text:
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Jonathan Pryce and Adam Driver in THE MAN WHO KILLED DON QUIXOTE (2018)
Don Qui-OK
THE MAN WHO KILLED DON QUIXOTE, directed by Terry Gilliam
Published on Apr 22, 2019 4:12PM EDT REVIEWS Primary category in which blog post is published
Joe MacLeod	@JOEMACLEOD666
Creative Director for HMM DAILY. Once and future game show contestant.
Terry Gilliam, who directed Monty Python and the Holy Grail (1975), Time Bandits (1981), The Meaning of Life (1983), Brazil (1985), The Fisher King (1991), Twelve Monkeys (1995), and Fear and Loathing in Las Vegas (1998), among several other films, is the reason I recently bought a ticket for a "one night only" screening of The Man Who Killed Don Quixote, because I will support whatever he decides to release as a movie, and I will tell others about the latest Terry Gilliam movie. At this point it's like supporting a friend who has a band or something, and they put on a show, and you have to go, and it's gonna be great, because, you know, it's your friend's show, and sometimes not very many people make it out to the show.
Right now though, you really want a win for your friend, because it's been awhile for him, and you know the history of this latest film, an almost 30-year struggle of Mr. Gilliam trying to get this movie made, a project that has been in progress for so long that it has had a generational series of different actors, as the movie fell apart several times for several reasons. Now, it's completed, with Jonathan Price and the new Millennial Darth Vader-wannabe from Star Wars, Adam Driver, as the stars, but you can't help but think of the others who might have been in this film if you saw it almost 30 years ago.
This film is an exploration of El Ingenioso Hidalgo Don Quijote de la Mancha, Miguel de Cervantes metafictional novel of a man whose fantastical life as a knight-errant slops over onto reality, a beloved theme for director Gilliam, who folds the tale in upon itself with additional layers of meta involving a director making a Don Quixote movie, who somehow enables a real-life agent of chivalry.
I got so outside of this film trying to keep my expectations in check for something so long delayed, that all I could do was is try to not look forward to seeing it, and then once I got in the theater, I failed at not being disappointed. It was fine, but it's surrounded by the haze of what it could have, should have, might have been.
Adam Driver in The Man Who Killed Don Quixote (2018)
The film starts by doing something I think everybody loves, which is to shit on the industry of Advertising, as we witness Mr. Driver's jaded commercial director being an asshole making pretentious crap to sell stuff. He's unhappy and empty, and by happenstance is presented with a DVD of the student film he made years ago, when he cast a local cobbler in the role of Don Quixote. Then he has an identity crisis as he meets with more people we love to hate, clients, but it's easy to see them as Mr. Gilliam's idea of movie producers, the money people, and I got happy thinking the film was going to be a whole bunch of showing how shitty the movie business is, and Terry Gilliam should know, because he produced a bunch of visionary hits, but lately he can't ever seem to scrape up enough dough to fully realize his projects, which would be confirmed by anyone who saw his film The Zero Theorem (2013), and watched the money run out, on screen.
As we trip from flashback to dream to fantasy to movie, the menace of the money people stays in the picture, well manifested by Stellan Skarsgård and Jordi Mollà, big-money goons leaning on the director, but Mr. Driver is miscast, he was great at being a tortured spoiled director, but he's too vigorous, I kept not believing he was in any kind of jeopardy. There's no way I could watch this film and not think what it would have been with Johnny Depp playing the character, he's the guy I would believe as being afraid of physical violence and afraid of the police, one of Gilliam's favorite themes. Mr. Driver is the big doofus from Girls, he's Darth Vader Jr., for fuck's sake, he can project weakness, but not the right kind.
Jonathan Pryce in The Man Who Killed Don Quixote (2018)
Meanwhile, Jonathan Pryce, a guy I know from a few Terry Gilliam movies, plus recently he was the High Sparrow on Game of Thrones, is the cobbler who would be the Ingenious Gentleman Sir Quixote of La Mancha, a chivalrous knight-errant, and he is doeeng an ethSpanish acksenn', and it sucked me out of the movie. It's a solid emotional performance that goes from nuanced to appropriately overblown, all within the scope of the delusional Quixote's diagnosis, but the accent ripped me out of the movie, yes, it's all made up, and actors are people who act like other people from other places and sometimes other planets, but this one, this time, it just seemed a little shticky. I only have so much disbelief to suspend, and I have been suspending my disbelief for over 20 years about this flick ever getting made, and I came to the theater for Terry Gilliam, and he put on a show, and I'll be at the next one.
The Great Firewall Stands in the Open Now
JiaJing Liu
Published on Jul 15, 2019 4:38PM EDT
THE WORST THING WE READ™ Primary category in which blog post is published
They Makes a Good Case
The Best Thing We Read Yesterday™
Published on Jul 11, 2019 11:59PM EDT
FEATURES Primary category in which blog post is published
A True Story About Visas
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text:
<?php defined('BX_DOL') or die('hack attempt');

bx_import('BxDolCron');

class BxDolCronUpgradeModulesCheck extends BxDolCron
{
    public function processing()
    {
        if ('on' != getParam('sys_autoupdate_modules'))
            return;

        $o = bx_instance('BxDolUpgraderModules');
        if(!$o->prepare()) {
            sendMailTemplateSystem('t_UpgradeFailed', array (
                'error_msg' => $o->getError(),
            ));
            setParam('sys_autoupdate_modules', ''); // disable auto-update if it is failed
        }
    }
}

/** @} */

redpajama_set_name: RedPajamaGithub


text:
import common._

class User(val username: String, var password: String) {
  override def toString: String = s"$username / $password"
}

class ApplicationUser(username: String, password: String, val fullName: String, var address: String = "None",
                      var profession: String = "Unknown")
  extends User(username, password) {

  // example of secondary constructor
  def this(username: String, password: String, fullName: String, address: String, number: Int) = {
    this(username, password, fullName, address + " " + number)
    profession = "Not disclosed"
  }

  override def toString: String = s"${super.toString}, $fullName, $address, profession = ${profession}"
}

object Classes extends App {

  bigTitle("Classes")

  headLine("Concise syntax for class creation")

  val superman = new User("superman", "superpassword")
  println(superman)
  superman.password = "newpassword"
  println(superman)
  // would give error: "reassignment to val"
  // superman.username = "new username"

  headLine("Inheritance")

  val clarkKent = new ApplicationUser("ckent", "34jh3khk34", "Clark Kent", "Some street", "Under cover superhero")
  println(clarkKent)
  clarkKent.address = "A new address"
  println(clarkKent)

  val bobbyBrown = new ApplicationUser("bbrown", "goingdown", "Bobby Brown", "FZ Avenue")
  println(bobbyBrown)

  var jesseJames = new ApplicationUser("jjames", "vcs9dwm568f", "Jesse James", "Some Address", 355)
  println(jesseJames)

}


I would like to officially announce the rebirth of my independent Mac consulting career: Jake Fowler Consulting. I am now in business for myself again and it feels great. What do you think of the new look?
This entry was posted in Blog on May 4, 2005 by jakefowler.
I like it! Yeah you!


One year ago today, we moved home. We have moved a number of times since we left Birmingham in June of 2000. But this was the only one without any real emotional difficulty. We left a church where I was doing youth ministry and was pretty much paid to leave (severance…thankful for that). In my first few months there I had to fire my secretary and was told the lead pastor – the fundamental reason I felt comfortable accepting a call to that church – was leaving for another position. I'm not bitter.
OK, maybe I'm a little bitter.
We left behind a number of close friends and a city we liked. But there has not been one moment in which we have not been passionately glad to be back in the city where we were raised.
All four seasons have passed through our time here. We have watched them come and go like old friends who we long to see again soon. Though summer can take the long way back and spring should stick around a while longer, we yearn for many more and look forward to watching the seasons turn into years and the years into decades. All here.
Speaking of old friends, we have spent a good bit of time with many. It is incredible that I can now go to lunch with friends who have known me for so long and love me still. What is just as wonderful is all the new friends we now have – so easy to be with, laugh with and rejoice with. We have no money to speak of but are wealthy in friendships to the point of not having the time to spend them all. Our cup runs over daily.
Our family is close. Birthdays are spent with them and our kids can get to know them well. All was out of grasp before. Now, our children beg to spend the day with their grandparents and vice versa.
Only someone from Birmingham will understand. But every turn down a mountain lane and every road climbing out of one of the valleys is a reason for wonder. Right now the trees have burst into the usual colors of Autumn we never really get used to. The morning fog is likely to cut off the tops of the trees and block any hope for a view of the valley below. The air is crisp and bursts with nostalgia.
Those who know me well will understand the salvific and noetic effect of my being able to enjoy all my favorite restaurants. Mr. P's Deli and New China in Bluff Park, both about 3 minutes from where we live. Jim-N-Nick's and Surin West on Southside. The Purple Onion at the bottom of the mountain. The Fish Market. These are important landmarks in my life. No really, when not eating there, I am dreaming about them.
I love this place. Nothing could induce me to move again. The desire for Emma, Knox and Dylan to grow up here is strong. I know it may sound crazy for a pastor to say but place seems important right now. Especially this place, this place so crammed like a dusty attic full of memories, good and bad – reminders of God's grace and reminders of my indwelling sin. It's impossible to explain. But, well, the desire to stay here forever feels like a gift. We were willing to go anywhere for so long. And did. And so this gladness to be here till buried here feels like a gracious present given by a loving Father.


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The Right-to-Sue: Privacy's Stand-or-Fall
Federal Trade Commission is the main federal agency that ensures that companies follow federal data privacy standards. However, even FTC cannot compare in manpower with thousands (or millions, depending on the company's scale) of vigilant consumers. That brings up an important issue – for what exactly can consumers sue?
Consumers' ability to sue is a powerful tool. It may force many companies to spend hundreds of thousands of dollars on legal fees. Amplified by the class action potential, such right-to-sue will be able to put many start-ups out of business.
That is why the right-to-sue is a point of discussion in privacy law bills that may soon pass into federal law. However, even if the federal law grants consumers the right-to-sue, which is as yet unclear, the Constitution has the ultimate say, and this say is called the standing requirement.
Let us imagine a company provided computerized search and information about people. It issued a report about a person that turned out to be incorrect. The person found it out and sued the company that issued the report. Will the person be able to sue the company under federal laws? The U.S. Supreme Court says, "it depends." Spokeo, Inc. v. Robins, 136 S.Ct. 1540 (2016).
In the Spokeo case, plaintiff Robins sued a company that provided computerized search information about him. The company reported him as a married man in his 50s, employed and with children. Robins did not like this misinformation because it made him appear overqualified and made it hard for him to find a job. The Supreme Court addressed the question of "standing."
"Standing" is a constitutional requirement that one coming to court with a claim must have (1) an injury in fact, (2) that is fairly traceable to the defendant's conduct, and (3) that is likely to be redressed by a favorable judicial decision. What is important, the injury-in-fact requirement cannot be erased by a federal law's grant of the right to sue. So, even if federal law says that Robins can sue, that does not mean that the Constitution says the same.
What does the Constitution say?
The Supreme Court said that Constitution requires something more than just a procedural violation.
"Robins could not, for example, allege a bare procedural violation, divorced from any concrete harm, and satisfy the injury-in-fact requirement of Article III."
Spokeo at 1549.
So, if a person alleged a "bare procedural violation" – without any harm – that alone will not satisfy the constitutional standing requirement.
"For example, even if a consumer reporting agency fails to provide the required notice to a user of the agency's consumer information, that information regardless may be entirely accurate. In addition, not all inaccuracies cause harm or present any material risk of harm."
Thus, the Spokeo case teaches us that there should be more than just a procedural violation like a zip code typo or a harmless omission of consumer notice. The violation of privacy should result in concrete and particularized harm.
Additional Note: This article talks only about federal law and the U.S. Constitution. Please note that state laws may already provide for the private right-to-sue. For example, California Civil Code Sec. 1798.150 states that consumers can institute civil actions. Whether state laws will be preempted (trumped over) by the new federal law in this regard, is still unclear.
EN RUS KOR Business Law

redpajama_set_name: RedPajamaCommonCrawl


We would definitely recommend HIP Communications. We had a few issues in the beginning but they listened to what we needed and solved everything! Their customer service is very good. With their high-speed internet service, we are able to upload photos and information for the families and that adds value to our center.


