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TITLE: Geometry question on ratios
QUESTION [0 upvotes]: In an acute triangle ABC, points D, E, F are located on the sides BC, CA, AB respectively such that CD/CE = CA/CB , AE/AF = AB/AC , BF/BD = BC/BA . how to prove that AD, BE, CF are the altitudes of ABC.
REPLY [2 votes]: Each of the given ratios serves as the condition for “the converse of the power of a point”. Therefore, for example, the converse of the power of point C yields a circle (in red) passing through A, B, D, E. The green and the blue circles can be similarly formed.
Another fact needed is "With respect to 3 intersecting circles, the 3 common chords are concurrent at the radical center". This can be proved by considering the powers with respect to each circles. Thus, AD, BE and CF intersect at H.
[Note that $CHF$ is a straight line and Please ignore the black line being dotted. The dotted line comes from an old drawing which I am too lazy to replace.]
From the green circle, $\alpha = \beta$
From the red circle, $\beta = \gamma$
This means $C, E, H, D$ are con-cyclic. That is, the purple broken line is a circle.
From the purple circle, $\theta = \phi$
From the red circle, $\phi = \omega$
This means $\theta = \omega = 90^0$ implying $AD$ is an altitude. The other 2 altitudes can be found in the similar fashion. | 34,358 |
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This eye catching Retro Parrot Paradise Insulated Picnic Bag displays the gorgeous vintage illustrations of parrots in a warm colourway, making this design mature and stylish. So, keep your lunch fresh in this cool, insulated picnic bag which is large enough to hold a substantial lunch and drinks for all. Our insulated picnic bags will keep your picnic fresh until lunch time, so are ideal for day trips, festivals and days by the beach and made of 50% recycled materials it is eco-friendly as well.
Measures: 25.5 x 31.5 x 17.5 cm
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Here's a great gift for a current Sailor or for a USN Veteran. This 12" by 18" metal sign is a reproduction of a vintage recruitment poster for the U.S. Navy. It would look great hanging in your favorite room, shop, office, or garage. This USA-made metal sign has eyelets for ease in hanging on a wall. | 381,653 |
A.
Industrial Cleaning Using Paper Towels and Elbow Grease.
Clean Restaurant Equipment with Natural Products
Purchase a box of baking soda and a bottle of apple cider vinegar to remove stubborn grease and grime from the commercial grill. Make a thick paste by using water and the baking soda. The baking soda paste must resemble toothpaste to work effectively in cleaning the industrial equipment. Smear the paste all over the equipment and leave the commercial grill alone for 15 to 25 minutes. Take a wash cloth and rub circular motions over the restaurant equipment until the surface forms a haze. Follow by cleaning the commercial grill by applying a generous amount of apple cider vinegar all over the cooking surface. Let the vinegar sit for a few minutes until it bubbles and then wipe off the surface. According to Foodservicewarehouse.com, bleach has a tendency to discolor stainless steel. Never use bleach for cleaning a stainless steel surface.
Finish off with a Mild Detergent
Take a spray bottle and add a teaspoon of a mild detergent, such as dish soap. Fill the bottle with lukewarm water and begin spraying the commercial grill surface. Let the soap sit on the surface for up to 15 minutes. Use a dish cloth and wipe the equipment all over. Take paper towels and thoroughly dry the area. Use clean water to rinse any soap residue twice by dousing the food cooking area and then wiping it dry with towels. Thoroughly drying the restaurant cooking equipment ensures that no streaking or water marks will occur. | 235,635 |
By ADVANCED DESERT DERMATOLOGY
May 05, 2021
>
How do I treat a rash?
When should I see a dermatologist about my rash?
Dealing with a rash that is painful or causing your concern? If in doubt, don’t hesitate to call your dermatologist. We can discuss your symptoms over the phone and determine whether you should come in for a consultation. | 182,089 |
So, I guess that should read “exemplary teacher [pictured] threatened by student after asking class to step on piece of paper with Jesus’s name in harmless exercise on cultural sensitivity”. And he’s been receiving death threats from those ever-lovin’ Christians, of course.
I was in kinda-sorta-loose agreement with Sam Harris right until he says “[t]here is no such thing as Islamophobia” and that “[it] is a term of propaganda”. Now that’s how you shipwreck an argument, folks. (If he wants to contest the conflation of genuine criticism with bigotry, he should stress that ‘Islamophobia’ only applies to the latter, not that it isn’t actually a thing.)
Reality: Hoaxer dude calls domestic violence counseling line with phony abuse story, repeatedly refuses offered assistance to find free shelter or contact authorities. MRAs: “Abused man ‘denied help’”!
Public Policy Polling does conspiracy theories, reveals an average 10–15% of Americans are effin’ nuts. Not sure why “Bush lied about Iraqi WMDs” is counted as a myth, though.
(via @BuzzFeedAndrew)
And finally, it’s getting almost impossible to tell when Vox Day is being either seriously or facetiously stupid.
If you have any story suggestions, feel free to leave them in the comments or send them in. | 193,437 |
Photos Taken by Kirstie Dunston Photography
Photos by Basically Bronwyn
Happy Wednesday, everyone! I know I probably say this a lot, but I am SO excited to share today's post with you all. A couple of weeks ago I had the pleasure of getting together with Lexington and local blogger/boss babes for the most epic, Pinterest worthy brunch you've ever seen! How gorgeous is this setup?? I'm still trying to figure out how I can convince Justin that a four year vow renewal is a "thing" so we can have a massive party here at The Barn at Springhouse! Between the gorgeous tablescape and setting, amazing food, goodie bags filled with local products, and the beautiful company, the morning was perfect. It's a shame we can't do this every week, am I right?!
What makes this post so fun is that not only will I be sharing style inspiration for you all, but also the links to the other bloggers who attended the event for you to check out their posts on it. Everyone brought a dish to try (along with their fab styles), so be sure to click through the links for more recipes and fashion! I, of course, brought my overnight oats that I shared with you all on the blog a few weeks ago. It's super easy to make PLUS it's delicious; win-win. I figured the best way was to make individual servings in mason jars, and they turned out so cute (and yummy of course)! If you missed my post on the recipe, you can find it HERE!
I went back and forth between two dresses to wear for the brunch, and with a little help from my Instagram family, I went with this gorgeous blue midi dress! I love the simple, floral print and the pleated detail at the bottom. It's very comfortable, fits true to size (I'm wearing a medium), and is less than $80. It can easily be dressed up with a pair of heels or down with sneakers, like I did with this white pair. You know me- I'm all about that comfort life and these sneakers were MADE for me! They have a great cushion and obviously look cute with dresses or your favorite pair of jeans. Bonus: they're only $30! I would go up AT LEAST 1/2 size or even one full size. This look would be great for any weddings or showers you have coming up. Just throw on a pair of fun sunglasses or statement purse, and it's such an easy, feminine look for any occasion!
Be sure to check out everyone else's blog post and vendor information below! Scheibel | Fabulous in Fayette
Ella Rutledge | Girl Meets Lex. | 229,607 |
Alexandria City Council approved Saturday the redevelopment of the Shell station located at 5740 Edsall Road.
The redevelopment plan includes a 3,000-square-foot convenience store, a 1,100-square-foot car wash and an increase in the number of gas pumps from four to six. The underground gas tanks on the site, which has housed a gas station since the early 1960s, will be reoriented to accommodate the renovation. New canopies will also be installed.
The redevelopment of the Shell site also aims to alleviate issues at the busy intersection of Edsall Road and South Van Dorn Street with a new layout and streetscape improvements. The plan offers 30 feet of right-of-way on South Van Dorn Street to be dedicated to the city for future roadway improvements, including the potential installation of high-capacity transit.
The city’s planning staff said it believed the proposal is an appropriate interim use of the site until the full redevelopment of the Landmark-Van Dorn Plan is contemplated.
In a March meeting with the Cameron Station Civic and Community Association, residents expressed concerns that the addition of a car wash and convenience store at the site may increase congestion in the area.
Councilmembers said Saturday that the carwash is an ancillary use. Gas customers will be able to purchase a wash at the pump. Ingress and egress to the carwash will be constructed to enable vehicles to circulate without disrupting the function of the convenience store or fuel pump customers.
“It’s a much more attractive facility with the landscaping you’re going to do,” Vice Mayor Kerry Donely said. “It’s an improvement. … People need to buy gas.”
Staff said it believed the addition of the carwash and convenience store would create small impacts on traffic.
The station will be marked with a monument-style sign no greater than 6 feet. | 156,248 |
What Houzzers are commenting on:
vicjeannewin added this to vicjeannewin's Ideas
March 16, 2014
March 16, 2014
This would work with an old atlas and/ or map
nmorello added this to nmorello's ideas
August 13, 2013
August 13, 2013
I like the vintage look
landman12 added this to landman12's Ideas
July 22, 2013
July 22, 2013
Cover old lamp shades with maps or posters
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April 3, 2013
April 3, 2013
Great material idea.
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March 17, 2013
March 17, 2013
Maps!
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December 13, 2012
December 13, 2012
Great way to add a local personal touch to a guest room.
jae1 added this to jae1's Ideas
December 12, 2012
December 12, 2012
Shade for lamp..salmon beach.
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November 25, 2012
November 25, 2012
One of savannah?
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October 7, 2012
map lampshade
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Lampshade of nyc
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September 20, 2012
Hand crafted from map
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Cheeky
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lao
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Lampshadei
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Fun for Quogue
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Love it
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Map shade
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map
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For Kimmie
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Another project?
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Maps can even help light up your home. This funky lampshade is handmade from a vintage New York map.
Vintage New York Map Lampshade
GBP 28
Maps can even help light up your home! This funky lampshade is handmade from a vintage New York map.
Price varies by size £28 to £65. — Rachel Newcombe
Price varies by size £28 to £65. — Rachel Newcombe
Product Specifications
- Sold By
- Rosie's Vintage Lampshades
- Category
- Lamp Shades
- Style
- Eclectic | 225,627 |
Shop Online
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WASHINGTON -- Attorney General Alberto Gonzales faced more pressure to resign Thursday as new evidence suggested that he and presidential adviser Karl Rove played bigger roles in developing plans to fire U.S. attorneys than they've acknowledged.
Rove has been President Bush's chief political adviser since Bush's first campaign for Texas governor, and the latest developments angered members of Congress, created new credibility problems for the administration and increased calls for Gonzales' resignation.
Democrats cited Rove's involvement as more evidence that the firings were intended to purge prosecutors who refused to let partisan politics influence criminal investigations.
Administration e-mails from early January 2005 show that Rove and Gonzales were directly involved in the initial planning to oust prosecutors who'd fallen out of favor. Recounting in an e-mail a conversation he'd had with Gonzales, aide Kyle Sampson said that they'd decided to replace 15 percent to 20 percent of the 93 U.S. attorneys while retaining those who "are doing a great job, are loyal Bushies, etc."
At the time, Gonzales was serving as White House counsel while awaiting confirmation to become attorney general. Sampson became his chief of staff at the Justice Department and continued to oversee planning for a mass firing. Sampson resigned earlier this week amid the growing controversy over the dismissals of eight U.S. attorneys last year.
An e-mail from another White House aide said that Rove wanted to know "how we planned to proceed regarding US Attorneys, whether we were going to allow all to stay, request resignations from all and accept only some of them, or selectively replace them, etc."
The e-mails, which the Justice Department released after the contents were leaked to ABC News, call into question Gonzales' assertion that he was essentially in the dark about the plans to dismiss federal prosecutors.
Despite expressions of support from Bush, prominent Republicans openly discussed the possibility of Gonzales' resignation. On Capitol Hill, House of Representatives and Senate committees pressed ahead with their investigations into the firings.
"If the president is damaged politically by some of the things that are going on, keeping his friend in as attorney general isn't the right decision," said Rep. Dana Rohrabacher, R-Calif.
Sen. Mel Martinez, R-Fla., the Republican Party's national chairman, said Bush might have to ask Gonzales to step down "at some point."
"Let's let the attorney general explain himself. Let's let all of the facts come out and then we can make judgments," Martinez told CNN.
The White House downplayed the significance of the e-mails but backed away from earlier statements that the plan to fire all 93 U.S. attorneys originated with former White House counsel Harriet Miers and was swiftly rejected by Gonzales and Rove.
"I do not have the specific answer for you as to whose idea it was," said White House spokeswoman Dana Perino.
The Justice Department issued a statement saying that Gonzales "has no recollection of any plan or discussion to replace U.S. attorneys while he was still White House counsel."
In a speech earlier in the day at Troy University in Troy, Ala., Rove dismissed the controversy as "a lot of politics" and said the U.S. attorney firings were proper.
"We're at a point where people are playing politics with it, and that's fine," Rove said. | 203,217 |
Real Life Drama: <b>The Importance of Being Allston:</b> Allston James’ own brushes with disaster imbue his playwrighting with substance. <small><i>Angelina Shamrock</i></small>
Real Life Drama
Allston James’ wild journeys bring depth and soul to his playwrighting.
Thursday, May 19, 2005
“Boys will apply all sorts of pressures on young girls,” says a father in Allston James’ new one-act Garden Talk. “Exploit every stripe of their emotion, every category of natural reflexes, whet precisely the wrong—and best—appetites, provoke the worst strains of curiosity. Take advantage, break hearts, fertilize eggs, deposit diseases, smash hopes, destroy pride. Make her hate herself. Maybe even make her want to die…rattle her soul like the cold dead branches of a pear tree in winter.”
Breathtaking dialogue drives Allston James’ plays. His ear for rhythm and eye for imagery, coupled with his compelling characters, have put James in the enviable position of premiering two separate plays this year, one in New York and one here in Monterey County.
Not bad for a guy who began thinking of himself as a playwright just two years ago.
“I’ve always kind of danced around the idea of dramatic writing,” James says. “The idea of collaboration did not appeal to me. I felt I’d have to give something up, so I historically avoided it.”
James, who has taught literature at Monterey Peninsula College for the past 25 years, says that eventually two things led him to try his hand writing for the stage.
“First off, I’m an unabashed Shakespeare addict, and I knew eventually that was going to take its toll,” he says. “The second thing is, I’ve had editors who’ve published my stories over the years and told me that dialogue is my strong suit.”
And having seen enough theater over the years to know the difference between bad and good plays, James began working with MPC’s resident drama diva, the inimitable Ms. Lee Brady, on some short plays. After only two short years of concentrated work, James’ first big break came last summer when the New York Collective for the Arts produced Drive Time, his snappy modern one-act about life after divorce. While in New York overseeing the production, he had an epiphany about the dramatic process.
“Contrary to what I’d thought, that I’d have to give something up to collaborate, I found the creative stream of drama to be much more sustained. With a story you finish it and it’s done. With theater it’s this drawn out, organic, immensely satisfying thing. Not only did I get to write it, but I got to witness the production be built from the ground up. It was really amazing.”
James’s own story is no less amazing. Born in Atlanta, Ga. in the late ‘40s, he moved to Montgomery, Ala. and then wound up in Florida in the ‘60s, where he discovered his life’s first passion: surfing.
After receiving a bachelor’s degree in government from Florida State in 1969, James joined the military and went to Vietnam as an artillery forward observer in the 1st Infantry.
“I was hoping against hope the war would be over by the time I got out of officer’s training,” he says.
No such luck. James went to war.
Four months into his tour, James was returning from a mission along the Cambodian border when the jeep he was riding in overturned. He would have been killed if not for the fact that the man riding behind him absorbed the impact. James escaped with his life, but his right foot was left a mashed and twisted hunk of flesh, muscle and bone, and doctors told him he would never be able to walk again.
After 15 months in various hospitals, months of physical rehab and a nightmarish slow-motion tussle with Demerol addiction, James recovered much of the use in his foot, a process that he poignantly recounts in his short story “Waterbed.”
“The damage to the foot played hell with surfing,” James says.
James returned to school, receiving an master’s in journalism from the University of Georgia in 1972. Upon graduation, he moved to the French Quarter in New Orleans, wrote for a weekly newspaper, and started working on a novel called Attic Light.
Then in 1973, in an effort to escape the summer heat, James got in his car and drove west, coming through Monterey at the age of 26.
“I was very, very influenced by Henry Miller,” he says. “He was a real icon to me and Big Sur left a huge imprint on me. I went back to New Orleans, packed everything up, moved out here, and rented a house for $120 a month.”
In Monterey, James finished Attic Light, his partly autobiographical novel set against the backdrop of the Vietnam War.
In addition to teaching at MPC, James wrote fiction for surf magazines. While on assignment in Hawaii in the ‘90s, he was nearly killed by a tidal surge that spun off Hurricane Iwa and smashed into the condo he was staying in.
“All of sudden this wall of whitewater hit the building and went up across the highway behind us,” James says. “There was this pause, this dead silence, then the water started receding—it was coming back, rushing back through the building. I remember the sound of exploding doors and TVs more than anything. It was a really powerful experience.”
Then in 2003, James began thinking about drama. He’d tried his hand at writing for the stage as an undergraduate back at Florida State and had “dabbled” a little over the years, but after a particularly rewarding stint at the Oregon Shakespeare Festival two years ago, he decided to approach it in earnest. With the help of Lee Brady, who showed him the basics of dramatic writing, he began working on one-acts. Then last year, inspired by the success of Drive Time, he began work on his first full-length play, The Pink Brothers.
This summer, his short, potent drama Garden Talk will be produced by the New York Collective for the Arts, and this fall, his full-length comedy The Pink Brothers will premiere at the Carl Cherry Center stage, under the direction of Rosemary Lukes.
The Pink Brothers is the story of two brothers who are, each in his way, trapped in a zone between ignorance and enlightenment in regards to women.
“This play is like a carnival ride,” James says. “It don’t stop ‘til we been ‘round the last twist of track.”
Not unlike James’ own life.
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Description
Gorgeous vintage 1960s silk kimono robe in a striking blush / copper / cream ombre with a floral pattern. Fully lined. It comes with a sash that I made from vintage kimono silk. It is in excellent condition, apart from one small faint spot, see last photo.
Measurements are approximate, taken flat and then doubled where necessary, see diagram on last image to understand how measurements are taken.
Around body: 48 inches (121.92 cm) when flat and overlapped as in the diagram
Length from shoulder to hem: 62 inches (157.48 cm)
Sleeve to sleeve width: 52 inches (132.08 cm)
Sleeve length: 19 inches (48.26 cm)
Sash: 116 x 4.5 inches (294.64 x 11.43 cm)
Kimono Care: Professional Dry Clean Only | 198,349 |
First published in the Western People on Monday.
In Rocky Ros Muc, Mac an Iomaire looks at the sport of boxing, life in Connemara, the Irish emigrant experience and the life of a man who could have been a contender. He weaves all those threads together to present an invaluable record of a way of Irish emigrant life and of a sport at a time when it was still a big deal.
Seán Mannion boxed as an amateur in Ireland before taking the plane to Massachusetts to make his fortune, like so many before him. He worked for his brother’s construction company by day and by night Mannion boxed in Connolly’s Gym.
Connolly’s Gym was on Broadway Street in Dorchester, the famous “Southie” area of Boston that is famous for its Irish and its hoodlums. Mac an Iomaire excels in portraying the atmosphere of the place at the time, when everyone knew who Whitey Bulger was and nobody wanted to get on the wrong side of him.
This is one of the many marvellous features of the book, how Mac an Iomaire is able to place you in the time and at the place. You’re in the Irish bars celebrating St Patrick’s Day, you’re ringside at the fights, and sometimes you can even hear the thock! thock! thock! of the punches hitting the heavy bag in the gym as Mannion hones his trade with a line of made guys, wiseguys and plain old two-bit hoodlums looking on and hanging out.
As a boxer, Mannion had several gifts. Firstly, he was lefthanded, a southpaw. Most boxers, like most people, are right-handed, which makes fighting a left-hander an oddity in itself. That puts opponents at a disadvantage straight away.
Secondly, Mannion had excellent ringcraft – he was a proper boxer, rather than just a brawler. But best of all, Mannion could take what was thrown at him. Seán Mannion fought fifty-seven professional fights, and was never knocked down in any of them. There are very few fighters about whom that can be said.
But for all that, Mannion had one fatal flaw. When he’s in training, a boxer has to live almost like a monk. He has to exercise right, train right, eat right. He has to go to bed early and be up before the dawn, running miles, skipping rope, sparring, hitting the heavy bag, hitting the light bag.
What he’s not meant to do is to live on fried food and booze, which commodities Mannion found hard to resist. There’s an amazing story in the book that illustrates just how far off the pace Mannion was in terms of training, and just how good he could have been if he’d been better managed.
On the 20th of August, 1982, Mannion was to fight Hector Figuerora at welterweight. Welterweight boxers weigh not less than 140 pounds and not more than 147. At the weigh-in on the day of the fight, Mannion weighed in at 156. Nine pounds overweight.
Figuerora’s seconds demanded a forfeit, but Mannion was given a chance to see if he could sweat the weight off. They ran the shower in his hotel room until hands couldn’t be seen in front of faces from steam, and then in Mannion went, dressed in a rubber suit and carrying a skipping rope.
After one solid hour’s skipping, Mannion was weighted again. Four pounds lost, but still five overweight. Back into the rubber suit with Mannion, and he started running up and down three flights of stairs. Up and down, up and down.
After half-an-hour, he was back on the scales. Another four pounds gone, but still one left. Figuerora’s corner wouldn’t concede the pound, even though Figuerora’s opponent was surely, surely spent after all this.
Finally, a last resort. Mannion was given a raw rubdown by one of his trainers – a massage without oil. The trainer more or less skinned Mannion to lose that extra pound. After the rubdown Mannion, naked and red as a lobster, climbed onto the scales and made the weight. Then he beat Figuerora on points over ten rounds. After all that working out, Seán Mannion was still able to box ten rounds and win.
Mannion got his shot at the World Title eventually, against Mike McCallum. It didn’t go well. Mannion had been injured in training before the fight but even if he hadn’t, McCallum would still have been too good. The great names of middleweight boxing in the 1980s were Roberto Duran, Sugar Ray Leonard, Tommy Hearns and Marvin Hagler, and not one of those four ever got in the ring with McCallum.
Seán Mannion’s is an extraordinary story of wins and losses inside and outside the ring and this book is a treasure. So much so that there may be people reading this who will wonder why, if it’s so good, Mac an Iomaire wrote it in Irish.
Firstly, Irish was very important to Mannion. He insisted on one of his brothers being one of his cornermen so they could speak in Irish during fights, and also insisted that Amhráin na bhFiann be sung, in Irish, before he fought McCallum. And secondly, why shouldn’t it be written in Irish?
Books written in Irish are not always good, and the currently ill-judged emphasis on ‘spoken’ Irish doesn’t do much to help. The market of books written in Irish, what gets published and what doesn’t, is a debate for another day. Don’t begrudge us our treat.
Besides; people often say they would warm up their school Irish if only they got a chance. The chance is here now with the publication of Rocky Ros Muc. Seconds out. | 252,173 |
TITLE: Does the mass of photon affect its size or vice-versa?
QUESTION [0 upvotes]: Does the wavelength of photon affect its size, as when the wavelength is big the photon has big size and it small it has small size?
In the electron microscope the electron should poses a small wavelength, does the wavelength of electron affect also on its size? I know when the electron accelerate its mass increase due to its velocity but its wavelength is small how it can penetrate the bodies and form a more qualified image than photon if it has more mass and thus it will have a big size, is there any relation between the wavelength and the size of both electron and photon ?
REPLY [3 votes]: Does the wavelength of photon affect on it's size as when the wavelength is big the photon has big size and it small it has small size??
No, photons are point particles in the standard model of particle physics, a quantum mechanical model. They have zero mass and energy equal to $h*ν$, where ν is the frequency of the classical wave that will be built up from a large number of the same energy photons. Here is a record of single photons , they have the footprint of a point in the detecting plane.
What the wavelength does is define the probability of detecting the photon, so accumulated measurements under same boundary conditions are needed to see the wave nature of a photon.
In the electronmicroscope the electron should poses a small wavelength ,does the wavelength of electron affect also on its size ?
Again, for quantum entities as the elementary particles, no size exists. For an electron it is the de broglie wavelength,
dependent on the electron momentum, which will control the probability of measuring the electron at an (x,y,z) at time t. It is the probability that may show wave effects, depending on the boundary conditions.
and I know when the electron accelerate its mass increase due to its velocity
This is a misconception, the mass you are describing is the relativistic mass and is not to be used in the de Broglie formula. The momentum there is with the invariant mass which does not change with energy as the name says.
but it's wavelength is small how it can penetrate the bodies and form a more qualified image than photon if it has more mass and thusit will have a big size
No, the size of the electron is always a point, what changes with momentum is the probability of interacting with the atoms and molecules under observation. The higher the energy the smaller the wavelength and accumulation of electrons will show the probability of scattering from the lattices under examination, which will depend on the wavelength.
is there any relation between the wavelength and the size of both electron and photon ?
No, see above. | 32,329 |
TITLE: Regular orbits on primitive module
QUESTION [0 upvotes]: Let $W$ be a quasi-primitive faithful and irreducible $H$-module with $H$ and $W$ of odd order. Suppose that the Fitting subgroup $F(H)$ is cyclic (so $H \le \Gamma(W)$ the semilinear group). Then each element of $W \setminus \{0\}$ generates a regular orbit under the action of $F(H)$.
I'm struggling with this, any idea?
Of course $H$ must be solvable by Feit-Thompson Theorem. I don't figure out how to prove a subgroup of prime order in $F(H)$ that centralizes one element
$x \in W\setminus \{0\}$ must centrlize every other non zero element, if this is the way to preceed.
Edit: quasi-primitive means that whenever $N$ is a normal subgroup of $H$, the restriction $W_N$ is homogeneous.
REPLY [1 votes]: Here's my solution. Call $\mathbb{F}$ the base field of $V$. The restriction $V_{F(G)}$ is homogeneus and we can write $V_{F(G)}=fU$, so $U$ a faithful irreducible $F(H)$-module.
But $F(H)$ is cyclic and so $\dim_{\mathbb{F}}(U)=1$, this means that $U \lesssim \mathbb{F}^{\times}$ and $F(G)$ acts as scalar multiplication on each irreducible constituent. So, if $v \in \mathbb{F}$ and $F(G)=<g>$ then $x^g=xc$ for a $c \in \mathbb{F}$. Being $V_{F(G)}$ homogeoneus $c$ is constant over all $F(G)$-irreducible constituent and then $F(G)$ acts as scalar multiplication on $V$ by a constant $c$ that has order $|F(G)|$ in $\mathbb{F}$. From this follows that every non zero element for a regular $F(G)$-orbit.
Once did it, I realize that this problem maybe isn't worth to be published to SE. However I cannot find a quicker way to prove it, as suggested by Derek Holt in a comment. | 119,058 |
- Room 3rd Floor, Marine Building, Drake Circus, Plymouth, PL4 8AA
- +44 7929 303997
- [email protected]
Profiles
Dr Adam Rees
Assistant Project Manager
School of Biological and Marine Sciences (Faculty of Science and Engineering)
Biography
Biography
Postdoctoral researcher - Sheehan Research Group:
Researcher - Blue Marine Foundation:
I am a marine ecologist undertaking research focussed on the impacts of various anthropogenic activities on protected marine habitats. My primary focus is fisheries, having been involved with the Lyme Bay reef recovery monitoring since 2010. I have also contributed to research on the impacts of marine renewable installations and marine litter. I completed my PhD which assessed the ecosystem impacts of commercial potting on both reef habitats the associated commercial fishery, within the Lyme Bay MPA. Currently, I am working as an assistant project manager at the University of Plymouth and as a consultant to the Blue Marine Foundation. I am coordinating research projects across multiple study sites throughout the UK, with the aim of providing conservation benefits and improving the sustainability of small-scale fisheries.
Qualifications
2018 - present: Postdoctoral researcher with Plymouth University and the Blue Marine Foundation
2014 - 2018: PhD researcher at Plymouth University
Attrill, M. J. (DoS), Sheehan, E. V. The ecological effects of increasing potting density in the Lyme Bay Marine Protected Area
2013 - 2014: Research assistant Plymouth University
2009 - 2011: BSc (hons) Marine Biology and Coastal Ecology
Professional membership
Marine Biological Association
Marine Conservation Society
British Ecological Society
Research
Research
Research interests
Marine ecology
Benthic ecosystems & recovery
Small-scale fisheries
Fishing impacts
Fisheries spatial data
Creative practice & artistic projects
British Ecological Society photography competition category winner
Publications
Publications
Key publications
Key publications are highlightedJournals
Articles
(2021) 'Rewilding of Protected Areas Enhances Resilience of Marine Ecosystems to Extreme Climatic Events' Frontiers in Marine Science 8, , DOI Open access
(2021) 'Optimal fishing effort benefits fisheries and conservation' Scientific Reports 11, (1) , DOI Open access
(2020) 'An evaluation of the social and economic impact of a Marine Protected Area on commercial fisheries' Fisheries Research 235, 105819-105819 , DOI Open access
(2019) 'Acoustic Complexity Index to assess benthic biodiversity of a partially protected area in the southwest of the UK' Ecological Indicators 111, , DOI Open access
(2017) 'Strandings of NE Atlantic gorgonians' Biological Conservation 209, 482-487 , DOI Open access
Chapters
Reports
(2019) The Lyme Bay experimental potting study. Department for Environment, Food and Rural Affairs Department for Environment, Food and Rural Affairs Open access | 319,538 |
Also see:
Ivan Suzman's commentary on the videos in the collection.
Biographical information given by Ivan Mfowethu Suzman
20 minutes.
Produced by the Washington Office on Africa, 1989
Interviews with Angolan leaders and people fourteen years after Independence from Portugal in 1975. Shows the devastation caused by the reactionary military organization, UNITA, covertly funded by your American taxes.
23 minutes.
Produced by the University of Southern Maine TV and Ivan Suzman
Directed by Jeffrey Phillips, 1989
Documentary of the Maine Project on Southern Africa's 1987-1989 18-month vigil at Citibank in Portland, Maine. Featuring MPOSA members and friends. The vigil caused at least $90,000.00 in withdrawals from citibank, which continues its banking services for South Africa.
31 minutes.
Produced by Randy Visser, 1987
Ivan Suzman's interview of anti-apartheid activist and African National Congress Representative, Victor Mashebela, during his 1987 tour of Maine. Filmed at Southern Maine Vocational and Technical College studio in South Portland.
2 hours 37 minutes.
Produced and Directed by Richard Attenborough, 1988
The story of the friendship between Steve Biko and Donald Woods. Biko's death in prison, and Woods' escape through Lesotho to England. Kevin Kline, Denzel Washington.
57 minutes.
Music by Abdullah Ibrahim
Produced by Robert Bilheimer and Ron Mix, 1988
The transformation of Beyers Naude, the Dutch Reformed Church leader, from trusted elite pastor to staunch supporter fo the struggle for freedom. naude is jailed, defrocked, harrassed, and banned. Academy Award nominee.
29 minutes.
Produced by Grenada TV
Directed by Caveat Briton, 1981
The tearful story of the forced relocation of four million black South Africans out of the so-called "Black Spots" and into the "Homelands" of Bantustans. Filmed clandestinely in the Ciskei and in Capetown and smuggled out. Includes an interview of anti-apartheid resistance leaders, and features many people forced into the Bantustans.
120 minutes.
Produced and Directed by Spike Lee, 1989
Lee's drama about predominantly black Bedford-Stuyvesant, New York. Includes a powerful closing scene comparing the messages of Malcolm X and Dr. Martin Luther King.
107 minutes.
Produced by Paula Weinstein
Directed by Euzhan Palcy, 1987
Thriller about the transformation of a white South African schoolteacher and his family, their harrassment by the South African police, the murder of his gardener's son. Starring Zakes Mokae, Donald Sutherland, Susan Sarandon, and Marlon Brando.
56 minutes.
Narrated by Ester Rolle
Produced and Directed by Marlon Riggs, 1987
A still-frame sequence demonstrating the pervasive racial sterotypes in American advertizing, consumer packaging, and books and newspapers, from the 1820s to the present.
39 minutes.
Produced and Directed by Barbara Brown, 1982
A slide and cassette show by Boston University African Studies specialist, Barbara Brown, about the lives of black women inSouth Africa. With special emphasis on the role of American companies in oppressing South African women.
39 minutes.
Produced by Chris Shepard
Directed by Claude Sauvageot, 1987
Contrasting stories of a 16-year-old black girl in Soweto and a 16-year-old white girlin suburban Johannesburg. Scenes of their homes, families and separate worlds are dramatically contrasted.
90 minutes.
Produced by Ian Hoblyn
Directed by Michael Lindsay-Hogg, 1987
Paul Simon's Graceland concert to 45,000 fans in Harare, Zimbabwe. Includes Miriam Makeba, Ladysmith Black Mambazo, and many more.
55 minutes.
1984
Archbishop Desmond M. Tutu is interviewed by church and standard media at the World Council of Churches assembly in Vancouver, British Columbia, Canada, after being nominated for the Nobel Peace Prize. Powerful and rare footage.
51 minutes.
1980
The story of the adoption of the Freedom Charter at Kliptown, South Africa, 26 June 1955. Includes rare historical footage of the Congress of the People. Black and white color sequences.
90 minutes.
February 11, 1988
Jesse Jackson's presidential race address at the University of Southern Maine, Portland, to a packed gymnasium.
98 minutes.
Produced by Howard Koch, Jr. and Dave Bell
Directed by Richard Pearce, 1991
The story of the 1955 Montgomery, Alabama, bus boycott and the friendship that developed between a housekeeper (Whoopi Goldberg) and her employer (Sissy Spacek).
79 minutes.
Produced by Globalvision
Directed by Rory O'Connor and Danny Schecter, 1990
Superb footage of the1990 release of Nelson Mandela and of his 27 years in prison.
50 minutes.
An ABC News production, 1990
A chronicle of the life of Nelson Mandela from his early years through his 1990 release from prison.
104 minutes.
Produced by Max Montocchio, with Oliver Schmitz
The first ever black South African feature film. Tells the story of a Soweto pickpocket, Panic, who becomes swept up in the resistance movement. In English, Zulu, Sotho and Afrikaans with English subtitles. Banned in South Africa and filmed in Soweto.
115 minutes.
1988
A stirring collection of all of Dr. Martin Luther King's great public speeches, including "I Have a Dream" and "I've Been to the Mountaintop." Includes Jimmy Carter, Andrew Young, Coretta Scott King, Bill Cosby, Ted Kennedy, and many more.
47 minutes.
Produced by William A. Elwood
Directed by mykola Kulish, 1990
The story of the desegregation of the American South, highlighted by a historical review of the 1954 Supreme Court case, Brown vs. Board of Education of Topeka, Kansas. Also gives the great black lawyer, Charles Hamilton Houston, the credit he deserves.
59 minutes.
Produced by University of Southern Maine TV and Ivan Suzman
Directed by Jeffrey Phillips, 1989
Compelling interview of South African exile and women's movement leader Rev. Motlapula Chabaku by Ivan Suzman. Tells the story of her early life, and of her dramatic three day escape from South Africa in 1979.
30 minutes.
Produced by Gerhard Schmidt, Chris Austin, and Ruth Weiss, 1980
Clandestine interviews with nine great South African women. Emphasizes the destruction of the black family under apartheid. Includes the first fimed interview with Winnie Mandela since 1960, and interviews Black Consciousness leader Numisi Khuzwayo and others.
29 minutes.
Produced by Afravision, 1986
The British filmmakers Afravision assembled this powerful footage about South Africa's townships from film smuggled out of Soweto and elsewhere. Documents police torture against township residents.
45 minutes.
Produced by Hart Perry
Directed by Godley & Creme, 1985
benefit film made by Artists United Against Apartheid. Features over 40 great musicians from Miles Davis to Bonnie Raitt singing and explaining their refusal to perform in South Africa's Sun City casino in Bophuphatswana.
29 minutes.
Produced by the South African Government, 1987
Propaganda film attacking the African National Congress, and associating it with communism.
58 minutes.
Narrated by Harry Belafonte
Produced by Jim Brown, Ginger Brown, Harold Leventhal, and George Stoney
Directed by Jim Brown, 1989
The inspiring true story of "We Shall Overcome," emphasizing the song's develoopment into the worldwide anthem of freedom. Includes historic and comtemporary footage. Starring Pete Seeger, Sweet Honey in teh Rock, Joan Baez, and many more. Emmy award winner for best documentary.
55 minutes.
Produced and directed by Sharon Sopher, with Kevin Harris, 1986
A painful documentary of the police terrorism and brutality directed against those who speak out against apartheid in South Africa. Includes an interview with Archbishop Desmond Tutu.
114 minutes.
Produced by Sarah Radclyffe
Directed by Chris Menges, 1988
Electrifying thriller about the tension between the South African heroine, Ruth First, and her daughter. Starring Barbara Hershey. A multiple award winner at the Cannes Film Festival.
55 minutes.
Produced and Directed by David Thompson, 1982
A fim adaptation of teh famous international hit satire about the return of Morena (Jesus Christ) to modern South Africa. Written by and starring Percy Mtwa and Mbongeni Ngena.
28 minutes.
Produced by Deborah May, 1981
The powerful story of the South African Women's Resistance Movement that developed after the 1952 attempt to extend the pass system to women. Features Lillian Ngoy, Helen Joseph, Frances Baard and other leading women. | 149,170 |
TITLE: Prove that the sequence $\{a_n \}_{n \geq 1}$ is convergent.
QUESTION [1 upvotes]: Suppose that a bounded sequence $\{a_n \}_{n \geq 1}$ is such that $$a_{n + 2} \leq \dfrac {1} {3} a_{n+1} + \dfrac {2} {3} a_n,\ \ \ \ \text {for}\ n \geq 1$$ Prove that the sequence $\{a_n \}_{n \geq 1}$ is convergent.
What I find is that for all $n \geq 1,$ we have $$a_{n + 2} - a_2 \leq \dfrac {2} {3} (a_1 - a_{n+1}).$$
Also it can't be eventually monotone increasing since for otherwise for all $n \geq 1,$ we have $$a_{n+2} - a_{n+1} \leq \dfrac {2} {3} (a_n - a_{n+1}) \leq 0 \implies a_{n+2} \leq a_{n+1},$$ a contradiction. So if the sequence is eventually monotone it has to be eventually monotone decreasing.
Is it of any importance? Thanks.
REPLY [4 votes]: With $c>0$ determined below, let $b_n=a_{n+1}+ca_n$. Then $\{b_n\}_{n\ge1}$ is also bounded from below and we obtain
$$b_{n+1}=a_{n+2}+ca_{n+1}\le \left(\frac13+c\right)a_{n+1}+\frac23 a_n=\frac{(1+3c)a_{n+1}+2a_n}{3}.$$
If we smartly pick $c=\frac23$, this amounts to
$$ b_{n+1}\le b_n.$$
We conclude that sequence $b_n$ converges to some limit $b$.
By applying $\liminf$ to both sides of the equation $a_{n+1}=b_n-\frac23 a_n$, we find
$$\tag1\liminf a_n=b-\frac23\limsup a_n $$
and similarly
$$\tag2\limsup a_n=b-\frac23\liminf a_n.$$
Solving the linear equations $(1)$ and $(2)$, we arrive at
$$ \liminf a_n=\limsup a_n=\frac 53 b,$$
i.e.,
$$\lim a_n=\frac 53 b. $$ | 83,986 |
TITLE: Prove that if $G$ is a group of order $39$ then $G$ has a subgroup of order $3$
QUESTION [3 upvotes]: I was able to show this by first proving $G$ requires and element of order $3$. However I am looking for alternative proofs without the use of Sylow theorems or Cauchy's theorem. Any hints would be appreciated.
REPLY [11 votes]: If there is no subgroup of order 3, then there are no elements of order 3. The only other possible orders are 1, 13, and 39, and there can be no elements of order 39 (otherwise there is an element of order 3). Thus, every element has order 1 or 13. Since 13 is prime, the distinct subgroups generated by the elements of the group intersect trivially, and these subgroups cover the group. Thus, there are $12n+1$ elements, for some $n$. Since 39 is not of this form, this is a contradiction.
REPLY [2 votes]: Let $g$ be an element of $G$ other than $e$. It has order $3$, $13$ or $39$.
If the order is $3$, then you are done. If it's $39$ then $g^{13}$ has order $3$.
If the order is $13$ then $g^k$ is a generator of $\langle g\rangle$ for any $k=1,\ldots,12$.
Let $h$ be an element of $G$ that is not in $\langle g\rangle$. Again it's order is $3$, $13$ or $39$ and the only concerning case is if the order is $13$. Then $h^k$ is a generator of $\langle h\rangle$ for $k = 1,\ldots, 12$ and since $h$ is not in $\langle g\rangle$ then it follows that $\langle g\rangle \cap \langle h\rangle = \{e\}$.
We consider now an element $i$ that is not in $\langle g\rangle \cup \langle h\rangle$. We have the same scenario as before and if its order is $13$ then the only element of $\langle i\rangle$ in common with $\langle g\rangle \cup \langle h\rangle$ is $e$. That means that $\langle g\rangle \cup \langle h\rangle\cup \langle i\rangle$ has $37$ elements. The two remaining elements cannot have order $13$ or $1$, and therefore have order $3$ or $39$ and we are done. | 210,641 |
TITLE: What, if anything, goes wrong in this supposed counterexample to antecedent strengthening?
QUESTION [1 upvotes]: According to Approach0, this is new to MSE.
In classical logic, there is the notion of antecedent strengthening; namely:
$$(A\to B)\to((A\land C)\to B)$$
is valid. A proof via tableau is given below.
It was generated here.
However, in a document on nonclassical logic (which can be found here), the following "counterexample" is given:
If Romney wins the election, he'll be sworn in in January. Therefore, if Romney wins the election and dies of a heart attack the same night, he'll be sworn in in January.
Here $A$ is "Romney wins the election", $B$ is "Romney will be sworn in in January", and $C$ is "Romney dies of a heart attack the same night as he wins the election".
My question is:
How does classical logic handle this supposed counterexample? What, if anything, is wrong with it?
I have a longstanding interest in nonclassical logics. See here for instance.
I doubt I could answer this myself. I don't want to end up a crank or anything, so I'm looking for an answer with full proofs or at least references.
I hope I have provided enough context.
Please help :)
REPLY [1 votes]: You are questioning the validity of $(A\to B)\to((A\land C)\to B)$ when $C$ implicitly contradicts $B$. Well, let's consider when it explicitly does so; namely when $C$ is $\lnot B$.
For any statement $\varphi$ we consider $\varphi\to(\lnot\varphi\to B)$ to be valid, by way of the Principle of Explosion (aka ex falso quodlibet, EFQ). When the premises contain a contradiction, then anything may be derived.
$$\begin{split}\varphi\,,\lnot\varphi&\vDash B\\[3ex]\therefore\qquad &\vDash \varphi\to(\lnot\varphi\to B)\end{split}$$
Well, since this holds for any statement $\varphi$, let's consider when $\varphi$ is $(A\to B)$. Then $\lnot\varphi$ is equivalent to $A\land\lnot B$. So by substitution we have : $$\vDash (A\to B)\to((A\land\lnot B)\to B)$$ | 34,140 |
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The meat we get from factory farms is a pandemic risk, too - Vox, posted 23 Apr by peter in environment food health.”
Defying International Community, Japan Resumes Commercial Whaling, posted Jul '19 by peter in environment food japan.
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Raze, rebuild, repeat: why Japan knocks down its houses after 30 years | Cities | The Guardian, posted 2017.. | 321,370 |
TITLE: On a matrix problem in the field $\mathbb F_2$
QUESTION [7 upvotes]: Given $M$ a symmetric matrix in $\mathbb F_2^{n\times n}$ having $\mathsf{det}_\mathbb R(M)\neq0$ (non-singular in reals) and satisfying $PMP'=(M+J+I)$ or $P(M+J+I)P'=M$ where $P$ is a permutation matrix in $\mathbb F_2^{n\times n}$ and where $J$ is all $1$s and $I$ is identity matrix in $\mathbb F_2^{n\times n}$ ($'$ is transpose and so $P'=P^{-1}$ holds).
$\mathsf{Det}(M)=\mathsf{Det}(M+J+I)$ and $\mathsf{Det}(M+I)=\mathsf{Det}(M+J)$ are satisfied as $\mathsf{Det}(P)\equiv1\bmod2$.
Can $M,M+I\not\in\mathsf{SL}(n,\mathbb F_2)$ be simultaneously impossible?
Is there an example of $\mathsf{det}(M)+\mathsf{det}(M+I)\equiv1\bmod2$?
Update Antoine Labelle's comments below suggests either $M,M+I\in\mathsf{SL}(n,\mathbb F_2)$ or $M,M+I\not\in\mathsf{SL}(n,\mathbb F_2)$ holds and he has no situation $\mathsf{det}(M)+\mathsf{det}(M+I)\equiv1\bmod2$.
REPLY [1 votes]: Most of the question has already been addressed. As regards the last part, I just poit out that if ${\rm det} M = 0$ and ${\rm det}(M+I) = 1$, then we must have ${\rm rank}(M) = n-1$ ( I mean here the $\mathbb{F}_{2}$ rank). This follows since $M + I + J$ is (by assumption) similar to $M$, and hence has the same rank. If that rank is $n-2$ or less, then the null space of $M+I+J$ is at least two-dimensional, and the null space of $J$ is $n-1$ dimensional. Hence there is a non-zero column vector $v$ with $(M+I+J)v = 0$ and $Jv = 0.$ Hence $(M+I)v = 0,$ so that ${\rm det}(M+I) = 0.$ | 157,060 |
Open iTunes to preview, buy, and download music.
Customer Reviews
Zazie is Ze Best French Singer
I'm very glad we can finally find Zazie on ITunes. I'm french but live in the US and have been a fan since her second album 'Zen' in 1996. She has numerous CDs out there, the first one was 'Je, Tu, Ils' (she got a music award with it) which is more accoustic to me, then 'Zen', 'Made in Love', 'La Zizanie', 'Rodeo' and 'Totem'. La Zizanie is definitely one of the best CDs she's made and showcases her music very well. This album and Zen, are actually my favorites so far (I haven't heard enough songs from Totem yet to change my mind ;-)). It combines less serious songs like 'Rue de la paix' and more engaged songs like 'Adam et Yves', 'Qui m'aime me fuit' and 'si j'etais moi'. The live version of 'Si j'etais moi' (on her live album) just gives me chills !. Zazie is a very genuine person, works with various artists and has made several duets with great (and sometimes unknown) artists in live performances. She also performs every year with other artists for 'Les restos du coeur', a charity gathering food for poor or homeless people. Hopefully we'll see more of Zazie on Itunes Us, so everybody can enjoy this great artist, but if you want to know about her very good french female singer, you have to get this album.
Welcome to America Zazie!
Download this record now. Zazie is a French pop star. I'm glad iTunes is starting to sell her music. Her best album is Rodeo, watch for it. Her new album is called Totem. It's not available in the US but who knows. Zazie is freaking cool.
GIVE US MORE!!!
finally! except I already own this cd!!! please release more of Zazie's music!!
Biography
Born: April 18, 1964 in Boulogne-Billancourt, France
Genre: French Pop
Years Active: '90s, '00s, '10s
Top Albums and Songs by Zazie
- $9.99
- Genres: French Pop, Music, Electronic, Rock, World, Europe, Pop, Britpop, Dance
- Released: Apr 15, 2002
- ℗ 2001 Mercury Music Group | 214,408 |
At Children's Hospital of Philadelphia .
Since 1855, CHOP.
At CHOP, we provide an environment that's as nurturing as it is dynamic. Our team-oriented approach allows for ample learning and career growth opportunities, while our hospital offers the ideal atmosphere for using. | 49,686 |
21 Marzo 2017 A SWISS WEDDING Last summer I had the pleasure to take photos in my first Swiss Wedding. It was a simple and elegant marriage in the hills among Lugano, Mandrisio and Varese. Everything has been planned by talented and nice Laura Dova Wedding Planner that handled every moment with serenity and | 320,192 |
\begin{document}
\title{
Spin covers of maximal compact subgroups of Kac--Moody groups and spin-extended Weyl groups}
\author{David Ghatei}
\author{Max Horn}
\author{Ralf K\"ohl}
\author{Sebastian Wei\ss}
\address{D.G.: University of Birmingham, School of Mathematics, Edgbaston, Birmingham, B15 2TT, United Kingdom}
\address{M.H., R.K., S.W.: JLU Giessen, Mathematisches Institut, Arndtstrasse 2, 35392 Giessen, Germany}
\email{[email protected]}
\email{[email protected] }
\maketitle
\begin{abstract}
Let $G$ be a split real Kac--Moody group of arbitrary type and let $K$ be its maximal compact subgroup, i.e. the
subgroup of elements fixed by a Cartan--Chevalley involution of $G$.
We construct non-trivial spin covers of $K$, thus confirming a conjecture by Damour and Hillmann \cite{DamourHillmann}. For irreducible simply laced diagrams and for all spherical diagrams these spin covers are two-fold central extensions of $K$. For more complicated irreducible diagrams these spin covers are central extensions by a finite $2$-group of possibly larger cardinality.
Our construction is amalgam-theoretic and makes use of the generalized spin representations of maximal compact subalgebras of split real Kac--Moody algebras studied in \cite{Hainke/Koehl/Levy}.
Our spin covers contain what we call spin-extended Weyl groups which admit a presentation by generators and relations obtained from the one for extended Weyl groups by relaxing the condition on the generators so that only their eighth powers are required to be trivial.
\end{abstract}
\section{Introduction} \label{sec:intro}
In \cite[Section~3.5]{DamourHillmann} it turned out that the existence of a spin-extended Weyl group $\Wspin(E_{10})$ would be very useful for the study of fermionic billards. Lacking a concrete mathematical model of that group $\Wspin(E_{10})$, Damour and Hillmann in their article instead use images of $\Wspin(E_{10})$ afforded by various generalized spin representations as described in \cite{DamourKleinschmidtNicolai}, \cite{deBuylHenneauxPaulot}, which can be realized as matrix groups.
In \cite[Section~3.5, footnote 18, p.\ 24]{DamourHillmann}, Damour and Hillmann conjecture that the spin-extended Weyl group $\Wspin(E_{10})$ can be constructed as a discrete subgroup of a double spin cover $\Spin{E_{10}}$ of the subgroup $K(E_{10})$ of elements fixed by the Cartan--Chevalley involution of the split real Kac--Moody group of type $E_{10}$.
The purpose of this article is to confirm this conjecture, and to generalize it to arbitrary diagrams resp.
arbitrary generalized Cartan matrices
In the simply laced case our result is as follows:
\begin{mainthm} \label{mainthm:sl-spincover}
Let $\Pi$ be an irreducible simply laced Dynkin diagram, i.e., a Dynkin diagram affording only single edges, let $I = \{ 1, \ldots, n \}$ be a set of labels of the vertices of $\Pi$, and let $K(\Pi)$ be the subgroup of elements fixed by the Cartan--Chevalley involution of the split real Kac--Moody group of type $\Pi$. For each $i \in I$ let $G_i \cong \Spin{2}$ and for each $i \neq j \in I$ let
\[G_{ij} \cong
\begin{cases}
\Spin{3}, & \text{if $i$, $j$ form an edge of $\Pi$,} \\
(\Spin{2} \times \Spin{2})/\gen{(-1,-1)}, &
\text{if $i$, $j$ do not form an edge of $\Pi$.}
\end{cases}\]
Moreover, for $i < j \in I$, let $\phi_{ij}^i : G_i \to G_{ij}$ be the standard embedding as ``upper-left diagonal block'' and $\phi_{ij}^j : G_j \to G_{ij}$ be the standard embedding as ``lower-right diagonal block''.
Then up to isomorphism there exists a uniquely determined group, denoted $\Spin{\Pi}$, whose multiplication table extends the partial multiplication provided by $\left( \bigsqcup_{i < j \in I} G_{ij} \right) / \sim$ where $\sim$ is the equivalence relation determined by $\phi_{ij}^i(x) \sim \phi_{ik}^i(x)$ for all $i \neq j, k \in I$ and $x \in G_i$.
Furthermore, there exists a canonical two-to-one central extension $\Spin\Pi \to K(\Pi)$.
\end{mainthm}
\noindent
The system $\{ G_i, G_{ij}, \phi_{ij}^i \}$ is called an \Defn{amalgam of groups}, the pair consisting of the group $\Spin\Pi$ and the set of canonical embeddings $\tau_{i} : G_{i} \hookrightarrow \Spin\Pi$, $\tau_{ij} : G_{ij} \hookrightarrow \Spin\Pi$ a \Defn{universal enveloping group}; the canonical embeddings are called \Defn{enveloping homomorphisms}. Formal definitions and background information concerning amalgams can be found in Section~\ref{sec:amalgams}. Since all $G_i\cong\Spin 2$ are isomorphic to one another, it in fact suffices to fix one group $U\cong \Spin 2$ instead with connecting homomorphisms $\phi_{ij}^i : U \to G_{ij}$.
The formalization of the concept of standard embedding as ``upper-left/lower right diagonal block'' can be found in Section~\ref{sec:spin2amalgams}. Note that, since the $G_i$ are only given up to isomorphism, these standard embeddings are only well-defined up to automorphism of $G_i$, which leads to some ambiguity. Since by \cite{Hartnick/Koehl/Mars} the group $K(\Pi)$ (and therefore each of its central extensions by a finite group) is a topological group, one may assume the $\phi_{ij}^i$ to be continuous, thus restricting oneself to the ambiguity stemming from the two continuous automorphisms of $\Spin 2$, the identity and the inversion homomorphisms. This ambiguity is resolved in Section~\ref{sec:spin2amalgams}.
\smallskip
Theorem~\ref{mainthm:sl-spincover} provides us with the means of characterizing $\Wspin(\Pi)$.
\begin{mainthm} \label{mainthm:sl-weyl}
Let $\Pi$ be an irreducible simply laced Dynkin diagram, let $I = \{ 1, \ldots, n \}$ be a set of labels of the vertices of $\Pi$, and for each $i \in I$ let
\begin{itemize}
\item $\tau_i : G_i \cong \Spin 2 \hookrightarrow \Spin\Pi$ be the canonical enveloping homomorphisms,
\item $x_i \in G_i$ elements of order eight whose polar coordinates involve
the angle $\frac{\pi}{4}$, and
\item $r_i := \tau_i(x_i)$.
\end{itemize}
Then $\Wspin(\Pi) := \langle r_i \mid i \in I \rangle$ satisfies the defining relations
\begin{align}
r_i^8&=1, \tag{R1} \\
r_i^{-1} r_j^2 r_i &= r_j^2 r_i^{2n(i,j)}
\ \qquad\text{ for } i\neq j\in I, \tag{R2} \\
\underbrace{r_i r_j r_i \cdots}_{m_{ij} \text{ factors}} &=
\underbrace{r_j r_i r_j \cdots}_{m_{ij} \text{ factors}}
\qquad\text{ for } i\neq j\in I, \tag{R3}
\end{align}
where \[
m_{ij} = \begin{cases}
3, & \text{if $i$, $j$ form an edge,} \\
2, & \text{if $i$, $j$ do not form an edge,}
\end{cases}
\quad \text{and} \quad
n(i,j) =
\begin{cases}
1, & \text{if $i$, $j$ form an edge,} \\
0, & \text{if $i$, $j$ do not form an edge.}
\end{cases}
\]
\end{mainthm}
To be a set of defining relations means that any product of the $r_i$ that in $\Wspin(\Pi)$ represents the identity can be written as a product of conjugates of ways of representing the identity via (R1), (R2), (R3).
\medskip
Our results in fact can be extended to arbitrary diagrams as discussed in Sections~\ref{sec:adm-amalgams}, \ref{sec:tametypes}, and \ref{extended}.
\medskip
As a by-product of our proof of Theorem~\ref{mainthm:sl-spincover} we show in Section~\ref{sec13} that for non-spherical diagrams $\Pi$ the groups $\Spin\Pi$ and $K(\Pi)$ are never simple; instead they always admit a non-trivial compact Lie group as a quotient via the generalized spin representation described in \cite{Hainke/Koehl/Levy}. The generalized spin representations of $\Spin\Pi$ is continuous, so that the obtained normal subgroups are closed.
Similar non-simplicity phenomena as abstract groups have been observed in \cite{CapraceHume}. Furthermore, we observe that for arbitrary simply laced diagrams the image of $\Wspin$ under the generalized spin representation is finite, generalizing \cite[Lemma~2, p.~49]{DamourHillmann}.
\medskip
Sections~\ref{sec:amalgams}, \ref{sec:cartan-dynkin}, \ref{sec:SO-O}, \ref{sec:spin-pin}, \ref{sec:7} are introductory in nature; we revise the notions of amalgams, Cartan matrices and Dynkin diagrams and fix our notation for orthogonal and spin groups. Sections~\ref{sec:so2amalgams} and \ref{sec:spin2amalgams} deal with the classification theory of amalgams and, as a blueprint for Theorem~\ref{mainthm:sl-spincover}, identify $\SO n$ and $\Spin n$ as universal enveloping groups of $\SO 2$-, resp.\ $\Spin 2$-amalgams of type $A_{n-1}$. In Section~\ref{sec:spin-cover-simply-laced} we prove Theorem~\ref{mainthm:sl-spincover}.
Sections~\ref{sec:g2}, \ref{sec:bc2}, \ref{sec:rank-2-residues} provide us with the necessary tools for generalizing our findings to arbitrary diagrams; they deal with equivariant coverings of the real projective plane by the split Cayley hexagon and the symplectic quadrangle and with coverings of the real projective plane and the symplectic quadrangle by trees. In Section~\ref{sec:adm-amalgams} we study $\SO 2$- and $\Spin 2$-amalgams for this larger class of diagrams. Section~\ref{sec:tametypes} deals with the general version of Theorem~\ref{mainthm:sl-spincover}. Section~\ref{sec14} deals with the proof of Theorem~\ref{mainthm:sl-weyl} and its generalization. In Section~\ref{sec13} we observe that our findings provide epimorphisms from $\Spin\Pi$ and $K(\Pi)$ onto non-trivial compact Lie groups.
\medskip
\textbf{Acknowledgements.} We thank Thibault Damour for pointing out his conjecture to us and Arjeh Cohen for very many valuable discussions concerning maximal compact subgroups of split real Lie groups of rank two. We also thank Guntram Hainke and Paul Levy for their comments and ideas concerning spin covers and Pierre-Emmanuel Caprace for several comments on a preliminary version of this article. The third named author moreover gratefully acknowledges the hospitality of the IHES in Bures-sur-Yvette and the Albert Einstein Institute in Golm. This research has been partially funded by the EPRSC grant
EP/H02283X.
\tableofcontents
\part{Basics}
\section{Conventions}
\begin{notation}
$\NN:=\{1,2,3,\ldots\}$ denotes the set of positive integers.
\end{notation}
\begin{notation}
Throughout this article we use the convention $ij:=\{i,j\}$ if the set $\{i,j\}$ is used as an index. For example, if $G_{ij}$ is a group, then $G_{ji}$ is the same group.
Note that this does not apply to superscripts, so $G^{ij}$ and $G^{ji}$ may differ.
\end{notation}
\begin{notation}
For any group $G$, consider the following maps:
\begin{alignat*}{2}
\inv:&\ G\to G : x\mapsto x^{-1}, &&\quad\text{the inverse map}, \\
\sq:&\ G\to G : x\mapsto x^2, &&\quad\text{the square map}.
\end{alignat*}
Both maps commute with any group homomorphism.
\end{notation}
\begin{notation}
For any group $G$, we denote by $Z(G)$ the \Defn{centre} of $G$.
\end{notation}
\section{Amalgams} \label{sec:amalgams}
In this section we recall the concept of amalgams. More details concerning this concept can, in various formulations, be found in \cite[Part~III.$\mathcal{C}$]{Bridson/Haefliger:1999}, \cite[Section~1.3]{Ivanov/Shpectorov:2002}, \cite[Section~1]{Gloeckner/Gramlich/Hartnick:2010}.
\begin{definition} Let $U$ be a group, and $I\neq\emptyset$ a set.
A \Defn{$U$-amalgam over $I$} is a set
\[\AAA=\{ G_{ij},\; \phi_{ij}^i \mid i\neq j\in I \}\]
such that $G_{ij}$ is a group and $\phi_{ij}^i:U\to G_{ij}$ is a monomorphism for all $i\neq j\in I$.
The maps $\phi^i_{ij}$ are called \Defn{connecting homomorphisms}.
The amalgam is \Defn{continuous} if $U$ and $G_{ij}$ are topological groups, and
$\phi_{ij}^i$ is continuous for all $i\neq j\in I$.
\end{definition}
\begin{definition} \label{def:ama-iso}
Let $\wt\AAA=\{ \wt{G}_{ij},\; \wt{\phi}_{ij}^i \mid i\neq j\in I \}$ and
$\AAA=\{ G_{ij},\; \phi_{ij}^i \mid i\neq j\in I \}$
be $U$-amalgams over $I$. An \Defn{epimorphism}, resp.\ \Defn{isomorphism} $\alpha:\wt\AAA\to\AAA$
of $U$-amalgams is a system
\[\alpha=\{ \pi,\; \alpha_{ij} \mid i\neq j \in I \}\]
consisting of a permutation $\pi\in\Sym(I)$ and group epimorphisms, resp.\ isomorphisms
\[
\alpha_{ij}:\wt{G}_{ij}\to G_{\pi(i)\pi(j)}
\]
such that for all $i \neq j \in I$
\[ \alpha_{ij}\circ \wt{\phi}_{ij}^i=\phi_{\pi(i)\pi(j)}^{\pi(i)}\ , \]
that is, the following diagram commutes:
\[
\xymatrix{
& \wt{G}_{ij} \ar[dd]^{\alpha_{ij}} \\
U \ar[rd]_{\phi_{\pi(i)\pi(j)}^{\pi(i)}} \ar[ru]^{\wt{\phi}_{ij}^i} \\
& G_{\pi(i)\pi(j)}
}
\]
More generally, let $\wt\AAA=\{ \wt{G}_{ij},\; \wt{\phi}_{ij}^i \mid i\neq j\in I \}$ be a $U$-amalgam and let
$\AAA=\{ G_{ij},\; \phi_{ij}^i \mid i\neq j\in I \}$ be a $V$-amalgam.
An \Defn{epimorphism} $\alpha : \wt\AAA \to \AAA$ is a system
\[\alpha=\{ \pi,\;\rho^i,\;\alpha_{ij} \mid i\neq j \in I \}\]
consisting of a permutation $\pi\in\Sym(I)$, group epimorphisms $\rho^i:U\to V$ and group epimorphisms
\[
\alpha_{ij}:\wt{G}_{ij}\to G_{\pi(i)\pi(j)}
\]
such that for all $i \neq j \in I$
\[ \alpha_{ij}\circ \wt{\phi}_{ij}^i=\phi_{\pi(i)\pi(j)}^{\pi(i)} \circ \rho^{\pi(i)} \ , \]
that is, the following diagram commutes:
\[
\xymatrix{
U \ar[d]_{\rho^{\pi(i)}} \ar[r]^{\wt{\phi}_{ij}^i} & \wt{G}_{ij} \ar[d]^{\alpha_{ij}} \\
V \ar[r]_{\phi_{\pi(i)\pi(j)}^{\pi(i)}} & G_{\pi(i)\pi(j)}
}
\]
\end{definition}
\begin{notation}
If (and only if) in the epimorphism $\alpha : \wt{\AAA} \to \AAA$ each $\rho^i : U \to V$ is an isomorphism, then one obtains an epimorphism $\alpha' : \wt{\AAA} \to \AAA'$ of $U$-amalgams by defining $\alpha' = \{ \pi, \alpha_{ij} \mid i \neq j \in I \}$ and $\AAA' = \{ G_{ij}, (\phi^i_{ij})' \}$ via \[(\phi^i_{ij})' : U \to G_{ij} : u \mapsto (\phi^{i}_{ij} \circ \rho^i)(u) ..\]
If this $\alpha'$ turns out to be an isomorphism of $U$-amalgams, by slight abuse of terminology we also call the epimorphism $\alpha$ an \Defn{isomorphism} of amalgams.
\end{notation}
\begin{remark} \label{altiso}
More generally, an amalgam can be defined as a collection of groups $G_i$ and a collection of groups $G_{ij}$ with connecting homomorphisms $\psi^i_{ij} : G_i \to G_{ij}$. Since in our situation for all $i$ there exist isomorphisms $\gamma_i : U \to G_i$, it suffices to consider the connecting homomorphisms $\phi^i_{ij} = \psi^i_{ij} \circ \gamma_i$.
In the more general setting, an isomorphism of amalgams consists of a permutation $\pi$ of the index set $I$ and isomorphisms $\alpha_i : G_i \to \ol{G}_{\pi(i)}$ and $\alpha_{ij} : G_{ij} \to \ol{G}_{\pi(i)\pi(j)}$ such that
\[ \alpha_{ij} \circ \psi^i_{ij} = \ol{\psi}^{\pi(i)}_{\pi(i)\pi(j)} \circ \alpha_i\ .\]
A routine calculation shows that $U$-amalgams and isomorphisms of $U$-amalgams are special cases
of amalgams and isomorphisms of amalgams as found in the literature.
\end{remark}
\begin{definition}
Given a $U$-amalgam $\AAA=\{ G_{ij},\; \phi_{ij}^i \mid i\neq j\in I \}$,
an \Defn{enveloping group} of $\mathit{\AAA}$ is a pair $(G,\tau)$ consisting of a group $G$ and a set
\[\tau = \{ \tau_{ij}\mid i\neq j\in I \}\]
of \Defn{enveloping homomorphisms} $\tau_{ij}:G_{ij}\to G$ such that
\begin{align*}
G=\gen{\tau_{ij}(G_{ij}) \mid i\neq j\in I}\ , &&\ \forall\ i\neq j\neq k\in I:\ \tau_{ij}\circ \phi_{ij}^j=\tau_{kj}\circ \phi_{kj}^j\ ,
\end{align*}
that is, for $i\neq j\neq k\in I$ the following diagram commutes:
\[
\xymatrix{
& G_{ij} \ar[dr]^{\tau_{ij}} \\
U \ar[dr]_{\phi_{kj}^j} \ar[ur]^{\phi_{ij}^j} && G \\
& G_{kj} \ar[ur]_{\tau_{kj}}
}
\]
We write $\tau : \AAA \to G$ and call $\tau$ an \Defn{enveloping morphism}.
An enveloping group $(G,\tau)$ and the corresponding enveloping morphism are \Defn{faithful} if $\tau_{ij}$ is a monomorphism for all $i\neq j\in I$.
\end{definition}
\begin{definition}
Given a $U$-amalgam $\AAA=\{ G_{ij},\; \phi_{ij}^i\}$, an enveloping group
$\big(G,\tau\big)$ is called a \Defn{universal enveloping group} if,
given an enveloping group $(H,\tau')$ of $\AAA$, there is a unique
epimorphism $\pi:G\to H$ such that for all $i \neq j \in I$ one has
$\pi\circ \tau_{ij}=\tau'_{ij}$. We write $\tau : \AAA \to G$ and call $\tau$ a \Defn{universal enveloping morphism}. By universality, two universal enveloping groups $(G_1,\tau_1)$ and $(G_2,\tau_2)$ of a $U$-amalgam $\AAA$ are (uniquely) isomorphic.
The \Defn{canonical universal enveloping group} of the $U$-amalgam $\AAA$
is the pair $\big(G(\AAA),\wh\tau\big)$, where
$G(\AAA)$ is the group given by the presentation
\[G(\AAA):=\left\langle \bigcup_{i\neq j\in I} G_{ij} \mid \text{all relations in } G_{ij}, \text{ and }
\forall\ i\neq j\neq k\in I, \forall x\in U:
\phi_{ij}^j(x)=\phi_{kj}^j(x) \right\rangle\]
and where $\wh\tau=\{\wh\tau_{ij}\mid i\neq j\in I\}$ with the canonical homomorphism $\wh\tau_{ij}:G_{ij}\to G(\AAA)$ for all $i\neq j\in I$. The canonical universal enveloping group of a $U$-amalgam is a universal enveloping group (cf.\ \cite[Lemma~1.3.2]{Ivanov/Shpectorov:2002}).
\end{definition}
\begin{lemma}
Let $U$ and $V$ be groups and $I$ an index set. Suppose
\begin{itemize}[itemsep=1mm plus 0.5mm minus 0.5mm]
\item $\wt\AAA=\{ \wt{G}_{ij},\; \wt{\phi}_{ij}^i \mid i\neq j\in I \}$ is a $U$-amalgam over $I$,
\item $\AAA=\{ G_{ij},\; \phi_{ij}^i \mid i\neq j\in I \}$ is a $V$-amalgam over $I$,
\item $\alpha=\{ \pi,\;\rho^i,\;\alpha_{ij} \mid i\neq j\in I \}$ is an amalgam epimorphism $\wt\AAA\to\AAA$,
\item $(G,\tau)$ with $\tau=\{ \tau_{ij} \mid i\neq j\in I \})$ is an enveloping group of $\AAA$.
\end{itemize}
Then the following hold:
\begin{enumerate}
\item\label{lem:ama-envelope-lift}
There is a unique enveloping group $(G, \wt \tau)$, $\wt \tau = \{ \wt\tau_{ij} \mid i\neq j\in I \}$, of $\wt\AAA$ such that the following
diagram commutes for all $i\neq j\in I$:
\[
\xymatrix{
\wt{G}_{ij} \ar@{-->}[rrd]^{ \wt\tau_{ij}} \ar[d]_{\alpha_{ij}} \\
G_{ij} \ar[rr]^{\tau_{ij}} && G
}
\]
\item \label{lem:ama-epi-induces-envelope-epi}
Suppose $(\wt{G}, \wt \tau)$, $\wt \tau = \{ \wt\tau_{ij} \mid i\neq j\in I \}$, is a universal enveloping group of $\wt\AAA$.
Then there is a unique epimorphism $\wh\alpha:\wt{G}\to G$
such that the following diagram commutes for all $i\neq j\in I$:
\[
\xymatrix{
\wt{G}_{ij} \ar[rr]^{\wt\tau_{ij}} \ar[d]_{\alpha_{ij}} &&
\wt{G} \ar@{-->}[d]^{\wh\alpha} \\
G_{ij} \ar[rr]^{\tau_{ij}} && G
}
\]
\item If $\alpha$ is an isomorphism and $(G,\tau)$ is a universal enveloping group,
then also $\wh\alpha$ is an isomorphism.
\end{enumerate}
\end{lemma}
\begin{proof}
\begin{enumerate}
\item
Let $i\neq j\in I$.
Since $\alpha_{ij}$ is an epimorphism, we must have $\wt\tau_{ij}:=\tau_{ij}\circ \alpha_{ij}$
for the diagrams to commute; the claimed uniqueness follows.
The fact that $\alpha_{ij}$ is an epimorphism also implies
\[ \wt\tau_{ij}(\wt{G}_{ij})=(\tau_{ij}\circ \alpha_{ij})(\wt{G}_{ij}) = \tau_{ij}(G_{ij})\ ,
\qquad\text{ and so }\qquad
G=\gen{\tau_{ij}(\wt{G}_{ij})}=\gen{\wt\tau_{ij}(\wt{G}_{ij})}\ ..\]
Moreover, for $i\neq j\neq k\in I$ we find
\begin{align*}
\wt\tau_{ij}\circ \wt{\phi}_{ij}^j
= \tau_{ij}\circ \alpha_{ij} \circ \wt{\phi}_{ij}^j
&= \tau_{ij}\circ \phi_{\pi(i)\pi(j)}^{\pi(j)} \circ \rho^{\pi(j)} \\
&= \tau_{kj}\circ \phi_{\pi(k)\pi(j)}^{\pi(j)} \circ \rho^{\pi(j)}
= \tau_{kj}\circ \alpha_{kj} \circ \wt{\phi}_{kj}^j
= \wt\tau_{kj}\circ \wt{\phi}_{kj}^j\ .
\end{align*}
Hence $(G,\{ \wt\tau_{ij} \})$ is indeed an enveloping group of $\wt\AAA$.
\item
On the one hand, by (a) the lower left triangle in the following diagram commutes:
\[
\xymatrix{
\wt{G}_{ij} \ar[rr]^{\tau_{ij}} \ar@{-->}[rrd]^{\ol\tau_{ij}\circ\alpha_{ij}} \ar[d]_{\alpha_{ij}} &&
G \ar@{-->}[d]^{\exists!\wh\alpha} \\
G_{ij} \ar[rr]^{\ol\tau_{ij}} && \ol{G}
}
\]
On the other hand, by the definition of universal enveloping group
there is a unique epimorphism $\wh\alpha$ making the upper right triangle
commute. The claim follows.
\item follows from (b) by exchanging the roles of $G_{ij}$, $G$ and $\widetilde G_{ij}$, $\widetilde G$.
\qedhere
\end{enumerate}
\end{proof}
\begin{notation} \label{nota:amalgamcomm}
We denote the situation in Lemma~\ref{lem:ama-envelope-lift} by the commutative diagram
\[
\xymatrix{
\wt{\mathcal{A}} \ar[rrd]^{ \wt\tau} \ar[d]_{\alpha} \\
\mathcal{A} \ar[rr]^{\tau} && G
}
\]
and the situation in Lemma~\ref{lem:ama-epi-induces-envelope-epi} by the commutative diagram
\[
\xymatrix{
\wt{\mathcal{A}} \ar[rr]^{\wt\tau} \ar[d]_{\alpha} &&
\wt{G} \ar[d]^{\wh\alpha} \\
\mathcal{A} \ar[rr]^{\tau} && G
}
\]
\end{notation}
The following proposition will be crucial throughout this article. The typical situation in our applications will be $U=\SO{2}$, $\wt{U}=\Spin{2}$, $\wt{V}=\{\pm1\}$.
\begin{proposition} \label{prop:env-grp-central-cover}
Let $U$, $\wt{U}$ and $\wt{V}\leq\wt{U}$ be groups and $I$ an index set. Suppose
\begin{itemize}[itemsep=1mm plus 0.5mm minus 0.5mm]
\item $\wt\AAA=\{ \wt{G}_{ij},\; \wt{\phi}_{ij}^i \mid i\neq j\in I \}$ is a $\wt{U}$-amalgam over $I$ such that $\wt{G}_{ij}=\gen{\wt\phi_{ij}^i(\wt{U}), \wt\phi_{ij}^j(\wt{U})}$,
\item $\AAA=\{ G_{ij},\; \phi_{ij}^i \mid i\neq j\in I \}$ is a $U$-amalgam over $I$,
\item $\alpha=\{ \pi,\;\rho^i,\;\alpha_{ij} \mid i\neq j\in I \}$ is an amalgam epimorphism $\wt\AAA\to\AAA$,
\item $(\wt{G},\wt\tau)$ with $\wt\tau=\{ \wt\tau_{ij} \mid i\neq j\in I \})$ is a universal enveloping group of $\wt\AAA$,
\item $(G,\tau)$ with $\tau=\{ \tau_{ij} \mid i\neq j\in I \})$ is a universal enveloping group of $\AAA$,
\item $\wh\alpha:\wt{G}\to G$ is the epimorphism induced by $\alpha$
via the commutative diagrams ($i \neq j \in I$)
\[
\xymatrix{
\wt{G}_{ij} \ar[rr]^{\wt\tau_{ij}} \ar[d]_{\alpha_{ij}} &&
\wt{G} \ar@{-->}[d]^{\wh\alpha} \\
G_{ij} \ar[rr]^{\tau_{ij}} && G
}
\]
as in Lemma~\ref{lem:ama-epi-induces-envelope-epi}.
\end{itemize}
For $i\neq j\in I$ define $Z_{ij}^i := \wt\phi_{ij}^i(\wt{V})$ and
$Z_{ij}^{\widetilde\phi} := \gen{ Z_{ij}^i, Z_{ij}^j}$, as well as
$A_{ij}:=\ker(\alpha_{ij})$.
Then if \[A_{ij} \leq Z_{ij}^{\widetilde\phi} \leq Z(\wt{G}_{ij}),\]
it follows that $\wt{G}$ is a central extension of $G$ by
$N:= \gen{ \wt\tau_{ij}(A_{ij}) \mid i\neq j\in I}$.
In this situation the epimorphism $\alpha:\wt\AAA\to\AAA$ is called an \Defn{$|N|$-fold central extension of amalgams}.
\end{proposition}
\begin{proof}
We proceed by proving the following two assertions:
\begin{enumerate}
\item $N \leq Z(\wt{G})$,
\item $\wt{G}/N \cong G$.
\end{enumerate}
Consider the following commutative diagram:
\[\xymatrix{
\wt{V} \ar[ddrr] \ar[rr]^{\wt{\phi}^i_{ij|\wt{V}}} &&
Z_{ij}^{\widetilde\phi} \ar[dr] &&
1 \ar[d] &&
1 \ar@{-->}[d] \\
&&& Z(\wt{G}_{ij}) \ar[dr] &
A_{ij} \ar[l] \ar[d] \ar[rr]^{\wt{\tau}_{ij|A_{ij}}} \ar[ull] &&
N \ar[d] \\
&&\wt{U} \ar[d]^{\rho^i} \ar[rr]^{\wt{\phi}^i_{ij}} &&
\wt{G}_{ij} \ar[d]^{\alpha_{ij}} \ar[rr]^{\wt{\tau}_{ij}} &&
\wt{G} \ar[d]^{\widehat \alpha}\\
&& U \ar[rr]^{{\phi}^i_{ij}} &&
G_{ij} \ar[rr]^{{\tau}_{ij}} \ar[d] &&
G \ar@{-->}[d] \\
&&&& 1 && 1
}
\]
For $i\neq j\in I$ set
\[
Z_i := \wt\tau_{ij}(Z_{ij}^i)
= (\wt\tau_{ij} \circ \wt\phi_{ij}^i)(\wt{V})
\qquad\text{ and }\qquad
\wt{G}_i := (\wt\tau_{ij} \circ \wt\phi_{ij}^i)(\wt{U}) \ .
\]
The hypothesis implies
\[
\wt\tau_{ij}(\wt{G}_{ij})=\gen{\wt{G}_i, \wt{G}_j}
\qquad\text{ and }\qquad
\wt{G}
= \gen{ \wt\tau_{ij}(\wt{G}_{ij}) \mid i\neq j\in I}
= \gen{\wt{G}_i \mid i\in I}\ .
\]
Moreover,
\[
Z_i = \wt\tau_{ij}(Z_{ij}^i)
\leq \wt\tau_{ij}(Z(\wt{G}_{ij}))
\leq Z(\wt\tau_{ij}(\wt{G}_{ij}))
= Z(\gen{\wt{G}_i, \wt{G}_j})\ ,
\]
whence $Z_i$ centralizes $\wt{G}_j$ for all $i,j\in I$. Since $\wt{G}$
is generated by the $\wt{G}_i$, one has
\[
\gen{Z_i\mid i\in I}
\leq Z(\gen{\wt{G}_i\mid i\in I})
= Z(\wt{G})\ .
\]
Therefore
\[
N = \gen{ \wt\tau_{ij}(A_{ij}) \mid i\neq j\in I}
\leq \gen{ \wt\tau_{ij}(Z_{ij}^i) \mid i\neq j\in I}
= \gen{ Z_i \mid i\in I}
\leq Z(\wt{G})\ ,
\]
i.e., (a) holds.
\medskip
Commutativity of the diagram implies $N\leq \ker(\wh\alpha)$ and so the homomorphism theorem yields an epimorphism $\wt{G}/N\to G : gN\mapsto\wh\alpha(g)$.
In order to show that this epimorphism actually is an isomorphism we construct an inverse
map by exploiting that $G$ and $\wt{G}$
are universal enveloping groups of $\AAA$, resp.\ $\wt\AAA$.
Indeed, for $g\in G_{ij}$, let $\wt{g}\in\alpha^{-1}_{ij}(g)$. Then
\[
\wt\tau_{ij}(\alpha^{-1}_{ij}(g))
= \wt\tau_{ij}(\wt{g} A_{ij})
\leq \wt\tau_{ij}(\wt{g}) N
= \wt\tau_{ij}(\alpha^{-1}_{ij}(g)) N
\in \wt{G}/N\ .
\]
Thus we obtain a well-defined homomorphism
\[
\wh\tau_{ij} : G_{ij} \to \wt{G}/N : g \mapsto \wt\tau_{ij}(\alpha^{-1}_{ij}(g)) N\ .
\]
Then $(\wt{G}/N, \{\wh\tau_{ij}\})$ is an enveloping group
for $\AAA$. In particular for $u\in U$ and $i\neq j\neq k\in I$ we have
\begin{align*}
\left(\wh\tau_{ij}\circ \phi_{ij}^j\right)(u)
&= \wt\tau_{ij}\left(\alpha^{-1}_{ij}\left(\phi_{ij}^j(u)\right)\right)
= \wt\tau_{ij}\left(\wt\phi_{ij}^j\left(\left(\rho^j\right)^{-1}(u)\right)\right) \\
&= \wt\tau_{kj}\left(\wt\phi_{kj}^j\left(\left(\rho^j\right)^{-1}(u)\right)\right)
= \wt\tau_{kj}\left(\alpha^{-1}_{kj}\left(\phi_{kj}^j(u)\right)\right)
= \left(\wh\tau_{kj}\circ \phi_{kj}^j\right)(u)\ .
\end{align*}
Since $(G,\{\tau_{ij}\})$ is a universal enveloping group of $\AAA$,
there exists a unique epimorphism $\beta:G\to\wt{G}/N$ such that
for $i\neq j\in I$ we have
\[ \beta\circ\tau_{ij} = \wh\tau_{ij}\ . \]
By the definition of $\wh\alpha$ and $\wh\tau_{ij}$ we find
\[ \wh\alpha\circ\wh\tau_{ij} = \tau_{ij}\ . \]
Therefore
\[ (\beta\circ\wh\alpha)\circ\wh\tau_{ij} = \wh\tau_{ij}
\qquad\text{ and }\qquad
(\wh\alpha\circ\beta)\circ\tau_{ij} = \tau_{ij}\ . \]
But $(G,\tau)$ and $(\wt{G},\wt\tau)$ are universal enveloping groups;
their uniqueness property implies that
$\beta\circ\wh\alpha=\id_{\wt{G}/N}$ and $\wh\alpha\circ\beta=\id_{G}$
and hence as claimed $\wt{G}/N\cong G$. We have shown assertion (b).
\end{proof}
\section{Cartan matrices and Dynkin diagrams} \label{sec:cartan-dynkin}
In this section we recall the concepts of Cartan matrices and Dynkin diagrams. For a thorough introduction see \cite[Chapter~4]{kac1994infinite}, \cite[Section~7.1]{Remy:2002}.
\begin{definition}
Let $I$ be a non-empty set.
A \Defn{generalized Cartan matrix} over $I$ is a matrix $A=(a(i,j))_{i,j\in I}$
such that for all $i\neq j\in I$,
\begin{enumerate}
\item $a(i,i)=2$,
\item $a(i,j)$ is a non-positive integer,
\item if $a(i,j)=0$ then $a(j,i)=0$.
\end{enumerate}
$A$ is of \Defn{two-spherical type} if $a(i,j)a(j,i)\in\{0,1,2,3\}$
for all $i\neq j\in I$.
\end{definition}
\begin{definition}
A \Defn{Dynkin diagram} (or short: \Defn{diagram}) is
a graph $\Pi$ with vertex set $V(\Pi)$ and edge set $E(\Pi)\subseteq
\binom{V(\Pi)}{2}$ such that each edge has an \Defn{edge valency} of $1$, $2$, $3$ or
$\infty$ and, in addition, edges with valency $2$ or $3$ are directed.
If $\{v,w\}\in E(\Pi)$ is directed from $v$ to $w$, we write $v\to w$.
Let $E_0(\Pi):=\binom{V(\Pi)}{2}\sm E(\Pi)$, and let
$E_1(\Pi)$, $E_2(\Pi)$, $E_3(\Pi)$, resp. $E_\infty(\Pi)$ be the subsets of
$E(\Pi)$ of edges of valency $1$, $2$, $3$, resp.\ $\infty$.
The elements of $E_1(\Pi)$, $E_2(\Pi)$, $E_3(\Pi)$ are called \Defn{edges of type} $A_2$, $\mathrm{C}_2$
resp.\ $\mathrm{G}_2$.
The diagram $\Pi$ is \Defn{irreducible} if it is connected as a graph, it is \Defn{simply laced} if all edges have valency $1$, it is \Defn{doubly laced} if all edges have valency $1$ or $2$, and it is \Defn{two-spherical} if no edge has valency $\infty$.
If $V(\Pi)$ is finite, then a \Defn{labelling} of $\Pi$ is a bijection $\sigma:I \to V(\Pi)$, where $I:=\{ 1,\ldots, \abs{V(\Pi)}\}$.
\end{definition}
Throughout this text, we assume all diagrams to have finite vertex set.
\begin{remark} \label{longandshort}
Let $I$ be a non-empty set and $A=(a(i,j))_{i,j\in I}$ a two-spherical
generalized Cartan matrix. Then this induces a two-spherical Dynkin diagram
$\Pi(A)$ with vertex set $V:=I$
as follows: For $i\neq j\in I$, there is an edge between $i$
and $j$ if and only if $a(i,j)\neq 0$. The valency of the edge then is
$q_{ij}:=a(i,j)a(j,i)\in\{1,2,3\}$. If $q_{ij}>1$, then the edge is
directed $i\leftarrow j$ if and only if $a(i,j) = -q < -1 = a(j,i)$.
Conversely, given a two-spherical Dynkin diagram $\Pi$ with vertex set $V$,
we obtain a two-spherical generalized Cartan matrix
$A(\Pi):=(a(i,j))_{i,j\in I}$ over $I:=V$ by setting for $i\neq j\in I$:
\[
a(i,i):=2,\qquad
a(i,j):=
\begin{cases}
0, & \text{if }\{i,j\}\notin E(\Pi), \\
-2, & \text{if }\{i,j\}\in E_2(\Pi)
\text{ and }i\leftarrow j, \\
-3, & \text{if }\{i,j\}\in E_3(\Pi)
\text{ and }i\leftarrow j, \\
-1, & \text{otherwise}.\\
\end{cases}
\]
These two operations are inverse to each other, i.e.,
$\Pi(A(\Pi))=\Pi$ and $A(\Pi(A))=A$.
\end{remark}
\begin{notation} \label{augmented}
If the generalized Cartan matrix $A$ is not of two-spherical type, it is nevertheless possible to associate a Dynkin diagram $\Pi(A)$ to it by labelling the edge between $i$ and $j$ with $\infty$ whenever $a(i,j)a(j,i) \geq 4$. In this case it is, of course, not possible to reconstruct the values of $a(i,j)$ and $a(j,i)$ from the diagram $\Pi$.
Therefore, by convention, in this article for each edge between $i$ and $j$ with label $\infty$ we consider the values of $a(i,j)$ and $a(j,i)$ as part of the \Defn{augmented} Dynkin diagram: write $-a(i,j)$ between the vertex $i$ and the $\infty$ label and $-a(j,i)$ between the vertex $j$ and the $\infty$ label.
In addition, an edge with $\infty$ label such that $a(i,j)$ and $a(j,i)$ have different parity gets directed $i \leftarrow j$, if $a(i,j)$ is even, and $i \to j$, if $a(i,j)$ is odd. See Figure~\ref{fig:annotated-dynkin} for an example.
\end{notation}
\begin{figure}[h]
\centering
$A=\begin{pmatrix}
2 & -2 & 0 & 0 \\
-2 & 2 & -1 & 0 \\
0 & -4 & 2 & -1 \\
0 & 0 & -1 & 2 \\
\end{pmatrix}\ \leadsto\ $
\begin{tikzpicture}[baseline]
\node[dnode,label=below:1,label={above right:2}] (1) at (0,0) {};
\node[dnode,label=below:2,label={above left:2},label={above right:1}] (2) at (2,0) {};
\node[dnode,label=below:3,label={above left:4}] (3) at (4,0) {};
\node[dnode,label=below:4] (4) at (6,0) {};
\path (1) edge[sedge] node[above=1.5mm] {$\infty$} (2)
(2) edge[sedge,middlearrow] node[above=1.5mm] {$\infty$} (3)
(3) edge[sedge] (4)
;
\end{tikzpicture}
\caption{An augmented Dynkin diagram.}
\label{fig:annotated-dynkin}
\end{figure}
\section{The groups ${\SO{n}}$ and $\O{n}$} \label{sec:SO-O}
In this section we fix notation concerning the compact real orthogonal groups.
\begin{definition} \label{qn}
Given a quadratic space $(\KK,V,q)$ with $\dim_\KK V<\infty$, we set
\begin{align*}
\O{q}:=\{ a\in \mathrm{GL}(V) \mid \forall\ v\in V:\ q(av)=q(v)\}\ , && \SO{q}:=\O{q}\cap \mathrm{SL}(V)\ .
\end{align*}
Given $n\in \NN$, let $\q{n}:\RR^n\to \RR : x\mapsto \sum_{i=1}^n x_i^2$ be the standard quadratic form on $\RR^n$, and
\begin{align*}
\O{n}&:=\{ a\in \GL{n}\mid aa^t=E_n\}\cong \O{\q{n}} = \O{-\q{n}}\ , \\
\SO{n}&:=\O{n}\cap \SL{n} \cong \SO{\q{n}}=\SO{-\q{n}}\nt \O{\q{n}} = \O{-\q{n}}\ .
\end{align*}
Since an element of $\O{n}$ has determinant $1$ or $-1$, we have $[\O{n}:\SO{n}]=2$.
\end{definition}
\begin{notation} \label{nota:EI-VI-qI}
Let $n\in \NN$ and let $\EEE=(e_1,\ldots,e_n)$ be the standard basis of $\RR^n$. Given a subset $I\subseteq \{1,\ldots,n\}$, we set
\begin{align*}
\EEE_I:=\{ e_i\mid i\in I\}\ , &&
V_I:=\gen{\EEE_I}_\RR\leq \RR^n\ , &&
q_I:={\q{n}}_{|V_I}:V_I\to \RR\ .
\end{align*}
There are canonical isomorphisms
\[M_\EEE:\End(\RR^n)\to \mathrm{M}_n(\RR) : a\mapsto M_\EEE(a)
\quad\text{and}\quad
M_{\EEE_I}:\End(V_I)\to M_{|I|}(\RR): a\mapsto M_{\EEE_I}(a)\]
that map an endomorphism into its transformation matrix with respect to the standard basis $\EEE$, resp.\ the basis $\EEE_I$.
Moreover, there is a canonical embedding \[\eps_I:\O{q_I}\to \O{\q{n}},\]
inducing a canonical embedding
\[M_\EEE\circ \eps_I\circ M_{\EEE_I}^{-1}:\O{|I|}\to \O{n}\ ,\]
which, by slight abuse of notation, we also denote by $\eps_I$.
We will furthermore use the same symbol for the (co)restriction of $\eps_I$ to $\SO{\cdot}$. The most important application of this map in this article is for $|I|=2$ with $I = \{ i, j \}$ providing the map \[\eps_{ij} : \SO{2} \to \SO{n}.\]
\end{notation}
\section{The groups ${\Spin{n}}$ and ${\Pin{n}}$} \label{sec:spin-pin}
In this section we recall the compact real spin and pin groups. For a thorough treatment we refer to \cite{Lawson/Michelsohn:1989}, \cite{Gallier}, \cite{Meinrenken:2013}.
\begin{definition} \label{def:defcliffordconj}
Let $(\RR,V,q)$ be a quadratic space and let $T(V) = \bigoplus_{n \geq 0} V^{\otimes n}$ be the tensor algebra of $V$. The identity $V^{\otimes 0} = \RR$ provides a ring monomorphism $\RR \to T(V)$, the identity $V^{\otimes 1}=V$ a vector space monomorphism $V \to T(V)$ that allow one to identify $\RR$, $V$ with their respective images in $T(V)$. For
\[\I(q):=\langle v\otimes v-q(v) \mid v\in V \rangle\] define the \Defn{Clifford algebra of ${q}$} as \[\Cl{q}:=T(V)/\I(q).\]
Moreover, let
\[\Cl{q}^*:=\{ x\in \Cl{q} \mid \exists\ y\in \Cl{q}:\ xy=1\}\ .\]
The \Defn{transposition map} is the involution
\[ \tau:\Cl{q}\to \Cl{q} \quad\quad \text{ induced by } \quad\quad v_1\cdots v_k\mapsto v_k\cdots v_1, \quad v_i \in V, \]
cf.\ \cite[Section~2.2.6]{Meinrenken:2013}, \cite[Proposition~1.1]{Gallier}.
The \Defn{parity automorphism} is the map
\[\Pi:\Cl{q}\to \Cl{q} \quad\quad \text{ given by } \quad\quad v_1\cdots v_k \mapsto (-1)^k \cdot v_1\cdots v_k, \quad v_i \in V, \]
cf.\ \cite[Section~2.2.2, Section~3.1.1]{Meinrenken:2013}, \cite[Proposition~1.2]{Gallier}.
We set
\[
\Cle{q}{0}:=\{ x\in \Cl{q} \mid \Pi(x)=x \}
\quad\text{ and }\quad
\Cle{q}{1}:=\{ x\in \Cl{q} \mid \Pi(x)=-x\},
\]
which yields a $\ZZ_2$-grading of $\Cl{q}$, i.e.,
\[ \Cl{q}=\Cle{q}{0}\oplus \Cle{q}{1}
\quad\text{ and }\quad
\Cle{q}{i}\Cle{q}{j}\subseteq \Cle{q}{i+j}
\quad\text{ for } i,j\in \ZZ_2.
\]
Furthermore, following \cite[Section~3.1]{Gallier}, we define the \Defn{Clifford conjugation}
\[ \sigma:\Cl{q}\to\Cl{q} : x\mapsto \ol{x}:=\tau\Pi(x)=\Pi\tau(x), \]
and the \Defn{spinor norm}
\[ N:\Cl{q}\to \Cl{q} : x\mapsto x\ol{x}. \]
\end{definition}
\begin{notation}
In the following, $(\RR,V,q)$ is an anisotropic quadratic space such that $\dim_\RR V<\infty$.
\end{notation}
\begin{definition} Given $x\in \Cl{q}^*$, the map
\[ \rho_x:\Cl{q}\to\Cl{q} : y\mapsto \Pi(x)yx^{-1} \]
is the \Defn{twisted conjugation with respect to ${x}$}.
Using the canonical identification of $V$ with its image in $\Cl{q}$,
we define
\[ \Gamma(q):=\{ x\in \Cl{q}^*\mid \forall\ v\in V:\ \rho_x(v)\in V\} \]
to be the \Defn{Clifford group with respect to ${q}$}, cf.\ \cite[Section~3.1.1]{Meinrenken:2013}, \cite[Definition~1.4]{Gallier}. We obtain a representation
\[ \rho: \Gamma(q)\to \mathrm{GL}(V) : x\mapsto \rho_x, \]
which is the \Defn{twisted adjoint representation}.
\end{definition}
\begin{definition}
Given $n\in \NN$ and $V = \RR^n$, we set
\[ \Cl{n}:=\Cl{-\q{n}}
\quad\text{ and }\quad
\Gamma(n):=\Gamma(-\q{n}).
\]
Recall that $\q{n}$ is defined to be the standard quadratic form on $\RR^n$, cf.\ Definition~\ref{qn}.
\end{definition}
Note that the literature one can also find the opposite sign convention.
\begin{remark}\label{rem:cl3=quaternions}
\begin{enumerate}
\item Let $n \in \NN$ and let $e_1,\ldots,e_n$ be the standard basis of $\RR^n$. Then the following hold in $\Cl{n}$ for $1 \leq i \neq j \leq n$:
\begin{align*}
e_i^2 &= -1, \\
e_ie_j &= -e_je_i, \\
(e_ie_j)^2 &= -1.
\end{align*}
The first identity is immediate from the definition. The second identity follows from polarization, as in the tensor algebra $T(\RR^n)$ one has
\begin{align*}
\I(\q{n}) &\ni (e_i+e_j) \otimes (e_i+e_j) - q(e_i+e_j) \\ &= e_i \otimes e_i + e_i \otimes e_j + e_j \otimes e_i + e_j \otimes e_j - q(e_i) - q(e_j) - 2b(e_i,e_j) \\ &= e_i \otimes e_j + e_j \otimes e_i,
\end{align*}
where $b(\cdot,\cdot)$ denotes the bilinear form associated to $\q{n}$. The third identity is immediate from the first two.
\item One has
$\Cle{3}{0}\cong \HH$, where $\HH$ denotes the quaternions. Indeed, given a basis $e_1$, $e_2$, $e_3$ of $\RR^3$, a basis of $\Cle{3}{0}$, considered as an $\RR$-vector space, is given by $1$, $e_1e_2$, $e_2e_3$, $e_3e_1$. By (a) the latter three basis elements square to $-1$ and anticommute with one another. Note, furthermore, that under this isomorphism the Clifford conjugation is transformed into the standard involution of the quaternions and, consequently, the the spinor norm into the norm of the quaternions.
\end{enumerate}
\end{remark}
\begin{lemma}\label{1}
The map $N:\Cl{q}\to \Cl{q}$ induces a homomorphism
\[N:\Gamma(q)\to \RR^*\] such that
\[\forall\ x\in \Gamma(q):\qquad N\big(\Pi(x)\big)=N(x)\ .\]
\end{lemma}
\begin{proof}
Cf. \cite[Proposition~1.9]{Gallier}.
\end{proof}
\begin{definition}\label{defspinpin}
The group
\[\Pin{q}:=\{ x\in \Gamma(q) \mid N(x)=1\}\leq \Gamma(q)\]
is the \Defn{pin group with respect to ${q}$}, and
\[\Spin{q}:=\Pin{q}\cap \Cle{q}{0}\leq \Pin{q}\]
is the \Defn{spin group with respect to ${q}$}. By Lemma~\ref{1} and the $\ZZ_2$-grading of $\Cl{q}$, the sets $\Pin{q}$ and $\Spin{q}$ are indeed subgroups of $\Gamma(q)$.
Given $n\in \NN$, define \[\Pin{n}:=\Pin{-q_n} \quad\quad \text{ and } \quad\quad \Spin{n}:=\Spin{-q_n}.\]
\end{definition}
\begin{theorem} \label{rho}
The following hold:
\begin{enumerate}
\item One has $[\Pin{q}:\Spin{q}]=2$ and $\Spin{q}=\rho^{-1}\big(\SO{q}\big)$.
\item The twisted adjoint representation $\rho:\Gamma(q)\to \mathrm{GL}(V)$ induces an epimorphism $\rho : \Pin{q} \to \O{q}$. In particular, given $n\in \NN$, we obtain epimorphisms
\begin{align*}
\rho_n:=M_\EEE\circ \rho:\Pin{n}\to \O{n}\ , && \rho_n:=M_\EEE\circ \rho:\Spin{n}\to \SO{n}
\end{align*}
with $\ker(\rho_n)=\{\pm1\}$ in both cases.
\item The group $\Spin{q}$ is a double cover of the group $\SO{q}$.
\end{enumerate}
\end{theorem}
\begin{proof}
See \cite[Theorem~1.11]{Gallier}.
\end{proof}
\begin{remark}\label{rem:3}\
\begin{enumerate}
\item By slight abuse of notation, suppressing the choice of basis, we will also sometimes denote the map $\rho_n$ by $\rho$.
\item Let $H_1\leq \Spin{n}$ and $H_2\leq \Pin{n}$ be such that $-1\in H_1$ and $-1\in H_2$, respectively, and let $\tilde{H}_i:=\rho_n(H_i)$. Then we have
$H_i=\rho_n^{-1}(\tilde{H}_i)$.
We will explicitly determine these groups for some canonical subgroups of $\SO{n}$ and $\O{n}$.
\end{enumerate}
\end{remark}
\begin{lemma}\label{2}
Let $n\in \NN$ and $I\subseteq \{1,\ldots,n\}$. Then there is a canonical embedding
$\tilde\eps_I:\Pin{-q_I}\to \Pin{-q_n}$
satisfying \[\forall x \in \Cl{-q_I} : \qquad {\rho_{\tilde\eps_I(x)}}_{|V_I} = \eps_I \circ \rho_x\]
such that the following diagram commutes:
\[\xymatrix{
\Pin{-q_I} \ar[d]^\rho\ar[rr]^{\tilde\eps_I} & & \Pin{-q_n} \ar[d]^{\rho} \\
\O{-q_I} \ar[rr]^{\eps_I} & & \O{-q_n}
}
\]
\end{lemma}
In analogy to Notation~\ref{nota:EI-VI-qI}
we will use the same symbol for the (co)restriction of $\tilde\eps_I$ to $\Spin{\cdot}$. The most important application of this map in this article is for $|I|=2$ with $I = \{ i, j \}$ providing the map \[\tilde\eps_{ij} : \Spin{2} \to \Spin{n}.\]
\begin{proof}
Let $x \in \Gamma(-q_I)$. By definition, \[\forall v \in V_I : \qquad \rho_{\tilde\eps_I(x)}(v) \in V_I \subseteq \RR^n.\]
Since $e_ie_j = -e_je_i$ for all $i \neq j \in I$ by Remark~\ref{rem:cl3=quaternions}(a), for each $\RR$-basis vector $y = e_{j_1}\cdots e_{j_k}$ of $\tilde\eps_I(\Cl{-q_I})$ and all $i \in \{ 1, \ldots, n \} \backslash I$ one has \[\Pi(y)e_i = e_iy.\] Hence \[\Pi(\tilde\eps_I(x))e_i=e_i \tilde\eps_I(x)\] and, thus, for all $i \in \{ 1, \ldots, n \} \backslash I$ \[\rho_{\tilde\eps_I(x)}(e_i) = \Pi(\tilde\eps_I(x))e_i\tilde\eps_I(x)^{-1} = e_i \in \RR^n.\]
As $\tilde\eps_I(\Cl{-q_I})$ is generated as an $\RR$-algebra by the set $\{ e_i \mid i \in I \}$, we in particular have \[\rho \circ \tilde\eps_I = \eps_I \circ \rho.\]
Therefore $\eps_I(x) \in \Gamma(-q_n)$. Finally, \[N(\tilde\eps_I(x)) = \tilde\eps_I(N(x)) = \tilde\eps_I(1)=1,\] whence \[\tilde\eps_I(x) \in \Pin{-q_n}. \qedhere\]
\end{proof}
\begin{remark}
Since $\tilde\eps_I(\Spin{-q_I}) = \langle e_ie_j \mid i \neq j \in I \rangle \subseteq \Cle{-q_n}{0}$, one has \[\tilde\eps_I(\Spin{-q_I}) \subseteq \Pin{-q_n} \cap \Cle{-q_n}{0} = \Spin{-q_n}.\]
\end{remark}
\begin{consequence}\label{7}
Let $n\in \NN$ and $I\subseteq \{1,\ldots,n\}$. Then \[\rho_n^{-1}\big(\eps_I(\O{|I|})\big)=\tilde\eps_I\big(\Pin{-q_I}\big) \qquad \text{and} \qquad \rho_n^{-1}\big(\eps_I(\SO{|I|})\big)=\tilde\eps_I\big(\Spin{-q_I}\big).\]
\end{consequence}
\begin{proof}
By Lemma~\ref{2}, one has
$\rho_n\tilde\eps_I\big(\Pin{-q_I}\big)=\eps_I\big(\O{|I|}\big)$ and $\rho_n\tilde\eps_I\big(\Spin{-q_I}\big)=\eps_I\big(\SO{|I|}\big)$,
thus the assertion results from Remark~\ref{rem:3}(b).
\end{proof}
\begin{remark}\label{6}
Let $n \in \NN$, let $I \subseteq \{ 1, \ldots, n \}$ and let $m:=\abs{I}$. Then there exists an isomorphism $i : \Pin{m} \to \Pin{-q_I}$ such that the following diagram commutes:
\[\xymatrix{
\Pin{m}\ar[d]_{\rho_m} \ar[rr]^{i} && \Pin{-q_I} \ar[rr]^{\tilde\eps_I} \ar[d]^\rho \ar[lld]^{M_{\eps_I} \circ \rho} && \Pin{-q_n} \ar[rr]^{\id} \ar[d]_\rho \ar[drr]_{\rho_n} && \Pin{n} \ar[d]^{\rho_n} \\
\O{m} \ar[rr]_{{M_{\eps_I}}^{-1}} && \O{-q_I} \ar[rr]_{\eps_I} && \O{-q_n} \ar[rr]_{M_\eps} && \O{n}
}
\]
As in \ref{nota:EI-VI-qI} we slightly abuse notation and also write $\tilde\eps_I$ for the map $\id \circ \tilde\eps_I \circ i : \Pin{m} \to \Pin{n}$ and $\eps_I$ for the map $M_\eps \circ \eps_I \circ {M_{\eps_I}}^{-1} : \O{m} \to \O{n}$. Consequently, we obtain the following commutative diagram:
\[\xymatrix{
\Pin{m} \ar[d]^{\rho_m}\ar[rr]^{\tilde\eps_I} & & \Pin{n} \ar[d]^{\rho_n} \\
\O{m} \ar[rr]^{\eps_I} & & \O{n}
}
\]
\end{remark}
\begin{remark}\label{coordinatesforspin}
According to \cite[Corollary~1.12]{Gallier}, the group $\Pin{n}$ is generated by the set \[\{ v \in \RR^n \mid N(v) = 1 \}\] and each element of the group $\Spin{n}$ can be written as a product of an even number of elements from this set.
That is, each element $g \in \Spin{2}$ is of the form \[g = \prod^{2k}_{i=1} (a_ie_1 + b_ie_2) = \prod^k_{i=1} \big((a_{2i-1}a_{2i} + b_{2i-1}b_{2i}) + (a_{2i-1}b_{2i} - a_{2i}b_{2i-1})e_1e_2\big) =: a + be_1e_2.\]
The requirement $a_ie_1 + b_ie_2 \in \{ v \in \RR^n \mid N(v) = 1 \}$ is equivalent to \[a_i^2+b_i^2 = (a_ie_1+b_ie_2)(-a_ie_1-b_ie_2)=(a_ie_1+b_ie_2)\overline{(a_ie_1+b_ie_2)}=N(a_ie_1+b_ie_2)= 1.\] Morever, \[1=N(g)=N(a+be_1e_2)=(a+be_1e_2)\overline{(a+be_1e_2)}=(a+be_1e_2)(a+be_2e_1)=a^2+b^2.\]
Certainly, $\Spin{2}$ contains all elements of the form $a+be_1e_2$ with $a^2+b^2=1$, i.e., one obtains \[\Spin{2} = \{ \cos(\alpha) + \sin(\alpha) e_1e_2 \mid \alpha \in \RR \}.\]
One has \[(\cos(\alpha) + \sin(\alpha) e_1e_2)^{-1} = \cos(\alpha)-\sin(\alpha)e_1e_2= \cos(-\alpha) + \sin(-\alpha) e_1e_2,\] i.e., the map \[\RR \to \Spin{2} : \alpha \mapsto \cos(\alpha) + \sin(\alpha) e_1e_2\] is a group homomorphism from the real numbers onto the circle group.
The twisted adjoint representation $\rho_2$ maps the element $\cos(\alpha)+\sin(\alpha)e_1e_2 \in \Spin{2}$ to the transformation
\begin{eqnarray*}
x_1e_1+x_2e_2 & \mapsto & (\cos(\alpha)+\sin(\alpha)e_1e_2)(x_1e_1+x_2e_2)(\cos(\alpha)-\sin(\alpha)e_1e_2) \\
& = & x_1\left(\cos(\alpha)^2-\sin(\alpha)^2\right)e_1 - 2x_2\cos(\alpha)\sin(\alpha)e_1 \\ & & + 2x_1\cos(\alpha)\sin(\alpha)e_2 + x_2\left(\cos(\alpha)^2-\sin(\alpha)^2\right)e_2 \\
& = & (x_1\cos(2\alpha)-x_2\sin(2\alpha))e_1 + (x_1\sin(2\alpha)+x_2\cos(2\alpha))e_2,
\end{eqnarray*}
i.e., the rotation of the euclidean plane $\RR^2$ by the angle $2\alpha$. In other words, $\rho_2$ is the double cover of the circle group by itself, cf.\ Theorem~\ref{rho}(b).
Similarly, each element $g \in \Spin{3}$ is of the form
\[g = \prod^{2k}_{i=1} (a_ie_1 + b_ie_2 + c_ie_3) = a + be_1e_2 + ce_2e_3 + de_3e_1\]
and each element $h \in \Spin{4}$ of the form
\[h = \prod^{2k}_{i=1} (a_ie_1 + b_ie_2 + c_ie_3 + d_ie_4) = h_1 + h_2e_1e_2 + h_3e_2e_3 + h_4e_3e_1 + h_5e_1e_2e_3e_4 + h_6e_4e_3 + h_7e_4e_1 + h_8e_4e_2.\]
\end{remark}
\section{The isomorphism $\Spin{4}\cong\Spin{3}\times\Spin{3}$}
In this section we recall special isomorphisms admitted by the groups $\Spin{3}$ and $\Spin{4}$. This structural information will only become relevant in Part III (Sections~\ref{strategy} and \ref{sec:g2}) of this article.
\begin{definition}
Denote by \[\HH := \{ a+bi+cj+dk \mid a, b, c, d \in \RR \}\] the \Defn{real quaternions}, identify $\RR$ with the centre of $\HH$ via $\RR \to \HH : a \mapsto a$, let \[\bar{\cdot} : \HH \to \HH : x = a +bi +cj + dk \mapsto \overline{x} = a -bi -cj -dk\] be the \Defn{standard involution}, and let \[\mathrm{U}_1(\HH) := \{ x \in \HH \mid x\overline{x} = 1_\HH\}\] be the group of unit quaternions.
\end{definition}
\begin{remark}\label{Spin4Spin3}
By \cite[Section~1.4]{Gallier} one has \[\Spin{3} \cong \mathrm{U}_1(\HH) \quad\quad \text{and} \quad\quad \Spin{4} \cong \Spin{3} \times \Spin{3} \cong \mathrm{U}_1(\HH) \times \mathrm{U}_1(\HH).\]
The isomorphism $\Spin 3 \cong \mathrm{U}_1(\HH)$ in fact is an immediate consequence of the isomorphism $\Cle{3}{0}\cong \HH$ from Remark~\ref{rem:cl3=quaternions}(b) plus the observation that this isomorphism transforms the spinor norm into the norm of the quaternions.
A canonical isomorphism $\Spin{4} \cong \Spin{3} \times \Spin{3} \cong \mathrm{U}_1(\HH) \times \mathrm{U}_1(\HH)$ can be described as follows (see \cite[Section~1.4]{Gallier}). By Remark~\ref{coordinatesforspin} each element of $\Spin 4$ is of the form \[a + be_1e_2 + ce_2e_3 + d e_3e_1 + a'e_1e_2e_3e_4 + b'e_4e_3 + c'e_4e_1 + d'e_4e_2.\]
For \[i:=e_1e_2, \quad j:=e_2e_3, \quad k:=e_3e_1, \quad \II:=e_1e_2e_3e_4, \quad i':=e_4e_3, \quad j':=e_4e_1, \quad k':=e_4e_2\] one has
\begin{align*}
ij &= k\ ,& jk &= i\ , & ki &= j\ , \\
i\II &= \II i = i'\ , &
j\II &= \II j = j'\ , &
k\II &= \II k = k'\ , \\
i^2 &= j^2 = k^2 = -1\ , &
\II^2 &= 1\ , &
\sigma(\II) &= \II\ ,
\end{align*}
where $\sigma(\II)$ denotes the Clifford conjugate of $\II$, cf.\ Definition~\ref{def:defcliffordconj}. We conclude that for every $x \in \Spin 4$ there exist uniquely determined $u = a + bi + cj + dk, v = a' + b'i + c'j + d'k \in \HH$ such that \[x = u + \II v.\]
One computes
\[N(x) = N(u+\II v) = (u+\II v)(\overline{u}+\II\overline{v}) = u\overline{u}+v\overline{v}+\II(u\overline{v}+v\overline{u}),\] i.e., \[N(x) = 1 \Longleftrightarrow u\overline{u} + v\overline{v} = 1 \text{ and } u\overline{v}+v\overline{u}=0.\]
Hence, for $1=N(x)=N(u+\II v)$, one has
\begin{eqnarray*}
N(u+v) & = & (u+v)(\overline{u}+\overline{v}) = 1, \\
N(u-v) & = & (u-v)(\overline{u}-\overline{v}) = 1.
\end{eqnarray*}
That is, the map \[\Spin{4} \to \Spin 3 \times \Spin 3 : u+\II v \mapsto (u+v,u-v)\] is a well-defined bijection and, since
\begin{eqnarray*}
(u+\II v)(u'+\II v') & = & uu'+vv'+\II(uv'+vu'), \\
(u+v,u-v)(u'+v',u'-v') & = & \left(uu'+vv'+uv'+vu',uu'+vv'-(uv'+vu')\right),
\end{eqnarray*}
in fact an isomorphism of groups.
Consequently, there exist a group epimorphism \[\tilde\eta : \Spin{4} \to \Spin{3} : u+\II v \mapsto u+v. \]
\end{remark}
\begin{remark}\label{mapsso4}
Using this isomorphism $\Spin{4} \cong \Spin{3} \times \Spin{3} \cong \mathrm{U}_1(\HH) \times \mathrm{U}_1(\HH)$ there exists a natural homomorphism \[\Spin{4} \to \SO{\HH} \cong \SO{4} : (a,b) \mapsto \left(x \mapsto axb^{-1} \right).\]
Note that the restrictions $(a,1) \mapsto \left( x \mapsto ax \right)$ and $(1,b) \mapsto \left( x \mapsto xb^{-1} \right)$ both are injections of $\Spin{3} \cong \mathrm{U}_1(\HH)$ into $\mathrm{GL}(\HH) \cong (\HH \backslash \{ 0 \}, \cdot) $, in fact into $\SO{\HH}$, as the norm is multiplicative. Since the kernel of this action has order two, the homomorphism $\Spin{4} \to \SO{\HH} \cong \SO{4}$ must be onto by Proposition~\ref{rho}. We conclude that the group $\SO{4}$ is isomorphic to the group consisting of the maps \[\HH \to \HH : x \mapsto axb^{-1} \qquad \text{for $a, b \in \mathrm{U}_1(\HH)$;}\] for an alternative proof see \cite[Lemma~11.22]{Salzmann:1995}.
A similar argument (or a direct computation using the twisted adjoint representation) shows that the natural homomorphism \[\Spin{3} \to \SO{\langle i, j, k \rangle_\RR} \cong \SO{3} : a \mapsto \left(x \mapsto axa^{-1} \right)\] is an epimorphism and, thus, that the group $\SO{3}$ is isomorphic to the group consisting of the maps \[\HH \to \HH : x \mapsto axa^{-1} \qquad \text{for $a \in \mathrm{U}_1(\HH)$;}\]
see also \cite[Lemma~11.24]{Salzmann:1995}.
\end{remark}
\begin{remark} \label{mapisrho}
There also exists a group epimorphism \[\eta : \SO{4} \to \SO{3}\] induced by the map
\begin{align*}
\SO{4} \cong \{ \HH \to \HH : x \mapsto axb^{-1} \mid a, b \in \mathrm{U}_1(\HH) \}
&\ \to\ \{ \HH \to \HH \mid x \mapsto axa^{-1} \mid a \in \mathrm{U}_1(\HH) \} \cong \SO{3} \\
(x \mapsto axb^{-1}) &\ \mapsto\ (x \mapsto axa^{-1}).
\end{align*}
Altogether, one obtains the following commutative diagram:
\[\xymatrix{
\Spin{4} \ar[rr]_{\tilde\eta} \ar[d]^{\rho_4} & & \Spin{3} \ar[d]_{\rho_3} \\
\SO{4} \ar[rr]^\eta & & \SO{3}
}\]
\end{remark}
\section{Lifting automorphism from $\SO{n}$ to $\Spin{n}$} \label{sec:7}
\begin{notation} \label{nota:Dalpha-Salpha}\label{notationiota}\label{iotaspin}
For $\SO{2}\times\SO{2} = \{ (a,b) \mid a,b\in\SO{2} \}$ let
\[\iota_1:\SO{2}\to \SO{2}\times\SO{2} : x \mapsto (x,1)\ ,\qquad
\iota_2:\SO{2}\to \SO{2}\times\SO{2} : x \mapsto (1,x)\ .\]
Similarly, for $\Spin{2}\times\Spin{2} = \{ (a,b) \mid a,b\in\Spin{2} \}$ let
\[\tilde\iota_1:\Spin{2}\to \Spin{2}\times\Spin{2} : x \mapsto (x,1)\ ,\qquad
\tilde\iota_2:\Spin{2}\to \Spin{2}\times\Spin{2} : x \mapsto (1,x)\ .\]
Moreover, define
\[\rho_2\times\rho_2:\Spin{2}\times\Spin{2}\to\SO{2}\times\SO{2} :
(a,b)\mapsto (\rho_2(a),\rho_2(b))\ .\]
Hence
\[
(\rho_2\times\rho_2)\circ\tilde\iota_1=\iota_1\circ\rho_2\ ,\qquad
(\rho_2\times\rho_2)\circ\tilde\iota_2=\iota_2\circ\rho_2\ .
\]
Furthermore, let \[\pi : \Spin{2} \times \Spin{2} \to \Spin{2} \times \Spin{2}/ \langle (-1,-1) \rangle\] be the canonical projection. By the homomorphism theorem of groups the map $\rho_2 \times \rho_2$ factors through $\Spin{2} \times \Spin{2}/ \langle (-1,-1) \rangle$ and induces the following commutative diagram:
\[
\xymatrix{
\Spin{2} \times \Spin{2} \ar[rr]^{\rho_2\times\rho_2} \ar[d]^{\pi} && \SO{2}\times\SO{2} \\
\Spin{2} \times \Spin{2}/ \langle (-1,-1) \rangle \ar[urr]^{\rho_2.\rho_2}
}
\]
For $\alpha\in\RR$ let
\[D(\alpha):=\begin{pmat} \cos(\alpha) & \sin(\alpha) \\
-\sin(\alpha) & \cos (\alpha)\end{pmat} \in \SO{2}
\quad\text{ and }\quad
S(\alpha):= \cos(\alpha)+\sin(\alpha)e_1e_2 \in \Spin{2}.\]
Then $\Spin{2}=\{ S(\alpha) \mid \alpha\in \RR\}$
and $\SO{2}=\left\{ D(\alpha) \mid \alpha\in\RR\right\}$ and there is a continuous group isomorphism
\[\psi:\SO{2}\to \Spin{2} : D(\alpha)\mapsto S(\alpha).\]
By the computation in Remark~\ref{coordinatesforspin} the epimorphism $\rho_2$ from Theorem~\ref{rho}
satisfies $\rho_2=\sq\circ\psi^{-1}$, i.e.
\[ \rho_2:\Spin{2}\to \SO{2} : S(\alpha)\mapsto D(2\alpha). \]
\end{notation}
\begin{proposition} \label{prop:lift-aut-so2}
Given an automorphism $\gamma\in \Aut(\SO{2})$, there is a unique automorphism $\tilde\gamma\in \Aut(\Spin{2})$ such that $\rho_2\circ \tilde\gamma=\gamma\circ \rho_2$.
Moreover, $\gamma$ is continuous if and only if $\tilde\gamma$ is continuous.
\end{proposition}
\begin{proof}
Define $\tilde\gamma:= \psi\circ \gamma\circ\psi^{-1}$. Then
\[ \rho_2\circ \tilde\gamma
= (\sq\circ\psi^{-1})\circ (\psi\circ \gamma\circ\psi^{-1})
= \sq\circ\gamma\circ\psi^{-1}
= \gamma\circ\sq\circ\psi^{-1}
= \gamma\circ\rho_2\ .
\]
Uniqueness follows as $\Aut(\SO{2})\to\Aut(\Spin{2}) : \gamma\mapsto \psi\circ \gamma\circ\psi^{-1}$
is an isomorphism.
\end{proof}
\begin{corollary}\label{prop:lift-aut-so2xso2}
Given an automorphism $\gamma\in \Aut(\SO{2}\times\SO{2})$,
there is a unique automorphism $\tilde\gamma\in \Aut(\Spin{2}\times\Spin{2})$ such that
\[(\rho_2\times\rho_2)\circ \tilde\gamma=\gamma\circ (\rho_2\times\rho_2).\]
\end{corollary}
\begin{proof}
Let $\tilde\gamma := \psi\circ\gamma\circ\psi^{-1}$, where
\[ \psi: \SO{2}\times\SO{2}\to\Spin{2}\times\Spin{2},\
(D(\alpha),D(\beta)) \mapsto (S(\alpha), S(\beta))\ , \]
and observe that $\rho_2\times\rho_2 = \sq\circ\psi^{-1}$. Proceed
as in the proof of Proposition~\ref{prop:lift-aut-so2}.
\end{proof}
\begin{proposition}\label{prop:lift-aut-soN}
Let $n\geq 3$. Given an automorphism $\gamma\in \Aut\big(\SO{n}\big)$,
there is a unique automorphism $\tilde\gamma\in \Aut\big(\Spin{n}\big)$ such that
\[\rho_n\circ \tilde\gamma=\gamma\circ \rho_n.\]
\end{proposition}
\begin{proof}
For $n\geq 3$, both $\SO{n}$ and $\Spin{n}$ are perfect, cf.\
\cite[Corollary~6.56]{Hofmann/Morris:1998}. By Theorem~\ref{rho}(b) the group $\Spin{n}$ is a central extension of $\SO{n}$. Since $\Spin{n}$ is simply connected (see, e.g., \cite[Section~1.8]{Gallier}, it in fact is the universal central extension of $\SO{n}$.
The universal property of universal central extensions (cf.\ e.g.\ \cite[Section~1.4C]{Hahn/OMeara:1989}) yields the claim: Indeed, there are unique homomorphisms $\tilde\gamma, \tilde\gamma' : \Spin{n} \to \Spin{n}$ such that \[\gamma \circ \rho_n = \rho_n \circ \tilde \gamma \quad \text{and} \quad \gamma^{-1} \circ \rho_n = \rho_n \circ \tilde\gamma'.\]
Hence
\[\rho_n \circ \tilde\gamma \circ \tilde\gamma' = \gamma \circ \rho_n \circ \tilde\gamma' = \gamma \circ \gamma^{-1} \circ \rho_n = \rho_n\] and, similarly, \[\rho_n \circ \tilde\gamma' \circ \tilde\gamma = \rho_n.\]
The universal property therefore implies $\tilde\gamma \circ \tilde\gamma' = \id = \tilde\gamma' \circ \tilde\gamma$, i.e., $\tilde\gamma$ is an automorphism.
In fact, all automorphisms are continuous by van der
Waerden's Continuity Theorem, cf.\ \cite[Theorem~5.64]{Hofmann/Morris:1998}.
\end{proof}
For the following proposition recall the definitions of $\eps_{ij}$ in Notation~\ref{nota:EI-VI-qI} and of $\tilde\eps_{ij}$ in Lemma~\ref{2}.
\begin{proposition} \label{prop:zusatz}
Let $\phi : \Spin{2} \to \Spin{n}$ be a homomorphism such that
\[\ker(\rho_n \circ \phi) = \{ 1, -1 \} \qquad \text{and} \qquad \rho_n \circ \phi = \eps_{ij} \circ \rho_2\] for some $i \neq j \in I$. Then $\phi = \tilde\eps_{ij}$.
\end{proposition}
\begin{proof}
By Consequence~\ref{7} one has \[\phi(\Spin{2}) \subseteq ({\rho_n}^{-1} \circ \eps_{ij} \circ \rho_2)(\Spin{2}) = {\rho_{n}}^{-1}(\eps_{ij}(\SO{2})) = \tilde\eps_{ij}(\Spin{2}).\]
By hypothesis $\ker\phi \subseteq \{ 1, -1\}$. If $-1 \in \ker\phi$, then $1 = \phi(-1) = \phi(S(\pi)) = \phi(S(\frac{\pi}{2}))^2$, i.e., $\phi(S(\frac{\pi}{2})) \in \{ 1, -1\}$, whence $S(\frac{\pi}{2}) \in \ker(\rho_n \circ \phi)$, a contradiction. Consequently, $\phi$ is a monomorphism.
Consider the following commuting diagram:
\[
\xymatrix{ && \Spin{n} \\
\Spin{2} \ar[rr]^\phi \ar[d]_{\rho_2} && \phi(\Spin{2}) \ar@{^{(}->}[u] \ar[rr]^{{\tilde\eps_{ij}}^{-1}} \ar[d]_{\rho_n} && \Spin{2} \ar[d]_{\rho_2} \\
\SO{2} \ar[rr]_{\eps_{ij}} && \eps_{ij}(\SO{2}) \ar@{^{(}->}[d] \ar[rr]_{{\eps_{ij}}^{-1}} && \SO{2} \\ && \SO{n}
}
\]
One has $\rho_2 \circ {\tilde\eps_{ij}}^{-1} \circ \phi = \rho_2 = \id \circ \rho_2$.
Since $\phi$ is injective, the map ${\tilde\eps_{ij}}^{-1} \circ \phi$ is an automorphism of $\Spin{2}$. Hence Proposition~\ref{prop:lift-aut-so2} implies ${\tilde\eps_{ij}}^{-1} \circ \phi = \id$.
\end{proof}
\part{Simply laced diagrams}
\section{$\SO{2}$-amalgams of simply laced type} \label{sec:so2amalgams}
In this section we discuss amalgamation results for compact real orthogonal groups. The results and exposition are similar to \cite{Borovoi:1984}, \cite{Gramlich:2006}. The key difference is that the amalgams in the present article are constructed starting with the circle group $\SO{2}$ instead of the perfect group $\mathrm{SU}(2)$. This leads to some subtle complications that we will need to address below.
Recall the maps $\eps_{12}, \eps_{23} : \SO{2} \to \SO{3}$ from Notation~\ref{nota:EI-VI-qI} and the maps $\iota_1, \iota_2 : \SO{2}\to \SO{2}\times \SO{2}$ from Notation~\ref{notationiota}.
\begin{definition} \label{defstandardso}
Let $\Pi$ be a simply laced diagram with labelling $\sigma:I\to V$.
An \Defn{$\SO{2}$-amalgam with respect to $\Pi$ and $\sigma$} is an amalgam
$\AAA=\{ G_{ij},\; \phi_{ij}^i \mid i\neq j\in I \}$
such that
\[\forall\ i\neq j\in I: \qquad
G_{ij}=\begin{cases}
\SO{3}, & \text{if }\{i,j\}^\sigma\in E(\Pi), \\
\SO{2}\times \SO{2}, & \text{if }\{i,j\}^\sigma\notin E(\Pi),
\end{cases}\]
and for $i<j\in I$,
\begin{align*}
\phi_{ij}^i\big(\SO{2}\big)=\begin{cases}
\eps_{12}\big(\SO{2}\big), & \text{if $\{i,j\}^\sigma\in E(\Pi)$}, \\
\iota_1\big(\SO{2}\big), & \text{if }\{i,j\}^\sigma\notin E(\Pi),
\end{cases} &&
\phi_{ij}^j\big(\SO{2}\big)=\begin{cases}
\eps_{23}\big(\SO{2}\big), & \text{if $\{i,j\}^\sigma\in E(\Pi)$}, \\
\iota_2\big(\SO{2}\big), & \text{if }\{i,j\}^\sigma\notin E(\Pi).
\end{cases}
\end{align*}
The \Defn{standard $\SO{2}$-amalgam with respect to $\Pi$ and $\sigma$} is the $\SO{2}$-amalgam
\[\AAA\big(\Pi,\sigma,\SO{2}\big):=\{ G_{ij},\; \phi_{ij}^i \mid i\neq j\in I \}\]
with respect to $\Pi$ and $\sigma$ with
\begin{align*} \forall\ i< j\in I: &&
\phi_{ij}^i=\begin{cases}
\eps_{12},&\text{if }\{i,j\}^\sigma\in E(\Pi), \\
\iota_1, &\text{if }\{i,j\}^\sigma\notin E(\Pi).
\end{cases} &&
\phi_{ij}^j=\begin{cases}
\eps_{23},&\text{if }\{i,j\}^\sigma\in E(\Pi), \\
\iota_2, &\text{if }\{i,j\}^\sigma\notin E(\Pi).
\end{cases}
\end{align*}
\end{definition}
\begin{remark} \label{rem:continuous}
The key difference between the standard $\SO{2}$-amalgam and an arbitrary $\SO{2}$-amalgam $\AAA=\{ G_{ij},\; \phi_{ij}^i \mid i\neq j\in I \}$ with respect to $\Pi$ and $\sigma$ is that, for instance, $\eps_{12}^{-1} \circ \phi_{ij}^i$ can be an arbitrary automorphism of $\SO{2}$.
Automatic continuity (like van der Waerden's Continuity Theorem, cf.\ \cite[Theorem~5.64 and Corollary~6.56]{Hofmann/Morris:1998}) fails for automorphisms of the circle group $\SO{2}$ whereas it does hold for the group $\SO{3}$.
Hence, obviously, not every automorphism of $\SO{2}$ is induced by an automorphism of $\SO{3}$ and so it is generally not possible to undo the automorphism $\eps_{12}^{-1} \circ \phi_{ij}^i$ inside $\SO{3}$.
Therefore Goldschmidt's Lemma (see \cite[Lemma~2.7]{Goldschmidt:1980}, also \cite[Proposition~8.3.2]{Ivanov/Shpectorov:2002}, \cite[Lemma~6.16]{Gloeckner/Gramlich/Hartnick:2010}) implies that for each diagram $\Pi$ there exist plenty of pairwise non-isomorphic abstract $\SO{2}$-amalgams.
However, by \cite[Section~4.G]{Kac/Peterson:1983}, \cite[Corollary~7.16]{Hartnick/Koehl/Mars}, a split real Kac--Moody group and its maximal compact subgroup (i.e., the group of elements fixed by the Cartan--Chevalley involution) both carry natural group topologies that induce the Lie group topology on their respective fundamental subgroups of ranks one and two and make the respective embeddings continuous.
It is therefore meaningful to use \emph{continuous} $\SO{2}$-amalgams for studying these maximal compact subgroups.
Such continuous amalgams are uniquely determined by the underlying diagram $\Pi$, as we will see in Theorem~\ref{thm:uniqueness-so-sl} below.
\end{remark}
\begin{convention} \label{fix}
For each group isomorphic to one of $\SO{2}$, $\SO{3}$, $\SO{2} \times \SO{2}$, we fix a matrix representation that allows us to identify the respective groups accordingly. Our study of amalgams by Goldschmidt's Lemma then reduces to the study of automorphisms of these groups.
\end{convention}
\begin{lemma}\label{10}
Let
\[D:=\begin{pmatrix} & & 1 \\ & -1 & \\ 1 & & \end{pmatrix}\in \SO{3}\ .\]
Then the map
$\gamma_D:\SO{3}\to \SO{3} : A\mapsto D\cdot A\cdot D^{-1}=D\cdot A\cdot D$
is an automorphism of $\SO{3}$ such that \[\gamma_D\circ \eps_{12}=\eps_{23} \qquad \text{and} \qquad \gamma_D\circ \eps_{23}=\eps_{12}.\]
\end{lemma}
\begin{proof}
Given $\begin{pmat} x & y \\ -y & x\end{pmat}\in \SO{2}$, we have
\begin{align*}
\begin{pmatrix} & & 1 \\ & -1 & \\ 1 & & \end{pmatrix}\cdot \begin{pmatrix} x & y & \\
-y & x & \\
& & 1 \end{pmatrix}\cdot \begin{pmatrix} & & 1 \\ & -1 & \\ 1 & & \end{pmatrix}=\begin{pmatrix} & & 1\\
y& -x & \\
x& y& \end{pmatrix}\cdot \begin{pmatrix} & & 1 \\ & -1 & \\ 1 & & \end{pmatrix}=\begin{pmatrix}1 & & \\ & x & y \\ & -y& x\end{pmatrix}\ .
\end{align*}
The second assertion follows analogously.
\end{proof}
The only influence of the labelling $\sigma$ of an amalgam is the choice which of the vertices $i^\sigma$, $j^\sigma$ corresponds to which subgroup of $G_{ij}$. We now show that this choice does not affect the isomorphism type of the amalgam.
\begin{consequence}\label{14}
Let $\Pi$ be a simply laced diagram with labellings $\sigma_1,\sigma_2:I\to V$. Then
\[\AAA\big(\Pi,\sigma_1,\SO{2}\big)\cong \AAA\big(\Pi,\sigma_2,\SO{2}\big).\]
\end{consequence}
\begin{proof}
Denote $\AAA:=\AAA\big(\Pi,\sigma_1,\SO{2}\big)$ and $\ol{\AAA}:= \AAA\big(\Pi,\sigma_2,\SO{2}\big)$.
Let $D\in \SO{3}$ be as in Lemma~\ref{10} and let $\pi:=\sigma_2^{-1}\circ\sigma_1\in \Sym(I)$. Notice that
\begin{align*}
\ol{G}_{\pi(i)\pi(j)}=\SO{3}\ &\Leftrightarrow\ \{\pi(i),\pi(j)\}^{\sigma_2}\in E(\Pi)\ \Leftrightarrow\ \{ i,j\}^{\pi\sigma_2}\in E(\Pi) \\
&\Leftrightarrow\ \{ i,j\}^{\sigma_1\sigma_2^{-1}\sigma_2}\in E(\Pi)\ \Leftrightarrow\ \{ i,j\}^{\sigma_1}\in E(\Pi)\ \Leftrightarrow\ G_{ij}=\SO{3}\ ..
\end{align*}
Given $i<j\in I$ with $\{ i,j\}^{\sigma_1} \in E(\Pi)$, let
\[\alpha_{ij}:=\begin{cases}
\id_{\SO{3}},&\text{if }\pi(i)<\pi(j), \\
\gamma_D,&\text{if }\pi(i)>\pi(j),
\end{cases}\] and given $i<j \in I$ with $\{ i,j\}^{\sigma_1} \not\in E(\Pi)$, let
\[\alpha_{ij}: \SO{2} \times \SO{2} \to \SO{2} \times \SO{2},
(x,y) \mapsto
\begin{cases}
(x,y),&\text{if }\pi(i)<\pi(j), \\
(y,x),&\text{if }\pi(i)>\pi(j).
\end{cases}
\]
Then the system
$\alpha:=\{ \pi, \alpha_{ij} \mid i\neq j\in I \}:\AAA\to\ol{\AAA}$
is an isomorphism of amalgams. Indeed, given $i<j\in I$ with $\{ i,j\}^{\sigma_1} \in E(\Pi)$, one has
\begin{align*}
\alpha_{ij}\circ \phi_{ij}^i=\alpha_{ij}\circ \eps_{12}=
\begin{cases}
\id_{\SO{3}}\circ\eps_{12}=\eps_{12}=\ol{\phi}_{\pi(i)\pi(j)}^{\pi(i)},&\text{if }\pi(i)<\pi(j), \\
\gamma_D\circ \eps_{12}=\eps_{23}=\ol{\phi}_{\pi(i)\pi(j)}^{\pi(j)} ,&\text{if }\pi(i)>\pi(j),
\end{cases}
\end{align*}
and
\begin{align*}
\alpha_{ij}\circ \phi_{ij}^j=\alpha_{ij}\circ \eps_{23}=
\begin{cases}
\id_{\SO{3}}\circ\eps_{23}=\eps_{23}=\ol{\phi}_{\pi(i)\pi(j)}^{\pi(i)},&\text{if }\pi(i)<\pi(j), \\
\gamma_D\circ \eps_{23}=\eps_{12}=\ol{\phi}_{\pi(i)\pi(j)}^{\pi(j)},&\text{if }\pi(i)>\pi(j).
\end{cases}
\end{align*}
The case $i<j\in I$ with $\{ i,j\}^{\sigma_1} \not\in E(\Pi)$ is verified similarly.
\end{proof}
\begin{definition}\label{def:std-ama-SO2}
As we have just seen, the labelling of a standard $\SO{2}$-amalgam is
irrelevant for its isomorphism type. Hence,
for a simply laced diagram $\Pi$, we write
$\AAA\big(\Pi,\SO{2}\big)$ to denote this isomorphism type and, moreover, by slight abuse of notation to denote any representative $\AAA\big(\Pi,\sigma,\SO{2}\big)$ of this isomorphism type. It is called
the \Defn{standard $\SO{2}$-amalgam with respect to $\Pi$}.
\end{definition}
\begin{lemma}\label{lem:rank1-inv-A2}
Let $B:=\begin{pmat} 1 & \\ & -I_2\end{pmat}$, $C:=\begin{pmat} -I_2 & \\ & 1\end{pmat}\in \SO{3}$. Then the following holds:
\begin{enumerate}
\item The map $\gamma_B:\SO{3}\to \SO{3} : A\mapsto B\cdot A\cdot B^{-1}$
is an automorphism of $\SO{3}$ such that
\[\gamma_B\circ \eps_{12}=\eps_{12}\circ \inv \qquad \text{and} \qquad \gamma_B\circ \eps_{23}=\eps_{23}.\]
\item The map $\gamma_C:\SO{3}\to \SO{3} : A\mapsto C\cdot A\cdot C^{-1}$
is an automorphism of $\SO{3}$ such that
\[\gamma_C\circ \eps_{12}=\eps_{12} \qquad \text{and} \qquad \gamma_C\circ \eps_{23}=\eps_{23}\circ\inv.\]
\end{enumerate}
\end{lemma}
\begin{proof}
\begin{enumerate}
\item Given $A:= \begin{pmat} x & y \\ -y & x\end{pmat}\in \SO{2}$, we have
\begin{align*}
\gamma_B( \eps_{12}(A))
= B \cdot \begin{pmatrix} x & y & \\
-y & x & \\
& & 1 \end{pmatrix}\cdot B^{-1}
=\begin{pmatrix} x& -y & \\ y& x & \\ & & 1 \end{pmatrix}
= \eps_{12}(A^{-1})\ .
\end{align*}
The second statement follows analogously.
\item is shown with a similar computation. \qedhere
\end{enumerate}
\end{proof}
\begin{theorem}\label{thm:uniqueness-so-sl}\label{cons:std-ams-SO2-iso}
Let $\Pi$ be a simply laced diagram with labelling $\sigma:I\to V$ and
let $\AAA=\{ G_{ij},\; \phi_{ij}^i \mid i\neq j\in I \}$
be a continuous $\SO{2}$-amalgam with respect to $\Pi$ and $\sigma$. Then
$\AAA\cong \AAA\big(\Pi,\sigma,\SO{2}\big)$.
\end{theorem}
\begin{proof}
Denote $\ol{\AAA}:=\AAA\big(\Pi,\sigma,\SO{2}\big)$.
The only continuous automorphisms of the circle group $\SO{2}$
are $\id$ and the inversion $\inv$.
Since $\AAA$ is continuous by hypothesis, for all $i<j \in I$ with $\{i,j\}^\sigma \in E(\Pi)$ we have
\begin{align*}
\phi_{ij}^i\in \{ \eps_{12}, \eps_{12}\circ \inv\}\ , && \phi_{ij}^j\in \{ \eps_{23}, \eps_{23}\circ \inv\}\ .
\end{align*}
Let $B,C\in \SO{3}$ be as in Lemma~\ref{lem:rank1-inv-A2}, let $\pi:=\id_{I}$, and given $i>j\in I$ such that $\{i,j\}^\sigma\in E(\Pi)$, let
\[\alpha_{ij}:=\begin{cases}
\id_{\SO{3}},&\text{if }\phi_{ij}^i=\eps_{12},\hphantom{{}\circ\inv}\ \phi_{ij}^j=\eps_{23}, \\
\gamma_B, &\text{if }\phi_{ij}^i=\eps_{12}\circ\inv,\ \phi_{ij}^j=\eps_{23}, \\
\gamma_C, &\text{if }\phi_{ij}^i=\eps_{12},\hphantom{{}\circ\inv}\ \phi_{ij}^j=\eps_{23}\circ \inv, \\
\gamma_B\circ \gamma_C, &\text{if }\phi_{ij}^i=\eps_{12}\circ\inv,\ \phi_{ij}^j=\eps_{23}\circ \inv.
\end{cases}\]
For $i<j \in I$ with $\{i,j\}^\sigma \not\in E(\Pi)$, define $\alpha_{ij} := \big(\iota_1 \circ( \phi_{ij}^i)^{-1}\big) \times \big(\iota_2 \circ( \phi_{ij}^j)^{-1}\big)$.
Then the system
$\alpha:=\{\pi, \alpha_{ij} \mid i\neq j\in I\}:\AAA\to \ol{\AAA}$
is an isomorphism of amalgams.
\end{proof}
The following is well-known, e.g.\ \cite[Theorem~1.2]{Medts/Gramlich/Horn}.
\begin{theorem} \label{thm:univso}
For $n\geq3$, the group $\SO{n}$ is a universal enveloping group of $\AAA\big(A_{n-1},\SO{2}\big)$.
\end{theorem}
\begin{proof}
Let $I:=\{1,\ldots,n-1\}$. The group $\SO{n}$ acts flag-transitively on the simply connected projective geometry $\mathcal{G} := \PP_{n-1}(\RR)$; simple connectedness follows from \cite[Theorem~13.32]{Tits:1974}, \cite[Theorem~2]{Tits:1981}, \cite[Proposition~11.1.9, Theorem~11.1.13]{Buekenhout/Cohen:2013}, flag-transitivity of the action from the Iwasawa/$QR$-decomposition of $\SL{n}$. A maximal flag is given by
\[\gen{e_1}_\RR \leq \gen{e_1,e_2}_\RR \leq \cdots \leq \gen{e_1,\ldots,e_{n-1}}_\RR.\]
Let $T$ be the subgroup of $\SO{n}$ of diagonal matrices; it is isomorphic to $C_2^{n-1}$ (where $C_2$ is a cyclic group of order 2). Moreover, for $1 \leq i \leq n-1$, let $H_i \cong \SO{2}$ be the circle group acting naturally on $\langle e_i, e_{i+1}\rangle_\RR$ and, for $1 \leq i \leq n-2$, let $H_{i,i+1} \cong \SO{3}$ be the group acting naturally on $\langle e_i,e_{i+1},e_{i+2}\rangle_\RR$. Furthermore, for $1 \leq i < j-1 \leq n-2$, let $H_{ij} := H_i H_j \cong \SO{2} \times \SO{2}$.
Then the stabilizer of a sub-flag of co-rank one is of the form $H_iT$, $1 \leq i \leq n-1$, and the stabilizer of a sub-flag of co-rank two is of the form $H_{ij}T$, $1 \leq i < j \leq n-1$.
The group $T \cong C_2^{n-1}$ admits a presentation with all generators and relations contained in the rank two subgroups $H_{ij}$ of $\SO{n}$: Indeed, $T$ is generated by the groups $C_2 \cong T_i := \gen{-1} \leq H_i \cong \SO{2}$ and for each $1 \leq i \neq j \leq n-1$ the relation $T_iT_j = T_jT_i$ is visible within $H_{ij}$.
