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{"url":"https:\/\/www.numerade.com\/questions\/ssm-a-pitcher-throws-a-curveball-that-reaches-the-catcher-in-060-s-the-ball-curves-because-it-is-spi\/","text":"\ud83d\udea8 Hurry, space in our FREE summer bootcamps is running out. \ud83d\udea8Claim your spot here.\n\n# ssm A pitcher throws a curveball that reaches the catcher in 0.60 s. The ball curves because it is spinning at an average angular velocity of 330 rev\/min (assumed constant) on its way to the catcher\u2019s mitt. What is the angular displacement of the baseball (in radians) as it travels from the pitcher to the catcher?\n\n## 21 $\\mathrm{rad}$\n\n### Discussion\n\nYou must be signed in to discuss.\n##### Andy C.\n\nUniversity of Michigan - Ann Arbor\n\nLB\n##### Jared E.\n\nUniversity of Winnipeg\n\nLectures\n\nJoin Bootcamp\n\n### Video Transcript\n\nWhat important thing to keep in mind with this problem is that when we have rotational motion, that's coupled with translational motion. The pitcher throws the ball and it moves sideways. But it's also rotating Is that that translational motion and the rotation are completely separate from each other. A lot like when we do to the emotion problems where motion along each dimension is independent of each other. Like if we have ballistic trajectories along X and Y, the rotation is completely separate from the translation, so we can ignore the fact that it's a baseball pitcher throwing the ball across some distance and only pay attention to the rotation of the ball. So we're given a time that the ball is rotating of 0.6 seconds, which is good. We want everything to be in S I u nits as much as possible and then the ball has a rotational velocity of 330 revolutions per second. We're sorry revolutions per minute, and that's something that we're gonna need to convert in order to be able to multiply. Ultimately, we want the total angle displaced by the ball, which is gonna be from this equation analysis too, if we had a translational velocity that I had some ex displacement. So it's good to make connections between formulas that we already know in love when we're moving into rotational motion. So we need to convert Omega into as I units in order to move on. So when you convert a value to different units, you need to use unit conversion. So omega we can write as 330 revolutions per minute times. Now we want to multiply by something that is equivalent to one. So it has the same thing on the top and the bottom of a fraction that has the same number on the top of the bottom. So we want to cancel revolutions. So in one revolution we have two pi radiance. So that's just the same, is multiplying by one. Now we also need to convert the minutes into seconds. So we have another fraction and to to cancel out the minutes, we have to have something in the numerator. So one minute is equal to 60 seconds, and now the minutes cancel the Revolution's cancel and we're gonna be left with a number that's ingredients per second. And when we multiply this out 330 times two pi divided by 60. What we end up getting is 34.6 radiance per second. Now, if we multiply this by the time 0.6 seconds, we end up getting a number in radiance. So for our final answer, we get 20 point. It's don't 0.8 radiance. But since we were only given two significant figures in the statement of the problem, we should only write our solution to the same number of significant figures. So this is 21 radiance.\n\nUniversity of Washington\n##### Andy C.\n\nUniversity of Michigan - Ann Arbor\n\nLB\n##### Jared E.\n\nUniversity of Winnipeg\n\nLectures\n\nJoin Bootcamp","date":"2021-06-15 20:00:41","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7116454839706421, \"perplexity\": 447.09909783775265}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-25\/segments\/1623487621519.32\/warc\/CC-MAIN-20210615180356-20210615210356-00267.warc.gz\"}"}
| null | null |
package com.bleyl.recurrence.adapters;
import android.content.Context;
import android.graphics.Color;
import android.graphics.drawable.GradientDrawable;
import android.support.v7.widget.RecyclerView;
import android.view.LayoutInflater;
import android.view.View;
import android.view.ViewGroup;
import android.widget.ImageView;
import android.widget.TextView;
import com.bleyl.recurrence.ui.fragments.TabFragment;
import com.bleyl.recurrence.models.Notification;
import com.bleyl.recurrence.R;
import com.bleyl.recurrence.utils.DateAndTimeUtil;
import java.util.Calendar;
import java.util.List;
public class NotificationAdapter extends RecyclerView.Adapter<NotificationAdapter.ViewHolder> {
private int mRowLayout;
private Context mContext;
private List<Notification> mNotificationList;
public class ViewHolder extends RecyclerView.ViewHolder {
public TextView mTitle;
public TextView mContent;
public TextView mTextSeparator;
public ImageView mIcon;
public ImageView mCircle;
public View mView;
public ViewHolder(final View view) {
super(view);
mView = view;
mTitle = (TextView) view.findViewById(R.id.title);
mContent = (TextView) view.findViewById(R.id.content);
mTextSeparator = (TextView) view.findViewById(R.id.header_separator);
mIcon = (ImageView) view.findViewById(R.id.image);
mCircle = (ImageView) view.findViewById(R.id.circle);
}
}
public NotificationAdapter(Context context, int rowLayout, List<Notification> notificationList) {
mContext = context;
mRowLayout = rowLayout;
mNotificationList = notificationList;
}
@Override
public ViewHolder onCreateViewHolder(ViewGroup viewGroup, int i) {
View v = LayoutInflater.from(viewGroup.getContext()).inflate(mRowLayout, viewGroup, false);
return new ViewHolder(v);
}
@Override
public void onBindViewHolder(ViewHolder viewHolder, final int position) {
// Show header for item if it is the first in date group
if (position > 0 && mNotificationList.get(position).getDate().equals(mNotificationList.get(position - 1).getDate()) ) {
viewHolder.mTextSeparator.setVisibility(View.GONE);
} else {
// Parse date and get appropriate date format
Calendar DateAndTime = DateAndTimeUtil.parseDateAndTime(mNotificationList.get(position).getDateAndTime());
String appropriateDate = DateAndTimeUtil.getAppropriateDateFormat(mContext, DateAndTime);
viewHolder.mTextSeparator.setText(appropriateDate);
viewHolder.mTextSeparator.setVisibility(View.VISIBLE);
}
viewHolder.mTitle.setText(mNotificationList.get(position).getTitle());
viewHolder.mContent.setText(mNotificationList.get(position).getContent());
int iconResId = mContext.getResources().getIdentifier(mNotificationList.get(position).getIcon(), "drawable", mContext.getPackageName());
viewHolder.mIcon.setImageResource(iconResId);
GradientDrawable bgShape = (GradientDrawable) viewHolder.mCircle.getDrawable();
bgShape.setColor(Color.parseColor(mNotificationList.get(position).getColour()));
viewHolder.mView.setOnClickListener(new View.OnClickListener() {
@Override
public void onClick(View view) {
TabFragment fragment = new TabFragment();
fragment.startViewerActivity(view, mNotificationList.get(position));
}
});
}
@Override
public int getItemCount() {
return mNotificationList.size();
}
}
|
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"redpajama_set_name": "RedPajamaGithub"
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| 7,338
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Q: l3build in directories that have spaces in their names I am getting into l3build for testing and installing my homegrown packages. It looks very useful. I hit a snag today when I put a bundle inside a directory that had a space in the name.
I keep all my source-controlled packages in repositories on github, and I check them out onto my local machine inside a ~/github directory. If I change the name of ~/github to ~/git hub, the install script won't work anymore:
$ cd ~/git\ hub/mypkg
$ l3build install --dry-run
sh: line 0: export: `hub/mypkg/build/local:': not a valid identifier
sh: /Users/matthew/git: No such file or directory
I don't know lua well enough to find exactly where this error comes from. But I can see that it's caused by a path with a space that's treated like two separate paths.
The obvious workaround is don't do that, and I'm open to changing my setup to guarantee no spaces in directory names. But I'd rather not—the directory in question is a Dropbox folder that's shared with somebody else, and if I change the name of the folder they might need to change some stuff too. Also, I'm a Mac user and I think it's pretty common practice to have folder names with spaces in them. So it would be nice if l3build could work with that.
A: Updated answer
Thanks for the report. I believe that this should now be fixed in the development version, and will arrange a CTAN release today.
Original answer
Whilst there is no special treatment of spaces inside l3build, there are places that an absolute path is returned. These may end up containing spaces, as you've observed, and currently there is no quoting/escaping for such spaces. The one in question here is relatively easy to track down, but there are likely (lots) of others (most obviously, the user can set the various relative directories to anything). I've logged an https://github.com/latex3/l3build/issues/76: I suspect a full fix will need a bit of work.
|
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| 5,924
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by Bobby Kosser in Cards, Pinback, Posters, Souvenirs, Sports 0 comments tags: cards, memorabilia, pinback, poster, Sheik, souvenir, Wrestling, Wrestling Heros
Vintage Early Old School Wrestling Poster. The Sheik. McGuire Twins. Tag Teams and Midgets. 1951 Wrestling Cards. Gene Stanley and the Dusek Brothers
My Forever Treasures wrestling fan clients have ached for vintage professional wrestling treasures. Today, we'll celebrate this sport, that's not really a sport. Professional wrestling's egotistical gladiators are never who they appear to be. They've all been created, solely for your ENTERTAINMENT! Their feuds and grudges are choreographed. The public is offered nothing honest. The McMahon family, even dictates the predetermined FINAL MATCH RESULTS! Yet, the American public is overwhelmingly entertained by Professional Wrestling. Last year, this entertainment/sport turned a $215 Million profit! Our vintage treasures await you. We're offering fans of pretend vain wrestling heroes and heels a vintage old school 1970's original Professional Wrestling Promotional Match Card Poster 22″ x 14″ heavy card stock. Pro wrestlers are always politically incorrect. Tag teams such as "The Assassins," "Beer Money Inc.,""The Spanish Fleas," Midget Team "The Mini Vans," & Native American Tag Team "The Wahoos" all have names meant to offend fans. Names like "The Sheik," "Andre The Giant" "The Big Boss Man" & "The Rock" aren't their real names. When born, weighing just 8 pounds, did their parents really name them "HULK HOGAN!" or "The Undertaker!" The public xxx's FAKE Wrestling. Our rare "2nd Golden Era" […]
by Bobby Kosser in Autographs, Posters, Souvenirs 0 comments tags: Ella Autograph, Ella Fitzgerald, Esther Phillips, Hubert Laws, Jazz Memorabilia, Jazz Music, Jazz Poster, Les McCann, Nancy Wilson, Stanley Turrentine
Iconic Jazz Original Ella Fitzgerald Autograph and Vintage 1975 Jelly Roll Jazz Festival Poster Paramount Theater Oakland
Jazz is a music genre that originated in the African-American communities of New Orleans in the late 19h and early 20th centuries,and developed from roots in blues and ragtime. Jazz is seen by many as 'America's Classical' music. Since the 1920's "Jazz Age" jazz has become recognized as a major form of musical expression. My Forever Treasures has opened our iconic vintage jazz vault. We're offering a dual jazz treasure. An Ella Fitzgerald autograph. There is only one "Queen of Jazz." Plus, our 1975 Jelly Roll Jazz Festival Poster. Welcome Nancy Wilson, Les McCann, Hubert Laws, Esther Phillips and Stanley Turrentine. Ella Fitzgerald was an American jazz singer often referred to as the First Lady of Song, the Queen of Jazz and Lady Ella. She was noted for her purity of tone, impeccable phrasing,and improvisational ability, especially in her scat singing. Her collaborations with Louie Armstrong attracted much attention over the years. Her classics include "A-Tisket, A-Tasket," "Autumn in New York," "Begin the Beguine," and "Bewitched Bothered and Bewildered." She recorded 200 albums and near 2,000 songs, many considered American treasures. Ella's voice spanned 3 octaves, offering very high & low notes for her scat singing. She won 14 […]
by Bobby Kosser in Autographs, Basketball Card, Souvenirs, Sports 0 comments tags: autographed baseball, autographed photo, Basketball Card, Berra, Lew Alcindor, Mantle, memorabilia, Rod Laver, souvenir, Tennis
Iconic Sports Heroes Rod Laver Autographed Photo Mantle and Berra Signed Baseball and Lew Alcindor Milwaukee Basketball Card Second Year
My Forever Treasures is pleased to offer our first combination 3 iconic sports hero treasures. Tennis great Rod Laver, basketball's Lew Alcindor and baseball's Mickey and Yogi. We start with an autographed photo of Rod Laver. Tennis's first king of the courts. Rod is an Australian tennis player, widely regarded as the greatest in the history of the sport. He won 11 singles grand slams. He won a record 200 tournaments. Held the #1 world ranking from 1964-1970. The only male or female tennis player to win two grand slams in single calendar years. Laver was the first to exceed more than $1 million on tour. He won 11 major titles in just 16 tries. Rod Laver's 2 French Opens, 4 Wimbledon Championships, 2 U.S. National Titles and 3 Australian Titles set the bar for Federer and all who follow him. Today Kareem Abdul Jabbar aka Lew Alcindor still holds the N.B.A.'s lifetime scoring record. Jordan and the others including Lebron James might never catch his incredible 38,387 points. In his second year, this 1970-71 Alcindor Milwaukee Bucks basketball card has him scoring 31 points a game with 16 rebounds as well.He changed his name to Kareem […]
by Bobby Kosser in Autographs, Photo, Souvenirs 0 comments tags: Academy Award, autographed photo, Best Director, film, Frank Capra, memorabilia, souvenir
Vintage Frank Capra Autographed Studio Publicity Photo
Frank Capra was an Italian-American film director, producer and screenwriter who became the creative force behind some of the major award winning motion pictures of the 1930's and 1940's. Nominated for 6 Best Director Oscars and winning 3 in just a 5 year span, separate Capra from the rest. Frank Capra's Best Director Academy Awards for "It Happened One Night,"1934 "Mr. Deeds Goes To Town,"1936 and "You Can't Take It With You,"1938 are just indications of his brilliance. He claimed "I made some mistakes with drama." "I thought drama was when the actor cried." "Drama is when the audience cries!" A Lifetime Achievement Oscar shows Frank figured it out. Capra wanted his films "to let every man, woman and child know that God loves them, that I love them and that peace and salvation will become a reality when they learn to love each other." The messages he conveyed to his audience of "everyman" heroes triumphing over mercenary big business and big government have his classics resonate more than 80 years later. His 1934 first Best Director Oscar for "It Happened One Night," was also the first film to win the top 5 awards. Best Film, Director, Screenwriting, Best […]
by Bobby Kosser in Memorabilia, Music, People, Posters 0 comments tags: Bob Dylan, concert poster, concerts, Japan, memorabilia, rock music, Tom Petty, videos
Vintage Bob Dylan & Tom Petty and the Heartbreakers 1986 True Confessions Tour Tokyo Japan Poster March 5 and 10.
Bob Dylan and Tom Petty performed at the 1st Farm Aid in 1985. Their chemistry was so incredible they decided to tour together in 1986. They called this the "True Confessions Tour." After New Zealand shows and 13 dates in Australia, they electrified audiences in Japan for another 4 concerts.This March 5 and March 10 ~ Nippon Budokan Tokyo,Japan Concert Poster is the treasure that publicized these events. The two music icons and R. & R. Hall of Fame members played classics from the opening number to their encores. Highlights included both Dylan and Petty solos and duets together. Bob's band and The Heartbreakers were at their best! This tour caught Tom Petty and The Heartbreakers at the top of their game. Bob was in a place where he wondered why he was still touring.The fans Bob D. For Your FANS! The Tokyo audiences celebrated every Bob Dylan song. He included "Just Like A Woman," "Positively 4th Street," "Like A Rolling Stone," "Blowin In The Wind," countless other favorites and closed with "Knockin On Heaven's Door." Tom Petty and The Heartbreakers overwhelmed the S.R.O. Stadium concert goers with "American Girl" "Refugee," "Don't Do Me Like That." "The Waiting," and […]
by Bobby Kosser in Autographs, Posters, Souvenirs, Sports 0 comments tags: Ali Autograph, boxing, memorabilia, Muhammad Ali, poster
Happy Birthday Champ Muhammad Ali The Greatest of All Time
It's Muhammad Ali's birthday. Today we celebrate the life of the most loved and recognized man on earth. The Heavyweight Champion who possessed a rare combination of love, respect, generosity, smart as could be, with a magical sense of humor above and beyond what made the world smile! In and out of the boxing ring Ali was the fastest. He'd always be the first to declare he was "The Greatest Champion of All Time!" It's true! How fortunate for his fans Muhammad Ali was also "The Prettiest of All Time" as well. On his birthday remember Muhammad Ali said, "After me there will never be another." Champ I agree. That would be impossible. All my xxx on your birthday. The lunch and 3 hours I spent with Muhammad Ali has been the highlight of my life. I wish everyone could experience his magic. Date that forever.:)Our photo together shows how wonderful it is to be in his presence. Top 10 Muhammad Ali Best Knockouts HD CLICK HERE FOR POSTERS: Muhammad Ali/Cassius Clay Posters
by Bobby Kosser in Advertisement, Pinback, Souvenirs 0 comments tags: advertising, Bally's Pinback, Dangerfield's Comedy Club, Matchbook, No Respect, postcard, Rodney Dangerfield, souvenirs
Classic Rodney Dangerfield Souvenir Respect – Bally's Las Vegas Publicity Pinback – Dangerfield's NYC Comedy Club Bonus Postcard and Matches
Rodney Dangerfield was an American comedian, writer, movie, television, stage, and radio voice actor. He was known for his catch-phrase "I don't get no respect." Rodney was the bug-eyed comic whose self-deprecating one-liners brought him stardom in clubs, television and movies. He was a professional joke teller not a guy looking for psychoanalysis from a night club audience. So you got jokes & gags. Many are among comedy's best. Rodney, "I don't get no respect." "My wife likes to talk to me during sex." "Last night she called me from a motel." R.D."What a doctor I've got." I said "Doc I broke my arm in 2 places." He told me "keep out of those places." Rodney was a major star at top entertainment venues. His Las Vegas act was a definitive comic event. He set fear on parade and all its consequences. Rodney, "Women show me no respect." "A belly dancer said I turned her stomach." Johnny Carson said "Quite simply no one was funnier doing stand up comedy than Rodney Dangerfield." Jay Leno said "Dangerfield was simply the greatest." Howard Stern said "Rodney was incredible." "His stand-up was spectacular." "His command of the stage was unequaled." "His films, […]
by Rich Benvin in Memorabilia, Music, People, Stamps 0 comments tags: artists, Bob Marley, Bobby Kosser, collection, Jamaica, memorabilia, music, musicians, people, reggae, stamp collecting, stamps, videos
Reggae Star Bob Marley Official First Day Cover Stamps From Jamaica 1981
Bob Marley is the "Reggae King." He's the Jamaican musician, singer & songwriter most responsible for reggae's international success. A pioneer who's earliest fame came with good friends Peter Tosh & Bunny Wailer as Bob Marley & The Wailers. "Simmer Down" was one of their ska classics. Bob drove a BMW not because he loved the car. It's because he loved the letters BMW. They stood for Bob Marley & the Wailers. Bob resurrected an old time 60's tune and turned it into a soft stoned love poem. "We'll be together with a roof right over our head." "We'll share the shelter of my single bed." This would be a popular theme as several Marley songs lead to his single bed. He's fathered 11 children. Rumor is there's many more. But no one's done "The Math!" After solo careers for Bob and these Wailers, he achieved fortune, fame and an idolization that's as powerful today as when he was gone in 1981. He's sold 15 million albums in the U.S. and 25 million worldwide. He continues to outsell most reggae artists 30 years after his passing. Bob Marley's estate earned $21 million last year Marley's huge reggae influence […]
by Bobby Kosser in Autographs, Photo, Souvenirs 0 comments tags: Academy Award Winner, Actor, autographed photo, Clint Eastwood, Director, Dirty Harry, Jack Nicholson, Joker Batman Card, memorabilia, videos
Vintage Jack Nicholson Autographed Chinatown Photo And Batman Trading Card And Clint Eastwood Autographed Publicity Photo
Most of My Forever Treasures revered clients, have requested an idolized icon treasure! Our only choice, was to open our ultimate icon vault. We're thrilled to offer two autographed photos of Hollywood's biggest legends. They're multiple Academy Award winners. Jack Nicholson & Clint Eastwood have both received Lifetime Achievement Academy Awards. Adjusted for inflation, their films have grossed more than $6.7 Billion. They've won 7 Oscars between them! They are loved worldwide. Jack Nicholson's reputation as the coolest cat in Hollywood is the most long-lived and undisputed since James Dean. Cinema has pretty cool actors from Johnny Depp to Sean Penn but in his era, late 1960's-2012 Jack reigns supreme.He's not only chosen wisely in his films but maintained the persona that blurs the lines between studied intellectual and Hell's Angel. His body of work as actor, producer, director and writer has almost defined American film art in the latter 20th century. When they talk about Jack in Hollywood, there is "no other." He's done it all authentically with an overall accomplishment of elegant discretion. From his career-altering role in "Five Easy Pieces"1969, "Chinatown," "Cuckoo's Nest," "As Easy As It Gets," "A Few Good Men," "Terms of Endearment," […]
by Bobby Kosser in Advertisement, Memorabilia, Souvenirs, Sports 0 comments tags: advertsing, cigarette companies, Crisp Crunch, memorabilia, Michael Jordan, Muhammad Ali, My Forever Treasures, Space Jam
American Classic Merchandising Lucky Strike Ad – Muhammad Ali Crisp Crunch Candy Bar – Michael Jordan Space Jam Bandages
The, purpose of all products, is to convince the consumer. Show the public, reasons why they want, need & should buy their product. Always focus on the consumer's desires. The 1930's offered cigarette companies an unusual circumstance. Their, objective was to influence consumers, a dangerous product (cigarettes) was safe and desirable. Today, cigarette packages have medical warnings! They're banned in most restaurants, theaters, federal buildings, radio and television commercials just to bring us up to date! Our, vintage 1930's Lucky Strike Cigarette ad, shows why "Luckies" was that decade's most successful cigarette advertising campaign. Lucky Strike Cigarettes, gained the consumer's trust by offering Doctors assuring the public. Their, brand "Luckies" "tastes great," and was not only safe, but "Toasted" assuring a smooth smoke."Luckies" was protecting their throat, from irritating coughing. Doctors, assurances and scientific survey evidence, convinced the 1930's American consumer! Even sugar cravings would disappear, if they lit up a Lucky Strike. Weight loss was possible. This, brought a 200% increase in female Lucky Strike purchases. Our, vintage Smoke "Luckies" Cigarette ad says it all. Vintage Lucky Strike Cigarette Commercial In 1978, Muhammad Ali Crisp Crunch All Natural Ingredients Candy Bar was created. Consumers are […]
|
{
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| 9,021
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The China Page Newsletter
Spotlight Coronavirus; A New U.S. Central Asia Strategy; And More...
Another person with symptoms of the Coronavirus has been identified. (RFE/RL Russian Service)
U.S. Announces 'New Possibilities' Amid 'Sea Change' in Central Asia
The United States on February 5 declared "an era of new possibilities" in Central Asia in remarks announcing an updated regional strategy that seeks to address new developments over the past five years and strengthen U.S. influence as a counterweight to Russia and China.
WATCH: Pompeo Talks Central Asia, China, Press Freedom
RFE/RL interviewed U.S. Secretary of State Mike Pompeo during his visit to Kazakhstan's capital, Nur-Sultan, on February 2. He discussed U.S. policies on Central Asia and China's oppression of minorities, but also journalistic freedom in the world in the wake of the State Department's controversial exclusion of a reporter from his traveling press pool.
WATCH: Countries On China's 'New Silk Road' Face Coronavirus Fears
Countries in and around China's "Belt and Road" trade routes are taking steps to protect themselves from the coronavirus. Pakistan has halted all flights to and from China, Kazakhstan has closed a flagship border free-trade zone, and Tajikistan is screening arrivals with thermal scanners.
LISTEN: Majlis Podcast: Central Asia Warily Monitors Spread Of Coronavirus From China
For the governments and people of Central Asia, the Coronavirus that is spreading from China is a source of great concern, given the countries' shared borders, exchange students, and migrant workers.
INFOGRAPHIC: The World's Largest Arms Producers
A new report suggests that China is now the second-largest arms producer in the world, behind the United States but ahead of Russia.
CHINA AND RUSSIA
Russia Joins China In Search For Vaccine As Virus Outbreak Spreads
Russia says it has received the genome of the coronavirus from China and is working jointly with its neighbor to develop a vaccine against the illness as the number of deaths and confirmed cases continues to jump.
St. Petersburg Court Bans Book Criticizing Chinese Communist Party
A St. Petersburg court has upheld a lower court decision banning Nine Commentaries on the Communist Party, a volume of opinions on the Chinese Communist Party's treatment of dissidents, contract killings, and imprisonment of Muslims. The decision says the texts are aimed at fomenting "social discord" in relation to Communism. (RFE/RL Russian Service)
CHINA AND CENTRAL ASIA
Tajik Claim Of Pipeline Progress Is Welcome News In Turkmenistan
Reports this week that Tajikistan was constructing its part of the Turkmenistan-China gas pipeline were welcome news on a project that was feared to be moribund.
Locked Up In China: The Plight Of Xinjiang's Muslims
Amnesty Reports Resistance And Repression Across Asia In 2019
A new Amnesty International report called 2019 "a year of repression in Asia, but also of resistance." "Online and offline, youth-led popular protests are challenging the established order," notably in China, Pakistan, and India, according to the report.
INFOGRAPHIC: The Spread Of The Coronavirus
PRESSROOM: Washington Post Cites RFE/RL Kazakh Service Interview With U.S. Secretary Of State Pompeo
PRESSROOM: RFE/RL Seeks To Expand In Russia, Despite 'Foreigin Agent' Law
About The China Page
The China Page is a bi-weekly collection of RFE/RL reporting on China-related developments from the Balkans to Central Asia and the Bering Strait.
Click here to receive the newsletter in your inbox.
|
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| 5,206
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John Radford, född 22 februari 1947 i Hemsworth, Yorkshire, är en engelsk före detta fotbollsspelare.
Radford spelade större delen av sin karriär i Arsenal, där han bildade ett radarpar med Ray Kennedy. Han började som högerytter, men bytte sedermera till centerforward, och gjorde 149 mål på 481 matcher för Arsenal. Han kom till klubben som lärling 1963 och gjorde A-lagsdebut i mars 1964. I matchen mot Wolverhampton den 2 januari 1965 blev han med sina 17 år och 315 dagar Arsenals yngste spelare att göra hattrick.
1970 var han med om att vinna Mässcupen och gjorde då ett mål i finalreturen mot Anderlecht på Highbury. 1968 och 1969 hade Radford varit med om att förlora två ligacupfinaler, men 1971 fick han lämna Wembley Stadium som segrare. Han bidrog starkt med sina två målgivande passningar till att Arsenal kunde besegra Liverpool i FA-cupfinalen. Man hade därmed vunnit "dubbeln" för första gången – ligasegern hade säkrats fem dagar tidigare.
Radford fortsatte i Arsenal under 1970-talet, men skador och det faktum att Frank Stapleton började blomma ut, gjorde att han förlorade sin ordinarie plats. Han såldes till West Ham 1976 och stannade där i två år innan han avslutade proffskarriären med två säsonger i Blackburn Rovers. Han spelade sedan i amatörlaget Bishop's Stortford FC, en klubb han även tränade i slutet av 1980-talet och början av 90-talet. Radford spelade även två landskamper för England, men gjorde inget mål. Efter fotbollskarriären drev han under en period en pub i Essex.
Engelska fotbollsspelare
Engelska fotbollstränare
Spelare i Arsenal FC
Spelare i West Ham United FC
Spelare i Blackburn Rovers FC
Spelare i Bishop's Stortford FC
Män
Födda 1947
Levande personer
Personer från Yorkshire
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{
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| 2,882
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Marcus dit Marius Baggers, né le à Amsterdam et mort le dans le 4e arrondissement de Paris, est un compositeur, arrangeur, pianiste et chef d'orchestre français d'origine hollandaise.
Biographie
Fils et petit-fils de tailleurs de diamants amstellodamois, Marcus Baggers s'installe à Lyon avec ses parents après la guerre de 1870. Ancien élève dans la classe de piano du Conservatoire de musique de Lyon, il commence sa carrière comme chef d'orchestre aux Fantaisies-Parisiennes avant d'être engagé le 1er septembre 1880 au théâtre des Bouffes-Parisiens, puis comme directeur de la musique au Théâtre du Châtelet.
Professeur au Conservatoire national de musique de Paris, on lui doit près de 90 créations dont des musiques pour chansons, des valses pour piano, des marches et de la musique de scène. Parmi les musiques de scène, il crée les musiques de deux reprises de pièces de Jules Verne : Michel Strogoff en 1904 et Le Tour du monde en 80 jours en 1927 sur une mise en scène de Georges de La Renaudie.
Mort à l'âge de 83 ans, il était veuf depuis février 1929 de Lucie Halbronn, une modiste parisienne d'origine alsacienne qu'il avait épousé en mai 1886. Son éloge funèbre au cimetière du Montparnasse a été prononcé par le revuiste Valentin Tarault (1880-1960), alors vice-président de la Société des auteurs et compositeurs dramatiques.
Marius Baggers était le frère de Joseph Baggers (1858-1938), également musicien, compositeur et professeur au Conservatoire, le neveu du comédien et administrateur de théâtre Abraham Stréliski (1832-1928), et le cousin de Maurice Stréliski (1870-1950), acteur et metteur en scène de théâtre et de
Louis Stréliski (1878-1961), chanteur et metteur en scène d'opéra.
Œuvres
1880 : La Mascotte, opéra-comique en trois actes d'Edmond Audran, sur un livret d'Henri Chivot et d'Alfred Duru, chef d'orchestre Marius Baggers, Théâtre des Bouffes-Parisiens, 29 décembre
1900 : Le Petit Chaperon rouge, opérette de Paul Ferrier, Ernest Blum et Pierre Decourcelle, musique de Marius Baggers, Paris, Théâtre du Châtelet, 22 décembre
1905 : Tom Pitt, le roi des pickpockets, pièce à grand spectacle en 4 actes et 18 tableaux, de Victor Darlay et Victor de Cottens, musique de Marius Baggers, théâtre du Châtelet, 2 mars
1906 : Pif ! Paf ! Pouf ! ou le Voyage endiablé, féérie en trois actes et trente-huit tableaux de Victor de Cottens et Victor Darlay, musique de Marius Baggers, Théâtre du Châtelet, 6 décembre
1908 : La Revue du Châtelet, revue en 3 actes et 28 tableaux, de Georges Nanteuil et Henry de Gorsse, musique de Marius Baggers, Théâtre du Châtelet, mars
1909 : Les aventures de Gavroche, pièce en 4 actes, de Victor Darlay et Gaston Marot, musique de Marius Baggers, théâtre du Châtelet, 27 janvier
1910 : Arsène Lupin contre Herlock Sholmès, pièce en 4 actes et 15 tableaux tirée des romans de Maurice Leblanc par Henry de Gorsse et Victor Darlay, musique de Marius Baggers, Théâtre du Châtelet, 28 octobre
1913 : Non !... pas les mains !, revue en 2 actes de P.-L. Flers et Eugène Héros, musique de Marius Baggers, Théâtre des Ambassadeurs, 31 mai
1916 : La Revue des Étoiles, revue en trois actes de Rip, musique de Marius Baggers et Émile Lassailly, Théâtre du Châtelet, 29 avril
1920 : En l'an 2020 ou la Merveilleuse Aventure de Benjamin Pirouette, pièce à grand spectacle en trois actes et 21 tableaux, de Henry de Gorsse, musique de Marius Baggers.
Distinctions
Officier d'Académie (1900)
Officier de l'Instruction publique (arrêté du ministre de l'Instruction publique, des Beaux-Arts et des Cultes du janvier 1907).
Notes et références
Bibliographie
La Revue musicale, volume 20, 1939, (nécrologie)
Kurt Gänzl, The Encyclopedia of the Musical Theatre, vol. 2, 1994,
Liens externes
Compositeur français
Chef d'orchestre français
Naissance en octobre 1855
Naissance à Amsterdam
Décès en mai 1939
Décès dans le 4e arrondissement de Paris
Décès à 83 ans
Personnalité inhumée au cimetière du Montparnasse
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WELLINGTON, New Zealand (AP) — The Sharks' 42-5 win over the Lions in the weekend's eighth round of Super Rugby threw light on a South African conference in which switchback changes of form have been the norm this season.
The Lions started the round atop the conference table, with the Sharks in third place. But after a series of surprising results, the two sides swapped positions, with the inconsistent Bulls separating them.
By the end of the round only seven points separated the table-topping Sharks from the last-placed Stormers. By contrast, 13 points separated the first and last teams in Australia, and there were 17 points between the first-placed Crusaders and fifth-placed Chiefs in New Zealand.
Every conference has contained intrigue this season. In New Zealand, the Auckland-based Blues have won four matches in a row for the first time since 2013. In Australia, the Melbourne Rebels have a become a dominant force. But the South African conference has had more plot twists than even the most action-packed potboiler, with predicting winners a testing chore.
For example, the Bulls beat the Stormers 40-3 in the first round of the seasons while the Lions — finalists in 2016 and 2017 — beat the Jaguares 25-16 in Buenos Aires. A week later the Stormers beat the Lions in Cape Town while the Bulls lost to the Jaguares in Argentina.
In week three, the Stormers beat the Sharks and the Bulls beat the Lions in away matches, dismissing to some degree the influence of home advantage on the changes in fortune.
The Bulls then beat the Sharks twice, 37-14 in Petoria in round five and 19-16 in Durban in round seven before the Sharks achieved an improbable 37-point win over the Lions in the round just ended.
Form fluctuations can be explained by injuries, by the imperative in World Cup seasons to rest top players, by home advantage and even by the weather. But even Lions coach Swys de Bruin was at a loss to explain how his team, which has risen so high in recent seasons, fell so low against the Sharks.
"I'm very disappointed," said de Bruin, who signed a two-year contract extension in the week before the match. "It was the same against the Bulls. Physically, they had us from the word go.
The Lions now face a tough tour to New Zealand and Australia in which they will play the ACT Brumbies, Hamilton-based Chiefs and the defending champion Crusaders. Wins may be hard to come by; in the first venture by a South African team to Australasia this season, the Stormers lost to the Hurricanes, Blues and Queensland Reds.
De Bruin is more interested in seeing his players demonstrate more character than they did against the Sharks.
The Jaguares also pulled off a major upset in round eight, beating the Bulls 22-20 in Pretoria, though that was a result in part of the Bulls being reduced by yellow cards to 13 players in the second half.
Domingo Miotto, who made his debut off the bench, scored two tries while the Bulls were under-manned to lead the Jaguares to a win which bounced them off the bottom of the South Africa conference.
Bulls coach Pote Human had no difficulty in putting his finger on the cause of his team's defeat.
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\section{Introduction}
Higher derivative corrections to the Einstein-Hilbert action have
received much notice in recent years, as such terms naturally show
up in the $\alpha'$ expansion of effective actions derived from string
theory. In general, the first non-trivial terms arise at the four
derivative level, corresponding to curvature-squared corrections to
classical Einstein theory of the form
\begin{equation}
e^{-1}\delta\mathcal L=\alpha_1R^2 + \alpha_2R_{\mu\nu}R^{\mu\nu}
+ \alpha_3R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma},
\label{eq:la1a2a3}
\end{equation}
where the coefficients $\alpha_1$, $\alpha_2$ and $\alpha_3$ are
determined by the underlying theory. It was suggested in
\cite{Zwiebach:1985uq} that the natural form of such terms would be given
by the Gauss-Bonnet combination
\begin{equation}
e^{-1}\delta\mathcal L_{\rm GB}=\alpha(R^2-4R_{\mu\nu}R^{\mu\nu}
+R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}),
\label{eq:lgb}
\end{equation}
as this is the unique combination that avoids
introducing ghosts in the effective theory. It was subsequently
argued, however, that in the absence of an off-shell formulation such
as string field theory, the $\alpha_1$ and $\alpha_2$ coefficients are
physically indeterminate as they may be eliminated by an on-shell field
redefinition of the form $g_{\mu\nu}\to g_{\mu\nu}+aRg_{\mu\nu}+bR_{\mu\nu}$.
In this sense, only the Riemann-squared term parameterized by $\alpha_3$
carries physical information from the underlying string theory.
The form of the higher derivative corrections are further constrained by
supersymmetry. Explicit computations for the uncompactified closed
superstring indicate that the first corrections enter at the $R^4$
order \cite{Gross:1986iv,Grisaru:1986vi,Freeman:1986zh}. This
is a feature of maximal supersymmetry, as curvature-squared terms are
present in, for example, the uncompactified heterotic theory
\cite{Gross:1986mw,Metsaev:1987zx}. An alternate route to obtaining
supersymmetric higher derivative corrections is to make use of supersymmetry
itself to construct higher derivative invariants that may show up in the
action. This was applied in the heterotic supergravity by supersymmetrizing
the Lorentz Chern-Simons form responsible for the modified Bianchi
identity $dH=\alpha'\mbox{Tr\,}(F\wedge F-R\wedge R)$ \cite{Bergshoeff:1989de};
the result agrees with the explicit calculations, once field redefinitions
are properly taken into account \cite{Chemissany:2007he}. More recently,
the supersymmetric completion of the $A\wedge \mbox{Tr\,} R\wedge R$ term
in five-dimensional $\mathcal N=2$ supergravity (coupled to a number of
vector multiplets) was obtained in \cite{Hanaki:2006pj}. This result has
led to new progress in the study of black hole entropy and precision
microstate counting in five dimensions (see {\it e.g.}~\cite{Castro:2008ne}
and references therein).
The supersymmetric four-derivative terms given in \cite{Hanaki:2006pj}
were obtained using conformal supergravity methods. Thus it should be
no surprise that they involve the square of the five-dimensional
Weyl tensor \cite{Hanaki:2006pj}
\begin{eqnarray}
e^{-1}\delta\mathcal L_{\rm sugra}&=&\ft{c_I}{24}[\ft18M^IC_{\mu\nu\rho\sigma}
C^{\mu\nu\rho\sigma}+\cdots]\nonumber\\
&=&\ft{c_I}{24}[\ft18M^I(\ft16R^2-\ft43R_{\mu\nu}
R^{\mu\nu}+R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma})+\cdots],
\label{eq:hanaki}
\end{eqnarray}
as opposed to the Gauss-Bonnet combination, (\ref{eq:lgb}). In principle,
an appropriate field redefinition may be performed to bring this into the
Gauss-Bonnet form. However, this is usually not done, as it would obscure
the overall supersymmetric structure of the theory. Thus in practice two
somewhat complimentary approaches have been taken to investigating the
curvature-squared corrections to the Einstein-Hilbert action. The first,
which applies whether the underlying theory is supersymmetric or not, is
to use a parameterized action of the form (\ref{eq:la1a2a3}), with special
emphasis on the Gauss-Bonnet combination. The second is to focus
directly on supergravity theory, and hence to use explicitly supersymmetric
higher-derivative actions of the form (\ref{eq:hanaki}). In principle,
these two approaches are related by appropriate field redefinitions. However,
in practice this is complicated by the fact that additional matter
fields ({\it e.g.}~$\mathcal N=2$ vector multiplets) as well as auxiliary
fields may be present, thus making any field redefinition highly non-trivial.
In this letter, we investigate and clarify some of the issues surrounding
field redefinitions in the presence of additional fields. In particular,
we take the bosonic sector of five-dimensional $\mathcal N=2$ gauged
supergravity and extend it with four-derivative terms built from the
Riemann tensor $R_{\mu\nu\rho\sigma}$ as well as the graviphoton
field-strength tensor $F_{\mu\nu}$. Although we introduce eight such
terms, we demonstrate that only four independent combinations remain
physical once field redefinitions are taken into account. To be explicit,
we construct the higher-derivative corrections to the spherically symmetric
$R$-charged AdS$_5$ black holes of \cite{Behrndt:1998ns,Behrndt:1998jd},
working to linear order in the higher-derivative terms, and then investigate
the effect of field redefinitions on these black hole solutions.
To some extent, our solutions generalize the Gauss-Bonnet black holes
originally constructed in \cite{Boulware:1985wk,Wheeler:1985nh} and
extended to Einstein-Maxwell theory in \cite{Wiltshire:1985us} and,
with the inclusion of Born-Infeld terms, in \cite{Wiltshire:1988uq}.
One advantage that the Gauss-Bonnet combination has over the generic
form of (\ref{eq:la1a2a3}) is that it leaves the graviton propagator
unmodified, and also yields a modified Einstein equation involving at
most second derivatives of the metric. With an appropriate metric ansatz,
the resulting Gauss-Bonnet black holes are then obtained by solving a
simple quadratic equation. Furthermore, this feature of the Gauss-Bonnet
term leads to a good boundary variation and natural generalization of
the Gibbons-Hawking surface term \cite{Myers:1987yn}. This is a primary
reason behind the popularity of applying Gauss-Bonnet (and more generally
Lovelock) extensions to braneworld physics (see
{\it e.g.}~\cite{Charmousis:2008kc}).
Our interest in studying the higher order corrections to $R$-charged
AdS$_5$ black holes is also motivated by our desire to explore finite
't~Hooft coupling corrections in AdS/CFT. Using the relation $\alpha'
=L^2/\sqrt\lambda$, we see that each additional factor of
$\alpha' R_{\mu\nu\rho\sigma}$ in the string effective action gives rise
to a $1/\sqrt\lambda$ factor in the strong coupling expansion of the
dual gauge theory. Since supersymmetry ensures that the leading correction
terms in IIB theory are of order $\alpha'^3$, this indicates that the
$\mathcal N=4$ super-Yang Mills theory dual to AdS$_5\times S^5$ will first
receive such corrections at the $\lambda^{-3/2}$ order. The effect of
these finite 't~Hooft coupling corrections on both the thermodynamics
\cite{Gubser:1998nz,Pawelczyk:1998pb} and hydrodynamics
\cite{Buchel:2003tz,Buchel:2004di,Benincasa:2005qc,Buchel:2008ac,Buchel:2008wy,Buchel:2008sh}
of the $\mathcal N=4$ plasma have received much attention in the context of
extrapolations between the strong and weak coupling limits of the
$\mathcal N=4$ theory.
In principle, it would be greatly desirable to extend the finite coupling
analysis to $\mathcal N=1$ gauge theories dual to AdS$_5\times Y^5$ where
$Y^5$ is Sasaki-Einstein. This is of particular interest in resolving
conjectures on the nature of the shear viscosity bound $\eta/s$
\cite{Policastro:2001yc,Kovtun:2003wp,Buchel:2003tz,Kovtun:2004de,Kats:2007mq,Brigante:2007nu,Brigante:2008gz}.
One difficulty in doing so, however, lies in the fact that the higher
derivative corrections involving the Ramond-Ramond five-form have not yet
been fully explored (but see \cite{Paulos:2008tn}). While it may be
argued that these terms will not contribute in the maximally supersymmetric
case, there is no reason to expect this to continue to hold for the reduced
supersymmetric backgrounds dual to $\mathcal N=1$ super-Yang Mills. For
this reason, recent investigations of the shear viscosity
\cite{Kats:2007mq,Brigante:2007nu,Brigante:2008gz}
(and drag force \cite{Fadafan:2008gb,VazquezPoritz:2008nw})
have assumed a
parameterized set of curvature-squared corrections of the form indicated
above in (\ref{eq:la1a2a3}). Our present construction of higher-derivative
corrected $R$-charged black holes allows for a generalization of the
finite coupling shear viscosity calculation to backgrounds dual to turning
on a chemical potential \cite{Benincasa:2006fu}.
We start with the two-derivative bosonic action of $\mathcal N=2$ gauged
supergravity and in Section~2 we introduce a parameterized set of four
derivative terms involving both curvature and graviphoton field strengths.
Then, in Section~3, we obtain the linearized corrections to the spherically
symmetric $R$-charged AdS$_5$ black holes. As one of the aims of this letter
is to clarify the use of field redefinitions, we take a closer look at this
in Section~4. Finally, we conclude with a discussion of our results in
Section~5.
\section{The higher-derivative theory}
Our starting point is the bosonic sector of pure $\mathcal N=2$ gauged
supergravity in five dimensions, with Lagrangian given by
\begin{equation}
e^{-1}{\mathcal L}_0 = R - \ft14F_{\mu\nu}F^{\mu\nu}+ 12g^2 +\ft1{12\sqrt3}
\epsilon^{\mu\nu\rho\sigma\lambda}F_{\mu\nu}F_{\rho\sigma}A_{\lambda}.
\label{eq:lag0}
\end{equation}
Although the Chern-Simons term is important from a supergravity point of view,
it will not play any role in the electrically charged solutions that are
investigated below.
In general, higher-derivative corrections to $\mathcal L_0$ may be expanded
in the number of derivatives. We are mainly interested in the first non-trivial
corrections, which arise at the four-derivative level. In a pure gravity
theory, this would correspond to the addition of $R^2$ terms to the Lagrangian. However, for the Einstein-Maxwell system, we may also consider higher-order
terms in the Maxwell field, such as $F^4$ and $RF^2$ terms. We thus introduce
the higher-derivative Lagrangian
\begin{equation}
{\cal L} = {\cal L}_0 + {\cal L}_{R^2} + {\cal L}_{F^4} + {\cal L}_{RF^2},
\label{eq:lhd}
\end{equation}
where $ {\cal L}_0 $ is given in (\ref{eq:lag0}), while the additional
terms are
\begin{eqnarray}
e^{-1}{\cal L}_{R^2} &=& \alpha_1R^2 + \alpha_2R_{\mu\nu}R^{\mu\nu}
+ \alpha_3R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma},\nonumber\\
e^{-1}{\cal L}_{F^4} &=& \beta_1(F_{\mu\nu}F^{\mu\nu})^2 +
\beta_2F^{\mu}{}_{\nu}F^{\nu}{}_{\rho}F^{\rho}{}_{\sigma}F^{\sigma}{}_{\mu},
\nonumber\\
e^{-1}{\cal L}_{RF^2} &=& \gamma_1RF_{\mu\nu}F^{\mu\nu} +
\gamma_2R_{\mu\nu}F^{\mu\rho}F_{\rho}{}^{\nu} +
\gamma_3R^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}.
\label{eq:hdlag}
\end{eqnarray}
Note that we have not considered terms such as $F_{\mu\nu}\Box F^{\mu\nu}$
that would in principle enter at the same order. Although we are not
complete in this regard, the terms that enter in $\mathcal L_{F^4}$ are
nevertheless sufficient for capturing the expansion of the Born-Infeld action.
\subsection{Equations of Motion}
Both the Maxwell and Einstein equations pick up corrections from the
higher-derivative terms in (\ref{eq:lhd}). The modified Maxwell equation
is straightforward
\begin{eqnarray}
\nabla_\mu F^{\mu\nu}+\ft1{4\sqrt3}\epsilon^{\nu\rho\lambda\sigma\delta}
F_{\rho\lambda}F_{\sigma\delta}&=&\nabla_\mu\bigl(8\beta_1F^2 F^{\mu\nu}
-8\beta_2F^{\mu\lambda}F_{\lambda\sigma}F^{\sigma\nu}\nonumber\\
&&\qquad+4\gamma_1 RF^{\mu\nu}+4\gamma_2(R^{[\mu}{}_\lambda F^{\nu]\lambda})
+4\gamma_3R^{\mu\nu\lambda\sigma}F_{\lambda\sigma}\bigr).\qquad
\label{eq:max}
\end{eqnarray}
The Einstein equation is somewhat cumbersome, but can be expressed in Ricci
form as
\begin{eqnarray}
&&R_{\mu\nu}+4g^2g_{\mu\nu}-\ft12F_{\mu\lambda}F_\nu{}^\lambda
+\ft1{12}g_{\mu\nu}F^2=\nonumber\\
&&\kern4em(2\alpha_1+\alpha_2+2\alpha_3)\nabla_\mu\nabla_\nu R
-(\alpha_2+4\alpha_3)\Box R_{\mu\nu}\nonumber\\
&&\kern4em-2\alpha_1RR_{\mu\nu}+4\alpha_3R_{\mu\lambda}R_\nu{}^\lambda
-2(\alpha_2+2\alpha_3)R_{\mu\lambda\nu\sigma}R^{\lambda\sigma}
-2\alpha_3R_{\mu\rho\lambda\sigma}R_\nu{}^{\rho\lambda\sigma}\nonumber\\
&&\kern4em+\ft13g_{\mu\nu}[(2\alpha_1+\alpha_2+2\alpha_3)\Box R+\alpha_1R^2
+\alpha_2R_{\lambda\sigma}^2+\alpha_3R_{\rho\lambda\sigma\delta}^2]\nonumber\\
&&\kern4em-4\beta_1F^2F_{\mu\lambda}F_\nu{}^\lambda
-4\beta_2F_{\mu\rho}F^{\rho\lambda}F_{\lambda\sigma}F^\sigma{}_\nu
+g_{\mu\nu}[\beta_1(F^2)^2+\beta_2F^4]\nonumber\\
&&\kern4em+\gamma_1(\nabla_\mu\nabla_\nu F^2-R_{\mu\nu}F^2
-2RF_{\mu\lambda}F_\nu{}^\lambda)\nonumber\\
&&\kern4em+\gamma_2(-\nabla_\lambda\nabla_{(\mu}F_{\nu)\rho}F^{\lambda\rho}
+\ft12\Box F_{\mu\lambda}F_\nu{}^\lambda+2R_{(\mu}{}^\lambda F_{\nu)}{}^\rho F_{\lambda\rho}+R_{\lambda\sigma}F_\mu{}^\lambda F_\nu{}^\sigma)\nonumber\\
&&\kern4em-\gamma_3(2\nabla^\lambda\nabla^\sigma F_{\mu\lambda}F_{\nu\sigma}+3R_{\mu\rho\lambda\sigma}F_\nu{}^\rho F^{\lambda\sigma})\nonumber\\
&&\kern4em+\ft13g_{\mu\nu}[(\gamma_1-\ft12\gamma_2)\Box F^2+2\gamma_3\nabla_\lambda\nabla_\sigma F^{\lambda\rho}F^\sigma{}_\rho\nonumber\\
&&\kern7em+2\gamma_1RF^2-2\gamma_2R_{\lambda\sigma}
F^{\lambda\rho}F^\sigma{}_\rho
+2\gamma_3R^{\rho\lambda\sigma\delta}F_{\rho\lambda}F_{\sigma\delta}].
\label{eq:eins}
\end{eqnarray}
Since we are mainly interested in obtaining corrections {\it linear} in the
parameters ($\alpha_1$, $\alpha_2$, $\alpha_3$, $\beta_1$, $\beta_2$,
$\gamma_1$, $\gamma_2$, $\gamma_3)$ of the higher derivative terms, we may
substitute the lowest order equations of motion, given by the left-hand-sides
of (\ref{eq:max}) and (\ref{eq:eins}) into the right-hand-side of
(\ref{eq:eins}) to obtain a slightly simpler form of the
Einstein equation
\begin{eqnarray}
&&R_{\mu\nu}+4g^2g_{\mu\nu}-\ft12F_{\mu\lambda}F_\nu{}^\lambda
+\ft1{12}g_{\mu\nu}F^2=\nonumber\\
&&\kern4em4g^2(5\alpha_1+\alpha_2-2\alpha_3+10\gamma_1-2\gamma_2)F_{\mu\lambda}
F_\nu{}^\lambda\nonumber\\
&&\kern4em-2\alpha_3R_{\mu\rho\lambda\sigma}R_\nu{}^{\rho\lambda\sigma}
-(\alpha_2+2\alpha_3-\gamma_2)R_{\mu\lambda\nu\sigma}F^{\lambda\rho}
F^\sigma{}_\rho-3\gamma_3R_{(\mu}{}^{\rho\lambda\sigma}F_{\nu)\rho}
F_{\lambda\sigma}\nonumber\\
&&\kern4em+\ft1{12}(2\alpha_1+\alpha_2+2\alpha_3+12\gamma_1-3\gamma_2)
\nabla_\mu\nabla_\nu F^2-\ft12(\alpha_2+4\alpha_3-\gamma_2)\Box F_{\mu\lambda}
F_\nu{}^\lambda\nonumber\\
&&\kern4em-2\gamma_3\nabla^\lambda\nabla^\sigma F_{\mu\lambda}F_{\nu\sigma}
-\ft1{12}(\alpha_1-\alpha_2+2\alpha_3+48\beta_1+8\gamma_1+2\gamma_2)
F^2F_{\mu\lambda}F_\nu{}^\lambda\nonumber\\
&&\kern4em+(\alpha_3-4\beta_2+\gamma_2)
F_{\mu\rho}F^{\rho\lambda}F_{\lambda\sigma}F^\sigma{}_\nu\nonumber\\
&&\kern4em+\ft13g_{\mu\nu}[-16g^4(5\alpha_1+\alpha_2)
-\ft23g^2(17\alpha_1+7\alpha_2+42\gamma_1-12\gamma_2)F^2\nonumber\\
&&\kern7em+\ft16(\alpha_1+2\alpha_2+7\alpha_3+6\gamma_1-3\gamma_2+3\gamma_3)
\Box F^2\nonumber\\
&&\kern7em+\ft1{144}(7\alpha_1-13\alpha_2+432\beta_1+60\gamma_1
+24\gamma_2)(F^2)^2\nonumber\\
&&\kern7em+\ft14(\alpha_2+12\beta_2-4\gamma_2)F^4
+\alpha_3R_{\rho\lambda\sigma\delta}^2+2\gamma_3R_{\rho\lambda\sigma\delta}
F^{\rho\lambda}F^{\sigma\delta}]\nonumber\\
&&\kern4em+\cdots.
\label{eq:einslin}
\end{eqnarray}
This is valid to first order in the four-derivative corrections.
Numerous previous studies higher-derivative corrections in five dimensions
have concentrated on the purely gravitational sector of the theory. In this
case, the first order Einstein equation simplifies to
\begin{equation}
R_{\mu\nu}+4g^2g_{\mu\nu}=-2\alpha_3R_{\mu\rho\lambda\sigma}
R_\nu{}^{\rho\lambda\sigma}
+\ft13g_{\mu\nu}[-16g^4(5\alpha_1+\alpha_2)
+\alpha_3R_{\rho\lambda\sigma\delta}^2].
\end{equation}
Working to this same order, we may define an effective cosmological constant
\begin{equation}
g_{\rm eff}^2=g^2[1+\ft23(10\alpha_1+2\alpha_2+\alpha_3)g^2],
\label{eq:geff}
\end{equation}
so that
\begin{equation}
R_{\mu\nu}+4g_{\rm eff}^2g_{\mu\nu}=\alpha_3(-2C_{\mu\rho\lambda\sigma}
C_\nu{}^{\rho\lambda\sigma}+\ft13g_{\mu\nu}C_{\rho\lambda\sigma\delta}^2),
\label{eq:rsqlin}
\end{equation}
where we made the substitution $R_{\mu\nu\lambda\sigma}=C_{\mu\nu\lambda\sigma}
-g^2(g_{\mu\lambda}g_{\nu\sigma}-g_{\mu\sigma}g_{\nu\lambda})+\cdots$
which is a consequence of the zeroth order Einstein equation, $R_{\mu\nu}
=-4g^2g_{\mu\nu}+\cdots.$
We see that the coefficients $\alpha_1$ and $\alpha_2$ of $R^2$ and
$R_{\mu\nu}^2$, respectively, do not enter at linear order, so long as
we use the effective cosmological constant given by $g_{\rm eff}$. This
is related to the fact that these two terms may be removed by a field
redefinition of the form $g_{\mu\nu}\to g_{\mu\nu}+a g_{\mu\nu}R+b R_{\mu\nu}$
with appropriate constants $a$ and $b$.
Although neutral black hole solutions may be obtained directly from
(\ref{eq:rsqlin}), we are mainly interested in $R$-charged solutions which
may be obtained from the full equations (\ref{eq:max}) and (\ref{eq:einslin}).
We turn to this in the next section.
\section{$R$-charged black holes}
The two-derivative Lagrangian, (\ref{eq:lag0}), admits a well-known
two-parameter family of static, stationary AdS$_5$ black hole solutions,
given by \cite{Behrndt:1998ns,Behrndt:1998jd}
\begin{eqnarray}
&&ds^2 = -H^{-2}fdt^2 + H(f^{-1}dr^2 + r^2d\Omega_{3}^{2}),\nonumber\\
&&A=\sqrt3\coth\beta\left(\fft1H-1\right)dt,
\label{eq:adsbh}
\end{eqnarray}
where the functions $H$ and $f$ are
\begin{eqnarray}
f&=&1-\fft\mu{r^2}+g^2r^2H^3,\nonumber\\
H&=&1 + \frac{\mu\sinh^2\beta}{r^2}.
\end{eqnarray}
The parameter $\mu$ is a non-extremality parameter, while $\beta$ is related
to the electric charge of the black hole.
The extremal (BPS) limit is obtained by taking $\mu\to\infty$ and $\beta\to0$ with $Q\equiv\mu\sinh^2\beta$ fixed, so that $f=1+g^2r^2H^3$ with $H=1+Q/r^2$.
These extremal solutions are naked singularities, and may be interpreted
as `superstars' \cite{Myers:2001aq}. In the absence of higher-derivative
corrections, the BPS solutions may be smoothed out by turning on angular
momentum to form true black holes
\cite{Gutowski:2004ez,Gutowski:2004yv,Chong:2005hr,Kunduri:2006ek}
\subsection{The first order solution}
We wish to find the first order corrections to the $R$-charged black hole
solution given by (\ref{eq:adsbh}). To do so, we treat the coefficients
($\alpha_1$, $\alpha_2$, $\ldots$, $\gamma_3$) of the four-derivative terms
in (\ref{eq:hdlag}) as small parameters, and make the ansatz
\begin{eqnarray}
&&ds^2 = -H^{-2}fdt^2 + H(f^{-1}dr^2 + r^2d\Omega_{3}^{2}),\nonumber\\
&&A=\sqrt3\coth\beta\left(\fft{1+a_1}H-1\right)dt,
\label{eq:ansz}
\end{eqnarray}
where
\begin{eqnarray}
H&=&1+\fft{\mu\sinh^2\beta}{r^2}+h_1,\nonumber\\
f&=&1-\fft{\mu}{r^2}+g^2r^2H^3+f_1.
\end{eqnarray}
Here, we treat $h_1$, $f_1$ and $a_1$ as small corrections, and will solve
for them to linear order in the parameters of the higher-derivative Lagrangian.
Note that this ansatz was designed so that the zeroth order equations are
automatically satisfied in the absence of $h_1$, $f_1$ and $a_1$.
Even after linearization in the small parameters, the individual equations of
motion, (\ref{eq:max}) and (\ref{eq:einslin}), yield complicated coupled
equations for the first order corrections. However, the use of certain
symmetries of these equations yields tractable equations. In particular,
the difference between the $tt$ and $rr$ components of the Einstein equation,
$R_t^t-R_r^r$, gives a second order equation involving only $h_1$, which is
easily solved. The solution for $h_1$ can then be inserted into the Maxwell
equation, (\ref{eq:max}), to obtain a solution for $a_1$. Finally, the
remaining components of the Einstein equation can be solved for $f_1$, thus
yielding the full solution. The result is
\begin{eqnarray}
\label{eq:h1soln}
h_1&=&\fft{\mu^2\sinh^22\beta}{6H_0^2r^6}\bigl(7\alpha_1 + 5\alpha_2
+ 13\alpha_3 + 42\gamma_1 -12\gamma_2 + 12\gamma_3\bigr),\\
\label{eq:a1soln}
a_1&=&\fft{\mu^2\sinh^22\beta}{6H_0^3r^6}\biggl[\bigl(7\alpha_1 + 5\alpha_2
+13\alpha_3+42\gamma_1-12\gamma_2-12\gamma_3\tanh^2\beta\bigr)\nonumber\\
&&\kern6em +\fft{\mu\sinh^2\beta}{2r^2}\bigl(7\alpha_1+5\alpha_2+13\alpha_3
\nonumber\\
&&\kern11em+24(6\beta_1+3\beta_2+2\gamma_1-\gamma_2
+\gamma_3(1+\mathrm{sech}^2\beta))\bigr)\biggr],\qquad\\
\label{eq:f1soln}
f_1&=&\ft23g^4\bigl(10\alpha_1+2\alpha_2+\alpha_3\bigr)r^2H_0^3\nonumber\\
&&+\frac{g^2\mu^2\sinh^22\beta}{r^4}
\bigl(10\alpha_1-\alpha_2-13\alpha_3+20\gamma_1-\gamma_2-6\gamma_3\bigr)
\nonumber\\
&&+\fft{\mu^2}{r^6H_0}\left[\sinh^22\beta\bigl(3\alpha_1-\alpha_3
+18\gamma_1-3\gamma_2\bigr)+2\alpha_3\right]\nonumber\\
\nonumber\\
&&-\fft{\mu^3\sinh^22\beta\cosh^22\beta}{2r^8H_0^2}
\bigl(5\alpha_1+\alpha_2+\alpha_3+30\gamma_1-6\gamma_2\bigr)\nonumber\\
&&+\fft{\mu^4\sinh^42\beta}{96r^{10}H_0^3}
\bigl(47\alpha_1+13\alpha_2+17\alpha_3-144\beta_1-72\beta_2+276\gamma_1
-48\gamma_2-24\gamma_3\bigr),\nonumber\\
\end{eqnarray}
where $H_0=1+\mu\sinh^2\beta/r^2$ is the zeroth order solution for $H$.
(Since $h_1$, $a_1$ and $f_1$ are already linear in the parameters of the
higher order corrections, we may use $H$ and $H_0$ interchangeably in
the above expressions.) Note that the first line in $f_1$ reproduces the
shift of the cosmological constant $g^2\to g_{\rm eff}^2$ given in
(\ref{eq:geff}). This allows us to write
\begin{equation}
f=1-\fft\mu{r^2}+g_{\rm eff}^2r^2H^3+\bar f_1,
\label{eq:feff}
\end{equation}
where $\bar f_1$ is given by the remaining terms in (\ref{eq:f1soln}).
In obtaining the above solution, we have imposed the boundary conditions
that $h_1$ and $a_1$ both fall off faster than $1/r^2$ as $r\to\infty$ so
that the $R$-charge is not modified from its zeroth order value. For
$f_1$, the boundary condition is taken as (\ref{eq:feff}), with $\bar f_1$
falling off faster than $1/r^2$.
\section{Field Redefinitions}
As given in (\ref{eq:hdlag}), we have parameterized the four-derivative terms
in the Lagrangian in terms of the eight coefficients ($\alpha_1$, $\alpha_2$,
$\ldots$, $\gamma_3$). However, not all of these coefficients are physical.
This is because some of the terms in the higher derivative Lagrangian can be
removed by field redefinition.
To proceed, we consider transformations of the form
\begin{eqnarray}
g_{\mu\nu}&\to&g_{\mu\nu}+a(R+20g^2)g_{\mu\nu}+b(R_{\mu\nu}+4g^2g_{\mu\nu})
+cF_{\mu\lambda}F^\lambda{}_\nu + d F^2g_{\mu\nu},\nonumber\\
A_\mu&\to&(1+g^2(25a+5b-12c+60d)) A_\mu.
\label{eq:fredef}
\end{eqnarray}
Note that the first two terms in the metric shift incorporate the cosmological
constant; this corresponds to the zeroth order Einstein equation in the absence
of gauge excitations. While this shift by the cosmological constant is not
strictly speaking necessary in performing the field redefinition, we
nevertheless find it convenient, as this avoids a shift in the effective
cosmological constant $g_{\rm eff}$ after the field redefinition. In addition,
the scaling of the gauge field is chosen so that it will remain canonically
normalized after the shift of the metric. The result of this transformation
is to shift the original Lagrangian (\ref{eq:lhd}) into
\begin{eqnarray}
e^{-1}\mathcal{L} &=& \left(1+12g^2(5a+b)\right)\biggl[R-\ft14F_{\mu\nu}^2
+ 12g^2\left(1-2g^2(5a+b)\right)\nonumber\\
&&\kern8em+\ft1{12\sqrt3}\left(1+3g^2(5a+b-12c+60d)\right)
\epsilon^{\mu\nu\rho\lambda\sigma}F_{\mu\nu}F_{\rho\lambda}A_\sigma\nonumber\\
&&\kern8em+\left(\alpha_1+\ft12(3a+b)\right)R^2+(\alpha_2-b)R_{\mu\nu}R^{\mu\nu}
+\alpha_3R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\nonumber\\
&&\kern8em+\left(\beta_1+\ft18(c-d)\right)(F_{\mu\nu}F^{\mu\nu})^2
+(\beta_2-\ft12c)
F^{\mu}{}_{\nu}F^{\nu}{}_{\rho}F^{\rho}{}_{\sigma}F^{\sigma}{}_{\mu}
\nonumber\\
&&\kern8em+\left(\gamma_1-\ft18(a+b+4c-12d)\right)RF^2\nonumber\\
&&\kern8em+\left(\gamma_2-\ft12(b+2c)\right)R_{\mu\nu}
F^{\mu\rho}F_{\rho}{}^{\nu} +
\gamma_3R_{\mu\nu\rho\sigma}F^{\mu\nu}F^{\rho\sigma}\biggr],
\end{eqnarray}
where, as usual, we only work to linear order in the shift parameters
($a$, $b$, $c$, $d$).
Up to an overall rescaling, this new Lagrangian can almost be brought back
to the original form, provided we shift the various coefficients as follows:
\begin{eqnarray}
&&g^2\to g^2\left(1+2g^2(5a+b)\right),\nonumber\\
&&\alpha_1\to\alpha_1-\ft12(3a+b),\kern6.1em\alpha_2\to\alpha_2+b,
\kern5.2em\alpha_3\to\alpha_3,\nonumber\\
&&\beta_1\to\beta_1-\ft18(c-d),\kern6.8em\beta_2\to\beta_2+\ft12c,\nonumber\\
&&\gamma_1\to\gamma_1+\ft18(a+b+4c-12d),\qquad\gamma_2\to\gamma_2+\ft12(b+2c),\qquad\gamma_3\to\gamma_3.
\label{eq:frtrans}
\end{eqnarray}
One difference remains, however, and that is the coefficient of the
$F\wedge F\wedge A$ Chern-Simons term. This suggests that, when considering
higher derivative corrections in gauged supergravity, there is in fact a
preferred field redefinition frame where this Chern-Simons term remains
uncorrected. (Such a preferred frame also shows up when considering the
supersymmetric completion of the mixed $\mbox{Tr}\,R\wedge R\wedge A$ term
\cite{Hanaki:2006pj}.) This $F\wedge F\wedge A$ term is unimportant, however,
for the spherically symmetric $R$-charged black holes considered above in
Section~3.
Ignoring the $F\wedge F\wedge A$ term, the freedom to perform field
redefinitions of the form (\ref{eq:fredef}) indicates that at most four
of the eight coefficients of the higher derivative terms will be physical.
Clearly $\alpha_3$ and $\gamma_3$ are physical, as they cannot be removed
by the transformation of (\ref{eq:frtrans}). The additional two physical
coefficients can be taken to be a linear combination of
\begin{equation}
\hat\beta_1\equiv\beta_1+\ft1{144}(\alpha_1-7\alpha_2)
+\ft1{12}(\gamma_1+\gamma_2)
\qquad\hbox{and}\qquad
\hat\beta_2\equiv\beta_2+\ft14\alpha_2-\ft12\gamma_2.
\label{eq:hb1b2}
\end{equation}
In addition, although $g^2$ is shifted by the field redefinition, the physical
cosmological constant, $g^2_{\rm eff}$, as defined in (\ref{eq:geff}),
remains invariant.
The use of field redefinitions allows us to rewrite the four-derivative
Lagrangian in various forms. A common choice would be to use the Gauss-Bonnet
combination $R^2-4R_{\mu\nu}^2+R_{\mu\nu\lambda\sigma}^2$ for the
curvature-squared terms. This system has been extensively studied in
the absence of higher-derivative gauge field corrections, and has the feature
that it admits {\it exact} spherically symmetric black hole solutions, both
without \cite{Boulware:1985wk,Wheeler:1985nh} and with \cite{Wiltshire:1985us}
$R$-charge. An alternate choice, which is perhaps more natural
from a supersymmetric point of view \cite{Hanaki:2006pj}, would be to use the
Weyl-squared combination $C_{\mu\nu\lambda\sigma}^2=\fft16R^2
-\fft43R_{\mu\nu}^2+R_{\mu\nu\lambda\sigma}^2$. Either one of these choices
would fix two of the coefficients ({\it i.e.}~$\alpha_1$ and $\alpha_2$ in
terms of $\alpha_3$). The additional freedom to perform field redefinitions
may then be used to eliminate the mixed $RF^2$ and $R_{\mu\nu}F^{\mu\lambda}
F^\lambda{}^\nu$ terms parameterized by $\gamma_1$ and $\gamma_2$.
\subsection{Field redefinitions and the first order solution}
Given the above field redefinition, it is instructive to examine its effect
on the first order black hole solution of (\ref{eq:h1soln}), (\ref{eq:a1soln})
and (\ref{eq:f1soln}). In this case, it is straightforward to see that
the coefficient shift of (\ref{eq:frtrans}) results in
\begin{eqnarray}
h_1&\to&\tilde h_1=h_1+\fft{\mu^2\sinh^22\beta}{8H_0^2r^6}
(-7a+b+12c-84d),\nonumber\\
a_1&\to&\tilde a_1=a_1+\fft{\mu^2\sinh^22\beta}{8H_0^3r^6}\left[
(-7a+b+12c-84d)-\fft{3\mu\sinh^2\beta}{r^2}(a+b-4c+12d)\right],\nonumber\\
f_1&\to&\tilde f_1=f_1-2g^4(5a+b)r^2H_0^3-\fft{g^2\mu^2\sinh^22\beta}{2r^4}
(25a+8b-18c+60d)\nonumber\\
&&\kern4em
+\fft{3\mu^2\sinh^22\beta}{8r^6H_0}\biggl[-2(3a+b-8c+36d)
+\fft{\mu\cosh^22\beta}{r^2H_0}(5a+b-12c+60d)\nonumber\\
&&\kern11em-\fft{\mu^2\sinh^22\beta}{r^4H_0^2}(a-2c+12d)\biggr].
\label{eq:cssol}
\end{eqnarray}
At first, this result may appear somewhat surprising. After all, this
field redefinition is supposed to be `unphysical', and yet the form of the
solution has changed. The resolution of this puzzle lies in the fact that
the we have shifted the metric by terms that are not necessarily proportional
to the lowest order equations of motion. (While we have taken care to
incorporate the cosmological constant in (\ref{eq:fredef}), we have omitted
the gauge field stress tensor in the shift.) In this sense, while the
original and shifted metrics both solve the equations of motion, they
nevertheless correspond to physically distinct solutions. The field
redefinition of (\ref{eq:fredef}) is then more naturally thought of as
a mapping between solutions.
More explicitly, we note that the shift of the metric given in
(\ref{eq:fredef}) takes the black hole solution away from the form of
the initial ansatz given by (\ref{eq:ansz}). In particular, shifting
the metric by (\ref{eq:fredef}) and using the zeroth order
solution gives
\begin{eqnarray}
g_{tt}&\to&\tilde g_{tt}=g_{tt}\left[1 - \fft{\mu^2\sinh^22\beta}{2r^6H_0^3}
(a + 2b -6c+12d)\right],\nonumber\\
g_{rr}&\to&\tilde g_{rr}=g_{rr}\left[1 - \fft{\mu^2\sinh^22\beta}{2r^6H_0^3}
(a + 2b -6c+12d)\right],\nonumber\\
g_{\alpha\beta}&\to&\tilde g_{\alpha\beta}=
g_{\alpha\beta}\left[1-\fft{\mu^2\sinh^22\beta}{2r^6H_0^3}
(a-b+12d)\right],
\end{eqnarray}
where $\alpha$ and $\beta$ refer to coordinates on $S^3$. It is now
possible to see that a coordinate transformation $r\to\tilde r$ is
necessary in order to restore the canonical form of the shifted metric.
By identifying
\begin{eqnarray}
d\tilde s^2 &=& \tilde{g}_{tt}dt^2 +
\tilde{g}_{rr}dr^2 +
\tilde{g}_{\theta\theta}d\Omega_3^2\nonumber\\
&=&-\tilde H^{-2}\tilde fdt^2+\tilde H(\tilde f^{-1}d\tilde r^2+\tilde r^2
d\Omega_3^2),
\end{eqnarray}
we end up with expressions for $\tilde H$ and $\tilde f$
\begin{equation}
\tilde H=\fft{\tilde g_{\theta\theta}}{\tilde r^2},\qquad
\tilde f=-\tilde g_{tt}\tilde g_{\theta\theta}^2{\tilde r^4},
\label{eq:hftmet}
\end{equation}
as well as a differential equation relating $\tilde r^2$ with $r^2$
\begin{equation}
\fft{d(\tilde r^2)}{d(r^2)}=\fft{\tilde g_{tt}\tilde g_{rr}
\tilde g_{\theta\theta}}{r^2}.
\end{equation}
Note that, in defining the angular coordinate $\theta$, we have taken the
metric on the unit $S^3$ to be of the form $d\Omega_3^2=d\theta^2+\cdots.$
The equation for $\tilde r^2$ is easily solved, and yields the relation
\begin{equation}
\tilde{r}^2 = r^2\left[1 +
\fft{3\mu^2\sinh^22\beta}{8r^6H_0^2}(3a +3b-12c+36d)\right],
\label{eq:rtrel}
\end{equation}
where we have set a possible integration constant to zero to preserve the
$r\to\infty$ asymptotics.
We are now able to explicitly compute the shifted metric functions $\tilde h_1$
and $\tilde f_1$ as well as the shifted gauge potential $\tilde a_1$. For
$\tilde h_1$, we use the definition
\begin{equation}
\tilde H=1+\fft{\mu\sinh^2\beta}{\tilde r^2}+\tilde h_1,
\end{equation}
along with (\ref{eq:hftmet}) and (\ref{eq:rtrel}) to obtain
\begin{equation}
\tilde h_1=h_1+\fft{\mu^2\sinh^22\beta}{8H_0^2r^6}(-7a+b+12c-84d),
\end{equation}
which is in perfect agreement with (\ref{eq:cssol}). For $\tilde f_1$, on
the other hand, we find
\begin{eqnarray}
\tilde f_1&=&f_1-2g^4(5a+b)r^2H_0^3
-\fft{3g^2\mu^2\sinh^22\beta}{2r^4}(b-2c)\nonumber\\
&&\qquad+\fft{3\mu^2\sinh^22\beta}{8r^6H_0}\biggl[
-2(3a+b-8c+36d)+\fft{\mu\cosh2\beta}{r^2H_0}(5a+b-12c+60d)\nonumber\\
&&\kern9em-\fft{\mu^2\sinh^22\beta}{r^4H_0^2}(a-2c+12d)\biggr].
\label{eq:ftcomp}
\end{eqnarray}
Note that we have defined $\tilde f_1$ by
\begin{equation}
\tilde f=1-\fft\mu{\tilde r^2}+\tilde g^2\tilde r^2\tilde H^3+\tilde f_1,
\label{eq:ftdef1}
\end{equation}
where $\tilde g^2=g^2(1+2g^2(5a+b))$ is the shifted cosmological constant
given in (\ref{eq:frtrans}).
Comparison of (\ref{eq:ftcomp}) with (\ref{eq:cssol}) clearly demonstrates
a difference in the $\mathcal O(g^2)$ term. The origin of this difference
is somewhat subtle, and is related to the choice of boundary conditions for
the shifted and unshifted solutions. To see this, we recall that the
gauge potential $A_\mu$ is also shifted by the field redefinition
(\ref{eq:fredef}) so that it maintains canonical normalization. The
implication of this shift on the black hole solution is that
\begin{equation}
A_t\to \bigl(1+g^2(25a+5b-12c+60d)\bigr)A_t,
\end{equation}
where
\begin{equation}
A_t=\sqrt3\coth\beta\left(\fft{1+a_1}H-1\right),\qquad
H=1+\fft{\mu\sinh^2\beta}{r^2}+h_1.
\end{equation}
In order to rescale the potential without adding any $\mathcal O(1/r^2)$
terms to $H_0$, $h_1$ or $a_1$, we must instead shift the two parameters
$\mu$ and $\beta$ of the black hole according to
\begin{equation}
\coth\beta\to\coth\beta\bigl(1+g^2(25a+5b-12c+60d)\bigr),\qquad
\mu\sinh^2\beta\to\mu\sinh^2\beta.
\label{eq:mubres}
\end{equation}
This corresponds to a rescaling of the nonextremality parameter $\mu$
\begin{equation}
\mu\to\tilde\mu=\mu(1+2g^2\cosh^2\beta(25a+5b-12c+60d)).
\end{equation}
In this case, the shifted metric function $\tilde f$, given in
(\ref{eq:ftdef1}), ought to more properly be written as
\begin{equation}
\tilde f=1-\fft{\tilde\mu}{\tilde r^2}+\tilde g^2\tilde r^2\tilde H^3
+\hat f_1,
\end{equation}
where
\begin{eqnarray}
\hat f_1&=&\tilde f_1+\fft{2g^2\mu\cosh^2\beta}{r^2}(25a+5b-12c+60d)
\nonumber\\
&=&f_1+\lambda\fft{H_0}{r^2}-2g^4(5a+b)r^2H_0^3
-\fft{g^2\mu^2\sinh^22\beta}{2r^4}(25a+8b-18c+60d)+\cdots.\nonumber\\
\end{eqnarray}
This now agrees with $\tilde f_1$ of (\ref{eq:cssol}) up to a solution
$\lambda H_0/r^2$ to the homogeneous differential equation for $f_1$, where
\begin{equation}
\lambda=2g^2\mu\cosh^2\beta(25a+5b-12c+60d).
\end{equation}
This is a modification of the $\mathcal O(1/r^2)$ term in $f_1$, which,
however, is subdominant in $f$, as the leading behavior of $f$ is given by
$f\sim g^2_{\rm eff}r^2$ for an asymptotically Anti-de Sitter background.
Finally, we may follow the effect of the field redefinition (\ref{eq:fredef})
on the gauge potential term $a_1$. Given the $\mu$ and $\beta$ rescaling of
(\ref{eq:mubres}), we obtain
\begin{equation}
\tilde a_1=(1+a_1)\fft{\tilde H}H-1.
\end{equation}
Working out the right hand side of this expression, we find that it
agrees with (\ref{eq:cssol}). We have thus seen that the first
order solution for the spherically symmetric $R$-charged black hole indeed
transforms as expected under field redefinitions.
\section{Discussion}
While we have considered general field redefinitions given by four
parameters ($a$, $b$, $c$, $d$), a preferred subset of this would be to
shift the metric by the full zeroth order equation of motion
\begin{equation}
R_{\mu\nu}+4g^2g_{\mu\nu}-\ft12F_{\mu\lambda}F_\nu{}^\lambda
+\ft1{12}g_{\mu\nu}F^2.
\end{equation}
In the above notation, this corresponds to taking
\begin{equation}
c=\ft12b,\qquad d=-\ft1{12}(a-b).
\end{equation}
In this case, we may redefine the coefficients of the higher derivative
terms according to
\begin{eqnarray}
\beta_1&=&\hat\beta_1-\ft1{12}(\hat\gamma_1+\hat\gamma_2)
+\ft1{144}(\alpha_1-7\alpha_2),\nonumber\\
\beta_2&=&\hat\beta_2+\ft12\hat\gamma_2+\ft14\alpha_2,\nonumber\\
\gamma_1&=&\hat\gamma_1-\ft16(\alpha_1-\alpha_2),\nonumber\\
\gamma_2&=&\hat\gamma_2+\alpha_2,
\end{eqnarray}
so that the set ($\alpha_3$, $\hat\beta_1$, $\hat\beta_2$, $\hat\gamma_1$,
$\hat\gamma_2$, $\gamma_3$) are invariant under the restricted field
redefinitions. Note that $\hat\beta_1$ and $\hat\beta_2$ are the physical
coefficients previously defined in (\ref{eq:hb1b2}).
It is illuminating to rewrite the higher derivative Lagrangian (\ref{eq:lhd})
in terms of the new parameters. Ignoring the Chern-Simons term, the result is
\begin{eqnarray}
\label{eq:plag}
e^{-1}\mathcal L&=&\bigl(1-8g^2(5\alpha_1+\alpha_2)\bigr)\biggl[
R-\ft14\hat F^2+12g_{\rm eff}^2+\alpha_1\mathcal E^2
+\alpha_2\mathcal E_{\mu\nu}^2
+\alpha_3\bigl(R_{\mu\nu\lambda\sigma}^2-8g^4\bigr)\nonumber\\
&&\kern6em+\hat\beta_1(\hat F^2)^2+\hat\beta_2\hat F^4
+\hat\gamma_1\mathcal E\hat F^2-\hat\gamma_2\mathcal E_{\mu\nu}
\hat F^{\mu\sigma}\hat F^\nu{}_\sigma+\gamma_3R^{\mu\nu\lambda\sigma}
\hat F_{\mu\nu}\hat F_{\lambda\sigma}\biggr],\nonumber\\
\end{eqnarray}
where
\begin{equation}
\mathcal E_{\mu\nu}\equiv R_{\mu\nu}+4g^2g_{\mu\nu}
-\ft12\hat F_{\mu\lambda}\hat F_\nu{}^\lambda+\ft1{12}g_{\mu\nu}\hat F^2,\qquad
\mathcal E=\mathcal E^\mu{}_\mu
\end{equation}
is the zeroth order equation of motion. Note that we have worked to linear
order in pulling out the overall factor $1-8g^2(5\alpha_1+\alpha_2)$
renormalizing Newton's constant. Furthermore, $\hat F=d\hat A$ is
a rescaled field strength defined by
\begin{equation}
\hat A_\mu=\left[1+8g^2\bigl(\ft13(5\alpha_1+\alpha_2)+5\hat\gamma_1
-\hat\gamma_2\bigr)\right]A_\mu,
\end{equation}
so that $\hat A_\mu$ remains invariant under the field redefinition of
(\ref{eq:fredef}). The structure of (\ref{eq:plag}) now clearly demonstrates
that, of the four-derivative terms, only those parameterized by
($\alpha_3$, $\hat\beta_1$, $\hat\beta_2$, $\gamma_3$) are physical, as the
remaining terms are manifestly proportional to the zeroth order equation of
motion.
In principle, the choice of field redefinitions allows us to go back and forth
between the Gauss-Bonnet and Weyl-squared parameterizations of the
higher-derivative terms in the Lagrangian. In this sense, it is perhaps not
a complete surprise to see that in some cases both parameterizations yield
the same results for the entropy of BPS black holes
\cite{Guica:2005ig,Castro:2007hc,Cvitan:2007en}, even though the bare
Gauss-Bonnet correction is not supersymmetric in itself. (Of course, the
bare Weyl-squared term is not supersymmetric by itself either.) What this
suggests is that the Riemann-squared term parameterized by $\alpha_3$ plays
a crucial and perhaps dominant role in the geometry of higher-derivative
black holes, and that the additional matter and auxiliary field terms may
contribute only indirectly through their effects on the geometry, at least
in the BPS case where there is additional symmetry at the horizon.
Finally, given the general higher-derivative corrected $R$-charged black holes,
it would be interesting to study their thermodynamics and hydrodynamics. One
outcome of this study ought to be a clear identification of physical versus
unphysical parameters of the theory. In particular, in the parameterization of
(\ref{eq:plag}), we would expect all dependence on ($\alpha_1$, $\alpha_2$,
$\hat\gamma_1$, $\hat\gamma_2$) to drop out of the thermodynamical quantities.
One difficulty in exploring the higher-derivative theory is that some care
must be taken in generalizing the Gibbons-Hawking surface term (which we have
ignored throughout this letter). This is because the general ({\it i.e.}~non
Gauss-Bonnet) combination of $R^2$ terms leads to higher than second-derivative
terms in the equations of motion, and hence necessitates specifying additional
boundary data \cite{Myers:1987yn}. As demonstrated in \cite{Buchel:2004di},
one way around this is to perturb in the higher-derivative terms and to
demand that the undesired boundary variations vanish when the lowest-order
equations of motion are imposed. We are currently applying this procedure
to the general parameterized four-derivative Lagrangian with a goal of
exploring higher-derivative black hole thermodynamics using holographic
renormalization.
\section*{Acknowledgments}
We wish to thank A. Castro, J. Davis and K. Hanaki for useful conversations.
This work was supported in part by the US Department of Energy under
grant DE-FG02-95ER40899.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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\section*{Acknowledgements}
This work was supported in part by the Italian Ministry of Research (MIUR) under grant PRIN 20172LNEEZ.
The work of J.B. has been supported by the FEDER/Junta de Andaluc\'ia project grant P18-FRJ-3735. The work of L.R. has been supported by the U.S. Department of Energy under grant DE-SC0010102.
\section*{Appendix on the \emph{conservative average} scenario}
In this appendix we present the results of our analysis in the \emph{conservative average} scenario for $m_t$ and $M_W$. Figure \ref{fig:mtmwsm_C} presents the posteriors for different fits in the $m_t$ vs $M_W$ and $\sin^2\theta_{\rm eff}^{\rm lept}$ vs $M_W$ planes in the SM. Results of SM fits are reported in Table \ref{tab:SM_c}, while Figure~\ref{fig:STU_C} and Table~\ref{tab:STU_C} present results obtained in the scenario with dominant oblique NP contributions, and Table~\ref{tab:SMEFT fit_c} presents the corresponding results for the SMEFT. Posteriors for all EWPO in the NP scenarios considered are reported in Table~\ref{tab:NP_fits_c}.
\begin{figure*}[htb]
\centering
\includegraphics[width=.425\textwidth]{MW_vs_mt_C.pdf}
\includegraphics[width=.425\textwidth]{MW_vs_sin2Eff_C.pdf}
\caption{Same as Figure~\ref{fig:mtmwsm} in the \emph{conservative average} scenario.}
\label{fig:mtmwsm_C}
\end{figure*}
\begin{figure*}[ht!]
\centering
\includegraphics[width=0.245\textwidth]{ST_C-each.pdf}
\includegraphics[width=0.245\textwidth]{STU_ST_C.pdf}
\includegraphics[width=0.245\textwidth]{STU_SU_C.pdf}
\includegraphics[width=0.245\textwidth]{STU_TU_C.pdf}
\caption{Same as Figure~\protect\ref{fig:STU} for the \emph{conservative average} scenario.}
\label{fig:STU_C}
\end{figure*}
\newpage
\begin{table*}[p!]
\centering\resizebox{\textwidth}{!}{
\begin{tabular}{l|c|c|cr|cr|cr}
\toprule
& Measurement & Posterior & Indirect/Prediction & Pull & Full Indirect & Pull & Full Prediction & Pull \\
\hline
$\alpha_{s}(M_{Z})$ & $ 0.1177 \pm 0.0010 $ & $ 0.11786 \pm 0.00095 $ & $ 0.11930 \pm 0.00281 $ & $ -0.5 $ & $ 0.12174 \pm 0.00473 $ & $ -0.8 $ & $ 0.1177 \pm 0.0010 $ &$ - $ \\
& & $[ 0.11603 , 0.11972 ]$ & $[ 0.11371 , 0.12482 ]$ & & $[ 0.1126 , 0.1311 ]$ & & $[ 0.1157 , 0.1197 ]$ \\
$\Delta\alpha^{(5)}_{\mathrm{had}}(M_Z) $& $ 0.02766 \pm 0.00010 $ & $ 0.027614 \pm 0.000097 $ & $ 0.026895 \pm 0.000394 $ & $ 1.9 $ & $ 0.027987 \pm 0.000699 $ & $ -0.5 $ & $ 0.02766 \pm 0.00010 $ &$ - $ \\
& & $[ 0.027422 , 0.027804 ]$ & $[ 0.026123 , 0.027677 ]$ & & $[ 0.02661 , 0.02935 ]$ & & $[ 0.02747 , 0.02786 ]$ \\
$M_Z$ [GeV] & $ 91.1875 \pm 0.0021 $ & $ 91.1887 \pm 0.0021 $ & $ 91.2227 \pm 0.0105 $ & $ -3.3 $ & $ 91.2111 \pm 0.0390 $ & $ -0.6 $ & $ 91.1875 \pm 0.0021 $ &$ - $ \\
& & $[ 91.1847 , 91.1927 ]$ & $[ 91.2024 , 91.2434 ]$ & & $[ 91.135 , 91.289 ]$ & & $[ 91.1834 , 91.1916 ]$ \\
$m_t$ [GeV] & $ 171.8 \pm 1.0 $ & $ 173.12 \pm 0.92 $ & $ 180.10 \pm 2.25 $ & $ -3.3 $ & $ 187.16 \pm 9.83 $ & $ -1.6 $ & $ 171.8 \pm 1.0 $ &$ - $ \\
& & $[ 171.30 , 174.92 ]$ & $[ 175.66 , 184.55 ]$ & & $[ 167.9 , 206.4 ]$ & & $[ 169.8 , 173.8 ]$ \\
$m_H$ [GeV] & $ 125.21 \pm 0.12 $ & $ 125.21 \pm 0.12 $ & $ 102.19 \pm 9.79 $ & $ 1.9 $ & $ 245.25 \pm 125.35 $ & $ -0.9 $ & $ 125.21 \pm 0.12 $ &$ - $ \\
& & $[ 124.97 , 125.45 ]$ & $[ 87.01 , 127.30 ]$ & & $[ 98.1 , 640.4 ]$ & & $[ 124.97 , 125.45 ]$ \\
\hline
$M_W$ [GeV] & $ 80.413 \pm 0.015 $ & $ 80.3634 \pm 0.0068 $ & $ 80.3505 \pm 0.0077 $ & $ 3.7 $ & $ 80.4116 \pm 0.0146 $ & $ 0.0 $ & $ 80.3497 \pm 0.0079 $ &$ 3.7 $ \\
& & $[ 80.3500 , 80.3769 ]$ & $[ 80.3355 , 80.3655 ]$ & & $[ 80.383 , 80.440 ]$ & & $[ 80.3342 , 80.3653 ]$ \\
\hline
$\Gamma_{W}$ [GeV] & $ 2.085 \pm 0.042 $ & $ 2.08859 \pm 0.00066 $ & $ 2.08859 \pm 0.00066 $ & $ -0.1 $ & $ 2.09426 \pm 0.00245 $ & $ -0.2 $ & $ 2.08743 \pm 0.00073 $ &$ 0.0 $ \\
& & $[ 2.08731 , 2.08988 ]$ & $[ 2.08732 , 2.08988 ]$ & & $[ 2.0894 , 2.0990 ]$ & & $[ 2.08601 , 2.08889 ]$ \\
\hline
$\sin^2\theta_{\rm eff}^{\rm lept}(Q_{\rm FB}^{\rm had})$ & $ 0.2324 \pm 0.0012 $ & $ 0.231491 \pm 0.000059 $ & $ 0.231490 \pm 0.000059 $ & $ 0.8 $ & $ 0.231461 \pm 0.000136 $ & $ 0.8 $ & $ 0.231558 \pm 0.000068 $ &$ 0.7 $ \\
& & $[ 0.231376 , 0.231608 ]$ & $[ 0.231374 , 0.231607 ]$ & & $[ 0.23119 , 0.23173 ]$ & & $[ 0.231426 , 0.231691 ]$ \\
\hline
$P_{\tau}^{\rm pol}=\mathcal{A}_\ell$ & $ 0.1465 \pm 0.0033 $ & $ 0.14725 \pm 0.00046 $ & $ 0.14727 \pm 0.00047 $ & $ -0.2 $ & $ 0.14750 \pm 0.00108 $ & $ -0.3 $ & $ 0.14674 \pm 0.00053 $ &$ -0.1 $ \\
& & $[ 0.14634 , 0.14817 ]$ & $[ 0.14635 , 0.14820 ]$ & & $[ 0.1454 , 0.1496 ]$ & & $[ 0.14570 , 0.14779 ]$ \\
\hline
$\Gamma_{Z}$ [GeV] & $ 2.4955 \pm 0.0023 $ & $ 2.49453 \pm 0.00066 $ & $ 2.49434 \pm 0.00070 $ & $ 0.5 $ & $ 2.49528 \pm 0.00205 $ & $ 0.1 $ & $ 2.49396 \pm 0.00072 $ &$ 0.6 $ \\
& & $[ 2.49324 , 2.49584 ]$ & $[ 2.49295 , 2.49572 ]$ & & $[ 2.4912 , 2.4993 ]$ & & $[ 2.49257 , 2.49538 ]$ \\
$\sigma_{h}^{0}$ [nb] & $ 41.480 \pm 0.033 $ & $ 41.4908 \pm 0.0077 $ & $ 41.4929 \pm 0.0080 $ & $ -0.4 $ & $ 41.4616 \pm 0.0304 $ & $ 0.4 $ & $ 41.4924 \pm 0.0080 $ &$ -0.4 $ \\
& & $[ 41.4757 , 41.5059 ]$ & $[ 41.4772 , 41.5087 ]$ & & $[ 41.402 , 41.522 ]$ & & $[ 41.4767 , 41.5083 ]$ \\
$R^{0}_{\ell}$ & $ 20.767 \pm 0.025 $ & $ 20.7491 \pm 0.0080 $ & $ 20.7458 \pm 0.0086 $ & $ 0.8 $ & $ 20.7589 \pm 0.0218 $ & $ 0.2 $ & $ 20.7470 \pm 0.0087 $ &$ 0.8 $ \\
& & $[ 20.7333 , 20.7649 ]$ & $[ 20.7287 , 20.7627 ]$ & & $[ 20.716 , 20.802 ]$ & & $[ 20.7297 , 20.7638 ]$ \\
$A_{\rm FB}^{0, \ell}$ & $ 0.0171 \pm 0.0010 $ & $ 0.01626 \pm 0.00010 $ & $ 0.01625 \pm 0.00010 $ & $ 0.8 $ & $ 0.01631 \pm 0.00024 $ & $ 0.8 $ & $ 0.01615 \pm 0.00012 $ &$ 1.0 $ \\
& & $[ 0.01606 , 0.01647 ]$ & $[ 0.01605 , 0.01646 ]$ & & $[ 0.01585 , 0.01679 ]$ & & $[ 0.01592 , 0.01638 ]$ \\
\hline
$\mathcal{A}_{\ell}$ (SLD) & $ 0.1513 \pm 0.0021 $ & $ 0.14725 \pm 0.00046 $ & $ 0.14728 \pm 0.00049 $ & $ 1.9 $ & $ 0.14750 \pm 0.00108 $ & $ 1.6 $ & $ 0.14674 \pm 0.00053 $ &$ 2.1 $ \\
& & $[ 0.14634 , 0.14817 ]$ & $[ 0.14632 , 0.14824 ]$ & & $[ 0.1454 , 0.1496 ]$ & & $[ 0.14570 , 0.14779 ]$ \\
$R^{0}_{b}$ & $ 0.21629 \pm 0.00066 $ & $ 0.21587 \pm 0.00010 $ & $ 0.21586 \pm 0.00011 $ & $ 0.7 $ & $ 0.21542 \pm 0.00037 $ & $ 1.2 $ & $ 0.21591 \pm 0.00011 $ &$ 0.6 $ \\
& & $[ 0.21566 , 0.21607 ]$ & $[ 0.21565 , 0.21607 ]$ & & $[ 0.21467 , 0.21613 ]$ & & $[ 0.21570 , 0.21611 ]$ \\
$R^{0}_{c}$ & $ 0.1721 \pm 0.0030 $ & $ 0.172210 \pm 0.000054 $ & $ 0.172210 \pm 0.000054 $ & $ 0.0 $ & $ 0.172400 \pm 0.000185 $ & $ -0.1 $ & $ 0.172190 \pm 0.000055 $ &$ -0.1 $ \\
& & $[ 0.172102 , 0.172316 ]$ & $[ 0.172103 , 0.172317 ]$ & & $[ 0.17205 , 0.17277 ]$ & & $[ 0.172082 , 0.172297 ]$ \\
$A_{\rm FB}^{0, b}$ & $ 0.0996 \pm 0.0016 $ & $ 0.10324 \pm 0.00033 $ & $ 0.10325 \pm 0.00035 $ & $ -2.2 $ & $ 0.10338 \pm 0.00076 $ & $ -2.1 $ & $ 0.10287 \pm 0.00037 $ &$ -2.0 $ \\
& & $[ 0.10259 , 0.10388 ]$ & $[ 0.10258 , 0.10393 ]$ & & $[ 0.10188 , 0.10489 ]$ & & $[ 0.10214 , 0.10361 ]$ \\
$A_{\rm FB}^{0, c}$ & $ 0.0707 \pm 0.0035 $ & $ 0.07377 \pm 0.00024 $ & $ 0.07377 \pm 0.00026 $ & $ -0.9 $ & $ 0.07391 \pm 0.00059 $ & $ -0.9 $ & $ 0.07348 \pm 0.00028 $ &$ -0.8 $ \\
& & $[ 0.07328 , 0.07425 ]$ & $[ 0.07327 , 0.07428 ]$ & & $[ 0.07275 , 0.07507 ]$ & & $[ 0.07293 , 0.07403 ]$ \\
$\mathcal{A}_b$ & $ 0.923 \pm 0.020 $ & $ 0.934746 \pm 0.000040 $ & $ 0.934746 \pm 0.000040 $ & $ -0.6 $ & $ 0.934594 \pm 0.000169 $ & $ -0.6 $ & $ 0.934721 \pm 0.000041 $ &$ -0.6 $ \\
& & $[ 0.934668 , 0.934825 ]$ & $[ 0.934668 , 0.934826 ]$ & & $[ 0.93426 , 0.93492 ]$ & & $[ 0.934640 , 0.934802 ]$ \\
$\mathcal{A}_c$ & $ 0.670 \pm 0.027 $ & $ 0.66789 \pm 0.00023 $ & $ 0.66789 \pm 0.00023 $ & $ 0.1 $ & $ 0.66816 \pm 0.00054 $ & $ 0.1 $ & $ 0.66766 \pm 0.00024 $ &$ 0.1 $ \\
& & $[ 0.66743 , 0.66834 ]$ & $[ 0.66743 , 0.66835 ]$ & & $[ 0.66712 , 0.66922 ]$ & & $[ 0.66718 , 0.66814 ]$ \\
\hline
$\mathcal{A}_s$ & $ 0.895 \pm 0.091 $ & $ 0.935663 \pm 0.000043 $ & $ 0.935663 \pm 0.000043 $ & $ -0.4 $ & $ 0.935714 \pm 0.000099 $ & $ -0.5 $ & $ 0.935622 \pm 0.000045 $ &$ -0.5 $ \\
& & $[ 0.935580 , 0.935746 ]$ & $[ 0.935580 , 0.935746 ]$ & & $[ 0.935522 , 0.935909 ]$ & & $[ 0.935533 , 0.935709 ]$ \\
BR$_{W\to\ell\bar\nu_\ell}$ & $ 0.10860 \pm 0.00090 $ & $ 0.108382 \pm 0.000022 $ & $ 0.108382 \pm 0.000022 $ & $ 0.2 $ & $ 0.108293 \pm 0.000110 $ & $ 0.3 $ & $ 0.108386 \pm 0.000023 $ &$ 0.2 $ \\
& & $[ 0.108339 , 0.108425 ]$ & $[ 0.108339 , 0.108425 ]$ & & $[ 0.10808 , 0.10851 ]$ & & $[ 0.108340 , 0.108432 ]$ \\
$\sin^2\theta_{\rm eff}^{\rm lept}$ (HC) & $ 0.23143 \pm 0.00025 $ & $ 0.231491 \pm 0.000059 $ & $ 0.231496 \pm 0.000061 $ & $ -0.2 $ & $ 0.231461 \pm 0.000136 $ & $ -0.1 $ & $ 0.231558 \pm 0.000068 $ &$ -0.5 $ \\
& & $[ 0.231376 , 0.231608 ]$ & $[ 0.231376 , 0.231616 ]$ & & $[ 0.23119 , 0.23173 ]$ & & $[ 0.231426 , 0.231691 ]$ \\
\hline
$R_{uc}$ & $ 0.1660 \pm 0.0090 $ & $ 0.172231 \pm 0.000033 $ & $ 0.172231 \pm 0.000033 $ & $ -0.7 $ & $ 0.172424 \pm 0.000180 $ & $ -0.7 $ & $ 0.172211 \pm 0.000034 $ &$ -0.7 $ \\
& & $[ 0.172167 , 0.172295 ]$ & $[ 0.172168 , 0.172296 ]$ & & $[ 0.17208 , 0.17279 ]$ & & $[ 0.172145 , 0.172277 ]$ \\
\bottomrule
\end{tabular}}
\caption{Same as Table \protect\ref{tab:SM_std} in the \emph{conservative average} scenario.}
\label{tab:SM_c}
\vspace{1.5cm}
\centering
\resizebox{0.47\textwidth}{!}{
\begin{tabular}{c|c|rr|c|rrr}
\hline
& Result & \multicolumn{2}{c|}{Correlation} & Result & \multicolumn{3}{c}{Correlation} \\\hline
& \multicolumn{3}{c|}{\scriptsize{(IC$_{\rm ST}$/IC$_{\rm SM}=24.5/37.1$)}} & \multicolumn{4}{c}{\scriptsize{(IC$_{\rm STU}$/IC$_{\rm SM}=25.3/37.1$)}} \\
\hline
$S$ & $ 0.086 \pm 0.077 $ & $1.00$ & & $ 0.004 \pm 0.096 $ & $1.00$ & & \\
$T$ & $ 0.177 \pm 0.070 $ & $0.89$ & $1.00$ & $ 0.040 \pm 0.120 $ & $0.90$ & $1.00$ & \\
$U$ & $-$ & $-$ & $-$ & $ 0.134 \pm 0.095 $ & $-0.60$ & $-0.81$ & $1.00$ \\ \hline
\end{tabular}}
\caption{Same as Table~\protect\ref{tab:STU} in the \emph{conservative average} scenario.}
\label{tab:STU_C}
\end{table*}
\begin{table*}[p]
\centering
\begin{tabular}{c|c|rrrrrrrrr}
\hline
& Result & \multicolumn{8}{c}{Correlation Matrix} \\
\hline
& \multicolumn{8}{c}{\scriptsize{(IC$_{\rm SMEFT}$/IC$_{\rm SM}=32.0/37.1$)}} \\
\hline
$\hat{C}_{\varphi l}^{(1)}$ & $ -0.007\pm 0.012 $ & $1.00$ & $ $ & $ $ & $ $ & $ $ & $ $ & $ $ & $ $ & $ $ \\
$\hat{C}_{\varphi l}^{(3)}$ & $ -0.042 \pm 0.018 $ & $-0.44$& $1.00$ & $ $ & $ $ & $ $ & $ $ & $ $ & $ $ & $ $ \\
$\hat{C}_{\varphi e}$ & $ -0.017 \pm 0.010 $ & $0.52$ & $0.31$ & $1.00$ & $ $ & $ $ & $ $ & $ $ & $ $ & $ $ \\
$\hat{C}_{\varphi q}^{(1)}$ & $ -0.018 \pm 0.045 $ & $-0.02$ & $-0.05$ & $-0.12$ & $1.00$ & $ $ & $ $ & $ $ & $ $ & $ $ \\
$\hat{C}_{\varphi q}^{(3)}$ & $ -0.114 \pm 0.044 $ & $0.02$ & $0.14$ & $-0.02$ & $-0.36$ & $1.00$& $ $ & $ $ & $ $ & $ $ \\
$\hat{C}_{\varphi u}$ & $ \phantom{+}0.090 \pm 0.150 $ & $0.05$ & $-0.04$ & $0.02$ & $0.61$ & $-0.76$ & $1.00$& $ $ & $ $ & $ $ \\
$\hat{C}_{\varphi d}$ & $ -0.630 \pm 0.250 $ & $-0.13$ & $-0.04$ & $-0.25$ & $0.40$ & $0.57$ & $-0.04$ & $1.00$& $ $ & $ $ \\
$\hat{C}_{ll}$ & $ -0.022 \pm 0.028 $ & $-0.72$ & $0.89$ & $0.01$ & $-0.06$ & $0.03$ & $-0.04$ & $-0.05$ & $1.00$ \\ \hline
\end{tabular}
\caption{Same as Table~\protect\ref{tab:SMEFT fit} for the \emph{conservative average} scenario.}
\label{tab:SMEFT fit_c}
\vspace{2.5cm}
\centering
\begin{tabular}{c|c|c|c|c}
\hline
& Measurement & ST & STU & SMEFT \\
\hline
$M_W$ [GeV] & $ 80.413 \pm 0.015 $ & $ 80.403 \pm 0.013 $ & $ 80.413 \pm 0.015 $ & $ 80.413 \pm 0.015 $ \\
$\Gamma_{W}$ [GeV] & $ 2.085 \pm 0.042 $ & $ 2.0916 \pm 0.0011 $ & $ 2.0925 \pm 0.0012 $ & $ 2.0778 \pm 0.0070 $ \\
$\sin^2\theta_{\rm eff}^{\rm lept}(Q_{\rm FB}^{\rm had})$ & $ 0.2324 \pm 0.0012 $ & $ 0.23143 \pm 0.00014 $ & $ 0.23147 \pm 0.00014 $ & -- \\
$P_{\tau}^{\rm pol}=\mathcal{A}_\ell$ & $ 0.1465 \pm 0.0033 $ & $ 0.1478 \pm 0.0011 $ & $ 0.1474 \pm 0.0011 $ & $ 0.1488 \pm 0.0014 $ \\
$\Gamma_{Z}$ [GeV] & $ 2.4955 \pm 0.0023 $ & $ 2.4976 \pm 0.0012 $ & $ 2.4951 \pm 0.0022 $ & $ 2.4955 \pm 0.0023 $ \\
$\sigma_{h}^{0}$ [nb] & $ 41.480 \pm 0.033 $ & $ 41.4909 \pm 0.0077 $ & $ 41.4905 \pm 0.0077 $ & $ 41.482 \pm 0.033 $ \\
$R^{0}_{\ell}$ & $ 20.767 \pm 0.025 $ & $ 20.7507 \pm 0.0084 $ & $ 20.7512 \pm 0.0084 $ & $ 20.769 \pm 0.025 $ \\
$A_{\rm FB}^{0, \ell}$ & $ 0.0171 \pm 0.0010 $ & $ 0.01637 \pm 0.00023 $ & $ 0.01630 \pm 0.00024 $ & $ 0.01660 \pm 0.00032 $ \\
$\mathcal{A}_{\ell}$ (SLD) & $ 0.1513 \pm 0.0021 $ & $ 0.1478 \pm 0.0011 $ & $ 0.1474 \pm 0.0011 $ & $ 0.1488 \pm 0.0014 $ \\
$R^{0}_{b}$ & $ 0.21629 \pm 0.00066 $ & $ 0.21591 \pm 0.00011 $ & $ 0.21591 \pm 0.00011 $ & $ 0.21632 \pm 0.00065 $ \\
$R^{0}_{c}$ & $ 0.1721 \pm 0.0030 $ & $ 0.172199 \pm 0.000055 $ & $ 0.172199 \pm 0.000055 $ & $ 0.17160 \pm 0.00099 $ \\
$A_{\rm FB}^{0, b}$ & $ 0.0996 \pm 0.0016 $ & $ 0.10359 \pm 0.00075 $ & $ 0.10337 \pm 0.00077 $ & $ 0.1009 \pm 0.0014 $ \\
$A_{\rm FB}^{0, c}$ & $ 0.0707 \pm 0.0035 $ & $ 0.07403 \pm 0.00059 $ & $ 0.07385 \pm 0.00059 $ & $ 0.0735 \pm 0.0022 $ \\
$\mathcal{A}_b$ & $ 0.923 \pm 0.020 $ & $ 0.934807 \pm 0.000097 $ & $ 0.934779 \pm 0.000100 $ & $ 0.903 \pm 0.013 $ \\
$\mathcal{A}_c$ & $ 0.670 \pm 0.027 $ & $ 0.66811 \pm 0.00052 $ & $ 0.66797 \pm 0.00053 $ & $ 0.658 \pm 0.020 $ \\
$\mathcal{A}_s$ & $ 0.895 \pm 0.091 $ & $ 0.935705 \pm 0.000096 $ & $ 0.935677 \pm 0.000097 $ & $ 0.905 \pm 0.012 $ \\
BR$_{W\to\ell\bar\nu_\ell}$ & $ 0.10860 \pm 0.00090 $ & $ 0.108385 \pm 0.000022 $ & $ 0.108380 \pm 0.000022 $ & $ 0.10900 \pm 0.00038 $ \\
$\sin^2\theta_{\rm eff}^{\rm lept}$ (HC) & $ 0.23143 \pm 0.00025 $ & $ 0.23143 \pm 0.00014 $ & $ 0.23147 \pm 0.00014 $ & -- \\
$R_{uc}$ & $ 0.1660 \pm 0.0090 $ & $ 0.172221 \pm 0.000034 $ & $ 0.172221 \pm 0.000034 $ & $ 0.17162 \pm 0.00099 $ \\ \hline
\end{tabular}
\caption{Same as Table~\protect\ref{tab:NP_fits} for the \emph{conservative average} scenario.}
\label{tab:NP_fits_c}
\end{table*}
\clearpage
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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Q: Is there a memory overhead using partitions in postgres vs. a single table? Background
I am ingesting time series data totalling to about 10M rows per day, where each row has a timestamp, a player id and some other columns.
That data is then queried via an API for a specific player and a time range (e.g. for last 90 days for player xxx), most cases require more than 1 concurrent request (normally around ~10 concurrent requests are executed)
I am running postgres 9.6.17, machine has 500GB hdd space (with about 15% available space at all time), 8 cores and 16GB of ram. work_mem is set to 2GB, cache_size to 12GB, max connections is set to 100 etc.
The API is 20 python gunicorn workers running flask and sqlalchemy+psycopg2 on a separate machine. Each worker has a pool of 2 connections to the DB allowing for an overflow of 5. Pool settings used to be higher but it turned out there is almost no benefit to using pools, hence the low numbers.
Naive approach
Initially, I put all the data into a single table and indexing both on timestamp and player. This approach worked fine until the amount of data started to make it slower and slower (for obvious reasons). This lead the API workers to timeout and return 500. A cost of a typical query (single player data over 6 months) as returned by explain would be around 1M, example below
Bitmap Heap Scan on player_data (cost=515553.98..585514.47 rows=80037 width=32)
Recheck Cond: (((player_id)::text = 'xxx'::text) AND (ts >= 1572566400) AND (ts < 1574899199))
-> BitmapAnd (cost=515553.98..515553.98 rows=62819 width=0)
-> Bitmap Index Scan on idx_player_id (cost=0.00..12749.35 rows=962837 width=0)
Index Cond: ((player_id)::text = 'xxx'::text)
-> Bitmap Index Scan on idx_ts (cost=0.00..502778.48 rows=37691480 width=0)
Index Cond: ((ts >= 1572566400) AND (ts < 1574899199))
Better approach with partitions
As an improvement I started storing data in partitions, one per day and then creating the player id and timestamp index on every partition instead. This reduced significantly the query times, improvements were visible also with explain:
Append (cost=0.00..85192.02 rows=80037 width=32)
-> Seq Scan on player_data (cost=0.00..0.00 rows=1 width=85)
Filter: ((ts >= 1572566400) AND (ts < 1574899199) AND ((player_id)::text = 'xxx'::text)
-> Index Scan using idx_player_id_20191104 on player_data_20191104 (cost=0.43..317.61 rows=280 width=32)
Index Cond: ((player_id)::text = 'xxx'::text)
Filter: ((ts >= 1572566400) AND (ts < 1574899199)
........<continued for a while>................
The cost was almost an order of magnitude lower and everything was going well, especially for queries of smaller time granularity.
Issues
After a while the users started complaining that the queries were not working, meaning the API started returning 500 errors. After checking the logs I noticed a lot of the following errors:
sqlalchemy.exc.OperationalError: (psycopg2.errors.OutOfMemory) out of memory
DETAIL: Failed on request of size NNNN.
Note that NNNN can be pretty much any integer between 1 and 10000. Once the error manifested itself it would happen to all queries executed concurrently, rendering the DB unresponsive for a while (few minutes). Normally a restart of the API service would restore things to normal. When the DB is unresponsive the same error is returned for any kind of queries and access, including psql connect attempts from my machine etc.
I should also note the queries that run in parallel mostly hit the same indexes and partitions, e.g. different player ids over same time range, sometimes the queries are identical except for offset/limit values.
Another note - same errors show up regardless of concurrency coming via API or directly psql.
Debugging and diagnostics
I've checked the resources on the DB machine (since it's supposed to run out of memory) while querying:
*
*available RAM never drops under 60%
*no swap is ever used, nothing gets written to disk (although, it should if postgres runs out of available mem)
*concurrent connection count to DB tops at 100 during testing, idles at 40
*single partition index size is about 5MB, the index size of the old table was almost 1GB
Also, I've tried the following changes to the DB:
*
*increase work_mem from 20MB to 2GB, no improvement
*increase RAM from 8GB to 12GB and finally to 16GB, no improvement
I've also compared the old vs. the new query, with the following results:
*
*more than 10 concurrent queries towards the partitioned table caused the error to manifest
*200 concurrent queries to the old, single table caused no issues whatsoever
Question(s)
So the questions I guess are:
*
*does the use of partitions create some memory overhead causing the queries to fail due to memory allocation failure?
*Is it the indexes? Is using multiple indexes worse than having one big one?
*Does this theory even make sense or am I missing something obvious?
A:
(normally around ~10 concurrent requests are executed)
... 16GB of ram. work_mem is set to 2GB,
Your work_mem setting seems nuts to me. With 10 concurrent queries and each one might use multiple instances of work_mem (especially common with partitioning--so to answer your title question, yes there is), it would be no surprise to run out of memory.
I've checked the resources on the DB machine (since it's supposed to run out of memory) while querying:
How did you do this? The data from different tools need to be interpreted in different ways.
Also, I've tried the following changes to the DB:
increase work_mem from 20MB to 2GB, no improvement
Right, increasing the amount of memory you use is unlikely to fix an out of memory problem. "The beatings will continue until morale improves."
But are you saying you had the exact same problem back when work_mem was set to only 20MB? Did you look in the database log file to see what it said about the problem directly (as opposed to what python passed along to you about it?)
From your description, it sounds like you might not need partitioning at all. Just a multicolumn index on (player_id, ts) would have likely fixed your problem without taking on the burden and overhead of partitioning.
Do you ever plan to delete/archive old data, or will in accumulate indefinitely?
A: Answering my own question since kinda makes sense:
*
*no config of 9.6 made the issues go away
*same issues manifested in PSQL12, although indeed partitioning was native there
*no configuration of PSQL12 made any difference
So in the end, just to keep the system in a working state I settled on (much) less partitions and longer query times.
I later migrated to custom encoding and aggregation solutions.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,773
|
{"url":"https:\/\/perimeterinstitute.ca\/video-library\/collection\/13\/14-psi-foundations-quantum-mechanics","text":"13\/14 PSI - Foundations of Quantum Mechanics\n\n13\/14 PSI - Foundations of Quantum Mechanics\n\n13\/14 PSI - Foundations of Quantum Mechanics - Lecture 15\n\nFriday Jan 24, 2014\nSpeaker(s):\n\n13\/14 PSI - Foundations of Quantum Mechanics - Lecture 14\n\nThursday Jan 23, 2014\nSpeaker(s):\n\n13\/14 PSI - Foundations of Quantum Mechanics - Lecture 13\n\nWednesday Jan 22, 2014\nSpeaker(s):\n\n13\/14 PSI - Foundations of Quantum Mechanics - Lecture 12\n\nTuesday Jan 21, 2014\nSpeaker(s):\n\n13\/14 PSI - Foundations of Quantum Mechanics - Lecture 11\n\nMonday Jan 20, 2014\nSpeaker(s):\n\n13\/14 PSI - Foundations of Quantum Mechanics - Lecture 10\n\nMonday Jan 20, 2014\nSpeaker(s):\n\n13\/14 PSI - Foundations of Quantum Mechanics - Lecture 9\n\nFriday Jan 17, 2014\nSpeaker(s):\n\n13\/14 PSI - Foundations of Quantum Mechanics - Lecture 8\n\nThursday Jan 16, 2014\nSpeaker(s):\n\n13\/14 PSI - Foundations of Quantum Mechanics - Lecture 7\n\nWednesday Jan 15, 2014\nSpeaker(s):\n\n13\/14 PSI - Foundations of Quantum Mechanics - Lecture 6\n\nTuesday Jan 14, 2014\nSpeaker(s):\n\nRECENT PUBLIC LECTURE\n\nAmber Straughn: A New Era in Astronomy: NASA\u2019s James Webb Space Telescope\n\nSpeaker: Amber Straughn","date":"2017-03-23 14:43:11","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8232711553573608, \"perplexity\": 7830.906253628341}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-13\/segments\/1490218187113.46\/warc\/CC-MAIN-20170322212947-00528-ip-10-233-31-227.ec2.internal.warc.gz\"}"}
| null | null |
package com.qlfg.miningcircle.chain.activity;
import org.apache.http.cookie.Cookie;
import com.example.miningcircle.R;
import com.qlfg.miningcircle.activity.BaseActivity;
import com.qlfg.miningcircle.application.MiningCircleApplication;
import com.qlfg.miningcircle.home.LoginActivity;
import com.qlfg.miningcircle.loan.activity.AlertDialog;
import com.qlfg.miningcircle.loan.activity.WebViewActiity;
import com.qlfg.miningcircle.usercenter.UserCenterActivity;
import com.qlfg.miningcircle.util.CkxTrans;
import com.qlfg.miningcircle.util.Contant;
import android.app.Activity;
import android.content.Intent;
import android.graphics.Bitmap;
import android.os.Bundle;
import android.util.Log;
import android.view.View;
import android.view.Window;
import android.view.View.OnClickListener;
import android.webkit.CookieManager;
import android.webkit.CookieSyncManager;
import android.webkit.WebChromeClient;
import android.webkit.WebView;
import android.webkit.WebViewClient;
import android.widget.ImageView;
import android.widget.TextView;
/**
* @author langkl
*
* 说明:矿业链加载网页界面
*
* @parm url 要加载的url
*
* @time 2015.10.12
*
* */
public class ChainHomeWebActivity extends BaseActivity implements OnClickListener{
private Activity mActivity;
private ImageView leftImage,titleImage,rightImage;
private TextView titleName;
private String title_name;
private WebView mWebView;
private String loadUrl;
private String userName;
@Override
protected void onCreate(Bundle savedInstanceState) {
super.onCreate(savedInstanceState);
requestWindowFeature(Window.FEATURE_NO_TITLE);
setContentView(R.layout.activity_miningcircle_dynamic);
mActivity = ChainHomeWebActivity.this;
loadUrl = Contant.server_url+"app2.do?lnidx";
initUI();
CookieSyncManager.createInstance(this);
CookieManager cookieManager = CookieManager.getInstance();
Cookie sessionCookie = Contant.APPCOOLIE;
if (sessionCookie != null) {
String cookieString = sessionCookie.getName() + "=" + sessionCookie.getValue() + "; domain=" + sessionCookie.getDomain();
cookieManager.setCookie(Contant.server_url, cookieString);
CookieSyncManager.getInstance().sync();
}
Log.e("lkl", "加载的URL="+loadUrl);
mWebView = (WebView) findViewById(R.id.dynamic_web);
mWebView.getSettings().setJavaScriptEnabled(true);
mWebView.loadUrl(loadUrl);
mWebView.setWebViewClient(new WebViewClient(){
@Override
public boolean shouldOverrideUrlLoading(WebView view, String url) {
if(url.contains("app2.do?lnprdinfo&pid=")) {
if(CkxTrans.isNull(MiningCircleApplication.getInstance().getUserPhone())){
final AlertDialog ad=new AlertDialog(mActivity);
ad.setTitle("抱歉,请您先登录。");
ad.setLeft("取消");
ad.setRight("登录");
ad.setPositiveButton("确定", new OnClickListener() {
@Override
public void onClick(View v) {
// TODO Auto-generated method stub
ad.dismiss();
Intent mintent = new Intent (mActivity,LoginActivity.class);
startActivity(mintent);
}
});
ad.setNegativeButton("取消", new OnClickListener() {
@Override
public void onClick(View v) {
// TODO Auto-generated method stub
ad.dismiss();
}
});
view.stopLoading();
}else{
Intent mintent = new Intent(mActivity , WebViewActiity.class);
mintent.putExtra("type", Contant.DETAIL);
mintent.putExtra("url",url);
mintent.putExtra("titlename","详情");
mintent.putExtra("type", "detail");
startActivity(mintent);
}
} else{
view.loadUrl(url);
}
return true;
}
@Override
public void onPageStarted(WebView view, String url, Bitmap favicon) {
// if (url.contains("call_appui=user_center")) {
// Intent mintent = new Intent(mActivity,UserCenterActivity.class);
// startActivity(mintent);
// finish();
//
// }
// super.onPageStarted(view, url, favicon);
}
});
mWebView.setWebChromeClient(new WebChromeClient() {
@Override
public void onProgressChanged(WebView view, int newProgress) {
super.onProgressChanged(view, newProgress);
}
});
}
private void initUI(){
View titleLayout = findViewById(R.id.home_title);
leftImage = (ImageView) titleLayout.findViewById(R.id.left_image);
leftImage.setBackgroundResource(R.drawable.back);
leftImage.setOnClickListener(this);
rightImage = (ImageView) titleLayout.findViewById(R.id.right_image);
rightImage.setOnClickListener(this);
}
@Override
protected void onResume() {
super.onResume();
userName = MiningCircleApplication.getInstance().getUserPhone();
if(CkxTrans.isNull(userName)){
rightImage.setBackgroundResource(R.drawable.user_out);
}else{
rightImage.setBackgroundResource(R.drawable.tabbar_my_bg);
}
}
@Override
public void onClick(View v) {
switch (v.getId()) {
case R.id.left_image:
finish();
break;
case R.id.right_image:
if(CkxTrans.isNull(userName)){
Intent login = new Intent(mActivity,LoginActivity.class);
startActivity(login);
}else{
Intent user = new Intent(mActivity,UserCenterActivity.class);
startActivity(user);
}
break;
default:
break;
}
}
@Override
protected void onActivityResult(int requestCode, int resultCode, Intent data) {
super.onActivityResult(requestCode, resultCode, data);
if(resultCode == 1){//确定
Intent mintent = new Intent (mActivity,LoginActivity.class);
startActivity(mintent);
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,516
|
\section{Introduction}
\label{1}
\IEEEPARstart{T}{arget} recognition on SAR images has been under research for many years \cite{wang2016robust,liu2018sar,zhang2018fast,zhao2018adaptive,8533426} due to its various applications in military and homeland security, such as friend and foe identification, battlefield surveillance, environmental monitoring, disaster relief, etc. And it can operate under all-weather and all-time conditions while producing high resolution images with a long standoff capability. Therefore, the SAR image interpretation is of critical importance and the development of automatic target recognition (ATR) system is practical and necessary.
The typical Synthetic Aperture Radar automatic target recognition (SAR-ATR) system can be divided into three parts: target detection, target discrimination and target classification \cite{dudgeon1993overview}. In the first part, a constant false alarm rate (CFAR) detector is used to extract potential targets from SAR images. These potential targets not only consist of true targets such as armored vehicles, rocket launcher and tanks, but also some background clutters such as trees, buildings and rivers. To reduce false alarm rate, the second discrimination part is designed to train a two-class (target and background) model into capturing the true targets by feature extraction. Finally, the third classification part helps decide which category the target belongs to. However, the traditional SAR-ATR system has several disadvantages \cite{morgan2015deep}. Firstly, it relies heavily on handcrafted features, needs large computational space and has poor robustness. Besides, the accuracy will degrade significantly while any of these three stages is not well designed. Lastly, when it comes to both localizing and classifying the multiple targets in the complex background, it is neither effective nor efficient.
To solve this problem, a novel Moving and Stationary Target Acquisition and Recognition (MSTAR) system was developed by the Air Force Research Laboratory and the Defense Advanced Research Projects Agency (AFRL/DARPA) \cite{article}. This dataset contains not only small target chips that are abstracted from the collected data but also simple and complex large-scene backgrounds since it is costly to directly acquire large-scene SAR images with targets. Based on this dataset, a lot of experiments have been conducted which can be summarized into two aspects: classification on small target chips and detection on synthesized large-scene SAR images \cite{cui2019d}.
With the emergence of deep learning methods, neural networks have been gradually applied to those two aspects owing to its superior performance on SAR image processing \cite{wen2018survey, 1202937}. Different from the traditional feature extraction methods which need to design the algorithms manually, neural networks are capable of capturing the inherent feature of the input images. As to the first aspect which is to classify the targets on small target chips, the most commonly used deep learning architecture CNN model \cite{inproceedings} is adopted to conduct ten-class classification on MSTAR target chips, which verifies the validity of deep neural network in the field of SAR target recognition. However, the sample number of each type is limited, thus the experimental results lack some commonality. To tackle with the problem of limited training data, domain-specific data augmentation operations combined with CNN \cite{ding2016convolutional} provides a new way to deal with the problem of translation of target, randomness of speckle noise and lack of pose images together. Since a large amount of data is necessary to train a CNN model, another way to deal with the problem of the limited data is to train a ConvNets model with fewer degrees of freedom by only using a sparsely connected convolution architecture \cite{chen2016target} and in the meanwhile randomly sampling relatively smaller patches from the original SAR images to expand the training set.
As in the above methods, the commonly used data augmentation approaches are horizontal flipping, randomly cropping, rotation, translation or randomly sampling, which means we need to manually control the variety of the additional images by randomly deciding on how many and which ways we are to use. Recently, a newly appeared Generative Adversarial Nets (GAN) proposed by Goodfellow \cite{goodfellow2014generative} is employed to produce more labeled SAR data \cite{he2019parallel}. Though thousands of data can be generated conveniently, not all of them are helpful for classification, so a certain number of generated samples should be carefully selected and it is difficult to find an objective standard to evaluate the quality of the generated images. To avoid the dilemma, another way to make full use of GAN is to train a super-resolution generative adversarial network (SRGAN) \cite{ShiAutomatic} directly to enhance the original images and improve the visual resolution and feature characterization ability of targets in the SAR images. These two methods verify the effective application of the adversarial networks in the SAR image recognition area.
However, GAN-based models have several disadvantages: first, they operate on observation space, which means a large number of parameters are needed during the training process, making it hard to converge; second, due to the high-noise characteristic of SAR dataset, the latent space is more able to capture the main feature of the target in the image which excludes the disturbance of the background. For the sake of solving these two problems, we use a new generative model called Adversarial Autoencoders (AAE) \cite{makhzani2015adversarial}. Different from GANs, AAE blazes a new trail by making the most of latent space. It absorbs the idea of autoencoder \cite{li2017prediction, xiao2020deep} and attempts to push the latent vector close to the distribution of the specific input sample. In this way, AAE is much easier to converge and consumes less space, and our experiment further shows that it also reaches higher quality on generated SAR images. Therefore, in this paper, the AAE network is used to realize data augmentation, and experiments are conducted for improvement on complex large-scene SAR images detection.
So far the above SAR-ATR algorithms are nearly all constructed on CNN framework and the main goal is to classify the targets after the corresponding small chips are abstracted from real large-scene images. In real conditions, however, the targets are randomly scattered into different areas in a real large-scene image with high resolution, and the complex background including trees, buildings, rivers and so on makes it rather hard to accurately detect and recognize them in real-time. Therefore it is under critical research for detecting and recognizing targets on complex large-scene SAR images. Two kinds of algorithms are widely used: two-stage ones, such as R-CNN series \cite{girshick2014rich,girshick2015fast,ren2015faster} and one-stage ones, e.g., SSD \cite{liu2016ssd} and YOLO series \cite{redmon2016you, redmon2017yolo9000,redmon2018yolov3}. The two-stage method Faster R-CNN generally reaches higher accuracy than one-stage methods SSD and YOLOv3 but is time-consuming, too computationally intensive for embedded systems and not suitable for real-time applications. Modified Faster R-CNN models and single shot multibox detector (SSD) are conducted to address SAR-ATR \cite{dong2019end}. It has been shown that MobileNet-SSD and SSD-Inception though have lower accuracy, perform hundreds of times faster than Faster R-CNNs. The work of \citenum{redmon2018yolov3} proposed an improved YOLO network which is known as YOLOv3. This network derives from the older version of YOLO with unique features such as bounding box prediction, class prediction, predictions across scales, feature extractor and training method. The experiment shows that it is three times faster than SSD on COCOs while reaching close detection accuracy. So far, the YOLOv3 network has proved its superiority in many fields such as novel landmark localization \cite{huang2015densebox}, 3-D human detection\cite{Tian2018Robust}, and thermal imaging \cite{ivavsic2019human}. Following aforementioned state-of-art works in the literature, in this paper, we adopt the YOLOv3 as the backbone for realizing effective and efficient SAR-ATR.
When it comes to detecting multiple objects in a large complex SAR background \cite{chang2010change, xu2019retrieval}, a fast sliding method can be used to segment the scene image into sub-images and then detection network is applied to locate the targets. The process of target segmentation and synthesis is of rather importance since it is costly to directly gain the large-scene SAR images with multiple targets inside, therefore this process plays a critical part in the final detection and recognition result.
In this paper, we propose a deep learning framework for detection and recognition on complex large-scene SAR images. Before training the network, AAE is firstly adopted to realize the data augmentation of small SAR chips. Such an operation is simple but useful for the extraction of key feature and enhancing the variety of generated images. In addition, instead of manual labeling, an automatic labeling method is then proposed to mark the targets. Due to the limited number of complex large-scene SAR images, we fully take advantage of small chips and then propose a target segmentation and synthesis method to establish a complex large-scene SAR database for study. After establishing the database, a fast sliding method on large-scene images is proposed to avoid obtaining abundant slices without targets or with incomplete targets. When training the YOLOv3 network, we pretrain the weights of the proposed deep learning method using the well-known COCO dataset by leveraging the advantages of the transfer learning \cite{wang2018sar}. At the training stage, the expanded small target chips and large-scene images after fast sliding are simultaneously fed into the network. Finally, non-maximum suppression on sub-images is conducted to obtain the unique bounding box for each target. The results show that our method exhibits superior accuracy on complex large-scene images and also demonstrates great real-time performance. Furthermore, numerical simulations demonstrate that the proposed method can accurately detect and recognize the targets with high anti-noise performance.
The remainder of this paper is organized as follows. Section \ref{2} elaborates a target detection and recognition framework for complex large-scene SAR images. In Sec. \ref{3}, we verify the effectiveness and efficiency of our proposed approach on a variety of experiments using the MSTAR dataset. The analysis and conclusions are drawn in Sec. \ref{4}.
\section{The ATR Framework}\label{2}
In this section, we will introduce our target detection and recognition framework on complex large-scene SAR images. Since we need to obtain small target chips for joint training, we will first introduce how to expand SAR target chips and conduct automatic labeling in Sec. \ref{2-1}. Then Sec. \ref{2-2} gives a further description of how to establish our large-scene SAR database, and use YOLOv3 for detection.
\subsection{Process on Small SAR Target Chips}\label{2-1}
The proposed ATR model on SAR target chips is shown in Fig. \ref{chips}. It is composed of three parts: data augmentation by AAE, automatic labeling, target detection and recognition. The last part is realized by YOLOv3 after automatically labeling these targets, which means we can not only detect the target but also recognize it with limited samples and without manual labeling. These expanded labeled small chips are fed into the network with large-scene images to enhance the detection accuracy on complex background.
In Fig. \ref{chips}, we use MSTAR four-target dataset, including 2S1, BTR60, BRDM2 and D7 as an example to illustrate the AAE augmentation method and automatic labeling.
\begin{figure}[]
\centering
\includegraphics[width=3.5in]{chip_detect}
\caption{An ATR framework or SAR target chips. \uppercase\expandafter{\romannumeral1}. Data augmentation method AAE is used to expand the training set; \uppercase\expandafter{\romannumeral2}. The training samples are then automatically labeled; \uppercase\expandafter{\romannumeral3}. The training images and labels are sent to the YOLOv3 network to complete target detection and recognition.}
\label{chips}
\end{figure}
\subsubsection{Data augmentation}
One of the key points in SAR image recognition is that SAR images suffer from the speckle noise due to the characteristic of the imaging system. And SAR images for training a robust ATR system is insufficient. For instance, in MSTAR four-target dataset, there are only 1152 images for training, which may lead to an overfitting problem and reduce the generalization effect.
To solve the problem of insufficient training samples, data augmentation is necessary. The classic methods of data augmentation are mostly operating on the original images through flipping, cropping, zooming, etc. This may result in data redundancy and therefore can not obviously enhance the variety of image characteristics.
The adversarial autoencoder (AAE) is a combination of autoencoder and GAN, and it achieves competitive performance on generating SAR target chips. As is shown in Fig. \ref{AAE}, the top row is a standard autoencoder that reconstructs an image $x$ from a latent code $z$.
\begin{equation}
\label{eqn_example}
q(z) = \int_x {q(z|x){p_d}(x)dx}
\end{equation}
The goal of the adversarial autoencoder is to match the aggregated posterior $q(z)$ to $p(z)$, which is an arbitrary prior (e.g. Gaussian distribution). The encoder of the autoencoder ${q(z|x)}$ acts as the generator of the adversarial network, attempting to fool the discriminative adversarial network into recognizing the hidden code $q(z)$ as the prior distribution $p(z)$. In the meanwhile, the autoencoder attempts to reconstruct the input image $x$ from the latent code vector $z$.
\begin{figure}[]
\centering
\includegraphics[width=3in]{aae}
\caption{The architecture of an adversarial autoencoder. The top row is a standard autoencoder that reconstructs an image $x$ from a latent code $z$. The bottom row is a network trained to discriminate whether the sample is from a prior distribution or from the latent vector $z$. }
\label{AAE}
\end{figure}
Different from GAN, in which the input noise lacks semantic information and the output distribution is uncontrollable, AAE largely increases the diversity of the output samples by making the latent code vector simulate the prior distribution. Therefore we can directly expand the training dataset through the generated samples by AAE without carefully selecting which image to use.
\subsubsection{Automatic labeling}
After collecting all the training images, an automatic labeling method is developed to avoid manual labeling, which reduces a large amount of redundant work. The detailed design is shown in Algorithm \ref{auto_label}, and the process is shown in Fig. \ref{auto-label}.
\renewcommand{\algorithmicrequire}{ \textbf{Input:}}
\renewcommand{\algorithmicensure}{ \textbf{Output:}}
\begin{algorithm}
\caption{Automatic labeling.}
\label{auto_label}
\begin{algorithmic}[1]
\REQUIRE ~~\\
The original SAR target chip.\\
\ENSURE ~~\\
The target coordinate $(x,y,w,h)$ and its corresponding category.
\STATE Binarize the original image;\\
\STATE Set a threshold of the white pixels' number and traverse the binarized image from 4 directions;\\
\STATE Count the number of white pixels for each row or column;\\
\STATE Stop traversing when reaching the threshold in each direction;\\
\STATE Form the corresponding rectangle;\\
\STATE Expand rectangle concentrically to a certain extent, e.g. 50\%, to produce a rectangle of proper size.
\RETURN The target label information.
\end{algorithmic}
\end{algorithm}
First, we need to binarize the original image using a thresholding method which will be introduced in Sec. \ref{2-2-1}. Though this binarization method can correctly segment the object, in a few cases there are still some small white spots in the background. To eliminate the effect of white spots in automatic labeling, we set a threshold of white pixel number to filter those spots and finally capture the object accurately. For each row or column, if the number of white pixels is lower than the threshold, we consider it not to constitute the target. We traverse the binarized image from 4 directions (up, down, left and right) to count the number of white pixels for each row or column, and stop traversing when reaching the threshold, formulating a rectangle which contains the center of the target. However, the edge of the target may also not reach the threshold thus may be filtered, so we need to expand the rectangle concentrically by a certain extent, which according to our experiment, is around 50\%, and the white pixel threshold of ten-class target usually lies in the interval [8,12].
\begin{figure}[]
\centering
\includegraphics[width=3in]{auto-label}
\caption{The method of automatic labeling.}
\label{auto-label}
\end{figure}
\subsection{The ATR Framework on Complex Large-scene SAR Images}\label{2-2}
After obtaining labeled small target chips with data augmentation, in this part, we will apply our ATR framework to detect multiple targets on complex large-scene SAR images. As is shown in Fig. \ref{large-scene}, firstly to prepare the large-scene database, we need to segment the target and its shadow from the speckle background; then the target is synthesized into the large-scene background which is acquired under the same depression degree; later a fast sliding method is used to divide the synthesized large-scene image into different sizes and a YOLOv3 network is adopted to train large-scene images and small target chips simultaneously to gain the final result on large-scene images with target categories, confidences and bounding boxes.
\begin{figure}[]
\centering
\includegraphics[width=3.5in]{large-scene}
\caption{An ATR framework for complex large-scene SAR images. \uppercase\expandafter{\romannumeral1}. Target chips are segmented from their backgrounds and then synthesized on the large-scene SAR images. \uppercase\expandafter{\romannumeral2}. Fast sliding is conducted to divide the large-scene images into different sizes. \uppercase\expandafter{\romannumeral3}. Both sliced large-scene images and small target chips are fed into YOLOv3 network simultaneously to gain the final result. \uppercase\expandafter{\romannumeral3}. Finally we map the detection result to the large-scene image and apply non-maximum suppression to gain target with single bounding box.}
\label{large-scene}
\end{figure}
\begin{figure}[]
\centering
\includegraphics[width=2.5in]{object_segmentation}
\caption{Object segmentation process. We sequentially adopt Gaussian blur, thresholding and morphological operation to generate the final segmented object. The threshold value is set as: $p=121$.}
\label{object_segmentation}
\end{figure}
\subsubsection{Database preparation}\label{2-2-1}
Since it is costly to directly obtain large-scene SAR images with multiple targets, a target segmentation and synthesis method is first proposed to establish a training database.
For the target segmentation part, we segment the object and its shadow in two steps, which are represented in Fig. \ref{object_segmentation}. The first step is to segment the object without its shadow. The image is firstly smoothed by Gaussian blur, with the convolution kernel determined adaptively by the image itself. A blurred image has fewer noises and then can be binarized. The selection of the threshold is semi-adaptive and it is the most critical step. Since there is only one target lying in the approximate center in the SAR target chip and it is obviously distinguished from the background, the binarization rule is set as:
\begin{equation}
(x',y') = \left\{ \begin{array}{l}
255\quad if\, (x,y) > p\\
0\quad\quad otherwise
\end{array} \right.
\end{equation}
where $p$ denotes the threshold.
The threshold can be determined by the pixel value proportion of the target in the whole image. The value $p$ is selected around 90\% in most circumstances, which can be estimated from the intuitive area ratio. As is shown in Fig. \ref{threshhold}, the object takes up about 10\% of the image, so we choose $p=121$ in this example as the threshold and apply the binarization rule. This method proves its effectiveness on MSTAR target chips since it can successfully separate more than 95\% objects in the original images.
\begin{figure}[]
\centering
\includegraphics[width=3in]{threshold}
\caption{The pixel value distribution of the SAR image. The left column is the original image and its histogram, and the right column is the result of applying our selected threshold $p=121$.}
\label{threshhold}
\end{figure}
\begin{figure}[]
\centering
\includegraphics[width=3in]{object_shadow_seg}
\caption{Object and shadow segmentation process. The object segmented from step one will be processed through four procedures: object shadowing, lightening and blurring, thresholding, and final morphological operations.}
\label{ob_shsyn}
\end{figure}
In the second step, the object and its shadow are segmented in the meantime, which is exhibited in Fig. \ref{ob_shsyn}. Firstly the segmented object from step one is shadowed to a comparatively small pixel value, then the binarization rule is adopted to highlight both the object and its shadow. After Gaussian blur, an adaptive threshold selection algorithm OTSU is adopted to segment them directly, since the present image has low noise. Then morphological operations are used to improve the segmentation result.
Usually the object and its shadow are separated after binarization, so the closing operation is first adopted for connection, and then the opening operation is applied to clear up small spots while keeping the main body unchanged, at last we add some edge details by dilation to produce the final segmentation result. It is clear to see that the object and its shadow are successfully segmented using our method.
After segmentation, it is of the same importance to synthesize the large-scene image with multiple targets naturally. As is shown in Fig. \ref{ob_sh}, the target synthesis process also goes through two steps. The first one is to design and record the target distribution in the large-scene SAR image. We randomly select 20 coordinates to put four-class targets and each class targets occupy five positions. Since there are some obstacles such as trees, buildings and rivers in the background, and the shadows of these obstacles have a certain direction due to the specific shooting angle and time, it is necessary to carefully select those targets which have the similar direction of shadow and finally design the location of the targets to ensure that they will not fall into the obstacles.
\begin{figure}[]
\centering
\includegraphics[width=3in]{target_synthesis}
\caption{Target synthesis process. Note that the original target chip and the cutted large-scene slice are of the same size.}
\label{ob_sh}
\end{figure}
After choosing the proper target and its coordinate, the second step is to cut a slice of the large-scene image in the corresponding designed position, which has the same shape as the target chip, then the mask produced by target segmentation and its inverse mask are used to perform the bitwise-and operation on the original image and the scene cut respectively. Finally these two operation results are combined to get the final natural synthesized large-scene image.
\subsubsection{Fast sliding}
Since the number of the large-scene images in the MSTAR dataset is limited, directly feed them into the YOLOv3 network will cause severe overfitting problems. Therefore, we can use a fast sliding method to expand the training dataset while still containing complex background information. However, if we randomly set the size of the sliding window and its stride, the target in the scene is likely to be divided into several parts and we may also obtain a large number of slices without targets which will increase the input data redundancy. Therefore, a fast sliding method is proposed which uses sliding windows to cut a synthesized large-scene image into small slices to expand the input data volume. Different from the method in work\cite{cui2019d}, we do not need to make sure that the size of the sliding window is fixed, so when the sliding window almost reaches the edge of the large-scene image, the remaining part which is smaller than the setted size can be directly cut from the image and saved for training. The fast sliding method is shown in Fig. \ref{fast_sliding}.
\begin{figure}[]
\centering
\includegraphics[width=3.5in]{fast_sliding}
\caption{The proposed fast sliding method.
$(x,y)$ represents the target position in the large-scene image; $(x',y')$ denotes the corresponding coordinate in the cropped slice.}
\label{fast_sliding}
\end{figure}
When calculating the corresponding target coordinate in the cropped slice, we respectively use
$height'$ and $width'$ to represent the number of slices that can be obtained from the vertical and horizontal direction (including the last possible incomplete slice). $(i,j)$ denotes the coordinate of a slice in the large-scene image, while $(x,y)$ along with $(x',y')$ respectively denote the coordinate of a target in the large-scene image and the cropped slice. For each sliding window $(i,j)$, we traverse the coordinates of the targets and if there is any target which could meet the following conditions simultaneously, we consider it falling into this sliding window completely.
\begin{equation}
\begin{aligned}
&{x_{\min }} > i * stride\\
&{x_{\max }} < i * stride + size\\
&{y_{\min }} > j * stride\\
&{y_{\max }} < j * stride + size
\end{aligned}
\end{equation}
Meanwhile, the slice will be automatically abandoned if there is no target falling into it or the target is incomplete and the sliding window will move on to the next one until the whole image is covered.
In this paper, we choose four sizes $128 \times 128$, $256 \times 256$, $512 \times 512$, $1024 \times 1024$ to apply fast sliding.
\subsubsection{Training on YOLOv3 network}
With expanded small target chips and large-scene image slices, we feed them into the YOLOv3 network using pre-trained weight on the COCO dataset. The main idea of YOLO is to divide the input image into
$S \times S$ grids and if the center of an object falls into one grid cell, then that grid cell is responsible for predicting the object.
\begin{figure}[]
\centering
\includegraphics[width=3.5in]{yolo}
\caption{The basic YOLO detection system.}
\label{yolo}
\end{figure}
Confidence is defined as
$\Pr (Object) * IOU_{pred}^{truth}$, which reflects how confident the model is that the bounding box contains an object and indicates the accuracy of the prediction.
$\Pr (Clas{s_i}\left| {Object)} \right.$ denotes the probabilities that each grid cell predicts $C$ classes.
\begin{equation}
\label{eqn_example2}
\begin{aligned}
\Pr (Clas{s_i}|Object) * \Pr (Object) * IOU_{pred}^{truth} \\
= \Pr (Clas{s_i}) * IOU_{pred}^{truth}
\end{aligned}
\end{equation}
By multiplying these two parts, we can obtain both the probability of that class appearing in the box and how well the predicted box fits the object. The loss function is defined as:
{\small
\begin{equation}
\centering
\begin{aligned}
loss = &{\lambda _{coord}}\sum\limits_{i = 0}^{{S^2}} {\sum\limits_{j = 0}^B { ]\kern-0.15em] _{ij}^{obj}[({x_i} - {{\hat x}_i}} } {)^2} + {({y_i} - {\hat y_i})^2}]\\
+ &{\lambda _{coord}}\sum\limits_{i = 0}^{{S^2}} {\sum\limits_{j = 0}^B { ]\kern-0.15em] _{ij}^{obj}[(\sqrt {{\omega _i}} - \sqrt {{{\hat \omega }_i}} } } {)^2} + {(\sqrt {{h_i}} - \sqrt {{{\hat h}_i}} )^2}]\\
+ &\sum\limits_{i = 0}^{{S^2}} {\sum\limits_{j = 0}^B { ]\kern-0.15em] _{ij}^{obj}{{({C_i} - {{\hat C}_i})}^2}} }+ \sum\limits_{i = 0}^{{S^2}} { ]\kern-0.15em] _i^{obj}\sum\limits_{c \in classes} {{{({p_i}(c) - {{\hat p}_i}(c))}^2}} } \\
+ &{\lambda _{noobj}}\sum\limits_{i = 0}^{{S^2}} {\sum\limits_{j = 0}^B { ]\kern-0.15em] _{ij}^{noobj}{{({C_i} - {{\hat C}_i})}^2}} }
\end{aligned}
\end{equation}
}
where ${ ]\kern-0.15em] _i^{obj}}$ denotes whether the object appears in cell $i$, and
${ ]\kern-0.15em] _{ij}^{obj}}$ denotes that the $jth$ bounding box in cell $i$ is responsible for the prediction. $\lambda _{coord}$ is used to increase the loss from bounding box coordinate predictions and $\lambda _{noobj}$ is to decrease the loss from confident predictions for boxes that don't contain objects. YOLOv3 has represented superior performance on both accuracy and speed. Compared with Faster R-CNN, which needs to repeatedly train the region proposal network (RPN) and Fast R-CNN, YOLOv3 is much faster without training on RPN and just need to "Look Once" to obtain both the location and classification of the object. As to SSD, which is also fast but inferior on small target detection due to low semantic value for the bottom layer, YOLOv3 is even faster based on the logistic loss while containing the competitive accuracy of detecting small targets since higher resolution layers can obtain higher semantic values.
When the training process is finished, each detected object's bounding box information, including (coordinates$(x,y,w,h)$, $class$ and $confidence$), is automatically set down for non-maximum suppression.
To conduct non-maximum suppression on sub-images, firstly, the bounding boxes information of targets on each sub-image is mapped to the large-scene image using coordinate conversion, and then non-maximum suppression is applied to the targets with multiple bounding boxes, which means the bounding box with the highest score remains and other boxes which have $IOU > 0.7$ with it are deleted. Fig. \ref{NMSS} shows how it works on one target. In the end, we can easily obtain large-scene images with multiple targets detected and recognized.
\begin{figure}[]
\centering
\includegraphics[width=3in]{NMSS}
\caption{Non-maximum suppression on sub-images for one target. The left column is the detection result without non-maximum suppression on sub-images; the right column is the detection result with non-maximum suppression on sub-images.}
\label{NMSS}
\end{figure}
\section{Experimental Results}
\label{3}
\subsection{MSTAR Dataset}
We use the MSTAR dataset to complete our experiments. The MSTAR dataset includes thousands of SAR images, including ten categories of ground military vehicles (armored personnel carrier: BMP2, BRDM2, BTR60, and BTR70; tank: T62 and T72; rocket launcher: 2S1; air defense unit: ZSU234; truck: ZIL131; and bulldozer: D7). They were collected under an X-band SAR sensor, in a 1-ft resolution spotlight mode, full aspect coverage (in the range of $0^{\circ}$ to $360^{\circ}$). The MSTAR dataset is widely used to test the performance of a SAR-ATR system. Fig. \ref{MSTAR} shows the optical images and the corresponding SAR images. The number of images for training in our experiment is summarized in Table \ref{mstar}. Besides small target chips, the MSTAR dataset also provides simple and complex scene images without targets, and these backgrounds include river, sea surface, forest and so on.
\begin{figure}[]
\centering
\includegraphics[width=3.5in]{MSTAR}
\caption{Types of military targets: (top) optical images versus (bottom) SAR images.}
\label{MSTAR}
\end{figure}
\begin{table}[]\footnotesize
\centering
\caption{MSTAR training and testing dataset.}
\label{mstar}
\begin{tabular}{ccccc}
\toprule[2pt]
\multirow{2}{*}{Targets} & \multicolumn{2}{c}{Train} & \multicolumn{2}{c}{Test} \\ \cline{2-5}
& No.Images & Depression & No.Images & Depression \\ \hline
2S1 & 299 & 17${\rm{^\circ }}$ & 274 & 15${\rm{^\circ }}$ \\
BRDM2 & 298 & 17${\rm{^\circ }}$ & 274 & 15${\rm{^\circ }}$ \\
BTR60 & 256 & 17${\rm{^\circ }}$ & 195 & 15${\rm{^\circ }}$ \\
D7 & 299 & 17${\rm{^\circ }}$ & 274 & 15${\rm{^\circ }}$ \\
T62 & 299 & 17${\rm{^\circ }}$ & 273 & 15${\rm{^\circ }}$ \\
ZIL131 & 299 & 17${\rm{^\circ }}$ & 274 & 15${\rm{^\circ }}$ \\
ZSU234 & 299 & 17${\rm{^\circ }}$ & 274 & 15${\rm{^\circ }}$ \\
BMP2 & 233 & 17${\rm{^\circ }}$ & 196 & 15${\rm{^\circ }}$ \\
BTR70 & 233 & 17${\rm{^\circ }}$ & 196 & 15${\rm{^\circ }}$ \\
T72 & 232 & 17${\rm{^\circ }}$ & 196 & 15${\rm{^\circ }}$ \\
\bottomrule[2pt]
\end{tabular}
\end{table}
\subsection{Experiment of Automatic Labeling}
We use the MSTAR ten-class training dataset to conduct the experiment of automatic labeling. The result of the automatic labeling method is shown in Table \ref{label-mstar}. We can see that the average error rate is 1.15\%, and six-class targets including D7, T62, ZSU234, BMP2, BTR70 and T72 are perfectly labeled without missing or not correctly marked, which proves the effectiveness and efficiency of our approach.
\begin{table}[]\footnotesize
\centering
\caption{automatic labeling result on MSTAR ten-class targets.}
\label{label-mstar}
\begin{tabular}{cccc}
\toprule[2pt]
\textbf{Class} & \textbf{Image Num} & \textbf{Not correctly labeled} & \textbf{Error rate/(\%)} \\ \hline
\textbf{2S1} & 299 & 10 & 3.34 \\
\textbf{BRDM2} & 298 & 6 & 2.01 \\
\textbf{BTR60} & 256 & 3 & 1.17 \\
\textbf{D7} & 299 & 0 & 0 \\
\textbf{T62} & 299 & 0 & 0 \\
\textbf{ZIL131} & 299 & 15 & 5.02 \\
\textbf{ZSU234} & 299 & 0 & 0 \\
\textbf{BMP2} & 233 & 0 & 0 \\
\textbf{BTR70} & 233 & 0 & 0 \\
\textbf{T72} & 232 & 0 & 0 \\
\textbf{Average} & - & - & \textbf{1.15} \\
\bottomrule[2pt]
\end{tabular}
\end{table}
\subsection{Detection and Recognition on SAR Target Chips}
In this part, we firstly conduct some experiments on the small SAR target chips to show the effectiveness of the AAE data augmentation method. Since the targets locate right in the center of the image chips, the result can only consist of three aspects: target not detected, target not correctly detected and target correctly detected. Therefore we will use accuracy (ACC) and False Negative Rate (FNR) as indicators, which shows how many targets we have missed and not correctly detected during detection. ACC and FNR are respectively defined as:
\begin{equation}
\label{ACC}
ACC = \frac{{TP}}{{TP + FN}}
\end{equation}
\begin{equation}
\label{FNR}
FNR = \frac{{FN}}{{TP + FN}}
\end{equation}
FN denotes the number of not correctly detected targets and the missing targets; TP denotes the number of correctly detected targets.
\subsubsection{Experiment without data augmentation}
The first experiment is conducted under the YOLOv3 framework without data augmentation, aiming to detect and classify these ten targets. The total 2747 image chips acquired under 17${\rm{^\circ }}$ depression angle are used for training and the 2426 image chips obtained under 15${\rm{^\circ }}$ are tested.
Our YOLOv3 network parameters are shown as follow: $anchors = 10$, $14$, $23$, $27$, $37$, $58$, $81$, $82$, $135$, $169$, $344$, $319$; $class = 10$; $ignore\_thresh = 0.7$;
$true\_thresh = 1$; $random = 1$, and the basic network parameters are set as: $batch\_size = 64$; $subdivisions = 2$; $momentum = 0.9$; $decay = 0.0005$; $learning\_rate = 0.001$; $epoch =200$. Besides that, we use the pretrained weights on COCO image set, and then feed SAR images into our network.
As is presented in Table \ref{without_aae}, the worst accuracy is 92.31\%, which is caused by T62 since 19 targets are recognized as ZSU234. Besides, targets 2S1, BTR60 and BMP2 are also not well correctly detected and classified.
\begin{table*}[]
\centering
\caption{Confusion matrix for ten-class SAR image detection and recognition without AAE.}
\label{without_aae}
\resizebox{\textwidth}{!}
{
\begin{tabular}{cccccccccccccc}
\toprule[2pt]
\textbf{class} & \textbf{2S1} & \textbf{BRDM2} & \textbf{BTR60} & \textbf{D7} & \textbf{T62} & \textbf{ZIL131} & \textbf{ZSU234} & \textbf{BMP2} & \textbf{BTR70} & \textbf{T72} & \textbf{None} & \textbf{ACC(\%)} & \textbf{FNR(\%)} \\ \hline
\textbf{2S1} & 269 & 0 & 0 & 0 & 2 & 3 & 0 & 0 & 0 & 0 & 0 & 98.18 & 1.82 \\
\textbf{BRDM2} & 0 & 271 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 98.91 & 1.09 \\
\textbf{BTR60} & 0 & 1 & 186 & 0 & 0 & 0 & 4 & 0 & 0 & 0 & 4 & 95.38 & 4.62 \\
\textbf{D7} & 0 & 0 & 0 & 268 & 0 & 0 & 6 & 0 & 0 & 0 & 0 & 97.81 & 2.19 \\
\textbf{T62} & 1 & 0 & 0 & 0 & 252 & 0 & 19 & 0 & 0 & 0 & 1 & 92.31 & 7.98 \\
\textbf{ZIL131} & 0 & 0 & 0 & 0 & 0 & 274 & 0 & 0 & 0 & 0 & 0 & 100 & 0 \\
\textbf{ZSU234} & 0 & 0 & 0 & 0 & 0 & 0 & 274 & 0 & 0 & 0 & 0 & 100 & 0 \\
\textbf{BMP2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 191 & 0 & 5 & 0 & 97.45 & 2.55 \\
\textbf{BTR70} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 196 & 0 & 0 & 100 & 0 \\
\textbf{T72} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 196 & 0 & 100 & 0 \\
\textbf{Average} & \multicolumn{11}{c}{} & \textbf{97.98} & \textbf{2.025} \\ \bottomrule[2pt]
\end{tabular}
}
\end{table*}
\subsubsection{Experiment on different generative networks}
In order to improve the classification accuracy, we adopt AAE to expand our dataset. We choose BTR60 (size $128 \times 128$) and set 200 training epochs in this experiment to illustrate the effectiveness of the AAE method. Fig. \ref{aae_gan} shows the generated SAR images under different generative models. To further illustrate the effectiveness of the AAE model, we use Fréchet Inception Distance (FID) \cite{Dowson1982The} to further evaluate the variety of the generated objection, which is defined in Eq. (\ref{gan}). The FID score of different generative models is shown in Table \ref{4}.
\begin{equation}\label{gan}
FID{\rm{ = ||}}{\mu _{\rm{r}}}{\rm{ - }}{\mu _{\rm{g}}}{\rm{|}}{{\rm{|}}^2}{\rm{ + }}Tr(\sum\nolimits_r { + \sum\nolimits_g { - 2(\sum\nolimits_r {\sum\nolimits_g {{)^{1/2}})} } } }
\end{equation}
where ${\mu _{\rm{r}}}$, ${\mu _{\rm{g}}}$ and $\sum\nolimits_r $, $\sum\nolimits_g $ are the respective means and covariance matrices of real and generated images.
\begin{figure}[]
\centering
\includegraphics[width=3.2in]{gan}
\caption{Generated SAR images based on different generative models.}
\label{aae_gan}
\end{figure}
\begin{table}[]\footnotesize
\centering
\caption{FID score on different generative models.}
\begin{tabular}{llllll}
\toprule[2pt]
\textbf{} & \textbf{AAE} & \textbf{GAN} & \textbf{DCGAN} & \textbf{WGAN} & \textbf{InfoGAN} \\ \hline
\textbf{FID} & 195.943 & 226.687 & 401.827 & 353.254 & 399.678 \\
\bottomrule[2pt]
\label{fid}
\end{tabular}
\end{table}
\begin{table*}[]
\centering
\caption{Confusion matrix for ten-class SAR image detection and recognition with AAE.}
\resizebox{\textwidth}{!}
{
\begin{tabular}{@{}cccccccccccccc@{}}
\toprule[2pt]
\textbf{class} & \textbf{2S1} & \textbf{BRDM2} & \textbf{BTR60} & \textbf{D7} & \textbf{T62} & \textbf{ZIL131} & \textbf{ZSU234} & \textbf{BMP2} & \textbf{BTR70} & \textbf{T72} & \textbf{None} & \textbf{ACC(\%)} & \textbf{FNR(\%)} \\ \midrule
\textbf{2S1} & 273 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 99.64 & 0.36 \\
\textbf{BRDM2} & 0 & 272 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 99.27 & 0.72 \\
\textbf{BTR60} & 1 & 2 & 184 & 0 & 0 & 0 & 5 & 0 & 0 & 0 & 3 & 95.83 & 5.64 \\
\textbf{D7} & 0 & 0 & 0 & 271 & 0 & 0 & 3 & 0 & 0 & 0 & 0 & 98.91 & 1.09 \\
\textbf{T62} & 2 & 0 & 0 & 0 & 268 & 1 & 1 & 0 & 0 & 0 & 1 & 98.17 & 1.83 \\
\textbf{ZIL131} & 0 & 0 & 0 & 1 & 0 & 273 & 0 & 0 & 0 & 0 & 0 & 99.64 & 0.36 \\
\textbf{ZSU234} & 0 & 0 & 0 & 0 & 0 & 0 & 274 & 0 & 0 & 0 & 0 & 100 & 0 \\
\textbf{BMP2} & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 193 & 0 & 2 & 0 & 98.47 & 1.53 \\
\textbf{BTR70} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 196 & 0 & 0 & 100 & 0 \\
\textbf{T72} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 195 & 0 & 99.49 & 0.51 \\
\textbf{Average} & \multicolumn{11}{c}{} & \textbf{98.89} & \textbf{1.204} \\ \bottomrule[2pt]
\label{with_aae}
\end{tabular}
}
\end{table*}
\subsubsection{Experiment with data augmentation}
To evaluate the effectiveness of this data augmentation method, we implement it in the ten-class SAR target task.
The number of images generated for training will strongly influence the classification results and a better classification result emerges when the number of generated images equals to half of the number of original images \cite{he2019parallel}. However, it is not sufficient enough to derive the most proper ratio in order to gain the best classification result.
In our experiment, the targets, which are more likely to be recognized by mistake, are 2S1, BTR60, D7 and T62. So we first separately generated 100 images to add to the four-target dataset. However, BTR60 and T62 still occupy a high error rate, thus additional 100 generated images are added into BTR60 and T62 dataset. The final result shows that the accuracy of BTR60 increases by 0.45\%, while the accuracy of T62 raises up by 5.86\%. When we continuously increase the generated image proportion, the outcome does not improve.
As is shown in Table \ref{with_aae}, the average accuracy rate reaches 98.89\%, while the FNR is controlled around 1.2\%. The main error is caused by ZSU234, since some images of BTR60 and D7 are recognized as ZSU234 by mistake.
\subsubsection{Comparison among different ATR methods}
Fig. \ref{compare} illustrates the performance of the proposed method compared to other SAR recognition methods. It has been shown that our method outperforms many methods, i.e., conditionally gaussian model (Cond Gauss) \cite{O2001SAR}, support vector machines (SVM) \cite{Zhao2001Support}, adaptive boosting (AdaBoost) \cite{Sun2007Adaptive}, sparse representation of monogenic signal (MSRC) \cite{NoneSparse}, monogenic scale space (MSS) \cite{DongClassification}, tri-task joint sparse representation (TJSR) \cite{Dong2017SAR}, supervised discriminative dictionary learning and sparse representation (SDDLSR) \cite{song2016sar}, joint dynamic sparse representation (JDSR) \cite{Sun2016SAR} and All-in-one CNN \cite{ding2016convolutional}. In addition, our method achieves a competitive performance compared to the state-of-the-art methods A-ConvNets \cite{chen2016target}, CNN-TL-bypass \cite{huang2017transfer}, discriminative statistical dictionary learning (DSDL) \cite{liu2018sar}, random convolution features and ensemble extreme learning machines (RCFEELM) \cite{Gu2018Fast}.
\begin{figure}[]
\centering
\includegraphics[width=3.5in]{method_compare}
\caption{The classification accuracy of proposed method versus some previous methods and state-of-the-art methods.}
\label{compare}
\end{figure}
\subsubsection{Experiment of anti-noise performance}
In order to test the anti-noise performance of our model, we randomly select a certain proportion of pixels in the test dataset and replace them with samples generated from a uniform distribution, which is shown in Fig \ref{noise}. This noise simulation method is consistent with the approach in \cite{chen2016target, NoneSparse} and the variance of noise can be easily obtained by many method \cite{ulfarsson2008dimension, chu2019eigen}. With the network previously trained by the ten-class SAR dataset, we feed these test images into our model. The anti-noise performance is shown in Fig \ref{noise_compare}, in which we compared the proposed method with four competing methods: SVM, A-ConvNets, MSRC, DSDL. The result shows that with noise proportion added to 20\%, the accuracy of our method is still beyond 98\%, while the other four methods have a comparatively significant drop.
\begin{figure}[]
\centering
\includegraphics[width=3.5in]{noise_corrup}
\caption{Illustration of random noise corruption. (a) is the original image. (b)-(f) are images respectively with 1\%, 5\%, 10\%, 15\% and 20\% noise corruption.}
\label{noise}
\end{figure}
\begin{figure}[]
\centering
\includegraphics[width=3.5in]{noise_compare.eps}
\caption{Average accuracy curves under different algorithms with different percentages of noise corruption.}
\label{noise_compare}
\end{figure}
\subsection{Detection and Recognition on Large-scene SAR Images}
The large-scene SAR images used in this paper were also collected under an X-band SAR sensor, in a 1-ft resolution spotlight mode, with a high resolution of $0.3\times0.3$ in both range and azimuth, which is the same as the small target chips. Therefore, it is reasonable to embed these targets from the $128\times128$ image chips into the large-scene SAR images.
When conducting experiments on complex large-scene SAR images, the evaluation indicators we use are accuracy (ACC), False Negative Rate (FNR), and False Positive Rate (FPR). FNR indicates how many targets we have missed or not correctly detected during detection, and FPR demonstrates the probability that we recognize clutters in the background as the true targets. FPR is defined in Eq. (\ref{FPR}):
\begin{equation}
\label{FPR}
FPR = \frac{{FP}}{{TN + FP}}
\end{equation}
\subsubsection{Experiments on target segmentation and synthesis method}For the purpose of selecting the appropriate threshold to segment the object without shadow from small target chips, we compare our thresholding method with the most popular adaptive thresholding method OTSU. These two methods are applied to the MSTAR ten-class target chips and the result is summarized in Table \ref{threshold_compare}.
The accuracy (ACC) denotes the percentage of the targets which can be correctly segmented from the target chips. Our proposed thresholding method leads to a better performance than the adaptive thresholding method OTSU. As to target D7, T62, ZSU234, BMP2, BTR70 and T72, which have comparatively lower noise, our proposed method does not have an obvious advantage in accuracy. However, when encountering with higher background noises, such as the target 2S1, BRDM2, BTR60 and ZIL131, the accuracy using OTSU drops significantly, especially in BRDM2 and BTR60, where OTSU cannot segment over a half of the total images. The segmentation result on low and high noise images based on these two methods is shown in Fig. \ref{highlowcom}. Besides, the average accuracy of our method reaches 96.28\%, 16.33\% higher than OTSU. Therefore, the proposed thresholding method has a more stable performance as to SAR target segmentation.
\begin{table*}[]
\centering
\caption{Thresholding method comparison on MSTAR ten-class target chips.}
\resizebox{\textwidth}{!}
{
\begin{tabular}{cc|ccccccccccc}
\toprule[2pt]
& & \textbf{2S1} & \textbf{BRDM2} & \textbf{BTR60} & \textbf{D7} & \textbf{T62} & \textbf{ZIL131} & \textbf{ZSU234} & \textbf{BMP2} & \textbf{BTR70} & \textbf{T72} & \textbf{Average} \\ \hline
\multicolumn{2}{c|}{\textbf{Number}} & 274 & 274 & 195 & 274 & 273 & 274 & 299 & 233 & 233 & 232 & - \\ \cline{1-2}
\multicolumn{1}{c|}{\multirow{2}{*}{\textbf{Our method}}} & \textbf{Threshold(\%)} & 92 & 88 & 90 & 92 & 90 & 90 & 95 & 95 & 95 & 95 & - \\ \cline{2-2}
\multicolumn{1}{c|}{} & \textbf{Acc(\%)} & 93.43 & 87.23 & 86.67 & 99.27 & 99.65 & 95.11 & 100 & 100 & 100 & 100 & 96.28 \\ \cline{1-2}
\multicolumn{1}{c|}{\textbf{OTSU}} & \textbf{Acc(\%)} & 53.65 & 32.85 & 43.08 & 98.91 & 88.64 & 79.06 & 100 & 100 & 100 & 100 & 79.95 \\ \bottomrule[2pt]
\label{threshold_compare}
\end{tabular}
}
\end{table*}
\begin{figure}[]
\centering
\includegraphics[width=3.5in]{highlowcom}
\caption{Comparison of OTSU and proposed method on low-noise SAR image and high-noise SAR image for object segmentation without shadow. (a) and (d) are respectively low-noise image and high-noise image; (b) and (e) are results of OTSU method; (c) and (f) are results of proposed method.}
\label{highlowcom}
\end{figure}
\begin{figure}[]
\centering
\includegraphics[width=3.5in]{syntarget}
\caption{Comparison of different target synthesis methods. (a) is the result of the method that we finally choose: bitewise operation and masking; (b) is the result of Poisson Fusion with segmented target; (c) is the result of Poisson Fusion with cropped target.}
\label{synthesistarget}
\end{figure}
As shown in Fig. \ref{synthesistarget}, the method bitwise operation and masking has better synthesis performance, since the object and its shadow are infused naturally into the background. The Poisson image fusion with segmented target softens the edges to a large extent so that it is hard to recognize the infused target. The third method which is to cut a small piece of the original SAR image chip and apply Poisson image fusion, solves the problem of over-softening, but the edge of the original image is still recognizable and it is hard to match the various background of target chips to the same large-scene image. In summary, the first method exhibits better generalization performance and is more suitable for the SAR target synthesis task.
\begin{figure*}[]
\centering
\includegraphics[width=1\textwidth]{large_eg}
\caption{Detection result on three complex large-scene SAR images.}
\label{large_eg}
\end{figure*}
\subsubsection{Detection and recognition on complex large-scene SAR images}
Four MSTAR targets including 2S1, BRDM2, BTR60 and D7 are chosen to be synthesized on the complex large-scene background. There are five same category targets on one large-scene image, which means 20 targets in each image.
In the training process, 35 synthesized large-scene images are randomly chosen and divided into four different sizes ($128 \times 128$, $256 \times 256$, $512 \times 512$, $1024 \times 1024$), and 5 synthesized images are divided for testing. The final size $1024 \times 1024$ is chosen to conduct fast sliding on the testing images, which has the highest detection accuracy rate in the validation part since it is of better robustness with more complex background information. After fast sliding, the total number of training images is 4338, which includes 1091 large-scene images and 3247 expanded small target chips. The number of testing images is 150. The detection results is shown in Fig. \ref{large_eg}. Fig. \ref{matrix} shows the normalized confusion matrix, which reflects four-class targets detection outcome. The accuracy rate raises from 93\% to 94\% after jointly training, and from 91\% to 94\% after data augmentation by AAE. In the meantime, by combining jointly training strategy and AAE method, FNR decreases by 1\%, and FPR drops from 1.33\% to 1\%.
\begin{figure*}[]
\centering
\includegraphics[width=1\textwidth]{matrix}
\caption{Normalized confusion matrix of large-scene SAR images. (a) is the result with AAE data augmentation method; (b) is the result with jointly training strategy; (c) is the result with AAE data augmentation method and jointly training strategy.}
\label{matrix}
\end{figure*}
Table \ref{largescene} shows the detection and recognition performance based on our proposed method and other widely used object detection methods. It is noted that the proposed method is a comprehensive detection framework including target segmentation and synthesis, data augmentation method AAE, automatic labeling, fast sliding and jointly training through YOLOv3 network. For large-scene image detection, it has been demonstrated by the experimental results that the proposed method has 23\% performance gain on accuracy when compared with the one directly applying YOLOv3 network. In addition, it is about 4.2 times faster than the well-established Faster R-CNN method, reaching a 5\% higher accuracy rate, and 3 times faster than SSD, which proves its superior real-time performance.
Thus, in respect of both effectiveness and efficiency, our method reaches 94\% on ACC and only cost 0.038s per image with 6\% FNR and 1\% FPR, which proves that it is a promising framework to deal with real-time detection and recognition on complex large-scene SAR images.
\begin{table}[]
\centering
\caption{Detection and recognition results on large-scene SAR images.}
\resizebox{0.5\textwidth}{!}{
\begin{tabular}{ccccc}
\toprule[2pt]
\textbf{Method} & \textbf{ACC(\%)} & \textbf{FNR(\%)} & \textbf{FPR(\%)} & \textbf{Time cost for detection/seconds} \\ \hline
\textbf{Our method} & \textbf{94} & \textbf{6} & \textbf{1} & \textbf{5.572} \\
\textup{YOLOv3 (applied with fast sliding)} & 93 & 7 & 1.33 & 5.627 \\
\textup{YOLOv3 (applied without fast sliding)} & 71 & 15 & 2.73 & 2.506 \\
\textup{Faster R-CNN (applied with fast sliding)} & 89 & 5 & 0 & 23.441 \\
\textup{Faster R-CNN (applied without fast sliding)} & 79 & 9 & 0 & 19.440 \\
\textup{SSD (applied with fast sliding)} & 85 & 7 & 1.45 & 16.329 \\
\textup{SSD (applied without fast sliding)} & 73 & 13 &3.63 & 9.267 \\
\bottomrule[2pt]
\label{largescene}
\end{tabular}}
\end{table}
To further prove the robustness of our method, a more challenging experiment is conducted. We selected three large-scene images with the most complex background, and laid the targets alongside the trees, as is shown in the following Fig. \ref{target-darkened}. Besides, we darkened the targets by 40\%, so there is lower contrast between the metallic targets and their background. Then we use our original trained weight to perform detection, and the results are shown in Fig. \ref{target-darkened_result} and Table \ref{darkenresult}.
\begin{figure*}[]
\centering
\includegraphics[width=1\textwidth]{target-darkened}
\caption{Synthesized images with darkened-targets and tricky position.}
\label{target-darkened}
\end{figure*}
\begin{figure*}[]
\centering
\includegraphics[width=1\textwidth]{target-darkened_result}
\caption{Detection results of synthesized images with darkened-targets and tricky position.}
\label{target-darkened_result}
\end{figure*}
\begin{table}[]
\centering
\caption{Confusion matrix for four-class large-scene SAR image detection and recognition.}
\resizebox{0.5\textwidth}{!}{
\begin{tabular}{ccccccccc}
\toprule[2pt]
\textbf{class} & \textbf{2S1} & \textbf{BRDM2} & \textbf{BTR60} & \textbf{D7} & \textbf{None} & \textbf{ACC(\%)} & \textbf{FNR(\%)} & \textbf{FPR(\%)} \\ \hline
\textbf{2S1} & 15 & 0 & 0 & 0 & 0 & 100 & 0 & 0 \\
\textbf{BRDM2} & 0 & 13 & 0 & 0 & 2 & 86.67 & 13.33 & 0 \\
\textbf{BTR60} & 2 & 0 & 13 & 0 & 0 & 86.67 & 0 & 13.33 \\
\textbf{D7} & 1 & 0 & 0 & 14 & 0 & 93.33 & 0 & 7.14 \\
\textbf{Average} & & & & & & \textbf{91.67} & \textbf{3.33} & \textbf{5.12} \\
\bottomrule[2pt]
\label{darkenresult}
\end{tabular}
}
\end{table}
It is seen from Table \ref{darkenresult} that low contrast leads to targets been treated as background, since being close to the tree brings much speckle noise interruption and these trees have nearly equal brightness with the targets. Besides, BTR60 and D7 have been recognized as 2S1. The reason may be that 2S1 target is darker as a whole, so the darkening process makes it hard to distinguish some objects from 2S1. However, the proposed method's average accuracy remains above 91\% even under such a tricky condition, proving the effectiveness and robustness of the proposed method.
\section{Discussions and Conclusions}\label{4}
\subsection{Discussions}
In this subsection, we will discuss several experimental results. \textbf{Firstly}, in data augmentation experiments, we found that the generated images by GANs are of far less variety when training epochs are under 200, and it costs much time and space to generate images when the size reaches $128 \times 128$. On the contrary, the images generated by the AAE framework are of high quality and rich diversity. Besides, the training process of the AAE framework can be completed in an efficient manner (the stable result can be obtained within 200 epochs, costing less than 3 seconds). What's more, the FID score of AAE is the lowest compared with GAN-based methods, which proves that the generated images are of richer diversity, thus we choose AAE as our data augmentation method.
\textbf{Secondly}, we conduct an experiment of target recognition on ten-class MSTAR dataset. From Table \ref{with_aae}, we can see that some targets in class BRDM, BTR60, D7, T62 and BMP2 are recognized as ZSU234, which leads to the decrease of final detection accuracy. The reason for that may be that the background of target ZSU234 is darker than other objects, thus the shadow around the targets may misguide the final judgment, as the recognition depends on the detected region. Besides, the FNR is largely caused by BTR60 since some targets in BTR60 have nearly the same pattern so the network treats the noise as the target, thus it is rather hard to tell them apart. But we can clearly see that the accuracy after data augmentation raises to nearly 99\%, and FNR drops from 2.025\% to 1.204\%, which proves the effectiveness of AAE data augmentation method on small target chips.
\textbf{Thirdly}, in the noise corruption experiment, we can see that our framework exhibits high noise immunity. As shown in Fig. \ref{noise_compare}, with noise proportion raises up to 20\%, the accuracy still remains above 98\%. Such superior performance can be explained by the fact that the proposed method is capable of telling the object from its background. As a result, the noise corruption in the background can be further learned as disturbance so that our method can significantly maintain the original object and recognize which category it belongs to.
\textbf{Finally}, the experiment conducted on large-scene images shows that the detection accuracy raises up by 1\% and FNR drops by 1\%, FPR drops by 0.33\% after we simultaneously train the expanded small SAR chips and sliced large-scene SAR images. This can be explained by noticing that the training process of small target chips can make the network learn more about the textural feature of SAR images since the target in small chips is comparatively a rather large object while in large-scene images is only a small object. Therefore, small chips act like a supplementary, assisting the recognition of SAR objects on large-scene images. Besides, the experiment on the more tricky dataset we provided proves that our network learns target textual feature rather than the pure edge information, therefore, it is much more robust. But the false positive cases still remain an intractable problem.
We suppose that the following ideas may be able to reduce the false positive cases. 1) Adopting more data augmentation methods on the objects which are easily detected incorrect, enabling the model to learn more diverse target features and enhance the model's robustness; 2) Methods like hard example mining \cite{shrivastava2016training} and focal loss \cite{lin2017focal} will increase the weight of hard example in training process, which may be beneficial to minimize the false positive cases. While in some situations, however, the shadow may share the same feature with the targets. Under such condition, pre-training can be an effective method to solve this problem. For instance, contrastive learning \cite{chen2020simple,momentum2020he}, targeting at learning an encoder that is able to map positive pairs to similar representations while push away those negative samples in the embedding space. In this way, the pre-trained model may have stronger generalization and feature extraction capability, thus effectively distinguishing target from its shadow.
\subsection{Conclusions}
In this paper, an efficient and robust deep learning based target detection method has been proposed based on a novel customized learning representations and multi-scale features of SAR images method. The framework of AAE has been employed for advanced data augmentation which was confirmed by a high variety of the generated samples. An automatic labeling method has been proposed to avoid the labor-intensive manual labeling. By jointly training the neural network with the small target chips and large-scene images, the proposed integrated target detector has been proposed to realized multiple targets detection and recognition. The experimental results confirmed our method reached competitive accuracy on complex large-scene SAR images with rapid speed. Besides, our method can obtain robust detection performance in terms of the different noise levels, even in the extreme case that the corrupted pixels reach 20\%.
It is noted that there are still some potential problems needed to be tackled in the future: 1) The detection accuracy varies among different categories, and some categories, such as BRDM2, are hard to recognize since their feature is similar to the background; 2) It was found that SAR targets have different rotation angles. Therefore, using rotated anchors to perform targets detection may enhance the final detection accuracy.
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
\bibliographystyle{IEEEtran}
\section{Introduction}
\label{1}
\IEEEPARstart{T}{arget} recognition on SAR images has been under research for many years \cite{wang2016robust,liu2018sar,zhang2018fast,zhao2018adaptive,8533426} due to its various applications in military and homeland security, such as friend and foe identification, battlefield surveillance, environmental monitoring, disaster relief, etc. And it can operate under all-weather and all-time conditions while producing high resolution images with a long standoff capability. Therefore, the SAR image interpretation is of critical importance and the development of automatic target recognition (ATR) system is practical and necessary.
The typical Synthetic Aperture Radar automatic target recognition (SAR-ATR) system can be divided into three parts: target detection, target discrimination and target classification \cite{dudgeon1993overview}. In the first part, a constant false alarm rate (CFAR) detector is used to extract potential targets from SAR images. These potential targets not only consist of true targets such as armored vehicles, rocket launcher and tanks, but also some background clutters such as trees, buildings and rivers. To reduce false alarm rate, the second discrimination part is designed to train a two-class (target and background) model into capturing the true targets by feature extraction. Finally, the third classification part helps decide which category the target belongs to. However, the traditional SAR-ATR system has several disadvantages \cite{morgan2015deep}. Firstly, it relies heavily on handcrafted features, needs large computational space and has poor robustness. Besides, the accuracy will degrade significantly while any of these three stages is not well designed. Lastly, when it comes to both localizing and classifying the multiple targets in the complex background, it is neither effective nor efficient.
To solve this problem, a novel Moving and Stationary Target Acquisition and Recognition (MSTAR) system was developed by the Air Force Research Laboratory and the Defense Advanced Research Projects Agency (AFRL/DARPA) \cite{article}. This dataset contains not only small target chips that are abstracted from the collected data but also simple and complex large-scene backgrounds since it is costly to directly acquire large-scene SAR images with targets. Based on this dataset, a lot of experiments have been conducted which can be summarized into two aspects: classification on small target chips and detection on synthesized large-scene SAR images \cite{cui2019d}.
With the emergence of deep learning methods, neural networks have been gradually applied to those two aspects owing to its superior performance on SAR image processing \cite{wen2018survey, 1202937}. Different from the traditional feature extraction methods which need to design the algorithms manually, neural networks are capable of capturing the inherent feature of the input images. As to the first aspect which is to classify the targets on small target chips, the most commonly used deep learning architecture CNN model \cite{inproceedings} is adopted to conduct ten-class classification on MSTAR target chips, which verifies the validity of deep neural network in the field of SAR target recognition. However, the sample number of each type is limited, thus the experimental results lack some commonality. To tackle with the problem of limited training data, domain-specific data augmentation operations combined with CNN \cite{ding2016convolutional} provides a new way to deal with the problem of translation of target, randomness of speckle noise and lack of pose images together. Since a large amount of data is necessary to train a CNN model, another way to deal with the problem of the limited data is to train a ConvNets model with fewer degrees of freedom by only using a sparsely connected convolution architecture \cite{chen2016target} and in the meanwhile randomly sampling relatively smaller patches from the original SAR images to expand the training set.
As in the above methods, the commonly used data augmentation approaches are horizontal flipping, randomly cropping, rotation, translation or randomly sampling, which means we need to manually control the variety of the additional images by randomly deciding on how many and which ways we are to use. Recently, a newly appeared Generative Adversarial Nets (GAN) proposed by Goodfellow \cite{goodfellow2014generative} is employed to produce more labeled SAR data \cite{he2019parallel}. Though thousands of data can be generated conveniently, not all of them are helpful for classification, so a certain number of generated samples should be carefully selected and it is difficult to find an objective standard to evaluate the quality of the generated images. To avoid the dilemma, another way to make full use of GAN is to train a super-resolution generative adversarial network (SRGAN) \cite{ShiAutomatic} directly to enhance the original images and improve the visual resolution and feature characterization ability of targets in the SAR images. These two methods verify the effective application of the adversarial networks in the SAR image recognition area.
However, GAN-based models have several disadvantages: first, they operate on observation space, which means a large number of parameters are needed during the training process, making it hard to converge; second, due to the high-noise characteristic of SAR dataset, the latent space is more able to capture the main feature of the target in the image which excludes the disturbance of the background. For the sake of solving these two problems, we use a new generative model called Adversarial Autoencoders (AAE) \cite{makhzani2015adversarial}. Different from GANs, AAE blazes a new trail by making the most of latent space. It absorbs the idea of autoencoder \cite{li2017prediction, xiao2020deep} and attempts to push the latent vector close to the distribution of the specific input sample. In this way, AAE is much easier to converge and consumes less space, and our experiment further shows that it also reaches higher quality on generated SAR images. Therefore, in this paper, the AAE network is used to realize data augmentation, and experiments are conducted for improvement on complex large-scene SAR images detection.
So far the above SAR-ATR algorithms are nearly all constructed on CNN framework and the main goal is to classify the targets after the corresponding small chips are abstracted from real large-scene images. In real conditions, however, the targets are randomly scattered into different areas in a real large-scene image with high resolution, and the complex background including trees, buildings, rivers and so on makes it rather hard to accurately detect and recognize them in real-time. Therefore it is under critical research for detecting and recognizing targets on complex large-scene SAR images. Two kinds of algorithms are widely used: two-stage ones, such as R-CNN series \cite{girshick2014rich,girshick2015fast,ren2015faster} and one-stage ones, e.g., SSD \cite{liu2016ssd} and YOLO series \cite{redmon2016you, redmon2017yolo9000,redmon2018yolov3}. The two-stage method Faster R-CNN generally reaches higher accuracy than one-stage methods SSD and YOLOv3 but is time-consuming, too computationally intensive for embedded systems and not suitable for real-time applications. Modified Faster R-CNN models and single shot multibox detector (SSD) are conducted to address SAR-ATR \cite{dong2019end}. It has been shown that MobileNet-SSD and SSD-Inception though have lower accuracy, perform hundreds of times faster than Faster R-CNNs. The work of \citenum{redmon2018yolov3} proposed an improved YOLO network which is known as YOLOv3. This network derives from the older version of YOLO with unique features such as bounding box prediction, class prediction, predictions across scales, feature extractor and training method. The experiment shows that it is three times faster than SSD on COCOs while reaching close detection accuracy. So far, the YOLOv3 network has proved its superiority in many fields such as novel landmark localization \cite{huang2015densebox}, 3-D human detection\cite{Tian2018Robust}, and thermal imaging \cite{ivavsic2019human}. Following aforementioned state-of-art works in the literature, in this paper, we adopt the YOLOv3 as the backbone for realizing effective and efficient SAR-ATR.
When it comes to detecting multiple objects in a large complex SAR background \cite{chang2010change, xu2019retrieval}, a fast sliding method can be used to segment the scene image into sub-images and then detection network is applied to locate the targets. The process of target segmentation and synthesis is of rather importance since it is costly to directly gain the large-scene SAR images with multiple targets inside, therefore this process plays a critical part in the final detection and recognition result.
In this paper, we propose a deep learning framework for detection and recognition on complex large-scene SAR images. Before training the network, AAE is firstly adopted to realize the data augmentation of small SAR chips. Such an operation is simple but useful for the extraction of key feature and enhancing the variety of generated images. In addition, instead of manual labeling, an automatic labeling method is then proposed to mark the targets. Due to the limited number of complex large-scene SAR images, we fully take advantage of small chips and then propose a target segmentation and synthesis method to establish a complex large-scene SAR database for study. After establishing the database, a fast sliding method on large-scene images is proposed to avoid obtaining abundant slices without targets or with incomplete targets. When training the YOLOv3 network, we pretrain the weights of the proposed deep learning method using the well-known COCO dataset by leveraging the advantages of the transfer learning \cite{wang2018sar}. At the training stage, the expanded small target chips and large-scene images after fast sliding are simultaneously fed into the network. Finally, non-maximum suppression on sub-images is conducted to obtain the unique bounding box for each target. The results show that our method exhibits superior accuracy on complex large-scene images and also demonstrates great real-time performance. Furthermore, numerical simulations demonstrate that the proposed method can accurately detect and recognize the targets with high anti-noise performance.
The remainder of this paper is organized as follows. Section \ref{2} elaborates a target detection and recognition framework for complex large-scene SAR images. In Sec. \ref{3}, we verify the effectiveness and efficiency of our proposed approach on a variety of experiments using the MSTAR dataset. The analysis and conclusions are drawn in Sec. \ref{4}.
\section{The ATR Framework}\label{2}
In this section, we will introduce our target detection and recognition framework on complex large-scene SAR images. Since we need to obtain small target chips for joint training, we will first introduce how to expand SAR target chips and conduct automatic labeling in Sec. \ref{2-1}. Then Sec. \ref{2-2} gives a further description of how to establish our large-scene SAR database, and use YOLOv3 for detection.
\subsection{Process on Small SAR Target Chips}\label{2-1}
The proposed ATR model on SAR target chips is shown in Fig. \ref{chips}. It is composed of three parts: data augmentation by AAE, automatic labeling, target detection and recognition. The last part is realized by YOLOv3 after automatically labeling these targets, which means we can not only detect the target but also recognize it with limited samples and without manual labeling. These expanded labeled small chips are fed into the network with large-scene images to enhance the detection accuracy on complex background.
In Fig. \ref{chips}, we use MSTAR four-target dataset, including 2S1, BTR60, BRDM2 and D7 as an example to illustrate the AAE augmentation method and automatic labeling.
\begin{figure}[]
\centering
\includegraphics[width=3.5in]{chip_detect}
\caption{An ATR framework or SAR target chips. \uppercase\expandafter{\romannumeral1}. Data augmentation method AAE is used to expand the training set; \uppercase\expandafter{\romannumeral2}. The training samples are then automatically labeled; \uppercase\expandafter{\romannumeral3}. The training images and labels are sent to the YOLOv3 network to complete target detection and recognition.}
\label{chips}
\end{figure}
\subsubsection{Data augmentation}
One of the key points in SAR image recognition is that SAR images suffer from the speckle noise due to the characteristic of the imaging system. And SAR images for training a robust ATR system is insufficient. For instance, in MSTAR four-target dataset, there are only 1152 images for training, which may lead to an overfitting problem and reduce the generalization effect.
To solve the problem of insufficient training samples, data augmentation is necessary. The classic methods of data augmentation are mostly operating on the original images through flipping, cropping, zooming, etc. This may result in data redundancy and therefore can not obviously enhance the variety of image characteristics.
The adversarial autoencoder (AAE) is a combination of autoencoder and GAN, and it achieves competitive performance on generating SAR target chips. As is shown in Fig. \ref{AAE}, the top row is a standard autoencoder that reconstructs an image $x$ from a latent code $z$.
\begin{equation}
\label{eqn_example}
q(z) = \int_x {q(z|x){p_d}(x)dx}
\end{equation}
The goal of the adversarial autoencoder is to match the aggregated posterior $q(z)$ to $p(z)$, which is an arbitrary prior (e.g. Gaussian distribution). The encoder of the autoencoder ${q(z|x)}$ acts as the generator of the adversarial network, attempting to fool the discriminative adversarial network into recognizing the hidden code $q(z)$ as the prior distribution $p(z)$. In the meanwhile, the autoencoder attempts to reconstruct the input image $x$ from the latent code vector $z$.
\begin{figure}[]
\centering
\includegraphics[width=3in]{aae}
\caption{The architecture of an adversarial autoencoder. The top row is a standard autoencoder that reconstructs an image $x$ from a latent code $z$. The bottom row is a network trained to discriminate whether the sample is from a prior distribution or from the latent vector $z$. }
\label{AAE}
\end{figure}
Different from GAN, in which the input noise lacks semantic information and the output distribution is uncontrollable, AAE largely increases the diversity of the output samples by making the latent code vector simulate the prior distribution. Therefore we can directly expand the training dataset through the generated samples by AAE without carefully selecting which image to use.
\subsubsection{Automatic labeling}
After collecting all the training images, an automatic labeling method is developed to avoid manual labeling, which reduces a large amount of redundant work. The detailed design is shown in Algorithm \ref{auto_label}, and the process is shown in Fig. \ref{auto-label}.
\renewcommand{\algorithmicrequire}{ \textbf{Input:}}
\renewcommand{\algorithmicensure}{ \textbf{Output:}}
\begin{algorithm}
\caption{Automatic labeling.}
\label{auto_label}
\begin{algorithmic}[1]
\REQUIRE ~~\\
The original SAR target chip.\\
\ENSURE ~~\\
The target coordinate $(x,y,w,h)$ and its corresponding category.
\STATE Binarize the original image;\\
\STATE Set a threshold of the white pixels' number and traverse the binarized image from 4 directions;\\
\STATE Count the number of white pixels for each row or column;\\
\STATE Stop traversing when reaching the threshold in each direction;\\
\STATE Form the corresponding rectangle;\\
\STATE Expand rectangle concentrically to a certain extent, e.g. 50\%, to produce a rectangle of proper size.
\RETURN The target label information.
\end{algorithmic}
\end{algorithm}
First, we need to binarize the original image using a thresholding method which will be introduced in Sec. \ref{2-2-1}. Though this binarization method can correctly segment the object, in a few cases there are still some small white spots in the background. To eliminate the effect of white spots in automatic labeling, we set a threshold of white pixel number to filter those spots and finally capture the object accurately. For each row or column, if the number of white pixels is lower than the threshold, we consider it not to constitute the target. We traverse the binarized image from 4 directions (up, down, left and right) to count the number of white pixels for each row or column, and stop traversing when reaching the threshold, formulating a rectangle which contains the center of the target. However, the edge of the target may also not reach the threshold thus may be filtered, so we need to expand the rectangle concentrically by a certain extent, which according to our experiment, is around 50\%, and the white pixel threshold of ten-class target usually lies in the interval [8,12].
\begin{figure}[]
\centering
\includegraphics[width=3in]{auto-label}
\caption{The method of automatic labeling.}
\label{auto-label}
\end{figure}
\subsection{The ATR Framework on Complex Large-scene SAR Images}\label{2-2}
After obtaining labeled small target chips with data augmentation, in this part, we will apply our ATR framework to detect multiple targets on complex large-scene SAR images. As is shown in Fig. \ref{large-scene}, firstly to prepare the large-scene database, we need to segment the target and its shadow from the speckle background; then the target is synthesized into the large-scene background which is acquired under the same depression degree; later a fast sliding method is used to divide the synthesized large-scene image into different sizes and a YOLOv3 network is adopted to train large-scene images and small target chips simultaneously to gain the final result on large-scene images with target categories, confidences and bounding boxes.
\begin{figure}[]
\centering
\includegraphics[width=3.5in]{large-scene}
\caption{An ATR framework for complex large-scene SAR images. \uppercase\expandafter{\romannumeral1}. Target chips are segmented from their backgrounds and then synthesized on the large-scene SAR images. \uppercase\expandafter{\romannumeral2}. Fast sliding is conducted to divide the large-scene images into different sizes. \uppercase\expandafter{\romannumeral3}. Both sliced large-scene images and small target chips are fed into YOLOv3 network simultaneously to gain the final result. \uppercase\expandafter{\romannumeral3}. Finally we map the detection result to the large-scene image and apply non-maximum suppression to gain target with single bounding box.}
\label{large-scene}
\end{figure}
\begin{figure}[]
\centering
\includegraphics[width=2.5in]{object_segmentation}
\caption{Object segmentation process. We sequentially adopt Gaussian blur, thresholding and morphological operation to generate the final segmented object. The threshold value is set as: $p=121$.}
\label{object_segmentation}
\end{figure}
\subsubsection{Database preparation}\label{2-2-1}
Since it is costly to directly obtain large-scene SAR images with multiple targets, a target segmentation and synthesis method is first proposed to establish a training database.
For the target segmentation part, we segment the object and its shadow in two steps, which are represented in Fig. \ref{object_segmentation}. The first step is to segment the object without its shadow. The image is firstly smoothed by Gaussian blur, with the convolution kernel determined adaptively by the image itself. A blurred image has fewer noises and then can be binarized. The selection of the threshold is semi-adaptive and it is the most critical step. Since there is only one target lying in the approximate center in the SAR target chip and it is obviously distinguished from the background, the binarization rule is set as:
\begin{equation}
(x',y') = \left\{ \begin{array}{l}
255\quad if\, (x,y) > p\\
0\quad\quad otherwise
\end{array} \right.
\end{equation}
where $p$ denotes the threshold.
The threshold can be determined by the pixel value proportion of the target in the whole image. The value $p$ is selected around 90\% in most circumstances, which can be estimated from the intuitive area ratio. As is shown in Fig. \ref{threshhold}, the object takes up about 10\% of the image, so we choose $p=121$ in this example as the threshold and apply the binarization rule. This method proves its effectiveness on MSTAR target chips since it can successfully separate more than 95\% objects in the original images.
\begin{figure}[]
\centering
\includegraphics[width=3in]{threshold}
\caption{The pixel value distribution of the SAR image. The left column is the original image and its histogram, and the right column is the result of applying our selected threshold $p=121$.}
\label{threshhold}
\end{figure}
\begin{figure}[]
\centering
\includegraphics[width=3in]{object_shadow_seg}
\caption{Object and shadow segmentation process. The object segmented from step one will be processed through four procedures: object shadowing, lightening and blurring, thresholding, and final morphological operations.}
\label{ob_shsyn}
\end{figure}
In the second step, the object and its shadow are segmented in the meantime, which is exhibited in Fig. \ref{ob_shsyn}. Firstly the segmented object from step one is shadowed to a comparatively small pixel value, then the binarization rule is adopted to highlight both the object and its shadow. After Gaussian blur, an adaptive threshold selection algorithm OTSU is adopted to segment them directly, since the present image has low noise. Then morphological operations are used to improve the segmentation result.
Usually the object and its shadow are separated after binarization, so the closing operation is first adopted for connection, and then the opening operation is applied to clear up small spots while keeping the main body unchanged, at last we add some edge details by dilation to produce the final segmentation result. It is clear to see that the object and its shadow are successfully segmented using our method.
After segmentation, it is of the same importance to synthesize the large-scene image with multiple targets naturally. As is shown in Fig. \ref{ob_sh}, the target synthesis process also goes through two steps. The first one is to design and record the target distribution in the large-scene SAR image. We randomly select 20 coordinates to put four-class targets and each class targets occupy five positions. Since there are some obstacles such as trees, buildings and rivers in the background, and the shadows of these obstacles have a certain direction due to the specific shooting angle and time, it is necessary to carefully select those targets which have the similar direction of shadow and finally design the location of the targets to ensure that they will not fall into the obstacles.
\begin{figure}[]
\centering
\includegraphics[width=3in]{target_synthesis}
\caption{Target synthesis process. Note that the original target chip and the cutted large-scene slice are of the same size.}
\label{ob_sh}
\end{figure}
After choosing the proper target and its coordinate, the second step is to cut a slice of the large-scene image in the corresponding designed position, which has the same shape as the target chip, then the mask produced by target segmentation and its inverse mask are used to perform the bitwise-and operation on the original image and the scene cut respectively. Finally these two operation results are combined to get the final natural synthesized large-scene image.
\subsubsection{Fast sliding}
Since the number of the large-scene images in the MSTAR dataset is limited, directly feed them into the YOLOv3 network will cause severe overfitting problems. Therefore, we can use a fast sliding method to expand the training dataset while still containing complex background information. However, if we randomly set the size of the sliding window and its stride, the target in the scene is likely to be divided into several parts and we may also obtain a large number of slices without targets which will increase the input data redundancy. Therefore, a fast sliding method is proposed which uses sliding windows to cut a synthesized large-scene image into small slices to expand the input data volume. Different from the method in work\cite{cui2019d}, we do not need to make sure that the size of the sliding window is fixed, so when the sliding window almost reaches the edge of the large-scene image, the remaining part which is smaller than the setted size can be directly cut from the image and saved for training. The fast sliding method is shown in Fig. \ref{fast_sliding}.
\begin{figure}[]
\centering
\includegraphics[width=3.5in]{fast_sliding}
\caption{The proposed fast sliding method.
$(x,y)$ represents the target position in the large-scene image; $(x',y')$ denotes the corresponding coordinate in the cropped slice.}
\label{fast_sliding}
\end{figure}
When calculating the corresponding target coordinate in the cropped slice, we respectively use
$height'$ and $width'$ to represent the number of slices that can be obtained from the vertical and horizontal direction (including the last possible incomplete slice). $(i,j)$ denotes the coordinate of a slice in the large-scene image, while $(x,y)$ along with $(x',y')$ respectively denote the coordinate of a target in the large-scene image and the cropped slice. For each sliding window $(i,j)$, we traverse the coordinates of the targets and if there is any target which could meet the following conditions simultaneously, we consider it falling into this sliding window completely.
\begin{equation}
\begin{aligned}
&{x_{\min }} > i * stride\\
&{x_{\max }} < i * stride + size\\
&{y_{\min }} > j * stride\\
&{y_{\max }} < j * stride + size
\end{aligned}
\end{equation}
Meanwhile, the slice will be automatically abandoned if there is no target falling into it or the target is incomplete and the sliding window will move on to the next one until the whole image is covered.
In this paper, we choose four sizes $128 \times 128$, $256 \times 256$, $512 \times 512$, $1024 \times 1024$ to apply fast sliding.
\subsubsection{Training on YOLOv3 network}
With expanded small target chips and large-scene image slices, we feed them into the YOLOv3 network using pre-trained weight on the COCO dataset. The main idea of YOLO is to divide the input image into
$S \times S$ grids and if the center of an object falls into one grid cell, then that grid cell is responsible for predicting the object.
\begin{figure}[]
\centering
\includegraphics[width=3.5in]{yolo}
\caption{The basic YOLO detection system.}
\label{yolo}
\end{figure}
Confidence is defined as
$\Pr (Object) * IOU_{pred}^{truth}$, which reflects how confident the model is that the bounding box contains an object and indicates the accuracy of the prediction.
$\Pr (Clas{s_i}\left| {Object)} \right.$ denotes the probabilities that each grid cell predicts $C$ classes.
\begin{equation}
\label{eqn_example2}
\begin{aligned}
\Pr (Clas{s_i}|Object) * \Pr (Object) * IOU_{pred}^{truth} \\
= \Pr (Clas{s_i}) * IOU_{pred}^{truth}
\end{aligned}
\end{equation}
By multiplying these two parts, we can obtain both the probability of that class appearing in the box and how well the predicted box fits the object. The loss function is defined as:
{\small
\begin{equation}
\centering
\begin{aligned}
loss = &{\lambda _{coord}}\sum\limits_{i = 0}^{{S^2}} {\sum\limits_{j = 0}^B { ]\kern-0.15em] _{ij}^{obj}[({x_i} - {{\hat x}_i}} } {)^2} + {({y_i} - {\hat y_i})^2}]\\
+ &{\lambda _{coord}}\sum\limits_{i = 0}^{{S^2}} {\sum\limits_{j = 0}^B { ]\kern-0.15em] _{ij}^{obj}[(\sqrt {{\omega _i}} - \sqrt {{{\hat \omega }_i}} } } {)^2} + {(\sqrt {{h_i}} - \sqrt {{{\hat h}_i}} )^2}]\\
+ &\sum\limits_{i = 0}^{{S^2}} {\sum\limits_{j = 0}^B { ]\kern-0.15em] _{ij}^{obj}{{({C_i} - {{\hat C}_i})}^2}} }+ \sum\limits_{i = 0}^{{S^2}} { ]\kern-0.15em] _i^{obj}\sum\limits_{c \in classes} {{{({p_i}(c) - {{\hat p}_i}(c))}^2}} } \\
+ &{\lambda _{noobj}}\sum\limits_{i = 0}^{{S^2}} {\sum\limits_{j = 0}^B { ]\kern-0.15em] _{ij}^{noobj}{{({C_i} - {{\hat C}_i})}^2}} }
\end{aligned}
\end{equation}
}
where ${ ]\kern-0.15em] _i^{obj}}$ denotes whether the object appears in cell $i$, and
${ ]\kern-0.15em] _{ij}^{obj}}$ denotes that the $jth$ bounding box in cell $i$ is responsible for the prediction. $\lambda _{coord}$ is used to increase the loss from bounding box coordinate predictions and $\lambda _{noobj}$ is to decrease the loss from confident predictions for boxes that don't contain objects. YOLOv3 has represented superior performance on both accuracy and speed. Compared with Faster R-CNN, which needs to repeatedly train the region proposal network (RPN) and Fast R-CNN, YOLOv3 is much faster without training on RPN and just need to "Look Once" to obtain both the location and classification of the object. As to SSD, which is also fast but inferior on small target detection due to low semantic value for the bottom layer, YOLOv3 is even faster based on the logistic loss while containing the competitive accuracy of detecting small targets since higher resolution layers can obtain higher semantic values.
When the training process is finished, each detected object's bounding box information, including (coordinates$(x,y,w,h)$, $class$ and $confidence$), is automatically set down for non-maximum suppression.
To conduct non-maximum suppression on sub-images, firstly, the bounding boxes information of targets on each sub-image is mapped to the large-scene image using coordinate conversion, and then non-maximum suppression is applied to the targets with multiple bounding boxes, which means the bounding box with the highest score remains and other boxes which have $IOU > 0.7$ with it are deleted. Fig. \ref{NMSS} shows how it works on one target. In the end, we can easily obtain large-scene images with multiple targets detected and recognized.
\begin{figure}[]
\centering
\includegraphics[width=3in]{NMSS}
\caption{Non-maximum suppression on sub-images for one target. The left column is the detection result without non-maximum suppression on sub-images; the right column is the detection result with non-maximum suppression on sub-images.}
\label{NMSS}
\end{figure}
\section{Experimental Results}
\label{3}
\subsection{MSTAR Dataset}
We use the MSTAR dataset to complete our experiments. The MSTAR dataset includes thousands of SAR images, including ten categories of ground military vehicles (armored personnel carrier: BMP2, BRDM2, BTR60, and BTR70; tank: T62 and T72; rocket launcher: 2S1; air defense unit: ZSU234; truck: ZIL131; and bulldozer: D7). They were collected under an X-band SAR sensor, in a 1-ft resolution spotlight mode, full aspect coverage (in the range of $0^{\circ}$ to $360^{\circ}$). The MSTAR dataset is widely used to test the performance of a SAR-ATR system. Fig. \ref{MSTAR} shows the optical images and the corresponding SAR images. The number of images for training in our experiment is summarized in Table \ref{mstar}. Besides small target chips, the MSTAR dataset also provides simple and complex scene images without targets, and these backgrounds include river, sea surface, forest and so on.
\begin{figure}[]
\centering
\includegraphics[width=3.5in]{MSTAR}
\caption{Types of military targets: (top) optical images versus (bottom) SAR images.}
\label{MSTAR}
\end{figure}
\begin{table}[]\footnotesize
\centering
\caption{MSTAR training and testing dataset.}
\label{mstar}
\begin{tabular}{ccccc}
\toprule[2pt]
\multirow{2}{*}{Targets} & \multicolumn{2}{c}{Train} & \multicolumn{2}{c}{Test} \\ \cline{2-5}
& No.Images & Depression & No.Images & Depression \\ \hline
2S1 & 299 & 17${\rm{^\circ }}$ & 274 & 15${\rm{^\circ }}$ \\
BRDM2 & 298 & 17${\rm{^\circ }}$ & 274 & 15${\rm{^\circ }}$ \\
BTR60 & 256 & 17${\rm{^\circ }}$ & 195 & 15${\rm{^\circ }}$ \\
D7 & 299 & 17${\rm{^\circ }}$ & 274 & 15${\rm{^\circ }}$ \\
T62 & 299 & 17${\rm{^\circ }}$ & 273 & 15${\rm{^\circ }}$ \\
ZIL131 & 299 & 17${\rm{^\circ }}$ & 274 & 15${\rm{^\circ }}$ \\
ZSU234 & 299 & 17${\rm{^\circ }}$ & 274 & 15${\rm{^\circ }}$ \\
BMP2 & 233 & 17${\rm{^\circ }}$ & 196 & 15${\rm{^\circ }}$ \\
BTR70 & 233 & 17${\rm{^\circ }}$ & 196 & 15${\rm{^\circ }}$ \\
T72 & 232 & 17${\rm{^\circ }}$ & 196 & 15${\rm{^\circ }}$ \\
\bottomrule[2pt]
\end{tabular}
\end{table}
\subsection{Experiment of Automatic Labeling}
We use the MSTAR ten-class training dataset to conduct the experiment of automatic labeling. The result of the automatic labeling method is shown in Table \ref{label-mstar}. We can see that the average error rate is 1.15\%, and six-class targets including D7, T62, ZSU234, BMP2, BTR70 and T72 are perfectly labeled without missing or not correctly marked, which proves the effectiveness and efficiency of our approach.
\begin{table}[]\footnotesize
\centering
\caption{automatic labeling result on MSTAR ten-class targets.}
\label{label-mstar}
\begin{tabular}{cccc}
\toprule[2pt]
\textbf{Class} & \textbf{Image Num} & \textbf{Not correctly labeled} & \textbf{Error rate/(\%)} \\ \hline
\textbf{2S1} & 299 & 10 & 3.34 \\
\textbf{BRDM2} & 298 & 6 & 2.01 \\
\textbf{BTR60} & 256 & 3 & 1.17 \\
\textbf{D7} & 299 & 0 & 0 \\
\textbf{T62} & 299 & 0 & 0 \\
\textbf{ZIL131} & 299 & 15 & 5.02 \\
\textbf{ZSU234} & 299 & 0 & 0 \\
\textbf{BMP2} & 233 & 0 & 0 \\
\textbf{BTR70} & 233 & 0 & 0 \\
\textbf{T72} & 232 & 0 & 0 \\
\textbf{Average} & - & - & \textbf{1.15} \\
\bottomrule[2pt]
\end{tabular}
\end{table}
\subsection{Detection and Recognition on SAR Target Chips}
In this part, we firstly conduct some experiments on the small SAR target chips to show the effectiveness of the AAE data augmentation method. Since the targets locate right in the center of the image chips, the result can only consist of three aspects: target not detected, target not correctly detected and target correctly detected. Therefore we will use accuracy (ACC) and False Negative Rate (FNR) as indicators, which shows how many targets we have missed and not correctly detected during detection. ACC and FNR are respectively defined as:
\begin{equation}
\label{ACC}
ACC = \frac{{TP}}{{TP + FN}}
\end{equation}
\begin{equation}
\label{FNR}
FNR = \frac{{FN}}{{TP + FN}}
\end{equation}
FN denotes the number of not correctly detected targets and the missing targets; TP denotes the number of correctly detected targets.
\subsubsection{Experiment without data augmentation}
The first experiment is conducted under the YOLOv3 framework without data augmentation, aiming to detect and classify these ten targets. The total 2747 image chips acquired under 17${\rm{^\circ }}$ depression angle are used for training and the 2426 image chips obtained under 15${\rm{^\circ }}$ are tested.
Our YOLOv3 network parameters are shown as follow: $anchors = 10$, $14$, $23$, $27$, $37$, $58$, $81$, $82$, $135$, $169$, $344$, $319$; $class = 10$; $ignore\_thresh = 0.7$;
$true\_thresh = 1$; $random = 1$, and the basic network parameters are set as: $batch\_size = 64$; $subdivisions = 2$; $momentum = 0.9$; $decay = 0.0005$; $learning\_rate = 0.001$; $epoch =200$. Besides that, we use the pretrained weights on COCO image set, and then feed SAR images into our network.
As is presented in Table \ref{without_aae}, the worst accuracy is 92.31\%, which is caused by T62 since 19 targets are recognized as ZSU234. Besides, targets 2S1, BTR60 and BMP2 are also not well correctly detected and classified.
\begin{table*}[]
\centering
\caption{Confusion matrix for ten-class SAR image detection and recognition without AAE.}
\label{without_aae}
\resizebox{\textwidth}{!}
{
\begin{tabular}{cccccccccccccc}
\toprule[2pt]
\textbf{class} & \textbf{2S1} & \textbf{BRDM2} & \textbf{BTR60} & \textbf{D7} & \textbf{T62} & \textbf{ZIL131} & \textbf{ZSU234} & \textbf{BMP2} & \textbf{BTR70} & \textbf{T72} & \textbf{None} & \textbf{ACC(\%)} & \textbf{FNR(\%)} \\ \hline
\textbf{2S1} & 269 & 0 & 0 & 0 & 2 & 3 & 0 & 0 & 0 & 0 & 0 & 98.18 & 1.82 \\
\textbf{BRDM2} & 0 & 271 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 98.91 & 1.09 \\
\textbf{BTR60} & 0 & 1 & 186 & 0 & 0 & 0 & 4 & 0 & 0 & 0 & 4 & 95.38 & 4.62 \\
\textbf{D7} & 0 & 0 & 0 & 268 & 0 & 0 & 6 & 0 & 0 & 0 & 0 & 97.81 & 2.19 \\
\textbf{T62} & 1 & 0 & 0 & 0 & 252 & 0 & 19 & 0 & 0 & 0 & 1 & 92.31 & 7.98 \\
\textbf{ZIL131} & 0 & 0 & 0 & 0 & 0 & 274 & 0 & 0 & 0 & 0 & 0 & 100 & 0 \\
\textbf{ZSU234} & 0 & 0 & 0 & 0 & 0 & 0 & 274 & 0 & 0 & 0 & 0 & 100 & 0 \\
\textbf{BMP2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 191 & 0 & 5 & 0 & 97.45 & 2.55 \\
\textbf{BTR70} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 196 & 0 & 0 & 100 & 0 \\
\textbf{T72} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 196 & 0 & 100 & 0 \\
\textbf{Average} & \multicolumn{11}{c}{} & \textbf{97.98} & \textbf{2.025} \\ \bottomrule[2pt]
\end{tabular}
}
\end{table*}
\subsubsection{Experiment on different generative networks}
In order to improve the classification accuracy, we adopt AAE to expand our dataset. We choose BTR60 (size $128 \times 128$) and set 200 training epochs in this experiment to illustrate the effectiveness of the AAE method. Fig. \ref{aae_gan} shows the generated SAR images under different generative models. To further illustrate the effectiveness of the AAE model, we use Fréchet Inception Distance (FID) \cite{Dowson1982The} to further evaluate the variety of the generated objection, which is defined in Eq. (\ref{gan}). The FID score of different generative models is shown in Table \ref{4}.
\begin{equation}\label{gan}
FID{\rm{ = ||}}{\mu _{\rm{r}}}{\rm{ - }}{\mu _{\rm{g}}}{\rm{|}}{{\rm{|}}^2}{\rm{ + }}Tr(\sum\nolimits_r { + \sum\nolimits_g { - 2(\sum\nolimits_r {\sum\nolimits_g {{)^{1/2}})} } } }
\end{equation}
where ${\mu _{\rm{r}}}$, ${\mu _{\rm{g}}}$ and $\sum\nolimits_r $, $\sum\nolimits_g $ are the respective means and covariance matrices of real and generated images.
\begin{figure}[]
\centering
\includegraphics[width=3.2in]{gan}
\caption{Generated SAR images based on different generative models.}
\label{aae_gan}
\end{figure}
\begin{table}[]\footnotesize
\centering
\caption{FID score on different generative models.}
\begin{tabular}{llllll}
\toprule[2pt]
\textbf{} & \textbf{AAE} & \textbf{GAN} & \textbf{DCGAN} & \textbf{WGAN} & \textbf{InfoGAN} \\ \hline
\textbf{FID} & 195.943 & 226.687 & 401.827 & 353.254 & 399.678 \\
\bottomrule[2pt]
\label{fid}
\end{tabular}
\end{table}
\begin{table*}[]
\centering
\caption{Confusion matrix for ten-class SAR image detection and recognition with AAE.}
\resizebox{\textwidth}{!}
{
\begin{tabular}{@{}cccccccccccccc@{}}
\toprule[2pt]
\textbf{class} & \textbf{2S1} & \textbf{BRDM2} & \textbf{BTR60} & \textbf{D7} & \textbf{T62} & \textbf{ZIL131} & \textbf{ZSU234} & \textbf{BMP2} & \textbf{BTR70} & \textbf{T72} & \textbf{None} & \textbf{ACC(\%)} & \textbf{FNR(\%)} \\ \midrule
\textbf{2S1} & 273 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 99.64 & 0.36 \\
\textbf{BRDM2} & 0 & 272 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 99.27 & 0.72 \\
\textbf{BTR60} & 1 & 2 & 184 & 0 & 0 & 0 & 5 & 0 & 0 & 0 & 3 & 95.83 & 5.64 \\
\textbf{D7} & 0 & 0 & 0 & 271 & 0 & 0 & 3 & 0 & 0 & 0 & 0 & 98.91 & 1.09 \\
\textbf{T62} & 2 & 0 & 0 & 0 & 268 & 1 & 1 & 0 & 0 & 0 & 1 & 98.17 & 1.83 \\
\textbf{ZIL131} & 0 & 0 & 0 & 1 & 0 & 273 & 0 & 0 & 0 & 0 & 0 & 99.64 & 0.36 \\
\textbf{ZSU234} & 0 & 0 & 0 & 0 & 0 & 0 & 274 & 0 & 0 & 0 & 0 & 100 & 0 \\
\textbf{BMP2} & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 193 & 0 & 2 & 0 & 98.47 & 1.53 \\
\textbf{BTR70} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 196 & 0 & 0 & 100 & 0 \\
\textbf{T72} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 195 & 0 & 99.49 & 0.51 \\
\textbf{Average} & \multicolumn{11}{c}{} & \textbf{98.89} & \textbf{1.204} \\ \bottomrule[2pt]
\label{with_aae}
\end{tabular}
}
\end{table*}
\subsubsection{Experiment with data augmentation}
To evaluate the effectiveness of this data augmentation method, we implement it in the ten-class SAR target task.
The number of images generated for training will strongly influence the classification results and a better classification result emerges when the number of generated images equals to half of the number of original images \cite{he2019parallel}. However, it is not sufficient enough to derive the most proper ratio in order to gain the best classification result.