The ceremony presenting the awards will recognise and celebrate professional excellence and best practice in the accountancy and finance profession.
Members, special guests and award winners will enjoy a networking reception and a 3 course dinner with live entertainment to celebrate accounting and finance leaders/organisations across the Middle East who have demonstrated superior excellence in their field – who are the very best in what they do.
There are ten awards categories in 2012 including: CFO of the Year, Business Leader of the Year, Young Accountant of the Year, Chartered Accountant of the Year, Excellence in Financial Journalism, Excellence in Financial Reporting, Firm of the Year and Corporate Finance Deal of the Year, Business Finance Team of the Year and Excellence in Training and Development of Finance Professionals.
The Middle East CFO Alliance (MECA) Conference is set to be held on 12/12/12 at Dubai Shangri-la Hotel.
Top Executives and Subject Matter Experts from American Express, SAP, Bayt.com, and other prominent organisations have been confirmed as speakers at the conference. The conference will end around 1300 hours with a networking lunch.
According to the association's coordinator Saleem Sufi, the core philosophy of MECA is to organise high quality in-person information sharing and networking events to its members at minimal or no cost to them.
Please note MECA conferences are invitation-only events and you must be a member of Middle East CFO Alliance to attend this conference. To become a member of MECA, you may send your request via the associations' LinkedIn forum.
Global financial markets demand international consistency in accounting practices.
The implementation of International Financial Reporting Standards (IFRS) is key to increasing the transparency and comparability of financial information and further enhancing the efficiency of international capital markets.
The standards themselves are principles–based and require a detailed knowledge of the underlying concepts, as well as the application of extensive professional judgment. This seminar will help delegates to understand key new IFRS and will focus on the implications of recent changes in IFRS. It will help delegates put both IFRS accounting principles into practice and understand and interpret IFRS financial statements.
This essential seminar will guide the participants through the current issues, changes and amendments of discussion papers, exposure drafts, and new IFRS standards.
The newly issued IFRS 10 'Consolidated Financial Statements', IFRS 11 'Joint Arrangements', IFRS 12 "Disclosure of Interests in Other Entities" and IFRS 13 'Fair Value Measurement' will be discussed and explained with practical examples. The status of all FASB–IASB projects will be reviewed, as well as the latest updates in the IASB and IFRIC.
The programme is designed for busy executives and will focus on the critical issues ahead of the forthcoming reporting season. This is Ernst & Young longest — established and most widely recognized annual IFRS event and is a programme that no finance professional in the region can afford to miss.


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text:
Afghan Girls Want A Cricket League Of Their Own
With no officially sanctioned women's league to play in, a 16-year-old in Kabul has set up her own cricket team to get on the pitch.
2018 Winter Olympics Open In South Korea
Host city Pyeongchang officially kicked off the 2018 Winter Olympic Games with a dazzling opening ceremony.
Polish Climbers Aim For Winter Scaling Of World's Second Highest Peak
A group of Polish mountaineers are attempting to be the first to scale K2, the world's second highest peak, in wintertime.
Iranian Wrestler Throws Match To Avoid Israeli Peer, Loses Medal
Iranians on social media are enraged after wrestler Alireza Karimi appeared to deliberately lose a match against his Russian opponent to avoid facing an Israeli rival in the next round.
Pakistan's 'Running Of The Bulls'
Near the border of the northern provinces of Punjab and Khyber Pakhtunkhwa, local villagers gather every few months to race. Bulls run the 100 meter distance with their owners and handlers hanging on for dear life. (RFE/RL's Radio Mashaal)
Sri Lankan Cricket Team In Pakistan, Eight Years After Attack
Security was tight as Sri Lanka's cricket squad arrived in Lahore to play the final of a three-match series, eight years after its players and team officials were attacked by terrorists in the city.
International Games Open In Ashgabat
The opening ceremony of the 2017 Asian Indoor and Martial Arts Games (AIMAG) is set to take place in Turkmenistan's capital, Ashgabat.
Cricket Craze Takes Afghanistan By Storm
Afghanistan's biggest success story is not in wars, politics or reconstruction, where most of international aid and Afghan energy have been focused for decades.
At Least Three Killed In Suicide Blast Outside Kabul Cricket Stadium
At least three people were killed when a suicide bomber blew himself up while attempting to enter a cricket stadium in the Afghan capital, Kabul, officials say.
Pakistan Holds First Top-Level Cricket Match Since 2009 Terror Attack
Play has concluded in a cricket match conducted under heavy security in Lahore, the first top-level competition held in Pakistan since a terror attack in 2009 scared off the sport's top teams.
Pakistan Hosts Big-Time Cricket For First Time Since 2009 Terror Attack
Amid extreme levels of security, Pakistan is set to reenter the world of international cricket, hosting world-class competition for the first time since a deadly terror attack in 2009 scared off the sport's top teams.
Kazakhs Defeat Kyrgyz For World's First Kokpar Championship
Kazakhstan has become the first world champion in an ancient equestrian sport known under different regional names -- kokpar, kok-boru, or buzkashi. (RFE/RL's Kazakh Service)


Martin Luther King Jr. addresses a crowd from the steps of the Lincoln Memorial, where he delivered his famous "I Have a Dream" speech, during the Aug. 28, 1963, march on Washington, D.C.
Martin Luther King Jr. had more than "a dream," but you might not notice that on Monday during observances for his birthday.
Somewhere between his assassination and today began an MLK-neutering campaign meant to turn the famed agitator's holiday into a national Day of Service, a generic mishmash of good feelings that contorts King's social-justice legacy into a blissful Hallmark card of post-racial nothingness.
This has not gone unnoticed, from scholar Cornel West—who has pushed back against the "Santa Clausification" of King—to the #ReclaimMLK campaign, currently being spearheaded by young activists involved in the Black Lives Matter campaign. Many realize it's time to stand up for what King actually stood for before his entire legacy is retconned into some "Chicken Soup for the Soul Brother," feel-good tripe.
Garza is part of the #ReclaimMLK campaign, which kicked off on King's actual birthday, Jan. 15 and involves acts of civil disobedience—marches, sit-ins and shut-ins—through Jan. 19. Protesters want to use the holiday to call for action against police brutality while reiterating King's legacy as a leader dedicated to defending black lives. Garza said that organizers are using the day off that many workers get to call for a "day on" to fight injustice.
"We really want to do everything that we can on this 'Day of Service' to lift up the needs and dreams of black lives," Garza said.
The work of this campaign is important, since MLK did not die because he wanted someone to paint a wall. While volunteering in and of itself is great and people should be encouraged to give back, it is not what got him killed. He was shot by an assassin for fighting against the racist, classist status quo, murdered while in Memphis, Tenn., to support striking black garbage workers. To make his birthday a day about giving back is to defang the legacy of a man who was both revered and reviled in his time for his tireless work confronting America's sins.
Calling out the ills of racism, capitalism and the Vietnam War didn't involve any of the warm fuzzies that revisionists have assigned to King by solely focusing on the fantastical end of his famed speech at the March on Washington.
Marshall Ganz, senior lecturer in public policy at Harvard's John F. Kennedy School of Government, pointed out that while most people know about King's "I Have a Dream" speech, they may not know that the bulk of the speech was about urging the Kennedy administration to act on protecting the rights of black citizens.
The "they" Ganz is referring to are those "who benefit from the status quo." He says that people who benefit from the way our society is structured like things the way they are and want to avoid controversy. Neutralizing King is more about "normalizing" a once-contentious situation to make it more palatable. In this case it involves shame surrounding the enforcement of racial segregation by violence and the disregard for African Americans' citizenship.
Ganz lauded the efforts of #ReclaimMLK, believing that it is necessary to "retrieve" MLK's "prophetic past as opposed to that Santa Claus, that sanitized past.


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Below are useful links to friends and supporters of IHFH
Idaho Conservation Officers Association
The Idaho Conservation Officer Association is comprised of nearly every game warden in Idaho. A voice for the present & future Idaho wildlife officers.
All wildlife, including all wild animals, wild birds, and fish, within the state of Idaho, is hereby declared to be the property of the state of Idaho. It shall be preserved, protected, perpetuated, and managed. It shall be only captured or taken at such times or places, under such conditions, or by such means, or in such manner, as will preserve, protect, and perpetuate such wildlife, and provide for the citizens of this state and, as by law permitted to others, continued supplies of such wildlife for hunting, fishing and trapping.
Ducks Unlimited is the world's leader in wetlands and waterfowl conservation. DU got its start in 1937 during the Dust Bowl when North America's drought-plagued waterfowl populations had plunged to unprecedented lows. Determined not to sit idly by as the continent's waterfowl dwindled beyond recovery, a small group of sportsmen joined together to form an organization that became known as Ducks Unlimited.
Idaho Food Bank
The Idaho Foodbank, established in 1984, is an independent nonprofit 501(c) (3) organization with warehouses in Boise, Lewiston and Pocatello. It is the largest hunger relief organization and the largest distributor of free food assistance in Idaho.
Food distribution is made possible by generous donors, thousands of volunteers and a network of more than 200 nonprofit partners that receive food from the Foodbank's warehouses and distribute it directly to people in need. Our partners include food pantries, senior centers, rescue missions, community kitchens and other feeding programs.
Nation Wild Turkey Federation
The NWTF — a national nonprofit organization — is the leader in upland wildlife habitat conservation in North America.
Through vital partnerships with state, federal and provincial wildlife agencies, the NWTF and our members have helped restore wild turkey populations throughout North America — from a mere 30,000 in the entire United States to more than 7 million across the United States, Canada and Mexico.
Help IHFH Feed the Hungry
We are accepting donations to help pay for the processing of donated wild game and we need your financial support.
(c) 2016 Idaho Hunters Feeding the Hungry


