez." "Maybe you can help me." "I'm looking for a Tru Davies." "I understand Miss Davies audits class in here?" "Yeah, in the morning she does, but it's over now." " So you're acquainted with her?" " Yeah, she's a medical student." "She's in my anatomy class." "What can you tell me?" "Um..." "May I ask why it is that you're looking for her?" "She's wanted for questioning in regard to a murder." "Oh, so you're looking for her as a witness?" "We'd just like to speak with her." "What's that smell?" "Oh, that's perfume." "I dropped a bottle." "It was a gift for my girlfriend." " What do you think?" " I think it's a little strong." "Why should I have to fork over my hard-earned taxes?" "Excuse me?" "Hey, can I help you?" "You mind if we check that?" "Wait right here." "Gomez here." "He did?" "No." "I wanna question him myself." "Make sure he doesn't leave." "I have to go." "Here's my card." "You can reach me at that number any time." "Was there something you wanted to tell me?" "Just if I hear anything, I'll call you." "Tru, what the hell is going on?" "Thank you." "Just thank you." "You did the right thing." "Hopefully, I'll be able to explain this to you someday or that I won't have to." "Tru, what are you...?" "Thanks, officer." "Mr. Davis." "It's actually just Davis." "Why don't we have a little chat." "Tru." "Tru, you gotta get back to the car." "The cops are looking for you." "You can't stay in one place too long." "Where the hell's Davis?" "He didn't show up with the body and he's not answering the phone." "May I?" "Davis, what's going on?" "It's Detective Gomez, Miss Davies." "Guess what." "You're number one on your boss's speed dial." "You need to think about turning yourself in." "You're an intelligent woman." "You're only getting yourself deeper in trouble." " And now your friends too." " Triangulating." "Detective, I didn't kill that man." "Hang up the phone." " That man was a good human being." " Tru." "I was meeting him because he told me about you." " What?" " You wanted to discuss a patient." "I was never his patient." "Tru, they can trace the call." "It's down to a 12-block radius." "Your name was in his book, you were seen arguing with him and you wanna tell me you didn't know him?" "Do yourself a favor and surrender." "Let your lawyer do the talking." "Give me that." "Look, you keep this." "I got a work phone in the car." " What are you gonna do with mine?" " I'm gonna drive away make a call and throw it in a truck." "If there's one thing I know, it's sneaking." "What about you?" "Davis got busted so there's no rewind." "All I've got is a note from the doctor." "Varden and DSM3." " So?" " Okay." "Can you do one more sneaky thing for me before you go?" "I need you to steal a book from the medical library." "No problem." "Stay here, keep an eye on our friend." "Watch in case Miss Davies returns." "You never know." "Detective, as I've told you," "I was taking Dr. Burrell's body to the security crypt." "Generally, that is where we keep high-profile cases." "I thought it was a good idea at the time because the suspect was a morgue employee." "And, yes, Miss Davies is on my speed dial." "But so is Funky Chicken and Charmed Pizza as well as three other morgue workers." "Now, you've got your body." "I have a morgue to run." "Excuse me." "Thank you so much, Harry." "I really appreciate it." "And a little bonus for you." "Tru, you watch yourself." "They're handing these out to everyone who passes the Union, okay?" "Thanks." "Hi." "Do you mind if I join you?" "No, go right ahead." "Oh, damn." "I forgot to ask for sugar." "I wanted to discuss patient-therapist confidentiality." "Oh." "Davis, as a psychologist, I'm employed by the city but I respect anything you might have shared during your evaluation." "Oh, I'm being hypothetical." "I'm writing a book." "Are you really?" "Wow." "Hands-on professional and a man of words, all in one package." "Tell me you can do laundry and we'll run away together." "Under what circumstances is it ethical to discuss a patient with the police?" "The laws vary from state to state but basically, when there's immediate threat." "Knowledge of a past crime wouldn't be enough?" "Only if it contributes to your belief that there's immediate threat." "And past crime can help predict future behavior." "You are a man of many surprises, Davis." " What's the book about?" " Hmm?" " Hello?" " It's me." "Harrison gave me your number." "He said something about putting your phone in the back of a truck?" "Oh, God, Davis, it's so good to hear your voice." "Are you all right?" "I'm fine, don't worry about me." "Did you find anything about the note?" "No." "And the word's spreading." "I don't know how much time I have left." "We still don't know what Varden means." "But if it is the killer's name, and he is a patient... he probably will be in Burrell's files." "I thought of that." "But breaking into the man's office?" "That's risky, and the police could still be there." "And if Varden's not in the files, then it was all for nothing." "Not necessarily." "I've been speaking with Carrie." "Search for someone the doctor would assume... would present an immediate danger." "Someone with a violent background." "I can't sit there all night going through files." "You're right." "We'll hope the right one is near the top." "There's one more thing." "Unless you have a background as a jewel thief..." "I don't know about, I don't see you just waltzing into a locked office." "Does the morgue still have Burrell's personal effects?" " I see where you're going with this." " Talk to you later." "I'll be right there." "Hey." "Messenger." " Package going to County Records?" " Make sure it gets there quickly." "Trust me, I got it." "You really came through today, Harry." "I'm good with emergencies." "It's that day-to-day stuff I can't hack." "Yesterday you were at a hockey game right now taking those pictures you wanted for Dad." "I'm sorry." "This is more important." "We stick together, sis." "And that way, Dad can be disappointed in both of us." "Okay." "Well, just get out of here." "I got my work phone." "If any cops even think about showing up, I'll call you." "Tru?" "Tru?" "Harrison." "Varden." "Varden." ""Donald Stuart Mitchell III."" "DSM3." ""Violent tendencies." "Narcissistic personality." "Sense of entitlement." "Possible involvement in the disappearance of girlfriend, Grace Varden. "" "Looks like my tip was right." "Detective, listen to me." "I'm holding the killer's file in my hands right now." "Donald Stuart Mitchell." "He's the patient Dr. Burrell wanted to talk to you about." "He must've found out what the doctor was gonna do." "Oh, my God." "If he was at the market early enough and saw you two talking he'll wanna kill you too." "Put the folder down." "You're not doing yourself any good." "Listen to me." "He'll wanna kill you too if he thinks you know he killed Grace Varden." "Grace Varden?" "Thanks." "I think that file's mine." " Relax, I'm not gonna kill you." " No." "Because the police would only look for another shooter." "This way I'm the end of the trail." "I go to prison for two murders." " And one a cop-killing." " I'll tell them the truth." "Feel free to tell them whatever you want." "Because without this file, it becomes a story you made up." "It's too bad the circumstances aren't a little different." "Because you... are a lot prettier than Grace was." "Jack." "He decked me." "Sorry, Harry." "There's nothing left to do." "Help me." "That's no way to handle a gallbladder, Mr. Lee." " Thank you." " Thank you." "Thank you." "Really." "But I really have to go." "Why are you taking that?" "I don't have a penknife." "Is it me, or does Tru seem a little abrupt?" " Day three?" " And two lives to save." "What I can find is that Grace Varden was a coed who disappeared three years ago." "The cops think she was a murder victim." "But there's no evidence of what became of her." "They never found her body." "And her family is devastated, but they have no idea what happened to her." "I have to make another call." "Are you clear on what I need you to do?" " Got it." " Okay." "Hey, sis." "I'm on my first solo gig for Dad." "No time." "It's a rewind day." "Listen." "Yesterday, I was arrested for murder." "You got decked by Jack." " I was what?" " Never mind." "Today, I'm gonna do things differently." "Jack's gonna do things differently too." "He'll throw a curve ball where we least expect it." "Here's what I need you to do." "Dr. Burrell." "You're here to meet Detective Gomez." "Yes." "Couldn't she make it?" "Are you with the police?" "You're in immediate danger from one of your patients." "Donald Stuart Mitchell." "We think he's here." "If you'll go with this man, he'll take you somewhere safe." "Thank you." "That was for yesterday." "I think." "Harrison, Harrison, get out of the way." "Your sister's in danger." "Whoa." "Right." "Coming from you?" "I don't think so." "You got two targets today." "You're trying to make sure that cop dies." "The killer thinks your sister is the cop." "Don't you see?" "He saw her talking to the doctor." "Your sister is the target." " Wouldn't that make you happy?" " No." "Tru didn't die yesterday." "She's not supposed to die today." "Whatever you think of me, I don't care." "But do you wanna take a chance with your sister's life?" " What are you doing?" " I'm saving your life." "Officer!" "Help!" "¡Police!" "¡Nobody move!" " Jack." " Watch this." "The detective's all mine." "Hey, that's Detective Gomez." "She's the one you want." "Drop it." "You're under arrest for the murder of Grace Varden." "I have a feeling Grace Varden's family... will finally be getting some closure." "Thank you." "Bye." "Well, it's been..." "Well, it's been confusing as usual." "I gotta get out of here." "I got a job to do." "Those incriminating photos don't take themselves." "Enjoy the hockey game." "And watch out for a sudden right hook." "I never thought I'd know anybody that was on the lam." "Yeah, it's more fun in the movies when you're with Cary Grant or Matt Damon." "Yeah." "Well, you just had Harrison and me." "Oh, Davis, please." "You guys were the best accomplices a girl could want." " Hey, Davis." " Oh, hi." "I'm so absent-minded, I was in the elevator... when I realized I forgot sugar." "Oh, here." " Thanks." " Sure." " Hi." "Tru." " Hi." "Carrie." "I've seen you around." "So seen any good cadavers lately?" "Uh..." "Yeah, actually I have." "This guy came in and his entire intestinal system was..." " It was orange." " She was kidding." "I..." "I was too." "It's, like joking around and stuff." " I should get back." "Bye." " See you." " Bye." " Enjoy the sugar." "Enjoy the sugar." "God." "My complete ineptitude." "Jeez." " You like her." " No, I don't." "Yeah, maybe." "I don't know." " Is there a day one for this?" " Didn't go well." " Day two?" " Don't know." "I was too busy to talk." "It probably didn't go very well." "There's some kind of weird connection I have, though." "I'm probably full of myself." "Well, as a henchman in my underworld gang you were the best." "I don't know what happened to etiquette." "It's been a week." "You don't call, you don't write." "Don't people still say thanks when someone saves their lives?" "I would thank you if you hadn't been setting up Detective Gomez to die at the same time." "Then there's that "framing for murder" part." "You take things too personally, Tru." "A word of advice:" "This med school thing is a nice dream." "But you can't have both worlds." "As long as you do what you do, you don't belong in there." "My advice to you:" "Work on being a gracious loser." "Tru can tell you I was right about the metatarsals." " You're not." " She's right about the metatarsals." "It's gonna be like that?" "It's gonna be like that for the rest of the year?" "You two ganging up?" "It's not fun." "He knows I was right." |
Tag Archive: Entertainment
When looking for some of my older articles, I realized that I’ve been mentioning determination quite a lot. Although it didn’t specifically mention about self determination definition, this one article about the importance of self determination has received a lot…Read more
Celebs are rich and famous. Their life is just like my or maybe your dream too. However, some of them were actually homeless before famous like now. They have overcame some failures to be there in their position right now…Read more
Are you familiar with the name Jennifer Lawrence? She is an actress who stars in the Hunger Games movie franchise as Katniss Everdeen. Not only is she famous for her on-screen roles, she is also popular because she is resilient….Read more
We know that the film industry has a big impact to influence the people. A movie with a better quality in the story can make a good impact to influence people especially in their mindset. We all know there are…Read more
“Follow your heart and change your life for the better”. Well, that is an old good advice that I believe still apply now. That is a basic advice for those who want to succeed in their lives. It is, therefore, important…Read more
Is hero to zero what you want your life to be? Of course not, but you may know people with such experience. Just to make it clear if you are a new reader to this website, zero means a failure…Read more
In the previous article, I’ve told you about the two rising star smartphone brands which turned out to be originated from Asia. One of them even outranks Apple’s market share domination in China. I would consider it as a solid…Read more |
Q:
Убрать повторяющиеся значения
В данном куске кода должно выводить текст с рандомным числом - это он выполняет.
public void FreePlace()
{
var rnd = new Random();
rnd.Next(1, 5);
int fp = 0;
for (int i = 1; i < rnd.Next(1, 5); i++) {
fp = rnd.Next(1, 5);
Console.WriteLine($"Свободно место: №{fp}");
}
}
Единственное что не могу понять - как сделать, чтобы не выводились повторные значения.
Например, вот: выводятся дважды строки "Свободное место: #2".
Предполагаю, что требуется что-то подобное Distinct()?
A:
Попробуйте это
var rnd = new Random();
foreach (var fp in Enumerable.Range(1, 5).OrderBy(x => rnd.Next()))
Console.WriteLine($"Свободно место: №{fp}");
И не забудьте, что Random лучше созать один раз и переиспользовать.
Вывод
Свободно место: №4
Свободно место: №3
Свободно место: №5
Свободно место: №2
Свободно место: №1
|
Earlier this week, Pew released a massive report called "AI, Robotics, and the Future of Jobs." The goal of the report was to determine just how much disruption artificial intelligence programs will cause to workplaces over the next decade. The report surveyed 1,896 experts and found that, while they agreed that "automation and intelligent digital agents" will permeate our daily lives by 2025, the respondents were split over how many workers will be displaced:
Half of these experts (48%) envision a future in which robots and digital agents have displaced significant numbers of both blue- and white-collar workers—with many expressing concern that this will lead to vast increases in income inequality, masses of people who are effectively unemployable, and breakdowns in the social order.
The other half of the experts who responded to this survey (52%) expect that technology will not displace more jobs than it creates by 2025. To be sure, this group anticipates that many jobs currently performed by humans will be substantially taken over by robots or digital agents by 2025. But they have faith that human ingenuity will create new jobs, industries, and ways to make a living, just as it has been doing since the dawn of the Industrial Revolution. My favorite line, however, comes from GigaOM Research head Stowe Boyd, who writes, "Robotic sex partners will become commonplace, although the source of scorn and division, the way that critics today bemoan selfies as an indicator of all that's wrong with the world."