Therefore, by an iteration of Tits's Lemma (see \cite[Corollary~1]{Tits:1986}, \cite[Corollary~1.4.6]{Ivanov/Shpectorov:2002}) with respect to the above maximal flag, the group $H:=\SO{n}$ is the universal enveloping group of the amalgam
$\AAA(\GGG, H)=\{ H_{ij},\; \Phi_{ij}^i \mid i\neq j\in I\}$,
where $\Phi_{ij}^i:H_i\to H_{ij}$ is the inclusion map for each $i\neq j\in I$.
One has
\[\forall\ i\in I:\qquad H_i=\eps_{\{i, i+1\}}\big(\SO{2}\big)\ ,\]
and
\[\forall\ i<j\in I:\qquad H_{ij}=\begin{cases}
\eps_{\{i,i+1,i+2\}}\big(\SO{3}\big),&\text{if }j=i+1, \\
\eps_{\{i,i+1\}}\big(\SO{2}\big)\times\eps_{\{j,j+1\}}\big(\SO{2}\big),&\text{if }j\neq i+1.
\end{cases}\]
As a consequence, the system
\[\alpha=\{ \id_I,\alpha_{ij},\alpha_i \mid i\neq j\in I\}: \AAA\big(A_{n-1},\SO{2}\big)\to \AAA(\GGG,H)\]
with
\[\forall\ i\in I:\qquad \alpha_i=\eps_{\{i,i+1\}}:\SO{2}\to H_i\]
and
\[
\forall\ i<j \in I:\qquad \alpha_{ij}=\begin{cases}
\eps_{\{i,i+1,i+2\}},&\text{if }j=i+1, \\
\eps_{\{i,i+1\}}\times\eps_{\{j,j+1\}}, &\text{if }j\neq i+1,
\end{cases}
\]
is an isomorphism of amalgams.
\end{proof}
\begin{remark}
The above proof mainly relies on geometric arguments in the Tits building of type $A_{n-1}$. We exploit this to generalize the above statements to other diagrams, see Theorems~\ref{thm:K-univ-sl} and \ref{thm:K-univ-adm}. The crucial observation to make is that --- via the local-to-global principle --- it basically suffices to understand the rank two situation in order to understand arbitrary types.
\end{remark}
\section{$\Spin{2}$-amalgams of simply laced type} \label{sec:spin2amalgams}
In analogy to Section~\ref{sec:so2amalgams} we now study the amalgamation of groups isomorphic to $\Spin{3}$, continuously glued to one another along circle groups. In particular, we describe $\Spin{n}$ as the universal enveloping group of its $\Spin{2}$-amalgam and relate the classification of continuous $\Spin{2}$-amalgams to the classification of continuous $\SO{2}$-amalgams via the lifting of automorphisms.
Recall the maps $\tilde\eps_{12}, \tilde\eps_{23} : \Spin{2} \to \Spin{3}$ from Lemma~\ref{2} and the maps $\tilde\iota_1, \tilde\iota_2 : \Spin{2} \to \Spin{2}\times \Spin{2}$ from Notation~\ref{iotaspin}.
\begin{definition}\label{18}
Let $\Pi$ be a simply laced diagram with labelling $\sigma:I\to V$.
A \Defn{$\Spin{2}$-amalgam with respect to $\Pi$ and $\sigma$} is an amalgam $\AAA=\{ G_{ij},\; \phi_{ij}^i \mid i\neq j\in I \}$ such that
\[\forall\ i\neq j\in I: \qquad
G_{ij}=\begin{cases}
\Spin{3},&\text{if }\{i,j\}^\sigma\in E(\Pi), \\
\Spin{2}\times\Spin{2},&\text{if }\{i,j\}^\sigma\notin E(\Pi),
\end{cases}\]
and for $i<j\in I$,
\begin{align*}
\phi_{ij}^i\big(\Spin{2}\big)
=\begin{cases}
\tilde\eps_{12}\big(\Spin{2}\big),&\text{if $\{i,j\}^\sigma\in E(\Pi)$}, \\
\iota_1(\Spin{2}),&\text{if }\{i,j\}^\sigma\notin E(\Pi),
\end{cases}
&& \phi_{ij}^j\big(\Spin{2}\big)
=\begin{cases}
\tilde\eps_{23}\big(\Spin{2}\big),&\text{if $\{i,j\}^\sigma\in E(\Pi)$}, \\
\iota_2(\Spin{2}),&\text{if }\{i,j\}^\sigma\notin E(\Pi).
\end{cases}
\end{align*}
The \Defn{standard $\Spin{2}$-amalgam with respect to $\Pi$ and $\sigma$} is the (continuous) $\Spin{2}$-amalgam \[\AAA\big(\Pi,\sigma,\Spin{2}\big):=\{ G_{ij},\; \phi_{ij}^i, \mid i\neq j\in I \}\] with respect to $\Pi$ and $\sigma$ with
\begin{align*} \forall\ i< j\in I: &&
\phi_{ij}^i=\begin{cases}
\tilde\eps_{12},&\text{if }\{i,j\}^\sigma\in E(\Pi), \\
\tilde\iota_1,&\text{if }\{i,j\}^\sigma\notin E(\Pi).
\end{cases} &&
\phi_{ij}^j=\begin{cases}
\tilde\eps_{23},&\text{if }\{i,j\}^\sigma\in E(\Pi), \\
\tilde\iota_2,&\text{if }\{i,j\}^\sigma\notin E(\Pi).
\end{cases}
\end{align*}
\end{definition}
\begin{notation} \label{rem:lift-so2-ama-to-spin2-ama}
Proposition~\ref{prop:lift-aut-so2} enables us to lift $\SO{2}$-amalgams to
$\Spin{2}$-amalgams: Let $\Pi$ be a simply laced diagram with labelling
$\sigma:I\to V$ and let $\AAA=\{ G_{ij}, \phi_{ij}^i \mid i\neq j\in I\}$ be
an $\SO{2}$-amalgam with respect to $\Pi$ and $\sigma$. Given $i>j\in I$,
there are $\gamma_{ij}^i,\gamma_{ij}^j\in \Aut(\SO{2})$ such that
\begin{align*}
\phi_{ij}^i=
\begin{cases}
\eps_{12}\circ \gamma_{ij}^j,&\text{if }\{i,j\}^\sigma\in E(\Pi), \\
\iota_1\circ \gamma_{ij}^j,&\text{if }\{i,j\}^\sigma\notin E(\Pi),
\end{cases}
&&
\phi_{ij}^j=
\begin{cases}
\eps_{23}\circ \gamma_{ij}^j,&\text{if }\{i,j\}^\sigma\in E(\Pi), \\
\iota_2\circ \gamma_{ij}^j,&\text{if }\{i,j\}^\sigma\notin E(\Pi).
\end{cases}
\end{align*}
We then lift $\gamma_{ij}^i,\gamma_{ij}^j$ as in Lemma~\ref{prop:lift-aut-so2}
to $\tilde\gamma_{ij}^i,\tilde\gamma_{ij}^j\in\Aut(\Spin{2})$ and set
\begin{align*}
\tilde\phi_{ij}^i:=
\begin{cases}
\tilde\eps_{12}\circ \tilde\gamma_{ij}^i,&\text{if }\{i,j\}^\sigma\in E(\Pi), \\
\tilde\iota_1\circ \tilde\gamma_{ij}^i,&\text{if }\{i,j\}^\sigma\notin E(\Pi),
\end{cases}
&&
\tilde\phi_{ij}^j:=
\begin{cases}
\tilde\eps_{23}\circ \tilde\gamma_{ij}^j,&\text{if }\{i,j\}^\sigma\in E(\Pi), \\
\tilde\iota_2\circ \tilde\gamma_{ij}^j,&\text{if }\{i,j\}^\sigma\notin E(\Pi),
\end{cases}
\end{align*}
and
\[\widetilde{G}_{ij}:=\begin{cases}
\Spin{3},&\text{if }\{i,j\}^\sigma\in E(\Pi), \\
\Spin{2}\times\Spin{2},&\text{if }\{i,j\}^\sigma\notin E(\Pi).
\end{cases}\]
\end{notation}
\begin{definition} \label{rhoij}
Let $\Pi$ be a simply laced diagram with labelling $\sigma:I\to V$ and let $\AAA=\{ G_{ij}, \phi_{ij}^i \mid i\neq j\in I\}$ be an $\SO{2}$-amalgam with respect to $\Pi$ and $\sigma$. Then
$\wAAA:=\{ \widetilde{G}_{ij}, \tilde\phi_{ij}^i \mid i\neq j\in I\}$
is the \Defn{induced $\Spin{2}$-amalgam with respect to $\Pi$ and $\sigma$}.
We also set
\[ \rho_{ij} :=
\begin{cases}
\rho_3 ,&\text{if }\{i,j\}^\sigma\in E(\Pi), \\
\rho_2\times\rho_2,&\text{if }\{i,j\}^\sigma\notin E(\Pi).
\end{cases}
\]
\end{definition}
\begin{lemma}\label{lem:ama-embed-and-rho-commute-An}
Let $\Pi$ be a simply laced diagram with labelling $\sigma:I\to V$, let $\AAA=\{ G_{ij}, \phi_{ij}^i \mid i\neq j\in I\}$ be an $\SO{2}$-amalgam with respect to $\Pi$ and $\sigma$, and
let $\tilde\phi_{ij}^i$ be as introduced in Notation~\ref{rem:lift-so2-ama-to-spin2-ama}.
Then for all $i \neq j \in I$
\[
\phi_{ij}^i\circ \rho_2 = \rho_{ij} \circ \tilde\phi_{ij}^i.
\]
\end{lemma}
\begin{proof}
Without loss of generality suppose $i<j\in I$.
Let $\gamma_{ij}^i,\gamma_{ij}^j\in \Aut\big(\SO{2})$,
$\tilde\gamma_{ij}^i,\tilde\gamma_{ij}^j\in \Aut\big(\Spin{2})$,
and $\tilde\phi_{ij}^i, \tilde\phi_{ij}^j$
be as in Notation~\ref{rem:lift-so2-ama-to-spin2-ama}.
Then, if $\{i,j\}^\sigma\in E(\Pi)$, we find that
\[
\phi_{ij}^i \circ \rho_2
= \eps_{12}\circ \gamma_{ij}^{i} \circ \rho_2
= \eps_{12} \circ \rho_2 \circ \tilde\gamma_{ij}^{i}
\overset{\ref{6}}{=} \rho_3 \circ \eps_{12} \circ \tilde\gamma_{ij}^{i}
= \rho_3 \circ \tilde\phi_{ij}^{i}\ .
\]
Similarly we also conclude $\phi_{ij}^j \circ \rho_2 = \rho_3\circ \tilde\phi_{ij}^{j}$.
Moreover, in case $\{i,j\}^\sigma\notin E(\Pi)$ we deduce
\[
\phi_{ij}^i \circ \rho_2
= \eps_{12}\circ \gamma_{ij}^{i} \circ \rho_2
= \eps_{12} \circ \rho_2 \circ \tilde\gamma_{ij}^{i}
= (\rho_2\times\rho_2)\circ\tilde\iota_1 \circ \tilde\gamma_{ij}^{i}
= (\rho_2\times\rho_2)\circ \tilde\phi_{ij}^{i}.
\]
Again we conclude by a similar argument that also
$\phi_{ij}^j \circ \rho_2 = (\rho_2\times\rho_2)\circ \tilde\phi_{ij}^{j}$.
\end{proof}
\begin{remark} \label{rem:spin-ama-to-so-ama-and-back}
Clearly the construction of an induced $\Spin{2}$-amalgam is symmetric and can also be applied backwards: Starting with a $\Spin{2}$-amalgam
$\hat\AAA$ one can construct an $\SO{2}$-amalgam $\AAA$, such that $\hat\AAA=\wAAA$. In particular, we obtain an epimorphism for the standard $\Spin{2}$- and $\SO{2}$-amalgams with respect to $\Pi$ and $\sigma$, which we denote by \[\pi_{\Pi,\sigma} = \{ \mathrm{id}_I, \rho_2, \rho_{ij} \} : \AAA\big(\Pi,\sigma,\Spin{2}\big) \to \AAA\big(\Pi,\sigma,\SO{2}\big).\]
\end{remark}
\begin{proposition}\label{prop:lift-ama-iso-sl}
Let $\Pi$ be a simply laced diagram with labelling $\sigma:I\to V$, let
$\AAA_1$ and $\AAA_2$ be $\SO{2}$-amalgams with respect to $\Pi$ and $\sigma$, and let $\alpha=\{ \pi, \alpha_{ij}\mid i\neq j\in I\}:\AAA_1\to \AAA_2$ be an isomorphism of amalgams.
Then there is a unique isomorphism $\tilde\alpha=\{ \pi, \tilde\alpha_{ij}\mid i\neq j\in I\}:\wAAA_1\to \wAAA_2$ such that for all $i \neq j \in I$
\[\rho_{\pi(i)\pi(j)}\circ \tilde\alpha_{ij}=\alpha_{ij}\circ \rho_{ij} .\]
\end{proposition}
\begin{proof}
Suppose $\AAA_1=\{ G_{ij}, \phi_{ij}^i \mid i\neq j\in I\}$
and $\AAA_2=\{ H_{ij}, \psi_{ij}^i \mid i\neq j\in I\}$.
Let $i\neq j\in I$. Since both amalgams are defined with respect to $\Pi$
and $\sigma$, and since $G_{ij}\cong H_{\pi(i)\pi(j)}$, we conclude that
$\{i,j\}^\sigma$ is an edge if and only if
$\{i,j\}^{\pi\sigma}$ is an edge. Hence up to relabeling and identifying $G_{ij}$ with its image under $\alpha_{ij}$, we
may assume $\pi=\id$ and $G_{ij}=H_{ij}$ and, thus,
$\alpha_{ij}\in\Aut(G_{ij})$ with $\alpha_{ij} \circ \phi_{ij}^i = \psi_{ij}^i$.
We distinguish two cases.
\begin{description}
\item[Case I: \boldmath$\{i,j\}^\sigma\in E(\Pi)$]
Then $\widetilde{G}_{ij}=\Spin{3}$.
Let $\tilde\alpha_{ij} \in \Aut(\Spin{3})$ be the unique automorphism from
Proposition~\ref{prop:lift-aut-soN} satisfying $\rho_3\circ
\tilde\alpha_{ij} = \alpha_{ij}\circ \rho_3$. It remains to verify that
this is compatible with the amalgam structure. Indeed,
\begin{align*}
\rho_3 \circ \tilde\alpha_{ij} \circ \tilde\phi_{ij}^i \circ (\tilde\psi_{ij}^i)^{-1}
&= \alpha_{ij} \circ \rho_3 \circ \tilde\phi_{ij}^i \circ (\tilde\psi_{ij}^i)^{-1}
\overset{\ref{lem:ama-embed-and-rho-commute-An}}{=}
\alpha_{ij} \circ \phi_{ij}^i \circ \rho_2 \circ (\psi_{ij}^i)^{-1} \\
&\overset{\ref{lem:ama-embed-and-rho-commute-An}}{=}
\underbrace{\alpha_{ij} \circ \phi_{ij}^i \circ (\psi_{ij}^i)^{-1}}_{=\id_{\SO{3}}} \circ\; \rho_3
= \rho_3 = \rho_3 \circ \id_{\Spin{3}}.
\end{align*}
Hence, by uniqueness in Proposition~\ref{prop:lift-aut-soN}, one has
$\tilde\alpha_{ij} \circ \tilde\phi_{ij}^i \circ (\tilde\psi_{ij}^i)^{-1} = \id_{\Spin{3}}$,
i.e., $\tilde\alpha_{ij} \circ \tilde\phi_{ij}^i = \tilde\psi_{ij}^i$.
\item[Case II: \boldmath$\{i,j\}^\sigma\notin E(\Pi)$]
Then $\widetilde{G}_{ij}=\Spin{2}\times\Spin{2}$.
Let $\tilde\alpha_{ij} \in \Aut(\Spin{2}\times\Spin{2})$ be the unique
automorphism from Corollary~\ref{prop:lift-aut-so2xso2} satisfying
$(\rho_2\times\rho_2)\circ \tilde\alpha_{ij} = \alpha_{ij}\circ
(\rho_2\times\rho_2)$. It remains to verify that this is compatible with
the amalgam structure. Indeed,
\begin{align*}
(\rho_2\times\rho_2) \circ \tilde\alpha_{ij} \circ \tilde\phi_{ij}^i \circ (\tilde\psi_{ij}^i)^{-1}
&= \alpha_{ij} \circ (\rho_2\times\rho_2) \circ \tilde\phi_{ij}^i \circ (\tilde\psi_{ij}^i)^{-1}
\overset{\ref{lem:ama-embed-and-rho-commute-An}}{=}
\alpha_{ij} \circ \phi_{ij}^i \circ \rho_2 \circ (\psi_{ij}^i)^{-1} \\
&\overset{\ref{lem:ama-embed-and-rho-commute-An}}{=}
\underbrace{\alpha_{ij} \circ \phi_{ij}^i \circ (\psi_{ij}^i)^{-1}}_{=\id_{\SO{2}\times\SO{2}}}
\circ\; (\rho_2\times\rho_2)
= (\rho_2\times\rho_2) ..
\end{align*}
By uniqueness in Corollary~\ref{prop:lift-aut-so2xso2}, one concludes as in the previous case that $\tilde\alpha_{ij} \circ \tilde\phi_{ij}^i = \tilde\psi_{ij}^i$.
\qedhere
\end{description}
\end{proof}
\begin{corollary}
Let $\Pi$ be a simply laced diagram with labellings $\sigma_1,\sigma_2$. Then $\AAA\big(\Pi,\sigma_1,\Spin{2}\big)\cong \AAA\big(\Pi,\sigma_2,\Spin{2}\big)$.
\end{corollary}
\begin{proof}
Let $\AAA_1:=\AAA\big(\Pi,\sigma_1,\SO{2}\big)$ and $\AAA_2:=\AAA\big(\Pi,\sigma_2,\SO{2}\big)$.
The definitions imply
$\wAAA_1=\AAA\big(\Pi,\sigma_1,\Spin{2}\big)$ and $\wAAA_2=\AAA\big(\Pi,\sigma_2,\Spin{2}\big)$.
Moreover, one has $\AAA_1\cong \AAA_2$ by Consequence~\ref{14}. The claim now follows by applying Proposition~\ref{prop:lift-ama-iso-sl}.
\end{proof}
\begin{definition} \label{def:std-ama-Spin2}
As before, for a simply laced diagram $\Pi$ with labelling $\sigma$, we write $\AAA\big(\Pi,\Spin{2}\big)$ to denote the isomorphism type of $\AAA\big(\Pi,\sigma,\Spin{2}\big)$ and, by slight abuse of notation, any representative of this isomorphism type.
It is called the \Defn{standard $\Spin{2}$-amalgam with respect to $\Pi$}.
\end{definition}
\begin{theorem} \label{thm:uniqueness-spin-sl}
Let $\Pi$ be a simply laced diagram with labelling $\sigma:I\to V$ and let
$\wAAA$ be a continuous $\Spin{2}$-amalgam with respect to $\Pi$ and $\sigma$.
Then $\wAAA\cong \AAA\big(\Pi,\Spin{2}\big)$.
\end{theorem}
\begin{proof}
Let $\AAA$ be the continuous $\SO{2}$-amalgam that induces $\wAAA$, which exists by Remark~\ref{rem:spin-ama-to-so-ama-and-back}.
By Theorem~\ref{thm:uniqueness-so-sl}, one has
$\AAA\cong \AAA\big(\Pi,\SO{2}\big)$.
Proposition~\ref{prop:lift-ama-iso-sl} yields the claim, since $\AAA\big(\Pi,\SO{2}\big)$ induces $\AAA\big(\Pi,\Spin{2}\big)$. \qedhere
\end{proof}
\begin{theorem}\label{thm:univspin}
For $n\geq3$, the group $\Spin{n}$ is the universal enveloping group of $\AAA\big(A_{n-1},\Spin{2}\big)$.
\end{theorem}
\begin{proof} The proof runs along the same lines as the proof of Theorem~\ref{thm:univso}.
Let $I:=\{1,\ldots,n-1\}$. The group $\Spin{n}$ acts flag-transitively via the twisted adjoint representation (cf.\ Theorem~\ref{rho}(b)) on the simply connected projective geometry $\GGG := \PP_{n-1}(\RR)$ with fundamental maximal flag $\langle e_1\rangle_\RR \leq \langle e_1,e_2 \rangle_\RR \leq \cdots \leq \langle e_1,\ldots,e_{n-1}\rangle_\RR$.
By an iteration of Tits's Lemma (\cite[Corollary~1]{Tits:1986}, \cite[Corollary~1.4.6]{Ivanov/Shpectorov:2002}) with respect to the above maximal flag, the group $H:=\Spin{n}$ is the universal enveloping group of the amalgam
\[\AAA(\GGG, H)=\{ H_{ij}, \phi_{ij}^i \mid i\neq j\in I\} ,\]
where the $H_{ij}$ are the ``block-diagonal'' rank two subgroups and $\Phi_{ij}^i:H_i\to H_{ij}$ is the inclusion map for each $i\neq j\in I$. By Consequence~\ref{7} and Remark~\ref{6}, one has
\[\forall\ i\in I:\qquad H_i=\tilde\eps_{\{i, i+1\}}\big(\Spin{2}\big)\ ,\]
and
\[\forall\ i<j\in I:\qquad H_{ij}=\begin{cases}
\hphantom{\big\langle}\tilde\eps_{\{i,i+1,i+2\}}\big(\Spin{3}\big),&\text{if }j=i+1, \\
\tilde\eps_{\{i, i+1\}}\big(\Spin{2}\big) \cdot \tilde\eps_{\{j,j+1\}}\big(\Spin{2}\big),&\text{if }j\neq i+1.
\end{cases}\]
As a consequence, the system
\[\alpha=\{ \id_I,\alpha_{ij},\alpha_i \mid i\neq j\in I\}: \AAA\big(A_{n-1},\Spin{2}\big)\to \AAA(\GGG,H)\]
with
\[\forall\ i\in I:\qquad \alpha_i=\tilde\eps_{\{i,i+1\}}:\Spin{2}\to H_i\]
and
\[
\forall\ i<j \in I:\qquad \alpha_{ij}=\begin{cases}
\tilde\eps_{\{i,i+1,i+2\}},&\text{if }j=i+1, \\
\eps_{\{i,i+1\}}\cdot\tilde\eps_{\{j,j+1\}},&\text{if }j\neq i+1,
\end{cases}
\]
is an epimorphism of $\Spin{2}$-amalgams. In fact, each $\alpha_i$ and each $\alpha_{ii+1}$ is an isomorphism, only the $\alpha_{ij} : \Spin{2} \times \Spin{2} \to \Spin{2}.\Spin{2}$, $j \neq i+1$, have a kernel of order two: the $-1$ in the left-hand factor gets identified with the $-1$ in the right-hand factor. We may conclude that $\Spin{n}$ is an enveloping group of $\AAA\big(A_{n-1},\Spin{2}\big)$. It remains to prove universality.
Let $(G,\tau_{ij})$ be an arbitrary enveloping group of $\AAA\big(A_{n-1},\Spin{2}\big)$ and let $1 \leq i < j \leq n-1$ with $j \neq i+1$.
By definition the following diagram commutes for $1 \leq a \neq b \neq c \leq n-1$:
\[
\xymatrix{
& G_{ab} \ar[dr]^{\tau_{ab}} \\
\Spin{2} \ar[dr]_{\phi_{bc}^{b}} \ar[ur]^{\phi_{ab}^{b}} && G \\
& G_{bc} \ar[ur]_{\tau_{bc}}
}
\]
In particular, one has:
\begin{align*}
(\tau_{i,i+2} \circ \phi_{i,i+2}^{i})(-1) & = (\tau_{i,i+1} \circ \phi_{i,i+1}^{i})(-1) & \text{(set $b=i$, $a=i+1$, $c=i+2$)}\\
&= (\tau_{i,i+1} \circ \phi_{i,i+1}^{i+1})(-1) & \text{(since $\tilde\eps_{12}(-1)=\tilde\eps_{23}(-1)$)} \\
&= (\tau_{i+1,i+2} \circ \phi_{i+1,i+2}^{i+1})(-1) & \text{(set $b=i+1$, $a=i$, $c=i+2$)}\\
&= (\tau_{i+1,i+2} \circ \phi_{i+1,i+2}^{i+2})(-1) & \text{(since $\tilde\eps_{12}(-1)=\tilde\eps_{23}(-1)$)} \\
&= (\tau_{i,i+2} \circ \phi_{i,i+2}^{i+2})(-1) & \text{(set $b=i+2$, $a=i$, $c=i+1$)}.
\end{align*}
We conclude by induction that $\tau_{ij} : \Spin{2} \times \Spin{2} \to G$, $j \neq i+1$, always factors through $\Spin{2}.\Spin{2}$ or, in other words, $\tau : \AAA\big(A_{n-1},\Spin{2}\big) \to G$ always factors through $\AAA(\GGG, H)$. That is, the universal enveloping group $\Spin{n}$ of $\AAA(\GGG, H)$ is also a universal enveloping group of $\AAA\big(A_{n-1},\Spin{2}\big)$.
\end{proof}
\begin{remark}
The proof of Theorem~\ref{thm:univspin} would become a bit easier if one replaced $\Spin{2} \times \Spin{2}$ by $\Spin{2}.\Spin{2}$ in Definition~\ref{18}. The setup we chose, on the other hand, makes it easier to deal with reducible diagrams. Of course, one could a priori try to just restrict oneself to the case of irreducible diagrams, which in the case of simply laced diagrams is unproblematic. However, when dealing with arbitrary diagams it will turn out that it is more natural to also allow reducible diagrams.
\end{remark}
\section{Spin covers of simply laced type} \label{sec:spin-cover-simply-laced}
\begin{notation}
Let $\Pi=(V,E)$ be a (finite) simply laced diagram with labelling $\sigma:I\to V$ and let $c(\Pi)$ denote the {number of connected components} of $\Pi$.
A \Defn{component labelling} of $\Pi$ is a map $\KKK:V\to\{1,\ldots,c(\Pi)\}$
such that $u,v\in V$ are in the same connected component of $\Pi$
if and only if $\KKK(u)=\KKK(v)$.
Throughout this section, let $\Pi$ be a (finite) simply laced diagram, $\sigma : I \to V$ and labelling and $\KKK$ a component labelling.
\end{notation}
Generalizing Theorem~\ref{thm:univso}, the universal enveloping group of a continuous $\SO{2}$-amalgam over an arbitrary simply laced diagram $\Pi$ is isomorphic to the maximal compact subgroup of the corresponding split real Kac--Moody group (cf.\ Theorem~\ref{thm:K-univ-sl}). The goal of this section is to construct and investigate its spin cover, which will arise as the universal enveloping group of the continuous $\Spin{2}$-amalgam over the same simply laced diagram $\Pi$. In the case of $E_{10}$ its existence has been conjectured by Damour and Hillmann in \cite[Section~3.5, p.\ 24]{DamourHillmann}.
Additional key ingredients, next to transitive actions on buildings and the theory of $\SO{2}$- and $\Spin{2}$-amalgams developed so far, will be the generalized spin representations constructed in \cite{Hainke/Koehl/Levy} and the Iwasawa decomposition of split real Kac--Moody groups studied, for example, in \cite{Medts/Gramlich/Horn}.
For definitions and details on Kac--Moody theory we refer the reader to \cite{kac1994infinite}, \cite{Remy:2002}, \cite[Section~2]{Hainke/Koehl/Levy}, \cite[Section~7]{Hartnick/Koehl/Mars}, \cite[Chapter~5]{Marquis:2013}.
\begin{theorem}\label{thm:K-univ-sl}
Let $\Pi$ be a simply laced diagram, let $G(\Pi)$ be the corresponding simply connected split real Kac--Moody group, and let $K(\Pi)$ be its maximal compact subgroup, i.e., the subgroup fixed by the Cartan--Chevalley involution. Then there exists a faithful universal enveloping morphism $\tau_{K(\Pi)} : \AAA\big(\Pi,\SO{2}\big) \to K(\Pi)$.
\end{theorem}
\begin{proof}
For $i \in I$ denote by $G_i$ the fundamental rank one subgroups of $G(\Pi)$ and, for $i \neq j \in I$, by $G_{ij}$ the fundamental rank two subgroups. The groups $G_i$ are isomorphic to $\SL{2}$ and the groups $G_{ij}$ are isomorphic to $\SL{3}$ or to $\SL{2} \times \SL{2}$, depending on whether the vertices $i^\sigma$, $j^\sigma$ of $\Pi$ are joined by an edge or not. The Cartan--Chevalley involution $\omega$ leaves the groups $G_i$, $G_{ij}$ setwise invariant and, in fact, induces the transpose-inverse map on these groups. Define $H_i := \Fix_{G_i}(\omega) \cong \SO{2}$ and $H_{ij} := \Fix_{G_{ij}}(\omega)$, the latter being isomorphic to $\SO{3}$ or to $\SO{2} \times \SO{2}$, and let $\psi_{ij}^i : H_i \to H_{ij}$ denote the (continuous) inclusion map for $i\neq j\in I$.
By
\cite[Theorem~1.2]{Medts/Gramlich/Horn}, the group $K(\Pi)$ is the
universal enveloping group of the amalgam $\AAA_1:=\{H_{ij},
\psi_{ij}^i, \mid i\neq j\in I\}$.
Given $i<j \in I$ such that $\{i,j\}^\sigma\in E(\Pi)$,
by Theorem~\ref{thm:uniqueness-so-sl} applied to the subdiagram of $\Pi$ of type $A_2$ consisting of the vertices $i^\sigma$, $j^\sigma$ there is a continuous isomorphism $\alpha_{ij}:H_{ij}\to
\SO{3}$ such that
\begin{align*}
(\alpha_{ij}\circ \psi_{ij}^i)\big(H_i)=\eps_{12}\big(\SO{2}\big)\ , &&
(\alpha_{ij}\circ \psi_{ij}^j)\big(H_j)=\eps_{23}\big(\SO{2}\big)\ ..
\end{align*}
Let $i\in I$ and choose $j\in I$ such that $\{i,j\}^\sigma\in E(\Pi)$. Define
\[\alpha_i:=\begin{cases}
\eps_{12}^{-1}\circ \alpha_{ij}\circ \psi_{ij}^i:H_i\to \SO{2},&\text{if }i>j, \\
\eps_{23}^{-1}\circ \alpha_{ij}\circ \psi_{ij}^i:H_i\to \SO{2},&\text{if }i<j.
\end{cases}\]
For $i<j \in I$ such that $\{i,j\}^\sigma\not\in E(\Pi)$, define $\alpha_{ij} := \alpha_i \times \alpha_j : H_{ij} = H_i \times H_j \to \SO{2} \times \SO{2}$. For arbitrary $i \neq j \in I$, let
\[
K_{ij}:=\begin{cases}
\SO{3},&\text{if }\{i,j\}^\sigma\in E(\Pi), \\
\SO{2}\times \SO{2},& \text{if }\{i,j\}^\sigma\notin E(\Pi),
\end{cases}
\]
and
\[\bar\phi_{ij}^i:=\alpha_{ij}\circ\psi_{ij}^i\circ \alpha_{i}^{-1}:\SO{2}\to K_{ij}.\]
Then
$\AAA_2:=\{ K_{ij}, \bar\phi_{ij}^i, \mid i\neq j\in I \}$
is an $\SO{2}$-amalgam with respect to $\Pi$ and $\sigma$. Moreover, the system
$\alpha=\{ \id_I,\alpha_{ij},\alpha_i \mid i\neq j\in I \}:\AAA_1\to \AAA_2$
is an isomorphism of amalgams. Indeed, given $i\neq j\in I$, one has
$\bar\phi_{ij}^i\circ \alpha_i=\alpha_{ij}\circ \psi_{ij}^i\circ \alpha_{i}^{-1}\circ \alpha_i=\alpha_{ij}\circ \psi_{ij}^i$.
Finally, $\alpha_i$ is continuous for each $i\in I$, whence $\bar\phi_{ij}^i$ is continuous for all $i\neq j\in I$. Therefore, $\AAA_2$ is a continuous $\SO{2}$-amalgam with respect to $\Pi$ and $\sigma$ so that
$\AAA_1\cong \AAA_2\cong \AAA\big(\Pi,\SO{2}\big)$
by Theorem~\ref{thm:uniqueness-so-sl}.
\end{proof}
\begin{notation} \label{nota:sl-ama-and-epi}
For consistency, we fix the groups and connecting morphisms in the
standard $\SO{2}$-amalgam with respect to $\Pi$ as follows (cf.\ Definition~\ref{def:std-ama-SO2}):
\[ \AAA\big(\Pi,\SO{2}\big)=\{ K_{ij}, \phi_{ij}^i,\mid i\neq j\in I\}\ . \]
Similarly for the standard $\Spin{2}$-amalgam with respect to $\Pi$ (cf.\ Definition~\ref{def:std-ama-Spin2}):
\[ \AAA\big(\Pi,\Spin{2}\big)=\{ \widetilde{K}_{ij}, \tilde\phi_{ij}^i \mid i\neq j\in I\}\ . \]
We denote the epimorphism of amalgams from Remark~\ref{rem:spin-ama-to-so-ama-and-back} by
\[ \pi_{\Pi} : \AAA\big(\Pi,\Spin{2}\big) \to \AAA\big(\Pi,\SO{2}\big)\ .\]
\end{notation}
\begin{remark}\label{addtopology}
As discussed in Remark~\ref{rem:continuous} the amalgam $\AAA\big(\Pi,\SO{2}\big)$ consists of compact Lie groups with continuous connecting homomorphisms. On the other hand, the group $K(\Pi)$ naturally carries a Hausdorff group topology that is $k_\omega$: Indeed, $K(\Pi)$ is the subgroup of the unitary form studied in \cite[Section~6]{Gloeckner/Gramlich/Hartnick:2010} fixed by complex conjugation and it is the subgroup of the real Kac--Moody group $G(\Pi)$ studied in \cite[Section~7]{Hartnick/Koehl/Mars} fixed by the Cartan--Chevalley involution; both ambient groups are $k_\omega$ (by \cite[Theorem~6.12]{Gloeckner/Gramlich/Hartnick:2010}, resp.\ \cite[Theorem~7.22]{Hartnick/Koehl/Mars}) and, hence, so is any subgroup fixed by a continuous involution (cf.\ \cite[Proposition~4.2(b)]{Gloeckner/Gramlich/Hartnick:2010}). Note that the $k_\omega$-group topologies on $K(\Pi)$ induced from the real Kac--Moody group $G(\Pi)$ and from the unitary form coincide, as both are induced from the $k_\omega$-group topology on the ambient complex Kac--Moody group (cf.\ \cite[Theorem~6.3]{Gloeckner/Gramlich/Hartnick:2010}, resp.\ \cite[Theorem~7.22]{Hartnick/Koehl/Mars}).
Furthermore, a straightforward adaption of the proof of \cite[Proposition~6.9]{Gloeckner/Gramlich/Hartnick:2010} implies that this $k_\omega$-group topology is the finest group topology with respect to the enveloping homomorphisms $\tau_{ij} : K_{ij} \to K(\Pi)$. In other words, the obvious analog of \cite[Theorem~6.12]{Gloeckner/Gramlich/Hartnick:2010}, \cite[Theorem~7.22]{Hartnick/Koehl/Mars} holds for $(K(\Pi),\tau_{\Pi})$. In particular, to any enveloping morphism $\phi = (\phi_{ij}) : \AAA\big(\Pi,\SO{2}\big) \to H$ into a Hausdorff topological group $H$ with continuous homomorphisms $\phi_{ij} : K_{ij} \to H$ there exists a unique {\em continuous} homomorphism $\phi : K(\Pi) \to H$ such that the following diagram commutes:
\[\xymatrix{
\AAA\big(\Pi,\SO{2}\big) \ar[drr]_\phi \ar[rr]^{\tau_{\Pi}} && K(\Pi) \ar[d]^\psi \\ && H
}\]
\end{remark}
Theorems \ref{thm:univso} and \ref{thm:univspin} state that the double cover $\Spin{n}$ of $\SO{n}$ is the universal enveloping group of the two-fold central extension $\AAA\big(A_{n-1},\Spin{2}\big)$ of the amalgam $\AAA\big(A_{n-1},\SO{2}\big)$ as defined in Proposition~\ref{prop:env-grp-central-cover}.
In view of Theorem~\ref{thm:K-univ-sl} it is therefore natural to introduce the following notion.
\begin{definition} \label{defn:sl-spin-group}
The \Defn{spin group ${\Spin\Pi}$ with respect to $\Pi$} is the
canonical universal enveloping group of the (continuous) amalgam
$\AAA\big(\Pi,\Spin{2}\big)$ with the canonical universal enveloping morphism
\[\tau_{\Spin\Pi} = \{ \tau_{ij} \mid i \neq j \in I \} : \AAA\big(\Pi,\Spin{2}\big) \to {\Spin\Pi}.\]
\end{definition}
\begin{lemma}\label{lem:sl-K-envelops-spin-amalgam}
$K(\Pi)$ is an enveloping group of the amalgam $\AAA\big(\Pi,\Spin{2}\big)$. There exists a unique central extension $\rho_\Pi : \Spin\Pi \to K(\Pi)$ that makes the following diagram commute (cf.\ Notation~\ref{nota:amalgamcomm}):
\[
\xymatrix{
\AAA\big(\Pi,\Spin{2}\big) \ar[rr]^{\tau_{\Spin\Pi}} \ar[d]_{\pi_\Pi} &&
\Spin{\Pi} \ar@{-->}[d]^{\rho_{\Pi}} \\
\AAA\big(\Pi,\SO{2}\big) \ar[rr]^{\tau_{K(\Pi)}} && K(\Pi)
},
\]
where \[\tau_{K(\Pi)} = \{ \psi_{ij} : K_{ij} \to
K(\Pi) \mid i \neq j \in I \} : \AAA\big(\Pi,\SO{2}\big) \to K(\Pi)\]
is the universal enveloping morphism whose existence is guaranteed by Theorem~\ref{thm:K-univ-sl}.
\end{lemma}
\begin{proof}
As in Definition~\ref{rhoij}, for $i \neq j \in I$ let $\rho_{ij} : \widetilde{K}_{ij} \to K_{ij}$ be the epimorphism
$\rho_3$ if $\{i,j\}^\sigma$ is an edge, and $\rho_2\times\rho_2$ otherwise.
Then, by Lemma~\ref{lem:ama-envelope-lift}, the group $K(\Pi)$ with the homomorphisms $\xi_{ij}:=\psi_{ij}\circ
\rho_{ij}:\widetilde{K}_{ij}\to K(\Pi)$ for all $i \neq j \in I$ is an enveloping
group of $\AAA\big(\Pi,\sigma,\Spin{2}\big)$.
By universality of $\tau_{\Spin{\Pi}} : \AAA\big(\Pi,\Spin{2}\big) \to {\Spin\Pi}$, Lemma~\ref{lem:ama-epi-induces-envelope-epi} provides a unique epimorphism $\rho_\Pi : \Spin\Pi \to K(\Pi)$ with the properties as claimed. This epimorphism is a central extension by Proposition~\ref{prop:env-grp-central-cover}.
\end{proof}
The following is a straightforward generalization of the observation we made towards the end of the proof of Theorem~\ref{thm:univspin}.
\begin{lemma} \label{lem:unique-minus-one-sl}
Let $i\neq j\in I$ and $k \neq \ell\in I$. If $i^\sigma$ and
$k^\sigma$ are in the same connected component of $\Pi$, then
\[\tau_{ij}( \tilde\phi_{ij}^i(-1_{\Spin{2}}))
=\tau_{k\ell}( \tilde\phi_{k\ell}^k(-1_{\Spin{2}}))\ .\]
\end{lemma}
\begin{proof}
As $i^\sigma$ and $k^\sigma$ are in the same connected component,
there exists a sequence $i_0:=i,i_1,\ldots,i_n:=k\in I$ such that
$\{i_r^\sigma,i_{r+1}^\sigma\}$ are edges for $0\leq r<n$.
Thus $\widetilde{K}_{i_r i_{r+1}}=\Spin{3}$ and by the definition of $\Spin\Pi$
as the canonical universal enveloping group of the amalgam $\AAA\big(\Pi,\Spin{2}\big)$, we have
\[
\tilde\phi_{i_r i_{r+1}}^{i_r}(-1_{\Spin{2}}))
= -1_{\Spin{3}}
= \tilde\phi_{i_r i_{r+1}}^{i_{r+1}}(-1_{\Spin{2}}))\ .
\]
Hence
\begin{align*}
\tau_{ij}( \tilde\phi_{ij}^i(-1_{\Spin{2}}))
&=\tau_{i_0i_1}( \tilde\phi_{i_0i_1}^{i_0}(-1_{\Spin{2}}))
=\tau_{i_0i_1}( \tilde\phi_{i_0i_1}^{i_1}(-1_{\Spin{2}}))
= \cdots \\
&=\tau_{i_{n-1}i_n}( \tilde\phi_{i_{n-1}i_n}^{i_n}(-1_{\Spin{2}}))
=\tau_{i_{n-1}k}( \tilde\phi_{i_{n-1}k}^{k}(-1_{\Spin{2}})) \\
&=\tau_{k\ell}( \tilde\phi_{k\ell}^k(-1_{\Spin{2}}))\ .
\end{align*}
where the first and last equality hold due to the definition
of enveloping homomorphisms.
\end{proof}
Thus the following is well-defined.
\begin{definition}\label{minus1}
For $i\neq j\in I$ define
\[-1_{\Spin\Pi,\KKK(i)}:=\tau_{ij}(\tilde\phi_{ij}^i(-1_{\Spin{2}}))
\quad\text{ and }\quad
Z:=\gen{ -1_{\Spin\Pi,1}, \ldots, -1_{\Spin\Pi,c(\Pi)}} \leq \Spin\Pi\ .\]
\end{definition}
\begin{observation} \label{finitecentralextension}
The following are true:
\begin{enumerate}
\item $Z$ is contained in the centre of $\Spin\Pi$.
\item $|Z|\leq 2^{c(\Pi)}$.
\end{enumerate}
\end{observation}
The first assertion is immediate from Proposition~\ref{prop:env-grp-central-cover} applied to the $\Spin{2}$-amalgam $\AAA\big(\Pi,\Spin{2}\big)$ and the $\SO{2}$-amalgam $\AAA\big(\Pi,\SO{2}\big)$ with $\wt U = \Spin{2}$, $\wt V = \langle -1 \rangle$ and $U = \SO{2}$. The second follows from the fact that $Z$ is abelian by assertion (a) and admits a generating system of $c(\Pi)$ involutions by definition.
\medskip
The remainder of this section is mostly devoted to proving the following result:
\begin{theorem} \label{Zmaximal}
One has $|Z|=2^{c(\Pi)}$.
\end{theorem}
We start the proof of this theorem by revisiting Remark~\ref{rem:cl3=quaternions}.
\begin{lemma}\label{24}
Let $V$ be an $\RR$-vector space and let $X_i,X_j\in \End(V)$ be such that
\begin{align*}
X_i^2=-\id_V=X_j^2\ , && X_iX_j=-X_jX_i\ .
\end{align*}
Then the map
\begin{align*}
\psi: \Spin{3} & \to \mathrm{GL}(V) \\
a + b e_1e_2 + c e_2e_3 + de_1e_3
& \mapsto a \id_V + b X_i + c X_j + d X_i X_j
\end{align*}
is a group monomorphism such that
\[
\psi(\tilde\eps_{12}(S(\alpha))) = \cos(\alpha)\id_V + \sin(\alpha) X_i\ ,\quad
\psi(\tilde\eps_{23}(S(\alpha))) = \cos(\alpha)\id_V + \sin(\alpha) X_j\ .
\]
\end{lemma}
\begin{proof}
The subspace $\HH:=\gen{\id_V, X_i,X_j,X_iX_j}_\RR$ is an $\RR$-subalgebra of $\End(V)$, the set $\{\id_V, X_i,X_j,X_iX_j\}$ is an $\RR$-basis of $\HH$, and the $\RR$-linear extension $\hat\psi: \Cl{3}^0\to \HH$ of
\begin{align*} 1\mapsto \id_V\ , && e_1e_2\mapsto X_i\ , && e_1e_3\mapsto X_iX_j\ , && e_2e_3\mapsto X_j\end{align*}
is an isomorphism of algebras:
Indeed, since $\id_V$, $X_i$, $X_j$ and $X_iX_j$ satisfy the same relations as $1$, $e_1e_2$, $e_2e_3$ and $e_1e_3$, the map $\hat\psi$ is a homomorphism of rings. Since $X_i\neq 0_{\End(V)}$, one has $\ker(\hat\psi)\neq \Cl{3}^0$. By Remark~\ref{rem:cl3=quaternions}, $\Cl{3}^0$ is a skew field and, thus, simple as a ring. Therefore, $\hat\psi$ is injective and, hence, bijective, because $\dim_\RR \HH\leq 4$, i.e., $\hat\psi$ is an isomorphism of algebras.
Consequently, the restriction $\psi$ of $\hat\psi$ to $\Spin{3}$ is injective with values in $\mathrm{GL}(V)$, i.e., $\psi : \Spin{3} \to \mathrm{GL}(V)$ is a group monomorphism.
The final statement is immediate from the definitions.
\end{proof}
\begin{lemma}\label{25}
Let $V$ be an $\RR$-vector space and let $X_i,X_j\in \End(V)$ be such that
\begin{align*}
X_j\notin \gen{\id_V, X_i}_\RR\ , && X_i^2=-\id_V=X_j^2\ , && X_iX_j=X_jX_i\ .
\end{align*}
Then
the map
\begin{align*}
\psi: \gen{ \tilde\eps_{12}\big(\Spin{2}\big), \tilde\eps_{34}\big(\Spin{2}\big) } \subseteq \Spin{4}
&\to \mathrm{GL}(V) \\
a + b e_1e_2 + c e_3e_4 + de_1e_2e_3e_4
&\mapsto a \id_V + b X_i + c X_j + d X_i X_j
\end{align*}
is a group monomorphism such that
\[
\psi(\tilde\eps_{12}(S(\alpha))) = \cos(\alpha)\id_V + \sin(\alpha) X_i\ ,\quad
\psi(\tilde\eps_{34}(S(\alpha))) = \cos(\alpha)\id_V + \sin(\alpha) X_j\ .
\]
\end{lemma}
\begin{proof}
The subspace $\AA:=\gen{\id_V, X_i,X_j,X_iX_j}_\RR$ is an $\RR$-subalgebra of $\End(V)$, the set $\{\id_V, X_i,X_j,X_iX_j\}$ is an $\RR$-basis of $\AA$, and the $\RR$-linear extension $\hat\psi: \tilde{\AA}:=\gen{1,e_1e_2,e_3e_4,e_1e_2e_3e_4}_\RR\subseteq \Cl{4}^0 \to \AA$ of
\begin{align*} 1\mapsto \id_V\ , && e_1e_2\mapsto X_i\ , && e_3e_4\mapsto X_j, && e_1e_2e_3e_4\mapsto X_iX_j\ \end{align*}
is an isomorphism of algebras: Indeed, since $\id_V$, $X_i$, $X_j$ and $X_iX_j$ satisfy the same relations as $1$, $e_1e_2$, $e_3e_4$ and $e_1e_2e_3e_4$, the map $\psi$ is a homomorphism of rings. The hypothesis $X_j\notin \gen{\id_V, X_i}_\RR$ implies that \[\{1,X_i,X_j,X_iX_j\}\] is $\RR$-linearly independent. Therefore, $\hat\psi$ is injective and, thus, bijective, because $\dim_\RR \AA\leq 4$.
Consequently, the restriction $\psi$ of $\hat\psi$ to $\langle \tilde\eps_{12}\big(\Spin{2}\big), \tilde\eps_{34}\big(\Spin{2}\big)\rangle\subseteq \Spin{4}$ is injective with values in $\mathrm{GL}(V)$, i.e., $\psi : \langle \tilde\eps_{12}\big(\Spin{2}\big), \tilde\eps_{34}\big(\Spin{2}\big)\rangle\subseteq \Spin{4} \to \mathrm{GL}(V)$ is a group monomorphism as claimed.
The final statement is immediate from the definitions.
\end{proof}
\begin{remark}
We are now in a position to use the results of \cite{Hainke/Koehl/Levy} in order to confirm the conjecture concerning $\Spin{\Pi}$ made in footnote 18 on page 24 of \cite{DamourHillmann}.
The definition of a generalized spin representation can be found in \cite[Definition~4.4]{Hainke/Koehl/Levy}, the definition and existence of a maximal one in \cite[Corollary~4.8]{Hainke/Koehl/Levy}.
We point out that \cite[Example~4.1]{Hainke/Koehl/Levy} uses a convention for Clifford algebras different from the one used in the present article; however, \cite[Corollary~4.8]{Hainke/Koehl/Levy} is formulated and proved without making any reference to Clifford algebras whatsoever.
\end{remark}
\begin{theorem} \label{m1}
Let
\begin{itemize}
\item $\Pi$ be an irreducible simply laced diagram with labelling $\sigma:I\to V$,
\item $\mathfrak{g}$ be the Kac--Moody algebra corresponding to $\Pi$ and $\mathfrak{k}$ its maximal compact subalgebra with Berman generators $Y_1$, \ldots, $Y_n$ (cf.\ \cite[Section~2.2]{Hainke/Koehl/Levy}),
\item $\mu: \mathfrak{k} \to \End(\CC^s)$, $s\in\NN$, be a maximal generalized spin representation (cf.\ \cite[Corollary~4.8]{Hainke/Koehl/Levy}), and
\item $X_i:=2\mu(Y_i)$ for each $i\in I$.
\end{itemize}
Then, for each $i \neq j \in I$, there exist subgroups $H_{ij} \leq \mathrm{GL_{s}}(\CC)$
and an enveloping morphism
\[\Psi_\AAA = \{ \psi_{ij} \mid i \neq j \in I \} : \AAA\big(\Pi,\Spin{2}\big) \to H:=\gen{ H_{ij} \mid i\neq j\in I }\]
with injective $\psi_{ij}$ whenever $\{i,j\}^\sigma \in E(\Pi)$.
\end{theorem}
\begin{proof}
According to \cite[Remark~4.5]{Hainke/Koehl/Levy}, given $i\neq j\in I$,
one has $X_i^2=-\id_V = X_j^2$, and
\[X_iX_j=\begin{cases}
-X_jX_i,&\text{if }\{i,j\}^\sigma\in E(\Pi)\ , \\
X_jX_i,&\text{if }\{i,j\}^\sigma\notin E(\Pi)\ .
\end{cases}\]
Moreover, $X_j\notin \gen{\id_V, X_i}_\RR$, as $\mu$ is maximal.
Thus Lemma~\ref{24} provides group monomorphisms $\psi_{ij} : \wt{K}_{ij} \to \mathrm{GL}_s(\CC)$, if $\{i,j\}^\sigma \in E(\Pi)$, and Lemma~\ref{25} provides group homomorphisms $\psi_{ij} : \wt{K}_{ij} \to \mathrm{GL}_s(\CC)$ with kernel $\langle -1, -1 \rangle$, if $\{i,j\}^\sigma \not\in E(\Pi)$. This allows one to define
\[ H_{ij} := \mathrm{im}(\psi_{ij}). \]
Restriction of the ranges of the maps $\psi_{ij}$ to $H_{ij}$ thus provides
\begin{align*}
\text{group isomorphisms }\quad&
\psi_{ij}:\wt{K}_{ij} = \Spin{3}\to H_{ij},&
\text{if }\{i,j\}^\sigma\in E(\Pi)\ , \\
\text{group epimorphisms }\quad&
\psi_{ij}:\wt{K}_{ij} = \Spin{2}\times\Spin{2}\to H_{ij},&
\text{if }\{i,j\}^\sigma\notin E(\Pi)\ ,
\end{align*}
satisfying
\[ \forall i \neq j \in I: \qquad \psi_{ij}\big(\tilde\phi_{ij}^j(\cos(\alpha)+\sin(\alpha)e_1e_2)\big)=\cos(\alpha)\id_V+\sin(\alpha)X_j\ .\]
In particular, one has
\[\forall\ i\neq j\neq k:\qquad \psi_{ij}\circ \tilde\phi_{ij}^j=\psi_{kj}\circ \tilde\phi_{kj}^j\ .
\]
The set $\Psi_\AAA := \{ \psi_{ij} \mid i \neq j \in I \}$ is the desired enveloping morphism.
\end{proof}
\begin{remark}\label{inparticular}
Let everything be as in Theorem~\ref{m1}.
By universality of $\tau_{\Spin\Pi} : \AAA\big(\Pi,\Spin{2}\big) \to \Spin\Pi$ (cf.\ Definition~\ref{defn:sl-spin-group}) there exists an epimorphism \[\Xi:\Spin\Pi\to H\]
such that the following diagram commutes:
\[
\xymatrix{
\Spin{\Pi} \ar[rr]^\Xi && H \\
\Spin{2} \ar[u]^{\tau_{ij} \circ \tilde\phi_{ij}^i} \ar[urr]_{\psi_{ij} \circ \tilde\phi_{ij}^i}
}
\]
The commutative diagram in Lemma~\ref{lem:sl-K-envelops-spin-amalgam} and the finiteness of the central extension $\Spin{\Pi} \to K(\Pi)$ by Observation~\ref{finitecentralextension} in fact allow one to lift the topological universality statement from Remark~\ref{addtopology} concerning $\tau_\Pi : \AAA\big(\Pi,\SO{2}\big) \to K(\Pi)$ to a topological universality statement concerning $\tau_{\Spin\Pi} : \AAA\big(\Pi,\Spin{2}\big) \to \Spin\Pi$. Moreover, the maps $\psi$ constructed in Lemmas~\ref{24} and \ref{25} are certainly continuous with respect to the Lie group topologies, if $\dim_\RR(V) < \infty$.
In particular, the enveloping morphism $\Psi_\AAA = \{ \psi_{ij} \}$ from the theorem consists of continuous maps, so that by universality $\Xi : \Spin\Pi\to H$ is continuous as well.
\end{remark}
As an immediate consequence we record:
\begin{corollary} \label{m1cor}
Let $\Pi$ be an irreducible simply laced diagram.
Then $1_{\Spin\Pi}\neq -1_{\Spin\Pi}$.
\end{corollary}
\begin{proof}
We conclude from Remark~\ref{inparticular}
\[\Xi(-1_{\Spin\Pi})=(\Xi \circ \tau_{12} \circ \tilde\phi_{12}^1)(-1_{\Spin{2}})=(\xi_{12}\circ\tilde\phi_{12}^1)(-1_{\Spin{2}})=\cos(\pi)\id_V+\sin(\pi)X_1=-\id_V, \]
and, hence, $1_{\Spin\Pi}\neq -1_{\Spin\Pi}$.
\end{proof}
\begin{theorem} \label{m2}
Let $\Pi$ be a simply laced diagram.
Then the universal enveloping group \[\left(\Spin\Pi,\tau_{\Spin\Pi} = \{ \tau_{ij} \mid i \neq j \in I \}\right)\]
of $\AAA\big(\Pi,\Spin{2}\big)$ is a $2^{c(\Pi)}$-fold central extension of
the universal enveloping group $K(\Pi)$ of $\AAA\big(\Pi,\SO{2}\big)$.
\end{theorem}
\begin{proof}
Let $\Pi_1,\ldots,\Pi_{c(\Pi)}$ be the connected components of $\Pi$. Then \[\Spin\Pi = \Spin{\Pi_1} \times \cdots \times \Spin{\Pi_{c(\Pi)}}.\]
Indeed,
\begin{align*}
\tau_{\Spin\Pi} : \AAA\big(\Pi,\Spin{2}\big) & \to \Spin{\Pi_1} \times \cdots \times \Spin{\Pi_{c(\Pi)}} \\
\tau_{ij} : \wt K_{ij} & \to \tau_{ij}(\wt K_{ij}) & \text{if $\KKK(i) = \KKK(j)$} \\
\tau_{ij} : \wt K_{ij} & \to (\tau_{ij} \circ \tilde\phi_{ij}^i)(\Spin{2}) \times (\tau_{ij} \circ \tilde\phi_{ij}^j)(\Spin{2}) : (x,y) \mapsto \tau_{ij}(x,y) & \text{if $\KKK(i) \neq \KKK(j)$}
\end{align*}
is an enveloping morphism.
It therefore suffices to prove the theorem for irreducible simply laced diagrams $\Pi$. In this case, however, it is immediate from Proposition~\ref{prop:env-grp-central-cover} applied to the $\wt U$-amalgam $\AAA\big(\Pi,\Spin{2}\big)$ and the $U$-amalgam $\AAA\big(\Pi,\SO{2}\big)$ with $\wt U = \Spin{2}$, $U=\SO{2}$ and $\wt V = \langle -1 \rangle$ combined with Lemma~\ref{lem:unique-minus-one-sl} and Corollary~\ref{m1cor}.
\end{proof}
We have proved Theorem~\ref{Zmaximal} and Theorem~\ref{mainthm:sl-spincover} from the introduction.
\part{Non-simply laced rank two diagrams}
\section{Strategies for reducing the general case to the simply-laced one} \label{strategy}
Until now we exclusively studied spin covers of maximal compact subgroups of split real Kac--Moody groups of simply laced type. Our next goal is to generalize this concept to arbitrary Dynkin diagrams resp.\ generalized Cartan matrices. We pursue this goal via two strategies: The first one is via epimorphisms between maximal compact subgroups induced by local epimorphisms on amalgam-level in rank two where we replace non-simple edges by non-edges, simple edges or double edges; the second one is via embeddings into larger groups by unfolding the diagrams resp.\ the Cartan matrices to simply-laced cover diagrams as in \cite{Hainke/Koehl/Levy}.
The first strategy will allow us to transform arbitrary Dynkin diagrams resp.\ generalized Cartan matrices into doubly laced ones. The second strategy will work for the resulting doubly laced generalized Cartan matrices. A combination of both strategies allows us to deal with arbitrary generalized Cartan matrices.
\begin{strategy} \label{simplelacing}
In order to deal with the two non-simply laced spherical diagrams of rank two -- $\mathrm{C}_2$ and $\mathrm{G}_2$ -- we consider point-line models of the Tits buildings of the split real Lie groups $\Sp{4}$ and $\mathrm{G}_2(2)$, the so-called symplectic quadrangle and the so-called split Cayley hexagon. As in the proof of Theorem~\ref{thm:univso}, the Iwasawa decomposition implies that the maximal compact subgroups $\U{2} \leq \Sp{4}$ and $\SO{4} \leq \mathrm{G}_2(2)$ act flag-transitively on the respective point-line geometries.
Their unique double covers $\SO{2} \times \SU{2} \onto \U{2}$ and $\Spin{4} \onto \SO{4}$ fit into the commutative diagrams
\begin{eqnarray*}
& \xymatrix{\SO{2} \times \SU{2} \ar[rr] \ar[d] && \mathrm{Spin}(3) \ar[d] \\
\U{2} \ar[rr] && \mathrm{SO}(3)
} & \text{(cf.\ Remark~\ref{coordinatesrev})}
\\ \text{and} \\
& \xymatrix{\mathrm{Spin}(4) \ar[rrr] \ar[d] & && \mathrm{Spin}(3) \ar[d] \\
\mathrm{SO}(4) \ar[rrr] & && \mathrm{SO}(3)
} & \text{(cf.\ Proposition~\ref{surjG2})}
\end{eqnarray*}
which allow one to transform point and line stabilizers in $\U{2}$ and $\SO{4}$ into point, resp.\ line stabilizers in $\SO{3}$ in a way that is compatible with the covering maps. This in turn will allow us to transform a $\Spin{2}$-amalgam for a given two-spherical diagram $\Pi$ into a $\Spin{2}$-amalgam for the simply laced diagram $\Pi^{\mathrm{sl}}$ that one obtains from $\Pi$ by replacing all edges by simple edges. As a consequence --- based on Theorem~\ref{m2} --- in Theorem~\ref{m3} below we will be able to prove that the spin cover $\Spin{\Pi}$ is a non-trivial central extension of $K(\Pi)$ for suitable two-spherical diagrams $\Pi$.
As a caveat we point out that the compatibility of the covering maps in the $\mathrm{C}_2$ case is quite subtle and actually fails under certain circumstances, due to the phenomena described in Lemma~\ref{lem:ama-embed-and-rho-commute-B2}. In order to control these subtleties we introduce the notion of admissible colourings of Dynkin diagrams in Definition~\ref{adtypedef}. These subtleties are also why we actually only replace certain double edges by single edges and additionally employ Strategy~\ref{unfolding} below.
When trying to deal with non-two-spherical diagrams further subtleties arise. The non-spherical Cartan matrices of rank two are of the form \[\begin{pmatrix} 2 & -r \\ -s & 2\end{pmatrix}\] for $r, s \in \NN$ such that $rs \geq 4$. The isomorphism type of the maximal compact subgroup $K$ of the corresponding split real Kac--Moody group depends (only) on the parities of $r$ and $s$. Indeed, in all cases $K$ is isomorphic to a free amalgamated product
\[
K \cong K_1T_K *_{T_K} K_2T_K.
\] where $K_1 \cong \SO{2} \cong K_2$ with $T_K = \{ 1, t_1, t_2, t_1t_2 \} \cong \ZZ/2\ZZ \times \ZZ/2\ZZ$ and $T_K \cap K_1 = \langle t_1 \rangle$, $T_K \cap K_2 = \langle t_2 \rangle$ and $K_i \unlhd K_iT_K$. We conclude that the isomorphism type of $K$ is known once the action of $t_1$ on $K_2$ and the action of $t_2$ on $K_1$ are known. It turns out that $t_1$ centralizes $K_2$ if and only if $r$ is even and inverts $K_2$ if and only if $r$ is odd; similarly, $t_2$ centralizes $K_1$ if and only if $s$ is even and inverts $K_1$ if and only if $s$ is odd (cf.\ Remark~\ref{tcentralizes}).
To these four cases of parities of $r$ and $s$ correspond three cases of epimorphisms from $K$ onto compact Lie groups: $K \onto \SO{2} \times \SO{2}$, if both $r$ and $s$ are even; $K \onto \SO{3}$, if both $r$ and $s$ are odd; $K \onto \U{2}$, if $r$ and $s$ have different parities.
A study of various double covers of $K$ will, in analogy to what we sketched above for diagrams of type $\mathrm{C}_2$ and $\mathrm{G}_2$, enable us to replace edges labelled $\infty$ by non-edges, simple edges, resp.\ double edges, thus allowing us to understand the non-two-spherical situation as well.
Again, the case in which $r$ and $s$ have different parities lead to some subtleties that we get control of with the concept of admissible colourings introduced in Definition~\ref{adtypedef}.
Following this strategy leads directly to Proposition~\ref{thm:Spin(Delta)-covers-Spin(Delta-sl)}.
\end{strategy}
\begin{strategy} \label{unfolding}
Let $\Pi$ with type set $I$ and (generalized) Cartan matrix $A = (a(i,j))_{i,j \in I}$ be an irreducible doubly laced diagram that admits two root lengths.
Then the unfolded Dynkin diagram is the simply laced Dynkin diagram $\Pi^{\mathrm{un}}$ with type set \[I^\mathrm{un} := \{ \pm i \mid i \in I, i \text{ short root} \} \cup \{ i \mid i \in I, i \text{ long root} \}\] and edges defined via the generalized Cartan matrix $A^{\mathrm{un}}= (a^{\mathrm{un}}(i,j))_{i,j \in I^{\mathrm{un}}}$ given by
\[a^{\mathrm{un}}(i,j) =
\begin{cases}
0, & \text{if $|i|$, $|j|$ have different lengths and $a(|i|,|j|) = 0$}, \\
-1, & \text{if $|i|$, $|j|$ have different lengths and $a(|i|,|j|) \neq 0$}, \\
a(|i|,|j|), & \text{if $|i|$, $|j|$ have the same length and $ij > 0$}, \\
0, & \text{if $|i|$, $|j|$ have the same length and $ij<0$};
\end{cases}\]
(cf.\ Definition~\ref{unfoldeddiagram}).
There exists an embedding of $K(\Pi)$ into $K(\Pi^{\mathrm{un}})$ that by Corollary~\ref{corunfolding} allows one to related the respective spin covers to one another.
\end{strategy}
\section{Diagrams of type $\mathrm{G}_2$} \label{sec:g2}
In this section we prepare Strategy~\ref{simplelacing} for diagrams of type $\mathrm{G}_2$.
\begin{defn}
Denote by $\HH:=\{ a+bi+cj+dk \mid a,b,c,d\in \RR\}$
the \Defn{real quaternions}.
Then the \Defn{standard involution} of $\HH$ is given by
\[\bar{\cdot}:\HH\to\HH: x=a+bi+cj+dk \mapsto \ol{x}=a-bi-cj-dk\ .\]
The set of \Defn{purely imaginary quaternions}, cf.\ \cite[11.6]{Salzmann:1995}, is
\[ \Pu \HH
:= \{ x\in\HH \mid x=-\ol{x} \}
= \{ bi+cj+dk \mid b,c,d\in \RR\}\subset \HH\ .
\]
\end{defn}
\begin{defn}
The \Defn{split Cayley algebra} $\OO$ is defined as the vector space
$\HH \oplus \HH$ endowed with the multiplication \[xy = (x_1,x_2)(y_1,y_2) =
(x_1y_1 + y_2 \ol{x_2},\ y_1x_2 + \ol{x_1} y_2),\] cf.\
\cite[Section~5.1]{Cohen:1995}.
The \Defn{real split Cayley hexagon} $\HHH(\RR)$ consists of the one-
and two-dimensional real subspaces of $\OO$ for which the restriction
of the multiplication map is trivial, i.e., $\HHH(\RR) = (\PPP, \LLL,
\subset)$ with the point set
\[\PPP := \{ \gen{x}_\RR \mid x \in \OO, x^2 = 0 \neq x \}\]
and the line set
\[\LLL := \{ \gen{x, y}_\RR \mid \gen{x}_\RR \neq \gen{y}_\RR \in \PPP, xy = 0 \},\] cf.\
\cite[Section~5.1]{Cohen:1995}, also \cite[Section~2.4.9]{Maldeghem:1998}.
\end{defn}
\begin{lemma}\label{lem:cayley-zero-square}
Let $x=(x_1,x_2)\in\OO$.
Then
$x^2 = 0 \text{ if and only if } x_1 \in \Pu \HH \text{ and } \ol{x_1} x_1 - \ol{x_2} x_2 = 0$.
\end{lemma}
\begin{proof}
Suppose $x_1 \in \Pu \HH$ and $\ol{x_1} x_1 - \ol{x_2} x_2 = 0$. Then $-x_1x_1 - x_2 \ol{x_2} = \ol{x_1} x_1 - \ol{x_2} x_2 = 0$ and $x_1x_2 + \ol{x_1} x_2 = x_1x_2 - x_1x_2 = 0$, and, thus, $x^2 = 0$. Conversely, suppose $x^2 = 0$. Then $0 = x^2 = (x_1x_1+x_2\ol{x_2},x_1x_2 + \ol{x_1} x_2)$, so if $x_2=0$, then $x_1=0$, and there is nothing to show. For $x_2 \neq 0$ multiplication of $x_1x_2 + \ol{x_1} x_2 = 0$ from the right with $x_2^{-1}$ gives $x_1 + \ol{x_1} = 0$ or, equivalently, $x_1 \in \Pu \HH$. But now $0 = x_1x_1 + x_2 \ol{x_2} = - \ol{x_1} x_1 + \ol{x_2} x_2$, and the claim follows.
\end{proof}
\begin{definition}
Let $N:\HH\to \RR : x\mapsto x\ol{x}$ be the norm map associated to the
standard involution of the real quaternions. Moreover, let
\[\UH:=\{x\in\HH \mid x\ol{x}=1 \}\]
be the group of real quaternions of norm one.
\end{definition}
By Remark~\ref{mapsso4} (see also \cite[Lemma 11.22]{Salzmann:1995}), the group $\SO{4}$ is isomorphic to
the group consisting of the maps
\[ \HH \to \HH : x \mapsto a x b^{-1} \quad\text{ for }\quad a,b\in \UH \ . \]
\begin{lemma} \label{flagtransg2}
The group $\SO{4} \cong \{ \HH \to \HH : x \mapsto a x b^{-1} \mid a,b\in \UH \} $ acts flag-transitively on the split Cayley hexagon $\HHH(\RR)$ via
\[ \gen{(x_1,x_2)}_\RR \mapsto \gen{(a x_1 a^{-1},\ a x_2 b^{-1})}_\RR\ . \]
\end{lemma}
\begin{proof}
We show that the map
\[ f_{a,b}:\OO\to\OO : (x_1,x_2) \mapsto (a x_1 a^{-1},\ a x_2 b^{-1})\]
is an algebra automorphism of $\OO$ for all $a, b \in \UH$. Then it
also induces an automorphism of $\HHH(\RR)$, because it is defined via the multiplication in $\OO$. We have
\begin{align*}
&\ f_{a,b}(x_1,x_2)\cdot f_{a,b}(y_1,y_2) \\
=&\ \left( a x_1 a^{-1},\ a x_2 b^{-1} \right) \cdot \left( a y_1 a^{-1},\ a y_2 b^{-1} \right) \\
=&\ \left( (a x_1 a^{-1}) (a y_1 a^{-1}) + (a y_2 b^{-1}) \ol{(a x_2 b^{-1})},\
(a y_1 a^{-1}) (a x_2 b^{-1}) + \ol{(a x_1 a^{-1})} (a y_2 b^{-1}) \right) \\
=&\ \left( a (x_1y_1 + y_2 \ol{x_2}) a^{-1},\ a (y_1x_2 + \ol{x_1} y_2) b^{-1} \right) \\
=& f_{a,b}\big((x_1,x_2)(y_1,y_2)\big) \, ,
\end{align*}
whence the map $f_{a,b}$ is multiplicative. Since it is certainly an
$\RR$-linear bijection, it is an algebra automorphism of $\OO$.
Flag-transitivity is an immediate consequence of the Iwasawa
decomposition and the fact that $\SO{4}$ is the maximal compact subgroup of the (simply connected semisimple) split real group
$\mathrm{G}_2(2)$ of type $\mathrm{G}_2$.
\end{proof}
\begin{remark}
There exists a nice direct proof of flag-transitivity without making use of the Iwasawa decomposition and the structure theory of $\mathrm{G}_2(2)$ that in particular illustrates how to
compute point and line stabilizers and, thus, helps our understanding how to
properly embed the circle group into $\SO{4}$ for our amalgamation problem.
Let \[\Pu \OO := \Pu \HH \oplus \HH\] be the set of \Defn{purely imaginary
split octonions} and consider the points of the real (projective) quadric
\[\ol{x_1} x_1 - \ol{x_2} x_2 = 0\] in $\Pu \OO$, i.e., the set of isotropic
one dimensional real subspaces of $\Pu \OO$. By Remark~\ref{mapsso4} (see also \cite[11.24]{Salzmann:1995}),
the group $\SO{3}$ is isomorphic to the group consisting of the maps \[\Pu
\HH \to \Pu \HH : x \mapsto a x a^{-1} \text{ for } a\in\UH\] and acts
transitively on the set $\{ \{ x, -x \} \subset \Pu\HH \mid x\ol{x} = 1 \}$. Moreover, for
each $a, x, z \in \UH$, there exists a unique solution $b \in \UH$ for the
equation $z = a xb^{-1}$.
Hence $\SO{4}$ acts transitively on the set \[\left\{ \big\{ (x,y), (-x,-y) \big\} \subset \Pu\HH \times \HH \mid x\ol{x} = 1 = y\ol{y} \right\}.\] But this
implies point transitivity on the projective real quadric $\ol{x_1} x_1 -
\ol{x_2} x_2 = 0$ in $\Pu \OO$, which, in turn, implies point transitivity
on $\HHH(\RR)$ by Lemma~\ref{lem:cayley-zero-square}.
Now choose one point of $\HHH(\RR)$, say $\langle (i,i) \rangle_\RR$. Then a
point $\langle y \rangle_\RR = \langle (y_1,y_2)\rangle_\RR$ is collinear to
this point if and only if \[(iy_1 - y_2i, y_1i - iy_2) = 0 \Longleftrightarrow
y_1 = -iy_2i.\] So the question of transitivity of the stabilizer of $\langle
(i,i)\rangle_\RR$ in $\SO{4}$ on the line pencil of $\gen{(i,i)}_\RR$ in
$\HHH(\RR)$ is equivalent to the question of transitivity of the stabilizer
of $\gen{i}_\RR$ in $\SO{3}$ on the line pencil of $\gen{i}_\RR$ in the
projective plane $\Pu \HH \cong \Pu \HH \oplus \{ 0 \} \subset \Pu \OO$.
But since the latter is transitive, so is the former, and hence $\SO{4}$
acts flag-transitively on $\HHH(\RR)$ by means of the maps given in
Lemma~\ref{flagtransg2}.
\end{remark}
The stabilizer of the point $\gen{(i,i)}_\RR$ contains the circle group $\SO{2}$ acting naturally diagonally on $\gen{j_1,k_1}_{\RR} \oplus \gen{j_2,k_2}_{\RR}$. The stabilizer of the line $\gen{(i,i) , (j,-j)}_\RR$ in $\SO{4}$ contains $\SO{2}$ acting naturally by rotations on $\gen{i_1,j_1}_{\RR}$ and by rotations in the opposite direction on $\gen{i_2,j_2}_{\RR}$ in such a way that an element with first coordinate $\lambda i_1 + \mu j_1$ has second coordinate $\lambda i_2 - \mu j_2$.
\begin{notation}
We denote these embeddings of the circle group into $\SO{4}$ by $\eta_p$ resp.\ $\eta_l$.
Concretely, one has
\begin{align*}
\eta_p &: \SO{2}\to \SO{4} : D(\alpha)\mapsto \begin{pmat} I_2 & \\ & D(\alpha)\end{pmat}
=\eps_{34}(D(\alpha))\ ,\\
\eta_l &: \SO{2}\to \SO{4} : D(\alpha)\mapsto \tilde{D}(\alpha):=
\begin{pmatrix}
\cos(2\alpha) & & & \sin(2\alpha) \\
& \cos(\alpha) & -\sin(\alpha) & \\
& \sin(\alpha) & \cos(\alpha) & \\
-\sin(2\alpha) & & & \cos (2\alpha)
\end{pmatrix} = \eps_{14}(D(2\alpha)) \cdot \eps_{23}(D(-\alpha))\ .
\end{align*}
\end{notation}
\begin{lemma} \label{lem:rank1-inv-G2}
Let $B:=\diag(-1,1,1,-1)$, $C:=\diag(-1,-1,1,1)\in \U{2}$. Then the following hold:
\begin{enumerate}
\item The map $\gamma_B: \SO{4}\to \SO{4} : A\mapsto B\cdot A\cdot B^{-1}=B\cdot A\cdot B$
is an automorphism of $\SO{4}$ such that
\[\gamma_B\circ \eta_p=\eta_p\circ \inv \qquad \text{and} \qquad \gamma_B\circ \eta_l=\eta_l.\]
\item The map $\gamma_C:\SO{4}\to\SO{4} : A\mapsto C\cdot A\cdot C^{-1}=C\cdot A\cdot C$
is an automorphism of $\SO{4}$ such that
\[\gamma_C\circ \eta_p=\eta_p \qquad \text{and} \qquad \gamma_C\circ \eta_l=\eta_l\circ\inv.\]
\end{enumerate}
\end{lemma}
\begin{proof}
Straightforward.
\end{proof}
\begin{notation} \label{etaspin}
In the following, let
\begin{align*}
\tilde\eta_p &: \Spin{2} \to \Spin{4} : S(\alpha)\mapsto \tilde\eps_{34}\big(S(\alpha)\big)\ , \\
\tilde\eta_l &: \Spin{2}\to \Spin{4} : S(\alpha)\mapsto \tilde\eps_{14}\big(S(2\alpha)\big)\cdot \tilde\eps_{23}\big(S(-\alpha)\big)\ ,
\end{align*}
and recall from Theorem~\ref{rho}(b) that for $n\geq 2$ the map
\[\rho_n:\Spin{n}\to \SO{n} \]
is the twisted adjoint representation.
\end{notation}
In order to generalize our definition of spin amalgams, we need
$\tilde\eta_p$ and $\tilde\eta_l$ to be injective. For the former this
is clear from its definition, for the latter we verify it now.
\begin{lemma} \label{lem:eta-l-inj}
The map $\tilde\eta_l$ is a monomorphism.
\end{lemma}
\begin{proof}
For $S(\alpha)\in \ker \tilde\eta_l$ one has
\begin{align*}
\tilde\eta_l(S(\alpha))
=\big(\cos(2\alpha)+\sin(2\alpha)e_1e_4\big)
\big(\cos(\alpha)-\sin(\alpha)e_2e_3\big)
=1
\end{align*}
and, thus, $\alpha\in \pi\ZZ$. As $\tilde\eta_l\big ( S(\pi)\big)=-1$ and $\tilde\eta_l\big(S(2\pi)\big)=1$, one obtains
\[\ker \tilde\eta_l=\{ S(\alpha) \mid \alpha\in 2\pi\ZZ \}=\{ 1\}\ . \qedhere\]
\end{proof}
\begin{lemma} \label{lem:ama-embed-and-rho-commute-G2}
One has
\begin{align*}
\rho_4\circ \tilde\eta_p= {\eta}_p\circ \rho_2\ , &&
\rho_4\circ \tilde\eta_l= {\eta}_l\circ \rho_2\ .
\end{align*}
\end{lemma}
\begin{proof}
For $\alpha\in \RR$,
\begin{align*}
(\rho_4\circ \tilde\eta_p)\big(S(\alpha)\big)
&=(\rho_4\circ \tilde\eps_{34})\big(S(\alpha)\big)
=(\eps_{34}\circ\rho)\big(S(\alpha)\big)
=(\eta_p\circ\rho)\big(S(\alpha)\big)
\end{align*}
and
\begin{align*}
(\rho_4\circ \tilde\eta_l)\big(S(\alpha)\big)
&=\rho_4\big(\tilde\eps_{14}\big(S(2\alpha)\big)
\cdot \tilde\eps_{23}\big(S(-\alpha)\big)\big)
=\eps_{14}\big(D(4\alpha)\big)\cdot \eps_{23}\big(D(-2\alpha)\big) \\
&=(\eta_l\circ \rho_2)\big(S(\alpha)\big)\ .
\qedhere
\end{align*}
\end{proof}
\begin{lemma} \label{lemmaleftright}
Let $V:=\HH$ and $\EEE:=\{ 1, i , j , k\}$. Then the following hold:
\begin{enumerate}
\item For $a,b\in \UH$ the
maps \[\ell_a: \HH\to \HH : x\mapsto ax \text{ and } r_b:\HH\to \HH : x\mapsto xb^{-1}\]
preserve the norm $N:\HH\to \RR : x\mapsto x\ol{x}$. In particular,
$\ell_a,r_b\in \SO{\HH}\cong \SO{4}$.
\item \label{lem:UHxUH-covers-SO4}
The map
\[ \UH\times \UH\to\SO{4} : (a,b)\mapsto \ell_{a}\circ r_{b} \]
is a group epimorphism with kernel $\{(1,1),(-1,-1)\}$.
\end{enumerate}
\end{lemma}
\begin{proof}
This has been discussed in Remark~\ref{mapsso4}. Alternatively, it also follows from \cite[Lemma~{11.22} to Corollary~{11.25}]{Salzmann:1995}.
\end{proof}
For $\EEE=\{ 1, i , j , k\}$ define
\begin{align*}
L_a:=M_\EEE(\ell_a)\in \SO{4}\ , && R_b:=M_\EEE(r_b)\in \SO{4}\ .
\end{align*}
\begin{remark} \label{remark5}
\begin{enumerate}
\item
The map $\UH\times\UH\to\SO{4}$ from Lemma~\ref{lem:UHxUH-covers-SO4} equals the covering map $\rho_4$, cf.\ Remark~\ref{mapisrho}.
\item Given $x=a+bi+cj+dk\in \UH$, a short computation shows
\begin{align*}
L_x=\begin{pmatrix} a & - b & -c & -d \\
b & a & -d & c \\
c & d & a & -b \\
d& -c & b & a \end{pmatrix}\ , &&
R_x=\begin{pmatrix} a & b & c & d \\
-b & a & -d & c \\
-c & d & a & -b \\
-d & -c & b & a \end{pmatrix}
\end{align*}
as $\RR$-linear maps via left action. Lemma~\ref{lem:UHxUH-covers-SO4} implies that for all $x,y \in \mathrm{U}_1(\HH)$ one has $L_xR_y = R_yL_x$ and that up to scalar multiplication with $-1$ the matrices $L_x$ and $R_y$ are uniquely determined by their product.
\item The action from Lemma~\ref{flagtransg2} translates into \[\omega: \SO{4}\times \HHH(\RR) \to \HHH(\RR) : \big( L_aR_{b} , (x,y)\big) \mapsto ( L_a R_{a}\cdot x, L_aR_{b}\cdot y).\]
\item \label{30a}
For $\alpha\in \RR$, one has $\eta_p\big(D(\alpha)\big)=L_a\cdot R_a$ with
$a=\cos(\tfrac{\alpha}{2})-\sin(\tfrac{\alpha}{2})i$, i.e.,
\begin{align*}
\begin{pmatrix} I_2 & \\ & D(\alpha) \end{pmatrix}
= \begin{pmatrix} D(\frac{\alpha}{2}) & \\ & D(\frac{\alpha}{2}) \end{pmatrix}
\cdot
\begin{pmatrix} D(-\frac{\alpha}{2}) & \\ & D(\frac{\alpha}{2})\end{pmatrix}\ .
\end{align*}
\item \label{30b}
For $\alpha\in \RR$, we have $\eta_l\big(D(\alpha)\big)=L_a\cdot R_b$ with $a=\cos(\frac{\alpha}{2})-\sin(\frac{\alpha}{2})k$ and $b=\cos(\frac{3\alpha}{2})+\sin(\frac{3\alpha}{2})k$, i.e.,
\begin{align*}
\eta_l\big(D(\alpha)\big)=
\begin{pmatrix}
\cos(\frac{\alpha}{2}) & & \sin(\frac{\alpha}{2}) \\
& D(\frac{\alpha}{2}) & \\
-\sin(\frac{\alpha}{2}) & & \cos(\frac{\alpha}{2})
\end{pmatrix}\cdot
\begin{pmatrix}
\cos(\frac{3\alpha}{2}) & & \sin(\frac{3\alpha}{2}) \\
& D(-\frac{3\alpha}{2}) & \\
-\sin(\frac{3\alpha}{2}) & & \cos(\frac{3\alpha}{2})
\end{pmatrix}\ .
\end{align*}
\end{enumerate}
\end{remark}
The subgroup of right translations $R_b$ is normal in $\SO{4}$, the resulting quotient is isomorphic to $\SO{3}$. This canonical projection induces a surjection from the split Cayley hexagon onto the real projective plane, given by the projection $\langle (x_1,x_2) \rangle_\RR \mapsto \langle x_1 \rangle_\RR$. (Cf.\ \cite[Section~5]{Gramlich:1998}. An alternative description of this surjection can be found in \cite{GramlichVanMaldeghem}.)
The following lemma describes how the corresponding embedded circle groups behave under this surjection.
\begin{lemma} \label{lemsurjG2} \label{lemsurjG2I}
\begin{enumerate}
\item There is an epimorphism $\eta_1:\SO{4}\to \SO{3}$ such that
\begin{align*}
\eta_1\circ \eta_p=\eps_{23}\ , && \eta_1\circ \eta_l=\eps_{12}\ .
\end{align*}
\item There is an epimorphism $\eta_2:\SO{4}\to \SO{3}$ such that
\begin{align*}
\eta_2\circ \eta_p=\eps_{12}\ , && \eta_2\circ \eta_l=\eps_{23}\ .
\end{align*}
\end{enumerate}
\end{lemma}
\begin{proof}
By Remark~\ref{mapsso4}, the map
\[ \psi: \SO{4}\to \eps_{\{2,3,4\}}\big(\SO{3}\big): L_aR_b\mapsto L_{a}R_{a}\]
is an epimorphism (see also \cite[Corollaries~11.23 and 11.24]{Salzmann:1995}). By Remark \ref{30a},
\[\psi\circ \eta_p=\eta_p=\eps_{34}\ ,\]
and, by Remark \ref{30b},
\begin{align*}
(\psi\circ \eta_l)\big(D(\alpha)\big)&=\begin{pmatrix}
\cos(\frac{\alpha}{2}) & & \sin(\frac{\alpha}{2}) \\
& D(\frac{\alpha}{2}) & \\
-\sin(\frac{\alpha}{2}) & & \cos(\frac{\alpha}{2})
\end{pmatrix}
\begin{pmatrix}
\cos(\frac{\alpha}{2}) & & -\sin(\frac{\alpha}{2}) \\
& D(\frac{\alpha}{2}) & \\
\sin(\frac{\alpha}{2}) & & \cos(\frac{\alpha}{2})
\end{pmatrix}=\begin{pmatrix}
1 & & \\
& D(\alpha) & \\
& & 1
\end{pmatrix} = \eps_{23}(D(\alpha)) \ .
\end{align*}
Therefore, the map $\eta_1:=\eps_{\{2,3,4\}}^{-1}\circ \psi$ has the desired properties. The existence of the map $\eta_2$ now follows from Lemma~\ref{10}.
\end{proof}
The final result of this section allows us to carry out Strategy~\ref{simplelacing} for edges of type $\mathrm{G}_2$ in Theorem~\ref{thm:Spin(Delta)-covers-Spin(Delta-sl)} below.
\begin{proposition} \label{surjG2}\label{surjG2I}
\begin{enumerate}
\item There is an epimorphism $\tilde\eta_1:\Spin{4}\to \Spin{3}$ such that
\begin{align*}
\tilde\eta_1\circ \tilde\eta_p=\tilde\eps_{23}\ , && \tilde\eta_1\circ \tilde\eta_l=\tilde\eps_{12}\ .
\end{align*}
\item There is an epimorphism $\tilde\eta_2:\Spin{4}\to \Spin{3}$ such that
\begin{align*}
\tilde\eta_2\circ \tilde\eta_p=\tilde\eps_{12}\ , && \tilde\eta_2\circ \tilde\eta_l=\tilde\eps_{23}\ .
\end{align*}
\end{enumerate}
\end{proposition}
\begin{proof}
The map $\tilde\eta_1 : \Spin{4} \to \Spin{3} : u+\II v \mapsto u+v $ makes the inner/right-hand quadrangle of the following diagram commute (see Remarks~\ref{Spin4Spin3} and \ref{mapisrho}):
\[\xymatrix{
\Spin{2} \ar[ddd]_{\rho_2} \ar[rd]_{\tilde\eta_p} \ar[rrrd]^{\tilde\eps_{23}} \\
& \Spin{4} \ar[rr]_{\tilde\eta_1} \ar[d]^{\rho_4} & & \Spin{3} \ar[d]_{\rho_3} \\
& \SO{4} \ar[rr]^\eta & & \SO{3} \\
\SO{2} \ar[ur]^{\eta_p} \ar[urrr]_{\eps_{23}}
}
\]
The lower triangle commutes by Lemma~\ref{lemsurjG2}. The left-hand quadrangle commutes by Lemma~\ref{lem:ama-embed-and-rho-commute-G2}.
Hence \[\eps_{23} \circ \rho_2 = \eta \circ \eta_p \circ \rho_2 = \eta \circ \rho_4 \circ \tilde\eta_p = \rho_3 \circ \tilde\eta_1 \circ \tilde\eta_p\] and \[\ker(\rho_3 \circ \tilde\eta_1 \circ \tilde\eta_p)=\ker(\eps_{23}\circ\rho_2)=\ker\rho_2 = \{\pm 1_{\Spin{2}} \}.\]
Therefore $\tilde\eps_{23} = \tilde\eta_1 \circ \tilde \eta_p$ by Proposition~\ref{prop:zusatz}. In particular, also the upper triangle of the diagram commutes.
The second claim concerning $\tilde\eta_1$ follows by an analogous argument
The claims concerning $\tilde\eta_2$ are now immediate by Lemma~\ref{10} and Proposition~\ref{prop:lift-aut-soN}
\end{proof}
\section{Diagrams of type $\mathrm{C}_2$} \label{sec:bc2}
In this section we prepare Strategy~\ref{simplelacing} for diagrams of type $\mathrm{C}_2$.
\begin{definition}
Let $\Sp{4}$ be the matrix group with respect to the $\RR$-basis $e_1$, $ie_1$, $e_2$, $ie_2$ of $\CC^2$ leaving the real alternating form \[\big( (x_1,x_2,x_3,x_4),(y_1,y_2,y_3,y_4)\big) \mapsto x_1y_2 - x_2y_1 + x_{3}y_{4} - x_{4}y_{3}\] invariant.
\end{definition}
\begin{remark} \label{coordinates}
The maximal compact subgroup of $\Sp{4}$ is the group $\U{2}$, embedded as follows. Let $e_1$, $e_2$ be the standard basis of $\CC^2$ and consider $\U{2}$ as the isometry group of the scalar product $\big((v_1,v_2),(w_1,w_2)\big) \mapsto \ol{v_1}w_1+\ol{v_2}w_2$. Defining
\begin{align*}
&& && x_1&:=\Rea(v_1), & x_2&:=\Ima(v_1), && && \\
&& && x_3&:=\Rea(v_2), & x_4&:=\Ima(v_2), && && \\
&& && y_1&:=\Rea(w_1), & y_2&:=\Ima(w_1), && && \\
&& && y_3&:=\Rea(w_2), & y_4&:=\Ima(w_2), && &&
\end{align*}
we compute
\begin{displaymath}
\ol{v_1} w_1 + \ol{v_2} w_2 = x_1y_1 + x_2y_2 + x_3y_3 + x_4y_4 + i(x_1y_2 - x_2y_1 + x_{3}y_{4} - x_{4}y_{3}).
\end{displaymath}
Since two complex numbers are equal if and only if real part and imaginary part coincide, the group $\U{2}$ preserves the form $x_1y_2 - x_2y_1 + x_{3}y_{4} - x_{4}y_{3}$ and we have found an embedding in $\Sp{4}$, acting naturally on the $\RR$-vector space $\CC^2$ with $\RR$-basis $e_1$, $ie_1$, $e_2$, $ie_2$.
As $\U{2}$ also preserves the form $x_1y_1 + x_2y_2 + x_3y_3 + x_4y_4$ with respect to the $\RR$-basis $e_1$, $ie_1$, $e_2$, $ie_2$ of $\CC^2$ it is at the same time also a subgroup of $\O{4}$, in fact of $\SO{4}$, since $\U{2}$ is connected with respect to its Lie group topology. We will give concrete coordinates for this embedding in Remark~\ref{coordinatesrev} below.
\end{remark}
The real symplectic quadrangle can be modelled as the point-line geometry consisting of the one-dimensional and two-dimensional subspaces of $\RR^4$ which are totally isotropic with respect to a nondegenerate alternating bilinear form, cf.\ \cite[Section~2.3.17]{Maldeghem:1998}, also \cite[Section~10.1]{Buekenhout/Cohen:2013}.
Using the $\RR$-basis $e_1$, $ie_1$, $e_2$, $ie_2$ and the $\RR$-alternating form on $\CC^2$ given in Remark~\ref{coordinates}, an incident point-line pair of the resulting symplectic quadrangle of $\Sp{4}$ is given by $\langle e_1 \rangle_\RR \subset \langle e_1, e_2 \rangle_\RR$. The stabilizer in $\U{2}$ of the point $\langle e_1 \rangle_\RR$ is isomorphic to $\mathrm{O}_1(\RR) \times \U{1}$ where the first factor acts diagonally on ${\langle e_1 \rangle_\RR \oplus \langle i e_1 \rangle_{\RR}}$ and the second factor acts naturally on ${\langle e_2 \rangle_{\CC}}$. The stabilizer of the line $\langle e_1, e_2 \rangle_\RR$ is isomorphic to $\mathrm{O}_2(\RR)$ acting diagonally on ${\langle e_1,e_2 \rangle_{\RR} \oplus \langle ie_1,ie_2 \rangle_{\RR}}$.
\begin{definition} \label{rank1embb2}
Let \[\zeta_p : \SO{2} \to \U{2} \subset \Sp{4} \cap \SO{4} : \begin{pmatrix} x & y \\ -y & x \end{pmatrix} \mapsto \begin{pmatrix} 1 \\ & 1 \\ & & x & y \\ & & -y & x \end{pmatrix}\] and let \[\zeta_l : \SO{2} \to \U{2} \subset \Sp{4} \cap \SO{4} : \begin{pmatrix} x & y \\ -y & x \end{pmatrix} \mapsto \begin{pmatrix} x && y \\ & x && y \\ -y & & x & \\ & -y & & x \end{pmatrix}\] be the embeddings of the circle group arising as point-stabilizing resp.\ line-stabilizing rank one groups as above with respect to the $\RR$-bases $e_1$, $e_2$ of $\RR^2$ and $e_1$, $ie_1$, $e_2$, $ie_2$ of $\CC^2$.
\end{definition}
Recall the definitions of $D(\alpha)$ and $S(\alpha)$ from Notation~\ref{nota:Dalpha-Salpha}.
\begin{notation} \label{su2acts}
In the following, identify $\CC=\{x+iy\mid x,y\in\RR\}$ with $\{ \begin{pmat} x & y \\ -y & x \end{pmat} \mid x, y\in \RR\}$. This identification, in particular, embeds $\SO{2}$ into $\CC$ as the unit circle group. For $\alpha\in \RR$, let
\[\tilde{D}(\alpha):=\zeta_l(D(\alpha))
=\begin{pmatrix} \cos(\alpha) && \sin(\alpha) \\ & \cos(\alpha) && \sin(\alpha) \\ -\sin(\alpha) & & \cos(\alpha) & \\ & -\sin(\alpha) & & \cos(\alpha) \end{pmatrix}
\in \U{2}\ ..\]
\end{notation}
\begin{lemma} \label{lem:rank1-inv-B2}
Let $B:=\diag(-1,1,-1,1)$, $C:=\diag(-1,-1,1,1)\in \U{2}$. Then the following hold:
\begin{enumerate}
\item The map $\gamma_B:\U{2}\to \U{2} : A\mapsto B\cdot A\cdot B^{-1}=B\cdot A\cdot B$
is an automorphism of $\U{2}$ such that
\[\gamma_B\circ \zeta_p=\zeta_p\circ \inv \qquad \text{and} \qquad \gamma_B\circ \zeta_l=\zeta_l.\]
\item The map $\gamma_C:\U{2}\to \U{2} : A\mapsto C\cdot A\cdot C^{-1}=C\cdot A\cdot C$
is an automorphism of $\U{2}$ such that
\[\gamma_C\circ \zeta_p=\zeta_p \qquad \text{and} \qquad \gamma_C\circ \zeta_l=\zeta_l\circ\inv.\]
\end{enumerate}
\end{lemma}
\begin{proof}
Straightforward.
\end{proof}
\begin{lemma}
Let $\alpha\in \RR$. Then $\tilde{D}(\alpha)\in \SU{2}$ and $\begin{pmat} D(-\alpha) & \\ & D(\alpha)\end{pmat}\in \SU{2}$.
\end{lemma}
\begin{proof}
Given $\alpha\in \RR$, one has the following (where the determinant is taken in $\mathrm{SL}_2(\CC)$):
\begin{align*}
\det\big(\tilde{D}(\alpha)\big)
=\begin{pmat} \cos(\alpha) & \\ & \cos(\alpha)\end{pmat}^2+
\begin{pmat} \sin(\alpha) & \\ & \sin(\alpha)\end{pmat}^2
=\begin{pmat} \cos(\alpha)^2+ \sin(\alpha)^2 & \\ & \cos(\alpha)^2+\sin(\alpha)^2\end{pmat}
=I_2=1_\CC
\end{align*}
and
\[\det \begin{pmat} D(-\alpha) & \\ & D(\alpha)\end{pmat}=D(-\alpha)\cdot D(\alpha)=I_2=1_\CC\ . \qedhere\]
\end{proof}
\begin{remark} \label{coordinatesrev}
Returning to the embedding $\U{2} \to \SO{4}$ mentioned in Remark~\ref{coordinates}, the group \[\SU{2} = \left\{ \begin{pmatrix} x_1+iy_1 & x_2+iy_2 \\ -x_2+iy_2 & x_1-iy_2 \end{pmatrix} \mid x_1, x_2, y_1, y_2 \in \RR, x_1^2+x_2^2+y_1^2+y_2^2=1 \right\} \leq \U{2}\] acts $\RR$-linearly on $\CC^2$ with transformation matrices \[\begin{pmatrix} x_1 & -y_1 & x_2 & -y_2 \\ y_1 & x_1 & y_2 & x_2 \\ -x_2 & -y_2 & x_1 & y_1 \\ y_2 & -x_2 & -y_1 & x_1 \end{pmatrix}\] with respect to the basis $e_1$, $ie_1$, $e_2$, $ie_2$.
Remark~\ref{remark5} implies that the map
\begin{eqnarray*}
\SU{2} & \to & \SO{4} \\
\begin{pmatrix} x_1+iy_1 & x_2+iy_2 \\ -x_2+iy_2 & x_1-iy_2 \end{pmatrix} & \mapsto & R_{x_1-y_1i+x_2j-y_2k} = \begin{pmatrix} x_1 & -y_1 & x_2 & -y_2 \\ y_1 & x_1 & y_2 & x_2 \\ -x_2 & -y_2 & x_1 & y_1 \\ y_2 & -x_2 & -y_1 & x_1 \end{pmatrix}
\end{eqnarray*}
injects $\SU{2}$ into $\SO{4}$. The restriction of this map to $\SO{2} \subset \SU{2}$ by setting $y_1=0=y_2$ provides the transformations $\tilde D$ from Notation~\ref{su2acts}.
The group \[\left \{ \begin{pmatrix} \cos(\alpha) + i\sin(\alpha) & 0 \\ 0 & \cos(\alpha) + i\sin(\alpha) \end{pmatrix} \right \} \cong \U{1} \cong \SO{2}\] acts with transformation matrices \[\begin{pmatrix} D(-\alpha) & 0 \\ 0 & D(-\alpha) \end{pmatrix} = \begin{pmatrix} \cos(\alpha) & -\sin(\alpha) & 0 & 0 \\ \sin(\alpha) & \cos(\alpha) & 0 & 0 \\ 0 & 0 & \cos(\alpha) & -\sin(\alpha) \\ 0 & 0 & \sin(\alpha) & \cos(\alpha) \end{pmatrix},\] i.e., Remark~\ref{remark5} implies that the map
\begin{eqnarray*}
\U{1} & \to & \SO{4} \\
\begin{pmatrix} \cos(\alpha) + i\sin(\alpha) & 0 \\ 0 & \cos(\alpha) + i\sin(\alpha) \end{pmatrix} & \mapsto & L_{\cos(\alpha)+i\sin(\alpha)} = \begin{pmatrix} \cos(\alpha) & -\sin(\alpha) & 0 & 0 \\ \sin(\alpha) & \cos(\alpha) & 0 & 0 \\ 0 & 0 & \cos(\alpha) & -\sin(\alpha) \\ 0 & 0 & \sin(\alpha) & \cos(\alpha) \end{pmatrix}
\end{eqnarray*}
injects $\U{1}$ into $\SO{4}$.
Altogether, using Remark~\ref{Spin4Spin3}, we obtain the following commutative diagram:
\[\xymatrix{
\U{1} \times \SU{2} \ar[rr] \ar[d]_{\widehat\rho} && \mathrm{U}_1(\HH) \times \mathrm{U}_1(\HH) \cong \Spin{3} \times \Spin{3} \cong \Spin{4} \ar[d]^{\rho_4} \\
\U{2} \ar[rr] && \SO{4}
}.
\]
\end{remark}
Our candidate for a spin cover of the group $\U{2}$
therefore is its double cover
\[ \U{1} \times\SU{2} \to \U{2} : (z,A) \mapsto zA.\]
Note that the fundamental group of $\U{2}$ equals $\ZZ$, as the determinant map $\det : \U{2} \to \U{1} \cong \SO{2}$ induces an isomorphism of fundamental groups; its simply connected universal cover is isomorphic to $\RR \times \SU{2}$. The above double cover is unique up to isomorphism, because $\ZZ$ has a unique subgroup of index two (cf.\ \cite[Theorem~1.38]{Hatcher:2002}).
\begin{notation} \label{zetaspin}
In the following, let
\begin{align*}
\tilde{\zeta}_p &: \Spin{2} \to \SO{2}\times\SU{2} \subset \Spin{4} : S(\alpha)\mapsto \left( D(\alpha), \begin{pmat} D(-\alpha) & \\ & D(\alpha)\end{pmat}\right)\ , \\
\tilde{\zeta}_l &: \Spin{2}\to \SO{2}\times\SU{2} \subset \Spin{4} : S(\alpha)\mapsto \big( \id , \tilde{D}(\alpha)\big)
\end{align*}
and let
\[
\widehat\rho:\SO{2}\times \SU{2}\to \U{2} : (z,A)\mapsto \begin{pmat} z & \\ & z\end{pmat}\cdot A\ .
\]
\end{notation}
Recall the maps
\begin{align*}
\rho_2:\Spin{2}\to \SO{2} & : S(\alpha)\mapsto D(2\alpha), \\
\mathrm{sq} : G \to G & : x \mapsto x^2, \\
\mathrm{inv} : G \to G & : x \mapsto x^{-1}.
\end{align*}
\begin{lemma} \label{concretedescriptiontilde}
One has
\begin{align*}
\tilde{\zeta}_p= \tilde\eps_{34} && \tilde{\zeta}_l\circ\sq = \tilde\eps_{23}\cdot\tilde\eps_{14}
\end{align*}
\end{lemma}
\begin{proof}
Using the identification $\Spin{4} \cong \Spin{3} \times \Spin{3}$ and considering the left hand factor as transformations by left multiplication and the right hand factor as transformations by right multiplication of unit quaternions we compute
\begin{align*}
\tilde\eps_{34}\big(\cos(\alpha)+\sin(\alpha)e_1e_2\big) &= \cos(\alpha)+\sin(\alpha)e_3e_4 \\
&= \cos(\alpha)-\II i\sin(\alpha) & \text{by Remark~\ref{Spin4Spin3}} \\
&= \big(\cos(\alpha)-i\sin(\alpha),\cos(\alpha)+i\sin(\alpha)\big) & \text{by Remark~\ref{Spin4Spin3}} \\
&=\left(L_{\cos(\alpha)-i\sin(\alpha)},R_{\cos(\alpha)-i\sin(\alpha)}\right) & \text{by Lemma~\ref{lemmaleftright}, Remark~\ref{remark5}} \\
&=\left( D(\alpha), \begin{pmatrix} D(-\alpha) & \\
& D(\alpha)\end{pmatrix}\right) & \text{by Remark~\ref{coordinatesrev}} \\
&=\tilde{\zeta}_p\big(\cos(\alpha)+\sin(\alpha)e_1e_2\big)
\end{align*}
and
\begin{align*}
& \tilde\eps_{23}\big(\cos(\alpha)+\sin(\alpha)e_1e_2\big)\tilde\eps_{14}\big(\cos(\alpha)+\sin(\alpha)e_1e_2\big) \\ =& \big(\cos(\alpha)+\sin(\alpha)e_2e_3\big)\big(\cos(\alpha)+\sin(\alpha)e_1e_4\big) \\
=&\big(\cos(\alpha)\big)^2+j\cos(\alpha)\sin(\alpha) + \II\left(\big(\sin(\alpha)\big)^2-j\cos(\alpha)\sin(\alpha)\right) & \text{by Remark~\ref{Spin4Spin3}} \\
=&\left(1,\big(\cos(\alpha)+j\sin(\alpha)\big)^2\right) & \text{by Remark~\ref{Spin4Spin3}} \\
=&\left(\id,R_{\big(\cos(\alpha)+j\sin(\alpha)\big)^2}\right) & \text{by Lemma~\ref{lemmaleftright}, Remark~\ref{remark5}} \\
=&\left(\id,\tilde D(2\alpha)\right) & \text{by Remark~\ref{coordinatesrev}} \\
=& \left(\tilde{\zeta}_l\circ\sq\right)\big(\cos(\alpha)+\sin(\alpha)e_1e_2\big). \qedhere
\end{align*}
\end{proof}
The following observation is the analog of Lemma~\ref{lem:ama-embed-and-rho-commute-G2}.
\begin{lemma} \label{lem:ama-embed-and-rho-commute-B2}
One has
\begin{align*}
\widehat\rho\circ \tilde{\zeta}_p= {\zeta}_p\circ \rho_2\ , &&
\widehat\rho\circ\tilde{\zeta}_l\circ\sq=\widehat\rho\circ\sq\circ \tilde{\zeta}_l=
{\zeta}_l\circ \rho_2\ .
\end{align*}
Moreover,
\begin{align*}
\left(\widehat\rho\right)^{-1}\left({\zeta}_p\right)(\SO{2}) &\cong \Spin{2} \quad \text{ and } \\
\left(\widehat\rho\right)^{-1}\left({\zeta}_l\right)(\SO{2}) &\cong \{ 1, -1 \} \times \SO{2}.
\end{align*}
\end{lemma}
\begin{proof}
For $\alpha\in \RR$,
\[
(\widehat\rho\circ \tilde{\zeta}_p)\big(S(\alpha)\big)
=\widehat\rho \left( D(\alpha), \begin{pmat} D(-\alpha) & \\ & D(\alpha)\end{pmat}\right)
=\begin{pmat} 1_{\SO{2}} & \\ & D(2\alpha)\end{pmat}
=\zeta_p\big(D(2\alpha)\big)
=(\zeta_p\circ \rho_2)\big(S(\alpha)\big)
\]
and
\[
(\widehat\rho\circ\sq\circ \tilde{\zeta}_l)\big(S(\alpha)\big)
=\widehat\rho\big( 1_{\SO{2}}, \tilde{D}(2\alpha)\big)
=\tilde{D}(2\alpha)
=\zeta_l\big(D(2\alpha)\big)
=(\zeta_l\circ \rho_2)\big(S(\alpha)\big)\ .
\]
For the second claim observe that
\begin{align*}
\left(\widehat\rho\right)^{-1}\left({\zeta}_p\right)(\SO{2}) = \left\{ \left( D(\alpha), \begin{pmat} D(-\alpha) & \\ & D(\alpha)\end{pmat}\right) \mid \alpha \in \RR \right\} &\cong \Spin{2} , \\
\left(\widehat\rho\right)^{-1}\left({\zeta}_l\right)(\SO{2}) = \left\{ \big( \pm \id , \tilde{D}(\alpha)\big) \mid \alpha \in \RR \right\} &\cong \{ 1, -1 \} \times \SO{2}. \qedhere
\end{align*}
\end{proof}
\begin{consequence} \label{concretedescription}
One has
\begin{align*}
{\zeta}_p= \eps_{34} && \zeta_l = \eps_{23}\cdot\eps_{14}
\end{align*}
\end{consequence}
\begin{proof}
By Remarks~\ref{6} and \ref{coordinatesrev} and Lemma~\ref{lem:ama-embed-and-rho-commute-B2} this is immediate from Lemma~\ref{concretedescriptiontilde}.
\end{proof}
\begin{proposition} \label{prop:lift-aut-B2}
Given $\tau\in \Aut\big(\U{2}\big)$, there is a unique $\tilde\tau\in \Aut\big( \SO{2}\times \SU{2}\big)$ such that
\[\widehat\rho\circ \tilde\tau=\tau\circ \widehat\rho\ .\]
\end{proposition}
\begin{proof}
Let $\tau\in \Aut\big(\U{2}\big)$. Consider the characteristic subgroups
\begin{align*}
H_1:=Z\big(\U{2}\big)=\{ z\cdot I_2 \mid z\in \U{1} \cong \SO{2} \}\ , &&
H_2:=[\U{2},\U{2}]=\SU{2}\ ,
\end{align*}
of $\U{2}$.
Note that $H_1$ and $H_2$ are also characteristic in $H:=H_1\times H_2$, as $H_1=Z(H)$ and $H_2=[H,H]$.
Hence $\Aut(H)=\Aut(H_1)\times\Aut(H_2)$.
Let $\tau_1:=\tau_{|H_1}\in\Aut(H_1)$ and $\tau_2:=\tau_{|H_2}\in\Aut(H_2)$. Then $\tilde\tau:=(\tau_1,\tau_2)\in\Aut(H)$ satisfies
\begin{align*}
(\widehat\rho\circ \tilde\tau)(z,A)=\widehat\rho\big( \tau_1(z), \tau_2(A) \big)=\tau_1(z)\tau_2(A)=\tau(z)\tau(A)=\tau(zA)=(\tau\circ\widehat\rho)(z,A)
\end{align*}
for all $(z,A)\in \SO{2}\times \SU{2}$.
Let $\psi=(\psi_1,\psi_2)\in \Aut \big( \SO{2}\times \SU{2}\big)$ be such that
$\widehat\rho\circ \psi=\tau\circ \widehat\rho$. Given $z\in \SO{2}$ and $A\in \SU{2}$, one has
\[
\psi_1(z)
=\psi_1(z)\psi_2(I_2)
=(\widehat\rho\circ \psi)(z,I_2)
=(\tau\circ \widehat\rho)(z,I_2)
=(\widehat\rho\circ \tilde\tau)(z,I_2)
=\tau_1(z)\tau_2(I_2)=\tau_1(z)
\]
and
\[
\psi_2(A)
=(\widehat\rho\circ \psi)(1_{\SO{2}},A)
=(\widehat\rho\circ \tilde\tau)(1_{\SO{2}},A)
=\tau_2(A)\ .
\qedhere
\]
\end{proof}
The final result of this section allows us to carry out Strategy~\ref{simplelacing} for edges of type $\mathrm{C}_2$ in Theorem~\ref{thm:Spin(Delta)-covers-Spin(Delta-sl)} below.
\begin{proposition} \label{surjB2}
There is an epimorphism $\tilde{\zeta}:\SO{2}\times \SU{2}\to \Spin{3}$ such that
\begin{align*}
\tilde{\zeta} \circ \tilde{\zeta}_p=\tilde\eps_{12}\circ \inv\ , && \tilde{\zeta}\circ \tilde{\zeta}_l=\tilde\eps_{23}\ .
\end{align*}
\end{proposition}
\begin{proof}
The map
\begin{eqnarray*}
\psi:\Spin{3} & \to & \SU{2} \\
a+be_1e_2+ce_2e_3+de_3e_1 & \mapsto & R_{a+bi+cj+dk} = \begin{pmatrix} a & b & c & d \\
-b & a & -d & c \\
-c & d & a & -b \\
-d & -c & b & a \end{pmatrix}
\end{eqnarray*}
is a group isomorphism by Remarks~\ref{rem:cl3=quaternions}(b) and \ref{remark5}(b) (see also \cite[11.26]{Salzmann:1995}), with the convention that the matrix group $\SU{2}$ acts $\RR$-linearly on $\CC^2 \cong \HH$ with respect to the $\RR$-basis $1$, $i=i1$, $j$, $k=ij$ as in Definition~\ref{rank1embb2} and Notation~\ref{su2acts}. Let
\[\tilde{\zeta}: \SO{2}\times \SU{2}\to \Spin{3} : (z,A)\mapsto \psi^{-1}(A)\ .\]
For $\alpha\in \RR$, one has
\begin{align*}
(\tilde{\zeta}\circ \tilde{\zeta}_p)\big(S(\alpha)\big)
&=
\tilde{\zeta}\left( D(\alpha), \begin{pmat} D(-\alpha) & \\ & D(\alpha)\end{pmat}\right)
= \cos(\alpha)-\sin(\alpha)e_1e_2
=(\tilde\eps_{12}\circ\inv)\big(S(\alpha)\big)
\end{align*}
and
\begin{align*}
(\tilde{\zeta}\circ \tilde{\zeta}_l)\big(S(\alpha)\big)
&= \tilde{\zeta}\big( 1_{\SO{2}}, \tilde{D}(\alpha)\big)
=\zeta\big(\tilde{D}(\alpha)\big)
= \cos(\alpha)+\sin(\alpha)e_2e_3
=\tilde\eps_{23}\big(S(\alpha)\big)\ .
\qedhere
\end{align*}
\end{proof}
\begin{corollary} \label{corsurjB2}\label{corsurjB2b}\label{corsurjB2I}\label{surjB2b}\label{surjB2I}
\begin{enumerate}
\item
There exist epimorphisms $\zeta_1, \zeta_2, \zeta_3 : \mathrm{U}_2(\CC) \to \SO{3}$ such that \begin{align*}
{\zeta_1} \circ {\zeta}_p=\eps_{12}\circ \inv\ , && {\zeta_1}\circ {\zeta}_l=\eps_{23}, \\
{\zeta_2} \circ {\zeta}_p=\eps_{12} , && {\zeta_2}\circ {\zeta}_l=\eps_{23}, \\
{\zeta_3} \circ {\zeta}_p=\eps_{23} , && {\zeta_3}\circ {\zeta}_l=\eps_{12}.
\end{align*}
\item There exist epimorphisms $\tilde{\zeta}_2, \tilde{\zeta}_3 :\SO{2}\times \SU{2}\to \Spin{3}$ such that
\begin{align*}
\tilde{\zeta}_2 \circ \tilde{\zeta}_p=\tilde\eps_{12} , && \tilde{\zeta}_2\circ \tilde{\zeta}_l=\tilde\eps_{23}, \\
\tilde{\zeta}_3 \circ \tilde{\zeta}_p=\tilde\eps_{23} , && \tilde{\zeta}_3\circ \tilde{\zeta}_l=\tilde\eps_{12}.
\end{align*}
\end{enumerate}
\end{corollary}
\begin{proof}
The kernel of the epimorphism $\SO{2} \times \SU{2} \to \U{2} : (z,A) \mapsto zA$ is equal to $\left\langle \left(D(\pi), \tilde D(\pi)\right) \right\rangle$.
One has $\tilde\zeta\left(D(\pi), \tilde D(\pi)\right) = \psi^{-1}(-I_4) = -1_{\Spin{3}}$, whence $\left\langle \left(D(\pi), \tilde D(\pi)\right) \right\rangle \subseteq \ker(\rho_3 \circ \tilde\zeta)$.
Therefore, the claim concerning $\zeta_1$ follows from Proposition~\ref{surjB2} and the homomorphism theorem of groups.
The claims about $\zeta_2$, $\zeta_3$ then follow from Lemma~\ref{lem:rank1-inv-A2}, resp.\ \ref{10}. A subsequent application of Proposition~\ref{prop:lift-aut-soN} yields the claims about $\tilde\zeta_2$, $\tilde\zeta_3$.
\end{proof}
\section{Non-spherical diagrams of rank two} \label{sec:rank-2-residues}
In this section we prepare Strategy~\ref{simplelacing} for non-spherical diagrams of rank two. For an introduction to the concept of a Kac--Moody root datum we refer the reader to \cite[Introduction]{Tits:1987}, \cite[7.1.1, p.~172]{Remy:2002}, \cite[Definition~5.1]{Marquis:2013}. For the definition of simple connectedness see \cite[7.1.2]{Remy:2002}. Note that for a given generalized Cartan matrix, up to isomorphism, there exists a uniquely determined simply connected Kac--Moody root datum.
\begin{definition} \label{krs}
Let $r, s \in \NN$ such that $rs \geq 4$ and consider the generalized Cartan matrix of rank two given by
\[ A := (a(i,j))_{i \in \{1, 2\}}
= \begin{pmatrix} 2 & -r \\ -s & 2\end{pmatrix}
\]
and a simply connected Kac--Moody root datum $\mathcal{D} = ( \{1, 2 \}, A, \Lambda, (c_i)_{i \in \{1, 2\}}, (h_i)_{i \in \{1, 2\}})$.
Let $G:=G(A):=G(\mathcal{D})$ be the corresponding (simply connected) real Kac--Moody group of rank two, let $T_0$ be the fundamental torus of $G$ with respect to the fundamental roots $\alpha_1$, $\alpha_2$, and let
\[K:=K^{r,s}:=K(A)\]
be the subgroup consisting of the elements fixed by the Cartan--Chevalley involution with respect to $T_0$ of $G$ (its maximal compact subgroup).
\end{definition}
\begin{remark}
Let $G_i \cong \SL{2}$ be the corresponding fundamental subgroups of rank one and define
\[
K_i := K^{r,s}_i := G_i \cap K^{r,s} \cong \SO{2}
\quad\text{ and }\quad
T:=T_0\cap K.
\]
By (KMG3) (see \cite[p.\ 545]{Tits:1987} or, e.g., \cite[p.\ 84]{Marquis:2013}), the torus $T_0$ is generated by $\mu^{h_i}$ for $i=1,2$ and $\mu\in \RR \backslash \{ 0 \}$
arbitrary. The action of the Cartan--Chevalley involution
on the torus is given by $\mu^{h_i} \mapsto (\mu^{-1})^{h_i}=\mu^{-h_i}$. Hence
\[
T = T_0\cap K =
\left\{
\id = 1^{h_1}=1^{h_2},
t_1:=(-1)^{h_1}, t_2:=(-1)^{h_2}, t_1 t_2 = (-1)^{h_1+h_2}
\right\}
\cong \ZZ/2\ZZ \times \ZZ/2\ZZ.
\]
\end{remark}
\begin{lemma} \label{kisfreeamalgam}
$K$ is isomorphic to a free amalgamated product
\[
K_1T *_{T} K_2T.
\]
\end{lemma}
\begin{proof}
The twin building of the Kac--Moody group $G$ is a twin tree (cf.\ \cite{Ronan/Tits:1987}, \cite{Ronan/Tits:1994}). The chambers are the edges; the panels are the sets of edges sharing one vertex and, hence, correspond to vertices. The group $K$ acts edge-transitively and without inversions on each half of the twin tree of $G$ by the Iwasawa decomposition (see, e.g., \cite{Medts/Gramlich/Horn}).
The Cartan--Chevalley involution $\omega$ interchanges the two halves of the twin tree,
mapping edges to opposite edges. Hence the stabilizer in $K$ of
the fundamental edge $c^+$ also stabilizes the opposite edge $c^-=\omega(c^+)$
and, thus, the unique twin apartment spanned by them. It follows that the edge stabilizer is $T$.
Since the panels correspond to the vertices of the tree, the stabilizers of the vertices of
the fundamental edge $c^+$ are equal to $K_1T$ and $K_2T$.
The claim follows from \cite[I, \S 5]{Serre:Trees}.
\end{proof}
\begin{remark} \label{tcentralizes}
Since $K_i \unlhd K_iT$, this free amalgamated product is fully determined by the intersections $K_i \cap T$ and the action of $T$ on each $K_i$.
Note that (KMG3) implies $T \cap K_1 = \{ 1, t_1\}$ and $T \cap K_2 = \{ 1, t_2 \}$.
The action of $T$ on each $K_i$ can be extracted from the action of $T_0$ on each $G_i$, which according to \cite[(4), p.\ 549]{Tits:1987} (or also \cite[(5.1), p.\ 86]{Marquis:2013}) is given by
\[ t x_i(\lambda) t^{-1} = x_i (t(c_i) \lambda) \]
for $t \in T_0$, $\lambda \in \RR$ and root group functor $x_i$.
According to \cite[Section~2, p.~544]{Tits:1987} (or also \cite[Definitions 5.1 and 5.5]{Marquis:2013}) one computes for $i,j\in\{1,2\}$ that
\[ t_i(c_j) = (-1)^{h_i}(c_j) = (-1)^{h_i(c_j)} = (-1)^{a(i,j)}. \]
We conclude that $t_i$ acts trivially on $K_j$, if and only if the entry $a(i,j)$ of the generalized Cartan matrix is even; conversely it acts non-trivially (and hence by inversion) if and only if $a(i,j)$ is odd.
In symbols, for
\[
n(i,j):=
\begin{cases}
0, & \text{if } a(i,j) \text{ is even}, \\
1, & \text{if } a(i,j) \text{ is odd}.
\end{cases}
\]
and $k_j \in K_j$ one has
\begin{eqnarray}
t_i^{-1}k_jt_i & = & k_j^{-2n(i,j)}k_j. \label{crucialidentity}
\end{eqnarray}
\end{remark}
\begin{notation}
Let
\[
\theta_1 := \theta_1^{r,s} : \SO{2} \to K_1^{r,s}, \qquad
\theta_2 := \theta_2^{r,s} : \SO{2} \to K_2^{r,s},
\]
be continuous isomorphisms.
\end{notation}
\begin{defn}
Let $r, s \in \NN$ such that $rs \geq 4$, let $A = \begin{pmat} 2 & -r \\ -s & 2\end{pmat}$. Then set
\[
H^{r,s}
:= \begin{cases}
\SO{2} \times \SO{2} & \text{ if } r\equiv s \equiv 0 \pmod 2\ ,\\
\SO{3} & \text{ if } r\equiv s \equiv 1 \pmod 2\ ,\\
\U{2} & \text{ otherwise}.\\
\end{cases}
\]
Also set
\[
\delta_1:=
\begin{cases}
\iota_1 & \text{ if } r\equiv s \equiv 0 \pmod 2\ ,\\
\eps_{12} & \text{ if } r\equiv s \equiv 1 \pmod 2\ ,\\
\zeta_l & \text{ if } r\equiv 0, s\equiv 1 \pmod 2,\\
\zeta_p & \text{ if } r\equiv 1, s\equiv 0 \pmod 2,
\end{cases}
\quad
\delta_2:=
\begin{cases}
\iota_2 & \text{ if } r\equiv s \equiv 0 \pmod 2
\qquad \text{(see \ref{notationiota})},\\
\eps_{23} & \text{ if } r\equiv s \equiv 1 \pmod 2
\qquad \text{(see \ref{nota:EI-VI-qI})},\\
\zeta_p & \text{ if } r\equiv 0, s\equiv 1 \pmod 2
\quad \text{(see \ref{rank1embb2})},\\
\zeta_l & \text{ if } r\equiv 1, s\equiv 0 \pmod 2.
\end{cases}
\]
\end{defn}
\begin{proposition} \label{keyproposition}
Let $r, s \in \NN$ such that $rs \geq 4$, let $A = \begin{pmat} 2 & -r \\ -s & 2\end{pmat}$, let $G(A)$ be the corresponding simply connected real Kac--Moody group, and let $K^{r,s}$ be its maximal compact subgroup. Then
there exists a group epimorphism $\theta : K^{r,s} \to H^{r,s}$ such that
\[\theta \circ \theta_1 = \delta_1 \quad \text{ and } \quad \theta \circ \theta_2 = \delta_2.\]
\end{proposition}
\begin{proof}
By Lemma~\ref{kisfreeamalgam} the group $K$ is isomorphic to $K_1T *_{T} K_2T$ with $T = \{ 1, t_1, t_2, t_1t_2\} \cong \ZZ/2\ZZ \times \ZZ/2\ZZ$ and $T_0 \cap K_1 = \{ 1, t_1 \}$, $T_0 \cap K_2 = \{ 1, t_2 \}$; in particular, $K$ is generated by $K_1 \cong \SO{2}$ and $K_2 \cong \SO{2}$. As $K$ is a free amalgamated product it therefore suffices to define $\theta$ on each of the $K_i$ and to verify that the actions of the $t_i$ on the $K_j$ are compatible with the actions of the images of the $t_i$ on the images of the $K_j$.
Define $\theta$ via
\[\theta_{|K_1} : K_1 \to \delta_1(\SO{2}) : x \mapsto \left(\delta_1 \circ {\theta_1}^{-1}\right)(x)
\quad\text{ and }\quad
\theta_{|K_2} : K_2 \to \delta_2(\SO{2}) : x \mapsto \left(\delta_2 \circ {\theta_2}^{-1}\right)(x).\]
Then this is compatible with the action of $T$. Indeed, using Remark~\ref{tcentralizes}, one observes:
\begin{enumerate}
\item
Since $r\equiv s \equiv 0 \pmod 2$, the elements $t_i$ centralize the groups $K_j$ which is compatible with the fact that $\SO{2} \times \SO{2}$ is an abelian group.
\item
Since $r\equiv s \equiv 1 \pmod 2$, the element $t_1$ inverts the group $K_2$ and the element $t_2$ inverts the group $K_1$ which is compatible with the situation in $\SO{3}$ by Lemma~\ref{lem:rank1-inv-A2}.
\item
Since $r\equiv 0 \pmod 2$, $s\equiv 1 \pmod 2$, the element $t_1$ centralizes $K_2$ and the element $t_2$ inverts the group $K_1$. This is compatible with the following computations (cf.\ Definition~\ref{rank1embb2}):
\[\forall g \in K_2 : \theta(t_1gt_1)
= \begin{pmat} -1 && 0 \\ & -1 && 0 \\ 0 & & -1 & \\ & 0 & & -1 \end{pmat}
\begin{pmat} 1 \\ & 1 \\ & & x & y \\ & & -y & x \end{pmat}
\begin{pmat} -1 && 0 \\ & -1 && 0 \\ 0 & & -1 & \\ & 0 & & -1 \end{pmat}
= \begin{pmat} 1 \\ & 1 \\ & & x & y \\ & & -y & x \end{pmat}
= \theta(g),\]
\[\forall g \in K_1 : \theta(t_2gt_2)
= \begin{pmat} 1 \\ & 1 \\ & & -1 & 0 \\ & & 0 & -1 \end{pmat}
\begin{pmat} x && y \\ & x && y \\ -y & & x & \\ & -y & & x \end{pmat}
\begin{pmat} 1 \\ & 1 \\ & & -1 & 0 \\ & & 0 & -1 \end{pmat}
= \begin{pmat} x && -y \\ & x && -y \\ y & & x & \\ & y & & x \end{pmat}
= \theta(g^{-1}).\]
\item This is dual to (c). \qedhere
\end{enumerate}
\end{proof}
\begin{defn} \label{thetaspin}
Let $r, s \in \NN$ such that $rs \geq 4$, let $A = \begin{pmat} 2 & -r \\ -s & 2\end{pmat}$. Then set
\[
\widetilde{H}^{r,s}
:= \begin{cases}
\Spin{2} \times \Spin{2} & \text{ if } r\equiv s \equiv 0 \pmod 2\ ,\\
\Spin{3} & \text{ if } r\equiv s \equiv 1 \pmod 2\ ,\\
\SO{2} \times \SU{2} & \text{ otherwise}.\\
\end{cases}
\]
Furthermore, set
\[
\tilde\delta_1:=
\begin{cases}
\tilde\iota_1 & \text{ if } r\equiv s \equiv 0 \pmod 2,\\
\tilde\eps_{12} & \text{ if } r\equiv s \equiv 1 \pmod 2,\\
\tilde\zeta_l & \text{ if } r\equiv 0, s\equiv 1 \pmod 2,\\
\tilde\zeta_p & \text{ if } r\equiv 1, s\equiv 0 \pmod 2,\\
\end{cases}
\quad
\tilde\delta_2:=
\begin{cases}
\tilde\iota_2 & \text{ if } r\equiv s \equiv 0 \pmod 2
\qquad \text{(see \ref{iotaspin})},\\
\tilde\eps_{23} & \text{ if } r\equiv s \equiv 1 \pmod 2
\qquad \text{(see \ref{2})},\\
\tilde\zeta_p & \text{ if } r\equiv 0, s\equiv 1 \pmod 2
\quad \text{(see \ref{zetaspin})},\\
\tilde\zeta_l & \text{ if } r\equiv 1, s\equiv 0 \pmod 2.\\
\end{cases}
\]
The central extension \[\bar\rho : \wt H^{r,s} \to H^{r,s}\] satisfies
\[\bar\rho =
\begin{cases}
\rho_2 \times \rho_2 & \text{ if } r\equiv s \equiv 0 \pmod 2
\quad \text{(see \ref{iotaspin})}, \\
\rho_3 & \text{ if } r\equiv s \equiv 1 \pmod 2
\quad \text{(see \ref{rho})},\\
\widehat \rho & \text{ otherwise }
\qquad\qquad\qquad\text{(see \ref{zetaspin})}.
\end{cases}
\]
Let $K^{r,s}=K(A) = K_1T *_{T} K_2T$ as in Lemma~\ref{kisfreeamalgam} and let $t_1 \in K_1 \cap T$, $t_2 \in K_2 \cap T$ as in Remark~\ref{tcentralizes}.
\medskip
Define
\begin{align*}
u_i &:= \theta(t_i) \quad\quad\quad\quad \text{(see \ref{keyproposition})}, \\
U &:= \langle u_1, u_2\rangle \cong \ZZ/2\ZZ \times \ZZ/2\ZZ.
\end{align*}
Furthermore, define
\begin{align*}
\wt U &:= \bar\rho^{\,-1}(U), \\
\wt K_i := \wt K_i^{r,s} &:= \bar\rho^{\,-1}\big(\theta(K_i)\big), \\
\intertext{and the \Defn{spin extension} }
\wt K := \wt K^{r,s} := \wt K(A) &:= \wt K_1\wt U *_{\wt U} \wt K_2\wt U
\quad\text{ of }\quad
K=K^{r,s}=K(A),
\end{align*}
let \[\hat{\hat\rho} : \wt K_1\wt U *_{\wt U} \wt K_2\wt U \to K_1T *_T K_2T\] be the epimorphism induced by $\bar\rho_{|\wt K_1}$, $\bar\rho_{|\wt K_2}$ and let \[\tilde\theta_1 : \Spin{2} \to \widetilde K_1 \quad\quad \text{and} \quad\quad \tilde\theta_2 : \Spin{2} \to \widetilde K_2\] be continuous monomorphisms such that the following diagrams commute for $i=1,2$:
\[
\xymatrix{
\Spin{2} \ar[d] \ar[rr]^{\tilde\theta_i} && \wt K_i \ar[d]_{\bar\rho_{|\wt K_i}} \\
\SO{2} \ar[rr]^{\theta_i} && K_i
}
\]
\end{defn}
\begin{remark} \label{dichotomy}
One has $\wt K_1 \cong \Spin{2}$ unless $r\equiv 0, s\equiv 1 \pmod 2$ and $\wt K_2 \cong \Spin{2}$ unless $r\equiv 1, s\equiv 0 \pmod 2$, in which case the respective group is isomorphic to $\{ 1, -1 \} \times \SO{2}$ (cf.\ Lemma~\ref{lem:ama-embed-and-rho-commute-B2}). Hence $\tilde\theta_1$ actually is a (continuous) isomorphism unless $r\equiv 0, s\equiv 1 \pmod 2$, in which case it is a (continuous) isomorphism onto the unique connected subgroup of index two of $\wt K_1$.
For $i=1$ the map on the left hand side of the above commutative diagram is
\[\begin{array}{rclll}
\Spin{2} \to \SO{2} &: & S(\alpha) \mapsto D(\alpha) & \text{if } r\equiv 0, s\equiv 1 \pmod 2 \quad\quad & \text{(see \ref{nota:Dalpha-Salpha})} \\
\Spin{2} \to \SO{2} &:& x \mapsto \rho_2(x) & \text{otherwise}.
\end{array}\]
The dual statement holds for $\tilde\theta_2$.
In particular, for $i=2$ the map on the left hand side of the above commutative diagram is
\[\begin{array}{rclll}
\Spin{2} \to \SO{2} &: &S(\alpha) \mapsto D(\alpha) & \text{if } r\equiv 1, s\equiv 0 \pmod 2 \quad\quad & \text{(see \ref{nota:Dalpha-Salpha})} \\
\Spin{2} \to \SO{2} &: & x \mapsto \rho_2(x) & \text{otherwise}.
\end{array}\]
Define
\begin{align*}
\tilde t_1 & := \begin{cases}
\tilde\theta_1(S(\pi)), & \text{if } r\equiv 0, s\equiv 1 \pmod 2 \\
\tilde\theta_1(S(\frac{\pi}{2})), & \text{otherwise}.
\end{cases}
\\
\tilde t_2 & := \begin{cases}
\tilde\theta_2(S({\pi})), & \text{if } r\equiv 1, s\equiv 0 \pmod 2 \\
\tilde\theta_2(S(\frac{\pi}{2})), & \text{otherwise}.
\end{cases}
\end{align*}
\end{remark}
The following is true by construction:
\begin{proposition} \label{keycorollary}
Let $r, s \in \NN$ such that $rs \geq 4$, let $A = \begin{pmat} 2 & -r \\ -s & 2\end{pmat}$,
let $G(A)$ be the corresponding simply connected real Kac--Moody group, and let $K^{r,s}$ be its maximal compact subgroup,
and let $\wt{K}^{r,s}$ be its spin extension.
Then there exists a group epimorphism $\tilde\theta : \wt{K}^{r,s} \to \widetilde{H}^{r,s}$ such that
\[\tilde\theta \circ \tilde\theta_1 = \tilde\delta_1 \quad \text{ and } \quad \tilde\theta \circ \tilde\theta_2 = \tilde\delta_2.\]
Moreover, the following diagram commutes:
\[
\xymatrix{
\Spin{2} \ar[dr]^{\tilde\theta_i} \ar@/^/[drrr]^{\tilde\delta_i} \ar[ddd]_{\rho_2 \text{ or } S(\alpha)\mapsto D(\alpha)} \\
&\wt K^{r,s} \ar[rr]^{\tilde\theta} \ar[d]_{\hat{\hat\rho}} && \widetilde{H}^{r,s} \ar[d]^{\bar\rho} \\
&K^{r,s} \ar[rr]^\theta && H^{r,s} \\
\SO{2} \ar[ur]_{\theta_i} \ar@/_/[urrr]_{\delta_i}
}
\]
where the epimorphism on the left hand side is one of $\rho_2$ or $S(\alpha) \mapsto D(\alpha)$ as described in Remark~\ref{dichotomy}.
Furthermore, for $\{ i, j \} = \{ 1, 2 \}$ and $\tilde k_j \in \wt K_j$ one has
\begin{eqnarray}
\tilde t_i^{-1} \tilde k_j \tilde t_i & = & \tilde k_j^{-2n(i,j)}\tilde k_j. \label{crucialidentity2}
\end{eqnarray}
\end{proposition}
Identity (\ref{crucialidentity2}) follows from identity (\ref{crucialidentity}) in Remark~\ref{tcentralizes}
\begin{remark} \label{longvsshort}
The Cartan matrix of type $\mathrm{C}_2$ over $\{ 1, 2 \}$ with short root $\alpha_1$ and long root $\alpha_2$ (i.e., $2\to 1$; see Remark~\ref{longandshort}) is \[\begin{pmatrix} 2 & -2 \\ -1 & 2 \end{pmatrix},\]
cf.\ \cite[p.~44]{Carter:1989}.
The group $H^{r,s}$ from Proposition~\ref{keycorollary} is of type $\mathrm{C}_2$ with short root $\alpha_1$ and long root $\alpha_2$ if and only if $r$ is even and $s$ is odd. We conclude that for $i=1$ the map on the left hand side of the commutative diagram is \[\Spin{2} \to \SO{2} : S(\alpha) \mapsto D(\alpha)\] which is in accordance with Notation~\ref{zetaspin} and Lemma~\ref{lem:ama-embed-and-rho-commute-B2}.
In other words, in this example the rank one group corresponding to the long root is doubly covered by its spin cover and the rank one group corresponding to the short root is singly covered by its spin cover.
The direction introduced for edges labelled $\infty$ in Notation~\ref{augmented} was chosen to fit this observation: the arrow points away from the doubly covered vertex of the diagram towards the singly covered vertex of the diagram.
We point out that \cite[Plate~III, p.~269]{Bourbaki:Lie4-6} incorrectly gives the transpose of the above matrix as the Cartan matrix of type $\mathrm{C}_2$ with short root $\alpha_1$ and long root $\alpha_2$.
\end{remark}
\begin{remark}\label{allodd}
If both $r$, $s$ are odd, then in $\wt K = \wt K_1 \wt U *_{\wt U} \wt K_2 \wt U$ one has \[\wt U \cong Q_8.\]
In other words, the element $S(\pi)$ of the group $\wt K_1 \cong \Spin{2}$ is identical to the element $S(\pi)$ of the group $\wt K_2 \cong \Spin{2}$, like for the groups $\tilde\eps_{12}(\Spin{2})$ and $\tilde\eps_{23}(\Spin{2})$ in $\Spin{3}$. The same is true for the groups $\tilde\eta_p(\Spin{2})$ and $\tilde\eta_l(\Spin{2})$ in $\Spin{4}$ by Proposition~\ref{surjG2I}.
\end{remark}
\begin{remark} \label{otherU}
If both $r$ and $s$ are even, then in $\wt K = \wt K_1 \wt U *_{\wt U} \wt K_2 \wt U$ one has \[\wt U \cong \ZZ/4\ZZ \times \ZZ/4\ZZ,\] if one of $r$, $s$ is even and the other odd, then one has \[\wt U \cong \ZZ/4\ZZ \times \ZZ/2\ZZ.\]
\end{remark}
\begin{lemma} \label{lem:automorphismsarenice}
Each automorphism $\alpha$ of $K = K_1U *_U K_2U$ and each automorphism $\wt \alpha$ of $\wt K= \wt K_1 \wt U *_{\wt U} \wt K_2 \wt U$ induces an automorphism of the Bruhat--Tits tree of $G$. The set $\{ \alpha(K_1U), \alpha(K_2U) \}$ is $K$-conjugate to $\{ K_1U, K_2U \}$, the set $\{ \wt\alpha(\wt K_1\wt U), \wt\alpha(\wt K_2 \wt U) \}$ is $\wt K$-conjugate to $\{ \wt K_1 \wt U, \wt K_2 \wt U\}$.
\end{lemma}
\begin{proof}
$K_1U$ and $K_2U$ are indecomposable as amalgamated products and do not admit $\ZZ$ as a quotient.
Moreover, $U$ is finite.
Therefore by \cite[Theorem 6]{Karrass/Solitar:1970}, each $\alpha(K_iU)$ is conjugate to $K_1U$ or to $K_2U$.
Hence $\alpha(K_1U)$, $\alpha(K_2U)$ each stabilize a vertex of the Bruhat--Tits tree $X$ of $G$.
Since $\alpha(K_1U)\cap\alpha(K_2U)=\alpha(U)$, these two vertices are adjacent and $\alpha$ acts on $X$.
Since $K$ acts edge-transitively on $X$ the set $\{ \alpha(K_1U), \alpha(K_2U) \}$ is conjugate to $\{ K_1U, K_2U \}$.
The same argument works for $\wt \alpha$ and $\wt K_i \wt U$.
\end{proof}
\begin{proposition} \label{prop:controlautoinfty}
For each automorphism $\alpha$ of $K$ there exists a unique automorphism $\wt \alpha$ of $\wt K$ such that
\[ \hat{\hat \rho} \circ \wt \alpha = \alpha \circ \hat{\hat \rho}.\]
\end{proposition}
\begin{proof}
Since $K$ modulo its centre is isomorphic to $\wt K$ modulo its centre, the claim holds for inner automorphisms. It therefore suffices to study the outer automorphisms groups $\Out(\cdot)=\Aut(\cdot)/\Inn(\cdot)$. By Lemma~\ref{lem:automorphismsarenice} it therefore suffices to investigate automorphisms that preserve the sets $K_1U \cup K_2U$, resp.\ $\wt K_1\wt U \cup \wt K_2 \wt U$. The claim follows from Proposition~\ref{prop:lift-aut-so2}.
\end{proof}
\part{Arbitrary diagrams}
\section{Admissible colourings} \label{sec:adm-amalgams}
In this section we extend the classification results for $\SO{2}$- and $\Spin{2}$-amalgams from Sections \ref{sec:so2amalgams} and \ref{sec:spin2amalgams} to arbitrary diagrams.
\begin{remark}
Throughout this section,
let $A=(a(i,j))_{i,j\in I}$ be a generalized Cartan matrix over the index set $I$ and let $\Pi$ be the augmented Dynkin diagram with vertex set $V$ induced by $A$ (see Notation~\ref{augmented})
with respect to a labelling $\sigma:I\to V$.
\end{remark}
We have seen in Proposition~\ref{keycorollary} that --- given two vertices $i^\sigma$, $j^\sigma$ of $\Pi$ --- some subtleties related to single versus double covers of the circle group arise in the theory of spin covers of rank two depending on the parities of $a(i,j)$ and $a(j,i)$. To this end we develop a theory of admissible colourings that will help us distinguish the respective vertices of the diagram from one another.
\begin{defn}\label{def:nij-mij}\label{adtypedef}
For $i\neq j\in I$, define
\[
n(i,j):=
\begin{cases}
0, & \text{if } a(i,j) \text{ is even}, \\
1, & \text{if } a(i,j) \text{ is odd}.
\end{cases}
\]
Let $\Pi^\mathrm{adm}$ be the graph on the vertex set $V$ with edge set
\[
\big\{ \{i,j\}^\sigma \in V \times V \mid i \neq j \in I, n(i,j)=n(j,i)=1 \big\}.
\]
An \Defn{admissible colouring}
of $\Pi$ is a map $\kappa:V\to\{1,2\}$ such that
\begin{enumerate}
\item $\kappa(i^\sigma)=1$ whenever there exists $j \in I \backslash \{ i \}$ with $n(i,j)=0$ and $n(j,i)=1$,
\item the restriction of $\kappa$ to any connected component of the graph $\Pi^\mathrm{adm}$ is a constant map.
\end{enumerate}
\end{defn}
Let $c(\Pi,\kappa)$ be the number of connected components of $\Pi^\mathrm{adm}$ on which $\kappa$ takes the value $2$.
An admissible colouring $\kappa$ is \Defn{elementary} if $c(\Pi,\kappa)=1$, i.e., if there exists a unique connected component of the graph $\Pi^\mathrm{adm}$ on which $\kappa$ takes the value $2$.
The admissible colouring $\kappa \equiv 1$ (i.e., the one with $c(\Pi,\kappa)=0$) is called the \Defn{trivial colouring}. An admissible colouring $\kappa$ is \Defn{proper} if every
connected component of $\Pi$ contains a vertex $v$ with $\kappa(v)=2$.
\begin{remark} \label{obstruction}
An elementary admissible colouring is given by assigning the value $2$ to exactly one connected component of $\Pi^\mathrm{adm}$, while all other connected components take value $1$. Therefore, in order to construct all elementary admissible colourings, it suffices to decide which connected components may be assigned the value $2$.
Let $k^\sigma$ be a vertex of $\Pi$. The only obstruction to being able to assign the value $2$ to $k^\sigma$ is being contained in the connected component of $\Pi^{\mathrm{adm}}$ of a vertex that necessarily has to be assigned value $1$. That is, there exists a vertex $i^\sigma$ in the same connected component of $\Pi^{\mathrm{adm}}$ as $k^\sigma$ such that there is a vertex $j^\sigma$ with $n(i,j)=0$ and $n(j,i)=1$.
\end{remark}
\begin{lemma} \label{lem:max-adm-colouring}
Consider the partial order $\preceq$ on the set of admissible colouring of $\Pi$
where $\kappa\preceq\kappa'$ if $\kappa(v)\leq\kappa'(v)$ for all $v\in V$.