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text:
Catholic Sexual Abuse / By TriMark
Phil Saviano, Survivor of Clergy Sex Abuse, Dies at 69
Category: Catholic Sexual Abuse
He played a pivotal role in helping The Boston Globe uncover the widespread scandal in the Catholic Church and was portrayed in the movie "Spotlight."
Phil Saviano was a relentless and determined activist who documented the actions of dozens of pedophile priests in the Boston area and coaxed other survivors to go public with their stories.
Phil Saviano carried a terrible secret for much of his life — that in the early 1960s, when he was 11, he was sexually molested by his parish priest in Massachusetts.
Nearly 30 years later, suffering from AIDS and believing he would soon die, he decided to go public about the abuse and disclosed his experience to The Boston Globe. As it happened, Mr. Saviano lived, and he went on to play a pivotal role in bringing to light the widespread pedophile priest scandal coursing through the Roman Catholic Church.
He provided key information and guidance for a series of articles by The Globe's Spotlight team, which won the Pulitzer Prize for Public Service in 2003. And his role was dramatized in the Academy Award-winning 2015 movie "Spotlight," which showed how the church had hidden its crimes and how The Globe had uncovered them.
Mr. Saviano died on Sunday in Douglas, Mass., at the home of his brother, Jim. He was 69.
Jim Saviano said that his brother had been in failing health for some time. He underwent heart bypass surgery this year, then had a stroke and was diagnosed with aggressive gall bladder cancer.
A relentless and determined activist, Phil Saviano documented the actions of dozens of pedophile priests in the Boston area and coaxed other survivors to go public with their stories. He helped educate the Spotlight team about how priests had groomed their victims for eventual seduction and how the church had knowingly shuttled rogue priests to different parishes, where they often went on to abuse other children.
"Think of the enormity of what the church had been hiding for so long," Walter V. Robinson, the former editor of the Spotlight team, said in an interview for this obituary in September. "That was a lot of combustible material, and it was Phil more than anyone else who set it ablaze."
See the full, original post here: https://www.nytimes.com/2021/11/28/us/phil-saviano-dead.html
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According to the Veep, by spending $50B + for high speed rail in economic times that put all of us and generations to come owing tons of money this administration spent and borrowed and spent to the point of (almost??) no return.
In other words – rocketing to the poorhouse.
Idiot. I hope everyone else picks up on the timing of this idiocy.
hhhmm, I wonder how much of that $50B could be used to seal the southern border …….
This entry was posted in Constitution, Corruption, Deficit, Economy, Energy, Ethics, Illegal Aliens, National Security, Politics, Rights, Rule of Law, Sovereignty, Taxes, Terrorism and tagged Control, Fraud, Incompetence, Jobs, Marxism, Spending, Unemployment. Bookmark the permalink.
And from what I'm hearing, the border with Canada isn't all that great since they all too many into their country who want to slip over the border to do damage to ours!


text:
The Hainan Island Operation (), or  in Japanese, was part of a campaign by the Empire of Japan during the Second World War to blockade the Guangdong mainland and prevent it from communicating with the outside world and from receiving imports of much-needed arms and materials.

Background
Hainan Island lies midway between French Indochina and Hong Kong, occupying a position south of the Leizhou Peninsula across the Strait of Hainan. It is also near Kwangchowan, a French-leased territory on the southern coast of China. The Hainan Island has an area of , and had a population of 2,200,000 at the time. The island was guarded by the 152nd Division, approximately 25,000 strong, under the command of Yu Hanmou, who was in charge of peace preservation in Kwangtung Province.

The Japanese Navy, after the capture of Canton (Guangzhou) the previous year, had maintained a formidable blockade all along the coast of south, central and north China. However, loopholes were found in the southern end of the blockade line. These included the supply route to Chiang Kai-shek with Hong Kong and Northern French Indochina as relay points and direct routes through Hainan Island and Kwangchowan. Because of these loopholes, as well as the necessity to conduct air operations deep into the interior of China, as far as the Kunming area, the Japanese Navy came to feel the necessity of establishing air bases on Hainan Island. The Central Authorities of the Navy advocated for this move. Operations were carried out by the Special Naval Landing Forces with Army elements supporting them.

Operation
Escorting a convoy, the South China Naval Force (Fifth Fleet) commanded by Vice Admiral Kondo Nobutake entered and anchored in Tsinghai Bay on the northern shore of Hainan Island at midnight on 9 February 1939 and carried out a successful landing. In addition, Navy land combat units effected a landing at Haikou at 1200 on 10 February. Thereafter, the Army and Navy forces acted in concert to mop up the northern zone. On 11 February, land combat units landed at Samah (Sanya) at the southern extremity of Hainan Island and occupied the key positions of Yulin and Yai-Hsien. Thereafter, the units engaged in the occupation and subjugation of the entire island.

Retreat to Wuzhi mountain range
Facing crisis, Nationalist forces evacuated all remaining civilians from Haikou to Qionghai to the safe Wuzhi mountain range in central Hainan. However, they faced fierce opposition by the ethnic Li highlanders there. An ethnic Li called Wang Guoxing started an uprising but was brutally crushed, and, in revenge, the Nationalists killed 7,000 of Wang Guoxing's family members in his village.

The Communists under Feng Baiju and the native Li people of Hainan fought a vigorous guerrilla campaign against the Japanese occupation, the Japanese killed large numbers of Li in western Hainan (e.g. Sanya, Danzhou). Furthermore, numerous foreign slave labourers were also killed. There are mass graves of tens of thousands of Korean slave labourers in Sanya and throughout the island. Of the 100,000 slave labourers from Hong Kong, only 20,000 survived the war.

Partial occupation of Hainan
Later, Japanese-occupied parts of Hainan Island became a naval administrative district with Hainan Guard District Headquarters established at Samah. Strategically, the island was built as a forward air base as well as an advance base for blockading Chiang. At the same time, the iron and copper resources of the island were exploited. Partial control of certain areas of Hainan Island provided a base of operations for the invasion of Guangdong province and French Indochina, as well as providing airbases that permitted long-distance air raids of routes into China from French Indochina and Burma.

The occupation of some parts of Hainan lasted until the surrender of Japan in September 1945, following the atomic bombings of Hiroshima and Nagasaki, in part  by the Enola Gay, that killed approximately 226,000 Japanese civilians in Hiroshima and Nagasaki.

See also
Order of Battle: Hainan Island Operation

References

  Japanese Monograph No. 144: POLITICAL STRATEGY PRIOR TO OUTBREAK OF WAR PART I, Prepared by MILITARY HISTORY SECTION HEADQUARTERS, ARMY FORCES FAR EAST, DISTRIBUTED BY OFFICE OF THE CHIEF OF MILITARY HISTORY DEPARTMENT OF THE ARMY, Chapter 4, Operations against China during 1939 Pg 46-47.
 中国抗日战争正面战场作战记 China's Anti-Japanese War Combat Operations
 Author: Guo Rugui, editor-in-chief Huang Yuzhang
 Press: Jiangsu People's Publishing House
 Date published: 2005-7-1

 Online in Chinese
  第七部分:相持阶段前期的作战海南岛作战 1 Hainan Island battles
 Phillips, R. T. "The Japanese Occupation of Hainan," Modern Asian Studies, Vol. 14, No. 1 (1980), pp. 93–109,
 Hainan Lawyers' group to help ex-'comfort women,' China Daily, 2006-03-23,

Battles of the Second Sino-Japanese War
Military history of Hainan
Second Sino-Japanese War
Conflicts in 1939
1939 in China
1939 in Japan
February 1939 events

redpajama_set_name: RedPajamaWikipedia


Made of vintage Japanese kimono silk, this silk square has a pale orange ground with floral prints in white. This fabric has a dry, bouncy handle and wears well as a puff and fold. Fantastic as a long scarf too.


THE union boss behind strikes causing misery to thousands of rail commuters has been pictured travelling to work by car.
When asked if he was enjoying an easier commute than strike-hit passengers, he said: "I do not even drive.
The Evening Standard found that Mr Whelan often rides to his office on trains where drivers open and close the doors – the very system he condemns as too unsafe.
Figures released under the Freedom of Information act have revealed there has been a fall in dangerous incidents since driver-only operated (DOO) trains were introduced on key London Overground lines, which Mr Whelan uses.
Transport For London gave details of the number of incidents in the year before DOO services and the year after on two key lines.
They showed the number of "signals passed at danger" incidents fell from 12 to seven and the number of times the emergency braking system was used fell from 30 to 23.
As well as this, the number of train doors on the wrong side were opened fell from 32 to 16 and incidents of doors released when the train was not fully in the station fell from five to zero.
This has cast questions over Aslef's claim that safety dears over the trains are behind the strikes that will halt services for 300,000 people next week.
Station over runs and speeding was also reduced, the only category that increased, from three to five cases, were incidents where drivers went through a station without stopping.
On Thursday, the independent rail safety watchdog approved Southern's DOO services as safe for passengers and rail staff.
The capital is bracing itself for a week of havoc as RMT union is planning a Tube strike on Sunday night and Monday – unless last ditch talks prevent them.
Southern Rail drivers are also due to go on strike for three days next week and three more later in the month due to the row over who should close train doors.
Business groups estimated the cost to London as tens of millions of pounds.


Here, in one book, are the full plans and instructions for the conversion to selective fire of 7 popular weapons. Includes plans for the HK 91, HK93, Mac-10, Mac-11, AR-15, Mini-14 & the M1 Carbine. 5 1/2 x 8 1/2, 71 pages, softcover, illus.


text:
\section{Introduction}
\label{s_intro}
This paper is the second in the series
initiated with Ref.~\cite{Isacker15}, henceforth referred to as I.
The overall purpose of this series
is the study of nuclear shapes with a higher-rank discrete symmetry,
in particular of the tetrahedral or octahedral type,
in the framework of a variety of interacting boson models.
For the general context of this study we refer the reader to the introduction in I,
of which only the essential points and references are mentioned here.

The paper by Li and Dudek~\cite{Li94}
pointed out the possibility of intrinsic nuclear shapes with a higher-rank discrete symmetry.
Subsequent publications by the Strasbourg group and their collaborators~\cite{Dudek02,Dudek03,Dudek06,Rouvel14}
showed the possible occurrence of tetrahedral, octahedral and icosahedral symmetries
through combinations of deformations of specific multipolarity.
In parallel with these theoretical developments,
a search was initiated for experimental manifestations
of such symmetries in nuclei~\cite{Curien10,Curien11,Jentschel10}.
In addition to these experimental searches cited in I,
a more recent study of this type appeared in 2018~\cite{Dudek18},
formulating criteria for the identification of tetrahedral and/or octahedral symmetries in nuclei.