I have little doubt that the moment robots become alluring and safe enough to have sex with, there will be a huge market for automated sexbots. For now, the closest thing we have is this:
And therein lies one problem -- people will build sex robots, and they'll continue to grow closer and closer to some acceptable level of simulated robot sex, but for the foreseeable future, they will be little more than glorified dolls, toys, and flesh-lights, only with more moving parts and therefore a greater risk for embarrassing hospital visits.
But even assuming robots overcome the so-called "uncanny valley," which posits that humans become repulsed by robots as they look and act more and more like us, but not quite like us, some think that a proliferation of sex robots will lead to an increase in human-on-human sex, at least prostitution. NUI Galway law school professor John Danaher recently published a paper suggesting that, with all the robot displacement going on in other fields, it could lead to these displaced employees to becoming commercial sex providers. After all, it's a field where humans undoubtedly have an advantage.
My bigger worry isn't "sex robots" per se, but the "emotional partner" robots from the television series "Black Mirror." In that show, a woman's husband dies and so a company creates a robot version of him out of his digital footprint -- all the emails, tweets, and Facebook messages he sent over his lifetime. The result is... well, you'll have to see it, but it isn't what anyone would call an emotionally healthy relationship.
In any case, I'm registering Articles of Incorporation as soon as possible for "Bender," the "Tinder for robots." And if you still can't wait for sex robots to become mainstream and affordable, just turn your shower head into a girlfriend like this guy did.
Enjoy your nightmares and have a good weekend!
[Illustration by Brad Jonas for Pando] |
184 Cal.App.4th 712 (2010)
109 Cal.Rptr.3d 270
S.M., a Minor, etc., et al., Plaintiffs and Appellants,
v.
LOS ANGELES UNIFIED SCHOOL DISTRICT, Defendant and Respondent.
No. B209178.
Court of Appeals of California, Second District, Division Eight.
May 13, 2010.
*714 Orren & Orren, Tyna Thall Orren; Richard R. Reyes; and Victor Jacobovitz for Plaintiffs and Appellants.
Carlson & Messer, Jeffery J. Carlson and Edgar N. De Vera for Defendant and Respondent.
*715 OPINION
RUBIN, J.
S.M., a minor, appeals from the summary judgment entered on her action against the Los Angeles Unified School District for negligent supervision of a teacher who sexually fondled her. Because the undisputed facts show that S.M. waited too long to file the required tort claim with the school district, we affirm.
FACTS AND PROCEDURAL HISTORY
S.M. sued the Los Angeles Unified School District (the district) for negligence after she was repeatedly fondled by Michael McMurray, her fourth grade teacher at Plainview Elementary School during the 2002-2003 school year. According to S.M., McMurray would rub her leg from ankle to thigh while kneeling by her desk to answer questions about her schoolwork. This happened regularly during the school year even though she would move her legs away or tell McMurray to stop. As a result, S.M. stopped asking questions about her schoolwork to keep McMurray away.
The undisputed facts showed that the school year ended on June 30, 2003, that S.M. had a different teacher the next school year and had no contact with McMurray, and that she switched to a different school for sixth grade. S.M. testified at her deposition that she felt what McMurray was doing was wrong, and that his actions made her scared and nervous. Therefore, the district contended, her cause of action accrued no later than June 30, 2003, when the school year ended. Instead of filing a tort claim by December 30, 2003, however, she did not do so until April 12, 2005, meaning her claim was barred. The district moved for summary judgment on that basis.[1]
S.M. was one of several girls who were sexually fondled by McMurray. Acting out of embarrassment and fear they might somehow be blamed, they agreed to keep quiet and not tell their parents what had happened. McMurray was arrested October 14, 2004, when one of his victims came forward and reported the incident to the police.[2] S.M.'s mother learned of the arrest that day, and asked S.M. what she knew about it. S.M. told her mother what McMurray had done to her, and her mother filed a tort claim with the district *716 on April 12, 2005. S.M. opposed the summary judgment motion on the ground that her cause of action did not accrue until October 14, 2004, when her mother discovered what had happened. As a result, her tort claim was timely, she argued. The trial court disagreed, and entered judgment for the district.
On appeal, S.M. contends her cause of action did not accrue until her mother learned what happened. She also raises an issue not raised below: that the district is equitably estopped from asserting the statutory time limits because it created an atmosphere of fear and intimidation that delayed her from telling her mother what had happened.
STANDARD OF REVIEW
Summary judgment is granted when a moving party establishes the right to the entry of judgment as a matter of law. (Code Civ. Proc., § 437c, subd. (c).) In reviewing an order granting summary judgment, we must assume the role of the trial court and redetermine the merits of the motion. In doing so, we must strictly scrutinize the moving party's papers. The declarations of the party opposing summary judgment, however, are liberally construed to determine the existence of triable issues of fact. All doubts as to whether any material, triable issues of fact exist are to be resolved in favor of the party opposing summary judgment. While the appellate court must review a summary judgment motion by the same standards as the trial court, it must independently determine as a matter of law the construction and effect of the facts presented. (Barber v. Marina Sailing, Inc. (1995) 36 Cal.App.4th 558, 562 [42 Cal.Rptr.2d 697].)
A defendant moving for summary judgment meets its burden of showing that there is no merit to a cause of action if that party has shown that one or more elements of the cause of action cannot be established or that there is a complete defense to that cause of action. (Code Civ. Proc., § 437c, subds. (o)(2), (p)(2).) If the defendant does so, the burden shifts back to the plaintiff to show that a triable issue of fact exists as to that cause of action or defense. In doing so, the plaintiff cannot rely on the mere allegations or denial of her pleadings, "but, instead, shall set forth the specific facts showing that a triable issue of material fact exists . . . ." (Id., subd. (p)(2).) A triable issue of material fact exists "if, and only if, the evidence would allow a reasonable trier of fact to find the underlying fact in favor of the party opposing the motion in accordance with the applicable standard of proof." (Aguilar v. Atlantic Richfield Co. (2001) 25 Cal.4th 826, 850 [107 Cal.Rptr.2d 841, 24 P.3d 493], fn. omitted.)
Our first task is to identify the issues framed by the pleadings. (Lennar Northeast Partners v. Buice (1996) 49 Cal.App.4th 1576, 1582 [57 Cal.Rptr.2d *717 435].) The moving party need address only those theories actually pled and an opposition which raises new issues is no substitute for an amended pleading. (Tsemetzin v. Coast Federal Savings & Loan Assn. (1997) 57 Cal.App.4th 1334, 1342 [67 Cal.Rptr.2d 726].)
DISCUSSION
1. S.M.'s Cause of Action Accrued by June 30, 2003
(1) Under the Tort Claims Act, a person may not sue a public entity for personal injury unless he or she first presents a written claim to the entity within six months of the time her cause of action accrues, and the entity then denies the claim. (Gov. Code, §§ 911.2, 945.4.)[3] If the public entity does not give written notice that the claim has been rejected (§ 913), the plaintiff has until two years from the date her cause of action accrued to sue the entity. (§ 945.6, subd. (a); K.J. v. Arcadia Unified School Dist. (2009) 172 Cal.App.4th 1229, 1233 [92 Cal.Rptr.3d 1] (K.J.).)[4] The claim filing requirement is not merely procedural, but is instead a condition precedent to maintaining a cause of action and is therefore an element of a plaintiff's cause of action. (K.J., at p. 1238.)
(2) The accrual date for presenting a government tort claim is determined by the rules applicable to determining when any ordinary cause of action accrues. (§ 901.) That date may be postponed under the delayed discovery doctrine. (K.J., supra, 172 Cal.App.4th at p. 1233.) Under this doctrine, a cause of action does not accrue until the plaintiff discovers, or has reason to discover, the cause of action. (Fox v. Ethicon Endo-Surgery, Inc. (2005) 35 Cal.4th 797, 807 [27 Cal.Rptr.3d 661, 110 P.3d 914] (Fox).) A plaintiff has reason to discover a cause of action when he or she has reason to at least suspect a factual basis for its elements. Suspicion of one or more of the elements, coupled with knowledge of any remaining elements, will generally trigger the applicable limitations period. (Ibid.) This refers to the "generic" elements of wrongdoing, causation, and harm and does not require a hypertechnical approach. Instead, "we look to whether the plaintiffs have reason to at least suspect that a type of wrongdoing has injured them." (Ibid.)
The district contends that because S.M. testified she knew what McMurray did was wrong, her cause of action accrued no later than the end of her fourth grade school year on June 30, 2003. Because her claim was not filed with the district until nearly two years later, the district contends her action was barred.
*718 S.M.'s summary judgment opposition did not dispute the content of her deposition testimony. Relying on Curtis T. v. County of Los Angeles (2004) 123 Cal.App.4th 1405 [21 Cal.Rptr.3d 208] (Curtis T.), she argued that because of her age and inexperience, her knowledge of wrongfulness was irrelevant, and her cause of action did not accrue until October 14, 2004, when her mother learned what happened. Under that scenario, she contends, her tort claim was timely filed, and her complaint was timely because she sued within two years of the time her cause of action accrued.
To the extent S.M. contends Curtis T. holds that a minor's sexual molestation cause of action does not accrue until a parent learns of the molestation, she has misread that decision. The plaintiff in Curtis T. was placed in foster care by Los Angeles County when he became the subject of a dependency proceeding under Welfare and Institutions Code section 300. (Curtis T., supra, 123 Cal.App.4th at p. 1411.) He lived in foster care between the ages of five and eight until October 1999, when the dependency case was terminated and he was returned to his mother. In March 2003, when the plaintiff was 12, he filed a claim with the county alleging that he was sexually molested by another child while living in the foster home. When the county denied the claim, the plaintiff sued, alleging that his foster parent knew about the molestation but did nothing to stop it, and that his mother did not learn about the molestation until September 2002. (Id. at p. 1412.) The trial court sustained without leave to amend a demurrer to the complaint because the minor did not file a claim with the county within six months of the time when the molestations ended. (Id. at p. 1414.)
The Curtis T. court reversed, holding that the delayed discovery rule applied to child molestation cases, and that the plaintiff should have been granted leave to amend, if he could truthfully allege that "given his youth, ignorance, and inexperience, as well as his foster parent's alleged complicity in the abusethat he lacked a real awareness, until his mother's discovery of the alleged molestation, that what happened to him between the ages of five and eight was wrong." (Curtis T., supra, 123 Cal.App.4th at pp. 1422-1423.)
As part of its analysis, the Curtis T. court discussed Whitfield v. Roth (1974) 10 Cal.3d 874 [112 Cal.Rptr. 540, 519 P.2d 588] (Whitfield), which the plaintiff relied on to argue that his cause of action did not accrue until his mother learned about the molestations. The plaintiff in Whitfield was a girl who, between the ages of 10 and 13, was treated by both private and county-run medical facilities, and was misdiagnosed with a psychiatric condition when she in fact had a brain tumor. Surgery to remove the tumor caused a stroke that left her paralyzed and with a greatly reduced life expectancy. She first sued the various private hospitals and doctors, but, during discovery, obtained documents showing that some of her county *719 doctors suspected a brain tumor but did nothing to follow up on those suspicions. She then filed a tort claim with the county, which was denied, and amended her complaint to add the county as a defendant. At trial, the county was granted a nonsuit on the ground that the plaintiff did not timely file her tort claim. Citing two other medical malpractice cases involving injured infantsWozniak v. Peninsula Hospital (1969) 1 Cal.App.3d 716 [82 Cal.Rptr. 84] and Myers v. Stevenson (1954) 125 Cal.App.2d 399 [270 P.2d 885] (Myers)the Whitfield court said, "Where the plaintiff is a minor, it is not the knowledge or lack thereof of the minor, but the knowledge or lack thereof of the minor's parents which determines the time of accrual of the cause of action." (Whitfield, supra, 10 Cal.3d at p. 885.) Even though the plaintiff's mother knew a year before filing her claim that the plaintiff had a brain tumor and that a misdiagnosis might have occurred, it was not until she obtained the county medical records during discovery that she in fact knew the negligent cause of the daughter's injuries. Accordingly, there was sufficient evidence to avoid the nonsuit. (Id. at pp. 886-887.)
In discussing Whitfield, the Curtis T. court rejected the notion that it stated a blanket delayed discovery rule applicable to all causes of action by plaintiffs who are minors: "While there is no blanket rule for always or never applying the delayed discovery rule to minors' molestation cases, we believe the courts may equitably apply the delayed discovery rule in appropriate child molestation cases. Whitfield . . . does not offer much, if any, guidance on when the courts should apply the delayed discovery rule in contexts other than medical malpractice." (Curtis T., supra, 123 Cal.App.4th at p. 1418.) The Whitfield court's statement that the knowledge of a minor plaintiff's parent controls accrual "does not, in our view, create a blanket delayed discovery rule applicable to all causes of action where a minor is the plaintiff." (Curtis T., at p. 1418.) Because earlier decisions had already applied the delayed discovery rule to child molestation victims who sued as adults, "it is all the more reasonably possible for a 12- or 13-year-old child such as plaintiff to allege he was unaware that the acts done to him between the ages of five and eight were wrongful, particularly when he also alleges that his foster parent saw the alleged molestation but failed to stop it." (Id. at p. 1422.) Therefore, it was "reasonable to believe this minor plaintiff can amend to allege that due to his youth, ignorance, and inexperience, coupled with his foster parent's alleged complicity in the abuse, he was unaware that what was done to him was wrongful prior to his mother's discovery of the abuse." (Ibid.)