In our experiment, the targets, which are more likely to be recognized by mistake, are 2S1, BTR60, D7 and T62. So we first separately generated 100 images to add to the four-target dataset. However, BTR60 and T62 still occupy a high error rate, thus additional 100 generated images are added into BTR60 and T62 dataset. The final result shows that the accuracy of BTR60 increases by 0.45\%, while the accuracy of T62 raises up by 5.86\%. When we continuously increase the generated image proportion, the outcome does not improve.
As is shown in Table \ref{with_aae}, the average accuracy rate reaches 98.89\%, while the FNR is controlled around 1.2\%. The main error is caused by ZSU234, since some images of BTR60 and D7 are recognized as ZSU234 by mistake.
\subsubsection{Comparison among different ATR methods}
Fig. \ref{compare} illustrates the performance of the proposed method compared to other SAR recognition methods. It has been shown that our method outperforms many methods, i.e., conditionally gaussian model (Cond Gauss) \cite{O2001SAR}, support vector machines (SVM) \cite{Zhao2001Support}, adaptive boosting (AdaBoost) \cite{Sun2007Adaptive}, sparse representation of monogenic signal (MSRC) \cite{NoneSparse}, monogenic scale space (MSS) \cite{DongClassification}, tri-task joint sparse representation (TJSR) \cite{Dong2017SAR}, supervised discriminative dictionary learning and sparse representation (SDDLSR) \cite{song2016sar}, joint dynamic sparse representation (JDSR) \cite{Sun2016SAR} and All-in-one CNN \cite{ding2016convolutional}. In addition, our method achieves a competitive performance compared to the state-of-the-art methods A-ConvNets \cite{chen2016target}, CNN-TL-bypass \cite{huang2017transfer}, discriminative statistical dictionary learning (DSDL) \cite{liu2018sar}, random convolution features and ensemble extreme learning machines (RCFEELM) \cite{Gu2018Fast}.
\begin{figure}[]
\centering
\includegraphics[width=3.5in]{method_compare}
\caption{The classification accuracy of proposed method versus some previous methods and state-of-the-art methods.}
\label{compare}
\end{figure}
\subsubsection{Experiment of anti-noise performance}
In order to test the anti-noise performance of our model, we randomly select a certain proportion of pixels in the test dataset and replace them with samples generated from a uniform distribution, which is shown in Fig \ref{noise}. This noise simulation method is consistent with the approach in \cite{chen2016target, NoneSparse} and the variance of noise can be easily obtained by many method \cite{ulfarsson2008dimension, chu2019eigen}. With the network previously trained by the ten-class SAR dataset, we feed these test images into our model. The anti-noise performance is shown in Fig \ref{noise_compare}, in which we compared the proposed method with four competing methods: SVM, A-ConvNets, MSRC, DSDL. The result shows that with noise proportion added to 20\%, the accuracy of our method is still beyond 98\%, while the other four methods have a comparatively significant drop.
\begin{figure}[]
\centering
\includegraphics[width=3.5in]{noise_corrup}
\caption{Illustration of random noise corruption. (a) is the original image. (b)-(f) are images respectively with 1\%, 5\%, 10\%, 15\% and 20\% noise corruption.}
\label{noise}
\end{figure}
\begin{figure}[]
\centering
\includegraphics[width=3.5in]{noise_compare.eps}
\caption{Average accuracy curves under different algorithms with different percentages of noise corruption.}
\label{noise_compare}
\end{figure}
\subsection{Detection and Recognition on Large-scene SAR Images}
The large-scene SAR images used in this paper were also collected under an X-band SAR sensor, in a 1-ft resolution spotlight mode, with a high resolution of $0.3\times0.3$ in both range and azimuth, which is the same as the small target chips. Therefore, it is reasonable to embed these targets from the $128\times128$ image chips into the large-scene SAR images.
When conducting experiments on complex large-scene SAR images, the evaluation indicators we use are accuracy (ACC), False Negative Rate (FNR), and False Positive Rate (FPR). FNR indicates how many targets we have missed or not correctly detected during detection, and FPR demonstrates the probability that we recognize clutters in the background as the true targets. FPR is defined in Eq. (\ref{FPR}):
\begin{equation}
\label{FPR}
FPR = \frac{{FP}}{{TN + FP}}
\end{equation}
\subsubsection{Experiments on target segmentation and synthesis method}For the purpose of selecting the appropriate threshold to segment the object without shadow from small target chips, we compare our thresholding method with the most popular adaptive thresholding method OTSU. These two methods are applied to the MSTAR ten-class target chips and the result is summarized in Table \ref{threshold_compare}.
The accuracy (ACC) denotes the percentage of the targets which can be correctly segmented from the target chips. Our proposed thresholding method leads to a better performance than the adaptive thresholding method OTSU. As to target D7, T62, ZSU234, BMP2, BTR70 and T72, which have comparatively lower noise, our proposed method does not have an obvious advantage in accuracy. However, when encountering with higher background noises, such as the target 2S1, BRDM2, BTR60 and ZIL131, the accuracy using OTSU drops significantly, especially in BRDM2 and BTR60, where OTSU cannot segment over a half of the total images. The segmentation result on low and high noise images based on these two methods is shown in Fig. \ref{highlowcom}. Besides, the average accuracy of our method reaches 96.28\%, 16.33\% higher than OTSU. Therefore, the proposed thresholding method has a more stable performance as to SAR target segmentation.
\begin{table*}[]
\centering
\caption{Thresholding method comparison on MSTAR ten-class target chips.}
\resizebox{\textwidth}{!}
{
\begin{tabular}{cc|ccccccccccc}
\toprule[2pt]
& & \textbf{2S1} & \textbf{BRDM2} & \textbf{BTR60} & \textbf{D7} & \textbf{T62} & \textbf{ZIL131} & \textbf{ZSU234} & \textbf{BMP2} & \textbf{BTR70} & \textbf{T72} & \textbf{Average} \\ \hline
\multicolumn{2}{c|}{\textbf{Number}} & 274 & 274 & 195 & 274 & 273 & 274 & 299 & 233 & 233 & 232 & - \\ \cline{1-2}
\multicolumn{1}{c|}{\multirow{2}{*}{\textbf{Our method}}} & \textbf{Threshold(\%)} & 92 & 88 & 90 & 92 & 90 & 90 & 95 & 95 & 95 & 95 & - \\ \cline{2-2}
\multicolumn{1}{c|}{} & \textbf{Acc(\%)} & 93.43 & 87.23 & 86.67 & 99.27 & 99.65 & 95.11 & 100 & 100 & 100 & 100 & 96.28 \\ \cline{1-2}
\multicolumn{1}{c|}{\textbf{OTSU}} & \textbf{Acc(\%)} & 53.65 & 32.85 & 43.08 & 98.91 & 88.64 & 79.06 & 100 & 100 & 100 & 100 & 79.95 \\ \bottomrule[2pt]
\label{threshold_compare}
\end{tabular}
}
\end{table*}
\begin{figure}[]
\centering
\includegraphics[width=3.5in]{highlowcom}
\caption{Comparison of OTSU and proposed method on low-noise SAR image and high-noise SAR image for object segmentation without shadow. (a) and (d) are respectively low-noise image and high-noise image; (b) and (e) are results of OTSU method; (c) and (f) are results of proposed method.}
\label{highlowcom}
\end{figure}
\begin{figure}[]
\centering
\includegraphics[width=3.5in]{syntarget}
\caption{Comparison of different target synthesis methods. (a) is the result of the method that we finally choose: bitewise operation and masking; (b) is the result of Poisson Fusion with segmented target; (c) is the result of Poisson Fusion with cropped target.}
\label{synthesistarget}
\end{figure}
As shown in Fig. \ref{synthesistarget}, the method bitwise operation and masking has better synthesis performance, since the object and its shadow are infused naturally into the background. The Poisson image fusion with segmented target softens the edges to a large extent so that it is hard to recognize the infused target. The third method which is to cut a small piece of the original SAR image chip and apply Poisson image fusion, solves the problem of over-softening, but the edge of the original image is still recognizable and it is hard to match the various background of target chips to the same large-scene image. In summary, the first method exhibits better generalization performance and is more suitable for the SAR target synthesis task.
\begin{figure*}[]
\centering
\includegraphics[width=1\textwidth]{large_eg}
\caption{Detection result on three complex large-scene SAR images.}
\label{large_eg}
\end{figure*}
\subsubsection{Detection and recognition on complex large-scene SAR images}
Four MSTAR targets including 2S1, BRDM2, BTR60 and D7 are chosen to be synthesized on the complex large-scene background. There are five same category targets on one large-scene image, which means 20 targets in each image.
In the training process, 35 synthesized large-scene images are randomly chosen and divided into four different sizes ($128 \times 128$, $256 \times 256$, $512 \times 512$, $1024 \times 1024$), and 5 synthesized images are divided for testing. The final size $1024 \times 1024$ is chosen to conduct fast sliding on the testing images, which has the highest detection accuracy rate in the validation part since it is of better robustness with more complex background information. After fast sliding, the total number of training images is 4338, which includes 1091 large-scene images and 3247 expanded small target chips. The number of testing images is 150. The detection results is shown in Fig. \ref{large_eg}. Fig. \ref{matrix} shows the normalized confusion matrix, which reflects four-class targets detection outcome. The accuracy rate raises from 93\% to 94\% after jointly training, and from 91\% to 94\% after data augmentation by AAE. In the meantime, by combining jointly training strategy and AAE method, FNR decreases by 1\%, and FPR drops from 1.33\% to 1\%.
\begin{figure*}[]
\centering
\includegraphics[width=1\textwidth]{matrix}
\caption{Normalized confusion matrix of large-scene SAR images. (a) is the result with AAE data augmentation method; (b) is the result with jointly training strategy; (c) is the result with AAE data augmentation method and jointly training strategy.}
\label{matrix}
\end{figure*}
Table \ref{largescene} shows the detection and recognition performance based on our proposed method and other widely used object detection methods. It is noted that the proposed method is a comprehensive detection framework including target segmentation and synthesis, data augmentation method AAE, automatic labeling, fast sliding and jointly training through YOLOv3 network. For large-scene image detection, it has been demonstrated by the experimental results that the proposed method has 23\% performance gain on accuracy when compared with the one directly applying YOLOv3 network. In addition, it is about 4.2 times faster than the well-established Faster R-CNN method, reaching a 5\% higher accuracy rate, and 3 times faster than SSD, which proves its superior real-time performance.
Thus, in respect of both effectiveness and efficiency, our method reaches 94\% on ACC and only cost 0.038s per image with 6\% FNR and 1\% FPR, which proves that it is a promising framework to deal with real-time detection and recognition on complex large-scene SAR images.
\begin{table}[]
\centering
\caption{Detection and recognition results on large-scene SAR images.}
\resizebox{0.5\textwidth}{!}{
\begin{tabular}{ccccc}
\toprule[2pt]
\textbf{Method} & \textbf{ACC(\%)} & \textbf{FNR(\%)} & \textbf{FPR(\%)} & \textbf{Time cost for detection/seconds} \\ \hline
\textbf{Our method} & \textbf{94} & \textbf{6} & \textbf{1} & \textbf{5.572} \\
\textup{YOLOv3 (applied with fast sliding)} & 93 & 7 & 1.33 & 5.627 \\
\textup{YOLOv3 (applied without fast sliding)} & 71 & 15 & 2.73 & 2.506 \\
\textup{Faster R-CNN (applied with fast sliding)} & 89 & 5 & 0 & 23.441 \\
\textup{Faster R-CNN (applied without fast sliding)} & 79 & 9 & 0 & 19.440 \\
\textup{SSD (applied with fast sliding)} & 85 & 7 & 1.45 & 16.329 \\
\textup{SSD (applied without fast sliding)} & 73 & 13 &3.63 & 9.267 \\
\bottomrule[2pt]
\label{largescene}
\end{tabular}}
\end{table}
To further prove the robustness of our method, a more challenging experiment is conducted. We selected three large-scene images with the most complex background, and laid the targets alongside the trees, as is shown in the following Fig. \ref{target-darkened}. Besides, we darkened the targets by 40\%, so there is lower contrast between the metallic targets and their background. Then we use our original trained weight to perform detection, and the results are shown in Fig. \ref{target-darkened_result} and Table \ref{darkenresult}.
\begin{figure*}[]
\centering
\includegraphics[width=1\textwidth]{target-darkened}
\caption{Synthesized images with darkened-targets and tricky position.}
\label{target-darkened}
\end{figure*}
\begin{figure*}[]
\centering
\includegraphics[width=1\textwidth]{target-darkened_result}
\caption{Detection results of synthesized images with darkened-targets and tricky position.}
\label{target-darkened_result}
\end{figure*}
\begin{table}[]
\centering
\caption{Confusion matrix for four-class large-scene SAR image detection and recognition.}
\resizebox{0.5\textwidth}{!}{
\begin{tabular}{ccccccccc}
\toprule[2pt]
\textbf{class} & \textbf{2S1} & \textbf{BRDM2} & \textbf{BTR60} & \textbf{D7} & \textbf{None} & \textbf{ACC(\%)} & \textbf{FNR(\%)} & \textbf{FPR(\%)} \\ \hline
\textbf{2S1} & 15 & 0 & 0 & 0 & 0 & 100 & 0 & 0 \\
\textbf{BRDM2} & 0 & 13 & 0 & 0 & 2 & 86.67 & 13.33 & 0 \\
\textbf{BTR60} & 2 & 0 & 13 & 0 & 0 & 86.67 & 0 & 13.33 \\
\textbf{D7} & 1 & 0 & 0 & 14 & 0 & 93.33 & 0 & 7.14 \\
\textbf{Average} & & & & & & \textbf{91.67} & \textbf{3.33} & \textbf{5.12} \\
\bottomrule[2pt]
\label{darkenresult}
\end{tabular}
}
\end{table}
It is seen from Table \ref{darkenresult} that low contrast leads to targets been treated as background, since being close to the tree brings much speckle noise interruption and these trees have nearly equal brightness with the targets. Besides, BTR60 and D7 have been recognized as 2S1. The reason may be that 2S1 target is darker as a whole, so the darkening process makes it hard to distinguish some objects from 2S1. However, the proposed method's average accuracy remains above 91\% even under such a tricky condition, proving the effectiveness and robustness of the proposed method.
\section{Discussions and Conclusions}\label{4}
\subsection{Discussions}
In this subsection, we will discuss several experimental results. \textbf{Firstly}, in data augmentation experiments, we found that the generated images by GANs are of far less variety when training epochs are under 200, and it costs much time and space to generate images when the size reaches $128 \times 128$. On the contrary, the images generated by the AAE framework are of high quality and rich diversity. Besides, the training process of the AAE framework can be completed in an efficient manner (the stable result can be obtained within 200 epochs, costing less than 3 seconds). What's more, the FID score of AAE is the lowest compared with GAN-based methods, which proves that the generated images are of richer diversity, thus we choose AAE as our data augmentation method.
\textbf{Secondly}, we conduct an experiment of target recognition on ten-class MSTAR dataset. From Table \ref{with_aae}, we can see that some targets in class BRDM, BTR60, D7, T62 and BMP2 are recognized as ZSU234, which leads to the decrease of final detection accuracy. The reason for that may be that the background of target ZSU234 is darker than other objects, thus the shadow around the targets may misguide the final judgment, as the recognition depends on the detected region. Besides, the FNR is largely caused by BTR60 since some targets in BTR60 have nearly the same pattern so the network treats the noise as the target, thus it is rather hard to tell them apart. But we can clearly see that the accuracy after data augmentation raises to nearly 99\%, and FNR drops from 2.025\% to 1.204\%, which proves the effectiveness of AAE data augmentation method on small target chips.
\textbf{Thirdly}, in the noise corruption experiment, we can see that our framework exhibits high noise immunity. As shown in Fig. \ref{noise_compare}, with noise proportion raises up to 20\%, the accuracy still remains above 98\%. Such superior performance can be explained by the fact that the proposed method is capable of telling the object from its background. As a result, the noise corruption in the background can be further learned as disturbance so that our method can significantly maintain the original object and recognize which category it belongs to.
\textbf{Finally}, the experiment conducted on large-scene images shows that the detection accuracy raises up by 1\% and FNR drops by 1\%, FPR drops by 0.33\% after we simultaneously train the expanded small SAR chips and sliced large-scene SAR images. This can be explained by noticing that the training process of small target chips can make the network learn more about the textural feature of SAR images since the target in small chips is comparatively a rather large object while in large-scene images is only a small object. Therefore, small chips act like a supplementary, assisting the recognition of SAR objects on large-scene images. Besides, the experiment on the more tricky dataset we provided proves that our network learns target textual feature rather than the pure edge information, therefore, it is much more robust. But the false positive cases still remain an intractable problem.
We suppose that the following ideas may be able to reduce the false positive cases. 1) Adopting more data augmentation methods on the objects which are easily detected incorrect, enabling the model to learn more diverse target features and enhance the model's robustness; 2) Methods like hard example mining \cite{shrivastava2016training} and focal loss \cite{lin2017focal} will increase the weight of hard example in training process, which may be beneficial to minimize the false positive cases. While in some situations, however, the shadow may share the same feature with the targets. Under such condition, pre-training can be an effective method to solve this problem. For instance, contrastive learning \cite{chen2020simple,momentum2020he}, targeting at learning an encoder that is able to map positive pairs to similar representations while push away those negative samples in the embedding space. In this way, the pre-trained model may have stronger generalization and feature extraction capability, thus effectively distinguishing target from its shadow.
\subsection{Conclusions}
In this paper, an efficient and robust deep learning based target detection method has been proposed based on a novel customized learning representations and multi-scale features of SAR images method. The framework of AAE has been employed for advanced data augmentation which was confirmed by a high variety of the generated samples. An automatic labeling method has been proposed to avoid the labor-intensive manual labeling. By jointly training the neural network with the small target chips and large-scene images, the proposed integrated target detector has been proposed to realized multiple targets detection and recognition. The experimental results confirmed our method reached competitive accuracy on complex large-scene SAR images with rapid speed. Besides, our method can obtain robust detection performance in terms of the different noise levels, even in the extreme case that the corrupted pixels reach 20\%.
It is noted that there are still some potential problems needed to be tackled in the future: 1) The detection accuracy varies among different categories, and some categories, such as BRDM2, are hard to recognize since their feature is similar to the background; 2) It was found that SAR targets have different rotation angles. Therefore, using rotated anchors to perform targets detection may enhance the final detection accuracy.
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
\bibliographystyle{IEEEtran}
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"redpajama_set_name": "RedPajamaArXiv"
}
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Clang!—from its red and yellow trumpets.
But the smoke was white—white!
And Myra sings "Yankee Doodle" at her milking.
Am I well painted to-day, 'caro Abate mio'?
You will be proud of me at the 'Ridotto', hey?
Proud of being 'Cavalier Servente' to such a lady?"
"Can you doubt it, 'Bellissima Contessa'?
And Venus herself shines less..."
I will read my letter in peace."
Or I shall die of laughing."
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"redpajama_set_name": "RedPajamaC4"
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I. Take the initiative to move beyond the crowd.
II. Make your prayer requests very specific.
1. SPEND TIME REMEMBERING THE GRACIOUSNESS OF GOD.
3. GO TO BED EVERY NIGHT DEPENDING ON GOD FOR GOODNESS, NOT MEN.
|
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"redpajama_set_name": "RedPajamaC4"
}
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Q: Getting started with the sport of competitive programming How do you get started with competitive programming and get well versed with various topics in it ?
What all things you can do ?
Get started directly or do some concepts first.
A: This is a very popular question on Quora, but is generally considered off-topic for Stack Overflow. The best way to use SO for competitive programming advice is to ask specific questions about problems you're having when coding a solution. For example, you might ask how a language feature works. Often you'll find that the question has already been asked.
Here are the 108+ Quora answers to your question: https://www.quora.com/How-do-I-become-a-competitive-programmer
The summary answer is: Get started solving problems. If you have any programming background, which I'm sure you do since you're asking this question, you'll get more benefit from just starting rather than reading a lot first. When you get stuck on something, that's the time to read books or online resources.
If you're having trouble deciding what to start on, here's a suggestion from my blog of how I would get started: http://www.redgreencode.com/about-project-462/
A: My 2 cents...
Best option is to get registered at the following coding sites..
+ topcoder.com
+ codechef.com
+ hackerrank.com
And, while you hack code here, you can build upon your programming foundation by learning more on
+ Data structures
+ Algorithms
+ Operating system concepts
+ Networking concepts and more ...
You could also start looking at the following books in this area...
+ The Algorithm Design Manual
+ Programming Challenges: The Programming Contest Training Manual
+ Competitive Programming 2
A: My advice would be to get registered with a Competitive Programming site if you know how to write simple codes in a particular programming language and solve the basic problems(the ones which does not require algorithms or require basic ones).
My suggestion on the choice of site is:-
1)HackerRank:-https://www.hackerrank.com/
Problems are well categorized here.Practice the questions marked 'easy' here followed by a few 'medium' ones.Complete the '30 Days of Coding Challenge' that would give you a good basic idea of the Java language.Also, check other's code and the editorial even if your code passes all the test cases.
2)SPOJ:-
Practice the problems categorized under 'ad-hoc' here.Start by solving the problems which are solved by most people...usually they will be easier.
3)Start implementing basic(or standard) algorithms. It is suggested to read them from Topcoder tutorials or Introduction to algorithms.Also, follow a standard book along with it.Now, practice problems based on those algorithms until you get familiar with them.
4)After you get acquainted with the most common algorithms start competing in coding challenges.Practice makes you better. Try to solve problems from previous contests. Solve as many problem A-s as you can until they give you some trouble. Then move on to B, then C etc.
Hope it helps :)
P.S. Also check the following answer on Quora:-
https://www.quora.com/What-is-the-best-strategy-to-improve-my-skills-in-competitive-programming-in-2-3-months/answer/Sachin-Gupta-6?srid=4gZ6
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\section{Introduction}
A complex circulant matrix is a special kind of Toeplitz matrix whose rows are composed of right cyclically shifted versions of a list $\boldsymbol{a}=(a_0,a_1,\dots,a_{n-1})\in \mathbb{C}^n$. Precisely, a complex circulant matrix is of the form \begin{align*} \left[\begin{array}{ccccc}
a_0&a_1&\dots&a_{n-2}&a_{n-1}\\
a_{n-1}&a_0&\dots&a_{n-3}&a_{n-2}\\
\vdots& \vdots& \ddots& \vdots& \vdots\\
a_2&a_3&\dots&a_{0}&a_{1}\\
a_1&a_2&\dots&a_{n-1}&a_{0}\\
\end{array
\right]\end{align*}
for some vector $\boldsymbol{a}=(a_0,a_1,\dots,a_{n-1})\in \mathbb{C}^n$. Such matrices have extensively been studied since their
first appearance in the paper by Catalan in 1946 (see \cite{c1846}). In 1994, the algebraic structures, properties and some applications of circulant matrices have been summarized in the book ``Circulant Matrices'' in \cite{D1994}. These matrices are interesting due to their rich algebraic structures and various applications (see \cite{D1994}, \cite{GG2009}, \cite{KS2012}, \cite{r1}, and references therein).
Circulant matrices have been applied to various disciplines such as such as image processing, communications, signal processing, networked systems and coding theory. Circulant matrices can be diagonalized by a discrete Fourier transform, and hence linear equations that contain them may be quickly solved using a fast Fourier transform.
Determinants are known for their applications in matrix theory and linear algebra, e.g.,
determining the area of a triangle via Heron's formula in \cite{K2004}, solving linear systems using Cramer's rule in \cite{CK2010}, and determining the singularity of a matrix. Therefore, properties matrices and determinants of matrices have been extensively studied (see \cite{CK2010}, \cite{MMMPSS2008}, and references therein). The determinants of circulant matrices have been studied (see, for example, \cite{D1994}, \cite{KJ2015}, \cite{r6}, and \cite{r5}).
In \cite{R2013}, the determinants of $n\times n$ circulant matrices whose first row consists of the coefficients in the Binomial expansion of $(x+y)^{n-1}$ have been completely determined.
In this paper, we focus on a more general set up. Precisely, we study $n\times n$ circulant matrices whose first row consists of the coefficients in the expansion of $(x+zy)^{n-1}$, where $z$ is a nonzero complex number and $n$ is a positive integer. Such matrices will be referred to as Binomial-related circulant matrices. Here, the determinants of Binomial-related circulant matrices are completely determined in the cases where $z\in \{1,-1,i,-i\}$. The determinant of the $1\times 1$ Binomial-related circulant matrix is always $1$. In this paper, $n$ is assumed to be a positive integer greater than $1$.
The paper is organized as follows. In Section 2, some basic results on matrices and trigonometric identities are recalled and proved. The determinants of $n\times n$ Binomial-related right circulant matrices are studied in Section 3. In Section 4, the analogous results for the determinants of $n\times n$ Binomial-related left circulant matrices are given. Some remarks and open problems are discussed in Section 5.
\section{Preliminaries}
In this section, some basic results on circulant matrices are recalled together with introduction to the concept of Binomial-related circulant matrices. Subsequently, some trigonometric identities required in the proofs of the main results are discussed.
\subsection{Left and Right Circulant Matrices}
Given a positive integer $n$, denote by $M_n(\mathbb{C})$ the set of all $n\times n$ complex matrices. A matrix $A \in M_n(\mathbb{C})$ is called a {\em right circulant matrix} if each row of $A$ is rotated one element to the right relative to the preceding row. Precisely, a complex right circulant matrix is of the form
\begin{align*}
\left[\begin{array}{ccccc}
a_0&a_1&\dots&a_{n-2}&a_{n-1}\\
a_{n-1}&a_0&\dots&a_{n-3}&a_{n-2}\\
\vdots& \vdots& \ddots& \vdots& \vdots\\
a_2&a_3&\dots&a_{0}&a_{1}\\
a_1&a_2&\dots&a_{n-1}&a_{0}\\
\end{array
\right]=:{\rm rcir}( \boldsymbol{a}),\end{align*}
where $\boldsymbol{a}=(a_0,a_1,\dots,a_{n-1})\in \mathbb{C}^n$.
The eigenvalues and the determinants of $n \times n$ circulant matrices have been determined in terms of the $n$th roots of unity and the elements in its first row (see \cite{D1994}, \cite{KJ2015}, and \cite{r5}).
\begin{Lemma}\label{1.3}
Let $\boldsymbol{a} = (a_{0},a_{1},\cdots,a_{n-1})\in \mathbb{C}^n$. Then the eigenvalues of ${\rm rcir}({\boldsymbol{a}})$ are of the form
\[\lambda_m = \sum_{k=0}^{n-1} a_ke^\frac{2km\pi i}{n} \] for all $m \in \{0,1,\cdots,n-1\}$.
\end{Lemma}
\begin{Lemma} \label{2.5}
Let $A \in M_n(\mathbb{C}) $. If the eigenvalues of $A$ are $\lambda_0,\lambda_1,\dots.,\lambda_{n-1}$, then \[\det(A)=\prod_{i=0}^{n-1} \lambda_i.\]
\end{Lemma}
Combining Lemmas \ref{1.3} and \ref{2.5}, the determinants of circulant matrices can be easily computed.
In a similar fashion, a matrix $A \in M_n(\mathbb{C})$ is called a {\em left circulant matrix} if
each row of $A$ is rotated one element to the left relative to the preceding row. Precisely, a complex left circulant matrix is of the form
\[\left[ {\begin{array}{ccccc}
a_0 & a_1 & \dots & a_{n-2} & a_{n-1} \\
a_1 & a_2 & \dots & a_{n-1} & a_0\\
\vdots & \vdots & \ddots & \vdots &\vdots\\
a_{n-2} & a_{n-1} & \dots & a_{n-4} & a_{n-3}\\
a_{n-1} & a_0 & \dots & a_{n-3} & a_{n-2}
\end{array} } \right]=:{\rm lcir }(\boldsymbol{a}) , \]
where $\boldsymbol{a}=(a_0 ,a_1 , \dots , a_{n-1})\in \mathbb{C}^n$.
The determinant of an $n\times n$ left circulant matrix can be determined in terms of the determinant of a right circulant matrix and $n$ as follows.
\begin{Lemma}[{\cite{rg}}] \label{col:1}
Let $\boldsymbol{a} \in \mathbb{C}^n$.
Then
\[ {\rm lcir}(\boldsymbol{a}) = H {\rm rcir}(\boldsymbol{a}) ,\]
where
$ H = \left[ {\begin{array}{cc}
1 & O_1 \\
O^T_1 & z \tilde{I}_{(n-1)}\\
\end{array} } \right],$
$\tilde{I}_{(n-1)} = {\rm adiag(1,1,\dots,1)_{(n-1)\times (n-1)} }
$ and $O_1= (\displaystyle\underbrace{0,0,\dots,0}_{n-1 \text{copies}})$.
\end{Lemma}
\begin{Corollary} \label{cor-left}
Let $\boldsymbol{a} \in \mathbb{C}^n$.
Then
\[\det( {\rm lcir}(\boldsymbol{a}) )= (-1)^{\lfloor \frac{n-1}{2}\rfloor} \det({\rm rcir}(\boldsymbol{a}) ),\]
where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$ for all real numbers $x$.
\end{Corollary}
\begin{proof}
From Lemma \ref{col:1}, we note that $\det( {\rm lcir}(\boldsymbol{a}) )= \det(H) \det({\rm rcir}(\boldsymbol{a}) )$. By permuting the $i$th and $(n+1-i)$th rows of $H$ for all $i=1,2,\dots, {\lfloor \frac{n-1}{2}\rfloor} $, the resulting matrix is $I_n$. It follows that $\det(H)= (-1)^{\lfloor \frac{n-1}{2}\rfloor} $ and hence $\det( {\rm lcir}(\boldsymbol{a}) )= (-1)^{\lfloor \frac{n-1}{2}\rfloor} \det({\rm rcir}(\boldsymbol{a}) )$ as desired.
\end{proof}
\subsection{Left and Right Binomial-Related Circulant Matrices}
In \cite{R2013}, $n\times n$ left and right circulant matrices whose first row is $\left( {n-1 \choose 0},{n-1 \choose 1}, \dots, {n-1 \choose n-1}\right)$, the coefficients in the Binomial expansion of $(x+y)^{n-1}$ have been introduced and the determinants of such matrices have been completely determined.
Here, we focus on a more general set up. For a positive integer $n$ and a nonzero complex number $z$, let $\boldsymbol{c}_n (z)$ denote the vector of the coefficients in the expansion of $(x+zy)^{n-1}$. Precisely, we have $\boldsymbol{c}_n (z)=\left( {n-1 \choose 0}z^0,{n-1 \choose 1}z^1, \dots, {n-1 \choose n-1}z^{n-1}\right) $. In this paper, right and left Binomial-related circulant matrices $ {\rm rcir}(\boldsymbol{c}_n (z) )$ and $ {\rm lcir}(\boldsymbol{c}_n(z)) $ are studied.
\begin{Example} Consider $(x+iy)^4=x^4+4ix^3y-6x^2y^2-4ixy^3+y^4 $. Then we have
{\small \[
{\rm rcir}(\boldsymbol{c}_5(i)) =\left[\begin{array}{ccccc}
1 &4i&-6&-4i&1\\
1&1 &4i&-6&-4i\\
-4i&1&1 &4i&-6\\
-6 &-4i&1&1 &4i\\
4i& -6 &-4i&1&1 \\
\end{array}\right] \text{~~and~~}
{\rm lcir}(\boldsymbol{c}_5(i)) =\left[\begin{array}{ccccc}
1 &4i&-6&-4i&1\\
4i&-6&-4i&1&1\\
-6&-4i&1&1&4i\\
-4i&1&1&4i&-6\\
1&4i&-6&-4i\\
\end{array}\right].\]
}
\end{Example}
The eigenvalues of can be determined in the following lemma.
\begin{Lemma} \label{Eigen}
Let $n\geq 2$ be a positive integer and let $z$ be a nonzero complex number. Then the eigenvalues of ${\rm rcir}({\boldsymbol{c}_n(z)})$ are of the form \[\lambda_m=(1+ze^\frac{2m\pi i}{n})^{n-1}\] for all $m = 0,1,2,\dots,n-1$.
\end{Lemma}
\begin{proof} Let $m\in \{ 0,1,2,\dots,n-1\}$.
From $\boldsymbol{c}_n (z)=\left( {n-1 \choose 0}z^0,{n-1 \choose 1}z^1, \dots, {n-1 \choose n-1}z^{n-1}\right) $ and Lemma \ref{1.3}, we have
\begin{equation}\label{2.2.1}
\lambda_m
= \sum_{k=0}^{n-1} \begin{pmatrix} n-1\\k \end{pmatrix}z^k (e^\frac{2m \pi i}{n})^k.
\end{equation}
Substituting $y=e^\frac{2m\pi i}{n}$ in
\begin{equation*}
(1+zy)^{n-1} = \sum_{k=0}^{n-1}\begin{pmatrix} n-1\\k \end{pmatrix} z^k y^k,
\end{equation*}
we have
\begin{equation}\label{2.2.3}
\sum_{k=0}^{n-1} \begin{pmatrix} n-1\\k \end{pmatrix}z^k(e^\frac{2m\pi i}{n})^k
= (1+ze^\frac{2m\pi i}{n})^{n-1}
\end{equation}
From \eqref{2.2.1} and \eqref{2.2.3},
it follows that $
\lambda_m = (1+ze^\frac{2m\pi i}{n})^{n-1}$.
\end{proof}
For each $A=[a_{ij}]_{n\times n}\in M_n(\mathbb{C})$, we write $\overline{A}=[\overline{a_{ij}}]_{n\times n}$. We have the following relation for the determinants of Binomial-related circulant matrices.
\begin{Lemma}\label{conj} Let $n\geq 2$ be a positive integer and let $z$ be a nonzero complex number. Then
\[ \det({\rm rcir}(\boldsymbol{c}_n (\overline{z}) ))= \overline{\det({\rm rcir}(\boldsymbol{c}_n (z) ))}\text{~~~~and~~~~} \det({\rm lcir}(\boldsymbol{c}_n (\overline{z}) ))= \overline{\det({\rm lcir}(\boldsymbol{c}_n (z) ))}.\]
\end{Lemma}
\begin{proof} It is not difficult to see that
$ {\rm rcir}(\boldsymbol{c}_n (\overline{z}) )= \overline{{\rm rcir}(\boldsymbol{c}_n (z) )}\text{~~~~and~~~~} {\rm lcir}(\boldsymbol{c}_n (\overline{z}) )= \overline{{\rm lcir}(\boldsymbol{c}_n (z) )}$. Since $\det(\overline{A})=\overline{\det(A)} $ for all $A\in M_n(\mathbb{C})$, the result follows.
\end{proof}
The determinants of right and left Binomial-related circulant matrices $ {\rm rcir}(\boldsymbol{c}_n (z) )$ and $ {\rm lcir}(\boldsymbol{c}_n (z)) $ with $z\in \{1,-1,i,-i\}$ will be determined in Sections 3 and 4.
\subsection{Trigonometric Identities}
In order to determine the determinants of Binomial-related circulant matrices in the next section, the following trigonometric identities are required.
\begin{Lemma} \label{lemEi1} Let $k$ be a positive integer. Then
\[\prod_{m=1}^{k} \cos \frac{2m\pi}{2k+1} =(-1)^k \prod_{m=1}^{k} \cos \frac{(2m-1)\pi}{2k+1}.\]
\end{Lemma}
\begin{proof} Since $\{1,2,\dots, k\} =\{k-m+1\mid k\in \{ 1,2,\dots, k\}\}$, it follows that
\begin{align}
\label{eq1}
\prod_{m=1}^k \cos \frac{2m\pi}{2k+1} = \prod_{m=1}^k \cos \frac{2(k-m+1)\pi}{2k+1}.
\end{align}
For each $m\in \{1,2,3,\dots, k\}$, we have
\begin{align*} \cos \frac{(2m-1)\pi}{2k+1}&= \cos \frac{-(2m-1)\pi}{2k+1} \\
&=-\cos \frac{(2k+1-(2m-1))\pi}{2k+1} \\
&= -\cos \frac{2(k-m+1)\pi}{2k+1} .
\end{align*}
Hence,
\begin{align*} \prod_{m=1}^k \cos \frac{(2m-1)\pi}{2k+1} =(-1)^k \prod_{m=1}^k \cos \frac{2(k-m+1)\pi}{2k+1} .
\end{align*}
Together with \eqref{eq1}, we have
\[\prod_{m=1}^{k} \cos \frac{2m\pi}{2k+1} =(-1)^k \prod_{m=1}^{k} \cos \frac{(2m-1)\pi}{2k+1}\]
as desired.
\end{proof}
\begin{Lemma}[{\cite[Equation 4.12]{Trig}}] \label{l2.9} Let $k$ be a positive integer. Then
\[\prod_{m=1}^{k} \cos \frac{m\pi}{k+1} = \frac{\sin \frac{(k+1)\pi}{2}}{2^{k}}.\]
\end{Lemma}
\begin{Lemma} \label{l2.10} Let $k$ be a positive integer. Then
\[\left(\prod_{m=1}^{k} \cos \frac{2m\pi}{2k+1} \right)^2= \left( \frac{1}{4} \right)^k.\]
\end{Lemma}
\begin{proof} By Lemma \ref{lemEi1}, it can be deduced that
\begin{align*} (-1)^k\left(\prod_{m=1}^{k} \cos \frac{2m\pi}{2k+1} \right)^2 &=\prod_{m=1}^{k} \cos \frac{(2m-1)\pi}{2k+1} \prod_{m=1}^{k} \cos \frac{2m\pi}{2k+1} \\
&=\prod_{m=1}^{2k} \cos \frac{m\pi}{2k+1}\\
&= \frac{\sin\frac{(2k+1)\pi}{2}}{2^{2k}} ~~~~~~~~~~~~~~~~~~~~~~\text{by Lemma \ref{l2.9},}\\
&=\left(-\frac{1}{4}\right)^k.
\end{align*}
Hence, we have \[\left(\prod_{m=1}^{k} \cos \frac{2m\pi}{2k+1} \right)^2 =\left(\frac{1}{4}\right)^k\]
as desired.
\end{proof}
\section{Determinants of Binomial-Related Right Circulant Matrices}
In this section, we focus on the determinants of Binomial-related right circulant matrices in the cases where $z\in \{-1,i,-i\}$. In the case where $z=1$, the result has been given in \cite{R2013}.
\subsection{Right Circulant Matrices from the Coefficients of $(x+y)^{n-1}$ and $(x-y)^{n-1}$}
By substituting $z=1$ and $z=-1$ in Lemma \ref{Eigen}, the next lemma follows.
\begin{Lemma} \label{1_-1}
Let $n\geq 2$ be a positive integer. Then the following statements hold.
\begin{enumerate}
\item The eigenvalues of ${\rm rcir}({\boldsymbol{c}_n(1)})$ are of the form $\lambda_m=(1+e^\frac{2m\pi i}{n})^{n-1}$ for all $m = 0,1,2,\dots,n-1$.
\item The eigenvalues of ${\rm rcir}({\boldsymbol{c}_n(-1)})$ are of the form $\lambda_m=(1-e^\frac{2m\pi i}{n})^{n-1}$ for all $m = 0,1,2,\dots,n-1$.
\end{enumerate}
\end{Lemma}
Based on Lemma \ref{1_-1}, the determinant of ${\rm rcir}({\boldsymbol{c}_n(1)})$ has been given in \cite{R2013}.
\begin{Proposition}[{\cite[Theorem 2.1]{R2013}}] \label{prop1} Let $n\geq 2$ be a positive integer. Then
\begin{equation*}
\det({\rm rcir}({\boldsymbol{c}_n(1)}))= (1+(-1)^{n-1})2^{n-2}.
\end{equation*}
\end{Proposition}
From Lemma \ref{1_-1}, observe that the eigenvalue $\lambda_0$ of ${\rm rcir}({\boldsymbol{c}_n(-1)})$ is $\lambda_0=(1-e^0)^{n-1}=0$. By Lemmas \ref{1.3} and \ref{2.5}, the determinant of ${\rm rcir}({\boldsymbol{c}_n(-1)})$ follows immediately.
\begin{Proposition}\label{prop-1} Let $n\geq 2$ be a positive integer. Then
\begin{equation*}
\det({\rm rcir}({\boldsymbol{c}_n(-1)}))= 0.
\end{equation*}
\end{Proposition}
\subsection{Right Circulant Matrices from the Coefficients of $(x+iy)^{n-1}$}
In this subsection, we focus on the determinant of $\boldsymbol{c}_n (i)$. The results are given in terms of the residues of $n$ modulo $4$.
By setting $z=i$ in Lemma \ref{Eigen}, we have the following lemma.
\begin{Lemma} \label{2.2}
Let $n\geq 2$ be a positive integer. Then the eigenvalues of ${\rm rcir}({\boldsymbol{c}_n(i)})$ are of the form $\lambda_m=(1+ie^\frac{2m\pi i}{n})^{n-1}$ for all $m = 0,1,2,\dots,n-1$.
\end{Lemma}
The following properties of the eigenvalues of ${\rm rcir}({\boldsymbol{c}_n(i)})$ are key to prove the main results.
\begin{Lemma} \label{l3.5} Let $n\geq 2$ be a positive integer. Then
\[\lambda_m\lambda_{n-m}= \left( 2i \cos \frac{2m\pi }{n}\right)^{n-1} \]
for all $1\leq m <n$, where $\lambda_m$ and $\lambda_{n-m}$ are given in Lemma \ref{2.2}
\end{Lemma}
\begin{proof}
Let $m$ be an integer such that $1\leq m<n$. Then
\begin{align*}
\lambda_m\lambda_{n-m}&=(1+ie^\frac{2m\pi i}{n})^{n-1}(1+ie^\frac{2(n-m)\pi i}{n})^{n-1}\\
&=\left((1+ie^\frac{2m\pi i}{n})(1+ie^\frac{2(n-m)\pi i}{n})\right)^{n-1}\\
&=\left((1+i(e^\frac{2m\pi i}{n}+e^\frac{2(n-m)\pi i}{n}) -1\right)^{n-1}\\
&=\left(i(e^\frac{2m\pi i}{n}+e^\frac{-2m\pi i}{n}) \right)^{n-1}\\
&=\left( 2i \cos \frac{2m\pi }{n}\right)^{n-1}
\end{align*}
by the Euler's formula.
\end{proof}
\begin{Lemma} \label{lem:propSpecLam}Let $n\geq 2$ be a positive integer. Then the following statements hold.
\begin{enumerate}
\item If $n$ is odd, then
\[\lambda_0=\left(2i\right)^{\frac{n-1}{2}}.\]
\item If $n$ is even, then
\[\lambda_0\lambda_{\frac{n}{2}} = 2^{n-1}.\]
\item If $n\equiv 0\,{\rm mod}\,4$, then
\[ \lambda_{\frac{n}{4}} =0.\]
\end{enumerate}
\end{Lemma}
\begin{proof} Assume that $n$ is odd. Then $n-1$ is even and hence
\begin{align*}
\lambda_0&=(1+i)^{n-1}\\
&=\left((1+i)^2\right)^{\frac{n-1}{2}}\\
&=\left(1+2i-1\right)^{\frac{n-1}{2}}\\
&=\left(2i\right)^{\frac{n-1}{2}}.
\end{align*}
Assume that $n$ is even. Then
\begin{align*} \lambda_0\lambda_{\frac{n}{2}}&=(1+i)^{n-1} (1+ie^ {\pi i})^{n-1} \\
&= (1+i)^{n-1} (1-i)^{n-1} \\
&=2^{n-1}.
\end{align*}
Assume that $n\equiv 0\,{\rm mod }\, 4$. Then
\begin{align*} \lambda_{\frac{n}{4}}&= (1+ie^ {\frac{\pi i}{2}})^{n-1} \\
&= (1+i^2)^{n-1} \\
&=0.
\end{align*}
The lemma is proved.
\end{proof}
Next, the determinant of ${\rm rcir}({\boldsymbol{c}_n(i)})$ is determined in the following four cases.
\begin{Proposition} If $n \equiv 0\,{\rm mod}\, 4$, then
\[\det({\rm rcir}({\boldsymbol{c}_n(i)}))=0.\]
\end{Proposition}
\begin{proof}
Assume that $n\geq 2$ is a positive integer such that $n \equiv 0\,{\rm mod}\, 4$. By Lemma \ref{lem:propSpecLam}, we have $\lambda_{\frac{n}{4}}=0$ and hence
\[\det({\rm rcir}({\boldsymbol{c}_n(i)}))=\prod_{m=0}^{n-1} \lambda_m=0\]
by Lemma \ref{2.5}.
\end{proof}
\begin{Proposition} If $n \equiv 1\,{\rm mod}\, 4$, then
\[\det({\rm rcir}({\boldsymbol{c}_n(i)}))=(2i)^{\frac{n-1}{2}} .\]
\end{Proposition}
\begin{proof} Let $n\geq 2$ be a positive integer such that $n \equiv 1\,{\rm mod}\, 4$. Then $n=4a+1$ for some positive integer $a$.
By Lemmas \ref{1.3} and \ref{2.5}, we have
\begin{align*}
\det ({\rm rcir}({\boldsymbol{c}_n(i)}))&= \prod_{m=0}^{n-1} \lambda_m\\
&=\lambda_0 \prod_{m=1}^{\frac{n-1}{2}} \lambda_m\lambda_{n-m}\\
&= \left(2i\right)^{\frac{n-1}{2}} \prod_{m=1}^{ \frac{n-1}{2}} \left( 2i \cos \frac{2m\pi }{n}\right)^{n-1}~~~~\text{ by Lemmas \ref{l3.5} and \ref{lem:propSpecLam},}\\
&= \left(2i\right)^{\frac{n-1}{2}} \left( \prod_{m=1}^{ 2a} 2i \cos \frac{2m\pi }{4a+1}\right)^{4a}\\
&= \left(2i\right)^{\frac{n-1}{2}} \left( (2i)^{2a}\prod_{m=1}^{ 2a} \cos \frac{2m\pi }{4a+1}\right)^{4a}\\
&= \left(2i\right)^{\frac{n-1}{2}} \left( (2i)^{4a} \left(\prod_{m=1}^{ 2a} \cos \frac{2m\pi }{4a+1}\right)^2\right)^{2a}\\
&= \left(2i\right)^{\frac{n-1}{2}} \left( 2^{4a} \left(\frac{1}{4} \right)^{2a}\right)^{2a} ~~~~~~~~~~~~~\text{ by Lemma \ref{l2.10},}\\
&= \left(2i\right)^{\frac{n-1}{2}} .
\end{align*}
Therefore, we have $\det({\rm rcir}({\boldsymbol{c}_n(i)}))=(2i)^{\frac{n-1}{2}} $.
\end{proof}
\begin{Proposition} If $n \equiv 2\,{\rm mod}\, 4$, then
\[\det({\rm rcir}({\boldsymbol{c}_n(i)}))=2^{n-1}.\]
\end{Proposition}
\begin{proof}
Let $n\geq 2$ be a positive integer such that $n \equiv 2\,{\rm mod}\, 4$. Then $n=2(2a+1)$ for some positive integer $a$. By Lemmas \ref{1.3} and \ref{2.5}, it can be deduced that
\begin{align*}
\det ({\rm rcir}({\boldsymbol{c}_n(i)}))&= \prod_{m=0}^{n-1} \lambda_m\\
&=\lambda_0 \lambda_{\frac{n}{2}} \prod_{m=1}^{\frac{n}{2}-1} \lambda_m\lambda_{n-m}\\
&= 2^{n-1}\prod_{m=1}^{ \frac{n}{2}-1} \left( 2i \cos \frac{2m\pi }{n}\right)^{n-1} ~~~~~~~~~~~~~\text{ by Lemmas \ref{l3.5} and \ref{lem:propSpecLam},}\\
&= 2^{n-1} \left( \prod_{m=1}^{ 2a} 2i \cos \frac{2m\pi }{4a+2}\right)^{4a+1} \\
&= 2^{n-1} \left( (2i )^{2a}\prod_{m=1}^{ 2a} \cos \frac{m\pi }{2a+1}\right)^{4a+1} \\
&= 2^{n-1} \left( (2)^{2a}(-1)^a \frac{\sin\frac{(2a+1)\pi}{2}}{2^{2a}} \right)^{4a+1} ~~~ ~\text{ by Lemma \ref{l2.10},}\\
&= 2^{n-1} \left( (2)^{2a}(-1)^a \frac{(-1)^{a}}{2^{2a}} \right)^{4a+1} \\
&= 2^{n-1}.
\end{align*} Hence, $\det({\rm rcir}({\boldsymbol{c}_n(i)}))=2^{n-1}$ as desired.
\end{proof}
\begin{Proposition} If $n \equiv 3\,{\rm mod}\, 4$, then
\[\det({\rm rcir}({\boldsymbol{c}_n(i)}))=- (2i)^{\frac{n-1}{2}} .\]
\end{Proposition}
\begin{proof} Let $n\geq 2$ be a positive integer such that $n \equiv 3\,{\rm mod}\, 4$. Then $n=4a+3$ for some positive integer $a$. By Lemmas \ref{1.3} and \ref{2.5}, we have
\begin{align*}
\det ({\rm rcir}({\boldsymbol{c}_n(i)}))&= \prod_{m=0}^{n-1} \lambda_m\\
&=\lambda_0 \prod_{m=1}^{\frac{n-1}{2}} \lambda_m\lambda_{n-m}\\
&= \left(2i\right)^{\frac{n-1}{2}} \prod_{m=1}^{ \frac{n-1}{2}} \left( 2i \cos \frac{2m\pi }{n}\right)^{n-1} ~~~~\text{ by Lemmas \ref{l3.5} and \ref{lem:propSpecLam},}\\
&= \left(2i\right)^{\frac{n-1}{2}} \left( \prod_{m=1}^{ 2a+1} 2i \cos \frac{2m\pi }{4a+3}\right)^{4a+2}\\
&= \left(2i\right)^{\frac{n-1}{2}} \left( (2i)^{2a+1}\prod_{m=1}^{ 2a+1} \cos \frac{2m\pi }{4a+3}\right)^{4a+2}\\
&= \left(2i\right)^{\frac{n-1}{2}} \left( (2i)^{4a+2} \left(\prod_{m=1}^{ 2a+1} \cos \frac{2m\pi }{4a+3}\right)^2\right)^{2a+1}\\
&= \left(2i\right)^{\frac{n-1}{2}} \left( -2^{4a+2} \left(\frac{1}{4} \right)^{2a+1}\right)^{2a+1}~~~~~~~ ~~\text{ by Lemma \ref{l2.10},}\\
&= -\left(2i\right)^{\frac{n-1}{2}} .
\end{align*}
Therefore,
$\det({\rm rcir}({\boldsymbol{c}_n(i)}))=- (2i)^{\frac{n-1}{2}}$ for all positive integers $n$ such that $n \equiv 3\,{\rm mod}\, 4$.
\end{proof}
The results can be summarized as follows.
\begin{Theorem}\label{thm_i} Let $n\geq 2$ be a positive integer. Then
\[\det({\rm rcir}({\boldsymbol{c}_n(i)}))= \begin{cases}
0 & \text{ if } n\equiv 0\,{\rm mod}\, 4\\
(2i)^{\frac{n-1}{2}} & \text{ if }n\equiv 1\,{\rm mod}\, 4\\
2^{n-1} & \text{ if }n\equiv 2\,{\rm mod}\, 4\\
-(2i)^{\frac{n-1}{2}} & \text{ if }n\equiv 3\,{\rm mod}\, 4.
\end{cases}\]
\end{Theorem}
\subsection{Right Circulant Matrices from Coefficients of $(x-iy)^{n-1}$}
In this subsection, we focus on the determinant of ${\rm rcir}({\boldsymbol{c}_n(-i)})$. The formula for ${\rm rcir}({\boldsymbol{c}_n(-i)})$ can be given in based on the determinant of ${\rm rcir}({\boldsymbol{c}_n(i)})$ given in Subsection 3.2.
By setting $z=-i$ in Lemma \ref{Eigen}, the eigenvalues of ${\rm rcir}({\boldsymbol{c}_n(-i)})$.
\begin{Lemma}
Let $n\geq 2$ be a positive integer. Then the eigenvalues of ${\rm rcir}({\boldsymbol{c}_n(-i)})$ are of the form $\lambda_m=(1-ie^\frac{2m\pi i}{n})^{n-1}$ for all $m = 0,1,2,\dots,n-1$.
\end{Lemma}
Using the analysis as in Subsection 2.2, the determinant of ${\rm rcir}({\boldsymbol{c}_n(-i)})$ can be determined.
Alternatively, we have $-i=\overline{i}$. From Lemma \ref{conj}, we have \[\det({\rm rcir}({\boldsymbol{c}_n(-i)}))= \det({\rm rcir}({\boldsymbol{c}_n(\overline{i})})) =\overline{ \det({\rm rcir}({\boldsymbol{c}_n({i})}))} .\]
Based on Theorem \ref{thm_i}, the formula for $\det({\rm rcir}({\boldsymbol{c}_n(-i)}))$ can be derived in the following theorem.
\begin{Theorem} \label{thm_-i}Let $n\geq 2$ be a positive integer. Then
\[\det({\rm rcir}({\boldsymbol{c}_n(-i)}))= \begin{cases}
0 & \text{ if } n\equiv 0\,{\rm mod}\, 4\\
(2i)^{\frac{n-1}{2}} & \text{ if }n\equiv 1\,{\rm mod}\, 4 \text{ or }n\equiv 3\,{\rm mod}\, 4.\\
2^{n-1} & \text{ if }n\equiv 2\,{\rm mod}\, 4 .
\end{cases}\]
\end{Theorem}
\begin{proof}
From Theorem \ref{thm_i}, we have
\begin{align*}
\det({\rm rcir}({\boldsymbol{c}_n(-i)}))
=\overline{\det({\rm rcir}({\boldsymbol{c}_n({i})}))}
= \begin{cases}
0 & \text{ if } n\equiv 0\,{\rm mod}\, 4\\
(-2i)^{\frac{n-1}{2}} & \text{ if }n\equiv 1\,{\rm mod}\, 4\\
2^{n-1} & \text{ if }n\equiv 2\,{\rm mod}\, 4\\
-(-2i)^{\frac{n-1}{2}} & \text{ if }n\equiv 3\,{\rm mod}\, 4.
\end{cases}
\end{align*}
If $n\equiv 1\,{\rm mod}\, 4$, then $\frac{n-1}{2}$ is even and hence $(-1)^{\frac{n-1}{2}}=1 $. For $n\equiv 3\,{\rm mod}\, 4$, then $\frac{n-1}{2}$ is odd and $(-1)^{\frac{n-1}{2}}=-1 $. The result is therefore follows.
\end{proof}
\section{Determinants of Binomial-Related Left Circulant Matrices}
In this section, a brief summary on the determinant of $ {\rm lcir}({\boldsymbol{c}_n(z)})$ is given based on $\det({\rm rcir}({\boldsymbol{c}_n(z)}))$ determined in Section 3 and Corollary \ref{cor-left}.
From Corollary \ref{cor-left}, we have \[\det( {\rm lcir}(\boldsymbol{a}) )= (-1)^{\lfloor \frac{n-1}{2}\rfloor} \det({\rm rcir}(\boldsymbol{a}) )\]
for all $\boldsymbol{a}\in \mathbb{C}^n$.
Note that $ (-1)^{\lfloor \frac{n-1}{2}\rfloor} =1 $ if and only if $ n\equiv 1\,{\rm mod}\, 4 $ or $ n\equiv 2\,{\rm mod}\, 4 $; and $ (-1)^{\lfloor \frac{n-1}{2}\rfloor} =-1 $ if and only if $ n\equiv 0\,{\rm mod}\, 4 $ or $ n\equiv 3\,{\rm mod}\, 4 $. Together with Proposition \ref{prop1}, Proposition \ref{prop-1}, Theorem \ref{thm_i} and Theorem \ref{thm_-i}, the following results concerning the determinants of Binomial-related left circulant matrices can be concluded.
\begin{Proposition}[{\cite[Theorem 2.3]{R2013}}] Let $n\geq 2$ be a positive integer. Then
\begin{equation*}
\det({\rm lcir}({\boldsymbol{c}_n(1)}))=(-1)^{\lfloor \frac{n-1}{2}\rfloor} (1+(-1)^{n-1})2^{n-2}.
\end{equation*}
\end{Proposition}
\begin{Proposition} Let $n\geq 2$ be a positive integer. Then
\begin{equation*}
\det({\rm lcir}({\boldsymbol{c}_n(-1)}))= 0.
\end{equation*}
\end{Proposition}
\begin{Theorem} \label{t4.3} Let $n\geq 2$ be a positive integer. Then
\[\det({\rm lcir}({\boldsymbol{c}_n(i)}))= \begin{cases}
0 & \text{ if } n\equiv 0\,{\rm mod}\, 4\\
(2i)^{\frac{n-1}{2}} & \text{ if }n\equiv 1\,{\rm mod}\, 4 \text{ or }n\equiv 3\,{\rm mod}\, 4.\\
2^{n-1} & \text{ if }n\equiv 2\,{\rm mod}\, 4
\end{cases}\]
\end{Theorem}
Observe that $\det({\rm rcir}({\boldsymbol{c}_n(-i)}))= \det({\rm lcir}({\boldsymbol{c}_n(i)}))$ by Theorems \ref{thm_-i} and \ref{t4.3}.
\begin{Theorem} \label{thm4.4} Let $n\geq 2$ be a positive integer. Then
\[\det({\rm lcir}({\boldsymbol{c}_n(-i)}))= \begin{cases}
0 & \text{ if } n\equiv 0\,{\rm mod}\, 4\\
(-1)^{ \frac{n-1}{2} } (2i)^{\frac{n-1}{2}} & \text{ if }n\equiv 1\,{\rm mod}\, 4 \text{ or }n\equiv 3\,{\rm mod}\, 4.\\
2^{n-1} & \text{ if }n\equiv 2\,{\rm mod}\, 4
\end{cases}\]
\end{Theorem}
Observe that $\det({\rm rcir}({\boldsymbol{c}_n(i)}))= \det({\rm lcir}({\boldsymbol{c}_n(-i)}))$ by Theorems \ref{thm_i} and \ref{t4.4}.
\section{Conclusion and Remarks} The concept of complex Binomial-related left and right circulant matrices has been introduced. Such matrices are complex $n\times n$ circulant matrices whose first row consists of the coefficients in the expansion of $(x+zy)^{n-1}$. In the case where $z=1$, the determinants of Binomial left and right circulant matrices have been determined in \cite{R2013}. In this paper, the eigenvalues and the determinants of Binomial-related left and right circulant matrices have been completely determined for $z\in \{-1,i,-1\}$. It is of natural interest to investigate the eigenvalues and the determinants of such matrices for other forms of $z$. This issue is remained as an open problem.
\section*{Acknowledgments}
{S. Jitman was supported by the Thailand Research Fund under Research
Grant MRG6080012.}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 3,337
|
A toggable clock that switches between timezone.
Call init with a table containing the various TZ you want to be able to list:
```lua
{
{
clock = 3600,
prefix = "🇫🇷"
},
{
clock = -5 * 3600,
prefix = "🇺🇸"
}
}
```
If your current timezone is the one you are viewing
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,453
|
The Technical Ecstasy Tour was a concert tour by English heavy metal band Black Sabbath. It began on 22 October 1976 and ended on 22 April 1977.
Overview
Background
North America leg
Having toned down the band's 'black magic' image for Technical Ecstasy, Geezer Butler assured Circus, "Parents can take their kids to our shows now." Opening acts included Ted Nugent and, promoting their Next album, Journey.
Europe leg
A notorious encounter occurred between Geezer Butler and Malcolm Young of support band AC/DC when the tour reached Europe. "Flick-knives were banned in England," Butler recalled, "but, when we were playing in Switzerland, I bought one. I was just flicking it, when Malcolm Young came up to me and started slagging Sabbath… He came over and said, 'You must think you're big, having a flick-knife.' I said, 'What are you talking about?' And that was it. Nobody got hurt."
AC/DC's support slot had, in any case, begun inauspiciously. "All the gear was blowing up," reported Angus Young of their first show, in Paris. "We played about twenty minutes then destroyed the stage."
Setlist
Songs played overall
"Supertzar" [Audio Introduction]
"Symptom of the Universe"
"Snowblind"
"All Moving Parts Stand Still"
"War Pigs"
"Gypsy"
"Megalomania"
"Black Sabbath"
"Dirty Women"
Bill Ward drum solo
Instrumental band jam
Tony Iommi guitar solo
"Rock 'N' Roll Doctor"
"Electric Funeral"
"Bassically" Geezer Butler bass solo
"N.I.B."
"Iron Man"
"Fairies Wear Boots"
"Embryo" and "Children of the Grave"
Encore
"Supernaut" [Intro] and "Paranoid"
Typical setlist
"Supertzar" [Audio Introduction]
"Symptom of the Universe"
"Snowblind"
"All Moving Parts Stand Still"
"War Pigs"
"Gypsy"
"Black Sabbath"
"Dirty Women"
Bill Ward drum solo
Instrumental band jam
Tony Iommi guitar solo
"Rock 'N' Roll Doctor"
"Electric Funeral"
"Bassically" [Geezer Butler bass solo]
"N.I.B."
"Iron Man"
"Embryo" and "Children of the Grave"
Encore
"Supernaut" [Intro] and "Paranoid"
Tour dates
Box office score data
Personnel
Ozzy Osbourne – vocals
Tony Iommi – guitar
Geezer Butler — bass
Bill Ward – drums
Gerald Woodroffe – keyboards
References
Black Sabbath concert tours
1976 concert tours
1977 concert tours
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 1,339
|
Q: usage of definite article "the" Can anyone please tell me if I can leave the word the out in the following sentences? Can any teacher on anyone who has good grasp on the language please explain it to me?
1- Nothing is impossible in this world. Everything can be achieved. It depends on the mindsets of people how they cope with tough situations in their lives and how hard they work to reach their goals.
2- The teeth of tigers are very sharp.
Thanks in advance.
A: In both examples, you could remove the by restructuring to use possessives.
*
*It depends on people's mindsets how they cope...
*Tigers' teeth are very sharp.
As for just removing the from the original sentence:
*
*It would still make grammatical sense, but might not flow as well.
*A native English speaker wouldn't say teeth of tigers. You'd either need the, or use a possessive as above.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 6,125
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\section{Introduction}
BL Lac objects are extragalactic sources characterised by
weak or undetectable optical line emission and strong and variable
flux and polarization in a wide range of wavebands from optical through
radio (for review, e.g. Kollgaart 1994).
In the radio band, very long baseline interferometry
(VLBI) images show that the pc-scale
structures of BL Lac objects are dominated by compact, flat-spectrum cores
from which (apparently) one-sided jets emerge, often showing knots of
enhanced brightness with apparent superluminal speeds
(e.g Gabuzda, Pushkarev \& Cawthorne 2000).
These properties are explained by Doppler boosted synchrotron radiation
from a relativistic jet closely aligned to
our line of sight (Blandford \& Rees 1974).
In the framework of the unified model
of radio-loud active galactic nuclei (AGNs)
(see e.g., Urry \& Padovani 1995), the observed
differences in the properties of quasars, radio galaxies,
and BL Lac objects are due to the orientation of the
relativistic beam and an obscuring torus with respect to
our line of sight.
The BL Lac object PKS~0003$-$066 (NRAO~005, J0006$-$0623) at a redshift of
$z$ = 0.347 (Stickel, Fried \& K\"{u}hr 1989) is a bright compact radio source
and has been observed with VLBI
for many projects (e.g. Fey \& Charlot 1997).
The core is unresolved
at arcsecond resolution (Ulvestad et al. 1981, Perley 1982).
Gabuzda, Pushkarev \& Cawthorne (2000) observed PKS~0003$-$066 with a global
VLBI array and showed that
the pc-scale structure is resolved, with the major components strongly polarised,
but couldn't identify any superluminal motions from comparison to previous VLBI
observations. Kellermann et al. (2004) analysed the results
of a 15 GHz (2 cm) multiepoch Very Long Baseline Array (VLBA) program
begun in 1994, to study the jets in 110 quasars and active galaxies.
PKS~0003$-$066 is one of 18 BL Lac objects in that sample,
Kellermann et al. (2004) identifing three jet components with no
significant proper motions.
Total flux density monitoring of this source with the ATCA at 1.4, 2.5, 4.8
and 8.6 GHz shows rapid variability over a 2 - 3 year period,
with a higher degree of variability at higher frequencies (Tingay et al. 2003),
suggesting variability of the pc-scale structures.
To investigate this possibility,
we have made new global VLBI observations of PKS~0003$-$066 at 2.3 GHz.
We have also analysed 25 other VLBI observations at 2.3 and 8.6 GHz
from a 10 year series of observations to study the flux density and structural
variability of the source in more detail than has previously been possible.
In this paper we describe these results and a comparison to the results
of previous multi-frequency and multi-epoch VLBI experiments.
In contrast to the results in Kellermann et al. (2004),
we find evidence for fast jet components ejected from the core which
interact strongly with slower jet
components further from the core and cause local jet brightening.
In section 2.1, we describe new VLBI observations at 2.3 GHz
using a global VLBI array,
new disk-based recording techniques, and software correlation techniques.
In section 2.2 we summarize the results obtained from
archival VLBI data. In section 3 we discuss the pc-scale structure and
evolution of this source.
Throughout this paper we use the following cosmological parameters
as used by Kellermann et al. (2004):
$H_0$ = 70 kms$^{-1}$Mpc$^{-1}$, $\Omega_{m} = 0.3$, $\Omega_{\Lambda} = 0.7$.
The corresponding luminosity distance is 1.939 Gpc
at the redshift of PKS~0003$-$066, $z = 0.347$, and
the linear scale is therefore 4.91 pc/mas.
A proper motion of 1 mas/yr corresponds to an apparent speed
of $16c$.
\section{Observations and Results}
\subsection{Global VLBI observations at 2.3 GHz}
\subsubsection{Global disk-based VLBI observations}
PKS~0003$-$066 was observed with a global array of antennas on 2004 April 18,
using 5 Australian telescopes, one South African telescope (Hartebeeestoek Radio Astronomy Observatory),
and one Japanese telescope (Kashima),
as summarised in Table 1, for a period of 20 minutes. This experiment
was designed as a test observation for the software correlator described
in section 2.1.2.
The observational setup utilised a number of different disk-based
recording systems (as briefly summarised in Table 1) to record
the Nyquist-sampled and digitised telescope output as a function of time.
At the antennas that utilised the LBA Data Recorder (LBADR)
\footnote{The LBADR recorders use the VLBI Standard Interface I/O Board
(VSIB) and the VLBI Standard Interface Converter(VSIC) cards
developed by the Mets\"{a}hovi Radio Observatory
(Ritakari \& Mujunen 2002), along with Apple XRaid units for data storage,
except Ceduna which used internally raided drives for this experiment.
For further details, see Phillips et al. (2006, in preparation)}
recording system (West 2004), an aggregate data rate of 256 Mbps was recorded,
corresponding to four 16 MHz bands, each
sampled at 32 MHz and 2-bit digitised. The four 16 MHz bands were
arranged with two bands at right circular polarisation (RCP) (2274 - 2290 MHz)
and two bands at left circular polarisation
(LCP) (2274 - 2290 MHz). Duplicate bands were recorded at RCP and LCP
for redundancy.
At the antennas that utilised the Mark5 recording system, an aggregate
data rate of 128 Mbps was recorded, in matching 16 MHz bands.
At the Kashima antenna, which utilized the Japanese K5 recording
system (Koyama et el. 2003), an aggregate data rate of 1 Gbps was recorded.
The 2274-2306 MHz frequency range of the RCP signal was converted
to a 0-32MHz baseband signal, sampled at the rate of 512 MHz
and then digitized with 2-bit precision. The 1 Gbps data was digitally processed to
obtain the time series data corresponding to
a single 16 MHz band sampled at 32 MHz using 2-bit digitization.
This digitally filtered 16 MHz band ranged from 2274 - 2290 MHz,
corresponding to one of the 16 MHz bands recorded using the LBADR
and Mark5 systems at the other antennas.
An account of the software used to perform the digital
filtering can be found in Takeuchi (2004).
This experiment illustrates the power and flexibility of disk-based
recording for VLBI.
Traditional tape-based VLBI requires considerable effort to translate
different recorder systems prior to correlation. Our software correlator
is capable of reading different disk-based recorder formats without
conversion, improving the compatibility of antennas with
different recording systems.
The data recorded at each antenna were shipped to the Swinburne University
of Technology, for correlation using the software correlator described in
section 2.1.2. Since the Kashima
antenna only produced a single 16 MHz band (RCP) using the software digital
filter, only this band was correlated with a view to full data reduction
and imaging.
\subsubsection{Description of correlation in software}
A software correlator for VLBI has been developed using the Beowulf
cluster at the Swinburne University of Technology Centre for Astrophysics
and Supercomputing. The Beowulf cluster consists of a large number
($\sim$300) of Pentium based PCs
(for detail, see http://astronomy.swin.edu.au/supercomputing/).
The software correlator is an $XF$ style correlator
\footnote{This is a proto-type software correlator (West 2004).
The correlator has more recently been superceded with a more capable
$FX$ software correlator (Deller et al. 2006 in preparation).}.
Data recorded to hard disk
at each individual telescope are mounted
on the supercomputer and the data for each antenna pair (baseline) is transfered
to a processing node for correlation, in segments of up to 25 ms at a
time. High level control of the data flow is provided under the standard
Message Passing Interface (MPI). The data to be transfered is selected
from the data on disk according to the geometric delay model and antenna
clock models at the selected time in the observation; the geometric delay model
is based on an imaginary telescope located at the centre of the Earth,
allowing the two data streams to be aligned to +/- 1 sample accuracy (the fractional
sample error is retained for use in post-correlation corrections).
The geometric delay model is generated using the CALC package
(Himwich 1988, see also http://gemini.gsfc.nasa.gov/solve/solve.html).
Once the selected data from two antennas (a single baseline) are
transfered to a processing node for correlation, the data are unpacked into
32-bit floating point numbers. The fringe rotation function
derived from the predicted delay between the two antennas is then formed
and applied to one of the two data streams. The two data streams are then
cross correlated to form the lag spectrum (expressed in terms of
the correlator coefficient at each lag), and the lag spectrum Fourier
transformed to give the cross power spectrum. In order to recover the
full signal to noise (e.g. D'Addario 1989), a second fringe rotation
function is generated, $\pi/2$ out of phase from the first fringe rotation
function. The correlation is repeated using the second fringe rotation
function. The two resulting cross power spectra are then coherently
averaged, after correction for the $\pi/2$ phase shift, to give the final
cross power spectrum. A fractional sample error correction is applied to
the cross power spectrum, to correct for the phase slope across the band
due to the limited accuracy in the initial alignment of the two data streams.
Finally, the measured system temperatures and gains from each antenna are
applied to calibrate the visibility amplitudes into Janskys (see Table 1).
The correlated data are reported back to a master node for collation and are
exported to the FITS format, for further reduction in
standard processing packages. This correlation scheme closely follows that
of the Australian Long Baseline Array
(LBA) correlator (Wilson et al. 1992; Roberts et al. 1997).
An account of a prototype version of the software correlator can be found
in West (2004). This new system for global VLBI is available to all users
through the Australia Telescope National Facility (ATNF)
(for detail, see http://www.atnf.csiro.au/vlbi/).
\subsubsection{Post-correlation data reduction and imaging analysis}
Following correlation, the data were imported into the AIPS
package for fringe-fitting and phase calibration,
then into the DIFMAP package (Shepherd 1997) for editing,
imaging, and evaluation of the source structure using modelfitting in the
($u,v$) plane.
The ($u,v$) coverage for the observation is thus shown in Fig. 1.
The coverage is reasonably good for a 20 minute observation (essentially
a single snapshot) due to the long east-west baselines from Australia to
South Africa, the long north-south baselines from Australia to Japan, and
the longest baseline from Japan to South Africa, which has large
components in both the east-west and north-south directions.
Fig. 2 shows the visibility amplitude as a function of distance in the
($u,v$) plane for the observation, projected onto the $u$ axis. Structure in
the source is immediately apparent from the variations in the visibility
amplitude.
The data were model-fitted in the ($u,v$) plane using a three component model
(Table 2) and self-calibrated in amplitude and phase. Amplitude
corrections of less than 5\% were derived from this procedure, showing
that the {\it a priori} calibration (Table 1) was of a reasonable quality.
The image resulting from the modelfit is shown in Fig. 3.
As shown in section 2.2
and discussed in section 3, the structure of PKS~0003$-$066 derived from
these observations agrees well with the structure found from other VLBI
array/correlator combinations such as VLBA+geodetic antennas
at 2.3 and 8.6 GHz (Fey \& Charlot 1997, see also below), the VLBA at 5 GHz
(Fomalont et al. 2000, Gabuzda, Pushkarev \& Cawthorne 2000),
and the VLBA at 15 GHz (Kellermann et al. 1998).
The core component is marginally resolved on the Australia-Japan-South Africa
baselines, which are the longest baselines yet used to observe this source,
apart from the VSOP observation for this source where three ground telescopes were used
for two hours in conjunction with the HALCA space telescope (Scott et al. 2004).
The brightness temperature for the core component, $T_b$, is derived to be
5.3$\times 10^{11}$K from Table 2, being consistent with the lower limits of
$>4.8\times 10^{11}$K derived by Scott et al. (2004) from the VSOP observations
(see also Horiuchi et al. 2004) and $>4.60\times 10^{11}$K derived by Kovalev et al.
(2005) from a VLBA 15 GHz observation.
\subsection{Analysis of VLBA/geodetic archive data}
Additional to the data described above, to study the structural variability of PKS~0003$-$066, we have used
data from the Radio Reference Frame Image Database (RRFID) of the
US Naval Observatory (USNO), at 2.3 and 8.6 GHz (Fey \& Charlot 1997).
Observations were made using an array consisting of
the 10 antennas of the Very Long Baseline Array (VLBA)
of the National Radio Astronomical Observatory (NRAO)
along with geodetic antennas. In total 13 epochs of data
are available from 1995 October to 2002 January (see Table 3).
Each observation was undertaken with the
dual-frequency VLBA receivers.
The sources were observed using short duration ($\sim$ 3 minutes)
``snapshots" over a number of different hour angles.
We analysed and imaged using DIFMAP the original datasets for PKS~0003$-$066
kindly provided to us by Alan Fey (private communication).
As images from the RRFID are
already published
\footnote{All images are available from the USNO-RRFID
website (see http://rorf.usno.navy.mil/RRFID/)
and the NASA/IPAC Extragalactic Database (NED)},
we do not reproduce them in full here.
Full results of our model-fitting
analysis of the data are presented, however, in Table 4
for observations listed in Table 3.
We have also obtained two datasets from the VLBA online archive for
PKS~0003$-$066. One is a part of a series of coordinated
RDV (Research and Development VLBA)
astrometric/geodetic experiments, numbering in total 9 epochs from 2002
July to 2004 July. These experiments use the full 10-station VLBA plus
up to 10 Mark 4 geodetic stations currently capable of recording VLBA modes.
These experiments were coordinated by the geodetic VLBI programs
of three agencies: USNO, NASA, and NRAO to monitor source structure
and to determine a high accuracy terrestrial reference frame (Gordon 2000).
Another dataset consists of three epoch observations at 2.3 and 8.6 GHz
with the VLBA only (obs. code BP 118) in May, June, and July 2004.
These archival data are exported to AIPS and DIFMAP for imaging
using standard procedures.
Since these images from the VLBA archive data have not been previously presented,
in Figs. 4, 5 and 6 we show images obtained from our analysis of the 9 epoch
RDV data and the 3 epoch BP 118 VLBA data.
Parameters of the images are given in Table 5.
We thus have a total of 25 epochs of data at 2.3 and
8.6 GHz in addition to the global VLBI observation at 2.3 GHz
described in Section 2.1.
Figures 7 and 8 show subsets of the RRFID and RDV images as montages at
both 2.3 and 8.6 GHz over a 10 year period.
The image from year 2003.35 is from the RDV data,
images at all other epochs are from the RRFID data.
From a model-fit analysis of the 8.6 GHz data we identified 3 components
in the pc-scale jet (as did Kellermann et al. 2004) that are
persistently recognisable until 1998,
in addition to the core. Then a new, 4th, component emerged from
the core in late 2002.
Table 4 lists the source models we obtained at each epoch.
Fig. 9 shows the angular distance of each component from the core as
a function of time. Fig. 10 shows the offsets of the jet
components in right ascension and declination
from the core position, as a function of time.
Finally, the flux density variability of PKS~0003$-$066 is shown
in Fig. 11 for 8.6 GHz and Fig. 12 for 2.3 GHz, using the all VLBI data analysed,
decomposing the source into its different pc-scale components.
An interpretation of the flux density monitoring observations
such as the Australia Telescope Compact Array (ATCA) (Tingay et al. 2003)
and the University of Michigan 26m (UMRAO) database
(see e.g. Aller et al. 1985 for description of the monitoring program)
will appear elsewhere (Horiuchi \& Tingay 2006, in preparation).
\section{Discussion}
\subsection{Proper motions and variability of the jet components}
From their analysis of 5 epochs of data over 5.5 years (from July 1995 to January 2001)
at 15 GHz, Kellermann et al. (2004) found no siginificant
proper motions for any pc-scale jet components in PKS~0003$-$066.
In contrast, from an examination of the montages in Figs. 6 and 7,
substantial evolution of the source structure is
apparent when considering a large amount of data over a ten year period.
We fit the radial distances from the core versus time for each component
of the 8 GHz images with the linear motion that minimizes the ${\chi}^2$ (Fig. 9 (a)).
The uncertainty of the proper motion, defined by a significance
of $\ge 3 \sigma$, is estimated using the ${\chi}^2$
statistics that includes both scattering of the data points around the proper
motion-model and uncertainties of individual observations
(one quarter of the beam sizes are adopted).
We show that the inner jet components,
denoted C2 and C4 (see Fig. 9 (a)), have proper motions
of 0.58$\pm$0.02 and 1.35$\pm$0.05 mas/yr, corresponding to
apparent speeds of 9.3$\pm$0.3 $c$ and 21.6$\pm$0.8 $c$.
These inner components are much faster than the outer component, C1,
which is moving at 0.12$\pm$0.01 mas/yr or approximately 2.0$\pm$0.2 $c$
(Table 8).
In Fig. 9 (b), the 15 GHz model fit components by Kellermann et al. (2004)
based on their image analysis (no error estimates on their component position
were provided) are superposed on the 8.6 GHz fits for the linear motions
together with the 2.3 GHz model components. The 2.3 GHz components
follow the similar trend of proper motions as seen at 8.6 GHz.
Although there is a hint
of shifts within one milliarcsecond or so between the paths at
2.3 and 8.6 GHz, the positional acuracy of 2.3 GHz data,
typically about 0.5 mas, is comparable to this.
The reason why Kellermann et al. (2004) didn't identify such high
proper motions may be attributed to the misidentification of the components
due to the large gaps between observations.
We have data points almost
every two months from 1997 to 1998, hence we clearly see the fast proper
motion of C2 continuously over that period.
The fact that the results of proper motion studies of core-dominated radio
sources can be so affected by the sampling frequency and overall period of
monitoring is well known (e.g. G\'{o}mez et al. 2001).
Moreover, Fig. 9 (a),(b) indicates that our data point for C2 at the epoch of
16 January 2002 (MJD 52291) is consistent with those for the component C
of Kellermann et al. (2004) around that epoch.
Also our images are at a lower frequency, hence steep spectrum components such
as C1, C2 and C4 are brighter (see Fig 12(b)) and easier to identify in spite of the
lower spatial resolution.
In contrast, at 15 GHz some jet components are resolved and are weaker,
making a consistent identification of the overall jet structure more difficult.
Note also that Kellermann et al. (2004) derived positions
in the image plane while we derived positions in the UV plane. This may lead
to some differences in the derived positions for the components.
\subsubsection{Kinematics of the fast jet components}
Adopting the relation between the intrinsic jet speed $\beta = v/c <$ 1,
the apparent speed $\beta_{{\rm app}}$, and the viewing angle of the
jet to our line of sight $\theta$
$$ \beta_{{\rm app}} = \frac{\beta \sin\theta}{1-\beta \cos\theta}, $$
we find that the jet inclination that maximises the apparent speed is
$\theta_{{\rm max}} = 5.3^\circ$, using the fastest apparent motion,
$\beta_{{\rm app}} = 21.6$, of C4 or $12.3^\circ$ for the $9.3c$ apparent motion of C2.
One can then derive a minimum Lorentz factor, $\gamma_{{\rm min}} = 21.6$ for C4,
or 9.35 for C2,
which corresponds to a Doppler factor of
$\delta_{\rm min} =
\gamma_{\rm min}^{-1}(1-\beta_{\rm min}\cos\theta_{\rm max})^{-1}$ = 8.63
for C4 and 3.74 for C2,
where $\beta_{{\rm min}} = \sqrt{1-\gamma_{{\rm min}}^{-2}}$ = 0.9989 for C4
or 0.9942 for C2.
For smaller viewing angles the Doppler factor approaches $\delta \sim 2\gamma$
as $\theta \sim 0$, which yields $\delta_{\rm max} = 43.2$ for C4 and 18.7 for C2.
Although Kellermann et al. (2004) didn't measure superluminal
motions in PKS~0003$-$066, in thier sample of 110 sources for the VLBA 2cm suevey
they measued superluminal motions ranging up to $\beta_{app} \sim 25$.
Our results suggest that the superluminal motion of PKS 0003-066 we measure
corresponds to the middle to high end of the distribution of superluminal motions
in the sample of Kellermann et al. (2004, Fig.10).
\subsubsection{The interaction between the moving and quasi-stationary components}
Also clear from the montage of images and Fig. 9 is that the fast components
C2 and C4 catch up to the slower component C1 and interact strongly near epoch 1999.
As this interaction occurs, the
C1 region clearly gets brighter. The component C3, closest to the core, appears stationary.
Coexistence of faster and slower or quasi-stationary components are
seen in many sources such as
4C 39.25 (Alberdi et al. 2000),
Centaurus~A (Tingay et al. 2001), 3C 120 (G\'{o}mez et al. 2001), 3C~279 (Jorstad et el.
2004) and 1803+784 (Britzen et al. 2005), which may be
explained within relativistic
time-dependent hydrodynamic models of ``trailing shocks" (Agudo et al. 2001).
In this scenario it is expected that the motion of inner components
is slower than that of the outer primary component, as clearly seen in 3C~120
(G\'{o}mez et al. 2001) and Centaurus~A (Tingay et al. 2001).
Our analysis of PKS~0003$-$066 suggests that C3 may be trailing shock for C2.
Another explanation of the stationary features could be as bends of the jet
trajectory in a plane different to the observers plane, as suggested for
4C~39.25 by Alberdi et al. (2000).
In PKS~0003$-$066 the component C1 has a non-radial motion, from NE to SW, while
brightning in total flux density. The component C3 is stationary in the distance from
the core but not stationary in the direction perpendicular to the radial direction.
The interaction between C1 and C2 may be also evident in the variability
of the polarization profile for C1.
The VLBA polarization images at 15 GHz from 2003 February to 2005 June
(Lister \& Homan 2005, MOJAVE database website) show that the component C1 is
significantly polarized, $\sim$20\% average and up
to 60\% around the outer edge of the component, suggesting compression and
enhancement of the local magnetic field at the interface with the ambient media.
Such high fractional polarization
is not seen for the C1 component in the 5 GHz global VLBI image from 1995 May
(Gabuzda, Pushkarev \& Cawthorne 2000), with only an upper limit of $<$11\%,
possibly because the component is not compressed enough at that stage.
The brightning of a jet caused by internal shocks with a slightly
varying jet speed is seen also in the well studied galactic X-ray binary jet
SS~433. Migliari, Fender \& M\'{e}ndez (2002) show evidence for a hot continuum and
Doppler-shifted iron emission lines from spatially resolved regions,
suggesting that {\it in situ} reheating of the jet component takes place in a
flow that moves with relativistic bulk velocity more than 100 days after
launch from the binary core of the SS~433 jet.
\subsubsection{Relation to large-scale structure}
As discussed earlier, the pc-scale VLBI jet kinematics possibly
supports jet bending.
An interesting question is whether the structure on kpc-scales as seen
with the VLA would be related to the pc-scale structure with an
$\sim$90$^\circ$ misalignment, as commonly
seen in many compact core-dominant sources (e.g. Conway \& Murphy 1993).
In a report of a VLA calibrator position survey, Perley (1982) noted a single secondary
component of PKS~0003$-$066 located at $r=1.7$\arcsec~ (8.5 kpc), p.a=30$^\circ$, but to
our knowledge no kpc-scale image for this source has ever been published.
We made two VLA images from archived data at 4.9 and 8.5 GHz
(Figs. 13(a) and (b)). We clearly detect the secondary component as pointed out by
Perley (1982), although it appears very weak, approximately 0.3 percent
of the flux density from the unresolved core.
The inner VLBI jet ($<$ 0.8 mas) has an initial position angle centered at 7$^\circ$ with
an oscillation of $\pm$15$^\circ$. On scales of 2-6 mas, the position angle
is close to $\sim$90$^\circ$. On arcsec scales, the jet appears close to
0$^\circ$ once again.
A natural explanation of the extreme curvature of the jet is that
we are looking at a part of a spiral-like trajectory,
its appearance affected by projection onto the sky plane.
An example of such orthogonal bending in a kpc-scale jet of a BL Lac object
can be seen in 1803$+$784 as discussed by Tateyama et el. (2002), modeling a
helical jet structure.
\section{Conclusion}
The main results of our multi-epoch VLBI image analysis of the BL Lac PKS~0003$-$066
are as follows:
1. In contrast to previous studies of this source, VLBI components of the jet
are highly variable, with proper motions of two components, C2 and C4, found
to be as high as 0.6 mas/yr and 1.4 mas/yr ($\sim$ 9 $c$ and 22 $c$) respectively.
The jet component C1 is moving very slowly with $\sim$0.1 mas/yr
or $\sim$2.0 $c$. As the inner component C2 approaches C1, the flux density of C1
increases 3 times, indicating an interaction between the components.
2. A new disk-based recording system and software correlator for VLBI were verified
by comparing imaging results to other established recorder/correlator systems.
\acknowledgments
This work is supported by the Australian Federal Governments
Major National Research Facility (MNRF) program.
The Australia Telescope Compact Array is funded by the
Australian Commonwealth Government for operation as a national
facility managed by CSIRO.
The VLBA is an instrument of the National Radio Astronomical Observatory,
which is a facility of the National Science Foundation operated under
cooperative agreement by Associated Universities, Inc.
This research has made use of the United States Naval Observatory (USNO)
Radio Reference Frame Image Database (RRFID).
This research has made use of data from the University of Michigan Radio
Astronomy Observatory, which is supported by the National Science
Foundation and by funds from the University of Michigan.
This research has made use of the NASA/IPAC Extragalactic Database (NED)
which is operated by the Jet Propulsion Laboratory, California Institute
of Technology, under contract with the National Aeronautics and Space Administration.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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\section{\label{S.Intro}Introduction}
State-space models (SSMs) are a popular modeling framework for analyzing ecological time-series data. They are commonly used to model population dynamics \citep{Newman-etal-2014},
including metapopulation dynamics \citep{Ward-etal-2010}, they have a long history in fisheries stock assessment \citep{Aeberhard-etal-2018}, and have been recently proposed as a means of analyzing sparse biodiversity data \citep
{Kindsvater-etal-2018}. Moreover, they have been a favored approach in movement ecology for more than a decade \citep{Patterson-etal-2008}, and are increasingly used with biologging data \citep{Jonsen-etal-2013}. In addition, the flexibility of SSMs is advantageous when modeling complex capture-recapture data \citep{King-2012}. SSMs are also used in epidemiology \citep{Dukic-etal-2012, Fasiolo-etal-2016} and disease ecology \citep{Hobbs-etal-2015}. These common uses of SSMs, and their many unique applications \citep[e.g., investigating animal health from photographs, \citealt{Schick-etal-2013}; plant invasion, \citealt{Damgaard-etal-2011}; and host-parasitoid dynamics,][]{Karban-and-deValpine-2010}, demonstrate their wide-spread importance in ecology.
SSMs are popular for time series in part because they directly model temporal autocorrelation in a way that helps differentiate process variation from observation error.
SSMs are a type of hierarchical model \citep[see Table \ref{t.terms} for definitions;][]{Cressie-etal-2009} and their hierarchical structure accommodates the modeling of two time
series: 1) a state, or process, time series that is unobserved and attempts to reflect the true, but hidden, state of nature; and 2)
an observation time series that consists of observations of,
or measurements related to, the state time series. For example,
actual fish population size over time would be the state time series,
while incomplete and imprecise counts of fish sampled in a survey,
or caught in a fishery, would be the observation time series. Process variation represents the stochastic processes
that changes the population size of a fish stock through time (e.g., the
birth and death processes), while observation error
reflects differences between the hidden state and the observed data due to randomness or imprecision in the sampling or
survey methodology. These two stochastic components act
at different levels of the model hierarchy, and
the SSM framework allows them to be modeled separately. The assumptions that the hidden states are autocorrelated (e.g., that a large population in year $t$ will likely lead to a large population in year $t+1$), and that
observations are independent once we account for their dependence on the states (Fig. \ref{fig:structure}a), allow SSMs to separate these two levels of stochasticity. When we fit a SSM to time series, we can often estimate the process and observation parameters, as well as the hidden states. These estimates
of the hidden states generally reflect the true state of nature better than the original observations (Fig. \ref{fig:structure}b). For example, the estimates of the hidden states will generally reflect the true fish population size better than the survey- or fisheries-based counts.
The first SSMs, often referred as Normal Dynamic Linear
Models (NDLMs), were a special case where the state and the observation time
series were modeled with linear equations and normal distributions. Two seminal papers on NDLMs, \citet{Kalman-1960}
and \citet{Kalman-and-Bucy-1961}, provided an algorithmic
procedure, the now famous Kalman filter, for making inferences
about the hidden states given imperfect observations and known parameters. These papers led to developments that revolutionized
aerospace engineering in the 1960s and allowed the Apollo mission
to correct the trajectory of a spacecraft going to the moon,
given inaccurate observations of its location through time
\citep{Grewal-Mohinder-2010}. The earliest applications of SSMs
to ecological data, which used NDLMs and the Kalman filter,
were in the 1980-90s and focused primarily on
fisheries \citep{Mendelssohn-1988, Sullivan-1992} and
animal movement \citep{Anderson-Sprecher-1991}.
The first animal movement SSMs were closely analogous
to the original aerospace application in that they recreated
the trajectory of an animal based on inaccurate observations.
However, these ecological models required parameters to be estimated.
Unlike a planned mission to the moon, we rarely have \textit{a
priori} knowledge of the intended speed and direction of an animal.
Developments in the time-series literature made use of the Kalman filter to evaluate the likelihood function for unknown parameters, thus allowing
calculation of maximum likelihood parameter estimates in addition to state estimates \citep{Harvey-1990}. NDLMs, however, are a restricted class of SSMs and
their applicability to many ecological time series, which have
nonlinear and non-Gaussian structure, is limited.
Since their initial development, there have been important advancements in SSMs, and in their
application in ecology. In the 1990s, the simultaneous popularization of
Markov chain Monte Carlo methods \citep[MCMC,][]{Gilks-etal-1995}, including the freely
available BUGS software \citep{Lunn-etal-2009}, and high speed desktop computing considerably expanded the diversity of possible SSMs
to include non-Gaussian and nonlinear formulations
\citep[e.g.,][]{Meyer-Millar-1999}. As a result, Bayesian ecological SSMs were developed for a variety of applications in the following decades, including capture-recapture models \citep[e.g.,][]{Dupuis-1995,Gimenez-etal-2007, Royle-2008} and formulations structured around matrix population models \citep{Buckland-etal-2004}. Further developments have
advanced fitting procedures in both Bayesian and frequentist
frameworks \citep{deValpine-2004, Ionides-etal-2015, Kristensen-etal-2016, Monnahan-etal-2017}.
These methods provide the means to fit increasingly complex
SSMs with multiple hierarchical levels
\citep[e.g.,][]{Jonsen-etal-2005} and integrate disparate datasets
\citep[e.g.,][]{Hobbs-etal-2015}.
However, while advancements in fitting SSMs have changed how we
model time series in ecology, the computational burden required
to fit some of these models is often high enough that
comparisons between multiple SSMs can be difficult, and the
complex structure of some SSMs complicates model validation
and diagnostics. In the ecological literature, there has been a recent interest in
model comparison and validation for hierarchical
models or models for datasets with complex dependence structure \citep{Hooten-Hobbs-2015, Roberts-etal-2017, Conn-etal-2018}. In line with this, new validation tools for SSMs are being developed
\citep{Thygesen-etal-2017}.
While SSMs are powerful tools for modeling ecological time series,
the fitting procedures may seem prohibitively complex to
many practitioners. The variety
of inference procedures and tools that can be used to fit SSMs
\citep{Harvey-1990, Doucet-etal-2001, Durbin-and-Koopman-2012,
Ionides-etal-2015, Kristensen-etal-2016}
may bewilder all but the most quantitative ecologists, thus limiting the ability of many researchers to formulate, fit, and evaluate
their own SSMs. While there are some popular application-specific \texttt{R} \citep{R} packages with functions to fit specialized SSMs
\citep[e.g., \texttt{MARSS} for multivariate
NDLMs, \citeauthor{Holmes-etal-2012}, \citeyear{Holmes-etal-2012}; \texttt{bsam}, for
animal movement,][]{Jonsen-etal-2005}, few ecologists are aware of the full range of SSMs that can be fitted with such packages. In addition, these packages may be inadequate for the data-at-hand, especially when using SSMs to answer novel questions or with new data types. A further complication with the application of SSMs
is the potential for estimability issues where some states or
parameters cannot be estimated well, or at all,
given the available data \citep{Dennis-etal-2006, Knape-2008, AugerMethe-etal-2016}.
For example, such estimability issues may arise because the formulation of a SSM is too complex for the data (e.g., the time resolution of the process model is too
fine relative to the time resolution of the observations). While
there has been some effort to provide a general, and easy to use,
set of tools for ecologists to fit SSMs to their data \citep[e.g.,][]{King-etal-2016, deValpine_et_al:2017}, the available tools and array of
choices may be overwhelming to those with little familiarity with SSMs. Given these challenges and the recent advancements in inference methods and model diagnostics, we believe the time is ripe to
provide a review of these developments
for scientists wanting to fit SSMs to ecological time series.
In this review, we first demonstrate the flexibility of SSMs through a set of examples (Section \ref{S.examples}) and discuss how ecologists should consider SSMs as a default modeling technique for many of their time series (Section \ref{S.OTHER}). Next, we review the different inference methods that can be used to fit a SSM to data (Section \ref{S.Fitting}). We then discuss how one can assess whether a SSM suffers from estimability
or identifiability issues (Section \ref{S.Estimability}). Lastly, we describe model selection procedures (Section \ref{S.Model.Comparison}) and diagnostic tools that can be used to
verify whether a model is adequate (Section \ref{S.Diagnostics}), crucial steps that are often
ignored. This review is accompanied by an in-depth tutorial that provides examples of how one can use \texttt{R} \citep{R} to fit, and validate, SSMs with various inference methods. We believe this review will give a strong foundation to ecologists interested in learning about SSMs and hope it will provide new tools to veteran SSM users interested in inference methods and model validation techniques.
\section{\label{S.examples}Examples of ecological SSMs}
SSMs are flexible hierarchical models for time series, where observations are imperfect measures of temporally evolving hidden states. Through examples, we demonstrate that SSMs can model univariate or multivariate observations, as well as biological processes that evolve in discrete or continuous time steps. We also show that SSMs can be linear or nonlinear, and can use a variety of statistical distributions (e.g., normal, Poisson, multinomial). To show the structural flexibility of SSMs, we chose many examples from population and movement ecology, two fields that have been crucial in the development of these models. However, SSMs can be used to model time series from all branches of ecology.
\subsection{A toy example: normal dynamic linear model \label{S.M.toy}}
To formalize the description of SSMs, we start by describing a simple, toy example. It models a time series of univariate observations, denoted $y_t$, made at discrete and evenly-spaced points in
time $t$ $(t = 1, 2, \ldots, T)$. The time series of states, denoted $z_t$, is defined at the same time points $t$ as the observations. Our model is a simple normal dynamic linear model (NDLM), thus process variance and observation error are modeled with Gaussian distributions and both time series are modeled with linear equations.
SSMs make two main assumptions. First, SSMs assume that the state time series evolves as a Markov process \citep{Aeberhard-etal-2018}. This Markov process, which is generally of first-order, is a relatively simple way to incorporate temporal dependence. For our toy model, this means that the state at time $t$, $z_t$, depends only on the state at the previous time step, $z_{t-1}$. Second, SSMs assume that the observations are independent of one another once we account for their dependence on the states. More formally, we say that, given
the corresponding state $z_t$, each observation $y_t$ is conditionally independent of all other observations, $y_s$, $s \ne t$. Thus, any dependence between observations is the result of the dependence between hidden states \citep{Aeberhard-etal-2018}. For our toy model, this means that $y_t$ is independent of $y_{t-1}$, and all other
observations, once we account for the dependence of $y_t$ on $z_t$ (Fig. \ref{fig:structure}a). In a population dynamics context, this could be interpreted to mean that the values of observations are autocorrelated because the process driving them (i.e., the true population size of the animal) is autocorrelated through time. In contrast, the discrepancy between the true population size and the observation is not correlated in time. We can see this structure in the equations for our toy SSM:
\begin{linenomath*}
\begin{align}
\label{E.state.NDLM}
z_t &= \beta z_{t-1} + \epsilon_t, & \epsilon_t \sim \text{N} \left( 0, \sigma^2_p \right ), \\
\label{E.obs.NDLM}
y_t & = \alpha z_t + \eta_t, & \eta_t \sim \text{N} \left( 0, \sigma^2_o \right ).
\end{align}
\end{linenomath*}
The autocorrelation in the states is captured by the parameter $\beta$. The observations are a function of the states only and the parameter $\alpha$ allows the observation at time $t$ to be a biased estimate of the state at time $t$. The process variation ($\epsilon_t$) and observation error ($\eta_t$) are both modeled with normal distributions but have different standard deviations ($\sigma^2_p$ and $\sigma^2_o$). We have not defined the state at time $0$, $z_0$, and many authors will provide an additional equation, often referred as the initialization equation, which describes the probability of different values of $z_0$ (e.g., $z_0 \sim \text{N}(0, \sigma_{z_0}^2)$). For our toy example, we view $z_0$ as a fixed and unknown parameter.
The terminology used to refer to the process and observation equations varies in the literature. A process equation can be referred as a process model, state equation, state model, or transition equation. An observation equation can be referred as an observation model,
measurement equation, or measurement model. In this paper, we generally use the terms `process equation' and `observation equation' respectively, and we often describe SSMs with equations that combine a deterministic function with a stochastic component (e.g., Eqs. \ref{E.state.NDLM}-\ref{E.obs.NDLM}).
To further reveal the dependence structure and understand how to fit SSMs to data, it can help to additionally characterize a SSM in terms of probability distributions
for the states and the observations, e.g.:
\begin{linenomath*}
\begin{align}
\label{E.state.general}
& f(z_t|z_{t-1},\boldsymbol{\theta}_p), & t=1,\ldots, T, \\
\label{E.obs.general}
& g(y_t|z_t,\boldsymbol{\theta}_o), & t=1,\ldots, T.
\end{align}
\end{linenomath*}
In the case of our toy model, $f$ and $g$ are two normal probability density functions, while $\boldsymbol{\theta}_p$ and $\boldsymbol{\theta}_o$ are
vectors of parameters associated with each equation (i.e., $\boldsymbol{\theta}_p = (\beta, \sigma_p^2)$, $\boldsymbol{\theta}_o = (\alpha, \sigma_o^2)$). Eq. \ref{E.state.general} describes the autocorrelation in state values as a first-order Markov process, while Eq. \ref{E.obs.general} describes how observations depend simply on the states. This definition also demonstrates that states are random variables and thus that SSMs are a type of hierarchical model.
One of the goals of fitting a SSM to data is typically to estimate unknown parameters. Here, to contrast them with the states, we refer to these as the fixed parameters and denote them together as $\boldsymbol{\theta}$. For example, here $\boldsymbol{\theta} = (\boldsymbol{\theta}_p, \boldsymbol{\theta}_o, z_0)$, thus $\alpha$, $\beta$, $\sigma^2_p$,
$\sigma^2_o$, $z_0$ in Eqs. \ref{E.state.NDLM}-\ref{E.obs.NDLM}. A second important goal is to estimate the unobserved states, $\mathbf{z}_{1:T} = (z_1, z_2, ..., z_T)$, where $T$ is the length of the time series. The notation $1{:}t$, which we use throughout, refers to the sequence 1, 2, $\ldots$, $t$. Fig. \ref{fig:structure}b shows how close estimates of the states ($\hat{\mathbf{z}}_{1:T}$) can be to their true values.
SSMs can be fitted using frequentist or Bayesian approaches to statistical inference. When using a Bayesian approach, a third level is added to the model hierarchy: the prior distribution(s) for the fixed parameters denoted by the probability density function, $\pi(\boldsymbol{\theta} |\boldsymbol{\lambda})$, where $\boldsymbol{\lambda}$ are known values called hyperparameters. While we refer to $\boldsymbol{\theta}$ as fixed parameters to differentiate them from the states, in Bayesian inference $\boldsymbol{\theta}$ is a vector of random variables. In Section \ref{S.Fitting} and Appendix S2, we discuss how we can use these probabilistic descriptions of the model for inference.
This simple linear and normal model is a toy example that we will use throughout to explain the concepts associated with fitting and validating SSMs. We will also use this model in Appendix S1 to demonstrate how to use \texttt{R} to fit a SSM to data. While the simplicity of this toy example makes it a helpful teaching tool, it is not a particularly useful model for ecology. We now turn to the description of a set of ecological SSMs.
\subsection{Handling nonlinearity \label{S.M.simple.pop}}
We use a set of simple univariate SSMs for population dynamics to demonstrate that even simple ecological models can rarely be blindly modeled as NDLMs. \citet{Jamieson-Brooks-2004} applied multiple SSMs to abundance estimates from North American ducks obtained through annual aerial counts. We start with one of their simplest models, for which the process equation is the stochastic logistic model of \citet{dennis1994}. This model allows for density dependence, i.e., a change in growth rate dependent on the abundance in the previous year:
\begin{linenomath*}
\begin{align}
z_t &= z_{t-1} \exp\left(\beta_0 + \beta_1 z_{t-1} + \epsilon_t \right),
& \epsilon_t \sim \text{N}(0, \sigma_p^2). \label{eq:ducks_process}
\end{align}
\end{linenomath*}
As in the toy example above, $z_t$ denotes the true hidden state, in this case the number of ducks in year $t$. The parameter $\beta_0>0$ determines the median rate of population growth when population size is 0. The parameter $\beta_1\leq 0$ determines how much the growth rate decreases with an increase in population size, with $\beta_1 = 0$ indicating no density dependence. The process variation, $\epsilon_t$, is normally distributed and represents the random change in growth rate each year. The observations $y_t$ are modeled as unbiased estimates of the true population size with a normally distributed error:
\begin{linenomath*}
\begin{align}
y_t &= z_t + \eta_t,
& \eta_t \sim \text{N}(0, \sigma_o^2).
\end{align}
\end{linenomath*}
Even though the observation equation is linear with a Gaussian error, the SSM is not a NDLM because of the exponent in the process equation (Eq. \ref{eq:ducks_process}). \citet{Jamieson-Brooks-2004} also modeled the population size on a logarithmic scale, $w_t=\log(z_t)$, which resulted in the following formulation:
\begin{linenomath*}
\begin{align}
w_t &= w_{t-1} + \beta_0 + \beta_1 \exp(w_{t-1}) + \epsilon_t,
& \epsilon_t \sim \text{N}(0, \sigma_p^2),
\\
y_t &= \exp(w_t) + \eta_t,
& \eta_t \sim \text{N}(0, \sigma_o^2) \label{eq:untrans1Pop}.
\end{align}
\end{linenomath*}
While such reconfiguration can sometimes linearize the model, in this case the model remains nonlinear.
\citet{Jamieson-Brooks-2004} use a Bayesian framework to fit this model, see their original paper for the description of the priors.
The modeling of density dependence has been extensively debated in the literature, and \citet{Jamieson-Brooks-2004} also explored an alternative
process equation, a stochastic Gompertz model:
\begin{linenomath*}
\begin{align}
z_t &= z_{t-1} \exp\left(\beta_0 + \beta_1 \log z_{t-1} + \epsilon_t \right),
& \epsilon_t \sim \text{N}(0, \sigma_p^2),
\end{align}
\end{linenomath*}
which assumes that the per-unit-abundance growth rate depends on the log abundance, $\log(z_{t-1})$, instead of the abundance, $z_{t-1}$ \citep{dennis1994}. Such a model is often linearized as follows:
\begin{linenomath*}
\begin{align}
w_t &= \beta_0 + (1 + \beta_1) w_{t-1} + \epsilon_t,
& \epsilon_t \sim \text{N}(0, \sigma_p^2),
\\
g_t &= w_t + \eta_t,
& \eta_t \sim \text{N}(0, \sigma_o^2),
\end{align}
\end{linenomath*}
where $w_t = \log (z_t)$ and $g_t = \log (y_t)$ are the logarithms of the states and observations, respectively \citep[e.g.,][]{Dennis-etal-2006}. The linear version of this model is a NDLM that can be fitted with tools such as the Kalman filter \citep{Dennis-etal-2006}. This statistical convenience may have contributed to the uptake of the stochastic Gompertz SSM in the literature. However, it may not always be adequate to assume that the growth rate depends logarithmically on population density \citep{dennis1994}.
Many papers have extended these models to incorporate external covariates \citep[e.g.][]{viljugrein2005,Saether-etal-2008,Linden-Knape-2009}. For example, one could account for the influence of fluctuating availability of wetlands on the population size of ducks by including the number of ponds in year $t$, $p_t$, as a covariate in the process equation and modifying the Gompertz stochastic model as follows:
\begin{linenomath*}
\begin{align}
w_t &= \beta_0 + (1 + \beta_1) w_{t-1} + \beta_2 p_t + \epsilon_t,
& \epsilon_t \sim \text{N}(0, \sigma_p^2).
\end{align}
\end{linenomath*}
This set of examples shows how easy it is to adapt and extend models in the SSM framework. While even simple ecological models may only be linear with transformations and assumptions, \citet{Jamieson-Brooks-2004}, \citet{viljugrein2005}, and \citet{Linden-Knape-2009} showed that accounting for observation error improved the inference regardless of the process equation. For example, \citet{viljugrein2005} demonstrated that using a SSM, rather than a model that ignores observation error, decreased the size of the bias in the estimates of density dependence. This decreased bias, and a better quantification of uncertainty, reduced the cases where one would erroneously conclude the presence of density dependence.
\subsection{Joining multiple data streams \label{S.M.stock.assessment}}
Integrating multiple sources of data, often referred as data streams, into a single model can offset their individual limitations and reveal more complex ecological relationships \citep{Mcclintock-etal-2017}. To showcase how SSMs can extract the information provided by multiple data streams, we present a simplified version of a state-space stock assessment model described by \citet{Nielsen-Berg-2014}. SSMs are often used in fisheries stock assessments \citep{Aeberhard-etal-2018}, where the first data stream, $C_{a,t}$, represents how many fish from each age class $a$ are caught in the commercial fishery in each year $t$, and the second data stream, $I_{a,t,s}$, includes age-specific indices from distinct scientific surveys, $s$, which can occur in different years and only capture some portion of the age classes.
The hidden state in each year $t$ is a vector combining the log-transformed stock sizes, $N_{a,t}$, and fishing mortality rates, $F_{a,t}$, for each age class: $\mathbf{z}_t$ $=$ $(\log{N_{1,t}},\ldots,\log{N_{{\textsc{a}},t}},$ $\log{F_{1,t}},\ldots, \log{F_{\textsc{a},t}})'$, where $A$ represents the oldest age class. Just as for the toy example, the process equations describe the state in year $t$ as a function of the state in year $t-1$. However, unlike the toy model, we no longer have a single process equation. We have instead a set of equations describing recruitment, survival, and mortality:
\begin{linenomath*}
\begin{align}
\log (N_{1,t}) &=\log (N_{1,t-1}) + \epsilon_{N_{1, t}}, \label{E.fsa.p.N1}\\
\log (N_{a,t}) &=\log(N_{a-1,t-1}) - F_{a-1,t-1} - M_{a-1,t-1} + \epsilon_{N_{a,t}}, \hspace{3.5cm} 2\leq a \leq A, \label{E.fsa.p.Na}\\
\log (F_{a,t}) &=\log (F_{a,t-1}) + \epsilon_{F_{a,t}}, \hspace{7.8cm} 1\leq a \leq A, \label{E.fsa.p.F}
\end{align}
\end{linenomath*}
where age and year specific log fishing mortality rates, $\log F_{a,t}$, are considered states that evolve as a random walk through time, but the equivalent natural mortality rate, $\log M_{a,t}$, is assumed known from outside sources. While the main equation describing the population growth (Eq. \ref{E.fsa.p.Na}) is based on demographic processes, the other equations (Eqs. \ref{E.fsa.p.N1} and \ref{E.fsa.p.F}) are simply assuming that recruitment and fishing mortality are each correlated across years. The formulation of Eq. \ref{E.fsa.p.Na}, and Eqs. \ref{E.fsa.obs.C}-\ref{E.fsa.obs.I} below, are based on the well-known Baranov catch equation, which states that a cohort continuously decreases in size through time according to two sources of mortality \citep[i.e., fishing and natural, see][for detail]{Aeberhard-etal-2018}. Derived from a continuous-time equation, the Baranov equation maps the surviving cohort size as depending exponentially on fishing and natural mortality rates. Thus, as shown here, the SSM can be modeled by expressing the age-specific stock size and mortality rates on the logarithmic scale.
The process variation for all of these equations are assumed to be Gaussian with zero mean, but they differ in their variance and covariance parameters. For recruitment and survival, the variation is assumed to be uncorrelated, i.e., $\epsilon_{N_{1, t}} \sim \text{N}(0, \sigma^2_{N_{a=1}})$, and $\epsilon_{N_{a,t}}$ and $\epsilon_{N_{\textsc{a},t}} \sim \text{N}(0,\sigma^2_{N_{a>1}})$. However, for fishing mortality, the yearly variation is assumed to be correlated across age classes (i.e., $\boldsymbol{\varepsilon}_{F_t} = (\epsilon_{F_1,t}, ..., \epsilon_{F_{\textsc{a}},t})' \sim \text{N} (\mathbf{0}, \boldsymbol{\Sigma}_F)$) due to age/size correlations in capture probability. The covariance matrix, $\boldsymbol{\Sigma}_{F}$, is assumed to have an auto-regressive order 1, AR(1), correlation structure (i.e., each element $\Sigma_{a,\tilde{a}} = \rho^{|a-\tilde{a}|} \sigma_a \sigma_{\tilde{a}}$, where $\rho^{|a-\tilde{a}|} = \mbox{cor}(\epsilon_{F_{a_t}}, \epsilon_{F_{\tilde{a} _t}})$).
The two different sets of data streams (i.e., the observed age-specific log-catches, $\log C_{a,t}$, and the age-specific log-indices from scientific surveys, $\log I_{a,t,s}$) are related to the time series of the unobserved states, $\mathbf{z}_t$, with the following observation equations:
\begin{linenomath*}
\begin{align}
\log C_{a,t} &=
\log\left(\frac{F_{a,t}}{K_{a,t}}(1-e^{-K_{a,t}})N_{a,t}\right)+\eta_{a,t,c} \label{E.fsa.obs.C},\\
\log I_{a,t,s} &= \log\left(Q_{a,s}
e^{-K_{a,t}\frac{D_s}{365}}N_{a,t}\right)+\eta_{a,t,s} \label{E.fsa.obs.I},
\end{align}
\end{linenomath*}
where $K_{a,t}$ is the total mortality rate of age class $a$ in year
$t$ (i.e., $K_{a,t}=M_{a,t}+F_{a,t}$), $D_s$ is the number of days
into the year when the survey $s$ is conducted, and each $Q_{a,s}$ is a model parameter describing the catchability coefficient. The observation error terms, $\eta_{a,t,c}$ and $\eta_{a,t,s}$, are assumed to be Gaussian distributed and their variances are designed such that the catch data, and each scientific survey have their own covariance matrix. We can use different covariance structures for each matrix \citep[e.g., independent catches across ages, but each survey index has an AR(1) correlation structure across ages; see][for other examples]{Berg-Nielsen-2016}.
This example depicts how to harness more information from independent data streams. The observation equations (Eqs. \ref{E.fsa.obs.C}-\ref{E.fsa.obs.I}) account for the differences in how each data stream is related to a common of set of states (i.e., stock sizes, $N_{a,t}$). In addition, the potentially more biased data stream (i.e., the fisheries catch data) provides direct information on the other set of states (i.e., fishing mortality rate, $F_{a,t}$), which would otherwise be difficult to estimate. This type of structure provides the opportunity to model more complex ecological mechanisms (e.g., Eqs. \ref{E.fsa.p.N1}-\ref{E.fsa.p.F}). SSMs that integrate multiple data streams have been used in other fields of ecology, including movement ecology \citep{Mcclintock-etal-2017} and disease ecology \citep{Hobbs-etal-2015}.
\subsection{Accounting for complex data structure \label{S.M.movement.model}}
SSMs are well suited to handle the complex structure of many ecological datasets. For example, the first difference correlated random walk model \citep[DCRW,][]{Jonsen-etal-2005}, one of the earliest SSMs for animal movement, was developed to account for the peculiarities of Argos doppler shift location data \citep{Jonsen-etal-2005}. Argos tags are often used to track marine animals because they overcome some of the challenges associated with using conventional GPS units in an aquatic environment. However, unlike most GPS datasets, Argos locations, $\mathbf{y}_i = \left[\begin{smallmatrix} y_{i,lon}\\ y_{i,lat} \end{smallmatrix}\right]$, data have large observation errors \citep[mean error ranging from 0.5-36km,][]{Costa-etal-2010}, including large outliers. In addition, they are collected at irregular time intervals, $i$, (i.e., when the animal is at the surface and the satellites are overhead), and have a quality rating that classifies each location into one of six categories, $q_i$. All of these aspects of the data are incorporated in the simplified version of the DCRW presented below.
While the observations are taken at irregular time intervals, the process equation models the true locations of the animal at regular time intervals $t$, $\mathbf{z}_t = \left[\begin{smallmatrix} z_{t,lon} \\ z_{t,lat}\end{smallmatrix}\right]$, for $T$ time steps. The process equation assumes that the animal's location at time $t$ is not only dependent on the previous location, $\mathbf{z}_{t-1}$, but also on the animal's previous displacement in each coordinate, $\mathbf{z}_{t-1} - \mathbf{z}_{t-2}$:
\begin{linenomath*}
\begin{align}
\mathbf{z}_t &= \mathbf{z}_{t-1} +
\gamma(\mathbf{z}_{t-1} - \mathbf{z}_{t-2}) +
\boldsymbol{\epsilon}_t, & \boldsymbol{\epsilon}_t &\sim \text{N}(0,\mathbf{\Sigma}), & 1 \leq t \leq T, \label{e.dcrw.p}
\end{align}
\end{linenomath*}
where
\begin{linenomath*}
\begin{align}
\boldsymbol{\Sigma} &= \begin{bmatrix}
\sigma_{\epsilon,lon}^2 & \rho\sigma_{\epsilon,lon}\sigma_{\epsilon,lat} \\
\rho\sigma_{\epsilon,lat}\sigma_{\epsilon,lon} & \sigma_{\epsilon,lat}^2
\end{bmatrix}.
\end{align}
\end{linenomath*}
The parameter $\gamma$ can take values between 0 and 1 (i.e., $0 \le \gamma \le 1$), and controls the degree of correlation between steps. Values close to 0 mean that the movement only depends on the previous location. Values close to 1 reflect strong correlation in both latitudinal and longitudinal displacements, and thus mean that the animal has a tendency to move at the same speed and in the same direction as the previous step. The covariance matrix for the process variation, $\boldsymbol{\Sigma}$, allows for covariance between longitude and latitude, but in many instances it is simpler to assume that $\rho = 0$.
The observation equation accounts for various characteristics of the Argos data:
\begin{linenomath*}
\begin{align}
\mathbf{y}_i &= (1-j_i)\mathbf{z}_{t-1} + j_i\mathbf{z}_t + \boldsymbol{\eta}_i, & \boldsymbol{\eta}_i &\sim \text{T}(\boldsymbol{\Psi} \circ \mathbf{S}_i, \mathbf{D}_i), & 1 \leq i \leq N,
\label{e.dcrw.o}
\end{align}
\end{linenomath*}
where
\begin{linenomath*}
\begin{align}
\boldsymbol{\Psi} &= \begin{bmatrix}
\psi_{lon}\\
\psi_{lat}
\end{bmatrix},\\
\mathbf{S}_i &= \begin{bmatrix}
s_{lon,q_i}\\
s_{lat,q_i}
\end{bmatrix},\\
\mathbf{D}_i &= \begin{bmatrix}
df_{lon,q_i}\\
df_{lat,q_i}
\end{bmatrix},
\label{e.dcrw.v}
\end{align}
\end{linenomath*}
and $N$ is the number of observed Argos locations.
Because data are taken at irregular time intervals, the true location of the animal is linearly interpolated to the time of the observation, with $j_i$ representing the proportion of the regular time interval between $t-1$ and $t$ when the observation $\mathbf{y}_i$ was made. Because the data often have outliers, the measurement errors are modeled with t-distributions, which have fat tails. Finally, to model the differences in error size between the six quality categories, each category, $q_i$, is associated with unique bivariate t-distributions: $\text{T}(\boldsymbol{\Psi} \circ \mathbf{S}_i, \mathbf{D}_i)$. In particular, each category is associated with a unique scale parameter, $s_{c,q_i}$, and degrees of freedom, $df_{c,q_i}$, for each coordinate (i.e., $c = lon$ or $lat$). Instead of estimating these 24 parameters, many researchers fix them to known values derived from field experiments \citep[e.g.,][]{Jonsen-etal-2005}. To allow for discrepancies between these fixed values and the ones that may fit the data best, we can add a correction factor for each coordinate, $\psi_c$. Note that the Hadamard product, $\circ$, simply states that we perform entrywise multiplication of the correction factors to the scale parameters, i.e., $\psi_c s_{c,q_i}$, for $c= (lon,lat)$. Fig. \ref{fig:polarbear} shows the DCRW fitted to a polar bear track, and Appendix S1: Section S1 2.3 provides the code to fit the model.
Datasets with unexplained outliers and data points with differing quality ratings are common in ecology, and the flexibility of SSMs allow to directly account for these characteristics in the model, rather than arbitrarily discarding data.
\subsection{Accommodating continuous-time processes }\label{S.cont.mov.model}
So far we have only described SSMs where the hidden state evolves in discrete time steps. However, many biological processes occur in continuous time and modeling them as such can facilitate the use of irregularly-timed observations \citep{Dennis-etal-2014, McClintock-etal-2014}. Using a simplified version of the movement model of \citet{Johnson-etal-2008}, we showcase how SSMs can accommodate continuous-time process equations.
The SSM of \citet{Johnson-etal-2008} models the movement of an animal with a continuous-time correlated random walk. The process equation is formulated in terms of how changes in velocity $v$ through time affect the location $\mu$ of an animal. While the model describes an animal moving in two dimensions (e.g., latitude and longitude), for simplicity, we assume the velocity processes in each coordinate to be independent and only describe the process for one coordinate. Velocities at time $t$, denoted $v(t)$, are the first set of states. Change in velocity over time is described using a type of diffusion model called an Ornstein-Uhlenbeck (OU) process. At time $t + \Delta$, velocity is:
\begin{linenomath*}
\begin{align}
v(t+\Delta) &= e^{-\beta\Delta} v(t) + \zeta(\Delta), & \zeta(\Delta) \sim \text{N}\left(0, \sigma^2_{OU} (1-e^{-2\beta\Delta})/2\beta\right), &\;\;\;\;\;\; \beta > 0
\label{eq:OU}
\end{align}
\end{linenomath*}
where $\beta$ represents how fast the temporal correlation in velocity tends towards 0, and $\zeta(\Delta)$ is a random perturbation. As both increases, the autocorrelation in velocity decreases. In addition, as the time difference ($\Delta$) increases, the velocity value at time $t+\Delta$ depends less on the previous velocity value and more on the random perturbation. This assumption is often reasonable as we expect an animal to continue at the same speed during a short period of time and be more likely to change speed over long time periods.
While the core of the process model describes changes in velocity, the observations are locations. Thus, we have a second set of states, the locations $\mu(t)$, which are related
to velocities as follows:
\begin{linenomath*}
\begin{align}
\mu(t+\Delta) &= \mu(t) + \int_{t}^{t+\Delta}v(u) du.
\label{eq:muInt}
\end{align}
\end{linenomath*}
Integrating the rate of change, here speed, over the time interval is often key to link continuous-time processes to ecological observations \citep[e.g., to model oxygen concentration,][]{Appling-etal-2018}. Such integration can be difficult to handle, but \citet{Johnson-etal-2008} solved Eq. \ref{eq:muInt} to show that the change in location in time $\Delta$ is simply:
\begin{linenomath*}
\begin{align}
\mu(t+\Delta) &= \mu(t) + v(t)\left(\frac{1-e^{-\beta \Delta}}{\beta}\right) + \xi(\Delta), & \xi(\Delta) \sim \text{N}\left(0, \frac{\sigma^2_{OU}}{\beta^2}\right).
\label{eq:mu}
\end{align}
\end{linenomath*}
Because $\Delta$ can take any non-negative value, we can keep track of the states at any time intervals, thus easily accommodating observations, $y_i$, collected at irregular-spaced times, $t_i$. For the state, $\mathbf{z}_i$, the final process equations in matrix notation form are:
\begin{linenomath*}
\begin{align}
\mathbf{z}_i &= \begin{bmatrix}
\mu_i \\ v_i
\end{bmatrix}
= \begin{bmatrix}
1 & (1-e^{\beta\Delta_i})/\beta \\ 0 & e^{-\beta\Delta_i}
\end{bmatrix} \begin{bmatrix}
\mu_{i-1} \\ v_{i-1}
\end{bmatrix} + \boldsymbol{\eta}_i,
& \boldsymbol{\eta}_i \sim \text{N}\left(\mathbf{0}, \boldsymbol{\Sigma}^2_p\right),
\label{eq:process}
\end{align}
\end{linenomath*}
where $u_i$ and $v_i$ are $u(t)$ and $v(t)$ at the time when the $i^{th}$ observation occurred, $\Delta_i = t_i-t_{i-1}$, and the variance-covariance matrix was solved to be:
\begin{linenomath*}
\begin{align}
\boldsymbol{\Sigma}^2_p &= \begin{bmatrix}
\frac{\sigma^2_{OU}}{\beta^2}&\frac{\sigma^2_{OU}}{2\beta^2}\left(1-e^{-\beta\Delta_i}\right)^2 \\
\frac{\sigma^2_{OU}}{2\beta^2}\left(1-e^{-\beta\Delta_i}\right)^2&\sigma^2_{OU} (1-e^{-2\beta\Delta_i})/2\beta
\end{bmatrix}.
\end{align}
\end{linenomath*}
The observation equation can be chosen as usual, for example as simply adding normal error to the true location:
\begin{linenomath*}
\begin{align}
y_i &= \mu_i + \epsilon_i, & \epsilon_i \sim \text{N}\left(0, \sigma^2_o\right).
\label{eq:obs}
\end{align}
\end{linenomath*}
The SSM defined by Eqs. \ref{eq:process}-\ref{eq:obs} is a linear Gaussian SSM and can therefore be fitted with a Kalman filter.
This model allows various extensions to include different aspects of animal movement. For example, \citet{Johnson-etal-2008} show how haul-out behavior of tagged seals can be incorporated using data on how long the tag has been dry (e.g., by making $\beta$ an increasing function of dry time). To account for the large outliers associated with Argos data one can use a t-distribution (see Section \ref{S.M.movement.model}), in which case the Kalman filter will no longer be adequate and other fitting methods will be required \citep{Albertsen-etal-2015}. While continuous-time models can be more complex to understand, they are useful in a variety of contexts where data is collected at unequal time intervals and when ecological processes are intrinsically continuous \citep[e.g., population dynamics,][]{Dennis-etal-2014}.
\subsection{Integrating count and categorical data streams}
The SSM framework provides the flexibility to create joint models that integrate different data types and link biological processes. Here, we use the model of \citet{Schick-etal-2013} to demonstrate how count and categorical data can be integrated in a single SSM for the health, monthly movement, and survival of North Atlantic right whales (\textit{Eubalaena glacialis}).
\citet{Schick-etal-2013} extracted two types of data from photographic observations of individual whales. The first type, denoted $y_{i,t,k}$, is the number of sightings of individual $i$ in geographic zone $k$ and month $t$. The second type, denoted $H_{q,i,t}$, is the value for the $q^{th}$ visual health metric for individual $i$ in month $t$. The six visual health metrics (e.g., skin condition, entanglement status) are on ordinal scales, each with two or three levels. In addition, ancillary data (e.g., search effort, whale age) are used.
Three process equations model the health, survival, and monthly movement of each individual whale. Whale $i$ in month $t$ is characterized by its age $a_{i,t}$, health status $h_{i,t}$ (defined on an arbitrary, but positive, scale: $(0,100)$), and location $k_{i,t}$ (one of nine geographic zones). Health status, $h_{i,t}$, is modeled as a function of previous health status and age:
\begin{linenomath*}
\begin{align}
h_{i,t} &= \beta_0 + \beta_1 h_{i,t-1} + \beta_2 a_{i,t-1} + \beta_3 a_{i,t-1}^2 + \epsilon_{i,t}, & \epsilon_{i,t} \sim \text{N}(0, \sigma^2).
\end{align}
\end{linenomath*}
When $\beta_2 > 0$ and $\beta_3 < 0$, the quadratic age term allows for the fact that health status, and thus survival probability, initially increases but declines with advanced age. Survival from month $t$ to $t+1$ is modeled as a
Bernoulli process, with survival probability modeled with a logit link function:
\begin{linenomath*}
\begin{align}
\text{logit}(s_{i,k,t}) & = \alpha_{0,k} + \alpha_1 h_{i,t}.
\end{align}
\end{linenomath*}
Here, $\alpha_{0,k}$ denotes the fixed effect for zone $k$ and $\alpha_1$ the relationship with health. Hence, survival probability depends on health and on the occupied zone, allowing researchers to identify the geographic zones associated with reduced survival.
While a whale is assumed to stay in a single zone during the month, it can move between zones each month. The monthly location of each individual, $z_{i,t}$, is only known when the individual is sighted that month. The subscript $t$, throughout, represents the number of months since the beginning of the time series. For each month of the year (January, ..., December), denoted $t^{(u)}$, the movement between zones is modeled with a transition matrix, where each element, $m_{j,k,t^{(u)}}$, describes the probability of moving from zone $j$ to zone $k$ (i.e. $m_{j,k,t^{(u)}} = \text{Pr}(z_{i,t^{(u)}+1} = k |z_{i,t^{(u)}} = j)$). As the complete geographic range of the whales is assumed to be covered by these zones, a living whale will be in one of the nine distinct zones at time $t+1$, $\sum_{k=1}^9 m_{j,k,t^{(u)}} = 1$. The changes in transition probabilities between the months of the year, $t^{(u)}$, allow for the modeling of seasonal migration.
The model has two sets of observation equations. First, the number of sightings of whale $i$ in location $k$ and month $t$ is modeled as a Poisson random variable.
\begin{linenomath*}
\begin{align}
y_{i,k,t} &\sim \text{Pois}(\lambda_i E_{k,t}),
\end{align}
\end{linenomath*}
where $E_{k,t}$ denotes the search effort in zone $k$ and month $t$ and $\lambda_i$ denotes the expected number of sightings of individual $i$ per unit effort. The number of sightings of whale $i$ is only modeled in months where the individual is alive based on state $s_{i,k,t}$ and in the appropriate monthly geographical zones according to state $z_{i,t}$. Second, each visual health metric is modeled as coming from a multinomial logit distribution. The probability of being in each level of the $q^{th}$ health metric depends on the true health status, $h_{i,t}$, and the model is structured so as to ensure that the ordinal aspect of the variables is respected (i.e., that lower values means lower health). For example, if the health metric $ H_{q,i,t}$ has a three-level ordinal scale, the observation equations for this metric are: \begin{linenomath*}
\begin{align}
H_{q,i,t}& \sim \text{Multinom}(1, \boldsymbol{p}_{q, i,t})
\\
\text{logit}(p_{q, i,t,1})&= \log\left(\frac{p_{q, i,t,1}}{p_{q, i,t,2}+p_{q,i,t,3}}\right) = c_{q,0,1} + c_{q,1,1} h_{i,t}
\\
\text{logit}(p_{q,i,t,1} + p_{q, i,t,2})&= \log\left(\frac{p_{q,i,t,1} + p_{q, i,t,2}}{p_{q,i,t,3}}\right) = c_{q,0,2} + c_{q,1,2} h_{i,t} \label{eq:summultnomlink}
\\
p_{q,i,t,3} &= 1- p_{q,i,t,1}-p_{q,i,t,2}.
\end{align}
\end{linenomath*}
The vector $\boldsymbol{p}_{q,i,t}$ contains the probabilities with which an individual with true health $h_{i,t}$ is assigned a specific health level. Since $h_{i,t}$ is positive, forcing the parameters $c_{q,0,1} < c_{q,0,2}$ and $c_{q,1,1} < c_{q,1,2}$, and modeling cumulative probabilities (Eq. \ref{eq:summultnomlink}) ensure that the order of the levels is accounted for. The probabilistic nature of the model allows health metrics to depend on the true health status but to be observed with error.
By integrating different data types, this SSM allows inference about various aspects of North Atlantic right whales. For example, we can learn which visual health metrics show the strongest links to underlying health, whether geographic regions (and thereby human activity) have an impact on survival, and at which times of the year movement to certain zones occurs. While this joint model may seem complex at first sight, each of the individual hierarchical levels are relatively straightforward.
\subsection{\label{S.CRM}Capturing heterogeneity with random effects}
SSMs can account for additional dependencies and heterogeneity in parameter values with random effects. This feature has been used to incorporate individual variation in capture-recapture models \citep{Royle-2008, King-2012}. Capture-recapture models, such as the Cormack-Jolly-Seber model, are often used to estimate survival probabilities and gain insight on the factors that may affect survival. They model data, where individuals are uniquely identifiable via artificial (e.g., rings) or natural marks (e.g., coloring) \citep{King-2012}. One of the first applications of SSMs to such data was by \citet{Royle-2008} to demonstrate how to model variation in survival and capture probabilities. \citet{Royle-2008} applied the model to a 7-year study of European dippers (\textit{Cinclus cinclus}).
\citet{Royle-2008} presents a SSM parametrization of a Cormack-Jolly-Seber model, where the observation $y_{i,t}$ represents whether individual $i$ was capture during the $t^{th}$ sampling occasion (i.e., $y_{i,t} = 1$ means the individual was captured at time $t$) and the state $z_{i,t}$ describes whether individual $i$ is dead or alive at time $t$ (i.e., $z_{i,t} = 1$ means the individual was alive at time $t$). At the time of first capture, $f_i$, the state is considered fixed: $z_{i,f_i} = 1$. Afterward, the process and observation equations are both Bernoulli trials, representing the survival and capture processes for each individual $i$:
\begin{linenomath*}
\begin{align}
z_{i,t} &\sim \text{Bernoulli}(\phi_{i,t-1}z_{i,t-1}), & f_i < t \leq T, \label{E.crmp}\\
y_{i,t} &\sim \text{Bernoulli}(p_{i,t} z_{i,t}), & f_i < t \leq T, \label{E.crmo}
\end{align}
\end{linenomath*}
where $T$ is the total number of sampling occasions, and $\phi_{i,t}$ is the probability of survival of individual $i$ over the interval $(t, t+1)$ if the individual was alive at time $t-1$, and $p_{i,t}$ is the probability of capturing individual $i$ during the $t^{th}$ sampling occasion if it is alive. These probabilities are multiplied to the state values. Thus, an individual's probability of surviving to time $t$ becomes 0 if the animal was dead at time $t-1$ (i.e., $z_{t-1}=0$) regardless of the value of $\phi_{i,t-1}$, which means that once the animal is dead it remains dead for the rest of the time series. Similarly, the probability of being captured at time $t$ becomes 0 if the individual is dead at that time.
We could simplify the model by having a single overall survival probability (i.e., $\phi_{i,t} = \phi$) and a single capture probability ($p_{i,t} = p$). However, differences between sampling occasions and individuals (e.g., due to variations in environmental and body conditions) often warrant for temporal and individual variations in survival and capture probabilities. \citet{Royle-2008} modeled the variations in these probabilities as follows:
\begin{linenomath*}
\begin{align}
\text{logit}(\phi_{i,t}) &= b_t + \beta_i, & \beta_i &\sim \text{N}(0, \sigma^2_{\beta}) \\
\text{logit}(p_{i,t}) &= a_t + \alpha_i, & \alpha_i &\sim \text{N}(0, \sigma^2_{\alpha})
\end{align}
\end{linenomath*}
where $a_t$ and $b_t$ are the fixed temporal effects (i.e., effects associated with each sampling occasion), $\alpha_i$ and $\beta_i$ are the latent individual effects, and $\sigma_{\alpha}^2$ and $\sigma_{\beta}^2$ are the variances for the random effects. Just as for generalized linear model, the logit link function ensures that probability parameters stay between 0 and 1. The fixed temporal effects require that we estimate $(T-1) + (T-2)$ additional parameters \citep[for details, see][]{Royle-2008}. In contrast, the individual random effects allow one to model heterogeneity in survival and capture probabilities with only two additional parameters.
SSMs are now commonly used to model capture-recapture data because their mechanistic structure allows one to incorporate additional complexity \citep{King-2012}. Using random effects to model variation in parameter values can be used in many other ecological applications.
\subsection{\label{S.HMM}Modeling discrete state values with hidden Markov models}
Hidden Markov models (HMMs) are a special class of SSMs, where the states are discrete rather than continuous \citep[generally categorical with a finite number of possible values;][]{Langrock-etal-2012}. HMMs have gained popularity in ecology, where they are used to model capture-recapture data \citep[e.g.,][]{Choquet-Gimenez-2012, Johnson-etal-2016} and animals that switch between distinct behavioral modes \citep{Langrock-etal-2012}. Recently, \citet{McClintock-etal-2020} have demonstrated that HMMs are widely applicable in ecology. Having discrete states in a SSM becomes important when choosing fitting procedures (see Section \ref{S.discrete.states}), and thus we provide a few examples.
The two main characteristics of HMMs are: 1) each observation is assumed to be generated by one of $N$ distributions, and 2) the hidden state sequence that determines which of the $N$ distributions
is chosen at time $t$ is modeled as a Markov chain, where the
probability of being in each mode at time $t$ depends only on the
state value at the previous time step \citep{Langrock-etal-2012}. The capture-recapture model presented in Section \ref{S.CRM} is an HMM, because state $z_{i,t}$ can only have one of two discrete values: 0 if the individual is dead or 1 if alive. The state value directly affects the observation equation (Eq. \ref{E.crmo}), and the observation, $y_{i,t}$, is generated by one of two distributions: $y_{i,t} \sim \text{Bernoulli}(0)$ if $z_{i,t} = 0$ or $y_{i,t} \sim \text{Bernoulli}(p_{i,t})$ if $z_{i,t} = 1$. As seen in the process equation (Eq. \ref{E.crmp}), the probability of being in each state at time $t$ depends only on the state value at the previous time step. This process can be viewed as a Markov chain with the following transition probability matrix:
\begin{linenomath*}
\begin{align}
\boldsymbol{\Gamma} & = \begin{bmatrix}
1 & 0 \\
1-\phi_{i,t-1} & \phi_{i,t-1}
\end{bmatrix}, \label{E.tpm}
\end{align}
\end{linenomath*}
where for each individual, the probability of staying dead (i.e., $\text{Pr}(z_{i,t} = 0 |z_{i,t-1} = 0)$) is 1, that of resurrecting (i.e., $\text{Pr}(z_{i,t} = 1 |z_{i,t-1} = 0)$) is 0, that of dying (i.e., $\text{Pr}(z_{i,t} = 0 |z_{i,t-1} = 1)$) is the probability that it did not survive (i.e., $1-\phi_{i,t-1}$), and that of surviving (i.e., $\text{Pr}(z_{i,t} = 1 |z_{i,t-1} = 1)$) is $\phi_{i,t-1}$.
In other contexts, the transition probabilities may be more flexible, allowing for transition between all states, and the SSM may include both discrete and continuous states. For example, the model presented in Section \ref{S.M.movement.model} was originally developed to model the movement of animals tracked with Argos data that switched between two behavioral modes \citep{Jonsen-etal-2005}. Instead of having a single $\gamma$ parameter that controls how correlated the steps are (e.g., Eq. \ref{e.dcrw.p}), this model has two parameters, $\gamma_{b_t}$, each one associated with one of the behavioral modes, $b_t = 1$ or $b_t = 2$. When $\gamma_1$ is close to 0 and $\gamma_2$ is close to 1, the movement path switches between tortuous and directed movement. The switch between the behavioral modes is modeled with a simple Markov chain (i.e., $\text{Pr}(b_t = j | b_{t-1} = i) = \alpha_{ij}$). While here we could allow the animal to switch back and forth between the behavioral modes (i.e., no $\alpha_{ij}$ is set to 0), the transition out of a given mode always need to sum to one (i.e., $\sum_{j=1}^2 \alpha_{ij}= 1$ for $i, j = 1,2$).
\section{\label{S.OTHER}SSMs as a framework for ecological time series}
Time is one of the fundamental axes that shape ecological systems \citep{Wolkovich-etal-2014} and time series are crucial to understand the complex processes and interactions that govern all aspects of ecology \citep[e.g.,][]{Boero-etal-2015,Damgaard-2019}. While SSMs have a long history in only a few fields of ecology, the breath of their applications has been expanding and could be extended to most ecological time series. SSMs provide a framework that can be used to understand the mechanisms underlying complex ecology systems and handle the large uncertainties associated with most ecological data and processes.
SSMs have been increasingly used in plant ecology. \citet{Damgaard-2012} showed the usefulness of using SSMs to analyze plant cover data collected through quadrats (specifically through pin-point methods). \citet{Bell-etal-2015} demonstrated how SSMs could be used to estimate canopy processes (e.g., conductance and transpiration) using imperfectly monitored stem sap flux data. \citet{Clark-etal-2011} used the SSM approach to model the growth, fecundity, and survival of more than 27,000 individual trees. They showed how these processes are linked with light competition and spatiotemporal variation in climate.
SSMs are now used for paleoecological research. For example, \citet{Tome-etal-2020} used SSMs to identify the drivers of changes in the mass and diet of a small mammal during the late Pleistocene. They use three separate linear Gaussian SSMs to model temporal changes in mass (as estimated from molar size) and in two stable isotopes (extracted from jaw bone collagen) as responses to each other and of a set of covariates related to climate (e.g., maximum temperature) and community structure (e.g., species richness). \citet{Einarsson-etal-2016} developed a SSM for sediment core data. One process equation modeled the change in abundance of midges egg capsule. The other modeled the change in pigment concentration characterizing potential resources (e.g., diatoms). These process equations modeled the abundance of each group with a Gompertz population model (similar to Section \ref{S.M.simple.pop}), modified to add the effect of the other group's abundance at the previous time step. The measurement equations modeled sediment mixing and its associated uncertainty. They use their model to show that the cyclic fluctuations in midges are likely driven by consumer-resource (a.k.a exploiter-victim or predator-prey) interactions.
Detecting cyclicity in ecological time series can be challenging due to temporal autocorrelation, and \citet{Louca-Doebeli-2015} showed that SSMs can outperform statistical tests for cyclicity. For example, they showed that simpler models would often lead to erroneous conclusions that cycles are present; while the SSM generally had an appropriate 5\% rate of Type I errors, simpler tests had rates as high as 79\%. The midge example of \citet{Einarsson-etal-2016} further demonstrates the usefulness of SSMs to identify the mechanisms behind cycles in ecological time series. Similar SSMs have been used to investigate fluctuations in other ecological fields \citep[e.g., host-parasitoid systems,][]{Karban-and-deValpine-2010}.
SSMs have been used in ecosystem ecology and biogeochemisty. For example, \citet{Appling-etal-2018} used a SSM to model changes in oxygen concentration in an aquatic ecosystem as a function of three important processes: ecosystem's gross primary production, respiration, and gas exchange rate with the atmosphere. The process equation of \citet{Appling-etal-2018} predicts the oxygen concentration at time $t$ as a function of its previous value and its instantaneous rate of change (similar to the model in Section \ref{S.cont.mov.model}). The rate of change is modeled through a mechanistic equation, which sums the three processes of interest. The study compared various versions of the model, including versions that were not SSMs (i.e., versions without measurement error or process stochasticity), and showed that the best SSM formulation significantly improved the accuracy and reduced the bias of estimates of gross primary productivity, respiration, and gas exchange. In some cases, the magnitude of the bias of the SSM was half as large as that of simpler models. The study of \citet{Jia-etal-2011} is one of the many examples of applications of SSMs in soil science. \citet{Jia-etal-2011} used linear SSMs with normally distributed error to model the effects of elevation and the physical and chemical properties of soil (e.g., clay content and organic carbon) on the total net primary productivity of managed grasslands. They showed that the SSMs described the spatial patterns of soil total net primary productivity better than classical regression methods.
The term SSM has been used broadly in ecology to represent various types of hierarchical models with complex dependence structure. In particular, the term has been used for occupancy models that are based on capture-recapture SSMs similar to the one described in Section \ref{S.CRM} \citep[e.g.,][]{Kery-etal-2009, Mordecai-etal-2011}. While they have similar structure, many of them lack the specific temporal autocorrelation in process equation that we generally ascribe to SSM (Fig. \ref{fig:structure}a) and may be better thought as a related, but different, type of hierarchical model. For some of these models, it may be worth adding the Markovian dependence of the state in the process equation. However, to our knowledge, there is no studies that compare these related hierarchical models to SSMs.
The ubiquity of SSMs in ecology may have been obscured as some complex SSMs that combine various statistical techniques have not been identified as SSMs. For example, \citet{Thorson-etal-2016} present a joint species distribution model that has temporal dynamics. Although not called a SSM, their model has the essential structure of a SSM (Fig. \ref{fig:structure}A). We view their model as a Gompertz SSM (similar to Section \ref{S.M.simple.pop}) combined to dynamic factor analysis to reduce dimensions and Gaussian random fields to account for spatial autocorrelation. This complex multi-species model was used to demonstrate that the spatiotemporal patterns of butterfly from the same genus were significantly correlated and to identify dominant patterns in community dynamics of marine fish.
The complexity of SSMs may incite ecologists to ask: could we use a simpler alternative? In fisheries science, early papers on SSMs showed that they were particularly superior to simpler models when both the process variance and the observation error are large \citep[e.g.,][]{deValpine-Hastings-2002}. When one of the sources of stochasticity is small, and the model dynamics are not too complex, simpler models that account for either just the process variance or observation error give adequate results \citep{deValpine-Hastings-2002}. A key point, however, is that the simpler model performs adequately only if the model is well specified with regards to which source of stochasticity is most important. Thus, using a process variance-only model, is only suitable if we are certain that the observation error is negligible. Similarly, using an observation error-only model, is only suitable if we are certain that the process variation is small. While simpler alternatives can be adequate in some contexts \citep[see Chapter 11 of][]{Bolker-2008}, many studies have shown that SSMs provided better inference than easier models \citep[e.g.,][]{Jamieson-Brooks-2004, Jia-etal-2011, Louca-Doebeli-2015, Appling-etal-2018}. For example, \citet{Linden-Knape-2009} showed that, unlike SSMs, simpler models often had unreliable point and uncertainty estimates for environmental effects, and that the 95\% confidence intervals excluded the true simulated value much more than 5\% of the time (up to 30\%). They showed that the SSMs always outperformed the simpler alternatives. As such, we believe that SSMs, and their extensions, should be a default statistical modeling technique for many ecological time series. In the rest of the paper, we provide the tools that allow ecologists to apply these complex models adequately.
\section{\label{S.Fitting}Fitting SSMs}
The goals of fitting a SSM to data include estimating the parameters, $\boldsymbol{\theta}$, the states, $\mathbf{z}$, or both. In ecology, we regularly need to
estimate both, as we rarely know the value of $\boldsymbol{\theta}$ \textit{a priori} and estimating the states is often a primary goal of the analysis. In movement
ecology, researchers often fit SSMs similar to that described in
Section \ref{S.M.movement.model} because the states provide better
estimates of the true locations of the animal than the data. In the
SSM literature, a distinction is often drawn between three different
types of state estimation processes based on the amount of observations used to inform the estimates \citep{Shumway-Stoffer-2016}.
Using all of the observations, $y_{1:T}$, to estimate the states is
referred to as `smoothing'. Smoothing is common with ecological SSMs,
as we often have the complete dataset in hand when we start the
analysis. We denote the smoothed state estimate as
$\hat{z}_{t|1:T}$, with the subscript $t|1{:}T$ identifying that
the state at time $t$ is estimated using the observations from time
$1$ to $T$. In the original engineering application and in other fields, states are often estimated while data continues to be collected, so only observations
up-to and including time $t$, $y_{1:t}$, are used to estimate the
state $\hat{z}_{t|1:t}$. This ubiquitous estimation procedure is
referred to as `filtering'. Finally, we can use a subset of the
observations that ends $s$ time steps before time $t$, $y_{1:t-s}$,
to predict the state at time $t$, $\hat{z}_{t|1:t-s}$, a procedure
we refer to as `forecasting'. A common forecast is the
one-step-ahead prediction, $\hat{z}_{t|1:t-1}$, which is also
used within fitting algorithms (Appendix S2) and to validate models (Section \ref{S.Diagnostics}). While these three types of state estimation processes are useful, the uncertainty associated with the state estimates tend to decrease for processes that use more observations \citep[e.g.,][]{Shumway-Stoffer-2016}.
The states are random variables, and thus have probability distributions. The states are sometimes referred to as random effects or latent variables. The fundamental differences in the procedures used to estimate states, as opposed to parameters (see Section \ref{S.Freq}), means that although we use estimation as an all-purpose term for both states and parameters, state estimation procedures are often referred as prediction, even when smoothing and filtering are used. The inferences about the states can include a variety of summary measures of their probability distributions. Above, the state estimates (e.g., $\hat{z}_{t|1:T}$) referred to point estimates such as the expected value. However, one can also calculate interval estimates (e.g., 95\% confidence intervals) and single measures of uncertainty (e.g., standard deviations or variances).
Methods for fitting SSMs can be divided into the two main
inferential approaches: frequentist and Bayesian. These approaches differ in their philosophies, see \citet{Bolker-2008} for a discussion. In brief, frequentist methods determine the probability of the data for a set of particular conditions (i.e., the hypothesis is fixed, but the data have a probability distribution). In contrast, Bayesian methods determine the probability that particular conditions exist given the data-at-hand (i.e., the data are fixed, but the hypothesis/parameters have probability distributions). The Bayesian approach requires the specification of prior beliefs for these distributions. Because of the early development of Bayesian computational methods for hierarchical models, historically it was easier to fit complex SSMs with a Bayesian approach and frequentist methods were limited to simple models \citep{DeValpine-2012}. As we will show, this is no longer true. There are now many accessible methods that allow to fit complex SSMs with a frequentist approach (e.g., see Sections \ref{S.Laplace}-\ref{S.Iterative.Filtering}). Thus, researchers can choose to work with their favored philosophical approach and/or based on the advantages of the algorithms available within each approach (see Section \ref{S.method.comp}).
In terms of fitting procedures, frequentists maximize the likelihood, while Bayesians focus on the posterior density. As we show below, despite these differences, both approaches involve high-dimensional integration, which is at the crux of the difficulties associated with fitting SSMs to data. The many tools developed for fitting SSMs are essentially different solutions to this high-dimensional integration problem.
\subsection{Frequentist approach}
\label{S.Freq}
When we fit a SSM with a frequentist approach, we search for the parameter values that maximize the likelihood, a method called maximum likelihood estimation with the resulting estimates called maximum likelihood estimates (MLEs). For our toy SSM (Eqs. \ref{E.state.NDLM}-\ref{E.obs.NDLM}), the joint likelihood for $\boldsymbol{\theta}$ and $\mathbf{z}_{1:T}$ would be defined as:
\begin{linenomath*}
\begin{align}
L_{\textsc{j}}(\boldsymbol{\theta}, \mathbf{z}_{1:T} | \mathbf{y}_{1:T})=\prod_{t=1}^T g(y_t | z_t, \boldsymbol{\theta}_o) f(z_t | z_{t-1}, \boldsymbol{\theta}_p),
\label{E.Joint.Likelihood}
\end{align}
\end{linenomath*}
where $T$ is the length of our time series and $\boldsymbol{\theta}$ is a vector of (unknown) model parameters that contains the parameters for the process equation, $\boldsymbol{\theta}_p$, and the observation equation, $\boldsymbol{\theta}_o$, and in this example the initial state, $z_0$. Maximizing the joint likelihood with respect to both parameters and the states is challenging. Instead, one can use a process with two interrelated steps, each focused on estimating either the parameters or the states.
To estimate the parameters, we maximize the marginal likelihood, $L_{\textsc{m}}(\boldsymbol{\theta} | \mathbf{y}_{1:T})$:
\begin{linenomath*}
\begin{align}
\boldsymbol{\hat{\theta}} = \argmax_{\boldsymbol{\theta} \in \boldsymbol{\Theta}} L_{\textsc{m}}(\boldsymbol{\theta} | \mathbf{y}_{1:T}),
\label{E.argmax.par}
\end{align}
\end{linenomath*}
where
\begin{linenomath*}
\begin{align}
L_{\textsc{m}}(\boldsymbol{\theta} | \mathbf{y}_{1:T})=\int L_\textsc{j}(\boldsymbol{\theta}, \mathbf{z}_{1:T} | \mathbf{y}_{1:T}) \text{d} \mathbf{z}_{1:T}.
\label{E.Marginal.Likelihood}
\end{align}
\end{linenomath*}
Here, the key is that we integrate out the hidden states and thus have a function that only depends on the observations. The parameter estimates that result from maximizing the marginal likelihood have desired statistical properties \citep[consistency and asymptotic normality, see][]{Douc-etal-2011, DeValpine-2012}, where the estimates are anticipated to improve with increasing sample size. Such properties would be hard to achieve when maximizing the joint likelihood, because the number of states to estimate generally increase with the number of observations.
To estimate the hidden states, we can use the conditional distribution of the states given the observations and the estimated parameter values, for example:
\begin{linenomath*}
\begin{align}
p(\mathbf{z}_{1:T}|\mathbf{y}_{1:T},\boldsymbol{\hat{\theta}}) = \frac{L_\textsc{j}(\mathbf{z}_{1:T} | \mathbf{y}_{1:T},\boldsymbol{\hat{\theta}})}{\int L_\textsc{j}(\mathbf{z}_{1:T} | \mathbf{y}_{1:T},\boldsymbol{\hat{\theta}}) \text{d} \mathbf{z}_{1:T}},
\label{E.cond.dist.state}
\end{align}
\end{linenomath*}
where $L_\textsc{j}(\mathbf{z}_{1:T} | \mathbf{y}_{1:T},\boldsymbol{\hat{\theta}})$ is similar to the right-hand side of Eq. \ref{E.Joint.Likelihood}, except that we use the MLEs for the parameters. Conditional distributions of the states, in particular the filtering distributions ($p(z_{t}|\mathbf{y}_{1:t}, \boldsymbol{\hat{\theta}})$, see Appendix S2 for an example), are at the base of filtering methods, such as the Kalman filter (Section \ref{S.Kalman.Filter}) and particle filter (Section \ref{S.Iterative.Filtering}). The means and variances of filtering densities can provide good point estimates and measures of uncertainty for state values (Appendix S2). As an approximation of the state estimates, one can also maximize $L_\textsc{j}(\mathbf{z}_{1:T} | \mathbf{y}_{1:T},\boldsymbol{\hat{\theta}})$ with respect to $\hat{\mathbf{z}}_{1:T}$:
\begin{linenomath*}
\begin{align}
\hat{\mathbf{z}}_{1:T} = \argmax_{\mathbf{z}_{1:T} \in \mathbf{Z}^T}L_\textsc{j}(\mathbf{z}_{1:T} | \mathbf{y}_{1:T}, \boldsymbol{\hat{\theta}}),
\label{E.argmax.states}
\end{align}
\end{linenomath*}
where $\mathbf{Z}^T$ is the set of all possible values
for the states. This maximization treats the states as if they were equivalent to parameters in an ordinary likelihood \citep[see][for more details]{Aeberhard-etal-2018} and is often used when the marginal likelihood is estimated with the Laplace approximation (see Section \ref{S.Laplace}). While Eq. \ref{E.argmax.states} treats the parameters as known when estimating the states, one can propagate the estimation variability when reporting the state estimate variance \citep[e.g., see \texttt{TMB} function \texttt{sdreport},][]{Kristensen-etal-2016}.
The marginal likelihood used to estimate the parameters, and thus the states, requires the computation of the high-dimensional integral found in Eq. \ref{E.Marginal.Likelihood}. This computation is difficult to achieve for most SSMs and the frequentist inference methods discussed below are different ways to either evaluate the marginal likelihood (e.g., Kalman filter) or to approximate it (e.g., Laplace and simulation-based approximations).
\subsubsection{Kalman filter \label{S.Kalman.Filter}}
For simple linear SSMs with Gaussian errors (i.e., NDLMs), the state estimates and marginal likelihood can be directly calculated using the Kalman filter \citep{Kalman-1960}. The Kalman filter provides an algorithm that, using only elementary linear algebra operations, sequentially updates the filtering mean and variance of the states \citep{Harvey-1990,Durbin-and-Koopman-2012}. While the Kalman filter was developed to estimate the state values for models with known parameter values, its output can be used to evaluate the marginal likelihood and thus to find the MLE. The Kalman smoother is an analogous algorithm that uses backward recursion in time to obtain the mean and variance of each smoothing distribution \citep[i.e., distribution of $z_{t|1:T}$;][]{Harvey-1990,Durbin-and-Koopman-2012}. See Appendix S2 for a detailed example of the Kalman filter as applied to our toy model.
In Appendix S1: Section S1 1.3.1, we demonstrate how to use the \texttt{R} package \texttt{dlm} \citep{petris2010} to perform Kalman filtering and smoothing, as well as forecasting. It can also be used to find MLEs of unknown fixed parameters. The package is flexible enough to allow univariate and multivariate NDLMs, accounting for constant or time-varying distributions of states and observations. More details about Kalman filter and smoother and \texttt{dlm} can be found in \citet{DLMwR} and \citet{petris2010}. See also Chapter 6 of \citet{Shumway-Stoffer-2016} for description of filtering, smoothing, forecasting and maximum likelihood estimation.
The Kalman filter is among the most broadly used algorithms to fit SSMs to ecological data. For example, it has been used in population ecology \citep[e.g.,][]{Dennis-etal-2006}, movement ecology \citep[e.g.,][]{Johnson-etal-2008}, community ecology \citep{Ives-etal-2003}, and plant ecology \citep{Hooten-etal-2009}. The main advantage of the Kalman filter is that it is fast and easy to calculate \citep{DeValpine-2012}. In addition, unlike most other methods that provide an approximation of the likelihood, the Kalman filter provides an exact evaluation of the marginal likelihood for linear and Gaussian SSMs \citep[e.g., toy model;][]{DeValpine-2002}.
While the Kalman filter is an important algorithm for fitting SSMs to data, it does not work with nonlinear and non-Gaussian SSMs. Approximate
techniques based on the Kalman filter are available for linear models whose observations follow an exponential family distribution \citep[e.g., Poisson, see][Ch.\ 9]{Durbin-and-Koopman-2012}. Other approximate filtering and smoothing methods based on the Kalman filter, such as the extended Kalman filter and the unscented Kalman filter \citep[e.g.,][Ch.\ 10]{Durbin-and-Koopman-2012} are useful for some nonlinear and/or non-Gaussian SSMs. Such related methods have been used in ecology \citep[e.g.,][]{Einarsson-etal-2016}. However, for more complex, nonlinear, and non-Gaussian models, one must use one of the methods described below.
\subsubsection{Laplace approximation methods}
\label{S.Laplace}
The Laplace approximation is a commonly used tool for obtaining an approximation of the marginal likelihood of a SSM \citep{Fournier-etal-2012,Kristensen-etal-2016}. The general idea is that if the marginal likelihood (Eq. \ref{E.Marginal.Likelihood}) is a well-behaved unimodal function, it can be approximated with a Normal density function. We can use this approximation to find the MLE. For a given set of parameter values, the Laplace approximation of the marginal likelihood requires the maximization of the joint likelihood (Eq. \ref{E.Joint.Likelihood}) with respect to the states. Thus, the parameter estimation process also returns an approximation of the state estimates. See Appendix S2 for detail.
This method is flexible, and a variety of SSMs can be fitted using the Laplace approximation. However, the method assumes that the states can be locally-approximated with a Gaussian distribution, which means that the states are assumed to have an unimodal distribution. Because the method uses the second derivative of the log likelihood (Appendix S2), we cannot use the Laplace approximation with categorical states or other state distributions that are not twice differentiable. An important advantage of the Laplace approximation, over the simulation-based approaches described below, is the speed at which SSMs are fitted to data \citep[see][]{AugerMethe-etal-2017}. Many software use the Laplace approximation approach \citep[e.g.,][]{Fournier-etal-2012}. We demonstrate in Appendix S1: Section S1 1.3.2 how \texttt{TMB} \citep{Kristensen-etal-2016} is a particularly useful \texttt{R} package for SSMs. The Laplace approximation has been used in ecology, including in movement ecology \citep[e.g.,][]{AugerMethe-etal-2017} and fisheries science \citep[e.g.,][]{Aeberhard-etal-2018}.
\subsubsection{Sequential Monte Carlo methods}
\label{S.Iterative.Filtering}
Monte Carlo methods can be used to estimate the states and evaluate the integral needed to obtain the marginal likelihood. Monte Carlo methods are computer intensive sampling procedures that generate random samples from specific probability distributions, which can then be used to evaluate integrals. While in this section we discuss Monte Carlo methods in the context of a frequentist inference approach, we will see in Section \ref{S.Bayesian} that Monte Carlo methods are commonly used for Bayesian inference.
Sequential Monte Carlo methods, also referred to as particle filters, approximate the filtering distribution through simulated sampling \citep{DeValpine-2012}. In the context of SSMs fitted with a frequentist approach, these Monte Carlo methods generally sample the state space by generating samples using the process equation and weighting the samples with the observation equation. Sequential importance sampling \citep{Doucet-etal-2001} is a general procedure that can be used to generate $N$ time series of the states, referred to as particles, and using their weighted average as the state estimates (see Appendices S1-S2 for detail). However, sequential importance sampling is impractical for even moderately long time series (e.g., $T$=20) because only a small proportion of the $N$ randomly generated particles are generally supported by the observations. The reduced support for many of the particles, known as particle depletion,
is a serious problem with sequential importance sampling that leads to state estimates with unacceptably large variances.
The bootstrap filter \citep{Gordon-etal-1993}
is a procedure designed to remedy particle depletion. The bootstrap filter assesses the weight of a particle through time and iteratively removes
particles with low weights and replaces them with duplicates of
particles with higher weights. There are various algorithms for the bootstrap filter, see Appendices S1 and S2 for an example. While simple bootstrap filters can reduce particle depletion, they do not completely solve the problem particularly for long time series. There are various remedies aimed at reducing
particle depletion \citep{DeValpine-2012}, including more sophisticated importance sampling distributions that include information from the observations
\citep{Pitt-Shephard-1999} or changing the resampling methods
\citep{Liu-Chen-1998}. Sequential Monte Carlo methods are also used for Bayesian inference and these methods are often built to reduce particle depletion \citep[e.g., particle Markov chain Monte Carlo methods,][]{Andrieu-etal-2010, Michaud-etal-2020}.
Sequential Monte Carlo methods, such as sequential importance sampling, can be used to estimate the likelihood. However, the likelihood maximization required for frequentist inference comes with additional challenges (e.g., to maximize the likelihood, one must explore what is often a complicated likelihood surface). In principle, it is possible to use a general-purpose optimization algorithm such as Nelder-Mead to maximize the likelihood computed by a simple particle filter. However, such an approach is usually prohibitively expensive. In addition, the stochastic ingredients of a particle filter make each of its runs different, making it hard to identify the precise peak of the likelihood surface \citep{DeValpine-2012}. Several methods have been proposed to overcome this difficulty \citep{DeValpine-2012, Michaud-etal-2020}.
Iterated filtering is an attractive method for maximizing the likelihood using particle filter \citep{Ionides-etal-2015}. This method repeatedly applies the particle filter but perturbs the fixed parameters of the model at each observation time step.
These random perturbations enhance performance and forestall particle depletion by continually re-injecting random variability into the filter. However, because it applies artificial perturbations to parameters, iterative filtering is not learning about the model of interest (i.e., model with fixed parameters), but about a modified model (i.e., model where fixed parameters have been transformed into state variables).
Therefore, as filtering iterations proceed, one gradually cools (i.e., reduces the magnitude of) the artificial perturbations, so that the modified model approaches the model of interest as the iterations proceed.
Because statistical inference hinges on identification of the global likelihood maximum, it is usually advisable to perform many independent iterative filtering computations, starting from widely dispersed starting points. See Appendix S2 for more detail.
Iterative filtering, and other similar sequential Monte Carlo methods, can be easily implemented using the \texttt{R} packages \texttt{pomp} and \texttt{nimble} \citep[see Appendix S1: Section S1 1.3.4;][]{King-etal-2016, Michaud-etal-2020, deValpine_et_al:2017}.
The main advantage of sequential Monte Carlo methods is that they are flexible, thus can be used to conduct inference on any SSM \citep{Michaud-etal-2020}. Frequentist sequential Monte Carlo methods have been used in ecological fields such as bioenergetics \citep[e.g.,][]{Fujiwara-etal-2005} and movement ecology \citep[e.g.,][]{Breed-etal-2012}. The main disadvantage of sequential Monte Carlo methods is that they can be computationally expensive.
\subsubsection{Other methods}
The methods described above represent, in our view, the most
commonly used methods to fit SSMs to ecological data in a frequentist
framework. These methods are associated with comprehensive \texttt{R}
packages that facilitate their implementation. However, many other
methods exist \citep[see][for a review of frequentist methods]{DeValpine-2012}. Of note, \citet{Kitagawa-1987} provided a general algorithm for non-Gaussian SSMs similar to the Kalman
filter, but that approximates the non-normal distributions by discretizing them (e.g., through piecewise linear functions). It can be viewed as discretizing the continuous
state space and reformulating the model as a HMM
\citep{Pedersen-etal-2011}. \citet{deValpine-Hastings-2002}
demonstrated how flexible this approach was to fit nonlinear
non-Gaussian population dynamics models. The main advantages of this approach are that it can be computationally efficient for models with a few state dimensions and does not require Monte Carlo methods \citep{DeValpine-2012}. This approach appears particularly promising for population modeling, where the states are counts, and thus the state space is already discretized \citep{Besbeas-Morgan-2019}. \citet{Pedersen-etal-2011} demonstrated that while this method is general and can provide results similar to the Laplace approximation and Bayesian methods, it is computationally limited to problems with only a few state dimensions. This limitation arises from the curse of dimensionality, where even if each dimension has manageable number of cells (e.g., 1,000 cells), the number of values needed to be stored become impractical as the number of dimension increases \citep[e.g., three-dimension would results in $1,000^3 = 10^9$ cells, see][]{DeValpine-2012}.
\subsection{Bayesian framework}
\label{S.Bayesian}
When we fit a SSM with a Bayesian approach, the function of interest (also known as the target distribution) is the posterior distribution for the states and parameters given the observations:
\begin{linenomath*}
\begin{align}
p(\boldsymbol{\theta},\mathbf{z}_{1:T}|\mathbf{y}_{1:T},\boldsymbol{\lambda}) = \frac{ L_{\textsc{j}}(\boldsymbol{\theta}, \mathbf{z}_{1:T} | \mathbf{y}_{1:T}) \pi(\boldsymbol{\theta} |\boldsymbol{\lambda})}{\int \int L_{\textsc{j}}(\boldsymbol{\theta}, \mathbf{z}_{1:T} | \mathbf{y}_{1:T})\pi(\boldsymbol{\theta} |\boldsymbol{\lambda}) \text{d}\mathbf{z}_{1:T} \text{d}\boldsymbol{\theta}},
\label{E.posterior}
\end{align}
\end{linenomath*}
where $L_\textsc{j}(\boldsymbol{\theta}, \mathbf{z}_{1:T} | \mathbf{y}_{1:T})$ is the joint likelihood (i.e., $p(\mathbf{y}_{1:T} | \boldsymbol{\theta}, \mathbf{z}_{1:T})$, see for example Eq. \ref{E.Joint.Likelihood}), and $\pi(\boldsymbol{\theta}|\boldsymbol{\lambda})$ is the prior distribution(s) for the parameters with fixed hyperparameters, $\boldsymbol{\lambda}$. Eq. \ref{E.posterior} is an application of Bayes' theorem ($p(\theta | \mathbf{y}) = \frac{p(\mathbf{y}| \theta) p(\theta)}{p(\mathbf{y})}$) and the denominator of Eq. \ref{E.posterior} represents the probability of the data (i.e., the marginal likelihood, which is the probability of the data for all possible values of the states and parameters). In Bayesian analyses, both the states, $\mathbf{z}_{1:T}$, and what we have been referring to as fixed parameters, $\boldsymbol{\theta}$, are considered random variables. The posterior distribution is a complete characterization of these random variables given the data and prior information. As such, the first inferential goal of a Bayesian analysis is often to evaluate the posterior distribution. While point estimates for the parameters and the states are not necessarily the primary goal of a Bayesian analysis, they can be obtained by summarizing the center of the posterior distribution (e.g., mean or mode of the posterior distribution). Similarly we can use the posterior distribution to obtain interval estimates and single measures of variation.
As for the frequentist framework, the fitting procedures are complicated by high-dimensional integrals and it is common to avoid calculating the integral and the posterior distribution explicitly. Instead, quantities of interest are generally approximated using Monte Carlo methods (see also Section \ref{S.Iterative.Filtering}), where large samples of states and parameters are randomly drawn from the posterior distribution. For example, one can approximate the point estimate of a parameter with the sample mean of the draws from the posterior distribution (often referred as the posterior mean). Simulating independent draws from Eq. \ref{E.posterior} is typically impossible. However, there are various algorithms that can approximate the posterior distribution with large samples of dependent draws. In particular, Markov Chain Monte Carlo (MCMC) methods are a broad class of algorithms that obtain samples from the target distribution (here the posterior distribution Eq. \ref{E.posterior}), by sampling from a Markov chain rather than sampling from the target itself. This Markov chain needs to have an invariant distribution (i.e., the probability distribution remains unchanged as samples are drawn) equal to the target distribution \citep{Geyer:2011}, a quality which is dependent on the initial condition of the chain and the transition probabilities, and relates to the importance of chain convergence as a diagnostic in MCMC sampling. MCMC algorithms fall into two broad families: Metropolis-Hastings samplers (which include Gibbs samplers) and Hamiltonian Monte Carlo.
\subsubsection{Metropolis-Hastings samplers}
\label{S.Metropolis-Hastings}
Metropolis-Hastings samplers are at the base of most MCMC algorithms used to sample the posterior distribution in a Bayesian analysis. Metropolis-Hastings samplers are iterative algorithms that construct an appropriate Markov chain to sample the target distribution. The general idea is that for each step $j$ of the chain, we use a proposal distribution to generate a candidate value for the variable of interest (e.g., a parameter value). The probability that this candidate value is used for that step rather than the previous value of the chain is based on the relative fit of the model with that candidate value compared to the previous value of the chain (see Appendix S2 for detail).
In the context of SSMs, we have a multivariate posterior distribution for the states and the parameters. Using Metropolis-Hastings algorithms to sample for more than one random variables is complex, but there are various implementation tools to do so. For example, for each iteration $j$ of the chain, one can first sample sequentially all parameter values, and then sequentially sample the state values \citep[][see also Appendix S2]{Newman-etal-2014}. If groups of variables are related, they can be sampled simultaneously from a multivariate distribution rather than sequentially. In practice, states and parameters are often correlated, and thus it may be difficult to implement an efficient MCMC sampler that does not require very long simulations before convergence \citep{Newman-etal-2014}.
Gibbs samplers are commonly-used Metropolis-Hastings samplers for multivariate distributions, where the proposal distributions are conditional distributions of the target distribution and thus the candidate values are always accepted \citep[][see also Appendix S2]{Geyer:2011}. For NDLMs, the entire sequence $\mathbf{z}_{0:T}$ can be simulated at once from its conditional distribution, given the data $\mathbf{y}_{1:T}$ and the time-invariant parameter $\boldsymbol{\theta}$, using the Forward Filtering Backward Sampling algorithm described in \citet{Carter+Kohn:1994}. The Forward Filtering Backward Sampling algorithm can also be used to conduct inference for the SSMs that are conditionally linear and Gaussian. However, Gibbs samplers for nonlinear and non Gaussian models often require sampling from each conditional distribution sequentially \citep[see chapter 4.5 of][for an overview]{Prado-West-2010}. A drawback of this particular Gibbs sampler design is that consecutive draws of $z_{0:T}^{j}$ and $z_{0:T}^{j-1}$ tend to be highly correlated, slowing the convergence and deteriorating the quality of the Monte Carlo approximations. Despite these drawbacks, Metropolis-Hasting samplers, including Gibbs samplers, are commonly used to fit ecological SSMs because they are flexible and freely available software to implement these algorithms have been available since the 1990s \citep{Meyer-Millar-1999}. They have been used to fit many of the original models described in Section \ref{S.examples}, including the population models of \citet{viljugrein2005}, the movement model of \citet{Jonsen-etal-2005}, the health and survival model of \citet{Schick-etal-2013}, and the capture-recapture model of \citet{Royle-2008}.
Combining sequential Monte Carlo methods (Section \ref{S.Iterative.Filtering}) within MCMC algorithms can help alleviate some of the efficiency problems produce by generic MCMC algorithms \citep{Michaud-etal-2020}. In these combined algorithms, a sequential Monte Carlo algorithm draws the states, while a MCMC algorithm draw the parameters. Particle MCMC methods \citep{Andrieu-etal-2010} are particularly useful for SSMs \citep{Michaud-etal-2020}. Some particle filters, such as the bootstrap filter (Section \ref{S.Iterative.Filtering} and Appendix S2), can return unbiased estimates of the marginal likelihood (Eq. \ref{E.Marginal.Likelihood}). At each iteration $j$, a particle MCMC algorithm will estimate the marginal likelihood and use it to draw a full state sequence (i.e., one sample particle will be used for $\mathbf{z}_{1:T}^j$). While particle MCMC may still suffer from poor mixing when the likelihood estimates are highly variable, these algorithms tend to reduce the correlations between successive draws of the states \citep{Michaud-etal-2020}. Custom-made particle MCMC algorithms have been used to fit different ecological SSMs, including population models \citep[e.g.,][]{Knape-deValpine-2012, White-etal-2016} and complex models for range expansion \citep{Osada-etal-2019}. The recent implementation of such algorithms in \texttt{R} packages such as \texttt{pomp} and \texttt{nimble} will facilitate their uptake \citep{Michaud-etal-2020}.
There are a few important general Bayesian software and \texttt{R} packages that can be easily used to fit ecological SSMs using Metropolis-Hastings samplers. Generating draws from the posterior distributions can done using software from the BUGS \citep[Bayesian analysis Using Gibbs Sampling, see][]{Lunn_et_al:2013} project and their associated \texttt{R} packages: \texttt{WinBUGS} can be called in \texttt{R} via \texttt{R2WinBUGS} \citep{Lunn-et-al-2000}, \texttt{OpenBUGS} via \texttt{BRugs} \citep{Lunn-etal-2009}, while \texttt{MultiBUGS} \texttt{R} interface is in development \citep{Goudie-etal-2017}. \citet{gimenez2009} provide a tutorial on how to fit ecological models (including some of the SSMs of Section \ref{S.M.simple.pop}) with \texttt{WinBUGS} in \texttt{R}. \texttt{JAGS} \citep[Just Another Gibbs Sampler;][]{Plummer:2003} is an alternative to BUGS project software that is written for UNIX, thus preferred by Mac and Linux users. \texttt{JAGS} is available through the \texttt{R} package
\texttt{rjags} \citep{rjags}. The \texttt{R} package \texttt{nimble} \citep{deValpine_et_al:2017} is a recent alternative to \texttt{JAGS} and BUGS software that is more transparent in how the sampling is performed. \texttt{nimble} allows users to write custom Gibbs samplers that perform block updating or implement a variety of other techniques including particle MCMC \citep{deValpine_et_al:2017, Michaud-etal-2020}. All these software allow one to write general models in a language based on BUGS. The user can set up the sampler in \texttt{R}, and once compiled, can use it to simulate draws to make inference about states and parameters. See Appendix S1: Sections S1 1.3.5 and S1 1.3.6 for detailed examples in \texttt{JAGS} and \texttt{nimble}.
\subsubsection{Hamiltonian Monte Carlo}
\label{S.HMC}
An efficient alternative to Metropolis-Hastings sampling is provided by Hamiltonian Monte Carlo (HMC) methods, which have gained popularity in recent years thanks in part to their
implementation in the \texttt{Stan} software \citep{STAN:2012}. These methods are inspired by analogies drawn from physics and rely heavily on deep differential geometric
concepts, which are beyond the scope of this review. HMC can be a more efficient sampler than Metropolis-Hastings as fewer iterations are typically required and fewer rejections occur. This is achieved by the addition of a momentum variable that helps the Markov chain to remain within the typical set of the target distribution, rather than conducting random walk to explore the target distribution as is frequently done by Metropolis-Hastings samplers. Interested readers can read the introduction for ecologists by \citet{Monnahan-etal-2017} and explore the statistical details in \citet{Neal:2011} or \citet{Betancourt:2017}. Conducting inference for general SSMs via HMC is possible when all parameters and states are continuous or when the posterior distribution can be marginalized over any discrete parameters or states. Continuous distributions are required because density gradients of the target distribution are required to direct the sampling through the typical set of the target distribution \citep{Betancourt:2017, Monnahan-etal-2017}. Unlike Metropolis-Hastings samplers, HMC methods draw samples from the joint posterior distribution directly and can scale well to high dimensional spaces. General SSMs can be fitted either by defining the posterior as in Eq. \ref{E.posterior} or by marginalization over the state process to derive the posterior distribution of the time-invariant parameters only, $p(\boldsymbol{\theta} | \mathbf{y}_{1:T}, \boldsymbol{\lambda})$.
One of the most popular software that uses Hamiltonian Monte Carlo is \texttt{Stan}, available in \texttt{R} through the package \texttt{rstan} \citep{rstan:2018}. See Appendix S1: Sections S1 1.3.7 and S1 2.3.2 for detailed examples using \texttt{rstan}. \citet{Monnahan-etal-2017} showed that \texttt{Stan} can fit ecological SSMs more efficiently than Gibbs software like \texttt{JAGS}. Although the parameterization of the SSM affects \texttt{Stan}'s efficiency, it can reduce computing time by orders of magnitude \citep{Monnahan-etal-2017}. Other advantages of \texttt{Stan} over \texttt{JAGS} include better diagnostics for when the algorithms is unable to explore the entire posterior, which could results in biased inference \citep{Monnahan-etal-2017}. The main disadvantage of HMC is that one cannot easily work with discrete parameters, which makes it harder to have SSMs with discrete latent states \citep[e.g., counts, categories;][]{Monnahan-etal-2017}. We discuss methods to work around this limitation in Section \ref{S.discrete.states}. The use of HMC is increasing in ecology \citep{Monnahan-etal-2017}, and HMC has been recently used to fit ecological SSMs \citep[e.g., in ecosystem ecology,][and in fisheries science, \citealt{Best-Punt-2020}]{Appling-etal-2018}.
\subsubsection{Other algorithms}
The algorithms and software discussed above are the most commonly used to fit SSMs to ecological data in a Bayesian framework. For a more general introduction on how to develop statistical algorithms to fit Bayesian ecological models, please refer to \citet{Hooten-Hefley-2019}. However, the development of Bayesian sampling algorithms is an active field of research. New methods, such as variational inference, appear particularly promising for fitting SSMs \citep[e.g.][]{Ong-etal-2018}.
\subsubsection{Convergence diagnostics}
\label{S.Convergence.Diagnostic}
Regardless of the sampling method, it is important to assess whether it has reached the target posterior distribution. Convergence between multiple chains usually indicates that they have reached the invariant distribution. As such, multiple approaches have been developed to assess whether chain convergence has been achieved. In general, samples from the first iterations are discarded, as these likely occurred before the chain has reached the target distribution \citep[][but see \citealt{Geyer:2011}]{Gelman-Shirley-2011}. In the Metropolis-Hastings setting, this period is referred to as `burn-in'. A somewhat similar initial period, referred as the `warm-up', is discarded with HMC. Then, as a first step, convergence within and between chains can be assessed visually via traceplots (see Appendix S1). More formal metrics exist. The Gelman-Rubin metric, $\hat{R}$ \citep[][see \citealt{Brooks-Gelman-1998} for the multivariate analogue]{Gelman-Rubin-1992}, is one of the most popular multi-chain diagnostics. Although $\hat{R} <1.1$ generally indicates convergence \citep{Gelman-etal-2013}, recent research indicates that a threshold closer to one may be more suitable in some scenarios \citep{Vats-Knudson-2018}. Note that pseudo-convergence can occur in many different scenarios. For example, the sampler can get caught in one mode if the target distribution has multiple modes that are not well connected by the Markov chain dynamics \citep{Geyer:2011}. Running the chain for a long period can help limit these pseudo-convergence problems \citep{Geyer:2011}. A detailed summary of convergence methods is available in \citet{Cowles-Carlin-1996} and further research on convergence diagnostics includes \citet{Boone-etal-2014}, \citet{VanDerwerken-Schmidler-2017}, and \citet{Vats-Knudson-2018}. Both \texttt{JAGS} and \texttt{BUGS} project software, as well as the \texttt{R} package \texttt{coda} \citep{coda}, provide several methods to assess convergence.
\subsubsection{Priors}
\label{S.priors}
Selection of priors is a significant part of a Bayesian analysis because priors affect the resulting posterior distribution \citep{Robert-2007}. Several approaches can be taken depending on the information available about the model parameters and the philosophy of the modeler. Ecologists often use `noninformative' priors. These priors (e.g., a uniform distribution over the parameter space) are often thought to be objective and are generally chosen with the goal of maximizing the influence of the data on the posterior. However, noninformative priors may still have important effects on the posterior, and they should not be used naively \citep{Gelman-etal-2017, Lemoine-2019}. For example, \citet{Lele-2020} showed that noninformative priors could significantly influence the parameter and state estimates of ecological SSMs. Alternatively, ecologists can use informative priors, which are created using knowledge of the parameters or previously collected data \citep[e.g.,][]{Meyer-Millar-1999, Dunham-Grand-2016}. As there are many advantages to using informative priors, they are increasingly used in ecological models \citep{Hooten-Hobbs-2015}. For example, informative priors can be used to supplement SSMs with limited time-series data \citep{Chaloupka-Balazs-2007} and can improve state estimates \citep{Dunham-Grand-2016}. In most cases, noninformative and informative priors are used in the same model on different parameters. For technical reasons, it can be sometime advantageous to use conjugate priors (i.e., priors with the same distribution as the conditional posterior distribution or the posterior distribution). \citet{Kass-Wasserman-1996} and \citet{Millar-2002} have summarized priors typically used in fisheries models, including many SSMs. \citet{Lemoine-2019} advocates for the use of weakly informative priors as default in ecology and provides a guide to their implementation. More generally, \citet{Robert-2007} and \citet{Gelman-etal-2013} provide a thorough review of available priors, selection and examples for a variety of models.
\subsection{Information reduction approaches}
While uncommonly used with SSMs, information reduction approaches, such as synthetic likelihood or Approximate Bayesian Computation (ABC), appear promising to fit complex, highly nonlinear, ecological SSMs \citep{Fasiolo-etal-2016}. These methods bypass the calculation of the exact likelihood \citep{Csillery-etal-2010, Fasiolo-etal-2016}. Instead, these methods generate samples from the model and transform them into a vector of summary statistics that describe the data in the simplest manner possible \citep{Csillery-etal-2010, Fasiolo-etal-2016}. The simulated summary statistics are then compared to observed summary statistics using a predefined distance measure \citep{Fasiolo-etal-2016}. Information reduction approaches smooth the likelihood, reducing some of the common implementation problems encountered with other fitting methods. However, the results from information reduction approaches are often imprecise and, thus, may be most useful in the model development phase \citep{Fasiolo-etal-2016, Fasiolo-Wood-2018}. Interested readers are referred to \citet{Csillery-etal-2010}, \citet{Fasiolo-etal-2016}, and \citet{ Fasiolo-Wood-2018}.
\subsection{Fitting models with discrete states}
\label{S.discrete.states}
Depending on the complexity of the SSM and ones favored inferential approach, having discrete states can either facilitate or complicate the fitting process. A SSM with a single time series of categorical states, generally referred as an HMM (Section \ref{S.HMM}), can be relatively easily fitted with a frequentist approach. The key advantage of these HMMs is their mathematical simplicity: what
would be a high-dimensional integration in a SSM with continuous
state values (see Section \ref{S.Fitting}) is now a simple sum. As
such, having a finite number of possible state values (i.e., discrete
states) significantly simplifies the analysis \citep{Langrock-etal-2012}. The mathematical simplicity of HMMs makes
them highly attractive, and various efficient tools and \texttt{R} packages have been developed to fit HMMs to data. We refer readers interested in HMMs to \citet{McClintock-etal-2020} and \citet{Zucchini-etal-2016}.
While one can use Metropolis-Hastings samplers (Section \ref{S.Metropolis-Hastings}) to fit HMMs with a Bayesian approach \citep[e.g.,][]{Zucchini-etal-2016}, these algorithms are far less efficient than those used to fit HMMs in a frequentist framework. In addition, HMC algorithms (Section \ref{S.HMC}) do not generally allow to sample discrete states. However, recent work has demonstrated the gain in speed that can be made by marginalizing the latent states and how this can be implemented with Gibbs sampling (e.g., \texttt{JAGS}) and HMC software \citep[e.g., \texttt{Stan};][]{Leos-barajas-Michelot-2018, Betancourt-etal-2020, Yackulic-etal-2020}. Marginalizing the states means that when we estimate the parameter values, we do not sample the hidden states at each iteration, but rather track the likelihood of being in any given state \citep{Yackulic-etal-2020}. One can then estimate the states values using the conditional distribution (Eq. \ref{E.cond.dist.state}) or approximations of it \citep[see][for more detail]{Yackulic-etal-2020}, or algorithms that are commonly used with frequentist HMMs, such as the Viterbi algorithm \citep{Zucchini-etal-2016, Leos-barajas-Michelot-2018}. This two-step approach used when marginalizing the states has many parallels with the frequentist approach described in Section \ref{S.Freq}, where we first estimate the parameters using the marginal likelihood and subsequently estimate the hidden states based on the estimated parameter values.
While there are many efficient tools to fit simple HMMs with a frequentist approach, it can be more challenging to fit SSMs that combined both continuous and discrete states. Just as for Bayesian methods, some of the computationally efficient methods (e.g., Laplace approximation method described in Section \ref{S.Laplace}) do not allow for discrete states. One can use instead frequentist methods that rely on sampling the states (e.g., Sequential Monte Carlo methods described in Section \ref{S.Iterative.Filtering}). One could potentially develop algorithms that marginalize the discrete and continuous states with different approaches.
For Bayesian SSMs with discrete states, one additional consideration is label-switching \citep{Jonsen-etal-2013}. The labels given to the $N$ discrete states are arbitrary, and thus there are $N!$ potential label assignments \citep{Zucchini-etal-2016}. The different label permutations result in the same model. Thus, when the MCMC chains have reached convergence, all possible labels will have been assigned to each state and inference on the states will be difficult. For example, we will no longer be able to take the mean of the posterior distribution to estimate the states because all $ \hat{z}_{t,1:T} \approx N/2$. One solution is to impose constraints on the parameters that would be violated when labels are permuted \citep{Zucchini-etal-2016}. For example, in the two-behavior movement model described in Section \ref{S.HMM} we would constrain $\gamma_1 \leq \gamma_2$.
\subsection{When to use each method?}
\label{S.method.comp}
Choosing from this multitude of fitting methods can appear daunting, but can be guided by a choice of inference framework and the limitations of each methodology. In Table \ref{t.methods}, we list the methods discussed above, with some pros and cons. We simply state the associated inferential framework (frequentist vs. Bayesian), and we let the readers decide their favorite inferential framework. In general, there are more computationally efficient methods for simple models in the frequentist framework (e.g., Kalman filter and Laplace approximation), but such generalization cannot be made for more complex models.
Note that in some cases, it may also be easier to use one of the more specific ecological SSM \texttt{R} packages. For example, the package \texttt{MARSS} \citep[which stands for Multivariate Auto-Regressive State-Space,][]{Holmes-etal-2012, MARSS} can be useful to model multiple populations, if these can be reasonably formulated with a linear and normal SSM. Those interested in fisheries stock assessment SSMs should look at the package \texttt{stockassessment} \citep[available on GitHub at {https://github.com/fishfollower/SAM},][]{Nielsen-Berg-2014}. Those interested in SSMs for animal movement should explore \texttt{bsam} \citep{Jonsen-etal-2005, Jonsen-2016}, \texttt{crawl} \citep{Johnson-etal-2008, crawl}, and \texttt{momentuHMM} \citep{McClintock-Michelot-2018}.
\section{\label{S.Estimability}Formulating an appropriate SSM for your data}
SSMs are powerful tools, but their inherent flexibility can tempt ecologists to formulate models that are far too complex for the available data. The model structure or the characteristics of the specific dataset may make it impossible to estimate every parameter reliably. In such cases, parameter estimates will no longer provide key information on the underlying biological process and state estimates may become unreliable \citep[e.g.,][]{AugerMethe-etal-2016}. Formulation of SSMs needs to be guided by the inference objectives and the available data. In this section, we discuss how to assess whether a model is adequate for your data and how one can alleviate potential estimation problems.
\subsection{Identifiability, parameter redundancy and estimability}
\label{parredidentest}
When we estimate the parameters of a model, denoted here as $M(\boldsymbol{\theta})$, we often want to find the set of parameter values, $\boldsymbol{\theta}$, that results in the best fit to the data. For this to be possible, the model needs to be identifiable. Identifiability refers to whether or not there is a unique representation of the model. A model is globally identifiable if $M(\boldsymbol{\theta}_1) = M(\boldsymbol{\theta}_2)$ implies that $\boldsymbol{\theta}_1 = \boldsymbol{\theta}_2$. For example, in a frequentist framework, an identifiable model would have only a single $\boldsymbol{\theta}$ value that would maximise the likelihood (Fig. \ref{fig:profile}a). A model is locally identifiable if there exists a neighbourhood of $\boldsymbol{\theta}$ where this is true (Fig. \ref{fig:profile}b). Otherwise a model is non-identifiable \citep[Fig.
\ref{fig:profile}c;][]{Rothenberg1971,Coleetal2010}.
An obvious case of non-identifiability is when a model is overparameterised and can be reparameterised with a smaller set of parameters. For example, if two parameters only appear as a product in a model (e.g., $y = \alpha \beta x$); that model could be reparameterised with a single parameter replacing that product (e.g., $y = \gamma x$, where $\gamma = \alpha \beta$). The parameter redundancy of the original model will result in non-identifiability \citep{CatchpoleandMorgan1997} and non-identifiability caused by the inherent structure of a model is referred to as intrinsic parameter redundancy \citep{Gimenez-etal-2004} or structural non-identifiability \citep{CobelliDiStefano1980}. Regardless of the amount or quality of data, it is impossible to estimate all the parameters in such a model.
Having a structurally identifiable model does not guarantee that one can estimate its parameters with the data at hand. Non-identifiability can be caused by a specific dataset with, for example, missing or sparse data \citep{Gimenez-etal-2004}. This problem is known as extrinsic parameter redundancy \citep{Gimenez-etal-2004} or practical non-identifiability \citep{Raue-etal-2009}. A parameter is defined as practically non-identifiable if it has a confidence interval that is infinite \citep{Raue-etal-2009}. It is also possible for a dataset to create estimation problems with an otherwise structurally and practically identifiable model, a phenomenon referred to as statistical inestimability \citep{Campbellele2014}. If a model is statistical inestimable, a confidence interval for a parameter will be extremely large but not infinite. This often occurs because the model is very similar to a submodel that is parameter redundant for a particular dataset, which is known as near redundancy \citep{Catchpoleetal2001,Coleetal2010}.
Having a non-identifiable model (either structurally or practically) leads to several problems. First, there will be a flat ridge in the likelihood of a parameter redundant model \citep{CatchpoleandMorgan1997}, resulting in more than one set of MLEs. However, despite the parameter redundancy, numerical methods for parameter estimation usually converge to a single set of MLEs. Therefore, without further diagnostics, one may not realise that
the MLEs are not unique. Second, the Fisher information matrix will be singular \citep{Rothenberg1971} and therefore the standard errors will be undefined in a non-identifiable model. However, the exact Fisher information matrix is rarely known and standard errors are typically approximated using a Hessian matrix. The Hessian describes the local curvature of a multi-parameter likelihood surface. The Hessian is generally evaluated numerically, which can lead to explicit (but incorrect) estimates of standard errors. Third, many model selection methods (see Section \ref{S.Model.Comparison}) are based on the assumption that a model is identifiable and that the penalty for complexity is a function of the number of unique and estimable parameters \citep{Gimenez-etal-2004}. If a model is statistically inestimable or near redundant, these three problems may also occur, as the model is close to being non-identifiable. For example the log-likelihood profile will be almost flat.
Checking for identifiability and estimability should become part of the model fitting process and several methods are available to do so. A clear sign of problems is a flat log-likelihood profile (Fig. \ref{fig:profile}c), and plotting the log-likelihood profile for each parameter can serve as a diagnostic for this \citep[Fig. \ref{fig:profile};][]{Dennis-etal-2006, Raue-etal-2009, AugerMethe-etal-2016}. Correlation between parameters can also be indicative of estimation problems, and it may be useful to inspect the log-likelihood or posterior surface of pairs of parameters \citep{Campbellele2014, AugerMethe-etal-2016}. Depending on model complexity and computation time, simulations can be an easy way to investigate the estimability of SSMs \citep{AugerMethe-etal-2016}. For a specified SSM and a known set of parameters, one simulates the state process and observation time
series, and then estimates the parameters and states. One then compares estimated parameter and state values with the known
true values. Parameter estimates from non-identifiable models will usually be biased with large variances.
In addition to these simple checks, three advanced methods to assess estimability and identifiability problems exist. First, data cloning has been shown to be useful with ecological models \citep{Peacock-etal-2016}. Data cloning involves using Bayesian methodology with a likelihood based on $K$ copies of the data (clones).
The posterior variance of a parameter will tend towards $K$ times the asymptotic variance of the parameter, so that if a parameter is identifiable the posterior variance will tend to zero as $K$ tends to infinity. If a parameter is not identifiable, the posterior variance will tend to a fixed (non-zero) value \citep{Leleetal2010}. \cite{Campbellele2014} show how this method can be extended to find estimable parameter combinations in non-identifiable models.
Second, one can use the fact that the Hessian matrix in a non-identifiable model will be singular at the MLE. As a singular matrix has at least one zero eigenvalue, the Hessian method involves finding the eigenvalues of the Hessian matrix. If the Hessian matrix is found numerically, the eigenvalues for a singular matrix may be close to zero rather than exactly zero. Therefore, if any of the eigenvalues are zero or close to zero, the model is deemed non-identifiable or parameter redundant, at least for that particular dataset \citep{Viallefontetal1998}. The Hessian matrix will also have eigenvalues close to zero if the model is statistically inestimable or near redundant \citep{Catchpoleetal2001}.
Third, one can use the symbolic method. This method uses the concept that a model can be represented by an exhaustive summary, which is a vector of parameter combinations that uniquely define the model. For example, this vector could be $\mathbf{k} = (L_{\textsc{m}}(\boldsymbol{\theta} | y_1), L_{\textsc{m}}(\boldsymbol{\theta} | \mathbf{y}_{1:2}), \hdots, L_{\textsc{m}}(\boldsymbol{\theta} | \mathbf{y}_{1:T}))'$, where the first element is the marginal likelihood (Eq. \ref{E.Marginal.Likelihood}) for the first observation ($y_1$), the second element is the marginal likelihood for the first two observations ($\mathbf{y}_{1:2}$), etc. This straightforward exhaustive summary works well for HMMs \citep{Cole-2019}, but can be impractical for SSMs with continuous states as it involves integration. Suitable, but more complex to derive, exhaustive summaries for SSMs are given in \cite{ColeandMcCrea2016}. To investigate identifiability, we form a derivative matrix by differentiating each term of the vector with respect to each parameter. Then, we find the rank of this matrix. The rank of a matrix is the number of columns that are linearly independent. Since each column of the derivative matrix is associated with one of the parameters, the rank is the number of estimable parameters (or parameter combinations). If the rank is less than the number of parameters, then the model is non-identifiable or parameter redundant \citep{CatchpoleandMorgan1997,Coleetal2010}. This method can be used to investigate practical identifiability as well as structural identifiability by choosing an exhaustive summary that includes the specific dataset \citep{Coleetal2012}. In some more complex models, the computer can run out of memory calculating the rank of the derivative matrix. \cite{Coleetal2010} and \cite{ColeandMcCrea2016} provide symbolic algebra methods for overcoming this issue. The alternative is a hybrid symbolic-numerical method, which involves finding the derivative matrix using symbolic algebra, but then finding the rank at five random points in the parameter space \citep{ChoquetandCole2012}.
Each of the numerical methods (log-likelihood profile, simulation, data cloning, Hessian method) can be inaccurate. They are also not able to distinguish between estimability, practical identifiability and structural identifiability when applied to a specific dataset, although in some cases a large simulated dataset could be used to test structural identifiability. Being able to distinguish between these problems is useful as it can help us assess whether gathering more data will help. The symbolic method is accurate, but is more complicated to use as it involves using a symbolic algebra package. Code for assessing estimability using simulations and the Hessian method is given in Appendix S1: Section S1 1.4. Code for the symbolic algebra method is given in Appendix S3.
In Bayesian analysis, identifiability and estimability issues have a different focus because priors can affect our capacity to differentiate between parameters \citep{Cressie-etal-2009}. In general, parameters are said to be weakly indentifiable when the posterior distribution significantly overlaps with the prior \citep{Garrett-Zeger-2000, Gimenez-etal-2009-indentifiability}. If priors are well informed by previous data or expert knowledge, their strong influence on the posterior distribution is no longer an identifiability/estimability issue but one of the benefits of Bayesian analysis. However, misusing informed priors (e.g., when the information is not reliable) may hide identifiability issue or cause the estimability problems \citep{Yin-etal-2019}. Thus, one should choose priors with great care. To help ensure that the data inform the model and that the posterior is well behaved, \citet{Gelfand-Sahu-1999} suggested to use informative priors that are not too precise (see Section \ref{S.priors} for other considerations). Weak identifiability can result in multiple implementation issues, including slow convergence \citep{Gimenez-etal-2009-indentifiability}. Diagnostics for parameter identifiability in the Bayesian framework include some of the tools described above and the visual or numerical assessment of the overlap between priors and posterior distributions \citep{Garrett-Zeger-2000, Gimenez-etal-2009-indentifiability}.
\subsection{Remedies for identifiability issues}
When we fit a SSM to our data, we hope that it will provide accurate and precise estimates of our parameters and states. But how can we achieve these goals? First, we need to have a structurally identifiable model. Second, one needs a dataset appropriate for the model and vice-versa, otherwise one can face estimation problems even with structurally identifiable models. Generally, we assume that having more data will allow us to better estimate parameters and states. However, as discussed below, increasing the length of the time series may not be the best way to improve estimation.
\subsubsection{Reformulate the SSM}
To create a structurally identifiable model, one should start by avoiding overparametrization. As mentioned above, models where some
parameters only appear as products of each other should be simplified. The same holds for models where parameters only appear as sums (e.g., $y = (\alpha + \beta)x$), or differences, or fractions. Models where the magnitude of two sources of error are simply additive are also problematic (e.g., $Y \sim N(X, \sigma^2)$ and $X \sim N(\mu, \tau^2)$, will result in $Y \sim N(\mu, \sigma^2 + \tau^2)$ where $\sigma$ and $\tau$ cannot be uniquely identified). As such, one needs to check that none of the parameters are confounded and carefully inspect the combination of the sources of variability in all hierarchical models, including SSMs (see below). Some of the tools discussed above can help construct structurally identifiable models. In particular, the symbolic method can be used to identify the parameters that are confounded in a non-identifiable model, and thus can be used to select estimable parameter combinations. This involves solving a set of partial differential equations formed from the same derivative matrix used to check identifiability \citep{Catchpoleetal1998,Coleetal2010}.
\subsubsection{Make simplifying assumptions when data are limited}
A full model may be too complex for the data-at-hand and it may be advantageous to make simplifying assumptions. For example, when the data available for older age classes are limited, researchers can have difficulties fitting the fisheries stock assessment model presented in Section \ref{S.M.stock.assessment}. To help the estimation process, one can create a cumulative age class, $\text{A}^+$, that accounts for all fish older than a certain age \citep{Nielsen-Berg-2014}. To allow fish to remain in the cumulative age class, we need to add the following equation to the model:
\begin{linenomath*}
\begin{align}
\log (N_{\textsc{a}^+,t}) &=\log( N_{\textsc{a}^+-1,t-1}e^{- F_{\textsc{a}^+-1,t-1} - M_{\textsc{a}^+-1,t-1}} + N_{\textsc{a}^+,t-1}e^{- F_{\textsc{a}^+,t-1} - M_{\textsc{a}^+,t-1}}) + \epsilon_{N_{\textsc{a}^+, t}}.
\end{align}
\end{linenomath*}
The similar size of these older fish makes them more likely to be caught by the same type of fishing gear, and thus their catchability and fishing mortality can be further assumed to equal that of the previous age class ($Q_{\textsc{a}^+,s} = Q_{\textsc{a}^+-1,s}$ and $F_{\textsc{a}^+,t} = F_{\textsc{a}^+-1,t}$). While this appears to add complexity, creating this cumulative age class and equating some terms reduces the number of states and parameters to estimate. However, some simplifying assumptions may result in estimation problems. For example, the original DCRW model of \citet{Jonsen-etal-2005} has a single correction factor rather than one per coordinate ($\psi = \psi_{lon} = \psi_{lat}$, see Section \ref{S.M.movement.model}). The common correction factor can results in estimation problems because longitude and latitude often differ in the degree of correction they need \citep{AugerMethe-etal-2017}. As long as they are biologically reasonable, such simplifying assumptions can be useful in a wide range of fields, including in community ecology where SSMs can link multiple species to common latent variables and thus reduce the dimension of the model \citep{Thorson-etal-2016}.
\subsubsection{Estimate of measurement errors externally}
SSMs can be associated with significant estimability problems, particularly when trying to estimate the two main sources of variability \citep{Knape-2008, AugerMethe-etal-2016}. As a result, researchers often fix some of the parameters to known values, or use informed priors if they are working in a Bayesian framework. In particular, many use fixed values for the measurement errors and use for them independent estimates of measurement errors \citep[e.g.,][]{Jonsen-etal-2005}. While such method can alleviate estimation problems \citep{Knape-2008}, one must be careful not to use biased or misspecified values.
\subsubsection{Integrate additional data}
Covariates that provide additional information about a state or a process (e.g., survival) may be a means of overcoming identifiability problems. \citet{Polansky-etal-2019} showed that non-identifiability in the estimation of a fecundity and observation correction parameter could be overcome by including a covariate in the model for fecundity.
Similarly, identifiability issues can be overcome by combining a SSM with a model for another data set that has parameters in common with the SSM. For example, in integrated population models, SSMs for time series of census data are combined with capture-recapture data \citep{Besbeas-etal-2002,Abadi-etal-2010}. Adding additional data sources can be extremely useful but may not remove all identifiability issues. Methods for checking identifiability in integrated models are discussed in \citet{ColeandMcCrea2016}.
\subsubsection{Use replicated observations}
Having replicated observations through time (e.g., two independent population surveys) can help differentiate process variation from observation error, improve the parameter estimates accuracy, and improve the capacity of model selection methods to identify the correct observation distribution \citep{Dennis-etal-2010, Knape-etal-2011}. In many instances, such replicated observations have already been collected, but are aggregated. For example, in population monitoring studies, subsamples (e.g., transect portions) are often aggregated into one overall estimate of abundance. \citet{Dennis-etal-2010} demonstrated that using these as replicates, rather than aggregating them, can improve the estimates. One can also take advantages of time series with multiple data sources to estimate the errors of each data source \citep[e.g., double tagged individuals in movement SSMs,][]{Winship-eta-2012}. For animal movement models, individuals can be also seen as replicates of the same process, but often the SSMs are fitted separately to each individual track. To improve inference, one can create a population model, where each individual track is linked to a distinct state time series but all share the same parameters \citep{Jonsen-2016}. While the gains that can be made with replications are significant, one must understand the assumptions of models for replicated data. Simple population models for replicated datasets may assume that the replicates are independent \citep{Dennis-etal-2010}. However, many temporally varying factors (e.g., weather) may affect the sampling conditions and/or the behaviour of animals and result in correlations between replicates. \citet{Knape-etal-2011} demonstrated how to account for such dependence in population dynamics models. For animal movement, one may want to consider whether it is appropriate to assume that the behavioral mechanism driving movement is identical across individuals and, if not, may want to modify the model accordingly. However, as the gains that can be made with replications far surpasses those that could be made with longer time-series \citep{Dennis-etal-2010}, one should consider using replication in their models and when designing their studies. For example, \citet{Knape-etal-2011} suggested that in some cases managers may want to sample a population twice every second year rather than once a year. As SSMs are becoming the prime method to fit ecological time series, such study design issues should be explored further.
\subsubsection{Match temporal resolution for states and observations}
The temporal resolution of the data can affect the parameter and state estimates and it is important to define a model at a resolution that is appropriate for the data. In many cases, adequate temporal span or resolution is more important than increased data quantity. For example, if a model describes a long-term cycle, then collecting data from more individuals is unlikely to make parameters estimable if the dataset is not long enough to span the cycle being described \citep{Peacock-etal-2016}. If developing a model to classify a movement path into distinct behavioral modes, one must sample the movement track at a high enough frequency so that multiple locations are recorded in each movement bout \citep{Postlethwaite-Dennis-2013}. If one has a dataset with locations every 8 hrs, it would be challenging to estimate behavioral states lasting less than 16-24 hrs. One can use pilot data, simulations, and data cloning to identify the temporal (and spatial) scale of sampling appropriate for the model, in something akin to a power analysis \citep{Peacock-etal-2016}. Overall, finding an appropriate model for your data, or collecting the appropriate data for your questions, can be an iterative process where one assess the estimability of different models under different data conditions.
\section{\label{S.Model.Comparison}Computationally-efficient model comparison methods}
Model comparison (or selection) can be used to compare the relative fit of models representing multiple working hypotheses, and to identify the model amongst these that best describes the data (see Section \ref{S.Diagnostics} for methods to evaluate the absolute fit of a model). Because different model structures can affect the estimated states and parameters \citep{Knape-etal-2011}, model comparison can be extremely useful in helping to refine state estimates \citep{AugerMethe-etal-2017}. Model comparison is common in ecology and has been used to compare SSMs \citep[e.g.,][]{Siple-Francis-2016}. However, it is not uncommon for users to fit only a single SSM, likely due to the computational burden of fitting complex SSMs and some of the known limitations of applying model selection methods to SSMs \citep{Jonsen-etal-2013}. With the improved efficiency of fitting algorithms and advancements in model selection measures, model comparison of SSMs is becoming more attainable.
One common view is that ecological systems are so complex that it is impossible to develop a model that truly describes them, and that the goal of model selection is to find the best approximation of the truth \citep{Burnham-Anderson-2002}. Under this paradigm, a useful way to compare models is to assess how well they can be used to predict new data. Comparing the out-of-sample predictive accuracy of models can be done with cross-validation. However, it is rarely done with ecological SSMs because it requires fitting the same model multiple times, and thus can add significant computational burden to the analysis. Many advocate cross-validation as the best method for model selection \citep{Gelman-etal-2014, Link-etal-2017}, and gains in efficiency of fitting algorithms are making its use increasingly feasible. We discuss cross-validation as a model selection , and validation, method in Section \ref{S.Diagnostics}. Here, we focus on what can be considered approximations of predictive accuracy. In particular, we discuss information criteria measures used with frequentist and Bayesian approaches.
\subsection{\label{S.Model.Comparison.ML}Frequentist approach}
The most common model comparison measure in ecology is Akaike's Information Criterion \citep[AIC;][]{Aho-etal-2014}. AIC was derived to estimate the expected and relative distance between the fitted model and the unknown true data-generating mechanism \citep{Burnham-Anderson-2002}, and can be viewed as $-2$ times an approximation of the predictive accuracy of the model \citep{Gelman-etal-2014}:
\begin{linenomath*}
\begin{equation}
\text{AIC} = - 2 \log L(\boldsymbol{\hat{\theta}}_\textsc{mle} | \mathbf{y}) + 2k,
\label{E.AIC}
\end{equation}
\end{linenomath*}
where $L(\boldsymbol{\hat{\theta}}_\textsc{mle} | \mathbf{y})$ is the likelihood of the model at the MLE (i.e., the probability of the observed data given the model) and $k$ is the number of parameters estimated. The model with the lowest AIC, thus the shortest distance from the truth, is considered the best model. Models with more parameters will be more flexible and will tend to fit the existing data better by chance alone. Thus, AIC penalizes a model for its number of estimated parameters to compensate for overfitting.
There are many issues related to using AIC with SSMs, and some have cautioned against this practice \citep[e.g.,][]{Jonsen-etal-2013}. We identified five different concerns. The first three concerns are related to the fact that the states of a SSM can be considered as random effects. First, using AIC to understand whether including random effects improves the model is difficult because some of the models may have parameters at the boundary of parameter space
\citep{Bolker-etal-2009}. For example, testing whether or not there is process variance in SSMs (e.g., comparing our toy model to a model with no process variance, where $\sigma_p = 0$) could result in boundary problems, and is not recommended. Second, when you have random effects it is difficult to quantify the effective number of parameters \citep{Bolker-etal-2009}. For SSMs, it is difficult to know to what extent the states should be counted as estimated parameters and contribute to $k$. However, if all of the compared SSMs have the same number of states and no additional random effects, these two issues should be less problematic. In such cases, we would expect any bias in the penalty $k$ to be the same across models and thus have little effect on the difference in AIC across models. Third, one must decide whether the marginal likelihood or the conditional likelihood should be used when calculating AIC of a model with random effects \citep{Muller-etal-2013}. In contrast to the marginal likelihood, where we integrate out the states (Eq. \ref{E.Marginal.Likelihood}), the conditional likelihood considers the states as known: $L_{\textsc{C}} (\boldsymbol{\theta}_o| \mathbf{z}_{1:T}, \mathbf{y}_{1:T}) = \prod_{t=1}^T g(y_t | z_t, \boldsymbol{\theta}_o)$. When the conditional likelihood is used in the AIC framework, both the parameter and state estimates are plugged in and different approaches can be used to account for the number of states \citep{Vaida-Blanchard-2005, Muller-etal-2013}. This conditional AIC is a measure of the model's ability to predict new observations that share the same latent states, while the marginal AIC does not assume that the latent states are shared with the new observations and measures the model's ability to predict new observations from the same process \citep{Vaida-Blanchard-2005}. For example, for a SSM describing the population dynamics of a fish species, we would interpret the conditional AIC as assessing the ability to predict another survey of the same population during the same time period. The marginal AIC would be assessing the ability of the model to predict a survey from a similar population of the same species. To our knowledge the marginal likelihood has always been used with SSMs fitted in a frequentist framework. In most SSMs, the number of states increases with the sample size (i.e., with the length of the time series). Because frequentist model selection methods rely on asymptotic properties, which can be attained when the sample size is large compared to the number of quantities estimated, conditional AIC may be unreliable for most SSMs. This characteristic may explain why potential advantages of using the conditional likelihood remain uninvestigated in the frequentist SSM literature (the conditional likelihood is used in Bayesian information criteria, see Section \ref{S.Model.Comparison.B} for a discussion). The fourth source of concern is related to the problems associated with using AIC to choose the number of components in mixture models, which are particularly relevant for choosing the number of states in HMMs \citep{Jonsen-etal-2013}. \citet{Pohle-etal-2017} outline solutions to this HMM-specific problem.
The final concern, which is specific to cases with small sample size, is one that has been studied in the SSM literature. When the sample size, $n$, is small and the number of parameters, $k$, is relatively large (e.g., when $k \approx n/2$), the $2k$ penalty is inadequate and AIC has a tendency to favor more complex SSMs \citep{Cavanaugh-Shumway-1997}. Many use the corrected AIC (AICc) for small sample size \citep{Burnham-Anderson-2002}. However, \citet{Cavanaugh-Shumway-1997} noted that AICc may be inadequate for many SSMs, and suggested an alternative: the bootstrap-corrected measure, AICb. AICb has been used for ecological SSMs \citep{Ward-etal-2010, Siple-Francis-2016}, especially by users of the \texttt{R} package \texttt{MARSS} \citep{Holmes-etal-2012}. This package for estimating the parameters of linear multivariate auto-regressive SSMs with Gaussian errors (i.e., multivariate dynamic linear models) has a function that calculates various versions of AICb. AICb was developed in the context of linear Gaussian SSMs, but is thought to be relatively robust to violations to normality \citep{Cavanaugh-Shumway-1997}. We can describe AICb as:
\begin{linenomath*}
\begin{equation}
\text{AICb} = - 2 \log L_{\textsc{m}}(\boldsymbol{\hat{\theta}_\textsc{mle}} | \mathbf{y}_{1:T}) + 2\left(\frac{1}{N} \sum_{i=1}^{N} - 2\ \log \frac{L_{\textsc{m}}(\boldsymbol{\hat{\theta}}^i | \mathbf{y}_{1:T})}{L_{\textsc{m}}(\boldsymbol{\hat{\theta}_\textsc{mle}} | \mathbf{y}_{1:T})} \right),
\end{equation}
\end{linenomath*}
where $\boldsymbol{\hat{\theta}}^i$ is the $i^{th}$ bootstrap replicate of $\boldsymbol{\hat{\theta}}$, $N$ is the number of replicates, and $L_{\textsc{m}}(\boldsymbol{\hat{\theta}}^i | \mathbf{y}_{1:T})$ is the marginal likelihood of the model with the bootstrapped parameter sets given the original data. This bootstrap replicate can be achieved by simulating a time series from our model with $\boldsymbol{\hat{\theta}_{\textsc{mle}}}$ and estimating the parameters using this new time series. AICb was shown to outperform AIC and AICc when used with SSMs that had relatively small sample size for the number of parameter estimated \citep{Cavanaugh-Shumway-1997}. The disadvantage of AICb is that it requires fitting the model $N$ times. In the case of models that are computationally demanding to fit, one may need to continue to rely on AICc when sample sizes are small. While AICc tends to erroneously choose more complex models compared to AICb, it is better than AIC and many other metrics for SSMs with small sample size \citep{Cavanaugh-Shumway-1997}. Another similar computationally-intensive AIC variant for SSMs fitted to small samples has been developed by \citet{Bengtsson-Cavanaugh-2006}, but its use in ecology has been limited by some of its constraints \citep[e.g.,][]{Ward-etal-2010}. For large datasets, some ecologists prefer to use BIC over AIC because AIC tends to choose more complex models as sample size increases. However, these two measures are used to achieve different inferential goals, and choosing between them is largely a philosophical question \citep[see][]{Aho-etal-2014, Hooten-Hobbs-2015}.
Overall, AIC and its small-sample alternatives can be used with SSMs in many instances, especially when the number of states and random effects are the same. AIC has been used for decades with SSMs \citep{Harvey-1990}, and simple simulation studies have shown that AIC can be used to reliably select between SSMs \citep{AugerMethe-etal-2017}. Further research on the capacity of AIC to compare the predictive abilities of SSMs when the number of states or random effects vary, and research on how to account for the number of states in the penalty term, would be useful. In the meantime, one should be aware of the limitations outlined above, and interpret the results accordingly.
Other frequentist methods may be used to select between SSMs. For example, likelihood ratio tests can be used to select between nested models, especially when conducting planned hypothesis testing \citep[e.g.,][]{Karban-and-deValpine-2010}. However, likelihood ratio tests will suffer from some of the same issues as the those highlighted for AIC. \citet{Newman-etal-2014} also highlighted the potential use of score tests, transdimensional simulated annealing, and other methods. To our knowledge, these alternative methods have not been used in the SSM literature, but may be the focus of future research.
\subsection{\label{S.Model.Comparison.B}Bayesian approaches}
Two Bayesian information criteria, the Deviance Information Criteria (DIC, see Appendix S4) and the Watanabe-Akaike information criterion (WAIC), are popular with hierarchical models, and have been used with SSMs. They replace the information criteria based on MLEs, such as AIC, which do not have a clear interpretation for Bayesians \citep{Hooten-Hobbs-2015}. DIC and WAIC are similar to AIC, but they both use information from the posterior and estimate the effective number of parameters using data-based bias correction rather than a fixed rule. These data-based methods attempt to account for the effects of priors and the hierarchical structure (e.g., the characteristics of the random effects) on the flexibility of the model.
While DIC has been used to select ecological SSMs \citep[e.g.,][]{Michielsens-etal-2006}, and MCMC sampler software \citep[e.g., JAGS,][]{Plummer:2003} and \texttt{R} packages like \texttt{rjags} \citep{rjags} have functions that compute it easily, this information criterion is known to have many drawbacks that hinder its suitability for SSMs. DIC performs better when the number of effective parameters is much smaller than the sample size \citep{Hooten-Hobbs-2015}, a condition likely uncommon with SSMs because the number of latent states scales with the sample size. In addition, DIC is known to be problematic for mixture models, can poorly estimate the effective number of parameters (e.g., can return negative numbers), relies on approximate posterior normality, and is not fully Bayesian because its measure of fit relies on the posterior mean
of $\boldsymbol{\theta}$ (i.e., a point estimate, see Appendix S4) instead of the entire posterior distribution \citep{Gelman-etal-2014, Hooten-Hobbs-2015, Kai-Yokoi-2019}. These limitations may explain why \citet{Chang-etal-2015}, in contrast to \citet{Wilberg-Bence-2008}, showed that DIC had difficulties selecting amongst ecological SSMs.
\sloppy Many now favor WAIC, a recently developed Bayesian information criterion \citep{Gelman-etal-2014, Hooten-Hobbs-2015}:
\begin{linenomath*}
\begin{equation}
\text{WAIC} = -2 \sum_{i=1}^T \log \int p(y_i | \boldsymbol{\theta}) p(\boldsymbol{\theta} | \mathbf{y}) \text{d} \boldsymbol{\theta} + 2 p_\textsc{waic}.