From the theory side most work related to higher-rank discrete symmetries
has been carried out in the context of mean-field models.
The aim of this series of papers is to address the question
of the possible occurrence of higher-rank discrete symmetries
from a different theoretical perspective.
Our study is carried out in the context of algebraic collective models
inspired by the interacting boson model (IBM)
of Arima and Iachello~\cite{Arima76,Arima78,Arima79},
which proposes a description of quadrupole collective nuclear states
in terms of $s$ and $d$ bosons with angular momentum $\ell=0$ and $\ell=2$.
The application of this idea to collective states of different nature,
notably of octupole and hexadecapole character,
requires the introduction of other bosons~\cite{Iachello87},
in particular $f$ and $g$ bosons with angular momentum $\ell=3$ and $\ell=4$.
Since shapes with tetrahedral symmetry
arise in lowest order through a particular octupole deformation
and those with octahedral symmetry
emanate from a combination of hexadecapole deformations,
the study of such shapes in an algebraic context
requires the introduction of $f$ and $g$ bosons, respectively.

We initiated this program in I with the study of octahedral shapes in the \mbox{$sdg$-IBM}.
We considered the most general rotationally invariant Hamiltonian
with up to two-body interactions between the bosons
and derived the conditions for this Hamiltonian
to have in its classical limit a minimum with an intrinsic shape with octahedral symmetry.
Owing to the general nature of the analysis in I,
only qualitative conclusions could be drawn
with regard to the (non-)existence of such minima in the \mbox{$sdg$-IBM}.
In the present paper we arrive at more concrete conclusions
by considering a subset of all possible Hamiltonians of the \mbox{$sdg$-IBM},
namely those that have a dynamical symmetry and are analytically solvable.
Although none of the symmetry Hamiltonians leads to a stable shape with octahedral symmetry,
a subset can be used to propose a generalization
with additional interactions between the $g$ bosons
that drive the system toward such a shape.

The paper is structured as follows.
To avoid repeatedly referring to equations in I,
we list in Section~\ref{s_sdgibm} formulas from I that are needed in this paper.
In Section~\ref{s_limits} we focus on two dynamical symmetries of the \mbox{$sdg$-IBM}
of particular relevance in our quest for octahedral shapes
and in Section~\ref{s_clas} the classical limit is derived for the Hamiltonian
that is transitional between these two limits.
The central results of this paper are presented in Section~\ref{s_octa},
with a catastrophe analysis of the energy surface
associated with the transitional symmetry Hamiltonian,
and suitable generalizations thereof,
to uncover the existence of minima at stable shapes with octahedral symmetry.
Finally, in Section~\ref{s_conc} the conclusions
of the second paper in this series are summarized.

\section{The \mbox{$sdg$-IBM} and its classical limit}
\label{s_sdgibm}
A boson-number-conserving, rotationally invariant Hamiltonian of the \mbox{$sdg$-IBM}
with up to two-body interactions is of the form
\begin{eqnarray}
\hat H&=&\epsilon_s\hat n_s+
\epsilon_d\hat n_d+
\epsilon_g\hat n_g
\nonumber\\&&
+\sum_{\ell_1\leq\ell_2,\ell'_1\leq\ell'_2,L}
\frac{(-)^Lv^L_{\ell_1\ell_2\ell'_1\ell'_2}}{\sqrt{(1+\delta_{\ell_1\ell_2})(1+\delta_{\ell'_1\ell'_2})}}
[b^\dag_{\ell_1}\times b^\dag_{\ell_2}]^{(L)}\cdot
[\tilde b_{\ell'_2}\times\tilde b_{\ell'_1}]^{(L)},
\label{e_ham}
\end{eqnarray}
where $\hat n_\ell$ is the number operator for the $\ell$ boson,
$\epsilon_\ell$ is the energy of the $\ell$ boson
and $v^L_{\ell_1\ell_2\ell'_1\ell'_2}$ is a boson--boson interaction matrix element.

A geometric understanding of this quantum-mechanical Hamiltonian
is obtained by considering its expectation value in a coherent state.
As discussed in I,
a study of shapes with octahedral symmetry in \mbox{$sdg$-IBM}
requires a coherent state of the form
\begin{equation}
|N;\beta_2,\beta_4,\gamma_2,\gamma_4,\delta_4\rangle=
\sqrt{\frac{1}{N!(1+\beta_2^2+\beta_4^2)^N}}
\Gamma(\beta_2,\beta_4,\gamma_2,\gamma_4,\delta_4)^N|{\rm o}\rangle,
\label{e_coh}
\end{equation}
with
\begin{eqnarray}
\Gamma(\beta_2,\beta_4,\gamma_2,\gamma_4,\delta_4)&=&
s^\dag+
\beta_2\Bigl[\cos\gamma_2d^\dag_0+
\sqrt{\textstyle{\frac 1 2}}\sin\gamma_2(d^\dag_{-2}+d^\dag_{+2})\Bigr]
\label{e_coh1}\\&&+
\beta_4\Bigl[\Bigl(\sqrt{\textstyle{\frac{7}{12}}}\cos\delta_4+
\sqrt{\textstyle{\frac{5}{12}}}\sin\delta_4\cos\gamma_4\Bigr)g^\dag_0
\nonumber\\&&\phantom{+\beta_4\Bigl[}-
\sqrt{\textstyle{\frac 1 2}}\sin\delta_4\sin\gamma_4(g^\dag_{-2}+g^\dag_{+2})
\nonumber\\&&\phantom{+\beta_4\Bigl[}+
\Bigl(\sqrt{\textstyle{\frac{5}{24}}}\cos\delta_4-
\sqrt{\textstyle{\frac{7}{24}}}\sin\delta_4\cos\gamma_4\Bigr)
(g^\dag_{-4}+g^\dag_{+4})\Bigr],
\nonumber
\end{eqnarray}
in terms of the deformation parameters
$\beta_2$, $\beta_4$, $\gamma_2$, $\gamma_4$ and $\delta_4$,
following the convention of Rohozi\'nski and Sobiczewski~\cite{Rohozinski81}.
A shape with octahedral symmetry is obtained for $\beta_2=0$, $\beta_4\neq0$
and $\delta_4=0$ (octahedron) or $\delta_4=\pi$ (cube) with arbitrary $\gamma_4$.
Another solution with octahedral symmetry
exists for $\gamma_4=0$ and $\delta_4=\arccos(1/6)$,
corresponding to a rotated octahedron,
i.e. with the same intrinsic shape as for $\delta_4=0$.
The parameterization introduced in Ref.~\cite{Rohozinski81} therefore
does {\em not} define a unique intrinsic state.

The expectation value of the Hamiltonian~(\ref{e_ham})
in the coherent state~(\ref{e_coh}) (or the classical limit of $\hat H$)
leads to the energy surface
\begin{eqnarray}
\langle\hat H\rangle&\equiv&
E(\beta_2,\beta_4,\gamma_2,\gamma_4,\delta_4)
\nonumber\\&=&
\frac{N(N-1)}{(1+\beta_2^2+\beta_4^2)^2}
\sum_{kl}\beta_2^k\beta_4^l
\left[c'_{kl}+\sum_{ij}c^{ij}_{kl}\cos(i\gamma_2+j\gamma_4)\phi_{kl}^{ij}(\delta_4)\right],
\label{e_climit}
\end{eqnarray}
where $\phi_{kl}^{ij}(\delta_4)$ are trigonometric functions defined in I.
The coefficients $c'_{kl}$ are known
in terms of the scaled single-boson energies $\epsilon'_\ell\equiv\epsilon_\ell/(N-1)$
and the interaction matrix elements $v^L_{\ell_1\ell_2\ell'_1\ell'_2}$,
\begin{eqnarray}&&
\textstyle
c'_{00}={\frac 1 2}v_{ssss}^0+\epsilon'_s,
\nonumber\\&&
\textstyle
c'_{20}=\sqrt{\frac 1 5}v_{ssdd}^0+v_{sdsd}^2+\epsilon'_s+\epsilon'_d,
\nonumber\\&&
\textstyle
c'_{02}={\frac 1 3}v_{ssgg}^0+v_{sgsg}^4+\epsilon'_s+\epsilon'_g,
\nonumber\\&&
\textstyle
c'_{40}=\frac{1}{10}v_{dddd}^0+{\frac 1 7}v_{dddd}^2+\frac{9}{35}v_{dddd}^4+\epsilon'_d,
\nonumber\\&&
\textstyle
c'_{22}=
\frac{1}{\sqrt{45}}v_{ddgg}^0
+\frac{7}{\sqrt{715}}v_{ddgg}^4
+\frac{1}{6}v_{dgdg}^2
+\frac{4}{11}v_{dgdg}^4
+\frac{1}{6}v_{dgdg}^5
+\frac{10}{33}v_{dgdg}^6+\epsilon'_d+\epsilon'_g,
\nonumber\\&&
\textstyle
c'_{04}=
\frac{1}{18}v_{gggg}^0
+\frac{38}{693}v_{gggg}^2
+\frac{89}{1001}v_{gggg}^4
+\frac{62}{495}v_{gggg}^6
+\frac{1129}{6435}v_{gggg}^8+\epsilon'_g.
\label{e_coef1}
\end{eqnarray}
Only a single coefficient $c^{ij}_{kl}$ is needed in the subsequent analysis, {\it viz.}
\begin{equation}
\textstyle
c^{00}_{04}=
-\frac{2}{693}v_{gggg}^2
+\frac{4}{3003}v_{gggg}^4
+\frac{2}{495}v_{gggg}^6
-\frac{16}{6435}v_{gggg}^8,
\label{e_coef2}
\end{equation}
introducing a $\delta_4$ dependence
in the energy surface~(\ref{e_climit}) since
\begin{equation}
\phi^{00}_{04}(\delta_4)=2\cos2\delta_4+17\cos4\delta_4.
\label{e_phi}
\end{equation}
It is assumed in the following that the boson Hamiltonian is Hermitian
and therefore that $v^L_{\ell_1\ell_2\ell'_1\ell'_2}=v^L_{\ell'_1\ell'_2\ell_1\ell_2}$.
With this assumption the expressions for the coefficients $c_{kl}$ and $c^{ij}_{kl}$,
given in Eqs.~(20) and~(21) of I,
are valid with $v^L_{\ell_1\ell_2\cdot\ell'_1\ell'_2}=v^L_{\ell_1\ell_2\ell'_1\ell'_2}$.
Note that for a general Hamiltonian one has
$v^L_{\ell_1\ell_2\cdot\ell'_1\ell'_2}=(v^L_{\ell_1\ell_2\ell'_1\ell'_2}+v^L_{\ell'_1\ell'_2\ell_1\ell_2})/2$,
which corrects by a factor 2 the expression given in I.