(3) In short, Curtis T. did not hold that a minor's cause of action for sex abuse accrues only when a parent learns what happened. Instead, it adopted a circumstance-heavy approach, pegged to the unique facts of each case, and *720 held that, given the right circumstances, a minor suing for sexual abuse is entitled to show that the cause of action did not accrue until a parent learned what happened or some other date after the abuse occurred. The court in V.C. v. Los Angeles Unified School Dist. (2006) 139 Cal.App.4th 499 [43 Cal.Rptr.3d 103] (V.C.), applied Curtis T. to hold that a demurrer was properly sustained to a minor's complaint for sexual molestation. The plaintiff in V.C., while between the ages of 11 and 13, was allegedly molested by her teacher. The district's demurrer to the complaint was sustained without leave to amend because the plaintiff did not file a tort claim until more than a year after the molestations ended. While the record included a psychological assessment that cast doubt on whether the plaintiff truly appreciated what had been done to her, it also showed that the plaintiff's mother had long harbored suspicions that the teacher was molesting her daughter. As a result, the plaintiff could not plead facts supporting a delayed discovery theory. Even though it was "dismayed by the result," Division Two of this court affirmed because the plaintiff failed to plead facts supporting her claim of delayed discovery. (Id. at pp. 504, 515-516.)
(4) Applying Curtis T. and V.C. here, we too are constrained to conclude that summary judgment was proper. S.M.'s complaint incorrectly alleged that the molestations occurred on October 14, 2004, and that a tort claim was filed with the district on April 12, 2005.[5] It was silent on the delayed discovery issue, making no mention of factors that might have prevented S.M. from becoming aware she had been wronged, or about her mother's discovery of what had happened. The district's summary judgment motion cited the portions of S.M.'s deposition testimony that she knew what McMurray did was wrong, that she repeatedly tried to avoid his advances, and that his conduct made her scared and nervous. This evidence, if believed, shows that S.M. knew the generic elements of her claimthat she had been injured by McMurray's wrongdoing. (Fox, supra, 35 Cal.4th at p. 807 [cause of action accrues when plaintiff at least suspects that a type of wrongdoing has injured them]; Marsha V. v. Gardner (1991) 231 Cal.App.3d 265, 272-273 [281 Cal.Rptr. 473] [a young child sexually molested against her will suffers an actual and appreciable injury at that time and would be entitled to more than nominal damages].) It also placed the burden on S.M. to raise triable fact issues that she had not actually discovered her cause of action at the time the molestation occurred. She did not, relying solely on the mistaken belief that, as a matter of law, it was her mother's knowledge that counted, not hers. If S.M. had submitted a declaration that explained her deposition testimony and cast doubt on whether she appreciated the wrongfulness of McMurray's *721 conduct"lacked a real awareness" in the words of Curtis T. (Curtis T., supra, 123 Cal.App.4th at p. 1422)or if she had submitted a declaration from a child psychologist or other expert that put her testimony in context beyond her literal words, then under Curtis T. and V.C., she might have raised a triable fact issue to support her claim of delayed discovery. She did neither.
(5) S.M. tries to avoid this result by way of Code of Civil Procedure section 340.1, which sets the limitations period for childhood sexual molestation claims. Under that statute, a plaintiff can sue an entity if it bears legal responsibility for childhood molestation committed by one of its agents or employees. If the entity was on notice that its agent posed a risk of molesting children, the plaintiff may sue up to the later of age 26 or three years after discovery that psychological injury occurring after adulthood is the result of the childhood molestation. (§ 340.1, subds. (a)(2), (b)(1), (2).) According to S.M., it is the manifestation of this adult-onset psychological injury that starts the accrual date of a cause of action for childhood molestation, and, when the molestations occurred, there was no way she could have possibly anticipated the extent or magnitude of that type of injury. We disagree that section 340.1 has any direct application here. That statute extends the time during which a victim of childhood sexual abuse may sue, but it does not alter the cause of action's accrual date, which is when the molestation occurred subject to any applicable delayed discovery. (V.C., supra, 139 Cal.App.4th at pp. 509-510.) It is the date of accrual that triggers the government tort claim filing requirement, a predicate not addressed by section 340.1.
As a final observation, we do not intend to suggest a 10 year old, or a child of any age, necessarily has a real awareness of a wrong at the moment child sexual abuse occurs, or that abused children must as a matter of law report child abuse immediately to their parents upon penalty of losing their legal claims. Even in those cases in which the child has a vague appreciation that something is "wrong" because he or she experiences fear, discomfort or other emotion often associated with sexual abuse, the child may not have the real awareness to which Curtis T. refers. Conversely, it may very well be true, as the court in Myers, supra, 125 Cal.App.2d at pages 402-403, pointed out, that a child of six years old or less "could not in the nature of things know of his injury or the cause thereof . . . ." The present case does not lend itself to the conclusion that as a matter of law a 10 year old is or is not aware that the acts done to the child were wrongful. Our holding is that in this case, like many others, this is a factual question. Here, no triable issue of fact on that point was presented.[6]
*722 2. S.M. May Not Rely on Equitable Estoppel
S.M. contends the district is equitably estopped from asserting noncompliance with the claim filing requirement because conduct by both McMurray and the school principal deterred her from coming forward earlier. A public entity may be estopped from asserting noncompliance with the statutory claim filing deadline by some affirmative act of intimidation, such as threats or violence. (V.C., supra, 139 Cal.App.4th at pp. 516-517.) To support this claim, S.M.'s appellate brief points to evidence submitted with her summary judgment opposition brief in the trial court that the principal had yelled at her mother for an unrelated matter and had dismissed or failed to follow up on reports about misconduct by McMurray. She also relies on her deposition testimony that she was afraid of reporting what happened because McMurray was a teacher.
(6) We do not dismiss the possibility that a child might perceive that authority figures such as teachers, school counselors, or principals, will present a united front to defend against the child's accusations, and might fear reprisals should she come forward with those accusations. However, there must be proof of an affirmative act of intimidation or violence that was intended to deter the child from speaking up. (V.C., supra, 139 Cal.App.4th at pp. 516-517.) Although S.M.'s evidence might show her apprehension about reporting what happened because the principal was generally hostile or appeared protective of McMurray, it does not establish an affirmative act, such as an expressed or implied threat, specifically intended to deter S.M. from coming forward and filing her claim. (See K.J., supra, 172 Cal.App.4th at p. 1240.) Regardless, the issue was not raised below, either as part of S.M.'s points and authorities or responsive separate statement. Although the district does not mention this, and instead responds to S.M.'s claim on the merits, we deem the issue waived for two reasons: (1) it was not pleaded in the complaint (Lackner v. North (2006) 135 Cal.App.4th 1188, 1201, fn. 5 [37 Cal.Rptr.3d 863]); and (2) because it was not raised below, either in the points and authorities or the opposition separate statement of disputed facts (Saville v. Sierra College (2005) 133 Cal.App.4th 857, 872-873 [36 Cal.Rptr.3d 515]; Mills v. Forestex Co. (2003) 108 Cal.App.4th 625, 640-641 [134 Cal.Rptr.2d 273]).
*723 DISPOSITION
The judgment is affirmed. Respondent shall recover its appellate costs.
Bigelow, P. J., and Grimes, J., concurred.
NOTES
[1] S.M. also had a cause of action for sexual battery against the district on a vicarious liability theory. The trial court granted summary judgment on that claim because the district could not be held liable under that theory. (John R. v. Oakland Unified School Dist. (1989) 48 Cal.3d 438, 441 [256 Cal.Rptr. 766, 769 P.2d 948].) S.M. does not challenge that ruling on appeal.
[2] S.M. has asked us to judicially notice newspaper reports that McMurray was later convicted of sexually abusing several girls and sentenced to 16 years in state prison. We decline to do so.
[3] All further undesignated section references are to the Government Code.
[4] S.M. did not sue until June 20, 2006.
[5] S.M. contends, and we agree, that the allegation was nothing more than technical error about when the molestations occurred.
[6] In apparent recognition of the dilemma faced by families of children abused by public school officials, the law has changed. For claims described in Code of Civil Procedure section 340.1 for the recovery of damages suffered due to childhood sexual abuse occurring after January 1, 2009, the tort claim presentation requirement no longer applies. (Gov. Code, § 905, subd. (m); see Historical and Statutory Notes, 32 West's Ann. Gov. Code (2010 supp.) foll. § 905, p. 154.) However, the Legislature did not see fit to include an earlier cutoff date that would have preserved S.M.'s claims, and we have no power to rewrite the statute.
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Snellville Trash& Recycling Update
The Snellville City Council recently approved Advanced Disposal as the contractor to handle trash and recyclable pickup for city residents and businesses, beginning July 1st. Of the four companies submitting bids, Advanced was the lowest bidder that met city requirements, and was given a two-year contract.
For city residents, the biggest changes will be the name on the refuse carts and recycling bins, and new guidelines for acceptable recyclable items. Advanced Disposal will deliver new 65-gallon trash carts and 18-gallon recycling bins (the same sizes as those currently in use) prior to July 1st; 65-gallon recycling carts are available on request, at no additional charge. However, the existing carts and bins will be used throughout the month of June. On the last pick-up day of the month, the current trash hauler will collect the bins and carts when they pick up trash. The new carts and bins from Advanced Disposal should not be taken to the curb until after July 1st. (Advanced has subcontracted with Latham Home Sanitation to handle residential service.)
On occasions when a 65-gallon cart isn’t sufficient to hold the week’s trash, residents can still purchase the infamous blue bags, each of which holds approximately 33 gallons. And as is currently the situation, large bulky items will be hauled off if three bags are attached to each item. Blue bags are available at City Hall, Public Works, Kroger and Publix supermarkets.
The new recycling guidelines are not the result of the new contract, but because of changes within the recycling industry. The biggest change will be that glass will no longer be accepted in curbside recycling bins or carts. For a variety of reasons, the sorting facilities to which mixed recyclables are taken have reduced or eliminated the capability to separate glass from cardboard, plastic, newspapers, magazines and metal cans. It’s not just Snellville that will be eliminating glass as an acceptable curbside recyclable- virtually all cities will be following suit, if they haven’t already.
As with a host of other items, glass will still be accepted at the Snellville recycling center, which maintains separate bins for specific types of items, thereby eliminating the need for sorting. The recycling center also accepts cardboard, yard trimmings, steel and aluminum and a variety of items classified as “junk”.
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Good information, thank you! I like that plastics #3-#7 will be accepted. For people who may not know, those are the numbers inside the triangular shape on the bottom of most plastic items. Just look at the numbers.
Is Advanced Disposal the company Brett Harrell owns or did own? Dropping Robertson’s & going with the company we’ve most recently had was the first thing over which I’ve ever disagreed with Tom Witts…but I could understand why a less expensive contract with a company with new trucks would be attractive. It was worth a shot to see how they would do. The problem with the one person truck was, in my opinion, that there wasn’t a 2nd employee to make sure the person’s job was done well, and if pieces of trash blew down the street, there was nobody to do anything about it. Sure we have litterers too, but the amount of waste that ended up blowing around & eventually getting to our creeks & rivers was definitely increased.
Dave, please check out RecycleBank.org – it is free & educates people about recycling & people can earn reward points for recycling, that they can redeem for all kinds of great stuff – and this does NOT cost citizens, the city, or our hauler anything!
I think kids with parental guidance would especially get more interested in putting out recycling to earn rewards. It increases the amount of recycling!
Brett Harrell never owned Advanced Disposal. He worked for the company for a time but I don’t believe he does any more.
It was never Tom Witts’ idea to switch trash hauling companies. The city contracts with private companies to provide trash service. When a contract expires, it has to be put out for bid again and in most cases, the contract is awarded to the lowest bidder. (There are exceptions if it can be demonstrated that the low bidder can’t or is highly unlikely to be able to provide the required service level.) The entire Council votes on the awarding of the contract. Good idea bout recyclebank.org. |
/* crypto/cryptall.h */
/* Copyright (C) 1995-1998 Eric Young (eay@cryptsoft.com)
* All rights reserved.
*
* This package is an SSL implementation written
* by Eric Young (eay@cryptsoft.com).
* The implementation was written so as to conform with Netscapes SSL.
*
* This library is free for commercial and non-commercial use as long as
* the following conditions are aheared to. The following conditions
* apply to all code found in this distribution, be it the RC4, RSA,
* lhash, DES, etc., code; not just the SSL code. The SSL documentation
* included with this distribution is covered by the same copyright terms
* except that the holder is Tim Hudson (tjh@cryptsoft.com).
*
* Copyright remains Eric Young's, and as such any Copyright notices in
* the code are not to be removed.
* If this package is used in a product, Eric Young should be given attribution
* as the author of the parts of the library used.
* This can be in the form of a textual message at program startup or
* in documentation (online or textual) provided with the package.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* are met:
* 1. Redistributions of source code must retain the copyright
* notice, this list of conditions and the following disclaimer.
* 2. Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in the
* documentation and/or other materials provided with the distribution.
* 3. All advertising materials mentioning features or use of this software
* must display the following acknowledgement:
* "This product includes cryptographic software written by
* Eric Young (eay@cryptsoft.com)"
* The word 'cryptographic' can be left out if the rouines from the library
* being used are not cryptographic related :-).
* 4. If you include any Windows specific code (or a derivative thereof) from
* the apps directory (application code) you must include an acknowledgement:
* "This product includes software written by Tim Hudson (tjh@cryptsoft.com)"
*
* THIS SOFTWARE IS PROVIDED BY ERIC YOUNG ``AS IS'' AND
* ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
* ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
* FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
* DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
* OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
* LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
* OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
* SUCH DAMAGE.
*
* The licence and distribution terms for any publically available version or
* derivative of this code cannot be changed. i.e. this code cannot simply be
* copied and put under another distribution licence
* [including the GNU Public Licence.]
*/
#ifndef HEADER_CRYPTOALL_H
#define HEADER_CRYPTOALL_H
#include "buffer.h"
#include "stack.h"
#include "lhash.h"
#include "err.h"
#ifdef NO_MD2
#include <md2.h>
#else
#include "md2.h"
#endif
#ifdef NO_MD5
#include <md5.h>
#else
#include "md5.h"
#endif
#include "sha.h"
#ifdef NO_DES
#ifndef GEOS_CLIENT
#include <des.h>
#endif
#else
#include "des.h"
#endif
#include "rc2.h"
#include "rc4.h"
#include "idea.h"
#include "bn.h"
#ifndef NO_DH
#include "dh.h"
#endif
#include "rsa.h"
#include "dsa.h"
#include "rand.h"
#ifndef NO_CONF
#include "conf.h"
#endif
#ifndef NO_TXT_DB
#include "txt_db.h"
#endif
#include "err.h"
#include "evp.h"
#include "meth.h"
#include "x509.h"
#include "pkcs7.h"
#include "pem.h"
#include "asn1.h"
#include "objects.h"
#include "crypto.h"
#endif
|
Q:
ask underlying papers of MEAN SHIFT, OPTICAL FLOW, KALMAN FILTER
I need 3 underlying papers / most top tree in regard to MEAN SHIFT, OPTICAL FLOW, KALMAN FILTER.