\label{E.WAIC}
\end{equation}
\end{linenomath*}
The first component of WAIC is also a measure of fit, but unlike DIC it uses the entire posterior distribution
for $\boldsymbol{\theta}$ rather than a point estimate. As such, we can consider this measure of fit as truly Bayesian. There are different ways to estimate the effective number of parameters, $p_\textsc{waic}$. \citet{Gelman-etal-2014} recommend using $\sum_{i=1}^T \mathrm{Var}_{\textsc{post}}(\log p(y_i | \boldsymbol{\theta}))$ as it gives results closer to the leave-one-out cross validation. In our formulation of WAIC (Eq. \ref{E.WAIC}), we used a $-2$ multiplier as it helps highlight the similarity to AIC (Eq. \ref{E.AIC}). However, this multiplier may obscure how WAIC is a measure of the predictive accuracy of the model, and some researchers prefer not using it \citep[e.g.,][]{Vehtari-etal-2017}. See Appendix S4 for how Eq. \ref{E.WAIC} is calculated in practice.
WAIC has been used to compare ecological SSMs \citep[e.g.,][]{Baldwin-etal-2018, Ferretti-etal-2018} and can be computed using the R package \texttt{loo} \citep{Vehtari-etal-2017}. Recent reviews of Bayesian model comparison methods favor WAIC over DIC \citep{Gelman-etal-2014, Hooten-Hobbs-2015} because it is a fully Bayesian metric, it is not affected by parametrization, and will not return negative values for the effective number of parameters. However, WAIC has a few shortcomings, and new approximations of predictive accuracy have been recently proposed \citep[e.g., Pareto-smoothed importance sampling leave-one-out cross validation,][]{Vehtari-etal-2017}. Both parts of WAIC are computed by using the sum over each data point $i$, and thus rely on partitioning the data into disjoint, ideally conditionally independent, pieces \citep{Gelman-etal-2014}. Naively partitioning can be problematic with SSMs since the time-series nature of the data generally results in dependence structures (see Appendix S4 for a potential solution). While AIC and DIC rely on a point estimate rather than summing over each data point, they also assume conditional independence.
Just as for AIC, we could use either the conditional or marginal likelihood with DIC and WAIC \citep{Kai-Yokoi-2019,Merkle-etal-2019}. With the Bayesian approach, the likelihood is generally defined as fully conditional on both parameters and latent states and both are generally sampled when sampling the posterior. Thus, the conditional likelihood is usually used with Bayesian metrics even though this is rarely specified \citep{Millar-2018, Merkle-etal-2019}. While computing the marginal likelihood version of these Bayesian metrics is more computationally expensive, their conditional counterparts are often unreliable \citep{Millar-2009, Millar-2018, Merkle-etal-2019}. In particular, DIC and WAIC were shown to more reliably select the true underlying SSM when the marginal likelihood is used \citep{Kai-Yokoi-2019}.
As \citet{Gelman-etal-2014} noted, we are asking close to the impossible from these information criteria measures: an unbiased estimate of out-of-sample prediction error based on data used to fit the model that works for all model classes and requires minimum computation. As such, metrics such as WAIC can be unreliable estimates of the predictive ability of ecological models \citep{Link-etal-2017}. While further research is needed to assess when WAIC is appropriate for SSMs and to identify data partitioning schemes that resolve some of the potential biases, WAIC based on the marginal likelihood is likely the best information criterion for Bayesian SSMs at this point. Future work should explore how promising new approximation methods \citep[see][]{Vehtari-etal-2017,Burkner-etal-2020} perform with ecological SSMs. If the models are relatively inexpensive to fit, then one can bypass many of the shortcomings of WAIC, and other approximations of predictive ability, by comparing models using more computer intensive cross-validation methods \citep{Gelman-etal-2014, Link-etal-2017, Vehtari-etal-2017}. Cross-validation will also require one to partition data intelligently, but this may be more easily implemented with blocking \citep{Gelman-etal-2014, Roberts-etal-2017}.
Other methods could be used to compare models in a Bayesian framework \citep[e.g.][]{Newman-etal-2014}. For example, reversible-jump MCMC has been used to compare SSMs \citep{McClintock-etal-2012}, but is known to be difficult to implement \citep{Hooten-Hobbs-2015}. The importance of multiple covariates in a model (e.g., the effect of temperature and precipitation on bird survival) can be assessed
by multiplying coefficients in a model by indicator variables which
when equal to one include the covariate and when equal to zero exclude the covariate \citep{OHara-Sillanpaa-2009}. Such techniques have
been used to compare ecological SSMs \citep{Sanderlin-etal-2019}, but such an
approach is designed for nested models only. Posterior predictive loss approaches appear to be suitable for time-series data \citep{Hooten-Hobbs-2015} and have been used to compare ecological SSMs \citep{MillsFlemming-etal-2010}. While these alternative approaches may not be as commonly used to compare ecological SSMs, and will have drawbacks, many of them warrant further exploration.
\subsection{Model averaging}
Model averaging can combine the strength of several models and account for model uncertainty, something model selection cannot offer \citep{Buckland-etal_1997, Hooten-Hobbs-2015}. \citet{Wintle-etal-2003} argued against using a single model to make predictions because uncertainty about model structure is often high in ecology, and alternative models can have prediction differences with important repercussions for management decisions. When one selects
a single model, and presents the parameter and state estimates based on this best model, one implicitly assumes that the model is true and that the uncertainty is only in the estimation process \citep{Buckland-etal_1997, Wintle-etal-2003}. One can instead use model averaging, where, for example, each model is weighted and the predictions are a weighted sum across the plausible models \citep{Wintle-etal-2003}. Both the parameters and the predictions could be averaged, but this must be done with care and we would generally caution against averaging parameters. In many cases, differences in model structure result in changing the meaning of parameters, thus making their average nonsensical \citep{Dormann-etal-2018}. Model averaging has been used in a few studies applying SSMs to ecological data \citep[e.g.,][]{Maunder-Deriso-2011, Moore-and-Barlow-2011} and was shown to provide unbiased estimates \citep{Wilberg-Bence-2008}. However, simulations studies have shown that model averaging may not always provide more accurate point estimates than the best SSMs \citep{Wilberg-Bence-2008, Chang-etal-2015}. In addition, while model averaging generally reduces prediction errors compared to each of the contributing models, these gains can be counteracted by factors such as uncertainty in the model weights and covariance between models \citep{Dormann-etal-2018}. In addition, calculating weights using parametric methods such as AIC can perform poorly \citep{Dormann-etal-2018}. We refer interested readers to a recent review by \citet{Dormann-etal-2018}, which provides an in-depth discussion of model averaging in ecology.
\section{\label{S.Diagnostics}Diagnostics and model validation for SSMs}
While model selection can help us identify which of the fitted models best describes the data, it rarely provides an assessment of the absolute fit of that model. As such, the selected model could be a poor representation of the data generating process (i.e., could poorly describe the ecological process and/or measurement process) and relative measures of fit, such as AIC, do not quantify how closely the model matches the data. Thus, before interpreting model results, it is crucial to carry out some of the following model diagnostics. First, it is essential to examine whether estimated parameters
seem biologically reasonable. For example, our understanding
of the system may stipulate that a response variable should increase with a covariate.
A model with parameter estimates inconsistent with such \textit{a priori}
understanding or with unrealistic effect sizes will be suspect. Second, it is important to assess the influence
of individual observations on estimated parameters. For example, outliers can have a strong influence on parameter estimates. Third, one should examine whether the model assumptions
are reasonable. For example, with SSMs, assumptions
are made about the probability distributions for states and observations (e.g., Eqs. \ref{E.state.NDLM}-\ref{E.obs.NDLM} assume both are normal). Fourth, it is important to examine the goodness of fit, which defines how well the model describes the data used to fit the model. At the individual observation level, goodness of fit measures how far an observation is from its predicted value (e.g., $|y_t-\hat{y}_t|$, $t$=1,$\ldots$, $T$). At the model level, it
summarizes the overall fit of a model to all observations (e.g., the
average squared errors). Fifth, one ought to assess the model's predictive
accuracy, or how well the model predicts an outcome for an observation that was not used to fit the model (e.g., via cross-validation). With time-series models, including SSMs, one can use the first $t$ observations
to fit the model, and then use the model to predict the $t+1$ observation, or fit the model to all $T$ observations
and see how well future observations are predicted.
\subsection{Challenges with SSMs}
For simple statistical
models, such as linear regression, diagnostics
for most of the above features are well established. Diagnostics for SSMs, however, can be challenging for two reasons. First, observations are temporally dependent.
Many diagnoses rely on response or conventional residuals, which we define as follows for our toy model:
\begin{linenomath*}
\begin{align}
\label{E.naive.SSM.residual}
e_{t|1:T} &= y_t - \hat{y}_{t|1:T},
\end{align}
\end{linenomath*}
where $\hat{y}_{t|1:T}$ is the predicted observation at time $t$ given all observations. This predicted value depends on the smoothed state estimate at time $t$, $\hat{z}_{t|1:T}$, and the observation equation. For example, for our toy model (Eqs. \ref{E.state.NDLM}-\ref{E.obs.NDLM}), $\hat{y}_{t|1:T} = E[y_t|y_{1:T}] = \alpha\hat{z}_{t|1:T}$. \citet{Harvey-1990}
notes that these response residuals are not serially
independent. Their use can
impair one's capacity to identify model misspecification
\citep{Harvey-1990}, and can have negative consequences for model inference and further model diagnosis \citep[e.g., inflated goodness of fit, ][]{Thygesen-etal-2017}.
Second, as for most hierarchical
models, we generally do
not have direct observations of the hidden states, $z_t$, thus
one cannot directly compare predicted states with their ``true'' values.
Because of these challenges, researchers often fail to check the absolute
fit of SSMs, and thus risk making conclusions based on
a misspecified model or risk having biased parameter and state estimates. Here, we
provide a list of tools to help researchers perform this essential
model-checking step. We start with the tools commonly used to assess Bayesian hierarchical models. These tools can be easily used with frequentist and Bayesian SSMs alike, but have important limitations. We then discuss the tools that have been the focus of model validation developments for SSMs, which specifically address the issue of temporal dependence in the residuals. We end with methods relying on out-of-sample validation (e.g., cross-validation), which we believe is the gold standard for assessing the predictive ability of a model, and we hope will become the focus of future developments for SSMs. This order also reflects an increased division between the data used to estimate the model parameters and hidden states and the data used to perform the diagnostics.
\subsection{Posterior predictive measures}
Posterior predictive checking is a common Bayesian method to quantify the discrepancies between the data and the model \citep{Gelman-etal-2013, Conn-etal-2018}. It has been used to verify the fit of SSMs to ecological data \citep[e.g.,][]{Hobbs-etal-2015}. The idea behind posterior predictive checking is that if the model fits the data well, then data generated from the model should have characteristics similar to those of the observed data \citep{Gelman-etal-2013}. These posterior predictive checks often involve calculating a posterior predictive p-value, $p_B$:
\begin{linenomath*}
\begin{equation}
p_B = Pr(T(\mathbf{y}^i, \boldsymbol{\theta}) \geq T(\mathbf{y}, \boldsymbol{\theta}) |\mathbf{y}),
\end{equation}
\end{linenomath*}
where each $\mathbf{y}^i$
is a time series that has been simulated from the fitted model (i.e., representing a replicate time series that could have been observed from the model), $\mathbf{y}$ is the observed data, $\boldsymbol{\theta}$ contains the model parameters, and $T(\mathbf{y}, \boldsymbol{\theta})$
is a test quantity summarizing the data (e.g., the mean) or a discrepancy function (e.g., $\chi^2$
measure). This p-value is similar to the one used in frequentist inference. It measures the probability, under the model of interest, of finding a test quantity as extreme as that associated with the data. Posterior predictive checks use three steps: 1) sample a set of posterior $\boldsymbol{\theta}$ values, 2) simulate one $\mathbf{y}^i$ from each, and 3) calculate the test quantity for each $\mathbf{y}^i$. We estimate the p-value with the proportion of the replicates that have a test quantity value greater or equal to that of the real data. Posterior predictive p-values near 0 or 1 indicate that the pattern observed with the data would be unlikely if the model were true. Thus, unlike p-values associated with classic statistical tests used to reject null hypotheses (e.g., t-test), we are seeking a posterior predictive p-value close to 0.5 not smaller than 0.05. The relevance of the p-value largely depends on the choice of test quantity. \citet{Hobbs-etal-2015} used the mean and standard deviation of the observed data, as well as a discrepancy function ($T(\mathbf{y,\boldsymbol{\theta}}) = \sum_{t=1}^T (y_t - \hat{y}_{t|1:T})^2 = \sum_{t=1}^T e_{t|1:T}^2$) that measures the disagreement between the SSM and the data. \citet{Newman-etal-2014} and \citet{Conn-etal-2018} provide lists of important alternative functions. Although we described posterior predictive checks in a Bayesian framework and have defined the test quantity as a function of $\mathbf{y}$, \citet{King-etal-2015} have applied similar concepts in a frequentist framework, using test quantities that describe characteristics of the estimated hidden states, $\mathbf{z}$ (e.g., autocorrelation function at lag 1 of the states).
Although common, posterior predictive p-values have important limitations \citep{Conn-etal-2018}. Because they use the data twice, once to fit the model and once to test the model fit, they tend to be conservative (i.e., tend to return value closer to 0.5 than
to 0 or 1), and often have insufficient power to detect lack of fit. One can alter the method described above and generate all the observation replicates using only a single sample from the posterior parameter distribution. This method was shown to have better theoretical properties (e.g., better Type I error rate control), and to detect lack of fit more reliably for some ecological hierarchical models \citep{Conn-etal-2018}. Following \citet{King-etal-2015}, we recommend assessing discrepancies between the SSM and the data by looking at where $T(\mathbf{y},\boldsymbol{\theta})$ falls in the frequency distribution of $T(\mathbf{y}^i,\boldsymbol{\theta})$ (Fig. \ref{fig:diag}a,d). This graphical method is also more useful in assessing the ecological importance of the discrepancies than looking at the p-value, and can provide a better sense of why the model may be inappropriate for the data \citep{Conn-etal-2018}.
Posterior predictive checks are also useful to assess the validity of model assumptions \citep{Gelman-etal-2013}. We can use a single sample from the posterior distribution of the hidden states to assess the assumptions associated with the process equation \citep{Thygesen-etal-2017}. For example, we can sample a time series of state, $\mathbf{z}^i$, from the posterior state distribution of our toy model to calculate the process variation as $\epsilon_t^i = z_t^i - \beta z_{t-1}^i$, and verify whether the $\epsilon_t^i$ are normally distributed with a mean of 0 as assumed by Eq. \ref{E.state.NDLM}. Departures from the assumed distribution (e.g., if the mean of the process variation is far from 0), indicate that the model is not adequate for the data. This method is generally recommended for assessing the assumptions of Bayesian hierarchical models \citep{Gelman-etal-2013}, but \citet{Thygesen-etal-2017} used the Laplace approximation implemented in \texttt{TMB} to create a posterior distribution of the states for non-Bayesian models.
\subsection{One-step-ahead residuals and their extensions}
\label{S.One.step.ahead}
The model diagnostic that has received the most attention in the SSM literature is the one-step-ahead residuals \citep{Harvey-1990, Thygesen-etal-2017}, also known
as recursive residuals \citep{Fruhwirth-Schnatter-1996}. Unlike the response residuals
(Eq. \ref{E.naive.SSM.residual}), the one-step-ahead residuals should not have temporal dependence when the model is adequate because the residual
for the $t^{th}$ observation uses the expected observation at time $t$ given
observations only up to time $t-1$:
\begin{linenomath*}
\begin{align}
\label{E.one.step.ahead.residual}
e_{t|1:t-1} &= y_t - \hat{y}_{t|1:t-1}.
\end{align}
\end{linenomath*}
Effectively, for response residuals (Eq. \ref{E.naive.SSM.residual}) we use the smoothed estimates of states, $\hat{z}_{t|1:T}$, to predict the observation at time $t$, while for one-step-ahead residuals, we use the prediction of the states, $\hat{z}_{t|1:t-1}$. In the context of a Kalman filter, we can calculate $\hat{y}_{t|1:t-1}$ using the one-step-ahead forecast prediction that is already calculated as part of
the recursive algorithm. As more information is available for fitting the model as $t$ increases, the variance of prediction residuals will tend to decrease with $t$. To account for this change in variance, it is useful to scale
the prediction residuals by their standard deviations (a procedure equivalent to calculating standardized Pearson residuals):
\begin{linenomath*}
\begin{align}
\tilde{e}_{t|1:t-1} =
\frac{e_{t|1:t-1}}{sd(e_{t|1:t-1})}.
\end{align}
\end{linenomath*}
For the special case of SSMs with normally
distributed states and observations, such standardized residuals are independent
and identically distributed with a standard normal distribution and can be used to test a variety of assumptions. Diagnostic procedures include
qq-normal plots to check for normality, auto-correlation function plots
to see if the residuals are independent, and plots of the
residuals against observed values to check for non-constant
variance.
For non-normal SSMs, the probability distribution of these standardized residuals are not standard normal, making
the exploration of residuals harder. Probability scores (P-scores), and their transformed version, prediction quantile residuals, are useful alternatives \citep{Fruhwirth-Schnatter-1996, Thygesen-etal-2017}. A P-score, $u_t$, is the
cumulative distribution function for the predicted
observations evaluated at the $t^{th}$ observed value:
\begin{linenomath*}
\begin{align}
u_t = F_{Y_t|y_{1:t-1}}(y_t) &= \text{Pr}(Y_t \leq y_t | Y_{1:t-1} = y_{1:t-1}).
\end{align}
\end{linenomath*}
If $F_{Y_t|y_{1:t-1}}$ describes the cumulative distribution function of the true model, then the resulting
$u_t$ are uniformly distributed \citep{Conn-etal-2018}. Deviations from uniformity suggest model misspecification. As this is simply an
application of the probability integral transformation \citep[i.e.,
if $Y$ has the cumulative density function $F_Y$, then $F_Y(Y)$ is distributed with Uniform(0,1),][]{Smith-1985}, these are a specific
case of probability integral transform (PIT) residuals \citep{Warton-etal-2017}. To get normally distributed residuals, we can transform the P-scores to
prediction quantile residuals, $v_t$,
as follows:
\begin{linenomath*}
\begin{equation}
v_t = \Phi^{-1}(u_t),
\end{equation}
\end{linenomath*}
where $\Phi^{-1}$ is the inverse of the standard normal
cumulative distribution function (also known as the standard normal
quantile function). When the model is true,
$v_t$ should
be an independent sample from a standard normal. Thus, we can assess whether the data fits the
model assumptions using the same
diagnostic procedures available for standardized one-step-ahead
prediction residuals in the case of normally distributed SSMs \citep[see Fig. \ref{fig:diag}b-c,e-f; and][]{Newman-etal-2014,Thygesen-etal-2017}.
P-scores and prediction quantile residuals can be difficult to estimate for non-normal SSMs because their calculation requires knowledge of the cumulative distribution function (cdf) for $Y_t|y_{1:t-1}$, which
in many cases will not be known nor have an analytical
form. However, \citet{Thygesen-etal-2017} developed methods for approximating the cdf based on the Laplace approximation that can be implemented easily in \texttt{TMB}. Because this method depends on the Laplace approximation, it is important to assess the accuracy of this approximation (see Appendix S1: Section S1 1.2.3). The quantile residuals of \citet{Thygesen-etal-2017} are applicable to a broad range of frequentist SSMs, although there are some limitations in using them with multivariate time series. We are not aware of equivalent methods for as broad a range of Bayesian SSMs, although some exist for a limited class \citep{Fruhwirth-Schnatter-1996, Gamerman-etal-2013}.
\subsection{Cross-validation}
While one-step-ahead residuals and their extensions remove data when calculating the expected value of the observation at time $t$, they use the complete dataset to estimate the model parameters. Thus, these residuals cannot be used to fully assess the predictive ability of the model. Assessing the predictive ability of a model is thought to be best achieved with out-of-sample data, where two independent datasets are used: one to fit (or train) the model and one to validate (or test or evaluate) it \citep{Hooten-Hobbs-2015}. While rarely done with SSMs, there are examples where independent information on the true values of the hidden states was collected \citep[e.g., ][and Appendix S1: Section S1 2.2]{AugerMethe-etal-2017}, a data stream was used as validation data \citep[e.g.,][]{Hobbs-etal-2015}, or part of a time series was selected as a validation time period \citep[e.g.,][]{Holdo-etal-2009}.
When using a single subset of the data as validation, we can only assess the predictive ability for those specific observations. Instead, one can use cross-validation methods that look at the predictive ability of all data points by sequentially leaving out small subsets of the data \citep{Hooten-Hobbs-2015}. $k$-fold cross validation is a ubiquitous statistical method, where $k$ groups of similar size sequentially serve as the validation dataset while the remaining $k-1$ groups are collectively used as the training set. Leave-one-out is a common version that
leaves each of the data points out sequentially. To assess the predictive ability of the model, we can use score or discrepancy
functions, such as the mean squared prediction error (MSPE) for group $k$:
\begin{linenomath*}
\begin{align}
\text{MSPE}_k &= \sum_{i=1}^{n} (\mathbf{y}_{i,k,\text{oos}}- \hat{\mathbf{y}}_{i,k,\text{oos}})^2/n,
\end{align}
\end{linenomath*}
where we assume that $T/k$ is an integer $n$, oos means out of sample, $\mathbf{y}_{i,k,\text{oos}}$ is the $i^{th}$ observation in subsample $k$, and $\hat{\mathbf{y}}_{i,k,\text{oos}}$ is the expected observation based on the model fitted to the dataset without sample $k$. As an overall value, we can then average the $k$ $\text{MSPE}_k$. Such functions directly assess the predictive ability of the model and thus are intuitive measures of how good a model is.
As mentioned in Section \ref{S.Model.Comparison}, cross-validation is also often deemed the preferred method for model comparison \citep{Gelman-etal-2014, Hooten-Hobbs-2015}. The model set can be ranked based on their predictive accuracy, with better models having lower prediction error \citep[e.g., lower MSPE or its square root, RMSPE,][]{Hooten-Hobbs-2015}. While cross-validation can be implemented relatively easily, it can be computationally demanding \citep{Link-Sauer-2016, Vehtari-etal-2017}. Cross-validation generally requires refitting the models $k$ times, which can be a daunting task with Bayesian models \citep[but see][for suggested solutions]{Hooten-Hobbs-2015}. In addition, cross-validation assumes that the training and evaluation datasets are independent \citep{Roberts-etal-2017}. The main challenge with using cross-validation with SSMs is that, due to the temporal dependency in the data, removing only a few data points will underestimate the prediction error and removing many will lead to propagation of error \citep{Newman-etal-2014}.
Despite these drawbacks cross-validation is a powerful tool, which has been promoted for use with complex ecological models \citep{Link-etal-2017}. At present, there are few cross-validation methods specifically designed to handle the dependency structure of SSMs \citep{Ansley-Kohn-1987, DeJong-1988}. These are appropriate for only a restricted set of SSMs and appear to be rarely used. However, the time-series literature \citep[e.g.,][]{Tashman-2000, Bergmeir-Benitez-2012, Burkner-etal-2020} and the suggestions of \citet{Roberts-etal-2017} on block cross-validation methods to account for dependence structure in ecological data are useful starting points for the development and evaluation of such methods for SSMs. Cross-validation methods for time series include procedures analogous to the one-step-ahead residuals (Section \ref{S.One.step.ahead}), but where model parameters differ across the $k$ folds and are estimated using only observations prior to the expected values \citep{Hyndman-Athanasopoulos-2018}. One may need to consider additional modifications, such as whether one should use a rolling window for the training dataset \citep{Tashman-2000}.
The topic of model validation for SSM is one that has been relatively poorly studied, with a few notable exceptions \citep[e.g.,][]{Fruhwirth-Schnatter-1996,King-etal-2015, Thygesen-etal-2017}. Because of the additional parameter identifiability and estimability problems discussed in Section \ref{S.Estimability}, we believe this topic deserves more attention. Beyond the tools we have outlined above, SSM developers and users can gain inspiration from the tools developed for hierarchical models \citep[e.g., PIT-trap residuals,][]{Warton-etal-2017}. For researchers using Bayesian SSMs, we point readers towards the review of \citet{Conn-etal-2018} on model checking methods for Bayesian hierarchical models. Finally, we would like to remind readers that, while it is crucial to perform a model validation step, passing this step does not mean that the model is representing the truth. It simply means that one could not find difference between the data generating system and the model. This could be due to a low sample size or the conservative nature of some of the methods described above.
\section{\label{S.Conclusion}Conclusion}
Through a diverse set of examples, we have demonstrated that SSMs are flexible models for time series that can be used to
answer a broad range of ecological questions. They can be used to model univariate or multivariate time series. SSMs can be linear or nonlinear, and have discrete or continuous time steps. They can have normal or non-normal sources of stochasticity, and thus can model continuous, count, binary, or categorical data. They are particularly useful when one
has significant process variation and observation error. Accounting for these sources of uncertainty can substantially affect management decisions, making SSMs the perfect modeling tool in many contexts \citep[e.g.,][]{Jamieson-Brooks-2004, Hobbs-etal-2015}.
As we have outlined, a variety of tools to fit SSMs to data exist. Historically, many researchers wrote SSMs so they could be fitted with the Kalman filter and its extensions. However, the diversity of fitting procedures available now allows researchers to create models that are more representative of the structure of their data and the ecological processes they are interested in. In addition, flexible fitting tools now exist in both the frequentist and Bayesian frameworks, allowing researchers to choose their preferred inferential framework rather than have their model dictate the framework they can use. Within each inferential framework, the choice of a fitting procedure will be a compromise between flexibility and efficiency. In particular, highly efficient fitting methods (e.g., Laplace approximation and Hamiltonian Monte Carlo) have more restrictions than their slower alternatives (e.g., particle filter and Gibbs).
While these tools provide the means to fit complex SSMs, it is crucial to appropriately formulate the model. As discussed, SSMs can suffer from parameter estimability problems, but various tools exist to assess whether this is the case and to identify the type of study design or model simplification that will resolve these problems. In general, making use of replication or including covariates can help reduce some of the common estimation problems.
Researchers often forgo doing model selection and validation with SSMs, but we advocate that these should become part of every SSM user's workflow. Model mispecification can affect ecological inferences and the accuracy of state estimates. While no model selection measure is perfect for SSMs, AIC and WAIC, can be useful. While model validation is also difficult with SSMs, posterior predictive measures, and one-step-ahead residuals and their extensions are relatively easy ways to assess whether the model describes the data well and whether some of the model assumptions are met. Cross-validation methods are often computationally expensive, but provides one of the best ways to select and evaluate models when correlation is handled appropriately.
While there are many tools already available to fit, compare, and validate SSMs, five topics warrant further research. First, while we advocate that SSMs be a default framework to model many ecological time series, it is important to pinpoint the conditions under which simpler alternatives perform adequately (e.g., when do models without observation error provide reliable parameter and state estimates?). Such research should account for the additional identifiability and estimability issues that comes with fitting SSMs and the types of datasets that allow SSMs to return reliable estimates. Second, as SSMs are often the primary tools used to analyze time series, it is important to explore the data collection designs that optimize the estimation of SSMs, so that the best data possible are collected. Third, there is a need for further developments of computationally efficient model selection procedures for SSMs. Using the marginal likelihood with AIC and WAIC appears most adequate for SSMs, especially if one has a single observation time series. However, we should explore when the conditional likelihood can be used and whether it affects the predictive accuracy of the states and parameters differently. To facilitate the uptake of WAIC based on the marginal likelihood, new \texttt{R} functions that automatically calculate this information criterion should be written. In addition, we it would be helpful to explore how newer tools to approximate predictive ability \citep[e.g.,][]{Vehtari-etal-2017, Burkner-etal-2020} perform with SSMs. Fourth, while there have been a few important advances in model validation methods for SSMs, this remains a relatively untouched research area. Given the increasing use of SSMs in management, it is crucial that a broader range of validation methods be developed for these complex models. Fifth, with the increasing efficiency of fitting procedures, cross-validation is becoming a feasible procedure to assess predictive accuracy and compare models. As such, the time is ripe to start developing proper cross-validation procedures that will account for dependencies in the data.
Overall, we provided a review of the topics needed to formulate and fit SSMs to ecological data, and Appendix S1 provide an extensive set of examples of methods to facilitate this process. We hope this guide will help researchers develop and apply SSMs to
their own data, and foster the development of SSMs in multiple fields of ecology.
\section{Acknowledgments}
This paper was instigated during a Banff International Research Station (BIRS) workshop hosted at the Casa Matem\'atica Oaxaca (CMO) entitled \textit{New perspectives on state-space models}. We thank BIRS and CMO for their support and the lead organizer of the workshop, David Campbell, and all participants for their insights. This effort was also supported by the Canadian Statistical Sciences Institute through a Collaborative Research Team Project led by JMF. MAM thanks the Natural Sciences and Engineering Research Council of Canada and the Canada Research Chairs Program. We thank Andrew Derocher for the polar bear track used in Fig. \ref{fig:polarbear} and Appendix S1. We also thank Devin Lyons, Perry de Valpine, Aki Vehtari, and three anonymous reviewers for their insightful comments on previous versions of the manuscript.
\bibliographystyle{apalike}
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{
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\section{Introduction and main results}
Let $(M,g)$ be a compact Riemannian manifold (connected and $\partial M=\emptyset$) and $p:\tilde M\to M$ its universal
Riemannian covering, saving $\pi:TM\to M$ for the canonical projection. In \cite{Ma}, Manning introduced the volume
entropy (also called volume growth) $h(g)$ of $(M,g)$ defined by
$$ h(g):=\lim _{r \rightarrow +\infty }\frac{1}{r}\log \vol B(p,r),$$
where $p\in \tilde M$ and $B(p,r)$ denotes the open ball with center $p$ and
radius $r$. He proved that this limit exists and is independent of
$p$. Let $\h(\phi ^t)=\h(\phi^t_{SM})$ denote the topological entropy of the
geodesic flow $\phi ^t$ on the unit tangent bundle $SM$. Manning
proved the following estimate:
\[ \h(\phi^t_{SM}) \geq h(g) . \]
In the case of nonpositive curvature he showed that
equality holds. Subsequently this was generalized by Freire and
Ma\~n\'e \cite{FM} to metrics without conjugate points. Let $\tilde \M$ be the closed and $\phi^t$-invarint subset of $S \tilde M$ consisting of all $v\in S\tilde M$ such that
the geodesic $ c_v $ with $\dot c_v(0) = v$ is
globally minimizing. We denote by $\M=Dp(\tilde \M)$ the projection of $\tilde \M$ to $SM$ and by $ \phi ^t_\M,\phi ^t_{\tilde \M}$ the geodesic flow restricted to $\M,\tilde \M$, respectively. In
\cite{KH} Katok and Hasselblatt stated the following theorem, saying that it is enough to consider minimal geodesics to generate exponential complexity (provided $h(g)>0$).
\begin{theorem}
Let $(M,g)$ be a compact Riemannian manifold and $ \phi^t_\M$ be the geodesic flow $\phi ^t$ restricted to the minimal geodesics $\M\subset SM$. Then
$$
\h(\phi ^t_\M) \geq h(g).
$$
\end{theorem}
Following Klingenberg \cite{Kl} we call a compact manifold $M$ to be of hyperbolic type, if there exists
a metric of strictly negative curvature $g_0$ on $M$.
We hope to prove an inequality of the kind $\h(\phi ^t_\M) \leq h(g)$, i.e. that equality holds in the above theorem. A first result in this direction is the following. We will introduce the notation $h_{top}( \phi ^t,F, \beta)$ and the notion of entropy expansiveness in section \ref{def h_top}.
\begin{theorem}
Let $(M,g)$ be a compact Riemannian manifold of hyperbolic type.
There is some constant $\beta$ depending only on $(M,g)$ such that for each compact set
$K \subset \tilde M$ we have
$$ h_{top}( \phi ^t, \pi^{-1}(K) \cap \tilde\M, \beta) \leq h(g). $$
\end{theorem}
Using a result of Bowen \cite{Bo}, which we shall prove below in the non-compact setting, we obtain the following.
\begin{cor} \label{h=h(g) expansive}
Let $(M,g)$ be a compact Riemannian manifold of hyperbolic type. If $\phi ^t_{\tilde \M}$ is $\beta$-entropy-expansive for $\beta$ from the above theorem, we have
$$ \h(\phi ^t_\M) = h(g).$$
\end{cor}
Presently we do not know if $\phi^t_{\tilde \M}$ for Riemannian manifolds $(M,g)$ of hyperbolic type of arbitrary dimension is $\beta$-entropy-expansive. We shall prove, however, that in the two-dimensional case, $\beta$-entropy-expansiveness holds in the non-wandering set of $\M$. This gives the following result.
\begin{theorem}
Let $(M,g)$ be closed Riemannian surface. Then
$$ \h( \phi ^t_\M) = h(g). $$
\end{theorem}
This paper is organized as follows. In the second section we study topological entropy and local topological entropy
for homeomorphisms of metric spaces and following the ideas of Bowen \cite{Bo} we provide an estimate for the topological entropy.
In section 3, we give a complete proof using the ideas provided by Katok and Hasselblatt
that the topological entropy of the minimal geodesics is bounded
below by the volume growth (theorem 1.1). Moreover, we study
topological entropy of minimal geodesics on manifolds of
hyperbolic type and give the proof of theorem 1.2. Finally, in section 4 we show that for surfaces the topological entropy of $\phi^t_\M$ equals the
volume growth of $g$ (theorem 1.4).
\section{Topological Entropy for homeomorphisms of metric spaces}
In this section we study discrete dynamical systems. In order to apply our results to geodesic flows $\phi^t, t\in \R,$ observe that the topological entropy of $\phi^t$ defined in the continuous setting coincides with that of the discrete system $\phi^n, n\in\Z$, cf. \cite{KH}.
\subsection{Bowen's definition}\label{def h_top}
Here we recall Bowen's definition of topological entropy. Let $f : V \to V$ be a homeomorphism of a not necessarily compact metric space $(V,d)$. For each $n\in \N$, a
metric on $V$ is defined by
$$
d_n(x,y) := \max_{0 \le j <n} d(f^j(x), f^j(y)) .
$$
Let $F$ be a subset of $V$. We say that a set $Y\subset V$ is $(n,\e)$-spanning for $F$ if the closed balls
$\bar B_n(y,\e) = \{y\in V: d_n(x, y) \le \e \}, y\in Y$ cover $F$.
If $Y \subset F$ and $\bar B_n(y,\e)\cap Y = \{y\}$ for all $y\in Y$, we say that $Y$ is an $(n,\e)$-separated subset of $F$.
Let $r_n(F,\e)$ denote the minimal cardinality of $(n,\e)$-spanning sets for $F$ and let $s_n(F,\e)$ denote the maximal cardinality
of $(n,\e)$-separated subsets of $F$. It is easy
to see that for any $\e > 0$ we have
$$
r_n(F,\e ) \leq
s_n(F,\e ) \leq r_n(F,\e /2) .
$$
Note that $r_n(F,\e)<\infty$, if $F$ is compact.
We define the following notions of topological entropy.
\begin{align*}
\h(f,F, \e) &:= \varlimsup_{n\to+\infty}\frac{1}{n}\log r_n(F,\e), \\
\h(f,F) &:=\lim_{\e\to 0}\h (f,F, \e) , \\
\h(f) &:= \sup_{ F \subset V \text{ compact}} \h(f, F).
\end{align*}
Note that for any $\e>0$ we have $\h(f,F, \e) \leq \h(f,F)$ and if $V$ is itself compact, we get $\h(f)=\h(f,V)$. If we use $s_n(F,\e)$ instead of $r_n(F,\e)$, we obtain the same value for $\h(f,F)$. For details on topological entropy we refer to \cite{W}.
We need the following less known
concept of local entropy introduced by Bowen \cite{Bo}.
For $x\in V$ and $\beta >0$ set
$$Z_\beta(x):=\{y\in V : d(f^n(x), f^n(y))\leq \beta \; \forall n\in \mathbb{Z}\}.$$
Then we call
$$
\hloc(f,\beta):=\sup_{x\in V}\h(f, Z_\beta (x))$$
the $\beta$-local entropy of $f$.
We say that $f$ is $\beta$-entropy-expansive for $\beta>0$ if
$$\hloc(f,\beta) =0.$$
\subsection{An upper bound for the topological entropy of homeomorphisms}\label{bowen bounds}
In order to make use of the local entropy it will be important to compute entropy on coverings.
We consider the following setting.
Let $(\tilde V, \tilde d)$ be a metric space and $\Gamma$ a subgroup of isometries of $\tilde V$ acting
on $\tilde V$. Assume that the quotient $V := \tilde V/\Gamma$ is compact and equipped
with a metric $d$ such that the projection $p: \tilde V \to V$ is a local isometry. Let $ \tilde f: \tilde V \to \tilde V$ be a homeomorphism
which commutes with the group $\Gamma$ and let $f: V \to V$ be the projection defined by $f(x)=p\tilde f p^{-1}(x)$ (this is well-defined since $\tilde f,\Gamma$ commute). $f$ is a homemorphism as well. Recall the following result.
\begin{prop}[theorem 8.12 in \cite{W}] \label{walters}
For each compact set $K \subset \tilde V$ we have
$$
h_{top}(\tilde f, K) = h_{top}( f, p(K))
$$
In particular, if $p(K)=V$, then
\[ h_{top}(\tilde f, K) = h_{top}( f). \]
\end{prop}
We shall prove the following theorem which is a slight extension of a result of Bowen
(see \cite{Bo}). It allows to estimate the topological entropy using coverings and will be crucial for our applications.
\begin{theorem}\label{bowen expansive}
Let $K \subset \tilde V$ be a compact set such that $p(K) = V$. Then for any
$\beta >0$ we have
$$
\h( f) \le \h( \tilde f, K, \beta) + \hloc( \tilde f, \beta).
$$
\end{theorem}
The proof of \ref{bowen expansive} rests of the following estimate.
\begin{lemma} \label{lemma bowen}
Let $a = \hloc( \tilde f, \beta)$. For any $\e >0, \delta>0, \beta>0$ there exists a constant $c>0$, s.th.
\[ r_n\left(\bar B_n(x,\beta) ,\delta \right) \leq c e^{(a+\e)n} \quad \forall ~ x\in K, n\in\N. \]
\end{lemma}
We need the following elementary lemma (see \cite{Bo}).
\begin{lemma}\label{lemmabowen1}
Let $F\subset \tilde V$ and consider integers $0 = t_0 < t_1 < ... < t_r =n$. For $\al>0$ and $0\leq i <r$ let $E_i $ be a $(t_{i+1} -t_i, \al)$-spanning set
for $\tilde f^{t_i}(F)$. Then
\[ r_n(F, 2 \alpha) \leq \prod_{i=0}^{r-1} \# E_i . \]
\end{lemma}
\begin{proof}[Proof of \ref{lemmabowen1}]
For $(x_0, \ldots, x_{r-1}) \in E_0 \times \cdots \times E_{r-1}$ set
\begin{align*}
& B(x_0, \ldots, x_{r-1}) \\
& := \{ x \in F \mid ~ d(\tilde f^{t+t_i}(x), \tilde f^t(x_i)) \le \alpha ~~ \forall ~ 0 \le i <r, t \in [0, t_{i+1} -t_i]\cap\Z \}.
\end{align*}
By assumption the $B(x_0,...,x_{r-1})$ cover $F$ and using the triangle inequality we have $d_n(x,y) \le 2 \alpha $ for all $x,y \in B(x_0, ..., x_{r-1})$. Choosing from each nonempty set $B(x_0, \ldots, x_{k-1})$ one element we obtain a $(n,2 \alpha)$-spanning set. This yields the estimate.
\end{proof}
\begin{proof}[Proof of \ref{lemma bowen}]
In the following fix positive numbers $\e,\delta,\beta>0$, a point $x\in K$, an integer $n\in\N$ and set $F:= \bar B_n(x,\beta)$. We shall try to describe the orbit $\{x, \tilde f x, ... , \tilde f^{n-1} x\}$ by a finite collection of $y$'s in $K$ and their sets $Z_\beta(y)$.
\underline{Step 1.} (choice of $y_1,...,y_s\in K$ and appropriate neighborhoods $V(y_i)$) By definition of $a$ we find for all $y\in K$ some integer $m(y)\in\N$ and a $(m(y),\delta/2)$-spanning set $E(y)$ for $Z_\beta(y)$ with
\[ \frac{1}{m(y)} \log \# E(y) \leq a+\e . \]
Define the open neighborhoods
\[ U(y) := \bigcup_{z\in E(y)} B_{m(y)}(z,\delta/2) ~\supset ~ Z_\beta(y), \quad y\in K .\]
For $N\to\infty, R \searrow \beta$ the compact sets
\[ W_N(y,R) := \bigcap_{|j|\leq N} \tilde f^{-j} \bar B(\tilde f^jy, R) \]
decrease to the compact set $Z_\beta(y)$, so we find $N(y)\in\N,R(y)>\beta$, s.th. $W_{N(y)}(y,R(y))$ is contained in the neighborhood $U(y)$ of $Z_\beta(y)$. Define
\[V(y) := \Int W_{N(y)}(y,R(y)-\beta) , \quad y\in K. \]
The triangle inequality implies that
\[ (*) \qquad \forall z\in V(y): \quad W_{N(y)}(z,\beta) \subset W_{N(y)}(y,R(y)) \subset U(y). \]
By the compactness of $K$ we find $y_1,...,y_s\in K$ with
\[ \tilde V = \bigcup_{\gamma\in\Gamma}\bigcup_{i=1}^s \gamma V(y_i). \]
Set
\[n_0 := \max_{1\leq i\leq s}\max \{ N(y_i),m(y_i) \} \in \N. \]
\underline{Step 2.} (describtion of $F$ by the $y_i$'s) We claim the following:
\[ (**) \qquad \forall t\in [n_0,n-n_0)\cap \Z ~~\exists i\in\{1,...,s\}, \gamma\in\Gamma: \quad \tilde f^t (F) \subset \gamma U(y_i). \]
{\it Proof of the claim}. We find $\gamma,i$ with $\tilde f^tx\in \gamma V(y_i)$, and hence
\begin{align*}
\tilde f^t(F) & = \bigcap_{j=0}^{n-1} \tilde f^{t-j} \bar B(\tilde f^jx,\beta) = \bigcap_{j=-t}^{n-t-1} \tilde f^{-j} \bar B(\tilde f^j\tilde f^tx,\beta) \subset \bigcap_{j=-n_0}^{n_0} \tilde f^{-j} \bar B(\tilde f^j\tilde f^tx,\beta) \\
& = W_{n_0}(\tilde f^tx,\beta) = \gamma W_{n_0}(\gamma^{-1}\tilde f^tx,\beta) \subset \gamma U(y_i),
\end{align*}
where in the second line we used $\Gamma\subset \Iso(\tilde V,d)$, $[\tilde f,\Gamma]=0$ and $(*) , n_0 \geq N(y_i)$.
\underline{Step 3.} (application of lemma \ref{lemmabowen1}) As a consequence of $(**)$, the set $\gamma E(y_i)$ is $(m(y_i),\delta/2)$-spanning for $\tilde f^t (F)$. We want to apply \ref{lemmabowen1}, so we define integers $0=t_0 < ... < t_r = n$ as follows.
\begin{enumerate}
\item If $n\leq n_0$ take $r=1$ and $t_1=n$.
\item If $n>n_0$, take $t_1=n_0$ and choose $i_1\in\{ 1,...,s \}, \gamma_1 \in \Gamma$ with $\tilde f^{t_1}(x)\in \gamma_1V(y_{i_1})$. Suppose now we have already choosen $t_1, ... , t_k$ with $t_k<n$ together with $i_1,...,i_k, \gamma_1,...,\gamma_k$.
\begin{enumerate}
\item If $t_k \geq n-n_0$, set $r=k+1$ and $t_r=n$.
\item If $t_k < n-n_0$, set $t_{k+1}=t_k+m(y_{i_k})$ and choose $i_{k+1},\gamma_{k+1}$ with $\tilde f^{t_{i_{k+1}}}(x)\in \gamma_{k+1}V(y_{i_{k+1}})$.
\end{enumerate}
Eventually we are in case (a) and the process stops. Moreover we have $t_{r-2} <n-n_0 \leq t_{r-1} < n = t_r$ by $m(y_{i_{r-2}})\leq n_0$.
\end{enumerate}
Note that $t_{k+1}-t_k\leq n_0$ for $k=0,r-1$ and by $(**)$ the set $\gamma_k E(y_{i_k})$ is $(t_{k+1}-t_k, \delta/2)$-spanning for $\tilde f^{t_k}(F)$ for $k=1,...,r-2$. Choose $E_0,E_{r-1}$ to be $(n_0,\delta/2)$-spanning for $\bar B(x,\beta), \bar B(\tilde f^{t_{r-1}}x,\beta)$, respectively of minimal cardinality, so $E_0$ is also $(t_1-t_0,\delta/2)$-spanning for $F$ and $E_{r-1}$ is also $(t_r-t_{r-1},\delta/2)$-spanning for $\tilde f^{t_{r-1}}(F)$. Apply \ref{lemmabowen1} to
\[ E_0, ~ E_1 := \gamma_1 E(y_{i_1}) , ~..., ~ E_{r-2} := \gamma_{r-2} E(y_{i_{r-2}}), ~ E_{r-1} \]
and define
\[ \sqrt c := \sup_{y\in K} r_{n_0}(\bar B(y,\beta), \delta/2 ) ~~<\infty. \]
We obtain using the definition of $m(y_i)$ and $\sum_{k=1}^{r-2} m(y_{i_k}) \leq n-n_0 \leq n$ that
\begin{align*}
r_n(F,\delta) &\leq \# E_0 \cdot \left( \prod_{k=1}^{r-2} \# E_k \right) \cdot \# E_{r-1} \leq c \cdot \prod_{k=1}^{r-2} \# E(y_{i_k}) \\
&\leq c \cdot \prod_{k=1}^{r-2} e^{(a+\e)m(y_{i_k})} \leq c \cdot e^{(a+\e)n}.
\end{align*}
Observe that $c$ depends only on $\delta, n_0, \beta$ and $n_0$ in turn is indepenent of $x,n$.
\end{proof}
Now we are able to prove the theorem.
\begin{proof}[Proof of \ref{bowen expansive}]
Let $E_n$ be a minimal $(n,\beta)$-spanning set for $K$ and let $\e,\delta >0$. Then
\[ K \subset \bigcup_{x\in E_n} \bar B_n(x,\beta) \]
and by \ref{lemma bowen} each of the sets in the above union can be $(n,\delta)$-spanned by using only $c e^{(a+\e)n}$ elements where $a=\hloc(\tilde f, \beta)$. Hence
\[ r_n(K,\delta) \leq \# E_n \cdot c e^{(a+\e)n} \leq r_n(K,\beta) \cdot c e^{(a+\e)n} \]
and
\[ \h(\tilde f, K, \delta)\leq \h(\tilde f, K, \beta)+ a+\e. \]
Letting $\e,\delta\to 0$, the claim follows using \ref{walters}.
\end{proof}
\section{Bounds for topological entropy }
\subsection{Lower Bound}
We need the following theorem stated in the book \cite{KH} of Katok and Hasselblatt on the topological entropy of minimal geodesics on Riemannian manifolds. For the convenience of the reader we will provide here a complete proof of the result, which differs from the one in \cite{KH} in small details. Recall the notation $p:\tilde M \to M$ for the universal cover of $M$ and
\begin{align*}
\tilde \M & = \{ v \in S\tilde M \mid c_v \mbox{ is a minimizing geodesic } \} \subset S\tilde M , \\
\M & = Dp(\tilde \M)\subset SM .
\end{align*}
\begin{theorem}\label{katok hasselblatt}
Let $(M,g)$ be a compact Riemannian manifold and $ \phi^t_\M$ be the geodesic flow $\phi ^t$ restricted to $\M\subset SM$. Then
$$
\h(\phi ^t_\M) \geq h(g).
$$
\end{theorem}
For the proof of \ref{katok hasselblatt} we need a lemma similar to lemma \ref{lemmabowen1}. Recall that $s_T(A, \delta)$ denotes the maximal cardinality of a $(T,\delta)$-separated subset of $A$.
\begin{lemma}\label{lemmabowen2}
Let $(V,d)$ be a metric space, $\phi^t : V \to V$ a continuous flow and $A\subset V$. For times $0 = t_0 < t_1 < \cdots < t_m = T$ and $\delta > 0$ we have
\begin{eqnarray*}
\prod\limits_{i = 1}^m s_{t_i - t_{i-1}} (\phi^{t_{i-1}} A,
\delta) \geq s_T (A, 2 \delta),
\end{eqnarray*}
\end{lemma}
\begin{proof}[Proof of \ref{lemmabowen2}]
Let $L$ be a maximal $(T, 2\delta)$-separated subset of $A$ and let $L_i$ be maximal $(t_i-t_{i-1}, \delta)$-separated subsets of $\phi^{t_{i-1}} (A)$ for $i=1,...,m$ . For $(x_1, \ldots, x_m) \in L_1 \times \cdots \times L_m$ set
\begin{align*}
& B(x_1, \ldots, x_m) := \\
& \{z \in L \mid ~ d (\phi^{t+ t_{i-1}}z, f^t x_i) \leq \delta ~~ \forall 1\leq i \leq m, t\in [0,t_i - t_{i-1}]\}.
\end{align*}
Since $L$ is $(T,2\delta)$-separated, the triangle inequality implies $\# B(x_1, \ldots, x_m) \leq 1$. Therefore, since the cardinalities of the $L_i$ are maximal implying that they are also $(t_i-t_{i-1}, \delta)$-spanning,
\begin{eqnarray*}
\# L = \# \left (\bigcup_{(x_1, ...., x_m) } B(x_1, ..., x_m) \right ) \leq \prod\limits_{i=1}^m \# L_i.
\end{eqnarray*}
\end{proof}
\begin{proof}[Proof of \ref{katok hasselblatt}]
Fix $x \in \tilde M$, $\e>0$ and write
\[ \delta := \inj(M) >0, \quad h := h(g), \quad a := \sup_{y\in \tilde M} \vol B(y,2\delta), \quad b :=\h(\phi^t_{SM}). \]
We have the following: there exists a sequence $T_k\to \infty$ such that
\[ \vol B(x,T_k + \delta/2)-\vol B(x,T_k) \geq e^{h(1-\e)T_k}, \]
for otherwise adding up the volume of the annuli $B(x,T_k + \delta(2) \setminus B(x,T_k)$ with $T_{k+1} = T_k + \delta/2$ starting at $T_0$ sufficiently large would yield that the exponential growth rate is less than $h\cdot (1-\e)$.
Let $N_k$ be a maximal $2\delta$-separated set in the annulus $\bar B(x,T_k+ \delta/2) \setminus B(x,T_k)$, then we have for all $k\in \N$
\[ a \cdot \# N_k \geq \vol \left (\bigcup_{y\in N_k} B(y,2\delta)\right ) \geq \vol B(x,T_k + \delta/2)-\vol B(x,T_k) \geq e^{h(1-\e)T_k} .\]
For $y\in N_k$ let $c_y:[0,d(x,y)]\to\tilde M$ be a minimal geodesic segment with $c(0)=x$ and $c(d(x,y))=y$. Now, if $y_1,y_2\in N_k$ with $y_1\neq y_2$ we have
\[ d(c_{y_1}(T_k),c_{y_2}(T_k)) \geq d(y_1,y_2) -d(y_1,c_{y_1}(T_k))-d(y_2,c_{y_2}(T_k)) > \delta, \]
so the sets
\[ \tilde S_k :=\{\dot c_y(0) : y \in N_k\} \]
are $(T_k,\delta)$-separated w.r.t. the metric $d_1$ on $S\tilde M$, defined as
\[d_1(v,w) = \max_{t\in[0,1]} d(c_v(t),c_w(t)). \]
In $SM$ the sets $S_k := Dp(\tilde S_k)$ are $(T_k,\delta/2)$-separated. Define the decreasing sequence of compact sets
\[ \M_k := Dp \left\{v\in S\tilde M : c_v:[-\sqrt{T_k},\sqrt{T_k}]\to \tilde M \text{ is minimal}\right\}, \quad \bigcap_{k\in\N} \M_k = \M. \]
In order to find large separated sets in $\M$ we shall find them in the sets $\M_k$, observing that for $t\in [\sqrt{T_k}, T_k-\sqrt{T_k}]$ we have
\[\phi^tS_k \subset \M_k. \]
Assume $k$ is large enough, s.th.
\[ s_{\sqrt{T_k}}(S_k,\delta/4) \leq e^{2b\sqrt{T_k}}, \quad \sqrt{T_k} \geq \frac{2b}{\e h} . \]
We apply lemma \ref{lemmabowen2} and obtain
\begin{align*}
& s_{T_k-\sqrt{T_k}}(\phi^{\sqrt{T_k}} S_k , \delta/4) \cdot s_{\sqrt{T_k}}(S_k,\delta/4) \geq s_{T_k}(S_k, \delta/2) \geq \# N_k \geq \frac{1}{a} e^{h(1-\e)T_k} \\
\Rightarrow \quad & s_{T_k-\sqrt{T_k}}(\phi^{\sqrt{T_k}} S_k , \delta/4) \geq \frac{1}{a} e^{h(1-\e)T_k - 2b\sqrt{T_k}} \geq \frac{1}{a} e^{h(1-2\e)T_k}.
\end{align*}
Let now
\[ T \in (0, T_k-\sqrt{T_k} ] , \quad m_k = \left\lfloor \frac{T_k-\sqrt{T_k}}{T}\right\rfloor \in \N. \]
Applying lemma \ref{lemmabowen2} again gives
\begin{align*}
& \left( \prod_{i = 0}^{m_k-1} s_T(\phi^{iT + \sqrt{T_k}} S_k, \delta/8) \right) \cdot s_{T_k-\sqrt{T_k} - m_kT}(\phi^{m_kT + \sqrt{T_k}} S_k, \delta/8) \\
& \geq s_{T_k-\sqrt{T_k}} (\phi^{\sqrt{T_k}} S_k, \delta/4) \geq \frac{1}{a} e^{h(1-2\e)T_k} \\
\Rightarrow \quad & \prod_{i = 0}^{m_k-1} s_T(\phi^{iT + \sqrt{T_k}} S_k, \delta/8) \geq \frac{\frac{1}{a} e^{h(1-2\e)T_k}}{s_{T_k-\sqrt{T_k} - m_kT}(\phi^{m_kT + \sqrt{T_k}} S_k, \delta/8)} \\
& \geq \frac{\frac{1}{a} e^{h(1-2\e)T_k}}{s_T(SM, \delta/8)} \geq \frac{1}{a} e^{h(1-2\e)T_k - 2bT} ,
\end{align*}
where in the last step we assumed that $T$ is large, so that $s_T(SM, \delta/8) \leq e^{2bT}$. Hence one of the factors in the last product has to be ''large'', i.e. for some $i\in \{0,...,m_k-1\}$ we have
\[ s_T(\phi^{iT + \sqrt{T_k}} S_k, \delta/8) \geq \frac{1}{a} e^{\frac{h(1-2\e)T_k - 2bT}{m_k}} \geq \frac{1}{a} e^{h(1-2\e)T} e^{ - \frac{2bT}{m_k}} . \]
Note also that $\phi^{iT + \sqrt{T_k}} S_k \subset \M_k$, so when letting $k\to\infty$ while fixing $T$ and using $m_k\to\infty$, we find a $(T,\delta/8)$-separated set in $\M=\cap_k \M_k$ of cardinality at least
\[ \frac{1}{a} e^{h(1-2\e)T} \cdot \lim_{k\to\infty} e^{ - \frac{2bT}{m_k}} = \frac{1}{a} e^{h(1-2\e)T} . \]
This proves the theorem:
\[ \h(\phi^t_\M) \geq \h (\phi^t_\M, \delta/8) \geq h - 2\e . \]
\end{proof}
\subsection{Upper Bound for manifolds of hyperbolic type} \label{upbound}
Following Klingenberg \cite{Kl} we call a compact Riemannian manifold $(M,g)$ of hyperbolic type, if there exists a metric of strictly negative curvature $g_0$ on $M$. From now on we assume the existence of such $g_0$ on the compact Manifold $M$. When we lift objects such as $g, g_0$ from $M$ to the universal cover $\tilde M$ we will frequently denote them by the same letters. In the following we write $d$ for the metric on $\tilde M$ induced by $g$ and $d_{g_0}$ for the one induced by the background metric $g_0$. Due to the compactness of $M$ the two metrics on $\tilde M$ are equivalent, i.e. there exists a constant $C>0$ such that
$$\frac{1}{C}d(p,q)\leq d_{g_0}(p,q) \leq Cd(p,q) \qquad \forall p, q \in \tilde M.$$
We write $d_1$ for the metric on $S\tilde M$ defined by
$$
d_1(v,w) := \max_{t\in [0,1 ]} d(c_v(t), c_w(t))
$$
and $d_H(A,B)$ for the Hausdorff metric on sets $A,B\subset\tilde M$ w.r.t. $d$.
The following theorem is fundamental for the study of $\M$ in manifolds of hyperboldic type. It has been proven by Morse in dimension 2 and by Klingenberg in arbitrary dimensions.
\begin{theorem}[Morse lemma, cf. \cite{Kl} or \cite{Kn}] \label{theoremmorse}
Let $(M,g)$ be a manifold of hyperbolic type. Then there is a constant $r_0=r_0(g,g_0) >0$ with the following properties. \begin{itemize}
\item[(i)] If $c :[a,b] \rightarrow \tilde M$ and $\alpha : [ a_0, b_0] \rightarrow \tilde M$ are minimizing geodesic segments w.r.t. $g, g_0$, respectively, joining $c(a)=\al(a_0)$ to $c(b)=\al(b_0)$, then
$$d_H(c[a,b], \alpha[a_0, b_0]) \leq r_0.$$
\item[(ii)] For any minimizing $g$-geodesic $c :\R \to \tilde M$ there is a $g_0$-geodesic $\alpha :\R \to \tilde M$ and conversely for any $g_0$-geodesic $\alpha :\R \to \tilde M$ a minimizing $g$-geodesic $c :\R \to \tilde M$ with
$$
d_H(\alpha (\R) , c(\R )) \leq r_0 .
$$
\end{itemize}\end{theorem}
In this subsection we prove the following theorem stated in the introduction. As a consequence we immediately obtain corollary \ref{h=h(g) expansive} in the introduction using the results in section \ref{bowen bounds}.
\begin{theorem}\label{upper bound beta}
Let $(M,g)$ be a compact Riemannian manifold of hyperbolic type and $K \subset \tilde M$ a compact set in the universal cover $\tilde M$.
Let
$${\cal F}=SK \cap {\tilde {\cal M}},$$
where $SK=\pi^{-1}(K)$. Then there is some constant $\beta$ such that
$$ \h(\phi^t , {\cal F}, \beta) \leq h(g).$$
\end{theorem}
In order to prove the theorem, we construct spanning sets for $\cal F$. Let $K\subset {\tilde M}$ be a compact set with $\diam K=a$. For $r >a$ consider
$$
K_r:=\{ z\in {\tilde M} \; | \; r - a \leq d(z,K) \leq r\}
$$
Let $K^\e, K_r^\e$ be minimal $\e$-spanning sets for $K,K_r$, respectively. For $y\in K^\e, z \in K_r^\e$, let $\alpha_{yz} : \mathbb{R} \to \tilde M$ be the $g_0$-geodesic connecting $y$ and $z$
such that $\alpha_{yz}(0) =y$ and $\alpha_{yz}(d_{g_0}(y,z)) =z$. By the Morse lemma, there exists a minimizing $g$-geodesic $c_{yz} : \mathbb{R} \to \tilde M$ $r_0$-close to $\al_{yz}(\R)$. Set
\[ P_r:=\{ \dot c_{yz}(0) : y\in K^\e, z\in K_r^\e \} \subset \tilde \M . \]
\begin{lemma} \label{P_r spanning}
$P_r$ is a $(r-1, \beta)$-spanning set for ${\cal F}$ with respect to the metric $d_1$ where $\beta$ is given by
$\beta:=5r_0+(2C^2+1)\e$.
\end{lemma}
\begin{proof}[Proof of \ref{P_r spanning}]
Let $c:\R \to \tilde M$ be a minimizing $g$-geodesic with $c(0) \in K$. Then $c(r)\in K_r$ and we can choose $y\in K^\e,z\in K_r^\e$ with
$$
d(y,c(0)) \leq \e ,\quad d(z,c(r)) \leq \e.
$$
Let $\al$ be the $g_0$-geodesic connecting $c(0)$ and $c(r)$ parametrized such that $\al(0) = c(0) $ and $\al(d_{g_0}(y,z)) = c(r)$. Using the convexity of the function $t \mapsto d_{g_0}(\alpha(t), \alpha_{yz}(t))$ due to negative curvature we find
$$
d_{g_0}(\alpha (t) ,\alpha_{yz}(t)) \leq \max\{ d_{g_0}(c(0),y), d_{g_0}(c(r),z) \} \leq C\e \quad \forall ~ t \in [0, d_{g_0}(y,z)].
$$
Let $A=c[0,r]$ and $B =c_{yz}[0, r'] $ be the subsegment of $c_{yz}$ lying $r_0$-close to $\al_{yz}[0, d_{g_0}(y,z)]$ w.r.t. the $g$-Hausdorff metric $d_H$. Using the Morse lemma we find (omitting for the moment the intervals $[0, d_{g_0}(y,z)]$ for $\al,\al_{yz}$)
$$
d_H(A,B)\leq d_H(A,\al)+d_H(\al,\al_{yz})+d_H(\al_{yz}, B) \leq 2r_0+C^2\e .
$$
By definition of the Hausdorff distance, for $t\in[0,r]$ there is some $t'\in \R$ with $d(c(t),c_{yz}(t'))\leq 2r_0+C^2\e$.
Using the minimality of $c,c_{yz}$ we find with $d(c(0),c_{yz}(0))\leq r_0+\e$ that $|t-t'| \leq 3r_0 + (C^2+1)\e$ and hence
\[ d(c(t),c_{yz}(t)) \leq d(c(t),c_{yz}(t')) + d(c_{yz}(t'),c_{yz}(t)) \leq 2r_0+C^2\e + 3r_0 + (C^2+1)\e. \]
Therefore, taking $\beta:=5r_0+(2C^2+1)\e$ we obtain
$$
d_1(\dot{c}(t), \dot {c}_{yz}(t)) = \max \limits _{s\in [0,1]} d(c(t+s),c_{yz}(t+s)) \leq \beta \quad \forall t\in [0, r-1].
$$
\end{proof}
We can now prove the theorem.
\begin{proof}[Proof of \ref{upper bound beta}]
We have
\begin{align*}
& \# K_r^\e \leq C_\e \cdot \vol B\left(x, r+a+\e/2\right) , \quad C_\e := \left( \inf_{y\in M}\vol B(y,\e/2) \right)^{-1}, \\
\Rightarrow \quad & \# P_r \leq \# K^\e \cdot \# K_r^\e \le \# K^\e \cdot C_\e \cdot \vol B\left(x, r+a+\e/2\right) .
\end{align*}
Hence
\begin{align*}
& \h(\phi^t , {\cal F}, \beta) \leq \varlimsup_{r\to \infty }\frac{1}{r-1}\log \# P_r \leq \varlimsup_{r\to \infty }\frac{1}{r-1}\log \vol B \left(x, r+a+ \e/2 \right) \\
&= \lim_{r \rightarrow \infty }\frac{ r+a+\e/2} {r-1}\frac{1}{r+a+\e/2}\ \log \vol B\left(x, r+a+\e/2 \right ) = h(g) .
\end{align*}
\end{proof}
\section{The two-dimensional case}
We use the notation introduced at the beginning of section \ref{upbound}. Morse \cite{morse} studied the structure of minimal geodesics in the universal cover $\tilde M$ (called ''class A geodesics'' there), where $M=\tilde M/\Gamma$ is a closed orientable surface of genus $\geq 2$. Apart from the Morse lemma in section \ref{upbound}, which is valid in any dimension, the assumption $\dim M=2$ provides additional information since in $\tilde M$ the minimizing geodesics intersect quite easily. As a background metric for $M$ we can choose by the uniformisation theorem a metric of constant negative curvature $-1$ and we use for $\tilde M$ the Poincar\'e model given by
\[ \tilde M=\{ z\in\C: |z|<1 \}, \quad (g_0)_z=\frac{4}{(1-|z|^2)^2}\skp_{euc}. \]
This model has a simple boundary at infinity, namely $\tilde M(\infty)=S^1$. Using the Morse lemma, for pairs $\xi_-,\xi_+\in S^1$ with $\xi_-\neq \xi_+$ we distinguish the minimal $g$-geodesics lying in bounded distance from the $g_0$-geodesic in $\tilde M$ joining $\xi_-,\xi_+$. Write
\begin{align*}
& c(\pm\infty):=\lim_{t\to\pm\infty}c(t) = \xi_\pm \qquad \text{(the limit in the euclidean sense in $\C$)} \\
& B := \{ \xi= (\xi_-,\xi_+): \xi_-, \xi_+ \in S^1, \xi_-\neq \xi_+ \} = \tilde M(\infty)\times\tilde M(\infty)-\diag, \\
& \tilde \M_\xi := \{ \dot c(0) \mid \text{$c:\R\to \tilde M$ is an arc-length $g$-minimal with $c(\pm\infty)=\xi_\pm$} \} ,
\end{align*}
then $\tilde \M = \cup_{\xi \in B} \tilde\M_\xi$ and each class $\tilde\M_\xi$ is non-empty. In the sequel \emph{minimal} refers to $g$-minimizing arc-length geodesics $c:\R\to \tilde M$.
\subsection{Structure of the minimals}
\begin{definition}\label{M^pm}
For $v\in \tilde \M$ let $\tilde M^+(v),\tilde M^-(v)$ be the open connected components (half discs) of $\tilde M-c_v(\R)$, where $\tilde M^+(v)$ contains $\pi v + t \cdot i v$ for small $t>0$. For $\xi \in B$ set
\begin{align*}
\tilde\M_\xi^+ & := \left \{v \in \tilde\M_\xi ~\big|~ \forall w\in\tilde\M_\xi : \quad \pi w \in c_v(\R) \quad\Rightarrow\quad c_w[0,\infty) \subset \overline{\tilde M^-(v)} ~ \right \}, \\
\tilde\M_\xi^- & := \left \{v \in \tilde\M_\xi ~\big|~ \forall w\in\tilde\M_\xi : \quad \pi w \in c_v(\R) \quad\Rightarrow\quad c_w[0,\infty) \subset \overline{\tilde M^+(v)} ~ \right \}, \\
\tilde\M_\xi^0 & := \tilde\M_\xi^+\cup \tilde\M_\xi^- , \quad \tilde \M^0 := \bigcup_{\xi\in B} \tilde\M_\xi^0 \subset S\tilde M, \quad \M^0 := Dp(\tilde\M^0) \subset SM.
\end{align*}
\end{definition}
\begin{remark}\label{remark M^pm} \begin{itemize}
\item[(i)] The sets $\tilde\M_\xi^\pm$ are never empty. In fact, the intersection $\tilde\M_\xi^- \cap \tilde\M_\xi^+$ contains the velocity vectors of the bounding geodesics of $\tilde\M_\xi$ (cf. theorem 8 in \cite{morse}).
\item[(ii)] It is easy to see that no two geodesics from $\tilde\M_\xi^+$ (resp. $\tilde\M_\xi^-$) intersect transversely. We shall refer to this as the graph property of $\tilde\M^\pm_\xi$.
\item[(iii)] The sets $\tilde\M_\xi^\pm$ and hence $\tilde\M^0$ and $\M^0$ are closed and $\phi^t$-invariant.
\end{itemize}\end{remark}
By (ii) in \ref{remark M^pm} the sets $\tilde\M^0_\xi$ have a simple structure in $\tilde M$, so when calculating $\h(\phi_\M^t)$ we would like to stick to $\M^0$. For this it is important that $\M^0$ is ''sufficiently large''. Let $\Om\subset SM$ denote the non-wandering set of $\phi^t$ restricted to $\M$. The following proposition is the key observation to obtain $\h(\phi^t_\M)=h(g)$ in the two-dimensional case.
\begin{prop}\label{M^0 recurrent}
$\M^0\subset SM$ contains the non-wandering set $\Om$ of $\phi_\M^t$.
\end{prop}
\begin{proof}
Let $v\in Dp^{-1}(\Om)\cap \tilde\M_\xi$ and $U_n=B(v,1/n)\cap \tilde\M\subset S\tilde M$ for $n\in\N$. By definition of $\Om$ there exists $\gamma_n\in\Gamma-\{\id\}$ and $t_n>0$ such that $ D\gamma_n\phi^{t_n} U_n \cap U_n \neq \emptyset$. In particular there is some $v_n \in U_n$ such that $w_n := D\gamma_n \phi^{t_n} v_n \in U_n$. Assume $v\notin\tilde\M_\xi^0$, so there are two minimals $c^\pm :\R \to \tilde M$ in $\tilde\M_\xi$ with $c^\pm(0)=c_v(t^\pm)$ and $c^\pm(0,\infty)\subset \tilde M^\pm(v)$.
First suppose $c_{v_n}(\infty)=c_{w_n}(\infty)=c_v(\infty)=\xi_+\in\tilde M(\infty)$ for some $n$. Then $\xi_+=c_{w_n}(\infty) =\gamma_n c_{v_n}(\infty)=\gamma_n \xi_+$, so $\xi_+$ is the point at $+\infty$ for some periodic minimal axis of $\gamma_n$. If $c_v$ is itself periodic, theorems 10 and 13 in \cite{morse} show that in fact there are no minimal geodesics in $\tilde \M_\xi$ intersecting $c_v$ transversely, i.e. $v\in \tilde\M_\xi^0$. If $c_v$ is not periodic, it is asymptotic to some periodic minimal $c_0$ in $+\infty$, approaching its limit from ''below'' (i.e. from $\tilde M^-(c_0(\R))$), say, by theorem 10 in \cite{morse}. Now $c^+$ is also asymptotic in $+\infty$ to that same minimal $c_0$ (theorem 13 in \cite{morse}). But two asymptotic minimals in $\tilde M$ cannot intersect transversely (theorem 6 in \cite{morse}), so we obtain a contradiction.
Assume now that $c_{v_n}(\infty) \neq c_{w_n}(\infty)$ for all $n\in\N$. Interchanging $v_n, w_n$ and maybe taking a subsequence, we may assume that $v_n\to v$ and $c_{v_n}(\infty)\neq \xi_+$ for all $n$. Moreover we can assume that the $c_{v_n}(\infty)$ lie in one connected component of $\tilde M(\infty)-\{ \xi_-,\xi_+ \}$, say $c_{v_n}(\infty) \in \tilde M(\infty) \cap \overline{\tilde M^+(v)}$. Now, by $\dot c_{v_n}(t^+)\to \dot c_v(t^+)$ and the assumptions on the points at infinity of $c_{v_n},c^+$, there have to be two intersections of $c_{v_n},c^+$ for large $n$, contradicting the minimality of both geodesics.
\end{proof}
\subsection{Entropy in strips of finite width}
In this section we will show that the local entropy of the geodesic flow in the non-wandering set $\Om\subset \M^0$ of $\phi^t_\M$ is vanishing. We work in the universal cover and write $\tilde \Om := Dp^{-1}(\Om)\subset S\tilde M$ for the lifted non-wandering set of $\phi^t_\M$. Recall
\begin{align*}
& d_1(v,w)=\max_{t\in[0,1]}d(c_v(t),c_w(t)) \quad v,w\in S\tilde M , \\
& Z_\beta(v) = \{ w \in \tilde \Om : d(c_v(t),c_w(t)) \leq \beta ~ \forall t\in\R \} \subset S\tilde M, \quad v\in\tilde \Om.
\end{align*}
\vspace*{20pt}
\begin{prop}\label{entropy-exp}
For any $v_0 \in \tilde \Om$ and any $\beta>0$ we have
\[ \h(\phi^t_{\tilde\Om}, Z_\beta(v_0))=0. \]
Hence the geodesic flow restricted to $\tilde \Om$ is $\beta$-entropy-expansive for any $\beta>0$.
\end{prop}
\begin{proof}
Fix $v_0\in\tilde \Om, \beta>0$ and some small $\delta >0$. By \ref{M^0 recurrent} we find $\xi\in B$ with $v_0\in\tilde \M_\xi^0$ and hence $Z_\beta(v_0) \subset \tilde\M_\xi^0$. We shall prove that $(T-1,2\delta)$-spanning sets $E$ of minimal cardinality for $Z_\beta(v_0)\cap \tilde\M_\xi^+$ have cardinality growing at most linearly in $T$. The same arguments work for $Z_\beta(v)\cap \tilde\M_\xi^-$ and hence give the proposition.
Write
\begin{align*}
A & := Z_\beta(v_0)\cap \tilde\M_\xi^+ , \\
K_T & :=\{x\in\tilde M: d(x,c_{v_0}[0,T])\leq\beta\} , \\
\TT(c_v,\delta) & := \{ x \in \tilde M : d(x,c_v(\R)) < \delta \} , \quad v\in A.
\end{align*}
The sets $A,K_T$ are compact.
\underline{Step 1.} For $\delta>0$ and $v,w\in A$ with $c_w[0,T]\subset \TT(c_v,\delta/3)$ there exists
$s_0 \in \R$ such that
\[ d(c_v(t+s_0),c_w(t)) \leq \delta \quad \forall t\in[0,T]. \]
Proof. By assumption for any $t\in [0,T]$ there is some $s(t)\in \R$ with
\[d(c_v(s(t)),c_w(t))\leq \delta_0 := \delta/3. \]
Using the minimality of $c_v,c_w$ one finds
\[ s(t)-s(0) \leq 2\delta_0 +t , \quad t \leq 2\delta_0 + s(t)-s(0). \]
Hence with $s_0:= s(0)$ we have
\begin{align*}
d(c_v(t+s_0),c_w(t)) & \leq d(c_v(t+s(0)),c_v(s(t))) + d(c_v(s(t)),c_w(t)) \\
& \leq |s(t)-s(0)-t|+\delta_0 \leq 3\delta_0 = \delta.
\end{align*}
\underline{Step 2.} Let $F(v,\delta) := \{ \phi^{j\delta}v : j\in\Z, |j| \leq 2(1+\beta/\delta) \}$ for $v\in A$. Then for $v,w\in A$ with $c_w[0,T]\subset \TT(c_v,\delta/3)$ there exists $v_j =\phi^{j\delta}v\in F(v,\delta)$ such that
\[ d(c_{v_j}(t),c_w(t)) \leq 2\delta \quad \forall t\in[0,T]. \]
Proof. Let $s_0$ be as in step 1 and $j\in \Z, r\in[0,\delta)$ with $s_0 = j\delta + r$. Then
\begin{align*}
d(c_v(t+j\delta) , c_w(t)) & \leq d(c_v(t+j\delta) , c_v(t+s_0)) + d(c_v(t+s_0) , c_w(t)) \\
& \leq | t+j\delta - t-s_0 | + \delta \leq 2\delta.
\end{align*}
By definition of $A$ we have $d(\pi v,\pi w)\leq 2\beta$ and hence again by step 1
\[ |s_0| = d(c_v(s_0),\pi v) \leq d(c_v(s_0),\pi w) + d(\pi w,\pi v) \leq \delta + 2\beta, \]
showing
\[ |j| \leq \frac{|s_0|+|r|}{\delta} \leq \frac{\delta + 2\beta+\delta}{\delta} = 2(1+\beta/\delta). \]
\underline{Step 3.}
\[ \h(\phi^t,A)=0 . \]
Proof. Consider the family of (oriented) unparametrised curves
\[ \A : = \{c_v(\R) \subset \tilde M : v\in A \}. \]
$\A$ is ordered by the graph property of $\tilde\M_\xi^+$ ($c<c'$ iff $c'\subset \tilde M^+(c)$) and we construct a sequence of geodesics $c_1< ... < c_n< ...$ in $\A$.
By closedness of $A$ we find a $<$-smallest geodesic $c_1$ in $\A$. If $c_1,...,c_n$ are already chosen, take $c_{n+1}\in \A$ to be the $<$-smallest geodesic $c_{n+1}>c_n$, such that the compact segment
$c_n\cap K_T$ is not entirely contained in the open tube $\TT(c_{n+1},\delta/3)$. By construction, there is some $p_n\in c_n\cap K_T$ with $d(p_n,c_{n+1}(\R))\geq \delta/3$, hence the upper open half disc
\[ D_n := \tilde M^+(c_n) \cap B(p_n,\delta/3) \]
lies in the open strip between $c_n<c_{n+1}$ (for $\delta$ small, s.th. $c_n$ does not return to $D_n$ by minimality).
Moreover all half discs $D_n$ are contained in a $\delta/3$-neighborhood of $K_T$ and disjoint, since the $c_i$ are ordered.
As the volume of $D_n$ is bounded from below by some constant $C(\delta)$ using standard comparison theorems and the compactness of $M$,
and the volume of the $K_T$-neighborhood is finite, growing linearly with $T$, the above construction stops at some finite $N(T)$, again $N(T)$ growing at most linearly.
On the other hand, by construction for any $c\in\A$ we find some $i\in\{1,...,N(T)\}$ such that $c\cap K_T \subset \TT(c_i,\delta/3)$. Choose the parameterization of the $c_i$ such that $v_i := \dot c_i(0) \in A$. Now by step 2 the set
\[ E(T,\delta) := \bigcup_{i=1}^{N(T)} F(v_i,\delta) \]
is $(T-1,2\delta)$-spanning for $A$ w.r.t. $d_1$ with cardinality
\[ \# E(T,\delta) = N(T) \cdot \# F(v_i , \delta) = N(T) \cdot ( 4(1+\beta/\delta) + 1). \]
Hence for any $\delta>0$ we have
\[ \h(\phi^t,A,2\delta) \leq \lim_{T\to\infty}\frac{\log \#E(T,\delta)}{T-1} = \lim_{T\to\infty}\frac{\log N(T)}{T-1} = 0 \]
and by letting $\delta\to 0$, the claim follows.
\end{proof}
We now have the following result.
\begin{theorem}
If $M$ is a closed orientable surface with genus $\geq 2$, then
\[ \h(\phi^t_\M) = h(g) \]
for any Riemannian metric $g$ on $M$.
\end{theorem}
\begin{proof}
By \ref{katok hasselblatt} we have $\h(\phi^t_\M)\geq h(g)$. To show the reverse inequality take any compact set $K\subset \tilde M$
with $p(K)=M$. By \ref{upper bound beta} there is some $\beta>0$ with
\[ \h(\phi^t,\tilde \Om\cap SK, \beta) \leq \h(\phi^t,\tilde \M\cap SK, \beta) \leq h(g). \]
Using \ref{bowen expansive} and \ref{entropy-exp} we find
\begin{align*}
\h(\phi^t_\Om) & \leq \h(\phi^t,\tilde \Om \cap SK, \beta) + \hloc(\phi^t,\tilde \Om \cap SK, \beta) \\
& = \h(\phi^t,\tilde \Om \cap SK, \beta) \leq h(g).
\end{align*}
But since $\Om$ is has full measure w.r.t. any invariant probability on $\M$, we find (cf. 8.6.1 (ii) in \cite{W})
\[ \h(\phi_\M^t) = \h(\phi_{\Om}^t) \leq h(g). \]
\end{proof}
\begin{remark}\begin{itemize}
\item[(i)] The theorem is trivial for the 2-sphere $M=S^2$ and also holds in the case where $M$ is the 2-torus $M=T^2$. This can be shown using the same ideas presented in this paper and is implemented in \cite{glasmachers}. If $M$ is non-orientable, the theorem holds for the orientable double cover $\hat M$ of $M$ and $M,\hat M$ have the same universal cover. Hence:
For any closed surface $M$ with any Riemannian metric $g$ we have
\[ \h(\phi^t_\M) = h(g) \]
as stated in the introduction.
\item[(ii)]
Theorem 4.5. also holds if one replaces the Riemann metric $g$ by a Finsler metric $F$ (not necessarily reversible).
The Morse lemma only requires that the two norms $F, F_0 = \sqrt{g_0}$ are equivalent. The volume entropy of $F$ can be defined by
$$
h(F) = \lim_{r \to \infty} \frac {\log \vol_{g_0} B(p,r)}{r}
$$ where $ \vol_{g_0}$ is the $g_0$-volume (in fact one could take the lift of an arbitrary Riemannian metric to compute the volume)
and $B(p,r)$ is the ball defined by the endpoints of Finsler geodesics rays
of length $\le r$ initiating form $p$.
The result \ref{katok hasselblatt} of Katok and Hasselblatt holds also in this setting, just as the arguments for \ref{upper bound beta}. Arguing along these lines
yields Theorem 4.5 in the Finsler case. This has implications for Tonelli Lagrangian systems, as on high enough energy levels the arising Euler-Lagrange flow is a reparametrisation of a Finsler geodesic flow by Maupertuis' principle.
\end{itemize}\end{remark}
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Q: C# Interface Debug Information not linked to sources I'm trying to re-jig the layout of a very large solution which has become impossibly hard (and s l o w) to work with. My plan is to create a number of solutions containing related projects, and then use binary references where necessary to link to libraries produced by the other solutions.
The thing we rely on to make this usable is Resharper's Navigate to External Sources functionality, so we can easily browse the source of the projects we are referencing from other solutions. Quite why VS can't do this out of the box is beyond me.
This is all working very nicely for classes with implementation. However, for C# interfaces and classes containing only auto-implemented properties, Resharper isn't able to browse to the sources, and falls back to cruddy metadata viewer.
I used srctool.exe, which comes with the Symbol Server tools in MS Debugging Tools For Windows, to browse the sources listed in the .pdb file, and it's clear that the sources for these interfaces and empty(ish) classes are not referenced in the pdb file. If I switch the auto-implemented properties to those with backing fields, then the source link appears in the pdb.
I'm guessing the sources are excluded because there are no places you could set breakpoints on interfaces and auto-implemented properties.
I'm wondering, though, if there is some exotic compiler option or workaround we can employ to force the PDB file to include references to the source of C# interfaces.
Thanks,
Mark
A: The question doesn't have enough detail. Shooting off the hip, I'd guess that you tackled the problems with the slow massive solution by converting project references to assembly references. And used the Release build of those projects as the reference.
And yes, that stumps any tool that tries to find source code files from the PDB. The release build of a .NET project uses a stripped version of the PDB, all the source code file and line number info has been removed from it. That's a pretty normal thing to do with real release builds. Release built code normally is optimized. That causes code to be re-ordered, no longer matching the logical position of the code in the source file. Any info you get from the source+line PDB info now tends to get between harmful and useless, you start looking in the wrong place for a problem.
That is however not a concern for IDE tooling or while debugging your app. The optimizer is automatically disabled in a case like this. Actually a configuration item in VS: Tools + Options, Debugging, General, "Suppress JIT optimization on module load" option. Turned on by default.
Clearly any tooling that uses the PDB is going to catatonic when they don't have a chance to find source files. The fix is to go back to the original project, select the Release configuration again and change a setting: Project + Properties, Build tab, scroll down, Advanced button. Change the "Debug info" combo from "pdb-only" to "full". Rebuild the project.
Should fix your problem. Also revives the debugger, you can step into the source code again.
Don't move files around too much btw, you might stump the chumps again. At least keep the PDB with the DLL in the same directory. If the source code is no longer present in the same directory but you checked it out again in another one then you have to tell the IDE about it. Right-click the solution, Properties, Debug Source Files setting.
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"redpajama_set_name": "RedPajamaStackExchange"
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De Universiteitsbibliotheek van de Technische Universiteit Delft op de campus van de universiteit in de Wippolder werd geopend op 15 mei 1998.
Geschiedenis
De bibliotheek bevond zich sinds 1843 in in het hoofdgebouw van de Koninklijke Akademie op de Oude Delft 95. Vanwege ruimtegebrek werd in 1915 een nieuw gebouw geopend aan de Doelenstraat (Delft). Het gebouw was een ontwerp van Rijksbouwmeester Jan Vrijman. De bibliotheek bestond uit twee delen; het hoofdgebouw met studiezalen en werkkamers in Neo-Renaissance stijl, en het depotgebouw in Neo-Classicistische stijl. In de tweede helft van de twintigste eeuw was de collectie verspreid geraakt over meerdere dependances. In 1998 verhuisde de bibliotheek daarom naar de Wippolder. Het oude bibliotheeksgebouw werd omgebouwd tot appartementencomplex.
Het Wippolder-complex
Het complex werd ontworpen door het Delftse architectenbureau Mecanoo. De bibliotheek ontving in 1998 de Nationale Staalprijs in de categorie Gebouwen met een stalen of hybride draagconstructie vooral vanwege het bijzondere gebruik van staal in alle verschillende onderdelen van de constructie.
De bibliotheek is ingedeeld in ruimtes met verschillende functies en activiteiten. De centrale hal biedt ruimte om laatste nummers van tijdschriften en boeken te raadplegen. Oudere uitgaven en verzamelingen worden in de halfverzonken lagere verdieping opgeslagen en kunnen worden opgevraagd. Díe verdieping is niet voor het publiek toegankelijk. Faculteitspecifieke boeken en tijdschriften worden beheerd op satellietbibliotheken in faculteiten.
Op de publiekelijk toegankelijke verdieping van de bibliotheek is een uitleenbalie, en er zijn computers beschikbaar voor algemeen gebruik: óf voor het raadplegen van de databank van de bibliotheek, óf om bijvoorbeeld via internet informatie te zoeken.
In de bibliotheek is een aparte ruimte ingericht met computers voor studenten, en er zijn verschillende studie- of vergaderruimtes die voor studenten gratis te reserveren zijn. In het midden van de bibliotheek bevindt zich "de kegel": een vide, naar boven open in het midden, met vier verdiepingen eromheen, met op elke verdieping ruimte met tafels en stoelen om te studeren. De vide loopt kegelvormig toe naar boven, samen met de puntvorm van het dak.
De oostwand van de centrale hal is bedekt met kasten, van vloer tot plafond over vier verdiepingen, met boeken. De noordkant is een muur van glas om buitenlicht binnen te laten. De zuidkant zijn kantoren. De hoofdingang is aan de westkant.
Het gebouw heeft een schuin dak dat van het maaiveld oploopt naar boven. Het is bedekt met gras, en met enige banen van grote rechthoekige tegels ter decoratie. Dit dak kan vrijelijk betreden worden.
Trivia
In tegenstelling tot wat er vaak beweerd wordt, staat TU Delft Library het niet toe om de helling als skihelling te gebruiken. Omdat dit toch gebeurt wanneer er sneeuw gevallen is worden er doorgaans dranghekken geplaatst om toegang tot het dak te ontnemen.
Galerij
Plattegronden
Externe links
Website Bibliotheek TU Delft
Bronnen
Delft
Delft
Technische Universiteit Delft
Bouwwerk van Mecanoo architecten
Bouwwerk van de Technische Universiteit Delft
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"redpajama_set_name": "RedPajamaWikipedia"
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\section*{Acknowledgments}
The authors gratefully acknowledge the support by the Graduiertenkolleg 2339 IntComSin of the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) -- Project-ID 321821685.
The work of Bal\'azs Kov\'acs is funded by the Heisenberg Programme of the DFG -- Project-ID 446431602.
\addcontentsline{toc}{section}{References}
\printbibliography
\end{document}
\section{Derivation and modelling aspects}
\label{sec:derivation}
In this section, we present the derivation of the general viscoelastic model for tumour growth \ref{P}.
The outline of this section is as follows.
We first present basic balance laws, before we use an energy inequality, a Lagrange multiplier approach and several constitutive assumptions to derive the general viscoelastic model.
We then reformulate the pressure and derive a general energy identity, before we specify the initial and boundary conditions. Then, we give the most relevant examples for the source terms and specify the nutrient and elastic energy densities. After that, we present several variants and limit cases of the model.
\subsection{Conservation laws}
\subsubsection{Balance law of mass}
We consider a mixture consisting of healthy and tumour cells. We denote their difference of volume fractions by $\phi$, with $\{\phi=1\}$ representing the unmixed tumour tissue and $\{\phi=-1\}$ representing the surrounding healthy tissue.
We assume the existence of an unspecified species acting as a nutrient for the tumour whose concentration is denoted by $\sigma$.
Moreover, we assume that $\phi$ and $\sigma$ are transported by a volume-averaged velocity $\pmb{\mathrm{v}}$ and some diffusive fluxes $\pmb{\mathrm{J}}_\phi$ and $\pmb{\mathrm{J}}_\sigma$, respectively.
Based on these assumptions, the balance laws of mass read
\begin{align}
\label{eq:balance_phi}
\partial_t \phi + \divergenz{\phi \pmb{\mathrm{v}}} + \divergenz{\pmb{\mathrm{J}}_\phi}
&= \Gamma_\phi,
\\
\label{eq:balance_sigma}
\partial_t \sigma + \divergenz{\sigma \pmb{\mathrm{v}}} + \divergenz{\pmb{\mathrm{J}}_\sigma}
&= - \Gamma_\sigma,
\end{align}
where $\Gamma_\phi$ and $\Gamma_\sigma$ denote the source and sink terms of the phase field variable and the nutrient.
Moreover, mass exchange in terms of the divergence of $\pmb{\mathrm{v}}$ is explicitly given by
\begin{align}
\label{eq:balance_div_v}
\divergenz{\pmb{\mathrm{v}}} &= \Gamma_{\pmb{\mathrm{v}}}.
\end{align}
The specific motivation for \eqref{eq:balance_phi} and \eqref{eq:balance_div_v} is based on mass balance laws for the two components of the mixture, i.e.~the tumour and healthy cells, and we refer the reader to \cite{ebenbeck_garcke_nurnberg_2020} for more details.
In the general case, the mass densities of the tumour cells $\bar{\rho}_{1}$ and the healthy cells $\bar{\rho}_{-1}$ can differ, which yields for the mass density $\rho$ of the mixture,
\begin{align}
\label{eq:mass_density}
\rho = \hat\rho(\phi) =
\tfrac{1}{2} \bar{\rho}_{1} (1+\phi)
+ \tfrac{1}{2} \bar{\rho}_{-1} (1-\phi).
\end{align}
For simplicity reasons, we consider matching mass densities of the pure components in this work, which results in $\rho = \bar{\rho}_{1} = \bar{\rho}_{-1}$.
\subsubsection{Balance law of linear momentum}
Motivated by, e.g., \cite{AbelsGG_2012, ebenbeck_garcke_nurnberg_2020}, we assume that the mixture is a single viscoelastic fluid that fulfills the balance law of linear momentum of continuum mechanics.
We further neglect any gravity effects or body forces and suppose that contact forces are represented by a viscoelastic stress tensor $\overline\T$.
Moreover, we assume that the viscoelastic stress tensor is symmetric, isotropic and can depend on $\nabla\pmb{\mathrm{v}}, \phi,\mu,\sigma,\nabla\phi$ and $\B_e$, where $\B_e$ is the left Cauchy--Green tensor associated with the elastic part of the total mechanical response which will be defined later (see \eqref{eq:B_e}).
With these assumptions, the balance law of linear momentum is given by
\begin{align}
\label{eq:balance_momentum}
\rho \partial_t \pmb{\mathrm{v}}
+ \rho (\pmb{\mathrm{v}}\cdot\nabla)\pmb{\mathrm{v}}
= \divergenz{\overline\T},
\end{align}
where $\overline\T$ has to specified by constitutive assumptions.
\subsubsection{Concept of viscoelasticity}
\label{sec:concept_viscoelasticity}
In the literature \cite{HuLinLiu_2018, liu_2008_viscoelastic_incompr, LinLiuZhang_2005}, a popular approach for viscoelasticity is in terms of the deformation gradient $\F: \Omega \times (0,T) \to \R^{d\times d}$ between the initial configuration and the current configuration of a viscoelastic body. Writing $\F$ in Eulerian coordinates, we obtain the hyperbolic evolution equation
\begin{align}
\label{eq:F}
\partial_t \F + (\pmb{\mathrm{v}}\cdot\nabla) \F = \nabla\pmb{\mathrm{v}} \F.
\end{align}
Hence, it is easily deduced from \eqref{eq:F} that the left Cauchy--Green tensor $\tilde \B \coloneqq \F\F^T$ satisfies the evolution equation
\begin{align}
\label{eq:B0_infinite_weissenberg}
\partial_t \tilde\B + (\pmb{\mathrm{v}}\cdot\nabla) \tilde\B= \nabla\pmb{\mathrm{v}}\tilde\B + \tilde\B(\nabla\pmb{\mathrm{v}})^T.
\end{align}
This is the so-called Oldroyd-B equation with infinite Weissenberg number which is a common way to describe viscoelastic materials of Kelvin--Voigt type, where stress relaxation is neglected.
However, stress relaxation is a typical behaviour of living tissues \cite{ambrosi_2009}.
For this reason, we follow the approach of M{\'a}lek and Pr{\r u}{\v s}a \cite{malek_prusa_2018} in order to derive a viscoelastic approach that accounts for stress relaxation.
We assume a \textit{virtual} framework consisting of three configurations:~the initial configuration, the current configuration at time $t>0$ and the natural configuration which would be taken by the considered body at time $t>0$ after immediate relaxation, see Figure \ref{fig:configurations}.
Therefore, we assume a \textit{virtual} multiplicative decomposition of the full deformation gradient $\F$ by
\begin{align}
\label{eq:F_decomposition}
\F = \F_{e} \F_{d},
\end{align}
where $\F_{d}$ describes the deformation gradient between the initial and the natural configuration, taking into account only the dissipative processes of the fluid, which, in the biological context, can arise from, e.g., cell reorganizations, birth and death of cells \cite{ambrosi_2009}.
Besides, $\F_{e}$ measures only the elastic part of the deformation, which is the deformation gradient between the natural and the current configuration.
Then, the sought measure of our main interest is the left Cauchy--Green tensor $\B_{e} \coloneqq \F_{e} \F_{e}^T$ associated with the elastic part of the deformation.
\begin{figure}[ht]
\small
\centering
\includegraphics[width=0.7\linewidth]{figures/states.PNG}
\caption{Configurations of a viscoelastic cell-mixture within a fixed domain, described by a virtual decomposition of the total deformation. Adapted from \cite{malek_prusa_2018}.}
\label{fig:configurations}
\end{figure}
Following the ideas of M{\'a}lek and Pr{\r u}{\v s}a \cite{malek_prusa_2018}, we define $\L \coloneqq \nabla\pmb{\mathrm{v}}$ and its symmetric part $\D\coloneqq \frac{1}{2}(\L+\L^T)$. From \eqref{eq:F} we see that $\L = (\partial_t^\bullet \F) \F^{-1}$, where the material derivative of $\F$ is defined by $\partial_t^\bullet \F \coloneqq \partial_t \F + (\pmb{\mathrm{v}} \cdot \nabla) \F$.
This motivates to introduce the tensorial quantity $\L_{d}$ by
\begin{align}
\label{eq:L_e}
\L_{d} \coloneqq (\partial_t^\bullet \F_{d}) \F_{d}^{-1},
\end{align}
and its symmetric part by $\D_{d} \coloneqq \frac{1}{2} ( \L_{d} + \L_{d}^T)$.
Together with the formula $\partial_t^\bullet (\F_{d}^{-1}) = -\F_{d}^{-1} (\partial_t^\bullet\F_{d}) \F_{d}^{-1}$ we see that the material derivative of the relative deformation gradient $\F_{e}$ is given by
\begin{align}
\label{eq:F_e}
\partial_t^\bullet \F_{e}
= \partial_t^\bullet ( \F \F_{d}^{-1} )
= (\partial_t^\bullet \F) \F_{d}^{-1} + \F \partial_t^\bullet (\F_{d}^{-1})
= \L\F\F_{d}^{-1} - \F\F_{d}^{-1}(\partial_t^\bullet\F_{d})\F_{d}^{-1}
= \L\F_{e} - \F_{e} \L_{d},
\end{align}
which implies, as $\B_{e} = \F_{e} \F_{e}^T$, that
\begin{align}
\label{eq:B_e}
\partial_t^\bullet \B_{e} = \L \B_{e} + \B_{e} \L^T
- 2 \F_{e} \D_{d} \F_{e}^T.
\end{align}
This is the sought formula for the evolution of the left Cauchy--Green tensor $\B_{e}$. The right-hand side of \eqref{eq:B_e} depends on the quantities $\F_{e}$, $\D_{d}$ and $\L=\nabla\pmb{\mathrm{v}}$. Later, the dependency on the tensor $\D_{d}$ will be removed by constitutive assumptions.
\subsection{Energy inequality and the Lagrange multiplier method}
In order to derive the system \ref{P} from thermodynamical principles, we apply the Lagrange multiplier method by Liu and M{\"u}ller developed in \cite{liu_1972}. We remark that the mass density $\rho$ is assumed to be constant. In the case of a non-constant mass density given by formula \eqref{eq:mass_density}, the derivation of a system of equations can be performed with methods from Abels, Garcke and Gr{\"u}n \cite{AbelsGG_2012}.
We postulate a general energy density of the form
\begin{align}
e = \hat e(\phi,\nabla\phi,\sigma, \B_{e}) + \frac{1}{2} \rho \abs{\pmb{\mathrm{v}}}^2,
\end{align}
where $\hat e$ denotes the free energy density of the system which can depend on $\phi,\nabla\phi,\sigma, \B_{e}$, whereas $\frac{1}{2} \rho \abs{\pmb{\mathrm{v}}}^2$ denotes the kinetic energy density.
Let $V(t)\subset \Omega$ be an arbitrary volume which is transported with the fluid velocity. We now consider the following energy inequality resulting from the second law of thermodynamics:
\begin{align}
\label{eq:energy_law}
\begin{split}
\underbrace{\ddv{}{t} \int_{V(t)}
e(\phi,\nabla\phi,\sigma, \pmb{\mathrm{v}}, \B_{e}) \dv{x}}_{\text{ change of energy}}
&\leq
\underbrace{- \int_{\partial V(t)} \pmb{\mathrm{J}}_e \cdot \pmb{\mathrm{n}} \ \mathrm d \calH^{d-1}}_{ \substack{\text{ energy flux across} \\ \text{ the boundary}}}
+ \underbrace{\int_{\partial V(t)} (\overline\T\pmb{\mathrm{n}})\cdot\pmb{\mathrm{v}} \ \mathrm d \calH^{d-1}}_{ \substack{\text{ work due to } \\ \text{ macroscopic stresses}}}
\\
&\quad
+ \underbrace{\int_{V(t)} c_{\pmb{\mathrm{v}}} \Gamma_{\pmb{\mathrm{v}}} + c_\phi \Gamma_\phi + c_\sigma (-\Gamma_\sigma) \dv{x}}_{\text{\normalfont supply of energy}},
\end{split}
\end{align}
where $\pmb{\mathrm{n}}$ is the outer unit normal to $\partial V(t)$, $\pmb{\mathrm{J}}_e$ is an energy flux yet to be determined and $c_{\pmb{\mathrm{v}}}$, $c_\phi$ and $c_\sigma$ are unknown multipliers which have to be specified.
Following the arguments in, e.g., \cite{AbelsGG_2012, ebenbeck_garcke_nurnberg_2020, GarckeLSS_2016}, we introduce Lagrange multipliers $\lambda_{\pmb{\mathrm{v}}}$, $\lambda_\phi$ and $\lambda_\sigma$ for \eqref{eq:balance_div_v}, \eqref{eq:balance_phi} and \eqref{eq:balance_sigma}, respectively.
Using the momentum balance equation \eqref{eq:balance_momentum} and the Gauss theorem, we can reformulate the boundary integral describing work due to macroscopic stresses by
\begin{align*}
- \int_{\partial V(t)} (\overline\T\pmb{\mathrm{n}})\cdot\pmb{\mathrm{v}} \ \mathrm d \calH^{d-1}
= - \int_{V(t)} \divergenz{\overline\T} \cdot \pmb{\mathrm{v}} + \overline\T : \nabla\pmb{\mathrm{v}} \dv{x}
= - \int_{V(t)} \rho \partial_t^\bullet \pmb{\mathrm{v}} \cdot \pmb{\mathrm{v}}
+ \overline\T : \nabla\pmb{\mathrm{v}} \dv{x}.
\end{align*}
Therefore, using Reynold's transport theorem \cite{eck-garcke-knabner} and the fact that $V(t)$ is arbitrary,
we obtain the following local dissipation inequality
\begin{align}
\begin{split}
\label{eq:dissipation_1}
-\ensuremath{\mathcal{D}}
&\coloneqq
\partial_t^\bullet e + e \divergenz{\pmb{\mathrm{v}}} + \divergenz{\pmb{\mathrm{J}}_e} - \overline\T:\nabla\pmb{\mathrm{v}}
- \rho \partial_t^\bullet\pmb{\mathrm{v}} \cdot \pmb{\mathrm{v}}
- c_{\pmb{\mathrm{v}}}\Gamma_{\pmb{\mathrm{v}}} - c_\phi\Gamma_\phi + c_\sigma\Gamma_\sigma
\\
&\quad - \lambda_{\pmb{\mathrm{v}}} (\divergenz{\pmb{\mathrm{v}}} - \Gamma_{\pmb{\mathrm{v}}})
- \lambda_{\phi} (\partial_t^\bullet\phi + \phi\divergenz{\pmb{\mathrm{v}}} + \divergenz{\pmb{\mathrm{J}}_\phi}- \Gamma_{\phi})
- \lambda_{\sigma} (\partial_t^\bullet\sigma + \sigma\divergenz{\pmb{\mathrm{v}}} + \divergenz{\pmb{\mathrm{J}}_\sigma}+ \Gamma_{\sigma})
\\
&\leq 0,
\end{split}
\end{align}
which has to hold for arbitrary values of $\phi$, $\sigma$, $\nabla\phi$, $\nabla\sigma$, $\rho$, $\pmb{\mathrm{v}}$, $\B_e$, $\Gamma_{\pmb{\mathrm{v}}}$, $\Gamma_\phi$, $\Gamma_\sigma$, $\partial_t^\bullet \phi$, $\partial_t^\bullet \sigma$ and $\partial_t^\bullet \pmb{\mathrm{v}}$.
By the chain rule, we then have
\begin{align*}
\partial_t^\bullet e &= \fracdel{e}{\phi} \partial_t^\bullet\phi + \fracdel{e}{\nabla\phi} \cdot \partial_t^\bullet(\nabla\phi) + \fracdel{e}{\sigma} \partial_t^\bullet\sigma
+ \fracdel{e}{\pmb{\mathrm{v}}} \cdot \partial_t^\bullet \pmb{\mathrm{v}}
+ \fracdel{e}{\B_{e}} : \partial_t^\bullet \B_{e}.
\end{align*}
Therefore, on noting \eqref{eq:B_e}, we obtain
\begin{align}
\begin{split}
\label{eq:dissipation_2}
- \ensuremath{\mathcal{D}}
&=
\divergenz{\pmb{\mathrm{J}}_e - \lambda_\phi \pmb{\mathrm{J}}_\phi - \lambda_\sigma \pmb{\mathrm{J}}_\sigma }
+ \nabla\lambda_\phi \cdot \pmb{\mathrm{J}}_\phi + \nabla\lambda_\sigma \cdot \pmb{\mathrm{J}}_\sigma
+ \fracdel{e}{\nabla\phi} \cdot \partial_t^\bullet(\nabla\phi)
\\
&\quad
+ \partial_t^\bullet\phi \Big( \fracdel{e}{\phi}-\lambda_\phi \Big)
+ \partial_t^\bullet\sigma \Big( \fracdel{e}{\sigma}-\lambda_\sigma \Big)
+ \partial_t^\bullet\pmb{\mathrm{v}} \cdot \Big( \fracdel{e}{\pmb{\mathrm{v}}}- \rho\pmb{\mathrm{v}} \Big)
\\
&\quad
- \T:\nabla\pmb{\mathrm{v}}
+ \fracdel{e}{\B_{e}} : (\L \B_{e} + \B_{e} \L^T
- 2 \F_{e} \D_{d} \F_{e}^T)
\\
&\quad
+ (c_\sigma - \lambda_\sigma) \Gamma_\sigma
+ (\lambda_{\pmb{\mathrm{v}}} - c_{\pmb{\mathrm{v}}}) \Gamma_{\pmb{\mathrm{v}}}
+ (\lambda_\phi - c_\phi) \Gamma_\phi
+ (e - \lambda_\phi \phi - \lambda_\sigma \sigma - \lambda_{\pmb{\mathrm{v}}}) \divergenz{\pmb{\mathrm{v}}}
\\
&\leq 0.
\end{split}
\end{align}
Together with $\partial_{x_j} (\partial_t^\bullet\phi)
= \partial_t \partial_{x_j} \phi
+ \pmb{\mathrm{v}} \cdot \nabla(\partial_{x_j}\phi)
+ \partial_{x_j} \pmb{\mathrm{v}} \cdot \nabla\phi
= \partial_t^\bullet(\partial_{x_j}\phi) + \partial_{x_j}\pmb{\mathrm{v}} \cdot \nabla\phi$,
we calculate
\begin{align*}
\divergenz{\partial_t^\bullet\phi \fracdel{e}{\nabla\phi}}
= \partial_t^\bullet \phi \divergenz{\fracdel{e}{\nabla\phi}}
+ \partial_t^\bullet(\nabla\phi) \cdot \fracdel{e}{\nabla\phi}
+ \nabla\pmb{\mathrm{v}} : \Big( \nabla\phi \otimes \fracdel{e}{\nabla\phi} \Big).
\end{align*}
Then, using $\fracdel{e}{\pmb{\mathrm{v}}} = \rho\pmb{\mathrm{v}}$, we can reformulate \eqref{eq:dissipation_2} as
\begin{align}
\begin{split}
\label{eq:dissipation_3}
- \ensuremath{\mathcal{D}}
&=
\divergenz{\pmb{\mathrm{J}}_e - \lambda_\phi \pmb{\mathrm{J}}_\phi - \lambda_\sigma \pmb{\mathrm{J}}_\sigma + \partial_t^\bullet\phi \fracdel{e}{\nabla\phi}}
+ \nabla\lambda_\phi \cdot \pmb{\mathrm{J}}_\phi
+ \nabla\lambda_\sigma \cdot \pmb{\mathrm{J}}_\sigma
\\
&\quad
+ \partial_t^\bullet\phi \Big( \fracdel{e}{\phi} - \divergenz{\fracdel{e}{\nabla\phi}} - \lambda_\phi \Big)
+ \partial_t^\bullet\sigma \Big( \fracdel{e}{\sigma}-\lambda_\sigma \Big)
\\
&\quad
- \Big( \overline\T+ \Big( \nabla\phi \otimes \fracdel{e}{\nabla\phi} \Big) \Big) : \nabla\pmb{\mathrm{v}}
+ \fracdel{e}{\B_{e}} : (\L \B_{e} + \B_{e} \L^T
- 2 \F_{e} \D_{d} \F_{e}^T)
\\
&\quad
+ (c_\sigma - \lambda_\sigma) \Gamma_\sigma
+ (\lambda_{\pmb{\mathrm{v}}} - c_{\pmb{\mathrm{v}}}) \Gamma_{\pmb{\mathrm{v}}}
+ (\lambda_\phi - c_\phi) \Gamma_\phi
+ (e - \lambda_\phi \phi - \lambda_\sigma \sigma - \lambda_{\pmb{\mathrm{v}}}) \divergenz{\pmb{\mathrm{v}}}
\\
&\leq 0.
\end{split}
\end{align}
In the following we denote the chemical potential of the order parameter $\phi$ as
\begin{align*}
\mu \coloneqq \fracdel{e}{\phi} - \divergenz{\fracdel{e}{\nabla\phi}}.
\end{align*}
\subsection{Constitutive relations}
To fulfill the last inequality for $-\ensuremath{\mathcal{D}}$, we can argue similarly to \cite{AbelsGG_2012, ebenbeck_garcke_nurnberg_2020, GarckeLSS_2016} and we make the following constitutive assumptions for the fluxes $\pmb{\mathrm{J}}_e$, $\pmb{\mathrm{J}}_\sigma$, $\pmb{\mathrm{J}}_\phi$, for the constants $c_{\pmb{\mathrm{v}}}$, $c_\phi$, $c_\sigma$ and for the Lagrange multipliers $\lambda_{\pmb{\mathrm{v}}}$, $\lambda_\phi$, $\lambda_\sigma$:
\begin{align}
\label{eq:constitutive_1}
\begin{split}
\pmb{\mathrm{J}}_e &= \lambda_\sigma \pmb{\mathrm{J}}_\sigma + \lambda_\phi \pmb{\mathrm{J}}_\phi
- \partial_t^\bullet\phi \fracdel{e}{\nabla\phi} ,
\quad
\pmb{\mathrm{J}}_\phi = - m(\phi) \nabla\lambda_\phi,
\quad
\pmb{\mathrm{J}}_\sigma = -n(\phi) \nabla\lambda_\sigma,
\\
c_{\pmb{\mathrm{v}}} &= \lambda_{\pmb{\mathrm{v}}},
\quad
c_\phi = \lambda_\phi = \fracdel{e}{\phi} - \divergenz{\fracdel{e}{\nabla\phi}} = \mu,
\quad
c_\sigma = \lambda_\sigma = \fracdel{e}{\sigma},
\end{split}
\end{align}
where $m(\phi)$ and $n(\phi)$ are non-negative mobilities corresponding to a generalised Fick's law (see \cite{AbelsGG_2012}).
Actually, $m(\phi)$ and $n(\phi)$ could also depend on the chemical potential $\mu$ and the nutrient $\sigma$. As pointed out in \cite{malek_prusa_2018}, one could also consider cross effects by assuming that the relations between the fluxes $\pmb{\mathrm{J}}_\phi$, $\pmb{\mathrm{J}}_\sigma$ and the gradients of the Lagrange parameters $\nabla\lambda_\phi$, $\nabla\lambda_\sigma$ take the form
$\pmb{\mathrm{J}}_\alpha = - \sum_{\beta\in\{\phi,\sigma\}} \mathrm{M}_{\alpha,\beta} \nabla\lambda_\beta$ for $\alpha\in\{\phi,\sigma\}$,
where $(\mathrm{M}_{\alpha,\beta})$ is a symmetric positive definite matrix which could depend on, e.g., $\phi$, $\mu$ and $\sigma$.
This would of course lead to a more complex system of equations, and is hence not explored here.
Under these constitutive assumptions, the inequality for the local dissipation \eqref{eq:dissipation_3} holds if
\begin{align}
\label{eq:constitutive_3a}
\begin{split}
&- \Big( \overline\T+ \Big( \nabla\phi \otimes \fracdel{e}{\nabla\phi} \Big) \Big) : \nabla\pmb{\mathrm{v}}
+ \fracdel{e}{\B_{e}} : (\nabla\pmb{\mathrm{v}} \B_{e} + \B_{e} (\nabla\pmb{\mathrm{v}})^T
- 2 \F_{e} \D_{d} \F_{e}^T)
\\
&\quad + (e - \lambda_\phi \phi - \lambda_\sigma \sigma - \lambda_{\pmb{\mathrm{v}}}) \divergenz{\pmb{\mathrm{v}}}
\\
&\leq 0.
\end{split}
\end{align}
Using the properties of the trace and the symmetry
of $\B_{e}$ and $\fracdel{e}{\B_{e}}$,
we calculate
\begin{align*}
&\fracdel{e}{\B_{e}} : (\nabla\pmb{\mathrm{v}} \B_{e})
= \trace\Big( \fracdel{e}{\B_{e}}
\B_{e}^T (\nabla\pmb{\mathrm{v}})^T \Big)
= \Big( \fracdel{e}{\B_{e}} \B_{e}\Big) : \nabla\pmb{\mathrm{v}},
\\
&\fracdel{e}{\B_{e}} : \big(\B_{e}(\nabla\pmb{\mathrm{v}})^T \big)
= \trace\Big( \Big(\fracdel{e}{\B_{e}}\Big)^T
\B_{e} (\nabla\pmb{\mathrm{v}})^T \Big)
= \Big( \fracdel{e}{\B_{e}} \B_{e}\Big) : \nabla\pmb{\mathrm{v}},
\end{align*}
and
\begin{align*}
& \fracdel{e}{\B_{e}} : ( \F_{e} \D_{d} \F_{e}^T)
= \trace\Big(\fracdel{e}{\B_{e}} \F_{e} \D_{d}^T \F_{e}^T \Big)
= \trace\Big( \F_{e}^T \fracdel{e}{\B_{e}} \F_{e} \D_{d}^T \Big)
= \Big( \F_{e}^T \fracdel{e}{\B_{e}} \F_{e} \Big) : \D_{d}.
\end{align*}
Therefore, we can reformulate \eqref{eq:constitutive_3a} as
\begin{align}
\label{eq:constitutive_3b}
\begin{split}
& \Big( - \overline\T- \Big( \nabla\phi \otimes \fracdel{e}{\nabla\phi}\Big)
+ 2 \fracdel{e}{\B_{e}} \B_{e} \Big) : \nabla\pmb{\mathrm{v}}
- 2 \Big( \F_{e}^T \fracdel{e}{\B_{e}} \F_{e} \Big) : \D_{d}
+ (e - \lambda_\phi \phi - \lambda_\sigma \sigma - \lambda_{\pmb{\mathrm{v}}}) \divergenz{\pmb{\mathrm{v}}}
\leq 0.
\end{split}
\end{align}
Introducing the unknown pressure $p$, we can rewrite the stress tensor $\T$ as follows:
\begin{align}
\label{eq:constitutive_3}
\overline\T= \S - p \I, \quad \text{ i.e. } \quad \S = \overline\T+ p\I.
\end{align}
Similar arguments as in \cite{ebenbeck_garcke_nurnberg_2020} imply that
\begin{align}
\label{eq:constitutive_4}
\fracdel{e}{\nabla\phi
= a(\phi,\nabla\phi,\sigma,\B_{e}) \nabla\phi,
\end{align}
where $a(\phi,\nabla\phi,\sigma,\B_{e})$ is a real valued function. Since $\S$ is symmetric, we have
\begin{align*}
\S:\nabla\pmb{\mathrm{v}} = \S : \frac{1}{2}(\nabla\pmb{\mathrm{v}} + (\nabla\pmb{\mathrm{v}})^T)
+ \S : \frac{1}{2} (\nabla\pmb{\mathrm{v}} - (\nabla\pmb{\mathrm{v}})^T)
= \S : \D,
\end{align*}
and similarly, as $\fracdel{e}{\B_{e}}$ and $ \B_{e}$ are symmetric, we also obtain
\begin{align*}
\Big( \fracdel{e}{\B_{e}} \B_{e} \Big) : \nabla\pmb{\mathrm{v}}
= \Big( \fracdel{e}{\B_{e}} \B_{e} \Big) : \D.
\end{align*}
Using the identity $\I : \nabla\pmb{\mathrm{v}} = \I:\D = \divergenz{\pmb{\mathrm{v}}}$, we have
\begin{align*}
\overline\T: \nabla\pmb{\mathrm{v}} = (\S - p \I) : \nabla\pmb{\mathrm{v}}
= \S : \D - p \divergenz{\pmb{\mathrm{v}}}.
\end{align*}
This yields
\begin{align}
\label{eq:constitutive_5a}
\begin{split}
& \Big( - \S - a(\phi,\nabla\phi,\sigma,\B_{e}) (\nabla\phi \otimes \nabla\phi) + 2 \fracdel{e}{\B_{e}} \B_{e} \Big) : \D
\\
&\quad
- 2 \Big( \F_{e}^T \fracdel{e}{\B_{e}} \F_{e} \Big) : \D_{d}
+ (e - \lambda_\phi \phi - \lambda_\sigma \sigma + p - \lambda_{\pmb{\mathrm{v}}}) \divergenz{\pmb{\mathrm{v}}}
\leq 0.
\end{split}
\end{align}
The quantities $\D$ and $\divergenz{\pmb{\mathrm{v}}}$ appear in the first and in the last term on the left-hand side of \eqref{eq:constitutive_5a} as it holds $\trace\D = \divergenz{\pmb{\mathrm{v}}}$. Therefore, these two quantities are not independent. As pointed out in \cite{malek_prusa_2018}, it would actually be necessary to split $\D$ to mutually independent quantities consisting of the traceless part of $\D$ and the part containing $\divergenz{\pmb{\mathrm{v}}}$, which requires one to split the quantities in the first brackets in \eqref{eq:constitutive_5a} in a similar manner.
However, following the strategy of \cite{ebenbeck_garcke_nurnberg_2020, GarckeLSS_2016}, we choose the constitutive assumption
\begin{align}
\label{eq:constitutive_5}
\lambda_{\pmb{\mathrm{v}}} \coloneqq e - \lambda_\phi \phi - \lambda_\sigma \sigma + p,
\end{align}
for the Lagrange parameter $\lambda_{\pmb{\mathrm{v}}}$ in order to control the mass exchange term even though $\D$ and $\divergenz{\pmb{\mathrm{v}}}$ are not independent quantities. The reason for this is that we can reformulate the unknown pressure $p$ and therefore adapt the constitutive assumption for the Lagrange parameter $\lambda_{\pmb{\mathrm{v}}}$ afterwards.
Hence, it remains to fulfill the inequality
\begin{align}
\label{eq:constitutive_6a}
\begin{split}
& - \Big( \S + a(\phi,\nabla\phi,\sigma,\B_{e}) (\nabla\phi \otimes \nabla\phi)
- 2 \fracdel{e}{\B_{e}} \B_{e} \Big) : \D
- 2 \Big( \F_{e}^T \fracdel{e}{\B_{e}} \F_{e} \Big) : \D_{d}
\leq 0.
\end{split}
\end{align}
At this point we make the constitutive assumptions
for $\S$ and $\D_{d}$ as follows
\begin{align}
\label{eq:constitutive_6}
\S
&= 2 \eta(\phi)\D + \lambda(\phi)\divergenz{\pmb{\mathrm{v}}}\I
+ 2 \fracdel{e}{\B_{e}} \B_{e}
- a(\phi,\nabla\phi,\sigma,\B_{e}) (\nabla\phi \otimes \nabla\phi),
\\
\label{eq:constitutive_7}
\D_{d} &= \frac{1}{\tau(\phi)} \Big( \F_{e}^{-1} \fracdel{e}{\B_{e}} \F_{e} \Big),
\end{align}
with non-negative viscosities $\eta(\cdot)$, $\lambda(\cdot)$ and a non-negative viscoelastic relaxation function $\tau(\cdot)$, which could also depend on $\mu$ and $\sigma$.
Noting that $\lambda(\phi) \divergenz{\pmb{\mathrm{v}}} \I : \D = \lambda(\phi) \divergenz{\pmb{\mathrm{v}}}^2$ and
\begin{align*}
\Big( \F_{e}^T \fracdel{e}{\B_{e}} \F_{e} \Big) :
\Big( \F_{e}^{-1} \fracdel{e}{\B_{e}} \F_{e} \Big)
= \trace\Big( \F_{e}^T \fracdel{e}{\B_{e}} \F_{e}
\F_{e}^T \Big(\fracdel{e}{\B_{e}}\Big)^T \F_{e}^{-T} \Big)
= \abs{\fracdel{e}{\B_{e}} \F_{e}}^2,
\end{align*}
we obtain that the local dissipation inequality is fulfilled, i.e.
\begin{align}
\label{eq:constitutive_10}
\begin{split}
\ensuremath{\mathcal{D}}
&= 2\eta(\phi) \abs{\D}^2 + \lambda(\phi) (\divergenz{\pmb{\mathrm{v}}})^2
+ m(\phi) \abs{\nabla\mu}^2
+ n(\phi) \abs{\nabla \fracdel{e}{\sigma}}^2
+ \frac{2}{\tau(\phi)} \abs{\fracdel{e}{\B_{e}} \F_{e}}^2
\geq 0.
\end{split}
\end{align}
So, dissipation can be divided into the following processes: viscosity effects on the velocity (i.e.~$2\eta(\phi) \abs{\D}^2$),
changes in volume (i.e.~$\lambda(\phi) (\divergenz{\pmb{\mathrm{v}}})^2$),
transport along $\nabla\mu$ and $\nabla \fracdel{e}{\sigma}$, and dissipation caused by viscoelastic relaxation (i.e.~$\frac{2}{\tau(\phi)} \abs{\fracdel{e}{\B_{e}} \F_{e}}^2$).
We remark that multiplying \eqref{eq:constitutive_7} with $\F_{e}$ from the left and with $\F_{e}^T$ from the right yields a formula for $\F_{e} \D_{d} \F_{e}^T$, i.e.
\begin{align}
\label{eq:constitutive_14}
& \F_{e} \D_{d} \F^T_{e}
= \frac{1}{\tau(\phi)} \fracdel{e}{\B_{e}} \F_{e} \F^T_{e}
= \frac{1}{\tau(\phi)} \fracdel{e}{\B_{e}} \B_{e}.
\end{align}
Combining \eqref{eq:constitutive_14} and \eqref{eq:B_e} leads to the following constitutive equation for the left Cauchy--Green tensor:
\begin{align}
\partial_t^\bullet \B_{e}
+ \frac{1}{\tau(\phi)} \fracdel{e}{\B_{e}} \B_{e}
= \nabla\pmb{\mathrm{v}} \B_{e}
+ \B_{e} (\nabla\pmb{\mathrm{v}})^T.
\end{align}
This can be seen as a generalized viscoelastic model of Oldroyd-B type \cite{Oldroyd_1950}.
Instead of the constitutive assumption \eqref{eq:constitutive_7}, it is also possible to assume
\begin{subequations}
\begin{alignat}{2}
\label{eq:constitutive_7b}
&\D_{d} = \frac{1}{\tau(\phi)} \Big( \F_{e}^T \fracdel{e}{\B_{e}} \F_{e} \Big),
\end{alignat}
\end{subequations}
which, after multiplication with $\F_{e}$ from the left and with $\F_{e}^T$ from the right, using $\B_{e} = \F_{e} \F_{e}^T$ and noting \eqref{eq:B_e}, leads to
\begin{align}
\partial_t^\bullet \B_{e}
+ \frac{2}{\tau(\phi)} \B_{e} \fracdel{e}{\B_{e}} \B_{e}
= \nabla\pmb{\mathrm{v}} \B_{e}
+ \B_{e} (\nabla\pmb{\mathrm{v}})^T.
\end{align}
This can be seen as a generalized version of the viscoelastic model of Giesekus \cite{giesekus_1982}.
In this case, the local dissipation is given by
\begin{align}
\label{eq:constitutive_10b}
\begin{split}
\ensuremath{\mathcal{D}}
&= 2\eta(\phi) \abs{\D}^2 + \lambda(\phi) (\divergenz{\pmb{\mathrm{v}}})^2
+ m(\phi) \abs{\nabla\mu}^2
+ n(\phi) \abs{\nabla \fracdel{e}{\sigma}}^2
+ \frac{2}{\tau(\phi)} \abs{\F_{e}^T \fracdel{e}{\B_{e}} \F_{e}}^2
\geq 0,
\end{split}
\end{align}
instead of \eqref{eq:constitutive_10}.
However, the focus of this work lies on the constitutive relation \eqref{eq:constitutive_7} leading to the viscoelastic model of Oldroyd-B type.
We now summarize all the constitutive assumptions from this section:
\begin{align}
\label{eq:constitutive_9}
\begin{split}
&\pmb{\mathrm{J}}_e = \lambda_\sigma \pmb{\mathrm{J}}_\sigma + \lambda_\phi \pmb{\mathrm{J}}_\phi
- \partial_t^\bullet\phi \fracdel{e}{\nabla\phi} ,
\quad\quad
\pmb{\mathrm{J}}_\phi = - m(\phi) \nabla\lambda_\phi,
\quad\quad
\pmb{\mathrm{J}}_\sigma = -n(\phi) \nabla\lambda_\sigma,
\\
&c_{\pmb{\mathrm{v}}} = \lambda_{\pmb{\mathrm{v}}} = e - \lambda_\phi \phi - \lambda_\sigma \sigma + p,
\quad\quad
c_\phi = \lambda_\phi = \fracdel{e}{\phi} - \divergenz{\fracdel{e}{\nabla\phi}} = \mu,
\quad\quad
c_\sigma = \lambda_\sigma = \fracdel{e}{\sigma},
\\
&\Big( \S + a(\phi,\nabla\phi,\sigma,\B_{e}) (\nabla\phi \otimes \nabla\phi)
- 2 \fracdel{e}{\B_{e}} \B_{e} \Big)
= 2 \eta(\phi)\D + \lambda(\phi)\divergenz{\pmb{\mathrm{v}}}\I,
\\
& \D_{d} = \frac{1}{\tau(\phi)} \Big( \F_{e}^{-1} \fracdel{e}{\B_{e}} \F_{e} \Big),
\quad \text{or} \quad
\D_{d} = \frac{1}{\tau(\phi)} \Big( \F_{e}^T \fracdel{e}{\B_{e}} \F_{e} \Big).
\end{split}
\end{align}
\subsection{Further aspects of modelling}
\subsubsection{The model equations}
From now on we suppress the index of $\B_{e}$, i.e.~we write $\B$ instead of $\B_e$, and we also write $\D(\pmb{\mathrm{v}})$ instead of $\D$ to point out the dependency on $\pmb{\mathrm{v}}$.
In the following, we assume a general energy density of the form
\begin{align}
\label{eq:constitutive_energy}
e(\phi,\nabla\phi,\sigma,\pmb{\mathrm{v}},\B)
= f(\phi,\nabla\phi)
+ N(\phi,\sigma)
+ W(\phi,\B)
+ \frac{1}{2} \rho \abs{\pmb{\mathrm{v}}}^2.
\end{align}
The first term $f(\phi,\nabla\phi)$ in \eqref{eq:constitutive_energy} accounts for interfacial energy of the diffuse interface \cite{cahn_hilliard_1958}
which we assume to be of Ginzburg--Landau type:
\begin{align}
f(\phi,\nabla\phi) = A \psi(\phi) + \frac{B}{2} \abs{\nabla\phi}^2,
\end{align}
where $\psi(\cdot)$ is a non-negative potential with equal minima at $\phi=\pm1$, and $A,B>0$ are constants. Usually we set $A=\frac{\beta}{\epsilon}$ and $B=\beta\epsilon$, where the constants $\beta, \epsilon>0$ are related to the surface tension and the interfacial thickness, respectively.
The second term $N(\phi,\sigma)$ in \eqref{eq:constitutive_energy} describes the energy contribution due to the presence of the nutrient and the interaction between the tumour tissues and the nutrients, also see \cite{GarckeLSS_2016}. The third term $W(\phi,\B)$ in \eqref{eq:constitutive_energy} represents the elastic part of the energy which we additionally assume to depend on the type of material and hence on $\phi$.
For the moment, both the nutrient and the elastic energy density are kept in a general form, but later, possible choices are given.
The last term in \eqref{eq:constitutive_energy} is the kinetic part of the energy.
With these choices we calculate
\begin{align}
\begin{split}
&\fracdel{e}{\phi} = A \psi'(\phi) + N_{,\phi}
+ W_{,\phi},
\quad
\fracdel{e}{\nabla\phi} = B \nabla\phi,
\quad
\fracdel{e}{\sigma} = N_{,\sigma},
\quad
\fracdel{e}{\B}
= W_{,\B},
\quad
a(\phi,\nabla\phi,\sigma,\B) = B,
\end{split}
\end{align}
where $N_{,\phi}$, and $N_{,\sigma}$ denote the partial derivatives of $N(\phi,\sigma)$ with respect to $\phi$ and $\sigma$. For better readability, note that we sometimes suppress the arguments of $N_{,\phi}(\phi,\sigma), N_{,\sigma}(\phi,\sigma)$ and we write $N_{,\phi}, N_{,\sigma}$ instead. Similarly, we adopt the notation for $W$.
Next, we specify the constitutive relation for the full stress tensor $\overline\T= - p\I + \S$:
\begin{align}
\label{eq:constitutive_11}
\begin{split}
\overline\T&= -p\I + 2\eta(\phi)\D(\pmb{\mathrm{v}}) + \lambda(\phi) \divergenz{\pmb{\mathrm{v}}}\I
+ 2 W_{,\B}(\phi,\B) \B
- B \nabla\phi \otimes\nabla\phi.
\end{split}
\end{align}
Collecting all equations from above, the general viscoelastic model of Oldroyd-B type reads:
\begin{subequations}
\begin{align}
\label{eq:phi0}
\partial_t \phi + \divergenz{\phi \pmb{\mathrm{v}}}
&= \divergenz{m(\phi) \nabla\mu} + \Gamma_\phi,
\\
\label{eq:mu0}
\mu &= A \psi'(\phi) - B \Delta\phi + N_{,\phi}(\phi,\sigma)
+ W_{,\phi}(\phi,\B),
\\
\label{eq:sigma0}
\partial_t \sigma + \divergenz{\sigma \pmb{\mathrm{v}}}
&= \divergenz{n(\phi) \nabla N_{,\sigma}(\phi,\sigma)} - \Gamma_\sigma,
\\
\label{eq:div_v0}
\divergenz{\pmb{\mathrm{v}}} &= \Gamma_{\pmb{\mathrm{v}}},
\\
\label{eq:v0}
\rho \partial_t \pmb{\mathrm{v}} + \rho (\pmb{\mathrm{v}}\cdot \nabla)\pmb{\mathrm{v}}
&= \divergenz{\T(\phi,\pmb{\mathrm{v}}, p, \B)}
- \divergenz{B \nabla\phi \otimes \nabla\phi},
\\
\label{eq:B0}
\partial_t \B + (\pmb{\mathrm{v}}\cdot\nabla)\B
+ \frac{1}{\tau(\phi)} \T_{\mathrm{el}}(\phi,\B)
&= \nabla\pmb{\mathrm{v}} \B + \B (\nabla\pmb{\mathrm{v}})^T.
\end{align}
\end{subequations}
For future reference, the full viscoelastic stress tensor is denoted by
\begin{align}
\label{eq:T_viscoelastic0}
\T(\phi,\pmb{\mathrm{v}}, p, \B)
\coloneqq \T_{\mathrm{visc}}(\phi,\pmb{\mathrm{v}},p) + \T_{\mathrm{el}}(\phi,\B),
\end{align}
where the viscous and the elastic parts of the stress tensor are defined as
\begin{align}
\T_{\mathrm{visc}}(\phi,\pmb{\mathrm{v}},p)
&\coloneqq
\eta(\phi) \big( \nabla\pmb{\mathrm{v}} + (\nabla\pmb{\mathrm{v}})^T \big)
+ \lambda(\phi) \divergenz{\pmb{\mathrm{v}}}\I - p\I,
\\
\label{eq:T_elastic0}
\T_{\mathrm{el}}(\phi,\B) &\coloneqq 2W_{,\B}(\phi,\B) \B.
\end{align}
Note that $\T(\phi,\pmb{\mathrm{v}}, p, \B)$ corresponds to $\overline\T$ without the last term in \eqref{eq:constitutive_11}.
As remarked in the derivation, a viscoelastic description of Giesekus type is also possible, which then leads to the system of equations \eqref{eq:phi0}--\eqref{eq:v0} together with the constitutive equation
\begin{align}
\label{eq:giesekus0}
\partial_t \B + (\pmb{\mathrm{v}}\cdot\nabla)\B
+ \frac{1}{\tau(\phi)} \B \T_{\mathrm{el}}(\phi,\B)
&= \nabla\pmb{\mathrm{v}} \B + \B (\nabla\pmb{\mathrm{v}})^T.
\end{align}
However, the focus in this work lies on the viscoelastic description of Oldroyd-B type.
\subsubsection{Reformulations of the pressure}
We consider the following two reformulations of the pressure leading to a variant of \eqref{eq:v0}. For more examples, see \cite{ebenbeck_garcke_nurnberg_2020, GarckeLSS_2016}.
\begin{itemize}
\item
Using the fact that $\nabla\big( \frac{B}{2} \abs{\nabla\phi}^2 \big) = \divergenz{B \nabla\phi \otimes \nabla\phi} - B \Delta\phi\nabla\phi$ and defining $q \coloneqq p + f(\phi,\nabla\phi)+N(\phi,\sigma)$ yields
\begin{align*}
\nabla q
= \nabla p + (\mu - W_{,\phi}) \nabla\phi
+ N_{,\sigma}\nabla\sigma
+ \divergenz{B\nabla\phi\otimes\nabla\phi}.
\end{align*}
We can hence write \eqref{eq:v0} as
\begin{align}
\label{eq:v0b}
\rho \partial_t \pmb{\mathrm{v}} + \rho (\pmb{\mathrm{v}}\cdot\nabla)\pmb{\mathrm{v}}
= \divergenz{\T(\phi,\pmb{\mathrm{v}},q,\B)}
+ (\mu-W_{,\phi}) \nabla\phi + N_{,\sigma}\nabla\sigma.
\end{align}
Let us mention that the system \eqref{eq:phi0}--\eqref{eq:B0} with \eqref{eq:v0} replaced by \eqref{eq:v0b} matches with the general viscoelastic model \ref{P}. Moreover, replacing \eqref{eq:v0} by \eqref{eq:v0b} makes it possible that the convection terms in \eqref{eq:phi0} and \eqref{eq:sigma0} cancel out within specific testing procedures.
\item The following reformulation is of great importance when dealing with quasi-static nutrient equations, see \cite{ebenbeck_garcke_nurnberg_2020}.
Setting $q \coloneqq p + f(\phi,\nabla\phi)$ yields
\begin{align*}
\nabla q
= \nabla p + (\mu-N_{,\phi} - W_{,\phi}) \nabla\phi
+ \divergenz{B\nabla\phi\otimes\nabla\phi},
\end{align*}
so that \eqref{eq:v0} becomes
\begin{align}
\begin{split}
\label{eq:v0a}
\rho \partial_t \pmb{\mathrm{v}} + \rho (\pmb{\mathrm{v}}\cdot\nabla)\pmb{\mathrm{v}}
= \divergenz{\T(\phi,\pmb{\mathrm{v}},q,\B)}
+ (\mu - N_{,\phi} - W_{,\phi})\nabla\phi.
\end{split}
\end{align}
\end{itemize}
\subsubsection{A general energy identity}
In the following, we derive a general energy identity for the viscoelastic model of Oldroyd-B type \eqref{eq:phi0}--\eqref{eq:B0},
where we write \eqref{eq:v0b} instead of \eqref{eq:v0} and we write $p$ instead of $q$, such that the convection terms in \eqref{eq:phi0} and \eqref{eq:sigma0} cancel out within the following testing procedure.
Let us temporarily assume that there exists a sufficiently smooth solution of the above system. We multiply \eqref{eq:phi0} with $\mu$, \eqref{eq:mu0} with $-\partial_t\phi$ and \eqref{eq:sigma0} with $N_{,\sigma}$, integrate over $\Omega$ and use Green's formula. We then obtain:
\begin{align}
\label{eq:gen_eq_1}
0&=\int_\Omega \partial_t\phi \mu
+ \mu\nabla\phi\cdot\pmb{\mathrm{v}} + \phi\Gamma_{\pmb{\mathrm{v}}}\mu
+ m(\phi) \abs{\nabla\mu}^2
- \Gamma_\phi \mu \dv{x}
- \int_{\partial\Omega} m(\phi) \mu \nabla\mu\cdot\pmb{\mathrm{n}} \ \mathrm d \calH^{d-1} ,
\\
\label{eq:gen_eq_2}
0&= - \int_\Omega \partial_t\phi(\mu-N_{,\phi}-W_{,\phi}) \dv{x}
+ \ddv{}{t} \int_\Omega A\psi(\phi) + \frac{B}{2}\abs{\nabla\phi}^2 \dv{x}
- \int_{\partial\Omega} B\partial_t\phi \nabla\phi\cdot\pmb{\mathrm{n}} \ \mathrm d \calH^{d-1},
\\
\label{eq:gen_eq_3}
\begin{split}
0&= \int_\Omega \partial_t\sigma N_{,\sigma}
+ N_{,\sigma} \nabla\sigma \cdot \pmb{\mathrm{v}}
+ N_{,\sigma} \sigma \Gamma_{\pmb{\mathrm{v}}}
+ n(\phi) \abs{\nabla N_{,\sigma}}^2
+ \Gamma_\sigma N_{,\sigma} \dv{x}
\\
&\quad - \int_{\partial\Omega} n(\phi) N_{,\sigma} \nabla N_{,\sigma} \cdot\pmb{\mathrm{n}} \ \mathrm d \calH^{d-1} .
\end{split}
\end{align}
Next, we multiply \eqref{eq:v0b} with $\pmb{\mathrm{v}}$ and integrate over $\Omega$ and use Green's formula so that we have
\begin{align}
\label{eq:gen_eq_4}
\begin{split}
0 &=
\int_\Omega \ddv{}{t}\left(\frac{1}{2} \rho \abs{\pmb{\mathrm{v}}}^2 \right)
+ \rho (\pmb{\mathrm{v}}\cdot\nabla)\pmb{\mathrm{v}} \cdot \pmb{\mathrm{v}}
+ 2\eta(\phi) \abs{\D(\pmb{\mathrm{v}})}^2 + \lambda(\phi) (\divergenz{\pmb{\mathrm{v}}})^2
- p\Gamma_{\pmb{\mathrm{v}}} \dv{x}
\\
&\quad + \int_\Omega(- (\mu-W_{,\phi})\nabla\phi - N_{,\sigma} \nabla\sigma) \cdot\pmb{\mathrm{v}}
+ 2 (W_{,\B} \B) : \nabla\pmb{\mathrm{v}} \dv{x}
- \int_{\partial\Omega} \big(\T(\phi,\pmb{\mathrm{v}},p,\B)\pmb{\mathrm{n}} \big)\cdot \pmb{\mathrm{v}} \ \mathrm d \calH^{d-1}.
\end{split}
\end{align}
Here we used that $\D(\pmb{\mathrm{v}}) :\nabla\pmb{\mathrm{v}} = \D(\pmb{\mathrm{v}}) :\D(\pmb{\mathrm{v}})$ and $\divergenz{\pmb{\mathrm{v}}}\I : \nabla\pmb{\mathrm{v}} = (\divergenz{\pmb{\mathrm{v}}})^2$.
After that, we multiply \eqref{eq:B0} with $W_{,\B}$ and we integrate over $\Omega$ and apply Green's formula. This yields
\begin{align}
\label{eq:gen_eq_5}
0 &= \int_\Omega \Big( \partial_t\B
+ (\pmb{\mathrm{v}}\cdot\nabla) \B
- \nabla\pmb{\mathrm{v}}\B - \B(\nabla\pmb{\mathrm{v}})^T
+ \frac{2}{\tau(\phi)} W_{,\B} \B \Big) : W_{,\B} \dv{x}.
\end{align}
For the reader's convenience, we now note some useful identities concerning the velocity and the Cauchy--Green tensor:
\begin{align*}
\int_\Omega \rho (\pmb{\mathrm{v}}\cdot\nabla)\pmb{\mathrm{v}} \cdot \pmb{\mathrm{v}}
&=
- \int_\Omega \Gamma_{\pmb{\mathrm{v}}} \Big(\frac{1}{2}\rho\abs{\pmb{\mathrm{v}}}^2\Big) \dv{x}
+ \int_{\partial\Omega} \pmb{\mathrm{v}}\cdot\pmb{\mathrm{n}} \Big(\frac{1}{2}\rho\abs{\pmb{\mathrm{v}}}^2\Big)\dv{x}
\\
\big(\nabla\pmb{\mathrm{v}}\B + \B(\nabla\pmb{\mathrm{v}})^T\big) : W_{,\B}
&= 2 (W_{,\B} \B) : \nabla\pmb{\mathrm{v}},
\\
\partial_t^\bullet W(\phi,\B)
&= \partial_t^\bullet \phi W_{,\phi}
+ \partial_t^\bullet \B : W_{,\B},
\\
\int_\Omega (\pmb{\mathrm{v}}\cdot\nabla) W(\phi,\B) \dv{x}
&= - \int_\Omega \Gamma_{\pmb{\mathrm{v}}} W(\phi,\B) \dv{x}
+ \int_{\partial\Omega} \pmb{\mathrm{v}}\cdot\pmb{\mathrm{n}} W(\phi,\B)\dv{x},
\\
(W_{,\B} \B) : W_{,\B} & \geq 0, \quad \text{ if } \B \text{ is positive definite.}
\end{align*}
Collecting all equations \eqref{eq:gen_eq_1}--\eqref{eq:gen_eq_5}, we obtain the general energy identity for the viscoelastic model for tumour growth:
\begin{align}
\label{eq:energy0}
\begin{split}
0 &=
\ddv{}{t} \Big( \int_\Omega A\psi(\phi) + \frac{B}{2} \abs{\nabla\phi}^2 + N(\phi,\sigma) + W(\phi,\B)
+ \frac{1}{2} \rho \abs{\pmb{\mathrm{v}}}^2 \dv{x} \Big)
\\
&\quad
+ \int_\Omega m(\phi) \abs{\nabla\mu}^2
+ n(\phi)\abs{\nabla N_{,\sigma}}^2
+ 2 \eta(\phi) \abs{\D(\pmb{\mathrm{v}})}^2
+ \lambda(\phi)(\divergenz{\pmb{\mathrm{v}}})^2
+ \frac{2}{\tau(\phi)} (W_{,\B} \B):W_{,\B} \dv{x}
\\
&\quad
+ \int_\Omega - \Gamma_\phi \mu + \Gamma_\sigma N_{,\sigma}
+ \Big(\mu\phi + N_{,\sigma}\sigma - p - W(\phi,\B)
- \frac{1}{2} \rho \abs{\pmb{\mathrm{v}}}^2 \Big) \Gamma_{\pmb{\mathrm{v}}} \dv{x}
\\
&\quad
- \int_{\partial\Omega} m(\phi) \mu\nabla\mu \cdot\pmb{\mathrm{n}}
+ n(\phi) N_{,\sigma} \nabla N_{,\sigma} \cdot\pmb{\mathrm{n}}
+ B \partial_t\phi \nabla\phi\cdot\pmb{\mathrm{n}} \ \mathrm d \calH^{d-1}
\\
&\quad
+ \int_{\partial\Omega} \pmb{\mathrm{n}}\cdot\pmb{\mathrm{v}} \Big(
W(\phi,\B) + \frac{1}{2} \rho \abs{\pmb{\mathrm{v}}}^2 \Big)
- \big(\T(\phi,\pmb{\mathrm{v}},p,\B)\pmb{\mathrm{n}} \big) \cdot\pmb{\mathrm{v}} \ \mathrm d \calH^{d-1}.
\end{split}
\end{align}
Note that in order to study existence theory, there are several difficulties that arise from this general identity and heavily depend on the choices for the potential $\psi(\phi)$, the energy densities $N(\phi,\sigma), W(\phi,\B)$, the source terms $\Gamma_\phi, \Gamma_\sigma, \Gamma_{\pmb{\mathrm{v}}}$, the functions $m(\phi), n(\phi), \eta(\phi), \lambda(\phi), \tau(\phi)$ and the initial and boundary conditions.
\subsubsection{Initial and boundary conditions}
For $\phi$, $\sigma$, $\pmb{\mathrm{v}}$ and $\B$, we impose the initial conditions
\begin{align}
\phi(\cdot,0) = \phi_0, \quad
\sigma(\cdot,0) = \sigma_0, \quad
\pmb{\mathrm{v}}(\cdot,0) = \pmb{\mathrm{v}}_0, \quad
\B(\cdot,0) = \B_0 \quad \text{ a.e.~in } \Omega.
\end{align}
We prescribe homogeneous Neumann boundary conditions on $\partial\Omega$ for the phase field variable and the chemical potential, i.e.
\begin{align}
\nabla\phi\cdot\pmb{\mathrm{n}} = \nabla\mu\cdot\pmb{\mathrm{n}} = 0
\quad \text{ a.e.~on } \partial\Omega \times (0,T).
\end{align}
For the nutrient we prescribe Robin-type boundary conditions
\begin{align}
n(\phi) \nabla N_{,\sigma} \cdot \pmb{\mathrm{n}} = K(\sigma_\infty - \sigma)
\quad \text{ a.e.~on } \partial\Omega \times (0,T),
\end{align}
where the constant $K\geq0$ is referred to as the boundary permeability and $\sigma_\infty$ denotes a given nutrient supply at the boundary.
The last boundary condition is depending on the choice of $\Gamma_{\pmb{\mathrm{v}}}$. In this work, we consider $\Gamma_{\pmb{\mathrm{v}}}=0$. Hence, we prescribe no-slip (homogeneous Dirichlet) boundary conditions for the velocity, i.e.
\begin{align}
\pmb{\mathrm{v}} = \pmb 0
\quad \text{ a.e.~on } \partial\Omega \times (0,T).
\end{align}
In the case $\Gamma_{\pmb{\mathrm{v}}}\not=0$, we prescribe the following boundary condition for $\T$,
\begin{align}
\Big(\T(\phi,\pmb{\mathrm{v}},p,\B) - W(\phi,\B) \I - \frac{1}{2} \rho \abs{\pmb{\mathrm{v}}}^2 \I \Big) \pmb{\mathrm{n}} = \pmb 0
\quad \text{ a.e.~on } \partial\Omega \times (0,T),
\end{align}
so that the last line in the general energy identity \eqref{eq:energy0} vanishes.
In the case of no-slip boundary conditions for the velocity, we recall that no further boundary conditions for the Cauchy--Green tensor $\B$ are needed, as the evolution equation \eqref{eq:B0} is a hyperbolic partial differential equation of first order and has no incoming characteristics at the boundary.
\subsubsection{Specific choices for the source terms}
Now, we explain possible specifications for the source terms $\Gamma_\phi,\Gamma_\sigma,\Gamma_{\pmb{\mathrm{v}}}$.
\begin{itemize}
\item Usually the source terms $\Gamma_\phi$ and $\Gamma_{\pmb{\mathrm{v}}}$ are closely related. In particular,
\begin{align}
\Gamma_\phi \coloneqq \frac{1}{\bar\rho_1}\Gamma_{1} - \frac{1}{\bar\rho_{-1}}\Gamma_{-1},
\qquad
\Gamma_{\pmb{\mathrm{v}}} \coloneqq \frac{1}{\bar\rho_1}\Gamma_{1} + \frac{1}{\bar\rho_{-1}}\Gamma_{-1},
\end{align}
where $\bar\rho_1,\bar\rho_{-1}$ are the mass densities of the tumour cells and healthy cells, respectively, and $\Gamma_{1}, \Gamma_{-1}$ are source or sink terms in the mass balance laws for the single components of the mixture, see \cite{ebenbeck_garcke_nurnberg_2020, GarckeLSS_2016}. By the assumption of matching mass densities, we have $\rho= \bar\rho_1 = \bar\rho_{-1}$ and hence
\begin{align}
\Gamma_\phi = \frac{1}{\rho} (\Gamma_{1} - \Gamma_{-1}), \qquad \Gamma_{\pmb{\mathrm{v}}} = \frac{1}{\rho}( \Gamma_{1} + \Gamma_{-1}).
\end{align}
A common choice of $\Gamma_{\pmb{\mathrm{v}}}$ is obtained by assuming no gain or loss of mass locally, i.e.~$\Gamma_1 \coloneqq - \Gamma_{-1}$, which implicates
\begin{align}
\label{source_1a}
\Gamma_\phi = \frac{2}{\rho} \Gamma_1, \qquad \Gamma_{\pmb{\mathrm{v}}} = 0.
\end{align}
On the other hand, setting $\Gamma_{-1} \coloneqq 0$ yields
\begin{align}
\label{source_1b}
\Gamma_\phi = \Gamma_{\pmb{\mathrm{v}}} = \frac{1}{\rho} \Gamma_1.
\end{align}
\item
Motivated by linear kinetics, Garcke and co-authors \cite{GarckeLSS_2016} suggested
\begin{align}
\label{eq:source_2}
&\Gamma_\phi(\phi,\sigma)
\coloneqq ( \ensuremath{\mathcal{P}} \sigma - \mycal{A} ) h(\phi),
\qquad
\Gamma_\sigma(\phi,\sigma)
\coloneqq \mycal{C} \sigma h(\phi)
+ \mycal{B} (\sigma_B - \sigma),
\end{align}
where $\ensuremath{\mathcal{P}}, \mycal{A}, \mycal{C}$ denote the proliferation rate, apoptosis rate and consumption rate.
Moreover, $h(\cdot)$ is an interpolation function with $h(-1) = 0$ and $h(1) = 1$ which ensures that proliferation, apoptosis and nutrient consumption only take place in the tumour phase. The simplest example ist $h(\phi) = \tfrac{1}{2} (1+\phi)$.
Besides, $\mycal{B} (\sigma_B - \sigma)$ models the nutrient supply from an existing vasculature.
\item To account for the influence of mechanical stresses on tissue growth, the authors of \cite{garcke_lam_signori_2021_optimal_control} proposed to scale the proliferation term $ \ensuremath{\mathcal{P}} \sigma h(\phi)$ in \eqref{eq:source_2} with $\tilde f(\T_{\mathrm{el}}) \coloneqq (1 + \abs{\T_{\mathrm{el}}}^2)^{-1/2}$,
which decreases when elastic stresses increase.
This motivates to introduce the choice
\begin{align}
\label{eq:source_3}
\Gamma_\phi(\phi,\sigma,\B) \coloneqq \big( \ensuremath{\mathcal{P}} \sigma f(\phi,\B) -\mycal{A} \big) h(\phi) ,
\quad \text{where} \quad
f(\phi,\B) \coloneqq \left(1 + \abs{\T_{\mathrm{el}}(\phi,\B)}^2\right)^{-\frac{1}{2}},
\end{align}
where $h(\cdot), \ensuremath{\mathcal{P}}, \mycal{A}$ are as in \eqref{eq:source_2}.
\item
Based on linear phenomenological laws for chemical reactions, the authors of \cite{hawkins_2012} proposed to take
\begin{align}
\label{eq:source_4}
&\Gamma_\phi(\phi,\mu,\sigma)
= \Gamma_\sigma(\phi,\mu,\sigma)
\coloneqq P(\phi) \big( N_{,\sigma}(\phi,\sigma) - \mu \big),
\end{align}
with a non-negative proliferation function $P(\cdot)$, e.g., $P(\phi) = \max\left\{0, \delta P_0 (1+\phi)\right\}$, where $\delta, P_0$ are positive constants.
For a different example of $P(\cdot)$, we refer to, e.g., \cite{ebenbeck_garcke_nurnberg_2020}.
\end{itemize}
\subsubsection{Specific choices for the nutrient and elastic energy density}
In the following, we specify the nutrient energy density and give several examples for the elastic energy density.
\begin{itemize}
\item
In the literature, the nutrient energy density usually takes the form
\begin{align}
\label{eq:energy_nutrient}
N(\phi,\sigma) \coloneqq \frac{\chi_\sigma}{2} \abs{\sigma}^2
+ \chi_\phi \sigma(1-\phi),
\end{align}
with
\begin{align*}
N_{,\phi}(\phi,\sigma) = -\chi_\phi \sigma,
\qquad
N_{,\sigma}(\phi,\sigma) = \chi_\sigma \sigma - \chi_\phi \phi.
\end{align*}
The first term $\frac{\chi_\sigma}{2} \abs{\sigma}^2$ increases the energy in the presence of nutrients, where $\chi_\sigma > 0$ denotes the diffusivity of the nutrient.
The second term $\chi_\phi \sigma(1-\phi)$ can be regarded as chemotaxis energy which accounts for interactions between the tumour and the nutrient. Here, the constant $\chi_\phi \geq 0$ can be seen as a sensitivity parameter for chemotaxis and active uptake mechanisms which favours unstable tumour growth \cite{GarckeLSS_2016}. Let us point out that the nutrient energy density can have a negative sign in general if $\chi_\phi \not= 0$, which is one difficulty in the derivation of suitable \textit{a priori} estimates from the general energy identity \eqref{eq:energy0}.
\item
Physically motivated by the theory of constitutive relations for isotropic compressible elastic materials, where large elastic stresses are penalized, the elastic energy density is supposed to satisfy
\begin{align}
W(\phi,\B) \to +\infty, \quad \text{if} \quad
\begin{cases}
\abs{\B} \to +\infty, \\
\det(\B) \to 0.
\end{cases}
\end{align}
Hence, an infinite amount of energy is required such that the material can be expanded to infinite volume or compressed to a single point \cite{horgan_2004_constitutive}.
Note that $\B\coloneqq \B_e = \F_e \F_e^T$ is always symmetric and positive semi-definite by definition while the elastic part of the deformation gradient $\F_e$ is not symmetric in general.
\item
An example for the elastic energy density we have in mind is
\begin{align}
\label{eq:energy_oldroyd}
W(\phi,\B)
\coloneqq \frac{1}{2} \kappa(\phi) \trace (\B)
- \frac{1}{2} \kappa_0(\phi)\ln(\det\B),
\end{align}
with
\begin{align*}
W_{,\phi}(\phi,\B) = \frac{1}{2} \kappa'(\phi) \trace (\B) - \frac{1}{2} \kappa_0'(\phi) \ln(\det\B),
\qquad
W_{,\B}(\phi,\B) = \frac{1}{2} \kappa(\phi) \I - \frac{1}{2} \kappa_0(\phi) \B^{-1},
\end{align*}
where $\kappa(\phi), \kappa_0(\phi) > 0$ denote elasticity parameter functions
depending on the material and $\I\in\R^{d\times d}$ denotes the identity matrix.
Hence, the elastic stress tensor is $\T_{\mathrm{el}}(\phi,\B) = \kappa(\phi) \B - \kappa_0(\phi) \I $ and \eqref{eq:B0} is specified by
\begin{align}
\label{eq:oldroyd}
\partial_t \B + (\pmb{\mathrm{v}}\cdot\nabla)\B
+\frac{1}{\tau(\phi)} \left( \kappa(\phi) \B - \kappa_0(\phi) \I \right)
&= \nabla\pmb{\mathrm{v}}\B + \B (\nabla\pmb{\mathrm{v}})^T.
\end{align}
In the case $\kappa(\phi)=\kappa_0(\phi)$ and for fixed $\phi$, \eqref{eq:oldroyd} is exactly the classical viscoelastic Oldroyd-B equation \cite{barrett_boyaval_2009, malek_prusa_2018} for the left Cauchy--Green tensor $\B$.
\item
For a given $b\gg1$, the elastic energy density for the viscoelastic FENE-P model \cite{barrett_2018_fene-p} reads
\begin{align}
\label{eq:energy_fene-p}
W(\B) \coloneqq - \frac{b}{2} \ln\left( 1-\frac{\trace\B}{b}\right)
- \frac{1}{2} \trace (\ln\B),
\qquad \text{with}
\qquad W_{,\B}(\B) = \frac{1}{2} \left( 1 - \frac{\trace\B}{b} \right)^{-1} \I - \frac{1}{2} \B^{-1}.
\end{align}
Note that $W(\B) \to +\infty$ if $\trace(\B) \to b$ or if $\B$ becomes singular.
Here, the corresponding elastic stress tensor is $\T_{\mathrm{el}}(\B) = \left( 1 - \frac{\trace\B}{b} \right)^{-1} \B - \I$ and the constitutive law \eqref{eq:B0} reads
\begin{align}
\label{eq:fene-p}
\partial_t \B + (\pmb{\mathrm{v}}\cdot\nabla)\B
+\frac{1}{\tau(\phi)} \left( \left( 1 - \frac{\trace\B}{b} \right)^{-1} \B - \I \right)
&= \nabla\pmb{\mathrm{v}}\B + \B (\nabla\pmb{\mathrm{v}})^T.
\end{align}
Moreover, the Oldroyd-B equation \eqref{eq:oldroyd} with $\kappa(\phi)=\kappa_0(\phi) = 1$ can be recovered by formally sending $b\to+\infty$.
\item The authors of \cite{Brunk_Lukacova_2021} study a generalized viscoelastic Peterlin model for phase separation which is based on the elastic energy density
\begin{align}
\label{eq:energy_gen_peterlin}
W(\B) \coloneqq \frac{1}{4} \trace(\B)^2 - \frac{1}{2} \trace(\ln\B)
\qquad \text{with} \qquad
W_{,\B}(\B) = \frac{1}{2} \trace(\B) \I - \frac{1}{2} \B^{-1},
\end{align}
and the elastic stress tensor $\T_{\mathrm{el}}(\B) = \trace(\B) \B - \I$. Moreover, the tensor $\B$ satisfies a generalized evolution equation of the form
\begin{align}
\label{eq:gen_peterlin}
\partial_t \B + (\pmb{\mathrm{v}}\cdot\nabla)\B
+ f(\phi) g(\trace\B) \left( \trace(\B) \B - \I \right)
&= \nabla\pmb{\mathrm{v}}\B + \B (\nabla\pmb{\mathrm{v}})^T,
\end{align}
where $(f(\phi)g(\trace\B))^{-1}$ denotes a generalized relaxation time depending on the phase field variable $\phi$ and the trace of $\B$.
Besides, a more generalized approach has been studied in \cite{Lukacova_2017} which includes several viscoelastic models.
\end{itemize}
The elastic energy densities \eqref{eq:energy_oldroyd}, \eqref{eq:energy_fene-p}, \eqref{eq:energy_gen_peterlin} in the one dimensional case are visualized in Figure \ref{fig:elastic_energy}.
\begin{figure}[ht]
\centering
\begin{tikzpicture}[scale=\textwidth/15.2cm]
\begin{axis}[
axis x line=middle,
axis y line=middle,
xmin=0,xmax=4,
ymin=0,ymax=3,
xlabel=$x$,
xtick={1, 2, 3},
xticklabels={1, 2, 3},
ytick={1, 2},
yticklabels={1, 2},
]
\addplot[samples=400, smooth, sharp plot
{0.5*x - 0.5*ln(x)}
node [pos=0.65, below right] {Oldroyd-B};
\addplot[samples=400, smooth, blue, sharp plot
{ - 0.5*3.5* ln(1-x/3.5) - 0.5*ln(x)}
node [pos=0.34, left] {FENE-P};
\addplot[samples=400, smooth, red, sharp plot
{0.25*x*x - 0.5*ln(x)}
node [pos=0.55, right] {Peterlin};
\end{axis}
\end{tikzpicture}
\caption{The elastic energy densities for the Oldroyd-B model \eqref{eq:energy_oldroyd} with $\kappa=\kappa_0=1$ (black), the FENE-P model \eqref{eq:energy_fene-p} with $b=3.5$ (blue) and the generalized Peterlin model \eqref{eq:energy_gen_peterlin} (red) in the one dimensional case.}
\label{fig:elastic_energy}
\end{figure}
\subsubsection{Including growth in the equation for \texorpdfstring{$\B$}{B}}
Mechanical stresses increase when tumour cells proliferate \cite{ambrosi_2009}. Therefore, instead of \eqref{eq:B0}, we may consider
\begin{align}
\label{eq:B_growth}
\partial_t \B + (\pmb{\mathrm{v}}\cdot\nabla)\B
+ \frac{1}{\tau(\phi)} \T_{\mathrm{el}}(\phi,\B)
&= \nabla\pmb{\mathrm{v}} \B + \B (\nabla\pmb{\mathrm{v}})^T
- \gamma(\phi,\mu,\sigma,\B) \B,
\end{align}
where the scalar function $\gamma(\phi,\mu,\sigma,\B)$ acts as a source or sink term for the left Cauchy--Green tensor and can depend on $\phi$, $\mu$, $\sigma$ and $\B$.
This source term can be derived from the multiplicative decomposition
\begin{align}
\F = \F_{e} \F_{d} \F_{g},
\end{align}
where $\F_{g}$ describes deformation by growth \cite{ambrosi_2009}. Assuming spherical growth, i.e.~$\F_{g} = g \I$, then, analogously to \eqref{eq:F_e}, we obtain
\begin{align}
\partial_t^\bullet \F_{e} = \L \F_{e} - \F_{e} \L_{d} - \F_{e} (\partial_t^\bullet g) \frac{1}{g},
\end{align}
and, as $\B_{e}=\F_{e}\F_{e}^T$, we have
\begin{align}
\partial_t^\bullet \B_{e} = \L \B_{e} + \B_{e} \L^T
- 2 \F_{e} \D_{d} \F_{e}^T
- 2 \B_{e} (\partial_t^\bullet g) \frac{1}{g} ,
\end{align}
which coincides with \cite[eq.~(3.29)]{ambrosi_2009}. Then, \eqref{eq:B_growth} is recovered with
\begin{align}
\D_{d} = \frac{1}{\tau(\phi)} \Big( \F_{e}^{-1} \fracdel{e}{\B_{e}} \F_{e} \Big),
\qquad
\partial_t^\bullet g = \frac{1}{2} g \gamma(\phi,\mu,\sigma,\B_{e}).
\end{align}
A different way to obtain \eqref{eq:B_growth} can be motivated by the constitutive choice
\begin{align}
\label{eq:constitutive_growth}
\D_{d} &= \frac{1}{\tau(\phi)} \Big( \F_{e}^{-1} \fracdel{e}{\B_{e}} \F_{e} \Big)
+ \frac{1}{2} \gamma(\phi,\mu,\sigma,\B_{e}) \I,
\end{align}
instead of \eqref{eq:constitutive_7}, and by inserting this into \eqref{eq:B_e}.
On the right-hand side of \eqref{eq:constitutive_growth}, the first term accounts for stress relaxation while the second term is responsible for growth induced stresses.
In \cite{ambrosi_2009},
it has been suggested that $\gamma$ is proportional to the source function $\Gamma_\phi$.
Therefore, we propose the choice
\begin{align}
\gamma(\phi,\mu,\sigma,\B) = c \Gamma_\phi(\phi,\mu,\sigma,\B),
\end{align}
where $\Gamma_\phi$ can be given by \eqref{eq:source_2}, \eqref{eq:source_3} or \eqref{eq:source_4}, and $c\in\R$.
\subsection{Variants of the model}
Now, we present several variants of the model \eqref{eq:phi0}--\eqref{eq:B0} and exemplify the motivation of these variants with strategies of related works in the literature.
\subsubsection{Limit of a small Reynolds number}
In biological processes, the Reynolds number is often very small. Then, a non-dimensionalization argument motivates to neglect the terms $\rho \partial_t\pmb{\mathrm{v}} + \rho (\pmb{\mathrm{v}}\cdot\nabla)\pmb{\mathrm{v}}$ in the momentum equation. Hence, we introduce the viscoelastic model with quasi-static momentum equation, which is the system \eqref{eq:phi0}--\eqref{eq:B0} with \eqref{eq:v0} replaced by
\begin{align}
\label{eq:v0_Brinkman}
- \divergenz{2\eta(\phi) \D(\pmb{\mathrm{v}})
+ \lambda(\phi) \divergenz{\pmb{\mathrm{v}}}\I}
+ \nabla p
= \divergenz{\T_{\mathrm{el}}(\phi,\B)}
+ (\mu-W_{,\phi}) \nabla\phi + N_{,\sigma}\nabla\sigma.
\end{align}
In absence of the elastic effects, this model corresponds to a special case of the Cahn--Hilliard--Brinkman model for tumour growth which has been extensively studied in, e.g., \cite{ebenbeck_2019_analysis, ebenbeck_2019_singular_limit, ebenbeck_garcke_nurnberg_2020, ebenbeck_2020_optimal_control, ebenbeck_2019_medication, ebenbeck_2021_singular_potentials}.
\subsubsection{Limit of a short nutrient diffusion timescale}
From the modelling point of view, sometimes a quasi-static nutrient equation instead of \eqref{eq:sigma0} seems realistic since the timescale of nutrient diffusion can be quite small compared to the tumour doubling timescale.
Such approaches have been introduced for related models in the literature, e.g., for the Cahn--Hilliard--Brinkman model \cite{ebenbeck_garcke_nurnberg_2020} or the Cahn--Hilliard--Darcy model \cite{GarckeLSS_2016}.
Hence, we introduce the viscoelastic model with quasi-static nutrients which corresponds to \eqref{eq:phi0}--\eqref{eq:B0} with \eqref{eq:sigma0} replaced by
\begin{align}
\label{eq:sigma0_quasistatic}
0&= \divergenz{n(\phi) \nabla N_{,\sigma}(\phi,\sigma)} - \Gamma_\sigma(\phi,\sigma).
\end{align}
\subsubsection{Interpolation between different rheologies}
The main concept of viscoelastic models is that both viscous and elastic effects are taken into account. In the context of tumour growth, Bresch and co-authors \cite{bresch_2009} proposed a viscoelastic multiphase tumour model of Oldroyd-B type in presence of healthy cells, tumour cells and extracellular liquids, where the material parameters depend on the phases. For example, healthy cells are considered to be more elastic, extracellular liquids are supposed to be fully viscous and tumour cells are assumed to combine both elastic and viscous properties.
We now illustrate the idea of different material laws on the basis of the model \eqref{eq:phi0}--\eqref{eq:B0} with the help of suitable choices of the viscosities $\eta(\phi), \lambda(\phi)$ and the relaxation time $\tau(\phi)$. We can account for a Newtonian fluid without elastic stresses by sending the relaxation time $\tau(\phi)$ to zero, which leads to \eqref{eq:phi0}--\eqref{eq:B0} with \eqref{eq:v0}--\eqref{eq:B0} replaced by
\begin{align}
\rho \partial_t \pmb{\mathrm{v}} + \rho (\pmb{\mathrm{v}}\cdot \nabla)\pmb{\mathrm{v}}
&= \divergenz{\T_{\mathrm{visc}}(\phi,\pmb{\mathrm{v}}, p)}
- \divergenz{B \nabla\phi \otimes \nabla\phi},
\\
\T_{\mathrm{el}}(\phi,\B)
&= 0.
\end{align}
Besides, we can allow a viscoelastic description of Maxwell type by neglecting the viscosities $\eta(\phi),\lambda(\phi)$. Hence, the model corresponds to \eqref{eq:phi0}--\eqref{eq:B0} with \eqref{eq:v0}--\eqref{eq:B0} replaced by
\begin{align}
\rho \partial_t \pmb{\mathrm{v}} + \rho (\pmb{\mathrm{v}}\cdot \nabla)\pmb{\mathrm{v}} + \nabla p
&= \divergenz{\T_{\mathrm{el}}(\phi, \B)}
- \divergenz{B \nabla\phi \otimes \nabla\phi},
\\
\partial_t \B + (\pmb{\mathrm{v}}\cdot\nabla)\B
+ \frac{1}{\tau(\phi)} \T_{\mathrm{el}}(\phi,\B)
&= \nabla\pmb{\mathrm{v}} \B + \B (\nabla\pmb{\mathrm{v}})^T.
\end{align}
Moreover, by sending the relaxation time $\tau(\phi)$ to infinity, we obtain the viscoelastic material law of Kelvin--Voigt type and hence recover the Oldroyd-B equation with infinite Weissenberg number \eqref{eq:B0_infinite_weissenberg} from \eqref{eq:B0} in the limit $\tau(\phi)\to\infty$, i.e.~the model corresponds to \eqref{eq:phi0}--\eqref{eq:div_v0} with
\begin{align}
\rho \partial_t \pmb{\mathrm{v}} + \rho (\pmb{\mathrm{v}}\cdot \nabla)\pmb{\mathrm{v}}
&= \divergenz{\T(\phi,\pmb{\mathrm{v}}, p, \B)}
- \divergenz{B \nabla\phi \otimes \nabla\phi},
\\
\partial_t \B + (\pmb{\mathrm{v}}\cdot\nabla)\B
&= \nabla\pmb{\mathrm{v}} \B + \B (\nabla\pmb{\mathrm{v}})^T.
\end{align}
Further, the material law for an elastic solid can be obtained by neglecting the viscosities and assuming an infinite relaxation time. Hence, the model reads \eqref{eq:phi0}--\eqref{eq:div_v0} combined with
\begin{align}
\rho \partial_t \pmb{\mathrm{v}} + \rho (\pmb{\mathrm{v}}\cdot \nabla)\pmb{\mathrm{v}} + \nabla p
&= \divergenz{\T_{\mathrm{el}}(\phi,\B)}
- \divergenz{B \nabla\phi \otimes \nabla\phi},
\\
\partial_t \B + (\pmb{\mathrm{v}}\cdot\nabla)\B
&= \nabla\pmb{\mathrm{v}} \B + \B (\nabla\pmb{\mathrm{v}})^T.
\end{align}
Of course, we can handle different material laws for the respective phases $\phi=1$ and $\phi=-1$ at once by specifying the viscosities and the relaxation time for the respective phases.
An overview can be found in Table \ref{tab:material_parameters}, which has been adapted from \cite{mokbel_abels_aland_2018}.
\begin{table}[ht]
\centering
\begin{tabular}{|c|c|c|c|}
viscosities
& relaxation time
& material law
& stress tensor
\\
$\eta(\phi),\lambda(\phi)$
& $\tau(\phi)$
&
& $\T(\phi,\pmb{\mathrm{v}},p,\B)$
\\ \hline
$*$ & 0
& Newtonian fluid
& $\T_{\mathrm{visc}}(\phi,\pmb{\mathrm{v}},p)$
\\
0 & $*$
& Maxwell (viscoelastic)
& $- p \I + \T_{\mathrm{el}}(\phi,\B)$
\\
$*$ & $+\infty$
& Kelvin--Voigt (viscoelastic)
& $\T_{\mathrm{visc}}(\phi,\pmb{\mathrm{v}},p) + \T_{\mathrm{el}}(\phi,\B)$
\\
0 & $+\infty$
& elastic solid
& $- p \I + \T_{\mathrm{el}}(\phi,\B)$
\end{tabular}
\caption{Different material laws can be obtained by a different choice of the viscosities and the relaxation time, where `$*$' marks parameters that are given by the physical problem itself; adapted from \cite{mokbel_abels_aland_2018}.}
\label{tab:material_parameters}
\end{table
\subsubsection{Evolution of the elastic stress tensor}
In the literature, viscoelastic models related to the works of Oldroyd \cite{Oldroyd_1950} or Giesekus \cite{giesekus_1982} are sometimes stated in terms of the elastic stress tensor $\T_{\mathrm{el}}(\phi,\B) = 2 W_{,\B}(\phi,\B)\B$ instead of the left Cauchy--Green tensor $\B$.
Therefore, we shortly explain how the evolution of the elastic stress tensor is resulting from the evolution equation of the left Cauchy--Green tensor $\B$ for the case $W(\B) = \frac{1}{2} \kappa \trace(\B - \ln\B)$, where, for simplicity, $\kappa$ is constant.
Then, in the Oldroyd-B model, the evolution equation \eqref{eq:B0} for the Cauchy--Green tensor $\B$ is equivalent to the following evolution equation for the elastic stress tensor $\T_{\mathrm{el}} = \kappa(\B-\I)$:
\begin{subequations}
\begin{align}
\label{eq:B0c}
\partial_t \T_{\mathrm{el}} + (\pmb{\mathrm{v}}\cdot\nabla) \T_{\mathrm{el}}
+ \frac{\kappa}{\tau(\phi)} \T_{\mathrm{el}} - 2\kappa\D(\pmb{\mathrm{v}})
&= \nabla\pmb{\mathrm{v}} \T_{\mathrm{el}} + \T_{\mathrm{el}} (\nabla\pmb{\mathrm{v}})^T,
\end{align}
while, in the Giesekus model, \eqref{eq:giesekus0} is equivalent to the following evolution equation for $\T_{\mathrm{el}}$:
\begin{align}
\label{eq:giesekus_c}
\partial_t \T_{\mathrm{el}} + (\pmb{\mathrm{v}}\cdot\nabla)\T_{\mathrm{el}}
+ \frac{1}{\tau(\phi)} \T_{\mathrm{el}}^2
+ \frac{\kappa}{\tau(\phi)} \T_{\mathrm{el}} - 2\kappa\D(\pmb{\mathrm{v}})
&= \nabla\pmb{\mathrm{v}} \T_{\mathrm{el}} + \T_{\mathrm{el}} (\nabla\pmb{\mathrm{v}})^T.
\end{align}
\end{subequations}
For more details concerning the calculation, we refer to \cite[eq.~(205)]{malek_prusa_2018} for the Oldroyd-B model and to \cite[eq.~(187)]{malek_prusa_2018} for the Giesekus model.
\section{Finite element approximation of the model with stress diffusion}
\label{sec:fem}
In this section, we provide a proof for Theorem \ref{theorem:weak_solution} with the following strategy.
First, we attend some ideas of \cite[Sec.~5]{barrett_boyaval_2009} and introduce a finite element approximation of the problem \ref{P_alpha_delta}, which helps us to mimic the inequality \eqref{eq:delta_4} on the fully discrete level (see Section \ref{sec:stability}) and to show that there exist stable $\delta$-regularized discrete solutions in abritrary dimensions $d\in\{2,3\}$, see Section \ref{sec:existence}.
In Section \ref{sec:delta_to_zero}, we pass to the limit $\delta\to 0$ and obtain the existence of discrete functions for $d\in\{2,3\}$, including a positive definite discrete left Cauchy--Green tensor, which solve a finite element approximation of the problem \ref{P_alpha}.
After that, we first improve the regularity of discrete solutions in arbitrary dimensions $d\in\{2,3\}$ in Section \ref{sec:regularity} and then restrict to $d=2$ to improve the regularity of the discrete Cauchy--Green tensor and the discrete velocity in Section \ref{sec:regularity_2D}.
Finally, in Section \ref{sec:convergence}, we send the discretization parameters to zero in order to obtain existence of a global-in-time weak solution to the problem \ref{P_alpha} in two dimensions.
Let us introduce the notation for the fully-discrete finite element approximation.
From now on, we throughout assume that \ref{A1} holds, i.e., suppose that $T>0$ and $\Omega \subset\R^d$, $d\in\{2,3\}$, is a convex, polygonal domain with boundary $\partial\Omega$. We split the time interval $[0,T)$ into intervals $[t^{n-1},t^n)$ with $t^n = n \Delta t$ and $t^{N_T}=T$, where $\Delta t>0$ and $n=0,...,N_T$.
We require $\{\ensuremath{\mathcal{T}}_h\}_{h>0}$ to be a quasi-uniform family of conforming triangulations with mesh parameter $h>0$ (in the sense of \cite{bartels_2016}).
We also require that the family of meshes $\{\ensuremath{\mathcal{T}}_h\}_{h>0}$ consists only of non-obtuse simplices.
For a given partitioning of meshes $\ensuremath{\mathcal{T}}_h$, we denote the simplices by $K_k$ with $k\in\{1,...,N_K\}$.
The set of internal edges of triangles ($d=2$) in the mesh $\ensuremath{\mathcal{T}}_h$ or facets of tetrahedra ($d=3$) is denoted by $\partial\ensuremath{\mathcal{T}}_h = \{E_j\}_{j=1}^{N_E}$. The set of all the vertices of $\ensuremath{\mathcal{T}}_h$ is denoted by $\{P_p\}_{p=1}^{N_p}$.
Let us consider the problem \ref{P_alpha_delta}.
We approximate the scalar variables $\phi$, $\mu$ and $\sigma$ and the matrix valued quantity $\B$ with continuous and piecewise linear functions.
Hence, we define the following scalar $\ensuremath{\mathcal{P}}_1$-finite element space
\begin{subequations}
\begin{align}
\ensuremath{\mathcal{S}}_h &\coloneqq \left\{ q_h \in C(\overline\Omega) \mid q_h|_{K} \in \ensuremath{\mathcal{P}}_1(K) \ \forall \ K\in\ensuremath{\mathcal{T}}_h \right\}
\subset H^1(\Omega),
\end{align}
and the matrix valued $\ensuremath{\mathcal{P}}_1$-finite element space
\begin{align}
\ensuremath{\mathcal{W}}_h &\coloneqq \left\{ \B_h \in C(\overline\Omega;\R^{d\times d}_{\mathrm{S}}) \mid \B_h|_{K} \in \ensuremath{\mathcal{P}}_1(K; \R^{d\times d}_{\mathrm{S}}) \ \forall \ K\in\ensuremath{\mathcal{T}}_h \right\}
\subset H^1(\Omega;\R^{d\times d}_{\mathrm{S}}).
\end{align}
Moreover, we define
\begin{align}
\ensuremath{\mathcal{W}}_{h,\mathrm{PD}} &\coloneqq \left\{ \B_h \in \ensuremath{\mathcal{W}}_h \mid \B_h(P_p) \in \R^{d\times d}_{\mathrm{SPD}} \ \forall \ p=1,...,N_p \right\}.
\end{align}
For the velocity vector $\pmb{\mathrm{v}}$ and the pressure $p$, we use the $\ensuremath{\mathcal{P}}_2$--$\ensuremath{\mathcal{P}}_1$-Taylor--Hood element \cite{girault_raviart_2012} given by
\begin{align}
\ensuremath{\mathcal{V}}_h &\coloneqq \left\{ \pmb{\mathrm{v}}_h \in C(\overline\Omega;\R^d) \cap H^1_0(\Omega;\R^d) \mid \pmb{\mathrm{v}}_{h}|_{K} \in \ensuremath{\mathcal{P}}_2(K;\R^d) \ \forall \ K\in\ensuremath{\mathcal{T}}_h \right\},
\end{align}
for the discrete velocity and $\ensuremath{\mathcal{S}}_h \cap L^2_0(\Omega)$ for the discrete pressure.
We also introduce
\begin{align}
\ensuremath{\mathcal{V}}_{h,\mathrm{div}} &\coloneqq \left\{ \pmb{\mathrm{v}}_h \in \ensuremath{\mathcal{V}}_h \mid \int_\Omega \divergenz{\pmb{\mathrm{v}}_h} q_h \dv{x} = 0 \ \forall \ q_h \in \ensuremath{\mathcal{S}}_h \right\},
\end{align}
which approximates the space $\mathbf{V}$.
\end{subequations}
It is well-known (cf.~\cite{girault_raviart_2012}) that this choice for the discrete velocity--pressure space satisfies the discrete Ladyzhenskaya--Babu{\v s}ka--Brezzi (LBB) stability condition
\begin{align}
\label{eq:LBB}
\inf_{q_h\in \ensuremath{\mathcal{S}}_h}
\sup_{\pmb{\mathrm{v}}_h\in \ensuremath{\mathcal{V}}_h}
\frac{\int_\Omega \divergenz{\pmb{\mathrm{v}}_h}q_h \dv{x}}{ \norm{q_h}_{L^2} \norm{\pmb{\mathrm{v}}_h}_{H^1} }
\geq C > 0,
\end{align}
where, unless otherwise stated, $C>0$ always denotes a generic constant which is independent of $h,\Delta t, \alpha, \delta$.
At this point, let us mention that also other choices for the discrete velocity--pressure space can be used instead of the $\ensuremath{\mathcal{P}}_2$--$\ensuremath{\mathcal{P}}_1$-Taylor--Hood element as long as the discrete LBB stability condition \eqref{eq:LBB} is fulfilled. For example, the mini-element \cite{girault_raviart_2012} is also a suitable choice.
Moreover, we denote the standard nodal interpolation operator by $\mycal{I}_h: C(\overline{\Omega})\to \ensuremath{\mathcal{S}}_h$ such that $(\mycal{I}_h \eta)(P_p) = \eta(P_p)$ for all $p\in\{1,...,N_p\}$ and $\eta\in C(\overline\Omega)$, which is naturally extended to $\mycal{I}_h: C(\overline\Omega;\R^{d\times d}_{\mathrm{S}}) \to \ensuremath{\mathcal{W}}_h$.
As we use \textit{mass lumping}, we introduce the following semi-inner products and the induced semi-norms on $C(\overline\Omega)$ and $C(\partial\Omega)$, respectively, by
\begin{alignat}{5}
&\skp{\eta_1}{\eta_2}_h &&\coloneqq
\int_\Omega \mycal{I}_h \big[ \eta_1 \eta_2 \big] \dv{x},
\quad\quad
&&\norm{\eta_1}_h &&\coloneqq \sqrt{\skp{\eta_1}{\eta_1}_h},
\quad\quad
&&\forall \ \eta_1, \eta_2 \in C(\overline\Omega),
\\
&\skp{\eta_3}{\eta_4}_{h,\partial\Omega} &&\coloneqq
\int_{\partial\Omega} \mycal{I}_h \big[ \eta_3 \eta_4 \big] \ \mathrm d \calH^{d-1},
\quad\quad
&&\norm{\eta_3}_{h,{\partial\Omega}} &&\coloneqq \sqrt{\skp{\eta_3}{\eta_3}_{h,{\partial\Omega}}}
\quad\quad
&&\forall \ \eta_3, \eta_4 \in C({\partial\Omega}).
\end{alignat}
Below, we state some well-known properties concerning $\ensuremath{\mathcal{S}}_h$ and the interpolant $\mycal{I}_h$. Let $K\in \ensuremath{\mathcal{T}}_h$, $0 \leq s \leq m \leq 1$ and $1 \leq r \leq p \leq \infty$.
Then, as the family of triangulations is quasi-uniform, it holds for all $\eta\in H^2(\Omega)$ and all $q_h \in \ensuremath{\mathcal{S}}_h$ that
\begin{alignat}{2}
\label{eq:interp_H2}
\norm{\eta - \mycal{I}_h \eta}_{L^2}
+ h \norm{\nabla(\eta-\mycal{I}_h\eta)}_{L^2}
&\leq C h^2 \abs{\eta}_{H^2},
\\
\label{eq:inverse_estimate}
\abs{q_h}_{W^{m,p}(K)}
&\leq
C h^{s - m + \frac{d}{p} - \frac{d}{r}} \abs{q_h}_{W^{s,r}(K)},
\end{alignat}
see, e.g., \cite[Thm.~3.3]{bartels_2016} and \cite[Lem.~4.5.3]{brenner_scott_2008}, respectively.
It follows from an $L^\infty(\Omega)$-error estimate for $\mycal{I}_h$ (see \cite[Thm.~4.4.20]{brenner_scott_2008}) and an approximation argument by the Stone--Weierstrass theorem, that
\begin{align}
\label{eq:interp_continuous}
\lim_{h\to 0} \norm{\eta - \mycal{I}_h\eta}_{L^\infty}
&= 0
\quad\quad
\forall \ \eta \in C(\overline\Omega).
\end{align}
As the basis functions associated with $\ensuremath{\mathcal{S}}_h$ are non-negative and sum to one everywhere, it follows from a Cauchy--Schwarz inequality, that
\begin{align}
\label{eq:interp_estimate}
\abs{\mycal{I}_h \eta(x)}^2
&\leq \mycal{I}_h\big[ \abs{\eta(x)}^2 \big]
\quad\quad
\forall \ x\in K, \ K \in \ensuremath{\mathcal{T}}_h, \ \eta\in C(\overline\Omega).
\end{align}
We deduce from \eqref{eq:interp_estimate}, \eqref{eq:inverse_estimate} and H{\"o}lder's inequality, that, for all $q_h\in \ensuremath{\mathcal{S}}_h$,
\begin{alignat}{3}
\label{eq:norm_equiv}
&\norm{q_h}_{L^2}^2
&&\leq \norm{q_h}_h^2
&&\leq C \norm{q_h}_{L^2}^2,
\\
\label{eq:norm_equiv_Gamma}
&\norm{q_h}_{L^2({\partial\Omega})}^2
&&\leq \norm{q_h}_{h,{\partial\Omega}}^2
&&\leq C \norm{q_h}_{L^2({\partial\Omega})}^2.
\end{alignat}
Applying \eqref{eq:interp_H2} elementwise and then summing over all simplices yields the \textit{mass lumping} error estimate
\begin{align}
\label{eq:lump_Sh_Sh}
\abs{ \skp{q_h}{\zeta_h}_h
- \skp{q_h}{\zeta_h}_{L^2} }
&\leq
C h^2 \norm{\nabla q_h}_{L^2} \norm{\nabla \zeta_h}_{L^2}
\quad\quad
\forall \ q_h, \zeta_h \in \ensuremath{\mathcal{S}}_h.
\end{align}
Now, we provide a technical result concerning the \textit{mass lumping} errors on the boundary ${\partial\Omega}$.
\begin{lemma}
Let $q_h, \zeta_h\in \ensuremath{\mathcal{S}}_h$. Then, as $\{\ensuremath{\mathcal{T}}_h\}_{h>0}$ is a conforming family of quasi-uniform partitionings, it holds
\begin{align}
\label{eq:lump_Gamma_Sh_Sh}
\abs{ \skp{q_h}{\zeta_h}_{h,{\partial\Omega}}
- \skp{q_h}{\zeta_h}_{L^2({\partial\Omega})} }
&\leq
C h \norm{\nabla q_h}_{L^2} \norm{\nabla \zeta_h}_{L^2}.
\end{align}
\end{lemma}
\begin{proof}
Let $E \in \partial\ensuremath{\mathcal{T}}_h$ be a side simplex with diameter $h_E$ and let $K_E\in \ensuremath{\mathcal{T}}_h$ such that $E\subset \partial K_E \cap {\partial\Omega}$. Then, it holds with H{\"o}lder's inequality and a local trace inequality (i.e.~\cite[Lem.~4.2]{bartels_2016}), that
\begin{align*}
\begin{split}
\abs{ \int_E (\mycal{I}_h - \mycal{I})[ q_h \zeta_h] \ \mathrm d \calH^{d-1} }
&\leq
C \abs{E}^\frac{1}{2} \norm{ (\mycal{I}_h - \mycal{I})[ q_h \zeta_h] }_{L^2(E)}
\\
&\leq
C \abs{E}^\frac{1}{2} \Big(
h_E^{-1} \norm{ (\mycal{I}_h - \mycal{I})[ q_h \zeta_h] }_{L^2(K_E)}^2
+ h_E \norm{ \nabla (\mycal{I}_h - \mycal{I})[ q_h \zeta_h] }_{L^2(K_E)}^2 \Big)^\frac{1}{2},
\end{split}
\end{align*}
which gives us, on noting \eqref{eq:interp_H2} and as the family of triangulations is quasi-uniform, that
\begin{align*}
\abs{ \int_E (\mycal{I}_h - \mycal{I})[ q_h \zeta_h] \ \mathrm d \calH^{d-1} }
&\leq C h^{\frac{d}{2} + 1} \abs{ q_h \zeta_h }_{H^2(K_E)}.
\end{align*}
As $q_h, \zeta_h \in \ensuremath{\mathcal{S}}_h$, we obtain with a product rule and with \eqref{eq:inverse_estimate}, that
\begin{align*}
\abs{ \int_E (\mycal{I}_h - \mycal{I})[ q_h \zeta_h] \ \mathrm d \calH^{d-1} }
&\leq C h^{\frac{d}{2} + 1}
\norm{\nabla q_h}_{L^\infty(K_E)}
\norm{\nabla\zeta_h}_{L^2(K_E)}
\leq C h \norm{\nabla q_h}_{L^2(K_E)}
\norm{\nabla\zeta_h}_{L^2(K_E)}.
\end{align*}
Summing over all $E\in \partial\ensuremath{\mathcal{T}}_h$ with $E\subset {\partial\Omega}$ and using the fact that each element $K_E$ occurs at most $d+1$ times imply \eqref{eq:lump_Gamma_Sh_Sh}.
\end{proof}
The results \eqref{eq:interp_H2}--\eqref{eq:lump_Gamma_Sh_Sh} can also be established with the corresponding matrix valued functions. The inverse inequality \eqref{eq:inverse_estimate} also holds for $\pmb{\mathrm{v}}_h\in\ensuremath{\mathcal{V}}_h$ instead of $q_h\in\ensuremath{\mathcal{S}}_h$.
Furthermore, we recall the quasi-interpolation operator $\mycal{I}_h^{\mathrm{Cl}}: L^2(\Omega)\to \ensuremath{\mathcal{S}}_h$ from Cl{\'e}ment \cite{clement_1975}, which is defined by local averages instead of nodal values. The following properties are taken from \cite[Chap.~3]{ciarlet}:
\begin{subequations}
\begin{alignat}{3}
\label{eq:clement_error}
\abs{\eta - \mycal{I}_h^{\mathrm{Cl}} \eta }_{W^{k,2}}
&\leq C h^{m-k} \abs{\eta}_{W^{m,2}}
\quad
&& \forall \ \eta \in W^{m,2}(\Omega),\ &&0 \leq k \leq m \leq 2,
\\
\label{eq:clement_conv}
\lim\limits_{h\to 0} \norm{\eta-\mycal{I}_h^{\mathrm{Cl}} \eta}_{W^{k,2}} &= 0,
\quad
&& \forall \ \eta \in W^{k,2}(\Omega), \ &&0 \leq k \leq 1.
\end{alignat}
Moreover, if only a finite number of patch shapes occur in the sequence of triangulations, then
\begin{alignat}{2}
\label{eq:clement_Gamma}
\norm{\eta - \mycal{I}_h^{\mathrm{Cl}} \eta }_{L^2({\partial\Omega})}
&\leq C h^{1/2} \norm{\nabla\eta}_{L^2}
\quad
&& \forall \ \eta \in H^1(\Omega),
\end{alignat}
\end{subequations}
see \cite[Thm.~4.2]{bartels_2016}. In practice, this assumption seems to be not that restrictive. Hence, we suppose it to hold.
\subsection{Approximation of the initial and boundary values}
In this work, we require the following assumptions for the discrete initial and boundary values.
\begin{assumptions}
Suppose that the discrete initial data $( \phi_h^0, \sigma_h^0, \pmb{\mathrm{v}}_h^0, \B_h^0 ) \in (\ensuremath{\mathcal{S}}_h)^2 \times \ensuremath{\mathcal{V}}_{h,\mathrm{div}} \times \ensuremath{\mathcal{W}}_{h,\mathrm{PD}}$ and discrete boundary data $\sigma_{\infty,h}\in L^2(0,T;\ensuremath{\mathcal{S}}_h)$ fulfill the following bounds uniformly in $h,\Delta t,\alpha,\delta$:
\begin{subequations}
\label{eq:init_bounds}
\begin{align}
\label{eq:init_phi}
\int_\Omega \mycal{I}_h\big[ \psi(\phi_h^0)\big] \dv{x}
+ \norm{\phi_h^0}_{H^1}^2
+ \Delta t \norm{\Delta_h \phi_h^0}_{L^2}^2
&\leq C,
\\
\label{eq:init_sig}
\norm{\sigma_h^0}_{L^2}^2
+ \Delta t \norm{\nabla \sigma_h^0}_{L^2}^2
+ \Delta t \norm{\sigma_h^0}_{L^2(\partial\Omega)}^2
&\leq C,
\\
\label{eq:init_v}
\norm{\pmb{\mathrm{v}}_h^0}_{L^2}^2
+ \Delta t \norm{\nabla\pmb{\mathrm{v}}_h^0}_{L^2}^2
&\leq C,
\\
\label{eq:init_B}
\norm{\B_h^0}_{L^2}^2
+ \Delta t \norm{\nabla\B_h^0}_{L^2}^2
&\leq C,
\\
\label{eq:bc_sig}
\norm{\sigma_{\infty,h}}_{L^2(0,T;L^2(\partial\Omega))}
&\leq C,
\end{align}
and, with constants $0<\tilde b^0_{\min}\leq \tilde b^0_{\max}$,
\begin{align}
\label{eq:init_B_spd}
\tilde b^0_{\min} \abs{\pmb\xi}^2
\leq \pmb\xi^T \B_h^0(P_p) \pmb\xi
&\leq \tilde b^0_{\max} \abs{\pmb\xi}^2
\quad\quad \forall \ \pmb\xi\in \R^d, \ \forall \ p\in\{1,...,N_p\}.
\end{align}
\end{subequations}
\end{assumptions}
Here, $\Delta_h: \ensuremath{\mathcal{S}}_h \to \left\{z_h \in \ensuremath{\mathcal{S}}_h \mid \int_\Omega z_h \dv{x} = 0 \right\}$ denotes the discrete Neumann--Laplacian such that $\Delta_h q_h$ is the unique solution of
\begin{align}
\label{eq:discr_laplace}
\skp{\Delta_h q_h}{\zeta_h}_{h}
= \int_\Omega \mycal{I}_h\big[ (\Delta_h q_h) \zeta_h\big] \dv{x}
= - \int_\Omega \nabla q_h \cdot \nabla\zeta_h \dv{x}
= - \skp{\nabla q_h}{\nabla\zeta_h}_{L^2}
\quad\quad \forall \ \zeta_h\in \ensuremath{\mathcal{S}}_h.
\end{align}
We note for future reference, as $\{\ensuremath{\mathcal{T}}_h\}_{h>0}$ is a quasi-uniform family of partitionings, and,
as the domain $\Omega$ is convex, that
\begin{align}
\label{eq:discr_laplace_bound}
\abs{q_h}_{W^{1,s}}
\leq C \norm{\Delta_h q_h}_{L^2}
\qquad \forall \ q_h\in \ensuremath{\mathcal{S}}_h, \ \forall \ s\in\left[1,\tfrac{2d}{d-2}\right),
\end{align}
see, e.g., \cite[Lem.~3.1]{barrett_langdon_nuernberg_2004} or \cite[Thm.~6.4]{gruen_2013}.
Moreover, we define for all $t\in[t^{n-1},t^n)$ and
$n\in\{1,...,N_T\}$ the piecewise constant in time approximation of $\sigma_{\infty,h}$ by
\begin{align}
\label{eq:def_bc}
\sigma_{\infty,h}^{\Delta t, +} (t,\cdot)
\coloneqq \sigma_{\infty,h}^n(\cdot)
\coloneqq
\frac{1}{\Delta t} \int_{t^{n-1}}^{t^n} \sigma_{\infty,h}(\tilde t,\cdot) \dv{\tilde t} \ \in \ensuremath{\mathcal{S}}_h,
\end{align}
which fulfills
\begin{subequations}
\begin{align}
\label{eq:bounds_bc}
\Delta t \sum_{n=1}^{N_T} \norm{\sigma_{\infty,h}^n}_{L^2({\partial\Omega})}^2
= \norm{\sigma_{\infty,h}^{\Delta t, +}}_{L^2(0,T;L^2({\partial\Omega}))}^2
\leq \nnorm{\sigma_{\infty,h}}^2_{L^2(0,T;L^2(\partial\Omega))},
\\
\sigma_{\infty,h}^{\Delta t, +} \to \sigma_{\infty,h}
\quad \text{strongly in } L^2(0,T;\ensuremath{\mathcal{S}}_h),
\quad \text{as } \Delta t\to 0.
\end{align}
\end{subequations}
Furthermore, we make the following assumption on the discrete initial and boundary data which is needed for the limit process $(h,\Delta t)\to(0,0)$.
\begin{assumptions}
Let \ref{A5} hold true. Then, in the limit $(h,\Delta t)\to (0,0)$, we assume
\begin{subequations}
\label{eq:init_conv}
\begin{alignat}{3}
\label{eq:init_phi_conv}
\phi_h^0 &\to \phi_0
\quad &&\text{ weakly }
\quad &&\text{ in } L^2(\Omega),
\\
\label{eq:init_sig_conv}
\sigma_h^0 &\to \sigma_0
\quad &&\text{ weakly }
\quad &&\text{ in } L^2(\Omega),
\\
\label{eq:init_v_conv}
\pmb{\mathrm{v}}_h^0 &\to \pmb{\mathrm{v}}_0
\quad &&\text{ weakly }
\quad &&\text{ in } \mathbf{H},
\\
\label{eq:init_B_conv}
\B_h^0 &\to \B_0
\quad &&\text{ weakly }
\quad &&\text{ in } L^2(\Omega;\R^{d\times d}),
\\
\label{eq:bc_conv}
\sigma_{\infty,h}|_{\partial\Omega} &\to \sigma_\infty|_{\partial\Omega}
\quad && \text{ strongly }
\quad &&\text{ in } L^2(0,T;L^2(\partial\Omega)).
\end{alignat}
\end{subequations}
\end{assumptions}
\begin{remark}
The assumptions \eqref{eq:init_bounds} and \eqref{eq:init_conv} are no severe constraints in practice.
For example, let \ref{A5} hold true and $\psi:\R\to\R$ be continuous. Then, the following choices for $\phi_h^0,\sigma_h^0,\pmb{\mathrm{v}}_h^0,\B_h^0, \sigma_{\infty,h}$ are in accordance with \eqref{eq:init_bounds} and \eqref{eq:init_conv}:
\begin{subequations}
\label{eq:def_initial}
\begin{alignat}{2}
\phi_h^0 &= \mycal{I}_h \phi_0,
\\
\int_\Omega \mycal{I}_h\big[ \sigma_h^0 q_h \big] \dv{x}
+ \Delta t \int_\Omega \nabla\sigma_h^0 \cdot \nabla q_h \dv{x}
+ \Delta t \int_{\partial\Omega} \mycal{I}_h\big[ \sigma_h^0 q_h \big] \ \mathrm d \calH^{d-1}
&= \int_\Omega \sigma_0 q_h \dv{x}
\qquad
&&\forall \ q_h\in \ensuremath{\mathcal{S}}_h,
\\
\int_\Omega \pmb{\mathrm{v}}_h^0\cdot\pmb{\mathrm{w}}_h \dv{x}
+ \Delta t \int_\Omega \nabla\pmb{\mathrm{v}}_h^0 : \nabla\pmb{\mathrm{w}}_h \dv{x}
&= \int_\Omega \pmb{\mathrm{v}}_0 \cdot \pmb{\mathrm{w}}_h \dv{x}
\qquad
&&\forall \ \pmb{\mathrm{w}}_h \in \ensuremath{\mathcal{V}}_{h,\mathrm{div}},
\\
\int_\Omega \mycal{I}_h\big[ \B_h^0 : \C_h \big] \dv{x}
+ \Delta t \int_\Omega \nabla\B_h^0 : \nabla\C_h \dv{x}
&= \int_\Omega \B_0 : \C_h \dv{x}
\qquad
&&\forall \ \C_h\in \ensuremath{\mathcal{W}}_h,
\\
\sigma_{\infty,h} &= \mycal{I}_h^{\mathrm{Cl}} \sigma_\infty.
\end{alignat}
\end{subequations}
We note that \eqref{eq:init_phi} follows from \eqref{eq:interp_H2}, \cite[eq.~(3.16)]{barrett_nurnberg_styles_2004} and \ref{A5}. Moreover, \eqref{eq:init_sig}--\eqref{eq:init_B} are a direct consequence of Hölder's and Young's inequalities, \eqref{eq:norm_equiv}, \eqref{eq:norm_equiv_Gamma} and \ref{A5}.
As we have a triangulation with non-obtuse simplices, \eqref{eq:init_B_spd} follows from \ref{A5} and \cite[Lem.~5.2]{barrett_boyaval_2009}.
Furthermore, \eqref{eq:clement_Gamma} and \ref{A5} yield \eqref{eq:bc_sig}.
Moreover, \ref{A5} and the error estimates \eqref{eq:interp_H2} and \eqref{eq:clement_Gamma} imply \eqref{eq:init_phi_conv} and \eqref{eq:bc_conv}, respectively.
Besides, \eqref{eq:init_sig_conv} follows from \ref{A5}, \eqref{eq:init_sig}, \eqref{eq:lump_Sh_Sh}, \eqref{eq:lump_Gamma_Sh_Sh}, the denseness of $H^1(\Omega)$ in $L^2(\Omega)$ and the fact that for all $q\in H^1(\Omega)$ there exists a sequence $\{q_h\}_{h>0}\subset \ensuremath{\mathcal{S}}_h$ such that $\norm{q_h-q}_{H^1}\to 0$, as $h\to 0$.
Similarly, \eqref{eq:init_v_conv} follows from \ref{A5}, \eqref{eq:init_v}, the denseness of $\mathbf{V}$ in $\mathbf{H}$ and the fact that for all $\pmb{\mathrm{w}}\in \mathbf{V}$ there exists a sequence $\{\pmb{\mathrm{w}}_h\}_{h>0} \subset \ensuremath{\mathcal{V}}_{h,\mathrm{div}}$ such that $\norm{\pmb{\mathrm{w}}_h-\pmb{\mathrm{w}}}_{H^1}\to 0$, as $h\to 0$, which is due to \eqref{eq:LBB}. The remaining identity \eqref{eq:init_B_conv} follows with similar arguments.
\end{remark}
\subsection{A regularized fully discrete finite element approximation}
Now, for given $\delta\in(0,\frac{1}{2}]$, we introduce a fully discrete approximation of \ref{P_alpha_delta}.
There are several difficulties on the fully discrete level which have to be taken into account.
One of the most important issues arises
from the fact that $\B_h \in \ensuremath{\mathcal{W}}_h$ only implies $\mycal{I}_h[G_\delta'(\B_h)] \in \ensuremath{\mathcal{W}}_h$, as in general $G_\delta'(\B_h) \not\in \ensuremath{\mathcal{W}}_h$. For that reason, it is not clear that the analogues of \eqref{eq:formal_delta_1}--\eqref{eq:delta_3} can be performed on the discrete level, especially controlling the convective term in \eqref{eq:B2_delta}.
Here, the approach of Barrett and Boyaval \cite[Sec.~5]{barrett_boyaval_2009} is very helpful.
We recall the fourth order tensorial function $\Lambda_\delta: \ensuremath{\mathcal{W}}_h \to \R^{d^4}$, where the symmetric $(d\times d)$-matrix $\Lambda_{\delta,i,j}(\B_h)$ approximates $\delta_{i,j} \beta_\delta(\B_h)$ in a certain sense, where $i,j\in\{1,...,d\}$ and $\B_h\in\ensuremath{\mathcal{W}}_h$ and $\delta_{i,j}$ denotes the Kronecker delta. The reason for introducing this nonlinear quantity is to control the discrete version of the convective term from \eqref{eq:B2_delta}, which is due to the following property:
\begin{align}
\label{eq:Lambda1}
\sum\limits_{j=1}^d
\Lambda_{\delta,i,j}(\B_h) : \partial_{x_j} \mycal{I}_h\big[ G_\delta'(\B_h) \big]
= \partial_{x_i} \mycal{I}_h\big[ \trace\big(
H_\delta( G_\delta'(\B_h)) \big) \big]
\quad \text{ on } K_k,
\end{align}
for $k\in\{1,...,N_k\}$ and $i\in\{1,...,d\}$, see \cite[eq.~(5.17)]{barrett_boyaval_2009},
which will make it possible to derive an \textit{a priori} estimate on the fully discrete level.
As the family of partitionings $\{\ensuremath{\mathcal{T}}_h\}_{h>0}$ is quasi-uniform,
it follows from the definition of $\Lambda_{\delta,i,j}$ (cf.~\cite[Sec.~5.1]{barrett_boyaval_2009}) that
\begin{align}
\label{eq:Lambda2}
\norm{ \Lambda_{\delta,i,j}(\B_h) }_{L^\infty(\Omega)}
\leq C \norm{\beta_\delta(\B_h)}_{L^\infty(\Omega)}
\quad \forall \ \B_h\in \ensuremath{\mathcal{W}}_h.
\end{align}
Next, we present an approximation of \ref{P_alpha_delta} for which we explain the motivation afterwards.
\subsubsection*{Problem \ref{P_alpha_delta_FE}:}
\mylabelHIDE{P_alpha_delta_FE}{$(\pmb{\mathrm{P}}_{\alpha,\delta,h}^{\Delta t})$}
Let $\delta\in(0,\frac{1}{2}]$. For given discrete initial and boundary data satisfying \eqref{eq:init_bounds} and $n\in\{1,...,N_T\}$, find the discrete solution $(\phi_h^{n}, \mu_h^n, \sigma_{h}^{n}, p_h^n, \pmb{\mathrm{v}}_{h}^{n}, \B_{h}^{n}) \in (\ensuremath{\mathcal{S}}_h)^4 \times \ensuremath{\mathcal{V}}_h \times \ensuremath{\mathcal{W}}_h$ which satisfies, for any $(\zeta_h, \rho_h, \xi_h, q_h, \pmb{\mathrm{w}}_h, \C_h) \in (\ensuremath{\mathcal{S}}_h)^4 \times \ensuremath{\mathcal{V}}_h \times \ensuremath{\mathcal{W}}_h$:
\begin{subequations}
\begingroup
\allowdisplaybreaks
\begin{align}
\label{eq:phi_FE_delta}
0 &= \int_\Omega \mycal{I}_h \Big[ \Big(\frac{\phi_h^n-\phi_h^{n-1}}{\Delta t}
- \Gamma_{\phi,h}^n \Big) \zeta_h \Big]
+ \mycal{I}_h[m(\phi_h^{n-1})] \nabla\mu_h^n \cdot \nabla \zeta_h
- \phi_h^{n-1} \pmb{\mathrm{v}}_h^{n} \cdot \nabla \zeta_h \dv{x},
\\
\label{eq:mu_FE_delta}
0 &= \int_\Omega \mycal{I}_h \Big[ \Big( - \mu_h^n
+ A \psi_1'(\phi_h^n)
+ A \psi_2'(\phi_h^{n-1})
- \chi_\phi \sigma_h^n \Big) \rho_h \Big]
+ B \nabla\phi_h^n \cdot \nabla\rho_h\dv{x},
\\
\nonumber
\label{eq:sigma_FE_delta}
0 &= \int_\Omega \mycal{I}_h \Big[ \Big(\frac{\sigma_h^n-\sigma_h^{n-1}}{\Delta t}
+ \Gamma_{\sigma,h}^n \Big) \xi_h \Big]
+ \mycal{I}_h[n(\phi_h^{n-1})]
\nabla (\chi_\sigma \sigma_h^n - \chi_\phi \phi_h^n) \cdot \nabla \xi_h
- \sigma_h^{n-1} \pmb{\mathrm{v}}_h^{n} \cdot \nabla\xi_h \dv{x}
\\
&\qquad + \int_{\partial\Omega} \mycal{I}_h\Big[ K \big(\sigma_h^n - \sigma_{\infty,h}^n \big) \xi_h \Big] \ \mathrm d \calH^{d-1},
\\
\label{eq:div_v_FE_delta}
0 &= \int_\Omega \divergenz{\pmb{\mathrm{v}}_h^{n}} q_h \dv{x},
\\
\label{eq:v_FE_delta}
\nonumber
0 &= \int_\Omega \frac{\pmb{\mathrm{v}}_h^n-\pmb{\mathrm{v}}_h^{n-1}}{\Delta t} \cdot \pmb{\mathrm{w}}_h
+ \frac{1}{2} \left( \left(\pmb{\mathrm{v}}_h^{n-1}\cdot \nabla\right) \pmb{\mathrm{v}}_h^n\right) \cdot \pmb{\mathrm{w}}_h
- \frac{1}{2} \pmb{\mathrm{v}}_h^n \cdot \left(\left(\pmb{\mathrm{v}}_h^{n-1} \cdot \nabla\right) \pmb{\mathrm{w}}_h \right)
+ 2\mycal{I}_h[\eta(\phi_h^{n-1})] \D(\pmb{\mathrm{v}}_h^n) : \D(\pmb{\mathrm{w}}_h)\dv{x}
\\
&\qquad + \int_\Omega
\kappa \mycal{I}_h\big[ \beta_\delta(\B_h^n) - \I \big] : \nabla\pmb{\mathrm{w}}_h
- \divergenz{\pmb{\mathrm{w}}_h} p_h^n
+ \big( \phi_h^{n-1} \nabla\mu_h^n
+ \sigma_h^{n-1} \nabla (\chi_\sigma \sigma_h^n - \chi_\phi \phi_h^n) \big) \cdot \pmb{\mathrm{w}}_h \dv{x},
\\
\label{eq:B_FE_delta}
\nonumber
0 &= \int_\Omega \mycal{I}_h \Big[
\Big(\frac{\B_h^n - \B_h^{n-1}}{\Delta t}
+ \frac{\kappa}{\tau(\phi_h^{n-1})} (\B_h^n - \I) \Big): \C_h \Big]
- 2 \nabla\pmb{\mathrm{v}}^n_h : \mycal{I}_h\big[ \C_h \beta_\delta(\B_h^n) \big]
+ \alpha \nabla\B_h^n : \nabla\C_h \dv{x}
\\
&\qquad - \int_\Omega \sum\limits_{i,j=1}^d
[\pmb{\mathrm{v}}_h^{n-1}]_i \Lambda_{\delta,i,j}(\B_h^n) : \partial_{x_j} \C_h \dv{x},
\end{align}
\endgroup
\end{subequations}
where we define
$\Gamma_{\phi,h}^n \coloneqq\Gamma_\phi(\phi_h^n,\sigma_h^n,\B_h^n)$ and $\Gamma_{\sigma,h}^n \coloneqq \Gamma_\sigma(\phi_h^n,\sigma_h^n)$.
Let us now motivate the idea for \ref{P_alpha_delta_FE} by explaining the derivation from the weak formulation of \ref{P_alpha} in the sense of Definition \ref{def:weak_solution}.
First, the $\delta$-regularization strategy from Section \ref{sec:regularization} is applied and, as we use finite element functions for the approximation in space, we also use the fourth order tensor $\Lambda_\delta$ to control the convection term for $\B$ on the discrete level in \eqref{eq:B_FE_delta}. Besides, a semi-implicit time discretization of first order is chosen where linear terms are treated fully implicitly and most of the nonlinear terms are treated explicitly. In \eqref{eq:mu_FE_delta}, a convex-concave splitting for the potential $\psi=\psi_1+\psi_2$ is chosen, which allows the inequality
\begin{align}
\label{eq:convex_concave}
\left(\psi_1'(\phi_h^n) + \psi_2'(\phi_h^{n-1}) \right) (\phi_h^n- \phi_h^{n-1}) \geq \psi(\phi_h^n) - \psi(\phi_h^{n-1}).
\end{align}
Besides, the nonlinear source terms $\Gamma_\phi,\Gamma_\sigma$ are treated fully implicitly and the nonlinear functions $n, m, \eta, \tau$ are treated explicitly, but also a different time approximation can be chosen for these terms.
The remaining terms are approximated in a way such that stability of the scheme \ref{P_alpha_delta_FE} can be shown, see Lemma \ref{lemma:bounds_FE_delta}.
Furthermore, we make use of numerical integration in terms of the nodal interpolation operator $\mycal{I}_h$. On the one hand, this can reduce the computational effort as the mass matrices are diagonal, whereas on the other hand, the nodal interpolation operator in \eqref{eq:B_FE_delta} and in the second line in \eqref{eq:v_FE_delta} is required for stability of the scheme.
\begin{remark}
In the literature, the velocity field in Navier--Stokes systems is sometimes approximated with finite element functions where the constraint \eqref{eq:div_v_FE_delta} is directly included in the finite element space $\ensuremath{\mathcal{V}}_{h,\mathrm{div}}$. Hence, the velocity field and the test functions in equation \eqref{eq:v_FE_delta} would belong to the finite element space $\ensuremath{\mathcal{V}}_{h,\mathrm{div}}$, and \eqref{eq:v_FE_delta} would be replaced by
\begin{align}
\label{eq:v_FE_delta_b}
\nonumber
0 &= \int_\Omega \frac{\pmb{\mathrm{v}}_h^n-\pmb{\mathrm{v}}_h^{n-1}}{\Delta t} \cdot \pmb{\mathrm{w}}_h
+ \frac{1}{2} \left( \left(\pmb{\mathrm{v}}_h^{n-1}\cdot \nabla\right) \pmb{\mathrm{v}}_h^n\right) \cdot \pmb{\mathrm{w}}_h
- \frac{1}{2} \pmb{\mathrm{v}}_h^n \cdot \left(\left(\pmb{\mathrm{v}}_h^{n-1} \cdot \nabla\right) \pmb{\mathrm{w}}_h \right)
+ 2\mycal{I}_h[\eta(\phi_h^{n-1})] \D(\pmb{\mathrm{v}}_h^n) : \D(\pmb{\mathrm{w}}_h) \dv{x}
\\
&\qquad + \int_\Omega
\kappa \mycal{I}_h\big[ \beta_\delta(\B_h^n) - \I \big] : \nabla\pmb{\mathrm{w}}_h
+ \big( \phi_h^{n-1} \nabla\mu_h^n
+ \sigma_h^{n-1} \nabla (\chi_\sigma \sigma_h^n - \chi_\phi \phi_h^n) \big) \cdot \pmb{\mathrm{w}}_h \dv{x},
\end{align}
for all $\pmb{\mathrm{w}}_h\in\ensuremath{\mathcal{V}}_{h,\mathrm{div}}$, where $\pmb{\mathrm{v}}_h^{n-1}, \pmb{\mathrm{v}}_h^n \in\ensuremath{\mathcal{V}}_{h,\mathrm{div}}$ are the solution from the previous time step and the unknown solution from the current time step, respectively.
The unknown pressure $p_h^n\in \ensuremath{\mathcal{S}}_h$, which is unique up to an additive constant, can be reconstructed afterwards as the discrete LBB stability condition \eqref{eq:LBB} is fulfilled, see, e.g., \cite[Chap.~I, Lem.~4.1]{girault_raviart_2012} or \cite[Lem.~4.2]{braess_2007}.
However, it is rather hard to construct test functions $\pmb{\mathrm{w}}_h\in \ensuremath{\mathcal{V}}_{h,\mathrm{div}}$ in practice.
This is the reason why we use \eqref{eq:v_FE_delta} instead of \eqref{eq:v_FE_delta_b}.
\end{remark}
\subsection{Stability of the regularized discrete system}
\label{sec:stability}
We now introduce the discrete energy $\ensuremath{\mathcal{F}}_{\delta,h}: \ensuremath{\mathcal{S}}_h\times \ensuremath{\mathcal{S}}_h \times \ensuremath{\mathcal{V}}_h \times \ensuremath{\mathcal{W}}_h \to \R$ of the problem \ref{P_alpha_delta_FE} given by
\begin{align}
\label{eq:def_energy_FE_delta}
\begin{split}
\ensuremath{\mathcal{F}}_{\delta,h}(\phi_h,\sigma_h,\pmb{\mathrm{v}}_h,\B_h)
&=
\int_\Omega
\mycal{I}_h\Big[ A \psi(\phi_h)
+ \frac{\chi_\sigma}{2} \abs{\sigma_h}^2
+ \chi_\phi \sigma_h (1-\phi_h)
+ \frac{\kappa}{2} \trace\big( \B_h - G_\delta(\B_h) \big)\Big] \dv{x}
\\
&\quad + \int_\Omega
\frac{B}{2} \abs{\nabla\phi_h}^2
+ \frac{1}{2} \abs{\pmb{\mathrm{v}}_h}^2 \dv{x},
\end{split}
\end{align}
for all $(\phi_h,\sigma_h,\pmb{\mathrm{v}}_h,\B_h)\in \ensuremath{\mathcal{S}}_h \times \ensuremath{\mathcal{S}}_h \times \ensuremath{\mathcal{V}}_h \times \ensuremath{\mathcal{W}}_h$, where $\delta\in(0,\frac{1}{2}]$.
We remark that it is not guaranteed that $\ensuremath{\mathcal{F}}_{\delta,h}$ is non-negative as the term $\sigma_h(1-\phi_h)$ can have a negative sign. This is one of the main difficulties we have to handle in the derivation of useful \textit{a priori} estimates.
For future reference, we note the elementary identity
\begin{align}
\label{eq:elementary_identity}
2x(x-y) = x^2-y^2 + (x-y)^2 \quad\quad \forall \ x,y\in\R.
\end{align}
Moreover, we recall the following discrete version of Gronwall's inequality, i.e.~Lemma \ref{lemma:gronwall}. For a proof, we refer to, e.g., \cite[pp.~401--402]{dahmen_reusken_numerik}.
\begin{lemma}
\label{lemma:gronwall_discrete}
Assume that $e_n, a_n, b_n \geq 0$ for all $n\in \N_0$. Then
\begin{align}
\label{eq:gronwall_discrete}
e_n \leq a_n + \sum\limits_{i=0}^{n-1} b_i e_i
\quad \forall \ n\in \N_0
\quad
\Longrightarrow \quad
e_n \leq a_n \cdot
\exp\Big( \sum\limits_{i=0}^{n-1} b_i \Big)
\quad \forall \ n\in \N_0.