\section{The U$_g$(9) and SO$_{sg}$(10) limits}
\label{s_limits}
Since the Hamiltonian of the \mbox{$sdg$-IBM} conserves the total number of bosons,
it can be written in terms of the $(1+5+9)^2=225$ operators $b_{\ell m}^\dag b_{\ell' m'}$.
The 225 operators generate the Lie algebra U(15) with a substructure
that determines the dynamical symmetries of the \mbox{$sdg$-IBM}.

A comprehensive list of the dynamical symmetries of the \mbox{$sdg$-IBM}
is given by De Meyer {\it et al.}~\cite{Meyer86},
and their group-theoretical properties
are extensively discussed by Kota {\it et al.}~\cite{Kota87}.
It is found that the model has seven major dynamical symmetries,
four of strong coupling, SU(3), SU(6), SU(5) and SO(15),
and three of weak coupling,
${\rm U}_s(1)\otimes{\rm U}_{dg}(14)$,
${\rm U}_{sd}(6)\otimes{\rm U}_g(9)$
and ${\rm U}_{sg}(10)\otimes{\rm U}_d(5)$.
The question treated in this paper is
whether any of the dynamical symmetries of the \mbox{$sdg$-IBM}
corresponds to a shape with octahedral symmetry.
In this section a choice of the limits that possibly have such property
is made on the basis of intuitive arguments.
Subsequently, in Sections~\ref{s_clas} and~\ref{s_octa},
the conditions for the existence of a shape with octahedral symmetry
are derived rigorously on the basis of the results obtained in I.

A minimum with octahedral shape
requires mixing of $s$ and $g$ bosons,
so as to induce hexadecapole deformation,
and no or weak mixing of these with the $d$ boson
to ensure zero quadrupole deformation.
These conditions rule out all limits
where $s$, $d$ and $g$ bosons are strongly mixed on an equal footing,
that is, they discard the SU(3), SU(6), SU(5) and SO(15) limits~\cite{Bouldjedri05}.
A strict decoupling of the $s$ and $g$ from the $d$ bosons
is obtained by the reduction
\begin{equation}
\begin{array}{ccccc}
{\rm U}(15)&\supset&{\rm U}_d(5)&\otimes&{\rm U}_{sg}(10)\\
\downarrow&&\downarrow&&\downarrow\\[0mm]
[N]&&n_d&&n_{sg}
\end{array},
\label{e_ds}
\end{equation}
and, furthermore, zero quadrupole deformation
follows from a U(5) classification for the $d$ bosons,
\begin{equation}
\begin{array}{ccccc}
{\rm U}_d(5)&\supset&{\rm SO}_d(5)&\supset&{\rm SO}_d(3)\\
\downarrow&&\downarrow&&\downarrow\\[0mm]
n_d&&\upsilon_d&\nu_d&L_d
\end{array},
\label{e_u5}
\end{equation}
where underneath each algebra the associated quantum number is given.
Specifically, $N$ is the total number of bosons
while $n_\ell$ is the number of $\ell$ bosons
and $n_{\ell\ell'}$ is the number of $\ell$ plus $\ell'$ bosons.
The seniority label associated with an $\ell$ boson is denoted as $\upsilon_\ell$
and corresponds to the number of $\ell$ bosons
not in pairs coupled to angular momentum zero.
Additional (or missing) labels, not associated with any algebra, are indicated with $\nu_\ell$.
Finally, the angular momentum generated by the $\ell$ bosons is denoted as $L_\ell$.

The ${\rm U}_{sg}(10)$ algebra in Eq.~(\ref{e_ds}) allows two classifications of interest.
The first is obtained by eliminating from the generators of ${\rm U}_{sg}(10)$
those that involve the $s$ boson, leading to
\begin{equation}
({\rm I})\qquad
\begin{array}{ccccccc}
{\rm U}_{sg}(10)&\supset&{\rm U}_g(9)&\supset&{\rm SO}_g(9)&\supset&{\rm SO}_g(3)\\
\downarrow&&\downarrow&&\downarrow&&\downarrow\\
n_{sg}&&n_g&&\upsilon_g&\nu_g&L_g
\end{array}.
\label{e_u9}
\end{equation}
In this limit, which for brevity shall be referred to as ${\rm U}_g(9)$ or limit I,
the separate boson numbers $n_s$ and $n_g$ are conserved.
The resulting spectrum is vibrational-like with a spherical ground state
and excited states that correspond to oscillations in the hexadecapole degree of freedom.

The second classification of ${\rm U}_{sg}(10)$ is specified by the following chain of nested algebras:
\begin{equation}
({\rm II})\qquad
\begin{array}{ccccccc}
{\rm U}_{sg}(10)&\supset&{\rm SO}_{sg}(10)&\supset&{\rm SO}_g(9)&\supset&{\rm SO}_g(3)\\
\downarrow&&\downarrow&&\downarrow&&\downarrow\\
n_{sg}&&\upsilon_{sg}&&\upsilon_g&\nu_g&L_g
\end{array},
\label{e_so10}
\end{equation}
which for brevity shall be referred to as ${\rm SO}_{sg}(10)$ or limit II.
The defining feature of the reduction~(\ref{e_so10})
is the appearance of the algebra ${\rm SO}_{sg}(10)$ and its label $\upsilon_{sg}$,
associated with the pairing of $s$ and $g$ bosons.
As is shown in Section~\ref{s_octa},
the ground state in this limit acquires a permanent hexadecapole deformation
and the limit is therefore of interest in our quest for octahedral shapes.
On its own, however, limit II implies degenerate energies of the $s$ and $g$ boson,
and as such it is not realistic.
It is therefore necessary to study a combination of the two limits I and II.

In Sections~\ref{s_clas} and~\ref{s_octa}
we investigate to what extent non-degenerate energies
can be taken for the $s$ and $g$ boson
that still lead to a hexadecapole-deformed minimum
and whether that minimum can have octahedral symmetry.
In the remainder of this section we list some of the properties of limits I and II
that are necessary to carry out this analysis.

The classification of limits I and II can be summarized with the algebraic lattice
\begin{equation}
\begin{array}{ccccc}
{\rm U}(15)&\supset&{\rm U}_d(5)&\otimes&{\rm U}_{sg}(10)\\
&&|&&\swarrow\quad\searrow\\
&&\downarrow&&\quad {\rm U}_g(9)\quad {\rm SO}_{sg}(10)\\
&&{\rm SO}_d(5)&&\searrow\quad\swarrow\\
&&|&&{\rm SO}_g(9)\\
&&\downarrow&&\downarrow\\
&&{\rm SO}_d(3)&&{\rm SO}_g(3)\\
&&\qquad\searrow&&\swarrow\qquad\\
&&&{\rm SO}(3)&
\end{array},
\label{e_lat}
\end{equation}
where SO(3) is associated with the total angular momentum $L$,
which results from the coupling of $L_d$ and $L_g$.
The generators of the different algebras in the lattice~(\ref{e_lat})
are as follows:
\begin{eqnarray}
{\rm U}_d(5)&:\quad&
\{[d^\dag\times\tilde d]^{(\lambda)}_\mu,\lambda=0,\dots,4\},
\nonumber\\
{\rm SO}_d(5)&:\quad&
\{[d^\dag\times\tilde d]^{(\lambda)}_\mu,\lambda=1,3\},
\nonumber\\
{\rm SO}_d(3)&:\quad&
\{\hat L_{d,\mu}\equiv\sqrt{10}[d^\dag\times\tilde d]^{(1)}_\mu\},
\nonumber\\
{\rm U}_{sg}(10)&:\quad&
\{[s^\dag\times\tilde s]^{(0)}_0,[s^\dag\times\tilde g]^{(4)}_\mu,[g^\dag\times\tilde s]^{(4)}_\mu,
[g^\dag\times\tilde g]^{(\lambda)}_\mu,\lambda=0,\dots,8\},
\nonumber\\
{\rm U}_g(9)&:\quad&
\{[g^\dag\times\tilde g]^{(\lambda)}_\mu,\lambda=0,\dots,8\},
\nonumber\\
{\rm SO}_{sg}(10)&:\quad&
\{[s^\dag\times\tilde g+g^\dag\times\tilde s]^{(4)}_\mu,
[g^\dag\times\tilde g]^{(\lambda)}_\mu,\lambda=1,3,5,7\},
\nonumber\\
{\rm SO}_g(9)&:\quad&
\{[g^\dag\times\tilde g]^{(\lambda)}_\mu,\lambda=1,3,5,7\},
\nonumber\\
{\rm SO}_g(3)&:\quad&
\{\hat L_{g,\mu}\equiv\sqrt{60}[g^\dag\times\tilde g]^{(1)}_\mu\},
\nonumber\\
{\rm SO}(3)&:\quad&
\{\hat L_\mu\equiv\hat L_{d,\mu}+\hat L_{g,\mu}\}.
\label{e_gens}
\end{eqnarray}