I've searched in ieee xplore, it showed many related papers.
Any idea?
Thanks in advance.
A:
Do you know about CiteSeerX?
For Mean Shift I get Mean shift: A robust approach toward feature space analysis, which is a very good paper on that topic.
For the other topics I cannot help you, but you generally find good papers by reading papers and looking at the references.
|
About 600 people attended the service at Glenormiston College, which was originally part of the land owned by the pioneering Black family.
Mr Black, 77, died on November 4 when his ute rolled down a steep embankment on his Mount Noorat property.
Duncan Morris, who served on Terang’s DemoDAIRY with Mr Black, said he was a champion of the dairy cooperative movement, helping found the Noorat Artificial Breeders Co-operative and the DemoDAIRY and was active in Co-operatives Victoria.
When Mr Black was recently listed on the Great South West Dairy Industry honour board, he made a call for more support for co-operatives and expressed concern at the rise of investor-focused multi-national dairy companies that sought to pay milk suppliers the lowest price.
Mr Black was concerned the shift away from co-operatives, which focused on suppliers’ interests, was allowing dairy profits to be taken overseas, Mr Morris said.
Paul Ford, who worked with Mr Black on the Bonlac Foods dairy co-operative, said Mr Black made a massive contribution to the leadership of the “dairy value chain.”
Mr Black was a humble man who was known for the “quiet way he could make things happen” such as his advocacy with his late wife Josie to establish the South West Community Leadership Program, the audience heard.
“Niel didn’t subscribe to the concept that he was born to rule but rather built his technical proficiency in dairy by attending Melbourne University and Dookie Agricultural College,” Mr Ford said.
“He then travelled to the USA to study dairying.
“The building of diverse networks quickly followed so that rather than just regurgitate facts, he could draw together threads from a range of sources to create insights that meant for Niel, one plus one defied the laws of mathematics and equalled three or five.”
While Mr Black’s work in the Victorian dairy industry gave him a public profile, it was his contribution to his family about which many of eulogists spoke.
Mr Black’s cousin, Maggie Black, from the UK, said he had great goodwill towards everyone.
“He was always positive, he was a great listener,” she said.
Mr Black’s stepsons said he was a good father to them after they moved to Mount Noorat from Melbourne in their teens following his marriage to their mother Josie.
Marcus Hunt said Mr Black was his moral compass.
“I didn’t want to be like him but I wanted a pinch of his generosity and a spoonful of his dedication,” he said.
Tara Reid, a daughter of Mr Black’s second wife, Eve, said Mr Black was an “interesting and interested man”. |
Students protest Bernie Sanders at Purdue
Here's why these Purdue students are protesting Senator Bernie Sanders, of Vermont, on his presidential campaign stop on Wednesday. |
define(["sugar-web/graphics/palette"], function (palette) {
'use strict';
describe("palette", function () {
it("should start down", function () {
var invoker = document.createElement('button');
var myPalette = new palette.Palette(invoker);
expect(myPalette.isDown()).toBe(true);
});
it("should toggle", function () {
var invoker = document.createElement('button');
var myPalette = new palette.Palette(invoker);
myPalette.toggle();
expect(myPalette.isDown()).toBe(false);
myPalette.toggle();
expect(myPalette.isDown()).toBe(true);
});
it("if one palette in a group popups, the others popdown", function () {
var invokerA = document.createElement('button');
var invokerB = document.createElement('button');
var myPaletteA = new palette.Palette(invokerA);
var myPaletteB = new palette.Palette(invokerB);
myPaletteA.toggle();
expect(myPaletteA.isDown()).toBe(false);
expect(myPaletteB.isDown()).toBe(true);
myPaletteB.toggle();
expect(myPaletteA.isDown()).toBe(true);
expect(myPaletteB.isDown()).toBe(false);
});
});
});
|
Lemon Pie Popsicles
Transform pie into a frozen treat with cool and refreshing Lemon Pie Popsicles! McCormick Pure Lemon Extract helps to pump up the lemony flavor of the Greek yogurt base. Sprinkle pops with crushed shortbread cookies and freeze to set. What a tasty summer treat!Recipe and photo courtesy of Dorothy Kern of Crazy for Crust. |
Q:
Get Formatted Value of Cell in Excel
How do you get the formatted value of a cell when using =CONCATENATE(E1, " - " , D1)?
E1 = 08/21/2014 8:00 PM EST (Formatted value = 08/21)
D1 = Task Item 1
Wanted output: = 08/21 - Task Item 1
A:
Use the TEXT() function:
TEXT(value, format_text)
So if the value is 23.5 and you pass =TEXT(A1, "$0.00") it will return $23.50
Source: http://office.microsoft.com/en-us/excel-help/text-function-HP010062580.aspx
|
Effect of chronically induced thermotolerance on thermosensitization in CHO cells.
Chronic thermotolerance was induced in Chinese hamster ovary (CHO) cells by pretreatment at 40 degrees C for various times ranging from 15 min to 16 h. The thermotolerant cells were either exposed to single heat treatments at 43 degrees C or subjected to step-down heating consisting of a priming treatment at 43 degrees C for 90 min immediately followed by a graded test treatment at 40 degrees C. The results showed that chronic thermotolerance affected the thermal sensitivity of step-down-heated CHO cells in two ways: by lowering the effectiveness of the priming treatment at 43 degrees C and by reducing the response to the test treatment at 40 degrees C. The effect on the priming treatment corresponds to a reduction in the effective heating time, i.e. the thermotolerant cells respond as if they were exposed to 43 degrees C for times shorter than 90 min. It was further shown that, for a given conditioning treatment, the effectiveness of both the priming and the test treatment was reduced by the same factor; the thermotolerance ratios determined for 43 degrees C and 40 degrees C showed an identical dependence on the duration of the thermotolerance-inducing conditioning treatment. Since thermotolerance development did not reverse heat sensitization by step-down heating, it is concluded that thermotolerance and thermosensitization are distinct phenomena which act independently. |
Forces loyal to Libyan renegade general Khalifa Haftar have said that they had taken one of the last remaining strongholds of an al-Qaeda-linked group in the eastern city of Benghazi.
Haftar's self-declared Libyan National Army (LNA) "liberated all of Qanfouda", an area 15km west of the centre of Benghazi, spokesman Colonel Ahmed al-Mesmari posted on Facebook on Wednesday.
Two other LNA officials confirmed to AFP news agency that Qanfouda, the scene of fierce fighting since June against Ansar al-Sharia, had fallen.
In Photos: Libyan forces retake Sirte from ISIL
Jamal al-Zahawi, an LNA commander, told broadcaster Libya Channel that his forces have freed more than 60 people from captivity, following the fighting.
No casualty numbers were provided in the latest fighting.
Another official, however, said there were still pockets of resistance from Ansar al-Sharia, which is linked to al-Qaeda in the Islamic Maghreb, in the districts of Al-Saberi and Souq al-Hout.
Regaining Qanfouda is part of a military campaign launched in mid-2014 under Haftar, who repeatedly said his aim is "to drive radical Islamists" from Benghazi.
Benghazi, Libya's second largest city, was the birthplace of the 2011 armed revolt that toppled longtime Libyan leader Muammar Gaddafi.
But Libya descended into chaos after Gaddafi fell, with several armed groups, including the Islamic State of Iraq and the Levant, gaining a foothold in the north African country.
The UN-backed Government of National Accord based in the capital, Tripoli, has failed to assert its authority over the country, while a rival group backed by Haftar runs a separate administration from the eastern city of Tobruk. |
Rory McIlroy will not compete in the upcoming Rio Olympics - with the risk of the Zika virus one he is “unwilling to take”.
The World number four golfer joins a growing list of golfers including Vijay Singh, Marc Leishman, Adam Scott, Louis Oosthuizen and Charl Schwartzel who have already said they will not feature in Rio.
On Wednesday morning, McIlroy released the following statement:
“After much thought and deliberation, I have decided to withdraw my name from consideration for this summer’s Olympic Games in Rio de Janeiro.
“After speaking with those closest to me, I’ve come to realise that my health and my family’s health comes before anything else. Even though the risk of infection from the Zika virus is considered low, it is a risk nonetheless and a risk I am unwilling to take.
“I trust the Irish people will understand my decision. The unwavering support I receive every time I compete in a golf tournament at home or abroad means the world to me.
“I will continue to endeavour to make my fans and fans of golf proud with my play on the course and my actions off it.”
Initial concerns
Last month McIlroy expressed his initial concerns over the situation in Rio, and said he would be “monitoring” developments.
The 27-year-old told reporters back then that - “as it gets closer, I am relishing the thought of going down there and competing for gold. But I have been reading a lot of reports about Zika and there have been some articles coming out saying that it might be worse than they are saying. I have to monitor that situation.”
In response to McIroy’s decision, the Olympic Council of Ireland (OCI) said it is “extremely disappointed” with the news.
“However, as we have always said, it is down to the individual and of course we respect his decision, which he has taken for personal reasons.
“Rory was set to be one of the big stars of Rio 2016, but now there is an opportunity for another Irish golfer to take up the chance to become an Olympian and participate in golf’s historic return to the Olympic Games after a 112-year absence.
‘Total confidence’
“The OCI and our medical team have taken our lead from the IOC on the Zika situation, as we do in all matters. They have provided us with every assurance, and we have total confidence that the Games will be safe for all athletes.
“We are now following the IOC’s recommendations, as well as the recommendations of the Rio 2016 organisers, the World Health Organisation and national health authorities, to ensure that Team Ireland’s athletes are kept fully updated with the latest and best advice, and that they are equipped to take all necessary precautions.”
The mosquito-borne Zika virus has been linked to causing birth defects and, in adults, has been linked to the neurological disorder Guillain-Barre. |
TEN banks slipped out from under the TARP last week, striking deals with the Treasury Department to repay taxpayer money they received last fall. Many bailout-weary investors and taxpayers welcomed this sign that government intervention in the financial sector may finally be receding.
It has been a long slog through the credit morass, and investors are understandably eager to think they are emerging onto higher ground. Still, the bad debt that was amassed by consumers and companies in recent years hasn’t been fully purged, even with the help of the Troubled Asset Relief Program. And the debt on financial companies’ balance sheets must do a lot more shrinking before we can move on from this ugly chapter in financial history.
To get a fix on how much work remains to be done, consider the substantial amount of short-term debt coming due at financial companies in the next year or two. As you absorb these figures, keep in mind that many of the entities that bought this debt when it was issued aren’t around now they’ve either left the market or are gone, casualties of the crisis.
As a result, they’re not around to step up and buy the debt again. So issuers can’t roll it over. They’ll be forced to buy back the debt, at a time when they’re already wallowing in other forms of troublesome debt and short on liquidity. |
The prospect of using alternative medical care facilities in an influenza pandemic.
Alternative care facilities (ACFs) have been widely proposed in state, local, and national pandemic preparedness plans as a way to address the expected shortage of available medical facilities during an influenza pandemic. These plans describe many types of ACFs, but their function and roles are unclear and need to be carefully considered because of the limited resources available and the reduced treatment options likely to be provided in a pandemic. Federal and state pandemic plans and the medical literature were reviewed, and models for ACFs being considered were defined and categorized. Applicability of these models to an influenza pandemic was analyzed, and recommendations are offered for future ACF use. ACFs may be best suited to function as primary triage sites, providing limited supportive care, offering alternative isolation locations to influenza patients, and serving as recovery clinics to assist in expediting the discharge of patients from hospitals. |
Blog
Mother’s Day is this Sunday and somebody better DANG WELL rise up and call us blessed. Churches have their own unique ways of doing this—some are great , some are very strange, and most are somewhere in between.
God is in the business of developing His character in us. He is not, I believe, in the business of overhauling the way He has already uniquely imprinted us with His divine image. Sanctification is not His process of cloning us into the Model Christian Woman.
Dwelling in my anger was hurting me, hindering me from seeking good, and making me irritated, indifferent, and angsty in my closest relationships. I am changing, but I’m not still not equipped to talk about moving past anger. The best I can do is talk about what I’m learning as I deal with it.
We have to talk about anger, for two reasons: For one, feeling anger is inevitable. I don’t think we’re going to suddenly stop feeling the angst that comes along with being squeezed into a box that we don’t belong in. We also need to talk about anger because we feel alone in it sometimes, and that can lead us to even more frustration. Healing often comes in realizing that we’re all dealing with the same issues, feeling the same hurt, and working towards the same goal.
When I inevitably proved not to be the kind of woman I was supposed to be over and over again, it wasn’t something wrong with what I was doing; itwas something fundamentally wrong with me as a person. I always felt like the black sheep, I was passed over a lot because there was something about me that wasn’t quite right.It sometimes felt as though calling me a “Christian woman” was as ironic as calling me a name that means “peaceful and quiet.”
I’m realizing that all the things about being a woman that used to make me feel weak have actually revealed the strength I have. This is true for all the women that I encounter every day. We consistently show up for our jobs, families, friends, and communities, often while dealing with additional and unseen physical, emotional, societal, and systemic realities that simply come along with being a woman.
Most of us realize that taking one day of rest a week is important. We may realize on a soul level that we were not designed to go hard seven days a week—and between jobs, errands, housework, church, family schedules, and even technology, that is precisely what many of us are doing.
First of all, the answer to any pregnancy ailment is to drink lots of water & eat small meals throughout the day. Me: "I think I'm pregnant with an octopus, not a human" Doctor: "Drink lots of water & eat small meals throughout the day"
…Also, does anyone happen to have strong opinions on the book “Baby Wise”?