Then there is a unique
maximal admissible colouring $\kappa_\mathrm{max}$ of $\Pi$ with respect to $\preceq$.
\end{lemma}
\begin{proof}
Suppose $X$ is a set of admissible colourings of $\Pi$. Then one readily
checks that $\kappa:V\to\{1,2\}:v\mapsto\max\{\kappa'(v) \mid \kappa'\in
X\}$ is again an admissible colouring, satisfying $\kappa'\preceq\kappa$ for
all $\kappa'\in X$. Since $V$ is finite, so is its set of admissible
colourings, i.e., there exists a maximal element.
\end{proof}
\begin{remark} \label{combinatoricscolouring}
Given two admissible colourings $\kappa_1$, $\kappa_2$ of $\Pi$, define the \Defn{sum} of $\kappa_1$ and $\kappa_2$ as \[\kappa_1+\kappa_2 : V\to\{1,2\}:v\mapsto\max\{\kappa_i(v) \mid i = 1, 2 \}.\]
By the preceding discussion, this is again an admissible colouring.
Each non-trivial admissible colouring of a Dynkin diagram is the sum of (finitely many pairwise distinct) elementary colourings. The maximal admissible colouring $\kappa_{\mathrm{max}}$ is the sum of all elementary colourings.
\end{remark}
\begin{notation}
Throughout this section,
let $\kappa:V\to\{1,2\}$ be an admissible colouring of $\Pi$. For $i\neq j\in I$, set
\[
v_{ij}:=v\ :\Leftrightarrow\ \{i,j\}^\sigma\in E_v(\Pi)\ ,
\qquad
\kappa_{ij}:=\frac{\kappa(i^\sigma)+\kappa(j^\sigma)}{2}\in \{1,1.5,2\}\ .
\]
Note that $v_{ij}$ is well-defined since the sets $E_v(\Pi)$ for $v\in\{0,1,2,3,\infty\}$ form
a partition of $\binom{V}{2}$.
Moreover, note that $\kappa_{ij}=1.5$ implies $v_{ij}\in\{0,2,\infty\}$ by Definition~\ref{adtypedef}(b). Furthermore,
$\kappa_{ij}=2$ implies $v_{ij}\in\{0,1,3,\infty\}$ by Definition~\ref{adtypedef}(a).
\end{notation}
\begin{remark}\label{orientation}
The labelling $\sigma$ and the colouring $\kappa$ allow one to extend the direction relation on the augmented Dynkin diagram $\Pi$ to a direction relation between all pairs of vertices $u\neq v\in V$ (cf.\ Remark~\ref{longvsshort}):
\begin{itemize}
\item If $\{u,v\}$ is a directed edge, retain the direction $u \to v$.
\item If $\kappa(u)\neq\kappa(v)$,
then set $u\to v$ whenever $\kappa(u)>\kappa(v)$.
\item For the remaining cases, use the labelling to introduce a direction as follows:
\[
\forall\ \{i,j\}^\sigma \text{ non-directed with } \kappa(i^\sigma)=\kappa(j^\sigma):
\qquad i^\sigma\to j^\sigma\ :\Leftrightarrow\ i>j\ .
\]
\end{itemize}
This direction relation is called the \Defn{orientation} of $\Pi$ induced by the labelling $\sigma$ and the colouring $\kappa$.
\end{remark}
In the following we will quite freely use the notation introduced in the sections in which we studied the rank two situation, such as Sections \ref{sec:spin-pin}, \ref{sec:7}, \ref{sec:g2}, \ref{sec:bc2}, \ref{sec:rank-2-residues}.
\begin{notation}\label{31}
For $i\neq j\in I$, let $r:=a(i,j)$ and $s:=a(j,i)$.
Define
\[
G^{r,s}:=\begin{cases}
\SO{2}\times\SO{2}, & \text{if }rs=0, \\
\SO{3}, & \text{if }rs=1, \\
\U{2}, & \text{if }rs=2, \\
\SO{4}, & \text{if }rs=3, \\
K^{r,s} = K_1T *_{T} K_2T & \text{if }rs\geq 4,
\end{cases}
\]
where in the final case --- as discussed in Remark~\ref{tcentralizes} --- the action of $T$ on $K_i$ depends on the parities of the entries of the generalized Cartan matrix.
Furthermore, define homomorphisms from $\SO{2}$ into $G^{r,s}$ as follows:
\begin{align*}
\eps_1^{r,s}:=\begin{cases}
\iota_1, & \text{if }rs=0, \\
\eps_{12}, & \text{if }rs=1, \\
\zeta_p, & \text{if }rs=2, \\
\eta_p, & \text{if }rs=3, \\
\theta_1^{r,s}, & \text{if }rs\geq 4,
\end{cases}
\qquad\text{and}\qquad
\eps_2^{r,s}:=\begin{cases}
\iota_2, & \text{if }rs=0, \\
\eps_{23}, & \text{if }rs=1, \\
\zeta_l, & \text{if }rs=2, \\
\eta_l, & \text{if }rs=3, \\
\theta_2^{r,s}, & \text{if }rs\geq 4.
\end{cases}
\end{align*}
Next, define various covering groups of these:
\[
\widetilde{G}^{r,s,k}:=\begin{cases}
G^{r,s}, & \text{if }k=1, \\
\Spin{2}\times\Spin{2}, & \text{if }rs=0 \text{ and } k>1, \\
\Spin{3}, & \text{if }rs=1\text{ and } k=2, \\
\SO{2}\times\SU{2}, & \text{if }rs=2\text{ and } k=1.5, \\
\Spin{4}, & \text{if }rs=3\text{ and } k=2, \\
\wt K^{r,s} = \widetilde K_1 \wt{U} *_{\wt{U}} \widetilde K_1 \wt{U} & \text{if }rs\geq 4\text{ and } k>1.
\end{cases}
\]
Recall from Notation~\ref{nota:Dalpha-Salpha} the isomorphism
\[\psi:\SO{2}\to \Spin{2} : D(\alpha)\mapsto S(\alpha).\]
Using this, define the following homomorphisms from $\Spin{2}$ into $\widetilde{G}^{(r,s,k)}$:
\[
\tilde\eps_1^{(r,s,k)}:=
\begin{cases}
\eps_1^{r,s}\circ\psi^{-1}
& \text{if }k=1 \\
\tilde\iota_1, & \text{if }rs=0\text{ and } k>1, \\
\tilde\eps_{12}, & \text{if }rs=1\text{ and } k=2, \\
\tilde\zeta_p, & \text{if }rs=2\text{ and } k=1.5, \\
\tilde\eta_p, & \text{if }rs=3\text{ and } k=2, \\
\tilde\theta_1^{r,s}, & \text{if }rs\geq 4\text{ and } k>1,
\end{cases}
\quad\text{and}\quad
\tilde\eps_2^{r,s,k}:=
\begin{cases}
\eps_2^{r,s}\circ\psi^{-1}
& \text{if }k=1 \\
\tilde\iota_2, & \text{if }rs=0\text{ and } k>1, \\
\tilde\eps_{23}, & \text{if }rs=1\text{ and } k=2, \\
\tilde\zeta_l, & \text{if }rs=2\text{ and } k=1.5, \\
\tilde\eta_l, & \text{if }rs=3\text{ and } k=2, \\
\tilde\theta_2^{r,s}, & \text{if }rs\geq 4\text{ and } k>1.
\end{cases}
\]
Finally, define epimorphisms \[\rho^{r,s,k} : \widetilde{G}^{r,s,k} \to G^{r,s}\] via
\begin{align*}
\rho^{r,s,1}&:=\id,
&
\rho^{r,s,1.5}&:=\begin{cases}
\rho_2\times\psi^{-1}, & \text{if }rs=0\text{ and } i^\sigma\to j^\sigma, \\
\psi^{-1} \times \rho_2, & \text{if }rs=0\text{ and } i^\sigma\leftarrow j^\sigma, \\
\widehat\rho, & \text{if }rs=2, \\
\hat{\hat\rho}, & \text{if }rs\geq 4,
\end{cases}
&
\rho^{r,s,2}&:=\begin{cases}
\rho_2\times\rho_2, & \text{if }rs=0, \\
\rho_3, & \text{if }rs=1, \\
\rho_4, & \text{if }rs=3, \\
\hat{\hat\rho}, & \text{if }rs\geq 4.
\end{cases}
\end{align*}
\end{notation}
\begin{definition}\label{lotsofdef}
\begin{enumerate}
\item
An \Defn{$\SO{2}$-amalgam with respect to $\Pi$ and $\sigma$} is an amalgam
\[\AAA=\{ G_{ij}, \phi_{ij}^i, \mid i\neq j\in I \}\]
such that for all $i\neq j\in I$, we have
\[
G_{ij}=G^{a(i,j),a(j,i)}
\quad\text{ and }\quad
\phi_{ij}^i\big(\SO{2}\big)=
\begin{cases}
\eps_1^{a(i,j),a(j,i)}\big(\SO{2}\big), & \text{if }i^\sigma \to j^\sigma, \\
\eps_2^{a(i,j),a(j,i)}\big(\SO{2}\big), & \text{if }i^\sigma\leftarrow j^\sigma.
\end{cases}
\]
\item
The \Defn{standard $\SO{2}$-amalgam with respect to $\Pi$ and $\sigma$}
is the (continuous) $\SO{2}$-amalgam
\[\AAA\big(\Pi,\sigma,\SO{2}\big):=\{ G_{ij}, \phi_{ij}^i, \mid i\neq j\in I \}\]
with respect to $\Pi$ and $\sigma$
such that for all $i\neq j\in I$, we have
\[
G_{ij}=G^{a(i,j),a(j,i))}
\quad\text{ and }\quad
\phi_{ij}^i=\begin{cases}
\eps_1^{a(i,j),a(j,i)}, & \text{if }i^\sigma \to j^\sigma, \\
\eps_2^{a(i,j),a(j,i)}, & \text{if }i^\sigma\leftarrow j^\sigma.
\end{cases}
\]
\item
A \Defn{$\Spin{2}$-amalgam with respect to $\Pi$, $\sigma$ and $\kappa$} is an amalgam
\[\AAA=\{ G_{ij}, \phi_{ij}^i, \mid i\neq j\in I \}\]
such that for all $i\neq j\in I$, we have
\[
G_{ij}=\widetilde{G}^{a(i,j),a(j,i),\kappa_{ij}}
\quad\text{ and }\quad
\phi_{ij}^i\big(\Spin{2}\big)=
\begin{cases}
\tilde\eps_1^{a(i,j),a(j,i),\kappa_{ij}}\big(\Spin{2}\big), & \text{if }i^\sigma \to j^\sigma, \\
\tilde\eps_2^{a(i,j),a(j,i),\kappa_{ij}}\big(\Spin{2}\big), & \text{if }i^\sigma\leftarrow j^\sigma.
\end{cases}
\]
\item
The \Defn{standard $\Spin{2}$-amalgam with respect to $\Pi$, $\sigma$ and $\kappa$}
is the (continuous) $\Spin{2}$-amalgam
\[\AAA\big(\Pi,\sigma,\kappa,\Spin{2}\big):=\{ G_{ij}, \phi_{ij}^i, \mid i\neq j\in I \}\]
with respect to $\Pi$ and $\sigma$
such that for all $i\neq j\in I$, we have
\[
G_{ij}=\widetilde{G}^{a(i,j),a(j,i),\kappa_{ij}}
\quad\text{ and }\quad
\phi_{ij}^i=\begin{cases}
\tilde\eps_1^{a(i,j),a(j,i),\kappa_{ij}}, & \text{if }i^\sigma \to j^\sigma, \\
\tilde\eps_2^{a(i,j),a(j,i),\kappa_{ij}}, & \text{if }i^\sigma\leftarrow j^\sigma.
\end{cases}
\]
\item
Let
$\AAA=\{ G_{ij}, \phi_{ij}^i \mid i\neq j\in I\}$ be an $\SO{2}$-amalgam
with respect to $\Pi$ and $\sigma$. Given $i\neq j\in I$,
there is $\tau_{ij}^i\in \Aut(\SO{2})$ such that
\begin{align*}
\phi_{ij}^i=\begin{cases}
\eps_1^{a(i,j),a(j,i),\kappa_{ij}}\circ \tau_{ij}^i,&\text{if }i^\sigma \to j^\sigma, \\
\eps_2^{a(i,j),a(j,i),\kappa_{ij}}\circ \tau_{ij}^i,&\text{if }i^\sigma\leftarrow j^\sigma.
\end{cases}
\end{align*}
Define $\tilde\tau_{ij}^i\in\Aut(\Spin{2})$ as in Lemma~\ref{prop:lift-aut-so2}
and set
\[
\widetilde{G}_{ij}:=\widetilde{G}^{a(i,j),a(j,i),\kappa_{ij}}
\quad\text{ and }\quad
\tilde\phi_{ij}^i:=
\begin{cases}
\tilde\eps_1^{a(i,j),a(j,i),\kappa_{ij}}\circ \tilde\tau_{ij}^i,&\text{if }i^\sigma \to j^\sigma, \\
\tilde\eps_2^{a(i,j),a(j,i),\kappa_{ij}}\circ \tilde\tau_{ij}^i,&\text{if }i^\sigma\leftarrow j^\sigma.
\end{cases}
\]
Then
$\wAAA:=\{ \widetilde{G}_{ij}, \tilde\phi_{ij}^i \mid i\neq j\in I\}$
is the \Defn{induced $\Spin{2}$-amalgam with respect to $\Pi$, $\sigma$ and $\kappa$}.
\end{enumerate}
\end{definition}
\begin{remark} \label{rem:spin-ama-to-so-ama-and-back-2}
In analogy to Remark~\ref{rem:spin-ama-to-so-ama-and-back}, the construction in Definition~\ref{lotsofdef}(e) is symmetric and can be applied backwards: Starting with a $\Spin{2}$-amalgam
$\hat\AAA$, one can construct an $\SO{2}$-amalgam $\AAA$ such that $\hat\AAA=\wAAA$. As before, denote the corresponding epimorphism of amalgams by \[\pi_{\Pi,\sigma,\kappa} = \{ \id, \rho^i, \rho_{ij} \} : \AAA\big(\Pi,\sigma,\kappa,\Spin{2}\big) \to \AAA\big(\Pi,\sigma,\SO{2}\big),\] where $\rho_{ij} = \rho^{a(i,j),a(j,i),\kappa_{ij}}$ and $\rho^i = \psi^{-1}$ for all $i \in I$ with $\kappa(i^\sigma)=1$ and $\rho^i = \rho_2$ for all $i \in I$ with $\kappa(i^\sigma)=2$.
\end{remark}
\begin{proposition}\label{prop:lift-ama-iso-adm}
Let
$\AAA_1$ and $\AAA_2$ be $\SO{2}$-amalgams with respect to $\Pi$ and
$\sigma$, and let $\alpha=\{ \pi, \alpha_{ij}\mid i\neq j\in I\}:\AAA_1\to \AAA_2$
be a colouring-preserving isomorphism of amalgams, i.e., for all $i \in I$ one has $\kappa(i^\sigma) = \kappa(i^{\pi\sigma})$. Then there is a unique isomorphism
$\tilde\alpha=\{ \pi, \tilde\alpha_{ij}\mid i\neq j\in I\}:\wAAA_1\to \wAAA_2$
such that for all $i \neq j \in I$, one has
\[\rho_{\pi(i)\pi(j)}\circ \tilde\alpha_{ij}=\alpha_{ij}\circ \rho_{ij}.\]
\end{proposition}
\begin{proof}
This result generalizes Proposition~\ref{prop:lift-ama-iso-sl}. Its proof relies on the references
\begin{enumerate}
\item Lemma~\ref{lem:ama-embed-and-rho-commute-An},
\item Proposition~\ref{prop:lift-aut-soN}, resp.\ Corollary~\ref{prop:lift-aut-so2xso2}.
\end{enumerate}
We indicate how one may adapt the proof of \ref{prop:lift-ama-iso-sl} in order to deal with the general situation.
The cases $(k_{ij},v_{ij})=(2,0)$ and $(k_{ij},v_{ij})=(2,1)$ are covered by Proposition~\ref{prop:lift-ama-iso-sl}.
For $(k_{ij},v_{ij})=(2,3)$ the proof is virtually identical to the proof for $(k_{ij},v_{ij})=(2,1)$,
except that one uses $\rho_4$ instead of $\rho_3$, and
Lemma~\ref{lem:ama-embed-and-rho-commute-G2}
instead of Lemma~\ref{lem:ama-embed-and-rho-commute-An}.
Similarly, for $(k_{ij},v_{ij})=(2,\infty)$ we use $\hat{\hat\rho}$ instead of $\rho_3$, Proposition~\ref{keycorollary} instead of Lemma~\ref{lem:ama-embed-and-rho-commute-An} and Proposition~\ref{prop:controlautoinfty} instead of Proposition~\ref{prop:lift-aut-soN}.
For $k_{ij}=1$ there is nothing to show.
The case $(k_{ij},v_{ij})=(1.5,0)$ is very similar to $(k_{ij},v_{ij})=(2,0)$ (Case II of the proof of Proposition~\ref{prop:lift-ama-iso-sl}) and uses straightforward adaptions of Lemma~\ref{lem:ama-embed-and-rho-commute-An} and Corollary~\ref{prop:lift-aut-so2xso2}.
Finally, the cases $(k_{ij},v_{ij})=(1.5,2)$ and $(k_{ij},v_{ij})=(1.5,\infty)$ are again similar to the case $(k_{ij},v_{ij})=(2,1)$ (Case I of the proof of Proposition~\ref{prop:lift-ama-iso-sl}):
replace $\rho_3$ by $\widehat\rho$, resp.\ $\hat{\hat\rho}$, Proposition~\ref{prop:lift-aut-soN}
by Proposition~\ref{prop:lift-aut-B2}, resp.\ Proposition~\ref{prop:controlautoinfty} and Lemma~\ref{lem:ama-embed-and-rho-commute-An} by Proposition~\ref{surjB2}, resp.\ Proposition~\ref{keycorollary}.
\end{proof}
\begin{proposition} \label{prop:adm-ama-labelling-irrelevant}
Let $\sigma_1$, $\sigma_2$ be two labellings of $\Pi$.
Then the following hold:
\begin{enumerate}
\item
$\AAA\big(\Pi,\sigma_1,\SO{2}\big)\cong \AAA\big(\Pi,\sigma_2,\SO{2}\big)$.
\item
$\AAA\big(\Pi,\sigma_1,\kappa,\Spin{2}\big)\cong \AAA\big(\Pi,\sigma_2,\kappa,\Spin{2}\big)$.
\end{enumerate}
\end{proposition}
\begin{proof}
Assertion (a) follows by generalizing the proof of Consequence~\ref{14},
and setting $\alpha_{ij}:=\id_{G_{ij}}$ whenever
$a(i,j)$ and $a(j,i)$ have different parities or $\kappa(i^\sigma) \neq \kappa(j^\sigma)$; otherwise, set $\alpha_{ij}:=\id_{G_{ij}}$, if $\pi$ preserves the order relation between $i$ and $j$, and define $\alpha_{ij}$ via $K_1 \to K_2 : D(\phi) \mapsto D(\phi)$ and $K_2 \to K_1 : D(\phi) \mapsto D(\phi)$ in analogy to Lemma~\ref{10}, if $\pi$ switches the order relation between $i$ and $j$.
Observe that the permutation $\pi \in \Sym(I)$ with $\pi\sigma_1=\sigma_2$ preserves $\kappa$, as for $i_1, i_2 \in I$ with $i_1^{\sigma_1}=i_2^{\sigma_2}$ certainly $\kappa(i_1^{\sigma_1})=\kappa(i_2^{\sigma_2})$.
Therefore, since $\AAA\big(\Pi,\sigma_i,\SO{2}\big)$ induces $\AAA\big(\Pi,\sigma_i,\kappa,\Spin{2}\big)$ for $i=1,2$, assertion (b) is an immediate consequence of (a) and Proposition~\ref{prop:lift-ama-iso-adm}.
\end{proof}
We now generalize Definitions \ref{def:std-ama-SO2} and \ref{def:std-ama-Spin2}.
\begin{definition}\label{def:std-ama-adm}
We write
$\AAA\big(\Pi,\SO{2}\big)$ to denote the isomorphism type of $\AAA\big(\Pi,\sigma,\SO{2}\big)$ and $\AAA\big(\Pi,\kappa,\Spin{2}\big)$ to denote the isomorphism type of $\AAA\big(\Pi,\sigma,\kappa,\Spin{2}\big)$. By slight abuse of notation, we also denote any representative of the respective isomorphism types by these symbols.
They are called the \Defn{standard $\SO{2}$-amalgam with respect to $\Pi$}, resp.\ the \Defn{standard $\Spin{2}$-amalgam with respect to $\Pi$ and $\kappa$}.
Accordingly, we denote the epimorphism of amalgams from Remark~\ref{rem:spin-ama-to-so-ama-and-back-2} by \[\pi_{\Pi,\kappa} : \AAA\big(\Pi,\kappa,\Spin{2}\big) \to \AAA\big(\Pi,\SO{2}\big).\]
\end{definition}
Theorems \ref{thm:uniqueness-so-sl} and \ref{thm:uniqueness-spin-sl} generalize from simply laced diagram to arbitrary admissible diagrams.
\begin{theorem} \label{thm:uniquenessadmissible}
The following hold:
\begin{enumerate}
\item Any continuous $\SO{2}$-amalgam $\AAA$ with respect to $\Pi$ and $\sigma$ is isomorphic to the standard amalgam $\AAA\big(\Pi,\SO{2}\big)$.
\item Any continuous $\Spin{2}$-amalgam $\wAAA$ with respect to $\Pi$, $\sigma$ and $\kappa$ is isomorphic to the standard amalgam $\AAA\big(\Pi,\kappa,\Spin{2}\big)$.
\end{enumerate}
\end{theorem}
\begin{proof}
\begin{enumerate}
\item
The following is an adaption of the proof of Consequence~\ref{cons:std-ams-SO2-iso},
which will imply the claim via Proposition~\ref{prop:adm-ama-labelling-irrelevant}.
The only continuous automorphisms of the circle group $\SO{2}$
are $\id$ and the inversion $\inv$.
Since $\AAA$ continuous by hypothesis, for all $i \neq j \in I$,
if $i^\sigma\to j^\sigma$, one has
\begin{align*}
\phi_{ij}^i\in \{ \eps_1^{a(i,j),a(j,i)}, \eps_1^{a(i,j),a(j,i)}\circ \inv\}\ , &&
\phi_{ij}^j\in \{ \eps_2^{a(i,j),a(j,i)}, \eps_2^{a(i,j),a(j,i)}\circ \inv\}\ .
\end{align*}
For $m=1,2,3$ respectively, set $\gamma_B$, $\gamma_C$ as in
Lemma~\ref{lem:rank1-inv-A2}, \ref{lem:rank1-inv-B2} or \ref{lem:rank1-inv-G2}, respectively.
If $m=0$, then set $\gamma_B=\inv\times\id$ and $\gamma_C=\id\times\inv$. If $m=\infty$, then set $\gamma_B$ to be the automorphism induced by the inversion on $K_1$ and the identity on $K_2$ and $\gamma_C$ to be the automorphism induced by the identity on $K_1$ and the inversion on $K_2$.
Using this, for $i^\sigma\to j^\sigma$ let
\[\alpha_{ij}:=\begin{cases}
\id_{G_{ij}}, &\text{if }\phi_{ij}^i=\eps_1^{a(i,j),a(j,i)},\hphantom{{}\circ\inv}\ \phi_{ij}^j=\eps_2^{a(i,j),a(j,i)}, \\
\gamma_B, &\text{if }\phi_{ij}^i=\eps_1^{a(i,j),a(j,i)}\circ\inv,\ \phi_{ij}^j=\eps_2^{a(i,j),a(j,i)}, \\
\gamma_C, &\text{if }\phi_{ij}^i=\eps_1^{a(i,j),a(j,i)},\hphantom{{}\circ\inv}\ \phi_{ij}^j=\eps_2^{a(i,j),a(j,i)}\circ \inv, \\
\gamma_B\circ \gamma_C,
&\text{if }\phi_{ij}^i=\eps_1^{a(i,j),a(j,i)}\circ\inv,\ \phi_{ij}^j=\eps_2^{a(i,j),a(j,i)}\circ \inv.
\end{cases}\]
Then the system
$\alpha:=\{\pi, \alpha_{ij} \mid i\neq j\in I\}:\AAA\to \AAA\big(\Pi,\sigma,\SO{2}\big)$
is an isomorphism of amalgams.
\item
Let $\AAA$ be the continuous $\SO{2}$-amalgam that induces $\wAAA$ (cf.\ Remark~\ref{rem:spin-ama-to-so-ama-and-back-2}).
Assertion (a) implies $\AAA\cong \AAA\big(\Pi,\SO{2}\big)$.
Proposition~\ref{prop:lift-ama-iso-adm} yields the claim, since $\AAA\big(\Pi,\SO{2}\big)$ induces $\AAA\big(\Pi,\kappa,\Spin{2}\big)$.
\qedhere
\end{enumerate}
\end{proof}
We now generalize Theorem~\ref{thm:K-univ-sl}.
\begin{theorem} \label{thm:K-univ-adm}
Let $G(\Pi)$ be the simply connected split real Kac--Moody group associated to $\Pi$, and let $K(\Pi)$ be its maximal compact subgroup, i.e., the subgroup fixed by the Cartan--Chevalley involution. Then there exists a faithful universal enveloping morphism \[\tau_{K(\Pi)} : \AAA\big(\Pi,\SO{2}\big) \to K(\Pi).\]
\end{theorem}
\begin{proof}
The same proof as for Theorem~\ref{thm:K-univ-sl} applies; the uniqueness of
amalgams then follows from Theorem~\ref{thm:uniquenessadmissible} (instead
of Theorem~\ref{thm:uniqueness-so-sl}).
\end{proof}
In view of Theorem~\ref{thm:K-univ-adm}, we generalize
Definition~\ref{defn:sl-spin-group} as follows:
\begin{definition} \label{defspingrp}
The \Defn{spin group $\Spin{\Pi,\kappa}$ with respect to $\Pi$ and $\kappa$} is the canonical
universal enveloping group of the (continuous) amalgam
$\AAA\big(\Pi,\kappa,\Spin{2}\big)=\{ \widetilde{K}_{ij}, \tilde\phi_{ij}^i \mid i\neq j\in I\}$ with the canonical universal enveloping morphism \[\tau_{\Spin{\Pi},\kappa} : \AAA\big(\Pi,\kappa,\Spin{2}\big) \to \Spin{\Pi,\kappa}.\]
The \Defn{maximal spin group $\Spin{\Pi}$ with respect to $\Pi$} is $\Spin{\Pi,\kappa_\mathrm{max}}$ (cf.\ Lemma~\ref{lem:max-adm-colouring}).
\end{definition}
\begin{observation} \label{naturalhomo}
Whenever $\kappa\preceq\kappa'$ are admissible colourings of $\Pi$, by construction there exists a canonical central extension
\[ \Spin{\Pi,\kappa'}\onto\Spin{\Pi,\kappa}. \]
\end{observation}
\begin{lemma} \label{lem:adm-K-envelops-spin-amalgam}
$K(\Pi)$ is an enveloping
group of the amalgam $\AAA\big(\Pi,\kappa,\Spin{2}\big)$. There exists a canonical central extension $\rho_{\Pi,\kappa} : \Spin{\Pi,\kappa} \to K(\Pi)$ that makes the following diagram commute:
\[
\xymatrix{
\AAA\big(\Pi,\kappa,\Spin{2}\big) \ar[rr]^{\tau_{\Spin{\Pi,\kappa}}} \ar[d]_{\pi_{\Pi,\kappa}} &&
\Spin{\Pi} \ar@{-->}[d]^{\rho_{\Pi,\kappa}} \\
\AAA\big(\Pi,\SO{2}\big) \ar[rr]^{\tau_{K(\Pi)}} && K(\Pi)
}
\]
\end{lemma}
\begin{proof}
Essentially the same proof as for Lemma~\ref{lem:sl-K-envelops-spin-amalgam}
works, after appropriately substituting definitions and results from
the present Section~\ref{sec:adm-amalgams}.
\end{proof}
\section{Spin covers of arbitrary type} \label{sec:tametypes}
In this section we prove the following theorem concerning the central extension $\rho_{\Pi,\kappa} : \Spin{\Pi,\kappa} \to K(\Pi)$ from Lemma~\ref{lem:adm-K-envelops-spin-amalgam}.
\begin{theorem} \label{m3}
Given a diagram $\Pi$ and an admissible colouring $\kappa$, the universal enveloping group
$\Spin{\Pi,\kappa}$ of $\AAA\big(\Pi,\kappa,\Spin{2}\big)$ is a $2^{c(\Pi,\kappa)}$-fold central extension of the
universal enveloping group $K(\Pi)$ of $\AAA\big(\Pi,\SO{2}\big)$.
\end{theorem}
Recall from Definition~\ref{adtypedef} that the number $c(\Pi,\kappa)$ counts the connected components of $\Pi^\mathrm{adm}$ on which $\kappa$ takes the value $2$.
\begin{remark}\label{simplylacedalreadyknown}
Let $\kappa$ be an admissible colouring of a simply laced diagram $\Pi$. One has $\Pi = \Pi^\mathrm{adm}$ and $\kappa$ is constant on each connected component of $\Pi$. The number $c(\Pi,\kappa)$ counts the components on which it has value $2$. A reduction to the irreducible case as in the proof of Theorem~\ref{m2} therefore immediately implies Theorem~\ref{m3} for simply laced diagrams.
\end{remark}
Our strategy of proof in the general case is based on a reduction to the simply-laced case. We start our investigation with carrying out Strategy~\ref{unfolding} for doubly laced diagrams.
\begin{definition} \label{unfoldeddiagram}
Let $n \in \NN$, let $I = \{ 1, \ldots, n \}$ and let $\Pi$ be an irreducible doubly laced Dynkin diagram that is not simply laced with generalized Cartan matrix $A=(a(i,j))_{i,j}$, labelling $\sigma : I \to V(\Pi)$ and admissible colouring $\kappa : V \to \{ 1, 2 \}$ such that for each pair of vertices $i^\sigma$, $j^\sigma$ with $i^\sigma \to j^\sigma$ that form a diagram of type $\mathrm{C}_2$ one has $\kappa(i^\sigma)=2$.
Then the \Defn{unfolded} Dynkin diagram is the Dynkin diagram $\Pi^{\mathrm{un}}$ with labelling \[\sigma^{\mathrm{un}} : I^\mathrm{un} := \{ \pm i \mid i \in I, \kappa(i^\sigma) = 1 \} \cup \{ i \mid i \in I, \kappa(i^\sigma)=2 \} \to V(\Pi^{\mathrm{un}})\] and edges defined via the generalized Cartan matrix $A^{\mathrm{un}}=(a^{\mathrm{un}}(i,j))_{i,j}$ given by \[a^{\mathrm{un}}(i,j) = \begin{cases}
0, & \text{if }\kappa(|i|^\sigma) \neq \kappa(|j|^\sigma)
\text{ and }a(|i|,|j|) = 0, \\
-1, & \text{if }\kappa(|i|^\sigma) \neq \kappa(|j|^\sigma)
\text{ and }a(|i|,|j|) \neq 0, \\
a(|i|,|j|), & \text{if }\kappa(|i|^\sigma) = \kappa(|j|^\sigma)
\text{ and }ij > 0, \\
0, & \text{if }\kappa(|i|^\sigma) = \kappa(|j|^\sigma)
\text{ and }ij<0.
\end{cases}\]
Note that the unfolded Dynkin diagram $\Pi^{\mathrm{un}}$ is simply laced. Define the admissible colouring $\kappa^{\mathrm{un}} :\equiv 2$.
For a reducible Dynkin diagram $\Pi$ with admissible colouring $\kappa : V \to \{ 1, 2 \}$ such that for each pair of vertices $i^\sigma$, $j^\sigma$ with $i^\sigma \to j^\sigma$ that form a diagram of type $\mathrm{C}_2$ one has $\kappa(i^\sigma)=2$, define the \Defn{unfolded} Dynkin diagram $\Pi^{\mathrm{un}}$ and its admissible colouring $\kappa^{\mathrm{un}}$ componentwise: replace each connected component that is not simply laced by its unfolded Dynkin diagram and colouring with constant value $2$ as defined above and retain each simply laced component and its colouring.
\end{definition}
\begin{remark}
Note that for an {\em irreducible} doubly laced Dynkin diagram $\Pi$ that is not simply laced with labelling $\sigma : I \to V(\Pi)$ there is at most one admissible colouring $\kappa : V \to \{ 1, 2 \}$ such that for each pair of vertices $i^\sigma$, $j^\sigma$ with $i^\sigma \to j^\sigma$ that form a diagram of type $\mathrm{C}_2$ one has $\kappa(i^\sigma)=2$ and, if it exists, actually is equal to the maximal admissible colouring $\kappa_{\mathrm{max}}$ from Lemma~\ref{lem:max-adm-colouring}.
This is not necessarily true for reducible such Dynkin diagrams as any simply laced connected components allows the two colourings constant one and constant two.
\end{remark}
\begin{proposition} \label{propunfolding}
Let $\Pi$ be a doubly laced Dynkin diagram with labelling $\sigma : I \to V$ and admissible colouring $\kappa : V \to \{ 1, 2 \}$ such that for each pair of vertices $i^\sigma$, $j^\sigma$ with $i^\sigma \to j^\sigma$ that form a diagram of type $\mathrm{C}_2$ one has $\kappa(i^\sigma)=2$, let $\Pi^{\mathrm{un}}$ be the unfolded Dynkin diagram, let $G(\Pi)$ and $G(\Pi^{\mathrm{un}})$ be the corresponding simply connected split real Kac--Moody groups, let $K(\Pi)$ and $K(\Pi^{\mathrm{un}})$ be their maximal compact subgroups, let \[\tau_{K(\Pi)} : \AAA\big(\Pi,\SO{2}\big) \to K(\Pi)\] and \[\tau_{K(\Pi^{\mathrm{un}})} : \AAA\big(\Pi^{\mathrm{un}},\SO{2}\big) \to K(\Pi^{\mathrm{un}})\] be the respective (faithful) universal enveloping morphisms (cf.\ Theorem \ref{thm:K-univ-adm}), for each $i \in I$ let \[K_i := (\tau_{K(\Pi)} \circ \phi^i_{ij})(\SO{2}) \leq K(\Pi),\] and for each $i \in I^{\mathrm{un}}$ let \[K_i^{\mathrm{un}} := (\tau_{K(\Pi^{\mathrm{un}})} \circ (\phi^i_{ij})^{\mathrm{un}})(\SO{2}) \leq K(\Pi^{\mathrm{un}}),\] where $(\phi^i_{ij})^{\mathrm{un}}$ denote the connecting homomorphisms of the amalgam $ \AAA\big(\Pi^{\mathrm{un}},\SO{2}\big)$.
Then the assignment
\begin{align*}
\forall i \in I \text{ with } \kappa(i^\sigma)=2: \quad\quad K_i &\to K_i^{\mathrm{un}} \\ g &\mapsto (\tau_{K(\Pi^{\mathrm{un}})} \circ (\phi^i_{ij})^{\mathrm{un}}) \circ (\tau_{K(\Pi)} \circ \phi^i_{ij})^{-1}(g) \\
\forall i \in I \text{ with } \kappa(i^\sigma)=1: \quad\quad K_i &\to K_i^{\mathrm{un}} \times K_{-i}^{\mathrm{un}} \\ g &\mapsto \Big(\big(\tau_{K(\Pi^{\mathrm{un}})} \circ (\phi^i_{ij})^{\mathrm{un}}\big) \times \big(\tau_{K(\Pi^{\mathrm{un}})} \circ (\phi^{-i}_{-ij})^{\mathrm{un}}\big) \Big) \circ (\tau_{K(\Pi)} \circ \phi^i_{ij})^{-1}(g)
\end{align*}
induces a monomorphism \[\Omega_{K(\Pi)} : K(\Pi) \to K(\Pi^{\mathrm{un}}).\]
\end{proposition}
\begin{proof}
The existence of the homomorphism $\Omega_{K(\Pi)} : K(\Pi) \to K(\Pi^{\mathrm{un}})$ is straightforward using Consequence~\ref{concretedescription} and the universal property of $\tau_{K(\Pi)} : \AAA\big(\Pi,\SO{2}\big) \to K(\Pi)$. Injectivity follows from the faithfulness of the universal enveloping morphism $\tau_{K(\Pi^{\mathrm{un}})} : \AAA\big(\Pi^{\mathrm{un}},\SO{2}\big) \to K(\Pi^{\mathrm{un}})$.
\end{proof}
\begin{corollary} \label{corunfolding}
Retain the notation and hypotheses from Proposition~\ref{propunfolding}.
Moreover, let \[\tau_{\Spin{\Pi,\kappa}} : \AAA\big(\Pi,\kappa,\Spin{2}\big) \to \Spin{\Pi,\kappa}\] and \[\tau_{\Spin{\Pi^{\mathrm{un}},\kappa^{\mathrm{un}}}} : \AAA\big(\Pi^{\mathrm{un}},\kappa^{\mathrm{un}},\Spin{2}\big) \to \Spin{\Pi^{\mathrm{un}},\kappa^{\mathrm{un}}}\] be the respective universal enveloping morphisms (cf.\ Definitions~\ref{defn:sl-spin-group} and \ref{defspingrp}), for each $i \in I$ let
\begin{align*}
\widetilde K_i & := (\tau_{\Spin{\Pi,\kappa}} \circ \wt\phi^i_{ij})(\Spin{2}) \leq \Spin{\Pi,\kappa},
\end{align*}
and for each $i \in I^{\mathrm{un}}$ let
\begin{align*}
\widetilde K_i^{\mathrm{un}} &:= (\tau_{\Spin{\Pi^{\mathrm{un}},\kappa^{\mathrm{un}}}} \circ (\wt\phi^i_{ij})^{\mathrm{un}})(\Spin{2}) \leq \Spin{\Pi^{\mathrm{un}},\kappa^{\mathrm{un}}},
\end{align*}
where $(\wt\phi^i_{ij})^{\mathrm{un}}$ denote the connecting homomorphisms of the amalgam $ \AAA\big(\Pi^{\mathrm{un}},\kappa^{\mathrm{un}},\Spin{2}\big)$.
Then the assignment
\begin{align*}
\forall i \in I \text{ with } \kappa(i^\sigma)=2: \quad\quad \wt K_i &\to \wt K_i^{\mathrm{un}} \\ g &\mapsto (\tau_{\Spin{\Pi^{\mathrm{un}},\kappa^{\mathrm{un}}}} \circ (\wt\phi^i_{ij})^{\mathrm{un}}) \circ (\tau_{\Spin{\Pi,\kappa}} \circ \wt\phi^i_{ij})^{-1}(g) \\
\forall i \in I \text{ with } \kappa(i^\sigma)=1: \quad\quad \wt K_i &\to \wt K_i^{\mathrm{un}} \cdot \wt K_{-i}^{\mathrm{un}} \\ g &\mapsto \Big(\big(\tau_{\Spin{\Pi^{\mathrm{un}},\kappa^{\mathrm{un}}}} \circ (\wt\phi^i_{ij})^{\mathrm{un}}\big) \cdot \big(\tau_{\Spin{\Pi^{\mathrm{un}},\kappa^{\mathrm{un}}}} \circ (\wt\phi^{-i}_{-ij})^{\mathrm{un}}\big) \Big) \circ (\tau_{\Spin{\Pi,\kappa}} \circ \wt\phi^i_{ij})^{-1}(g)
\end{align*}
induces a homomorphism $\Omega_{\Spin{\Pi,\kappa}} : \Spin{\Pi,\kappa} \to \Spin{\Pi^{\mathrm{un}},\kappa^{\mathrm{un}}}$ that makes the following diagram commute:
\[
\xymatrix{
\AAA\big(\Pi,\kappa,\Spin{2}\big) \ar[rrd]^{\tau_{\Spin{\Pi,\kappa}}} \ar[ddd]_{\pi_{\Pi,\kappa}} && && && \AAA\big(\Pi^{\mathrm{un}},\kappa^{\mathrm{un}},\Spin{2}\big) \ar[ddd]^{\pi_{\Pi^{\mathrm{un}},\kappa^{\mathrm{un}}}} \ar[dll]_{\tau_{\Spin{\Pi^{\mathrm{un}},\kappa^{\mathrm{un}}}}} \\
&&
\Spin{\Pi,\kappa} \ar[d]^{\rho_{\Pi,\kappa}} \ar@{-->}[rr]^{\Omega_{\Spin{\Pi,\kappa}}}&& \Spin{\Pi^{\mathrm{un}},\kappa^{\mathrm{un}}} \ar[d]_{\rho_{\Pi^{\mathrm{un}},\kappa^{\mathrm{un}}}}\\
&& K(\Pi) \ar[rr]_{\Omega_{K(\Pi)}} && K(\Pi^{\mathrm{un}}) \\
\AAA\big(\Pi,\SO{2}\big) \ar[urr]_{\tau_{K(\Pi)}} &&&&&& \AAA\big(\Pi^{\mathrm{un}},\SO{2}\big) \ar[ull]^{\tau_{K(\Pi^{\mathrm{un}})}}
}
\]
In particular, if the admissible colouring $\kappa$ is non-trivial, then \[\rho_{\Pi,\kappa} : \Spin{\Pi,\kappa} \to K(\Pi)\] is a non-trivial central extension.
\end{corollary}
\begin{proof}
The existence of the homomorphism $\Omega_{\Spin{\Pi,\kappa}} : \Spin{\Pi,\kappa} \to \Spin{\Pi^{\mathrm{un}},\kappa^{\mathrm{un}}}$ is straightforward using Lemma~\ref{concretedescriptiontilde} and the universal property of $\tau_{\Spin{\Pi},\kappa} : \AAA\big(\Pi,\kappa,\Spin{2}\big) \to \Spin{\Pi,\kappa}$.
Let $i \in I$ with $\kappa(i^\sigma) = 2$ and define
\begin{align*}
z_i &:= (\tau_{\Spin{\Pi,\kappa}} \circ \wt\phi^i_{ij})(S(\pi)) \in \widetilde K_i, \\
z_i^{\mathrm{un}} &:= (\tau_{\Spin{\Pi^{\mathrm{un}},\kappa^{\mathrm{un}}}} \circ (\wt\phi^i_{ij})^{\mathrm{un}})(S(\pi)) \in \widetilde K_i^{\mathrm{un}}.
\end{align*}
Then $\Omega_{\Spin{\Pi,\kappa}}(z_i) = z_i^{\mathrm{un}} = -1_{\Spin{\Pi^{\mathrm{un}},\kappa^{\mathrm{un}}},\mathcal{K}(i)}$ (cf.\ Definition~\ref{minus1}).
Since the central extension $\rho_{\Pi^{\mathrm{un}}} : \Spin{\Pi^{\mathrm{un}},\kappa^{\mathrm{un}}} \to K(\Pi^{\mathrm{un}})$ is non-trivial by the simply-laced version of Theorem~\ref{m3} (see also Corollary~\ref{m1cor}), so is the central extension $\rho_{\Pi,\kappa} : \Spin{\Pi,\kappa} \to K(\Pi)$.
\end{proof}
Next we carry out Strategy~\ref{simplelacing} in order to reduce arbitrary diagrams to doubly laced ones.
\begin{definition} \label{doublylacing}
For an augmented Dynkin diagram $\Pi$ with labelling $\sigma : I \to V$ and an admissible colouring $\kappa$, let $\Pi^{\mathrm{dl}\kappa}$ be the doubly
laced diagram with identical orientation (cf.\ Remark~\ref{orientation}) obtained by
\begin{itemize}
\item removing all edges $\{i,j\}^\sigma \in E_\infty(\Pi)$ with $a(i,j)$, $a(j,i)$ even,
\item replacing all edges $\{i,j\}^\sigma \in E_\infty(\Pi)$ with $a(i,j)$ odd, $a(j,i)$ even and $\kappa(i^\sigma) = 2$ by double edges,
\item retaining all edges $\{i,j\}^\sigma \in E_2(\Pi)$ with $a(i,j)$ odd, $a(j,i)$ even and $\kappa(i^\sigma) = 2$,
\item replacing all other edges in $\Pi$ by simple edges.
\end{itemize}
\end{definition}
\begin{remark}
The diagram $\Pi^{\mathrm{dl}\kappa}$ is a doubly laced with admissible colouring $\kappa$ such that for each pair of vertices $i^\sigma$, $j^\sigma$ with $i^\sigma \to j^\sigma$ that form a diagram of type $\mathrm{C}_2$ one has $\kappa(i^\sigma)=2$.
Note that $c(\Pi,\kappa)=c(\Pi^{\mathrm{dl}\kappa},\kappa)$.
The orientation of $\Pi^{\mathrm{dl}\kappa}$ induced by its labelling and colouring in general differs from the one that $\Pi^{\mathrm{dl}\kappa}$ inherits from the orientation of $\Pi$ induced by its labelling and colouring. Note that, of course, it is possible to change the labellings of $\Pi$ and $\Pi^{\mathrm{dl}\kappa}$ so that the two orientations coincide.
\end{remark}
The correspondence of the indices $i$, $j$ of the epimorphisms $\alpha_{ij}$ and $\tilde\alpha_{ij}$ in the following proposition to the indices $p$ and $l$ in Definition~\ref{lotsofdef} depends on the latter orientation of $\Pi^{\mathrm{dl}\kappa}$ not the former.
\begin{proposition}\label{hom-amalgams}
Given a diagram $\Pi$ and an admissible colouring $\kappa$, there exist epimorphisms of amalgams $\tilde\alpha = (\id,\tilde\alpha_{ij}) : \AAA\big(\Pi,\kappa,\Spin{2}\big) \to \AAA\big(\Pi^{\mathrm{dl}\kappa},\kappa,\Spin{2}\big)$ and $\alpha = (\id,\alpha_{ij}) : \AAA\big(\Pi,\SO{2}\big) \to \AAA\big(\Pi^{\mathrm{dl}\kappa},\SO{2}\big)$ making the following diagram commute:
\[
\xymatrix{
\AAA\big(\Pi,\kappa,\Spin{2}\big) \ar@{-->}[rr]^{\tilde\alpha} \ar[d]_{\pi_{\Pi,\kappa}} && \AAA\big(\Pi^{\mathrm{dl}\kappa},\kappa,\Spin{2}\big) \ar[d]^{\pi_{\Pi^{\mathrm{dl}\kappa},\kappa}} \\
\AAA\big(\Pi,\SO{2}\big) \ar@{-->}[rr]^{\alpha} && \AAA\big(\Pi^{\mathrm{dl}\kappa},\SO{2}\big)
}
\]
\end{proposition}
\begin{proof}
There exist the following group epimorphisms that provide the epimorphism $\alpha$:
\begin{align*}
\mathrm{U}_2(\CC) & \to \SO{3} & \text{(see Corollary~\ref{corsurjB2b})} \\
\SO{4} & \to \SO{3} & \text{(see Lemma~\ref{lemsurjG2})} \\
K(A_1) & \to \SO{2} \times \SO{2} & \text{(see Proposition~\ref{keyproposition})} \\
K(A_2) & \to \SO{3} & \text{(see Proposition~\ref{keyproposition})} \\
K(A_3) & \to \mathrm{U}_2(\CC) & \text{(see Proposition~\ref{keyproposition})}
\end{align*}
where $A_1 = \begin{pmatrix} 2 & -r_1 \\ -s_1 & 2 \end{pmatrix}$ with $r_1$, $s_1$ even, $A_2 = \begin{pmatrix} 2 & -r_2 \\ -s_2 & 2 \end{pmatrix}$ with $r_2$, $s_2$ odd, $A_3 = \begin{pmatrix} 2 & -r_3 \\ -s_3 & 2 \end{pmatrix}$ with one of $r_3$, $s_3$ even, the other odd.
The respective lifts to the spin covers exist by Corollary~\ref{surjB2b}, Proposition~\ref{surjG2}, Corollary~\ref{keycorollary}, yielding the epimorphism $\tilde\alpha$.
By \ref{lemsurjG2}, \ref{surjG2} and \ref{corsurjB2b} and \ref{keyproposition}, \ref{keycorollary} the above diagram commutes.
\end{proof}
\begin{proposition} \label{thm:Spin(Delta)-covers-Spin(Delta-sl)}
Given a diagram $\Pi$ and an admissible colouring $\kappa$, there exist (uniquely determined) epimorphisms
\[\Xi_{\Spin{\Pi,\kappa}}: \Spin{\Pi,\kappa} \to \Spin{\Pi^{\mathrm{dl}\kappa},\kappa} \qquad
\text{and} \quad \quad
\Xi_{K(\Pi)} : K(\Pi) \to K(\Pi^{\mathrm{dl}\kappa})\]
resulting in the following commutative diagram:
\[
\xymatrix{
\AAA\big(\Pi,\kappa,\Spin{2}\big) \ar[rrrrrr]^{\tilde\alpha} \ar[drr]_{\tau_{\Spin{\Pi,\kappa}}} \ar[ddd]_{\pi_{\Pi,\kappa}} && && && \AAA\big(\Pi^{\mathrm{dl}\kappa},\kappa,\Spin{2}\big) \ar[lld]^{\tau_{\Spin{\Pi^{\mathrm{dl}\kappa},\kappa}}} \ar[ddd]^{\pi_{\Pi^{\mathrm{dl}\kappa},\kappa}} \\
&& \Spin{\Pi,\kappa} \ar@{-->}[rr]^{\Xi_{\Spin{\Pi,\kappa}}} \ar[d]^{\rho_\Pi} &&
\Spin{\Pi^{\mathrm{dl}\kappa},\kappa} \ar[d]^{\rho_{\Pi^{\mathrm{dl}\kappa},\kappa}} \\
&& K(\Pi) \ar@{-->}[rr]^{\Xi_{K(\Pi)}} && K(\Pi^{\mathrm{dl}\kappa}) \\
\AAA\big(\Pi,\SO{2}\big) \ar[rrrrrr]^{\alpha} \ar[urr]^{\tau_{K(\Pi)}} && && && \AAA\big(\Pi^{\mathrm{dl}\kappa},\SO{2}\big) \ar[llu]_{\tau_{K(\Pi^{\mathrm{dl}\kappa})}}
}
\]
\end{proposition}
\begin{proof}
The left-hand and the right-hand commutative squares exist by Lemma~\ref{lem:adm-K-envelops-spin-amalgam}. The outer commutative square exists by Proposition~\ref{hom-amalgams}. The composition \[{\tau_{\Spin{\Pi^{\mathrm{dl}\kappa},\kappa}}} \circ \tilde\alpha : \AAA(\Pi,\kappa,\Spin{2}) \to \Spin{\Pi^{\mathrm{dl}\kappa},\kappa}\] is an enveloping morphism, whence the universal property of $\Spin{\Pi,\kappa}$ yields the uniquely determined epimorphism $\Xi_{\Spin{\Pi},\kappa}: \Spin{\Pi,\kappa} \to \Spin{\Pi^{\mathrm{dl}\kappa},\kappa}$, which makes the upper square commute. Similarly, one obtains the uniquely determined epimorphism $\Xi_{K(\Pi)} : K(\Pi) \to K(\Pi^{\mathrm{dl}\kappa})$, that makes the lower square commute. Also \[\tau_{K(\Pi^{\mathrm{dl}\kappa})} \circ \alpha \circ \pi_{\Pi,\kappa} = \tau_{K(\Pi^{\mathrm{dl}\kappa}),\kappa} \circ \pi_{\Pi^{\mathrm{dl}\kappa},\kappa} \circ \tilde\alpha : \AAA(\Pi,\kappa,\Spin{2}) \to K(\Pi^{\mathrm{dl}\kappa})\] is an enveloping morphism. By the universal property of $\tau_{\Spin{\Pi,\kappa}} : \AAA(\Pi,\kappa,\Spin{2}) \to \Spin{\Pi,\kappa}$ in connection with the outer commuting squares one has $\rho_{\Pi^{\mathrm{dl}\kappa},\kappa} \circ \Xi_{\Spin{\Pi},\kappa} = \Xi_{K(\Pi)} \circ \rho_{\Pi,\kappa}$. That is, also the inner square commutes.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{m3}.]
Lemma~\ref{lem:adm-K-envelops-spin-amalgam} provides a canonical central extension
$\rho_{\Pi,\kappa} : \Spin{\Pi,\kappa} \to K(\Pi)$.
By Proposition~\ref{thm:Spin(Delta)-covers-Spin(Delta-sl)}, there exist an epimorphism $\Xi_{\Spin{\Pi,\kappa}} : \Spin{\Pi,\kappa} \to
\Spin{\Pi^{\mathrm{dl}\kappa},\kappa}$ resulting in the commutative diagram:
\[
\xymatrix{\Spin{\Pi,\kappa} \ar[rr]^{\Xi_{\Spin{\Pi,\kappa}}} \ar[d]^{\rho_{\Pi,\kappa}} && \Spin{\Pi^{\mathrm{dl}\kappa},\kappa} \ar[d]^{\rho_{\Pi^{\mathrm{dl}\kappa},\kappa}} \\
K(\Pi) \ar[rr]^{\Xi_{K(\Pi)}} && K(\Pi^{\mathrm{dl}\kappa})
}
\]
Observe first that the claim is obvious if $\kappa$ is trivial (i.e., $c(\Pi,\kappa)=0$). Moreover, it is also true if $\kappa$ is elementary (i.e., $c(\Pi,\kappa)=1$). Indeed, since $\ker(\rho_{\Pi^{\mathrm{dl}\kappa},\kappa}) \neq \{ 1 \}$ by Corollary~\ref{corunfolding}, necessarily also $\ker(\rho_{\Pi,\kappa}) \neq \{ 1 \}$ by the homomorphism theorem of groups.
By Remark~\ref{allodd} each connected component of $\Pi^{\mathrm{adm}}$ contributes at most a factor $2$ to the order of $\ker(\rho_{\Pi,\kappa})$. Since only those connected components that admit a vertex $i^\sigma$ with $\kappa(i^\sigma)=2$ actually can contribute, one has $|\ker(\rho_{\Pi,\kappa})| \leq 2^{c(\Pi,\kappa)}$. The claim follows for elementary colourings.
Since $c(\Pi,\kappa)=c(\Pi^{\mathrm{dl}\kappa},\kappa)$, once more the homomorphism theorem of groups implies that is suffices to show the claim for the central extension \[\rho_{\Pi^{\mathrm{dl}\kappa},\kappa} : \Spin{\Pi^{\mathrm{dl}\kappa},\kappa} \to K(\Pi^{\mathrm{dl}\kappa}).\] Since the upper bound that has just been established also holds for $|\ker(\rho_{\Pi^{\mathrm{dl}\kappa},\kappa})|$, it suffices to prove $2^{c(\Pi^{\mathrm{dl}\kappa},\kappa)} \leq |\ker(\rho_{\Pi^{\mathrm{dl}\kappa},\kappa})|$.
To this end decompose $\kappa = \kappa_1 + \cdots + \kappa_{c(\Pi^{\mathrm{dl}\kappa},\kappa)}$ into a sum of pairwise distinct elementary admissible colourings of $\Pi^{\mathrm{dl}\kappa}$.
Let \[\left\{ i_t \in I \mid 1 \leq t \leq c\left(\Pi^{\mathrm{dl}\kappa},\kappa\right), \kappa_t(i_t)=2 \right\}\] be a set of representatives of the connected components of $(\Pi^{\mathrm{dl}\kappa})^\mathrm{adm}$ on which $\kappa$ takes value $2$.
By Observation~\ref{naturalhomo}, for each $1 \leq t \leq c(\Pi^{\mathrm{dl}\kappa},\kappa)$ there exists a central extension \[\pi_t : \Spin{\Pi^{\mathrm{dl}\kappa},\kappa} \to \Spin{\Pi^{\mathrm{dl}\kappa},\kappa_t}.\] Since for each $1 \leq t \leq c(\Pi^{\mathrm{dl}\kappa},\kappa)$ the colouring $\kappa_t$ is elementary, the central extension \[\rho_{\Pi^{\mathrm{dl}\kappa},\kappa_t} : \Spin{\Pi^{\mathrm{dl}\kappa},\kappa_t} \to K(\Pi^{\mathrm{dl}\kappa})\] has kernel of order two. The kernel of the natural homomorphism \[\Spin{\Pi^{\mathrm{dl}\kappa},\kappa} \longrightarrow \prod_{t=1}^{c(\Pi^{\mathrm{dl}\kappa},\kappa)} \Spin{\Pi^{\mathrm{dl}\kappa},\kappa_t} \longrightarrow \prod_{t=1}^{c(\Pi^{\mathrm{dl}\kappa},\kappa)} K(\Pi^{\mathrm{dl}\kappa})\] equals $\ker(\rho_{\Pi^{\mathrm{dl}\kappa},\kappa})$.
By the proof of Corollary~\ref{corunfolding} one has $\pi_t(z_{i_s}) \neq 1$ in $\Spin{\Pi^{\mathrm{dl}\kappa},\kappa_t}$ if and only if $s=t$. We conclude that \[2^{c(\Pi^{\mathrm{dl}\kappa},\kappa)} \leq |\langle z_{i_t} \mid 1 \leq t \leq c(\Pi^{\mathrm{dl}\kappa},\kappa) \rangle| \leq |\ker(\rho_{\Pi^{\mathrm{dl}\kappa},\kappa})|.\]
The claim follows.
\end{proof}
\part{Applications}
\section{Spin-extended Weyl groups} \label{extended} \label{sec14}
Let $A$ be a generalized Cartan matrix with corresponding augmented Dynkin diagram $\Pi:=\Pi(A)$.
In this section we formally construct the spin cover $\Wspin(\Pi,\kappa)$ whose existence has been postulated in \cite[Section~3.5]{DamourHillmann}, for arbitrary diagrams $\Pi$ with admissible colouring $\kappa$ (cf.\ Definition~\ref{adtypedef}). In fact, we provide both a concrete embedding of $\Wspin(\Pi,\kappa)$ into $\Spin{\Pi,\kappa}$ and a presentation by generators and relations.
\begin{notation}
Throughout this section, let $\Pi$ be an augmented Dynkin diagram with vertex set $V$, labelling $\sigma:I\to V$ and
admissible colouring $\kappa:V\to\{1,2\}$.
Let $A=(a(i,j))_{i,j\in I}$
be the generalized Cartan matrix associated to $\Pi$. Let $n:=\abs{V}$,
i.e., $I=\{1,\ldots,n\}$, and let \[J:=\{i\in I \mid \kappa(i^\sigma)=1 \}\subset I.\]
\end{notation}
\begin{defn}
For $i\neq j\in I$, recall from \ref{adtypedef}
\[
n(i,j)=
\begin{cases}
0, & \text{if } a(i,j) \text{ is even}, \\
1, & \text{if } a(i,j) \text{ is odd}.
\end{cases}
\]
Define
\[
m_{ij} :=
\begin{cases}
2, &\text{ if } a(i,j)a(j,i)=0, \\
3, &\text{ if } a(i,j)a(j,i)=1, \\
4, &\text{ if } a(i,j)a(j,i)=2, \\
6, &\text{ if } a(i,j)a(j,i)=3, \\
0, &\text{ if } a(i,j)a(j,i)\geq 4. \\
\end{cases}
\]
\end{defn}
\begin{defn} \label{defwspin}
Consider the standard amalgams with respect to $\Pi$ and $\kappa$
\[
\AAA\big(\Pi,\kappa,\Spin{2}\big)=\{\widetilde{G}_{ij},\tilde\phi_{ij}^i \mid i\neq j\in I \}
\quad\text{and}\quad
\AAA\big(\Pi,\SO{2}\big)=\{ G_{ij},\phi_{ij}^i \mid i\neq j\in I\}
\]
with enveloping homomorphisms $\tilde\psi_{ij} : \widetilde{G}_{ij} \to \Spin{\Pi,\kappa}$ and $\psi_{ij} : G_{ij} \to K(\Pi)$ to the respective universal enveloping groups.
For $i\neq j\in I$ let
\[ \wh{s}_i:=
\begin{cases}
\tilde\psi_{ij}(\tilde\phi_{ij}^i(S(\tfrac{\pi}{4}))), & \text{ if } i\notin J, \\
\tilde\psi_{ij}(\tilde\phi_{ij}^i(S(\tfrac{\pi}{4})))^2=\tilde\psi_{ij}(\tilde\phi_{ij}^i(S(\tfrac{\pi}{2}))), & \text{ if } i\in J,
\end{cases}
\]
and set
\[ \wh W:=\wh{W}(\Pi,\kappa):=\cgen{ \wh{s}_i }{ i\in I } \leq \Spin{\Pi,\kappa}. \]
Similarly, let
\[ \wt{s}_i:= \psi_{ij}(\phi_{ij}^i(D(\tfrac{\pi}{2})))
\qquad\text{ and }\qquad
\wt W:=\wt{W}(\Pi):=\cgen{ \wt{s}_i }{ i\in I } \leq K(\Pi). \]
Note that the elements $\wh{s}_i$ and $\wt{s}_i$ are well-defined --- in particular, independent of the
choice of $j$ --- due to the definition of enveloping groups.
\end{defn}
\begin{defn} \label{preswspin}
The \Defn{Weyl group $W(\Pi)$} associated to $\Pi$ is given by the finite presentation
\begin{align*}
W :=W(\Pi):=
\Big\langle s_1,\ldots,s_n
\,\vert\,
& s_i^2=1 \text{ for } i \in I, \\
& \underbrace{s_i s_j s_i \cdots}_{m_{ij} \text{ factors}} =
\underbrace{s_j s_i s_j \cdots}_{m_{ij} \text{ factors}} \text{ for } i\neq j\in I
\Big\rangle.
\end{align*}
The \Defn{extended Weyl group $\Wext(\Pi)$} associated to $\Pi$ is given by the finite presentation
\begin{align}
\Wext :=\Wext(\Pi) :=
\Big\langle t_1,\ldots,t_n
\,\vert\,
& t_i^4=1 \text{ for } i \in I, \label{rels:Wext-ri8} \tag{T1} \\
& t_j^{-1} t_i^2 t_j = t_i^2 t_j^{2n(i,j)} \quad \text{ for } i\neq j\in I, \label{rels:Wext-D-conj} \tag{T2} \\
& \underbrace{t_i t_j t_i \cdots}_{m_{ij} \text{ factors}} =
\underbrace{t_j t_i t_j \cdots}_{m_{ij} \text{ factors}} \label{rels:Wext-braid} \tag{T3} \text{ for } i\neq j\in I
\Big\rangle.
\end{align}
The \Defn{spin-extended Weyl group $\Wspin(\Pi)$} associated to $\Pi$ is given by the finite presentation
\begin{align}
\Wspin :=\Wspin(\Pi):=
\Big\langle r_1,\ldots,r_n
\,\vert\,
& r_i^8=1, \text{ for } i \in I,
\label{rels:Wspin-ri8} \tag{R1} \\
& r_j^{-1} r_i^2 r_j = r_i^2 r_j^{2n(i,j)} \quad\text{ for } i\neq j\in I,
\label{rels:Wspin-D-conj} \tag{R2} \\
& \underbrace{r_i r_j r_i \cdots}_{m_{ij} \text{ factors}} =
\underbrace{r_j r_i r_j \cdots}_{m_{ij} \text{ factors}}
\label{rels:Wspin-braid} \tag{R3}
\text{ for } i\neq j\in I \Big\rangle.
\end{align}
\end{defn}
\begin{remark}\label{differentrelations}
The group $\Wext(\Pi)$ is studied in \cite{Tits:1966} and \cite{Kac/Peterson:1985}. By \cite[Corollary 2.4]{Kac/Peterson:1985} and its proof there exists an isomorphism \[\Wext(\Pi) \cong \wt W(\Pi)\] induced by $t_i \mapsto \wt s_i$.
The elements $(\wt s_i)^2$ are torus elements of $K(\Pi)$ of order two, so that the relation (\ref{rels:Wext-D-conj}) \[t_j^{-1} t_i^2 t_j = t_i^2 t_j^{2n(i,j)} \quad\quad \Longleftrightarrow \quad\quad t_i^{-2} t_j^{-1} t_i^2 = t_j^{2n(i,j)}t_j^{-1}\] is equivalent to the centralize-or-invert relation (\ref{crucialidentity}) for torus elements of order two discussed in Remark~\ref{tcentralizes}. If one prefers left-conjugation, one can therefore use one of the relations
\begin{align}
t_j t_i^2 t_j^{-1} = t_i^2 t_j^{-2n(i,j)} \quad\quad & \Longleftrightarrow \quad\quad t_i^{-2} t_j t_i^2 = t_j^{-2n(i,j)}t_j \tag{T2'}\\ \text{or} \quad\quad\quad\quad\quad \notag \\
t_j t_i^2 t_j^{-1} = t_j^{2n(i,j)} t_i^2 \quad\quad & \Longleftrightarrow \quad\quad t_i^{2} t_j^{-1} t_i^{-2} = t_j^{-1}t_j^{2n(i,j)} \tag{T2''}
\end{align}
instead of (\ref{rels:Wext-D-conj}).
This accounts for the missing minus sign in relation (\ref{rels:Wext-D-conj}) of our presentation of $\Wext(\Pi)$ compared to relation (n1) given in \cite[Corollary 2.4]{Kac/Peterson:1985}. Since, because of relation (\ref{rels:Wext-ri8}) $t_i^4=1$, only the parity of $a(i,j)$ matters, there is no harm in replacing $a(i,j)$ in relation (n1) from \cite[Corollary 2.4]{Kac/Peterson:1985} with $n(i,j)$, as we have done in our relation (\ref{rels:Wext-D-conj}).
The version of the relations of $\Wext(\Pi)$ we chose can then be directly generalized -- simply by weakening relation (\ref{rels:Wext-ri8}) to (\ref{rels:Wspin-ri8}) -- in order to obtain suitable relations of $\Wspin(\Pi)$ to make it the appropriate spin-extended Weyl group for the spin group $\Spin{\Pi}$.
Valid alternatives for relation (\ref{rels:Wspin-D-conj}) are
\begin{align}
r_j r_i^2 r_j^{-1} = r_i^2 r_j^{-2n(i,j)} \quad\quad & \Longleftrightarrow \quad\quad r_i^{-2} r_j r_i^2 = r_j^{-2n(i,j)}r_j \tag{R2'}\\ \text{or} \quad\quad\quad\quad\quad \notag \\
r_j r_i^2 r_j^{-1} = r_j^{2n(i,j)} r_i^2 \quad\quad & \Longleftrightarrow \quad\quad r_i^{2} r_j^{-1} r_i^{-2} = r_j^{-1}r_j^{2n(i,j)} \tag{R2''}
\end{align}
as all are equivalent to the centralize-or-invert relation (\ref{crucialidentity2}) from Proposition~\ref{keycorollary}.
\end{remark}
\begin{lemma} \label{manyrelations}
In $\Wspin(\Pi)$ for all $i,j\in I$ the following relations hold:
\begin{enumerate}
\item $[r_i^2,r_j^2] = r_j^{4n(i,j)}$ and $r_i^{4n(j,i)} = r_j^{4n(i,j)}$,
\item if $n(i,j)=1=n(i,j)$, then $r_i^4=r_j^4$; otherwise $[r_i^2,r_j^2] = 1$,
\item if $n(i,j)=0$, $n(j,i)=1$, then $r_i^4=1$,
\item $r_j r_i^4r_j^{-1}
= \begin{cases}
r_i^4 & \text{ if } n(i,j)=0, \\
r_i^2 r_j^2 r_i^2 r_j^2 & \text{ if } n(i,j)=1,
\end{cases}$
\item $[r_j,r_i^4]=1$.
\end{enumerate}
\end{lemma}
\begin{proof}
One computes
\begin{align*}
& r_j^{-1}r_i^2r_j = r_i^2r_j^{2n(i,j)} \\
\Longleftrightarrow \quad & r_j^{-2}r_i^2r_j^{2} = r_j r_i^2r_j^{-1+2n(i,j)} = r_i^2r_j^{2n(i,j)}r_j^{2n(i,j)}\\
\Longleftrightarrow \quad & [r_i^{2},r_j^2] = r_j^{4n(i,j)}
\end{align*}
Moreover:
\begin{eqnarray}
r_i^{4n(j,i)} = r_i^{-4n(j,i)} = [r_j^2,r_i^{2}]^{-1} = [r_i^{2},r_j^2] = r_j^{4n(i,j)}. \label{chainofequalities}
\end{eqnarray}
If $n(i,j)=1=n(j,i)$ this immediately implies $r_i^4=r_j^4$. Furthermore, if one of $n(i,j)$, $n(j,i)$ is $0$, then each term in (\ref{chainofequalities}) is equal to $1$, in particular $[r_i^2,r_j^2] = 1$. In case $n(i,j)=0$ and $n(j,i)=1$ one obtains $r_i^4=r_i^{4n(j,i)}=r_j^{4n(i,j)}=1$.
Also:
\[r_j^{-1} r_i^4r_j = \left(r_j^{-1}r_i^2r_j\right)^2
= \left(r_i^2 r_j^{2n(i,j)}\right)^2
= \begin{cases}
r_i^4 & \text{ if } n(i,j)=0, \\
r_i^2 r_j^2 r_i^2 r_j^2 & \text{ if } n(i,j)=1.
\end{cases}\]
If $n(i,j)=1$ and $n(j,i)=0$, then $[r_i^2,r_j^2]=1$ and $r_j^4=1$,
whence $r_j^{-1} r_i^4r_j = r_i^2 r_j^2 r_i^2 r_j^2 = r_i^4r_j^4 = r_i^4$.
Finally, if $n(i,j)=1=n(j,i)$, then $r_i^4=r_j^4$, so we also have $r_j^{-1} r_i^4r_j = r_j^{-1} r_j^4r_j = r_j^4 = r_i^4$.
Therefore always $[r_j,r_i^4]=1$.
\end{proof}
\begin{consequence} \label{Zabelian}
The subgroup $Z:=\gen{r_i^4\mid i\in I}$
of $\Wspin(\Pi)$ is central.
\end{consequence}
\begin{definition}
The \Defn{coloured spin-extended Weyl group $\Wspin(\Pi,\kappa)$} associated to $\Pi$ and $\kappa$ is given by the finite presentation
\begin{align}
\Wspin(\Pi,\kappa):=
\Big\langle r_1,\ldots,r_n
\,\vert\,
& r_i^8=1, \text{ for } i \in I,
\tag{R1} \\
& r_j^{-1} r_i^2 r_j = r_i^2 r_j^{2n(i,j)} \quad\text{ for } i\neq j\in I,
\tag{R2} \\
& \underbrace{r_i r_j r_i \cdots}_{m_{ij} \text{ factors}} =
\underbrace{r_j r_i r_j \cdots}_{m_{ij} \text{ factors}}
\tag{R3}
\text{ for } i\neq j\in I \\
& r_i^4=1, \text{ for } i \in J \label{rels:Wspin-ri4} \tag{R4} \Big\rangle.
\end{align}
\end{definition}
\begin{proposition} \label{orderZ}
\begin{enumerate}
\item For all $i^\sigma$, $j^\sigma$ in the same connected component of $\Pi^{\mathrm{adm}}$ one has $r_i^4=r_j^4$ in $\Wspin(\Pi)$.
\item One has $\Wspin(\Pi) = \Wspin(\Pi,\kappa_{\mathrm{max}})$.
\end{enumerate}
\end{proposition}
\begin{proof}
Assertion (a) is immediate from Lemma~\ref{manyrelations}(b).
By Remark~\ref{combinatoricscolouring}, the maximal admissible colouring $\kappa_{\mathrm{max}}$ is the sum of all elementary admissible colourings. By Remark~\ref{obstruction} the only obstruction to being able to assign the value $2$ to a vertex $k^\sigma$ of $\Pi$ is the existence of a vertex $i^\sigma$ in the same connected component of $\Pi^{\mathrm{adm}}$ as $k^\sigma$ such that there is a vertex $j^\sigma$ with $n(i,j)=0$ and $n(j,i)=1$. This implies $r_k^4=r_i^4=1$ by assertion (a) and Lemma~\ref{manyrelations}(c).
We conclude that for all $i$ with $\kappa_{\mathrm{max}}(i^\sigma)=1$ one has $r_i^4=1$ in $\Wspin(\Pi)$. Assertion (b) follows.
\end{proof}
\begin{definition}
Let $\Dext(\Pi):=\cgen{ t_i^2 }{ i\in I } \leq \Wext(\Pi)$
and $\Dspin(\Pi):=\cgen{ r_i^2 }{ i\in I } \leq \Wspin(\Pi)$.
\end{definition}
In case $\Pi$ is the Dynkin diagram of type $E_{10}$, the group
$\Dspin(\Pi)$ is isomorphic to the groups $\mathcal{D}^{vs}$ and
$\mathcal{D}^s$ from \cite[Propositions~1 and 2]{DamourHillmann}.
\medskip
The similarities of the presentations of $\Wspin(\Pi)$, $\Wext(\Pi)$, $W(\Pi)$ immediately yield the following:
\begin{observation} \label{upperrowexact}
The following hold:
\begin{enumerate}
\item There is a unique epimorphism $\Wspin(\Pi)\to \Wext(\Pi)$ mapping $r_i$ to $t_i$
and with kernel $Z$.
\item There is a unique epimorphism $\Wspin(\Pi)\to W(\Pi)$ mapping $r_i$ to $s_i$
and with kernel $\Dspin$.
\item There is a unique epimorphism $\Wext(\Pi)\to W(\Pi)$ mapping $t_i$ to $s_i$
and with kernel $\Dext$.
\end{enumerate}
\end{observation}
Moreover, from the literature one can extract:
\begin{proposition} \label{KacPeterson}
The following hold:
\begin{enumerate}
\item There is a unique isomorphism $\Wext(\Pi)\to\wt{W}(\Pi)$ mapping $t_i$ to $\wt{s}_i$.
\item There is a unique epimorphism $\wt{W}(\Pi)\to W(\Pi)$ mapping $\wt{s}_i$ to $s_i$.
\item $\Dext\cong\ker\left(\wt{W}(\Pi)\to W(\Pi)\right)$ is an elementary abelian group of order $2^{\abs{I}}$.
\end{enumerate}
\end{proposition}
\begin{proof}
\begin{enumerate}
\item This follows from \cite[Corollary 2.4]{Kac/Peterson:1985} and its proof.
\item
This follows from \cite[Corollary 2.3]{Kac/Peterson:1985}
or, alternatively, by combining (a) with Observation~\ref{upperrowexact}.
\item
This follows from \cite[Lemma 2.2(a) and Corollary 2.3]{Kac/Peterson:1985}.
\qedhere
\end{enumerate}
\end{proof}
\begin{remark}\label{straightforwardcomputation}
Let $n\geq2$ and let $e_1,\ldots,e_n$ be the standard basis of $\RR^n$. In the following, we frequently use variations of the following basic computations in $\Cl{n}$ (cf.\ Section~\ref{sec:spin-pin}):
\[
(e_1e_2)^2=-1,
\qquad
\left(\tfrac{1}{\sqrt{2}}(1+e_1e_2)\right)^2
=\tfrac{1}{2} (1 + 2 e_1e_2 + (e_1e_2)^2)
= e_1 e_2,
\]
\[
\left(\tfrac{1}{\sqrt{2}}(1+e_1e_2)\right)^{-1}
= \tfrac{1}{\sqrt{2}}(1+e_1e_2) \cdot (e_1e_2)^{-1}
= \tfrac{1}{\sqrt{2}} (1-e_1e_2).
\]
Moreover, one has
\[ \frac{1}{\sqrt{2}}(1+e_1e_2)\frac{1}{\sqrt{2}}(1+e_2e_3)+\frac{1}{\sqrt{2}}(1+e_2e_3)\frac{1}{\sqrt{2}}(1+e_1e_2) = 1+e_1e_2 + 1+e_2e_3 - 1.
\]
\end{remark}
\begin{proposition} \label{uniqueepi}
For each admissible colouring $\kappa$ there is a unique epimorphism $f : \Wspin(\Pi) \to \wh{W}(\Pi,\kappa)$ mapping $r_i$ to $\wh{s}_i$ for $i\in I$, which factors through $\Wspin(\Pi,\kappa)$:
\[\xymatrix{
\Wspin(\Pi) \ar[drr]^f \ar[d] \\
\Wspin(\Pi,\kappa) \ar[rr]_{\overline{f}} && \wh{W}(\Pi,\kappa)
}\]
\end{proposition}
\begin{proof}
The $r_i$ generate $\Wspin$, so if $f$ exists, it is unique. Moreover, the $\wh{s}_i$
generate $\wh{W}$, so if $f$ exists, it is also surjective.
It remains to show
that $f$ maps every relator of $\Wspin$ to $1_{\wh{W}}$.
Since all relators of $\Wspin$ involve at most two generators it suffices to consider the local situation in all rank 2 subgroups, i.e., to look
at all pairs $i\neq j \in I$ and verify that $r_i$, $r_j$ and the relations
between them are mapped correctly.
For $i\neq j \in I$
define
\[
x_j:= \begin{cases}
\tilde\phi_{ij}^j(S(\tfrac{\pi}{4})), & \text{if }j\in I\backslash J, \\
\tilde\phi_{ij}^j(S(\tfrac{\pi}{2})), & \text{if }j\in J.
\end{cases}
\]
Thus for all $j \in I$ one has $\wh{s}_j = \tilde\psi_{ij}(x_j)$ according to Definition~\ref{defwspin}.
Certainly, the relations \ref{rels:Wspin-ri8} (and \ref{rels:Wspin-ri4}) hold for the $\wh{s}_j$; one has \[x_j^8 = \left\{ \begin{array}{cc} \tilde\phi_{ij}^j(S(2\pi)), & \text{if $j\in I\backslash J$}, \\
\tilde\phi_{ij}^j(S(4\pi)), & \text{if $j\in J$} \end{array} \right\} = 1.\]
Define $v:=v_{ij}$, i.e., $\{i,j\}^\sigma\in E_v(\Pi)$, and $k:=k_{ij}$.
We check the relations for each of the possible values of $(v,k)$ in Notation~\ref{31}.
\begin{description}
\item[$\mathbf{k=1}$] This case is well-known, see e.g.\ Proposition~\ref{KacPeterson}(a).
\item[$\mathbf{v=0,\ k>1}$]
We have $G_{ij}=\Spin{2} \times \Spin{2}$. Recall Notation~\ref{iotaspin}. Then
\begin{align*}
x_i &=\tilde\iota_1(S(\tfrac{\pi}{4}))\ , \\
x_j &=\tilde\iota_2(S(\tfrac{\pi}{4}))\ .
\end{align*}
Relation \ref{rels:Wspin-D-conj}:
\begin{align*}
x_i^{-1} x_j^2 x_i
& = x_j^2 = x_j^2 x_i^{2n(i,j)}\ , \\
x_j^{-1} x_i^2 x_j
&= x_i^2 = x_i^2 x_j^{2n(j,i)}\ .
\end{align*}
Relation \ref{rels:Wspin-braid}:
\begin{align*}
x_j x_i
&= x_i x_j\ .
\end{align*}
\item[$\mathbf{(v,k)=(1,2)}$]
We have $G_{ij}=\Spin{3}$. Recall Remark~\ref{6}. Then
\begin{align*}
x_i
&=\tilde\eps_{23}(S(\tfrac{\pi}{4}))
= \tfrac{1}{\sqrt{2}}(1+e_2e_3)\ ,
\\
x_j
&=\tilde\eps_{12}(S(\tfrac{\pi}{4}))
= \tfrac{1}{\sqrt{2}}(1+e_1e_2)\ .
\end{align*}
Relation \ref{rels:Wspin-D-conj}:
\begin{align*}
x_i^{-1} x_j^2 x_i
&= \tfrac{1}{\sqrt{2}} (1-e_2e_3) \cdot e_1e_2 \cdot \tfrac{1}{\sqrt{2}} (1+e_2e_3) \\
&=\tfrac{1}{2} (e_1 e_2 + e_1e_2e_2e_3 - e_2e_3e_1e_2 - e_2e_3e_1e_2e_2e_3) \\
&=\tfrac{1}{2} (e_1 e_2 - e_1e_3 - e_1e_3 - e_1e_2) \\
&= -e_1e_3 = e_1 e_2 e_2 e_3 = x_j^2 x_i^2
= x_j^2 x_i^{2 n(i,j)}\ , \\
x_j^{-1} x_i^2 x_j
&= x_i^2 x_j^2 = x_i^2 x_j^{2 n(j,i)}\ .
\end{align*}
Relation \ref{rels:Wspin-braid}:
\begin{align*}
x_j x_i x_j
&= x_j \tfrac{1}{\sqrt{2}}(1 + e_2e_3) x_j
= \tfrac{1}{\sqrt{2}}(x_j^2 + x_j x_i^2 x_j) \\
&= \tfrac{1}{\sqrt{2}}(x_j^2 + x_j (x_i^2 x_j^2) x_j^{-1}) \\
&= \tfrac{1}{\sqrt{2}}(x_j^2 + x_j (x_j^{-1} x_i^2 x_j) x_j^{-1}) \\
&= \tfrac{1}{\sqrt{2}}(x_j^2 + x_i^2)\ .
\end{align*}
By symmetry, we conclude $x_j x_i x_j = x_i x_j x_i$.
\item[$\mathbf{(v,k)=(2,1.5),\ i^\sigma\to j^\sigma}$]
We have $G_{ij}=\SO{2}\times \SU{2}$ and $j\in J$. Then
\begin{align*}
x_i
&=\tilde{\zeta}_p(S(\tfrac{\pi}{4}))
=\tfrac{1}{\sqrt{2}} \left(
\begin{pmat} 1 & 1 \\ -1 & 1\end{pmat},
\begin{pmat}
1 & -1 & & \\
1 & 1 & & \\
& & 1 & 1 \\
& & -1 & 1
\end{pmat}
\right)\ , & (\text{cf.\ Notation~\ref{zetaspin}}) \\
x_j
&=\tilde{\zeta}_l(S(\tfrac{\pi}{2}))
=\left( 1_{\SO{2}}, \begin{pmat} &&1 \\ &&&1 \\ -1&&& \\ &-1&& \end{pmat}
\right)\ .
\end{align*}
Relation \ref{rels:Wspin-D-conj}:
\begin{align*}
x_i^{-1} x_j^2 x_i
&= (1_{\SO{2}},-1_{\SU{2}}) = x_j^2
= x_j^2 x_i^{2\cdot 0}
= x_j^2 x_i^{2n(i,j)}
\ , \\
x_j^{-1} x_i^2 x_j
&= \left(
\begin{pmat} & 1 \\ -1 & \end{pmat},
\begin{pmat}
& 1 & & \\
-1 & & & \\
& & & -1 \\
& & 1 &
\end{pmat}
\right)
= x_i^2 x_j^2
= x_i^2 x_j^{2\cdot 1}
= x_i^2 x_j^{2n(j,i)}
\ .
\end{align*}
Relation \ref{rels:Wspin-braid}:
\begin{align*}
x_j x_i x_j x_i = (x_j x_i)^2
&= \left(
\begin{pmat} & 1 \\ -1 & \end{pmat},
-1_{\SU{2}}
\right)
=(x_i x_j)^2 = x_i x_j x_i x_j\ ..
\end{align*}
\item[$\mathbf{(v,k)=(3,2),\ i^\sigma\to j^\sigma}$]
We have $G_{ij}=\Spin{4}$. Recall Notation~\ref{etaspin}. Then
\begin{align*}
x_i
&=\tilde\eta_p(S(\tfrac{\pi}{4}))
=\tilde\eps_{34}\big(S(\tfrac{\pi}{4})\big)
= \tfrac{1}{\sqrt{2}}(1+e_3e_4)\ ,
\\
x_j
&=\tilde\eta_l(S(\tfrac{\pi}{4}))
=\tilde\eps_{14}\big(S(2\cdot\tfrac{\pi}{4})\big)\cdot \tilde\eps_{23}\big(S(-\tfrac{\pi}{4})\big)
=e_1e_4 \cdot \tfrac{1}{\sqrt{2}}(1-e_2e_3) \\
&=\tfrac{1}{\sqrt{2}}(e_1e_4 + e_1e_2e_3e_4)\ .
\end{align*}
Relation \ref{rels:Wspin-D-conj}:
\begin{align*}
x_i^{-1} x_j^2 x_i
&= e_2 e_4 = x_j^2 x_i^2
= x_j^2 x_i^{2 \cdot 1}
= x_i^2 x_j^{2n(i,j)}
\ , \\
x_j^{-1} x_i^2 x_j
&= -e_2 e_4 = x_i^2 x_j^2
= x_j^2 x_i^{2 \cdot 1}
= x_j^2 x_i^{2n(j,i)}
\ .
\end{align*}
Relation \ref{rels:Wspin-braid}:
$
x_j x_i x_j x_i x_j x_i = (x_j x_i)^3
= -e_1e_2e_3e_4 =
(x_i x_j)^3 = x_i x_j x_i x_j x_i x_j\ .
$
\item[$\mathbf{v=\infty,\ k>1,\ i^\sigma \to j^\sigma}$]
We have $G_{ij}=\wt K_1 \wt U *_{\wt U} \wt K_2 \wt U$. Recall Definition~\ref{thetaspin}. Then
\begin{align*}
x_i &= \tilde\theta_1(S(\tfrac{\pi}{4}))\ ,
\\
x_j &=\begin{cases}
\tilde\theta_2(S(\tfrac{\pi}{2})), & \text{if }j\in J
\qquad (\iff n(i,j)=1,\ n(j,i)=0) , \\
\tilde\theta_2(S(\tfrac{\pi}{4})), & \text{if }j\in I\backslash J.
\end{cases}
\end{align*}
Relation \ref{rels:Wspin-D-conj}: As discussed in Remark~\ref{differentrelations} one has
\begin{align*}
x_i^{-1} x_j^2 x_i = x_j^2 x_i^ {2n(j,i)} & \Longleftrightarrow x_j^{-2} x_i^{-1} x_j^{2} = x_i^{2n(j,i)}x_i^{-1}, \\
x_j^{-1} x_i^2 x_j = x_i^2 x_j^ {2n(i,j)} & \Longleftrightarrow x_i^{-2} x_j^{-1} x_i^{2} = x_j^{2n(i,j)}x_j^{-1},
\end{align*}
Since $x_i \in \wt K_1$, $x_j \in \wt K_2$, $x_i^2 = \tilde t_1$, $x_j^2 = \tilde t_2$ (cf.\ Remark~\ref{dichotomy}), the respective equalities on the right hand sides hold by identity (\ref{crucialidentity2}) in Proposition~\ref{keycorollary}.
\qedhere
\end{description}
\end{proof}
\begin{theorem} \label{constwspin}
The group homomorphism
\[
\overline{f} : \Wspin(\Pi,\kappa) \to \wh{W}(\Pi,\kappa)
\]
mapping $r_i \mapsto \wh{s}_i$
is an isomorphism. In particular \[\Wspin(\Pi) \cong \wh{W}(\Pi,\kappa_{\mathrm{max}}).\]
\end{theorem}
\begin{proof}
We already established that $\overline{f}$ is an epimorphism.
It remains to prove that it is injective.
\medskip
Let $\rho:\Spin{\Pi,\kappa}\to K(\Pi)$ be the central extension
from Theorem~\ref{m3}.
Then, using
Lemmas \ref{lem:ama-embed-and-rho-commute-An}, \ref{lem:ama-embed-and-rho-commute-G2}, \ref{lem:ama-embed-and-rho-commute-B2} and Proposition~\ref{keycorollary},
it follows that
\begin{align*}
\rho(\wh{s}_i)
&=
\begin{cases}
(\rho\circ\tilde\psi_{ij}\circ\tilde\phi_{ij}^i\circ S)(\tfrac{\pi}{4}) & \text{ if } i\notin J \\
(\rho\circ\tilde\psi_{ij}\circ\tilde\phi_{ij}^i\circ\sq\circ S)(\tfrac{\pi}{4}) & \text{ if } i\in J
\end{cases} \\
&=
\begin{cases}
(\psi_{ij}\circ\rho^{(v_{ij})}\circ\tilde\phi_{ij}^i\circ S)(\tfrac{\pi}{4}) & \text{ if } i\notin J \\
(\psi_{ij}\circ\rho^{(v_{ij})}\circ\tilde\phi_{ij}^i\circ\sq\circ S)(\tfrac{\pi}{4}) & \text{ if } i\in J
\end{cases} \\
&= (\psi_{ij}\circ\phi_{ij}^i\circ\rho_2\circ S)(\tfrac{\pi}{4})
= (\psi_{ij}\circ\phi_{ij}^i\circ D)(\tfrac{\pi}{2})
= \wt{s}_i .
\end{align*}
Since $\ker\rho=\langle (\wh{s}_i)^4 \mid i \in I \rangle \subseteq \wh{W}(\Pi,\kappa)$,
the restriction of $\rho$ to an epimorphism $\wh{W}(\Pi,\kappa)\to\wt{W}(\Pi)$
is a $2^{c(\Pi,\kappa)}$-fold central extension by Theorem~\ref{m3}.
Consider the following diagram:
\[\xymatrix{
1 \ar[rr] & & \langle (r_i)^4 \mid i \in I \rangle \ar[d]^{(r_i)^4 \mapsto (\wh{s}_i)^4} \ar[rr] &
& \Wspin(\Pi,\kappa) \ar[d]^{\overline{f} : r_i\mapsto \wh{s}_i}\ar[rr]^{r_i\mapsto t_i} &
& \Wext(\Pi) \ar[d]^{t_i\mapsto\wt{s}_i}\ar[rr] & & 1 \\
1 \ar[rr] & & \langle (\wh{s}_i)^4 \mid i \in I \rangle \ar[rr] &
& \wh{W}(\Pi,\kappa) \ar[rr]^{\wh{s}_i\mapsto \wt{s}_i} &
& \wt{W}(\Pi) \ar[rr] & & 1
}
\]
The upper row of this diagram is exact by Lemma~\ref{upperrowexact}(a), the lower row by the above discussion.
It is not hard to verify that the diagram is also commutative.
By Consequence~\ref{Zabelian} the group $\langle (r_i)^4 \mid i \in I \rangle$ is abelian and by Proposition~\ref{orderZ} it has at most $2^{c(\Pi,\kappa)}$ elements. Since $|\langle (\wh{s}_i)^4 \mid i \in I \rangle| = 2^{c(\Pi,\kappa)}$ by the above discussion, the left-hand arrow is an isomorphism.
The right-hand arrow is an isomorphism by Proposition~\ref{KacPeterson}(a). Thus $\overline{f}$ is also an isomorphism.
The final statement follows from Proposition~\ref{orderZ}(b).
\end{proof}
We have proved Theorem~\ref{mainthm:sl-weyl} from the introduction.
\section{Finite-dimensional compact Lie group quotients of $\Spin{\Pi}$ and $K(\Pi)$} \label{sec13}
In this section we prove that for each simply laced diagram $\Pi$ the group $\Spin{\Pi}$ admits an epimorphism onto a non-trivial compact Lie group afforded by the representation constructed in Theorem~\ref{m1}. By virtue of Theorem~\ref{thm:Spin(Delta)-covers-Spin(Delta-sl)} such an epimorphism then in fact exists for any diagram $\Pi$ and any admissible colouring $\kappa$.
\begin{theorem}
Let $\Pi$ be a simply laced diagram. Then the target of the (continuous) group epimorphism $\Xi : \Spin{\Pi} \to X$ from Remark~\ref{inparticular} is a non-trivial compact Lie group, as is the target of the induced (continuous) group epimorphism $\overline \Xi : K(\Pi) \to X / \left\langle \Xi(Z) \right\rangle$.
\end{theorem}
\begin{proof}
By \cite[Theorem~4.11]{Hainke/Koehl/Levy} the image of the generalized spin representation $\mu : \mathfrak{k} \to \End(\CC^s)$ from Theorem~\ref{m1} is compact. Hence the claim for $\Xi$, and by the homomorphism theorem for groups also for $\overline\Xi$, follows.
\end{proof}
\begin{corollary} [{cf.\ \cite[Lemma~2, p.~49]{DamourHillmann}}] \label{DamourHillmann}
Let $\Pi$ be a simply laced diagram. Then the image $\Xi\left(\widehat W(\Pi)\right)$ is finite.
\end{corollary}
\begin{proof}
As in the proof of \cite[Lemma~2, p.~49]{DamourHillmann} the key observation is that $\Xi\left(\widehat W(\Pi)\right) \leq X$ is a discrete subgroup of the compact group X. Consider the commutative diagram obtained from Theorem~\ref{m1}
\[
\xymatrix{
\{ k \frac{\pi}{4} \mid k \in \ZZ/8\ZZ \} \ar@{^{(}->}[dd]_S \ar@{^{(}->}[rr]^{\tilde\psi_{ij} \circ \tilde\phi_{ij}^i \circ S} && \Spin\Pi \ar[drr]^\Xi \\
&& \AAA(\Pi,\Spin 2) \ar[rr]^{\Psi_\AAA} \ar[u]^{\widetilde\Psi} && X \\
\Spin{2} \ar@/^1pc/[uurr]^{\tilde\psi_{ij} \circ \tilde\phi_{ij}^i} \ar[urr]^{\tilde\phi_{ij}^i} \ar@/_/[urrrr]_{\psi_{ij} \circ \tilde\phi_{ij}^i}
}
\]
where \[\widetilde\Psi = \{ \tilde \psi_{ij} : \widetilde G_{ij} \to \Spin\Pi \} := \tau_{\Spin\Pi}= \{ \tau_{ij} : \widetilde G_{ij} \to \Spin\Pi \}\] is the canonical universal enveloping morphism of the amalgam $\AAA(\Pi,\Spin 2)= \{ \widetilde G_{ij}, \tilde\phi_{ij}^i \mid i \neq j \in I \}$.
Then \[\widehat W(\Pi) = \langle \widehat s_i \mid i \in I \rangle\] where $\widehat s_i := \tilde\psi_{ij}(\tilde\phi_{ij}^i(S(\frac{\pi}{4})))$ and \[\Xi\left(\widehat W(\Pi) \right) = \left\langle \psi_{ij}(\tilde\phi_{ij}^i(S(\frac{\pi}{4}))) \mid i \in I \right\rangle = \left\langle \cos(\frac{\pi}{4}) + \sin(\frac{\pi}{4})X_i \mid i \in I \right\rangle.\]
For $R_i := \cos(\frac{\pi}{4}) + \sin(\frac{\pi}{4})X_i=\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}X_i$ the commutator relations
\[X_iX_j=\begin{cases}
-X_jX_i,&\text{if }\{i,j\}^\sigma\in E(\Pi)\ , \\
X_jX_i,&\text{if }\{i,j\}^\sigma\notin E(\Pi)\
\end{cases}\]
established in \cite[Remark~4.5]{Hainke/Koehl/Levy} (cf.\ the proof of Theorem~\ref{m1}) imply that
\[R_iR_j = \begin{cases}
-R_jR_i + \sqrt{2}R_i + \sqrt{2}R_j - 1, &
\text{if }\{i,j\}^\sigma \in E(\Pi), \\
R_jR_i, & \text{if }\{i,j\}^\sigma \not\in E(\Pi),
\end{cases}\]
as in Remark~\ref{straightforwardcomputation} or in the proof of \cite[Lemma~2, p.~49]{DamourHillmann};
moreover, $X_i^2 = -\id$ implies $R_i^2 = \sqrt{2}R_i-1$.
We conclude that any product of the above generators of $\Xi\left(\widehat W(\Pi)\right)$ can be written as a polynomial in the $R_i$, $i \in I$, such that each $R_i$ appears at most linearly in each polynomial, with coefficients in $\ZZ[\sqrt{2}]$. Hence $\Xi\left(\widehat W(\Pi)\right)$ is a discrete set. As it is a subgroup of the compact group $X$, it has to be finite.
\end{proof}
Not much is known about the kernels of the maps $\Xi$ and $\overline{\Xi}$. However, Pierre-Emmanuel Caprace pointed out to us that these kernels generally cannot be abstractly simple. The argument relies on the concept of acylindrical hyperbolicity (see \cite{Osin:2013}, also \cite{CapraceHume}).
\begin{corollary}
Let $\Pi$ be an irreducible non-spherical, non-affine simply laced Dynkin diagram. Then the kernels of the generalized spin representations $\Xi : \Spin{\Pi} \to X$ and $\overline \Xi : K(\Pi) \to X / \left\langle \Xi(Z) \right\rangle$ are acylindrically hyperbolic and, in particular, not abstractly simple.
\end{corollary}
\begin{proof}
$\Spin{\Pi}$ and $K(\Pi)$ naturally act on each half of the twin building of the associated Kac--Moody group $G(\Pi)$ and, thus, so do $\ker(\Xi)$ and $\ker(\overline{\Xi})$. The stabiliser in $K(\Pi)$ of the fundamental chamber consists of the elements of the standard torus of $G(\Pi)$ that square to the identity, the stabiliser in $\Spin{\Pi}$ is a double extension of this group. In particular, chamber stabilisers in $K(\Pi)$ and in $\Spin{\Pi}$ are finite and, hence, so are chamber stabilisers in $\ker(\overline{\Xi})$ and $\ker(\Xi)$. By Corollary~\ref{DamourHillmann} the intersection $\ker(\overline{\Xi}) \cap \wt W(\Pi)$ has finite index in $\wt W(\Pi)$ and the intersection $\ker(\Xi) \cap \wh W(\Pi)$ has finite index in $\wh W(\Pi)$ (see Section~\ref{extended} for definitions). Hence \cite[Theorem~1.4]{CapraceHume} applies and $\ker(\Xi)$ and $\ker(\overline{\Xi})$ are acylindrically hyperbolic. Therefore by \cite[Theorem~8.5]{Dahmani/Guirardel/Osin} both groups contain a non-trivial proper free normal subgroup.
\end{proof}
\begin{corollary}
Let $\Pi$ be a Dynkin diagram with admissible colouring $\kappa$. Then the target of the group homomorphism $\Spin{\Pi,\kappa} \stackrel{\Xi_{\Spin{\Pi,\kappa}}}{\longrightarrow} \Spin{\Pi^{\mathrm{dl\kappa}},\kappa} \stackrel{\Omega_{\Spin{\Pi^{\mathrm{dl\kappa}},\kappa}}}{\longrightarrow} \Spin{(\Pi^{\mathrm{dl}})^{\mathrm{un}},\kappa^{\mathrm{un}}} \stackrel{\Xi}{\longrightarrow} X$ is a non-trivial compact Lie group, as is the target of the induced group epimorphism $K(\Pi) \stackrel{\Xi_{K(\Pi)}}{\longrightarrow} K((\Pi^{\mathrm{dl}\kappa}) \stackrel{\Omega_{K(\Pi^{\mathrm{dl\kappa}})}}{\longrightarrow} K(\Pi^{\mathrm{dl}})^{\mathrm{un}}) \stackrel{\overline{\Xi}}{\longrightarrow} X/ \left\langle \Xi(Z) \right\rangle$.
\end{corollary}
\begin{proof}
It suffices to observe that $\Spin{\Pi^{\mathrm{dl\kappa}},\kappa} \stackrel{\Omega_{\Spin{\Pi^{\mathrm{dl\kappa}},\kappa}}}{\longrightarrow} \Spin{(\Pi^{\mathrm{dl}})^{\mathrm{un}},\kappa^{\mathrm{un}}}$ and $K(\Pi^{\mathrm{dl}\kappa}) \stackrel{\Omega_{K(\Pi^{\mathrm{dl\kappa}})}}{\longrightarrow} K((\Pi^{\mathrm{dl}})^{\mathrm{un}})$ have closed images with respect to the Kac--Peterson topology, as they can be realized as fixed points of continuous involutions.
\end{proof}
\begin{remark}
For $\Pi=E_n$ for some $n \in \NN$, the isomorphism type of $X$ can be extracted from the Cartan--Bott periodicity described in \cite[Theorem~A]{Horn/Koehl}.
\end{remark}
\bibliographystyle{alpha}
\bibliography{References}
\end{document} | 168,369 |
TITLE: Prove that $\int_{-\infty}^{\infty} x^2 e^{-\alpha x^2} dx = \frac{1}{2} \sqrt{\frac{\pi}{\alpha^3}}$
QUESTION [3 upvotes]: Prove that $\displaystyle\int_{-\infty}^{\infty} x^2 e^{-\alpha x^2} dx = \frac{1}{2} \sqrt{\frac{\pi}{\alpha^3}}$.
You're allowed to use the formula $\displaystyle\int_{-\infty}^{\infty} e^{-\alpha x^2} dx = \sqrt{\frac{\pi}{\alpha}}$.
REPLY [5 votes]: This requires Integration by parts. Note that
$$ \int u(x)v'(x) dx = u(x)v(x) - \int u'(x)v(x) dx $$
If we let $$ v'(x) = -2axe^{-ax^2}$$
And we let
$$ u(x) = -\frac{1}{2a}x $$
Then it is easy to see that
$$ u(x)v'(x) = x^2e^{-ax^2} $$
Note that by the chain rule
$$ v(x) = e^{-ax^2} $$
Thus
$$ \int_{-\infty}^{\infty} x^2e^{-ax^2} = -\frac{x}{2a}e^{-ax^2} | _{-\infty}^{\infty} - \frac{1}{2a} \int_{-\infty}^{\infty} e^{ax^2} dx $$
This is equal to
$$ 0+\frac{1}{2a} \sqrt{\frac{\pi}{a}} = \frac{1}{2} \sqrt{\frac{\pi}{a^3}}$$ | 50,975 |
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\begin{document}
\maketitle
\begin{abstract}
There are many papers about the proof of Abel-Ruffini theorem, but, in this paper, we will show a way to solve any quintic equation by radicals which will be a outstandig result because of work the mathematicians Abel and Ruffini who showed that there is no formula to quintic equation. We need to mention the mathematician Évariste Galois that also showed the same conclusion of Abel and Ruffini using permutations,group theory etc. However, in this article we will present a method to split any quintic equation into two equation (one of degree 3 and another of degree 2) where both equations can be solved by radicais using the quadratic and the cubic formula.
\end{abstract}
\section{Introduction}
This paper contributes to a new way to solve quintic equation using a polynomial (Martinelli's polynomial) that can be proved using some simple steps. Since the 1500's when the cubic and quartic formula was discovered the world waited centuries for the next step: to find a method or general formula to solve the quintic equation. Although Abel and Ruffini showed the impossibility of a closed formula to solve general quintic equation the search for a formula to solve quintic equation ends. But the mathematician Joseph-Louis Lagrange proved the inversion theorem that could solve specific cases of the quintic equation and higher degree. However using hypergeometric function will be possible to solve any quintic equation if reducing the general quintic equation to the Bring-Jerrard form (but it is a very complex process to transform the general quintic equation to that form). At the end of this paper there is an example that checks if the method that will be presented works. Finally, to end the abstract we will not proof or talk about the Lagrange inversion theorem and Bring-Jerrard form to get more understanding this paper.
\section{About Galois theory and Abel-Ruffini theorem}
\label{sec:headings}
We will see some concepts about Galois' theory and Abel-Ruffini's theorem.
The theorem, in a succinct way, consists of the proof that there is no "closed" formula for all equations of degree greater or equal to 5. That is, there is no way to algebraically arrive at a formula for those equations greater or equal to 5. As Zoladek (2000) reported, "A general algebraic equation of degree $\geq$ 5 cannot be solved in radicals. This means that there does not exist any formula
which would express the roots of such equation as functions of the coefficients by means of the algebraic operations and roots of natural degrees." (p. 254).
Thus, Galois theory comes to confirm, through the study of the relationship of the roots that it is impossible to solve, algebraically, equations of degree equal to or greater than 5. In its most basic form, the theorem states that given a
E / F field extension which is finite and Galois, there is a one-to-one correspondence between its intermediate fields and subgroups of its Galois group. (Intermediate fields are K fields that satisfy $F \subseteq K \subseteq E$; they are also called sub-extensions of E / F.).
So, according to Abel-Ruffini's theorem and Galois theory, there are cases where equations of degree 5 or higher are solvable by radicals or algebraically.
\section{Proof of Martinelli's polynomial}
To demonstrate the Martinelli's Polynomial it is necessary to consider the roots of any fifth degree equation.
Each root must be associated with a single other root. We have then that the combination of the roots of a fifth degree equation is ten (due that the polynomial is tenth degree). Let us then consider the following equations:
\begin{equation}
(x^2-kx+n)(x^3+kx^2+kx+m)=0.
\end{equation}
\begin{equation}
(x^2-kx+n)(x^3+kx^2+lx+m)=0.
\end{equation}
Equation (1) have a relation with equation (2). If we expand both equations (1) and (2), each corresponding term can be factorized. Let put those terms into a table:
\def\arraystretch{2.0}
\begin{table}[htb]
\centering
\caption{Equation (1) expanded}
\begin{tabular}{|c|c|c|c|}
\hline
$x^3$ & $x^2$ & $x$ & Term Independent \\
\hline
$k+n-k^2=C_2$ & $-k^2+kn+m=D_2$ & $kn-km=E_2$ & $ mn=F$ \\
\hline
\end{tabular}
\end{table}
\def\arraystretch{2.0}
\begin{table}[htb]
\centering
\caption{Equation (2) expanded}
\begin{tabular}{|c|c|c|c|}
\hline
$x^3$ & $x^2$ & $x$ & Term Independent \\
\hline
$l+n-k^2=C$ & $nk-lk+m=D$ & $nl-km=E$ & $ mn=F$ \\
\hline
\end{tabular}
\end{table}
Andrade (2019) consider the follow equation to better understand the proof of Martinelli's Polynomial:
\begin{equation}
x^2+3x+2=0
\end{equation}
If we plug in x the sum of the roots of equation (3) we have 2=0, that would be absurd, but with that idea, we can create a second degree equation using the terms $C, C_2 , D, D_2, E$ e $E_2$ as shown in Tables 1 and 2 above. Matching the equation that will be created with $(-k^2+n+k-C)n$ we will have, on the right side of the equation, the same thing $E_2- E$, because $E_2$ is the same as $kn-km$ and $-E$ is the same as $-nl+km$. The sum of $E_2-E$ will result $kn-nl$ then $l=C-n+k^2$. So we have $kn -n(C-n+k^2)$ that is the same as $(-k^2+n+k-C)n$.
To create a second degree equation where one of the solutions of the equation will be the sum of two solutions of a fifth degree equation we must follow this logic: We have the general form of the second degree equation that is
$ax^2 + bx + c = 0$, the coefficients of the second degree equation that will be created will be $a = C_2- C$, $b = D_2- D$, $c=E_2- E$ and on the right side of the equation we have to add $(-k^2+n+k-C)n$ because on the left side of the equation will remain $E_2- E$.
This idea will give us the possibility to know n from equation (2). The goal is to form a tenth degree equation with the unknown k. So $a = C_2 - C$ and $b=D_2-D$ according to tables 1 and 2, it follows that $a=k+n-k^2-C$, $b=-k^2+kn+m-D$, $c=nk -km-E$ and the right side of the equation will be $(-k^2+n+k-C)n$. Thus, making the substitutions in $ak^2+bk+c=0$ we will have:
\begin{align*}
(-k^2+n+k-C)k^2 + (-k^2+nk+m-D)k + nk-km-E &= (-k^2+n+k-C)n.\\
(-k^4+nk^2+k^3-Ck^2)+(-k^3+nk^2+km- Dk)+ nk-km- E &= (-k^2+n+k-C)n. \\
-k^4+2nk^2-Ck^2-Dk+ nk- E &= -nk^2+n^2+nk-nC.\\
-k^4-Ck^2-Dk-E&=-3nk^2+n^2-nC.\\
\end{align*}
Putting n on the left side of the equation, we have:
\begin{equation}
n = \frac{k^4+Ck^2+Dk+E}{3k^2-n+C}.
\end{equation}
Since k is the sum of two roots of a fifth degree equation, C, D, E and F the coefficients of a fifth degree equation and n is the product of the roots of a second degree equation, so to arrive at Martinelli's polynomial, we have to replace the variable n with the use of an algebraic manipulation. This algebraic manipulation consists of taking the equality referring to the term D in table (2). Thus:
\begin{equation}
nk-(k^2-n+C)k+\frac{F}{n}=D.
\end{equation}
\begin{equation}
n^2=\frac{nk^3+Cnk-F+Dn}{2k}.
\end{equation}
Now just replace $ n^2$ in equation (4) and get n:
\begin{equation*}
n = \frac{k^4+Ck^2+Dk+E}{3k^2-n+C}.
\end{equation*}
\begin{equation*}
-n^2+3nk^2+Cn=k^4+Ck^2+Dk+E.
\end{equation*}
\begin{equation*}
\frac{-nk^3-Cnk+F-Dn}{2k} + 3nk^2 + Cn = k^4+Ck^2+Dk+E.
\end{equation*}
\begin{equation}
n = \frac{2(k^5+Ck^3+Dk^2+Ek)-F}{5k^3+Ck-D}.
\end{equation}
If we have n, then we can create the Martinelli's polynomial that will be very important to solve any quintic equation
\begin{equation*}
\frac{2(k^5+Ck^3+Dk^2+Ek)-F}{5k^3+Ck-D} = \frac{k^4+Ck^2+Dk+E}{3k^2-\frac{2(k^5+Ck^3+Dk^2+Ek)-F}{5k^3+Ck-D}+C}.
\end{equation*}
Arranging the equation on both sides and equaling 0 we arrive at the Martinelli's polynomial:
\begin{eqnarray*}
{(2(k^5+Ck^3+Dk^2+Ek)-F)(13k^5+6Ck^3-5Dk^2+(-2E+C^2)k+F-DC)}
\end{eqnarray*}
\begin{equation}
-(k^4+Ck^2+Dk+E)(5k^3+Ck-D)^2 = 0.
\end{equation}
\section{Martinelli's Polynomial expanded}
\begin{equation*}
k^{10} + 3Ck^8 + Dk^7 + (3C^2-3E)k^6 + (2DC-11F)k^5 + (C^3-D^2-2CE)k^4 + (DC^2-4DE-4CF)k^3
\end{equation*}
\begin{equation}
+ (7DF-CD^2-4E^2+EC^2)k^2 + (4EF-FC^2-D^3)k-F^2+FDC-D^2E=0.
\end{equation}
\section{Solving a quintic equation as an example}
Let's begin with the quintic equation:
\begin{equation}
x^{5}+x+3=0
\end{equation}
Using the Martinelli's Polynomial (9) we get the follow polynomial:
\begin{equation}
k^{10}-3k^{6}-33k^{5}-4k^{2}+12k-9=0
\end{equation}
The roots of the equation (11) are the combination of the sums of the equation (10). So, wouldn't be necessary to know all roots of the equation (11). But, solving the equation (10) we will see that the sum of the complex roots give us one of the real root: 2.0837590792241645736... that is represented using hypergeometric function as follows:
\begin{equation}
k = \sqrt{2}\pFq{4}{3}{-\frac{1}{20},\frac{3}{20},\frac{7}{20},\frac{11}{20}}{\frac{1}{4},\frac{1}{2},\frac{3}{4}}{-\frac{253125}{256}}
-\frac{45\pFq{4}{3}{-\frac{9}{20},\frac{13}{20},\frac{17}{20},\frac{21}{20}}{\frac{3}{4},\frac{5}{4},\frac{3}{2}}{-\frac{253125}{256}}}{16\sqrt{2}}
+\frac{3}{2}\pFq{4}{3}{\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5}}{\frac{1}{2},\frac{3}{4},\frac{5}{4}}{-\frac{253125}{256}}
\end{equation}
\begin{equation}
\Bigg(x^2-kx+\frac{2(k^5+k)-3)}{5k^3}\Bigg)\Bigg(x^3+kx^2+\Bigg(k^2-\frac{2(k^5+k)-3)}{5k^3}\Bigg)x+\frac{15k^3}{2(k^5+k)-3}\Bigg)=0
\end{equation}
Now, plugging the k root, represented by hypergeometric function (12), we will have the equation (10), but, in this case, the equation (10) was factorized into two equations of degree 2 and another of degree 3. Using the quadratic and cubic formula we will able to solve in terms of radicals the equation (13) which it is the same as the equation (10).
\section{Remark}
As seen the Martinelli's polynomial provides the sum of two roots of any quintic equation and this is very important to split a quintic equation into two smaller degree equations. This method solves quintic equation using quadratic and cubic formulas but in the most cases it is necessary to use the hypergeometric function to represent the variable that splits the equation. However, a general method has been achieved to solve any quintic equation. But, it's no make sense, because we need to know one of the real root of the Martinelli's polynomial once that is the sum of quintic roots equation. We need to know before the roots of the quintic equation in the Bring-Jerrard form to split the fifth degree equation and solve in terms of radicals. So, Abel-Ruffini theorem and Galois theory didn't give us a clear answer about this way to solve quintic equation in terms of radicals and in the Bring-Jerrard form. Maybe it's a new discussion about equations theory field. | 150,648 |
Regular.
7 Things to Do With 48 Hours in Austin, Texas
1. Barton Springs Pool
When it came to activities, I knew little about Barton Springs Pool, but from what I gleaned, I knew to put it top of my list. First of all, I’m quite partial to pools. And this one sounded incredible: Located within Austin’s Zilker Park, it’s a three-acre (three acres!) spring-fed pool that is open to the public (for $4 apiece). And it far, far exceeded my expectations to become one of the most favorite public urban offerings I’ve ever visited. I’d say it’s Austin’s No. 1 can’t-miss activity if the weather is right.
First of all, look at this gorgeous thing:
We were advised to arrive “early” to avoid crowds, but we were on vacation schedule and took our sweet time, arriving at 11:30 or so. And we only waited behind a few people to pay our admission and go inside. As you can see in the photos, folks of all ages and categories — families, couples, teens — were out enjoying the pool, which is an awesomely refreshing 68 to 70 degrees compared to the 100-degree air. (Think that sounds cold? For comparison, I got my scuba diving certification in the dead of winter in 50-degree water when the air was the same.)
Around 1 p.m. on a gorgeous hot Saturday and still no crowds! (We did see them arriving later in the afternoon as we were leaving, and we were told the congestion and parking situation starts to get messy around mid-afternoon.)
You’re not allowed to bring outside food in, but people bring their towels to soak up the sun on the grass, and relax between dips. There’s even a diving board!
And forget whatever (germy) images you conjure when you think “public pool.” Evidently, Barton Springs closes every Thursday for a top-to-bottom cleaning. This is truly a spectacular gem I’d visit every summer weekend if I lived in Austin. So amazing!
2. Graffiti Park
Here’s something I’d never heard about — I think most people even in Austin haven’t — and wouldn’t have known about except for the W Austin. The W Insider (a.k.a. concierge) had put together a list of cool things worth checking out arund town and placed it in our room (next to a bottle of welcome bubbly rosé, where it was sure to get our attention). This was on that list (cited there as “Hope Gallery,” but I have since seen it referred to as “Graffiti Park” more frequently).
Basically, it’s an abandoned building site that’s been turned into a living, vibrant canvas by local graffiti artists. There are gorgeous and thought-provoking images and quotes. But visitors are also welcomed to buy paint on the spot and express their own impromptu creativity (ideally not over the murals, but wherever else they carve out space).
So, so cool and worth seeing — and, of course, also a selfie magnet. (Hubby compared it to Lacma’s Urban Light installation in our hometown of L.A.)
3. Kayaking on Lady Bird Lake
From our room at the W Austin, we could easily see people recreating in all kinds of ways on what appears to be a river but is actually called Lady Bird Lake.
So we asked the W Insider about our options for renting kayaks, and he let us know about Congress Avenue Kayaks, easy walking distance from the hotel.
Off we went to rent our double kayak for $15 per hour (or $25 for a half day), just to get the flavor. It’s such a peaceful experience, it’s almost hard to believe you’re right in a downtown area and not some kind of park (except, of course, for the high rises in view). Totally recommend!
Passing beneath Congress Avenue Bridge (and making my standard bad judgement call re: iPhone and water)
4. Ashiatsu massage at Away Spa
I know from massage. I’ve been lucky to experience all manner of hot stones, aromatherapies, outdoor treatments, specific cultural specialties around the world like Turkish hammam — all kinds of blissful massages. But I’ve honestly never had one like the ashiatsu massage at the W Austin’s Away Spa.
It’s a brand new offering, and it uses the therapist’s… feet! She holds onto bars on the ceiling to do this. Yes, the pressure was fairly deep (I wanted it that way), but it doesn’t have to be a crazy-intense massage. My therapist, who was a dancer, said the pressure would be easily variable based on people’s preferences.
I don’t mean to sound like a complainer, but I find myself sore all the time these days. It’s from carrying children (both on the inside and then on the outside), and from lugging a 15-inch laptop (rue the day I made that purchase decision). And I’ve got near-constant core soreness from my continued work on my postpartum diastasis recti through specialized programs like the postpartum healing Mutu System and Pilates reformer classes (ClassPass, I’m obsessed with you). So when I have a chance to have a massage this epic, it can be a real game changer for a whole weekend.
5. Watching bat flight at Congress Avenue Bridge
In working on a recent U.S. travel piece that appeared in Country Living and Esquire and other such pubs earlier this month, I researched a place I’ve never personally visited but sounds so amazing: Carlsberg Caverns in New Mexico, where people gather to watch every night as bats take flight from the cave into the night sky. It sounded like a dramatic and unique spectacle of nature, so I included it in that story.
Well, turns out, Austin — yes, urban Austin — has its very own version of that! Apparently the world’s largest urban bat colony (because anyone knew that was such a thing?) lives under downtown’s Congress Avenue Bridge, and people gather every night to watch them take flight.
The bridge is within easy walking distance from our hotel, and we watched from up top with other pedestrians lined up against the railing. But people also gather on the grass down below near the water.
What I liked even more than watching the urban bats (which are themselves weird/interesting/unique, of course) was this evening ritual of people all gathered in anticipation. It reminded me of gathering with everyone in Oia on Santorini in Greece, part of a ritual to watch the sun set. Sunsets and bats are cool, obviously, but getting to be a part of a ritual like this actually feels like the main attraction to me. I’m a sucker for community.
6. W Austin Wet Deck
OK, I’ve already mentioned that I rarely meet a pool I don’t like. But the W Austin’s pool — pardon, its Wet Deck — is a special scene. It’s open from sunrise to sunset, and is basically an all-day party requiring a wristband for hotel guests (or for purchase). In addition to the long fourth-floor pool itself, there are private cabanas, lounge chairs, tanning shelves, a fireplace, and outdoor showers.
Yes, this pool gets crowded, and that’s the point: It’s a party. But it worked in my favor that I am an early person and Austin is a sleep-in town, so I never had any trouble getting a place to set up shop and feel comfortable.
I took the below photos on an ordinary Saturday. But just as we were checking out the next day, the place was getting packed for the ginorm party, Shockwave, which takes place the last Sunday of every month in the summer.
Anyway, despite all the beautiful people, I found the vibe very welcoming and down to earth. I actually had a long, semi-heart-to-heart chat with a fellow twin mom, who was also on a minibreak from her kids. And there’s no better place for a heart-to-heart than in a spectacular pool, I say. But that’s me.
And this is on a Saturday. The following day, the pool was even more packed for the monthly summertime bash known as Shockwave.
7. Just actually (wait for it!) relaxing
Thanks to the aforementioned FOMO, I sometimes have a hard time relaxing when I’m on vacation. Who has time for that?! Sunrises and sunsets to catch on a schedule! Ferries and puddle-jumpers and tours departing! This one restaurant you have to go at one particular time to put your name down six hours in advance!
Well, I loved Austin for not making me feel like that. Its relaxed vibe is perfect for carving out the particular pace that suits you.
So, in the evenings, after dinner, hubs and I went back to our gorgeous, comfy 15th-floor suite at the W Austin with its river and downtown views and high-tech everything — and just chilled. The eff. Out. We were content to watch the stream of people over the Congress Avenue Bridge, or on the boats in the river, while actually allowing ourselves a bit of time to regroup and relax amid a powerful AC and a pair of massive televisions. And that was the perfect counterpoint to our well-paced Austin days.
So those are a few of my favorite things in Austin!
[What? No food, you ask? Of course Austin was full of superlative food options — and I’m reserving the whole next post for that. So check back!]
Disclosure: Starwood and the W Austin paid for this travel with the expectation that I would write about my experience in town here on this blog. My itinerary.
1 Comment
[…] after a morning of sightseeing, we stopped at the Brown’s Bar-B-Que trailer for hubby to eat his meaty lunch. (Pause to […] | 268,761 |
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no allergens highlighted or any may contain warnings listed. On opening the box the teabags were in a sealed foil packaging. This is always a positive trying to keep the tea fresh and seal the both flavour and aroma in. On opening the foil packaging I was immediately hit with.
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\begin{document}
\begin{center}
{\huge Joint weak type interpolation on Lorentz-Karamata spaces}\\\vskip7mm
{\Large Michal Bathory}\footnote{Mathematical Institute of the Charles University 186 75 Praha 8, Sokolovsk\'{a} 83, Czech Republic\\
e-mail: \texttt{[email protected].}\\ The author was supported by the grant SVV-2016-260335 and the project UNCE 204014.\\
Mathematics subject classification (2010): 26D10, 46E30, 46B70, 47B38, 47G10\\
Keywords: real interpolation, joint weak type operators, Lorentz-Karamata spaces, Hardy inequalities}
\end{center}
\begin{abstract}
We present sharp interpolation theorems, including all limiting cases, for a class of quasilinear operators of joint weak type acting between Lorentz-Karamata spaces over $\s$-finite measure. This class contains many of the important integral operators. The optimality in the scale of Lorentz-Karamata spaces is also discussed. The proofs of our results rely on a characterization of Hardy-type inequalities restricted to monotone functions and with power-slowly varying weights. Some of the limiting cases of these inequalities have not been considered in the literature so far.
\end{abstract}
\section{Introduction}
The concept of Lorentz-Karamata (LK) spaces is a natural generalization of the generalized Lorentz-Zygmund (GLZ) spaces, that has been proven to be very useful when one needs to find a precise description of the boundedness of the given operator, especially in the limiting cases (cf. \cite{BR}, \cite{EOP}, for example). It also seems that the LK spaces lead to an optimal balance between the generality and explicitness of the resulting theorems.
The main results of this paper are formulated in Section~\ref{S3} and proved in Section~\ref{S5}. Those are the interpolation theorems for quasilinear operators of joint weak type, i.e.\!\! operators, which are, in certain sense, dominated by the Calder\'{o}n operator (see \eqref{S} below). This class contain many important operators (e.g.\! convolution or singular integral operators) and thus, our results are widely applicable. We will illustrate this on several examples in Section~\ref{S7}. The assumption that some operator is of joint weak type allows to reduce the question of its boundedness to the question of the validity of certain Hardy-type inequality, restricted to non-increasing functions. Thus, the essential part of this paper is to find necessary and sufficient conditions for this kind of inequalities to hold - this is the content of Section~\ref{S4}. Moreover, since weights appearing in those inequalities are of a special (and yet very general) form ($w(x)=x^{\alpha}b(x)$, where $\alpha\in\R$ and $b$ is a slowly varying function) we are able to pinpoint the cases, where the restriction of these inequalities to monotone functions plays any role. In fact, we will show that in most of the cases it is sufficient to apply the known criteria for non-restricted weighted Hardy inequalities (Theorem~\ref{TH} below) and some rather elementary arguments. However, there are certain limiting cases in which one requires a different approach to obtain sharp results. It turns out that these problematic cases can occur only for certain subclass of considered operators; the Hilbert transform is the canonical example. Thus, the characterization of its boundedness in the limiting cases is, in a sense, the most challenging and this will be our ultimate goal.
Our work extends the results of several papers. In \cite{BR} the authors already use the notion of joint weak type and develop an interpolation theory for operators acting between Lorentz-Zygmund spaces over $\s$-finite measure. We, on the other hand, work with the more general scale of spaces and also we clarify the connection with corresponding Hardy-type inequalities, which we characterize fully and which are certainly of independent interest. This allows us to prove also the necessity of obtained conditions. In \cite{Saw} the author gives necessary and sufficient conditions for the boundedness of several important operators acting between the classical Lorentz spaces. However, some of the limiting cases (when the Lorentz space index $r$ is $1$ or $\infty$) are missing there and the used methods does not apply to them (the case $0<r\leq 1$ was eventually described by M. Carro and J. Soria in \cite{CS}). Moreover, unlike in both articles \cite{Saw} and \cite{BR}, we discuss also the optimality (or sharpness) of obtained results (see Section~\ref{S6}). Finally, we extend the theory presented in \cite{EOP} by considering more general spaces over only $\s$-finite measure and consequently, by proving more general Hardy-type inequalities for the whole interval $(0,\infty)$.
\section{Preliminaries}\label{S2}
The following conventions are used throughout this paper: $\infty:=+\infty$, $\tf00:=0$, $\tf{c}{\infty}:=0$, $\tf{c}{0}:=\infty$, for $c\in(0,\infty]$. We also put $\infty\cdot0=0\cdot\infty:=0$. The conjugate index $p'$ to $p\in[1,\infty]$ is defined by $\tf1p+\tf1{p'}=1$. The symbol $\chi_I$ stands for the characteristic function of an interval $I\subseteq\R$. The abbreviations $\text{LHS}(\#)$ or $\text{RHS}(\#)$ are used for the left-hand side or the right-hand side of the relation $(\#)$.
For two non-negative expressions $E,F$, we shall write $E\ls F$ or equivalently $F\gs E$ if there is a~constant $c\in(0,\infty)$ such that $E\leq cF$ and $c$ is independent of appropriate quantities involved in $E$, $F$. Typically, $c$ will always be independent of functions $f,g,h$ and variables $x,t,u,\tau$, but can depend on any other symbol. When $E\ls F\ls E$, we say that $E$ is equivalent to $F$ and we will denote this by $E\es F$.
\subsection*{The decreasing rearrangement}
Let $(R,\mu)$ be a measure space with $\sigma$-finite measure $\mu$. If $\mu(R)<\infty$, we will suppose $\mu(R)=1$ without loss of generality. We denote by $\Meas(R,\mu)$ the set of all scalar valued (real or complex) $\mu$-measurable functions defined on $R$. The symbol $\Meas^+(A,B)$ stands for the set of non-negative, measurable (with respect to Lebesgue measure on $\R$) functions defined on the interval $(A,B)$, which is always one of the intervals $(0,1),(1,\infty),(0,\infty)$. Moreover, the symbols $\Dupa{A,B}$ and $\Upa{A,B}$ denote the sets of all functions from $\Pos{A,B}$ which are non-increasing and non-decreasing, respectively. By $\norm{\cdot}_{r,(A,B)}$, $1\leq r\leq\infty$, we shall denote the usual Lebesgue space norm over $(A,B)$.
The distribution function $d$ of $f$ with respect to $\mu$ is defined by
$$d(\mu,f)(h)=\mu(\set{x\in R:\abs{f(x)}>h}),\quad h\geq0.$$
The decreasing rearrangement of $f$ is then given by
$$f^*(t)=f^*_{(R,\mu)}(t)=\inf\set{h>0:d(\mu,f)(h)\leq t},\quad t\in(0,\infty).$$
We say that functions $f\in(R_1,\mu_1)$ and $g\in(R_2,\mu_2)$ are equimeasurable if their distribution functions are the same, i.e. if $d(\mu_1,f)=d(\mu_2,g)$.
See \cite[Chapter 2, Section 1]{BS} for details.
\subsection*{The Calder\'{o}n operator}
Suppose $1\leq p_1<p_2\leq\infty$, $1\leq q_1,q_2\leq\infty$, $q_1\neq q_2$. The Calder\'{o}n operator $S_\sigma$ associated with the interpolation segment $\sigma=[(\f1{p_1},\f1{q_1});(\f1{p_2},\f1{q_2})]$ is defined for every $g\in\Pos{0,\infty}$ and all $x\in(0,\infty)$ as
\begin{equation}\label{S}
S_\sigma g(x)=x^{-\f1{q_1}}\intt{0}{x^m}{t^{\f1{p_1}-1}g(t)}{t}+x^{-\f1{q_2}}\intt{x^m}{\infty}{t^{\f1{p_2}-1}g(t)}{t},
\end{equation}
where $m=(\f1{q_1}-\f1{q_2})(\f1{p_1}-\f1{p_2})^{-1}$ denotes the slope of the segment $\sigma$.
Throughout the paper we consider only such operators $T$ which take some linear subspace $\mathcal{D}$ of $\Meas(R_1,\mu_1)$ into $\Meas(R_2,\mu_2)$. The operator $T$ is quasilinear if there is $k\geq1$ such that
$$\abs{T(f+g)}\leq k(\abs{Tf}+\abs{Tg})\quad\text{and}\quad\abs{T(\alpha f)}=\abs{\alpha}\,\abs{Tf},$$
$\mu_2\text{-a.e. on }R_2$, for every $f,g\in\mathcal{D}$ and all $\alpha\in\mathbb{C}$. Let us denote $\mathcal{D}_{S}$ the set of all functions $f\in\Meas(R_1,\mu_1)$ which satisfy $S_{\sigma}f^*(1)<\infty$. The quasilinear operator $T$ is said to be of joint weak type $(p_1,q_1;p_2,q_2)$ (notation $T\in\JW(p_1,q_1;p_2,q_2)$) if $\mathcal{D}_{S}\subseteq \mathcal{D}$ and
$$(Tf)^*(x)\ls S_\sigma f^*(x)\quad\forall x\in(0,\infty)\quad\forall f\in\mathcal{D}_{S}.$$
We write $T\in\LBl(p_1,q_1;m)$, or $T\in\LBr(p_2,q_2;m)$, for a~quasilinear operator $T$ if, for any $f\in\Dupa{0,\infty}$, there is a~function $g\in\Meas(R_1,\mu_1)$ equimeasurable with $f$ such that, for all $x\in(0,\infty)$,
$$(Tg)^*(x)\gs x^{-\f1{q_1}}\intt{0}{x^m}{t^{\f1{p_1}-1}f(t)}{t},\quad
\text{or}\quad(Tg)^*(x)\gs x^{-\f1{q_2}}\intt{x^m}{\infty}{t^{\f1{p_2}-1}f(t)}{t},$$
respectively.
If $X$ and $Y$ are two (quasi-) normed spaces, then the symbol $T:X\lra Y$ means that $T$ is bounded from $X$ to $Y$ (i.e. $\norm{Tf}_Y\ls \norm{f}_X$ for all $f\in X$). Furthermore, the symbol $X\embl Y$ stands for $id:X\lra Y$.
\subsection*{Slowly varying functions}
The function $a\in\Pos{A,B}$, $0\not\equiv a\not\equiv\infty$, is said to be slowly varying (s.v.) on $(A,B)$ if, for each $\eps>0$, there exist functions $g_{\eps}\in\Upa{A,B}$, $g_{-\eps}\in\Dupa{A,B}$ such that
\begin{equation}\label{eq8}
t^{\eps}a(t)\es g_{\eps}(t)
\quad\text{and}\quad
t^{-\eps}a(t)\es g_{-\eps}(t)
\quad\forall t\in(A,B).
\end{equation}
We denote by $\SV(A,B)$ the set of all slowly varying functions on $(A,B)$.
We shall now review some important properties of the slowly varying functions. The most basic ones contained in the following proposition are used in the paper without reference.
\begin{proposition}\label{Prop}
Let $a,b\in \SV(A,B)$.
\begin{itemize}
\item[{\rm(i)}] All of the functions $ab$, $\tf1a$, $a^r$, $t\mapsto a(t^r)$, $r\geq0$, are slowly varying on $(A,B)$.
\item[{\rm(ii)}] Let $[C,D]\subseteq[A,B]\cap(0,\infty)$. Then there exist constants $c_1,c_2\in(0,\infty)$ such that $c_1\leq a(x)\leq c_2$ for all $x\in[C,D]$.
\item[{\rm(iii)}] If $c>0$, then $a(ct)\es a(t)$ for every $t\in(0,\infty)$.
\end{itemize}
\end{proposition}
\begin{proof}
For (i) and (iii), see \cite[Proposition 2.2 (i), (ii), (iii)]{GOT}.
Clearly, it is sufficient to prove the assertion (ii) in the case $(A,B)=(0,\infty)$. Let $\eps>0$. By \eqref{eq8}, there exists a function $g_{-\eps}\in\Dupa{0,\infty}$ which is equivalent to the function $x\mapsto x^{-\eps}a(x)$. Then, for all $x\in[C,D]$,
$$a(x)=x^{\eps}x^{-\eps}a(x)\es x^{\eps}g_{-\eps}(x)\leq D^{\eps}g_{-\eps}(C)=:c_2.$$
The existence of the lower bound then follows from (i) ($\tf1a$ is also slowly varying).
\end{proof}
\begin{lemma}\label{LSV}
Let $\la\in\SV(0,\infty)$ and $r\in[1,\infty]$.
\begin{itemize}
\item[\rm{(i)}] If $\eps>0$, then
$$\norm{t^{\eps-\f1r}\la(t)}_{r,(0,x)}\es x^{\eps}\la(x)\quad\text{and}\quad\norm{t^{-\eps-\f1r}\la(t)}_{r,(x,\infty)}\es x^{-\eps}\la(x)\quad\forall x\in(0,\infty).$$
\item[\rm{(ii)}] Then
$$\norm{t^{-\f1r}\la(t)}_{r,(0,x)}\gs\la(x)\quad\text{and}\quad\norm{t^{-\f1r}\la(t)}_{r,(x,\infty)}\gs\la(x)\quad\forall x\in(0,\infty).$$
Furthermore, if $\norm{t^{-\f1r}\lambda(t)}_{r,(0,1)}\!\!\!<\infty$ and $\norm{t^{-\f1r}\lambda(t)}_{r,(1,\infty)}\!\!\!<\infty$, then the functions $x\mapsto\norm{t^{-\f1r}\la(t)}_{r,(0,x)}$ and $x\mapsto\norm{t^{-\f1r}\la(t)}_{r,(x,\infty)}$ belong to $\SV(0,\infty)$, respectively.
\end{itemize}
\end{lemma}
\begin{proof}
For (i) see \cite[Proposition 2.2 (iv)]{GOT}. An important consequence of (i) is that every slowly varying function is equivalent to some continuous function. This fact implies (ii) in the case $r=\infty$. When $r<\infty$, we can write
$$\norm{t^{-\f1r}\lambda(t)}_{r,(0,x)}
=\left(\intt{0}{x}{t^{-1}\lambda(t)^r}{t}\right)^{\f1r}
\gs\left(x^{-1}\lambda(x)^r\intt{0}{x}{1}{t}\right)^{\f1r}
=\lambda(x)$$
and
$$\norm{t^{-\f1r}\lambda(t)}_{r,(x,\infty)}
=\left(\intt{x}{\infty}{t^{-1-\eps}\;t^{\eps}\lambda(t)^r}{t}\right)^{\f1r}
\gs\left(x^{\eps}\lambda(x)^r\intt{x}{\infty}{t^{-1-\eps}}{t}\right)^{\f1r}
\es\lambda(x)$$
for all $x\in(0,\infty)$. For the last assertion of (ii) see \cite[Lemma 2.1. (v)]{GO}.
\end{proof}
\begin{lemma}\label{L}
Let $R\in[1,\infty)$, $S\in[1,\infty]$, $\la\in\SV(A,B)$ and set
$$\Lambda_1(x)=\intt{A}{x}{t^{-1}\lambda(t)^R}{t}\quad\text{and}\quad\Lambda_2(x)=\intt{x}{B}{t^{-1}\lambda(t)^R}{t},\quad x\in(A,B).$$
\begin{itemize}
\item[{\rm (i)}] Suppose that
\begin{equation}\label{1001}
\intt{A}{B}{t^{-1}\lambda(t)^R}{t}=\infty.
\end{equation}
Then
\begin{align}\label{eq34}
\norm{t^{-\f1R}\la(t)}_{R,(A,x)}
&\es\norm{t^{-\f1S}\la(t)^{\f{R}{S}}\Lambda_1(t)^{-\f1R-\f1S}}_{S,(x,B)}^{-1}
\end{align}
and
\begin{align}\label{eq35}
\norm{t^{-\f1R}\la(t)}_{R,(x,B)}
&\es\norm{t^{-\f1S}\la(t)^{\f{R}{S}}\Lambda_2(t)^{-\f1R-\f1S}}_{S,(A,x)}^{-1}
\end{align}
for all $x\in(A,B)$.
\item[{\rm(ii)}] Suppose $\delta\in(0,1)=(A,B)$. Then \eqref{eq34} holds for all $x\in(0,\delta)$.
\item[{\rm(iii)}] Suppose $\delta\in(1,\infty)=(A,B)$. Then \eqref{eq35} holds for all $x\in(\delta,\infty)$.
\end{itemize}
\end{lemma}
\begin{proof}
\mbox{}
\textit{Case} (i). We prove relation \eqref{eq34} here, the proof of \eqref{eq35} is analogous.
If $S=\infty$, then \eqref{eq34} is in fact an equality. It can also happen that both sides of \eqref{eq34} are identically infinite. In other cases, we use the change of variables $\tau=\Lambda_1(t)$ and \eqref{1001} to get, for all $x\in(A,B)$, that
$$\RHS{eq34}=\left(\intt{x}{B}{t^{-1}\la(t)^{R}\Lambda_1(t)^{-\f SR-1}}{t}\right)^{-\f1S}
=\left(\intt{\Lambda_1(x)}{\infty}{\tau^{-\f{S}{R}-1}}{\tau}\right)^{-\f1S}
\es\LHS{eq34}.$$
\textit{Case} (ii). We proceed in the same way as in (i) to get
$$\RHS{eq34}\es(\Lambda_1(x)^{-\f SR}-\Lambda_1(1)^{-\f SR})^{-\f1S}\geq\LHS{eq34}$$
for all $x\in(0,1)$. Moreover, since the function $t\mapsto \Lambda_1(t)^{-\f SR}$ is strictly decreasing on $(0,1)$, it follows that
$$(\Lambda_1(x)^{-\f SR}-\Lambda_1(1)^{-\f SR})^{-\f1S}\leq (c(\delta)\Lambda_1(x)^{-\f SR})^{-\f1S}\es\LHS{eq34}$$
for all $x\in(0,\delta)$, where $c(\delta)=1-\Lambda_1(\delta)^{\f SR}\Lambda_1(1)^{-\f SR}>0$.
\textit{Case} (iii) can be proven analogously as case (ii).
\end{proof}
\subsection*{The Lorentz-Karamata spaces}
\begin{definition}
Let $0<p,r\leq\infty$, $a\in\SV(A,B)$ and put
$$\norm{f}_{p,r;a;(A,B)}:=\norm{t^{\f1p-\f1r}a(t)f^*(t)}_{r,(A,B)},\quad f\in\Meas(R,\mu).$$
Let $B=1$ if $\mu(R)=1$ and $B=\infty$ if $\mu(R)=\infty$. Then, the Lorentz-Karamata (LK) space $L_{p,r;a}(R,\mu)\equiv L_{p,r;a}$ is defined as the set of all functions $f\in\Meas(R,\mu)$ such that $\norm{f}_{p,r;a;(0,B)}<\infty$.
\end{definition}
Using the monotonicity of $f^*$ and Lemma~\ref{LSV}~(i), one can observe that $L_{p,r;a}$ is the trivial space if and only if $p=\infty$ and $\norm{t^{-\f1r}a(t)}_{r,(0,1)}=\infty$. Moreover, if we set $d(t)=\norm{a}_{\infty,(0,t)}$, $t\in(0,B)$, and if the space $L_{\infty,\infty;a}$ is non-trivial, then $d\in\SV(0,B)$ (see Lemma~\ref{LSV}~(ii)) and
$$\norm{af^*}_{\infty,(0,B)}
\ls\norm{\norm{a}_{\infty,(0,t)}f^*(t)}_{\infty,(0,B)}
\leq\norm{\norm{af^*}_{\infty,(0,t)}}_{\infty,(0,B)}
=\norm{af^*}_{\infty,(0,B)}.$$
Thus, $L_{\infty,\infty;a}=L_{\infty,\infty;d}$ and, consequently, in the case $p=\infty$ it is natural to assume that
\begin{equation}\label{210}
\text{if}\quad r=\infty,\quad\text{then}\quad a\in\Upa{0,B}.
\end{equation}
LK spaces contain many of familiar spaces as particular cases. For example, let $\el(t)=1+\abs{\log t}$, $t\in(0,\infty)$, and $\ell_i=\el(\ell_{i-1})$ for all $i\in\set{2,3,\ldots}$ and set $\mathcal{L}=\prod_{i=1}^{n}\ell_i^{\alpha_i}$, where $\alpha_i\in\R$, $n\in\N$.
Then $\mathcal{L}\in\SV(0,\infty)$ and $L_{p,r;\mathcal{L}}$ is the~generalized Lorentz-Zygmund (GLZ) space with the $n$-th tier of logarithm. In particular, if $\alpha,\beta,\gamma\in\R$, then $L_{p,r;\alpha,\beta,\gamma}:=L_{p,r;\el^{\alpha}\eld^{\beta}\elt^{\gamma}}$ and $L_{p,r;\alpha,\beta}:=L_{p,r;\el^{\alpha}\eld^{\beta}}$ are the GLZ spaces of Edmunds, Gurka and Opic (cf. \cite{EGO}) and $L_{p,r;\el^{\alpha}}$ is the Lorentz-Zygmund space of Bennett and Rudnick (cf. \cite{BS}). The LK spaces also cover the (generalized) Lorentz-Zygmund spaces $L_{p,r;\mathbb{A}}$, $\mathbb{A}=(\alpha_0,\alpha_{\infty})\in\R^2$, with ``broken-logartmic'' function, which were introduced in \cite{EOP2}. Furthermore, if $\mathfrak{1}=\chi_{(0,\infty)}$, then $L_{p,r}:=L_{p,r;\mathfrak{1}}$ is the Lorentz space. Moreover, the space $L_p(\log{L})^{\alpha}:=L_{p,p;\el^{\alpha}}$ is the Zygmund space, and $L_p:=L_{p,p}$ is the~Lebesgue space (original definitions and properties of these classical spaces can be found also in \cite{BS}). In the literature also spaces, which are close to $L_{\infty}$, such as $L_{\exp}^{\alpha}$, appear. These spaces are covered by the LK spaces as well ($L^{\alpha}_{\exp}=L_{\infty,\infty;\el^{-\alpha}}$). Since the special spaces mentioned above were introduced by different authors at various times, there is slight inconsistency in their definitions (many functionals can be used to define the same space). This is resolved by \cite[Lemma~2.2.]{EOP} (the definitions of the GLZ space from \cite{EOP} and of the LK space given here are consistent).