\end{align}
\end{lemma}
With the help of Lemma \ref{lemma:gronwall_discrete}, we derive stability bounds for solutions of \ref{P_alpha_delta_FE}.
\begin{lemma}[Stability]
\label{lemma:bounds_FE_delta}
Let \ref{A1}--\ref{A5} hold true and let $\delta\in(0,\frac{1}{2}]$. Suppose that the discrete initial and boundary data satisfy \eqref{eq:init_bounds} and assume that $\Delta t < \Delta t_*$, where the constant $\Delta t_*$ depends only on the model parameters and is defined in \eqref{eq:dt}. Then, for $n\in\{1,...,N_T\}$, a solution $(\phi_h^{n}$, $\mu_h^n$, $\sigma_{h}^{n}$, $p_h^n$, $\pmb{\mathrm{v}}_{h}^{n}$, $\B_{h}^{n}) \in (\ensuremath{\mathcal{S}}_h)^4 \times \ensuremath{\mathcal{V}}_h \times \ensuremath{\mathcal{W}}_h$ to the problem \ref{P_alpha_delta_FE}, if it exists, satisfies
\begin{align}
\begin{split}
\label{eq:bounds_FE_delta}
& \max_{n=1,...,N_T} \Big(
\norm{\phi_h^n}_{H^1}^2
+ \norm{\sigma_h^n}_{L^2}^2
+ \norm{\pmb{\mathrm{v}}_h^n}_{L^2}^2
+ \nnorm{ \mycal{I}_h\big[ \abs{\B_h^n} \big] }_{L^1}
+ \frac{1}{\delta} \nnorm{ \mycal{I}_h\big[ \abs{ [\B_h^n]_- } \big] }_{L^1} \Big)
\\
&\quad
+ \sum_{n=1}^{N_T} \Big(
\norm{\nabla\phi_h^n - \nabla\phi_h^{n-1}}_{L^2}^2
+ \norm{\sigma_h^n - \sigma_h^{n-1}}_{L^2}^2
+ \norm{\pmb{\mathrm{v}}_h^n - \pmb{\mathrm{v}}_h^{n-1}}_{L^2}^2 \Big)
\\
&\quad
+ \Delta t \sum_{n=1}^{N_T} \Big(
\norm{\mu_h^n}_{H^1}^2
+ \norm{\nabla \sigma_h^n}_{L^2}^2
+ \norm{\sigma_h^n}_{L^2({\partial\Omega})}^2
+ \norm{\nabla \pmb{\mathrm{v}}_h^n}_{L^2}^2 \Big)
\\
&\quad
+ \Delta t \sum_{n=1}^{N_T} \bigg(
\alpha\delta^2 \nnorm{ \nabla \mycal{I}_h\big[ G'_\delta(\B_h^n)\big] }_{L^2}^2
+ \int_\Omega
\mycal{I}_h\Big[ \trace\big(\beta_\delta(\B_h^n) + [\beta_\delta(\B_h^n)]^{-1}-2\I \big) \Big] \dv{x} \bigg)
\\
&\leq
C(T) \Big( 1 + \abs{\ensuremath{\mathcal{F}}_{h,\delta}(\phi_h^0, \sigma_h^0, \pmb{\mathrm{v}}_h^0, \B_h^0)}
+ \Delta t \sum_{n=1}^{N_T} \norm{\sigma_{\infty,h}^n}_{L^2(\partial\Omega)}^2 \Big)
\leq C(T),
\end{split}
\end{align}
where the constants $C(T)$ are independent of $h, \Delta t, \alpha, \delta$, but depend exponentially on $T$.
\end{lemma}
\input{proofs/proof_stability}
\subsection{Existence of regularized discrete solutions}
\label{sec:existence}
In the next theorem, we apply a strategy based on Brouwer's fixed point theorem \cite[Chap.~8.1.4, Thm.~3]{evans_2010} in order to prove existence of discrete solutions to \ref{P_alpha_delta_FE}.
Here, one of the main difficulties is to construct specific mappings on a finite dimensional Hilbert space such that Brouwer's fixed point theorem can be applied in the right way.
It turns out that the testing procedure of Lemma \ref{lemma:bounds_FE_delta}
is very helpful. However, we need to deal with similar difficulties as in Lemma \ref{lemma:bounds_FE_delta}, which explains the minor constraint on the time step size.
\begin{theorem}[Existence]
\label{theorem:existence_FE_delta}
Let \ref{A1}--\ref{A5} hold true and let $\delta\in(0,\frac{1}{2}]$. Suppose that the discrete initial and boundary data satisfy \eqref{eq:init_bounds} and assume that $\Delta t < \Delta t_*$, where $\Delta t_*$ is defined in \eqref{eq:dt}.
Then, for all $n\in\{1,...,N_T\}$, there exists at least one solution $(\phi_h^{n},\mu_h^n,\sigma_{h}^{n},p_h^n,\pmb{\mathrm{v}}_{h}^{n},\B_{h}^{n}) \in (\ensuremath{\mathcal{S}}_h)^4 \times \ensuremath{\mathcal{V}}_h \times \ensuremath{\mathcal{W}}_h$ to the problem \ref{P_alpha_delta_FE} which is stable in the sense of \eqref{eq:bounds_FE_delta}.
\end{theorem}
\input{proofs/proof_existence}
\subsection{Existence of unregularized discrete solutions}
\label{sec:delta_to_zero}
Now let us consider a finite element approximation of \ref{P_alpha} without the regularization parameter $\delta$ and with a positive definite discrete Cauchy--Green tensor.
\subsubsection*{Problem \ref{P_alpha_FE}:}
\mylabelHIDE{P_alpha_FE}{$(\pmb{\mathrm{P}}_{\alpha,h}^{\Delta t})$}
For given discrete initial and boundary data satisfying \eqref{eq:init_bounds} and $n\in\{1,...,N_T\}$, find the discrete solution $(\phi_h^{n}, \mu_h^n, \sigma_{h}^{n}, p_h^n, \pmb{\mathrm{v}}_{h}^{n}, \B_{h}^{n}) \in (\ensuremath{\mathcal{S}}_h)^4 \times \ensuremath{\mathcal{V}}_h \times \ensuremath{\mathcal{W}}_{h,\mathrm{PD}}$ which satisfies, for any $(\zeta_h, \rho_h, \xi_h, q_h, \pmb{\mathrm{w}}_h, \C_h) \in (\ensuremath{\mathcal{S}}_h)^4 \times \ensuremath{\mathcal{V}}_h \times \ensuremath{\mathcal{W}}_h$:
\begin{subequations}
\begingroup
\allowdisplaybreaks
\begin{align}
\label{eq:phi_FE}
0 &= \int_\Omega \mycal{I}_h \Big[ \Big(\frac{\phi_h^n-\phi_h^{n-1}}{\Delta t}
- \Gamma_{\phi,h}^n \Big) \zeta_h \Big]
+ \mycal{I}_h[m(\phi_h^{n-1})] \nabla\mu_h^n \cdot \nabla \zeta_h
- \phi_h^{n-1} \pmb{\mathrm{v}}_h^{n} \cdot\nabla \zeta_h \dv{x},
\\
\label{eq:mu_FE}
0 &= \int_\Omega \mycal{I}_h \Big[ \Big( - \mu_h^n
+ A \psi_1'(\phi_h^n) + A \psi_2'(\phi_h^{n-1})
- \chi_\phi \sigma_h^n \Big) \rho_h \Big]
+ B \nabla\phi_h^n \cdot \nabla\rho_h\dv{x},
\\
\nonumber
\label{eq:sigma_FE}
0 &= \int_\Omega \mycal{I}_h \Big[ \Big(\frac{\sigma_h^n-\sigma_h^{n-1}}{\Delta t}
+ \Gamma_{\sigma,h}^n \Big) \xi_h \Big]
+ \mycal{I}_h[n(\phi_h^{n-1})] \nabla(\chi_\sigma\sigma_h^n - \chi_\phi\phi_h^n) \cdot \nabla \xi_h
- \sigma_h^{n-1} \pmb{\mathrm{v}}_h^{n} \cdot\nabla\xi_h \dv{x}
\\
& \qquad + \int_{\partial\Omega} \mycal{I}_h\big[ K (\sigma_h^n - \sigma_{\infty,h}^n) \xi_h \big] \ \mathrm d \calH^{d-1},
\\
\label{eq:div_v_FE}
0 &= \int_\Omega \divergenz{\pmb{\mathrm{v}}_h^{n}} q_h \dv{x},
\\
\label{eq:v_FE}
\nonumber
0 &= \int_\Omega \frac{\pmb{\mathrm{v}}_h^n-\pmb{\mathrm{v}}_h^{n-1}}{\Delta t} \cdot \pmb{\mathrm{w}}_h
+ \frac{1}{2} \left( \left(\pmb{\mathrm{v}}_h^{n-1}\cdot \nabla\right) \pmb{\mathrm{v}}_h^n\right) \cdot \pmb{\mathrm{w}}_h
- \frac{1}{2} \pmb{\mathrm{v}}_h^n \cdot \left(\left(\pmb{\mathrm{v}}_h^{n-1} \cdot \nabla\right) \pmb{\mathrm{w}}_h \right)
+ 2\mycal{I}_h[\eta(\phi_h^{n-1})] \D(\pmb{\mathrm{v}}_h^n) : \D(\pmb{\mathrm{w}}_h)\dv{x}
\\
&\qquad + \int_\Omega
\kappa ( \B_h^n - \I ) : \nabla\pmb{\mathrm{w}}_h
- \divergenz{\pmb{\mathrm{w}}_h} p_h^n
+ \big( \phi_h^{n-1} \nabla\mu_h^n
+ \sigma_h^{n-1} \nabla(\chi_\sigma\sigma_h^n - \chi_\phi\phi_h^n) \big) \cdot \pmb{\mathrm{w}}_h \dv{x},
\\
\label{eq:B_FE}
\nonumber
0 &= \int_\Omega \mycal{I}_h \Big[
\Big(\frac{\B_h^n - \B_h^{n-1}}{\Delta t}
+ \frac{\kappa}{\tau(\phi_h^{n-1})} (\B_h^n - \I) \Big): \C_h \Big]
- 2 \nabla\pmb{\mathrm{v}}^n_h : \mycal{I}_h\big[ \C_h \B_h^n \big]
+ \alpha \nabla\B_h^n : \nabla\C_h \dv{x}
\\
&\qquad - \int_\Omega \sum\limits_{i,j=1}^d
[\pmb{\mathrm{v}}_h^{n-1}]_i \Lambda_{i,j}(\B_h^n) : \partial_{x_j} \C_h \dv{x}.
\end{align}
\endgroup
\end{subequations}
Here, the nonlinear function $\Lambda_{i,j}(\C_h)$ for $\C_h\in\ensuremath{\mathcal{W}}_{h,\mathrm{PD}}$ is defined similarly to $\Lambda_{\delta,i,j}(\tilde\C_h)$ for $\tilde\C_h\in\ensuremath{\mathcal{W}}_h$, see \cite[Rem.~5.1]{barrett_boyaval_2009}.
Moreover, the analogues of \eqref{eq:Lambda1},\eqref{eq:Lambda2} without $\delta$-regularization follow with the same arguments. In particular, for $k\in\{1,...,N_k\}$, it holds
\begin{alignat}{2}
\label{eq:Lambda3}
\sum\limits_{j=1}^d
\Lambda_{i,j}(\B_h) : \partial_{x_j} \mycal{I}_h\big[ \B_h^{-1} \big]
&= - \partial_{x_i} \mycal{I}_h\big[ \trace (
\ln \B_h ) \big]
\qquad &&\text{on } K_k,
\\
\label{eq:Lambda4}
\norm{ \Lambda_{i,j}(\B_h) }_{L^\infty(\Omega)}
&\leq C \norm{\B_h}_{L^\infty(\Omega)}
\qquad &&\forall \ \B_h\in \ensuremath{\mathcal{W}}_{h,\mathrm{PD}}.
\end{alignat}
Here we note that $H_\delta(G_\delta'(s)) \to \ln(s^{-1}) = - \ln(s)$ and $\beta_\delta(s) \to s$ for all $s>0$, as $\delta\to0$.
Now we define the unregularized energy $\ensuremath{\mathcal{F}}_{h}: \ensuremath{\mathcal{S}}_h\times \ensuremath{\mathcal{S}}_h \times \ensuremath{\mathcal{V}}_h \times \ensuremath{\mathcal{W}}_{h,\mathrm{PD}} \to \R$ of the problem \ref{P_alpha_FE} by
\begin{align}
\label{eq:def_energy_FE}
\begin{split}
\ensuremath{\mathcal{F}}_h(\phi_h,\sigma_h,\pmb{\mathrm{v}}_h,\B_h)
&=
\int_\Omega
\mycal{I}_h\Big[ A \psi(\phi_h)
+ \frac{\chi_\sigma}{2} \abs{\sigma_h}^2
+ \chi_\phi \sigma_h (1-\phi_h)
+ \frac{\kappa}{2} \trace\big( \B_h - \ln(\B_h) \big)\Big] \dv{x}
\\
&\quad + \int_\Omega
\frac{B}{2} \abs{\nabla\phi_h}^2
+ \frac{1}{2} \abs{\pmb{\mathrm{v}}_h}^2 \dv{x},
\end{split}
\end{align}
for all $(\phi_h,\sigma_h,\pmb{\mathrm{v}}_h,\B_h)\in \ensuremath{\mathcal{S}}_h \times \ensuremath{\mathcal{S}}_h \times \ensuremath{\mathcal{V}}_{h} \times \ensuremath{\mathcal{W}}_{h,\mathrm{PD}}$.
Next, we obtain existence and stability of solutions to the problem \ref{P_alpha_FE} by passing to the limit $\delta\to0$ in the regularized discrete problem \ref{P_alpha_delta_FE} and in the \textit{a priori} bounds \eqref{eq:bounds_FE_delta}.
This can be achieved analogously to \cite[Thm.~5.2]{barrett_boyaval_2009}.
We remark that the positive definiteness of the discrete Cauchy--Green tensor for the problem \ref{P_alpha_FE} is guaranteed as we can control the negative eigenvalues and the inverse of the discrete Cauchy--Green tensor from the $\delta$-regularized problem \ref{P_alpha_delta_FE}, which is due to \eqref{eq:bounds_FE_delta}.
Moreover, as we have no control over the pressure of the regularized problem \ref{P_alpha_delta_FE}, the existence of a pressure for the problem \ref{P_alpha_FE} can still be established with the discrete LBB stability condition \eqref{eq:LBB}.
\begin{theorem}[Solutions to the unregularized discrete problem]
\label{theorem:existence_FE}
Let \ref{A1}--\ref{A5} hold. Suppose that the discrete initial and boundary data satisfy \eqref{eq:init_bounds} and assume that $\Delta t < \Delta t_*$, where $\Delta t_*$ is defined in \eqref{eq:dt}.
Then, for all $n\in\{1,...,N_T\}$, there exists at least one solution $(\phi_h^{n},\mu_h^n,\sigma_{h}^{n},p_h^n,\pmb{\mathrm{v}}_{h}^{n},\B_{h}^{n}) \in (\ensuremath{\mathcal{S}}_h)^4 \times \ensuremath{\mathcal{V}}_h \times \ensuremath{\mathcal{W}}_{h,\mathrm{PD}}$ to the unregularized discrete problem \ref{P_alpha_FE} with $\B_{h}^{n}$ being positive definite.
Moreover, all solutions of \ref{P_alpha_FE} are stable in the sense that
\begin{align}
\begin{split}
\label{eq:bounds_FE}
& \max_{n=1,...,N_T} \Big(
\norm{\phi_h^n}_{H^1}^2
+ \norm{\sigma_h^n}_{L^2}^2
+ \norm{\pmb{\mathrm{v}}_h^n}_{L^2}^2
+ \nnorm{ \mycal{I}_h\big[ \abs{\B_h^n} \big] }_{L^1}\Big)
\\
&\quad
+ \sum_{n=1}^{N_T} \Big(
\norm{\nabla\phi_h^n - \nabla\phi_h^{n-1}}_{L^2}^2
+ \norm{\sigma_h^n - \sigma_h^{n-1}}_{L^2}^2
+ \norm{\pmb{\mathrm{v}}_h^n - \pmb{\mathrm{v}}_h^{n-1}}_{L^2}^2 \Big)
\\
&\quad
+ \Delta t \sum_{n=1}^{N_T} \Big(
\norm{\mu_h^n}_{H^1}^2
+ \norm{\nabla \sigma_h^n}_{L^2}^2
+ \norm{\sigma_h^n}_{L^2({\partial\Omega})}^2
+ \norm{\nabla \pmb{\mathrm{v}}_h^n}_{L^2}^2
+ \int_\Omega
\mycal{I}_h\Big[ \trace\big(\B_h^n + [\B_h^n]^{-1} -2\I \big) \Big] \dv{x} \Big)
\\
&\leq
C(T) \Big( 1 + \abs{\ensuremath{\mathcal{F}}_h(\phi_h^0, \sigma_h^0, \pmb{\mathrm{v}}_h^0, \B_h^0)}
+ \Delta t \sum_{n=1}^{N_T} \norm{\sigma_{\infty,h}^n}_{L^2(\partial\Omega)}^2 \Big)
\leq C(T),
\end{split}
\end{align}
where the constants $C(T)>0$ are independent of $h, \Delta t, \alpha$ but depend exponentially on $T$.
\end{theorem}
\subsection{Improving the regularity results in arbitrary dimensions}
\label{sec:regularity}
In the following, we derive higher order estimates for discrete solution of \ref{P_alpha_FE} in arbitrary dimensions $d\in\{2,3\}$.
For the next steps, we require the $L^2$ projectors $\ensuremath{\mathcal{P}}_h: \mathbf{V}\to \ensuremath{\mathcal{V}}_{h,\mathrm{div}}$ and $\mycal{Q}_h: H^1(\Omega) \to \ensuremath{\mathcal{S}}_h$ defined by
\begin{alignat}{2}
\label{eq:projector_Ph_def}
\int_\Omega \ensuremath{\mathcal{P}}_h \pmb{\mathrm{v}} \cdot \pmb{\mathrm{w}}_h \dv{x}
&= \int_\Omega \pmb{\mathrm{v}}\cdot \pmb{\mathrm{w}}_h \dv{x}
\quad\quad &&\forall \ \pmb{\mathrm{w}}_h \in \ensuremath{\mathcal{V}}_{h,\mathrm{div}},
\\
\label{eq:projector_Qh_def}
\int_\Omega \mycal{I}_h \big[\mycal{Q}_h \rho \zeta_h\big] \dv{x}
&= \int_\Omega \rho \zeta_h \dv{x}
\quad\quad &&\forall \ \zeta_h \in \ensuremath{\mathcal{S}}_h,
\end{alignat}
which fulfill, as $\Omega$ is convex and the family $\{\ensuremath{\mathcal{T}}_h\}_{h>0}$ is quasi-uniform, that
\begin{align}
\label{eq:projector_Ph_bound}
\norm{\ensuremath{\mathcal{P}}_h\pmb{\mathrm{v}}}_{H^1} &\leq C \norm{\pmb{\mathrm{v}}}_{H^1}
\quad\quad \forall \ \pmb{\mathrm{v}}\in \mathbf{V},
\\
\label{eq:projector_Qh_bound}
\norm{\mycal{Q}_h\zeta}_{H^1} &\leq C \norm{\zeta}_{H^1}
\quad\quad \forall \ \zeta\in H^1(\Omega),
\end{align}
see, e.g., \cite{barrett_boyaval_2009} and references therein. Analogously to \eqref{eq:projector_Qh_def}, we also introduce a matrix valued projection operator $\mycal{Q}_h: H^1(\Omega;\R^{d\times d}_{\mathrm{S}}) \to \ensuremath{\mathcal{W}}_h$ which fulfills a stability estimate corresponding to \eqref{eq:projector_Qh_bound}, see \cite{barrett_boyaval_2009}.
Now, we improve the regularity for the order parameter and the nutrient.
\begin{lemma}
Let \ref{A1}--\ref{A5} hold. Suppose that the discrete initial and boundary data satisfy \eqref{eq:init_bounds} and let $\Delta t < \Delta t_*$, where $\Delta t_*$ is defined in \eqref{eq:dt}. Then, in addition to \eqref{eq:bounds_FE}, all solutions of \ref{P_alpha_FE} fulfill for any $l \in \{1,...,N_T\}$,
\begin{subequations}
\begin{align}
\label{eq:bounds_FE_phi_dtphi_dtsigma}
\Delta t \sum\limits_{n=1}^{N_T} \left(
\norm{\Delta_h \phi_h^n}_{L^2}^2
+ \nnorm{\frac{\phi_h^n - \phi_h^{n-1}}{\Delta t} }_{(H^1)'}^2
+ \nnorm{\frac{\sigma_h^n - \sigma_h^{n-1}}{\Delta t} }_{(H^1)'}^{4/d}
\right)
&\leq C(T),
\\
\label{eq:bounds_FE_phi_translation}
\Delta t
\sum_{n=0}^{N_T - l}
\norm{ \phi_h^{n+l}
- \phi_h^n}_{L^2}^2
&\leq C(T) l \Delta t,
\end{align}
\end{subequations}
where the constants $C(T)$ are independent of $\alpha, h,\Delta t$, but depend exponentially on $T$.
\end{lemma}
\input{proofs/regularity_phi_sigma}
\subsection{Improving the regularity results in two dimensions}
\label{sec:regularity_2D}
The next result contains ideas of \cite[Thm.~7.1]{barrett_boyaval_2009}.
We provide a regularity result for the left Cauchy--Green tensor in two space dimensions, supposed that a CFL condition for the time step size is fulfilled.
The restriction to two space dimensions is due to a Gagliardo--Nirenberg inequality for $d=2$.
\begin{lemma}
\label{lemma:regul_B}
Let \ref{A1}--\ref{A6} hold true. Suppose that the discrete initial and boundary data satisfy \eqref{eq:init_bounds} and assume
\begin{align}
\label{eq:dt2}
\Delta t \leq \min\{ \Delta t_*, \ c_*(T) \alpha^2 h^2 \},
\end{align}
where $\Delta t_*$ is defined in \eqref{eq:dt} and $c_*(T)>0$ is a (probably very small) constant which is independent of $\alpha, h, \Delta t$ but can depend on $T$. Then, in addition to \eqref{eq:bounds_FE}, \eqref{eq:bounds_FE_phi_dtphi_dtsigma}, \eqref{eq:bounds_FE_phi_translation}, the following bound holds for all solutions of \ref{P_alpha_FE}:
\begin{align}
\label{eq:bounds_FE_B_dtB}
\begin{split}
&\max\limits_{n=1,...,N_T} \norm{\B_h^n}_{L^2}^2
+ \sum\limits_{n=1}^{N_T} \left(
\norm{\B_h^n - \B_h^{n-1}}_{L^2}^2
+ \Delta t \norm{\nabla \B_h^n}_{L^2}^2
+ \Delta t \nnorm{ \frac{\B_h^n - \B_h^{n-1}}{\Delta t} }_{(H^1)'}^{4/3}
\right)
\leq C(T, \alpha^{-1}),
\end{split}
\end{align}
where the constant $C(T,\alpha^{-1})>0$ is independent of $h,\Delta t$, but depends exponentially on $T, \alpha^{-1}$.
\end{lemma}
\input{proofs/regularity_B}
Now we have more control over the left Cauchy--Green tensor. This makes it possible to prove a regularity result for the discrete time derivative for the velocity.
First, we introduce the Helmholtz--Stokes operator $\ensuremath{\mathcal{S}}: \mathbf{V}' \to \mathbf{V}$ such that $\ensuremath{\mathcal{S}}\pmb{u}$ is the unique solution to the Helmholtz--Stokes problem
\begin{align}
\label{eq:helmholtz-stokes}
\int_\Omega (\ensuremath{\mathcal{S}}\pmb{u})\cdot \pmb{\mathrm{w}}
+ \nabla (\ensuremath{\mathcal{S}}\pmb{u}) : \nabla\pmb{\mathrm{w}} \dv{x}
= \dualp{\pmb{u}}{\pmb{\mathrm{w}}}_{\mathbf{V}}
\quad\quad \forall \ \pmb{\mathrm{w}}\in \mathbf{V},
\end{align}
where $\dualp{\cdot}{\cdot}_{\mathbf{V}}$ denotes the duality pairing between $\mathbf{V}'$ and $\mathbf{V}$. We remark that $\norm{\ensuremath{\mathcal{S}} \cdot}_{H^1}$ and $\norm{\cdot}_{\mathbf{V}'}$ are equivalent norms on $\mathbf{V}'$, see, e.g., \cite{barrett_sueli_2007}.
\begin{lemma}
\label{lemma:regul_v}
Let \ref{A1}--\ref{A6} hold. Suppose that the discrete initial and boundary data satisfy \eqref{eq:init_bounds} and that the CFL constraint \eqref{eq:dt2} holds. Then, in addition to \eqref{eq:bounds_FE}, \eqref{eq:bounds_FE_phi_dtphi_dtsigma}, \eqref{eq:bounds_FE_phi_translation}, \eqref{eq:bounds_FE_B_dtB}, all solutions of \ref{P_alpha_FE} fulfill
\begin{align}
\label{eq:bounds_FE_dtv_p}
& \Delta t \sum\limits_{n=1}^{N_T}
\nnorm{\ensuremath{\mathcal{S}}\left( \frac{\pmb{\mathrm{v}}_h^n - \pmb{\mathrm{v}}_h^{n-1}}{\Delta t} \right)}_{H^1}^{\frac{4}{3}}
\leq C(T, \alpha^{-1}),
\end{align}
where the constant $C(T,\alpha^{-1})>0$ is depending exponentially on $T, \alpha^{-1}$, but is independent of $h, \Delta t$.
\end{lemma}
\input{proofs/regularity_v_p}
At this point, we note that the bound \eqref{eq:bounds_FE_dtv_p} is not useful to apply common compactness techniques based on Aubin--Lions.
The problem is that the discrete velocity belongs to $\ensuremath{\mathcal{V}}_{h,\mathrm{div}} \subset H^1_0(\Omega;\R^d)$, but $\ensuremath{\mathcal{V}}_{h,\mathrm{div}}$ is no subspace of $\mathbf{V}$, as the discrete velocity is only divergence-free with respect to the ansatz space $\ensuremath{\mathcal{S}}_h$.
Moreover, $H^1_0(\Omega;\R^d)$ is compactly embedded in $L^2(\Omega;\R^d)$, but there exists no injective mapping from $L^2(\Omega;\R^d)$ into $\mathbf{V}'$. This is why Aubin--Lions cannot be applied here.
However, there are other techniques one can use for the velocity, see, e.g., \cite{barrett_boyaval_2009, barrett_2018_fene-p, guillen_2013_liquid_crystal}.
In this work, we follow the strategy of Metzger \cite{metzger_2018} which is based on \cite{azerad_guillen_2001} and we introduce the orthogonal Stokes projector $\mycal{R}_h: \ensuremath{\mathcal{V}}_{h,\mathrm{div}} \to \mathbf{V}$ by
\begin{align}
\label{eq:projector_Stokes_def}
\int_\Omega \nabla \mycal{R}_h \pmb{\mathrm{v}}_h : \nabla \pmb{\mathrm{w}} \dv{x}
= \int_\Omega \nabla \pmb{\mathrm{v}}_h : \nabla\pmb{\mathrm{w}} \dv{x}
\quad\quad \forall \ \pmb{\mathrm{w}} \in \mathbf{V}.
\end{align}
For any $\pmb{\mathrm{v}}_h \in \ensuremath{\mathcal{V}}_{h,\mathrm{div}}$, it holds (cf.~\cite{guillen_2013_liquid_crystal})
\begin{subequations}
\begin{align}
\label{eq:projector_Stokes_H1}
\norm{\mycal{R}_h \pmb{\mathrm{v}}_h}_{H^1}
&\leq C \norm{\pmb{\mathrm{v}}_h}_{H^1},
\\
\label{eq:projector_Stokes_L2}
\norm{\mycal{R}_h \pmb{\mathrm{v}}_h - \pmb{\mathrm{v}}_h}_{L^2}
&\leq C h \norm{\divergenz{\pmb{\mathrm{v}}_h}}_{L^2},
\\
\label{eq:projector_Stokes_H1'}
\norm{\mycal{R}_h \pmb{\mathrm{v}}_h}_{\mathbf{V}'}
&\leq C \big( h \norm{\divergenz{\pmb{\mathrm{v}}_h}}_{L^2}
+ \norm{\pmb{\mathrm{v}}_h}_{\mathbf{V}'} \big).
\end{align}
\end{subequations}
The following result is based on the bounds \eqref{eq:projector_Stokes_H1'}, \eqref{eq:bounds_FE} and \eqref{eq:bounds_FE_dtv_p}.
\begin{lemma}
Let \ref{A1}--\ref{A6} hold. Suppose that the discrete initial and boundary data satisfy \eqref{eq:init_bounds} and that the CFL constraint \eqref{eq:dt2} holds. Then, there exists a constant $C(T,\alpha^{-1})>0$ depending exponentially on $T, \alpha^{-1}$ but not on $h,\Delta t$, such that, in addition to \eqref{eq:bounds_FE}, \eqref{eq:bounds_FE_phi_dtphi_dtsigma}, \eqref{eq:bounds_FE_phi_translation}, \eqref{eq:bounds_FE_B_dtB}, \eqref{eq:bounds_FE_dtv_p}, all solutions of \ref{P_alpha_FE} satisfy for all $l\in\{1,...,N_T\}$,
\begin{align}
\label{eq:bounds_FE_v_translation}
\Delta t \sum_{n=0}^{N_T - l}
\norm{\mycal{R}_h \pmb{\mathrm{v}}_h^{n+l} - \mycal{R}_h \pmb{\mathrm{v}}_h^{n}}_{\mathbf{V}'}^2
\leq C(T,\alpha^{-1}) \big( l^\frac{3}{4} \Delta t + h^2 \big).
\end{align}
\end{lemma}
\input{proofs/regularity_v_translation}
\subsection{Passage to the limit and convergence to a weak solution}
\label{sec:convergence}
In the following, we prove that there exists a weak solution of \ref{P_alpha} in the sense of Definition \ref{def:weak_solution} which is obtained from converging subsequences of discrete solutions of \ref{P_alpha_FE} by passing to the limit $(h,\Delta t) \to (0,0)$.
For future reference, we recall the following compactness results from \cite[Sect.~8, Cor.~4 and Thm.~5]{simon_1986}.
Let $X, Y, Z$ be Banach spaces with a compact embedding $X \hookrightarrow \hookrightarrow Y$ and a continuous embedding $Y \hookrightarrow Z$. Let $1\leq p < \infty$ and $r>1$. Then, the following embeddings are compact:
\begin{subequations}
\begin{alignat}{3}
\label{eq:compact_Lp}
&\{\eta \in L^p(0,T;X)
&&\mid \partial_t\eta \in L^1(0,T;Z) \}
&&\hookrightarrow \hookrightarrow L^p(0,T;Y),
\\
\label{eq:compact_C}
&\{\eta \in L^\infty(0,T;X)
&&\mid \partial_t\eta \in L^r(0,T;Z) \}
&&\hookrightarrow \hookrightarrow C([0,T];Y).
\end{alignat}
Moreover, let $F$ be a bounded subset in $L^p(0,T;X)$ with
\begin{align}
\label{eq:compact_translation}
\lim_{\theta\to 0} \norm{\eta(\cdot+\theta) - \eta(\cdot)}_{L^p(0,T;Z)} = 0
\quad
\text{ uniformly for } \eta\in F.
\end{align}
Then $F$ is relatively compact in $L^p(0,T;Y)$ if $1\leq p < \infty$ and in $C([0,T];Y)$ if $p=\infty$, respectively.
Furthermore, we recall the following ``compactness by perturbation'' result from Azérad and Guillén-González \cite{azerad_guillen_2001} which provides strong convergence for subsequences of the discrete velocity.
Let $X,Y,Z$ be like before. Let $\{f_\epsilon\}_{\epsilon>0}$ be a family of functions which is bounded in $L^p(0,T;X)$ with $1\leq p < \infty$ such that
\begin{align}
\label{eq:compact_perturbation}
\norm{f_\epsilon(\cdot+\theta) - f_\epsilon(\cdot)}_{L^p(0,T;Z)}
\leq g_1(\theta) + g_2(\epsilon),
\end{align}
with $g_1(\theta)\to 0$ as $\theta\to 0$ and $g_2(\epsilon)\to 0$ as $\epsilon\to 0$. Then, the family $\{f_\epsilon\}_{\epsilon>0}$ possesses a cluster point in $L^p(0,T;Y)$ as $\epsilon\to 0$.
\end{subequations}
\bigskip
Let us introduce the following notation for affine linear and piecewise constant extensions of time discrete function $a^n(\cdot)$, $n=0,...,N_T$:
\begin{alignat}{2}
\label{def:fun_Delta_t}
a^{\Delta t}(\cdot, t)
&\coloneqq
\frac{t - t^{n-1}}{\Delta t} a^n(\cdot)
+ \frac{t^n - t}{\Delta t} a^{n-1}(\cdot)
\quad\quad
&& t\in [t^{n-1},t^n], n\in \{1,...,N_T\},
\\
\label{def:fun_Delta_t_pm}
a^{\Delta t,+}(\cdot, t)
&\coloneqq a^n(\cdot),
\quad\quad
a^{\Delta t,-}(\cdot, t)
\coloneqq a^{n-1}(\cdot)
\quad\quad
&& t\in (t^{n-1},t^n], n\in \{1,...,N_T\}.
\end{alignat}
Let us note that we write $a^{\Delta t,\pm}$ for results that hold true for both $a^{\Delta t,+}$ and $a^{\Delta t,-}$, and we write $a^{\Delta t(,\pm)}$ for results that hold true for $a^{\Delta t}$, $a^{\Delta t,+}$ and $a^{\Delta t,-}$, respectively.
Using this notation, we reformulate the problem \ref{P_alpha_FE} continuously in time. Multiplying each equation by $\Delta t$ and summing from $n=1,...,N_T$, we obtain for any test functions $(\zeta_h$, $\rho_h$, $ \xi_h$, $ q_h$, $ \pmb{\mathrm{w}}_h$, $ \C_h) \in (L^2(0,T;\ensuremath{\mathcal{S}}_h))^4 \times L^2(0,T;\ensuremath{\mathcal{V}}_h) \times L^2(0,T;\ensuremath{\mathcal{W}}_h)$ that
\begin{subequations}
\begingroup
\allowdisplaybreaks
\begin{align}
\label{eq:phi_FE_time}
0 &= \int_{\Omega_T} \mycal{I}_h \Big[ \big(\partial_t \phi_h^{\Delta t}
- \Gamma_{\phi,h}^{\Delta t,+} \big) \zeta_h \Big]
+ \mycal{I}_h[m(\phi_h^{\Delta t,-})] \nabla\mu_h^{\Delta t,+} \cdot \nabla \zeta_h
- \phi_h^{\Delta t,-} \pmb{\mathrm{v}}_h^{\Delta t,+} \cdot\nabla \zeta_h \dv{x} \dv{t},
\\
\label{eq:mu_FE_time}
0 &= \int_{\Omega_T} \mycal{I}_h \Big[ \Big( - \mu_h^{\Delta t,+}
+ A \psi_1'(\phi_h^{\Delta t,+})
+ A \psi_2'(\phi_h^{\Delta t,-})
- \chi_\phi \sigma_h^{\Delta t,+} \Big) \rho_h \Big]
+ B \nabla\phi_h^{\Delta t,+} \cdot \nabla\rho_h\dv{x} \dv{t},
\\
\nonumber
\label{eq:sigma_FE_time}
0 &= \int_{\Omega_T} \mycal{I}_h \Big[ \big( \partial_t \sigma_h^{\Delta t}
+ \Gamma_{\sigma,h}^{\Delta t,+} \big) \xi_h \Big]
+ \mycal{I}_h[n(\phi_h^{\Delta t,-})] \nabla \big(\chi_\sigma\sigma_h^{\Delta t,+} - \chi_\phi\phi_h^{\Delta t,+} \big) \cdot \nabla \xi_h
- \sigma_h^{\Delta t,-} \pmb{\mathrm{v}}_h^{\Delta t,+} \cdot\nabla \xi_h \dv{x} \dv{t}
\\
&\qquad + \int_0^T \int_{\partial\Omega} \mycal{I}_h\Big[ K \big( \sigma_h^{\Delta t,+} - \sigma_{\infty,h}^{\Delta t,+} \big) \xi_h \Big] \ \mathrm d \calH^{d-1} \dv{t},
\\
\label{eq:div_v_FE_time}
0 &= \int_{\Omega_T} \divergenz{\pmb{\mathrm{v}}_h^{\Delta t,+}} q_h \dv{x} \dv{t},
\\
\label{eq:v_FE_time}
\nonumber
0 &= \int_{\Omega_T} \partial_t \pmb{\mathrm{v}}_h^{\Delta t} \cdot \pmb{\mathrm{w}}_h
+ \frac{1}{2} \left( \left(\pmb{\mathrm{v}}_h^{\Delta t,-}\cdot \nabla\right) \pmb{\mathrm{v}}_h^{\Delta t,+}\right) \cdot \pmb{\mathrm{w}}_h
- \frac{1}{2} \pmb{\mathrm{v}}_h^{\Delta t,+} \cdot \left(\left(\pmb{\mathrm{v}}_h^{\Delta t,-} \cdot \nabla\right) \pmb{\mathrm{w}}_h \right) \dv{x} \dv{t}
\\
\nonumber
&\qquad + \int_{\Omega_T} 2\mycal{I}_h[\eta(\phi_h^{\Delta t,-})] \D(\pmb{\mathrm{v}}_h^{\Delta t,+}) : \D(\pmb{\mathrm{w}}_h)
+ \kappa ( \B_h^{\Delta t,+} - \I ) : \nabla\pmb{\mathrm{w}}_h
- \divergenz{\pmb{\mathrm{w}}_h} p_h^{\Delta t,+} \dv{x} \dv{t}
\\
&\qquad + \int_{\Omega_T} \Big( \phi_h^{\Delta t,-} \nabla\mu_h^{\Delta t,+}
+ \sigma_h^{\Delta t,-} \nabla \big(\chi_\sigma\sigma_h^{\Delta t,+} - \chi_\phi\phi_h^{\Delta t,+}\big) \Big) \cdot \pmb{\mathrm{w}}_h \dv{x} \dv{t},
\\
\label{eq:B_FE_time}
\nonumber
0 &= \int_{\Omega_T} \mycal{I}_h \Big[
\Big( \partial_t \B_h^{\Delta t}
+ \frac{\kappa}{\tau(\phi_h^{\Delta t,-})} (\B_h^{\Delta t,+} - \I) \Big): \C_h \Big]
- 2 \nabla\pmb{\mathrm{v}}_h^{\Delta t,+} : \mycal{I}_h\big[ \C_h \B_h^{\Delta t,+} \big] \dv{x} \dv{t}
\\
&\qquad + \int_{\Omega_T} \alpha \nabla\B_h^{\Delta t,+} : \nabla\C_h
- \sum\limits_{i,j=1}^d
[\pmb{\mathrm{v}}_h^{\Delta t,-}]_i \Lambda_{i,j}(\B_h^{\Delta t,+}) : \partial_{x_j} \C_h \dv{x} \dv{t},
\end{align}
\endgroup
\end{subequations}
subject to the initial conditions $\phi_h^{\Delta t}(0) = \phi_h^0$, $\sigma_h^{\Delta t}(0) = \sigma_h^0$, $\pmb{\mathrm{v}}_h^{\Delta t}(0) = \pmb{\mathrm{v}}_h^0$ and $\B_h^{\Delta t}(0) = \B_h^0$, where we write $\Gamma_{\phi,h}^{\Delta t,+} = \Gamma_{\phi}(\phi_h^{\Delta t,+}, \sigma_h^{\Delta t,+}, \B_h^{\Delta t,+})$ and similarly for $\Gamma_{\sigma,h}^{\Delta t,+}$.
The following result is a direct consequence of \eqref{def:fun_Delta_t}, \eqref{def:fun_Delta_t_pm}, Theorem \ref{theorem:existence_FE}, \eqref{eq:bounds_FE}, \eqref{eq:bounds_FE_phi_dtphi_dtsigma}, \eqref{eq:bounds_FE_phi_translation}, \eqref{eq:bounds_FE_B_dtB}, \eqref{eq:bounds_FE_dtv_p}, \eqref{eq:bounds_FE_v_translation} and \eqref{eq:init_bounds}.
\begin{corollary}
Let \ref{A1}--\ref{A5} hold. Suppose that the discrete initial and boundary data satisfy \eqref{eq:init_bounds}. Moreover, assume that $\Delta t < \Delta t_*$, where $\Delta t_*$ is defined in \eqref{eq:dt}. Then, there exist functions $\phi_h^{\Delta t(,\pm)}$,
$\mu_h^{\Delta t,+}$,
$\sigma_h^{\Delta t(,\pm)}$,
$p_h^{\Delta t,+}$,
$\pmb{\mathrm{v}}_h^{\Delta t(,\pm)}$,
$\B_h^{\Delta t(,\pm)}$ solving \eqref{eq:phi_FE_time}--\eqref{eq:B_FE_time} and constants $C(T)>0$ depending on $T$ but not on $\alpha,h,\Delta t$, such that
\begin{subequations}
\begin{align}
\label{eq:bounds_time1}
\nonumber
&\norm{\phi_h^{\Delta t(,\pm)}}_{L^\infty(0,T;H^1)}
+ \norm{\Delta_h \phi_h^{\Delta t(,\pm)}}_{L^2(0,T;L^2)}
+ \norm{\partial_t \phi_h^{\Delta t} }_{L^2(0,T;(H^1)')}
+ \tfrac{1}{\sqrt{\Delta t}} \norm{\phi_h^{\Delta t} - \phi_h^{\Delta t,\pm}}_{L^2(0,T;H^1)}
\\
\nonumber
&\quad
+ \norm{\mu_h^{\Delta t,+}}_{L^2(0,T;H^1)}
+ \norm{\sigma_h^{\Delta t(,\pm)}}_{L^\infty(0,T;L^2)}
+ \norm{\sigma_h^{\Delta t(,\pm)}}_{L^2(0,T;H^1)}
+ \norm{\partial_t \sigma_h^{\Delta t} }_{L^{4/d}(0,T;(H^1)')}
\\
\nonumber
&\quad
+ \tfrac{1}{\sqrt{\Delta t}} \norm{\sigma_h^{\Delta t} - \sigma_h^{\Delta t,\pm}}_{L^2(0,T;L^2)}
+ \norm{\sigma_h^{\Delta t(,\pm)}}_{L^2(0,T;L^2({\partial\Omega}))}
+ \norm{\pmb{\mathrm{v}}_h^{\Delta t(,\pm)}}_{L^\infty(0,T;L^2)}
+ \norm{\pmb{\mathrm{v}}_h^{\Delta t(,\pm)}}_{L^2(0,T;H^1)}
\\
\nonumber
&\quad
+ \tfrac{1}{\sqrt{\Delta t}} \norm{\pmb{\mathrm{v}}_h^{\Delta t} - \pmb{\mathrm{v}}_h^{\Delta t,\pm}}_{L^2(0,T;L^2)}
\\
&\leq C(T),
\end{align}
and, for any $l\in\{1,...,N_T\}$,
\begin{align}
\label{eq:bounds_time_phi_translation}
\int_0^{T- l \Delta t}
\nnorm{\phi_h^{\Delta t(,\pm)}(\cdot,t+ l \Delta t)
- \phi_h^{\Delta t(,\pm)}(\cdot,t) }_{L^2}^2 \dv{t}
&\leq C(T) l\Delta t.
\end{align}
Further, if in addition \ref{A6} and the CFL constraint \eqref{eq:dt2} hold true, then there exist constants $C(T,\alpha^{-1})>0$ depending on $T,\alpha^{-1}$ but not on $h,\Delta t$, such that
\begin{align}
\label{eq:bounds_time2}
\nonumber
& \norm{\ensuremath{\mathcal{S}} \partial_t \pmb{\mathrm{v}}_h^{\Delta t} }_{L^{4/3}(0,T;H^1)}
+ \norm{\B_h^{\Delta t(,\pm)}}_{L^\infty(0,T;L^2)}
+ \norm{\B_h^{\Delta t(,\pm)}}_{L^2(0,T;H^1)}
+ \norm{\partial_t \B_h^{\Delta t} }_{L^{4/3}(0,T;(H^1)')}
\\
&\quad
+ \tfrac{1}{\sqrt{\Delta t}} \norm{\B_h^{\Delta t} - \B_h^{\Delta t,\pm}}_{L^2(0,T;L^2)}
\leq C(T,\alpha^{-1}),
\end{align}
and, for any $l\in\{1,...,N_T\}$,
\begin{align}
\label{eq:bounds_time_v_translation}
\int_0^{T- l \Delta t}
\nnorm{\mycal{R}_h \pmb{\mathrm{v}}_h^{\Delta t(,\pm)}(\cdot,t+ l \Delta t)
- \mycal{R}_h \pmb{\mathrm{v}}_h^{\Delta t(,\pm)}(\cdot,t) }_{\mathbf{V}'}^2 \dv{t}
&\leq C(T,\alpha^{-1}) \Big( l^\frac{3}{4} \Delta t + h^2 \Big).
\end{align}
\end{subequations}
\end{corollary}
We now show that there exist subsequences of discrete solutions which converge to some limit functions, as $(h,\Delta t)\to 0$.
\begin{lemma}[Converging subsequences]
\label{lemma:convergence}
Let \ref{A1}--\ref{A5} hold. Suppose that the discrete initial and boundary data satisfy \eqref{eq:init_bounds} and \eqref{eq:init_conv}. Moreover, assume that $\Delta t < \Delta t_*$, where $\Delta t_*$ is defined in \eqref{eq:dt}.
Then, there exists a (non-relabeled) subsequence of $\left\{ \phi_h^{\Delta t(,\pm)},
\mu_h^{\Delta t,+},
\sigma_h^{\Delta t(,\pm)},
p_h^{\Delta t,+},
\pmb{\mathrm{v}}_h^{\Delta t(,\pm)},
\B_h^{\Delta t(,\pm)} \right\}_{h,\Delta t>0}$, such that \eqref{eq:phi_FE_time}--\eqref{eq:B_FE_time} is fulfilled, and functions
\begin{align*}
\phi &\in L^\infty(0,T;H^1)
\cap L^2(0,T;H^2)
\cap H^1(0,T; (H^1)'),
\quad
\mu \in L^2(0,T; H^1),
\\
\sigma &\in L^\infty(0,T; L^2)
\cap L^2(0,T; H^1)
\cap W^{1,\frac{4}{d}}(0,T; (H^1)'),
\quad
\pmb{\mathrm{v}} \in L^\infty(0,T; \mathbf{H}) \cap L^2(0,T; \mathbf{V}),
\end{align*}
with $\phi(0) = \phi_0 \text{ in } L^2(\Omega)$ and $\sigma(0) = \sigma_0 \text{ in } L^2(\Omega)$ exist,
such that, as $(h,\Delta t) \to (0,0)$,
\begin{alignat}{3}
\setcounter{subeq}{0}\refstepcounter{equation}
\refstepcounter{subeq}\tag{\thesubeq}
\label{eq:conv_phi_Linf_H1}
\phi_h^{\Delta t(,\pm)} &\to \phi \quad &&\text{weakly-$*$} \quad && \text{in } L^\infty(0,T;H^1),
\\
\refstepcounter{subeq}\tag{\thesubeq}
\label{eq:conv_dtphi_L2_H1'}
\partial_t \phi_h^{\Delta t} &\to \partial_t\phi \quad
&&\text{weakly} \quad && \text{in } L^2(0,T;(H^1)'),
\\
\refstepcounter{subeq}\tag{\thesubeq}
\label{eq:conv_phi_L2_H2}
\Delta_h \phi_h^{\Delta t(,\pm)} &\to \Delta \phi \quad &&\text{weakly} \quad && \text{in } L^2(0,T;L^2),
\\
\refstepcounter{subeq}\tag{\thesubeq}
\label{eq:conv_phi_L2_W1s}
\phi_h^{\Delta t(,\pm)} &\to \phi \quad &&\text{weakly} \quad && \text{in } L^2(0,T;W^{1,s}),
\\
\refstepcounter{subeq}\tag{\thesubeq}
\label{eq:conv_phi_strong}
\phi_h^{\Delta t(,\pm)} &\to \phi \quad &&\text{strongly} \quad && \text{in } L^2(0,T;C^{0,\gamma}(\overline\Omega)),
\\[2ex]
\label{eq:conv_mu_L2_H1}
\mu_h^{\Delta t,+} &\to \mu \quad &&\text{weakly} \quad && \text{in } L^2(0,T;H^1),
\\[2ex]
\setcounter{subeq}{0}\refstepcounter{equation}
\refstepcounter{subeq}\tag{\thesubeq}
\label{eq:conv_sigma_Linf_L2}
\sigma_h^{\Delta t(,\pm)} &\to \sigma \quad &&\text{weakly-$*$} \quad && \text{in } L^\infty(0,T; L^2),
\\
\refstepcounter{subeq}\tag{\thesubeq}
\label{eq:conv_sigma_L2_H1}
\sigma_h^{\Delta t(,\pm)} &\to \sigma \quad &&\text{weakly} \quad && \text{in } L^2(0,T;H^1),
\\
\refstepcounter{subeq}\tag{\thesubeq}
\label{eq:conv_dtsigma_L4/3_H1'}
\partial_t \sigma_h^{\Delta t} &\to \partial_t\sigma \quad
&&\text{weakly} \quad && \text{in } L^{4/d}(0,T;(H^1)'),
\\
\refstepcounter{subeq}\tag{\thesubeq}
\label{eq:conv_sigma_strong}
\sigma_h^{\Delta t(,\pm)} &\to \sigma \quad &&\text{strongly} \quad && \text{in } L^2(0,T; L^q),
\\[2ex]
\setcounter{subeq}{0}\refstepcounter{equation}
\refstepcounter{subeq}\tag{\thesubeq}
\label{eq:conv_v_Linf_L2}
\pmb{\mathrm{v}}_h^{\Delta t(,\pm)} &\to \pmb{\mathrm{v}} \quad
&& \text{weakly-$*$} \quad &&\text{in }
L^\infty(0,T; \mathbf{H}),
\\
\refstepcounter{subeq}\tag{\thesubeq}
\label{eq:conv_v_L2_H1}
\pmb{\mathrm{v}}_h^{\Delta t(,\pm)} &\to \pmb{\mathrm{v}} \quad
&& \text{weakly} \quad &&\text{in }
L^2(0,T;\mathbf{V}),
\end{alignat}
where $s\in\left[2, \frac{2d}{d-2} \right)$, $\gamma\in \left(0, \frac{4-d}{2}\right)$ and $q\in \left[1,\frac{2d}{d-2}\right)$, respectively.
Moreover, if in addition \ref{A6} and the CFL constraint \eqref{eq:dt2} holds, then additionally $\pmb{\mathrm{v}} \in W^{1,\frac{4}{3}}(0,T; \mathbf{V}')$ with $\pmb{\mathrm{v}}(0) = \pmb{\mathrm{v}}_0 \text{ in } \mathbf{H}$ and there exists a function
\begin{align*}
\B &\in L^\infty\left(0,T; L^2(\Omega;\R^{2\times2}_{\mathrm{SPD}})\right)
\cap L^2\left(0,T; H^1(\Omega;\R^{2\times2}_{\mathrm{S}})\right)
\cap W^{1,\frac{4}{3}}\left(0,T; (H^1(\Omega;\R^{2\times2}_{\mathrm{S}}))'\right)
\end{align*}
with $\B(0) = \B_0 \text{ in } L^2(\Omega;\R^{2\times 2}_{\mathrm{S}}) \text{ and } \B \text{ positive definite a.e.~in } \Omega\times(0,T)$, such that, as $(h,\Delta t)\to (0,0)$,
\begin{alignat}{3}
\refstepcounter{subeq}\tag{\thesubeq}
\label{eq:conv_dtv_L4/3_H1'}
\partial_t \pmb{\mathrm{v}}_h^{\Delta t} &\to \partial_t \pmb{\mathrm{v}} \quad
&& \text{weakly} \quad &&\text{in }
L^{4/3}(0,T;\mathbf{V}'),
\\
\refstepcounter{subeq}\tag{\thesubeq}
\label{eq:conv_v_strong}
\pmb{\mathrm{v}}_h^{\Delta t(,\pm)} &\to \pmb{\mathrm{v}} \quad
&& \text{strongly} \quad &&\text{in }
L^2(0,T;L^r(\Omega;\R^2)),
\\[2ex]
\setcounter{subeq}{0}\refstepcounter{equation}
\refstepcounter{subeq}\tag{\thesubeq}
\label{eq:conv_B_Linf_L2}
\B_h^{\Delta t(,\pm)} &\to \B \quad
&& \text{weakly-$*$} \quad &&\text{in }
L^\infty \left(0,T;L^2(\Omega;\R^{2\times2}_{\mathrm{S}})\right),
\\
\refstepcounter{subeq}\tag{\thesubeq}
\label{eq:conv_B_L2_H1}
\B_h^{\Delta t(,\pm)} &\to \B \quad
&& \text{weakly} \quad &&\text{in }
L^2 \left(0,T;H^1(\Omega;\R^{2\times2}_{\mathrm{S}})\right),
\\
\refstepcounter{subeq}\tag{\thesubeq}
\label{eq:conv_dtB_L4/3_H1'}
\partial_t \B_h^{\Delta t} &\to \partial_t \B \quad
&& \text{weakly} \quad &&\text{in }
L^{4/3}\left(0,T;(H^1(\Omega;\R^{2\times2}_{\mathrm{S}}))'\right),
\\
\refstepcounter{subeq}\tag{\thesubeq}
\label{eq:conv_B_strong}
\B_h^{\Delta t(,\pm)} &\to \B \quad
&& \text{strongly} \quad &&\text{in }
L^2\left(0,T;L^p(\Omega;\R^{2\times2}_{\mathrm{S}})\right),
\end{alignat}
where $r,p\in [1,\infty)$, respectively.
\end{lemma}
\input{proofs/proof_lemma_convergence}
For the main result, we recall the following technical result taken from \cite[Lem.~5.3]{barrett_boyaval_2009}.
\begin{lemma}
For all $K_k\in\ensuremath{\mathcal{T}}_h$, and for all $\C_h\in \ensuremath{\mathcal{W}}_{h,\mathrm{PD}}$, it holds
\begin{align}
\label{eq:error_Lambda}
\max\limits_{i,j=1,...,d}
\int_{K_k} \abs{ \Lambda_{i,j}(\C_h) - \C_h \delta_{ij} }^2 \dv{x}
\leq C h^2 \int_{K_k} \abs{\nabla\C_h}^2 \dv{x}.
\end{align}
\end{lemma}
We also recall the following result from \cite[Lem.~6.8]{barrett_2018_fene-p}.
\begin{lemma}
Let $g\in C^{0,1}(\R)$ with Lipschitz constant $L_g$. For all $K_k \in \ensuremath{\mathcal{T}}_h$ and for all $q_h\in \ensuremath{\mathcal{S}}_h$, $\C_h\in \ensuremath{\mathcal{W}}_h$, it holds
\begin{align}
\label{eq:interp_Lipschitz}
\begin{split}
\int_{K_k} \abs{\mycal{I}_h\big[ g(q_h) \big] - g(q_h) }^2 \dv{x}
&\leq C L_g^2 h^2 \int_{K_k} \abs{\nabla q_h}^2 \dv{x},
\\
\int_{K_k} \abs{\mycal{I}_h\big[ g(\C_h) \big] - g(\C_h) }^2 \dv{x}
&\leq C L_g^2 h^2 \int_{K_k} \abs{\nabla \C_h}^2 \dv{x}.
\end{split}
\end{align}
\end{lemma}
We now pass to the limit $(h,\Delta t) \to (0,0)$ in \ref{P_alpha_FE} and show that the function tuple $( \phi, \mu, \sigma, \pmb{\mathrm{v}}, \B )$ from Lemma \ref{lemma:convergence} forms a weak solution to \ref{P_alpha} in the sense of Definition \ref{def:weak_solution}, which finally proves Theorem \ref{theorem:weak_solution}.
For the reader's convenience, we state this result in the following theorem.
Note that in comparison to Lemma \ref{lemma:convergence} we need the additional assumption \ref{A7} on the source terms $\Gamma_\phi,\Gamma_\sigma$, as otherwise the limit passing would not be possible in presence of $\mycal{I}_h$. However, this assumption is not strict in practice and can be dropped if, e.g., mass lumping is not considered.
\begin{theorem}[Limit passing in \ref{P_alpha_FE}]
\label{theorem:convergence}
Let \ref{A1}--\ref{A7} hold true. Suppose that the discrete initial and boundary data satisfy \eqref{eq:init_bounds} and \eqref{eq:init_conv}, and that the CFL constraint \eqref{eq:dt2} holds.
Then the functions $\phi,\mu,\sigma, \pmb{\mathrm{v}}, \B$ from Lemma \ref{lemma:convergence} form a weak solution to \ref{P_alpha} in the sense of Definition \ref{def:weak_solution}.
\end{theorem}
\input{proofs/proof_theorem_convergence}
\section{Introduction}
\label{sec:introduction}
In the past few years, the study of mathematical models for tumour growth has become a popular topic of research.
Even though many biological processes with regard to tissue growth are very complicated and still not fully understood, mathematical models try to give an insight into the qualitative behaviour of the most significant processes.
Yet, the main difficulty is to choose the model in a way such that the individual properties of the respective biological material are described as good as possible.
Here, material laws play a decisive role and several different approaches have been proposed in the literature.
Detailed comparisons with \textit{in vivo} experiments indicate that neglecting the elastic effects completely would be too restrictive, as mechanical stresses have a noticeable impact on the growth behaviour \cite{lima_2016}. Hence, living tissues are sometimes modelled as an elastic solid where the behaviour is described with linear or nonlinear elasticity.
Moreover, there are models that refer to very short time scales for stress relaxation and thus propose viscous approaches, as they allow to consider the random and directional movement of the cells qualitatively, which is a well-known behaviour of tumour cells \cite{chemotaxis_in_cancer_2011}.
On the other hand, the behaviour of tumour cells within the extracellular matrix resembles granular material for which usually Darcy's law is prescribed \cite{ambrosi_2009}.
A popular ansatz in the literature is to combine multiple material laws at once.
For example, Brinkman's law is used to describe material featuring properties of granular material and viscous fluids \cite{ebenbeck_garcke_nurnberg_2020}.
To account for viscous and elastic properties, viscoelastic approaches are very helpful and they are mostly studied in the context of polymeric fluids \cite{barrett_boyaval_2009, barrett_lu_sueli_2017, Lukacova_2017}. Although there exist viscoelastic models for tumour growth \cite{ambrosi_2009, bresch_2009, lowengrub_2021_viscoelastic}, there is still a huge gap in the literature concerning the derivation and mathematical analysis, especially for phase field approaches.
A Cahn--Hilliard model coupled to viscoelasticity with a Neo-Hookean finite elasticity, which is different to the one in the present paper, has been derived and analyzed in \cite{garcke_2022_viscoelastic}.
The goal of this work is to introduce and study a new mathematical model for tumour growth where viscoelasticity is taken into account.
The general mathematical model of our interest is given by the following nonlinear system of partial differential equations.
\subsubsection*{Problem \ref{P}:}
\mylabelHIDE{P}{$(\pmb{\mathrm{P}})$} Find $\phi,\mu,\sigma,p: \Omega\times (0,T) \to \R$, $\pmb{\mathrm{v}}:\Omega\times(0,T)\to \R^d$, $\B:\Omega\times(0,T)\to\R^{d\times d}$ such that in $\Omega\times (0,T)$:
\begin{subequations}
\begin{align}
\label{eq:phi}
\partial_t \phi + \divergenz{\phi \pmb{\mathrm{v}}}
&= \divergenz{m(\phi) \nabla\mu} + \Gamma_\phi(\phi,\mu,\sigma,\B),
\\
\label{eq:mu}
\mu &= A \psi'(\phi) - B \Delta\phi
+ N_{,\phi}(\phi,\sigma)
+ W_{,\phi}(\phi,\B),
\\
\label{eq:sigma}
\partial_t \sigma + \divergenz{\sigma \pmb{\mathrm{v}}}
&= \divergenz{n(\phi) \nabla N_{,\sigma}(\phi,\sigma)}
- \Gamma_\sigma(\phi,\mu,\sigma),
\\
\label{eq:div}
\divergenz{\pmb{\mathrm{v}}} &= \Gamma_{\pmb{\mathrm{v}}}(\phi,\mu,\sigma,\B),
\\
\label{eq:v}
\rho \partial_t \pmb{\mathrm{v}} + \rho (\pmb{\mathrm{v}}\cdot \nabla)\pmb{\mathrm{v}}
&= \divergenz{\T(\phi,\pmb{\mathrm{v}},p,\B)}
+ \big(\mu - W_{,\phi}(\phi,\B)
\big) \nabla\phi
+ N_{,\sigma}(\phi,\sigma) \nabla\sigma,
\\
\label{eq:B}
\partial_t \B + (\pmb{\mathrm{v}}\cdot\nabla)\B
+ \frac{1}{\tau(\phi)} \T_{\mathrm{el}}(\phi,\B)
&= \nabla\pmb{\mathrm{v}} \B + \B (\nabla\pmb{\mathrm{v}})^T,
\end{align}
\end{subequations}
where $\Omega\subset \R^d$, $d\in\{2,3\}$, is a bounded domain and $T>0$ is a fixed time. Here, the viscoelastic stress tensor is given by
\begin{align}
\label{eq:T_viscoelastic}
\T(\phi,\pmb{\mathrm{v}}, p, \B)
\coloneqq
\T_{\mathrm{visc}}(\phi,\pmb{\mathrm{v}},p) + \T_{\mathrm{el}}(\phi,\B),
\end{align}
where the viscous and the elastic parts of the stress tensor are defined as
\begin{align}
\T_{\mathrm{visc}}(\phi,\pmb{\mathrm{v}},p)
&\coloneqq
\eta(\phi) \big( \nabla\pmb{\mathrm{v}} + (\nabla\pmb{\mathrm{v}})^T \big)
+ \lambda(\phi) \divergenz{\pmb{\mathrm{v}}}\I - p\I,
\\
\T_{\mathrm{el}}(\phi,\B) &\coloneqq 2W_{,\B}(\phi,\B) \B.
\end{align}
The above system is composed of a convected Cahn--Hilliard system \eqref{eq:phi}--\eqref{eq:mu} for the order parameter $\phi\in[-1,1]$ denoting the difference of volume fractions, with $\{\phi=1\}$ representing the unmixed tumour tissue and $\{\phi=-1\}$ representing the surrounding healthy tissue, and the chemical potential $\mu$ related to the phase field variable $\phi$. This system is coupled to a convected parabolic diffusion equation \eqref{eq:sigma} where $\sigma$ denotes the concentration of an unknown species serving as a nutrient for the tumour.
We include hydrodynamic effects through the viscoelastic system \eqref{eq:div}--\eqref{eq:v} with constant mass density $\rho$ for the volume-averaged velocity $\pmb{\mathrm{v}}$, the pressure $p$ and the viscoelastic stress tensor $\T$. Here, $\B$ denotes the left Cauchy--Green tensor associated with the elastic part of the total mechanical response of the viscoelastic fluid and it is given by the constitutive equation \eqref{eq:B} of Oldroyd-B type \cite{Oldroyd_1950}, but other constitutive equations for $\B$ are also possible, e.g., a constitutive equation of Giesekus type \cite{giesekus_1982}, i.e.
\begin{align}
\label{eq:giesekus}
\partial_t \B + (\pmb{\mathrm{v}}\cdot\nabla)\B
+ \frac{1}{\tau(\phi)} \B \T_{\mathrm{el}}(\phi,\B)
&= \nabla\pmb{\mathrm{v}} \B + \B (\nabla\pmb{\mathrm{v}})^T.
\end{align}
By $N_{,\phi}(\phi,\sigma)$ and $N_{,\sigma}(\phi,\sigma)$, we denote the variations of a general nutrient energy density $N(\phi,\sigma)$ with respect to $\phi$ and $\sigma$, respectively. Similarly, by $W_{,\phi}(\phi,\B)$ and $W_{,\B}(\phi,\B)$, we denote the variations of a general elastic energy density $W(\phi,\B)$ with respect to $\phi$ and $\B$. These energies will be specified later.
The positive constants $A,B$ usually have the form $A = \frac{\beta}{\epsilon}$ and $B=\beta\epsilon$, where $\epsilon$ is proportional to the thickness of the diffuse interface and $\beta$ represents the surface tension. By $m(\cdot)$ and $n(\cdot)$, we denote the non-negative mobilities for the order parameter $\phi$ and the nutrient $\sigma$, respectively, and $\psi(\cdot)$ is a non-negative potential with two equal minima at $\pm 1$.
Biological effects like proliferation, apoptosis and nutrient consumption are taken into account through the source and sink terms $\Gamma_\phi$ and $\Gamma_\sigma$. Moreover, $\Gamma_{\pmb{\mathrm{v}}}$ denotes a source for the velocity divergence and is often related to $\Gamma_\phi$.
The non-negative functions $\eta(\cdot)$ and $\lambda(\cdot)$ denote the shear and bulk viscosities, respectively. The non-negative function $\tau(\cdot)$ is the viscoelastic relaxation time accounting for dissipation.
We now present the outline of this work. We end Section \ref{sec:introduction} by introducing our notation.
Then, in Section \ref{sec:derivation}, we present the derivation of the general viscoelastic model for tumour growth \ref{P} using basic thermodynamical principles and we give several examples of constitutive laws. Moreover, we highlight further important aspects of modelling like a dissipation law for a general energy of the system, reformulations of the pressure leading to variants of the velocity equation \eqref{eq:v}, and initial and boundary conditions. We also give relevant examples for the source functions $\Gamma_\phi,\Gamma_\sigma,\Gamma_{\pmb{\mathrm{v}}}$.
Further, we specify the nutrient energy density $N(\phi,\sigma)$ and the elastic energy density $W(\phi,\B)$.
Besides, we handle the case of possible source or sink terms due to growth in the equation for $\B$, and we present several limit cases of our model \ref{P} which were introduced for other models in the literature.
In Section \ref{sec:Section3}, we consider a special variant of the problem \ref{P} which is additionally regularized with a dissipative term $\alpha\Delta\B$ in the Oldroyd-B equation, and the regularized problem is denoted by \ref{P_alpha}. This regularization improves the mathematical properties of the governing equations while it has a minor impact on the dynamical behaviour of the model, supposed that the viscoelastic diffusion constant $\alpha$ is small.
For \ref{P_alpha}, we give the definition of weak solutions and provide an existence result in two spatial dimensions in Section \ref{sec:weak_solution}. To highlight the difficulties and to better understand the techniques in the proof of the existence result, we present the derivation of formal \textit{a priori} estimates in Section \ref{sec:formal_bounds}, the need for the restriction to two dimensions in Section \ref{sec:formal_bounds_2d} and a regularization strategy from Barrett and Boyaval \cite{barrett_boyaval_2009} in Section \ref{sec:regularization} which is needed to show positive definiteness of the Cauchy--Green tensor $\B$.
The existence result itself will be proved in Section \ref{sec:fem} by the limit passing in a fully-discrete finite element scheme in two dimensions where a CFL condition, i.e.~$\Delta t \leq C h^2$, is required, whereby Section \ref{sec:fem} is organized as follows. First, a regularized fully-discrete finite element scheme is introduced in arbitrary dimensions $d\in\{2,3\}$, where the regularization strategy from Barrett and Boyaval \cite{barrett_boyaval_2009} is applied on the fully-discrete level. For the regularized discrete scheme, stability and existence in arbitrary dimensions $d\in\{2,3\}$ are shown in Sections \ref{sec:stability} and \ref{sec:existence}, respectively. Then, in Section \ref{sec:delta_to_zero}, the regularization parameter is sent to zero which guarantees that the discrete Cauchy--Green tensor is positive definite. After that, the regularity of the discrete solutions is improved in arbitrary dimensions $d\in\{2,3\}$ in Section \ref{sec:regularity} and in also in two dimensions $d=2$ in Section \ref{sec:regularity_2D}, where a CFL condition, i.e.~$\Delta t\leq C h^2$, is needed. Then, Section \ref{sec:convergence} is devoted to the limit process $(\Delta t,h)\to (0,0)$ in two space dimensions, where existence of global-in-time weak solutions is provided by converging subsequences of the discrete solutions. Finally, in Section \ref{sec:numeric}, we present numerical results for the fully-discrete tumour model from Section \ref{sec:fem} in two spatial dimensions.
\subsection{Notation}
In this work, vector or matrix valued quantities are represented with a bold or blackboard bold font, respectively.
For $d\in\{2,3\}$, we define the scalar product of two vectors $\pmb{\mathrm{v}},\pmb{\mathrm{w}}\in\R^d$ by $\pmb{\mathrm{v}}\cdot\pmb{\mathrm{w}} \coloneqq \pmb{\mathrm{v}}^T \pmb{\mathrm{w}} = \pmb{\mathrm{w}}^T \pmb{\mathrm{v}}$, and the scalar product of two matrices $\ensuremath{\mathbb{A}},\B\in\R^{d\times d}$ by $\ensuremath{\mathbb{A}}:\B \coloneqq \trace(\ensuremath{\mathbb{A}}^T \B) = \trace(\B^T \ensuremath{\mathbb{A}})$, where $\trace(\ensuremath{\mathbb{A}})$ denotes the trace of a matrix $\ensuremath{\mathbb{A}}\in\R^{d\times d}$.
Moreover, $\R^{d\times d}_{\mathrm{S}}$ and $\R^{d\times d}_{\mathrm{SPD}}$ are the sets of symmetric $\R^{d\times d}$ and symmetric positive definite $\R^{d\times d}$ matrices, respectively.
For a vector or matrix valued quantity, we denote the induced norm by $\abs{\cdot}$, and for a scalar quantity, we denote by $\abs{\cdot}$ the Euclidean norm.
For a real Banach space $X$, we denote by $\norm{\cdot}_{X}$ its norm, by $X'$ its dual space, and by $\dualp{\cdot}{\cdot}_{X}$ the duality pairing between $X$ and $X'$.
For $p\in[1,\infty]$, an integer $m\geq 0$ and a bounded domain $\Omega\subset\R^d$, $d\in\{2,3\}$, we use the standard notation from, e.g., \cite{alt_2016}, and we write $L^p\coloneqq L^p(\Omega)$, $W^{m,p}\coloneqq W^{m,p}(\Omega)$ and $H^m\coloneqq H^m(\Omega) \coloneqq W^{m,2}(\Omega)$, where $W^{0,p}\coloneqq L^p$ in the case $m=0$. We also define $L^2_0 \coloneqq L^2_0(\Omega) \coloneqq \{q \in L^2 \mid \int_\Omega q \dv{x} = 0\}$ and $H^1_0 \coloneqq H^1_0(\Omega) \coloneqq \{q \in H^1 \mid q|_{\partial\Omega}=0$ a.e.~on $\partial\Omega\}$, where $q|_{\partial\Omega}$ should be interpreted in the sense of the trace theorem.
We sometimes use the same notation for vector valued or matrix valued spaces.
For instance, $L^p$ can mean $L^p(\Omega)$, $L^p(\Omega;\R^d)$ or $L^p(\Omega;\R^{d\times d})$, which of course depends on the context.
Moreover, we define $\mathbf{H} \coloneqq \left\{\pmb{\mathrm{w}} \in L^2(\Omega;\R^d) \mid \divergenz{\pmb{\mathrm{w}}} = 0 \text{ a.e.~in }\Omega, \ \pmb{\mathrm{w}}\cdot\pmb{\mathrm{n}} = 0 \text{ on } \partial\Omega \right\}$ and $\mathbf{V}\coloneqq \left\{ \pmb{\mathrm{w}} \in H^1_0(\Omega;\R^d) \mid \divergenz{\pmb{\mathrm{w}}} = 0 \text{ a.e.~in }\Omega\right\}$, where $\pmb{\mathrm{n}}$ denotes the outer unit normal on $\partial\Omega$.
The norms and seminorms of the Sobolev spaces are denoted by $\norm{\cdot}_{W^{m,p}}$ and $\abs{\cdot}_{W^{m,p}}$, respectively, and similarly for the spaces $L^p$ and $H^m$.
We denote the inner product of the spaces $L^2$ and $L^2({\partial\Omega})$ by $\skp{\cdot}{\cdot}_{L^2}$ and $\skp{\cdot}{\cdot}_{L^2({\partial\Omega})}$, respectively.
For $\alpha\in[0,1]$, we write $C^{0,\alpha}(\overline\Omega)$ for the H{\"o}lder spaces.
For a real Banach space $X$, a real number $p\in[1,\infty]$ and an integer $m\geq 0$, we denote the Bochner spaces by $L^p(0,T;X)$ and $W^{m,p}(0,T;X)$ and they are equipped with the norms $\norm{\cdot}_{L^p(0,T;X)}$ and $\norm{\cdot}_{W^{m,p}(0,T;X)}$. For $p=2$, we will also write $H^m(0,T;X)\coloneqq W^{m,2}(0,T;X)$ and $\norm{\cdot}_{H^m(0,T;X)} \coloneqq \norm{\cdot}_{W^{m,2}(0,T;X)}$. Sometimes, $L^p(0,T;L^p)$ will be identified with $L^p(\Omega_T)$, where $\Omega_T\coloneqq \Omega\times (0,T)$ with $\Omega\subset\R^d$, $d\in\{2,3\}$, and $T>0$.
\section{Numerical results}
\label{sec:numeric}
In this section, we present numerical results for the scheme \ref{P_alpha_FE} that was analyzed in Section \ref{sec:fem}.
\subsection{Computational aspects}
\subsubsection{Description of the solution algorithm}
Before presenting the numerical results, we first discuss the solution strategy.
On one side, one could think of applying Newton's method to solve the nonlinear system of equations \ref{P_alpha_FE} as it can provide good error reduction rates.
However, Newton's method would be too expensive and would require too much memory, as the coupled system of equations \ref{P_alpha_FE} is very large, and is hence not useful in practice.
On the other side, making use of a fixed point iteration allows to decouple the system of equation \ref{P_alpha_FE} into linear subsystems \eqref{eq:phi_FE}--\eqref{eq:mu_FE}, \eqref{eq:sigma_FE}, \eqref{eq:div_v_FE}--\eqref{eq:v_FE} and \eqref{eq:B_FE} that can be solved separately. But here, a very small time step size $\Delta t$ is required such that the fixed point iteration can converge. Therefore, a fixed point iteration would need too much computing time in practice.
Moreover, numerical experiments indicate that the subsystem \eqref{eq:phi_FE}--\eqref{eq:mu_FE} requires additional consideration and the most precision which is because of the scaling with $B = \beta\epsilon$ and $A= \frac{\beta}{\epsilon}$ with $\epsilon>0$ very small.
For these reasons, we propose an inner-outer type algorithm to solve the nonlinear coupled scheme \ref{P_alpha_FE}.
For the outer iteration, we apply a fixed point-like strategy, where \ref{P_alpha_FE} is decoupled into the subsystems \eqref{eq:phi_FE}--\eqref{eq:mu_FE}, \eqref{eq:sigma_FE}, \eqref{eq:div_v_FE}--\eqref{eq:v_FE} and \eqref{eq:B_FE}, where all nonlinear terms are treated explicitly except of $A \psi_1'(\phi_h^n)$ in \eqref{eq:mu_FE} which is treated implicitly.
Hence, we first solve the nonlinear subsystem \eqref{eq:phi_FE}--\eqref{eq:mu_FE} with Newton's method, where the resulting linear systems are solved with a preconditioned BICGSTAB-method. After that, we solve the linear subsystems \eqref{eq:sigma_FE}, \eqref{eq:div_v_FE}--\eqref{eq:v_FE} and \eqref{eq:B_FE} separately with an AMG-preconditioned MINRES-solver, an AMG-preconditioned GMRES-method and an AMG-preconditioned MINRES-solver, respectively. The algorithm is implemented with the finite element toolbox FEniCS \cite{fenics_book_2012} which also provides the iterative linear solvers and the preconditioners.
However, due to limited computational possibilities with the finite element toolbox FEniCS, we consider \eqref{eq:B_FE} with $\sum_{i,j=1}^d [\pmb{\mathrm{v}}_h^{n-1}]_i \Lambda_{i,j}(\B_h^n) : \partial_{x_j} \C_h$ replaced by $\B_h^{n} : ((\pmb{\mathrm{v}}_h^{n-1}\cdot\nabla) \C_h)$, which however is a good approximation due to \eqref{eq:error_Lambda}, and, we replace $\nabla\pmb{\mathrm{v}}_h^n : \mycal{I}_h\big[\C_h\B_h^n\big]$ by $\nabla\pmb{\mathrm{v}}_h^n : (\C_h\B_h^n)$.
To increase the accuracy of our numerical solutions, we make use of a mesh refinement strategy, similarly to \cite{trautwein_2021}, where the mesh is locally refined near the interface where $\abs{\phi_h^{n-1}} \leq 1-\delta$, for a small $\delta>0$, where the local mesh size corresponds to a uniform $N_f \times N_f$ grid. Away from the interface, where $\abs{\phi_h^{n-1}} > 1-\delta$, a coarse mesh is used with a local mesh size corresponding to a uniform $N_c \times N_c$ grid.
\subsubsection{Specification of the parameters, model functions and initial data}
Now we specify the parameters, model functions and initial data, where our choices are motivated by, e.g., \cite{ebenbeck_garcke_nurnberg_2020, GarckeLSS_2016}.
We perform the calculations on the domain $\Omega = (-5,5)^2 \subset \R^2$ and we use the model functions
\begin{alignat*}{3}
&\Gamma_\phi(\phi,\sigma,\B) = h(1.1\phi) \big( \ensuremath{\mathcal{P}} \sigma f(\B) -\mycal{A} \big) ,
\qquad &&\Gamma_\sigma(\phi,\sigma) = \mycal{C} h(\phi) \sigma,
\qquad &&m(\phi) = 2 \left( h(\phi) \right)^2 + m_0,
\\
&\eta(\phi) = \eta_{-1} h(-\phi) + \eta_{1} h(\phi),
\qquad
&&\tau(\phi) = \tau_{-1} h(-\phi) + \tau_{1} h(\phi),
\qquad
&&n(\phi) = n_0,
\end{alignat*}
where the interpolation function with cut-offs $h(\cdot)$ is defined in \eqref{eq:h_phi} and $f(\cdot)$ is defined in \eqref{eq:f_B}.
Unless otherwise stated, we choose the parameters
{\small
\begin{equation}
\begin{aligned}
\label{eq:num_parameters}
& N_c = 32,\quad
&& N_f = 1024,\quad
&&\delta = 0.075,\quad
&&\Delta t= 5 \cdot 10^{-4}, \quad
&&\epsilon=0.01,\quad
&&\beta=0.1,\quad
\\
& \ensuremath{\mathcal{P}} = 2, \quad
&&\mycal{A} = 0, \quad
&&\mycal{C} = 10, \quad
&&\chi_\sigma = 500, \quad
&&\chi_\phi = 10, \quad
&&\alpha=10^{-3},\quad
\\
& m_0 = 10^{-12}, \quad
&& n_0 = 0.002,\quad
&& K = 1000,\quad
&&\eta_{1}=5000,\quad
&&\eta_{-1}=5000,\quad
&&\kappa=10^{4},\quad
\\
&\frac{\tau_{1}}{\kappa} = 1,\quad
&& \frac{\tau_{-1}}{\kappa} = 1.
\end{aligned}
\end{equation}
Therefore, $h_{\max} = 10\cdot 2^{-5} = 0.3125$ is the maximal diameter and $h_{\min} = 10 \cdot 2^{-10} \approx 0.009766$ is the minimal diameter of all triangular elements.
Moreover, note that the assumptions \ref{A2}--\ref{A3} are fulfilled with these choices.
Actually, the (modified) potential with quadratic growth from \eqref{eq:psi_modified} should be used such that the growth assumptions in \ref{A4} are fulfilled.
However, for simplicity, we use the (unmodified) potential
\begin{align*}
&\psi(\phi) = \tfrac{1}{4} (1-\phi^2)^2,
\quad \text{where} \quad
\psi'(\phi) = \psi_1'(\phi) + \psi_2'(\phi) = \phi^3 - \phi,
\end{align*}
as the order parameter $\phi$ always stays very close to the interval $[-1,1]$ in our numerical experiments. Also note the scaling with $1.1\phi$ in the source term $\Gamma_\phi$, as $\phi\approx{-1}$ in practice in the pure healthy phase and we want to exclude proliferation effects there.
For the initial tumour profile, we set $\phi_h^0 = \mycal{I}_h \phi_0$, where $\phi_0 \in C(\overline\Omega)$ is a slightly perturbed sphere given by
\begin{align}
\label{eq:num_phi0}
\phi_0(x) &=
- \tanh \Big(\frac{r(x)}{\sqrt{2}\epsilon} \Big),
\quad \text{where} \quad
r(x) = \abs{x} - \frac{5}{12}(2 + 0.2\cos(2\theta)),
\end{align}
where $x = \abs{x} ( \cos(\theta),\sin(\theta) )^T \in \Omega$.
We choose $\sigma_h^0 \in\ensuremath{\mathcal{S}}_h$ as the solution of the quasi-static equation
\begin{align}
\label{eq:num_sigma_FE_quasistatic}
\int_\Omega
n_0 \big(\chi_\sigma \nabla\sigma_h^0 - \chi_\phi \nabla \phi_h^0 \big) \cdot \nabla \xi_h
+ \mycal{I}_h \big[ \Gamma_\sigma(\phi_h^0,\sigma_h^0) \xi_h \big]\dv{x}
+ \int_{\partial\Omega} \mycal{I}_h\Big[ K(\sigma_h^0 - \sigma_\infty) \xi_h \Big] \ \mathrm d \calH^{d-1}
= 0,
\end{align}
for all $\xi\in\ensuremath{\mathcal{S}}_h$, where $\sigma_\infty \coloneqq 1$ in the numerical experiments unless otherwise stated.
Moreover, we assume no initial velocity and no initial elastic stresses.
More precisely, we start with
\begin{align}
\label{eq:num_v0_B0}
\pmb{\mathrm{v}}_0^h(x) = (0,0)^T, \qquad \B_0^h(x) &= \text{diag}(1, 1),
\end{align}
where $x\in\Omega$. The initial profile $\phi_h^0$, the initial nutrient $\sigma_h^0$ and the initial mesh are shown in Figure \ref{fig:init}.
It is easy to verify that the initial and boundary values satisfy the assumptions \eqref{eq:init_bounds}.
\begin{figure}[H]
\centering
\subfloat
{\includegraphics[width=0.24\textwidth]{figures/init_phi.jpeg}}
\hspace{-0.8em}
\subfloat
{\includegraphics[width=0.24\textwidth]{figures/init_sig.jpeg}}
\hspace{-0.8em}
\subfloat
{\includegraphics[width=0.24\textwidth]{figures/init_mesh.jpeg}}
\caption{Initial tumour (left), initial nutrient (center) and initial mesh (right).}
\label{fig:init}
\end{figure}
In the following, we will systematically interpret the influence of different parameters in our model.
The influence of the parameters $\epsilon, \beta, \ensuremath{\mathcal{P}}, \mycal{A}, \mycal{C}, \chi_\phi, \chi_\sigma$ and the mobility $m(\cdot)$ in related models has been extensively studied, see, e.g., \cite{ebenbeck_garcke_nurnberg_2020, GarckeLSS_2016, trautwein_2021}, and we observed similar behaviour for our model. For that reason, we focus the presentation of the numerical tests on the effects arising from viscoelasticity.
The main difficulty is to find a good choice of parameters. To observe an unstable growth, i.e.~the development of fingers, the chemotaxis parameter $\chi_\phi$ has to be in the same scale as $\frac{\beta}{\epsilon}$.
Choosing $\chi_\phi$ too large or $\beta$ too small reduces the forming of the pure phases $\phi=\pm1$.
On the other side, we observe a jump of the nutrient $\sigma$ along the interface which is proportional to $\chi_\sigma^{-1} \chi_\phi$, hence $\chi_\sigma$ is chosen large compared to $\chi_\phi$.
However, this can result in very large velocities if the viscosities $\eta_1, \eta_{-1}$ are not large enough.
\subsection{Comparison with the fully viscous model}
We now investigate the time evolution in the viscoelastic model and compare it to the fully viscous model where $\B=\I$.
The parameters are chosen as in \eqref{eq:num_parameters} and the goal is to observe an unstable growth.
In absence of initial elastic stresses, i.e.~$\B_0=\I$, any changes in the left Cauchy--Green tensor $\B$ are induced by the velocity field $\pmb{\mathrm{v}}$, see \eqref{eq:B_FE}. As the viscosities $\eta_1, \eta_{-1}$ are chosen very large, we can expect small velocities and hence $\B\approx\I$, such that the elastic stress tensor is approximately zero, i.e.~$\T_{\mathrm{el}}(\B)=\kappa (\B-\I)\approx 0$.
Therefore, we expect that the qualitative behaviour of both models is very similar, which can be observed in Figure \ref{fig:2}. Here, we show the numerical solutions for both models at time $t=2$. In the first row from left to right, we show the order parameter $\phi$, the nutrient $\sigma$, the velocity magnitude $\abs\pmb{\mathrm{v}}$ and the final mesh for the fully viscous model. In the second row from left to right, the order parameter $\phi$, the nutrient $\sigma$, the velocity magnitude $\abs\pmb{\mathrm{v}}$ and the magnitude of the elastic stress tensor $\abs{\T_{\mathrm{el}}(\B)}$ of the viscoelastic model are visualized.
Indeed, the qualitative behaviour for both models is very similar, as $\abs{\T_{\mathrm{el}}(\B)} \approx 0$ is close to machine precision for the viscoelastic model.
In both cases, the tumour has developed fingers showing towards regions with higher concentration of the nutrient which can be interpreted as the chemotaxis effect, i.e.~the cell movement in response to an extracellular chemical gradient. This behaviour has also been observed for other models \cite{ebenbeck_garcke_nurnberg_2020, GarckeLSS_2016, trautwein_2021}.
After that, in Figure \ref{fig:2b}, we show the time evolution of the tumour for the viscoelastic model at the times $t\in\{0.5, 1, 1.5, 2\}$.
\begin{figure}[H]
\centering
\subfloat
{\includegraphics[width=0.22\textwidth]{figures/viscous_viscoelastic/3b1_NS_phi.jpeg}}
\hspace{-0.8em}
\subfloat
{\includegraphics[width=0.22\textwidth]{figures/viscous_viscoelastic/3b1_NS_sig.jpeg}}
\hspace{-0.8em}
\subfloat
{\includegraphics[width=0.22\textwidth]{figures/viscous_viscoelastic/3b1_NS_v.jpeg}}
\hspace{-0.8em}
\subfloat
{\includegraphics[width=0.22\textwidth]{figures/viscous_viscoelastic/3b1_mesh.jpeg}}
\\[-2.7ex]
\subfloat
{\includegraphics[width=0.22\textwidth]{figures/viscous_viscoelastic/3b1_phi.jpeg}}
\hspace{-0.8em}
\subfloat
{\includegraphics[width=0.22\textwidth]{figures/viscous_viscoelastic/3b1_sig.jpeg}}
\hspace{-0.8em}
\subfloat
{\includegraphics[width=0.22\textwidth]{figures/viscous_viscoelastic/3b1_v.jpeg}}
\hspace{-0.8em}
\subfloat
{\includegraphics[width=0.22\textwidth]{figures/viscous_viscoelastic/3b1_Tel.jpeg}}
\caption{Comparison of the fully viscous model (first row) to the viscoelastic model (second row) at time $t=2$. In the first three columns, $\phi$, $\sigma$ and $\abs\pmb{\mathrm{v}}$ are visualized. In the last column, the final mesh and $\abs{\T_{\mathrm{el}}(\B)}$ are shown.}
\label{fig:2}
\end{figure}
\begin{figure}[H]
\centering
\subfloat
{\includegraphics[width=0.22\textwidth]{figures/viscous_viscoelastic/3b1_phi_t=0.5.jpeg}}
\hspace{-0.8em}
\subfloat
{\includegraphics[width=0.22\textwidth]{figures/viscous_viscoelastic/3b1_phi_t=1.0.jpeg}}
\hspace{-0.8em}
\subfloat
{\includegraphics[width=0.22\textwidth]{figures/viscous_viscoelastic/3b1_phi_t=1.5.jpeg}}
\hspace{-0.8em}
\subfloat
{\includegraphics[width=0.22\textwidth]{figures/viscous_viscoelastic/3b1_phi.jpeg}}
\caption{Time evolution of the tumour $\phi$ for the viscoelastic model at times $t\in\{0.5, 1, 1.5, 2\}$.}
\label{fig:2b}
\end{figure}
\subsection{Influence of the viscosity}
In the next experiment, we illustrate how the choice of the viscosity function $\eta(\cdot)$ can affect the elastic stress tensor and hence the evolution of the tumour. We increase the proliferation rate, i.e.~$\ensuremath{\mathcal{P}}=5$, and decrease the chemotactic sensitivity, i.e.~$\chi_\phi=7.5$. Moreover, we increase the relaxation times such that $\frac{\tau_{1}}{\kappa}=\frac{\tau_{-1}}{\kappa} = 100$ and choose the viscosities $\eta_1 = 2000$ and $\eta_{-1}\in\{500, 1000, 1500\}$.
The numerical solutions $\phi$ (first row) and $\abs{\T_{\mathrm{el}}(\B)}$ (second row) at time $t=1.6$ are visualized in Figure \ref{fig:large_stress} for the cases $\eta_{-1}\in\{500, 1000, 1500\}$. From left to right, the elastic stresses decrease and the size of the tumours increase with increasing viscosity $\eta_{-1}$. However, note that the elasticity parameter $\kappa=10^{4}$ is chosen very large in order to see the influence of the elastic stresses.
\begin{figure}[H]
\centering
\subfloat
{\includegraphics[width=0.24\textwidth]{figures/large_stresses/2f1_phi.jpeg}}
\hspace{-0.8em}
\subfloat
{\includegraphics[width=0.24\textwidth]{figures/large_stresses/2f2_phi.jpeg}}
\hspace{-0.8em}
\subfloat
{\includegraphics[width=0.24\textwidth]{figures/large_stresses/2f3_phi.jpeg}}
\\[-4.5ex]
\subfloat
{\includegraphics[width=0.24\textwidth]{figures/large_stresses/2f1_Tel.jpeg}}
\hspace{-0.8em}
\subfloat
{\includegraphics[width=0.24\textwidth]{figures/large_stresses/2f2_Tel.jpeg}}
\hspace{-0.8em}
\subfloat
{\includegraphics[width=0.24\textwidth]{figures/large_stresses/2f3_Tel.jpeg}}
\caption{From left to right: $\eta_{-1}\in\{500, 1000, 1500\}$, first row: $\phi$, second row: $\abs{\T_{\mathrm{el}}}$ at time $t=1.6$.}
\label{fig:large_stress}
\end{figure}
\subsection{Mechanical stresses generated by growth}
Now we consider a variant of the model with an additional source term in the equation of $\B$ like in \eqref{eq:B_growth}. For this reason we replace \eqref{eq:B_FE} by
\begin{align}
\label{eq:B_FE_growth}
\nonumber
0 &= \int_\Omega \mycal{I}_h \Big[
\Big(\frac{\B_h^n - \B_h^{n-1}}{\Delta t}
+ \frac{\kappa}{\tau(\phi_h^{n-1})} (\B_h^n - \I) \Big): \C_h \Big]
- 2 \nabla\pmb{\mathrm{v}}_h^n : (\C_h\B_h^n)
+ \alpha \nabla\B_h^n : \nabla\C_h \dv{x}
\\
&\qquad + \int_\Omega
\mycal{I}_h\Big[ \gamma(\phi_h^{n-1},\sigma_h^{n-1}) \B_h^n : \C_h \Big]
- \B_h^n : ((\pmb{\mathrm{v}}_h^{n-1}\cdot\nabla) \C_h)
\dv{x},
\end{align}
where $\gamma(\phi,\sigma) = \ensuremath{\mathcal{G}} \sigma h(\phi)$ with a constant $\ensuremath{\mathcal{G}}\geq0$.
For the first numerical test, the parameters are chosen as in \eqref{eq:num_parameters} but with
$\ensuremath{\mathcal{P}}=5$, $\chi_\phi=7.5$, $\eta_1 = 2000$, $\eta_{-1}=1500$, $\frac{\tau_{1}}{\kappa}=\frac{\tau_{-1}}{\kappa} = 0.01$ and $\ensuremath{\mathcal{G}}\in\{0, 0.1, 0.5, 1\}$.
In Figure \ref{fig:stress_source}, we visualize the tumour (upper row) and the magnitude of the elastic stress tensor (lower row) where $\ensuremath{\mathcal{G}}\in\{0, 0.1, 0.2, 0.5\}$ from left to right at time $t=2.25$. From left to right, the size of the tumours decrease as the elastic stresses become larger with increasing $\ensuremath{\mathcal{G}}$. Moreover, we observe that the elastic stresses are particularly large in the fingers of the tumour.
\begin{figure}[H]
\centering
\subfloat
{\includegraphics[width=0.24\textwidth]{figures/stress_source/2h4_phi.jpeg}}
\hspace{-0.8em}
\subfloat
{\includegraphics[width=0.24\textwidth]{figures/stress_source/2h3_phi.jpeg}}
\hspace{-0.8em}
\subfloat
{\includegraphics[width=0.24\textwidth]{figures/stress_source/2h7_phi.jpeg}}
\hspace{-0.8em}
\subfloat
{\includegraphics[width=0.24\textwidth]{figures/stress_source/2h5_phi.jpeg}}
\\[-4.5ex]
\subfloat
{\includegraphics[width=0.24\textwidth]{figures/stress_source/2h4_Tel.jpeg}}
\hspace{-0.8em}
\subfloat
{\includegraphics[width=0.24\textwidth]{figures/stress_source/2h3_Tel.jpeg}}
\hspace{-0.8em}
\subfloat
{\includegraphics[width=0.24\textwidth]{figures/stress_source/2h7_Tel.jpeg}}
\hspace{-0.8em}
\subfloat
{\includegraphics[width=0.24\textwidth]{figures/stress_source/2h5_Tel.jpeg}}
\caption{From left to right: $\ensuremath{\mathcal{G}}\in\{0, 0.1, 0.2, 0.5\}$. First row: $\phi$, second row: $\abs{\T_{\mathrm{el}}}$ at time $t=2.25$.}
\label{fig:stress_source}
\end{figure}
Next, we present the influence of the relaxation times $\tau_{-1},\tau_{1}$ on the growth behaviour in presence of source terms for $\B$.
On one side, we expect the elastic stresses to vanish if the relaxation time is small enough. On the other side, large elastic stresses can build up if the relaxation time is large, which then reduces the proliferation effect.
Therefore, the parameters are chosen as in \eqref{eq:num_parameters} but with
$\ensuremath{\mathcal{P}}=5$, $\chi_\phi=7.5$, $\ensuremath{\mathcal{G}}=0.2$ and with $\frac{\tau_{1}}{\kappa} = \frac{\tau_{-1}}{\kappa} \in \{10^{-4}, 10^{-2}, 1\}$.
In Figure \ref{fig:relaxation}, we visualize the tumour (upper row) and the magnitude of the elastic stress tensor (lower row) at time $t=2.4$, where $\frac{\tau_{1}}{\kappa} = \frac{\tau_{-1}}{\kappa} \in \{10^{-4}, 10^{-2}, 1\}$ from left to right. Here, no elastic stresses occur if the relaxation time is very small, and the elastic stresses can become very large if the relaxation time is large.
\begin{figure}[H]
\centering
\subfloat
{\includegraphics[width=0.24\textwidth]{figures/relaxation_time/3a2_phi.jpeg}}
\hspace{-0.8em}
\subfloat
{\includegraphics[width=0.24\textwidth]{figures/relaxation_time/3a1_phi.jpeg}}
\hspace{-0.8em}
\subfloat
{\includegraphics[width=0.24\textwidth]{figures/relaxation_time/3a3_phi.jpeg}}
\\[-4.5ex]
\subfloat
{\includegraphics[width=0.24\textwidth]{figures/relaxation_time/3a2_Tel.jpeg}}
\hspace{-0.8em}
\subfloat
{\includegraphics[width=0.24\textwidth]{figures/relaxation_time/3a1_Tel.jpeg}}
\hspace{-0.8em}
\subfloat
{\includegraphics[width=0.24\textwidth]{figures/relaxation_time/3a3_Tel.jpeg}}
\caption{From left to right: $\frac{\tau_{1}}{\kappa} = \frac{\tau_{-1}}{\kappa} \in \{10^{-4}, 10^{-2}, 1\}$. First row: $\phi$, second row: $\abs{\T_{\mathrm{el}}}$ at time $t=2.4$.}
\label{fig:relaxation}
\end{figure}
\subsection{Numerical results for a phase-dependent elastic energy density}
In the following, we want to illustrate the impact of a phase-dependent elastic energy density on the evolution of the tumour. In particular, we now consider the phase-dependent elasticity parameter function $\kappa(\phi) = \tfrac{1}{2} \kappa_1 (1+\phi) + \tfrac{1}{2} \kappa_{-1} (1-\phi)$, which leads to the elastic energy density $W(\phi,\B) = \frac{1}{2} \kappa(\phi) \trace (\B - \ln \B)$.
Hence, we adapt the system of equations \eqref{eq:phi_FE}--\eqref{eq:v_FE}, \eqref{eq:B_FE_growth} as follows.
First, we replace $\kappa$ with $\kappa(\phi_h^n)$ in \eqref{eq:v_FE}, \eqref{eq:B_FE_growth} and in the source term $\Gamma_\phi$.
To be consistent with \eqref{eq:mu}, we add the term $+ \int_\Omega \frac{1}{4} (\kappa_1 - \kappa_{-1}) \mycal{I}_h\left[\trace(\B_h^{n} - \ln\B_h^{n}) \rho_h\right] \dv{x}$ to the right-hand side of \eqref{eq:mu_FE}.
Lastly, due to \eqref{eq:v0b}, we include the term $- \int_\Omega \frac{1}{4} (\kappa_1 - \kappa_{-1}) \phi_h^{n-1} \nabla \trace(\B_h^{n}-\ln\B_h^n) \cdot \pmb{\mathrm{w}}_h \dv{x}$ on the right-hand side of \eqref{eq:v_FE}.
The goal is now to study the growth behaviour of the tumour. In particular, it is of main interest whether the chemotactic development of fingers is intensified, weakened or completely changed.
Now, as the term $W_{,\phi} = \frac{1}{4}(\kappa_{1}-\kappa_{-1}) \trace (\B - \ln\B)$ enters the equation for the chemical potential $\mu$, we vary the elasticity parameters $\kappa_{1},\kappa_{-1}$ and we make sure that $\trace (\B - \ln\B)$ changes near the tumour region.
For these reasons, the parameters in the following experiments are chosen as in \eqref{eq:num_parameters} but with $\chi_\phi=7.5$, $\ensuremath{\mathcal{G}}=10$, $\frac{\tau_{1}}{\kappa_{1}}=10, \frac{\tau_{-1}}{\kappa_{-1}}=0.1$ and with varying $\kappa_{1},\kappa_{-1}$.
First, we show the time evolution of the tumour with matched elasticity parameters $\kappa_{1}=\kappa_{-1}=1$ at times $t\in\{1, 1.5, 2.3\}$ and the magnitude of the elastic stress tensor at the final time. Here, the tumour growths and develops four thick fingers.
\begin{figure}[H]
\centering
\subfloat
{\includegraphics[width=0.24\textwidth]{figures/phi_dependent_energy/3e8c_phi_t=1.0.jpeg} }
\hspace{-0.8em}
\subfloat
{\includegraphics[width=0.24\textwidth]{figures/phi_dependent_energy/3e8c_phi_t=1.5.jpeg}}
\hspace{-0.8em}
\subfloat
{\includegraphics[width=0.24\textwidth]{figures/phi_dependent_energy/3e8c_phi_t=2.3.jpeg}}
\hspace{-0.8em}
\subfloat
{\includegraphics[width=0.24\textwidth]{figures/phi_dependent_energy/3e8c_Tel_t=2.3.jpeg}}
\caption{Numerical solution $\phi$ with $\kappa_{1}=\kappa_{-1}=1$ at times $t\in\{1, 1.5, 2.3\}$, and $\abs{\T_{\mathrm{el}}}$ at the final time.}
\label{fig:3a}
\end{figure}
Now, we visualize the tumour with unmatched elasticity parameters, where $\kappa_{1}=1$, $\kappa_{-1}=2$ and $t\in\{1,2,3\}$ in Figure \ref{fig:3b}, and $\kappa_{1}=1$, $\kappa_{-1}=5$ and $t\in\{1,2,3\}$ in Figure \ref{fig:3c}. In comparison to the case with matched elasticity parameters, we observe that the invasive growth of the tumour needs more time when $\kappa_{-1}$ is large. In addition, the shapes of the tumours are more elongated and the development of fingers is barely recognizable.
\begin{figure}[H]
\centering
\subfloat
{\includegraphics[width=0.24\textwidth]{figures/phi_dependent_energy/3e7_phi_t=1.jpeg}}
\hspace{-0.8em}
\subfloat
{\includegraphics[width=0.24\textwidth]{figures/phi_dependent_energy/3e7_phi_t=2.jpeg}}
\hspace{-0.8em}
\subfloat
{\includegraphics[width=0.24\textwidth]{figures/phi_dependent_energy/3e7_phi_t=3.jpeg}}
\hspace{-0.8em}
\subfloat
{\includegraphics[width=0.24\textwidth]{figures/phi_dependent_energy/3e7_Tel_t=3.jpeg}}
\caption{Numerical solution $\phi$ with $\kappa_{1}=1, \kappa_{-1}=2$ at times $t\in\{1,2,3\}$, and $\abs{\T_{\mathrm{el}}}$ at the final time.}
\label{fig:3b}
\end{figure}
\begin{figure}[H]
\centering
\subfloat
{\includegraphics[width=0.24\textwidth]{figures/phi_dependent_energy/3e1_phi_t=1.jpeg}}
\hspace{-0.8em}
\subfloat
{\includegraphics[width=0.24\textwidth]{figures/phi_dependent_energy/3e1_phi_t=2.jpeg}}
\hspace{-0.8em}
\subfloat
{\includegraphics[width=0.24\textwidth]{figures/phi_dependent_energy/3e1_phi_t=3.jpeg}}
\hspace{-0.8em}
\subfloat
{\includegraphics[width=0.24\textwidth]{figures/phi_dependent_energy/3e1_Tel_t=3.jpeg}}
\caption{Numerical solution $\phi$ with $\kappa_{1}=1, \kappa_{-1}=5$ at times $t\in\{1,2,3\}$, and $\abs{\T_{\mathrm{el}}}$ at the final time.}
\label{fig:3c}
\end{figure}
Next, we show the tumour with unmatched elasticity parameters, where $\kappa_{1}=2$, $\kappa_{-1}=1$ and $t\in\{0.5,1,1.5\}$ in Figure \ref{fig:3d}, and $\kappa_{1}=5$, $\kappa_{-1}=1$ and $t\in\{0.5,0.75,1\}$ in Figure \ref{fig:3e}. Here we observe that the invasive growth of the tumour takes less time when $\kappa_{1}$ is large and the development of fingers increases.
\begin{figure}[H]
\centering
\subfloat
{\includegraphics[width=0.24\textwidth]{figures/phi_dependent_energy/3e9_phi_t=0.5.jpeg}}
\hspace{-0.8em}
\subfloat
{\includegraphics[width=0.24\textwidth]{figures/phi_dependent_energy/3e9_phi_t=1.0.jpeg}}
\hspace{-0.8em}
\subfloat
{\includegraphics[width=0.24\textwidth]{figures/phi_dependent_energy/3e9_phi_t=1.5.jpeg}}
\hspace{-0.8em}
\subfloat
{\includegraphics[width=0.24\textwidth]{figures/phi_dependent_energy/3e9_Tel_t=1.5.jpeg}}
\caption{Numerical solution $\phi$ with $\kappa_{1}=2, \kappa_{-1}=1$ at times $t\in\{0.5, 1, 1.5\}$, and $\abs{\T_{\mathrm{el}}}$ at the final time.}
\label{fig:3d}
\end{figure}
\begin{figure}[H]
\centering
\subfloat
{\includegraphics[width=0.24\textwidth]{figures/phi_dependent_energy/3e2_phi_t=0.5.jpeg}}
\hspace{-0.8em}
\subfloat
{\includegraphics[width=0.24\textwidth]{figures/phi_dependent_energy/3e2_phi_t=0.75.jpeg}}
\hspace{-0.8em}
\subfloat
{\includegraphics[width=0.24\textwidth]{figures/phi_dependent_energy/3e2_phi_t=1.jpeg}}
\hspace{-0.8em}
\subfloat
{\includegraphics[width=0.24\textwidth]{figures/phi_dependent_energy/3e2_Tel_t=1.jpeg}}
\caption{Numerical solution $\phi$ with $\kappa_{1}=5, \kappa_{-1}=1$ at times $t\in\{0.5, 0.75, 1\}$, and $\abs{\T_{\mathrm{el}}}$ at the final time.}
\label{fig:3e}
\end{figure}
\section{A viscoelastic tumour model with stress diffusion}
\label{sec:Section3}
In this section, we consider a variant of the system \ref{P}, where we fix the mass density of the mixture as $\rho\coloneqq 1$ and we neglect local exchange of mass, i.e.~$\Gamma_{\pmb{\mathrm{v}}}\coloneqq0$, see \eqref{source_1a}.
From the modelling point of view, the term $\Gamma_\phi$ usually describes biological effects like proliferation or apoptosis of the tumour, whereas the term $\Gamma_\sigma$ models nutrient consumption of the tumour \cite{GarckeLSS_2016}. Moreover, elastic stresses are supposed to influence growth.
Hence, it makes sense to assume $\Gamma_\phi$ to depend on $\phi,\sigma,\B$, and $\Gamma_\sigma$ to depend on $\phi,\sigma$, respectively.
Moreover, we choose the nutrient energy density \eqref{eq:energy_nutrient} and assume that the elastic energy is hence given by $W(\B) = \frac{1}{2} \kappa \trace\big( \B - \ln\B \big)$, which corresponds to \eqref{eq:energy_oldroyd} with the elasticity parameters $\kappa=\kappa_0$ not depending on $\phi$.
Besides, we assume small stress diffusion effects, i.e., we add the dissipative term $+\alpha\Delta \B$
to the right-hand side of the Oldroyd-B equation, which improves the mathematical properties of the system. This mathematical regularization can physically be motivated from a nonlocal energy storage mechanism or a nonlocal entropy production mechanism, see, e.g., \cite{malek_2018_Oldroyd_diffusive}.
Then, the mathematical system of our interest reads:
\medskip
\subsubsection*{Problem \ref{P_alpha}:}
\mylabelHIDE{P_alpha}{$(\pmb{\mathrm{P}}_\alpha)$}
For a given constant $\alpha>0$, consider the system in $\Omega\times (0,T)$
\begin{subequations}
\label{eq:system2}
\begin{align}
\label{eq:phi2}
\partial_t \phi + \pmb{\mathrm{v}} \cdot\nabla\phi
&= \divergenz{m(\phi) \nabla\mu} + \Gamma_\phi(\phi,\sigma,\B),
\\
\label{eq:mu2}
\mu &= A \psi'(\phi) - B \Delta\phi - \chi_\phi \sigma,
\\
\label{eq:sigma2}
\partial_t \sigma + \pmb{\mathrm{v}} \cdot\nabla\sigma
&= \divergenz{n(\phi) \nabla (\chi_\sigma \sigma - \chi_\phi \phi)} - \Gamma_\sigma(\phi,\sigma),
\\
\label{eq:div_v2}
\divergenz{\pmb{\mathrm{v}}} &= 0,
\\
\label{eq:v2}
\partial_t \pmb{\mathrm{v}} + (\pmb{\mathrm{v}}\cdot\nabla) \pmb{\mathrm{v}}
-\divergenz{2\eta(\phi) \D(\pmb{\mathrm{v}}) }
+ \nabla p
&= \divergenz{\kappa(\B-\I)}
+ \mu\nabla\phi
+ (\chi_\sigma \sigma - \chi_\phi \phi) \nabla\sigma,
\\
\label{eq:B2}
\partial_t \B + (\pmb{\mathrm{v}} \cdot\nabla)\B
+ \frac{\kappa}{\tau(\phi)}\big( \B-\I \big)
&= \nabla\pmb{\mathrm{v}} \B
+ \B (\nabla\pmb{\mathrm{v}})^T
+ \alpha \Delta\B,
\end{align}
together with the boundary conditions on $\partial\Omega \times (0,T)$
\begin{align}
\label{eq:bc_phimu2}
\nabla\phi\cdot\pmb{\mathrm{n}} &= \nabla\mu\cdot\pmb{\mathrm{n}} = 0,
\\
\label{eq:bc_sigma2}
\chi_\sigma n(\phi) \nabla \sigma \cdot \pmb{\mathrm{n}} &= K(\sigma_\infty - \sigma),
\\
\label{eq:bc_v2}
\pmb{\mathrm{v}}&= 0,
\\
\label{eq:bc_B2}
(\pmb{\mathrm{n}}\cdot\nabla)\B &= 0,
\end{align}
\end{subequations}
and the initial data $\phi(0) = \phi_0$, $\sigma(0) = \sigma_0$, $\pmb{\mathrm{v}}(0)=\pmb{\mathrm{v}}_0$ and $\B(0) = \B_0$.
\subsection{Assumptions and existence of weak solutions}
\label{sec:weak_solution}
In this section, we state the definition of a weak solution of \ref{P_alpha} and provide an existence result in two space dimensions. First, we state our assumptions.
\begin{assumptions}
\label{as:weak_solution}~
\begin{itemize}
\item[\mylabel{A1}{$(\mathrm{A}1)$}] Let $T>0$ and suppose that $\Omega \subset\R^d$, $d\in\{2,3\}$, is a convex, polygonal domain with boundary $\partial\Omega$.
\item[\mylabel{A2}{$(\mathrm{A}2)$}]
For $d\in\{2,3\}$, the source functions $\Gamma_\phi: \R\times\R\times\R^{d\times d} \to \R$ and $\Gamma_\sigma: \R\times\R\to\R$ are continuous and there exists a constant $R_0>0$ such that, for all $\phi,\sigma\in\R$, and $\B\in \R^{d\times d}$,
\begin{align*}
\abs{\Gamma_\phi(\phi,\sigma,\B)} + \abs{\Gamma_\sigma(\phi,\sigma)} \leq R_0 (1+\abs{\phi}+\abs{\sigma}).
\end{align*}
\item[\mylabel{A3}{$(\mathrm{A}3)$}]
Let $\chi_\phi\geq 0$ and $\chi_\sigma, A, B, K, \kappa, \alpha>0$ be constants. Moreover, let $m, n, \eta, \tau \in C^0(\R)$ and suppose there exist constants $m_0, m_1, n_0, n_1, \eta_0, \eta_1, \tau_0, \tau_1>0$ such that, for all $t\in\R$,
\begin{align*}
m_0 &\leq m(t) \leq m_1,
\quad
n_0 \leq n(t) \leq n_1,
\quad
\eta_0 \leq \eta(t) \leq \eta_1,
\quad
\tau_0 \leq \tau(t) \leq \tau_1.
\end{align*}
\item[\mylabel{A4}{$(\mathrm{A}4)$}] The potential $\psi$ is non-negative and belongs to $C^{1}(\R)$ with
\begin{align}
\label{A4_1}
\psi(t) &\geq R_1 \abs{t}^2 - R_2 \qquad \forall \ t\in\R,
\end{align}
where $R_1,R_2>0$. Additionally, the potential can be decomposed as $\psi=\psi_1 + \psi_2$ with $\psi_1$ convex and $\psi_2$ concave such that
\begin{align}
\label{A4_2}
\abs{\psi_i^\prime(t)} &\leq R_3 (1+\abs t) \qquad \forall \ t\in\R,
\end{align}
where $i=1,2$ and $R_3 >0$. Moreover, we assume
\begin{align}
\label{A4_3}
A > \frac{4\chi_\phi^2}{\chi_\sigma R_1}.
\end{align}
\item[\mylabel{A5}{$(\mathrm{A}5)$}]
For the initial and boundary data, assume
\begin{align*}
&\phi_0 \in H^2_{\mathrm{N}}(\Omega) \coloneqq \{q \in H^2(\Omega) \mid \nabla q\cdot \pmb{\mathrm{n}} = 0 \text{ on }\partial\Omega\},
\quad \sigma_0 \in L^2(\Omega),
\quad \pmb{\mathrm{v}}_0 \in \mathbf{H},
\quad \sigma_\infty \in L^2(0,T;H^1(\Omega)),
\\
& \B_0 \in L^\infty(\Omega;\R^{d\times d}_{\mathrm{SPD}})
\quad
\text{with} \quad
b^0_{\min} \abs{\pmb\xi}^2
\leq \pmb\xi^T \B_0(x) \pmb\xi
\leq b^0_{\max} \abs{\pmb\xi}^2
\quad \forall \ \pmb\xi\in \R^d
\quad \text{for a.e. } x\in\Omega,
\end{align*}
where $d\in\{2,3\}$, $b^0_{\min},b^0_{\max}\in\R$ with $0<b^0_{\min}\leq b^0_{\max}$ and $\pmb{\mathrm{n}}$ denotes the outer unit normal on ${\partial\Omega}$.
\item[\mylabel{A6}{$(\mathrm{A}6)$}] The spatial dimension is restricted to $d=2$.
\item[\mylabel{A7}{$(\mathrm{A}7)$}] The source functions $\Gamma_\phi,\Gamma_\sigma$ from \ref{A2} are Lipschitz continuous.
\end{itemize}
\end{assumptions}
Unter these assumptions, we provide an existence result for weak solutions to \ref{P_alpha}.
However, note that \ref{A1}--\ref{A5} are stated for arbitrary dimensions $d\in\{2,3\}$. This is sufficient when studying stability and existence of discrete solutions to a fully-discrete finite element approximation of \ref{P_alpha} in Section \ref{sec:fem}. Later, also \ref{A6} is needed to improve the regularity of discrete solutions. Moreover, \ref{A7} is needed for the limit passing in the discrete scheme in presence of mass lumping but can be dropped if the terms containing $\Gamma_\phi, \Gamma_\sigma$ are integrated exactly.
Possible choices for the source functions which fulfill the assumptions can be constructed as follows. Let
\begin{align}
\label{eq:source_terms}
\Gamma_\phi(\phi,\sigma,\B) \coloneqq h(\phi) \big( \ensuremath{\mathcal{P}} g(\sigma) f(\B) -\mycal{A} \big) ,
\qquad \Gamma_\sigma(\phi,\sigma) \coloneqq \mycal{C} h(\phi) g(\sigma),
\end{align}
where $\ensuremath{\mathcal{P}},\mycal{A}, \mycal{C}\geq 0$ are non-negative constants accounting for proliferation, apoptosis and nutrient consumption, and
\begin{subequations}
\begin{alignat}{3}
\label{eq:h_phi}
h(\phi) &\coloneqq \max\left\{ 0, \ \min\left\{ \tfrac{1}{2} (1+\phi), 1 \right\} \right\}
\qquad && \forall \ \phi\in\R,
\\
g(\sigma) &\coloneqq \max\left\{ 0, \ \min\left\{ \sigma , 1 \right\} \right\}
\qquad && \forall \ \sigma\in\R,
\\
\label{eq:f_B}
f(\B) &\coloneqq \left(1 + \abs{\kappa (\B-\I)}^2 \right)^{-1/2}
\qquad && \forall \ \ \B\in\R^{d\times d}.
\end{alignat}
\end{subequations}
Then, as $f,g,h$ are non-negative, Lipschitz-continuous and bounded, the source functions $\Gamma_\phi,\Gamma_\sigma$ satisfy the assumptions from above, i.e.~\ref{A2} and \ref{A7}.
The mobility functions $m,n$, the viscosity $\eta$ and the relaxation time $\tau$ can be defined with similar cut-offs outside of the interval $[-1,1]$ such that \ref{A3} holds.
In practice, the polynomial double-well potential $\tilde \psi(t) = \tfrac{1}{4} (1- t ^2)^2$ is a common choice. However, in order to fulfill \ref{A4}, the growth of the polynomial double-well potential shall be restricted to be at most quadratic for, e.g., $t\not\in[-1,1]$, i.e.
\begin{align}
\label{eq:psi_modified}
\psi(t) =
\begin{cases}
t^2 - 2 t + 1
&\mbox{if } t>1,
\\
\frac{1}{4} (1-t^2)^2
&\mbox{if } t \in [-1, 1],
\\
t^2 + 2 t + 1
& \mbox{if } t < -1,
\end{cases}
\qquad
\psi'(t) =
\begin{cases}
2 t - 2
&\mbox{if } t>1,
\\
t^3-t
&\mbox{if } t \in [-1, 1],
\\
2 t + 2
& \mbox{if } t < -1.
\end{cases}
\end{align}
Besides, the parameter $A$ is often chosen as $A=\frac{\beta}{\epsilon}$ with $\beta>0$ and a small constant $\epsilon>0$ relating to the thickness of the diffuse interface.
Therefore, \eqref{A4_3} is not a severe constraint.
\begin{definition}[Weak solution]
\label{def:weak_solution}
Under the assumptions \ref{A1}--\ref{A7}, the quintuple $(\phi,\mu,\sigma,\pmb{\mathrm{v}},\B)$ for $d=2$ is called a weak solution of \ref{P_alpha} if
\begin{subequations}
\begin{align}
\label{eq:conv_phi}
\phi &\in L^\infty(0,T;H^1)
\cap L^2(0,T;H^2)
\cap H^1(0,T; (H^1)'),
\\
\label{eq:conv_mu}
\mu &\in L^2(0,T; H^1),
\\
\label{eq:conv_sigma}
\sigma &\in L^\infty(0,T; L^2)
\cap L^2(0,T; H^1)
\cap H^1(0,T; (H^1)'),
\\
\label{eq:conv_v}
\pmb{\mathrm{v}} &\in L^\infty(0,T; \mathbf{H})
\cap L^2(0,T; \mathbf{V})
\cap W^{1,\frac{4}{3}}(0,T; \mathbf{V}'),
\\
\label{eq:conv_B}
\B &\in L^\infty\left(0,T; L^2(\Omega;\R^{2\times2}_{\mathrm{SPD}})\right)
\cap L^2\left(0,T; H^1(\Omega;\R^{2\times2}_{\mathrm{S}})\right)
\cap W^{1,\frac{4}{3}}\left(0,T; (H^1(\Omega;\R^{2\times2}_{\mathrm{S}}))'\right),
\end{align}
\end{subequations}
such that
\begin{align}
\begin{split}
\label{eq:weak_init}
&\phi(0) = \phi_0 \text{ in } L^2(\Omega),
\quad
\sigma(0) = \sigma_0 \text{ in } L^2(\Omega),
\quad
\pmb{\mathrm{v}}(0) = \pmb{\mathrm{v}}_0 \text{ in } \mathbf{H},
\\
&\B(0) = \B_0 \text{ in } L^2(\Omega;\R^{2\times 2}_{\mathrm{S}})
\quad \text{and} \quad \B \text{ positive definite a.e.~in } \Omega\times(0,T),
\end{split}
\end{align}
and
\begin{subequations}
\begin{align}
\label{eq:phi_weak}
0 &= \int_0^T \dualp{\partial_t\phi}{\zeta}_{H^1} \dv{t}
+ \int_{\Omega_T} m(\phi)\nabla\mu\cdot\nabla\zeta
- \Gamma_\phi(\phi,\sigma,\B) \zeta
- \phi \pmb{\mathrm{v}}\cdot \nabla\zeta
\dv{x}\dv{t} ,
\\
\label{eq:mu_weak}
0 &= \int_{\Omega_T} \mu\rho
- A \psi^\prime(\phi)\rho
- B\nabla\phi\cdot\nabla\rho
+ \chi_\phi \sigma \rho \dv{x} \dv{t},
\\
\nonumber
\label{eq:sigma_weak}
0 &= \int_0^T \dualp{\partial_t\sigma}{\xi}_{H^1} \dv{t}
+ \int_{\Omega_T} n(\phi) \nabla (\chi_\sigma\sigma - \chi_\phi\phi)
\cdot\nabla\xi
+ \Gamma_\sigma(\phi,\sigma)\xi
- \sigma \pmb{\mathrm{v}} \cdot \nabla\xi \dv{x} \dv{t}
\\
&\qquad + \int_0^T \int_{\partial\Omega} K(\sigma-\sigma_\infty)\xi \ \mathrm d \calH^{d-1} \dv{t},
\\
\label{eq:v_weak}
\nonumber
0 &= \int_0^T \dualp{\partial_t\pmb{\mathrm{v}}}{\pmb{\mathrm{w}}}_{\mathbf{V}} \dv{t}
+ \int_{\Omega_T} (\pmb{\mathrm{v}}\cdot\nabla)\pmb{\mathrm{v}} \cdot \pmb{\mathrm{w}}
+ 2 \eta(\phi) \D(\pmb{\mathrm{v}}) : \D(\pmb{\mathrm{w}})
+ \kappa (\B-\I) : \D(\pmb{\mathrm{w}}) \dv{x}\dv{t}
\\
&\qquad + \int_{\Omega_T}
\big(\phi \nabla\mu + \sigma \nabla(\chi_\sigma\sigma - \chi_\phi\phi) \big) \cdot \pmb{\mathrm{w}} \dv{x}\dv{t},
\\
\label{eq:B_weak}
0 &= \int_0^T \dualp{\partial_t\B}{\C}_{H^1} \dv{t}
+ \int_{\Omega_T} \frac{\kappa}{\tau(\phi)} (\B-\I): \C
- 2 (\nabla\pmb{\mathrm{v}} \B) : \C
+ \alpha \nabla\B : \nabla\C
- \B : (\pmb{\mathrm{v}}\cdot\nabla)\C \dv{x}\dv{t},
\end{align}
\end{subequations}
for all $\zeta,\rho,\xi \in L^2(0,T;H^1)$, $\pmb{\mathrm{w}}\in L^4(0,T;\mathbf{V})$ and $\C \in L^4\left(0,T;H^1(\Omega;\R^{2 \times 2}_{\mathrm{S}})\right)$.
\end{definition}
\begin{theorem}[Existence of weak solutions]
\label{theorem:weak_solution}
Let \ref{A1}--\ref{A7} hold. Then, there exists a weak solution $(\phi,\mu,\sigma,\pmb{\mathrm{v}},\B)$ of \ref{P_alpha} in the sense of Definition \ref{def:weak_solution}. Moreover, there exist positive constants $C_1(T), C_2(T,\alpha^{-1})$, both depending exponentially on $T$ and $C_2(T,\alpha^{-1})$ depending additionally on $\alpha^{-1}$, such that
\begin{subequations}
\begin{align}
\nonumber
&\nnorm{\phi}_{L^\infty(0,T;H^1)}
+ \nnorm{\phi}_{L^2(0,T;H^2)}
+ \nnorm{\partial_t\phi}_{L^2(0,T;(H^1)')}
+ \nnorm{\mu}_{L^2(0,T;H^1)}
\\
&\quad
+ \nnorm{\sigma}_{L^\infty(0,T;L^2)}
+ \nnorm{\sigma}_{L^2(0,T;H^1)}
+ \nnorm{\partial_t\sigma}_{L^2(0,T;(H^1)')}
+ \nnorm{\pmb{\mathrm{v}}}_{L^\infty(0,T;L^2)}
+ \nnorm{\pmb{\mathrm{v}}}_{L^2(0,T;H^1)} \leq C_1(T),
\\[1ex]
& \nnorm{\partial_t\pmb{\mathrm{v}}}_{L^{4/3}(0,T;\mathbf{V}')}
+ \nnorm{\B}_{L^\infty(0,T;L^2)} + \nnorm{\B}_{L^2(0,T;H^1)} + \nnorm{\partial_t \B}_{L^{4/3}(0,T;(H^1)')}
\leq C_2(T,\alpha^{-1}).
\end{align}
\end{subequations}
\end{theorem}
\begin{remark}~
\label{remark:weak_solution}
\begin{enumerate}[(i)]
\item
This existence result will be proved in Section \ref{sec:fem} by the passage to the limit in a fully-discrete finite element scheme in two dimensions, where a CFL condition is necessary, i.e.~$\Delta t\leq C h^2$ with a possibly very small positive constant $C$, see Theorem \ref{theorem:convergence}. Further, the additional Lipschitz assumption \ref{A7} on the source terms is needed for the limit passing in presence of mass lumping, but it can be dropped if the integrals containing $\Gamma_\phi,\Gamma_\sigma$ are evaluated exactly.
\item As $\pmb{\mathrm{v}}\in L^2(0,T;\mathbf{V})$, integration by parts over $\Omega$ in the convection term in \eqref{eq:phi_weak} leads to \eqref{eq:phi_weak} with $-\phi\pmb{\mathrm{v}}\cdot\nabla\zeta$ replaced by $\pmb{\mathrm{v}}\cdot\nabla\phi \zeta$, which is consistent with \eqref{eq:phi2}. One can argue similarly for the convection terms in \eqref{eq:sigma_weak}, \eqref{eq:B_weak} and the last term in \eqref{eq:v_weak}.
\item Moreover, one can obtain \eqref{eq:B_weak} with $2\nabla\pmb{\mathrm{v}}\B$ replaced by $\nabla\pmb{\mathrm{v}}\B + \B (\nabla\pmb{\mathrm{v}})^T$, which is consistent with \eqref{eq:B2}, by choosing the test function $\C = \frac{1}{2} (\mathbb{G} + \mathbb{G}^T)$, where $\mathbb{G} \in L^4(0,T;H^1(\Omega;\R^{2\times 2}))$, and using the symmetry of $\B$.
\item This existence result still holds true if source or sink terms for $\B$ are included like in \eqref{eq:B_growth}, i.e.~if \eqref{eq:B2} is replaced by
\begin{align}
\partial_t \B + (\pmb{\mathrm{v}} \cdot\nabla)\B
+ \frac{\kappa}{\tau(\phi)}\big( \B-\I \big)
&= \nabla\pmb{\mathrm{v}} \B
+ \B (\nabla\pmb{\mathrm{v}})^T
- \gamma(\phi,\sigma)\B
+ \alpha \Delta\B,
\end{align}
where $\gamma\colon \R^2\to\R$ is continuous and bounded. The additional term $\gamma(\phi,\sigma)\B$ on the right-hand side can be controlled with a Gronwall argument similarly to \eqref{eq:formal_4}.
\end{enumerate}
\end{remark}
\subsection{Formal a priori estimates}
\label{sec:formal_bounds}
To better understand the strategy for the proof of Theorem \ref{theorem:weak_solution}, we temporarily assume that \ref{A1}--\ref{A5} hold and that $(\phi,\mu,\sigma,p,\pmb{\mathrm{v}},\B)$ is a sufficiently smooth solution of \ref{P_alpha} with $\B$ positive definite in $\Omega_T \coloneqq \Omega\times (0,T)$. The first step is to provide the formal derivation of \textit{a priori} estimates based on the energy
\begin{align}
\ensuremath{\mathcal{F}}(\phi,\sigma,\pmb{\mathrm{v}},\B) \coloneqq \int_\Omega
A\psi(\phi) + \frac{B}{2} \abs{\nabla\phi}^2
+ \frac{\chi_\sigma}{2} \abs{\sigma}^2
+ \chi_\phi \sigma (1-\phi)
+ \frac{1}{2} \abs{\pmb{\mathrm{v}}}^2
+ \frac{\kappa}{2}\trace(\B- \ln\B) \dv{x}.
\end{align}
Note that this energy is not finite if $\B$ is not positive definite, which is due to the logarithmic term.
Later, with the help of suitable regularization techniques, it turns out that the positive definiteness of the left Cauchy--Green tensor $\B$ is preserved for all $t>0$ if $\B(t=0)$ is positive definite. This has also been observed for other viscoelastic systems in the literature,
see, e.g., \cite[Lem.~2.1]{Hu_Lelievre_2007} or \cite[Rem.~3.4]{Lukacova_2017}.
Moreover, the energy $\ensuremath{\mathcal{F}}(\phi,\sigma,\pmb{\mathrm{v}},\B)$ can become negative due to the term $\sigma(1-\phi)$. This is one reason why the derivation of reasonable \textit{a priori} estimates must be performed carefully.
From the general energy identity \eqref{eq:energy0}, we have
\begin{align}
\begin{split}
\label{eq:formal_1}
&\ddv{}{t} \ensuremath{\mathcal{F}}(\phi, \sigma, \pmb{\mathrm{v}}, \B)
+ \int_\Omega m(\phi) \abs{\nabla\mu}^2
+ n(\phi) \abs{\nabla (\chi_\sigma \sigma - \chi_\phi\phi)}^2
+ 2 \eta(\phi) \abs{\D(\pmb{\mathrm{v}})}^2 \dv{x}
\\
&\quad
+ \int_{\partial\Omega} K \chi_\sigma \abs{\sigma}^2 \ \mathrm d \calH^{d-1}
+ \int_\Omega \frac{\kappa^2}{2\tau(\phi)} \trace\big(\B + \B^{-1}-2 \I\big)
- \frac{\alpha\kappa}{2} \nabla \B : \nabla \B^{-1} \dv{x}
\\
&=
\int_\Omega \mu \Gamma_\phi(\phi,\sigma,\B)
- (\chi_\sigma \sigma + \chi_\phi (1-\phi)) \Gamma_\sigma(\phi,\sigma) \dv{x}
+ \int_{\partial\Omega} K (\chi_\sigma \sigma + \chi_\phi(1-\phi)) \sigma_\infty
- K \chi_\phi(1-\phi) \sigma \ \mathrm d \calH^{d-1}.
\end{split}
\end{align}
We recall that this is obtained by formally multiplying \eqref{eq:phi2} with $\mu$, \eqref{eq:mu2} with $\partial_t \phi$, \eqref{eq:sigma2} with $\chi_\sigma \sigma + \chi_\phi (1-\phi)$, \eqref{eq:v2} with $\pmb{\mathrm{v}}$ and \eqref{eq:B2} with $\frac{\kappa}{2} (\I - \B^{-1})$, integrating over the domain $\Omega$, using Green's formula and then summing up the resulting equations.
Apart form the energy, the terms on the left-hand side of \eqref{eq:formal_1} are non-negative as $\B$ is positive definite and as the functions $m(\cdot), n(\cdot), \eta(\cdot), \tau(\cdot)$ are continuous, uniformly positive and bounded due to \ref{A3}. As $\B$ is symmetric positive definite, we note that it holds
\begin{align*}
- \int_\Omega \nabla \B : \nabla \B^{-1} \dv{x}
&\geq
\int_\Omega \frac{1}{d} \abs{\nabla \trace(\ln\B)}^2 \dv{x},
\quad
\text{ and }
\quad
\trace\big(\B + \B^{-1}-2 \I\big)
= \abs{ (\I - \B^{-1}) \sqrt{\B}}^2
\geq 0,
\end{align*}
see \cite[Lem.~3.1]{barrett_lu_sueli_2017} for the first inequality.
Now we estimate the terms on the right-hand side of \eqref{eq:formal_1}. First, for the terms involving the boundary integrals on the right-hand side of \eqref{eq:formal_1}, we apply H{\"o}lder's and Young's inequalities and the trace theorem to obtain
\begin{align}
\begin{split}
\label{eq:formal_boundary_term}
&\abs{ \int_{\partial\Omega} K (\chi_\sigma \sigma + \chi_\phi(1-\phi)) \sigma_\infty
- K \chi_\phi(1-\phi) \sigma \ \mathrm d \calH^{d-1} }
\\
&\leq
\frac{3}{4} K \chi_\sigma \norm{\sigma}_{L^2(\partial\Omega)}^2
+ C(K, C_{\mathrm{tr}}, \chi_\phi, \chi_\sigma) \norm{\phi}_{H^1}^2
+ C(K, \chi_\phi, \chi_\sigma) \big( \abs{\partial\Omega}
+ \norm{\sigma_\infty}_{L^2(\partial\Omega)}^2 \big) .
\end{split}
\end{align}
The terms on the right-hand side of \eqref{eq:formal_1} involving the source terms $\Gamma_\phi$ and $\Gamma_\sigma$ are bounded as follows,
\begin{align}
\begin{split}
\label{eq:formal_source_terms}
&\abs{ \int_\Omega
\mu \Gamma_\phi(\phi,\sigma,\B)
- \big(\chi_\sigma \sigma + \chi_\phi (1-\phi)\big) \Gamma_\sigma(\phi,\sigma)
\dv{x} }
\\
&\leq
\frac{1}{2} \norm{\mu}_{L^2}^2
+ C(R_0, \chi_\sigma, \chi_\phi) \Big(
\norm{\phi}_{L^2}^2
+\norm{\sigma}_{L^2}^2 \Big)
+ C(R_0, \chi_\phi, \Omega),
\end{split}
\end{align}
where we used \ref{A2} such as Hölder's and Young's inequalities.
However, one now needs an $L^2(\Omega)$-bound for the chemical potential in order to control the source terms.
This is obtained by multiplying \eqref{eq:mu2} with $\mu$ and integrating over the domain $\Omega$ and applying Green's formula, which yields
\begin{align*}
& \norm{\mu}_{L^2}^2
=
\int_\Omega \big(
A\psi'(\phi)
- \chi_\phi \sigma \big) \mu
+ B \nabla\phi \cdot \nabla\mu \dv{x}.
\end{align*}
At this point, we also need that the elasticity parameter $\kappa$ is independent of $\phi$, otherwise we would have to control additional $\B$-dependent terms, see \eqref{eq:mu0},
for which we do not have any \textit{a priori} knowledge.
In absence of any \textit{a priori} estimate for $\phi$, we can control $\norm{\mu}_{L^2}^2$ only if $\psi'(\cdot)$ has at most linear growth.
Hence, one obtains with H{\"o}lder's and Young's inequalities that
\begin{align}
\label{eq:formal_mu}
\norm{\mu}_{L^2}^2
\leq
\frac{m_0}{4} \norm{\nabla\mu}_{L^2}^2
+ C(A, B, R_3, \chi_\phi, m_0) \Big(
1 + \norm{\phi}_{L^2}^2
+ \norm{\nabla\phi}_{L^2}^2
+ \norm{\sigma}_{L^2}^2 \Big).
\end{align}
Moreover, we use
\begin{align}
\begin{split}
\label{eq:formal_nabla_sigma}
\chi_\sigma^2 \norm{\nabla \sigma}_{L^2}^2
&\leq 2 \nnorm{\chi_\sigma \nabla \sigma - \chi_\phi \nabla\phi}_{L^2}^2
+ 2 \chi_\phi^2 \norm{\nabla \phi}_{L^2}^2,
\end{split}
\end{align}
such that \eqref{eq:formal_1} becomes
\begin{align}
\begin{split}
\label{eq:formal_2}
&\ddv{}{t} \ensuremath{\mathcal{F}}(\phi, \sigma, \pmb{\mathrm{v}}, \B)
+ C \Big(
\norm{\mu}_{H^1}^2
+ \norm{\nabla \sigma}_{L^2}^2
+ \norm{\D(\pmb{\mathrm{v}})}_{L^2}^2
+ \norm{\sigma}_{L^2(\partial\Omega)}^2 \Big)
\\
&\quad
+ C\Big( \norm{\trace\big( \B + \B^{-1}-2 \I \big)}_{L^1}
+ \alpha \norm{\nabla \trace(\ln\B)}_{L^2}^2 \Big)
\\
&\leq
C\Big( 1 + \norm{\phi}_{H^1}^2
+ \norm{\sigma}_{L^2}^2
+ \norm{\sigma_\infty}_{L^2(\partial\Omega)}^2 \Big).
\end{split}
\end{align}
As the term $\chi_\phi \sigma (1-\phi)$ in the energy can have a negative sign, the next step is to absorb it with the help of the non-negative terms in the energy. In particular, we first apply Hölder's and Young's inequalities
\begin{align*}
\abs{\int_\Omega \chi_\phi \sigma (1-\phi) \dv{x} }
\leq
\frac{\chi_\sigma}{4} \norm{\sigma}_{L^2}^2
+ \frac{2 \chi_\phi^2}{\chi_\sigma} \norm{\phi}_{L^2}^2,
\end{align*}
so that we obtain by integrating over $t\in(0,s)$, where $s\in(0,T)$,
\begin{align}
\begin{split}
\label{eq:formal_3}
& A \norm{\psi(\phi(s))}_{L^1}
+ \frac{B}{2} \norm{\nabla\phi(s)}_{L^2}^2
+ \frac{\chi_\sigma}{4} \norm{\sigma(s)}_{L^2}^2
+ \frac{1}{2} \norm{\pmb{\mathrm{v}}(s)}_{L^2}^2
+ \frac{\kappa}{2} \norm{\trace(\B(s) - \ln\B(s))}_{L^1}
\\
&\quad
+ C \Big(
\norm{\mu}_{L^2(0,s;H^1)}^2
+ \norm{\nabla \sigma}_{L^2(0,s;L^2)}^2
+ \norm{\D(\pmb{\mathrm{v}})}_{L^2(0,s;L^2)}^2
+ \norm{\sigma}_{L^2(0,s;L^2(\partial\Omega))}^2 \Big)
\\
&\quad
+ C\Big( \norm{\trace\big( \B + \B^{-1}-2 \I \big)}_{L^1(0,s;L^1)}
+ \alpha \norm{\nabla \trace(\ln\B)}_{L^2(0,s;L^2)}^2 \Big)
\\
&\leq
\frac{2 \chi_\phi^2}{\chi_\sigma} \norm{\phi(s)}_{L^2}^2
+ C\Big( 1 + \abs{\ensuremath{\mathcal{F}}(\phi_0, \sigma_0, \pmb{\mathrm{v}}_0, \B_0)}
+ \norm{\phi}_{L^2(0,s;H^1)}^2
+ \norm{\sigma}_{L^2(0,s;L^2)}^2
+ \norm{\sigma_\infty}_{L^2(0,s;L^2(\partial\Omega))}^2 \Big).
\end{split}
\end{align}
Then, by \eqref{A4_1}, we have
\begin{align}
\begin{split}
\label{eq:formal_4}
& \left( A R_1 - \frac{2 \chi_\phi^2}{\chi_\sigma} \right) \norm{\phi(s)}_{L^2}^2
+ \frac{B}{2} \norm{\nabla\phi(s)}_{L^2}^2
+ \frac{\chi_\sigma}{4} \norm{\sigma(s)}_{L^2}^2
+ \frac{1}{2} \norm{\pmb{\mathrm{v}}(s)}_{L^2}^2
+ \frac{\kappa}{2} \norm{\trace(\B(s) - \ln\B(s))}_{L^1}
\\
&\quad
+ C \Big(
\norm{\mu}_{L^2(0,s;H^1)}^2
+ \norm{\nabla \sigma}_{L^2(0,s;L^2)}^2
+ \norm{\D(\pmb{\mathrm{v}})}_{L^2(0,s;L^2)}^2
+ \norm{\sigma}_{L^2(0,s;L^2(\partial\Omega))}^2 \Big)
\\
&\quad
+ C\Big( \norm{\trace\big( \B + \B^{-1}-2 \I \big)}_{L^1(0,s;L^1)}
+ \alpha \norm{\nabla \trace(\ln\B)}_{L^2(0,s;L^2)}^2 \Big)
\\
&\leq
C\Big( 1 + \abs{\ensuremath{\mathcal{F}}(\phi_0, \sigma_0, \pmb{\mathrm{v}}_0, \B_0)}
+ \norm{\phi}_{L^2(0,s;H^1)}^2
+ \norm{\sigma}_{L^2(0,s;L^2)}^2
+ \norm{\sigma_\infty}_{L^2(0,s;L^2(\partial\Omega))}^2 \Big).
\end{split}
\end{align}
Note that $AR_1 - \frac{2 \chi_\phi^2}{\chi_\sigma}$ is positive due to \eqref{A4_3}, which is not a severe constraint in practice, as $A = \frac{\beta}{\epsilon}$ with a small $\epsilon>0$. Hence, we apply a Gronwall argument (see below for Lemma \ref{lemma:gronwall}), to obtain the inequality
\begin{align}
\begin{split}
\label{eq:formal_5}
& \norm{\phi(s)}_{H^1}^2
+ \norm{\sigma(s)}_{L^2}^2
+ \norm{\pmb{\mathrm{v}}(s)}_{L^2}^2
+ \norm{\trace(\B(s) - \ln\B(s))}_{L^1}
\\
&\quad
+
\norm{\mu}_{L^2(0,s;H^1)}^2
+ \norm{\nabla \sigma}_{L^2(0,s;L^2)}^2
+ \norm{\D(\pmb{\mathrm{v}})}_{L^2(0,s;L^2)}^2
+ \norm{\sigma}_{L^2(0,s;L^2(\partial\Omega))}^2
\\
&\quad
+ \norm{\trace\big( \B + \B^{-1}-2 \I \big)}_{L^1(0,s;L^1)}
+ \alpha \norm{\nabla \trace(\ln\B)}_{L^2(0,s;L^2)}^2
\\
&\leq
C\Big(
1 + \abs{\ensuremath{\mathcal{F}}(\phi_0, \sigma_0, \pmb{\mathrm{v}}_0, \B_0)}
+ \norm{\sigma_\infty}_{L^2(0,T;L^2(\partial\Omega))}^2 \Big),
\end{split}
\end{align}
for almost all $s\in(0,T)$. This leads to formal \textit{a priori} estimates as the right-hand side of \eqref{eq:formal_5} is bounded due to \ref{A5}.
For completeness, we recall the following Gronwall inequality from \cite[Lem.~3.1]{garcke_lam_2017}.
\begin{lemma}
\label{lemma:gronwall}
Let $\alpha, \beta, u$ and v be real-valued functions defined on $I=[0,T]$. Assume that $\alpha$ is integrable, $\beta$ is non-negative and continuous, $u$ is continuous, $v$ is non-negative and continuous. Suppose $u$ and $v$ satisfy the integral inequality
\begin{align*}
u(s) + \int_0^s v(t) \dv{t} \leq \alpha(s) + \int_0^s \beta(t) u(t) \dv{t} \quad \forall \ s\in I.
\end{align*}
Then it follows
\begin{align*}
u(s) + \int_0^s v(t) \dv{t}
\leq \alpha(s) + \int_0^s \alpha(t)\beta(t) \exp\Big(\int_t^s \beta(r) \dv{r} \Big) \dv{t}.
\end{align*}
\end{lemma}
\subsection{Stronger bounds in two spatial dimensions}
\label{sec:formal_bounds_2d}
The bounds on $\B$ are not sufficiently strong to establish existence of a solution. However, we get an estimate in a stronger norm if we restrict to two spatial dimensions. Hence, suppose that \ref{A6} holds true in addition to \ref{A1}--\ref{A5}.
To derive higher order estimates for $\B$, we formally multiply \eqref{eq:B2} with $\B$, integrate over $\Omega$ and apply Green's formula to obtain
\begin{align}
\label{eq:formal_B_1}
\ddv{}{t} \frac{1}{2} \norm{\B}_{L^2}^2
+ \frac{\kappa}{\tau(\phi)} \norm{\B}_{L^2}^2
+ \alpha \norm{\nabla\B}_{L^2}^2 \dv{x}
&=
\int_\Omega \frac{\kappa}{\tau(\phi)} \trace\B
+ 2 \nabla\pmb{\mathrm{v}} : \B^2
- (\pmb{\mathrm{v}} \cdot\nabla)\B : \B \dv{x}.
\end{align}
On noting \eqref{eq:div_v2}, the last term in \eqref{eq:formal_B_1} vanishes by integration by parts.
Then, with H{\"o}lder's and Young's inequalities and a Gagliardo--Nirenberg interpolation inequality (see, e.g., \cite{barrett_boyaval_2009}) for $d=2$, it holds
\begin{align}
\label{eq:formal_B_2}
&\ddv{}{t} \norm{\B}_{L^2}^2
+ C \norm{\B}_{L^2}^2
+ \alpha \norm{\nabla\B}_{L^2}^2
\leq
C(\alpha^{-1}) \big( 1 + \norm{\B}_{L^2}^2
+ \norm{\nabla\pmb{\mathrm{v}}}_{L^2}^2 \norm{\B}_{L^2}^2 \big),
\end{align}
where $C(\alpha^{-1})$ denotes a constant that depends on the inverse of the viscoelastic diffusion parameter $\alpha$.
It follows with integration in time and Lemma \ref{lemma:gronwall} that
\begin{align}
\label{eq:formal_B_3}
&\norm{\B(s)}_{L^2}^2
+ \norm{\B}_{L^2(0,s;L^2)}^2
+ \alpha \norm{\nabla\B}_{L^2(0,s;L^2)}^2
\leq C \big(\alpha^{-1}, \norm{\nabla\pmb{\mathrm{v}}}_{L^2(0,T;L^2)} \big) \norm{\B_0}_{L^2}^2,
\end{align}
for almost all $s\in(0,T)$. The right-hand side of \eqref{eq:formal_B_3} is bounded due to \eqref{eq:formal_5} and \ref{A5}.
\subsection{Formal estimates for a regularized problem}
\label{sec:regularization}
Showing the positive definiteness of the left Cauchy--Green tensor $\B$ is one of the main difficulties we have to deal with.
Here, we apply a regularization strategy of Barrett and Boyaval \cite{barrett_boyaval_2009} and introduce a regularized problem with a cut-off on the left Cauchy--Green tensor on certain terms in the system \ref{P_alpha}.
First, we introduce the following concave regularized approximations of the logarithm function $G(s)=\ln(s)$ and of the identity $\beta(s) = [G'(s)]^{-1} = s$ for all $s>0$ similarly to \cite[Sec.~2.1]{barrett_boyaval_2009}:
\begin{alignat}{5}
&G_\delta(s) &&=
\begin{cases}
\frac{s}{\delta} + \ln(\delta) - 1,
& s < \delta,
\\
\ln(s),
& s \geq \delta,
\end{cases}
\quad\quad\quad
&&\beta_\delta(s) &&=
\big[G_\delta'(s)\big]^{-1}
&&= \max\{s,\delta\}
\quad \forall \ s\in\R,
\\
&G^L(s) &&=
\begin{cases}
\ln(s),
& s\in(0,L),
\\
\frac{s}{L} + \ln(L) - 1,
& s\geq L,
\end{cases}
\quad\quad\quad
&&\beta^L(s) &&=
\big[G^{L'}(s)\big]^{-1}
&&= \min\{s,L\}
\quad \forall \ s>0,
\end{alignat}
where $0 < \delta < 1 < L$, also see Figure \ref{fig:regularizations}.
We also define the concave $C^1(\R)$ function
\begin{align}
H_\delta(s) \coloneqq G^{\delta^{-1}}(s)
\quad \forall \ s\in\R_{>0}.
\end{align}
\begin{figure}[ht]
\centering
\subfloat
{
\begin{tikzpicture}
\begin{axis}[
axis x line=middle,
axis y line=middle,
xlabel=$s$,
xtick={0.4, 1, 3},
xticklabels={$\delta$, 1, $L$},
legend style={at={(axis cs:3,-3)},anchor=south west},
ymajorticks=false,
]
\addplot[samples=400, smooth, sharp plot, domain=0:7]
{ln(x)};
\addplot[samples=5, dashdotted, sharp plot, domain=3:7, mark=square, every mark/.append style={solid}]
{x/3+ln(3)-1};
\addplot[samples=3, dotted, sharp plot, domain=-0.8:0.4, mark=*]
{x/0.4 + ln(0.4) - 1};
\addplot[samples=3, only marks, domain=0.1:2, mark=square, every mark/.append style={solid}]
{ln(x)};
\addplot[samples=7, only marks, domain=0.4:7, mark=*]
{ln(x)};
\addplot[dashed, sharp plot] coordinates {(0.4,0) (0.4, -1 )};
\addplot[dashed, sharp plot] coordinates {(3,0) (3, 1.2 )};
\addlegendentry{$G(s)$}
\addlegendentry{$G^L(s)$}
\addlegendentry{$G_\delta(s)$}
\end{axis}
\end{tikzpicture}
}
\hspace{0.5cm}
\subfloat
{
\begin{tikzpicture}
\begin{axis}[
axis x line=middle,
axis y line=middle,
xlabel=$s$,
xtick={0.5,1.2,3},
xticklabels={$\delta$,1,$L$},
ymajorticks=false,
legend style={at={(axis cs:0.5,3)},anchor=south west},
]
\addplot[samples=100, smooth, sharp plot, domain=-1:4]
{x};
\addplot[samples=2, dashdotted, sharp plot, domain=3:4, mark=square, every mark/.append style={solid, fill=gray}]
{min(x,3)};
\addplot[samples=3, dotted, sharp plot, domain=-1:0.5, mark=*]
{max(x,0.5)};
\addplot[samples=5, only marks, domain=0:3, mark=square, every mark/.append style={solid}]
{min(x,3)};
\addplot[samples=5, only marks, domain=0.5:4, mark=*]
{max(x,0.5)};
\addplot[dashed, sharp plot] coordinates {(0.5,0) (0.5, 0.5 )};
\addplot[dashed, sharp plot] coordinates {(3,0) (3, 3 )};
\addlegendentry{$\beta(s)$}
\addlegendentry{$\beta^L(s)$}
\addlegendentry{$\beta_\delta(s)$}
\end{axis}
\end{tikzpicture}
}
\caption{The functions $G$ (left) and $\beta$ (right) and their regularizations.}
\label{fig:regularizations}
\end{figure}
We recall the following result from \cite[Lem.~2.1]{barrett_boyaval_2009}. Let us note that the domain of definition of scalar functions is naturally extended to symmetric matrices in terms of the eigenvalues.
\begin{lemma}
For all $\Phi,\Psi\in\R^{d\times d}_{\mathrm{S}}$ and for any $\delta \in (0,1)$, it holds
\begin{subequations}
\begin{align}
\label{eq:lemma_reg1a}
\beta_\delta(\Phi) G_\delta'(\Phi)
&= G_\delta'(\Phi) \beta_\delta(\Phi)
= \I,
\\
\label{eq:lemma_reg1b}
\trace\big( \beta_\delta(\Phi)
+ [\beta_\delta(\Phi)]^{-1} - 2\I \big)
&\geq 0,
\\
\label{eq:lemma_reg1c}
\trace\big( \Phi - G_\delta(\Phi) - \I \big)
&\geq 0,
\\
\label{eq:lemma_reg1d}
\big( \Phi - \beta_\delta(\Phi)\big)
: \big( \I - G_\delta'(\Phi) \big)
&\geq 0,
\\
\label{eq:lemma_reg1e}
(\Phi-\Psi) : G_\delta'(\Psi)
&\geq \trace\big( G_\delta(\Phi) - G_\delta(\Psi) \big),
\\
\label{eq:lemma_reg1f}
- (\Phi-\Psi) : \big( G_\delta'(\Phi) - G_\delta'(\Psi) \big)
&\geq \delta^2 \abs{G_\delta'(\Phi) - G_\delta'(\Psi)}^2.
\end{align}
In addition, if $\delta\in(0,\frac{1}{2}]$, it holds
\begin{align}
\label{eq:lemma_reg1g}
\trace\big( \Phi - G_\delta(\Phi) \big)
&\geq
\begin{cases}
\frac{1}{2} \abs{\Phi},
\\
\frac{1}{2\delta} \abs{ [\Phi]_{-} },
\end{cases}
\\
\label{eq:lemma_reg1h}
\Phi : \big( \I - G_\delta'(\Phi) \big)
&\geq \frac{1}{2}\abs{\Phi} - d,
\end{align}
\end{subequations}
where $[\cdot]_{-}$ denotes the negative part function defined by $[s]_{-} \coloneqq \min\{s,0\}$ $\forall \ s\in\R$.
\end{lemma}
Let us return to the problem \ref{P_alpha} and introduce the reguralized problem with a cut-off on the left Cauchy--Green tensor on certain terms in the system.
\subsubsection*{Problem \ref{P_alpha_delta}:}
\mylabelHIDE{P_alpha_delta}{$(\pmb{\mathrm{P}}_{\alpha,\delta})$}
Let $\delta\in(0,\frac{1}{2}]$. The regularized problem \ref{P_alpha_delta} corresponds to \ref{P_alpha} with \eqref{eq:v2}--\eqref{eq:B2} replaced by
\begin{align}
\label{eq:v2_delta}
\partial_t \pmb{\mathrm{v}} + (\pmb{\mathrm{v}}\cdot\nabla) \pmb{\mathrm{v}}
-\divergenz{2\eta(\phi) \D(\pmb{\mathrm{v}}) }
+ \nabla p
&= \divergenz{\kappa(\beta_\delta(\B)-\I)}
+ \mu\nabla\phi
+ (\chi_\sigma \sigma - \chi_\phi \phi) \nabla\sigma,
\\
\label{eq:B2_delta}
\partial_t \B + (\pmb{\mathrm{v}} \cdot\nabla)\beta_\delta(\B)
+ \frac{\kappa}{\tau(\phi)}\big( \B-\I \big)
&= \nabla\pmb{\mathrm{v}} \beta_\delta(\B)
+ \beta_\delta(\B) (\nabla\pmb{\mathrm{v}})^T
+ \alpha \Delta\B.
\end{align}
Now again, we temporarily assume that \ref{A1}--\ref{A5} hold and that $(\phi,\mu,\sigma,p,\pmb{\mathrm{v}},\B)$ is a sufficiently smooth solution of \ref{P_alpha_delta} for a given $\delta \in (0,\frac{1}{2}]$. The first step is again to provide the formal derivation of \textit{a priori} estimates based on the regularized energy
\begin{align}
\ensuremath{\mathcal{F}}_\delta(\phi,\sigma,\pmb{\mathrm{v}},\B) \coloneqq \int_\Omega
A\psi(\phi) + \frac{B}{2} \abs{\nabla\phi}^2
+ \frac{\chi_\sigma}{2} \abs{\sigma}^2
+ \chi_\phi \sigma (1-\phi)
+ \frac{1}{2} \abs{\pmb{\mathrm{v}}}^2
+ \frac{\kappa}{2}\trace(\B- G_\delta(\B)) \dv{x}.
\end{align}
Let us remark that $\B$ does not necessarily have to be positive definite in presence of the regularization parameter $\delta$ as the term $\trace(\B-G_\delta(\B))$ is well-defined even if $\B$ is not positive definite.
We perform a similar testing procedure as in \eqref{eq:formal_1}. More concretely, we formally multiply \eqref{eq:phi2} with $\mu$, \eqref{eq:mu2} with $\partial_t \phi$, \eqref{eq:sigma2} with $\chi_\sigma \sigma + \chi_\phi (1-\phi)$, \eqref{eq:v2_delta} with $\pmb{\mathrm{v}}$ and \eqref{eq:B2_delta} with $\frac{\kappa}{2} (\I - G_\delta'(\B))$, integrate over the domain $\Omega$, apply Green's formula, and then sum up the resulting equations, so that
\begin{align}
\begin{split}
\label{eq:formal_delta_1}
&\ddv{}{t} \ensuremath{\mathcal{F}}_\delta(\phi, \sigma, \pmb{\mathrm{v}}, \B)
+ \int_\Omega m(\phi) \abs{\nabla\mu}^2
+ n(\phi) \abs{\nabla (\chi_\sigma \sigma - \chi_\phi\phi)}^2
+ 2 \eta(\phi) \abs{\D(\pmb{\mathrm{v}})}^2 \dv{x}
\\
&\quad
+ \int_{\partial\Omega} K \chi_\sigma \abs{\sigma}^2 \ \mathrm d \calH^{d-1}
+ \int_\Omega \frac{\kappa^2}{2\tau(\phi)} (\B-\I) : (\I - G_\delta'(\B))
- \frac{\alpha\kappa}{2} \nabla \B : \nabla G_\delta'(\B)) \dv{x}
\\
&=
\int_\Omega \mu \Gamma_\phi(\phi,\sigma,\B)
- (\chi_\sigma \sigma + \chi_\phi (1-\phi)) \Gamma_\sigma(\phi,\sigma) \dv{x}
+ \int_{\partial\Omega} K (\chi_\sigma \sigma + \chi_\phi(1-\phi)) \sigma_\infty
- K \chi_\phi(1-\phi) \sigma \ \mathrm d \calH^{d-1}.
\end{split}
\end{align}
On noting \eqref{eq:lemma_reg1a} and \eqref{eq:lemma_reg1d} we have
\begin{align}
\label{eq:delta_2}
\int_\Omega (\B - \I) : \big(\I - G_\delta'(\B) \big) \dv{x}
&\geq \int_\Omega \trace\big(\beta_\delta(\B) + [\beta_\delta(\B)]^{-1}-2\I \big) \dv{x} \geq 0,
\end{align}
and similarly to \eqref{eq:lemma_reg1f}, see \cite[Sec.~4.2]{barrett_boyaval_2009}, it holds
\begin{align}
\label{eq:delta_3}
- \int_\Omega \nabla\B : \nabla G_\delta'(\B) \dv{x}
\geq \delta^2 \int_\Omega \abs{\nabla G_\delta'(\B)}^2 \dv{x}.
\end{align}
Then, with arguments that are similar to \eqref{eq:formal_5}, the following inequality can be derived, for almost all $s\in(0,T)$,
\begin{align}
\begin{split}
\label{eq:delta_4}
& \norm{\phi(s)}_{H^1}^2
+ \norm{\sigma(s)}_{L^2}^2
+ \norm{\pmb{\mathrm{v}}(s)}_{L^2}^2
+ \norm{\trace(\B(s) - G_\delta(\B(s))}_{L^1}
\\
&\quad
+
\norm{\mu}_{L^2(0,s;H^1)}^2
+ \norm{\nabla \sigma}_{L^2(0,s;L^2)}^2
+ \norm{\D(\pmb{\mathrm{v}})}_{L^2(0,s;L^2)}^2
+ \norm{\sigma}_{L^2(0,s;L^2(\partial\Omega))}^2
\\
&\quad
+ \nnorm{\trace\big(\beta_\delta(\B) + [\beta_\delta(\B)]^{-1}-2\I \big)}_{L^1(0,s;L^1)}
+ \alpha \delta^2 \norm{\nabla G_\delta'(\B)}_{L^2(0,s;L^2)}^2
\\
&\leq
C\Big(
1 + \ensuremath{\mathcal{F}}_\delta(\phi_0, \sigma_0, \pmb{\mathrm{v}}_0, \B_0)
+ \norm{\sigma_\infty}_{L^2(0,T;L^2(\partial\Omega))}^2 \Big),
\end{split}
\end{align}
which holds uniformly in $\delta\in(0,\frac{1}{2}]$. Moreover, with \eqref{eq:lemma_reg1g}, it additionally holds
\begin{align}
\label{eq:delta_5}
\norm{\B(s)}_{L^1}
+ \frac{1}{\delta} \nnorm{ [\B(s)]_- }_{L^1}
\leq \norm{\trace(\B(s) - G_\delta(\B(s))}_{L^1},
\end{align}
for almost all $s\in(0,T)$, which, together with \eqref{eq:delta_4}, makes sure that the eigenvalues of $\B$ are positive in the formal limit $\delta\to 0$.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 510
|
Published 04/21/2019 03:01:11 am at 04/21/2019 03:01:11 am in Pirate Wall Decor.
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|
{
"redpajama_set_name": "RedPajamaC4"
}
| 2,232
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## CHARLES BUKOWSKI
## BETTING ON THE MUSE
POEMS & STORIES
for Linda Lee
## TABLE OF CONTENTS
splash
the women
the monkey
Whistler
the pleasures of the damned
those marvelous lunches
panties
the dead flowers of myself
me against the world
the snails
again
the World War One movies
to hell and back in a buggy carriage
stages
escape
woman on the street
CONFESSION OF A COWARD
the secret
somebody else
A View from the Quarter, March 12th, 1965:
drink
black and white
and all the snow melted
an empire of coins
A NICKEL
nature poem
warning
answer to a note on the dresser:
you don't know
let not
the death of a roach
the unwritten
right now
the sheep
piss
last fight
defining the magic
writing
views
the strong man
the terror
the kiss-off
betting on the muse
THE UNACCOMMODATING UNIVERSE
met a man on the street
hell is now
the kid
"To Serve and Protect"
bad day
the dick
fall of the Roman Empire
people
RANSOM
it's difficult for them
think of it
chicken giblets
the lover
no win
THE STAR
an evaluation
neon
they think this is the way it's done
the pile-up
12 minutes to post
as the poems go
the telephone
HIDEAWAY
this dirty, valiant game
stay out of my slippers, you fool
the voice
the bard of San Francisco
on biographies
a real break
avoiding humanity
WHAT HAPPENED TO THE LOVING, LAUGHING GIRL IN THE GINGHAM DRESS?
the luck of the word
bad form
last call
the shape of the Star
upon reading a critical review
Paris, what?
a social call
the girls we followed home
slow starter
barstool
look back, look up
Paris
the good soul
lousy mail
THE SUICIDE
confession of a genius
traffic report
hands
final score
the misanthrope
putting it to bed
the trash can
block
storm
the similarity
MY MADNESS
pastoral
finis
that rare good moment
doesn't seem like much
strange luck
until it hurts
DEATH IN THE AFTERNOON
the gods
floss, brush and flush
a great show
epilogue
Fante
it got away
the luck of the draw
let it enfold you
the 13th month
finis, II
the observer
August, 1993
this night
betting on now
decline
in the mouth of the tiger
the laughing heart
a challenge to the dark
so now?
About the Author
Other Books by Charles Bukowski
Copyright
About the Publisher
## splash
the illusion is that you are simply
reading this poem.
the reality is that this is
more than a
poem.
this is a beggar's knife.
this is a tulip.
this is a soldier marching
through Madrid.
this is you on your
death bed.
this is Li Po laughing
underground.
this is not a god-damned
poem.
this is a horse asleep.
a butterfly in
your brain.
this is the devil's
circus.
you are not reading this
on a page.
the page is reading
you.
feel it?
it's like a cobra.
it's a hungry eagle
circling the room.
this is not a poem.
poems are dull,
they make you
sleep.
these words force you
to a new
madness.
you have been blessed,
you have been pushed
into a
blinding area of
light.
the elephant dreams
with you
now.
the curve of space
bends and
laughs.
you can die now.
you can die now as
people were meant to
die:
great,
victorious,
hearing the music,
being the music,
roaring,
roaring,
roaring.
## the women
my uncle Ben was interested in the
ladies
and many a time he would drive up
in his Model-A,
get out and come in with his new
lady.
they'd sit on the couch and chatter
away,
then my Uncle Ben would follow
my father into another
room.
"come on, Henry," he'd say to my
father,
"let me have a couple of bucks..."
"you're nothing but a bum," my
father would answer, "get yourself
a job!"
"Henry, I'm trying!
I've been to 6 places already
today!"
"you haven't, you just want
money for that whore!"
the going rate in those days
was two dollars.
"listen, dear brother, I'm
hungry!"
"you're hungry to go to bed
with that whore!
where do you find them
all?"
"shhh...she's a lady, an
actress!"
"get her out of my house!
we don't allow those kinds
of women in here!"
"Henry, just two bucks..."
"get her out of here before
I throw her out of
here!"
my uncle would walk back into
the other room.
"come on, Clara, let's go..."
they would leave the house
together
and we would hear the
Model-A starting up and
driving off.
my mother would run about
opening all the windows
and doors.
"she stinks!
that cheap perfume, that
awful cheap perfume!"
"we're going to have to
fumigate this place!"
my father would scream.
it would be the same
scene over and over
again,
in a few days or a week
the Model-A would pull
up and in would walk
my uncle Ben with
another woman.
"come on, Henry, just
two bucks!"
I never saw my
uncle Ben get his two
bucks
but he tried again and
again.
"those women are so
ugly," my mother would
say.
"I don't know where he
finds them," my father
would say, "and I don't
know where he gets the
gas for his car!"
they would sit down
then and a great gloom
would fall over them
for the remainder of
the day.
they would stop talking
and just sit there,
there would be nothing
else to do
but just sit there
thinking how terrible it
had been—
that woman actually
daring to enter their
lives,
to leave her smell,
and the remembrance
of
her laughter.
## the monkey
one summer Saturday afternoon
during the depression
an organ grinder came into the
neighborhood.
he stopped on each
block
and played his organ
and while he played
the monkey did a little
dance.
it was an awkward dance.
the monkey was on a leash
which sometimes hindered
his movements.
but as we watched
it did a little somersault
or stuck its tongue out
at us.
it was dressed in a vest
and pants and had a
little hat strapped to its
head.
when the music stopped
the man gave it a tin
cup
and the monkey went
from person to
person
holding out its
cup.
we children gave it
pennies
but some of the adults
gave it nickels,
dimes and
quarters.
then the man would
take the cup and
empty it of the
money.
the man was fat,
needed a
shave
and wore a red
Sultan's hat
badly faded by
the sun.
the man and the
monkey went from
house to
house.
we followed him.
the monkey had
tiny dark
unhappy
eyes.
then they got to
my father's
house and stood in
the driveway.
the man began to
play his organ
and the monkey
danced.
the door was
flung open and my
father rushed
out.
"what's all the god-damned
noise?"
he stood angrily next to the
man.
"that ape is probably
diseased!
if he shits on my lawn
you clean it
up!"
"he's got a rubber
diaper on,"
said the man,
continuing to
play the
organ.
"that's unnatural!
how'd you like to
wear
rubber
diapers?"
"they'd look better
on you,"
the man said,
continuing to play
the organ
as the monkey
pirouetted,
then did a
flip.
"what did you
say?" my father
asked.
"you heard me,"
said the
man.
"why don't you
get a decent job
and put that stinking
animal in the
zoo?" my father
screamed.
the loud screaming
upset the monkey
and he leaped on
top of the
organ.
he had fang-like
yellow teeth
his lips curled back
and he bit the
organ grinder
on the hand,
hard,
grabbed the tin
cup, leaped to the
cement and began
wildly circling with
it.
the man was bleeding
badly.
he took out a handkerchief
and wrapped it around
his hand.
the blood soaked
through.
the monkey took the
cup and hurled it
into the
street.
the man sighed
heavily.
then carrying the
organ
and dragging the
monkey
he walked out
into the street and
picked up the
cup.
"you stay out of this
neighborhood!"
my father
yelled.
"this is a free
country, I can go
anywhere!"
the man yelled
back.
"yeah?
get your ass out of
here or I'm going to
kick it
out!"
"you and whose
army?" the organ
grinder
asked.
"my army! I
served in World War
One!
where were
you?"
the monkey was
straining at the
end of his
leash, pulling
against it,
he was
choking.
the man picked
it up, kissed it,
put it on his
shoulder.
"you've upset
my monkey,"
he said.
"be glad that's
all,"
said my
father.
the organ grinder
walked off
with the monkey
on his
shoulder.
my father walked
back into the
house,
slamming the
door.
we watched
the man and the
monkey.
they reached
the end of
the block.
then they turned
the corner and
were
gone.
we all just stood
there.
nobody said
anything.
then somebody
said, "well, the
monkey's gone,
let's do something
else."
"what?"
"I don't know..."
there were five of
us.
we turned and
began walking
down the
sidewalk, the
other
way.
something would
turn
up.
## Whistler
she said, "all of a sudden
someone arrived.
he was called just
'Edgar'...
he was a post-Impressionist
painter,
dressed all in black.
it was stunning.
he was wearing a black
hat with a large
brim.
he was wearing a
rather high collar and a
lavaliere, the kind
that only artists
wear.
and he had a black
cape, was dressed
like Whistler.
he was probably in his
60s
but he was a most
handsome man.
he was bringing a huge
bouquet—c'était à la mode
des violettes de Palmes—
the violets from Palma—
which are pale violets,
and he cut a
fantastic figure."
when everybody left
I said to my grandmother,
"Who was that man?"
and she said, "Ah,
he is an
Artist."
when my grandmother said
that,
she meant "Ah,
mais oui, c'était une artiste!"
and I answered right away,
"Ah, moi aussi."
oh, Jesus or somebody
help us, help us, help
us,
save us from
these,
the centuries have
reeked with them.
no wonder the animals
are what we consort
with,
no wonder we sleep
away the
nights.
## the pleasures of the damned
the pleasures of the damned
are limited to brief moments
of happiness:
like the eyes in the look of a dog,
like a square of wax,
like a fire taking the city hall,
the county,
the continent,
like fire taking the hair
of maidens and monsters;
and hawks buzzing in peach trees,
the sea running between their claws,
Time
drunk and damp,
everything burning,
everything wet,
everything fine.
## those marvelous lunches
when I was in grammar school
my parents were
poor
and in my lunch bag there was
only a peanut butter sandwich.
Richardson didn't have a
lunch bag,
he had a lunch pail with
compartments, a
thermos full of
chocolate milk.
he had ham sandwiches,
sliced beef sandwiches,
apples, bananas, a
pickle and a large bag of
potato chips.
I sat next to Richardson
as we ate.
his potato chips looked
so good—
large and crisp as the
sun blazed upon
them.
"you want some potato
chips?" he would
ask.
and each day
I would eat some.
as I went to school each
day
my thoughts
were on Richardson's
lunch, and especially
those chips.
each morning as we
studied in class
I thought about
lunch time.
and sitting next to
Richardson.
Richardson was the
sissy and the other
boys looked down
on me
for eating with
him
but I
didn't care.
it was the potato
chips, I couldn't
help myself.
"you want some
potato chips, Henry?"
he would
ask.
"yes."
the other boys got
after me
when Richardson
wasn't
around.
"hey, who's your
sissy friend?
you one
too?"
I didn't like that
but the potato
chips were more
important.
after a while
nobody spoke to
me.
sometimes I ate
one of Richardson's
apples
or I got half a
pickle.
I was always
hungry.
Richardson was
fat,
he had a big
belly
and fleshy
thighs.
he was the only
friend I had in
grammar
school.
we seldom spoke
to each
other.
we just sat
together at
lunch time.
I walked home with
him after school
and often some of
the boys would
follow us.
they
would gather around
Richardson,
gang up on him,
push him around,
knock him
down
again and
again.
after they were
finished
I would go
pick up his lunch
pail,
which was
spilled on its
side
with the lid
open.
I would place the
thermos back
inside,
close the
lid.
then I would
carry the pail as
I walked Richardson
back to his
house.
we never spoke.
as we got to his door
I would hand him
the lunch
pail.
then the door would
close and he would
be gone.
I was the only friend
he had.
sissies live a hard
life.
## panties
hell, I don't know how old I was,
maybe 7,
and Lila lived next door to me,
she was, maybe 6, and one day
she was standing in her yard
and she looked at me
and lifted her dress and showed
me her panties.
something about it looked good
to me and I stared
and then she let her dress
fall back down and she walked
off.
"Lila," I yelled, "come back!"
she didn't.
but thereafter
every day when she
saw me
she would lift her dress and
show me her panties.
they were a nice clean white
and fitted snugly.
then she would let her dress
fall back down and walk off
again.
one day I was in the back
yard and 3 kids
I had never seen before
came running in
and started swinging their
fists at me.
I surprised myself, I
fought back well, in
fact I gave 2 of them
bloody noses and they
ran off.
but the bigger kid
remained and we
kept fighting.
he began to slowly
wear me down.
he backed me up against
the fence
and I was catching
3 punches to each
one I threw.
his hands were much
larger than mine
and he was very
strong.
then there was a
dull thump.
somebody had hit
him over the
head with something,
a large bottle.
it was Lila.
she hit him
again
and he ran from the
yard
yowling and holding
his head.
"thanks, Lila," I said,
"show me your
panties."
"no," she said.
she walked
back to her house
and went inside.
I saw her many times
after that in her
yard.
I'd ask her,
"show me your
panties, Lila."
but she always
said, "no."
then her family
sold their house and
moved away.
I never quite
understood what it all
meant
and still
don't.
## the dead flowers of myself
bulls strut in pinwheel glory,
rockets stun the sky,
but I don't know
quite what to make
of the dead flowers
of myself,
whether to dump them
out of the bowl
or
press them between
these blank pages
and go on;
well, all grief comes down
to hard death
and weeping finally ends.
thank the god
who made
it.
## me against the world
when I was a kid
one of the questions asked was,
would you rather eat a bucket of shit
or drink a bucket of piss?
I thought that was easy.
"that's easy," I said, "I'll take the
piss."
"maybe we'll make you do both,"
they told me.
I was the new kid in the
neighborhood.
"oh yeah," I said.
"yeah!" they said.
there were 4 of them.
"yeah," I said, "you and whose
army?"
"we won't need no army," the
biggest one said.
I slammed my fist into his
stomach.
then all 5 of us were down on
the ground fighting.
they got in each other's way
but there were still too many
of them.
I broke free and started
running.
"sissy! sissy!" they yelled.
"going home to mama?"
I kept running.
they were right.
I ran all the way to my house,
up the driveway and onto the
porch and into the
house
where my father was beating
my mother.
she was screaming.
things were broken on the floor.
I charged my father and started swinging.
I reached up but he was too tall,
all I could hit were his
legs.
then there was a flash of red and
purple and green
and I was on the floor.
"you little prick!" my father said,
"you stay out of this!"
"don't you hit my boy!" my mother
screamed.
but I felt good because my father
was no longer hitting my
mother.
to make sure, I got up and charged
him again, swinging.
there was another flash of colors
and I was on the floor
again.
when I got up again
my father was sitting in one chair
and my mother was sitting in
another chair
and they both just sat there
looking at me.
I walked down the hall and into
my bedroom and sat on the
bed.
I listened to make sure there
weren't any more sounds of
beating or screaming
out there.
there weren't.
then I didn't know what to
do.
it wasn't any good outside
and it wasn't any good
inside.
so I just sat there.
then I saw a spider making a web
in the window.
I found a match, walked over,
lit it and burned the spider.
then I felt better.
much better.
## the snails
my mother stood at the
window
watching my father
in the back
yard.
he was bent over in the
flower garden,
very still, very
intense.
"what's he doing out
there?" my mother
asked me.
"I don't know."
"look, he hasn't moved,
he's like a
statue!"
"yes."
"I'm going to see what
he's doing!"
I watched her walk out
into the yard,
she walked up very
quietly
behind him.
then she screamed.
she came running
into the house,
screaming,
"my god, my god,
my god!"
"what's wrong?"
I asked.
"What's wrong?
What's wrong?
He was watching
two snails doing it
to each other!"
she screamed a long
and horrible scream.
the tears were rolling
down her face.
my father walked in.
"oh, shut up!"
he said.
"Why did you do
that?
Why did you watch?"
"I told you to shut up!"
I walked out of the room
and into the
bedroom and closed the
door.
I could still hear them
screaming,
it went on and on.
then there was the sound
of
breaking glass,
then the slamming of a
door.
I walked out into the front
room.
my mother was sitting on
the couch,
the tears were still running
down her face.
she looked at me.
"why did he do that?
my god, why did he do
that?"
"I don't know," I told
her.
then I turned and
walked back to the
bedroom.
## again
now the territory is taken,
the sacrificial lambs have been slain,
as history is scratched again on the sallow walls,
as the bankers scurry to survive,
as the young girls paint their hungry lips,
as the dogs sleep in temporary peace,
as the shadow gets ready to fall,
as the oceans gobble the poisons of man,
as heaven and hell dance in the anteroom,
it's begin again and go again,
it's bake the apple,
buy the car,
mow the lawn,
pay the tax,
hang the toilet paper,
clip the nails,
listen to the crickets,
blow up the balloons,
drink the orange juice,
forget the past,
pass the mustard,
pull down the shades,
take the pills,
check the air in the tires,
lace on the gloves,
the bell is ringing,
the pearl is in the oyster,
the rain falls
as the shadow gets ready to fall again.
## the World War One movies
were best, the aviators drank at the bar
every night, fighting over the one or two blondes,
and it was gallant because in the dawn they
might die going after those Fokkers with their
Spads, so they lined up along that bar
and slugged them down.
we kids loved those movies, the men weren't
like our fathers, those men laughed and fought
and loved slinky blondes in long tight dresses.
each dawn was glorious, they'd go to their Spads,
pulling on their goggles, a quick wave of the hand and
a long white scarf flowing out behind them. They
grinned and flew off into the blue.
and then came the Germans high above the
clouds.
they'd spot the Spads, the leader would give the
signal and they'd dive downward with a roar,
coming down through the clouds, their machine guns
spitting fire,
and the Spads would see them
but not before one of the planes would be hit
and roar down in flames—usually
the guy with the sense of humor, the guy who
had made everybody laugh at the bar—
there he'd go, his hands rising in the
flames, then oil splashing his goggles, he'd
wiggle trying to free himself to parachute to safety
but it was always too late—
you'd see the Spad crash into a hill
exploding in a mass of flame.
the dogfight was a real spectacle, the hero
would have a Fokker on his tail, have to pull
an Immelman to get him off.
then he'd be on the other guy's tail
and the bullets would rip through
the German, his mouth would open, a
spurt of blood and his plane would head
toward the earth with a WHINING roar.
the dogfights were exciting and lasted a
long time but the Germans always lost
and one or two of their remaining planes
would limp off and that would be it.
then the Spads would begin their
journey back to the airfield.
this was always very dramatic because
one or two of them would be shot up,
crippled, being nursed back, often
the pilot hit by 3 or 4 bullets but
determined to bring the plane back
in and land it safely.
the ground crew would be
waiting and they would count the Spads
as they came in: one, two...6, 7,
8...but there had been ten...
the ground crew would be
badly shaken.
the crippled planes would come in first,
followed by the
others.
it was a very sad time.
but that night the remaining pilots would
be back at the bar with the slinky blondes,
even the aviators who had been shot were
there.
they had their arms in slings, their heads
bandaged but they were drinking and
making the slinky blondes
laugh.
outside the movie theaters they displayed
parts of a Spad, a huge wing, a
propeller, and at night there was a
searchlight probing the skies, you could
see it for miles.
all we boys loved those World War One
movies
and we built our own balsa wood
model airplanes, Spads and
Fokkers.
most kits cost 25 cents
which was a lot of money in the
1930s but somehow
every kid had his own
plane.
we were in a hurry to grow
up.
we all wanted to be
fighter pilots,
we wanted those slinky
blondes, we wanted to lean
against that bar and gulp
down a straight whiskey
like nothing had
happened.
we had dogfights with our
model planes and they
sometimes developed into
fist fights.
we fought until we were
bloody and
torn.
we fought for our
honor
while
our fathers watched us
and
yawned.
## to hell and back in a buggy carriage
that was one of the popular sayings, I didn't know
what it
meant, standing on a corner in the mid-thirties
with a cigarette dangling from my mouth like the
tough guys in the movies, scoring for some beer
was the big thing and once in a while
some whiskey but there was no money anywhere
for fathers or sons or anybody and we were all
bluffing, tough, nothing else to be, we stood
around flexing our muscles, getting down to the
beach now and then but the young girls ran with
the rich guys with cars (even in bad times
there were rich guys), kids driving canary yellow
convertibles, pulling up to corners, opening doors,
laughing, I could kick any of that ass but it meant
nothing to the girls, they were off with those richies,
their hair flying in the wind, it was a crappy time
for us, standing there on the street corners, our
cigarettes dangling, nothing to be tough about,
nothing near enough to fight and hating our
fathers who sat in chairs or read newspapers
all day, they couldn't find work, their guts hanging
out and their lives hanging out—dried, dead, useless.
dinners of beans and canned meats, still we
grew, inching out of our old clothes, leaving our
homes late at night to stand under street lamps or
sit on park benches sucking at wine, beer, gin,
talking, smoking, going to hell and back in a
buggy carriage.
we were tough with nothing to be tough about,
we were the depression kids
and we swore we'd never be like our fathers
or our fathers' fathers.
we'd break through the crap and the
fakery.
we knew something.
we knew something, sitting in the dark,
drinking and smoking.
it was all a matter of which one of us
got there first.
the ends of our cigarettes glowing in the
dark.
as perfect as we could get.
the laughter like knives cutting the
stupid air.
Los Angeles 1935.
## stages
back then, you'd go through stages,
one of them being that you'd get so
deeply tanned it was almost horrifying,
and you'd lift weights, learn
acrobatic techniques,
and all of this was done with
a demonic zest—it was a matter of
fighting back against the stifling
forces everywhere and you had
huge tanned muscles
and you walked like an ape
trying to hold a load in his buttocks.
when you walked into a room, all
conversation stopped, you looked
dangerous, indeed, and you had a
way of staring at people with an
off-hand disdain, and you were not
the only monster from hell, there
were usually one or two others with
you.
you would walk down the street
as if your very feet could break the
sidewalks.
you would work little routines, like
walking up to a fruit stand with the
clerk watching,
you would pick up an apple with
one hand and crush it,
then smile at him and
replace the crushed apple on the
stack.
you ripped phone books in half,
picked up cars by their front
bumpers.
the stronger you got the more
you wanted to use it.
and you not only had strength
but an ultra-quickness—
you caught flies in mid-flight,
shadow-boxed with frightening
speed—left jab, left jab, zip, zip,
right lead, right hook, left hook,
uppercut, you had a pair of red
boxing gloves and you
laced them on with great calm
as your opponent waited, his
eyes jumping with fear.
that was the first stage, the
other was when you gave it
all up, the muscles shrank,
you paled, slouched,
assuming the worst
posture imaginable, smoking
cigarette after cigarette, coughing,
masturbating, drinking
endless coffee and all the
booze you could steal.
you had more friends that
way, now you really looked
dangerous and people hung
on your every word, you were
now the ultimate discontent,
your mind a dirty saber
which cut through all the world's
crap.
you found that this stage
garnered you far more
attention, not only from your
peers but from your parents,
the neighbors, the girls and
the teachers.
you were always in the
principal's office, not because
you had done anything
heinous but because you
looked like you might and,
actually, you felt like you
might.
"It's your ATTITUDE, Mr.
Chinaski, it's horrible, in
and out of class."
"huh?"
"Do you want to
graduate?"
"I dunno..."
"Don't you care?"
"'bout what?"
"Mr. Chinaski, you will now go
and sit in the phone
booth and you will remain
there
until I tell you to come
out!"
"o.k."
it was his phone booth
torture chamber.
I'd go in there, rack my
knees against one wall,
loll my head back and
pretend to go to
sleep.
it pissed him something
awful.
I graduated, still in the
2nd stage,
and I think that I have
been stuck there
ever
since.
## escape
the day you were starving and watching the
swans in the park,
it was truly not a bad day
watching them circle,
it was quiet,
you looked at their feathers, their necks,
their eyes.
for a moment you thought of
catching one, killing it, eating it.
but
you had nothing to cook
one on.
and you knew you couldn't do
it anyway.
there were many things you
couldn't do.
that's why you were starving
in a public park.
then there were voices, a
young lady in her summer
dress, and she was with her
young man and they were
laughing.
you looked at them and made
them dead,
you placed them in their
grave,
you saw their bones,
the skulls.
then you got up from the
grass and left them there with
the swans.
you walked out of the park,
you were on the boulevard,
you began walking,
walking seemed sensible
and it wasn't a bad
day,
just another day,
walking the sidewalk,
the world slanting through
your brain—
a white shot of
light.
being alone you decided, was a
magnificent
miracle.
nothing else made any
sense at
all.
## woman on the street
her shoes themselves
would light my room
like many candles.
she walks like all things
shining on glass,
like all things
that make a difference.
she walks away.
## CONFESSION OF A COWARD
God, she thought, lying in bed naked and re-reading Aldington's Portrait of a Genius, But..., he's an imposter! Not D. H. Lawrence, but her husband—Henry—with his bauble of a belly and all the hair he never combed and the way he stood around in his shorts, and the way he stood naked before the window like an Arabian and howled; and he told her that he was turning into a toad and that he wanted to buy a Buddha and that he wanted to be old and drown in the sea, and that he was going to grow a beard and that he felt as if he was turning into a woman.
And Henry was poor, poor and worthless and miserable and sick. And he wanted to join the Mahler Society. His breath was bad, his father was insane and his mother was dying of cancer.
And besides all this, the weather was hot, hot as hell.
"I've got a new system," he said. "All I need is four or five grand. It's a matter of investment. We could travel from track to track in a trailer."
She felt like saying something blasé like, "We don't have four or five grand," but it didn't come out. Nothing came out; all the doors were closed and all the windows were down, and it was in the middle of the desert—not even vultures—and they were about to drop the Bomb. She should have stayed in Texas, she should have stayed with Papa—this man is a goon, a gunnysack, a gutless no-nothing in a world of doers. He hides behind symphonies and poetic fancies; a weak and listless soul.
"Are you going to take me to the museum?" she asked.
"Why?"
"They're having an Art Exhibit."
"I know."
"Well, don't you want to see Van Gogh?"
"To hell with Van Gogh! What's Van Gogh to me?"
The doors closed again and she couldn't think of an answer.
"I don't like museums," he continued. "I don't like museum-people."
The fan was going but it was a small apartment and the heat held as if enclosed in a kettle.
"In fact," he said, peeling off his T-shirt and standing in just his shorts, "I don't like any kind of people."
Amazingly, he had hair on his chest.
"In fact," he continued, pulling his shorts down and over the end of one foot, "I'm going to write a book some day and call it Confession of a Coward."
The doorbell rang like a rape, or the tearing of ripe flesh.
"Jesus Christ!" he said like something trapped.
She jumped off the bed, looking very white and unpeeled. Like a candy banana. Aldington and D. H. Lawrence and Taos fell to the floor.
She ran to the closet and began stuffing herself inside the flying cloth of female necessaries.
"Never mind the clothes," he said.
"Aren't you going to answer?"
"No! Why should I?"
It rang again. The sound of the bell entered the room and searched them out, scaled and scalded their skins, pummeled them with crawling eyes.
Then it was silent.
And the feet turned with their sound, turning and guiding some monster, taking it back down the stairwell, one two three, 1, 2, 3; and then gone.
"I wonder," he said, still not moving, "what that was?"
"I don't know," she said, bending double at the waist and pulling her petticoat back over her head.
"Here!" she yelled. "Here!" holding her arms out like feelers.
He finished yanking the petticoat off over her head with some distaste.
"Why do you women wear this crap?" he asked in a loud voice.
She didn't feel an answer was necessary and went over and pulled Lawrence out from under the bed. Then she got into bed with Lorenzo and her husband sat on the couch.
"They built a little shrine for him," he said.
"Who?" she asked irritably.
"Lawrence."
"Oh."
"They have a picture of it in that book."
"Yes, I've seen it."
"Have you ever seen a dog-graveyard?"
"What?"
"A dog-graveyard."
"Well, what about it?"
"They always have flowers. Every dog always has flowers, fresh, all in neat little clusters on each grave. It's enough to make you cry."
She found her place in the book again, like a person searching for solitude in the middle of a lake: So the bitter months dragged by miserably, accompanied by Lorenzo's tragic feeling of loss, his—
"I wish I had studied ballet," he said. "I go about all slumped over but that's because my spirit is wilted. I'm really lithe, ready to tumble on spring mattresses of some sort. I should have been a frog, at least. You'll see. Someday I'm going to turn into a frog."
Her lake rippled with the irritating breeze: "Well, for heaven's sake, study ballet! Go at night! Get rid of your belly! Leap around! Be a frog!"
"You mean after WORK?" he asked woefully.
"God," she said, "you want everything for nothing." She got up and went to the bathroom and closed the door.
She doesn't understand, he thought, sitting on the couch naked, she doesn't understand that I'm joking. She's so god-damned serious. Everything I say is supposed to carry truth or tragic import, or insight or something. I've been through all that!
He noticed a pencil-scrawled piece of paper, in her handwriting, on the side table. He picked it up:
My husband is a poet published alongside Sartre and Lorca;
he writes about insanity and Nietzsche and Lawrence,
but what has he written about me?
she reads the funnies
and empties garbage
and makes little hats
and goes to Mass at 8 AM
I too am a poet and an artist, some discerning critics
say, but my husband wrote about me:
she reads the funnies...
He heard the toilet flush, and a moment later, out she came.
"I'd like to be a clown in a circus," he greeted her.
She got back on the bed with her book.
"Wouldn't you like to be a tragicomic clown stumbling about with a painted face?" he asked her.
She didn't answer. He picked up the Racing Form:
Power 114 B.g.4, by Cosmic Bomb—
Pomayya, by Pompey
Breeder, Brookmeade Stable.
1956 12 241 $12,950
July 18-Jan 1 1/16 1:45 1/5 ft. 3 122 2
1/2 3 2h GuerinE' Alw 86
"I'm going to Caliente next Sunday," he said.
"Good. I'll have Charlotte over. Allen can bring her in the car."
"Do you believe she really got propositioned by the preacher in that movie like she claimed?"
She turned the page of her book.
"God damn you, answer me!" he screamed, angry at last.
"What about?"
"Do you think she's a whore and making it all up? Do you think we're all whores? What are we trying to do, reading all these books? Writing all the poems they send back, and working in some dungeon for nothing because we're not really interested in money?"
She put the book down and looked back over her shoulder at him. "Well," she said in a low voice, "do you want to give it all up?"
"Give WHAT all up? We don't have anything! Or, do you mean Beethoven's Fifth or Handel's Water Music? Or do you mean the SOUL?"
"Let's not argue. Please. I don't want to argue."
"Well, I want to know what we are trying to do!"
The doorbell rang like all the bells of doom sweeping across the room.
"Shhh," he said, "shhh! Be quiet!"
The doorbell rang again, seeming to say, I know you are in there, I know you are in there.
"They know we're in here," she whispered.
"I feel that this is it," he said.
"What?"
"Never mind. Just be quiet. Maybe it will go away."
"Isn't it wonderful to have all these friends?" she took up the joke-cudgel.
"No. We have no friends. I tell you, this is something else!"
It rang again, very short, flat and spiritless.
"I once tried to make the Olympic swimming team," he said, getting completely off the point.
"You make more ridiculous statements by the minute, Henry."
"Will you get off my back? Just for that!" he said, raising his voice, "WHO IS IT?"
There was no answer.
Henry rose wide-eyed, as if in a trance, and flung the door open, forgetting his nakedness. He stood there transfixed in thought for some time, but it was obvious to her that nobody was there—in his state of undress there would have been quite a commotion or, at the very least, some sophisticated comment.
Then he closed the door. He had a strange look on his face, a round-eyed almost dull look and he swallowed once as he faced her. His pride, perhaps?
"I've decided," he announced, "that I'm not going to turn into a woman after all."
"Well, that will help matters between us considerably, Henry."
"And I'll even take you to see Van Gogh. No, wait, I'll let you take me."
"Either way, dear. It doesn't matter."
"No," he said, "you'll have to take me!"
He marched into the bathroom and closed the door.
"Don't you wonder," she said through the door, "who that was?"
"Who what was?"
"Who that was at the door? Twice?"
"Hell," he said, "I know who it was."
"Who was it, then?"
"Ha!"
"What?"
"I said, 'Ha!' I'm not telling!"
"Henry, you simply don't know who it was, anymore than I do. You're simply being silly again."
"If you promise to take me to see Van Gogh, I'll tell you who was at the door."
"All right," she humored him along, "I promise."
"O.K., it was me at the door!"
"You at the door?"
"Yes," he laughed a silly little laugh, "me looking for me! Both times."
"Still playing the clown aren't you, Henry?"
She heard the water running in the basin and knew he was going to shave.
"Are you going to shave, Henry?"
"I've decided against the beard," he answered.
He was boring her again and she simply opened her book at a random page and began reading:
You don't want any more of me?
I want us to break off—you be free of me, I free of you.
And what about these last months?
I don't know. I've not told you anything but what I thought was true.
Then why are you different now?
I'm not—I'm the same—only I know it's no good going on.
She closed the book and thought about Henry. Men were children. You had to humor them. They could take no hurt. It was a thing every woman knew. Henry tried—he was just so—all this playing the clown. All the poor jokes.
She rose from the bed as if in a dream, walked across the floor, opened the door and stared. Against the basin stood a partly soaped shaving brush and his still wet shaving mug. But the water in the basin was cold and at the bottom—against the plug, green and beyond her reach at last and the size of a crumpled glove—stared back the fat, living frog.
## the secret
don't worry, nobody has the
beautiful lady, not really, and
nobody has the strange and
hidden power, nobody is
exceptional or wonderful or
magic, they only seem to be.
it's all a trick, an in, a con,
don't buy it, don't believe it.
the world is packed with
billions of people whose lives
and deaths are useless and
when one of these jumps up
and the light of history shines
upon them, forget it, it's not
what it seems, it's just
another act to fool the fools
again.
there are no strong men, there
are no beautiful women.
at least, you can die knowing
this
and you will have
the only possible
victory.
## somebody else
he had long thin
arms,
sat always in a
white t-shirt,
no gut at all,
he was in his
mid-40s
cheeks hollowed
in,
an x-con,
he rolled a
cigarette with
one hand,
skin burned
brown,
he had crazy
gray
eyebrows,
never looked
right at
you,
he had no
luck with
women,
was always in
love with some
number
who disdained
him,
he coughed too
often,
talked about
all his terrible
jobs of the
past,
sitting in a
chair
he drank wine
out of tall
water glasses,
preferred port,
said muscatel
made him
crazy.
each time
we drank
it was about the
same...
"come on, Hank,
let's fight!
you've got guts,
let's fight!"
"I don't want to
fight you,
Lou."
I wasn't afraid
of him.
in fact, he
bored
me.
there wasn't
anybody else
to drink with
in that
hotel
except a lady
I knew down
the
hall.
"you banging
her, Hank?"
"maybe."
"can you fix
me up?"
"I don't think
so."
"come on, Hank,
let's fight!"
"go on, drink
your wine."
"I got in a fight
with a guy once,
we used pick
handles.
he broke my
arm on the
first swing.
I still got him.
I busted him
up
good."
he poured the
wine down.
he always got
sick.
he could seldom
make it to the
hall
bathroom.
he'd let it go
in my
sink.
"all right, Lou,
clean up that
fucking
sink!"
"sorry, Hank,
sorry, I think I
got an
ulcer."
"clean the
sink!"
he was like a
17 year old
boy,
nothing had
developed.
I preferred to
drink
alone
but I didn't want
to hurt his
feelings.
one time
he didn't come
around for a
couple of
nights.
that was all
right but he
owed me
ten bucks
and I needed the
money.
I went down to
his door and
knocked.
no answer.
I pushed the
door open.
he was on the
bed
and the gas
heater was
hissing loudly.
it wasn't lit
and all the
windows
were closed.
I shut the
heater off,
opened the
windows
and stood at the
door
swinging it
back and forth
to get air
into the
room.
then I shook
him.
he was still
alive.
he gave me
a stupid
smile.
"Hank, you
saved my
life!
you saved my
life!"
he sat up
in bed,
put his feet
on the
floor.
"you saved
my life!
you're my
buddy
forever!"
"next time
you want to
kill yourself,
lock your
door."
I walked out
of there
and back to
my room.
then he was
knocking on
my door.
I told him
to come
in.
he sat in
the chair.
"I'm in
love,"
he said.
"yeah?"
"it's the
manager.
you ever notice
her body,
her eyes,
her hair?
and she's
intelligent."
"Lou, you owe
me ten
bucks."
"all I got is
a five."
"let me have
it."
he took a
5 from his
wallet.
that's all that
was in
there.
I took it.
"I wrote her a
long love
letter, 4 pages,
I slipped
it under her
door."
"did you
sign it?"
"no."
"don't worry
about
it."
"all right,
Hank.
but I think
she'll know
it's me.
I'm afraid
to face
her.
you got any
wine?"
"one bottle."
"can I have a
drink?"
I got the bottle
and put the
corkscrew to
the
cork.
Lou sat there
and rolled a
cigarette with
one
hand.
## A View from the Quarter, March 12th, 1965:
we are in a terrible hurry to die
as large Negroes break the
pavement
our fingers tremble on dark
coffee cups
as this city
all the cities
lie spread-legged
dipped into with
beak,
I awaken to pull a shade
open
I awaken to black men and
white men and no
men—
they rape everything
they walk into churches and
churches burn down
they pet dogs and dogs heave
yellow saliva and
die
they buy paintings that they
don't understand
they buy women that they
don't understand
they buy everything and
what they can't buy
they kill
their women approach me
they wiggle in the sacrament of
their flesh
they sway before me upon the towers
of their high-heels
the whole sum of them wanting
to make me scream
in some idiot's glory
but I look again
and I know that they are
dead
that it is useless
and I cross the street
to buy a loaf of
bread
at night
the sweetest sound I hear is
the dripping of the
toilet
or some unemployed Jazzman
practicing his runs—
a wail of martyrdom to an
always
incomplete
self
we only pretend to live
while we wait on something
we wait on something
and look at diamond wrist watches
through plate glass windows
as a spider sucks the guts out of a
fly
we pay homage to Marshal Foch's
granddaughter bending over a
tub of laundry,
we walk down St. Peter St.
hoping to find a
dime in the gutter
the dogs know us
the dogs know us
best
the Jazzman sends it home to
me through the blue glass of a
4 p.m. Friday
afternoon
he wants me to know how he
feels
as feet run over my
head
as the dead men suck in
spaghetti
as the dead men machinegun the
bridge
and in moments of rest
pray and drink
good scotch
I have watched the artists
rotting in their chairs
while the tourists took pictures
of an old iron railing not yet made
into guns
I have seen you, New Orleans,
I have seen you, New York,
Miami, Philly, Frisco, St. Louie,
L.A., Dago, Houston, and
most of the rest. I have
seen nothing. your best men are
drunks and your worst men are
locking them
up,
your best men are killers and
your worst men are
selling them
bullets
your best men die in alleys
under a sheet of paper
while your worst men
get statues in parks
for pigeons to shit upon for
centuries
the Jazzman stops. My god, it's
quiet, that's all I can say now!
it's quiet. it's quiet. let me think
if I feel like thinking and if
I don't, mama, let me not
think.
4:26 p.m.
the Quarter
I look down on the floor—
a beer carton
busted open and empty
says
"Don't litter!
Keep America
Beautiful!"
and like the Jazzman:
don't wanta think
no more.
## drink
the saddest bar I was ever
in was in New Orleans,
a place west of Canal
Street.
I still remember the
name of it
but for now
let's just call it Bar
Zero.
it was across from
my room,
a mouse-infested
hole on the
second
floor.
I walked into Bar
Zero one night
around eleven
p.m.
and
asked for a
beer.
it took the bartender
an eternity to get it
to me.
the poor devil had
a club foot.
the people
sat at old round
wooden
tables.
the overhead lights
were glaringly
bright.
I was 20 years old,
not too keen on
living
and the place
immediately
brought me
down.
I looked over
at one table.
a lady was sitting
with 3
men.
the poor dear had
a glass eye.
it was bright green,
no sign of a
pupil.
the glass eye
gleamed silently
in the impossible
light.
the men seemed
almost as
one, they looked
so similar,
they were skeletal
with sagging
almost snow-white
skin.
their toothless mouths
hung open.
one of the men was
a bit younger:
a toothpick hung from
his mouth.
he was the liveliest of
that
group.
at another table
a man sat alone in
pin-striped
coveralls.
his beer glass had
tipped upon its
side.
there was a pool
of beer on the
table.
he was
still, he never
moved.
he didn't appear
to be
breathing.
but
out of each
corner of his mouth
oozed two streams
of spittle.
the new spittle
slowly
ran over the old
spittle which had
dried white.
there was a total
silence.
I gulped my beer
down and ordered
another.
an old black and
white dog
sat in the
corner.
his ribs showed
through
as he continued
to bite at his
body,
he never stopped,
the fleas were
eating him
alive.
his teeth were
gone,
so he just gummed
his flesh,
doing what he
could, a gallant
battle—
you heard the
continuous
sucking,
the only
sound in the
place.
then from somewhere
an old dame
appeared,
straight white
hair,
she was dressed
all in black,
looked a
hundred years
old,
she walked up,
stuck her face
into mine,
"HEY!" she
said.
some speech
at last.
"HEY!"
she attempted to
mount the bar
stool next to
mine,
wheezing.
I helped her up
on the
stool,
asked the barkeep
for two
beers.
she put the glass
to her lips, chugged most
of it down,
the rest running
down her face and
into her black
lap.
she made no
attempt to
dry herself.
I ordered her
another
beer.
then one of the
three men at the
other table began
singing:
"Somebody bet
on the bob-tailed
nag, I'm gonna
bet on the
grey!"
he sang the same
line three times,
then
stopped.
I asked for a glass
of wine.
when it finally
arrived
there was
dust floating
on the
top.
I drank it
down.
there was the
faint taste of
turpentine.
I ordered
another.
I drank there a couple
of hours.
nothing really
happened.
the bright lights
remained
bright and the
poor dog
kept
gumming at
himself.
"HEY!" the old dame
would yell
and I'd order her
another
beer.
then I remembered I
had something to
drink in my
room.
I got off my stool
and
walked
out.
I walked across
the street,
went to my room,
found the bottle,
sat in a chair,
in the dark,
drinking
and looking
across the street
and into
the bar.
the old dame
had not moved,
the people at the
tables were as
before
as the dog
continued to
chomp.
I heard the mice
moving around
behind me
in the
dark.
where before
they had always
irritated me
with their bold
sharing of my
space,
I now felt the
sound of them,
the presence
of them
almost
endearing.
I drank
from the
bottle
looking down
at the
bar.
I lived in that
room for two
more months
but only once
went back
to that
place.
as I walked
in
the man was
singing:
"Somebody bet
on the bob-tailed
nag, I'm gonna
bet on the
grey!"
and I turned
around and
walked out
and that was
that.
## black and white
I must have checked in drunk
because I awakened in the
morning
in a small bed in an old
hotel room.
I wasn't even sure of the
city.
I walked to the window
and looked down.
I was on one of the
upper floors.
the movement of the
people and the automobiles
down there
almost took on a dream-
like
quality.
I had a suicide complex
or I thought I had
one.
I tried to open the window,
it would make a great
jump
down.
the window wouldn't open,
I'd have to try something
else.
there was a knock on the
door.
"come in," I said.
it was a buxom black
maid.
I was standing in my
underwear.
she didn't say
anything, just went about
changing the
sheets.
"what's a good way to
kill yourself?" I asked
her.
"you want to kill yourself?"
she asked.
"yeah."
"you look like you need
a drink."
"yeah."
"I'll order something," she
said.
she got on the
telephone.
I heard her ordering whiskey
and beer.
"what city is this?"
I asked.
"St. Louis."
"you been working here
long?" I asked.
"2 years..."
she had a duster.
she was dusting things.
the duster was made up of
black and white
feathers.
"forget that," I said.
"forget what?"
"dusting."
she walked over with the
duster and dusted me
up the front.
then she dusted my
rear.
there was a knock at
the door.
I went to my pants and
got my
wallet.
I opened the door,
got the drinks, tipped
him a dollar.
"you sure this is
St. Louis?" I asked.
she took the tray,
uncapped the
whiskey, poured two glasses,
half full, added seltzer
water.
she uncapped 2 bottles
of beer.
we sat on the edge
of the bed,
clicked glasses, went for
it.
"the first one's best,"
she said.
"damn right..."
we sat there drinking.
"don't you have to work?"
I asked.
"what do you mean?"
"I mean, the rooms, don't
you have to do the
rooms?"
"they won't fire me.
listen, do you really want to
kill yourself?" she asked.
"I think so."
"you're not sure?"
"sometimes I'm more sure
than other times."
"my sister killed herself."
I poured 2 more drinks.
the clock radio said
10: 37 a.m.
"what do you do?"
she asked.
"I'm unemployed."
"you ever worked?"
"many times."
we sat there drinking.
sometimes she poured,
sometimes I did.
soon it was close to
noon.
we ended up in bed
together.
we must have
slept.
when I awakened it was
evening going into
night.
I saw her getting
dressed.
then she was finished.
she walked to the door,
opened it, then walked
out and was
gone.
I got up and sat in a
chair and looked out the
window.
I watched the headlights
of the tiny cars
moving down
there.
and I still didn't know
what to do with
myself.
## and all the snow melted
she was a
German girl with a figure like quicksilver
quick something
anyhow
I'd say, "I want to fuck you"
and she'd smile and say
"So?"
we'd be sitting in some cheap nightclub
and the "So?" meant
go ahead
rip my clothes off now
but you won't do that—
so what are you going to do about it?
dear old Gertrude
a design in Sex
in dear old St. Louis
her quicksilver jumping up and down
inside my god-damned soul.
screwing her was like going to heaven
on a drunken trolley
but first it meant
a walk through the snow
watching her ride those haunches
like all the magic in the universe
on those high heels
and up to her vast bed flocked with the
toy animals—stuffed bears, giraffes, elephants, whatever—
all looking at us
and my sweeping them to the floor
and the biggest toy animal of them all
taking over
with those bastards on the rug
with their sawdust hard-ons
and dripping cotton tongues, ah
we rode all the way out and
never came back, really,
any of us.
## an empire of coins
the legs are gone and the hopes—the lava of outpouring,
and I haven't shaved in sixteen days
but the mailman still makes his rounds and
water still comes out of the faucet and I have a photo of
myself with glazed and milky eyes full of simple music
in golden trunks and 8 oz. gloves when I made the
semi-finals
only to be taken out by a German brute who should have
been
locked in a cage for the insane and allowed to drink blood.
Now I am insane and stare at the wallpaper as one would
stare
at a Dali (he has lost it) or an early Picasso, and I send
the girls out for beer, the old girls who barely bother to wipe
their asses and say, "well, I guess I won't comb my hair
today:
it might bring me luck." well, anyway, they wash the dishes
and
chop the wood, and the landlady keeps insisting "let me in,
I can't
get in, you've got the lock on, and what's all that singing
and
cussing in there?" but she only wants a piece of ass while
she pretends
she wants the rent
but she's not going to get either one of 'em.
meanwhile the skulls of the dead are full of beetles and Shakespeare
and old football scores like S.C. 16, N.D. 14 on a John
Baker field goal.
I can see the fleet from my window, the sails and the guns,
always
the guns poking their eyes in the sky looking for trouble like
young
L.A. cops too young to shave, and the younger sailors out
there sex-hungry, trying to act tough, trying to act like men
but really closer to their mother's nipples than to a true evaluation
of existence. I say god damn it, that
my legs are gone and the outpourings too. inside my brain
they cut and snip and
pour oil
to burn and fire out early dreams.
"darling," says one of the girls, "you've got to snap out of it,
we're running out of MONEY. how do you want
your toast?
light or dark?"
a woman's a woman, I say, and I put my binoculars between
her
kneecaps and I can see where
empires have fallen.
I wish I had a brush, some paint, some paint and a brush, I
say.
"why?" asks one of the
whores.
BECAUSE RATS DON'T LIKE OIL! I scream.
(I can't go on. I don't belong here.) I listen to radio programs
and people's voices talking and I marvel that they can get
excited
and interested over nothing and I flick out the lights, I
crash out the lights, and I pull the shades down, I
tear the shades down and I light my last cigar imagining
the dreamjump off the Empire State Building
into the thickheaded bullbrained mob with the hard-on
attitude.
already forgotten are the dead of Normandy, Lincoln's
stringy beard,
all the bulls that have died to flashing red capes,
all the love that has died in real women and real men
while fools have been elevated to the trumpet's succulent
sneer
and I have fought red-handed and drunk
in slop-pitted alleys
the bartenders of this rotten land.
and I laugh, I can still laugh, who can't laugh when the
whole thing
is so ridiculous
that only the insane, the clowns, the half-wits,
the cheaters, the whores, the horseplayers, the bankrobbers,
the
poets...are interesting?
in the dark I hear the hands reaching for the last of my
money
like mice nibbling at paper, automatic feeders on inbred
helplessness, a false drunken God asleep at the wheel...
a quarter rolls across the floor, and I remember all the faces
and
the football heroes, and everything has meaning, and an
editor
writes me, you are good
but
you are too emotional
the way to whip life is to quietly frame the agony,
study it and put it to sleep in the abstract.
is there anything less abstract
than dying day by day?
The door closes and the last of the great whores are gone
and somehow no matter how they have
killed me, they are all great, and I smoke quietly
thinking of Mexico, the tired horses, of Havana and Spain
and Normandy, of the jabbering insane, of my dear
friends, of no more friends
ever; and the voice of my Mexican buddy saying, "you won't die
you won't die in the war, you're too smart, you'll take care
of yourself."
I keep thinking of the bulls. the brave bulls dying every day.
the whores are gone. the bombing has stopped for a minute.
fuck everybody.
## A NICKEL
I.
It was a lazy day and a lousy day to work. It seemed that even the spiders hadn't thrown out their webs. And when I finally got to my job down at the railroad yards I found out that shithead Henderson was the new foreman.
I learned that the old Mexican, Al or Abe or somebody, had retired or died or gone insane. Too bad. Now Henderson was boss. The boys were matching pennies down by the barn when Henderson called me over.
"Gaines," he said, "Gaines, I understand you're somewhat of a playboy. Well, that's all right. I don't mind a little horseplay now and then, but we'll get our work done first and then we'll play."
"Just like recess at school, eh coach?"
Henderson put his face real close to mine. I put mine real close to his—
"Or haven't you been to school, Hendy?"
I could look right down into his red mouth and his frog jaws as he spoke: "I can tie the can to you, boy."
"Proving what?" I asked.
"Proving you are out of position."
Which was a pretty good answer, and a pretty good criticism: I was always out of position.
I took a nickel out of my pocket and flipped it to the cement where the boys were lagging to the line. They stood back stunned, looking from the nickel back to me. I turned around and walked the hell out of there. For good.
II.
I laid up in my room and studied the Racing Form for a couple of hours and knocked off half a bottle of left-over wine. Then I got into my 1958 Ford and headed for the track.
I wrote the morning line down on my program and walked over to the bar where I noticed a big blonde, about 35 and alone—well, about as alone as a big babe like that can get amongst 8,000 men. She was trying her damnedest to burst and pop out of her clothes, and you stood there watching her, wondering which part would pop out first. It was sheer madness, and every time she moved you could feel the electricity running up the steel girders. And perched on top of all this madness was a face that really had some type of royalty in it. I mean, there was a kind of stateliness, like she was beyond it all. I mean, there are some women who could simply make damned fools out of men without making any type of statement, or movement, or demand—they could simply stand there and the men would simply feel like damned fools and that was all there was to it. This was one of those women.
I looked up from my drink as if it didn't matter and as if she was just anybody, and as if I was a pretty jaded type (which, to tell the truth, I was) and said, "How you been doin', with the ponies, I mean?"
"All right," she said.
I'd expected something else. I don't know what. But the "all right" sounded good enough.
I was about half-gone on the liquor and felt I owned the world, including the blonde.
"I used to be a jockey," I told her.
"You're pretty big for a jock."
"210, solid muscle," I said.
"And belly," she said, looking right above my belt.
We both kind of laughed and I move closer.
"You want the winner of the first race? To kinda start you off right?"
"Sure," she said, "sure," and I felt that big hip-flank touch the side of my leg and I felt like I was on fire.
I smelled perfume, and imagined waterfalls and forests and throwing scraps of venison to fine dogs, and furniture soft as clouds and never again awakening to an alarm clock.
I drained my drink. "Try six," I said. "Number six: Cat's Head."
"Cat's Head?"
Just then somebody tapped me, I should say—rapped me hard on one of my shoulder blades.
"Boy," the voice said, "get lost!"
I stared down into my drink waiting for her to send this stranger away.
"I said," the voice got a little louder, "run along and play with your marbles!"
As I stared down into my drink I realized the glass was empty.
"I don't like to play marbles," I told the voice.
I motioned to the bartender. "Two more—for the lady and myself."
I felt it in my back then: what seemed to be the sure, superior nudge of a no doubt highly efficient switchblade.
"Learn," said the voice, "learn to like to play marbles!"
"I'm going right away," I said. "I brought my agate. I hear there's a big game under the grandstand."
I turned and caught a look at him as he slid into my seat. And I'd always thought I was the meanest-looking-son-of-a-bitch in the world.
"Tommy," I heard her tell him, "I want you to play a hundred on the nose for me."
"Sure. On who?"
"Number six."
"Number SIX?"
"Yes, six."
"But that stiff is 10 to 1!"
"Play it."
"O.K., baby, O.K. but..."
"Play it."
"Can I finish my drink?"
"Sure."
After a while I walked over to the two dollar window.
"Number six," I said, "once."
It was my last two dollars.
Number six paid $23.40.
I watched my horse go down into the Winner's Circle like I do all my winners, and I felt as proud of him as if I had ridden him or raised him. I felt like cheering and telling everybody he was the greatest horse that had ever lived, and I felt like reaching out and hugging him around the neck, even though I was two or three hundred feet away.
Instead I lit my cigarette and pretended I was bored.
Then I headed back to the bar, kind of to see how she took it, intending to stay pretty far away. But they weren't there.
I ordered a double backed by a beer, drank both, ordered up again and drank at my leisure, studying the next race. When the 5 minute warning blew, they still hadn't shown up and I went off to place my bet.
I blew it. I blew them all. And the woman and her boyfriend never showed. At the end of the last race I had 35 cents, a 1958 Ford, about two gallons of gas and one night's rent left.
I went into the men's room and stared in the mirror at my face in disgust. I looked like I knew something, but it was a lie, I was a fake and there's nothing worse in the world than when a man suddenly realizes and admits to himself that he's a phoney, after spending all his time up to then trying to convince himself that he wasn't. I stared at all the sinks and pipes and bowls and I felt like them, worse than them: I'd rather be them.
I swung out the door and stood there feeling like a hare or a tortoise or somebody needing a good bath, and then I felt her pressing against me like the good part of myself suddenly coming back with a rush. I noticed how green her dress was, and I didn't care what happened next: seeing her again had made it O.K.
"Where've you been?" she asked hurriedly. "I've been looking all over for you!"
What the hell is this? I started to say, you've been looking for me?
"Here comes Tommy!" She hesitated, and I felt her push something into my hand. Then she walked out, carefully, slowly to meet him. I jammed whatever it was into my pocket and walked out to the parking lot. I got into my car, lit my next-to-last cigarette, leaned back and dropped my hand into my pocket.
I unfolded 5 one hundred dollar bills, one fifty, 2 tens and a five. "Your half," the note said, "with thanks. Nicki." And then I saw the phone number.
I sat there and watched all the cars leave; I sat there and watched the sun completely disappear; I sat there and watched a man change a flat tire; and then I drove out of there slowly, like an old man, letting it hit me, little by little, and scared to death I'd run somebody over or be unable to stop for a red light. Then I thought about the nickel I'd thrown away and I started to laugh like crazy. I laughed so hard I had to park the car. And when the guy who'd changed his flat passed me and I saw his white blob of a face staring back I had to laugh all over again. I even honked my horn and hollered at him.
Poor devil: he had no soul.
## nature poem
you are 50,000 Light Years
running through my brain in
tracksuits or
you are like sitting in a bar
with enough money
with a good drink
and looking through the window
at the snow
you are the dead fish of miracle
moving
you are the love-god of ice cream
phantasy
you have diminished the screaming of
children as they drink my
blood
I think that you have killed landlords
wanting rent
and also bad
tigers
there is a white flower laying against
my screen
like a whore
like a cat
like a white flower
I could not go to work
tonight because I could not
stop living
and now I am lying in bed
looking at the white flower.
## warning
upon your darkened red mouth wild birds scream
and bowls of fish swim their jungles,
a China morning, a withered noon of axes and
witches;
you desire a man-plagued sun and strands of
fiber calling my name;
beware, I am not your silly husband,
I am your silly lover
and of all your silly lovers,
the last one here.
## answer to a note on the dresser:
the price of the sun is the tulip rotting black
and the prince on his knees
and a boy born without eyes
and a kitten without a bird,
nothing but twine
and waiting
and whores dipping hearts in poison,
and exhaust and exhaustion
and the bliss and the kiss of syphilis,
drag down the vines
the broken-foot bottles,
I keep saying
ha ha ha the giants
the giant sun
am I, the giant. our sun
tonight
without sun
your shoes alone without you in them
and I alone frying steaks and drinking beer
and listening to Wagner
the price of the sun,
the price of the sun,
and I don't give a damn if you never come back.