Most of treat our bodies like collectibles. We reach some kind of "ideal" body around our early twenties and from that point on, we live just to keep our bodies in that condition--so no one can tell that they've been taken out of the box. We constantly try to keep off the pounds. We dye, shave, pluck, and tan. We try everything to get rid of freckles, wrinkles, stretch marks, acne.
It's really easy to pull out the "context" card when you run into a difficult passage, or one that is not your favorite. You could pull out some sort of context argument for every chapter and verse of Scripture, arguing that the Bible never actually says what it means. Which can lead us to ask: Is context just a cop out? Is it just an excuse for ignoring parts of the Bible we don't like?
When we eliminate 50% of our possible leaders, teachers, writers, counselors, and speakers from the majority of our church ministries, we are hurting ourselves. We are pushing men to step into roles that just don't fit them and concentrating the skills of women into a very small amount of ministries.
I've always felt that talk of hospitality is akin to taking an Emily Post course where you should get the cleanness of your house, the decor, the food, etc, all perfect. And then the whole Spiritual aspect is tied into everything, so if things aren't perfect, it kind of makes you not as good of a Christian woman.
Too often the Christian circle wants to talk about where we should land on the feminist spectrum. Around whether Jesus was a feminist or not and what those hard female Biblical passages mean. Maybe the best question isn't what word we pin to our vests, but what we plan to do with those beliefs.
1 Timothy 6:10, the verse about the love of money being the root of all kinds of evil, is always assigned to rich people. It's a rich people verse because all of those rich people need to know that they better not get too attached to their massive wealth. But the truth is, you don't have to have money to love money.
In trying to be thrifty, I've got to be careful that my motivation isn't just to have money. Am I being wise with my money so I can be generous, debt-free, and prepared for the future, or just so that I can have more wealth and possessions? Being frugal is often seen as virtuous, and it often is, but I have to look past appearances and into my own heart and ask myself what my motivations are.
When you're single, sermons, books, and Bible studies are focused on telling you that only the love of Jesus can fulfill you. When you're married, the same sources are focused on teaching how a husband and wife can love each other in a fulfilling way. For all our talk when we were single about how no person could fill the place of God in our lives, we sure started acting like it was possible when we got married.
Politicians, businesses and teachers readily disappoint you, and sometimes Pastors, Deacons, and Seminary Professors add to the burden. Purposefully and pointedly, beyond simply having a sinful moment, they take that swing that leaves you staggering. And it just hurts.
So where's the line? That's what we always want to know. Fingertip length? Cover the shoulders? One piece swimsuits? No cleavage ever? Where is the line between abaya and hooker? Because all of us have an unspoken line somewhere between the two and deep down we want an actual rulebook to give validity to what we've always thought.
I replace prayer with guilt over the fact that I haven't prayed. When I do pray for something, I feel like one time is not good enough. This guilt is toxic to what my communication with God could be right now. I don't feel the freedom to just come to Him and talk...or listen. I feel like when I come to Him, I have a backlog of 10 year's worth of prayers. |
The unstoppable cultural phenomenon that is craft beer has just ratcheted up even more mainstream support.
Supermarket giant Asda has agreed a listings deal with 13 breweries to put 25 new craft beers on its shelves in Scotland, in a deal worth more than £850,000.
An industry source told City A.M. that if sales in Scotland are robust and the demand is strong, the distribution deal could be rolled out across the rest of the UK.
As a result of the deal, Asda will become the largest single retailer of craft beers and ales in Scotland, where there are more than 100 craft breweries, stocking a total of 75 Scottish beers, ales and ciders at its 61 stores, the company said in a statement.
Read more: Try not to panic: Craft beer prices have been threatened by hop shortages
Brian O’Shea, Asda’s regional buying manager for Scotland, said: "The craft beer culture continues to grow, and it’s clear customers are trading up to more premium beers, particularly ones which come from local brands.
"The new lines we have launched are all premium quality and will give customers a new more local choice, with a variety of flavours to suit every palate and occasion."
Asda will work with Isle of Arran, Isle of Skye, Loch Ness and Kelburn breweries, while the supermarket has also joined forces with wholesaler Craft Beer Clan to recruit the nine additional craft beer lines from: Deeside Brewery, Eden Mill Brewery, Jaw Brewery, Knops Brewery, Lerwick Brewery, Stewart Brewing, Loch Lomond Brewing, Wooha Brewing and WEST Brewery.
"Scottish provenance and taste is important, but what makes our lager and ales stand out from the rest is that we put a modern twist on traditional brewing by using scientific knowledge of microbiology to create modern hop strains, alongside state-of-the-art equipment," Heather McDonald, head brewster and owner of WooHa Brewing Company, said.
Read more: London toasts boom in craft beer start-ups
Snapped up
Despite the appeal of craft beer companies as independent, upstart brewers, in reality there has been a recent slew of takeovers from drinks giants.
In December, the world's largest brewer, Anheuser-Busch InBev bought the London craft beer producer Camden Town Brewery, while SABMiller snapped up fellow London craft brewer Meantime in May last year.
Last week, AB InBev also gulped down JD Wetherspoon's craft beer favourite – US-based Devils Backbone Brewing.
To protect against the move to merge and acquire popular craft beer producers, Scottish craft heavyweight BrewDog last week voted to change its constitution to specifically protect itself against a buyout by a "monolithic purveyor of industrial beer". |
Arrrrgh ya ready to be the finest lass in all the seven seas? Ye may think ya are, but we think you’re just missing one thing. This Women’s Plus Size Wicked Wench Costume is just what you need to start your endless reign over the sea!
If you want to sail off into the horizon on a grand adventure this Halloween this is the costume for you. Once you toss on this sassy, yet elegant, dress you’ll have scurvy seadogs on their hands and knees begging to join your crew. From the top of the neckline to the bottom of the hemline of the skirt, this dress just screams out Pirate Queen. Look through our pirate accessories and pick out the cutest captain’s hat you can find along with a pair of high heel boots that say you mean business. Then you just need to arm yourself with a trusty sword and a pistol to get you out a tight situation in a pinch. If you want to live the life of a pirate then you’ve got to expect trouble around every corner and beyond every wave. But with this amazing dress and the right accessories, we know that you will prove to be victorious when faced with any challenge.
If you’re ready to live life on the edge this Halloween then get your hands on this Women’s Plus Size Wicked Wench Costume. You’ll be a whole new face of fear. Only the unwisest of pirates will cross sails with you! |
Björke Church
Björke Church () is a medieval church in Björke on the Swedish island of Gotland, in the Diocese of Visby.
The nave is the oldest part of Björke Church, dating from the mid-13th century. The choir dates from the middle of the 14th century and replaced an earlier choir, half the size of the present one. A church tower was planned but never built. The sacristy is the last part to be added the church; it dates from 1860.
Internally, the church ceiling is supported by six vaults – two in the choir and four, resting on a central column, in the nave. Among the furnishings, the crucifix dates from 1160 and the pulpit from 1594; it is one of the oldest on Gotland. Of later date are the altarpiece (1911) and the votive ship, donated to the church in 2004.
The church underwent a renovation in 1910–12.
Björke Church belongs to the Church of Sweden and lies within the Diocese of Visby.
References
External links
Category:Gothic architecture in Sweden
Category:Churches in Gotland County
Category:Churches in the Diocese of Visby
Category:Churches converted from the Roman Catholic Church to the Church of Sweden |
[Recent advances in the treatment of advanced breast cancer: therapeutic modalities at the National Cancer Center Hospital].
A retrospective analysis was performed comparing the survival in 483 patients with advanced breast cancer during the period from 1962 to 1986. Recently, the five-year survival rate at the start of therapy to the primary tumor was 63% and the five-year survival rate after the recurrence was 25%. The survival has been significantly prolonged as opposed to the figure of less than 5% among patients of 15 years ago. Multidisciplinary treatment has been utilized as adjuvant and recurrent therapies, such as combined chemotherapy, chemo-endocrine therapy and irradiation. Within these patients, 13 cases (3%) survived for more than ten-years from recurrence. The patients were recurrent cases with a small amount of metastatic sites and responsive to long-term endocrine therapy. |
Admissions & Aid
Whether you are attending college for the first time or are changing careers, no other
institution offers you the advantages of a Collin College education. As you will see,
Collin College is among the finest in the nation.
Resources
Whether you are a current student or a prospective student we hope you will take advantage
of the many resources that are available within the Student and Enrollment Services
Division. Our goal is to support students in achieving their academic and career goals.
About
Since offering its first classes at area high schools in 1985, Collin College has
expanded to serve about 55,000 credit and continuing education students each year.
The only public college based in the county, the college offers more than 100 degrees
and certificates in a wide range of disciplines.
Business
If you are interested in a career in business or plan to pursue a bachelor’s degree
in accounting, business administration, finance, international business, management
or marketing, the business field of study (FOS) curriculum at Collin College is a
great starting point.
A Field of Study is a set of courses that will transfer and apply to a corresponding
bachelor’s - level degree at a Texas college or university. Both the Field of Study
and the Core Curriculum courses are transferrable for full academic credit to any
public college or university in Texas. Students who complete the Field of Study block
of courses will earn a certificate in that Field of Study.
Did you know that you can also earn an AA or AS degree in addition to the Field of
Study certificate? Contact an academic advisor today to learn more about how to develop
an educational plan that is right for you. |
\#1[[$\backslash$\#1]{}]{}
Introduction
============
Orbital ordering and concomitant Jahn-Teller (JT) distortions are observed in some perovskite-type 3$d$ transition-metal compounds such as KCuF$_3$ and LaMnO$_3$. [@KK; @KCuF3; @LaMnO3; @LaMnO3n] In the perovskite-type lattice, there are two possible JT distortions depending on the stacking of the elongated octahedra along the $c$-axis as shown in Fig. \[distortion\](a). [@KCuF3] In the $d$-type JT distortion, the elongated axes of the octahedra are parallel along the $c$-axis. On the other hand, the elongated axes are rotated by 90$^\circ$ along the $c$-axis in the $a$-type JT distortion. While LaMnO$_3$ ($d^4$), YVO$_3$ ($d^2$) and YTiO$_3$ ($d^1$) have the $d$-type JT distortion, [@LaMnO3; @YVO3; @YTiO3] LaVO$_3$ ($d^2$) has the $a$-type JT distortion. [@LaVO3] In KCuF$_3$, both the $d$-type and the $a$-type JT distortions are observed. [@KCuF3] Hartree-Fock calculations which consider the hybridization between the transition-metal 3$d$ and oxygen 2$p$ orbitals predict that the orbital ordered state compatible with the $a$-type JT distortion is lower in energy than that with the $d$-type JT distortion for $d^1$ and $d^2$ systems and that the two states are degenerate for $d^4$ and $d^9$ systems. [@Mizokawa] Therefore, one cannot explain why the $d$-type JT distortion is realized in LaMnO$_3$, YVO$_3$ and YTiO$_3$ by considering the energy gain due to orbital ordering alone.
Perovskite-type $AB$O$_3$ compounds with relatively small $A$-site ions undergo the GdFeO$_3$-type distortion which is caused by tilting of $B$O$_6$ octahedra as shown in Fig. \[distortion\](b). [@GdFeO3] While LaMnO$_3$, YVO$_3$, LaVO$_3$ and YTiO$_3$ are accompanied by the GdFeO$_3$-type distortion, KCuF$_3$ has no GdFeO$_3$-type distortion. In addition, the magnitude of the GdFeO$_3$-type distortion becomes larger in going from LaVO$_3$ and LaMnO$_3$ to YVO$_3$ and YTiO$_3$. Here, one can notice that the compounds with the larger GdFeO$_3$-type distortion tend to have the $d$-type JT distortion. It has been pointed out by Goodenough that the covalency between the $A$-site and oxygen ions ($A$-O covalency) is important in the GdFeO$_3$-type distortion. [@covalency] In this paper, we have studied the relationship between the GdFeO$_3$-type and JT distortions considering the $A$-O covalency and explored the reason why orbital ordering compatible with the $d$-type JT distortion are favored in LaMnO$_3$, YVO$_3$ and YTiO$_3$ in terms of the interaction between the two distortions.
Method of calculation
=====================
We have employed lattice models for the perovskite-type structure in which the transition-metal 3$d$, the oxygen 2$p$ and the $A$-site cation $d$ orbitals are included. The on-site Coulomb interaction between the transition-metal 3$d$ orbitals, which is essential to make the system insulating and to cause orbital ordering, are expressed using Kanamori parameters $u$, $u'$, $j$ and $j'$. [@Kanamori] The charge-transfer energy $\Delta$ is defined by $\epsilon^{0}_{d}-\epsilon_{p}+nU$, where $\epsilon^{0}_{d}$ and $\epsilon_{p}$ are the energies of the bare transition-metal 3$d$ and oxygen 2$p$ orbitals and $U$ (=$u$-20/9$j$) is the multiplet averaged $d$-$d$ Coulomb interaction energy. The hybridization between the transition-metal 3$d$ and oxygen 2$p$ orbitals is expressed by Slater-Koster parameters ($pd\sigma$) and ($pd\pi$). The ratio ($pd\sigma$)/($pd\pi$) is fixed at -2.16. [@Mat] $\Delta$, $U$, and ($pd\sigma$) can be deduced from cluster-model analysis of photoemission spectra. [@Mizokawa] Although the error bars of these parameters estimated from photoemission spectra are not so small \[$\sim \pm$1 eV for $\Delta$ and $U$ and $\sim \pm$0.2 eV for $(pd\sigma)$\], the conclusions obtained in the present calculations are not changed if the parameters are varied within the error bars.