The choice of slowly varying function $a$ is, of course, not restricted to composite logarithmic functions as $\mathcal{L}$. For complete information on how various examples of slowly varying functions can be constructed, see \mbox{\cite[Section 1.3, p. 12]{Reg}}. Note that a~general slowly varying function can also exhibit oscillations of infinite amplitude at zero. An example of such slowly varying function is $a(x)=\exp\left(\el(x)^{\f13}\cos(\el(x)^{\f13})\right)$, $x\in(0,\infty)$, which is taken from \cite[Exercise 1.11.3, p. 58]{Reg}.
Similarly as in \cite{BR}, we shall also consider sums and intersections of the LK spaces.
\begin{definition}\label{DSum}
Let $p_1,p_2,r_1,r_2\in[1,\infty]$, $p_1\neq p_2$, $a\in\SV(0,\infty)$, $f\in\Meas(R,\mu)$. Then, we define
$$\norm{f}_{(p_1,r_1)+(p_2,r_2);a}:=\left\{
\begin{array}{cc}
\norm{f}_{p_1,r_1;a;(0,1)}+\norm{f}_{p_2,r_2;a;(1,\infty)} & \text{if } p_1<p_2\\
\norm{f}_{p_2,r_2;a;(0,1)}+\norm{f}_{p_1,r_1;a;(1,\infty)} & \text{if } p_1>p_2
\end{array}\right.$$
and
$$\norm{f}_{(p_1,r_1)\cap(p_2,r_2);a}:=\norm{f}_{(p_2,r_2)+(p_1,r_1);a}.$$
The spaces $L_{p_1,r_1;a}+L_{p_2,r_2;a}$ and $L_{p_1,r_1;a}\cap L_{p_2,r_2;a}$ consist of all functions $f\in\Meas(R,\mu)$, such that $\norm{f}_{(p_1,r_1)+(p_2,r_2);a}<\infty$ and $\norm{f}_{(p_1,r_1)\cap(p_2,r_2);a}<\infty$, respectively.
\end{definition}
Obviously, this definition enables us to control the behaviour of $f^*$ near $0$ and $\infty$ independently. Also, it agrees with the usual definition of sum and intersection of spaces (therefore the notation) up to one exception, when one of the spaces of the sum is trivial. This exception allows us to properly define, e.g., the space \mbox{$L(\log L)+L_{\infty,1}$}, which is particularly important for Hilbert transform and which, using the usual definition of the sum, would be trivial (since $L_{\infty,1}=\set{0}$). For detailed explanation of this problematic, see \cite{BR}.
\section{The statement of the main results}\label{S3}
If not stated otherwise, we shall assume in this section that $1\leq p_1<p_2\leq\infty$, $1\leq q_1,q_2\leq\infty$, $q_1\neq q_2$, $1\leq r,s,r_1,s_1,r_2,s_2\leq\infty$ and $a,b\in\SV(A,B)$. If $r>s$, then the number $\rh$ is defined by
\begin{equation*}
\f1{\rh}=\f1s-\f1r.
\end{equation*}
To formulate our main results conveniently, we introduce the following quantities. Set
$$N(r,s,a,b;A,B)=\left\{\begin{array}{cc}
\sup\limits_{A<x<B}b(x)a(x)^{-1} & \text{if }r\leq s\\
\norm{x^{-\f1{\rh}}b(x)a(x)^{-1}}_{\rh,(A,B)} & \text{if }r>s
\end{array}\right.,$$
$$L(r,s,a,b;A,B)=\left\{\begin{array}{cc}
\sup\limits_{A<x<B}\norm{t^{-\f1s}b(t)}_{s,(x,B)}\norm{t^{-\f1{r'}}a(t)^{-1}}_{r',(A,x)} & \text{if }r\leq s\\
\norm{x^{-\f1{\rh}}a(x)^{-\f{r'}{\rh}}\norm{t^{-\f1s}b(t)}_{s,(x,B)}\norm{t^{-\f1{r'}}a(t)^{-1}}_{r',(A,x)}^{\f{r'}{s'}}}_{\rh,(A,B)} & \text{if }r>s
\end{array}\right.,$$
and
$$R(r,s,a,b;A,B)=\left\{\begin{array}{cc}
\sup\limits_{A<x<B}\norm{t^{-\f1s}b(t)}_{s,(A,x)}\norm{t^{-\f1{r'}}a(t)^{-1}}_{r',(x,B)} & \text{if }r\leq s\\
\norm{x^{-\f1{\rh}}a(x)^{-\f{r'}{\rh}}\norm{t^{-\f1s}b(t)}_{s,(A,x)}\norm{t^{-\f1{r'}}a(t)^{-1}}_{r',(x,B)}^{\f{r'}{s'}}}_{\rh,(A,B)} & \text{if }r>s
\end{array}\right..$$
Furthermore, we put
\begin{align*}
R_1(r,s&,a,b;A,B)\!=\!\left\{\begin{array}{cc}
\sup\limits_{A<x<B}\norm{t^{-\f1s}b(t)\log\tf xt}_{s,(A,x)}\norm{t^{-\f1{r}}a(t)}^{-1}_{r,(A,x)} & \text{if }r\leq s\\
\norm{x^{-\f1{\rh}}a(x)^{\f r{\rh}}\norm{t^{-\f1s}b(t)\log \tf xt}_{s,(A,x)}\norm{t^{-\f1{r}}a(t)}_{r,(A,x)}^{-\f rs}}_{\rh,(A,B)} & \text{if }r>s
\end{array}\right.,\\
R_2(r,s&,a,b;A,B)\\
&=\left\{\begin{array}{cc}
\sup\limits_{A<x<B}\norm{t^{-\f1s}b(t)}_{s,(A,x)}\norm{t^{-\f1{r'}}a(t)^{\f r{r'}}V(t)^{-1}\log\tf tx}_{r',(x,B)} & \text{if }r\leq s\\
\norm{x^{-\f1{\rh}}b(x)^{\f{s}{\rh}}\norm{t^{-\f1s}b(t)}_{s,(A,x)}^{\f sr}\norm{t^{-\f1{r'}}a(t)^{\f r{r'}}V(t)^{-1}\log\tf tx}_{r',(x,B)}}_{\rh,(A,B)} & \text{if }r>s
\end{array}\right.,
\end{align*}
where $V(t)=\intt{A}{t}{u^{-1}a(u)^r}{u}$, $t\in(A,B)$, and
$$R_3(r,s,a,b;A,B)=\norm{x^{-\f1s}b(t)\intt{x}{B}{t^{-1}\norm{a}_{\infty,(A,t)}^{-1}}{t}}_{s,(A,B)}\quad\text{if }r=\infty.$$
Finally, let
$$R_{\infty}(r,s,a,b;A,B)=\left\{\begin{array}{cl}
R_1(r,s,a,b;A,B)+R_2(r,s,a,b;A,B) & \text{if } 1<r,s<\infty\\
R_1(r,s,a,b;A,B) & \text{if }r=1 \text{ or } 1=s<r<\infty\\
R_2(r,s,a,b;A,B) & \text{if }1<r<s=\infty\\
R_3(r,s,a,b;A,B) & \text{if }r=\infty.
\end{array}\right.$$
Whenever the context is clear, we shall write just $N$ instead of $N(r,s,a,b;A,B)$ and similarly for all the other quantities above.
Now we are almost ready to formulate our interpolation theorems. We recall that we work with the operators acting between (subspaces of) $\Meas(R_1,\mu_1)$ and $\Meas(R_2,\mu_2)$. We shall suppose that $\mu_1(R_1)=\mu_2(R_2)=\infty$; for the finite measure spaces see Remark~\ref{Rem}~(i) below. If $b\in\SV(A,B)$, we put
$$b_*(t)=b(t^{\f1m}),\quad t\in(A,B),$$
where $m$ denotes the slope of the interpolation segment $\sigma=[(\f1{p_1},\f1{q_1});(\f1{p_2},\f1{q_2})]$, associated with the Calder\'{o}n operator $S_{\sigma}$.
The following theorem is a~generalization of the classical Marcinkiewicz interpolation theorem (cf. \cite[Chapter 4, Theorem 4.13]{BS}) to the LK spaces.
\begin{theorem}\label{Int}
Let $T\in\JW(p_1,q_1;p_2,q_2)\cap(\LBl(p_1,q_1;m)\cup\LBr(p_2,q_2;m))$. Suppose that $\theta\in(0,1)$ and $p,q$ satisfy
$$\f1p=\f{1-\theta}{p_1}+\f{\theta}{p_2},\quad\f1q=\f{1-\theta}{q_1}+\f{\theta}{q_2}.$$
Then
\begin{equation*}
T:L_{p,r;a}\lra L_{q,s;b}
\end{equation*}
if and only if
$$N(r,s,a,b_*;0,\infty)<\infty.$$
\end{theorem}
The parameter $\theta$ from the previous theorem was restricted to $(0,1)$, therefore we refer to this case as to the non-limiting case. The next theorem describes the limiting case $\theta=0$.
\begin{theorem}\label{IntL}
Let $T\in\JW(p_1,q_1;p_2,q_2)\cap\LBl(p_1,q_1;m)$. Then
\begin{equation*}
T:L_{p_1,r;a}\lra L_{q_1,s;b}
\end{equation*}
if and only if
$$L(r,s,a,b_*;0,\infty)<\infty.$$
\end{theorem}
The following theorem describes the limiting case $\theta=1$ and is completely analogical to the previous theorem as long as $p_2<\infty$.
\begin{theorem}\label{IntR}
Let $T\in\JW(p_1,q_1;p_2,q_2)\cap\LBr(p_2,q_2;m)$, $p_2<\infty$. Then
\begin{equation*}
T:L_{p_2,r;a}\lra L_{q_2,s;b}
\end{equation*}
if and only if
$$R(r,s,a,b_*;0,\infty)<\infty.$$
\end{theorem}
When $p_2=\infty$, the situation turns out to be more delicate.
\begin{theorem}\label{IntRR}
Let $T\in\JW(p_1,q_1;\infty,q_2)\cap\LBr(\infty,q_2;m)$ and suppose that \eqref{210} is satisfied. Then
$$T:L_{\infty,r;a}\lra L_{q_2,s;b}$$
if and only if
\begin{equation}\label{1234}
\norm{t^{-\f1r}a(t)}_{r,(0,\infty)}=\infty
\end{equation}
and
$$R_{\infty}(r,s,a,b_*;0,\infty)<\infty.$$
\end{theorem}
Next, we state results concerning the sums and intersections of LK spaces. We concentrate on the limiting cases only as the situation in the non-limiting case is obvious.
\begin{theorem}\label{IntLim}
Let $T\in\JW(p_1,q_1;p_2,q_2)\cap\LBl(p_1,q_1;m)\cap\LBr(p_2,q_2;m)$ with $p_2<\infty$.
\begin{itemize}
\item[{\rm(i)}] Then
\begin{equation*}
T:L_{p_1,r_1;a}+ L_{p_2,r_2;a}\lra L_{q_1,s_1;b}+L_{q_2,s_2;b}
\end{equation*}
if and only if
$$L(r_1,s_1,a,b_*;0,1)+R(r_2,s_2,a,b_*;1,\infty)<\infty.$$
\item[{\rm(ii)}] Then
\begin{equation*}
T:L_{p_1,r_1;a}\cap L_{p_2,r_2;a}\lra L_{q_1,s_1;b}\cap L_{q_2,s_2;b}
\end{equation*}
if and only if
$$L(r_1,s_1,a,b_*;1,\infty)+R(r_2,s_2,a,b_*;0,1)<\infty.$$
\end{itemize}
\end{theorem}
\begin{theorem}\label{IntHil}
Let $T\in\JW(p_1,q_1;\infty,q_2)\cap\LBl(p_1,q_1;m)\cap\LBr(\infty,q_2;m)$ and suppose that \eqref{210} is satisfied.
\begin{itemize}
\item[{\rm(i)}] Then
\begin{equation*}
T:L_{p_1,r_1;a}+ L_{\infty,r_2;a}\lra L_{q_1,s_1;b}+L_{q_2,s_2;b}
\end{equation*}
if and only if
$$\norm{t^{-\f1{r_2}}a(t)}_{r_2,(1,\infty)}=\infty$$
and
$$L(r_1,s_1,a,b_*;0,1)+R_{\infty}(r_2,s_2,a,b_*;1,\infty)<\infty.$$
\item[{\rm(ii)}] Then
\begin{equation*}
T:L_{p_1,r_1;a}\cap L_{\infty,r_2;a}\lra L_{q_1,s_1;b}\cap L_{q_2,s_2;b}
\end{equation*}
if and only if
$$L(r_1,s_1,a,b_*;1,\infty)+R_{\infty}(r_2,s_2,a,b_*;0,1)<\infty.$$
\end{itemize}
\end{theorem}
Let us now make some remarks about the theorems above.
\begin{remark}\label{Rem} \hspace{1pt}
(i) When the underlying measure spaces are finite and $q_1<q_2$, then Theorems~\ref{Int}, \ref{IntL}, \ref{IntR} continue to hold, provided we replace the interval $(0,\infty)$ by $(0,1)$. The same is true for Theorem~\ref{IntRR}, provided that condition \eqref{1234} is dropped. This is a consequence of the fact that the Hardy-type inequalities which will be used to prove the mentioned theorems hold on $(0,1)$ and $(0,\infty)$ in the same form (cf. Lemmas~\ref{TLH} -- \ref{LSaw} below).
(ii) It will be apparent from the proofs in Section~\ref{S5} that the existence of the lower bounds for the operator $T$ is only used to prove the necessity of the corresponding conditions. In other words, if we omit the assumptions $T\in\LBl(p_1,q_1;m)$ and $T\in\LBr(p_2,q_2;m)$, the theorems above still provide sufficient conditions for the boundedness of $T$.
(iii) There exist $r,s\in[1,\infty]$ and $a,b\in\SV(A,B)$ such that
$$R_1(r,s,a,b;A,B)+R_2(r,s,a,b;A,B)<\infty\quad\text{and}\quad R(r,s,a,b;A,B)=\infty.$$
Indeed, let $1<r<s<\infty$ and
$$a(t)=\el(t)^{-\f1r}\eld(t)^{\theta}\quad\text{and}\quad b(t)=\el(t)^{-1-\f1s}\eld(t)^{\theta+\gamma},\quad t\in(0,1),$$
where $\theta<-\f1r-\f1s$ and $0<\gamma<\f1r-\f1s$ (we prove the statement for $(A,B)=(0,1)$; other cases are similar). Then, using the substitution $\tau=\eld(t)$, we get
\begin{align*}
\norm{t^{-\f1s}b_*(t)\log\tf xt}_{s,(0,x)}&\norm{t^{-\f1r}a(t)}_{r,(0,x)}^{-1}\\
&\ls\norm{t^{-\f1s}\el(t)^{-\f1s}\eld(t)^{\theta+\gamma}}_{s,(0,x)}\norm{t^{-\f1r}\el(t)^{-\f1r}\eld(t)^{\theta}}_{r,(0,x)}^{-1}\\
&=\left(\intt{\eld(x)}{\infty}{\tau^{\theta s+\gamma s}}{\tau}\right)^{\f1s}\left(\intt{\eld(x)}{\infty}{\tau^{\theta r}}{\tau}\right)^{-\f1r}\\
&\es\eld(x)^{\theta+\gamma+\f1s}\eld(x)^{-\theta-\f1r}
=\eld(x)^{\gamma+\f1s-\f1r}\ls1
\end{align*}
for all $x\in(0,1)$ and thus $R_1(r,s,a,b;0,1)<\infty$. Now observe that in our case
$$A(t)=\intt{0}{t}{u^{-1}a(u)^r}{u}\es\eld(t)^{\theta r+1}\quad\forall t\in(0,1).$$
Consequently, using the substitutions $u=\el(t)$ and $\tau=\eld(t)$, we obtain
\begin{align}\label{190}
&\norm{t^{-\f1s}b_*(t)}_{s,(0,x)}\norm{t^{-\f1{r'}}a(t)^{\f r{r'}}A(t)^{-1}\log\tf tx}_{r',(x,1)}\nonumber\\
&\qquad\ls\norm{t^{-\f1s}\el(t)^{-1-\f1s}\eld(t)^{\theta+\gamma}}_{s,(0,x)}\norm{t^{-\f1{r'}}\el(t)^{-\f1{r'}}\eld(t)^{\theta\f r{r'}-\theta r-1}}_{r',(x,1)}\el(x)\nonumber\\
&\qquad=\left(\intt{\el(x)}{\infty}{u^{-s-1}\el(u)^{\theta s+\gamma s}}{u}\right)^{\f1s}\left(\intt{1}{\eld(x)}{\tau^{-r'(\theta+1)}}{\tau}\right)^{\f1{r'}}\el(x)\nonumber\\
&\qquad\es\el(x)^{-1}\eld(x)^{\theta+\gamma}\eld(x)^{-\theta-1+\f1{r'}}\el(x)=\eld(x)^{\gamma-\f1r}\ls1
\end{align}
for all $x\in(0,1)$ and thus, $R_2(r,s,a,b;0,1)<\infty$ holds as well. It remains to show that $R(r,s,a,b;0,1)=\infty$. We can see from \eqref{190} that
$$\norm{t^{-\f1s}b_*(t)}_{s,(0,x)}\es\el(x)^{-1}\eld(x)^{\theta+\gamma}\quad\forall x\in(0,1).$$
This, together with
\begin{align*}
\norm{t^{-\f1{r'}}a(t)^{-1}}_{r',(x,1)}
&=\norm{t^{-\f1{r'}}\el(t)^{\f1r}\eld(t)^{-\theta}}_{r',(x,1)}
=\left(\intt{1}{\el(x)}{u^{r'-1}\el(u)^{-\theta r'}}{u}\right)^{\f1{r'}}\\
&\es\el(x)\eld(x)^{-\theta}
\end{align*}
for all $x\in(0,1)$, gives
$$\norm{t^{-\f1s}b_*(t)}_{s,(0,x)}\norm{t^{-\f1{r'}}a(t)^{-1}}_{r',(x,1)}\es\eld(x)^{\gamma},$$
which tends to infinity as $x\to0_+$. When $r=1$ or $s=\infty$, the given example (with the usual modifications) works as well.
\end{remark}
\section{Weighted inequalities for integral operators}\label{S4}
We will show in Section~\ref{S5} that the boundedness of $T$ is fully determined by the validity of certain Hardy-type inequalities that are restricted to non-increasing functions. The aim of this section is to characterize weights for which these inequalities hold. By weights, we mean functions from $\Meas^+(A,B)$ that are positive and finite almost everywhere on $(A,B)$. We shall denote the set of all weights by~$\Wg(A,B)$.
First of all, we are going to state Lemmas~\ref{TLH}, \ref{LRed}, \ref{LRedR} and \ref{LSaw}, which form an essential part of the paper (they are applied to prove the main results). After that, we review some general criteria and use them to prove the lemmas. The following assertion will be used to prove Theorem~\ref{Int} (the non-limiting case).
\begin{lemma}\label{TLH}
Let $r,s\in[1,\infty]$, $a,b\in\SV(A,B)$. Suppose $\kappa\in\R$ and $\nu>\mu>0$. Then the following five conditions are equivalent:
\begin{itemize}
\item[{\rm(i)}] $N(r,s,a,b;A,B)<\infty;$
\item[{\rm(ii)}]
$\displaystyle\norm{t^{-\mu-\f1s}b(t)\intt{A}{t}{u^{\kappa-1}g(u)}{u}}_{s,(A,B)}
\!\!\!\!\!\ls\norm{t^{-\mu+\kappa-\f1r}a(t)g(t)}_{r,(A,B)}\quad\forall g\in\Meas^+(A,B);$
\item[{\rm(iii)}]
$\displaystyle\norm{t^{-\mu-\f1s}b(t)\intt{A}{t}{u^{\nu-1}f(u)}{u}}_{s,(A,B)}
\!\!\!\!\!\ls\norm{t^{-\mu+\nu-\f1r}a(t)f(t)}_{r,(A,B)}\;\forall f\!\in\!\Dupa{A,B};$
\item[{\rm(iv)}]
$\displaystyle\norm{t^{\mu-\f1s}b(t)\intt{t}{B}{u^{\kappa-1}g(u)}{u}}_{s,(A,B)}
\!\!\!\ls\norm{t^{\mu+\kappa-\f1r}a(t)g(t)}_{r,(A,B)}\quad\forall g\in\Pos{A,B};$
\item[{\rm(v)}]
$\displaystyle\norm{t^{\mu-\f1s}b(t)\intt{t}{B}{u^{\nu-1}f(u)}{u}}_{s,(A,B)}
\!\!\!\ls\norm{t^{\mu+\nu-\f1r}a(t)f(t)}_{r,(A,B)}\quad\forall f\!\in\!\Dupa{A,B}.$
\end{itemize}
\end{lemma}
The parameter $\kappa$ can be, of course, eliminated by a suitable substitution; we keep it there just to emphasise that the inequalities above share the same structure. The following two lemmas describe the case, where $\mu$ from Lemma~\ref{TLH} is zero. They will be used to prove Theorems~\ref{IntL}, \ref{IntR} and \ref{IntLim} (limiting cases $\theta=0$, $\theta=1$ with $p_2<\infty$).
\begin{lemma}\label{LRed}
Let $r,s\in[1,\infty]$, $a,b\in\SV(A,B)$, $\kappa\in\R$ and $\nu>0$. Then the following three conditions are equivalent:
\begin{itemize}
\item[\rm{(i)}] $L(r,s,a,b;A,B)<\infty;$
\item[\rm{(ii)}] $\displaystyle\norm{t^{-\f1s}b(t)\intt{A}{t}{u^{\kappa-1}g(u)}{u}}_{s,(A,B)}
\ls\norm{t^{\kappa-\f1r}a(t)
g(t)}_{r,(A,B)}\quad\forall g\in\Pos{A,B};$
\item[\rm{(iii)}]
$\displaystyle\norm{t^{-\f1s}b(t)\intt{A}{t}{u^{\nu-1}f(u)}{u}}_{s,(A,B)}
\ls\norm{t^{\nu-\f1r}a(t)f(t)}_{r,(A,B)}\quad\forall f\in\Dupa{A,B}.$
\end{itemize}
\end{lemma}
\begin{lemma}\label{LRedR}
Let $r,s\in[1,\infty]$, $a,b\in\SV(A,B)$, $\kappa\in\R$ and $\nu>0$. Then the following three conditions are equivalent:
\begin{itemize}
\item[\rm{(i)}] $R(r,s,a,b;A,B)<\infty;$
\item[\rm{(ii)}] $\displaystyle\norm{t^{-\f1s}b(t)\intt{t}{B}{u^{\kappa-1}g(u)}{u}}_{s,(A,B)}
\ls\norm{t^{\kappa-\f1r}a(t)
g(t)}_{r,(A,B)}\quad\forall g\in\Pos{A,B};$
\item[\rm{(iii)}] $\displaystyle\norm{t^{-\f1s}b(t)\intt{t}{B}{u^{\nu-1}f(u)}{u}}_{s,(A,B)}
\ls\norm{t^{\nu-\f1r}a(t)
f(t)}_{r,(A,B)}\quad\forall f\in\Dupa{A,B};$
\end{itemize}
\end{lemma}
Finally, in the following lemma we consider the remaining and most interesting case, that occurs when $\mu=\nu=0$ and the inequality is restricted to non-increasing functions. It will be used to prove Theorems~\ref{IntRR} and \ref{IntHil} (limiting case $\theta=1$ with $p_2=\infty$).
\begin{lemma}\label{LSaw}
Let $r,s\in[1,\infty]$ and $a,b\in\SV(A,B)$. Then
\begin{equation}\label{36}
\norm{t^{-\f1s}b(t)\intt{t}{B}{u^{-1}f(u)}{u}}_{s,(A,B)}
\ls\norm{t^{-\f1r}a(t)f(t)}_{r,(A,B)}
\end{equation}
holds for every $f\in\Dupa{A,B}$ if and only if
$$R_{\infty}(r,s,a,b;A,B)<\infty$$
and
\begin{equation}\label{A}
\norm{t^{-\f1r}a(t)}_{r,(A,B)}=\infty\quad\text{when}\quad B=\infty.
\end{equation}
\end{lemma}
Note that, due to the monotonicity of $f$, the analogy of inequality \eqref{36} for $\int_A^t$ is non-trivial only if $(A,B)=(1,\infty)$ and then it can be converted to an inequality of the same form as \eqref{36} on $(0,1)$, but restricted to non-decreasing functions. Since we will have no use for such an inequality and since the resulting characterization is not as interesting, we shall omit it.
To prove the first three of the four lemmas above, we will use the following well known characterization of weighted Hardy inequalities, for which we refer to \cite[Theorems~5.9, 5.10, 6.2, 6.3, Remark~5.5]{OK} or to \cite{Sin}.
\begin{theorem}\label{TH}
Let $v,w\in \Wg(A,B)$, $r,s\in[1,\infty]$ and let $\f1{\rh}=\f1s-\f1r$.
\begin{itemize}
\item[{\rm(i)}] Then
$$\norm{w(t)\int_{A}^{t}g}_{s,(A,B)}\ls\norm{v\,g}_{r,(A,B)}\quad\forall g\in\Pos{A,B}$$
if and only if
\begin{align*}
\text{either}\quad r\leq s\quad\text{and}\quad& \sup_{A<x<B}\norm{w}_{s,(x,B)}\norm{v^{-1}}_{r',(A,x)}<\infty,\\
\text{or}\quad r>s\quad\text{and}\quad&\norm{\norm{w}_{s,(x,B)}\norm{v^{-1}}_{r',(A,x)}^{\f{r'}{s'}}v(x)^{-\f{r'}{\rh}}}_{\rh,(A,B)}<\infty.
\end{align*}
\item[{\rm(ii)}] Then
$$\norm{w(t)\int_{t}^{B}g}_{s,(A,B)}\ls\norm{v\,g}_{r,(A,B)}\quad\forall g\in\Pos{A,B}$$
if and only if
\begin{align*}
\text{either}\quad r\leq s\quad\text{and}\quad& \sup_{A<x<B}\norm{w}_{s,(A,x)}\norm{v^{-1}}_{r',(x,B)}<\infty,\\
\text{or}\quad r>s\quad\text{and}\quad&\norm{\norm{w}_{s,(A,x)}\norm{v^{-1}}_{r',(x,B)}^{\f{r'}{s'}}v(x)^{-\f{r'}{\rh}}}_{\rh,(A,B)}<\infty.
\end{align*}
\end{itemize}
\end{theorem}
\begin{proof}[Proof of Lemma~\ref{TLH}.] \textit{Equivalence of} (i) \textit{and} (ii) follows from Theorem \ref{TH} with $v(t)=t^{-\mu+\f1{r'}}a(t)$ and $w(t)=t^{-\mu-\f1s}b(t)$, $\mu>0$, $t\in(A,B)$. Indeed, using Lemma~\ref{LSV} (i), we obtain this way that
\begin{equation}\label{112}
\norm{t^{-\mu-\f1s}b(t)\int_A^th}_{s,(A,B)}\ls\norm{t^{-\mu+\f1{r'}}a(t)h(t)}_{r,(A,B)}\quad\forall h\in\Pos{A,B}
\end{equation}
if and only if $N<\infty$. Condition (i) follows from \eqref{112} on substituting $h(u)=u^{\kappa-1}g(u)$, $u\in(A,B)$, where $\kappa\in\R$ and \mbox{$g\in\Pos{A,B}$},
\textit{Equivalence of} (i) \textit{and} (iv) can be proved analogously as that of (i) and (ii).
\textit{Implications} $({\rm ii})\Rightarrow({\rm iii})$ \textit{and} $({\rm iv})\Rightarrow({\rm v})$ are trivial.
\textit{Implication} $({\rm iii})\Rightarrow({\rm ii})$. Let $g\in\Pos{A,B}$ and
\begin{equation}\label{133}
f(t)=\int_{t}^{B}g,\quad t\in(A,B).
\end{equation}
Then $f\in\Dupa{A,B}$ and we obtain from (iii) that
\begin{equation}\label{33}
\norm{t^{-\mu-\f1s}b(t)\intt{A}{t}{u^{\nu-1}\left(\int_{u}^{B}g\right)}{u}}_{s,(A,B)}
\ls\norm{t^{-\mu+\nu-\f1r}a(t)\int_{t}^{B}g}_{r,(A,B)}
\end{equation}
for every $g\in\Pos{A,B}$. To estimate the integral on $\LHS{33}$, we use the Fubini's theorem to get
\begin{align}\label{34}
\intt{A}{t}{u^{\nu-1}\intt{u}{B}{g(\tau)}{\tau}}{u}
&\geq\intt{A}{t}{\intt{u}{t}{u^{\nu-1}g(\tau)}{\tau}}{u}
=\intt{A}{t}{\intt{A}{\tau}{u^{\nu-1}g(\tau)}{u}}{\tau}\nonumber\\
&\es\intt{A}{t}{(\tau^{\nu}-A^{\nu})g(\tau)}{\tau}
\end{align}
for all $t\in(A,B)$. If $A\neq0$, i.e. if $(A,B)=(1,\infty)$, then we continue with the estimate as follows:
\begin{align*}
\intt{1}{t}{(\tau^{\nu}-1)g(\tau)}{\tau}
\geq\intt{2}{t}{(\tau^{\nu}-1)g(\tau)}{\tau}
\geq\intt{2}{t}{(\tau^{\nu}-(\tf{\tau}2)^{\nu})g(\tau)}{\tau}
\es\intt{2}{t}{\tau^{\nu}g(\tau)}{\tau}
\end{align*}
for all $t\in(2,\infty)$. This, together with \eqref{34} gives (after simple substitutions)
\begin{equation}\label{1102}
\LHS{33}\gs\norm{t^{-\mu-\f1s}b(t)\intt{A}{t}{u^{\nu}g(u)}{u}}_{s,(A,B)}.
\end{equation}
Now we estimate $\RHS{33}$ from above. We put $\alpha=-\mu+\nu>0$ and $\beta=1$. We are going to apply weighted Hardy inequality (iv) with $\alpha,\beta$ instead of $\mu,\kappa$, respectively, and with $s=r$ and $b=a$, so that $N(r,r,a,a;A,B)<\infty$. Thus, by the equivalence of (i) and (iv), which we have already proved, we get
$$\norm{t^{-\mu+\nu-\f1r}a(t)\int_t^Bg}_{r,(A,B)}
\ls\norm{t^{-\mu+\nu+\f1{r'}}a(t)g(t)}_{r,(A,B)}\quad\forall g\in\Pos{A,B}.$$
This, \eqref{33} and \eqref{1102} give
$$\norm{t^{-\mu-\f1s}b(t)\intt{A}{t}{u^{\nu}g(u)}{u}}_{s,(A,B)}
\ls\norm{t^{-\mu+\nu+\f1{r'}}a(t)g(t)}_{r,(A,B)}\quad\forall g\in\Pos{A,B},$$
which can be rewritten (using the substitution $g(u)=u^{\kappa-\nu-1}h(u)$, $u\in(A,B)$) as (i).
\textit{Implication} $({\rm v})\Rightarrow({\rm iv})$ can be proved similarly as implication ${\rm(iii)}\Rightarrow{\rm(ii)}$. Indeed, using test function \eqref{133} in (iv), we arrive at
\begin{equation}\label{116}
\norm{t^{\mu-\f1s}b(t)\intt{t}{B}{u^{\nu-1}\left(\int_{u}^{B}g\right)}{u}}_{s,(A,B)}
\ls\norm{t^{\mu+\nu-\f1r}a(t)\int_{t}^{B}g}_{r,(A,B)}
\end{equation}
for every $g\in\Pos{A,B}$. The estimate of \LHS{116}, corresponding to \eqref{34}, now takes the form
\begin{align*}
\intt{t}{B}{u^{\nu-1}\intt{u}{B}{g(\tau)}{\tau}}{u}
&=\intt{t}{B}{\intt{t}{\tau}{u^{\nu-1}g(\tau)}{u}}{\tau}
\es\intt{t}{B}{(\tau^{\nu}-t^{\nu})g(\tau)}{\tau}\nonumber\\
&\geq\intt{2t}{B}{(\tau^{\nu}-t^{\nu})g(\tau)}{\tau}
\geq\intt{2t}{B}{(\tau^{\nu}-(\tf{\tau}2)^{\nu})g(\tau)}{\tau}
\es\intt{2t}{B}{\tau^{\nu}g(\tau)}{\tau}
\end{align*}
for all $t\in(A,\tf B2)$. The rest of the proof is analogous to the proof of implication \mbox{${\rm(iii)}\Rightarrow{\rm(ii)}$}.
\end{proof}
\begin{proof}[Proofs of Lemmas~\ref{LRed} and \ref{LRedR}.]
One can repeat the proof of Lemma~\ref{TLH} (the equivalence of (i), (ii), (iii), or (i), (iv), (v), respectively) with $\mu=0$.
\end{proof}
The proof of Lemma~\ref{LSaw} is the most difficult and it will require different approach than the proof of Lemmas~\ref{TLH}, \ref{LRed}, \ref{LRedR}. The problem is that the characterizing conditions for inequality \eqref{36} restricted to non-increasing functions can be actually weaker than the characterizing conditions for the same inequality considered for all non-negative functions (cf. Lemma~\ref{LRedR}~(ii) with $\kappa=0$ and Remark~\ref{Rem}~(iii)). In other words, the restriction of \eqref{36} to non-increasing functions has a significant effect on its characterizing conditions (cf. \cite[p.129]{EOP} and \cite[Remarks~10.5. and 10.8.]{EOP}). This in turn means that one cannot prove the sufficiency of those weaker conditions using Theorem~\ref{TH} and thus, more suitable results are needed - we are going to use the reduction theorem.
Probably the first result of this kind appeared in \cite{Saw}. Sawyer's result can be very well used in our situation; we will, however, use another result by A. Gogatishvili and V. D. Stepanov, which is more recent (and easier to prove). We are going to state it here for an integral operator with general kernel given by
\begin{equation}\label{eq65}
Sg(t)=\intt{A}{B}{k(t,u)g(u)}{u},\quad g\in\Pos{A,B},\quad t\in(A,B),
\end{equation}
where $k$ is non-negative measurable function on $(A,B)\times(A,B)$.
\begin{theorem}\label{AGRed}
Let $1\leq r<\infty$, $0<s\leq\infty$, $v,w\in\Wg(A,B)$ and $V(t)=\int_A^tv$, $t\in(A,B)$. Let $S$ be the integral operator \eqref{eq65} with the kernel $k$. Set $$K(t,u)=\intt{A}{u}{k(t,\tau)}{\tau}\quad\text{and}\quad\con{S}f(t)=\intt{A}{B}{K(t,u)f(u)}{u},\quad t,u\in(A,B).$$
Then
$$\norm{w\,Sf}_{s,(A,B)}\ls\norm{v\,f}_{r,(A,B)}\quad\forall f\in\Dupa{A,B}$$
if and only if
$$\norm{w\,K(\cdot,B)}_{s,(A,B)}\ls\norm{v}_{r,(A,B)}$$
and
$$\norm{w\,\con{S}g}_{s,(A,B)}\ls\norm{v^{1-r}\,Vg}_{r,(A,B)}\quad\forall g\in\Pos{A,B}.$$
\end{theorem}
\begin{proof}
The theorem is an easy consequence of \cite[Theorem~2.1]{AG}.
\end{proof}
We shall also need a characterization of the boundedness of Volterra integral operators defined by
\begin{equation}\label{eq68}
Vg(t)=\intt{0}{t}{k(t,u)g(u)}{u},\quad g\in\Pos{0,\infty},\quad t\in(0,\infty),
\end{equation}
where the kernel $k$ satisfies:
\begin{itemize}
\item[(i)] the function $(t,u)\mapsto k(t,u)$ is non-decreasing in $t$ or non-increasing in $u$;
\item[(ii)] $k(t,u)\geq0$ for all $t>u>0$;
\item[(iii)] $k(t,\tau)\es k(t,u)+k(u,\tau)$ for all $t>u>\tau>0$.
\end{itemize}
\begin{theorem}\label{TVolt}
Let the $V$ be Volterra integral operator \eqref{eq68} with kernel $k$ satisfying {\rm(i)}, {\rm(ii)}, {\rm(iii)}. Suppose $v,w\in\Wg(0,B)$, where $B=1$, or $B=\infty$.
Then
\begin{equation}\label{1101}
\norm{w\,Vg}_{s,(0,B)}\ls\norm{v\,g}_{r,(0,B)}\quad\forall g\in\Pos{0,B}
\end{equation}
if and only if one of the following conditions hold:
\begin{itemize}
\item[{\rm(i)}] $1<r\leq s<\infty$,
$$\sup_{0<x<B}\norm{w\,k(\cdot,x)}_{s,(x,B)}\norm{v^{-1}}_{r',(0,x)}<\infty,$$
$$\sup_{0<x<B}\norm{w}_{s,(x,B)}\norm{v^{-1}\,k(x,\cdot)}_{r',(0,x)}<\infty;$$
\item[{\rm(ii)}] $1<s<r<\infty$,
$$\norm{v(x)^{-\f{r'}{\rh}}\norm{w\,k(\cdot,x)}_{s,(x,B)}\norm{v^{-1}}^{\f{r'}{s'}}_{r',(0,x)}}_{\rh,(0,B)}<\infty,$$
$$\norm{w(x)^{\f s{\rh}}\norm{w}^{\f sr}_{s,(x,B)}\norm{v^{-1}\,k(x,\cdot)}_{r',(0,x)}}_{\rh,(0,B)}<\infty.$$
\end{itemize}
\end{theorem}
\begin{proof}
If $B=\infty$, then the result can be found in \cite[Theorems 1, 2]{Step}.
In the case $B=1$, we can prove the sufficiency of conditions (i), (ii) by using the theorem with $B=\infty$, $w=\chi_{(0,1)}\con{w}$ and by considering \eqref{1101} for every $g\in\Pos{0,\infty}$, such that $g=0$ on $(1,\infty)$. To prove that conditions (i), (ii) are also necessary in this case, use the same test functions as in \cite{Step}.
\end{proof}
Finally, we can start with a proof of Lemma~\ref{LSaw}.
\begin{proof}[Proof of Lemma~\ref{LSaw}.]\mbox{}
\textit{Case $r=\infty$.} To prove the necessity of the condition $R_{\infty}<\infty$, we test \eqref{36} by
$$f(u)=\norm{a}_{\infty,(A,u)}^{-1},\quad u\in(A,B),$$
which is clearly a non-increasing function on $(A,B)$. In this way, we obtain
\begin{align}\label{160}
\norm{t^{-\f1s}b(t)\intt{t}{B}{u^{-1}\norm{a}_{\infty,(A,u)}^{-1}}{u}}_{s,(A,B)}
&\ls\norm{a(t)\norm{a}_{\infty,(A,t)}^{-1}}_{\infty,(A,B)}\nonumber\\
&\ls\norm{\norm{a}_{\infty,(A,t)}\norm{a}_{\infty,(A,t)}^{-1}}_{\infty,(A,B)}=1,
\end{align}
which we wanted to show.
To prove the sufficiency, we use $R_{\infty}<\infty$ (i.e. \eqref{160}) and the monotonicity of $f$ to get
\begin{align*}
\LHS{36}
&=\norm{t^{-\f1s}b(t)\intt{t}{B}{u^{-1}\norm{a}_{\infty,(A,u)}^{-1}\;\norm{a}_{\infty,(A,u)}f(u)}{u}}_{s,(A,B)}\\
&\leq\norm{t^{-\f1s}b(t)\norm{\norm{a}_{\infty,(A,u)}f(u)}_{\infty,(t,B)}\intt{t}{B}{u^{-1}\norm{a}_{\infty,(A,u)}^{-1}}{u}}_{s,(A,B)}\\
&\ls\norm{\norm{a}_{\infty,(A,t)}f(t)}_{\infty,(A,B)}
\leq\norm{\norm{af}_{\infty,(A,t)}}_{\infty,(A,B)}
=\RHS{36},
\end{align*}
hence the case $r=\infty$ is proved.
In the remaining cases, Theorem~\ref{AGRed} with $k(t,u)=\chi_{(t,B)}(u)u^{-1}$, $v(t)=t^{-\f1r}a(t)$ and $w(t)=t^{-\f1s}b(t)$, $t,u\in(A,B)$, yields that \eqref{36} holds for all $f\in\Dupa{A,B}$ if and only if
\begin{equation}\label{1114}
\norm{t^{-\f1s}b(t)\log\tf Bt}_{s,(A,B)}\ls\norm{t^{-\f1r}a(t)}_{r,(A,B)}
\end{equation}
and
\begin{equation}\label{1113}
\norm{t^{-\f1s}b(t)\intt{t}{B}{g(u)\log\tf ut}{u}}_{s,(A,B)}\!\!\!\!\!\!\!\!\ls\norm{t^{\f1{r'}}a(t)^{-\f r{r'}}V(t)g(t)}_{r,(A,B)}\;\forall g\in\Pos{A,B},
\end{equation}
where $V(t)=\intt{A}{t}{u^{-1}a(u)^r}{u}$, $t\in(A,B)$. Condition \eqref{1114} translates as \eqref{A} if $B=\infty$. When $B=1$, then \eqref{1114} means that if $\RHS{1114}$ is finite, then $\LHS{1114}$ is as well. We will now show that this is, in fact, a consequence of $R_{\infty}(r,s,a,b,0,1)<\infty$. Indeed, this is obvious in all cases but $1<r<s=\infty$, i.e. when $R_{\infty}$ is defined only by $R_2$. In this case, using the Lemma~\ref{L} and the assumption $\RHS{1114}<\infty$, we obtain
\begin{align*}
\norm{t^{-\f1{r'}}a(t)^{\f r{r'}}V(t)^{-1}\log\tf tx}_{r',(x,1)}
&\geq\norm{t^{-\f1{r'}}a(t)^{\f r{r'}}V(t)^{-1}}_{r',(\sqrt{x},1)}\log\tf{\sqrt{x}}{x}\\
&\es\norm{t^{-\f1r}a(t)}_{r,(0,\sqrt{x})}^{-1}\log\tf1x
\gs\log\tf1x
\end{align*}
for all $x\in(0,\tf12)$. Thus, if $R_{\infty}(r,\infty,a,b;0,1)<\infty$, then
\begin{align*}
\infty&>R_{\infty}(r,\infty,a,b;0,1)=R_2(r,\infty,a,b;0,1)
\nonumber\\
&\qquad=\sup_{0<x<1}\norm{b}_{\infty,(0,x)}\norm{t^{-\f1{r'}}a(t)^{\f r{r'}}V(t)^{-1}\log\tf tx}_{r',(x,1)}\gs\sup_{0<x<1}\norm{b}_{\infty,(0,x)}\log\tf1x.
\end{align*}
From that we finally get
$$\LHS{1114}=\norm{b(t)\log\tf1t}_{\infty,(0,1)}
\ls\norm{\norm{b}_{\infty,(0,t)}\log\tf1t}_{\infty,(0,1)}<\infty.$$
Now it remains to prove that, under condition \eqref{A}, inequality \eqref{1113} holds if and only if $R_{\infty}(r,s,a,b;A,B)<\infty$.
\textit{Case $1<r,s<\infty$.} Note that, by the duality (or more precisely, by the sharp H\"{o}lder's inequality, cf. \cite[Chapter 1, Theorem 2.5.]{BS}), inequality \eqref{1113} holds if and only if
\begin{equation}\label{1115}
\norm{t^{-\f1{r'}}a(t)^{\f r{r'}}V(t)^{-1}\intt{A}{t}{g(u)\log\tf tu}{u}}_{r',(A,B)}\ls
\norm{t^{\f1s}b(t)^{-1}g(t)}_{s',(A,B)}
\end{equation}
holds for all $g\in\Pos{A,B}$. Now we apply Theorem~\ref{TVolt} with $w(t)=t^{-\f1{r'}}a(t)^{\f r{r'}}V(t)^{-1}$, $v(t)=t^{\f1s}b(t)^{-1}$, $t\in(A,B)$, with $r,s$ replaced by $s',r'$ and we use Lemma~\ref{L} to get that \eqref{1115} holds for every $g\in\Pos{A,B}$ if and only if $R_{\infty}<\infty$.
\textit{Case $r=1$.} We can rewrite \eqref{1113} as
\begin{equation}\label{140}
\norm{\intt{A}{B}{k(t,u)g(u)}{u}}_{s,(A,B)}\ls\norm{g}_{1,(A,B)}\quad\forall g\in\Pos{A,B},
\end{equation}
where
\begin{equation}\label{125}
k(t,u)=t^{-\f1s}b(t)V(u)^{-1}\chi_{(t,B)}(u)\log\tf ut,\quad t,u\in(A,B).
\end{equation}
We claim that \eqref{140} holds if and only if
\begin{equation}\label{138}
\sup_{A<u<B}\norm{k(\cdot,u)}_{s,(A,B)}<\infty.
\end{equation}
Indeed, the general result for arbitrary kernels \cite[Chapter XI, Theorem 4]{Ak} implies that \eqref{140} is equivalent to $\esssup_{A<x<B}\norm{k(\cdot,u)}_{r',(A,B)}<\infty$, which, in our case, is equivalent to \eqref{138}. However, let us also give an explicit proof of this claim, using the properties of $k$.
To prove the sufficiency of \eqref{138}, we take $g,h\in\Meas^+(A,B)$ and write
\begin{align}\label{1010}
\intt{A}{B}{\intt{A}{B}{k(t,u)g(u)}{u}\;h(t)}{t}
&=\intt{A}{B}{\intt{A}{B}{k(t,u)h(t)}{t}\;g(u)}{u}\nonumber\\
&\leq\intt{A}{B}{\norm{k(\cdot,u)}_{s,(A,B)}\norm{h}_{s',(A,B)}g(u)}{u}\nonumber\\
&\ls\esssup_{A<u<B}\norm{k(\cdot,u)}_{s,(A,B)}\norm{h}_{s',(A,B)}\norm{g}_{1,(A,B)}.
\end{align}
Inequality \eqref{140} then follows from \eqref{1010} by taking the supremum over all $h$ with $\norm{h}_{s',(A,B)}\leq1$, using the sharp H\"{o}lder's inequality and \eqref{138}.
Conversely, suppose that \eqref{140} holds. Fix $x\in(A,B)$ and test \eqref{140} with
$$g_{x,n}=n\chi_{(x,x+\f1n)},\quad n\in\N,$$
to get
\begin{equation}\label{1003}
\norm{n\intt{x}{x+\f1n}{k(\cdot,u)}{u}}_{r',(A,B)}\ls1\quad\forall n\in\N.
\end{equation}
We see from \eqref{125} that $k$ is continuous in the second variable in $(A,B)$. Therefore,
$$\lim_{n\to\infty}n\intt{x}{x+\f1n}{k(t,u)}{u}=k(t,x)\quad\forall t\in(A,B)$$
by the fundamental theorem of calculus. Thus, using \eqref{1003} and Fatou's lemma, we obtain
\begin{equation}\label{1005}
1\gs\liminf_{n\to\infty}\,\norm{n\intt{x}{x+\f1n}{k(\cdot,u)}{u}}_{r',(A,B)}
\geq\norm{k(\cdot,x)}_{r',(A,B)}.
\end{equation}
Since the multiplicative constant in \eqref{1005} does not depend on $x$, we get \eqref{138}.
It is easy to see that the condition \eqref{138} with $k$ given by \eqref{125} coincides with $R_{\infty}<\infty$, hence the proof of the case $r=1$ is finished.
\textit{Case $1=s<r<\infty$.} Instead of \eqref{1113}, we will characterize its equivalent dual version \eqref{1115}, which can be rewritten as
\begin{equation}\label{1200}
\norm{t^{-\f1{r'}}a(t)^{\f r{r'}}V(t)^{-1}\intt{A}{t}{u^{-1}b(u)g(u)\log\tf tu}{u}}_{r',(A,B)}\ls\norm{g}_{\infty,(A,B)},
\end{equation}
for all $g\in\Pos{A,B}$. It is now obvious that the condition
$$\norm{t^{-\f1{r'}}a(t)^{\f r{r'}}V(t)^{-1}\intt{A}{t}{u^{-1}b(u)\log\tf tu}{u}}_{r',(A,B)}\ls1$$
is both sufficient and necessary for \eqref{1200} and also that it coincides with $R_{\infty}<\infty$ in this case.
\textit{Case $1<r<s=\infty$.} Now \eqref{1115} can be rewritten as
$$\norm{\intt{A}{B}{k(t,u)g(u)}{u}}_{r',(A,B)}\ls\norm{g}_{1,(A,B)}\quad\forall g\in\Pos{A,B},$$
where
\begin{equation}\label{1201}
k(t,u)=t^{-\f1{r'}}a(t)^{\f r{r'}}V(t)^{-1}b(u)\chi_{(A,t)}(u)\log\tf tu,\quad t,u\in(A,B),
\end{equation}
and we can use the same technique of proof as in the case $r=1$. There is one slight difference that instead of $b$ itself we need to take its continuous representation (from Lemma~\ref{LSV}~(i)). This way, we obtain condition \eqref{138} again, only with $r'$ instead of $s$. To see that this condition coincides with $R_{\infty}<\infty$, we use \eqref{1201} to get
\begin{equation}\label{1202}
\infty>\sup_{A<x<B}\norm{k(\cdot,x)}_{r',(A,B)}
=\sup_{A<x<B}b(x)f(x),
\end{equation}
where $x\mapsto f(x):=\norm{t^{-\f1{r'}}a(t)^{\f r{r'}}V(t)^{-1}\log\tf tx}_{r',(x,B)}$ is a decreasing function in $(A,B)$. Now observe that
$$b(x)f(x)
\ls\norm{b}_{\infty,(A,x)}f(x)\leq\norm{bf}_{\infty,(A,x)}
\leq\norm{bf}_{\infty,(A,B)}\quad\forall x\in(A,B),$$
and hence, using \eqref{1202}, we obtain
$$\infty>\sup_{A<x<B}\norm{k(\cdot,x)}_{r',(A,B)}
\es\norm{bf}_{\infty,(A,B)}
\es\sup_{0<x<\infty}\norm{b}_{\infty,(A,x)}f(x),$$
which is precisely $R_{\infty}<\infty$. This finishes the proof of the case $1<r<s=\infty$ and of the lemma.
\end{proof}
\section{The proofs of the main results}\label{S5}
First we shall prove the following simple lemma.
\begin{lemma}\label{LNer}
Let $r,s\in[1,\infty]$, $a,b\in\SV(A,B)$ and suppose that \eqref{210} is satisfied. Then
$$N(r,s,a,b;A,B)\ls\min\left(L(r,s,a,b;A,B),R(r,s,a,b;A,B),R_{\infty}(r,s,a,b;A,B)\right).$$
\end{lemma}
\begin{proof}
The inequalities $N\ls L$ and $N\ls R$ follow easily from Lemma~\ref{LSV}~(ii). The inequality $N\ls R_{\infty}$ is a consequence of the estimates
\begin{align*}
\norm{t^{-\f1s}b(t)\log\tf xt}_{s,(A,x)}
&\geq\norm{t^{-\f1s}b(t)}_{s,(A,\tf x2)}\log2 &\forall x\in(2A,B),\\
\norm{t^{-\f1{r'}}a(t)^{\f r{r'}}V(t)^{-1}\log\tf tx}_{r',(x,B)}
&\geq\norm{t^{-\f1{r'}}a(t)^{\f r{r'}}V(t)^{-1}}_{r',(2x,B)}\log2 &\forall x\in(A,\tf B2),
\end{align*}
Lemma~\ref{L}, assumption \eqref{210} and of Lemma~\ref{LSV}~(ii).
\end{proof}
\begin{proof}[Proof of Theorem~\ref{IntLim}.]
First of all, we are going to show that the assumptions
\begin{equation}\label{1104a}
T\in\JW(p_1,q_1;p_2,q_2)
\end{equation}
and
\begin{equation}\label{1104b}
T\in\LBl(p_1,r_1;m)\cap\LBr(p_2,q_2;m)
\end{equation}
imply that
\begin{equation}\label{1105}
\norm{(Tg)^*}_B\ls\norm{g^*}_A\quad\forall g\in\Meas(R_1,\mu_1)
\end{equation}
is equivalent to
\begin{equation}\label{1106}
\norm{S_{\sigma}f}_B\ls\norm{f}_A\quad\forall\Dupa{0,\infty},
\end{equation}
where $\norm{\cdot}_A$ and $\norm{\cdot}_B$ are some rearrangement-invariant quasi-norms on $\Pos{0,\infty}$ (that we specify later on). Indeed, using \eqref{1104b} and \eqref{1105}, we get
$$\norm{S_{\sigma}f}_B\ls\norm{(Tg)^*}_B\ls\norm{g^*}_A=\norm{f}_A\quad\forall f\in\Dupa{0,\infty},$$
where $g\in\Meas(R_1,\mu_1)$ is equimeasurable with $f$. To prove the opposite implication, use \eqref{1104a} and \eqref{1106} to obtain
$$\norm{(Tg)^*}_B\ls\norm{S_{\sigma}g^*}_B\ls\norm{g^*}_A\quad\forall g\in\Meas(R_1,\mu_1).$$
Thus, the question of the boundedness of $T$ is reduced to the characterization of \eqref{1106}.
\textit{Case} (i). Now $\norm{\cdot}_A=\norm{\cdot}_{(p_1,r_1)+(p_2,r_2);a}$ and $\norm{\cdot}_B=\norm{\cdot}_{(q_1,s_1)+(q_2,s_2);b}$. If we make a temporary assumption that $m>0$ (i.e. $q_1<q_2$) and use the substitution $\tau=t^m$, then
\begin{align}\label{1233}
&\norm{S_{\sigma}f}_{(q_1,s_1)+(q_2,s_2);b}
=\norm{t^{\f1{q_1}-\f1{s_1}}b(t)S_{\sigma}f(t)}_{s_1,(0,1)}\!\!\!\!\!\!\!\!\!+\norm{t^{\f1{q_2}-\f1{s_2}}b(t)S_{\sigma}f(t)}_{s_2,(1,\infty)}\nonumber\\
&\quad\es\norm{t^{-\f1{s_1}}b(t)\intt{0}{t^m}{u^{\f1{p_1}-1}f(u)}{u}}_{s_1,(0,1)}
\!\!\!\!\!\!\!\!\!+\norm{t^{\f1{q_1}-\f1{q_2}-\f1{s_1}}b(t)\intt{t^m}{\infty}{u^{\f1{p_2}-1}f(u)}{u}}_{s_1,(0,1)}\nonumber\\
&\qquad+\norm{t^{\f1{q_2}-\f1{q_1}-\f1{s_2}}b(t)\intt{0}{t^m}{u^{\f1{p_1}-1}f(u)}{u}}_{s_2,(1,\infty)}
\!\!\!\!\!\!\!\!\!+\norm{t^{-\f1{s_2}}b(t)\intt{t^m}{\infty}{u^{\f1{p_2}-1}f(u)}{u}}_{s_2,(1,\infty)}\nonumber\\
&\quad\es\norm{t^{-\f1{s_1}}b_*(t)\intt{0}{t}{u^{\f1{p_1}-1}f(u)}{u}}_{s_1,(0,1)}
\!\!\!\!\!\!\!\!\!+\norm{t^{\f1{p_1}-\f1{p_2}-\f1{s_1}}b_*(t)\intt{t}{\infty}{u^{\f1{p_2}-1}f(u)}{u}}_{s_1,(0,1)}\nonumber\\
&\qquad+\norm{t^{\f1{p_2}-\f1{p_1}-\f1{s_2}}b_*(t)\intt{0}{t}{u^{\f1{p_1}-1}f(u)}{u}}_{s_2,(1,\infty)}
\!\!\!\!\!\!\!\!\!+\norm{t^{-\f1{s_2}}b_*(t)\intt{t}{\infty}{u^{\f1{p_2}-1}f(u)}{u}}_{s_2,(1,\infty)}\nonumber\\
&=:N_1+N_2+N_3+N_4
\end{align}
for all $f\in\Dupa{0,\infty}$. Now observe that for $m<0$, the role of the intervals $(0,1)$ and $(1,\infty)$ in the computation above is interchanged at the initial stage (cf. Definition~\ref{DSum}), but then the substitution swaps the intervals once more. Therefore, the resulting expression is the same and the assumption $m>0$ can be removed. In the rest of the proof we apply the weighted inequalities of Section~\ref{S4} to show that
\begin{align}\label{1108}
N_1+N_2+N_3+N_4&\ls\norm{t^{\f1{p_1}-\f1{r_1}}a(t)f(t)}_{r_1,(0,1)}+\norm{t^{\f1{p_2}-\f1{r_2}}a(t)f(t)}_{r_2,(1,\infty)}\nonumber\\
&=\norm{f}_{(p_1,r_1)+(p_2,r_2);a}
\end{align}
for every $f\in\Dupa{0,\infty}$ if and only if
\begin{equation}\label{1107}
L(r_1,s_1,a,b_*;0,1)+R(r_2,s_2,a,b_*;1,\infty)<\infty.
\end{equation}
Lemma~\ref{LRed} with $\nu=\f1{p_1}$ implies that
\begin{equation}\label{1220}
N_1\ls\norm{t^{\f1{p_1}-\f1{r_1}}a(t)f(t)}_{r_1,(0,1)}\leq\norm{f}_{(p_1,r_1)+(p_2,r_2);a}\quad\forall f\in\Dupa{0,\infty}
\end{equation}
if and only if $L(r_1,s_1,a,b_*;0,1)<\infty$. Similarly, we get from Lemma~\ref{LRedR} with $\nu=\f1{p_2}$ that
\begin{equation}\label{1221}
N_4\ls\norm{t^{\f1{p_2}-\f1{r_2}}a(t)f(t)}_{r_2,(1,\infty)}\leq\norm{f}_{(p_1,r_1)+(p_2,r_2);a}\quad\forall f\in\Dupa{0,\infty}
\end{equation}
if only if $R(r_2,s_2,a,b_*;1,\infty)<\infty$. Now we estimate the expressions $N_2$ and $N_3$. By Lemma~\ref{LNer}, condition \eqref{1107} implies
\begin{equation}\label{1110}
N(r_1,s_1,a,b_*;0,1)+N(r_2,s_2,a,b_*;1,\infty)<\infty
\end{equation}
and also (using the properties of s.v. functions)
\begin{equation}\label{1109}
\norm{t^{-\f1{r_1'}}a(t)^{-1}}_{r_1',(0,1)}
+\norm{t^{-\f1{r_2'}}a(t)^{-1}}_{r_2',(1,\infty)}<\infty.
\end{equation}
Thus, using \eqref{1110}, Lemma~\ref{TLH} with $\mu=\f1{p_1}-\f1{p_2},\kappa=\f1{p_2}$, Lemma~\ref{LSV}~(i), H\"{o}lder's inequality and \eqref{1109}, we obtain
\begin{align}\label{1223}
N_2&\es\norm{t^{\f1{p_1}-\f1{p_2}-\f1{s_1}}b_*(t)\intt{t}{1}{u^{\f1{p_2}-1}f(u)}{u}}_{s_1,(0,1)}\nonumber\\
&\mkern100mu+\norm{t^{\f1{p_1}-\f1{p_2}-\f1{s_1}}b_*(t)}_{s_1,(0,1)}\intt{1}{\infty}{u^{\f1{p_2}-1}f(u)}{u}\nonumber\\
&\ls\norm{t^{\f1{p_1}-\f1{r_1}}a(t)f(t)}_{r_1,(0,1)}
\!\!+\norm{t^{\f1{p_2}-\f1{r_2}}a(t)f(t)}_{r_2,(1,\infty)}\norm{t^{-\f1{r_2'}}a(t)^{-1}}_{r_2',(1,\infty)}\nonumber\\
&\es\norm{f}_{(p_1,r_1)+(p_2,r_2);a}
\end{align}
for all $f\in\Dupa{0,\infty}$. Analogically, we can estimate $N_3$:
\begin{align}\label{1224}
N_3&\es\norm{t^{\f1{p_2}-\f1{p_1}-\f1{s_2}}b_*(t)\intt{1}{t}{u^{\f1{p_1}-1}f(u)}{u}}_{s_2,(1,\infty)}\nonumber\\
&\mkern100mu+\norm{t^{\f1{p_2}-\f1{p_1}-\f1{s_1}}b_*(t)}_{s_2,(1,\infty)}\intt{0}{1}{u^{\f1{p_1}-1}f(u)}{u}\nonumber\\
&\ls\norm{t^{\f1{p_2}-\f1{r_2}}a(t)f(t)}_{r_2,(1,\infty)}
+\norm{t^{\f1{p_1}-\f1{r_1}}a(t)f(t)}_{r_1,(0,1)}\norm{t^{-\f1{r_1'}}a(t)^{-1}}_{r_1',(0,1)}\nonumber\\
&\es\norm{f}_{(p_1,r_1)+(p_2,r_2);a}
\end{align}
for every $f\in\Dupa{0,\infty}$. The inequality \eqref{1108} then follows from \eqref{1220}, \eqref{1221}, \eqref{1223} and \eqref{1224}, hence the proof of part (i) is complete.
\textit{Case} (ii). We can proceed in the same way as in the case (i) to obtain that
$$T:L_{p_1,r_1;a}\cap L_{p_2,r_2;a}\lra L_{q_1,s_1;b}\cap L_{q_2,s_2;b}$$
holds if and only if the inequality
\begin{align}\label{1230}
N_1&+N_2+N_3+N_4:=\nonumber\\
&\norm{t^{-\f1{s_1}}b_*(t)\intt{0}{t}{u^{\f1{p_1}-1}f(u)}{u}}_{s_1,(1,\infty)}
\!\!\!\!\!\!\!\!\!+\norm{t^{\f1{p_1}-\f1{p_2}-\f1{s_1}}b_*(t)\intt{t}{\infty}{u^{\f1{p_2}-1}f(u)}{u}}_{s_1,(1,\infty)}\nonumber\\
&\quad+\norm{t^{\f1{p_2}-\f1{p_1}-\f1{s_2}}b_*(t)\intt{0}{t}{u^{\f1{p_1}-1}f(u)}{u}}_{s_2,(0,1)}
\!\!\!\!\!\!\!\!\!+\norm{t^{-\f1{s_2}}b_*(t)\intt{t}{\infty}{u^{\f1{p_2}-1}f(u)}{u}}_{s_2,(0,1)}\nonumber\\
&\ls\norm{t^{\f1{p_1}-\f1{r_1}}a(t)f(t)}_{r_1,(1,\infty)}+\norm{t^{\f1{p_2}-\f1{r_2}}a(t)f(t)}_{r_2,(0,1)}=\norm{f}_{(p_1,r_1)\cap(p_2,r_2);a}
\end{align}
holds for all $f\in\Dupa{0,\infty}$ (the only difference from the corresponding inequality in the case (i) is that the intervals $(0,1)$ and $(1,\infty)$ were interchanged, cf. Definition~\ref{DSum}). As in the case (i), the (parts of) terms $N_1$ and $N_4$ represent the limiting case of interpolation and hence, the Lemmas~\ref{LRed}, \ref{LRedR} imply that the condition
\begin{equation}\label{1231}
L(r_1,s_1,a,b_*;1,\infty)+R(r_2,s_2,a,b_*,0,1)<\infty
\end{equation}
is necessary for \eqref{1230} to hold for every $f\in\Dupa{0,\infty}$. Now, it remains to prove that \eqref{1231} is also sufficient to estimate all the remaining terms in $\LHS{1230}$ by $\RHS{1230}$.
Note that \eqref{1231} implies
\begin{equation}\label{1232}
\norm{t^{-\f1{s_1}}b_*(t)}_{s_1,(1,\infty)}+\norm{t^{-\f1{s_2}}b_*(t)}_{s_2,(0,1)}<\infty.
\end{equation}
The expression $N_1$ contains the following term, which, using \eqref{1232}, H\"{o}lder's inequality and Lemma~\ref{LSV}~(i), can be estimated as
\begin{align*}
\norm{t^{-\f1{s_1}}b_*(t)\intt{0}{1}{u^{\f1{p_1}-1}f(u)}{u}}_{s_1,(1,\infty)}
\!\!\!\!\!\!\!&\ls\norm{t^{\f1{p_2}-\f1{r_2}}a(t)f(t)}_{r_2,(0,1)}\norm{t^{\f1{p_1}-\f1{p_2}-\f1{r_2'}}a(t)^{-1}}_{r_2',(0,1)}\\
&\ls\norm{t^{\f1{p_2}-\f1{r_2}}a(t)f(t)}_{r_2,(0,1)}.
\end{align*}
The corresponding term in expression $N_4$ can be estimated in the same way as
\begin{align*}
\norm{t^{-\f1{s_2}}b_*(t)\intt{1}{\infty}{u^{\f1{p_2}-1}\!\!f(u)}{u}}_{s_2,(0,1)}
\!\!\!\!\!\!\!\!\!&\ls\norm{t^{\f1{p_1}-\f1{r_1}}a(t)f(t)}_{r_1,(1,\infty)}\norm{t^{\f1{p_2}-\f1{p_1}-\f1{r_1'}}a(t)^{-1}}_{r_1',(1,\infty)}\\
&\ls\norm{t^{\f1{p_1}-\f1{r_1}}a(t)f(t)}_{r_1,(1,\infty)}.
\end{align*}
By Lemma~\ref{LNer}, \eqref{1231} implies
$$N(r_1,s_1,a,b_*,1,\infty)+N(r_2,s_2,a,b_*,0,1)<\infty$$
and hence, using Lemma~\ref{TLH}, the remaining (non-limiting) terms $N_2$, $N_3$ can be estimated by $\RHS{1230}$ as well.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{IntHil}.] One can repeat the proof of Theorem~\ref{IntLim} with $p_2=\infty$, the only difference being that we use Lemma~\ref{LSaw} instead of Lemma~\ref{LRedR}.
\end{proof}
The proofs of Theorems~\ref{IntLim} and \ref{IntHil} contain all the possible difficulties which one can encounter. More precisely, in the proofs of the remaining theorems of Section~\ref{S3} the inequality \eqref{1233} (or \eqref{1230}) will have just two terms - either $N_1+N_2$, or $N_3+N_4$. Therefore, we can obtain the remaining proofs as fragments of the proof of Theorem~\ref{IntLim}.
\section{The optimality of results}\label{S6}
Let $X,Y,W,Z$ be LK spaces, or their sum, or intersection in the sense of Definition~\ref{DSum} and let $T$ be a quasilinear operator acting between $X$ and $Y$. We say that the result
$$T:X\lra Y$$
is optimal in the scale of LK spaces if
$$Y\embl Z\text{ for every }Z\text{ satisfying }T:X\lra Z$$
and
$$W\embl X\text{ for every }W\text{ satisfying }T:W\lra Y.$$
Embeddings of LK spaces are characterized by the following lemma.
\begin{lemma}\label{Temb}
Let $p,q,r,s\in(0,\infty]$ and $a,b\in\SV(A,B)$. Then
\begin{equation}\label{146}
\norm{t^{\f1q-\f1s}b(t)f(t)}_{s,(A,B)}\ls\norm{t^{\f1p-\f1r}a(t)f(t)}_{r,(A,B)}\quad\forall f\in\Dupa{A,B},
\end{equation}
if and only if one of the following conditions hold\,{\rm:}
\begin{itemize}
\item[\rm{(i)}] $(A,B)=(0,1)$, $p>q${\rm;}
\item[\rm{(ii)}] $p=q$, $0<r\leq s\leq\infty$,
\begin{equation}\label{142}\sup_{A<x<B}\norm{t^{\f1{p}-\f1s}b(t)}_{s,(A,x)}\norm{t^{\f1{p}-\f1r}a(t)}_{r,(A,x)}^{-1}<\infty{\rm;}
\end{equation}
\item[{\rm(iii)}] $p=q$, $0<s<r\leq\infty$,
\begin{equation}\label{143}
\norm{x^{\f s{\rh}\f 1p-\f1{\rh}}b(x)^{\f s{\rh}}\norm{t^{\f1p-\f1s}b(t)}_{s,(A,x)}^{\f sr}\norm{t^{\f1p-\f1r}a(t)}_{r,(A,x)}^{-1}}_{\rh,(A,B)}<\infty.
\end{equation}
\end{itemize}
When $p<\infty$, conditions \eqref{142} and \eqref{143} can be simplified to
\begin{equation}\label{148}
N(r,s,a,b;A,B)<\infty.
\end{equation}
\end{lemma}
\begin{proof}
The simplification of \eqref{142} and \eqref{143} in the case $p<\infty$ follows easily from Lemma~\ref{LSV}~(i).
\textit{Case $p>q$.} See \cite[Theorem 3.4]{N}.
\textit{Case $p=q$, $0<r,s<\infty$.} The inequality \eqref{146} can be further rewritten as
\begin{equation}\label{145}
\sup_{f\in\Dupa{A,B}}\f{\norm{wf}_{s,(A,B)}}{\norm{vf}_{r,(A,B)}}<\infty,
\end{equation}
where $v(t)=t^{\f1p-\f1r}a(t)$ and $w(t)=t^{\f1p-\f1s}b(t)$, $t\in(A,B)$. The problem of characterization of \eqref{145} with general weights is fully resolved for $0<r,s<\infty$ and leads directly to conditions \eqref{142} and \eqref{143}. The first result of this kind is due to E. Sawyer for the range $1<r,s<\infty$ (he applied his reduction theorem for the identity operator - see \cite[p.148]{Saw}). This result was extended to $0<r,s<\infty$ by, for example, M. Carro and J. Soria, or V. D. Stepanov in \cite[Proposition~1]{Step2}, who also provided estimates (independent of $v,w$) for \LHS{145}. However, Stepanov's proof relies on the approximation of non-increasing functions by absolutely continuous functions, which was left unjustified. For a rigorous and yet very elegant treatment of this topic, we refer to \cite[Section 2]{Sin}.
We are going to prove the cases which are missing in the literature cited above, that is, cases where $r=\infty$ or $s=\infty$.
\textit{Case $p=q$, $0<r\leq s=\infty$.} To prove the necessity of \eqref{142} for \eqref{146}, it is enough to test \eqref{146} with $f=\chi_{(A,x)}$, where $x\in(A,B)$.
Now we prove the sufficiency of \eqref{142} for \eqref{146}. Using the estimate
$$t^{\f1p}b(t)\ls\norm{u^{\f1p}b(u)}_{\infty,(A,t)}\quad\forall t\in(A,B)$$
together with \eqref{142} and the monotonicity of $f$, we obtain
\begin{align*}
\norm{t^{\f1p}b(t)f(t)}_{\infty,(A,B)}
&\ls\norm{\norm{u^{\f1p}b(u)}_{\infty,(A,t)}f(t)}_{\infty,(A,B)}
\ls\norm{\norm{u^{\f1p-\f1r}a(u)}_{r,(A,t)}f(t)}_{\infty,(A,B)}\\
&\leq\norm{\norm{u^{\f1p-\f1r}a(u)f(u)}_{r,(A,t)}}_{\infty,(A,B)}
=\RHS{146}
\end{align*}
for every $f\in\Dupa{A,B}$, which proves \eqref{146}.
\textit{Case $p=q<\infty$, $0<s<r=\infty$.} The necessity of \eqref{148} follows by testing \eqref{146} with $f(t)=t^{-\f1p}a(t)^{-1}$, $t\in(A,B)$.
For the sufficiency, we use \eqref{148} to obtain
\begin{align*}
\LHS{146}
&=\norm{t^{\f1p-\f1s}b(t)f(t)}_{s,(A,B)}
=\norm{t^{-\f1s}b(t)a(t)^{-1}\;t^{\f1p}a(t)f(t)}_{s,(A,B)}\\
&\leq\norm{t^{-\f1s}b(t)a(t)^{-1}}_{s,(A,B)}\norm{t^{\f1p}a(t)f(t)}_{\infty,(A,B)}
\ls\RHS{146}
\end{align*}
for every $f\in\Dupa{A,B}$.
\textit{Case $p=q=\infty$, $0<s<r=\infty$.} To prove the necessity of \eqref{143} for \eqref{146}, we test \eqref{146} with $f(t)=\norm{a}^{-1}_{\infty,(A,t)}$, $t\in(A,B)$. In this way, we get
\begin{align}\label{147}
\norm{t^{-\f1s}b(t)\norm{a}_{\infty,(A,t)}^{-1}}_{s,(A,B)}\ls\norm{a(t)\norm{a}_{\infty,(A,t)}^{-1}}_{\infty,(A,B)}
\ls1,
\end{align}
which is indeed \eqref{143} with $p=r=\infty$.
To show the sufficiency, we use \eqref{147} and the monotonicity of $f$ to obtain
\begin{align*}
\norm{t^{-\f1s}b(t)f(t)}_{s,(A,B)}
&\leq\norm{t^{-\f1s}b(t)\norm{a}^{-1}_{\infty,(A,t)}}_{s,(A,B)}\norm{\norm{a}_{\infty,(A,t)}f(t)}_{\infty,(A,B)}\\
&\ls\norm{\norm{af}_{\infty,(A,t)}}_{\infty,(A,B)}\!=\RHS{146}
\end{align*}
for every $f\in\Dupa{A,B}$ and thus, the proof is finished.
\end{proof}
Since every non-increasing function on $(A,B)$ arises as a (restriction of) decreasing rearrangement of a function from $\Meas(R_1,\mu_1)$ (see \cite[p.86, Corollary 7.8.]{BS}), the Lemma~\ref{Temb} characterizes the embedding $L_{p,r;a}\embl L_{q,s;b}$. For the embeddings of the sums and intersections of the LK spaces in the sense of Definition~\ref{DSum}, we apply Lemma~\ref{Temb} on the two parts of the corresponding quasi-norm separately.
Now we turn our attention to the optimality itself, which, in the non-limiting case, is simple to describe. To save some space, we illustrate the idea only in the setting $\mu_1(R_1)=\mu_2(R_2)=\infty$ without considering sums and intersections of spaces. For the other settings analogous assertions to the following one hold as well and proofs are similar.
\begin{theorem}\label{Opt}
Let the assumptions of Theorem~\ref{Int} be satisfied. Then
\begin{equation}\label{181}
T:L_{p,s;b_*}\lra L_{q,s;b}
\end{equation}
is an optimal result in the scale of LK spaces.
\end{theorem}
\begin{proof}
It follows immediately from Theorem~\ref{Int} that \eqref{181} holds. Next we shall prove that the choice of the target space $L_{q,s;b}$ is optimal in the scale of LK spaces (the optimality of the source space can be proved analogously). Suppose that
\begin{equation}\label{193}
T:L_{p,s;b_*}\lra L_{Q,R;\lambda}
\end{equation}
for some $Q,R\in[1,\infty]$ and $\lambda\in \SV$. This together with $T\in\LBl(p_1,q_1;m)$ gives, for all $f\in\Dupa{0,\infty}$, that
$$\norm{t^{(\f1Q-\f1q)+(\f1q-\f1{q_1})-\f1R}\la(t)\intt{0}{t^m}{u^{\f1{p_1}-1}f(u)}{u}}_{R,(0,\infty)}
\ls\norm{t^{\f1p-\f1s}b_*(t)f(t)}_{s,(0,\infty)},$$
which can be rewritten (using the change of variables) as
\begin{equation}\label{195}
\norm{t^{\gamma+\f1p-\f1{p_1}-\f1R}\la_*(t)\intt{0}{t}{u^{\f1{p_1}-1}f(u)}{u}}_{R,(0,\infty)}
\ls\norm{t^{\f1p-\f1s}b_*(t)f(t)}_{s,(0,\infty)},
\end{equation}
where $\gamma=\f1m(\f1Q-\f1q)$. First we are going to show that \eqref{195} implies $\gamma=0$, i.e. $Q=q$. Suppose to the contrary that $\gamma\neq0$ and choose $\eps$ satisfying
\begin{equation}\label{196}
0<\eps<\min(\tf1p,\tf1{p_1}-\tf1p,\abs{\gamma}).
\end{equation}
\textit{Case $\gamma<0$.} Put
$$f(t)=t^{\eps-\f1p}\chi_{(0,1)}(t),\quad t\in(0,\infty).$$
Then $f\in\Dupa{0,\infty}$ and, using \eqref{196}, we obtain
\begin{align*}
\LHS{195}
&\gs\norm{t^{\gamma+\f1p-\f1{p_1}-\f1R}\la_*(t)\intt{0}{t}{u^{\eps+\f1{p_1}-\f1p-1}}{u}}_{R,(0,1)}\es\norm{t^{\gamma+\eps-\f1R}\la_*(t)}_{R,(0,1)}=\infty,
\end{align*}
while
$$\RHS{195}=\norm{t^{\eps-\f1s}b_*(t)}_{s,(0,1)}\ls1,$$
which gives the contradiction.
\textit{Case $\gamma>0$.} Now put
$$f(t)=\chi_{(0,1)}(t)+t^{-\eps-\f1p}\chi_{[1,\infty)}(t),\quad t\in(0,\infty).$$
Then $f\in\Dupa{0,\infty}$ and, using \eqref{196}, we obtain
\begin{align*}
\LHS{195}
&\gs\norm{t^{\gamma+\f1p-\f1{p_1}-\f1R}\la_*(t)\intt{1}{t}{u^{-\eps+\f1{p_1}-\f1p-1}}{u}}_{R,(1,\infty)}\es\norm{t^{\gamma-\eps-\f1R}\la_*(t)}_{R,(1,\infty)}=\infty,
\end{align*}
while
$$\RHS{195}\es\norm{t^{\f1p-\f1s}b_*(t)}_{s,(0,1)}+\norm{t^{-\eps-\f1s}b_*(t)}_{s,(1,\infty)}\ls1,$$
which is the contradiction.
Thus, under the assumption $T\in\LBl(p_1,q_1;m)$, we have proved $Q=q$. When $T\in\LBr(p_2,q_2;m)$, one can proceed analogously.
Now, using Theorem~\ref{Int} on \eqref{193} with $Q=q$, we obtain $N(s,R,\la_*,b_*;0,\infty)<\infty,$ which, by Lemma~\ref{Temb} ($q<\infty$), implies that $L_{q,s;b}\embl L_{q,R;\la},$ and the proof is complete.
\end{proof}
In the limiting cases one can prove the optimality in the sense mentioned above only in some special cases. This is caused by the fact that, in general, the optimal target or source spaces lie outside the scale of LK spaces (cf. \cite[Section 5]{GOT}). However, we shall mention at least some partial (sharp) results in this direction. Similar results for the sharp embeddings of Bessel-potential-type spaces into LK spaces appeared in \cite{GNO}, for example. For brevity, we shall state the following results only in the case where $\mu_1(R_1)=\mu_2(R_2)=1$. The next theorem describes the limiting case $\theta=0$.
\begin{theorem}\label{OptL}
Let $T\in\JW(p_1,q_1;p_2,q_2)\cap\LBl(p_1,q_1;m)$ be a quasilinear operator.
\begin{itemize}
\item[{\rm(i)}] Let $1<r\leq s\leq\infty$ and suppose $a\in\SV(0,1)$ is such that
\begin{equation}\label{156}
\intt{0}{1}{t^{-1}a(t)^{-r'}}{t}<\infty.
\end{equation}
Define
\begin{equation}\label{eq40}
\beta(t)=a(t^m)^{-\f{r'}{s}}\left(\intt{0}{t^m}{u^{-1}a(u)^{-r'}}{u}\right)^{-\f1{r'}-\f1s},\quad t\in(0,1).
\end{equation}
Then $\beta\in\SV(0,1)$ and
\begin{equation}\label{153}
T:L_{p_1,r;a}\lra L_{q_1,s;\beta}.
\end{equation}
Moreover, if $\lambda\in SV(0,1)$ is such that
\begin{equation}\label{154}
T:L_{p_1,r;a}\lra L_{q_1,s;\lambda}
\end{equation}
and the limit
\begin{equation}\label{158}
\lim_{x\to0_+}\f{\lambda_*(x)}{\beta_*(x)}
\end{equation}
exists when $s<\infty$, then
\begin{equation}\label{159}
L_{q_1,s;\beta}\embl L_{q_1,s;\lambda}.
\end{equation}
\item[{\rm(ii)}] Let $1\leq r\leq s<\infty$ and suppose $b\in\SV(0,1)$ is such that
$$\intt{0}{1}{t^{-1}b_*(t)^s}{t}=\infty.$$
Define
\begin{equation}\label{eq45}
\alpha(t)=b_*(t)^{-\f{s}{r'}}\left(1+\intt{t}{1}{u^{-1}b_*(u)^s}{u}\right)^{\f1{r'}+\f1s},\quad t\in(0,1).
\end{equation}
Then $\alpha\in \SV(0,1)$ and
\begin{equation*}
T:L_{p_1,r;\alpha}\lra L_{q_1,s;b}.
\end{equation*}
Moreover, if $\lambda\in \SV(0,1)$ is such that
$$T:L_{p_1,r;\lambda}\lra L_{q_1,s;b}$$
and the limit
$$\lim_{x\to0_+}\f{\alpha(x)}{\lambda(x)}$$
exists when $r>1$, then
$$L_{p_1,r;\lambda}\embl L_{p_1,r;\alpha}.$$
\end{itemize}
\end{theorem}
\begin{proof}
We prove part (i) here; the proof of part (ii) is analogous.
\textit{Case $s<\infty$.} By Lemma~\ref{L} (ii) ($r>1$), the function $\beta$ defined by \eqref{eq40} satisfies
\begin{equation}\label{155}
\norm{t^{-\f1s}\beta_*(t)}_{s,(x,1)}\es\norm{t^{-\f1{r'}}a(t)^{-1}}_{r',(0,x)}^{-1}\quad\forall x\in(0,\tf12).
\end{equation}
Therefore, Theorem~\ref{IntL} (and Remark~\ref{Rem}~(i)) yields \eqref{153}. Moreover, Theorem~\ref{IntL} and \eqref{154} implies that
$$\norm{t^{-\f1s}\lambda_*(t)}_{s,(x,1)}\norm{t^{-\f1{r'}}a(t)^{-1}}_{r',(0,x)}\ls1\quad\forall x\in(0,\tf12).$$
Together with \eqref{155}, this gives
\begin{equation}\label{157}
\f{\intt{x}{1}{t^{-1}\lambda_*(t)^s}{t}}{\intt{x}{1}{t^{-1}\beta_*(t)^s}{t}}\ls1\quad\forall x\in(0,\tf12).
\end{equation}
Since the denominator of $\LHS{157}$ tends to infinity as $x\to0_+$ (see \eqref{155} and \eqref{156}) and we assume that limit \eqref{158} exists, we can apply L'Hospital's rule to $\LHS{157}$ to get
$$1\gs\lim_{x\to0_+}\f{\intt{x}{1}{t^{-1}\lambda_*(t)^s}{t}}{\intt{x}{1}{t^{-1}\beta_*(t)^s}{t}}
=\lim_{x\to0_+}\f{\lambda_*(x)^s}{\beta_*(x)^s}.$$
Thus, by Lemma~\ref{Temb}, we obtain \eqref{159}.
\textit{Case $s=\infty$.} Now \eqref{eq40} reads as $\beta_*(x)=\norm{t^{-\f1{r'}}a(t)^{-1}}_{r',(0,x)}^{-1}$, $x\in(0,1)$. This, \eqref{154} and Theorem~\ref{IntL} yield
$$1\gs\norm{\lambda_*}_{\infty,(x,1)}\norm{t^{-\f1{r'}}a(t)^{-1}}_{r',(0,x)}\gs\lambda_*(x)\beta_*(x)^{-1}$$
for all $x\in(0,1)$, which, by Lemma~\ref{Temb}, implies \eqref{159}.
\end{proof}
It is obvious that the requirement about the existence of the limit in Theorem~\ref{OptL} may be dropped in many situations. For example, this assumption is redundant if $a$ and $b$ are products of composite logarithmic functions.
Next we consider the limiting case $\theta=1$ that is analogous to the previous theorem, except for the case $p_2=\infty$ (thus we shall prove only this case). In order to keep the presentation brief, let us make a convention that if we say that some result is sharp, then we mean it in the sense of the previous theorem (assuming the existence of the corresponding limits when needed).
\begin{theorem}\label{OptR}
Let $T\in\JW(p_1,q_1;p_2,q_2)\cap\LBr(p_2,q_2;m)$ be a quasilinear operator.
\begin{itemize}
\item[{\rm(i)}] Let $p_2<\infty$, $1<r\leq s\leq\infty$, $a\in\SV(0,1)$ and suppose that $\intt{0}{1}{t^{-1}a(t)^{-r'}}{t}=\infty$. Then
$$T:L_{p_2,r;a}\lra L_{q_2,s;\beta},$$
where
\begin{equation}\label{00}
\beta(t)=a(t^m)^{-\f{r'}{s}}\left(1+\intt{t^m}{1}{u^{-1}a(u)^{-r'}}{u}\right)^{-\f1{r'}-\f1s},\quad t\in(0,1),
\end{equation}
is a sharp result.
\item[{\rm(ii)}] Let $p_2<\infty$, $1\leq r\leq s<\infty$, $b\in\SV(0,1)$ and suppose that $\intt{0}{1}{t^{-1}b_*(t)^s}{t}<\infty$. Then
$$T:L_{p_2,r;\alpha}\lra L_{q_2,s;b},$$
where
$$\alpha(t)=b_*(t)^{-\f{s}{r'}}\left(\intt{0}{t}{u^{-1}b_*(u)^{s}}{u}\right)^{\f1{r'}+\f1s},\quad t\in(0,1),$$
is a sharp result.
\item[{\rm(iii)}] Suppose that \eqref{210} holds. The assertions in {\rm(i)} and {\rm(ii)} remain true if $p_2=\infty$, provided that $r=s=\infty$ and $r=s=1$, respectively.
\end{itemize}
\end{theorem}
\begin{proof}\mbox{}
\textit{Case $r=s=\infty$.} By Theorem~\ref{IntRR} and \eqref{210}, the result $T:L_{\infty,\infty;a}\lra L_{q_2,\infty;\lambda}$ implies
$$\norm{\lambda_*}_{\infty,(0,x)}\norm{t^{-1}a(t)^{-1}}_{1,(x,1)}\ls1\quad\forall x\in(0,1),$$
and then we can argue similarly as in the case $s=\infty$ of the proof of Lemma~\ref{OptL} (i).
\textit{Case $r=s=1$.} In this case we can write
\begin{align}\label{1240}
\norm{t^{-\f1s}b_*(t)\log\tf xt}_{s,(0,x)}
=\intt{0}{x}{t^{-1}b_*(t)\intt{t}{x}{u^{-1}}{u}}{t}
&=\intt{0}{x}{u^{-1}\intt{0}{u}{t^{-1}b_*(t)}{t}}{u}\nonumber\\
&=\norm{t^{-1}\alpha(t)}_{1,(0,x)}
\end{align}
for all $x\in(0,1)$, hence, by Theorem~\ref{IntRR}, the result
$$L_{\infty,1;\alpha}\lra L_{\infty,1;b}$$
holds. Now if $\la\in\SV(0,1)$ is such that $L_{\infty,1;\la}\lra L_{\infty,1;b}$, then Theorem~\ref{IntRR} and \eqref{1240} imply
$$1\gs\norm{t^{-\f1s}b_*(t)\log\tf xt}_{s,(0,x)}\norm{t^{-1}\la(t)}_{1,(0,x)}^{-1}
=\norm{t^{-1}\alpha(t)}_{1,(0,x)}\norm{t^{-1}\la(t)}_{1,(0,x)}^{-1},$$
for all $x\in(0,1)$, therefore, by Lemma~\ref{Temb}, we get $L_{\infty,1;\la}\embl L_{\infty,1;\alpha}$.
\end{proof}
Analogous assertions can be formulated on the interval $(0,\infty)$ (i.e. if $\mu_1(R_1)=\mu_2(R_2)=\infty$) for the sums and intersections of the LK spaces. However, since there are many possible configurations to cover and the formulas connecting the s.v. functions remain essentially the same, we shall skip this. When needed, we can extract the sharp results directly from the conditions of our interpolation theorems by assuming that the functions appearing under the supremum in $N$, $L$, $R$, $R_\infty$ are equivalent to $1$ and by using Lemma~\ref{L}.
\begin{remark}
There are situations in which the sharp results of Theorem~\ref{OptR} are also optimal. For example, suppose that the assumptions of Theorem~\ref{OptR}~(i) are satisfied and let $b_s=\beta$, where $\beta$ is defined by \eqref{00}. Furthermore, suppose that $q_2=\infty$. Then, from Lemmas~\ref{L} and \ref{Temb}, we deduce that $L_{\infty,s;b_s}\embl L_{\infty,S,b_S}$ whenever $s\leq S\leq\infty$. Therefore, in this situation, the target space $L_{\infty,r,b_r}$ is optimal (cf. \cite[Remark~3.2.~(iii)]{GNO}).
\end{remark}
\section{Applications of the results}\label{S7}
Our main results can be applied to many familiar operators - we will now give several examples. We shall mention the sharp (or optimal) results only.
In the following, the symbol $\Meas(\Omega)$ stands for the set of all Lebesgue measurable functions on $\Omega\subseteq\R^n$, $n\in\N$, and $|Q|$ for the Lebesgue measure of the set $Q\subseteq\R^n$.
In the next lemma we recall the definitions of several familiar operators and corresponding well known estimates that are required by our interpolation theorems.
\begin{lemma}\label{Oper}
\begin{itemize}
\item[{\rm(i)}] Let $|\Omega|=1$. \textsc{The Hardy-Littlewood maximal operator} $M_{\Omega}$ is defined for a locally integrable function $f\in\Meas(\Omega)$ by
$$M_{\Omega}f(x)=\sup_{Q\ni x}\f1{|Q|}\int\limits_{Q\cap\Omega}|f|,\quad x\in\Omega,$$
where the supremum extends over all cubes containing $x$ which have sides parallel to coordinate axes. The operator $M_{\Omega}$ satisfies
\begin{equation}\label{1100}
(M_{\Omega}f)^*(t)\es\f1t\intt{0}{t}{f^*(u)}{u}\quad\forall \text{\rm loc. int. } f\in\Meas(\Omega)\quad\forall t\in(0,1),
\end{equation}
and thus $M_{\Omega}\in\JW(1,1;\infty,\infty)\cap\LBl(1,1;1)$.
\item[{\rm(ii)}] \textsc{The conjugate function}, defined for a locally integrable $2\pi$-periodic function $f\in\Meas(\R)$ by
$$\CC f(x)=\f1{\pi}\lim_{\eps\to0_+}\int\limits_{\eps<|t|\leq\pi}\!\!\!\!(2\cot\tf{t}2)f(x-t)\,\mathrm{d}t,\quad x\in\R,$$
satisfies $\CC\in\JW(1,1;\infty,\infty)\cap\CC\in\LBl(1,1;1)\cap\LBr(\infty,\infty;1)$.
\item[{\rm(iii)}] \textsc{The Riesz potential} $I_{\gamma}$, $0<\gamma<n$, defined for a locally integrable function $f\in\Meas(\R^n)$ by
$$I_{\gamma}f(x)=c(n,\gamma)\intt{\R^n}{}{|t|^{\gamma-n}f(x-t)}{t},\quad x\in\R^n,$$
satisfies $I_{\gamma}\in\JW(1,\tf{n}{n-\gamma};\tf{n}{\gamma},\infty)\cap\LBl(1,\tf{n}{n-\gamma};1)\cap\LBr(\tf{n}{\gamma},\infty;1)$.
\item[{\rm(iv)}] \textsc{The Hilbert transform}, defined for every function $f\in\Meas(\R)$ such that $f\in L_1+L_{\infty,1}$ by
\begin{equation*}
Hf(x)=\f1{\pi}\lim_{\eps\to0_+}\int\limits_{\eps<|t|}t^{-1}f(x-t)\,\mathrm{d}t,\quad x\in\R,
\end{equation*}
satisfies $H\in\JW(1,1;\infty,\infty)\cap\LBl(1,1;1)\cap\LBr(\infty,\infty;1)$.
\item[{\rm(v)}] \textsc{The Riesz transforms} $R_i$, $1\leq i\leq n$, defined for all functions $f\in\Meas(\R^n)$ such that $f\in L_1+L_{\infty,1}$ by
$$R_if(x)=c(n)\lim_{\eps\to0_+}\int\limits_{\eps<|t|}{\f{t_i}{|t|^{n+1}}f(x-t)}\,\mathrm{d}t,\quad x\in\R^n,$$
satisfy $R_i\in\JW(1,1;\infty,\infty)\cap\LBl(1,1;1)\cap\LBr(\infty,\infty;1)$.
\end{itemize}
\end{lemma}
\begin{proof}\mbox{}
\textit{Case} (i). See \cite[Chapter 3, Theorem 3.8]{BS}.
\textit{Cases} (ii), (iv). That $\CC,H\in\JW(1,1;\infty,\infty)$ follows from \cite[Chapter 3, Theorem~6.8]{BS} and \cite[Chapter~3, Theorem~4.8.]{BS}. The corresponding lower bound for $H$ is proved in \cite[Proposition~4.10.]{BS} and this proof works as well for $\CC$ (cf. also \cite[Theorem~10.2.~(ii)]{EOP}).
\textit{Case} (iii). See \cite[p.150]{Saw} and references there.
\textit{Case} (v). The Riesz transforms satisfy essentially the same rearrangement inequality as $H$ does (cf. \cite[p.150]{Saw}).
\end{proof}
The following theorem concerns the boundedness of operators $M_{\Omega}$ and $\CC$, which are acting between function spaces over finite measure spaces.
\begin{theorem}\label{TM}
Let $T$ be $M_{\Omega}$ or $\CC$. Then
\begin{align}
T&: & L_{p,s;b} &\lra L_{p,s;b},\quad1<p<\infty, & 1&\leq s\leq\infty,\quad b\in\SV(0,1);\label{M1}\\
T&: & L_{1,1;b}&\lra L_{1,\infty;b}, &b&\in\SV(0,1)\cap\Dupa{0,1};\label{MCL}\\
T&: & L_{1,r;\f1{r'},\f1{r'},\f1{r'}+\alpha}&\lra L_{1,s;-\f1s,-\f1s,-\f1s+\alpha}, & 1&\leq r\leq s\leq\infty,\quad\alpha>0;\label{M5}\\
M_{\Omega}&: & L_{\infty}&\lra L_{\infty};& &\label{MN}\\
\CC&: & L_{\infty} &\lra L_{\infty,\infty;\el^{-1}};& &\label{C1}\\
\CC&: & L_{\infty,1;-1,-1,-1-\alpha}&\lra L_{\infty,1;0,0,-\alpha},& \alpha&>0;\label{C2}\\
\CC&: & L_{\infty,\infty;\,\exp(-\sqrt{\el})}&\lra L_{\infty,\infty;\,\exp(-\sqrt{\el})/\sqrt{\el}}.& &\label{C3}
\end{align}
\end{theorem}
\begin{proof}
Result \eqref{M1} is a consequence of Theorem~\ref{Opt}.
Result \eqref{MCL} follows easily from Theorem~\ref{IntL} (and Remark~\ref{Rem}~(i)), since, in this case, one has
$$L(1,\infty,a,b;0,1)=\norm{b}_{\infty,(x,1)}\norm{a^{-1}}_{\infty,(0,x)}.$$
Result \eqref{M5} follow from Theorem~\ref{OptL}. Indeed, observe that $\beta$ and $\alpha$ from \eqref{eq40} and \eqref{eq45} now take the form
\begin{align*}
\beta(t)&=\el(t)^{-\f1s}\eld(t)^{-\f1s}\elt(t)^{-\f1s-\alpha\f{r'}{s}}\left(\intt{0}{t}{u^{-1}\el(u)^{-1}\eld(u)^{-1}\elt(u)^{-1-\alpha r'}}{u}\right)^{-\f1{r'}-\f1s}\\
&\es\el(t)^{-\f1s}\eld(t)^{-\f1s}\elt(t)^{-\f1s-\alpha\f{r'}{s}}\elt(t)^{\alpha+\alpha\f{r'}{s}}=\el(t)^{-\f1s}\eld(t)^{-\f1s}\elt(t)^{-\f1s+\alpha}
\end{align*}
and
\begin{align*}
\alpha(t)&=\el(t)^{\f1{r'}}\eld(t)^{\f1{r'}}\elt(t)^{\f1{r'}-\alpha\f{s}{r'}}\left(1+\intt{t}{1}{u^{-1}\el(u)^{-1}\eld(u)^{-1}\elt(u)^{-1+\alpha s}}{u}\right)^{\f1{r'}+\f1s}\\
&\es\el(t)^{\f1{r'}}\eld(t)^{\f1{r'}}\elt(t)^{\f1{r'}-\alpha\f{s}{r'}}\elt(t)^{\alpha\f{s}{r'}+\alpha}=\el(t)^{\f1{r'}}\eld(t)^{\f1{r'}}\elt(t)^{\f1{r'}+\alpha}
\end{align*}
for all $t\in(0,1)$.
Result \eqref{MN} follows either from the definition of $M_{\Omega}$, or from \eqref{1100} and Lemma~\ref{TLH} with $\mu=\kappa=1$, $r=s=\infty$, $a=b$.
Results \eqref{C1} and \eqref{C3} follow from Theorem~\ref{OptR}~(iii) with $r=s=\infty$. Indeed, this is obvious for \eqref{C1} and for \eqref{C3} we set $a(t)=\exp\sqrt{-\el(t)}$, $t\in(0,1)$, use the substitution $\tau=\sqrt{\el(u)}$ and integration by parts to get
\begin{align*}
\intt{t}{1}{u^{-1}a(u)^{-r'}}{u}
=2\intt{1}{\sqrt{\el(t)}}{\tau\exp(\tau)}{\tau}
\es\sqrt{\el(t)}\exp\sqrt{\el(t)}\quad\forall t\in(0,\tf12).
\end{align*}
Result \eqref{C2} is a consequence of Theorem~\ref{OptR}~(iii) with $r=s=1$.
\end{proof}
\begin{remark}\label{RInt}
It is obvious from the proof above that results \eqref{M5} and \eqref{C2} hold analogously for arbitrary tier of logarithms.
\end{remark}
Result \eqref{M5} together with Remark~\ref{RInt} yield many particular results (in fact, also the result \eqref{MCL} with $b\equiv1$ can be seen as the limiting case $\alpha\to0$ of \eqref{M5} with $r=1$, $s=\infty$). Some of these are stated in the following corollary.
\begin{cor}
Let $T$ be $M_{\Omega}$ or $\CC$. Then
\begin{align*}
T&: & L(\log L)&\lra L_{1};& &\\
T&: & L(\log\log\log L)&\lra L(\log L)^{-1}(\log\log L)^{-1};& &\\
T&: & L_{1,\infty;1,1,1+\alpha}&\lra L_{1,1;0,0,\alpha},&\quad\alpha&>0.
\end{align*}
\end{cor}
The result \eqref{M1} for $M_{\Omega}$ with $b\equiv1$ is a well known result of Hardy, Littlewood. The non-limiting case for operator $\CC$ was resolved by F. Riesz. The limiting cases with single logarithm for operator $\CC$ are due to Zygmund. Analogous results for GLZ spaces with second tier of logarithms were proven in \cite{EOP}. The results \eqref{M5}, \eqref{C2}, \eqref{C3} (and their versions for higher tiers of logarithms) are new. The spaces in \eqref{C3} are not GLZ spaces.
Now we shall present some results for operators $I_{\gamma}$, $H$ and $R_i$, acting between function spaces over $\R$ or $\R^n$. We start with $I_{\gamma}$, since its behaviour near the right endpoint is easier to describe than for the other two operators ($p_2=\f{n}{\gamma}<\infty$).
\begin{theorem}\label{TRP}
Let $\g=\f{n}{n-\gamma}$. The operator $I_{\gamma}$ satisfies
\begin{align}
I_{\gamma}&: &L_{p,s;b} &\lra L_{q,s;b},& 1&<p<\tf{n}{n-\gamma},\quad\tf1p=\tf1q+\tf{\gamma}n,\quad b\in\SV(0,\infty);\label{I1}\\
I_{\gamma}&: &L_1+L_{\f{n}{\gamma},1}&\lra L_{\f{n}{n-\gamma},\infty}+L_{\infty};& &\label{I2}\\
I_{\gamma}&: &L_1\cap L_{\f{n}{\gamma},1}&\lra L_{\f{n}{n-\gamma},\infty}\cap L_{\infty};& &\label{I2b}
\end{align}
\vspace{-0.75cm}
\begin{align}\label{I3}
I_{\gamma}&: &L_{1,r_1;\f1{r_1'},\f1{r_1'}+\alpha}+L_{\f{n}{\gamma},r_2;\f1{r_2'},\f1{r_2'}+\beta} &\lra L_{\f{n}{n-\gamma},s_1;-\f1{s_1},-\f1{s_1}+\alpha}+L_{\infty,s_2;-\f1{s_2},-\f1{s_2}+\beta},& &\mkern45mu \nonumber\\
& & 1\leq r_1\leq s_1&\leq\infty,\quad 1\leq r_2\leq s_2\leq\infty,\quad \alpha,\beta>0;& &
\end{align}
\vspace{-0.7cm}
\begin{align}\label{I4}
I_{\gamma}&: &L_{1,r_1;\f1{r_1'},\f1{r_1'}-\alpha}\cap L_{\f{n}{\gamma},r_2;\f1{r_2'},\f1{r_2'}-\beta} &\lra L_{\f{n}{n-\gamma},s_1;-\f1{s_1},-\f1{s_1}-\alpha}\cap L_{\infty,s_2;-\f1{s_2},-\f1{s_2}-\beta},& & \mkern45mu\nonumber\\
& & 1\leq r_1\leq s_1&\leq\infty,\quad 1\leq r_2\leq s_2\leq\infty,\quad \alpha,\beta>0;& &
\end{align}
\end{theorem}
\begin{proof}
The non-limiting case \eqref{I1} follows from Theorem~\ref{Opt}. Result \eqref{I3} follows from Theorem~\ref{IntLim}~(i). Indeed, we know from this theorem that \eqref{I3} holds if and only if
\begin{align*}
\infty&>L(r_1,s_1,a,b;0,1)+R(r_2,s_2,a,b;1,\infty)\\
&=\norm{t^{-\f1{s_1}}\el(t)^{-\f1{s_1}}\eld(t)^{-\f1{s_1}+\alpha}}_{s_1,(x,1)}\norm{t^{-\f1{r_1'}}\el(t)^{-\f1{r_1'}}\eld(t)^{-\f1{r_1'}-\alpha}}_{r_1',(0,x)}\\
&\qquad+\norm{t^{-\f1{s_2}}\el(t)^{-\f1{s_2}}\eld(t)^{-\f1{s_2}+\beta}}_{s_2,(1,x)}\norm{t^{-\f1{r_2'}}\el(t)^{-\f1{r_2'}}\eld(t)^{-\f1{r_2'}-\beta}}_{r_2',(x,\infty)}\\
&\es\eld(x)^{\alpha}\eld(x)^{-\alpha}+\eld(x)^{\beta}\eld(x)^{-\beta}\es1
\end{align*}
for all $x\in(0,\infty)$. The proof of \eqref{I4} is analogous (use Theorem~\ref{IntLim}~(ii)). Results \eqref{I2}, \eqref{I2b} can be seen as the limiting cases $\alpha,\beta\to0$ of \eqref{I3}, \eqref{I4}, respectively, and their proof is similar to the proof of \eqref{MCL}.
\end{proof}
Similarly as for Theorem~\ref{TM}, results \eqref{I3} and \eqref{I4} (and their versions for other tiers of logarithms) yield many particular results. These generalize some of those given in \cite{EOP} to measure spaces with $\mu_1(R_1)=\mu_2(R_2)=\infty$.
\begin{cor}
Let $\g=\f{n}{n-\gamma}$. The operator $I_{\gamma}$ satisfies
\begin{align*}
I_{\gamma}&: &L_{1,1;1,0}+L_{\g ',\g';1,0} &\lra L_{\g,\g;\f1{\g'},0}+L_{\infty,\infty;\f1{\g'},0};\\
I_{\gamma}&: &L_{1,1;\f1{\g},0}+L_{\g',\g';1,0}&\lra L_{\g}+L_{\infty,\g'};\\
I_{\gamma}&: &L_{1,1;-1,0}\cap L_{\g',\g';\f1{\g},0}&\lra L_{\g,\infty;-1,0}\cap L_{\infty,\infty;0,-\f1{\g}};\\
I_{\gamma}&: &L_{1,1;0,\f1{\g}}+L_{\g',\g';\f1{\g},1}&\lra L_{\g,\g;-\f1{\g},0}+L_{\infty,\infty;0,\f1{\g'}};\\
I_{\gamma}&: &L_{1,1;0,-\f1{\g'}}\cap L_{\g',\g';\f1{\g},0}&\lra L_{\g,\g;-\f1{\g},-1}\cap L_{\infty,\infty;0,-\f1{\g}}.
\end{align*}
\end{cor}
We shall conclude the paper with the application to the operators $H$ and $R_i$.
\begin{theorem}\label{THil}
Let $T$ be one of the operators $H$, $R_i$. Then
\begin{align*}
T&: &L_{p,s;b}&\lra L_{p,s;b} & 1&<p<\infty,\quad b\in\SV(0,\infty);\\
T&: &L_1+L_{\infty,1}&\lra L_{1,\infty}+L_{\infty}; & &\\
T&: &L_{1,1;1,0}+L_{\infty,1}&\lra L_{1}+L_{\infty}; & &\\
T&: &L_{1,1;1,0}+L_{\infty,1;1,0}&\lra L_{1}+L_{\infty,1};& &\\
T&: &L_{1,1;0,1}+L_{\infty,1;0,1}&\lra L_{1,1;-1,0}+L_{\infty,1;-1,0};& &\\
T&: &L_{1,1;1,0}\cap L_{\infty,1;0,-\alpha}&\lra L_{1}\cap L_{\infty,1;-1,-1-\alpha}& \alpha&>0;\\
T&: &L_{1,1;1,0}+L_{\infty,\infty;1,1+\alpha}&\lra L_{1}+L_{\infty,\infty;0,\alpha}& \alpha&>0;\\
T&: &L_{1}\cap L_{\infty}&\lra L_{1,\infty}\cap L_{\infty,\infty;-1,0};& &\\
T&: &L_{1}\cap L_{\infty,\infty;1,0}&\lra L_{1,\infty}\cap L_{\infty,\infty;0,-1}.& &
\end{align*}
\begin{proof}
The proof can be done using similar ideas as in the proof of Theorem~\ref{TRP}. Instead of Theorem~\ref{IntLim}, we use Theorem~\ref{IntHil}.
\end{proof}
\end{theorem}
The results contained in Theorem~\ref{THil} again generalize some of the results of \cite{EOP} and extend those of \cite{BR}. | 95,397 |
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This event occurs before submitting of an insert form. It allows you to detect errors on the client side before the form is submitted to the server to avoid the round trip of information necessary for server-side validation. Live example.
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function OnInsertFormValidate (fieldValues, errorInfo)
Parameters:
fieldValues
An associative array of values containing user input. To access the value of the editor of the column_name column, use the field_values['column_name'] syntax.
errorInfo
The object that provides interface (the SetMessage method) to set a validation error message.
Example 1:
The code below demonstrates the verification of the percent number on the record inserting. inserted game:
if (fieldValues['home_team_id'] == fieldValues['away_team_id'])
errorInfo.SetMessage('Home and away teams must be different');
return false;
See also: OnEditFormValidate | 234,217 |
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Sweetwater, TX – Fabian Torrez ID’d as Victim of I-20 Crash
Sweetwater, TX (August 29, 2019) – On August 25, emergency crews responded to the scene of a motor vehicle accident on Interstate 20.
Reports from authorities state that 33-year-old Fabian Torrez was crossing the road when he was struck by a motor vehicle.
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At this time, the incident remains under investigation.
We would like to offer our sincere condolences to the family of Fabian Torrez. Our thoughts are with them at this time.
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For nearly two decades, Texas personal injury attorneys at The Benton Law Firm have fought aggressively to ensure families are able to obtain restitution to cover medical expenses, funeral arrangements, and other damages. | 153,523 |
TITLE: Is my proof showing $A$ and $B$ are invertible if $AB$ is invertible correct?
QUESTION [2 upvotes]: $V$ is a finite dimensional vector space and $A, B \in \mathcal{L} (V)$ where $\mathcal{L}$ is the space of linear operators. The question precisely asks to prove the statement either ways (iff statement). I have no problem proving that if $A^{-1}$ and $B^{-1}$ exist $\implies$ $(AB)^{-1}$ exists and that $(AB)^{-1}=B^{-1}A^{-1}$. My question is for the backward direction, for which I have used to following argument:
If $(AB)^{-1}$ exists then $(AB)^{-1}AB=AB(AB)^{-1}=I$
Assume $S, T \in \mathcal{L}$ such that $AS=SA=I$ and $BT=TB=I$, then $S=A^{-1}$ and $T=B^{-1}$ since the inverse is unique.
From the first equation we have $S=B(AB)^{-1}$ and $T=(AB)^{-1}A$, then
$A^{-1}$ and $B^{-1}$ exist, completing the proof in the backward direction.
Is my logic ok? Because I can't seem to find this answer anywhere.
REPLY [2 votes]: Note that this fails for infinite-dimensonal spaces $V$, e.g. take $V= \mathbb{R}^\mathbb{N}$ and define $A(x_0, x_1, x_2, \ldots) = (0, x_0, x_1, x_2, \ldots)$ and $B(x_0, x_1, x_2, x_3, \ldots) = (x_1 ,x_2, x_3, \ldots)$. Then $B \circ A$ is the identity but $A$ and $B$ are both not invertible, as $B$ is not onto and $A$ is not injective.
For finite dimensional spaces this canot happen as then $\dim(\operatorname{ker}(A)) + \dim(\operatorname{Im}(A)) = n$ for $A: V \to V$ linear and $\dim(V) = n$.
$A$ is injective iff $\operatorname{ker}(A) = \{0\}$ iff $\dim(\operatorname{Im}(A)) = n$ iff $A$ is onto.
And $AB$ invertible means $AB$ is injective and so $B$ is injective. (if $x \in \operatorname{ker}(B)$ then $x \in \operatorname{ker}(AB) = \{0\}$.) So $B$ is onto as well by the finite-dimensionality argument, hence invertible. So $A = (AB)B^{-1}$ is invertible as the product of invertible maps. | 153,496 |
\begin{document}
\title[Pairwise Semiregular Properties]{Pairwise Semiregular Properties on
Generalized Pairwise Lindel\"{o}f Spaces}
\author{Zabidin Salleh}
\address{School of Informatics and Applied Mathematics, Universiti Malaysia
Terengganu, 21030 Kuala Nerus, Terengganu, Malaysia.}
\email{[email protected]}
\date{January 28, 2019}
\subjclass[2010]{54A05, 54A10, 54D20, 54E55 }
\keywords{Bitopological space, $\left( i,j\right) $-nearly Lindel\"{o}f,
pairwise nearly Lindel\"{o}f, $\left( i,j\right) $-almost Lindel\"{o}f,
pairwise almost Lindel\"{o}f, $\left( i,j\right) $-weakly Lindel\"{o}f,
pairwise weakly Lindel\"{o}f, pairwise semiregular property.}
\begin{abstract}
Let $\left( X,\tau _{1},\tau _{2}\right) $ be a bitopological space and $
\left( X,\tau _{\left( 1,2\right) }^{s},\tau _{\left( 2,1\right)
}^{s}\right) $ its pairwise semiregularization. Then a bitopological
property $\mathcal{P}$\ is called pairwise semiregular provided that $\left(
X,\tau _{1},\tau _{2}\right) $\ has the property $\mathcal{P}$\ if and only
if $\left( X,\tau _{\left( 1,2\right) }^{s},\tau _{\left( 2,1\right)
}^{s}\right) $\ has the same property. In this work we study pairwise
semiregular property of $\left( i,j\right) $-nearly Lindel\"{o}f, pairwise
nearly Lindel\"{o}f, $\left( i,j\right) $-almost Lindel\"{o}f, pairwise
almost Lindel\"{o}f, $\left( i,j\right) $-weakly Lindel\"{o}f and pairwise
weakly Lindel\"{o}f spaces. We prove that $\left( i,j\right) $-almost Lindel
\"{o}f, pairwise almost Lindel\"{o}f, $\left( i,j\right) $-weakly Lindel\"{o}
f and pairwise weakly Lindel\"{o}f are pairwise semiregular properties, on
the contrary of each type of pairwise Lindel\"{o}f space which are not
pairwise semiregular properties.
\end{abstract}
\maketitle
\section{\textbf{Introduction}}
\noindent Semiregular properties in topological spaces have been studied by
many topologist. Some of them related to this research studied by Mrsevic et
al. \cite{MrsevicReillyVamanamurthy85, MrsevicReillyVamanamurthy86} and
Fawakhreh and K\i l\i \c{c}man \cite{FawaAdem04}. The purpose of this paper
is to study pairwise semiregular properties on generalized pairwise Lindel
\"{o}f spaces, that we have studied in \cite{BidiAdem, AdemBidi,
BidiAdem2014, BidiAdem1}, namely, $\left( i,j\right) $-nearly Lindel\"{o}f,
pairwise nearly Lindel\"{o}f, $\left( i,j\right) $-almost Lindel\"{o}f,
pairwise almost Lindel\"{o}f, $\left( i,j\right) $-weakly Lindel\"{o}f and
pairwise weakly Lindel\"{o}f spaces.
The main results is that the Lindel\"{o}f, $B$-Lindel\"{o}f, $s$-Lindel\"{o}
f and $p$-Lindel\"{o}f spaces are not pairwise semiregular properties. While
$\left( i,j\right) $-almost Lindel\"{o}f, pairwise almost Lindel\"{o}f, $
\left( i,j\right) $-weakly Lindel\"{o}f and pairwise weakly Lindel\"{o}f
spaces are pairwise semiregular properties. We also show that $\left(
i,j\right) $-nearly Lindel\"{o}f and pairwise nearly Lindel\"{o}f spaces are
satisfying pairwise semiregular invariant properties.
\section{\textbf{Preliminaries}}
\noindent Throughout this paper, all spaces $\left( X,\tau \right) $ and $
\left( X,\tau _{1},\tau _{2}\right) $ $($or simply $X)$ are always mean
topological spaces and bitopological spaces, respectively unless explicitly
stated. If $\mathcal{P}$ is a topological property, then $\left( \tau
_{i},\tau _{j}\right) $-$\mathcal{P}$ denotes an analogue of this property
for $\tau _{i}$ has property $\mathcal{P}$ with respect to $\tau _{j}$, and $
p$-$\mathcal{P}$ denotes the conjunction $\left( \tau _{1},\tau _{2}\right) $
-$\mathcal{P}\wedge \left( \tau _{2},\tau _{1}\right) $-$\mathcal{P}$, i.e.,
$p$-$\mathcal{P}$ denotes an absolute bitopological analogue of $\mathcal{P}$
. As we shall see below, sometimes $\left( \tau _{1},\tau _{2}\right) $-$
\mathcal{P}\Longleftrightarrow \left( \tau _{2},\tau _{1}\right) $-$\mathcal{
P}$ (and thus $\Leftrightarrow p$-$\mathcal{P}$) so that it suffices to
consider one of these three bitopological analogue. Also sometimes $\tau
_{1} $-$\mathcal{P}\Longleftrightarrow \tau _{2}$-$\mathcal{P}$ and thus $
\mathcal{P}\Longleftrightarrow \tau _{1}$-$\mathcal{P}\wedge \tau _{2}$-$
\mathcal{P}$, i.e., $\left( X,\tau _{i}\right) $ has property $\mathcal{P}$
for each $i=1,2$.
Also note that $\left( X,\tau _{i}\right) $ has a property $\mathcal{P}
\Longleftrightarrow \left( X,\tau _{1},\tau _{2}\right) $ has a property $
\tau _{i}$-$\mathcal{P}$. Sometimes the prefixes $\left( \tau _{i},\tau
_{j}\right) $- or $\tau _{i}$- will be replaced by $\left( i,j\right) $- or $
i$- respectively, if there is no chance for confusion. By $i$-open cover of $
X$, we mean that the cover of $X$ by $i$-open sets in $X$; similar for the $
\left( i,j\right) $-regular open cover of $X$ etc. By $i$-$\limfunc{int}
\left( A\right) $ and $i$-$\limfunc{cl}\left( A\right) $, we shall mean the
interior and the closure of a subset $A$ of $X$ with respect to topology $
\tau _{i}$, respectively. In this paper always $i,j\in \left\{ 1,2\right\} $
and $i\neq j$. The reader may consult \cite{Dvalish05} for the detail
notations.
The following are some basic concepts.
\begin{definition}
\cite{KhedShib91, SingAry71}\textbf{\ }A subset $S$\ of a bitopological
space $\left( X,\tau _{1},\tau _{2}\right) $\ is said to be $\left(
i,j\right) $-regular open $($resp. $\left( i,j\right) $-regular closed$)$\
if $i$-$\limfunc{int}\left( j\text{-}\limfunc{cl}\left( S\right) \right) =S$
\ $($resp. $i$-$\limfunc{cl}\left( j\text{-}\limfunc{int}\left( S\right)
\right) =S)$, where $i,j\in \left\{ 1,2\right\} ,i\neq j$. $S$\ is called
pairwise regular open $($resp. pairwise regular closed$)$\ if it is both $
\left( 1,2\right) $-regular open and $\left( 2,1\right) $-regular open $($
resp. $\left( 1,2\right) $-regular closed and $\left( 2,1\right) $-regular
closed$)$.
\end{definition}
\begin{definition}
\cite{KhedShib91, SingSing70}\textbf{\ }A bitopological space $\left( X,\tau
_{1},\tau _{2}\right) $\ is said to be $\left( i,j\right) $-almost regular
if for each point $x\in X$\ and for each $\left( i,j\right) $-regular open\
set $V$\ containing $x$, there exists an $\left( i,j\right) $-regular open
set $U$\ such that $x\in U\subseteq j$-$\limfunc{cl}\left( U\right)
\subseteq V$. $X$\ is called pairwise almost regular if it is both $\left(
1,2\right) $-almost regular\ and $\left( 2,1\right) $-almost regular.
\end{definition}
In any bitopological space $\left( X,\tau _{1},\tau _{2}\right) $, the
family of all $\left( i,j\right) $-regular open sets is closed under finite
intersections. Thus the family of $\left( i,j\right) $-regular open sets in
any bitopological space $\left( X,\tau _{1},\tau _{2}\right) $ forms a base
for a coarser topology called $\left( i,j\right) $-semiregularization of $
\left( X,\tau _{1},\tau _{2}\right) $, which is defined as follows.\
\begin{definition}
\cite{BidiAdem} The topology generated by the $\left( i,j\right) $-regular
open subsets of $\left( X,\tau _{1},\tau _{2}\right) $\ is denoted by $\tau
_{\left( i,j\right) }^{s}$\ and it is called $\left( i,j\right) $
-semiregularization of $X$. The topologies is pairwise semiregularization of
$X$ if the first topology is $\left( 1,2\right) $-semiregularization of $X$\
and the second topology is $\left( 2,1\right) $-semiregularization of $X$.
If $\tau _{i}\equiv \tau _{\left( i,j\right) }^{s}$, then $X$\ is said to be
$\left( i,j\right) $-semiregular. $\left( X,\tau _{1},\tau _{2}\right) $\ is
called pairwise semiregular if it is both $\left( 1,2\right) $-semiregular
and $\left( 2,1\right) $-semiregular, that is, whenever $\tau _{i}\equiv
\tau _{\left( i,j\right) }^{s}$\ for each $i,j\in \left\{ 1,2\right\} $\ and
$i\neq j$. In other words, $\left( X,\tau _{1},\tau _{2}\right) $ is $\left(
i,j\right) $-semiregular if the family of $\left( i,j\right) $-regular open
sets form a base for the topology $\tau _{i}$.
\end{definition}
It is very clear that $\tau _{\left( i,j\right) }^{s}\subseteq \tau _{i}$,
but it is not necessary $\tau _{i}\subseteq \tau _{\left( i,j\right) }^{s}$.
Thus with every given bitopological space $\left( X,\tau _{1},\tau
_{2}\right) $ there is associated another bitopological space $\left( X,\tau
_{\left( 1,2\right) }^{s},\tau _{\left( 2,1\right) }^{s}\right) $ in the
manner described above (see \cite{SingAry71}). We provide the following
example in order to understand the concept of pairwise semiregular spaces
clearly.
\begin{example}
For the set of all real numbers $
\mathbb{R}
$, let $\tau _{u}$\ denotes the usual topology and $\tau _{s}$\ denote the
Sorgenfrey topology, i.e., topology generated by right half-open intervals $
( $see \cite{Steen78}$)$. Then $\left(
\mathbb{R}
,\tau _{u},\tau _{s}\right) $\ is $\left( \tau _{u},\tau _{s}\right) $
-semiregular since $\tau _{u}=\tau _{\left( \tau _{u},\tau _{s}\right) }^{s}$
, i.e., $\tau _{u}$\ generated by $\left( \tau _{u},\tau _{s}\right) $
-regular open subsets of $
\mathbb{R}
$. $\left(
\mathbb{R}
,\tau _{u},\tau _{s}\right) $\ is also $\left( \tau _{s},\tau _{u}\right) $
-semiregular since $\tau _{s}=\tau _{\left( \tau _{s},\tau _{u}\right) }^{s}$
\ because any set $E\in \tau _{s}$ is the union of a collection of $\left(
\tau _{s},\tau _{u}\right) $-regular open sets in $
\mathbb{R}
$. Thus $\left(
\mathbb{R}
,\tau _{u},\tau _{s}\right) $\ is pairwise semiregular.
\end{example}
Khedr and Alshibani \cite{KhedShib91} defined the equivalent definition of $
\left( i,j\right) $-semiregular spaces as follows.
\begin{definition}
A bitopological space $X$\ is said to be $\left( i,j\right) $-semiregular if
for each $x\in X$\ and for each $i$-open subset $V$\ of $X$\ containing $x$,
there is an $i$-open set $U$\ such that $x\in U\subseteq i$-$\limfunc{int}
\left( j\text{-}\limfunc{cl}\left( U\right) \right) \subseteq V$. $X$\ is
called pairwise semiregular if it is both $\left( 1,2\right) $-semiregular
and $\left( 2,1\right) $-semiregular.
\end{definition}
\begin{definition}
\cite{Datta72} A bitopological space $\left( X,\tau _{1},\tau _{2}\right) $\
is said to be $\left( i,j\right) $-extremally disconnected if the $i$
-closure of every $j$-open set is $j$-open. $X$\ is called pairwise
extremally disconnected if it is both $\left( 1,2\right) $-extremally
disconnected and $\left( 2,1\right) $-extremally disconnected.
\end{definition}
Recall that a property $\mathcal{P}$ will be called bitopological property
(resp. $p$-topological property) if whenever $\left( X,\tau _{1},\tau
_{2}\right) $ has property $\mathcal{P}$, then every space homeomorphic
(resp. $p$-homeomorphic) to $\left( X,\tau _{1},\tau _{2}\right) $ also has
property $\mathcal{P}$ (see \cite{AdemBidi07-1}). If a bitopological space $
X $ has bitopological (or $p$-topological) property $\mathcal{P}$, one may
ask, does the pairwise semiregularization of $X$ satisfies the property $
\mathcal{P}$ also? Now we arrive to the concept of pairwise semiregular
property.\
\begin{definition}
\label{Def 2.6}Let $\left( X,\tau _{1},\tau _{2}\right) $\ be a
bitopological space and let $\left( X,\tau _{\left( 1,2\right) }^{s},\tau
_{\left( 2,1\right) }^{s}\right) $\ its pairwise semiregularization. A
bitopological property $\mathcal{P}$\ is called pairwise semiregular
provided that $\left( X,\tau _{1},\tau _{2}\right) $\ has the property $
\mathcal{P}$\ if and only if $\left( X,\tau _{\left( 1,2\right) }^{s},\tau
_{\left( 2,1\right) }^{s}\right) $\ has the property $\mathcal{P}$.
\end{definition}
\begin{lemma}
\label{Lem 2.1}\cite{BidiAdem} Let $\left( X,\tau _{1},\tau _{2}\right) $\
be a bitopological space and let $\left( X,\tau _{\left( 1,2\right)
}^{s},\tau _{\left( 2,1\right) }^{s}\right) $\ its pairwise
semiregularization. Then
\end{lemma}
\noindent $\left( a\right) $\ $\tau _{i}$-$\limfunc{int}\left( C\right)
=\tau _{\left( i,j\right) }^{s}$-$\limfunc{int}\left( C\right) $\ for every $
\tau _{j}$-closed set $C;$
\noindent $\left( b\right) $\ $\tau _{i}$-$\limfunc{cl}\left( A\right) =\tau
_{\left( i,j\right) }^{s}$-$\limfunc{cl}\left( A\right) $\ for every $A\in
\tau _{j};$
\noindent $\left( c\right) $\ the family of $\left( \tau _{i},\tau
_{j}\right) $-regular open sets of $\left( X,\tau _{1},\tau _{2}\right) $\
are the same as the family of $\left( \tau _{\left( i,j\right) }^{s},\tau
_{\left( j,i\right) }^{s}\right) $-regular open sets of $\left( X,\tau
_{\left( 1,2\right) }^{s},\tau _{\left( 2,1\right) }^{s}\right) ;$
\noindent $\left( d\right) $\ the family of $\left( \tau _{i},\tau
_{j}\right) $-regular closed sets of $\left( X,\tau _{1},\tau _{2}\right) $\
are the same as the family of $\left( \tau _{\left( i,j\right) }^{s},\tau
_{\left( j,i\right) }^{s}\right) $-regular closed sets of $\left( X,\tau
_{\left( 1,2\right) }^{s},\tau _{\left( 2,1\right) }^{s}\right) ;$
\noindent $\left( e\right) $ $\left( \tau _{\left( i,j\right) }^{s}\right)
_{\left( i,j\right) }^{s}=\tau _{\left( i,j\right) }^{s}$.
\section{\textbf{Pairwise Semiregularization of Pairwise Lindel\"{o}f Spaces
\label{Sect 3}}}
\begin{definition}
\cite{ForaHdeib83, AdemBidi07}. A bitopological space $\left( X,\tau
_{1},\tau _{2}\right) $\ is said to be $i$-Lindel\"{o}f if the topological
space $\left( X,\tau _{i}\right) $\ is Lindel\"{o}f. $X$\ is called Lindel
\"{o}f if it is $i$-Lindel\"{o}f for each $i=1,2$. In other words, $\left(
X,\tau _{1},\tau _{2}\right) $\ is called Lindel\"{o}f if the topological
space $\left( X,\tau _{1}\right) $\ and $\left( X,\tau _{2}\right) $\ are
both Lindel\"{o}f.
\end{definition}
Note that $i$-Lindel\"{o}f property as well as Lindel\"{o}f property is not
a pairwise semiregular property by the following example.\
\begin{example}
\label{Ex 3.1}Let $X$\ be a set with cardinality $2^{c}$, where $c=\limfunc{
card}\left(
\mathbb{R}
\right) $. Let $\tau _{1}$\ be a co-$c$\ topology on $X$\ consisting of $
\emptyset $\ and all subsets of $X$\ whose complements have cardinality at
most $c$\ and let $\tau _{2}$\ be a cofinite topology on $X$. Then $\left(
X,\tau _{1},\tau _{2}\right) $\ is $\tau _{2}$-Lindel\"{o}f but not $\tau
_{1}$-Lindel\"{o}f and hence not Lindel\"{o}f. Observe that $\left( X,\tau
_{\left( 1,2\right) }^{s},\tau _{\left( 2,1\right) }^{s}\right) $\ is $\tau
_{\left( 1,2\right) }^{s}$-Lindel\"{o}f and $\tau _{\left( 2,1\right) }^{s}$
-Lindel\"{o}f since $\tau _{\left( 1,2\right) }^{s}$\ and $\tau _{\left(
2,1\right) }^{s}$\ are indiscrete topologies. Hence $\left( X,\tau _{\left(
1,2\right) }^{s},\tau _{\left( 2,1\right) }^{s}\right) $\ is Lindel\"{o}f
\textit{.}
\end{example}
\begin{definition}
A bitopological space $\left( X,\tau _{1},\tau _{2}\right) $\ is called $
\left( i,j\right) $-Lindel\"{o}f \cite{ForaHdeib83, AdemBidi07}\ if for
every $i$-open cover of $X$\ there is a countable\ $j$-open subcover. $X$\
is called $B$-Lindel\"{o}f \cite{ForaHdeib83}\ or $p_{1}$-Lindel\"{o}f \cite
{AdemBidi07}\ if it is both $\left( 1,2\right) $-Lindel\"{o}f\ and $\left(
2,1\right) $-Lindel\"{o}f.
\end{definition}
An $\left( i,j\right) $-Lindel\"{o}f property as well as $B$-Lindel\"{o}f
property is not pairwise semiregular property by the following example.
\begin{example}
Let $\left( X,\tau _{1},\tau _{2}\right) $\ be a bitopological space as in
Example \ref{Ex 3.1}. Then $\left( X,\tau _{1},\tau _{2}\right) $\ is not $
\left( \tau _{1},\tau _{2}\right) $-Lindel\"{o}f but it is $\left( \tau
_{2},\tau _{1}\right) $-Lindel\"{o}f and hence not $B$-Lindel\"{o}f. Observe
that $\left( X,\tau _{\left( 1,2\right) }^{s},\tau _{\left( 2,1\right)
}^{s}\right) $\ is $\left( \tau _{\left( 1,2\right) }^{s},\tau _{\left(
2,1\right) }^{s}\right) $-Lindel\"{o}f and $\left( \tau _{\left( 2,1\right)
}^{s},\tau _{\left( 1,2\right) }^{s}\right) $-Lindel\"{o}f since $\tau
_{\left( 1,2\right) }^{s}$\ and $\tau _{\left( 2,1\right) }^{s}$\ are
indiscrete topologies. Hence $\left( X,\tau _{\left( 1,2\right) }^{s},\tau
_{\left( 2,1\right) }^{s}\right) $\ is $B$-Lindel\"{o}f.
\end{example}
\begin{definition}
A cover $U\ $of a bitopological space $\left( X,\tau _{1},\tau _{2}\right) $
\ is called $\tau _{1}\tau _{2}$-open \cite{Swart71} if $U\subseteq \tau
_{1}\cup \tau _{2}$. If, in addition, $U$\ contains at least one nonempty
member of $\tau _{1}$\ and at least one nonempty member of $\tau _{2}$, it
is called $p$-open \cite{Fletcher69}.
\end{definition}
\begin{definition}
\cite{ForaHdeib83}\textbf{\ }A bitopological space $\left( X,\tau _{1},\tau
_{2}\right) $\ is called $s$-Lindel\"{o}f \ $($resp. $p$-Lindel\"{o}f$)$\ if
every $\tau _{1}\tau _{2}$-open $($resp. $p$-open$)$\ cover of $X$\ has a
countable\ subcover.
\end{definition}
A $p$-Lindel\"{o}f property is not pairwise semiregular property by the
following example. Thus the $s$-Lindel\"{o}f property is also not pairwise
semiregular property.
\begin{example}
Let $\left( X,\tau _{1},\tau _{2}\right) $\ be a bitopological space as in
Example \ref{Ex 3.1}. Then $\left( X,\tau _{1},\tau _{2}\right) $\ is not $p$
-Lindel\"{o}f and hence not $s$-Lindel\"{o}f. Observe that $\left( X,\tau
_{\left( 1,2\right) }^{s},\tau _{\left( 2,1\right) }^{s}\right) $\ is $p$
-Lindel\"{o}f and $s$-Lindel\"{o}f since $\tau _{\left( 1,2\right) }^{s}$\
and $\tau _{\left( 2,1\right) }^{s}$\ are indiscrete topologies.
\end{example}
\section{\textbf{Pairwise Semiregularization of Generalized Pairwise Lindel
\"{o}f Spaces}}
\begin{definition}
\cite{BidiAdem, AdemBidi, BidiAdem1}\textbf{\ }A bitopological space $X$\ is
said to be $\left( i,j\right) $-nearly Lindel\"{o}f $($resp. $\left(
i,j\right) $-almost Lindel\"{o}f, $\left( i,j\right) $-weakly Lindel\"{o}f$)$
\ if for every $i$-open cover $\left\{ U_{\alpha }:\alpha \in \Delta
\right\} $\ of $X$, there exists a countable subset $\left\{ \alpha
_{n}:n\in
\mathbb{N}
\right\} $\ of $\Delta $\ such that $X=\dbigcup\limits_{n\in
\mathbb{N}
}i$-$\limfunc{int}\left( j\text{-}\limfunc{cl}\left( U_{\alpha _{n}}\right)
\right) $\ $\left( \text{\textit{resp.}}\mathit{\ }X=\dbigcup\limits_{n\in
\mathbb{N}
}j\text{\textit{-}}\limfunc{cl}\left( U_{\alpha _{n}}\right) \text{\textit{,
}}X=j\text{\textit{-}}\limfunc{cl}\left( \dbigcup\limits_{n\in
\mathbb{N}
}\left( U_{\alpha _{n}}\right) \right) \right) $.\ $X$\ is called pairwise
nearly Lindel\"{o}f $($resp. pairwise almost Lindel\"{o}f, pairwise weakly
Lindel\"{o}f$)$\ if it\ is both $\left( 1,2\right) $-nearly Lindel\"{o}f $($
resp. $\left( 1,2\right) $-almost Lindel\"{o}f, $\left( 1,2\right) $-weakly
Lindel\"{o}f$)$\ and $\left( 2,1\right) $-nearly Lindel\"{o}f $($resp. $
\left( 2,1\right) $-almost Lindel\"{o}f, $\left( 2,1\right) $-weakly Lindel
\"{o}f$)$.
\end{definition}
Our first result is analogue with the result of Mr\v{s}evi\'{c} et al. \cite[
Theorem 1]{MrsevicReillyVamanamurthy86}.
\begin{theorem}
\label{Th 4.1}A bitopological space $\left( X,\tau _{1},\tau _{2}\right) $
is $\left( \tau _{i},\tau _{j}\right) $-nearly Lindel\"{o}f if and only if $
\left( X,\tau _{\left( 1,2\right) }^{s},\tau _{\left( 2,1\right)
}^{s}\right) $\ is $\tau _{\left( i,j\right) }^{s}$-Lindel\"{o}f.
\end{theorem}
\begin{proof}
Let $\left( X,\tau _{1},\tau _{2}\right) $ be a $\left( \tau _{i},\tau
_{j}\right) $-nearly Lindel\"{o}f and let $\left\{ U_{\alpha }:\alpha \in
\Delta \right\} $ be a $\tau _{\left( i,j\right) }^{s}$-open cover of $
\left( X,\tau _{\left( 1,2\right) }^{s},\tau _{\left( 2,1\right)
}^{s}\right) $. For each $x\in X$, there exists $\alpha _{x}\in \Delta $
such that $x\in U_{\alpha _{x}}$ and since for each $\alpha _{x}\in \Delta
,U_{\alpha _{x}}\in $ $\tau _{\left( i,j\right) }^{s}$, there exists a $
\left( \tau _{i},\tau _{j}\right) $-regular open set $V_{\alpha _{x}}$ in $
\left( X,\tau _{1},\tau _{2}\right) $ such that $x\in V_{\alpha
_{x}}\subseteq U_{\alpha _{x}}$. So $X=\dbigcup\nolimits_{x\in X}V_{\alpha
_{x}}$ and hence $\left\{ V_{\alpha _{x}}:x\in X\right\} $ is a $\left( \tau
_{i},\tau _{j}\right) $-regular open cover of $X$. Since $\left( X,\tau
_{1},\tau _{2}\right) $ is $\left( \tau _{i},\tau _{j}\right) $-nearly Lindel
\"{o}f, there exists a countable subset of points $x_{1},\ldots
,x_{n},\ldots $ of $X$ such that $X=\dbigcup\nolimits_{n\in
\mathbb{N}
}V_{\alpha _{x_{n}}}\subseteq \dbigcup\nolimits_{n\in
\mathbb{N}
}U_{\alpha _{x_{n}}}$. This shows that $\left( X,\tau _{\left( 1,2\right)
}^{s},\tau _{\left( 2,1\right) }^{s}\right) $ is $\tau _{\left( i,j\right)
}^{s}$-Lindel\"{o}f.
Conversely, suppose that $\left( X,\tau _{\left( 1,2\right) }^{s},\tau
_{\left( 2,1\right) }^{s}\right) $\ is $\tau _{\left( i,j\right) }^{s}$
-Lindel\"{o}f and let $\left\{ V_{\alpha }:\alpha \in \Delta \right\} $ be a
$\left( \tau _{i},\tau _{j}\right) $-regular open cover of $\left( X,\tau
_{1},\tau _{2}\right) $. Since $V_{\alpha }\in \tau _{\left( i,j\right)
}^{s} $ for each $\alpha \in \Delta ,\left\{ V_{\alpha }:\alpha \in \Delta
\right\} $ is a $\tau _{\left( i,j\right) }^{s}$-open cover of $\left(
X,\tau _{\left( 1,2\right) }^{s},\tau _{\left( 2,1\right) }^{s}\right) $.
Since $\left( X,\tau _{\left( 1,2\right) }^{s},\tau _{\left( 2,1\right)
}^{s}\right) $\ is $\tau _{\left( i,j\right) }^{s}$-Lindel\"{o}f, there
exists a countable subcover such that $X=\dbigcup\nolimits_{n\in
\mathbb{N}
}V_{\alpha _{n}}$. This implies that $\left( X,\tau _{1},\tau _{2}\right) $
is $\left( \tau _{i},\tau _{j}\right) $-nearly Lindel\"{o}f.
\end{proof}
\begin{corollary}
A bitopological space $\left( X,\tau _{1},\tau _{2}\right) $ is pairwise
nearly Lindel\"{o}f if and only if $\left( X,\tau _{\left( 1,2\right)
}^{s},\tau _{\left( 2,1\right) }^{s}\right) $\ is Lindel\"{o}f.
\end{corollary}
\begin{proposition}
\label{Prop 4.1}A bitopological space $\left( X,\tau _{\left( 1,2\right)
}^{s},\tau _{\left( 2,1\right) }^{s}\right) $ is $\left( \tau _{\left(
i,j\right) }^{s},\tau _{\left( j,i\right) }^{s}\right) $-nearly Lindel\"{o}f
if and only if $\left( X,\tau _{\left( 1,2\right) }^{s},\tau _{\left(
2,1\right) }^{s}\right) $\ is $\tau _{\left( i,j\right) }^{s}$-Lindel\"{o}f.
\end{proposition}
\begin{proof}
The sufficient condition is obvious by the definitions. So we need only to
prove necessary condition. Suppose that $\left\{ U_{\alpha }:\alpha \in
\Delta \right\} $ is a $\tau _{\left( i,j\right) }^{s}$-open cover of $
\left( X,\tau _{\left( 1,2\right) }^{s},\tau _{\left( 2,1\right)
}^{s}\right) $. For each $x\in X$, there exists $\alpha _{x}\in \Delta $
such that $x\in U_{\alpha _{x}}$. Since $\left( X,\tau _{\left( 1,2\right)
}^{s},\tau _{\left( 2,1\right) }^{s}\right) $ is $\left( \tau _{\left(
i,j\right) }^{s},\tau _{\left( j,i\right) }^{s}\right) $-semiregular, there
exists a $\tau _{\left( i,j\right) }^{s}$-open set $V_{\alpha _{x}}$ in $
\left( X,\tau _{\left( 1,2\right) }^{s},\tau _{\left( 2,1\right)
}^{s}\right) $ such that $x\in V_{\alpha _{x}}\subseteq \tau _{\left(
i,j\right) }^{s}$-$\limfunc{int}\left( \tau _{\left( j,i\right) }^{s}\text{-}
\limfunc{cl}\left( V_{\alpha _{x}}\right) \right) \subseteq U_{\alpha _{x}}$
. Hence $X=\dbigcup\nolimits_{x\in X}V_{\alpha _{x}}$ and thus the family $
\left\{ V_{\alpha _{x}}:x\in X\right\} $ forms a $\tau _{\left( i,j\right)
}^{s}$-open cover of $\left( X,\tau _{\left( 1,2\right) }^{s},\tau _{\left(
2,1\right) }^{s}\right) $.\ Since $\left( X,\tau _{\left( 1,2\right)
}^{s},\tau _{\left( 2,1\right) }^{s}\right) $ is $\left( \tau _{\left(
i,j\right) }^{s},\tau _{\left( j,i\right) }^{s}\right) $-nearly Lindel\"{o}
f, there exists a countable subset of points $x_{1},\ldots ,x_{n},\ldots $
of $X$ such that $X=\dbigcup\nolimits_{n\in
\mathbb{N}
}\tau _{\left( i,j\right) }^{s}$-$\limfunc{int}\left( \tau _{\left(
j,i\right) }^{s}\text{-}\limfunc{cl}\left( V_{\alpha _{x_{n}}}\right)
\right) \subseteq \dbigcup\nolimits_{n\in
\mathbb{N}
}U_{\alpha _{x_{n}}}$. This shows that $\left( X,\tau _{\left( 1,2\right)
}^{s},\tau _{\left( 2,1\right) }^{s}\right) $ is $\tau _{\left( i,j\right)
}^{s}$-Lindel\"{o}f.
\end{proof}
\begin{corollary}
A bitopological space $\left( X,\tau _{\left( 1,2\right) }^{s},\tau _{\left(
2,1\right) }^{s}\right) $ is pairwise nearly Lindel\"{o}f if and only if $
\left( X,\tau _{\left( 1,2\right) }^{s},\tau _{\left( 2,1\right)
}^{s}\right) $\ is Lindel\"{o}f.
\end{corollary}
From the Definition \ref{Def 2.6}, if the property $\mathcal{P}$ is not
bitopological property but it satisfies the condition $\left( X,\tau
_{1},\tau _{2}\right) $\ has the property $\mathcal{P}$\ if and only if $
\left( X,\tau _{\left( 1,2\right) }^{s},\tau _{\left( 2,1\right)
}^{s}\right) $\ has the property $\mathcal{P}$, then the property $\mathcal{P
}$ will be called pairwise semiregular invariant property. The following
theorem prove that $\left( i,j\right) $-nearly Lindel\"{o}f as well as
pairwise nearly Lindel\"{o}f property satisfying the pairwise semiregular
invariant property since $\left( i,j\right) $-nearly Lindel\"{o}f and
pairwise nearly Lindel\"{o}f are not $i$-topological property \cite
{AdemBidi07-1} and bitopological property, respectively. This is because the
$i$-continuity and $\left( i,j\right) $-$\delta $-continuity (resp.
continuity and $p$-$\delta $-continuity) are independent notions (see \cite
{AdemBidi2010}).
\begin{theorem}
A bitopological space $\left( X,\tau _{1},\tau _{2}\right) $\ is $\left(
\tau _{i},\tau _{j}\right) $-nearly Lindel\"{o}f if and only if $\left(
X,\tau _{\left( 1,2\right) }^{s},\tau _{\left( 2,1\right) }^{s}\right) $\ is
$\left( \tau _{\left( i,j\right) }^{s},\tau _{\left( j,i\right) }^{s}\right)
$-nearly Lindel\"{o}f.