## you don't know
you don't know how good it
can get
being in a strange city,
nobody knowing who you
are,
coming in from the low-paying
job,
forgetting dinner,
taking off your shoes,
climbing onto the bed,
lights out
in that cheap dark
room
living with the roaches
or the mice,
hearing the crackling of
the wallpaper
or the rush of small
feet darting
across the floor.
lifting the wine bottle
there in the moonlight
or in the light of the
street lamps and the
neon signs,
the wine entering your
body,
the flare of your match
lighting a
cigarette.
you don't know how good
it can get
without women,
without a telephone,
without a tv set,
without a car.
with the bathroom down
the hall.
relaxed in the dark
hearing the voices of the
other roomers,
hearing pans rattling,
food frying,
toilets flushing,
arguments,
occasional
laughter.
you don't know
the names of the
streets,
who the mayor is
or how long you
will remain.
you will remain
until the next city,
the next room,
the next low-paying
job.
the mice will become
bolder.
one will come up on
the dresser,
climb up on the handle
of the coffee cup,
hang there,
looking at you.
you will get up and
approach the mouse.
you are the
intruder.
as you get closer
he still will not
move.
his eyes and your eyes
will intermix.
it is the clash of
centuries.
then he will leap
through the air
in the darkness and
be gone.
you will return to
the bed, smiling,
thinking, he's lucky,
he doesn't have to
pay the rent.
you will drink some more
from the wine
bottle,
then rise, take off your
clothing, stack it on
the chair.
you will sit up against
the pillow,
listening to the cars
passing below.
you will get up,
check the alarm clock,
see that it is set for
7:30 a.m.
then, foolishly, you'll
have to put your pants
on again
to make a bathroom
run.
the hall will be quiet
and empty,
the lights will be out,
there will only be
darkness under each
doorway.
the roomers are
sleeping.
your face
in the bathroom mirror
will grin at
you.
then you will walk
back to your room,
get the pants off
again, hang them over
the back of the
chair that is possibly
older than
you.
the last drink is
best, the last flare of
the match
lighting the last
cigarette.
you hold the match,
still burning,
up against the palm
of your right
hand.
long life line.
too bad.
then to stretch out,
the covers up
against your
neck.
warm covers.
rented covers.
covers of love.
the day seeps slowly
back through your
consciousness.
not much.
then, like the other
roomers, you are
asleep.
you are equal to the
side of a
triangle,
to a mountain in
Peru,
to a tiger
licking its
paw.
you don't know
how good it can be
until you've been
there.
## let not
let not the people be your
foundation,
not the young girls,
not the old girls,
not the young men,
not the old men,
not those in-between,
not any of these,
let not the people be your
foundation.
rather
build on sand
build on landfills,
build over cesspools,
build over graveyards,
build even over water,
but don't build on the
people.
they are a bad bet,
the worst bet you can make.
build it elsewhere,
anywhere else,
anywhere
but on the people,
the headless, heartless
mass
mucking up the
centuries,
the days,
the nights,
the towns, the cities, the
nations,
the earth,
the stratosphere,
mucking up the
light,
mucking up
all chance,
here,
totally mucking
it up
then
now
tomorrow.
anything,
compared to the people,
is a foundation worth
searching for.
anything.
## the death of a roach
...when the last fig falls and we are pruned from light,
our golden ladies gleaned of love—
infest us with the mercy
of stone.
calisthenic tempest, kingly pain
the flowers held kisses and blossoms
crackling with lightning power against our
pinioned brain; I watch the roach
as prophets of exile drink
and break their cups.
the grasses held long and green their secrets.
now, old ladies cassocked like monks
treadmill the slow poor stairs
bumping their angry canes: solatium! solatium!
and they close themselves in shawls
as the sun rallies new buds to color,
and they think...of onions and biscuits
(beautiful day, isn't it?)
(did you hear Father Francis? Sunday?)
the roach climbs
(the mirrors of love are broken)
blind yet begotten with life, a dedicated wraith
of pus and antennae.
I take him from his task
with a stab of a finger that wretches
like a stomach against the sick black twisted
death; no bandores here, or philosophical canvas to color
with tantamounts.
I hide him in some hasty packet and flush his ugliness away,
and above me in the mirror, consumed and listening there:
a crevice, a demon declaring his hand:—
all about me the old ladies cackle enraged, infirm
and bleeding
violate,
lepisma,
they attack my tired guts with canes and
pins,
with scrolls and bibles, with celebrations
of
witchcraft
they maim my brain with mercy until I fall witless and ill,
shouting
shouting roominghouses and grass,
shouting apes and horses,
shouting
flowers and kisses: the insects are
suspect—
man can only destroy himself.
## the unwritten
it's been months now: the most
horrible thing I have ever
felt.
and I might have avoided
it.
might have.
maybe not.
but I didn't and in a way I
couldn't.
it occurred more quickly than
I could respond.
I should have been more
able,
more ready.
and for some
what was a horror for me
might have been
trivial to them.
but I have never been
"them."
it's over now.
the pain of that should be
finished.
but it stays with me.
and that I did not act in
time to prevent it—
but that moment is
gone.
and
I truly hate myself
for the first
time.
I will never recover.
it comes back to me
again and
again.
and in its aftermath,
nothing will ever be
quite right
again—
walking down a
hill,
getting out of
bed,
common tasks,
celebrations,
just
happenings
are
reshaped
by that occurrence.
I was gored
by my own
stupidity.
it was an animal.
it was an animal,
caused by some
human
thing?
would that it was
human.
so I could have
considered it
trivial.
## right now
the party's over, the rooster is
crowing and they've called in
the dice, the dancing girls are
snoring, the mice are crawling
in the paper cups, the donkey is
pinned to the tail, the fable has
crawled away to die, love is
covered with dust, the temples
are empty, the bird has flown
the cage, the cage encloses a
midget heart weeping, the dream
has taken a dive and I sit
looking at my hands, looking at
my hands
empty of the sound of the
moment.
## the sheep
in centuries past
audiences at symphony concerts
were not afraid to act out their
displeasure at works which
offended
them.
in our time
I have either attended or
listened to
hundreds of concerts
and never have I heard an
audience
express even the mildest displeasure
with any
work.
have our musical artists improved
to such an
extent?
or is it the decay of courage,
the inability of the
mass mind to
reach its own
decisions?
not only in the world of
music
but in the other
world?
the next time you hear
a symphony concert
note
the obedient applause,
the death of the bluebird,
the shading of the sun;
the hooves of the horses from
hell
pounding on the barren
ground
of the human
spirit.
## piss
remember once I was sitting in this hotel
room when my woman came in drunk and said,
"Christ, I couldn't hold it, I had to piss in the
elevator!"
I was drunk too, I was barefoot and in
my shorts.
I got up and walked out the door and down
the hall and pushed the elevator
button.
it came up.
the door opened.
the elevator was empty but sure enough
there in the corner was the
puddle.
as I was standing there a man and a
woman came out of their place
and walked toward the
elevator.
the door was beginning to close
so I held it open with my hand
so they could get
on.
as the door began to close I heard the
woman say,
"that man was in his shorts."
and just as it closed I heard the man say,
"and he pissed in the elevator."
I went back to the room and told her,
"they think I pissed in the elevator."
"who?" she asked.
"people."
"what people?"
"the people who saw me standing
in my shorts."
"well, screw them," she said.
she was sitting there drinking a glass
of wine.
"take a bath," I said.
"you take a bath," she said.
"at least take a shower," I said.
"you take a shower," she said.
I sat down and poured a glass of
wine.
we were always arguing about
something.
## last fight
he's just a handler
now.
he's in the gym
watching the young
boxers spar.
he knows all the
moves,
watches the footwork,
the counter-
punching, the leads,
the hooks, the
timing, the
will.
he was a fighter
once,
went a number of
ten rounders.
now he watches
the action,
squinting,
analyzing.
he's got a gut
now
it bulges out
under his old
sweat shirt.
it's an afternoon
in the gym.
he can hear them
grunt,
he can hear the
shots, the
big gloves
landing.
inside his head
he can see
himself in the
ring,
he can hear the
screams of the
young girls
again,
the yelling of
the men,
he can feel the
lights,
the canvas
under his feet,
the ropes
squaring him
into
battle.
son-of-a-bitch,
what a
time,
son-of-a-bitch,
what a
life!
then he returns
to reality.
son-of-a-bitch,
he's old.
he's got a bucket
and a
towel.
well, it beats
sucking buttermilk
through a
straw.
the rounds are
finished,
something else
now waits.
yeah.
there'll be
no more split
decisions for
that
son-of-a-bitch.
## defining the magic
a good poem is like a cold beer
when you need it,
a good poem is a hot turkey
sandwich when you're
hungry,
a good poem is a gun when
the mob corners you,
a good poem is something that
allows you to walk through the streets of
death,
a good poem can make death melt like
hot butter,
a good poem can frame agony and
hang it on a wall,
a good poem can let your feet touch
China,
a good poem can make a broken mind
fly,
a good poem can let you shake hands
with Mozart,
a good poem can let you shoot craps
with the devil
and win,
a good poem can do almost anything,
and most important
a good poem knows when to
stop.
## writing
often it is the only
thing
between you and
impossibility.
no drink,
no woman's love,
no wealth
can
match it.
nothing can save
you
except
writing.
it keeps the walls
from
falling.
the hordes from
closing
in.
it blasts the
darkness.
writing is the
ultimate
psychiatrist,
the kindliest
god of all the
gods.
writing stalks
death.
it knows no
quit.
and writing
laughs
at itself,
at pain.
it is the last
expectation,
the last
explanation.
that's
what it
is.
## views
my friend says, how can you write so many poems
from that window? I write from the womb,
he tells me. the dark thing of pain,
the featherpoint of pain...
well, this is very impressive
only I know that we both receive a good many
rejections, smoke a great many cigarettes,
drink too much and attempt to steal each other's
women, which is not poetry at all.
and he reads me his poems
he always reads me his poems
and I listen and do not say too much,
I look out of the window,
and there is the same street
my street
my drunken, rained-on, sunned-on,
childrened-on street,
and at night I watch this street
sometimes
when it thinks I am not looking,
the one or 2 cars moving quietly,
the same old man, still alive, on his
nightly walk,
the shades of houses down,
love has failed but
hangs on
then lets go
as the tomcats chase it,
but now it is daylight and children
who will some day be old men and women
walking through last moments,
these children run around a red car
screaming their good nothings,
then my friend puts down his poem...
well, what do you think? he asks.
try so and so, I name a magazine,
and then oddly
I think of guitars under the sea
trying to play music;
it is sad and good and quiet.
he sees me at the window.
what's out there?
look, I say,
and see...
he is eleven years younger than I.
he turns from the window: I need a beer,
I'm out of beer.
I walk to the refrigerator
and the subject is closed.
## the strong man
I went to see him, there in that place in
Echo Park
after my shift at the
post office.
he was a huge bearded fellow
and he sat in his chair like a
Buddha
and he was my Buddha, my guru,
my hero, my roar of
light.
sometimes he wasn't kind
but he was always more than
interesting.
to come from the post office
a slave
to that explosion of light
confounded me,
but it was a remarkable and
delightful
confusion.
thousands of books upon
hundreds of subjects
lay rotting in his
cellar.
to play chess with him was
to be laughed off the
board.
to challenge him
physically or
mentally was
useless.
but he had the ability to
listen to your
persiflage
patiently
and then the ability
to sum up its
weaknesses,
its delusions in
one sentence.
I often wondered how
he put up with my
railings; he was kind,
after all.
the nights lasted 7,
8 hours.
I had myself.
he had himself
and a beautiful woman
who quietly smiled as she
listened to
us.
she worked at a drawing
board,
designing things.
I never asked what and
she never
said.
the walls and the ceilings
were pasted over
with hundreds of odd
legends,
like the last words of
a man in an electric
chair,
or gangsters on their
death beds,
or a murderer's instructions
to her children;
photos of Hitler, Al Capone,
Chief Sitting Bull,
Lucky Luciano.
it was an endless honeycomb
of strange faces
and
utterances.
it was darkly refreshing.
and at odd rare times
even I was interesting.
then the Buddha would
nod.
he recorded everything on
tape.
sometimes on another
night he would play a
tape back for
me.
and then I would
realize how pitiful, how
cheap, how
inept I sounded.
he seldom did.
at times I wondered why
the world had not
discovered
him.
he made no effort to be
discovered.
he had other
visitors,
always wild, original
refreshing
folk.
it was crazier than the
sun burning up the
sea,
it was the bats of hell
whirling about the
room.
that was decades ago
and he is still
alive.
he made a place when
there was no
place.
a place to go when all
was closing in,
strangling, crushing,
debilitating,
when there was no
voice, no sound,
no sense,
he lent his easy
saving
natural
grace.
I feel that I owe him
one,
I feel that I owe him
many.
but I can hear him
now, that same
voice
as when he sat
so huge
in that same
chair:
"Nothing is owed,
Bukowski."
you're finally wrong,
this time,
John Thomas, you
bastard.
## the terror
the terror is in viewing the human
face
and then hearing it talk
and watching the creature
move.
the terror is in knowing its
motives.
the terror is in seeing it
skinned,
opened
for the internal view of the
spirit.
the terror is looking at the
eyes.
the terror is knowing of the
centuries of its
doings.
the terror is the unchangeability
of it.
and its multiplicity,
its duplicity, it's
everywhere, a giant mass
of it
self-revered,
self-serving,
self-destructive,
the terror of no selves
spreading from here and now into
space,
cluttering the universe,
marring pure space,
poisoning hope,
raping chance,
going on,
this massive zero of
life
labeled
Humanity.
the terror, the
horror,
the waste of them
and you and
me
through and
through.
## the kiss-off
it was one of those
half-ass
literary gatherings
and this girl dropped to her
knees on the rug and
said to
him:
"O, Mr. C., let me kiss
that thumb
that great amputated thumb
that appeared in that great American novel
On the Road!"
Mr. C. held out the amputated thumb
and she kissed
it
and we all came
all around all
around, we all came all
around.
## betting on the muse
Jimmy Foxx died an alcoholic
in a skidrow hotel
room.
Beau Jack ended up shining
shoes,
just where he
began.
there are dozens, hundreds
more, maybe
thousands more.
being an athlete grown old
is one of the cruelest of
fates,
to be replaced by others,
to no longer hear the
cheers and the
plaudits,
to no longer be
recognized,
just to be an old man
like other old
men.
to almost not believe it
yourself,
to check the scrapbook
with the yellowing
pages.
there you are,
smiling;
there you are,
victorious;
there you are,
young.
the crowd has other
heroes.
the crowd never
dies,
never grows
old
but the crowd often
forgets.
now the telephone
doesn't ring,
the young girls are
gone,
the party is
over.
this is why I chose
to be a
writer.
if you're worth just
half-a-damn
you can keep your
hustle going
until the last minute
of the last
day.
you can keep
getting better instead
of worse,
you can still keep
hitting them over the
wall.
through darkness, war,
good and bad
luck
you keep it going,
hitting them out,
the flashing lightning
of the
word,
beating life at life,
and death too late to
truly win
against
you.
## THE UNACCOMMODATING UNIVERSE
Carl sat at the end of the bar where he wouldn't have to deal with anybody. He kept his head down and didn't look at anybody. He was on his second drink, a vodka-7. Then he heard two girls behind him talking. He hadn't heard them walk in.
"Well, we can't sit at the bar," one said, "no two empty stools together."
"Maybe we can get a table?"
"No, the tables are full..."
"Shit."
"Well, let's go someplace else."
"No, this is where the action is!"
Carl felt a finger explore under and around his collar. Then he felt it tickle his ear. One of the girls giggled. Carl didn't move. Then he said, without looking around, "Didn't we know each other in Toledo?"
"Athens, Georgia," came the answer. The finger withdrew.
"I'm Toni," one of the girls said.
"I'm Cristina," said the other girl.
"I'm Carl," said Carl, still not looking around.
"Could you move down one stool?" said Toni. "We can't find a place to sit together."
"Too fucking bad," said Carl.
He drained his drink and nodded Blinky the Barkeep in for a refill.
"Blinky," said Carl, "I need a ticket to the Laker's game."
"When?"
"Tonight."
"I'll see what I can do." Blinky walked off.
Toni leaned against Carl, pressing her breasts against his back.
"Tell us something about yourself," she said.
"I've got AIDS."
"Bullshit!"
Toni pulled away.
"Hey, we don't have to fuck around with this asshole! There are plenty of NICE men around here!"
"Yeah, he's an asshole!" Cristina said.
The girls walked down to the other end of the bar. They were in their mid-twenties, well-dressed. Toni was the redhead, Cristina was the blonde. They had nice buttocks, were slim-hipped, long of leg. They had bright healthy eyes, clever smiles. They were...attractive.
They stood behind Barney the Hump, talking to him.
Then the phone rang. Blinky answered it and then brought the phone down and placed it in front of Carl. Carl picked it up.
"Hello?"
It was Rissy. Rissy was crying.
"I gotta see ya, Jesus, I gotta see ya!"
"Rissy, there is nobody you got to see unless it's a shrink."
"The son-of-a-bitch beat me, Carl! I'm all bruises and lumps, I can't go out on the street!"
"Good. You need a rest."
Carl hung up. He went for his drink. The phone rang again. Carl winked at Blinky and picked it up.
"Lion's Nuts Bar."
She was still crying. "I gotta see ya, don't ya understand? Don't ya have no compassion?"
"Our marriage has been annulled. I like the sound of that word: ANNULLED."
He hung up.
There was a scream down at the end of the bar. It was Toni. Then Carl saw the girls moving briskly back toward him and the exit. They stopped at his stool. Toni stood in front and Cristina stood behind her as they faced Carl.
Toni was in a fury. "THAT SON-OF-A-BITCH SLAPPED ME! NO SON-OF-A-BITCH SLAPS ME! NO SON-OF-A-BITCH SLAPS TONI EBERT! NOBODY! NOBODY! I NEVER SEEN A BAR SO FULL OF ASSHOLES! YOU GUYS FAGS? ARE YOU AFRAID OF WOMEN? OR ARE YA FUCKIN' STUPID?"
"We're just fuckin' stupid," somebody said.
"YOU CAN SURE AS SHIT SAY THAT AGAIN!"
"We're just fuckin' stupid," somebody said again.
Blinky walked down to the end of the bar.
"Girls, I'm sorry..."
"SORRY AIN'T ENOUGH, ASSHOLE. I'M GOING TO HAVE THIS DUMP TRASHED!"
"I thought it already was," said Blinky.
"YOU PRICK!"
The girls turned on their heels and were gone into the night.
Blinky walked up to Carl. He slid the Laker's ticket at him.
Carl reached for his wallet. Blinky waved him off and walked down to Barney the Hump.
"Why'd you slap that girl, Barney?"
"WHY? HEY, WHY, HUH? WHY, HUH?"
"Yeah, why?"
"That whore stuck her finger in my ear!"
"What's the matter? You got a problem with that?"
"I just don't like girls who jerk me around," Barney said with a grin.
The phone rang again. Carl picked it up.
"Lion's Nuts..."
"I'll kill myself, that's what I'll do, I'LL KILL MYSELF!"
"No chance," said Carl and hung up.
The hardest thing about life, he thought, was dealing with other people's problems. You could be consumed with other people's problems: they were always having car crashes or going mad or forgetting to pay the rent, or they left the butter out, fucked strangers, had insomnia, or—if they slept—had unhappy dreams. And they never considered the fact that you had your own miseries to unravel. Ah, well...
Carl nodded Blinky in for another refill.
"You gonna make the game?" Blinky asked.
"Sure. I always arrive late to beat the traffic and leave early to beat the traffic."
"Why go at all?"
"What do you want me to do? Sit around and listen to Chopin?"
"Carl, those two girls were fine looking. How come you passed?"
"I don't know. Fucking to me is like shaving. I guess it's something I have to do now and then but I feel like putting it off."
"You getting old?"
"Maybe just wise. You know, fucking is nature's idea."
"A good idea, I think."
"Yeah, but overrated."
"You're putting me on..."
Blinky moved off...
It was maybe ten minutes later that the girls came back. They stood just inside the door. And in front of them stood their pimp. Big and dark. But he was different than most. He wasn't one of those slick pimps. He wasn't dressed to shine. He had on an old overcoat and heavy workman's shoes. He was very big with a razor scar curling down the left side of his face. He looked like a good natured guy who could get very mean and he looked ready to get very mean.
"Gentlemen, I hear my girls have been having some trouble in here."
Nobody answered.
"It makes me unhappy when somebody makes one of my girls unhappy. And I don't like them or me to be unhappy."
Blinky moved forward a bit, then stopped.
"Listen, man, it was just a mistake. One of those things, you know."
"No, I don't know."
The pimp just stood there.
He stood there and stood there. It was very quiet. The girls waited behind the big guy. It was an agony of tension. Every small sound could be heard. The dripping of the bar faucet, the slight hum of the electric clock and the almost soothing sound of the street traffic.
Then Mickey the Bookie, the drunkest of them all, sitting at bar center said, "Yeah. So shit. What ya gonna do?"
The pimp moved at once. He moved in behind Mickey before Mickey could react. Mickey was working on a draft beer. His glass was half full. The pimp took the glass and spilled the contents on the bar.
"What I'm going to do, I'm going to do. But the first thing YOU'RE going to do is lap that up!"
"Kiss my ass," Mickey said.
Mickey had on a blue Dodger's baseball cap. The pimp flipped it off, grabbed Mickey by the hair and then he had the razor at his throat.
"Get it! Lap it up! Every last motherfucking drop! NOW!"
He pushed Mickey's head down and Mickey's tongue came out. He began lapping at the bar.
"Hey, man," said Blinky, "you..."
"SHUT UP!"
The pimp held Mickey's head down and Mickey's tongue worked up the beer. Then he let him go. He stepped back. Mickey straightened up and lit a cigarette. The cigarette trembled in his mouth. He inhaled, then exhaled a pitiful curl of smoke.
"You guys," said the pimp, "got to learn that my ladies are real ladies and must be treated accordingly. They offer a service that keeps mankind contented and I don't want them pushed around."
Carl turned on his stool.
"All right, whatever we did, it's done. Maybe it was wrong. It probably was. We're sorry for that. But you're making too much of it."
"I'll decide what's too much," the pimp said. "I intend to see that this kind of shit doesn't continue."
"So what are you going to do?" asked Carl, looking at the razor in his hand. "Kill somebody? You want somebody's balls in a sack?"
"I wouldn't mind that, I might arrange that."
"Come on, Jason," said Toni, "let's get out of here. We don't need any more. We don't need this shit."
The pimp nodded her off.
"I want to know which guy hit my woman. Now, whoever hit my woman, I want him to speak up."
There was silence.
"You might as well speak up. All I gotta do is ask my woman."
There was more silence. Barney the Hump drained his drink and stood up.
"I hit your whore. She stuck her finger in my ear and messed with me and if she did the same thing again I'd hit her again."
"Mister," said the big pimp, "it's evident your mother never taught you manners."
The pimp moved forward. Barney the Hump squared off in front of the crapper. Barney missed with a right as the pimp came in and they both crashed through the crapper door. It splintered like balsa wood. There was a scramble in the crapper and the pimp came out holding Barney in a death grip. He spun him once, then lifted him and threw him across the bar and into the bar mirror. The mirror shattered, bottles fell and smashed as Barney fell behind the bar and lay motionless, face down. Then a full quart of gin came sailing from somewhere and caught the pimp behind the ear. He staggered a moment, then righted himself.
Then he roared, "I'LL GET ALL YOU MOTHERFUCKERS!"
Patrons were running out the front and out the back. The big pimp had his razor out and he sliced through the motion, sliced part of an ear from Mickey the Bookie. Suddenly the lights went out. The girls screamed, ran. There was the flash of a gunshot and the pimp dropped his razor and grabbed his belly.
"Christ, you chickenshit..."
Carl ran out the back way and into the alley and then out of there and west down 6th Street. People were just strolling along and he slowed to a fast walk. He circled the corner and went down to where his car was parked. He got in, kicked it over, looking back at the bar. Nobody was coming out of there. Then the pimp walked out. He looked powerful in the early night. He stood there a moment like a man looking for a cab. Then he fell forward not able to put out his hands to break the fall. His head hit first, bounced, then he was still. Carl drove off to the sound of an approaching siren.
Carl unlocked the door, put the chain on and flicked on the light. Rissy was sitting on the couch. There was a half-a-fifth of scotch on the table and Rissy was drunk, hair down in her face. She was smoking a king-sized cigarette, a red glow on the end of it. She coughed.
"Hey, where ya been, lover boy? Out fuckin'?"
"Christ, what are you doing here?"
"I wanna talk. I told you he hit me! I wanna talk!"
Carl sat down, took a hit straight from the bottle.
"There's nothing to talk about."
"Hey, that's been our PROBLEM, lover boy! We never talked about things!"
"We don't have any problem. Our marriage is annulled."
Carl sat to her left. She reached out a hand, touched him, and as she did so she spilled some of her drink in her lap. The long glowing cigarette was in her mouth and she smiled around it.
"Hey, what do you think? I'm NEVER going to let you go! It's love! True love!"
"Ah, shit," said Carl. He lifted the fifth and had another hit.
Rissy put her cigarette out in the ashtray, tossed off her drink, filled it again, lit another cigarette.
"That son-of-a-bitch beat me up, can you imagine? That son-of-a-bitch BEAT me!"
"What did you do? Were you screwing around?"
She looked at him, hair still down in her face. Her speech was slurred. She sat with her cigarette in one hand, her drink in the other:
"What's THAT got to do with it? You don't BEAT people! People have their rights! Don't ya think?"
Carl didn't answer. He picked up a cigarette and the lighter. He bent over the lighter, flicked it. The flame was too full. As he lit the cigarette he burned his nose.
"God damn it," he said.
Rissy reached out and touched him again.
"Whatsa matter, honey?"
Then she picked up the remote control, switched on the tv set and they both sat waiting for the screen to come to life.
## met a man on the street
who said, "you've kept me going for two
years, it's really amazing to meet you."
"thank you," I answered, "but who's
going to keep me going?"
I've asked this question before and
all I ever get back is a gentle
smile.
but it's a good question.
they have no notion that I may consider
suicide several times a
week.
they've read some of my books
and that's enough for
them.
but I only write that stuff,
I can't read
it.
## hell is now
the sun was rather diminished,
the dog came in low,
11:32 a.m.
Wednesday in the year of
our Lord,
all the man heard was the
low gurgling growl,
then the beast had ripped
his thigh,
it was summertime,
the scream parted the
air,
the beast
pirouetted,
leaped powerfully,
sailed toward the
man's
throat,
flowers grew in the
flower beds,
the lawn was newly
mowed,
the man threw up
his hands
against the bared
fangs,
shrank away,
the beast bounced
off,
landed on all
fours,
the small finger
of the man's
right hand
in his
mouth.
the dog stood
dumbly,
then dropped the
finger.
it was a majestic
and beautiful
animal.
its fur rose
along its back
and about the
neck.
it began circling
the man
rapidly.
"JESUS CHRIST!
JESUS CHRIST,
HELP ME!"
two men came
running from the neighboring
back yard.
one was fat and
bald
with a face like
an owl.
the other was
thin with a very
white face
with a large
birthmark,
purple-black,
shaped like a
walnut.
"BRIGGS!" they
yelled,
"BRIGGS!
STOP THAT!"
Briggs paused, then
trotted off into the
back yard.
the man held his
hand
up against his
chest
and covered it
with his
other hand.
the man was
sobbing, sobbing
choking
sobs.
"I'll KILL that
fucking dog!
I'll KILL both
of you!
what's the matter?
are you CRAZY?
ARE YOU
CRAZY?"
then the fat man
with the face
like an
owl
saw something
on the
lawn.
he walked over
and looked down
at it.
it was the
finger.
"what's this?"
he asked.
"what's this?"
an old man on a
bicycle rode past
on the sidewalk
he was in red
and white shorts,
wore goggles
and a yellow
helmet.
on the back of
his sweat shirt
it said,
MEAT ME,
BABY.
he rode on
by.
it was 11:39 a.m.
in the year of our
Lord.
## the kid
had trouble hitting left
handers so I got him to
switch hit,
then I shifted him from
left to center,
dropped him from
lead-off to the 6th
spot,
also had him work
on the bunt.
I had long talks with
him about his
career,
told him that
concentration was
essential.
I worked hard with
the kid,
had him take
extra batting
practice,
had him switch
to a lighter
bat,
work on
contact,
the power would
come by
itself.
I had him stand
closer to the
plate,
be more
selective at
what he
swung
at.
I worked hard
with the
kid,
played him
every day
but his average
dipped to
.229 and I had
to ship him
to the
minors.
all that talent
and he couldn't get
it
together.
he acted confused,
disoriented.
my guess was
it's some
broad.
poor bastard.
all that
natural talent
shot to
shit.
I've seen it
happen so many
times.
well, I've got
Sunderson out
there now.
he's hitting
.289,
lots of line
drives,
he's adequate
in the
field,
steady.
we oughta be
right in the
race,
come
September.
## "To Serve and Protect"
there were two policemen on motorcycles.
there was a policelady and a policeman
from a squad car.
the car was angled crosswise in the
driveway to the parking lot
of the cafe.
one policeman was calling in
downtown.
there was a man about
23.
he was facing the wall of a
building.
he was obviously an
indigent.
his clothes were greasy and
ill-fitting.
and he had shit his
pants.
the stain was showing
through the back.
he was not cuffed
and he was not directly
facing the
wall.
he was turned a little to
one side,
peeking at his
captors.
the police seemed to be
hardly
watching him.
they were
indifferent,
talking among
themselves.
it was a beautiful winter
afternoon.
I walked past the scene
on the way to the
cafe.
as I did, the lady policeman
gave me a hateful look
that said, buzz off, this is
none of your
business.
it was and it
wasn't.
I went into the cafe and had
lunch.
when I came out
everybody was
gone
and it was still a
beautiful winter
afternoon.
poor bastard had shit his
pants.
my car hadn't been
stolen.
I got in and drove
off.
## bad day
the jellyfish has a purpose,
the hyena,
the tick,
the rat,
the roach
each filled with their
swollen
light.
my light is
out.
who did this to
me?
## the dick
I was sitting in my office in the dark
not thinking about much, well, maybe a little about
the Barker caper
when the door opened real slow—
I was not expecting any visitors—
I slid my hand slowly into my pocket and fondled
my 45.
hell, it's a dame, a looker, dressed to kill, she's rocking
there on her high heels and long legs, one garter belt
showing through her slitted dress. She said, lighting a
cigarette, "remember me?"
"well, no," I said. "I've got a metal plate in my head and
I drink too much vodka."
"cut the crap," she hissed like a tigress, "we made love 7 times
the night before last!"
"you cut the crap," I told her. "I haven't had it up since Gettysburg."
I saw her reach into her purse, I saw the glint of metal in the moonlight
coming through the dirty Venetian blinds.
I tossed the vodka bottle at her head quicker than you can say
better to cheat on your wife than on your income tax.
I got up, walked around, bent over the vodka bottle, it was all right,
no breakage.
then I looked at her, out, cold and beautiful.
I began to get ideas.
I lifted her dress.
but then she opened her eyes.
"you killed Eddie," she said.
"who?" I asked.
"Eddie Blankenship."
"wait," I told her, "I'm Eddie Blankenship."
"Christ," she said.
"No, not him. Eddie Blankenship."
I went around behind the desk, uncapped the vodka and had a
good hit.
the whole thing didn't make sense.
she sat in a chair, crossing her legs high.
"I'll solve the case for you," I told her, "but I don't come cheap."
"money's no object," she said, her pure gold earrings shining in
the moonlight coming through the dirty Venetian blinds. "I'm Marcy Peats
Booty the 3rd, I've got billions."
"20 bucks," I said.
"you're on."
she threw back her beautiful head and laughed.
I fondled my gun under the desk.
"well, it's like this," I told her. "I couldn't have killed Eddie
Blankenship because I'm him."
I paused.
"so!" she smiled a smile that would melt a steel
gate.
"so," I said, "there have to be TWO Eddie Blankenships."
"sounds like crap to me," she said.
"baby," I said, "crap don't sound. I'm giving you the frigging facts."
just then the door swung open and here stood this ragged looking guy,
no class, not much ass, not much of anything.
"I'm Eddie Blankenship," he said.
"well, suck a rabbit's tits," I said.
"hello, Eddie," the doll said to him.
"hi, baby," he said, "what's this punk doing sitting behind my desk?"
"your desk and what Army's desk?" I snarled.
"me and the Canadian Royal Mounties!" he snarled.
"that's no Army!" I yelled.
"your mother's armpits!" he screamed.
he reached.
I reached.
two bright roaring flashes.
his bullet bounced off my steel plate.
he crumpled.
I went around, bent over him, took his wallet, then felt his
pulse.
I looked up at her.
"this man is dead," I told her.
"you killed Eddie Blankenship!" she screamed.
I saw her reach into her purse.
there was another bright roaring flash.
she pitched forward off her chair.
I bent over her.
in her right hand was a fingernail file.
I emptied her purse.
I felt her breasts, her legs.
I felt her pulse.
she was dead.
I walked around behind my desk, had a hit of vodka and
sat there.
the moon came in through the dirty Venetian blinds.
I had 2 dead bodies and half-a-bottle of
vodka.
it was time to do some thinking.
I was in some hell of a jam.
I had to do something.
I reached for the phone.
I got the operator.
I asked her to connect me with my mother in
Iowa City.
collect.
then I sat there listening to the phone
ring.
## fall of the Roman Empire
car on its side in the moonlight,
wheels toward the sky still spinning.
a man crawls out of the broken window
of the door.
he is wearing a white shirt splotched
with blood.
inside the car the radio is still playing
loudly.
the man walks across the street, sits down
on
the curbing.
he was on his way to pick up a girlfriend
for dinner.
he will be late, very late.
in fact, there will be no dinner.
the wheels have stopped spinning.
it was just one of those things which
happen
like the fall of the Roman Empire.
somebody puts a blanket around the man.
he asks for a cigarette, gets one.
somebody lights it for him, he inhales,
exhales.
then the ambulance is there.
the police cruiser.
"he ran a red light," said the man to the cop.
"I hit the brakes,
clipped his rear end and somehow flipped.
that son of a bitch."
"he left the scene?" the cop asked.
"yeah," said the man, "the son of a bitch."
the people stood off a little in the distance,
staring.
their night had become interesting.
all of them were glad they weren't the man
sitting on the
curbing.
it was better than tv.
"you been drinking?" asked the cop,
"I smell liquor."
"I had a few beers..."
"how many?"
"2...3..."
it was getting interesting.
the car radio was still playing.
bad rap music.
a boy of about 6 started dancing to
the music.
two ambulance drivers walked up.
one of them needed a shave.
the one who needed a shave asked
the cop,
"can he walk or will he need a stretcher?"
"can you walk to the ambulance?"
the cop asked
the man.
"sure," he said.
he stood up and began walking toward the
ambulance.
he took a misstep, seemed to twist to
the right,
then lost his balance and fell.
he hit the street hard.
his head bounced up once, then fell back.
he was still.
it looked ugly.
the ambulance driver who needed a shave
knelt down over him.
it was a hot July night in a decent
neighborhood.
then the radio in the car stopped.
a few of the people turned and walked off.
they had seen enough.
the others waited
in the brilliant and lovely
moonlight.
## people
look at the people: elbows, knees,
earlobes, crotches, feet,
noses, lips, eyes, all the parts
usually clothed, and they are
engaged
in whatever they usually do
which is hardly ever
delightful,
their psyches stuffed with
used matter and propaganda,
advertising propaganda, religious
propaganda, sexual propaganda,
political propaganda, assorted
propagandas, and they
themselves are
dull and vicious.
they are dull because they have been
made dull and they are
vicious because they are
fearful of losing what they have.
the people are the biggest
horror show on earth,
have been for
centuries.
you could be sitting in a
room with one of them
now
or with many of
them.
or you could be one
of them.
every time the phone
rings or there is a knock on
the door
I'm afraid it will be one of
the disgusting,
spiritually destroyed
useless
babbling
ugly
fawning
hateful
humans.
or worse, on picking up the
phone the voice I hear
might be my
own,
or upon opening the
door
I will see myself
standing there,
a remnant of the
wasted centuries,
smiling a
false smile,
having learned well,
having forgotten
what I am here
for.
## RANSOM
Marty drove up the unpaved lane, parked the car and got out. He walked to the small run-down house, opened the door and walked into the kitchen. The Kid was still tied to the chair. Kell was reading an old copy of Playboy. Marty sat down and looked at the Kid across the table. Then he got up, went to the refrigerator and got a beer. He looked at the Kid, "You got a tight old man, Kid, I've heard that rich guys are tight, tighter than a virgin."
Kell put the magazine down, "What happened?"
"What happened? The old bastard said 'no' and hung up. Just like that. He likes his money better than his bloodlines. This is his only son."
"Maybe we ought to ask for less."
"Shit, no. I asked for two million and at two million it stays."
"What are we going to do?" asked Kell.
"We're going to get rough. We're going to cut off one of the Kid's ears and mail it to the old man."
"Suppose he don't pay then?"
"Then we send the other ear."
"Listen, fellows," said the Kid, "I..."
"You shut up," Marty said.
"Listen," said Kell, "I don't like to go around cutting people's ears off."
"I'll do the cutting."
"Suppose he don't pay after two ears?"
"Then we send his balls."
"Listen," said the Kid, "just..."
"Shut up! I've got to cut off your goddamned ear tonight. Do you think I like doing that sort of thing?"
"Let's not do it, Marty."
"We've got to. We don't have any choice. Untie the Kid's hands and give him a beer."
The Kid rubbed his wrists where the rope had bound him. His legs were still tied. He lifted the beer.
"I'm sorry, Kid," said Marty. "I told your old man that we were going to lop off one of your ears if he didn't pay up. Know what he said?"
"No."
"He said, 'Go ahead.' Now you might kind of say we got his blessing."
"Dad never cared much for me."
"We're going to have to shame him into caring. We'll ship him your eyeballs if necessary."
"You two guys are worse than my old man! You're bloody filthy cowards!"
"Maybe so. And your old man's tight with his money. So you're caught in the middle."
"It's hard to believe that there are people as cold as you bastards are!"
"There are. We're just two of them. There's plenty more, plenty. All members of the human race."
"Isn't there some other way out?" asked Kell, "I don't want to see the Kid lose his ear."
"Get me and the Kid another beer. You're too soft. How'd you ever get into this business?"
"I don't know, Jesus, I just kind of looked around and I was in it. I started with the numbers racket in Philly and..."
"All right. That's enough history. One way or the other the Kid's ear goes—tonight."
"You're a chickenshit bastard!" said the Kid.
"Now is that any way to talk to a man who has given you two free beers?"
"Fuck you, you swine!"
"You live in a country whose president was murdered during your lifetime and then whose brother was murdered. You live in a country where people are afraid to walk the streets after dark. Taking one of your ears just about fits the scene."
"It doesn't take any guts to do that."
"Who's talking about guts? If I had any guts I'd be a linebacker for the Chicago Bears. All we want is a little advantage, an edge, something like two million bucks."
Then they were all quiet. Kell got up and got himself a beer. He twisted the cap off and sat down. "This is a nice little place up here in the hills. I'd like to live here instead of always being on the fucking run."
"Yeah. But even with that million in your sock, Kell, you're still going to have to keep running."
"Yeah, but the women will be better."
"Women are all pretty much the same inside. What you call a better woman, well she just has a better facade. It doesn't mean that much."
"I'll take that better facade."
"We're going to have to sterilize a butcher knife."
"How we going to do that?"
"On the stove. Over the flame. We gag the Kid and lop it off. Zip! It will be over fast."
"Could he bleed to death?"
"He's not that lucky."
"Do you think we really stand a chance to get that ransom?"
"A damn good chance but we're going to have to make some tough moves. For two million you've got to do a few extra things."
"I still don't like it. It makes me sick to think about it."
"Kell, I'm not as hard as I pretend to be either. Get me another beer."
"Shit, let's not do it."
"The old man is calling our hand. We've got to. We've got no choice."
The Kid bent his head down on the table. He vomited. It was mostly the beer but there were bits of undigested food.
"Now, Kid, that wasn't nice. That was really unsanitary. But you're scared so I'll forgive you."
Marty got up and found a dish towel and cleaned the tabletop.
"Tie his hands again. Let's get this fucking thing over with!"
"You pricks," said the Kid, "you chickenshit pricks!"
"And gag him so I don't have to hear that dirty language."
Marty walked to the drawer and found the butcher knife. He walked to the stove and turned a burner on. He held the knife over the flame.
"We can go to South America, Kell. We can live there the rest of our lives. Some of the Nazis went down there after the war and they've never found them. A man can pay for protection just like he pays for pussy." He turned the knife over the flame. "And you're right. I'll take that better facade too. I've been to bed with some real hags."
He took the knife from the flame. The Kid was fully tied and gagged. He walked around behind the Kid so he wouldn't have to look into his eyes. He took the left ear gently between his fingers and pulled it away from the Kid's head. "Hey Kell, hold this son-of-a-bitch still!"
The knife was still pink from the heat. He held it between the ear and the skull. He held it there. Then he threw the knife into a corner of the kitchen, hard. It clattered and bounced, then was still.
"Shit! I can't do it! Come on, let's get the hell out of here!"
Marty walked right out of the kitchen and Kell followed him. They walked through the front room and out the front door and to the car. They got in and Marty backed it out of the drive, took a left on the unpaved lane that led down out of the hills. He looked at Kell. "Got a cigarette?" Kell dug out the pack, pulled out two, lit them both and handed one to Marty.
"Thanks, I'll let the old man know where the Kid is as soon as we get a few hundred miles away. And don't say a fucking thing to me. I don't want to hear a fucking thing out of you!"
It was 9:30 p.m. It was September. The gas tank read full. Marty turned on the radio. Of all things it was Ray Charles. Marty winced. Kell didn't say a fucking thing.
## it's difficult for them
some university profs
find me crude, crass, obvious,
repetitive and pornographic
and I often am,
I sometimes deliberately
am
but this should not concern
them,
they have their friends, their
compatriots, their peers
writing the poesy
which they find
admirable.
but why they rage
against me
in their critical essays
is what I find
strange.
now, I don't like their work
either,
find it pale,
contrived, overworked
and a century behind the
times
but
I don't attack them
critically,
I just stop reading them
and I don't hate them,
I don't care how many books
they publish or who does or
doesn't read
them.
yet, they are very concerned
about my existence
and my large readership,
and almost hysterically
upset
that in some places
I am accepted as an
original writer of some
power.
I tend to ignore this, why
can't they?
if they want their place in
literary history,
fine, they can have
it,
I don't give a damn.
all I want to do is
my work
anyway
I choose to do it,
all I want is the next line
and the line after
that.
what they do and who they
are and what they want
and what they say and what
they write
has no interest for me
and, unfortunately for
them, no interest to most others
living, dying or about to be
born, uh
huh.
## think of it
think of it, there were fellows like
Kierkegaard and Sartre
who found existence
absurd,
who battled against
anxiety and anguish,
nothingness,
nausea,
and death hanging over them
like a
Damocles sword
while there are other men
now
so empty of concern
that their first thought of the
day is
when are they going to have
lunch?
granted, it could be more
comfortable
to live, say, as a fly, an
ant, a mugwump,
but as a human,
just think,
as a human
to live
thusly,
as millions do
again and again.
of course, hell is other
people,
the waste, the waste,
all flushed away
like
it, like
that.
the garage mechanic
walking toward you
with dead
eyes.
## chicken giblets
he's like you, she said, he locks himself in
his basement room and he doesn't want
to see anybody.
I want you to meet him.
I don't want to meet him, I said.
we were driving south down Western.
I want some chicken giblets, she said.
god damn it, I said.
what's the matter? she asked.
I want a drink, I said.
well, I want some chicken giblets,
she said.
I pulled into an all-night drive-in,
opened the door, gave her some
money and she went to the
counter and ordered.
it was 3 in the morning.
she stood there eating her chicken
giblets.
two men walked up.
she started talking to them.
she was smiling.
then they all were laughing.
she had finished eating her
giblets
they kept talking and
laughing.
5 minutes, I thought.
then I looked at my watch.
after 5 minutes I backed my car out of
there and drove off.
I was sitting back in my apartment
having scotch with a beer
chaser when there was a knock
on the door.
I got up and opened it.
it was her.
what the hell happened to you?
she asked.
nothing, I said.
well, pay the cabby, she
said.
there was a cabby standing
behind her.
yeah, he said, pay me.
hey buddy, I said, step closer.
he did.
yeah, he said.
go fuck yourself, I said.
hey, man, he said, I gotta get paid!
I didn't ride in your cab, buddy.
but she's yours, he said.
she's not mine, I said.
whose is she then? he asked.
you take her.
I closed the door.
about ten minutes passed.
there was a knock on the door.
I opened it,
it was her.
she pushed her way in.
gimme a drink, she said.
pour your own, I said.
she did.
she sat in a chair with her drink.
my brother stole my purse,
she said, he took all my
money.
he's on drugs, I said.
so am I, she said.
it was another 3:45 a.m. in
east Hollywood
and the black sky came in like a
knife
and if you were alive you were
lucky
and if you were dead
you never knew
it.
## the lover
at that apartment in east Hollywood
I was often with the hardest numbers
in town.
I don't speak as a misogynist.
I had other people ask me,
"what the hell are you doing, anyhow?"
they were floozies, killers, blanks.
they had bodies, hair, eyes, legs,
parts
and often it was like
sitting with a shark dressed in a
dress, high heels, smoking, drinking,
swallowing pills.
the nights melted into days and the days
collapsed into nights
as we babbled on, sometimes
bedding down, badly.
because of the drink, the uppers, the
downers, I often imagined
things—say, that this one was the
golden girl of the golden heart and
the golden way of laughter and love
and hope.
in the dim smokey light the long hair
looked better than it was, the legs
more shapely, the conversation not as
bare, not as vicious.
I fooled myself pretty well, I even
got myself to thinking that I loved
one of them, the worst one.
I mean, why the hell be negative?
we drank, drugged, stayed
together through sunset,
sunrise, played Scrabble for 8
or ten hours at a
stretch.
each time I went to piss she
stole the money she needed.
she was a survivor, the
bitch.
after one marathon session
of 52 hours of whatever we
were doing
she said, "let's drive to
Vegas and get married?"
"what?" I asked.
"let's drive to Vegas and
get married before we
change our minds!"
"suppose we get married,
then what?"
"then you can have it any
time you want it," she told
me.
I went in to take a piss
to let her steal the money
she needed.
and when I came out I opened
a new bottle of wine
and spoke no more of the
subject.
she didn't come around as
much after that
but there were others.
about the same.
sometimes there were
more than one.
they'd come in twos.
the word got out that
there was an old sucker
in the back court, free
booze and he wasn't
sexually demanding.
(although at times something
would overtake me and I
would grab a body and throw
in a sweaty horse copulation,
mostly, I guess, to see if
I could still do it.)
and I confused the mailman.
there was an old couch on
the porch and many a morning
as he came by I'd be sitting
there with, say, two of them,
we'd be sitting there,
smoking and
laughing.
one day he found me alone.
"pardon me," he said, "but can
I ask you something?"
"sure."
"well, I don't think you're
rich..."
"no, I'm broke."
"listen," he said, "I've been
in the army, I've been around
the world."
"yeah?"
"and I've never seen a man with
as many women as you have.
there's always a different one,
or a different pair..."
"yeah?"
"how do you do it?
I mean, pardon me, but you're kind
of old and you're not exactly a
Casanova, you know?"
"I could be ugly, even."
he shifted his letters from one hand to the
other.
"I mean, how do you do it?"
"availability," I told him.
"what do you mean?"
"I mean, women like a guy who is always
around."
"uh," he said, then walked off to continue his
rounds.
his praise didn't help me.
what he saw wasn't as good as he thought.
even with them around there were unholy periods
of
drab senselessness, despair,
and worse.
I walked back into my place.
the phone was ringing.
I hoped that it would be a female
voice.
## no win
to live in a jungle
where each face is a face of
horror,
where each voice grates,
where bodies walk
without grace,
where the only communion
is between the dead and
the dead.
to live in a place
where empty faces
and common bodies
win
beauty contests.
to live in a place
where being alone
is always better than being
with someone.
to live a lifetime
with just your
fingernails
more real than
the multitudes,
to roll a 7 in hell
with nothing in the
pot,
that's what this life
is.
## THE STAR
He sat in the garden chair watching the birds dig into the freshly watered lawn. He was James Stagler, 81, ex-movie star. He was remembered for his major roles in such epic movies as Skies Over Bermuda, The Brooklyn Kid, Son of the Devil, A Big Kill, and The Ten Count. Those were his principal films, although he had appeared in hundreds of others and had also starred in a Broadway musical, Kickin' High.
"Lunch!" He heard the woman's voice, and he rose slowly from his chair, made his way gingerly across the lawn toward the house. James entered from the yard door and walked to the dining room table. He still somewhat resembled the leading man from the 1940s, except his hair was white and his eyes seemed to have disappeared into his face. His eyes stared out as if he was hiding within himself. As he neared the table the woman, Wanda, screamed at him:
"For Christ's sake, how many times have I told you to wipe your feet? Now, take your shoes off and put them outside!"
James did as he was told. Then walked back to the table in his stocking feet, sat down. Wanda had come to his 75th birthday party one evening with some of his friends and she had simply stayed. Now he didn't see much of his friends anymore. Wanda, who was 34 years younger, now handled his social affairs and his financial affairs. There had been sex between them at first but that had stopped years ago. James sat down to a plate of eggs and fried potatoes. Wanda sat across from him with a glass of sherry and lit a cigarette. She glared at James.
"Christ, I couldn't sleep last night! You were snoring again! I don't know what I'm going to do!"
The phone rang. It was there on the table next to Wanda. Wanda always answered the phone.
"Yeh? This is the James Stagler residence. You're talking to Wanda Bradley, Mr. Stagler's agent. No, you can't speak to Mr. Stagler. What do you want? An interview for what magazine? What do you pay? I thought so, we don't give unpaid interviews."
Wanda banged the phone back into the cradle, glared at James again.
"Don't put so much butter on your toast! How many times do I have to tell you?"
James wasn't hungry. He liked to eat when it was quiet. It was seldom quiet. The phone rang again. Wanda snatched it up as if she were angry at it.
"Yes? Oh, Mr. Stanhouse. Listen, I told you, 500 grand if you want him in your movie...yes, I know it's a cameo role! No, you can't speak to Jimmy! Yes, he's all right, he's fine, I see to that! Now, if you agree to the 500 thousand, bring over the papers and we'll dust him off."
Wanda put the phone down again, took a drink of her sherry.
"Eat your eggs! I didn't cook them for nothing!"
"I don't want to eat, Wanda."
"Eat those eggs!"
"No!"
"God damn you!"
Wanda stood up. She took her napkin and slapped James' face once and then again, hard.
James looked down into his plate of uneaten eggs. He spoke softly.
"I want you out of my house. I don't want you here..."
Wanda just stood there. Then she laughed.
"Why, you old fuck! After all these years of taking care of you, you think I'm just going to walk out of here?"
"I'll give you the money..."
"You'll give me the money? I handle the money around here."
"I don't want you here..."
Wanda walked around the table and stood over him.
"Why you big baby! That's what you are, a big baby!" she laughed.
"I hate you," he said.
"You hate me, you ungrateful old man? Who cuts your hair, your toenails, pays your bills? Who makes your dental appointments? Who protects you from people? Who washes the shit out of your shorts? Who feeds you? You'd be dead in a week without me!"
James sat there over his eggs as Wanda stood there.
"I want to die," he said, "I don't care anymore..."
"No use dying, old boy, you can still make us some money. I know Stanhouse is going to give us that half million. And all you have to do is say a few lines, or mumble a few. Anyhow, if you die now, you'll only go to hell."
"This is hell..."
"Yeah, for me. Now, Jimmy, I'm telling you for the last time. Eat those eggs!"
James hated those eggs. They were dry and burned. He only felt like eating when he felt good and Wanda just stood there not understanding how or why he felt like he did. When he had first met her she had seemed so nice. She had laughed at everything he said, she had sat with him in the projection room while they watched his old films and she had said, "You were really better than Brando and a hell of a lot more man!" After his four wives and his endless girlfriends, Wanda had finally seemed the answer. But it had changed, it had changed all around.
He picked up the plate of eggs and threw them on the floor.
"I won't eat these eggs!"
Wanda stepped back a moment. She was a large woman with straight black hair, cut short. She stiffened and she smiled.
"Well, well, well. Look here, we have a bad boy here today, a very bad boy!"
Wanda walked over and finished off her sherry. Her cigarette had gone out. She lit her cigarette. Then she walked to the kitchen closet. She came back with a whisk broom, a dustpan and a wastebasket. She stood over James with them and then suddenly threw them at him. They struck him, then clattered to the floor.
"Now!" she said, "you clean up that mess!"
James just sat there staring at the table. She stood over him. He could feel her there. Like something impossible. A pain gripped his throat, then his head. He sat there.
"Well," she said, "get going!"
Still, he sat there.
"Well, I'm not going to wait much longer!"
Then he said it:
"Go to hell!"
"What? What did you say?"
"I said, go to hell!"
Wanda leaped on him like a leopard. His chair fell backwards. She had a grip on his head and they rolled on the floor. She was partly on top of him, an arm locked around his head. Her strength surprised him. He could hardly breathe, but he could hear her:
"You old fool, you don't know the misery it's been living with you..."
James couldn't breathe. It was getting worse. He felt that it was over for him and he didn't mind that except somehow he really resented it that it was at the hands of Wanda. Then he saw the fork on the floor. Then he had the fork in his hand and he plunged it into her back as hard as he could. Wanda screamed and leaped up. James scrambled to his feet. Wanda stood there trying to reach the fork in her back, screaming. It was in a place that she couldn't quite reach with either hand. She looked awful with that fork stuck in there and the blood coming down. Then she stopped screaming and just looked at him. She had the look of an animal in a trap.
"It's not going to kill you, Wanda," he said, "it's just a fork."
"Pull it out, Jimmy!" she commanded.
She turned her back to him and he stared at the fork sticking out there. It was firmly in place and the blood was flowing. He was surprised at all the blood. The blood made Wanda real again. It was like when they first met: she was human after all.
"Pull it out, Jimmy!"
"I will, Wanda, if you will promise me something..."
"Just pull the fork out!"
He looked at the fork in her back. He remembered how they used to make love. How every day was a good day. How it felt so good to care for somebody again and how it felt so good to be loved again. How everything had seemed funny, there were so many things to laugh at. Why did it go away? He had never wanted it to go away.
"You've got to promise me something..."
"All right, I promise! What is it?"
"If I pull the fork out will you go away and leave me alone?"
"I promise! Now pull it out!"
James grabbed the fork with both hands and pulled.
"Christ," he said, "it's really in there!"
"Pull, you son-of-a-bitch! You're the leading man, you're the movie star, remember?"
James remembered his movies and it gave him strength. The fork came out and he had it in his hand and he looked at it. Wanda whirled, furious, grabbed the fork and they stared at one another. Then she suddenly plunged it into his stomach. She pulled it out and jammed it in again and pulled it out. James fell to the floor holding his gut.
"Now we're even," he said helplessly, looking at her.
"You senile asshole!" she screamed. "I always hated you and your movies!"
She moved over him and jabbed the fork at his face. She pulled it back as he grabbed at his mouth with both hands. She stuck the fork into his stomach again. She leaped on him and rolled him over screaming, "I hate you, I hate you!"
Once more she jammed the fork into his stomach, pulled it out. Then she stopped. James lay very still, not looking at her, almost not breathing. She dropped the fork, got up and walked back to the table, sat down. She then saw his plate, his eggs, his potatoes on the floor. When she saw that, the anger left her. Her eyes became very wide and almost beautiful. With a rush a sudden remorse came over her. It was odd. Now, she cared for him. He had been a strange and a wonderful and famous man. He had gotten old. But that wasn't his fault. Now she didn't want the money. She only wanted him alive. She wanted him there with her. Far off she heard a dog barking. That dog was alive. When something was alive it was unique, exceptional, no matter what the circumstances.
Wanda inhaled, exhaled, very conscious of doing so. She didn't dare think of James.
The dog barked again.
She took the bottle, poured another sherry. She drank it down. She looked around. It was a beautiful house.
The phone rang. Wanda picked it up.
"Hello?"
It was Stanhouse. Stanhouse said it was okay about the half million. He was ready to come over with the papers when James could see him.
"I'm sorry, Mr. Stanhouse," Wanda said, "we've talked it over and James has decided to give up acting."
She hung up quietly.
Off in the distance the same dog barked again.
## an evaluation
I've seen 70,000 horse races
and often
like this afternoon
as the horses slowly approached
the gate,
I thought, this is insanity,
I am murdering the hours,
I am tearing my heart out and
stamping on it with my
feet,
this is a madhouse,
this is towering stupidity,
this is death laughing at
me.
this is just another 8 hour
job.
they put them in the gate,
the sun came down,
a bell rang and they broke
from the gate
and were off down the
track,
and I thought, does it
really matter?
where's the glory here?
it's just repeat and
repeat and
repeat,
the grinding hours,
the routine.
it was a
business,
it was a
fake.
the game was getting old,
I was getting old.
they came around and into
the stretch,
the son-of-a-bitch, it was
the 7 horse, my horse,
drawing away at about 9-to-one.
I had a ten on it.
it paid $90.20.
I decided to stay for one more
race.
what would I do at home
at 3:30 in the
afternoon?
sleep?
I strolled toward the
payoff
window.
a fellow had to keep his
hand in the
action.
## neon
today at the track they gave
all the patrons
neon caps.
the caps glowed and
said
HOLLYWOOD PARK.
some of those jerk-offs
wore their caps
backwards.
25 thousand neon
heads.
faces of
greed.
stone
faces.
faces of
horror.
blank wall
faces.
idiot eyes
under
neon.
fat white
stupefied
husbands and
wives.
Oakies with
blond hair.
screechers
preachers
poachers
punks...
left-overs,
half-dead,
part
warm.
neon
neon.
cement
faces.
blithering
voices.
nothing.
neon over
nothing.
I thought I was
in hell.
maybe I was in
hell.
a day-glow
inferno of
festering
hell.
## they think this is the way it's done
he saw me walking into the track and he stood
waiting, he was a jockey's agent and I only knew
him slightly
but then he moved toward me,
"Hey, Hank, I want to ask you something..."
I stopped.
he said, "Listen, I know this fellow, he's a friend
of mine and he writes poems, really wonderful
poems..."
"I can't help him," I said and began walking
off.
"Yes, you can, all you have to do is to get on the
telephone!"
"No, I can't..."
I walked further off.
"WHAT IS HE SUPPOSED TO DO THEN?" he yelled.
"SEND HIS WORK TO A GOD-DAMNED PUBLISHER!" I
yelled back.
then I was up the escalator and that was
over.
if I ever owned a horse I would never use one of
his jocks.
meanwhile, I checked the tote.
my selection was reading 5-to-one.
nice way to right a day that had started
wrong.
## the pile-up
the 3 horse clipped the heels of
the 7, they both went down and
the 9 stumbled over them,
jocks rolling, horses' legs flung
skyward.
then the jocks were up, stunned
but all right
and I watched the horses
rising in the late afternoon,
it had not been a good day for
me
and I watched the horses rise,
please, I said inside, no broken
legs!
and the 9 was all right
and the 7
and the 3 also,
they were walking,
the horses didn't need the van,
the jocks didn't need the
ambulance.
what a beautiful day,
what a perfectly beautiful day,
what a wondrously lovely
day—
3 winners in a
single race.
## 12 minutes to post
as we stand there before the purple mountains
in our stupid clothing, we pause, look
about: nothing changes, it only solidifies,
our lives crawl slowly, our wives deprecate
us.
then
we awaken a moment—
the animals are entering the track:
Quick's Sister, Perfect Raj, Vive le Torch,
Miss Leuschner, Keepin' Peace, True to Be,
Lou's Good Morning.
now, it's good for us: the lightning flash
of hope, the laughter of the hidden gods.
we were never meant to be what we are or where
we are, we are looking for an out, some music
from the sun, the girl we never found.
we are betting on the miracle again
there before the purple mountains
as the horses parade past
so much more beautiful than
our lives.
## as the poems go
as the poems increase into the thousands you
realize that you've created very
little.
it all comes down to rain, the sunlight,
the traffic, the nights and the days of the
years, the faces.
leaving this will be easier than living
it.
typing one more line now as
a man plays a piano through the radio.
the best writers have said very
little
and the worst,
far too much.
## the telephone
many women I have known have
been very much connected to
the telephone.
they can talk virtually for
hours.
it is their manner of
measuring where they
are or are
not.
some women have major
problems with aging
and with
men.
on the telephone they
speak of
real and imagined
injustice,
they let loose their
poison,
they justify their
beliefs and
positions.
my wife has been
speaking to one of her
gender
back east.
the conversation is
now proceeding
into its second
hour.
if a psychiatrist or
a psychologist
were listening
their notes would be
bulging with
references to
trenchant
instability and
gratuitous masturbation
of the
psyche.
but I am neither psychiatrist
nor psychologist.
I am just the poor son-of-a-bitch
who has to pay
the
phone bill.
a misogynist who
writes these
poems.
## HIDEAWAY
Harry walked into the bar and found a stool alone. Nobody on either side of him. The bartender dragged his bloated body up and Harry ordered a scotch and water. The barkeep waddled off. He was wearing dark brown pants. His butt was wide, gross. Harry stared at the sagging buttocks, watched the wrinkles in the back of his pants. Then Harry glanced around. Nothing but lonely middle-aged guys who wanted to talk about the Rams or the Dodgers or something equally senseless.
The bartender came back with the drink. Harry paid him but the bartender kept standing there. He was wearing a faded blue t-shirt with a hole near the left shoulder. He leaned against the bar and his belly flopped over the wood. He kept looking at Harry and Harry could hear him breathing.
"What do you want?" Harry asked him.
"I wanna welcome ya to the Hideaway." The bartender grinned through his greasy lips.
"Thanks," said Harry.
The bartender reached under the bar and came up with a wooden cup. He grinned foolishly at Harry, shook the container up near his ear, lowered it and flipped out a pair of dice. "All the boys," he said, "are going to roll to see who buys the next round of drinks. Low number buys. You wanna join us?"
The conversation in the bar stopped. The juke box was silent. Harry noted that most of the patrons were dressed in dirty white t-shirts. Some of them were skinny, with long thin arms and the t-shirts hung from them like dirty rags. Others were fat or muscular and the t-shirts gripped them snugly, creeping up toward their armpits leaving their hairy bellies and bellybuttons exposed. One guy was dressed in a heavy jacket that was much too large for him. They all seemed to be waiting for his answer.
"No," Harry said, "count me out."
The barkeep turned and waddled back down to the guy at the end. They whispered a moment, then the bartender turned his head and looked back at Harry. The look was noncommittal. The first guy rolled the dice.
Harry belted his drink down.
The barkeep was moving from man to man. There was a high sense of glee in the place as each man rattled the container and spilled the dice out.
I wonder if a woman ever comes in here? thought Harry.
"Hey, barkeep!" Harry hollered.
The barkeep looked at Harry.
Harry raised his empty glass, winked, "How about a refill?"
The barkeep looked at Harry, inhaled, held it, then let it slowly come out. As he waddled toward Harry he snatched a bottle of scotch as if irritated. Then he stood there, pouring. Some of the scotch ran over his fat brown fingers as it poured into the shot glass. He dumped the shot in, added the water, then said to Harry, "You know, we got a great place here, everybody knows each other, everybody gets along."
"What do I owe you?" Harry asked.
"Same as before."
The barkeep took the money, made it down to the register, banged it open, slammed it shut. Then he went back to the dice. He moved along the bar, announcing the results of each roll. Finally he came down to the last patron, the guy dressed in the large jacket.
"Now, David," said the barkeep, "all ya gotta do is beat a 4, because Pee Wee threw a 4. Roll 'em, David!"
David rattled the dice in the wooden cup and let them go.
"Holy shit!" screamed the barkeep, "SNAKE EYES!"
It busted up the whole bar: fat guys and thin guys started whooping it up and beating on the wood. One guy got going so bad he started to gag, couldn't get his breath. He bent over the bar and they beat on his back until he could breathe again.
Then it got quiet and the guy in the jacket reached into his wallet and flipped out some bills.
"It's all right," he said, "next time somebody else will be whistling Dixie out of his butthole."
The barkeep went about pouring refills. One of the fellows, one of the very thin ones, got up and put some money into the juke box. It was a song about "Bette Davis' Eyes."
"That Bette Davis, she was some woman," said one of the fellows.
"She's still alive," said another.
"Oh yeah?"
"She still was some woman."
"Yeah, but there was something evil about her."
"She was still a great actress."
"Maybe so."
The barkeep walked down to Harry, stood there.
"You all right?" he asked Harry.
"Yes."
"You had a fight with your woman?"
"Not really."
"What do you mean?"
"Nothing."
"I got to tell you something, mister. We don't like unhappy people around here. We get along."
"I'm not unhappy."
"Then what is it?"
"What do you mean?"
"I mean, you don't seem to be a friendly fellow."
"I'm sorry, I didn't mean to give that impression."
"We get along here. We all know each other."
"How about another drink?"
The barkeep waddled off, came back with the bottle: "You know, we don't want trouble here. We're all peaceful people."
"O.K.," said Harry, "only this time don't add so much water."
"O.K.," said the barkeep, "by the way, what do you do?"
"What do I do? Right now, I'm drinking."
The barkeep leaned back a little from the bar.
"HEY FELLOWS!" he yelled.
All the white t-shirts looked toward them, plus the big jacket.
"I asked this gentleman what he did and you know what he told me? He said he drank!"
One of the white t-shirts applauded. The others joined in.
"All right!" one of them yelled, "He's one of us!"
The barkeep leaned toward Harry: "You play pool?"
"No, I was never any good at pool."
The bartender leaned closer. His belly was almost crawling across the bar and into Harry's drink.
"What're you good at?"
Harry laughed. "Well, hell, I guess I just don't excel at anything."
The bartender leaned closer: "Where you from? Newark? Kansas City?"
"Santa Fe."
"Wow! Santa Fe!"
The barkeep leaned back and raised his walrus head: "HEY YOU GUYS, THIS GUY IS FROM SANTA FUCK!"
The fellows didn't seem to pay so much attention to that.
The barkeep leaned forward again. "How come you came to this bar tonight?"
"No real reason. Give me a refill."
The barkeep poured it right into the glass, forgetting the water.
Harry drained the glass.
"O.K., I had a fight with my woman."
"You told me earlier that you didn't have a fight with your woman."
"I said, 'not really.'"
"What's that mean?"
"I mean, not really."
"So you just came in here because there was nowhere else to go?"
"I'm not knocking your place. I just didn't feel like going right home tonight."
Then the barkeep leaned back and stood there. He didn't look at Harry. He appeared to be looking at some place over Harry's head and to the left. He seemed to be in a reverie.
Then he leaned forward, leaned against the wood and looked at Harry.
"You been in the service?"
With that question it seemed as if the entire bar became very quiet.
"You mean the armed forces?"
"Yes."
"No."
"Everybody here's been in the service. Except for Pee Wee. He was too small."
Harry didn't answer.
The barkeep reared back and looked at the same spot over Harry's head. Then he leaned forward again.
"How come you didn't go?"
"I don't know. I guess I fell somewhere between Korea and Vietnam. I was never the proper age. Besides, what does it matter?"
The barkeep's stomach left the wood and he stood almost straight.
"Hey fellows!" he said in a loud voice. "HERE'S A GUY WHO SAYS ALL THE WARS WE FOUGHT IN DIDN'T MATTER!"
"He's got a pussy for brains," said one of the white t-shirts.
"All right," said Harry, "I'm leaving."
He got off his stool and started walking toward the rear exit. His car was in the parking lot back there. He was feeling all right. The drinks had helped.
As he neared the end of the bar, one of the white t-shirts stuck out a foot and tripped him. Harry lost his balance and almost crashed into the pinball machine. But he slammed his palms against the glass and righted himself.
Harry turned and walked over to the man who had tripped him. The man had nice blue eyes. On one of his thin arms was tattooed the message: BORN TO DIE. On the bar in front of the man stood half a drink. Harry reached over, picked the drink up, pulled the fellow's t-shirt open at the neck and poured the drink in.
He was drunker than he thought. He found the car, got the key, opened the door, got in, locked the door and here they came. The white t-shirts and the big jacket. The bartender was not with them.
Harry started the engine. They were all over his car like a swarm of drunken killer bees. Two were on the hood. One was on the roof. Two were attempting to roll the car over.
Harry put it into reverse and slowly backed out toward the alley. Several of the drunks were now pushing against the rear of the car. In the rearview mirror Harry saw one of them fall under the wheels. He hit the brakes and rolled down the window on the driver's side.
"Jesus Christ, get out of the way!"
A long thin arm came in through the window and tried to pull the keys from the ignition. Harry took the arm and bent it hard against the steering wheel. He heard the snap, there was a scream and the arm vanished back out of the window. Harry rolled the window up and continued backing out.
He backed and made a left turn toward the boulevard. There was a face pressed against the windshield, eyes leering in. He saw the hands, their fingers, clutching at the glass, frog-like useless things. Harry knew that once he was on the boulevard he could shake him free.
He roared up the alley. The man fell off the hood. At the last moment he spotted the sacrificial lamb, a fat white t-shirt spreading its arms and blocking the alley exit. Harry veered to the right, ramming a brown slat fence. The fence broke apart. There were slats and pieces of wood flying everywhere...
Harry got back to his apartment, took off his clothing, his shoes. He sat there in his shorts for a few minutes and then walked to the refrigerator. Luck: 4 cans of beer left. He cracked one, brought it out and sat back down on the couch. He flicked the remote control, he got Johnny Carson.
Now, thought Harry, there is a man. If the whole world was like Johnny Carson there might be a chance.
Then he thought, that's wrong, Carson gets along too well with just anyone. He likes everybody.
Harry swallowed the last gulp of that can of beer and then the phone rang.
It was Lisa.
"Where have you been? I've been phoning you for hours! Where have you been?"
"Nowhere, really."
"You've been with some slut! I'm a woman! Women have a way of knowing these things! You've been with some slut!"
Harry hung up the phone, took the thing off the hook.
He had three cans of beer left.
With them and if he was careful he might make it to morning.
## this dirty, valiant game
I see e. e. cummings drinking a
rum and tonic while sitting on
the front porch of a white
house.
I see Ezra at St. Liz
accepting visitors as a confirmation
of his existence.
I see Hart Crane on an
ocean steamer
rejecting the advances of
literary ladies while
lusting for the cabin
boy.
I see Hemingway cleaning
his shotgun
while thinking of his
father.
I see Dostoevsky at the
roulette wheel
losing everything to
Christ.
I see Carson McCullers
dunking her beautiful
soul
in
whiskey.
I see Li Po
that wino
laughing at the
futility of word
following
word.
I see Sherwood
Anderson
swallowing the
toothpick that killed
him.
I see William
Saroyan
written-out,
sitting in his Malibu
beachfront home
waiting
vainly
for the luck to
return.
I see Timothy
Leary
going from
table to table
at parties
hoping to be
recognized.
I see Chatterton
purchasing the
rat poison,
I see Pascal
getting into the bath-tub
of warm water
with the
razor.
I see Ginsberg
gone
from Howling to
mewing
as a professor in
Brooklyn.
I see Henry
Miller
long stopped
writing,
putting advertisements
in a
college newspaper
for
secretaries.
I see Richard
Brautigan,
the age he high-
lighted past,
his books no
longer selling,
his love affairs
rotting, I can
see him blowing
himself away in
that mountain
cabin.
I see the
necessity of
creation, the love
of it, the danger of
it.
I can see where
creation often
stops while the
body still lives
and often
does not care
to.
the death of life
before life
dies.
Tolstoy sitting alone
in the
road.
all days night
forever.
flowers frozen in
blood
urine
wine.
## stay out of my slippers, you fool
it's not good, some of the days we have, horrible
dead-dog-in-the-
street days.
son-of-a-bitch, going on sometimes seems rather
useless.
read in the paper the other day,
a man fell into a meat grinder and was ground
up.
makes you think a bit about the gods.
like some things seem almost planned, worked out
on some
drawing board.
it's fate, they say.
this man was born to die being ground to bits in
a meat
grinder.
that was his main purpose.
they allowed him to do a few things first.
he'll be replaced.
somebody will take his job.
somebody will take your job
and mine.
your place and mine.
and the trees will shed their leaves
and the whores will sing in their showers
and the cats will sleep throughout the day
and the 20th century will click into the 21st
and somebody will throw away your shoes
and your belt and your old clothes and your
new clothes.
somebody will sleep in your bed.
somebody will throw a handful of dirt upon
you.
I get like this when I read about a man being
ground to death in a meat
grinder.
how do you feel?
what do you know?
get the hell out of my face!
## the voice
we had a table outside
by the water,
it was a Saturday night,
all the tables were
filled.
we had finished eating,
we were drinking and
watching the freighters
and passenger ships
going by on their way to the sea
and Frankel was
talking.
I became very
conscious of his loud
voice.
I wasn't too
interested in what
he was saying
and neither were
the others,
but Frankel kept on,
he even got
louder,
he laughed, waved
his hands;
little pieces of
saliva flew from
his mouth.
heads were turning,
looking at us.
Frankel had been
told
in some distant
past
that he had a
great sense of
humor,
that he should
have been a
stand-up
comic.
he had 3 or 4
good lines but
we had
heard them all
before.
I finished my
drink, set it
down, managed
to reach out,
grab one of
Frankel's
flying hands.
I interrupted him
in mid-speech.
"listen, your voice,
can you lower it
just a
bit?"
"huh?
oh sure..."
then he went
on.
he kept it low
for some
moments,
then,
something he
was saying
excited him,
and he was
back at full
volume.
we paid the bill
and got him
out of
there.
going back
Frankel
was in another
car
following us.
"I hope I didn't
hurt his feelings,"
I said to my
wife.
"I was about to
tell him
myself," she
answered.
back at our
place
Frankel
began talking
again.
there were 4
other people
and we
listened.
it wasn't so bad
because we
all knew him
and the house
was set far
back,
not too close
to the
neighbors.
but we had
6 cats and they
all ran off,
out through the
door,
or they jumped
out of the
window.
the night went
on and Frankel
expounded loudly upon
the strange and
funny things in
his life, what
he said to
somebody and
what they
replied.
he used different
voices for the
different
people.
well, the night
finally wound
down
and we said
goodbye to
Frankel and his
friend
at the doorway.
they both said
they had had
a good
time.
then they were
in their car
and backing
out the
drive.
we sat down
for a quiet
nightcap.
the silence was
glorious.
it seeped through
us and we began
to recover.
then the cats
returned
one by one,
looking around
cautiously,
lifting their feet
delicately.
life was returning
to normal.
nobody said
anything.
enough (had been)
said.
## the bard of San Francisco
don't old poets ever
die?
this one fellow,
you can see him every
morning
in the coffeehouse
at his own table
sipping a white wine and
reading The New York
Times.
then he'll go down to
the pool for a
swim.
they say he has the most
beautiful blue eyes in
America.
he dashes off on little
trips to Paris and
Madrid,
then returns.
he still gives poetry
readings, reads
well, has no fear of
his audience.
he can impress them,
does, just for something
to do.
he is not embittered,
refuses to
gossip.
he wears all manner
of hats, caps, head
gear,
and whatever he
puts on,
he never looks
ridiculous.
rather, he looks
dashing, he looks
like royalty.
he's thin, he's
straight, he's
tall,
and if the sun is
shining anywhere,
it shines on
him.
and his books
still sell,
handsomely.
the male poets
talk about him,
they use much of
their time
talking about him
and
rather
unkindly.
the lady poets
adore him.
and the other
ladies
adore him.
he is often seen
with a new
woman.
he is very composed
about it
all.
and with death
looking over his
shoulder
he still manages
to write
decent
poetry.
## on biographies
if you're dead
they don't
matter.
most biographers,
of course,
imagine things
about their
subjects
that aren't
true.
worse, they take
your jokes as
fact
and the other
way
around.
and in interviewing
ladies from your
past
they will accept
their
pronouncements
without
question.
biographies
about writers
are mostly
tomes of literary
gossip.
and if it is about
a living writer,
by then
he is often
almost physically
dead
and
in most cases
absolutely
spiritually
dead.
he will accept any
amount of praise,
ignore any
criticism,
congratulate his
biographer
on a job
well
done
and wonder
what
took them
so god-damned
long
to do
it,
anyhow.
## a real break
I've heard it said that you
give a real lively
performance
and there really isn't
much going on
in this
town,
so we'll fly you
down,
put you up in a nice
hotel,
you can have
all you want to
drink,
we can rent this
hall,
it holds a real
bunch,
and you'd be
surprised
how many people
around here
know about
you,
we'll pack them
in
and we promise
you
25% of the
gate.
we love you,
man!
how about
it,
huh?
## avoiding humanity
much of my life has been dedicated
to just that.
and still is.
even today at the track,
I was sitting alone between races,
in a dumb dream-state
but dumb or not,
it was mine.
then I heard a voice.
some fellow had seated himself
right behind me.
"I've come where it's nice and
quiet," he said.
I got up, walked about 150 yards
away and sat down
again.
I felt no guilt, only the return of a
more pleasant state of
being.
for decades I have been
bothered by door-knockers,
phone-ringers, letter-writers; and
strangers in airports and bars,
boxing matches, cafes, concerts,
libraries, supermarkets, jails,
hospitals, hotels, motels,
pharmacies, post offices,
etc.
I am not a lonely person.
I don't want to be embraced, cajoled,
told jokes to, I don't want to share
opinions or talk about the
weather and/or etc. and
etc.
I have never met a lively, original
interesting soul by accident and
I don't expect to.
all I have ever met are a herd of
dullards who have wanted to project
their petty frustrations upon me.
for some time women fooled
me.
I would see a body, a face, a
seeming aura of peace and
gentleness, a cool refreshing lake
to splash in,
but once they spoke
there was a voice like
chalk scratching a blackboard,
and what came forth as
speech
was a hideous and crippled
mind.
I lived with dozens of these.
wait.
the phone is ringing now.
but I have a message
machine.
they are leaving
one.
this one wants to see
me.
it wants to invite
itself over.
a reason is given,
some pretense.
it is hardly a worthy
one.
the last words are,
"Please let me know."
why do they want to see
me?
I don't want to see
them.
can't they sense
this?
am I the only one in the
world who finds being
alone to be a blessing, a
miracle?
must I always be kind to
those who would wallow
in my hours?
am I an ugly soul?
unkind?
unappreciative?
misanthropic?
a misogynist?
a crackpot?
a bastard?
a murderer of hope?
do I torture animals?
am I without love?
do I reek of bitterness?
am I unfair?
am I the wrecking ball of dreams?
am I the devil's encore?
do I put glass in the sandbox?
am I without morals or mercy?
if so, why do they want to keep
seeing me?
I would never want to see
anybody like that.
especially
when I am
shaving.
## WHAT HAPPENED TO THE LOVING, LAUGHING GIRL IN THE GINGHAM DRESS?
Harry reached over and switched off the table lamp. It had been a wasted night: nothing on tv as usual, nothing to read. It was 12:30 a.m. At least, he hadn't gotten drunk. But maybe he should have. At least that would have been an accomplishment. But some nights you just wasted, and some days and some weeks and some years. He'd had some rough years but here he was, still alive, and some might even consider him a financial success but money meant little to him. He had no desire for possessions, trinkets, travel. One thing he liked was solitude and another thing he liked was the absence of trouble of any kind. Harry had had more than his fill of trouble. At times, when he looked back, it was amazing to him that he was still alive. But there were many lives such as his, he was sure of that.
Well, sleep had always been one of his favorite escapes. Sleep was the grand healer, the equalizer. Harry slept well, he slept almost with a vengeance.
Harry noted the full moon through the window, closed his eyes, inhaled, exhaled. A man didn't really need too much. Just some ease of mind, a gentleness for the spirit. He was almost asleep when the phone rang. He turned on the table lamp, picked up the receiver. It was Diana.
"I've got a flat tire! Jesus Christ, I don't know what to do! I've got a flat tire! I decided to go to the 7-11 for some cat food and I got this god-damned flat!"
"Listen," Harry said, "you've got your Auto Club card. Phone them and they'll come out and change your tire."
"I've tried, I've tried!" Diana screamed. "I keep getting a busy signal or they put me on hold! And when you finally get through to them it takes them hours to come! I'm terrified! A gang of guys drove by in a car and hollered at me! I might get raped!"
"Look," Harry said, "just phone the Auto Club once more. I've always had luck with them. Ten or fifteen minutes at the most. Meanwhile, I'll get dressed and come over."
"I'm not going to call them again! I've used up all my change! This is the last call I can make!"
There was some further cursing interspersed by screams. At the first opportunity Harry spoke.
"Listen, I told you I was coming over. It will be all right. Please calm down."
"But you don't know where I am! How are you going to find me?"
"Tell me where you are."
"But you have no sense of direction! You're always getting lost! How are you going to find me?"
"I'll find you. Tell me where you are."
"I'm on Ocean Street!"
"I know where that's at. That's where you live."
"I'm not near where I live! I'm on a different part of Ocean Street!"
"What's the nearest cross street?"
"Sepulveda! Do you know where Sepulveda is?"
"Of course."
"You asshole, you've been living in this area for years and you probably don't know where Sepulveda is!"
"I'll get there. Sepulveda and Ocean. I'll find you."
"But you don't know what corner I'm on!"
"Don't worry. I'll see your car."
"Tell me exactly how you're going to get here!"
"I'll take Western to Pacific Coast Highway, take a left, then take a right on either Crenshaw or Hawthorne, drive until I hit Sepulveda, take a left and go until I hit Ocean."
"Do you know where Lomita is?"
"The street or the city?"
"The street, you asshole!"
"I thought you were at Sepulveda and Ocean?"
"I am! But Lomita is the first street you come to before you get to Sepulveda!"
For a moment Harry felt like hanging up. Instead he said, "All right, I'm coming over but after I get you out of this one, I never want to see you again. You got that? This is it!"
There was a long scream. Then:
"No, no, no! I'm going to kill myself! I'll kill myself right now!"
Diana screamed again. When she finished and began to sob Harry said, "All right, I didn't mean it. I'm sorry. I'll be right out. I have to get dressed first."
Diana reverted right back to her old self. "All right, do you know exactly where I am?"
"Yes, I'll find you. Now, calm down. We can fix this whole thing."
"Oh, you asshole!"
"Now what is it?"
"It's just that you're so fucking calm!"
"Listen, Diana, I'll be right over. I'm going to hang up. I'm on the way."
Harry picked his shorts up off the floor, got into them, got into his pants, his shoes without stockings, then stopped at the refrigerator, got a beer, uncapped it, drank it. It went down like a thimbleful. Then he went in and forced a piss so that he wouldn't have to piss on Sepulveda, made his way to the car and drove off.
As he drove up Western he looked at the people in other cars. They seemed quite rational. It was all very strange. Almost every woman he had ever dated had done time in a madhouse, or had madness in the family, brothers in jail, sisters who suicided. Harry drew these types to him. Even in the schoolyards, the mad and the strange and the misfits had been drawn to him. It was his curse. But he didn't have the cure, he just had the problem. And Diana was an extremist. Each time she got ill, she thought she was dying. She would scream and rant. "Jesus Christ," Harry had told her once, "when I was on my god-damned deathbed I didn't make all this fuss. All you can do is die." The message had been wasted.
Finally he was on Sepulveda. That was a relief. Sometimes Diana almost had him believing his own assholeness. Harry drove along, watching for Ocean. Then he saw the car. An Alfa Romeo. He had purchased it for Diana. Sky blue. Diana loved sky blue. He pulled up and parked behind the Alfa Romeo. There was no movement within the car. He opened his door, got out, walked up to the car. Diana was sitting there, staring straight ahead. Harry knocked on the window. Diana rolled it down.
"O.K.," Harry said, "I'm going to phone the Auto Club. I'll be right back."
"You're not going to leave me here! I'm going with you!"
She leaped from the car door, stood on the pavement, hair in eyes, hands dangling oddly.
"No, wait! We're not going to phone the Auto Club. It takes them hours! We can do it ourselves!"
Diana ran to the back of the car, came back with a tiny jack, plus a lug wrench about the size of an ordinary can opener. Harry tried the lug wrench, knowing ahead of time that it was useless. The nuts were frozen. They'd probably been tightened with an electric lug wrench. Harry got his own lug wrench and tried it on the wheel. It didn't fit.
"We're going to have to phone the Auto Club," Harry said.
"Why the fuck do they make stupid wrenches like that? Why is everybody so fucking stupid?"
"Come on, let's try a phone booth."
They started to cross Sepulveda when an old car with four young guys waving beer cans drove by and let out a yell. So Diana hadn't imagined it after all. Harry only hoped that they would come back so he could bang their heads together. But they didn't. It wasn't Harry's lucky night.
Harry got the Auto Club on the phone. He had Diana's card in his hand. He gave the lady the location of the car, the problem and the Auto Club identification number.
"Is the lady there?" Harry was asked.
"She's here but I'm phoning for her."
"I can hear her," said the Auto Club lady. "Would you mind putting her on the phone?"
Diana had been cursing and offering instructions from the background.
"Is that necessary?" Harry asked the Auto Club operator.
"Yes, I wish to speak to the lady..."
Harry handed the phone to Diana, thinking, oh shit, they'll never come now. We're finished.
"He told you where we were! How many fucking times do we have to tell you? No, I don't know the number! There aren't any street numbers! It's a deserted area! Where am I now? I'm outside a Thrifty drugstore in a phone booth! No, I don't know the number of the Thrifty drugstore! Your driver can find it! Thrifty Drugs! No, I'm not going to stay here! It's too cold! I'm going to wait in the car!"
Diana let go of the receiver and it dangled from the cord. Harry picked up the receiver in order to pacify the Auto Club. The line was dead.
"That cunt!" Diana screamed.
"Come on," Harry said, "let's go back to the car."
They crossed Sepulveda and Harry put Diana in the car. She was still ranting about the Auto Club. Harry walked out to the curb and lit a cigarette, waiting, somehow, for an Auto Club tow truck which might never arrive. All the dispatcher had to do was to take offense and not put in the call. Harry hoped the lady had a good soul. As for Harry, he'd give anything to be sitting in front of his tv with a beer, watching a replay of "The Honeymooners." If only a man could pack off to some city in Canada and never be seen or heard from again. But it was never that easy. You were destroyed by what you befriended.
Harry lit another cigarette and walked up and down. Then came a great surprise! An Auto Club tow truck came rolling along! Harry jumped and waved. The guy saw him and pulled up. Such a beautiful sight. If there was proof of God it was the arrival of an Auto Club truck in the middle of the night.
The man got out of the truck and approached the Alfa Romeo. Diana leaped out.
"We couldn't get the wheels off with this stupid wrench! Why do they make stupid wrenches like this?"
The man didn't answer. Then he said, "You've got two flat tires."
"Oh," Diana said, "I hadn't noticed. When I hit that fucking traffic island I felt the tire blow. I didn't know it was two."
Ah, Christ, Harry thought, this nightmare is endless.
"Well, I don't know what I can do," said the Auto Club man.
"Just go ahead and put on the spare," said Harry. "Maybe I'll think of something. Better one flat than two."
Then Diana couldn't find her car keys. There was more major hysteria. Then she found the keys—in her purse.
The Auto Club man found the spare in the trunk, brought it out, said, "There's no air in this spare. Somebody has let it go down."
The man brought out an air tank and inflated the spare. The spare went down again.
"This spare is flat," said the man.
For once, Diana was silent.
Three flat tires.
"Well, shit," Harry laughed, "let's just blow up the fucking car and leave it here."
"No, wait," said the Auto Club man, turning to Diana.
"You live near here?"
"Yes, about a mile."
"Well, I can tow your car to your place and leave it there."
"Can you do that?" said Diana. "That would be just fine."
More endless nightmare upon endless nightmare, thought Harry, no, no, no, no.
"No," said Harry, thinking, "there's a tire place about 4 or 5 blocks from here. Let's just haul the car down there and we'll fix it in the morning."
"That's O.K. with me," said the Auto Club man. "I know where that place is."
"Shit," said Harry, "let's do it." He and Diana climbed into the tow truck.
The Auto Club man towed the car to the tire shop and then he left. The Alfa Romeo with its flat tires was parked directly in front of the building.
"Now," Harry said, "we can leave one note on the windshield under the wiper and we'll leave another note under the office door. Then they can't miss it."
"What'll I write?" Diana asked.
"Tell them you need tires. That we'll be back in the morning. Leave your phone number and mine."
Diana got some sheets of paper out of her car and laid them on the hood of the Alfa Romeo and began writing. She wrote for a long time. Then she handed the sheets to Harry. Each sheet was 18 or 20 lines long. Harry had no idea what she had written. He took one sheet and placed it under the wiper and then walked to the office door with the other sheet.
"What are you doing?" Diana screamed. "Put it in the mail slot!"
"No," said Harry, and he slid it under the door, face up, so they would see it. Every edge against possible misunderstanding was needed.
Harry got Diana back to her place. He told her that in the morning he'd be back, he'd get her some new tires, and then everything would be all right.
When he got back to his place it was 4:35 a.m. Not too late. He uncorked a bottle of good wine and had a large glassful. Then he had another. It went down well and it was needed. It was cowardly, of course, to attempt to forget the incomprehensible, but nevertheless it was necessary.
In the morning Harry phoned the tire shop and told them he'd be over to purchase tires for the Alpha Romeo. "Fine," said the man, "we got your letters."
Harry got to Diana's about ten a.m.
As he approached her open doorway she must have heard him coming.
"Oh my God! My God! I can't stand it! I want to die!"
He walked in.
"What is it, Diana?"
"I can't leave this place like this!"
"What is it?"
"Can't you see? There's piss and shit all over the floor! The toilet backed up!"
"Well, we'll clean it up."
"The toilet's stopped up and I don't have a plunger! And I've got nothing to clean the floor with! I can't leave!"
This is Saturday, thought Harry, if I don't get her car fixed it'll be there until Monday and there will be further complications.
"I'll get you some stuff," said Harry.
"Where are you going? Where are you going?"
"I'll be right back," said Harry.
Where the hell can I buy some towels? he thought as he drove along.
He saw a large department store, parked, got out. The doors were just opening. He walked in with the customers who had been waiting.
Harry found the towel department. He grabbed three of the largest towels and put them on his VISA card. He had a hangover.
He asked the lady where he could get a plunger.
"Hardware," she said. "Two aisles to the left and one down..."
Harry walked around to the Hardware section. There were no plungers on display. There were no clerks in the Hardware section. He went over to Automotive where the clerk saw him coming, turned his back and walked off. He cornered the clerk at the dead end of an aisle.
"Listen, don't they have a clerk in the Hardware section?"
"I don't think so."
"Don't they have any plungers in this store?"
"They should be in Hardware."
"There's nothing there."
"They must be on order."
Harry left the store and drove around some more. Then he saw Thrifty Drugs. He parked and went in. It was a hot morning and the hangover made him sweat excessively. He saw some plungers. But it was madness. They were tiny. They only cost a dollar. Maybe I can make it do until I find another, he thought. He purchased the little plunger and went back to Diana's.
"Here," he said, "some towels and a plunger."
"Oh, Jesus Christ, I can't use that plunger! Oh, I feel like dying!"
Then she screamed. When she finished Harry said, "I'll be right back."
"Where are you going?"
"I'll be right back."
"Oh, all this mess! What will I do?"
"I'll be right back."
Harry jumped into his car and drove off again. He saw a home appliance store, parked, went in. He found a plunger! A beautiful black plunger! He paid cash and took it back to the car.
Back at Diana's he said, "Here's a real plunger! Look!"
Diana grabbed it.
"Wait, I'll do it," suggested Harry.
But Diana already had the plunger and she was working at the toilet. She sobbed as the water splashed about. She stopped to flush the toilet, then worked the plunger again. The dark water rose to the edge of the toilet and Harry thought, oh, my God. Then, at the last moment, the water whirled down and away. The toilet was clear.
"There," he said, "we've solved that."
"I can't go!" Diana screamed. "I can't clean this floor! I don't know what to do! What will I do?"
"You've got the towels."
"I can't use those beautiful towels on the floor!"
"What do you need?"
"Paper towels!"
"I'll be right back..."
Harry jumped into his car and went back to Thrifty Drugs. He found the paper towels. He got several different types of paper towels. Then he went back to Diana's. What can she say now? he thought.
"I can't clean the floor, oh God, I can't clean the floor!"
"Why, what's wrong?"
"I don't have any soap! How can you clean the floor without soap?"
"I thought maybe you had some soap."
"I don't have any soap!"
"I'll be right back."
The hangover seemed to be getting worse. He jumped into the car, lit a cigarette, gagged. Then he drove back to Thrifty Drugs. He got three different brands of soap. The same girl was at the cash register, but she didn't recognize him. Or maybe she did and thought he was mad.
Then he was back at Diana's with the soap.
"I'm going to get a newspaper," he told her.
He jumped back into his car, went back to Thrifty Drugs and got a newspaper out of the rack in front. Then he returned. He sat in a chair outside and read the newspaper. His mouth was very dry and he was ill to the stomach. He read the front section, the feature section, the sports section.
Then he heard Diana. "As soon as I shower we'll go."
"O.K." he said...
The Alfa Romeo sat there with its flat tires and Harry went to the office to get things moving. There would be 3 new tires needed and put the most worn tire in the trunk for the spare, thank you.
The clerk seemed very understanding.
"Come back in an hour and your car will be ready."
They walked down to the Sizzler and they got the Hibachi chicken, the Double-Hibachi chicken. Diana also had a salad and an iced tea. Harry had a coffee. The place was crowded.
"Eat slowly," said Harry, "we've got an hour."
Somehow they managed to kill an hour. Harry drank much more coffee than he felt like drinking. He felt as if he was going to puke.
They walked back to the tire shop. The car stood there, untouched, still on its flat tires.
Harry went back to the nice clerk.
"They haven't touched the car," Harry told him.
"They haven't?"
The clerk left the counter and shouted through the door, "HEY, EDDIE, BRING THAT ALFA UP HERE, WILL YOU?"
The clerk turned to Harry, "Sorry sir, we'll get right on it!"
"Let's wait in my car," Harry suggested to Diana.
They walked to the car and sat and waited. Still, nobody moved the Alfa. There were various men about in their white uniforms. Some drank coffee. Others stood and smoked, talking to each other. Another was on the telephone.
Then from out of nowhere came a fat man in his white uniform. He got into the Alfa Romeo and started the motor.
"What's he going to do?" Diana screamed.
"He's going to move your car over to the rack," said Harry.
"He can't drive it like that! He can't drive it on those tires. He'll ruin the rims! Tell him to stop!"
"It's just a short distance. The rims will hold up."
The fat man slowly drove the car toward the rack.
"He's ruining my car! Make him stop!"
Harry put his head down and stared at the floorboard. He didn't want to stop him.
When he looked up the car was parked near the rack. He saw the fat man get out and walk off. The fat man was gone for 5 minutes. When he came back he had a sandwich in one hand and a large Coke in the other. He walked past the Alfa and out a side door and was gone. Harry started to open the door to go back to see the clerk in the office.
"Don't bother them, they might resent it," said Diana.
"Maybe you're right."
They sat there. In another ten minutes a thin man appeared. He rolled up 3 new tires.
"Tell him not to use the electric lug wrench to put the tires on," said Diana.
Harry walked over and told the thin man that.
"O.K." said the man.
Harry walked back to the car. The thin man changed one wheel, then walked off.
Oh my God, thought Harry. This is most surely the day I am being tested, to see if I am ready for the other world.
Then the thin man was back smoking a very long cigarette.
"Hey, Monty," he yelled to somebody, "what are you doing tonight?"
"We're double-dating," came the answer from somewhere. "We're going to Orion's. Do you know where Orion's is?"
"Sure, I know where Orion's is!"
Then, suddenly, music came out over the loudspeakers. It was loud, quite loud. A woman was singing, only you couldn't make out the words. The music stayed on. The song ended and a man began singing. Harry really felt like puking.
Twenty minutes of song. Then the thin man yelled at them over the music.
"O.K., IT'S READY!"
This is it, thought Harry, victory at last! We have endured. We have come through. We have surmounted all.
Harry walked into the office and paid the bill. He felt great. He joked with the clerk. He loved the clerk. All men were brothers. The world was fine. He was free.
He walked back to Diana.
"Well, you've got a new car. 3 new tires. And you've got a new paint job from last week and a new top from the week before. Your car looks great."
Diana got into her car and started it up.
"Thanks," she said, "and I'm sorry for everything. Things have been so fucked-up lately."
"Forget it. Everything's straight now. Happy driving. I'll phone you later. I'm going home to sleep for a couple of hours."
"Thanks again..."
"O.K., kid, see you later."
Diana drove off toward the exit. She gave a little wave. Harry waved back.
Then she stopped at the street exit. She started honking her horn and staring tearfully at him through the driver's side window.
Harry ran up.
"What's wrong?"
"I'm sorry but I can't drive this car this way!" she said through the window.
"What is it?"
"There's this scraping sound! Listen!" She drove forward a few feet.
It was true. You could hear it over the music.
"Back it in again," Harry told her.
Harry went to the thin man and explained the scraping sound.
"Oh, we'll fix that right up," he said. "It's a minor adjustment."
The thin man took off the wheel that had been scraping, looked at it, put it back on. No scraping now.
Diana got in again and drove toward the exit. She waved, he waved. Harry inhaled and waited. The Alfa Romeo pulled into traffic and was gone.
Harry got back to his place, took a bath and had a beer. He got lucky. There was a good middleweight fight on tv. He was still alive. The late afternoon sun came through the window and bathed him in its glory. Things were coming together. He made an egg sandwich with green peppers. In an hour or so he phoned Diana.
"Everything all right?" he asked.
"Yes," she said, "but I've been worried about my cat. Those males have been terrorizing her, those sons-of-bitches. But she's here now. She's all right."
"Great..."
"Mother just called. She's coming to visit next week just as planned. And she says to thank you again for letting her stay in your spare bedroom. You have such a lovely home."
"It's all right..."
"She's only going to stay 3 days, then she's going north."
"O.K."
"I've got her arrival time at the airport and all that. You know she's getting along in years. Last week she broke a bone in her foot coming down a stairway. She might be in a wheelchair."
"We'll take care of her," said Harry.
"I want to get some blinds for her bedroom. You can see right in there from the street, that's not right."
"O.K."
"And thanks again."
"Forget it."
They said goodbye for the time being. Harry went and got another beer, then went outside and sat on the steps and smoked a cigarette. It was getting dark. Harry liked it dark. The darker the better. He smoked his cigarette and gulped at his beer. For the first time in 18 or 19 hours he felt pretty good. Pretty damned good. And he allowed himself the full enjoyment of that. He felt he had it coming. Darkness and peace. Ah, ah, ah...
## the luck of the word
throughout the years
I have gotten letters
from men
who say
that reading my
books
has helped them
get through,
go on.
this is high praise
indeed
and I know what
they mean:
my nerve to go
on was helped
by reading
Fante, Dostoevsky,
Lawrence, Celine, Hamsun
and others.
the word
raw on the page,
the similarities of
our hells,
when it all comes
through with
special
force,
those words and
what they speak
of
do help
get our asses
through the
fire.
a good book
can make an almost
impossible
existence,
liveable
for the reader
and
the writer.
## bad form
the famous actor sat at the table with
his friends and the friends of the owner
of the horse
who was to run in the big race.
everybody had purchased tickets on the
owner's horse.
they sat together and watched the
race.
the owner's horse ran
badly, he ran
last.
some moments passed,
then the famous actor took his
stack of tickets
and tossed them down in front of the
owner.
they were spread there upon the white
tablecloth.
I no longer liked any of the movies
I had seen the famous actor
in.
I no longer liked the famous
actor.
I left the table.
I left the Director's Room.
I took the elevator down and out of
there.
I walked across to the
grandstand area
to where the non-famous
poor people were
and they were beautiful,
they had faces like
flowers
and I stared at them,
drinking in their
voluptuous
normalness.
## last call
this is it, sucker, the dead nightingale
in your lap, the final circle around
the mirage, the bones of your dreams
buried, laughter caught in the specimen
bottle, the caked blood of your
little paintings, the Hunter sighs,
the lynx huddles in the dark,
parsnip fingers grip the bottle,
old ladies mail you postcards from
Illinois,
as one fly circles the room and one room
circles the fly.
phone messages from the persistent:
old memories crushed in your brain
with hanging tongues;
the hammerhead shark dressed as a
nun;
2,000 years like a spider sucking at a
webbed insect;
the sodomized headless horse of
History;
the grandmother's smile;
Persistent Madness Syndrome
as a spiritual occupation;
mares eating oats and oats eating me
as the fleas play tambourines;
suicide as the last serenade to the
curse of Time;
the legless spirit flung against the
wall like
a bottle of vinegar;
the cat with 3 eyes walking through
the nightmare melody;
roasted pigs that cry in the heart
of a dog
walking north;
my aunt spitting out her paperclip
soul through the open window of
a 1938 Ford driving along Colorado
Boulevard;
Brahms talking to me as I lay a
20 dollar bet on the
6 horse;
the majesty of the club-footed duck
looking for the blocked
exit;
the applause of the terrified masses;
the last torn card upside down
in the ringing of an empty
room;
the last bluebird flying from the
burning
funhouse;
an apricot seed challenging the
sun;
the sheets of the whore raised
as a flag by political
centipedes;
zero times zero times zero
times zero;
the face in your mirror is love
drowned alone;
eating an apple is eating
yourself standing on a corner;
the paperclip speaking;
an onion more beautiful than
you;
Spain in your coffee cup;
the white horse standing on
the hill;
the dream stuffed in the
trash and the trash stuffed
in
you;
the beginning and the end
are the same;
the new gods imagined and the
old gods re-invented;
the human voice being the most
ugly instrument;
the falcon swirling and the vulture
swirling and the girls dancing with
eyes so blank;
everywhere the trees and plants
and flowers watching us
as their sadness towers tall
in the mighty night;
they weep and they weep
and they
weep;
the horse running last into
snow-covered mountains
as Li Po smiles
and bitter people
tear up their paper tickets
and blame the horse
and blame the life
and blame the blame
as the mountains weep
and the cross comes down
and lifts the sun;
the great white shark sniffing
the dark purple sea
as the mouse
alone
stares through its eyes at
all the
terror;
we burn separately and
together
in the December of our
undoing;
the walking blood of our
screams unrecorded
anywhere
but in our singular
private hells;
we dance when we can
we dig for worms and
coffins
we swim
we walk
we talk
we fornicate,
we gag
we gargle
we fish and
are
fished
hooked
caught
cleaned
fried
baked
broiled
simmered
eaten
digested
expelled;
it's a long wash
in and out of shore
through small lights and long darkness;
the bluebird
the bluebird
the bluebird
the chair in the center of the room with
nobody in
it;
everything waiting for the silver sword;
a piano playing somewhere
one small note at a time
a bluebird on each key;
my 6 cats asleep in the other room
waiting for me;
death only means something to
death;
it's late now
as the walls kiss me and hold me
and you
and you
and you
this terrible glory
as the Hunter himself almost wearies of
the hunt
but not
quite
not quite
not
not
quite.
## the shape of the Star
well, you know, he started out as a
comedian
and then it was decided to make
him into a serious
actor,
the public always like that.
and then we decided to make him
politically aware,
we got him to pitch
all the right causes.
then Publicity sent out a story:
how he pulled a woman from a
wrecked car,
how he contributed large sums
to various charities while asking
that his name not be
revealed,
how he was going to give this
Benefit or that Benefit,
donating his time and
talent,
how he saved a child from
drowning,
how he did this and that.
we worked our asses black
and blue to create his
Public Image,
we were just starting to reap
a profit,
then, what happens?
the son of a bitch gets
drugged,
runs his Mercedes off a
cliff near Malibu
and kills
himself.
we couldn't do much with
that one.
we claimed some communists
who disliked some of his
causes
had messed with his
brake cables.
that took pretty well
but all in all
we finally had to write him
off
as a dead loss.
we got a new one now,
found some boy
working behind a fish
counter.
Tom is perfect:
totally bland features,
even a few
freckles,
large empty eyes
and a dog-like
grin.
he's a bit
addled,
but the clay's all there,
we'll shape him into
what they think they
need.
only with this one
we're going to use a
new twist, we are going to
start him as a serious
actor
and then turn him into
a comedian.
we're thinking all the time
here,
that's what makes
Hollywood
what it
is.
## upon reading a critical review
it's difficult to accept
and you look around the room
for the person they are talking
about.
he's not there
he's not here.
he's gone.
by the time they get your book you
are no longer your
book.
you are on the next page,
the next
book.
and worse,
they don't even get the old books right.
you are given credit for things you don't
deserve, for insights that aren't
there.
people read themselves into books, altering
what they need and discarding what they
don't.
good critics are as rare as good
writers.
and whether I get a good review or a
bad one
I take neither
seriously.
I am on the next page.
the next book.
## Paris, what?
you want to get stiffed? he asked
me, well, just send something to
the Paris Review, they have
their own select crowd of boys and
girls, it's a special club, you've
got to stink just right.
is that so? I sneered.
he drove off in his lambskin
Caddy
and I walked into the next
room,
looked at my 6 cats asleep
on the bed,
there was enough Power there
to crack the Universe
like a
walnut
shell.
I could taste it with the tips
of my ears,
I could see it through my
dark-stained
shorts.
the Paris Review ain't crap
to me,
I thought.
I was at the track today and
I picked 6 out of
nine
with agony stuffed in my
pockets
and the sun
behind a film of
pain.
I took a crap, then put
on Brahms'
2nd,
sent
this
one.
## a social call
to suffer the fanged indifference of the
interloper
slurping beers at your
coffeetable,
if you asked this unquestionable
bore
to leave the premises
then your wife would forever
brand you as a mean and ugly
human
and so you measure your
choices
and decide to wait out the
boor
as he lights his cigarettes and
slurps his beer
talking on and on about
absolutely nothing
as the very walls yawn
as the rugs twist in agony
as the good hours are
uselessly murdered
as you consider,
this is what it must be like in
hell.
not flames and the devil
but just some fellow
fair of heart
and good enough in his own
way
talking about the mundane
variables,
going on,
caught in the mystery of his own
voice,
slurping the beer,
lighting the cigarettes
while Time is taking the 8-count,
while Time is being mugged.
some day you will be on
your deathbed
wondering why you
wasted it
all
as you now listen and
listen and listen,
in a hell before hell,
the palaver seeping to
your marrow.
when you are unkind
to yourself
you will know no
worse.
and deserve no
better.
## the girls we followed home
the girls we once followed home are
now the bag ladies,
or one of them is that white-haired
old crone who
whacked you with her
cane.
the girls we once followed home
sit on bedpans in nursing
homes,
play shuffleboard at the public
park.
they no longer dive into the
white-capped waves,
those girls we followed home,
no longer rub their bodies with oil
under the sun,
no longer primp before the
beautiful mirror,
those girls we followed home,
those girls we followed home
have gone somewhere,
some forever,
and we who followed them?
dead in wars, dead of heart
attack,
dead of yearning,
thick of shoe and slow of
speech,
our dreams are tv dreams,
the few of us,
so few of us remember
the girls we followed home.
when the sun always seemed to
be shining.
when life moved so new and
strange and wonderful
in
bright dresses.
I remember.
## slow starter
by the time I got good with things
other people were into
something else.
from the worst baseball player
I became the best,
unbelievably swift in the field,
tremendous power at the
plate
but by then the others were into
schooling, books, getting ready
for the future.
from a sissy I developed into
one of the best fighters
around
but by then
there was nobody left to
fight.
the girls took me even longer.
by the time I became an
expert lover
all of my compatriots were
either married
or disillusioned by the
chase.
all that was left for me were
the leftovers, the uglies,
the divorced, the mad, the
ladies of the
streets.
I always became the best
at things when those things
no longer counted:
football, high-speed driving,
drinking, gambling, clowning,
debating, bullshitting, going
to jail, going crazy, lifting
weights, shadow boxing with
fate.
but I was alone.
the others had become sedate,
had become responsible
citizens with
children, jobs, mortgages,
life insurance and pet
dogs.
the very things which terrorized
me.
I was the retarded child
still looking for more
childhood.
I still wanted to play but
there were no
playmates.
I bummed the country,
prowled the avenues,
the bars.
I found nothing, I
found
nobody.
I searched the skid
rows
thinking that something
could be hiding
there.
I thought
wrong.
being a late starter
also makes you late for heaven
or hell,
you are always trying to
catch something,
catch up to something,
some tangent, some
invisible thing,
it has to be there,
I can feel it there,
I see it sometimes in the eyes
of a tired old waitress,
or the round spot on a pillow
where the cat has
slept.
it's there and it beats the
funeral parlors
and the millions of feet
walking in their
shoes
and the way it seems to
be,
the cities, the faces, the
newspapers, the sidewalks,
the stop signs, the churches,
the flags and the
calendars, the whole
unholy act.
this childhood on the
hunt,
this late starter,
this slugger, this drunkard
is still on the
look-out
and I know it's there,
unfound,
waiting,
centuries late,
boiling,
swirling,
I've got the fix on
it,
it's coming into
focus,
don't you almost feel it
now?
I do.
## barstool
the longer I live the more I realize
that I knew exactly what I was doing
when I didn't seem to be doing
anything
but watching a wet fly on the
bar
nuzzling a pool of
spilled beer.
I was quitting the game,
tossing in my hand
early,
it felt grand, I tell you,
it even felt dramatic, I mean
to cough it up and out,
to give way,
to sit there
the dirty Venetian blinds
behind me,
nothing to do but get my
wits up enough
to cage another free
drink.
I had zeroed out, I was
the Grand Marshal of
Nowhere,
still young,
I realized that there was
no place to go,
ever,
I was already there.
I was the Clown of the
Patrons.
I was the Nut.
I was the Heart of a
Heartless bar.
the drinks came.
the days and nights
went.
the years went.
I lived by my addled
crushed wits,
sometimes
ended up bloodied in
some alley, given up
for dead,
only to rise again.
I knew exactly what I
was doing: I was
doing nothing.
because I knew there
was nothing
to do.
I know now
that I knew then all that there
was to
know,
and tonight
sitting alone here,
nobody about,
I am still fixed in this
floating
perfect
aspect.
my wits have gotten me
from nowhere to
nowhere
and death like life
is lacking,
and I know so well
I did right
watching that fly
nuzzle the beer
suds
as the others
hustled their butts,
circled in the
tenebrous
light.
## look back, look up
was Celine married?
did Hemingway have 6
cats?
why did Bogart smoke
himself to
death?
was Ty Cobb as mean
as they claim?
whatever happened to
Clark Gable's
ears?
did Van Gogh ever
ice skate?
where were you in
1929?
Nijinski was a
madman.
remember Admiral
Byrd?
Joe Louis was a
cobra.
remember a-dime-a-
dance?
Pearl Harbor?
Mutt and Jeff?
The Katzenjammer
Kids?
gluing together
balsa wood
airplanes?
a bagful
of candy for
7 cents?
remember the
iceman?
Slapsy Maxy
Rosenbloom?
garter belts?
garters?
all night movies?
marathon dance
contests?
Al Jolson?
Mickey Walker?
a nickel beer?
a nickel phone call?
a 3 cent stamp?
Primo Carnera?
a good ten cent
cigar?
Bull Durham?
fuse boxes?
ice boxes?
the ruler against
the open
palm?
the Indian head
penny?
Tom Mix?
Buck Rogers?
jaw breakers?
the WPA?
the NRA?
Jack Benny?
the Hit Parade?
movie houses with
ushers?
cigarettes called
Wings?
zoot suiters?
geeks?
grandmothers who
baked apple
pies?
gold-fish-eating
contests?
Red Grange?
the Babe holding
out for
80 grand?
Man of War?
flagpole-sitting
marathons?
I could go on
and on...
but, Christ, if
you remember
all of these things
you must be
at least as old
as I am.
listing these things
on my
Macintosh
computer
with a 50-50 shot
of seeing the
21st
century,
betting the horse
instead of
riding it,
we're lucky to be
here and we'll
be lucky when we
leave.
see you in
St. Louis.
see you behind
that last curtain,
see you at another
time,
baby.
## Paris
was just like not being there.
Celine was gone.
there was nobody there.
Paris was a bite of bluegrey air.
the women rushed by as if you would never
DARE to go to bed with
them.
there were no armies around.
everybody was rich.
there were no poor in view.
there were no old in view.
to sit at a table in a cafe
would get you careful stares from the other
patrons
who were certain that they were
more important than
you.
food was too expensive to eat.
a bottle of wine would cost you
your left hand.
Celine was gone.
the fat men smoked cigars and became
gloried puffs of smoke.
the thin men sat very straight and spoke
only to each other.
the waiters had big feet and were sure
that they were more important than
anything or
anybody.
Celine was gone.
and Picasso was dying.
Paris was absolutely nothing.
I did see a dog that looked like a
white wolf.
I don't remember leaving
Paris.
but I must have been
there.
it was somewhat like leaving
a fashion magazine in a
train station.
## the good soul
it's not enough that he's one of
the richest men on
television,
he has to reappear on the
tube
and complain that many other
programs are not
decent,
they are full of obscene
words and
gestures,
or that people are
"anti-social,"
that they should look up
to things that
will inspire
and purify
them.
his own program is
full of cute
children,
well-dressed, well-
fed,
overlooked by a
very understanding
father
and a mother
who understands the
father better than
he does
himself.
they live in a
luxurious home
and at times
certain members of
this family
have little
programmed arguments,
but they all work it
out,
become instruments
for a more
loving and understanding
togetherness.
all that I can say
to this
is:
shit, fuck, bullshit,
crap,
come here and
bite
this.
## lousy mail
drinking up here, looking out at the lights of
the city, the rows of headlights snaking down
the Harbor Freeway south
forever,
Sibelius working on the radio.
there is a small refrigerator in the room.
I get up now, reach in there, crack a
beer as
Sibelius continues to work.
about 3 times a week now I get manuscripts
in the mail from young men
who seem to think that I can get them
published.
they tell me that their work is good.
I read it and find it astonishingly
bad.
they don't want to write, they want
fame.
they probably read their stuff to
their mothers, their girl
friends.
they probably give poetry readings
at poetry holes.
they will go on and on
typing dead work for decades
never believing that their failure is
simply the result of a lack of
talent.
as I sit tonight 3 such manuscripts
are on the desk in front of
me.
I don't know what to tell these
men.
they have no self-doubt.
I probably won't answer.
what would you tell them?
would you send them to hell
with a cruel comment?
would you give them
undeserved praise?
how can you be true and
kind at the same
time?
how?
## THE SUICIDE
Contemplating suicide was standard practice for Marvin Denning. Sometimes his thinking about it disappeared for days, even for weeks, and he felt nearly normal, normal enough to continue living comfortably for a while. Then the urge would return. At those times life became too much for him, the hours and the days dragged along uselessly. The voices, the faces, the behavior of people sickened him.
Now, driving in from work the urge to suicide was fully there. He turned off the car radio. He had been listening to Beethoven's 3rd and the music had seemed all wrong, pretentious, forced.
"Shit," he said.
Marvin was driving over the bridge that took him back to his apartment. It was a bridge which spanned one of the largest harbors in the world.
Marvin stopped his car near the middle of the bridge, switched on the hazard light and got out of the machine. There was a ledge next to the bridge's rail and he stepped up on it.
Above him stretched a wire fence a good 10 feet tall. He'd have to climb that wire fence in order to get over the side.
Below him was the water. It looked peaceful. It looked just fine.
Rush hour traffic was building up. Marvin's car blocked the outer lane. The cars in that lane were trying to make a lane change. Traffic was backing up.
Some of the cars honked as they swung by. Drivers cursed Marvin as they drove by.
"Hey, you nuts or what?"
"Take a dive! The water's warm!"
Marvin continued to stare down at the water. He decided to climb the fencing and go over. Then he heard another voice.
"Sir, are you all right?"
A police car had parked behind Marvin's car. Red lights flashed. One officer approached him as the other remained in the car.
The officer moved quickly toward him. He was young with a thin white face.
"What's the problem, sir?"
"It's my car, officer, it has stalled, won't start."
"What are you doing up on the ledge?"
"Just looking."
"Looking at what?"
"The water."
The officer came closer.
"This is not a sightseeing area."
"I know. It's the car. I was just standing here, waiting."
Marvin stepped down from the ledge. The officer was next to him. He had a flashlight.
"Open your eyes wide, please!"
He shined the flashlight into Marvin's left eye, then his right, then he re-hooked the flashlight on his belt.
"Let me see your license."
The cop took the license.
"Stay where you are."
The cop walked back to the squad car. He stuck his head in the window and spoke with the other cop. Then he straightened up and waited. After a few minutes he walked back to Marvin, handed him back his license.
"Sir, we are going to have to move your car from the bridge."
"You mean you're going to call a tow truck? Thank you."
Marvin's car was parked on a slight incline near the center of the bridge.
"No, we are going to give you a push. Maybe when you get rolling you can get it started."
"That's very good of you, officer."
"Please get in your car, sir."
Marvin got in his car and waited. When the police car bumped his, he took off the hand brake and put it into neutral. They rolled up over the center of the bridge and down the other side. He put it into 2nd, stepped on the gas and, of course, the car started. He waved to the police and drove along.
They followed him. They followed him off the bridge and down the main boulevard. The blocks went by. They continued to follow. Then Marvin saw a cafe: The Blue Steer. He pulled into the parking lot, found a space.
The police car had pulled in behind him, a few yards to one side, between Marvin and the cafe. Marvin got out of his car, locked it and walked toward The Blue Steer. As he passed the cops in the squad car he gave them another little wave, "Thank you again, officers."
"Better get that car checked out, sir."
"I will, of course."
Marvin walked into the cafe without looking back. The restaurant was packed. All the faces almost made him sick. There was a sign:
PLEASE WAIT TO BE SEATED
Marvin didn't wait. He walked to the last empty booth, sat down. He wasn't hungry.
A huge waitress floated up in a pink outfit. She had a very round head and her lips were painted a bright raspberry. She handed him a shiny menu.
"How are you today?" she asked.
"Fine. And you?"
She didn't answer. Then she spoke.
"Coffee, sir?"
"No."
"Are you ready to order?"
"No. For the moment, bring me a glass of wine."
"What kind?"
"The house wine will do. Do you have port?"
The waitress left and he watched as her oversized buttocks worked away.
Maybe I can go back to the bridge tonight when there is nobody around, Marvin thought.
Two men had a table behind Marvin. He could hear them talking.
"The Dodgers are sure looking good, aren't they?"
"Yeah. And the Angels are right up there too. Just think of it. Maybe we can have a Freeway Series."
"That would be a hell of a hoot, wouldn't it?"
Then the waitress was back with Marvin's wine. She sat it down hard and some of the drink leaped out and splashed on the table.
"Sorry, sir."
"It's all right."
"Are you ready to order yet?"
"No, not yet."
"We have a sirloin steak special tonight."
"No, thank you."
Then she cranked up her buttocks and moved off. Marvin had a sip of the wine. It tasted dusty, somehow made him think of spiders. Then he heard the piped-in music. "I don't have to say I love you," a male voice sang.
Then he heard the men behind him.
"I'm going to say something right now that you're not going to believe."
"Like what?"
"Ronald Reagan was the greatest president this nation ever had."
"Come on now, we've had a lot of them. That's a big statement."
"Without Reagan those fucking Russians would be all over the world, they'd be climbing over the fence and into our backyard. He stopped them where they should be stopped. They knew he meant business!"
"Well, yeah, he was a good man."
"I'll tell you something else. There's going to be a war in SPACE! Between the Russians and us! We're going to be fighting over the moon, over Mars, over all the planets!"
"We already got our flag on the moon."
Marvin finished his wine and got the attention of the waitress. She trundled over.
"Ready to order now, sir?"
"Another wine, please."
"We've got a sirloin steak special..."
"Just the wine, please."
Marvin heard the piped-in music again. Another man was singing, he sang, "If you don't answer the telephone soon, I'm gonna come to your room."
Then the waitress was back with his wine. She set it down.
"You see, I didn't spill it this time!"
She let loose an utterly false cackle of laughter.
"I'm getting better, you see?"
"You're all right..."
"Diana's the name."
"You're all right, Diana."
Then she struggled off to her other duties. Evening had rapidly dissolved into night. Marvin sipped his wine.
When he hit that water it would be like hitting cement. Except he would slide into that blue cold—one leg like this, another like that—and the hair on his head floating out. Dumb shoes on dumb feet. Out of it. Zero minus zero. As ultimate as you could get, from here to nowhere. Fine enough. You couldn't have it all.
Suddenly there was a crash, the breaking of glass. The front door was kicked open and two men entered wearing stocking masks. A woman screamed.
"Shut the fuck up or you're dead!" the shorter man screamed. "I mean it! No bullshit! Shape up or you're all dead!"
Each man carried a canvas sack. The taller man moved to the cash register, hit a key, the drawer sprang open. He began scooping bills and change into his sack.
Each man had what appeared to be a .357 Magnum.
"Don't anybody move!" yelled the shorter man.
He waved the Magnum over his head in a wild circle, then brought it down and pointed it around the cafe.
"O.K., all wallets and purses on the tables! Rings too! Watches! Everything! Anybody try any shit, it's your ass, got it?"
Then he began to move among the tables scooping everything into his sack.
The taller man was finished at the register. He saw the fat waitress cowering a few yards off. He ran up to her, said, "Where's the money box?"
"What?"
"The fucking money box! Where they keep the big bills!"
The fat waitress just stood there. The short man spun her around, jammed the Magnum against her neck.
"I'll blow your fucking head off! Where's the cash box?"
The fat waitress was sobbing, gulping for air. She said, "It's in the kitchen! Under the sink!"
"Don't anybody move!"
The tall man ran into the kitchen.
The short man pushed the frightened waitress to one side. He resumed clearing valuables off the tables, scooping them into his sack.
The tall man came running out of the kitchen.
"I got the fucking money! Let's go!"
The short man was busy.
"You watch the door! Nail anybody who comes in! Watch the door!"
"Come on, let's go, we got enough!"
"No, I'm going to get it all!"
He moved along until he got to Marvin's booth.
"Hey, fucker, where's your wallet?"
Marvin looked up at the stocking face. He rather liked it. The less you could see of the human face the more pleasant it was.
"I've decided to keep my wallet."
"You ain't deciding shit!"
"Of course I am."
"O.K., baby, you want it, you get it!"
Marvin felt the Magnum against his temple.
"Now, get out the wallet, O.K.?"
"Not O.K. I am keeping my wallet."
"Hey," yelled the tall man, "let's get out of here!"
The short man jammed the Magnum hard against Marvin's temple.
"You want this to be your last moment?"
"Go ahead and shoot," said Marvin.
Marvin waited. The safety catch went back on. Marvin saw the man switch his grip to the barrel of the Magnum. He saw the gun rise, sat there waiting. It smashed down on the top of his skull. There was an explosion of yellow, blue and red light but Marvin felt no pain. For a moment he couldn't move. Then he felt as if he could move. He tried it. He kicked out savagely and caught the man in the stomach with his right foot.
"Oooh..."
The hold-up man dropped the sack, grabbed his groin, almost sank to one knee.
"Oh, God-damn it..."
Marvin heard the safety catch go off again. The man aimed the Magnum, squeezed the trigger. The bullet whizzed past Marvin's left ear and broke an overhanging light fixture apart further down the room.
"Let's get out of here!" yelled the tall man.
The short man straightened up and walking half bent, and carrying his Magnum and his sack, he followed the tall man out the door. Then they were gone.
With that, the customers all started walking around and talking at once.
The cafe manager who had been hiding in the kitchen was on the telephone.
Marvin Denning finished his glass of wine and motioned to the fat waitress who was standing just a few feet away, trembling. Marvin got up, walked over to her. "Diana, another glass of wine, please..."
"Oh," she said, "oh...yes...of course..."
Marvin went back and sat down. The noise of the patrons had risen to a sickening pitch as they talked about the hold-up.
Marvin waited, then Diana was back with his wine.
"Thank you, Diana."
He took a sip.
"That was a brave thing you did, sir. By doing that you saved the belongings of many of the customers."
"Oh...yeah..."
"You're bleeding poor man!"
"It's all right."
Diana ran off as well as she could. Denning heard the sound of the police siren. He took a napkin and held it up to the top of his head. Then he pulled it away and looked at it. Blood. The stupid simplicity of blood.
Then Diana was back.
"Here. All I could find was this dish towel but it's clean."
"Thanks."
He folded the towel and to please her he held it to the top of his head.
"You better get that sewed up."
"It's all right. Main thing: get me that steak you were talking about and maybe some french fries!"
Diana went back to the kitchen and Denning sipped his wine.
In another minute the police entered. They came running through the door, hands on holsters.
"Everybody stay where you are!"
One of the officers was the one with the thin white face, the same one who had stopped Denning on the bridge. Their eyes met. Thin white face stared at him.
"What're you doing here?"
"Waiting for a steak. You followed me over here, remember?"
Two more cops entered.
"Waiting for a steak?"
"Yes, any law against that?"
"Officer," said a patron who was standing nearby, "this is the man who almost captured one of the bandits. He kicked him to the floor."
Diana walked up with Denning's steak and fries, set it down.
"Officer, this is a very brave man," she said.
One of the patrons began to applaud. The others joined in.
Denning raised his wine glass to them, drained it.
Thin white face asked, "Did you know the hold-up men?"
"Can't say that I did."
Then Denning heard another siren. The patrons were pressing around his table.
The cop, irritated, said, "Stand back!"
A stocky, dumb-looking fellow in need of a shave came through the door followed by another cop. The stocky man pushed up to Denning's table.
"What's going on?"
"I've been held up, this place has been held up!" said the manager.
"Who are you?"
"Richard Fouts, manager of The Blue Steer."
The stocky man pulled out his badge. "Marsh Hutchinson, Hillside Division," he said.
Then he looked at Denning. Marsh took out his pen and pad.
"Who are you?"
"Marvin Denning, customer."
"He knocked one of those robbers right to the floor," said Diana.
"That right?" the stocky man asked Denning.
"Yeah, I kicked him in the balls."
"Why?"
"Is there a better place?"
"What'd he look like?"
"He looked like a man wearing a stocking mask."
"Height?"
"About 5-7."
"Weight?"
"Say, 145."
"Anything to distinguish him?"
"What do you mean?"
"What was the most outstanding feature you noticed?"
"He was carrying a .357 Magnum."
The stocky man inhaled, exhaled. "Denning, there's something I don't like about you."
"Hutchinson, we're even. There's something I don't like about you."
"O.K. You stay where you are."
He began questioning the manager of The Blue Steer.
Diana looked at Denning.
"Mind if I sit down? This whole thing has been too much for me."
"Sit down, sure."
Denning felt the whole booth give way as Diana put her buttocks down.
"You're brave," she said, "you're a brave man. I saw what you did."
"O.K." said Denning.
"I know this may shock you, and I know it will sound weird and crazy but...I'd like to do something nice for you. Are you shocked?"
"No."
"Will you let me do something nice?"
"Sure."
"After all this is over we'll go to my place. Leave the steak. I'll cook you something better. Do you think I'm bold?"
"No."
"You know," Diana laughed, "when he put that gun to my head, I thought, I might die and I've...I've never had a man. Isn't that terrible?"
"I guess it happens sometimes."
"I know I'm fat...I'm embarrassed."
"It's all right."
"I should get you another wine."
"Why don't you?"
Diana struggled up and worked her way toward the kitchen.
Later, in the dark at Diana's place, he worked away. Denning hadn't worked so strenuously since he had been on a construction gang after high school and before college. Diana was groaning and moaning.
"Hold still, for Christ's sake!" he implored her.
Denning worked on, a good four minutes more, substituting fantasy after fantasy in his mind. Finally, he rolled off. He was in a sweat, inhaling and exhaling heavily. His head wound had broken open and he could feel a trickle of blood running down the back of his neck.
"Marvin," she said, "I love you."
"Thank you, Diana."
He got up and walked to the bathroom. He wetted a towel, cleaned off, then took the dry part of the towel and worked at the blood on his neck and head.
Well, many a man went to his death without having had a virgin. He wouldn't be one.
He threw the towel on the floor, walked out of the bathroom, through the bedroom and into the kitchen. He poured himself a glass of water at the sink and drank it down.
He looked around. Diana had a nice place. Maybe she got a lot of tips out of sympathy.
He found a can of beer in the refrigerator, cracked it, and sat at the breakfastnook table, sipping and smoking a cigarette he had found in a pack on the table. He finished the beer and the cigarette, walked back to the bedroom. Diana was in the bathroom. He began getting dressed. He heard her singing in the bathroom. Then the door opened and she walked out dressed in her nightwear. She saw him dressing and the happiness on her face vanished.
"Oh, you're leaving?"
"Yes."
"Will I see you again?"
"No."
"Oh, my God..." She walked slowly over to the bed. She sat on the edge of the bed, her back to him. She just sat there, looking very large. The lights were out in the bedroom and just the light from the half-open bathroom door shone in.
Denning sat on a chair lacing his shoes.
The vision of the bridge now sat in the center of his brain, it beckoned, how it beckoned him once more. The water pulled at him as if it were a magnet.
Denning finished lacing his shoes, stood up.
"Goodbye, Diana."
She didn't answer. She just sat there. Denning could see little shivers running through her body. She was sobbing very quietly, trying to hold it back. It was almost obscene. Diana's head was bent forward. As Denning looked it seemed almost as if he were staring at the back of a large headless body.
"Listen," he asked after a long pause, "you got anything to eat around here?"
"What?"
"I asked if you had anything to eat around here."
She raised her head, turned.
"Oh. Oh yes, Marvin, I have a bottle of wine and a couple of steaks and some vegetables."
"Shall we have dinner?" Denning asked.
Diana rose from the bed as if she were weightless. It was very strange. Then she went off to the kitchen.
Denning took off his coat, sat back down in the chair, took off his shoes, stockings, his pants and when she came back he was still in his shirt and shorts.
Diana walked through the doorway carrying a wine bottle, two glasses, the wine opener. She was having a little struggle carrying all that and she was laughing, not a loud laugh, but a continuous little joyous crazy laugh.
The light from the half-open bathroom framed her body, her face, the two glasses, the wine bottle, the wine opener.
Never before in the 46 years of his life had Marvin Denning seen a more beautiful woman.
## confession of a genius
during world war two
some of the worst
writing of our time
appeared in books
and magazines,
it was truly
regrettable.
I lived
alone and
insane in tiny
rooms
being neither a
soldier nor a
writer.
it is possible to
be truly mad
and to still
exist
upon scraps
of
life.
I knew my
name,
was able to
dress myself,
was able to
speak the
language
but I was
entirely
inept,
without design,
I was a
meaningless
conglomeration
of
ideas.
I was an
idiot.
the army didn't
want me,
women didn't
want me
and I didn't want
myself.
I was a
husk.
yet twice
I found myself
with a typewriter.
I wrote a short
story which was
accepted by
a leading
magazine.
and I wrote
another which
appeared in an
intercontinental
journal
along with
Henry Miller
and
Camus.
then I hocked the
typewriter and
stopped
writing.
I felt that what I
had written was
meaningless.
I went from
city to city
from room to
room
from bar to
bar.
the war
ended and I
continued
existing in that
manner.
I read the
successful writers
and decided that
they too
were
meaningless.
I really didn't
begin writing
again
until I started
living with
women.
they startled
me
out of my
stupor,
dropped me
splashing and
thrashing into a
new
confusion.
my work began
to appear
in literary magazines.
people hated me
for the way
I wrote about
women.
but these people
never met the
women I
lived
with.
I was only
photographing
in words
the reality of
it all.
I wrote of my
horrible women
and my
horrible jobs
and the first damn
thing you knew
I had
half-a-fame.
I noticed that the
sycophants and
weaklings were
writing poetry.
so,
I tried that
too.
it was
easy.
the whole game
was just a matter
of tossing your
stuff at
them.
I gave readings,
packed them in,
I drank throughout,
insulting them,
tossing the
crap.
they hated it
and loved
it,
they ate up
my crap.
and through it
all
I had this
feeling of
bored
disinterest.
but then I
noticed that
the women I went
with were getting
younger,
with better bodies,
longer hair,
more light to their
eyes.
it was
paying off.
I no longer had to
hock typewriters
or work horrible
jobs.
I had become
something to
some
people.
others had
better sense.
but I was the
same
half-shot
asshole that
I had
always
been,
I was nothing
at all
but somehow
I had stumbled
into a lucky and
easy
game,
a shell game,
a hustle,
a lark,
a sunny
midnight,
a stance,
an
out,
an
in,
and yes I've been
there
ever
since.
## traffic report
here in Los Angeles
on the freeways
it's like the Wild West
again.
many of the drivers carry guns
and if you cut them off
or irritate them in any manner
with your driving,
they simply pull up, point their
guns and begin
firing.
life has gotten to be too much
for many of us out
here,
the razor's edge is always
up
and any slight, slight as it might
be
becomes the ultimate and final
challenge.
many wait for it, many even hope
for it.
but out of it all, something else
has emerged:
far more polite driving habits.
who the hell wants to catch a
.32 caliber bullet in order to gain
3 car lengths in
heavy traffic?
me?
I'm so polite I'd make a nun
puke.
I prefer to die by my own
hand.
## hands
I'm not even drinking
and I look down at my
hands and they look
large.
unfortunately for me
I've always had
small hands.
the hands are the
tools
for fist fights,
in gripping an
ax,
in strangling
and
related
exercises
I have always been
disadvantaged.
but now
my hands look
large.
I look down at
them
and they grow
larger.
they keep growing
it's
marvelous.
now I can
beat hell out of
some guy.
I decide to go
downstairs and
show my wife
my new
hands.
"look!" I'll say.
"look!"
and I'll hold
out my
hands.
and she'll say,
"what?
what is it?"
I decide not to
go downstairs.
I just sit here
and look at
my hands.
it is one of my
better
evenings.
yesterday I was
very
depressed.
## final score
at the track today
read where Kosinski
did it in the bathtub
with a bag over his
head.
bad health was
inferred
but loss of
stature and literary fame
are very unhealthy
to some.
plus New York
publisher's parties,
power plays,
and
the hint that
he had outside
help writing
his books.
he had friends
at The New York
Times,
enemies at the
Village Voice.
not killed by the
Holocaust,
he couldn't live
with the
critics.
bag over his
head
in a bathtub
full of
water.
what Hitler
couldn't do,
he did to
himself.
happy
journey.
## the misanthrope
I've been accused of being
one.
well, I'm the ruins of Athens,
you know.
I'm always working to
rebuild, I'm on the
mend.
when I am with people
something gets subtracted
from me.
most people are hardly
joyous and seldom
interesting.
I listen to their complaints,
take note of their
braggadocio,
their unoriginal
insights.
they yawn my life
away.
you ask me to embrace
them?
I don't hate them,
I don't want to defeat
them or kill them.
I just want to get away
from them.
it is when I am alone
that I feel at my
best.
it is my normal
way,
it is when I smooth
out, float,
it is when whatever
light there is
enters
me.
the ruins of Athens.
the old bum.
the cockroach in the
cathedral.
the good wine.
the mental conversations
with Mrs. Death.
the dream of golden
windmills.
the inhaling of
life.
the soaring confinement.
the gentle walls.
if preferring this to
Humanity makes me a
misanthrope
then I
am
to the hilt,
gladly
now
here
tonight
tomorrow
next year
alone with
aloneness
finally.
## putting it to bed
the first poem is the last poem is the
best poem
pulling its stockings off
late in the night of the
morning
the best poem is the last
poem
the poem poem poem
as nine tenths of the people of
this city are
asleep
I am up with the murderers and the
thieves and the cab drivers
and some of the
prostitutes
and many of the drunks
and the mad
and the insomniacs
and the etc.
I murder the language
I steal the language,
I drink the language,
I am mad with the language
in the cab of my mind,
I am a whore.
the last poem
running out of my fingers
soon I will be asleep with
my wife and my
cats.
we will be all in the same
room,
still,
except for some wheezings
and turnings
and this last poem will
sit in this room
and I will be in the other
room
and some day you will
read this poem,
perhaps,
and think,
that guy makes too much
of it.
the last poem
the last poem
the best for me.
## the trash can
this is great, I just wrote two
poems I didn't like.
there is a trash can on this
computer.
I just moved the poems
over
and dropped them into
the trash can.
they're gone forever, no
paper, no sound, no
fury, no placenta
and then
just a clean screen
awaits you.
it's always better
to reject yourself before
the editors do.
especially on a rainy
night like this with
bad music on the radio.
and now—
I know what you're
thinking:
maybe he should have
trashed this
misbegotten one
also.
ha, ha, ha,
ha.
## block
in the past two months the poems have
riveted themselves to paper in ungodly
numbers
and if a poet may judge—
most of them were of high quality.
now I have become spoiled,
I walked into here tonight expecting
more luck
but the night has been slow.
and rightfully so—
occurrence must precede action,
the tank must refill.
writing, at its best, is not a contest,
it's not even an occupation,
it's a hazardous madness
that arrives at its own
behest.
prod it and you lose it.
pretend, and the words fall
ill.
when the lulls arrive there is
nothing to do but
wait,
do other things.
the writing must leap upon you
like a wild beast.
there are none of those in this
room with me
tonight.
they are elsewhere.
they are with somebody
else.
so all I can do is sit in this chair
tonight
and tell you that I can't
write.
there are other things to do.
like now I am going downstairs
to see my wife
and my 6 cats
and they will see me
and we will look at each
other.
it will be all right.
I'm sure it
will.
they might even remember
me.
## storm
a storm at last in this damned Los Angeles
desert,
even the lights went out in the neighborhood,
most of the people asleep,
the drunks just pour another drink,
I poured another drink,
1:42 a.m.
the lights go back on,
Brahms begins to play on the radio again,
I think of Turgenev, just for the hell of it,
just because I like his name.
there are good names: Mozart, Celine,
Artaud, Bach.
some names ring through and stick.
anyhow, it's raining and raining and raining.
and Joe Louis is dead and Ty Cobb is dead
and it's been a long time since the Waner brothers
patrolled the outfield in Pittsburgh
and whatever happened to Smith Brothers cough
drops?
I used to eat them like candy.
we need the rain.
we need the rain.
we need it.
I used to eat those cough drops like candy and I had
a dot-and-dash set and I knew the Morse code and I
sent out S.O.S.s for years but help never
came.
Turgenev.
I wish my name was Turgenev.
hello, I am Ivan Turgenev and it's raining and I'm writing
about the rain
it rains hard here in Russia and the nights are black and
the days are black
and my girlfriend keeps telling me about our leader who has
arching eyebrows.
and I say, "oh, yes, very interesting..."
my name is Turgenev and it's raining and we need the
rain.
ran into Gorky the other day and he said rain was just so
much capitalist bullshit.
crazy guy, crazy.
well, it's 1:58 a.m. and I am sleepy.
sleeping in the rain helps me forget things like I am going
to
die and you are going to die and the cats are going to die
but it's still good to stretch out and know you have arms
and
feet and a head, hands, all the parts, even eyes to close
once
more, it really helps to know these things, to know your
advantages
and your limitations, but why do the cats have to die, I
think that the
world should be full of cats and full of rain, that's all, just
cats and
rain, rain and cats, very nice, good
night.
## the similarity
lost another 3 page poem to this computer,
reminds me of the past,
you know, with some women
you leave them in bed
before going off to the warehouse
to work
and you ask them,
"Baby, you going to be here when I
get back?"
"sure, Hank, I love you..."
and you come back to find the bed cover
flipped back, they slipped out right after
you drove off,
didn't even empty an
ashtray.
well, you're a fool but you don't give up
on women on account of
that.
the next one might be
better.
and this poem can't replace the one
lost
but it's a good shot in the dark
which beats
none at
all
maybe.
## MY MADNESS
There are degrees of madness, and the madder you are the more obvious it will be to other people. Most of my life I have hidden my madness within myself but it is there. For instance, some person will be speaking to me of this or that and while this person is boring me with their stale generalities, I will imagine this person with his or her head resting on the block of the guillotine, or I will imagine them in a huge frying pan, frying away, as they look at me with their frightened eyes. In actual situations such as these, I would most probably attempt a rescue, but while they are speaking to me I can't help imagining them thus. Or, in a milder mood, I might envision them on a bicycle riding swiftly away from me. I simply have problems with human beings. Animals, I love. They do not lie and seldom attempt to attack you. At times they may be crafty but this is allowable. Why?
Most of my young and middle-aged life was spent in tiny rooms, huddled there, staring at the walls, the torn shades, the knobs on dresser drawers. I was aware of the female and desired her but I didn't want to jump through all the hoops to get to her. I was aware of money, but again, like with the female, I didn't want to do the things needed to get it. All I wanted was enough for a room and for something to drink. I drank alone, usually on the bed, with all the shades pulled. At times I went to the bars to check out the species but the species remained the same—not much and often far less than that.
In all the cities, I checked out the libraries. Book after book. Few of the books said anything to me. They were mostly dust in my mouth, sand in my mind. None of it related to me or how I felt: where I was—nowhere—what I had—nothing—and what I wanted—nothing. The books of the centuries only compounded the mystery of having a name, a body, walking around, talking, doing things. Nobody seemed stuck with my particular madness.
In some of the bars I became violent, there were alley fights, many of which I lost. But I wasn't fighting anybody in particular, I wasn't angry, I just couldn't understand people, what they were, what they did, how they looked. I was in and out of jail, I was evicted from my rooms. I slept on park benches, in graveyards. I was confused but I wasn't unhappy. I wasn't vicious. I just couldn't make anything out of what there was. My violence was against the obvious trap, I was screaming and they didn't understand. And even in the most violent fights I would look at my opponent and think, why is he angry? He wants to kill me. Then I'd have to throw punches to get the beast off me. People have no sense of humor, they are so fucking serious about themselves.
Somewhere along the way, and I have no idea where it came from, I got to thinking, maybe I should be a writer. Maybe I can put down the words that I haven't read, maybe by doing that I can get this tiger off my back. And so I started and decades rolled by without much luck. Now I was a mad writer. More rooms, more cities. I sunk lower and lower. Freezing one time in Atlanta in a tar paper shack, living on one dollar and a quarter a week. No plumbing, no light, no heat. I sat freezing in my California shirt. One morning I found a small pencil stub and I began writing poems in the margins of old newspapers on the floor.
Finally, at the age of 40, my first book appeared, a small chapbook of poems, Flower, Fist and Bestial Wail. The package of books had arrived in the mail and I opened the package and here were the little chapbooks. They spilled on the sidewalk, all the little books and I knelt down among them, I was on my knees and I picked up a Flower Fist and I kissed it. That was 30 years ago.
I'm still writing. In the first four months this year I have written 250 poems. I still feel the madness rushing through me, but I still haven't gotten the word down the way I want it, the tiger is still on my back. I will die with that son-of-a-bitch on my back but I've given him a fight. And if there is anybody out there who feels crazy enough to want to become a writer, I'd say go ahead, spit in the eye of the sun, hit those keys, it's the best madness going, the centuries need help, the species cry for light and gamble and laughter. Give it to them. There are enough words for all of us.
## pastoral
listening to a piano and a
trumpet
mix it up
on the radio,
the express purpose of
existence remains
unsolved.
all 6 cats are asleep
now,
12:30 a.m.,
my wife is across the
street visiting with a
neighbor lady.
good, they need
it.
the racetrack was
closed today
and I was a lost
fat
butterfly.
most days go
nowhere
but the avoidance
of pain and
dissolution are
lovely.
they will arrive
soon enough,
fecund,
recharged,
valiant,
evermost.
now there is a
chorus on the radio,
they sing to me
as I clean my
fingernails with a
toothpick.
no thunder
tonight.
no tiger roaring
in my brain.
I am resting.
I rub my face with
my fingers.
I am waiting for
war.
the centuries have
trained me
well.
I lean back in the
chair
and smile
to myself,
for myself,
for everything,
for nothing.
this is absolutely
great.
this is as good as
it is ever
going to
get.
## finis
those times are gone now
but I remember the 50s
at the track, people crushed
around the bars, laughing,
wise cracking and there were
fist fights, there were crowds
of 50 and 60 thousand people
on the weekends, it seemed
everybody had money and
even the mutuel clerks were
happy; good-looking prostitutes
were everywhere and
Willie Shoemaker was young,
even Johnny Longden was
young and Ralph Neves
smoked cigarettes in the
walking ring, you saw George
Raft, and there were 8 races
instead of 9 and there was
the feeling that you were
going to make money and if
you didn't, what the hell,
they were running the next
day.
and there was always a
woman with you and if there
wasn't there would be
that night.
it was gamble and drink
and forget
tomorrow.
those were the 50s.
go out there now, it's sparse
and drab, it's like a home for
the mentally deficient.
nobody's laughing,
the rent money's up
for grabs and
the ladies are old, white-
haired, they sit together,
bet two dollars to
show.
they are terrified of
everything.
they should be.
the bartenders have
nothing to do.
the track gives away
prizes, trinkets
trying to draw the
crowds.
the track offers
exotic betting.
the crowd does not
arrive
and what there is
begins leaving
after each race.
there are now 9 races,
it doesn't matter—
there is no money to
bet,
the track is a funeral
parlor, it is the end
of life.
the sun can't make it
through the filthy
air.
it gets dark soon.
the people move
slowly toward the
exits.
their faces are
unhappy, their faces
are
murdered.
it is a procession of
the dead.
it's the 90s.
it's 40 years back to the
50s,
it's centuries back.
it's the 90s.
nobody's laughing.
tomorrow is all too
close.
the last race is here.
## that rare good moment
when the gods relent
when the dogs back
off,
you are sitting in a
Sushi joint
working the chopsticks
between two tall bottles of
Kirin
and you are quietly thinking
about any number of Hells
you have
survived,
probably no more than
anybody else
but they're yours to
remember.
survival is a very
funny thing,
and it's weird,
passing safely through all the
wars,
the women,
the hospitals, the jails,
youth,
middle-age,
suicide dances,
decades of
nothingness.
now in a Sushi joint
on a side street
in a small town,
it all passes before
you
quickly
like a bad/good
movie.
there is this
strange feeling of
peace.
not a car passing
in the street,
not a sound.
you hold the chopsticks
as if you have used
them for
centuries,
note a tiny piece of
coleslaw at the
edge of your
plate.
there, you have it,
all that style,
grace,
god damn it's so
strange
to feel good to
be alive,
doing nothing
exceptional
and feeling
the glory of
that,
like a full
choir behind
you,
like the
sidewalks,
like the
doorknobs.
grass grows in Greece
and even ducks
sleep.
## doesn't seem like much
my editor-publisher who is about
60
writes me,
"let's go another ten years.
you up to it?"
I'm 70.
ten years?
that's just a walk around the
block.
I feel almost
insulted.
how about 30
years?
a man can get a little
work done in that
time.
I don't answer my editor-
publisher.
is he getting
tired?
what else would he do
if he wasn't publishing
me?
work his garden?
play golf?
travel?
well, in a sense I do
answer him
by sitting down to the
keyboard
and typing out
poems
in different type faces,
on different
colored papers,
just to pep up the
show,
and the content is
good too—ripely
burning and also
laughing a
bit.
ten years?
this is 1991.
the year 2,000 will
come and go
in the blink of an
eye.
hey, editor-publisher,
how about the year
2020?
then we can putter in
our gardens and write
our goddamned auto-
biographies.
you up to it?
## strange luck
slapped across the face with a
shit brick
he stopped at Biff's Bar
for a quick one and stayed
five years.
he survived through and with
a half-witted
guile.
he was evicted from room after
room.
within a four block area he
had lived in nine
rooms.
each was about the same:
dirty, small, gloomy.
he lived on loaves of bread
alone.
at rare times he added
bologna or peanut
butter.
in the bar it was beer,
beer, beer
and at rare times,
whiskey or vodka or
scotch or gin.
gin didn't do much for
him but he
welcomed it.
nobody knew where he had
come from, what he wanted.
the others accepted him
as a fixture, an oddball
fixture.
the women, largely, ignored
him.
he was neither bitter, angry
or displeased
he was just there.
then, one day, after 5 years
he just walked out and was
never seen there
again.
now he owns a large home, a
late model car,
there is a spa, a swimming
pool, a vast garden, a
wife.
sometimes you will read of
him in the
metropolitan
dailies.
he still drinks,
but moderately.
beer, wine or an occasional
vodka.
he drinks alone
in an upstairs room.
he sits at the keyboard of an
expensive
computer.
those few who remember him
can't believe the
transition.
he knows that is all
just game-playing by the
gods.
he feels no different than
he ever
did.
he is no less or no more
than he was
then.
he drinks at the computer
and waits for death
as he has always
done.
it's hard but it's
fair.
and strange and strange and
strange and
strange.
## until it hurts
you have to wait until it
hurts, until it clangs in
your ears like the bells
of hell, until nothing
else counts but it, until
it is everything,
until you can't do anything
else
but.
then sit down and write
or stand up and
write
but write
no matter what
the other people are
doing,
no matter what
they will do to
you.
lay the line down,
a party of one,
what a party,
swarmed by the
light,
the time of the
times,
out of the tips of
your
fingers.
## DEATH IN THE AFTERNOON
We are in Musso's Restaurant around 2 p.m., it's the best time there, the tablecloths aren't on the dinner tables yet and it's quiet. The tourists are all at Disneyland. I'm having a turkey sandwich with a side order of fries. I don't know what Blackwell is eating. It's a large rectangle of meat very well done (almost black) but inside it's a bright red. He slices very thin portions and chews each piece with great reverence. Outside, Hollywood Boulevard has disintegrated into skid row. Just Musso's stands there as it has since 1919, the last bit of class in sight. It is a good place to be when you are feeling down and I am usually feeling down.
"Well, what ya gonna do?" Blackwell asks me.
"Do? I'll just get rid of the girl. I'm too old now to take any more gorings. I feel like an old matador who wants to hang it up."
"You've lived with a dozen women in the last 15 years. How ya gonna break the habit?"
"How can you eat that raw meat?" I ask Blackwell. "Don't you feel as if you're eating something alive?"
"Better that than the other way around."
"Pardon me, I've got to piss. Order me another beer, will you?"
I get up and walk toward the rear. There is Fellini leaning against the wall. Not that Fellini. This one is a waiter. Whenever Fellini sees me he unfurls this great big smile but it's almost always as if he was laughing at me.
"How are the ponies going, buddy?" he asks me.
"Night harness racing right now..."
"I know, but there is also the thoroughbreds down at Del Mar. I was there last Sunday. Didn't make much. $280. Had my wife along. She spoiled my concentration."
Fellini always wins, he says.
I go in to piss, I do, then wash my claws, come out. Fellini is still standing there. Still smiling like a blazing sunset.
I stop.
"Reminds me," I tell him. "Damndest thing happened at the harness races the other night. Got a lot of things on my mind, you know. For example, I got these 3 creatures in my front hedge, large as cats. They come out every night and raid my vegetable garden. Anyhow, it's the last race, I'm a few bucks in the hole, maybe 5, and I decide to go $50 win, and besides being distracted by the hookers with no panties on, I get a toothache. I'm also trying to get the late action, I'm watching my horse, and at the last flash my horse drops from 5/2 to two-to-one and I run up to the window and bet $50 win."
"What happens?" Fellini asks, still smiling.
"What happens? I look down at my ticket later and I realize I'm really fucked!"
"Oh yeah?" he smiles.
"Yeah. I had gone up and hollered out, 'Fifty-to-win on the 2!' I had been thinking odds, you know what I mean? I had mistakenly bet on the two horse and he was reading fifty-to-one on the board!"
"A guy will always find a way to lose," smiles Fellini.
"Only," I say, "the 2 gets up in the last jump and pays $108.40. I get back $2,710.00."
Fellini's face darkens. The smile jumps from that physiognomy, runs into the men's room and slithers down the nearest latrine.
I walk back to the table feeling good, sit down and Blackwell is still slicing at his red death lunch. I take a pull of beer.
"The old matador returns," chews Blackwell.
"What?"
"You called yourself the old matador, said you didn't want to be gored anymore."
"Don't worry. I'll get rid of her. Just finish your kill."
"Reminds me," he says, "I had a horrible hangover the other morning. Been drinking red wine and scotch. I can't get out of bed. I kick on the tv. And there's one of those old movies they've shown over and over. Anyhow, I watched. It was about an old matador..."
"Uh..."
"I watch, and the way I get it, the old matador had been or was, the greatest."
"Huh..."
Then Blackwell looks at me, "Aren't you gonna finish your turkey sandwich?"
"Not today..."
"Can I have it?"
I shove the sandwich toward him.
"How about the fries?" he asks.
"No, I'm keeping my fries."
"Oh," says Blackwell. "...Anyhow, where I come in on this film the old matador is very upset. He's in his dressing room, sitting in front of the mirror, arranging himself, getting ready, you know. His handlers are running around like sissies. Suddenly the old matador rips off his fake pigtail and throws it to the floor. 'What the hell's the matter?' one of his handlers asks him."
Blackwell stops. "Hey, listen, buddy, isn't that Jonathan Winters over there, sitting at that table?"
I look: "Yes, it is...Don't stare. He's been in the funny farm, you know. Don't stare. Let him eat in peace."
Blackwell sighs, "Well, anyhow the old matador says, 'I'm not going on!' 'What? What? What?' the 3 or 4 handlers ask. 'I'm getting out of here!' the old matador screams. He knocks down his handlers and runs out the door."
I look up. It's Fellini. He's still not smiling. He looks at me: "I don't believe that story you told me about the 50-to-one shot."
"Are you our waiter?" I ask him.
"No."
"Then, will you please inform our waiter that I wish another beer and that my friend here would like a glass of Corvo Salaparuta White, and if you don't have that, then please, the nearest thing..."
Fellini walks off to find Swanney, our waiter. Swanney is a real nice fellow, he's always consoling me about those animals in my front hedge who eat the red cabbage, the carrots, the zucchini and the eggplant.
"Where was I?" asks Blackwell.
"The way I see it, the old matador has decked a few of his boys and is running out the door..."
"Oh, yeah, he has decided not to fight at the arena that day with the rising young matador on the same card. There's been so much said about the young matador, and on top of that the old matador had just recently seen his best friend killed in the ring, another old matador..."
"You must have been really sick to keep watching that movie."
"Yeah. Mixing the drinks like that."
"Here come our drinks. Good old Swanney!"
He puts down the drinks, looks at me. "Are those animals still eating your celery stalks?"
"Yes, Swanney. I am considering Capital Punishment."
"Anything else, sir?"
"Isn't that enough?"
"All right," continues Blackwell, "the old matador leaps into his car and drives away, but guess what?"
"What?"
"He's followed by...Jonathan Winters is leaving."
"We all must, at some time, do that."
"You're right. Anyhow, the old matador is being followed by this lovely rich redhead. They met casually one time down by the bull stables, the rich redhead turning it on and the old matador hardly noticing. I mean, why should he? Don't those guys get a gift of a virgin after every great performance?"
"Here," I say, "take my fries..."
"Oh. All right. So, the rich redhead follows him. Her car is faster. The old matador can't elude her. He stops his car. He gets out. 'Why are you following me?' he asks."
Fellini is back. "Listen," he says to me, "I wasn't meaning to be impolite. What I was inferring was that maybe we both exaggerate about the horses..."
"Fellini," I say, "show me a horseplayer who doesn't and I'll show you a liar..."
Fellini leaves.
"So," says Blackwell, "she switches on her car radio while the old matador is standing there and he hears the mob at the arena, they are going crazy with sorrow and anger because the old matador has run off..."
"He rushes back to the arena?" I suggest.
"No. She looks at him. She says, 'We need to talk. Follow me!' And then she leaps into her sports car, spins it around in the dirt road as he watches her. Then he leaps into his car and follows..."
I flag Swanney for refills as Blackwell consumes my last fry and continues. "They get to her place, a mansion. They walk through the mansion and go out to a garden patio, sit at a table. The servant arrives with refreshments."
"Now," I suggest, "they will begin to commiserate with each other about his tormented soul and that commiseration will lead to further torment..."
"Do you think everybody has bad luck with women like you do?"
After that we fall into 4 minutes of silence. Swanney comes with more drinks and Blackwell orders a plate of fries. He looks at me. "Eating is better than fucking, it takes longer and you can do it more often."
"Do tell me more about the old matador..."
"O.K. They are in the patio and the old matador looks around. 'You own all this?' he asks. The redhead nods in the affirmative. He explains, 'I admire wealth.'"
"That's when you turn the set off?"
"Right. I get up, puke. Then I mix half a bottle of beer with the same amount of tomato juice, sprinkle in a touch of paprika and ground pepper, drink some and switch the set back on..."
"They're drunk?" I ask, "and she's holding a red tablecloth and he's charging it like a bull?"
"No, there's been a passage of time. The old matador has been living there 3 or 4 days when his new rival, the young matador, arrives. The rich redhead asks the young matador what he wants. 'I know that he is here, Senora!' he replies. And he goes on to make a speech about how he has worshipped the old matador since he was a boy and he has dreamed of fighting on the same card with him..."
"How terribly dull. Can I have one of your french fries when they arrive?" I ask.
"Sure..."
"The young matador and the rich redhead stare at each other. Then the young matador says, 'I must go!' He seems to be a dull fellow but I guess all you need to be a bullfighter is a lack of imagination and good reflexes..."
"Oh," I say, "please tell me what happens next!"
"Sure. Before the young matador can leave the old matador steps up and tells the redhead, 'I must go back!'"
"It is a great moment," I say.
We fall into another 4 minutes of reflective silence. The skid row of Hollywood Boulevard bakes in the sun outside as we sit lost in the heart of Mexico. The fries arrive. Blackwell passes the plate. I spear the biggest, fattest, yellowest of them all, bite off a hot end as Blackwell continues.
"So, of course, the next scene we are there. The bull ring. The young matador goes on first. He makes glorious and impossible movements as the bull charges—such innovative classicism. Again and again. And then—the perfect kill."
"One more fry and I won't bug you anymore."
Blackwell passes the plate. "Say, wasn't that Allen Ginsberg who just walked in?"
"No, that was Andy Warhol."
"Well," says Blackwell, "next scene. On walks the old matador to a chorus of boos, pure hatred."
"Is there any other kind?" I ask.
"Hell, I don't know. Anyhow, the old matador just stands there. He looks pitiful like he can't get off the dime. His buttocks are all bunched up and quivering..."
"On a woman that wouldn't be bad."
"I know," says Blackwell. "Anyhow, the old matador draws the meanest bull of them all: 'Muerto.'"
I flag Swanney for a new set of drinks. (When I want to get a waiter's attention I always wrap a napkin around a fork and wave. When I am with the ladies it always disgusts them, but waiters respond when they see it.)
"Anyhow," continues Blackwell, "the old matador draws Muerto but the picadors screw up the banderilla job—very sloppy about it. When Muerto makes his first charge at the old matador, the picadors hardly touch him as Muerto rushes past the old matador, who almost fertilizes his shorts."
"No shit?"
"The old matador shakes the cape through the laughter of the crowd and Muerto charges again. This time the old matador is a bit more graceful."
"Ah..."
"Yes. The crowd grows quiet. As Muerto moves in again the old matador seems to find his legs, his youth, his courage...he executes a perfect Digaxxello!"
"A what?"
"Forgive me. It's been 40 years since I've read Barnaby Conrad or Hemingway..."
"Do you know that Faulkner used to drink here at Musso's?"
"Yeah, anyhow, the old matador has Muerto charmed. Muerto moves in again to be baffled by the soundless Tearasouloh..."
"As the crowd roars?"
"...wildly, remembering the old matador at his best, but never...like this! The massive and beautiful bull, an instrument of the old matador's will..."
"Andy Warhol just left," I say. "I think we've been here a long time too..."
"He's probably going back to New York," says Blackwell.
"I hope," I say, "so."
"Anyhow," says Blackwell, "there are more brave and symphonic moves by the old matador. Now, Muerto the magnificent bull is helpless. The time for the kill has come."
"And here," I say, "come our drinks."
They are set down before us. We nod, pick up our drinks, click a toast in the Spanish manner.
"...Up high in the stands, sitting in a box with the President of Mexico, the rich redhead's eyes glisten with love for the old matador."
"He know where she sits?"
"Yes. And in the midst of a Figeralla he looks up and catches her eye, smiles, waves, and that's all the opportunity Muerto needs. He gets the left horn in, guts the matador, lifts him high, shakes him like a sawdust doll, shows him to the sun..."
"Shit..."
"But he's not quite dead. Don't you go to the movies?"
"Mostly just to eat popcorn in the dark."
"Well, the next scene is in the infirmary. The old matador is stretched there on a table with many people milling around. The old matador raises his hand and gestures for them to leave...and they do...and he's left alone with the redhead. She looks into his eyes. She says, 'You were beautiful!'"
"The old matador," I ask, "smiles?"
"Yes, and she kisses him on the mouth, hard. Then she straightens and looks sadly down at him as the people file back in."
"Great timing."
"She turns, tells them, 'The matador is dead...'"
"You know," I tell Blackwell, "when I'm in a real depressed mood—which is most of the time—it's always great to listen to you tell some long story which fails to make me laugh."
"I'm sorry. Maybe we can try again sometime?"
"Sure. But what was it you wanted to see me about today?"
"Hey," says Blackwell, "I thought you wanted to see me..."
Out in the parking lot I can't quite find my car. I've lost my parking ticket. I feel like the old matador, I am surely much older than the old matador.
I find my car, get in. It starts.
The sun is going down.
I drive out of there more depressed and alienated than ever. The beautiful people are useless and everybody else is dull.
I cut south on Cherokee, wait at the red light as some dried-up, worked-over, unimaginative 8 or 9 helpless citizens walk this way and that. I get the green light, move through the warm evening, get onto the freeway where I immediately incite a challenge from 3 kids in a souped-up Chevy. So I step on it, and here they come after me, leering, giving me the finger, as a shitty afternoon turns into a shitty evening. I luck out. They run into a traffic jam. I find the free lane inside, jump up to 85, 90, then check the rearview mirror, see them caught back there, and I am in San Pedro.
I find my place, pull into the driveway, park it, get out, just another old matador. But inside, as I open the door, my favorite white cat, The Jinx, leaps up into my arms and suddenly I am in love again.
## the gods
I sit here on the 2nd floor
hunched over in yellow
pajamas
still pretending to be
a writer.
some damned gall,
at 71,
my brain cells eaten
away by
life.
rows of books
behind me,
I scratch my thinning
hair
and search for the
word.
for decades now
I have infuriated the
ladies,
the critics,
the university
suck-toads.
they all will soon have
their time to
celebrate.
"terribly overrated..."
"gross..."
"an aberration..."
my hands sink into the
keyboard
of my
Macintosh,
it's the same old
con
that scraped me
off the streets and
park benches,
the same simple
line
I learned in those
cheap rooms,
I can't let
go,
sitting here
on this 2nd floor
hunched over in yellow
pajamas
still pretending to be
a writer.
the gods smile down,
the gods smile down,
the gods smile down.
## floss, brush and flush
sitting, talking through the
night, it's a
malfeasance trying to
feel good, the empty
beer and wine bottles
gathering, the ashtrays
runneth over,
twice-told jokes are
told again,
somebody's religiosity
is hurt,
politics limp in and out,
death comes in with heavy
shoes and kicks holes in
the air,
somebody complains of
bad luck,
forgotten movies are
discussed that would
rather have remained
forgotten.
nobody talks of books,
of paintings,
of the stock market or
the life of the
inch worm.
each person quietly
mocks the other person,
in a wholesome, good-
natured way
(of course).
some heads fall,
others
laugh.
it is an evening of
friends and
relatives.
the hours inch-worm
along.
they and we are in the
trenches
of hell,
throwing mud at the
fates.
then they grow weary of
the absurd battle and
leave
one by
one.
then there is just the
wife and
myself.
soon she goes up the
stairway
and I am left with
myself,
right back where I
began.
I sit there
lighting
cigars.
there are still things
to be
resolved
but what are
they?
I turn out the lights
and sit in the
dark.
then I see a strange headless
thing walk up to the
glass door.
it places its paws
high
upon the
door and
leans there.
its eyes are in its
belly.
one is gold and
glowing.
the other is green
with shots of
red.
I walk up the stairway,
climb into
bed.
my wife snores
peacefully.
the night is finished.
I am still alive.
the bluebird swallows
the worm.
the harbor tangles with
the fog.
morning swarms the
window.
I am a joke told
again.
I sleep.
## a great show
when I went to visit my friend
at the Motion Picture Hospital,
it was full of actors and
freaks and directors and
assistant directors and grips
and cameramen and film editors
and script writers and sound
men and etc.
some of them were sick
some of them were dying
but somehow it wasn't like a
regular hospital,
that special heavy darkness
wasn't there,
everything was:
"LIGHTS! CAMERA! ACTION!"
everybody still
on the
set.
at least, it seemed like that
to me.
as bad as most Hollywood
movies had been, were and
still are,
there remained the touch of
the brave and dramatic in
the air.
when I went to the cafeteria,
everything was on cue:
even the people in wheelchairs made
dramatic gestures, spoke in
senatorial tones; they had
fierce blue eyes,
white, carefully cropped
beards,
deliberate enunciations,
there was blithe bullshit,
a whole Shakespearean
afterglow.
dwarfs sitting at tables
eating blueberry
pie.
old script writers, all
looking Faulknerian
musing about their drunken
afternoons at Musso and
Frank's.
old dolls, once beautiful
now toothlessly munching
soft toast, poking at
peaches.
and almost all the rooms
were private,
arranged to bring in the
light of hope.
the nurses, as in all
hospitals
worked their asses
off,
and the doctors were
congenial,
good actors in a bad
scene.
and my friend, who was
dying, spoke to me
not of his death
but of his idea for
his next
novel.
he also spoke of the
crazies and geniuses
or would-be
geniuses
running
loose.
"we've got one of the
original Tarzans here,"
he told me.
"every now and then he
runs all over the
place
giving his Tarzan
yodel and looking for
his Jane."
"they let him run
loose?"
"oh, yes, he doesn't
harm anybody.
we rather like it."
well, my friend
died, so I didn't go
there anymore.
but it was a very odd
visitation.
death was there but
death was on camera
as He was so often in
Hollywood.
it was as if
everybody was ready for
the last scene,
having practiced it so
often
already.
and about a month
later
I read a small bit in
the paper:
Tarzan had
died,
perhaps he has gone
on to find his
Jane.
there are still happy
endings, aren't
there?
like my friend who
died
his books have become
famous throughout much
of the
world.
which is only half a
happy ending
but at least his widow
in Malibu
won't have to baby
sit
to have bacon with
her
eggs.
## epilogue
Fante gone to Hollywood,
Fante on the golf course,
Fante at the gambling tables,
Fante in a home in Malibu,
Fante a friend of William
Saroyan.
But Fante, I remember you
best,
in the 1930s
living in that hotel next to
Angel's Flight,
struggling to be a writer,
sending stories and letters
to Mencken.
the scream came from
the gut
then.
I heard it.
I still hear it.
and I refuse to imagine you
on a golf course
or in Hollywood.
but now it doesn't matter.
you're dead
but the good writing
remains
and the way you helped
me get the line down
the way I
wanted it.
I'm glad I finally met you
even though you were
dying
and remember when I
asked you,
"listen, John, whatever
happened to that
Mexican girl in
Ask the Dust?"
and you answered,
"she turned out to be
a goddamned
lesbian!"
and then the nurse
came in with your
big white
pill.
## Fante
every now and then it comes back to
me,
him in bed there, blind,
being slowly chopped away,
the little bulldog.
the nurses passing through, pulling
at curtains, blinds, sheets.
seeing if he was still alive.
the Colorado Kid.
the scourge of the American
Mercury.
Mencken's Catholic bad boy.
gone Hollywood.
and tossed up on shore.
being chopped away.
chop, chop, chop.
until he was gone.
he never knew he would be
famous.
I wonder if he would have given
a damn.
I think he would have.
John, you're big time now.
you've entered the Books of
Forever
right there with Dostoevsky,
Tolstoy, and your boy
Sherwood Anderson.
I told you.
and you said, "you wouldn't
shit an old blind man,
would you?"
ah, no need for that,
bulldog.
## it got away
lost another poem
in this computer,
it's like reeling in
a fish
and then it
escapes the hook
just as you reach
for it.
only this poem
wasn't a very big
fish.
the world won't
miss it.
it has swum
away to the
Netherlands.
and I'm baiting
my hook
again.
waiting for
the big
one.
## the luck of the draw
after decades and decades of poverty
as I now approach the lip of the
grave,
suddenly I have a home, a new car, a
spa, a swimming pool, a computer.
will this destroy me?
well, something is bound to destroy
me soon enough.
the boys in the jails, the slaughterhouses,
the factories, on the park benches, in the
post offices, the bars
would never believe me
now.
I have a problem believing myself.
I am no different now
than I was in the tiny rooms of
starvation and madness.
the only difference
is that I am
older.
and I drink better
wine.
all the rest is
nonsense,
the luck of the
draw.
a life can change in a tenth of
a second.
or sometimes it can take
70
years.
## let it enfold you
either peace or happiness,
let it enfold you.
when I was a young man
I felt that these things were
dumb, unsophisticated.
I had bad blood, a twisted
mind, a precarious
upbringing.
I was hard as granite, I
leered at the
sun.
I trusted no man and
especially no
woman.
I was living a hell in
small rooms, I broke
things, smashed things,
walked through glass,
cursed.
I challenged everything,
was continually being
evicted, jailed, in and
out of fights, in and out
of my mind.
women were something
to screw and rail
at, I had no male
friends,
I changed jobs and
cities, I hated holidays,
babies, history,
newspapers, museums,
grandmothers,
marriage, movies,
spiders, garbagemen,
English accents, Spain,
France, Italy, walnuts and
the color
orange.
algebra angered me,
opera sickened me,
Charlie Chaplin was a
fake
and flowers were for
pansies.
peace and happiness
were to me
signs of
inferiority,
tenants of the weak
and
addled
mind.
but as I went on with
my alley fights,
my suicidal years,
my passage through
any number of
women—it gradually
began to occur to
me
that I wasn't different
from the
others, I was the
same.
they were all fulsome
with hatred,
glossed over with petty
grievances,
the men I fought in
alleys had hearts of
stone.
everybody was nudging,
inching, cheating for
some insignificant
advantage,
the lie was the
weapon and the
plot was
empty,
darkness was the
dictator.
cautiously, I allowed
myself to feel good
at times.
I found moments of
peace in cheap
rooms
just staring at the
knobs of some
dresser
or listening to the
rain in the
dark.
the less I needed
the better I
felt.
maybe the other
life had worn me
down.
I no longer found
glamour
in topping somebody
in conversation.
or in mounting the
body of some poor
drunken female
whose life had
slipped away into
sorrow.
I could never accept
life as it was,
I could never gobble
down all its
poisons
but there were parts,
tenuous magic parts
open for the
asking.
I reformulated,
I don't know when,
date, time, all
that
but the change
occurred.
something in me
relaxed, smoothed
out.
I no longer had to
prove that I was a
man,
I didn't have to prove
anything.
I began to see things:
coffee cups lined up
behind a counter in a
cafe.
or a dog walking along
a sidewalk.
or the way the mouse
on my dresser top
stopped there,
really stopped there
with its body,
its ears,
its nose,
it was fixed,
a bit of life
caught within itself
and its eyes looked
at me
and they were
beautiful.
then—it was
gone.
I began to feel good,
I began to feel good
in the worst
situations
and there were plenty
of those.
like say, the boss
behind his desk,
he is going to have
to fire me.
I've missed too many
days.
he is dressed in a
suit, necktie, glasses,
he says, "I am going
to have to let you go."
"it's all right," I tell
him.
he must do what he
must do, he has a
wife, a house, children,
expenses, most probably
a girlfriend.
I am sorry for him.
he is caught.
I walk out into the blazing
sunshine.
the whole day is
mine.
temporarily,
anyhow.
(the whole world is at the
throat of the world,
everybody feels angry,
short-changed, cheated,
everybody is despondent,
disillusioned.)
I welcomed shots of
peace, tattered shards of
happiness.
I embraced that stuff
like the hottest number,
like high heels, breasts,
singing, the
works.
(don't get me wrong,
there is such a thing as
a cockeyed optimism
that overlooks all
basic problems just for
the sake of
itself—
this is a shield and a
sickness.)
the knife got near my
throat again,
I almost turned on the
gas
again
but when the good
moments arrived
again
I didn't fight them off
like an alley
adversary.
I let them take me,
I luxuriated in them,
I bade them welcome
home.
I even looked into
the mirror
once having thought
myself to be
ugly,
I now liked what
I saw, almost
handsome, yes,
a bit ripped and
ragged,
scars, lumps,
odd turns,
but all in all,
not too bad,
almost handsome,
better at least than
some of those movie
star faces
like the cheeks of
a baby's
butt.
and finally I discovered
real feelings for
others,
unheralded,
like lately,
like this morning,
as I was leaving
for the track,
I saw my wife in bed,
just the shape of
her head there, covers
pulled high, just the
shape of her
head there
(not forgetting
centuries of the living
and the dead and
the dying,
the pyramids,
Mozart dead
but his music still
there in the
room, weeds growing,
the earth turning,
the toteboard waiting for
me)
I saw the shape of my
wife's head,
she so still,
I ached for her life,
just being there
under the
covers.
I kissed her on the
forehead,
got down the stairway,
got outside,
got into my marvelous
car,
fixed the seatbelt,
backed out the
drive.
feeling warm to
the fingertips,
down to my
foot on the gas
pedal,
I entered the world
once
more,
drove down the
hill
past the houses
full and empty
of
people,
I saw the mailman,
honked,
he waved
back
at
me.
## the 13th month
in the November of our hell
the birds still fly
or are murdered by the
cats.
in the November of our hell
the boxers hear the bell
and rise to do
what they must do.
in the November of our hell
in the November of our hell,
December
approaches.
in the November of our hell
I walk down the stairway
an old man now.
I reach the bottom,
walk outside
into a world millions of
years old,
I bend down to pet my cat,
his eyes look into mine
and past the
sun
in the November of our hell,
December coming
for both of us
for all of us.
I leave the cat,
climb into my automobile,
the engine starts,
I go out the driveway
backing carefully,
swing into the street
toward the mass of the
living
in the November of their hell,
December coming,
December coming,
look, look, look,
such effrontery!
can you believe it?
and after December?
what month?
what time?
what?
## finis, II
we all falter, give way, want to
toss it in.
the bad days come.
the bad days come more often.
we sit and wait, thinking, it will
pass.
but the day will come when it
will not pass.
it will stay.
you will sit in a garden chair
breathing the thick
air.
and an old cat will come and
lay at your feet.
he will wait with you.
death comes slowly some
times.
sometimes much too
slowly.
you will reach down and
pet the cat.
thinking again of the mad and
drunken
years.
## the observer
every time I drove past the hospital
I looked at it and thought, some day
I'll be in there.
and eventually I was in there,
sometimes sitting at this long
narrow window
and watching the cars pass on the
street below, as I once had
done.
it was a stupid window,
I had to sit on two folded blankets so that
I could see out.
they had built the window so that part of
the wooden frame
was eye-height
so you either had to look over or
under it.
so I sat on the blankets and looked
over.
well, the window wasn't stupid,
the designers
were.
so I sat there and watched the cars
pass on the street and I thought,
those lucky sons of bitches don't
know how lucky they
are
just to be dumb and driving through
the air
while I sit here on top of my
years
trapped,
nothing but a face in the window
that nobody ever
saw.
## August, 1993
easy, go easy, you can't outlast the mountain,
you've just come back from another
war,
go easy.
they are clamoring for you to do it for them once
again,
let them wait.
sit in the shade, wait for your strength to
return.
you'll know when the time is here.
then you'll arrive
for yourself and for them.
a bright sun.
a new fire.
a new gamble.
but
for now
go easy.
let them wait.
let them watch the new boys, the old
boys
meanwhile, you'll need a day or two
to sharpen the
soul,
musing through these D. H. Lawrence
afternoons,
those horseless days,
these nights of music trickling from the
walls,
this waiting for the fullness and the
charge.
## this night
I sit in a chair on the balcony
and drink natural spring
water.
the large palms run down the
hill with their dark
heads.
I can see the lights of this
city, of several
cities.
I sit in this balcony chair
where a high voltage wire runs
down and connects underneath
here
where I can reach out and
touch it.
(we can go very fast around
here.)
I hold a bottle of natural
spring water.
a plane flies high in the
overcast, I can't see him,
he can't see
me.
he is very fast.
I can't catch him but I can
pass him by
stretching out
my hand.
it's a cool summer night.
hell trembles nearby,
stretches.
I sit in this chair.
my 6 cats are
close by.
I lift the bottle of water,
take a large
swallow.
things will be far worse than
they are
now.
and far
better.
I wait.
## betting on now
I am old enough to have died several
times and I almost have,
now I drive my car through the sun
and over the freeway and past
Watts and to the racetrack
where the parking lot attendants
and the betting clerks
throw garlands of flowers at
me.
I've reached the pause before the full
stop and they are celebrating
because it just seems proper.
what the hell.
the hair I've lost to chemotherapy
is slowly growing
back but my feet are numb
and I must concentrate on my
balance.
old and battered, olden
matter,
I am still lucky with the
horses.
the consensus is that I
have a few seasons
left.
you would never believe
that I was once young
with a narrow razor face
and crazy eyes of
gloom.
no matter, I sit at my
table
joking with the waiters.
we know it's a fixed
game.
it's funny, Christ, look
at us:
sitting ducks.
"what are you having?"
asks my waiter.
"oh," I say and
read him something
from the menu.
"o.k.," he says
and walks away
between the earthquake,
the volcano and the
leopard.
## decline
sitting naked behind the house,
8 a.m., spreading sesame seed oil
over my body, jesus, have I come
to this?
I once battled in dark alleys for a
laugh,
now I'm not laughing.
I splash myself with oil and wonder,
how many years do you want?
how many days?
my blood is soiled and a dark
angel sits in my brain.
things are made of something and
go to nothing.
I understand the fall of cities, of
nations.
a small plane passes overhead.
I look upward as if it made sense to
look upward.
it's true, the sky has rotted:
it won't be long for any of
us.
## in the mouth of the tiger
the rivers of hell are well
peopled with the living.
this is what I write tonight,
a metallic taste in my mouth,
my wife and 6 cats in this
house, I am so sorry for them
because I am not bright with
life for them.
I had no idea that all this
would come so slowly,
running up from my feet
to my brain,
no trumpets blaring
here, no flags of
victory.
I can't even find the
courage to accept my
fate.
I once felt myself greater
than any trap.
nobody is.
damn it, where has the
music gone?
and myself?
pale as mountain light.
damn it, why?
I would have nobody be
me
now.
## the laughing heart
your life is your life.
don't let it be clubbed into dank
submission.
be on the watch.
there are ways out.
there is light somewhere.
it may not be much light but
it beats the
darkness.
be on the watch.
the gods will offer you
chances.
know them, take them.
you can't beat death but
you can beat death
in life,
sometimes.
and the more often you
learn to do it,
the more light there will
be.
your life is your life.
know it while you have
it.
you are marvelous
the gods wait to delight
in
you.
## a challenge to the dark
shot in the eye
shot in the brain
shot in the ass
shot like a flower in the dance
amazing how death wins hands down
amazing how much credence is given to idiot forms of
life
amazing how laughter has been drowned out
amazing how viciousness is such a constant
I must soon declare my own war on their war
I must hold to my last piece of ground
I must protect the small space I have made that has
allowed me life
my life not their death
my death not their death
this place, this time, now
I vow to the sun
that I will laugh the good laugh once again
in the perfect place of me
forever.
their death not my life.
## so now?
the words have come and gone,
I sit ill.
the phone rings, the cats sleep.
Linda vacuums.
I am waiting to live,
waiting to die.
I wish I could ring in some bravery.
it's a lousy fix
but the tree outside doesn't know:
I watch it moving with the wind
in the late afternoon sun.
there's nothing to declare here,
just a waiting.
each faces it alone.
Oh, I was once young,
Oh, I was once unbelievably
young!
## About the Author
CHARLES BUKOWSKI is one of America's best-known contemporary writers of poetry and prose and, many would claim, its most influential and imitated poet. He was born in Andernach, Germany to an American soldier father and a German mother in 1920, and brought to the United States at the age of three. He was raised in Los Angeles and lived there for fifty years. He published his first story in 1944 when he was twenty-four and began writing poetry at the age of thirty-five. He died in San Pedro, California on March 9, 1994 at the age of seventy-three, shortly after completing his last novel, Pulp (1994).
During his lifetime he published more than forty-five books of poetry and prose, including the novels Post Office (1971), Factotum (1975), Women (1978), Ham on Rye (1982), and Hollywood (1989). His most recent books are the posthumous editions of What Matters Most Is How Well You Walk Through the Fire (1999), Open All Night: New Poems (2000), Beerspit Night and Cursing: The Correspondence of Charles Bukowski & Sheri Martinelli, 1960-1967 (2001) and The Night Torn Mad with Footsteps: New Poems (2001).
All of his books have now been published in translation in over a dozen languages and his worldwide popularity remains undiminished. In the years to come Black Sparrow will publish additional volumes of previously uncollected poetry and letters.
Visit www.AuthorTracker.com for exclusive information on your favorite HarperCollins author.
## also by CHARLES BUKOWSKI
The Days Run Away Like Wild Horses Over the Hills (1969)
Post Office (1971)
Mockingbird Wish Me Luck (1972)
South of No North (1973)
Burning in Water, Drowning in Flame: Selected Poems 1955-1973 (1974)
Factotum (1975)
Love Is a Dog from Hell: Poems 1974-1977 (1977)
Women (1978)
You Kissed Lily (1978)
Play the Piano drunk Like a percussion instrument Until the fingers begin to bleed a bit (1979)
Shakespeare Never Did This (1979)
Dangling in the Tournefortia (1981)
Ham on Rye (1982)
Bring Me Your Love (1983)
Hot Water Music (1983)
There's No Business (1984)
War All the Time: Poems 1981-1984 (1984)
You Get So Alone at Times That It Just Makes Sense (1986)
The Movie: "Barfly" (1987)
The Roominghouse Madrigals: Early Selected Poems 1946-1966 (1988)
Hollywood (1989)
Septuagenarian Stew: Stories & Poems (1990)
The Last Night of the Earth Poems (1992)
Screams from the Balcony: Selected Letters 1960-1970 (Volume 1) (1993)
Pulp (1994)
Living on Luck: Selected Letters 1960s-1970s (Volume 2) (1995)
Betting on the Muse: Poems & Stories (1996)
Bone Palace Ballet: New Poems (1997)
The Captain Is Out to Lunch and the Sailors Have Taken Over the Ship (1998)
Reach for the Sun: Selected Letters 1978-1994 (Volume 3) (1999)
What Matters Most Is How Well You Walk Through the Fire: New Poems (1999)
Open All Night: New Poems (2000)
The Night Torn Mad with Footsteps: New Poems (2001)
Beerspit Night and Cursing: The Correspondence of Charles Bukowski & Sheri Martinelli (2001)
Sifting through the madness for the Word, the line, the way: New Poems (2003)
## Copyright
BETTING ON THE MUSE. Copyright © 2007 by Linda Lee Bukowski. All rights reserved under International and Pan-American Copyright Conventions. By payment of the required fees, you have been granted the non-exclusive, non-transferable right to access and read the text of this e-book on-screen. No part of this text may be reproduced, transmitted, down-loaded, decompiled, reverse engineered, or stored in or introduced into any information storage and retrieval system, in any form or by any means, whether electronic or mechanical, now known or hereinafter invented, without the express written permission of HarperCollins e-books.
EPub Edition © JUNE 2007 ISBN: 9780061860690
10 9 8 7 6 5 4 3 2 1
## About the Publisher
Australia
HarperCollins Publishers (Australia) Pty. Ltd.
25 Ryde Road (PO Box 321)
Pymble, NSW 2073, Australia
http://www.harpercollinsebooks.com.au
Canada
HarperCollins Publishers Ltd.
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http://www.harpercollinsebooks.co.uk
United States
HarperCollins Publishers Inc.
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New York, NY 10022
http://www.harpercollinsebooks.com
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{"url":"http:\/\/math.stackexchange.com\/questions\/270155\/how-to-evaluate-this-complex-integral","text":"# how to evaluate this complex integral\n\nI need to evaluate $$\\int_{|z|=2}\\frac{1}{(z-1)^3}dz.$$ At $z=1$, it has a pole of order $3$. I can not remember how to find the residue when there are poles with multiplicity, could any one tell me?\n\n-\nen.wikipedia.org\/wiki\/\u2026 \u2013\u00a0anon Jan 4 '13 at 7:13\nDo you know how to integrate by finding antiderivatives? \u2013\u00a0Jonas Meyer Jan 4 '13 at 7:17\n@JonasMeyer Na. \u2013\u00a0La Belle Noiseuse Jan 4 '13 at 7:23\n\nRecall Cauchy integral formula, which goes as follows. Let $f(z)$ be analytic inside an open set $\\Omega$. Let $\\Gamma$ be a closed curve inside $\\Omega$. Let $z_0$ lie inside the closed curve $\\Gamma$, we then have that $$f^{(n)}(z_0) = \\int_{\\Gamma} \\dfrac{f(z)}{(z-z_0)^{n+1}} dz$$ In your case, $f(z) = 1$, $n=2$, $z_0 = 1$ and $\\Gamma$ is the circle $\\vert z \\vert = 2$. Hence, $f^{(2)}(z) = 0$. Hence, the integral is $0$.\n\n-\nWould it be $z_0\\in\\Gamma$? \u2013\u00a0La Belle Noiseuse Jan 4 '13 at 7:28\n@Kuttus Yes. The closed curve is $\\Gamma$ and not $\\Omega$. I have changed it. \u2013\u00a0user17762 Jan 4 '13 at 7:32\n@Kuttus: $\\Gamma$ denotes the closed curve, not the region it bounds. Note that $z_0$ is in the interior of this region, so $z_0 \\not\\in \\Gamma$. \u2013\u00a0Michael Albanese Jan 4 '13 at 7:34\nwill there be a $n!\\over 2\\pi i$ in the formula u have written? and I think something wrong with your answer \u2013\u00a0La Belle Noiseuse Jun 3 '13 at 17:28\n@TaxiDriver Yes. There should be $n!\/(2 \\pi i)$. But that won't change the fact that the answer is $0$. \u2013\u00a0user17762 Jun 3 '13 at 17:35\n\nThe residue is the coefficient $a_{-1}$ in the Laurent expansion $$\\frac{1}{(z-1)^3}=\\cdots+\\frac{a_{-3}}{(z-1)^3}+\\frac{a_{-2}}{(z-1)^2}+\\frac{a_{-1}}{z-1}+a_0+a_1(z-1)+a_2(z-1)^2+\\cdots.$$\n\nCan you find a sequence $(a_n)$ making this equation hold?\n\nAlternatively, the reason residues determine the integrals is that all terms of the form $a_n(z-a)^n$ with $n\\neq -1$ have antiderivatives on $\\mathbb C\\setminus\\{a\\}$, hence integrate to $0$ on any closed curve not containing $a$. In particular, the function you present has an antiderivative on $\\mathbb C\\setminus\\{1\\}$.\n\n-","date":"2016-05-29 08:16:02","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9860529899597168, \"perplexity\": 283.83753455412386}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2016-22\/segments\/1464049278417.79\/warc\/CC-MAIN-20160524002118-00115-ip-10-185-217-139.ec2.internal.warc.gz\"}"}
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Q: Vote to close titles missing The close vote dialog on the iOS app has no titles on the options:
Unless there is an existing vote where that specific title shows (including the current vote count indicator):
Since it only shows when there is a vote count to display, I'm assuming it is a side effect of this fix.
For reference, this is what it should look like (old image I took from elsewhere, but you get the idea):
*
*App Version: 1.6.1.1
*Device: iPhone 6s
*OS Version: Version 10.0.1 (Build 14A403)
A: This will be fixed in 1.6.1.2.
As you could guess, I was incorrectly returning nil if I didn't need to append a vote count, rather than just returning the unmodified title.
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layout: partner
lang: en
permalink: /principal/
id: principal
name: principal
logo: principal.png
contact: coleman.jesse.c@principal.com
links:
- title: Individual Mapping Materials
link: https://cdn.hotosm.org/leaderboard/Principal/Principal+Mapping+how+to+guide.pdf
- title: Mapathon Event Materials
link: https://cdn.hotosm.org/leaderboard/Principal/Mapathon+Event+Materials.zip
primary-hashtag: principal
subhashtags:
- principal*
tm-projects:
- id: 13796
desc: "This remote mapping of buildings will support the identification and characterization of settlements, as well as the implementation of planned activities and largely the generation of data for humanitarian activities"
---
|
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"redpajama_set_name": "RedPajamaGithub"
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| 6,686
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Q: how to maintain session in android I am new to android .In my application i need to maintain a session when first connection of web service(i need to implement java server web service urls when i call the first url i got response. after that second url call it gives the false response .thing is i need to call the url with in the same session as i called my first url) .I found different answers but i don't under stand those.what i am expecting is how to get the session id when my first call and how can i keep those in further activities. please share sample code or related docs or relate answers.I hope you understand my intention .
A: to get cookies or session from the server if using http client use this:
List<Cookie> cookies = ((AbstractHttpClient) httpclient).getCookieStore().getCookies();
if (cookies.isEmpty()) {
Log.d("TAG","no cookies received");
} else {
for (int i = 0; i < cookies.size(); i++) {
if(cookies.get(i).getName().contentEquals("ASP.NET_SessionId")) {
asp = cookies.get(i).getValue();
}
}
Log.e("this is the cookiee", asp);
}
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Альфред Райзенауэр (; , Кёнигсберг — , Либава) — немецкий пианист и музыкальный педагог.
Биография
Учился в своём родном городе у Луи Кёлера, затем начиная с 1876 г. занимался в Веймаре у Франца Листа; Лист с одобрением отнёсся к выполненной Райзенауэром оркестровке Третьего Мефисто-вальса. В 1881 г. впервые выступил с концертом в Риме. Затем на протяжении года изучал право в Лейпцигском университете, однако отказался от мысли стать юристом и вернулся к исполнительской карьере.
Интенсивно гастролировал по всему миру. Рецензент «Нью-Йорк Таймс» отзывался о сольном выступлении Райзенауэра как о «совершенно восхитительном», особенно отмечая его исполнение концерта Моцарта, однако дал резко отрицательную оценку участию пианиста в камерном ансамбле (с Квартетом Кнайзеля), поскольку темперамент не позволял ему умерить пыл и на равных сотрудничать с партнёрами по ансамблю. В то же время Бернард Шоу полагал, что присущий Райзенауэру избыток технического мастерства заставляет темп торжествовать над мыслью. Особенно масштабными были гастроли Райзенауэра по Российской империи: на рубеже 1880-90-х гг. он, как утверждал его импресарио, дал в России более 300 концертов. Всего же, по некоторым сведениям, Райзенауэр на протяжении всей карьеры выступил более чем в 2000 концертах. 10 апреля 1905 г. он записал для фирмы Welte-Mignon десять пьес.
С 1885 г. преподавал в Зондерсхаузене, с 1900 г. профессор Лейпцигской консерватории. Среди его учеников, в частности, был Сергей Борткевич, вспоминавший впоследствии:
Рейзенауэр был фортепианным гением. Он не нуждался в изнуряющих упражнениях, мастерство приходило к нему как бы само собой… Он очень мало обучал и говорил о технических проблемах.
Борткевич посвятил памяти «моего дорогого учителя Альфреда Райзенауэра» Десять этюдов Op. 15 (1911). У Райзенауэра учился также Зигфрид Карг-Элерт.
Умер от разрыва сердца в ходе гастрольного тура, незадолго до концерта. По утверждению исследователей, жизнь Райзенауэра была сокращена тяжёлым пристрастием к спиртным напиткам.
Источники
Литература
Alfred Reisenauer. Systematic musical training // Great pianists on piano playing / Ed. James Francis Cooke. — NY, 1917.
Родившиеся в Кёнигсберге
Пианисты Германии
Музыкальные педагоги Германии
Академические музыканты Германии
Выпускники кёнигсбергской гимназии Вильгельма
|
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\section{Introduction}
The study of centrality in networks goes back to the late forties. Since then,
several measures of centrality with different properties have been proposed---see~\cite{BoVAC} for a survey.
To sort out which measures are more apt for a specific application, one can try
to classify them through some axioms that they might satisfy or not.
In~\cite{BoVAC,BLVRMCM}, two of the authors have studied in particular
\emph{score monotonicity} and \emph{rank monotonicity} on directed graphs.
The first property says that when an arc $x\to y$ is added to the graph, the
score of $y$ strictly increases~\cite{SabCIG}.
Rank monotonicity~\cite{CDKLEAA} states that after adding an
arc $x\to y$ all nodes with a score smaller than (or equal to) $y$ have still a
score smaller than (or equal to) $y$.
Score and rank monotonicity complement themselves: score monotonicity tells us
that ``something good happens''; rank monotonicity that ``nothing bad happens''.
In some way, both axioms aim at answering the following question: is it always
worth it for a node in a directed social network (say, Twitter) to have a new
incoming arc (in Twitter parlance, a new follower)? The two monotonicity axioms
introduced above have a different interpretation of what ``worth'' means.
``Score monotonicity'' interprets it simply as an increase of score: if you get
a new follower, does your score always increase? ``Rank monotonicity''
interprets it with respect to the score of other nodes: if you get a new
follower, do you still dominate (have a larger score than) the same nodes you used
to dominate before, and possibly more? As we said, for most notions of
importance (i.e., centrality measures) the answer to both questions is ``yes'',
under very mild assumptions~\cite{BLVRMCM}.
Once we move to undirected graphs, however, previous definitions and results are
no longer applicable. Thus, in this paper, we aim at answering a subtly different question:
is it always worth it for an actor in an \emph{undirected} social network (say,
Facebook) to have a new friend? Again, ``worth'' can be taken to refer
to the score or to the rank.
In this paper, we propose more precise definitions that are natural extensions
of score and rank monotonicity to the undirected case, and prove results about
classical centrality measures: closeness~\cite{BavMMGS}, harmonic centrality~\cite{BeaIIC}, betweenness~\cite{AntRG,FreSMCBB}, and
four variants of spectral ranking~\cite{VigSR}---eigenvector
centrality~\cite{LanZRWT,BerTGA}, Katz's index~\cite{KatNSIDSA}, Seeley's
index~\cite{SeeNRI}, and PageRank~\cite{PBMPCR}.
As we will see, while in some cases we can witness some score increase,
except for Seeley's index \emph{none of the centrality measures we consider
is rank monotone}. This is somehow surprising, and will yield some reflection.
Note that adding a single edge to an undirected graph is equivalent to adding
\emph{two} opposite arcs in a directed graph, which may suggest why the
situation is so different, at least from the mathematical viewpoint.
Understanding under which conditions a centrality measure does not satisfy
an axiom will be a theme that we
will try to pursue in the course of the discussion.
We provide classes of counterexamples of arbitrary size; moreover, we
always provide both a counterexample in which the loss of rank happens in the
less important node of the new edge and a counterexample in which the loss of rank happens in the
more important node of the new edge. In this way, we will show that it is impossible
for the two actors in the social network creating the new edge to predict
whether the edge will be beneficial \emph{even knowing their relative importance}.
The results obtained in this paper are resumed in Table~\ref{tab:summ}.
To prove general results in the case of spectral rankings,
we exploit the connection between spectral rankings and
graph fibrations~\cite{BoVGF,BLSGFGIP}, which makes us able to reduce
computations on graphs with a variable number of nodes to similar
computations on graphs with a fixed number of nodes.
This approach to proofs, which we believe is of independent interest, makes it
possible to use analytic techniques to control the values assumed by
eigenvector centrality, Katz's index, and PageRank.
We conclude the paper with some anecdotal evidence from a medium-sized real-world
network, showing that violations of monotonicity do happen also in practice.
With respect to the conference paper~\cite{BFVSRMUN}, all results on geometric
centralities and betweenness are new, as well as all general results on eigenvector
centrality, and all results about Katz's index. The second PageRank counterexample
is also new. All results about demotion, for all centralities, are also new. The proofs
for the first PageRank example have been significantly simplified.
Most of the computations in this paper (in particular, the manipulation of complex rational functions)
have been performed using Sage~\cite{Sage}. All our Sage worksheets are available at
\url{https://github.com/vigna/monotonicity}, and will be badged on the Zenodo platform after the reviewing process.
\begin{table}
\centering
\begin{tabular}{l||c|c}
& score monotonicity & rank monotonicity\\\hline
Closeness & yes & no\\
Harmonic centrality & yes & no\\
Betweenness & no & no\\
Eigenvector centrality & no & no\\
Seeley's index & yes & yes\\
Katz's index & yes & no\\
PageRank & no & no\\
\end{tabular}
\vspace*{3mm}
\caption{\label{tab:summ}Summary of the results of this paper for the case of \emph{connected undirected graphs}.
For comparison, recall that~\cite{BLVRMCM} all the
centrality measures listed are both score and rank monotone on \emph{strongly connected directed graphs},
with the only exception of betweenness that is neither.}
\end{table}
\section{Graph-theoretical preliminaries}
\label{sec:defs}
While we will focus on simple undirected graphs, we are going to make use of
some proof techniques that require handling more general types of graphs.
A \emph{(directed multi)graph} $G$ is defined by a set $N_G$ of nodes, a set
$A_G$ of arcs, and by two functions $s_G,t_G:A_G\to N_G$ that specify the
source and the target of each arc; a \emph{loop} is an arc with the same source and target;
the main difference between this definition and the standard definition of a
directed graph is that we allow for the presence of multiple arcs between a pair of nodes.
When we do not need to distinguish between multiple arcs, we write $x\to y$ to denote
an arc with source $x$ and target $y$.
Since we do not need to discriminate between graphs that only differ because of
node names, we will often assume that $N_G=\{\,0,1,\dots,n_G-1\,\}$ where $n_G$
is the number of nodes of $G$.
Every graph $G$ has an associated $n_G \times n_G$ \emph{adjacency matrix}, also
denoted by $G$, where $G_{xy}$ is the number of arcs from $x$ to $y$.
A \emph{(simple) undirected graph} is a loopless\footnote{Note that our negative results are \emph{a fortiori} true if we consider
undirected graphs with loops. Our positive results are still valid in the same case
using the standard convention that each loop increases the degree by two.} graph $G$ such that for all $x,y \in N_G$
we have $G_{xy}=G_{yx}\leq 1$. In other words, there is at most one arc between any two nodes and if there is an
arc from $x$ to $y$ there is also an arc in the opposite direction.
In an undirected graph, an \emph{edge} between $x$ and $y$ is a pair of arcs
$x\to y$ and $y\to x$, and it is denoted by $x\scalebox{0.5}[1.0]{-} y$. This definition is equivalent to the more common
notion that an edge is an unordered set of nodes, but it makes it
possible to mix undirected and directed graphs: indeed, even in drawings
we will freely mix arcs and edges. For undirected graphs,
we prefer to use the word ``vertex'' instead of ``node''.
\section{Score and rank monotonicity axioms on undirected graphs}
One of the most important notions that researchers have been trying to capture
in various types of graphs is ``node centrality'':
ideally, every node (often representing an
individual) has some degree of influence or importance within the social domain
under consideration, and one expects such importance to be reflected in the
structure of the social network; centrality is a quantitative measure that
aims at revealing the importance of a node.
Formally, a \emph{centrality} (measure or index) is any function $c$ that, given a graph $G$, assigns a
real number $c_G(x)$ to every node $x$ of $G$; countless notions of centrality have been proposed over time, for
different purposes and with different aims; some of them were originally defined only for a specific category of graphs. Later some of
these notions of centrality have been extended to more general classes; all centrality measures discussed in this
paper can be defined properly on all undirected graphs (even disconnected ones).
We assume from the beginning that
the centrality measures under examination are invariant by isomorphism, that is, that they depend just on the
structure of the graph, and not on a particular name chosen for each node. In particular, all nodes exchanged
by an autorphism necessarily share the same centrality score, and we will use this fact to simplify our computations.
Axioms are useful to isolate properties of different centrality measures and make it possible to compare them. One
of the oldest papers to propose this approach is~\cite{SabCIG}, which introduced score monotonicity,
and many other proposals have appeared in the
last few decades.
In this paper we will be dealing with two properties of centrality measures:
\begin{defi}[Score monotonicity]
Given an undirected graph $G$,
a centrality $c$ is said to be \emph{score monotone on $G$} iff for every pair of non-adjacent vertices $x$ and $y$ we have that
\[
c_G(x) < c_{G'}(x) \quad\text{and}\quad c_G(y) < c_{G'}(y),
\]
where $G'$ is the graph obtained adding the new edge $x \scalebox{0.5}[1.0]{-} y$ to $G$.
We say that $c$ is \emph{score monotone on undirected graphs} iff it is score monotone on all
undirected graphs.
\end{defi}
\begin{defi}[Rank monotonicity]
Given an undirected graph $G$,
a centrality $c$ is said to be \emph{rank monotone\footnote{We remark that in~\cite{BFVSRMUN}
rank monotonicity was defined incorrectly, using an apparently (but not effectively)
equivalent condition to stated in~\cite{BLVRMCM} and~\cite{CDKLEAA}.} on $G$} iff for every pair of non-adjacent vertices $x$ and $y$ we have that for all vertices $z\neq x,y$
\[
c_G(z) < c_{G}(x) \Rightarrow c_{G'}(z) < c_{G'}(x) \quad\text{and}\quad
c_G(z) < c_{G}(y) \Rightarrow c_{G'}(z) < c_{G'}(y),
\]
and moreover
\[
c_G(z) \leq c_{G}(x) \Rightarrow c_{G'}(z)\leq c_{G'}(x) \quad\text{and}\quad
c_G(z)\leq c_{G}(y) \Rightarrow c_{G'}(z) \leq c_{G'}(y),
\]
where $G'$ is the graph obtained adding the new edge $x \scalebox{0.5}[1.0]{-} y$ to $G$.
It is said to be \emph{strictly rank monotone on $G$} if instead
\[
c_G(z) \leq c_{G}(x) \Rightarrow c_{G'}(z) < c_{G'}(x) \quad\text{and}\quad
c_G(z) \leq c_{G}(y) \Rightarrow c_{G'}(z) < c_{G'}(y)
\]
We say that $c$ is \emph{(strictly) rank monotone on undirected graphs} iff it is (strictly) rank monotone on all
undirected graphs.
\end{defi}
Score monotonicity tells us that in absolute terms the new edge is beneficial to $x$ and $y$. Rank monotonicity tells us
that in relative terms the new edge is not hurting them, in the sense that nodes that were (strictly) dominated by $x$ or $y$ are still (strictly) dominated. Finally,
strict rank monotonicity is a stronger property that implies, besides preservation of dominance, an improvement, as additionally all
nodes in a score tie with $x$ or $y$ will have a strictly smaller score after adding the new edge. As a sanity
check, we note that degree, the simplest centrality measure, is both score monotone and strictly rank monotone.
These three properties can be studied on the class of all undirected graphs or only on the class of connected graphs, giving rise to six possible ``degrees
of monotonicity'' that every given centrality may satisfy or not. This paper studies these different degrees of monotonicity for some of the most popular
centrality measures, also comparing the result obtained with the corresponding properties in the directed case.
With respect to the directed case, there is an important difference: violation of the axioms may happen on one of
the nodes involved, or on both. While we never witnessed the latter situation, there is in the first case
a distinction that we feel important enough to deserve a name:
\begin{defi}
A violation of score monotonicity is a \emph{top violation} if the endpoint of the new edge whose scores decreases is more
important than the other. It is a \emph{bottom violation} otherwise. The same distinction applies to violations of rank monotonicity.
\end{defi}
Top violations are somewhat sociologically natural: if a network superstar becomes friend with a nobody, it is not surprising that
the nobody increases their popularity, whereas the superstar loses a bit of charm. Bottom violations, however, are
much less natural: in the same context, the nobody sees their importance decrease,
nurturing in a bizarre inversion of flow the superstar popularity.
As we already anticipated, and differently from the directed case, all
centrality measures we consider, except for Seeley's index (which however is trivial in this context---see Section~\ref{sec:seeley}) will turn out to be not rank monotone.
Moreover, most centralities are not score monotone.
As a consequence, this paper
is a sequence of counterexamples (to score monotonicity and to rank monotonicity, hence \emph{a fortiori} to its strict version):
all counterexamples exhibit an undirected graph $G$ and two non-adjacent vertices $x$ and $y$ such that when
you add the edge $x \scalebox{0.5}[1.0]{-} y$ to $G$, $x$ decreases its score, or its rank with respect to some other vertex $z$. We may call $x$ the ``losing endpoint'' (i.e., the one that is hurt by the addition of the edge).
Not all counterexamples are equally good, though. We will make an effort to have the theoretically
strongest counterexamples we can find, and we will also look for properties that have a practical interpretation. More in
detail:
\begin{itemize}
\item all our counterexamples are connected;
\item all our counterexamples are parametric graphs that can be instantiated in graphs of arbitrarily large size;
\item we always give both top and bottom violation counterexamples; thus, even knowing whether you are more or less
important than your new neighbor will not help in knowing if you will gain or lose from the new edge;
\item in all our counterexamples the losing endpoint of the new edge is also \emph{demoted}, that is,
the number of nodes with a larger score than the losing endpoint increases after adding the new edge.
\end{itemize}
The last point is particularly important because demotion \emph{is not implied
by the lack of rank monotonicity}:
it may be the case that $x$ used to be more important than $z$ and it becomes
less important than $z$ after the addition of the edge $x\scalebox{0.5}[1.0]{-} y$, but still the
number of nodes that are more important than $x$ becomes smaller with the
addition of $x \scalebox{0.5}[1.0]{-} y$. The lack of demotion might suggest a weaker notion of rank monotonicity,
in which the number of nodes whose score dominates $x$ (or $y$) decreases (such a notion is strictly
weaker as it is implied by rank monotonicity). However, this weaker notion
is not very appealing from a practical viewpoint, as it is not \emph{locally testable}---it
has no immediate consequence for the relative importance of an endpoint of the edge and another vertex.
Proving demotion implies that the counterexamples in this paper are strong enough to violate
also the weaker notion of monotonicity described above.
\section{Geometric centralities}
Since adding a new edge can only shorten existing shortest paths or create new
ones, it is immediate to show that harmonic centrality is score monotone; for the same reason,
closeness centrality is score monotone on connected graphs, whereas
counterexamples similar to those of the directed case of~\cite{BoVAC} prove
that closeness is not score monotone in the general case.
Less intuitively, neither closeness nor harmonic centrality are rank monotone in the undirected case.
The family of counterexamples we found shows that adding an edge can shorten distances in ways
that are much more useful for some vertices not incident on the new edge than on its endpoints.
Our counterexample for rank monotonicity of closeness and harmonic centrality
is shown in Figure~\ref{fig:closeness}.
The idea behind the graph is that the edge $0\adj1$ reduces the distance between
vertex $0$ and the vertices labeled with $4$, but does not reduce the distance between vertex $0$ and vertex $3$ (and more
importantly between vertex $0$ and the star around vertex $3$). Thus the vertices labeled with $4$ will
gain more centrality from the new edge than vertex $0$, and for appropriate values of $j$ and
$k$ we will be able to prove a violation of rank monotonicity (all vertices labeled with $4$ share
the same centrality). The stars of size $r$ around vertex $1$
and vertex $2$ will instead be useful by giving us some more space to play with the
relative importance of the endpoints of the new edge, tuning the graph in Figure~\ref{fig:closeness} to be an example
of top or bottom violation.
\subsection{Closeness}
We recall that closeness of a vertex $x$ is defined as the reciprocal of its \emph{peripherality}
\[
p(x)=\sum_{y\in N_G}d(x,y),
\]
where $d(x,y)$ is the \emph{distance} (i.e., the length of a shortest path) between $x$ and $y$.
\begin{figure}
\centering
\includegraphics{fig/rankund-403-mps}
\caption{\label{fig:closeness}A counterexample to rank monotonicity for closeness and harmonic centrality.
There is a star with $j$ leaves
around vertex $0$, a star with $k$ leaves around vertex $3$,
a star with $r$ leaves around vertex $1$ and a star with $r$ leaves around vertex $2$.
Before adding the edge $0\adj1$, the score of vertex $0$ is larger than the score of the vertices
labeled with $4$; after, it is smaller.}
\end{figure}
We denote
for simplicity with $\operatorname{pre}(-)$ and $\operatorname{post}(-)$ the peripherality of the graph in Figure~\ref{fig:closeness} before and after adding the edge $0\scalebox{0.5}[1.0]{-} 1$. Then,
\begin{align*}
\operatorname{pre}(0)&=15 + j + 4 k + 11r & \operatorname{post}(0)&=9 + j + 4 k + 5r \\
\operatorname{pre}(1)&=15 +6j + 3 k + 3r & \operatorname{post}(1)&=9 + 2j + 3 k + 3r \\
\operatorname{pre}(4)&=15 + 6j + 3 k + 5r & \operatorname{post}(4)&=13 + 4j + 3 k + 5r. \\
\end{align*}
We are interested in finding solutions, if they exists, to the set of inequalities
\[
\operatorname{pre}(0) > \operatorname{pre}(1), \operatorname{pre}(0)<\operatorname{pre}(4), \operatorname{post}(0)> \operatorname{post}(4),
\]
which specify that vertex $0$ violates rank monotonicity with respect to vertices labeled with $4$, and that it is less important than vertex $1$ (recall we are
manipulating the reciprocal of closeness), and
\[
\operatorname{pre}(0) < \operatorname{pre}(1), \operatorname{pre}(0)<\operatorname{pre}(4), \operatorname{post}(0)> \operatorname{post}(4),
\]
that correspond to the analogous case in which vertex $0$ is more important than vertex $1$. There
are infinite solutions for both sets of inequalities, and in particular
$j=5r$, $k=18r$ ($r\geq 2$), and $j=4r+4$, $k=12r+17$ ($r\geq 1$) satisfy the
first and second set, respectively.
\begin{thm}
Closeness is not rank monotone on the graphs of Figure~\ref{fig:closeness} for $r\geq 2$,
$j=5r$, and $k=18r$ (bottom violation) and for $r\geq 1$,
$j=4r+4$, and $k=12r+17$ (top violation).
\end{thm}
While the family of graphs we consider contains graphs of unbounded size, each graph has just
ten distinct peripherality scores. We can thus
compare exactly the peripherality of all vertices with that of vertex $0$ before and after
adding the new edge. It is easy to see that for the parameter
sets of the previous theorem all vertices, except the $j$ vertices labeled with $4$ and sometimes vertex $1$, maintain the same relative position
to vertex $0$ after adding the edge $0\scalebox{0.5}[1.0]{-} 1$. Thus, in both cases vertex $0$ is demoted by at least $j-1$
positions.
\subsection{Harmonic centrality}
The counterexample in Figure~\ref{fig:closeness} works also for harmonic centrality, which is
not surprising as the only difference between closeness and harmonic centrality is the usage
of a harmonic mean instead of an arithmetic mean.
Denoting this time with $\operatorname{pre}(-)$ and $\operatorname{post}(-)$ the harmonic centrality of the graph in Figure~\ref{fig:closeness} before and after adding the edge $0\scalebox{0.5}[1.0]{-} 1$,
we have
\begin{align*}
\operatorname{pre}(0)&=\frac{137}{60} + j + \frac14k + \frac{11}{30}r & \operatorname{post}(0)&=\frac{10}{3} + j + \frac14k + \frac56r \\
\operatorname{pre}(1)&=\frac{137}{60} + \frac16j + \frac13k +\frac32r & \operatorname{post}(1)&=\frac{10}{3} + \frac12j + \frac13k + \frac32r\\
\operatorname{pre}(4)&= \frac{137}{60}+ \frac16j + \frac13k + \frac56r
& \operatorname{post}(4)&=\frac{29}{12} + \frac14j + \frac13k + \frac56r . \\
\end{align*}
This time we are interested in finding solutions, if they exists, to the set of inequalities
\[
\operatorname{pre}(0) < \operatorname{pre}(1), \operatorname{pre}(0)>\operatorname{pre}(4), \operatorname{post}(0)< \operatorname{post}(4)
\]
and
\[
\operatorname{pre}(0) >\operatorname{pre}(1), \operatorname{pre}(0)>\operatorname{pre}(4), \operatorname{post}(0)< \operatorname{post}(4).
\]
There
are again infinite solutions for both sets of inequalities, and in particular
$j=26r$, $k=247r$ ($r\geq 1$) and $j=26r$, $k=246r$ ($r\geq 1$), satisfy the
first and second set, respectively.
\begin{thm}
Harmonic centrality is not rank monotone on the graphs of Figure~\ref{fig:closeness} for $r\geq 1$,
$j=26r$, and $k=247r$ (bottom violation) and for $r\geq 1$,
$j=26r$, and $k=246r$ (top violation).
\end{thm}
Also in this case, for
the same parameter sets, all vertices, except the $j$ vertices labeled with $4$ and sometimes vertex $1$, maintain the same relative position
to vertex $0$ after adding the edge $0\scalebox{0.5}[1.0]{-} 1$. Thus, vertex $0$ is demoted by at least $j-1$
positions.
\section{Betweenness}
\begin{figure}
\centering
\includegraphics{fig/rankund-501-mps}
\caption{\label{fig:betweenness}A counterexample to score and rank monotonicity for betweenness.
There is a star with $k$ leaves around vertex $0$, a star with $h$ leaves around vertex $1$, and $j$ vertices
labeled with $4$ with the same neighborhood.
Before adding the edge $0\scalebox{0.5}[1.0]{-} 1$, the score of vertex $0$ is larger than the score of vertex $2$; after the addition, it becomes smaller.
Moreover, the score of vertex $0$ does not change when the edge is added.}
\end{figure}
Betweenness is neither score nor rank monotone on directed graphs~\cite{BLVRMCM}; the same is true in the undirected case, as
shown in the graph of Figure~\ref{fig:betweenness}.
Intuitively, the new edge puts $2$ on many shortest paths (e.g., those between any
replica of $3$ and any replica of $4$) that before needed to pass on the upper route
of the rectangle. Vertex $0$, instead, does not gain as much by the addition of the edge.
Denoting
with $\operatorname{pre}(-)$ and $\operatorname{post}(-)$ the value of betweenness before and after adding the edge $0\scalebox{0.5}[1.0]{-} 1$, we have
\begin{align*}
\operatorname{pre}(0)&=\frac{k(2h+2j+k+11)}2 &
\operatorname{post}(0)&=\frac{k(2h+2j+k+11)}2\\
\operatorname{pre}(1)&=\frac{h^2+(2j+2k+11)h+3k+7}2 &
\operatorname{post}(1)&=\frac{h^2+(2j+2k+11)h+(k+1)(j+4)+4}2\\
\operatorname{pre}(2)&=\frac{(2h+2)j+3h+k+5}2 &
\operatorname{post}(2)&=\frac{(2h+k+2)j+3h+2k+6}2 . \\
\end{align*}
Observe that $\operatorname{pre}(0)=\operatorname{post}(0)$, showing that score monotonicity is violated.
To prove that also rank monotonicity does not hold, we are interested in finding solutions to the set of inequalities
\[
\operatorname{pre}(0) < \operatorname{pre}(1), \operatorname{pre}(0)>\operatorname{pre}(2), \operatorname{post}(0)< \operatorname{post}(2)
\]
and
\[
\operatorname{pre}(0) >\operatorname{pre}(1), \operatorname{pre}(0)>\operatorname{pre}(2), \operatorname{post}(0)< \operatorname{post}(2).