In the present model, unoccupied $d$ orbitals of the $A$-site cation such as Y 4$d$ and La 5$d$ are taken into account. The hybridization term between the oxygen 2$p$ orbitals and $A$-site cation $d$ orbitals is expressed by $(pd\sigma)_A$ and $(pd\pi)_A$. The ratio $(pd\sigma)_A$/$(pd\pi)_A$ is also fixed at -2.16. The hybridization term between the oxygen 2$p$ orbitals is given by ($pp\sigma$) and ($pp\pi$) and the ratio ($pp\sigma$)/($pp\pi$) is fixed at -4. [@Mat] It is assumed that the transfer integrals $(pd\sigma)$ and $(pd\sigma)_A$ are proportional to $d^{-3.5}$ and $(pp\sigma)$ is to $d^{-2}$, where $d$ is the bond length. [@Mat] Without the JT and GdFeO$_3$-type distortions, the bond length between two neighboring oxygens and that between the oxygen and the $A$-site cation are $\sqrt{2} a$, where $a$ is the bond length between the transition-metal ion and the oxygen. ($pp\sigma$) and $(pd\sigma)_A$ for bond length of $\sqrt{2} a$ are assumed to be -0.60 and -1.0 eV, respectively.
In the GdFeO$_3$-type distortion, the four octahedra in the unit cell are rotated by angle of $\omega$ around the axes in the (0,1,1) plane in terms of the orthorombic unit cell. Here, we model the GdFeO$_3$-type distortion by rotating the octahedra around the (0,1,0)-axis or the $b$-axis \[see Fig. \[distortion\](b)\]. The subsequent small rotation around the $a$-axis is required to retain the corner-sharing network of the octahedra. The magnitude of the GdFeO$_3$-type distortion is expressed by the tilting angle $\omega$ around the $b$-axis. It is important that the $A$-site ions are shifted approximately along the $b$-axis to decrease the distance from the $A$-site ion to the three closest oxygen ions and increase the distance to the three next closest oxygen ions as shown in Fig. \[distortion\](b). Here, it is assumed that the shift is along the ($\pm$1/8,7/8,0)-direction. The magnitude of the shift is proportional to the tilting angle and is assumed to be $\sim$ 0.05$a$, 0.1$a$ and 0.15$a$ for the tilting angles of 5, 10 and 15$^\circ$, respectively, which are realistic values for the compounds studied in the present work. [@LaMnO3; @LaVO3] As for the Jahn-Teller distortion, it is assumed that the longest bond is by 0.1$a$ longer than the shortest bond which is reasonable for LaMnO$_3$ and is relatively large for LaVO$_3$ and YTiO$_3$. [@LaMnO3; @LaVO3; @YTiO3]
Results and Discussion
======================
LaMnO$_3$
---------
In the high-spin $d^4$ system, in which one of the $e_g$ orbitals is occupied at each site, the $A$-type antiferromagnetic (AFM) states with $3x^2-r^2$/$3y^2-r^2$-type orbital ordering with considerable mixture of $3z^2-r^2$ are predicted to be stable by theoretical calculations [@KK; @Mizokawa; @Theory] and are studied by x-ray and neutron diffraction measurements. [@LaMnO3n] Here, the $z$-direction is along the $c$-axis. The amount of the $3z^2-r^2$ component decreases with the JT distortion. [@Mizokawa] Different ways of stacking the orbitals along the $c$-axis give two types of orbital ordering: the one compatible with the $d$-type JT distortion and the other with the $a$-type JT distortion. These two types of orbital ordering are illustrated in Fig. \[orbital\]. While, in the orbital ordering of the $a$-type, the sites 1, 2, 3, and 4 are occupied by $3y^2-r^2$, $3x^2-r^2$, $3x^2-r^2$, and $3y^2-r^2$ orbitals, the sites 1, 2, 3, and 4 are occupied by $3y^2-r^2$, $3x^2-r^2$, $3y^2-r^2$, and $3x^2-r^2$ orbitals in the orbital ordering compatible with the $d$-type JT distortion.
In Fig. \[LaMnO3\], the energy difference between the orbital ordered states compatible with the $d$-type and $a$-type JT distortions is plotted as a function of the tilting angle of the octahedra, i.e., the magnitude of the GdFeO$_3$-type distortion. $\Delta$, $U$, and $(pd\sigma)$ are 4.0, 5.5, and -1.8 eV, respectively, for LaMnO$_3$. [@Mizokawa] Without the JT distortion and the shift of the $A$-site ion, the two states are degenerate within the accuracy of the present calculation ($\pm$ 1 meV/formula unit cell). This degeneracy is lifted when the JT distortion is included. With the JT distortion and the shift of the $A$-site ion, the orbital ordered state with the $d$-type JT distortion is lower in energy than that with the $a$-type JT distortion. If we tentatively switch off the shift of the $A$-site ion and include only the JT distortion, the orbital ordered state with the $a$-type JT distortion becomes slightly lower than that with the $d$-type JT distortion as shown in Fig. \[LaMnO3\]. Therefore, one can conclude that the shift of the $A$-site ion driven by the GdFeO$_3$-type distortion is essential to stabilize the orbital ordered state with the $d$-type Jahn-Teller distortion. [@Made]
The qualitative explanation of this behavior is as follows. In the $d$-type JT distortion, the four oxygen ions nearest to the $A$-site ion \[shaded in Fig. \[distortion\] (a) and (b)\] shift approximately in the same direction and, consequently, the system can gain the hybridization energy between the $A$-site and oxygen ions effectively. On the other hand, in the $a$-type JT distortion, the two of the four oxygen ions move in the other direction and the energy gain due to the hybridization is small compared to the $d$-type JT distortion. Another possible picture is that, in the $d$-type JT distortion, these four oxygen ions can push the $A$-site ion in the same direction since the JT distortion along the $c$-axis is in phase. On the other hand, in the case of the $a$-type JT distortion, the two oxygen ions in the upper plane push the $A$-site ion in the other direction than the two in the lower plane as shown in Fig. \[push\]. Therefore, the stronger is the GdFeO$_3$-type distortion, the more does it stabilize the $d$-type Jahn-Teller distortion and corresponding orbital ordering.
In the charge-ordered state of Pr$_{0.5}$Ca$_{0.5}$MnO$_3$, the Mn$^{3+}$ and Mn$^{4+}$ sites are arranged like a checkerboard within the $c$-plane and the same arrangement is stacked along the $c$-axis. [@Jirak] The Mn$^{3+}$ sites are accompanied by the JT distortion and the elongated axes are parallel along the $c$-axis just like the $d$-type JT distortion. Since the $A$-sites are occupied by Pr and Ca ions in Pr$_{0.5}$Ca$_{0.5}$MnO$_3$, we cannot simply apply the present model calculation to it. However, it is reasonable to speculate that the stacking along the $c$-axis in Pr$_{0.5}$Ca$_{0.5}$MnO$_3$ is also determined by the interaction between the JT distortion and the shift of the $A$-site ion in the same way as in LaMnO$_3$.
YVO$_3$
-------
In the $d^2$ system, the $C$-type AFM state in which the sites 1, 2, 3, and 4 are occupied by $xy$ and $yz$, $xy$ and $zx$, $xy$ and $zx$, and $xy$ and $yz$ orbitals and the $G$-type AFM state in which the sites 1, 2, 3, and 4 are occupied by $xy$ and $yz$, $xy$ and $zx$, $xy$ and $yz$, and $xy$ and $zx$ orbitals are competing. While the $C$-type AFM state is favored by the orbital ordering which is compatible with the $a$-type JT distortion, the $G$-type AFM state is favored by the orbital ordering of the $d$-type. The relative energy of the $G$-type AFM state with the $d$-type JT distortion to the $C$-type AFM state with the $a$-type JT distortion, $E_d-E_a$, is plotted as a function of the tilting angle in Fig. \[YVO3\]. $\Delta$, $U$, and $(pd\sigma)$ are 6.0, 4.5, and -2.2 eV, respectively, for LaVO$_3$ and YVO$_3$. [@Mizokawa] Without the GdFeO$_3$-type distortion, the $C$-type AFM state with the $a$-type JT distortion is lower in energy than the $G$-type AFM state, indicating that the energy gain due to the orbital ordering is larger in the $C$-type AFM state than in the $G$-type AFM state. The energy difference becomes smaller with the tilting or the GdFeO$_3$-type distortion. Finally, with the tilting of 15$^\circ$, the $G$-type AFM state with the $d$-type JT distortion becomes lower in energy than the $C$-type AFM state. The present calculation is in good agreement with the experimental result that the less distorted LaVO$_3$ is $C$-type AFM below 140 K [@LaVO3] and the more distorted YVO$_3$ is $G$-type AFM below 77 K. [@YVO3]
This situation is illustrated in Fig. \[YVO3s\]. When the GdFeO$_3$-type distortion is large, the interaction between the $d$-type JT distortion and the shift of the $A$-site ion, namely, the energy gain due to $A$-O covalency becomes dominant just like in LaMnO$_3$ and, consequently, the $G$-type AFM with the $d$-type JT distortion is favored. On the other hand, when the GdFeO$_3$-type distortion is small, the energy gain due to orbital ordering becomes dominant and the $a$-type JT distortion and the orbital ordering compatible with it are realized. The JT distortion may be suppressed if the system is located near the crossing point where the $a$-type and $d$-type JT distortions are almost degenerate. An interesting experimental result related to this point is that YVO$_3$ becomes $C$-type AFM between 77 K and 118 K.[@YVO3] The present model calculation suggests that, if the JT distortion is switched off, the $C$-type AFM state is favored because of the orbital ordering. [@Mizokawa] Therefore, YVO$_3$ is expected to be close to the crossing point and become $C$-type AFM when the $d$-type JT distortion is suppressed at elevated temperature. In this sense, between 77 K and 118 K, YVO$_3$ may be an ideal orbitally ordered system without JT distortion.
YTiO$_3$
--------
For the $d^1$ system, the ferromagnetic (FM) states with orbital ordering are favored in the model HF calculations. There are two possible orbital orderings compatible with the $a$-type and $d$-type JT distortions. The model HF calculation without the covalency between the $A$-site cation and the oxygen ion predicted that the orbital ordering of the $a$-type is lower in energy. However, the recent neutron diffraction measurement by Akimitsu [*et al.*]{} have shown that the orbital ordering compatible with the $d$-type JT distortion is realized in the FM insulator YTiO$_3$. [@YTiO3] YTiO$_3$ has the considerable GdFeO$_3$-type distortion and the tilting angle is expected to be larger than 15$^\circ$. In Fig. \[YTiO3\], the relative energy of the FM and orbital ordered state of the $d$-type to that of the $a$-type is plotted as a function of the tilting. $\Delta$, $U$, and $(pd\sigma)$ are 7.0, 4.0, and -2.2 eV, respectively, for YTiO$_3$. [@Mizokawa] With the tilting of 15$^\circ$, the orbital ordered state of the $d$-type is lower in energy, in agreement with the experimental result. In this state, the sites 1, 2, 3, and 4 are occupied by $c_1yz+c_2xy$, $c_1zx+c_2xy$, $c_1yz-c_2xy$, and $c_1zx-c_2xy$ orbitals ($c_1 \sim 0.8$ and $c_2 \sim 0.6$). However, experimentally, less distorted LaTiO$_3$ has no or very small JT distortion and has a $G$-type AFM state. [@LaTiO3] The present calculation cannot explain why the $G$-type AFM state can be stable compared to the FM state in LaTiO$_3$.
Conclusion
==========
In conclusion, we have studied the relationship between orbital ordering and the JT and GdFeO$_3$-type lattice distortions. It has been found that the covalency between the $A$-site cations and oxygen makes the $d$-type JT distortion (same orbitals along the $c$-direction) lower in energy than the $a$-type JT distortion (alternating orbitals along the $c$-direction) in the presence of the large GdFeO$_3$-type distortion. As a result, the orbital ordered states compatible with the $d$-type JT distortion are favored in LaMnO$_3$, YVO$_3$, and YTiO$_3$ which have the relatively large GdFeO$_3$-type distortion. On the other hand, in less distorted LaVO$_3$, the orbital ordering compatible with the $a$-type JT distortion is favored because of the pure superexchange effect.
Acknowledgment {#acknowledgment .unnumbered}
==============
The authors would like to thank useful discussions with K. Tomimoto, J. Akimitsu, Y. Ren and P. M. Woodward. This work was supported by the Nederlands foundation for Fundamental Research of Matter (FOM).
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H. Kawano, H. Yoshizawa, and Y. Ueda, J. Phys. Soc. Jpn. [**63**]{}, 2857 (1994).
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We have evaluated the Madelung energy of the model perovskite lattices with the JT and GdFeO$_3$-type distortions and have found that the $a$-type JT distortion is slightly lower in energy than the $d$-type. This is opposite to the effect of the covalency between the $A$-site cations and oxygens.
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|
New ideas can make strange bedfellows: The world's biggest brick-and-mortar game retailers are teaming up with the online game marketplace Steam to bring Valve's gaming hardware to their stores.
GameStop will be the exclusive brick-and-mortar retailer of Steam Machines and the Steam Controller in the US for this holiday season, with GAME UK selling them in the United Kingdom and EB Games in Canada.
Walk into participating stores on or after November 10 and you'll spy areas devoted to Steam's long-anticipated lineup of Steam-angled PC-based gaming boxes, set-top streaming devices, funky-looking game controllers and prepaid software cards.
With the rise of e-tail, traditional retail game sales have plummeted. GameStop took a stab at challenging Steam when it snatched e-tail service Impulse from Stardock in 2011, but the idea—briefly renamed GameStop App—never took, and the company shuttered the service in April 2014.
Retailers like GameStop have their PC gaming sections to tiny areas that mostly look like shrines to Blizzard. If you want access to most of the 6,000 extant PC games Valve currently stocks, it's pretty much Steam or nothing.
On the other hand: GameStop, GAME UK, and EB Games (owned by GameStop) still have thousands of physical stores when combined worldwide, and Valve's been selling Steam prepaid cards in some of these locales for years (the company claims they've been "increasing each year," though doesn't specify how much).
Valve, whose Steam-centric gaming PCs, game-streaming device Steam Link and unique touch-driven controller have been both lauded and criticized, needs a retail partner that speaks fluent enthusiast-ese. For the modern PC gamer, governed by Steam's iron grip on PC games distribution, that's clearly not the big box stores.