\end{theorem}
\begin{proof}
It is obvious by Theorem \ref{Th 4.1} and Proposition \ref{Prop 4.1}. \
\end{proof}
\begin{corollary}
A bitopological space $\left( X,\tau _{1},\tau _{2}\right) $\ is pairwise
nearly Lindel\"{o}f if and only if $\left( X,\tau _{\left( 1,2\right)
}^{s},\tau _{\left( 2,1\right) }^{s}\right) $\ is pairwise nearly Lindel\"{o}
f\textit{.}
\end{corollary}
\begin{theorem}
\label{Th 4.3}\cite{SingAry71}\textbf{\ }If $\left( X,\tau _{1},\tau
_{2}\right) $\ is pairwise semiregular, then $\left( X,\tau _{1},\tau
_{2}\right) =\left( X,\tau _{\left( 1,2\right) }^{s},\tau _{\left(
2,1\right) }^{s}\right) $.\
\end{theorem}
The converse of Theorem \ref{Th 4.3} is also true by the definitions.
\begin{proposition}
Let $\left( X,\tau _{1},\tau _{2}\right) $\ be a pairwise semiregular space.
Then $\left( X,\tau _{1},\tau _{2}\right) $ is $\left( i,j\right) $-nearly
Lindel\"{o}f if and only if it is $i$-Lindel\"{o}f.
\end{proposition}
\begin{proof}
By Theorem \ref{Th 4.3}, $\left( X,\tau _{1},\tau _{2}\right) =\left( X,\tau
_{\left( 1,2\right) }^{s},\tau _{\left( 2,1\right) }^{s}\right) $. The
result follows immediately by Proposition \ref{Prop 4.1}.
\end{proof}
\begin{corollary}
Let $\left( X,\tau _{1},\tau _{2}\right) $\ be a pairwise semiregular space.
Then $\left( X,\tau _{1},\tau _{2}\right) $ is pairwise nearly Lindel\"{o}f
if and only if it is Lindel\"{o}f.
\end{corollary}
Unlike all types of pairwise Lindel\"{o}f properties, the $\left( i,j\right)
$-almost Lindel\"{o}f, pairwise almost Lindel\"{o}f, $\left( i,j\right) $
-weakly Lindel\"{o}f and pairwise weakly Lindel\"{o}f properties are
pairwise semiregular properties as we prove in the following theorems.
\begin{theorem}
\label{Th 4.4}A bitopological space $\left( X,\tau _{1},\tau _{2}\right) $\
is $\left( \tau _{i},\tau _{j}\right) $-almost Lindel\"{o}f if and only if $
\left( X,\tau _{\left( 1,2\right) }^{s},\tau _{\left( 2,1\right)
}^{s}\right) $\ is $\left( \tau _{\left( i,j\right) }^{s},\tau _{\left(
j,i\right) }^{s}\right) $-almost Lindel\"{o}f.
\end{theorem}
\begin{proof}
Let $\left( X,\tau _{1},\tau _{2}\right) $ be a $\left( \tau _{i},\tau
_{j}\right) $-almost Lindel\"{o}f and let $\left\{ U_{\alpha }:\alpha \in
\Delta \right\} $ be a $\tau _{\left( i,j\right) }^{s}$-open cover of $
\left( X,\tau _{\left( 1,2\right) }^{s},\tau _{\left( 2,1\right)
}^{s}\right) $. Since $\tau _{\left( i,j\right) }^{s}\subseteq \tau _{i}$, $
\left\{ U_{\alpha }:\alpha \in \Delta \right\} $ is a $\tau _{i}$-open cover
of the $\left( \tau _{i},\tau _{j}\right) $-almost Lindel\"{o}f space $
\left( X,\tau _{1},\tau _{2}\right) $. Then there is a countable subset $
\left\{ \alpha _{n}:n\in
\mathbb{N}
\right\} $ of $\Delta $ such that $X=\dbigcup\nolimits_{n\in
\mathbb{N}
}\tau _{j}$-$\limfunc{cl}\left( U_{\alpha _{n}}\right) $. By Lemma \ref{Lem
2.1}, we have $X=\dbigcup\nolimits_{n\in
\mathbb{N}
}\tau _{\left( j,i\right) }^{s}$-$\limfunc{cl}\left( U_{\alpha _{n}}\right) $
, which implies $\left( X,\tau _{\left( 1,2\right) }^{s},\tau _{\left(
2,1\right) }^{s}\right) $ is $\left( \tau _{\left( i,j\right) }^{s},\tau
_{\left( j,i\right) }^{s}\right) $-almost Lindel\"{o}f.
Conversely suppose that $\left( X,\tau _{\left( 1,2\right) }^{s},\tau
_{\left( 2,1\right) }^{s}\right) $\textit{\ }is $\left( \tau _{\left(
i,j\right) }^{s},\tau _{\left( j,i\right) }^{s}\right) $-almost Lindel\"{o}f
and let $\left\{ V_{\alpha }:\alpha \in \Delta \right\} $ be a $\tau _{i}$
-open cover of $\left( X,\tau _{1},\tau _{2}\right) $. Since $V_{\alpha
}\subseteq \tau _{i}$-$\limfunc{int}\left( \tau _{j}\text{-}\limfunc{cl}
\left( V_{\alpha }\right) \right) $ and $\tau _{i}$-$\limfunc{int}\left(
\tau _{j}\text{-}\limfunc{cl}\left( V_{\alpha }\right) \right) \in $ $\tau
_{\left( i,j\right) }^{s}$, we have $\left\{ \tau _{i}\text{-}\limfunc{int}
\left( \tau _{j}\text{-}\limfunc{cl}\left( V_{\alpha }\right) \right)
:\alpha \in \Delta \right\} $ is a $\tau _{\left( i,j\right) }^{s}$-open
cover of the $\left( \tau _{\left( i,j\right) }^{s},\tau _{\left( j,i\right)
}^{s}\right) $-almost Lindel\"{o}f space $\left( X,\tau _{\left( 1,2\right)
}^{s},\tau _{\left( 2,1\right) }^{s}\right) $. So there is a countable
subset $\left\{ \alpha _{n}:n\in
\mathbb{N}
\right\} $ of $\Delta $ such that $X=\dbigcup\nolimits_{n\in
\mathbb{N}
}$ $\tau _{\left( j,i\right) }^{s}$-$\limfunc{cl}\left( \tau _{i}\text{-}
\limfunc{int}\left( \tau _{j}\text{-}\limfunc{cl}\left( V_{\alpha
_{n}}\right) \right) \right) $. By Lemma \ref{Lem 2.1}, we have $
X=\dbigcup\nolimits_{n\in
\mathbb{N}
}$ $\tau _{j}$-$\limfunc{cl}\left( \tau _{i}\text{-}\limfunc{int}\left( \tau
_{j}\text{-}\limfunc{cl}\left( V_{\alpha _{n}}\right) \right) \right)
\subseteq \dbigcup\nolimits_{n\in
\mathbb{N}
}$ $\tau _{j}$-$\limfunc{cl}\left( V_{\alpha _{n}}\right) $. This implies
that $\left( X,\tau _{1},\tau _{2}\right) $ is $\left( \tau _{i},\tau
_{j}\right) $-almost Lindel\"{o}f.
\end{proof}
\begin{corollary}
\label{Cor 4.5}A bitopological space $\left( X,\tau _{1},\tau _{2}\right) $\
is pairwise almost Lindel\"{o}f if and only if $\left( X,\tau _{\left(
1,2\right) }^{s},\tau _{\left( 2,1\right) }^{s}\right) $\ is pairwise almost
Lindel\"{o}f.
\end{corollary}
Note that, the $\left( i,j\right) $-almost Lindel\"{o}f property and the
pairwise almost Lindel\"{o}f property are both bitopological properties (see
\cite{BidiAdem2}). Utilizing this fact, Theorem \ref{Th 4.4} and Corollary
\ref{Cor 4.5}, we easily obtain the following corollary.
\begin{corollary}
The $\left( i,j\right) $-almost Lindel\"{o}f property and the pairwise
almost Lindel\"{o}f property are both pairwise semiregular properties.
\end{corollary}
\begin{proposition}
Let $\left( X,\tau _{1},\tau _{2}\right) $\ be a $\left( \tau _{i},\tau
_{j}\right) $-almost regular space. Then $\left( X,\tau _{1},\tau
_{2}\right) $\ is $\left( \tau _{i},\tau _{j}\right) $-almost Lindel\"{o}f
if and only if $\left( X,\tau _{\left( 1,2\right) }^{s},\tau _{\left(
2,1\right) }^{s}\right) $\ is $\tau _{\left( i,j\right) }^{s}$-Lindel\"{o}f.
\end{proposition}
\begin{proof}
Let $\left( X,\tau _{1},\tau _{2}\right) $ be a $\left( \tau _{i},\tau
_{j}\right) $-almost Lindel\"{o}f and let $\left\{ U_{\alpha }:\alpha \in
\Delta \right\} $ be a $\tau _{\left( i,j\right) }^{s}$-open cover of $
\left( X,\tau _{\left( 1,2\right) }^{s},\tau _{\left( 2,1\right)
}^{s}\right) $. For each $x\in X$, there exists $\alpha _{x}\in \Delta $
such that $x\in U_{\alpha _{x}}$ and since $U_{\alpha _{x}}\in $ $\tau
_{\left( i,j\right) }^{s}$, there exists a $\left( \tau _{i},\tau
_{j}\right) $-regular open set $V_{\alpha _{x}}$ in $\left( X,\tau _{1},\tau
_{2}\right) $ such that $x\in V_{\alpha _{x}}\subseteq U_{\alpha _{x}}$.
Since $\left( X,\tau _{1},\tau _{2}\right) $\textit{\ }is $\left( \tau
_{i},\tau _{j}\right) $-almost regular, there is a $\left( \tau _{i},\tau
_{j}\right) $-regular open set $C_{\alpha _{x}}$ in $\left( X,\tau _{1},\tau
_{2}\right) $ such that $x\in C_{\alpha _{x}}\subseteq \tau _{j}$-$\limfunc{
cl}\left( C_{\alpha _{x}}\right) \subseteq V_{\alpha _{x}}$. Hence $
X=\dbigcup\nolimits_{x\in X}C_{\alpha _{x}}$ and thus the family $\left\{
C_{\alpha _{x}}:x\in X\right\} $ forms a $\left( \tau _{i},\tau _{j}\right) $
-regular open cover of $\left( X,\tau _{1},\tau _{2}\right) $. Since $\left(
X,\tau _{1},\tau _{2}\right) $ is $\left( \tau _{i},\tau _{j}\right) $
-almost Lindel\"{o}f, there exists a countable subset of points $
x_{1},\ldots ,x_{n},\ldots $ of $X$ such that $X=\dbigcup\nolimits_{n\in
\mathbb{N}
}\tau _{j}$-$\limfunc{cl}\left( C_{\alpha _{x_{n}}}\right) \subseteq
\dbigcup\nolimits_{n\in
\mathbb{N}
}V_{\alpha _{x_{n}}}\subseteq \dbigcup\nolimits_{n\in
\mathbb{N}
}U_{\alpha _{x_{n}}}$. This shows that $\left( X,\tau _{\left( 1,2\right)
}^{s},\tau _{\left( 2,1\right) }^{s}\right) $ is $\tau _{\left( i,j\right)
}^{s}$-Lindel\"{o}f. Conversely, let $\left( X,\tau _{\left( 1,2\right)
}^{s},\tau _{\left( 2,1\right) }^{s}\right) $ be a $\tau _{\left( i,j\right)
}^{s}$-Lindel\"{o}f and let $\left\{ U_{\alpha }:\alpha \in \Delta \right\} $
be a $\tau _{i}$-open cover of $\left( X,\tau _{1},\tau _{2}\right) $. Since
$U_{\alpha }\subseteq \tau _{i}$-$\limfunc{int}\left( \tau _{j}\text{-}
\limfunc{cl}\left( U_{\alpha }\right) \right) $ and $\tau _{i}$-$\limfunc{int
}\left( \tau _{j}\text{-}\limfunc{cl}\left( U_{\alpha }\right) \right) \in $
$\tau _{\left( i,j\right) }^{s}$, $\left\{ \tau _{i}\text{-}\limfunc{int}
\left( \tau _{j}\text{-}\limfunc{cl}\left( U_{\alpha }\right) \right)
:\alpha \in \Delta \right\} $ is $\tau _{\left( i,j\right) }^{s}$-open cover
of the $\tau _{\left( i,j\right) }^{s}$-Lindel\"{o}f space $\left( X,\tau
_{\left( 1,2\right) }^{s},\tau _{\left( 2,1\right) }^{s}\right) $. Then
there exists a countable subset $\left\{ \alpha _{n}:n\in
\mathbb{N}
\right\} $ of $\Delta $ such that $X=\dbigcup\nolimits_{n\in
\mathbb{N}
}\tau _{i}$-$\limfunc{int}\left( \tau _{j}\text{-}\limfunc{cl}\left(
U_{\alpha _{n}}\right) \right) \subseteq \dbigcup\nolimits_{n\in
\mathbb{N}
}\tau _{j}$-$\limfunc{cl}\left( U_{\alpha _{n}}\right) $. This implies that $
\left( X,\tau _{1},\tau _{2}\right) $ is $\left( \tau _{i},\tau _{j}\right) $
-almost Lindel\"{o}f.\bigskip
\end{proof}
\begin{corollary}
Let $\left( X,\tau _{1},\tau _{2}\right) $\ be a pairwise almost regular
space. Then $\left( X,\tau _{1},\tau _{2}\right) $\ is pairwise almost Lindel
\"{o}f if and only if $\left( X,\tau _{\left( 1,2\right) }^{s},\tau _{\left(
2,1\right) }^{s}\right) $\ is Lindel\"{o}f.
\end{corollary}
\begin{proposition}
\label{Prop 4.4}Let $\left( X,\tau _{\left( 1,2\right) }^{s},\tau _{\left(
2,1\right) }^{s}\right) $\ be a $\left( \tau _{\left( j,i\right) }^{s},\tau
_{\left( i,j\right) }^{s}\right) $-extremally disconnected space. Then $
\left( X,\tau _{\left( 1,2\right) }^{s},\tau _{\left( 2,1\right)
}^{s}\right) $ is $\left( \tau _{\left( i,j\right) }^{s},\tau _{\left(
j,i\right) }^{s}\right) $-almost Lindel\"{o}f if and only if $\left( X,\tau
_{\left( 1,2\right) }^{s},\tau _{\left( 2,1\right) }^{s}\right) $\ is $\tau
_{\left( i,j\right) }^{s}$-Lindel\"{o}f.
\end{proposition}
\begin{proof}
The sufficient condition is obvious by the definitions. So we need only to
prove necessary condition. Suppose that $\left\{ U_{\alpha }:\alpha \in
\Delta \right\} $ is a $\tau _{\left( i,j\right) }^{s}$-open cover of $
\left( X,\tau _{\left( 1,2\right) }^{s},\tau _{\left( 2,1\right)
}^{s}\right) $. For each $x\in X$, there exists $\alpha _{x}\in \Delta $
such that $x\in U_{\alpha _{x}}$. Since $\left( X,\tau _{\left( 1,2\right)
}^{s},\tau _{\left( 2,1\right) }^{s}\right) $ is $\left( \tau _{\left(
i,j\right) }^{s},\tau _{\left( j,i\right) }^{s}\right) $-semiregular, there
exists a $\tau _{\left( i,j\right) }^{s}$-open set $V_{\alpha _{x}}$ in $
\left( X,\tau _{\left( 1,2\right) }^{s},\tau _{\left( 2,1\right)
}^{s}\right) $ such that $x\in V_{\alpha _{x}}\subseteq \tau _{\left(
i,j\right) }^{s}$-$\limfunc{int}\left( \tau _{\left( j,i\right) }^{s}\text{-}
\limfunc{cl}\left( V_{\alpha _{x}}\right) \right) \subseteq U_{\alpha _{x}}$
. Hence $X=\dbigcup\nolimits_{x\in X}V_{\alpha _{x}}$ and thus the family $
\left\{ V_{\alpha _{x}}:x\in X\right\} $ forms a $\tau _{\left( i,j\right)
}^{s}$-open cover of $\left( X,\tau _{\left( 1,2\right) }^{s},\tau _{\left(
2,1\right) }^{s}\right) $.\ Since $\left( X,\tau _{\left( 1,2\right)
}^{s},\tau _{\left( 2,1\right) }^{s}\right) $ is $\left( \tau _{\left(
i,j\right) }^{s},\tau _{\left( j,i\right) }^{s}\right) $-almost Lindel\"{o}f
and $\left( \tau _{\left( j,i\right) }^{s},\tau _{\left( i,j\right)
}^{s}\right) $-extremally disconnected, there exists a countable subset of
points $x_{1},\ldots ,x_{n},\ldots $ of $X$ such that $X=\dbigcup
\nolimits_{n\in
\mathbb{N}
}\tau _{\left( j,i\right) }^{s}$-$\limfunc{cl}\left( V_{\alpha
_{x_{n}}}\right) =\dbigcup\nolimits_{n\in
\mathbb{N}
}\tau _{\left( i,j\right) }^{s}$-$\limfunc{int}\left( \tau _{\left(
j,i\right) }^{s}\text{-}\limfunc{cl}\left( V_{\alpha _{x_{n}}}\right)
\right) \subseteq \dbigcup\nolimits_{n\in
\mathbb{N}
}U_{\alpha _{x_{n}}}$. This shows that $\left( X,\tau _{\left( 1,2\right)
}^{s},\tau _{\left( 2,1\right) }^{s}\right) $ is $\tau _{\left( i,j\right)
}^{s}$-Lindel\"{o}f.
\end{proof}
\begin{corollary}
Let $\left( X,\tau _{\left( 1,2\right) }^{s},\tau _{\left( 2,1\right)
}^{s}\right) $\ be a pairwise extremally disconnected space. Then $\left(
X,\tau _{\left( 1,2\right) }^{s},\tau _{\left( 2,1\right) }^{s}\right) $ is
pairwise almost Lindel\"{o}f if and only if $\left( X,\tau _{\left(
1,2\right) }^{s},\tau _{\left( 2,1\right) }^{s}\right) $\ is Lindel\"{o}f
\end{corollary}
\begin{proposition}
Let $\left( X,\tau _{1},\tau _{2}\right) $\ be a pairwise semiregular and $
\left( j,i\right) $-extremally disconnected space. Then $\left( X,\tau
_{1},\tau _{2}\right) $ is $\left( i,j\right) $-almost Lindel\"{o}f if and
only if it is $i$-Lindel\"{o}f.
\end{proposition}
\begin{proof}
By Theorem \ref{Th 4.3}, $\left( X,\tau _{1},\tau _{2}\right) =\left( X,\tau
_{\left( 1,2\right) }^{s},\tau _{\left( 2,1\right) }^{s}\right) $. The
result follows immediately by Proposition \ref{Prop 4.4}.
\end{proof}
\begin{corollary}
Let $\left( X,\tau _{1},\tau _{2}\right) $\ be a pairwise semiregular and
pairwise extremally disconnected space. Then $\left( X,\tau _{1},\tau
_{2}\right) $ is pairwise almost Lindel\"{o}f if and only if it is Lindel
\"{o}f.
\end{corollary}
\begin{theorem}
\label{Th 4.5}A bitopological space $\left( X,\tau _{1},\tau _{2}\right) $\
is $\left( \tau _{i},\tau _{j}\right) $-weakly Lindel\"{o}f if and only if $
\left( X,\tau _{\left( 1,2\right) }^{s},\tau _{\left( 2,1\right)
}^{s}\right) $\ is $\left( \tau _{\left( i,j\right) }^{s},\tau _{\left(
j,i\right) }^{s}\right) $-weakly Lindel\"{o}f.
\end{theorem}
\begin{proof}
The proof is similar to the proof of Theorem \ref{Th 4.4} by using the fact
that
\begin{eqnarray*}
\tau _{\left( j,i\right) }^{s}\text{-}\limfunc{cl}\left(
\dbigcup\nolimits_{n\in
\mathbb{N}
}\tau _{i}\text{-}\limfunc{int}\left( \tau _{j}\text{-}\limfunc{cl}\left(
V_{\alpha _{n}}\right) \right) \right) &=&\tau _{j}\text{-}\limfunc{cl}
\left( \dbigcup\nolimits_{n\in
\mathbb{N}
}\tau _{i}\text{-}\limfunc{int}\left( \tau _{j}\text{-}\limfunc{cl}\left(
V_{\alpha _{n}}\right) \right) \right) \\
&\subseteq &\tau _{j}\text{-}\limfunc{cl}\left( \dbigcup\nolimits_{n\in
\mathbb{N}
}\tau _{j}\text{-}\limfunc{cl}\left( V_{\alpha _{n}}\right) \right) \\
&\subseteq &\tau _{j}\text{-}\limfunc{cl}\left( \dbigcup\nolimits_{n\in
\mathbb{N}
}V_{\alpha _{n}}\right) \text{.}
\end{eqnarray*}
Thus we choose to omit the details.
\end{proof}
\begin{corollary}
\label{Cor 4.10}A bitopological space $\left( X,\tau _{1},\tau _{2}\right) $
\ is pairwise weakly Lindel\"{o}f if and only if $\left( X,\tau _{\left(
1,2\right) }^{s},\tau _{\left( 2,1\right) }^{s}\right) $\ is pairwise weakly
Lindel\"{o}f.
\end{corollary}
Note that, the $\left( i,j\right) $-weakly Lindel\"{o}f property and the
pairwise weakly Lindel\"{o}f property are both bitopological properties (see
\cite{BidiAdem2}). Utilizing this fact, Theorem \ref{Th 4.5} and Corollary
\ref{Cor 4.10}, we easily obtain the following corollary.
\begin{corollary}
The $\left( i,j\right) $-weakly Lindel\"{o}f property and the pairwise
weakly Lindel\"{o}f property are both pairwise semiregular properties.
\end{corollary}
Recall that, a bitopological space $X$\ is called $\left( i,j\right) $-weak $
P$-space \cite{BidiAdem1} if for each countable family $\left\{ U_{n}:n\in
\mathbb{N}
\right\} $\ of $i$-open sets in $X$, we have $j$\textit{-}$\limfunc{cl}
\left( \dbigcup\limits_{n\in
\mathbb{N}
}U_{n}\right) =\dbigcup\limits_{n\in
\mathbb{N}
}j$\textit{-}$\limfunc{cl}\left( U_{n}\right) $\textit{. }$X$\ is called
pairwise weak $P$-space if it is both $\left( 1,2\right) $-weak $P$-space
and $\left( 2,1\right) $-weak $P$-space.
\begin{proposition}
Let $\left( X,\tau _{1},\tau _{2}\right) $\ be a $\left( \tau _{i},\tau
_{j}\right) $-almost regular and $\left( \tau _{i},\tau _{j}\right) $-weak $
P $-space. Then $\left( X,\tau _{1},\tau _{2}\right) $\ is $\left( \tau
_{i},\tau _{j}\right) $-weakly Lindel\"{o}f if and only if $\left( X,\tau
_{\left( 1,2\right) }^{s},\tau _{\left( 2,1\right) }^{s}\right) $\ is $\tau
_{\left( i,j\right) }^{s}$-Lindel\"{o}f.
\end{proposition}
\begin{proof}
Necessity: Let $\left\{ U_{\alpha }:\alpha \in \Delta \right\} $ be a $\tau
_{\left( i,j\right) }^{s}$-open cover of $\left( X,\tau _{\left( 1,2\right)
}^{s},\tau _{\left( 2,1\right) }^{s}\right) $. For each $x\in X$, there
exists $\alpha _{x}\in \Delta $ such that $x\in U_{\alpha _{x}}$ and since $
U_{\alpha _{x}}\in $ $\tau _{\left( i,j\right) }^{s}$ for each $\alpha
_{x}\in \Delta $, there exists a $\left( \tau _{i},\tau _{j}\right) $
-regular open set $V_{\alpha _{x}}$ in $\left( X,\tau _{1},\tau _{2}\right) $
such that $x\in V_{\alpha _{x}}\subseteq U_{\alpha _{x}}$. Since $\left(
X,\tau _{1},\tau _{2}\right) $\textit{\ }is $\left( \tau _{i},\tau
_{j}\right) $-almost regular, there is a $\left( \tau _{i},\tau _{j}\right) $
-regular open set $C_{\alpha _{x}}$ in $\left( X,\tau _{1},\tau _{2}\right) $
such that $x\in C_{\alpha _{x}}\subseteq \tau _{j}$-$\limfunc{cl}\left(
C_{\alpha _{x}}\right) \subseteq V_{\alpha _{x}}$. Hence $
X=\dbigcup\nolimits_{x\in X}C_{\alpha _{x}}$ and thus the family $\left\{
C_{\alpha _{x}}:x\in X\right\} $ forms a $\left( \tau _{i},\tau _{j}\right) $
-regular open cover of $\left( X,\tau _{1},\tau _{2}\right) $. Since $\left(
X,\tau _{1},\tau _{2}\right) $ is $\left( \tau _{i},\tau _{j}\right) $
-weakly Lindel\"{o}f and $\left( \tau _{i},\tau _{j}\right) $-weak $P$
-space, there exists a countable subset of points $x_{1},\ldots
,x_{n},\ldots $ of $X$ such that $X=\tau _{j}$-$\limfunc{cl}\left(
\dbigcup\nolimits_{n\in
\mathbb{N}
}C_{\alpha _{x_{n}}}\right) =\dbigcup\nolimits_{n\in
\mathbb{N}
}\tau _{j}$-$\limfunc{cl}\left( C_{\alpha _{x_{n}}}\right) \subseteq
\dbigcup\nolimits_{n\in
\mathbb{N}
}V_{\alpha _{x_{n}}}\subseteq \dbigcup\nolimits_{n\in
\mathbb{N}
}U_{\alpha _{x_{n}}}$. This shows that $\left( X,\tau _{\left( 1,2\right)
}^{s},\tau _{\left( 2,1\right) }^{s}\right) $ is $\tau _{\left( i,j\right)
}^{s}$-Lindel\"{o}f.
Sufficiency: Let $\left\{ U_{\alpha }:\alpha \in \Delta \right\} $ be a $
\tau _{i}$-open cover of $\left( X,\tau _{1},\tau _{2}\right) $. Since $
U_{\alpha }\subseteq \tau _{i}$-$\limfunc{int}\left( \tau _{j}\text{-}
\limfunc{cl}\left( U_{\alpha }\right) \right) $ and $\tau _{i}$-$\limfunc{int
}\left( \tau _{j}\text{-}\limfunc{cl}\left( U_{\alpha }\right) \right) \in $
$\tau _{\left( i,j\right) }^{s}$, $\left\{ \tau _{i}\text{-}\limfunc{int}
\left( \tau _{j}\text{-}\limfunc{cl}\left( U_{\alpha }\right) \right)
:\alpha \in \Delta \right\} $ is $\tau _{\left( i,j\right) }^{s}$-open cover
of the $\tau _{\left( i,j\right) }^{s}$-Lindel\"{o}f space$\left( X,\tau
_{\left( 1,2\right) }^{s},\tau _{\left( 2,1\right) }^{s}\right) $. Then
there exists a countable subset $\left\{ \alpha _{n}:n\in
\mathbb{N}
\right\} $ of $\Delta $ such that $X=\dbigcup\nolimits_{n\in
\mathbb{N}
}\tau _{i}$-$\limfunc{int}\left( \tau _{j}\text{-}\limfunc{cl}\left(
U_{\alpha _{n}}\right) \right) \subseteq \dbigcup\nolimits_{n\in
\mathbb{N}
}\tau _{j}$-$\limfunc{cl}\left( U_{\alpha _{n}}\right) =\tau _{j}$-$\limfunc{
cl}\left( \dbigcup\nolimits_{n\in
\mathbb{N}
}U_{\alpha _{n}}\right) $. This implies that $\left( X,\tau _{1},\tau
_{2}\right) $ is $\left( \tau _{i},\tau _{j}\right) $-weakly Lindel\"{o}f.
\end{proof}
\begin{corollary}
Let $\left( X,\tau _{1},\tau _{2}\right) $\ be a pairwise almost regular and
pairwise weak $P$-space. Then $\left( X,\tau _{1},\tau _{2}\right) $\ is
pairwise weakly Lindel\"{o}f if and only if $\left( X,\tau _{\left(
1,2\right) }^{s},\tau _{\left( 2,1\right) }^{s}\right) $\ is Lindel\"{o}f.
\end{corollary}
\begin{proposition}
\label{Prop 4.7} Let $\left( X,\tau _{\left( 1,2\right) }^{s},\tau _{\left(
2,1\right) }^{s}\right) $\ be a $\left( \tau _{\left( j,i\right) }^{s},\tau
_{\left( i,j\right) }^{s}\right) $-extremally disconnected and $\left( \tau
_{\left( i,j\right) }^{s},\tau _{\left( j,i\right) }^{s}\right) $-weak $P$
-space. Then $\left( X,\tau _{\left( 1,2\right) }^{s},\tau _{\left(
2,1\right) }^{s}\right) $ is $\left( \tau _{\left( i,j\right) }^{s},\tau
_{\left( j,i\right) }^{s}\right) $-weakly Lindel\"{o}f if and only if $
\left( X,\tau _{\left( 1,2\right) }^{s},\tau _{\left( 2,1\right)
}^{s}\right) $\ is $\tau _{\left( i,j\right) }^{s}$-Lindel\"{o}f.
\end{proposition}
\begin{proof}
The sufficient condition is obvious by the definitions. So we need only to
prove necessary condition. Suppose that $\left\{ U_{\alpha }:\alpha \in
\Delta \right\} $ is a $\tau _{\left( i,j\right) }^{s}$-open cover of $
\left( X,\tau _{\left( 1,2\right) }^{s},\tau _{\left( 2,1\right)
}^{s}\right) $. For each $x\in X$, there exists $\alpha _{x}\in \Delta $
such that $x\in U_{\alpha _{x}}$. Since $\left( X,\tau _{\left( 1,2\right)
}^{s},\tau _{\left( 2,1\right) }^{s}\right) $ is $\left( \tau _{\left(
i,j\right) }^{s},\tau _{\left( j,i\right) }^{s}\right) $-semiregular, there
exists a $\tau _{\left( i,j\right) }^{s}$-open set $V_{\alpha _{x}}$ in $
\left( X,\tau _{\left( 1,2\right) }^{s},\tau _{\left( 2,1\right)
}^{s}\right) $ such that $x\in V_{\alpha _{x}}\subseteq \tau _{\left(
i,j\right) }^{s}$-$\limfunc{int}\left( \tau _{\left( j,i\right) }^{s}\text{-}
\limfunc{cl}\left( V_{\alpha _{x}}\right) \right) \subseteq U_{\alpha _{x}}$
. Hence $X=\dbigcup\nolimits_{x\in X}V_{\alpha _{x}}$ and thus the family $
\left\{ V_{\alpha _{x}}:x\in X\right\} $ forms a $\tau _{\left( i,j\right)
}^{s}$-open cover of $\left( X,\tau _{\left( 1,2\right) }^{s},\tau _{\left(
2,1\right) }^{s}\right) $.\ Since $\left( X,\tau _{\left( 1,2\right)
}^{s},\tau _{\left( 2,1\right) }^{s}\right) $ is $\left( \tau _{\left(
i,j\right) }^{s},\tau _{\left( j,i\right) }^{s}\right) $-weakly Lindel\"{o}
f, $\left( \tau _{\left( j,i\right) }^{s},\tau _{\left( i,j\right)
}^{s}\right) $-extremally disconnected and $\left( \tau _{\left( i,j\right)
}^{s},\tau _{\left( j,i\right) }^{s}\right) $-weak $P$-space, there exists a
countable subset of points $x_{1},\ldots ,x_{n},\ldots $ of $X$ such that $
X=\tau _{\left( j,i\right) }^{s}$-$\limfunc{cl}\left(
\dbigcup\nolimits_{n\in
\mathbb{N}
}V_{\alpha _{x_{n}}}\right) =\dbigcup\nolimits_{n\in
\mathbb{N}
}\tau _{\left( j,i\right) }^{s}$-$\limfunc{cl}\left( V_{\alpha
_{x_{n}}}\right) =\dbigcup\nolimits_{n\in
\mathbb{N}
}\tau _{\left( i,j\right) }^{s}$-$\limfunc{int}\left( \tau _{\left(
j,i\right) }^{s}\text{-}\limfunc{cl}\left( V_{\alpha _{x_{n}}}\right)
\right) \subseteq \dbigcup\nolimits_{n\in
\mathbb{N}
}U_{\alpha _{x_{n}}}$. This shows that $\left( X,\tau _{\left( 1,2\right)
}^{s},\tau _{\left( 2,1\right) }^{s}\right) $ is $\tau _{\left( i,j\right)
}^{s}$-Lindel\"{o}f.
\end{proof}
\begin{corollary}
Let $\left( X,\tau _{\left( 1,2\right) }^{s},\tau _{\left( 2,1\right)
}^{s}\right) $\ be a pairwise extremally disconnected and pairwise weak $P$
-space. Then $\left( X,\tau _{\left( 1,2\right) }^{s},\tau _{\left(
2,1\right) }^{s}\right) $ is pairwise weakly Lindel\"{o}f if and only if $
\left( X,\tau _{\left( 1,2\right) }^{s},\tau _{\left( 2,1\right)
}^{s}\right) $\ is Lindel\"{o}f.
\end{corollary}
\begin{proposition}
Let $\left( X,\tau _{1},\tau _{2}\right) $\ be a pairwise semiregular, $
\left( j,i\right) $-extremally disconnected and $\left( i,j\right) $-weak $P$
-space. Then $\left( X,\tau _{1},\tau _{2}\right) $ is $\left( i,j\right) $
-weakly Lindel\"{o}f if and only if it is $i$-Lindel\"{o}f.
\end{proposition}
\begin{proof}
By Theorem \ref{Th 4.3}, $\left( X,\tau _{1},\tau _{2}\right) =\left( X,\tau
_{\left( 1,2\right) }^{s},\tau _{\left( 2,1\right) }^{s}\right) $. The
result follows immediately by Proposition \ref{Prop 4.7}.
\end{proof}
\begin{corollary}
Let $\left( X,\tau _{1},\tau _{2}\right) $\ be a pairwise semiregular,
pairwise extremally disconnected and pairwise weak $P$-space. Then $\left(
X,\tau _{1},\tau _{2}\right) $ is pairwise weakly Lindel\"{o}f if and only
if it is Lindel\"{o}f.
\end{corollary} | 185,296 |
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TITLE: Prove that $n^4 \mod 8$ is identically equal to either 0 or 1, $\forall \ n\in \mathbb{N}$.
QUESTION [2 upvotes]: I feel pretty confident about the first half of my solution for this, however I don't like how I used the induction hypothesis on the case for even integers, it feels like it isn't doing anything useful since it is really easy to show that $P(2x+2) \mod 8 \equiv 0$ without using the inductive hypothesis.
This is also the first semester I've been doing proofs, so am I going about this right?
Proof. Since $1^4 \mod 8 = 1$, $P(1)$ holds. Also, $2^4 \mod 8 = 16 \mod 8 = 0$ so, $P(2)$ holds.
We claim that $P(2x-1)$ holds, then $(2x-1)^4 \mod 8\equiv 1$ for any odd integer $n=2x-1$. So,
$$(2x-1)^4 = 16x^4-32x^3+24x^2-8x+1 \mod 8\equiv 1,$$
now consider
$$(2x+1)^4 = 16x^4+32x^3+24x^2+8x+1 = (16x^4-32x^3+24x^2-8x+1)+64x^3+16x,$$
then since
$$64x^3+16x \mod 8 = 8(8x^3+2x) \mod 8 \equiv 0,$$
we conclude that
$$(16x^4-32x^3+24x^2-8x+1)+64x^3+16x \mod 8 \equiv 1+0 = 1.$$
Thus, $P(2x+1)$ holds, and all odd integers $n$ hold.
Next, we claim that $P(2x)$ holds, then $(2x)^4 \mod 8\equiv 0$ for any even integer $n=2x$. So,
$$(2x)^4 \mod 8 = 16x^4 \mod 8 \equiv 0,$$
now consider
$$(2x+2)^4 = 16x^4+64x^3+96x^2+64x+16,$$
then
$$64x^3+96x^2+64x+16 \mod 8 = 8(8x^3+12x^2+8x+2) \mod 8 \equiv 0,$$
we conclude that
$$16x^4+64x^3+96x^2+64x+16 \mod 8 \equiv 0+0 = 0.$$
Thus, $P(2x)$ holds, and all even integers $n$ hold.
By mathematical induction, if $n=2x-1$ and $n=2x$ hold for the given statement, $n=2x+1$ and $n=2x+2$ also hold. Therefore, the statement holds for all $n \in \mathbb{N}$.
$\mathbb{QED}$
REPLY [2 votes]: Your proof is unnecessarily complicated.
If $n$ is even, then we can say $n = 2k$
$(2k)^4 = 16k^4 = 8(2k^4)$ which is divisible by $8.$
$(2k)^4 \equiv 0 \pmod 8$
If $n$ is odd, then $n = 2k+1$
$(2k+1)^4 = 16k^4 + 4(2k)^3 + 6(2k)^2 + 4(2k)+ 1$
Each of the first $4$ terms is divisible by $8$
$(2k+1)^4 \equiv 1 \pmod 8$
REPLY [1 votes]: The proof looks good, although as Doug M suggests, there are much easier ways to reach the same result. In particular, you're right to observe that you don't actually need the inductive hypothesis.
If you do want to apply induction, my advice is to start with a summary of your argument. Something like: "Let $P(n)$ be the claim. We will prove $P(1)$, $P(2)$, $P(2x-1)\implies P(2x+1)$, and $P(2x)\implies P(2x+2)$, completing a proof by induction."
REPLY [1 votes]: Your idea is okay, but $(2x+2)^4=16 + 64 x + 96 x^2 + 64 x^3 + 16 x^4$,
unlike what you initially put in the question.
Alternatively, you could argue that if $2|n$ then $8|n^3|n^4$, so $n^4\equiv0\pmod8$,
and if $2\nmid n$ then $2|n-1, n+1, $ and $n^2+1$, so $8|n^4-1$; i.e., $n^4\equiv1\pmod8$. | 126,438 |
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TITLE: Cyclic error correcting code
QUESTION [2 upvotes]: Notation: I denote the field with $2$ elements by $\mathbb{F}_2$. For a vector $u\in\mathbb{F}_2^m$, I write $w(u)$ for the Hamming weight of $u$ (the number of components equal to $1$ in $u$).
Problem: I am looking for a pair $(n,v)$, where $n$ is a positive integer and $v$ is a vector in $\mathbb{F}_2^n$, such that:
every vector $u\in\mathbb{F}_2^n$ with $w(u)\leq\lceil\sqrt{2n}\rceil$ is not-orthogonal to at least $\lceil\sqrt{2n}\rceil+1-w(u)$ cyclic shifts of the vector $v$.
Motivation: This is related to coding theory. In particular, for such pair $(n,v)$, the subspace of $\mathbb{F}_2^n$ of all vectors orthogonal to all cyclic shifts of $v$, is a cyclic code with distance at least $\lceil\sqrt{2n}\rceil$ (but it is more than that).
Attempt: Since what I'm looking for induces a cyclic code, I looked a little into the theory of such codes. In particular, I considered BCH codes. These codes allow control of the distance of a code by choosing a generating polynomial in a certain way (every cyclic code can be defined in terms of a generating polynomial). Now, the orthogonal complement of a cyclic code is itself a cyclic code and we can find a generating polynomial for it and take $v$ to be the vector corresponding to the coefficients of this generating polynomial. I am not claiming that this method always solves my problem, but I'm trying to see if maybe it solves my problem for one particular choice of $n$ and a BCH code of length $n$.
REPLY [1 votes]: Progress report:
Unless I committed a Mathematica-programming error, a brute force check tells me that the combination $n=23$,
$$v=(1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)$$
works. This vector and its cyclic shifts span the famous binary Golay of code of length $23$, dimension $12$ and minimum distance $7$. It is contained in its dual code (= the standard coding theory term for what you call the orthogonal complement).
I have this vague notion that this might always work similarly for self-orthogonal cyclic codes of a high enough minimum distance, but I need to jump start my brain to see, if that goes through. Anyway, there aren't very many such codes. Quadratic residue codes are probably worth checking, Golay code is an example of those.
Meanwhile, does this help you? Do double-check my claim that this actually works!
AFAICT all the words of length $23$ and weight $\le 7$ are non-orthogonal to at least five cyclic shifts of $v$. | 192,106 |
TITLE: How to rewrite $\int\limits_A^B \frac{x^n \exp(-\alpha x)}{\small(x + \beta\small)^m} \, dx$?
QUESTION [9 upvotes]: Currently, I am a post-graduate researcher in Telecommunications. During the process of evaluating the transmission error probability, I need to evaluate the following integral $I = \int\limits_A^B \frac{x^n \exp(-\alpha x)}{\small(x + \beta\small)^m} \, dx$?
How to rewrite this improper integral in terms of special function (for example $Ei(x)$, Bessel,...)?
Notice that $A, B, \alpha,\beta > 0$ (positive real number) and $m,n$ are two positive integers.
I have tried to compute this integral with different values of $A, B, \alpha,\beta > 0$ and $m,n$ by using Wolfram Mathematica.
It seems that the results of this integral have a form of the exponential integral function $Ei\left( x \right) = \int\limits_{t = - x}^\infty {\frac{{{e^{ - t}}}}{t}dt} = \int\limits_{t = - \infty }^{t = x} {\frac{{{e^t}}}{t}dt}$ as:
$I = C_1\bigg[C_2 + C_3\big[ {\rm Ei}\big(- \alpha(\beta+ A)\big) - {\rm Ei}\big(- \alpha(\beta+ B)\big) \big] \bigg]$.
Are there any way to find out the correct values of $C_1$, $C_2$, and $C_3$.
Thank you for your enthusiasm!
REPLY [1 votes]: Thank you for considering my concern.
I have found out myself.
For the given integral above, I firstly change variable $y = x + \beta \to dx = dy$. Now, $I$ can be rewritten as
$I = \int_{y_{\min}}^{y_{\max}}\frac{\small(y - \beta\small)^n \exp\big(- \alpha\small(y-\beta\small)\big)}{y^m}dy$, where $y_{\min} = A +\beta$ and $y_{\max} = B + \beta$.
By using the binomial theorem for $\small(y - \beta\small)^n$, we can be modified $I$ as
$I = \exp(\alpha\beta)\sum_\limits{k=0}^n\binom{k}{n}\small(-\beta\small)^{n-k}
\int_{y_{\min}}^{y_{\max}}\frac{\exp\big(- \alpha y\big)}{y^{m-k}}dy$.
Based on the condition of $(m-k)$ and thanks to the help of Wolfram Mathematica, I can achieve the final result of $I$ as follows:
$\begin{array}{l}
I = \exp \left( {\alpha \beta } \right)\sum\limits_{k = 0}^n {\left( {\begin{array}{*{20}{c}}
n\\
k
\end{array}} \right){{\left( { - \beta } \right)}^{n - k}}} \\
\,\,\,\,\,\,\, \times \left\{ \begin{array}{l}
{\alpha ^{ - \left( {m - k} \right) - 1}}\left[ {\Gamma \left( {\left( {m - k} \right) + 1,{y_{\min }}\alpha } \right) - \Gamma \left( {\left( {m - k} \right) + 1,{y_{\max }}\alpha } \right)} \right],m - k = 0\\
{\alpha ^{\left( {m - k} \right) - 1}}\left[ {\Gamma \left( {1 - \left( {m - k} \right),{y_{\min }}\alpha } \right) - \Gamma \left( {1 - \left( {m - k} \right),{y_{\max }}\alpha } \right)} \right],m - k \ne 0
\end{array} \right.
\end{array}$,
where $\Gamma \left(\cdot,\cdot\right)$ is upper bound incomplete Gamma function. | 154,117 |
\begin{document}
\maketitle
\begin{abstract}
We look at sequences of positive integers that can be realized as degree sequences of iterates of rational dominant maps of smooth projective varieties over arbitrary fields. New constraints on the degree growth of endomorphisms of the affine space and examples of degree sequences are displayed. We also show that the set of all degree sequences of rational maps is countable; this generalizes a result of Bonifant and Fornaess.
\end{abstract}
\tableofcontents
\section{Introduction and results}
\subsection{Groups of birational transformations and degree sequences}
Let $X_k$ be a projective variety defined over a field $k$; denote by $\Bir(X_k)$ the group of birational transformations of $X_k$. A group $\Gamma$ is called a {\it group of birational transformations} if there exists a field $k$ and a projective variety $X_k$ over $k$ such that $\Gamma\subset\Bir(X_k)$. More generally, one can consider $\Rat(X_k)$, the monoid of dominant rational maps of $X_k$. Accordingly, we call a monoid $\Delta$ a {\it monoid of rational dominant transformations}, if there exists a field $k$ and a projective variety $X_k$ over $k$ such that $\Delta\subset\Bir(X_k)$.
If $X_k$ is a smooth projective variety, an interesting tool to study the structure of monoids of rational dominant transformations are {\it degree functions}. Fix a polarization of $X_k$, i.e. an ample divisor class $H$ of $X_k$. Then one can associate to every element $f\in\Rat(X_k)$ its degree $\deg_H(f)\in\Z^+$ with respect to $H$, which is defined by
\[
\deg_H(f)=f^*H\cdot H^{d-1},
\]
where $d$ is the dimension of $X_k$ and $f^*H$ is the total transform of $H$ under $f$. For a smooth projective variety $X_k$ over a field of characteristic zero $k$, one has for $f,g\in\Bir(X_k)$
\[
\deg_H(f\circ g)\leq C(X_k, H)\deg_H(f)\deg_H(g),
\]
where $C(X_k,H)$ is a constant only depending on $X_k$ and the choice of polarization $H$ (see \cite{MR2180409}). For a generalization of this result to fields of positive characteristic, see \cite{Truong:2015aa}.
Let $f$ be a rational self map of $\p_k^d$. With respect to homogeneous coordinates $[x_0:\dots:x_d]$ of $\p_k^d$, $f$ is given by $[x_0:x_1:\dots:x_d]\mapsto [f_0:f_1:\dots:f_d]$, where $f_0,\dots, f_d\in k[x_0,\dots,x_d]$ are homogeneous polynomials of the same degree and without a common factor. We define $\deg(f)=\deg(f_i)$. Note that if $f$ is dominant, then $\deg(f)=\deg_H(f)$ for $H=\mathcal{O}(1)$. So in case $X_k=\p_k^d$ we can extend the notion of degree to all rational self maps. Note that if $f$ is an endomorphism of $\A_k^d$ defined by $(x_1,\dots, x_d)\mapsto (f_1,\dots, f_d)$ with respect to affine coordinates $(x_1,\dots, x_d)$ of $\A_k^d$, where $f_1,\dots, f_d\in k[x_1,\dots, x_d]$, then $\deg(f)$ is the maximal degree of the $f_i$.
Let $\Delta\subset\Bir(X_k)$ be a finitely generated monoid of rational dominant transformations of a smooth projective variety $X_k$ with a finite set of generators $S$. We define
\[
D_{S, H}\colon\Z^+\to\Z^+\]
by
\[
D_{S,H}(n):=\max_{\delta\in B_S(n)}\{\deg_H(\gamma)\},
\]
where, $B_S(n)$ denotes all elements in $\Delta$ of word length $\leq n$ with respect to the generating set $S$. We call a map $\Z^+\to\Z^+$ that can be realized for some field $k$ and some $(X_k, H, \Delta, S)$ as such a function a {\it degree sequence}.
Note that our definition of degree sequences includes in particular degree sequences that are given by finitely generated groups of birational transformations $\Gamma\subset\Bir(X_k)$.
In this paper we show that there exist only countably many degree sequences, display certain constraints on their growth and give some new examples.
\subsection{Countability of degree sequences}
In \cite{MR1793690}, Bonifant and Fornaess proved that the set of sequences $\{d_n\}$ such that there exists a rational self map $f$ of $\p^d_{\C}$ satisfying $\deg(f^n)=d_n$ for all $n$, is countable, which answered a question of Ghys. We generalize the result of Bonifant and Fornaess to all degree sequences over all smooth projective varieties, all fields, all polarizations and all finite generating sets $S$ of finitely generated monoids of rational dominant maps:
\begin{theorem}\label{rationalcount}
The set of all degree sequences is countable.
\end{theorem}
\subsection{Previous results}
In dimension 2 the degree growth of birational transformations is well understood and is a helpful tool to understand the group structure of $\Bir(S_k)$ for projective surfaces $S_k$ over a field $k$.
\begin{theorem}[Gizatullin; Cantat; Diller and Favre]\label{dim2}
Let $k$ be an algebraically closed field, $S_k$ a projective surface over $k$ with a fixed polarization $H$ and $f\in\Bir(S_k)$. Then one of the following is true:
\begin{itemize}
\item the set $\{\deg_H(f^n)\}$ is bounded;
\item $\deg_H(f^n)\sim cn$ for some positive constant $c$ and $f$ preserves a rational fibration;
\item $\deg_H(f^n)\sim cn^2$ for some positive constant $c$ and $f$ preserves an elliptic fibration;
\item $\deg_H(f^n)\sim \lambda^n $, where $\lambda$ is a Pisot or Salem number.
\end{itemize}
\end{theorem}
For more details on this rich subject and references to the proof of Theorem \ref{dim2} see \cite{cantat2012cremona}.
In the case of polynomial automorphisms of the affine plane, the situation is even less complicated. Let $f\in\aut(\A_k^2)$. Then the sequence $\{\deg(f^n)\}$ is either bounded or it grows exponentially in $n$. See \cite{MR1667603} for this and more results on the degree growth in $\aut(\A_k^2)$
In higher dimensions there are only few results on the degree growth of birational transformations. In particular, the following questions are open:
\begin{question}
Does there exist a birational transformation $f$ of a projective variety $X_k$ such that $\deg_H(f^n)$ is of intermediate growth, for instance $\deg_H(f^n)\sim e^{\sqrt{n}}$?
\end{question}
\begin{question}
Does there exist a birational transformation $f$ such that $\deg_H(f^n)$ is unbounded, but grows ''slowly``? For instance, can we have $\deg_H(f^n)\sim \sqrt{n}$? Or do unbounded degree sequences grow at least linearly?
\end{question}
\begin{question}
If there is a birational transformation $f$ such that $\deg_H(f^n)\sim \lambda^n$, is $\lambda$ always an algebraic number?
\end{question}
\begin{question}
Do birational transformations of polynomial growth always preserve some non-trivial rational fibration?
\end{question}
In \cite{MR3210135} Lo Bianco treats the case of automorphisms of compact K\"ahler threefolds.
\subsection{Degree sequences of polynomial automorphisms}
A good place to start the examination of degree sequences seems to be the group of polynomial automorphisms of the affine $d$-space $\aut(\A_k^d)$. In Section \ref{proofaut1} we will show the following observation (the proof of which can be found as well in \cite{Deserti:2016zl}):
\begin{proposition}\label{aut1}
Let $k$ be a field and $f\in\aut(\A^d_k)$ a polynomial automorphism such that $\deg(f^d)=\deg(f)^d$, then $\deg(f^n)=\deg(f)^n$ for all $n\in\Z^+$.
\end{proposition}
The monoid $\End(\A^d_k)$ has the additional structure of a $k$-vector space, on which the degree function induces a filtration of finite dimensional vector spaces. This gives rise to a new technique, which we will use to prove that unbounded degree sequences of groups of polynomial automorphisms diverge and can not grow arbitrarily slowly:
\begin{theorem}\label{degaut}
Let $f\in\End(\A_k^d)$ be an endomorphism such that the sequence $\{\deg(f^n)\}$ is unbounded. Then for all integers $K$
\[
\#\left\{m\mid \deg(f^m)\leq K\right\}< C_d\cdot K^{d},
\]
where $C_d=\frac{(1+d)^{d}}{(d-1)!}$. In particular, $\deg(f^n)$ diverges to~$\infty$.
\end{theorem}
By a result of Ol'shanskii (\cite{MR1714850}), Theorem \ref{degaut} shows that an unbounded degree sequence of a polynomial automorphism behaves in some ways like a word length function.
The following corollary is immediate:
\begin{corollary}\label{degautcor}
Let $\Gamma\subset\End(\A^d_k)$ be a finitely generated monoid with generating system $S$. If $D_S(n)< C_d\cdot n^{1/d}$ for infinitely many $n$ then $\Gamma$ is of bounded degree.
\end{corollary}
Unfortunately our methods to prove Theorem \ref{degaut} do not work for arbitrary birational transformations of $\p^d_k$. However, if we assume the ground field to be finite, we obtain a similar result:
\begin{theorem}\label{degfinite}
Let $\F_q$ be a finite field with $q$ elements and let $f\in\Rat(\p^d_{\F_q})$ such that the sequence $\{\deg(f^n)\}$ is unbounded. Then, for all integers $K$,
\[
\#\left\{m\mid \deg(f^m)\leq K\right\}\leq q^{C(K,d)},
\]
where $C(K,d)=(d+1)\cdot\binom{d+K}{K}$. In particular, $\deg(f^n)$ diverges to $\infty$.
\end{theorem}
This implies the following for degree sequences in $\Rat(\p^d_{\F_q})$:
\begin{corollary}\label{maincor}
Let $\Gamma\subset\Rat(\p^d_{\F_q})$ be a finitely generated monoid with generating system $S$. There exists a positive constant $C_{d,q}$ such that if $D_S(n)<C_{d,q}\cdot\log(n)^{1/d}$ for all $n$, then $\Gamma$ is of bounded degree.
\end{corollary}
\subsection{Types of degree growth}
\begin{definition}
Let $X_k$ be a smooth projective variety with polarization $H$ over a field $k$ and let $f\in\Bir(X_k)$. We denote the {\it order of growth} of $\deg_H(f^n)$ by
\[
\dpol(f):=\limsup_{n\to\infty}\frac{\log(\deg_H(f^n))}{\log(n)}.
\]
The order of growth can be infinite.
\end{definition}
By a result of Dinh and Sibony, the order of growth does not depend on the choice of polarization if we work over the field of complex numbers (see Section~\ref{complex}):
\begin{proposition}\label{indeppolar}
Let $X_{\C}$ be a smooth complex projective variety and let $f\in\Bir(X_{\C})$. Then $\dpol(f)$ does not depend on the choice of polarization.
\end{proposition}
Let $f$ be a birational transformation of a surface. As recalled above, in that case $\dpol(f)=0,1, 2$ or $\infty$. This gives rise to the following question:
\begin{question}
Does there exist a constant $C(d)$ depending only on $d$ such that for all varieties $X_k$ of dimension $d$ we have $\dpol(f)<C(d)$ for all $f\in \Bir(X_k)$ with $\dpol(f)$ finite?
\end{question}
We give some examples of degree sequences that indicate that the degree growth in higher dimensions is richer than in dimension 2.
First of all, note that elements in $\aut(\A^d_k)$ can have polynomial growth:
\begin{example}\label{linearex}
Let $k$ be any field and define $f,g,h\in\aut(\A^d_k)$ by $g=(x+yz, y, z)$, $h=(x, y+xz, z)$ and
\[
f=g\circ h=(x+z(y+xz), y+xz,z).
\]
One sees by induction that $\deg(f^n)=2n+1$; in particular, $\dpol(f)=1$.
\end{example}
\begin{example}\label{exaut}
More generally, for all $l\leq d/2$ there exist elements $f_l\in\aut(\A_k^d)$ such that $\dpol(f)=l$ (Section \ref{proofexaut}).
\end{example}
Other interesting examples of degree sequences of polynomial automorphisms and the dynamical behavior of the corresponding maps are described in~\cite{Deserti:2016zl}.
For birational transformations of $\p^d_k$ we can obtain even faster growth (see \cite{MR2917145} for more details):
\begin{example}
The birational transformation $f=(x_1, x_1x_2,\dots, x_1x_2\cdots x_n)$ of $\p_k^d$ defined with respect to local affine coordinates $(x_1,\dots, x_d)$ satisfies {$\deg(f^n)=n^{d-1}$}, i.e. $\dpol({f})=d-1$.
\end{example}
The following interesting observation is due to Serge Cantat:
\begin{example}\label{oguiso}
Define $\omega:=\frac{-1+\sqrt{-3}}{2}$ and the elliptic curve $E_\omega:=\C/(\Z+\Z\omega)$. Let
\[
X:=E_\omega\times E_\omega\times E_\omega
\]
and $s\colon X\to X$ the automorphism of finite order given by diagonal multiplication with $-\omega$. In \cite{MR3329200} Oguiso and Truong prove that the quotient $Y:=X/s$ is a rational threefold. Let $f\colon X\to X$ be the automorphism defined by $(x_1,x_2,x_3)\mapsto (x_1, x_1+x_2, x_2+x_3)$. Since $f$ commutes with $s$, it induces an automorphism on $Y$, which we denote by $\hat{f}$. Let $\phi_1\colon\tilde{Y}\to Y$ be a resolution of the singularities of $Y$ and define $\tilde{f}\in\Bir(\tilde{Y})$ by
\[
\tilde{f}:=\phi_1^{-1}\circ\hat{f}\circ\phi_1.
\]
We will show in Section \ref{oguisoproof} that $\dpol(\tilde{f})=4$.
\end{example}
\subsection{Acknowledgements}
I thank my advisors J\'er\'emy Blanc and Serge Cantat for many interesting and helpful discussions and their constant support. I would also like to thank Junyi Xie for inspiring conversations and for giving me, together with Serge Cantat, some of the main ideas for the proof of Theorem \ref{rationalcount}, as well as Federico Lo Bianco for showing me Proposition~\ref{propfede}. Many thanks also to Bac Dang and Julie D\'eserti for their comments and helpful references.
\section{Preliminaries}
\subsection{Monoids of rational dominant transformations}
Let $X_k$ be a projective variety over a field~$k$. There is a one to one correspondence between rational dominant self maps of $X_k$ and $k$-endomorphisms of the function field $k(X_k)$. The field $k(X_k)$ is the field of fractions of a $k$-algebra of finite type $k[x_1,\dots, x_n]/I$, where $I\subset k[x_1,\dots, x_n]$ is a prime ideal generated by elements $f_1,\dots, f_l\in k[x_1,\dots, x_n]$. A field extension $k\to k'$ induces a base change $X_{k'}\to X_k$. The function field of $X_{k'}$ is the field of fractions of the $k'$-algebra $k'[x_1,\dots,x_n]/I'$, where $I'$ is the ideal generated by $f_1,\dots, f_l$. Note that $k(X_k)\subset k'(X_{k'})$. We say that a $k'$-endomorphism of $k'(X_{k'})$ is defined over $k$, if it restricts to a $k$-endomorphism of $k(X_k)$. Consider a $k'$-endomorphism $G$ of $k'(X_{k'})$ sending generators $(x_1,\dots, x_n)$ of $k'(X_{k'})$ to $(g_1,\dots, g_n)$, where $g_i\in k'(X_{k'})$. Then $G$ is defined over $k$ if and only if $g_i\in k(X_k)$ for all $i$. On the other hand, let $g_i,\dots, g_n\in k(X_k)$ and let $(x_1,\dots, x_d)\mapsto(g_1,\dots, g_d)$ be a $k$-endomorphism of $k(X_k)$. Then $(x_1,\dots, x_d)\mapsto(g_1,\dots, g_d)$ defines as well a $k'$-endomorphism of $k'(X_{k'})$. So a $k$-endomorphism $(x_1,\dots, x_n)\mapsto (g_1,\dots, g_n)$ of $k(X_k)$ extends uniquely to a $k'$-endomorphism of $k'(X_{k'})$. This yields the following observation:
\begin{lemma}
Let $X_k$ be a projective variety over a field $k$ and $\varphi\colon k\to k'$ a homomorphism of fields. Then $\varphi$ induces a natural injection of monoids $\Psi_{\varphi}\colon\Rat(X_k)\to\Rat(X_{k'})$.
\end{lemma}
Recall that there are uncountably many finitely generated groups and thus in particular uncountably many finitely generated monoids. The following observation by de Cornulier shows that being a monoid of rational dominant transformations is in some sense a special property (cf. \cite{MR3160544} and \cite{cantatremark}).
\begin{proposition}\label{countablesubgroups}
There exist only countably many finitely generated isomorphism classes of monoids of rational dominant transformations.
\end{proposition}
\begin{proof}
Let $\Delta\subset\Rat(X_k)$ be a monoid of rational dominant transformations with a finite generating set $f_1,\dots, f_n\in\Delta$ , where $X_k$ is a projective variety defined over a field $k$. Denote by $S\subset k$ the finite set of coefficients necessary to define $X_k$ and the rational dominant transformations $f_i$. Let $k'$ be the field $\F_p(S)$, where $p=\car(k)$, or $\Q(S)$ if $\car(k)=0$.
We consider the function field $k'(X_{k'})$ as a subfield of the function field $k(X_{k})$. Note that the action of the elements of $\Delta$ on $k'(X_{k})$ preserves $k'(X_{k'})$ and that $f_{i_1}f_{i_2}\cdots f_{i_{k-1}}=f_{i_k}$ in $\Rat(X_k)$ if and only if $f_{i_1}f_{i_2}\cdots f_{i_{k-1}}=f_{i_k}$ in $\Rat(X_{k'})$. So without loss of generality we may assume $\Delta\subset\Rat(X_{k'})$, where $k'$ is a finitely generated field extension of some $\F_p$ or of $\Q$.
A rational dominant transformation of a given variety $X_k$ is defined by finitely many coefficients in $k$. So the cardinality of the set of all finitely generated monoids of rational dominant transformations of a variety $X_k$ is at most the cardinality of~$k$.
Recall that the cardinality of the set of all finitely generated field extensions of $\F_p$ and $\Q$ is countable. Since a projective variety is defined by a finite set of coefficients, we obtain that there are only countably many isomorphism classes of projective varieties defined over a field $k'$ that is a finitely generated field extension of $\F_p$ or $\Q$. The claim follows.
\end{proof}
\subsection{Intersection form}
Let $X_k$ be a smooth projective variety of dimension $d$ over an algebraically closed field $k$ and let $D$ be a Cartier divisor on $X_k$. The {\it Euler characteristic of $D$} is the integer
\[
\chi(X_k, D)=\sum_{i=0}^{\infty}(-1)^i\dim_k H^i(X_k, D).
\]
Let $D_1,\dots, D_d$ be Cartier divisors on $X_k$. The function
\[
(m_1,\dots, m_d)\mapsto \chi(X_k, m_1D_1+\dots+m_dD_d)
\]
is a polynomial in $(m_1,\dots, m_d)$. The {\it intersection number} of $D_1,\dots, D_d$ is then defined to be the coefficient of the term $m_1m_2\cdots m_d$ in this polynomial and we denote it by $D_1\cdot D_2\cdots D_d$. One can show that intersection multiplicities are always integers and that the intersection form is symmetric and linear in all $d$ arguments. Moreover, if $D_1,\dots, D_d$ are effective and meet properly in a finite number of points, $D_1\cdots D_d$ is the number of points in $D_1\cap\cdots\cap D_d$ counted with multiplicities and the intersection form is the unique bilinear form with this property. For the details on this construction we refer to \cite{MR1841091} and the references in there. The intersection number is preserved under linear equivalence, therefore it is well defined on classes of Cartier divisors. Note as well that an isomorphism between algebraically closed fields does not change the cohomology dimensions and hence that the intersection multiplicities are invariant under such an isomorphism.
Let $X_k$ be a smooth projective variety of dimension $d$ over an arbitrary field $k$ and let $D_1,\dots, D_d$ be Cartier divisors on $X_k$. Denote by $\overline{k}$ an algebraic closure of $k$. We define the intersection multiplicity $D_1\cdot D_2\cdots D_d$ as the intersection multiplicity of $D_1, D_2,\dots, D_d$ on $X_{\overline{k}}$ after base extension $k\to\overline{k}$. By the remark above, this does not depend on the choice of the algebraic closure $\overline{k}$. Every field isomorphism $k\to k'$ extends to an isomorphism between the algebraic closures of $k$ and $k'$, hence the intersection number is invariant under field isomorphisms. Since the intersection form is unique, it also does not change under a base extension $k\to k'$ between algebraically closed fields.
We summarize these properties in the following proposition:
\begin{proposition}\label{polarprop}
Let $X_k$ be a smooth projective variety of dimension $d$ over a field $k$. Then there exists a symmetric $d$-linear form on the group of divisors of $X_k$:
\[
\Div(X_k)\times\cdots\times \Div(X_k)\to\Z,\hspace{1.5mm}(D_1,\dots, D_d)\mapsto D_1\cdot D_2\cdots D_d,
\]
such that if $D_1,\dots, D_d$ are effective and meet properly in a finite number of points, $D_1\cdots D_d$ is the number of points in $D_1\cap\cdots\cap D_d$ counted with multiplicity.
Moreover, this intersection form is invariant under base change $k\to k'$ of fields.
\end{proposition}
\subsection{Polarizations and degree functions}\label{polarizations}
A polarization on a smooth projective variety $X_k$ of dimension $d$ is an ample divisor class $H$. This implies in particular that $Y\cdot H^{d-1}>0$ for all effective divisors $Y$.
Let $f\in\Rat(X_k)$ and denote by $\dom(f)$ the maximal open subset of $X_k$ on which $f$ is defined. The {\it graph} $\Gamma_f$ of $f$ is the closure of $\{(x,f(x)\mid x\in\dom(f))\}\subset X_k\times X_k$. Let $p_1$ and $p_2\colon\Gamma_f\to X$ be the natural projections on the first respectively second factor, then $f=p_2\circ p_1^{-1}$. Note that $p_1$ is birational. The {\it total transform} of a divisor $D$ under $f$ is the divisor ${p_1}_*(p_2^*D)$, where ${p_2}^*D$ is the pullback of $D$ as a Cartier divisor and ${p_1}_*(p_2^*D)$ is the pushforward of ${p_2}^*D$ as a Weil divisor. Note that if $k$ is algebraically closed and $D$ is effective, then $f^*D$ is the closure of all the points in $\dom(f)$ that are mapped to $D$ by $f$.
\subsection{Degrees in the case of complex varieties}\label{complex}
Let $X_{\C}$ be a smooth complex projective variety of dimension $d$. The following constructions and results can be found in \cite{MR2095471}. Recall that two divisors $D_1$ and $D_2$ are numerically equivalent if $D_1\cdot\gamma=D_2\cdot\gamma$ for all curves $\gamma$ on $X$. Denote by $N^1(X)$ the Neron-Severi group, which is the group of divisors modulo numerical equivalence. The intersection number of divisors $D_1,\dots, D_d$ is invariant under numerical equivalence. By taking the first Chern class one can embed $N^1(X_{\C})$ into $H^2(X;\Z)_{t.f.}$, which is $H^2(X_{\C};\Z)$ modulo its torsion part. It turns out that $N^1(X_{\C})=H^2(X_{\C};\Z)_{t.f.}\cap H^{1,1}(X_{\C})$. Let $D_1,\dots, D_d$ be divisors and $\omega_1,\dots, \omega_d$ the corresponding $(1,1)$-forms, then
\[
D_1\cdot D_2\cdots D_d=\int_{X_{\C}}\omega_1\wedge \omega_2\wedge\cdots\wedge\omega_d.
\]
A divisor is ample if and only if the corresponding $(1,1)$-form is cohomologous to a K\"ahler form. Hence a polarization of a smooth complex projective variety can be seen as a K\"ahler class. In this section we will take this point of view.
Let $f$ be a birational transformation of $X$. With the help of currents, one can define the pullback $f^*\omega$ of a $(1,1)$-form $\omega$ (see \cite{MR2851870}). If $H$ is a divisor of $X$ and $\omega$ the corresponding $(1,1)$-form then $f^*\omega$ is the form corresponding to $f^*H$.
If we fix a polarization $\omega_{X_{\C}}$ of smooth complex projective variety $X_{\C}$, then the degree of a birational map $f\in\Bir(X_{\C})$ is defined by
\[
\deg_{\omega_{X_{\C}}}(f)=\int_{X_{\C}}f^*\omega_{X_{\C}}\wedge\omega_{X_{\C}}\wedge\cdots\wedge \omega_{X_{\C}}.
\]
So the degree does not depend on whether we look at an ample divisor class or its corresponding K\"ahler class.
In \cite{MR2180409} the following is shown (see also \cite{MR2851870}):
\begin{proposition}\label{dinhnguyen}
Let $X_{\C}$ be a smooth complex projective variety with polarization $\omega_{X_{\C}}$ and $f\in\Bir(X_{\C})$. There exists a constant $A>0$ such that
\[
A^{-1}\deg_{\omega_{X_{\C}}}(f^n)\leq \|(f^n)^*\|\leq A\deg_{\omega_{X_{\C}}}(f^n),
\]
where $\|\,\cdot\,\|$ denotes any norm on $\End(H^{1,1}(X_{\C}))$.
\end{proposition}
The following corollary follows directly from Proposition \ref{dinhnguyen} and proves Proposition \ref{indeppolar}:
\begin{corollary}\label{cordinhnguyen}
Let $X_{\C}$ be a smooth complex projective variety, then
\[
\dpol(f)=\limsup_{n\to\infty}\frac{\log(\|(f^n)^*\|)}{\log(n)}.
\]
In particular, $\dpol(f)$ does not depend on the choice of polarization.
\end{corollary}
\begin{proposition}\label{propfede}
Let $X_{\C}$ and $Y_{\C}$ be complex smooth projective varieties of dimension $d$, let $f\in\Bir(X_{\C})$, $g\in\Bir(Y_{\C})$ and $\pi\colon X\to Y$ a dominant morphism of finite degree $k$, such that $\pi\circ f=g\circ \pi$. Then $\dpol(f)=\dpol(g)$.
\end{proposition}
\begin{proof}
Let ${\omega_{X_{\C}}}$ be a K\"ahler form on $X_{\C}$ and $\omega_{Y_{\C}}$ be a K\"ahler form on $Y_{\C}$. We have
\begin{equation*}
\begin{split}
\deg_{\omega_{X_{\C}}}(f^n)&=\int_{X_{\C}}(f^n)^*\omega_{X_{\C}}\wedge\omega_{X_{\C}}^{d-1}=\deg(\pi)\int_{Y_{\C}}\pi_*(f^n)^*\omega_{X_{\C}}\wedge\pi_*\omega_{X_{\C}}^{d-1}\\
&=\deg(\pi)\int_{Y_{\C}}(g^n)^*\pi_*\omega_{X_{\C}}\wedge\pi_*\omega_{X_{\C}}^{d-1}\\
&\leq K_1\deg(\pi)\cdot\|(g^n)^*\|\cdot\|\pi_*\omega_{X_{\C}}\|\cdot\|\pi_*\omega_{X_{\C}}^{d-1}\|\leq K_2\cdot \|(g^n)^*\|,
\end{split}
\end{equation*}
for some positive constants $K_1, K_2$ not depending on $n$. By Corollary \ref{cordinhnguyen}, this yields $\dpol(f)\leq\dpol(g)$.
On the other hand, we have
\begin{equation*}
\begin{split}
\deg_{\omega_{Y_{\C}}}(g^n)&=\int_{Y_{\C}}(g^n)^*\omega_{Y_{\C}}\wedge\omega_{Y_{\C}}^{d-1}=\frac{1}{\deg(\pi)}\int_{X_{\C}}\pi^*(g^n)^*\omega_{Y_{\C}}\wedge\pi^*\omega_{Y_{\C}}^{d-1}\\
&=\frac{1}{\deg(\pi)}\int_{X_{\C}}(f^n)^*\pi^*\omega_{Y_{\C}}\wedge\pi^*\omega_{Y_{\C}}^{d-1}\\
&\leq K_1'\frac{1}{\deg(\pi)}\|(f^n)^*\|\cdot\|\pi^*\omega_{Y_{\C}}\|\cdot\|\pi^*\omega_{Y_{\C}}^{d-1}\|\leq K_2'\cdot \|(f^n)^*\|,
\end{split}
\end{equation*}
for some positive constants $K_1', K_2'$ not depending on $n$ and therefore, by Corollary~\ref{cordinhnguyen}, $\dpol(g)\leq\dpol(f)$.
\end{proof}
\section{Proofs}
\subsection{Proof of Theorem \ref{rationalcount}}
Let $k$ be a field, $X_k$ a smooth projective variety defined over $k$, $H$ a polarization of $X_k$ and $\Delta\subset\Rat(X_k)$ a finitely generated monoid of rational dominant transformations with generating set $T$. Then $X_k$, $H$ and $T$ are defined by a finite set $S$ of coefficients from $k$. Let $k'\subset k$ be the field $\F_p(S)$, where $p=\car(k)$, or $k'=\Q(S)$ if $\car(k)=0$. By Proposition \ref{polarprop}, the degree of elements in $\Gamma$ considered as rational dominant transformations of $X_k$ with respect to the polarization $H$ is the same as the degree of elements in $\Delta$ considered as rational dominant transformations of $X_{k'}$. So without loss of generality, we may assume that $\Delta$ is a submonoid of $\Rat(X_{k'})$, where $k'$ is a finite field extension of $\F_p$ or of~$\Q$.
As in the proof of Proposition \ref{countablesubgroups}, we use the fact that there are only countably many isomorphism classes of such varieties. Polarizations and rational self maps are defined by a finite set of coefficients, so the cardinality of the set of all $(k',X_{k'},H,T)$ up to isomorphism, where $k'$ is a finitely generated field extension of $\F_p$ or $\Q$, $X_{k'}$ a smooth projective variety over $k'$, $H$ a polarization of $X_{k'}$ and a finite set of elements in $\Rat(X_{k'})$ is countable. It follows in particular that the set of all degree sequences is countable.
\subsection{Proof of Proposition \ref{aut1}}\label{proofaut1}
Let $k$ be a field and $f\in\Bir(\p_k^d)$ a birational map that is given by $[f_0:\dots:f_d]$, with respect to homogeneous coordinates $[x_0:\dots:x_d]$, where the $f_i$ are homogenous polynomials of the same degree without common factors. There are two important closed subsets of $\p_k^d$ associated to $f$, the indeterminacy locus $\Ind(f)$, consisting of all the points where $f$ is not defined and the exceptional divisor $\exc(f)$, the set of all the points where $f$ is not a local isomorphism. If $f$ is not an automorphism, the indeterminacy locus is a closed set of codimension $\geq 2$ and the exceptional divisor a closed set of codimension 1. Note that $\Ind(f)$ is exactly the set of points, where all the $f_i$ vanish. Let $X\subset \p_k^d$ be an irreducible closed set that is not contained in $\Ind(f)$. We denote by $f(X)$ the closure of $f(X\setminus \Ind(f))$ and we say that $f$ {\it contracts} $X$ if $\dim(f(X))<\dim(X)$.
The following lemma is well known (see for example \cite{zbMATH00663864}):
\begin{lemma}\label{deglemma}
Let $k$ be a field and $g,f\in\Bir(\p_k^d)$. Then $\deg(f\circ g)\leq\deg(f)\deg(g)$ and $\deg(f\circ g)<\deg(f)\deg(g)$ if and only if $g$ contracts a hypersurface to a subset of $\Ind(f)$.
\end{lemma}
\begin{proof}
Let $f=[f_0:\dots:f_d]$ and $g=[g_0:\dots:g_d]$. Then $\deg(f\circ g)<\deg(f)\deg(g)$ if and only if the polynomials $f_0(g_0,\dots, g_d),\dots, f_d(g_0,\dots, g_d)$ have a non constant common factor $h\in k[x_0,\dots, x_d]$. Let $M\subset \p^d_k$ be the hypersurface defined by $h=0$. Then $f$ is not defined at $g(M)$. This implies that the codimension of $g(M)$ is $\geq 2$ and therefore that $g$ contracts $M$ to a subset of $\Ind(f)$.
On the other hand, let $M$ be an irreducible component of a hypersurface that is contracted by $g$ to a subset $\Ind(f)$. Assume that $M$ is the zero set of an irreducible polynomial $h$. Since $g(M)\subset \Ind(f)$, we obtain that $f_0(g_0,\dots, g_d),\dots, f_d(g_0,\dots, g_d)$ vanish all on $M$ and therefore that $h$ divides $f_0(g_0,\dots, g_d),\dots, f_d(g_0,\dots, g_d)$. This implies $\deg(f\circ g)<\deg(f)\deg(g)$.
\end{proof}
In order to prove Proposition \ref{aut1}, we consider an element $f\in\aut(\A_k^d)$ as a birational transformation of $\p^d_k$ whose exceptional divisor is the hyperplane at infinity $H:=\p_k^d\setminus\A_k^d$ and whose indeterminacy points are contained in $H$.
Note that $\deg(f^d)=\deg(f)^d$ implies $\deg(f^l)=\deg(f)^l$ for $l=1,\dots, d$. We look at $f$ as an element of $\Bir(\p_k^d)$. If $f$ is an automorphism of $\p_k^d$, its degree is $1$ and the claim follows directly. Otherwise, $f$ contracts the hyperplane $H$. By Lemma \ref{deglemma}, $\deg(f^{l+1})=\deg(f^l)\deg(f)$ is equivalent to $f^l(H)$ not being contained in $\Ind(f)$. In particular, if $\deg(f^l)=\deg(f)^l$ and $f^l(H)=f^{l+1}(H)$ for some $l$ then $\deg(f^i)=\deg(f)^i$ for all $i\geq l$.
Let $1\leq l\leq d$. By Lemma \ref{deglemma}, $f^l(H)$ is not contained in $\Ind(f)$. Observe that $f^l(H)$ is irreducible and $f^{l+1}(H)\subset f^{l}(H)$. This implies that $\dim(f^{l+1}(H))\leq \dim(f^l(H))$ and $\dim(f^{l+1}(H))= \dim(f^l(H))$ if and only if $f^{l+1}(H)= f^{l}(H)$. It follows that the chain $H\supset f(H)\supset f^2(H)\supset\cdots$ becomes stationary within the first $d$ iterations. In particular, $f^d(H)=f^{d+i}(H)$ for all $i$ and therefore $\deg(f^i)=\deg(f)^i$ for all $i$.
\subsection{Proof of Theorem \ref{degaut} and Theorem \ref{degfinite}}
We start by proving Theorem~\ref{degaut}. Let $f\in\End(\A_k^d)$ be an endomorphism such that the sequence $\{\deg(f^n)\}$ is unbounded. Our first remark is that the elements of $\{f^n\}$ are linearly independent in the vector space of polynomial endomorphisms $\End(\A_k^d)$. Indeed, if for any $n$ we have
\[
f^n=\sum_{l<n}c_lf^l,
\]
for some $c_1,\dots, c_{n-1}\in k$. It follows by induction that $\deg(f^{n+i})$ is smaller or equal to $\max_{l<n}\deg(f^l)$ for all~$i$ .
Denote by $\End(\A^d)_{\leq K}$ the $k$-vector space of polynomial endomorphisms of degree $\leq K$, which has dimension $d\cdot\binom{d+K}{K}$. One calculates
\[
d\cdot\binom{d+K}{K}< \frac{1}{(d-1)!}(K+d)^{d}\leq \frac{(1+d/K)^{d}}{(d-1)!}K^{d}\leq C_dK^{d},
\]
where $C_d=\frac{(1+d)^{d}}{(d-1)!}$. Since the elements in $\{f^k\}\cap\End(\A^d)_{\leq K}$ are linearly independent, the cardinality of $\{f^k\}\cap\End(\A^d)_{\leq K}$ is at most $C_dK^{d-1}$.
For the proof of Theorem \ref{degfinite} note that in the case of finite fields, there are only finitely many birational transformations of a given degree. If $f^l=f$ for some $l>1$, then $\{\deg(f^n)\}$ is bounded. There are $\binom{d+K}{K}$ monomials of degree $\leq K$. A birational transformation of degree $\leq K$ is given by $d+1$ polynomials of degree $\leq K$, so by $C(K,d)=(d+1)\binom{d+K}{K}$ coefficients from $\F_q$. Hence there are less than $q^{C(K,d)}$ birational transformations of degree $\leq K$. This proves the claim.
\subsection{Proof of Example \ref{exaut}}\label{proofexaut}
Let $d$ be an integer and $l=\lfloor d/2 \rfloor$. For $d=3$ the automorphism $f_3:=(x+z(y+xz), y+xz,z)$ from Example \ref{linearex} satisfies $\deg(f_3^n)\sim n$. Moreover, the first coordinate of $f_3^n$ is the coordinate with highest degree. Assume now that $d\geq 5$ and that we are given an automorphism $f_{d-2}\in\aut(\A^{d-2})$ such that $\deg(f_{d-2}^n)$ grows like $n^{l-1}$ and that the first coordinate of $f_{d-2}^n$ is the entry with highest degree. Let
\[
f_d:=(x_1+x_3(x_2+x_1x_3), x_2+x_1x_3, f_{d-2}(x_3,\dots,x_{d})).
\]
One sees by induction that the degree of $f_d^n$ grows like $n^l$ and that the first coordinate of $f_{d}^n$ is the coordinate of $f_d^n$ with highest degree.
\subsection{Proof of Example \ref{oguiso}}\label{oguisoproof}
In this section we use the notation introduced in Example \ref{oguiso}.
Let $dz_1, dz_2, dz_3$ be a basis of $H^{1,0}(X)$. Then $d\bar{z_1},d\bar{z_2}, d\bar{z_3}$ is a basis of $H^{0,1}(X)$. The automorphism $f^n$ induces an action on both $H^{1,0}(X)$ and $H^{0,1}(X)$ whose norm grows like $n^2$. Since $H^{1,1}(X)=H^{1,0}(X)\otimes H^{0,1}(X)$, this implies that the norm of the induced action of $f^n$ on $H^{1,1}(X)$ grows like $n^4$. By Corollary \ref{cordinhnguyen}, we obtain $\dpol(f)=4$.
Denote by $\pi\colon X\to Y=X/s$ the quotient map. Let $\tilde{X}$ be a smooth projective variety, $\phi_2\colon\tilde{X}\to X$ a birational morphism and $\psi\colon\tilde{X}\to \tilde{Y}$ a dominant morphism such that the following diagram commutes:
\begin{center}
\begin{tikzcd}
\tilde{X} \arrow{r}{\psi} \arrow{d}{\phi_2}&\tilde{Y} \arrow{d}{\phi_1}
\\X \arrow{r}{\pi}&Y.
\end{tikzcd}
\end{center}
Note that $\psi$ is generically finite. By Proposition \ref{propfede}, we have $\dpol(\phi_2^{-1}\circ f\circ\phi_2)=\dpol(f)=4$ and hence, again by Proposition \ref{propfede}, $\dpol(\tilde{f})=\dpol(\psi\circ \phi_2^{-1}\circ f\circ\phi_2\circ\psi^{-1})=4$, which proves the claim of Example \ref{oguiso}.
\section{Remarks}
\subsection{Other degree functions}
One can define more general degree functions. Let $X_k$ be a smooth projective variety over a field $k$ with polarization $H$ and $1\leq l\leq d-1$, then
\[
\deg_H^l(f):=(f^*H)^l\cdot H^{d-l}.
\]
These degree functions play an important role in dynamics. In characteristic 0, we still have $\deg_H^l(fg)\leq C\deg_H^l(f)\deg_H^l(g)$ for a constant $C$ not depending on $f$ and $g$ (see \cite{MR2180409} and \cite{Truong:2015aa} for generalizations to fields of positive characteristic).
Our proof of Theorem \ref{rationalcount} works as well if we replace the function $\deg_H$ by $\deg_H^l$ for any $l$. Let $\Gamma\subset\Bir(X_k)$ be a finitely generated group of birational transformations with a finite symmetric set of generators $S$. We define
$D^l_{S, H}\colon\Z^+\to\Z^+$
by
$D^l_{S,H}(n):=\max_{\gamma\in B_S(n)}\{\deg_H(\gamma)\}$ and we call a map $\Z^+\to\Z^+$ that can be realized for some $(X_k, H, \Gamma, S,l)$ as such a function a {\it general degree sequence}.
\begin{theorem}
The set of all general degree sequences is countable.
\end{theorem}
\begin{proof}
Analogous to the proof of Theorem \ref{rationalcount}.
\end{proof}
\bibliographystyle{amsalpha}
\bibliography{/Users/christian/Dropbox/Literatur/bibliography_cu}
\end{document} | 126,431 |
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REVIEW: Zen Mafia, California (RCA)
- Scott Hudson
Victor Murgatroyd worked as an A&R executive with Epic Records shepherding such acts as Ian Astbury, Masters of Reality, Buckcherry and American Pearl. Multi-instrumentalist Chris Johnson worked as a producer for a wide variety of acts, with his most notable credits including producing the Trey Parker/Matt Stone and Goldfinger cuts on the BASEketball soundtrack as well as John ("Bisexual Chick") Oszejca's debut album for Interscope.
Murgatroyd and Johnson's paths crossed six years ago when the two worked together on Epic recording artist Kate Gibson's album. From that experience, a professional relationship turned into a personal friendship, and Zen Mafia was born.
When the two recorded their first song together, "Sweet Home California," it was given to a friend, who played it for her friends - friends who just happened to be in the record industry. Zen Mafia were promptly signed and sent to the studio to record their debut album, California.
Murgatroyd and Johnson enlisted members of Blind Melon, Immature, Buckcherry, Goldfinger and American Pearl to complete the project. The finished product is a work that defies categorization.
California is a conglomeration of dance music, hip-hop and rock that meshes perfectly with Murgatroyd's Lou Reed narrative-styled vocals.
The aforementioned "Sweet Home California" is a shuffling blues-rock/hip-hop anthem that utilizes Lynyrd Skynyrd's "Sweet Home Alabama" as it's infectious chorus. Three other songs include re-creations of classic choruses, like the Willie Dixon/Foghat hit, "I Just Want To Make Love To You," Free's "All Right Now" and Bad Company's "Feel Like Makin' Love."
"Another Way" is an acoustic-driven rocker replete with Latin American horns, while "I Keep Comin'" is propelled by classic wah-wah rhythms amidst a turntable frenzy. The record is also graced with two outstanding tracks in "Hold On" and "Show Me" that showcase the bands' penchant for writing great hooks behind a scrim of wonderfully layered harmonies.
All in all, California is a record that features top-notch production, clever, yet very personal, lyrics, and brisk arrangements which makes for a strong freshman effort for Zen Mafia.
Because it defies categorization, it may be difficult to decide which bin your favorite record store will house this one in. But when you find it - Grab it! You won't be sorry! | 237,370 |
Matthew Kennedy grabbed an assist as Hibernian won and moved up to sixth in the Scottish Championship.
The on-loan Everton winger crossed for Jordon Foster to head home the first goal of the game against Cowdenbeath.
The score was locked at 2-2 going into injury time, but Hibs’ Jason Cummings netted with almost the last kick of the game to make it 3-2.
Kennedy played the full match for Alan Stubbs’ side.
John Lundstram helped Blackpool register their first point of the season on Saturday.
The on-loan Everton midfielder played the full 90 minutes for the Tangerines as they drew 0-0 with visitors Wolves in the Championship.
Boss Jose Riga later praised the atmosphere the home fans created, saying: “It was incredible and it helped us a lot.
“The way the team behaved and the commitment between the fans, players and the staff was great.”
Hallam Hope played no part in Sheffield Wednesday's 0-0 draw at Botlon. | 281,724 |
ScienceDaily (Feb. 29, 2008) — Natural History Museum in Oslo, Norway has announced the discovery of one of the largest dinosaur-era marine reptiles ever found – an enormous sea predator known as a pliosaur estimated to be almost 15 meters (50 feet) feet long.
The 150 million year-old Jurassic fossil was discovered on the remote Norwegian archipelago of Svalbard, at 78 degrees north latitude, approximately 1300 km (800 miles) from the North Pole. It was found in the summer of 2006 by a team of Norwegian paleontologists and volunteers from the University of Oslo Natural History Museum, led by Dr. Jørn Hurum. The fossil was excavated in the summer of 2007 and has until now been prepared and conserved by a team at the Natural History Museum in Oslo .
A pliosaur is a type of plesiosaur, a group of extinct reptiles that lived in the world's oceans during the age of dinosaurs. Pliosaurs had a tear-drop shaped body and two sets of powerful paddles that it used to “fly” through the water. Their short neck supported a massive skull full of an impressive set of teeth. Pliosaurs were the top predators in the sea at the time, preying upon squid-like animals, fish, and even other marine reptiles.
Pliosaurs were large reptiles that averaged 5-6 meters (16-20 feet) in length. One of the largest pliosaurs known is the Australian giant Kronosaurus , which measures in at 10-11 meters (33-36 feet) long. The new Norwegian find, named “The Monster” by team members, is estimated to be about 15 meters (50 feet) long, making it one of the longest and most massive plesiosaurs yet found.
“Not only is this specimen significant in that it is one of the largest and relatively complete plesiosaurs ever found, it also demonstrates that these gigantic animals inhabited the northern seas of our planet during the age of dinosaurs” said Dr. Patrick Druckenmiller, a plesiosaur specialist at the University of Alaska Museum, and a member of the expedition that found and excavated the fossil.ørn Hurum.
A larger crew returned in August of 2007 to excavate the fossil. After removing about hundred tons of rock by hand the team was rewarded by uncovering a significant portion of the skeleton.
ørn Hurum comments.
The bones collected last summer are currently undergoing the slow process of cleaning at the Natural History Museum in Oslo .
“From the bones we have finished stabilizing so far this absolutely looks like a new species” Jørn Hurum tells enthusiastically.
Last summer's field work ended on a high note; after the team finished excavating the new find, parts of a skull and skeleton from another gigantic pliosaur, possibly of the same new species, were found weathering out at a different site. The team is currently planning to return this upcoming summer in order to excavate the skull during Svalbard 's short summer field season.
“All together we have GPS-coordinates on 40 skeletons of different marine reptiles in the area. We will have work for many years to come” Jørn Hurum comments with a smile.
The scientific description of the monster and other new marine reptiles from the locality will be parts of a Ph.D. study by one of the members of the crew, Espen M. Knutsen.
Adapted from materials provided by University of Oslo.
=====
SD
| 67,536 |
Level 2 October 2019
Mondays at 7:30pm starting October 28th
Time to take it up the pole! Once you’ve completed our Level 1 course (or if you have prior pole experience from another studio) you can move on up to Level 2 and work on your climbing skills! We will also learn some more complex static pole spins and transitions, seated aerial poses, as well as a healthy dose of conditioning and strength work to start prepping us for inversions.
Level 2 is a step up from Level 1, definitely more challeging but that just means the rewards are bigger when you finally get those moves! Many students repeat Level 2, in pole dance it just takes time as we’re not used to holding ourselves up by our arms. However, this is the best (and I think most fun) way to develop that upper body and grip strength! If you want to do some extra strength training I suggest also attending the off-pole Conditioning classes on Tuesday nights at 6:15pm, or contacting Angela at Aria Fitness (she also teaches the Conditioning class) for a customized plan you can do at the gym for cross-training.
What to wear/bring
You will need to wear a pair of shorts that leave the upper thigh exposed as you need skin contact for climbing and sitting on the pole. I suggest wearing a pair of sweat pants over top for warm-up, especially in the colder winter months.
Please bring a water bottle, because we love the earth and are doing our part to reduce plastic waste! We have a water filter available for filling/refilling your bottle.
Registration & Pricing
Just like Level 1 we can take a maximum of 10 students per class (2 to a pole).. The short version is we need at least 72 hours notice if you need to cancel to be able to issue a refund. We promise it’s not so that Lisa can pay off the mortgage on her beach house, it’s so we can keep the lights on and continue bringing pole dancing fun and community to Nanaimo! | 140,438 |
How to Determine on the Perfect Anime Gifts and Merch
Buying or rather purchasing aNaruto gift tends to be one thing that a lot of individuals tend to find so stressful. The fact that getting to be able to settle on the gift that the individual finds idealMy Hero Acadamia to buy is one thing and the other tends to be getting to be able to choose the one that is one... Licensing stands to be a major challenge that has not been solved. There is tendency of the main reason for this to be due to the fact that there is negligent when it comes to the dealings of the buyers or rather the sellers where.. Getting the cost of the anime gifts and merch as well as their value getting to be equivalent is what tends to make them to be termed as being the right ones. More to this, they should ensure that the seller is not profit oriented. | 357,756 |
Aepodia integrates Ipsen's R&D Campus in Paris-Saclay biotechnology hub
Aepodia integrates Ipsen’s R&D Campus in Paris-Saclay biotechnology hub
Christophe Thurieau, Senior Vice-President, Global Scientific Affairs & President Ipsen Innovation stated :
We are pleased to announce this partnership with Aepodia, as it paves the way to build new relationships and agreements with other leading innovators in the life sciences. It also gives new capabilities at our largest R&D center located in the leading Paris-Saclay biotech hub. This “Campus initiative” will lead and catalyze our early stage research and clinical studies programs.
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We are very pleased to be the first company to participate in Ipsen’s Open Innovation initiative in Les Ulis. The agreement represents and important recognition of Aepodia’s expertise in the early clinical development of novel drug candidates. This partnership between two patient-focused companies with deep clinical and discovery expertise will address patient needs by leveraging the cutting edge competencies of both companies in the effective development of innovative new therapies. Aepodia’s participation as the first partner in the Open Innovation Campus supports our strong position as a leader in early stage clinical drug development.
Find out more information on: | 70,950 |
\begin{document}
\newcommand{\bigslant}[2]{{\left.\raisebox{.2em}{$#1$}\middle/\raisebox{-.2em}{$#2$}\right.}}
\global\long\def\quotient#1#2{\bigslant{#1}{#2}}
\global\long\def\Span{\operatorname{Span}}
\global\long\def\vol{\operatorname{vol}}
\global\long\def\id{\operatorname{id}}
\global\long\def\Mat{\operatorname{Mat}}
\global\long\def\re{\operatorname{Re}}
\global\long\def\im{\operatorname{Im}}
\global\long\def\rank{\operatorname{rank}}
\global\long\def\img{\operatorname{im}}
\global\long\def\diam{\operatorname{diam}}
\global\long\def\dist{\operatorname{dist}}
\global\long\def\Length{\operatorname{Length}}
\global\long\def\Count{\operatorname{Count}}
\global\long\def\Clopen{\operatorname{Clopen}}
\global\long\def\var{\operatorname{var}}
\global\long\def\cov{\operatorname{cov}}
\global\long\def\median{\operatorname{Median}}
\global\long\def\td{\mathbb{T}^{d}}
\global\long\def\Td{\mathcal{T}^{d}}
\global\long\def\zd{\mathbb{Z}^{d}}
\global\long\def\rd{\mathbb{R}^{d}}
\global\long\def\rk{\mathbb{R}^{k}}
\global\long\def\rn{\mathbb{R}^{n}}
\global\long\def\rm{\mathbb{R}^{m}}
\global\long\def\R{\mathbb{R}}
\global\long\def\c{\mathbb{C}}
\global\long\def\z{\mathbb{Z}}
\global\long\def\q{\mathbb{Q}}
\global\long\def\n{\mathbb{N}}
\global\long\def\t{\mathbb{T}}
\global\long\def\p{\mathbb{P}}
\global\long\def\E{\mathbb{E}}
\global\long\def\Z{\mathcal{Z}}
\global\long\def\D{\mathcal{D}}
\global\long\def\N{\mathcal{N}}
\global\long\def\F{\mathbf{F}}
\global\long\def\e{\mathrm{e}}
\global\long\def\i{\mathrm{i}}
\global\long\def\Empty{\varnothing}
\global\long\def\epsilon{\varepsilon}
\global\long\def\tilde#1{\widetilde{#1}}
\global\long\def\ll{\lesssim}
\global\long\def\gg{\gtrsim}
\global\long\def\sums{\Lambda_{L}}
\global\long\def\sumsplus{\Lambda_{L}^{+}}
\global\long\def\sumsnorm{\widetilde{\Lambda}_{L}}
\global\long\def\hilbert{\mathcal{H}_{L}}
\global\long\def\sphere{\mathcal{S}^{d-1}}
\global\long\def\nsphere{\mathcal{S}^{n-1}}
\global\long\def\trigs{\mathcal{P}_{D}}
\global\long\def\laplacedom{\mathcal{D}_{\Delta}}
\global\long\def\matrices{\Mat_{n}\left(\R\right)}
\newcommandx\Int[4][usedefault, addprefix=\global, 1=, 2=]{\int_{#1}^{#2}#3\,\mathrm{d}#4}
\global\long\def\indic{\mathbbm{1}}
\title{\noindent \textbf{\Huge{}The Number of Nodal Components of Arithmetic
Random Waves}}
\author{Yoni Rozenshein\thanks{School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel.
Email: \protect\href{mailto:[email protected]}{[email protected]}}}
\maketitle
\begin{abstract}
We study the number of nodal components (connected components of the
set of zeroes) of functions in the ensemble of arithmetic random waves,
that is, random eigenfunctions of the Laplacian on the flat $d$-dimensional
torus $\td$ ($d\ge2$). Let $f_{L}$ be a random solution to $\Delta f+4\pi^{2}L^{2}f=0$
on $\td$, where $L^{2}$ is a sum of $d$ squares of integers, and
let $N_{L}$ be the random number of nodal components of $f_{L}$.
By recent results of Nazarov and Sodin, $\E\left\{ N_{L}/L^{d}\right\} $
tends to a limit $\nu>0$, depending only on $d$, as $L\to\infty$
subject to a number-theoretic condition - the equidistribution on
the unit sphere of the normalized lattice points on the sphere of
radius $L$. This condition is guaranteed when $d\ge5$, but imposes
restrictions on the sequence of $L$ values when $2\le d\le4$. We
prove the exponential concentration of the random variables $N_{L}/L^{d}$
around their medians and means (unconditionally) and around their
limiting mean $\nu$ (under the condition that it exists).
\end{abstract}
\section{Introduction and presentation of the results}
\subsection{Toral eigenfunctions and arithmetic random waves}
Let $\hilbert\subset L^{2}\left(\td\right)$ be the real Hilbert space
of Laplacian eigenfunctions on the torus, i.e.\ functions $f\colon\td\to\R$
satisfying the partial differential equation:
\[
\Delta f+4\pi^{2}L^{2}f=0.
\]
We consider $d\ge2$ to be a fixed dimension (all ``constants''
mentioned below may depend on $d$); $L$ may vary. It is known that
the spectrum of eigenvalues is discrete; eigenfunctions exist whenever
$L^{2}$ can be expressed as a sum of $d$ squares of integers, and
then,
\[
\hilbert=\Span\left\{ \cos\left(2\pi\lambda\cdot x\right),\sin\left(2\pi\lambda\cdot x\right):\lambda\in\sums\right\} ,
\]
where $\sums=\left\{ \lambda\in\zd:\left|\lambda\right|=L\right\} $.
Each $\lambda$ generates the same functions as $-\lambda$, so $\dim\hilbert=\#\sums$.
For any $f\colon\td\to\R$, we denote by $Z\left(f\right)$ its \emph{nodal
set }(the subset of $\td$ where $f$ vanishes), and by $N\left(f\right)$
the number of its \emph{nodal components }(the connected components
of the nodal set). In this paper, we address the question: \emph{What
is the typical behavior of $N\left(f\right)$ for $f\in\hilbert$,
with fixed $d$ and large $L$?}
Typically (when $f$ and $\nabla f$ do not vanish simultaneously),
the number of nodal components almost equals the number of \emph{nodal
domains} (the connected components of $\td\setminus Z\left(f\right)$)
- they cannot differ by more than $d-1$. Thus, Courant's nodal domain
theorem gives a general upper bound $N\left(f\right)\ll L^{d}$, with
an explicit constant. Unfortunately, a general, non-trivial lower
bound cannot be obtained, as there are classical counterexamples with
arbitrarily large $L$ and only two nodal domains, originally shown
in \cite{Stern-thesis} (see also \cite{Berard-Helffer-Dirichlet-eigenfunctions},
\cite{Bruning-Fajman-On-the-nodal-count-for-flat-tori}).
It is expected, however, that such eigenfunctions with high eigenvalue
but few nodal components are outliers, and $N\left(f\right)$ is in
the order of magnitude of $L^{d}$ for ``most'' $f\in\hilbert$.
To study the typical case, we refer to a probabilistic model that
was introduced and investigated in \cite{Oravecz-Rudnick-Wigman-the-Leray-measure-of-nodal-sets,Rudnick-Wigman-the-volume-of-nodal-sets}.
Consider the random function $f_{L}\colon\td\to\R$:
\begin{equation}
f_{L}\left(x\right)\coloneqq\sqrt{\frac{2}{\dim\hilbert}}\sum_{\lambda\in\sumsplus}\left(a_{\lambda}\cos\left(2\pi\lambda\cdot x\right)+b_{\lambda}\sin\left(2\pi\lambda\cdot x\right)\right),\label{eq:fL}
\end{equation}
where the set $\sumsplus=\quotient{\sums}{\pm}$ is half of the set
$\sums$ (representatives of the equivalence $\lambda\sim\pm\lambda$),
and $a_{\lambda},b_{\lambda}$ are random variables, i.i.d.\ $\N\left(0,1\right)$.
The sequence of functions $\left\{ f_{L}\right\} $ is called the
\emph{ensemble of arithmetic random waves}. The random function $f_{L}$
may be viewed as a random element of the finite-dimensional space
$\hilbert$, or as a centered, stationary Gaussian process, normalized
such that $\E\left\{ \left|f_{L}\left(x\right)\right|^{2}\right\} =1$,
with covariance kernel:
\begin{equation}
K_{L}\left(x,y\right)\coloneqq\E\left\{ f_{L}\left(x\right)f_{L}\left(y\right)\right\} =\frac{1}{\dim\hilbert}\sum_{\lambda\in\sums}\cos\left(2\pi\lambda\cdot\left(x-y\right)\right).\label{eq:KL}
\end{equation}
Note that due to rotation invariance, the definition of $f_{L}$ does
not depend on the choice of basis for $\hilbert$.
Under this probabilistic model, the number of nodal components $N_{L}\coloneqq N\left(f_{L}\right)$
becomes a random variable (we discuss its measurability in detail
in Section \ref{sec:measurability}) and the question of its behavior
may be formulated in terms of expected value and concentration as
$L\to\infty$.
\subsection{Asymptotic law for \texorpdfstring{$\protect\E\left\{ N_{L}\right\} $}{E\{NL\}}}
Nazarov and Sodin \cite{Nazarov-Sodin-asymptotic-laws} (see also
lecture notes \cite{Sodin-SPB-Lecture-Notes}) proved, in a much more
general setting of ensembles of Gaussian functions on Riemannian manifolds,
an asymptotic law for the expected value of $N\left(f\right)$. Our
first theorem, Theorem \ref{thm:asymptotic-law}, is simply a formulation
of the Nazarov-Sodin theorem, applied to our case. There is one obstacle:
The theorem requires the existence of a limiting spectral measure
satisfying certain properties. In our case, this limiting spectral
measure does not necessarily exist, and it depends on the following
number-theoretic equidistribution condition:
\begin{defn}
\label{def:admissible-sequence}A sequence of values of $L$ that
tends to infinity, with $L^{2}$ always a sum of $d$ squares, is
called an \emph{admissible sequence of $L$ values} if the integer
points on the sphere of radius $L$, when projected onto the unit
sphere, become equidistributed as $L\to\infty$. In other words,
\begin{equation}
\frac{1}{\dim\hilbert}\sum_{\lambda\in\sums}\delta_{\lambda/L}\Rightarrow\sigma_{d-1},\label{eq:weak-convergence-of-measures}
\end{equation}
where ``$\Rightarrow$'' indicates weak-{*} convergence of measures,
and $\sigma_{d-1}$ is the uniform measure on the unit sphere, with
the normalization $\sigma_{d-1}\left(\sphere\right)=1$.
\end{defn}
This equidistribution condition depends on the dimension $d$. When
$d\ge5$, any sequence of $L$ values is admissible, whereas in the
low dimensions $2\le d\le4$, some conditions must be satisfied. For
more on the subject, see Appendix \ref{sec:appendix-equidistribution}.
\begin{thm}
\label{thm:asymptotic-law}There is a constant $\nu>0$ such that:
\[
\begin{array}{cc}
\E\left\{ N_{L}\right\} \sim\nu L^{d} & \text{ as }L\to\infty\text{ through any admissible sequence}.\end{array}
\]
\end{thm}
\subsection{Main result: Exponential concentration of \texorpdfstring{$N_{L}$}{NL}}
Our main result is that $N_{L}$ concentrates around its median, mean,
and limiting mean, exponentially in $\dim\hilbert$, i.e.\ the number
of independent random variables. This is similar to a previous result
by Nazarov and Sodin in the case of random spherical harmonics \cite{Nazarov-Sodin-spherical-harmonics}.
\begin{thm}
\label{thm:exponential-concentration}Let $\varepsilon>0$. There
exist constants $C\left(\varepsilon\right),c\left(\varepsilon\right)>0$
such that:
\begin{enumerate}
\item For any $L$:
\[
\p\left\{ \left|\frac{N_{L}}{L^{d}}-\median\left\{ \frac{N_{L}}{L^{d}}\right\} \right|>\varepsilon\right\} \le C\left(\epsilon\right)\e^{-c\left(\epsilon\right)\dim\hilbert}.
\]
\item For any $L$:
\[
\p\left\{ \left|\frac{N_{L}}{L^{d}}-\E\left\{ \frac{N_{L}}{L^{d}}\right\} \right|>\varepsilon\right\} \le C\left(\epsilon\right)\e^{-c\left(\epsilon\right)\dim\hilbert}.
\]
\item If $\mathbb{E}\left\{ N_{L}L^{-d}\right\} $ tends to a limit $\nu$
through some sequence of $L$ values, then for any large enough $L$
in this sequence:
\[
\p\left\{ \left|\frac{N_{L}}{L^{d}}-\nu\right|>\varepsilon\right\} \le C\left(\varepsilon\right)\e^{-c\left(\varepsilon\right)\dim\hilbert}.
\]
\end{enumerate}
In all cases, our proof yields $c\left(\varepsilon\right)\gg\epsilon^{\left(d+2\right)^{2}-1}$.\end{thm}
\begin{rem}
\label{rem:understanding-the-main-result}In the low dimensions $2\le d\le4$,
it is possible to have sequences of $L$ values for which $\dim\hilbert$
stays bounded as $L$ grows. However, if $\dim\hilbert$ is bounded,
then all parts of Theorem \ref{thm:exponential-concentration} are
completely trivial, and say nothing. This is unlike the case of random
spherical harmonics discussed in \cite{Nazarov-Sodin-spherical-harmonics},
in which the dimension is a simple ascending function of the eigenvalue.
The second and third parts of Theorem \ref{thm:exponential-concentration}
are straightforward consequences of the first part. This is proven
in Subsection \ref{sub:concentration-implies-concentration}. When
$d\ge3$, the third part of the theorem only makes sense when $\nu$
is the same $\nu$ from Theorem \ref{thm:asymptotic-law}. This is
because under the assumption that $\dim\hilbert$ is not bounded from
below (without which, the theorem says nothing anyway), the limit
(\ref{eq:weak-convergence-of-measures}) holds and the sequence is
admissible. However, when $d=2$, we could have a limiting measure
other than $\sigma_{d-1}$ in (\ref{eq:weak-convergence-of-measures}),
so value of $\nu$ in the third part of Theorem \ref{thm:exponential-concentration}
truly depends on the chosen sequence of $L$ values. See also Appendix
\ref{sec:appendix-equidistribution} and \cite{Kurlberg-Wigman-Non-universality}.
\end{rem}
\subsection{Outline of the paper}
In Section \ref{sec:measurability}, we prove the Borel measurability
of the random variable $N_{L}$ - the number of nodal components.
We deduce it from a more general result - the measurability of the
number of nodal components of more general random functions (Proposition
\ref{prop:measurability-generalization}), which may be of independent
interest.
In Section \ref{sec:proof-of-usability-of-Nazarov-Sodin-theorem},
we show that Theorem \ref{thm:asymptotic-law} follows directly from
the Nazarov-Sodin theorem.
In Section \ref{sec:trigonometric-polynomials-and-nodal-sets}, we
treat trigonometric polynomials in general, and give algebraic proofs
to bounds on the sum of diameters of their connected components.
In Section \ref{sec:proof-of-main-theorem}, we present a proof of
Theorem \ref{thm:exponential-concentration}.
In Appendix \ref{sec:appendix-equidistribution}, we provide background
and quote the known results on the problem of equidistribution of
lattice points on spheres.
In Appendix \ref{sec:appendix-additional-proofs}, we provide proofs
for some of the claims used in the paper.
\subsection{Notation}
We reserve the letters $C$ and $c$ for positive constants (usually
upper and lower bounds, respectively) which may vary from line to
line; all constants may depend on the dimension $d$. When $A,B$
are positive quantities, we denote by $A\ll B$, $A\gg B$ and $A\simeq B$
that $A\le CB$, $A\ge cB$ and $cB\le A\le CB$, respectively.
The notation $K_{+\delta}$ indicates the set of all points of distance
at most $\delta$ from the compact set $K$, which may be a set in
$\td$ or $\rd$ with Euclidean distance or $L^{2}\left(\td\right)$
with the norm-induced distance.
\subsection{Acknowledgments}
This work was written following and based on my master's thesis in
Tel Aviv University. I thank my advisor, Mikhail Sodin, for providing
me with the opportunity to work on such a diverse project; his invaluable
guidance and patience has made this work possible. I also thank Zeév
Rudnick, Lior Bary-Soroker, Eugenii Shustin, and my fellow students,
for many discussions and helpful advice. I also thank the anonymous
referee, whose fruitful comments made it possible to improve the presentation
of the paper. This work, and the author's master degree studies, were
partly supported by grant N\textsuperscript{\underline{o}} 166/11
of the Israel Science Foundation of the Israel Academy of Sciences
and Humanities.
\section{\label{sec:measurability}Measurability of \texorpdfstring{$N_{L}$}{NL}}
The random process $f_{L}$ is, formally, a function $f_{L}\colon\td\times\Omega\to\R$,
where $\Omega$ is the Gaussian probability space. In this point of
view, $N_{L}$ is the function on $\Omega$ given by $\omega\mapsto N\left(f_{L}\left(\cdot,\omega\right)\right)$.
\begin{prop}
\label{prop:NL-is-measurable}$N_{L}$ is a random variable. In other
words, the mapping $N_{L}\colon\Omega\to\n\cup\left\{ 0,\infty\right\} $
is measurable.
\end{prop}
Naturally, the countable set $\n\cup\left\{ 0,\infty\right\} $ is
equipped with the discrete (power set) $\sigma$-algebra.
There is an ``analytic'' way to prove that $N_{L}$ is at least
Lebesgue measurable. Skipping some details, this proof is as follows:
By Bulinskaya's lemma, the event that $f_{L}$ does not have a ``stable
nodal set'' is a subset of $\Omega$ of Lebesgue measure zero (see
Definition \ref{def:stable-nodal-set} and Proposition \ref{prop:a.s.-stable-nodal-set}
ahead). The complement of this event is open because, since $\dim\hilbert$
is finite, the norm $\left\Vert g\right\Vert $ of any small perturbation
$g\in\hilbert$ bounds both $\max\left|g\right|$ and $\max\left|\nabla g\right|$.
$N_{L}$ is locally constant in this open set (see Proposition \ref{prop:smooth-perturbation}
ahead). Thus, any event of the form $\left\{ N_{L}=n\right\} $ is
a union of an open set and a subset of an event of Lebesgue measure
zero.
The above proof has two weaknesses: The first is that it makes assumptions
on the function space. To use Bulinskaya's lemma, we require the smoothness
of the functions, the finite dimension of $\td$, and a condition
on the probability density of the random process, and to continue
the proof, we need the finite dimension of $\hilbert$. The second
weakness is that the proof only yields Lebesgue measurability.
In this section, we present a strong generalization of Proposition
\ref{prop:NL-is-measurable} and prove it using only the most basic
definitions in topology and measure theory:
\begin{prop}
\label{prop:measurability-generalization}Let $X$ be a compact metric
space, let $\Omega$ be a (not necessarily complete) probability space,
and let $f\colon X\times\Omega\to\R$ be a random real-valued function
on $X$ that is a.s.\ continuous. Then the number of nodal components
of $f$ is a random variable, i.e.\ a measurable mapping $\Omega\to\n\cup\left\{ 0,\infty\right\} $.
\end{prop}
This immediately implies Proposition \ref{prop:NL-is-measurable}
(where $X=\td$) with Borel measurability. To prove Proposition \ref{prop:measurability-generalization},
observe that the number of nodal components of $f$ is the composition
of three maps:
\[
\Omega\xrightarrow{\quad f\quad}C\left(X\right)\xrightarrow{\quad Z\quad}{\bf F}\left(X\right)\xrightarrow{\quad\Count\quad}\mathbb{N}\cup\left\{ 0,\infty\right\}
\]
\begin{itemize}
\item The first map $\omega\mapsto f\left(\cdot,\omega\right)$ sends almost
every $\omega\in\Omega$ to a corresponding function in $C\left(X\right)$.
\item The second map $f\mapsto Z\left(f\right)$ sends a continuous real
function to its (closed) zero set.
\item The third map $F\mapsto\Count\left(F\right)$ counts the number of
connected components of a given closed set $F\subset X$.
\end{itemize}
In the remainder of this section, we show that all three maps are
measurable with respect to the following (standard) $\sigma$-algebras
on $C\left(X\right)$ and $\F\left(X\right)$:
\begin{itemize}
\item The standard $\sigma$-algebra on $C\left(X\right)$ is generated
by the \emph{point-evaluation maps} $\left\{ f\mapsto f\left(x\right)\right\} _{x\in X}$;
that is, it is generated by the family of sets $\left\{ f\in C\left(X\right):f\left(x\right)\in B\right\} $,
where $x$ varies over all points in $X$ and $B$ varies over all
Borel subsets of $\R$.
\item The standard $\sigma$-algebra on $\F\left(X\right)$ is given by
the following equivalent definitions (see also chapters 2.4 and 3.3
of \cite{Srivastava-book-A-course-on-Borel-sets}):
\begin{enumerate}
\item The $\sigma$-algebra $\mathcal{F}_{1}$ generated by the family of
sets $\left\{ F\in{\bf F}\left(X\right):F\cap U\ne\Empty\right\} $,
where $U$ varies over all open subsets of $X$.
\item The $\sigma$-algebra $\mathcal{F}_{2}$ generated by the family of
sets $\left\{ F\in\mathbf{F}\left(X\right):F\subset U\right\} $,
where $U$ varies over all open subsets of $X$.
\item The $\sigma$-algebra $\mathcal{F}_{3}$ generated by the family of
sets $\left\{ F\in\mathbf{F}\left(X\right):F\cap K\ne\Empty\right\} $,
where $K$ varies over all compact subsets of $X$.\end{enumerate}
\begin{proof}[Proof that $\mathcal{F}_{1}=\mathcal{F}_{2}=\mathcal{F}_{3}$]
Any subset of $X$ is compact if and only if it is closed, so the
sets $\left\{ F\in\mathbf{F}\left(X\right):F\subset U\right\} $ and
$\left\{ F\in\mathbf{F}\left(X\right):F\cap K\ne\Empty\right\} $
are complementary when taking $K=X\setminus U$, and we get $\mathcal{F}_{2}=\mathcal{F}_{3}$.
Any open subset of $X$ is a countable union of compact subsets, so
any set $\left\{ F\in{\bf F}\left(X\right):F\cap U\ne\Empty\right\} $
is a countable union of sets $\left\{ F\in\mathbf{F}\left(X\right):F\cap K\ne\Empty\right\} $,
and we have $\mathcal{F}_{1}\subset\mathcal{F}_{3}$. Similarly, any
compact subset is a countable intersection of open subsets, so any
set $\left\{ F\in\mathbf{F}\left(X\right):F\cap K\ne\Empty\right\} $
is a countable intersection of sets $\left\{ F\in{\bf F}\left(X\right):F\cap U\ne\Empty\right\} $,
and we have $\mathcal{F}_{3}\subset\mathcal{F}_{1}$.
\end{proof}
\end{itemize}
It is evident that the first map $\Omega\to C\left(X\right)$ is measurable:
By the definition of the $\sigma$-algebra on $C\left(X\right)$,
this is equivalent to the map $\Omega\to\R$ given by $\omega\mapsto f\left(x,\omega\right)$
being measurable for any $x\in X$, which is precisely the definition
of $f$ being a random function.
The second map is measurable by the following:
\begin{prop}
Let $X$ be a compact metric space. The map $Z\colon C\left(X\right)\to{\bf F}\left(X\right)$
given by $f\mapsto f^{-1}\left(\left\{ 0\right\} \right)$ is measurable.\end{prop}
\begin{proof}
We will show that for any compact $K\subset X$, the set $\left\{ F\in{\bf F}\left(X\right):F\cap K\ne\Empty\right\} $
has a measurable preimage under $Z$ - that is, that the set $\left\{ f\in C\left(X\right):\exists x\in K:f\left(x\right)=0\right\} $
is measurable in $C\left(X\right)$. Let $A\subset K$ be countable
and dense in $K$. By a standard continuity argument, we have:
\begin{equation}
\left\{ f\in C\left(X\right):\exists x\in K:f\left(x\right)=0\right\} =\bigcap_{\varepsilon>0}\bigcup_{x\in A}\left\{ f\in C\left(X\right):\left|f\left(x\right)\right|<\varepsilon\right\} .\label{eq:all-f-with-zero-in-K}
\end{equation}
The sets on the right hand side are generating sets in the $\sigma$-algebra
of $C\left(X\right)$, so the set on the left hand side is measurable,
proving the proposition.
\end{proof}
The measurability of the third map - the component counting function
- is a little trickier. We first show a couple of lemmas that will
help translate the number of components to a property of covers by
open sets, which is more easily expressed by generators of the $\sigma$-algebra
on $\F\left(X\right)$.
For any topological space $Y$, we denote by $\Clopen\left(Y\right)$
the Boolean algebra of clopen (that is, closed and open) subsets of
$Y$. Note that $Y$ is connected if and only if $\Clopen\left(Y\right)=\left\{ \Empty,Y\right\} $,
and that if $A\in\Clopen\left(Y\right)$ then $\Clopen\left(A\right)\subset\Clopen\left(Y\right)$.
Recall that any clopen set is a union of connected components, and
that connected components are always closed.
\begin{lem}
\label{lem:counting-components-in-the-clopen-algebra}Let $Y$ be
a topological space and let $N$ be a positive integer. The following
are equivalent:
\begin{enumerate}
\item $Y$ has strictly fewer than $N$ connected components.
\item For any $Y_{1},\ldots,Y_{N}\in\Clopen\left(Y\right)$, if they are
pairwise disjoint then one of them is empty.
\end{enumerate}
\end{lem}
\begin{proof}
The second condition follows from the first simply by the pigeonhole
principle, since clopen sets are unions of connected components. Conversely,
assume the second condition holds. Suppose that $\Clopen\left(Y\right)$
is an infinite collection of sets. Let $Y_{1}\in\Clopen\left(Y\right)$.
Any clopen set $A$ is the union of $A\cap Y_{1}$ and $A\setminus Y_{1}$,
both clopen, so either $\Clopen\left(Y_{1}\right)$ is infinite or
$\Clopen\left(Y\setminus Y_{1}\right)$ is infinite. We may thus iteratively
construct a sequence $Y_{1},Y_{2},\ldots\in\Clopen\left(Y\right)$
of nonempty pairwise disjoint sets, contradicting the assumed condition.
Therefore, $\Clopen\left(Y\right)$ must be finite, and being a finite
Boolean algebra, it is generated by a finite number of atoms - clopen
sets with no clopen subsets. These atoms are precisely the connected
components, and applying the assumed condition on the atoms, there
must be fewer than $N$ of them.
\end{proof}
For the following, recall that any compact metric space $X$ admits
a countable collection $\mathcal{U}$ of open sets which separates
closed sets: For any closed, pairwise disjoint $F_{1},\ldots,F_{n}\subset X$,
there are pairwise disjoint $U_{1},\ldots,U_{n}\in\mathcal{U}$ with
$F_{i}\subset U_{i}$. We skip the proof of this, which follows easily
from the fact that any compact metric space is second-countable and
normal.
\begin{lem}
\label{lem:N-components-separated-by-open-sets}Let $X$ be a compact
metric space, let $\mathcal{U}$ be a collection of open sets in $X$
which separates closed sets, let $F\subset X$ be closed and let $N$
be a positive integer. The following are equivalent:
\begin{enumerate}
\item $F$ has strictly fewer than $N$ connected components.
\item For any pairwise disjoint $U_{1},\ldots,U_{N}\in\mathcal{U}$ such
that $F\subset\bigcup_{i=1}^{N}U_{i}$, there is a proper subcover
(i.e.\ one of the sets $U_{i}$ does not intersect $F$).
\end{enumerate}
\end{lem}
\begin{proof}
Suppose the first condition holds. Let $U_{1},\ldots,U_{N}\in\mathcal{U}$
be pairwise disjoint sets that cover $F$. Each $F\cap U_{i}$ is
relatively open in $F$, and its $F$-complement $F\cap\left(\bigcup_{j\ne i}U_{j}\right)$
is also relatively open in $F$. Therefore, $F\cap U_{i}\in\Clopen\left(F\right)$.
By Lemma \ref{lem:counting-components-in-the-clopen-algebra}, one
of the $F\cap U_{i}$ must be empty.
Conversely, suppose the second condition holds. We will show that
for any $F_{1},\ldots,F_{N}\in\Clopen\left(F\right)$ that are pairwise
disjoint, one of them must be empty, and then we are done by Lemma
\ref{lem:counting-components-in-the-clopen-algebra}.
Without loss of generality, we assume $F=\bigcup_{i=1}^{N}F_{i}$
(otherwise, we replace $F_{1}$ with $F\setminus\bigcup_{i=2}^{N}F_{i}$).
Since $F_{1},\ldots,F_{N}$ are relatively closed in $F$, they are
closed in $X$. Let $U_{1},\ldots,U_{N}\in\mathcal{U}$ be pairwise
disjoint such that $F_{i}\subset U_{i}$. By the hypothesis of the
second condition, one of the $U_{i}$ does not intersect $F$, meaning
one of the $F_{i}$ is empty.\end{proof}
\begin{prop}
Let $X$ be a compact metric space. The map $\Count\colon\F\left(X\right)\to\mathbb{N}\cup\left\{ 0,\infty\right\} $,
that counts the number of connected components in the given closed
set, is measurable.\end{prop}
\begin{proof}
The sets $\left\{ 0\right\} ,\left\{ 0,1\right\} ,\left\{ 0,1,2\right\} ,\ldots$
generate the $\sigma$-algebra on $\mathbb{N}\cup\left\{ 0,\infty\right\} $,
so it is enough to show that given a positive integer $N$, the following
set is measurable:
\[
\F_{<N}\coloneqq\left\{ F\in{\bf F}\left(X\right):F\text{ has strictly fewer than }N\text{ connected components}\right\} .
\]
Let $\mathcal{U}$ be a countable collection of open sets separating
closed sets. By Lemma \ref{lem:N-components-separated-by-open-sets},
$\F_{<N}$ may be written as the set of all $F\in\F\left(X\right)$
such that for any pairwise disjoint $U_{1},\dots,U_{N}\in\mathcal{U}$,
if $F\subset\bigcup_{i=1}^{N}U_{i}$ then one of the $U_{i}$ does
not intersect $F$:
\begin{align}
{\bf F}_{<N} & =\bigcap_{\substack{U_{1},\ldots,U_{N}\in\mathcal{U}\\
\text{pairwise disjoint}
}
}\left\{ F\in\mathbf{F}\left(X\right):\left(\left(F\subset\bigcup_{i=1}^{N}U_{i}\right)\implies\left(\exists i:F\cap U_{i}=\Empty\right)\right)\right\} \nonumber \\
& =\bigcap_{\substack{U_{1},\ldots,U_{N}\in\mathcal{U}\\
\text{pairwise disjoint}
}
}\left\{ F\in\mathbf{F}\left(X\right):\left(F\not\subset\bigcup_{i=1}^{N}U_{i}\right)\text{ or }\left(\exists i:F\cap U_{i}=\Empty\right)\right\} \nonumber \\
& =\bigcap_{\substack{U_{1},\ldots,U_{N}\in\mathcal{U}\\
\text{pairwise disjoint}
}
}\left(\left\{ F\in\mathbf{F}\left(X\right):F\not\subset\bigcup_{i=1}^{N}U_{i}\right\} \cup\bigcup_{i=1}^{N}\left\{ F\in\mathbf{F}\left(X\right):F\cap U_{i}=\Empty\right\} \right).\label{eq:final-form-of-F<N}
\end{align}
Subsets of $\F\left(X\right)$ of the form $\left\{ F\in\mathbf{F}\left(X\right):F\not\subset U\right\} $
and $\left\{ F\in\mathbf{F}\left(X\right):F\cap U=\Empty\right\} $,
where $U\subset X$ is open, are basic measurable sets in the $\sigma$-algebra
on $\F\left(X\right)$. Thus, the expression under the (countable)
intersection in (\ref{eq:final-form-of-F<N}) evaluates to a measurable
set, and we get that $\mathbf{F}_{<N}$ is measurable.
\end{proof}
\section{\label{sec:proof-of-usability-of-Nazarov-Sodin-theorem}Proof of
Theorem \ref{thm:asymptotic-law}}
The Nazarov-Sodin theorem \cite{Nazarov-Sodin-asymptotic-laws} (see
also lecture notes \cite{Sodin-SPB-Lecture-Notes}) gives an asymptotic
law for the expected number of connected components of Gaussian functions
under very general conditions. In this section, we show that Theorem
\ref{thm:asymptotic-law} is a specialization of the Nazarov-Sodin
theorem for our case.
We begin by computing the objects $K_{x,L}\left(u,v\right)$ and $C_{x,L}\left(u\right)$,
as they are defined in \cite[Sections 2.2-2.3]{Sodin-SPB-Lecture-Notes}.
In our case, $K_{L}$ is given by (\ref{eq:KL}). First, the scaled
covariance kernel $K_{x,L}\left(u,v\right)$:
\begin{equation}
K_{x,L}\left(u,v\right)=K_{L}\left(x+\frac{u}{L},x+\frac{v}{L}\right)=K_{L}\left(\frac{u-v}{L}\right)=\frac{1}{\dim\hilbert}\sum_{\lambda\in\sums}\cos\left(2\pi\frac{\lambda}{L}\cdot\left(u-v\right)\right).\label{eq:KxL}
\end{equation}
Note that this expression for $K_{x,L}\left(u,v\right)$ satisfies
the definition for $C^{3,3}$-smoothness of the ensemble (see \cite[Definition 2]{Sodin-SPB-Lecture-Notes},
where it is called ``separate $C^{3}$-smoothness''). To see this,
it is enough to show that the partial derivative $\partial_{u}^{i}\partial_{v}^{j}K_{x,L}\left(u,v\right)$
with $0\le i,j\le3$ remains uniformly bounded. Note that this partial
derivative is given by an expression similar to (\ref{eq:KxL}), where
the function in the sum is either $\pm\cos$ or $\pm\sin$ (depending
on $i+j$), and the addend corresponding to $\lambda$ is multiplied
by $\pm2\pi\lambda_{m}L^{-1}\in\left[-2\pi,2\pi\right]$ every time
a derivative is taken with respect to $u_{m}$ or $v_{m}$; therefore,
$\left|\partial_{u}^{i}\partial_{v}^{j}K_{x,L}\left(u,v\right)\right|\le\left(2\pi\right)^{6}$.
Second, the scaled covariance matrix $C_{x,L}\left(u\right)$. By
the following computation, we have that $C_{x,L}\left(u\right)$ is
simply a constant multiple of the identity matrix:
\begin{align*}
\left(C_{x,L}\left(u\right)\right)_{ij} & =\left.\partial_{u_{i}}\partial_{v_{j}}K_{x,L}\left(u,v\right)\right|_{v=u}=\left.\frac{1}{\dim\hilbert}\sum_{\lambda\in\sums}\left(2\pi\right)^{2}\frac{\lambda_{i}\lambda_{j}}{L^{2}}\cos\left(2\pi\lambda\cdot\frac{u-v}{L}\right)\right|_{v=u}\\
& =\frac{1}{\dim\hilbert}\frac{\left(2\pi\right)^{2}}{L^{2}}\sum_{\lambda\in\sums}\lambda_{i}\lambda_{j}=\begin{cases}
\hfill0\hfill & \text{if }i\ne j\\
\hfill{\displaystyle \frac{4\pi^{2}}{d}}\hfill & \text{if }i=j
\end{cases}
\end{align*}
In the last step we have used an orthogonality relation that can be
seen easily by observing symmetries within the set $\sums$. Thus,
it clearly satisfies the definition for non-degeneracy of the ensemble
(see \cite[Definition 3]{Sodin-SPB-Lecture-Notes}).
Finally, we introduce our target limiting spectral measure for the
process: $\sigma_{d-1}$, the normalized Lebesgue measure on the sphere
$\sphere$, with $\sigma_{d-1}\left(\sphere\right)=1$. The target
translation-invariant local limiting covariance kernel $k$ is thus
the Fourier (cosine) transform of $\sigma_{d-1}$:
\[
k\left(x\right)\coloneqq\Int[\sphere]{\cos\left(2\pi x\cdot\zeta\right)}{\sigma_{d-1}\left(\zeta\right)}.
\]
Under the assumption that $L\to\infty$ through an admissible sequence
of $L$ values (Definition \ref{def:admissible-sequence}), we have
$K_{x,L}\left(u,v\right)\to k\left(x\right)$ pointwise in $x\in\rd$.
Pointwise convergence implies compact convergence in $\rd$ by a standard
application of the Arzelà-Ascoli theorem, and we get translation-invariant
local limits as in \cite[Definition 1]{Sodin-SPB-Lecture-Notes}.
Thus, Theorem \ref{thm:asymptotic-law} follows from \cite[Theorem 4]{Sodin-SPB-Lecture-Notes},
where the positivity of the constant $\nu$ for the spectral measure
$\sigma_{d-1}$ follows from condition $\left(\rho4\right)$ in \cite[Theorem 1]{Sodin-SPB-Lecture-Notes}.
\section{\label{sec:trigonometric-polynomials-and-nodal-sets}Trigonometric
polynomials and their nodal sets}
\subsection{Stability of nodal sets under small perturbations}
We begin by introducing some notation and definitions for the discussion
of the topological stability of a function's nodal set under small
perturbations. Let $f\colon\td\to\R$ be any continuous function.
We define $Z\left(f\right),\Z\left(f\right)$ and $N\left(f\right)$
by:
\begin{align*}
Z\left(f\right) & \coloneqq\left\{ x\in\td:f\left(x\right)=0\right\} =f^{-1}\left(\left\{ 0\right\} \right)\\
\Z\left(f\right) & \coloneqq\left\{ \text{connected components of }Z\left(f\right)\right\} \\
N\left(f\right) & \coloneqq\text{\# connected components of }Z\left(f\right)=\#\Z\left(f\right)
\end{align*}
$Z\left(f\right)$ is called the \emph{nodal set} of $f$, and its
connected components (which comprise $\Z\left(f\right)$) are called
the \emph{nodal components} of $f$. The connected components of $\td\setminus Z\left(f\right)$
are called the \emph{nodal domains} of $f$.
\begin{defn}
\label{def:stable-nodal-set}We say that a $C^{1}$-smooth function
$f\colon\td\to\R$ \emph{has a stable nodal set} if $\nabla f\left(x\right)\ne0$
for all $x\in Z\left(f\right)$.\end{defn}
\begin{rem}
\label{rem:about-nodal-stability}By compactness, the condition that
$f$ has a stable nodal set is equivalent to the existence of $\alpha,\beta>0$
such that for any $x\in\td$, $\left|f\left(x\right)\right|>\alpha$
or $\left|\nabla f\left(x\right)\right|>\beta$, and $\alpha,\beta$
may be chosen under a constraint $\alpha/\beta<\delta$ for any arbitrary
$\delta>0$. Note that if $f$ has a stable nodal set then $Z\left(f\right)$
is a $\left(d-1\right)$-dimensional smooth compact submanifold of
$\td$ having finitely many connected components.
\end{rem}
By the following two propositions, stable nodal sets are indeed stable
under small perturbations. Proposition \ref{prop:perturbation-in-U}
discusses ``local'' stability in an open subset of the torus, and
Proposition \ref{prop:smooth-perturbation} is a ``global'' version
(cf.\ \cite[Corollary 4.3]{Nazarov-Sodin-spherical-harmonics}).
These propositions may be proven in a standard way, by studying the
flow of the vector field $\left|\nabla f\right|^{-2}\nabla f$, and
for completeness, we provide their proofs in Appendix \ref{sec:appendix-additional-proofs}.
\begin{prop}
\label{prop:perturbation-in-U}Let $\alpha,\beta>0$ and let $U$
be an open subset of $\td$. Let $f\colon U\to\R$ be a smooth function
such that $\left|f\left(x\right)\right|>\alpha$ or $\left|\nabla f\left(x\right)\right|>\beta$
for any $x\in U$.
Let $g\colon U\to\R$ be a continuous function such that $\left|g\left(x\right)\right|<\alpha$
for any $x\in U$ (this is the ``small perturbation'').
Then for each connected component $\Gamma$ of $\left\{ x\in U:f\left(x\right)=0\right\} $
that satisfies $\Gamma_{+\alpha/\beta}\subset U$ , there is a connected
component $\tilde{\Gamma}\subset\Gamma_{+\alpha/\beta}$ of $\left\{ x\in U:f\left(x\right)+g\left(x\right)=0\right\} $.
Furthermore, the mapping $\Gamma\mapsto\tilde{\Gamma}$ is injective
(that is, different components $\Gamma$ generate different components
$\tilde{\Gamma}$).
\end{prop}
\begin{prop}
\label{prop:smooth-perturbation}Let $\alpha,\beta>0$ and let $f\colon\td\to\R$
be a smooth function such that $\left|f\left(x\right)\right|>\alpha$
or $\left|\nabla f\left(x\right)\right|>\beta$ for any $x\in\td$.
Let $g\colon\td\to\R$ be a smooth function such that $\left|g\left(x\right)\right|<\alpha/2$
and $\left|\nabla g\left(x\right)\right|<\beta/2$ for any $x\in\td$.
Then there is a bijection $\Z\left(f\right)\to\Z\left(f+g\right)$
mapping each $\Gamma\in\Z\left(f\right)$ to a corresponding $\tilde{\Gamma}\in\Z\left(f+g\right)$,
which satisfies:
\[
\diam\Gamma\le\frac{2\alpha}{\beta}+\diam\tilde{\Gamma}.
\]
\end{prop}
\begin{lyxcode}
\end{lyxcode}
The following proposition allows us to change our focus from the number
of nodal components to the number of nodal domains and vice versa,
by showing that their difference is very small. It is proven in a
standard way using singular homology theory. A proof is presented
in Appendix \ref{sec:appendix-additional-proofs}.
\begin{prop}
\label{prop:nodal-components-similar-to-nodal-domains}Let $f\colon\td\to\R$
be smooth with a stable nodal set.
\begin{enumerate}
\item If $f$ has $k$ nodal components and $r$ nodal domains, then $r-1\le k\le r+d-1$.
\item If $f$ has $k'$ nodal components and $r'$ nodal domains lying completely
inside some open ball of radius less than $\frac{1}{2}$, then $k'\le r'$.
\end{enumerate}
\end{prop}
\subsection{The number and sum of diameters of nodal components of trigonometric
polynomials}
We denote by $\trigs$ the linear space of trigonometric polynomials
on $\td$ of degree at most $D$:
\[
\trigs\coloneqq\Span\left\{ \cos\left(2\pi\lambda\cdot x\right),\sin\left(2\pi\lambda\cdot x\right):\lambda\in\zd,\left\Vert \lambda\right\Vert _{1}\le D\right\} .
\]
\begin{prop}
\label{prop:trig-polynomials-nodal-set-result}If $T\in\trigs$ has
a stable nodal set then $N\left(T\right)\ll D^{d}$ and $\sum_{\Gamma\in\Z\left(T\right)}\diam\Gamma\ll D^{d-1}$.
\end{prop}
The first result, the bound on the number of nodal components of $T$,
is a trigonometric version of a classical bound on the sum of the
Betti numbers of the nodal hypersurface of a polynomial due to Oleinik
and Petrovsky, Milnor, and Thom. We obtain this by result by algebraizing
(that is, writing the trigonometric polynomials as algebraic ones)
and applying elimination theory and Bézout's theorem to count the
number of critical points. This result is obtained along the way of
proving the second result, the bound on the sum of diameters, which
is shown using simple integral-geometric tools: The diameter of $\Gamma\in\Z\left(T\right)$
is comparable to its average width, which may be computed by measuring
the set of hypersurfaces that intersect it (a Crofton-type formula).
In \cite{Nazarov-Sodin-spherical-harmonics}, a different approach
is taken to bound the sum of diameters of nodal components in dimension
$d=2$ - it is bounded by a well-known estimate on the total length
of the nodal set. However, although this estimate may be generalized
to higher dimensions (where length is replaced by hypersurface volume),
when $d>2$ it fails to bound the sum of diameters; a nodal component
might be a long, thin ``noodle'' having large diameter and small
hypersurface volume.
It is likely that the stability condition in Proposition \ref{prop:trig-polynomials-nodal-set-result}
may be lifted, but we assume it as it makes the proof a little simpler,
and for $f_{L}$, this condition is almost surely satisfied (see Proposition
\ref{prop:a.s.-stable-nodal-set} in the next section).
\paragraph*{Algebraic background.}
Given polynomials $P_{1},\ldots,P_{n}\colon\c^{k}\to\c$, denote by
$Z\left(P_{1},\ldots,P_{n}\right)$ their common zero set. Subsets
of $\c^{k}$ of this form are called \emph{algebraic}, and their complements
\emph{coalgebraic}. The family of algebraic sets (thus also coalgebraic
sets) is closed under finite unions and finite intersections, and
any coalgebraic set is either empty or dense in $\c^{k}$ (since a
nonzero polynomial cannot vanish in an open set).
Any homogeneous polynomial of positive degree $P\colon\c^{n+1}\to\c$
has a trivial zero at the origin; other zeroes are called \emph{nontrivial
zeroes}. Note that if $P\left(z_{0,}\ldots,z_{n}\right)=0$, then
$P\left(\lambda z_{0},\ldots,\lambda z_{n}\right)=0$ for any $\lambda\in\c$,
so nontrivial zeroes extend at least to their complex spans, called
\emph{solution rays}.
In what follows, we consider polynomials in which some (possibly all)
of the coefficients are \emph{indeterminate}; that is, they are parameters
which may be assigned complex values, and on which we may impose conditions.
When considering polynomials with indeterminate coefficients, they
have a \emph{formal degree}, that is, the degree of the polynomial
with all the coefficients written explicitly; the actual degree may
become lower if some coefficients become zero after assigning values.
We recall two classical theorems. See, for instance, \cite[Sections 80 and 83]{van-der-Waerden-book-Modern-Algebra-vol-II}.
\begin{itemize}
\item \textbf{The fundamental theorem of elimination theory:} Given a system
of homogeneous polynomials of positive formal degree, the existence
of a nontrivial common zero is an algebraic condition on the coefficients.
\item \textbf{Bézout's theorem:} Given $n$ homogeneous polynomials $P_{1},\ldots,P_{n}\colon\c^{n+1}\to\c$,
if they have finitely many common solution rays, then the number of
common solution rays is at most $\prod_{i=1}^{n}\deg P_{i}$.
\end{itemize}
It is easy see that Bézout's\textbf{ }theorem continues to hold under
the weakened hypothesis that the polynomials have at most countably
many common solution rays; we will later name this result ``Bézout's
theorem'' as well.
\paragraph*{Coalgebraic condition for finiteness.}
We begin by presenting a coalgebraic condition for the finiteness
of the set of solutions to a system of polynomials which we will use
later in a few settings.
First, we define the \emph{homogenization }of a polynomial $P\colon\c^{n}\to\c$
of formal degree $D$ as the homogeneous polynomial $\tilde P\left(z_{0},\ldots,z_{n}\right)\coloneqq z_{0}^{D}P\left(\frac{z_{1}}{z_{0}},\ldots,\frac{z_{n}}{z_{0}}\right)$.
Note that plugging in $z_{0}=1$ yields the original polynomial, so
any zero $\left(\zeta_{1},\ldots,\zeta_{n}\right)$ of $P$ yields
a solution ray of $\tilde P$: $\Span_{\c}\left\{ \left(1,\zeta_{1},\ldots,\zeta_{n}\right)\right\} $.
However, $\tilde P$ may have more solution rays not obtained this
way - solution rays where $z_{0}=0$.
Second, we recall that the \emph{Jacobian} of $n$ polynomials $P_{1},\ldots,P_{n}\colon\c^{n}\to\c$
is the polynomial given by $\det\left(\frac{\partial P_{i}}{\partial z_{j}}\right)_{1\le i,j\le n}$.
In case that the Jacobian is nonzero wherever $P_{1},\ldots,P_{n}$
are zero, we get by the implicit function theorem that there are at
most countably many common zeroes of $P_{1},\ldots,P_{n}$.
\begin{condition}[Condition for finiteness]
\label{cond:finiteness}Polynomials $P_{1},\ldots,P_{n}\colon\c^{n}\to\c$
are said to satisfy the \emph{condition for finiteness} if both of
the following hold:
\begin{enumerate}
\item The homogenizations $\tilde P_{1},\ldots,\tilde P_{n}$ have no common
nontrivial zero with $z_{0}=0$.
\item At any common zero of $P_{1},\ldots,P_{n}$, their Jacobian is nonzero.
\end{enumerate}
\end{condition}
\begin{lem}
\label{lem:cond-finiteness-results}Condition \ref{cond:finiteness}
is a coalgebraic condition on the coefficients of the polynomials
$P_{1},\ldots,P_{n}$, and whenever it holds, the number of common
zeroes of $P_{1},\ldots,P_{n}$ is finite and bounded by $\prod_{i=1}^{n}\deg P_{i}$.\end{lem}
\begin{proof}
Let $P_{n+1}$ be the Jacobian of $P_{1},\ldots,P_{n}$. Replacing
the second part of the condition with ``$\tilde{P_{1}},\ldots,\tilde{P_{n}},\tilde{P_{n+1}}$
have no nontrivial common zero'' yields exactly the same condition,
since by the first part of the condition, there cannot exist such
a solution ray with $z_{0}=0$, and solution rays with $z_{0}\ne0$
are in correspondence with zeroes of $P_{1},\ldots,P_{n},P_{n+1}$.
By the fundamental theorem of elimination theory, this is a coalgebraic
condition on the coefficients of $P_{1},\ldots,P_{n}$ (and also $P_{n+1}$,
but the coefficients of $P_{n+1}$ are themselves polynomials of the
coefficients of $P_{1},\ldots,P_{n}$). Bézout's theorem then gives
the required bound for the number of common solution rays of $\tilde{P_{1}},\ldots,\tilde{P_{n}}$
with $z_{0}\ne0$, which are in correspondence with the zeroes of
$P_{1},\ldots,P_{n}$.