\]
There
are infinite solutions for both sets of inequalities, and in particular
$h=k$, $j=\bigl\lfloor(k^2-4k-15)/2\bigr\rfloor$, $k\geq 13$ and $k=2+h$, $j=4h$, $h\geq 12$ satisfy the
first and second set, respectively.
\begin{thm}
Betweenness is not rank monotone on the graph of Figure~\ref{fig:betweenness},
for $k=2+h$, $j=4h$, $h\geq 12$, (top violation) and
for $h=k$, $j=\bigl\lfloor(k^2-4k-15)/2\bigr\rfloor$, $k\geq 13$ (bottom violation).\end{thm}
Also in this case we have just nine different betweenness scores, which makes it possible
to show that in both cases vertex $0$ is demoted by at least one position.
\section{Eigenvector centrality}
Eigenvector centrality is probably the oldest attempt at deriving a centrality
from matrix information: a first version was proposed by~\cite{LanZRWT} for
matrices representing the results of chess tournaments, and it was defined in
full generality by~\cite{BerTGA}; it was rediscovered many times since then.
One considers the adjacency matrix of the graph and computes its left or right
dominant eigenvector (in our case, the two eigenvectors coincide): the result is thus defined
modulo a scaling factor, and if the graph is (strongly) connected, the result is
unique (again, modulo the scaling factor) by the Perron--Frobenius theorem~\cite{BePNMMS}.
It is not difficult to find anecdotal examples of violation of rank (and even score, fixing a normalization)
monotonicity in simple examples.
\begin{figure}
\centering
\includegraphics{fig/rankund-203-mps}
\caption{\label{fig:ec}A counterexample to score monotonicity for eigenvector centrality. After adding the edge $0\adj1$, the score of vertex $0$ decreases:
in norm $\ell_1$, from $0.30656$ to $0.29914$; in norm $\ell_2$, from $0.65328$ to $0.63586$; and when projecting the
constant vector $\mathbf1$ onto the dominant eigenspace, from $1.39213$ to $1.35159$.}
\end{figure}
In Figure~\ref{fig:ec} we show a very simple graph that does not satisfy score monotonicity under the most obvious forms of normalization.
In particular, the score of vertex $0$ decreases after adding the edge $0\adj1$ both
in norm $\ell_1$ and norm $\ell_2$, and even when projecting the constant vector $\mathbf1$ onto the dominant eigenspace, which is an alternative way of circumventing the
scaling factor~\cite{VigSR}. The intuition
is that initially vertex $0$ has a high score because of its largest degree (three). However, once we close the triangle
we create a cycle that absorbs a large amount of rank, effectively decreasing the score of vertex $0$.
\begin{figure}
\centering
\includegraphics{fig/rankund-204-mps}
\caption{\label{fig:ec2}A counterexample to rank monotonicity for eigenvector centrality.
Before adding the edge $0\scalebox{0.5}[1.0]{-} 1$, the score of vertex $1$ is larger than the score of vertex $3$; after, it is smaller.}
\end{figure}
A similar counterexample, shown in Figure~\ref{fig:ec2}, proves that eigenvector centrality does not satisfy rank monotonicity.
Before adding the edge $0\scalebox{0.5}[1.0]{-} 1$, the score of vertex $1$ used to be larger than the score of vertex $3$; the converse is true after the addition
of the edge. This counterexample,
however, is not very satisfactory as vertex $1$ is not demoted---in fact, the opposite happens; on the other hand,
the set of vertices that dominate it changes completely with the addition of the new edge, showing that eigenvector
centrality can undergo turbulent modifications upon a simple perturbation.
We are now going to prove that eigenvector centrality does not satisfy
rank monotonicity on a class of graphs of arbitrarily large size in which we will also experience demotion.
Proving analytical results will require combining a few techniques from spectral
graph theory and analysis, as we would otherwise not be able to perform exact
computations, as in the previous cases.
\section{Interlude: graph fibrations}
\label{sec:fib}
Proving analytical results about graphs of arbitrary size requires in principle manipulating matrices
of arbitrary size, and obtaining closed-form expressions for eigenvalues and eigenvectors of such
matrices would be difficult, if not impossible. We thus turn to ideas going back to the results
obtained in the '60s in the context of the theory of \emph{graph divisors}~\cite{SacUTFCPG}, restating
them in the more recent language of \emph{graph fibrations}~\cite{BoVGF}.
A \emph{(graph) morphism} $\phi:G\to H$ is given by a pair of functions
$f_N:N_G\to N_H$ and $f_A:A_G\to A_H$ commuting with the source and
target maps, that is, $s_H(f_A(a))=f_N(s_G(a))$ and $t_H(f_A(a))=f_N(t_G(a))$ for all
$a \in A_G$. In other
words, a morphism maps nodes to nodes and arcs to arcs in such a way to
preserve the incidence relation.
The definition of morphism we give is the obvious extension to the case of multigraphs of the standard notion the
reader may have met elsewhere.
\begin{defi}
\label{def:fibration}
A \emph{fibration}~\cite{BoVGF,GroTDTEGAI} between the graphs $G$ and $B$ is a morphism $\phi: G\to B$ such
that for each arc $a\in A_B$ and each node $x\in N_G$ satisfying
$\phi_N(x)=t_B(a)$ there is a unique arc $\lift ax\in A_G$ (called the \emph{lifting of
$a$ at $x$}) such that $\phi_A(\lift ax)=a$ and $t_G(\lift ax)=x$.
\end{defi}
If $\phi:G\to B$ is a fibration, $G$
is called the \emph{total graph} and $B$ the \emph{base} of $\phi$.
We shall also say that $G$ is \emph{fibered (over $B$)}. The \emph{fiber over a
node $x\in N_B$} is the set of nodes of $G$ that are mapped to $x$.
A verbal restatement of the definition of fibration
is that each arc of the base lifts uniquely to each node in the fiber of its target;
moreover, we remark that Definition~\ref{def:fibration} is just an elementary restatement
of Grothendieck's notion of fibration between categories applied to the
free categories generated by $G$ and $B$.
In Figure~\ref{fig:exfib}, we show two graph morphisms; the morphisms are
implicitly described by the colors on the nodes and in the only possible way on the arcs. The morphism displayed on the
left is not a fibration, because the loop
on the base has no counterimage ending at the lower gray node, and
moreover the other arc has two counterimages with the same target. The
morphism displayed on the
right, on the contrary, is a fibration. Observe that loops are not necessarily
lifted to loops.
\begin{figure}[htbp]
\begin{center}
\includegraphics{fig/rankund-5-mps.pdf}\qquad\qquad\qquad\qquad\includegraphics{fig/rankund-6-mps}
\end{center}
\caption{\label{fig:exfib}On the left, an example of graph morphism that is
not a fibration; on the right, a fibration. Colors on the nodes are used to
implicitly specify the morphisms (arcs are mapped in the only possible way).}
\end{figure}
\begin{defi}
If $\phi:G\to B$ is a fibration, given a (row) vector $\bm u$ of size $n_B$, define its \emph{lifting along $\phi$} as
the vector $\bm u^\phi$ of size $n_G$ given by
\[
\left(\bm u^\phi\right)_i=u_{\phi(i)}.
\]
\end{defi}
Otherwise said, $\bm u^\phi$ is the vector obtained by copying $\bm u$ along the fibers of $\phi$.
\begin{thm}[\cite{SacUTFCPG}]
\label{th:sachs}
If $\phi:G\to B$ is a fibration surjective on the nodes, given a (row) vector $\bm u$ of size $n_B$ we have
\[
\bm u^\phi G = (\bm u B)^\phi.
\]
\end{thm}
In other words, one can lift and multiply by $G$, or equivalently multiply by $B$ and then lift: the base
$B$ ``resumes'' the graph $G$ well enough that the multiplication of fiberwise constant vectors by $G$
can be carried on (usually smaller) $B$. The proof of Theorem~\ref{th:sachs} is in fact immediate
once one realizes that Definition~\ref{def:fibration} implies that $\phi$ induces \emph{a local isomorphism} between
the in-neighborhood of a node $x$ of $G$ and the in-neighborhood of $\phi_N(x)$~\cite{BoVGF}.
Theorem~\ref{th:sachs} has the important consequence that every left eigenvector $\bm e$ of $B$ can be lifted to
a left eigenvector $\bm e^\phi$ of $G$, so every eigenvalue of $B$ is an eigenvalue of $G$, and
thus the characteristic polynomial of $B$ divides that of $G$ (hence the name \emph{graph divisor}).
In our case, by the Perron--Frobenius theorem~\cite{BePNMMS},
if $B$ is strongly connected the dominant eigenvector of $B$
is strictly positive, so its lifting is strictly positive, and thus (applying again the Perron--Frobenius theorem) it is the dominant eigenvector of $G$; moreover,
$G$ and $B$ share the same dominant eigenvalue (and thus spectral radius).
\section{Back to eigenvector centrality}
\label{sec:eigen}
We now get back to eigenvector centrality: Figure~\ref{fig:eig} shows a family of total graphs $G_k$ depending
on an integer parameter $k$, and an associated family of bases $B_k$, with fibrations defined on the nodes following the node labels,
and on the arcs in the only possible way. We will show that when the edge $0\scalebox{0.5}[1.0]{-} 1$ is added to the graphs (obtaining new graphs $G_k'$ and
$B_k'$), all vertices
labeled with $4$, which used to have a smaller score than vertex $1$ in $G_k$, will become more important than vertex $1$ in $G_k'$.
The intuitive idea behind the graphs $G_k$ is that the new edge makes the vertices labeled with $4$ much closer
to vertex $1$, a high-degree vertex; at the same time, the new edge doubles the number of paths from the vertices
labeled with $6$ to the vertices labeled with $4$. The advantage for vertex $1$ is to get much closer to the
vertices labeled with $4$, but those have a much smaller degree. All in all, the new edge will turn out to be much more advantagous
for the vertices labeled with $4$ than for vertex $1$.
The fundamental property of our counterexample is that albeit $G_k$ is a simple undirected graph
with $k^2-k-6$ vertices, $B_k$ is a general directed multigraph with seven nodes, independently of $k$,
so its adjacency matrix, shown
in Figure~\ref{fig:eig}, is a fixed-sized matrix containing a parameter $k$ due
to the variable number of arcs. Thus, fibrations make it possible to move our
proof from matrices of arbitrary size to a parametric matrix of fixed size.
\begin{figure}
\centering
\begin{tabular}{cc}
\raisebox{3cm}{$G_k$\qquad}&\includegraphics{fig/rankund-601-mps}\\
\raisebox{2cm}{$B_k$\qquad}&\includegraphics{fig/rankund-602-mps}
\end{tabular}
\[
\setlength\arraycolsep{2ex}
\renewcommand\arraystretch{1.5}
B_k=\left(\begin{matrix}
0 & \fcolorbox{gray}{gray}{0} & 0 & 1 & 1 & 0 & 0\\
\fcolorbox{gray}{gray}{0} & 0 & 1 & 0 & 0 & 1 & 0\\
0 & 1 & 0 & 1 & 0 & 0 & 1\\
1 & 0 & 1 & 0 & 0 & 0 & 0\\
k & 0 & 0 & 0 & k-1 & 0 & 0\\
0 & (k-1)(k-2) & 0 & 0 & 0 & 0 & 0\\
0 & 0 & k & 0 & 0 & 0 & k-1
\end{matrix}\right)
\]
\caption{\label{fig:eig}The parametric counterexample graph for eigenvector centrality:
when adding the edge $0\adj1$ vertex $1$ violates rank monotonicity (top).
The $k$ vertices labeled with $4$
form a $(k+1)$-clique with vertex $0$, and the $k$ vertices labeled with $6$ form a $(k+1)$-clique
with vertices $2$; finally, there is a star with $(k-1)(k-2)$ leaves around vertex $1$.
Arc
labels represent multiplicity. The matrix displayed is the adjacency matrix of $B_k$, with the grayed entries
to be set to $1$ when $0\adj1$ is added to the graph. Table~\ref{tab:eig} shows a set of values for
the size of the cliques and the size of the star causing vertex $1$ to be less important than vertex $0$.}
\end{figure}
\subsection{Sturm polynomials}
There is no way to compute exactly the eigenvalues and eigenvectors of $B_k$. However,
we will be able to control their behavior using \emph{Sturm polynomials}~\cite{RaSATP}, a standard, powerful
technique to analyze and locate real roots of polynomials.
\begin{defi}
If $p(x)$ is a polynomial with real coefficients and $p'(x)$ its derivative, the \emph{Sturm sequence} of polynomials
associated with $p(x)$ is defined by
\begin{align*}
S_0(x) &= p(x)\\
S_1(x) &= p'(x)\\
S_{i+1}(x) &= - S_{i}(x) \bmod S_{i-1}(x)\qquad \text{for $i\geq 1$,}
\end{align*}
where $S_{i}(x) \bmod S_{i-1}(x)$ is the remainder of the Euclidean division of $S_i(x)$ by $S_{i-1}(x)$.
The sequence stops when $S_{i+1}(x)$ becomes zero, and it is long at most as the degree of $p(x)$.
\end{defi}
Given a real number $a$, the number of \emph{sign variations} $V(a)$ of a Sturm sequence is the number of
sign changes, ignoring zeros, of the sequence $S_0(a)$,~$S_1(a)$,~$S_2(a)$, $\dots\,$. Finally,
if $p(x)$ is \emph{squarefree} (i.e.,
it is not divisible by the square of a noncostant polynomial),
the number of distinct roots of $p(x)$ in the interval $(a\..b]$ is $V(a)-V(b)$; all polynomials we will study will be squarefree.
\subsection{Bounding the dominant eigenvalue}
We now discuss how to bound the dominant eigenvalue $\rho_k$ of $B_k$ (and thus $G_k$); the same results
hold for the dominant eigenvalue $\rho'_k>\rho_k$ of $B'_k$ (and thus $G'_k$).
The approach we describe will be used
throughout the rest of the paper.
Consider the characteristic polynomial of $B_k$
\[
p_k(\lambda) = \det(1- \lambda B_k).
\]
We can compute its Sturm polynomials and evaluate them at the points
$k+\frac1{k^2}$ and $k+\frac3{4k}$.
This evaluation leaves us with a pair of rational functions in $k$ for each
Sturm polynomial in the sequence, and such functions have a defined sign
for $k\to\infty$ that depends on the sign of the ratio of the leading
coefficients of their numerator and denominator:
in other words, for large enough $k$ we can count the number of zeroes of
$p_k(\lambda)$ in the interval
$(k+\frac1{k^2}\..k+\frac3{4k}]$, and indeed $p_k(\lambda)$
has exactly one zero in that interval for $k\geq 24$.
If we apply the same technique to the interval $\left(k+\frac3{4k}\..2k\right]$, we find no zeroes. Since $2k$ is
an upper bound for the dominant eigenvalue of both matrices (as it is larger than the geometric mean
of indegree and outdegree of all vertices~\cite{KwaSRDG}), we conclude that the spectral radius $\rho_k$ of $B_k$
lies in $\left(k+\frac1{k^2}\..k+\frac3{4k}\right]$.
\subsection{Bounding the dominant eigenvector}
Armed with this knowledge, we approach the study of the dominant eigenvectors of $B_k$ and $B'_k$.
There is no way to compute them exactly: thus, we resort to the study of
$\mathbf 1(1 -\alpha B_k\bigr)^{-1}$,
because the dominant eigenvector $\bm e$ of $B_k$ and $\bm e'$ of $B'_k$ can be expressed as~\cite{VigSR}
\begin{align}
\label{eq:limit}
\bm e &= \lim_{\alpha\to1/\rho_k}\bigl(1-\alpha\rho_k\bigr)\mathbf1\bigl(1 -\alpha B_k\bigr)^{-1}.\\
\bm e' &= \lim_{\alpha\to1/\rho'_k}\bigl(1-\alpha\rho'_k\bigr)\mathbf1\bigl(1 -\alpha B'_k\bigr)^{-1}.
\end{align}
In fact,
$(1 -\alpha B_k\bigr)^{-1}$ is a slightly different
way (up to a constant factor) to define the \emph{resolvent} of $B_k$~\cite{DuSLO1}, but the formulation we use here will make it easier to apply the results
we will develop in the sections on Katz's index and PageRank.
While we have no way to compute exactly the eigenvectors of $B_k$, we can
compute symbolically $\mathbf1\bigl(1 -\alpha B_k\bigr)^{-1}$, thus obtaining for each node of $B_k$
a rational function in $\alpha$ whose coefficients are polynomials in $k$, and do the same for $B'_k$.
We will be interested in comparing eigenvector centralities, that is, in proving statements (for nodes $x$ and $y$ of $B_k$) of the form
\[
\frac{e_x}{e_y}=\lim_{\alpha\to1/\rho_k}\frac{\left[\bigl(1-\alpha\rho_k\bigr)\mathbf1\bigl(1 -\alpha B_k\bigr)^{-1}\right]_x}{\left[ \bigl(1-\alpha\rho_k\bigr)\mathbf1\bigl(1 -\alpha B_k\bigr)^{-1}\right]_y}>1.
\]
However,
\[
\frac{e_x}{e_y}=\lim_{\alpha\to1/\rho_k}\frac{\left[\mathbf1\bigl(1 -\alpha B_k\bigr)^{-1}\right]_x}{\left[ \mathbf1\bigl(1 -\alpha B_k\bigr)^{-1}\right]_y}
=\lim_{\alpha\to1/\rho_k}\frac{\left[\mathbf1\cdot\adjugate{1-\alpha B_k}\right]_x}{\left[\mathbf1\cdot\adjugate{1-\alpha B_k}\right]_y}
=\frac{\left[\mathbf1\cdot\adjugate{1- B_k/\rho_k}\right]_x}{\left[\mathbf1\cdot\adjugate{1- B_k/\rho_k}\right]_y},
\]
where we used the fact that the inverse is the \emph{adjugate matrix}~\cite{GanTM} divided by the determinant
\[
\adjugate{1-\alpha B_k}=\bigl(1 -\alpha B_k\bigr)^{-1}\cdot \det(1 -\alpha B_k).
\]
The final substitution can be performed safely because the column-sums of the adjugate
must be nonzero in a neighborhood of $\rho_k$, or the limits~(\ref{eq:limit}) would not be finite and positive.
The advantage is that the entries of $\adjugate{1-\alpha B_k}$ are just polynomials. The same considerations hold for $B'_k$.
We thus define, for every node $x$,
\begin{align*}
\operatorname{pre}_\alpha(x)&= \left[\mathbf1\cdot\adjugate{1 -\alpha B_k}\right]_x\\
\operatorname{post}_\alpha(x)&= \left[\mathbf1\cdot\adjugate{1 -\alpha B'_k}\right]_x.\\
\end{align*}
For example,
\begin{multline*}
\operatorname{pre}_\alpha(0) = (-2k^3 + 7k^2 - 7k + 2)\alpha^6 + (2k^2 - 7k + 5)\alpha^5 + (2k^3 - 6k^2 + 6k)\alpha^4\\ + (k^3 - 5k^2 + 9k - 7)\alpha^3 + (-k^2 + k - 3)\alpha^2 + (-k + 2)\alpha + 1.
\end{multline*}
Note that in the adjacency matrix of $B_k$ just three rows contain $k$: as a consequence, the degree in $k$
of the coefficients of the polynomials in $\alpha$ is at most three.
Since $k+\frac3{4k}>\rho_k$, we start by showing that
\[
\operatorname{pre}_{1/\left(k+\frac3{4k}\right)}(1)>\operatorname{pre}_{1/\left(k+\frac3{4k}\right)}(4)
\]
and once again, since we are dealing with rational functions in $k$, for enough large $k$ the difference
\[
\operatorname{pre}_{1/\left(k+\frac3{4k}\right)}(1)-\operatorname{pre}_{1/\left(k+\frac3{4k}\right)}(4)
\]
has a constant sign: in particular, for $k\geq 53$ it is positive. The same analysis, however, shows that
\[
\operatorname{post}_{1/\left(k+\frac3{4k}\right)}(1) < \operatorname{post}_{1/\left(k+\frac3{4k}\right)}(4)
\]
when $k\geq 3$.
We are now going to extend our inequalities to a range comprising $1/\rho_k$. If we consider the Sturm polynomials (in $\alpha$)
of \[
\operatorname{pre}_\alpha(1)- \operatorname{pre}_\alpha(4),\] we find no zero between $\alpha = 1/\left(k+\frac3{4k}\right) <1/ \rho_k$
and $\alpha = 1/\left(k+\frac1{k^2}\right) >1/ \rho_k$ for $k\geq 53$. Hence, for $1/\left(k+\frac3{4k}\right)< \alpha\leq 1/\left(k+\frac1{k^2}\right)$
\[
\operatorname{pre}_\alpha(1)>
\operatorname{pre}_\alpha(4),
\]
so, in particular,
\[
\operatorname{pre}_{1/\rho_k}(1)>\operatorname{pre}_{1/\rho_k}(4),
\]
showing that the eigenvector centrality of node $1$ is larger than that of node $4$ for $k\geq 53$.
A similar analysis for $\operatorname{post}$ shows that
\[
\operatorname{post}_{1/\rho'_k}(1)<\operatorname{post}_{1/\rho'_k}(4)
\]
for $k\geq 1$.
Thus, in the graph $G_k$ the addition of the edge $0\adj1$ causes
vertex $1$ to violate rank monotonicity. Further analysis of the same kind on the remaining nodes show
that only the vertices labeled with $4$ change their importance relatively to vertex $1$,
which implies that vertex $1$ is demoted by $k$ positions.
Finally, studying the polynomial $\operatorname{pre}_\alpha(1)-\operatorname{pre}_\alpha(0)$ it is easy to see that in our example
vertex $1$ is more important than vertex $0$ for $k\geq 54$.
While all the previous discussions are valid for $k\geq 54$, numerical computations show that the result indeed extends to all $k\geq 7$.
Hence:
\begin{thm}
Eigenvector centrality is not rank monotone (top violation) on the graphs $G_k$ of Figure~\ref{fig:eig} for $k\geq 7$.
\end{thm}
By gaging accurately the size of the star around $1$ it is possible to
find also bottom violations of rank monotonicity. We have
tabulated the first few values of $k$ for which there is a suitable star, and we
show them in Table~\ref{tab:eig}: we conjecture that there is a function of $k$ of
order $\Theta(k^2)$ which gives a correct real value for $s$, and examples emerge when
such value is very close to an integer.
\begin{table}
\centering
\begin{tabular}{ll|ll|ll|ll|ll|ll}
\multicolumn{1}{c}{$k$}&\multicolumn{1}{c|}{$s$}&\multicolumn{1}{c}{$k$}&\multicolumn{1}{c|}{$s$}&\multicolumn{1}{c}{$k$}&\multicolumn{1}{c|}{$s$}&\multicolumn{1}{c}{$k$}&\multicolumn{1}{c|}{$s$}&\multicolumn{1}{c}{$k$}&\multicolumn{1}{c|}{$s$}&\multicolumn{1}{c}{$k$}&\multicolumn{1}{c}{$s$}\\
\hline
8 & 40 & 17 & 217 & 30 & 733 & 40 & 1344 & 57 & 2815 & 68 & 4059 \\
9 & 53 & 18 & 246 & 31 & 786 & 43 & 1564 & 59 & 3024 & 69 & 4184 \\
10 & 67 & 19 & 276 & 32 & 840 & 44 & 1641 & 61 & 3241 & 70 & 4310 \\
11 & 83 & 24 & 456 & 34 & 955 & 45 & 1720 & 62 & 3352 & 72 & 4569 \\
12 & 101 & 26 & 541 & 35 & 1015& 48 & 1968 & 63 & 3465 & 73 & 4701 \\
14 & 142 & 27 & 586 & 36 & 1077 & 50 & 2143 & 64 & 3580 & 74 & 4835 \\
15 & 165 & 28 & 633 & 37 & 1141 & 51 & 2233 & 65 & 3697 & 75 & 4971 \\
16 & 190 & 29 & 682 & 38 & 1207 & 56 & 2713 & 66 & 3816 & 76 & 5109 \\
\end{tabular}
\vspace*{2mm}
\caption{\label{tab:eig}Pairs of values providing bottom violations of
rank monotonicity for eigenvector centrality: $k$ is the same as in Figure~\ref{fig:eig},
and $s$ is the size of the star around $1$ (in Figure~\ref{fig:eig}, $s=(k-1)(k-2)$).}
\end{table}
\section{Seeley's index}
\label{sec:seeley}
A natural variant of eigenvector centrality is Seeley's index~\cite{SeeNRI}, the steady state of the (uniform)
random walk on the graph (for more details, see~\cite{BoVAC}). The situation here is quite different: it is a
well-known fact that if the graph is connected the steady-state probability of
vertex $x$ is simply $d(x)/2m$, where $d(x)$ is the degree of $x$---essentially, the centrality of a vertex
is just its $\ell_1$-normalized degree.
As a consequence:
\begin{thm}
\label{thm:seeleyrank}
Seeley's index is strictly rank monotone on undirected graphs.
\end{thm}
The situation is almost the same for score monotonicity if we assume $\ell_1$-normalization:
\begin{thm}
\label{thm:seeleyscore}
Seeley's index ($\ell_1$-normalized degree) is score monotone on undirected graphs, except in the case of a graph formed by a star graph and one or more additional isolated vertices.
\end{thm}
\begin{proof}
When we add an edge between $x$ and $y$ in a graph with $m$ edges, the score of $x$ changes from $d(x)/2m$ to $(d(x)+1)/(2m+2)$. If we require
\[
\frac{d(x)+1}{2m+2}> \frac{d(x)}{2m}
\]
we obtain $d(x)< m$. Since obviously $d(x)\leq m$, the condition is always true except when $d(x)=m$, which corresponds to
the case of a disconnected graph formed by a star graph and by additional isolated vertices.
Indeed, in that case adding an edge between an isolated vertex and the center of the star will not change the score
of the center.
\end{proof}
\section{Interlude: graph fibrations and damped spectral rankings}
The key observation used to build the counterexample for eigenvector centrality was Theorem~\ref{th:sachs},
stating that lifting of vectors commutes with matrix multiplication.
The theorem is true also for \emph{weighted} graphs, as long as the fibration preserves weights
and adjacency matrices are defined by adding the weights of all arcs between two nodes.
An interesting consequence of this fact is the following:
\begin{thm}
\label{th:sachsex}[\cite{BLSGFGIP}]
Let $G$ and $B$ be weighted graphs, and $\phi:G\to B$ be a surjective weight-preserving fibration; then,
given a (row) vector $\bm v$ of size $n_B$ we have
\[
\bm v^\phi (1-\alpha G)^{-1} = \bigl(\bm v (1-\alpha B)^{-1}\bigr)^\phi.
\]
\end{thm}
The proof is simple:
\begin{multline*}
\bm v^\phi (1-\alpha G)^{-1} = \bm v^\phi \sum_{i=0}^\infty (\alpha G)^i =
\sum_{i=0}^\infty\bm v^\phi (\alpha G)^i \\= \sum_{i=0}^\infty\bigl(\bm v (\alpha B)^i\bigr)^\phi =
\Bigl(\bm v\sum_{i=0}^\infty(\alpha B)^i\Bigr)^\phi = \bigl(\bm v (1-\alpha B)^{-1}\bigr)^\phi.
\end{multline*}
Theorem~\ref{th:sachsex} makes it possible to apply the techniques we used for eigenvector centrality to general \emph{damped spectral rankings}, as defined
in~\cite{VigSR}, of which both Katz's index and PageRank are special instances. Both centralities can be defined, up to a constant multiplying factor, as
\[
\bm v (1-\alpha M)^{-1}
\]
for suitable preference vector $\bm v$ and for a matrix $M$ derived from the adjacency matrix of the graph.
\section{Katz's index}
\label{sec:katz}
Recall that Katz's index~\cite{KatNSIDSA} is defined as
\[
\mathbf1\sum_{i=0}^\infty \alpha^i G^i = \mathbf1(1-\alpha G)^{-1},
\]
where $0\leq\alpha<1/\rho(G)$ (here, $\rho(G)$ is the spectral radius of $G$).
It is trivially score monotone, but we will prove that it is not rank monotone.
First of all, we note that if $\alpha$ is small enough Katz's index will be strictly rank monotone:
\begin{thm}
\label{th:katzzero}
Let $G$ be a graph and $\rho$ its spectral radius.
Then there is an $\bar\alpha<1/\rho$ such that for $\alpha\leq\bar\alpha$
Katz's index is strictly rank monotone on $G$.
\end{thm}
\begin{proof}
We remark that
\[
\mathbf1\sum_{i=0}^\infty \alpha^i G^i = \mathbf1 + \alpha\mathbf1G + \alpha^2 \mathbf1 G^2\sum_{i=0}^\infty \alpha^i G^i.
\]
The relative node importance in $\mathbf1 + \alpha\mathbf1G$ is exactly that defined by degree, and score differences are $O(\alpha)$ for $\alpha\to0$. However,
\[
\Bigl\|\alpha^2\mathbf1 G^2 \sum_{i=0}^\infty \alpha^iG^i\Bigr\|_\infty \leq \alpha^2\bigl\|G^2\bigr\|_\infty\Bigl\|\sum_{i=0}^\infty \alpha^i G^i\Bigr\|_\infty,
\]
and given any $0<\alpha'<1/\rho_k$ for $\alpha\leq\alpha'$ the last expression is $O\bigl(\alpha^2\bigr)$ for $\alpha\to 0$. Thus, there is an $\alpha_G$ such that,
for $\alpha\leq\alpha_G$, the relative importance of a node of $G$ is that defined by its degree. If we minimize over all such $\alpha$'s for all
graphs obtained by adding an edge to $G$, we obtain the value $\bar\alpha$ of the statement.
\end{proof}
On the other hand, we are now going to provide an example on which
rank monotonicity is not satisfied when we go sufficiently close to $1/\rho$.
We can use the same counterexample as for eigenvector centrality (Figure~\ref{fig:eig}):
in view of Theorem~\ref{th:sachsex}
the analysis performed
in Section~\ref{sec:eigen} already shows that Katz's index is not rank monotone on $G_k$ for sufficiently large $k$ and for all
\[
\alpha \in \biggl[\frac1{k+\frac3{4k}}\..\frac1{\rho_k'}\biggr).
\]
In other words,
\begin{thm}
\label{thm:katz}
Let $\rho_k'$ be the spectral radius of the graph $G_k'$ in Figure~\ref{fig:eig}.
For $k\geq 54$, there exists some $\nu_k<\frac1{k+\frac3{4k}}$ such that
Katz's index is not rank monotone (top violation) on $G_k$ for all $\alpha\in \left(\nu_k\..\frac1{\rho_k'}\right)$.
\end{thm}
Note that the theorem above claims that the violation happens in a left neighborhood of the upper bound of $\alpha$; moreover,
on the left we can get as close as desired to $0$ given a suitable $k$. This is the best possible scenario, in view of Theorem~\ref{th:katzzero}.
Also our considerations about demotion in Section~\ref{sec:eigen} transfer immediately to the present setting.
Further analysis by Sturm polynomials in the interval $\left(\frac1{k+\frac1k}\..\frac1{k+\frac3{4k}}\right]$ shows the following:
\begin{itemize}
\item the relative importance of node $1$ and node $4$ in $\operatorname{pre}_\alpha(-)$ flips (node $4$ is more important than node $1$ at the beginning of the interval
and then becomes less important, after some value of $\alpha$, say $\alpha'$);
\item the relative importance of node $0$ and node $1$ in $\operatorname{pre}_\alpha(-)$ flips (node $0$ is more important than node $1$ at the beginning of the interval
and then becomes less important, after some value of $\alpha$, say $\alpha''$);
\item $\operatorname{pre}_\alpha(1)-\operatorname{pre}_\alpha(4)$ always dominates $\operatorname{pre}_\alpha(1)-\operatorname{pre}_\alpha(0)$.
\end{itemize}
The latter observation implies $\alpha'<\alpha''=\nu_k$, in the notation of Theorem~\ref{thm:katz}; since the relative importance of node $1$ and node $4$ remains the same in $\operatorname{post}_\alpha(-)$
(node $4$ is always more important than node $1$ in the interval after the addition of the edge), in the interval $\bigl(\alpha'\.. \nu_k\bigr)$
we can observe a bottom violation of rank monotonicity.
Moreover, the interval $(\alpha'\..\nu_k)$ gets closer to the upper bound $\frac1{\rho'_k}>\nu_k$ as $k$ gets larger:
\[
\frac{\frac1{\rho'_k} - \alpha'}{\frac1{\rho'_k}}\leq \frac{\frac1{\rho'_k} - \frac1{k+\frac1k}}{\frac1{\rho'_k}} = 1-\frac{\rho'_k}{k+\frac1k}\leq1-\frac{k+\frac1{k^2}}{k+\frac1k}\to 0 \quad\text{for $k\to\infty$}.
\]
\begin{thm}
\label{thm:katz2}
For every $k$, there is an interval of values of $\alpha$ contained in $\left(\frac1{k+\frac1k}\..\frac1{k+\frac3{4k}}\right]$
in which Katz's index is not rank monotone (bottom violation). The interval gets arbitrarily close to $\frac1{\rho'_k}$ as $k\to \infty$.
\end{thm}
As a final consideration, there is another range of validity of Theorem~\ref{thm:katz}: if we further analyze with Sturm polynomials the relative
importance of node $1$ and node $4$ in the interval $\left(\frac1{k+\frac2k}\..\frac1{k+\frac3{4k}}\right]$, we find two sign changes
in $\operatorname{pre}_\alpha(1)- \operatorname{pre}_\alpha(4)$, two sign changes
in $\operatorname{pre}_\alpha(1)- \operatorname{pre}_\alpha(0)$ and zero sign changes in $\operatorname{post}_\alpha(1)- \operatorname{post}_\alpha(4)$: thus, there is an interval
comprising $\frac1{k+\frac2k}$ in which the violation of rank monotonicity happens again. Also in this interval
$\operatorname{pre}_\alpha(1)-\operatorname{pre}_\alpha(4)$ always dominates $\operatorname{pre}_\alpha(1)-\operatorname{pre}_\alpha(0)$, hence, we have both top violations and bottom violations; it
is also immediate to show demotion.
Figure~\ref{fig:katz} resumes graphically the results proved in this section.
\begin{figure}
\centering
\includegraphics{fig/rankund-1-mps}
\caption{\label{fig:katz}A graphical resume of the results about Katz's index proved in Section~\ref{sec:katz}.
Dashed lines (dotted lines, resp.) represent intervals
of values of $\alpha$ in which we proved top (bottom, resp.) violations of rank monotonicity (Theorems~\ref{thm:katz} and~\ref{thm:katz2}).
The thick interval represents a region where rank monotonicity is guaranteed (Theorem~\ref{th:katzzero}).}
\end{figure}
\section{PageRank}
PageRank~\cite{PBMPCR} can be defined as
\[
(1-\alpha)\bm v\sum_{i=0}^\infty \alpha^i\bar G^i = (1-\alpha) \bm v(1-\alpha \bar G)^{-1},
\]
where $\alpha\in[0\..1)$ is the damping factor,
$\bm v$ is a non-negative preference vector with unit $\ell_1$-norm,
and $\bar G$ is the row-normalized version\footnote{Here we are assuming that $G$ has no \emph{dangling nodes} (i.e., nodes with outdegree $0$).
If dangling nodes are present, you can still use this definition (null rows are left untouched in $\bar G$), but then to obtain PageRank you
need to normalize the resulting vector~\cite{BSVPFD,DCGRFPCSLS}. So all our discussion can also be applied
to graphs with dangling nodes, up to $\ell_1$-normalization.} of $G$; that is,
$\bar G$ is just the (adjacency matrix of the) weighted version of $G$ defined by letting $w(a)=1/\sum_{x\in N_G}G_{s_G(a)\, x}$.
Hence, if you have a weighted graph $B$,
a weight-preserving fibration $\phi: \bar G \to B$ that is surjective on the nodes, and a vector $\bm u$ of size $n_B$ such that
$\bm u^\phi$ has unit $\ell_1$-norm, you can deduce from Theorem~\ref{th:sachsex} that
\begin{equation}
\label{eqn:resumepr}
(1-\alpha) \bm u^\phi (1-\alpha \bar G)^{-1}=\left(\bm (1-\alpha) \bm u (1-\alpha B)^{-1}\right)^\phi.
\end{equation}
On the left-hand side you have the actual PageRank of $G$ for a preference vector that is fiberwise constant;
on the right-hand side you have a damped spectral ranking of $B$.
Note that $B$ is not necessarily row-stochastic, and $\bm u$ has not unit $\ell_1$-norm,
so technically the right-hand side of the equation in Theorem~\ref{th:sachsex} is
not PageRank anymore.
\smallskip
We first observe that
\begin{thm}
\label{th:prlim}
Given an undirected graph $G$, there is a value of $\alpha$ for which PageRank is strictly rank monotone on $G$.
The same is true for score monotonicity, except when $G$ is formed by a star graph and one or more additional isolated vertices.
\end{thm}
\begin{proof}
We know that for $\alpha\to 1$, PageRank tends to Seeley's index~\cite{BSVPFDF}. Since Seeley's index is strictly rank monotone (Theorem~\ref{thm:seeleyrank}), for each non-adjacent pair
of vertices $x$ and $y$ there is a value $\alpha_{xy}$ such that for $\alpha\geq\alpha_{xy}$ adding the edge $x\scalebox{0.5}[1.0]{-} y$ is strictly
rank monotone. The proof is completed by taking $\alpha$ larger than all $\alpha_{xy}$'s.
The result for score monotonicity is similar, using Theorem~\ref{thm:seeleyscore}.
\end{proof}
It is interesting to remark that this result is dual to Theorem~\ref{th:katzzero}: Katz's index is approximated by
degree for values of the damping factor close to the lower bound (zero), whereas PageRank is approximated by degree for values of
the damping factor close to the upper bound (one).
On the other hand, we will now show that
\emph{for every possible value of the damping factor $\alpha$} there is a graph on which PageRank is neither
rank nor score monotone.
Our proof strategy will be identical to the one we used for Katz's index, except that now we expect our example to
satisfy rank monotonicity when $\alpha$ is close
to its upper bound, instead of its lower bound, because of Theorem~\ref{th:prlim}.
\begin{figure}
\centering
\begin{tabular}{cc}
\raisebox{.5cm}{$G_k$\qquad}&\includegraphics{fig/rankund-201-mps}\\
\raisebox{.5cm}{$B_k$\qquad}&\includegraphics{fig/rankund-202-mps}
\end{tabular}
\[
B_k = \left(\begin{matrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac1{k} & \frac1{k} \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & \frac12 & 0 & \frac12 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & \frac1{k + 2} & 0 & \frac1{k + 2} & \frac1{k + 2} & 0 & 0 & 0 \\
0 & 0 & 0 & \frac{k}{k} & \frac{k - 1}{k} & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & \frac12 & 0 & 0 & \frac12 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & \frac12 & 0 & \frac12 & 0 \\
\frac1{k + 1} & 0 & 0 & 0 & 0 & 0 & \frac1{k + 1} & 0 & \frac1{k + 1} \\
\frac{k - 1}{k} & 0 & 0 & 0 & 0 & 0 & 0 & \frac{k - 1}{k} & \frac{k - 2}{k}
\end{matrix}\right)
\]
\[
B'_k = \left(\begin{matrix}
0 & \frac1{k + 1} & 0 & 0 & 0 & 0 & 0 & \frac1{k + 1} & \frac1{k + 1} \\
\frac12 & 0 & \frac12 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & \frac12 & 0 & \frac12 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & \frac1{k + 2} & 0 & \frac1{k + 2} & \frac1{k + 2} & 0 & 0 & 0 \\
0 & 0 & 0 & \frac{k}{k} & \frac{k - 1}{k} & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & \frac12 & 0 & 0 & \frac12 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & \frac12 & 0 & \frac12 & 0 \\
\frac1{k + 1} & 0 & 0 & 0 & 0 & 0 & \frac1{k + 1} & 0 & \frac1{k + 1} \\
\frac{k - 1}{k} & 0 & 0 & 0 & 0 & 0 & 0 & \frac{k - 1}{k} & \frac{k - 2}{k}
\end{matrix}\right)
\]
\caption{\label{fig:pr}A parametric counterexample graph for PageRank: when adding the edge $0\scalebox{0.5}[1.0]{-} 1$,
vertex $1$ violates score and rank monotonicity (bottom violation). The $k$ vertices labeled with $4$
form a $(k+1)$-clique with vertex $3$, and the $k-1$ vertices labeled with $8$ form a $(k+1)$-clique
with vertices $0$ and $7$. Arc
labels represent multiplicity; weights are induced by the uniform distribution on the upper graph.
The matrices displayed are the adjacency matrix of $B_k$ and $B'_k$; differently from Figure~\ref{fig:eig}, we show them both explicitly to
highlight how the addition of the new edge influences row normalization.}
\end{figure}
In Figure~\ref{fig:pr} we show a family of total graphs $G_k$ depending
on an integer parameter $k$, and an associated family of bases $B_k$, with fibrations defined on the nodes following the node labels,
and on the arcs in the only possible way.\footnote{Note that in the conference version of this paper~\cite{BFVSRMUN}
nodes are numbered differently, and the denominators of the second row of the adjacency matrix
displayed therein are $k-1$, mistakenly, instead of $k+1$.} Weights are defined by
normalizing the adjacency matrix of $G_k$, and then using the fibration to transfer the weights on the arcs $B_k$
(it is easy to see that no conflict arises when multiple arcs of $G_k$ are mapped to the same arc of $B_k$).
As usual, $G'_k$ and $B'_k$ are the same graphs with the additional edge $0\scalebox{0.5}[1.0]{-} 1$.
The basic intuition behind the graphs $G_k$ is that when you connect a high-degree vertex $x$ with a low-degree vertex $y$,
$y$ will pass to $x$ a much larger fraction of its score than in the opposite direction. This phenomenon is caused by the stochastic normalization of the
adjacency matrix: the arc from $x$ to $y$ will have a low coefficient, due to the high degree of $x$, whereas the arc from $y$ to $x$
will have a high coefficient, due to the low degree of $y$.
While $G_k$ has $2k+6$ vertices, $B_k$ has $9$ vertices, independently of $k$, and
thus its PageRank can be computed analytically as rational functions of $\alpha$ whose coefficients are rational functions
in $k$ (since the number of arcs of each $B_k$ is different).
We thus define, for every node $x$,
\begin{align*}
\operatorname{pre}_\alpha(x)&= \left[(1-\alpha)\mathbf1\bigl(1 -\alpha B_k\bigr)^{-1}\right]_x\\
\operatorname{post}_\alpha(x)&= \left[(1-\alpha)\mathbf1\bigl(1 -\alpha B'_k\bigr)^{-1}\right]_x.\\
\end{align*}
Note when discussing score monotonicity we cannot use the adjugate matrix
to simplify our computations, as we did in Section~\ref{sec:eigen}, but we can use without loss of generality
an arbitrary constant vector as preference vector. When discussing rank monotonicity,
however, we will switch silently to the adjugate (because the denominator cannot change its sign anywhere in $[0\..1)$).
For example,
\[
\operatorname{post}_\alpha(1)= \frac{\frac{2 k^{2} - 6 k + 4}{k^{2} - 6 k + 4} \alpha^{5} +
\frac{-14 k^{2} + 12 k + 4}{k^{2} - 6 k + 4} \alpha^{4} +\cdots+
\frac{6 k^{4} + 6 k^{3} - 24 k^{2} - 24 k}{k^{2} - 6 k + 4} \alpha +
\frac{-4 k^{4} - 12 k^{3} - 8 k^{2}}{k^{2} - 6 k + 4}}
{\alpha^{5} + \frac{-2 k^{3} - 10 k^{2} + 12 k + 4}{k^{2} - 6 k + 4} \alpha^{4} + \cdots +
\frac{4 k^{4} + 4 k^{3} - 20 k^{2} - 24 k}{k^{2} - 6 k + 4} \alpha +
\frac{-4 k^{4} - 12 k^{3} - 8 k^{2}}{k^{2} - 6 k + 4}},
\]
where we omitted part of the terms for lack of space.
Once again, in the adjacency matrix of $B_k$ just four rows contain $k$: as a consequence, the degree in $k$
of numerators and denominators of coefficients of the rational functions in $\alpha$ is at most four.
\subsection{Score monotonicity}
We start by considering node $1$: evaluating $\operatorname{post}_\alpha(1)-\operatorname{pre}_\alpha(1)$ in
$\alpha=2/3$ we obtain a negative value for $k\geq 11$, showing there is
a value of $\alpha$ for which node $1$ violates score monotonicity.
Then, we use again Sturm polynomials to show that
for $k\geq 13$ the numerator of $\operatorname{post}_\alpha(1)-\operatorname{pre}_\alpha(1)$ never changes its sign
in $\bigl(a_k\..b_k\bigr]$, where
\[
a_k = \frac23 - \frac{2k}{3k+ 100} < \frac23 < \frac23+\frac{k}{3k+100} = b_k,
\]
while the denominator of $\operatorname{post}_\alpha(1)-\operatorname{pre}_\alpha(1)$ cannot have zeros in $[0\..1)$.
The interval $\bigl(a_k\..b_k\bigr]$ approaches $(0\..1]$ as $k$ grows,
so we conclude that
the interval of values of $\alpha$ for which the score of node $1$ decreases reaches the whole unit interval as $k$ grows.
Finally, by studying (as in the case of Katz's index) the polynomial $\operatorname{pre}_\alpha(1)-\operatorname{pre}_\alpha(0)$ it is easy to see that in our example
node $0$ is always more important than node $1$ as long as $k\geq 1$.
\begin{thm}
\label{th:prscore}
For every value of $\alpha\in(0\..1)$, for sufficiently large $k$ PageRank with damping factor $\alpha$ is not score monotone (bottom violation) on the
graphs $G_k$ of Figure~\ref{fig:pr}.
\end{thm}
It is also interesting to count the sign changes of $\operatorname{post}_\alpha(1)-\operatorname{pre}_\alpha(1)$ in $\bigl(0\..a_k\bigr]$ (one) and $\bigl(b_k\..1\bigr)$ (one), as they describe the behavior
of the score change for limiting values: initially, the score increases;
then, it starts to decrease somewhere before $a_k$ and stops decreasing somewhere after $b_k$, as expected from Theorem~\ref{th:prlim}.
\subsection{Rank monotonicity}
We now use the same example to prove the lack of rank monotonicity. In this case, we study in a similar way
$\operatorname{pre}_\alpha(1)-\operatorname{pre}_\alpha(5)$, which is positive in $\alpha=2/3$ if $k\geq 13$.
To extend our results about rank monotonicity to every $\alpha$, we use again Sturm polynomials to show that
the numerator of $\operatorname{pre}_\alpha(1)-\operatorname{pre}_\alpha(5)$ never changes its sign in $\bigl(a_k\..b_k\bigr]$ for $k\geq 14$.
Again, it is interesting to count the sign changes of $p(\alpha)$ in $(0\..a_k]$ (one) and $(b_k\..1)$ (one):
initially, node $1$ has a smaller PageRank than node $5$; then, somewhere before $a_k$,
node $1$ starts having a larger PageRank than $5$; somewhere after $b_k$, we return to the initial condition, as expected from Theorem~\ref{th:prlim}.
Finally, we study $\operatorname{post}_\alpha(1)-\operatorname{post}_\alpha(5)$ which, is negative in $\alpha=2/3$
and again has no sign changes in $\bigl(a_k\..b_k\bigr]$ for $k\geq 5$. More precisely, we study $p(\alpha)$, where
$\operatorname{post}_\alpha(1)-\operatorname{post}_\alpha(5) = (1-\alpha)^2p(\alpha)$, as $\operatorname{post}_\alpha(1)-\operatorname{post}_\alpha(5)$ is not squarefree, but $p(\alpha)$ is.
In this case, there are two sign changes in $(0\..a_k]$ and no sign change in $(b_k\..1)$, so
initially, node $1$ is less important than node $5$; then, in an interval of values before $a_k$ it is more important;
then, it starts to be again less important before $a_k$; and it becomes as important as node $5$
only in the limit for $\alpha\to 1$.
\begin{thm}
\label{th:prrank}
For every value of $\alpha\in(0\..1)$, for sufficiently large $k$ PageRank with damping factor $\alpha$ is not rank monotone (bottom violation)
on the graphs $G_k$ of Figure~\ref{fig:pr}.
\end{thm}
Recall that in~\cite{BLVRMCM} PageRank was proven to be both score and
(strictly) rank monotone for all \emph{directed} graphs and all
$\alpha\in[0\..1)$, given that the preference vector is positive:
comparing those results with Theorems~\ref{th:prscore} and~\ref{th:prrank}, we
see once more that in the undirected case the behavior is radically different.
For sufficiently large $k$, almost all nodes are more important (i.e., have larger PageRank score) than node $1$
both before and after edge addition, with the only exception of nodes $5$ and $6$: as we said, node $5$ is less important than node $1$ before
but more important after the edge addition; whereas node $6$ is also less important than node $1$ before, and becomes as important as node $1$
after the edge addition (as node $6$ and node $1$
become equivalent modulo an automorphism). As a result, node $1$ is demoted.
Finally, we provide in Figure~\ref{fig:pr-new} a counterexample in which the more important node violates rank monotonicity.
In this case, the intuition is that we connect two nodes with the same degree but different scores.
As in the previous case, the counterexample works for any chosen $\alpha$, up to an appropriate choice of the parameter $k$.
The proof follows the same line of attack, and detailed computations can be found in the Sage worksheets. The main
difference is that the relevant interval $\bigl(a_k\..b_k\bigr]$ is now
\[
a_k = \frac23-\frac{2\sqrt k}{3\sqrt k+10}<\alpha\leq \frac23 + \frac k{3k+1} = b_k.
\]
\begin{thm}
\label{th:prrank-new}
For every value of $\alpha\in(0\..1)$, for sufficiently large $k$ PageRank with damping factor $\alpha$ is neither score nor rank monotone (top violation)
on the graphs $G_k$ of Figure~\ref{fig:pr-new}.
\end{thm}
\begin{figure}
\centering
\begin{tabular}{cc}
\raisebox{1cm}{$G_k$\qquad}&\includegraphics{fig/rankund-701-mps}\\
\raisebox{1cm}{$B_k$\qquad}&\includegraphics{fig/rankund-702-mps}
\end{tabular}
\caption{\label{fig:pr-new}
A parametric counterexample graph for PageRank: when adding the edge $0\scalebox{0.5}[1.0]{-} 1$,
vertex $0$ violates score and rank monotonicity (top violation).
There is a star with $k$ leaves around vertex $0$, a star with $k$ leaves around vertex $4$,
and the $k$ vertices labeled with $6$ form a $(k+2)$-clique with vertices $1$ and $2$.
Arc labels represent multiplicity; weights are induced by the uniform distribution on the upper graph.}
\end{figure}
\subsection{Anecdotal Evidence: PageRank on the IMDB}
\begin{table}[t]
\renewcommand{\arraystretch}{1.2}
\renewcommand{\tabcolsep}{1ex}
\begin{tabular}{lll}
Score increase & Score decrease & Violations of rank monotonicity\\
\hline
Meryl Streep & Yasuhiro Tsushima & Anne--Mary Brown, Jill Corso,~\ldots\\
Denzel Washington & Corrie Glass & Patrice Fombelle, John Neiderhauser,~\ldots\\
Sharon Stone & Mary Margaret (V) & Dolores Edwards, Colette Hamilton,~\ldots\\
John Newcomb & Robert Kirkham & Brandon Matsui, Evis Trebicka,~\ldots
\end{tabular}
\vspace*{.5em}
\caption{\label{tab:rank}A few examples of violations of score monotonicity and rank monotonicity in the Hollywood co-starship graph
\texttt{hollywood-2011}. If we add an edge between the actors in the first and second column,
the first actor has a score increase, the second actor has a score decrease,
and the actors in the third column, which were less important than the second actor, become more important after the edge addition.
The first three examples are bottom violations, whereas the last one is a top violation.}
\end{table}
To show that our results are not only theoretical, we provide a few interesting anecdotal examples from
the PageRank scores ($\alpha=0.85$) of the Hollywood co-starship graph,
whose vertices are actors/actresses in the Internet Movie Database, with an edge connecting them if played in the same movie.
In particular, we used the \texttt{hollywood-2011} dataset from the Laboratory for Web Algorithmics,\footnote{\url{http://law.di.unimi.it/}}
which contains approximately two million vertices and $230$ million edges.
To generate our examples, we picked two actors either at random, or considering
the top $1/10000$ of the actors of the graph in PageRank order and the bottom
quartile, looking for a collaboration that would hurt either actor (or
both).\footnote{Note that for this to happen, the collaboration should be a
two-person production. A production with more people would add more
edges.} About $4$\% of our samples yielded a violation of monotonicity, and in
Table~\ref{tab:rank} we report a few funny examples.
The first three cases are bottom violations: it is the less-known actor that loses score (and rank) by the collaboration
with the star, and not the other way round, as it happens also in the counterexample of Figure~\ref{fig:pr}.
In the last case, instead, we hava top violation:
a collaboration would damage the more important vertex, like in the counterexample of Figure~\ref{fig:pr-new}.
We found no case in which both actors would be hurt by the collaboration, and it is in fact an open problem whether this situation
can happen.
\section{Conclusions}
We have studied score and rank monotonicity on undirected graphs for some popular notions of centrality.
Our results show that except for Seeley's index (on connected graphs) there are always cases in which rank monotonicity
does not hold, and in the case of Katz's index and PageRank we can find range of values of the parameters where the
violation occurs;
moreover, some centralities are also not score monotone. We provide examples of both top and bottom violations
to highlight that even the knowledge of whether one is the more important or less important node is insufficient
to decide whether the new edge will be beneficial. A possible direction for future research
is to show that top and bottom violations cannot happen at the same time, that is, that the new
edge is beneficial for at least one endpoint.
This lack of monotonicity is opposite to that we observed in the directed case, and it can also be seen in real-world graphs (at least for PageRank).
It is interesting to note that even centrality indices that were designed for undirected graphs (e.g., closeness) are
not rank monotone in the undirected case (even under a connectedness assumption).
Our results show that common knowledge and intuitions about the behavior of centrality measures in the directed case cannot
be applied to the undirected case.
|
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| 1,692
|
{"url":"https:\/\/app.mschool.xyz\/courses\/Math\/library\/308e838a-c12f-4f9d-a037-5fef463cb2b5\/ee2f6af9-9e20-4268-a231-d3af37128718\/","text":"##### Indices - part 2\n###### Which of the following is NOT equivalent to $\\frac{1}{8}$?\n\nAll of the given expressions are valid forms of $\\frac{1}{8}$. If you are unsure about one of the expressions, you may need to check earlier videos.","date":"2019-02-23 18:50:58","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 6, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7242339849472046, \"perplexity\": 340.7102622452474}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-09\/segments\/1550249530087.75\/warc\/CC-MAIN-20190223183059-20190223205059-00393.warc.gz\"}"}
| null | null |
Q: Inherit color and underline color on links I'm having problems getting the green color applied to the anchor element (a) within #container.
In this fiddle:
http://jsfiddle.net/lasseedsvik/DnhHb/2/
HTML
<p>
<a href="">Green</a>
</p>
<div id="container">
<p>
<a href="">Green also</a>
</p>
<p>
<div><div><a href="">Also Green</a></div></div>
</p>
<p>
<h3><a href="">Red</a></h3>
</p>
<p>
<span style="color:yellow"><a href="">Yellow</a></span>
</p>
</div>
CSS
* {
font-size: inherit;
}
a {
color: green;
}
#container a {
color: inherit;
font-size: inherit;
}
p {
font-size: 18px;
}
h3 {
color: red;
font-size: 28px;
}
I've managed to get close by using the inherit property on #container a but the problem is that the 2nd and 3rd link dont "fall back" to the green color.
My original problem which is almost solved here, was that links that had <h2> or other colors that differed from green but always had a green text-decoration.
The content in the container is generated by a WYSIWYG-editor so I don't have much control over its content.
A: The original problem looked like this:
By specifying the following CSS:
a { color: green; }
#container a { color: inherit; }
The OP wanted to have a default green color for <a> elements, unless its ancestors have a set color, in which case the <a> element should inherit.
<a> was colored green
<.. id="container"> <*> <a> was colored black
<h3 style="color: red"> <a> should be colored red
<span style="color: yellow"> <a> should be colored yellow.
The problem was that the second <a> above should be colored green, instead of black. The reason it is rendered black, is because there is no such thing as an HTML element with undefined color, since the browser's default CSS, adds a computed style which makes it black.
What the OP initially wanted could be named Selective CSS Inheritance, which after a lot of research, seems impossible with the current CSS implementation.
The above could be easily done if CSS implemented a keyword/property value similar to inherit, possibly called inherit-user-defined, which would only inherit values from user-set, non browser-set styles.
I know that this doesn't actually answer the question, but I think it will be useful for readers that will search for this, since the question's accepted answer was unexpected for some of us.
A: Just remove the rule for
#container a {
color: inherit;
font-size: inherit;
}
A: See this link
use
* {
font-size: inherit;
}
a {
color: green;
}
#container a {
}
p {
font-size: 18px;
}
h3 a{
color: red;
font-size: 28px !important;
}
span a{
color: yellow;
}
A: Check this http://jsfiddle.net/DnhHb/5/
I just Removed the color:inherit from Container a Class
A: add this css:
#container p a {
color: green;
}
#container div a {
color: green;
}
#container span a {
color: yellow;
}
Now remove the inline css since it is not needed.
Reason: since you told it to inherit, you need to explicitly declare each instance, i.e. you need to say #container p a needs to be green.
See it fixed:
http://jsfiddle.net/DnhHb/6/
Also see a better formatted version:
http://jsfiddle.net/DnhHb/9/
|
{
"redpajama_set_name": "RedPajamaStackExchange"
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| 5,081
|
America's Lawyer
Environment | Health
David Pakman
Mike Malloy
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The Ring of Fire Network
Halliburton Gets Away with Misdemeanor Charge in Gulf Oil Spill
Ring of Fire Staff
Oil giant Halliburton pled guilty on Thursday to destroying evidence related to the 2010 BP oil spill. However, unlike the other companies involved in the oil spill, Halliburton, the company responsible for cementing the well, was not charged with a crime related to the causes of the disaster.
One man, a former Halliburton cementing technology director, was charged with destroying evidence Thursday. "Halliburton believes that this closure holds significant positive impacts for the company, its employees and shareholders," a company statement reads.
Sixty-one-year-old Anthony Badalamenti from Katy, Texas is accused of instructing two Halliburton employees to delete computer simulation data showing how BP constructed their Macondo well where the 2010 blowout occurred, leading to the deaths of 11 men, and a devastating oil spill.
According to Justice Department filings, "The computer simulation didn't bear out Halliburton's contention that BP erred by not following its advice on using certain equipment," the Wall Street Journal reports.
Badalamenti was told to preserve any data related to the well, as the government was conducting an investigation into the incidents that caused the Gulf oil spill. He is the fifth individual to face criminal charges for their involvement in the blowout.
Halliburton agreed to plead guilty to a misdemeanor count of unauthorized destruction of evidence. US District Judge Jane Triche-Milazzo in New Orleans accepted Halliburton's plea agreement, and charged the company with the maximum-allowable fine of $200,000 and a 3-year probation term.
The company also agreed to make a $55 million contribution to the National Fish and Wildlife Foundation.
Halliburton was contracted by BP to cement and seal the casing in the borehole, a process that should make any release of oil or gas impossible. But the cement pumped into the Macondo well just one day before the blowout was not an appropriate cement blend for the job, and was not given time to set before a negative pressure test was conducted, which allowed oil and gas to travel up the drill pipe where it exploded, according to an oil-field cementing expert's testimony.
According to Glen Benge's testimony in the civil trial against BP and its partner companies, there were at least 9 errors made during the cementing of BP's Macondo well, which contributed to the blowout. Benge laid most of the blame on BP and Halliburton, the company that provided the cementing material, The Times-Picayune reports.
Halliburton has always insisted that BP is to blame for the failed cement job because BP did not follow Halliburton's recommendation to use 21 centralizers, instead using only 6. Centralizers help seal the well. However, according to court documents, Badalamenti directed two employees to run two separate computer simulations to compare the performance of 21 centralizers versus 6.
When both simulations showed little difference of outcome between the use of 21 and 6 centralizers, Badalamenti ordered both employees to delete the simulation test results from their computers, Reuters reports.
Alisha is a writer and researcher with Ring of Fire. Follow her on Twitter @childoftheearth.
Papantonio Wins MLK Civil Rights Award
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Papantonio Wins MLK Civil Rights Award January 19, 2021
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Bill Barr Warned Trump That His Lawyers Were Lying About Election Fraud January 19, 2021
Boeing Agrees To Pay $2.5 Billion In Settlement Over Aircraft Safety Lies January 19, 2021
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Mega Banks Decide To STOP Funding Politicians Following Capitol Riot January 18, 2021
Law Professor Who Taught Cruz And Hawley Says They Clearly Didn't Pay Attention January 18, 2021
Pelosi Says She'll Fine Republicans Thousands Of Dollars For Avoiding Metal Detectors January 18, 2021
This Week on Ring of Fire Radio Podcast
Scott Millican - December 3, 2020
Episode 568: This week on Ring of Fire; a new contributor to the Ring of Fire Podcast, we will welcome political analyst and progressive superstar,...
Copyright © 2004-2018, Ring of Fire Radio, LLC and The Ring of Fire Broadcasting, LLC
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 6,832
|
Q: Downloading file with a href tag makes the file illegible I have the following code
<a href="../../assets/all-work/proto/pdf-test.pdf" download>Download</a>
It's inside a vue component which is located in src/components/Home/Proto.vue and the file is located in src/assets/all-work/proto/pdf-test.pdf, when I click download it downloads something completely illegible, why's that and how could I fix it?
A: I'm going to assume you're using Vue CLI.
The default configuration generated by the Vue CLI has special handling for images. To get the same thing working for PDFs you need to configure it yourself.
Firstly, you'll need to explicitly use require on your href:
<a :href="require('../../assets/all-work/proto/pdf-test.pdf')" download>Download</a>
In some scenarios, such as the src attribute of an <img>, the wrapping with require is performed automatically. For an <a href> it isn't.
You'll then run into a Webpack loader error because Webpack has no idea what to do with a .pdf file.
To fix that you'll need to configure a file-loader in vue.config.js:
module.exports = {
chainWebpack (config) {
config.module.rule('pdf')
.test(/\.pdf$/)
.use('file-loader')
.loader('file-loader')
}
}
If you don't already have a vue.config.js file then you'll need to create one. It goes at the top level, next to your package.json.
If you do already have a vue.config.js then you'll need to merge the chainWebpack property in, alongside your existing configuration.
By default the file will be renamed to a hash based on its content. It you want to retain the original file name you can configure that instead:
chainWebpack (config) {
config.module.rule('pdf')
.test(/\.pdf$/)
.use('file-loader')
.loader('file-loader')
.options({
name: '[name].[ext]'
})
}
More details on the name option can be found at:
https://webpack.js.org/loaders/file-loader/#name
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,720
|
{"url":"https:\/\/stats.stackexchange.com\/questions\/307165\/regressors-vs-conditioning-variables-in-glmtree","text":"# Regressors vs. conditioning variables in glmtree\n\nI have a dataset with ~800K samples, ~300 features and I'm trying to predict a binary outcome. I've started with sklearn's SGDClassifier (using log loss and l1 penalty), and I got a nice 0.67 auc score on my validation set; now I'm trying to improve this.\n\nI have a reason to believe that different subspaces of features space \"behave\" differently, so I thought fitting a glmtree would make sense (splitting the features space to different subspaces and fitting a logistic regression in each of them). However, when trying to write the formula, I don't have any prior knowledge regarding which features should be used as predictors for the glm, and which should be used as conditioning variables.\n\nIs there any automated way to select which is which?\n\nglmtree() is particularly strong in situations where you have a GLM that you would typically fit to the whole sample, e.g., a voter targeting model (Rusch et al. 2013), a treatment effect model (Seibold et al. 2016), or an economic growth model (Wagner & Zeileis 2017). Then you can detect heterogeneity in the model parameters depending on the partitioning variables.\n\nIf you have no such \"base\" model and have no idea which variables are important, then you can fit an intercept-only GLM in each node. However, then the resulting tree is often rather similar to other classification trees (CART, CTree, etc.).\n\nI have also seen situations where all numeric variables have been used in the regression and the categorical variables in the partitioning part. Possibly the numeric variables can also appear in both parts. However, this is only likely to yield good results if the number of numeric variables is small to moderate.\n\nIn your case with ~800K observations and ~300 variables and no prior information I personally would not bother with single trees anyway. I would probably use a random forest. You have enough observations to approximate potentially linear (or partially linear) effects sufficiently well through the forest.\n\nReferences\n\n\u2022 Thomas Rusch, Ilro Lee, Kurt Hornik, Wolfgang Jank, Achim Zeileis (2013). Influencing Elections with Statistics: Targeting Voters with Logistic Regression Trees. The Annals of Applied Statistics, 7(3), 1612\u20131639. doi:10.1214\/13-AOAS648\n\u2022 Heidi Seibold, Achim Zeileis, Torsten Hothorn (2016). Model-Based Recursive Partitioning for Subgroup Analyses. The International Journal of Biostatistics, 12(1), 45\u201363. doi:10.1515\/ijb-2015-0032\n\u2022 Martin Wagner, Achim Zeileis (2017). Heterogeneity and Spatial Dependence of Regional Growth in the EU: A Recursive Partitioning Approach. German Economic Review. Forthcoming. doi:10.1111\/geer.12146\n\u2022 Thanks for the detailed response! The reason I'm not using a RF approach is that I'm specifically looking for the partitioning rules; the ability to interpret the rules is roughly as important to me as the prediction accuracy. I understand from your answer that there's no principled way to select which is which - is that correct? \u2013\u00a0Adam Haber Oct 16 '17 at 13:11\n\u2022 I wouldn't know of a solution that will automatically and reliably separate linear, partially linear, nonlinear effects in the presence of high-order interactions and subgroup effects. There are search heuristics, however, e.g., performing AIC or BIC selection for the model in each node of the tree. But so far I have just tried these on a couple of data sets but never evaluated them properly. As for interpreting the rules: Note that these may change dramatically if you introduce linear effects in the subgroups! This model fits may still be similar but the structure could look very different. \u2013\u00a0Achim Zeileis Oct 16 '17 at 23:44","date":"2019-05-21 08:49:56","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7036173343658447, \"perplexity\": 1326.311708659335}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-22\/segments\/1558232256314.25\/warc\/CC-MAIN-20190521082340-20190521104340-00135.warc.gz\"}"}
| null | null |
Rybník Zrcadlo o rozloze vodní plochy 0,5 ha se nalézá v přírodní rezervaci Obora Choltice asi 100 m jihozápadně od zámku Choltice v okrese Pardubice. Rybník je využíván pro sportovní rybolov a zároveň představuje spolu s rybníky Červený a Chrtnickým rybníkem významné biocentrum pro rozmnožování obojživelníků.
Galerie
Externí odkazy
Rybníky v okrese Pardubice
Rybníky ve Svitavské pahorkatině
Povodí Struhy
Choltice
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
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Pilea auricularis är en nässelväxtart som beskrevs av C.J. Chen. Pilea auricularis ingår i släktet pileor, och familjen nässelväxter. Inga underarter finns listade i Catalogue of Life.
Källor
Pileor
auricularis
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{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 2,128
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Мењам жену () британска је ријалити-емисија аутора Стивена Ламбера која се емитује на каналу Еј-Би-Си. Мењам жену је током времена постала познат ријалити формат и добила више међународних издања.
Радња
Током прве недеље, жена се мора прилагодити задацима и стилу живота који је направила жена коју она мења. Свака жена објашњава задатке које друга мора испунити као и породичне обавезе.
Током друге недеље, новим женама је допуштено да напишу своја нова правила, док се њихове нове породице морају прилагодити новим правилима која су постављена. Обично треба више времена породицама како би се навикли на нови начин живота, док жене бирају суму новца коју ће добити и касније потрошити са својом породицом.
Након две недеље, два пара се први пут срећу сви заједно, док жене заједно са мужевима разговарају о томе како су се осећали током периода док су били раздвојени. Ово често проузрокује личне увреде као и физичко насиље које се десило минимално двапут. Али на крају обе породице појасне нову лекцију коју су научили. Неколико недеља касније камере се врате како би снимили промене које су се десиле након промене жена.
Мењам жену у Србији
Мењам жену се у Србији емитовала у два наврата, први пут од 23. јула 2006. до 6. септембра 2011. на телевизији Пинк а затим од 9. јула 2013. до септембра 2015. на телевизији Хепи. Продукцију емисије радила је Емошон продукција.
Референце
Телевизијске емисије
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
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Achyranthes sicula é uma espécie de planta com flor pertencente à família Amaranthaceae.
A autoridade científica da espécie é (L.) All., tendo sido publicada em Auctarium ad Synopsim Methodicam Stirpium Horti Reg. Taurinensis 41. 1773.
Portugal
É uma espécie presente no território português, nativa do Arquipélago da Madeira e introduzida no Arquipélagos dos Arquipélago dos Açores.
Protecção
Não se encontra protegida por legislação portuguesa ou da Comunidade Europeia.
Referências
Achyranthes sicula - Checklist da Flora de Portugal (Continental, Açores e Madeira) - Sociedade Lusitana de Fitossociologia
Checklist da Flora do Arquipélago da Madeira (Madeira, Porto Santo, Desertas e Selvagens) - Grupo de Botânica da Madeira
Achyranthes sicula - Portal da Biodiversidade dos Açores
Tropicos.org. Missouri Botanical Garden. 21 de dezembro de 2013 <http://www.tropicos.org/Name/50074813>
Achyranthes sicula - The Plant List (2010). Version 1. Published on the Internet; http://www.theplantlist.org/ (consultado em 21 de dezembro de 2013).
Achyranthes sicula - International Plant Names Index
Ligações externas
Achyranthes sicula - Flora Digital de Portugal. jb.utad.pt/flora.
Achyranthes sicula - Flora-on
Achyranthes sicula - The Euro+Med PlantBase
Achyranthes sicula - Flora Vascular
Achyranthes sicula - Biodiversity Heritage Library - Bibliografia
Achyranthes sicula - JSTOR Global Plants
Achyranthes sicula - Flora Europaea
Achyranthes sicula - NCBI Taxonomy Database
Achyranthes sicula - Global Biodiversity Information Facility
Achyranthes sicula - Encyclopedia of Life
Flora de Portugal
sicula
Flora da Madeira
Flora dos Açores
Flora introduzida nos Açores
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{
"redpajama_set_name": "RedPajamaWikipedia"
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| 7,874
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China's order for 300 Airbus jets signed during a state visit last week was bolstered by repeat announcements of dozens of existing deals and advance approval for deals that have yet to be struck, said Reuters.
The Airbus deal was reported to have been worth around $35 billion at list prices but the amount of new business is lower, said Reuters sources. Duplicate announcements included a deal for 10 A350 aircraft to an unnamed buyer, which represents a repeat announcement of an order for 10 jets by Sichuan Airlines at an air show last year.
The deal was considered an economic highlight of a trip to Europe by Chinese President Xi Jinping but Reuter's sources explained plane orders typically take months to negotiate, disclosing the overall figure of 300 was introduced late in the process and after Xi's visit was well underway.
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{
"redpajama_set_name": "RedPajamaC4"
}
| 4,294
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Q: Why install a Linux dist inside a Docker container? I am starting to understand Docker and as far as I am aware the docker container runs on the default Linux dist where the container is installed - in my case it's a Mac OS X lightweight dist that comes with docker toolbox.
So why do I see many Docker files actually installing a distrib inside the container, does this not defeat the object of keeping things light?
For example, here is one Docker file starting with:
FROM debian:jessie
so this is installing a Docker image inside the container which is based on Debian.
I also see many others using Ubuntu, for example.
Can this step not be bypassed and software installed directly in the container use the underlining Linux dist where the container is installed?
A: Because, just as for physical or virtual machines, setting up a userland environment is going to be a pain without a distribution.
This is, IMO, one of the strong benefits of docker: Pick the most suitable distribution for a particular application.
A containerized application is probably going to have dependencies. To install these dependencies, it helps a lot to have a package manager. Some dependencies are also included by default in many distributions, which makes it a good idea for the container creator (application) to choose its own distribution.
Additionally, remember that packaging a whole distribution does not necessarily waste a lot of resources:
*
*Docker images are stored as deltas against a common baseline, meaning that two images based on debian:jessie could reuse the same data for the baseline.
*Distributions are actually not that large, as they are usually minified versions of the full system images.
If you really want to create a minimal image, try busybox. However, you will oftentimes find yourself outgrowing it quite fast for any real world container image.
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{
"redpajama_set_name": "RedPajamaStackExchange"
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Q: Recommended approaches for making my code swiggable? I'm currently refactoring a Tcl plugin library written in C++. Originally the code was hand-written. A second library exists that does the same thing for Java.
The refactored library will be a single C++ library that can be used to create bindings to different languages.
My first tests with SWIG are promising. However, a lot of junk is generated as well. Various base classes and utilities are all exported. These don't make sense from the scripting point of view and only increase clutter.
Possible solutions that I can think of are:
*
*Use #ifndef SWIG in the original codebase to filter out unneeded code
*Create a SWIG-compatible wrapper around the API classes.
*Differentiate between public and private headers. Public headers are pure abstract base classes that contain no implementation. The private headers inherit and implement them. Only SWIG the public headers.
*The opposite of the above solution: inherit a SWIG-compatible class for each API class.
I'm currently leaning towards solution 3 at the moment. However, I'm not really certain so I'd like to know the SO community's opinion on this. Feel free to share your thoughts.
Update
I forgot to list one solution:
*
*Code that should not exported by SWIG should probably not be in the public section of your class.
Perhaps this is the answer. I'll have another look on Monday.
Update
I settled with a solution. See my answer.
A: Any approach that means that the C++ library becomes less useful to the C++ user is not the ideal solution.
*
*#ifdef SWIG in the middle of .hpp files: Muddies up your C++ with unnecessary cruft, so it's not ideal
*SWIG Specific Interface: This is a viable option, but only makes sense if the code you want to expose to SWIG is significantly higher level then the base C++ API.
*Public vs Private interface: Might make sense, but again you have to ask at what cost to the C++ user of the API? Are you limiting the public interface too much? Who has access to the private interface? Should the pImpl idiom be considered instead?
*SWIG Compatible Class for each interface: Probably more work than necessary.
First and foremost, to keep your SWIG related code separate from the rest of the API.
You probably don't want to import the .hpp files directly into SWIG (if SWIG wasn't considered during the initial design of the library), but if you do, you want to use a SWIG .i file to help you clean up the mess. There are three basic approaches we use, each with different use cases.
First, direct inclusion. This is useful if you know your API is nice and clean and well suited for parsing by SWIG:
// MyClass.i
%{
#include "MyClass.hpp" // included for the generated .cxx file
%}
%include "MyClass.hpp" // included and parsed directly by SWIG
The second case is for code that is most of the way there. This is code that had SWIG taken into consideration, but really needed some stuff for the C++ user that we didn't want to expose to SWIG:
// MyClass.i
%{
#include "MyClass.hpp" // included for the generated .cxx file
%}
%ignore MyClass::someFunction(); // This function really just causes us problems
%include "MyClass.hpp" // included and parsed directly by SWIG
The third case, and probably the one you want to use, is to directly choose which functions you want to expose to SWIG.
// MyClass.i
%{
#include "MyClass.hpp" // included for the generated .cxx file
%}
// With this example we provide exactly as much information to SWIG as we want
// it to have. Want it to know the base class? Add it. Don't want it to know about
// a function? Leave it out. want to add a new function? %extend it.
class MyClass
{
void importantFunction();
void importantFunction2();
}
A: I'd use apprach #3 too. I'm using a similar approach in my projects, and it is used by COM too (interfaces inherithed by private implementation class).
It is really easy to detect errors and maintain code in that way! Unfortunately you will end implementing all functions as virtual, but it should not be a big issue...
Separating the interface will keep it really clean and understandable!
A: My final solution: simply SWIG the original code base. In order to avoid generation of non-relevant code I use the following techniques. In order of preference:
*
*Make non-swig code private or protected. If it doesn't need to be swigged, then it probably doesn't need to be public.
*If possible, change the original code to make it more compatible with SWIG. I replaced a curiously recurring template pattern with abstract base classes. I was willing to make that sacrifice for SWIG :)
*Add %ignore statements to the interface file.
*Use #ifndef SWIG to filter it out. I don't like to pollute my original code so I only use this as a last resort.
Concerning my previous ideas:
*
*Create a SWIG-compatible wrapper around the API classes.
*Differentiate between public and private headers. Public headers are
pure abstract base classes that
contain no implementation. The private
headers inherit and implement them.
Only SWIG the public headers.
*The opposite of the above solution: inherit a SWIG-compatible class for
each API class.
All these solutions require writing SWIG-compatible wrapper code. This is a bit silly because you are ditching SWIG's the strongest selling point: automatic generation of wrapper code. If I write my own wrapper code for SWIG then I might just as well write regular JNI code.
That said, I realize that for some projects writing wrapper code may the most cost-efficient solution. However, I this was not the case in my situation.
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{
"redpajama_set_name": "RedPajamaStackExchange"
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| 2,611
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package gw.internal.gosu.parser;
import gw.config.AbstractPlatformHelper;
import gw.lang.reflect.module.IModule;
public class DefaultPlatformHelper extends AbstractPlatformHelper {
public static boolean IN_IDE = false;
@Override
public boolean isInIDE() {
return IN_IDE;
}
@Override
public boolean shouldCacheTypeNames() {
return false;
}
@Override
public void refresh(IModule module) {
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,828
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# organic cooking
on a budget
# HOW TO GROW ORGANIC, BUY LOCAL,
WASTE NOTHING, AND EAT WELL
## Arabella Forge
with illustrations by Genna Campton
Skyhorse Publishing
Copyright © 2014 by Arabella Forge
All Rights Reserved. No part of this book may be reproduced in any manner without the express written consent of the publisher, except in the case of brief excerpts in critical reviews or articles. All inquiries should be addressed to Skyhorse Publishing, 307 West 36th Street, 11th Floor, New York, NY 10018.
Skyhorse Publishing books may be purchased in bulk at special discounts for sales promotion, corporate gifts, fund-raising, or educational purposes. Special editions can also be created to specifications. For details, contact the Special Sales Department, Skyhorse Publishing, 307 West 36th Street, 11th Floor, New York, NY 10018 or info@skyhorsepublishing.com.
Skyhorse® and Skyhorse Publishing® are registered trademarks of Skyhorse Publishing, Inc.®, a Delaware corporation.
www.skyhorsepublishing.com
10 9 8 7 6 5 4 3 2 1
Library of Congress Cataloging-in-Publication Data is available on file.
Print ISBN: 978-1-62914-540-2
ebook ISBN: 978-1-63220-078-5
Printed in China
## Contents
INTRODUCTION
How to spot a frugavore
Frugavore nutrition
Home-cooked meals
Straight from the farm
Is good food more expensive?
EAT AS A PEASANT WOULD EAT
Waste not, want not
How to minimize waste in your home
SOURCING YOUR FOOD
Some terminology
Straight from the farm
Grassroots movements
Feel at loose ends?
THE FRUGAVORE KITCHEN
Cleaning
Buying ready-made cleaning products
Making your own
Kitchen utensils
Meal planning
Lunch options for the nine-to-fiver
Make it a team effort
STOCKING YOUR PANTRY
Beans, legumes, & peas
Fats & oils
Flour
Rice
Polenta
Sago or tapioca
Coconut milk & coconut cream
Seasonings
Vinegars
Preserves, pickles, & condiments
THE VEGGIE PATCH
Planting in pots
Nature's recycling systems
Compost heaps
Worm farms
Bokashi buckets
Garden burials
Growing from seed
Some easy seeds to sow
What to plant when
Building & maintaining your soil
Long-term maintenance
Heirloom roast vegetables
Humble baked potato
Bubble & squeak
Wild greens pie
Tomato & onion pie
Cauliflowers with bacon
Garden salad.
. . .
THE CHICKEN & THE EGG
A good egg
Choosing your chickens
What your hens will need
Chicken feed
Scratch that
To rooster or not to rooster?
Beyond the chickens: ducks, quail, & geese
Superbly scrambled eggs
Egg mayonnaise
Egg & greens pie
Omelets
Zucchini & basil omelet
Potato & nutmeg omelet
Leek & sour cream omelet
. . .
Poultry Basics
Chicken soup
Frugal roast chicken
Roast duck with orange & sage
Spanish-style chicken casserole
Chicken & leek pie
. . .
STOCKS & SOUPS
Buying stock bones
Making stock
Storing your stock
Chicken or poultry stock
Chicken-feet stock
Beef, lamb, or pork stock
Vegetable stock
Stock-broth cold cure
French onion soup
Pea & ham soup
Minestrone
Potato & leek soup
Chicken & corn soup
Saffron stracciatella
Chickpea & rosemary soup
Pumpkin soup
Kale & zucchini soup
Lentil soup
Bean & green soup
. . .
BEANS, LENTILS, & LEGUMES
Preparing your legumes
A note on soybeans
Cannellini bean salad with pumpkin & beets
Treacle baked beans
Chickpea salad with greens
Lentil salad
Split-pea purée (fava)
Cannellini bean dip
Sweet potato hummus
. . .
MEAT
Fresh from the farm
Butchers
Supermarkets
What to buy?
Certified produce
Buying in bulk
Thrifty bits
Cooking with offal
Beef casserole
Irish stew
Pot-au-feu
Home-cured bacon
Homecured bacon served with lentils
Mutton curry
Oxtail stew with apples & spices
Baked sweetbreads with butter & sage
Steak & kidney pie
Brined tongue with salsa verde
Crispy liver with caramelized onion
Liver pâté
Meatloaf with red sauce
Pork-mince apples
Marrow on toast
Moroccan rabbit hot pot
Baked meatballs with nutmeg
. . .
FRESH FROM THE SEA
Buying fish
Cooking fish
Preserves & pickles
Seaweed.
Oven-baked sardines with oregano
Baked whole fish with tomatoes, herbs, & fennel
Fish pie
Poached fish with sabayon sauce
Minced fish cakes
Fish soup
Fish stock
Fish broth with lemon & rice
Beautiful bouillabaisse
Bermuda fish chowder
Salmon gravlax (cured salmon)
Quick-pickled sardines
. . .
GOOD GRAINS
Wholegrains
Bread
Choosing a flour
Bread-makers
Prepare your kitchen
Making your starter
Maintaining your starter
Basic sourdough bread
Sourdough fruit loaf or fruit buns
Porridge
Porridge cake
Coconut slice
Oatmeal pastry
Oatmeal pikelets
Polenta
Hearty brown rice
GOT MILK?
Allergies & intolerance
Accessing fresh milk
Champagne milk (kefir)
Curds & whey
Maple-syrup yogurt
Lemon curd cheese
. . .
COOKING WITH NUTS: SPROUTING AND SAVING
Sprouting nuts
Nut flour
Almond butter.
THE SWEET STUFF
A glossary of sweeteners
Summer icy-poles
Zesty raspberrymint
Bread & butter pudding
Home-made organic chocolate
Coconut sago
Apple & nectarine shortcrust tart
Baked fruits stuffed with ricotta & honey
Chocolate mousse
Baked custard with rum
Stewed pears with cinnamon syrup
Peachy mint salad
. . .
PRESERVES FOR THE PANTRY
Preparing your kitchen
Fermented foods
The dos & don'ts of fermentation
Starter cultures
Blueberry chia jam
Cultured beets with cabbage
Zesty pear & ginger relish
Preserved lemons
Preserved globe artichokes
Quick & easy tomato preserve
Fermented tomato salsa
Turmeric tonic
Traditional sauerkraut
. . .
RESOURCES
BIBLIOGRAPHY
ACKNOWLEDGMENTS
INDEX OF RECIPES
# INTRODUCTION
I STARTED WRITING THIS BOOK AFTER I found myself trying to juggle two seemingly opposing things: I wanted to provide good, nutritious food for myself and my family, while also watching my dollars when I went to the supermarket.
At first, like many people today, I took steps toward healthy eating by shopping at organic food stores and occasionally at the local farmers' market. Where possible, I tried to buy organic meat, fresh seafood, and good quality fruit and vegetables. Living in a busy household, I was meticulous about throwing things out when they got moldy or stale. This made life expensive. And, looking back on it, more than a little wasteful.
Eventually, though, I started to feel frustrated. I was fed up with the high prices at small organic stores, and by the poor quality of the produce at my local supermarket. I started to look for other ways to access good food, and I began to live and cook a lot more frugally. I still wanted to buy the most nutritious food I could, and to enjoy good quality meat, fish, and vegetables. But I learned to shop more wisely, to make the most of what I bought, to waste less, and to connect more closely to where my food came from.
Along the way, more than a few changes took place at our house. The front lawn was taken up and replaced with a veggie patch (much to the horror of anyone who liked to kick a soccer ball!). We got two little hens and converted an old warehouse container into a henhouse. I also bought a second-hand freezer and started to buy food in bulk when it was in season. As well as saving us money, this made my life much simpler and easier: I always had frozen produce on hand if I needed to whip up dinner in a hurry during the week, and I could easily source fresh vegetables and herbs from the garden.
I also started to ask more questions about the food I bought. I quizzed my local butcher about the cheapest cuts of meat. I scoured the supermarket shelves for low-cost, easy to prepare foods like lentils, chickpeas, and legumes. As I experimented in the kitchen and used my family as guinea pigs, I also read more and more about traditional cooking techniques and peasant-style cuisines. Somewhere along the way I fell upon a word that I soon became very fond of. It was the magic word _frugal,_ meaning to make the most of what we already have and also, wherever possible, to use _less._ People living off the land have employed frugal cooking and harvesting techniques for millennia. They did so not just to save costs and eliminate waste but also as a means of staying healthy. As I explored new ways of buying and preparing food, this notion of frugality made more and more sense.
As a nutritionist, I was lucky enough to have contacts in the food world, who were a great source of help and advice. Of all of these relationships, by far the most valuable were those with local farmers. From the farm, I could buy meat in bulk, fresh milk, and a wonderful array of other produce. I learned how to use all the thrifty bits—pork belly could be cured into bacon, chicken feet could be used to make stock for soup, and extra milk could be turned into homemade cheese or yogurt. By visiting local farms, I also gained a much better understanding of how my food was produced and what was fresh and in season.
In many respects, this approach to food represents a return to traditional peasant ways of eating. Only a few generations ago, almost everyone with a patch of land grew their own vegetables, kept a few chickens, and preserved their own fruit. What you couldn't produce yourself, you could usually find nearby: a neighbor with a lemon tree, a friend with a fresh catch of fish, a local baker selling freshly made loaves. Your diet would reflect what was locally and seasonally abundant, and you would have a clear understanding of where the food on your plate came from. Being a frugavore is all about rediscovering these peasant habits of frugality, returning to simple, fresh, seasonal food and reconnecting with the farm.
It's not just a matter of nostalgia, though: the time is ripe for a more frugal approach to food. We live in an age of profligacy. Never before have we had so much food available, and never before have we wasted so much. Celebrity chefs entertain us with elaborate meals and trendy ingredients, but most of us don't eat the sort of food we see on TV. We buy our food ready-made from the supermarket or as take-out, carry it home in plastic containers, and zap it in the microwave (doing away with any nutrients that managed to survive that far). Most of us have got into the habit of shopping only at supermarkets, and many people don't know how to prepare meals at home from scratch, or don't think they have time. When food is available immediately it's easy to take it for granted; it's no wonder that around one third of the food we buy is wasted every year.
Many people would like to eat differently but aren't sure where to start or don't think they can afford it. There can be a big price discrepancy between quality, chemical-free produce and conventional supermarket fodder. A leg of grass-fed lamb from the local organic butchery is going to cost you a lot more than the no-name equivalent from the supermarket. There are cheaper ways to access good-quality, organic produce, but very few of us know the way to our local farm or how to grow our own vegetables. That's where this book comes in: as a frugavore, you'll be equipped with the skills and knowledge to enjoy the best produce at a much better price. So instead of buying lean, organic chicken breasts at your local organic store, go buy the whole bird (and I mean the whole bird—head and feet included). Use the meat in a roast, keep the leftovers for school sandwiches, and make a good hearty soup with all the bones. You now have three meals instead of one— and if you shop at the same places I do, you'll probably find that a whole chicken costs about the same as those two skinless fillets. What's more, as you will learn on the following pages, all those extra bits—the bones, the heads, and the skin—are far better for you than the cardboard-flavored chicken breasts you might be accustomed to.
Of course, traditional peasant groups had the advantage of living close to the land; reconnecting with the farm can seem a daunting prospect for those of us living in modern cities and working long hours. But times are a-changing; more people are demanding healthier, tastier, more sustainable food, and the inner-city food market has had to adapt. In an attempt to rebuild the connection between farm and table, modern-day urban peasants are staging a food revolution. They are joining co-ops or buying clubs, or driving out to nearby farms to collect fresh food. Suburban and inner-city gardens are a great untapped resource, and more and more communities are finding ways to grow fresh vegetables on public and private land. Our food culture is changing as people seek out creative ways to improve their diets. This book will help you to be a part of this, whether you live on a farm, in a high-rise apartment, or on a suburban block.
## HOW TO SPOT A FRUGAVORE
So what is a frugavore? A frugavore makes the most of what they have, supports best practices in farming, wastes nothing, and grows their own food when they can. As a frugavore, you may find yourself:
* Sourcing food locally and seasonally;
* Buying food in bulk, be it meat direct from the farm or grains, vegetables, and fruits from co-ops, markets, or other sources;
* Stocking your pantry wisely with staple items that can be used as a basis for simple, healthy meals;
* Learning to cook as a peasant would, using frugal ingredients such as lentils, legumes, offcuts, cheaper varieties of meat and fish, and seasonal fruits and vegetables;
* Letting nothing go to waste—using scraps for a compost heap or worm farm, making stock with old vegetables and bones, and recycling glass jars for storage and preserves; and
* Connecting to local, grassroots food movements and exploring food resources in your local community.
How will you feel after all of this? You may notice that you have no room in your garden, as it has been taken over by pots of herbs and climbing tomatoes. Your kitchen cupboards will be overflowing with spare jam jars. You'll start contemplating replacing your coffee table with a small second-hand freezer, and you may catch yourself going to work with dirt underneath your fingernails. Don't worry: it's not a bad thing! It's all part of becoming a frugavore.
## FRUGAVORE NUTRITION
There's no point spending money on food that's not going to do you any good; "cheap" food is hardly good value if it's bad for you. Instead, if you want to eat and live frugally, buy the best quality produce possible and make the most of it. By "best quality," I don't mean the finest French cheeses or the most expensive bottled water. Quality food is food that is full of nutrients, grown locally, and prepared fresh at home.
Technically speaking, the term "nutrient-density" refers to the quantity of nutrients in a food compared to the number of calories or kilojoules it contains. A nutrient-dense diet will promote health and stamina and help your body to ward off disease.
Nutrient-density is also fundamental to the most important aspect of our eating pleasure—taste. I used to wonder why home-grown fruit and vegetables tasted so good compared to commercially grown supermarket produce. I've seen people swoon over the first apple of autumn, or over ripe asparagus sprouting from the ground in spring, but never over a tired navel orange, puffed up and yellowing, that has been in transit for weeks before reaching the supermarket shelf. Nature cleverly organizes itself so that when food is at its ripest, it is also at its most nutritious peak. When food is ripe and rich in nutrients, it will be alive-tasting and bubbling with flavor and bite. This may explain why we are tempted to pick fruit from over a neighbor's fence when a tree is dripping with plums, ripe and ready to be eaten, but don't feel similarly tempted by something that has been sitting in a fruit bowl for days.
## HOME-COOKED MEALS
The best way to ensure you're eating nutrient-dense food is to cook your meals at home, using raw ingredients, rather than relying on processed or pre-packaged foods. Seek out foods that have come from rich and fertile soils, and try to grow some of your produce yourself or connect to a local farm or community garden.
You don't need to be a arugula scientist to know that home-cooking is better for you than processed or ready-made meals. Processed food is treated to ensure a long shelf life and a neat appearance. The food industry uses methods such as canning, pasteurizing, refining, and irradiating to create products that can sit on supermarket shelves for months, sometimes years. Additives and preservatives are added to ensure that your food won't crumble, age, or lose its color and shape before it reaches you. Unfortunately, these methods also strip nutrients from food. That's how they work: by removing nutrients, processing eliminates the food source for micro-organizms and insects, and so prevents the product from deteriorating on the shelf.
The food industry is well aware of what is taken out of food when it is processed. That's why companies "fortify" foods with missing vitamins and minerals such as fiber, B vitamins, and iron. When a food has been broken down and stripped of its nutrients, however, it cannot be restored simply by adding nutrients later on. Nutrients require special enzymes and co-enzymes for proper assimilation. These natural combinations are only found in whole foods; they can't be recreated in a factory.
This isn't to say that all processed foods are unhealthy. Traditional food preservation methods maintained or even enhanced the nutrients in foods. Salting and air-drying can be used to preserve meat for long periods as delicious salami and bacon. Dairy products can be turned into yogurt, kefir, and cheese, which are rich in healthy bacteria and can be even more nutritious than fresh milk. Adding salt to cabbage produces sauerkraut, an excellent source of healthy bacteria and vitamin C. Foods preserved using these traditional methods can be delicious and highly nutritious (in fact, it was the addition of sauerkraut to the diet of Captain Cook's crew that made his voyage so successful). To ensure you're getting the most nutrition for your buck, however, steer clear of foods processed using artificial additives and preservatives. They may seem cheap and convenient, but you'll be missing out on the best bits—the nutrients and the flavor.
## STRAIGHT FROM THE FARM
When it comes to fresh produce, nutrient-density can be traced back to the farm. What we feed growing plants, and what they encounter in the ground, will affect the nutrients in the produce that we eventually eat. If the soil is biodynamic or organic, it will contain slow-release fertilizers in the form of manure, compost, and some minerals, providing the plant with a range of nutrients and creating robust root systems. Biodynamic and organic produce can take longer to grow than industrially farmed vegetables, but this extra time allows their roots to develop and run deep. The plant gathers extra nutrients and we end up with a healthier final product. And because plants that are rich in nutrients have a natural immunity and can repel pests by themselves, they can be grown without chemical pesticides.
Over the past fifty years, agricultural companies have worked to develop larger crops, greater yields, and hardier species. This has been achieved through selective breeding methods and high-yield fertilizers. Today's fruits and vegetables can be dropped, bounced, and stacked on the supermarket shelf without damage; they can be stored for months without any visible change. Fresh produce is cheaper and easier to access than it was fifty years ago, especially in city areas.
But greater yield and bulk have come at a cost: there are now fewer nutrients in our food. Plants are pushed to grow bigger by accumulating more water and starch, but there is not a corresponding increase in nutrients. Vegetables tested in 1980 contained significantly lower levels of calcium, magnesium, copper, and sodium than vegetables tested in 1930. As Michael Pollan has observed, "you now have to eat three apples to get the same amount of iron as you would have gotten from a single 1940 apple." We might have more food in the twenty-first century, but it is worth less in terms of nutrient-density.
With the growth of large-scale commercial agriculture, we have also lost many traditional varieties of fruit and vegetable. In our endless quest for the reddest tomato or the hardiest pear, much diversity has been lost. There were once thousands of varieties of tomato available, yet you'll be lucky to find more than two or three in most modern supermarkets. Carrots once came in white, purple, red, and yellow. They could be small, tall, stumpy, or slim. Somewhere along the line, the orange Dutch variety became the most popular (most likely a tribute to Dutch Nationalism). This variety became universally available, while the others dropped out of the food chain. Many of these traditional varieties (often called "heritage" or "heirloom") are now experiencing a rebirth, helped along by local food communities, small farms, heritage seed sellers, and home gardeners. They may look small and mis-shapen next to the uniform produce at the supermarket, but these heritage vegetables are packed full of flavor and nutrients. If you grow them yourself, you'll be able to nourish them with natural fertilizers in your garden soil. And you'll get to enjoy the novel pleasure of discovering different colors, shapes, and flavors. A home-grown carrot with two legs, baked with a little goose fat and fresh herbs; a black tomato, speckled with tiny red spots, drizzled with some fresh olive oil; violet gnocchi made with "purple congo" potatoes. We miss out on this diversity if we depend on uniform supermarket produce. Luckily, it's much easier than you might think to reclaim it.
The animal products we eat, whether dairy, eggs, or meat, are also affected by the nutrients in the soil and the plants we feed them. Traditionally, all livestock were grazed on fresh grass, with only minimal supplementary feeding during the drier months. Poultry, being omnivores, pecked at mixed pastures and enjoyed insects, grubs, and a small amount of grain. Their lives included plenty of exercise and sunlight.
In the push to achieve cheaper food production during the twentieth century, more concentrated feeding practices were developed. Commercially farmed livestock and poultry are now often fed a diet exclusively of grains, corn, or soy meal. They are kept in close confinement, without access to sunshine or space. Poultry, who have a strong natural instinct to run around, play in the dirt, and establish their own pecking order, are kept in small cages, have their beaks clipped, and are fed an unnatural (and monotonous) diet.
Nutritional tests have shown that animals raised this way are less healthy than their free-range, natural-living counterparts. Because they live in cramped conditions and eat an unnatural diet, they are prone to infection and more likely to need antibiotics, which end up in the products we eat. Meat, milk, and eggs from grass-fed and free-range animals, meanwhile, are rich in healthy fatty acids including Omega 3 and conjugated linoleic acid (CLA)—both healthy fats known to benefit heart health and aid in the prevention of inflammatory and autoimmune disorders. Grass-fed and free-range products also contain higher levels of antioxidants and important fat-soluble vitamins such as A, D, and E. Grass-fed livestock do not suffer from many of the afflictions faced by their grain-fed counterparts such as acidosis, rumenitis, liver abscesses, and bloat, and they are at less risk of E-coli contamination.
The surest way to find the most nutritious produce is to reconnect with local farmers. As you'll discover later in this book, there are all sorts of ways to do this. Whether you buy directly from a farm, at a farmers' market, or through a co-op or buyers' club, a clearer understanding of where your food came from and how it was grown will help you to choose the healthiest fresh ingredients.
## IS GOOD FOOD MORE EXPENSIVE?
The health benefits of good food are clear. But people are always telling me that it costs too much to eat well. They quip that buying processed foods or take-out is cheaper than cooking at home, and complain that organic produce is overpriced and overrated.
There is some truth in this. The great price difference between organic or biodynamic and conventional produce is undeniable. This morning, I bought a bunch of celery at my local supermarket for around a quarter of the price I would pay for the biodynamic equivalent. Grass-fed organic meat costs more than conventional meat; a 6-pound leg of lamb at my local gourmet butcher is usually 30 to 40 percent more expensive than at the supermarket.
So if you ask me whether it costs more to buy the best quality produce through conventional retail outlets, my answer, in short, is yes it does. But there are shortcuts and backstreets you can take to access good food without going broke. By tapping into unconventional food resources and being frugal with what you buy, you can stick to a low budget while enjoying quality produce; that is what being a frugavore is all about.
For example, when it comes to my celery and grass-fed lamb, you could consider growing some of your own celery. A packet of six seedlings retails for around the cost of a bag of potato chips, and will last you for most of the season. These can be grown in pots, or in a small patch of your garden.
Similarly, to obtain grass-fed organic meat at a better price, you could consider purchasing your meat in bulk, directly from the farm, or opting for the cheaper cuts—such as casserole, mince, or offal—and exploring different ways to cook them. Every section of this book details creative ways to make the most of what you purchase, and to access good food at a reasonable price.
While I was writing this book, I wanted to test the premise myself, to find out if it was in fact easier and cheaper to eat out than to cook at home. So I conducted a little experiment. On a Monday night, I drove to my sister's place, where she lives with her husband and three young kids. I picked them up and took them out for a fast-food dinner.
Ordering three kids' servings, plus two adult meals, our bill came to around thirty-five dollars. It was a fun night out; it took no more than half an hour to drive to the restaurant and back, and we avoided all the dirty dishes. But from a nutritional perspective, I couldn't quite believe what we were eating. It was rich in salt, sugars, trans-fats, preservatives, and flavorings. It was low in nutrients, and high in calories.
The following Monday night, I thought we'd try something different. I live three blocks from my sister's place, so I invited them round for dinner. Because I didn't have a lot of time after work, I made lentil soup. When you're in a hurry, this is a pretty simple exercise: I grabbed a packet of lentils from the pantry, a few bay leaves from the garden, some stock from the freezer, and some carrots and celery from the fridge. It took me ten minutes to get the ingredients ready and forty minutes to cook. While the soup was cooking, I picked some arugula from a pot outside my kitchen door and threw together a salad, and then put a sago and coconut pudding on to cook for dessert. The entire preparation took me half an hour. The total cooking time was about an hour. The meal, all ingredients included, cost less than half of what we paid for our meal at the fast-food outlet—plus, there were leftovers.
As we ate, we chatted about the concepts of price and convenience. I calculated that of the recipes I cook on an average weeknight, nearly all are cheaper and more convenient than eating out—cheaper even than the cheapest of drive-thrus. People are always telling me that they choose to eat fast or processed food because it is cheaper, but this is just not true! By being frugal with food—making the most of what you have, growing some herbs or veggies yourself, and connecting to local food resources—you can save money on groceries, and spend less time shopping or picking up ready-made meals. Don't believe me? Well, you'd better keep reading then.
# EAT AS A PEASANT WOULD EAT
"The old peasant kitchen habits of frugality [involved] making stock out of bones, pickling and salting in times of glut, stocking the pantry, using diet to care for the sick and the elderly, making good food out of few and simple ingredients."
—Elisabeth Luard, _European Peasant Cookery_
PEASANT-STYLE COOKING EMBODIES ALL that it is to be a frugavore: choosing nutrient-dense foods from local sources, making do with what is available at different times of the year, and being as self-sufficient as possible.
More and more, we are realizing that we have a lot to learn from peasant diets. The food we eat today is not nourishing us as it should. We are largely unaware of where it comes from, but we are beginning to understand that many modern food-production methods come at a great cost to the environment.
Some might see the word _peasant as_ a derogatory term: maybe a starving farmer living in impoverished conditions with too many children to feed and not enough food. In fact, the word is derived from the fifteenth-century French word _païsant,_ meaning a person from the local _pays,_ or countryside. By definition a peasant is any person who lives or works close to the land.
The peasant diet has been described as "simple and nourishing," and their health was all the better for it. When they weren't affected by wars or food shortages, peasants lived surprisingly healthy lives. Elisabeth Luard has described peasant communities in Spain in the 1920s as "communal, supportive and hardworking," and Dr. Weston A. Price, in his travels around the globe during the 1930s, found them to display extraordinary health, strength, and vigour.
Existing exclusively on a traditional diet drawing only on local resources, peasants relied on their food to keep them strong and protect them from disease. Their food preparation methods were well considered and thrifty, designed to get as much value and nourishment as possible from every dish.
Most importantly, nothing was ever wasted. Food had to be harvested with the utmost diligence during times of scarcity, and stored with the utmost care during times of abundance. Peasants were thrifty and frugal with food because they had to be; their diets were dictated by limitations such as the fertility of the soil, the seasons, the weather, and their region's habitat.
So we have a lot to learn from peasant cultures, and food choices are just the beginning. Peasants lived a lifestyle that was supremely self-sufficient. They made the most of what they had on-hand and wasted nothing. All waste produced on the farm was used for some other purpose: food scraps went to the chickens or were used as compost; chicken poop fertilized the plants. Only the bare minimum was purchased or traded and nothing was carelessly discarded.
Today, when it comes to food, we are inundated with choices. At the supermarket we find foods out of season, imported from other countries, fresh, frozen, processed, pasteurized, and irradiated. Meals can be bought ready-made and reheated in seconds with the click of a button. Food might be more readily and abundantly available, but it is worth less in nutritional terms and lacks the flavor of locally grown, homemade fare.
It's no surprise that many people now yearn for a simple and frugal diet, and for the freshness and flavor of plain old peasant cooking: fresh bread with lashings of butter and homemade jam; a hock of bacon simmering in a soup of split peas and spices; tender tomatoes bursting with flavor at the beginning of the season.
Peasant fare does not need to revolve around plain or bland ingredients. In fact, traditional peasant foods were full of richness, flavor, and diversity. European peasants living near the ocean enjoyed such scrumptious foods as salmon, oysters, and crayfish, while those living inland had slow-cooked snails, raw-milk cheeses, superb truffles, and foie gras. Even the simplest of dishes tasted deliciously good. Unfortunately for the food industry, this cannot be recreated with additives, preservatives, or modern processing methods. The joy of this sort of food lies in its freshness, and the integrity of the ingredients used.
Not surprisingly, as world leaders in gastronomy, the French have always valued their traditional peasant cooking as much as they do their _cuisine bourgeoise._ Curnonsky, editor of _Larousse Gastronomique_ and founder of the Michelin Star rating system, believed that regional peasant-style cooking possessed some of the richest of gastronomic treasures, and that French cooking should always strive to respect "the taste of things as they are." The French have gone to great lengths to preserve their traditional food and farming culture, and it is for this reason that they've been able to maintain such a flavorsome, nutritious cuisine.
So, what did peasants eat? Most foods were bought fresh from their source and any processing, cooking, or preserving was done in the home or communal kitchen. As much food as possible was grown at home: no garden space was wasted. All parts of the animal were used, and any animal or vegetable scraps were put back into the land as compost.
One-pot meals such as soups and stews were common, as they required only a single heat source and were a good way to use up tougher cuts of meat. Traditional diets included a range of bread and other wholegrain products, which were prepared using traditional methods to render them optimally nutritious and easy to digest. Dairy products were consumed both raw and fermented in the form of yogurt, butter, kefir, and cheese. Meat and other animal products came from both wild and domesticated animals. Livestock and poultry were grass-fed and free-range; they were not fed large quantities of grains or corn, nor any soy meal.
Peasant food also depended on seasonal availability. Food that was in season would feature as the main aspect of a meal: the latest garden peas, fresh cheeses locally made, or a new batch of eggs from the chickens. The season's ingredients would be cooked from scratch or preserved for later use using traditional methods. Place was everything, and each local area had its own specialties. As they say in France, "The triumph of Marseille, it is only good when eaten in Marseille. Don't try to eat it in Paris." Wherever you were, great respect was shown for the food on the table. The traditional prayer before a meal demonstrated gratitude for the food, and for the farmers and animals who produced it.
It is also telling to consider what peasants _didn't_ eat. Refined sugars, high-fructose corn syrup, and artificial sweeteners did not exist. Natural sweeteners such as honey, molasses, malt, and maple syrup were used when they were available. Seed oils like cottonseed, safflower, and sunflower oil were burned for light and used in traditional remedies, but were not generally used for cooking. Instead, animal fats in the form of eggs, cream, butter, and oil were highly prized when they were available, as was cold-pressed olive oil. Hydrogenated fats such as margarine did not yet exist. The only soy foods consumed were fermented: soy sauce, tempeh, miso, nato, and tofu. Soy "milk," soy-based imitation foods, and soy additives were not known. Needless to say, genetically modified foods did not feature in traditional diets.
## WASTE NOT, WANT NOT
Peasant frugality was not confined to the kitchen. Out of necessity, traditional cultures kept waste to a minimum and squeezed as much use as possible out of what they had. Even a generation or two ago, this attitude was common. Growing up, I was constantly reminded that food should be used frugally and enjoyed to the last morsel. I remember my mother removing a soured pumpkin from the compost heap (after I had sneakily thrown it there), trimming off the moldy edges, and using it for dinner. It was surprisingly enjoyable. She always kept an eye out at the supermarket for food that was on sale because it was nearing its use-by date, and all our leftovers went to our backyard chickens or onto the compost heap. For my parents' generation, these habits were fairly standard. Growing up after the war, they and their parents appreciated the energy required to produce a loaf of bread, a stick of celery, or a can of beans.
Today, we have forgotten how to be frugal. If electrical goods break, it is often cheaper to buy new ones than to have them repaired. We buy food in take-out containers and disposable plastic bags without a thought. We buy and discard food carelessly, knowing there will always be more for us to purchase tomorrow. Along the way, we have forgotten many of the clever tricks used by earlier generations to reduce waste.
But with rising food prices and overflowing landfills, we are seeing a quiet backlash against this profligacy. "Dumpster divers" scour garbage skips for food that would otherwise go to waste. I haven't dumpster-dived myself, and I can't say I want to, but I can appreciate what they do and why they do it. In Australia, food waste is estimated to be 30 million tons annually.
During the past decade, we have become increasingly environmentally aware; we are conscious of our carbon footprint and of the need to save energy and use fewer natural resources. But we also need to generate less waste. Our landfills are constantly growing. In Victoria, Australia, where I live, food and green waste account for almost half of the municipal waste we send to landfills. This type of waste produces methane, one of the worst greenhouse gases. Obviously, recycling is an integral part of reducing the impact of landfills, but recycling also requires energy, and only a proportion of what we put in our recycling bins ends up being properly reused. To actually make a difference, we need to use less and be frugal with what we purchase.
Happily, in a kitchen environment, this is not as hard as it sounds, and this is where those rediscovered peasant habits come in handy. Sending organic waste (any plant or animal matter) to a landfill is a missed opportunity for frugavore cooks and gardeners. We could reduce our waste by almost half by recycling organic matter in our own backyards. For instance, you can take your kitchen scraps or lawn clippings and put them in your compost. Start a worm farm and use your stale vegetables and newspaper scraps as worm-food (the liquid from a worm farm is an extraordinary fertilizer and can produce stupendous results in your vegetable garden, season after season). Re-use glass jars to store homemade jam or preserved fruits and vegetables. Use tired vegetables and meat scraps to make stock. Not only will these things reduce waste, they'll also save you money by letting you squeeze much more value out of the things you've bought, grown, or made. Similarly, by buying fewer processed foods and purchasing more nutrient-dense raw ingredients, you will be using less plastic and packaging. So less waste is good for us, really and truly.
## HOW TO MINIMIZE WASTE IN YOUR HOME
* Keep some chickens, they can eat up your kitchen scraps and provide you with delicious, nutrient-dense eggs. Their droppings also make superb garden fertilizer when they are mixed with hay or compost.
* Reduce packaging waste and food miles by growing some of your own produce. It could be a few leafy green vegetables in pots or a large vegetable bed on your front lawn. Even a small amount of edible produce grown at home can reduce your food bill and lessen your waste output.
* Get a compost bin, worm farm, or bokashi bucket for your kitchen waste. Use this to fertilize your garden and grow a blossoming vegetable patch.
* Consume less: only buy products that you really need. Try to avoid resorting to "retail therapy."
* Shop locally and avoid products that are processed or have a lot of plastic packaging.
* Recycle old clothing as rags for cleaning.
* Recycle old jars for jams and preserves. Use plastic containers to store leftovers and pantry staples, or for cleaning products.
* Stop using paper and plastic bags: purchase a recycled cotton, hemp, or string bag.
* Bottle your own filtered tap water instead of buying plastic bottled water.
* Practice "positive pilfering": if someone is throwing something out that could be used, grab it!
We have a lot to learn from the peasant habits of frugality—choosing delicious and nutrient-dense food, making the most of what we have, and wasting less. Obviously we can't return completely to the pre-industrialized way of life. But we can do the little things— and these little things can make a huge difference. Understand where your food comes from, choose food with less packaging, waste less, and recycle everything you can.
# SOURCING YOUR FOOD
"Locating better food is not something that you do once and forget. It becomes important to continually learn which fruit stand has the best items and which farm or farmette is worth a little 'drive in the country.'"
—Rex Harrill, farmer, Keedysville, Maryland
PEOPLE ARE ALWAYS TELLING ME THAT IT costs too much to eat well. I used to think the same thing. Every week I would drive to an inner-city organic food store where the fruit and vegetables were limp-looking and expensive. If I didn't feel like blowing the weekly budget, I could go to the supermarket next door, where they sold plumped-up fruit and vegetables for a quarter of the price, but with very little flavor or personality and possibly some mild pesticide residue. I knew what nutritious food was, but accessing it was another matter.
It was around this time that I started to dabble in gardening. Growing some of my own food, even in a modest way, changed how I thought about fresh produce. I found that a small patch of arugula and silverbeet could provide a fresh green accompaniment to any meal. Fresh herbs were easy (bordering on idiot-proof) to grow and added flavor to any dish. With time and a little help from some good gardening books, we were able to grow up to half of our own produce in our standard suburban backyard. This saved us money and time (no more last-minute trips to the greengrocer) and provided us with the most nutritious source of food possible. We also acquired a few chickens, happy little hens that provided an egg or two every day. Living in a suburban block, this was all we could fit, but it saved us considerable time, money, and effort in putting food on the table. We now get our food from a combination of our own backyard, a local farm, and a few organic food stores. If dinner isn't organized we can always throw together something nutritious and easy like a tasty omelet or toad-in-the-hole, garnished with a fresh green salad.
Becoming more self-sufficient made me realize how expensive and unnecessarily stressful my old approach to grocery shopping was. I used to think that I had two choices when it came to fresh food: healthier but expensive organic produce, or cheap but flavorless supermarket vegetables. Growing some of my own made me realize that tasty, nutritious food doesn't have to be expensive, and prompted me to seek out alternative sources for those things I couldn't grow myself. Most of us won't ever be completely self-sufficient. Not everyone has the time and space to have a garden, and not everyone can keep their own chickens. But there's no need to feel restricted to supermarkets and organic shops. There is an increasing demand for alternative ways to access good food. New avenues are opening up, allowing consumers to buy fresh, healthy food at much more reasonable prices. This section of the book outlines some of these different options. You might shop at a farmers' market, join a co-op or buying club, or even drive out to a local farm. You may also want to explore some of the grassroots movements described later in this chapter. They can be a great source of information about where to find local, sustainable food. You might be surprised to learn what is already happening in your area.
## SOME TERMINOLOGY
Whether you're buying directly from the farm or through a market or co-op, you'll want to know a little about the farm your food is coming from. A good farm employs traditional methods to grow nutritious, natural produce. It might use organic or biodynamic methods, or simply employ traditional techniques to avoid the need for chemicals and pesticides. Before we look at different places to buy your food, here's a guide to some of the terms you might encounter along the way.
Organic: Organic farming is based on the idea that food should be produced using natural methods and natural additives. It does not use synthetic chemicals, pesticides, or fertilizers, either on the produce itself or on the soil. It does use natural fertilizers such as animal manures, blood and bone, composts, rock phosphate, lime, gypsum, and dolomite.
Advantages of organically grown produce include:
* A reduced environmental impact. Organic farms are better at sustaining diverse ecosystems, are more energy-efficient, produce less waste, and do not consume or release synthetic pesticides into the environment.
* Better tasting, more nutritious produce. Studies of organic fruit and vegetables have shown that they have higher levels of many nutrients and antioxidants. Blind taste tests have found that organic produce often tastes better.
* Better living conditions for animals. All food fed to organically farmed animals must be certified organic. Furthermore, animals are not treated with unnecessary antibiotics, hormones, or genetically modified organizms.
Different organic certification systems exist in different countries. Keep in mind that thanks to hype surrounding organic labeling, and the higher prices farmers can charge for organically certified produce, there is the occasional flaw in the system, particularly when it comes to animal products (see the "Meat" chapter for more information). In some countries, particularly in the United States, organic certification is expensive and difficult for small family farms to obtain. If you are able to connect directly with a local farmer and find out how your food was grown, this will be far better reassurance than any certification label. Personally, I'd prefer to buy my food from a local farm that grows good quality produce that I know and trust, with or without certification, than from an organic supplier who I have never met
Biodynamic: Biodynamic agriculture is an "enhanced organic" method, inspired by a series of lectures given by Dr. Rudolf Steiner in 1924. It has developed most widely and successfully in Australia. It involves a range of techniques including biodynamic soil preparations, composting, and cultivation patterns based on the lunar calendar, all of which enliven the soil with energy and nutrients. For fresh fruit, vegetables, and animal products, biodynamics is probably the best assurance of quality, as the animal or plant will have been raised in optimal conditions with fertile soils and grass-based feed. Biodynamic farming has been shown to produce more friable, fertile soils and healthier plants and animals. Biodynamic products can be expensive, but it's worth it for the added nutritional value. If you want to save money, try to use your produce frugally, become a biodynamic home-gardener, or join a local community co-op to share the costs.
Free-range: This certification system applies to poultry and some livestock (especially pigs). A "free-range" label indicates that the animals have lived outside a conventional feedlot and have had access to fresh air and fresh pasture, however, in some localities, the free-range certification standard can be variable and the label does not always guarantee that the animal was raised on an optimum diet. Also the label does not guarantee that the animal was raised on a healthy diet. Pellet-feed, grains, and antibiotics are sometimes fed to "free-range" animals, so look for other certification systems in addition to this one if you can.
## STRAIGHT FROM THE FARM
By buying directly from a local producer, you can often save money on excellent, nutrient-dense food. Most retailers bump their food prices up by 100 percent, so if you can buy directly from the farmer you can avoid many added costs. Many social and environmental benefits also spring from developing a direct relationship with producers. You will be helping local farmers to maintain and grow their farms, while reducing food miles by purchasing unprocessed local food. So much of our inner-city lifestyle depends on the quality of the produce from our farms. Similarly, the farmer's survival depends on his ability to sell his wares consistently and at a good price. If we can connect directly, the benefits are not just financial—we will also improve communication and understanding, and help to ensure the quality of our food into the future.
Farmers' markets: A good place to start is by shopping at a local farmers' market. Farmers' markets provide a direct link between the farmer and the consumer. They are a terrific source of local and seasonal produce (and remember—food that's in season will be cheaper and better for you) and a great way to meet the people who grow your food. You can develop long-term relationships with local farmers and sometimes (if your farmer is a flexible one) order produce to suit your particular needs. Farmers' markets are popping up everywhere. See the "Resources" section for some help getting started.
Farmgate sales: Another alternative is to drive to the farm itself and purchase food onsite. This can require a bit of effort and you may have to drive for several hours, depending on where you live. You could share the drive with friends and take turns making the trip, or purchase your food in bulk and freeze it. Even if you only try it once or twice, farmgate sales are a great way to see for yourself where your food comes from.
From a legal perspective, different regulations exist governing direct farm-to-consumer sales, so you'll need to find out what is legally available in your local area. In some instances, you may have to join a buyers' club, co-op, or community-supported agriculture group.
Buyers' clubs and co-operatives: Buyers' clubs and food co-ops are another way to purchase food directly from the farm or wholesaler. Members usually pay an entry fee to join, and can then purchase food from any of the co-op's participating vendors. Clubs and coops are often based in a warehouse or retail shop. Others have a central spot from which members collect their purchases but conduct all their transactions online. These clubs enable members to purchase food below the normal retail price while supporting small-scale vendors.
The co-ops I've visited over the years have been started by passionate, health-conscious foodies who are keen to establish a direct link with local farms or wholesalers. The range of products available through co-ops is impressive. They often sell such things as homemade sauerkraut, jam, and raw milk—typical peasant fare that was widely available only a few generations ago, but has been lost with the shift to large-scale food production and supermarkets. Members can also buy products at wholesale prices from major food suppliers (the same companies that supply large supermarkets). I recently visited a friend in Manhattan who is part of a buying club of over 900 people. They do all their ordering online and meet the farmer (an Amish man from Pennsylvania) at a pick-up location once a week. According to the members I spoke to, it's easy to take part and gives them access to fresh milk, vegetables, and meat, and homemade preserves, all direct from the farm. The prices are significantly lower than those in organic food stores, and members enjoy the chance to meet the farmer and his family. Clubs may also showcase products made by their members. My friend Becca makes batches of her famous Beccaroons—a delicious almond-flavored macaroon—in her home kitchen and supplies them to her local club.
So clubs and co-ops can give you access to wholesale produce, homemade goods, and produce direct from the farm. There is also a wonderful feel-good factor; you'll be part of a local community, and directly supporting local producers.
Community-supported agriculture: A community-supported agriculture group or CSA consists of a group of individuals who pledge support to a local farm by buying a share of its ownership. The farm then becomes the community's farm, with growers and consumers sharing the risks and benefits of food production. Most CSAs involve a weekly delivery or pick-up of vegetables and fruit, and sometimes dairy products and meat. The term CSA is more common in the United States, but similar systems are found worldwide.
To get a CSA started, you need a group of interested, like-minded friends and a willing farmer. There are several websites (see the "Resources" section) dedicated to helping people to establish CSAs in their local areas.
Community gardens: Of course, buying from a farmer isn't the only way to access good produce: you can also grow your own. A community garden is a single piece of land gardened collectively by a group of people. Gardeners usually rent a plot of land, often from the local council, on which to grow their own fruit and vegetables; sometimes there is space for small animals such as chickens or quail. This movement has been extremely popular in Australia, with over 350 community gardens in Melbourne alone, ranging from local-council establishments to "guerilla gardening," whereby people plant on nature strips and in the centers of roundabouts. I would love one day to see every extra pocket of land taken up by edible gardens, fruit trees, and climbing sugar-snap peas. Victoria could become the "edible state" rather than the "garden state," and we could take pride in our delicious self-sufficiency rather than in wide expanses of dull green lawns.
Community gardens are an excellent resource for people who don't have enough space to grow their own vegetables at home. They also foster an important sense of community among people from different cultural backgrounds, and allow people to well and truly get to know their neighbors. Other programs often spring up around community gardens. "Urban orchard" and "food swap" schemes, for example, allow people to swap produce they have grown themselves and don't need. Last week I ventured to a Melbourne community garden with an excess of lemons from my tree at home. I took them to the food-swap table and traded them for a few bulbs of home-grown garlic. I _love_ garlic, so I went home very satisfied.
Land shares: Land-share agreements enable people to rent vegetable-growing land from private owners; they are an extension of the community garden concept to privately owned land. This is happening on a local level as communities develop different and innovative ways to share land between them.
One of the first land-share projects in Melbourne started close by to where I live. A woman living on a reasonably large suburban block decided to divide her backyard into garden plots and rent these out, for a minimal price, to her neighbors. She now has a back garden overflowing with fresh produce. Happy-go-lucky chickens roam around in the lap of luxury, producing delicious eggs for all the tenants.
In the UK, Hugh Fearnley-Whittingstall, founder of the well-known River Cottage, has created an online meeting spot where interested land-owners and land-renters can find each other and join forces. We need more meeting spots such as this! If you're interested in establishing a land-share agreement, start inquiring in your local area, or find an interested group of people via the internet. Alternatively, try posting notes in health-food stores, community centers or libraries, and you may be surprised by what you find.
## GRASSROOTS MOVEMENTS
The groups listed here are just a few of the many food-related networks developing around the world. They are a great way to meet like-minded people, share information about buying and cooking good food, and find out about foodie-related events. If you feel like you're in a good-food no-man's-land, joining one of these grassroots networks can be a good place to start.
Slow Food: The Slow Food movement began in 1986 in Italy and has since inspired a worldwide movement against "the universal folly of Fast Life," including fast food. Slow Food now has over 800 local _convivia_ or chapters throughout the world. There is a membership fee—usually around sixty or seventy dollars per year—which allows you to take part in local events and meetings. Slow Food has done great things to champion traditional food production, local food economies, and biodiversity. They also host fabulous dinners, showcasing the latest seasonal produce and traditional cooking methods. These are often pricey, but are well worth it.
Localism: The local food movement encourages consumers to buy food produced in their local area. The intention is to build more self-reliant food economies and to encourage people to have a closer connection to local producers. Eating locally also reduces transport, packaging, and processing, contributing to more sustainable food production and fresher, healthier products. A "hundred-mile diet," whereby people only consume food produced within a hundred-mile radius of their homes, has become popular in the United States, Australia, and the UK. This movement has done a lot to promote local food culture and traditional indigenous cooking methods. Locavores don't just shop locally, they also cook locally, with kitchen co-ops and cooking classes designed to foster community spirit. Anyone can be a locavore—have a look online and see if there is an active network of support near you.
Local food currencies: Some communities have developed local "food currencies" for use in specific towns or regions. Local groups print their own money (for instance, Berkshire in the United States has the "Berkshare" and Lewes in the United Kingdom the "Lewes Pound"). Local vendors can choose to trade with the local currency in addition to the official national currency. As well as the feel-good factor, there is often a monetary benefit for consumers. In Berkshire, for example, shoppers using Berkshares receive a 5 percent discount. This encourages consumers to shop locally, while in theory buffering the local food economy against global economic upheavals.
The Weston A. Price Foundation: The Weston A. Price Foundation (WAPF) is a grassroots movement active in over seventeen countries. It promotes traditional, nutrient-dense foods, home-cooking methods, and the pioneering work of Dr. Weston A. Price, a dentist who studied the diets of traditional cultures in the 1930s.
WAPF members have been active in starting many of the schemes mentioned earlier in this section, such as CSAs and co-ops. Local chapters provide listings of food resources in the area and enable like-minded people to exchange information and ideas. Some branches also run cooking classes and workshops. Some WAPF information is available free online; other resources are available only to paid members. Go to www.westonaprice. org to find out more.
## FEEL AT A LOOSE END?
If you feel like nothing is happening where you live, the first step is to get a group of likeminded, enthusiastic friends together. Local community groups or public noticeboards can be a good way to find one another. Once you have a few interested people together, you can start forging links with local farms or food networks. Remember, there's power in numbers! Just keep in mind that rules can vary depending on where you live, so you'll need to check your local regulations before sourcing any of the foods mentioned in this book.
Our relationship with food is changing. Consumers no longer have to depend on what supermarkets and organic shops choose to offer them. People are finding creative new ways to access the best quality produce straight from the farm, and at more frugal prices. In the city, more and more local governments are accommodating demands from residents who want to create community gardens, farmers' markets, or land-share arrangements. Whether you buy from a farmers' market, a co-op, a community garden, or straight from a farm gate, you'll have the satisfaction of knowing exactly where your food has come from. You'll be supporting your local economy, and saving money at the same time. You'll also, I promise you, have a lot of fun. So, what are you waiting for? Go find out what's happening near you!
**6 simple ways to connect to a local food network:** \- Search online for fresh food markets in your local area and spend a month shopping at a different location every week. The website www.localharvest.org is a good place to start. We call this the 'market trip challenge'; it forces you to seek out new shopping locales and learn about developing markets in your area. - Contact your local chapter leader from The Weston A. Price Foundation and request a list of local resources. Visit www.westonaprice.org for chapter leader listings. In particular, ask for the details of farmers who sell direct, or in bulk with a discount. - Start a social media group with your suburb name and food interest. You can link in with like-minded friends to share ideas and seek out resources in your area. - Connect with a farmer by searching online or shopping at a farmers market. Find out if they do discount sales and pitch in with friends and family to make a bulk order. -Start a buying club. All you need is a willing farmer, some friends, a drop-off point and a basic working agreement for the farmer. I have friends that organize this through their children's kindergarten program and use the kindergarten premise as a drop-off point. You can also use a church hall, a café, or someone's garage and drive-way as a drop-off point. -Place a flyer at your local organic store or Church notice board stating that you wish to start a buying club, anyone who is interested can contact you directly.
# THE FRUGAVORE KITCHEN
"Before enlightenment—washing dishes, carrying water; after enlightenment—washing dishes, carrying water."
—Lao Tsu
GOOD FOOD ISN'T JUST ABOUT THE FINAL product you put on the table. It is a reflection of your entire cooking environment—your kitchen, bench tops, cooking equipment, and garden, and even the quality of the air. The kitchen is the heart of the house. How you clean it, chop up your food, and even heat your meals will not only affect the nutrients in your food, but also the flavor of every bite. So invest in some good old-fashioned cleaning products that won't spread harsh chemicals, grab a simple timber cutting board, look for some old pots and pans at the thrift store, and take your microwave to the dump. Air out your kitchen, clean out your freezer, and keep some pots of herbs by the back door. Your frugavore cooking experience is about to begin.
## CLEANING
Only a few generations ago, housewives were able to keep a clean and tidy home with just a few staples: some bicarbonate of soda, vinegar, lemon juice, cleaning cloths, and good old elbow grease. We have been led to believe that anti-bacterial chemicals and bubbly soaps are necessary to get homes really clean, but this is just not true. Bacteria are everywhere around us—on our skin, in the air, on every surface that we touch. If we try to completely eliminate these bacteria from our environment, it disrupts the natural balance of nature and chemical-resistant strains can develop. Many studies are now showing that children require some exposure to bacteria to assist in their natural immune development.
What substances should you be wary of? For starters, any cleaning product labeled "danger," "hazardous," or "poison" should not be used around food. The chemicals that have been found to cause the most damage include the concentrated form of chlorine, which is found in many dishwashing detergents. This can have lasting or even fatal effects if ingested or swallowed. The lye and ammonia found in many oven cleaners can cause damage to the respiratory system and skin. Traces of these chemicals can enter the food we eat, either via residues on our plates and cutlery or through the air in the oven. The chemicals in cleaning products can also have lasting effects on the ecosystem after they've been washed down the drain. The phosphates used in many automatic dishwasher detergents, for instance, have been shown to kill fish and other organizms in our waterways.
So, what to use instead? Plenty of trendy, eco-friendly cleaning products are now popping up in supermarkets and health-food stores. These products are usually a better alternative than conventional products, but they can be expensive. The truth is you don't have to spend loads of money on expensive cleaning products. Natural, safe, inexpensive alternatives, just like our parents and grandparents relied on, are cost-effective and easy to use. Instead of having a different cleaning product for every surface, you just need a few simple basics and some good recycled cloths.
## BUYING READY-MADE CLEANING PRODUCTS
* Be wary of any product labeled "danger," "warning," or "poison": you don't want anything toxic in your kitchen!
* When interpreting "eco-friendly" or "organic" claims on a label, look for the product's degree of biodegradability. The more specific, the better: "Biodegradable in five to seven days" is much more reassuring than just "natural" or "earth-loving."
* Look for "phosphate-free" and "solvent-free" wherever possible.
* Look for products that are plant-based. If you are unfamiliar with the chemicals listed on the label, either look them up or don't buy it.
* To reduce waste, re-use your cleaning bottles. Many food stores and co-ops now provide refillable containers.
## MAKING YOUR OWN
Making your own cleaning products is easy, effective, healthy, and incredibly cheap. To get started, you'll need:
* Bicarbonate of soda
* Vinegar
* Borax
* Salt
* Lemons
* Eucalyptus oil
* Castile liquid soap
* A spray bottle
* Some recycled rags or old microfiber cloths
With the recipes in this section you can turn these basic ingredients into cleaning products for every surface. Always try to use the smallest amount of cleaning product possible. Even inoffensive products such as bicarbonate of soda can cause damage to the environment if used in excess. Water alone is an excellent solvent and good cleaning can be done with a strong arm and a damp cloth.
To clean bench tops, sinks, taps, and floorboards: Use a 50:50 solution of vinegar and water in a spray bottle; or a 3 percent saltwater solution (¼ tablespoon of salt for every 6 ¾ tablespoons of water); or sprinkle some baking soda on a damp cloth, wipe the surface to be cleaned, then rinse with clean water; or mix a small amount of baking soda with liquid Castile soap to get countertops extra shiny. To finish, add a few drops of lemon juice for a lovely aroma.
Casserole dishes: To remove stubborn baked-on leftovers, put 2 tablespoons of salt into the dish and fill it with boiling water. Let it stand until the water cools, then wash the dish as usual.
Dishcloths: Soak dishcloths and muslin cloths overnight in a 1:3 vinegar and hot water solution (1 cup of vinegar for every 3 cups of water). In the morning, scrub them if necessary, then rinse them. Air-dry them in the sunshine or rest them on an indoor heater.
Dishwashers: Put some bicarbonate of soda in the soap dispenser and run the wash cycle. This will clean your dishwasher, and can also be used for dishwashing instead of dishwashing powder.
Ovens: Make a solution of 4 ¼ cups of warm water, 1 ½ teaspoons of borax, and 1 ½ tablespoons of liquid soap. Spray the solution onto the oven's interior, wait twenty minutes, then clean with a rag and rinse with clean water. For tough stains, sprinkle a thick coating of bicarb soda and leave it to sit for an hour or two. Wipe off with a damp cloth.
Pots and pans: Bicarbonate of soda works well for scrubbing pots and pans, and is especially fantastic on burnt pots.
Trash cans: Splash some vinegar or lemon juice into the bottom of the can. Leave it for twenty minutes then rinse it clean. This should neutralize any bad smells.
Stainless-steel stovetops: On a damp rag, combine a dab of liquid Castile soap with a sprinkle of bicarb soda. Wipe your stovetop with the rag to remove stains and grease. Then, with a fresh damp cloth, wipe the stovetop again. Sprinkle any really hard-to-shift marks with bicarb soda, leave it for ten minutes, then scrub it off.
Wooden cutting boards: Wash the cutting board with warm soapy water, then rub it down with a handful of salt and leave it for five or ten minutes before rinsing it clean. If your cutting board starts to feel dry and brittle, rub it down with almond oil or olive oil and leave it overnight so that the oil can soak in.
## KITCHEN UTENSILS
Too many cooks these days get hung up on having the latest equipment—shiny pots and pans, non-stick coatings, and the newest technology. But food doesn't need to be created with the latest appliances to look and taste good. If you want a healthy cooking environment and a nourishing finished product, stick to the basics, just like our grandparents did.
Cookware: Where possible, try to avoid cookware made with "non-stick" coating as this has been found to contain the chemical polytetrafluoroethylene (PTFE). PTFE has been associated with numerous chronic and acute health problems. If you do choose to use nonstick cookware, never heat it to high temperatures, as this appears to release more of the toxic emissions.
A better alternative is heavy stainless steel, cast-iron, stoneware, or enamel-coated pots and pans. Stainless steel or copper (not aluminum) work well for everyday steaming, frying, or boiling. If you can, it is worth investing in one or two large, heavy enamel pots. They are excellent for long, slow, gentle cooking as they retain the heat in their thick, cast-iron walls. These old-fashioned, heavyset pots aren't always cheap, but they do last a lifetime. Many thrift shops sell good quality old pots and pans, and I've seen some good buys on eBay. My own favorite slow-roasting stew-pot was bought ten years ago from a thrift shop; it must be about forty years old. It cooks odd cuts of beef and lamb second-to-none and could easily out-do its counterparts in modern cookware departments. So choose your pots wisely—and don't worry about superficial scratches or chips; it's what's inside that counts.
Cutting boards: Wooden cutting boards that haven't been sealed or treated are the best choice for the home kitchen. Natural timber breathes easily, so repels bacteria and bugs better than plastic boards can. Clean your timber cutting board regularly and always allow it to breath when you are not using it. Keep it on the bench top or propped up against the wall, rather than tucked away in a stuffy drawer.
Microwaves: Microwaves are a convenient appliance for busy households. But microwaves do more than just reheat food quickly. They radiate heat using a form of electromagnetic energy, which destroys and deforms the molecules in food and depletes many of the important nutrients. As an alternative, choose an oven with a fan-force option. We have at home a "double oven"; one oven is 35.4 inches wide and the other is 11.85 inches wide. The smaller oven heats up quickly and uses less energy to reheat meals. With the fan-forced heat, it takes only ten or so minutes to reheat a dish—nearly as quick as a microwave, and much healthier.
Freezers: The best investment we ever made was a large freezer, which we keep in the garage or back porch. We store meat and grains that we've purchased in bulk, as well as stock and leftovers. The freezer saves me much time and worry. If I come home late at night, I can defrost something quickly and easily—some sausages, or some mincemeat for hamburgers or meatballs. I combine this with a few fresh greens from the garden and any vegetables that happen to be in the fridge, and ta-da! dinner is made. No shopping and very little planning required.
## MEAL PLANNING
If you'd like to cook from scratch and have a busy job or chaotic family, you'll need to start planning.
Start by choosing a day that is relatively stress-free—perhaps a Saturday afternoon or Sunday morning. Use this time to take stock of what's already in your cupboards and plan your meals for the week. Once your meals are planned, the rest is fairly straightforward. Cooking as a frugavore is not actually time-consuming, it just requires a little thinking ahead. So if you are cooking chickpea soup on Wednesday night, you need to put the chickpeas on to soak on Tuesday night or Wednesday morning. Most of the dishes in this book have been adapted to be work-friendly. If you stock your fridge and freezer wisely, you can easily whip up a tasty dish in a matter of minutes on any given weekday.
I try to plan three hearty meals per week that take a little more time and effort. For the other four nights, I use up the leftovers or opt for simple meals such as soups, salads, or omelets. For instance, you could prepare a whole roast chicken on Sunday night, then leftover chicken salad and some roast potatoes on Monday night. On Tuesday night try another fiddly dish—like Irish stew, or steak and kidney pie, then use the leftovers to invent something again on Wednesday night.
When leftovers run thin, try leaning toward pantry staples like lentil or pumpkin soup, basic omelets with garden salads or any array of beans and pulses. These 'last-minute' meals made with only a few basic ingredients are inexpensive to put together and require very little cooking time. Make sure your pantry is well-stocked at the beginning of each week and you'll never run out of nutritious meal ideas.
I make Saturday my market day, when I visit the local food market with my shopping list for the week. Saturday afternoon is my cooking and organizing time. Before I put away my shopping, I clear out the fridge. Any stale vegetables—floppy celery or tired carrots—can be used to make stock with the fresh bones I've purchased at the market. Anything else that looks a bit spoiled goes to the compost or the chickens (our chickens love Saturdays!). I leave the stock on to simmer for twenty-four hours and load the freezer up with it on Sunday. Having stock on hand means soups can be prepared quickly and easily throughout the week. I also use the weekend to check on the garden, reload the compost, and ensure that the hens are happy. After this, the rest of the week is fairly smooth sailing (well, from a cooking perspective, at least).
## LUNCH OPTIONS FOR THE NINE-TO-FIVER
It's easy to forget about lunch until you start to feel hungry at noon and find yourself rushing out to buy an overpriced sandwich. But with a little planning, you can save heaps of money and time by packing your own. If you are short of lunch ideas, try to think beyond the obvious. You might buy a small thermos (these are usually cheap; I picked one up at a thrift store for two dollars), in which you can store hot soup or broth. Recycle plastic containers and bring in leftovers from the night before. Dishes such as casserole and baked beans taste better if they are reheated. If you don't have reheating facilities at work, leftovers stored at room temperature will be tasty by 1 p.m.
Another favorite of mine is a mixture of fresh veggies—carrots, celery, and cucumber, for instance—placed raw in the lunchbox with a small container of olive oil and vinegar or some homemade mayonnaise. This is very quick to prepare in the morning—you don't need to chop up the vegetables, just whack them in your lunchbox—and you can prepare the mayonnaise or dressing the night before.
## MAKE IT A TEAM EFFORT
Lastly, try to get your whole household involved. We all know what it's like to live in a busy home: kids, pets, neighbors, friends, roommates. Life can be chaotic. Trying to cook, garden, and keep chickens can create more madness if you aren't organized, or if one person alone is trying to make everything work. Include everyone in your mission: ask your kids to feed the hens (they'll love it), get your partner interested in the compost, make gardening a family activity. If you live with roommates, try to get them excited about an outing to a farmers' market or joining a co-op. Most people find that they get a lot of pleasure from doing these tasks, so it needn't be burdensome or arduous. Working together to get things started will make your frugavore experiment all the more productive and enjoyable.
# STOCKING YOUR PANTRY
"It is thrifty to prepare today for the wants of tomorrow."
—Aesop
A TRADITIONAL KITCHEN PANTRY WAS STOCKED with all manner of preserves as well as ready supplies of herbs, vinegars, oils, legumes, and natural sweeteners. A well-planned pantry will save you so much time, worry, and money. You will be able to come home and prepare a healthy, inexpensive meal using just a few ingredients—no exhausting (and expensive) last-minute supermarket trips required. A basic lentil soup can be made with some dried lentils and herbs and some stock from the freezer. Chickpea soup can be thrown together with some dried chickpeas, some canned tomatoes, and a sprig of fresh herbs from the garden.
When stocking your pantry, don't feel you have to buy everything all at once. Start with the familiar, less expensive items first, but don't be afraid to experiment as you go. And when storing your purchases, remember that all pantry items are best kept in glass or ceramic containers and stored in a cool, dry place with minimal light.
## BEANS, LEGUMES, & PEAS
Beans, legumes, and peas are extremely cheap and wonderfully nutritious. As a starting point try some dried chickpeas, lentils, red kidney beans, and split peas. With a stash of these in your pantry you will always be ready to make a wide range of soups, stews, and salads. See the "Legumes" section for recipes. Dried beans are the most cost-effective. They last for longer periods of time in the pantry and come with much less packaging. But if you are not the kind of cook who likes to plan ahead, it might be worth keeping a few cans of chickpeas and legumes on hand. They are a little bit more expensive, but they are quicker to prepare as they do not require pre-soaking or pre-cooking.
## FATS & OILS
Oil is essential for cooking as well as for dressing salads and other cold dishes. I recommend stocking your pantry with some good-quality cold-pressed olive and coconut oil.
Olive oil: This is the food of the gods. Enjoy it and purchase the best quality you can. Always look for extra-virgin, cold-pressed olive oil, and buy organic wherever possible. Olive oil should be stored in dark opaque glass bottles as it is sensitive to light.
Coconut oil: Coconut oil or coconut butter can be used for all types of cooking; it is especially well suited to high-temperature cooking such as frying and baking. Look for extra-virgin, cold-pressed, organic oil.
## FLOUR
Whether you are an avid bread-maker or a very occasional baker, keeping a few different flours on hand will give you many more options when cooking. For truly fresh grains, you could consider buying your wheat berries whole and grinding them with a small kitchen flour mill. Alternatively, if you are purchasing ground flour, look for "stoneground" and "organic" wherever possible. I try always to have a finely ground rye and spelt flour on hand. Because flour is sensitive to heat and light, it is best to keep it in the fridge or freezer. See the "Grains" chapter for more information.
Arrowroot powder, also known as tapioca flour: Made with the tapioca plant, this flour is an excellent thickener in soups and stews.
## RICE
Rice is perfect for last-minute dinners, to flesh out leftovers, or complement the latest pickings from your garden. There are many sorts of rice available, but I generally prefer cooking with brown rice. This can be bought in bulk and stored in large containers. If you are worried about weevils, keep it in the freezer.
## POLENTA
Polenta is essentially ground-up corn, and has been a peasant staple since the Middle Ages. Incredibly quick to cook, it's a wonderful last-minute dish and can accompany meat dishes, casseroles, and vegetables.
## SAGO OR TAPIOCA
These tiny balls are extracted from the pith of sago palm stems. It's an excellent staple and has a very long shelf life. Sago pudding is a favorite dessert at our house.
## COCONUT MILK & COCONUT CREAM
These are very useful ingredients for people who suffer from milk allergies. They can be added to custards or desserts in place of milk.
## SEASONINGS
Salt: Every pantry should include sea salt, which is very different from normal table salt. It provides a wide range of minerals and stimulates digestion. It can be found at most health-food shops and supermarkets.
Dried herbs: Fresh herbs add the best flavor, but a supply of dried herbs is useful for when fresh herbs aren't available. Parsley, rosemary, thyme, oregano, and bay leaves are all indispensable.
Seaweed: Seaweed varieties including kombu, arame, and dulse flakes can be added to grain and seafood dishes and to bone broths and stocks. They are an excellent way to add exotic flavor and extra minerals. In some organic food stores they can be quite expensive, but you can find them at much better prices in small Asian supermarkets.
Spices: Dried cinnamon, nutmeg, ginger, allspice, and paprika are invaluable for both sweet and savory cooking.
Sweeteners: For day-to-day cooking and for when you crave a treat, you'll want a few sweeteners in your pantry. There are many healthier alternatives to refined sugar, including honey, molasses, rapadura sugar, and stevia. See the "Sweet Stuff" chapter for more details about their different uses. I like to keep a selection of these on hand, along with some brown sugar (which is cheaper than the healthier alternatives) for big cooking sessions.
## VINEGARS
Vinegar can be used to flavor salad dressings, is an essential ingredient in stock (it helps to release the calcium from the bones), and can be added to many soup recipes for a slightly tart flavor. It's also very handy for household cleaning. There are countless types of vinegar available, but the few listed here will get your pantry started.
Apple-cider vinegar: This is my favorite variety of vinegar, thanks to its tangy flavor and excellent nutritional value. Apple-cider vinegar can be used for all types of dishes, from salad dressings to soups and stews. Be sure to choose a brand that contains a "live mother culture" (the label may just say "with mother," which I always think is rather funny), and try to buy organic wherever possible.
Balsamic vinegar: This is a very popular vinegar, and useful for salad dressings.
Red or white-wine vinegar: These can be used for all the same purposes as apple-cider vinegar, but are not as nutritious. Wine vinegar is also good for poaching eggs and for adding flavor to salad dressings.
## PRESERVES, PICKLES, & CONDIMENTS
A selection of jams, pickles, preserves, and condiments will enrich any pantry. The "Preserves" section of this book includes recipes for making your own. In addition to homemade jams and pickles, these store-bought preserves can come in handy.
Tomato passata or canned tomatoes: Preserved tomatoes can be added to soups, stews, and seafood dishes, and are especially handy when tomatoes are not in season. Ideally, look for good-quality glass bottles of organic passata or try the recipe for preserved tomatoes on page 312.
Fermented soy products (miso, natto, tempeh, and soy sauce): These traditional soy products can be found in Asian grocery shops, healthfood shops, and supermarkets. Miso can be used to flavor soups and stocks, while soy sauce can be added to meat and vegetable dishes. Natto goes well in soups or with Asian noodle and vegetable dishes. When buying these products, look for "traditionally fermented" or "traditionally brewed" on the label. Also try to buy organic, as many soy products are now genetically engineered. Organic and traditionally fermented brands may be a little more expensive, but remember that a good bottle of soy sauce or packet of miso should last you several months, if not years.
Worcestershire sauce: This is useful for flavoring rich stews and bean dishes. Look for organic wherever possible, and check the label to be sure it contains no odd-sounding additives or preservatives.
Tomato ketchup: You can make your own ketchup (see the recipe on page 320), or buy it ready-made. Try to choose one that contains no preservatives or artificial ingredients. It goes beautifully with mince dishes such as hamburgers and meatloaf.
# THE VEGGIE PATCH
"It is said that the effect of eating too much lettuce is 'soporific.'"
—Beatrix Potter
IT WASN'T ALL THAT LONG AGO THAT ALMOST every person who owned his or her own land also grew his or her own food. Every inch of backyard was used to its fullest capacity to grow anything and everything edible, or to support animals like poultry or pigs. You could walk past your neighbors' front yards and witness all manner of blossoming fruits, odd-looking speckled vegetables, and rabbit traps and pigeon huts ready to catch the evening meal. I sometimes dream of walking back in time into this world, where my own small patch of vegetables would be complemented by my neighbors'. All I'd have to do at dinner time is turn on my oven, get out my clippers, and walk out the back door.
Like many people today, I started my own vegetable garden as a way to deal with constraints of time and money. I was exhausted, shopping at organic food stores on the other side of town and coming home late to cook dinner. Our grocery bills were always high and I would arrive home late and frazzled. I might have a basket full of produce, but it would be too late or I would simply be too tired to cook.
Growing your own produce has a number of advantages. It will save you time, it is cost-effective, and (the best benefit of all) you will have access to the freshest, most nutritious produce.
Vegetable gardening does take time in the initial stages. You need to plan your plot, plant your produce, fertilize the soil, and keep an eye on its water and nutrient levels. But once your garden is established, it is actually very little work. And as I came to learn, the work that is required is generally enjoyable and often very sociable. At our place, it can easily become a morning-long affair with my mother, a cup of tea or two and some music blaring from the kitchen. Gardening isn't arugula science; it's a combination of knowledge and intuition, both equally important. Be prepared to give everything a go and you'll be a very good gardener.
If you are just starting out or have limited space, some herbs are the first thing you'll need. You don't need a garden bed: you can plant them in pots, in a window box, or even in a recycled container (we recently rescued an old sink from someone's dumpster and turned it into a herb garden—moveable and cost effective, and a conversation starter too!). A few fresh herbs growing on a windowsill or on the back doorstep are invaluable for home-cooking. Even the simplest of dishes can be improved with a few sprigs of fresh parsley or basil, and growing your own is much cheaper than buying them fresh every week. A bunch of fresh rosemary or thyme at our local supermarket retails for three or four dollars, whereas a seedling from the nursery sells for a similar price and will last indefinitely.
If you have the space, a few pots of fresh greens—lettuce, arugula, silverbeet, or spinach—are also wonderful to have. These are super easy to grow, and with a handful of arugula and some pickled sardines or a fresh egg from your hens, you can have dinner organized in a flash. Greens really need to be fresh for optimum flavor; even if you have the time and money to shop for them regularly, they will taste much better freshly picked from your own backyard. You can buy greens as seedlings, if you want to keep things easy, or as seeds, which require a little more care. They will provide your garden with color and life and your kitchen with a convenient and highly nutritious food source.
Once you have a few basics under your belt, you can try venturing into a bigger garden plot, either in your own backyard or through a community garden. This only needs to be the size of a bathtub or two; in just a few square feet, you'll be able to grow a good proportion of your household's produce. An average suburban front lawn could supply a family of four throughout the year.
When it comes to choosing what to grow, there are countless options, but the easiest place to start is your local nursery. See what is most appealing and what is most suitable for your climate and season. Try to keep your garden as varied as possible and rotate your crops each season. Nature is built around the principles of biodiversity, so by regularly changing your crops and planting different species of plants together, you'll add nutrients to the soil and decrease the risk of pest attacks. Including some animal life in your backyard—be it by keeping chickens, or adding manure to your compost heap—is another good way to replenish the soil.
If you build a healthy soil, water regularly (and do some clever drought-proofing), and plant seasonally, before you know it your garden will be thriving.
## PLANTING IN POTS
If you only have a small garden, courtyard, or balcony, planting in pots may be your best bet. This could set you back a couple of hundred bucks at many nurseries, with their wide array of pots and potting mixes. There's no need to spend a fortune on designer containers and fancy soils, however; here are some frugal tips for getting things started more cheaply.
Choosing your pots: Wherever possible, choose pots as big as you can fit in your space. The more room your plants have, the further their roots can grow. Heavy pots (such as ceramic, terracotta, or metal) and plentiful soil will protect your plants against extreme temperatures. Ceramic or terracotta pots can be expensive, however. To save money, look for pots at thrift shops and recycled hardware stores, and experiment with what you already have or can scavenge. My most treasured pot is a bright orange barbeque base that I removed from a dumpster. My boyfriend delicately drilled two holes in the bottom, and it now holds a few varieties of lettuce and several clumps of parsley. We've also collected large stainless steel containers, old pig troughs, and a baby's bath, each of which we've filled with homemade potting mixture and turned into a thriving vegetable bed. If you can get your hands on them, halved wine barrels make excellent pots—they are big and deep, and the timber provides insulation (and, dare I say it, a nice winey aroma!). Wine barrels are often sold at nurseries or hardware stores, and can be considerably cheaper than ceramic pots. Even better, vineyards often sell used ones for next to nothing, or may even be willing to give them away for free.
Plastic pots are another option for the low-budget gardener. They don't insulate the plant as well as ceramic or steel, and I tend to be anti-plastic as a general rule—but a plastic pot is better than no pot and no garden. Some nurseries give away used plastic pots for free.
Make sure you keep your pots in a sunny spot. As a general rule, plants need at least six hours of sunshine per day, particularly during winter. Also try to elevate them off the ground (for instance with feet or a tray) so that the water can easily drain from the bottom. You'll need to water your pots regularly and keep them protected during hot, dry summers.
Potting mixture: Vegetables don't tend to do so well in pots with just standard garden soil. You'll need to make up a potting mixture to help your plants thrive and provide adequate drainage. Some gardeners prefer to make their own potting mixture, or to use a combination of bought mixture and a few boosters from their own backyard. Here's a rough recipe I use regularly. The basic ingredients can be bought at most nurseries or hardware stores:
2 parts commercial potting mixture or healthy topsoil from your garden
1 part compost or worm castings, mixed with bark or garden clippings in a 50:50 blend
1 part coarse river sand
If you're trying to make your commercially bought mixture or topsoil go further, you can reduce the amount so that your recipe is 1 part potting mixture, 1 part compost, and 1 part sand.
Potting fruit trees: Fruit trees can grow well in pots. Look for dwarf varieties, as they provide an abundance of fruit and don't grow too big. You should easily be able to fit a dwarf citrus tree in a large pot or wine barrel. Potting mixture for citrus trees is different from that for vegetables. They require additional sand to provide extra drainage and encourage root growth. Here's my recipe:
1 part garden topsoil
1 part compost or worm castings, mixed with bark or garden clippings in a 50:50 blend
2 parts coarse river sand
## NATURE'S RECYCLING SYSTEMS
My own garden only started to thrive when I began to recycle our kitchen waste back into the soil. When my brother moved, he gave me a worm farm he didn't want anymore. I dutifully put all our kitchen waste in it, and it soon started to produce an amber-colored worm juice. My brother had advised me to put this on the garden, so I diluted it in a watering can and poured it onto a couple of silverbeets I was growing (unenthusiastically, I'll admit: I was still a bad gardener at this stage!). Within a week, the plants looked like they had been given a dose of anabolic steroids. My confidence grew quickly: I started planting more seeds and feeding the worm farm more regularly. I used the solid worm castings as potting mixture for new plants and spread them on the garden. I knew very little about proper soil ecology or plants generally, so I was relying on intuition. The worm farm seemed to make sense and the nutrients allowed the plants to really thrive. Within six months, with a growing sense of confidence, I had ripped out our front lawn and was growing zucchini, pumpkins, and tall stalks of artichokes and leeks. I am not a clever gardener and didn't expect things to go this well; it really all started with the worm farm and a good healthy soil. The rest was remarkably easy.
Nature has a clever way of recycling itself; the process of plant death, breakdown, and regrowth is an intrinsic part of our natural environment. Plants have a well-defined life cycle: they grow, blossom, ripen, then break down and fertilize the ground around them. Composting and worm farming are not only great ways to dispose of waste; they also let you harness these natural processes to improve your soil. If vegetable and fruit scraps end up at the land fill, this plant cycle creates one of the most environmentally damaging gases— methane—which is produced as plant matter breaks down and is released directly back into the atmosphere. So by composting, you'll be helping the environment as well as your own little patch of earth. In fact, it has been estimated that if every person had their own worm farm or compost heap, it would reduce our waste output by 1 ton per person per year. Imagine what we could achieve if more communities, corporations, and local councils embraced this—we could see local drop-off points where people discarded their kitchen waste, and it could be used to fertilize parks, local gardens, and community vegetable patches.
Composting and worm farms allowed me to create a mini-ecosystem in our home and garden. Our fruit and vegetable scraps go to our worm farm, along with layers of shredded newspaper and tea leaves. Onions and citrus fruit (which the worms don't like) go to a bokashi bucket, which uses micro-organizms to ferment them; I can later plant them in the garden or at the bottom of pots. Meat scraps and bones are given to our dog or buried in the garden (deep enough that the dog can't dig them up!), where they fertilize the soil. The chickens also love leftovers— porridge, stale bread soaked in water, and greens—so there are always treats for them. I often see people struggling to nurture plants that won't thrive or spending loads of money on mulch and fertilizers. Gardening doesn't need to be so complicated! Start simple: recycle your waste, get a good composter or worm farm, and the rest should go from there. Below, I've outlined a few different ways to get started with composting.
## COMPOST HEAPS
Compost heaps are the most time-tested and traditional way to recycle your waste and fertilize your garden. They take up more space than a worm farm and are generally best suited for medium to large backyards. Don't be dissuaded by this, however, as it's well worth the space it requires; compost can be used as a mulch, potting mixture, or liquid fertilizer, or to prepare a garden bed for a fresh season of planting.
The two most popular designs for compost bins are the traditional plastic bin, which sits upright in a corner of your garden, and the tumbler bin, which looks like a plastic wine barrel supported by two timber posts and is easier to rotate and swing. Tumbler bins are rat-proof and easier to move. The only drawback is that they are usually a lot more expensive, often a couple hundred dollars. If you are handy with a tool kit, you could construct your own (have a look around online for instructions). More traditional compost bins, which sit upright and stationary on the ground, are much cheaper (usually about fifty dollars from gardening or hardware stores). I've also seen some good buys on eBay and through local councils.
Handy home-gardeners, however, can easily create their own compost heap with a few basic components. With some timber logs, a two-yard square area can be sectioned off and used for your compost heap. Any large, weatherproof container will also work well. My mother uses a large recycled corrugated-iron water tank; she can easily get inside and stir the compost with a pitchfork. Another friend of mine has converted a large bathtub into a compost heap; it's affectionately known as "the coffin."
When it comes to choosing a spot for your bin, remember that compost needs plenty of warmth and sunlight for fermentation to take place. It's also a good idea to situate your bin as close to the kitchen as possible, so that you actually use it!
Developing your compost: For a healthy compost heap, you need to balance wet and dry ingredients to ensure the right level of moisture. This is easier than it sounds—really it boils down to common sense. Dry ingredients include any type of dried mulch (garden leaves, shredded paper, hay, or straw) and garden cuttings. Wet ingredients include freshly dug weeds, manures, fruit and veggie scraps, and lawn clippings. A healthy balance of these will provide the right environment for micro-organisms to thrive and prevent the heap from becoming wet and stinky or dry and dusty.
If you are using a tumbler bin, you can simply monitor the ingredients you put in; they will all be mixed together when you spin the plastic tub. If your heap is based on the ground, try to layer the different ingredients like lasagna, so that they are all spread evenly throughout the heap. You can have a layer of vegetable scraps, followed by sheets of newspaper, followed by manure, then some more paper, and then some straw or mulch. For some extra nutrients, add a bit of basalt rock or a few spadefuls of garden soil—this will reinvigorate your compost with a good dose of minerals and healthy microorganisms.
For it to break down into compost, the waste needs to be tossed or rotated regularly. Tumbler bins can simply be spun whenever you happen to walk past. If you make your own bin, just use a pitchfork to toss the compost at least once a month.
Your compost is ready for your garden when it is a rich, dark color, crumbly in texture, with a warm, earthy smell. If your compost is on the ground, this can take up to eighteen months. If it is in a tumbler, it should only take three to four weeks. Some keen gardeners like to have two separate bins on the go; while one is fermenting and breaking down, the other can be filled with new ingredients.
Compost ingredients: All of the following things can be added to your compost heap. If you cut up your ingredients as small as possible, they'll break down more quickly:
* Uncooked fruit and veggie scraps
* Coffee grounds and tea leaves
* Shredded newspaper
* Old garden mulch, straw, lawn clippings, autumn leaves, and prunings
* Fireplace ash
* Cow, sheep, horse, or chicken manure
* Dolomite lime and basalt rock
* Garden soil
Ingredients to avoid:
* Cat or dog droppings
* Cooked food (I do occasionally add small amounts of cooked vegetable matter, as it will break down fairly easily if the compost is strong and healthy. As a general rule, however, cooked scraps are best avoided)
* Meat matter, whether cooked or raw, which can breed bad bacteria and parasites
Using your compost:
* As mulch: spread your compost around your garden plants, applying it up to 4 inches deep.
* As liquid fertilizer: in a bucket or tub, add one part compost to three parts water. Give it a good stir, then leave it for two to four days to ferment. Apply this liquid to the plants as a tonic.
* As potting mixture: compost can be combined with other ingredients to produce a rich and healthy potting mixture.
* To prepare a garden bed: reinvigorate your soil with plenty of compost just before a new season of planting.
## WORM FARMS
I love worm farms. They are usually small— less than 3 feet in diameter—which means that they can fit on even the smallest apartment balcony. They can be fed uncooked kitchen waste (vegetables and fruit peelings) as well as some garden waste and small quantities of newspaper scraps.
Worms are integral to a healthy soil environment; they recycle and transform organic waste matter into useable, nutrient-dense soil humus. You want to encourage as much worm activity in your garden as possible, and having a worm farm is usually the easiest way to achieve this.
You can buy a worm-farm kit from most hardware stores and nurseries. They usually retail for between eighty and one hundred dollars, but you may find them cheaper on eBay or at recycled hardware outlets. Some local councils sell them at a reduced price as an incentive for people to recycle their waste.
A worm farm consists of several layers of plastic trays. The worms migrate between the different levels as they work their way through your scraps. Most ready-made worm farms contain three moveable shelves. Farms can be circular or rectangular shaped, and they are usually on stilts so that they are out of reach of mice and rats. As the worms eat their way through the scraps in the bottom tray, you start adding your scraps to the next and they will gradually migrate upwards to the new food source. The worms in worm farms are different from earth worms—they are usually Indian blues, tiger worms, and red wrigglers. They can be bought from nurseries and hardware stores. Alternatively, if you have a friend with an already thriving farm, ask if you can take a container of worms from their farm to start your own.
If you are strapped for cash, a homemade worm farm is cheap and easy to make. All you need are two large polystyrene boxes (about 20 by 20 inches) and a well-fitting lid for one of the boxes. Make sure that the boxes can fit neatly one on top of the other. In one corner of the bottom box, drill a hole so that the liquid can drain out. Poke at least twenty holes in the bottom of the top box, so that the worms can travel between boxes without difficulty. On the bottom of the farm, place a layer of newspaper or hessian material. Start your worms on the bottom layer, with a little bit of food, then build up their intake as their population grows. When they are eating well on the bottom layer, transfer half of the material from the bottom layer to the top layer and continue feeding on both levels.
Worm farms produce a special juice (commonly referred to as "liquid gold" by home gardeners), which comes out of a tap on the bottom layer of the farm. You can dilute this juice to use as fertilizer on your garden. You can also use the worms' solid waste (or "castings") as potting mixture. Anecdotally, biodynamic farmers have told me that worms can be used to correct imbalances in a plant's nutrition. For instance, if you are growing cabbage and it's not looking too healthy, place a bunch of the ailing plant in your worm farm. The worms will break it down and create the nutrients the plant is lacking; add their "worm juice" to your sickly cabbage plant and it should improve.
Getting started: When starting a new worm farm, it's a good idea to provide some healthy bedding at the bottom, so that your worms can settle in to their new environment. I recommend placing a few layers of newspaper or a hessian sack on the bottom tray of the farm. Then add the worms, some feed, and a spadeful of your garden soil.
Temperature and positioning: Worms like to be kept in a cool, dark place. If they get too hot (above 98.6°F) they can die. You will also find that their activity will markedly increase during wintertime, but slow down during summer. During the peak of summer, keep your worm farm in a shady spot and regularly tip cold water in to prevent them from drying out. A cold wet hessian sack on top of the farm will also help to keep them cool.
Feeding: Worms like to be fed regularly (at least once a month) and they like to be able to get through their feed at their own pace. If you feed them too much too quickly, they may not be able to keep up and some of your waste may go bad, disrupting the ecology of the farm. Worms love to eat uncooked kitchen waste (fruit and vegetable scraps), tea leaves, coffee grounds, newspaper clippings, and any other organic matter you can drum up. If you can, cut up your scraps into small pieces before adding it to the farm. It will be eaten and broken down much more quickly, preventing it from going bad.
Like compost heaps, worms prefer to get through their food in "layers"; this protects them against extreme temperatures and balances the wet and dry ingredients within the farm. I usually alternate layers of kitchen waste with thin layers of newspaper (one or two sheets should do it). Every now and then I also throw in some basalt rock and perhaps even a little garden soil. Using my intuition, I add food depending on how well they are going. If their activity is high and the worms seem to be thriving, I'll throw in as much food as I can. But if they are struggling, reduced in numbers, or their activity is slow (as in the peak of summer), I'll add only a small amount of food, plus thin layers of newspaper, some soil, and some basalt rock.
Worms cannot eat any meat or oily foods. They also don't like dairy products or acidic foods such as onions, leeks, and citrus fruits.
Troubleshooting: The two main problems people encounter with worm farms are the worms dying out or the worm farm "rotting" and stinking. The first problem occurs if the temperature is too hot for the worms to survive, or if they are not fed appropriate food. The second problem occurs if you feed the worms too much, too quickly. To prevent this, avoid inappropriate foods and measure your worms' food intake, feeding them greater quantities as their population grows. If you encounter fruit flies or bad smells, it might be that the farm has become too acidic. This can be corrected by adding some potash, lime, or dolomite. You will also need to add a few more dry ingredients, such as newspaper or garden soil, to balance things out.
## BOKASHI BUCKETS
Bokashi buckets are small enough to sit on your kitchen bench, usually about 1 foot by 1 foot. They can break down anything—food scraps, meat, dairy products, and vegetable matter. They come with a special type of sawdust that contains active micro-organisms, which break down your waste. You can plant the broken-down scraps in your garden or pot plants to nourish the soil. Bokashis only break down scraps; they don't create the same rich fertilization you get from a compost bin or worm farm. They are, however, a very compact solution, and can take just about anything. I would suggest using a bokashi bucket if you are short on space or time, or as a way to dispose of meat and citrus scraps if you already have a worm farm or compost heap.
Bokashis are at the more expensive end of home recycling, usually selling for eighty or ninety dollars. The bokashi sawdust will set you back about ten dollars per refill. Some of my more innovative home-gardening friends have made their own bokashi buckets by drilling holes in large plastic containers, securing a lid, and buying just the special sawdust—so don't be afraid to experiment.
## GARDEN BURIALS
If you don't have the space to compost or don't have the money to invest in any of the aforementioned composting systems, there is another alternative. Collect all your kitchen scraps, dig a hole, and bury them in your garden. If the hole is deep enough they will not be dug up by animals and they will nourish your soil in the long term. Uncooked kitchen scraps work best for this method. If you are using cooked scraps or meat, you may want to throw in some sawdust or bokashi flakes to ensure they break down properly and don't get stinky. Similarly, if you have pot plants, I would suggest planting your food scraps at the bottom of your pots, adding some sawdust or bokashi flakes, then covering them with a thick layer of soil before you plant your seeds or seedlings.
## GROWING FROM SEED
I'll never forget the first time I discovered a large cantaloupe sprouting uninvited from our veggie patch, or a pumpkin vine unexpectedly edging its way across the lawn. Self-sown seeds spring frequently from the nurturing environments of worm farms and compost heaps, but the joy of witnessing those first surprise buds never ceases to amaze me.
Almost all fruits and vegetables can be grown straight from the seed, and they'll add another dimension of creativity to your backyard veggie patch. In true frugavore style you can throw a rotten tomato into a greenhouse basket and watch it spring tall shoots, or allow the leftover seeds from pumpkin soup to germinate in the warm summer soil and creep up a fence, creating ripe, orange fruit. Or simply plant the seeds in your garden during a warm period (preferably spring) with some worm solids or compost around each seed and let nature take its course. If you buy heritage varieties of seeds you can grow interesting and odd-looking plants such as black tomatoes, purple carrots, and five-color silverbeet. You will also, in your own clever way, be promoting our planet's biodiversity, as well as saving money and reducing waste.
You can grow plants from seed either by sowing them directly into the ground or by planting them in planters in a greenhouse. Direct sowing works best for plants that don't mind cooler temperatures, such as root vegetables. Planters are required for seeds that need warmer temperatures to germinate. Because they are small and shallow, planters tend to warm up faster than the ground, while the greenhouse provides a protective environment free of wind, birds, and dryness.
A mini-greenhouse is a good guarantee that your seeds will grow: it provides the most nurturing environment possible. At large hardware stores you can buy yard-wide greenhouses with three or five shelves for less than fifty dollars. If you don't have the money or space for one of these, you can plant your seeds in recycled polystyrene boxes, which will insulate them from the heat. However if you grow your veggies in planters, try to keep them in a warm, protected area.
## SOME EASY SEEDS TO SOW
PUMPKIN: You can use the seeds from any store-bought organic pumpkin. They germinate easily, except during heavy frosts.
TOMATOES: Seeds from any organic variety of tomato will work. Sow your seeds in containers six to eight weeks before you intend to plant them in your garden. In warm, frost-free areas where soil temperatures are above 59°F seeds can be sown directly into the ground.
MELONS: You can use seeds from homegrown or organic store-bought melons. Melon seeds require lots of heat to germinate. They sprout easily in containers or in warm, well-composted soil.
GARLIC: Replant unused garlic cloves, pointy-end up, about 2 ¾ inches deep in well-composted soil. They do not require containers and can be sown in autumn.
ONIONS: Onion seeds can be sown directly into the garden, but it is generally easier to sow them in containers. Most onion varieties can be sown at any time of the year.
POTATOES: Old, saggy potatoes can be transformed into new potatoes by planting them in your garden bed. Each planted potato should yield ten new potatoes. Plant them in spring, about 4 inches deep, with plenty of fresh manure. They will take about 120 days to produce a new crop. All parts of the potato plant are poisonous except for the tubers (the part we eat), which should have no green tinge.
## WHAT TO PLANT WHEN
Choose vegetables and plants according to your local climate and soil conditions. Here is a general guide to what to plant when:
SUMMER (June through August): tomato, eggplant, peppers, melons, cucumber.
AUTUMN (September through November): silverbeet, greens, beetroot, garlic, celery, fava beans.
WINTER (December through February): potatoes, broccoli, cabbage, fava beans, leek.
SPRING (March through May): string beans, peas, zucchini, lettuce, spring onions, pumpkin.
## BUILDING & MAINTAINING YOUR SOIL
If you want to give your plants the best start, you need to begin with healthy soil ecology. This can be done by mulching, fertilizing, and removing surrounding weeds, as well as consistent upkeep and watering. Don't feel you need to go out and buy a lot of products to achieve this; it's actually a lot simpler than you might think. If you have a compost heap, worm farm, or bokashi bucket, you're off to an excellent start—many gardens get by beautifully with one of these and nothing else. Listed here are some other ways to build and maintain healthy soil.
Each garden is different, and different fertilizers will suit different soils. An easy way to test whether a particular fertilizer is suitable for your soil is to take a teacup of worms and soil from your garden bed and mix in the new fertilizer. If the worms continue to wriggle around at the bottom of the teacup, they probably like the fertilizer. If they don't like it they'll go crazy, trying to get out of the teacup as quickly as possible. This test is especially handy if you're buying commercially sold fertilizer, which may contain traces of chemicals.
Basalt rock: This is a rock mineral sold at most nurseries. It contains important trace minerals necessary for plant growth. It can also be added to compost heaps or worm farms to facilitate growth and fermentation and balance out pH levels.
Blood and bone: This is full of minerals, including slow-release nitrogen, calcium, and phosphorus. A frugal alternative to buying commercial blood and bone is to ask your local butcher for some bone shavings from the back of his workroom floor. This will do the trick just as well.
Dolomite lime: Dolomite lime is a naturally occurring rock mineral that boosts plant growth and alkalizes the soil. Dolomite can be added in moderate amounts throughout the plant's life cycle. Read up on the plants you're growing first, however, as some plants—namely those that prefer an acidic soil, including azaleas and blueberries—don't like it.
Manure: Sheep, chicken, and livestock manure are all useful fertilizers, but they need to be added to your compost heap first so that they can break down and ferment before being added to your soil. You can buy them from nurseries or from the side of the road in rural areas or near stables, usually for two or three dollars per bag. Some plants really thrive on animal products. Rhubard is a classic example—a little manure and you'll have stems as thick and strong as tree stumps. The best time to add animal products to your soil is at the beginning of spring or autumn when you are about to plant new seedlings.
Mulch: This can include pea straw, straw, lucerne hay, autumn leaves, or even newspaper clippings. Mulch provides a protective covering for your plants against extreme temperatures and other climatic conditions. It also adds organic matter to the soil and encourages worms. Pea straw and hay are probably the best choices for their rich organic matter. Autumn leaves collected from your lawn can also be added straight onto your garden bed, or mixed into the compost bin. If you are worried about autumn leaves or other forms of mulch flying away, you can weigh them down with a couple of sheets of newspaper and a brick. Otherwise, just alternate sheets of newspaper with soil from your garden. It won't have the same nutritional properties as straw or garden leaves, but it will provide protection for the worms and encourage them to grow and breed. Mulch is best added during planting time, or at the beginning of extreme temperature periods such as summer or winter.
Mushroom compost: This is the residual waste generated by mushroom farmers. It's a great source of nutrients and is also a good soil conditioner. Only add mushroom compost in moderation as it can raise the soil's pH levels and create too alkaline an environment.
Seaweed: This provides the soil with important minerals, including iodine. When I was growing up, we used to rake this from the side of the beach, but I'm not sure if this is still legal. You can buy seaweed from most nurseries.
Lead testing: One thing I didn't take into consideration when I started my own garden, but which I now understand to be important, is lead testing. If you are establishing a garden for the first time and live in an inner-city area, it can be worthwhile testing your soil for lead levels. If your soil has high levels of lead, you will need to build your garden elevated from the soil or choose large pots to avoid contamination of your produce. Lead-testing kits can be bought from most major hardware stores.
## LONG-TERM MAINTENANCE
Once you've got your vegetable garden established, there is very little you need to do, other than providing your plants with ongoing care and maintenance.
Regular weeding: This shouldn't be a difficult job. Clear out all the weeds surrounding your plants, but be careful not to put them back in your compost or worm heap, as they may sprout and re-grow. We usually leave ours on the footpath to shrivel up in the sun, and only then put them back on the garden bed. Weeding is most important during wintertime, when plants are competing for sunlight.
Regular fertilization: The best time to add nutrients to your soil is just before you plant—i.e., the beginning of autumn or spring, when you are about to plant your new harvest for the upcoming season. After this, a little bit here and there whenever your plants look like they need a bit of love is always a good thing.
Keeping bug-free: There are many different ways to get rid of bugs from your veggie patch. Here are a few I've learned along the way.
Bugs can flourish when the soil becomes too acidic. Throw on some dolomite lime and some compost or fertilizer to boost your plants' immunity.
Let your chickens loose in your veggie patch for a day or two. They'll get rid of any bugs, but you may lose a few vegetables as casualties. Consider it a two-day blitz, after which you can re-plant and recover.
To exterminate slugs, place a ring of salt around the plants they seem to favor. You can also fill a small bowl with beer. The slugs will crawl into the bowl, become intoxicated, and die.
Placing netting over plants can often solve the problem of larger bugs (and also ward off birds).
Protection against heat and cold: Some plants can't cope in extreme climates—be it the heat of summer or the frost of winter. You can provide protection by draping them with a hessian cloth (which can block out sunlight, or insulate against the cold) or bringing them indoors.
Watering and drought-proofing: As our climate is becoming hotter and drier, we need to think of innovative ways to keep plants hydrated and cool. Plants love to be watered regularly. In the heat of summer, vegetables really need a drink at least every couple of days.
Pea straw, straw, or mulch are an excellent investment as they allow the plants to retain moisture around the roots, and hence not require too much watering.
You can also save your shower and bath water using a bucket and tip this over the vegetables after you've washed. Water tanks and recycling systems are also excellent investments.
Lastly, get a citrus tree! If you are a frugavore with a backyard, a citrus tree is an essential water-saving measure. Not only do they look lovely and require only minimal watering, they also provide a staple ingredient for home-cooking. Imagine all those ripe blood-oranges you could grow, or those lustrous lemon trees dripping with fruit. Citrus trees thrive on nitrogen, and the best source of nitrogen is . . . wait for it . . . urine. If you can convince the male members of your household to pee on your tree, you will save a lot of water and get beautiful lemons for your cooking. Just think, if every flush of the toilet uses between ¾ and 1 ½ gallons of water, imagine how much we could save just by peeing on our citrus trees every day. In no time, I'm sure, we'd be the marmalade and lemon-tart capital of the world . . .
## HEIRLOOM ROAST VEGETABLES
I love the heirloom varieties of carrots—red, purple, white, and orange. They take a little longer to cook but are worth the extra wait. You can buy heirloom varieties at some organic shops, but the easiest way to enjoy them is to purchase the seeds and grow them yourself. Of course, plain old orange carrots will also do just fine.
_Preparation time:_ 10 minutes
_Cooking time:_ 45 minutes
_Serves: 4_
_Ingredients:_
Olive or coconut oil for frying
6 to 8 medium carrots 3 onions
¾ tablespoon fresh rosemary
¾ tablespoon fresh thyme
salt and pepper
Preheat your oven to 350°F. Drizzle some of the oil into a baking tray.
Scrub the vegetables and remove the stalks from the carrots. Cut the onions into wedges. Boil the carrots in a little water for about 10 minutes or until they are slightly soft.
Drain the carrots and arrange them in the baking tray. Add the onions and toss through the fresh herbs and seasonings. Drizzle with oil. Bake for 30 to 45 minutes, or until cooked through and crispy.
## HUMBLE BAKED POTATO
Potatoes are one of the most cost-effective vegetables to grow at home. Even when I was renting a house with no garden, I found a large plastic pot, filled it a quarter full with soil, added some old organic potatoes that were beginning to sprout, and watched them grow.
Baked potatoes can be a hearty meal in themselves. Try them with melted cheese, bean salad, sauerkraut, or yogurt and dill. The possibilities are endless, but this is my personal favorite.
_Preparation time:_ 5 minutes
_Cooking time: 50_ minutes
_Serves:_ 2
_Ingredients:_
_2_ large potatoes
2 or 3 medium onions fat for frying
2 slices bacon, cut into bitesized cubes
1 generous handful flat-leaf parsley
¾ tablespoon freshly chopped dill
½ cup plain yogurt
1 tablespoon grated parmesan cheese
fresh butter to serve
Preheat the oven to 390°F.
Wash the potatoes and poke a skewer through their centers a few times. These airways will help the potatoes to cook evenly. Place them in the oven and bake for 50 minutes or so (depending on the size of the potato).
While the potatoes are baking, caramelize your onion. Put 1 teaspoon of fat in a frying pan and place it over low heat. Thinly slice the onions, then add them to the pan and simmer, stirring occasionally, for 25 minutes. They should become sweet and caramel-colored, but shouldn't burn. When they are ready, remove them from the pan and set them aside.
While the pan is still hot, add the bacon and fry it until it lightly browns.
Next, finely chop the parsley and place it in a small bowl.
Finely chop the dill and combine it with the yogurt.
When your spuds are ready, you can layer the toppings however you like. I usually like to add a dollop of butter first, followed by the onions and bacon, then a dollop of yogurt, a spoonful of the parsley, and a sprinkling of parmesan cheese. Do as you see fit!
## BUBBLE & SQUEAK
Bubble and squeak is a traditional English dish made with shallow-fried leftover vegetables. It became popular during World War II as it was an easy way to reuse leftovers during food rationing. In has also been referred to as "bubble and scrape," as it can be made using whatever leftovers you can scrape together. Don't be put off by its wartime origins, though: brussels sprouts, potatoes, and duck fat are a match made in heaven, and you don't have to be on food rations to enjoy it.
_Preparation time:_ 10 minutes
_Cooking time:_ 15 minutes
_Serves:_ 6
_Ingredients:_
1⅓ pounds waxy potatoes
1 medium onion, finely chopped
1 clove of garlic, crushed and sliced
1 small handful sage, finely chopped
1 spoonful of duck or goose fat
2¼ pounds brussels sprouts
1½ cups water
salt and pepper
olive oil
Clean the potatoes and chop them into small rectangles (about ¼ inch by ¾ inch).
Heat a large, heavy saucepan over medium heat. Fry the onion, garlic, sage, and duck fat for a few minutes, then add the potatoes. You may need to reserve some of the duck fat and add it to the pan gradually, to keep the potatoes from sticking. Cook them for about 10 minutes, until they are crispy on the outside but still firm in the middle.
While the potatoes are cooking, remove the outer skins from the brussels sprouts and cut the sprouts into quarters.
When the potatoes are ready, add the sprouts to the pan and stir them through. Ideally, the sprouts should go to the bottom of the pan, where they'll soak up the most liquid, but don't panic if they go everywhere.
Turn up the heat, pour in the water, and secure the lid of the frying pan. Perform the "fry-pan shuffle," shaking the pan so that the sprouts move around a bit and cook evenly. They should take 4 or 5 minutes to cook in the steam from the water. Take care to stir the mixture once or twice to make sure it's cooking evenly and not sticking to the bottom. If there is any excess liquid once the brussels sprouts are cooked, simply remove the lid and let the excess water evaporate.
Season with salt, pepper, and a dash of olive oil.
## WILD GREENS PIE
I became obsessed with spinach pie after traveling through Greece, where horta—their trademark bitter greens—can be found at most restaurants and delis. I am sure this recipe is nothing like the original, but I've tried to capture the things I love about traditional Greek pie as nearly as possible (any Greeks reading this, you can stop laughing now). You can use any garden greens for this pie: silverbeet, spinach, kale, cavolo nero, or the traditional Greek horta. Leafy greens are easy to grow at home, and should be top of the list for any first-time gardener. They are just about foolproof and you should only need to plant them once a season—if you pick at them regularly, they will last you a good part of the year.
_Preparation time:_ 15 minutes
_Cooking time:_ 45 minutes
_Serves:_ 8
_Ingredients:_
2 quantities oatmeal pastry (see the recipe on page 252)
3 large bunches of greens
2 large zucchini
1 head of broccoli 8 ounces feta cheese
3 eggs
1 teaspoon freshly grated nutmeg
juice of 1 lemon
¾ tablespoon olive oil
salt and pepper
Preheat the oven to 350°F.
Prepare your pastry. Use ⅔ of it to line a 9-inch baking tin and bake "blind" for 15 minutes. Remove the "bake blind" filling and return the pastry to the oven for a further 5 minutes, allowing it to lightly crisp. Keep the rest of the pastry aside for later; it will become the lid.
Chop the greens into 1-inch pieces. Dice the zucchini and cut the broccoli into small florets. Sauté these vegetables in batches in a large frying pan with a dash of water, then drain them of any excess fluid.
In a large mixing bowl, combine the feta cheese, two of the eggs, nutmeg, lemon juice, and salt and pepper to taste. Add the cooked vegetables and stir well.
Transfer the filling to your pastry shell and spread it out evenly.
Roll out the remaining pastry to make a lid and place it over the pie. Whisk the remaining egg with a fork and brush it over the lid.
Bake the pie for 35 to 40 minutes, or until the lid is golden brown.
## TOMATO & ONION PIE
This is a terrific last-minute dish, which I've borrowed from my mom. The idea is to use whatever you have on hand, so don't worry if the bread is a little stale or if you have some other variety of cheese in the fridge. Provided you use the ripest, most succulent tomatoes you can find, your pie will be delicious.
_Preparation time:_ 10 minutes
_Cooking time:_ 25 minutes
_Serves:_ 6
_Ingredients:_
4 or 5 medium onions
4 or 5 large, ripe tomatoes
2 slices sourdough bread
1 teaspoon finely chopped fresh thyme or rosemary
1 tablespoon finely chopped parsley
¼ cup grated parmesan cheese
¼ cup grated tasty cheese
¾ tablespoon olive oil
salt and pepper
Preheat the oven to 350°F.
Thinly slice the onions and place them in the bottom of a deep baking dish. You don't need to add any fat; just pop them in the oven and bake for 10 to 15 minutes while you prepare the remaining ingredients.
While the onions cook, thinly slice the tomatoes. Rip the bread into tiny breadcrumbs, or whir it in the food processor for a minute or two. In a small bowl, combine the breadcrumbs with the fresh herbs, cheese, olive oil, and salt and pepper to taste.
Remove the baking dish from the oven and layer the tomatoes over the onions. Sprinkle the bread and cheese mixture over the top, then return the dish to the oven and cook for a further 30 minutes, or until the tomatoes are soft and the cheese is lightly browned and crispy.
## CAULIFLOWERS WITH BACON
I used to think of cauliflower as a rather dull vegetable, but exploring different varieties made me realize how wrong I was. Heirloom cauliflowers are a real treat—on a recent trip to a local farmers' market I came home with a purple, white and yellow beauty. Cauliflower goes deliciously with bacon and this is a very easy and impressive side dish, especially if you can find a cauliflower in an unusual color. If you don't have access to heirloom varieties, use the standard white one; it's still delicious when cooked this way.
_Preparation time:_ 5 minutes
_Cooking time:_ 10 minutes
_Serves:_ 6 (as a side dish)
_Ingredients:_
1 large cauliflower
2 slices of bacon, finely chopped
1 small handful flat-leaf parsley, finely chopped
juice of ½ lemon
sea salt
olive oil
Cut the cauliflower into quarters and boil or steam it for 5 minutes, or until lightly cooked and "al dente." Remove it from the heat and drain it of any liquid. When it has cooled a little, cut the cauliflower into bite-sized florets.
Heat a frying pan over medium-to-high heat and add the bacon. Allow it to brown, turning it once or twice. When the bacon has browned, add the cauliflower and stir well so that it is coated in the juices from the meat.
Transfer the cauliflower and bacon to a serving bowl. Add the parsley, lemon juice, salt to taste, and a little olive oil. Toss well and serve immediately.
## GARDEN SALAD
With a few pots of this and that or a small garden bed, a fresh salad is never far away. When you're growing your own produce, salads are a magnificent way to celebrate every twist, turn, and oddity in the plants from your garden.
_Preparation time:_ 5 minutes
_Cooking time:_ 30 minutes
_serves:_ 6
_Ingredients:_
fat for frying
1 plump beetroot sea salt
2 large heads of lettuce
2 large fennel
2 preserved artichoke hearts
1 large carrot
1 teaspoon white-wine or apple-cider vinegar
1½ tablespoons olive oil
Preheat your oven to 350°F. Lightly grease a baking tray with a little cooking fat.
Peel the beetroot and cut it into bite-sized cubes. Place them on the baking tray and sprinkle with sea salt. Bake in the preheated oven for 25 to 30 minutes, or until golden and crispy.
Meanwhile, shred the lettuce into a large salad bowl. Finely slice the fennel, cut the artichokes into small slivers, and roughly grate the carrot. Add these to the lettuce, then toss through the vinegar and olive oil.
Remove the beetroot from the oven and allow it to cool to room temperature. When it has cooled, arrange it on top of the salad and serve.
# THE CHICKEN & THE EGG
"Regard it as just as desirable to build a chicken house as to build a cathedral."
—Frank Lloyd Wright
I WOULD LOVE TO SEE A DAY WHEN EVERY household with its own backyard has a few chickens running loose, every apartment block has its own henhouse, and every farm has fresh eggs available for its neighbors. Chickens can play an important role in your garden's ecosystem: plants thrive when fertilized by hen droppings. Hens also provide excellent eggs and meat and are easy-to-handle pets, the perfect choice for inner-city dwellers.
But you don't have room? Don't think you could give them a good life? Well, think again. Consider the life of a commercially farmed chicken. They are given unnatural feed, little room to run around, and a shortened lifespan. I am sure any hen would prefer even a small backyard with fresh food and room to move to a lifetime in a small metal cage under UV lights.
Fresh eggs from healthy hens are highly nutritious. The healthiest eggs come fresh from pasture-raised chickens that have been allowed to run around. Their droppings and henhouse straw are excellent fertilizers for the garden, and they can recycle your kitchen scraps, pick at your lawn, and rid your garden of bugs and grubs. What's more, they are gorgeous and hilarious creatures that will add character to any backyard or veggie patch.
Chickens are also wonderful pets for children. Some of my most precious childhood memories revolve around the chickens we kept in our backyard. I remember racing around the yard, and sometimes even through the house, determined to catch a hen or pat at one of our roosters' tails.
My chicken memories aren't entirely happy-go-lucky. We had one rooster that got so big and full of himself that he attacked my sister and ripped a large hole in her jeans. We were all petrified to go out to the backyard after that, but my clever mother somehow smuggled him out of the house in a box, took him to the Chinese woman up the street, and we all enjoyed a hearty meal of chicken soup the following evening. Mom was thrifty enough not to be too sentimental about his demise. He'd had a good life and he came to a good end. My memory of what he looked like is a little vague, but I will never forget that soup—it was second to none.
But with the exception of the odd rowdy rooster, chickens make wonderful pets and often bond quite closely with humans. The year I broke up with a boyfriend, I moved to the outer suburbs of Melbourne and lived by myself with nothing but a dog and a hen to keep me company. It was a depressing year to say the least, but this darling bird—whom I'd bought from a nearby battery farm for less than the price of a hamburger—became one of my closest companions. She'd lay everyday—usually on the front doorstep, but sometimes in the house (she came in via the dog door, the clever thing). She particularly liked the cushions on the sofa, and was known to lay one beautiful egg in the center of the featherdown couch while I was out for the day. Her eggs were incredibly good, and they provided me with a wonderful sense of wholesomeness that I desperately needed at the time.
One day, while the front gate was open, poor little Egna ventured a little too far over our front fence and was bitten by the neighbor's dog. She had a broken wing and a pierced ribcage, and I knew there was only one option. I sped to the local vet, talking to her all the way, and had Egna put down. Had I been a bit more frugal, of course, I would have opted for the traditional nip and twist method used by farmers to finish off their hens. But I couldn't bear to. Hens are special creatures. So you've been warned: you may get attached. Despite her untimely death, Egna had six good months living with me—probably the best six months of her life. So despite my sadness at her passing, I was also very pleased that she had that period of freedom. Would her life have been any better or longer if I'd left her at the battery farm? I doubt it.
## A GOOD EGG
The benefits of having your own hens go way beyond their beauty and friendship; they also produce glorious eggs and wonderful meat.
The best eggs come from hens that have had an omnivorous diet—lots of fresh greens, occasional kitchen scraps, and minimal grains—along with plenty of exercise. This is akin to their natural diet. Commercial chickens, however, are reared to produce more eggs and to yield meat as quickly as possible. They are fed an unnatural diet of grains, corn, or soymeal. Consequently, the chickens grow faster, and they produce more eggs—but the nutritional profile of their eggs and meat is different. Chickens fed a traditional diet and given room to run around develop healthier muscle meat. Their eggs and meat contain greater quantities of vitamins A, D, and E, as well as more omega-3 fatty acids, which are known to help prevent coronary-artery disease, hypertension, arthritis, cancer, and other inflammatory and autoimmune disorders. The yolks of their eggs will be a brighter orange and contain a higher level of health-promoting carotenoids. Their bones and cartilage contain more minerals, and they are not fed antibiotics or growth hormones.
By keeping your own hens or sourcing locally produced eggs, you can see what their diet and lifestyle are like, which is the best guarantee of good eggs. It will also mean that your eggs are fresh. I can't quite describe to you the beauty of fresh eggs. They are lighter, softer, and easier to poach. If you can't keep your own chickens, look for a local community garden that keeps chickens and has fresh eggs for sale.
If you can only buy eggs from a retail outlet, look for free-range _and_ organic. There are sometimes loopholes in our systems of chicken certification. In some cases, hens can be certified as organic, but be fed a diet predominantly of grains. They can be labeled free-range but have relatively little room to run around in. So if you look for both certification ticks, that is your best assurance of quality.
Raising your own chickens or sourcing them from a local farm also gives you access to quality meat, and to all the traditional bits. Chicken feet are considered the magic ingredient in chicken soup (also known as "Jewish penicillin"). They are incredibly cost-effective—one bag of chicken feet can make a large pot of wholesome and nourishing stock. There's also a certain feel-good factor in knowing that nothing has been wasted from the rooster or hen that ended up on the chopping block.
## CHOOSING YOUR CHICKENS
Chickens come in all shapes, sizes, temperaments, and laying capacities. You can buy them from markets, wholesalers, or even at roadside stalls. Places that breed battery hens will often let you purchase their breeding hens for next to nothing when they are finished with them. Sometimes they give them away for free. These hens will be tame and quiet. You will get the satisfaction of giving them a second life and watching their personalities develop, as they go from being shy and timid to friendly and even a little boisterous.
If you are living in an inner-city or suburban area where noise is a problem, it is probably easiest to buy one of the more domesticated breeds such as Isa Brown or Australorp. These lay well, are very tame, and reasonably quiet. Silky Bantams (commonly referred to as "fluffy-bums") are also very good-natured, but tend to lay only a few eggs per week. Their eggs are well worth it, however—they are smaller than normal eggs and exquisitely delicious.
If you have a little more room, and more tolerant neighbors, you might try some of the heritage varieties of chickens such as Light Sussex, Leghorns, or Rhode Island Reds. The list is endless, and we should be supporting more of these rare breeds to keep diversity and flavor in our chicken market. They are a bit more expensive (you could be looking at forty or fifty dollars per hen) and their personalities may be less domesticated. But they are beautiful to look at, and you might just find yourself mooning over them most mornings, watching them rummage around your backyard.
## WHAT YOUR HENS WILL NEED
First, you'll need a secure henhouse that is impenetrable by foxes. If your chickens are out and about everyday, their henhouse doesn't need to be very big. For a while, when living in the inner-city, we had a henhouse that was 3 feet by 20 inches for two friendly Isa Brown hens. This was all the space they needed, as they only used the henhouse at night. You can buy henhouses online, or convert an old play house or shed. Hens like to be elevated, so if your henhouse isn't high off the ground, be sure it includes an elevated perch. This makes them feel protected from predators. They'll also need somewhere to lay, either nesting boxes or a pile of straw in their henhouse, and somewhere to shelter in extreme heat. You can insulate their henhouse with vines and straw. Chickens like to bury themselves in the dirt to keep cool, so let them do this.
Chickens need a constant supply of fresh water. Hens like their water cool (no warmer than 40°F) and they prefer a dripping tap or some other source of running water. It makes them feel like they are drinking from a stream rather than a stagnant pond. Adding a few drops of apple-cider vinegar to their drinking water every now and then will help to keep their digestive systems clean and free of parasites.
As omnivores, chickens like a diverse supply of food on demand. The easiest way to provide this is with a chicken feeder stocked with grains, supplemented by kitchen scraps. Chickens love their food fresh, especially their greens. When you are pruning your silverbeet or clipping your lettuces, be kind and throw some to your hens.
Clipping your chickens' wings (the feathers, obviously—not the bone!) will stop them from escaping. You need only clip one wing, and it isn't painful for them.
Very occasionally, you may find that your chickens have started pecking at their own eggs. This must be stopped immediately or they will make it a habit. You can buy fake eggs at pet shops. Put some of these in your henhouse and gather the real eggs as soon as they lay them. Within a few days they should lose interest.
## CHICKEN FEED
Chickens love kitchen scraps, so this should be one of the first things you feed them. Chickens are omnivores (not vegetarians, as many people believe), so they require a varied diet of vegetables, grasses, worms, and some grains. They are also clever. They will pick through your kitchen scraps and take what they like. Very rarely will they eat anything that would make them ill. A tiny bit of meat in their diet is a good thing, but only feed it to them in moderation. In a natural environment they are always hunting for worms and snails, as these are the most nutritious food source. But in the wild, these are only available in small numbers, so form only a tiny part of their diet.
In addition to kitchen scraps, you should always have grains available for your hens, so that they can access food on demand. Various chicken feeders are available on the market, many of which are rat-proof and pigeon-proof. Wheat is the most commonly available chicken feed, but you can also feed them oats, maize, and sunflower seeds.
"Shellgrit," the leftover sea shells that wash up on the beach, is also essential in your chickens' diet, as it provides an important source of calcium to keep their bones and egg shells strong. You can collect it from the beach or purchase it from an animal feed store.
If you are buying chickens straight from a battery farm, you might find that they will only eat pellet feed for the first couple of days. Buy some of this pellet feed so that they can ease into their new diet. It may take them a while, but with time they should adjust to a natural diet of grasses, worms, and grubs.
## SCRATCH THAT
Chickens love to scratch, and they love to peck and mow at grasses. They will keep your lawn nicely trimmed and eat any old lettuces and greens you have springing loose. However, be warned that if you let them loose in your vegetable patch they will get rid of any grubs and weeds, but they may also destroy some of your plants with their reckless pecking. Just be prepared!
An ideal set-up for chickens is to have them in a movable pen or within temporary fencing. That way you can rotate their position, moving them around your backyard so that they always have access to fresh grubs, dirt, and grasses. Another clever idea is to grow hardy fresh greens such as silverbeet or spinach around the outside of their henhouse. They'll always have a source of fresh greens, which they can peck at easily every day (ensuring very nutritious eggs), but they won't be able to destroy the whole plant.
## TO ROOSTER OR NOT TO ROOSTER?
If you have the space, and the right council regulations, roosters are a huge amount of fun. They strut around the backyard, cavort with the hens, and make plenty of noise in the morning. But if you live anywhere in the inner city, roosters will not be a good idea. They do crow at sunrise and no amount of coaxing can prevent this (putting them in a small box at night can stop it, but I wouldn't advise this as a long-term solution!). In a farm environment, roosters protect the hens from predators during the daytime (nighttime is another story—they are useless) and establish a pecking order among the flock. They also make excellent chicken soup.
## BEYOND THE CHICKENS: DUCKS, QUAIL, & GEESE
Chickens are the best food-providing animal for suburban environments. But if you have a bit more space and more distance from your neighbors, you can think beyond chickens. If you have room for a pond, you could consider getting a few ducks or geese. If you have room in your garden for a large cage, you could also consider getting some quail. Like chickens, ducks and quail provide wholesome, healthy eggs, as well as fertilizer for the garden. The drawback is that they can make more noise, and potentially a lot more mess.
## SUPERBLY SCRAMBLED EGGS
For the best results, eggs need gentle cooking. This method of scrambling only _lightly_ cooks them, so that they stay light and fluffy. They go deliciously well with caramelized onion and some crunchy sourdough toast.
_Preparation time:_ 5 minutes
_Cooking time:_ 12 minutes
_Serves:_ 1
_Ingredients:_
1 small onion
Olive or coconut oil for frying
3 eggs
salt and pepper
Finely chop the onion. Put a little oil in a frying pan and place it over low heat. Cook the onion for 5 to 10 minutes, stirring intermittently, until it lightly caramelizes but does not burn.
While the onion cooks, whisk the eggs with a fork in a small bowl. Add salt and pepper to taste.
When the onion is done, transfer it to the bowl containing the eggs and wipe the frying pan clean.
Increase the temperature and add a small dollop of fresh oil to the pan. Pour in the egg and onion mixture.
Now it's time to scramble. The ideal technique is to constantly fold the egg mixture from the bottom of the pan to the top. This will keep it from overcooking and the middle section will stay beautifully soft and fluffy. Do this constantly for 1 or 2 minutes, or until the eggs are cooked to your liking.
Serve with some crunchy toast.
## EGG MAYONNAISE
When I was growing up, egg mayonnaise was the ultimate "waste nothing" dish. Mom and I used to team up: I would make meringues using the egg whites and she would make mayonnaise using the yolks. Unfortunately, as a result, I was always hopeless at making mayonnaise, and she was rather bad at making meringues. This is therefore an extremely easy recipe with very few ingredients. If you have your own birds to provide the yolks, this mayonnaise is as cheap as chickens (excuse the pun).
_Preparation time:_ 4 minutes
_Cooking time:_ none
_Serves:_ 6 (as a condiment)
_Ingredients:_
2 egg yolks
1 or 2 teaspoons apple-cider or wine vinegar
1 teaspoon honey
½ teaspoon mustard
juice of half a lemon
¾ cup olive oil
salt and pepper
to season, finely chopped fresh dill or 1 teaspoon Dijon mustard (optional)
_Tip:_
If your mayonnaise curdles during preparation, don't despair: it can be saved! In a clean bowl, work an extra egg yolk into a smooth paste. Slowly add the failed mayonnaise, whisking well after each spoonful.
Place the egg yolks in a medium bowl and whisk well or beat with an electric beater.
Gradually, in a very thin stream, add the olive oil, whisking all the time. The whisking will be easier if the bowl is held at a slight angle, so you may want to make this a two-person job; one person can hold the bowl and whisk while the other gradually adds the oil in a steady stream. Or, if you're working solo, try this trick: layer two tea towels on the kitchen counter and prop the bowl against them, so that it sits at an angle. Your second hand will be free to pour the olive oil.
Once all the oil has been added, gradually add the remaining ingredients, one at a time. Season to taste. Test the mayonnaise for acidity. If necessary, adjust by adding drops of lemon juice, salt, and pepper.
_Variations:_
AIOLI: Add 2 small cloves of crushed garlic to the egg yolks before you start whisking. Aioli is delicious drizzled on baked potatoes or with boiled eggs and salad greens.
SMOKED PAPRIKA MAYONNAISE: For every cup of mayonnaise, add 1 teaspoon of smoked paprika and a squeeze of lime juice at the end. This works well with poached eggs and bacon, or in a bacon and lettuce sandwich.
## EGG & GREENS PIE
This pie is the perfect way to show off the latest produce from your backyard—against the green of the silverbeet, the orange egg yolks stand out like traffic lights. It's also a wonderfully simple dish, ideal for Sunday brunch or even, made the night before, for a weekday lunch. Whenever you serve it, it will definitely give you something to talk (or crow) about.
_Preparation time:_ 20 minutes
_Cooking time:_ 30 minutes
_Serves:_ 4
_Ingredients:_
1 quantity of oatmeal pastry (see recipe on page 252)
1 small onion olive or coconut oil for frying
2 or 3 large silverbeet leaves, stems removed
7 eggs
salt and pepper
_Variation:_
A few slices of ham or prosciutto go beautifully with the combination of silverbeet and onion.
Preheat the oven to 350°F and grease a 10-inch pie tin.
Use your fingers to roll out the pastry into the pie tin. Bake the pastry blind (see note on page 253) in the oven for 15 minutes, or until the sides turn golden brown. Remove the bake-blind filling and return the pastry to the oven for another 5 minutes, just enough to make the bottom nice and crispy.
While the pastry bakes, finely chop the onion and gently cook it over low heat with a little oil. Let it simmer, stirring intermittently, for 5 to 10 minutes, or until it sweetens and turns golden. Don't let it brown.
Finely chop the silverbeet. Add it to the onion and let it simmer until it wilts. This usually takes only a minute or two. When the silverbeet has wilted, transfer the vegetables to your pie crust.
Carefully break each egg into a glass and pour them onto the pie one by one (you don't want any broken yolks, as they won't look as striking). The recipe calls for 7 eggs, but for just the right balance of colors and flavors I usually use 5 whole eggs and 2 egg yolks, saving the 2 extra whites for other cooking.
Bake the pie for 20 minutes or until the eggs are set but still a little moist.
## OMELETS
With all those chickens running around, you are going to have to think of creative ways to use up all the eggs! With a few fresh salad leaves and some crusty bread, a delicious, nourishing omelet can be ready in minutes. Omelets are a great way to celebrate whatever is currently fresh, whether it's a ripe zucchini or a tasty potato.
Omelets should be a special, simple delicacy—but things can go wrong. If the eggs overcook, they will be dry and tough. The best omelets strike a balance, so that the bottom is crispy but the middle and top are still delicately soft. You can achieve this by pan-frying first for a few minutes, then oven-baking. To do this you'll need a frying pan with an ovenproof handle, preferably one with a heavy base. I use a cast-iron Le Creuset pan that I bought from a thrift store—it dishes out superb, foolproof omelets. But don't worry if you haven't found your thrift-store Le Creuset yet: a basic stainless steel pan will be just fine. A small pan (about 8 inches in diameter) is best. If you're using a larger pan, you may need to adjust the quantities. The omelets described here are about an inch thick, crisp on the bottom, and soft on top.
## ZUCCHINI & BASIL OMELET
Preheat the oven to its highest setting.
_Preparation time:_ 5 minutes
_Cooking time:_ 10 minutes
_Serves:_ 2
_Ingredients:_
1 small zucchini
olive or coconut oil for frying
3 eggs
1 handful of fresh basil, chopped
salt and pepper
Slice the zucchini into thin rounds. Over medium heat, heat a teaspoon of oil and fry the zucchini until it is just tender.
Whisk the eggs, then add the basil and salt and pepper to taste.
When the zucchini is ready, transfer it to the egg mixture. Wipe any crumbs from the pan and return it to the stove and turn up the heat.
Add a small dollop of oil to the pan, then pour in the egg mixture and cook for 2 to 3 minutes, allowing the edges of the omelet to curl up.
Remove the pan from the stove and place it in the oven. Cook for 5 to 10 minutes, keeping a watchful eye on it to ensure that the eggs do not overcook. Remove the pan from the oven when the top of the omelet is set but still soft; it should be moist but not too runny.
## POTATO & NUTMEG OMELET
Preheat the oven to its highest setting.
_Preparation time:_ 5 minutes
_Cooking time:_ 10 minutes
_Serves:_ 2
_Ingredients:_
2 or 3 small potatoes fat for frying
4 eggs
salt and pepper
¼ teaspoon freshly ground nutmeg
¼ cup sour cream
Wash the potatoes and chop them up into small cubes (about ⅓ or ¾ inch wide). Heat your frying pan over a medium heat. Add the fat, then fry the potatoes for 3 to 5 minutes or until they are lightly browned and cooked all the way through.
Whisk the eggs in a bowl with the salt, pepper, nutmeg, and sour cream.
Pour the cooked potatoes into the egg mixture. Wipe out the frying pan and return it to the heat. Let it heat up for about 30 seconds, then add some more fat.
Pour the egg and potato mixture into the pan, making sure the potatoes are evenly dispersed. Let the bottom and sides get crispy and the edges curl up.
Transfer the pan to the oven and bake for 5 to 10 minutes or until the top of the omelet is cooked to your liking.
## LEEK & SOUR CREAM OMELET
Preheat the oven to its highest setting.
_Preparation time:_ 5 minutes
_Cooking time:_ 10 minutes
_Serves: 2_
_Ingredients:_
1 medium leek fat for frying
4 eggs
¼ cup sour cream
salt and pepper
Wash and finely slice the leek. Add some fat to the frying pan and sauté the leek over low heat for 5 minutes. It should soften but not brown.
Whisk the eggs with the sour cream and add salt and pepper to taste.
When the leeks are ready, add them to the eggs. Wipe the frying pan clean and return it to the heat. Let it heat up for about 30 seconds, then add some fresh fat and pour in the omelet mixture. Cook for 3 to 4 minutes or until the omelet curls up at the edges and lightly browns.
Remove the pan from the heat and transfer it to the oven. Cook for 5 to 10 minutes or until the top of the omelet is just cooked (or perhaps a little gooey, if you prefer).
## POULTRY BASICS
Cooking with the whole bird is so much more economical than buying individual breast and thigh fillets. Not only do you get more meat for your money, you also get all the extra bits: gelatinous bones and feet for stock, and delicious poultry fat for home-cooking. These are some of the most nutritious parts of the bird, and are very handy to have on hand. To get the most from your bird, here are a few tricks you should have up your sleeve.
Slaughter: All this talk of chickens and cooking brings me to a tricky question: how to finish off your hen or rooster when the time comes. A generation or two ago, people slaughtered their chickens themselves or enlisted a neighbor. We were lucky when I was growing up; the infamous Chinese lady three doors down did the whole job—axe, pluck, and gut, all for a very small fee. But where is she now? Long gone, I suspect.
To finish off a bird, you can opt for either a traditional twist and pull or an axe and chopping block (the latter can be less confusing for the inexperienced chicken-slaughterer). It may sound confronting, but this process needn't be beyond suburban gardeners.
You'll find that your home-reared birds make wonderful chicken soup. I prefer to use older hens rather than the plumped-up youngsters, as they give the soup much more depth and flavor, and you have the satisfaction of knowing that the birds had long and happy lives.
Jointing a bird: Jointing a bird involves cutting up a whole bird into portions suitable for a crock-pot or casserole. Don't worry: it's easy! You'll need a sharp knife, a clean cutting board, and three bowls at the ready. As you work, use one bowl for the meat, one for the fat, and the other for bones and offcuts, which you can use for stock.
To start, place the bird on the cutting board, breast side up. If you are jointing a duck or goose, they have quite a lot of fat around the neck and bottom areas. Trim this off and place it in the fats bowl. Chickens have relatively little fat, but it's still worth keeping whatever fat you can find.
To start jointing your bird, first find the wing and cut off the wing tip. Put it in the offcuts bowl. If the bird still has its feet (this is unlikely; most butchers remove them), cut them off and add them to the offcuts bowl too.
Next, have a look for the neck. Your butcher may have removed it, but if it's still on the carcass you will need to remove it with the knife at the lowest point you can reach.
Now make a cut in the skin between one of the legs and the bird's body. Pull the leg towards you so that the thigh bone pops completely out of its socket. Cut the leg at the joint and remove it, placing it in the meat bowl. Repeat this process with the other leg, and then with the wings, popping the joints from the sockets before cutting them away from the carcass.
Slice along the backbone and remove the breasts, placing them in the meat bowl.
Lastly, trim any remaining meat from the carcass and place it in the meat bowl.
Congratulations: you've jointed a bird! You now have a whole animal's worth of meat, fat for frying or pastry-making, and a supply of bones and offcuts for stock.
If you're squeamish about jointing a chicken, you can still enjoy the benefits of purchasing a whole bird. Just ask your butcher if he's willing to "joint" the chicken for you. The hard work of snapping and pulling the bones will be done, and you'll just need to remove the meat.
To render the fat from your bird, see the recipes in the "Fats" chapter.
Giblets and gizzards: Giblets include the internal organs—the heart, liver, and any other thrifty bits. The gizzards include the stomach and intestines. The liver is best eaten immediately, pan-fried with some fresh herbs and enjoyed on toast. Alternatively it can be turned into pâté. Everything else can be added to the stockpot or tossed with some herbs and breadcrumbs and used as a stuffing if you are making a roast.
Reusing the bones from a bird after a meal: When you've finished a meal of roast poultry, gather the bones—and I mean all the bones. It doesn't matter if people have held them in their hands or chewed on them; retain everything you can. The bones will be boiled at a very high temperature, so any bacteria will be killed off.
Put the bones in a stockpot, cover them with cold water, and add the head, neck, or wings if you have them. Throw in any vegetables or herbs you have on hand and make the stock according to the "Poultry Stock" recipe on page 128.
If you don't have time to make the stock right away, throw the carcass, plus any liquid that has accumulated at the bottom of the pan, the neck, and any other bits and pieces in the freezer, and make the stock whenever the mood takes you.
## CHICKEN SOUP
This dish is a friend to come home to, a wholesome meal and a dose of Jewish penicillin all in one. One whole chicken can feed six people generously and will fill your kitchen with warmth and nourishment. Wherever possible, try to include the chicken feet when you make the stock. They add extra gelatin, which, according to folklore, is what gives chicken soup its famed medicinal powers. I think it's a real shame that when we buy a "whole chicken" from the butcher or supermarket we seldom get more than the body and the bones. The head and the feet are well worth using if you can get your hands on them.
_Preparation time:_ 10 minutes
_Cooking time:_ At least 1 hour
_Serves:_ 4 (generously)
_Ingredients:_
1 whole chicken (including the head and feet if possible)
⅛ cup apple-cider or wine vinegar
4 medium carrots or parsnips (or a combination)
4 sticks celery, including the leaves
1 large leek
1 onion
1 heaped teaspoon black peppercorns
1 stick kombu seaweed (optional)
1 small handful fresh thyme and/or fresh rosemary
2 cups additional chicken stock (only if you opt for the shorter cooking time)
Place the chicken and the vinegar in a large stockpot. Fill the pot with enough cold water to cover the bird. Bring the water to a boil, then reduce to a simmer. Chop up 2 of the carrots and 2 of the celery sticks and add them to the pot along with the leek, onion, peppercorns, kombu, and herbs. Simmer gently, partially covered, for 45 minutes or until the meat appears to be loosening from the bones.
Remove the chicken carcass from the pot, but leave the vegetables and any loose bones or feet to gently simmer.
Using your hands, remove the meat from the carcass. Drain the meat of any excess fluid and place it in a small bowl. Sprinkle a layer of sea salt over the top and splash the chicken with olive oil. Put the bowl in the refrigerator.
Return the carcass to the cooking pot. You can make the soup now and it will be ready in 10 minutes. Or, you can let the stock continue to simmer, which will give it more richness and flavor. I sometimes let mine simmer for another hour or even, if I'm not in a hurry to use it, overnight. If you are going to make the soup right away, it can be a good idea to add some extra chicken stock for extra depth of flavor, but this is not essential.
When you're ready to make the soup, finely chop the remaining carrots and celery and the leek. Sauté them with a little cooking fat for 3 to 5 minutes or until they are lightly browned.
Transfer a few ladlefuls of stock from the stockpot into the saucepan with the vegetables. Simmer for a further 5 minutes, or until the vegetables are cooked.
Remove the chicken meat from the fridge and shred it into individual serving bowls. Ladle spoonfuls of the soup into each bowl and garnish with a generous drizzle of olive oil.
There is usually some leftover liquid, sometimes as much as 2 quarts of useful cooking stock. This can be stored in the fridge or freezer for later use.
## FRUGAL ROAST CHICKEN
Whether you buy your hen from your local butcher, a farmer, or the supermarket, as a frugavore you'll want to make the most of every last morsel. This classic roast chicken makes use of every last scrap, from the wings to the feet.
_Preparation time:_ 10 minutes
_Cooking time:_ 1 hour
_Serves:_ 4 to 6 (depending on the size of the chicken)
_Ingredients:_
1 whole chicken
2 teaspoons butter
1 small bunch thyme, finely chopped
1 small bunch rosemary, finely chopped
1 lemon
1 onion
½ cup white wine (optional)
Preheat the oven to 350°F.
If you have a whole chicken with head, neck, and feet still attached, chop these off and set them aside for stock-making. To remove the head, cut through the base of the neck where it reaches the top of the breast. If your chicken is already headless, you'll still need to remove the neck (it doesn't taste terribly good in a roast, so keep it for stock-making). Remove the "wingettes"—the end joint of the wings—by cutting at the joint and snapping them off (these can be kept for stockmaking, too).
Using your fingertips, poke the butter and a few snippets of the herbs into the space between the breast flesh and the skin.
Cut the lemon in half and squeeze the juice over the whole bird. Rub the skin down with the remaining chopped herbs. Coarsely chop the onion and place half of it in the cavity of the bird. Save the other half for stock-making, or for use in a side dish.
Put your bird, breast-side up, into a baking dish with a tight-fitting lid. The pot should be big enough to fit the bird snugly, without squashing it at the top or the sides; the lid will help to keep the meat moist. Place the dish, uncovered, in the oven and roast for 15 to 20 minutes or until the bird is nicely browned.
Remove the dish from the oven. Add the white wine if you are using it, plus enough boiling water so the dish is filled ¾ to ¼ inches deep, then put the lid on. Return the dish to the oven and cook for a further 45 minutes. To test whether the bird is ready, poke one of the drumsticks with a skewer. When the bird is ready the juices will be clear, not pink.
Don't forget that at the end of the meal, the scraps, carcass, and offcuts can all be used for stock, while any excess fat can be rendered to make cooking fat.
## ROAST DUCK WITH ORANGE & SAGE
Duck has long been prized for its rich, dark flesh and delicious fat. Although ducks are not cheap, they can still be economical, as a single duck contains ingredients that can be put to good use in multiple dishes. Duck fat is perfect for cake-making and frying, and duck bones make a beautiful stock (lentil soup with duck stock is a favorite at our house).
_Preparation time:_ 15 minutes
_Cooking time:_ 1 hour and 20 minutes
_Serves:_ 5 (approximately, depending on the size of the duck)
_Ingredients:_
1 whole duck
1 slice sourdough bread
1 medium onion
1 small handful fresh sage, finely chopped
1 small handful fresh thyme, finely chopped
olive oil
2 oranges
_Note:_
To prepare a side dish of vegetables, arrange an assortment of pumpkins, carrots, and potatoes in a separate baking tray. Cut off ¾ teaspoon of fat from the rear of the bird and add this to the tray along with some fresh sage and thyme. Bake in the oven alongside the duck for 50 to 60 minutes or until golden and crispy.
Preheat your oven to 425°F.
Take a look at your bird. If your duck has come with a head, neck, or feet, you need to remove these by snapping them and then cutting them off with a sharp knife. Don't throw them away; stash them in your fridge or freezer for the next time you make stock. Trim off any excess fat from the neck of the duck. Use a spoonful for roasting vegetables (see side note) or save it to be rendered for cooking fat.
Rub the duck down with salt and pepper to taste. To make a thrifty stuffing, cut a few slices of onion and orange and place these in the cavity of the duck. Some breadcrumbs and herbs also work well. If your duck has come with giblets or gizzards, you can finely chop them and add them to the stuffing too.
Truss the duck by tying together the legs with a piece of string to keep them close to the body. With a sharp-tipped knife or fine needle, carefully poke a few holes on the surface of the duck to allow the fat to permeate throughout the skin. Squeeze half of one orange over the duck and rub thoroughly.