Whether Valve's Steam Machines can capture PC gamers' hearts and wallets is another matter. The company's been quietly touting the idea of selling Steam-focused hardware and a special controller with circular thumb-driven touch pads for years now. But anyone can build their own "Steam machine" to taste, and probably for less money.
Valve's shied away from making SteamOS (basically Debian Linux) its de facto operating system, let alone attempting to cajole other software makers into supporting it. Steam machines will run whatever operating system you care to, just like any other PC.
Valve and GameStop thus look to have an up-mountain extreme sports climb ahead, convincing a demographic that already skews do-it-yourself and bargain-hunter that paying the inexorable retail markup is worth it. |
[Effects of tourniquet on cardiac function in total knee arthroplasty with trans-esophageal echocardiography].
To employ trans-esophageal echocardiography (TEE) to observe the tourniquet's influence on cardiac function in total knee arthroplasty (TKA). Twenty ASA I-II patients undergoing TKA under general anesthesia from September 2011 to February 2012 at Department of Orthopedics, Beijing Jishuitan Hospital were selected. Tourniquet was loosened at 30 minutes after the start of TKA. And TEE was employed to observe the occurrence rate and degree of intra-cardiac embolus and the changes of cardiac function at 5, 10 and 15 min post-loosening respectively. Different degrees of "meteor-shaped" microemboli images appeared in right atriums of all patients. The "meteor shape" image was the most obvious after loosening for 5 min. It started to decrease after 10 min, persisted generally and declined further after 15 min. The left ventricular areas at end diastole (LVAd) after loosening tourniquet for 5, 10 and 15 min were (7.62 ± 0.54), (7.86 ± 0.46) and (8.55 ± 0.56) cm(2) respectively. And there were remarkable changes compared with the area (9.80 ± 0.48) cm(2) pre-loosening (P < 0.01). The fractional area change (FAC) showed obvious decrease since loosening for 5 min (P < 0.01). Massive meteor-shaped microembolis appear on TEE with weakened cardiac function. |
Q:
jquery validation rules
My phone validation depends on a checkbox (no, don't contact me via phone). If this is checked, then I will not need run the phone validation. I googled around and found 'depends'
function.
I have
$("#myForm").validate({
....
rules: {
phone1: {
required: {
depends: "!#pri_noPhone:checked"
},
number: true,
minlength:3,
}
It doesn't throw an error, but it still tries to validate the phone number.
Under the rules: how do i make sure that email and confirmEmail are the same? I have rules, and messages separate.
A:
try something like this:
"#phone1": {
number: true,
minlength:3,
required: function(element){
return ($('#pri_noPhone_wrapper input:checked').val() == 'True');
}
}
The HTML (after looking at this I forgot to add the wrapper HTML)
<span id='pri_noPhone_wrapper'>
Phone:
<input type="checkbox" name="pri_noPhone" value="what ever" />
</span>
|
|By Valerie Volcovici
|By Valerie Volcovici
|By Valerie Volcovici
|By Valerie Volcovici
|By Valerie Volcovici
WASHINGTON (Reuters) - U.S. Environmental Protection Agency chief Scott Pruitt may not survive long in his job after reports that he paid below market rate to live in a condo owned by a lobbyist who deals with issues overseen by his agency, lawmakers and a former Trump official said in television interviews on Sunday.
On Friday, ABC News and Bloomberg News reported that during Pruitt's first six months in Washington last year, he made a deal to pay about $50 a night - less than a third the price of similar properties - to rent a room in a Capitol Hill neighborhood condo building co-owned by energy industry lobbyist Steven Hart and his wife.
"I don’t know how you survive this one, and if he has to go, it’s because he never should have been there in the first place, said Chris Christie, the Republican former governor of New Jersey, on Sunday on ABC News' This Week program.
Christie was for a short time the head of President Donald Trump's transition team and has previously raised concerns that many political appointments had not been vetted for such conflicts of interest.
ABC later reported that Pruitt’s daughter also used the apartment in 2017 during her tenure as a White House summer intern, which Hart's wife said was not agreed in their lease.
Pruitt declined comment on the reports. The White House referred reporters to the EPA, which said the arrangement had been cleared by the agency's ethics officials.
Pruitt has already faced public criticism for his frequent use of first-class flights, which is under investigation by the EPA Inspector General, and installing a $43,000 secure phone booth to conduct confidential calls.
He "may be on his way out" after the latest reports, said Democratic Senator Doug Jones of Alabama later on the same ABC News show.
"I think he’s in real trouble," said Jones. "People are just frustrated with Cabinet members who seem to want to use taxpayer dollars to fund their own personal lifestyle.”
EPA spokesman Jahan Wilcox defended the arrangement in an email on Friday.
“As EPA career ethics officials stated in a memo, Administrator Pruitt’s housing arrangement for both himself and family was not a gift and the lease was consistent with federal ethics regulations,” he said.
That memo by Kevin Minoli, the designated EPA ethics official, said such arrangements were not considered "gifts" if a federal official pays market value for them.
"Under the terms of the lease, if the space was utilized for one 30-day month, then the rental cost would be $1,500, which is a reasonable market price," the memo said.
Local real estate websites show that the average market price for a similar property in the area is at least three times as much.
Travel records obtained through a public records request show that Pruitt spent over $17,000 in taxpayer money for a December trip to Morocco to promote U.S. exports of LNG. Marketing U.S. LNG is not the jurisdiction of the EPA administrator. |
Q:
How to get GoogleMap object from a SupportMapFragment created in an XML? API level 8
I'm trying to get the GoogleMap object from the SupportMapFragment object. Because I want this work on API level 8 too, solution like findFragmentById wont work for me..
I have this XML:
<?xml version="1.0" encoding="utf-8"?>
<RelativeLayout xmlns:android="http://schemas.android.com/apk/res/android"
xmlns:tools="http://schemas.android.com/tools"
android:layout_width="match_parent"
android:background="#fff"
android:id="@+id/map_view"
android:layout_height="match_parent" >
<fragment xmlns:android="http://schemas.android.com/apk/res/android"
xmlns:map="http://schemas.android.com/apk/res-auto"
android:id="@+id/map"
android:layout_width="match_parent"
android:layout_height="match_parent"
android:name="com.google.android.gms.maps.SupportMapFragment"
map:cameraZoom="11"
map:cameraTargetLat="32.1275701"
map:cameraTargetLng="34.7983432"
map:uiZoomControls="false"/>
<ImageView
android:id="@+id/button_image"
android:layout_width="match_parent"
android:layout_height="50dp"
android:gravity="bottom"
android:contentDescription="@string/back"
android:layout_alignParentBottom="true"
android:clickable="true"
android:onClick="returnToMain"
android:background="@drawable/return_button" />
</RelativeLayout>
UPDATE:
The Java code:
public void category(View v){
if(isNetworkAvailable()){
if(viewMap==null){
setContentView(R.layout.map_view);
viewMap=(RelativeLayout)findViewById(R.id.map_view);
Fragment ssmf=getFragmentManager().findFragmentById(R.id.map);
mapObject=((SupportMapFragment) getFragmentManager().findFragmentById(R.id.map)).getMap();
mapObject.addMarker(new MarkerOptions()
.position(new LatLng(32.1275701, 34.7983432))
.title("Hello world"));
}
else{
setContentView(viewMap);
}
}
}
I got this error:
Cannot cast form Fragment to SupportMapFragment.
A:
Use getSupportFragmentManager() when you use, SupportFragments.
A:
Because I want this work on API level 8 too, solution like findFragmentById wont work for me.
Yes, it will. From your FragmentActivity, after you inflate the above layout, call findFragmentById(R.id.map), cast the result to a SupportMapFragment, and call getMap() on that to retrieve your GoogleMap object.
|
Hong Kong/Singapore: The resignation of the Reserve Bank of India (RBI) governor on 11 December follows a period of government pressure on the central bank to spur economic growth, and highlights risks to the RBI’s policy priorities, says Fitch Ratings. The RBI’s efforts to address bad loan problems have the potential to improve banking-sector health over the long term and its commitment to inflation targeting has supported a more stable macroeconomic environment in recent years. Increased government influence on the central bank could undermine this progress.
The full implications of Urjit Patel’s resignation will only become clearer once there is some indication of the RBI’s policy approach under his replacement, Shaktikanta Das, an experienced government bureaucrat. The central bank’s stance may still remain unchanged. Mr Patel cited personal reasons for leaving the RBI, rather than government interference. Moreover, there was no obvious break in policy continuity after the last governor, Raghuram Rajan, decided not to seek a second term in 2016, which also sparked market concerns.
Nevertheless, Mr Patel’s decision comes after months of escalating government pressure on the RBI to ease some of the strains created by its clean-up of the banking sector. Increased bad loan recognition has led to large credit costs – particularly for state banks – and weaker capitalisation in recent years. Capital constraints have, in turn, held back lending, while 11 state banks have fallen under the RBI’s “prompt corrective action” (PCA) framework, which allows the central bank to directly restrict their lending. Problems in the non-bank financial sector following the recent default of Infrastructure Leasing & Financial Services (IL&FS) have further reduced credit availability.
The government has unsuccessfully pushed the RBI to relax the PCA thresholds to allow some troubled banks to step up lending. Calls to dilute provisions in a new regulatory NPL framework that has accelerated bad loan recognition this year and to provide emergency liquidity to non-bank financial institutions (NBFIs) have also been dismissed. The introduction of a 0.625% counter-cyclical buffer (CCB) that was set to kick in from April 2019 has been delayed, but the RBI has so far resisted pressure to push back the implementation of other Basel III minimum capital requirements.
A roll-back of measures that address long-standing bad-loan problems and restrict the growth of weakly capitalised banks could have a negative impact on the credit profiles of affected banks and increase risks in the financial system. Most state banks are in a poor position to ramp up lending, with their common equity Tier 1 ratios well below the 7.375% that will apply from April 2019 under Basel III implementation. Some banks are also likely to continue reporting losses, further adding to capitalisation challenges.
In terms of monetary policy, the establishment of a Monetary Policy Committee (MPC) in October 2016 and recent introduction of inflation targeting has underpinned our view that the RBI’s macroeconomic policy framework is credible and effective. However, that assessment could change if government influence pushes the RBI away from its mandate. We affirmed India’s ‘BBB-‘ sovereign rating with a Stable Outlook in November.
General elections due by May 2019 will create a political incentive for the government to push for more supportive RBI policies. India’s economy remains one the fastest-growing in the world, but GDP growth slowed to 7.1% yoy in 3Q18 (calendar year), from 8.2% in the previous quarter. We recently lowered our growth forecast for the fiscal year ending March 2019 to 7.2% from 7.8%, due to the weak data, higher financing cost and reduced credit availability. |
Hillary Clinton Hillary Diane Rodham ClintonFox News poll: Biden ahead of Trump in Nevada, Pennsylvania and Ohio Trump, Biden court Black business owners in final election sprint The power of incumbency: How Trump is using the Oval Office to win reelection MORE on Wednesday fired back at criticism from President Trump using his now-infamous "covfefe" tweet.
"People in covfefe houses shouldn't throw covfefe," Clinton tweeted.
People in covfefe houses shouldn't throw covfefe. https://t.co/M7oK5Z6qwF — Hillary Clinton (@HillaryClinton) June 1, 2017
The original phrase Clinton appears to have changed is "people in glass houses shouldn't throw stones." Clinton inserted "covfefe" in reference to Trump's tweet that went viral just after midnight Wednesday.
"Despite the constant negative press covfefe," Trump tweeted.
ADVERTISEMENT
Clinton's message jabbing at the president was a quote tweet of Trump, who earlier in the evening had hit Clinton for refusing "to say she was a terrible candidate."
"Crooked Hillary Clinton now blames everybody but herself, refuses to say she was a terrible candidate. Hits Facebook and even Dems and DNC," Trump wrote on Twitter.
Trump's attack came after Clinton criticized the Democratic National Committee's data operation earlier Wednesday during a Q&A session at Recode's Code Conference in Rancho Palos Verdes, Calif., saying that she "inherited nothing" from the party after she became its presidential nominee.
"I mean, it was bankrupt. It was on the verge of insolvency. Its data was mediocre to poor, nonexistent, wrong," she said. "I had to inject money into it." |
Main menu
Breeze, 2006
Peadar Lamb
DC:2007.11
‘Breeze’, panel of stained glass in blue and green with a vertical line of black and red. Upper half of illustration broken by three curved black lines running horizontally. Lead lines visible throughout. |
Hank Hall
Hank Hall is a fictional character in the DC Comics universe who first appeared in Showcase #75 as Hawk of Hawk and Dove. He later became the supervillain Monarch in the crossover event limited series Armageddon 2001. He later became known as Extant, and appeared in the Zero Hour limited series (as well as some related tie-ins). Hawk was restored, and in the final issue of Blackest Night, he was finally returned to life.
Hawk has appeared in numerous cartoon television shows and films. He appears in his first live adaptation in the DC Universe series Titans, played by Alan Ritchson.
Fictional character biography
Hawk and Dove
Hank Hall was originally the superhero Hawk of Hawk and Dove. Hawk represented "chaos", while Dove represented "order". His brother Don Hall died during Crisis on Infinite Earths and was replaced with Dawn Granger.
Armageddon 2001: Monarch
Monarch was an oppressive tyrant from a bleak, dystopian Earth fifty years in the future. The people were unhappy with his rule, particularly a scientist named Matthew Ryder, an expert on temporal studies, who was convinced he could use his technology to travel back in time and prevent the maniacal ruler from ever coming to power. He learned that forty years ago, one of Earth's strongest and most powerful heroes would eventually turn evil and become Monarch, and ten years from that event he would conquer the world. During a time travel experiment, Matthew was transformed into a being called "Waverider", and began searching the timestream for the hero who would become Monarch, not knowing that Monarch was following him. When Monarch came into battle with the heroes of the present day, he killed Dove, and her enraged partner killed him for it. Removing the villain's mask, Hank discovered that he was Monarch, and donned the armor.