\end{proof}
\paragraph*{Algebraization of trigonometric polynomials.}
Laplacian eigenfunctions on the torus are trigonometric polynomials,
so it would be fruitful to consider trigonometric polynomials in general
in order to analyze them. Recall that trigonometric polynomials are
linear combinations of trigonometric monomials - cosines or sines
of $2\pi\lambda\cdot x$, where $\lambda\in\zd$. The degree of a
trigonometric monomial is $\left\Vert \lambda\right\Vert _{1}=\sum_{i=1}^{d}\left|\lambda_{i}\right|$,
and the degree of a trigonometric polynomial is the maximal degree
among its monomials. The polynomial is said to be homogeneous when
its monomials all have the same degree.
We would like to use algebraic tools, such as Bézout's theorem, to
analyze trigonometric polynomials. Therefore, we \emph{algebraize}
them - convert them to simple algebraic polynomials in a different
space.
In the following definition, we denote by $x_{1},\ldots,x_{d}$ the
coordinates of $\td$ and by $c_{1},s_{1},\ldots,c_{d},s_{d}$ the
coordinates of $\R^{2d}$.
\begin{defn}[algebraization of trigonometric polynomials]
$ $
\begin{enumerate}
\item Given a trigonometric polynomial $T\colon\td\to\R$, its \emph{algebraization}
is the polynomial $P\colon\R^{2d}\to\R$ such that $\deg P=\deg T$
and
\[
T\left(x_{1},\ldots,x_{d}\right)=P\left(\cos\left(2\pi x_{1}\right),\sin\left(2\pi x_{1}\right),\ldots,\cos\left(2\pi x_{d}\right),\sin\left(2\pi x_{d}\right)\right).
\]
\item Let $T_{1},\ldots,T_{r}$ be a system of $r$ trigonometric polynomials
$\td\to\R$. The \emph{algebraization }of this system is a system
of $r+d$ polynomials $\R^{2d}\to\R$, the first $r$ being the algebraizations
of $T_{1},\ldots,T_{r}$ and the last $d$ being the polynomials $c_{i}^{2}+s_{i}^{2}-1$
for $i=1,\ldots,d$.
\end{enumerate}
\end{defn}
\begin{rem*}
Given a trigonometric polynomial $T$, it is easy to see that its
algebraization $P$ exists and is unique. Moreover, the coefficients
of $P$ are linear combinations of the coefficients of $T$, and if
$T$ is homogeneous then $P$ is homogeneous. Regarding systems of
polynomials, it is easy to see that the assignments $c_{i}=\cos\left(2\pi x_{i}\right)$
and $s_{i}=\sin\left(2\pi x_{i}\right)$ for $i=1,\dots,d$ yield
a bijective correspondence between the set of zeroes of a system of
trigonometric polynomials and the set of (real) zeroes of the algebraized
system.
\end{rem*}
\paragraph*{Coalgebraic conditions for regularity of trigonometric polynomials.}
Let $T$ be a real trigonometric polynomial in $d$ variables $x_{1},\ldots,x_{d}$.
We present two regularity conditions, viewed as conditions on the
coefficients of $T$.
\begin{condition}[Condition for nodal regularity]
\label{cond:nodal-regularity}The homogenization of the algebraization
of the system $T,\frac{\partial T}{\partial x_{1}},\ldots,\frac{\partial T}{\partial x_{d}}$
has no common nontrivial complex zeroes.
\end{condition}
Condition \ref{cond:nodal-regularity} is coalgebraic by the fundamental
theorem of elimination theory, and it implies that $T$ has a stable
nodal set (recall Definition \ref{def:stable-nodal-set}).
\begin{condition}[Condition for critical set regularity]
\label{cond:critical-regularity}The algebraization of the system
$\frac{\partial T}{\partial x_{1}},\ldots,\frac{\partial T}{\partial x_{d}}$
satisfies the \emph{condition for finiteness} (Condition \ref{cond:finiteness}).
\end{condition}
By Lemma \ref{lem:cond-finiteness-results}, Condition \ref{cond:critical-regularity}
is coalgebraic and it implies that $\left\{ \nabla T=0\right\} $
is a finite set.
\begin{condition}[Condition for full regularity]
\label{cond:regular}Suppose $d>1$. \emph{$T$} is said to be \emph{fully
regular} if:
\begin{enumerate}
\item $T$ satisfies both the condition for nodal regularity and the condition
for critical set regularity.
\item For all $j=1,\ldots,d$, the restriction of $T$ to the hyperplane
$\left\{ x_{j}=0\right\} $, viewed as a trigonometric polynomial
in the remaining $d-1$ variables, satisfies both the condition for
nodal regularity and the condition for critical set regularity.
\end{enumerate}
\end{condition}
It is clear that Condition \ref{cond:regular} is coalgebraic in the
coefficients of $T$. We will later need our trigonometric polynomial
$T$ to have regular restrictions to almost any hyperplane of the
form $\left\{ x_{j}=t\right\} $, not just $\left\{ x_{j}=0\right\} $.
The following lemma shows that the full regularity condition is enough.
\begin{lem}
\label{lem:improved-regularity}Suppose $T$ is fully regular. Let
$1\le j\le d$. For all but finitely many values of $t\in\t$, the
restriction of $T$ to the hyperplane $\left\{ x_{j}=t\right\} $,
viewed as a trigonometric polynomial in the remaining $d-1$ variables,
satisfies both the condition for nodal regularity and the condition
for critical set regularity.\end{lem}
\begin{proof}
For any $t\in\t$, let $T_{t}$ be the restriction of $T$ to the
hyperplane $H\coloneqq\left\{ x_{j}=t\right\} $.
Put $\kappa\coloneqq\cos\left(2\pi t\right)$ and $\sigma\coloneqq\sin\left(2\pi t\right)$.
In what follows, $\kappa$ and $\sigma$ are treated as indeterminate
parameters, like $t$. The gradient $\nabla T_{t}$ comprises $d-1$
trigonometric polynomials on $\t^{d-1}$ (since the coordinate $x_{j}$
is omitted in the restriction to $H$), so the algebraization of the
system $T_{t},\nabla T_{t}$ is a system of $2d-1$ polynomials in
$2d-2$ variables ($c_{1},s_{1},\ldots,c_{d},s_{d}$ without $c_{j},s_{j}$)
whose coefficients are polynomial expressions in the coefficients
of $T$ (which are \emph{not }indeterminate) and the parameters $\kappa$
and $\sigma$.
Condition \ref{cond:nodal-regularity} and Condition \ref{cond:critical-regularity}
are coalgebraic; therefore, there exists a system of polynomials $P_{1}\left(\kappa,\sigma\right),P_{2}\left(\kappa,\sigma\right),\ldots$
that vanishes whenever $\kappa$ and $\sigma$ are assigned values
for which the conditions are not satisfied. At least one of these
polynomials (w.l.o.g.\ $P_{1}$) is not identically zero, because
both conditions are satisfied for $t=0$ (i.e.\ for $\kappa=1$ and
$\sigma=0$).
Consider the function $Q\left(t\right)\coloneqq P_{1}\left(\cos\left(2\pi t\right),\sin\left(2\pi t\right)\right)$.
Whenever $Q\left(t\right)\ne0$, $t$ is a value for which $T_{t}$
satisfies both conditions. We also know that $Q\left(0\right)\ne0$,
so $Q$ is an analytic function that isn't identically zero. Therefore,
$Q$ has at most finitely many zeroes in $\t$, proving the lemma.
\end{proof}
\paragraph*{Abundance of fully regular trigonometric polynomials.}
Since the condition for full regularity is coalgebraic, it is enough
to show it is nonempty to see that it is, in fact, dense in $\trigs$.
This is the content of the following:
\begin{lem}
There exists a $T\in\trigs$ that is fully regular.\end{lem}
\begin{proof}
We will show that
\[
T\left(x_{1},\dots,x_{d}\right)\coloneqq\sum_{j=1}^{d}\sin\left(2\pi Dx_{j}\right)+A,
\]
where $A>d$ is constant, satisfies the condition for nodal regularity
and the condition for critical set regularity. Since the restriction
of $T$ to a hyperplane of the form $\left\{ x_{j}=0\right\} $ takes
exactly the same form as $T$ with dimension smaller by $1$, this
is enough to imply that $T$ is fully regular.
Denote by $C_{D}\left(c,s\right)$ and $S_{D}\left(c,s\right)$ the
algebraizations of $\cos\left(2\pi Dx\right)$ and $\sin\left(2\pi Dx\right)$.
These are homogeneous polynomials of degree $D$ for which it is a
simple exercise to prove:
\selectlanguage{british}
\begin{enumerate}[labelindent=\parindent,leftmargin=*,label=({\alph*})]
\item \foreignlanguage{american}{\label{enu:C^2+S^2=00003Dc^2+s^2}$\left(C_{D}\left(c,s\right)\right)^{2}+\left(S_{D}\left(c,s\right)\right)^{2}=\left(c^{2}+s^{2}\right)^{D}$.}
\selectlanguage{american}
\item \label{enu:C=00003DS=00003D0->c=00003Ds=00003D0}The only solution
in $c,s\in\c$ of $C_{D}\left(c,s\right)=S_{D}\left(c,s\right)=0$
is $c=s=0$.
\item \label{enu:determinant-property}$\det\left(\begin{array}{cc}
\frac{\partial C_{D}}{\partial c}\left(c,s\right) & \frac{\partial C_{D}}{\partial s}\left(c,s\right)\\
c & s
\end{array}\right)=D\,S_{D}\left(c,s\right)$.
\end{enumerate}
\selectlanguage{american}
The algebraization of $T,\nabla T$ is the following system of $2d+1$
polynomials in $2d$ variables $c_{1},s_{1},\dots,c_{d},s_{d}$:
\begin{equation}
\begin{array}{ccc}
\sum_{j=1}^{d}S_{D}\left(c_{j},s_{j}\right)+A, & \underbrace{2\pi D\,C_{D}\left(c_{j},s_{j}\right)}_{1\le j\le d}, & \underbrace{c_{j}^{2}+s_{j}^{2}-1}_{1\le j\le d}\end{array}\label{eq:nodal-regularity-system}
\end{equation}
The condition for nodal regularity (Condition \ref{cond:nodal-regularity})
requires this system to have no common nontrivial zeroes after homogenization.
We introduce a homogenizing variable $z_{0}$, and split to two cases:
$z_{0}=0$ and $z_{0}\ne0$. Since $S_{D}$ and $C_{D}$ are already
homogeneous of degree $D$, letting $z_{0}=0$, system (\ref{eq:nodal-regularity-system})
becomes:
\begin{equation}
\begin{array}{ccc}
\sum_{j=1}^{d}S_{D}\left(c_{j},s_{j}\right), & \underbrace{2\pi D\,C_{D}\left(c_{j},s_{j}\right)}_{1\le j\le d}, & \underbrace{c_{j}^{2}+s_{j}^{2}}_{1\le j\le d}\end{array}\label{eq:nodal-regularity-system-homogeneous}
\end{equation}
If $c_{1},s_{1},\ldots,c_{d},s_{d}$ is any zero of the system (\ref{eq:nodal-regularity-system-homogeneous}),
then $C_{D}\left(c_{j},s_{j}\right)=0$ and $c_{j}^{2}+s_{j}^{2}=0$
for $j=1,\ldots,d$, so by property \ref{enu:C^2+S^2=00003Dc^2+s^2}
above, $S_{D}\left(c_{j},s_{j}\right)=0$; by property \ref{enu:C=00003DS=00003D0->c=00003Ds=00003D0},
$c_{j}=s_{j}=0$ for $j=1,\ldots,d$, i.e.\ the zero must be trivial.
The other case is $z_{0}\ne0$, and it suffices to search for solutions
with $z_{0}=1$, i.e.\ solutions to the original system (\ref{eq:nodal-regularity-system}).
If $c_{1},s_{1},\ldots,c_{d},s_{d}$ is any zero of the system (\ref{eq:nodal-regularity-system}),
then $C_{D}\left(c_{j},s_{j}\right)=0$ and $c_{j}^{2}+s_{j}^{2}=1$
for $j=1,\ldots,d$. By property \ref{enu:C^2+S^2=00003Dc^2+s^2},
$\left|S_{D}\left(c_{j},s_{j}\right)\right|=1$ for $j=1,\ldots,d$.
But then $\sum_{j=1}^{d}S_{D}\left(c_{j},s_{j}\right)+A$ cannot be
zero, since $A>d$. Therefore there are no solutions with $z_{0}\ne0$.
To check the condition for critical set regularity (Condition \ref{cond:critical-regularity}),
we first write the polynomials in the algebraization of $\nabla T$
(as in system (\ref{eq:nodal-regularity-system}), excluding the first
polynomial) in the following order:
\begin{equation}
\begin{array}{ccccc}
2\pi D\,C_{D}\left(c_{1},s_{1}\right), & c_{1}^{2}+s_{1}^{2}-1, & \ldots, & 2\pi D\,C_{D}\left(c_{d},s_{d}\right), & c_{d}^{2}+s_{d}^{2}-1\end{array}\label{eq:algebraization-of-gradient}
\end{equation}
This system should satisfy the condition for finiteness (Condition
\ref{cond:finiteness}). We have already seen that its homogenization
has no zero with $z_{0}=0$, and it remains to check that it has no
zero common with its Jacobian. This Jacobian may be computed using
property \ref{enu:determinant-property}, and it equals $\left(4\pi D^{2}\right)^{d}\prod_{j=1}^{d}S_{D}\left(c_{j},s_{j}\right)$.
Thus, for any such common zero $\left(c_{1},s_{1},\ldots,c_{d},s_{d}\right)$,
there exists some $j$ such that $S_{D}\left(c_{j},s_{j}\right)=0$,
but also $C_{D}\left(c_{j},s_{j}\right)=0$ and $c_{j}^{2}+s_{j}^{2}=1$;
this is impossible by property \ref{enu:C^2+S^2=00003Dc^2+s^2}.
\end{proof}
\paragraph*{Proof of the main proposition for fully regular $T$.}
We may now prove a weak version of Proposition \ref{prop:trig-polynomials-nodal-set-result},
assuming the full regularity condition; this assumption will later
be lifted.
Background for the proof: Denote by $\Td$ the ``round'' torus -
the $d$-dimensional submanifold of $\mathbb{R}^{2d}$ defined by
the equations $c_{j}^{2}+s_{j}^{2}=1$, $j=1,\ldots,d$. Let $\varphi\colon\td\to\Td$
be the natural diffeomorphism $\varphi\left(x_{1},\ldots,x_{d}\right)\coloneqq\left(\cos\left(2\pi x_{1}\right),\sin\left(2\pi x_{1}\right),\ldots,\cos\left(2\pi x_{d}\right),\sin\left(2\pi x_{d}\right)\right)$.
This diffeomorphism is not an isometry, but it induces a strongly
equivalent metric; that is, there exist constants $\alpha,\beta>0$
such that for any $x,y\in\mathbb{T}^{d}$, $\alpha\dist\left(x,y\right)\le\left|\varphi\left(x\right)-\varphi\left(y\right)\right|\le\beta\dist\left(x,y\right).$
We will need a little background in integral geometry. For any compact,
connected set $K\subset\rk$, denote by $W_{j}\left(K\right)\coloneqq\max_{x,y\in K}\left|x_{j}-y_{j}\right|$
the \emph{width of $K$ along the $j$th axis}, and by $W\left(K\right)\coloneqq\frac{1}{k}\sum_{j=1}^{k}W_{j}\left(K\right)$
the \emph{average width}\footnote{This is a simpler version of the \emph{mean width}, defined similarly
as an integral average of support functions.}. It is an easy exercise to show that $\diam K\ll W\left(K\right)$,
and also that $W_{j}\left(K\right)=\Int[-\infty][\infty]{I\left(K,\left\{ x_{j}=\gamma\right\} \right)}{\gamma}$,
where $I\left(A,B\right)$ is the intersection indicator function
($1$ if $A\cap B\ne\varnothing$, $0$ otherwise). This is all we
need; for more on the subject of integral geometry, see \cite{Klain-Rota-book-Introduction-to-geometric-probability}.
\begin{lem}
\label{lem:fully-regular-trig-polys-admit-results-of-main-proposition}Suppose
$T\in\trigs$ is fully regular. Then $N\left(T\right)\ll D^{d}$ and
$\sum_{\Gamma\in\Z\left(T\right)}\diam\Gamma\ll D^{d-1}$.\end{lem}
\begin{proof}
By Lemma \ref{lem:cond-finiteness-results}, the number of critical
points of $T$ in $\td$ is at most $O\left(D^{d}\right)$. There
is at least one critical point in each nodal domain, since each nodal
domain has a (necessarily nonzero) minimum or a maximum point by compactness.
Therefore there are $O\left(D^{d}\right)$ nodal domains, and by Proposition
\ref{prop:nodal-components-similar-to-nodal-domains}, $O\left(D^{d}\right)$
nodal components, proving the first part.
For the second part, let $\Gamma\in\Z\left(T\right)$. We have:
\[
\diam\Gamma\ll\diam\varphi\left(\Gamma\right)\ll W\left(\varphi\left(\Gamma\right)\right)=\frac{1}{2d}\sum_{j=1}^{2d}\Int[-\infty][\infty]{I\left(\varphi\left(\Gamma\right),\left\{ x_{j}=\gamma\right\} \right)}{\gamma}.
\]
Since intersections are preserved by the bijection $\varphi$, the
integrand $I\left(\varphi\left(\Gamma\right),\left\{ x_{j}=\gamma\right\} \right)$
can be written equivalently as $I\left(\Gamma,H_{j,\gamma}\right)$,
where:
\[
H_{j,\gamma}\coloneqq\varphi^{-1}\left(\left\{ x\in\R^{2d}:x_{j}=\gamma\right\} \right)=\begin{cases}
\left\{ x\in\td:\cos\left(2\pi x_{j}\right)=\gamma\right\} & j\text{ odd}\\
\left\{ x\in\td:\sin\left(2\pi x_{j}\right)=\gamma\right\} & j\text{ even}
\end{cases}
\]
We may restrict the integral's limits to $\left|\gamma\right|<1$
and get: $\diam\Gamma\ll\sum_{j=1}^{2d}\Int[-1][1]{I\left(\Gamma,H_{j,\gamma}\right)}{\gamma}$.
Summing over all $\Gamma\in\Z\left(T\right)$ (a finite sum), we get:
\[
\sum_{\Gamma\in\Z\left(T\right)}\diam\Gamma\ll\sum_{j=1}^{2d}\Int[-1][1]{\left(\text{\# of }\Gamma\in\Z\left(T\right)\text{ which intersect }H_{j,\gamma}\right)}{\gamma}.
\]
The number of nodal components that intersect some hyperplane is clearly
bounded by the number of nodal components of the restriction of $T$
to this hyperplane. $H_{j,\gamma}$ is the union of two hyperplanes
of the form $\left\{ x_{j}=t\right\} $, and by Lemma \ref{lem:improved-regularity}
and the first part of this lemma, for almost all $\gamma$ the number
of nodal components in the restriction is $O\left(D^{d-1}\right)$.
This gives the required bound.
\end{proof}
\paragraph*{Proof of the main proposition without the regularity condition.}
\begin{proof}[Proof of Proposition \ref{prop:trig-polynomials-nodal-set-result}.]
Let $\alpha,\beta>0$ be such that $\left|T\left(x\right)\right|>\alpha$
or $\left|\nabla T\left(x\right)\right|>\beta$ for any $x\in\td$,
under the constraint that $\alpha/\beta<\left(2D\right)^{-1}$ (see
Remark \ref{rem:about-nodal-stability}). Since the condition for
full regularity is dense in $\trigs$, it is possible to choose a
perturbation $P\in\trigs$ with $\max_{\td}\left|P\right|<\alpha/2$
and $\max_{\td}\left|\nabla P\right|<\beta/2$ such that $T+P$ is
fully regular.
Applying Proposition \ref{prop:smooth-perturbation} with the function
$T$ and the perturbation $P$ gives $N\left(T\right)=N\left(T+P\right)$.
By Lemma \ref{lem:fully-regular-trig-polys-admit-results-of-main-proposition},
$N\left(T+P\right)\ll D^{d}$; therefore, $N\left(T\right)\ll D^{d}$.
By Proposition \ref{prop:smooth-perturbation}, each component $\Gamma\in\Z\left(T\right)$
generates a component $\tilde{\Gamma}\in\Z\left(T+P\right)$ that
satisfies:
\[
\diam\Gamma\le\frac{2\alpha}{\beta}+\diam\tilde{\Gamma}\le\frac{1}{D}+\diam\tilde{\Gamma}.
\]
Therefore:
\[
\sum_{\Gamma\in\Z\left(T\right)}\diam\Gamma\le\frac{N\left(T\right)}{D}+\quad\sum_{\mathclap{\Gamma\in\Z\left(T+P\right)}}\quad\diam\Gamma.
\]
We've established $N\left(T\right)\ll D^{d}$, and by Lemma \ref{lem:fully-regular-trig-polys-admit-results-of-main-proposition}
we have $\sum_{\Gamma\in\Z\left(T+P\right)}\diam\Gamma\ll D^{d-1}$;
therefore, $\sum_{\Gamma\in\Z\left(T\right)}\diam\Gamma\ll D^{d-1}$.
\end{proof}
\section{\label{sec:proof-of-main-theorem}Proof of Theorem \ref{thm:exponential-concentration}}
\subsection{Tools used in the proof}
\begin{prop}
\label{prop:nodal-domain-area}Any nodal domain $\Omega$ of any $f\in\hilbert$
satisfies $\vol\left(\Omega\right)\gg L^{-d}$\textup{.}
\end{prop}
Proposition \ref{prop:nodal-domain-area} follows immediately from
the classical Faber-Krahn inequality (note that the first Dirichlet
eigenvalue of any nodal domain $\Omega$ is $4\pi^{2}L^{2}$). See,
for instance, \cite[Chapter IV]{Chavel-book-Eigenvalues-in-Riemannian-geometry}.
\begin{prop}
\label{prop:local-bounds}Suppose $f\colon\rd\to\R$ is smooth and
satisfies $\Delta f+4\pi^{2}L^{2}f=0$. Let $x_{0}\in\rd$ and $r>0$.
Then there is a constant $C=C\left(r,d\right)>0$ such that:
\begin{align}
\left|f\left(x_{0}\right)\right|^{2} & \le CL^{d}\Int[B^{d}\left(x_{0},r/L\right)]{\left|f\left(x\right)\right|^{2}}x\label{eq:bound-function}\\
\left|\nabla f\left(x_{0}\right)\right|^{2} & \le CL^{d+2}\Int[B^{d}\left(x_{0},r/L\right)]{\left|f\left(x\right)\right|^{2}}x\label{eq:bound-gradient}\\
\left|\nabla\nabla f\left(x_{0}\right)\right|^{2} & \le CL^{d+4}\Int[B^{d}\left(x_{0},r/L\right)]{\left|f\left(x\right)\right|^{2}}x\label{eq:bound-hessian}
\end{align}
\end{prop}
Proposition \ref{prop:local-bounds} is a local property of functions
on $\rd$, and as such, it may be applied directly to functions in
$\hilbert$, viewed as functions on $\rd$ periodic extension. These
local bounds are special cases of very general classical local bounds
on solutions of PDEs (see, for instance, \cite[Chapter 8]{Gilbarg-Trudinger-book}),
but for the sake of completeness, we provide a simple and easily readable
proof for our case in Appendix \ref{sec:appendix-additional-proofs}.
\begin{prop}
\label{prop:concentration-of-norm-of-fL}$\p\left\{ \left\Vert f_{L}\right\Vert >2\right\} \le\e^{-c\dim\hilbert}$
for some absolute constant $c>0$.
\end{prop}
Considering $f_{L}$ as a random vector in $\rn$ (where $n=\dim\hilbert$)
that is distributed like $\frac{X}{\sqrt{n}}$, where $X$ has standard
multivariate normal distribution in $\rn$, Proposition \ref{prop:concentration-of-norm-of-fL}
may be formulated equivalently as $\p\left\{ \left|X\right|>2\sqrt{n}\right\} \le\e^{-cn}$.
This is a special case of Bernstein's classical inequalities and it
may be proven by applying Chebyshev's inequality on $\exp\left(\frac{1}{4}\left|X\right|^{2}\right)$.
We omit the details.
Again considering $f_{L}$ as a random vector in $\rn$, the next
proposition is a form of the Gaussian isoperimetric inequality (see
\cite{Sudakov-Tsirelson}, \cite{Borell}).
\begin{prop}
\label{prop:isoperimetric-fL}Let $F\subset\hilbert$ and for any
$\rho>0$, denote $F_{+\rho}\coloneqq\left\{ f\in\hilbert:\dist\left(f,F\right)\le\rho\right\} $.
Suppose that $\mathbb{P}\left(F_{+\rho}\right)\le\frac{3}{4}$. Then
$\mathbb{P}\left(F\right)\le C\e^{-c\rho^{2}\dim\hilbert}$ for some
absolute constants $C,c>0$.
\end{prop}
Next, recall that a smooth function is said to have a \emph{stable
nodal set} if it doesn't have zeroes in common with its gradient (Definition
\ref{def:stable-nodal-set}). The following proposition follows either
from Bulinskaya's lemma \cite[Lemma 11.2.10]{Adler-Taylor-book-Random-fields-and-geometry}
or from \cite[Lemma 2.3]{Oravecz-Rudnick-Wigman-the-Leray-measure-of-nodal-sets}.
\begin{prop}
\label{prop:a.s.-stable-nodal-set}Almost surely, $f_{L}$ has a stable
nodal set.
\end{prop}
\subsection{\label{sub:concentration-implies-concentration}Concentration around
the median implies concentration around the mean and limiting mean}
We begin by showing that in Theorem \ref{thm:exponential-concentration},
the first part implies the second and third parts. Throughout the
remainder of this section, we denote $m_{L}\coloneqq\median\left\{ N_{L}/L^{d}\right\} $.
Let $X_{L}\coloneqq N_{L}/L^{d}-m_{L}$. By the first part of the
theorem, $\mathbb{P}\left\{ \left|X_{L}\right|>\varepsilon\right\} \le C\left(\varepsilon\right)\e^{-c\left(\varepsilon\right)\dim\hilbert}$.
The random variables $X_{L}$ are a.s.\ uniformly bounded, since
$N_{L}/L^{d}$ are a.s.\ uniformly bounded by Courant's nodal domain
theorem (or alternatively by Proposition \ref{prop:trig-polynomials-nodal-set-result}).
By the law of total expectation:
\begin{align*}
\left|\E\left\{ X_{L}\right\} \right| & \le\left|\E\left\{ X_{L}|X_{L}\ge\varepsilon\right\} \right|\cdot\p\left\{ X_{L}\ge\varepsilon\right\} \\
& \phantom{\le\left|\E\left\{ X_{L}|X_{L}\ge\varepsilon\right\} \right|}+\left|\E\left\{ X_{L}|\left|X_{L}\right|<\varepsilon\right\} \right|\cdot\p\left\{ \left|X_{L}\right|<\varepsilon\right\} +\left|\E\left\{ X_{L}|X_{L}\le-\varepsilon\right\} \right|\cdot\p\left\{ X_{L}\le-\varepsilon\right\} .
\end{align*}
The first and third terms are bounded by $C\left(\varepsilon\right)\e^{-c\left(\varepsilon\right)\dim\hilbert}$
multiplied by the a.s.\ uniform bound on $X_{L}$, while the second
term is bounded by $\varepsilon$. We may assume that $\dim\hilbert$
is large enough (see Remark \ref{rem:understanding-the-main-result})
such that the sum of the first and third terms is smaller than $\varepsilon$,
and get $\left|\E\left\{ X_{L}\right\} \right|\le2\varepsilon$. Therefore,
by the triangle inequality:
\[
\p\left\{ \left|\frac{N_{L}}{L^{d}}-\E\left\{ \frac{N_{L}}{L^{d}}\right\} \right|>3\varepsilon\right\} =\p\left\{ \left|X_{L}-\E\left\{ X_{L}\right\} \right|>3\varepsilon\right\} \le\p\left\{ \left|X_{L}\right|>\varepsilon\right\} \le C\left(\varepsilon\right)\e^{-c\left(\varepsilon\right)\dim\hilbert}.
\]
This proves the second part of the theorem. For the third part, let
$L$ be large enough such that $\left|\E\left\{ \frac{N_{L}}{L^{d}}\right\} -\nu\right|\le\varepsilon$.
Then, by the triangle inequality:
\[
\p\left\{ \left|\frac{N_{L}}{L^{d}}-\nu\right|>4\varepsilon\right\} \le\p\left\{ \left|\frac{N_{L}}{L^{d}}-\E\left\{ \frac{N_{L}}{L^{d}}\right\} \right|>3\varepsilon\right\} \le C\left(\varepsilon\right)\e^{-c\left(\varepsilon\right)\dim\hilbert}.
\]
\subsection{The exceptional set of instability}
We now show that the concentration around the median follows from
the existence of a small \emph{exceptional set of instability}, which
we will later construct. This is an exponentially small set $E\subset\hilbert$
such that outside this set, the number of nodal components is stable
under sufficiently small perturbations.
\begin{prop}
\label{prop:exceptional-set-implies-theorem}Suppose that for every
$\epsilon>0$, there exist $\rho=\rho\left(\epsilon\right)>0$ and
$\tau=\tau\left(\varepsilon\right)>0$ such that for every $L$, there
exists an ``exceptional set of instability'' $E=E\left(\epsilon,L\right)\subset\mathcal{H}_{L}$
satisfying two conditions:
\begin{enumerate}
\item ($f_{L}$ has exponentially small probability to be exceptional.)
For some constants $C\left(\varepsilon\right),c>0$,
\begin{equation}
\mathbb{P}\left(E\right)\le\min\left\{ \frac{1}{4},C\left(\epsilon\right)\e^{-c\tau^{2}\dim\hilbert}\right\} .\label{eq:bound-on-P(E)}
\end{equation}
\item ($N$ is lower semi-continuous for non-exceptional functions.) For
any $f\in\hilbert\setminus E$ and $g\in\hilbert$ such that $\left\Vert g\right\Vert \le\rho$,
\begin{equation}
N\left(f+g\right)\ge N\left(f\right)-\epsilon L^{d}.\label{eq:N-is-lsc-outside-E}
\end{equation}
\end{enumerate}
Then the first part of Theorem \ref{thm:exponential-concentration}
holds with constant $c\left(\varepsilon\right)$ proportional to $\min\left\{ \rho^{2},\tau^{2}\right\} $.\end{prop}
\begin{proof}
Notice that $\left\{ f\in\hilbert:\left|N\left(f\right)/L^{d}-m_{L}\right|>\epsilon\right\} =F\cup G$,
where:
\begin{align*}
F & =\left\{ f\in\mathcal{H}_{L}:N\left(f\right)>\left(m_{L}+\epsilon\right)L^{d}\right\} \\
G & =\left\{ f\in\mathcal{H}_{L}:N\left(f\right)<\left(m_{L}-\epsilon\right)L^{d}\right\}
\end{align*}
First, we bound $\p\left(F\right)$. Let $h\in\left(F\setminus E\right)_{+\rho}$;
that is, $h=f+g$ where $f\in F\setminus E$ and $g\in\hilbert$ satisfies
$\left\Vert g\right\Vert \le\rho$. Then:
\[
N\left(h\right)=N\left(f+g\right)\stackrel{\left(f\notin E,\left\Vert g\right\Vert \le\rho\right)}{\ge}N\left(f\right)-\varepsilon L^{d}\stackrel{\left(f\in F\right)}{>}m_{L}L^{d}.
\]
Therefore $\left(F\setminus E\right)_{+\rho}\subset\left\{ h\in\mathcal{H}_{L}:N\left(h\right)/L^{d}>m_{L}\right\} $,
the latter set having probability at most $\frac{1}{2}$.
Proposition \ref{prop:isoperimetric-fL} gives $\mathbb{P}\left(F\setminus E\right)\le C\e^{-c\rho^{2}\dim\hilbert}$.
Together with $\p\left(E\right)\le C\left(\varepsilon\right)\e^{-c\tau^{2}\dim\hilbert}$,
we have $\mathbb{P}\left(F\right)\le C\left(\epsilon\right)\e^{-c\left(\epsilon\right)\dim\hilbert}$
with $c\left(\varepsilon\right)$ proportional to $\min\left\{ \rho^{2},\tau^{2}\right\} $.
Next, we bound $\p\left(G\right)$. Let $h\in G_{+\rho}\setminus E$;
that is, $h=f+g$ where $f\in G$, $g\in\hilbert$ satisfies $\left\Vert g\right\Vert \le\rho$,
and $h\notin E$. Then:
\[
\left(m_{L}-\varepsilon\right)L^{d}\stackrel{\left(f\in G\right)}{>}N\left(f\right)=N\left(h-g\right)\stackrel{\left(h\notin E,\left\Vert g\right\Vert \le\rho\right)}{\ge}N\left(h\right)-\varepsilon L^{d}.
\]
Therefore $G_{+\rho}\setminus E\subset\left\{ h\in\mathcal{H}_{L}:N\left(h\right)/L^{d}<m_{L}\right\} $,
the latter set having probability at most $\frac{1}{2}$. Thus $\mathbb{P}\left(G_{+\rho}\right)\le\frac{3}{4}$,
and applying Proposition \ref{prop:isoperimetric-fL} again gives
$\mathbb{P}\left(G\right)\le C\e^{-c\rho^{2}\dim\hilbert}$.
\end{proof}
\subsection{Construction of the exceptional set \texorpdfstring{$E$}{E}}
We now present the construction of the set $E=E\left(\epsilon,L\right)\subset\hilbert$,
so throughout this part of the paper, $\epsilon$ and $L$ are fixed.
We introduce new small parameters $0<\alpha,\beta,\delta<1$ and one
large parameter $R>2$ that all depend only on $\epsilon$ in a way
that will be determined later.
Cover the torus $\mathbb{T}^{d}$ by as few as possible closed balls
$\left\{ B_{j}\right\} $ of radius $RL^{-1}$ (their amount is $O\left(L^{d}R^{-d}\right)$).
We will later refer to $2B_{j}$, $3B_{j}$ and $4B_{j}$; these are
balls with the same center as $B_{j}$ and radius multiplied by $2$,
$3$ and $4$ respectively. We require the cover to satisfy a bounded
multiplicity condition: For any point $x\in\td$, the amount of balls
$4B_{j}$ that cover $x$ is $O\left(1\right)$.
For any $f\in\mathcal{H}_{L}$, we say that $3B_{j}$ is \emph{an
unstable ball with respect to $f$} if there exists a point $x\in3B_{j}$
such that $\left|f\left(x\right)\right|\le\alpha$ and $\left|\nabla f\left(x\right)\right|\le\beta L$.
Finally, define $E\subset\hilbert$ as the set of functions for which
the number of unstable balls exceeds $\delta L^{d}$. It is easy to
verify that $E$ is measurable.
\subsection{Proof that \texorpdfstring{$E$}{E} has exponentially small probability}
We will now prove (\ref{eq:bound-on-P(E)}) under certain assumptions
that will arise from the proof. We introduce two new small parameters
$0<\gamma,\tau<1$ that depend only on $\varepsilon$ in a way that
will be determined later.
It suffices to prove (\ref{eq:bound-on-P(E)}) for $\p\left(\tilde E\right)$
where $\tilde E\coloneqq E\cap\left\{ f\in\hilbert:\left\Vert f\right\Vert \le2\right\} $
instead of $E$. The leftover set $E\setminus\tilde E$ may be discarded
by Proposition \ref{prop:concentration-of-norm-of-fL} as it has comparatively
negligible probability.
Thus, let $f\in E$ with $\left\Vert f\right\Vert \le2$. In each
ball $3B_{j}$ that is unstable with respect to $f$, fix a point
$x_{j}\in3B_{j}$ such that $\left|f\left(x_{j}\right)\right|\le\alpha$
and $\left|\nabla f\left(x_{j}\right)\right|\le\beta L$.
Using the local bound from Proposition \ref{prop:local-bounds} with
arbitrary $x_{0}\in B\left(x_{j},\gamma L^{-1}\right)$ and $r=1$,
we have:
\begin{equation}
\sup_{x_{0}\in B\left(x_{j},\gamma L^{-1}\right)}\left|\nabla\nabla f\left(x_{0}\right)\right|^{2}\ll\sup_{x_{0}\in B\left(x_{j},\gamma L^{-1}\right)}L^{d+4}\Int[B\left(x_{0},L^{-1}\right)]{\left|f\left(x\right)\right|^{2}}x\ll L^{d+4}\Int[B\left(x_{j},2L^{-1}\right)]{\left|f\left(x\right)\right|^{2}}x.\label{eq:E-is-small-1}
\end{equation}
Since $R>2$, we have $B\left(x_{j},2L^{-1}\right)\subset4B_{j}$,
and the amount of balls $B\left(x_{j},2L^{-1}\right)$ that any point
$x\in\td$ belongs to is $O\left(1\right)$. Therefore, summing over
all balls:
\begin{equation}
\sum_{j}\Int[B\left(x_{j},2L^{-1}\right)]{\left|f\left(x\right)\right|^{2}}x\ll\left\Vert f\right\Vert ^{2}\ll1.\label{eq:E-is-small-2}
\end{equation}
Plugging (\ref{eq:E-is-small-2}) into (\ref{eq:E-is-small-1}) yields
$\sum_{j}\sup_{B\left(x_{j},\gamma L^{-1}\right)}\left|\nabla\nabla f\right|^{2}\ll L^{d+4}$.
There are at least $\delta L^{d}$ summands, so the average summand
is $O\left(L^{4}\delta^{-1}\right)$. Multiplying the hidden constant
by $4$, at least a proportion $\frac{3}{4}$ of all indexes $j$
satisfy:
\begin{equation}
\sup_{B\left(x_{j},\gamma L^{-1}\right)}\left|\nabla\nabla f\right|\ll L^{2}\delta^{-1/2}.\label{eq:bound-on-sup-grad-grad-f}
\end{equation}
Now, introduce a perturbation $g\in\hilbert$ with $\left\Vert g\right\Vert \le\tau$.
By the same argument as above, using Proposition \ref{prop:local-bounds}
and summing over $j$, we have $\sum_{j}\sup_{B\left(x_{j},\gamma L^{-1}\right)}\left|g\right|^{2}\ll L^{d}\tau^{2}$,
and at least a proportion $\frac{3}{4}$ of all indexes $j$ satisfy:
\begin{equation}
\sup_{B\left(x_{j},\gamma L^{-1}\right)}\left|g\right|\ll\tau\delta^{-1/2}.\label{eq:bound-on-sup-g}
\end{equation}
Using this argument for the third time, now with $\nabla g$, we have
$\sum_{j}\sup_{B\left(x_{j},\gamma L^{-1}\right)}\left|\nabla g\right|^{2}\ll L^{d+2}\tau^{2}$,
so at least a proportion $\frac{3}{4}$ of all indexes $j$ satisfy:
\begin{equation}
\sup_{B\left(x_{j},\gamma L^{-1}\right)}\left|\nabla g\right|\ll\tau L\delta^{-1/2}.\label{eq:bound-on-sup-grad-g}
\end{equation}
Thus, at least a proportion $\frac{1}{4}$ of all indexes $j$, i.e.\ at
least $\frac{1}{4}\delta L^{d}$ indexes, satisfy all three (\ref{eq:bound-on-sup-grad-grad-f}),
(\ref{eq:bound-on-sup-g}) and (\ref{eq:bound-on-sup-grad-g}). For
such indexes $j$, applying Taylor's formula on $f$ and on $\nabla f$,
we get:
\begin{align}
\sup_{B\left(x_{j},\gamma L^{-1}\right)}\left|f\right| & \le\alpha+\beta\gamma+C\gamma^{2}\delta^{-1/2}\label{eq:bound-on-sup-f}\\
\sup_{B\left(x_{j},\gamma L^{-1}\right)}\left|\nabla f\right| & \le\left(\beta+C\gamma\delta^{-1/2}\right)L\label{eq:bound-on-sup-grad-f}
\end{align}
Here, $C$ is the hidden constant from (\ref{eq:bound-on-sup-grad-grad-f}).
By summing (\ref{eq:bound-on-sup-g}) + (\ref{eq:bound-on-sup-f})
and (\ref{eq:bound-on-sup-grad-g}) + (\ref{eq:bound-on-sup-grad-f}),
we get:
\begin{align*}
\sup_{B\left(x_{j},\gamma L^{-1}\right)}\left|f+g\right| & \le\underbrace{\alpha+\beta\gamma+C\delta^{-1/2}\left(\gamma^{2}+\tau\right)}_{A}\\
\sup_{B\left(x_{j},\gamma L^{-1}\right)}\left|\nabla f+\nabla g\right| & \le\underbrace{\left(\beta+C\delta^{-1/2}\left(\gamma+\tau\right)\right)}_{B}L
\end{align*}
Therefore, $f+g\in U$, where:
\[
U\coloneqq\left\{ h\in\mathcal{H}_{L}:\vol\left\{ x\in\td:\left|h\left(x\right)\right|\le A,\left|\nabla h\left(x\right)\right|\le BL\right\} \ge c\delta\gamma^{d}\right\} .
\]
For every fixed $x\in\td$, we have $\p\left\{ \left|f_{L}\left(x\right)\right|\le A,\left|\nabla f_{L}\left(x\right)\right|\le BL\right\} \ll AB^{d}$
due to the independence of the random variable $f_{L}\left(x\right)$
and the random vector $\nabla f_{L}\left(x\right)$ and the fact that
they have bounded densities in $\R$ and $\rd$, respectively. By
Fubini's theorem, we also have:
\[
\E\left\{ \vol\left\{ x\in\td:\left|h\left(x\right)\right|\le A,\left|\nabla h\left(x\right)\right|\le BL\right\} \right\} \ll AB^{d}.
\]
Thus, if we assume $AB^{d}\le C\delta\gamma^{d}$ for an appropriate
constant $C$, we get by Chebyshev's inequality that $\p\left(U\right)\le\frac{1}{2}$
(or any other constant smaller than $\frac{1}{2}$; since $\tilde E\subset U$,
we may decrease this constant to ensure $\p\left(E\right)\le\frac{1}{4}$).
Since furthermore $\p\left(\tilde E_{+\tau}\right)\le\frac{1}{2}$,
by Proposition \ref{prop:isoperimetric-fL} we get $\mathbb{P}\left(\widetilde{E}\right)\le C\e^{-c\tau^{2}\dim\hilbert}$.
To conclude, we have showed (\ref{eq:bound-on-P(E)}) under the following
assumption:
\begin{assumption}
$\left(\alpha+\beta\gamma+C\delta^{-1/2}\left(\gamma^{2}+\tau\right)\right)\left(\beta+C\delta^{-1/2}\left(\gamma+\tau\right)\right)^{d}\ll\delta\gamma^{d}$.
\end{assumption}
\subsection{Proof that \texorpdfstring{$N$}{N} is lower semi-continuous outside
of \texorpdfstring{$E$}{E}}
Next, we prove (\ref{eq:N-is-lsc-outside-E}) for $f\in\hilbert\setminus E$
and $g\in\hilbert$ with $\left\Vert g\right\Vert \le\rho$, where
$0<\rho<1$ is a new parameter depending only on $\varepsilon$ in
a way that will be determined later. Again, this proof will require
a few assumptions.
We may assume $f$ has a stable nodal set (this is a.s.\ true). For
each nodal component $\Gamma\in\Z\left(f\right)$, we pick one ball
$B_{j}$ that intersects it, and call it \emph{the intersecting ball
of $\Gamma$}. Now, assume three conditions: \emph{(a)} $\diam\Gamma\le RL^{-1}$;
\emph{(b)} $3B_{j}$ is a stable ball for $f$; and \emph{(c)} $\sup_{3B_{j}}\left|g\right|<\alpha$.
A component $\Gamma\in\Z\left(f\right)$ that satisfies these conditions
(with $B_{j}$ being its intersecting ball) is said to be a \emph{controllable
component}.
Since the radius of $B_{j}$ if $RL^{-1}$, we have $\Gamma\subset2B_{j}$,
thus $\Gamma$ is $RL^{-1}$-separated from the boundary of $3B_{j}$;
under an additional assumption that $R\ge\alpha/\beta$, we have $\Gamma_{+\alpha/\left(\beta L\right)}\subset3B_{j}$.
Since $3B_{j}$ is a stable ball for $f$, we have $\left|f\left(x\right)\right|>\alpha$
or $\left|\nabla f\left(x\right)\right|>\beta L$ for any $x\in3B_{j}$,
and also $\left|g\left(x\right)\right|<\alpha$. By Proposition \ref{prop:perturbation-in-U},
$\Gamma$ generates a component $\tilde{\Gamma}\in\Z\left(f+g\right)$
that is also contained in $3B_{j}$, and the mapping $\Gamma\mapsto\tilde{\Gamma}$
is injective for all $\Gamma$ satisfying condition (a) above with
intersecting ball $B_{j}$. Thus, all controllable components $\Gamma$
generate different components $\tilde{\Gamma}$ under perturbation
by $g$, and their number does not decrease.
Thus, to prove (\ref{eq:N-is-lsc-outside-E}) it remains to show that
the number of components that are not controllable can be bounded
by $\epsilon L^{d}$.
First, by Proposition \ref{prop:trig-polynomials-nodal-set-result},
the number of components $\Gamma$ with $\diam\Gamma>RL^{-1}$ is
a.s.\ $O\left(L^{d}R^{-1}\right)$, and we get the required bound
with an additional assumption: $R^{-1}\ll\epsilon$.
Second, suppose $\diam\Gamma\le RL^{-1}$ (thus $\Gamma\subset2B_{j}$),
but $3B_{j}$ is an unstable ball for $f$. The number of components
$\Gamma$ that can fit into $2B_{j}$ is at most $O\left(R^{d}\right)$:
If the radius of $2B_{j}$ is greater than half, then this is trivial
(as there are no more than $O\left(L^{d}\right)$ components in general);
otherwise, by Proposition \ref{prop:nodal-components-similar-to-nodal-domains}
it suffices to bound the number of nodal domains, which has the required
bound by Proposition \ref{prop:nodal-domain-area}. Since $f\notin E$,
there are at most $\delta L^{d}$ unstable balls, so there are $O\left(\delta L^{d}R^{d}\right)$
components of this type, which has the required bound with an additional
assumption: $\delta R^{d}\ll\epsilon$.
Finally, suppose $k$ of the balls $3B_{j}$ satisfy $\sup_{3B_{j}}\left|g\right|\ge\alpha$.
For each such ball, fix $x_{j}\in3B_{j}$ for which $\left|g\left(x_{j}\right)\right|\ge\alpha$.
By Proposition \ref{prop:local-bounds}, we have:
\[
\alpha^{2}\le\left|g\left(x_{j}\right)\right|^{2}\ll L^{d}\Int[B\left(x_{j},2L^{-1}\right)]{\left|g\left(x\right)\right|^{2}}x.
\]
Since a constant proportion of the balls $B\left(x_{j},2L^{-1}\right)$
may be dropped leaving a disjoint collection of balls, we get by summing
over the remaining balls:
\[
k\ll L^{d}\alpha^{-2}\rho^{2}.
\]
We have seen that at most $O\left(R^{d}\right)$ components have $B_{j}$
as their intersecting ball, so the number of components for which
the intersecting ball satisfies $\sup_{3B_{j}}\left|g\right|\ge\alpha$
is at most $O\left(L^{d}R^{d}\alpha^{-2}\rho^{2}\right)$, which has
the required bound with an additional assumption: $R^{d}\alpha^{-2}\rho^{2}\ll\epsilon$.
To summarize, (\ref{eq:N-is-lsc-outside-E}) is proven assuming the
following assumptions:
\begin{assumption}
$R\ge\alpha/\beta$.
\end{assumption}
\begin{assumption}
$R^{-1}\ll\epsilon$.
\end{assumption}
\begin{assumption}
$\delta R^{d}\ll\epsilon$.
\end{assumption}
\begin{assumption}
$R^{d}\alpha^{-2}\rho^{2}\ll\epsilon$.
\end{assumption}
\subsection{Choice of parameters}
It remains to choose values for the small parameters $\alpha,\beta,\delta,\gamma,\tau,\rho$
and large parameter $R$ that were used in the previous three subsections;
firstly, to satisfy the five assumptions that arose from the proofs,
and secondly, to maximize the value $c\left(\varepsilon\right)\simeq\min\left\{ \rho^{2},\tau^{2}\right\} $
that appears in Theorem \ref{thm:exponential-concentration}.
The conditions to satisfy are asymptotic inequalities, so we express
each parameter asymptotically as a power of $\varepsilon$: $\alpha\simeq\varepsilon^{a}$,
$\beta\simeq\epsilon^{b}$, $\delta\simeq\epsilon^{2k}$, $\gamma\simeq\epsilon^{g}$,
$\tau\simeq\epsilon^{t}$, $\rho\simeq\epsilon^{h}$ and $R\simeq\epsilon^{-r}$.
Each of $a,b,k,g,t,h,r$ must be a positive real number, and the five
asymptotic inequalities above can be expressed as the following constraints
on the exponents:
\begin{align}
2k+dg & \le\min\left\{ a,b+g,2g-k,t-k\right\} +d\cdot\min\left\{ b,g-k,t-k\right\} \label{eq:ineq1}\\
b & \le a+r\label{eq:ineq2}\\
r & \ge1\label{eq:ineq3}\\
2k & \ge1+rd\label{eq:ineq4}\\
2h & \ge1+2a+rd\label{eq:ineq5}
\end{align}
That is, we want to find positive values that satisfy the above, and
\emph{minimize} $\max\left\{ h,t\right\} $.
First, note that $h$ is minimized simply by changing (\ref{eq:ineq5}),
the only inequality it appears in, to an equation, and we get $2h=1+2a+rd$.
Next, observe that in (\ref{eq:ineq1}), any choice of two minimal
values on the right hand side creates a simple inequality. There are
12 such inequalities, and the inequality (\ref{eq:ineq1}) is equivalent
to all 12 occurring simultaneously. Of those 12 inequalities, 6 are
constraints on $t$ (in the other 6, $t$ does not appear). In the
6 constraints on $t$, we find that increasing $b$, decreasing $k$
or decreasing $g$ either weakens or doesn't change the constraint
on minimizing $t$. Therefore, $b$ must be maximized and $k$ and
$g$ must be minimized. To maximize $b$, we may change the inequality
(\ref{eq:ineq2}) into an equation $b=a+r$, and to maximize $k$,
(\ref{eq:ineq4}) gives $2k=1+rd$.
Since $b+g=a+r+g>a$, we may drop the term $b+g$ from the first minimum
in the right hand side of (\ref{eq:ineq1}). This leaves only one
inequality that bounds $g$ from below - the one obtained by choosing
$2g-k$ as the first minimum and $g-k$ as the second. From this,
we get $2g=\left(3+d\right)k$.
Finally, decreasing $r$ decreases $h$ and does not affect any constraint
on $t$, so $r$ must be minimized. The only remaining constraint
on $r$ is (\ref{eq:ineq4}), so we get $r=1$.
Collecting and simplifying all of the above equations, we have:
\begin{align*}
b & =a+1\\
2k & =d+1\\
4g & =\left(d+1\right)\left(d+3\right)\\
2h & =d+1+2a\\
r & =1
\end{align*}
Our target is to minimize $\max\left\{ h,t\right\} $. Decreasing
$a$ decreases $h$, but tightens the constraints in (\ref{eq:ineq1})
on minimizing $t$. Therefore, we set $t=h$ and find minimal value
of $a$ by the constraint (\ref{eq:ineq1}), which may now be written
as:
\[
\left(d+1\right)\left(d^{2}+3d+4\right)\le\min\left\{ 4a,2\left(d+1\right)\left(d+2\right)\right\} +d\cdot\min\left\{ 4a,\left(d+1\right)^{2}\right\} .
\]
The minimal solution to this inequality is:
\[
2a=\left(d+1\right)\left(d+2\right),
\]
and we can see that all constraints are satisfied, with $2h=2t=\left(d+2\right)^{2}-1$;
so, $c\left(\epsilon\right)\simeq\epsilon^{\left(d+2\right)^{2}-1}$.
\appendix
\section{\label{sec:appendix-equidistribution}Equidistribution of lattice
points on spheres}
For completeness, we present the known results on equidistribution
of lattice points on spheres (as in Definition \ref{def:admissible-sequence})
in various dimensions $d$.
In dimension $d\ge5$, it was shown in \cite{Pommerenke} that we
have equidistribution unconditionally, and any sequence $L\to\infty$
(with $L^{2}\in\z$) is admissible.
When $d=4$, any natural number is a sum of four squares, but there
are arbitrarily large values of $L$ with few representations, and
$\dim\hilbert$ may remain bounded as $L\to\infty$. Requiring $\dim\hilbert\to\infty$
(for instance, by bounding the multiplicity of the prime $2$ in $L^{2}$)
yields equidistribution, again by \cite{Pommerenke} (see also \cite{Malyshev-four-dimensional-equidistribution}).
In dimension $d=3$, a congruence condition $L^{2}\not\equiv0,4,7\pmod8$
ensures $\dim\hilbert\to\infty$ (thus, bounding the multiplicity
of the prime $2$ in $L^{2}$ also ensures this). The question of
whether $\dim\hilbert\to\infty$ implies equidistribution is very
difficult, and was answered affirmatively in \cite{Golubeva-Fomenko}
and \cite{Duke} following a breakthrough by Iwaniec \cite{Iwaniec-Fourier-coefficients-of-modular-forms-of-half-integral-weight}.
See also \cite{Duke-Introduction}.
The equidistribution question in dimension $d=2$ is trickier than
higher dimensions. Any condition that simply ensures $\dim\hilbert\to\infty$
must strongly depend on the prime decomposition of $L^{2}$, as can
be concluded from Gauss's classical formula for the number of representations
of integers as sums of two squares. Furthermore, it turns out that
the condition $\dim\hilbert\to\infty$ is not strong enough to ensure
equidistribution, and the limit measure may even be a sum of 4 atoms,
as shown in \cite{Cilleruelo-1993}. On the positive side, equidistribution
can be proven for a subsequence of relative density $1$ in the sequence
of sums of two squares, as shown in \cite{Katai-Kornyei} and \cite{Erdos-Hall}
(see also \cite{Fainsilber-Kurlberg-Wennberg}).
In case that $d=2$ and the integer points on the circle accumulate
according to a non-uniform limiting measure, if it has no atoms, then
the result of Theorem \ref{thm:asymptotic-law} still holds. However,
the value of $\nu$ depends on the limiting measure. This situation
is further investigated in \cite{Kurlberg-Wigman-Non-universality};
see also \cite{Buckley-Wigman-On-the-number-of-nodal-domains-of-toral-eigenfunctions}.
Regarding the number of lattice points, when $d\ge4$ we have $\dim\hilbert\gg L^{d-2}$
(under the correct assumptions when $d=4$) by the classical Hardy-Littlewood
circle method - see, for instance, \cite[Chapter 12]{Grosswald-book-Representations-of-integers-as-sums-of-squares}.
When $d=3$, we have (under the correct assumptions) $\dim\hilbert\gg c\left(\delta\right)L^{1-\delta}$
for any fixed $0<\delta<1$, due to Siegel - see \cite[Chapter 21]{Davenport-book-Multiplicative-number-theory}.
Finally, when $d=2$, we may find equidistributed subsequences of
relative density $1$ that satisfy $\dim\hilbert\gg\left(\log L\right)^{\gamma}$
for any fixed $0<\gamma<\frac{1}{2}\log\frac{\pi}{2}\approx0.226$,
by \cite{Erdos-Hall}.
\section{\label{sec:appendix-additional-proofs}Additional proofs}
\subsection{Stability of nodal sets - the ``shell lemma'' and Propositions
\ref{prop:perturbation-in-U} and \ref{prop:smooth-perturbation}}
Let $\alpha,\beta>0$ and let $U$ be an open subset of $\td$. Let
$f\colon U\to\R$ be a smooth function such that $\left|f\left(x\right)\right|>\alpha$
or $\left|\nabla f\left(x\right)\right|>\beta$ for any $x\in U$.
The ``shell lemma'', given below (cf.\ \cite[Claim 4.2]{Nazarov-Sodin-spherical-harmonics}),
shows that each connected component of $\left\{ x\in U:f\left(x\right)=0\right\} $
which is not too close to $\partial U$ is contained in a ``shell'',
which is a connected component of $\left\{ x\in U:\left|f\left(x\right)\right|<\alpha\right\} $,
and the shells satisfy certain properties. Proposition \ref{prop:perturbation-in-U}
follows immediately from this lemma, and the proof of Proposition
\ref{prop:smooth-perturbation}, given below, also follows from it.
Before presenting the lemma and its proof, we construct a vector field
whose integral curves are used in the proof. See, for instance, \cite[Chapter 9]{Lee-book-Introduction-to-Smooth-Manifolds-second-ed}
for the necessary background in the theory of integral curves and
flows on smooth manifolds.
Let $M\coloneqq\left\{ x\in U:\left|\nabla f\left(x\right)\right|>\beta\right\} $.
On the open submanifold $M$, define the following vector field:
\[
V\coloneqq\frac{\nabla f}{\left|\nabla f\right|^{2}}.
\]
For any $p\in M$, let $\theta^{\left(p\right)}\colon\D^{\left(p\right)}\to M$
be the integral curve starting at $p$ with respect to the vector
field $V$ (where $\D^{\left(p\right)}$ is an open interval containing
zero, the curve's maximal domain). It is easy to see that this integral
curve has the following three properties (see Figure \ref{fig:integral-curve}):
\begin{enumerate}
\item $f\left(\theta^{\left(p\right)}\left(t\right)\right)=f\left(p\right)+t$
for any $t\in\D^{\left(p\right)}$ (because the left side has constant
derivative $1$).
\item $\dist\left(p,\theta^{\left(p\right)}\left(t\right)\right)\le t/\beta$
(because $\left|V\right|\le\beta^{-1}$).
\item If $p$ is such that $f\left(p\right)=0$ and $\overline{B}\left(p,\alpha/\beta\right)\subset U$,
then $\left[-\alpha,\alpha\right]\subset\D^{\left(p\right)}$.
\end{enumerate}
\noindent
\begin{figure}[h]
\centering{}\input{Drawings/integral_curve.tex}\caption{\label{fig:integral-curve}Starting at a point $p\in M$, the parameterization
of the integral curve $\theta^{\left(p\right)}$ corresponds to the
change in value of $f$. The distance between $p$ and $q=\theta^{\left(p\right)}\left(t\right)$
is at most $t/\beta$.}
\end{figure}
Following the above construction, we now formulate and prove the shell
lemma. The shells described by the lemma are illustrated in Figure
\ref{fig:shell-lemma}.
\begin{lem}[the shell lemma]
\label{lem:shell}$ $
\selectlanguage{british}
\begin{enumerate}[labelindent=\parindent,leftmargin=2em,label=({\roman*})]
\item \foreignlanguage{american}{\label{enu:shell-prop-1}Each connected
component $\Gamma$ of $\left\{ x\in U:f\left(x\right)=0\right\} $
that satisfies $\Gamma_{+\alpha/\beta}\subset U$ is contained in
an open, connected ``shell'' $S_{\Gamma}\subset\left\{ x\in U:\left|f\left(x\right)\right|<\alpha\right\} $
whose boundary consists of two components, with $f=\alpha$ on one
and $f=-\alpha$ on the other.}
\selectlanguage{american}
\item \label{enu:shell-prop-2}$S_{\Gamma}\subset\Gamma_{+\alpha/\beta}$,
and for any point $p\in\Gamma$, the ball $\overline{B}\left(p,\alpha/\beta\right)$
contains a path through $p$ from one boundary component of $S_{\Gamma}$
to the other.
\item \label{enu:shell-prop-3}Given two such components $\Gamma_{1}\ne\Gamma_{2}$,
the shells $S_{\Gamma_{1}},S_{\Gamma_{2}}$ are disjoint.
\item \label{enu:shell-prop-4}$S_{\Gamma}$ may be decomposed as $\Gamma\cup S_{\Gamma}^{+}\cup S_{\Gamma}^{-}$,
where $S_{\Gamma}^{+}=\left\{ x\in S_{\Gamma}:f\left(x\right)>0\right\} $,
$S_{\Gamma}^{-}=\left\{ x\in S_{\Gamma}:f\left(x\right)<0\right\} $.
In this decomposition, $S_{\Gamma}^{+}$ and $S_{\Gamma}^{-}$ are
connected open sets.
\item \label{enu:shell-prop-5}In case $U=\td$, the shells $\left\{ S_{\Gamma}\right\} _{\Gamma\in\Z\left(f\right)}$
are precisely the connected components of $\left\{ x\in\td:\left|f\left(x\right)\right|<\alpha\right\} $.
\end{enumerate}
\end{lem}
\noindent
\begin{figure}[h]
\begin{centering}
\input{Drawings/shell_lemma.tex}
\par\end{centering}
\centering{}\caption{\label{fig:shell-lemma}Illustration of two connected components of
$\left\{ x\in U:f\left(x\right)=0\right\} $ and their corresponding
shells.}
\end{figure}
\begin{proof}
Let $\Gamma$ be a connected component of $\left\{ x\in U:f\left(x\right)=0\right\} $
that satisfies $\Gamma_{+\alpha/\beta}\subset U$. Define:
\[
S_{\Gamma}\coloneqq\left\{ \theta^{\left(x_{0}\right)}\left(t\right):\left|t\right|<\alpha,x_{0}\in\Gamma\right\} .
\]
For any $p\in S_{\Gamma}$, if $p=\theta^{\left(x_{0}\right)}\left(t\right)$,
then $t=f\left(p\right)$ and $x_{0}=\theta^{\left(p\right)}\left(-t\right)$.
Therefore, the smooth map $\left(-\alpha,\alpha\right)\times\Gamma\to S_{\Gamma}$
given by $\left(t,x_{0}\right)\mapsto\theta^{\left(x_{0}\right)}\left(t\right)$
has a smooth inverse, and we have that $\left(-\alpha,\alpha\right)\times\Gamma$
is diffeomorphic to $S_{\Gamma}$. Therefore $S_{\Gamma}$ is open,
connected, and has exactly two boundary components, both diffeomorphic
to $\Gamma$, with $f=\alpha$ on one and $f=-\alpha$ on the other,
proving \ref{enu:shell-prop-1}. The path described in \ref{enu:shell-prop-2}
is given by $\theta^{\left(p\right)}\left(\left[-\alpha,\alpha\right]\right)$.
For \ref{enu:shell-prop-3}, note that for each $p\in S_{\Gamma}$
we have $\theta^{\left(p\right)}\left(-f\left(p\right)\right)\in\Gamma$,
so $p$ cannot be simultaneously in $S_{\Gamma_{1}}$ and $S_{\Gamma_{2}}$.
\ref{enu:shell-prop-4} follows from the fact that $S_{\Gamma}^{+}$
and $S_{\Gamma}^{-}$ are continuous images of the connected sets
$\left(0,\alpha\right)\times\Gamma$ and $\left(-\alpha,0\right)\times\Gamma$,
respectively.
Finally, we prove \ref{enu:shell-prop-5}. Suppose $U=\td$; the shells
$S_{\Gamma}$ are among the connected components of $\left\{ x\in\td:\left|f\left(x\right)\right|<\alpha\right\} $,
and it remains to show that there is no other connected component
$S$. We have $f=\pm\alpha$ on $\partial S$, but $f$ cannot be
constant on $\partial S$ or there would have to be a point inside
$S$ where $\nabla f=0$, a contradiction. Therefore there must be
some $\Gamma\in\Z\left(f\right)$ lying inside $S$, and we have $S=S_{\Gamma}$.
\end{proof}
\begin{proof}[Proof of Proposition \ref{prop:smooth-perturbation}]
By applying Proposition \ref{prop:perturbation-in-U} directly, we
get $N\left(f\right)\le N\left(f+g\right)$, and by applying Proposition
\ref{prop:perturbation-in-U} on the function $f+g$ with perturbation
$-g$, we get $N\left(f+g\right)\le N\left(f\right)$. Therefore,
the mapping $\Gamma\mapsto\tilde{\Gamma}$ given in Proposition \ref{prop:perturbation-in-U}
is a bijection, and each $\tilde{\Gamma}$ is the \emph{only} component
of $\Z\left(f+g\right)$ lying inside $S_{\Gamma}$. Let $p,q\in\Gamma$
be a pair of points realizing the diameter: $\diam\Gamma=\dist\left(p,q\right)$.
By property \ref{enu:shell-prop-2} in Lemma \ref{lem:shell}, the
ball $\overline{B}\left(p,\alpha/\beta\right)$ contains a point $\tilde p\in Z\left(f+g\right)$,
which must then belong to $\tilde{\Gamma}$. Similarly, the ball $\overline{B}\left(q,\alpha/\beta\right)$
contains a point $\tilde q\in\tilde{\Gamma}$. By the triangle inequality,
\[
\diam\Gamma=\dist\left(p,q\right)\le\dist\left(p,\tilde p\right)+\dist\left(\tilde p,\tilde q\right)+\dist\left(\tilde q,q\right)\le\frac{\alpha}{\beta}+\diam\tilde{\Gamma}+\frac{\alpha}{\beta}.\qedhere
\]
\end{proof}
\subsection{Counting nodal components vs.\ counting nodal domains - Proposition
\ref{prop:nodal-components-similar-to-nodal-domains}}
We prove Proposition \ref{prop:nodal-components-similar-to-nodal-domains}
using an elementary concept in singular homology theory - the Mayer-Vietoris
sequence (see, for instance, \cite[Section 2.2]{Hatcher-book-Algebraic-Topology}).
We denote by $H_{n}\left(X\right)$ the $n$th singular homology group
of the topological space $X$, and by $\cong$ an isomorphism of groups.
\begin{proof}[Proof of Proposition \ref{prop:nodal-components-similar-to-nodal-domains}]
Let $\alpha,\beta>0$ be such that $\left|f\left(x\right)\right|>\alpha$
or $\left|\nabla f\left(x\right)\right|>\beta$ for any $x\in\td$
(see Remark \ref{rem:about-nodal-stability}). Define two sets $A,B\subset\td$
by:
\begin{align*}
A & \coloneqq\left\{ x\in\td:f\left(x\right)\ne0\right\} \\
B & \coloneqq\left\{ x\in\td:\left|f\left(x\right)\right|<\alpha\right\}
\end{align*}
$A$ and $B$ are both open, and $A\cup B=\td$. We count the connected
components of $A,B$ and $A\cap B$:
\begin{itemize}
\item The connected components of $A$ are precisely the nodal domains,
so $A$ has $r$ components.
\item The connected components of $B$ are precisely the shells $S_{\Gamma}$
defined in Lemma \ref{lem:shell}, which are in correspondence with
the nodal components, so $B$ has $k$ components.
\item By Lemma \ref{lem:shell} \ref{enu:shell-prop-4}, excluding $\Gamma$
from any shell $S_{\Gamma}$ leaves it with exactly two components,
so $A\cap B$ has $2k$ components.
\end{itemize}
Consider the last four terms of the Mayer-Vietoris sequence, and name
the nonzero maps $\phi_{1},\phi_{2},\phi_{3}$:
\[
\cdots\longrightarrow H_{1}\left(\td\right)\overset{\phi_{1}}{\longrightarrow}H_{0}\left(A\cap B\right)\overset{\phi_{2}}{\longrightarrow}H_{0}\left(A\right)\oplus H_{0}\left(B\right)\overset{\phi_{3}}{\longrightarrow}H_{0}\left(\td\right)\longrightarrow0.
\]
Written explicitly:
\[
\cdots\longrightarrow\zd\overset{\phi_{1}}{\longrightarrow}\z^{2k}\overset{\phi_{2}}{\longrightarrow}\z^{r+k}\overset{\phi_{3}}{\longrightarrow}\z\longrightarrow0.
\]
$\phi_{1},\phi_{2},\phi_{3}$ are group homomorphisms, and they may
be extended naturally to linear maps between $\q$-vector spaces,
so the rank-nullity theorem applies and we get:
\begin{equation}
\begin{aligned}d & =\rank\img\phi_{1}+\rank\ker\phi_{1}\\
2k & =\rank\img\phi_{2}+\rank\ker\phi_{2}\\
r+k & =\rank\img\phi_{3}+\rank\ker\phi_{3}
\end{aligned}
\label{eq:rank-nullity}
\end{equation}
By the exactness of the Mayer-Vietoris sequence:
\begin{equation}
\begin{aligned}\img\phi_{1} & =\ker\phi_{2}\\
\img\phi_{2} & =\ker\phi_{3}\\
\img\phi_{3} & =\z
\end{aligned}
\label{eq:by-exactness}
\end{equation}
Plugging equations (\ref{eq:by-exactness}) into equations (\ref{eq:rank-nullity}),
we get:
\[
\rank\ker\phi_{1}=d-k+r-1.
\]
Since $\ker\phi_{1}$ is a subgroup of $\zd$, we have $0\le\rank\ker\phi_{1}\le d$,
leading to the required conclusion.
For the second part, let $\Gamma_{1},\ldots,\Gamma_{k'}\in\Z\left(f\right)$
be the nodal components of $f$ that lie completely inside some open
ball $U$ with radius less than $\frac{1}{2}$. Suppose the set $A\coloneqq U\setminus\left(\Gamma_{1}\cup\dots\cup\Gamma_{k'}\right)$
has $s$ components. Exactly one touches the (connected) boundary
of $U$. The other $s-1$ have $f=0$ on their boundary, and each
of them must contain at least one nodal domain of $f$ that lies completely
inside $U$, so $s-1\le r'$.
We conclude by showing that $s=k'+1$. Let $\alpha,\beta>0$ be such
that $\left|f\left(x\right)\right|>\alpha$ or $\left|\nabla f\left(x\right)\right|>\beta$
for any $x\in\td$, and $\dist\left(\Gamma_{i},\partial U\right)\ge\alpha/\beta$
for any $1\le i\le k'$ (see Remark \ref{rem:about-nodal-stability}).
By Lemma \ref{lem:shell}, each $\Gamma_{i}$ is contained in a shell
$S_{\Gamma_{i}}\subset U$. Define $B\coloneqq S_{\Gamma_{1}}\cup\dots\cup S_{\Gamma_{k'}}$.
As in the first part, $A,B$ are open and $A\cup B=U$, $A$ has $s$
components, $B$ has $k'$ components, and $A\cap B$ has $2k'$ components.
$U$ is simply-connected (because its radius is less than $\frac{1}{2}$),
so the last four terms of the Mayer-Vietoris sequence form the short
exact sequence $0\longrightarrow\z^{2k'}\longrightarrow\z^{s+k'}\longrightarrow\z\longrightarrow0$,
and we get $s=k'+1$.
\end{proof}
\subsection{Local bounds on eigenfunctions - Proposition \ref{prop:local-bounds}}
\begin{proof}[Proof of Proposition \ref{prop:local-bounds}]
W.l.o.g.\ we assume that $x_{0}=0$, and also that $L=1$ - for
the latter, simply apply the result to $\tilde f\left(x\right)\coloneqq f\left(x/L\right)$.
Thus, $f$ satisfies $\Delta f+4\pi^{2}f=0$.
We utilize a trick to view $f$ as a harmonic function in one more
dimension. Define $u\colon\R^{d+1}\to\R$ by:
\[
u\left(x_{1},\ldots,x_{d},x_{d+1}\right)\coloneqq f\left(x_{1},\ldots,x_{d}\right)\cosh\left(2\pi x_{d+1}\right).
\]
Clearly, $u$ is harmonic and $u\left(0\right)=f\left(0\right)$,
$\frac{\partial u}{\partial x_{j}}\left(0\right)=\frac{\partial f}{\partial x_{j}}\left(0\right)$
and $\frac{\partial^{2}u}{\partial x_{i}\partial x_{j}}\left(0\right)=\frac{\partial^{2}f}{\partial x_{i}\partial x_{j}}\left(0\right)$
for $1\le i,j\le d$. Using the mean value property in the ball $B^{d+1}\left(0,r/\sqrt{d}\right)$,
we get:
\[
f\left(0\right)=C\underset{B^{d+1}\left(0,r/\sqrt{d}\right)}{\idotsint}f\left(x_{1},\ldots,x_{d}\right)\cosh\left(2\pi x_{d+1}\right)\,\mathrm{d}x_{1}\cdots\mathrm{d}x_{d+1},
\]
where $C$ is a positive constant depending only on $r,d$. Taking
the absolute value and using the inclusions
\[
B^{d+1}\left(0,\frac{r}{\sqrt{d}}\right)\subset\left[-\frac{r}{\sqrt{d}},\frac{r}{\sqrt{d}}\right]^{d+1}\qquad\text{and}\qquad\left[-\frac{r}{\sqrt{d}},\frac{r}{\sqrt{d}}\right]^{d}\subset B^{d}\left(0,r\right),
\]
we get:
\begin{align*}
\left|f\left(0\right)\right| & \ll\underset{\left[-r/\sqrt{d},r/\sqrt{d}\right]^{d+1}}{\idotsint}\left|f\left(x_{1},\ldots,x_{d}\right)\right|\left|\cosh\left(2\pi x_{d+1}\right)\right|\,\mathrm{d}x_{1}\cdots\mathrm{d}x_{d+1}\ll\Int[B^{d}\left(0,r\right)]{\left|f\left(x\right)\right|}x,
\end{align*}
where the implied constant in $\ll$ depends only on $r,d$. By the
Cauchy-Schwarz inequality, we get (\ref{eq:bound-function}).
Next, we turn to (\ref{eq:bound-gradient}) and (\ref{eq:bound-hessian}).
Note that for any $\left|x\right|<\frac{r}{2}$ in $\R^{d+1}$, we
have:
\[
u\left(x\right)=\left(\frac{r}{2}\right)^{d-1}\Int[\left|\zeta\right|=1]{u\left(\frac{1}{2}r\zeta\right)P\left(x,\frac{1}{2}r\zeta\right)}{\sigma_{d}\left(\zeta\right)},
\]
where ${\displaystyle P\left(x,y\right)\coloneqq\frac{\left|x\right|^{2}-\left|y\right|^{2}}{\left|x-y\right|^{d+1}}}$
is the $\left(d+1\right)$-dimensional Poisson kernel. Differentiating
under the integral sign with respect to $x_{j}$, we get:
\[
\frac{\partial f}{\partial x_{j}}\left(0\right)=\frac{\partial u}{\partial x_{j}}\left(0\right)=\left(\frac{r}{2}\right)^{d-1}\Int[\left|\zeta\right|=1]{u\left(\frac{1}{2}r\zeta\right)\frac{\partial P}{\partial x_{j}}\left(0,\frac{1}{2}r\zeta\right)}{\sigma_{d}\left(\zeta\right)}.
\]
The Poisson kernel is smooth when its two parameters are separated,
so $\zeta\mapsto\frac{\partial P}{\partial x_{j}}\left(0,\frac{1}{2}r\zeta\right)$
is a continuous function on $\left\{ \left|\zeta\right|=1\right\} $.
Its integral depends only on $r,d$. Therefore:
\[
\left|\frac{\partial f}{\partial x_{j}}\left(0\right)\right|\ll\sup_{\left|\zeta\right|=1}\left|u\left(\frac{1}{2}r\zeta\right)\right|\le\sup_{\left|x\right|\le r/2}\left|f\left(x\right)\right|\sup_{\left|t\right|\le r/2}\left|\cosh\left(2\pi t\right)\right|\ll\sup_{\left|x\right|\le r/2}\left|f\left(x\right)\right|.
\]
Using (\ref{eq:bound-function}) applied on $f$ at point $x$, this
gives:
\[
\left|\nabla f\left(0\right)\right|^{2}=\sum_{j=1}^{d}\left|\frac{\partial f}{\partial x_{j}}\left(0\right)\right|^{2}\ll\sup_{\left|x\right|\le r/2}\left|f\left(x\right)\right|^{2}\ll\sup_{\left|x\right|\le r/2}\Int[B^{d}\left(x,r/2\right)]{\left|f\left(y\right)\right|^{2}}y\le\Int[B^{d}\left(0,r\right)]{\left|f\left(y\right)\right|^{2}}y.
\]
This proves (\ref{eq:bound-gradient}). Similarly, applying (\ref{eq:bound-gradient})
on $\frac{\partial f}{\partial x_{j}}$ at point $0$ and then applying
(\ref{eq:bound-function}) on $\frac{\partial f}{\partial x_{j}}$
at point $x$, we get:
\[
\left|\nabla\frac{\partial f}{\partial x_{j}}\left(0\right)\right|^{2}\ll\Int[B^{d}\left(0,r/2\right)]{\left|\frac{\partial f}{\partial x_{j}}\left(x\right)\right|^{2}}x\ll\sup_{\left|x\right|\le r/2}\left|\frac{\partial f}{\partial x_{j}}\left(x\right)\right|^{2}\ll\Int[B^{d}\left(0,r\right)]{\left|f\left(y\right)\right|^{2}}y.
\]
Summing over $j$ gives (\ref{eq:bound-hessian}).
\end{proof}
\bibliographystyle{amsalpha-initials}
\phantomsection\addcontentsline{toc}{section}{\refname}\bibliography{Article}
\end{document} | 129,842 |
\begin{document}
\author{Kenneth L. Baker}
\author{Jesse E. Johnson}
\author{Elizabeth A. Klodginski}
\address{Department of Mathematics\\ University of Georgia\\ Athens GA 30602}
\email{[email protected]}
\address{Department of Mathematics\\ University of California\\ Davis, CA 95616}
\email{[email protected]}
\address{Department of Mathematics\\ University of California\\ Davis, CA 95616}
\email{[email protected]}
\title{Tunnel number one, genus one fibered knots
}
\thanks{The first author was supported by NSF VIGRE grant DMS-0089927. The second and third authors were supported by NSF VIGRE grant DMS-0135345.}
\subjclass[2000]{Primary 57M50, Secondary 57M12, 57M25}
\keywords{braid, two bridge link, lens space, genus one fibered
knot, double branched cover, tunnel number one}
\begin{abstract}
We determine the genus one fibered knots in lens spaces that
have tunnel number one. We also show that every tunnel number one, once-punctured torus bundle is the result of Dehn filling a component of the Whitehead link in the 3-sphere.
\end{abstract}
\maketitle
\section{Introduction}
A null homologous knot $K$ in a $3$--manifold $M$ is a {\em genus
one fibered knot}, GOF-knot for short, if $M-N(K)$ is a
once-punctured torus bundle over the circle whose monodromy is the
identity on the boundary of the fiber and $K$ is ambient isotopic in
$M$ to the boundary of a fiber.
We say the knot $K$ in $M$ has {\em tunnel number one} if there is a
properly embedded arc $\tau$ in $M-N(K)$ such that $M-N(K)-N(\tau)$
is a genus $2$ handlebody. An arc such as $\tau$ is called an {\em
unknotting tunnel}. Similarly, a manifold with toroidal boundary is
{\em tunnel number one} if it admits a genus-two Heegaard splitting.
Thus a knot is tunnel number one if and only if its complement is
tunnel number one.
In the genus $1$ Heegaard surface of $L(p,1)$, $p \neq 1$, there is
a unique link that bounds an annulus in each solid torus. This two-component link
is called the {\em $p$--Hopf link} and is fibered with
monodromy $p$ Dehn twists along the core curve of the fiber.
We refer to the fiber of a $p$--Hopf link as a {\em $p$--Hopf band}.
In this terminology, the standard positive and negative Hopf bands
in $S^3 = L(+1, 1)=L(-1,1)$ are the $(+1)$-- and $(-1)$--Hopf bands,
respectively.
Gonz\'alez-Acu\~na~\cite{ga:dcok} shows that the trefoil (and its
mirror) and the figure eight knot are the only GOF-knots in $S^3$.
These knots arise as the boundary of the plumbing of two
$(\pm1)$--Hopf bands. For both knots, a transverse arc on one of the
plumbed Hopf bands is an unknotting tunnel, so both have tunnel
number one.
GOF-knots in lens spaces were first studied by Morimoto in \cite{morimoto:gofkils}, and were classified by the first author of the present article in \cite{baker:cgofkils}. Unlike in $S^3$, these knots do not all have tunnel number one. In particular we prove the following:
\begin{thm}\label{thm:main}
Every tunnel number one, genus one fibered knot in a lens space is
the plumbing of an $r$--Hopf band and a $(\pm 1)$--Hopf band which is contained in the
lens space $L(r,1)$, with one exception. Up to mirroring, this
exception is the genus one fibered knot in $L(7,2)$ that arises as
$(-1)$--Dehn surgery on the plumbing of a $7$--Hopf band and a
$(+1)$--Hopf band.
\end{thm}
Every GOF-knot has tunnel number 1 or 2. Thus Theorem~\ref{thm:main} determines the tunnel number of every GOF-knot in a lens space.
We also obtain
\begin{thm}\label{thm:main2}
Every tunnel number one, once-punctured torus bundle is the complement of
a GOF-knot in a lens space of the form $L(r,1)$.
\end{thm}
\begin{thm}\label{thm:main3}
Every tunnel number one, once-punctured torus bundle is the $(r/1)$--Dehn filling of a boundary component of the exterior of the Whitehead link for some integer $r$.
\end{thm}
This article begins by determining the monodromy of a tunnel number
one, once-punctured torus bundle in Section~\ref{sec:2}. We then
translate these monodromies into the language of closed $3$--braids
in Section~\ref{sec:3}. In particular we determine which
$3$--braids in a solid torus have double branched covers producing
tunnel number one, once-punctured torus bundles.
In Section~\ref{sec:4} we discuss the presentations of two bridge links as closed $3$--braids in $S^3$. In Section~\ref{sec:5} we determine which of the braids from Section~\ref{sec:3} are two bridge links. The proof of Theorem~\ref{thm:main} is presented in Section~\ref{sec:6}, and Theorems~\ref{thm:main2} and \ref{thm:main3} are proved in Section~\ref{sec:7}.
We refer the reader to \cite{bz:knots} for background regarding
fibered knots, braids, two bridge links, lens spaces, and double
coverings of $S^3$ branched over a link.
Also, recall that plumbing is a local operation generalizing the connect sum.
In particular, the plumbing of a fibered link in $S^3$ with a fibered link in another $3$--manifold $M$ produces a new fibered link in $M$.
See for example Gabai's geometric description of the yet more generalized Murasugi sum in \cite{gabai:msiango}.
The authors would like to thank Alan Reid for a helpful conversation
regarding the Whitehead link.
\section{The monodromy of a tunnel number one, once-punctured torus bundle.}\label{sec:2}
Heegaard splittings of closed torus bundles over the circle were studied in
detail by Cooper and Scharlemann~\cite{cooper-scharlemann}. In
particular, their work characterized genus two Heegaard splittings of such
bundles. Their method transfers almost directly to the case of
once-punctured torus bundles which are tunnel number one.
\begin{lemma}
\label{monodromylem}
Let $M$ be a once-punctured torus bundle over the circle with once-punctured torus fiber $T$ and monodomy $\phi$. Further assume $M$ allows a genus two Heegaard splitting.
Then there is a pair of simple closed curves $\alpha_1$,
$\alpha_2$ in $T$ such that $\alpha_1 \cap \alpha_2$ is a single
point and $\phi$ sends $\alpha_1$ onto $\alpha_2$. The map $\phi$
is isotopic to $(s_2 s_1 s_2)^{\pm 1} s_1^n$ where $s_1$ is a Dehn
twist along $\alpha_1$ and $s_2$ is a Dehn twist along $\alpha_2$.
\end{lemma}
\begin{proof}
We will show that the proof for closed torus bundles in Theorem~4.2
of \cite{cooper-scharlemann} works equally well in the once-punctured
case. Because this method has been described in detail elsewhere,
we will give only an outline of the setup and leave many of the
details to the reader. A similar exposition for general surface bundles
can also be found in \cite{bachman-schleimer}.
Let $(\Sigma, H_1, H_2)$ be a genus-two Heegaard splitting for $M$.
Assume $H_1$ is a compression body and $H_2$ a handlebody. A spine
$K_1$ for $H_1$ consists of $\partial_- H_1$ and an arc properly embedded in $H_1$ such that
the complement in $H_1$ of $K_1$ is homeomorphic to $\partial_+ H_1
\times (-1,0]$. A spine $K_2$ for $H_2$ is a graph whose complement
is homeomorphic to $\partial H_2 \times [0,1)$. The Heegaard
splitting determines a continuous one-parameter family of embedded, pairwise disjoint
surfaces $\{\Sigma_x : x \in (-1,1)\}$ such that as $x$ approaches $-1$, the
surfaces limit to $K_1$ and as $x$ approaches $1$, the surfaces limit onto
$K_2$. This family of surfaces is called a \textit{sweep-out}. For each $x$,
the surface $\Sigma_x$ separates $M$ into a compression body $H^x_1$
and a handlebody $H^x_2$.
The fibers of the bundle structure on $M$ form a continuous one-parameter family of embedded, pairwise disjoint essential surfaces $\{T_y : y \in S^1\}$. Assume
the surfaces $\{\Sigma_x\}$ and $\{T_y\}$ are in general position.
The {\em Rubinstein-Scharlemann graphic} is the subset $R$ of
$(-1,1) \times S^1$ consisting of pairs $(x,y)$ such that $\Sigma_x$ and $T_y$
are tangent at some point in $M$. General position implies that this set will be a
graph.
At each point $(x,y)$ in the complement of the graphic, the corresponding
surfaces $\Sigma_x$ and $T_y$ are transverse. Label each point $(x,y)$ (in $(-1,1) \times S^1$) with a $1$ if some loop component of $\Sigma_x \cap T_y$ is essential in $\Sigma_x$ and is the boundary component of a disk or an essential annulus in $H^x_1$. Note that for such an essential annulus, its other boundary component is a curve on $\bdry M$ isotopic to $\bdry T_y$. Label $(x,y)$ with a $2$ if a loop is essential in $\Sigma_x$ and bounds a disk in $H^x_2$. Given a second point $(x',y')$ in the same component of the complement of the graphic as $(x,y)$, a piecewise vertical an horizontal path from $(x,y)$ to $(x',y')$ determines an ambient isotopy of $M$ taking $\Sigma_x$ to $\Sigma_{x'}$ and $T_y$ to $T_{y'}$. Thus any two points in the same component of the complement of the graphic have the same labels.
\begin{claim} \label{claim:distinctlabelheight}
For a fixed $x$, there cannot be values $y,y'$ such that $(x,y)$ is labeled
with a $1$ and $(x,y')$ is labeled with a $2$. (In particular, a point cannot have
both labels $1$ and $2$.)
\end{claim}
\begin{proof}
Let $M'$ be the result of Dehn filling $\partial M$ along the slope of the boundary of a level surface $T_y$. Then
$M'$ is a closed torus bundle and the image of a loop which is
boundary parallel in $T_y$ will be trivial in $M'$. The
induced Heegaard splitting of $M'$ will be irreducible because a
torus bundle cannot be a lens space. Moreover, an irreducible,
genus-two Heegaard splitting is strongly irreducible so the induced
Heegaard splitting of $M'$ is strongly irreducible.
Now assume for contradiction the point $(x,y)$ is labeled with a $1$ and $(x,y')$ is labeled with a $2$. Then there is a loop in $\Sigma_x \cap T_y$ that bounds an essential disk in the filling of $H_1^x$ (after the filling, an essential annulus is capped off and becomes a disk) and a loop $\Sigma_x \cap T_{y'}$ that bounds an essential disk in $H_2^x$. Because $T_y$ and $T_{y'}$ are disjoint, each loop in $\Sigma_x \cap T_{y'}$ is disjoint from each loop in $\Sigma_x \cap T_y$. Thus the Heegaard surface $\Sigma_x$ for $M'$ is weakly reducible and therefore
reducible. This contradicts the assumption that $M'$ is not a lens space.
\end{proof}
\begin{claim}\label{claim:TynotinH1}
No fiber $T_y$ can be made disjoint from $H^x_2$.
\end{claim}
\begin{proof}
If a fiber $T_y$ were disjoint from $H^x_2$, then it would be
contained in $H^x_1$. Since $T_y$ is essential, a non-separating
compressing disk of $H^x_1$ must be disjoint from $T_y$. Therefore
compressing $H^x_1$ along such a disk yields a manifold which is
homeomorphic to $T^2 \times I$ and contains a properly embedded,
essential, once-punctured torus, which cannot occur.
\end{proof}
\begin{claim}\label{claim:nonemptylabels}
In a component of $(-1,1) \times S^1 - R$ that intersects $(-1, -1+\epsilon) \times S^1$ for suitably small $\epsilon$, each point is labeled with a $1$. In a component of $(-1,1) \times S^1 - R$ that intersects $(1-\epsilon, 1) \times S^1$ for suitably small $\epsilon$, each point is labeled with a $2$.
\end{claim}
\begin{proof}
Fix $y$. Because $\Sigma_x$ limits onto $K_1$, if $x$ is near $-1$,
then $T_y \cap H^x_1$ will be a collection of disks and an essential
annulus. Hence $(x,y)$ has label $1$.
If $x$ is near $1$, then $T_y \cap H^x_2$ consists of disks. If
none of these disks are essential in $H^x_2$,
then $T_y$ can be pushed into $H_1^x$,
contradicting Claim~\ref{claim:TynotinH1}. Hence $(x,y)$ has label
$2$.
\end{proof}
\begin{claim}\label{claim:unlabeledpoints}
For some value $x$, the point $(x,y)$ is unlabeled for every $y$.
\end{claim}
\begin{proof}
The complement of the graphic is an open set, as is each component of the
complement. The union of all the components labeled with a 1 projects to
an open set in $(-1,1)$ as does the union of the components labeled with a 2.
By Claim~\ref{claim:nonemptylabels} the projection of each set is non-empty. By Claim~\ref{claim:distinctlabelheight} the images of these projections are disjoint. Because $(-1,1)$ is connected, it cannot be written as the union of two non-empty open sets. Thus there is a point $x$ in the complement of the projections. This $x$ has the desired property.
\end{proof}
Fix an $x$ such that for every $y$, $(x,y)$ is unlabeled as
guaranteed by Claim~\ref{claim:unlabeledpoints}. If for some $y$,
$T_y$ is transverse to $\Sigma_x$ and each loop of $\Sigma_x \cap
T_y$ is trivial in $T_y$ then $T_y$ can be isotoped into $H^x_1$. As
noted in Claim~\ref{claim:TynotinH1} above, this is impossible.
Therefore for each $y \in S^1$, if $T_y$ and $\Sigma_x$ intersect
transversely, then the intersection $\Sigma_x \cap T_y$ must contain
a loop which is essential in $T_y$. Since two disjoint, essential
loops in a once-punctured torus are parallel, the essential loops in
$\Sigma_x \cap T_y$ are all parallel in $T_y$, for each $y$.
For each point $p \in \Sigma_x$ there is a $y \in S^1$ such that $p$ is
contained in $T_y$. Define the function $f_x : \Sigma_x \rightarrow S^1$
such that $p$ is contained in $T_{f(p)}$ for each $p \in \Sigma_x$. By general
position, either $f_x$ is a circle-valued Morse function or $f_x$ is a near-Morse function such that the critical points consist of either a single isolated degenerate critical point or just two critical points at the same level.
Assume for contradiction $f_x$ is a Morse function. Then for each
$y \in S^1$, the level set $f_x^{-1}(y)$ contains a loop which is
essential in $\Sigma_x$. Thus $\Sigma_x \cap T_y$ contains a loop
which is essential in $\Sigma_x$ and therefore essential in $T_y$.
By continuity, the isotopy class in $\Sigma_x$ of the essential
loops cannot change as $y$ varies, so the monodromy $\phi$ must send
this essential loop onto itself. A quick calculation shows the
homology of this manifold would have rank 3, contradicting the fact
that $M$ admits a genus two Heegaard splitting.
If $f_x$ has an isolated degenerate critical point, then once again
$f_x^{-1}(y)$ must contain an essential loop for each $y$. This leads to the
same contradiction as when $f_x$ is a Morse function, see~\cite{cooper-scharlemann}. Therefore we conclude that $f_x$ must have two
critical points on the same level. As described in~\cite{cooper-scharlemann},
this implies that $\Sigma_x$ is embedded in $M$ as shown in
Figure~\ref{fig:twocrits}.
\begin{figure}[htb]
\centering
\includegraphics[width=3.5in]{twocrits.eps}
\caption{$\Sigma_x$ in the once-punctured torus bundle $M$ cut open
along a fiber $T_y$, and the sequence of intersections of $\Sigma_x$
with the fibers $T_y$ as $y$ passes through the critical value of
$f_x$.} \label{fig:twocrits}
\end{figure}
The right side of the figure shows how the loops of $\Sigma_x \cap T_y$ sit in each $T_y$. As $y$ passes through the critical value $c$ of $f_x$ from $c+\epsilon$ to $c-\epsilon$, the isotopy class $\ell_{c+\epsilon}$ of the essential loops of $\Sigma_x \cap T_{c+\epsilon}$ is replaced by a new isotopy class $\ell_{c-\epsilon}$ of the loops of $\Sigma_x \cap T_{c-\epsilon}$ which, under the projection $T \times (c-\epsilon, c+\epsilon) \to T$, intersects the original in a single point. Because this is the only level at which this happens, we conclude that there are essential
loops $\alpha_1, \alpha_2 \subset T$ such that $\alpha_1 \cap \alpha_2$ is a
single point and $\phi(\alpha_1) = \alpha_2$.
Let $s_1$ be a Dehn twist along $\alpha_1$ and $s_2$ a Dehn twist
along $\alpha_2$. Either composition $(s_2 s_1 s_2)^{\pm 1}$ takes $\alpha_1$ to $\alpha_2$ and $\alpha_2$ to $\alpha_1$. By choosing $+1$ or $-1$ appropriately, we can ensure that the map sends $\alpha_1$ to $\alpha_2$ with the same orientation as $\phi$. Composing further with $s_1^n$ for some $n$ will take $\alpha_2$ to $\phi(\alpha_2)$. Thus $(s_2 s_1 s_2)^{\pm 1}s_1^n$ is isotopic to $\phi$.
\end{proof}
\section{Genus one fibered knots via closed $3$--braids}\label{sec:3}
A genus one fibered knot may be described in terms of a double
branched covering of a closed 3-braid whose word encodes the
monodromy of the fibering. This viewpoint allows us to describe
tunnel number one once-punctured torus bundles.
\begin{lemma}\label{lem:corresp}
Every genus one fibered knot is the image of the braid axis of a
closed $3$--braid in $S^3$ in the double branched cover of the
closed $3$--braid. Conversely, the image of the braid axis of a closed
$3$--braid in the double branched cover of the closed $3$--braid is
always a genus one fibered knot.
\end{lemma}
By considering the standard involution of the once-punctured torus
$T$, we will show that the set of genus one fibered knots in
$3$--manifolds and the set of braid axes of closed $3$--braids in $S^3$ are in
one-to-one correspondence. We sketch the passage from genus one fibered knots
to axes of closed $3$--braids in $S^3$ below. The return passage is
then clear. (See Section 5 of \cite{birman-hilden:hsobcoS3},
Sections 4 and 5 of \cite{mr:gfm}, and Section 2 of
\cite{baker:cgofkils}.)
\begin{proof}[Sketch]
The standard involution $\iota$ with three fixed points of the
once-punctured torus $T$ extends to an involution of the
once-punctured torus bundle $M$ that takes a fiber to a fiber. Since
$\iota$ commutes with the monodromy of $M$, quotienting $M$ by this
involution yields a closed $3$--braid $\hat{\beta}$ in a solid torus
$W$ where the braid $\hat{\beta}$ is the image of the fixed set of
$\iota$ and a meridional disk of $W$ is the image of a fiber.
The extension of $\iota$ is a fixed point free involution on the
boundary of $M$, so $\iota$ further extends to an involution of the
filling $M'$ of $M$ by a solid torus $V'$ whose meridians intersect
fibers of $M$ just once. In $M'$ the core of $V'$ is a genus one
fibered knot $K$. Since the extension of $\iota$ acts as a free
involution of $V'$, in the quotient $V'$ descends to a solid torus
$V$. The meridian of $V$ intersects the meridian of $W$ once, as
the meridian of $V'$ intersects the boundary of a fiber of $M$ once.
Hence under this quotient, $K$ descends to the core of $V$ which is
the axis of the closed $3$--braid $\hat{\beta}$ in $V \cup W \cong
S^3$. This defines the correspondence.
\end{proof}
\begin{lemma}\label{lem:tnobraid}
Every tunnel number one, genus one fibered knot is the lift of the
braid axis of the closure $\hat{\beta}_{k,n}$ of the braid
$\beta_{k,n} = (\sigma_2 \sigma_1 \sigma_2)^k \sigma_1^n$ in the
double cover of $S^3$ branched over $\hat{\beta}_{k,n}$, where $k$
is odd and $n$ is an arbitrary integer.
\end{lemma}
A depiction of the braid $\beta_{k,n} = (\sigma_2 \sigma_1 \sigma_2)^k \sigma_1^n$ is shown in Figure~\ref{fig:betakn}.
\begin{figure}
\centering
\input{betakn.pstex_t}
\caption{The braid $\beta_{k,n} = (\sigma_2 \sigma_1 \sigma_2)^k \sigma_1^n$.}
\label{fig:betakn}
\end{figure}
\begin{proof}
Let $\widehat{M}$ be a $3$--manifold that contains a tunnel number
one, GOF-knot $K$. Let $M$ be the once-punctured torus bundle
exterior $\widehat{M}-N(K)$ with fiber $T$.
By Lemma~\ref{monodromylem}, the monodromy of $M$ may be given by
$(s_2 s_1 s_2)^{\pm 1}s_1^n$ up to Dehn twists along the boundary of
a fiber $T$. Write a single Dehn twist along $\bdry T$ as $(s_2 s_1
s_2)^{4}$. Then the monodromy $T \to T$ fixing $\bdry T$ with an
orbit on $\bdry M$ bounding a meridional curve for $K$ is $(s_2 s_1
s_2)^{4\ell \pm 1}s_1^n$ for some $\ell \in \Z$.
By Lemma~\ref{lem:corresp}, $\widehat{M}$ is the double branched
cover of a closed $3$--braid in $S^3$ with braid axis lifting to
$K$. Accordingly, $M$ is the double branched cover of the closed
$3$--braid in the solid torus exterior of the braid axis. A
meridian of $K$ in $\bdry M$ corresponds to a meridian of the braid
axis in the solid torus.
Under the quotient of the covering involution, the Dehn twist $s_i$
along $\alpha_i$ corresponds to the braid $\sigma_i$, a right-handed
crossing between the $i$th and $(i+1)$th strands. Thus $K$ in
$\widehat{M}$ corresponds to the braid axis of the closed braid
$(\sigma_2 \sigma_1 \sigma_2)^{k} \sigma_1^n$ where $k=4\ell \pm1$.
\end{proof}
\begin{remark}\label{rem:dbcsurgery}
A meridian of the braid axis is a longitudinal curve on the solid
torus containing the closed $3$--braid. In the double branched
cover it lifts to two meridians of the genus one fibered knot. The
longitude of the braid axis is the meridian of the solid torus
containing the closed $3$--braid. It lifts to the longitude of the
genus one fibered knot.
In these coordinates for the braid axis, a slope of $p/q$ lifts to
the slope $2p/q$. If $q$ is even (and $p$ and $q$ are coprime) this
slope is to be interpreted as two parallel curves of slope
$p/(q/2)$. It follows that $1/q$ surgery on a genus one fibered
knot corresponds to inserting $2q$ full twists (right-handed if
$q<0$, left-handed if $q>0$) into the $3$--braid.
If $q$ is odd then a $p/q$ slope lifts to a single curve of slope
$2p/q$. Hence $p/q$ surgery on the braid axis cannot lift to
surgery on the genus one fibered knot in the double branched cover
unless the core of the surgery solid torus is added to the branch
locus.
\end{remark}
\section{Genus one fibered knots in lens spaces}\label{sec:4}
By Corollary 4.12 of \cite{hodgson-rubinstein:iaiols}, the lens
space $L(\alpha, \beta)$ is the double cover of $S^3$ branched over
a link $L$ if and only if $L$ is equivalent to the two bridge link
$\B(\alpha, \beta)$. Thus to understand GOF-knots in lens spaces
using Lemma~\ref{lem:corresp} we must consider representations of
two bridge links as closed $3$--braids; see
\cite{baker:cgofkils}.
Murasugi and later Stoimenow describe which two bridge
links admit such closed braid representations. These two descriptions may be seen to be equivalent by working out their corresponding continued fractions. Let ${\bf b}(L)$ denote the braid index of the link $L$.
\begin{prop}[Proposition 7.2, \cite{murasugi:otbioal}] \label{prop:murasugi}
Let $L$ be a two bridge link of type $\B(\alpha, \beta)$, where
$0<\beta<\alpha$ and $\beta$ is odd. Then
\begin{enumerate}
\item ${\bf b}(L) = 2$ iff $\beta = 1$.
\item ${\bf b}(L) = 3$ iff either
\begin{enumerate}
\item \label{item:one} for some $p,q >1$, $\alpha = 2pq+p+q$ and $\beta = 2q+1$, or
\item \label{item:two} for some $q>0$, $\alpha =2pq+p+q+1$ and $\beta = 2q+1$.
\end{enumerate}
\end{enumerate}
\end{prop}
\begin{lemma}[Corollary 8, \cite{stoimenow:tspoc3b}]\label{lemma:stoimenow}
If $L$ is a two bridge link of braid index 3, then $L$ has Conway notation $(p,1,1,q)$ or $(p,2,q)$ for some $p,q>0$.
\end{lemma}
The two bridge link with Conway notation $(p,1,1,q)$ is shown in Figure~\ref{fig:twobridgeequiv}. The link with Conway notation $(p,2,q)$ is shown in Figure~\ref{fig:twobridgethreebraid}.
In the other direction, we can determine which $3$--braids have closures that are two bridge links.
\begin{lemma}\label{lem:braidconj}
The closure of a $3$--braid $\beta$ is a two bridge link if
and only if $\beta$ or its mirror image is conjugate to $\sigma_2^{-1} \sigma_1^p \sigma_2^2 \sigma_1^q$ for some $p, q \in \Z$.
\end{lemma}
\begin{proof}
Assume the closure $\hat{\beta}$ of a $3$--braid $\beta$ is a two bridge link. By Lemma~\ref{lemma:stoimenow} $\hat{\beta}$ may be described with Conway notation $(p,1,1,q)$ or $(p,2,q)$ for some integers $p,q$. Figure~\ref{fig:twobridgeequiv} shows that a two bridge link with Conway notation $(p,1,1,q)$ is equivalent to one with Conway notation $(p,2,-q-1)$. Figure~\ref{fig:twobridgethreebraid} shows the expression of a two bridge link with Conway notation $(p,2,q)$ as the closure of the braid $\sigma_2^{-1} \sigma_1^p \sigma_2^2 \sigma_1^q$. By Theorem~2.4 in~\cite{baker:cgofkils} the braid axis is unique in Case (2) of Proposition~\ref{prop:murasugi}. For Case (1), the braid axes are classified and have the desired form (see the proof of Theorem~2.4 in~\cite{baker:cgofkils}).
If a $3$--braid $\beta$ is conjugate to $\sigma_2^{-1} \sigma_1^p
\sigma_2^2 \sigma_1^q$ for some $p, q \in \Z$, then its closure can readily be identified with a two bridge link, cf.\ Figure~\ref{fig:twobridgethreebraid}.
\end{proof}
\comment{
\begin{remark}
Figure~\ref{fig:twobridgethreebraid} shows the two bridge link with Conway notation $(p,2,q)$ can be expressed as closures of the
(often non-conjugate) braids $\sigma_2^{-1} \sigma_1^p \sigma_2^2
\sigma_1^q$ and $\sigma_2^{-1} \sigma_1^q \sigma_2^2 \sigma_1^p$.
\end{remark}
}
\begin{lemma}\label{lem:3braidlensspace}
The double branched cover of the closure of the braid $\sigma_2^{-1}
\sigma_1^p \sigma_2^2 \sigma_1^q$ is the lens space
$L(2pq+p+q,2q+1)$.
\end{lemma}
\begin{proof}
Since the closure of the braid $\sigma_2^{-1} \sigma_1^p \sigma_2^2
\sigma_1^q$ is a two bridge link with Conway notation $(p,2,q)$ (see Figure~\ref{fig:twobridgethreebraid}), it
corresponds to the two bridge link $\B(\alpha, \beta)$ where
$\alpha/\beta = p+\cfrac{1}{2+\frac{1}{q}}=\frac{2pq+p+q}{2q+1}$.
Since the double branched cover of the two bridge link $\B(\alpha,
\beta)$ is the lens space $L(\alpha, \beta)$ the result follows.
\end{proof}
\begin{figure}
\centering
\input{twobridgeequiv.pstex_t}
\caption{The two bridge link $(p,1,1,q)$ is equivalent to $(p,2,-q-1)$.} \label{fig:twobridgeequiv}
\end{figure}
\begin{figure}
\centering
\input{p2qknot.pstex_t}
\caption{Closed $3$--braid representatives of the two bridge
link $(p,2,q)$.} \label{fig:twobridgethreebraid}
\end{figure}
\section{Tunnel number one, genus one fibered knots in lens spaces}\label{sec:5}
The monodromy of tunnel number one once-punctured torus bundle has a
special form, which can now be connected with lens spaces via
two bridge links.
\begin{lemma}\label{lem:braidform}
The closure $\hat{\beta}_{k,n}$ of the braid $\beta_{k,n} =
(\sigma_2 \sigma_1 \sigma_2)^k \sigma_1^n$, where $k$ is odd, is a
two bridge link if and only if $k=\pm1$, $(k,n) = \pm(-3,3)$, or $(k,n)=\pm(-3,5)$.
\end{lemma}
\begin{proof}
If $k = \pm1$ then
\[\beta_{\pm1,n} = (\sigma_2 \sigma_1 \sigma_2)^{\pm1} \sigma_1^n = (\sigma_1 \sigma_2 \sigma_1)^{\pm1} \sigma_1^n \equiv \sigma_2^{\pm1} \sigma_1^{n\pm2}.\]
Thus $\hat{\beta}_{\pm1,n}$ is the two bridge link $\B(n\pm2, 1)$
(the $(2,n\pm2)$--torus link). The reader may check that $\hat{\beta}_{k,n}$ is the two bridge knot $\B(\mp5,1)$ for $(k,n) = \pm(-3,3)$ and the two bridge knot $\B(\pm7,2)$ for $(k,n)=\pm(-3,5)$.
If $n$ is even then $\hat{\beta}_{k,n}$ has one unknotted component
and one potentially knotted component (a $(2,k)$--torus knot).
Since a two bridge link has either one component or two unknotted
components, the $(2,k)$--torus knot must be the unknot. Hence $k =
\pm 1$.
Assume that $n$ is an odd integer. By taking mirror images, we
only need consider the case that $k \equiv 1 \mod 4$. We will
deal with the three cases $n=1, n=-1,$ and $|n|>2$ separately.
Since $k = 4 \ell + 1$ for some $\ell \in \Z$ we have $\beta_{k,n} =
(\sigma_2 \sigma_1 \sigma_2)^{4\ell} \beta_{1,n}$. By
Remark~\ref{rem:dbcsurgery}, the double branched cover of
$\hat{\beta}_{k,n}$ may be obtained from the double branched cover
$L(n+2,1)$ of $\hat{\beta}_{1,n}$ by $(1/\ell)$--Dehn surgery on the
lift of its braid axis.
If $n=-1$, then the braid axis of $\hat{\beta}_{1,n}$ lifts to the
right handed trefoil. A $(1/\ell)$--Dehn surgery on this knot this
is a lens space only when $\ell=0$ \cite{moser:esaatk}; hence $k=1$.
To handle the cases when $n=+1$ and $|n|>2$, we must further
understand the relationship between the order of the lens space and
the braid structure of $\hat{\beta}_{k,n}$.
The exponent sum of a braid is the sum of the exponents of a word in the letters $\{\sigma_1, \sigma_2\}$ representing the braid.
By Lemma~\ref{lem:braidconj}, $\hat{\beta}_{k,n}$ is a two bridge
link if and only if it is conjugate to $\sigma_2^{-1} \sigma_1^p
\sigma_2^2 \sigma_1^q$. Since an exponent sum is invariant under conjugation, we must have:
\begin{equation}\label{eqn:1}
3k+n = p+q+1.
\end{equation}
Since the double branched cover of $\hat{\beta}_{4\ell+1,n}$ is obtained from
$L(n+2,1)$ by $(1/\ell)$--Dehn surgery on the lift of the braid axis, its first homology group must be cyclic of order $n+2$.
By Lemma~\ref{lem:3braidlensspace}, the double branched cover of
closure of the braid $\sigma_2^{-1} \sigma_1^p \sigma_2^2
\sigma_1^q$ is the lens space $L(2pq+p+q,2q+1)$. Thus the orders of
first homology of the double branched cover of each
$\hat{\beta}_{4\ell+1,n}$ and the closure of $\sigma_2^{-1}
\sigma_1^p \sigma_2^2 \sigma_1^q$ must agree:
\begin{equation}\label{eqn:2}
|n+2| = |2pq+p+q|.
\end{equation}
Since $n$ is assumed to be odd, Equation~(\ref{eqn:2}) implies that
the integers $p$ and $q$ have different parity.
Putting Equations~(\ref{eqn:1}) and (\ref{eqn:2}) together, we
obtain $|p+q+3-3k| = |2pq+p+q|$. Hence either
\begin{equation}\label{eqn:3}
3k-3 = 2(pq+p+q) \mbox{ or } 3k-3 = -2pq.
\end{equation}
Now consider the case when $n=+1$.
For $\hat{\beta}_{4\ell+1,1}$ to be a two bridge knot,
Equation~(\ref{eqn:2}) implies that $\{p,q\}=\{0,3\}, \{0,-3\},
\{-1, 2\}$, or $\{-1, -4\}$. Since $n=+1$, Equation~(\ref{eqn:1})
then implies that $k = 1, -1, 1/3$, or $-5/3$ respectively. Because
$k$ must be an integer, $k = \pm1$.
Lastly consider the case when $|n| > 2$. Denote the lift of the
braid axis of $\hat{\beta}_{1,n}$ by $K_{1,n}$. The action of the
monodromy on the homology of the fiber of $K_{1,n}$ is given by
\[ \left(\begin{array}{cc} 1 & 0 \\ -1 & 1\end{array} \right)
\left(\begin{array}{cc} 1 & n+2 \\ 0 & 1\end{array} \right) =
\left(\begin{array}{cc} 1 & n+2 \\ -1 & -n-1\end{array} \right). \]
This has trace $-n$. By \cite{cb:aosanat} this monodromy is
pseudo-Anosov for $|n|>2$ and therefore by \cite{Th2} the
complement of $K_{1,n}$ is hyperbolic. According to the Cyclic
Surgery Theorem \cite{cgls:dsok}, $1/\ell$ surgery on $K_{1,n}$ is a
lens space only if $\ell = 0$, $\ell = +1$, or $\ell = -1$. Thus
only for $k=1$, $k=5$ or $k=-3$ respectively could
$\hat{\beta}_{k,n}$ with $|n|
> 2$ be a two bridge link. We complete the proof with an examination of the latter two possibilities.
{\bf Case A: $k=5$}
By Equation~(\ref{eqn:3}) either $-6 = pq$ or $6 = pq+p+q$. Thus
$-6=pq$ or $7=(p+1)(q+1)$. Since $p$ and $q$ have opposite parity,
$-6=pq$, and so $\{p, q\} = \{-2, 3\}, \{2, -3\}, \{1, -6\},$ or $\{-1,
6\}$. Hence by Equation~(\ref{eqn:1}) $n=-13, -15, -19,$ or $-9$ respectively. {\tt Braid Group Calculator} \cite{bgc} shows that none of the braids $\beta_{5,n}$ are conjugate to $\sigma_2^{-1} \sigma_1^{p} \sigma_2^2 \sigma_1^{q}$ for corresponding choices of $n$ and $\{p,q\}$. By Lemma~\ref{lem:braidconj} this implies
the braid closure $\hat{\beta}_{5, n}$ with $|n|>2$ is not a two
bridge link.
{\bf Case B: $k=-3$}
By Equation~(\ref{eqn:3}) either $6=pq$ or $-6=pq+p+q$. Thus
$6=pq$ or $-5=(p+1)(q+1)$. Since $p$ and $q$ have opposite parity,
$6=pq$, and so $\{p, q\} = \{2,3\}, \{-2, -3\}, \{1, 6\}$, or $\{-1,
-6\}$. Hence by Equation~(\ref{eqn:1}) $n= 15, 5, 17$, or $3$ respectively. {\tt Braid Group Calculator} \cite{bgc} shows that the braids $\beta_{-3,n}$ are not conjugate to $\sigma_2^{-1} \sigma_1^{p} \sigma_2^2 \sigma_1^{q}$ for $(n,\{p,q\}) = (15,\{2,3\})$ and $(n,\{p,q\}) = (17, \{1,6\})$. However these braids are conjugate if $(n,\{p,q\}) = (5,\{-2,-3\})$ or $(n,\{p,q\}) = (3, \{-1,-6\})$.
By Lemma~\ref{lem:braidconj} this implies the braid closure $\hat{\beta}_{-3, n}$ with $|n|>2$ is not a two bridge link unless $n = 3$ or $n = 5$.
\end{proof}
\begin{lemma}\label{lem:exception}
The braid $\beta_{-3,5}$ is not conjugate to a braid $\beta_{\epsilon,n}$ for any choice of $\epsilon = +1$ or $-1$ and integer $n$; neither is $\beta_{3,-5}$.
\end{lemma}
\begin{proof}
The proof of Lemma~\ref{lem:braidform} shows that $\beta_{-3,5}$ is conjugate to $\sigma_2^{-1} \sigma_1^{-2} \sigma_2^2
\sigma_1^{-3}$. By Lemma~\ref{lem:3braidlensspace}, the double branched cover of its closure is the lens space $L(7,2)$. For $\epsilon = \pm1$ the double branched cover of the closure of $\beta_{\epsilon,n}$ is the lens space $L(n+2\epsilon,1)$. If $\beta_{-3,5}$ were conjugate to $\beta_{\epsilon,n}$ then the double branched covers of their closures would be equal, but this is not the case. Similarly, $\beta_{3,-5}$ is not conjugate to $\beta_{\epsilon,n}$ for any choice of $\epsilon$ and $n$.
\end{proof}
\begin{remark}\label{rem:conjbraids}
As one may check, the braids $\beta_{-3,3}$ and $\beta_{-1,-3}$ are conjugate as are $\beta_{3,-3}$ and $\beta_{1,3}$.
\end{remark}
\section{Proof of Theorem~\ref{thm:main}}\label{sec:6}
Using our understanding of the braid structure associated to the
knot, we can now determine the precise lens space in which the knot
lies, as well as describe the structure of the knot.
\begin{thmmain}
Every tunnel number one, genus one fibered knot in a lens space is
the plumbing of a $r$--Hopf band and a $(\pm 1)$--Hopf band in the
lens space $L(r,1)$ with one exception. Up to mirroring, this exception is the genus one fibered knot in $L(7,2)$ that arise as $(-1)$--Dehn surgery on the plumbing of a $7$--Hopf band and a $(+1)$--Hopf band.
\end{thmmain}
\begin{proof}
Let $K$ be a tunnel number one, genus one fibered knot in a lens space. By Lemma~\ref{lem:corresp} $K$ corresponds to the braid axis of the closure of a $3$--braid $\beta$. As $K$ lies in a lens space, $\hat{\beta}$ must be a $2$--bridge link \cite{hodgson-rubinstein:iaiols}. Since $K$ is tunnel number one, Lemma~\ref{lem:tnobraid}, Lemma~\ref{lem:braidform}, and Remark~\ref{rem:conjbraids} together imply that $\beta$ must be conjugate to $\beta_{k,n}$ for either $k = \pm1$ and $n \in \Z$ or $(k,n) = \pm(-3,5)$.
Let us first assume $\beta = \beta_{\pm1, n}$. Then $\beta$ is conjugate to $\sigma_2^{\pm 1} \sigma_1^{n \pm 2}$. Setting $r = n \pm 2$, the closure $\hat{\beta}$ is then the $(2, r)$--torus link and its double branched cover is the lens space $L(r,1)$.
We may now view $K$ as the plumbing of an $r$--Hopf band and a
$(\pm1)$--Hopf band as follows. First observe that $\beta$ is a
Markov stabilization of the $2$--braid $\sigma_1^r$: a third
string and a single crossing ($\sigma_2^{\pm 1}$) is added to
$\sigma_1^r$ to form the $3$--braid $\beta$ preserving its
closure.
As in Proposition 12.3 of \cite{bz:knots}, the lens space $L(r,1)$ can be decomposed along a torus arising as the double branched cover of a bridge sphere $S^2$ for the closure $\widehat{\sigma_1^r}$. Isotope the braid axis $h$ of $\sigma_1^r$ to lie on $S^2$, and let $D$ be a component of $S^2 - h$.
The link $\widehat{\sigma_1^r}$ punctures $D$ twice, and thus in the double branched covering, $D$ lifts to an annulus $A$ bounded by the preimage of $h$. By Lemma~11.8 of \cite{bz:knots}, the braid $\sigma_1^r$ corresponds to $r$
Dehn twists about the core of $A$. In other words, the preimage of $h$ lifts to an $r$--Hopf link.
As demonstrated in Theorem 5.3.1 of \cite{montesinosmorton:flfcb}, the Markov stabilization $\beta$ of $\sigma_1^r$ corresponds to the plumbing of a Hopf band onto $A$ in the double branched cover. Thus we see $K$ as the boundary of the plumbing of an $r$--Hopf band and a $(\pm1)$--Hopf band in the lens space $L(r,1)$.
Now assume $\beta$ is conjugate to $\beta_{-3,5}$. Since $\hat{\beta}_{-3,5} = \B(7,2)$, $K$ is a knot in $L(7,2)$.
By Lemma~\ref{lem:exception}, $\beta$ is not conjugate to $\beta_{\pm1,n}$. Therefore $\beta$ is not a Markov stabilization of any $2$--braid, and so $K$ is not the boundary of the plumbing of an $r$--Hopf band and a $(\pm1)$--Hopf band. (Indeed, $K$ is not the Murasugi sum of any two fibered links.) Nevertheless, observe that adding two full positive twists to $\beta$ gives the braid $(\sigma_2 \sigma_1 \sigma_2)^4 \beta_{-3,5} = \beta_{1,5}$. Remark~\ref{rem:dbcsurgery} thus implies that $K$ is the core of a $(-1)$--Dehn surgery on the lift of the braid axis of $\hat{\beta}_{1,5}$ to the double branched cover, i.e.\ the boundary of the plumbing of a $7$--Hopf band and a $(+1)$--Hopf band.
\end{proof}
\section{Punctured torus bundles as knot complements.}\label{sec:7}
\begin{main2}
Every tunnel number one once-punctured torus bundle is the complement of
a GOF-knot in a lens space of the form $L(r,1)$.
\end{main2}
\begin{proof}
Let $M$ be a once-punctured torus bundle with tunnel number one. Filling
$\bdry M$ along a slope that intersects each fiber once produces a
closed $3$--manifold $\widehat{M}$ in which the core of the filling
solid torus is a tunnel number one GOF-knot $K$. By
Lemma~\ref{lem:tnobraid}, $K$ is the lift of the braid axis of the
closure of the braid $(\sigma_2 \sigma_1 \sigma_2)^k \sigma_1^n$
where $k=4\ell \pm1$ is odd. Following Remark~\ref{rem:dbcsurgery}, we
may perform $-1/\ell$ surgery on $K$ to produce a manifold
$\widehat{M}'$ in which the core of the surgery solid torus is a
tunnel number one GOF-knot $K'$. Hence $K'$ is the lift of the
braid axis of the closure of the braid $(\sigma_2 \sigma_1
\sigma_2)^{\pm1} \sigma_1^n$. This closed braid is equivalent to
the two bridge link $\B(r,1)$ where $r=n\pm2$. Therefore
$\widehat{M}'$, the double cover of this link, is the lens space
$L(r,1)$, and $M$ is the complement of $K'$.
\end{proof}
\begin{main3}
Every tunnel number one once-punctured torus bundle is the $(r/1)$--Dehn filling of a boundary component of the exterior of the Whitehead link for some integer $r$.
\end{main3}
\begin{proof}
By Theorem~\ref{thm:main2}, a tunnel number one once-punctured torus bundle is the complement of a GOF-knot $K$ in a lens space $L(r,1)$. By Theorem~\ref{thm:main}, $K$ is the plumbing of a $r$--Hopf band and a $(\pm 1)$--Hopf band.
The $r$--Hopf band is obtained by $r/1$ surgery on a circle $C$ that links an annulus $A$ whose boundary is the unlink in $S^3$. Plumbing a $(\pm1)$--Hopf band onto $A$ produces the unknot $U$ whose union with $C$ forms the Whitehead link. The $r/1$ surgery on $C$ transforms $U$ into the plumbing of a $(\pm1)$--Hopf band onto the $r$--Hopf band in the lens space $L(r,1)$.
Thus the image in $L(r,1)$ of $U$ is our knot $K$.
Therefore the complement of $K$ in $L(r,1)$ is obtained from the complement of the Whitehead link $U \cup C$ by the filling of the boundary component coming from $C$.
\end{proof}
\bibliographystyle{abbrv} | 137,824 |
TITLE: Status of particles in interacting QFT
QUESTION [4 upvotes]: From my readings in QFT and answers such as this, I've read that the concept of particles and particle-number in interacting systems becomes ill-defined in QFT.
Of course, in the real world, a number of experiments allow me to observe a countably finite number of particles interacting, with that finite number being well-defined throughout the entire experiment. If I take a laboratory measurement that allows me to observe individual atoms (e.g., for concreteness's sake, single-molecule AFM), I'm interacting with a finite number of particles through different fields (in this case, the EM field) without the number of particles ever being fuzzy
So, when people say that particle number is not well defined for interacting fields, is their claim just that current QFT formalism cannot recover/ calculate a finite interacting particles in the way that classical mechanics and "ordinary" QM can? This seems hardly satisfying.
Or do they really mean to say that an observation of single-atoms such as this are not observations of countably many atoms at all?
REPLY [4 votes]: If you have a free particle theory then the number of particles is well behaved because they are just the Fock states.
The trouble is that when you turn on interactions between the particles then the states of the interacting field are not the Fock states i.e. they are not eigenstates of the particle number operator. In fact we don't know what the states of the interacting field are.
But even for an interacting field, when the particles are far apart they are effectively non-interacting so once again we have states that are to a good approximation Fock states.
So if you're considering a typical scattering calculation then initially when the particles are far apart we have well defined states with a well defined number of particles. And after the scattering event when the particles head back out to infinity we also have well defined states with a well defined number of particles. The problem is that when the particles are near to each other and interacting strongly with each other. That's when the number of particles isn't well defined. | 153,736 |
We’re launching a new and ongoing blog series, “Making Waves”, in which our Publisher Performance Manager, Callum Ridley, will highlight innovative and exciting publishers that are making waves in the affiliate industry.
We’re kicking off with a spotlight on Love the Sales...
Launched in 2013, LovetheSales.com is an innovative publisher in the affiliate space which has grown to become one of the leading destinations for shopping the sales online. As an aggregator, they use machine-learning to scrape products from 100’s of retailers worldwide and sort them into categories that you can shop with ease. The main focus is on sales, however, it’s also possible to ‘favourite’ full price items and get alerted as soon as they go on sale.
In 2017, Love The Sales had over 1.7 million shoppers on the site - in 2018 they were on track to see over 3 million. The Love The Sales team have received many awards and nominations since launching, with one of the most recent wins being named Tech Platform of the Year (2018), in addition to being ranked 17 in Startups 100.
Love The Sales also offer retailers the opportunity to run a sale, without having to show the sale on their own site. This works well for high-end retailers that don’t want to show last season stock on site, yet want to sell the stock at a reduced price. This can also work well for preview sales, before the full sale goes live on the retailer's website.
We are always on the lookout for new and exciting publishers that are going to add value and help achieve goals and objectives for our client's affiliate programmes. So when we were introduced to Love The Sales, we knew they would make a great addition to our publisher portfolio.
We started working with Love The Sales in 2016 and have seen continual growth in performance since. Through our partnership, Love The Sales have been able to add over 500,000 additional products to the site, increasing the product range offered to Love The Sales users, which is crucial for a large aggregator.
Over the Black Friday period, we saw exceptional performance increases for our clients, leading to over 9% of all conversions through Love The Sales coming from Visualsoft. This high level of performance continued with many clients going into Winter Sale, ahead of the Christmas period.
Stuart McClure, Founder and CMO of Love The Sales said:
"We've achieved some phenomenal growth since working with Visualsoft. Their retailer network is really impressive and they have given us access to some of the best up and coming retailers in the UK.
As an aggregator of discounts, LovetheSales.com needs to have a really wide breadth of retailers and brands, working with the Visualsoft team to onboard their retailers has given us a much wider catalogue of products to promote.
One of the benefits of working with the team is their knowledge of the industry. Because the Visualsoft business is not just an affiliate platform, they have a much wider understanding of every facet of retail."
Love The Sales work on a performance model (CPA) through the affiliate channel, with the retailers awarding commission for each approved conversion. The level of commission will be decided between both parties, before integration. Love The Sales will utilise the retailers feed, in addition to them scraping the retailer's site, to ensure product information is up to date.
Love The Sales also offer additional exposure on a tenancy basis, this is to help increase performance and brand presence. Retailers can take advantage of; homepage exposure, product & sale boosting, newsletters, promo bar and inclusion in the navigation.
As a team, we are really excited for 2019, to see further performance increases through our partnership with Love The Sales. Love The Sales have big plans to grow their user base - for example, a tube advertisement campaign which went live this month. We strongly advise that Q1 2019 would be the perfect time to start working with them if you’re not already!
If you don’t have an Affiliate Programme but are interested in working with Love The Sales and other publishers like them, get in touch! We would be more than happy to discuss Affiliate Management in more detail, simply contact our team direct on 01642 988 416 or via our contact form.
-
Alternatively, if you’re a publisher and would like to know more about Visualsoft Affiliates and our portfolio of clients, contact our Publisher Performance Manager, Callum Ridley, via [email protected] or [email protected]. | 97,516 |
Promoting heart health and fighting against CVD
at the Women's Euro 2017
Home > Partnering with UEFA > UEFA A Healthy Heart Your Goal
UEFA, the World Heart Federation, the Dutch Heart Foundation and the Royal Netherlands Football Association (KNVB) came together at the UEFA Women’s Euro 2017 to promote heart health and fight against cardiovascular disease, the number one killer in the world.
Our programme theme is ‘A healthy heart your goal‘, focusing on encouraging children, women and all fans to lead an active, healthy lifestyle and take up sports such as football to help keep their heart healthy.
On the occasion of the UEFA women’s EURO, we encouraged women, kids and all the football fans to embrace a healthy, active lifestyle – because we know it makes a difference to our hearts and our health.
You can do something now to keep your and your friends’ and loved ones’ hearts healthy!
Visit the #MatchFitWoman challenge page
We hosted a 28 day fitness challenge starting on 19th June to encourage women to be more active. The challenge is over, but you can still access all the inspirational “challenge of the day” and tips on the challenge Facebook page
Download the Active Match app
Fans and staff traveling to UEFA Women’s EURO 2017 fixtures in the Netherlands were encouraged to walk and cycle to host stadia in a bid to boost heart health, burn calories and protect the environment. This free mobile app developed by the Healthy Stadia Network provides users with directions and mapping to host venues plus information on where bicycle locking facilities are located at each stadium. The tournament is over, but you can still download the app and get inspired to walk or cycle when attending sport events!
Download the app:
Tell all your friends and contacts
Post a message on your social network and encourage your friends to take steps to protect their heart (see sample social media messages below).
Access the #MatchFitWoman challenge terms and conditions here
Submit your pictures and videos to celebrate your #MatchFitWoman achievements and favorite activities!
Discover the #MatchFitWoman challenge – get active 30 minutes a day, 5 days a week
Care for your heart? Love football? Get your heart pumping with the #MatchFitWoman challenge bit.ly/2qW7mGs
We can all be #MatchFitWoman ! Our goal: be active 30 mins a day. For daily tips: bit.ly/2snWxRE
31% of women we surveyed said they are too embarrassed to exercise, 42% are too busy and 36% find it too expensive. But 74% would like to be more active! What is holding you back?
Your #MatchFitWoman Challenge of the Day: never stay seated for longer than one hour – move regularly, even if it’s just to go get a glass of water!
You can find more pictures and resources to share with your messages in the section below
Click to access more #MatchFitWoman sample messages
Access all the “challenge of the day”: goo.gl/6SnTju
The #MatchFitWoman video
The campaign key image
The Active Match app
The #MatchFitWoman challenge key image
The #MatchFitWoman infographic
The movie introducing the key image
More images from the campaign:
The Women’s EURO 2017 featured ‘A healthy heart your goal’ activities organized at both the national and European levels, with the objective of encouraging women, kids and all the football fans to embrace a healthy, active lifestyle.
The programme included: activities in local schools and in fan zones; the #MatchFitWoman online fitness challenge; campaigns and activities during the competition; healthy assessments of stadia; an app to encourage fans to walk or bike to stadia for the Euro (download on Google Play or App Store); movies, PR activities and much more.
View healthy tips on the #MatchFitWoman challenge facebook page
Learn more about the Speel Je Fit/Play Fitter activities in the Netherlands and view the short movies presenting exercises (in Dutch)
Protect your heart and the heart of your loved ones. Learn more about how you can get involved in the campaign by clicking above.
2 August 2017: UEFA Women’s EURO fans get active to make a healthy heart their goal
Download the full press release
14 June 2017: Women across Europe unaware of heart health risk
Speel Je Fit launch
6 June 2017: Dutch Heart Foundation and KNVB to get children to exercise more
Download the full press release (in English)
Active travel app launch
30 June 2017: Fans get moving at UEFA Women’s EURO 2017 with the Active Match App
Download the press release in English in Dutch
Articles on UEFA website
New active app at Women’s EURO
Healthy hearts the goal at Women’s EURO
Contact: World Heart Federation team at Barley communications
Catherine Jordan Jones
+44 (0) 7917 664648
[email protected]
Sam Williams
+44 (0) 7949 607029
[email protected] took place from 16 July to 6 August 2017 in the Netherlands. It was won by the host country, the Netherlands.
A Healthy Heart Your Goal: Join the #MatchFitWoman challenge ahead of the UEFA Women’s EURO 2017
News
Make a healthy heart your goal campaign video - 2013
Videos and Webinars
Children in the City campaign promotes physical activity for children in Romania
News-type
A healthy heart your goal
Toolkits
Infographic on physical activity
Infographics
#MatchFitWoman Infographic | 60,219 |
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THE FILM.
A madman without mercy, serial killer Kyung-chul (Choi Min-sik) has been stalking rural Korea, preying upon vulnerable young women, taking great delight in the torture and murder of his victims. As the fiancé of Kyung-chul's recent target, secret agent Soo-hyun (Lee Byung-hun) is emotionally destroyed, vowing to bring down the mass murderer. Instead of simply pursuing revenge, Soo-hyun turns the tables on Kyung-chul, implanting a tracking device inside the killer, setting up a surveillance scenario where the force of justice can easily thwart the force of evil's wicked plans, elongating the vengeful suffering. However, Kyung-chul is not easily disturbed, with his vile mind taking pleasure in the contortion of Soo-hyun's eroding moral code.
At 140 minutes in length, "I Saw the Devil" angles for an epic wipe of ugly behavior, examining the futility of revenge by digging around every last moment of despair, cleverly toying with the conventions of the serial killer genre. Director Kim Ji-woon ("The Good, The Bad, The Weird") establishes a wicked atmosphere of punishment early on, forgoing a routine of suspense to jump to the brutal game as quickly as possible, creating a blood-drenched chess board for the cop and killer to do battle upon. At least for the first half of the feature, there's a fiery pace here that keeps the film alert and devious, anxious to spring the next trap on viewers to summon disgust, bridging the divide between the two forces of fury.
"I Saw the Devil" is a visceral viewing experience, though one of style and some patience. The director keeps the story bubbling along through procedural efforts from Soo-hyun, who tracks his prey by listening to his every word, following the killer not only to potential victims, but also to another monster, and this one has developed a taste for human flesh. So yeah, this movie's not for kids. The violence is unflinching, with stabbings, beheadings, and rape competing for screentime, making Kyung-chul the ugliest character of 2011 -- a boastful ghoul who gamely defiles anything in his path, yet retains an unnerving clarity of madness to help guide his delivery of pain. "I Saw the Devil" is a repulsive feature, not always easy to watch, but the gruesomeness holds a thematic purpose, though I'll freely admit the film leans into a celebration of the macabre one too many times.
Rich with texture and concentration (flecked with a few bravura camera shots as well), the feature is ultimately too long, dragging out the cat-and-mouse game to a point of exhaustion. The climax is nearly rendered unusable due to its glacial pace, excessively stewing in the moment, looking to draw out frazzled mental states as long as it possibly can. The whole film feels like a rough cut in need of some slimming, with a few superfluous moments halting the flow and a mid-movie reveal concerning Kyung-chul's internal tracking device that's a little far-fetched, even for a gore zone movie like this.
THE BLU-RAY
Visual:
The AVC encoded image (1.78:1 aspect ratio) presentation is quite erratic, with quality ranging from scene to scene. On the plus side, clarity is quite strong when the transfer cooperates, allowing for plenty of gruesome detail and facial response, while dank set interiors are viewed in full. Colors are capable, satisfactorily separated, with blood red a primary element that never disappoints. When the transfer goes wrong, it looks terribly washed out, with troubling contrast issues and softness that pulls all texture out of the frame. Skintones are generally quite good, with virginal female faces retaining their ethereal porcelain quality. It's not consistent, but the BD provides most of the elements this film needs to penetrate the senses.
Audio:
The 5.1 DTS-HD sound mix run on the tinny side, detailing murderous happenings and verbal exchanges without a compelling heft. The suspense is left a little saggy by the thin voices, though scoring is a more consistent element, holding up well in the surrounds without stepping on the dialogue. Drippy, squishy sound effects are interesting, with atmospherics served well in the mix, capturing cavernous death chambers and crowd activity, bringing some life to the circular event. Low-end is sparse, lacking a needed punch to reinforce the violence. An English dub track is also included.
Subtitles:
English, English Narrative, English SDH, and Spanish subtitles are offered.
Extras:
"Deleted Scenes" (24:50) supply a few more moments of connective tissue to the story, fleshing out supporting characters and the opening of the picture. The major additions include a sex scene between Kyung-chul and the cannibal's mistress (her role is nearly deleted from the final film), and there's an alternate ending that keeps Soo-hyun conflicted about his future duty.
"Raw and Rough" (27:06) is an extremely sedate making-of journey, with cast and crew interviews intermixed with B-roll footage, capturing the production in motion. A bit more technical than the average BTS experience (the concentration here is on stunt work), the package is enlightening, just not enthusiastic.
A Theatrical Trailer has not been included.
FINAL THOUGHTS
Overflowing with stunning acts of depravity, churning pits of emotion, and a few traditional genre twists, "I Saw the Devil" casts quite a spell. Even if the whole endeavor limps to a close, there's enough vivid imagery and teary passion within to fuel several movies. I'm not suggesting the film is pleasurable to watch, but when it fires in full, it's a compelling, distressing ride. | 242,470 |
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Get email updates for differential in Virginia, United States | 300,440 |
Auckland Transport has definitely been improving some of their communications (not all) Late last year they released a new City Rail Link video which finally addressed many of the issues we had with how they were explaining the project in the past
And they did a good job with with the video for the new bus network.
The one downside with these two videos is that they do require someone to sit down and watch a few minutes of video – something not everyone will do, especially those with little interest in transport. So perhaps Auckland Transport could to do something similar to what has been done in Singapore to explain their new transport master plan and make a few quick and catchy images that explain what they are aiming to do. Note: they are also good for highlighting the diversity of the population in Singapore.
Of course they won’t get through to everyone – like this guy.
With AT reviewing their Integrated Transport Programme and it potentially being quite different to the current one then perhaps they could do something like this help explain it.
We know that Auckland’s transport plans are completely unaffordable, a more interesting question is “why?” Much of the answer to that questions comes from what I refer to as “overkill”. Essentially, a solution that’s vastly oversized compared to the problem it’s trying to solve. There are a large number of examples of “overkill” when it comes to transport projects currently being planned:
It seems like good transport planning should flush out what projects are overkill and what projects aren’t. An interesting comparison against the above projects is the process that the City Rail Link has gone through over the past few years – especially in the form of the City Centre Future Access Study, which looked in detail at a range of “smaller options” for resolving issues with access to the city centre – outlining which of these would be necessary anyway, which could occur prior to CRL being built but also the point at which the ‘small scale’ interventions need to become so significant you might as well do the job properly – in this case by building CRL.
Throughout the ITP there are a vast number of projects which are obviously “overkill”. Examples include $665m on Albany Highway (surely a typo?), around $800m on a section of Great South Road, a $150m motorway bypass of Kumeu, the $240m Mill Road corridor project and many others. Strip back these overkill projects so they really focus on the problems they’re designed to resolve and we’ve probably gone a long way towards solving our future funding shortfalls.
Before last week’s distractions, there was an NZ Herald article which discussed key issues with Len Brown over the coming three years. Perhaps the most interesting part of that article was the focus that Len put on looking again at the Council’s budget..
Somewhat strangely, he then goes off to discuss how we apparently need the completely idiotic $5 billion Additional Harbour Crossing project, but let’s leave that issue aside for just a minute or two.
During the first term as Mayor there was obviously a lot of pressure to ensure that rates didn’t increase very much – particularly as combining the rates system was always going to mean serious winners and losers (funny how we only ever hear about the losers). Yet there continued to be some big spending items: fixing up the electric trains deal after the government got rid of the regional fuel tax, getting started on AMETI and many other non-transport projects. It seems like the “out” in the equation for the first three years was for the Council to increase its debt levels – taking advantage I suppose of the greater financial muscle the new combined Council now has.
If there is an increased desire to limit the growth of Council’s debt during the next three years, some pretty tough calls will need to be made around what projects happen and what projects don’t (since it’s capital expenditure rather than operating expenditure that is funded from debt). While that applies to all types of Council spending, as we know transport is by far the largest area of Council capital expenditure.
This is where the Congestion Free Network comes in. We first created this network because we were horrified by how expensive the transport plan outlined in the Integrated Transport Programme is and by how badly it missed achieving the goals and targets set by the Auckland Plan. Remember that for $68 billion in the current plan you get:
More Congestion:
A heap more transport greenhouse gas emissions (rather than the sought decline):
Basically no increase in the proportion of people catching public transport into the City Centre:
Nowhere near the PT, walking and cycling modeshare target set in the Auckland Plan:
I could go on but this is getting boring. The basic story the Integrated Transport Programme tells is that if you spend a lot of money on stupid projects, you don’t actually achieve what you want to achieve.
As we explained recently, the real beauty of the Congestion Free Network is that because you’ve provided a complete rapid transit network that is free of congestion, you can stop wasting money on unnecessary roading projects which aim for (but of course never deliver thanks to induced demand) reductions in congestion. This means you can build the Congestion Free Network and still save at least $10 billion from what’s in our current 30 year transport plans.
$10 billion of course sounds like a LOT of money. And it is. Surely ripping $10 billion out of a transport budget – even over 20 years – is going to hurt due to what can no longer be done? Well only if your transport budget isn’t full of excessive projects to begin with. The table below outlines our initial thoughts about how you can reduce the spend on new roading projects over the next 20 years from $21.6 billion to $7.0 billion:
To show the savings a bit clearer it’s quite useful to look at them in pie chart form:
As you can see the biggest chunk of the savings just comes from not doing a project that’s not only a waste of money but will actually make things worse. Then we shave money off big slush funds highlighted for motorways to sprawl that won’t be required (unless someone truly is planning to build the Karaka-Weymouth bridge) or widening arterial roads which don’t need to be widened (they probably just need a bus lane). Puhoi-Wellsford gets chopped back to Operation Lifesaver, AMETI’s cost gets fixed so it doesn’t double-count the East-West Link, we scrub out the unnecessary six laning of the Northern Motorway between Albany and Orewa (why would you six lane when you’ve just built a busway?) and many other projects which are generally not completely stupid ideas but where the solution proposed is vastly bigger than the problem that exists (e.g. East West Link). I suspect the only people who really want everything on that list built are the NZCID and their members who stand to benefit hugely from it all.
The scary thing about ripping nearly $15 billion out of the roading budget was just how easy it was. We barely felt mean at all with many of the cutbacks. We’re still proposing Penlink to happen some time before 2030, we still propose to spend money on widening the southern motorway south of Manukau, we’re still spending $1.3 billion on AMETI, we’re still completing the Western Ring Route, we’re still grade separating the Kirkbride Road/SH20A intersection, we’re still spending $800 million on upgrading arterial roads, $700 million on new arterial roads in greenfield areas, $350 million for rail grade separations – the list goes on.
Auckland Transport is supposed to be preparing a second version of the Integrated Transport Programme in the upcoming months – including consideration of the Congestion Free Network as one of the scenarios looked at in this process. We await with great interest to see whether the Mayor’s enhanced focus on the budget extends to chopping out some of the extraordinary waste of money proposed in the first version of the ITP.:
The main areas where we’ve saved money come from a variety of locations, where we really think that we’re cutting out the poor value aspects of many projects yet still doing what makes sense: was a great article yesterday by Brian Rudman in the NZ Herald – and not just because it seems that he’s a fan of the Congestion Free Network. He starts off by arguing that the year of work and the $1 million+ spent on the Auckland transport funding Consensus Building Group has actually achieved relatively little:.
Personally I think that it hasn’t been a complete waste of time. It got a lot of important different groups around a table to discuss transport issues in Auckland and I’m sure that influenced associations like the Chamber of Commerce and the NZ Council for Infrastructure Development to place pressure on the government to change its opinion of the City Rail Link. That alone was worth the exercise in some respects. Furthermore, it also probably led to one of the more intelligent discussions about road pricing and its benefits that’s been had in Auckland – with the majority of people providing feedback supporting some sort of road pricing scheme ahead of increased fuel taxes and rates.
Yet, of course, the elephant in the room issue was that members of the Consensus Building Group weren’t able to question the merits and make-up of the transport package that they set out to.
In a nutshell, the argument seems to be that if we spend $12 billion more on transport over the next 30 years compared to what we can afford, things will still be much worse than today – but not quite as bad as if we didn’t spend that money.
If we look at AM peak congestion levels, the ITP seems like it’s trying to argue that we should spend $12 billion to reduce the percentage of excessive congested travel from 30% to about 27%. That’s four billion dollars for each percentage point of lower congestion!