Place the duck on its side in a baking dish. If necessary, secure it in place by chopping an onion in half and placing the pieces on either side of the bird. Roast the duck for 20 minutes, then turn it over and cook for a further 20 minutes on the other side.
Remove the tray from the oven and turn the duck breastup. Slice the orange very thinly and place the slices on the up-turned breast of the bird. Reduce the oven temperature to 400°F and return the duck to the oven. Cook for a further 20 to 30 minutes or until the skin is crispy. Duck is properly cooked when the temperature of the meat in the thickest part of the thigh or breast reaches 165°F (this can be checked with a meat thermometer). Alternatively, just look for crispy skin and pink juices from the thickest portion of meat. When the duck is ready, transfer it to a serving tray and enjoy with roasted vegetables.
## SPANISH-STYLE CHICKEN CASSEROLE
This recipe uses up all the bits—the bones can be set aside for stock, the meat goes into the casserole, and the fat can be rendered for later cooking. It's also an excellent way to use up any extra tomatoes that are lurking in the back of the fridge (soft and squishy is fine for this recipe). Serve with roasted potatoes or toasted polenta and a fresh garden salad.
_Preparation time:_ 5 minutes
_Cooking time:_ 1 hour
_Serves:_ 4 (depending on the size of the chicken)
_Ingredients:_
1 large free-range chicken rind of 1 lemon
2 cups soft tomatoes (or 1 can preserved tomatoes)
1 large onion, finely chopped
2 cups stock
2 bay leaves
½ teaspoon paprika
¼ teaspoon cayenne pepper
6 large cloves garlic, crushed fat for frying
½ cup pitted olives
Preheat the oven to 350°F.
If your chicken has come with feet, a neck, or "wingettes" (the bone at the end of the wing), remove these and set them aside to use in stock. Trim off any excess fat and save it to be rendered into cooking fat. Joint the chicken, cut the meat into casserole-sized chunks, and put the carcass aside for stock.
Have ready a heavy-based cooking pot with a tight-fitting lid. Peel a long piece of lemon rind and add this to the pot along with the tomatoes, onion, stock, bay leaves, paprika, cayenne pepper, and garlic. Place the pot on the stove and bring the stock to a gentle simmer.
In a frying pan over a high heat, fry each piece of chicken with a little cooking fat for about a minute on each side to seal in the flavor, then transfer them to the cooking pot, ensuring that there is enough liquid to cover the meat.
Bake in the preheated oven for 40 minutes. To test if the chicken is ready, poke a skewer through one of the drumsticks. When the meat is no longer pink, remove the pot from the oven and transfer the pieces of chicken to a serving tray.
Return the cooking pot to the stovetop. Add the olives, then bring the liquid to a boil over high heat and simmer for a good 10 minutes to allow the sauce to thicken.
Season the sauce to taste, then pour it over the roasted chicken and serve.
## CHICKEN & LEEK PIE
There is nothing quite as special as the combination of chicken, leeks, tarragon, and dill. If you grow the herbs yourself, the flavor will be all the richer for it. I use a whole chicken for this dish, as it's significantly cheaper to buy the whole bird. However, you can make this recipe using pre-cut chicken breasts or thighs—a less frugal approach, but quicker. I've included both options in this recipe.
_Preparation time:_ 1 hour
_Cooking time:_ 1 hour
_Serves:_ 6
_Ingredients:_
1 quantity oatmeal pastry (see recipe on page 252)
1 whole chicken (about 3⅓ pounds), or 1¾ pounds chicken breast or thigh
fat for frying
½ pound mushrooms
2 leeks
1 cup sour cream
2 eggs
¾ tablespoon finely chopped tarragon
½ tablespoon finely chopped dill
1 teaspoon mustard
salt and pepper
¼cup chicken stock or water
3 teaspoons arrowroot powder
First, prepare the pastry and preheat your oven to 350°F.
If you are using a whole chicken, cut off any excess fat and debone the bird according to the instructions on page 108. You can also get the butcher to do this for you. Cut the meat into bite-sized chunks. Alternatively, if you are using ready-cut thighs and breasts, cut them into bite-sized pieces.
Heat a saucepan over medium heat, add a little fat and fry the chicken pieces in batches for 30 to 60 seconds on each side. They should seal but not cook through. Transfer them to a separate dish and wipe the pan clean.
Slice the mushrooms into ¼-inch pieces. Fry these in the pan with some fat for a couple of minutes or until they lose much of their moisture. Set them aside with the chicken pieces and again wipe the pan clean.
Finely chop the leeks, then fry them gently with a little fat, stirring occasionally, for 10 to 15 minutes or until they are soft but not browned.
While the leeks cook, prepare the sauce by combining the sour cream, 1 of the eggs, and the tarragon, dill, and mustard. Season with salt and pepper to taste, then stir well and set aside.
In a small saucepan, heat the stock until it is warm but not boiling. If you don't have any stock, use the equivalent quantity of water. When the liquid is warm, add the arrowroot powder and stir to produce a smooth paste.
To the pan containing the cooked leeks, add the chicken, mushrooms, sour-cream mixture, and arrowroot paste. Simmer gently, stirring regularly, for 5 minutes. The mixture should be thick and gluey; if it's not, boil it down further and add some more arrowroot powder. When it is ready, transfer the filling to a large casserole dish.
Roll out the pastry to form a lid the size of your casserole dish and place it over the chicken filling. Prepare a glaze by whisking the remaining egg in a small bowl and brushing it over the pastry.
Place the pie in the oven and cook for 45 minutes, or until the crust is golden and crispy and the chicken has cooked through. Serve with a crispy green salad or cooked vegetables.
# STOCKS & SOUPS
"To make a good soup, the pot must only simmer, or 'smile.'"
—French proverb
SOUP HAS ALWAYS BEEN A MAINSTAY OF PEASANT fare, the perfect way to combine the cheapest ingredients or even scraps (bones, knuckles, some chicken feet) to make a delicious and nourishing broth. A good pot of soup should restore and nourish. In France, the word _restoratif_ described soups and stews sold at roadside taverns to weary travelers. They were a source of sustenance for those who were unwell and an antidote to physical exhaustion. The concept of a _restoratif_ bowl of soup eventually expanded to describe any place people could stop for a sustaining meal—that is, a restaurant.
Stock is the basis of all good soup-making. Stock can be made from almost anything—bones, heads, tails, vegetable scraps, and cooking water. Making stock out of odds and ends reduces the cost of homecooking and also decreases kitchen waste. Bone stocks made using the gelatinous parts of the animal are the most nutritious. Many traditional groups placed enormous value on bone stock. It was inexpensive and easy to make, was an important means of keeping well nourished, and ensured that no part of the animal went to waste. Gelatin-rich stock has many powerful properties. It can strengthen the skin, cartilage, bones, heart, muscles, and immune system. Animal feet, marrow, and shank are particularly rich in gelatin, so don't throw those chicken feet out—use them! Oxtail is also exceptionally good and is often very cheap, as not many people seem to buy them.
In addition to animal sources, you can also use vegetable scraps, the juice from lentils, beans, and legumes, or the cooking water left over after boiling vegetables. These do not have the same powerful properties as bone broth, but will still add flavor and nutrients to soups, stews, and casseroles.
## BUYING STOCK BONES
When you are shopping, make sure you ask your butcher for "stock bones." If he or she doesn't understand what you're after, explain that you need cartilage and gelatin, for instance shank, marrow, or feet. If your butcher is a friendly one, request that the bones be chopped into smaller bits; this will make cooking them all the easier.
From a price perspective, bones are a good point to haggle on. Make sure you ask for anything they won't be needing and you'll most likely get a good price. You can also ask for "dog bones" or "scraps," as they are essentially the same thing.
## MAKING STOCK
Making stock is all about extracting the nutrients and flavors from animal bones or vegetable scraps so that you can use them in later cooking. You can use any odds and ends: heads, tails, feet, and other offcuts such as shank, marrow, and rib. These offcuts are usually sold at a reduced price and are sometimes given away for free.
The golden rule when making stock with animal bones is to adjust the cooking time to suit the size of the bones. Fish bones are fairly small and thin, so you only need to cook them for two or three hours. Rabbit, chicken, and poultry bones are slightly bigger, so stock using these ingredients can be gently simmered for eight to ten hours. Beef and lamb bones take the longest and should be cooked for at least twenty hours, or overnight.
In all honesty, all you need for a good basic stock are some bones, water, and a small amount of vinegar. The other ingredients in the following recipes are really optional extras. They make the stock delicious, but don't go busting your chops if you don't have them. It's definitely not worth a trip to the supermarket: just make do with what you've got. And don't forget, stock-making is a great way to use leftovers. Stale and floppy carrots, that tired-looking celery that's sitting at the back of your fridge—these are all perfect ingredients for stock. You definitely don't need vegetables that are super fresh, as after a couple of hours on a gentle simmer, you won't know the difference anyway.
Whatever ingredients you're using, coarsely chop them and place them in a large stockpot with cold water. Bring to a boil, then simmer, partially covered, for several hours. Drain the juice and discard the solids, then refrigerate or freeze your stock in an airtight container. If you like, you can place the stock in the fridge, allow a layer of fat to form, then skim the fat from the surface and discard it.
## STORING YOUR STOCK
Stock will keep in the freezer for months and is incredibly handy for last-minute dinners. If you store it in meal-sized portions, you can defrost them as you need them. It can also be boiled down to a thick, jelly-like consistency so that it takes up less space in your fridge or freezer. I store mine in recycled plastic ice-cream and yogurt containers. Alternatively, you can use zip-lock bags. Some people prefer to use glass jars. If using glass, only fill the jar three-quarters full, as the liquid will expand when it freezes.
## CHICKEN OR POULTRY STOCK
If your butcher is willing to sell them to you, chicken heads, feet, and wings make excellent additions to stock. If you are buying your bird directly from the farm, it shouldn't be a problem to get the feet as well as the whole bird. If you can't get the feet, ask your butcher for the wingettes, which are also rich in gelatine and cartilage. This recipe is a master recipe for chicken stock, but it can be made using any poultry bird such as duck, goose, turkey, or pheasant. Larger birds with bigger bones require a longer cooking time and a few extra herbs.
_Preparation time:_ 10 minutes
_Cooking time:_ at least 5 hours
_Makes:_ 1 gallon
_Ingredients:_
4½ pounds chicken carcasses and/or offcuts such as wingettes or feet
1 or 2 onions
2 or 3 carrots, parsnips, or turnips (or a combination)
2 sticks celery, including the leaves
1 bunch fresh thyme or rosemary
1 gallon cold water, or enough to cover all your bones
10 peppercorns ½ cup apple-cider or wine vinegar
_Optional:_
2 large fennel
1 or 2 leeks
A knob of fresh ginger
Roughly chop the vegetables and crack the bones. Place all the ingredients in a large pot and cover them with cold water. Bring the stock to a boil, then allow it to simmer, partially covered, for 5 to 12 hours (if you are using a larger bird such as turkey or goose, let it simmer for a bit longer). When you've finished cooking, drain and discard the solids and retain the liquid.
## CHICKEN-FEET STOCK
Chicken feet are the magic ingredient in this rich and beautiful stock. They are full of gelatin, which imparts a special quality to soups and other dishes. If you are shopping directly from a farm or a knowledgeable butcher, you should be able to buy chicken feet in bulk—many good butchers sell them by the bag for less than the cost of a whole chicken. Making stock from chicken feet is very similar to standard chicken stock, but a bit of extra spice adds extra warmth.
_Preparation time:_ 10 minutes
_Cooking time:_ at least 5 hours
_Makes:_ 1⅓ 1gallons
_Ingredients:_
42/5 pounds chicken feet
½ cup apple-cider or wine vinegar
1 large onion
2 or 3 carrots, parsnips, or turnips (or a combination)
3 or 4 sticks celery, including the leaves
1 bunch fresh thyme or rosemary
1 heaped teaspoon black peppercorns
a few slices of fresh ginger
_Optional:_
2 large fennel
flat-leaf parsley, added 10 minutes before you finish
nettle herbs or roots (dried or fresh)
a pinch of cayenne pepper
Put the chicken feet into a large pot and cover them with cold water. Add the vinegar and slowly bring to a simmer.
Meanwhile, chop the vegetables and the herbs. When the water is simmering, add them to the pot.
Simmer, partially covered, for at least 5 hours. Check the water levels from time to time; if the chicken feet are not covered with water, add a little more. When you've finished cooking, drain and discard the solids and retain the liquid.
## BEEF, LAMB, OR PORK STOCK
Crack the meat bones and roughly chop all the vegetables. Place the bones and the vinegar in a large pot. Cover them with cold water and slowly bring to a boil. Add the herbs, peppercorns, and water then simmer, partially covered. The stock should not be boiling; it only needs to simmer or "smile."
_Preparation time:_ 10 minutes
_Cooking time:_ at least 3 hours
_Makes:_ 1⅓ gallons
_Ingredients:_
42/5 pounds soup bones
2 large carrots, parsnip, or turnips (or a combination)
2 stalks celery, including the leaves
1 or 2 onions
¼ cup apple-cider or wine vinegar
1 bunch fresh thyme or rosemary (or a combination)
10 black peppercorns or 1 tea spoon black pepper
1 gallon cold water
_Optional:_
nettle herbs or roots (dried or fresh)
dried arame or kombu seaweed
flat-leaf parsley, added 10 minutes before you finish
You can cook your stock for anywhere between 3 and 48 hours; the longer, the better. As it cooks, skim off and discard any residue that accumulates on the surface.
When you have finished cooking, drain and discard the solids and retain the liquid. If you'd like to reduce the amount of fat, place the stock in the fridge, allow a layer of fat to form, then skim it off and discard it.
## VEGETABLE STOCK
This is really an exercise in cleaning out your fridge or freezer. You can add whatever you like, in whatever quantity suits you. The recipe below is just a guide.
_Preparation time:_ 10 minutes
_Cooking time:_ 1 to 3 hours
_Makes:_ 2 quarts
_Ingredients:_
1 onion
2 leeks
3 stalks celery, including the leaves
4 parsnips or carrots 1 turnip
¾ inch ginger
10 black peppercorns or 1 teaspoon black pepper
1 teaspoon sea salt
8½ cups water
1 bouquet garni
_Optional:_
2 large fennel
flat-leaf parsley (added 10 minutes before you finish)
nettle herbs or roots (dried or fresh)
dried arame or kombu seaweed flakes
Coarsely chop all the ingredients and place them in a large pot. Cover with cold water. Bring to a boil, then reduce the heat and simmer, partially covered, for 1 to 3 hours. When you've finished cooking, drain the juice and discard the solids.
_Note:_
The water left over after cooking chickpeas or beans can also be used as a vegetable stock. After cooking your beans, retain the juice and use it as a stock in casseroles, soups, or sauces. Bean juice does not contain all of the nutrients found in bonebased stock, but it does add flavor to any dish and is rich in vegetable-based nutrients.
## STOCK-BROTH COLD CURE
This concoction will defeat any cold, flu, or cooties that might be plaguing your household. Cheaper than acetaminophen and quicker than antibiotics, it will have the bugs literally flying out from under your nose. You can use any stock for this recipe, but I find that a rich beef stock works best. Feel free to add more garlic if you wish, and try not to cook the garlic too much, if at all.
_Preparation time:_ 2 minutes
_Cooking time:_ 3 minutes
_Serves:_ 1
_Ingredients:_
1½ cups strong stock
½ cup water
3 large cloves garlic, crushed and finely chopped
2 teaspoons sea salt
Bring the stock and the water to a boil. Reduce the heat and add the garlic and sea salt. Serve in a large mug. After drinking it, have a rest for 20 minutes. On awakening, your cold should be gone, or your money back . . .
## FRENCH ONION SOUP
This is a very handy recipe if you don't have many fresh ingredients on hand. All you need is some stock, onions, wine, and a little bit of crusty bread and cheese. Gruyere is traditional, but a quality parmesan or mozzarella can be used instead.
_Preparation time:_ 10 minutes
_Cooking time:_ 1 hour, 10 minutes
_Serves:_ 4
_Ingredients:_
4 or 5 medium onions fat for frying
2 cloves garlic
2 quarts beef stock
½ cup white wine
⅛ cup brandy (optional) a few pieces of crusty bread
5⅓ ounces Gruyere, parmesan, or mozzarella cheese, grated
salt and pepper
Preheat your oven to 390°F, using the grill setting.
Finely slice the onions. In a large frying pan, heat a teaspoon of fat over low heat. Add the onions and cook them gently for 20 to 30 minutes, stirring occasionally. The onions should caramelize, turning a gentle golden color, but should not brown or burn. When they are almost ready, crush the garlic, chop it finely, and add it to the pan.
In a large pot, bring the stock to a boil, then add the white wine, brandy, and onions. Simmer gently for 25 minutes.
Thickly slice the bread and lightly toast the slices in the toaster.
Pour the soup into individual soup bowls. Garnish each bowl with a few slices of toast and a generous sprinkling of cheese. Season to taste, then place the bowls in the preheated oven for 5 to 10 minutes or until the cheese has melted.
## PEA & HAM SOUP
You can use either fresh or smoked ham bone (or hock) for this dish. Smoked ham has a wonderfully charismatic odor as it simmers on the stove. We usually use the leftover ham from Christmas dinner—a good pot can last us several days, which means investing in a healthy, free-range pig is well worth it.
_Preparation time:_ 10 minutes
_Cooking time:_ 2 hours
_Serves:_ 8
_Ingredients:_
1 large onion
3 large carrots
3 large sticks celery, including the leaves
3 cloves garlic
1 ham bone, with or without meat (roughly 1¾ pounds)
2 cups green or yellow split peas
⅛ cup apple-cider or wine vinegar
1 bay leaf
2 teaspoons fresh thyme, finely chopped
1 handful fresh parsley, finely chopped
1 handful fresh mint, finely chopped
Finely chop the onion. Dice the carrots and celery and crush and finely chop the garlic.
Place your ham bone in a large pot and completely cover it with water. Add the split peas, vinegar, bay leaf, thyme, and onion and bring to a boil. Reduce the heat and simmer gently for about 90 minutes, or until the meat begins to fall off the bone. Add the carrots, celery, and garlic and simmer for a further 30 minutes.
When the vegetables are soft and well cooked, remove them from the pot and place them in a food processor. Purée them until they are smooth, then return them to the pot. Alternatively, you can keep everything in the pot but push the meat to one side and use a hand-held blender to purée the vegetables.
Pull the meat from the bone and stir it through the soup. If necessary, adjust the water levels by adding a little more or by boiling down the stock. When you are happy with how it's looking, add salt and pepper to taste.
Pour the soup into individual bowls and sprinkle generously with freshly chopped parsley and mint. Drizzle with olive oil and serve with crusty bread.
## MINESTRONE
Nothing beats a good minestrone, with its rich stock, fresh vegetables, and nourishing legumes. Your minestrone will reflect what you have on hand in your pantry or back garden. Don't be afraid to play around with the ingredients, swapping carrots for parsnip or onions for shallots if it suits you. And if you happen to have something fresh and exciting from the garden, be it crunchy green beans, an unusual variety of turnip, or a handful of spinach, don't be afraid to throw it in as well.
_Preparation time:_ 10 minutes
_Cooking time:_ 1 hour
_Makes:_ 1 gallon (serves 8 to 10)
_Ingredients:_
1 cup chickpeas (dry weight), soaked overnight
½ cup white beans (dry weight), such as cannellini or haricot, soaked overnight
1 large onion
1 small handful fresh rosemary, finely chopped
1 small handful fresh thyme, finely chopped
1 quart chicken stock
3 medium carrots
1 cup green beans
1 leek
3 medium potatoes
1 can diced or whole tomatoes, or 4 large and juicy fresh tomatoes
2 cups fresh greens such as silverbeet, cavolo nero, or spinach
2 cups shredded cabbage
salt and pepper
olive oil
freshly grated parmesan
Rinse your pre-soaked beans with cold running water. Place them in a large soup pot and cover generously with water. Finely chop the onion. Add the onion, fresh herbs, and stock to the pot and bring to a boil. Reduce the heat and simmer, partially covered, for 45 minutes or until the beans are soft.
Cut the carrots, green beans, leek, and potatoes into bitesized pieces. Add them and the tomatoes to the pot and simmer for a further 15 minutes or until the vegetables are soft.
Finely chop the greens and finely shred the cabbage. Add them to the pot and cook them for just a few minutes. Season your soup to taste, then pour it into individual serving bowls. Drizzle each bowl with some extra-virgin olive oil and sprinkle with parmesan cheese.
## POTATO & LEEK SOUP
Two simple ingredients, a few dried herbs, dinner ready in thirty minutes. Enough said.
_Preparation time:_ 10 minutes
_Cooking time:_ 15 minutes
_Serves:_ 5
_Ingredients:_
3 large leeks fat for frying
1½ pounds potatoes
2 teaspoons fresh thyme
1 teaspoon dried nutmeg, freshly grated
1½ quarts chicken stock
salt and pepper
olive oil
_To serve:_
I like a generous dollop of Greek yogurt or sour cream, some fresh parsley, and a good drizzle of olive oil. My partner recommends some fried bacon, cut into small pieces and sprinkled over each bowl.
Slice the leeks into pieces about half an inch thick.
In a large pot, heat a teaspoon of fat over low heat. Add the leeks and half a cup of water. Cover and simmer gently for 5 minutes with the lid on, then 10 minutes without the lid. Allow the leeks to soften and sweat, but do not let them brown.
Meanwhile, coarsely slice the potatoes. When the leeks are ready, add the potatoes to the pot along with the thyme, nutmeg, chicken stock, and two cups of water. Bring to a boil then simmer, partially covered, for 10 minutes, or until the potatoes are soft.
Transfer ¾ of the soup to a blender and purée it, then return it to the pot. Alternatively, leave all the soup in the pot and purée it with a hand-held blender, leaving a few chunky bits.
Season to taste and serve.
## CHICKEN & CORN SOUP
This recipe is a godsend for busy nine-to-fivers. If you have a whole chicken, simply slice off the breast for this recipe and use the rest of the bird for chicken soup or chicken and leek pie.
_Preparation time:_ 10 minutes
_Cooking time:_ 20 minutes
_Serves:_ 2
_Ingredients:_
12/5 quarts chicken stock
2 fresh corn cobs
2½ tablespoons arrowroot powder
1 medium chicken breast or thigh, skinned and boned
1 teaspoon umeboshi plum vinegar
½ teaspoon soy sauce
2 spring onions
In a small saucepan, bring the chicken stock to a boil. Reduce the heat until it is gently simmering, then add the whole corn cobs. Let them cook for 10 minutes or until they are soft.
Remove the corn from the stock and scrape the kernels from the cobs. Place the corn kernels in a food processor with ⅓ cup of the stock and all of the arrowroot powder. Purée to a fine paste, then return the mixture to the pot and stir well. Season to taste. The soup should now be thick and gluey. If it doesn't seem thick enough, add a little more arrowroot powder.
Finely slice the chicken meat and add it to the pot. Reduce the temperature and cook gently until the chicken is cooked through. Remove it from the heat and add the vinegar and soy sauce. Stir well. Serve with sprinklings of finely chopped spring onion.
## SAFFRON STRACCIATELLA
Remember all those tupper ware containers filled with chicken stock that are sitting in your freezer? Well, now is the time to use them up. Even if you are exhausted, hungover, or only have one functioning hand, you can make this soup. It's like a fluffy rug and hotwater bottle, only tastier.
_Preparation time:_ 5 minutes
_Cooking time:_ 5 minutes
_Serves:_ 2
_Ingredients:_
1 quart chicken stock
½ teaspoon saffron threads
3 eggs
¼ cup finely grated parmesan, plus extra for serving
¼ cup flat-leaf parsley, finely chopped
1 slice of bread, crusts removed, ripped into small breadcrumbs
salt and pepper
olive oil
In a medium saucepan, bring the chicken stock to a gentle simmer. Add the saffron threads and stir well.
In a small bowl, combine the eggs, cheese, parsley, breadcrumbs, and salt and pepper to taste. Whisk until the eggs are well combined.
In a steady stream, pour half of the chicken stock into the bowl, whisking as you pour. Whisk well for a further 30 seconds, then pour the mixture back into the saucepan. Return to the boil and cook for 2 minutes, gently whisking all the time. The eggs and breadcrumbs should clump together.
_Voilà._ Serve, garnished with parmesan cheese and a dash of olive oil.
## CHICKPEA & ROSEMARY SOUP
This soup is best made with ripe tomatoes and home-grown rosemary freshly plucked from your backyard pots. I have on occasion resorted to stealing rosemary from over a neighbor's fence. But the soup is so good, this little theft seems worth it; just think of it as positive pilfering . . .
_Soaking time:_ overnight
_Preparation time:_ 10 minutes
_Cooking time:_ 1 hour
_Serves:_ 8
_Ingredients:_
2 cups chickpeas (dry weight), soaked overnight
3 onions, finely chopped
4 medium potatoes butter for frying
2 garlic cloves, crushed and finely chopped
3 large stems of rosemary, about the length of your hand
8 ripe tomatoes, roughly chopped, or 2 cans of preserved tomatoes (fresh is better)
1 quart water
1 quart chicken stock, vegetable stock, or leftover bean juice
salt and pepper
1 large handful flat-leaf parsley
olive oil
Soak your chickpeas overnight.
Finely chop the onions and dice the potatoes.
Heat a little butter in a large pot and gently fry the onions for about 5 minutes, or until they are soft and brown. Finely chop the garlic and add it to the pot, along with the rosemary leaves. Turn down the heat and simmer for 3 minutes.
Drain the chickpeas and discard the soaking water. Add the chickpeas to the pot along with the tomatoes, water, and stock.
Bring to a boil and simmer for one hour, or until the chickpeas are soft. When they are beginning to soften, add the potatoes and stir well. Cook for a further 10 minutes or until the potatoes are cooked.
Season to taste and serve with a generous sprinkling of finely chopped flat-leaf parsley and a drizzle of olive oil.
## PUMPKIN SOUP
Pumpkins manage to pop up in all sorts of peculiar places in our garden. Because all of our vegetable scraps are composted and ultimately end up back on the veggie patch as fertilizer, it's no surprise that little pumpkin sprouts seem to appear here, there, and everywhere when spring comes. If you have a wide open space, pumpkins can go a bit bananas; this is an excellent way to use them up.
_Preparation time:_ 10 minutes
_Cooking time:_ 30 minutes
_Serves:_ 8
_Ingredients:_
1 large onion
fat for frying
1⅓ pounds Japla pumpkin (roughly half of one large pumpkin)
1 pound butternut pumpkin (roughly half of one large pumpkin)
2 medium carrots
1 quart chicken stock
2 teaspoons allspice (or 1 teaspoon dried cinnamon and 1 teaspoon nutmeg)
½ teaspoon dried ginger
1 clove garlic
to serve, sour cream or yogurt, and fresh parsley or chives
Uncut and unopened, pumpkins can sit happily in a cool dark place such as a pantry or cupboard for weeks, if not months. Store them there until you feel like making soup. They also look fabulous stacked on your kitchen table or mantelpiece.
Chicken stock is traditionally used as a base for pumpkin soup, but I have also had success with lamb stock. They make for a soup that is richer and heavier, but still delicious. Coarsely chop the onion. Heat a little fat in a soup pot over a medium heat and sauté the onion for 5 to 10 minutes or until it turns a rich golden color.
Meanwhile, coarsely chop the other vegetables. When the onion is ready, add the vegetables, stock, spices, and garlic to the pot, plus a little extra water if necessary to completely immerse everything. Bring to a boil, then reduce the heat and simmer for 20 minutes, or until the vegetables are soft.
Purée the vegetables, either with a handheld blender or in a food processor.
Season to taste, then serve with dollops of sour cream or yogurt and a sprinkling of freshly chopped parsley or chives.
## KALE & ZUCCHINI SOUP
This is an excellent mid-week pick-me-up when you are feeling fried and tired after one too many deli coffees. In one bowl of soup you will be getting about five whole vegetables. The seaweed adds extra minerals and a subtle salty flavor.
_Preparation time:_ 2 minutes
_Cooking time:_ 10 minutes
_Serves:_ 2
_Ingredients:_
2 small carrots
3 medium zucchini
4 large leaves of dark leafy greens (kale or cavolo nero), stems removed
2 strips kombu seaweed
¼ cup finely chopped basil
¼ cup finely chopped coriander
a few generous dollops of plain yogurt to serve
sea salt
olive oil
Slice the carrot into 1-inch pieces and cut the zucchini in half. Rip the leafy greens into large chunks. Place the vegetables in a small saucepan with just enough water to cover them. Add the seaweed, then bring to a boil and simmer for 5 minutes or until the vegetables are soft.
Transfer the mixture to a blender or food processor and process until it forms a fine purée; it should be thick, like heavy cream. As it is blending, add the basil and coriander and season to taste.
Serve with a dollop of plain yogurt and a dash of olive oil for each bowl.
## LENTIL SOUP
Lentil soup is the ultimate frugavore food. Lentils and stock bones, both wonderfully cheap, form the hearty base, complemented by whatever vegetables are fresh in your garden or at the market. Enjoy this soup as a mid-week meal, or take it to work for lunch in a thermos.
_Preparation time:_ 10 minutes
_Cooking time:_ 1 hour
_Serves:_ 4
_Ingredients:_
1 large onion
2 cloves garlic
2 sticks celery, including the leaves
2 medium carrots
1 cup lentils
1½ quarts stock
½ teaspoon dried thyme leaves
1 bay leaf
salt and pepper
olive oil or coconut oil
1 tablespoon tomato paste
Place the lentils in a bowl and cover them with filtered boiling water. This will soften them, and shorten your cooking time. Soak them for as long as possible—a few minutes is fine, but 30-60 minutes is ideal.
Finely chop the leek, and cut the celery and carrots into small (1/2 inch) pieces. Place a large soup pot on a medium heat and gently fry the leek with coconut or olive oil. Stir regularly with a wooden spoon. When the leek begins to soften, add the carrots and celery. Crush the garlic and add it in with the herbs and bay leaf. Cook only for another 2-3 minutes.
Drain the lentils and rinse them from their soaking water. Add these into the pot along with the stock. Increase the heat, and simmer gently for 5-15 minutes, or until the lentils soften (the time will be shorter if the lentils have been pre-soaked for longer).
Add the tomato paste and season to taste. Stir well and serve.
## BEAN & GREEN SOUP
Silverbeet is one of the easiest vegetables to grow yourself. With some fresh leaves and a few staples from your pantry, you can have this nourishing soup ready in less than an hour.
_Soaking time:_ overnight
_Preparation time:_ 10 minutes
_Cooking time:_ 1 hour
_Serves:_ 8
_Ingredients:_
2 medium onions
2 cloves garlic
1 cup chickpeas, soaked overnight
1 cup white beans (such as cannellini or haricot), soaked overnight
1½ quarts chicken or vegetable stock
1 bay leaf
½ teaspoon cumin
½ teaspoon paprika
1 large bunch silverbeet
salt and pepper
7 ounces fresh ricotta
olive oil
Finely chop the onions and garlic. Drain the beans and discard the soaking water. Place them in a large pot with the stock, bay leaf, onions, cumin, paprika, and garlic. Add two cups of water and bring to a boil. Reduce the heat and simmer, partially covered, for 40 to 50 minutes, or until the beans soften.
Chop the silverbeet into bite-sized pieces and add these to the pot. Cook for 5 to 7 minutes, or until the leaves are soft and wilted. Season to taste, then serve with a dollop of ricotta cheese and a dash of olive oil.
# BEANS, LENTILS, & LEGUMES
"Red Beans and Ricely Yours."
—Sign-off used by Louis Armstrong
MANY YEARS AGO, AS A YOUNG WHIPPER-snapper university student, I embraced lentils and legumes purely because of their cost. I remember walking past them in the supermarket aisle (in a dank and lonely corner of the supermarket, I might add) and noticing that a 17½-ounce packet of lentils cost less than a candy bar. It hit me that with a single packet, I could make several meals for less than the cost of a pizza.
Beans, lentils, and legumes are not only cheap; they are also extremely good for you. These foods have nourished peasant populations for millennia, and are still a staple food in many third-world countries. Throughout history, they have been considered a "poorman's food" and have often replaced meat and fish when these ingredients were not available or too expensive. Poor Catholics who could not afford fish during Lent ate lentils instead.
To be truly delicious, most legumes need a bit of extra flavor, usually in the form of herbs, spices, garlic, or bay leaves. Traditional dishes often combine beans and legumes with some kind of protein or fat—ham hock with split-pea soup, lentil soup with black pudding, or bean soup with bacon and silverbeet. They also go beautifully with fresh herbs and produce from your garden.
There are many varieties of legumes available, wherever you travel. But for the purposes of the following pages, we'll consider the varieties most commonly found in supermarkets and organic foodstores. With a small bag of any of these, you can make cost-effective and nourishing soups, stews, and salads. There is a wonderful variety of dishes to be made with each of these legumes, so let's get cracking on soaking, sprouting, souping, salading, blanching—and, of course, eating . . .
## PREPARING YOUR LEGUMES
Most beans and legumes require a period of pre-soaking before they are cooked, as they contain large amounts of carbohydrates called oligosaccharides, which human digestive enzymes can't readily convert into absorbable sugars. Pre-soaking causes these sugars to be broken down and released into the soaking water. Lentils and split peas do not require presoaking, but all of the larger beans—kidney, chickpea, adzuki, and white beans—do.
Pre-soaking beans is easy—place the beans in a bowl of cold water and leave them there for twelve to twenty-four hours. The longer the pre-soaking time, the shorter the cooking time, and the more tender the bean is once it is cooked. If you pre-soak for more than twenty-four hours, it's a good idea to change the water once or twice. To speed up the pre-soaking process, you can use hot or boiling water and/or make the pre-soaking water alkaline by adding a small quantity (about a teaspoon) of bicarbonate of soda for every quart of water. Make sure that you drain and rinse your beans well before you cook them (the soaking water must be discarded). You can keep the cooking water to use as a vegetable stock in other dishes. If legumes are not soaked properly, you may notice a bit of bloating or gas following your meal (yes, that's right. Also known as a fart).
Of course, for any of these recipes, canned beans can replace dried and pre-soaked beans. Just keep in mind that canned beans are not nearly as cost effective and for some reason, just don't taste as good.
## A NOTE ON SOYBEANS
Soybeans are an anomaly in the world of legumes. These beans were traditionally fermented for a period of months, sometimes years, to make old-style condiments such as miso, natto, and soy sauce. Asian cultures knew that the beans required fermentation to naturally break down components that were indigestible and caused ill health effects. Not surprisingly, soybeans have been found to be rich in phytates, which work as anti-nutrients (these attach themselves to nutrients and draw them from your body). They also contain large amounts of protease inhibitors and oligosaccharides, which interfere with digestion. So try to enjoy soybeans as they were traditionally consumed—that is, fermented to produce such products as miso, soy sauce, natto, and tempeh.
## CANNELLINI BEAN SALAD WITH PUMPKIN & BEETS
I love the combination of baked beetroot and pumpkin, and this is a terrifically easy salad to put together. With a few beets from your garden and some preserved artichokes and dried beans from your pantry, you are ready to go.
_Preparation time:_ 10 minutes
_Cooking time:_ 45 minutes
_Serves:_ 6
_Ingredients:_
1 cup cannellini beans, soaked overnight
1 bay leaf (optional)
1 clove garlic (optional)
3 artichoke hearts preserved in olive oil or brine
½ pound fresh green beans (about 2 cups)
1 medium beetroot
1⅓ pound Japla pumpkin
1 handful fresh basil salt and pepper
1 teaspoon white-wine vinegar
1½ tablespoons olive oil
Pre-soak the beans as described earlier in this chapter.
When you're ready to start cooking, preheat your oven to 350°F. Grease a baking tray with a little cooking fat.
Rinse the beans and place them in a medium saucepan. Cover them with water and bring to a boil. Add the bay leaf and garlic, then reduce to a gentle simmer and cook, partially covered, for 45 minutes or until the beans are soft.
Meanwhile, scrub the beetroots and remove the tops, tails, and any gritty parts from the skin. Chop the beetroot and pumpkin into bite-sized chunks and place them on a baking tray. Sprinkle with salt, then bake for 20 to 30 minutes or until the pumpkin is soft and lightly browned. Remove the tray from the oven and leave to cool.
When the cannellini beans are soft, remove them from the heat and drain the cooking water. Put the beans in a large salad bowl. Sprinkle with sea salt and a dash of olive oil, toss well, and leave to cool.
While the other ingredients cool, top and tail your green beans. Use a vegetable steamer to steam them for 5 minutes or until they are lightly cooked. Rinse them in cold water and cut them into thirds.
The salad can be served either warm or chilled. When everything has cooled to the desired temperature, combine the green beans, beetroot, pumpkin, and cannellini beans. Finely slice the artichoke hearts and stir them through, along with the chopped basil, vinegar, and olive oil. Season to taste and serve.
## TREACLE BAKED BEANS
These are cheap as chips to make, and always a winner on a cold winter's night. Serve them with a generous leafy salad and some crusty bread, and you have a delicious meal-in-one.
_Soaking time:_ overnight
_Preparation time:_ 10 minutes
_Cooking time:_ 2¼ hours
_Serves:_ 8
_Ingredients:_
1½ cups (17½ ounces) white beans, such as haricot or cannellini, soaked overnight
1 large onion
2 cups stock
1 bay leaf
2 cloves garlic, crushed
½ teaspoon whole cloves
½ teaspoon whole allspice berries
1 tablespoon treacle
2 teaspoons Worcestershire sauce
6 large juicy tomatoes, cut into quarters, or 1 can (16 ounces) whole tomatoes
2 teaspoons soy sauce
1 handful fresh flat-leaf parsley, finely chopped
olive oil
Soak your beans overnight, as described earlier in this chapter.
After soaking, rinse the beans and place them in a large soup pot. Finely chop the onion and add it to the pot along with the stock, bay leaf, and 2 cups of water. Turn the heat up and gently simmer for 1 hour or until the beans are tender. They will double in size, so you'll need to have plenty of liquid in the pot for them to soak up; make sure all the ingredients are completely immersed, and check on the pot occasionally as it cooks.
Meanwhile, prepare the allspice and cloves by pounding them in a mortar and pestle to form fine crumbs.
Add the spices to the dish, along with the garlic, treacle, Worcestershire sauce, soy sauce, and tomatoes. Continue to simmer for another hour and 10 minutes. The beans should be tender and the liquid should boil down to form a thick and delicious paste.
Season with salt and pepper to taste, then serve with a generous handful of parsley and a dash of olive oil.
## CHICKPEA SALAD WITH GREENS
This is one of my favorite take-it-to-work lunches. Even after bumping along in my lunchbox on the back of my bike, it still tastes wonderful.
_Soaking time:_ overnight
_Preparation time:_ 10 minutes
_Cooking time:_ 1 hour
_Serves:_ 5
_Ingredients:_
1 cup chickpeas, soaked overnight
1 large broccoli
1 cup cavolo nero
3½ ounces goat's cheese or soft feta cheese
2 teaspoons apple-cider or white-wine vinegar
¾ tablespoon olive oil
Pre-soak the beans as described earlier in this chapter.
When you are ready to start cooking, drain and rinse the chickpeas and place them in a small saucepan. Cover them with water and place the pot over medium heat. Gently simmer for 50 to 60 minutes, or until the chickpeas are soft.
Meanwhile, break the broccoli into large chunks and slice the cavolo nero into bite-sized pieces. In a small vegetable steamer, cook them for 5 minutes or until they are just tender. Rinse them under cold water and pat them dry, then cut the broccoli into small florets.
When the chickpeas are ready, drain them from their cooking water and place them in a salad bowl. Add the vegetables and cheese, season to taste, and toss through the vinegar and olive oil. This salad can be served warm or cold.
## LENTIL SALAD
Alone, this is a satisfying light lunch or mid-morning snack. It's also a delicious accompaniment to a main meal.
_Preparation time:_ 10 minutes
_Cooking time:_ 1½ hours
_Serves:_ 8
_Ingredients:_
2 cups French green lentils
1 bay leaf
1 clove garlic
2 teaspoons white-wine or apple-cider vinegar
¾ tablespoon olive oil
1 red capsicum
2 large celery sticks, including some of the leaves
1 small handful mint, finely chopped
1 small handful basil, finely chopped
5⅓ ounces feta cheese
Rinse the lentils, then place them in a medium saucepan with the bay leaf and garlic. Cover them with water and bring to a boil, then reduce the heat and simmer for 45 minutes, or until the lentils are soft.
Drain the lentils and put them in a salad bowl. Add the vinegar and olive oil. Toss well and season to taste. Put the bowl in the refrigerator and leave the lentils to cool.
Finely dice the capsicum and celery and finely chop the fresh herbs. When the lentils have cooled, add the herbs and vegetables and toss well. Crumble the feta and stir it through. Serve at room temperature.
## SPLIT-PEA PURÉE (FAVA)
The ancient Greeks used to purée split peas into a dip much like this one here. Accompanied by a green leafy salad and an array of meat and fish dishes, it made for a wholesome meal. Split peas do not require any pre-soaking, so this is an easy dish to whip up at the last minute.
_Preparation time:_ 10 minutes
_Cooking time:_ 50 minutes
_Serves:_ 8
_Ingredients:_
2 cups yellow split peas
2 cloves garlic
½ teaspoon paprika
¼ teaspoon cayenne pepper
1 large lemon
¼ cup olive oil
salt and pepper
Put the split peas, garlic, paprika, and cayenne pepper in a saucepan. Cover them with water and bring to a boil. Reduce the heat and simmer for 50 minutes, stirring occasionally and adding extra water if necessary. The peas should turn to mush and develop a nice gooey texture.
When the peas have cooked, allow them to cool to room temperature, then stir through the lemon juice and olive oil. Season to taste and drizzle with a little extra olive oil to serve.
## CANNELLINI BEAN DIP
Dips are for more than dipping—they can be used on sandwiches, or served on toast with avocado or ham. I love making up a large batch when we are having people over for dinner. The leftovers can be relished for days, livening up work lunches or picked at with bread or crackers.
_Soaking time:_ overnight
_Preparation time:_ 10 minutes
_Cooking time:_ 45 minutes
_Makes:_ 1 generous bowl
_Ingredients:_
1 cup cannellini beans, soaked overnight
1 bay leaf (optional)
1 garlic clove (optional)
1 medium lemon
3 anchovy fillets in extra virgin olive oil, or 1 preserved sardine
½ cup fresh chopped parsley salt and pepper
½ cup olive oil, and perhaps a little extra
After pre-soaking the beans, rinse them, then put them in a large pot, and cover them with water. Add the bay leaf and garlic. Bring to a boil, then reduce the heat and simmer for 40 minutes or until the beans are soft.
Strain the beans from their cooking water. In a food processor, combine them with the juice of the lemon, a thin curl of lemon rind (about an inch long), the anchovies, parsley, and salt and pepper to taste. Gradually add the olive oil in a thin and steady stream, while pulsing the food processor on a high setting. The beans should clump together to form a smooth paste. Add a little extra olive oil if you need to.
Season to taste, then transfer the dip to a bowl or airtight container and refrigerate. The dip will keep for a good couple of days in the fridge.
## SWEET POTATO HUMMUS
I love this dip. I like to make up a big batch and use it throughout the week in sandwiches and as a snack with sliced carrots and celery. It's easy and economical, and can keep me going through a long week.
_Soaking time:_ overnight
_Preparation time:_ 5 minutes
_Cooking time:_ 1 hour
_Serves:_ 8
_Ingredients:_
1 cup dried chickpeas, or 2 cans (16 ounces each) chickpeas
1 large sweet potato
1 medium carrot
1 clove garlic (or more if you particularly like garlic)
½ teaspoon paprika or Ras el Hanout spice mix
juice of 1 large lemon sea salt
½ cup olive oil
If you've soaked your own chickpeas, drain them, and put them into a medium saucepan. Cover them with water, bring to a boil, then simmer for 30 minutes. If you're using canned chickpeas, you can skip this step.
Coarsely chop the carrot and sweet potato and add them to the pot along with the garlic. Simmer for a further 20 to 30 minutes, or until the chickpeas and vegetables are soft.
Drain the chickpeas and vegetables and, using a food processor, combine the chickpeas and vegetables with the spices, lemon juice, and salt to taste. Add the olive oil gradually, puréeing the mixture at a high speed until a smooth paste forms. This quantity of olive oil is only approximate—you may need a little more or less to get the desired consistency.
Season to taste and serve, or refrigerate in an airtight container.
# MEAT
"The only thing wasted on a pig is his squeal."
—Mrs. Burns, dairy farmer, Gippsland, Australia.
She died aged ninety-four while still running
her own farm and tending her backyard tomatoes.
HOW DO YOU FEEL ABOUT THE MEAT YOU buy? Are you happy to buy something that is anonymously labeled, packed in plastic, and stacked under lights at your local supermarket? Or would you prefer to know a bit about your animal—where it comes from, how it lived, and the journey it took to get to your dinner plate?
As consumers, we have a great power to influence how our meat is produced, packaged, and distributed. We can demand better quality meat by connecting to our local farms, supporting good butchers, and telling retailers what we would like to see on their shelves. Purchasing from local, small-scale farms is the best way to support your local food economy. If you shop at farmers' markets or buy directly from the farm, try to buy your meat in bulk and look for unusual cuts and different breeds. For instance, why not try some mutton instead of lamb, or look for boiling fowl instead of the typical plumped-up hen next time you shop?
## FRESH FROM THE FARM
When you buy your meat straight from the farm, you can see directly where your food is coming from. You can also get a better price, have access to a range of different cuts, and be directly involved in your local food economy. For me, the knowledge that an animal has lived well is far more important than any organic or biodynamic certification tick.
By shopping directly from the farm you can also save money, as you cut out the transportation and retail costs. Buying from the farmer is generally 50 percent cheaper than shopping at a butcher's shop or organic store. The farmer is also getting a better price; for most farmers, it is much more profitable to sell directly to consumers than through retail outlets.
I also enjoy seeing the diversity that is available at the farm gate whenever I visit—a fresh rooster, older hens, mutton, fresh spring lamb. Before I started dealing directly with farmers, I never realized how many ways there were to cook meat, using different breeds or animals of different ages and different cuts. What a shame that more of this diversity is not available in the mainstream retail market!
Shopping directly from the farm often requires that you buy your meat in bulk. You can store any excess in your freezer and explore cooking with the whole animal, including all the "thrifty bits."
## BUTCHERS
Good butchering is a highly specialized artisanal craft. Traditionally, it took years of training and butchers were respected pillars of small communities. A good butcher will have a clear understanding of where his meat came from and will cut, carve, and age his meat onsite at the back of his shop.
Unfortunately, there aren't many good butchers left. They've been replaced by "meat carvers," who lack the intimate knowledge required to choose the best quality meat and prepare it to its full potential. The blame for this lies with consumers, who increasingly seem to want only certain "easy" cuts of meat—racks of lamb, loin chops, beef steaks, and chicken breasts. We've forgotten how to cook many of the more economical cuts—animal feet, bones, offal, fat, and stewing steaks. Fifty years ago, housewives sought out these cuts and nothing went to waste. They were also fussy about the quality of what they purchased; they noticed if a slice of liver wasn't fresh or a stewing steak lacked gristle. If the quality was lacking, the butcher wouldn't get their business the following week.
Nowadays, with consumers buying only a few of the leaner cuts, many butchers get their meat ready-cut from the slaughterhouse. The process of dry-aging (where the meat is hung out to mature behind the shop, giving it a richer flavor) has been abandoned by many butchers, replaced by "cryo-vaccing" to save time and money. At the same time, stricter regulations mean butchers are no longer permitted to make their own salami or air-dry their own hams in the traditional way.
My interest in finding a good butcher started when I embarked on a quest to cook with traditional Jewish-style sweetbreads—thymus and pancreas glands—but couldn't find them at my local store. I tried everything. I attempted an "offal drive" with several like-minded friends; we all pitched in and tried to buy them in bulk from a local slaughterhouse. No go. I tried a local farmer, but he couldn't sell them to me either. Finally, I found a good butcher. And when I say "good," I mean really good. Roger processes whole carcasses in the back room of his shop, so I can access all the different cuts, including other hard-to-find items like suet, chicken feet, and pork fat. He is a godsend.
These were all once kitchen essentials, but they've been utterly lost from our food supply. As consumers, we've become unadventurous. Consequently, there is now a much narrower supply of meat on the market. For this to change, we need to be prepared to buy more of our food in bulk, and to source our meat directly from the farm or from an excellent butcher. A good butcher is one who knows how to prepare an animal from top to toe. Nothing is wasted at a good butcher's shop, and you should be able to buy all the cuts—not just the popular ones. A good butcher may have his own farm, or at the very least have a close relationship with wherever his meat comes from. He will be able to tell you clearly about the origins of your meat, how it was raised, and how it was processed—was it wet or dry-aged, and for how long? He will do most of his butchering and processing onsite. If he dry-ages his own meat and makes his own salami and hams, he is a true gem. So when you find one, don't let him go!
## SUPERMARKETS
Supermarkets are not known for careful processing, or for sourcing the best produce. That's a given. But I have noticed that some smaller supermarkets, especially locally run co-ops, provide detailed information about the quality and history of their meat via information sheets on display next to the produce. This is not as good as a proper chat with your butcher or farmer, but I would say it is the next best thing.
On a larger commercial scale, supermarkets are supplying more of the frugal cuts—such as lambs' necks, chicken carcasses, pigs' feet, and stewing steak—in response to consumer demand. They are also stocking a more diverse range of produce from organic and grass-fed farms. This is good. But most of the meat they sell is still anonymously labeled, with no details about where it came from. It always makes me want to ask: What was the farm like? What did the animal eat? Did the farmer get a good deal? If you are shopping at supermarkets, look for thrifty cuts, organic or grass-fed, and try to support local farms wherever you can.
## WHAT TO BUY?
Many people are confused as to what constitutes good meat. Should you buy grain-fed or grass-fed? Free-range or organic? From a nutritional perspective, the healthiest animal products come from animals that have lived as close to their natural environment as possible. For livestock, this means a diet of grass and hay, preferably involving "salad-bar grazing" on a diverse range of grass varieties. For poultry, the ideal diet includes an omnivorous range of grasses, insects, and grubs for protein, and only minimal grains.
Nutritional tests have shown that animals raised on natural diets have fewer medical problems and provide better-quality produce. Their products contain more omega-3 and conjugated linoleic (CLA) fatty acids, both healthy fats known to benefit heart health and aid in the prevention of inflammatory and autoimmune disorders. Grass-fed and free-range products also contain higher levels of antioxidants and important fat-soluble vitamins such as A, D, and E. Grass-fed livestock do not suffer from many of the afflictions faced by their grain-fed counterparts such as acidosis, rumenitis, liver abscesses, and bloat, and are at a reduced risk of _E. coli_ contamination. And when the animals are healthier, they require fewer antibiotics, again resulting in healthier produce.
So if you are looking for the healthiest meat, look first at what the animals have eaten. Check that they had access to appropriate food, and also that they enjoyed fresh air and exercise. Foods that are not appropriate for animal consumption include large amounts of corn and grains (for livestock), or any soy meal, brewery waste, and what the food industry likes to call "miscellaneous food scraps."
## CERTIFIED PRODUCE
Certification systems are really a secondary measure, used to ensure quality when we cannot see for ourselves how the animal has lived. Personally, I would much prefer to buy fresh produce from a farmer whom I know, rather than an anonymous package with an organic stamp.
But if you are buying meat from a retail outlet, the first thing you should try to determine is whether the animal was raised using natural feeding practices (i.e., look for "grass-fed," "free-range," or "pasture-raised"). If this is well established, then you can look for organic or biodynamic certification. Organic and biodynamic farming methods don't rely on chemicals or pesticides. Instead, nutrient dense grasses and feed are used both to nourish the animals and enliven the soil.
The bottom line is, find out where your food comes from. Certification systems are a great way to do this, but not the only way, and often not the cheapest way. If you can connect with your local supplier and see that their animals have enjoyed a natural diet and a decent dose of air and sunshine, don't worry too much about whether or not they are officially "certified"—just use your own judgment and common sense.
## BUYING IN BULK
If you have a good freezer, purchase a whole or half animal, or buy in bulk. This applies to all types of meat: it might involve buying a whole chicken or duck, rather than just the drumsticks or breasts, or purchasing half a sheep carcass and having it butchered. This is cost-effective and less wasteful, and gives you access to a wider range of cuts. Buying the whole animal also gives you access to all the wonderfully nutritious "extra bits." I am not a vegetarian (obviously!). But I do believe that if we are going to eat meat, we should at least respect the life of the animal and what it has given to us. Wasting no part of it is an important aspect of this.
To buy in bulk directly from the farm, you can negotiate an arrangement at local farmers' markets, or tap into local grassroots movements such as buying clubs and co-ops. I have an arrangement with a local farmer whereby I purchase a whole animal (if it's a larger animal, I usually go in with a few friends). We usually go home with a few coolers full of meat, enough to last us for a few months.
When buying half a cow (a top-of-the-range, grass-fed organic beast), we usually pay around half of what we would at a local retail outlet. Prices vary, so you'll have to do some research and see what your local farmers have to offer. You should also check your local regulations. Generally speaking, if you can get a number of people together and buy a large quantity in one go, it will be less trouble for the farmer, meaning a better price for you.
If you're shopping at supermarkets or organic retailers, look for a whole chicken rather than just the lean breasts or drumsticks. The price is often comparable, and you'll get three meals rather than one.
## THRIFTY BITS
A generation or two ago, it was standard to prepare a casserole or some other slow-cooked dish a couple of times a week. This could use up all those odd cuts of meat—the stewing steak, oyster blade, lambs' necks, or oxtail, or whatever happened to be cheap that week at the butcher's. Nowadays, with so many of us working longer hours, we have come to favor cuts like stir-fry, chops, and more expensive steaks, which don't need so much cooking time. The drawback is that these cuts are a lot more expensive, and much less rich in gelatin and nutrients. To get the best meat for the lowest price, look for the less popular pieces—the stewing steaks, offal, and odd bits (tails, bones, feet, etc. ) and make the most of them. These cuts are not only delicious; they are also extremely good for you. Not surprisingly, these odd bits have always been favored in traditional cuisines. Pigs' feet are a European delicacy. Rooster combs are _haute cuisine_ in southern France. And heads, feet, and wings provide gelatin, which is essential to stocks and stews.
Stewing steak (also called oyster-blade, chuck steak, and casserole steak): These cuts come from the muscular part of the animal such as the legs and neck. They are thick with muscle and rich in gelatin, which means they require long, slow, gentle cooking. They are highly nutritious and are delicious in casseroles, stews, and other slowly cooked dishes.
Mince: Mince is very cost-effective, and many mince recipes are extremely quick and easy: think hamburgers, meatballs, and meatloafs. If you are trying to save on grocery bills, have a few good mince recipes up your sleeve and buy it in bulk.
Offal and "odd pieces" (feet, tails, necks, and so on): Offal is significantly cheaper than the leaner cuts such as steak or shoulder roast. These cheaper cuts were considered the most nutritious parts of the animal by many traditional groups. Offal has the highest amount of fat-soluble vitamins (vitamins A, D, and K), while cuts such as liver are rich in iron and other important vitamins (pregnant mothers are still encouraged by nutritionists to eat liver pâté to keep their iron levels up). "Odd pieces" such as the tail are also rich in gelatin and important minerals, and are great in soups and stews. See "Cooking with Offal" later in this chapter for detailed information about different cuts.
Bones: Animals have a high ratio of bones to meat, so throwing the bones away means much of the animal goes to waste. Because it is rich in gelatin, bone stock is extremely nutritious. You can buy bones cheaply from your butcher, or in bulk from your local farm. See the "Soup" chapter of this book for a multitude of ways to prepare stocks and stews. If you make stock in advance, you can freeze it and use it to make cheap and delicious meals throughout the week.
## COOKING WITH OFFAL
Sweetbreads: These include the pancreas and thymus glands. To prepare, soak them in cold, salted water for two or three hours. Drain them, then place them in boiling water with a bay leaf, some peppercorns, and ¾ tablespoon of vinegar. Simmer for 15 minutes. Allow to cool, then remove any traces of skin or sinew. The sweetbreads can now be cooked in a creamy sauce, sautéed with mushrooms, or fried with some fresh herbs.
Brains: These need to be soaked in salted water for at least 3 or 4 hours. After soaking, drain away the liquid and remove any membrane or specks of blood. Place the brains in boiling water with a bay leaf, some peppercorns, and ¾ tablespoon of vinegar. Cook for 20 minutes on a gentle simmer. Like the sweetbreads, they can now be sauced with mushrooms, butter, or cream. A gentle fry-up with some butter and fresh herbs is also a treat.
Liver: The key to cooking liver is to make sure that it's fresh. Then try to slice it as thinly as possible—I often get my butcher to do this for me. I like to soak liver in some cold milk for 10 to 20 minutes before tossing it in flour and frying it with a little cooking fat on high heat. Fry for only a few seconds on each side; it should be only lightly cooked.
Kidneys: Lamb or calf kidneys are lighter tasting and only require gentle cooking. I've enjoyed these barbequed or lightly fried with some fresh herbs from the garden and a salad. Ox kidneys or the kidneys of mature cattle tend to be tougher, so do better with long, slow cooking (for instance in a steak and kidney pie) or sautéed with a rich red-wine sauce.
Heart: Trim away the veins and arteries so that you are left with just the muscle itself. Leave the fat on, as this helps to keep the meat tender as it cooks. I recommend cooking this just as you would a casserole, with plenty of herbs, stock, and wine. Gentle cooking at 350°F for 3 to 4 hours should do the trick.
Tongue: The tongue is also a muscle and needs to be brined for at least 24 hours, then gently cooked with bay leaves, peppercorns, vinegar, and herbs for 3 to 4 hours. It is wonderful served cold with a tangy sauce.
## BEEF CASSEROLE
This recipe comforts and warms, and turns a few simple ingredients into a hearty and wholesome meal. During the week, I often halve this recipe and use a smaller cooking pot to make a quick and easy dinner for two. It goes well with baked spuds and a fresh green salad from the garden.
_Preparation time:_ 10 minutes
_Cooking time:_ 1½ hours
_Serves:_ 4 generous servings
_Ingredients:_
1kg diced beef chuck or blade steak
2 medium/large onions, cut into thin wedges
2 garlic cloves, crushed
2 sticks celery, finely sliced
2 large carrots, diced
1 × 400g cans diced tomatoes
¼ cup red wine
1 cup chicken or beef stock
olive oil or coconut oil for frying
In a medium-sized cooking pot, heat the cooking oil and fry the beef in batches until browned. Set aside on a plate. Reduce the heat and add the onions, celery and carrots. Cook for 3-5 minutes, or until lightly browned. Add the wine and the garlic. Turn up the heat and let it sizzle for another minute. Return the beef to the pan and add the tomatoes, wine and stock. Season generously. The meat should be fully covered with liquid. If necessary, add a little water or additional stock. Reduce the heat to low and simmer, fully covered for up to one hour. Remove the lid and continue to simmer gently for an extra thirty minutes, stirring occasionally. The casserole will be ready when the beef becomes soft and tender. Serve with a generous baked potato, some sauerkraut and a leafy garden salad.
## IRISH STEW
For less than the cost of a delivered pizza, this dish can easily feed a family of four and provide leftovers the next day. Irish stew was traditionally made with lamb or mutton and the cheapest, most readily available cuts such as neckbones, shanks, and other offcuts. Arrowroot, cumin, and lemon juice are not traditional Irish ingredients; I've added them to thicken the sauce and provide a lighter, sweeter flavor. I hope Irish readers will forgive me!
_Preparation time:_ 15 minutes
_Cooking time:_ 3 hours
_Serves:_ 5 (generously)
_Ingredients:_
1 lamb neck, cut into chops (roughly ¾ pound)
2 teaspoons arrowroot powder
1 teaspoon cumin
fat for frying
¼ cup pearled barley
2 large onions
2 cloves garlic
1 cup beef stock
1 turnip
2 medium carrots
2 large potatoes
juice of 1 lemon
1 heaped cup finely chopped flat-leaf parsley
Preheat your oven to 350°F.
Toss the meat in the arrowroot powder and cumin. Put a frying pan over high heat and add a small dollop of cooking fat, then fry the meat on both sides until it browns.
Transfer the meat to an ovenproof pot with a tight-fitting lid (cast-iron is best). Add the barley, onions, garlic, and stock and plenty of water to cover them. Cook in the preheated oven for 2 hours.
Meanwhile, prepare the turnip, carrots, and potatoes by giving them a good scrub and chopping them into bite-sized pieces. (You can peel them if you prefer, but keep in mind that a lot of the nutrients are in the skin.) Add them to the pot and check that there is enough liquid to cover them.
Return the pot to the oven and cook the stew for a further hour. It's ready when the meat is tender (give it an extra 10 minutes if it doesn't seem quite soft enough).
Once the stew is out of the oven, loosely pick the meat from the bones and stir this through the stew. Season generously with salt and pepper and stir through the lemon juice and flat-leaf parsley. Serve immediately.
## POT-AU-FEU
Pot-au-feu translates literally as "pot on the fire." This recipe has a long history in French cuisine. Traditionally, poorer families would have had to make do with barley or rice, a few root vegetables, plenty of bones, and probably very little meat. Nowadays, this dish is still a cost-effective way to make the meat you buy go a little bit further, producing not just one but three courses. With long, slow cooking, even the toughest cuts become beautifully tender. Despite its humble origins, pot-aufeu is still a favorite in France and served at many top-notch restaurants throughout the countryside. Serve it with crusty bread and some mayonnaise, aioli, mustard, or horseradish sauce.
_Preparation time:_ 5 minutes
_Cooking time:_ 6 to 7 hours
_Serves:_ 6
_Ingredients:_
5½ pounds bony cuts of beef (I recommend 2¼ pounds beef brisket, and 2¼ pounds marrow bones. But you can use any cuts you like, including tongue, ox cheek, or oxtail). If you have room in your pot, add a few extra bones for additional flavor
1 bouquet garni (or 1 teaspoon dried thyme, rosemary, and basil)
2 bay leaves, fresh or dried
½ cup brown rice or barley (optional)
2 large tomatoes
4 large carrots, topped and tailed but left whole
2 large leeks, cut into 2-inch pieces
2 medium onions, cut in half
2 turnips, cut in half (optional)
1 bulb garlic
1 large bunch flat-leaf parsley
Place a large stockpot on the stove. Add the meat, excluding the marrow bones, and cover with water. If you are using osso bucco, cut the bone out from the steak and set it aside, and add just the meat to the pot. If you are adding any additional cooking bones, add them now. Cover them with water.
Bring the water to a boil, then reduce the heat and simmer very gently, partially covered. As it cooks, a layer of scum may form at the top; you can skim this off with a wooden spoon and discard it. Continue gently simmering for 3 to 4 hours so that the meat can soften.
Add the herbs, rice, and vegetables (excluding the parsley and garlic) and continue to cook gently for an additional 1½ hours.
Add the marrow bones and cook for 20 minutes or until the fat in their centers is soft and runny.
To serve your first course, remove the marrow bones from the pot and place them on a serving dish with some crusty bread, fresh garlic cloves, and a handful of fresh parsley. Encourage your guests to rub the bread with garlic, scoop out the marrow, spread it over the bread, and sprinkle with salt and parsley.
For your next course, drain the broth from the meat and vegetables. Serve the broth with a little rice and a few vegetables in each bowl. Season to taste.
For the main course, arrange the meat and vegetables on separate platters and serve with mayonnaise, aioli, mustard, and some more crusty bread.
This dish can also be prepared in advance—the meat and vegetables served cold, and the broth and marrow bones re-heated before serving. This allows you to skim off the fat that rises to the surface when the broth is cooled.
## HOME-CURED BACON
Home-curing bacon is easy and extremely cheap. Pork belly is a terrifically thrifty cut, and home-curing it will preserve it for a period of months. Snippets of bacon are delicious in soups and stews, or fried with some eggs for breakfast. You can also boil it to make a traditional French dish known as petit salé. All the quantities in this recipe are approximate. The idea is to use enough salt and sugar to draw the water out from the meat, so go ahead and add a bit more if it looks like it needs it.
_Preparation time:_ 10 minutes
_Resting time:_ 5 to 7 days
_Ingredients:_
½ of 1 large pork belly, cut into thirds
17½ ounces fine sea salt
3 or 4 bay leaves, finely chopped
1 large handful fresh thyme, finely chopped
½ cup brown or whole cane sugar
2 tablespoons black peppercorns, roughly crushed in a mortar and pestle
Use a sharp knife to cut some slits, about ¼ or ½ inch deep, in the pork meat. Combine the salt, bay leaves, thyme, sugar, and peppercorns in a large bowl and toss well. Rub this mixture all over the meat.
Layer the pork belly one piece on top of the other in the bowl. The meat should be completely covered by the salt, so spread the salt around and add a little more if you need to. Place a sheet of greaseproof paper over the top and weigh it down with something (I usually use a bag of dried beans or oatmeal). Place the bowl in the refrigerator and leave it for 48 hours.
Remove the bowl from the fridge and drain off any excess water. Rearrange the meat so that the bottom slice is on top and the top slice is on the bottom. Pack any excess salt back onto the meat (if the salt doesn't seem enough to cover the meat, make up another mixture of salt and sugar). Cover the meat with the paper again, return the bowl to the fridge, and leave it for another 48 hours.
After 5 days, your meat should be ready to nibble on. I like to cure mine for 6 or 7 days, checking on it every 2 or 3 days. The longer it is cured, the tougher and saltier it will become and the longer it will last in your fridge or pantry.
When you have finished the salting process, you can wrap the bacon in a natural fiber (such as cotton or muslin), or fold some greaseproof paper over it and store it in a cool place such as a fridge or cellar. It should keep for a good couple of months. Slice off thin slices whenever you need it. If you have salted it for a long period of time, you may have to soak it in fresh water for an hour or two before using it to soften it up.
## HOME-CURED BACON SERVED WITH LENTILS
This is a staple dish in northern France, where it is known as petit salé aux lentils. It is an excellent and thrifty way to use up home-cured pork belly. If your bacon seems too tough or salty, soak it in filtered water for a few hours before cooking this dish.
_Preparation time:_ 10 minutes
_Cooking time:_ 2 to 3 hours
_Serves:_ 2
_Ingredients:_
1⅓ pounds home-cured pork belly
2 or 3 carrots
2 or 3 celery sticks
1 onion
1 bouquet garni (or ½ teaspoon each of dried thyme and rosemary, plus 1 bay leaf)
1 cup French green lentils
Place the pork belly, vegetables, and herbs in a saucepan and cover them with cold water. Bring to a gentle simmer and cook for 2 hours or until the meat is tender and soft. Remove the pork belly from the pot and set aside to cool. Discard the vegetables but retain the cooking liquid.
Add the lentils to the pot with the cooking liquid and turn up the heat. Simmer for 40 minutes or until tender.
Drain the lentils, season them with salt and pepper to taste, then transfer them to a serving dish and place the pork belly on top.
## MUTTON CURRY
Mutton (the meat from older sheep) has fallen out of favor in recent times, as sweeter and more expensive cuts of lamb became more widely available. This is a shame, as well-prepared mutton has a delectably subtle flavor. It requires longer, gentler cooking than lamb and is best combined with plentiful spices and herbs. If you can't find mutton chops, you can substitute lamb loin chops, which are one of the cheapest cuts of lamb.
_Preparation time:_ 5 minutes
_Cooking time:_ 1¼ hours
_Serves:_ 4
_Ingredients:_
1¾ pounds mutton or lamb chops
fat for frying
4 teaspoons curry powder
1 medium onion, finely chopped
1¼ cups coconut milk
1 bouquet garni (or ½ teaspoon each of dried rosemary, thyme, and marjoram)
⅓ cup plain yogurt
Preheat the oven to 350°F.
Place a frying pan over high heat and fry the chops with a little cooking fat until they brown, sealing in the flavor. Transfer the meat to a medium-sized casserole dish and coat it with the curry powder.
In the same frying pan, fry the onions over low heat for 5 to 10 minutes, or until they caramelize. Add them to the pot with the meat, along with the coconut milk and bouquet garni. Stir well.
Place the cooking pot in the preheated oven and cook at 350°F for 1¼ hours. Remove from the oven and stir through the yogurt. Season to taste and serve with brown rice and a fresh leafy salad.
## OXTAIL STEW WITH APPLES & SPICES
This hearty stew might take a little longer to prepare than your average dish, but don't be put off; it's delicious and well worth the effort. Oxtail is a traditional cut that seems to be coming back into fashion. Even at my neighborhood middle-of-the-road supermarket, I've spotted locally sourced oxtail on sale for less than the cost of 2 pounds of mince. It requires slow, gentle cooking for the best results. I highly recommend cooking this the day before you eat it, so that you can skim off the fat and get a rich and gelatinous sauce. The vegetables don't need to be super fresh for this dish, so it's a good way to use up anything that's starting to look a little wilted. Serve with rice and a leafy green salad.
_Preparation time:_ 20 minutes
_Cooking time:_ at least 6 hours
_Serves:_ 4
Best prepared the day before
_Ingredients:_
2½ pounds oxtail, cut into 1-inch pieces
fat for frying
2 cups stock
½ cup red wine
2 bay leaves
1 large onion
2 large carrots
2 large parsnips
1½ tablespoons tomato ketchup
1½ teaspoons Ras el Hanout spice mix
3 medium cooking apples salt and pepper
Preheat your oven to 300°F. Put a frying pan over a high heat, add a little cooking fat, and fry the meat in batches to seal in the flavor.
Place the stock, wine, and bay leaves in a heavy-based ceramic dish with a tight-fitting lid. When the meat is ready, transfer it to this pot and add enough water to cover it entirely.
Cook the stew in the preheated oven for about 5 hours, or until the meat is tender and beginning to fall off the bone.
Remove the pot from the oven and allow it to cool. Drain the meat from the liquid and place the liquid in a jug in the fridge. After several hours the jug of liquid will cool and a layer of fat will form at the top. Scrape this off and discard it. Similarly, any excess fat on the oxtail meat should be trimmed off and discarded.
When you're ready to prepare your dinner, put the casserole dish on the stovetop and add the meat and cooking liquid and bring it to a gentle simmer.
Finely chop the onion and fry it in a frying pan on medium heat until it caramelizes and turns golden.
Chop the carrots and parsnip into bite-sized pieces. Add these to the cooking pot with the onion, ketchup, and spice mix. Place the pot in the oven, or allow to simmer on the stove top, and cook for an additional 1¼ hours, or until the vegetables are tender.
Cut the apples into quarters and add them to the pot. Cook for a further 15 minutes (preferably on the stovetop, so that you can keep an eye on things), or until the apples are soft. If necessary, remove the lid and turn up the heat to boil down any excess liquid. Season to taste and serve.
## BAKED SWEET BREADS WITH BUTTER & SAGE
I confess I was a bit freaked out the first time I brought these little thymus glands home from my butcher. They were soft, mushy, and very weird to the touch, and I did have some reservations about cooking (let alone eating!) them that first night. But I finally drummed up the courage to dip them in egg and breadcrumbs and bake them in the oven, and this is now my _favorite_ offal dish. Please don't be dissuaded by what they look like raw. Their taste is superb, and they go beautifully with a crisp green salad.
_Soaking time:_ at least 30 minutes
_Cooking time:_ 50 minutes
_Serves:_ 4
_Ingredients:_
1 thymus gland from a calf
sea salt
butter
3 teaspoons apple-cider or wine vinegar
1 teaspoon whole black peppercorns
1 bay leaf
1 egg
salt and pepper
½ cup finely chopped sage
½ cup breadcrumbs
Soak the sweetbreads in cold, salted water (use roughly ¾ tablespoon of sea salt for every quart of water). If you're in a real hurry, 30 minutes will suffice, but an hour or two is better.
When you're ready to start cooking, preheat the oven to 350°F and grease a baking tray with butter.
Put the sweetbreads in a saucepan of warm water with the vinegar, peppercorns, and bay leaf. Bring to a boil and gently simmer for 15 minutes. Drain the sweetbreads and let them cool, then slice them into bite-sized pieces.
Combine the egg with salt and pepper to taste and whisk well. Finely chop the sage and, in a separate bowl, toss it with the breadcrumbs.
Drop the sweetbreads into the egg mixture and let them sit for 2 minutes. Dip them in the breadcrumbs, then place them on the baking tray. Bake for 30 to 40 minutes or until they are lightly browned. Be careful not to let them overcook, as they can dry out.
Serve with a green salad.
## STEAK & KIDNEY PIE
This is a favorite at our house. If you are new to offal, this is a good place to start. You can adjust the ratio of kidney to steak if you want more or less kidney flavor. If you slice the kidneys finely enough, they may even be mistaken for mushrooms! Although it takes a while to cook, don't be put off, as this is a great leftover dish and can save you time during the week. It can be made on the weekend and reheated for weeknight dinners or packed in school lunches. The filling can also be made in advance and frozen or refrigerated until you need it.
_Preparation time:_ 20 minutes
_Cooking time:_ 3 hours
_Serves:_ 6 (generously)
_Ingredients:_
2 quantities oatmeal pastry (see recipe on page 252)
4/5 pound beef or lamb kidney
2¼ pounds slow-cooking beef, such as beef skirt or chuck steak
fat for frying
1 cup red wine
1½ teaspoons soy sauce
2½ cups stock
2 small bay leaves
1 onion, finely sliced
2 teaspoons finely chopped fresh thyme
¼ cup tomato paste
1½ teaspoons mustard
1 pound fresh mushrooms
3 to 3¾ tablespoons arrowroot powder
1 egg
First, prepare your oatmeal pastry.
If your butcher hasn't already done so, chop the steak and the kidney into bite-sized pieces. Fry the meat in batches over a high heat with a little cooking fat.
When the meat is sealed, transfer it to a cast-iron cooking pot. Add the wine, soy sauce, stock, bay leaves, onion, thyme, tomato paste, and mustard. Bring the liquid to a boil, then reduce it to a simmer. You may need to add an extra half cup of water to ensure that everything is submerged.
Let it simmer on the stove for 2 to 3 hours, stirring occasionally so that nothing sticks to the bottom. Alternatively, you can place the cooking pot, with lid, in the oven and bake it at 350°F for the same period of time. I prefer to use the stove, as it lets me keep an eye on the liquid level.
While the meat is cooking, cut your mushrooms in half and pan fry them with a little fat for 10 to 15 minutes, or until they are cooked through and reduced in size.
When the beef is tender, boil down any excess fluid so that the liquid only just covers the meat and add the mushrooms.
At this point, preheat your oven to 350°C and grease a large rectangular pie dish (about 11 inches by 6 inches) with a little fat.
Remove about half a cup of liquid from the pot and combine it with the arrowroot powder. Return the resulting paste to the pot and stir well. The filling should now be thick and gluey.
Tip the filling into the greased pie dish.
Using your fingers, roll out the pastry between two sheets of grease-proof paper to make a lid. Place this on top of the pie. Whisk the egg and brush it onto the lid as a glaze.
Bake your pie for 30 minutes or until the pastry is golden.
## BRINED TONGUE WITH SALSA VERDE
This is a delectable dish with a tangy green sauce that really complements the saltiness of the tongue. I've served it to guests who were nervous about eating tongue and heard them raving about it years later. If you prefer, you can buy brined tongue from most good butchers. Whether you buy it ready-brined or prepare it yourself, served with salsa verde it makes for a delicious Sunday lunch.
_Soaking time:_ at least 24 hours
_Preparation time:_ 5 minutes
_Cooking time:_ 2 to 3 hours
_Serves:_ 6 (generously)
_Ingredients_
FOR THE TONGUE:
1 beef tongue
1 gallon water
21 ounces sea salt
2 cups sugar
2 teaspoons black peppercorns
1 onion
2 carrots
a few sticks celery (optional)
3 bay leaves
5 cloves
FOR THE SALSA VERDE:
1 slice stale bread
¼ cup olive oil
1 cup parsley (either flat or curly)
¾ tablespoon white-wine vinegar
2 anchovy fillets
1 clove garlic, crushed
half a preserved artichoke heart
salt and pepper
_To brine the tongue:_
To make the brine, combine the water, salt, sugar, and 1 teaspoon of the peppercorns. Bring them to a boil so that the salt and sugar dissolve, then remove from the heat and refrigerate.
When the brine is no longer hot, add the tongue and leave it to soak. The longer you leave it, the stronger the salty flavor will be. If you only have 24 to 48 hours, I'd suggest using half of the brine as cooking water when it's time to cook the tongue. Alternatively, if you have more time, soak the tongue for up to 7 days and rinse it in fresh water for a few minutes before you cook it.
After soaking, put the tongue into a large pot with the onion, carrots, celery, bay leaves, cloves, and the remaining peppercorns. Simmer for 2 to 3 hours. When it's ready, the tongue should be tender and the outer skin fairly easy to remove.
Remove the tongue from the pot and remove the skin while it is still warm (it becomes much harder when it's cold). Some cooks peel off the skin with their fingers, but I find it easier to use a small knife.
Thinly slice the tongue and arrange the slices on a plate. Sprinkle them with sea salt and place the plate in the refrigerator to cool before serving with the salsa verde.
To make the salsa, soak the bread in the oil for ten minutes, then combine all the ingredients in a food processor and blend until smooth. Season to taste.
## CRISPY LIVER WITH CARAMELIZED ONION
Some people recall bad childhood experiences when I mention liver. I admit, the first time I cooked liver it was terrible—overcooked, tough, and dry. It took me years to want to try it again—but once I got it right, I became seriously hooked.
When properly cooked, liver can be exquisite. The trick is to buy it as fresh as you possibly can, preferably grass-fed and organic. If you're a first-time liver-eater, try to buy lamb's liver, as it is milder than beef or pork. If your butcher offers to slice it for you, take him up on it. Liver should be as thinly sliced as possible, and it can be difficult to do this at home if you don't have a very sharp knife and good knife skills. I like to soak the liver in milk for a few minutes before dusting it in flour and pan-frying it; this results in a subtle taste and tender texture. If you have any leftovers, you can use them in pâté.
_Preparation time:_ 10 minutes
_Cooking time:_ 15 minutes
_Serves:_ 5 (generously)
_Ingredients:_
butter or coconut oil for frying
3 onions
1 fresh lamb's liver (about 1½ pounds)
1 cup fresh milk
flour or arrowroot powder for dusting
salt and pepper
Start by caramelizing the onions: slice them as thinly as possible and place them in a frying pan with 1 teaspoon of butter. Fry over low heat, stirring occasionally, for 10 to 15 minutes. The onion should soften, become sweet and tender, but not brown.
While the onion cooks, prepare the liver. If the butcher hasn't already done so, slice the liver as thinly as possible, cutting against the grain of the meat. In a bowl, soak the liver in the milk for 5 to 10 minutes.
Dust each piece of milk-soaked liver in the flour and fry for 5 seconds on each side in a small amount of butter. The inside should still be pink, but the outside brown and crispy.
Season with salt and pepper. Serve with the caramelized onion. Fruit chutney also complements this dish nicely.
## LIVER PÂTÉ
Good pâté is a special treat. It can be served on thick slices of crusty bread, with crackers, or in a sandwich of fresh arugula and creamy avocado. Many people think of it as a luxury food, but in fact pâté is very quick and cost-effective to make at home.
_Preparation time /Cooking time:_ 10 minutes
_Refrigeration time:_ minimum 1 hour
_Makes:_ 4 × 200mL jars
_Ingredients:_
1 small leek
200g butter–roughly chopped into cubes
1 tablespoon coconut oil or butter for frying purposes 500g fresh pasture-raised chicken livers cut into small (1cm) pieces
⅓ clove nutmeg, finely grated
¼ cup brandy
1 generous bunch curly-leaf parsley, coarsely chopped salt and pepper to taste
Clean the livers using a small sharp knife to remove and discard any white sinew. Remove and discard any green patches (which will give it a bitter flavour).
Finely slice the leek and add it to frypan with a spoonful of butter or coconut oil. Cook on a low heat until it softens and turns a golden colour. Remove from the fry pan and set aside.
Slice the liver thinly and measure out your brandy. Place a large frypan on a high heat and add the liver, another dollop of butter and coconut oil and pour over the brandy. Light the brandy with a match. It should set off a flame over the meat. Shake the pan until the flame dies down.
When the liver is 90 percent cooked (it may still have a slightly pink centre), remove it from the pan (note that it will continue to cook after it has been removed from the pan, so take it off as soon as it is near done).
Allow the liver to cool to room temperature, then combine the leek, liver and all remaining ingredients in a food processor. Process at a high speed so that the mixture becomes a fine paste. Season to taste.
Transfer mixture to sterilized glass jars. To prevent oxidation on the top of the meat, add some coconut oil, melted butter or chicken fat as an interior seal. Refrigerate immediately.
## MEATLOAF WITH RED SAUCE
Once upon a time, just about every housewife would have had her own prized meatloaf recipe. The beauty of meatloaf is its simplicity: it uses very cost-effective ingredients and can be served in multiple ways. It's particularly good for school lunches; it's excellent in a sandwich with some fresh lettuce and arugula. This recipe is based on a traditional Jewish meatloaf with a red sauce. Some cooks bake a boiled egg in the middle; I prefer to omit the egg, but I adore the red sauce.
_Preparation time:_ 10 minutes
_Cooking time:_ 45 minutes
_Serves:_ 6
_Ingredients_
FOR THE LOAF:
1½ pounds minced beef
1 onion, finely chopped
1½ tablespoons tomato paste
1½ pounds breadcrumbs (or 2 small slices of bread whirred in a food processor)
1 egg, lightly whisked
1 cup finely chopped flat leaf parsley or celery leaves
salt and pepper
butter, for greasing
FOR THE SAUCE:
6 very ripe tomatoes, coarsely chopped, or 1 can of whole tomatoes
1 or 2 teaspoons sugar
salt and pepper
Preheat your oven to 350°F.
Combine the mince, onion, tomato paste, breadcrumbs, egg, and parsley in a food processor, or mix them together in a large mixing bowl. Season with salt and pepper to taste.
Grease a baking tin (preferably loaf-shaped) with a little butter. Pour in the meat mixture and bake for 45 minutes, or until the meatloaf is cooked right through.
While the loaf bakes, prepare the sauce. Put the tomatoes, sugar, and salt and pepper to taste in a small saucepan and bring to a boil. Simmer for 5 to 10 minutes, or until the mixture has boiled down to a sauce-like consistency.
To serve, slice the meatloaf and pour sauce over each slice.
## PORK-MINCE APPLES
There is nothing quite so beautiful as a whole apple, golden and crispy, emerging from your oven. Stuffed with delicious pork mince, currants, and spices, these are very easy to prepare, and never fail to impress. If you don't have an apple corer, a small sharp knife will do the trick.
_Preparation time:_ 15 minutes
_Cooking time:_ 25 to 30 minutes
_Makes:_ 12 stuffed apples (serves 4 to 6)
_Ingredients :_
⅓ cup split peas
12 medium cooking apples (heritage varieties work best)
1 large onion, finely chopped
fat for frying
1 pound pork mince
1 teaspoon dried cinnamon
½ teaspoon dried nutmeg
¼ cup currants
⅓ cup wine vinegar
2¼ tablespoons whole cane or brown sugar
Preheat the oven to 350°F.
Place the split peas in a small saucepan of water and cook for about 20 minutes or until tender.
Using a small knife, slice a lid from the top of each apple. Set these lids aside (you'll need them later). Using an apple corer or small knife, dig a large cavity in each apple, leaving between ¼ and 2/5 inch around the sides and on the bottom.
Over a low heat, fry the onions with a little fat for 5 to 10 minutes or until they caramelize.
In a mixing bowl, combine the onions, pork mince, spices, and currants. Drain the split peas and add them to the bowl. Season to taste.
Fill each apple with this mixture, then put the lids back on.
Arrange the apples in a large baking dish so that they fit snugly and stand upright. Add ½ cup of water to the base of the dish, then bake the apples for 10 minutes.
Meanwhile, in a small saucepan, combine the butter, vinegar, sugar, and ½ cup of water. Bring to a gentle simmer and stir well.
Remove the apples from the oven. Lift each apple's lid, pour in a little sauce, then replace the lid. Return the tray to the oven and cook for a further 15 minutes or until the apples are soft and lightly brown, but not split or cracked. Serve immediately.
## MARROW ON TOAST
Bones are filled with a delicious and nourishing fat called "marrow," which occupies the cavities of larger, longer bones in livestock. This was highly prized in hunter-gatherer communities and considered just as valuable as offal and brains (other favorites of our ancestors). Fortunately, you don't need a spear and club to enjoy this dish! The bones are baked and you can very elegantly scoop out the fat and spread it on toast with some fresh parsley and garlic.
_Soaking time:_ 24 hours
_Preparation time:_ 5 minutes
_Cooking time:_ 15 minutes
_Serves:_ 6 (as an entrée)
_Ingredients:_
6 beef marrow bones, 1 or 2 inches long
sea salt
1 generous bunch of flat-leaf parsley
2 lemons
6 slices of rye sourdough or other wholegrain bread
6 (or more) large garlic cloves
Prepare the bones by soaking them for 24 to 48 hours with 1½ tablespoons of sea salt in a bowl of water in the fridge. Change the water once or twice if you can.
When you're ready to cook the bones, preheat the oven to 350°F.
Grease a baking tray with a little cooking fat. Drain the bones and arrange them on the tray. Cook them in the oven for 10 to 15 minutes, or until the marrow is soft and beginning to rise. It should bubble and pop and become a light golden color.
Meanwhile, finely chop the flat-leaf parsley. Juice both lemons and grate the rind of one of them. Combine the parsley, lemon juice, and lemon rind in a serving bowl and toss well.
Lightly toast the sourdough bread. Serve the marrow on a platter with the parsley salad, toast, and garlic cloves in separate dishes on the side. Each person takes a slice of toast, rubs it with garlic, adds a scoop of marrow fat, then sprinkles some parsley salad on top.
## MOROCCAN RABBIT HOT POT
I envy people who live in the country. I'd love to be able to go out to the back field and catch my own dinner. When an animal is killed this way, you have the assurance that it lived as close to nature as possible, and that its ending was swift. Rabbits are a major threat to our native fauna and flora, so here is a dish that can fill your belly and clear your conscience at the same time. You can often spot young kids selling rabbits for next to nothing on the side of the road in rural areas. Alternatively, you should be able to pick a rabbit up at your local farmers' market. The Moroccan flavors of this dish go well with wet polenta or brown rice and a green salad.
_Preparation time:_ 10 minutes
_Marinading time:_ 6 hours
_Cooking time:_ 30 minutes
_Serves:_ 6
_Ingredients:_
2 fresh rabbits
2 teaspoons black peppercorns
2 cloves garlic, crushed
1 teaspoon sea salt
1½ teaspoons cinnamon
1 teaspoon ground ginger
¾ tablespoon olive oil
8 large, ripe tomatoes, or 2 cans diced tomatoes
rind of 1 small lemon, cut into strips
½ cup sherry, white wine, or verjuice
2 cups chicken stock
a few large sprigs of thyme
1 red onion, finely sliced
1 bunch fresh coriander, finely chopped
If your butcher hasn't already done it for you, you'll need to cut the rabbit into joints. Cut at each shoulder and hip joint, then slice down the middle of each rabbit's back. Break the resulting bits into casserole-sized pieces that will fit easily into your cooking pot.
Combine the peppercorns, garlic, salt, cinnamon, ginger, and olive oil. Rub the mixture over the meat, then cover and refrigerate for 6 to 24 hours.
Preheat the oven to 350°F.
In a frying pan over medium heat, fry the meat for a minute or so on each side until it seals.
Put the meat, tomatoes, lemon rind, sherry, stock, thyme, and onion in a heavy-based pot with a tight-fitting lid. Ideally, the ingredients should take up ¾ of the space in the pot. Check that there is enough liquid to just cover the meat, keeping in mind that if you are using fresh tomatoes, they will release a lot of juice.
Put the pot in the oven and cook for 30 minutes, or until the meat is soft. Be sure not to let it overcook, as rabbit meat can become dry and leathery if cooked for too long.
Remove the pot from the oven and season the stew to taste. If you wish to thicken the sauce, remove the meat from the pot and set it aside while you let the sauce simmer, uncovered, for 5 to 10 minutes. Finely chop the coriander or parsley and stir them into the liquid.
Serve each piece of meat with a generous pouring of sauce.
## BAKED MEATBALLS WITH NUTMEG
These meatballs can be whipped up with very little fuss. Serve them as a snack at a party, or combine them with some vegetables for a main meal. They also work well in sandwiches and make an excellent lunchbox snack.
_Preparation time:_ 15 minutes
_Cooking time:_ 15 minutes
_Makes:_ 24 meatballs
_Ingredients:_
1 pound minced meat
2 small onions, coarsely chopped
½ clove freshly ground nutmeg or ½ teaspoon dried nutmeg powder
1 egg, lightly whisked
1 2/5 ounces breadcrumbs (or 1 slice of bread whizzed in the food processor) *
2 garlic cloves
½ cup flat-leaf parsley or celery leaves
½ cup basil leaves
olive oil
*For a gluten-free variation, replace bread with 2 tbsp buckwheat flakes.
Preheat the oven to 350°F.
In a food processor, combine the meat, onions, nutmeg, egg, breadcrumbs, garlic, parsley, and basil. Pulse the mixture a few times until well combined, then season to taste.
Grease a shallow baking tray with a little olive oil and rub some oil onto your palms to stop them sticking. Roll the mince into golfball-sized balls and arrange them on the tray. Bake for 15 minutes, or until the meatballs are cooked right through and no longer pink.
# FRESH FROM THE SEA
"Give me a fish and I will eat for a day.
Teach me to fish and I will eat for a lifetime."
—Chinese proverb
WE ALL KNOW THAT FISH IS GOOD FOR US. Most nutritionists recommend that people eat it several times per week. Seafood is rich in the omega-3 fatty acids DHA and EPA, which are known to benefit heart health, brain development, and metabolic functioning. Seafood is also an excellent source of important minerals, most of which reside in the bones. Smaller varieties of fish such as sardines and mackerel can be eaten whole, while the bones of larger varieties can be added to soups and stews to produce a nourishing and gelatinous broth.
For a long time, however, I had very little knowledge about which fish to choose, or how to cook seafood healthily on a low budget. I purchased the odd salmon steak or whole snapper from my local fishmonger, but felt confused whenever I tried to venture beyond the realm of pan-frying or simple grilling. For a novice, the thought of yourself with a knife in one hand and a squirming little creature in the other can be beyond daunting, and the choices involved in buying fish can be very confusing.
Many people tend to stick to old habits when it comes to cooking fish. It's easy to favor the basic staples such as tuna steak and ignore some of the less popular—but fresher and cheaper—varieties. It never ceases to amuse me that one country's firm favorite may be another country's oddity, and seafood that is overlooked in one part of the world may be highly prized in another. Eels are rarely eaten in the United States but are much loved in Europe. In Australia, carp are considered a pest suitable only for use as a fertilizer, while in some Asian and European countries they are treated as a delicacy. With this in mind, why not try something different the next time you are at the fish market? Find out what is available, fresh, and sustainably caught at your local fish market or fishmonger's. This chapter will cover the basics of choosing fish; cost-effectiveness, freshness, and sustainability. I'm also very much in favor of making the most of every little flipper we purchase—using the bones for stock and the whole fish for soups and stews. Fish were also traditionally pickled, air-dried, or fermented to last through the seasons when it was not available. I have included a few recipes that cater for this (just in case anyone reading this book does not possess a refrigerator).
Lastly, don't be afraid of failure. If it doesn't work, or if your fish goes stale in your fridge before you remember to cook it, simply get out the shovel and bury it in your garden. Rich in iodine and other minerals, a fish buried deep under your vegetable patch will do wonders for whatever you are growing.
## BUYING FISH
Price: Since most seafood is still caught in its natural environment, its supply, and consequently its price, is more erratic than that of meat and vegetables. As a general rule, seafood that is in season will be less expensive—which, luckily for frugavores, means that the cheapest seafood is often also the freshest. Wherever possible, choose fish that is in season (your fishmonger will be able to tell you which these are).
Even so, fresh fish can be expensive. But with some traditional peasant know-how, you can obtain good, nutritious fish at a fraction of the cost. If you are going to spend money on seafood, buy the best quality you can afford and use every little morsel. Here are my frugavore tips for your next trip to the fish market:
* Buy the whole fish wherever possible. As well as getting a better price, you'll get the bones as well as the flesh, and the bones can be made into delicious and nourishing soups and stocks.
* Try smaller varieties such as sardines and herring. These are rich in nutrients, but are often a fraction of the cost of larger fish. If you find them at a good price, buy them in bulk and pickle or preserve them.
* Make fish stock. Fish bones, heads, and carcasses are often given away for free (particularly at markets with a high turnover). Stock made from them is rich in vital nutrients, minerals, and iodine, and is extremely cheap and easy to make.
Freshness: Whenever possible, try to buy your fish on the day you want to use it. If you can, shop at a market close to the water's edge; this can be a good assurance that the product is fresh and locally caught.
How can you tell that a fish is fresh? For starters, the eyes should be clear and protruding, with black pupils and transparent corneas. The gills should be a pinky-red color, not brown. Your fishmonger may let you touch the fish: it should be soft and springy, like the rubber on a trampoline. As for the smell, when fish is truly fresh, it should smell only of the ocean, a lovely clean and salty fragrance. If it smells "fishy," don't buy it. The only exceptions to this rule are sharks, skates, and rays, which are best cooked and eaten a few days after they were caught. They contain a chemical called urea, which breaks down after they die, producing ammonia. A fishy smell from them is good—it indicates that the urea is disappearing.
It is always preferable to choose a whole fish and have it filleted than to buy pre-cut fillets, which will lack flavor and probably not be as fresh. If you do buy fillets, look for translucent rather than milky flesh. Fillets that are dry around the edges or show signs of discoloration will most likely be stale.
Sustainability: Traditionally, fish was a staple food for anyone who lived near the sea or had access to a friendly fishmonger. The day's catch would be displayed for locals to pick and choose from, either at the water's edge or at a local market. Choices were fairly easy to make, as people had a clear understanding of what was locally available.
Nowadays, when I visit the supermarket and see a label that says "salmon," "tuna," or "monkfish," my mind draws a blank. Where has this fish come from? Is it healthy? Contaminated with mercury? Sustainably or unsustainably farmed? These questions are reasonable. Buying fish is an ethical issue. Overfishing is now widely acknowledged as the greatest single threat to marine habitats. Over 70 percent of the world's fish stocks are now fully or over-exploited. Further damage is caused by unsustainable modern fish farms and "by-catch" such as turtles and dolphins being killed by modern trawlers.
So what should the frugavore cook be looking for? This is where a good rapport with your local fishmonger comes in handy: frozen fillets from the supermarket will not usually come with any information about where the fish came from, but your fishmonger will be able to tell you which fish are local, fresh, and sustainably fished or farmed.
Sustainable seafood can be sourced either from the wild or from fish farms. If you are buying wild fish, choose fast-growing, highly productive species and avoid fish caught by methods (such as trawling) that damage ocean habitats. If you can't get this information from your fishmonger, try exploring the online resources at the back of this book.
If buying farmed fish, you will want to be sure that they have been farmed using sustainable practices. As a solution to overfishing, farming is not all it's cracked up to be. Just like farms on land, modern aquaculture can produce chemical runoff from antibiotics, pesticides, and detergents. Not surprisingly, fish grown on farms are often not as healthy as wild fish; they have traces of chemicals and antibiotics in their flesh, and contain less muscle tissue and fewer omega-3 fatty acids than their wild counterparts. Carnivorous species, including salmon, tuna, cod, and shrimp, require more food to grow than they produce: it takes roughly three pounds of wild fish to produce one pound of farmed salmon or shrimp.
Truly sustainable farming uses organic feeding practices to farm a wide variety of fish. It involves small, closed aquaculture systems that do not destroy coastal habitats or depend on wild fish as feed. Smaller varieties of fish such as sardines, mackerel, and trout, as well as molluscs such as oysters, mussels, and clams, are best suited to farming, as they do not need wild fish as food.
To find out more about the ecological impact of fishing methods used in your area, get in touch with your local marine conservation group. In the United States, The Natural Resources Defense Council offers information about choosing sustainable seafood online at www.nrdc.org.
Mercury toxicity: There has been concern during recent years about rising levels of mercury in fresh seafood. Mercury and other toxic metals accumulate in larger varieties of fish such as shark, tuna, marlin, and Spanish mackerel. To avoid ingesting mercury, try to purchase the smaller varieties of oilier fish such as sardines, mackerel, and shellfish, which will have accumulated only minimal amounts of mercury.
## COOKING FISH
It's a real shame that so many of us are unfamiliar with what to do with a fresh fish straight off the boat. We have become so accustomed to buying single fillets or even fish fingers, crumbed, frozen, and boxed. But nothing quite beats the beauty of a whole fish fresh from the hold of a fishing boat. Taking it home with you to clean and fillet yourself is a special experience; you get to see the whole process from start to finish. It is also much more economical, the flavor is supreme, and you have access to all the nutrients contained in the bones, whether you eat them along with the fish (in the case of smaller varieties), or cook them up into a broth.
Some of the recipes in this chapter call for fillets of fish rather than the whole animal. It's still a good idea to buy whole fish for these dishes, as the bones and head are full of valuable minerals and can be used to make stock (the heads are particularly gelatinous). Ask your fishmonger to fillet the fish for you, and take the bones and the heads home in a separate bag.
Cleaning your fish: If you are lucky enough to catch your own fish, you'll need to clean and fillet it as soon as possible after catching it. Use the back of a knife to scrape off the large scales, working from the tail to the head. Then slit the fish open along the length of the belly up to the gills. Remove the gills, empty the cavity of all the guts and scrape away any dark blood. Rinse the fish well before placing it in the fridge or freezer.
Filleting your fish: If your fishmonger won't fillet the fish for you, take the whole fish home and do it yourself. Don't be daunted by this; simply make a cut in the flesh as close to the tail as possible. Hold the knife parallel to the backbone and, pressing firmly, slice along the flesh alongside the backbone until just before the gills. Remove the ribcage and the small line of bones that runs about 1¼ inches in from the thick end of each fillet, then rinse the fish clean of any excess blood or guts.
Steaming: A vegetable or pasta steamer can be used to steam fish, but the best implement is a Japanese bamboo steamer, which you can usually find for a couple of dollars at an Asian grocer or market. Make sure you take your saucepan in with you, or at least its measurements, so that you get the right size.
With the right equipment, steaming is quick and easy. A small fillet (¼ or ¾ inches thick) should steam in less than 5 minutes, and a larger whole fish (2 or 2 ⅓ inches 9 thick) should take 10 to 12 minutes. Firm-fleshed fish such as salmon, trout, or flounder are the best candidates for steaming.
Baking: This method works best for whole fish. Preheat your oven to 350°F. On a large baking tray, place your scaled and gutted fish and add any herbs or flavorings you fancy. Bake for 20 minutes per 2 pounds.
Pan-frying: With some butter and fresh herbs, this should please even the fussiest of fish-eaters. In Australia we often use fillets of King George whiting, fried with a squeeze of lemon and some freshly chopped parsley, chives, or tarragon.
Poaching: This method requires a little more time, but is well worth the effort. In restaurants they use professional fish steamers (rectangular steel pots where the fish lies on a perforated platform) but I have achieved very good results using just a deep frying pan and poaching the thick fish fillets with stock, a little wine, vinegar, and lemon juice.
## PRESERVES & PICKLES
Traditionally, anyone living close to the sea would have had all sorts of preserved seafood in their pantry. Traditional methods of air-drying, salting, and fermenting filled the pantry for times when fresh fish was not available. These traditional methods might not be as essential today, when we are not faced with the same periods of scarcity. But these dishes are fun to make and can be very useful to have on hand for weekday sandwiches, hors d'oeuvre, or any of those can't-think-of-what-to-eat situations.
## SEAWEED
Seaweed is rich in important minerals, particularly iodine, which is a common nutritional deficiency. It also contains a form of gelatin, which has a soothing effect on the digestive system. All sea vegetables can be added to stocks or used to flavor fish or vegetable dishes. The price of seaweed varies depending on where you shop. Asian supermarkets and grocery stores stock them fairly cheaply, whereas fancy organic stores sell them for much higher prices.
The best varieties to use for home-cooking include kombu, wakame, arame, and dulse flakes. Kombu, arame, and wakame can be added to soups and stews for extra flavor, minerals, and gelatin. Dulse flakes can be used in place of sea salt to flavor dishes.
## OVEN-BAKED SARDINES WITH OREGANO
Sardines are one of the most underrated fish. They are always cheap and are surprisingly delicious. Being a small and oily variety of fish, sardines are also full of healthy fats, and their bones (which are edible) are rich in minerals. The first time I made this dish I got goosebumps, they were so good.
_Preparation time:_ 5 minutes
_Cooking time:_ 10 minutes
_Serves:_ 1
_Ingredients:_
7 small or medium sardines
juice of half a lemon
sea salt
1 small handful fresh oregano, finely chopped
1 large clove garlic, crushed and finely chopped
1 medium red onion, thinly sliced
olive oil
Preheat the oven to 350°F.
If you bought the sardines as whole fish (rather than fillets), you will need to fillet them. Slice each sardine along its belly, then butterfly it out (spreading each of the side fillets out) and remove the guts. Chop off the heads, then rinse off any excess blood and guts under cold running water. Discard the head and innards (I usually bury them in the garden; they are excellent fertilizers and will give your herbs extra bite. They do especially well buried under parsley or chives).
Place the fillets, innards-side up, on a baking dish. Drizzle them with lemon juice and sprinkle with sea salt, oregano, garlic, and onion. Add a good dash of olive oil. Put the tray in the oven and cook for 10 minutes, or until the fish are lightly browned but still juicy and soft. Serve with a green salad or on some fresh crusty bread.
## BAKED WHOLE FISH WITH TOMATOES, HERBS, & FENNEL
This is an Australian classic, usually done with one large fresh snapper, although you can also use a large bream or a reef fish such as red emperor. You can chop and change the seasonings depending on what you have handy.
_Preparation time:_ 5 minutes
_Cooking time:_ 30 minutes
_Serves:_ 4
_Ingredients:_
1 large whole snapper, scaled and gutted
1 large red onion
1 large fennel
¾ tablespoon fresh thyme, finely chopped
4 or 5 large tomatoes
2 bay leaves
¾ cup dry white wine
2 slices crusty sourdough bread
½ cup finely chopped flat-leaf parsley
¾ tablespoon olive oil
Preheat the oven to 350°F.
Place the whole fish in a large ceramic or stainless steel tray. Finely slice the onion and fennel. Place half of the onion inside the cavity of the fish with some of the fresh thyme and 1 of the bay leaves. Arrange the rest of the onion and all of the fennel around the fish, along with the remaining thyme and the second bay leaf.
Finely slice the tomato and layer it over the onion and fennel, then pour over the white wine.
Rip the bread into tiny pieces or pulse it in a food processor so that it forms neat breadcrumbs. Toss these in a small bowl with the flat-leaf parsley and olive oil and season to taste. Spread the breadcrumb mixture over the top of the fish.
Place the dish in the oven and bake for 30 minutes at 350°F. To check whether the fish is ready, pull a portion of the flesh away from the bone. It should be white-colored and separate easily from the bone. Serve immediately with a green leafy salad and some baked potatoes.
## FISH PIE
Fish pie has long been a staple for people living near the water's edge. Any sort of fish will do, but I recommend a white-fleshed fish. Before you start, make sure the fillets are free of bones.
_Preparation time:_ 20 minutes
_Cooking time:_ 25 minutes
_Serves:_ 6
_Ingredients:_
1 large onion
1 medium fennel
7 ounces mushrooms
4/5 pound potatoes
3 eggs
1 pound white-fleshed fish fillets
2 tablespoons butter
¼ cup arrowroot powder
1 cup flat-leaf parsley, finely chopped
1 handful fresh dill, finely chopped (optional)
Preheat the oven to 350°F.
Finely chop the onion and fennel and fry them over a gentle heat for 15 minutes or until they are soft and lightly browned. Add the mushrooms and cook for a further 10 minutes or until the mushrooms have cooked through and reduced in size.
Scrub the potatoes and put them into a cooking pot. Cover them with water, bring to a boil, and cook for 10 minutes or until soft. Drain them, then either purée them in a food processor or mash them well with a hand masher. Season with salt, pepper, and butter to taste. Add the eggs one at a time, stirring to form a smooth paste.
Pan-fry the fish with a little cooking fat so that it is evenly cooked on both sides. Remove it from the pan and cut it into bite-sized pieces. Stir this through the mushrooms and fennel.
To make the white sauce, place the butter and arrowroot powder in a small saucepan. Stir over high heat until the butter melts and a smooth paste forms. Gradually add 1 cup of hot water in a steady, even stream, stirring continuously. Allow the sauce to simmer and thicken for a further minute.
Stir the sauce through the mushrooms and fish, then add the fresh herbs.
Pour this mixture into an ovenproof pie dish. Spread the mashed potato over the top to form a lid and add a few extra dollops of butter to the top.
Cook in the preheated oven for 25 minutes or until the pie top is golden.
## POACHED FISH WITH SABAYON SAUCE
Poaching fish is a wonderful way to keep it tender and moist. You also retain all the important minerals and the herbs from the poaching liquid. Poaching produces delicious results and works well with flathead or any other firm, white-fleshed fish. Sabayon is a traditional French sauce that goes well with any type of seafood.
_Preparation time:_ 5 minutes
_Cooking time:_ 15 minutes
_Serves:_ 4
_Ingredients:_
2 or 3 cups fish stock
1 red onion, finely sliced
juice of 1 lemon
2 or 3 cloves garlic, crushed
½ cup dry white wine
1 pound filleted fresh white fish
FOR THE SAUCE:
2 egg yolks
½ cup leftover poaching liquid salt and pepper
⅔ cup butter, cut into small cubes
¾ tablespoon finely chopped dill
In a deep frying pan, bring the stock, onion, lemon juice, garlic, and wine to a boil. Reduce the heat and simmer for 3 to 5 minutes so that the liquid reduces a little. Make sure you have enough liquid left to fully cover the fish as it cooks. If you don't, you may need to add some extra.
Add the fish to the pan and cook for 3 minutes on each side, or until the centers of the fillets are no longer pink. When the fish is cooked, remove it from the pan with the sliced onions and place it on a serving dish. Be sure to retain the cooking liquid in the pan.
To make the sabayon sauce, pour the cooking liquid through a sieve.
Put a small or medium bowl inside a saucepan and add enough water to the pot so that the water comes halfway up the sides of the bowl. Turn the heat to its lowest temperature and allow the water to get hot but not boil.
Place the egg yolks in the bowl with half a cup of the strained poaching liquid. Whisk them together well, then add salt and pepper to taste. Continue to whisk, adding the butter as you go, one cube at a time. Keep whisking and adding butter until the sauce thickens and coats the back of a wooden spoon.
When the sauce has reached the desired consistency, take the pan off the heat and continue whisking for about 30 seconds as it cools. Pour the sauce into a warmed jug and stir through the finely chopped dill before serving.
## MINCED FISH CAKES
These little fish balls are delicious with sour cream and dill, or sauerkraut and a baked potato. If you make them small enough, they are also a wonderful hors d'oeuvre, or packed in a lunchbox and taken to school or work.
I'm not saying you should deliberately buy fish that isn't fresh—but keep in mind that you don't have to buy the most expensive fish for these cakes. Any white-fleshed fish that is free of bones will do.
_Preparation time:_ 5 minutes
_Cooking time:_ 15 minutes
_Serves:_ 4
_Ingredients:_
1 small potato
1 pound filleted fresh fish
⅓ cup plain full-cream yogurt
½ teaspoon freshly ground nutmeg
¼ cup finely chopped parsley
1 large clove garlic, crushed and finely sliced
½ teaspoon finely chopped chili flakes (optional)
1 teaspoon butter salt and pepper
Preheat the oven to 350°F. Generously grease a large steel or glass baking tray with butter.
Boil the potato until it's soft, then mash it.
Combine the mashed potato with all the other ingredients in a food processer or by pounding them together in a mixing bowl with a potato masher.
Roll the mixture into 1-inch round balls with your hands. Place each ball on the baking tray and bake for 15 minutes in the preheated oven. Serve hot or cold.
## FISH SOUP
Fish soup depends on a good hearty fish stock, made from bones or offcuts and some fresh vegetables, herbs, and seasonings. Fish stock is highly nutritious—it is rich in iodine, minerals, and gelatin—and is also very cost-effective to make at home. The heads, tails, and bones of larger fish, sometimes referred to as "scraps" by fishermen, often sell for next to nothing at fish markets.
Traditionally, the contents of fish soup would vary depending on the day's catch and the local region's preferred flavors. Bouillabaisse is a superb French Mediterranean dish made with tomatoes and plenty of saffron. In Greece, a variation involves rice, eggs, and lemon. In Portugal and Spain, spicy flavors such as red peppers, paprika, and chili dominate. The best soups are made with what is available locally, with the seasonings and spices that you enjoy the most.
## FISH STOCK
Many fishmongers give away carcasses and fish heads for next to nothing, so fish stock can be ridiculously cost-effective. Fish heads are particularly nutritious, as they are rich in gelatin and fatty tissue and produce a deliciously rich and healthy stock. Avoid using oily fish such as salmon or mackerel when making stock. Their fragile oils will oxidize and create a very smelly stock.
_Preparation time:_ 5 minutes
_Cooking time:_ 30 to 40 minutes
_Makes:_ 4 to 5⅓ quarts
_Ingredients:_
2¼ pounds fish carcasses, heads, tails, bones, and other offcuts
¼ cup apple-cider or wine vinegar
2 bay leaves
1 handful fresh or dried herbs, such as thyme or rosemary
1 large onion
2 or 3 sticks celery, including the leaves
2 large carrots, coarsely chopped
_Optional extras:_
a few strips seaweed, such as arame or dulse
1 or 2 leeks, coarsely chopped
1 fresh knob of ginger
A dash of white wine
Place all the ingredients in a large stockpot and fill it with cold water. Slowly bring it to a boil, then reduce the heat and gently simmer, partially covered, for 30 to 40 minutes, or until the liquid is thick and rich. Check the stock occasionally and crush the bones as they cook so that they break and drop to the bottom of the pot.
When the stock is nice and thick, drain the liquid from the solids and store it in the fridge.
After stock-making, the bones and vegetables can be buried in the garden. Fish bones are an excellent source of iodine and other minerals. If you are worried about dogs or birds digging them up before they have a chance to break down, just place a few rocks over the spot where they are buried.
## FISH BROTH WITH LEMON & RICE
This recipe was born when there seemed to be _literally_ nothing to eat in my kitchen. With only some frozen fish stock, a packet of rice, and some eggs from the henhouse, I whipped this up for a late breakfast on a Saturday morning. I was surprised how very good it was, and it's now a regular part of my repertoire, with the addition of some lemon juice and parsley. I cook it whenever I want something simple, quick, light, and nourishing.
_Preparation time:_ 5 minutes
_Cooking time:_ 30 minutes
_Serves:_ 3
_Ingredients:_
1 quart fish stock
¾ cup rice
2 cloves garlic (optional) juice of 2 lemons
3 eggs
salt and pepper
1 generous handful flat-leaf parsley, finely chopped
olive oil
Bring the fish stock to a boil in a large pot, then reduce the heat and let it simmer gently. Add the rice and garlic and cook for 30 minutes, or until the rice is soft.
In a large bowl, whisk together the lemon juice and eggs until they are light and fluffy (an electric beater makes this easier, but a hand whisk does the job too).
Allow the fish stock to cool until it is still hot but no longer boiling. Pour it through a sieve into the bowl with the egg mixture, leaving the rice in the pot.
Whisk together the eggs and stock for 30 seconds, and then return them to the pot with the rice. Season with salt and pepper; the soup is now ready to serve. Ladle it into individual bowls and garnish with the parsley and olive oil.
## BEAUTIFUL BOUILLABAISSE
This is such an easy and delicious dish to put together. If you buy whole fish, you can fillet them yourself and use the bones and heads for stock and the fillets for the bouillabaisse. Alternatively, buy them filleted and make your stock in advance, and you'll be able to whip up this beautiful dish in less than an hour.
_Preparation time:_ 10 minutes
_Cooking time:_ 30 minutes
_Serves:_ 6 (generously)
_Ingredients:_
1 medium onion
2 or 3 garlic cloves
3 or 4 medium potatoes
2 medium fennel
3¾ quarts fish stock
3 cans (16 ounces each) diced tomatoes
½ cup dry white wine
1 long curl orange peel
½ teaspoon saffron threads
1¾ pounds white-fleshed fish such as blue grenadier, rockling, or cod
2¼ pounds mussels or other crustaceans
2 teaspoons finely chopped fresh thyme
1 large handful fresh parsley, finely chopped
Finely chop the onions. Crush and finely chop the garlic. Cut the potatoes into cubes or quarters, and slice the fennel into thin slivers.
In a large pot, combine the onions, garlic, fish stock, canned tomatoes, wine, and orange peel. Bring to a boil, then reduce to a gentle simmer. Add the potatoes, fennel, and saffron and cook for a further 10 minutes, or until the potatoes are soft.
While the stock simmers, prepare your fish. Cut the fillets into bite-sized pieces. Clean the mussels and have them ready in a bowl next to the stove.
Increase the heat under the pot and bring the stock to a strong simmer. Check the liquid levels; the broth should be thick and gelatinous and not too watery before you add the fish (simmer it a little longer if you need to). Add the fish to the pot and let it cook. Stir well and season to taste.
Add the mussels 5 minutes before serving. Cover the pot and increase the heat to a steady boil for 3 minutes. Serve in individual bowls with generous scatterings of parsley and a dash of olive oil.
## BERMUDA FISH CHOWDER
Bermudians love rum. They also love pepper and cloves, and this chowder combines these flavors sublimely. This dish is rich and gelatinous and requires long, slow cooking. Enjoy it with some crusty bread—and an additional dash of rum or pepper, if you want an extra kick.
_Preparation time:_ 20 minutes
_Cooking time:_ 2 hours
_Serves:_ 8 to 10
_Ingredients:_
2 onions
3 garlic cloves
8 small to medium potatoes (about 1¾ pounds)
6 medium carrots
1 stalk celery
2 teaspoons whole peppercorns
1 heaped teaspoon whole cloves
4 quarts fish stock
6 bay leaves
¾ teaspoon fresh thyme
4 cans (16 ounces each) chopped tomatoes
1½ tablespoons tomato paste
1½ tablespoons Worcestershire sauce
¼ cup rum
1½ pounds fish fillets
salt and pepper
1 generous bunch flat-leaf parsley, finely chopped
olive oil
Finely chop the onion and crush and finely chop the garlic. Peel and dice the potatoes and carrots and thinly slice the celery. Grind the peppercorns and cloves with a mortar and pestle.
In a large pot, combine the stock, bay leaves, onions, garlic, celery, carrots, potatoes, thyme, tomatoes, tomato paste, and Worcestershire sauce. Bring the liquid to a boil. Add the ground peppercorns and cloves, then reduce the heat so that the stock simmers and gurgles. Let it simmer, partially covered, for 45 to 50 minutes.
At the end of this time, cut the fish fillets into bite-sized pieces and add them to the pot along with the rum. Simmer, partially covered, for a further hour, or until the stock is thick and gelatinous and the vegetables are tender and soft.
Season with salt and pepper to taste, then serve with a generous sprinkling of parsley and a good dash of olive oil for each bowl.
## SALMON GRAVLAX (CURED SALMON)
Gravlax was traditionally made by Scandinavian fishing communities using a combination of salt and herbs to preserve the fish for long periods. The fish was packed with salt and buried in the ground so that it lightly fermented. The word "gravlax" comes from a Scandinavian word for "grave," while "laks" means "salmon." Now, there is no need for a garden burial; you can easily prepare this dish in your kitchen. Gravlax goes beautifully with sour cream, dill, pickles, and a dark rye bread.
_Preparation time:_ 10 minutes
_Refrigeration time:_ 1 or 2 days
_Ingredients:_
1 large salmon (at least 2¼ pounds)
2 cups brown sugar
2½ cups rock salt
¾ cup freshly chopped dill
¼ cup vodka
Have ready a large ceramic or glass dish, big enough for the fish to sit comfortably without spilling over the sides.
Remove the small bones from the thickest part of the fillet using tweezers. Combine the sugar, salt, dill, and vodka in a small bowl and rub the mixture over the sides of the fish. Sprinkle some of the mixture into the bottom of a large dish and place the fish on top. Pack the rest of the mixture around the sides of the fish.
Cover the fish with a sheet of greaseproof paper and weigh the paper down using a bag of sugar or dried beans or something similar. This will ensure the fish is packed in tightly and that water is able to seep out as it is drawn from the fish.
Place the dish in the fridge for at least 12 hours. I usually refrigerate mine for about 24 hours. The longer it's in there, the stronger the flavor will be. Every 12 hours, turn the fillet over and drain out any excess water.
When it's ready, thinly slice the fillet and serve with bread, sour cream, and fresh dill. It will last for a good couple of days in the fridge.
## QUICK-PICKLED SARDINES
At our local organic health-food store, they sell neatly packaged sardines in olive oil for seven dollars per can. Each can contains three little sardines and a bit of olive oil. When I told my fishmonger I'd been buying these, he had a good laugh. For the price of one can you could buy a whole 2 pounds of fresh sardines. He thought it was an absolute joke that someone would pay so much for what she could easily make herself at home.
Pickled sardines can be thinly sliced and added to sandwiches or salads. I like to make a dip with finely chopped pickled fish, sour cream, dill, and finely chopped cucumber. They are also delicious served alongside oven-roasted cherry tomatoes and a thick slice of sourdough bread.
_Preparation time:_ 15 minutes
_Refrigeration time:_ 5 hours
_Makes:_ 1 jar
_Ingredients:_
1 pound fresh sardines, whole or filleted
sea salt
1¼ cups wine vinegar
olive oil
1 teaspoon thyme
A few slices fresh chili (optional)
¼ cup water
1 bay leaf
1 teaspoon peppercorns
lemon juice (for serving)
_Note:_
See the "Preserves" chapter for instructions for sterilizing glass jars.
If you bought the sardines whole, you will need to fillet them. Slice each sardine along its belly and remove the guts. Remove the spine from the center of each fish. Chop off the heads and fold open the flesh. Rinse off any excess blood or guts.
In a small saucepan, add the vinegar, onion, herbs, bayleaf and water. Bring to a boil, and reduce to a simmer. Continue simmering for ten minutes, then remove from heat and allow to cool to room temperature. Place the sardines in a small-sized, non-reactive dish and cover with the vinegar solution. Make sure the sardines are fully immersed in the liquid with no exposure to air. Place them in the refrigerator to pickle for a minimum of 5 hours (if you are using larger species of fish, I recommend leaving them for a minimum 12 hours). Remove the dish from the refrigerator and drain off any excess fluid. Squeeze a little lemon juice on each fillet before serving.
# GOOD GRAINS
"How can a nation be great if its bread tastes like Kleenex?"
—Julia Child
I LOVE COOKING WITH GRAINS, BE IT AN oatmeal slice filled with fruit or a freshly baked loaf of sourdough bread piping hot out of the oven. There are very few weekends when I don't have something rising or fermenting in my kitchen. Grain-based foods, when sourced and prepared correctly, can be highly nutritious and seriously good to eat.
Without the right preparation, however, grains lose most of their nutrients and fiber and can be difficult to digest. So if you are one of the many people who don't really like bread, porridge, or oatmeal, or react badly to the soft and fluffy loaves from the supermarket, keep reading. This section covers some time-honored staples—oats, polenta, rice, and bread. The recipes all employ traditional cooking techniques to deliver a tasty and highly nutritious product with minimal fuss. I hope they will make you fall in love with traditional bread-making and old-fashioned oatmeal slice, just like I did.
## WHOLEGRAINS
A wholegrain consists of a husk, germ, and endosperm. It is a highly nutritious ingredient, full of fiber and many essential nutrients. But when a wholegrain is processed to make an industrially produced bread or cereal, the germ and endosperm are removed—along with most of the fiber and protein and a significant proportion of the vitamins. The result is very different from traditional peasant-style breads and gruels.
Wholegrains are quite volatile and very sensitive to heat and light. When you are preparing them at home, they need to be stored in a fridge or freezer, or used very quickly to prevent them from rotting. In comparison, refined grains, which are stripped of many of their nutrients, can last for months on the supermarket shelf. You might see a "wholegrain" label on your packet of cereal or loaf of bread. Don't be fooled. These have usually been treated with a range of fungal inhibitors, additives, and preservatives to prevent them from rotting, so they don't have the natural goodness of a traditional wholegrain product.
Food companies are well aware of the many nutrients that are lost during grain refinement. To compensate, breads and cereals are often fortified with iron, B vitamins, and fiber. The problem with fortification is that many of these nutrients can only be assimilated with the help of naturally present enzymes and coenzymes that are not present when foods are fortified.
Peasant-style cooking methods, by contrast, not only used the whole grain; they also employed preparation methods such as soaking, sprouting, and fermenting that were designed to enhance the available nutrients and break down any indigestible components, such as phytates, that are naturally present in grains. To my mind, everyone should be preparing their own grains at home, or looking for a good baker to do the same job.
## BREAD
Do you love your bread? Do you relish every nuance of taste and texture? Does it digest well? And does it fill you up? A good loaf, rich in wholegrains and leavened for a period of days, should be filling and satisfying, giving you plenty of energy and vigor and keeping you going between meals.
Real bread, like our great-grandparents ate, was made from little other than flour and water. Commercial yeast was only developed during the late nineteenth century. Before this, bread-makers had to rely on the naturally occurring yeasts and lactobacilli found in flour to get their loaves to rise.
Peasant-style bread involves a traditional leavening process. A "starter" is made using flour and water, which turn into a bubbling concoction of live bacteria and yeast. You use this to ferment your loaf and to make your dough rise. Sourdough bread has its characteristic sour flavor as a result. During the leavening process, the healthy lactobacilli in the starter create an acidic environment, which ferments or pre-digests the proteins in the flour. Phytic acid, an "antinutrient" that inhibits the absorption of calcium, iron, magnesium, and zinc, is broken down during this process, as are protease inhibitors, which interfere with the digestion of proteins such as gluten. Anecdotally, many people who cannot tolerate commercially made bread find they have no trouble digesting traditionally made loaves.
Sourdough bread also has a lower glycaemic index than conventionally made bread. It is usually thicker and heavier, which means you feel full after only a few slices. If you buy ready-made bread, keep in mind that sourdough is much better value for money; in nutritional terms, one loaf of sourdough is equivalent to two loaves of conventional bread—it will fill you up more quickly and contains a greater range of important nutrients.
Bread-making is not hard. Once you get into a rhythm, it takes just ten or fifteen minutes in the evening. Then you leave the dough to rest overnight, switch the oven on in the morning, and allow your darling loaves to rise and bake. Easy. If you don't fancy doing this often, you can bake vigorously for a period of a week, then freeze your loaves in bulk. They do not dry out in the freezer, and in fact taste beautifully fresh if you defrost them slowly in a warm oven.
If you are really averse to bread-making and are thinking of skipping this section altogether, that's fine. Just make sure you find a good baker—one who uses organic flour with no odd-sounding additives to make their own sourdough loaves. That is the next best thing to baking your own. But I might add, every time I make bread I feel it should be a cause for celebration. Yeast and bacteria are everywhere—in our hair, on our skin, and in the air we breathe. Sourdough fermentation takes advantage of this, tapping into these naturally present ingredients in your home. Each loaf of bread will be different, reflecting the scents of your kitchen, the air of your neighborhood, and the touch of your fingers.
## CHOOSING A FLOUR
Flour is a volatile food product. It needs to be used quickly, or stored in a fridge or freezer. To avoid it going rancid, many home bakers like to buy their flour as whole wheat (also called "wheat berries") and use a flour mill to grind it themselves. If you are not grinding it yourself, look for stone-ground, organic varieties of wheat. Non-organic flours often do not work in sourdough bread-making because they contain various preservatives and mold inhibitors, which can prevent the fermentation process from taking place.
Commercial "plain flour" is made from wheat that has been specially bred to produce a higher yield and a more glutinous loaf. Many people therefore prefer traditional grains such as rye, spelt, and kamut for both taste and nutritional reasons. If you are a first-time bread-maker, however, I suggest that you use at least 50 percent plain or spelt flour to ensure a good rise in your first couple of loaves. You can experiment with different types of flour once you're a bit more confident. When it comes to your starter, I've found that rye flour works exceptionally well; try to use at least 50 percent rye flour in your starter if you can. Generally speaking, I use a variety of different flours and mix and match as I go. Plain flour is usually a lot cheaper than spelt. If you want to save on costs, you can also buy your flour in bulk and store it in the freezer.
## BREAD-MAKERS
I have never used a bread-maker and don't believe they are necessary to make good bread. Some of my sourdough-minded friends do have bread-makers and have adapted these recipes for their machines with great success.
## PREPARE YOUR KITCHEN
Make sure you have a clean kitchen that hasn't been cleaned with chemical cleaning products. A sterilized workbench will kill the healthy micro-organizms in the starter. Wash your hands thoroughly before you start and keep your fingernails short. You can also soak your hands in a 50:50 vinegar and water solution for ten minutes before you start.
## MAKING YOUR STARTER
The starter is a beautiful and diverse living product, a combination of wild yeast, fungi, and several strains of lactobacilli. It goes to work on the wheat flour, digesting the peptides and breaking down the gluten and phytic acid that are naturally present in wheat.
Making a rye-grain starter is easy. Get some fresh organic rye flour and filtered, non-chlorinated tap water and you are good to go. The chlorine in tap water will kill many of the micro-organizms required for fermentation. If you don't have access to filtered water, boil your tap water for at least ten minutes and allow it to cool before using it. When following the steps below, keep in mind that the exact time it takes for your starter to develop will depend on your kitchen environment, temperature, and choice of flour.
DAY 1: In a small bowl, combine ¼ cup flour with ¼ cup filtered water. Stir well to form a smooth paste. Cover the bowl with a tea towel and leave it on the kitchen counter. If you have a chance, give the starter an additional stir later in the day.
DAY 2: In the morning, add a heaped teaspoon of flour and a similar quantity of water to the bowl and stir well. The idea is to keep the same consistency, and the same proportion of flour to water, as you achieved on the first day. Repeat this process again in the afternoon if you can.
DAY 3: Now you can start feeding your starter a bit more. In the morning, add two heaped teaspoons of flour and an equal quantity of water and stir well. Repeat in the afternoon if possible.
DAY 4: Repeat as per day three.
DAY 5: You should be noticing some good bubbles by now. The next step involves a little guesswork. You want to add a quantity of flour equivalent to half the amount of starter. So if it looks like you have 1 cup of starter in your bowl, add ½ cup flour, plus enough water to maintain the same smooth consistency. In the afternoon, add ¼ cup flour and an equal quantity of water for some extra nourishment. Stir well.
DAY 6: Your starter should be ready by now, but you can leave it another day if you like. The bubbles should be big and rich, just like honeycomb. If this is the case, you can start your bread-making! If there are still no bubbles, throw it out (or bury it in your compost) and start again.
Kefir-sourdough starter:
A small amount of active, bubbling kefir (see page 264) will activate your starter very quickly. This is a great way to short-cut the starter making process, and it always delivers superb bread-making results. Day 1: Combine 1 cup active, bubbling kefir with 1 cup rye flour. Mix well. Place in a glass bowl on your benchtop, in a warm spot and cover with a tea towel. Keep it in a place that is easily accessible and stir it as often as possible. Day 2: Add 1 tablespoon kefir and 1 tablespoon rye flour to the starter culture. Stir well. Continue to rest it in a warm, dark place and stir frequently. Day 3: Repeat as per day 2. You may see some bubbles today! Day 4: Repeat as per day 3. The bubbles should be fully active by now. You can use this starter to start making bread!
## MAINTAINING YOUR STARTER
For good, flavorsome, well-risen loaves, you need to keep the sourdough culture bubbly and active (by giving it a lot of attention, in other words). In warm climates (between 68 and 86°F) it will grow quickly, bubble a lot, and require a lot of nourishment. In cooler temperatures, it will slow down and hibernate. If you are going away for a week, or won't be baking for a while, you can keep your friend in an airtight container in the refrigerator and it will lie dormant. You'll just need to build it up again with regular feedings and give it some warm air before your next bread-making session. A layer of light brown liquid (known as "hooch" ) may form on the top of the starter while it is in the fridge. Simply skim this off and discard it before getting started.
The starter likes to be constantly fed and used, so try to feed it every day. At the minimum, add ¼ cup flour and an equal quantity of water. If you really want to build it up, add half of its bulk in flour and an equal amount of water. Give it a good stir each time you feed it. If you bake every day, you can replenish your starter each time you use it. On days when you are not baking, just add a little flour and water to the starter to feed it and keep it happy. I often go on a baking blitz: I'll bake several loaves in one go and freeze every second loaf I make. You can keep your starter in the fridge from Monday through Wednesday, then take it out on Thursday and start building it up for a baking session on Saturday. If it has been in the fridge it will take a few days of feeding and warmth to get back to its peak of liveliness.
## BASIC SOURDOUGH BREAD
This recipe is for a basic sourdough—simple, wholesome and delicious. Once you've got this down pat, you can experiment with other flavors.
_Preparation time:_ 15 minutes
_Rising time:_ 24 hours
_Cooking time:_ 1 hour
_Makes:_ One 9¾-inch loaf
_Ingredients:_
1 cup starter, alive and bubbling
8 cups flour (spelt or plain)
filtered water
1 handful polenta
olive oil
sea salt
_Variation:_
Herb bread: When you add the salt and flour to the bubbling bread mixture, also add 1 teaspoon each of dried coriander, fennel, and cumin.
In a large bowl, combine the starter with a little more than 3 cups of the flour. Add just enough filtered water (usually no more than 1½ cups) to form a smooth paste, similar in consistency to the starter. The water quantity will vary depending on the moisture content of your starter. Stir well, then cover the bowl with a tea towel and leave it to sit at room temperature for 12 hours or overnight. It should be bubbling by the morning.
Now you're ready to make your bread. Before you get your hands dirty, have your equipment ready to go. You'll need a large mixing bowl, wide but not too deep. You'll also need a large cutting board—or clear and clean your kitchen counter so that you have room to knead your loaf directly on it. Have your flour ready and a jug of fresh, filtered water on hand, just in case you need extra. Dust your bread tins with polenta; this will prevent the bread from sticking. Lastly, clear plenty of room: things can get messy! As you work, use the olive oil to grease your hands and prevent stickiness.
Add the sea salt and 4 cups of the flour to the bubbling bread mixture. Mix well so that breadcrumbs form, gradually adding a little more water if necessary to maintain a smooth consistency. If the dough gets too sticky, add a little more flour, but no more than 1 more cup.
Work the dough into a round ball. Place it on your chopping block or bench and start kneading, using firm but gentle wrist motions. This is the only time you'll need to knead your loaf, so give it a good 10 to 15 minutes of your attention. The dough should feel soft and springy. If you poke it, it should slowly spring back to its original form. If it feels like hard pastry, either you haven't kneaded it enough or your starter wasn't active enough. If it's the latter, your loaf might not work.
Once you're done kneading, place your ball of dough into the baking tin. The loaf will double in size, so make sure there's room in the tin to accommodate this.
Next, you'll need to find a place for your tin to sit while the dough rises, preferably somewhere around 17 or 18°C and protected from drafts. I usually keep mine in the unheated oven, where it's safe from flies and moths. Leave your dough to sit for 3 to 4 hours, or until risen significantly.
The next morning, place your loaf in the oven (if it isn't already there) and _then_ turn the oven on to 350°F. As the oven slowly heats up, it will add an extra rise to your dough. Most modern ovens will take about 10 to 15 minutes to get to the desired temperature. Once the oven reaches 350°F, bake the bread for 45 minutes. So if you turn the oven on at 7.30 a.m., you can take your loaf out at 8.30 a.m.
A perfectly baked loaf will be well risen and come away from the tin at the sides. To remove it from the tin you should only need to tip it upside down. To check if your loaf is ready, tap it gently; if you hear a nice hollow sound, the loaf is well cooked.
Once it's out of the tin, let your bread rest on the counter for at least 10 or 15 minutes before you slice and eat it.
## SOURDOUGH FRUIT LOAF OR FRUIT BUNS
In a large bowl, combine the starter with 3 cups of the flour. Gradually add water, stirring as you go, until it forms a smooth paste, similar in consistency to the original starter. Cover the bowl with a tea towel and leave it to sit at room temperature for 12 hours or overnight. It should be bubbling by the morning.
_Preparation time:_ 15 minutes
_Resting time:_ 12 hours or overnight
_Cooking time:_ 1 hour
_Makes:_ One 9¾-inch loaf, or one tray of 16 buns
_Ingredients:_
1 cup bubbling starter
7 cups flour (spelt or plain) filtered water
1 handful polenta
1 teaspoon sea salt
2 teaspoons cinnamon
2 teaspoons ginger
1 teaspoon nutmeg
1 cup chopped dates
1 cup currants
1 cup rapadura sugar
2 teaspoons apple-cider vinegar
1 orange
olive oil
When you're ready to start baking, gather your equipment and ingredients and dust your baking tin with polenta.
Add 3 cups of the flour to the bubbling mixture, stirring with a wooden spoon.
Add the cinnamon, ginger, nutmeg, dates, currants, sugar, and vinegar. Juice the orange and add half the juice to the mixture (retain the rest of the juice for other cooking, or drink it later). Finely grate the orange rind and add it, too. Stir the dough well, adding a little water if necessary to maintain the smooth consistency.
Grease your hands with a little olive oil. Gradually add the last cup of flour to the dough, while kneading it in the bowl with your hands. Knead the dough in the bowl until it becomes thick and heavy and can easily be taken out as a single ball, then place it on your countertop and knead hard for 10 to 15 minutes. It should feel springy and light, and not in any way dense or hard like pastry. If you press your fingers into the dough, it should slowly spring back into shape.When you have finished kneading, place your loaf in the bread tin. Or, if you are making buns, roll the dough into small balls and arrange them on a baking tray. The dough will double in size, so be sure to leave enough room in the tin, or enough space between buns, to accommodate this.
Place the tin or tray in the unheated oven and leave it to rest for 2 to 3 hours. Once the dough has rested, turn the oven on to 350°F. When the oven reaches 350°F, bake the bread for a further 45 minutes, or 30 minutes if you are making buns. If you're baking a loaf, you should only need to tap on it to remove it from the tin. Once the bread is out of the oven, let it rest on the counter for at least 10 or 15 minutes before you slice and eat it.
## PORRIDGE
People who claim that they don't like porridge have not tasted the real thing. Soaking the oats before you cook them imparts a delicious, tangy flavor. Porridge can be soaked overnight or even for a period of days. The mixture won't go bad, and the texture will become softer and more delicious. Once you get into the habit, putting the oats on to soak before you go to bed becomes second nature and makes for a delicious and nourishing breakfast. I love my porridge drizzled with butter or smothered in cream. You can also sprinkle some fresh fruit or desiccated coconut over the top. Instead of buying artificially flavored oats, raid your pantry, fridge, or backyard; dried coconut, fresh fruit, and plain yogurt are all excellent additions.
_Soaking time:_ overnight
_Cooking time:_ 5 to 10 minutes
_Serves:_ 2
_Ingredients:_
1 cup rolled oats (you can also try rye, spelt, or barley oats)
½ teaspoon apple-cider vinegar or the juice of half a lemon
½ teaspoon sea salt
_Variations:_
¾ tablespoon of plain yogurt added to the soaking oats will give your porridge a smoother, creamier texture.
_Before serving, try adding:_
* ½ cup finely chopped dates or dried currants
* ½ cup desiccated coconut
* 1 apple, pear, or peach, finely chopped
* honey or maple syrup and some fresh cream or yogurt
Put the oats, vinegar (or lemon juice), and salt in a saucepan, cover them with water, and leave them to soak overnight.
In the morning, place the saucepan on a low-to-medium heat and cook for 5 minutes or until the porridge is smooth and creamy. Remove from the heat and add any flavorings you like.
_Note:_
Try to buy organic oats wherever possible. They are generally only marginally more expensive than conventional oats and are well worth the extra money. "Quick cooking" oats have been finely chopped; they are still nutritionally good for you, but whole oats are better. Oats are known to be good for the heart, as they are rich in indigestible carbohydrates called beta glucans that have been shown to lower blood cholesterol. They also contain a number of phenolic compounds that have been shown to have antioxidant effects. Neither oats nor barley have any gluten-producing proteins, but some people who suffer from gluten-sensitivity can still have a reaction to oatmeal products.
## PORRIDGE CAKE
Ever wonder what to do with the leftover porridge stuck to the bottom of your saucepan? You could always give it to your chickens (chickens love leftover porridge; it's like Christmas dinner for them). But after doing this one too many times, I had an idea: soft-centered, beautifully crisp porridge cake. Excellent with a dollop of cream or yogurt, or a cup of tea in the late afternoon . . . "high tea" in true frugavore style, turning leftovers from breakfast into an afternoon treat. Easy, frugal—and truly delicious.
_Preparation time:_ 15 minutes
_Cooking time:_ 60 minutes
_Serves :_ 6
_Ingredients:_
2 cups leftover porridge (or a combination of leftover and fresh porridge)
3 eggs
½ cup coconut oil
¾ cup brown sugar or 1 cup whole cane sugar
juice and rind of 1 lemon
¾ cup desiccated coconut
a few drops vanilla essence
1 teaspoon dried cinnamon
½ teaspoon dried ginger
Preheat your oven to 350°F and grease two 9¾-inch cake tins. Combine all the ingredients in a food processor or a large bowl and mix well. Place the mixture in the greased tins and bake for 50 to 60 minutes. The filling will remain moist but the top will turn golden and crispy.
Serve with cream and a drizzle of maple syrup.
## COCONUT SLICE
If I had to choose one dish to embody my entire university experience, it would be this one. I'd often put this slice together late at night, then spoon it into a pan in the morning and bake it while I had my shower. Piping coconut slice would accompany me either between my knees as I drove to work, or at the bottom of my backpack as I rode my bike to class. It was the perfect student dish: incredibly delicious, incredibly easy, and incredibly cheap. You can also use the recipe for stewed pears (see page 300) as a topping, instead of bananas.
_Preparation time:_ 10 minutes
_Cooking time:_ 35 minutes
_Serves:_ 8
1 ½ cups fine desiccated coconut
¼ cup buckwheat or coconut flour
½ cup coconut oil melted juice and rind of 1 orange
juice from ½ lemon
1 teaspoon vanilla essence
2-3 tablespoons honey or maple syrup
4 eggs, separated
¼ teaspoon baking powder
¼ teaspoon bicarbonate of soda
2 large ripe bananas, sliced into long, thin, strips
Beat the egg whites until stiff. Slowly stir through the dry ingredients, followed by the citrus fruits, vanilla, honey, coconut oil and egg yolks. Mix well. Pour mixture into a 20 x 20cm backing tray and layer with bananas on top. Drizzle with additional honey if desired. Bake in a pre-heated oven at 180 degrees Celsius for 35 minutes or until slice is lightly brown and crisp, and skewer comes out clean.
## OATMEAL PASTRY
This recipe makes a fluffy and delicious pastry, which you can use for sweet or savory dishes (I use this recipe elsewhere in this book). You can also experiment with certified gluten free oats and buckwheat flour to make it a gluten-free pastry.
_Preparation time:_ 15 minutes
_Resting time:_ overnight
_Makes :_ 1 quantity (enough to make a pastry base for a 12-inch pie)
_Ingredients:_
½ cup yogurt
1¼ cups plain flour (or buckwheat flower)
½ cup rolled oats
¼ cup butter or coconut oil
¼ teaspoon sea salt
½ tablespoon brown or whole cane sugar (optional, for sweet dishes)
juice and rind of one lemon (optional, for sweet dishes)
Combine all the ingredients in a food processor. Blend them until they are well combined; the oatmeal should be finely ground and the mixture sticky and gooey. Add a few teaspoons of cold water if necessary. Transfer the mixture to a bowl and leave it to rest overnight at room temperature.
The next day, grease a baking tin and, if you'll be baking the pastry now, preheat your oven to 350°F. If your pastry is too warm and sloppy to work with, place it in the refrigerator and allow it to set for 10 minutes.
Use your hands to fold the pastry out into the baking tin. The pastry should be supple enough for you to shape it by hand; you shouldn't need to add any more flour. If it starts to stick to you, you can grease your hands with a little olive oil and start again.
If you prefer a smoother, thinner pastry, you can roll out the pastry between two sheets of grease-proof paper and spread it with a rolling pin. Place it on the baking tray so that it sits flat.
_To bake blind:_
Some pie recipes call for you to bake the pastry "blind." This simply involves baking the pastry by itself for a short time before adding your pie filling. Once you have put the pastry in the tin, cover it with greaseproof paper and fill it with enough rice to weigh the paper down (you can buy some cheap rice for this purpose and use it more than once). Bake it at 350°F for 15 minutes, then remove it from the oven and take out the rice and greaseproof paper. Return the pastry to the oven and bake it for a further 5 minutes. Allow the pastry to cool to room temperature while you prepare the filling.
_To make an egg glaze:_
You can make a glaze for the pie top by whisking 1 egg in a small bowl with a fork. Pour half of the whisked egg onto the pastry and spread it with a pastry brush or the back of a spoon, then bake as normal.
## OATMEAL PIKELETS
These are a wonderful breakfast to wake up to on a Saturday morning. I like to serve them with plain yogurt and a drizzle of maple syrup, but they also go nicely with fresh jam, honey, or marmalade.
_Preparation time:_ 15 minutes
_Refrigeration time:_ overnight
_Cooking time:_ 10 minutes
_Makes :_ 20 pikelets
_Ingredients:_
1 cup flour
½ cup rolled oats
½ cup yogurt
1⅛ cup milk
¼ cup whole cane or brown sugar
1 pinch baking soda (no more than ¼ teaspoon)
juice and rind of half a lemon
½ teaspoon ground cinnamon
2 eggs
fat for frying
Combine the flour, oats, yogurt, and milk and stir to form a smooth paste. Cover the bowl with a tea towel and leave it to sit overnight at room temperature.
The next morning, add the sugar, baking soda, lemon juice, lemon rind, cinnamon, and eggs and mix well.
Heat a frying pan over medium heat and add a little cooking fat. Place a small ladleful of the mixture into the pan—enough to spread to about 2 inches wide. Repeat in the remaining space around the frying pan.
When the pikelets are lightly browned on the underside, flip them over. Cook them for a further minute or two on the second side, then remove them from the pan and cook your next batch.
Serve with plain yogurt and maple syrup (my favorite combination), or whatever condiments you prefer.
## POLENTA
Polenta is a traditional peasant dish made from finely ground corn. It is generally very cheap (hence the phrase 'poor-man's grain') and can be used as a flour replacement for people following a gluten-free diet. Traditionally, polenta was blended with a rich cheese such as parmesan and mozzarella, and served as a warm, and richly satisfying side-dish. I prefer to prepare it in a bread tin, and cut off slices to make crispy polenta toasts. You can also add seaweed flakes for extra nutrition (seaweed flakes are high in minerals) and an added salty flavour. Just like a traditional loaf, polenta bread can be sliced and cooked in a toaster, or placed under a grill. It's a fabulous side dish or bread replacement for people following a glutenfree diet.
_Preparation time: 2 minutes_
_Cooking time:_ 15 minutes
_Refrigeration time:_ overnight
_Serves:_ 6
_Ingredients:_
1 ½ cups polenta meal
3 cups water (or more)
1-2 teaspoons sea salt
1 handful arame seaweed flakes (optional)
Place the polenta and water in a medium-sized saucepan on a medium heat. With a heavy wooden spoon stir frequently as the mixture begins to heat, taking care not to let any lumps form. Continue for ten to fifteen minutes as the mixture starts to bubble. Stir frequently. You may need to add additional liquid. When the polenta becomes thick and gluey, add seaweed flakes (if you are using them), plus enough salt to taste. Continue mixing. Pour the mixture into a 20cm bread tin, lined with grease-proof paper. Place bread tin in the fridge and leave for 2-3 hours or until the mixture sets cold. Remove from fridge. Slice the bread thickly and drizzle with olive oil before placing it under the grill. Cook both sides until crispy and brown.
## HEARTY BROWN RICE
Rice is currently a staple food for around half of the world's population, including much of the developing world. It is cheap to produce, but does not constitute a meal in itself; it needs to be combined with protein and vegetables to make a complete meal.
There are thought to be more than 100,000 distinct types of rice throughout the world. The most nutritious are the unrefined varieties such as brown rice and wild rice, and this hearty brown rice is a great way to give leftovers another life.
Brown rice is unmilled and comes in long-grain, short-grain, and aromatic varieties. It takes longer to cook, as its outer layers are intact and need time to break down. Just like flour, it is more susceptible to spoiling than its refined counterparts, so if you are not going to use it quickly, it is best stored in the fridge or freezer.
_Preparation time:_ 2 minutes
_Cooking time:_ 1 hour
_Serves:_ 4 or 5
_Ingredients:_
1½ cups brown rice
1 cup stock
2 cups water
Put the rice, stock, and water in a medium saucepan. Bring the water to a gentle simmer and cook for 1 hour, stirring occasionally. You might need to add more water as you go.
When the rice is almost cooked, increase the temperature and boil for 3 to 5 minutes to evaporate the remaining water. Alternatively, you can drain the rice through a sieve. Toss through some salt, pepper, and olive oil and serve as an accompaniment to a main meal.
_Leftover roast chicken with rice:_
Steam a head of broccoli and chop it into small florets. Combine it with a handful of chopped spinach and the meat left over from a roast chicken. Stir this through 1 quantity of hearty brown rice, season to taste, and add a little extra olive oil for moisture.
_Leftover roast lamb with rice:_
Combine 1 quantity of hearty brown rice with 1 cup of leftover roast lamb, cut into bite-sized pieces. Add 2 cups of chopped spinach, 3 finely chopped spring onions, and a handful of finely chopped mint. Season with salt, pepper, and olive oil.
# GOT MILK?
"I asked the waiter, 'Is this milk fresh?' He said, 'Lady, three hours ago it was grass.'"
—Phyllis Diller
TO ME, GOOD MILK EPITOMIZES THE frugavore way of eating. It's all about getting back to basics: obtaining tasty, nutritious food straight from the source and supporting sustainable farming practices. I love cows, and I love seeing what they eat when I visit my local milkman every second weekend. So it always intrigues me when people say they don't like milk or can't drink it. What sort of milk are they drinking? And when was the last time they visited a real cow in a real bam? I sometimes feel the urge to drag them out of the city, introduce them to my bovine friends, and encourage them to try a glass or two.
Milk was a staple food in many traditional diets. The dairy cow occupied a central position in traditional small-scale farms, providing a highly valued product that could be sold locally. Dairy products were thought to have unique medicinal properties. During the early twentieth century, physicians in Europe and America were known to prescribe fresh milk for a range of conditions including asthma, neuralgia, gastric disorders, and diabetes (this was sometimes called "the milk cure" ).
Traditionally, most dairy products were fermented. Without refrigeration, raw milk would culture in a period of days or even hours in warm weather. The naturally occurring bacteria would transform it into yogurt, kefir, cheese, and cottage cheese. Many of the famous French cheeses, including roquefort, camembert, and brie, were first created as a means to preserve milk for long periods.
Most importantly, people loved drinking milk. It was part of their culture, depicted in art and pottery, and revered as a holy food source. In countries such as India, the cow is still considered a sacred and virtuous animal.
Our attitude to dairy changed at the beginning of the twentieth century, as cities grew and milk production became more centralized. Milk was being transported over long distances without proper sanitary practices, and many people became ill from drinking it. We can now see that most of these cases involved cows being fed an improper diet of brewery waste, food scraps, and grains, while living in close confinement without access to grass or fresh air. People had little idea about sanitation or how to safely transport milk, and they didn't have stainless steel storage or refrigeration; no wonder the milk was foul. As a reaction to the many outbreaks of contamination, milk came to be pasteurized and we lost our connection to the farm and to the fresh bovine wine we once enjoyed.
Today, when I pick up a bottle of milk at my local milkbar, the product is very different from its traditional counterpart. Most milk is now reduced in fat. It is homogenized and pasteurized. That's fine from a food-safety perspective. The milk is easier to transport and has a longer shelf life. But from a taste and nutritional perspective, it leaves a lot to be desired.
Pasteurization, which gets rid of dangerous pathogens and bacteria by boiling the milk, was introduced in the early twentieth century as a food-safety measure. Homogenization became popular somewhat later, during the 1950s. It is a process whereby fat particles in the milk are broken up and dispersed throughout the liquid. The only reason for homogenization is aesthetic, but many people believe it makes the milk far blander and less palatable. Personally, I love a good layer of cream on top.
## ALLERGIES & INTOLERANCE
During the past fifty years we have seen a sharp rise in milk allergies and many people, as part of their quest to live a healthier lifestyle, simply don't drink milk. Lactose intolerance is now one of the most common food allergies. Essentially an inability to digest the milk-sugar lactose, it is caused by the absence of the enzyme lactase in the digestive tract. This enzyme is naturally present in raw milk, but is killed off during pasteurization. Many people who are diagnosed as lactose intolerant find that they have no problem drinking their milk raw. What's more, during the traditional fermentation of products such as kefir, yogurt, and cheese, the lactase enzyme, along with the plentiful healthy bacteria that are present in fresh dairy products, breaks down the lactose in the milk even further, so that it is virtually extinguished before it is consumed.
## ACCESSING FRESH MILK
Fresh milk was traditionally considered a natural superfood. Unprocessed milk is teeming with live enzymes and healthy bacteria, which aid the digestive process. It is also more nutrient-dense than its modern processed counterpart. During the pasteurization process, many key vitamins and minerals are depleted. Fresh milk production also favors small-scale, sustainable farms and heritage breeds of cow (these produce a rich and creamy milk with a high butterfat content). Today, many people are rediscovering this and looking for ways to access unprocessed milk again. They are creating "cow shares," or driving for several hours to buy their milk straight from the farm. In some places, it is not legal to sell unpasteurized dairy products. In these areas, farmers may label their wares as "bath milk" or "pet milk" in order to meet consumer demand. Sometimes there will simply be a "no transaction" system in place, whereby people collect their milk from the farm and leave some fresh food or other goods in return. I have heard of one farmer who gets his clients to pay his electricity bills while he gives them "free" milk!
In other parts of the world, an alternative system has developed to ensure healthy, topquality milk without pasteurization. "Certified raw milk" must meet standards relating to the diet and lifestyle of the animals, sanitation, and transportation. Under this system, milk only needs to be pasteurized if it is not fit to be consumed raw. In California, for example, you can buy fresh milk, totally unprocessed and in plain glass bottles, from organic supermarkets. The largest raw-milk distributor in the United States of America, Organic Pastures Dairy Company, tests its milk for pathogens and posts the results on its website daily. They have never had any problem with microbial contamination or unhealthy milk. They attribute this to their clean pastures, healthy cows, organic farming methods, and sanitary milking conditions.
Similarly, in parts of Europe where small-scale local farming is highly valued, milk can be bought from "milk machines." These automatic milk-dispensing devices provide non-homogenized, non-pasteurized milk from cows in the local area. Above the machine is a description of the animals, the farm, and the nutritional contents of the milk. In Italy, there are now more than 250 such machines in schools, offices, hospitals, and supermarkets.
The best way to find a good source of quality milk is to tap into local food networks and find out where your milk comes from. Start by shopping at farmers' markets and getting a sense of what is available (and legal!) in your area. Websites such as www.realmilk.com give a good rundown of what to look for when you are buying milk fresh—ensure that the cows are healthy and pasture-fed, and that the correct sanitary practices are in place during processing and bottling. It is not expensive for dairy farms to test for pathogen counts, and this should be standard practice in all milk distribution.
If you're not keen to buy your milk bubbling fresh from the farm gate or local market, at least look for non-homogenized, organic milk and dairy products from grass-fed cows. This will ensure a healthier, creamier product, and you will also be supporting more sustainable farming practices. When you take your jug of milk home, be it fresh from the farm or from a local foodstore, you'll be able to make all sorts of delicious goodies, including kefir, yogurt, cream cheese, and whey. In this chapter, you'll find recipes for some of these traditional treats.
## CHAMPAGNE MILK (KEFIR)
Fermented milk products such as yogurt and kefir are traditional staples in all milk-drinking societies. Food historians have documented over forty different words to describe yogurt and kefir, which shows how widely they are enjoyed. As well as preserving milk, fermentation creates healthy lactic-acid bacteria and makes other nutrients in milk easier for the human body to use.
From a nutritional perspective, traditional kefir is superior to the yogurt you find at the supermarket. It contains a diverse range of micro-organizms and wild yeasts and has a tarter, sweeter flavor. It also has a delightful fizz, hence its "champagne" nickname. It's easy to make at home; all you need is some fresh, non-homogenized milk and some kefir grains, which you can buy at health-food stores or online. Kefir grains contain the yeast and bacteria essential for fermentation. They can be re-used and will last indefinitely—so as long as you have milk, you will have an endless supply of kefir.
_Preparation time:_ 5 minutes
_Resting time:_ 24 to 36 hours
_Makes:_ about 3¼ cups
_Ingredients:_
¼ cup kefir grains
1 quart non-homogenized organic milk or fresh milk
_Variations:_
To make a smoothie, combine 1½ cups kefir with 1 or 2 egg yolks and ¾ cup fresh berries in a blender. Add a little stevia, honey, or maple syrup if you like a sweeter flavor.
Combine the kefir grains and milk in a sterilized 1-quart glass jar. Place a muslin cloth or tea towel over the top and leave it in a warm, dark place. Stir or shake the glass once or twice while it sits, making sure the grains are dispersed throughout the milk, rather than accumulating at the surface. The optimal temperature is between 66 and 84°F; depending on the temperature, the time required varies between 24 and 36 hours (the warmer the temperature, the faster the milk will ferment). Don't worry too much about getting the temperature and the timing exactly right—making kefir is all about intuition. When it is ready it should be tart-tasting and sparkly, like drinking yogurt. If it isn't, it probably needs another couple of hours' fermentation.
Once fermentation has taken place, separate the milk from the grains using a wooden spoon, or by pouring the milk through a plastic sieve. Try to avoid any contact between the kefir and stainless steel. The milk can be drunk and the grains can be re-used in the same jar with a new quart of milk.
If you've had success and are going to use your kefir jar again right away, don't wash it! If your kefir worked the first time, it means the friendly bacteria have acclimatized and will thrive with subsequent batches. If a batch hasn't worked, or if you've left a long time between batches, however, it's advisable to sterilize your jar before you next use it.
## CURDS & WHEY
When you buy raw milk, you don't just get a healthier, tastier drink; you also get the ingredients you need to make other dairy products at home. Curds can be spread on bread or toast, while whey can be drunk or used as a starter in cheese-making or sauerkraut. This recipe is as old as milk, but I first encountered it in Sally Fallon's bestseller _Nourishing Traditions,_ a great book about traditional foods.
_Ingredients:_
1 quart fresh raw milk
Pour the milk into a sterilized glass jar. Place the jar in a warm, dark spot. As for kefir, the optimal temperature is between 66 and 84°F, and the time required will vary between 18 and 30 hours, depending on the temperature. Over time, the milk should separate into curds and whey; the whey will be a clear, yellowish liquid, while the curds are heavy white blobs.
When the curds and whey have separated, place two muslin cheese cloths (or a single cloth folded in half) over a bowl. Pour the curds and whey onto the cloth and leave it overnight; the whey will drip through to the bowl, leaving just the curds on the cloth. The whey can be drunk (it's highly nutritious) or used to make cheese or sauerkraut. The curds can be enjoyed as a spread, similar to cottage cheese.
## MAPLE-SYRUP YOGURT
Fermenting milk to produce yogurt is a traditional way to make it last longer. Natural yogurt also contains a wide range of healthy bacteria known to benefit digestion and improve the availability of the nutrients in the milk. This recipe is delicious at breakfast time, and a great way to make milk go a little further.
_Preparation time:_ 5 minutes
_Resting time:_ overnight
_Makes:_ 1 quart
_Ingredients:_
34/5 cups non-homogenized organic milk
½ cup natural yogurt
1 spoonful maple syrup
_Tip:_
To make a thicker yogurt, drain the yogurt through a muslin cloth to separate the whey from the milk solids. The whey can be re-used as a starter for sauerkraut or pineapple beer.
Put the milk into a saucepan over low heat. Allow it to warm to just above body temperature (104 to 107°F is ideal). To test it, stick a clean finger into the milk; it is warm enough when it is just warmer than lukewarm.
Rinse a large pouring jug with hot water, then pour in the lukewarm milk, yogurt, and maple syrup. Stir well. Pour the milk mixture into a sterilized 1-quart thermos and leave it in a warm spot overnight. The milk needs to stay at a constant warm temperature to turn into yogurt. If you don't have a thermos, pour it into a ceramic dish and leave it near a heater or in the oven with the oven light on.
The next morning, pour the yogurt from the thermos and enjoy with your cereal, porridge, or stewed fruit and honey.
## LEMON-CURD CHEESE
This is a simple way to turn fresh milk into a light and fluffy lemony cheese. This is delicious spread over fruit toast, or wrapped with prosciutto and served on rye bread. Ceramic and copper cooking pots are ideal for cheese-making, but stainless steel will also do just fine.
_Preparation time:_ 15 minutes
_Resting time:_ 6 hours
_Makes:_ 3 cups of cheese
_Ingredients:_
1 quart non-homogenized organic milk
3 lemons
½ cup whey (optional)
½ teaspoon finely ground sea salt
Pour the milk and whey into a cooking pot and place over low heat on the stovetop. Stir the milk every few minutes and bring it to just above body temperature (104°F is ideal).
Meanwhile, juice the lemons. When the milk is ready, pour in the lemon juice, stirring constantly with a wooden spoon.
Let the mixture sit undisturbed for 25 to 30 minutes. The milk should separate into clumps of curds and liquid whey.
Fold a cheesecloth in half and place it over a large bowl. Pour in the separated milk, then suspend the cheesecloth over the bowl for 1 hour. After an hour, remove the curds from the cloth, add the salt, and toss them with your hands, allowing clumps to form. Return the mixture to the cloth and leave it for a further 3 to 4 hours.
Remove the cheese from the cheese cloth. It can be eaten now, or will keep in the fridge for up to 3 days.
# COOKING WITH NUT5: SPROUTING AND SAVING
"Love is like butter. It is good with bread."
—Yiddish proverb
NNUTS ARE AN EXTREMELY NUTRITIOUS component of diet. They contain a variety of important nutrients such as calcium (found in almonds), magnesium, vitamin E and fibre. They are also an excellent source of protein and health-promoting fats—such as mono-unsaturated and poly-unsaturated fatty acids. Much attention has recently been paid to the health properties of nuts. And, they have gained quite some attention as a substitute food for people following a gluten-free or dairy-free diet. When ground-up into flour, they make an excellent grain replacement for baked goods, and almond milk, when properly prepared, is a great substitute for the creamy flavour of cows' milk. Most health food stores now stock a good range of nut snacks, nut flour, nut butter and nut milk, however buying them in bulk can be pricey. Nuts are best prepared sprouted or activated, which can add to their cost. A good way to save money is buy raw nuts in bulk (preferably through a co-op or wholesale outlet) and then prepare your own nut-based foods at home. These foods are easily stored in the fridge, and will last indefinitely in the freezer.
## SPROUTING NUTS
Studies of traditional eating habits have shown that nuts were soaked or sprouted prior to cooking. It is believed that this process can decrease levels of anti-nutrients such as phytic acid and also make them easier to digest. The sprouting process involves a period of pre-soaking, followed by dehydration. In hunter-gatherer groups, the nuts and seeds were soaked in seawater and then left to dry out in the sun. You can imitate this process at home by soaking your nuts in a salt-water solution overnight then place them in a dehydrator or on a baking tray in your oven to dry out for 12-24 hours. The best thing about sprouted nuts is that they taste so much better; they are crisper with a richer nut flavour. You can also flavour your nuts—with maple syrup, honey or spices—prior to the dehydration process. Simply toss through your flavours prior to spreading the nuts on the baking tray and cook as per the recipe below.
Cost-saving tips: Nuts, seeds, legumes and grains can be bought in bulk through co-ops and some - organic stores. It's worth seeking out places that sell them raw, and in bulk to save costs. At home, raw nuts can be stored in recycled glass jars and will last for several months in the pantry. They contain large quantities of antioxidants (such as vitamin E), which keeps them fresh. Freshly made nut milk and nut flour can be stored in the freezer for up to a year. Store them in well-cleaned, recycled plastic food containers and zip-lock bags and label them clearly. Sprouted nutsIt might sound complicated to sprout your own nuts, but after doing it once or twice, it quickly becomes second nature. The results will speak for themselves and you'll find yourself never eating plain, unsprouted nuts again! If you are using an oven in place of a dehydrator, make sure it is set at the right temperature. An oven thermometer (which usually around $2.00 online) is an excellent investment for long, slow sprouting sessions. 2 cups raw nuts*2-teaspoons sea salt-filtered water Place the nuts in a large glass or ceramic bowl. Cover with water. Add the salt and stir until dissolved. Cover the bowl with a tea towel and leave it to rest on your kitchen bench for 12 hours or overnight. Drain the nuts from the water and rinse well. Spread the nuts out on a baking tray or in a food dehydrator and place in the oven or dehydrator at 40-50 degrees. Leave to cook for 12-24 hours or until they become crispy without any moisture. Activated nuts can be stored in the cupboard for several weeks. If you grind them into a nut flour it is worth storing them in the fridge or freezer. *This recipe works well for all nuts excluding cashews. Cashews need only be soaked for 5-6 hours (they go slimy if soaked for longer). It is also difficult to source raw cashews.
Nut flour: If you follow a gluten-free or grain-free diet, you can make your own sprouted nut flour. This is easy to do with any food processor—you only need to grind the nuts into a smooth, flour-like consistency. The nut flour can then be stored in a zip lock bag in the fridge or freezer (where it will keep for several months).
Almond butter: 1½ cups ground almonds/ almond flour 1 tablespoon honey pinch sea salt ¼ teaspoon almond essence ¼ teaspoon vanilla essence coconut oil (melted) OR almond oil OR cold-pressed sunflower oil. In a food processor, combine the ground almonds, honey, sea salt, vanilla and almond essence. Mix on a high speed for one minute. Slowly, pour in a thin stream of the oil, keeping the mixer running at a high speed as you go. Mix well and scrape any excess nuts from the wall of the food processor. Taste for seasoning and sweetness. Pour into sterilized recycled glass jars and store.
# THE SWEET STUFF
"Life is uncertain. Eat dessert first."
—Ernestine Ulmer
THE WORD _DESSERT_ IS TAKEN FROM THE French verb _desservir,_ meaning to clear the table following a meal. As this suggests, the traditional purpose of dessert is to refresh and nourish guests after eating, not to fill them up with refined sweeteners and processed food. So in keeping with the frugavore ethos, I've tried to make these desserts as simple and nutritious as possible. The cakes are not the usual fluffy afternoon-tea variety. They are more like the English pound cakes our grandmothers used to make: denser, less sweet, but very satisfying and much better for you.
As humans, we naturally crave a sweet flavor in our food. Since the beginning of time, we have gone to great lengths to obtain natural sweeteners such as fruit, honey, molasses, or maple syrup. This might have involved braving a swarm of bees to get a few teaspoons of honey, or tapping a maple tree for a drizzle of syrup. Natural, traditional sweeteners have many nutritional benefits that are not found in their modern, refined counterparts. For instance, blackstrap molasses is sourced directly from sugar cane. It contains iron, B vitamins, calcium, and trace minerals and is known to have many health-promoting properties. Modern table sugar is also sourced from the sugar cane plant—but thanks to intense processing and refinement, it is devoid of nutrients and is implicated in modern health problems such as diabetes, obesity, and nutrient deficiencies.
Not surprisingly, as new sweeteners have become available, our dessert recipes have had to adapt. Back in 1400 AD, gingerbread was made by soaking breadcrumbs in a mixture of honey and spices. Chocolate was traditionally a spicy drink made with bitter-tasting cocoa beans. Nomadic wanderers through the desert would wrap dried figs in animal fat for a daily sugar fix. Eskimo cultures would whip together berries with seal fat to make ice cream. Today, we have pure sugar lollipops, richly flavored chocolates, and soft drinks aplenty. Our sweets are overly refined and stripped of most of their nutrients.
There are alternatives, however. Most health-food stores now stock a wide variety of natural sweeteners, which can be used in place of conventional sugar in most recipes. Alternatively, opt for old-fashioned sweeteners such as ripe fruit, fresh dates, or a few drops of honey or molasses.
## A GLOSSARY OF SWEETENERS
Many people are confused about which sweeteners to use for home-cooking. Here's a rundown of some good and bad options.
Agave syrup: Made from the sap of various species of agave desert plants, which are related to the cactus family. Just because it is "plant-based," however, doesn't mean it's any better for you than standard sugar. Many brands of agave syrup are highly processed and have a similar nutritional profile to conventional white sugar.
Brown sugar: The first industrially produced brown sugars were by-products of turning cane juice into unrefined sugars. These included demerara, turbinado, and muscovado. They had some nutritional value and were vastly superior to modern refined sugars. Nowadays, brown sugars are produced at the refinery using raw sugar. Molasses is added to the sugar to impart a more complex flavor.
From a nutritional perspective, brown sugar is only marginally better than white sugar, and is best avoided.
Chocolate: The chocolate we buy from our local candy store is rich in sugar and preservatives. For cooking purposes, look for dark chocolate (at least 70 percent cocoa) and free of preservatives (including soy lecithin), or consider making your own (see recipe).
Dates: Dates can be puréed in a food processor and used as a substitute for sugar—one cup of dates can replace a cup of sugar. They are a good alternative in many baking recipes.
Honey: The nectar of the delicate honey bee. There is nothing quite as special as honey. Look for raw honey wherever possible (many of the good nutrients are killed off during pasteurization). Many inner-city foodies are now starting their own hives, which can be kept on rooftops or apartment balconies in the inner city. Honey contains unique antibacterial properties and has been used as a traditional remedy by many cultures.
Lakanto: Although not widely available in the West, Lakanto is a natural sweetener derived from the _luo han guo_ fruit of China. Lakanto has no kilojoules and does not raise blood-glucose levels at all. It can be used in baking as a substitute for sugar.
Malt syrup: An ancient and versatile sweetener. Along with honey, this was the primary sweetener in China for 2,000 years. It is made from a combination of germinated cereal grains, especially barley, and ordinary cooked grains. Malt syrup is considerably less sweet than sugar syrup, but can be used in baking.
Maple syrup: The sap of the maple tree, boiled until it forms a smooth syrup. It is delicious on pancakes and porridge and can be used in place of normal sugar for baking purposes.
Molasses: Also known as treacle, this is the syrup left over after cane-sugar processing. Blackstrap molasses is the most nutritious of all the molasses varieties produced during sugar processing.
Rapadura or sucanat: This is dehydrated cane juice, and is a less refined version of conventional white sugar. Whole cane does raise blood sugars, but not nearly as quickly or intensely as conventional sugar. This is a useful alternative to sugar and can be used for baking. It can be expensive, so try to buy it in bulk or at a wholesale outlet.
Stevia: This is a natural herb, native to South America. Stevia can be grown as a plant in your own backyard: I have several pots sprouting around my kitchen door. Many health-food stores sell it in powdered or liquid form. These are extremely, exquisitely sweet: one drop equals one cup of sugar, but does not raise blood-glucose levels at all, so has none of the negative effects associated with sugar. It is useful as a sweetener in liquid drinks, but I do not recommend it for cake-making.
White sugar: Of all the sweeteners mentioned, this is the sweetener that I would go to great lengths to avoid. Because white sugar is highly refined, it is rich in kilojoules but contains no nutrients whatsoever.
## SUMMER ICY-POLES
There's nothing better than a cold, sticky icy-pole during the summer months. As a kid, it was a highlight of our summer to buy icy-poles from our local milk bar. Sadly, most brands of icy-poles and ice-creams these days are loaded with sugar and numerous preservatives to keep them looking fresh and at the right melting point. At least you know that when you make them at home, you can use simply, healthy ingredients and it won't cost you a fortune. These recipes are quick to prepare and a lot of fun for the kids. You can also layer the flavors in icy-pole moulds, making a staggered pattern; simply pour a layer into each mould, allow it set, then follow with a layer from a different flavor. Continue until the mould is completely full.
_Preparation time:_ 2 minutes
Refrigeration time: 4 hours minimum
Makes: 6 small icy-poles3 tablespoons cocoa or cacao powder1 cup unsweetened, plain yoghurt3 tablespoons coconut sugar or rapadura sugar½ cup water½ teaspoon vanilla essence
**Fool-proof, kid-friendly chocolate**
Combine all ingredients in a food processor or blender and process until smooth. Transfer into icy-pole moulds and freeze. Most moulds require a minimum of four hours' freezing time.
## ZESTY RASPBERRY-MINT
In a food processor or blender, combine all ingredients and process until smooth. Taste for sweetness—if the berries are too sour add 1-2 teaspoons coconut sugar or rapadura. Pour into icy-pole moulds and freeze. Most moulds require a minimum of four hours' freezing time.
_Preparation time:_ 10 minutes
_Refrigeration time:_ 5 hours minimum
_Makes:_ 6 small icy-poles
_Ingredients:_
1 well-compacted cup of fresh raspberries
1¾ cups coconut water
1 teaspoon lemon juice
generous handful of fresh mint, finely chopped
coconut or rapadura sugar (optional)
## BREAD & BUTTER PUDDING
When you realize late on a Sunday night that you have a hungry family roving your kitchen and nothing to feed them for dessert, this is the ultimate leftover dish, quick to prepare with a few simple ingredients. Any bread will do—I've made it with sourdough hot cross buns at Easter, and leftover panettone after Christmas. Here is the basic recipe, but you can also add fresh fruit, some jam from your last harvest, or whatever happens to be in season.
_Preparation time:_ 10 minutes
_Soaking time:_ 30 minutes
_Cooking time:_ 45 minutes
_Serves:_ 6
_Ingredients:_
butter
7 slices bread
¾ cup whole cane or brown sugar
4 eggs
¾ cup milk
1 pinch cinnamon
½ teaspoon vanilla essence
⅓ cup currants
_Variations:_
Replace the currants with 1 cup of fresh berries or thinly sliced nectarines, peaches, or plums. Or, add a few dollops of jam to each slice of bread. Apricot jam works well.
Preheat your oven to 350°F.
Butter the sides and bottom of a small or medium baking dish about 2 inches deep. Generously butter one side of each slice of bread and neatly pack the buttered bread into the dish.
In a small bowl, whisk together the sugar, eggs, milk, cinnamon, and vanilla. Pour this mixture over the bread so that the bread is three-quarters immersed, then leave the pudding to settle and soak for 30 minutes at room temperature. Sprinkle the currants between the layers of bread and bake in the preheated oven for 40 minutes, or until the liquid is set and the bread is lightly toasted.
## HOME-MADE, ORGANIC CHOCOLATE
The finest chocolate comes from good quality cocoa butter and cocoa powder. When you make chocolate at home, you can use these ingredients to make superb-tasting chocolate without any of the additives or preservatives of the major brands, and for a fraction of the price. To prepare this recipe, you only need a basic whisk, saucepan and wooden spoon. Good chocolate moulds are also worthwhile. You can find silicone ones at most cooking stores, or search online through outlets such as Amazon or eBay. As a cost-saving tip: try replacing the required amount of cocoa butter with coconut oil. Just be aware that the coconut oil has a lower melting point, so your chocolate will need to be served cold, and kept in the fridge.
_Preparation time:_ 10-15 minutes
_Refrigeration time:_ 1 hour minimum
_Makes:_ 1 dozen, coin-sized chocolates
_Ingredients:_
⅓ cup (55g) cocoa butter
1 ½ level tablespoons of coconut oil (at room temperature—it should be soft, but not liquid)
5 tablespoons raw cacao or cocoa powder
¼ teaspoon cinnamon
¼ teaspoon mixed spice
2 generous tablespoons honey
Gently melt the cocoa butter by placing it in a small bowl on top of a saucepan filled with hot (almost boiling) water. Stir frequently with a wooden spoon. When it has melted, remove it from the heat, set it aside, but make sure it doesn't harden again. The coconut oil needs to be soft; in a consistency akin to thick cream. If it is too hard, place it in a warm place (for instance your oven on a low heat) to soften. If it becomes runny (like oil), place it in the fridge to harden. Once you have the consistency right, place it in a deep mixing bowl. Mix well with a whisk. Slowly add the cocoa butter, and the remaining ingredients, one at a time, and keep whisking. As the temperature drops, it should thicken slightly. Remove a sample from the mixing bowl and check for flavor; for a richer, darker chocolate, add more cocoa, for a sweeter taste, add more honey. Next, pour mixture into silicone chocolate moulds and refrigerate immediately. It should take a minimum of four hours for the chocolate to set properly. When it's ready, remove from moulds and serve cold.
## COCONUT SAGO
Sago is also known as "seed tapioca"—tiny balls of starch extracted from the trunk of the sago palm in countries including New Guinea and the Moluccas. The little balls are the size of caviar, and kids will often refer to them as "fish eggs" when they are cooked. Sago combines nicely with coconut milk, and this warming recipe is delicious served with fresh or stewed fruit and some fresh cream.
_Preparation time:_ 10 minutes
_Soaking time:_ at least 30 minutes
_Cooking time:_ 30 minutes
_Serves:_ 5
_Ingredients:_
1 cup sago
2 cans (15¼ ounces each) coconut milk
½ cup milk, plus a little extra
1 teaspoon vanilla essence
⅓ cup whole cane or brown sugar
2 egg yolks
juice and rind of 1 lemon
cream and fresh or stewed fruit (optional, to serve)
Place the sago in a saucepan, pour over the coconut milk, and leave it to soak. The longer it soaks, the quicker it will cook: a couple of hours is ideal, but 30 minutes is plenty.
Add the milk and place the saucepan over low heat and gently cook for 30 to 45 minutes. Sago has a tendency to stick to the saucepan, so keep a watchful eye on it. When the sago is ready, the tiny balls will be clear and soft, not hard or crunchy. If it seems too firm, you may need to add a little more milk as it cooks.
When the sago is done, add the vanilla, sugar, egg yolks, lemon juice, and lemon rind, and stir until they are well dissolved. Remove from the heat and serve with fruit or cream.
## APPLE & NECTARINE SHORTCRUST TART
Use the freshest fruit you can find for this recipe. If you have a fresh batch of homemade jam, it will be even more delicious.
_Preparation time:_ 20 minutes
_Cooking time:_ 40 minutes
_Serves:_ 4
_Ingredients:_
1 quantity sweet oatmeal pastry (see recipe on page 252)
1½ tablespoons cream
3 drops vanilla essence
2 tablespoons coconut or rapa-dura sugar
2 egg yolks
⅔ cup almond meal
1 green apple
1 nectarine or plum
1½ tablespoons apricot or plum jam
Preheat your oven to 350°F.
Grease a 8-inch tart tin with a little cooking fat. Press the pastry into the tin using your fingers (you won't need a rolling pin). Push it up the sides so that it is ¼ or ¾ inch high. Place the tin in the oven and bake for 5 minutes or until lightly crisped.
Prepare the filling by combining the cream, vanilla, sugar, egg yolks, and almond meal in a small mixing bowl and stirring well. When the pastry is ready, pour the filling in.
Wash and core the apple and slice it as thinly as possible. Arrange the slices on the pastry base in a circular pattern. Do the same with the nectarine, layering it on top of the apple but letting slivers of apple show through.
Use a small sieve or tea strainer to dribble droplets of jam over the fruit. If the jam is too thick, you may need to combine it with ¾ tablespoon of boiling water first.
Place the tart in the oven and bake for an additional 30 minutes or until the fruit and filling are fully cooked and lightly browned. Serve with cream.
## BAKED FRUITS STUFFED WITH RICOTTA & HONEY
This is a wonderful summer dessert. Choose fruit that is well and truly ripe; if you are picking it yourself, it should be almost ready to fall from the tree. You can make the ricotta mixture in advance and simply combine it with the fruit just before you start dinner.
_Preparation time:_ 10 minutes
_Cooking time:_ 40 minutes
_Serves:_ 10
_Ingredients:_
4/5 cup ricotta cheese
juice and finely grated rind of 1 lemon
¾ tablespoon brown sugar or whole cane sugar
1 egg
½ teaspoon dried cinnamon
10 medium or 20 small fruits (nectarines, apricots, peaches, or pears)
Preheat your oven to 350°F.
In a small mixing bowl combine the ricotta, lemon, and sugar and stir well. Whisk in the egg to form a smooth paste.
Cut each piece of fruit in half using a sharp knife. Remove the pits or seeds from each piece and lay them hollow-side up on a baking tray.
Spoon a dollop of the ricotta mixture into the hollow of each piece of fruit and balance them carefully on the baking tray.
Bake in the preheated oven for 30 minutes or until the ricotta is lightly browned and the fruit is well cooked and soft.
## CHOCOLATE MOUSSE
What could be simpler and more enticing than a delicious chocolate mousse with only two ingredients? This is hands-down the easiest and most delicious chocolate mousse recipe you will ever try.
_Preparation time:_ 20 minutes
_Refrigeration time:_ at least 2 hours
_Serves:_ 10
_Ingredients:_
8 eggs
7 ounces organic dark chocolate (55 to 70 percent cocoa)
fresh fruit or summer berries (to serve)
Separate the eggs, putting the yolks and whites into separate bowls.
Break the chocolate into small pieces and place them in a small ceramic mixing bowl. Put the bowl in a saucepan with an inch of water and place the saucepan over medium heat. Increase the temperature until the water simmers but does not boil. Stir the chocolate until it melts to a smooth liquid.
Whisk the egg whites until soft peaks form. When they're ready, you should be able to turn the bowl upside down without them sliding out.
Combine the egg yolks with the melted chocolate. Gradually add this mixture to the egg whites, stirring gently with a whisk until well combined.
Pour the finished mixture into individual soufflé bowls or small glasses and refrigerate for at least 2 hours. Serve with fresh fruit or berries.
## BAKED CUSTARD WITH RUM
This cake is similar to the traditional "Far Breton" from Brittany in France. Every time I cook it I am reminded of the humble but delicate cooking methods that still exist in that region. You can bake this as a single cake, or divide the mixture into soufflé dishes and serve it as individual custards. The combination of prunes, rum, and custard is exquisite.
_Preparation time:_ 10 minutes
_Soaking time:_ 8 to 10 hours
_Cooking time:_ 45 minutes
_Serves:_ 6
_Ingredients:_
½ cup flour
½ cup rapadura or coconut sugar
2 cups milk
4 eggs
4/5 cup plain yogurt
1½ cups pitted prunes
¼ cup rum
butter for greasing
Place the flour, sugar, milk, eggs, and yogurt in a mixing bowl and stir to form a smooth batter. Place this mixture in the fridge and allow it to chill for 5 to 8 hours (the longer the better).
Place the prunes in a small bowl and cover them with the rum. Leave them to soak for a similar period.
When you are ready to make the cake, preheat your oven to 350°F. Combine the prunes with the milk mixture and stir well.
Grease a single cake tin or 8 individual soufflé dishes with butter, or line them with greaseproof paper. Pour in the custard mixture, making sure the prunes are evenly distributed throughout the dish.
Bake for 30 minutes if you are using soufflé dishes or 50 minutes if you are using a cake tin. The cake will be ready when the custard is lightly browned and a skewer comes out clean. Serve immediately.
## STEWED PEARS WITH CINNAMON SYRUP
My mother used to whip this up with whatever fruit she happened to have on hand, a little wine, and a fresh stick of cinnamon. She always served it with a hearty dollop of cream or yogurt. The key is not to smother the pears with liquid—add the minimum amount of juice and tightly pack all the ingredients into a small, heavy saucepan with a tight lid. Ideally, the pears should fit snugly and take up three quarters of the space in the pot.
_Preparation time:_ 10 minutes
_Cooking time:_ 1 hour
_Serves:_ 6
_Ingredients:_
3 large green or yellow pears
¼ cup honey
a few drops vanilla essence
¼ cup dry white wine
1 large stick cinnamon
1 lemon
1 cup fresh strawberries or raspberries (optional)
Rinse the pears and cut them in half. Using a small, sharp knife, remove the seeds and the stems.
Place the pears in a small saucepan and add the sugar, vanilla, and wine. Break the cinnamon stick in half and add it to the saucepan. Peel a long strip of rind from the lemon and throw this in too.
Add enough water to the saucepan to just cover the pears—don't add too much, or the sauce will be watery. Firmly secure the lid and bring the liquid to a gentle simmer. Cook for 45 to 60 minutes on low heat, stirring occasionally.
When the pears are soft and light brown, add the berries (if you're using them). Juice the lemon and add the juice to the pot. Check the water level; if the liquid doesn't cover three-quarters of the fruit, you might want to add a little extra. Remove the lid, increase the heat, and simmer for 5 to 10 minutes so that the sauce thickens and the berries soften.
Transfer the pears to individual serving bowls. Pour on some syrup and drop a dollop of cream or yogurt into the hollow of each piece of fruit.
_Note:_
You can make this dish sugar-free by replacing the sugar with 1 cup of dried dates. Purée the dates in a food processor until they form a smooth paste, and use this in place of the sugar.
## PEACHY MINT SALAD
This is a very simple dessert that never fails to please. Grab the ripest fruit from your or a neighbor's backyard. Add some maple syrup and a handful of finely chopped mint, squeeze on some lemon juice, and _voilà_! Serve with a little cream, some plainyou, or even coconut-flavored sago. A little dessert wine doesn't take away anything, either.
_Preparation time:_ 10 minutes
_Serves:_ 8
_Ingredients:_
6 nectarines or peaches
¾ tablespoon maple syrup
juice of 1 lemon
1 handful fresh mint, finely chopped
cream or yogurt (optional, to serve)
Thinly slice the nectarines or peaches and arrange them in a large serving bowl.
Combine the syrup and lemon juice in a small jug. Pour the mixture over the fruit, add the mint, and stir well. Serve with fresh cream or yogurt.
# PRESERVES FOR THE PANTRY
_"Food preservation techniques can be divided into two categories: the modern scientific methods that remove the life from food, and the natural 'poetic' methods that maintain or enhance the life in food."_
—Eliot Coleman
THE TRADITIONAL PEASANT PANTRY contained a wide range of preserves made from seasonal foods that were only available fresh at certain times of the year. Sheer necessity dictated that raw ingredients be stored for later use. We may not face the same shortages today, but preserves can still be incredibly cost-effective. Ever wonder what to do when your neighbor's lemon tree is overflowing into your back garden? Or when a pile of onions goes on sale at your local supermarket? Buy up big, I say, and preserve them for the future. Most preservation methods require only basic ingredients such as salt or vinegar. Other flavorings, like herbs and spices, can be added if you wish.
Preserves also make wonderful presents. Organic food stores sell stylishly packaged preserves, replete with frilly ribbons and raffia, for twenty dollars or more per jar. When you see how quick and inexpensive they are to make at home, you'll realize how ridiculous this is!
Before you start preserving, you need to remember one important thing: never throw anything out. Old jam jars, glass bottles, glass food containers—keep, keep, keep. When you are pounding your cabbage or bottling your lemons, you will make good use of them, and will be glad you kept them in your cupboards for all those years.
## PREPARING YOUR KITCHEN
Choosing containers: Always try to use glass if possible. In some instances, ceramic cookware can be used, but make sure that it is properly sterilized. Plastic should be avoided wherever possible.
Steriliszation: To sterilize glass jars and cooking equipment (excluding plastic), first place your containers in a large pot of cold or warm water. Make sure they are fully immersed. Put the pot over a gentle heat and bring the water to a boil. Reduce the heat and simmer for five minutes. Remove the containers from the water and allow them to cool. Please note: if you add glass jars to a pot of already boiling water, they will most likely crack!
Hygiene: Many of the most dangerous microbes are transferred via our hands, skin, and hair. Follow these golden rules to avoid any contamination. Many of the recipes in this chapter require a lot of hands-on stirring and mixing. With clean hands, you can feel free to handle food as much as you like.
* Wash your hands thoroughly with soap and water before getting started, and after any breaks or trips to the toilet.
* Keep your fingernails short. This is super important, as bacteria can sit under your nails and come out as you cook.
* Before a major cooking session, soak your hands in a bowl of warm water and vinegar for five minutes.
* Keep your hair off your face and avoid any nose blowing.
## FERMENTED FOODS
The traditional pantry included a wide array of fermented foods: sourdough bread, homemade beer, wine, sauerkraut, pickled beets, garlic, and carrots. Fermentation is a unique method of food preservation whereby microbes create a rich array of healthy lactic-acid bacteria, which make foods last longer. Fermented foods are what nutritionists call "functional foods"—that is, they have added nutritional value. Fermentation increases the nutrient-density of the raw ingredients.
Before refrigeration was invented, foods had to be fermented, preserved with vinegar, air-dried, salted—or eaten pretty darn quickly. Fermentation was an easy and inexpensive method of preserving food used by many traditional cultures. Dr. Weston A. Price recorded Eskimo cultures allowing their fish to "rot" for months before consuming them. Fermented cabbage in the form of sauerkraut, choucroute, or kimchi is a mainstay of Eastern European and Asian cuisines. In Australia, Aboriginal groups buried yams in the ground for several months, allowing them to ferment before eating them. Traditional fermentation was done simply, with just the basics—a container, some salt, a few herbs, and any vegetables not wanted for cooking. Often they were left for months and months, either in a cellar or buried in the ground.
Sadly, with the advent of modern refrigeration, along with changes in our lifestyle and climate, these fermentation practices have been largely lost or forgotten. Pollution, a dryer climate, and increased pesticide use mean that the growth of good bacteria can sometimes be inhibited. For this reason, when trying traditional fermentation techniques at home, it's important to use the best quality ingredients you can, and to make sure everything is properly sterilized.
People often ask me if they can use vinegar to ferment vegetables. In supermarkets you can buy sauerkraut made with vinegar and salt. But cabbage cooked with vinegar is not real sauerkraut. Vinegar works well as a preserving agent, but it does so by _inhibiting_ the growth of micro-organizms and bacteria; it stops harmful micro-organizms from growing, but it also prevents true fermentation from taking place. The recipes here therefore don't include any vinegar.
## THE DOS & DON'TS OF FERMENTATION
* Wherever possible, use organic, biodynamic, or home-grown produce. Commercial sprays and pesticides can have anti-fungal and anti-insect properties, which inhibit fermentation.
* The vegetables _must_ be completely submerged in the liquid. Problems can occur if they are left exposed to the air.
* You must use non-chlorinated water. Buy some filtered water, or simply boil your own tap water for ten minutes, then allow it to cool before you use it.
* Use good quality sea salt, which is rich in a wide variety of trace minerals. Avoid table salt, which is highly refined, contains few trace minerals, and contains additives.
## STARTER CULTURES
Starter cultures are used to kick start fermentation. Adding a starter is optional, as most traditional fermentation will happen naturally without one. If you wish to use a starter culture, you can use the fermented whey from cream cheese and whey, or a commercially made starter culture, which you can purchase online. A brand that I recommend is Caldwell's. www.caldwellbiofermentation.com
## BLUEBERRY CHIA JAM
I had previously assumed that all jams were laden with sugar and that jam making couldn't be done without large quantities of the white stuff. How wrong I was! Chia jam has all the wonderful attributes of traditional jam making; it makes a thick, smooth paste, it's deliciously sweet—thanks to the natural sugars present in fruit, and also spreads easily over bread. The good news is that it doesn't contain any refined sugar and is easy to make—in fact much quicker—than traditional jam making. This recipe calls for fresh (or frozen) blueberries, but you can easily substitute other berries such as raspberry, strawberries or blackberries. To make an apricot or rhubarb chia jam, simply cook the fruit for longer (try an extra 20 minutes, or cook until its very soft) you might also need to adjust for sweetness.
_Preparation time:_ 5 minutes
_Cooking time:_ 20 minutes
_Makes:_ 1 small jar
_Ingredients:_
1 cup blueberries (fresh or frozen)
2 teaspoons chia seeds
2 teaspoons honey or maple syrup
(optional - depending on the sweetness in the fruit)
Place the blueberries, and the sweetener (if you are using it) in a small saucepan on a medium heat. Stir frequently with a wooden spoon and let the mixture simmer for 15-20 minutes or until the blueberries are soft. Remove from the heat and pour into a food processor or blender. Mix for a minute or so, until the blueberries form a smooth paste. Taste for sweetness. Add the chia seeds and mix again. Pour mixture out of the food processor and let it set in a sterilized glass jar. The jam will be ready to eat immediately. It will also last in the fridge for up to ten days.
## CULTURED BEETS WITH CABBAGE
Beetroot ferments exceptionally well; it is rich in natural sugars and creates a wonderful, sour-beet taste in the final product. This dish is a great introduction for anyone starting out on fermented foods. It's easy to put together and almost impossible to get wrong. It's also very kid-friendly.
_Preparation time:_ 10 minutes
_Fermentation time:_ 3-4 days
_Makes:_ 1.5 quarts
_Ingredients:_
500g coarsely grated beetroot
700g finely sliced cabbage (green, purple or a combination of both)
1 tablespoon caraway seeds
1 tablespoon fennel seeds
1½ tablespoons sea salt
Remove and discard any outer cabbage leaves that have dirt on them. Working on the next layer, remove a couple of large leaves that you can use later as a 'cabbage lid.' Set these aside. Slice the cabbage as thinly as possible using a large, sharp knife. You can also use a mandolin or food processor with a shredding blade if you have it on hand. Grate the beetroot and measure out the salt and spices. In a large mixing bowl, combine all the ingredients together and mix them well with your hands. Squeeze the mixture in large clumps to release the liquid and break down the cell walls. After a few minutes, you should notice a small pool of liquid developing in the bottom of the bowl. Keep mixing and squeezing until the pool of water is a sizeable half cup or more. Pack the mixture into sterilized, glass jars and push it down to close off any air bubbles. Cover with a layer of cabbage leaves (that you set aside earlier) and weigh it down with a kitchen weight, or tall glass jar filled with water. As the mixture ferments, the vegetables will rise up in the jar and make contact with the air. You can push the mixture down using your hand (however you will need to do this everyday). Or simply, place a weight on top. The golden rule of fermentation is that the vegetables need to stay immersed in liquid. If they make contact with the air, harmful bacteria may start to grow. Leave the jars at room temperature for 3-5 days (depending on the heat in your kitchen, if it's the peak of summer, opt for less time). Then transfer to a refrigerator or cool cellar. The ferment will be ready to eat within a week, but will last for several months in a cool place. Check it regularly to ensure it has no contact with air.
## ZESTY PEAR & GINGER RELISH
During the autumnal months, when pear production is at its fullest, this relish recipe is a great way to preserve them. After a period of fermentation, the relish becomes zesty and salty, with the warm, nourishing flavours of ginger and pear. It works well with richly-flavoured meat, chicken and game dishes. It's also a great topping for thick slices of home-made chocolate or richly-flavoured chocolate mousse; whereby it makes perfect match of salty, sour and sweet.
_Preparation time:_ ten minutes
_Fermentation time:_ 5 days
_makes:_ 1 quart-size jar
2 cups pears, peeled, cored, and cut into ¼ inch cubes
2 teaspoons freshly, grated ginger
¼ cup raisins
1 tablespoon lemon juice
1 ½ teaspoons whey
30g sea salt
500ml water
Prepare a salt solution by heating the water in a small saucepan and stirring through the salt until it dissolves. Set aside and allow to cool. In a large mixing bowl combine the remaining ingredients. Toss them together using your hands. Place the mixture in a 1 litre glass jar and pour over the salt solution. Make sure all the fruit is completely immersed under the liquid. Cover with a cheesecloth. Leave the relish at room temperature for 3 days to ferment. It may start to lightly fizz, which is completely fine. After 3 days, remove it from the jar and drain it through a sieve. Discard the liquid (you can use this as a base for salad dressings, or as an extra ginger 'kick' to soft drinks and cocktails) and place the solids in a food processor. Mix the solids to form a smooth paste—alike thick, crunchy, peanut butter. Place it back in the glass jar, secure with a lid, and store it in the fridge. The relish is ready to be eaten at any time and will last for several months in the refrigerator.
## PRESERVED LEMONS
Preserved lemons are one of the easiest things to make in your home kitchen, and they are delicious added to roast chicken, lamb dishes, or sliced into thin slivers and added to salads or desserts for an intense lemony flavor. Don't worry too much about measuring the lemons or salt; it's easier just to add them in proportion to what you've got on hand. Many things can be added to the basic recipe for extra flavor and color, but the simple combination of lemons and salt always works a treat, and it can keep in your kitchen pantry for several months.
_Preparation time:_ 15 minutes
_Makes:_ Approximately one 1-cup jar for every 2 or 3 lemons
_Ingredients:_
lemons
sea salt (about 8 teaspoons per lemon)
_Variations:_
For added flavor, try adding one of these seasonings
* whole cloves
* bay leaves
* crushed cinnamon sticks
* whole chili seeds
Wash and dry the lemons to remove any surface dust or dirt. Scrub the skins to remove any grit. Cut the ends off each lemon, then slice each fruit into quarters (if they are small) or eighths (if they are large). Remove the pith from the inside of each slice.
Put the lemon slices into a large bowl with the sea salt and toss well. If you are adding herbs or seasonings, add them now. Roughly speaking, you will need 2 or 3 teaspoons of salt for each slice of lemon.
Push the lemon slices, one by one, into a sterilized jar. As you push, the juice should drain out and accumulate at the bottom of the jar. It is important that all the lemon pieces are completely immersed in fluid. If you find your jars are too full of liquid, drain a little from the top. You should not need to add any excess water as the juice will be squeezed out from each lemon.
When the jar is full, sprinkle the surface with salt, then seal the jar tightly. Leave it in a cool cupboard for 2 or 3 weeks before you start consuming the lemons. Each time you open the jar to use some of the lemon, sprinkle some salt on the surface when you're done. The lemons should last up to a year in your pantry.
## PRESERVED GLOBE ARTICHOKES
Artichokes grow in abundance during hot European summers, and most traditional Mediterranean pantries would have included a few jars of these. They can cost a startling amount at the local deli, but if you preserve them yourself using your backyard harvest, you will pay only for the oil, the herbs, and the salt.
_Preparation time:_ 5 minutes
_Cooking time:_ 30 minutes
_Makes:_ 1 large jar
_Ingredients:_
10 small globe artichokes
2 cups vinegar
sea salt
olive oil
fresh or dried herbs (oregano, thyme, or rosemary, or a combination)
Peel the artichokes and cut them into quarters, removing the tough outer leaves. Place the vinegar, salt, and 4¼ cups of water in a small saucepan, then add the artichokes. Bring to a boil, then simmer for 10 minutes. Remove the artichokes from the saucepan and pat them dry. Let them drain for a couple of hours or overnight.
Pack them into sterilized glass jars along with the herbs, then drizzle them with olive oil and tightly seal the jars. Let them sit for at least a week before consuming them. They will last several months in a cool pantry.
## QUICK & EASY TOMATO PRESERVE
Italian families often have a tomato bottling session as an annual ritual. They'll make enough to last the whole year, and no one goes home empty-handed. For those of us with just a few tomato plants, this quick and easy recipe will produce a couple of delicious jars of preserved tomatoes. They can be added to minestrone and other soups and stews and will last for several months in the freezer.
_Preparation time:_ 2 minutes
_Cooking time:_ 20-30 minutes
_Makes:_ ½ quart
_Ingredients:_
6 large ripe tomatoes
1 teaspoon dried oregano
¼ teaspoon dried thyme
sea salt
olive oil
Cut the tomatoes in half, and spread them on a baking tray. Sprinkle generously with the sea salt and dried herbs and drizzle with olive oil. Place them in a pre-heated oven at 480°F and bake for 20-30 minutes or until soft and lightly browned. Drain of any liquid or oil that has accumulated at the bottom of the pan and place tomatoes in a food processor. Process at a high speed until a thick paste forms. Serve immediately or store in the fridge for up to a week. This paste can also be frozen, in recycled plastic food containers and will last indefinitely.
## FERMENTED TOMATO SALSA
Okay—I know every supermarket stocks tomato sauce. This one, however, tastes far better than anything you will buy off the shelf. It contains no artificial colorings or preservatives and is an excellent way to use up over-ripe tomatoes during the summer months. When we buy large quantities of meat from our local farm, we usually get a few boxes of hand-made sausages. I devised this tomato salsa recipe specially to complement the sausages, but it goes equally well with any meat dish, or as a dipping sauce with homemade hot potatoes.
_Preparation time:_ 5 minutes
_Fermentation time:_ 3 days
_Makes:_ ½ quart
_Ingredients:_
¼ tsp dried chilli flakes
5 medium fresh tomatoes, skinned & de-seeded
1 purple onion, finely chopped
¾ tablespoon sea salt
2 cloves garlic, crushed
3 tablespoons lemon juice
¼ cup whey OR use an extra
1 teaspoon sea salt
1 ½ tbsp finely chopped fresh oregano
1 ½ tbsp finely chopped fresh coriander
Mix together all the ingredients. Process to a smooth paste in a food processor. Keep at room temperature for 3 days then refrigerate. The salsa can be eaten immediately. It should last in the fridge for 2-3 months.
## TURMERIC TONIC
Before the advent of sugar-laden soft drinks and fast foods, most households would regularly prepare their own home brew. Fermented soft drinks contain no artificial colors, flavors, or preservatives. The natural, wild fermentation process also develops healthy lactic-acid bacteria, which are good for the digestive tract.
_Preparation time:_ 10 minutes
_Fermentation time:_ 4 days
_Makes:_ 3 ½ cups concentrated liquid
_Ingredients:_
½ cup honey
2 tablespoons finely grated turmeric
¼ cup lemon juice
1 tablespoon finely grated ginger
2 tablespoons tamarind paste
The only drawback of natural fermentation is that the results can be a little unpredictable. Some ingredients ferment more quickly than others, depending on how much sugar they contain. You may find that your brew ferments more quickly than expected, or that it's extra bubbly or has extra bite!
Turmeric has long been revered for its medicinal properties; it contains compounds which exhibit anti-inflammatory and anti-oxidant effects on the body. It has also been shown to support healthy joint function, improve digestion and promote radiant skin.
In a saucepan, place the turmeric, tamarind, and half the grated ginger. Add two cups water and bring to boil. Reduce to a gentle simmer, and leave to cook, uncovered for 15-20 minutes. Gently mash the mixture with a wooden spoon every few minutes as it cooks. This will help to break up the tamarind. After cooking, remove from heat and allow it to cool to room temperature. Drain the liquid through a fine mesh sieve and discard the solids (tablespoons of the solids are excellent base to tea, just add 1 tbsp to a cup of boiling water). To the liquid, add the remaining ginger and other ingredients. Add additional filtered water so the total volume is 3 ½ cups. Place in a 1 litre glass jar. Cover with a cloth and leave at room temperature for 3-4 days (a warm part of the house is ideal). Stir once or twice daily in a circular direction with a clean metal spoon. This is very important. Don't forget to stir! When it starts to bubble—on day 4 or 5, it is ready to drink. It can be drunk immediately or stored in a refrigerator. The tonic is quite strong—so best diluted with water in a ratio of 1 part tonic, 3 parts water. Add sparkling mineral water for additional carbonation.
## TRADITIONAL SAUERKRAUT
Throughout history, in many parts of the world, cabbage has been preserved to create delectable fermented foods such as sauerkraut (Germany), choucroute (France), and kimchi (Korea, China and Japan). Archaeologists have found that even during the hunter-gatherer stage of human development, people fermented plants, and cabbage was a favorite.
A luscious and large cabbage, big enough to produce three quarts of sauerkraut, costs about four dollars at most organic stores. This quantity of sauerkraut will last you several months, and will work as an excellent source of convenient, ready-to-eat food in your cellar or refrigerator. Look for cabbage that is as fresh as possible, with tightly-packed leaves. It's great to use organic produce for fermentation, but this is by no means essential. A good quality, fresh and locally-sourced plant is fine.
_Preparation time:_ 15 minutes
_Fermentation time:_ 4-5 days
_Makes:_ 1.5 quarts
_Ingredients:_
1kg finely sliced or shredded cabbage
200g coarsely grated apple
1 tablespoon caraway seeds
1 teaspoon fennel seeds
¾ tablespoon juniper berries
1 ½ tablespoons sea salt
Remove and discard any outer cabbage leaves that have any visible dirt. Then, with the next, clean layer of leaves, remove and set aside a couple of large leaves that you can later use as a 'cabbage lid'.Slice the cabbage as thinly as possible using a large, sharp knife. You can also use a mandolin or food processor with a shredding blade if you have it on hand. Grate the apple and measure out the salt and spices. In a large mixing bowl, combine all the ingredients together and mix them well with your hands. Squeeze the mixture in large clumps to release the liquid and break down the cell walls. After a few minutes, you should notice a small pool of liquid developing in the bottom of the bowl.
Pack the cabbage into sterilized jars. Fill each jar about four-fifths full (don't fill them completely, or the cabbage may spill when it ferments). You need to create an anaerobic environment (that is, an environment with no air) for the cabbage to ferment. As you add every piece of sauerkraut to the jar, check to be sure that the cabbage is tightly packed and there are no air pockets.
Place a cabbage leaf over the top of each jar as a protective lid, to shield the sauerkraut from the air. Then place a weight, such as a smaller glass jar or bottle filled with water, on top of the leaf. As the mixture ferments, the vegetables will rise up in the jar and make contact with the air. You can push the mixture down using your hand (however you will need to do this regularly). Or simply, place a weight on top. The golden rule of fermentation is that the vegetables need to stay immersed in liquid. If they make contact with the air, harmful bacteria may develop. Check it in a few days' time to ensure that the cabbage is still immersed in liquid.
After 4 days, transfer the sauerkraut to the fridge, and check that the vegetables are still immersed under liquid. It will be ready to eat within 24 hours, but can last for several months if kept refrigerated.
## _Resources_
There is a wealth of information online for anyone interested in the frugavore approach to food. These sites below should help you to get started.
And of course, you can visit me at www.frugavore.com.
## SOURCING YOUR FOOD
Networking Association for Farm Direct Marketing and Agritourism (www.nafdma.com): A wonderful resource an direct producer-to-consumer agricultural relationships in the United States.
The Weston A. Price Foundation (www.westonaprice.org): The WAPF website provides information about nutrition and traditional cooking techniques, while local chapters can let you know what's happening nearby.
Slow Food (www.slowfood.com): Slow Food has branches all over the world. If you join your local convivium, you'll be kept up-to-date with news and events.
Real Milk (www.realmilk.com): A good place to start for information about whole milk, the philosophy behind it, and what's available in your area.
## GARDENING
Heirloom Organics (www.non-hybrid seeds.com): An online site that supplies information about non-hybrid seeds. You may also purchase your seeds through this site.
American Community Gardening Association (<http://communitygarden.org>) A wonderful resources for those interested in finding community gardens.
## FURTHER AFIELD
These additional sites are worth exploring, even from afar. Much of their advice is relevant wherever you are, and they provide an insight into what's happening elsewhere. Who knows: they might inspire you to start something similar in your neighborhood . . .
Local Harvest (USA) (www.localharvest.org): This site offers comprehensive information about sourcing organic food, growing heirloom vegetables, and getting involved in co-ops and community-supported agriculture. It also hosts a lively collection of food forums and blogs, so is a great place to ask questions and share ideas.
Locavores (USA) (www.locavores.com): If you're interested in learning more about the "locavore" movement, this is the place to start.
EatWild (USA) (www.eatwild.com): This site outlines the benefits—for humans and animals—of raising livestock on pasture.
The River Cottage (UK) (www.rivercottage.net): Through his _River Cottage_ series of television programs, Hugh Fearnley-Whittingstall has become a champion of local, sustainable eating. The River Cottage website includes recipes and advice and recently launched a program to help put willing land-owners in touch with would-be community-gardeners.
The Ecologist (UK) (www.ecologist.org): Established in 1972 and now published online, _The Ecologist_ was one of the first magazines to promote sustainable living. It provides practical tips and resources as well as news about events and campaigns.
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## _Acknowledgments_
This book would not have been possible without the encouragement and support of so many people. First and foremost, thank you to the innovative, and forward-thinking team at Black Inc Books (Australia) and Sky-horse Publishing (USA).
Thank you to Denise O'Dea for your continual support and editing assistance, Sally Fallon-Morell for your guidance on fats and oils, Will Winter and Becca Griffith for your knowledge on co-ops and buyers clubs, Ron Hull for your expert opinion on raw milk, Beverly and Ronald Smith for sharing your knowledge of biodynamic farming methods.
Thank you also to Marie Danvers and Anne O'Donovan for your support during the early stages of the manuscript development.
Thank you to my dear family; Justine, Rebecca, Alistair and Mum—for creating a household full of laughter and warmth, and fostering the growth of several chickens, ducks, geese, dogs and just about every other animal and plant in our inner-city home.
Thank you to Darryl and Barnaby; the two people who make every trip to the nursery, op-shop or junkyard all the more interesting and exciting. Thank you for all the wonderful times spent in our kitchen, backyard and veggie patch. It's been great fun.
## _Index of Recipes_
Aioli
Apple
cider vinegar
and nectarine tart
Artichokes, preserved
Bacon _see pork_
Beetroot
cannellini bean salad
garden salad
Beans
cannellini bean dip
cannellini bean salad
split-pea purée (fava)
treacle baked beans
_see also: chickpeas, lentils_
Beef
casserole
meatballs
meatloaf
oxtail stew
pot-au-feu
steak and kidney pie
stock
tongue
Beer
pineapple
Bermuda fish chowder
Bouillabaisse
Bread
basic sourdough
fruit loaf
herb
Bread and butter pudding
Bubble and squeak
Butter, almond
Cake
porridge
Casserole
beef
chicken
Cauliflower with bacon
Cavolo nero _see greens_
Cheese, lemon-curd
Chicken
casserole, Spanish
chicken-feet stock
and corn soup
fat
and leek pie
and rice
roast
soup
stock
Chickpeas
bean and green soup
hummus
minestrone
and rosemary soup
salad with greens
Chocolate mousse
Coconut sago
Curds and whey
Custard, baked
Dips
cannellini bean
hummus
liver pâté
split-pea purée (fava)
Duck
fat
roast with orange and sage
Egg
glaze
and wild greens pie
mayonnaise
omelets -
scrambled
Fava
Fish
baked
bouillabaisse
broth
cakes
chowder
pie
poached
pickled
salmon gravlax
soup -
stock
French onion soup
Fruit, baked
_see also: individual fruits_
Gravlax, salmon
Greens
bean and green soup
chickpea salad with greens
kale and zucchini soup
wild greens pie
Ham _see pork_
Hummus
Irish stew
Kale _see greens_
Kefir
Kidney and steak pie
Lamb
Irish stew
mutton curry
and rice
stock
Leek
and chicken pie
and potato soup
and sour cream omelet
Lemons, preserved
Lentils
with bacon
salad
soup
Liver
crispy
pâté
Marrow
on toast
pot-au-feu
Mayonnaise
Meatballs, baked
Meatloaf
Minestrone
Mousse, chocolate
Mutton curry
Oatmeal
pastry
pikelets
slice
_see also: porridge_
Offal
cooking with –
liver with caramelized onion
liver pâté
steak and kidney pie
sweetbreads, baked
tongue, brined 1
Omelet
leek and sour cream
potato and nutmeg
zucchini and basil
Onion
caramelized ,
French-onion soup
and tomato pie
Oxtail stew
Pastry, oatmeal
Pâté, liver
Pea and ham soup
Peachy mint salad
Pears
stuffed
stewed
Pie
chicken and leek
fish
steak and kidney
tomato and onion
wild greens
Pikelets
Pineapple beer
Polenta
Pork
bacon, home-cured
bacon with lentils
fat,
pea and ham soup
pork-mince apples
stock
Porridge –
Porridge cake
Pot-au-feu
Potato
baked
bubble and squeak
and leek soup
and nutmeg omelet
Poultry
fat
jointing a bird
stock
_see also: chicken, duck_
Preserves
artichokes
fish –
lemons
sauerkraut , –
tomatoes
Pumpkin
cannellini bean salad
soup
Rabbit, Moroccan hot pot
Rice
Sabayon sauce
Sago, coconut
Salad
cannellini bean
chickpea
garden
lentil
peachy mint
Salmon gravlax
Salsa verde
Sardines
baked
pickled
Sauerkraut
Silverbeet _see greens_
Soup
bouillabaisse
chicken
chicken and corn
chickpea and rosemary
bean and green
fish broth with rice
fish chowder
French onion
Irish stew
kale and zucchini
lentil
minestrone
oxtail stew
pea and ham
pot-au-feu
potato and leek
pumpkin
saffron stracciatella
Spanish-style chicken casserole
Spinach _see greens_
Split-pea purée (fava)
Steak and kidney pie
Stock
beef
chicken-feet
fish
lamb
poultry
vegetable
Stracciatella
Suet
Sweetbreads, baked
Sweet potato hummus
Tomato
and onion pie
preserved
sauce
Whey
Yogurt
Zucchini
and basil omelet
and kale soup
Vegetables, roast
_see also: individual vegetables_
|
{
"redpajama_set_name": "RedPajamaBook"
}
| 6,435
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Händelser
Efter plats
Romerska republiken
Scipio Aemilianus besegrar numantierna och erövrar Numantien, vilket avslutar det numantiska kriget.
Tiberius Sempronius Gracchus försöker genomdriva en ny lag, som omdistribuerar allmän mark, för att gynna småjordägare. Då han för detta får diverse fiender bland rikets mer välbeställda familjer i senaten dödas han av en grupp senatorer.
Attalos III överlämnar genom testamente Pergamon till Rom för att förhindra inbördeskrig. Pergamon blir provinsen Asia.
Födda
Avlidna
Attalos III, kung av Pergamon
Tiberius Sempronius Gracchus, romersk tribun (mördad)
Externa länkar
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,949
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\section{Introduction}
Deep reinforcement learning (DRL), which combines reinforcement learning algorithms and deep neural networks, has achieved great success in many domains, such as playing Atari games \cite{mnih2015human}, playing the game of Go \cite{silver2016mastering} and robotics control \cite{levine2016end}.
Although DRL is viewed as one of the most potential ways to General Artificial Intelligence (GAI), it is still criticized for its data inefficiency. Training an agent from scratch requires considerable numbers of interactions with the environment for a specific task.
One approach dealing with this problem is \textit{Transfer Learning} (TL) \cite{taylor2009transfer}, which makes use of prior knowledge from relevant tasks to reduce the consumption of samples and improve the performance of the target task.
Most of the current TL methods are essential to share the knowledge that is contained in the parameters of neural networks. However, this kind of knowledge cannot be directly transferred when faced with the cross-domain setting, where the source and target tasks have different state-action spaces. In order to overcome the domain gap, some researches focus on mapping state spaces into a common feature space, such as using manifold alignment \cite{ammar2015unsupervised,gupta2017learning}, mutual information \cite{wan2020mutual} and domain adaptation \cite{carr2019domain}.
However, none of them considers similar actions in different action spaces of related tasks share similar semantics.
To illustrate this insight, we take the massively multiplayer online role-playing game (MMORPG) as an example. An MMORPG usually consists of a variety of roles, each of which is equipped with a set of unique skills. However, these skills share some similarities since some of them cause similar effects, such as 'Damage Skill', 'Control Skill', 'Evasion Skill' and so on.
Several work have studied action representations and used them to improve function approximation \cite{TennenholtzM19}, generalization \cite{ChandakTKJT19,jain2020generalization} or transfer between tasks with the same state-action spaces \cite{whitney2019dynamicsaware}.
In contrast to these researches, we study the feasibility of leveraging action embeddings to transfer policy across tasks with different action spaces. Intuitively, similar actions will be taken when performing tasks with the same goal. Hence, if the semantics of actions is learned explicitly, there is a chance to utilize the semantic information to transfer policy.
The main challenge in the action-based transfer is how to learn meaningful action embeddings that captures the semantics of actions. Our key insight is that the semantics of actions can be reflected by their effects which are defined by state transitions in RL problems. Thus, we learn action embeddings through a transition model, which predicts the next state according to the current state and action embedding.
Another challenge is that how to transfer policy across tasks with different state and action spaces.
To this end, we propose a novel framework named TRansfer with ACtion Embeddings (TRACE) for DRL, where we leverage both state embeddings and action embeddings for policy transfer.
When transferring to the target task, we transfer both the policy model and the transition model learned from the source task. The nearest neighbor algorithm used by the policy model can select the most similar action in the embedding space. Meanwhile, the transition model helps to align action embeddings of the two tasks.
The main contributions are summarized as follows:
\begin{itemize}
\item We propose a method to learn action embeddings, which can capture the semantics of the actions.
\item We propose a novel framework TRACE which learns both state embeddings and action embeddings to transfer policy across tasks with different state and action spaces.
\item Our experimental results show that TRACE can
a) learn informative action embeddings;
b) effectively improve sample efficiency compared with vanilla DRL and state-of-the-art transfer algorithms.
\end{itemize}
\section{Related Work}
\subsection{Transfer in Reinforcement Learning}
Transfer learning is considered an important and challenging direction in reinforcement learning and has become increasingly important.
\cite{teh2017distral} proposes a method that uses a shared distilled policy for joint training of multiple tasks named Distral.
\cite{finn2017model} introduces a general framework of meta-learning that can achieve fast adaptation.
Successor features and generalized policy improvement are also applied to transfer knowledge \cite{barreto2019transfer,ma2018universal}.
\cite{yang2020efficient} proposes a Policy Transfer Framework (PTF) which can effectively select the optimal source policy and accelerate learning.
All these methods focus on the tasks that only differ in reward functions or transition functions.
To transfer across tasks with different state and action spaces, many research attempts to map state spaces into a common feature space.
\cite{gupta2017learning} learns common invariant features between different agents from a proxy skill and uses it as an auxiliary reward. However, it requires corresponding state pairs of two tasks, which may be difficult to acquire.
Adversarial training \cite{wulfmeier2017mutual,carr2019domain} is also utilized to align the representations of source and target tasks.
Additionally, MIKT \cite{wan2020mutual} learns an embedding space with the same dimension as the source task and uses lateral connections to transfer knowledge from teacher networks, which is similar to \cite{liu2019knowledge}.
These methods focus on the connection between state spaces but ignore the relationship between action spaces.
Recently, OpenAI Five \cite{raiman2019neural} introduces a technique named Surgery to determine which section of the model should be retrained when the architecture changes. The method is capable of continuous learning but is not suitable for tasks with totally different action spaces.
Inter-task mapping, which considers both state and action spaces, is used to describe the relationship between tasks through explicit task mappings.
\cite{taylor2007transfer} manually constructs an inter-task mapping and builds a transferable action-value function based on it.
Furthermore, Usupervised Manifold Alignment (UMA) is used to learn an inter-task mapping from trajectories of source and target tasks autonomously \cite{ammar2015unsupervised}.
\cite{zhang2021learning} learns state and action correspondence across domains using a cycle-consistency constraint to achieve policy transfer.
The main difference from our work is that we try to embed actions into a common space instead of learning a direct mapping.
\subsection{Action Embedding}
Action embedding is firstly studied by \cite{dulac2015deep}, aiming to solve the explosion of action space in RL.
However, the action embeddings are assumed to be given as a prior.
Act2Vec is introduced by \cite{TennenholtzM19}, in which a skip-gram model is used to learn action representations from expert demonstrations.
\cite{ChandakTKJT19} learns a latent space of actions by modeling the inverse dynamics of the environment, and a function is learned to map the embeddings into discrete actions.
While in this paper, we learn representations of discrete actions through a forward transition model.
Similarly, \cite{whitney2019dynamicsaware} simultaneously learns embeddings of states and action sequences that capture the environment's dynamic to improve sample efficiency.