The linear and quadratic Casimir operators
of the algebras appearing in the lattice~(\ref{e_lat})
can be expressed as follows
in terms of the generators~(\ref{e_gens}):
\begin{eqnarray}
\hat C_1[{\rm U}(15)]&=&\hat N=\hat n_s+\hat n_d+\hat n_g,
\nonumber\\
\hat C_2[{\rm U}(15)]&=&\hat N(\hat N+14),
\nonumber\\
\hat C_1[{\rm U}_d(5)]&=&\hat n_d,
\nonumber\\
\hat C_2[{\rm U}_d(5)]&=&\hat n_d(\hat n_d+4),
\nonumber\\
\hat C_2[{\rm SO}_d(5)]&=&
2\sum_{\lambda\;{\rm odd}}[d^\dag\times\tilde d]^{(\lambda)}\cdot[d^\dag\times\tilde d]^{(\lambda)},
\nonumber\\
\hat C_2[{\rm SO}_d(3)]&=&\hat L_d\cdot\hat L_d,
\nonumber\\
\hat C_1[{\rm U}_{sg}(10)]&=&\hat n_s+\hat n_g,
\nonumber\\
\hat C_2[{\rm U}_{sg}(10)]&=&(\hat n_s+\hat n_g)(\hat n_s+\hat n_g+9),
\nonumber\\
\hat C_1[{\rm U}_g(9)]&=&\hat n_g,
\nonumber\\
\hat C_2[{\rm U}_g(9)]&=&\hat n_g(\hat n_g+8),
\nonumber\\
\hat C_2[{\rm SO}_{sg}(10)]&=&
[s^\dag\times\tilde g+g^\dag\times\tilde s]^{(4)}\cdot
[s^\dag\times\tilde g+g^\dag\times\tilde s]^{(4)}+
\hat C_2[{\rm SO}_g(9)],
\nonumber\\
\hat C_2[{\rm SO}_g(9)]&=&
2\sum_{\lambda\;{\rm odd}}[g^\dag\times\tilde g]^{(\lambda)}\cdot[g^\dag\times\tilde g]^{(\lambda)},
\nonumber\\
\hat C_2[{\rm SO}_g(3)]&=&\hat L_g\cdot\hat L_g,
\nonumber\\
\hat C_2[{\rm SO}(3)]&=&\hat L\cdot\hat L.
\label{e_cas}
\end{eqnarray}
The expressions for the quadratic Casimir operators of unitary algebras
are not general but are valid in a symmetric irreducible representation.
A rotationally invariant Hamiltonian with up to two-body interactions
can be written in terms of the Casimir operators~(\ref{e_cas}):
\begin{eqnarray}
\hat H_{\rm sym}&=&
\epsilon_d\,\hat n_d+
a_d\,\hat C_2[{\rm U}_d(5)]+
b_d\,\hat C_2[{\rm SO}_d(5)]+
c_d\,\hat C_2[{\rm SO}_d(3)]
\nonumber\\&&+
\epsilon_s\,\hat n_s+\epsilon_g\,\hat n_g+
a_{sg}\,\hat C_2[{\rm U}_{sg}(10)]+
a_g\,\hat C_2[{\rm U}_g(9)]+
b_{sg}\,\hat P_{sg}^\dag\hat P_{sg}
\nonumber\\&&+
b_g\,\hat C_2[{\rm SO}_g(9)]+
c_g\,\hat C_2[{\rm SO}_g(3)]+
c\,\hat C_2[{\rm SO}(3)],
\label{e_hamlat}
\end{eqnarray}
where $\epsilon_\ell$, $a_\ell$, $a_{\ell\ell'}$,
$b_\ell$, $b_{\ell\ell'}$ and $c_\ell$ are parameters.
The quadratic Casimir operator of U(15) is omitted for simplicity
since it gives a constant contribution for a fixed boson number $N=n_s+n_d+n_g$.
Furthermore, it is convenient to define,
instead of the quadratic Casimir operator $\hat C_2[{\rm SO}_{sg}(10)]$,
the combination
\begin{equation}
\hat P^\dag_{sg}\hat P_{sg}=
\hat C_2[{\rm U}_{sg}(10)]-\hat C_1[{\rm U}_{sg}(10)]-\hat C_2[{\rm SO}_{sg}(10)],
\label{e_sfpairing}
\end{equation}
where $\hat P^\dag_{sg}\equiv s^\dag s^\dag-g^\dag\cdot g^\dag$
is the pairing operator for $s$ and $g$ bosons.
The symmetry Hamiltonian~(\ref{e_hamlat}) is less general than Eq.~(\ref{e_ham})
but it is the most general one that can be written
in terms of invariant operators of the lattice~(\ref{e_lat})
and as such it is intermediate between the limits I and II.

The ${\rm U}_g(9)$ limit occurs for $b_{sg}=0$,
leading to the eigenvalues
\begin{eqnarray}
E_{\rm I}&=&
\epsilon_d\,n_d+
a_d\,n_d(n_d+4)+
b_d\,\upsilon_d(\upsilon_d+3)+
c_d\,L_d(L_d+1)
\nonumber\\&&+
\epsilon_s\,n_s+\epsilon_g\,n_g+
a_{sg}\,n_{sg}(n_{sg}+9)+
a_g\,n_g(n_g+8)+
\nonumber\\&&+
b_g\,\upsilon_g(\upsilon_g+7)+
c_g\,L_g(L_g+1)+
c\,L(L+1).
\label{e_eigu9}
\end{eqnarray}
The ${\rm SO}_{sg}(10)$ limit is attained
for $\epsilon_s=\epsilon_g\equiv\epsilon_{sg}$ and $a_g=0$,
in which case the Hamiltonian's eigenstates have the eigenvalues
\begin{eqnarray}
E_{\rm II}&=&
\epsilon_d\,n_d+
a_d\,n_d(n_d+4)+
b_d\,\upsilon_d(\upsilon_d+3)+
c_d\,L_d(L_d+1)
\nonumber\\&&+
\epsilon_{sg}\,n_{sg}+
a_{sg}\,n_{sg}(n_{sg}+9)+
b_{sg}[n_{sg}(n_{sg}+8)-\upsilon_{sg}(\upsilon_{sg}+8)]
\nonumber\\&&+
b_g\,\upsilon_g(\upsilon_g+7)+
c_g\,L_g(L_g+1)+
c\,L(L+1).
\label{e_eigso10}
\end{eqnarray}
The eigenspectra are then determined
with the help of the necessary branching rules.
The reduction ${\rm U}(15)\supset{\rm U}_d(5)\otimes{\rm U}_{sg}(10)$
implies the relation $N=n_d+n_{sg}$ or the branching rule
\begin{equation}
[N]\mapsto(n_d,n_{sg})=(0,N),(1,N-1),\dots,(N,0).
\label{e_branch}
\end{equation}
The branching rules for the classification~(\ref{e_u5})
are known from the U(5) limit of the \mbox{$sd$-IBM}~\cite{Arima76}
and those for the classifications~(\ref{e_u9}) and~(\ref{e_so10})
can be found in Ref.~\cite{Kota87}.

\begin{figure}
\centering
\includegraphics[width=6.5cm]{fig1a.pdf}
\includegraphics[width=6.5cm]{fig1b.pdf}
\caption{Energy spectra in the ${\rm U}_g(9)$ and ${\rm SO}_{sg}(10)$ limits
of the \mbox{$sdg$-IBM} for $N=5$ bosons.
For the ${\rm U}_g(9)$ spectrum the non-zero parameters in the Hamiltonian~(\ref{e_hamlat})
are $\epsilon_d-\epsilon_s=800$, $\epsilon_g-\epsilon_s=1000$,
$b_d=40$, $c_d=10$, $b_g=25$ and $c_g=c=5$~keV.
For the ${\rm SO}_{sg}(10)$ spectrum the non-zero parameters
are $\epsilon_d-\epsilon_s=800$, $\epsilon_g-\epsilon_s=0$,
$b_d=40$, $c_d=10$, $b_{sg}=60$, $b_g=50$ and $c_g=c=5$~keV.}
\label{f_limits}
\end{figure}
Typical energy spectra in the ${\rm U}_g(9)$ and ${\rm SO}_{sg}(10)$ limits
are shown in Fig.~\ref{f_limits}.
The ${\rm U}_g(9)$ spectrum displays quadrupole- and hexadecapole-phonon multiplets
characterized by a fixed number of $d$  and $g$ bosons.
The multiplets are further structured by a seniority quantum number:
for example, the $n_d=2$ multiplet has $\upsilon_d=2$ except for the $0^+$ level,
which has $\upsilon_d=0$,
and similarly for the $n_g=2$ multiplet and the $\upsilon_g$ seniority.
Also combined quadrupole--hexadecapole multiplets occur in the spectrum.
The ${\rm SO}_{sg}(10)$ spectrum
contains sets of levels with $\upsilon_{sg}=N,N-2,\dots$ (for $n_d=0$)
with $\upsilon_{sg}=N-1,N-3,\dots$ (for $n_d=1$) etc.,
and $\upsilon_{sg}=N$ levels are lowest in energy due the repulsive $sg$-pairing.
Multiplets characterized by the seniority quantum number $\upsilon_g=0,1,\dots$
occur within each ${\rm SO}_{sg}(10)$ multiplet.
Note that in this limit, unless $b_{sg}$ is small, the first-excited state has $J^\pi=4^+$.
This is a consequence of the unrealistic condition
that the energies of the $s$ and $g$ bosons are degenerate, $\epsilon_s=\epsilon_g$.

\section{Classical limit of the symmetry Hamiltonian}
\label{s_clas}
The classical limit of the symmetry Hamiltonian~(\ref{e_hamlat})
can be obtained with the general procedure outlined in I.
First, a conversion to the standard representation~(\ref{e_ham}) is carried out.
A quadratic Casimir operator $\hat C_2(G)$
is given generically as an expansion over the generators,
\begin{equation}
\hat C_2(G)=
\sum_{\lambda r}a_{\lambda r}
\left(\sum_{\ell_1\ell_2}\alpha^{\lambda r}_{\ell_1\ell_2}
\left(b_{\ell_1}^\dag\times\tilde b_{\ell_2}\right)^{(\lambda)}\right)
\cdot
\left(\sum_{\ell'_1\ell'_2}\alpha^{\lambda r}_{\ell'_1\ell'_2}
\left(b_{\ell'_1}^\dag\times\tilde b_{\ell'_2}\right)^{(\lambda)}\right),
\label{e_casexp}
\end{equation}
with coefficients $\alpha^{\lambda r}_{\ell_1\ell_2}$ and $a_{\lambda r}$
that are specific to each algebra $G$, as given in Eq.~(\ref{e_cas}).
The overall sum in Eq.~(\ref{e_casexp})
is over the multipolarity $\lambda$ of the generators
and over an additional index $r$
to distinguish different generators with the same $\lambda$.
With use of the expansion~(\ref{e_casexp})
one finds the following expression
for the matrix element of $\hat C_2(G)$ between two-boson states~\cite{note1}:
\begin{eqnarray}
\lefteqn{\langle \ell_1\ell_2;L|\hat C_2(G)|\ell'_1\ell'_2;L\rangle}
\nonumber\\&=&
\left[f_{\ell_1}(G)+f_{\ell_2}(G)\right]
\delta_{\ell_1\ell_3}\delta_{\ell_2\ell_4}+
\frac{2(-)^{\ell_2+\ell_3}}{\sqrt{(1+\delta_{\ell_1\ell_2})(1+\delta_{\ell_3\ell_4})}}
\sum_{\lambda r}a_{\lambda r}(2\lambda+1)
\nonumber\\&&\times
\Biggl[\alpha^{\lambda r}_{\ell_1\ell_4}\alpha^{\lambda r}_{\ell_2\ell_3}
\Biggl\{\begin{array}{ccc}
\ell_1&\ell_4&\lambda\\
\ell_3&\ell_2&L
\end{array}\Biggr\}+
(-)^L\alpha^{\lambda r}_{\ell_1\ell_3}\alpha^{\lambda r}_{\ell_2\ell_4}
\Biggl\{\begin{array}{ccc}
\ell_1&\ell_3&\lambda\\
\ell_4&\ell_2&L
\end{array}
\Biggr\}\Biggr],
\label{e_convert}
\end{eqnarray}
where the symbol between curly brackets is a Racah coefficient~\cite{Talmi93}.
The quantity $f_\ell(G)$ is the expectation value
of the operator $\hat C_2(G)$ between single-boson states,
\begin{equation}
f_\ell(G)\equiv\langle \ell|\hat C_2(G)|\ell\rangle=
\sum_{\lambda r}\frac{2\lambda+1}{2\ell+1}a_{\lambda r}
\sum_{\ell'}(-)^{\ell+\ell'}
\alpha^{\lambda r}_{\ell\ell'}
\alpha^{\lambda r}_{\ell'\ell}.
\label{e_conone}
\end{equation}