As we have discussed many times in the last few weeks, it is this crazy situation that led us to create the Congestion Free Network. The ITP clearly tells us that we can’t build our way out of congestion – so the answer is to ensure that people have a choice of avoiding congested travel.
After describing the Congestion Free Network concept in excellent detail, Rudman concludes by noting the following:
It’s the sort of radical thinking that Mayor Len Brown and his transport planners should be engaged in. Producing a plan that isn’t programmed to fail.
We’re ready to help!
I think I speak for everyone on the blog by saying that the response to the Congestion Free Network has been beyond our wildest dreams. On Tuesday there were over 11,000 views of the blog – a new record eclipsing the day the government finally decided they could no longer oppose the City Rail Link. The main Congestion Free Network post has been viewed over 10,000 times (beyond people looking at it on the main page) already and we’ve had thousands upon thousands of visits via Facebook in particular pointing towards that post.
Radio NZ did a story on the Congestion Free Network (CFN) on Morning Report yesterday morning, including a couple of soundbites from me talking about the network.
Even Kiwiblog has picked up on the CFN, highlighting that while David Farrar doesn’t agree with every bit of the network, he appreciates the effort we’ve put into the CFN. In particular, he highlights a potential next step:
It would be good for an appropriate agency to independently cost their proposals, and estimate what impact on congestion their proposals would have.
Ultimately what pushed us to work with Generation Zero and create the Congestion Free Network was our despair at the Integrated Transport Programme, its gigantic cost and its utterly terrible outcomes. Remember, $60 billion over the next 30 years for this result:
When we dug into the modelling results of the ITP in more detail (aside from making us more sceptical about modelling than ever before), the key cause of this congestion in 2041 seemed to be that the public transport system was still comparatively rubbish:
The critical issue behind the results above seem to be that for most trips public transport is getting stuck in the same congestion as everyone else, plus the added time of walking to the bus/train/ferry, waiting for it and then stopping to pick up other passengers all the time. Hence the need for a congestion free alternative: to make public transport the fastest option – good enough to really encourage people out of their cars and by consequence ease pressure on the roading network.
What we want is to have the Congestion Free Network included as an option when Auckland Transport do their next iteration of the Integrated Transport Programme. We think this fits pretty well with Auckland Transport’s approach to the next ITP – as highlighted in their presentation to the Transport Committee in April:As Matt’s post this morning about roading projects that’ll still happen highlighted, in many respects the Congestion Free Network isn’t radically different to what’s in current long term transport planning documents – it just shifts forward some projects that provide really good improvements to public transport at relatively low cost (especially busway and bus priority projects), suggests a lot less gold-plating of many proposed motorway projects and suggests that we avoid projects which do more harm than good. Rail to the North Shore and Light-Rail along Dominion Road are really the only big projects in the Congestion Free Network which aren’t in the Auckland Plan and the ITP. And perhaps most importantly it shows how a vision of Auckland that is completely unlike any most of us have thought possible before, an Auckland with a vital piece of the jigsaw that we know makes other great cities great and enables them to function well, is completely within reach if only we work towards it clearly.
So we’re going to take the Congestion Free Network to the Council’s Transport Committee and to the AT Board over the next few months to really push for the CFN to be included as an option for testing in the next ITP. We’re also going to continue to promote the concept in innovative and exciting ways. And we’re always open to suggestions about how to refine the CFN further – we are certain that the idea is vital, we think the detail is pretty good, but we know it’s not perfect.
Over the next few weeks GenZed and ATB will try to get straight answers out of every candidate in the local elections on where they stand on the CFN and related transport, urban form, and environmental issues.
We will report back with some kind of list that should act as a voting guide for these issues.
Since the launch of our Congestion Free Network and how much it will cost we have had a few questions around what it means for the funding of improved roads. To try and clarify, we are not against funding going towards road improvements and our estimated budget for The Congestion Free Network is just over 40% of what is currently planned to be spent on transport over the same time frame. That means there is still scope for worthwhile roading projects to occur. In saying this government agreed too, are overkill and are akin to cracking a nut with a sledgehammer. So in this post I thought I would cover the projects we think should be considered to be built alongside the Congestion Free Network – Note: some of this is similar to what we discussed the other day following the government’s announcement.
There are a number of state highway projects either already under way, or being talked about that have useful components.
Waterview and WRR - This is obviously already under way and in my mind is the piece that finally “completes the motorway network”, despite the goal posts for that target being a consistently moving target. One of the massive benefits to this is that it provides an alternative for vehicles travelling north to bypass the central motorway junction and the Harbour Bridge. It is also for this reason that we need to hold off on any plans for another road based crossing of the harbour as the WRR really needs some time to settle in and have its impacts felt before moving ahead with the harbour crossing project. The causeway upgrade also provides for some improved bus lanes which will be useful as part of our Congestion Free Network.
Upper Harbour Upgrade SH1/SH18 interchange through to Greville Rd – Improving the SH1/SH18 junction does seem like a good idea but I’m not sure we need the full motorway to motorway ramps as proposed in the most recent study we have seen. There are also a few other small sections in the area that could be addressed, such as the small single lane section westbound between Paul Matthews Rd and Albany Highway. However general widening of the section between Constellation and Greville should really wait until after the busway has been completed through to Albany and we see the impacts of that investment.
Do we really need this motorway to motorway interchange?
SH1 Manukau to Papakura widening – Once again there are some aspects to this project that make a lot of sense, in particular upgrading the Takanini interchange (especially the northbound onramp) and addressing the number of lanes southbound between Manukau and Takanini. At this stage I’m not convinced that widening Northbound or the section between Takanini and Papakura is needed. Again it would also be worthwhile waiting to see what impacts the PT projects have first before embarking on wholesale widening.
SH20A upgrade – Once again there are some parts that seem to make sense, particularly around the grade separation of Kirkbride Rd however the ITP seems to suggest that the road would also be widened, something I don’t see as needed once that interchange is addressed. With the Kirkbride intersection sorted out, the Montgomerie Rd intersection could be cheaply closed. The other advantage to this project is it could be tied in with the proposal to extend the rail network to the airport.
There is one other State Highway project that I am aware of – that isn’t on the ITP list – which may have some value during this time and that is the additional northbound lane between Penrose and Greenlane. The Ellerslie station was narrowed last year to make way for this project. Other State Highway projects on the ITP, like the Additional Harbour Crossing and widening SH20 from Mangere to Puhinui should be at least moved back till after 2030.
There are some fairly sizeable local roading projects in the ITP, again some are worthwhile or at least have worthwhile components while others or specific parts of them make little sense.
AMETI – AMETI is one of those projects that improved with age, morphing from a version of the eastern highway proposal into a much more multi modal project. Crucially many of the roading elements of the project tend to focus on providing new road connections rather than just larger existing ones. They strongly focus on moving through traffic away from town centres, giving them a chance to develop more people friendly environments. The focus on new connections and bypassing town centres is a good thing and a strategy I will touch on more in later comments.
East-West Link – This is a project that we have looked at a recently. We definitely agree that some improvements are needed in the area, just not to the scale proposed. Instead smaller scale improvements should be considered first such as the ones below.
Do we really need a $600m mini motorway between Onehunga and East Tamaki?
Other Local Road Projects in the ITP – The ITP also contains other local road projects, such as widening Lake Rd on the North Shore. I’m sure that there are plenty of people in the area that would love to see it however in almost all instances I think we should avoid widening unless the new lanes are at least limited to T2/3 initially. This way there are still benefits that can be accrued from the works however not doing so in a way that undermines the bus network by encouraging more people to drive.
Other Local Road Projects outside the ITP - As mentioned earlier, focusing on providing new connections and bypassing town centres is a good strategy compared to just continually widening existing roads. As such I feel that there are a number of road projects not currently on the ITP should be considered instead of some of the overblown widening projects like what is being proposed for Mill Rd. We have highlighted some of these before including a long talked about bridge across the Whau River between the Rosebank Rd and Hepburn Rd which would provide a real alternative to either the Gt North Rd or Te Atatu Rd interchanges.
Or this route in Pakuranga between Hope Farm Ave and La Trobe St which would amongst other things, allow for a logical third main bus route from the east to pierce right through the middle of the wedge of housing in the area while also providing substantial benefits for drivers, cyclists and walkers.
Now you will notice that I haven’t set costs or timeframes for these projects like we have with The Congestion Free Network and that is for two reasons. The first is that we simply don’t know what some of our suggestions for scaling back projects would cost. As for the timeframes, we are happy for these road projects to be prioritised based on need to fit in with the funding available. There are also bound to be some other local road projects that we would potentially support so please don’t consider this an exhaustive list. Ultimately these projects need to go through a robust analysis, something they won’t all pass and as such may be dropped or scaled back further.
I guess the key point from this post is simply to point out that we are not opposed to roading projects that stack up but that the key is ensuring that the roads we do build are actually what we need.:
Last week’s transport announcements were in many ways a triumph for Auckland Council and Auckland Transport, as they finally got the government to buy into the transport direction that is set by the Auckland Plan and then given more detail by Auckland Transport’s Integrated Transport Programme (ITP). The ITP is a 30 year transport strategy for Auckland which gives effect to the transport chapter of the Auckland Plan, fills in the details of projects to be built over the next 30 years and then analyses the performance of the transport network over that time frame. Key elements of the ITP reflect the big ticket transport items in the Auckland Plan that were supported by the government last Friday: the City Rail Link, AMETI and the East-West Link and the Additional Harbour Crossing project.
So far this sounds like a great story: Auckland’s come up with a long-term transport plan involving a number of very expensive projects and the government has broadly agreed with that plan. Unfortunately, the ITP is complete rubbish – full of stupid projects which simply don’t make sense and lacking a true vision for Auckland’s transport future. And somewhat surprisingly, even with the eye-watering price tag of $60+ billion the transport network’s performance gets considerably worse over the next 30 years.
Congestion gets worse:Greenhouse Gas emissions get worse (the numbers on the x axis refer to land-use scenarios, with 3 indicating a medium growth scenario):There’s little increase from the current 50% of vehicular trips to the city centre in the AM peak being on public transport:We don’t get anywhere near the target for non-car modeshare during the AM peak period across the city:The prime reason for the failure to meet so many of these targets is that the public transport system remains a relatively poor choice for most trips compared to driving. Therefore people still drive, the roads still get clogged, the greenhouse gas emissions [and all the other dis-benefits of auto-depenancy] continue to increase, there’s little change in modeshare and so forth. Perhaps the situation is best summed up by the graph below – which highlights the continued relative unattractiveness of the PT system in the future:According to the graph above, pretty much no PT trips are less than half an hour long in the future – a pretty terrible outcome. Furthermore, it seems that hardly any employment is located within a 60 minute trip on public transport – another pretty terrible (although less plausible) outcome. Whether this reflects some serious errors in the transport modelling process or whether this is a true reflection of our future I’m not sure, but these results indicate that in 2040 we’re going to be in much the same situation we are now under the ITP: a relatively crap public transport system meaning that for most trips we’re still going to be car dependent. And no wonder, when you look at the funding in the ITP most of it goes to roads.
So much for a transformational shift.
I’m not sure whether the government’s advisors have highlighted the flaws in what they’ve just bought into. Perhaps the government felt that after giving the thumbs up to the City Rail Link it also needed to balance that with support for an eye-watering number of motorway projects, even though the big picture of what all this spending leads to doesn’t really make sense.
Over the next few weeks we’re going to discuss in detail an alternative to the government’s transport package and the Integrated Transport Programme. A far better alternative that’s around the same price but will deliver far superior outcomes. An alternative that will truly give Auckland an alternative to the congested roading network, an alternative to rising greenhouse gas emissions, an alternative that will make achieving the modeshare shift targets far more possible and which will finally deliver upon the promise to make public transport the ‘mode of choice’ for longer trips.
We call it the Congestion Free Network and we look forward to sharing it with you. | 328,164 |
Kansas May Be on Hook for More Costs in Voting Rights Case
WICHITA, Kan. (AP) — Kansas Secretary of State Kris Kobach's compliance with the latest ruling. This is separate from the more than $50,000 the American Civil Liberties Union seeks in attorney fees and other damages as punishment for Kobach violating an earlier order to fully register some voters. The judge has not yet ruled on the amount for contempt.
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Kansas Year-End Tax Receipts Exceed Expectations by $318 Million.
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Even in GOP Bastion Kansas, 2 Congressional Seats in Play
TOPEKA, Kan. (AP) — Competitive races for two of Kansas's four U.S. House seats are making Republicans sweat to keep their all-GOP state delegation, a twist in a state where President Donald Trump won by nearly 21 points and a leading candidate for governor is gun-rights and immigration hardliner Kris Kobach. In one case, the Republican incumbent who faced a tighter-than-expected race two years ago faces a field of Democrats energized by dislike of Trump on issues including immigration, health care and the environment. In the other, potential big-name candidates opted not to run for the open seat, leaving a Democrat with the best name recognition. Republicans say they can feel their opponents' energy and have been issuing warnings to their conservative base for months. "Both of those races are ones that we have known we have to be diligent in and work hard," state GOP Chairman Kelly Arnold said. "Who has the motivation to come out? The party that's not in power usually picks up seats and some wins. That's what we're fighting against." To boost Democrats' chances in both districts, the House Majority super PAC announced plans to reserve $900,000 in television ad time in the Topeka and Kansas City markets in the weeks before the election, far more attention than Kansas Democrats running for Congress have received in recent elections. The conservative Congressional Leadership Fund then promised nearly $3 million worth of ad time. Democrats need to pick up 23 seats nationally to flip the majority in the House. Incumbent Republican Kevin Yoder was destined to be a midterm target after Trump narrowly lost his Kansas City-area 3rd District and its urban neighborhoods and comfortable-to-posh suburbs. Yoder himself fared worse than expected. Democrats sensed Trump might be a liability and both sides poured money into the race at the last minute in 2016, giving Yoder an 11-point margin against an unknown Democrat — less than half his previous average. But Democrats' chances could be better in the neighboring 2nd District, which covers most of eastern Kansas from the Nebraska border in the north to Oklahoma in the south. Incumbent Republican Lynn Jenkins opted not to seek re-election. Democrats have their ideal candidate in former state legislative leader and governor candidate Paul Davis. The Democratic Congressional Campaign Committee included Davis on its first list of 11 candidates in promising races for its "Red to Blue" program. Seven lesser-known Republicans are vying for their party's nomination. "Republicans, generically, have the wind in their face," said Patrick Miller, a political scientist at the University of Kansas. "If this were 2010 or 2014, we wouldn't even be talking about the 2nd District." Davis carried Jenkins's district during his narrow statewide loss to Sam Brownback in the 2014 race for governor. Republicans who might have had equally strong support — Attorney General Derek Schmidt and State Treasurer Jake LaTurner, for example — opted out, citing family or professional reasons. The GOP field has four state lawmakers, an ex-Kansas House speaker, a new-to-politics military veteran and a small-town city councilman-nurse-criminologist. The GOP candidates have scrapped for attention and more than half of their dollars through March came from their own pockets. Davis has raised more than $1 million. "He has name ID," Arnold said. "We don't know where we're at with our candidate because we don't know who it is yet." Yoder's aides and supporters say he's working hard to sidestep any potential Democratic wave. He began April with nearly $2 million in campaign cash. Nevertheless, six Democrats want a crack at Yoder, including labor lawyer Brent Welder, who drew the endorsement of U.S. Sen. Bernie Sanders and is listed among the group of "Justice Democrats" in the mold of 28-year-old Alexandria Ocasio-Cortez, who defeated incumbent Rep. Joseph Crowley in the Democratic congressional primary in New York. Yoder also picked up two opponents for the Aug. 7 GOP primary. One, Trevor Keegan, an information technology consultant, describing himself as a moderate alternative. The heavy lift for the GOP in the congressional races comes even as the most talked-about candidate for governor is conservative Kobach, and state lawmakers have shown no sign of softening on issues that animate the base like immigration and guns. Wednesday's announcement that Supreme Court Justice Anthony Kennedy is retiring is expected to energize GOP voters across the country, which could ease the Republicans' path in Kansas. And Republican congressional candidates are not breaking with the president, arguing that voters like his handling of North Korea and Iran and the income tax cuts he pushed. Trump carried Jenkins' district by nearly 17 points two years ago, and Republicans expect a revived national economy to help. Davis avoided bashing Trump in a recent interview, saying that Washington is dysfunctional and both parties share the blame. "Donald Trump is president, and if I'm elected, I'm going to do my best to work with him when he's doing good things," Davis said. "I'm going to call him out when he's doing harmful things."
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Disturbance Reported at Troubled El Dorado Prison
EL DORADO, Kan. (AP) — Inmates broke dozens of windows and set fires during a weekend disturbance at a maximum-security prison in south-central Kansas that has been the scene of other incidents, officials said Monday. Nobody was hurt in the fracas that started around noon Sunday in the recreation yard at the prison in El Dorado and lasted about 4 ½ hours, said Kansas Department of Corrections spokesman Samir Arif. Between 75 and 150 inmates were involved. State union president Sarah LaFrenz says a corrections officer who witnessed what happened told her around 40 windows were broken. She said two classrooms had fire and smoke damage, and one other building was set on fire. The officer also told LaFrenz that inmates got access to radio communication devices from the classrooms and an office. There was "much more damage" than what occurred at the facility during a disturbance there last year, she said. The county sheriff's office and Highway Patrol established a perimeter, and prison officials were able to establish control inside the prison quicker than in the past, she said. "This was a very bad situation and we were very lucky that there weren't any injuries," LaFrenz said. "It was very significant and very frightening and incredibly dangerous for everyone involved." Arif confirmed there were broken windows, as well as smoke and fire damage, but provided no specifics because the investigation was ongoing. Prison officials are still investigating what set off the disturbance, but LaFrenz said it apparently began after an inmate on the yard who had refused to comply with directions was restrained and moved. The other inmates on the yard then started the disturbance. Several disturbances also were reported last year at the prison in El Dorado, which is 33 miles east of Wichita, including one that led to a five-day lockdown.
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Kansas Residents Still Concerned About Issues Raised by Proposed Poultry Plant
TONGANOXIE, Kan. (AP) — A city in eastern Kansas is still seeing political changes after plans for a massive chicken processing plant caused uproar last year. The Kansas News Service reports that the $300 million Tyson project planned for the outskirts of Tonganoxie was canceled after residents protested the pollution and the strains on local infrastructure they expected the facility would bring. Republican Rep. Jim Karleskint of Tonganoxie drafted a bill that would have given locals more say about proposed poultry operation. He says residents are still anxious about the issue. Democrat Stuart Sweeney says Tyson is one of the reasons he decided to run for the Statehouse. Stuart is challenging Republican Representative Willie Dove. Dove says he initially didn't take a stance on the project because he wanted more information. He ultimately opposed it.
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Judge Sets September Trial for Online Gamers in Kansas Case
WICHITA, Kan. (AP) — A federal judge has set a fall trial date for two online gamers whose alleged dispute over video game bet ultimately led police to fatally shoot a Kansas man while responding to a hoax call. U.S. District Judge Eric Melgren on Monday scheduled a Sept. 4 jury trial for 18-year-old Casey Viner of North College Hill, Ohio, and 19-year-old Shane Gaskill of Wichita. They are charged with conspiracy to obstruct justice, wire fraud and other counts. Prosecutors allege Viner became upset while playing the Call of Duty WWII video game and asked 25-year-old Tyler Barriss of Los Angeles to "swat" Gaskill, the practice of making a false report to get emergency responders to descend on an address. A police officer fatally shot 28-year-old Andrew Finch after he opened the door.
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Lawrence Officials to Consider Marijuana Law Changes
LAWRENCE, Kan. (AP) —.
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Ruling Revives Kansas Woman's Prayer Lawsuit
WICHITA, Kan. (AP) — The lawsuit filed by a Kansas woman who claims police ordered her to stop praying in her home has been revived after the U.S. Supreme Court last week summarily reversed a lower court ruling that had thrown it out. Mary Anne Sause sued several Louisburg, Kansas police officers and city officials in 2015 alleging her civil rights were violated while police were investigating a noise complaint two years earlier. The lawsuit alleges an officer told her she was going to jail, and when she knelt down to pray another officer told her to stop praying. First Liberty Institute, a nonprofit which advocates for religious liberty, says in a news release it had asked the high court to reverse an appellate court's ruling that the officers were entitled to qualified immunity.
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Man Accused of Dismembering Wife Found Competent for Trial
OLATHE, Kan. (AP) — A man arrested at a suburban Kansas City storage unit with two of his children and his dismembered wife's remains has been found competent to stand trial after undergoing a mental health evaluation. The ruling was made Monday during a hearing for Justin Rey in Johnson County, Kansas, where he's charged with child endangerment. He isn't charged in his wife's killing. Rey says she died after giving birth in October in a Kansas City, Missouri, hotel room. Investigators say he then went to the Lenexa, Kansas, storage unit where he was arrested with the newborn and the couple's toddler. He was preparing to catch a train. He's also charged in the death of a California man whose body hasn't been found. Rey told The Associated Press he didn't kill anyone.
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Wichita Police Investigating 3 Fatal Weekend Shootings
WICHITA, Kan. (AP) — Wichita police are investigating three fatal shootings that occurred within a couple of hours of each other. The Wichita Eagle reports that all three shootings occurred between 12:30 a.m. and 2 a.m. Saturday. In the first, 24-year-old Anthony Martinez was in a backyard when a cousin's gun discharged, fatally striking Martinez in the abdomen. The cousin was booked on suspicion of involuntary manslaughter. The other shootings occurred around 2 a.m. Officers found 29-year-old Michael Maxwell dead inside his vehicle. Police believe someone inside an SUV fired the fatal shots in a "targeted shooting." No arrests have been made. Police were called to an apartment and found 23-year-old Patrick Ball-Morse with a fatal gunshot wound to the head. Authorities are searching for the brother of Ball-Morse's girlfriend.
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Student Who Changed His Failing Grades to A-s Gets Probation
LAWRENCE, Kan. (AP) — A former University of Kansas student who hacked the school's computer system to changing his failing F grades to As has been sentenced to probation. The Lawrence Journal-World reports that.
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Report: Most of Kansas Winter Wheat Now Harvested
WICHITA, Kan. (AP) — The latest government snapshot shows the Kansas growers have harvested most of their winter wheat crops. The National Agricultural Statistics Service reported Monday that the Kansas wheat harvest was 71 percent complete. That is ahead of the five-year average. About 97 percent of the state's wheat left in the field is now mature. The agency also says about 15 percent of the corn in Kansas is in poor to very poor shape with 32 percent rated as fair, 47 percent as good and 7 percent as excellent. Other Kansas crops are also making progress. About 97 percent of the soybeans planted have now emerged and about 20 percent of them are blooming. About 5 percent of the sorghum in the state has now headed.
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Car Slams into Home After Kansas City Police Chase
KANSAS CITY, Mo. (AP) —.
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Ex-Kansas Teacher Gets Probation in Student Sex Case
OLATHE, Kan. (AP) — A former teacher at Gardner Edgerton High School has received a suspended prison sentence and three years' probation for having sexual contact with a student. The Kansas City Star reports that 45-year-old Todd Burd was ordered Friday to serve 30 days in jail as a condition of his probation. He was also required to register as a sex offender. Burd pleaded guilty in April to unlawful sexual relations. He had initially been charged with aggravated criminal sodomy after a 16-year-old boy reported being assaulted while sitting in Burd's pickup truck. Burd taught music and was the choir director at Gardner Edgerton High School in late 2016, when the incident happened. Burd received the Gardner-Edgerton district's Teacher of the Year honor in 2015.
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Immigration Rally Held in Dodge City.
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Police: 3 Dead in 3 Separate Overnight Shootings in Wichita
WICHITA, Kan. (AP) — Police in the southern Kansas city of Wichita are investigating the deaths of three people in separate shootings overnight. Police said Saturday that the first happened around 12:30 am, when officers were called to hospital for a man suffering gunshot wounds to his arm and abdomen. Police say the man was at his house when his cousin showed him a gun. Police say the gun fired, hitting the victim, who later died at the hospital. The man's cousin is currently in police custody. At 2 am, police say a drive-by shooting outside a house left a 29-year-old man with multiple gunshot wounds. The victim later died at a hospital. Also at 2 am, police responded to a shooting at an apartment during a fight among several people. A 23-year-old man was pronounced dead at the scene. No arrests had been announced by late Saturday morning in the 2 am shootings.
====================
Report: Many Child Trafficking Victims End Up Detained
WICHITA, Kan. (AP) — A newspaper investigation has found that one in five victims of possible child trafficking in Kansas are placed in juvenile detention, a process that experts say hampers the effort to heal them. The Wichita Eagle obtained data from the Kansas Department for Children and Families outlining a system that often can further traumatize victims. Experts say placing victims in detention, whatever the reason, can lead to future criminal behavior and add to the risk that they will be victims of trafficking again. Child victims of trafficking are placed in detention for various reasons. Police may suspect the child of committing a crime. He or she could be a runaway. They could be in detention because of a court order.
=====================
Kansas Mayor's Use of 'Tar Baby' Term Creating Outcry
SALINA, Kan. (AP) — Salina Mayor Karl Ryan's use of the phrase "tar baby" during a city commission meeting is creating an outcry in his central Kansas community. The Salina Journal reports the furor stems from a June 25 meeting during which Ryan said, "Can I just be a tar baby for a minute," during discussion about authorizing the city manager to exceed his spending authority to pay for care for seized animals. Ryan, who is white, says his mother would call him a "tar baby" when he being difficult. He says he first heard the phrase in Joel Chandler Harris's "Uncle Remus" tales of Br'er Rabbit and Tar Baby. But others say the term has taken a derogatory meaning and say Ryan is racially clueless. Several plan to voice their displeasure at Monday's meeting.
====================
Amtrak Explores Ending Passenger Service from Dodge City
HUTCHINSON, Kan. Senator Jerry Moran says he's disappointed in "the lack of commitment on the part of Amtrak to keep its word" on upgrading the Southwest Chief route. The nearly 2,300-mile passenger train service has run daily between Chicago and Los Angeles since 1974.
====================
Man Sentenced to Life in Woman's Shooting Death in Emporia
EMPORIA, Kan. (AP) — A man has been sentenced to life in prison in the fatal shooting of a woman in her apartment near the Emporia State University campus. The Emporia Gazette reports that Sony Uk, of Emporia, won't be eligible for parole for 50 years under the sentence ordered Monday for first-degree murder. Judge Merlin Wheeler said there was "absolutely no reason or justification" for the March 2017 death of 38-year-old Mahogany Brooks. Uk declined to speak at the hearing. During his trial, the defense argued that the killing wasn't premeditated, which is required for a first-degree murder conviction. But the prosecution argued it was, saying Uk arrived at Brooks's apartment with a loaded shotgun and pulled the trigger multiple times.
====================
Economic Growth Slows in Midwest States as Trade Fears Mount.
=====================
2 Dead, 4 Hurt in 3-Vehicle Crash in South Central Kansas
HAYSVILLE, Kan. (AP) — Authorities say two women have been killed and four other people injured in a three-vehicle crash in south-central Kansas. KSNW-TV reports that the Sedgwick County Sheriff's Office identified the two women as 42-year-old Michelle Lacey and 23-year-old Kayla Heim. Both are from Wichita. The crash happened around 10 p.m. Saturday in Haysville when the Nissan Murano in which the women were riding went through a stop sign and struck a Jeep. The Jeep then rolled over onto another SUV. The women were pronounced dead at the scene. Three others in the Murano also were injured — a 23-year-old man, a 5-year-old and a 16-month-old. The Jeep's driver sustained minor injuries, while the driver of the other SUV wasn't hurt.
====================
University of Kansas Museum Marks Panorama's Anniversary
LAWRENCE, Kan. (AP) — The University of Kansas Natural History Museum is celebrating the 125th anniversary of a panorama with special exhibits and programs. The Lawrence Journal-World reports that the wildlife panorama includes a painted backdrop and 121 taxidermy animals native to North America. Louis Lindsay Dyche unveiled the exhibit in 1893 at the Chicago World's Fair. Leonard Krishtalka is director of the university's Biodiversity Institute. He says the diorama gave visitors an immersive experience of America's wilderness and highlighted the importance of preservation and stewardship. The museum's new exhibit includes artifacts and photographs from Dyche's career. The museum also plans panorama-themed events. Krishtalka says the museum hopes to restore the panorama to its original condition. A 2014 assessment estimates it would cost $500,000 to repair deterioration caused by changing temperatures, light and humidity.
====================
Water.
===================== | 120,812 |
TITLE: $*$-homomorphism of $C^*$-algebras and representations
QUESTION [1 upvotes]: For $i=1,2$, let $\mathcal{A}_i$ be an abstract $C^*$-algebra and $\pi_i : \mathcal{A}_i \rightarrow \mathcal{B}(\mathcal{H}_i)$ a $C^*$-representation. Let $\alpha: \mathcal{A}_1 \rightarrow \mathcal{A}_2$ a *-homomorphism.
I'm looking for a theorem that may assert that there exists a unique $*$-homomorphism $\tilde{\alpha} : \pi_1 (\mathcal{A}_1)'' \rightarrow \pi_2 (\mathcal{A}_2)''$ of von Neumann algebras such $ \tilde{\alpha}(\pi_1(a))= \pi_2(\alpha(a))$ for all $a \in \mathcal{A}_1$. ¿In which topology $\tilde{\alpha}$ is continuous?
Please, I need references about such a theorem.
Thanks,
D
REPLY [1 votes]: Such extension does not usually exist. For instance take $\mathcal A_1=\mathcal A_2=UHF(2^\infty)$. Let $\pi_1:\mathcal A_1\to \mathcal B(\mathcal H_1)$ be a Powers representation such that $\pi_1(\mathcal A_1)''$ is a type III factor $M$. And let $\pi_2:\mathcal A_2\to \mathcal B(\mathcal H_2)$ be GNS from the trace, so $\pi_2(\mathcal A_2)''$ is the hyperfinite II$_1$-factor $R$.
Take $\alpha(x)=x$, the identity isomorphism.
Now you want a $*$-homomorphism $\bar\alpha: M\to R$. We may assume $\bar\alpha(1)=1$ (otherwise, replace $R$ with $\bar\alpha(1) R\bar\alpha(1)$, which is still a hyperfinite II$_1$). Given any projection $p\in M$, we have $p\simeq 1$, so $\bar\alpha(p)\simeq 1$; as this last equivalence happens in a II$_1$-factor, we have $\bar\alpha(p)=1$ for all projections $p\in M$. But then $1=\bar\alpha(1)=\bar\alpha(p+1-p)=\bar\alpha(p)+\bar\alpha(1-p)=2$. So $\bar\alpha$ does not exist. | 134,213 |
Good afternoon everyone and welcome to another octane fuelled National 3 Southwest clash as league leaders Barnstaple Chiefs welcome 4th place Hornets. We extend a warm welcome to their players, coaches, committee members and supporters, we hope you have an enjoyable day here at Pottington Road. Apologies for the confusion about kick off times but today's game will start at 2.30pm and not 2pm as advertised on the website. All the home games in December, January and February will, however, be 2pm starts due to the poor light.
We would also like to thank our sponsors for today's game SWM and Barricane Solutions for your continuing support which is hugely appreciated by the club and fans alike. We hope you enjoy a classic game today.
Well it had to happened sometime and The Chiefs five game winning streak was finally ended at Bournemouth last weekend. By half time Bournemouth were leading 11-0 but a second half try by James Bath converted by Luke Berry brought Barnstaple right back in it and with a bit of luck they might have won this fixture, but the bounce of the ball went Bournemouths way and two more tries from them sealed a 23-7 win. With 10 first team players out injured it was a good performance from the Chiefs unfortunately picking up more injuries including prop Kevin Angell. On a more positive note Ryan Carter started for the first time since a hernia operation and had a good game which will only improve with more game time.
This now leaves The Chiefs only two points clear of second place Brixham with Exmouth making it a Devon sweep of the top three places in the league. With today's opponents only four points behind there is absolutely no let up this year as only seven points separate the top 7 teams.
The Hornets come to Pottington Road today looking to reverse their fortunes from last seasons game where Barnstaple were comfortable winners. They do this with the confidence of a three game winning streak and the chance to overtake us if they can get a result here today. We know how good they can be and will need to be on top of our game today.
A much depleted injury hit Development XV went to Plymouth last week and playing in atrocious conditions were unable to overcome either a thunderous hail storm or Devonport losing the game 15-0. They look to reverse their fortunes next at home to Okehampton on December 5th.
Weather permitting i'm looking forward to another great game of rugby today. Good luck to both teams and i will catch with you all at the bar afterwards.
Welcome all,
Great to be back home in front of the Pottington Road faithful, and looking forward to hearing you in full voice this afternoon.
Bournemouth away was a bit of a mixed bag. We had good field position and possession but couldn’t find a way to put points on the board. Just seemed to be one of those days where a few too many players didn’t play particularly well…an opportunity missed perhaps.
Today will be a tight affair I’m sure, so it will come down to who does the simple things well and is clinical in the red zone.
I’m looking for an improved performance on last week, especially our backline. We have huge proven talent throughout the side, we just have to believe in our ability.
Thanks for your continued support
The U8s (otherwise known as the big red machine) have 23 players registered and more on the waiting list. Coaches Jonny Bowden, Jeremy Chugg (club legend), Joe Caley, Jon Limer and Gareth Hookway train the kids every Sunday. The team were unbeaten last season at the Under 7s Devon tournament at Ivybridge and came away with a trophy from the Westcoast Festival of Rugby. Progress thsi season is continuing with players making great strides in their development. The players are going on tour to the Leicester Tigers Rugby Festival in May. This week the team have found out that they have been successful in winning the AEG grassroots rugby competition which will provide the team with a full set of Canterbury kit, a top of the range washing machine for the club and gives them the opportunity to win a training session with Will Greenwood. Quita Chugg (Team Co-ordinator) said "The team and club will benefit greatly by this win. The players are great friends and despite their young age are very committed to their rugby. This was clear to the judges who chose us out of hundreds of entries." We would like to wish them all the best for the rest of the season.
The warmth of the clubhouse seemed much more inviting than the howling gale outside but, as kick off approached, the diehard fans reluctantly made their way to the main stand and wrapped themselves up inside ready for this National SW 3 encounter.
The rain came in sheets across the pitch, the wind whipped in from the North West and two brave rugby sides prepared to do battle. Hornets had the storm at their backs and soon pinned Barum in their 22. Kicks were sliced into touch, the ball was frequently dropped, line outs went askew but all thirty players did their best to construct a game of rugby with all elements set against them.
The pattern was set for virtually the whole forty minutes: Hornets working their phases, Barum defending like demons. Whenever possession was turned Pez looked to spin through to his backs but the conditions were just too tricky for handling and Hornets swooped to stifle every counter. Eventually the Somerset side worked a very good opening left and put the ball through hands for a touchdown out wide, converted expertly on the wind: 0-7.
With conditions deteriorating throughout the match it took until the 38th minute for Barum to get out of their own half. To the amused and ironic cheers from the stand, a penalty was kicked deep into the Hornets half. It was a courageous show of defiance from a home side decimated by injuries and missing eight regular first team members.
As the second half began the wind subsided slightly and Hornets were able to run from deep and counter more effectively than their hosts had been able. The young side from Weston showed an impressive range of skills in the mud but were unable to stem the power of a Barum maul following a series of penalties in the clubhouse corner. Gradually, as the rain reduced, a trickle of spectators had emerged at the rail to see the red shirts drive over. Those in the distance had no need of Mike Canon’s confirmation as to who was under the heap with the ball. Yes, surprise, surprise: captain Winston James, 7-7.
With one half of the excellent pitch now somewhat churned and the other half still pristine the game continued but neither side could make any headway. Barum’s scrum went backwards a few times but they were unable to sustain any continuity with the ball in hand. As play was frequently punctuated with handling errors the light faded and the match petered out in a sequence of scrums and knock-ons.
Credit to both sides for competing and working so hard in desperate circumstances. A talented Hornets team probably shaded the possession and stats honours but a draw was a fair result. Barum now need to consolidate league position and get some first team squad members back from the physio’s bench. | 180,979 |
\begin{document}
\title{\textbf{Stable Exact Solutions in Cosmological Models with Two Scalar Fields}}
\author{I.~Ya.~Aref'eva$^1$\footnote{[email protected]}, \
N.~V.~Bulatov$^2$\footnote{nick\[email protected]}, \
S.~Yu.~Vernov$^3$\footnote{[email protected]} \\[2.7mm]
${}^1$\small{Steklov Mathematical Institute, Russian Academy of Sciences,}\\
\small{Gubkina str. 8, 119991, Moscow, Russia}\\
${}^2$\small{Department of Quantum Statistics and Field Theory, Faculty
of Physics,}\\ \small{Moscow State University, Leninskie
Gory 1, 119991, Moscow, Russia}\\
${}^3$\small{Skobeltsyn Institute of Nuclear Physics, Moscow State University},\\
\small{Leninskie Gory 1, 119991, Moscow, Russia}\\
}
\date{ }
\maketitle
\begin{abstract}
The stability of isotropic cosmological solutions for two-field models in
the Bianchi I metric is considered. We prove that the sufficient
conditions for the Lyapunov stability in the
Friedmann--Robertson--Walker metric provide the stability with respect
to anisotropic perturbations in the Bianchi I metric and with respect
to the cold dark matter energy density fluctuations. Sufficient
conditions for the Lyapunov stability of the isotropic fixed points of
the system of the Einstein equations have been found. We use the
superpotential method to construct stable kink-type solutions and
obtain sufficient conditions on the superpotential for the Lyapunov
stability of the corresponding exact solutions. We analyze the
stability of isotropic kink-type solutions for string field theory
inspired cosmological models.
\end{abstract}
\section{Introduction}
To specify different components of the cosmic fluid one
typically uses a phenomenological relation $p=w\varrho$ between the
pressure (Lagrangian density) $p$ and the energy density $\varrho$
corresponding to each component of the fluid. The function $w$ is
called the state parameter. Contemporary
observations~\cite{cosmo-obser} give strong support that in the
Universe, the uniformly distributed cosmic fluid with negative
pressure, the so-called dark energy, currently dominates with a state
parameter value approximately equal to $-1$:
\begin{equation*}
w_{DE} = -1 \pm 0.2.
\end{equation*}
Strong restrictions on the anisotropy were found using
observations~\cite{Barrow,Bernui:2005pz}, and it was also shown that
the Universe is spatially flat at large distances. It has been shown
in~\cite{9,Nesseris,ZhangGui} (see also
reviews~\cite{Quinmodrev1,Quinmodrev2} and references therein), that
the recent analysis of the observation data indicates that the time
dependent state parameter gives a better fit than $w_{DE}\equiv -1$,
corresponding to the cosmological constant. It, in particular, gives
reasons for the interest in models with $w_{DE} < -1$. The field theory
with $w_{DE} < -1$ is also interesting as a possible solution of the
cosmological singularity problem~\cite{Hawking-Ellis,cyclic,GV}.
The standard way to obtain an evolving state parameter is to add scalar
fields into a cosmological model. Two-field models with the crossing
of the cosmo\-logical constant barrier $w_{DE}=-1$ are known as quintom
models and include one phantom scalar field and one ordinary scalar
field. Quintom models are being actively studied at present
time~\cite{Quinmodrev1,Quinmodrev2,Guo2004,AKV2,Lazkoz,Leon,QuintomModels}.
The cosmological models with $w_{DE}<-1$ violate the null energy
condition (NEC), this violation is generally related to the phantom
fields appearing. The standard quantization of these models leads to
instability, which is physically unacceptable.
In~\cite{AKV2}, the theory with $w_{DE}<-1$ was interpreted as an
approximation in the framework of the fundamental theory. Because the
fundamental theory must be stable and must admit quantization, this
instability can be considered an artefact of the approximation.
We note the problems of quantum instability that arise in effective
theories and are related to high order derivatives~\cite{AV-NEC,RAS}.
In~\cite{SW,Creminelli-eff} the instability problem has been reduced to
the problem of such choose of the effective theory parameters that the
instability turns out to be essential only at times that are not
described in the framework of the effective theory approximation. In
the mathematical language it means that the terms with higher-order
derivatives can be treated as corrections essential only at small
energies below the physical cut-off. This approach implies the
possibility to construct a UV completion of the theory and to consider
these effective theories physically acceptable with the presumption
that an effective theory admits immersion into a fundamental theory.
In the Friedmann--Robertson--Walker (FRW) metric the NEC violating
models can have classically stable solutions cosmology. In particular,
there are classically stable solutions for ghost models with minimal
coupling to gravity. Moreover, there exists an attractor behavior in a
class of the phantom cosmological
models~\cite{phantom-attractor,AKVCDM,Lazkoz} (attractor solutions for
inhomogeneous cosmological models were considered
in~\cite{Starobinsky}).
The stability of isotropic solutions in the Bianchi
models~\cite{Bianchi,Ellis98,Ellis} (see also~\cite{DSC}) has been
considered in inflationary models (see~\cite{Germani,Koivisto} and
references therein for details of anisotropic slow-roll
inflation). Assuming that the energy conditions are satisfied, it
has been proved that all initially expanding Bianchi models except
type IX approach the de Sitter space-time~\cite{Wald} (see
also~\cite{MossSahni,Wainwright,Kitada,Rendall04}). The Wald
theorem~\cite{Wald} shows that for space-time of Bianchi types
I--VIII with a positive cosmological constant and matter
satisfying the dominant and strong energy conditions, solutions
which exist globally in the future have certain asymptotic
properties at $t\rightarrow\infty$.
The standard way to analyse the stability of solutions for Friedmann
equations in quintom models uses the change of
variables~\cite{Guo2004,Lazkoz,Leon} (see also~\cite{Quinmodrev2}). In
the case of exponential potentials such a transformation is useful,
because it transforms the class of nontrivial solutions to fixed points
of the new system of equations~\cite{Guo2004}. We show that the
stability conditions for an arbitrary potential that were found
in~\cite{Lazkoz} can be obtained without introducing any new variables.
Studying the stability of solutions in the FRW metric, we specify a
form of fluctuations. It is interesting to know whether these
(isotropic) solutions are stable under the deformation of the FRW
metric to an anisotropic one, for example, to the Bianchi I metric. In
comparison with general fluctuations we can get an explicit form of
fluctuations in the Bianchi I metric, which can probably clarify some
nontrivial issues of theories with the NEC violation.
In this paper we consider the stability of isotropic solutions in the
Bianchi I metric. Interpreting the solutions of the Friedmann equation
as isotropic solutions in the Bianchi I metric, we include anisotropic
perturbations in our consideration. The stability analysis is
essentially simplified by a suitable choice of
variables~\cite{Ellis98,DSC,Pereira}. In this paper we show that for an
arbitrary potential the stability conditions, obtained
in~\cite{Lazkoz}, are sufficient for stability not only in the FRW
metric, but also in the Bianchi I metric. We also analyse the stability
with respect to small fluctuations of the initial value of the cold
dark matter energy density. We consider quintom cosmological models, as
well as models with two scalar (or two phantom scalar) fields.
The stability of a continuous solution tending to a fixed point implies
the stability of this fixed point. Using the Lyapunov theorem \cite{Lyapunov,Pontryagin} we find
conditions under which the fixed point and the corresponding kink (or
lump) solution are stable. For one field models the sufficient
conditions for stability of isotropic solutions, which tend to fixed
points, have been obtained in~\cite{ABJV09}.
In~\cite{AKV2,Vernov06} the superpotential method has been used to
construct quintom models with exact solutions. In this paper we get the
stability conditions in terms of the superpotential and use the
superpotential for construction of two-field models with stable exact
solutions. We also verify the stability of solutions, obtained in the
string field theory (SFT) inspired quintom models~\cite{AKV2,Vernov06}.
The paper is organized as follows. In Section 2 we consider two-field
model with an arbitrary potential and we find the sufficient stability
conditions in the FRW and Bianchi I metrics. In Section 3 we remind the
superpotential method and obtain conditions on the superpotential which
are sufficient for the stability of exact solutions. In Section 4 we
check the stability of kink-type solutions in a few SFT inspired
cosmological models. In Section~5 we summarize the results and make a
conclusion. In Appendix we present the stability conditions on the
superpotential in the case of one-field cosmological models.
\section{Sufficient stability conditions}
\subsection{The Einstein equations in the Bianchi I metric}
We consider a two-field cosmological model with the following action
\begin{equation}
S=\int d^4x \sqrt{-g}\left[\frac{R}{16\pi G_N}-
\frac{C_1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-
\frac{C_2}{2}g^{\mu\nu}\partial_{\mu}\xi\partial_{\nu}\xi
-V(\phi,\xi)\right], \label{action_2}
\end{equation}
where the potential $V(\phi,\xi)$ is a twice continuously
differentiable function, which can include the cosmological constant
$\Lambda$, $G_N$ is the Newtonian gravitational constant ($8\pi
G_N=1/M_P^2$, where $M_P$ is the Planck mass). Each of the fields
$\phi$ and $\xi$ is either scalar or phantom scalar fields in
dependence on signs of the constants $C_1$ and $C_2$.
Let us consider the Bianchi I metric
\begin{equation} \label{Bianchi}
{ds}^{2}={}-{dt}^2+a_1^2(t)dx_1^2+a_2^2(t)dx_2^2+a_3^2(t)dx_3^2.
\end{equation}
It is convenient to express $a_i$ in terms of new variables $a$ and
$\beta_i$ (we use the notation in~\cite{Pereira}):
\begin{equation}
a_i(t)= a(t) e^{\beta_i(t)}.
\end{equation}
Imposing the constraint
\begin{equation}
\label{restr1} \beta_1+\beta_2+\beta_3=0,
\end{equation}
one has the following relations
\begin{equation}
a(t)=(a_1(t)a_2(t)a_3(t))^{1/3},
\end{equation}
\begin{equation}\label{Hi}
H_i\equiv \dot a_i/a_i= H+\dot\beta_i, \qquad\mbox{and}\qquad H\equiv
\dot a/a=\frac{1}{3}(H_1+H_2+H_3),
\end{equation}
where the dot denotes the time derivative. To obtain (\ref{Hi}) we have
used the following consequence of (\ref{restr1}):
\begin{equation}
\label{restr2} \dot\beta_1+\dot\beta_2+\dot\beta_3=0.
\end{equation}
Note that $\beta_i$ are not components of a vector and, therefore,
are not subjected to the Einstein summation rule. In the case of
the FRW metric all $\beta_i$ are equal to zero and $H$ is the
Hubble parameter. Following~\cite{Ellis98,Pereira} (see
also~\cite{DSC}) we introduce the shear
\begin{equation}
\sigma^2\equiv \dot\beta_1^2+\dot\beta_2^2+\dot\beta_3^2.
\end{equation}
The Einstein equations have the following form:
\begin{equation}
\label{a2} 3H^2-\frac{1}{2}\sigma^2=8\pi G_N\varrho,
\end{equation}
\begin{equation}
\label{trequ} 2\dot H+3H^2+\frac{1}{2}\sigma^2={}-8\pi G_Np,
\end{equation}
\begin{equation}
\label{d2} \dot\phi=\psi,\qquad
\dot{\psi}={}-3H\psi-\frac{1}{C_1}\frac{\partial V}{\partial\phi},
\end{equation}
\begin{equation}
\label{e2} \dot\xi=\zeta,\qquad
\dot{\zeta}={}-3H\zeta-\frac{1}{C_2}\frac{\partial
V}{\partial\xi},
\end{equation}
where
\begin{equation}
\label{varrho_pressure} \varrho=
\frac{C_1}{2}\dot{\phi}^2+\frac{C_2}{2}\dot{\xi}^2+V(\phi,\xi),\qquad
p=\frac{C_1}{2}\dot{\phi}^2+\frac{C_2}{2}\dot{\xi}^2-V(\phi,\xi).
\end{equation}
For $\beta_i$ and $\sigma^2$ we obtain the following equations
\begin{equation}
\ddot\beta_i={}-3H\dot \beta_i, \label{equbeta}
\end{equation}
\begin{equation}
\label{equvartheta}
\frac{d}{dt}\left(\sigma^2\right)={}-6H\sigma^2.
\end{equation}
Functions $H(t)$, $\sigma^2(t)$, $\phi(t)$, $\xi(t)$, and $\rho(t)$ can
be obtained from equations (\ref{a2})--(\ref{e2}) and
(\ref{equvartheta}). If $H(t)$ is known then $\beta_i$ can be trivially
obtained from (\ref{equbeta}). We show in the next subsection that
functions $H(t)$, $\dot\beta_i(t)$ and $\sigma^2(t)$ are very suitable
to analyse the stability of isotropic solutions in the Bianchi I
metric.
\subsection{Sufficient conditions for the Lyapunov stability of a fixed
point}
Summing equations (\ref{a2}) and (\ref{trequ}) we obtain the following
system
\begin{equation}
\label{SYSTEM2}
\begin{array}{l}
\displaystyle \dot H={}-3H^2+8\pi G_NV(\phi,\xi),\\
\displaystyle \dot\phi=\psi,\\
\displaystyle \dot{\psi}={}-3H\psi-\frac{1}{C_1}\frac{\partial V}{\partial\phi},\\
\displaystyle \dot\xi=\zeta,\\
\displaystyle \dot{\zeta}={}-3H\zeta-\frac{1}{C_2}\frac{\partial
V}{\partial\xi}.
\end{array}
\end{equation}
Let the fields $\phi$ and $\xi$ tend to finite limits as $t\rightarrow
+\infty$. System (\ref{SYSTEM2}) has a fixed point
$y_f=(H_f,\phi_f,\psi_f,\xi_f,\zeta_f)$ if and only if
\begin{eqnarray}
H_f^2&=&\displaystyle \frac{8\pi G_N}{3} V(\phi_{f},\xi_{f}),
\label{l}\\
\psi_{f}&=&0,\label{h}
\\
\zeta_{f}&=&0, \label{i}
\\
V'_{\phi}& \equiv&\frac{\partial V}{\partial\phi}(\phi_{f},\xi_{f})=0,
\label{j}
\\
V'_{\xi}&\equiv& \frac{\partial V}{\partial\xi}(\phi_{f},\xi_{f})= 0,
\label{k}
\end{eqnarray}
All fixed points $y_f$ correspond to $\psi_{f}=0$ and $\zeta_{f}=0$. We
denote the fixed point $y_f=(H_f,\phi_f,\psi_f,0,0)$ as
$y_f=(H_f,\phi_f,\psi_f)$. To analyse the stability of $y_f$ we study
the stability of this fixed point for the corresponding linearized
system of equations and use the Lyapunov
theorem~\cite{Lyapunov,Pontryagin}. In the neighborhood of $y_f$ we
have
\begin{eqnarray}
H(t)&=&H_f + \varepsilon h_1(t) +{\cal O}(\varepsilon^2),
\label{m}
\\
\phi(t)&=&\phi_{f}+\varepsilon \phi_1(t)+{\cal O}(\varepsilon^2),
\label{n}
\\
\psi(t)&=&\varepsilon \psi_1(t)+{\cal O}(\varepsilon^2), \label{o}
\\
\xi(t)&=&\xi_{f}+\varepsilon \xi_1(t)+{\cal O}(\varepsilon^2),
\label{p}
\\
\zeta(t)&=&\varepsilon \zeta_1(t)+{\cal O}(\varepsilon^2)
\label{q},
\end{eqnarray}
where $\varepsilon$ is a small parameter.
To first order in $\varepsilon$ we obtain the following system of
equations
\begin{eqnarray}
\displaystyle \dot h_1(t)&=&\displaystyle {}-6H_fh_1(t), \label{r}
\\
\displaystyle \dot\phi_1(t)&=&\displaystyle \psi_1(t), \label{s}
\\
\displaystyle \dot\psi_1(t)&=&\displaystyle {}-3H_f\psi_1(t)-
\frac{1}{C_1}\left(V''_{\phi\phi} \phi_1(t) +V''_{\phi\xi}
\xi_1(t)\right), \label{t}
\\
\displaystyle \dot\xi_1(t)&=&\displaystyle \zeta_1(t), \label{u}
\\
\displaystyle \dot\zeta_1(t)&=&\displaystyle {}-3H_f\zeta_1(t)-
\frac{1}{C_2}\left(V''_{\xi\phi} \phi_1(t) +V''_{\xi\xi}
\xi_1(t)\right), \label{v}
\end{eqnarray}
where
\begin{equation*}
V''_{\phi\phi}\equiv \frac{\partial ^2V}{\partial \phi ^2}
(\phi_{f},\xi_{f}),\qquad V''_{\xi\xi}\equiv \frac{\partial
^2V}{\partial \xi ^2} (\phi_{f},\xi_{f}),\qquad V''_{\phi\xi}\equiv
\frac{\partial ^2V}{\partial \phi \partial \xi} (\phi_{f},\xi_{f}).
\end{equation*}
Equation (\ref{r}) has the following solution
\begin{equation}
h(t)=b_0e^{-6H_ft}, \label{w}
\end{equation}
where $b_0$ is a constant. For asymptotic stability of the fixed point
$y_f$ the function $h(t)$ should tend to zero at $t\rightarrow\infty$,
therefore, the asymptotic stability requires that the condition $H_f>0$
be satisfied.
The system of four first order equations (\ref{s})--(\ref{v}) can be
written as the following system of two second order equations
\begin{eqnarray}
&\displaystyle \ddot{\phi}_1(t)+3H_f\dot{\phi}_1(t)+
\frac{1}{C_1}\left(V''_{\phi\phi}\phi_1(t) +V''_{\phi\xi}
\xi_1(t)\right)=0, \label{x}
\\
&\displaystyle \ddot{\xi}_1(t)+3H_f\dot{\xi}_1(t)+
\frac{1}{C_2}\left( V''_{\phi\xi}\phi_1(t) +V''_{\xi\xi}\xi_1(t)
\right)=0 \label{y}.
\end{eqnarray}
In the case when $V_{\phi\xi}''=0$ the system of equations
(\ref{x})--(\ref{y}) becomes a system of two independent second order
equations. The general solution of this system is as follows:
\begin{itemize}
\item $\phi_1(t)=\tilde{D}_1e^{-\left(\frac{3}{2}H_f -
\frac{1}{2}\sqrt{9H_f^2-4\frac{V_{\phi\phi}''}{C_1}}\right)t}+\tilde{D}_2e^{-\left(\frac{3}{2}H_f
+ \frac{1}{2}\sqrt{9H_f^2-4\frac{V_{\phi\phi}''}{C_1}}\right)t}$ \
if $9H_f^2\neq 4\frac{V_{\phi\phi}''}{C_1}$,
\item
$\phi_1(t)=\left(\tilde{D}_1+\tilde{D}_2t\right)e^{-\frac{3}{2}H_ft}$ \
if $9H_f^2= 4\frac{V_{\phi\phi}''}{C_1}$,
\end{itemize}
\begin{itemize}
\item $\xi_1(t)=\tilde{D}_3e^{-\left(\frac{3}{2}H_f -
\frac{1}{2}\sqrt{9H_f^2-4\frac{V_{\xi\xi}''}{C_2}}\right)t}+\tilde{D}_4e^{-\left(\frac{3}{2}H_f
+ \frac{1}{2}\sqrt{9H_f^2-4\frac{V_{\xi\xi}''}{C_2}}\right)t}$ \ if
$9H_f^2\neq 4\frac{V_{\xi\xi}''}{C_2}$,
\item
$\xi_1(t)=\left(\tilde{D}_3+\tilde{D}_4t\right)e^{-\frac{3}{2}H_ft}$ \ if
$9H_f^2= 4\frac{V_{\xi\xi}''}{C_2}$,
\end{itemize}
where $\tilde{D}_1$, $\tilde{D_2}$, $\tilde{D_3}$ and $\tilde{D_4}$ are
arbitrary constants.
For the asymptotical stability of the considered fixed point the
functions $\phi_1(t)$ and $\xi_1(t)$ must converge to $0$ at
$t\rightarrow\infty$. We obtain that at $V_{\phi\xi}''=0$ the
sufficient conditions for the asymptotical stability are
\begin{equation}
H_f>0,\qquad \frac{V_{\xi\xi}''}{C_2} >0, \qquad
\frac{V_{\phi\phi}''}{C_1}>0. \label{V2fCondVFKVFK}
\end{equation}
Let us consider the case $V_{\phi\xi}''\neq 0$.
\begin{enumerate}
\item For
\begin{equation*}
{V_{\phi\xi}''}^2\neq\frac{C_1C_2}{16}\left(9H_f^2-\frac{4V_{\xi\xi}''}{C_2}\right)
\left(9H_f^2-\frac{4V_{\phi\phi}''}{C_1}\right)\quad\mbox{and}\quad
{V_{\phi\xi}''}^2 \neq{} -
\frac{C_1C_2}{4}\left(\frac{V_{\xi\xi}''}{C_2}-\frac{V_{\phi\phi}''}{C_1}\right)^2
\end{equation*}
the general solution of (\ref{x})--(\ref{y}) is
\begin{equation}
\begin{array} {c}
\displaystyle{\phi_1(t)}=\tilde{D}_1e^{-(\frac{3}{2}H_f +
\frac{1}{2}\sqrt{\Delta_1-2\sqrt{\Delta_2}})t}+\tilde{D_2}e^{-(\frac{3}{2}H_f
- \frac{1}{2}\sqrt{\Delta_1-2\sqrt{\Delta_2}})t}+{}\\
\displaystyle{}+\tilde{D_3}e^{-(\frac{3}{2}H_f+\frac{1}{2}\sqrt{\Delta_1+2\sqrt{\Delta_2}})t}+
\tilde{D_4}e^{-(\frac{3}{2}H_f-\frac{1}{2}\sqrt{\Delta_1+2\sqrt{\Delta_2}})t},
\end{array}
\label{z}
\end{equation}
\begin{equation}
\begin{array} {c}
\displaystyle{\xi_1(t)}=\frac{C_1V_{\xi\xi}''-C_2V_{\phi\phi}''+
\sqrt{(C_1V_{\xi\xi}''-C_2V_{\phi\phi}'')^2+4C_1C_2{V_{\phi\xi}''}^2}}{2C_2V_{\phi\xi}''}\times{} \\
\displaystyle{}\times\left(\tilde{D}_1e^{-(\frac{3}{2}H_f +
\frac{1}{2}\sqrt{\Delta_1-2\sqrt{\Delta_2}})t}+
\tilde{D_2}e^{-(\frac{3}{2}H_f -
\frac{1}{2}\sqrt{\Delta_1-2\sqrt{\Delta_2}})t}\right)+{} \\
\displaystyle{}+\frac{C_1V_{\xi\xi}''-C_2V_{\phi\phi}''-
\sqrt{(C_1V_{\xi\xi}''-C_2V_{\phi\phi}'')^2+4C_1C_2{V_{\phi\xi}''}^2}}{2C_2V_{\phi\xi}''}\times
\\{}\times \left(\tilde{D_3}e^{-(\frac{3}{2}H_f+\frac{1}{2}\sqrt{\Delta_1+2\sqrt{\Delta_2}})t}
+\tilde{D_4}e^{-(\frac{3}{2}H_f-\frac{1}{2}\sqrt{\Delta_1+2\sqrt{\Delta_2}})t}\right),
\end{array}
\label{aa}
\end{equation}
here and further $\tilde{D}_1$, $\tilde{D_2}$, $\tilde{D_3}$, and
$\tilde{D_4}$ are arbitrary constants and
\begin{equation}
\Delta_1=9H_f^2-2\frac{V_{\xi\xi}''}{C_2}-2\frac{V_{\phi\phi}''}{C_1},
\qquad \Delta_2=\left(\frac{V_{\xi\xi}''}{C_2}
-\frac{V_{\phi\phi}''}{C_1}\right)^2+4\frac{{V_{\phi\xi}''}^2}{C_1C_2}.
\label{abb}
\end{equation}
For asymptotical stability of the considered fixed point these
functions must converge to $0$. As we can see both functions are just
linear combinations of exponents to some degrees. To satisfy this
condition all these degrees must be negative. It is easy to obtain that
the sufficient conditions for asymptotic stability are
\begin{equation}
H_f>0, \qquad \frac{V_{\xi\xi}''}{C_2}+\frac{V_{\phi\phi}''}{C_1}>0,
\qquad
\frac{V_{\xi\xi}''V_{\phi\phi}''}{C_1C_2}>\frac{{V_{\phi\xi}''}^2}{C_1C_2}.
\label{V2fCond}
\end{equation}
\item In the case
\begin{equation*}
{V_{\phi\xi}''}^2=\frac{C_1C_2}{16}\left(9H_f^2-\frac{4V_{\xi\xi}''}{C_2}\right)
\left(9H_f^2-\frac{4V_{\phi\phi}''}{C_1}\right)\quad \mbox{and}\quad
{V_{\phi\xi}''}^2\neq -
\frac{C_1C_2}{4}\left(\frac{V_{\xi\xi}''}{C_2}-\frac{V_{\phi\phi}''}{C_1}\right)^2
\end{equation*}
the inequality $\Delta_1\neq 0$ is valid and the general solution of
(\ref{x})--(\ref{y}) is
\begin{equation}
\begin{array} {c}
\displaystyle{\phi_1(t)}=\tilde{D}_1e^{-\frac{3}{2}H_ft} +
\displaystyle\tilde{D_2}e^{-\frac{3}{2}H_ft}t+
\displaystyle\tilde{D_3}e^{(-\frac{3}{2}H_f+\frac{\sqrt{2}}{2}\sqrt{\Delta_1
})t}+
\displaystyle\tilde{D_4}e^{(-\frac{3}{2}H_f-\frac{\sqrt{2}}{2}\sqrt{\Delta_1
})t},
\end{array}
\label{ab}\
\end{equation}
\begin{equation}
\begin{array} {c}
\displaystyle{\xi_1(t)}=\frac{C_1}{4V_{\phi\xi}''}
\left(\left(9H_f^2-\frac{4V_{\phi\phi}''}{C_1}\right)\left(\tilde{D}_1e^{-\frac{3}{2}H_f t}
+
\displaystyle\tilde{D_2}e^{-\frac{3}{2}H_ft}t\right)\right.-{}
\\[2.7mm]
{}-\displaystyle\left.\left(9H_f^2-\frac{4V_{\xi\xi}''}{C_2}\right)
\left(\tilde{D_3}e^{(-\frac{3}{2}H_f+\frac{\sqrt{2}}{2}\sqrt{\Delta_1
})t}+
\displaystyle\tilde{D_4}e^{(-\frac{3}{2}H_f-\frac{\sqrt{2}}{2}\sqrt{\Delta_1
})t}\right)\right).
\end{array}
\label{ac}
\end{equation}
It is easy to show that in this case the sufficient conditions for
asymptotic stability of the considered fixed point coincide with
(\ref{V2fCondVFKVFK}).
\item In the case $V_{\phi\xi}^2={} -
\frac{C_1C_2}{4}\left(\frac{V_{\xi\xi}}{C_2}-\frac{V_{\phi\phi}}{C_1}\right)^2$,
in other words $\Delta_2=0$, and \\
$V_{\phi\xi}^2\neq\frac{C_1C_2}{16}\left(9H_f^2-\frac{4V_{\xi\xi}}{C_2}\right)
\left(9H_f^2-\frac{4V_{\phi\phi}}{C_1}\right)$ the inequality
$\Delta_1\neq 0$ holds and therefore
\begin{equation*}
\phi_1(t)=\left(\tilde{D}_1+\tilde{D_3}t\right)e^{-(3H_f -
\sqrt{\Delta_1})t/2}+\left(\tilde{D_2}+\tilde{D_4}t\right)e^{-(H_f +
\sqrt{\Delta_1})t/2}, \label{ad}
\end{equation*}
\begin{equation*}
\begin{array} {c}
\displaystyle{\xi_1(t)}=\frac{C_1}{\sqrt{-C_1C_2}}
\left\{\left(\tilde{D}_1+\left(1-\frac{C_1\sqrt{\Delta_1}}{V_{\phi\xi}''}\right)
\tilde{D}_3 t\right)e^{-(3H_f -
\sqrt{\Delta_1})t/2}\right.+{} \\[2.7mm]
{}+\displaystyle\left.
\left(\tilde{D}_2+\left(1+\frac{C_1\sqrt{\Delta_1}}{V_{\phi\xi}''}\right)
\tilde{D}_4t\right)e^{-(H_f + \sqrt{\Delta_1})t/2}\right\}.
\end{array}
\end{equation*}
The sufficient conditions for the asymptotic stability of the
considered fixed point are
\begin{equation}
H_f>0, \qquad \frac{V_{\xi\xi}}{C_2}+\frac{V_{\phi\phi}}{C_1}>0.
\label{V2fCond2}
\end{equation}
\item In the case
\begin{equation*}
{V_{\phi\xi}''}^2=\frac{C_1C_2}{16}\left(9H_f^2-\frac{4V_{\xi\xi}''}{C_2}\right)
\left(9H_f^2-\frac{4V_{\phi\phi}''}{C_1}\right)={} -
\frac{C_1C_2}{4}\left(\frac{V_{\xi\xi}''}{C_2}-\frac{V_{\phi\phi}''}{C_1}\right)^2
\end{equation*}
it is easy to check that the equality $\Delta_1= 0$ is valid
automatically and, therefore,
the solution of (\ref{x})--(\ref{y}) is
\begin{equation}
\begin{array} {c}
\displaystyle{\phi_1(t)}=\left(\tilde{D}_1+\tilde{D}_2t+\tilde{D}_3t^2+\tilde{D}_4t^3\right)e^{-\frac{3}{2}H_ft},
\end{array}
\label{af}\
\end{equation}
\begin{equation}
\begin{array} {c}
\displaystyle{\xi_1(t)}=\frac{C_1}{2V_{\phi\xi}''}\left\{\tilde{D_1}
\left(\frac{V_{\xi\xi}''}{C_2}-\frac{V_{\phi\phi}''}{C_1}\right)+
\tilde{D_2}\left(\frac{V_{\xi\xi}''}{C_2}-\frac{V_{\phi\phi}''}{C_1}\right)t\right.+\\
\displaystyle\left.\tilde{D_3}\left[\left(\frac{V_{\xi\xi}''}{C_2}
-\frac{V_{\phi\phi}''}{C_1}\right)t^2-4\right]+\tilde{D_4}\left[\left(\frac{V_{\xi\xi}''}{C_2}
-\frac{V_{\phi\phi}''}{C_1}\right)t^3-12t\right]\right\}e^{-\frac{3}{2}H_ft}.
\end{array}
\label{ag}
\end{equation}
For the asymptotic stability of the considered fixed point it is
sufficient that the inequality $H_f>0$ hold.
\end{enumerate}
So, we obtained that in the case $V_{\phi\xi}\neq0$ the general
sufficient conditions of asymptotical stability of the fixed point
$y_f=(H_f,\phi_f,\psi_f)$ of the system of equations~(\ref{SYSTEM2})
are as follows
\begin{equation}
H_f>0, \qquad \frac{V_{\xi\xi}''}{C_2}+\frac{V_{\phi\phi}''}{C_1}>0,
\qquad
\frac{V_{\xi\xi}''V_{\phi\phi}''}{C_1C_2}>\frac{{V_{\phi\xi}''}^2}{C_1C_2}.
\label{V2fCondVVV}
\end{equation}
It is easy to see that the conditions (\ref{V2fCondVFKVFK}) obtained in
the case $V_{\phi\xi}''=0$ are equivalent to the conditions
(\ref{V2fCondVVV}), if the equality $V_{\phi\xi}''=0$ is substituted in
them.
So, in the general case including all the above particular cases with
definite relations between the parameters, the sufficient conditions
for asymptotic stability of the fixed point $y_f=(H_f,\phi_f,\psi_f)$
of the system of equations~(\ref{SYSTEM2}) are (\ref{V2fCondVVV}).
Let us add the cold dark matter (CDM) to our model. One-field models
with the CDM in the Bianchi I metric have been considered
in~\cite{ABJV09}. The generalization for two-field models is
straightforward, so we point only the most important steps. If one adds
into consideration the CDM energy density $\rho_m$, then
system~(\ref{SYSTEM2}) should be modified as follows:
\begin{equation}
\begin{array}{l}
\displaystyle \dot H={}-3H^2+8\pi G_N\left(V(\phi,\xi)+\rho_m\right),\\
\displaystyle \dot\phi=\psi,\\
\displaystyle \dot{\psi}={}-3H\psi-\frac{1}{C_1}\frac{\partial V}{\partial\phi},\\
\displaystyle \dot\xi=\zeta,\\
\displaystyle \dot{\zeta}={}-3H\zeta-\frac{1}{C_2}\frac{\partial
V}{\partial\xi}\\
\displaystyle\dot\rho_m={}-3H\rho_m.
\end{array}
\label{eomcdm}
\end{equation}
Let us consider the possible fixed points of system (\ref{eomcdm}).
From the last equation of this system, it follows that at the fixed
point we have either $H_f=0$ or $\rho_{mf}=0$. Substituting
(\ref{m})--(\ref{q}) and
\begin{equation}
\rho_{m}(t)=\rho_{mf}+\varepsilon\tilde{\rho}_{m}(t)+{\cal
O}(\varepsilon^2),
\end{equation}
into the system (\ref{eomcdm}), we obtain the following system to
first order in $\varepsilon$:
\begin{eqnarray}
\displaystyle \dot h(t)&=&{}-6H_fh(t)+8\pi
G_N\tilde{\rho}_m(t),\label{sv1}
\\
\dot{\tilde{\rho}}_m(t)&=&{}-3H_f\tilde{\rho}_m(t)-3\rho_{mf}h(t),\\
\displaystyle \dot\phi_1(t)&=&\displaystyle \psi_1(t),\label{sv2} \\
\displaystyle \dot\psi_1(t)&=&\displaystyle {}-3H_f\psi_1(t)-
\frac{1}{C_1}\left(V''_{\phi\phi} \phi_1(t) +V''_{\phi\xi}
\xi_1(t)\right), \\
\displaystyle \dot\xi_1(t)&=&\displaystyle \zeta_1(t), \\
\displaystyle \dot\zeta_1(t)&=&\displaystyle {}-3H_f\zeta_1(t)-
\frac{1}{C_2}\left(V''_{\xi\phi} \phi_1(t) +V''_{\xi\xi}
\xi_1(t)\right),
\end{eqnarray}
It is easy to see that the last four equations of this system coincide
with the equations (\ref{s})--(\ref{v}). Therefore, the case $H_f=0$
can not be analysed with the Lyapunov theorem. Let us prove that
conditions (\ref{V2fCondVVV}) are sufficient for the stability of fixed
points for models with the CDM. First of all, it follows from $H_f\neq
0$ that $\rho_{mf}=0$. Solving equations (\ref{sv1})--(\ref{sv2}), we
obtain
\begin{equation}
\tilde{\rho}_m(t)=b_1e^{-3H_ft},\qquad
h(t)=b_0e^{-6H_ft}+\frac{b_1}{3H_f}e^{-3H_ft},
\end{equation}
where $b_0$ and $b_1$ are arbitrary constants.
We come to the conclusion that if conditions (\ref{V2fCondVVV}) hold,
then a solution of system of equations (\ref{eomcdm}) is stable. In
other words if conditions (\ref{V2fCondVVV}) are satisfied, then the
solution, which is stable in the model without the CDM, is stable with
respect to the CDM energy density fluctuations as well.
Let us compare the obtained results with the known results.
In~\cite{Lazkoz} the stability of fixed points in quintom models
($C_1=-1$, $C_2=1$) with an arbitrary potential has been considered in
the FRW metric. The authors have presented the Einstein equations in
the form of a dynamical system having introduced the Hubble-normalized
variables~\cite{Wainwright_Lim}
\begin{equation} x_\phi=\frac{\dot\phi}{\sqrt{6}H},\qquad
x_\xi=\frac{\dot\xi}{\sqrt{6}H},\qquad y=\frac{\sqrt V}{\sqrt 3
H},\label{vars}
\end{equation}
and considered a system of ordinary differential equations with
differentiation with respect to the new variable $\tau=\log a^3$
instead of $t$. In our paper we have demonstrated that such change of
variables is not necessary, because the stability conditions can be
easily found in the initial variables.
Proceeding from some physical considerations, the authors
of~\cite{Lazkoz} assumed that $H_f>0$, we proved that the condition
$H_f<0$ is sufficient for instability of the fixed point. Note that the
case $H_f=0$, or equivalently $V(\phi_f,\xi_f)=0$, has not been
analysed both in~\cite{Lazkoz} and in our paper. In the case $H_f>0$ we
and the authors of~\cite{Lazkoz} obtained the same conditions on the
potential, which guarantee the stability of the fixed point. In our
paper we have proved that the obtained conditions are sufficient for
the stability not only in the FRW metric, but also in the Bianchi I
metric.
\section{Construction of stable solutions via the superpotential method}
\subsection{The superpotential for two-field models}
Let us consider the superpotential method~\cite{DeWolfe} (see
also~\cite{Superpotential,AKV}) for two-field models. We consider the
FRW metric and assume that the Hubble parameter $H(t)$ is a function
(superpotential) of $\phi(t)$ and $\xi(t)$:
\begin{equation}H(t)=W(\phi(t),\xi(t)),
\end{equation}
and that functions $\phi(t)$ and $\xi(t)$ are solutions of the
following system of two ordinary differential equations
\begin{equation}
\label{FG}
\dot\phi=F(\phi,\xi), \qquad \dot\xi=G(\phi,\xi).
\end{equation}
Therefore,
\begin{equation}
\ddot\phi=\frac{\partial F}{\partial \phi}F+\frac{\partial F}{\partial
\xi}G,
\qquad
\ddot\xi=\frac{\partial G}{\partial \phi}F+\frac{\partial G}{\partial
\xi}G,
\end{equation}
and we can rewrite the Friedmann equations as follows:
\begin{equation}
\label{equH2W} 3W^2=8\pi
G_N\left(\frac{C_1}{2}F^2+\frac{C_2}{2}G^2+V\right),
\end{equation}
\begin{equation}
\frac{\partial W}{\partial\phi}F+\frac{\partial
W}{\partial\xi}G={}-4\pi G_N\left(C_1F^2+C_2G^2\right),
\end{equation}
\begin{equation}
\frac{\partial F}{\partial \phi}F+\frac{\partial F}{\partial
\xi}G+3WF+\frac{1}{C_1}\frac{\partial V}{\partial\phi}=0,
\end{equation}
\begin{equation}
\frac{\partial G}{\partial \phi}F+\frac{\partial G}{\partial
\xi}G+3WG+\frac{1}{C_2}\frac{\partial V}{\partial\xi}=0.
\end{equation}
From (\ref{equH2W}) we obtain
\begin{equation}
6W\frac{\partial W}{\partial \phi}=8\pi G_N\left(C_1F\frac{\partial
F}{\partial \phi}+C_2G\frac{\partial G}{\partial \phi}+\frac{\partial
V}{\partial \phi}\right),
\end{equation}
therefore,
\begin{equation}
G\left(\frac{C_2}{C_1}\frac{\partial G}{\partial
\phi}-\frac{\partial F}{\partial
\xi}\right)=3W\left(\frac{1}{4\pi G_NC_1}\frac{\partial
W}{\partial \phi}+F\right).
\end{equation}
Also, we have
\begin{equation}
F\left(\frac{C_1}{C_2}\frac{\partial F}{\partial
\xi}-\frac{\partial G}{\partial \phi}\right)=3W\left(\frac{1}{4\pi
G_NC_2}\frac{\partial W}{\partial \xi}+G\right).
\end{equation}
If the functions $F$ and $G$ satisfy the following condition:
\begin{equation}
\label{GF} \frac{\partial F}{\partial
\xi}=\frac{C_2}{C_1}\frac{\partial G}{\partial \phi},
\end{equation}
then
\begin{equation}
\frac{\partial W}{\partial \phi}={}-4\pi G_NC_1\,F,\qquad
\frac{\partial W}{\partial \xi}={}-4\pi G_NC_2\,G. \label{equW}
\end{equation}
Note that using condition (\ref{GF}) and formulas (\ref{equW}), one can
verify that
\begin{equation}
\frac{\partial^2 W}{\partial \phi\partial\xi}=\frac{\partial^2
W}{\partial \xi\partial\phi}.
\end{equation}
Therefore, to obtain particular solutions of system
(\ref{a2})--(\ref{e2}) it suffices to require that the relations
\begin{equation}
\frac{\partial W}{\partial\phi}={}-4\pi G_NC_1\dot\phi, \qquad
\frac{\partial W}{\partial\xi}={}-4\pi G_NC_2\dot\xi
\label{deWolfe_method},
\end{equation}
\begin{equation}
V =\frac{3}{8 \pi G_N}W^2-\frac{1}{32\pi^2
G_N^2}\left(\frac{1}{C_1}\left(\frac{\partial W}{\partial
\phi}\right)^2+\frac{1}{C_2}\left(\frac{\partial W}{\partial
\xi}\right)^2\right) \label{deWolfe_potential}
\end{equation}
be satisfied.
\subsection{Stability conditions in the superpotential method}
Let us obtain conditions on the superpotential $W$ that are equivalent
to conditions (\ref{V2fCondVVV}) for the corresponding potential $V$.
At the fixed point $y_f=(H_f,\phi_f,\psi_f)$ we get
\begin{equation}
W_f\equiv W(\phi_f,\xi_f)=H_f,\qquad W'_\phi\equiv \frac{\partial
W}{\partial \phi}(\phi_f,\xi_f)=0,\qquad W'_\xi\equiv\frac{\partial
W}{\partial \xi}(\phi_f,\xi_f)=0. \label{DW0}
\end{equation}
It is easy to see that from conditions (\ref{DW0}) it follows that
\begin{equation}
V'_\phi=0,\qquad V'_\xi=0 \label{DV0}
\end{equation}
and
\begin{equation}
\label{SDVWphi2}
V''_{\phi\phi}=\frac{1}{16C_1C_2\pi^2G_N^2}\left(12C_1C_2\pi G_N W_f
W''_{\phi\phi}-C_2{W''_{\phi\phi}}^2-C_1{W''_{\phi\xi}}^2\right),
\end{equation}
\begin{equation}
\label{SDVWxi2}
V''_{\xi\xi}=\frac{1}{16C_1C_2\pi^2G_N^2}\left(12C_1C_2\pi
G_NW_fW''_{\xi\xi}-C_1{W''_{\xi\xi}}^2-C_2{W''_{\phi\xi}}^2\right),
\end{equation}
\begin{equation}
\label{SDVWphixi}
V''_{\phi\xi}=\frac{1}{16C_1C_2\pi^2G_N^2}W''_{\phi\xi}\left(12C_1C_2\pi
G_NW_f-C_2W''_{\phi\phi}-C_1W''_{\xi\xi} \right).
\end{equation}
The condition $H_f>0$ can be rewritten as $W_f>0$.
If $W''_{\phi\xi}=0$, then $V''_{\phi\xi}=0$ and conditions
(\ref{V2fCondVFKVFK}) are
\begin{equation}
\left(12\pi G_NC_1 W_f -{W''_{\phi\phi}}\right)W''_{\phi\phi}
>0,\qquad
\left(12\pi G_NC_2W_f-{W''_{\xi\xi}}\right)W''_{\xi\xi}>0.
\end{equation}
In the general case conditions (\ref{V2fCondVVV}) are written in the
following form
\begin{equation}
12C_1C_2\pi
G_N(C_2W''_{\phi\phi}+C_1W''_{\xi\xi})W_f>C_2^2\left(W''_{\phi\phi})^2
+2C_1C_2(W''_{\phi\xi}\right)^2+C_1^2(W''_{\xi\xi})^2,
\end{equation}
\begin{equation}
\begin{array}{l}
\left( 144 C_1 C_2 \pi^2 G_N^2W_f^2-12C_1C_2\pi
G_N(C_2W''_{\phi\phi}+C_1W''_{\xi\xi})W_f+W''_{\xi\xi}W''_{\phi\phi}-(W''_{\phi\xi})^2\right)\times\\[2.7mm]
\times\left(W''_{\xi\xi}W''_{\phi\phi}-(W''_{\phi\xi})^2\right)>0.
\end{array}
\end{equation}
In the case of one-field models the stability conditions on the
superpotential are considered in Appendix.
\section{String field theory inspired cosmological models}
\subsection{Quintom models with the sixth degree potential}
The interest in cosmological models related to the
open string field theory~\cite{IA} is caused by the possibility to get
solutions describing transitions from a perturbed vacuum to the true
vacuum (so-called rolling solutions~\cite{IA_TMF}). After all massive
fields (or some of the lower massive fields) are integrated out by
means of equations of motion, the open string tachyon acquires a
potential with a nontrivial vacuum, corresponding to a minimum of the
energy . The dark energy model~\cite{IA} (see also~\cite{AKV,AJ,AKV2})
assumes that our Universe is a slowly decaying D3-brane and its
dynamics is described by the open string tachyon mode. For the
Neveu--Schwarz--Ramond (NSR) open fermionic string with the GSO$(-)$
sector~\cite{NPB} in a reasonable approximation, one gets the Mexican
hat potential for the tachyon field (see~\cite{SFT-review} for a
review). Rolling of the tachyon from the unstable perturbative
extremum towards this minimum describes, according to the Sen
conjecture~\cite{SFT-review}, the transition of an unstable D-brane to
a true vacuum. In fact one gets a nonlocal potential with a string
scale as a parameter of nonlocality. After a suitable field
redefinition the potential becomes local, meanwhile, the kinetic term
becomes non-local. This nonstandard kinetic term has a so-called
phantomlike behavior and can be approximated by a phantom kinetic
term. It has been found that the open string tachyon behavior is
effectively modelled by a scalar field with a negative kinetic
term~\cite{AJK}. The back reaction of the brane is determined by
dynamics of the closed string tachyon. This dynamic can be effectively
described by a local scalar field $\xi$ with an ordinary kinetic
term~\cite{Oh} and possibly a nonpolynomial self-action~\cite{BZ}. An
exact form of the open-closed tachyon interaction is not known. So,
following~\cite{AKV2}, we consider the simplest polynomial interaction.
In the papers~\cite{AKV2,Vernov06} quintom models ($C_1={}- C_2<0$)
with effective potentials $V(\phi,\xi)$ have been
considered\footnote{Note that quintom models naturally arise from
nonlocal cosmological models with quadratic
potential~\cite{AJV2,Mulruny,Vernov2010}.}. The form of these
potentials are assumed to be given from the SFT within the level
truncation scheme. We postulate that the potential is a polynomial.
Specifically, we assume that the potential $V(\phi,\xi)$ should be an
even sixth degree polynomial
\begin{equation}
\label{potenV}
V(\phi,\xi)=\sum_{k=0}^{6}\sum_{j=0}^{6-k}c_{kj}\phi^k\xi^j,\qquad
V(\phi,\xi)=V(-\phi,-\xi),
\end{equation}
therefore, if the sum $k + j$ is odd, then $c_{kj}=0$.
From the SFT we can also assume asymptotic conditions for
solutions~\cite{AKV2,Vernov06}. We assume that the phantom field
$\phi(t)$ smoothly rolls from the unstable perturbative vacuum
($\phi=0$) to a nonperturbative one, for example $\phi=A$, and stops
there. The field $\xi(t)$ corresponds to the closed string and is
expected to go asymptotically to zero in the infinite future. Namely,
we seek such a function $\phi(t)$ that $\phi(0)=0$ and it has a
non-zero asymptotic at $t\to +\infty$: $\phi(+\infty)=A$. The function
$\xi(t)$ should have zero asymptotic at $t\to +\infty$. In other words,
we analyse the stability of solutions, tending to a fixed point with
$\phi_f=A$ and $\xi_f=0$.
\subsection{Construction of stable solutions}
Let us assume that there exists a polynomial superpotential
$W(\phi,\xi)$, which determines potential (\ref{potenV}) with formula
(\ref{deWolfe_potential}). To construct an even sixth degree polynomial
potential $V(\phi,\xi)$ we should choose $W(\phi,\xi)$ as an odd third
degree polynomial. Obviously, the suitable form of the superpotential
is as follows:
\begin{equation}
\label{Wstring}
W_{3}(\phi,\xi)=4\pi G_N\bigl(a_{1,0}\phi+a_{3,0}\phi^3+a_{0,1}\xi+a_{0,3}\xi^3
+a_{2,1}\phi^2\xi+a_{1,2}\phi\xi^2\bigr),
\end{equation}
where $a_{i,j}$ are constants. For the superpotential $W_{3}$ system
(\ref{deWolfe_method}) is as follows:
\begin{equation}
\label{ODEstring}
\begin{split}
\dot\phi&=\frac{1}{C_2}\left(a_{1,0}+3a_{3,0}\phi^2+2a_{2,1}\phi\xi+a_{1,2}\xi^2\right),\\
\dot\xi&={}-\frac{1}{C_2}\left(a_{0,1}+3a_{0,3}\xi^2+a_{2,1}\phi^2+2a_{1,2}\phi\xi\right).
\end{split}
\end{equation}
Using asymptotic conditions: $\phi(+\infty)=A$, $\xi(+\infty)=0$,
$\dot\phi(+\infty)=\dot\xi(+\infty)=0$ we obtain
\begin{equation}
a_{1,0}={}-3a_{3,0}A^2, \qquad a_{0,1}={}-a_{2,1}A^2.
\end{equation}
So, we obtain the following system of equations:
\begin{equation}
\begin{split}
\dot\phi&=\frac{1}{C_2}\left(3a_{3,0}(\phi^2-A^2)+2a_{2,1}\phi\xi+a_{1,2}\xi^2\right),\\
\dot\xi&={}-\frac{1}{C_2}\left(a_{2,1}(\phi^2-A^2)+3a_{0,3}\xi^2+2a_{1,2}\phi\xi\right)
\end{split}
\label{equW2}
\end{equation}
and the corresponding superpotential:
\begin{equation}
\label{Wstring2}
W_3(\phi,\xi)=4\pi G_N\bigl({}-3a_{3,0}A^2\phi+a_{3,0}\phi^3-a_{2,1}A^2\xi+a_{0,3}\xi^3
+a_{2,1}\phi^2\xi+a_{1,2}\phi\xi^2\bigr).
\end{equation}
At the fixed point $\phi_f=A$, $\xi_f=0$
\begin{equation}
W_f={}-8\pi G_Na_{3,0}A^3.
\end{equation}
So, the condition $W_f>0$ is equivalent to $a_{3,0}A<0$. It is easy to
calculate, that
\begin{equation}
W_{\phi\phi}''=24 \pi G_N a_{3,0}A, \qquad W_{\xi\xi}''=8 \pi G_N
a_{1,2}A, \qquad W_{\phi\xi}''=8 \pi G_N a_{2,1}A.
\end{equation}
Using (\ref{SDVWphi2})--(\ref{SDVWphixi}), we obtain:
\begin{equation}
V''_{\phi\phi}=\frac{4A^2}{C_2}\left(9(1-4\pi
G_NC_2A^2)a_{3,0}^2-a_{2,1}^2\right),
\end{equation}
\begin{equation}
V''_{\xi\xi}=\frac{4A^2}{C_2}\left(a_{2,1}^2-a_{1,2}^2-12\pi
G_NC_2A^2a_{3,0}a_{1,2}\right),
\end{equation}
\begin{equation}
V''_{\phi\xi}=\frac{4A^2a_{2,1}}{C_2}\left(3(1-4\pi
G_NC_2A^2)a_{3,0}-a_{1,2}\right).
\end{equation}
If $a_{2,1}=0$, then $V''_{\phi\xi}=0$ and the sufficient
conditions for the stability are
\begin{equation}
a_{3,0}A<0, \qquad 4\pi G_NC_2A^2>1, \qquad a_{1,2}(a_{1,2}+12\pi G_NC_2 a_{3,0}A^2)<0.
\label{StabcondWSTF1}
\end{equation}
If $a_{2,1}\neq 0$, then conditions (\ref{V2fCondVVV}) are equivalent
to
\begin{equation}
\begin{split}
& a_{3,0}A<0,\\
&2a_{2,1}^2{}-a_{1,2}^2-12\pi G_N C_2A^2a_{3,0}a_{1,2}+9(4\pi
G_NC_2A^2-1)a_{3,0}^2>0, \\
&\left[3a_{3,0}a_{1,2}-a_{2,1}^2\right]\left[3(4\pi
G_NA^2C_2-1)a_{3,0}a_{1,2}-36\pi G_NA^2C_2(4\pi
G_NA^2C_2-1)a_{3,0}^2+a_{2,1}^2\right]<0.
\end{split}
\label{w3cond}
\end{equation}
\subsection{Examples of stable solutions}
The case of superpotential $W(\phi,\xi)$ with $a_{2,1}=0$ and
$a_{0,3}=0$ has been considered in~\cite{AKV2}. In this case the system
(\ref{equW2}) has the following form
\begin{equation}
\phi={}-\frac{C_2}{2a_{1,2}}\frac{\dot\xi}{\xi},
\end{equation}
\begin{equation}
\ddot\xi=\frac{(2a_{1,2}-3a_{3,0})}{2a_{1,2}\xi}\dot\xi^2
+\frac{2a_{1,2}}{C_2^2}\xi\left(3a_{3,0}A^2-a_{1,2}\xi^2\right).
\label{equxi}
\end{equation}
Equation (\ref{equxi}) can be integrated in quadratures:
\begin{equation}
\int\!\frac{\sqrt{\xi^{3 B-2}(3B+2)}C_2}{\sqrt{(12 B A^2+8 A^2-4
\xi^2)\xi^{3 B}a_{1,2}^2+(3 B+2) C_2^2D_1}}\:d\xi=\pm(t-t_0),
\end{equation}
where $D_1$ and $t_0$ are arbitrary constants and $B=a_{3,0}/a_{1,2}$.
At $B=-1/3$, $a_{2,1} = 0$, and $a_{0,3} = 0$, the general solution of
system (\ref{equW2}) can be obtained in explicit form~\cite{Vernov06}:
\begin{equation}
\phi_s(t)=\frac{A\left(C_2^2e^{4a_{1,2}At/C_2}-64a_{1,2}^4C_2^2A^2D_1^2-4a_{1,2}^2A^2D_2^2\right)}{\left(C_2e^{2
a_{1,2} A t/C_2}-2 D_2 a_{1,2} A\right)^2+64 D_1^2 a_{1,2}^4 C_2^2
A^2},
\end{equation}
\begin{equation}
\xi_s(t)=\frac{16 D_1 C_2^2 a_{1,2}^2 A^2e^{2 a_{1,2} A
t/C_2}}{\left(C_2e^{2 a_{1,2} A t/C_2}-2 D_2 a_{1,2} A\right)^2+64
D_1^2 a_{1,2}^4 C_2^2 A^2}.
\end{equation}
Let us analyse the stability of the exact solution. One can see
that $\phi_s(t)$ and $\xi_s(t)$ are continuous functions, which
tend to a fixed point at $t\rightarrow\infty$. Therefore, the
obtained exact solution is attractive if and only if the fixed
point is asymptotically stable. At $a_{3,0}={}-a_{1,2}/3$ we
obtain that three stability conditions (\ref{StabcondWSTF1})
transform into two independent conditions
\begin{equation}
a_{1,2}A>0,\qquad C_2>\frac{1}{4\pi G_NA^2}.
\end{equation}
From these conditions it follows that $a_{1,2}A/C_2>0$.
Let us check the stability of solutions, obtained in~\cite{Vernov06}.
In~\cite{Vernov06} the author has considered the quintom model with the
following energy density:
\begin{equation}
\rho=\frac{8\pi G_N}{m_p^2}\left({}-\frac12\dot\phi^2
+\frac12\dot\xi^2+V_1\right),
\end{equation}
where
\begin{equation} m_p^2=\frac{g_o}{8\pi G_N M_s^2},
\end{equation}
$g_o$ is the open string coupling constant, $M_s$ is the string mass,
\begin{equation}
V_1=\frac{\omega^2
}{8A^2}\left(\left(A^2-\phi^2+\xi^2\right)^2-4\phi^2\xi^2+
\frac{1}{6m_p^2}\phi^2\left(3A^2-\phi^2+3\xi^2\right)^2\right),
\end{equation}
where $\omega$ is a nonzero constant.
Therefore, we obtain $C_2=M_s^2/g_o$, the potential
\begin{equation}
V_s=\frac{1}{8\pi G_Nm_p^2}V_1=\frac{M_s^2}{g_o}V_1
\end{equation}
corresponds to the superpotential
\begin{equation}
W_s=4\pi
G_N\omega\phi\left(A\left(\frac{1}{2}-\frac{\phi^2}{6A^2}\right)+
\frac{\xi^2}{2A}\right).
\end{equation}
This choice corresponds to
\begin{equation}
a_{1,2}=\frac{\omega}{2A},
\end{equation}
so the stability conditions are
\begin{equation}
\omega>0,\qquad 4\pi G_NA^2M_s^2>g_o.
\end{equation}
So, we come to the conclusion that the exact solutions, obtained
in~\cite{Vernov06} are stable for sufficiently large~$A$.
\section{Conclusion}
We have analysed the stability of isotropic solutions for
two-field models in the Bianchi I metric. Using the Lyapunov
theorem we have found sufficient conditions of stability of
kink-type and lump-type isotropic solutions for two-field models
in the Bianchi I metric. The obtained results allow us to prove
that the exact solutions, found in string inspired phantom
models~\cite{AKV2,Vernov06}, are stable.
Our study of the stability of isotropic solutions for quintom models
in the Bianchi I metric shows that the NEC is not a necessary condition
for classical stability of isotropic solutions. In this paper we have
shown that the models~\cite{AKV2,Vernov06} have stable isotropic
solutions and that large anisotropy does not appear in these models.
It means that considered models are acceptable, because they do not violate
limits on anisotropic models, obtained from the observations~\cite{Barrow,Bernui:2005pz}.
We also have presented the algorithm for construction of kink-type and
lump-type isotropic exact stable solutions via the superpotential
method. In particular we have formulated the stability conditions in
terms of superpotential.
This work is supported in part by state contract of Russian Federal
Agency for Science and Innovations 02.740.11.5057 and by RFBR grant
08-01-00798. I.A. is supported in part by grant of the Program for
Supporting Leading Scientific Schools NSh-795.2008.1. S.V. is supported
in part by grant of the Program for Supporting Leading Scientific
Schools NSh-1456.2008.2.
\section*{Appendix. Stability conditions on superpotential in a one-field cosmological model}
The main goal of our paper is to consider stable solutions in two-field
models. At the same time it is convenient to remind the superpotential
method for a cosmological model with one scalar field $\tilde{\phi}$,
which is described with the action
\begin{equation}
S=\int d^4x \sqrt{-g}\left(\frac{R}{16\pi G_N}-
\frac{\tilde{C}}{2}g^{\mu\nu}\partial_{\mu}\tilde{\phi}\partial_{\nu}\tilde{\phi}
-\tilde{V}(\tilde{\phi})\right), \label{action_1}
\end{equation}
where the potential $\tilde{V}(\tilde{\phi})$ is a twice continuously
differentiable function and $\tilde{C}$ is a nonzero real constant.
It has been shown in~\cite{ABJV09} that to find sufficient conditions
for the stability of the isotropic fixed point in the Bianchi I metric
one can consider the spatially flat Friedmann--Robertson--Worker
Universe with
\begin{equation*}
ds^2={}-dt^2+\tilde{a}^2(t)\left(dx_1^2+dx_2^2+dx_3^2\right),
\end{equation*}
where $\tilde{a}(t)$ is the scale factor. In the FRW metric the field
$\tilde{\phi}$ depends only on time.
The Friedmann equations can be written in the following form:
\begin{equation}
\dot{\tilde{H}}={}-4\pi
G_N\tilde{C}\left(\dot{\tilde{\phi}}\right)^2,\qquad 3\tilde{H}^2=8\pi
G_N\left(\frac{\tilde{C}}{2}\left(\dot{\tilde{\phi}}\right)^2+\tilde{V}(\tilde{\phi})\right).\label{eom12}
\end{equation}
In~\cite{ABJV09} it has been proven that the fixed point
$\tilde{y}_f=(\tilde{H}_f,\tilde{\phi}_f)$ is asymptotically stable
and, therefore, the exact solution $(\tilde{\phi}(t),\tilde{H}(t))$,
which tends to this fixed point, is attractive if:
\begin{equation}
\frac{\tilde{V}''(\tilde{\phi}_f)}{\tilde{C}}>0\quad \mbox{and}\quad
\tilde{H}_f>0,\label{Stabcond}
\end{equation}
in this subsection a prime denotes a derivative with respect to
$\tilde{\phi}$.
System of equations (\ref{eom12}) with a polynomial potential
$\tilde{V}(\phi)$ is not integrable. At the same time it is possible to
construct the potential $\tilde{V}(\tilde{\phi})$ and to find
$\tilde{H}(t)$ if $\tilde{\phi}(t)$ is given explicitly.
Following~\cite{DeWolfe}, we assume, that $\tilde{H}(t)$ is a function
of $\tilde{\phi}(t)$, called superpotential (for details of the
Hamilton--Jacobi formulation of the Friedmann equations and the
superpotential method see
also~\cite{Superpotential,AKV,AKV2,Vernov06}), that is
$\tilde{H}(t)=\tilde{W}(\tilde{\phi}(t))$. Using equality
$\dot{\tilde{H}}=\tilde{W}^{\prime}\dot{\tilde{\phi}}$, where
$\tilde{W}^{\prime}\equiv\frac{\partial
\tilde{W}}{\partial\tilde{\phi}}$, we obtain from system (\ref{eom12}):
\begin{eqnarray}
\dot{\tilde{\phi}}&=&{}-\frac{1}{4\pi G_N\tilde{C}}
\tilde{W}^{\prime},\label{eom1W}\\
\tilde{V}&=&\frac{3}{8\pi
G_N}\tilde{W}^2-\frac{1}{32\pi^2G_N^2\tilde{C}}\left(\tilde{W}'\right)^2.
\label{eom2W}
\end{eqnarray}
The superpotential method is to choose $\tilde{W}(\tilde{\phi})$ in
such form that both $\tilde{\phi}(t)$ and $\tilde{V}(\tilde{\phi})$
have required properties. Equation (\ref{eom1W}) is always solvable at
least in quadratures. Formula (\ref{eom2W}) allows one to find the
potential $\tilde{V}$, provided the superpotential $\tilde{W}$ is
given.
Let $\tilde{\phi}(t)$ tend to a finite limit $\tilde{\phi}_f$ at
$t\rightarrow +\infty$. We assumed that $\tilde{V}(\tilde{\phi})$ is a
twice continuously differentiable function, therefore,
$\tilde{V}(\tilde{\phi}_f)$ is finite and
$\tilde{H}_f=\tilde{W}(\tilde{\phi}_f)$ is finite as well. So, system
(\ref{eom12}) has the fixed point
$\tilde{y}_f=(\tilde{H}_f,\tilde{\phi}_f)$. It is easy to see that
\begin{equation}
\tilde{V}'(\tilde{\phi}_f)=0, \qquad \tilde{H}_f^2=\frac{8}{3}\pi
G_N\left(\tilde{V}(\tilde{\phi}_f)\right).
\end{equation}
From (\ref{eom2W}) we get the condition
\begin{equation}
\tilde{V}'(\tilde{\phi}_f)=\frac{\tilde{W}'(\tilde{\phi}_f)}{16\pi^2G_N^2\tilde{C}}
\left(12\pi G_N
\tilde{C}\tilde{W}(\tilde{\phi}_f)-\tilde{W}''(\tilde{\phi}_f)\right)=0.
\end{equation}
If $\tilde{W}'(\tilde{\phi}_f^{\vphantom{27}})\neq 0$, then from
(\ref{eom1W}) it follows that $\tilde{\phi}_f$ is not a fixed point,
so, we analyse only the case
$\tilde{W}'(\tilde{\phi}_f^{\vphantom{27}})=0$ and obtain that
\begin{equation}
\tilde{V}''(\tilde{\phi}_f)
=\frac{\tilde{W}''(\tilde{\phi}_f)}{16\pi^2G_N^2\tilde{C}} \left(12\pi
G_N
\tilde{C}\tilde{W}(\tilde{\phi}_f)-\tilde{W}''(\tilde{\phi}_f)\right).
\end{equation}
Thus, we come to the conclusion that to construct a stable kink-type
solution one should find such $\tilde{W}(\tilde{\phi})$ that
$\tilde{\phi}(t)$ tends to a fixed point $\phi_f$ and the following
conditions are satisfied
\begin{equation}\label{StabcondW1}
\tilde{W}''(\tilde{\phi}_f^{\vphantom{27}})\left(12\pi G_N
\tilde{C}\tilde{W}(\tilde{\phi}_f)-\tilde{W}''(\tilde{\phi}_f)\right)>0\quad\mbox{and}\quad
\tilde{H}_f=\tilde{W}(\tilde{\phi}_f)>0.
\end{equation}
Note that the obtained conditions are sufficient for the stability of
the obtained isotropic solution in the Bianchi I metric as well as with
respect to small fluctuations of the CDM energy density~\cite{ABJV09}. | 34,877 |
\subsection{Galois categories}\label{GaloisCategories}
In this section we review the notion of Galois categories. We will use the definition found in \cite[\href{https://stacks.math.columbia.edu/tag/0BMQ}{Tag 0BMQ}]{stacks-project} and we refer the reader to \cite{SGA1}, \cite{Lenstra} and \cite{Cadoret2013} for more background on the subject. In this paper we will be mainly interested in the Galois category of finite tame coverings of an algebraic curve. We will show that the category of finite rigidified coverings of a metrized complex is equivalent to this category, giving an isomorphism of the corresponding profinite fundamental groups. This equivalence will be given in Section \ref{AlgebraicMetrizedCoverings}.
We will assume in this section that $\mathcal{C}$ is an essentially small category.
Let $\mathcal{C}$ be such a category. For any $\phi\in\mathrm{Ar}(\mathcal{C})$, we say that $\phi$ is a {\it{monomorphism}} if it is left cancellative. That is, $\phi:X\rightarrow{Y}$ is a monomorphism if for any two morphisms $g_{i}:Z_{i}\rightarrow{X}$ in $\mathcal{C}$ such that $f\circ{g_{1}}=f\circ{g_{2}}$, we have $g_{1}=g_{2}$. A {\it{connected}} object of $\mathcal{C}$ is a noninitial object $X$ such that any monomorphism $Y\rightarrow{X}$ is initial or an isomorphism. Let $F$ be a functor $\mathcal{C}\rightarrow{\mathcal{D}}$. We say that $F$ is {\it{exact}} if it commutes with finite limits and finite colimits. We say that $F$ {\it{reflects isomorphisms}} if for any $f\in\mathrm{Ar}(\mathcal{C})$, if $Ff$ is an isomorphism, then $f$ is an isomorphism.
\begin{mydef}\label{GaloisCategoryDefinition}
Let $\mathcal{C}$ be a category and let $F:\mathcal{C}\rightarrow{(\mathrm{Sets})}$ be a functor. The pair $(\mathcal{C},F)$ is a {\it{Galois category}} if
\begin{enumerate}
\item $\mathcal{C}$ has finite limits and finite colimits,
\item every object of $\mathcal{C}$ is a finite (possibly empty) coproduct of connected objects,
\item $F(X)$ is finite for every object $X\in\mathrm{Ob}(\mathcal{C})$ and
\item $F$ reflects isomorphisms and is exact.
\end{enumerate}
\end{mydef}
We refer to $F$ as the {\it{fundamental functor}} of the Galois category. The definition of a Galois category is completely categorical, so this structure is preserved under categorical equivalences. We can therefore define the notion of an {\it{induced Galois category structure}}.
\begin{mydef}\label{InducedStructure}{\it{(Induced Galois category structure)}}
Let $(\mathcal{C},F)$ be a Galois category and suppose that there is an equivalence of categories $G:\mathcal{D}\rightarrow{\mathcal{C}}$. Let $F'$ be the induced functor $FG:\mathcal{D}\rightarrow{(\mathrm{Sets})}$ defined by $FG(X)=F(G(X))$. Then $(\mathcal{D},F')$ is the {\it{induced Galois category structure}} on $\mathcal{D}$.
\end{mydef}
\begin{rem}\label{EquivalentDefinition}{\it{(An equivalent definition)}}
Let $\mathcal{C}$ be a category and let ${F}$ be a functor to $(\mathrm{Sets})$. In \cite{Lenstra}, \cite{Cadoret2013} and \cite{SGA1} a different definition for a Galois category is given. There, ${F}$ and $\mathcal{C}$ have to satisfy the following set of axioms:
\begin{enumerate}
\item $\mathcal{C}$ has a final object and and finite fiber products exist in $\mathcal{C}$.
\item Finite coproducts exist in $\mathcal{C}$ and categorical quotients by finite groups of automorphisms exist in $\mathcal{C}$.
\item Any morphism $u:Y\rightarrow{X}$ in $\mathcal{C}$ factors as $u"\circ{u'}:Y\rightarrow{X'}\rightarrow{X''}$, where $u'$ is a strict epimorphism and $u''$ is a monomorphism which is an isomorphism onto a direct summand of $X$.
\item ${F}$ sends final objects to final objects and commutes with fiber products.
\item ${F}$ commutes with finite coproducts and quotients by finite groups and sends strict epimorphisms to strict epimorphisms.
\item ${F}$ reflects isomorphisms.
\end{enumerate}
These definitions are in fact equivalent. To see this, let $\mathcal{C}$ be any category that is equivalent to $(\mathrm{Finite}\mhyphen{G}\mhyphen{\mathrm{Sets}})$, which is the category of finite sets with a continuous action by a profinite group $G$. Then $\mathcal{C}$ automatically satisfies the conditions in both definitions with respect to the composed functor $\mathcal{C}\rightarrow{}(\mathrm{Finite}\mhyphen{G}\mhyphen{\mathrm{Sets}})\rightarrow{(\mathrm{Sets})}$.
By Theorem \ref{MainTheoremGaloisCategories} and \cite[Theorem 2.8]{Cadoret2013}, we obtain an equivalence with $(\mathrm{Finite}\mhyphen{G}\mhyphen{\mathrm{Sets}})$ using either definition, so we see that the two definitions are equivalent.
\end{rem}
\begin{exa}
We will follow \cite[Section 3.7]{Lenstra} and \cite[Example 2.7]{Cadoret2013}. Let $X$ be a topological space and consider the category $\mathrm{Cov}(X)$ of finite {\it{coverings}} of $X$. Here a finite covering $\phi:X'\rightarrow{X}$ is a finite
continuous surjective map such that for every $x\in{X}$, there exists a neighborhood $U$ of $x$ such that $\phi^{-1}(U)$ is homeomorphic to a disjoint union of $n$ copies of $U$ for some $n\in\mathbb{N}$. The fundamental functor, denoted by $F_{x}$ for any point $x$ in $X$ is then given as follows:
one assigns to every covering $\phi:X'\rightarrow{X}$ the inverse image $\phi^{-1}(x)$ of $x$ under $\phi$.
\end{exa}
\begin{exa}\label{ExampleLenstra1}
We will follow \cite[Section 5]{Lenstra} and \cite[Section 5]{Cadoret2013}. Let $X$ be a connected scheme and consider the category $\mathrm{ECov}(X)$ of finite \'{e}tale coverings $X'\rightarrow{X}$. Let $x:\mathrm{Spec}(\Omega)\rightarrow{X}$ be any geometric point of $X$. Here $\Omega$ is some algebraically closed field. If $Y\rightarrow{X}$ is finite \'{e}tale, then so is $Y\times_{X}\mathrm{Spec}(\Omega)$. This gives a functor $\mathrm{ECov}(X)\rightarrow{\mathrm{ECov}(\mathrm{Spec}(\Omega))}$. But the fundamental group of $\mathrm{Spec}(\Omega)$ is trivial, so we obtain an equivalence $\mathrm{ECov}(\mathrm{Spec}(\Omega))\simeq{(\mathrm{Finite}\,\,\mathrm{Sets})}$. This gives the fundamental functor $F_{x}$ for any geometric point $x$. We denote the corresponding profinite group by $\pi(X,x)$.
If $X$ is a normal integral scheme, then there is a very concrete description of $\pi(X,x)$ in terms of the absolute Galois group corresponding to the function field of $X$. Let $K(X)$ be that function field, $\overline{K}$ an algebraic closure of $K(X)$ and consider the composite $M$ of all finite separable extensions $L\subset\overline{K}$ such that $X$ is unramified in $L$. By unramified, we mean that the normalization of $X$ in the field $L$ is unramified over $X$. The extension $M/K$ is then Galois and we have an isomorphism $\pi(X,x)\simeq{\mathrm{Gal}(M/K)}$. See \cite[Corollary 6.17]{Lenstra} for the details. We note that by taking $X:=\mathrm{Spec}(K)$ for some field $K$, we obtain the usual Galois group $\pi(X,x)\simeq{}\mathrm{Gal}(K^{sep}/K)$ where $x:\mathrm{Spec}(K)\rightarrow{\mathrm{Spec}(K)}$ is any morphism. The theory of Galois categories can thus be seen as a generalization of usual Galois theory.
\end{exa}
\begin{lemma}\label{GaloisObject}
Let $(\mathcal{C},F)$ be a Galois category. Then for any $X\in\mathrm{Ob}(\mathcal{C})$, we have that
\begin{equation}
|\mathrm{Aut}(X)|\leq{|F(X)|}.
\end{equation}
\end{lemma}
\begin{proof}
See \cite[\href{http://stacks.math.columbia.edu/tag/0BN0}{Lemma 0BN0}]{stacks-project}.
\end{proof}
\begin{mydef}
Let $(\mathcal{C},F)$ be a Galois category. We say that a connected object $X$ is a {\it{Galois object}} if
\begin{equation}
|\mathrm{Aut}(X)|={|F(X)|}.
\end{equation}
\end{mydef}
For any Galois object $X$, we let $X/\mathrm{Aut}(X)$ be the coequalizer of the arrows $\sigma:X\rightarrow{X}$, where $\sigma\in\mathrm{Aut}(X)$. This exists by condition $(1)$ in the definition of a Galois category. We then have the following Lemma:
\begin{lemma}
Let $X$ be a Galois object. Then the coequalizer $X/\mathrm{Aut}(X)$ is the terminal object of $\mathcal{C}$.
\end{lemma}
\begin{proof}
By using the fact that $F$ commutes with finite colimits and the fact that the action of $\mathrm{Aut}(X)$ is transitive on $F(X)$, we easily see that $F(X/\mathrm{Aut}(X))$ is the singleton set, implying that $X/\mathrm{Aut}(X)$ is the terminal object of $\mathcal{C}$ by \cite[\href{http://stacks.math.columbia.edu/tag/0BN0}{Lemma 0BN0, (5)}]{stacks-project}.
\end{proof}
\begin{exa}
For a topological space $X$, a Galois object in $\mathrm{Cov}(X)$ is a topological space $X'$ with a continuous group action of a finite group $G$ on it such that $X'/G=X$. For a connected normal scheme $X$ with function field $K(X)$, every Galois object is obtained as follows. We take a finite Galois extension $L$ of $K(X)$ with Galois group $G$ and consider the normalization $X'$ of $X$ in $L$. There is then a natural action of $G$ on $X'$ and we have $X'/G=X$.
\end{exa}
Let $F$ be any functor $F:\mathcal{C}\rightarrow{(\mathrm{Sets})}$. An automorphism of functors is an invertible morphism of functors $F\rightarrow{F}$.
We then have a canonical injective map
\begin{equation}\label{ProfiniteGroup}
\mathrm{Aut}(F)\rightarrow{\prod_{X\in\mathrm{Ob}(X)}\mathrm{Aut}(F(X))}.
\end{equation}
Assigning the discrete topology to every set on the righthand side of Equation (\ref{ProfiniteGroup}) and the product topology to the product, we see that $\mathrm{Aut}(F)$ naturally has the structure of a {\it{profinite}} group for a Galois category. In particular, we see that $\mathrm{Aut}(F)$ is a topological group. We then also have natural continuous maps
\begin{equation}
\mathrm{Aut}(F)\times{F(X)}\rightarrow{F(X)}
\end{equation}
for every $X\in\mathrm{Ob}(\mathcal{C})$. Letting $G:=\mathrm{Aut}(F)$, we have that the functor $F$ induces a functor
\begin{equation}
\tilde{F}:\mathcal{C}\rightarrow{(\mathrm{Finite}\mhyphen G \mhyphen \mathrm{Sets})},
\end{equation}
where $(\mathrm{Finite}\hyp G \hyp \mathrm{Sets})$ is the category of finite sets with a continuous $G$-action. We will often denote this functor by $F$ again.
We can now state the main theorem on Galois categories:
\begin{theorem}\label{MainTheoremGaloisCategories}
Let $(\mathcal{C},F)$ be a Galois category and let $\tilde{F}:\mathcal{C}\rightarrow{(\mathrm{Finite}\mhyphen G \mhyphen \mathrm{Sets})}$ be the induced functor. We then have
\begin{enumerate}
\item $\tilde{F}$ is an equivalence of categories.
\item If $\pi$ is any profinite group such that the categories $\mathcal{C}$ and $(\pi\hyp\mathrm{Sets})$ are equivalent by an equivalence that, composed with the forgetful functor $(\pi\hyp\mathrm{Sets})\rightarrow{(\mathrm{Sets})}$ yields the functor $F$, then $\pi$ and $\mathrm{Aut}(F)$ are canonically isomorphic.
\item If $F$ and $F'$ are two fundamental functors for $\mathcal{C}$, then they are isomorphic.
\item If $\pi$ is a profinite group such that $\mathcal{C}$ and $(\pi\hyp\mathrm{Sets})$ are equivalent, then there is an isomorphism of profinite groups $\mathrm{Aut}(F)\simeq{\pi}$ that is canonically determined up to an inner automorphism of $\mathrm{Aut}(F)$.
\end{enumerate}
\end{theorem}
\begin{proof}
See \cite[\href{http://stacks.math.columbia.edu/tag/0BN4}{Proposition 0BN4}]{stacks-project} or \cite[Theorem 3.5]{Lenstra}.
\end{proof}
\begin{rem}
Throughout this paper, we will often suppress the base-points in the fundamental functors $F_{x}$ and the profinite fundamental groups $\pi(X,x)$ since they are not of any significant relevance to us.
\end{rem}
\begin{exa}\label{Grothendieck}
We recall an important result by Grothendieck, which gives a finite representation of the {\it{tame}} profinite fundamental group of a punctured curve $(X,D)$, where $X$ is now a smooth, proper, connected curve over a separably closed field $k$ of characteristic $p$.
See \cite[D\'{e}finition 2.1.1]{SGA1} for the definition of the Galois category of finite \'{e}tale coverings that are tamely ramified.
\begin{theorem}
Let $(X,D)$ be a smooth proper connected punctured curve over a separably closed field $k$ of characteristic $p$ (we allow $p=0$) of genus $g:=g(X)$ with $d$ punctures $P_{i}\in{X(K)}$. Then the profinite fundamental group $\pi^{t}(X,D)$ corresponding to the Galois category of tamely ramified \'{e}tale coverings of the punctured curve $(X,D)$ is the profinite completion of the group $R=\langle{}x_{i},y_{i},z_{j}\rangle/I$ for $i=1,..,g$ and $j=1,..,d$, where $I$ is the group generated by the single element
\begin{equation}
\prod_{1\leq{i}\leq{g}}x_{i}y_{i}x_{i}^{-1}y_{i}^{-1}\cdot{}\prod_{1\leq{i}\leq{d}}z_{i}.
\end{equation}
Furthermore, every $z_{i}$ generates an inertia group above the puncture $P_{i}$.
\end{theorem}
\begin{proof}
See \cite[Corollaire 2.12]{SGA1}.
\end{proof}
In other words, the tame profinite fundamental group coincides with the profinite completion of the fundamental group of any genus $g$ curve $X/\mathbb{C}$ with $d$ punctures. In fact, the proof of Theorem \ref{Grothendieck} uses this analytic result, along with a theorem on the specialization of the fundamental group.
\end{exa} | 87,921 |
TITLE: Is it possible to continuously select a probability distribution over fixed points in Brouwer's fixed point theorem?
QUESTION [8 upvotes]: According to Brouwer's fixed point theorem, for compact convex $K\subset\mathbb{R}^n$, every continuous map $K\rightarrow K$ has a fixed point.
However, these fixed points cannot be chosen continuously, even for $K=[0,1]$, in the sense that there is no continuous map $fix:[0,1]^{[0,1]}\rightarrow[0,1]$ such that $\forall f:[0,1]\rightarrow[0,1]\;f(fix(f))=fix(f)$. To see this, consider a family of functions $f_x:[0,1]\rightarrow[0,1]$ ($x\in[0,1]$) such that $f_0(y)=\frac{1}{3}$; $f_{\frac{1}{2}}(y)=\frac{1}{3}$ for $y\leq\frac{1}{3}$, $f_{\frac{1}{2}}(y)=\frac{2}{3}$ for $y\geq\frac{2}{3}$, and linearly interpolates between those for $\frac{1}{3}<y<\frac{2}{3}$; $f_1(y)=\frac{2}{3}$; and $f_x$ linearly interpolates between $f_0$ and $f_\frac{1}{2}$ for $0<x<\frac{1}{2}$ and between $f_\frac{1}{2}$ and $f_1$ for $\frac{1}{2}<x<1$. In order for $fix$ to continuously select a fixed point, we would need $fix(f_x)=\frac{1}{3}$ for $x\leq\frac{1}{2}$ and $fix(f_x)=\frac{2}{3}$ for $x\geq\frac{1}{2}$, a contradiction.
But one could imagine gradually shifting probability from the lower fixed point of $f_x$ to the upper fixed point as $x$ increases. [Edit: actually, one couldn't do that; as Noam Elkies points out in the comments, this example answers my own question.]
Hence my question: For a compact convex $K\subset\mathbb{R}^n$, is there a continuous map $fix:K^K\rightarrow\Delta(K)$ such that $\forall f:K\rightarrow K$, $fix(f)$ is supported on fixed points of $f$? Here $K^K$ is given the compact-open topology and $\Delta(K)$ is the space of probability distributions over $K$, equipped with the weak topology.
REPLY [1 votes]: The problem here is not restricted to obtaining continuous choice functions, it already fails at the level of continuous multivalued functions.
Theorem The multivalued function $\mathrm{BFT}_k : \mathcal{C}([0,1]^k,[0,1]^k) \rightrightarrows [0,1]^k$ mapping a continuous function to some arbtirary fixed point is not continuous. (See https://arxiv.org/abs/1206.4809)
Theorem The multivalued function $\mathrm{SuppPoint} : \mathcal{PM}(\mathbf{X}) \rightrightarrows \mathbf{X}$ mapping a probability measure to an arbitrary point in its support is continuous (in fact, computable) for every "sufficiently complete" space $\mathbf{X}$, in particular it works for $\mathbf{X} = [0,1]^k$. (See http://logicandanalysis.org/index.php/jla/article/view/241/0)
By the latter, we see that getting probability distributions supported on the fixed points cannot be easier than getting a fixed point. The former tells us that this doesnt work. | 99,259 |
Sara Emley
Partner
Updated On : Sep 04, 2014
1250 24th Street, NW Suite 700
Washington, DC 20037
HQ Phone: 202-349-8000
Direct Phone : 202-349-8025
Type:
Private
Employees:
250 - 499
Revenue:
20 - 50 Million
Industry:
Services->Legal
SIC Code:
8111 - Legal Services
A Washington, DC-based organization, BuckleySandler LLP is just one of the many Lead411 profiles that you can find contact information, like phone numbers and emails. Their profile includes @buckleysandler.com email addresses, as well as details on Sara Emley's email, the organization's Partner. Their profile can be found in Legal category. If you also need twitter, facebook, linkedin, wiki, and biography details for Sara Emley, you can also find them in Lead411. Some possible email formats for Sara Emley are [email protected], [email protected], [email protected], and [email protected]. If you sign up for our free trial you will see our [email protected] addresses.
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Elizabeth Mcginn - Executive
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Alex Dempsey - Executive
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People Directory > Sara Emley | 299,584 |
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TITLE: Two definitions of topological terms in field theory
QUESTION [8 upvotes]: I've seen two distinct definitions for "topological" terms in the context of quantum field theory.
Topological terms don't depend on the metric $g_{\mu\nu}$. This makes sense since topology is 'geometry minus length and angle'. (One consequence of this is that they don't contribute to the stress-energy tensor.)
Topological terms are total derivatives, i.e. boundary terms. (One consequence of this is that they don't contribute at the classical level.)
Does the first statement imply the second? I can't see any clear reason for it to, but I've only seen examples where both are true.
REPLY [10 votes]: Topological terms of all types are always required not to depend on the metric, so their integrals will correspond to topological invariants, which serve as topological charges in quantum field theory.
However, it is important to distinguish between two the types of topological terms mentioned in the question, because they lead to different physical consequences. Please see the Deligne-Freed lectures on classical field theories.
The first type ($\theta$-terms) occurs when one takes a closed form on the target space of rank equal to the dimension of the base space $\mathcal{M}$:
$$ \omega(y) = \omega_{\alpha_1 …\alpha_n} dy^{\alpha_1}\wedge… dy^{\alpha_n}$$
pull it back to the base space and integrate:
$$\int_{\mathcal{M}}\omega_{\alpha_1 …\alpha_n} \frac{\partial y^{\alpha_1}}{ \partial x^{\beta_1}}… \frac{\partial y^{\alpha_n}}{ \partial x^{\beta_n}} dx^{\beta_1}\wedge… dx^{\beta_n}$$
The integration of this form does not require a metric.
An important subclass of this type of terms $\omega$ is a representative of a characteristic class (please see Nash and Sen section 7.22) of a fiber bundle over the target space. In this case, the topological term can be added to the Lagrangian on an even dimensional base space. $\theta$-terms are topological charges of instantons, and their inclusion in the Lagrangian is equivalent choosing a $\theta$-vacuum. Prototypes of this type of topological terms are the $\theta$- term of QCD and the winding number in the $\mathbb{C}P^1$ model.
The second type of topological terms constitute of pullbacks to the base manifold of secondary characteristic classes (please see Nash
page 223). These classes live in odd dimensions. They are closed only when the gauge connection is a pure gauge. In this case they constitute of holonomies (Berry's phases) of gauge connections and higher versions of which in higher dimensions.
In contrast to characteristic classes which classify fiber bundles over manifolds, secondary characteristic classes classify flat fiber bundles. The prototypes of topological terms associated with secondary characteristic classes are the electromagnetic interaction term of a charged particle (in 1D) and the Chern-Simons term (in 3D). The pure gauge case corresponds to an Aharonov-Bohm potential in 1D and a Wess-Zumino-Witten term in 3-D. | 150,694 |
Wealthy investors turn to ISAs to top up pension funds
With cuts to pensions tax relief for high earners hitting wealthy investors in the pocket, many are reportedly turning to Individual Savings Accounts (ISAs) as a means of boosting their retirement coffers. Acumen explores how these tax-free savings accounts have now become attractive alternatives to pensions for high earners.
It’s fair to say that ISAs have suffered a nosedive in popularity over the last few years. Stagnant interest rates, and more attractive ones offered by current account providers, led some to predict the death knell of ISAs. However, this year’s advent of a newly expanded £20,000 annual ISA limit, and government cuts to pensions tax relief for high earners, have rekindled high earners’ love for the humble ISA once more.
Pension tax relief for high earners slashed
Since 2012, wealthier pension savers have seen their lifetime allowance slashed from £1.8 million to the current £1 million figure. For those same high income earners, every £2 of adjusted income above £150,000 will reduce their annual allowance by £1. The maximum reduction is £30,000, reducing the annual allowance to £10,000 for anyone with an adjusted income of £210,000 or above.
An estimated 364,000 people have been affected by the tapered annual allowance on pensions, based on data from the Office for National Statistics (ONS). This has led many who have ‘maxed out’ their pension allowance to turn to ISAs as an alternative means of boosting their retirement income – particularly since the annual ISA allowance rose from £15,240 to £20,000 this current tax year.
Alternatives to pensions for high earners
Around 131,000 workers in the UK are thought to earn more than £150,000 a year, according to the ONS. However, when property, dividends and other sources of income are accounted for, it is estimated that many more individuals will fall foul of the tapered annual allowance.
This has led to some industry experts predicting that the rising trend in ISA contributions will continue unabated for some time to come. According to Hargreaves Lansdown, regular ISA contributors (as opposed to annual, one-off savers) increased by 40% between January 2016 and November 2017.
So, it seems that high income pension savers are increasingly turning to ISAs as a means of topping up their regular pension contributions.
Pension contributions for high earners
Angela Maher, Managing Director at Acumen, said: “Until very recently, high earners could comfortably put away £255,000 of annual pension contributions, towards an overall pot of £1.8 million. Now, these figures have been slashed dramatically; dropping to £40,000 of annual contributions and a £1 million maximum pot respectively.
“Is it any wonder then that higher earning pension savers have started to look elsewhere for alternative ways to save for retirement? The increase in the annual ISA allowance to £20,000 has proven to be fortuitously timed, allowing higher earners to access a viable alternative that enables them to maximise their savings in a legitimate and tax efficient manner,” Angela added.
If you’re worried about your pension allowance, or would like further advice on how to maximise your retirement prospects, speak to one of our expert financial advisers today. Call Acumen today on 0151 520 4353 or email us at [email protected] for further information or to arrange your free consultation meeting. | 22,518 |
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\begin{document}
\title{Two-layered numbers}
\author{\bfseries H. Behzadipour}
\address{School of Electrical and Computer Engineering\\ University College of Engineering\\ University of Tehran\\ Tehran\\ Iran}
\email{[email protected], [email protected]}
\subjclass[2010]{11R04.}
\keywords{two-layered numbers, weak two-layered numbers, perfect numbers, Zumkeller numbers}
\begin{abstract}
In this paper, first, I introduce two-layered numbers. Two-layered numbers are positive integers that their positive divisors except $1$ can be partitioned into two disjoint subsets. Similarly, I defined a half-layered number as a positive integer $n$ that its proper positive divisors excluding $1$ can be partitioned into two disjoint subsets. I also investigate the properties of two-layered and half-layered numbers and their relation with practical numbers and Zumkeller numbers.
\end{abstract}
\maketitle
\section{Introduction}
A perfect number is a positive integer $n$ that equals the sum of its proper positive divisors. Generalizing the concept of perfect numbers, Zumkeller in \cite{zumkeller} published a sequence of integers that their divisors can be partitioned into two disjoint subsets with equal sum. Clark et al. in \cite{clark} called such integers Zumkeller numbers and investigated some of their properties, and also suggested some conjectures about them. Peng and Bhaskara Rao in \cite{rao} introduced half-Zumkeller numbers and provided interesting results about Zumkeller numbers.
In the present paper, I define two-layered numbers based on the concept of perfect numbers and Zumkeller numbers. A two-layered number is a positive integer $n$ that its positive divisors excluding $1$ can be partitioned into two disjoint subsets of an equal sum. A partition $\set {A, B}$ of the set of positive divisors of $n$ except $1$ is a two-layered partition if each of $A$ and $B$ has the same sum.
In the first section, I investigate the properties of two-layered numbers. For a two-layered number $n$, that sum of its divisors is $\sigma(n)$, the following statements hold (See Proposition \ref{sigmaodd}):
Let $\sigma(n)$ be the sum of all positive divisors of $n$. If $n$ is a two-layered number, then
\begin{enumerate}
\item $\sigma(n)$ is odd.
\item Powers of all odd prime factors of $n$ should be even.
\item $\sigma(n) \geq 2n+1$, so $n$ is abundant.
\end{enumerate}
After that, In theorem \ref{twolayprop}, I prove that The integer $n$ is a two-layered number if and only if $\frac{\sigma(n)-1}{2}-n$ is a sum of distinct proper positive divisors of n excluding 1. I also introduce two methods of generating new two-layered numbers from known two-layered numbers. Suppose that $n$ is a two-layered number and $p$ is a prime number with $(n,p)=1$, then $np^\alpha$ is a two-layered number for any even positive integer $\alpha$ (See Theorem \ref{npalpha}). We can also generate two-layered numbers in another way. Let $n$ be a two-layered number and $p_1^{k_1}p_2^{k_2} \dots p_m^{k_m}$ be the prime factorization of $n$. Then for any nonnegative integers $\alpha_1, \dots \alpha_m$, the integer $$p_1^{k_1+\alpha_1(k_1+1)}p_2^{k_2+\alpha_2(k_2+1)} \dots p_m^{k_m+\alpha_m(k_m+1)}$$ is a two-layered number (See Theorem \ref{secondwaygenerate}).
In the second section of the present paper, I generalize the concept of practical numbers and define semi-practical numbers. A practical number is a positive integer $n$ that every positive integer less than $n$ can be represented as a sum of distinct positive divisors of $n$ \cite{practical}. A positive integer $n$ is a semi-practical number if every positive integer $x$ where $1<x<n$ can be represented as a sum of distinct positive divisors of $n$ excluding $1$ (See Definition \ref{defsemipractical}).
I investigate some properties of semi-practical numbers and their relations with two-layered numbers. For example, every semi-practical number is divisible by $12$ (See Proposition \ref{divis12}). I also proved that a positive integer $n$ is is a semi-practical number if and only if every positive integer $x$ where $1<x<\sigma(n)$, is a sum of distinct positive divisors of $n$ excluding $1$ (See Theorem \ref{spracsigma}). The most important relation between semi-practical numbers and two-layered numbers is that a semi-practical number $n$ is two-layered if and only if $\sigma(n)$ is odd (See Proposition \ref{keyprop}).
In section 3, I define a half-layered number. A positive integer $n$ is said to be a half-layered number if the proper positive divisors of $n$ excluding $1$ can be partitioned into two disjoint non-empty subsets of an equal sum (See Definition \ref{halflay}). A half-layered partition
for a half-layered number $n$ is a partition $\set {A, B}$ of the set of proper positive divisors of $n$ excluding $1$ so that
each of $A$ and $B$ sums to the same value (See Definition \ref{defhalflayeredpartition}).
After these definitions, I investigate the properties of half-layered numbers. For example, A positive integer $n$ is half-layered if and only if $\frac{\sigma(n)-n-1}{2}$ is the sum of some distinct positive proper positive divisors of $n$ (See Proposition \ref{sigma2n}). A positive even integer $n$ is half-layered if and only if $\frac{\sigma(n)-2n-1}{2}$ is the sum (possibly empty
sum) of some distinct positive divisors of $n$ excluding $n$, $\frac{n}{2}$, and $1$ (See Theorem \ref{halflay}). If $n$ is an odd half-layered number, then at least one of the powers of prime factors of $n$ should be even (See Proposition \ref{oddhalflayered}).
Using the definition of half-Zumkeller numbers, we can derive some of the interesting properties of half-layered numbers. A positive integer $n$ is said to be a half-Zumkeller number if the proper positive divisors of $n$ can be partitioned into two disjoint non-empty subsets of an equal sum. A half-Zumkeller partition for a half-Zumkeller number n is a partition $\set{A, B}$ of the set of proper positive divisors of $n$ so that each of $A$ and $B$ sums to the same value (Definition 3 in \cite{rao}). Based on these definition, I prove that if $m$ and $n$ are half-layered numbers with $(m, n)=1$, then $mn$ is half-layered (See Proposition \ref{mnhalflayered}).
After that, I investigate some relations between half-layered and two-layered numbers. For example, let $n$ be even. Then $n$ is half-layered if and only if $n$ admits a two-layered partition
such that $n$ and $\frac{n}{2}$ are in distinct subsets. Therefore, if $n$ is an even half-layered number then $n$ is two-layered (See Proposition \ref{halfzumnn2}). It is also proved that if $n$ is an even two-layered number and If $\sigma(n) < 3n$, then $n$ is half-layered (See Theorem \ref{itisalsoproved}). Let $n$ be even. Then, $n$ is two-layered if and only if either $n$ is half-layered or $\frac{\sigma(n)-3n-1}{2}$ is a sum
(possibly an empty sum) of some positive divisors of $n$ excluding $n$, $\frac{n}{2}$, and $1$ (See Proposition \ref{sigma3n}).
If $6$ divides $n$, $n$ is two-layered, and $\sigma(n) < \frac{10n}{3}$ , then $n$ is half-layered (See Proposition \ref{6dividesn}). If $n$ is two-layered, then $2n$ is half-layered (See Proposition \ref{n2ntwolayered}). Let $n$ be an even half-layered number and $p$ be a prime with (n, p) = 1. Then $np^{\ell}$ is half-
layered for any positive integer $\ell$ (See Proposition \ref{andprimefactorization}). Let $n$ be an even half-layered number and the prime factorization of $n$ be $ p_1^{k_1} p_2^{k_2} /dots p_m^{k_m} $ Then for nonnegative integers $\ell_1, \dots , \ell_m$, the integer
$$ p_1^{k_1+\ell_1(k_1+1)} p_2^{k_2+\ell_2(k_2+1)} \dots p_m^{k_m+\ell_m(k_m+1)} $$
is half-layered (See Theorem \ref{numberandtheprime}).
\section{two-layered numbers}
\label{section1}
\begin{definition}
A positive integer $n$ is a two-layered number if the positive divisors of $n$ excluding $1$ can be partitioned into two disjoint subsets of an equal sum.
\end{definition}
\begin{definition}
A two-layered partition for a two-layered number $n$ is a partition $\set {A,B}$ of the set of positive divisors of $n$ excluding $1$ so that each of $A$ and $B$ sums to the same value.
\end{definition}
\begin{example}
\label{example1}
The number 36 is a two-layered number and its two-layered partition is $\set{A,B}$, where $A = \set{2, 3, 4, 36}$ and $B=\set {6, 9, 12, 18}$. You can check that each of $A$ and $B$ has the sum of $45$. The numbers $72, 144,$ and $200$ are also two-layered. You can find the sequence of two-layered numbers in \cite{twosequenc}.
\end{example}
\begin{proposition}
\label{sigmaodd}
Let $\sigma(n)$ be the sum of all positive divisors of $n$. If $n$ is a two-layered number, then
\begin{enumerate}
\item $\sigma(n)$ is odd.
\item Powers of all odd prime factors of $n$ should be even.
\item $\sigma(n) \geq 2n+1$, so $n$ is abundant.
\end{enumerate}
\end{proposition}
\begin{proof}
$(1):$ If $\sigma(n)$ is even, then $\sigma(n)-1$ is odd, so it is impossible to partition the positive divisors of $n$ into two subset of equal sum.
$(2):$ using $(1)$, the number of odd positive divisors of $n$ is odd. Suppose that the prime factorization of $n$ is $2^{k_0}p_1^{k_1}p_2^{k_2}\dots p_m^{k_m}$. The number of odd positive divisors of $n$ is $(k_1+1)(k_2+1)\dots (k_m+1)$. All of $k_i$ should be even in order to make the product $(k_1+1)(k_2+1)\dots (k_m+1)$ odd.
$(3):$ Let $n$ be a two-layered number with two-layered partition $\set {A,B}$. Without loss of generality we may assume that $n \in A$, so the sum in $A$ is at least $n$ and we can conclude $\sigma(n)-1 \geq 2n$.
\end{proof}
\begin{theorem}
\label{twolayprop}
The integer $n$ is a two-layered number if and only if $\frac{\sigma(n)-1}{2}-n$ is a sum of distinct proper positive divisors of n excluding 1.
\end{theorem}
\begin{proof}
Let $n$ be a two-layered number and its two-layered partition is $\set{A,B}$. Without loss of generality we assume that $n \in A$, so the sum of the remaining elements of $A$ is $\frac{\sigma(n)-1}{2}-n$.
Conversely, if we have a set of proper divisors of $n$ excluding $1$ that its sum is $\frac{\sigma(n)-1}{2}-n$, we can augment this set with $n$ to construct a set of positive divisors of $n$ summing to $\frac{\sigma(n)-1}{2} $. The complementary set of positive divisors of $n$ sums to the same value, and so these two sets form a two-layered partition for $n$.
\end{proof}
With the help of the next two theorems, we can generate some new two-layered numbers by knowing a two-layered number.
\begin{definition} [Definition 1 in \cite{rao}]
A positive integer $n$ is said to be a Zumkeller number if the positive divisors of $n$ can be
partitioned into two disjoint subsets of equal sum. A Zumkeller partition for a Zumkeller number $n$ is
a partition $\set{A, B}$ of the set of positive divisors of $n$ so that each of $A$ and $B$ sums to the same value.
\end{definition}
\begin{theorem}
\label{npalpha}
Let $n$ be a two-layered number and $p$ be a prime number with $(n,p)=1$, then $np^\alpha$ is a two-layered number for any even positive integer $\alpha$.
\end{theorem}
\begin{proof}
Suppose that $\set {A,B}$ is a Zumkeller partition of $n$. Then $\set {(A \setminus \set{1}) \cup (pA)\cup (p^2A)\cup \dots \cup (p^\alpha A), (B \setminus \set{1}) \cup (pB) \cup(p^2B) \cup \dots \cup (p^\alpha B)}$ is a two-layered partition of $np^\alpha$.
\end{proof}
\begin{theorem}
\label{secondwaygenerate}
Suppose that $n$ is a two-layered number and $p_1^{k_1}p_2^{k_2} \dots p_m^{k_m}$ is the prime factorization of $n$. Then for any nonnegative even integers $\alpha_1, \dots \alpha_m$, the integer $$p_1^{k_1+\alpha_1(k_1+1)}p_2^{k_2+\alpha_2(k_2+1)} \dots p_m^{k_m+\alpha_m(k_m+1)}$$ is a two-layered number.
\end{theorem}
\begin{proof}
If we show that $p_1^{k_1+\alpha_1(k-1+1)}p_2^{k_2} \dots p_m^{k_m}$ the proof will be completed. Suppose that $\set {A,B}$ is a Zumkeller partition of $n$. If $D$ is the set of positive divisors of $n$, then $(D \setminus \set {1}) \cup (p_1^{k_1+1} D) \cup (p_1^{2(k_1+1)}D) \cup \dots \cup (p_1^{\alpha_1(k_1+1)}D))$ is the set of positive divisors of $p_1^{k_1+\alpha_1(k-1+1)}p_2^{k_2} \dots p_m^{k_m}$ excluding 1. Therefore a two-layered partition for $p_1^{k_1+\alpha_1(k-1+1)}p_2^{k_2} \dots p_m^{k_m}$ is $\set {A \setminus \set {1} \cup (p_1^{k_1+1}A) \cup (p_1^{2(k_1+1)}A) \cup \dots \cup (p_1^{\alpha_1(k_1+1)}A), B \setminus \set {1} \cup (p_1^{k_1+1}B) \cup (p_1^{2(k_1+1)}B) \cup \dots \cup (p_1^{\alpha_1(k_1+1)}B) }$ and the proof is complete.
\end{proof}
\section{semi-practical numbers and two-layered numbers}
Practical numbers have been introduced by Srinivasan in 1948 as what follows:
\begin{definition}
A positive integer $n$ is a practical number if every positive integer less than $n$ can be represented as a sum of distinct positive divisors of $n$.\cite{practical}
\end{definition}
Because of the structure of two-layered number, if we change the definition of practical numbers and call them semi-practical numbers, we can drive some useful relation between them and two-layered numbers, so I define semi-practical numbers as what follows:
\begin{definition}
\label{defsemipractical}
A positive integer $n$ is practical if every positive integer $x$ where $1<x<n$ can be represented as a sum of distinct positive divisors of $n$ excluding $1$.
\end{definition}
\begin{proposition}
\label{divis12}
Every semi-practical number is divisible by $12$.
\end{proposition}
\begin{proof}
Since we can not write $2,3,$ and $4$ as sums of more than one positive integer greater than $1$, they should be divisors of our semi-practical number.
\end{proof}
\begin{theorem}
\label{spracsigma}
A positive integer $n$ is is a semi-practical number if and only if every positive integer $x$ where $1<x<\sigma(n)$, is a sum of distinct positive divisors of $n$ excluding $1$.
\end{theorem}
\begin{proof}
Suppose that $n$ is a semi-practical number. I introduce an algorithm for writing all positive integer $x$ between $n$ and $\sigma(n)$ as sum of distinct positive divisors of $n$ excluding $1$.
First, let $x$ be $n+1$. Since $n$ is semi-practical, by Propositin \ref{divis12}, it is divisible by $n/2$ and $n/3$. Hence, $n+1 = n/2 + n/3 + r$, where $r$ is a positive integer. By Proposition \ref{divis12}, $n > 6$, so $n+1-n/2-n/3 < n/3$. On the other hand, since $n$ is a semi-practical number and $r<n/3<n$, $r$ is equal to some of distinct divisors of $n$ which are less than $n/3$ and greater than $1$.
For $n+1<x<\sigma(n)$, let the positive divisors of $n$ which are greater than $1$ be written in increasing order as $m_1<m_2 < \dots < m_k$. Now we can write $x= \sum_{i=\ell}^k m_i+r$ where $1\leq \ell \leq k$ and $0 \leq r < m_{\ell -1}$. If $r=0$ then $x$ is a sum of distinct divisors of $n$. If $1<r< m_{\ell -1}$, since $n$ is semi-practical and $r<n$, then we can write $r$ as a sum of distinct divisors of $n$ which are less than $m_{\ell-1}$, so $x$ is a sum of distinct divisors of $n$. If $r=1$, then we can write $x=\sum_{i=\ell+1}^k+r_1$ where $1<r_1<m_{\ell}$. since $n$ is semi-practical and $r<n$, then $r_1$ is sum of distinct divisors of $n$ which are less than $m_{\ell}$, so $x$ is a sum of distinct divisors of $n$.
Conversely, if every positive integer less than $\sigma(n)$ excluding $1$, is a some of distinct positive divisors of $n$ excluding $1$, it is clear that $n$ is semi-practical.
\end{proof}
\begin{proposition}
\label{keyprop}
A semi-practical number $n$ is two-layered if and only if $\sigma(n)$ is odd.
\end{proposition}
\begin{proof}
If $n$ is two-layered number, then $\sigma(n)$ is odd by Proposition \ref{sigmaodd}. Conversely, if $\sigma(n)$ is odd, then $\frac{\sigma(n)-1}{2}$ is a positive integer smaller than $\sigma(n)$. Since $n$ is a semi-practical number, using Proposition \ref{spracsigma}.
\end{proof}
\begin{theorem}
\label{somesum}
Let $n$ be a positive integer and $p$ be a prime with $(n, p) = 1$. Let $D$ be the set of all positive
divisors of $n$ including $1$. The following conditions are equivalent:
\begin{enumerate}
\item $np$ is two-layered.
\item There exist two partitions $\set {D_1,D_2}$ and $\set{D_3, D_4}$ of $D \setminus \set{1}$ such that $$p(\sum_{d \in D_1}d-\sum_{d \in D_2}d)=(\sum_{d \in D_3}d-\sum_{d \in D_4}d).$$
\item There exists a partition $\set {D_1,D_2}$ of $D\setminus \set{1}$ and subsets $A_1 \subseteq D_1$ and $A_2 \subseteq D_2$ such that $$\frac{p+1}{2}(\sum_{d \in D_1}d-\sum_{d \in D_2}d)=(\sum_{d \in A_1}d-\sum_{d \in A_2}d).$$
\end{enumerate}
\end{theorem}
\begin{proof}
It is clear that $(pD) \cup (D\setminus\set{1})$ is the set of all positive divisors of $np$ excluding $1$.