Armageddon 2001 is generally disliked by readers for what has been described as the dishonesty of its resolution. The frame story had been presented as a mystery - what superhero would go insane, kill all other heroes, and take over the world, and why? - and clues were provided. However, at some point during the mini series the future-culprit's identity was leaked. Captain Atom would be the one who became Monarch. In response to the leak, the surprise ending was changed at the last minute: Monarch was revealed to be, not Captain Atom, but rather Hank. The problem with this reveal as many fans pointed out is that Waverider had seen Monarch die and let Hank Hall kill him in which then he absorbed his powers thus allowing him to destroy the JLA.
Armageddon: The Alien Agenda
When hostile aliens encounter Monarch and Captain Atom in the past (sometime between 230 and 65 million years ago), they attempt to enlist both (with each figure having no knowledge of the other involved) to assist them in creating a wormhole. The wormhole's creation would destroy the universe in which the primitive Earth existed, but would allow the aliens to travel freely.
Zero Hour: Extant
Shortly after returning to the present, Monarch confronted Waverider and used his power to see the past and future to become aware of the power within him. It is explained at this point, that when Monarch killed Dove, her powers went directly into Hawk. Realising this, Monarch unleashes his hidden powers and becomes Extant. Extant then removes Waverider's timetravel device and joins forces with renegade Green Lantern Hal Jordan, now known as Parallax, in a plan to alter time as they saw fit.
His first act was to alter the future so that he could have a metahuman army at his disposal, mostly consisting of members of the Teen Titans; his plan was to amass an army so powerful that no one could interfere with his efforts to control time itself. Several armies of heroes banded together to stop his plans before they began in the 30th century, and altered history so that his followers never came to exist in the future.
Down, but not out, Extant began to strike back at the heroes at Ground Zero, the beginning of time. Parallax had warped several metahumans from various time periods together for the ultimate assault, and Extant hit the Atom with a chronal blast, de-aging him into a teenager. Sensing defeat was imminent, he escaped the fight, promising vengeance at a later date.
Extant would first reappear in the 1999 one-shot "Impulse: Bart Saves the Universe". In it, Extant picks a fight with the original Justice Society as a means of tricking the Linear Men into saving the life of an innocent bystander who was destined to die. The man they saved would now go on to develop a nuclear weapon that, when tested, would shift the Earth out of its proper orbit, causing massive changes in the timelines of some of Earth's greatest heroes. Among these changes, Hal Jordan never becomes Green Lantern, thus he never becomes Parallax, and never stops Extant from destroying all of time. Fortunately for the citizens of time, Impulse arrives and is barely able to defeat Extant and prevent the Linear Men from saving the doomed scientist.
He would engage the Justice Society again on a later date as he sought to acquire the reality-warping power of the Worlogog, recently dismantled by Hourman because he feared its power. Although Extant succeeded in his goal with the aid of Metron's stolen Mobius Chair, Doctor Fate learned from the imprisoned Mordru that when Hourman had dismantled the Worlogog, he had retained a small fragment of it, thus creating infinitesimal flaw in the prime Worlogog that the JSA could exploit.
After the resurrected Dove sacrificed herself to distract Extant, Hourman divided his Hour of Power amongst his teammates, thus granting them all immunity to Extant's reality warping powers for four minutes, each of them attacking him on a different temporal plane until they were able to separate him from the Worlogog. Following this setback, Extant again attempted to escape. Instead, Extant was teleported by Hourman and Metron, at Atom Smasher's behest, into the seat of an airplane whose crash Kobra had caused earlier (in his relative timestream). As a result of this, Atom Smasher's mother was saved (as she was on the plane when it crashed), but Atom Smasher replaced his mother with a weakened Extant, saving her life but murdering the super villain to stop his threat and ensure that the same number of people died on that plane who had died originally.
Hawk restored
In response to fan-criticism of Armageddon 2001, many of whose readers felt that the character of Hawk had been severely misused in the story's last-minute changes, DC Comics set about restoring the character as he had originally been intended; a hero. DC had already retconned Extant's portion of Hank Hall's timeline in issue 14 of JSA, dated September 2000, in which Metron announced his intention to erase the villain's "wretched timeline" with his Mobius Chair. This was the second issue of a 3-part story entitled The Hunt for Extant!, (the details of which are listed above). After this, DC also retconned Monarch's portion of Hall's timeline with the final issue of the 6-part miniseries The Battle for Blüdhaven, dated September 2006, which now depicted Captain Atom's transformation into Monarch, as had been DC's original intention back in 1991. Hawk was restored, but he would not be revived until the final issue of Blackest Night.
Blackest Night
In the Blackest Night crossover, Hank Hall is reanimated as a member of the undead Black Lantern Corps. The black power rings also try to reanimate his brother Don, but are denied, stating "Don Hall of Earth at Peace". Hank then tracks down and attacks Dawn and the new Hawk (Holly Granger). After a short battle, Hank rams his hand into Holly's chest, ripping her heart out, and using it to charge his ring. Holly's body is then revived by a black ring, and the two attack Dawn together. Severely outmatched, Dawn retreats, with Hank and Holly giving chase. Hank and Holly follow Dawn to Titans Tower, where more Black Lantern Titans are attacking the living heroes. The two eventually overwhelm Dawn, with Holly plunging her hand into Dawn's chest. Dawn suddenly radiates a white energy that completely destroys Holly's body and ring. The other Black Lanterns, seeing Dawn as their greatest threat, attack her. However, she turns the light on them, destroying all but Hank, Tempest and Terra who quickly retreat. While battling the Black Lanterns at Coast City, Hank is later brought back to life by the power of the white light. Dawn has a vision of Don who tells Dawn that she can save Hank, and to not give up on him.
Brightest Day/Birds of Prey
At the beginning of the Brightest Day event, Hank and Dawn begin working together again as a crime-fighting duo. Dawn expresses worries over Hank's increasingly violent demeanor, but he simply brushes off her concerns. While stopping an army of powerful teenaged super villains in Gotham City, Hank and Dawn are invited by Zinda Blake to join the Birds of Prey. The two are immediately called by Oracle to help Black Canary and Huntress during their battle with a dangerous villainess known as the White Canary. Dove attempts to defeat her herself, but is surprised when White Canary is somehow able to dodge her attack and then draw blood from her. Hank and Dawn later encounter Deadman who Hank asks to resurrect Don. At a crater in Silver City, New Mexico, Deadman attempts to revive Don, only to be prevented from doing so by the Entity. As a number of onlookers (including Jackson Hyde) watch the Entity speak to the heroes, it instructs Hank to catch the boomerang that Captain Boomerang will throw at Dove.
After being injured trying to kill himself because of depression, Hank Hall is sent to a hospital while his teammates plan their next move. During his hospital stay, Hank has a vision of himself, clad in a White Lantern uniform and talking to Don. Just before the dream ends, Don assures his brother that he is at peace. Later, Dawn is transported to the Star City forest by the Entity, Hawk unintentionally went with her, but when the "dark avatar" made his presence known, the Entity tells them that they must protect the forest and withstand the ultimate savior, which is Alec Holland.
It was revealed that Captain Boomerang's mission for throwing the boomerang was to free Hawk as an avatar of war from the Lords of Chaos because his act of saving Dove would have broken their hold on him to be his own self. However, he failed to catch the boomerang and instead it was caught by Boston Brand, who ended up dying in the process and used his final act to move his white power ring to Alec Holland and bring back the Swamp Thing in order to cleanse the Green of Nekron's influence.
Powers and abilities
As Hawk he possessed a "danger sense transformation" which allowed him to change into a super-human with the powers of super strength, unlimited stamina, enhanced speed, increased agility, enhanced body density, extreme durability and healing factor.
His partner Dove suppresses his violent nature, and without him or her Hank's rage becomes boundless.
As Monarch he possessed the same powers that he had as Hawk, along with a suit of highly durable armor that was crafted using advanced technology.
As Extant he had the powers of chronokinesis, energy projection, flight, and omniscience. After piecing together the Worlogog he became nigh-omnipotent.
While he was a member of the Black Lantern Corps, Hank wielded a black power ring which allowed him to generate black energy constructs. He was also able to perceive emotional auras. However, whilst he was able to perceive Holly's aura as red for rage, he saw Dawn's as a pure white that his ring could not identify. While wearing the black power ring it lowers his original power by over 50%.
Other versions
In the Elseworlds JLA: The Nail miniseries, and its sequel JLA: Another Nail, a version of Hank Hall exists, alongside the original Dove.
Justice League of America (vol. 2) #26 features an alternate reality created by the trickster god Anansi. In this reality, an armored version of Hawk is seen.
In other media
Television
Hawk (Hank Hall) is featured in Justice League Unlimited, voiced by Fred Savage. This version is depicted with a strong relationship with his brother Dove (Don Hall). In their self-titled episode, their fighting styles were thoroughly contrasted; Hawk employs brute-force, aggressive tactics, at times resembling a football player while Dove uses a blend of techniques reminiscent of aikido or perhaps judo, using his attacker's movements to fling aside. After defeating some thugs in a bar, Hawk and Dove are enlisted by Wonder Woman to help stop Ares from causing war in Kaznia. They are successful due to Dove's peaceful resistance against the rage-powered Annihilator. This is another example of how close the two are as Hawk struggles against Wonder Woman in an attempt to protect Dove. Hawk screaming for Dove as he feared for his brother's life closely resembles when Dove was killed in Crisis on Infinite Earths. In the episode "The Greatest Story Ever Told", Hawk and Dove are among the heroes that fight the forces of Mordru. In the episode "The Doomsday Sanction", Hawk and Dove assist in an evacuation of San Baquero before the island's volcano erupts. The brothers are last seen in the series finale "Destroyer" where they fight off Parademons alongside several other Justice League members. They later appear in the final scene running down the steps of the Metro Tower with the rest of the group. Fittingly enough, both in that fight scene and as they exit in the finale, they appear along with Steve Ditko's fellow creations: the Question, the Creeper and Captain Atom.
Hawk (Hank Hall) appears in Batman: The Brave and the Bold, voiced by Greg Ellis. In the teaser for "When OMAC Attacks", he and Dove help Batman stop an intergalactic war between the Controllers and the Warlords of Okaara. While Hawk and Dove take out the ground forces, Dove claims that it is better to settle things diplomatically while Hawk says that they have to hurt them or they will never stop. Regardless, Batman gets the two sides' leaders to sign the peace treaty and end the war. Hawk and Dove do manage to embarrass themselves, their bickering causing them to fight in front of the leaders. Batman invites the leaders to have a drink in his ship to draw their attention from the bickering brothers. Hawk and Dove also briefly appear in the two-part episode "The Siege of Starro" Pt. 1 amongst the heroes who were taken over by Starro. After Starro's defeat, the brothers are back to normal.
Hawk (Hank Hall) appears in the series Titans, portrayed by Alan Ritchson with Tait Blum as a younger version of the character. Instead of being depicted as having superpowers, Hank's physical prowess is as a football player. In the series, Hank and his half-brother Don Hall are the original Hawk and Dove team that hunt down sexual predators, motivated by abuse that Hank's football coach inflicted on him as a child. After Don is killed in the same accident that kills Dawn Granger's mother, Hank has a romantic pairing with Dawn while subsequently being Hawk and Dove respectively. Ritchson also appears as Hank in the crossover "Crisis on Infinite Earths"
Web series
Hawk (Hank Hall) appears in DC Super Hero Girls with Dove (Dawn Granger). They appear as background students of Super Hero High.
Film
Hawk (Hank Hall) appears in Superman/Batman: Public Enemies as one of the many individuals after Batman and Superman.
References
Category:Fictional mass murderers
Category:DC Comics superheroes
Category:DC Comics supervillains
Category:Comics characters introduced in 1967
Category:DC Comics characters with superhuman strength
Category:DC Comics characters who can move at superhuman speeds
Category:DC Comics characters with accelerated healing
Category:Characters created by Steve Ditko
Category:Characters created by Dennis O'Neil
Category:Characters created by Dan Jurgens
Category:Characters created by Archie Goodwin |
A commonly desired feature of video surveillance cameras is the ability to detect when something unusual happens and then issue an appropriate report or alarm. Historically, detection of unusual events in video surveillance has been performed by is having security professionals watching video footage of a scene on one or more video display monitors. More recently, the field of video analytics has allowed computers to perform automatic detection of objects in video. Security professionals can use these detected video objects to create rules that trigger alarms or events when certain criteria are met. For example, an object of a certain size entering a predefined area of a scene may trigger an alarm. These rules are used for a variety of purposes, such as intrusion detection, abandoned object detection, removed object detection, tailgating detection, speeding detection, and falling over detection.
While such rules are useful in a scene with requirements that are well understood and easily definable, sometimes a scene is more complicated and it is difficult to set up accurate rules, or the security professional just wants to be told when something unusual happens.
There are several existing systems for detecting abnormal events. One method uses motion detection to estimate velocity at each point of a scene captured in a video sequence, without associating that velocity with any particular object, in order to build up an average “flow map” of the scene over a period of time. If a current video sequence has velocities that are sufficiently different from the flow map, the method triggers an abnormal behaviour event. This method is limited to velocity vectors, however, because this method does not perform true object detection. This method cannot detect objects of unusual size, or objects in unusual positions in the scene, unless these objects are also accompanied by sufficiently unusual velocity vectors.
Another method uses background subtraction to build up statistics for a scene over time relating to how often a portion of the scene is part of the background. At a given time, a current foreground mask can be compared with an average background mask to detect whether the current frame has objects in abnormal positions. This method is limited to detecting abnormal positions of objects. This method is not able to detect an object moving at an unusual speed, or an object of an unusual size, unless that object was also in an unusual position.
A third method uses histograms to accumulate position and motion information about a scene, using point-feature extraction to obtain object and tracking data. Abnormal events are detected by comparing current positions and motions of objects with the histograms. This method has a disadvantage in that because input parameters are broken up into histogram bins, it is memory intensive to add extra parameters, each parameter contributing an additional dimension to the storage array. In addition, because this method uses point-feature extraction, it has no concept of object size.
Thus, a need exists to provide an improved method for classifying a behaviour of a detected video object in a video frame. |
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