The Hamiltonian~(\ref{e_hamlat}) can,
with use of Eqs.~(\ref{e_convert}) and~(\ref{e_conone}),
be converted into its standard representation
on which the classical-limit expression~(\ref{e_climit}) can be applied.
The procedure results in the energy surface
\begin{equation}
E(\beta_2,\beta_4)=
\frac{N(N-1)}{(1+\beta_2^2+\beta_4^2)^2}
\sum_{kl}c'_{kl}\beta_2^k\beta_4^l,
\label{e_climit1}
\end{equation}
where the non-zero coefficients $c'_{kl}$ are
\begin{eqnarray}
c'_{00}&=&
a_{sg}+b_{sg}+\Gamma_s,
\quad
c'_{20}=
\Gamma_s+\Gamma_d,
\quad
c'_{02}=
2a_{sg}-2b_{sg}+\Gamma_s+\Gamma_g,
\nonumber\\
c'_{40}&=&
a_d+\Gamma_d,
\quad
c'_{22}=
\Gamma_d+\Gamma_g,
\quad
c'_{04}=
a_{sg}+b_{sg}+a_g+\Gamma_g,
\label{e_coeflat}
\end{eqnarray}
in terms of the combinations
\begin{eqnarray}
\Gamma_s&\equiv&\frac{1}{N-1}(\epsilon_s+10a_{sg}),
\nonumber\\
\Gamma_d&\equiv&\frac{1}{N-1}(\epsilon_d+5a_d+4b_d+6c_d+6c),
\nonumber\\
\Gamma_g&\equiv&\frac{1}{N-1}(\epsilon_g+10a_{sg}+9a_g+8b_g+20c_g+20c).
\label{e_comb}
\end{eqnarray}
All coefficients $c^{ij}_{kl}$ of Eq.~(\ref{e_climit}) vanish identically
in the classical limit of the symmetry Hamiltonian~(\ref{e_hamlat}).

\section{Octahedral shapes}
\label{s_octa}
From the outset it should be clear that energy surface~(\ref{e_climit1})
cannot have an isolated minimum with octahedral shape
since it is independent of $\gamma_2$, $\gamma_4$ and $\delta_4$.
What can still happen, however, is the occurrence of a minimum
with zero quadrupole and non-zero hexadecapole deformation
($\beta^*_2=0$ and $\beta^*_4\neq0$),
which, given the instability in $\gamma_4$ and $\delta_4$,
{\em includes} a shape with octahedral symmetry.
Therefore, the goal of this section is
to establish the conditions on the parameters in the symmetry Hamiltonian~(\ref{e_hamlat})
such that its classical limit displays a minimum with $\beta^*_2=0$ and $\beta^*_4\neq0$,
and, subsequently, to identify the interactions in the general Hamiltonian~(\ref{e_ham})
that generate a dependence on $\delta_4$,
enabling the formation of an isolated minimum with octahedral shape.
This problem can be investigated with the procedure outlined in I.

According to the analysis of the previous section,
the classical energy~(\ref{e_climit1}) is a two-variable function
$E(\beta_2,\beta_4)$.
The conditions for this energy surface
to have an extremum are
\begin{equation}
\left.\frac{\partial E}{\partial\beta_2}\right|_{p^*}=
\left.\frac{\partial E}{\partial\beta_4}\right|_{p^*}=0,
\label{e_extr}
\end{equation}
where $p^*\equiv(\beta^*_2,\beta^*_4)$
is a short-hand notation for an arbitrary critical point.
Furthermore, a critical point at an extremum
with $\beta^*_2=0$ and $\beta^*_4\neq0$ shall be denoted as $h^*$.
The condition~(\ref{e_extr}) in $\beta_2$ is identically satisfied for $p^*=h^*$
and does not lead to any constraints on the coefficients $c'_{kl}$.
The condition in $\beta_4$ leads to a cubic equation with the solutions
\begin{equation}
\beta_4^*=0,
\qquad
\beta_4^*=\pm\sqrt{\frac{2c'_{00}-c'_{02}}{2c'_{04}-c'_{02}}}.
\label{e_extrb4}
\end{equation}
Only the last solution corresponds to an extremum $h^*$
and implies the following condition on the ratio of coefficients:
\begin{equation}
\frac{2c'_{00}-c'_{02}}{2c'_{04}-c'_{02}}>0.
\label{e_cond1}
\end{equation}

While the condition~(\ref{e_cond1}) is necessary and sufficient
to have an {\em extremum} at $\beta_2^*=0$ and $\beta_4^*\neq0$,
a {\it minimum} at these values implies further constraints.
They are obtained by requiring that the eigenvalues of the stability matrix
[{\it i.e.}, the partial derivatives of $E(\beta_2,\beta_4)$
of second order] are all positive.
Since the off-diagonal element of the stability matrix vanishes for the energy surface~(\ref{e_climit1}),
the existence of a minimum follows from the uncoupled conditions
\begin{equation}
\left.\frac{\partial^2 E}{\partial\beta_2^2}\right|_{h^*}>0,
\qquad
\left.\frac{\partial^2 E}{\partial\beta_4^2}\right|_{h^*}>0,
\label{e_stab1}
\end{equation}
or, in terms of the coefficients $c'_{kl}$ in Eq.~(\ref{e_climit1}),
\begin{eqnarray}
\frac{(2c'_{04}-c'_{02})[2c'_{00}(c'_{22}-2c'_{04})+c'_{02}(c'_{02}-c'_{20}-c'_{22})+2c'_{04}c'_{20}]}
{(c'_{00}-c'_{02}+c'_{04})^2}&>&0,
\nonumber\\
\frac{(2c'_{00}-c'_{02})(2c'_{04}-c'_{02})^3}{(c'_{00}-c'_{02}+c'_{04})^3}&>&0.
\label{e_stab2}
\end{eqnarray}
If we write Eq.~(\ref{e_cond1}) as $A/B>0$,
the second inequality in Eq.~(\ref{e_stab2}) becomes $AB^3/(A+B)^3>0$
and therefore both $A$ and $B$ should be positive,
$A\equiv2c'_{00}-c'_{02}>0$ and $B\equiv2c'_{04}-c'_{02}>0$,
leading to the constraint
\begin{equation}
-4b_{sg}-2a_g<\Gamma_g-\Gamma_s<4b_{sg}.
\label{e_cond2}
\end{equation}
The first inequality in Eq.~(\ref{e_stab2}) can be reduced to
\begin{equation}
2b_{sg}(2\Gamma_d-\Gamma_s-\Gamma_g-4a_{sg}-a_g)
+a_g(\Gamma_d-\Gamma_s-2a_{sg})>0.
\label{e_cond3}
\end{equation}

The conditions~(\ref{e_cond2}) and~(\ref{e_cond3}) are necessary and sufficient
for the energy surface $E(\beta_2,\beta_4)$
to have a minimum at zero quadrupole and non-zero hexadecapole deformation.
To obtain an intuitive understanding of them,
we note that $\Gamma_g-\Gamma_s$, for a reasonable choice of parameters, is positive.
The upper part of the inequality~(\ref{e_cond2}) therefore expresses
the need for $b_{sg}$ to be positive and sufficiently large,
corresponding to a repulsive $sg$-pairing interaction
that puts the configuration with {\em maximal} $sg$ seniority $\upsilon_{sg}=n_{sg}$ at lowest energy.
For $b_{sg}>0$ and $a_g>0$,
the lower part of the inequality~(\ref{e_cond2}) is automatically satisfied.
The condition~(\ref{e_cond3}) is easier to appreciate
if it is assumed that the coefficients in front of the quadratic Casimir operators
of the unitary algebras ${\rm U}_d(5)$, ${\rm U}_g(9)$ and ${\rm U}_{sg}(10)$ vanish,
$a_d=a_{sg}=a_g=0$.
This assumption is justified if anharmonicities are neglected in the various limits.
Given that $b_{sg}>0$, it then follows that
\begin{equation}
2\Gamma_d-\Gamma_s-\Gamma_g>0.
\label{e_cond4}
\end{equation}
In terms of the original parameters in the symmetry Hamiltonian~(\ref{e_hamlat})
(assuming $a_d=a_{sg}=a_g=0$)
the conditions to have a minimum at $\beta^*_2=0$ and $\beta^*_4\neq0$
can be summarized as
\begin{eqnarray}
-4(N-1)b_{sg}<\epsilon_g-\epsilon_s+8b_g+20(c_g+c)&<&4(N-1)b_{sg},
\nonumber\\
2\epsilon_d-\epsilon_s-\epsilon_g+8(b_d-b_g)+4(3c_d-5c_g-2c)&>&0.
\label{e_cond}
\end{eqnarray}

We now ask the question whether two-body interactions
can be added to the symmetry Hamiltonian~(\ref{e_hamlat}),
which lift the $(\gamma_4,\delta_4)$ instability
and create a minimum at $\delta_4=0$, $\delta_4=\arccos(1/6)$ or $\delta_4=\pi$.
To achieve this goal, we recall the result from I
that for a general Hamiltonian of the \mbox{$sdg$-IBM}
an extremum with $\beta_2^*=0$ and $\beta_4^*\neq0$ occurs for
\begin{equation}
\beta_4^*=\pm\sqrt{\frac{2c'_{00}-c'_{02}}{2c'_{04}-c'_{02}+38c^{00}_{04}}}.
\label{e_extrb4g}
\end{equation}
The term in $c^{00}_{04}$ introduces a dependence in $\delta_4$
that may lead to an isolated minimum with octahedral symmetry.
Given the expression~(\ref{e_coef2}) for $c^{00}_{04}$,
this argument suggests adding $g$-boson interactions $v_{gggg}^L$ to the Hamiltonian~(\ref{e_hamlat}).
The classical energy~(\ref{e_climit1}) then becomes a three-variable function
$E(\beta_2,\beta_4,\delta_4)$,
for which the above catastrophe analysis can be repeated.
With these additional two-body interactions
the extremum and stability conditions~(\ref{e_extr}) and~(\ref{e_stab1}) become
\begin{eqnarray}
&&2c'_{00}-c'_{02}>0,
\qquad
2c'_{04}-c'_{02}+38c^{00}_{04}>0,
\qquad
c^{00}_{04}<0,
\label{e_cong1}\\
&&2c'_{00}(c'_{22}-2c'_{04})+c'_{02}(c'_{02}-c'_{20}-c'_{22})+2c'_{04}c'_{20}-38c^{00}_{04}(2c'_{00}-c'_{20})>0.
\nonumber
\end{eqnarray}
For $a_d=a_{sg}=a_g=0$,
these conditions imply the inequalities
\begin{eqnarray}
&&-2b_{sg}-\tilde v<\Gamma_g-\Gamma_s<4b_{sg},
\nonumber\\
&&\bar v\equiv 65v_{gggg}^2-30v_{gggg}^4-91v_{gggg}^6+56v_{gggg}^8>0,
\nonumber\\
&&4b_{sg}(2\Gamma_d-\Gamma_s-\Gamma_g)
+(\Gamma_d-\Gamma_s-2b_{sg})(\tilde v-2b_{sg})>0,
\label{e_cong2}
\end{eqnarray}
where $\tilde v$ is the following linear combination of $g$-boson interaction matrix elements:
\begin{equation}
\textstyle
\tilde v=
\frac{1}{9}v_{gggg}^0
+\frac{98}{429}v_{gggg}^4
+\frac{40}{99}v_{gggg}^6
+\frac{10}{39}v_{gggg}^8.
\label{e_vtilde}
\end{equation}
For the Hamiltonian~(\ref{e_hamlat}) the linear combination $\bar v$ vanishes identically
and the second inequality in Eq.~(\ref{e_cong2}) is not fulfilled.
This expresses the $\delta_4$ independence of the symmetry Hamiltonian
and the fact that its classical limit does not acquire an isolated minimum with octahedral shape.
Furthermore, for the symmetry Hamiltonian one has $\tilde v=2b_{sg}$
and the conditions~(\ref{e_cong2}) reduce to Eq.~(\ref{e_cond}).