$(1) \Rightarrow$ (2). Suppose that $np$ is two-layered. Hence, there is a two-layered partition $\set {A,B}$ of $(pD) \cup (D\setminus\set{1})$. Let $D_1=\frac{1}{p}(A \cap (pD))$, $D_2=\frac{1}{p}(B \cap (pD))$, $D_3=B \cap (D\setminus\set{1})$, $A \cap (D\setminus \set{1})$, then $$p \sum_{d \in D_1}d + \sum_{d \in D_4}d = p \sum_{d \in D_2}d + \sum_{d \in D_3}d.$$ and the proof is complete.
$(2) \Rightarrow (3)$. Let $A_1 = D_1 \cap D_3$ and $A_2 = D_2\cap D_4$. We have
\begin{align}
\frac{p+1}{2}(\sum_{d \in D_1}d - \sum_{d \in D_2}d) & = \frac{1}{2}[ p (\sum_{d \in D_1}d - \sum_{d \in D_2}d)+(\sum_{d \in D_1}d- \sum_{d \in D_2}d)] \nonumber \\
& = \frac{1}{2} [\sum_{d \in D_3}d - \sum_{d \in D_4}d + \sum_{d \in D_1}d - \sum_{d \in D_2}d] \nonumber\\
& = \frac{1}{2}[2 (\sum_{d \in D_1 \cap D_3}d) - 2 (\sum_{d \in D_2 \cap D_4}d)] \nonumber \\
& = \sum_{d \in A_1}d - \sum_{d \in A_2}d. \nonumber
\end{align}
$(3) \Rightarrow (1) $. We can rewrite the equation in $(3)$ as follows:
$$\frac{p}{2} \sum_{d \in D_1}d + \frac{1}{2} \sum_{d \in A_2} + \frac{1}{2} \sum_{D_1 \setminus A_1}d = \frac{p}{2} \sum_{d \in D_2}d + \frac{1}{2}\sum_{d \in A_1}d + \frac{1}{2} \sum_{d \in D_2 \setminus A_2}d.$$
By multiplying this by $2$, we obtain the two-layered partition $\set{(pD_1)\cup A_2 \cup (D_1-A_1), (pD_2)\cup A_1 \cup (D_2-A_2)}$ for $np$, so $np$ is a two-layered number.
\end{proof}
\begin{proposition}
\label{orderdivisors}
Let the positive divisors of $n$ excluding $1$ be written in increasing order as follows: $ a_1 < a_2 < \dots <a_k =n.$ If $a_{i+1} < 2a_i$ for all $1 \leq i < k$ and $\sigma(n)$ is odd, then $n$ is two-layered.
\end{proposition}
\begin{proof}
Let $b_i = a_i$ or $−a_i$ for each $i$. I will explain how to chose the sign of $b_i$ precisely. Then I
show that $\sum_{i=1}^{k}b_k=0$. Hence, it will imply that $\sigma(n)-1$ can be partitioned into two equal-summed
subsets.
Let $b_k = a_k = n$ and let $b_{k−1} =−a_{k−1}$. Note that $0 < b_k + b_{k−1} < a_{k−1}$ since $a_k < 2a_{k−1}$. Since the
current sum $b_k + b_{k−1}$ is positive, we assign the negative sign to $b_{k−2}$. Then $b_{k−2} < b_k +b_{k−1} + b_{k−2} <
a_{k−1} − a_{k−2} < a_{k−2}$ since $a_{k−1} < 2 a_{k−2}$. If $b_k + b_{k−1} + b_{k−2} \geq 0$, we assign the negative sign to $b_{k−3}$; Otherwise, we assign the positive sign to $b_{k−3}$. Let $s_i$ be $\sum_{j=1}^{k}b_j$. In general, the sign assigned to $b_{i−1}$ is the opposite of the sign of $s_i$ . Let us show inductively that $|s_i | <a_i$ for $1 \leq i \leq k$. It is true for $i =k$. Assume that $|s_{i+1}| < a_{i+1}$. Since the sign of $b_i$ is opposite of the sign of $s_{i+1}$, $|s_i| = ||s_{i+1}| −a_i |$. Note
that $−a_i < |s_{i+1}| −a_i < a_{i+1} − a_i <a_i$ since $a_{i+1} < 2a_i$ . Therefore $|s_i| < a_i$. So $|s_1| <a_1 = 1$. Since $\sigma(n)-1$
is even, $s_1$, which is obtained by assigning a positive or negative sign to each of the terms in $\sigma(n)-1$
is even as well. So $s_1 = 0$. This implies that $\sigma(n)-1$ can be partitioned into two equal-summed subsets. Hence it is two-layered.
\end{proof}
\begin{proposition} [Proposition 1 in \cite{rao}]
\label{factorizationsigma}
Let the prime factorization of $n$ be $\prod_{i=1}^{m}p_i^{k_i}$. Then
$$\sigma(n)=\prod_{i=1}^{m}\frac{p_i^{k_i+1}-1}{p_i-1}$$
and
$$\frac{\sigma(n)}{n}= \prod_{i=1}^{m} \frac{p_i^{k_i+1}-1}{p_i^{k_i}(p_i-1)} < \prod_{i=1}^{m} \frac{p_i}{p_i-1}$$
\end{proposition}
\begin{proposition}
Let the prime factorization of an odd number $n$ be $p_1^k p_2^k \dots p_m^{k_m}$, where $3 \leq p_1 < p_2 < \dots < p_m$. If $n$ is two-layered, then
$$ \prod_{i=1}^{m} \frac{p_i}{p_i - 1} > 2 ,$$
and $m$ is at least $3$. In particular:
\begin{enumerate}
\item If $m \leq 6$, then $p_1=3$, $p_2=5$, $7$ or $11.$
\item If $m \leq 4$, then $p_1=3$, $p_2=5$ or 7.
\item If $m=3$, then $p_1=3$, $p_2=5$, and $p_3=7$ or $11$ or $13$.
\end{enumerate}
\end{proposition}
\begin{proof}
If $n$ is two-layered, then by Propositions \ref{sigmaodd} and \ref{factorizationsigma},
$$ 2 p_1^{k_1}p_2^{k_2} \dots p_m^{k_m} = 2n < \sigma(n) = \prod_{i=1}^{m}(\sum_{j=0}^{k_i}p_i^j).$$
Dividing both sides by $p_1^{k_1}p_2^{k_2} \dots p_m^{k_m}$, we get
$$ 2 < \prod_{i=1}^{m}(\sum_{j=0}^{k_i}p_i^{j-k_i}) < \prod_{i=1}^{m} \frac{p_i}{p_i-1}. $$
If $m \leq 2$, then
$$ \prod_{i=1}^{m} \frac{p_i}{p_i-1} \leq \frac{3}{2} \times \frac{5}{4} < 2 $$
Therefore $m \geq 3$. The parts of $1 - 3$ follows by verifying the condition $\prod_{i=1}^{m} \frac{p_i}{p_i-1} > 2$ directly as given below.
1. Let $m \leq 6$. If $p_1 \neq 3$, then $p_1 \geq 5$ and
$$ \prod_{i=1}^{m} \frac{p_i}{p_i-1} \leq \frac{5}{4} \times \frac{7}{6} \times \frac{11}{10} \times \frac{13}{12} \times \frac{17}{16} \times \frac{19}{18} < 2.$$
Therefore, $p_1 = 3$. If $p_2 > 11$, then $p2 \geq 13$ and
$$ \prod_{i=1}^{m} \frac{p_i}{p_i-1} \leq \frac{3}{2} \times \frac{13}{12} \times \frac{17}{16} \times \frac{19}{18} \times \frac{23}{22} \times \frac{29}{28} < 2.$$
Hence, $p_2 \leq 11$. This implies that $p_2 = 5$, $7$ or $11$.
2. Let $m \leq 4$. By $1$, $p_1 = 3$. If $p_2 > 7$, then $p_2 \geq 11$, so
$$ \prod_{i=1}^{m} \frac{p_i}{p_i-1} \leq \frac{3}{2} \times \frac{11}{10} \times \frac{13}{12} \times \frac{17}{16} < 2.$$
Therefore, $p_2 \leq 7$. This implies that $p_2 = 5$ or $7$.
3. Let $m = 3$. By $1$, $p_1 = 3$. If $p_2 \neq 5$, then $p_2 \geq 7$ and $p3 \geq 11$. So
$$ \prod_{i=1}^{3} \frac{p_i}{p_i-1} \leq \frac{3}{2} \times \frac{7}{6} \times \frac{11}{10} < 2.$$
Hence $p_2 = 5.$
If $p_3 \geq 17$, then
$$ \prod_{i=1}^{3} \frac{p_i}{p_i-1} \leq \frac{3}{2} \times \frac{5}{4} \times \frac{17}{16} < 2.$$
Hence, $p_3 < 17$ and consequently $p_3 = 7$, $11$ or $13$.
\end{proof}
\section{half-layered numbers}
\begin{definition}
\label{defhalflayered}
A positive integer $n$ is said to be a half-layered number if the proper positive divisors
of $n$ excluding $1$ can be partitioned into two disjoint non-empty subsets of equal sum.
\end{definition}
\begin{definition}
\label{defhalflayeredpartition}
A half-layered partition
for a half-layered number $n$ is a partition $\set {A, B}$ of the set of proper positive divisors of $n$ excluding $1$ so that
each of $A$ and $B$ sums to the same value.
\end{definition}
\begin{proposition}
\label{sigma2n}
A positive integer $n$ is half-layered if and only if $\frac{\sigma(n)-n-1}{2}$ is the sum of some distinct positive
proper positive divisors of $n$.
\end{proposition}
\begin{example}
In Example \ref{example1}, we saw that $36$ was a two-layered number. It is also a half-layered number and its half-layered partition is $\set{A,B}$, where $A = \set{2, 3, 4, 18}$ and $B=\set {6,9,12}$. You can check that each of $A$ and $B$ has the sum of $27$. The numbers $72, 105,$ and $144$ are also half-layered. You can find the sequence of half-layered numbers in \cite{halfsequenc}.
\end{example}
\begin{theorem}
\label{halflay}
A positive even integer $n$ is half-layered if and only if $\frac{\sigma(n)-2n-1}{2}$ is the sum (possibly empty
sum) of some distinct positive divisors of $n$ excluding $n$, $\frac{n}{2}$, and $1$.
\end{theorem}
\begin{proof}
An even number $n$ is half-layered if and only if there exists a which is the sum (possibly
empty sum) of some positive divisors of $n$ excluding $n$, $\frac{n}{2}$, and $1$ such that
$$\frac{n}{2}+a = \frac{\sigma(n)-n-1}{2}.$$
Therefore, $a = \frac{\sigma(n)-2n-1}{2}.$
\end{proof}
\begin{example}
The number $3^4 \times 2^4$ is a half-layered number, since
$$\frac{\sigma(3^4 \times 2^4)- 2 ( 3^4 \times 2^4)-1}{2}=579=432+108+36+3$$
is a sum of positive divisors of $3^4 \times 2^4$ excluding $3^4 \times 2^4$ , $3^4 \times 2^3$, and $1$. Hence, by Theorem \ref{halflay}, it is a half-layered number.
\end{example}
\begin{proposition}
\label{oddhalflayered}
If $n$ is an odd half-layered number, then at least one of the powers of prime factors of $n$ should be even.
\end{proposition}
\begin{proof}
If n is odd and half-layered, then $\sigma(n) − n - 1$ must be even and $\sigma(n)$ must be even. Let the prime factorization of $n$ be $\prod_{i=1}^{m}p_i^{k_i}$. Then $\sigma(n) = \prod_{i=1}^{m}(\sum_{j=0}^{k_i}p_i^j)$. If $\sigma(n)$ is odd, then there exists one $k-i$ which is odd.
\end{proof}
\begin{definition} [Definition 3 in \cite{rao}]
A positive integer $n$ is said to be a half-Zumkeller number if the proper positive divisors of $n$ can be partitioned into two disjoint non-empty subsets of an equal sum. A half-Zumkeller partition for a half-Zumkeller number n is a partition $\set{A, B}$ of the set of proper positive divisors of $n$ so that each of $A$ and $B$ sums to the same value.
\end{definition}
\begin{proposition}
\label{mnhalflayered}
If $m$ and $n$ are half-layered numbers with $(m, n)=1$, then $mn$ is half-layered.
\end{proposition}
\begin{proof}
Let $M$ be the set of proper positive divisors of $m$ and let $\set{M_1,M_2}$ be a half-Zumkeller partition
for $m$. Let $N$be the set of proper positive divisors of $n$ and let $\set{N1, N2}$ be a half-Zumkeller partition
for $n$. Since $(m,n) = 1$, then the set of proper positive divisors of $mn$ is $(MN) \cup (nM) \cup (mN)$. Observe
that $\set{(M_1N\setminus \set{1}) \cup (mN_1) \cup (nM_1), (M_2N \setminus \set{1}) \cup (mN_2) \cup (nM_2)}$ is a half-layered partition for $mn$. Therefore $mn$ is
half-layered.
\end{proof}
\begin{proposition}
\label{halfzumnn2}
Let $n$ be even. Then $n$ is half-layered if and only if $n$ admits a two-layered partition
such that $n$ and $\frac{n}{2}$ are in distinct subsets. Therefore, if $n$ is an even half-layered number then $n$ is two-layered.
\end{proposition}
\begin{proof}
Let $n$ be even. Let $D$ be the set of all positive divisors of $n$ excluding $1$. The number $n$ is half-layered if
and only if there exists $A \subset D \setminus \set{n, \frac{n}{2}}$ such that
$$ \frac{n}{2} + \sum_{a \in A}a = \sum_{b \in D, b \not\in \set{n, \frac{n}{2}} \cup A}b.$$
That is,
$$ n + \sum_{a \in A}a = \frac{n}{2} + \sum_{b \in D, b \not\in \set{n, \frac{n}{2}} \cup A}b.$$
This is equivalent to saying that $n$ admits a two-layered partition such that $n$ and $\frac{n}{2}$ are in distinct
subsets.
\end{proof}
\begin{theorem}
\label{itisalsoproved}
Let $n$ be an even two-layered number. If $\sigma(n) < 3n$, then $n$ is half-layered.
\end{theorem}
\begin{proof}
Since $n$ and $\frac{n}{2}$ together sum to more than $\frac{\sigma(n)}{2}$ , they must be in different subsets in any
two-layered partition for $n$. Therefore, by Proposition \ref{halfzumnn2}, $n$ is half-layered.
\end{proof}
\begin{proposition}
\label{sigma3n}
Let $n$ be even. Then, $n$ is two-layered if and only if either $n$ is half-layered or $\frac{\sigma(n)-3n-1}{2}$ is a sum
(possibly an empty sum) of some positive divisors of $n$ excluding $n$, $\frac{n}{2}$, and $1$.
\end{proposition}
\begin{proof}
Let $n$ be even. If $n$ is two-layered but not half-layered, then by Proposition \ref{halfzumnn2}, any two-layered
partition of the positive divisors of $n$ must have $n$ and $\frac{n}{2}$ in the same subsets. In other words, there
exists a which is a sum (possibly an empty sum) of some positive divisors of $n$ excluding $n$, $\frac{n}{2}$, and $1$ such that
$$ 2(n + \frac{n}{2}+a) = \sigma(n)-1$$
So, $a= \frac{\sigma(n)-3n-1}{2}$. Therefore, the number $ \frac{\sigma(n)-3n-1}{2}$ is a sum (possibly an empty sum) of some positive
divisors of $n$ excluding $n$, $\frac{n}{2}$, and $1$.
If $n$ is half-layered, then $n$ is two-layered by Proposition \ref{halfzumnn2}. If $ \frac{\sigma(n)-3n-1}{2}$ is a sum (possibly an empty
sum) of some positive divisors of $n$ excluding $n$, $\frac{n}{2}$, and $1$, then
$$ \frac{\sigma(n)-2n-1}{2} = \frac{\sigma(n)- 3n -1}{2} + \frac{n}{2}$$
is a sum of some positive divisors of $n$ excluding $n$, and $1$. By Theorem \ref{twolayprop}, the number $n$ is two-layered.
\end{proof}
\begin{proposition}
\label{6dividesn}
If $6$ divides $n$, $n$ is two-layered, and $\sigma(n) < \frac{10n}{3}$ , then $n$ is half-layered.
\end{proposition}
\begin{proof}
If $n$ is not half-layered, by Proposition \ref{sigma3n}, $\frac{\sigma(n)-3n-1}{2}$ is a sum (might be an empty sum)
of some positive divisors of $n$ excluding $n$, $\frac{n}{2}$, and $1$. Then,
$$ \frac{\sigma(n)-2n-1}{2} = \frac{\sigma(n)- 3n -1}{2} + \frac{n}{3} + \frac{n}{6}.$$
Since $\sigma(n)/n < \frac{10}{3}$ we have that $\frac{\sigma(n)-3n-1}{2} < \frac{n}{6}$. Hence $\frac{\sigma(n)-2n-1}{2}$ is a sum of some positive
divisors of $n$ excluding $n$, $\frac{n}{2}$, and $1$. By Proposition \ref{sigma2n}, $n$ is half layered. This is a contradiction.
\end{proof}
\begin{proposition}
\label{n2ntwolayered}
If $n$ is two-layered, then $2n$ is half-layered.
\end{proposition}
\begin{proof}
Let $n = 2^k L$ with $k$ a nonnegative integer and $L$ an odd number, be a two-layered number. Then
all positive divisors of $n$ excluding $1$ can be partitioned into two disjoint equal-summed subsets $D_1$ and $D_2$.
Observe that every positive divisor of $2n$ which is not a positive divisor of $n$ can be written as $2^{k+1} \ell$
where $\ell$ is a positive divisor of $L$. Observe that $2^k \ell$ is either in $D_1$ or $D_2$. Without loss of generality,
assume that $2^k \ell$ is in $D_1$. In this case, we move $2^k \ell$ to $D_2$ and add $2^{k+1} \ell$ to $D_1$. Perform this procedure
to all positive divisors of $2n$ which are not positive divisors of $n$ except $2n$ itself. This procedure will
yield an equal-summed partition of all positive divisors of $2n$ except $2n$ itself. This shows that $2n$ is
half-Zumkeller.
\end{proof}
\begin{corollary}
Let $n$ be even and the prime factorization of $n$ be $2^k p_1^{k_1} \dots p_m^{k_m}$. If $n$ is two-layered but not half-
layered, then $2^i p_1^{k_1} \dots p_m^{k_m}$ is not two-layered for any $i \leq k-1$, and $2^i p_1^{k_1} \dots p_m^{k_m}$ is half-layered for any
$i \geq k + 1$.
\end{corollary}
\begin{proposition}
\label{andprimefactorization}
Let $n$ be an even half-layered number and $p$ be a prime with (n, p) = 1. Then $np^{\ell}$ is half-
layered for any positive integer $\ell$.
\end{proposition}
\begin{proof}
Since $n$ is an even half-layered number, the set of all positive divisors of $n$, excluding $1$, denoted by $D_0$
can be partitioned into two disjoint subsets $A_0$ and $B_0$ so that the sums of the two subsets are equal
and $n$ and $\frac{n}{2}$ are in distinct subsets (by Proposition \ref{halfzumnn2}).
Group the positive divisors of $np^{\ell}$ except $1$ into $\ell +1$ groups $D_0, D_1,\dots D_{\ell}$ according to how many positive
divisors of $p$ they admit, i.e., $D_i$ consists of all positive divisors of $np^{\ell}$ admitting $i$ positive divisors
of $p$. Then each $D_i$ can be partitioned into two disjoint subsets so that the sums of the two subsets
are equal and $np^i$ and $\frac{np^i}{2}$ are in distinct subsets according to the two-layered partition of the set $D_0$. Therefore all positive divisors of $np^{\ell}$ excluding $1$ can be partitioned into two disjoint subsets so that the sum
of these two subsets equal and $np^{\ell}$ and $\frac{np^{\ell}}{2}$ are in distinct subsets. By Proposition \ref{halfzumnn2}, $np^{\ell}$ is half-
layered.
\end{proof}
\begin{corollary}
If $n$ is an even half-layered number and $m$ is a positive integer with $(n,m) = 1$, then $nm$ is
half-layered.
\end{corollary}
\begin{theorem}
\label{numberandtheprime}
Let $n$ be an even half-layered number and the prime factorization of $n$ be $ p_1^{k_1} p_2^{k_2} /dots p_m^{k_m} $ Then for nonnegative integers $\ell_1, \dots , \ell_m$, the integer
$$ p_1^{k_1+\ell_1(k_1+1)} p_2^{k_2+\ell_2(k_2+1)} \dots p_m^{k_m+\ell_m(k_m+1)} $$
is half-layered.
\end{theorem}
\begin{proof}
It is sufficient to show that $ p_1^{k_1+\ell_1(k_1+1)} p_2^{k_2} \dots p_m^{k_m} $ is half-layered if $ p_1^{k_1} p_2^{k_2} \dots p_m^{k_m} $ is an
even half-layered number. Assume that $ p_1^{k_1} p_2^{k_2} \dots p_m^{k_m} $ is even and half-layered, then the set
of all positive divisors of $n$ excluding $1$, denoted by $D_0$ can be partitioned into two disjoint subsets $A_0$ and $B_0$ so
that the sums of the two subsets are equal and $n$ and $\frac{n}{2}$ are in distinct subsets (by Proposition \ref{halfzumnn2}).
Note that the positive divisors of $ p_1^{k_1+\ell_1(k_1+1)} p_2^{k_2} \dots p_m^{k_m} $ excluding $1$ can be partitioned into $\ell_1 + 1$ disjoint groups
$D_i, 0 \leq i \leq \ell_1$, where elements in $D_i$ are obtained by multiplying $p_1^{i(k_1+1)}$ with elements in $D_0$. Using
the partition $A_0, B_0$ of $D0$ we can partition every $D_i$ into two disjoint subsets $A_i$ and $B_i$ so that
the sums of the corresponding subsets are equal and $ np_1^{i(k_1)+1}$ and $\frac{np_1^{i(k_1)+1}}{2}$ are in distinct subsets.
Therefore, the set of all positive divisors of $ p_1^{k_1+\ell_1(k_1+1)} p_2^{k_2} \dots p_m^{k_m} $ excluding $1$ can be partitioned into two disjoint equal-summed subsets and $ p_1^{k_1+\ell_1(k_1+1)} p_2^{k_2} \dots p_m^{k_m} $ and $\frac{p_1^{k_1+\ell_1(k_1+1)} p_2^{k_2} \dots p_m^{k_m}}{2} $ are in distinct subsets. By
Proposition \ref{halfzumnn2}, $ p_1^{k_1+\ell_1(k_1+1)} p_2^{k_2} \dots p_m^{k_m} $ is half-layered.
\end{proof}
\begin{theorem}
Let $n$ be an even integer and $p$ be a prime with $(n, p) = 1$. Let $D $ be the set of all positive
divisors of $n$ excluding $1$. Then the following conditions are equivalent:
\begin{enumerate}
\item $np$ is half-layered.
\item There exist two partitions $\set{D_1, D_2}$ and $\set{D_3, D_4}$ of $D$ such that $n$ is in $D_1$, $\frac{n}{2}$ is in $D_2$ and
$$ p (\sum_{d \in D_1}d - \sum_{d \in D_2} d) = \sum_{d \in D_3}d - \sum_{d \in D_4}d. $$
\item There exists a partition $\set{D_1, D_2}$ of $D$ and subsets $A_1 \subseteq D_1$and $A_2 \subseteq D_2$ such that $n$ is in $D_1$, $\frac{n}{2}$ is in $D_2$
and
$$ \frac{p+1}{2} (\sum_{d \in D_1}d - \sum_{d \in D_2} d) = \sum_{d \in A_1}d - \sum_{d \in A_2}d. $$
\end{enumerate}
\end{theorem}
\begin{proof}
By Proposition \ref{halfzumnn2}, $np$ is half-layered if and only if there is a two-layered partition $\set{A, B}$ of
$(pD) \cup D $such that $n \in A$ and $\frac{n}{2} \in B$. The rest of the proof follows along the lines of the proof of
Theorem \ref{somesum}.
\end{proof}
\begin{proposition}
If $a_1 < a_2 < \dots < a_k = n$ are all positive divisors of an even number $n$ excluding $1$ with $a_{i+1} < 2a_i$
for all $i$ and $\sigma(n)$ is odd, then $n$ is half-layered.
\end{proposition}
\begin{proof}
Note that in the proof of Proposition \ref{orderdivisors}, $b_k = n$ and $b_{k−1} = - \frac{n}{2}$ have different signs. So we
get a two-layered partition of $n$ such that $n$ and $\frac{n}{2}$ are in distinct subsets. By Proposition \ref{halfzumnn2}, $n$ is
half-layered.
\end{proof} | 42,894 |
The tempo of the city had changed sharply.
The buildings were higher.
The parties were bigger.
The morals were looser and the liquor was cheaper.
The restlessness approached hysteria.
... Who is this Gatsby?
Mr. Gatsby formally invites you to the Julia Morgan Ballroom for a Grand Affair.
During the time of Prohibition, the Speakeasy was born and all levels of society began to party behind closed doors. Join us as we take you back to the Roaring 20's and re-create the feeling of days gone by. Taking place at the historic Julia Morgan Ballroom, this event will feature beautiful Art Deco decor, specialty cocktails, music, dancing and a few surprises. Feel free to dress in theme for this event. Costumes are encouraged, but not required.
The early portion of the evening will feature traditional swing, jazz, and 1920's themed dance music. Later the night will take a turn bringing this 1920's themed event to 2012 with a DJ playing to hits from this era. The event will also feature other 1920's period decor and themes, including a photobooth with props and more.
See a trailer for the inspiration for this extravaganza below.
Feeling inspired to wear a costume? Check out some of these fine local establishments for rentals and purchases.
Relic Vintage (mention The Great Gatsby Party for 10% off)
In order to purchase these tickets in installments, you'll need an Eventbrite account. Log in or sign up for a free account to continue.
Share The Great Gatsby Party at Julia Morgan BallroomShare Tweet | 292,171 |
Phase I Cultural Resource Survey of a 3.5 Acre Borrow Site West of State Route 657, Hancock County, Survey of a 3.5 Acre Borrow Site West of State Route 657, Hancock County, Kentucky. Richard Stallings, Nancy Ross-Stallings. 1992 ( tDAR id: 142718)
Keywords
Investigation Types
Site Evaluation / Testing
General
Borrow Area • Cultural Resource • Cultural Resource Survey • No Sites • White Cemetery): Department of Transportation ; Brad Mar, Inc
Prepared By(s): Cultural Horizons, Inc.
Record Identifiers
NADB document id number(s): 579212
NADB citation id number(s): 000000073507
Notes
General Note: Sent from: Cultural Horizons, Inc.
General Note: Contract number: BRZ 0203(208)
General Note: Submitted to: Brad Mar, Inc | 243,826 |
You guys, I married my best friend on February 18th, 2012. Here are just a FEW of the photos from that day. I’ll post more as I share more about our day.
All photography taken by Katelyn James Photography. You can view our FULL wedding gallery here OR our wedding blog post on our photographer’s page well!.). | 372,509 |
TITLE: calculate the internally studentized residual
QUESTION [1 upvotes]: This is from my textbook:
it says that ...an ordinary residual divided by an estimate of its standard deviation $s(e_{i})$
As we can see from the example that mean for four residuals is 0, so $s(e_{i})=\sqrt{\frac{(-0.2-0)^2+(0.6-0)^2+(-0.6)^2+(0.2-0)^2}{4-1}}=\sqrt{\frac{0.8}{3}}\neq\sqrt{0.4(1-0.7)}$
where did I get it wrong?
REPLY [1 votes]: The studentized residuals are
$$t_i=\frac{\epsilon_i}{\hat\sigma\sqrt{1-h_{ii}}}$$
Where $\epsilon_i$ is the residual, $h_{ii}$ the leverage and $\hat\sigma$ is the estimate of the standard deviation of residuals, that is
$$\hat\sigma^2=\frac1{n-m}\sum_{i=1}^n\epsilon_i^2$$
Where $n$ is the number of observations (here $4$) and $m$ the number of estimated parameters (here $2$). The sum of square residuals is $0.8$, hence $\hat\sigma^2=\frac12\times0.8$, and
$$t_i=\frac{-0.2}{\sqrt{0.4\times(1-0.7)}}$$
See also studentized residuals on Wikipedia and this question on Cross Validated. | 136,846 |
Anyone visiting the Frome Medical Centre in the next few weeks is in for a special and pleasant surprise.
Located in the two large open spaces behind the building and in view of the main foyer and waiting areas are two large and stunning wooden pieces of sculpture from local sculptor Anthony Rogers.
Anthony, who has his workshops just outside Frome, will already be familiar to many local residents. His sculpted seat by the river walk at Welshmill Park has now been in situ for a couple of years.
Anthony said, “I really enjoyed creating these two sculptures. Both works were sourced from local wood. The tall upright piece was carved from an oak trunk from Longleat Forest. The other large circular construction was made from seven separate oak beams bolted together. Placed next to such a modern and impressive building as the new Frome Health Centre, I think that the large wood sculptures look absolutely right.”
The erection of the two sculptures is part of the ongoing commitment of the Frome Medical Practice to supporting local art. Community art classes and local schools have recently exhibited at the new health centre.
Dr Tina Merry, the senior partner at Frome Medical Practice said, “We are absolutely delighted to have Anthony’s sculptures here at the Health Centre. His very distinctive, flowing carving style really compliments the building, and they have already been a big hit, not only with our patients but also with everyone who works here.”
Pictured: The two sculptures by Anthony Rogers at the Frome Health Centre | 273,253 |
We are the largest supplier of OEM recycled Suzuki car/truck body parts in North America.
We reserve an extensive Suzuki Equator Header Panels, quarter panels, quarter windows, grilles, radiators supports, hoods,
tailgates, pillars, roofs, wheels and more. We offer up to three years warranty and 30-day low price guarantee on all
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Buy a recycled OEM Suzuki Equator body parts from a trustworthy salvage yard! For more information, contact us by phone or
browse our Suzuki Equator body parts inventory. | 168,399 |
BACHELOR OF PHYSIOTHERAPY ABOUT DEPARTMENT
ABOUT DEPARTMENT
Course: BPT (Bachelor of Physiotherapy)
Course Duration: 4 Years 6 months including 6 months Internship
Affiliation: RK University
Approved by:.
- Awarded the best college of the year for 2010 by IAP (Indian Association of Physiotherapists).
- 1st in Gujarat and 6th in India providing bachelor (BPT), Master (MPT), and Doctoral (PhD) level programs under one roof.
- Successfully accomplished 6th National Student Summit- Physiofest 2015.
- Research journal / magazine (ISSN/ISBN) publication rights for Quarterly Published Physiotherapy News Letter “physioforum”.
- Exposure to more than 1300 patients every month.
- More than 150 projects at undergraduate level.
- Clinical training in well known hospitals and physiotherapy centers like Gokul Super specialty Hospital, Madhuram Hospital, Civil Hospital, Ramkrishna Ashram, RK Physiotherapy and Rehabilitation Research Center etc.
- Active learning techniques to make students’ learning easy and engrossing.
- Lectures are delivered in an audio-visual mode with LCD projectors.
- Detailed lecture notes in easy language are given to students.
- Bio kinesiology projects in BPT
- Hands on practice of various physiotherapeutic techniques.
- Bedside Learning for clinical training
- Classroom Case presentations | 16,008 |
World’s first self-powered CO2 gas sensor
GSS (Gas Sending Solutions) from Scotland has collaborated with EnOcean to incorporate its wireless self-powering technology into the creation of the world’s first self-powered CO2 gas sensor. GSS showcased it at the ISH 2011 international tradefair in March at Frankfurt, Germany.
The COZIR CO2 sensor operates without batteries and asks for no maintenance at all. Apart from consuming lesser energy, it has a minimal warming up time that does not exceed 2 seconds and is an asset for various purposes such as heating, horticultural control, ventilation and air conditions systems (HVAC), monitoring of Indoor air quality (IAQ) and building control.
EnOcean’s unique wireless technology enables the COZIR to power itself from energy derived out of temperature, light or motion within its vicinity. If there is sufficient room light, then the sensor can derive enough power to provide three readings in every 10 minutes and send the information to a receiving device that sets off an alarm for opening the ventilation mechanism.
However, in case where there is no room light, then the sensor functions on its stored energy and sends values only when they appear to be critical.
The COZIR functions on the NDIR (non-dispersive infra-red configuration) technology that is patented under GSS and is available in three different ranges: 0-2 percent, 0-1 percent or 0 to 2000ppm. | 293,143 |
Home Video
MELISSA BLOCK, HOST:
Finally, this hour, a home video recommendation from our movie critic, Bob Mondello. This week, he's looking back a half century to a groundbreaking musical that won 10 Oscars, "West Side Story."
BOB MONDELLO, BYLINE: Clean cut delinquents and pressed jeans. Giggle all you want. You know the moves.
(SOUNDBITE OF MOVIE, "WEST SIDE STORY")
MONDELLO: Finger snaps, switchblades, Romeo and Juliet as a gang war pictured in colors that wouldn't be out of place in Bollywood. The streetscape's real, the acting and lip-syncing unbelievably artificial. In 1961, it was thought shattering. Today, the "Glee" version looks tough by comparison, but and a leg and, before you know it, they're all soaring.
(SOUNDBITE OF MOVIE, "WEST SIDE STORY")
MONDELLO: Believe it or not, after two years on Broadway, everyone was claiming Leonard Bernstein's music was unhummable, Stephen Sondheim's lyrics too tricky. But just weeks after the movie came out, the songs were all standards.
(SOUNDBITE OF SONG, "MARIA")
RICHARD BEYMER: (as Tony, Singing) Maria. I just met a girl named Maria and suddenly that name will never be the same to me.
MONDELLO: Among the four disc set's blu-ray extras, Stephen Sondheim knocking his own lyrics.
STEPHEN SONDHEIM: In "Tonight," I like to quote - today, the world was just an address, a place for me to live in. That's such a writer's line. You can't believe the guy, particularly an ex-leader of a gang, is saying - is singing that. You know, that's very writerly.
MONDELLO: Also, a documentary focusing on the film's legacy, everything from a solo harmonica version of the score to the spoof, "West Bank Story," which turns Anglos and Puerto Ricans into Arabs and Israelis. And a few people recall seeing "Punk Side Story."
UNIDENTIFIED MAN: You hear "One Hand, One Heart" at warp speed. Make it one hand, one hand, make it one heart, one heart. It's pretty incredible.
MONDELLO: Not to mention "Glee," spending most of the season so far in a multi-show buildup to a high school production of "West Side Story" cleverly timed to peak just as this 50th anniversary DVD is released, an unsubtle, but impressive bit of product placement for Fox Home Entertainment on Fox TV.
I'm Bob Mondello.
(SOUNDBITE OF MOVIE, "WEST SIDE STORY")
BEYMER: (as Tony, Singing) Play it cool, boy, real. | 118,704 |
\begin{document}
\begin{abstract}
We prove that the set of closed orbits in a real reductive representation contains a subset which is open with respect to the real Zariski topology if it has non-empty interior. In particular the set of closed orbits is dense.
\end{abstract}
\maketitle
\section{Introduction}
We consider the representation $G\times V\to V$ of a real Lie group on a finite dimensional real vector space $V$.
We say that the representation is real reductive, if there exists a complex reductive Lie group $U^\ce$ with Cartan involution $\theta$ such that $G$ is a $\theta$-stable subgroup of $U^\ce$ with finitely many connected components and if the representation $G\times V\to V$ extends to a holomorphic representation of $U^\ce$ on a complex vector space containing $V$ as a $G$-invariant subspace.
Denoting by $V_c$ the set of closed $G$-orbits in $V$,
our main result is the following.
\begin{mainTheorem}\label{theorem:closedOrbits}
Let $G\times V\to V$ be a real reductive representation and $W$ a subspace of $V$ such that $V_c\cap W$ has non-empty interior in $W$. Then $V_c\cap W$ contains a subset $\mathcal U$ of $W$ which is open in $W$ with respect to the real Zariski topology. In particular, $V_c\cap W$ is dense in $W$. If $W$ is $G$-invariant, then $\mathcal U$ can be chosen to be $G$-invariant.
\end{mainTheorem}
Theorem~\ref{theorem:closedOrbits} is a generalization of a result of Birkes for representations of real forms. If $G$ is a real form of $U^\ce$ and if the $G$-representation $V$ is given by restriction of a holomorphic representation of $U^\ce$ on $V^\ce$, then it is shown in \cite{Birkes} that the closed $G$-orbits in $V$ correspond to closed $U^\ce$-orbits in $V^\ce$. We will give the usual proof using moment maps for this result. Using a stratification of $V$ defined by orbit types of closed orbits, this enables us to prove Theorem~\ref{theorem:closedOrbits} for real forms.
In our general situation, where $G$ is not a real form, the idea is to complexify the $G$-representation $V$. If $G$ is semisimple, $G$ is a compatible real form of its universal complexification $G^\ce$ and the $G$-representation $V$ extends to a holomorphic $G^\ce$-representation $V^\ce$, so the proof for real forms applies. If $G$ is commutative, the weight space decomposition can be used in order to give an explicit description of the set of closed orbits.
The mixed case, where $G$ is a product of its center and a semisimple subgroup, is more sophisticated. We can not describe the closed $G$-orbits in terms of the closed orbits of the semisimple subgroup and the center. Instead, we will construct a group $H$ which contains $G$, which differs from $G$ only by a subgroup of its center and which is a real form of a complex reductive group $U^\ce$. Moreover $H$ admits a representation on $V$ which extends the $G$-representation and which is given by restriction of a holomorphic $U^\ce$-representation on $V^\ce$. Thus Theorem~\ref{theorem:closedOrbits} for real forms applies to the representation of $H$. Then it remains to establish a relation between the closed $G$-orbits and the closed $H$-orbits. Our main tool is the fact, that the closed $G$-orbits inside the set of semistable points $\mathcal S_G(\Mp):=\{z\in Z; \overline{G\cdot z}\cap\mu_\liep^{-1}(0)\neq\emptyset\}$ with respect to a restricted moment map $\mu_\liep$ (see Section~\ref{section:rep}) are parametrized by the orbits of a maximal compact subgroup of $G$ in the zero-fiber $\mu_\liep^{-1}(0)$ (\cite{HSch}, \cite{Kirwan}, \cite{Ness}). In our situation a $G$-orbit is closed inside the set of semistable points with respect to a standard moment map if the corresponding $H$-orbit is closed inside the set of semistable points with respect to a shifted standard moment map. Since we do not know if the latter set of semistable points is dense, we give an equivalent formulation of Theorem~\ref{theorem:closedOrbits} in projective space. Here the set of semistable points is dense by \cite{HMig}.
It should be noted that in the special case where $G$ is complex reductive and where its representation on $V$ is holomorphic, the situation is much simpler. Here the strata defined with respect to orbit types of closed orbits are locally closed with respect to the complex Zariski topology. Then the assumption that the set of closed orbits has non-empty interior in $W$ implies that $\mathcal U$ can be chosen to be the intersection of a stratum with $W$. In particular all orbits which intersect $\mathcal U$ are of the same orbit type. In the real case this is wrong. For this consider the adjoint representation of a real semisimple group. Here there exist finitely many Cartan subalgebras. An orbit through a regular element is closed and the isotropy equals the centralizer of the Cartan subalgebra. Therefore each Cartan subalgebra defines an open stratum which consists of closed orbits and the union of these open strata is dense.
Theorem~\ref{theorem:closedOrbits} may also be applied to actions on Kählerian manifolds $Z$ which admit a restricted moment map $\mu_\liep$ with respect to the action of $G$. If $z\in Z$ is contained in the zero fiber $\mu_\liep^{-1}(0)$, then there exists a geometric $G$-slice at $z$ (\cite{HSch}). In particular the $G$-action in a neighborhood of $z$ is described by the isotropy representation of $G_z$ on the tangent space $T_zZ$. Applying Theorem~\ref{theorem:closedOrbits} to the isotropy representation, we get
\begin{corollary}\label{corollary:manifolds}
Assume that there exists a restricted moment map $\mu_\liep$ for the action of $G$ on $Z$ and that the topological Hilbert quotient $\quot {\mathcal S_G(\Mp)}G$ is connected. If the set of closed $G$-orbits inside the set of semistable points $\mathcal S_G(\Mp)$ has non-empty interior, then its interior is dense in $\mathcal S_G(\Mp)$.
\end{corollary}
An application of Theorem~\ref{theorem:closedOrbits} is given in \cite{St}. If $H$ is a generic isotropy of the $G$-action on $Z$, define $Y:=\{z\in Z; G_z=H\text{ and $G\cdot z$ is closed}\}$ and let $\overline Y$ denote the closure of $Y$. Let $\mathcal N_G(H)$ denote the normalizer of $H$ in $G$. The main result in \cite{St} is a description of the topological Hilbert quotient $\quot ZG$ in terms of the quotient $\quot Y{\mathcal N_G(H)}$. Here Theorem~\ref{theorem:closedOrbits} applies to show that $Y$ is smooth.
The author would like to thank P.~Heinzner for
carefully reading the manuscript of this paper.
\section{Real reductive representations}
Let $U$ be a compact Lie group and $U^\ce$ its complexification. We have a Cartan decomposition $U^\ce=U\exp(\im\lieu)$ where $\lieu$ is the Lie algebra of $U$. The corresponding Cartan involution $\theta$ is given by $\theta(u\exp(\xi))=u\exp(-\xi)$. We call a closed real subgroup $G$ of $U^\ce$ \emph{compatible}, if it is invariant under the Cartan involution $\theta$ and consists of only finitely many connected components. Equivalently, a Lie subgroup $G$ is compatible if and only if $G=K\exp(\liep)$ for a compact subgroup $K$ of $U$ and a subspace $\liep$ of $\im\lieu$. The decomposition $G=K\exp(\liep)$ is called the \emph{Cartan decomposition} of $G$.
Note that in particular $U^\ce$ is a compatible subgroup of $U^\ce$ with Cartan decomposition $U^\ce=U\exp(\im\lieu)$ where $\lieu$ denotes the Lie algebra of $U$.
We call a representation $G\times V\to V$ of a real Lie group on a real vector space a \emph{real reductive representation}, if $G$ is a compatible subgroup of a complex reductive group $U^\ce$ and if $V$ is a $G$-invariant real subspace of a holomorphic $U^\ce$-representation space $\hat V$.
\section{Semistability in projective space}\label{section:rep}
The linear action $U^\ce\times\hat V\to\hat V$ induces a holomorphic action of $U^\ce$ on the complex projective space $\phatv$ such that the canonical projection $\pi\colon \hat V\setminus\{0\}\to\phatv$ is $U^\ce$-equivariant, i.\,e.~commutes with the actions of $U^\ce$ on $\hat V$ and on $\phatv$.
Our goal is to give an equivalent formulation of Theorem~\ref{theorem:closedOrbits} in terms of the induced $G$-action on projective space. For this, we describe the set of closed $G$-orbits in $V$ in terms of a moment map on $\phatv$.
We fix a Hermitian inner product on $\hat V$ such that $U$ acts by unitary operators on $\hat V$. Recall that the associated Fubini Study metric $\omega$ is a Kählerian metric on $\phatv$, which is invariant under the action of the unitary group and which is in particular invariant under the action of $U$. The corresponding moment map $\mu\colon \phatv\to\lieu^*$ is defined by the equation $d\mu^\xi=\iota_{\xi_*}\omega$. Here $\xi_*(x)=\ddt\exp(t\xi)\cdot x$ is the vectorfield induced by the action of the one-parameter group $\{\exp(t\xi);t\in\er\}$ and $\iota_{\xi_*}$ is the contraction of $\omega$ with $\xi_*$. Moreover, we require $\mu$ to be equivariant with respect to the coadjoint action of $U$ on $\lieu^*$ which is given by $(g\cdot \varphi)(\xi)=\varphi(\Ad(g)\xi)$.
Explicitly a moment map is given by $\mu^\xi(\pi(v))=\im\frac{\langle \xi_*v,v\rangle}{\langle v,v\rangle}$. It is unique up to addition of an element in the dual of the center of $\lieu^*$. The moment map contains much information on the geometry of the $U^\ce$-action on $\phatv$ (\cite{Kirwan}, \cite{Ness}). Analogously the geometry of the $G$-action, which we are interested in, can be described in terms of the restriction of the moment map to $(\im\liep)^*$ (\cite{HSch}). In order to simplify notation, we identify $(\im\liep)^*$ with $\liep$. Then the \emph{restricted moment map} $\mu_\liep\colon\phatv\to\liep$ is given by $\mu_\liep^\xi(x)=\langle\mu_\liep(x),\xi\rangle=\mu^{-\im\xi}(x)=\frac{\langle \xi_*v,v\rangle}{\langle v,v\rangle}$
For $\beta\in\liep$, we define $\Mp(\beta):=\mu_\liep^{-1}(\beta)$ and in order to shorten notation we set $\Mp:=\mu_\liep^{-1}(0)$. Moreover we define $\mathcal S_G(\Mp(\beta)):=\{x\in\phatv;\overline{G\cdot x}\cap\Mp(\beta)\neq\emptyset\}$ to be the set of $G$-orbits which intersect $\Mp(\beta)$ in their closure. For $\beta=0$ we call $\mathcal S_G(\Mp):=\mathcal S_G(\Mp(0))$ the set of \emph{semistable points} in $\phatv$ with respect to $\mu_\liep$. If $\beta\in\liep$ is contained in the center of $\lieu^\ce$, the shifted moment map $\mu+\im\beta$ is again a moment map and $\Mp(\beta)$ is the zero-fiber of the restricted moment map $\mu_\liep-\beta$, so $\sgmpb$ is the set of semistable points with respect to $\mu_\liep-\beta$. We have a relation between the closed orbits in $\mathcal S_G(\Mp(\beta))$ and $\Mp(\beta)$.
\begin{theorem}[\cite{HSch}]\label{theorem:semistable}
Let $\beta\in\liep$ be contained in the center of $\lieu^\ce$.
\begin{enumerate}\item A $G$-orbit $G\cdot x\subset\sgmpb$ is closed in $\sgmpb$ if and only if $G\cdot x\cap\Mp(\beta)\neq\emptyset$.\item Every non-closed orbit in $\sgmpb$ contains a unique closed orbit in its closure and has strictly larger dimension than that closed orbit.
\end{enumerate}
\end{theorem}
In particular, the closed $G$-orbits in the set of semistable points $\sgmp$ are exactly those orbits which intersect the zero-fiber $\Mp=\{\pi(v);<\xi_*v,v>=0\text{ for all }\xi\in\liep\}$ of $\mu_\liep$. It is shown in \cite{RS} that a $G$-orbit $G\cdot v$ in $\hat V$ is closed if and only if it intersects $\mathcal M:=\{v\in\hat V;<\xi_*v,v>=0\text{ for all }\xi\in\liep\}$. In our terminology, $\mathcal M$ is the zero-fiber of a restricted moment map on $\hat V$ for which the set of semistable points equals $\hat V$. We observe that $\mathcal M\setminus\{0\}=\pi^{-1}(\Mp)$. Since $\pi$ is $G$-equivariant, together with Theorem~\ref{theorem:semistable} this proves
\begin{lemma}\label{lemma:ClosedObenUnten}
An orbit $G\cdot v$, $v\neq 0$ is closed in $\hat V$ if and only if $G\cdot \pi(v)$ is closed in $\sgmp$.
\end{lemma}
The set of closed $G$-orbits in $\hat V$ is $\ce^*$-invariant. Therefore, applying Lemma~\ref{lemma:ClosedObenUnten}, Theorem~\ref{theorem:closedOrbits} can be reformulated as follows.
\medskip
\emph{Let $W$ be a linear subspace of $V$ and assume that the intersection of the set of closed orbits in $\sgmp$ with $\pi(W)$ has non-empty interior in $\pi(W)$. Then it contains a subset $\mathcal U$ of $\pi(W)$ which is open in $\pi(W)$ with respect to the real Zariski topology. If $\pi(W)$ is $G$-invariant, then $\mathcal U$ can be chosen to be $G$-invariant.}
\medskip
Now we give an explicit description of the set of semistable points. It is shown in \cite{RS} that each $G$-orbit in $\hat V$ contains exactly one closed $G$-orbit in its closure (compare Theorem~\ref{theorem:semistable}). We define $\mathcal N:=\{v\in \hat V;0\in\overline{G\cdot v}\}$ to be the nullcone of the $G$-representation $\hat V$, i.\,e.~the set of $G$-orbits which contain $0$ as the unique closed orbit in their closure. Note that $\mathcal N$ is $\ce^*$-stable. By a result in \cite{HSchuetz}, the nullcone $\mathcal N$ is a real algebraic subset of $\hat V$. Then $\pi(\mathcal N\setminus\{0\})$ is real algebraic in $\phatv$.
\begin{lemma}\label{lemma:semistabilZOffen}
The set of semistable points is given by $\sgmp=\pi(\hat V\setminus\mathcal N)$. In particular, it is open in $\phatv$ with respect to the real Zariski topology.
\end{lemma}
\begin{proof}
Let $x\in\sgmp\cap\pi(\mathcal N\setminus\{0\})$ and $y\in\overline{G\cdot x}\cap\Mp$. Then $y\in\pi(\mathcal N)$ since $\pi(\mathcal N)$ is closed and $G$-invariant. But $\Mp=\pi(\mathcal M\setminus\{0\})$ and $\mathcal M\cap\mathcal N=\{0\}$, a contradiction. This shows $\sgmp\subset\pi(\hat V\setminus\mathcal N)$.
Conversely, for $v\in\hat V\setminus\mathcal N$, there exists a $w\in\overline{G\cdot v}\cap(\mathcal M\setminus\{0\})$. Then $\pi(w)\in\overline{G\cdot\pi(v)}\cap\Mp$ by continuity of $\pi$ and thus $\pi(v)\in\sgmp$.
\end{proof}
There is no analogous statement for $\sgmpb$. But if $G$ is complex reductive, the following is known (\cite{HMig}).
\begin{proposition}\label{proposition:SemistabilBetaZOffen}
Assume $G=U^\ce$. Then $\sgmpb=\mathcal S_{U^\ce}(\mathcal M_{\im\lieu}(\beta))$ is open in $\phatv$ with respect to the complex Zariski topology.
\end{proposition}
\section{Actions of real forms on projective space}\label{section:realForms}
We call $G$ a real form of $U^\ce$ if its Lie algebra is a real form of $\lieu^\ce$ and if $G$ intersects every connected component of $U^\ce$. If $G$ is a real form of $U^\ce$ and if $V$ is a real form of $\hat V$, the sets of semistable points and the sets of closed orbits with respects to the actions of $G$ and $U^\ce$ are related. Observe that $\pi(V\setminus\{0\})\subset\phatv$ can be identified with the real projective space $\pv$ since $V$ is totally real in $\hat V$, i.\,e.~$V\cap\im V=\{0\}$ holds.
In order to shorten notation, we define $\mathcal S:=\mathcal S_{U^\ce}(\mathcal M_{\im\lieu}(\beta))\subset\phatv$. We denote by $\sgmpb_c$ and $\mathcal S_c$ the set of closed $G$-orbits and the set of closed $U^\ce$-orbits in $\sgmpb$ and $\mathcal S$, respectively. The following proposition bases on a more general argument which states that the moment map $\mu$ and the restricted moment map $\mu_\liep$ coincide up to a constant on $K$-stable Lagrangian submanifolds.
\begin{proposition}\label{proposition:semistabilGUce}
Let $\beta\in\liep$ be contained in the center of $\lieu^\ce$ and assume that $G$ is a real form of $U^\ce$ and that $V$ is a real form of $\hat V$.
\begin{enumerate}
\item
$\mathcal S_G(\Mp(\beta))\cap\pv=\mathcal S\cap\pv$.
\item $\sgmpb_c\cap\pv=\mathcal S_c\cap\pv$, i.\,e.~a $G$-orbit $G\cdot x$ in $\mathcal S_G(\Mp(\beta))\cap\pv$ is closed if and only if $U^\ce\cdot x$ is closed in $\mathcal S$.
\end{enumerate}
\end{proposition}
\begin{proof}
Recall that we are given a $U$-invariant Hermitian inner product $\langle \cdot,\cdot\rangle$ on $\hat V$. By \cite{RS} this inner product can be chosen such that $V$ is Lagrangian with respect to the Kähler structure which is given by its imaginary part. Since $K$ operates on $V$, this gives $\langle \xi_* v,v\rangle=0$ for $v\in V$ and $\xi\in\liek$. Therefore $\mu^\xi\equiv 0$ for $\xi\in\liek$. Since $G$ is a real form of $U^\ce$, we have a Lie algebra decomposition $\lieu=\liek\oplus\im\liep$ and we conclude $\Mp\cap\pv=\mathcal M_{\im\lieu}\cap\pv$. Since $\beta$ is contained in $\liep$ this also gives $\Mp(\beta)\cap\pv=\mathcal M_{\im\lieu}(\beta)\cap\pv$.
For the first part of the proposition, the inclusion $\mathcal S\subset \mathcal S_G(\Mp(\beta))$ is shown in \cite{HSch}. Conversely, for $x\in\mathcal S_G(\Mp(\beta))\cap\pv$, the closure $\overline{G\cdot x}$ intersects $\Mp(\beta)\cap\pv=\mathcal M_{\im\lieu}(\beta)\cap\pv$. In particular, $\overline{U^\ce\cdot x}$ intersects $\mathcal M_{\im\lieu}(\beta)\cap\pv$ and thus $x\in \mathcal S$.
For the second part, if $G\cdot x$ is closed in $\mathcal S_G(\Mp(\beta))$, then it intersects $\Mp(\beta)$ by Theorem~\ref{theorem:semistable}. But since $\Mp(\beta)\cap\pv=\mathcal M_{\im\lieu}(\beta)\cap\pv$, it follows that $U^\ce\cdot x$ is closed, again with Theorem~\ref{theorem:semistable}. Conversely, assume $U^\ce\cdot x$ is closed. The action of $U^\ce$ is holomorphic, $G$ is a real form of $U^\ce$ and $\pv$ is totally real in $\phatv$. Therefore the real dimension of each $G$-orbit in $U^\ce\cdot x\cap\pv$ equals the complex dimension of $U^\ce\cdot x$. In particular, all the $G$-orbits in the intersection have the same dimension. Moreover they are contained in $\sgmpb$ by the first part of the proposition. Then it follows from Theorem~\ref{theorem:semistable} that all these $G$-orbits, and in particular $G\cdot x$, are closed.
\end{proof}
\begin{remark}
The same proof applies in order to show the result of Birkes (\cite{Birkes}) that a $G$-orbit $G\cdot v$ in $V$ is closed if and only if $U^\ce\cdot v$ is closed.
\end{remark}
Let $\beta\in\liep$ be contained in the center of $\lieu^\ce$ and consider the shifted moment map $\mu+\im\beta$.
For its set of semistable points $\mathcal S\subset\phatv$, the GIT-Quotient $\pi\colon\mathcal S\to\quot{\mathcal S}{U^\ce}$ in the sense of Mumford (\cite{MFK}) exists and is an affine map. Each fiber of $\pi$ contains a unique closed orbit and every other orbit in the fiber contains the closed orbit in its closure. The quotient defines an orbit type stratification of $\mathcal S$ as follows (\cite{Lu73},\cite{Ri72},\cite{LR79}). For a subgroup $H$ of $U^\ce$, we denote by $\mathcal S^{<H>}:=\{x\in\mathcal S; U^\ce\cdot x\text{ closed, }(U^\ce)_x=H\}$ the set of points on closed orbits, for which the isotropy group is given by $H$. Then the saturation $I_H:=\pi^{-1}(\pi(\mathcal S^{<H>})):=\{x\in\mathcal S; \overline{U^\ce\cdot x}\cap \mathcal S^{<H>}\neq\emptyset\}$ is by definition the \emph{$H$-isotropy stratum}. The closed orbits in $I_H$ are exactly those which intersect $\mathcal S^{<H>}$ and they are the orbits of minimal dimension in $I_H$. The isotropy groups of two points on an orbit are conjugate. Therefore $I_H=I_{gHg^{-1}}$ for $g\in U^\ce$. Since each orbit in $\mathcal S$ contains a closed orbit in its closure, the isotropy strata cover $\mathcal S$. Moreover, two strata $I_{H}$ and $I_{H'}$ are either disjoint or equal, where the latter is the case if and only if $H$ and $H'$ are conjugate. Considering conjugacy classes of isotropy groups, $\mathcal S$ is the disjoint union of the strata and the union is locally finite. A stratum $I_H$ is open in its closure with respect to the complex Zariski topology and if the closure $\overline{I_H}$ intersects a stratum $I_{H'}$ with $I_H\neq I_{H'}$, then after conjugation, $H$ is contained in $H'$, i.\,e.~there exists a $g\in U^\ce$ such that $gHg^{-1}< H'$. This implies that for $n\in\mathbb N$ the set $\mathcal I_n:=\bigcup_{\codim H\geq n} I_H$ is open in $\mathcal S$ with respect to the complex Zariski topology.
For $n\in\mathbb N$, we define $\mathcal O_n:=\{x\in\phatv; \dim U^\ce\cdot x\geq n\}$ to be the set of orbits with complex dimension at least $n$. Since the elements of the Lie algebra of $U^\ce$ define holomorphic vectorfields on $\phatv$, the set $\mathcal O_n$ is open with respect to the complex Zariski topology.
Now, for actions of real forms, we prove a result which is slightly more general than the statement of Theorem~\ref{theorem:closedOrbits} since we do not assume $\beta=0$. We need this generality in order to prove Theorem~\ref{theorem:closedOrbits}. Recall that $\mathcal S_G(\mathcal M_\liep(\beta))_c$ denotes the set of closed $G$-orbits in $\mathcal S_G(\mathcal M_\liep(\beta))$.
\begin{proposition}\label{proposition:Projektiv}
Assume $G$ is a real form of $U^\ce$ and $\hat V=V^\ce$. Let $W$ be a subspace of $V$ and let $n$ be maximal such that $\mathcal O_n\cap\pw$ is non-empty. Then, for a $\beta\in\liep$ which is contained in the center of $\lieu^\ce$, the intersection $\mathcal S_G(\mathcal M_\liep(\beta))_c\cap\mathcal O_n\cap\pw$ is open in $\pw$ with respect to the real Zariski topology.
\end{proposition}
\begin{proof}
For $x\in\pw\subset\pv$, the orbit $G\cdot x$ is closed in $\mathcal S_G(\mathcal M_\liep(\beta))$ if and only if $U^\ce\cdot x$ is closed in $\mathcal S$ (Proposition~\ref{proposition:SemistabilBetaZOffen}). By definition of $n$ and $\mathcal O_n$, an orbit $U^\ce\cdot x$ with $x\in\mathcal O_n\cap\pw$ is of complex dimension $n$. So for $x\in\mathcal S_G(\mathcal M_\liep(\beta))_c\cap\mathcal O_n\cap\pw$ the orbit $U^\ce\cdot x$ is closed and of dimension $n$. Then we have $x\in I_H$ where $H$ is the isotropy group at $x$ and since the dimension of $U^\ce\cdot x$ equals the codimension of the isotropy group, $U^\ce\cdot x$ is contained in the union of isotropy strata $\mathcal I_n=\bigcup_{\codim H\geq n} I_H$.
Conversely, for $x\in \mathcal I_n\cap\mathcal O_n\cap\pw$ the orbit $U^\ce\cdot x$ is of dimension $n$ by the choice of $n$, so it is an orbit of minimal dimension in $\mathcal I_n$. Since the closure of $U^\ce\cdot x$ is contained in $\mathcal I_n$, it follows from Theorem~\ref{theorem:semistable} that $U^\ce\cdot x$ is closed. This shows \[\mathcal S_G(\mathcal M_\liep(\beta))_c\cap\mathcal O_n\cap\pw=\mathcal I_n\cap\mathcal O_n\cap\pw.\]
But $\mathcal I_n$ and $\mathcal O_n$ are open in $\phatv$ with respect to the complex Zariski topology, so the intersection $\mathcal I_n\cap\mathcal O_n\cap\pw$ is open in $\pw$ with respect to the real Zariski topology.
\end{proof}
\section{Proof of the main result}\label{section:enlarging}
We observed in Section~\ref{section:rep} that for the proof of Theorem~\ref{theorem:closedOrbits} it suffices to show that the intersection $\sgmp_c\cap\pi(W)$ of the set of closed orbits inside the set of semistable points with $\pi(W)$ contains a subset which is open in $\pi(W)$ with respect to the real Zariski topology. In Section~\ref{section:realForms} we proved a slightly more general result for actions of real forms. For the general case we will now give a construction of a representation of a real form which contains $G$ in a convenient way. This enables us to prove Theorem~\ref{theorem:closedOrbits} using Proposition~\ref{proposition:Projektiv}.
For our construction, we will make several assumptions on $G$ which do not affect the set $V_c$ of closed orbits.
First, replacing $G$ by its connected component of the identity $G^\circ$, we may assume that $G$ is connected. For this note that $G^\circ$ is compatible with Cartan decomposition $G^\circ=K^\circ\exp(\liep)$ and that a $G$-orbit $G\cdot v$ is closed if and only if $G^\circ\cdot v$ is closed.
The connected compatible subgroup $G$ of $U^\ce$ admits a decomposition $G=G_S\cdot G_Z$, where $G_S$ is a semisimple subgroup of $G$ and $G_Z$ is the center. Here the Lie algebra of $G_S$ is given by the Lie bracket $[\lieg,\lieg]$. Both, $G_S$ and $G_Z$ are compatible subgroups of $U^\ce$. Let $G_Z=K_Z\exp(\liep_Z)$ denote the Cartan decomposition of the center and define $P_Z:=\exp(\liep_Z)$. The group $G_S\cdot P_Z$ is compatible in $U^\ce$ and an orbit of $G_S\cdot P_Z$ is closed if and only if the corresponding orbit of $G$ is closed. Therefore we may assume $G=G_S\cdot P_Z$.
Dividing $P_Z$ by the ineffectivity of the $P_Z$-representation, we may furthermore assume that $P_Z$ acts effectively.
Since $\liep_Z$ acts by symmetric operators on $V$, we have a decomposition $V=\bigoplus_{i=1}^dV_i$ such that the action of the commutative group $P_Z$ on $V$ is given by a real character $\chi_i\colon P_Z\to\er^{>0}$ on each $V_i$. By assumption $P_Z$ acts effectively, so these characters define an injective homomorphism $\chi\colon P_Z\to(\er^{>0})^d$. Since $P_Z$ is contained in the center of $G$, each $V_i$ is invariant under the action of $G_S$. The universal complexification $G_S^\ce$ of $G_S$ in the sense of \cite{Hochschild} is complex reductive, contains $G_S$ as a compatible subgroup and by definition, it admits a holomorphic representation on $V_i^\ce$, which extends the $G_S$-representation. We define a representation of $(\ce^{*})^d$ on $V^\ce=\bigoplus_{i=1}^dV_i^\ce$ by letting the $i$-th component of $(\ce^*)^d$ act on $V_i^\ce$ by multiplication. The actions of $G_S^\ce$ and $(\ce^*)^d$ commute, so we obtain a holomorphic representation of the complex reductive group $G_S^\ce\times(\ce^*)^d$ on $V^\ce$. This group contains $G_S\times\chi(P_Z)$ as a compatible subgroup. Therefore we may assume $G=G_S\times\chi(P_Z)$, $U^\ce=G_S^\ce\times(\ce^*)^d$ and $\hat V=V^\ce$.
We define $H:=G_S\times (\er^{>0})^d$. Then $H$ is a compatible real form of $U^\ce$ which contains $G$. We denote its Cartan decomposition by $H=L\exp(\lieq)$. Note that by construction $L=K$ and that $\lieq$ is the direct sum of $\liep$ and a subspace $\liez$ of the center of $\lieu^\ce$. For an appropriate choice of the inner product, this sum can be assumed to be orthogonal.
Altogether, for the proof of Theorem~\ref{theorem:closedOrbits}, we may assume that $G$ is a compatible subgroup of a compatible real form $H=K\exp(\lieq)$ of $U^\ce$ and that $H=G\times \exp(\liez)$.
With these assumptions, we can complete the proof of Theorem~\ref{theorem:closedOrbits}.
\begin{proof}[Proof of Theorem~\ref{theorem:closedOrbits}]
We prove the equivalent formulation of Theorem~\ref{theorem:closedOrbits} given in Section~\ref{section:rep}. Note that here $\pi(W)=\pw$ since $V$ is totally real in $V^\ce$. Again, let $n$ be maximal such that $\mathcal O_n\cap\pw\neq\emptyset$. Since $\mathcal O_n\cap\pw$ is open in $\pw$ with respect to the real Zariski topology and since $\mathcal S_G(\Mp)_c\cap\pw$ has non-empty interior in $\pw$ by assumption, there exists an $x\in\mathcal S_G(\Mp)_c\cap\mathcal O_n\cap\pw$. Let $g\in G$ with $gx\in\Mp$. Then $\beta:=\mu_\lieq(gx)$ is contained in the orthogonal complement of $\liep$ in $\lieq$ and in particular $\beta$ is contained in the center of $\lieu^\ce$.
By Proposition~\ref{proposition:Projektiv} the set $\mathcal S_H(\mathcal M_\lieq(\beta))_c\cap\mathcal O_n\cap\pw$ is Zariski open in $\pw$. It contains $x$ since $gx\in H\cdot x\cap\mathcal M_{\lieq}(\beta)$. Therefore it is non-empty. Note that it is $G$-invariant if $\pw$ is $G$-invariant. So for the proof of the theorem it now suffices to show that $\mathcal S_H(\Mq(\beta))_c\subset\mathcal S_G(\Mp)_c$.
For this, let $x\in \mathcal S_H(\Mq(\beta))_c$ and $y\in H\cdot x\cap\Mq(\beta)$. Then $y\in\Mp$ since $\beta$ is contained in the orthogonal complement of $\liep$. Consequently $G\cdot y$ is closed. The $G$-nullcone in $\hat V$ is $H$-invariant since $H=G\cdot \exp(\liez)$ where $\exp(\liez)$ is contained in the center of $U^\ce$. Therefore $\mathcal S_G(\Mp)$ is $H$-invariant by Lemma~\ref{lemma:semistabilZOffen} and we have $x\in H\cdot y\subset \mathcal S_G(\Mp)$. Let $h\in \exp(\liez)$ with $G\cdot x=G\cdot hy$. Then $G\cdot x=h\cdot G\cdot y$ is closed in $\mathcal S_G(\Mp)$ since $G\cdot y$ is closed.
\end{proof}
\newcommand{\noopsort}[1]{} \newcommand{\printfirst}[2]{#1}
\newcommand{\singleletter}[1]{#1} \newcommand{\switchargs}[2]{#2#1}
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace} | 190,637 |
TITLE: Number of binary representations
QUESTION [1 upvotes]: Calculate the amount of numbers from $1$ to $N$ in the binary representation of which contains exactly $K$ zeroes.
For example:
for $N = 18$ which is $10010$ in binary representation and $K = 3$ the answer is $3$ $10001;10010;1000$
REPLY [0 votes]: Let $k = \lfloor \log_2{N} \rfloor$ and let $n = 2^ka_k + 2^{k-1}a_{k-1} + \dotsb + a_0$ in binary, where $n$ is a number satisfying the given conditions. We make two cases on $n$: $a_k = 0$ and $a_k = 1$.
If $a_k = 0$, then to get a number with $K$ significant zeroes in the digits $a_0, a_1, \dotsc, a_{k-1}$, we simply have to select $K$ places for the zeroes and fill the rest with ones. We cannot put a zero in the $k-1$th place, thus we can do this in $\binom{k-1}{K}$ ways. The number formed is obviously less than $N$.
If $a_{k-1} = 1$, then consider the number of zeroes in the binary representation of $N$. Call this number $c$. If $c > K$, there are no numbers with $a_{k-1} = 1$ and $K$ significant zeroes; else, out of the remaining $k - c$ digits which are 1, convert any $K - c$ to 0: there are $\binom{k - c}{K - c}$ ways to do this.
Thus, the final answer is:
If $c > K$, there are $\binom{k-1}{K}$ such numbers
If $c \leq K$, there are $\binom{k-1}{K} + \binom{k - c}{K - c}$ such numbers
where $c$ is the number of zeroes in the binary representation of $N$, and $k$ is $\lfloor \log_2{N} \rfloor$. | 189,905 |
TITLE: Is the Uncertainty Principle a logical consequence from the Wave-Particle duality?
QUESTION [3 upvotes]: I always thought of the Uncertainty Principle as a logical consequence that follows from the Wave-Particle duality, or more precise, from the fact that all particles behave as waves as long as they do not interact with other particles.
From basic wave properties, it's clear that you cannot know both the position and the speed/impulse of a wave exactly. A strongly localized wave dissipates fast, so it has a "blurry" impulse. Moreover, a wave with an exact impulse has no precise location. So, if a fundamental "particle" behaves like a wave, then the Uncertainty Principle (for the position/impulse variable pair) is a logical consequence.
Is that correct so far?
So I wonder, why do many physicists describe the Uncertainty Principle like it was some axiom that is built into this world, and that has no reason or explanation for why it exists? There are even physicists who describe it like Nature playing masquerade. "The more we look at the position, the more Nature conceals the impulse."
No, that's not what the Uncertainty Principle is about.
The fundamental fact is that everything behaves like waves. This is the axiom, the fact without a known reason behind it. The Uncertainty Principle is just a consequence of that fact, one of the first consequences discovered by mankind.
EDIT @ACuriousMind: Well that makes sense, but I have the impression that this just shifts the discussion over to the fact that there are non-commutating operators. It's clear that if two operators don't commutate, then there's the uncertainty that is quantitatively described by the formula you wrote. But I'm still convinced that the fact that there are non-commutating operators has a qualitative reason. After all, if classical mechanics is formulated using the Hilbert Spaces tools, your inequality becomes trivial because $[A, B] = 0$ for all operators in classical mechanics.
REPLY [6 votes]: The uncertainty principle is much more general than anything you might say about the wave-particle duality. In particular, wave-particle duality is a vague and imprecise statement about how certain types of quantum systems qualitatively behave, while the uncertainty principle is a very general and quantitative statement about the standard deviations of operators.
While, in settings like the double-slit, it is true that you may think about the quantum objects as being represented by a probability wave, this breaks down whenever one considers finite-dimensional Hilbert spaces, as they occur e.g. in the setting of quantum information and its qubits. There's no continuous set of generalized position operators - not ever a position operator at all - and hence no "wavefunction". Nevertheless, the relation
$$ \sigma_A(\psi)\sigma_B(\psi) \geq \frac{1}{2}\lvert\langle \psi \vert [A,B] \vert \psi \rangle\rvert$$
holds for all operators $A,B$ and all states $\psi$.
And even in the infinite-dimensional setting where you might claim that we have a "wave nature" and a "particle nature", this relation holds for all operators, not just position and momentum, and the proof just relies on basic properties of Hilbert spaces like the Cauchy-Schwarz inequality.
To stress this crucial fact: The uncertainty relation is a general consequence of the axioms that states are rays in a Hilbert space and the rule how these states give expectation values. No conception of "particle" or "wave" ever enters into the derivation, and the fact that waves also exhibit a type of uncertainty relation in their widths is a simple consequence of the properties of the Fourier transform. Since the Fourier transform is also intimately related to the position and momentum operators by the Stone-von Neumann theorem about their essentially unique representation as multiplication and differentiation, this explains the similarity without any reference to "wave-particle duality". | 183,690 |
How to Take Care of Your Dentures
Dentures are aesthetic replacements for missing teeth, usually almost similar to the color of your natural teeth that can be taken out and put back in your mouth anytime. Needless to say, it is one of the most economic and practical solutions for missing teeth. As compared to dental implants, dentures are cheaper and much more available to many dental clinics.
However, just like normal teeth, your dentures also need proper care in order to maintain not just the quality of the dentures but also the health of your mouth.
To help you out, here are some useful tips on taking care of your dentures:
- When handling your denture, make sure that you are standing over a folded towel or a sink full of water to make sure it will not break in case it drops.
- Clean your denture thoroughly with a soft-bristled brush but without toothpaste. The substance in the toothpaste can be abrasive and create microscopic scratches where food stains may build up causing plaque.
- Always make it a point that you rinse your denture every after eating your meals.
- Make use of a denture cleaner, a hand soap or a mild dish washing liquid to clean your dentures. Using bleach is not advised as it may whiten the pinkish portion of your denture, thus, wear down its strength. Ultrasonic cleaners are bathtub-like devices that contain cleaning solutions for your dentures.
- Remember the proper way of keeping your dentures if you will not wear it. It has to be kept moist when not being worn so it will not dry out or lose its shape. Place your dentures in a cleanser soaking solution or in water. But, if your dentures have some metal on it, it’s better to consult your dentist on the proper ways of storing it when not worn.
In addition, don’t forget that complete dentures can be used for 5 to 7 years only. Replacements can be done upon reaching this span of time wearing the denture you have.
Furthermore, wearing dentures also requires proper care for your mouth and gums. Brushing your teeth regularly is always a must and rinsing your mouth with lukewarm salted water is ideal to ensure the cleanliness of your mouth.
Need some repair or a replacement for your current denture? Consult Dr. Rowena Marzo of Marzo Dental Clinic now for better results! Her clinic specializes in various dental services including Orthodontics and Dental Implantology. Book a consultation now and see dental excellence at its finest!
Sources: | 17,409 |
Summit Hotel Properties, Inc. (the “Company”) today announced the commencement of an underwritten public offering of 15,000,000 shares of its common stock, par value $0.01 per share. The Company intends to grant the underwriters of the offering a 30-day option to purchase up to an additional 2,250 reduce the outstanding balances under the Company’s $150 million senior secured revolving credit facility and the Company’s $92 million senior secured interim loan and any remaining net proceeds will be used for general corporate purposes, including repayment of debt and acquisitions of additional hotel properties.
BofA Merrill Lynch, Deutsche Bank Securities, Baird, Raymond James, and RBC Capital Markets are acting as book-running relating to the offering will be filed with the Securities and Exchange Commission and, when available, can be obtained by contacting: BofA Merrill Lynch, Attention: Prospectus Department, 222 Broadway, New York, NY 10038, Email:[email protected]; Deutsche Bank Securities Inc., Attention: Prospectus Group, 60 Wall Street, New York, NY 10005, E-mail:[email protected]; Robert W. Baird & Co. Incorporated, Attention: Syndicate Department, 777 E. Wisconsin Avenue, Milwaukee, WI 53202, E-mail: [email protected]; Raymond James & Associates, Inc., Attention: Prospectus Department, 880 Carillon Parkway, St. Petersburg, Florida 33716, E-mail [email protected]; RBC Capital Markets, LLC, Attention: Prospectus Department, Three World Financial Center, 200 Vesey Street, 8th floor, New York, New York 10281-8098, Telephone: (877) 822-4089.
About Summit Hotel Properties
Summit Hotel Properties, Inc. is a publicly traded real estate investment trust focused primarily on acquiring and owning premium-branded select-service hotels in the upscale and upper midscale segments of the lodging industry. As of September 12, 2013, the Company’s portfolio consisted of 93 hotels with a total of 10,976 guestrooms located in 24 states. | 173,268 |
TITLE: Differential euqation for a pendulum: $ {d^2\alpha \over dt^2} + {g \over L} \cdot \alpha = 0 $
QUESTION [0 upvotes]: The differential equation $$ {d^2\alpha \over dt^2} + {g \over L} \cdot \alpha = 0 $$
describes a 1-dimensional mathematical pendulum, where $\alpha $ is the angle, $ g = 9.82 $, and $ L = 0.2 $ is the lentgh of the string. What is the position of the pendulum after 1 second if the velocity at time equals zero is $ 0 $ m/s and the angle at time equals zero is $ {\pi \over 60} $ radians.
To my understanding this is a homogenous equation, ergo
$$ y'' + Cy = 0$$
And with the help of the characteristic equation I get the complex roots $ Ci \, $ and $ -Ci $.
This is a farily simple differential equation, but I have a feeling my calculations so far are wrong (mainly because I didn't expect to see complex roots for this equation).
REPLY [2 votes]: As suggested in the comments above. As they say here is one I done earlier :).
Glad it has helped. | 198,908 |
In this section, we prove the superlinear convergence of \cref{alg4}.
The analysis is specific to the MT family.
In particular, its notable features are summarized as in the following remarks.
\begin{remark}\label{rem:1}
\begin{enumerate}
\item[(i)]\label{rem1} The function $\Xi^P_{\mu}$ is not normally defined
at a point satisfying $G(x)\in S^m\setminus S^m_{+}$ due to the presence of $G(x)^{\fr}$, and as a result, neither is its Jacobian on a boundary point, i.e., a point such that $G(x)\in S^m_+\setminus S^m_{++}$.
This fact causes difficulties on analyzing the limiting behavior of MT scaling directions.
\item[(ii)]\label{rem2} The analysis will be made under general assumptions (\cref{assum:A}) on scaling matrices used for producing MT scaling directions.
Whereas Yamashita and Yabe\,\cite{yamashita2012local} presented local convergence analysis for each of AHO, NT, and {\HKM} directions individually,
we can show the superlinear convergence result for a wider class of scaling matrices in a unified manner by virtue of these assumptions.
The assumptions are actually fulfilled by major MT members (\cref{prop:scaling}).
\end{enumerate}
\end{remark}
Henceforth, we assume that the functions $f$, $G$, and $h$ are three times continuously differentiable, and let $w^{\ast}:=(x^{\ast},Y_{\ast},z^{\ast})$ be a KKT point of NSDP\,\eqref{al:nsdp} that satisfies the following regularity conditions. These conditions are all adopted in the recent researches\,\cite{yamashita2012local,yamakawa2,okuno2018primal} on PDIPMs for NSDPs:\vspace{1em}\\
\noindent{\bf Nondegeneracy condition:}
Let
$r_{\ast}:={\rm rank}\,G(x^{\ast})$ and let $e_i\ (i=1,2,\ldots,m-r_{\ast})$ be an orthonormal basis of the null space of $G(x^{\ast})$.
Then, the vectors
$v_{ij}\in \R^n\ (1\le i\le j\le m-r_{\ast})$ and $\nabla h_i(x^{\ast})\ (i=1,2,\ldots,\ell)$ are linearly independent, where
$$
v_{ij}:=(e_i^{\top}\G_1(x^{\ast})e_j,\cdots, e_i^{\top}\G_n(x^{\ast})e_j)^{\top}\in \R^n\ (1\le i\le j\le m-r_{\ast}).
$$
\noindent{\bf Second-order sufficient condition}:
$\nabla_{xx}^2L(w^{\ast})+H(x^{\ast},Y_{\ast})$ is positive definite on the critical cone $C(x^{\ast})$ at $x^{\ast}$, which is defined by
\begin{equation*}
C(x^{\ast}):= \left\{d\in \R^n\mid \nabla f(x^{\ast})^{\top}d=0, \nabla h(x^{\ast})^{\top}d=0,
\mathcal{J}G(x^{\ast})d\in T_{S^m_+}(G(x^{\ast}))
\right\}, \label{eq:critical}
\end{equation*}
where $T_{S^m_+}(G(x^{\ast}))$ denotes the tangent cone of $S^m_+$ at $G(x^{\ast})$ and
$H(x^{\ast},Y_{\ast})$ is the matrix in $S^n$ whose entries are given by
$$
(H(x^{\ast},Y_{\ast}))_{i,j}:= 2Y_{\ast}\bullet \Gi(x^{\ast})G(x^{\ast})^{\dag}\G_j(x^{\ast})
$$
for $i,j=1,2,\ldots,n$.
Here, $G(x^{\ast})^{\dag}$ denotes the Moore-Penrose inverse matrix of $G(x^{\ast})$.\\
\noindent{\bf Strict complementarity condition}: It holds that $G(x^{\ast})+Y_{\ast}\in S^m_{++}$.\vspace{0.5em}\\
For detailed explanations of the above three conditions, we refer readers to, e.g., \cite{yamashita2012local, shapiro1997first} or \cite{bonnans2013perturbation}.
In what follows, we study the local convergence behavior of
\cref{alg4} in a sufficiently small neighborhood of $w^{\ast}$.
For the sake of analysis, we introduce the following functions
$\Phi^1_{\mu},\Phi^2_{\mu}:\mathcal{W}\to \mathcal{W}$:
\begin{equation}
{\Phi}^1_{\mu}(w):=\begin{bmatrix}
\nabla_xL(w)\\
G(x)Y-\mu I\\
h(x)
\end{bmatrix},\
{\Phi}^{2}_{\mu}(w):=\begin{bmatrix}
\nabla_xL(w)\\
{\rm Sym}\left(G(x)Y\right)-\mu I\\
h(x)
\end{bmatrix}.
\label{eq:defPhi}
\end{equation}
We often use the following relation: For any $\mu\ge 0$ and $w\in \mathcal{W}$,
\begin{equation}
\|\Phi_{\mu}^1(w)\|\ge \|\Phi_{\mu}^2(w)\|.\label{eq:0218}
\end{equation}
Under the above three regularity conditions of $w^{\ast}$, the Jacobian of ${\Phi}^2_{0}$ is nonsingular at $w^{\ast}$ by \cite[Corollary~1]{yamashita2012local}. Then, by the implicit function theorem along with the strict complementarity condition, we can ensure that there exist
a scalar $\bar{\mu}>0$ and a continuous path $v^{\ast}:[0,\bar{\mu}]\to \mathcal{W}_{+}$ such that $v^{\ast}(0)=w^{\ast}$, it is smooth in $(0,\bar{\mu})$,
and $\Phi^2_{\mu}(v^{\ast}(\mu))=0\ (\mu\in [0,\bar{\mu}])$, which indicates that $v^{\ast}(\mu)=(x^{\ast}(\mu),Y_{\ast}(\mu),z^{\ast}(\mu))$ is a BKKT point with a barrier parameter $\mu\in (0,\bar{\mu}]$.
Hereafter, we often refer to the smooth path $v^{\ast}$ as the central path that emanates from $w^{\ast}$.
Associated with the central path, we define the following set for $r>0$ and $\mu \in (0,\bar{\mu}]$ to measure the deviation of given iterates from the central path:
$$
\mathcal{N}^{r}_{\mu}:=\left\{w=(x,Y,z)\in \mathcal{W}_{++}\mid
\|
\Phi^1_{\mu}(w)
\|
\le r
\right\}.
$$
The following proposition concerns an error bound on $\|\tw^{\ell}-w^{\ast}\|$ that will play a crucial role in the convergence analysis.
It follows immediately from \cite[Theorem~1]{yamashita2012local}.
\begin{proposition}\label{prop:0219-1}
Let $\{\tmu_{\ell}\}\subseteq \R_{++}$
and $\{r_{\ell}\}\subseteq \R_{++}$
be a sequence such that $\lim_{\ell\to\infty}\tmu_{\ell}=0$ and $r_{\ell}=o(\tmu_{\ell})$.
Moreover, let $\{\tw^{\ell}\}\subseteq \mathcal{W}_{++}$ be a sequence satisfying $\tw^{\ell}\in \N^{r_{\ell}}_{\mu_{\ell}}$ for each $\ell$ and $\lim_{\ell\to\infty}\tw^{\ell}=w^{\ast}$.
Then,
$\|\tw^{\ell}-w^{\ast}\|=\Theta(\tmu_{\ell}).$
\end{proposition}
\subsection{Main results}
To make our goal clear, we first present our main results without proofs.
We shall make the following assumptions:
\begin{assumption}\label{assum:A}
For each $k$, scaling matrices $P_k$ and $P_{k+\fr}$ are set to $\P(w^k)$ and $\P(w^{k+\fr})$, respectively, where $\P:\mathcal{W}_{++}\to \R^{m\times m}$ is a function satisfying the following conditions,
where $w^{\ast}$ is the KKT point defined in the beginning of this \cref{sec:local}:
\begin{enumerate}
\item[{\rm (\bf P1)}] $\P(w)$ is nonsingular for any $w\in \mathcal{W}_{++}$.
\item[{\rm (\bf P2)}]
There exists $\xi\in (0,1)$ such that, for arbitrarily chosen sequences
$\{\tw^{\ell}\}:=\{(\tx^{\ell},\tY_{\ell},\tz^{\ell})\}\subseteq \mathcal{W}_{++}$,
$\{\tmu_{\ell}\}\subseteq \R_{++}$, and $\{r_{\ell}\}\subseteq\R_{++}$
satisfying
\begin{equation}
\lim_{\ell\to\infty}(\tw^{\ell},\tmu_{\ell})=(w^{\ast},0),\ \tw^{\ell}\in \N_{\tmu_{\ell}}^{r_{\ell}},\ r_{\ell}=\tau\tmu_{\ell}^{1+\xi},
\label{eq:P2}
\end{equation}
where $\tau>0$, the sequence
\begin{align*}
\{\zli\}:=\left\{\tmu_{\ell}{\P}(\tw^{\ell})^{-1}\mathcal{L}^{-1}_{\hat{G}_{\ell}^{\fr}}(\widehat{\G}_i(\tx^{\ell}))\hat{G}_{\ell}^{-\fr}{\P}(\tw^{\ell})\right\}\subseteq \R^{m\times m}
\end{align*}
is bounded for each $i=1,2,\ldots,n$, where
$$
G_{\ell}:=G(\tx^{\ell}),\ \hat{G}_{\ell}:=\P(\tilde{w}^{\ell})G_{\ell}\P(\tilde{w}^{\ell})^{\top},\ \widehat{\G}_i(\tx^{\ell}):=\P(\tilde{w}^{\ell})\Gi(\tx^{\ell})\P(\tilde{w}^{\ell})^{\top}.$$
\end{enumerate}
\end{assumption}
One may ask how general the above conditions~({\rm \bf P1}) and ({\rm \bf P2}) are.
We answer this question in the following proposition.
We defer its proof to Appendix.
In the proof, we often make use of \cref{prop:xyx}, which will be presented in \cref{subsec:prel}.
\begin{proposition}\label{prop:scaling}
If the function $\P$ attains at any $w\in \mathcal{W}$
either {\rm (i)} the identity matrix or {\rm (ii)} the scaling matrix for {\HKM} direction,
it satisfies conditions~{\rm({\bf P1})} and {\rm({\bf P2})} for any $\xi\in (0,1)$.
Moreover, it also does so for any $\xi\in [\frac{1}{2},1)$ when
it attains at any $w$ the scaling matrix corresponding to any one of {\rm (iii)} NT, {\rm (iv)}
{\HKM}-dual, and {\rm (v)} MTW directions.
\end{proposition}
The following theorem states the superlinear convergence of \cref{alg4} started from a point that stays in a small neighborhood of the central path and is furthermore sufficiently close to the KKT point $w^{\ast}$.
The proof will be given in \cref{sec:last} after preparing some preliminary results in the subsequent section.
In the theorem,
$\alpha$ is the constant chosen in the initial setting of \cref{alg4}.
In addition, let $\xi\in (0,1)$ and $\tau>0$ be the constants described in condition~{({\bf P2})} and also $\xi^{\prime}\in (0,1)$ be a constant satisfying
\begin{align}
\frac{\xi-\alpha}{1+\alpha}>\xi^{\prime}>\frac{\xi}{2},\ 0<\alpha<\frac{\xi}{\xi+2}.
\label{al:cond}
\end{align}
For example, the above conditions are fulfilled by $(\xi,\xi^{\prime},
\alpha)=(1/2,1/3,1/10)$.
\begin{theorem}\label{thm:main}
Suppose that \cref{assum:A} holds and $w^0$ is sufficiently close to $w^{\ast}$ and satisfies $w^0\in \N^{\tau\mu_0^{1+\xi}}_{\mu_0}$ with $\mu_0 = \tau \|\Xi_0^I(w^0)\|$.
Then, {\bf (i)} it holds that, for each $k\ge 0$, the linear equation~\eqref{eq:pre3} with
$(w,\mu,P)=(w^{k},\mu_k,P_k)$
and equation\,\eqref{eq:corr4} with
$(w,\mu,P)=(w^{k+\fr},\mu_{k+1},P_{k+\fr})$ are uniquely solvable and
\begin{eqnarray}
&\mu_{k+1}=\mu_k^{1+\alpha}<\mu_k,&\label{eq:0704-1721}\\
&s_k^{\rm t}=1-\mu_k^{\alpha},\ w^{k+\fr}=w^k+(1-\mu_k^{\alpha})\Deltap w^k\in
\N_{\mu_{k+1}}^{\tau\mu_{k+1}^{1+\xi^{\prime}}}&,\label{eq:0702-2329}\\
&s_{\kfr}^{\rm c}=1,\ w^{k+1} = w^{k+\fr}+\Deltac w^{\kfr}\in
\N_{\mu_{k+1}}^{\tau\mu_{k+1}^{1+{\xi}}}.&\label{eq:0704-1726}
\end{eqnarray}
{\bf (ii)} Furthermore, the generated sequence $\{w^k\}$ converges to $w^{\ast}$ superlinearly at the order of $1+\alpha\in (1,4/3)$, namely,
$$
\|w^{k+1}-w^{\ast}\|=O(\|w^k-w^{\ast}\|^{1+\alpha}).
$$
\end{theorem}
\subsection{Preliminary results for \cref{thm:main}}\label{subsec:prel}
In this section, we consider arbitrary sequences
$\{\tw^{\ell}\}\subseteq \Wcal_{++}$, $\{\tmu_{\ell}\}\subseteq \R_{++}$, and $\{r_{\ell}\}\subseteq\R_{++}$
satisfying \eqref{eq:P2} in condition~({\bf \rm P2}).
In addition, we define a sequence of nonsingular matrices $\{\tP_{\ell}\}\subseteq \R^{m\times m}$ by $\tP_{\ell}:=\P(\tw^{\ell})$ for each $\ell\ge 0$, where $\P:\mathcal{W}_{++}\to \R^{m\times m}$ is a matrix-valued function satisfying conditions~({\bf P1}) and ({\bf P2}).
We will discuss properties of these sequences.
Note that the above sequences are
not necessarily ones generated by \cref{alg4}.
They are just introduced for the sake of convergence analysis of the algorithm.
\subsubsection{Uniqueness and error bounds of the solutions of the linear equations~\eqref{eq:pre3} and \eqref{eq:corr4}}\label{sec:limit}
In this section, we aim to show the following proposition concerning
unique solvability of the linear equations~\eqref{eq:pre3} and \eqref{eq:corr4} near $w^{\ast}$
and error bounds of their solutions.
\begin{proposition}\label{prop:0613-2}
For any $\ell\ge \ell_0$ with $\ell_0$ sufficiently large, the linear equations~\eqref{eq:pre3} and \eqref{eq:corr4}
with $(P,w,\mu)=(\tP_{\ell},\tw^{\ell},\tmu_{\ell})$ have unique solutions, say
$\Deltac \tw^{\ell}$ and $\Deltap \tw^{\ell}$, respectively.
In particular, we have
\begin{enumerate}
\item\label{e1} $\|\Deltap \tw^{\ell}\|=O(\tmu_{\ell})$, and
\item\label{e2} $\|\Deltac \tw^{\ell}\|=O(\tmu_{\ell}^{1+\xi})$.
\end{enumerate}
\end{proposition}
Error bounds and uniqueness of solutions of the Newton equations play key roles in the local convergence analysis of the PDIPMs using the MZ family\cite{yamashita2012local,yamakawa2,okuno2018primal}, in which the proofs are based on the nonsingularity of the Jacobian of the function $\Phi_{0}^2$.
However, as explained in \cref{rem1}, the Jacobian of
$\Xi_0^P$ is unavailable at the KKT point $w^{\ast}$ in general and thus a different approach is necessary for proving \cref{prop:0613-2}.
To prove \cref{prop:0613-2}, we define the following linear mapping $T_{w,P}:\mathcal{W}\to \mathcal{W}$
for a triplet $w=(x,Y,z)\in \mathcal{W}$ and a nonsingular matrix $P\in \R^{m\times m}$:
$$
T_{w,P}(\Delta w):=
\begin{bmatrix}
&\nabla^2_{xx}L(w)\Delta x- \mathcal{J}{G}(x)^{\ast}\Delta Y+\nabla h(x)\Delta z\\
&
{\rm Sym}\left(G(x)\Delta Y+
\sum_{i=1}^n\Delta x_i
\mathcal{S}_{P}^i(x,Y)
\right)\\
&\nabla h(x)^{\top}\Delta x
\end{bmatrix},
$$
where $\Delta w=(\Delta x,\Delta Y,\Delta z)\in \mathcal{W}$ and
\begin{align}
\mathcal{S}_{P}^i(x,Y)&:=
P^{-1}\left(\hG(x)^{\fr}\hU_i\hat{Y}
+\hG(x)\hat{Y}\hU_i\hG(x)^{-\fr}\right)P,\notag \\
\hU_i&:=\mathcal{L}_{\hG(x)^{\fr}}^{-1}(\hGi(x))\label{al:ui2}
\end{align}
for $(x,Y)\in \R^n\times S^m$, $P\in \R^{m\times m}$, and $i=1,2,\ldots,n$.
We then consider the following linear equations:
\begin{align}
T_{w,P}(\Deltap w) &= \begin{bmatrix}
0\\
-\mu I\\
0
\end{bmatrix}\label{al:0608-1},\\
T_{w,P}(\Deltac w) &= \begin{bmatrix}
\nabla_xL(w)\\
-{\rm Sym}{\left(G(x)Y\right)}+{\mu} I\\
-h(x)
\end{bmatrix}. \label{al:0608-2}
\end{align}
\begin{lemma}\label{lem:0612}
Solving equation\,\eqref{eq:pre3} is equivalent to solving equation\,\eqref{al:0608-1}, and the same holds for solving equations\,\eqref{eq:corr4} and \eqref{al:0608-2}.
\end{lemma}
\begin{proof}
We show this lemma only for equations\,\eqref{eq:corr4} and \eqref{al:0608-2} because the equivalency of \eqref{eq:pre3} and \eqref{al:0608-1} can be shown in a similar manner.
To this end, it suffices to show that
the solution set of the second component equation of \eqref{eq:corr4} (see also the second line of \eqref{al:scalednewton-2})
is identical to that of \eqref{al:0608-2}, namely, we show
\begin{align}
&{\rm Sym}\left(G(x)\Delta Y+
\sum_{i=1}^n\Delta x_i
\mathcal{S}_{P}^i(x,Y)
\right)=\mu I - {\rm Sym}{\left(G(x)Y\right)} \label{al:0608-3}\\
&\Longleftrightarrow
\hG(x)^{\fr}\Delta \hY \hG(x)^{\fr} +
2{\rm Sym}\left(
\sum_{i=1}^n\Delta x_i\hU_i\hY\hG(x)^{\fr}
\right)
=\mu I-\hG(x)^{\fr}\hY\hG(x)^{\fr},
\label{eq:0422-2}
\end{align}
where $\Delta \hY:=P^{-\top}\Delta Y P^{-1}$.
We first show the part $(\Leftarrow)$.
Multiplying $P^{-1}\hG(x)^{\fr}$ and $\hG(x)^{-\fr}P$ from the left and right sides on both sides of \eqref{eq:0422-2},
respectively, we have
\begin{align*}
P^{-1}\hG(x)\Delta \hY P+
\sum_{i=1}^n
\Delta x_i\mathcal{S}_P^i(x,Y)=\mu I - P^{-1}\hG(x)\hY P,\notag
\end{align*}
which can be rewritten as $G(x)\Delta Y + \sum_{i=1}^n\Delta x_i\mathcal{S}_{P}^i(x,Y)=\mu I - G(x)Y$
via \eqref{eq:scalegxy} and $\Delta \hY=P^{-\top}\Delta Y P^{-1}$.
Summarizing this equation readily implies equation\,\eqref{al:0608-3}.
We next show the converse direction $(\Rightarrow)$.
For each $i=1,2,\ldots,n$,
let
$$Q_i:=
2P^{\top}
{\rm Sym}\left(
\hat{G}(x)^{-\fr}\hU_i\hat{Y}
\right)P\in S^m.$$
By taking into account \eqref{eq:scalegxy} and the fact that $G(x)P^{\top}\hG(x)^{-\fr}=P^{-1}\hG(x)^{\fr}$, it follows that,
$$
\mathcal{S}_{P}^i(x,Y)=
G(x)Q_i.
$$
Then, in terms of $\mathcal{L}_{G(x)}$, equation\,\eqref{al:0608-3} is equivalently transformed as
\begin{equation}
\mathcal{L}_{G(x)}\left(\Delta Y+\sum_{i=1}^n\Delta x_iQ_i+Y\right)=2\mu I.\label{eq:0608-4}
\end{equation}
Note the fact that given $A\in S^m_{++}$ and $B\in S^m$, the linear matrix equation $\mathcal{L}_AX=B$ has a unique solution in $X\in S^m$.
Then, equation \eqref{eq:0608-4} together with $G(x)\in S^m_{++}$
and $\mathcal{L}_{G(x)}^{-1}I=G(x)^{-1}/2$ implies
$\Delta Y = \mu G(x)^{-1}-\sum_{i=1}^n\Delta x_iQ_i-Y$. Multiplying $\hG(x)^{\fr}P^{-\top}$ and $P^{-1}\hG(x)^{\fr}$ from the left and right sides of both sides of this equation, respectively,
and recalling \eqref{eq:scalegxy} again yield the desired equation\,\eqref{eq:0422-2}.
This completes the proof.
\end{proof}
The above lemma indicates
that, to examine the linear equations \eqref{eq:pre3} and \eqref{eq:corr4},
we may consider \eqref{al:0608-1} and \eqref{al:0608-2} alternatively.
This is beneficial because, as stated in the following proposition, the limit operator of the sequence $\{T_{\tw^{\ell},\tP_{\ell}}\}_{\ell\ge 0}$
coincides with the Jacobian of $\Phi^2_{0}$ at $w^{\ast}$ (recall \eqref{eq:defPhi}), which is actually a one-to-one and onto mapping.
\begin{proposition}\label{prop:20190605-1}
It holds that (i)
$$
\lim_{\ell\to \infty}T_{\tw^{\ell},\tP_{\ell}} = \mathcal{T}_{\ast},
$$
where $\mathcal{T}_{\ast}:\mathcal{W}\to \mathcal{W}$ is the linear mapping defined by
$$
\mathcal{T}_{\ast}(\Delta w):=\begin{bmatrix}
\nabla^2_{xx} L(w^{\ast})\Delta x-\mathcal{J}G(x^{\ast})^{\ast}\Delta Y+\nabla h(x^{\ast})\Delta z \\
{\rm Sym}\left(G(x^{\ast})\Delta Y + \sum_{i=1}^n\Delta x_i\Gi(x^{\ast})Y_{\ast}\right)\\
\nabla h(x^{\ast})^{\top}\Delta x
\end{bmatrix}
$$
for any $\Delta w\in \mathcal{W}$.
(ii) Furthermore, $\mathcal{T}_{\ast}$ is a one-to-one and onto mapping.
\end{proposition}
\begin{proof}
(i) We show only the convergence on the second component of $T_{\tw^{\ell},\tP_{\ell}}$, namely,
for any $\Delta w=(\Delta x,\Delta Y,\Delta z)$, we show that
\begin{align}
\lim_{\ell\to \infty}
{\rm Sym}\left(G_{\ell}\Delta Y+
\sum_{i=1}^n\Delta x_i
\mathcal{S}_{P}^i(\tilde{x}^{\ell},\tY_{\ell})\right)={\rm Sym}\left(G(x^{\ast})\Delta Y+
\sum_{i=1}^n\Delta x_i\Gi(x^{\ast})Y_{\ast}\right).\notag
\end{align}
To prove this equation, since $\lim_{\ell\to\infty}\Gl=G(x^{\ast})$, it suffices to show that
\begin{align}
\lim_{\ell\to\infty}\mathcal{S}^i_{\tP_{\ell}}(\tilde{x}^{\ell},\tY_{\ell})=\Gi(x^{\ast})Y_{\ast}\label{eq:2342}
\end{align}
for an arbitrary index $i\in \{1,2,\ldots,n\}$.
Choose $i\in \{1,2,\ldots,n\}$ arbitrarily.
Note that, for any $w\in \mathcal{W}$, we have
\begin{align*}
\mathcal{S}_{P}^i(x,Y)&=
P^{-1}\left(
\hGi(x)\hY
-
\hU_i\hG(x)^{\fr}\hY
+
\hG(x)\hY\hU_i\hG(x)^{-\fr}
\right)P \\
=&
P^{-1}\left(
\hGi(x)\hY-\hU_i\hG(x)^{\fr}\hY+\mu\hU_i\hG(x)^{-\fr}+(\hG(x)\hY-\mu I)\hU_i\hG(x)^{-\fr}
\right)P\\
=&
P^{-1}\left(
-\hU_i\hG(x)^{-\fr}(\hG(x)\hY-\mu I)+
\hGi(x)\hY+(\hG(x)\hY-\mu I)\hU_i\hG(x)^{-\fr}
\right)P\\
=&-P^{-1}\hU_i\hG(x)^{-\fr}P(G(x)Y-\mu I)+\Gi(x) Y+(G(x)Y-\mu I)P^{-1}\hU_i\hG(x)^{-\fr}P,
\end{align*}
where the first equality follows because $\hG(x)^{\fr}\hU_i=\hGi(x)-\hU_i\hG(x)^{\fr}$ by \eqref{al:ui2}.
Since $\|\Gl\tY_{\ell}-\tmu_{\ell} I\|_F\le \tau \tmu_{\ell}^{1+\xi}$ from $\twl\in N_{\tmu_{\ell}}^{\tau \tmu_{\ell}^{1+\xi}}$, it holds that
\begin{align*}
\Phi_{i,\ell}
&:=\left\|-\tP_{\ell}^{-1}\mathcal{L}_{\hGl^{\fr}}^{-1}\left(\hGi(\txl)\right)\hGl^{-\fr}\tP_{\ell}\left(\Gl\tY_{\ell}-\tmu_{\ell} I\right)+\left(\Gl\tY_{\ell}-\tmu_{\ell} I\right)\tP_{\ell}^{-1}
\mathcal{L}_{\hGl^{\fr}}^{-1}\left(\hGi(\txl)\right)
\hGl^{-\fr}\tP_{\ell}\right\|_F\\
&\le
2\left\|\tP_{\ell}^{-1}
\mathcal{L}_{\hGl^{\fr}}^{-1}\left(\hGi(\txl)\right)
\hGl^{-\fr}\tP_{\ell}\right\|_F\left\|\Gl\tY_{\ell}-\tmu_{\ell} I\right\|_F\\
&\le 2\tmu_{\ell}\left\|\tP_{\ell}^{-1}
\mathcal{L}_{\hGl^{\fr}}^{-1}\left(\hGi(\txl)\right)
\hGl^{-\fr}\tP_{\ell}\right\|_F\cdot \frac{\tau \tmu_{\ell}^{1+\xi}}{\tmu_{\ell}}\\
&=O(\tmu_{\ell}^{\xi}),
\end{align*}
where the second equality holds from the boundedness of $\{\zli\}$ assumed in condition~({\bf P2}).
Hence, noting $\xi>0$ and $\lim_{\ell\to\infty}\tmu_{\ell}=0$, we have $\lim_{\ell\to
\infty}\Phi_{i,\ell} = 0$.
This fact readily yields $\|S^i_P(\txl,\tYl)-\Gi(x^{\ast})Y_{\ast}\|_F\le \Phi_{i,\ell}+\|\Gi(\txl)\tYl-\Gi(x^{\ast})Y_{\ast}\|_F\to 0$ as $\ell\to \infty$. This means that \eqref{eq:2342} is valid.
Since $i$ was arbitrary, we obtain the desired consequence.
(ii)
Notice that $\mathcal{T}_{\ast}$ is nothing but the Jacobian of
the function $\Phi^2_0(w)$ at the KKT point $w^{\ast}$.
Based on this fact, we can prove that $\mathcal{T}_{\ast}$ is a one-to-one and onto mapping in a manner similar to \cite[Theorem~1]{yamashita2012local} in the presence of the nondegeneracy, second-order sufficient, and strict complementarity conditions at $w^{\ast}$, which were assumed in the beginning of the section.
\end{proof}
Now, by means of \cref{prop:20190605-1},
let us prove \cref{prop:0613-2}.
\subsubsection*{Proof of \cref{prop:0613-2}}\label{sec:proof_thm}
By item~(ii) of \cref{prop:20190605-1},
by taking $\ell$ sufficiently large,
$\mathcal{T}_{\tw^{\ell},\tP_{\ell}}$ is a one-to-one and onto mapping and hence equations \eqref{al:0608-1} and \eqref{al:0608-2} have unique solutions. Therefore, by \cref{lem:0612},
equation~\eqref{eq:pre3} with $(P,w,\mu)=(\tP_{\ell},\tw^{\ell},\tmu_{\ell})$ turn out to posses the following solution\,\eqref{al:0415-22020} at $\tw^{\ell}$ uniquely, and the same relation holds between \eqref{eq:corr4} and \eqref{al:0415-12020}.
\begin{align}
\Deltap \tw^{\ell}&=T_{\tw^{\ell},\tP_{\ell}}(\tw^{\ell})^{-1}\eta^{\rm t}_{\ell},\label{al:0415-22020}\\
\Deltac \tw^{\ell}&=T_{\tw^{\ell},\tP_{\ell}}(\tw^{\ell})^{-1}\eta^{\rm c}_{\ell},\label{al:0415-12020}
\end{align}
where
$$
\eta^{\rm t}_{\ell}:=\begin{bmatrix}
0\\
- \tmu_{\ell}I \\
0
\end{bmatrix},\
\eta^{\rm c}_{\ell}:=\begin{bmatrix}
\nabla_{x}L(\tw^{\ell})\\
\tmu_{\ell}I - {\rm Sym}(G_{\ell}\tY_{\ell}) \\
- h(\tilde{x}^{\ell})
\end{bmatrix}
$$
for each $\ell$.
We next show the second-half claim.
It follows that
\begin{equation}
\|\eta^{\rm c}_{\ell}\|=O(\tmu_{\ell}^{1+\xi})\label{eq:0614-1}
\end{equation}
from
$\twl\in \N_{\tmul}^{\tau\tmul^{1+\xi}}$ and
$\|\tmu_{\ell}I - {\rm Sym}(G_{\ell}\tY_{\ell})\|_F\le
\|\tmu_{\ell}I -G_{\ell}\tY_{\ell}\|_F$.
Moreover, we have
\begin{equation}
\|\eta^{\rm t}_{\ell}\|=\sqrt{m}\tmu_{\ell}\label{eq:0614-2}
\end{equation}
by definition.
As the limit of $\{T_{\tw^{\ell},\tP_{\ell}}\}$ is a one-to-one and onto mapping by item~(ii) of \cref{prop:20190605-1},
equations \eqref{al:0415-22020} and
\eqref{al:0415-12020} derive
$\|\Deltap \tw^{\ell}\|=\Theta(\|\eta^{\rm t}_{\ell}\|)$ and $\|\Deltac \tw^{\ell}\|=\Theta(\|\eta^{\rm c}_{\ell}\|)$, which together with \eqref{eq:0614-1} and \eqref{eq:0614-2}
imply
\begin{equation*}
\|\Deltap \tw^{\ell}\|=O(\tmu_{\ell}),\
\|\Deltac \tw^{\ell}\|=O(\tmu_{\ell}^{1+\xi}).
\end{equation*}
The proof is complete.
\hfill $\square$
\subsubsection{Effectiveness of tangential and centering steps}
In this section, we give crucial properties holding at the next points we moved to in the tangential and centering steps.
Specifically, we show the following two propositions, where
$\alpha,\xi$, and $\xi^{\prime}$ are constants satisfying \eqref{al:cond}.
Henceforth,
$\ell_0$ is the positive integer defined in \cref{prop:0613-2} and moreover,
$\Deltap \tw^{\ell}$ and $\Deltac \tw^{\ell}$ are the unique solutions of the linear equations\,\eqref{eq:pre3} and \eqref{eq:corr4} with $(P,w,\mu)=(\tP_{\ell},\tw^{\ell},\tmu_{\ell})$, respectively, for each $\ell\ge \ell_0$.
\begin{proposition}\label{prop:0614}
The following properties hold:
\begin{enumerate}
\item\label{prop:0614-2}
Choose a sequence $\{s_{\ell}\}\subseteq (0,1]$ arbitrarily.
For $\ell\ge \ell_0$, we have
$
\|G(\tilde{x}^{\ell}+s_{\ell}\Deltap \tilde{x}^{\ell})(\tY_{\ell}+s_{\ell}\Deltap\tY_{\ell})-\mu(s_{\ell})I\|_F=O(\tmu_{\ell}^{1+\xi})
$
with $\mu(s_{\ell}):=(1-s_{\ell})\tmu_{\ell}$.
\item\label{prop:0614-1}
Choose $\alpha\in (0,\xi)$ arbitrarily. Then,
\begin{equation}
G(\tilde{x}^{\ell}+(1-\tmu_{\ell}^{\alpha})\Deltap \tilde{x}^{\ell})\in S^m_{++},\ \tY_{\ell}+(1-\tmu_{\ell}^{\alpha})\Deltap\tY_{\ell}\in S^m_{++} \label{eq:1517}
\end{equation}
hold for any sufficiently large $\ell\ge \ell_0$.
\item\label{prop:0614-3} Let $\twt^{\ell+\fr}:=\tw^{\ell} + (1-\tmu_{\ell}^{\alpha})\Deltap \tw^{\ell}$ and
$\tmu_{\ell+\fr}:=\tmu_{\ell}^{1+\alpha}$ for each $\ell$.
Then,
$\twt^{\ell+\fr}\in \N_{\tmu_{\ell+\fr}}^{\tau\tmu_{\ell+\fr}^{1+\xi^{\prime}}}$
holds for any sufficiently large $\ell\ge \ell_0$.
\end{enumerate}
\end{proposition}
\begin{proposition}\label{prop:0626}
The following properties hold:
\begin{enumerate}
\item\label{prop:0626-2}
Choose a sequence $\{s_{\ell}\}\subseteq (0,1]$ arbitrarily.
Then, we have
$$\|G(\tilde{x}^{\ell}+s_{\ell}\Deltac \tilde{x}^{\ell})(\tY_{\ell}+s_{\ell}\Deltac \tY_{\ell})-\tmu_{\ell}I\|_F=O((1-s_{\ell})\tmu_{\ell}^{1+\xi}+2s_{\ell}\tmu_{\ell}^{1+2\xi}).$$
\item\label{prop:0626-1}
For any sufficiently large $\ell\ge \ell_0$, we have
\begin{equation}
G(\tilde{x}^{\ell}+\Deltac \tilde{x}^{\ell})\in S^m_{++},\ \tY_{\ell}+\Deltac \tY_{\ell}\in S^m_{++}. \label{eq:2213}
\end{equation}
\item\label{prop:0626-3}
Choose $0<\kappa<\xi$ and let $\twc^{\ell+\fr}:=\tw^{\ell} +\Deltac \tw^{\ell}$ for each $\ell$. Then, $\twc^{\ell+\fr}\in
\N_{\tmu_{\ell}}^{\tau\tmu_{\ell}^{1+2\kappa}}$
holds for any sufficiently large $\ell\ge \ell_0$.
\end{enumerate}
\end{proposition}
The first proposition claims that
the next point after performing the tangential step is eventually accommodated by
$\N_{\tmu_{\ell+\fr}}^{\tau\tmu_{\ell+\fr}^{1+\xi^{\prime}}}$, which contains $\N_{\tmu_{\ell+\fr}}^{\tau\tmu_{\ell+\fr}^{1+\xi}}$ because $\xi>\frac{\xi-\alpha}{1+\alpha}>\xi^{\prime}$.
On the other hand, the second one claims that
the next point after the centering step is eventually accepted by
$\N_{\tmu_{\ell}}^{\tau\tmu_{\ell}^{1+2\kappa}}$, which is contained by
$\N_{\tmu_{\ell}}^{\tau\tmu_{\ell}^{1+\xi}}$ if $\kappa$ is chosen to be $\frac{\xi}{2}<\kappa$.
In order to prove the above propositions, we make use of the following two propositions.
\begin{proposition}\label{prop:order}
Let $F:\R^p\to \R^q$ be a twice continuously differentiable function
and $\{(v^{\ell},\Delta v^{\ell})\}\subseteq \R^p\times \R^p$ be a sequence converging to some point $(v^{\ast},0)\in \R^p\times \R^p$. Then,
$\left\|F(v^{\ell}+\Delta v^{\ell})-F(v^{\ell})-\nabla F(v^{\ell})^{\top}\Delta v^{\ell}
\right\|_2 = O(\|\Delta v^{\ell}\|_2^2)$.
\end{proposition}
\begin{proof}
By Taylor's theorem,
$
F_i(v^{\ell}+\Delta v^{\ell})-F_i(v^{\ell})-\nabla F_i(v^{\ell})^{\top}\Delta v^{\ell}
=\int_0^1 (1-t)(\Delta v^{\ell})^{\top}\nabla^2F_i(v^{\ell}+t\Delta v^{\ell})\Delta v^{\ell}dt
$
for $i=1,2,\ldots,p$, which yields the assertion.
\end{proof}
The second proposition is derived from the following lemma, whose proof is given in \,\cref{sec:appendix1}.
\begin{lemma}\label{lem:dx}
For matrices $A,B\in S^m$ and a scalar $\mu\in \R$, it holds that
$$
\|ABA-\mu I\|_F^2+\frac{\|A^2B-BA^2\|^2_F}{2}=\|A^2B-\mu I\|^2_F.
$$
\end{lemma}
\begin{proposition}\label{prop:xyx}
Let $X\in S^m_+$, $Y\in S^m$, and $\mu\in \R$.
Then, $\|X^{\fr}YX^{\fr}-\mu I\|_F\le \|XY-\mu I\|_F$.
\end{proposition}
\begin{proof}
The assertion follows from \cref{lem:dx} with $(A,B)=(X^{\fr},Y)$.
\end{proof}
\input{proofofprops}
\subsection{Proof of \cref{thm:main}}\label{sec:last}
In this section, we give a proof of our main result, \cref{thm:main}.
To this end, we first show the following lemma by invoking \cref{prop:0614} and \cref{prop:0626}.
We denote $\mathcal{B}_{r}(w):=\{v\in \W\mid \|v-w\|\le r\}$ for $r>0$ and $w\in \W$.
\begin{lemma}\label{thm:0710-1}
Choose $\delta\in (0,1)$ arbitrarily. Let $\P:\mathcal{W}_{++}\to \R^{m\times m}$
be a function satisfying conditions~{\rm({\bf P1})} and {\rm({\bf P2})}.
In addition, let $\xi$ be the constant in condition~{\rm ({\bf P2})} and $\alpha$ and $\xi^{\prime}$ be arbitrary constants satisfying \eqref{al:cond}.
Then, there exists some $\overline{u}>0$ such that,
for any $\mu\in (0,\overline{u}]$,
the following properties hold if $\olw\in \mathcal{B}_{\mu^{\delta}}(w^{\ast})\cap \N_{\mu}^{\tau\mu^{1+\xi}}$.: The linear equation~\eqref{eq:pre3} with $P=\P(\olw)$ has a unique solution $\Deltap w$ and
it holds that
\begin{align}
\olw_{\fr}:=\olw+(1-\mu^{\alpha})\Deltap w\in \mathcal{B}_{\mu_+^{\delta}}(w^{\ast})\cap \N_{\mu_+}^{\tau\mu_+^{1+\xi^{\prime}}},\label{al:0705-1}
\end{align}
where $\mu_+:=\mu^{1+\alpha}$.
Moreover, the linear equation~\eqref{eq:corr4} with $P=\P(\olw_{\fr})$ has a unique solution $\Deltac w$ and it holds that
\begin{align}
\olw_{\fr}+\Deltac w\in \mathcal{B}_{\mu_+^{\delta}}(w^{\ast})\cap \N_{\mu_+}^{\tau\mu_+^{1+\xi}}.\label{al:0705-2}
\end{align}
\end{lemma}
\begin{proof}
We first show the unique solvability of equation~\eqref{eq:pre3} by contradiction.
Suppose to the contrary that there exist sequences $\{\tmu_{\ellem}\}\subseteq \R_{++}$ and $\{\tw^{\ellem}\}\subseteq \mathcal{W}_{++}$ such that
\begin{equation}
\lim_{\ellem\to \infty}\tmu_{\ellem}=0,\ \ \tw^{\ellem}\in \mathcal{B}_{\tmu_{\ellem}^{\delta}}(w^{\ast})\cap \N_{\tmu_{\ellem}}^{\tau\tmu_{\ellem}^{1+\xi}}
\label{eq:0802-1}
\end{equation}
but equation~\eqref{eq:pre3} with $(w,\mu,P)=(\tw^{\ellem},\tmu_{\ellem},\P(\tw^{\ellem}))$
is not uniquely solvable for each $\ellem$. By the assumptions that $\tw^{\ellem}\in \mathcal{B}_{\tmu_{\ellem}^{\delta}}(w^{\ast})$ for each $\ellem$ and $\tmu_{\ellem}\to 0$ as $\ellem\to \infty$, we obtain $\lim_{\ellem\to \infty}\tw^{\ellem}=w^{\ast}$. Thus, from \cref{prop:0613-2},
equation\,\eqref{eq:pre3} must be uniquely solvable for any $\ellem$ sufficiently large. However, this is a contradiction and therefore the first assertion is ensured.
Next, we show \eqref{al:0705-1}. We derive a contradiction again by supposing the existence of sequences
$\{\tmu_{\ellem}\}\subseteq \R_{++}$ and $\{\tw^{\ellem}\}\subseteq \mathcal{W}_{++}$ such that \eqref{eq:0802-1} holds but \eqref{al:0705-1} with $(\olw,\mu,\mu_+)=(\tw^{\ellem},\tmu_{\ellem},\tmu_{\ellem}^{1+\alpha})$
is invalid for any $\ellem$, namely,
\begin{equation}
\tw^{\ellem}+(1-\tmu_{\ellem}^{\alpha})\Deltap \tw^{\ellem}\notin \mathcal{B}_{\tmu_{\ellem}^{(1+\alpha)\delta}}(w^{\ast})\cap \N_{\tmu_{\ellem}^{1+\alpha}}^{\tau\tmu_{\ellem}^{(1+\xi^{\prime})(1+\alpha)}},\label{eq:0710-1}
\end{equation}
where $\Deltap \tw^{\ellem}$ is the unique solution of equation\,\eqref{eq:pre3} with $(w,\mu,P)=(\tw^{\ellem},\tmu_{\ellem},\mathcal{P}(\tw^{\ellem}))$.
Noting that $\lim_{\ellem\to\infty}\tw^{\ellem}=w^{\ast}$ as in the former argument
and using item-\ref{prop:0614-3} of \cref{prop:0614}, we have
\begin{equation}
\tw^{\ellem}+(1-\tmu_{\ellem}^{\alpha})\Deltap \tw^{\ellem}\in \N_{\tmu_{\ellem}^{1+\alpha}}^{\tau\tmu_{\ellem}^{(1+\alpha)(1+\xi^{\prime})}} \label{eq:0711-1}
\end{equation}
for any $\ellem$ sufficiently large.
Since $\Deltap \tw^{\ellem}\to 0$ by item-\ref{e1} of \cref{prop:0613-2} and $w^{l}\in \N_{\tmu_{l}^{\tau\tmu_{l}^{1+\xi}}}$, $\lim_{\ellem\to\infty}\tw^{\ellem}=w^{\ast}$ implies
\begin{equation}
\lim_{\ellem\to\infty}\tw^{\ellem}+(1-\tmu_{\ellem}^{\alpha})\Deltap \tw^{\ellem}=w^{\ast}.\label{eq:0711-2}
\end{equation}
In addition, \eqref{eq:0711-1} together with \eqref{eq:0710-1} implies
$\tw^{\ellem}+(1-\tmu_{\ellem}^{\alpha})\Deltap \tw^{\ellem}\notin \mathcal{B}_{\tmu_{\ellem}^{\delta(1+\alpha)}}(w^{\ast})$ for any $\ellem$, from which we can derive
\begin{align*}
\tmu_{\ellem}^{\delta(1+\alpha)}&<\|\tw^{\ellem}+(1-\tmu_{\ellem}^{\alpha})\Deltap \tw^{\ellem}-w^{\ast}\|
=O(\tmu_{\ellem}^{1+\alpha})
=o(\tmu_{\ellem}^{\delta(1+\alpha)}),\notag
\end{align*}
where the first equality follows from \eqref{eq:0711-2} and \cref{prop:0219-1} with
$\{\tw^{\ell}\}$, $\{\tmu_{\ell}\}$, and $\{r_{\ell}\}$ replaced by $\{\tw^{\ellem}+(1-\tmu_{\ellem}^{\alpha})\Deltap \tw^{\ellem}\}$, $\{\tmu_{\ellem}^{1+\alpha}\}$, and $\{\tau\tmu_{\ellem}^{(1+\alpha)(1+\xi^{\prime})}\}$, respectively.
Therefore, we obtain an obvious contradiction $\tmu_{l}^{\delta(1+\alpha)}=o(\tmu_{\ellem}^{\delta(1+\alpha)})$ leading us to the desired relation~\eqref{al:0705-1}. Consequently, we ensure the existence of some $u_1>0$ such that \eqref{al:0705-1} together with the unique solvability of equation\,\eqref{eq:pre3} holds for $\mu\in (0,u_1]$.
In turn, we prove the second-half claim.
The unique solvability of \eqref{eq:corr4} can be proved similarly to the above arguments by replacing $(\olw,\mu)$ with $(\olw^{\fr},\mu_+)$ satisfying \eqref{al:0705-1}. So, we omit it.
We show \eqref{al:0705-2}.
Recall
$\xi^{\prime}>\frac{\xi}{2}$
from \eqref{al:cond} and choose $\xi^{\dprime}\in \R$ such that
\begin{equation}
\xi^{\prime}>\xi^{\dprime}>\frac{\xi}{2}.\label{eq:dprime}
\end{equation}
To prove \eqref{al:0705-2}, it actually suffices to show that, for a sufficiently small $\mu$, we have
\begin{equation}
\olw_{\fr}+\Deltac w\in \mathcal{B}_{\mu_+^{\delta}}(w^{\ast})\cap \N_{\mu_+}^{\tau\mu_+^{1+2\xi^{\dprime}}}.\label{eq:0102}
\end{equation}
Indeed,
note that $\mu_+=\mu^{1+\alpha}<1$ by taking a sufficiently small $\mu$.
Hence, the assumption that $\xi^{\dprime}>\xi/2$ yields
$\mu_+^{1+2\xi^{\dprime}}<\mu_+^{1+\xi}$ entailing $\N_{\mu_+}^{\tau\mu_+^{1+2\xi^{\dprime}}}\subseteq \N_{\mu_+}^{\tau\mu_+^{1+\xi}}$. Therefore, this fact together with \eqref{eq:0102} derives \eqref{al:0705-2}.
For the sake of proving \eqref{eq:0102},
take $\mu\in (0,u_1]$ with $u_1$ determined above. Then, \eqref{al:0705-1} holds and equation\,\eqref{eq:corr4} is uniquely solvable.
Moreover, for a contradiction, suppose that there exist sequences $\{\tw^{\ellem+\fr}\}$ and $\{\tmu_{\ellem}\}$
such that, by writing $p_{\ellem}:=\tmu_{\ellem}^{1+\xi}$ for each $\ellem$, it holds that
$$
\lim_{\ellem\to\infty}\tmu_{\ellem}=0,\
\tw^{\ellem+\fr}+\Deltac \tw^{\ellem+\fr}\notin \mathcal{B}_{p_{\ellem}^{\delta}}(w^{\ast})\cap\N_{p_{\ellem}}^{\tau p_{\ellem}^{1+2\xi^{\dprime}}},\
\tw^{\ellem+\fr}\in \mathcal{B}_{p_{\ellem}^{\delta}}(w^{\ast})\cap \N_{p_{\ellem}}^{\tau p_{\ellem}^{1+\xi^{\prime}}},
$$
where $\Deltac \tw^{\ellem+\fr}$ denotes the unique solution of \eqref{eq:corr4} with $(w,\mu,P)=(\tw^{\ellem+\fr},p_{\ellem},\mathcal{P}(\tw^{\ellem+\fr}))$.
A contradiction can be then derived by the same argument as the one for \eqref{al:0705-1}, where, in place of item-\ref{prop:0614-3} of \cref{prop:0614} and item-\ref{e1} of \cref{prop:0613-2},
we use item-\ref{prop:0626-3} of \cref{prop:0626} with $\{(\tw^{\ell},\Deltac \twl,\tmu_{\ell},\kappa)\}$ replaced
by $\{(\tw^{\ellem+\fr},\Deltac \tw^{\ellem+\fr},p_{\ellem},\xi^{\dprime})\}$ and also utilize item-\ref{e2} of \cref{prop:0613-2}
with $\{(\tw^{\ell},\Deltac \twl,\tmu_{\ell})\}$ replaced by $\{(\tw^{\ellem+\fr},\Deltac \tw^{\ellem+\fr},p_{\ellem})\}$.
We thus conclude \eqref{eq:0102}, and ensure the existence of some $u_2>0$ such that \eqref{eq:0102} holds and the unique solvability of equation\,\eqref{eq:corr4} is valid for $\mu\in (0,u_2]$.
Finally, by setting $\ol{u}:=\min(u_1,u_2)$, the whole proposition is proved.
\end{proof}
We are now ready to prove our main theorem.
\subsubsection*{Proof of \cref{thm:main}}
{Choose $\delta\in (0,1)$ arbitrarily.
By taking $w^0\in \N_{\mu_0}^{\tau \mu_0^{1+\xi}}$ with $\mu_0=\gamma \|\Xi_0^I(w^0)\|$ sufficiently close to $w^{\ast}$,
we obtain
$\|w^0-w^{\ast}\|<\|w^0-w^{\ast}\|^{\frac{\delta}{2}}<\mu_0^{\delta}$, where
the first inequality follows from $0<\delta<1$ and the last inequality does from \cref{prop:0219-1} and $\delta/2<\delta<1$.
We thus have
\begin{equation}
w^0\in \mathcal{B}_{\mu_0^{\delta}}(w^{\ast})\cap \N_{\mu_0}^{\tau \mu_0^{1+\xi}}.\label{eq:0123}
\end{equation}}
Now, let us prove assertion {\bf (i)} by mathematical induction.
To begin with, note that according to \cref{thm:0710-1}, there exists some $\overline{u}>0$ such that
\begin{equation}
\begin{array}{c}
\mu\in (0,\overline{u}];\\
w\in \mathcal{B}_{\mu^{\delta}}(w^{\ast})\cap \N_{\mu}^{\tau \mu^{1+\xi}}
\end{array}
\Longrightarrow
\begin{array}{c}
\mbox{\eqref{eq:pre3} with $P=\mathcal{P}(w)$: u.s., \eqref{al:0705-1}};
\\
\mbox{\eqref{eq:corr4} with $(w,\mu,P)=(w_{\fr},\mu_+,\mathcal{P}(w_{\fr}))$: u.s., \eqref{al:0705-2}},
\end{array}
\label{eq:0115}
\end{equation}
where
``u.s.'' stands for ``uniquely solvable'', $w_{\fr}:=w+(1-\mu^{\alpha})\Deltap w$,
and $\mu_+:=\mu^{1+\alpha}$. Let us prove \eqref{eq:0704-1721} for $k=0$.
Recall that $\xi$ is the constant chosen in the initial setting of \cref{alg4}.
Again, take $w^0$ so close to $w^{\ast}$ that $\mu_0=\gamma\|\Xi^I_0(w^0)\|<\min(0.95,\overline{u})$.
Under the setting $(w,\mu)=(w^0,\mu_0)$, we obtain \eqref{al:0705-1} and \eqref{al:0705-2} from \eqref{eq:0123} and \eqref{eq:0115}. This implies that $G(x^0+(1-\mu_0^{\alpha})\Deltap x^0)\in S^m_{++}$ and $Y_0+(1-\mu_0^{\alpha})\Deltap Y_0\in S^m_{++}$, and therefore
$1-\mu_0^{\alpha}$ is set to $\bar{s}^{\rm t}_0$ in Line\,\ref{al:skt} of \cref{alg4}.
Hence, $\mu_1=(1-\bar{s}^{\rm t}_0)\mu_0=\mu_0^{1+\alpha}$, i.e., \eqref{eq:0704-1721} with $k=0$.
In addition, by \eqref{eq:0115} with $(w,\mu)=(w^0,\mu_0)$ again,
the linear equation~\eqref{eq:pre3} with $(w,P,\mu)=(w^0,\P(w^0),\mu_0)$ and \eqref{eq:corr4} with
$(w,P,\mu)=(w^{\fr},\P(w^{\fr}),\mu_{\fr})$ are ensured to have unique solutions. Moreover, we obtain conditions~\eqref{eq:0702-2329} and \eqref{eq:0704-1726} for $k=0$. We thus conclude the desired conditions altogether for $k=0$.
Subsequently, suppose that conditions~\eqref{eq:0704-1721}, \eqref{eq:0702-2329}, and \eqref{eq:0704-1726} together with $\mu_k<\min(0.95,\ol{u})$ hold for some $k\ge 0$, which imply
$w^{k+1}\in \mathcal{B}_{\mu_{k+1}^{\delta}}(w^{\ast})\cap \N_{\mu_{k+1}}^{\tau\mu_{k+1}^{1+\xi}}$ and $\mu_{k+1}<\overline{u}$, that is, the left-hand side of \eqref{eq:0115} with $w=w^k$ and $\mu=\mu_k$.
We thus, by the right-hand side of \eqref{eq:0115}, obtain conditions\,\eqref{eq:0704-1721}, \eqref{eq:0702-2329}, and \eqref{eq:0704-1726} with $k$ replaced by $k+1$, and moreover establish the unique solvability of equations\,\eqref{eq:pre3} and \eqref{eq:corr4} at the $(k+1)$-th iteration.
By induction, we conclude assertion~{\bf (i)} for each $k\ge 0$.
We next prove assertion {\bf (ii)}. By the above proof, we see $w^{k}\in \mathcal{B}_{\mu_k^{\delta}}(w^{\ast})\cap N_{\mu_{k}}^{\tau\mu_{k}^{1+\xi}}$ for each $k\ge 0$ and ensure that $\{w^k\}$ converges to $w^{\ast}$ with fulfilling $w^k\in N_{\mu_{k}}^{\tau\mu_{k}^{1+\xi}}$. Hence, the assertion readily follows because \cref{prop:0219-1} together with \eqref{eq:0704-1721} yields
\begin{equation}
\|w^{k+1}-w^{\ast}\|=O(\mu_{k+1})=O(\mu_{k}^{1+\alpha})=O(\|w^k-w^{\ast}\|^{1+\alpha}).\notag
\end{equation}
Lastly, by \eqref{al:cond}, $1+\alpha$ is bounded from above as
$
1+\alpha<\frac{2(1+\xi)}{2+\xi}<\frac{4}{3}.
$
The proof is complete.
$ \hfill \Box $ | 153,314 |
TITLE: Spontaneous symmetry breaking by two scalar multiplets
QUESTION [2 upvotes]: Consider a theory with two multiplets of real scalar fields $\phi_i$ and $\epsilon_i$, where $i$ runs
from $1$ to $N$. The Lagrangian is given by: $$\mathcal L = \frac{1}{2} (\partial_{\mu} \phi_i) (\partial^{\mu} \phi_i) + \frac{1}{2} (\partial_{\mu} \epsilon_i) (\partial^{\mu} \epsilon_i) − \frac{m^2}{2}[\phi_i \phi_i+ \epsilon_i \epsilon_i] − \frac{g}{8}[(\phi_i \phi_i)(\phi_j \phi_j ) + (\epsilon_i \epsilon_i)(\epsilon_j \epsilon_j)] − \frac{λ}{2}(\phi_i \epsilon_i)(\phi_j \epsilon_j ),$$ where $m^2 < 0, g > 0 $ and $\lambda > −g/2.$ Summation over repeated indices is implied.
Is the following accurate? The lagrangian can be written in vector notation and we can see it is then invariant under a simultaneous transformation of $\phi$ and $\epsilon$ such that $\epsilon \rightarrow R_{ij}\epsilon_j$ and $\phi_i \rightarrow R_{ij} \phi_j$ if $R_{ik} R_{ij} = \delta_{kj}$ The symmetry group is then $O(N) \otimes O(N)$ with generators $T_a^{O(N) \otimes O(N)} = T_a^{O(N)} \otimes \text{Id}_{N \times N} + \text{Id}_{N \times N} \otimes T_a^{O(N)}$ so there are $\text{dim}O(N)$ number of generators.
The vacua of the theory can be found as the minimum of the potential $$V(\phi, \epsilon) = \frac{m^2}{2} ( \vec \phi^T \vec \phi + \vec \epsilon^T \vec \epsilon) + \frac{g}{8} ((\vec \phi^T \vec \phi)^2 + (\vec \epsilon^T \epsilon)^2) + \frac{\lambda}{2} (\vec \phi^T \vec \epsilon)^2$$ I am a bit confused here - to find the vacua I could demand $$\frac{\partial V}{\partial \phi^T \phi} = \frac{\partial V}{\partial \epsilon^T \epsilon} \overset{!}{=} 0$$ but what happens to the term proportional to $\lambda$?
REPLY [2 votes]: No, your "following" is not accurate. You wrote a SB Lagrangean invariant under O(N)×O(N) (⊂ O(2N)), except for the λ term, which is only invariant under its diagonal subgroup O(N), instead.
The N φs and the N εs fit into 2 N - vector, $(\vec{\phi},\vec{\epsilon})$, so the symmetry starts as O(2N) but the g term is only invariant under its O(N)×O(N) subgroup.The indices of the φs and the εs need not know about each other, except for the λ term contracting them together: think of synchronized swimming. The λ term is thus invariant under O(N), not O(N)×O(N).
To tease a stay against confusion, pick N =3, so six real scalar fields. Display the O(6) -invariant part, which the g term restricts to O(3)×O(3), and finally the λ term to O(3). Now study the further SSB built in---this is a popular problem I sometimes assign.
The key is always in the 2N×2N Goldstone mass matrix $\langle \delta_i \delta_j V \rangle $ and, in particular, its kernel consisting of the null eigenvectors.
Now simplify the algebra by defining
$$
\frac{-2m^2}{g} \equiv v^2 ,
$$
so that the positive overall scale of the potential, g/8, may be safely dropped.
Further shifting the potential by the innocuous constant terms to convert it to a sum of squares, obtain
$$
V=(\vec{\phi}^2 - v^2)^2+(\vec{\epsilon}^2 -v^2)^2+ \frac{4\lambda}{g} (\vec{\phi}\cdot\vec{\epsilon})^2.
$$
It is evident that, for λ =0, this is the two standard Goldstone hyper-sombrero potentials superposed, so their minima are at $\langle \vec{\phi}^2 \rangle= \langle \vec{\epsilon}^2 \rangle= v^2 $.
Naturally, $\langle \vec{\phi} \rangle$ may pick any orientation in the bottom of its sombrero hypersurface, and $\langle \vec{\epsilon} \rangle$ an arbirary, in general different one, in its own; so the group is SSBroken down to O(N-1)×O(N-1). Your 2N×2N Goldstone mass matrix will have 2(N-1) null vectors, so goldstons. For N = 3, you get 4 goldstons.
For λ ≠0, however, the symmetry is only O(N), as stated.
For λ >0, it is manifest from the potential sum of positive squares form that
$$
\langle \vec{\phi}^2 \rangle=v^2;\qquad \langle \vec{\epsilon}^2 \rangle=v^2; \qquad\langle \vec{\phi} \cdot \vec{\epsilon}\rangle =0~.
$$
That is to say, the vacuum orientations of $\vec{\phi}$ and $\vec{\epsilon}$ must be orthogonal. W.l.o.g, take $\langle \phi_1 \rangle=v =\langle \epsilon_2 \rangle$.
The surviving invariance is then only O(N-2), and the goldstons 2N-3, so 3 for N =3 -- can you see it in your Goldstone matrix? (Hint: Confirm only $\phi_1, \epsilon_2$ and $\phi_2+\epsilon_1$ are massive, for all N.)
The plot thickens for 0> λ > -g/2. Now the λ term is compelled to grow, not shrink, and, magnitudes being equal (dictated by the other terms), it presses to align $\langle \vec{\phi} \rangle$ with $\langle \vec{\epsilon} \rangle$, so then,
$\langle \vec{\phi} \rangle=\langle \vec{\epsilon} \rangle$.
Specifically, consider the first variation that apparently stymied you in the first place, (recall that stationarity is required for every component of the fields, not just the magnitudes of their group vectors!),
$$\langle \frac{\delta V}{\delta\vec{\phi}} \rangle =
\langle \frac{\delta V}{\delta\vec{\epsilon}} \rangle =0, $$
and thus
$$
0=-v^2\vec{\phi}+(\vec{\phi}\cdot\vec{\phi})\vec{\phi} +\frac{2\lambda}{g} (\vec{\phi}\cdot\vec{\epsilon})\vec{\epsilon} \\
0=-v^2\vec{\epsilon}+(\vec{\epsilon}\cdot\vec{\epsilon})\vec{\epsilon}+ \frac{2\lambda}{g} (\vec{\epsilon}\cdot \vec{\phi})\vec{\phi}.
$$
It is manifest that $ \vec{\epsilon}\propto \vec{\phi}$, so define $\vec{\epsilon}\equiv a \vec{\phi}$ for real nonvanishing a.
The extremizing conditions then reduce to just
$$
v^2=\langle \vec{\phi}\cdot \vec{\phi} \rangle \left(1+ \frac{2\lambda}{g} a^2\right); \qquad
v^2=\langle \vec{\phi}\cdot \vec{\phi} \rangle \left(a^2+ \frac{2\lambda}{g}\right),
$$
so, then, $a^2=1$, recalling the condition $\frac{2\lambda}{g}+1>0$.
Take a = 1, perfect alignment of $\langle \vec{\phi} \rangle$ with $\langle \vec{\epsilon} \rangle$, and $\langle \vec{\phi}^2 \rangle=\langle \vec{\epsilon}^2 \rangle=v^2/(1+2\lambda/g)$. You then have an unbroken residual subgroup O(N-1),
so only N-1 goldstons now, 2 for N =3. Observe the v.e.vs increase without bound with decreasing negative λ.
Given this alignment, you might go back to the potential and monitor how λ <- g/2, beyond the pale, would overwhelm the terms in the Sombrero potentials and flip them, destabilizing them, thus preventing SSB, among other calamities. | 51,101 |
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TITLE: Volume form is "always positive"
QUESTION [1 upvotes]: The Wikipedia article on volume forms states the following:
In mathematics, a volume form on a differentiable manifold is a nowhere-vanishing top-dimensional form (i.e., a differential form of top degree).
Later it is mentioned that:
Any oriented pseudo-Riemannian (including Riemannian) manifold has a natural volume form. In local coordinates, it can be expressed as $\omega ={\sqrt {|g|}}dx^{1}\wedge \dots \wedge dx^{n}$
Is this the only volume form that we can define on a Riemannian manifold or is it just the most natural (since it is constructed from the Jacobian) For example, could we prove that:
$$\omega = dx^1 \wedge \dots \wedge dx^n$$ may be vanishing ?
REPLY [4 votes]: If $\omega$ is a volume form on $M$, then $f \omega$ is also a volume form for any smooth positive function $f$ on $M$. So if there is one volume form, then there are infinite many volume forms!
The volume form $\sqrt{g} \ dx_1 \wedge \dots \wedge dx_n$ is the natural volume form in the context of general relativity because it actually corresponds to the physical volume.
[For example, if $M$ is a spherical surface, so the metric is $ds^2 = d\theta^2 + \sin^2 \theta \ d\phi^2$, then $\sqrt{g} \ d\theta \wedge d\phi = \sin \theta \ d\theta \ d\phi$ is the familiar volume form in spherical coordinates that actually describes the real physical volume.]
Another nice thing about the "natural" volume form is that it is given by the expression $\sqrt{g} \ dx_1 \wedge \dots \wedge dx_n$ in all local coordinate charts. Your suggestion of $dx_1 \wedge \dots dx_n$ will look very different in a different coordinate frame - and then you would have to decide how to define this outside your chosen coordinate patch as well, which may not even be possible.
[Back to the sphere example, picking $d\theta \wedge d\phi$ as the volume form wouldn't actually work because there would be no way of smoothly extending this to the north and south poles (which are not covered by the $(\theta, \phi)$ coordinate system).] | 120,984 |
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Anyway, here's part one:
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The wife of former Thai Prime Minister Thaksin Shinawatra has been found guilty of evading tax and sentenced to three years jail. Ron Corben reports for VOA from Bangkok.
The Thai criminal courts have found the wife of former Thai Prime Minister Thaksin Shinawatra guilty of tax evasion linked to a 1997 transfer of shares in the family company that cost the state $16 million in lost revenue.
The high-profile verdict against Potjaman Shinawatra, her brother-in-law and her secretary was broadcast nationally.
The judge said Potjaman and the co-accused faced jail terms of up to three years. In court to hear the verdict were the former prime minister and the couple's adult children - all of whom appeared grim-faced. Later, the three defendants were granted bail, pending an appeal.
Mrs. Thaksin had claimed the share transfer was a "gift" not a business transaction, as claimed by the prosecution.
Former Prime Minister Thaksin - who was ousted in a 2006 coup - is facing accusations of abuse of power and corruption. Soon after taking power, the military set up a special committee of auditors and judges to investigate cases of corruption during Thaksin's five years in power.
The committee has handed over several cases for prosecution through the National Counter-Corruption Commission.
Panitan Wattanayagorn, a political scientist from Chulalongkorn University, says the court's decision against Potjaman marks a step forward in Thai efforts to curb official corruption.
"If this case proceeds further, successfully and legally, and some ex-leaders, members of the leaders' families, if they are convicted it is going to be a new beginning of transparency, legitimacy and counter-corruption efforts in Thailand," Panitan said.
The Thai Supreme Court is also scheduled to hand down a verdict in September on abuse of power charges linked to the purchase of state-owned land by Potjaman when Thaksin was prime minister.
Other cases include charges connected to a $120-million loan to Burma's military government, a lottery scheme, an agriculture procurement program, bank loans to politicians and allegations of corruption linked to Bangkok's new $4-billion international airport.
A a political scientist from Chulalongkorn University, Thitinan Pongsudhirak, says the court cases alone may not settle Thailand's political climate.
"The court cases are critical," Thitinan said. "They are reshaping Thailand's political landscape. But I doubt that the judicial decisions that are coming up are going to resolve Thailand's political crisis."
Thailand's political scene remains fragile, despite the election of a new government last December. Prime Minister Samak Sundaravej is viewed as continuing to represent Thaksin, who remains politically influential in Thailand.
But Thaksin, together with more than 100 political party executives, is banned from politics for five years by a constitutional tribunal decision in 2007 ruled against his former Thai Rak Thai Party for election irregularities. | 31,754 |
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