There are clearly many ways to find matrix elements $v_{gggg}^L$
that satisfy all conditions~(\ref{e_cong2}) but one way is particularly simple.
Note that the quadrupole matrix element $v_{gggg}^2$
does not appear in the combination $\tilde v$.
By making this matrix element more repulsive,
the second inequality in Eq.~(\ref{e_cong2}) is satisfied
while the other two conditions are not modified
with respect those in Eq.~(\ref{e_cond}) valid for the symmetry Hamiltonian~(\ref{e_hamlat}).
A possible procedure to construct an \mbox{$sdg$-IBM} Hamiltonian
whose classical energy displays a minimum with octahedral shape
is therefore to add to a hexadecapole-deformed symmetry Hamiltonian~(\ref{e_hamlat})
a repulsive $v_{gggg}^2$ interaction.

\begin{figure}
\centering
\includegraphics[width=7cm]{fig2.pdf}
\caption{Energy spectrum of a ${\rm U}_g(9)$--${\rm SO}_{sg}(10)$ transitional Hamiltonian
of the \mbox{$sdg$-IBM} for $N=5$ bosons.
The non-zero parameters of the Hamiltonian~(\ref{e_hamlat})
are $\epsilon_d-\epsilon_s=1200$, $\epsilon_g-\epsilon_s=1500$,
$b_d=40$, $c_d=10$, $b_{sg}=150$, $b_g=25$ and $c_g=c=5$~keV.}
\label{f_u9so10}
\end{figure}
Let us illustrate this procedure with an example.
The starting point is a ${\rm U}_g(9)$--${\rm SO}_{sg}(10)$ transitional Hamiltonian
associated with the lattice~(\ref{e_lat}),
giving rise to the spectrum shown in Fig.~\ref{f_u9so10}.
Note that the choice of the single-boson energies $\epsilon_\ell$ for this figure
is realistic in the sense that the $g$-boson energy
is higher than that of the $d$ boson.
The $sg$-pairing strength $b_{sg}$,
of which little is known either empirically or microscopically,
is chosen such that a hexadecapole-deformed minimum occurs in the classical limit.
Other parameters in the Hamiltonian~(\ref{e_hamlat}) are of lesser importance
and are chosen as to lift degeneracies in the spectrum.
Note that with this choice of parameters
the resulting spectrum, as shown in Fig.~\ref{f_u9so10},
is rather closer to the ${\rm U}_g(9)$ and than to the ${\rm SO}_{sg}(10)$ limit.

\begin{figure}
\centering
\includegraphics[width=6cm]{fig3.pdf}
\caption{The energy surface $E(\beta_2,\beta_4)$
obtained in the classical limit
of a ${\rm U}_g(9)$--${\rm SO}_{sg}(10)$ transitional Hamiltonian of the \mbox{$sdg$-IBM}.
Parameters of the Hamiltonian~(\ref{e_hamlat})
are given in the caption of Fig.~\ref{f_u9so10}.
Black corresponds to low energies
and the lines indicate changes by 10~keV.}
\label{f_b2b4}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=6cm]{fig4a.pdf}
\includegraphics[width=6cm]{fig4b.pdf}
\caption{Energy surfaces $E(\beta_4,\delta_4)$
obtained in the classical limit
of two different Hamiltonians of the \mbox{$sdg$-IBM} for $N=5$ bosons.
The dependence on $\beta_4>0$ and $0\leq\delta_4\leq\pi$ is shown for $\beta_2^*=0$.
Black corresponds to low energies
and the lines indicate changes by 10~keV.
(a) The ${\rm U}_g(9)$--${\rm SO}_{sg}(10)$ transitional Hamiltonian is taken
with the parameters given in the caption of Fig.~\ref{f_u9so10}.
(b) The Hamiltonian of (a) is modified by taking a repulsive interaction $v_{gggg}^2=500$~keV.}
\label{f_b4d4}
\end{figure}
The parameters quoted in the caption of Fig.~\ref{f_u9so10}
satisfy both conditions~(\ref{e_cond}).
As a result, the energy surface in the classical limit of the corresponding Hamiltonian
has a minimum for $\beta_2^*=0$ and $\beta_4^*\approx0.34$,
as shown in Fig.~\ref{f_b2b4}.
According to the preceding discussion,
the surface is independent of $\delta_4$,
which is indeed confirmed by Fig.~\ref{f_b4d4}(a).
If the $v_{gggg}^2$ matrix element is modified,
a dependence in $\delta_4$ is introduced,
as illustrated in Fig.~\ref{f_b4d4}(b) for the value $v_{gggg}^2=500$~keV.
It is seen that the energy surface displays {\em three} isolated minima
that are exactly degenerate.
The three minima all have an octahedral symmetry,
corresponding to either an octahedron [$\delta_4^*=0$ and $\delta_4^*=\arccos(1/6)\approx84.4^{\rm o}$]
or a cube ($\delta_4^*=\pi$).

\begin{figure}
\centering
\includegraphics[width=3cm]{fig5a.pdf}
\includegraphics[width=7cm]{fig5b.pdf}
\includegraphics[width=3cm]{fig5c.pdf}
\caption{Energy spectrum of a general Hamiltonian of the \mbox{$sdg$-IBM} for $N=5$ bosons.
The same Hamiltonian is taken as in Fig.~\ref{f_u9so10}
but one $g$-boson two-body matrix element is modified to $v_{gggg}^2=500$~keV.
On the left- and right-hand sides are shown
the shapes at the minima in the energy surface
obtained in the classical limit of this Hamiltonian.
They have octahedral symmetry and correspond to either an octahedron or a cube.}
\label{f_u15}
\end{figure}
Although this analysis shows that
isolated minima with octahedral symmetry can be obtained
in the classical limit of an \mbox{$sdg$-IBM} Hamiltonian with reasonable parameters,
it can be expected that such minima are rather shallow.
Even for the fairly large value of the interaction matrix element
in the above example, $v_{gggg}^2=500$~keV,
the three minima are separated by a barrier of $\sim20$~keV,
inducing only very weak observable effects.
This point is illustrated with Fig.~\ref{f_u15},
which shows the spectrum of the ${\rm U}_g(9)$--${\rm SO}_{sg}(10)$ transitional Hamiltonian
with the modified $v_{gggg}^2$ matrix element.
Except for some minute changes the spectrum is essentially the same
as that shown in Fig.~\ref{f_u9so10}.

One subtle point made clear by the current study is that
it is not sufficient to carry out a catastrophe analysis
of the generic surface~(\ref{e_climit})
obtained in the classical limit of the most general \mbox{$sdg$-IBM} Hamiltonian~(\ref{e_ham})
with up to two-body interactions between the bosons.
The coefficients $c'_{kl}$ and $c^{ij}_{kl}$ cannot be treated as free parameters
but their expressions in terms of the single-bosons energies
and boson--boson interactions are an essential part of the analysis.
To illustrate this point consider the energy surface shown in Fig.~\ref{f_b4d4}(b).
The minima at $\delta_4^*=0$ and $\delta_4^*\approx84.4^{\rm o}$
correspond to the same intrinsic shape
(an octahedron, shown on the left-hand side of Fig.~\ref{f_u15})
and, as a consequence, the minima {\em must} be exactly degenerate.
This behavior is generally valid.
Therefore, whatever single-boson energies and boson--boson interactions
one adopts in the Hamiltonian~(\ref{e_ham}),
the energy surface in its classical limit must satisfy the constraint
that points on the surface with same intrinsic shape
(e.g., $\delta_4^*=0$ and $\delta_4^*\approx84.4^{\rm o}$)
are at the same energy.

\section{Conclusions}
\label{s_conc}
The main conclusion of this paper is
that no \mbox{$sdg$-IBM} Hamiltonian with a dynamical symmetry
that includes Casimir operators of up to second order
displays in its classical limit an isolated minimum with octahedral shape.
Nevertheless, a degenerate minimum that {\em includes} a shape with octahedral symmetry
can be obtained from a Hamiltonian transitional between two limits.
In the limits in question the $d$ boson is decoupled from $s$ and $g$ bosons.
Furthermore, limit I, ${\rm U}_g(9)$, has hexadecapole vibrational characteristics
while in limit II, ${\rm SO}_{sg}(10)$, $s$- and $g$-boson states
are mixed through an $sg$-pairing interaction.
A catastrophe analysis of the energy surface
obtained in the classical limit of this transitional symmetry Hamiltonian
indicates that a minimum with zero quadrupole and non-zero hexadecapole deformation
can be obtained with reasonable parameters.
However, this minimum is always $\delta_4$ independent,
meaning that it ranges from an octahedron to a cube
and includes intermediate shapes without octahedral symmetry.
Isolated minima with octahedral symmetry can be obtained
by adding two-body interactions between the $g$ bosons
to the transitional symmetry Hamiltonian.
The resulting energy surface displays in this case minima with octahedral symmetry,
with the shape of either an octahedron or a cube,
separated by a barrier with low energy
even for fairly strong interactions between the $g$ bosons.
The conclusion of this analysis in the context of the \mbox{$sdg$-IBM}
is therefore that it will be difficult to find
experimental manifestations of octahedral symmetry in nuclei.

\section*{Acknowledgements}
This work has been carried out
in the framework of a CNRS/DEF agreement,
project N 13760.

redpajama_set_name: RedPajamaArXiv


Iran and Morocco were locked into a tense scoreless draw for over an hour and a half until Moroccan striker Aziz Bouhaddouz broke the deadlock with a brilliant header to the near post in the fifth minute of injury time. Unfortunately, it was past his own keeper.
The 95th minute winner gave Iran their first World Cup victory since 1998. If there's any justice in this world, they'll sneak past Portugal and send Ronaldo packing in the most embarrassing possible way.


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