However, they focus on continuous control tasks and consider the effects of action sequences.
Action representations are also used to enable generalization to unseen actions \cite{jain2020generalization}. By contrast, we use it to transfer policy across different tasks.
\section{Problem Definition}
RL problems are often modeled as Markov Decision Processes (MDPs) which are defined as a tuple $\mathcal{M} = (\mathcal{S}, \mathcal{A}, \mathcal{T}, \mathcal{R}, \gamma)$, where $\mathcal{S}$ and $\mathcal{A}$ are sets of states and actions. In this work, we restrict our method on discrete action spaces, and $|\mathcal{A}|$ denotes the size of action set.
$\mathcal{T}: \mathcal{S} \times \mathcal{A} \times \mathcal{S} \mapsto [0, 1]$ is a state transition probability function, which can also be represented as the distribution of resulting states $p(s_{t + 1}|s_t, a_t)$ at time step $t$.
$\mathcal{R}: \mathcal{S} \times \mathcal{A} \mapsto \mathbb{R}$ is a reward function measuring the performance of agents and $\gamma$ is a discount factor for future rewards.
Additionally a policy $\pi: \mathcal{S} \times \mathcal{A} \mapsto [0, 1]$ is defined as a conditional distribution over actions for each state.
Given an MDP $\mathcal{M}$, the goal of the agent is to find an optimal policy $\pi^*$ that maximizes the expected discounted return $R = \sum_{t=0}^{\infty} \gamma^t r_{t}$.
In this paper, we consider the transfer problem between a source MDP $\mathcal{M}_S = (\mathcal{S}_S, \mathcal{A}_S, \mathcal{T}_S, \mathcal{R}_S, \gamma_S)$ and a target MDP $\mathcal{M}_T = (\mathcal{S}_T, \mathcal{A}_T, \mathcal{T}_T, \mathcal{R}_T, \gamma_T)$.
In this paper, we assume that the state and action spaces in the two MDPs are different, while there are some similarities in both the reward functions and the transition functions.
For example, in one of our experimental tasks, $\mathcal{M}_S$ and $\mathcal{M}_T$ correspond to two different roles in a MMORPG to fight against an enemy. While the dimensions of actions (skills) and states are completely different, the two agents both need to defeat the enemy, with a reward that depends on the final result: win, lose or tie. Besides, though the actions (skills) of the two roles are different, their effects can be similar, such as two agents both choose a damage skill which cause similar damage to the enemy.
\section{Transfer with Action Embeddings}
In this section, we introduce the TRACE framework.
We first discuss how to learn meaningful action embeddings.
Further, we describe how the action embeddings can be combined with RL algorithms and to facilitate policy transfer.
\subsection{Learning Action Embeddings}
To learn action embeddings that capture the semantics of actions, our main insight is that the semantics of actions can be reflected by their effects on the environment, which can be measured by the state transition probability in RL.
Thus, we aim to learn an action embedding $e(a) \in \mathbb{R}^d$ with dimension $d$ for each $a \in \mathcal{A}$, which should satisfy some properties: (1) The distance between action embeddings is adjacent if the actions have similar effects on the environment. (2) The embeddings should be sufficient so that the distribution conditioned on the embeddings approximates that conditioned on the actions $p(s_{t+1}|s_t, a_t) \approx p(s_{t+1}|s_t, e(a_t))$.
We approximate this by learning a transition model $f_{\theta^D}$, which predicts the next state $\tilde{s}_{t + 1}$ according to current state $s_t$ and action $a_t$ with parameter $\theta^D$, to capture the dynamics of the environment.
Figure \ref{fig: onestep} illustrates the learning process of action embeddings. At the beginning, an embedding matrix $W^{ae} \in \mathbb{R}^{|\mathcal{A}| \times d}$ is instantiated, in which the $i$-th row of the matrix denotes the embedding vector $e(a_i)$.
Given a state transition tuple $(s_t, a_t, s_{t+1})$, we first get action embedding $e(a_t)$ by lookup from the current matrix $W^{ae}$. Then a latent variable $z_t$ is sampled from $z_t \sim \mathcal{N}(\mu_{t}, \sigma_{t})$, where $[\mu_t, \sigma_t] = f_{\theta^{D_1}}(s_t, e(a_t))$, like variational autoencoder (VAE) \cite{kingma2013auto}.
The latent variable is introduced as a stochastic process to cope with stochastic environments, which is similar to \cite{goyal2017z}.
Further, the model predicts the next state $\tilde{s}_{t+1}$ by a multi-layered feed-forward neural network that conditions on $z_t$, $s_t$, and $e(a_t)$, i.e. $\tilde{s}_{t+1} = f_{\theta^{D_2}}(s_t, e(a_t), z_t)$.
The transition model and action embeddings are optimized to minimize the prediction error:
\begin{figure}
\centering
\includegraphics[height=100pt]{fig/onestep-transition2.pdf}
\caption{Illustration of the process of learning action embeddings. Given a state transition tuple, we first obtain action embedding $e(a_t)$ from $W_{ae}$. The transition model is used to predict the next state $\tilde{s}_{t+1}$. The red arrow denotes the gradients of the loss, and the grey grid denotes input variables.}
\label{fig: onestep}
\vspace{-10pt}
\end{figure}
\begin{figure*}
\centering
\includegraphics[height=160pt]{fig/arch.pdf}
\caption{The architecture of TRACE for tasks with different state spaces. When transferring to the target task, the parameters in grey grids are transferred as initialization. The brown dashes mean that the data is used by the other part, but the gradient does not propagate across the module. The number of circles denotes the size of the vector, and the subscripts $S$ and $T$ denote source and target task, respectively.}
\label{fig: arch}
\vspace{-1em}
\end{figure*}
\begin{equation}
\label{eq: loss}
\begin{aligned}
\mathcal{L}(\theta^D, W^{ae}) = & \mathop{\mathbb{E}}_{s_t, a_t, s_{t + 1}} \big{[} \ ||\tilde{s}_{t+1} - s_{t+1}||_2^2
\\ &+ \beta D_{KL} \big{(}\mathcal{N}(\mu_t,\sigma_t) \ || \ \mathcal{N}(0, I) \big{)} \big{]}
\end{aligned}
\end{equation}
where $\theta^D =\{\theta^{D_1},\theta^{D_2}\}$, $\beta$ is a scaling factor.
Note that, if the transition of tasks is non-markovian, we can apply a recurrent transition model (such as LSTM \cite{hochreiter1997long}) to learn the dynamics more accurately.
\begin{algorithm}[!t]
\caption{\label{alg: train} TRACE training algorithm on source task}
\begin{algorithmic}[1]
\STATE Randomly initialize the policy $f_{\theta^\pi}$, state
embedding $f_{\theta^{se}}$, transition model $f_{\theta^D}$, and action embeddings $W^{ae}$
/*\ \ State embedding is optional. In same-domain transfer, we set $f_{\theta^{se}}(s) = s$ \ \ */
\STATE Initialize replay buffer $\mathcal{B}$
\FOR{ $episode=1$ to $L$}
\STATE Receive initial state $s_1$ from environment
\FOR{timestep $t=1$ to $T$}
\STATE Select action $\hat{a}_t = f_{\theta^\pi}(f_{\theta^{se}}(s_t))$, $a_t = g(\hat{a}_t)$ according to current policy and action embeddings
\STATE Execute action $a_t$, receive reward $r_t$, and observe new state $s_{t+1}$
\STATE Add tuple $(s_t, \hat{a}_t, r_t, s_{t + 1})$ to $\mathcal{B}$
\STATE Sample random batch from $B \overset{i.i.d.}{\sim} \mathcal{B}$
\STATE Update $\theta^\pi$ and $\theta^{se}$ according to SAC training loss
\STATE Sample random batch from $B' \overset{i.i.d.}{\sim} \mathcal{B}$ and calculate state embedding $f_{\theta^{se}}(s)$ for each state in the batch
\STATE Update $\theta^D$ and $W^{ae}$ over Equation. (\ref{eq: loss})
\ENDFOR
\ENDFOR
\RETURN $\theta^D$, $\theta^\pi$, $\theta^{se}$, $W^{ae}$
\end{algorithmic}
\end{algorithm}
\begin{algorithm}[!t]
\caption{\label{alg: transfer} TRACE transfer algorithm on target task}
\textbf{Input:} Parameters $\theta_{S}^D$, $\theta_{S}^\pi$ from source task
\begin{algorithmic}[1]
\STATE Initialize the policy $f_{\theta^\pi}$ and transition model $f_{\theta^D}$ where $\theta^\pi = \theta_{S}^\pi$ and $\theta^D = \theta_{S}^D$, and randomly initialize state embedding $f_{\theta^{se}}$ and action embeddings $W^{ae}$
/*\ \ In same-domain transfer, we set $f_{\theta^{se}}(s) = s$ \ \ */
\STATE Train model according to line 2-15 in Algorithm \ref{alg: train}
\RETURN $\theta^D$, $\theta^\pi$, $\theta^{se}$, $W^{ae}$
\end{algorithmic}
\end{algorithm}
\subsection{Policy Training and Transfer}
\subsubsection{Train Policy with Action Embeddings}
In this section, we describe the training process of TRACE combined with SAC.
It is noteworthy that TRACE is not limited to SAC. It can be extended to any other RL algorithms with appropriate adaptions.
Algorithm \ref{alg: train} outlines the training process on the source task of TRACE (state embedding is introduced in Section Cross-Domain Transfer).
First, we initialize the action embeddings and the network parameters of the policy model and the transition model (Line 1). During the training process, to select an action at each timestep, the output of the policy model (continuous embedding space) should be mapped to the original discrete action space.
In this paper, we use a standard nearest neighbor algorithm to do the mapping(Line 6) \cite{dulac2015deep}.
Specifically, the policy parameterized by $\theta^\pi$ outputs a proto-action $\hat{a} = f_{\theta^\pi}(s)$ for a given state $s$, $\hat{a} \in \mathbb{R}^d$.
Then the real action performed is chosen by a nearest neighbor in the learned action embeddings:
\begin{equation*}
g(\hat{a}) = \mathrm{argmin}_{a \in \mathcal{A}} \ ||\hat{a} - e(a)||_2
\end{equation*}
where $g(\cdot)$ is a mapping from a continuous space to a discrete space.
It returns an action in $\mathcal{A}$ that is closest to proto-action $\hat{a}$ in embedding space by $L_2$ distance.
The agent executes action $a_t = g(\hat{a}_t)$, receives reward $r_t$, observes next state $s_{t+1}$, and stores the transition $(s_t, \hat{a}_t, r, s_{t+1})$ to the replay buffer $\mathcal{B}$ (Lines 7-8).
Then it updates the policy model and the action embeddings and transition model accordingly following SAC loss \cite{haarnoja2018soft} and Equation (\ref{eq: loss}) (Lines 9-12).
\subsubsection{Same-Domain Transfer}
With a trained source policy, we now describe how to transfer the policy to the target task.
For better understanding, we start from a simple setting where the state spaces of source and target tasks remain the same while the action spaces are different, and we call it the same-domain transfer.
For example, a character in a game carries different sets of skills to perform the same task. Each set of skills contains different skills in terms of number and type.
Within this setting, we can directly transfer policy $f_{\theta^\pi_S}$ and fine-tune on the target task.
The agent should behave similarly when facing the same state, and the nearest-neighbor algorithm will find the most relevant skills in the target task if action embeddings of the source and target tasks are well aligned.
To achieve that, we transfer the transition model's parameters $\theta^D_S$ learned from the source task and freeze them. Then the action embeddings of the target task $W^{ae}_T \in \mathbb{R}^{|\mathcal{A}_T| \times d}$ are optimized according to Equation \ref{eq: loss}. In this way, we can align action embeddings of the tasks, which is also validated in our experiments.
\vspace{-4pt}
\subsubsection{Cross-Domain Transfer}
In cross-domain transfer, where tasks differ in both state and action spaces, the transition model can not be reused since the dimensions of states are different between source and target tasks.
The premise of reusing the transition model is that states can be embedded into the same or similar space with the same size.
Thus, the input of the transition model becomes a tuple $(f_{\theta^{se}}(s_t), a_t, f_{\theta^{se}}(s_{t + 1}))$, where $f_{\theta^{se}}(\cdot)$ denotes a non-linear function with parameter $\theta^{se}$, mapping the original state space into a common space, called state embedding.
In this work, we train the state embedding along with the policy.
Note that the two modules (policy model and transition model) become interdependent --- transition model needs state embeddings as training input, and policy requires action embeddings to select actions.
Therefore, we train two modules together, which can also increase data utilization.
The architecture is shown in Figure \ref{fig: arch}
When training on the target task, we initialize the RL policy $f_{\theta_T^\pi}$ and the transition model $f_{\theta_T^D}$ with parameters $\theta_S^\pi$ and $\theta_S^D$ from the source task. At the same time, the state embedding $f_{\theta_T^{se}}$ and action embeddings $W_T^{ae}$ are randomly reinitialized.
Then we train the network as on the source task (Lines 2-15 in Algorithm \ref{alg: train}).
Algorithm \ref{alg: transfer} outlines the transfer process.
Unlike the same-domain transfer, the transition model parameters are not frozen but finetuned. This is because the inconsistency in the transition model increases when state dimensions are different, which leads to unstable training. We also verify this in our experiments shown in the appendix
\begin{figure}
\centering
\includegraphics[width=240pt]{fig/gridworld_embedding2.pdf}
\vspace{-10pt}
\caption{The learned embeddings projected into 2D space via PCA. Each dot in (a) represents an action embedding in gridworld. We show the action effect of each group. (b) Action embeddings of the source task (2-step gridworld) and the target task (1-step gridworld).}
\label{fig: embed}
\vspace{-10pt}
\end{figure}
\begin{figure*}
\centering
\includegraphics[height=15pt]{fig/new_legend2.pdf}
\vspace{-0.7em}
\subfigure[task $n=1$ (source task $n=3$)]{
\label{fig: gridworld-result:a}
\includegraphics[width=0.25\textwidth]{fig/gridworld-transfer-2-0-1.pdf}
}
\subfigure[task $n=2$ (source task $n=1$)]{
\label{fig: gridworld-result:b}
\includegraphics[width=0.25\textwidth]{fig/gridworld-transfer-0-1-1.pdf}
}
\subfigure[task $n=3$ (source task $n=2$)]{
\label{fig: gridworld-result:c}
\includegraphics[width=0.25\textwidth]{fig/gridworld-transfer-1-2-1.pdf}
}
\vspace{-10pt}
\caption{Experiment results on gridworld environments. The solid lines denote our method, and the dashed lines represent the compared methods. The shaded areas are bootstrapped 95\% confidence intervals.}
\label{fig: grid-result}
\vspace{-10pt}
\end{figure*}
\section{Experiments}
In this section, we empirically investigate the feasibility of learning action embeddings with the transition model and assess the effectiveness of TRACE.
We compare our method, abbreviated as \textit{TRACE-PT}, which means to transfer both the Policy model (P) and the Transition model (T), with three baselines:
a) \textit{SAC} \cite{haarnoja2018soft}, which learns from scratch on target tasks;
b) A basic transfer (\textit{BT}) strategy, which replaces the input and output layers of the neural network learned from the source task with new learnable layers that match the dimensions required of the target task and fine-tunes the whole network.
c) \textit{MIKT} \cite{wan2020mutual}, which leverages policies learned from source tasks as teachers to enhance the learning of target tasks;
Note that \textit{BT} can see as an ablation of our method that does not use action embedding.
We also report the result of our method that learns from scratch on the target task, denoted by \textit{TRACE(no transfer)}.
All the methods are based on SAC, and the results are averaged over 10 individual runs.
Appendix A provides detailed descriptions of environments and hyperparameters used in the experiments, and more experimental results are available in Appendix B.
\subsection{N-Step Gridworld}
We first validate our methods in an $11 \times 11$ gridworld, in which an agent needs to reach a randomly assigned goal position.
The agent could perform 4 atomic actions: \emph{Up}, \emph{Down}, \emph{Left}, and \emph{Right} at each step.
Once we consider combo moves in successive $n$-step, the number of actions becomes $4^n$.
We conduct experiments on three settings with $n \in \{1, 2, 3\}$.
The sizes of action spaces are 4, 16, 64, respectively.
The state of the tasks consists of the current position $(x, y)$ and the goal position $(\dot{x}, \dot{y})$.
The agent receives a -0.05 for each step and a +10 reward when the agent reaches the goal.
To investigate whether action embedding can capture the semantics of actions, we set $n = 3$, the dimension of action embedding $d = 4$, and sample 10,000 transition data based on a random policy.
Figure \ref{fig: embed}(a) shows the Principal Component Analysis (PCA) projections of the resulting embeddings.
In the figure, different colors denote different action effects.
For example, there are 9 blue dots (\textcolor{blue}{$\uparrow$}), including $\uparrow\uparrow\downarrow$, $\leftarrow\uparrow\rightarrow$, etc.
We see that the actions with the same effect are positioned closely and clustered into 16 separate groups.
What's more, those clusters show near-perfect symmetry along with the four directions in the gridworld, which means our method effectively captures the semantics of actions.
In word embeddings, the relationship between words is often discussed, such as \textit{Paris - France + Italy = Rome} \cite{mikolov2013efficient}.
In action embeddings, we can also get the similar property, such as $e(\uparrow \uparrow \leftarrow) + e(\uparrow \leftarrow \rightarrow) - e(\leftarrow \rightarrow \leftarrow) \approx e(\uparrow \uparrow \uparrow)$.
Further, we evaluate the policy transfer performance on tasks $n = \{1, 2, 3\}$.
Note that state spaces of these tasks are the same. So we freeze the parameters $\theta^D$ of the transferred transition model as described in Section 4.3.
As seen in Figure \ref{fig: grid-result}, the speed of training on target tasks with TRACE-PT outperforms that of all the other methods in all tasks.
MIKT learns faster than SAC, but slower than BT.
This is because that reusing previous parameters is more efficient than distilling regarding the same-domain transfer.
Besides, we find that SAC performs better than TRACE in all tasks because we map discrete actions into a continuous space, which makes it challenging to learn the policy, especially when the number of actions is small. Note that there are no jump-starts on those curves due to the action embeddings of target tasks are randomly initialized. However, the action embeddings adapt quickly, which results in a fast transfer.
Moreover, the action embeddings of the target task should align with the source task so that policy could have a promising performance on the target task.
To verify this, we project the embeddings of both source task $n=2$ and target task $n=1$ into 2D space. As shown in Figure \ref{fig: embed}(b), the embeddings of the source task and the target task are well aligned, and it is even observed that $e(\textcolor{red}{\uparrow}) \approx 0.5* e(\textcolor{blue}{\uparrow \uparrow}) + 0.5* e(\textcolor{blue}{\uparrow \downarrow})$.
\begin{figure*}
\centering
\includegraphics[height=15pt]{fig/new_legend2.pdf}
\vspace{-0.5em}
\subfigure[mP (source task mDP)]{
\label{fig: mujoco-result:a}
\includegraphics[width=0.25\textwidth]{fig/mujoco-transfer-2-0-1.pdf}
} \hspace{-1.15em}
\subfigure[rP (source task rDP)]{
\label{fig: mujoco-result:b}
\includegraphics[width=0.25\textwidth]{fig/mujoco-transfer-3-1-1.pdf}
} \hspace{-1.15em}
\subfigure[mDP (source task rP)]{
\label{fig: mujoco-result:c}
\includegraphics[width=0.25\textwidth]{fig/mujoco-transfer-1-2-1.pdf}
} \hspace{-1.15em}
\subfigure[rDP (source task mP)]{
\label{fig: mujoco-result:d}
\includegraphics[width=0.25\textwidth]{fig/mujoco-transfer-0-3-1.pdf}
}
\vspace{-10pt}
\caption{Experiment results on Mujoco and Roboschool tasks. The solid lines denote our method, and the dashed lines represent the compared methods. The shaded areas are bootstrapped 95\% confidence intervals.}
\label{fig: mujoco-result}
\vspace{-10pt}
\end{figure*}
\subsection{Mujoco and Roboschool}
Next, we consider a more difficult cross-domain transfer setting, in which state spaces are different as well as action spaces.
We conduct experiments among four environments, \textit{InvertedPendulum} and \textit{InvertedDoublePendulum} in \textbf{Mujoco} \cite{todorov2012mujoco} and \textbf{Roboschool}, respectively, denoted by \textit{mP} (mujoco Pendulum), \textit{mDP} (mujoco Double Pendulum), \textit{rP} (roboschool Pendulum) and \textit{rDP} (roboschool Double Pendulum) for short.
Currently, our methods are only suitable for discrete action spaces.
So we discretize the original $m$-dimension continuous control action space into $k$ equally spaced values on each dimension, resulting in a discrete action space with $|\mathcal{A}| = k^m$ actions.
The detailed configurations and descriptions of the environments are summarized in Appendix A.3 , and the learned action embeddings of the environments are shown in Figure 4 of Appendix.
In this experiment, our method is evaluated on four transfer tasks. We first try to transfer policy from \textit{mDP} to \textit{mP} and \textit{rDP} to \textit{rP}, where the source and the target tasks are still in the same underlying physical engine, and the target tasks are easier than the source ones. Further, we transfer policy from \textit{mP} to \textit{rDP} and \textit{rP} to \textit{mDP}, the source and the target tasks are in different physical engines, and the target tasks are more challenging than the source tasks.
Figure \ref{fig: mujoco-result} depicts the results of cross-domain transfer.
Overall, TRACE-PT still learns faster than SAC and BT in all tasks, and MIKT performs slightly better than SAC.
We can see that in Figure \ref{fig: mujoco-result:a} and \ref{fig: mujoco-result:b}, though BT has a promising performance, it fails to learn the target task in more challenging transfer tasks and even has negative transfer, shown in Figure \ref{fig: mujoco-result:c} and \ref{fig: mujoco-result:d}.
According to our extended experiments shown in Figure 3 of Appendix B, BT only performs well in the simplest cases. In most cases, it leads to a negative transfer. In contrast, TRACE-PT can handle all the transfer tasks and accelerate learning. This indicates that our approach is superior to BT and the advantages are more apparent when in more challenging transfer tasks.
\subsection{Combat Tasks in a Commercial Game}
To inspect our methods' potential in more practical problems, we validate them on a one-versus-one combat scenario in a commercial game where the agent can carry different sets of skills to fight against a build-in opponent.
In this experiment, we select two classes named \emph{She Shou} (SS) and \emph{Fang Shi} (FS).
The state representations are extracted manually and consist of information about the controlled agent and build-in opponent, forming two vectors with 48 and 60-dimensional, respectively. The sizes of action spaces are 10 for both classes, containing their unique skills and standard operations, such as move and attack.
The agent receives positive rewards for damaging and winning, and negative for taking damage and losing. Besides, the agent is punished for choosing unready skills.
We first verify whether our method can learn reasonable action embeddings in complex games.
\begin{wrapfigure}{r}{0.23\textwidth}
\centering
\includegraphics[width=0.25\textwidth]{fig/qnyh_embedding.png}
\caption{PCA projections of learned action embeddings. Each dot represents an action embedding of skill in the game.}
\vspace{-10pt}
\label{fig: qnyh_embed}
\end{wrapfigure}
we randomly sample 5 out of 15 skills and collect transition data. Table 4 of Appendix A lists the skill descriptions.
We sample 50,000 transition data to train action embeddings with $d=6$.
Figure \ref{fig: qnyh_embed} plots the result, and we notice that the skills with special effects, such as \textbf{Silence} and \textbf{Stun}, are distinguished clearly, and that damage skills are also closer to each other.
As annotated in the figure, the \textbf{DoT Damage} and \textbf{Instant Damage} are recognized as well.
\begin{figure}
\centering
\includegraphics[width=240pt]{fig/qnyh-res1.pdf}
\vspace{-10pt}
\caption{Experiment results on combat tasks.}
\label{fig: qnyh-result}
\vspace{-1em}
\end{figure}
For policy transfer, we first train policy individually for two roles and then transfer to each other.
The performance is measured by the average winning rate of the recent 100 episodes, as plotted in Figure \ref{fig: qnyh-result}.
For clarity, we show ablation results in Appendix B.
We can see that TRACE-PT achieves a better sample efficiency compared with MIKT, while BT may leads to the negative transfer.
It proves that our method can be applied to more practical problems.
\vspace{-5pt}
\subsection{Ablations}
To better understand our method, we analyze the contribution of the transition model and the policy model to performance promotion. The ablation experiments are designed as follows:
\begin{itemize}
\item \textit{TRACE-P}: Transfer policy model only, and randomly initialize transition model.
\item \textit{TRACE-T}: Transfer transition model only, and train policy from scratch.
\item \textit{BT}: Transfer policy, and do not use action embedding.
\end{itemize}
For same-domain transfer (Figure \ref{fig: grid-result}), we find that TRACE-P can also accelerate learning significantly. However, there still lies a gap between TRACE-PT and TRACE-P because it needs to learn the transition model and action embeddings anew, making the action embeddings of the target task may not align with the source task, and the policy requires more time to adapt. However, in the cross-domain transfer (Figure \ref{fig: mujoco-result}), TRACE-P achieves a competitive performance to TRACE-PT , especially in Figure \ref{fig: mujoco-result:b}.
It is easy to understand that the transition model and the action embeddings are learned from state embeddings, which are retrained on the target task. It limits the performance of transfer.
Besides, in both same-domain and cross-domain transfers, TRACE-T results in a similar performance to TRACE, which may indicate that the cost of training mainly lies in policy training and action representations can boost policy generalization.
Comparing TRACE-PT and BT, we find that the proposed action embeddings can indeed facilitate policy transfer.
\vspace{-7pt}
\section{Conclusion}
In this paper, we study how to leverage action embeddings to transfer across tasks with different action spaces and/or state spaces. We propose a method to effectively learn meaningful action embeddings by training a transition model. Further, we train RL policies with action embeddings by using the nearest neighbor in the embedding space. The policy and transition model are transferred to the target task, leading to a quick adaptation of policy. Extensive experiments demonstrate that it significantly improves sample efficiency, even with different state spaces and action spaces. In the future, we will try to extend our method to continuous action spaces and align the state embeddings with additional restrictions.
\small
\bibliographystyle{named}
|
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\section{SUPPLEMENTAL MATERIAL}
\section{Proof of theorem 1 and generalization to arbitrary priors}
Here we provide a lower and an upper bound on the probability of correct state discrimination, valid for a generic set of linearly independent pure states $\{ |\Psi_k\rangle\}_{k=1}^n$ and for a generic choice of prior probabilities $\{ p_k\}_{k=1}^n$.
The bounds are expressed in terms of the Gram matrix of the weighted states
\begin{align}
|\widetilde \Psi_k\rangle : = \sqrt {p_k}\, |\Psi_k\rangle \, ,
\end{align}
that is, the matrix $W$ with elements
\begin{align}W_{ij} : = \langle \widetilde \Psi_i| \widetilde \Psi_j\rangle \, .
\end{align}
The maximum success probability can be estimated with the following Theorem, which generalizes Theorem~1 of the main text to arbitrary prior distributions:
\begin{theo}
Let $\{ |\Psi_k\rangle\}_{k=1}^n$ be a set of linearly independent pure states.
The maximum probability of correctly identifying a state drawn from the set $\{ |\Psi_k\rangle\}_{k=1}^n$ with probability $\{p_k\}_{k=1}^n$ satisfies the bounds
\begin{align}\label{app:lower}
P_{\max} \ge \frac{\left( {{\rm tr}\, \sqrt W}\right)^2 } n
\end{align}
and
\begin{align}\label{app:upper}
P_{\max} \le \frac{\left( {{\rm tr}\, \sqrt W}\right)^2 } n + \sqrt{n\, \lambda_{\max }} \, \| {\bf q} - {\bf u} \|_1 \, ,
\end{align}
where $\lambda_{\max}$ is the maximum eigenvalue of $W$, \mbox{${\bf q} = \{q_k\}$} is the probability distribution defined by \mbox{$q_k : = \big( \sqrt W\,\big)_{kk}/{\rm tr}\,\big(\sqrt W\,\big)$}, $ {\bf u} = \{u_k\}$ is the uniform distribution, and $\| {\bf q} - {\bf u}\|_1 : =\sum_k |q_k-u_k|$ is the trace norm.
\end{theo}
{\bf Proof.} Since the states $\{ |\Psi_k\rangle\}_{k=1}^n$ are linearly independent, the optimal measurement consists of orthogonal rank-one projectors~\cite{belavkin}. Let us denote the projectors by $M_k = |m_k\rangle\langle m_k|$, for a suitable orthonormal basis $\{ |m_k\rangle\}_{k=1}^n$. Then, the probability of correct discrimination can be written as
\begin{align}
\nonumber P_{\rm succ} &= \sum_k p_k \, |\langle m_k| \Psi_k\rangle|^2 \\
\nonumber & = \sum_k |\langle m_k| \widetilde \Psi_k\rangle|^2\\
& = \sum_k \left| B_{kk} \right|^2\, ,
\end{align}
where $B$ is the matrix defined by the relation
\begin{align} |\widetilde \Psi_k \rangle = \sum_i \, B_{ik} \, |m_i \rangle \, .
\end{align}
By definition, one has $B^\dag B = W$.
Hence, the polar decomposition yields the relation
\begin{align}\label{polar}
B = U \sqrt W \, ,
\end{align}
for a suitable unitary matrix $U$.
Note that a generic change of orthonormal basis,
\begin{align}
|m_i\rangle \to |m_i' \rangle = \sum_j V_{ji} \, |m_j\rangle
\end{align}
(where $V$ is a generic unitary matrix), results into the change of matrix
\begin{align}\label{changeV}
B \to B' = V^\dag B \, .
\end{align}
Combining Eqs. (\ref{polar}) and (\ref{changeV}), the maximum probability of correct discrimination can be expressed as
\begin{align}\label{pmax}
P_{\max} = \max_U \, \sum_k \left| \left( U \sqrt W \right )_{kk} \right|^2 \, .
\end{align}
Setting $U=\openone$, one has the lower bound
\begin{align}
P_{\max} \ge \sum_k \left( \sqrt W \right )^2_{kk} \ge \frac{\left( {\rm tr}\, \sqrt W \right)^2}n \, ,
\end{align}
the second inequality following from the convexity of the function $f(x)= x^2$. This proves the lower bound (\ref{app:lower}).
Let us prove the upper bound (\ref{app:upper}). Using the Cauchy-Schwarz inequality in Eq.~(\ref{pmax}), we obtain the upper bound
\begin{align*}
P_{\max} & = \max_U \, \sum_{k} \left| \sum_{s}\left(U W^{\frac{1}{4}}\right)_{ks} \left(W^{\frac{1}{4}}\right)_{sk} \right|^2 \\
& \le \max_U \, \sum_k \left( \sqrt W\right)_{kk} \, \left( U \sqrt W U^\dag \right)_{kk} \\
& = {\rm tr}\, \left(\sqrt W\right) \times \, \max_U \left[ \sum_k q_k \, \left( U \sqrt W U^\dag \right)_{kk} \right] \, .
\end{align*}
Moreover, the argument of the maximum can be upper bounded as
\begin{align*}
\sum_k q_k \, \left( U \sqrt W U^\dag \right)_{kk} &\le \sum_k \frac 1n \, \left( U \sqrt W U^\dag \right)_{kk} \\
& \quad + \sum_k \left|q_k - \frac 1n \right| \left( U \sqrt W U^\dag \right)_{kk} \\
& \le \frac { {\rm tr}\, \sqrt W }n + \| {\bf q}- {\bf u} \|_1 \, \sqrt{ \lambda_{\max} }\, .
\end{align*}
Finally, from Eq. (\ref{app:lower}) we have the bound
\begin{align}
{\rm tr}\, \sqrt W \le \sqrt {n \, P_{\max}} \le \sqrt n \, .
\end{align}
Combining the above inequalities we obtain the desired upper bound (\ref{app:upper}). $\blacksquare$
\medskip
When the prior distribution is uniform, the weighted Gram matrix $W$ is given by $W= G/n$, where $G$ is the unweighted Gram matrix used in the main text. Substituting this relation into the bound (\ref{app:upper}) one obtains Eq.~(5) of the main text.
\section {\boldmath Eigenvalues and eigenvectors of $H$}
In this section we derive explicit expressions for the eigenvectors and eigenvalues of the matrix $H$.
To this purpose, it is useful to first recall some properties of
the Chebyshev polynomials of the second kind, denoted by $U_n(x)$. The Chebyshev polynomials of the second kind can be defined as the characteristic polynomial of the tridiagonal matrix $T$ of size~$n$ whose entries are $T_{ij}=\delta_{i\,j+1}+\delta_{j\,i+1}$. Specifically, \mbox{$U_n(x)=\det(2x\,\openone-T)$}, i.e., the eigenvalues of~$T$ are defined to be twice the roots, $x_l$, of~$U_n(x)$. By expanding the determinant by the first row one readily obtains the well known recursion relation $U_n(x)=2x \,U_{n-1}(x)-U_{n-2}(x)$~\cite{abramowitz}. One can check that this recursion relation along with the initial conditions in standard form, $U_0(x)=1$, $U_{-1}(x)=0$, give the right characteristic polynomial for {\em any} size of $T$. It suffices to check the~$n=1,2$ cases. One has~$U_1(x)=2x$ and $U_2(x)=4x^2-1$, which are indeed the characteristic polynomials of $T$ of sizes 1 and~2.
We now turn to the eigenvalues and eigenvectors of~$H$, which we will compute using a different approach. The matrix $H$ is nothing but the matrix $T$ with the addition of two extra entries at each end of the principal diagonal, namely,
$H_{ij}= T_{ij}+c\,( \delta_{i \,1}\delta_{j\,1}+ \,\delta_{i\, n}\delta_{j\, n})$. Let us denote by~$2x_l$ the $l$-th eigenvalue of $H$ and by $\vec{w}^{\, l}$ the corresponding
unnormalized eigenvector, chosen with the convention~$w^l_1=1$. The equation $H\vec{w}^{\, l}=2x_l \vec{w}^{\, l}$
is equivalent to the following system of linear equations:
\begin{eqnarray}
\label{eigenvectors}
c w_1^l+w^l_2&=& 2x_l\, w_1^l;
\nonumber \\
w^l_{j-1}+w^l_{j+1} &=& 2x_l\, w_j ^l, \qquad 2\le j\le n-1; \\
w_{n-1}^l+ c w^l_{n}&=& 2x_l\, w_n^l.
\nonumber
\end{eqnarray}
The second line of this system can be viewed as the recursion relation $w^l_{j+1} = 2x_l\, w_j ^l - w^l_{j-1}$,
which is the recursion relation of the Chebyshev polynomials given above (with $n\to j+1$). It follows that the first and second line of Eq.~(\ref{eigenvectors}), along with the convention $w^l_1=1$, imply
\begin{equation}
w^l_j=U_{j-1}(x_l)-c\,U_{j-2}(x_l),\qquad j=1,2,\dots, n.
\label{wl}
\end{equation}
Since all the components of $\vec{w}^{\, l}$ have been determined, the third line in Eq.~(\ref{eigenvectors}) must give the eigenvalues of~$H$.
By substituting Eq.~(\ref{wl}) in the third line of Eq.~(\ref{eigenvectors}) and using the Chebyshev recursion relation again, one obtains
\begin{equation}
0=U_n(x_l)-2c\,U_{n-1}(x_l)+c^2 U_{n-2}(x_l):=P_n(x_l),
\label{P_n again}
\end{equation}
which must hold for $l=1,\dots,n$. The polynomial~$P_n(x)$ has degree $n$ and its $n$ roots, $x_l$, give the eigenvalues of $H$ as $2x_l$. Note that $P_n(x)$ has to be proportional to the characteristic polynomial of $H$, i.e., $P_n(x)\propto \det(2x\openone-H)$, as both polynomials have the same degree and the same zeroes.
\section {\boldmath Distribution of the eigenvalues of $H$}
Here we analyze the distribution of the zeroes of the polynomial $P_n(x)$ defined in Eq. (\ref{P_n again}).
Setting $x= \cos \theta$, we recall that the Chebyshev polynomial $U_n (\cos \theta)$ can be expressed as \cite{abramowitz}
\begin{equation}
U_n(\cos\theta)={\sin(n+1)\theta\over \sin\theta} \, .
\label{U_n(cos)}
\end{equation}
Then, a little bit of trigonometry yields the relation
\begin{equation}
P_n(\cos\theta)=A(\theta)\sin\left[n\theta+\delta(\theta)\right],
\label{beat}
\end{equation}
with
\begin{eqnarray}
\left \{
\begin{array}{lll} A(\theta)&:=&\displaystyle{1-2c\cos\theta+c^2\over\sin\theta}\\
&&\\
\delta(\theta)&:=&\displaystyle\arctan{(1-c^2)\sin\theta\over(1+c^2)\cos\theta-2c}
\end{array}
\right. \, ,
\end{eqnarray}
where $0<\delta(\theta)<\pi$.
{}From Eq. (\ref{beat}) we can see that every zero of $P_n(\cos\theta)$ must be the solution to one of the equations
\begin{align} n\theta+\delta(\theta)= l \pi \, , \qquad l=1,2,\dots,n \, .
\end{align}
Denote by $\theta_l$ the angle that solves the $l$-th equation. Since $\delta (\theta_l)$ is contained in the interval $(0,\pi)$, we have the bound
\begin{equation}
{\pi \over n} l \le \theta_l \le {\pi \over n} \left(l+1\right) \, .
\end{equation}
In other words, the interval $(0,\pi)$ can be divided into intervals of length $\pi/n$, with the $l$-th interval containing the zero $\theta_l$. For large $n$, this means that the zeros are uniformly distributed in the interval $(0,\pi)$.
\iffalse
As a result, we can approximate sums with integrals, according to the relation
\begin{equation}
\label{sum to int}
\lim_{n\to \infty} \sum_{l=1}^{n} \frac{1}{n}f(\cos\theta _l) = \frac{1}{\pi}\int_0^\pi d \theta f(\cos\theta ) \, .
\end{equation}
valid for any function $f$ that is continuous in the interval $[0,\pi]$.
\fi
\iffalse To make this statement even more explicit, let us take~$a$ and~$b$ such that $0<a<b<\pi$. Let
$l_a$ be the smallest integer such that $a/\pi<l_a/n$. Analogously,
let $l_b$ be the largest integer such that $l_b/n<b/\pi$. Let~${\mathscr N}(a,b)$ be the number of zeroes $\theta_l$ in $(a,b)$, which satisfies
\begin{equation}
l_b-l_a-2\le{\mathscr N}(a,b)\le l_b-l_a+2.
\end{equation}
It immediately follows from the inequalities above that
\begin{equation}
\label{uniformity}
\left|{{\mathscr N}(a,b)\over n}-{b-a\over \pi}\right|<{2\over n}.
\end{equation}
In words, for large $n$, the ratio of the number of zeroes in the interval $(a,b)$ and $(0,\pi)$ approaches the ratio of the corresponding interval lengths for {\em any} interval $(a,b)$ in $(0,\pi)$, which proves that the zeroes are inded uniformly distributed on $(0,\pi)$.
\fi
\iffalse
Alternatively, a general proof that the zeros of a monic polynomial
are uniformly distributed in the asymptotic limit [WHAT IS $n$ IN THE GENERAL CASE?] was given by Erd\" os and Tur\'an in~\cite{erdos}. However, our proof shows that for the problem at hand uniformity holds up to order $1/n$, whereas Erd\"os and Tur\'an's result can only ensure order $\sqrt{\log n/n}$. Our result is a necessary condition to show in turn that the corrections to the asymptotic success probability are of order $1/n$.
\fi
\section{\boldmath The trace and diagonal matrix elements of $\sqrt G$}
Here we evaluate the normalized trace $\sqrt G/n$ and we quantify its deviation from the limit value $\gamma: = \lim_{n\to\infty } {\rm tr}\, \sqrt G / n$. In the process of computing the trace, we also evaluate the diagonal matrix elements of $\sqrt G$, which will become useful in the next section.
We proceed along the following steps:
\begin{enumerate}
\item construct the normalized eigenvectors of $\sqrt G$
\item evaluate the diagonal elements
\item evaluate the trace.
\end{enumerate}
\subsection{The normalized eigenvectors of $\sqrt G$}
The eigenvectors of $\sqrt G$ coincide with the eigenvectors of the matrix $H$, provided in Eq. (\ref{wl}).
Setting $x_l = \cos \theta_l$, we can use the trigonometric representation of the Chebyshev polynomials given in Eq.~(\ref{U_n(cos)}). In this way, we obtain
\begin{equation}
{w}^l_j={\sin( j\theta_l)-c \sin[(j-1)\theta_l]\over \sin\theta_l}.
\end{equation}
Now, the norm $\vec w^l$ can be evaluated explicitly as
\begin{align}
\|\vec w^l\|^2&:= \sum_j | w^l_j|^2 \nonumber \\
& ={n\over2\sin^2\theta_l}
\left\{
1-2c \cos\theta_l+c^2\phantom{{\sin(2n\theta_l)\over2n\sin\theta_l}}\right.\nonumber\\
& \quad +{1-c^2\over2n}\left[1-\cos(2n\theta_l)\right]\nonumber\\
&\quad- \left. {\sin(2n\theta_l)\over2n\sin\theta_l}\left[(1+c^2)\cos\theta_l-2c\right] \right\} \\
& = {n\over2\sin^2\theta_l}
\left\{
1-2c \cos\theta_l+c^2 + \frac{ f_n(\theta_l)}n \right\} \, , \nonumber
\end{align}
having defined the function
\begin{align}
\nonumber f_n(x) &: = {1-c^2\over2}\left[1-\cos(2n x)\right] \\
& \quad - {\sin(2n x)\over2\sin x}\left[(1+c^2)\cos x -2c\right] \, .
\end{align}
Defining the normalized eigenvectors $\vec v^l:=\vec w^l/\|\vec w^l\|$, we then have
\begin{align}\label{vl}
\left| v^l_j \right|^2 = \frac 2 n \frac{ \left [\, \sin( j\theta_l)-c \sin(j-1)\theta_l\, \right]^2 }{ 1-2c \cos\theta_l+c^2 + f_n(\theta_l)/n} \, .
\end{align}
\iffalse
\begin{align}
\|\vec w^l\|^2&:= \sum_j | w^l_j|^2 \nonumber \\
& ={n\over2\sin^2\theta_l}
\left\{
1-2c \cos\theta_l+c^2\phantom{{\sin(2n\theta_l)\over2n\sin\theta_l}}\right.\nonumber\\
& \quad +{1-c^2\over2n}\left[1-\cos(2n\theta_l)\right]\nonumber\\
&\quad- \left. {\sin(2n\theta_l)\over2n\sin\theta_l}\left[(1+c^2)\cos\theta_l-2c\right] \right\} \, . \nonumber
\end{align}
For $c\not = 1$, we can neglect the second and third term in the sum, obtaining
\begin{align}
\|\vec w^l\|^2 & ={n\over2\sin^2\theta_l}
\left\{
1-2c \cos\theta_l+c^2 + O\left( \frac 1n\right) \right\} \, .
\end{align}
Hence, the normalized eigenvectors $\vec v^l:=\vec w^l/\|\vec w^l\|$ have the simple asymptotic expression
\begin{equation}\label{vl}
v^l_j=\sqrt{2\over n} \left\{ {\sin j\theta_l-c\sin(j-1)\theta_l\over \sqrt{1-2c \cos\theta_l+c^2}} + O\left(\frac 1{n}\right)\right\} \, .
\end{equation}
\fi
\subsection{The diagonal elements of $\sqrt G$}
Having computed the eigenvalues and eigenvectors of the Gram matrix $G$, we can now evaluate the diagonal elements of its square root $\sqrt G$.
We start from the expression
\begin{align}
\sqrt G = \sum_l \sqrt {\lambda_l} \, |v^l\rangle\langle v^l| \, ,
\end{align} recalling that the eigenvalues are given by
\begin{align}\label{lambdal}\lambda_l = {\frac{1-c^2}{ 1-2c \cos\theta_l+c^2}} \, .
\end{align}
Then, the diagonal elements of $\sqrt G$ are
\begin{align}
\left( \sqrt G\right)_{kk} & = \sum_l \sqrt{\lambda_l} \, |v^l_k|^2 \, ,
\end{align}
with $v^l_k$ given as in Eq. (\ref{vl}). Explicitly, the matrix element $\big(\sqrt G\,\big)_{kk}$ is given by
\begin{align}
\nonumber \left( \sqrt G\right)_{kk}
\nonumber & = \frac 1 n \, \sum_l \sqrt{ {\frac{1-c^2}{ 1-2c \cos\theta_l+c^2}}} \\
\label{sqrtGkk}& \qquad \times \, {{\left[ \sin k\theta_l-c\sin(k-1)\theta_l\right]^2}\over { 1-2c \cos\theta_l+c^2 + f_n(\theta_l )/n }}
\end{align}
We now show that most of the matrix elements $\big(\sqrt G\,\big)_{kk}$ are approximately equal to the limit value
$ \gamma: =\lim_{n\to \infty} {\rm tr}\, \sqrt G /n$.
Note that $\gamma$ can be computed explicitly in terms the eigenvalues: indeed, one has
\begin{align} \nonumber
\gamma & = \lim_{n\to \infty} \frac 1 n \, \sum_l \sqrt \lambda_l \\
\nonumber & = \lim_{n\to \infty} \frac 1 n \, \sum_l \sqrt{\frac{1-c^2}{ 1-2c \cos\theta_l+c^2}} \\
& = \frac 1 \pi \, \int_0^\pi {\rm d} \theta\, \sqrt{\frac{1-c^2}{ 1-2c \cos\theta+c^2}} \, .
\end{align}
We now show that the deviation vanishes for all values of $k$ in the interval $[ n^\epsilon , n- n^\epsilon]$.
To this purpose, we evaluate the matrix element $\big( \sqrt G\,\big)_{kk}$ at the leading order of the large $n$ asymptotics, obtained by replacing the sum in Eq. (\ref{sqrtGkk}) by an integral and by dropping the term~$f_n( \theta_l)/n$ in the denominator. In this way, we obtain the approximate equality
\begin{align}\label{approx}
\left( \sqrt G\right)_{\!kk} \!\! \approx \frac {\sqrt{1\!-\!c^2}}\pi \! \int_0^\pi \!\! {\rm d} \theta \, {{\left[ \sin k\theta-c\sin(k-1)\theta \right]^2}\over { (1-2c \cos\theta +c^2 )^{3/2} }} \, .
\end{align}
Then, some elementary algebra gives
\begin{equation}
\left(\sqrt G\right)_{kk}-\gamma \approx
{\sqrt{1\!-\!c^2}\over\pi}\left( 2c I_{2k-1}\!-\!I_{2k}\!-\!c^2 I_{2k-2}\right) \, ,
\label{deviation}
\end{equation}
where the integrals $I_r$ are defined as
\begin{equation}
I_r\! :=\!\int_0^\pi\! {\cos r\theta\; d\theta\over\left(1-2c \cos\theta+c^2\right)^{3/2}} \, .
\label{I_m}
\end{equation}
We then show that the integrals $I_r$ vanish exponentially with $r$:
\begin{prop}\label{prop:integral}
For $c<1$, the leading order of the integral $I_r$ in Eq. (\ref{I_m}) is given by
\begin{equation}
I_r={2 c^r\sqrt{\pi r}\over(1-c^2)^{3/2}} \, .
\label{I_m asympt}
\end{equation}
\end{prop}
The proof can be found in the end of this subsection. Inserting the asymptotic expression (\ref{I_m asympt}) into Eq. (\ref{deviation}) we obtain the relation
\begin{equation}\label{skkminusave}
\left( \sqrt G\right)_{kk}\!-\gamma \approx {1 \over {4 (1-c^2)}} ~ {{c^{2k}} \over {\sqrt{2\pi k^3}}} \, ,
\end{equation}
valid in the interval $[ n^\epsilon , n- n^\epsilon]$. In conclusion, the deviation $\big( \sqrt G\,\big)_{kk} - \gamma$ decays exponentially with $k$.
We stress that the error introduced by the approximation (\ref{approx}) is negligible with respect to the leading order, quantified by the r.h.s. of Eq. (\ref{skkminusave}). This point is illustrated in Fig.~\ref{fig:3}, which compares the the r.h.s. of Eq. (\ref{skkminusave}) with the exact values of the deviation, computed by direct numerical evaluation of $\sqrt G$ from $G$. Log-scale plots are shown for various values of the overlap~$c$, setting $n=30$ and letting~$k$ vary from 1 to 15.
The agreement is extremely good and backs up the validity of the approximation (\ref{approx}) even for small values of $k$.
\begin{figure}[bhtp]
$$
\includegraphics[width=27em]{deviation_figure}
$$
\caption{ Log-scale plots of the deviation $\big( \sqrt G\,\big)_{kk}-\gamma$, for $n=30$ and $k$ varying from 1 to 15. The solid lines are the asymptotic approximation on r.h.s of Eq.~(\ref{skkminusave}) and the dots are the result of numerical evaluation of $\sqrt G$ from $G$.}
\label{fig:3}
\end{figure}
In the next subsection we will use Eq. (\ref{skkminusave}) to quantify the deviation between the diagonal of the matrix $\sqrt G/{\rm tr}\,(\sqrt G)$ and the uniform distribution.
\iffalse A similar situation occurs for $ n/2 \le k \ll n$. In this case, the matrix element are evaluated through the identity
\begin{align}
\left(\sqrt G\right)_{n-k,n-l}= \left(\sqrt G\right)_{k,l} \, ,
\end{align}
which yields the relation
\begin{equation}\label{reciprocal}
\left( \sqrt G\right)_{kk}\!-\!\gamma= {1 \over {4 (1-c^2)}} ~ {{c^{2 (n-k)}} \over {\sqrt{2\pi (n-k)^3}}} \, ,
\end{equation}
valid for $n/2\le k\ll n$.
\fi
\medskip
{\bf Proof of Proposition \ref{prop:integral}.}
We start by noticing that the integral on the r.h.s of Eq.~(\ref{I_m}) can be expressed as a contour integral over the unit circle~$C$ on the complex plane:
\begin{equation}
I_r=
{1\over2i}\oint_C dz\;{z^{r+1/2}\over\left[z-c(z^2+1)+c^2 z\right]^{3/2}} .
\label{I_m complex}
\end{equation}
\begin{figure}[bhtp]
$$
\includegraphics[width=2.1in]{cuts_contours}
$$
\caption{The figure shows (gray) the branch cuts of the integrand in Eq.~(\ref{I_m complex}) and the contour $C'$ used to obtain Eq.~(\ref{I_m 2 int}).}
\label{fig:2}
\end{figure}
We can choose the branch of the integrand so that its branch cuts are the intervals~$[0,c]$ and~$[c^{-1},\infty)$ on the real axis. Since this branch is analytic elsewhere~and the cut~$[c^{-1},\infty)$ is outside $C$, we can deform the contour~$C$ to a new contour $C'$ around $[0,c]$. One readily sees that the integrand in Eq.~(\ref{I_m complex}) behaves as $(z-c)^{-3/2}$, so care must be taken to evaluate the new contour integral near the end point $z=c$, as some divergencies may arise because of this singular behavior. Specifically, we choose~$C'$ as in Fig.~\ref{fig:2}, where $\epsilon>0$ and the limit $\epsilon\to0$ is implicit. As a result, $I_r$ has a contribution coming from the discontinuity of the integrand along the interval $[0,c-\epsilon]$ and a contribution coming from the integration around the circle $C_\epsilon$ of radius $\epsilon$ and center at $z=c$:
\begin{eqnarray}
I_r
&=&-{1\over c^{3/2}}\int_{0}^{c-\epsilon}dx\;{x^{r+1/2}\over\left\{(c-x)[(1/c)-x]\right\}^{3/2}}\nonumber\\
&+&{1\over2ic^{3/2}}\oint_{C_\epsilon}dz{c^{r+1/2}\over(z-c)^{3/2}[(1/c)-c]^{3/2}}.
\label{I_m 2 int}
\end{eqnarray}
Note that the limit $\epsilon\to0$ of each separate line is ill-defined, as they both diverge as $\epsilon^{-1/2}$. To circumvent this problem,
we write $(c-x)^{-3/2}=2(d/dx)(c-x)^{-1/2}$ and integrate the first line of~Eq.~(\ref{I_m 2 int}) by parts. In doing so, we see that the $\epsilon^{-1/2}$ terms cancel and we obtain the simple expression
\begin{equation}
I_r=2\int_{0}^{c}{dx\over (c-x)^{1/2}}{d\over dx}{x^{r+1/2}\over[1- c x]^{3/2}}.
\end{equation}
We can further simplify this expression using the change of variable $x= c t$, which enables us to express $I_r$ in terms of hypergeometric functions. However, we are just interested in the asymptotic behavior of $I_r$. Keeping only the leading contribution as $r$ goes to infinity, we have
\begin{equation}
I_r=2r c^r\int_0^1 dt\, t^{r-1/2}(1-t)^{-1/2}(1-c^2 t)^{-3/2}.
\label{I_m t-int}
\end{equation}
The asymptotic behaviour of this integral can be easily evaluated by noticing that the leading contribution comes from the region near the upper limit of integration, so we can set~$t=1$ in the last factor in Eq.~(\ref{I_m t-int}) and write
\begin{eqnarray}
I_r\!\!&=&\!\!{2rc^r\over(1-c^2)^{3/2}}\int_0^1\!\! dt\, t^{r-1/2}(1-t)^{-1/2}\nonumber\\[.5em]
\!\!&=&\!\!{2rc^r B(\mbox{${1\over2}$},r\!+\!\mbox{${1\over2}$})\over(1\!-\!c^2)^{3/2}}\nonumber\\[.5em]
&=&{2\sqrt\pi c^r\,\Gamma(r\!+\!\mbox{${1\over2}$})\over (r\!-\!1)!(1\!-\!c^2)^{3/2}},
\label{I_m Gamma}
\end{eqnarray}
where $B(a,b)$ is the Euler Beta function,
\begin{eqnarray}
B(a,b)=\int_0^1 dt\; t^{a-1}(1-t)^{b-1},\\[-.1em]\nonumber
\end{eqnarray}
and we have used the relation,
\begin{equation}
B(a,b)={\Gamma(a)\Gamma(b)\over\Gamma(a+b)}.
\end{equation}
Using the Stirling formula in the third line of Eq.~(\ref{I_m Gamma}), we finally obtain Eq.~(\ref{I_m asympt}). $\blacksquare$
\subsection{The trace of $\sqrt G$}
Here we show that the normalized trace ${\rm tr}\, \sqrt G/n$ is close to its limit value $\gamma$, up to an error of size $1/n^{1-\epsilon}$, where $\epsilon$ is an arbitrary constant in the interval $(0,1)$.
For this purpose, we divide the values of $k$ into two subsets, defined as
\begin{align}
\nonumber {\mathsf S} &: = \{ k \in \mathbb N : \lceil n^{\epsilon} \rceil \le k \le n-\lceil n^{\epsilon} \rceil \ \} \\
\label{sets} \overline{\mathsf S} & : = \{ 1,\dots, n\}\setminus \mathsf S \, .
\end{align}
The trace of $\sqrt G$ can be evaluated as
\begin{align}
\nonumber {\rm tr}\, \sqrt G & = \sum_{k \in \mathsf S} \, \left( \sqrt G\right)_{kk} + \sum_{k \in \overline {\mathsf S}} \, \left( \sqrt G\right)_{kk} \\
\nonumber & = \sum_{k \in \mathsf S} \, \left( \gamma + {1 \over {4 (1-c^2)}} ~ {{c^{2k}} \over {\sqrt{2\pi k^3}}} \right) + \sum_{k \in \overline {\mathsf S}} \, \left( \sqrt G\right)_{kk} \\
& = | \mathsf{S} \, | \gamma + \sum_{k \in \overline {\mathsf S}} \, \left( \sqrt G\right)_{kk} + O \left( n^{1-3\epsilon/2}c^{2n^\epsilon}\right) \, ,
\end{align}
the second equality following from Eq. (\ref{skkminusave}).
Using the above expression it is easy to produce upper and lower bounds on ${\rm tr}\, \sqrt G$. An upper bound is obtained as follows:
\begin{align}\label{upperboundtrsqrt}
\nonumber{\rm tr}\,\sqrt G & \le | \mathsf{S}| \, \gamma + \sqrt{ \lambda_{\max}} \, |\overline {\mathsf S}| + O \left( n^{1-3\epsilon/2}c^{2n^\epsilon}\right) \\
& \le n \, \gamma + \sqrt{\lambda_{\max}} |\overline {\mathsf S}| + O \left( n^{1-3\epsilon/2}c^{2n^\epsilon}\right) \, .
\end{align}
Similarly, we have the lower bound
\begin{align}
\nonumber{\rm tr}\,\sqrt G & \ge | \mathsf{S}| \, \gamma + O \left( n^{1-3\epsilon/2}c^{2n^\epsilon}\right) \\
& = n \, \gamma - \gamma |\overline {\mathsf S}| + O \left( n^{1-3\epsilon/2}c^{2n^\epsilon}\right) \, .
\end{align}
Using the relation $ 2 n^{\epsilon} \le |\overline {\mathsf S}| \le 2 \, (n^{\epsilon}+1) $ we finally obtain
the bounds
\begin{align}
\gamma - \frac{ 2\gamma}{n^{1-\epsilon}} \le \frac{{\rm tr}\, \sqrt G}n \le \gamma + 2 \sqrt{\lambda_{\max}} \left( \frac{ 1}{n^{1-\epsilon}} + \frac 1n \right) \, ,
\end{align}
valid up to an exponentially small correction of size~$ O \left( n^{-3\epsilon/2}c^{2n^\epsilon}\right)$.
More compactly, the above bounds can be written as
\begin{align}
\left| \frac{ {\rm tr}\, \sqrt G} n - \gamma \right| \le \frac { 2 \, \max \{ \gamma, \sqrt{ \lambda_{\max}}\} }{n^{1-\epsilon}} + O \left( \frac 1n\right) \, .
\end{align}
Now, direct inspection shows that $\sqrt{\lambda_{\max}}$ is always larger than $\gamma$. Hence, the bound becomes
\begin{align}\label{normtracebound}
\left| \frac{ {\rm tr}\, \sqrt G} n - \gamma \right| \le \frac { 2 \sqrt{ \lambda_{\max}} }{n^{1-\epsilon}} + O \left( \frac 1n\right) \, .
\end{align}
\section{\boldmath Deviation of ${\bf q}$ from the uniform distribution}
In this section we consider the probability distribution $ {\bf q} = \{ \big( \sqrt G \, \big)_{kk}/{\rm tr}\, \sqrt G\}$ for the change point problem and we quantify the deviation of $ \bf q$ from the uniform distribution.
Here we upper bound the trace distance between the probability distribution ${\bf q }=\{ q_k\} $ defined by
\[ q_k : = \frac{ \left( \sqrt G \right)_{kk}}{{\rm tr}\, \sqrt G}\] and the uniform distribution, denoted by $\bf u$. Our strategy is to separately analyze the contributions to the trace distance coming from the two sets $\mathsf S$ and $\overline {\mathsf S}$ defined in Eq.~(\ref{sets}).
Let us consider first the contribution of the set $\mathsf S$. For~$k\in\mathsf S$, we have
\begin{align}
\nonumber
\left| q_k - \frac 1n \right| & = \left| \frac { \left( \sqrt G\right)_{kk} - {\rm tr}\, \sqrt G /n} { {\rm tr}\, \sqrt G} \right| \\
& \le \frac { \left| \left( \sqrt G\right)_{kk} - \gamma \right| + \left| \gamma - {\rm tr}\, \sqrt G /n \right| } { {\rm tr}\, \sqrt G} \, .
\end{align}
Now, the first term is upper bounded as
\begin{align}
\nonumber \left| \left( \sqrt G\right)_{kk} - \gamma \right| & \le {1 \over {4 (1-c^2)}} ~ {{c^{2n^\epsilon}} \over {\sqrt{2\pi n^{3\epsilon}}}} \\
\label{primo} & = O \left( n^{-3\epsilon/2}c^{2n^\epsilon}\right) \, ,
\end{align} due to Eq. (\ref{skkminusave}). The second term is upper bounded by Eq. (\ref{normtracebound}).
Hence, the contribution of $\mathsf S$ to the trace distance can be upper bounded as
\begin{align}
\nonumber
\sum_{k\in\mathsf S} \left| q_k - \frac 1n \right| & \le \sum_{k\in\mathsf S} \frac{ { 2 \sqrt{ \lambda_{\max}} }/{n^{1-\epsilon}} + O \left( 1/n \right) } {{\rm tr}\, \sqrt G} \\
\nonumber
& \le \sum_{k\in\mathsf S} \frac{ { 2 \sqrt{ \lambda_{\max}} }/{n^{1-\epsilon}} + O \left( 1/n\right) } {n\sqrt {\lambda_{\min}}} \\
\nonumber
& \le \sqrt{\frac{ \lambda_{\max}} {\lambda_{\min}}} \, \frac{2}{n^{1-\epsilon}} + O \left( \frac 1n\right)
\, , \label{vanish1}
\end{align}
having used the relation
\begin{align}
{\rm tr}\, \sqrt G \ge n \, \sqrt{\lambda_{\min}} \, ,
\end{align}
where $\lambda_{\min}$ is the minimum eigenvalue of $G$.
In conclusion, Eq. (\ref{vanish1}) shows that the contribution of the set $\mathsf S$ vanishes in the large $n$ limit.
Let us consider the contribution of the set $\overline {\mathsf S}$. For $k\in \overline{\mathsf S}$, we have the inequality
\begin{align}
\nonumber
\left| q_k - \frac 1n \right| & = \left| \frac { \left( \sqrt G\right)_{kk} - {\rm tr}\, \sqrt G /n} { {\rm tr}\, \sqrt G} \right| \\
\nonumber & \le { \sqrt{\lambda_{\max}} \over { {\rm tr}\, \sqrt G }} \\
& \le \sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} \, \frac 1n \, ,
\end{align}
which leads to the upper bound
\begin{align}
\label{vanish2} \sum_{k\in \overline{\mathsf S}} \left| q_k - \frac 1n \right| &
\le \sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}} \, \frac2 {n^{1-\epsilon}} \, .
\end{align}
Using the bounds (\ref{vanish1}) and (\ref{vanish2}), the deviation between~$\bf q$ and the uniform distribution can be upper bounded as
\begin{align}
\nonumber \| {\bf q } - {\bf u}\|_1 & = \sum_{k\in {\mathsf S}} \left| q_k - \frac 1n \right| + \sum_{k\in \overline{\mathsf S}} \left| q_k - \frac 1n \right| \\
\nonumber & \le
\sqrt{\frac{ \lambda_{\max}} {\lambda_{\min}}} \, \frac{4}{n^{1-\epsilon}} + O \left( \frac1n\right) \\
& \le
{\frac{1+c} {1-c}} \, \frac{4}{n^{1-\epsilon}} + O \left( \frac 1n\right) \, ,
\end{align}
having used the bounds
\begin{align} \lambda_{\max} \le (1+c)/(1-c)
\end{align} and
\begin{align}\lambda_{\min} \ge (1-c)/(1+c) \, ,
\end{align} following from Eq. (\ref{lambdal}).
\section{Lower bound on the success probability of the square root measurement}
For a generic set of linearly independent pure states $\{ |\Psi_k\rangle\}_{k=1}^n$ and a generic choice of prior probabilities~$\{ p_k\}_{k=1}^n$, the success probability of the square root measurement can be expressed as \cite{pozza}
\begin{align}
P_{\rm SQ} = \sum_k \, \left( \sqrt W \right)_{kk}^2 \, .
\end{align}
The convexity of the function $f(x)= x^2$ then implies the bound $ P_{\rm SQ} \ge \big( {\rm tr}\, \sqrt W \, \big)^2/n $.
\section{Greedy strategy and Bayesian updating}
Here we show that Bayesian updating gives the optimal greedy strategy introduced in the main text. This follows from the observation that the optimal measurement (and the optimal guess) at step $s$ of the greedy strategy are determined solely by the posterior probability distribution after the measurement at step $s-1$, as will be explicitly shown at the end of this section.
To optimize the greedy strategy, we need to maximize ${\mathscr P}^{\rm G}_s=\sum_{r=1}^{n} \eta_{r}^{(s)} \langle \Psi_{r} |E_s(r) |\Psi_r\rangle$ over all POVM measurements on particle~$s$, $\{E_s(r)\}_{r=1}^n$.
Noticing that the source state $|\Psi_k\rangle$ restricted to particle~$s$ is $|\Psi_k\rangle_{\!s}=|0\rangle$ for $s< k$, and $|\Psi_k\rangle_{\!s}=|\phi\rangle$ for $s\ge k$, the following relations are self evident:
\begin{eqnarray}
\label{ps-t}
{\mathscr P}^{\mathrm{G}}_{s}\!\!\!&=&\!\!\!
\sum_{r=1}^{s} \eta_{r}^{(s)}\, \langle \phi | E_s(r) | \phi \rangle \! +\! \sum_{r=s+1}^n \eta_{r}^{(s)} \, \langle 0 | E_s(r)| 0 \rangle \nonumber\\
\!\!\!& \leq&\!\!p_{\phi}^{(s)} \!\sum_{r=1}^{s} \langle \phi | E_s(r) | \phi \rangle\! +\! p_{0}^{(s)}\! \sum_{r=s+1}^n \langle 0 | E_s(r)| 0 \rangle \nonumber \\[.5em]
\!\!\!&=&\!\! \! p_{\phi}^{(s)}\! \ \langle \phi | \Pi_s(\phi)| \phi \rangle + p_{0}^{(s)}\, \langle 0 | \Pi_s(0) | 0 \rangle ,
\end{eqnarray}
where
$p_{\phi}^{(s)} :=\max_r{\{\eta_{r}^{(s)}\}_{r=1}^{s}}$, $p_{0}^{(s)}=\max_r{\{\eta_{r}^{(s)}\}_{r=s+1}^n}$, $\Pi_{s}(\phi)=\sum_{r=1}^{s} E_s(r)$,
and $\Pi_s(0)=\openone -\Pi_s(\phi)$.
The inequality is saturated by choosing a new POVM $\{E'_s(r)\}_{r=1}^n$ whose elements are non-zero only in the two positions that maximize the prior probabilities:
\begin{equation}
r_\phi=\argmax_{r}{\{\eta_{r}^{(s)}\}_{r=1}^{s}},\; r_0=\argmax_{r}{\{\eta_{r}^{(s)}\}_{k=s+1}^n},
\end{equation}
so that
$E'_s ( r_0 ) = \Pi_s (0)$ and $E'_s ( r_1 ) = \Pi_s (\phi ) $. This justifies the choice of priors in Eq.~(12) of the main text.
The success probability can now we written in terms of the Helstrom matrix $\Gamma_s=p_{\phi}^{(s)} |\phi\rangle\langle\phi|-p_{0}^{(s)} |0\rangle\langle0|$ as:
\begin{eqnarray}
{\mathscr P}^{\mathrm{G}}_{s}&=&p_{0}^{(s)}+ {\rm tr}\,\left(\Pi_{s}(\phi) \Gamma_s\right)\leq p_{0}^{(s)}+ {\rm tr}\, \left(\Gamma_s^{+}\right)\nonumber \\
&=&\frac{1}{2}\left( p_{\phi}^{(s)}+p_{0}^{(s)}+{\rm tr}\,|\Gamma_{s}|\right),
\end{eqnarray}
where $\Gamma_s^{+}$ is the positive part of matrix $\Gamma$.
The inequality is saturated by choosing $\Pi_s(\phi)$ to be the projector onto the positive subspace of $\Gamma$~\cite{helstrom}.
We now show that the optimal measurement and guess at step $s$ of the greedy strategy
do not depend on the particular sequence measurement outcomes, but only on the posterior probability distribution after the measurement at step $s-1$.
Let us introduce the short-hand notation ${\mathbf{r}_{s}}:=\{r_{1},\ldots , r_{s}\}$ for a sequence of results obtained up to step~$s$.
The average success probability at each step $s$ is given by
\begin{equation}
\label{ps-tbayes}
\sum_{k=1}^n\sum_{\mathbf{r}_{s}}p(\mathbf{r}_{s},k) \delta_{k\,\hat{k}(\mathbf{r}_{s})}
\leq \sum_{\mathbf{r}_{s}}\max_{k}{p(\mathbf{r}_{s},k)},
\end{equation}
where $p(\mathbf{r}_{s},k)$ is the joint probability of obtaining the sequence $\mathbf{r}_{s}$ of results and the change point occuring at position $k$, and $\hat{k}(\mathbf{r}_{s})\in \{1,\ldots, n\}$ is the decision function that assigns to each $\mathbf{r}_{s}$ the guessed change point position~$k=\hat{k}(\mathbf{r}_{s})$. The inequality can be saturated by
$\hat{k}(\mathbf{r}_{s})=\argmax_{k} p(\mathbf{r}_{s},k)$. Since the source states $|\Psi_k\rangle$ are of product form, we can write
\begin{equation}
p(\mathbf{r}_{s},k)={1\over n}p(\mathbf{r}_{s-1}| k)\,\langle\Psi_k|E_{s}(r_s) |\Psi_k\rangle ,
\end{equation}
where we recall that $|\Psi_k\rangle$ restricted to particle $s$ is $|\Psi_k\rangle_{\!s}=|0\rangle$ for $s< k$, and $|\Psi_k\rangle_{\!s}=|\phi\rangle$ for $s\ge k$. The measurement over the $s$-th particle is represented by
the POVM $\{E_{s}(r)\}_{r=1}^n$ and it is understood that it may depend on the sequence~$\mathbf{r}_{s-1}$ of previous results. Hence,
the optimal greedy average success probability at step~$s$, can be written as
\begin{equation}
P^{\rm G}_{s}=
\sum_{\mathbf{r}_{s-1}}p(\mathbf{r}_{s-1}){\mathscr P}^{\rm G}_{s}({\mathbf{r}_{s-1}}),
\end{equation}
where the probability of successful identification of the change point at step $s$ conditioned to the occurrence of the sequence $\mathbf{r}_{s-1}$ [${\mathscr P}^{\rm G}_s$ in
Eq.~(\ref{ps-t}); we recall that the dependence on ${\bf r}_{s-1}$ is understood there]~is
\vspace{-.2cm}
\begin{eqnarray}
{\mathscr P}^{\rm G}_{s}({\mathbf{r}_{s-1}})\!\!&=&\!\!\max_{\{E_{s}(r)\}}\sum_{r=1}^n\max_{k} p(k|\mathbf{r}_{s-1})\nonumber
\\
\!\!&\times&\!\!\langle\Psi_k|E_{s}(r) |\Psi_k\rangle,
\label{only priors}
\\[-.5em]
\nonumber
\end{eqnarray}
\\[-1em]
and we have used Bayes' rule to obtain the relation $(1/n)\,p(\mathbf{r}_{s-1}|k)=p(k|\mathbf{r}_{s-1}) p(\mathbf{r}_{s-1})$. From Eq.~(\ref{only priors}), it is apparent that the optimal measurement can only depend on the updated priors $\eta_{k}^{(s)}:=p(k|\mathbf{r}_{s-1})$, rather than on the whole sequence of previous results, as the maximization is only subject to the POVM conditions $E_s(r)\ge0$ and $\sum_{r=1}^n E_s(r)=\openone$. Likewise, the optimal guess can only depend on $\eta_{r}^{(s)}$ [Eq.~(\ref{ps-tbayes}) and the paragraph below~it].
\bibliographystyle{unsrt}
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{
"redpajama_set_name": "RedPajamaArXiv"
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Success In Life and Relationships
AMBROSE Blog
Home News In his effort to break the Brexit deadlock, Johnson calls for new...
In his effort to break the Brexit deadlock, Johnson calls for new election in December
Ambrose Nwaopara
Boris Johnson, the UK Prime Minister has on Thursday called for a general election on December 12 to break Britain's Brexit impasse, conceding for the first time he will not meet his "do or die" deadline to leave the European Union next week.
Johnson said in a letter to opposition Labour leader Jeremy Corbyn that he would give parliament more time to approve his Brexit deal, insisting lawmakers must back a December election. This is Johnson's third attempt to try to force a snap vote.
Corbyn said he would wait to see what the EU decides on a Brexit delay before deciding which way to vote on Monday, repeating that he could only back an election when the risk of Johnson taking Britain out of the EU without a deal to smooth the transition was off the table.
With other opposition parties rejecting the election offer, it was increasingly unlikely that Johnson's latest bid to replace a parliament that has repeatedly put hurdles in his way would be successful.
Just a week before Britain was due to exit the EU, the bloc looks set to grant Johnson a Brexit delay, something he has repeatedly said he does not want but was forced by parliament to request.
An election is seen by his team as the only way of breaking the deadlock over Brexit after parliament voted in favour of his deal at the first stage, but then, minutes later, rejected his preferred timetable which would have met his Oct. 31 deadline.
But he has twice failed before to win the votes in parliament for an election, where he needs the support of two-thirds of its 650 lawmakers.
"This parliament has refused to take decisions. It cannot refuse to let the voters replace it with a new parliament that can make decisions," he wrote to Corbyn.
"Prolonging this paralysis into 2020 would have dangerous consequences for businesses, jobs and for basic confidence in democratic institutions, already badly damaged by the behaviour of parliament since the referendum. Parliament cannot continue to hold the country hostage."
Corbyn, a veteran critic of the EU, said he wanted to wait until Friday to see what Brussels had decided to do with Britain's request for a delay – something Johnson was forced to ask for by parliament.
"The principle is take 'no deal' off the table, the EU answers tomorrow, then we can decide," Corbyn told reporters.
The Scottish National Party and other smaller parties rejected the prime minister's attempt to force an election, casting doubt on whether the Conservative leader will be able to secure the votes needed to hold a ballot before Christmas.
More than three years after Britons voted 52%-48% to be the first sovereign country to leave the European project, the future of Brexit is unclear.
Brexit has increasingly dominated politics, pushing other pressing issues aside. On Thursday, Brexit delays were blamed for finance minister Sajid Javid postponing his first budget.
Johnson won the top job in July by staking his career on getting Brexit done by Oct. 31, though in Thursday's letter he makes clear he is ready to scrap his deadline. Last month, he said he would rather be "dead in a ditch" than ask for a delay.
But several of his aides think he can weather any criticism for failing to meet the deadline by arguing that he was thwarted by lawmakers, doubling down on his team's narrative of "people versus parliament".
At a meeting of Johnson's top ministers, some media reported disagreement over whether the government should try for an early election, fearing that doing so before Brexit was settled might damage the Conservatives.
Johnson seems to still hold out hope of securing a deal with Brussels, offering parliament until Nov. 6 to ratify an agreement he settled with the EU last week.
"This means that we could get Brexit done before the election on 12 December, if MPs (members of parliament) choose to do so," he said.
Labour has long said it cannot back an election until no- deal Brexit is off the table. But if the EU grants an extension until the end of January, that would appear to remove the threat of Johnson taking Britain out of the bloc without an agreement.
By proposing to dissolve parliament on Nov. 6, that would also be beyond the current October 31 deadline.
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US finally opens up on casualty situation after Iran's missile attacks
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Ambrose Nwaopara - January 18, 2020
The United States of America has finally opened up on the plight of its soldiers in Iraq after the January 8 missile attacks by...
Messi names his two best teams in the world as he...
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In an act of wickedness, man cuts off his aunty's head
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Success In Life and Relationships26
© 2019 Ambrosy Media. All rights reserved.
|
{
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Trawiszka bodziszówka (Agromyza nigrescens) – gatunek muchówki z podrzędu krótkoczułkich i rodziny miniarkowatych.
Gatunek ten opisany został w 1920 roku przez Friedricha Georga Hendela.
Larwy tych muchówek są beznogie i pozbawione puszki głowowej. Żerują minując liście bodziszków. Ponadto notowane z Erodium moschatum. Miny tworzone są na górnej stronie liścia. Mają barwę zielonkawobiałą. Początkowo mina jest korytarzowa i podąża mniej więcej wzdłuż brzegu blaszki liściowej. Później mina poszerza się, przybierając postać komorową. Granice między częścią pierwotną i wtórną pozostają dobrze widoczne. W części początkowej korytarza grudki odchodów układają się regularnie, natomiast w dalszych odcinkach chodnika bardziej nieregularnie.
Przepoczwarczenie następuje poza miną. Puparium jest ubarwione rudobrązowo, a tylne przetchlinki mają po trzy bulwki.
Pasożytują na niej błonkówki Chrysocharis amyite i Miscogaster hortensis.
Miniarka ta zasiedla Palearktykę od Wysp Kanaryjskich po Japonię. W Polsce wykazywana z dziko rosnących bodziszków.
Przypisy
Miniarkowate
Muchówki Afryki
Muchówki Europy
Muchówki Azji
Gatunki i podgatunki zwierząt nazwane w 1920 roku
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{
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{"url":"https:\/\/www.gradesaver.com\/textbooks\/math\/algebra\/intermediate-algebra-connecting-concepts-through-application\/chapter-6-logarithmic-functions-6-5-solving-exponential-equations-6-5-exercises-page-526\/2","text":"## Intermediate Algebra: Connecting Concepts through Application\n\n$x\\approx2.816$\nTake the log of both sides, and then use the power property. $7^x=240$ $x\\log7=\\log240$ $x=\\frac{\\log240}{\\log7}$ $x\\approx2.816$","date":"2018-06-20 15:48:41","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8804068565368652, \"perplexity\": 984.1754098780066}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-26\/segments\/1529267863650.42\/warc\/CC-MAIN-20180620143814-20180620163814-00172.warc.gz\"}"}
| null | null |
\section{Introduction}
In recent years, galaxy clusters have emerged as one of the most unique and powerful laboratories for cosmology and astrophysics. Being the largest and most magnificent structures in the Universe, clusters of galaxies serve as excellent tracers of the growth of cosmic structures. Current generation of X-ray cluster surveys have provided independent confirmation of cosmic acceleration and significantly tighten constraints on the nature of dark energy \citep{allen_etal08,vikhlinin_etal09} and alternative theories of gravity \citep[e.g.,][]{schmidt_etal09}. Several ongoing and new X-ray (e.g., {\it eROSITA}) and Sunyaev-Zel'dovich effect (SZE) cluster surveys (e.g., {\it SPT, ACT, Planck}) are underway to improve current cosmological constraints.
Outskirts of galaxy clusters have special importance for cluster cosmology, because they are believed to be much less susceptible to complicated cluster astrophysics, such as radiative gas cooling, star formation, and energy injection from active galactic nuclei. Dominant physical processes in the outskirts are limited to the gravity-driven collisionless dynamics of dark matter and hydrodynamics of the intracluster medium (ICM). In the hierarchical structure formation model, galaxy clusters grow by accreting clumps and diffuse gas from the surrounding large-scale structure in their outer envelope. Numerical simulations predict that the large-scale cosmic accretion and mergers give rise to internal gas motions and inhomogeneous gas distribution in the ICM. However, until very recently, observational studies of the ICM have been limited to radii considerably smaller than the virial radius of clusters.
Recently, {\it Suzaku} X-ray observations have extended X-ray measurements of the ICM profile out to and beyond the virial radius for several clusters \citep{bautz_etal09,george_etal09,reiprich_etal09,hoshino_etal10,kawaharada_etal10}. While these measurements are still quite uncertain, initial results suggested that the observed ICM profiles may deviate significantly from the prediction of hydrodynamical cluster simulations \citep[e.g.,][]{george_etal09}. In addition to testing models of structure formation, these new measurements will be important for controlling systematic uncertainties in cluster-based cosmological measurements.
In this work, I will present theoretical modeling of the outskirts of galaxy clusters based on cosmological simulations, with highlights on implications for the interpretation of forthcoming multi-wavelength observations of galaxy clusters. The simulations we present here are described in \citet{nagai_etal07a} and \citet{nagai_etal07b}, and we refer the readers to these papers for more details.
\begin{figure}[t]
\begin{center}
\vspace{-4mm}
\resizebox{\hsize}{!}{\includegraphics[clip=true]{f1.eps}}
\vspace{-8mm}
\caption{\footnotesize
{\it Top panel:} Ratio of pressure from random gas motions to total pressure as a function of radius. Relaxed clusters are represented by solid lines while unrelaxed clusters are represented by dashed lines. {\it Bottom panel:} Averaged mass profiles $M(< r)$ of the relaxed clusters, normalized by $M_{500}$. The solid line shows the actual mass profile from simulation, the long dashed line shows the mass profile from hydrostatic equilibrium including random gas and thermal pressure, and the short dashed line shows the mass profile from hydrostatic equilibrium including thermal pressure only. Hashed region shows the 1-$\sigma$ error of the mean. From \citet{lau_etal09}.}
\vspace{-5mm}
\label{fig:clump_phase}
\end{center}
\end{figure}
\section{Gas Motions in Cluster Outskirts}
Gas motions induced by cosmic accretion and mergers provide non-thermal pressure support in galaxy clusters. The top panel of Fig.~\ref{fig:clump_phase} shows that results of hydrodynamical cluster simulations, illustrating that the non-thermal pressure contributes to 10-20\% at $r=r_{500}$ of the total pressure. The fraction of non-thermal pressure support increases with radius, and it is larger for more dynamically active systems. The bottom panel of Fig.~\ref{fig:clump_phase} shows that the cluster mass profile based on hydrostatic assumption is biased low. We further demonstrate that it is possible correct the bias in the hydrostatic mass, if one could measure gas motions in clusters and hence correct for the bias. Upcoming {\it Astro-H} X-ray satellite mission (scheduled to be launched in 2014) will provide a first direct measurement of gas motions in clusters via doppler broadening of iron lines \citep[][]{inogamov_sunyaev03}.
\begin{figure}[t]
\begin{center}
\vspace{-4mm}
\resizebox{\hsize}{!}{\includegraphics[clip=true]{f2.eps}}
\vspace{-8mm}
\caption{\footnotesize
{\it Top panel:} Median clumping factor profiles of gas with different minimum temperature for the simulations with gas cooling and star formation. {\it Bottom panel:} Impact of gas clumping on X-ray measurements of the ICM entropy profile. The dashed line indicates the true entropy profile of the simulated clusters, while the solid line indicates the observed entropy profile, obtained by assuming no clumping ($C=1$). Black points are {\it Suzaku} observations of PKS0745-191 ({\it circles}), A1689 ({\it triangles}), A1413 ({\it stars}), and A1795 ({\it black dot-dashed lines}). From \citet{nagai_etal11}.}\label{fig:clump_phase}
\end{center}
\end{figure}
\section{Gas Clumping in Cluster Outskirts}
Recently, {\it Suzaku} X-ray observations revealed that the observed entropy profile of the ICM is significantly offset from the prediction of hydrodynamical simulations of galaxy clusters. Here, we point out that gas clumping is likely a major source of systematic bias in X-ray measurements of ICM profiles in the envelope of galaxy clusters ($r \gtrsim r_{200}$) \citep{nagai_etal11}. Using hydrodynamical simulations of cluster formation, we show that gas clumping introduces the overestimate of the observed gas density and causes flattening of the entropy profile at large radius. This is illustrated in Fig.~\ref{fig:clump_phase}. The top panel shows that the clumping factor of the X-ray emitting gas ($T\gtrsim 10^6$~K) is $C \equiv \langle \rho_{\rm gas}^2 \rangle / \langle \rho_{\rm gas} \rangle^2 \sim 1.3$ at $r=r_{200}$, and it increases with radius, reaching $C\sim 5$ at $r= 2 r_{200}$. In the bottom panel, the solid line indicates the true entropy profile, which is consistent with the self-similar prediction, $K \equiv T/n_e^{2/3}\propto r^{1.1}$ \citep{voit_etal05}. From the definition of entropy, the overestimate of gas density due to clumping causes an underestimate of the observed entropy profile by $C(r)^{1/3}$. Results of our analyses indicate that gas clumping causes the flattening of the observed entropy profiles at $r\gtrsim r_{200}$. While current {\it Suzaku} measurements are still uncertain, our results indicate that gas clumping is important for reducing the tension between recent {\it Suzaku} observations and theoretical prediction of the $\Lambda$CDM model.
\begin{figure}[t]
\begin{center}
\vspace{-2mm}
\resizebox{\hsize}{!}{\includegraphics[clip=true]{f3.eps}}
\vspace{-2mm}
\caption{\footnotesize
Astrophysical uncertainties in the SZ power spectrum. Different lines indicate theoretical uncertainties associated with (1) heating of gas by energy injection from stars and AGN (indicated with {\it solid} lines) and merger dynamics (indicated with {\it dashed} lines in {\it top-left} panel), (2) dark matter structures ({\it top-right}), and (3) non-thermal pressure by gas motions ({\it bottom-left}) and its time evolution ({\it bottom-right}). In each case, the thick red line represents our fiducial model. From \citet{shaw_etal10}.}\label{fig:SZpower_model}
\end{center}
\end{figure}
\section{Impact on the SZ power spectrum}
Recent measurements of the SZ power spectrum by {\it SPT} and {\it ACT} telescopes revealed that the SZ power is significantly below the signal predicted by the current cosmic structure formation model \citep{lueker_etal10,shirokoff_etal10,dunkley_etal10}.
In our recent work, we argued that the current SZ power spectrum template is overestimated by 50-100\%, due to lack of important astrophysical process in theoretical modeling of the SZ power spectrum \citep{shaw_etal10}. Fig.~\ref{fig:SZpower_model} illustrates theoretical uncertainties in the thermal SZ power spectrum. Our model accounts for star formation and energy feedback (from supernovae and active galactic nuclei) as well as radially dependent non-thermal pressure support due to random gas motions, the latter calibrated by recent hydrodynamical simulations. Varying the levels of feedback and non-thermal pressure support can significantly change both the amplitude and shape of the thermal SZ power spectrum. Increasing the feedback suppresses power at small angular scales, shifting the peak of the power spectrum to lower $l$. On the other hand, increasing the non-thermal pressure support significantly reduces power at large angular scales. In general, including non-thermal pressure at the level measured in simulations has a large effect on the power spectrum, reducing the amplitude by $\gtrsim 60$\% at angular scales of a few arcminutes compared to a model without a non-thermal component. Comparing with the recent measurements of the small-scale cosmic microwave background power spectrum, our model reduces the tension between the values of $\sigma_8$ measured from the SZ power spectrum and from cluster abundances.
\section{Future Prospects}
Modern numerical simulations predict that gas motions and clumping are ubiquitous in the outskirts of galaxy clusters. While they are generic predictions of the concordance $\Lambda$CDM model, we have had very little observational handle on these phenomenon until very recently.
New X-ray and SZE observations just coming online have significantly extended measurements of the ICM, out to and beyond the virial radius of clusters. For example, deep X-ray imaging of nearby clusters with current {\it Suzaku} and {\it Swift}-XRT have started to provide accurate measurements of the ICM profiles in the outer envelope of galaxy clusters. Comparison of current X-ray and SZE measurements should soon provide insights into the properties of the ICM in cluster outskirts. {\it Astro-H} X-ray mission promises to provide the first direct measurements of internal gas motions in clusters, and {\it eROSITA} and {\it WXRT} X-ray mission will help increase the number of clusters with deep imaging data extending out to large radii. Theoretical modeling based on detailed numerical simulations is also underway. A plethora of activities (in both theory and observation) will advance our understanding of cluster physics and provide foundation for the use of galaxy clusters as laboratories for cosmology and astrophysics.
\begin{acknowledgements}
I would like to thank my collaborators Andrey Kravtsov, Erwin Lau, Laurie Shaw for rewarding collaborations which produced results described here. This research was carried out at the Yale University and was supported in part by NSF under grant AST-1009811.
\end{acknowledgements}
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|
\section{Introduction}
Recent years have been marked by significant advances in understanding of
grain alignment processes (see Roberge 1996, Lazarian, Goodman \& Myers 1997).
A number of new alignment mechanisms have been suggested
(e.g. Draine \& Weingartner 1996, 1997, Lazarian 1995a) and traditional
mechanisms underwent serious revision (see Lazarian \& Draine 1997).
This process was motivated by new interstellar polarization data
(e.g. Goodman 1995, 1996) and, unfortunately, has not made the appropriate
inpact upon the areas beyond the interstellar domain.
At the same time the number of puzzling results is growing in the areas
of comet and circumstellar polarimetry, where it is customary to
believe that polarization arises from light scattering on randomly
oriented dust grains. In this paper we show that some of these puzzles
vanish if grain alignment is accounted for.
Although models of circumstellar regions that invoke aligned grains
have been occasionally discussed in the literature (see
Dolginov \& Mytrophanov 1976, Pudritz,
1986, 1988), their applicability was highly questionable in the
absence of the reliable grain alignment theory (see Bastien 1988).
On the contrary,
recent theoretical advances indicate that grain alignment is likely
to be ubiquitous and therefore must be accounted for while modeling
circumstellar polarization and the polarization from comets.
In what follows we identify mechanisms of alignment that
are most efficient in circumstellar regions and in comet atmospheres
(section~2), then touch upon the relation between grain alignment and linear and
circular polarization (section~3). In section~4
we discuss grain alignment in circumstellar regions, comets and in interplanetary space. Ways of separating the effects of multiple scattering and
those of grain alignment are discussed in section~5 and we summarize
our results in section~6. Important but more specialized
discussion of ferromagnetic versus superparamagnetic
relaxation is given in the Appendix.
\section{Grain Alignment}
Discovered half a century ago (see Hiltner 1949, Hall 1949), grain alignment
continues to be a tough problem for theorists. The dynamics of
rapidly rotating dust particles is being influenced both
by numerous processes that include gaseous and ion
bombardment, plasma effects, interactions with starlight etc.
(see more detail in Draine \& Lazarian 1998a, 1998b). Chemical processes, e.g.
H$_2$ formation that
take place on grain surfaces also influence grain dynamics (Purcell 1979,
Lazarian 1995b). Moreover, observations suggest a strong dependence of the
alignment efficiency on grain sizes. Indeed, interstellar observations
can be only explained if grains with sizes $>10^{-5}$~cm are aligned,
while smaller grains are not\footnote{This is not an exact statement
as a recent study by Lazarian \& Draine (1998) suggests that very small
grains with $a<10^{-7}$~cm may well be aligned.} (Kim \& Martin 1995).
In spite of all these difficulties substantial progress has been
recently achieved in understanding of grain alignment processes.
A list that includes six major mechanisms was presented in
Lazarian, Goodman \& Myers (1997) and a number of ``exotic'' mechanisms
have been described there as well
. Below we discuss only those of the mechanisms
that can be relevant for grain alignment in circumstellar regions and
in comet atmospheres.
We claim that to succeed in these environments
the process must be fast. Therefore
slow processes that may well work in the
interstellar medium are likely to fail in circumstellar
regions. For instance, we do not discuss paramagnetic
alignment of suprathermal grains (Lazarian \& Draine 1997) that slowly
but steadily alignes grains over many gaseous damping times.
To characterise the alignment we use the Raylegh reduction factor
(Greenberg 1968)
\begin{equation}
R=\frac{3}{2}\langle \cos^2\beta-\frac{1}{3}\rangle~~~,
\end{equation}
where $\langle ...\rangle$ denotes the ensemble average,
$\beta$ is the angle between grain axis of maximal inertia
and the direction of alignment. We show below that often
it is magnetic field that defines direction, even for non-magnetic
alignment mechanisms.
In general, grain alignment is non-equilibrium process. Therefore
in dark clouds where ``classical'' grains are in
thermodynamic
equilibrium with the ambient gas
no alignment is observed (Lazarian, Goodman \& Myers 1997).
To align grains,
i.e. to decrease the enthropy of their distribution, the enthropy
of some other system (systems) must increase.
\subsection{Paramagnetic alignment}
The oldest of the alignment mechanisms is the process
of paramagnetic relaxation suggested by Davis \& Greenstein (1951) and later
modified by Purcell (1979), who observed that grains may rotate much faster
that was originally thought. To understand
the essence of this mechanism it is sufficient to consider
a sherical grain which angular velocity makes angle $\beta$ with magnetic
field $\bf B$. The component of angular velocity perpendicular
to $\bf B$, i.e. $\omega \sin\beta$, will cause oscillating remagnetization
of the grain, while $\omega \cos\beta$ will not cause oscillations of
magnetization. As oscillating magnetization entails dissipation, the component
$\omega \sin \beta$ decreases, while $\omega \cos\beta$ stays the same.
As the result, $\beta$ decreases.
Thus magnetic field causes anisotropy in the distribution
of grain angular momenta. As non-spherical grains tend to rotate about
their axes of maximal moment of inertia
(Purcell 1979) the anisotropy in the distribution of angular
momentum is being translated into the anisotropy of the distribution of
grain longer axes.
Leaving aside the mathematical theory of alignment
(Lazarian 1997, 1998, Roberge \& Lazarian 1999), that accounts for grains
being non-spherical and internal relaxation being not complete,
we may claim that the
alignment happens on the time scale of paramagnetic relaxation, which
for ordinary paramagnetic grains is rather long, e.g.
\begin{equation}
t_{\rm al}=4\times 10^{12} K^{-1}_{-13} B^{-2}_{-5} a^2_{-5}~{\rm s}~~~,
\end{equation}
where the lower indexes used to denote the normailization values. For
instance, the $K$ function, which is the ratio of the imaginary part
of grain
magnetic susceptibility $\chi(\omega)$ to its angular velocity $\omega$,
was normalized to $10^{-13}$~s. In other words, $K_{-13}\equiv K/(10^{-13}~{\rm s})$.
Similarly, magnetic field is normalized by $10^{-5}$~G and grain size
$a_{-5}\equiv a/(10^{-5}~{\rm cm})$.
Grain rotation can be randomized by gaseous bombardment on time scales
\begin{equation}
t_{\rm gas}=6\times 10^{11} n_{10}^{-1} T_{\rm gas, 5000}^{-1/2} a_{-5}~{\rm s}~~~,
\end{equation}
where $n_{10}\equiv n/(3~{\rm cm}^{-3})$, $T_{\rm gas, 5000}\equiv
T_{\rm gas}/(5000~{\rm K})$. In the equation above the environmental
parameters are taken rather arbitrary
and for particular cases the more relevant values should be substituted.
Moreover the estimate for $t_{\rm gas}$ must be reduced nearly an order
of magntitude if gas is ionized (see Anderson \& Watson 1993,
Draine \& Lazarian 1998a). The latter effect is the consequence of
higher efficiency of plasma interactions with a charged grain compared
to gas-grain interactions.
To obtain efficient paramagnetic alignment $t_{\rm al}$ should be much less
than $t_{\rm gas}$.
Therefore grains with superparamagnetic and ferromagnetic inclusions
(Jones \&
Spitzer 1967, Mathis 1986, Martin 1995,
Draine 1996, Draine \& Lazarian 1998c) are to be considered.
How abundant ferro- and superparamagnetic
grains in comet environment and circumstellar regions is not clear.
The presence of small $FeNi$ and $FeNiS$ inclusions in
particles coming from the interplanetary space
has been recently reported (Bradley 1994) and this supports
the case for ``super'' grains (Goodman \& Whittet 1996). Our analysis
of the particle image in figure~1 in Goodman \& Whittet (1996)
indicates that most of the
inclusions are too large to exhibit superparamagnetic response for
$\omega>10^6$~s$^{-1}$\footnote{In circumstellar regions ``classical''
grains of 0.1~$\mu$m size rotate much faster due to the action of radiative
torques.}(see Appendix). However, our calculations in the Appendix prove
that the ferromagnetic response of grains with iron inclusions provides
enhancement of the paramagnetic relaxation by a factor $10^3-10^4$ if
the volume filling factor of inclusions is $\sim 0.01$ as we roughly
estimated from
the figure in Goodman \& Whittet (1996). The decrease of paramagnetic
alignment time $t_{\rm al}$
by the factor $10^4$ arising from ferromagnetic inclusions makes
$t_{\rm al}\sim 4\times 10^8$~s. This seems sufficient for circumstellar
alignment but may be slow for comet grain alignment.
\subsection{Mechanical Alignment}
Another mechanism of grain alignment stems from mechanical interaction
of grains with streaming gas. Suggested initially by Gold (1952) for
grains rotating with thermal velocities, the mechanical alignment has been
recently proved to be efficient for grains rotating with much higher
velocities (Lazarian 1994a,
Lazarian 1995a, Lazarian \& Efroimsky 1996, Lazarian,
Efroimsky \& Ozik 1996). Such high (suprathermal)
velocities arise from uncompensated
quasi-regular torques, e.g. from torques arising from H$_2$ formation
over catalytic sites on grain surface (Purcell
1975, 1979). These sites act as
rocket engines and their action spins up the grain. The number of sites over
grain opposing surfaces, in general, is different and this causes a regular
spin-up.
The original Gold's idea is based on the observation that when a flow
of gas interacts with an elongated grain the angular momentum
deposited with the grain tends
to be directed perpendicular to the flow.
Accounting for suprathermal rotation and the presence
of magnetic field makes the process of alignment a bit more involved
(see Lazarian 1995a).
The necessary condition for the mechanical alignment is the supersonic
relative motion of grains and gas. If this condition is not satisfied
isotropic gaseous bombardment randomizes grains (see eq.~(25) in
Lazarian 1997a). The rule of thumb for mechanical alignment is that the
process tends to minimize gas-grain cross section of
interaction\footnote{This is not true, however, for the process of
{\it alignment through friction} described in Lazarian (1995a). A
detailed discussion of the joint action of various alignment processes will
be given elsewhere.}.
It is easy to see that,
unlike paramagnetic alignment, the mechanical one is not directly
connected with the action of the ambient magnetic field. However, in
many cases mechanical processes align grains either parallel or
perpendicular to the direction of magnetic field. This is the concequence
of grain rapid precession about magnetic field. Indeed, a rotating
grain aquires a magnetic moment via the Barnett effect (Dolginov \& Mytrophanov
1976, Purcell 1979) and this magnetic moment precesses in the external magnetic
field with the period
\begin{equation}
t_{\rm L}=2\times 10^{5} B^{-1}_{-5} a^{2}_{-5}~{\rm s}~~~.
\end{equation}
If $t_{\rm L}$ is much shorter than the time of mechanical alignment
$t_{\rm mech}$,
external magnetic field defines the axis of alignment.
$t_{\rm mech}$ is different for thermally and suprathermally (much
faster than thermally)
rotating grains. In the former case $t_{\rm mech}$ can be defined as
a time during which the angular momentum of a grain changes by the value
of its thermal angular momentum $J_{\rm th}=(k T_{\rm gas}/I)^{1/2}$,
where $I$ is the grain moment of inertia. In the case of suprathermally
rotating grains $t_{\rm mech}$ is the time between crossovers, i.e. moments
when grain angular velocity approaches zero and the grain flips over (see
Lazarian \& Draine 1997)\footnote{Crossovers happen due to the occasional
change of the direction of quasi-regular torques. As this direction changes
a grain first spins down then flips over and spins up.}
The time between crossovers is approximately the sum of the
gaseous damping time $t_{\rm gas}$ and a rather uncertain
time of grain resurfacing (see Spitzer \& McGlynn 1979, Lazarian 1995a).
When $t_{\rm mech}\ll t_{\rm L}$ the alignment happens in respect to the
direction of gas-grain relative motion.
One could expect that in circumstellar regions both situations $t_{\rm mech}>
t_{\rm L}$ and $t_{\rm mech}< t_{\rm L}$ may be present. However,
in many cases violent
outflows of plasma are likely to deform magnetic field lines and therefore
the correlation of the magnetic field and the direction of alignment is
expected even for $t_{\rm mech}\ll t_L$. Also note that grains carry electric charge (Martin 1972)
and therefore tend to follow magnetic field lines.
A number of processes can cause the relative grains-gas drift. Stellar
winds, outflows are examples of processes that would tend to align grains
with long axis {\it along} magnetic field lines. Ambipolar diffusion
in Roberge \& Hanany (1990) and Alfven waves in Lazarian (1994a)
were suggested as the processes that can mechanically align grains
perpendicular to magnetic field lines. In circumstellar regions and comet
atmospheres we expect mechanical alignment to happen mostly along magnetic
field lines.
\subsection{Radiative Torques}
The third mechanism that is likely to be dominant in circumstellar regions
is based on the action of radiative torques. Although mentioned first
in Dolginov (1972) and Dolginov \& Mytrophanov (1976) this
process has not been considered seriously untill very recently. Draine
\& Weigartner (1996, 1997) rediscovered the mechanism and proved
using the DDA code that radiative torques (1) can be the dominant source
of grain suprathermal rotation and that (2) these torques can align
grains
with the longer directions perpendicular to magnetic
field. The origin of
the latter fact is not clear and this tendency contradits
to the conclusions in Dolginov \& Mytrophanov (1976)\footnote{Analytical
results in Dolginov \& Mytrophanov (1976) do not explain grain
spin-up when the radiation is isotropically distributed.
This fact was noted to me by Lyman Spitzer, Jr.}. Nevertheless,
treating the properties of radiative torques as established experimentally
we have to conclude that this alignment mechanism should be very
important in circumstellar regions where the radiation flux is orders
of magnitude higher than that in the interstellar environment.
Note, that even in the interstellar medium radiative torques constitue a
major mechanism of rotation for sufficiently large, e.g. $a>10^{-5}$~cm,
grains. Within circumstellar regions with enhanced UV flux smaller
grains can be aligned radiatively. This could present a possible solution
for the recently discovered anomalies of polarization in the 2175 \Angstrom
~~extinction feature (see Anderson et al 1996) that has been interpreted
as the evidence of graphite grain alginment (Wolff et al 1997). If
this alignment happens in the vicinity of particular
stars with enhanced UV flux
and having graphite grains in their circumstellar regions, this may
explain why no similar effect is observed along other lines of sight.
Radiative torques work in unison with paramagnetic relaxation.
The situation is less clear when mechanical alignment tends to align
grains along magnetic field lines, while radiative torques act
to align grains perpendicular to magnetic field lines.
It takes radiative torques at least a few
gaseous damping times to align grains\footnote{A peculiarity of the
radiative torque mechanism is that the gas acts as a cooling reservoir.}
while mechanical alignment can happen in one crossover time.
For particular angles between the direction of the incoming radiation
and magnetic field the grains perform numerous crossovers. This means
that in these situations the mechanical alignment should dominate.
The theory of crossovers in the presence of radiative torques is being
developed (Draine \& Lazarian, work in progress) and we hope to learn
soon at what conditions the mechanical alignment can win.
\section{Polarization}
Grain alignment theory can supply $R$. The observations can get Stocks
parameters. To compare observations and the theory one should related
$R$ to polarization. Because different definitions of $R$ have appeared
in the literature and confusing statements have been made in relation
to circular polarization of circumstellar origin, we find a brief discussion
of this subject appropriate.
\subsection{Linear Polarization from Aligned Grains}
For an ensemble of aligned grains the extinction perpendicular the direction
of alignment and parallel to it will be different. Therefore the
electromagnetic wave that initially was not polarized acquires polarization.
To characterize the process quantitatively one can consider
an electromagnetic wave propagating along the line of sight
\mbox{$\hat{\vecz}^{\bf\rm o}$}\ axis.
The transfer equations for the Stokes parameters
depend on the cross sections \mbox{$C_x^{\rm o}$}\ and \mbox{$C_y^{\rm o}$}\ for linearly polarized
waves with the electric vector, \vecE, along the \mbox{$\hat{\vecx}^{\bf\rm o}$}\ and \mbox{$\hat{\vecy}^{\bf\rm o}$}\ directions
that are in the plane perpendicular to \mbox{$\hat{\vecz}^{\bf\rm o}$}\
(see Martin 1974, Lee \& Draine 1985).
To calculate \mbox{$C_x^{\rm o}$}\ and \mbox{$C_y^{\rm o}$}\,
one transforms the components of \vecE\ to
a frame aligned with the principal axes of the grain and
takes the appropriately-weighted sum of the
cross sections, \mbox{$C_{\|}$}\ and \mbox{$C_{\bot}$}, for \vecE\ polarized along the grain
axes.
When the transformation is carried out and the resulting
expressions are averaged over precession angles, one finds that
the mean cross sections are
\begin{equation}
\mbox{$C_x^{\rm o}$} = \mbox{$C_{\rm avg}$} + \frac{1}{3}\,R\,\left(\mbox{$C_{\bot}$}-\mbox{$C_{\|}$}\right)\,
\left(1-3\cos^2\zeta\right)~~~,
\label{eq-2_5}
\end{equation}
where $\zeta$ is the angle between the polarization axis and the
\mbox{$\hat{\vecx}^{\bf\rm o}$}\ \mbox{$\hat{\vecy}^{\bf\rm o}$}\ plane,
\begin{equation}
\mbox{$C_y^{\rm o}$} = \mbox{$C_{\rm avg}$} + \frac{1}{3}\,R\,\left(\mbox{$C_{\bot}$}-\mbox{$C_{\|}$}\right)~~~,
\label{eq-2_6}
\end{equation}
where $\mbox{$C_{\rm avg}$}\equiv\left(2\mbox{$C_{\bot}$}+\mbox{$C_{\|}$}\right)/3$ is the effective
cross section for randomly-oriented grains.
\subsection{Circular Polarization from Aligned Grains}
One of the ways of obtaining circular polarization is to have magnetic field
that varies along the line of sight (Martin 1972). Passing through one
cloud with aligned dust the light becomes partially linearly polirized.
On passing the second cloud with dust aligned in a different direction the
light gets circular polarized.
Literature study shows that this effect that is well remembered
(see Menard et al 1988), while the other process that also creates
circular polarization is frequently forgotten. We mean the process
of single scattering of light on aligned particles. Electromagnetic
wave interacting with a single grain coherently excites dipoles parallel
and perpendicular to the grain long axis. In the presence of adsorption
these dipoles get phase shift giving rize to circular polarization.
This polarization can be observed from the ensemble of grains if
the grains are aligned. The intensity of circularly polarized
component of radiation emerging via scattering of radiation with
$\bf k$ wavenumber on small ($a\ll \lambda$) spheroidal
particles is (Schmidt 1972)
\begin{equation}
V( {\bf e}, {\bf e}_0, {\bf e}_1)=\frac{I_0 k^4}{2 r^2}i(\alpha_{\|}
\alpha^{\ast}_{\bot}-\alpha^{\ast}_{\|}\alpha_{\bot})\left([{\bf e_0}\times
{\bf e}_1] {\bf e}\right)({\bf e}_0 {\bf e}),
\end{equation}
where ${\bf e}_0$ and ${\bf e}_1$ are the unit vectors in the directions
of incident and scattered radiation, ${\bf e}$ is the direction along
aligned axes of spheroids; $\alpha_{\bot}$ and $\alpha_{\|}$ are particle
polarizabilities along ${\bf e}$ and perpendicular to it.
The intendity of the circular polarized radiation scattered in the
volume $\Delta \Gamma({\bf d}, {\bf r})$ at $|{\bf d}|$ from the star
and distances $|{\bf r}|$ from the observer is (Dolginov \& Mytrophanov 1978)
\begin{equation}
\Delta V ({\bf d}, {\bf r})=\frac{L_{\star} n_{\rm dust}\sigma_{V}}{6\pi |{\bf d}|^4
|{\bf r}||{\bf d}-{\bf r}|^2}R \left([{\bf d}\times {\bf r}] h\right)
({\bf d}{\bf r})\Delta \Gamma({\bf d}, {\bf r})~~~,
\end{equation}
where $L_{\star}$ is the stellar luminosity, $n_{\rm dust}$ is number
density of dust grains and $\sigma_V$ is the cross section for
producing circular polarization, which is for small grains
is $\sigma_V=i/(2k^4)(\alpha_{\|}\alpha^{\ast}_{\bot}-\alpha^{\ast}_{\|}\alpha_{\bot})$.
According to Dolginov \& Mytrophanov (1978) circular polarization arising
from single scattering on aligned grains
can be as high as several percent for metallic or graphite particles,
which is much more than one expects
from the process of varying magnetic field direction along the line of
sight.
\section{Particular cases}
\subsection{Circumstellar Regions}
Multiple scattering has been used to explain polarization
arising from circumstellar regions (see Bastien 1988, 1996). At the same
time it is obvious that in the presence of radiation and magnetic
field, grains in circumstellar envelops must be aligned perpendicular
to magnetic field. For the stars that exhibit outflows and intensive
stellar winds, numerical models (see Netzer \& Elitzur 1994) predict
a supersonic relative drift of grain and gas and this should result
in mechanical alignment. In circumstellar environments
the grain rotation temperature
is likely to be much higher than its body temperature. Therefore
results for mechanical obtained in
Lazarian (1994a) and Lazarian (1995a) are applicable.
This entails $R\sim -0.3$ for both prolate and oblate grains with
grain long axis along the outflow direction. The uncertainty involved,
as we have mentioned earlier, is related to the absence of the theory of
radiative crossovers. We may claim that our estimate of $R$ is valid
for sufficiently small (e.g. $a< 5\times 10^{-6}$~cm) grains, while
for larger grains the situation is unclear as yet.
If grains have superparamagnetic or ferromagnetic inclusions and for
radiative torques the alignment tends to be nearly perfect (i.e. $R\sim 1$)
with the logner grain dimensions perpendicular to magnetic field lines.
If, however, a grain with ferromagnetic inclusions (e.g.
``Goodman-Whittet grain'' discussed above) is subjected to
streaming along field lines, it will be aligned perpendicular
to magnetic field lines as the magnetic relaxation time is typically
shorter than that for mechanical alignment. We tend to believe
that grain alignment with grain longer axes perpendicular to
magnetic field and $R\sim 1$ can be a rule for circumstellar regions.
Future research should test this conjecture.
The examples above indicate that future modeling of circumstellar regions
should include aligned grains. Whether multiple scattering or
dichroic adsorption is dominant should be decided
by quantitative comparison of the simulations that include both
effects and observations. Submillimeter polarimetry will be helpful
for establishing grain alignment in circumstellar regions (see below).
\subsection{Comets}
Polarization from comets has been long known to exhibit anomalies
(see Martel 1960) that motivated a conjecture that grains may be aligned
in the comet atmospheres (see Dolginov \& Mytrophanov 1976). Later
studies of linear and circular polarization from Halley and Hale-Bopp
comet (Beskrovnaja et al 1987, Ganesh et al 1998) seem to support this
conclusion.
The alignment mechanism operating in comet heads should be really fast.
Indeed, dust particles spend only $\sim 10^5$~s crossing a comet head.
Unless magnetic field in the comet head is extremely high (e.g. $> 10^{-2}$~G)
the ferromagnetic relaxation fails to provide the alignment. In comet
heads grains are likely to disaggregate and change their shape rather
rapidly. This should mitigate the importance of raditative torques
that will change their direction with the change of grain shape. At the
same time, dramatic changes of grain shapes on the timescale $t_{\rm mech}$
wash out the distinction
between prolate and oblate grains and hinder the mechanical alignment as
well.
We believe that outflowing gases can be important for grain alignment at
the comet head.
Calculations in Probstein (1969) indicate that the relative velocities
of dust and gas are supersonic. We expect the alignment for
thermally rotating grains to be small (e.g. $R\sim -0.1$) and to happen
in respect to the outflow direction. Higher degrees of alignment
are possible (e.g. $R\sim -0.3$) if grains rotate suprathermally.
Indeed, both radiative torques and assymetry in the gas evaporation from grain
surface may contribute to suprathermal rotation. Very large dust particles
(e.g. $a>10^4$~cm) may be aligned by a weathercock mechanism discussed
in Lazarian (1994b).
Later, in the outer parts of comet
coma and in its tail the alignment via radiative torques and interaction
with solar wind should be important. $R$ approaching unity is
attainable in the former case.
Quantitative modeling of the
grain alignment in comets is under way (Bastien \& Lazarian, work in
progress).
\subsection{Zodiacal Light}
Zodiacal Light, i.e. solar light reflected from the interplanetary dust
particles, is partially polarized. Greenberg (1970) suggested that
interplanetary grains could be aligned. Later on similar ideas were
discussed by e.g.
Wolstencroft \& Kemp (1972) and Dolginov \& Mytrophanov (1978).
Greenberg (1970) worried that interplanetary particles can be sputtered
quicker than be aligned by solar wind. However, his arguments ignore
important plasma interactions and ion focusing effect (see Draine \& Lazarian
1998b) that make transfer of angular momentum from solar wind to grains much
more efficient. Thus mechanical alignment is concivable ($R\sim -0.3$)
with grain long axis along magnetic field lines.
The alignment by radiative torques and via ferromagnetic relaxation
are possible as well. If large silicate grains that produce most of the
linear polarization are aligned along magnetic field lines, while a
possible population of absorbtive iron grains that would account for
most of the circular polarization are aligned perpendicular to interplanetary
magnetic field, quite complex picture of polarization may arise.
However, it is likely that mechanical alignment is most important for
small ($a<5\times 10^{-6}$~cm) grains, while larger grains are being aligned
by radiative torques. Then both small iron grains and large silicate
ones are being aligned with long axes perpendicular to the direction of
the interplanetary magnetic field. Potentially,
studies of Zodiacal Light can bring a lot of information about magnetic
field structure and its variability in the Solar neighborhood.
The interplanetary magnetic field, as well as those of circumstellar regions
and comets, is not stationary. In fact it undergoes variations on a
whole range of time scales. If the variations are long compared to the
Larmor period $t_L$ they are adiabatic in the sence that the angle
between grain angular momentum and $\bf B$ is preserved. Therefore
time variations of the Zodiacal Light can provide important
information on the magnetic
variability up to the scale $t_L$.
\section{Future Work}
It is often difficult to separate the effects of multiple stattering from
the effects of grain alignment. One of the alluring possibilities is
to observe at longer wavelengths, where the effects of multiple stattering
are negligible. Polarimetry at submillimeter and longer wavelengths should
help constructing adequate models of polarized light transfer
in circumstellar regions and comets and unrevel magnetic field
structure in these regions.
Our discussion above was centered on the issue what ``classical'' or
sufficiently large grains can tell us. It looks, however, that very
small grains can make a valuable input as well. Recent experiments to map
cosmic microwave background, e.g. Kogut et al (1996), Oliveira-Costa
et al (1997) and Leitch et al (1997), have revealed a new component of
galactic microwave emission at 14 - 90 GHz. This component was identified by
Draine \& Lazarian (1998a) with the dipole emission from small
($a<10^{-7}$~cm) rotating grains. Lazarian \& Draine (1998) predicted
that such grains can be aligned and that this should result in anomalous
emission being partially polarized. This opens a new valuable window
for interstellar and circumstellar studies. An important feature of
the relaxation mechanism suggested is that it stays efficient even
when ``classical'' grains are in thermodynamic equilibrium with the
ambient gas and are randomly oriented. Thus the progress in grain alignment
theory presents new tools for observers.
\section{Conclusions}
The principal results of this paper are as follows:
The application of the results obtained in grain alignment theory to comets
and circumstellar regions suggest that the dust should be aligned there.
Three most important alignment mechanisms are (1) radiative torques,
(2) mechanical alignment, (3) ferromagnetic and superparamagnetic
relaxation. Observational data supports the conjecture that the dust is
aligned in circumstellar regions and comets. Therefore numerical codes that
describe radiation transfer in young
stellar objects and evolved stars should be modified to account for
dust alignment.
The analysis of the images of the dust
particles coming from the interplanetary space testify that the ferromagnetic
relaxation, rather that superparamagnetic relaxation is likely. The calculated
enhancement of the relaxation (compared to that in paramagnetic grains)
is $\sim 10^4$ and is sufficiently large to enable the efficient
alignment of circumstellar dust with ferromagnetic inclusions.
Mechanical alignment and radiative torques compete in aligning grains,
(along and perpendicular magnetic field lines, respectively) in the regions
of outflows. When streaming velocities are supersonic small grains
($a<5\times 10^{-6}$~cm) without
ferromagnetic inclusions are to be aligned with long axes
parallel to magnetic field
lines, while those with ferromagnetic inclusions are to be aligned
with long axes perpendicular to the field lines. The situation
is still unclear with large ($a>10^{-5}$~cm) grains, but we conjecture
that at least in circumstellar regiona and interplanetary space grains
are aligned with long axes perpendicular to magnetic field.
Both linear and circular polarization provide a valuable input on
magnetic fields in circumstellar regions, comet atmospheres and in the Solar
neighborhood. Measurements at submillimeter wavelenghs can disentangle
effects of multiple scattering from those of grain alignment. In particular
cases when large grains are not aligned it is advisable
to use microwave polarimetry that is sensitive to the alignment of
tiny ($a<10^{-7}$~cm) grains.
{\bf Acknowledgements}
\acknowledgements
I am grateful to Pierre Bastien, Bruce Draine, Alyssa Goodman and
Peter Martin for helpful discussions and happy to acknowledge the
support of NASA grant NAG5 2858 and CITA Senior Research Fellowship.
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{"url":"http:\/\/cstheory.stackexchange.com\/questions\/14218\/fast-deletion-contraction-in-combinatorial-embedding","text":"Fast deletion \/ contraction in combinatorial embedding\n\nI wonder if there is a sublinear algorithm to make deletion or contraction of an edge in a combinatorial embedding of, lets say, planar graph?\n\nSince in combinatorial embedding we have to maintain vertices of G and G* at the same time, taking in account that contraction in the primal is deletion in the dual, it's sufficient only to make deletions, updating primal permutation according to dual and vice-versa. But obvious way to do it is just recompute them, which takes linear time. Can we do any better?\n\nSecond question: is there any technique that helps to get rid of multiple edges between same vertices? (the only solution I see to the second problem is to postpone deletion of multiple edges until we will get graph with, for example, m=6n, where m - number of edges, n - number of vertices, this will make time amortized O(1)) Maybe there is some techniques, which can make this time not amortized? (I am also interested in just o(n) solutions, not necessarily O(1))\n\nThank you very much!\n\n-\nIn second question I meant that we want to get rid of multiple edges while doing contractions and deletions. \u2013\u00a0 Sergey Finsky Nov 5 '12 at 16:27\n\nThis question is incomplete without specifying what information about the graph as it changes you want your dynamic graph data structure to output or support queries for. But the following paper is likely relevant, even though it is described in a more general setting of combinatorial embeddings in arbitrary genus rather than just planar. It definitely supports both contractions and deletions, as well as their reverse operations, in logarithmic time per operation.\n\nDynamic generators of topologically embedded graphs. D. Eppstein. arXiv:cs.DS\/0207082. SODA 2003, pp. 599-608.\n\nAs for the second question: I don't see how to handle multiple adjacencies in general, but it's easy to get rid of bigons (multiple edges with nothing between them) as they can only come from the two faces that are on either side of a contracted edge or from the face that surrounds a deleted edge. That should be sufficient for many purposes since getting rid of the bigons ensures that the remaining graph has a number of edges proportional to its number of vertices.\n\n-\nTo summarize the technique in David's paper: Store the cyclic sequence of edges leaving each vertex, in both the primal graph and the dual graph, in a balanced binary tree that supports splits and joins in $O(\\log n)$ time (for example, a B-tree, a treap, or a splay tree), instead of a raw linked list. \u2013\u00a0 J\u025b\ufb00E Nov 5 '12 at 17:48","date":"2015-08-31 02:28:27","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6791746020317078, \"perplexity\": 581.4060227038422}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2015-35\/segments\/1440644065488.33\/warc\/CC-MAIN-20150827025425-00278-ip-10-171-96-226.ec2.internal.warc.gz\"}"}
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{"url":"https:\/\/socratic.org\/questions\/how-do-you-factor-x-6y-3-y-9","text":"# How do you factor x^6y^3 + y^9?\n\nFeb 25, 2017\n\n${x}^{6} {y}^{3} + {y}^{9} = {y}^{3} \\left({x}^{2} + {y}^{2}\\right) \\left({x}^{2} - \\sqrt{3} x y + {y}^{2}\\right) \\left({x}^{2} + \\sqrt{3} x y + {y}^{2}\\right)$\n\n#### Explanation:\n\nThe sum of cubes identity can be written:\n\n${a}^{3} + {b}^{3} = \\left(a + b\\right) \\left({a}^{2} - a b + {b}^{2}\\right)$\n\nSo we find:\n\n${x}^{6} {y}^{3} + {y}^{9} = {y}^{3} \\left({x}^{6} + {y}^{6}\\right)$\n\n$\\textcolor{w h i t e}{{x}^{6} {y}^{3} + {y}^{9}} = {y}^{3} \\left({\\left({x}^{2}\\right)}^{3} + {\\left({y}^{2}\\right)}^{3}\\right)$\n\n$\\textcolor{w h i t e}{{x}^{6} {y}^{3} + {y}^{9}} = {y}^{3} \\left({x}^{2} + {y}^{2}\\right) \\left({x}^{4} - {x}^{2} {y}^{2} + {y}^{4}\\right)$\n\nThen note that:\n\n$\\left({x}^{2} - a x y + {y}^{2}\\right) \\left({x}^{2} + a x y + {y}^{2}\\right) = {x}^{4} + \\left(2 - {a}^{2}\\right) {x}^{2} {y}^{2} + {y}^{4}$\n\nSo putting $a = \\sqrt{3}$ we find:\n\n${x}^{4} - {x}^{2} {y}^{2} + {y}^{4} = \\left({x}^{2} - \\sqrt{3} x y + {y}^{2}\\right) \\left({x}^{2} + \\sqrt{3} x y + {y}^{2}\\right)$\n\nPutting it all together:\n\n${x}^{6} {y}^{3} + {y}^{9} = {y}^{3} \\left({x}^{2} + {y}^{2}\\right) \\left({x}^{2} - \\sqrt{3} x y + {y}^{2}\\right) \\left({x}^{2} + \\sqrt{3} x y + {y}^{2}\\right)$","date":"2020-02-24 15:50:18","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 9, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9549122452735901, \"perplexity\": 3232.841117374747}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-10\/segments\/1581875145960.92\/warc\/CC-MAIN-20200224132646-20200224162646-00500.warc.gz\"}"}
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\section{Introduction}\label{sec intro}
Low rank matrix approximation is a central problem in numerical linear algebra (see \cite{Saad11}). It is well known that truncated singular value decompositions (SVD) of a matrix $ A\in\K^{m\times n}$
(for $\K=\R$ or $\K=\C$) produce optimal solutions to this problem (\cite{Bhatia,GVL13,HJ91,Saad11}). Indeed, let $ A= U \Sigma V^*$ be a SVD and let
$\sigma_1\geq \ldots\geq \sigma_p\geq 0$ be the singular values of $ A$, where $p=\min\{m,n\}$.
Given $1\leq h\leq \text{rank}( A)$, recall that
the truncated SVD of $ A$ is given by $ A_h= U_h \Sigma_h V_h^*$,
where the columns of $ U_h$, and $ V_h$ are the top $h$ columns of
$ U$, and $ V$ respectively, and $ \Sigma_h$ is the diagonal matrix with main diagonal given by $\sigma_1,\ldots,\sigma_h$. In this case, we have that
$\| A- A_h\|_{2,F}\leq \| A- B\|_{2,F}$ for every $ B\in\K^{m\times n}$ with $\text{rank}( B)\leq h$, where $\|\cdot\|_{2,F}$ stands for spectral and Frobenius norms respectively. Nevertheless, it is well known that (in general) computation of SVD of a matrix is expensive. In turn, this last fact is one of the motivations for the efficient numerical computation of approximations of truncated SVD of matrices \cite{D19,Gu15,HMT11,MM15,Saad11,WZZ15,Wood14}.
A closer look at these optimal approximations shows that
they can be described as $ A_h= P_h A$, where $ P_h\in\K^{m\times m}$ is the orthogonal projection onto the subspace $\cU_h$, spanned by the top $h$ columns of $ U$. Hence, one of the main strategies for computing low rank approximations is the computation of $h$-dimensional subspaces $\cS'\subset\K^m$ that are related (in some sense) to the {\it left dominant subspaces} $\cU_h$ of $A$ corresponding to SVD's of $A$.
There are several methods for efficient computation of low rank approximations of the form $PA$ for an orthogonal projection $P\in\K^{m\times m}$, based on the construction of convenient $h$-dimensional subspaces $R(P)=\cS'$ (equivalently, orthonormal sets of $h$ vectors); here $R(B)$ denotes the range of a matrix $B$.
Among others, implementations of the power and block Krylov methods have become very popular. The applications of these methods are based on deterministic and randomized approaches. Randomized methods \cite{DKM06b,DM16,Gu15,HMT11,MM15} typically draw a random $n\times r$ matrix $ X$ (a starting guess matrix) and consider the random subspace $R( X)\subset \K^n$ given by the range of $ X$.
One of the advantages of this approach is that it is possible to prove that, with high probability, $ X$ satisfies compatibility assumptions with the structure of $ A$, regardless of the particular choice of $ A$ (see \cite{DI19}).
Yet, even if $PA$ is a good low rank approximation of $A$, the range $R( P A)\subset \K^m$ might actually not be close to the subspaces
$\cU_h\subset \K^m$; here, the distance between subspaces is measured in terms of the principal angles between them.
Thus, in order to derive low rank approximations that also share some other features with $ A$, it seems natural to consider subspaces $\cS'$ that are close to the subspaces $\cU_h$. Moreover, these subspaces can be used to construct approximated truncated SVD and are also relevant in the study of principal component analysis \cite{Jolli86}. As opposed to the low rank approximation problem, there is an obstruction to consider the approximation of
the subspaces $\cU_h$, namely that they are not uniquely determined unless there is a singular gap $\sigma_h>\sigma_{h+1}$. In case there is a singular gap, then $\cU_h$ has structural relations with $ A$, and there are several positive results (both deterministic and randomized). Indeed, subspaces $\cS'$ that are close to $\cU_h$ can be obtained by the power and block Krylov methods and an starting guess matrix $ X\in\K^{n\times r}$ for $r\geq h$, that satisfies some compatibility assumptions in terms of $ V_h^*$ \cite{D19,Sai19,WZZ15}. In case there is no singular gap, there are also positive results related to approximation of $\cU_h$ in terms of an initial matrix $ X\in\K^{n\times r}$ for $r\geq h$ large enough (also satisfying compatibility conditions with $ A$) \cite{Gu15}.
In this work, we adopt a deterministic approach and adapt some of the main ideas of \cite{D19}, to deal with the approximation of $\cU_h$, in case there is no singular gap at the index $h$ (i.e. $\sigma_h=\sigma_{h+1}$). Thus, our results complement the convergence analysis in \cite{D19} (that was obtained under the assumption of the singular gap $\sigma_{h}>\sigma_{h+1}$). On the other hand, the no singular gap
case is of interest due to the common occurrence of repeated singular values in applications with some degree of symmetry (and the impact that these repeated singular values have on previous convergence analysis of deterministic block Krylov subspace methods).
In order to do this, we consider an starting guess matrix $ X\in\K^{n\times r}$ that satisfies some compatibility assumptions with $ A$, which can always be achieved with $r=h$ (i.e. for a minimal choice of $r$).
Our approach is based on enclosing $\sigma_j>\sigma_h=\sigma_{h+1}=\sigma_k>\sigma_{k+1}$ in such a way that $j<h$ and $k\geq h$ are the nearest indices for which there are singular gaps.
These gaps appear explicitly in the upper bounds related to our convergence analysis of block Krylov methods.
In this context, we show that block Krylov subspaces produce arbitrarily good $h$-dimensional approximations of left and right $h$-dominant subspaces. Moreover, we show that block Krylov spaces can also be used to compute (in terms of a well known proto-algorithm, see \cite{D19} for example) low rank approximations of $ A$ even if there is no singular gap (see Section \ref{sec dom subs} for a detailed description of the problems mentioned above).
The paper is organized as follows. In Section \ref{sec aux angles} we recall the notions of principal angles and principal vectors between subspaces, that play a central role in our work. In Section \ref{sec dom subs} we describe the context and main problems considered in this work. In Section \ref{sec DIKM} we include some of the main results from \cite{D19} related to the convergence analysis of the block Krylov methods, assuming the existence of a singular gap at a prescribed index.
In Sections \ref{sec main probs uno} and \ref{sec low rank approx} we state our main results (on dominant subspace approximations and low rank approximations) without proofs. In Section \ref{sec comentarios} we include some remarks and comments on the results herein and previous work on these matters. We also include a brief discussion of some open problems. In Section \ref{sec proofs} we present the proofs of the results described in Section \ref{sec main results}; some of these proofs require some technical facts that we consider in Section \ref{apendixity} (Appendix).
\section{Preliminaries and description of the main context}
We begin by recalling some geometric notions that play a central role
in the convergence analysis of iterative algorithms. Then, we describe
the context and problems that are the main motivation of our work.
Finally we include a description of some of the main results in \cite{D19}.
These results, that are obtained under the assumption of a singular gap at a prescribed index, also serve as a model for the type of convergence analysis that we are interested in.
\subsection{Principal angles between subspaces}\label{sec aux angles}
\noindent {\bf Principal angles between subspaces}. Let $\cS,\,\cT\subset \K^n$ be two subspaces of dimensions $s$ and $t$ respectively. Let $ S\in \K^{n\times s}$ and $ T\in\K^{n\times t}$ be isometries (i.e. matrices with orthonormal columns) such that $R( S)=\cS$ and $R( T)=\cT$. Following \cite{GVL13}, we define the principal angles between $\cS$ and $\cT$, denoted
$$0\leq \theta_1(\cS,\cT)\leq \ldots\leq \theta_k(\cS,\cT)\leq \frac{\pi}{2} \peso{where} k=\min\{s,\,t\}\,, $$
determined by the identities $\cos(\theta_{j}(\cS,\cT))=\sigma_j( S^* T)$, for $1\leq j\leq k$; in this case the roles of $ S$ and $ T$ are symmetric.
If we assume that $s\leq t$ (so $k=s$) the principal angles can be also determined in terms of the identities
\begin{equation}\label{eq sobre sen ang princ1}
\sin(\theta_{s-j+1}(\cS,\cT))=\sigma_j( ( I- T T^*) S)=\sigma_j( ( I- T T^*) S S^*)=\sigma_j(( I- P_\cT) P_\cS)
\end{equation}
for $1\leq j\leq s$, where $ P_\cH\in\K^{n\times n}$ denotes the orthogonal projection onto a subspace $\cH\subset \K^n$; it is worth noticing that in this last case the roles of $ S$ and $ T$ (equivalently the roles of $ P_\cT$ and $ P_\cT$) are not symmetric (unless $s=t$).
Principal angles can be considered as a vector valued measure of the distance between the subspaces $\cS$ and $\cT$.
Following \cite{SS90} we let $ \Theta(\cS,\cT)=\text{diag}(\theta_1(\cS,\cT),\ldots,\theta_s(\cS,\cT))$ denote the diagonal matrix with the principal angles in its main diagonal. In particular,
$$\|\sin \Theta(\cS,\cT)\|_{2,F}=\|( I- P_\cT) P_\cS\|_{2,F}
$$ are scalar measures of the (angular) distance between $\cS$ and $\cT$ (see
\cite{GVL13,SS90}).
We mention some properties of the principal angles between subspaces that we will need in what follows.
With the previous notation, we point out that if $\cS'\subset\cS$ and $\cT\subset \cT'$ are subspaces
with dimensions $s'$ and $t'$ respectively, then (recall that $s=\dim\cS\leq \dim\cT=t$)
$$
\| \Theta(\cS,\cT')\|_{2,F} \leq \| \Theta(\cS,\cT)\|_{2,F} \ \ , \ \
\| \sin \Theta(\cS,\cT')\|_{2,F} \leq \|\sin \Theta(\cS,\cT)\|_{2,F}
$$ and similarly
$$\| \Theta(\cS',\cT)\|_{2,F}\leq \| \Theta(\cS,\cT)\|_{2,F} \ \ , \ \
\|\sin \Theta(\cS',\cT)\|_{2,F}\leq \|\sin \Theta(\cS,\cT)\|_{2,F} \,,
$$
which follow from Eq. \eqref{eq sobre sen ang princ1}. On the other hand, $\dim\cS^\perp=n-s\geq n-t=\dim\cT^\perp$ and therefore,
$$
\sin(\theta_{(n-t)-j+1}(\cS^\perp,\cT^\perp))=\sigma_j(( I- P_{\cS^\perp}) P_{\cT^\perp})= \sigma_j( P_\cS( I- P_{\cT}))\ , \ \ 1\leq j\leq n-t\,.
$$
By comparing the previous identity with Eq. \eqref{eq sobre sen ang princ1},
if $\theta_1(\cS,\cT),\ldots,\theta_d(\cS,\cT)>0$ are the positive angles between
$\cS$ and $\cT$ (for some $0\leq d\leq \min\{s,n-t\}$) then these coincide with the
positive angles between
$\cS^\perp$ and $\cT^\perp$ i.e.
\beq\label{ang inv por com ort}
\theta_j(\cS,\cT)=\theta_j(\cS^\perp,\cT^\perp)\peso{for} 1\leq j\leq d\,.
\eeq
Notice that as a consequence of Eq. \eqref{ang inv por com ort} we get that
\beq\label{ang inv por com ort2}
\| \Theta(\cS,\cT)\|_{2,F}=\| \Theta(\cS^\perp,\cT^\perp)\|_{2,F}\,.
\eeq
\noindent {\bf Principal vectors between subspaces}. In what follows we shall also make use of the principal vectors associated to the subspaces $\cS$ and $\cT$: indeed, by construction of the principal angles, we get that there exist orthonormal systems
$\{ u_1,\ldots, u_s\}\subset \cS$ and $\{ v_1,\ldots, v_s\}\subset \cT$
such that
$$\langle u_i, v_j\rangle =\delta_{ij}\,\cos(\theta_{j}(\cS,\cT))\peso{for} 1\leq i,\,j\leq s\,,$$ where $\delta_{ij}$ is Kronecker's delta function.
We say that $\{ u_1,\ldots, u_s\}$ and $\{ v_1,\ldots, v_s\}$ are the {\it principal vectors (directions)} associated with the subspaces $\cS$ and $\cT$. Notice that the previous facts imply, in particular, that the subspaces $\cS_j=\text{Span}\{ u_1,\ldots, u_j\}\subset\cS$ and
$\cT_j=\text{Span}\{ v_1,\ldots, v_j\}\subset \cT$ are such that
$$ \Theta(\cS_j,\cT_j)=\text{diag}(\theta_1(\cS,\cT),\ldots,\theta_j(\cS,\cT))\in\R^{j\times j}\peso{for} 1\leq j\leq s\,.$$ Moreover, if $\tilde {\cS}\subset \cS$ and $\tilde{\cT}\subset\cT$ are two $j$-dimensional subspaces then, it follows that
$ \Theta(\cS_j,\cT_j)\leq \Theta(\tilde {\cS},\tilde {\cT})$; that is, $\cS_j$ and $\cT_j$ are $j$-dimensional subspaces of $\cS$ and $\cT$ respectively, that are at minimal angular (vector valued) length.
\subsection{Setting the context and problems}\label{sec dom subs}
We begin with a formal description of the class of dominant subspaces of a matrix, without assuming a singular gap. Then, we describe the context
and main problems considered in this work.
\medskip
\noindent {\bf Dominant subspaces and low rank approximations}. Let $A\in\K^{m\times n}$ and let $\sigma_1\geq \ldots \geq \sigma_p\geq 0$, where $p=\min\{m,\,n\}$, denote its singular values. Let $\cS'\subset \K^m$ be a subspace of dimension $1\leq h\leq \text{rank}( A)\leq p$. We say that $\cS'$ is a {\it left dominant subspace} for
$ A$ if $\cS'$ admits an orthonormal basis $\{ w_1,\ldots, w_h\}$ such that
$ A A ^* w_i=\sigma_i^2\, w_i$, for $1\leq i\leq h$.
Equivalently, $\cS'$ is a left dominant subspace for
$ A$ if the $h$ largest singular values of $ P_{\cS'} A$ are $\sigma_1\geq \ldots\geq \sigma_h$. Hence, in this case we have that
$$
\| P_{\cS'} A - A\|\leq \| Q A - A\|
$$ for every orthogonal projection $ Q\in\K^{m\times m}$ with rank$( Q)=h$ and every unitarily invariant norm; that is, $ P_{\cS'} A$ is an optimal low-rank approximation of $ A$ (see \cite[Section IV.3]{Bhatia}).
On the other hand, we say that $\cS\subset \K^n$ is a {\it right dominant subspace} for $ A$ if
$\cS$ admits an orthonormal basis $\{ z_1,\ldots, z_h\}$ such that
$ A^* A z_i=\sigma_i^2 z_i$, for $1\leq i\leq h$.
Similar remarks apply also to right dominant subspaces. It is interesting to notice that
the class of $h$-dimensional left dominant subspaces of $ A$ coincides with
the class of $h$-dimensional right dominant subspaces of $ A^*$; in what follows we will make use of this fact.
\medskip
\noindent {\bf Dominant subspaces and SVD}. Let $ A= U \Sigma V^*$ be a full SVD for $ A\in\K^{m\times n}$, where $\K=\R$ or $\K=\C$,
$ \Sigma\in \R^{m\times n}$ and $ U\in\K^{m\times m}$ and $ V\in\K^{n\times n}$ are unitary (orthogonal when $\K=\R$) matrices. In this case $\Sigma$ is a (rectangular) diagonal matrix, with diagonal entries given by the singular values of $A$. In what follows we let $ u_j$ (respectively $ v_j$) denote the columns of $ U$ (respectively of $ V$).
Given $1\leq h\leq m$, we define the subspace
$\cU_h=\text{Span}\{ u_1,\ldots, u_h\}\subset \K^m$; similarly, if $1\leq h\leq n$, we let
$\cV_h=\text{Span}\{ v_1,\ldots, v_h\}\subset \K^n$. Then, $\cU_h$ and $\cV_h$ are left and right dominant subspaces respectively.
In case $\sigma_h>\sigma_{h+1}$ then it is well known that the left (respectively right) dominant subspace for $ A$ of dimension $h$ is uniquely determined; hence, in this case $\cU_h$ and $\cV_h$ do not depend on our particular choice of SVD for $ A$.
On the other hand, if $\sigma_h=\sigma_{h+1}$ then we have a continuum class of
$h$-dimensional left dominant subspaces: indeed, let
$0\leq j=j(h)< h<k=k(h)$ be given by
$j(h)=\max\{ 0\leq j < h\ : \ \sigma_j> \sigma_h\}$, where we set $\sigma_0=\infty$ and
$k=k(h)=\max\{ 1\leq j\leq \text{rank}( A) \, : \, \sigma_j=\sigma_h\}$.
If we further let $\cU_0=\{0\}$ then, it is straightforward to check that an $h$-dimensional subspace $\cS'$ is a left dominant subspace for $ A$ if and only if there exists an $(h-j)$-dimensional subspace $\cU\subset \cU_k\ominus \cU_j:=\cU_k\cap \cU_j^\perp\subset \K^m$ such that
$$\cS'=\cU_j\oplus \cU\,.$$
Therefore, we have a natural parametrization of $h$-dimensional left dominant subspaces in terms of
subspaces $\cU$ that vary over
the Grassmann manifold of $(h-j)$-dimensional subspaces of $\cU_k\ominus \cU_j\subset \K^m$.
It is a basic fact in linear algebra that given $\cS'$ a left dominant subspace of dimension $h\geq 1$, there exists a SVD, $ A= U \Sigma V^*$ such that $\cS'=\cU_h$, i.e. the subspace spanned by the top $h$ columns of $ U$; and a similar fact also holds for
right dominant subspaces.
\medskip
\noindent {\bf Main problems}. Consider $ A$ as above and a matrix $ X\in\K^{n\times r}$ (an starting guess).
From $ A$ and $ X$ we construct the block Krylov space $\cK_q(A,X)$, for $q\geq 0$, that is
\begin{equation} \label{eq defi Krylov2}
\cK_q=\cK_q( A, X)=R(\ A X \quad ( A A^*) A X\quad \ldots\quad
( A A^*)^q A X\ )\subset \K^m\,,
\end{equation} (recall that $R( B)$ denotes the range of a matrix $ B$).
In this setting, our first main problem is to show the existence of some $h$-dimensional subspace $\cS'\subseteq \cK_q$ that is close to {\it some}
$h$-dimensional left dominant subspace $\cU_h$ of $A$. In this context, proximity between subspaces is measured by $\|\sin \Theta(\cK_q,\cS')\|_{2,F}$ i.e. in terms of (the spectral or Frobenius norm of) the sines of the principal angles between the subspaces $\cK_q$ and $\cS'$ (see Section \ref{sec main probs uno}). Once we establish the existence of $\cS'\subseteq \cK_q$ as above, we get the low rank approximation $P_{\cS'}A$ of $A$. We point out that our approach does not provide an effective way (algorithm) to compute $\cS'$.
Therefore our second main problem is to compute, in an algorithmic way, an $h$-dimensional subspace $ \cS''\subset \cK_q$ together with a corresponding upper bound for the approximation error
$$\|A-P_{\cS''} A\|_{2,F}\,.$$ Further, we require that the upper bound for the approximation error of $A$ by $P_{\cS''} A$ becomes arbitrarily close to $\|A-A_h\|_{2,F}$, i.e. the error in approximating $A$ by the (optimal) low rank matrix $A_h$ obtained from truncated SVD's of $A$ (as described at the beginning of Section \ref{sec intro}). Hence, by solving this second problem, we obtain (in an effective way) the low rank approximation $P_{ \cS''}A$ of $A$ (see Section \ref{sec low rank approx}) that behaves much like the optimal low rank approximations of $A$.
In case there is a singular gap i.e. $\sigma_h>\sigma_{h+1}$ these problems have been recently solved in \cite{D19} (see Section \ref{sec DIKM} below). In this work we adopt the approach considered in that work in order to construct approximations of dominant spaces and low rank approximation of $A$.
Hence, we focus on the {\it convergence analysis} of those methods in the case that there is no singular gap at the index $h$.
\subsection{DIKM-I theory with singular gaps: structural results}\label{sec DIKM}
In \cite{D19} P. Drineas, I.C.F. Ipsen, E.M. Kontopoulou and M. Magdon-Ismail merged a series of
techniques, tools and arguments that lead to structural results related to
the approximation of dominant subspaces from block Krylov spaces in the presence of a singular gap.
The convergence analysis obtained in \cite{D19} has a deep influence in our present work; indeed, we shall follow some of the lines developed in that work, that we refer to as the \textit{DIKM-I theory}. Of course, at some points we have to departure from those arguments to deal with the no-singular-gap case. Next we include some of the features of the DIKM-I theory that we need in what follows.
In this section we keep using the notation considered so far: $A\in\K^{m\times n}$, $A=U\Sigma V$ its SVD, its singular values $\sigma_1\geq \ldots\geq \sigma_p$, $p=\min\{m,n\}$, and so on. In case $1\leq k< \text{rank}( A)\leq p$ then
we consider the partitioning
\begin{equation} \label{eq particion eq1}
\Sigma=\begin{pmatrix} \Sigma_k & \\ & \Sigma_{k,\perp}\end{pmatrix}\ , \ \
U=\begin{pmatrix} U_k &
U_{k,\perp}\end{pmatrix}\ , \ \
V=\begin{pmatrix} V_k &
V_{k,\perp}\end{pmatrix}\,.
\end{equation}
The following is one of the main results of the DIKM-I theory. In what follows, given a matrix $ Z$ we let $ Z^\dagger$ denote its Moore-Penrose pseudo-inverse.
\begin{teo}[\cite{D19}] \label{DIKM-I theorem 1} \rm Assume that $\sigma_{k}>\sigma_{k+1}$, let $\phi(x)$ be a polynomial of degree at most $2q+1$ with odd powers only, such that $\phi( \Sigma_k)$ is non-singular. Let $\tilde { X}\in\K^{n\times r}$ be such that
$\text{rank}( V_k^*\tilde { X})=k$ (so $r\geq k$) and let $\tilde \cK_q=\cK_q( A,\tilde { X})$. Then,
$$
\|\sin \Theta(\tilde \cK_q,\cU_k)\|_{2,F}\leq \|\phi( \Sigma_{k,\perp})\|_2\,
\|\phi( \Sigma_k)^{-1}\|_2 \,\| V_{k,\perp}^*\tilde { X}( V_{k}^*\tilde { X})^\dagger\|_{2,F}\,.
$$
\qed
\end{teo}
Theorem \ref{DIKM-I theorem 1} provides an upper bound for the (angular) distance between the
subspaces $\tilde \cK_q$ and $\cU_k$. By choosing polynomials $\phi\in \K[x]$ as above in a convenient way, we can make the upper bound in Theorem \ref{DIKM-I theorem 1} arbitrarily small for sufficiently large $q\geq 0$ (see Theorem \ref{DIKM-I theorem 4} below). Thus we consider Theorem \ref{DIKM-I theorem 1}
as part of the convergence analysis of the block Krylov method.
\medskip
In the next result we make use of the following well known proto-algorithm (see for example \cite{D19}).
We will make use of this algorithm again in Section \ref{sec low rank approx}.
\smallskip
\begin{algorithm}
\caption{(Proto-algorithm for low rank approximation)}\label{algoalgo}
\centerline{
}
\begin{algorithmic}[1]
\REQUIRE $A\in\K^{m\times n}$, starting guess $ X\in \K^{n\times r}$; rank parameter $k\leq \text{rank}(A)$; power parameter $\ell\geq 0$.\\
$\quad~$ Set $ K_\ell=( A X\quad ( A A^*) A X
\quad \cdots \quad ( A A^*)^\ell A X)\in\K^{m\times (\ell+1)\cdot r}$;
\\
$\quad~$ Test that $d:=\dim R( K_\ell)d\geq k$. In this case:
\ENSURE $\hat{ U_k}\in\K^{m\times k}$ with orthonormal columns
\STATE Compute an orthonormal basis $ U_K\in\K^{m\times d}$ for $R( K_\ell)$.
\STATE Set $ W= U_K^* A\in \K^{d\times n}$ (notice that rank$( W)\geq k$).
\STATE Compute $ U_{W,k}\in \K^{d\times k}$ isometry, such that $R( U_{W,k})$ is a left dominant subspace of $ W$.
\STATE Return $\hat { U}_k= U_K\, U_{W,k}\in \K^{m\times k}$.
\end{algorithmic}
\end{algorithm}
\smallskip
Once the Algorithm \ref{algoalgo} is performed, we describe the output matrix in terms of its columns $\hat { U}_k=(\hat { u}_1,\ldots,\hat { u}_k)$. We also consider the matrices $\hat { U}_i=(\hat { u}_1,\ldots,\hat { u}_i)\in\K^{m\times i}$, for $1\leq i\leq k$.
\begin{teo}[\cite{D19}]\label{DIKM-I theorem 2} \rm
Assume that $\sigma_{k}>\sigma_{k+1}$, let $\phi(x)$ be a polynomial of degree at most $2q+1$ with odd powers only, such that
$\phi(\sigma_i)\geq \sigma_i$ for $1\leq i\leq k$. Consider the output of Algorithm \ref{algoalgo} with starting guess $\tilde { X}\in\K^{n\times r}$ such that
$\text{rank}( V_k^*\tilde { X})=k$, rank parameter $k$ and power parameter $q$. Then, for $1\leq i\leq k$,
\begin{eqnarray*}
\| A - \hat{ U_i}\hat{ U}_i^* A\|_{2,F}&\leq& \| A- A_i\|_{2,F}+\Delta\\
\sigma_i-\Delta&\leq& \|\hat{ u}_i^* A\|_2\leq \sigma_i
\end{eqnarray*}
where $ A_i\in\K^{m\times n}$ is a best rank-$i$ approximation of $ A$ and
$\Delta=\|\phi( \Sigma_{k,\perp})\|_2\,\| V_{k,\perp}^* \tilde X( V_{k}^* \tilde X)^\dagger\|_F$.
\qed
\end{teo}
The following result from \cite{D19} complements Theorems \ref{DIKM-I theorem 1} and \ref{DIKM-I theorem 2} above, in the sense that it implies that the upper bounds in those theorems can be made arbitrarily small.
This result corresponds to a generalization of the Chebyshev-based gap-amplifying polynomials developed in \cite{MM15} and \cite{WZZ15}.
\begin{lem}[\cite{D19}]\label{DIKM-I theorem 3} \rm
Assume that $k<\text{rank}( A)$, so that $\sigma_k>\sigma_{k+1}>0$, and let
$$\gamma_k=\frac{\sigma_k-\sigma_{k+1}}{\sigma_{k+1}}>0\,.$$
Then, there exists a polynomial $\phi(x)$ of degree at most $2q+1$ with odd powers only,
such that
\begin{eqnarray*}\phi(\sigma_1)\geq \ldots\geq \phi(\sigma_k) \quad ,\quad \phi(\sigma_i)\geq \sigma_i>0 \ &,& \ \peso{for} 1\leq i\leq k\,,\\
\py |\phi(\sigma_i)|\leq \frac{4\sigma_{k+1}}{2^{(2q+1)\min\{\sqrt{\gamma_k}\coma 1\}}}\ &,& \ \peso{for} i\geq k+1\,.
\end{eqnarray*}
Hence, $
\|\phi(\Sigma_k)^{-1}\|_2\leq \sigma_k^{-1}$ and $\|\phi(\Sigma_{k,\perp})\|_2\leq
\frac{4\sigma_{k+1}}{2^{(2q+1)\min\{\sqrt{\gamma_k}\coma 1\}}}\,.
$
\qed
\end{lem}
We point out that the inequalities $\phi(\sigma_1)\geq \ldots\geq \phi(\sigma_k)$ in the lemma above are a consequence of the super-linear growth for large input values (i.e. in this case for $x\geq \sigma_{k+1}$) of the gap amplifying Chebyshev polynomials (see \cite{D19}).
In order to describe the following result from the DIKM-I theory (that follows from Theorems \ref{DIKM-I theorem 1}, \ref{DIKM-I theorem 2} and Lemma \ref{DIKM-I theorem 3}), assume that
$\text{rank}(A)>k$; then we consider
\begin{equation} \label{defi gam y del1}
\gamma_k= \frac{\sigma_k-\sigma_{k+1}}{\sigma_{k+1}}>0 \py
\Delta( W,q,k)_{2,F} = 4\, \frac{\| V_{k,\perp}^* W( V_k^* W)^\dagger\|_{2,F}}{2^{(2q+1)\,\min\{\sqrt{\gamma_k}\coma 1\}}} \, ,
\end{equation} where $ W\in \K^{n\times \ell}$, for some $\ell\geq 1$ and $q\geq 1$.
Notice that in case $\sigma_k>\sigma_{k+1}$ then then
expressions in Eq. \eqref{defi gam y del1} do not depend on the particular SVD of $A$.
\begin{teo}[\cite{D19}]\label{DIKM-I theorem 4}\rm
Assume that $\sigma_{k}>\sigma_{k+1}>0$. Let $\tilde { X}\in\K^{n\times r}$ be such that
$\text{rank}( V_k^*\tilde { X})=k$ and let $\tilde \cK_q=\cK_q( A,\tilde { X})$. Consider the output of Algorithm \ref{algoalgo} with starting guess $\tilde X$ such that
$\text{rank}( V_k^*\tilde { X})=k$, rank parameter $k$ and power parameter $q$. Then
\begin{eqnarray*}
& &\|\sin \Theta(\tilde \cK_q,\cU_k)\|_{2,F}\leq \Delta(\tilde X,q,k)_{2,F}\ \frac{\sigma_{k+1}}{\sigma_k}\,,
\\
\\
& &\| A - \hat{ U_i}\hat{ U}_i^* A\|_{2,F}\leq \| A- A_i\|_{2,F}+
\Delta(\tilde X,q,k)_{F}\,\sigma_{k+1} \ \, , \, \ 1\leq i\leq k\,.
\end{eqnarray*}
\qed\end{teo}
Theorem \ref{DIKM-I theorem 4} shows the type of convergence analysis obtained in \cite{D19}
for the deterministic block Krylov methods. On the one hand, it
provides an upper bound for the distance between the subspaces $\tilde \cK_q$ and $\cU_k$, that becomes arbitrarily small for sufficiently large $q\geq 0$. Similarly, this result also provides an upper bound for the error in the approximation of $A$ by the low rank matrix $\hat{ U_k}\hat{ U}_k^* A$ that is arbitrarily close to the optimal error $\|A-A_k\|_{2,F}$ (obtained using exact truncated singular value decomposition) for sufficiently large $q\geq 0$.
\section{Main results}\label{sec main results}
In this section we state our main results related to dominant subspace approximations and low rank matrix approximations in terms of block Krylov subspaces. The proofs of these results are considered in Section \ref{sec proofs}. Our results are motivated by the structural results of the DIKM-I theory described in Section \ref{sec DIKM}. At the end of this section we include some comments and further research problems related to the present work.
\subsection{Approximation of dominant subspaces by block Krylov spaces}\label{sec main probs uno}
As before, let $ A\in\K^{m\times n}$ with singular values $\sigma_1\geq \ldots\geq \sigma_p$, for $p=\min\{m,n\}$. Given $1\leq h\leq \text{rank}( A)\leq p$, we let
$0\leq j(h)< h$ be given by
\beq\label{eq defi j}
j=j(h)=\max\{ 0\leq \ell < h\ : \ \sigma_\ell> \sigma_h\}\eeq where we set $\sigma_0=\infty$ and
\beq \label{eq defi k}
k=k(h)=\max\{ 1\leq \ell\leq \text{rank}( A) \, : \, \sigma_\ell=\sigma_h\}\,.\eeq
Since $h\leq \text{rank}( A)$, we get that $\sigma_k>0$.
As mentioned in the preceding sections, we will focus on the case when $h<k$ (i.e. when $\sigma_h=\sigma_{h+1}$).
In case $1\leq k< \text{rank}( A)\leq p$ then
we consider the partitioning as in Eq. \eqref{eq particion eq1} that is,
$$\Sigma=\begin{pmatrix} \Sigma_k & \\ & \Sigma_{k,\perp}\end{pmatrix}\ , \ \
U=\begin{pmatrix} U_k &
\\ &
U_{k,\perp}\end{pmatrix}\ , \ \
V=\begin{pmatrix} V_k &
V_{k,\perp}\end{pmatrix}\,.$$
\begin{fed}Given $ X\in\K^{n\times r}$ we say that $( A, X)$ is {\it $h$-compatible} if there is an $h$-dimensional right dominant subspace $\cS\subset \K^n$ for $ A$, with $$ \Theta(\cS,R( X))<\frac{\pi}{2}\, I\,,$$ where $ \Theta(\cS,R( X))\in\R^{h\times h}$ denotes the diagonal matrix with the principal angles between $\cS$ and $R( X)$ in the main diagonal (see Section \ref{sec aux angles}).
\EOE
\end{fed}
Given $ X\in\K^{n\times r}$ notice that $( A, X)$ is $h$-compatible if and only if
if $\dim ( X^*\cS)=h$, for some $h$-dimensional right dominant subspace $\cS$.
\medskip
We can now state our main results. We begin with the next technical result that will allow us to show
that block Krylov methods produce arbitrary good approximations of right and left dominant subspaces.
Recall that given a matrix $ Z$ we let $ Z^\dagger$ denote its Moore-Penrose pseudo-inverse.
Throughout the rest of the work, we fix $1\leq h\leq \text{rank}( A)\leq p=\min\{m,n\}$ and we let let $0\leq j=j(h)<h\leq k=k(h)\leq \text{rank}(A)$ be defined as in Eqs. \eqref{eq defi j} and \eqref{eq defi k}.
\begin{teo}\label{first main result}
Let $\phi(x)$ be a polynomial
of degree at most $2q+1$ with odd powers only,
such that
$\phi(\sigma_1)\geq \ldots\geq \phi(\sigma_k)>0$.
Let $( A, X)$ be $h$-compatible and let $\cK_q=\cK_q( A, X)$.
Then, there exists an $h$-dimensional left dominant subspace $\cS'$ for $ A$ such that
\begin{eqnarray*}
\|\sin \Theta(\cK_q,\cS')\|_{2,F}&\leq & 4\, \|\sin \Theta(R( V_k^* X), V_k^*\cV_j)\|_{2,F} + \\ \\ & &
\|\phi( \Sigma_{k,\perp})\|_2\|\phi( \Sigma_{k})^{-1}\|_2 \| V_{k,\perp}^* X
( V_k^* X) ^\dagger\|_{2,F}\,.
\end{eqnarray*}
In case $j=0$ (respectively $k=\text{rank}( A)$) the first term (respectively the second term) should be omitted in the previous upper bound.
Moreover, we have the inequality $$ \Theta(R( V_k^* X), V_k^*\cV_j)\leq \Theta(R( X),\cV_j)\,.$$
\end{teo}
\begin{proof}
See Section \ref{sec prueba teo 2.1}.
\end{proof}
We point out that Theorem \ref{first main result} above is related to
Theorem \ref{DIKM-I theorem 1} from the DIKM-I theory. In case $\sigma_h=\sigma_{h+1}$ (and hence $k>h$), the hypothesis in Theorem \ref{first main result} involves
a (continuum) class of matrices $ V_h^*$ corresponding to SVD of $A$; that is, we are allowed to consider any such matrix to test our assumptions. On the other hand,
since our assumptions are based on non-structural choices $ V_h^*$, there is a price to pay: we need {\it a priori} partial knowledge of the relative position of the subspaces $R( V_k^* X)$ and $R(V_k^*\cV_j)$ to have control on the upper bound above (notice that Theorem \ref{DIKM-I theorem 1} does not require such partial knowledge). We remark that the second inequality in Theorem \ref{first main result}
provides an alternative method to have a control of the relative position of the subspaces $R( V_k^* X)$ and $R(V_k^*\cV_j)$.
\medskip
In order to state the following result, we further assume that $\text{rank}(A)\geq k+1$ and recall the notation from Eq. \eqref{defi gam y del1} above from the DIKM-I theory; hence, given a SVD $A=U\Sigma V^*$ then
$$
\gamma_k= \frac{\sigma_k-\sigma_{k+1}}{\sigma_{k+1}}>0 \py \Delta( W,q,k)_{2,F} = 4\, \frac{\| V_{k,\perp}^* W( V_k^* W)^\dagger\|_{2,F}}{2^{(2q+1)\,\min\{\sqrt{\gamma_k}\coma 1\}}} \, ,
$$
where $ W\in \K^{n\times \ell}$, for some $\ell\geq 1$ and $q\geq 1$.
\begin{cor}\label{cor con gap amp}
Let $( A, X)$ be $h$-compatible and let $\cK_q=\cK_q( A, X)$.
Then, there exists an $h$-dimensional left dominant subspace $\cS'$ for $ A$ such that
\begin{eqnarray*}
\|\sin \Theta(\cK_q,\cS')\|_{2,F}&\leq & 4\, \|\sin \Theta(R( X),\cV_j)\|_{2,F} + \Delta( X,q,k)_{2,F}\, \frac{\sigma_{k+1}}{\sigma_k}\,.
\end{eqnarray*}
In case $j=0$ (respectively $k=\text{rank}( A)$) the first term (respectively the second term) should be omitted in the previous upper bound.
\end{cor}
\begin{proof}
See Section \ref{sec prueba teo 2.1}.
\end{proof}
The following result can be regarded as a convenient algorithmic augmentation process of the initial subspace $R( X)=\cX\subset \K^n$; that is, we begin with $ X$ that satisfies
a compatibility assumption with {\it some} $h$-dimensional right dominant subspace of $ A$ and we construct an associated (auxiliary) subspace $\cK^*_{q,t}\subset \K^n$ that is (arbitrarily) close to an $h$-dimensional {\it right} dominant subspace (see Theorem \ref{third main result S1} below). We remark that this result plays a central role in the construction of approximate left dominant subspaces and low-rank approximations of $ A$ from block Krylov methods in Theorems \ref{teo exist right dom1} and \ref{other main result} below.
\medskip
Let $(A,X)$ be $h$-compatible and consider $A=U\Sigma V^*$ a SVD of $A$. For the next result we consider the notation in Eq. \eqref{defi gam y del1}; we further introduce
\begin{equation}\label{defi gam y del2}
\Delta( X,q,j)_{2,F}=4\, \frac{\| V_{j,\perp}^* X( V_j^* X)^\dagger\|_{2,F}}{2^{(2q+1)\,\min\{\sqrt{\gamma_j}\coma 1\}}}
\ \ , \ \ \Delta^*( Y,t,k)_{2,F} := 4\, \frac{\| U_{k,\perp}^* Y( U_k^* Y)^\dagger\|_{2,F}}{2^{(2t+1)\,\min\{\sqrt{\gamma_k}\coma 1\}}}\,, \end{equation} where
$\gamma_j= \frac{\sigma_j-\sigma_{j+1}}{\sigma_{j+1}}>0$,
$ Y\in\K^{ m\times \ell}$, for some $\ell\geq 1$ and $t\geq 1$. Notice that expressions in Eq. \eqref{defi gam y del2} do not depend on the particular SVD of $A$, since by construction
$\sigma_j>\sigma_{j+1}$ and $\sigma_k>\sigma_{k+1}$.
\begin{teo}\label{second main result S1}
Let $(A,X)$ be $h$-compatible, let $\cK_q=\cK_q( A, X)\subset \K^m$ and let $ Y_q$ be such that $ Y_q Y_q^*$ is the orthogonal projection onto $\cK_q$. For $t\geq 0$ we let
$$\cK_{q,t}^*=
R(( A^* A) X) + R(( A^* A)^{2} X)+\ \dots +R( ( A^* A)^{q+t+1} X)\subset \K^n\,.$$
Then, there exists an $h$-dimensional right dominant subspace $\tilde \cS$ for $ A$ such that
\begin{equation} \label{eq teo sobre precond1}
\|\sin \Theta(\cK_{q,t}^*,\tilde \cS)\|_{2,F}\leq 4\, \Delta( X,q,j)_{2,F}\,\frac{\sigma_{j+1}}{\sigma_j}
+ \Delta^*( Y_q,t,k)_{2,F}\,\frac{\sigma_{k+1}}{\sigma_k}\,.
\end{equation}
In case $j=0$ (respectively $k=\text{rank}( A)$) the first term (respectively the second term) should be omitted in the previous upper bound.
\end{teo}
\begin{proof}
See section \ref{sec prueba del segundo}.
\end{proof}
Using the correspondence between left and right dominant subspaces of $ A$ we can derive the existence of (arbitrarily good) approximates of left dominant subspaces obtained from the block Krylov method in case there is no singular gap.
\pausa The next result complements Theorem \ref{first main result} (see the comments after Remark \ref{rem la cota puede ser tan chica como queramos} below); we consider the notation in Eq. \eqref{defi gam y del2}.
\begin{teo}\label{teo exist right dom1} Let $(A,X)$ be $h$-compatible. Given $q,\,t\geq 0$, consider
$\cK_{q+t+1}=\cK_{q+t+1}( A, X)\subset \K^m$. Then, there exists an $h$-dimensional left dominant subspace $\hat \cS$ for $ A$ such that
\begin{equation} \label{eq upper bounds x28}
\|\sin \Theta(\cK_{q+t+1},\hat\cS)\|_{2,F}\leq 4\, \Delta^*( A X,q,j)_{2,F}\,\frac{\sigma_{j+1}}{\sigma_j}
+ \Delta( W_q,t,k)_{2,F}\,\frac{\sigma_{k+1}}{\sigma_k}\,,
\end{equation} where $ W_q$ is such that $ W_q W_q^*$ is the orthogonal projection onto $\cK_q( A^*, A X)\subset\K^n$.
In case $j=0$ (respectively $k=\text{rank}( A)$) the first term (respectively the second term) should be omitted in the previous upper bound.
\end{teo}
\begin{proof}
Since the pair $( A, X)$ is $h$-compatible, there exists an $h$-dimensional right dominant subspace $\cS\subset \K^n$ for $ A$, such that $\dim ( X^*\cS)=h$. Set $ Z= A X$ and let $\cS'= A\cS\subset \K^m$. Hence, $\cS'$ is a left dominant subspace for $ A$ and then, a right dominant subspace of $ A^*$ with $\dim \cS'=h$. Moreover, $ Z^*\cS'= X^* A^* A\cS= X^*\cS$, since $ A^* A\cS=\cS$. In particular, $\dim Z^*\cS'=h$ and hence $ \Theta(R( Z),\cS')<\frac{\pi}{2}\, I$. Therefore, we can apply Theorem \ref{second main result S1} to $ A^*$ and $ Z$; in this case, we consider the (auxiliary) subspace
$$
\cK^*_{q,t}( Z)=R(( A A^*) Z) + R(( A A^*)^{2} Z)+\ \dots +R( ( A A^*)^{q+t+1} Z)\subset \K^n\,.
$$ It is clear that $\cK^*_{q,t}( Z)\subset \cK_{q+t+1}$. Then, by Theorem \ref{second main result S1} there exists an $h$-dimensional right dominant $\hat \cS$ for $ A^*$ (and therefore a left dominant subspace for $ A$) such that
\begin{eqnarray*}
\|\sin \Theta(\cK_{q+t+1},\hat \cS)\|_{2,F} &\leq& \|\sin \Theta(\cK_{q,t}^*( Z),\hat \cS)\|_{2,F}\\ &\leq& 4\, \Delta^*( A X,q,j)_{2,F}\,\frac{\sigma_{j+1}}{\sigma_j}
+ \Delta( W_q,t,k)_{2,F}\,\frac{\sigma_{k+1}}{\sigma_k}\,,
\end{eqnarray*} where we used that if $ A= U \Sigma V^*$ is a SVD for $ A$ then
$ A^*= V \Sigma U^*$ is a SVD for $ A^*$.
\end{proof}
With the notation of Theorems \ref{second main result S1} and \ref{teo exist right dom1}, it seems useful to obtain uniform upper bounds
for $\| U_{k,\perp}^* Y_q( U_k^* Y_q)^\dagger\|_{2,F}$
and $\| V_{k,\perp}^* W_q( V_k^* W_q)^\dagger\|_{2,F}$
at least for $q\geq \tilde q$, where $\tilde q$ is some fixed number. Theorem \ref{third main result S1} below shows that we can obtain such a uniform upper bound. In turn, this results will allow us to have a better control over the upper bound in Eq. \eqref{eq upper bounds x28}.
In what follows we let $\#\{\sigma_1,\ldots,\sigma_h\}$ denote
the number of {\it different} singular values of $ A$ between $\sigma_1$ and
$\sigma_h$.
\begin{teo}\label{third main result S1}
Let $q_0=\#\{\sigma_1,\ldots,\sigma_h\}-1< h$, for $h\leq \text{rank}( A)$.
Let $(A,X)$ be $h$-compatible, let $\cK_q=\cK_q( A, X)\subset \K^m$ and let $ Y_q$ be such that $ Y_q Y_q^*$ is the orthogonal projection onto $\cK_q$. Given $A=U\Sigma V^*$ a SVD of $A$,
$$
\| U_{k,\perp}^* Y_{q'}( U_k^* Y_{q'})^\dagger\|_{2,F}\leq
\| U_{k,\perp}^* Y_{q}( U_k^* Y_{q})^\dagger\|_{2,F}\peso{for} q_0\leq q\leq q'\,.
$$
\end{teo}
\begin{proof} See Section \ref{sec 4.3}.
\end{proof}
\begin{rem}\label{rem la cota puede ser tan chica como queramos}
Consider the notation in Theorem \ref{teo exist right dom1}. Let $q_0=\#\{\sigma_1,\ldots,\sigma_h\}-1<h$, where $\#\{\sigma_1,\ldots,\sigma_h\}$ denotes the number of different singular values of $ A^*$ (or equivalently of $ A$) between $\sigma_1$ and $\sigma_h$. Given $A=U\Sigma V^*$ a SVD (so that $A^*=V\Sigma^* U^*$ is a SVD of $A^*$), Theorem \ref{third main result S1} (applied to $A^*$ and $AX\in \K^{m\times r}$) shows that
$$
\Delta( W_q,t,k)_{2,F} \leq 4\, \frac{\| V_{k,\perp}^* W_{q_0}( V_k^* W_{q_0})^\dagger\|_{2,F}}{2^{(2t+1)\,\min\{\sqrt{\gamma_k}\coma 1\}}}\peso{for} q\geq q_0\py t\geq 1\,.
$$
Hence, the upper bound in Eq. \eqref{eq upper bounds x28} can be made arbitrarily small for sufficiently large $q,\,t\geq 0$. In this case, if we let $\{ v_1,\ldots, v_h\}\subset \cK_{q+t+1}$ be the principal vectors corresponding to the pair of subspaces $\cK_{q+t+1}$ and $\hat \cS$ (see Section \ref{sec aux angles}), then
we get that $\|\sin \Theta(\cT,\hat \cS)\|_{2,F}$ can be made arbitrarily small, where $\cT=\text{span}\{ v_1,\ldots, v_h\}\subset \cK_{q+t+1}$.
\EOE
\end{rem}
Consider the notation and hypothesis in Theorem \ref{teo exist right dom1}. Then Theorem \ref{teo exist right dom1} together with Remark \ref{rem la cota puede ser tan chica como queramos} above show the existence of
arbitrarily good approximations of a dominant subspace $\hat \cS$ of $A$ by {\it some} subspaces
$\cT\subseteq \cK_\ell(A,X)$, for sufficiently large $\ell\geq 0$. As opposed to Theorem \ref{first main result}, Theorem \ref{teo exist right dom1} does not require a priori knowledge of the relative position of the subspaces $R( V_k^* X)$ and $R(V_k^*\cV_j)$ (see the comments after Theorem \ref{first main result}). On the other hand, the speed at which the upper bound in
Eq. \eqref{eq upper bounds x28} decreases depends on the $\min\{\gamma_j\coma \gamma_k\}$; that is, this result warrants a better convergence speed when both singular gaps $\gamma_j$ and $\gamma_k$ are sufficiently large.
\subsection{Low rank approximations from block Krylov methods}\label{sec low rank approx}
As already noticed in Remark \ref{rem la cota puede ser tan chica como queramos},
the upper bound in Theorem \ref{teo exist right dom1} can be made arbitrarily small. Therefore the corresponding block Krylov subspace contains (arbitrarily good) approximate left dominant subspaces. Still, the previous results do not provide a practical method to compute such approximate dominant subspaces and the corresponding low rank approximations. In this section we revisit Algorithm \ref{algoalgo} without assuming a singular gap, as a practical way to construct such low rank approximations. Our approach to deal with this problem is based on approximate right dominant subspaces of a matrix $ A$; indeed, we follow arguments from \cite{WZZ15}.
For the next result, we consider Algorithm \ref{algoalgo} with input: $ A\in\K^{m\times n}$, starting guess $ X\in \K^{n\times r}$; moreover, we set our target rank to: $1\leq h\leq $ rank $( A)$.
Once the algorithm stops, we describe the output matrix in terms of its columns,
$\hat{ U}_h=(\hat{ u}_1,\ldots,\hat{ u}_h)\in\K^{m\times h}$. In this case we set
\begin{equation} \label{defi hatUi}
\hat{ U}_i=(\hat{ u}_1,\ldots,\hat{ u}_i)\in\K^{m\times i}\ , \peso{for} 1\leq i\leq h\,.
\end{equation}
As before, we let let $j=j(h)<h\leq k=k(h)$ be defined as in Eqs. \eqref{eq defi j} and \eqref{eq defi k},
and we consider the notation used so far; in particular, we consider the expressions defined in Eqs. \eqref{defi gam y del1} and \eqref{defi gam y del2}. Further, given $1\leq i\leq h$ we let $ A_i\in\K^{m\times n}$ denote a best rank-$i$ approximation of $ A$ (so that $\| A- A_i\|_2=\sigma_{i+1}$).
\begin{teo}\label{other main result}
Let $(A,X)$ be $h$-compatible, let $\cK_q=\cK_q( A, X)\subset \K^m$ and let $ Y_q$ be such that $ Y_q Y_q^*$ is the orthogonal projection onto $\cK_q$. Let $q,t\geq 0$ be such that
\begin{equation} \label{eq cond teo aprox}
4\, \Delta( X,q,j)_{2}\,\frac{\sigma_{j+1}}{\sigma_j}
+ \Delta^*( Y_q,t,k)_{2}\,\frac{\sigma_{k+1}}{\sigma_k}\leq \frac{1}{\sqrt 2}\,.
\end{equation}
Set the power parameter to $q+t+1$ in Algorithm \ref{algoalgo}. Then, for every $1\leq i\leq h$ we have that
\begin{equation}\label{eq conc teo aprox}
\| A-\hat{ U}_i\hat{ U}_i^* A\|_{2,F}\leq \| A- A_i\|_{2,F} + \delta_i
\end{equation}
where
$$ \delta_i:=\sqrt 2\, \| A- A_i\|_2\,
[\,4\, \Delta( X,q,j)_{F}\,\frac{\sigma_{j+1}}{\sigma_j}
+ \Delta^*( Y_q,t,k)_{F}\,\frac{\sigma_{k+1}}{\sigma_k}\,]
\,.
$$
In case $j=0$ (respectively $k=\text{rank}( A)$) the first term (respectively the second term) in the expression for $\delta_i$ should be omitted.
\end{teo}
\begin{proof} See Section \ref{sec4.4}.
\end{proof}
\begin{rem}\label{rem cotas chiquitas2}
Consider the notation in Theorem \ref{other main result}. We point out that Eq. \eqref{eq cond teo aprox} holds for sufficiently large $q,\,t\geq 0$.
On the other hand, the upper bound considered in Eq. \eqref{eq cond teo aprox} is rather arbitrary.
More generally, if we assume that the expression to the left in Eq. \eqref{eq cond teo aprox}
is bounded from above by $\sin(\theta_0)$ for some $0<\theta_0<\pi/2$, then we conclude that Eq. \eqref{eq conc teo aprox} holds
with $\delta_i$ replace by $\delta_i(\theta_0)=\frac{\delta_i}{\sqrt 2\, \cos(\theta_0)}$. This can be seen by inspection of the proof of this result (see Section \ref{sec4.4}). Notice that the statement of Theorem \ref{other main result} above corresponds to $\theta_0=\pi/4$.
\EOE
\end{rem}
Notice that Theorem \ref{other main result} can be compared with the second part of Theorem \ref{DIKM-I theorem 4} from the DIKM-I theory. As before, since our assumptions are based on non-structural choices $ V_h^*$, there is a price to pay: we need {\it an priori} control described in Eq. \eqref{eq cond teo aprox}. Nevertheless
this control is satisfied for sufficiently large $q,\,t\geq 0$, and our convergence analysis provides upper bounds that become arbitrarily close to $\|A-A_h\|_{2,F}$, i.e. the error in approximating $A$ by the (optimal) low rank matrix $A_h$ obtained from truncated SVD's of $A$.
(see Remark \ref{rem cotas chiquitas2} above).
\subsection{Comments and final remarks}\label{sec comentarios}
Our present work deals with two different (yet related) topics: dominant subspace approximation and low rank matrix approximation. On the one hand, there is a vast literature related to low rank approximation, both from a deterministic and randomized point of view, taking into account singular gaps, or disregarding this gaps.
\medskip
\noindent {\bf Algorithmic low rank approximations}.
We point out that our approach is deterministic in nature, and does not assume a singular gap.
Our convergence analysis decompose the problem of low rank approximation into two sub-problems (see Section \ref{sec proofs}): Assuming that $ A$, $ X$ and $j<h\leq k$ are as in Theorem \ref{other main result}
\begin{itemize}
\item We first apply Drineas, Ipsen, Kontopoulou and Magdon-Ismail theory from \cite{D19} to construct a block Krylov space $\cK_q=\cK_q( A, X)$ (for appropriate $q$), that now has a strong compatibility with the (uniquely determined) left dominant $j$-dimensional subspace $\cU_j$ of $ A$ (see the proof of Theorem \ref{second main result S1}).
\item Then we construct the auxiliary subspace $\cK_{q,t}^*=\cK_t( A^*, Y_q)$, where
$ Y_q Y_q^*$ is the orthogonal projection onto $\cK_q$, that has a strong compatibility with {\it some} $h$-dimensional right dominant subspace of $ A$.
\end{itemize}
The low rank approximation is now obtained from the best Frobenius approximation of $ A$ from $ A (\cK_{q,t}^*)\subseteq \cK_{q+t+1}$, as described in Algorithm \ref{algoalgo}. This factorization of the analysis is reflected in the upper bound obtained in Theorem \ref{other main result} for the convergence of the method. This new approach suggests possible numerical implementations including enlarging and re-starting techniques, by which we construct low rank approximations of dimension $h$ from approximate dominant subspaces of dimension $j$, for $j<h$. Such numerical implementations (including randomized initial matrices) would also have to deal with efficiency and stability; these matters are beyond the scope of our present work.
\medskip
\noindent {\bf Dominant subspace approximation by block Krylov subspaces}.
On the other hand, the results related to $h$-dimensional dominant subspace approximation without a singular gap at the index $h$ (in terms of initial compatible matrices $ X\in\K^{n\times r}$ with $r=h$) seem to be new; in this context,
we deal with the approximation of left and right dominant subspaces, adapting some of the main techniques of \cite{D19} to this setting.
In case there is no singular gap, then an interesting problem arises: namely, that there is no uniquely determined target subspace to approximate; in order to deal with his last fact, we are required to develop some geometric arguments related to subspace approximation. Our convergence analysis also decomposes this problem into two sub-problems. As before, this suggest possible numerical implementations that consider enlarging and re-starting techniques to build high quality approximations of dominant subspaces from high quality approximation of dominant subspaces of lower dimensions. We believe that this type of factorization of the analysis can also be of interest to deal with randomized methods. We plan to consider this analysis elsewhere.
\medskip
\noindent {\bf Further research directions}.
As a final remark, we point out that the present results together with the results from \cite{D19}
do not seem to cover the {\it complete picture} of the convergence analysis of deterministic block Krylov methods.
For example, assume that we are interested in computing an approximation of an $h$-dimensional dominant
subspace of the matrix $A$ in case there is a {\it small} singular gap $\sigma_{h}>\sigma_{h+1}$, for which
$\gamma_h\approx 0$ (so $\frac{\sigma_h}{\sigma_{h+1}}\approx 1$). This situation corresponds, for example, to the case where $\sigma_h$ lies in a cluster of singular values $\sigma_j\geq \ldots\geq \sigma_h\geq \ldots\geq \sigma_k$, for $j<h<k$ and $\frac{\sigma_j}{\sigma_{k}}\approx 1$. In this case the methods of our present work do not apply. On the other hand, Theorem \ref{DIKM-I theorem 1} from \cite{D19} does apply, but it shows a (possibly) rather slow speed of convergence with respect to $q\geq 0$. We will consider these and other related problems elsewhere.
\section{Proofs of the main results}\label{sec proofs}
In this section we present detailed proofs of our main results. Some of our arguments make use of some
basic facts from matrix analysis, that we develop in Section \ref{apendixity} (Appendix). We begin by recalling the notation introduced so far; then we consider some general facts about block Krylov spaces that are needed for developing the proofs below.
\medskip
\begin{nota}\label{nota21} \rm We keep the notation and assumptions introduced so far; hence, we consider:
\ben
\item $A\in\K^{m\times n}$ with singular values $\sigma_1\geq \ldots\sigma_p\geq 0$, with $p=\min\{m,n\}$.
\item $1\leq h\leq \text{rank}( A)\leq p$; moreover,
we let $0\leq j(h)< h$ be given by
$$j=j(h)=\max\{ 0\leq \ell < h\ : \ \sigma_\ell> \sigma_h\}<h$$ where we set $\sigma_0=\infty$ and
$$k=k(h)=\max\{ 1\leq \ell\leq \text{rank}( A) \, : \, \sigma_\ell=\sigma_h\}\geq h\,.$$
\item An starting guess $X\in\K^{n\times r}$ such that $(A,X)$ is $h$-compatible; that is, we assume that
there exists
an $h$-dimensional right dominant subspace $\cS$ of $ A$ such that $ \Theta(R( X),\cS)<\frac{\pi}{2} I$. In this case, $\dim X^*\cS=h$; in particular $r\geq \text{rank}(X)\geq h$.
\item The block Krylov space $\cK_q=\cK_q( A, X)$
constructed in terms of $ A$ and $ X$ as in Eq. \eqref{eq defi Krylov2}, that is
$$
\cK_q=\cK_q( A, X)=R(\ A X \quad ( A A^*) A X\quad \ldots\quad
( A A^*)^q A X\ )\subset \K^m\,.
$$ Moreover, we consider $Y_q$ an isometry (that is, a matrix with orthonormal columns) such that $Y_qY_q^*$ is the orthogonal projection onto $\cK_q$.
\item $A=U \Sigma V^*$ a SVD of $A$. Given $1\leq \ell\leq \text{rank}(A)$ we consider the partitions
$$ \Sigma=\begin{pmatrix} \Sigma_\ell & \\ & \Sigma_{\ell,\perp}\end{pmatrix}\ , \ \
U=\begin{pmatrix} U_\ell &
U_{\ell,\perp}\end{pmatrix}\ , \ \
V=\begin{pmatrix} V_\ell &
V_{\ell,\perp}\end{pmatrix}\,.
$$
\een
\end{nota}
Notice that the elements in $\cK_q$ can be described in terms of the elements of the range of matrices $\psi( A A^*) A X\in\K^{m\times r}$, where $\psi(x)\in \K[x]$ is a polynomial of degree at most $q$. In terms of SVD of $ A$, we get that
$$
\psi( A A^*) A X= U \psi( \Sigma^2 ) \Sigma V^* X= U \phi( \Sigma ) V^* X
$$ where $\phi(x)=x\,\psi(x^2)\in\K[x]$ is a polynomial of degree at most $2q+1$ with odd powers only, and represents a generalized matrix function (see \cite{ABF16,HBI73}). Here $ \Sigma =\text{diag}(\sigma_1,\ldots,\sigma_p)\in\R^{m\times n}$, where $p=\min\{m,n\}$; hence, $$\phi( \Sigma)=\text{diag}(\phi(\sigma_1),\ldots,\phi(\sigma_p))\in\K^{m\times n}\,.$$ In this case we note
\begin{equation}\label{eq defi super fi1}
\Phi:= U\phi( \Sigma) V^* X\in \K^{m\times r}\,,
\end{equation} so by the previous facts, $R( \Phi)\subset\cK_q$.
Let $\cS$ be an $h$-dimensional right dominant subspace $\cS$ of $ A$ such that $ \Theta(R( X),\cS)<\frac{\pi}{2} I$ as above. As already mentioned, we can consider a SVD of $ A=U\Sigma V^*$ in such a way that $\cS=\cV_h$. In this case, $R( V_h^* X)=R( V_h^*)$: if we assume further that $\phi(\sigma_\ell)\neq 0$ for $1\leq \ell\leq h$ (so $\phi(\Sigma_h)\in \K^{h\times h}$ is an invertible matrix) then $\dim R( \Phi)\geq h$, where $ \Phi$ is defined as in Eq. \eqref{eq defi super fi1}.
We will further consider similar facts related to convenient block decompositions of the SVD of $ A$.
\subsection{Proof of Theorem \ref{first main result} and Corollary \ref{cor con gap amp}}\label{sec prueba teo 2.1}
We begin this section with a proof of Theorem \ref{first main result}.
We present our arguments divided into steps.
\begin{proof} {\it Step 1: adapting the DIKM-I theory to the present context}.
Consider Notation \ref{nota21}.
By construction $\sigma_{j}> \sigma_{j+1}=\sigma_h= \sigma_{k}$. We first assume that $1\leq j$ and $k<\text{rank}( A)\leq p=\min\{m,n\}$. Since
$k<\text{rank}( A)$ then $\sigma_h= \sigma_{k}>\sigma_{k+1}>0$.
We consider $ X\in \K^{n\times r}$ such $\dim(\cX)=s\geq h$ and such that
there exists a right dominant subspace $\cS\subset\K^n$ of dimension $h$ with
$ \Theta(\cS,\cX)< \pi/2\, I$, where $\cX=R( X)\subset \K^n$ denotes the range of $ X$.
Consider
$ A= U \Sigma V^*$ a full SVD.
We now consider the partitioning corresponding to the index $1\leq k\leq \text{rank}(A)$,
\begin{equation}\label{eq decomp sigma y otros}
\Sigma=\begin{pmatrix} \Sigma_k & \\ & \Sigma_{k,\perp}\end{pmatrix}\ , \ \
U=\begin{pmatrix} U_k &
U_{k,\perp}\end{pmatrix}\ , \ \
V=\begin{pmatrix} V_k &
V_{k,\perp}\end{pmatrix}\,.
\end{equation}
It is worth to notice that $ \Sigma_k$, $R( U_k)=\cU_k$ and $R( V_k)=\cV_k$ do not depend on the particular choice of SVD of $ A$; also notice that the partition is also well defined since $k<\text{rank}( A)$.
Let $\phi(x)$ be a polynomial of degree $2q+1$ with odd powers only, such that $\phi(\sigma_1)\geq \ldots\geq \phi(\sigma_k)> 0$; hence $\phi( \Sigma_k)$ is invertible.
\smallskip
\noindent {\it Step 2: applying the DIKM-I theory to the adapted model}. We let $\cK_q=\cK_q( A, X)$ denote the block Krylov subspace and let $ P_q\in\K^{m\times m}$ denote the orthogonal projection onto $\cK_q$. Notice that if we let $ \Phi\in\K^{m\times r}$ be as in Eq. \eqref{eq defi super fi1}
then $R( \Phi)\subset \cK_q$.
Consider for now an arbitrary $h$-dimensional subspace $\cS'\subset\K^m$.
Then
\begin{equation} \label{eq teo 1 31}
\|\sin \Theta (\cK_q,\cS')\|_{2,F}=\|(I- P_q) P_{\cS'}\|_{2,F}\leq
\|(I- \Phi \Phi^\dagger ) P_{\cS'}\|_{2,F}
\,,\end{equation}
where we have used that $\dim\cK_q\geq \dim R( \Phi)\geq \dim{\cS'}=h$.
We now consider the decomposition $ \Phi= \Phi_k+ \Phi_{k,\perp}$, where
$$
\Phi_k\equiv U_k \phi( \Sigma_k) V_k^* X
\py
\Phi_{k,\perp}\equiv U_{k,\perp} \phi( \Sigma_{k,\perp}) V_{k,\perp}^* X\,.
$$
By \cite[Lemma 4.2]{D19} (see also \cite{Maher92}) we get that
$$\|(I- \Phi \Phi^\dagger ) P_{\cS'}\|_{2,F}\leq \| P_{\cS'}- \Phi B\|_{2,F}\peso{for} B\in \K^{r\times m}\,.$$
By the previous inequality we get that
\begin{equation}\label{eq comp con fik}
\|(I- \Phi \Phi^\dagger ) P_{\cS'}\|_{2,F}\leq
\|(I- \Phi \Phi_k^\dagger ) P_{\cS'}\|_{2,F}\,.
\end{equation}
We can further estimate
\begin{equation}\label{eq comp con fikx25}
\|(I- \Phi \Phi_k^\dagger ) P_{\cS'}\|_{2,F}\leq
\|(I- \Phi_k \Phi_k^\dagger ) P_{\cS'}\|_{2,F}+\| \Phi_{k,\perp} \Phi_k^\dagger P_{\cS'}\|_{2,F}\,.
\end{equation}
{\it Step 3: dealing with the fact that $R( V_k^* X)\neq R( V_k^*)$}.
We now consider the two terms to the right of Eq. \eqref{eq comp con fikx25}.
In our present case, we have to deal with the fact that $R( V_k ^* X)\neq R( V_k^*)$ when $h<k$.
Indeed, since $ \Theta(\cS, \cX)<\pi/2\, I$ and $\cS\subset R( V_k)$ we see that if we let
$$\cW\equiv R( V_k^* X)= V_k^*\cX\subset \K^k$$ then
$k\geq \dim(\cW):= t\geq h$.
Let $$\cT=\phi( \Sigma_k )\cW\subset \K^k\,.$$
Since, by hypothesis, $\phi( \Sigma_k)\in\R^{k\times k}$ is an invertible matrix then $\dim \cT=t$ and
\begin{equation}\label{eq ran fik}
\Phi_k \Phi_k^\dagger = U_k P_\cT U_k^*\,.
\end{equation}
We now consider $\cH'=\text{Span}\{ e_1,\ldots, e_j\}\subset \K^k$, where $j=j(h)\geq 1$ and $\{ e_1,\ldots, e_k\}$ denotes the canonical basis of $\K^k$; we also consider the principal angles
$$
\Theta(\cW,\cH')=\text{diag}(\theta_1(\cW,\cH'),\ldots,\theta_j(\cW,\cH'))\in\R^{j\times j}\,.
$$By Proposition \ref{pro app ang entre subs} we get that
$$
\Theta(\cW,\cH')\leq
\Theta(\cX,\cV_j) <\frac{\pi}{2}\, I\,,
$$
since $\cV_j\subset R( V_k)$ and $ V_k^* \cV_j=\cH'$, and the second inequality above follows from the fact that $\theta_i(\cX,\cV_j)\leq \theta_i(\cX,\cS)<\frac{\pi}{2}$, for $1\leq i\leq j$, since $\cV_j\subset \cS$ (see Section \ref{sec aux angles}).
\smallskip
\noindent {\it Step 4: computing the left dominant subspace $\cS'$}.
Let $\{ w_1,\ldots, w_j\}\subset\cW$ and $\{ f_1,\ldots, f_j\}\subset \cH'$ be the principal vectors associated to $\cW$ and $\cH'$ (as described in Section \ref{sec aux angles}).
Let $\cW'=\text{Span}\{ w_1,\ldots, w_j\}\subset \cW$; in this case, $ \Theta(\cW,\cH')= \Theta(\cW',\cH')$, by construction.
Consider the subspace $\cT'=\phi( \Sigma_k)\,\cW'\subset \cT$ so $\dim(\cT')=\dim(\cW')=j=\dim(\cH')$; since $\cH'$ is an invariant subspace of $\phi( \Sigma_k)$ then Proposition \ref{pro app 2} implies that
$$
\|\sin \Theta(\cT',\cH')\|_{2,F}\leq
\| \sin \Theta(\cW',\cH')\|_{2,F}=\| \sin \Theta(\cW,\cH')\|_{2,F}
$$
since
$\|\phi( \Sigma_k) ( I- P_{\cH'})\|_2\,\|\phi( \Sigma_k)^{-1}\|_2=1$, where we used that $\phi(\sigma_i)\geq \phi(\sigma_k)>0$, for $1\leq i\leq k$ and that $\phi(\sigma_{j+1})=\phi(\sigma_{k})$.
\smallskip
Let $\cT''=\cT\ominus\cT'$ so $\dim \cT''=t-j$ and $\cT''\subset (\cT')^\perp$.
Since $\dim((\cT')^\perp)= \dim((\cH')^\perp)$, by Eq. \eqref{ang inv por com ort2} we see that
$$
\| \Theta(\cT'',(\cH')^\perp)\|_{2,F}\leq
\| \Theta((\cT')^\perp,(\cH')^\perp)\|_{2,F}=\| \Theta(\cT',\cH')\|_{2,F}\leq
\| \Theta(\cW,\cH')\|_{2,F}\,.
$$
Let $\{ y_1,\ldots, y_{t-j}\}\subset \cT''$ and $\{ z_1,\ldots, z_{t-j}\}\subset (\cH')^\perp$ be the principal vectors associated to $\cT''$ and $(\cH')^\perp$.
Then, if we let $\cH''=\text{Span}\{ z_1,\ldots, z_{h-j}\}$ we have that $\dim\cH''=h-j$,
$$\|\sin \Theta (\cT'',\cH'')\|_{2,F}\leq \|\sin \Theta (\cT'',(\cH')^\perp)\|_{2,F}\leq \|\sin \Theta(\cW,\cH')\|_{2,F}\,. $$
On the one hand, we have that $\cT=\cT'\oplus \cT''$; on the other hand, we have that
$$
\cS':= U_k (\cH'\oplus \cH'' )=\cU_j\oplus U_k \cH'' \subseteq \cU_k \subseteq \K^m
$$is an $h$-dimensional left dominant subspace of $A$ (see Section \ref{sec dom subs}).
\medskip
\noindent{\it Step 5: obtaining some more upper bounds}.
Since
$$
\|\sin \Theta (\cT',\cH')\|_{2,F}\coma\| \sin \Theta (\cT'',\cH'')\|_{2,F}\leq\| \sin \Theta(\cW,\cH')\|_{2,F}
$$
then Proposition \ref{prop pegoteo de subespacios} implies that
$\|\sin \Theta( \cT\coma \cH'\oplus\cH'' )\|_{2,F} \leq 4\,
\| \sin \Theta(\cW,\cH') \|_{2,F}$.
Hence
\begin{equation}\label{eq teo1 41}
\|(I- \Phi_k \Phi_k^\dagger) P_{\cS'} \|_{2,F}=\|\sin \Theta( U_k\cT\coma \cS' )\|_{2,F} \leq 4\,
\| \sin \Theta(\cW,\cH') \|_{2,F}\,,
\end{equation}
since $ U_k$ is an isometry and $R( \Phi_k)= U_k\,\cT$ (see Eq. \eqref{eq ran fik}).
\smallskip
By Proposition \ref{pro app 1}, since $\cW=R( V_k^* X)$,
$$
\Phi_k^\dagger = ( V_k^* X) ^\dagger ( U_k \phi( \Sigma_k) P_\cW)^\dagger\,.$$
Since $ U_k\in \K^{m\times k}$ has trivial kernel, we get that
$$
( U_k \phi( \Sigma_k) P_\cW)^\dagger=(\phi( \Sigma_k) P_\cW)^\dagger ( U_k P_\cT)^\dagger=(\phi( \Sigma_k) P_\cW)^\dagger P_\cT U_k^*=(\phi( \Sigma_k) P_\cW)^\dagger U_k^*$$
where we have used Proposition \ref{pro app 1}, that
$( U_k P_\cT)^\dagger=( U_k P_\cT)^*= P_\cT U_k^*$ since
$ U_k P_\cT$ is a partial isometry and that $\ker((\phi( \Sigma_k) P_\cW)^\dagger)^\perp=\cT$. The previous facts show that
$$
\Phi_{k,\perp} \Phi_k^\dagger P_{\cS'} =
U_{k,\perp} \phi( \Sigma_{k,\perp}) V_{k,\perp}^* X
( V_k^* X) ^\dagger (\phi( \Sigma_k) P_\cW)^\dagger U_k^*
P_{\cS'}
$$ so then,
\begin{equation}\label{eq teo1 42}
\| \Phi_{k,\perp} \Phi_k^\dagger P_{\cS'} \|_{2,F}\leq \|\phi( \Sigma_{k,\perp})\|_2\,\|\phi( \Sigma_k)^{-1}\|_2\,\|
V_{k,\perp}^* X
( V_k^* X) ^\dagger \|_{2,F}\,.
\end{equation} The result now follows from the estimates in Eqs. \eqref{eq teo 1 31}, \eqref{eq comp con fik} and \eqref{eq comp con fikx25}
together with the bounds in Eqs. \eqref{eq teo1 41} and \eqref{eq teo1 42}.
The cases in which $j=0$ or $k=\text{rank}( A)$ can be dealt with similar arguments. Indeed, notice that if $j=0$ then we can take $\tilde \cT\subset \cT$ such that $\dim\tilde \cT=h$, and set $\cS'= U_k \tilde \cT$. By construction, $\cS'\subset R( \Phi_k)$ is a left dominant subspace of $ A$ (in this case any subspace of $\cU_k$ is a dominant subspace of $ A$). Finally, in case $k=\text{rank}( A)$ then $ \Sigma_{k,\perp}=0$ and then $\phi( \Sigma_{k,\perp})=0$,
so that we also get $ \Phi_{k,\perp}=0$.
\end{proof}
\medskip
Now we consider a brief proof of Corollary \ref{cor con gap amp}.
\begin{proof} By Lemma \ref{DIKM-I theorem 3}
we conclude that there exists a polynomial $\phi(x)$ satisfying the hypothesis of Theorem \ref{first main result} and such that
$$\|\phi( \Sigma_{k,\perp})\|_2\|\phi( \Sigma_{k})^{-1}\|_2 \| V_{k,\perp}^* X
( V_k^* X) ^\dagger\|_{2,F}\leq 4\, \frac{\| V_{k,\perp}^* X( V_k^* X)^\dagger\|_{2,F}}{2^{(2q+1)\,\min\{\sqrt{\gamma_k}\coma 1\}}}\,\frac{\sigma_{k+1}}{\sigma_k}\,.$$
The result now follows from the previous inequality and the definition of $\Delta( X,q,k)_{2,F}$.
\end{proof}
\begin{rem}
Some comments related to the previous proof are in order. We have followed the general lines of the proof of \cite[Theorem 2.1]{D19}. Nevertheless, the assumption in \cite{D19} (i.e., that $R( V_k^* X)= R( V_k^*)$) automatically implies that
$\|(I- \Phi_k \Phi_k^\dagger ) P_{\cS'}\|_{2,F}=0$ in Eq. \eqref{eq comp con fik}.
Since we are only assuming that the pair $( A, X)$ is $h$-compatible, our arguments need to include Steps 3, 4 and the first part of Step 5.
We can now see that the assumption that
the pair $( A, X)$ is $h$-compatible (for an arbitrary $1\leq h\leq \text{rank}( A)$) is weaker, at least from the point of view of our present approach, than the {\it structural}
assumption that the pair $( A, X)$ is $k$-compatible for an index $k$ such that $\sigma_k>\sigma_{k+1}$, as considered in \cite{D19}. \EOE
\end{rem}
\subsection{Proof of Theorem \ref{second main result S1}}\label{sec prueba del segundo}
\begin{proof}
Consider Notation \ref{nota21}; the cases $j=0$ or $k=\text{rank}( A)$ can be treated with similar arguments (the details are left to the reader). Notice that $\sigma_h=\sigma_k >0$, since $h\leq \text{rank}( A)$.
\medskip
\noindent \textit{Step 1: applying the DIKM-I theory using the singular gap $\sigma_{j}>\sigma_{j+1}$}. Let $ X\in \K^{n\times r}$ be such that $\dim(\cX)=s\geq h$ and $\Theta(\cS,\cX)< \frac{\pi}{2}\, I$,
where $\cX=R( X)\subset \K^n$ denotes the range of $ X$ and $\cS\subset \K^n$ is an $h$-dimensional right dominant subspace of $ A$.
Consider a full SVD $ A= U \Sigma V^*$ in such a way that $\cS=\cV_h$.
\smallskip
We can now consider decompositions as in Eq. \eqref{eq decomp sigma y otros}, using the index $j=j(h)$ that is,
\begin{equation}\label{eq decomp sigma y otros pero con j}
\Sigma=\begin{pmatrix} \Sigma_j & \\ & \Sigma_{j,\perp}\end{pmatrix}\ , \ \
U=\begin{pmatrix} U_j &
U_{j,\perp}\end{pmatrix}\ , \ \
V=\begin{pmatrix} V_j &
V_{j,\perp}\end{pmatrix}\,.
\end{equation}
As a consequence of our hypothesis, we get that $R( V_j^* X)=R( V_j^*)$ (notice that the subspace $R( V_j^*)$ is independent of our choice of SVD of $ A$, since $\sigma_j>\sigma_{j+1}$).
Hence, we can apply Theorem \ref{DIKM-I theorem 1} and Lemma \ref{DIKM-I theorem 3} (that correspond to the DIKM-I theory with singular gaps)
and conclude that if we let
$$
\gamma_j= \frac{\sigma_j-\sigma_{j+1}}{\sigma_{j+1}}\py \Delta( X,q,j)_{2,F} = 4\, \frac{\| V_{j,\perp}^* X( V_j^* X)^\dagger\|_{2,F}}{2^{(2q+1)\,\min\{\sqrt{\gamma_j}\coma 1\}}}
$$
then, we have that
\begin{equation} \label{eq primer est para j}
\|\sin \Theta(\cK_q,R( U_j))\|_{2,F}\leq \Delta( X,q,j)_{2,F}\,\frac{\sigma_{j+1}}{\sigma_j}
\,,
\end{equation} where $\cK_q=\cK_q( A, X)$ denotes the Krylov space of order $q\geq 0$.
\medskip
\noindent \textit{Step 2: applying Theorem \ref{first main result} to $ A^*$}.
Since $\sigma_h>0$ and $ \Theta(R( V_h),R( X))< \pi/2\, I$, we conclude that $R( V_h^* X)=R( V_h^*)$ and therefore
$\text{rank}( U_h^* A X)=\text{rank}( \Sigma_h V_h^* X)= \text{rank}( V_h^* X)=h$.
The previous facts show that
$$
\dim \cK_q\geq h \py \Theta(\cK_q,R( U_h))<\frac{\pi}{2}\, I\,.
$$
Let $ Y_q$ denote an isometry such that $ Y_q Y_q^*\in \K^{m\times m}$ is the orthogonal projection onto $\cK_q$. We now consider $\cK_{q,t}^*=\cK_t( A^*, Y_q)$ which is the block Krylov space of order $t$ constructed in terms of $ A^*$ and $ Y_q$.
Notice that $\cS^*=R( U_h)$ is an $h$-dimensional right dominant subspace of $ A^*$ such that
$ \Theta(R( Y_q),\cS^*)<\frac{\pi}{2}\, I$. Moreover,
the subspace $R( U_j)$ is a $j$-dimensional right dominant subspace of $ A^*$, such that $ \Theta(R( Y_q),R( U_j))= \Theta(\cK_q,R( U_j))$.
Hence, we can apply Theorem \ref{first main result} and Lemma \ref{DIKM-I theorem 3} to the matrices $ A^*$ and $ Y_q$
and conclude that if we let
$$
\gamma_k= \frac{\sigma_k-\sigma_{k+1}}{\sigma_{k+1}}>0\py \Delta^*( Y_q,t,k)_{2,F} = 4\, \frac{\| U_{k,\perp}^* Y_q( U_k^* Y_q)^\dagger\|_{2,F}}{2^{(2t+1)\,\min\{\sqrt{\gamma_k}\coma 1\}}}
$$
then, there exists an $h$-dimensional left dominant subspace $\tilde \cS$ of $ A^*$ such that
$$
\|\sin \Theta(\cK_{q,t}^*,\tilde \cS)\|_{2,F}\leq 4\, \Delta( X,q,j)_{2,F}\,\frac{\sigma_{j+1}}{\sigma_j}
+ \Delta^*( Y_q,t,k)_{2,F}\,\frac{\sigma_{k+1}}{\sigma_k}\,,
$$ where we have also applied Eq. \eqref{eq primer est para j}.
It is clear that $\tilde \cS$ is an $h$-dimensional right dominant subspace of $ A$.
\medskip
\noindent \textit{Step 3: computing $\cK_{q,t}^*$}.
We end the proof by noticing the following facts: recall that
\begin{equation}\label{eq sobre rang yq}
\cK_q= R( A X) + R(( A A^*) A X)+\ \dots +R( ( A A ^*)^q A X)=R( Y_q)\,.
\end{equation}
Similarly, notice that
$$
\cK_{q,t}^*= R( A^* Y_q) + R(( A^* A) A^* Y_q)+\ \dots +R( ( A^* A)^t A^* Y_q)\,.
$$
If we consider the identity in Eq. \eqref{eq sobre rang yq} and we let $0\leq \ell\leq t$ then
$$
R( ( A^* A)^\ell A^* Y_q)=
R(( A^* A)^{\ell+1} X) + R(( A^* A)^{\ell+2} X)+\ \dots +R( ( A^* A)^{\ell+q+1} X)\,.
$$The previous facts show that
$\cK_{q,t}^*=
R(( A^* A) X) + R(( A^* A)^{2} X)+\ \dots +R( ( A^* A)^{q+t+1} X)$.
\end{proof}
\subsection{Proofs of Theorem \ref{third main result S1}}\label{sec 4.3}
\begin{proof}[Proof of Theorem \ref{third main result S1}] Consider Notation \ref{nota21}.
We analyze the positive singular values of the matrix $ U_{k,\perp}^* Y_q( U_k^* Y_q)^\dagger$ as a function of $q\geq 0$. On the one hand, we have that
$$s( U_{k,\perp}^* Y_q( U_k^* Y_q)^\dagger)=
s(( U_{k,\perp}^* Y_q( U_k^* Y_q)^\dagger)^*)\,.$$
On the other hand, notice that
\begin{eqnarray*}
s^2(( U_{k,\perp}^* Y_q( U_k^* Y_q)^\dagger)^*)
&=&\la( U_{k,\perp}^* Y_q( Y_q^* U_k U_k^* Y_q)^\dagger
Y_q ^* U_{k,\perp})\\ &=&
\la(( Y_q^* U_k U_k^* Y_q)^\dagger
Y_q ^* U_{k,\perp} U_{k,\perp}^* Y_q)
\\ &=&
\la(( Y_q^* U_k U_k^* Y_q)^\dagger
( Y_q ^* Y_q - Y_q ^* U_k U_k^* Y_q))\,.
\end{eqnarray*}
where we have used that $ Z^\dagger( Z^\dagger)^*=( Z^* Z)^\dagger $ and that for matrices $ D,\, E$, the nonzero eigenvalues of $ D E$ coincide (counting multiplicities) with those of $ E D$. That is, there is some abuse of notation since the vectors above have different sizes in general, so
the identities are correct up to
some zero entries (which will not modify the norms $\| U_{k,\perp}^* Y_q( U_k^* Y_q)^\dagger\|_{2,F}$ in our argument).
Motivated by the last identity for singular values above, we now introduce the function
$$
f(x)=x^\dagger (1-x)\peso{for} x\in[0,1]\,,
$$ where $x^\dagger=x^{-1}$ for $x\neq 0$ and $0^\dagger=0$. Notice that $f(x)$
is decreasing in $(0,1]$. Hence, the previous facts show that
the positive singular values $ U_{k,\perp}^* Y_q( U_k^* Y_q)^\dagger$ are non-increasing as a function of the eigenvalues
$\la( Y_q^* U_k U_k^* Y_q)$, as long as the rank of $ Y_q^* U_k U_k^* Y_q$ is preserved.
Since $\la( Y_q^* U_k U_k^* Y_q)=\la( U_k^* Y_q Y_q^* U_k)$ (where the equality is up to some zero entries) we see that
the positive singular values $ U_{k,\perp}^* Y_q( U_k^* Y_q)^\dagger$ are non-increasing as a function of the projection $ Y_q Y_q^*$, where we consider the usual order (that is, range inclusion) between orthogonal projections, as long as the rank of $ Y_q^* U_k U_k^* Y_q$ is {\it preserved}. It is clear that $\cK_q\subseteq \cK_{q'}$ whenever $q\leq q'$ and therefore, $ Y_q Y_q^*\leq Y_{q'} Y_{q'}^*$; thus we are left to consider the problem of the stabilization of
$$
\text{rank}( Y_q^* U_k U_k^* Y_q)=
\text{rank}( U_k^* Y_q Y_q^* U_k)=\text{rank}( U_k^* Y_q)
$$
as a function of $q$. Since $R( Y_q)=\cK_q$ we consider a convenient description of the elements of this space: in this case, given $ u\in\cK_q$ there exists a polynomial $p(x)\in\K[x]$ of degree at most $q$ and $ v\in \K^r$ such that if
$\phi(x)=x \,p(x^2)\in\K[x]$ then
$$
u= U \phi( \Sigma) V^* X v\,,
$$ so then
$$
U_k^* u= \phi( \Sigma_k) V_k^* X v\,.
$$
Let $q_0=\#\{\sigma_1,\ldots,\sigma_h\}-1$, where $\#\{\sigma_1,\ldots,\sigma_h\}$ denotes the number of different singular values of $ A$ between $\sigma_1$ and
$\sigma_h$.
We can take a partition $\{C_1,\ldots,C_{q_0+1}\}$ of $\{1,\ldots,k\}$ in such a way that
$\ell,\,\tilde \ell\in C_t$ if and only if $\sigma_\ell=\sigma_{\tilde \ell}$.
We select a representative $g(t)\in C_t$, for $1\leq t\leq q_0+1$.
Consider the diagonal orthogonal projections $ P_1,\ldots, P_{q_0+1}\in\K^{k\times k}$
such that $ P_t$ projects onto the subspace $\text{Span}\{ e_\ell: \ell\in C_t\}$, where
$\{ e_1,\ldots, e_k\}$ denotes the canonical basis of $\K^k$. In this case,
$$
U_k^* u= \phi( \Sigma_k) V_k^* X v=\sum_{t=1}^{q_0+1} \phi(\sigma_{g(t)}) P_t V_k^* X v\subseteq \sum_{t=1}
^{q_0+1} R( P_t V_k^* X)=\cR\,.
$$
Hence, $R( U_k^* Y_q)\subset \cR$.
Assume that $q\geq {q_0}$; for $1\leq t\leq {q_0}+1$ let $p_t(x)\in\K[x]$ with degree at most $q$ such that $p_t(\sigma_{g(\ell)}^2)=\delta_{t,\ell}$, for $1\leq \ell\leq {q_0+1}$ (for instance, take the Lagrange polynomials). Let $\phi_t(x)=x\,p_t(x^2)$
and notice that
$$
U_k^*( U \phi_t( \Sigma) V^* X )=\sigma_{g(t)} P_t
V_k^* X \,.
$$The previous facts show that
$$
R( P_t V_k^* X)\subset R( U_k^* Y_q) \peso{for} 1\leq t\leq {q_0+1}\,,
$$ since $\sigma_1,\ldots,\sigma_k>0$ ($k\leq \text{rank}( A)$),
and then $R( U_k^* Y_q)= \cR$, for $q\geq {q_0}$.
As already explained, the fact that the range $R( U_k^* Y_q)$ stabilizes
for $q\geq {q_0}$ implies that the positive singular values $s( U_{k,\perp}^* Y_q( U_k^* Y_q)^\dagger)$ are (entry-wise) non-increasing functions of $q\geq q_0$. In particular,
$$
\| U_{k,\perp}^* Y_{q'}( U_k^* Y_{q'})^\dagger\|_{2,F}\leq \|
U_{k,\perp}^* Y_{q}( U_k^* Y_{q})^\dagger\|_{2,F}\peso{for} {q_0}\leq q\leq q'\,.
$$
\end{proof}
\subsection{Proof of Theorem \ref{other main result}}\label{sec4.4}
\begin{proof}[Proof of Theorem \ref{other main result}]
In what follows we consider Notation \ref{nota21} and the notation from Theorem \ref{other main result}.
In particular, we consider Algorithm \ref{algoalgo} with input: $ A\in\K^{m\times n}$, starting guess $ X\in \K^{n\times r}$; moreover, we set our target rank to: $1\leq h\leq $ rank $( A)$.
We let $U_{K}\in\K^{m\times d}$ denote the matrix whose columns form an orthonormal basis of the Krylov space $\cK_{q+t+1}$ constructed in terms of $ A$ and $ X$; further, we let $\hat{ U}_i$ denote the matrix whose columns are the top $i$ columns of the output of Algorithm \ref{algoalgo}.
\medskip
\noindent \textit{Step 1: applying Theorem \ref{second main result S1}}.
Let $\cK_{q,t}^*\subseteq \K^n$ be the subspace defined in Theorem \ref{second main result S1}, that is
$$
\cK_{q,t}^*=
R(( A^* A) X) + R(( A^* A)^{2} X)+\ \dots +R( ( A^* A)^{q+t+1} X)\,.
$$
By Theorem \ref{second main result S1}, there exists an $h$-dimensional right dominant subspace $\tilde \cS$ for $ A$ such that
\begin{equation} \label{eq aplic teo 1.3}
\|\sin \Theta(\cK_{q,t}^*,\tilde \cS)\|_{2,F}\leq 4\, \Delta( X,q,j)_{2,F}\,\frac{\sigma_{j+1}}{\sigma_j}
+ \Delta^*( Y_q,t,k)_{2,F}\,\frac{\sigma_{k+1}}{\sigma_k}\,.
\end{equation}
By the hypothesis in Eq. \eqref{eq cond teo aprox} and Eq. \eqref{eq aplic teo 1.3} we see that $\|\sin \Theta(\cK_{q,t}^*,\tilde \cS)\|_2\leq \frac{1}{\sqrt 2}$ and then,
$$
\Theta(\cK_{q,t}^*,\tilde \cS)\leq \frac{\pi}{4}\, I\,.
$$ On the other hand, notice that $ A(\cK_{q,t}^*)\subset \cK_{q+t+1}$.
\medskip
\noindent \textit{Step 2: applying the DIKM-I theory}. We now argue as in the proof of \cite[Theorem 2.3.]{D19}. Indeed, by \cite[Lemma 8]{BDMM14} we have that
$$
A-\hat{ U}_i\hat{ U}_i^* A= A- U_{K}( U_{K}^* A)_i\peso{for} 1\leq i\leq h\,,
$$ where $( U_{K}^* A)_i$ denotes a best rank-$i$ approximation of $ U_{K}^* A$. By the same result, we also get that $ U_{K}( U_{K}^* A)_i$ is the best rank $i$ approximation of $ A$ from $\cK_{q,t}^*$ in the Frobenius norm, i.e.
\begin{equation}\label{eqsec21}
\| A- U_{K}( U_{K}^* A)_i\|_F=\min_{\text{rank}( Y)\leq i}\|
A- U_{K} Y\|_F\,.
\end{equation}
We now consider a SVD, $ A= U \Sigma V^*$ such that
the top $h$ columns of $ V$ span the $h$-dimensional right dominant
subspace $R( V_h)=\cV_h=\tilde \cS$
(recall that this can always be done). We now set
$$
A= A_i+ A_{i,\perp}\peso{where} A_i= U_i \Sigma_i V_i^* \quad \text{and}\quad A_{i,\perp}= A- A_i\,.
$$ Then, by \cite[Lemma 7.2]{D19} we get that
\begin{equation}\label{eqsec22}
\| A-\hat{ U}_i\hat{ U}_i^* A\|_F^2\leq \| A- A_i\|_F^2+ \|
A_i-{ U_{K}} U_{K}^* A_i\|_F^2\,.
\end{equation}
\medskip
\noindent \textit{Step 3: bounding the second term in Eq. \eqref{eqsec22}}. Since
$ \Theta (R( V_h),\cK_{q,t}^*)\leq \frac{\pi}{4} \, I$ we have that
$ V_h^*(\cK_{q,t}^*)=R( V_h^*)$; then,
$$\text{rank}( V_i^* ( A^* A) X)=\text{rank}( \Sigma_i^2 V_i^* X)=\text{rank}( V_i^* X)=i$$
and we see that $ V_i^*(\cK_{q,t}^*)=R( V_i^*)$. Thus, we can apply Lemma \ref{lem C1} in this context. Hence,
we consider the principal vectors $\{ w_1,\ldots, w_i\}\subset \cK_{q,t}^*$ corresponding to the pair
$(\cK_{q,t}^*,R( V_i))$. Moreover, we let $ Q\in\K^{n\times i}$ be an isometry with
$R( Q)=\text{Span}\{ w_1,\ldots, w_i\}$ so that $R( A Q)\subset A(\cK_{q,t}^*)\subset \cK_{q+t+1}$.
The previous facts together with Lemma \ref{lem C1} show that
\begin{eqnarray*}
\| A_i-{ U_{K}} U_{K}^* A_i\|_F&\leq & \|( I- A Q ( A Q)^\dagger) A_i\|_F
=\| A_i - A Q ( A Q)^\dagger A_i\|_F
\\ &\leq &\| A- A_i\|_2\,\|\tan \Theta (\cK_{q,t}^*,R( V_i))\|_F \\ &\leq &
\| A- A_i\|_2\,\|\tan \Theta (\cK_{q,t}^*,R( V_h))\|_F\,.
\end{eqnarray*}
Since
$ \Theta (R( V_h),\cK_{q,t}^*)\leq \frac{\pi}{4}\, I$ then
\begin{eqnarray*}
\|\tan \Theta (\cK_{q,t}^*,R( V_h))\|_{F}&\leq &
\sqrt 2\, \|\sin \Theta (\cK_{q,t}^*,R( V_h))\|_{F}\\ &\leq &
\sqrt 2\,(4\, \Delta( X,q,j)_{F}\,\frac{\sigma_{j+1}}{\sigma_j}
+ \Delta^*( Y_q,t,k)_{F}\,\frac{\sigma_{k+1}}{\sigma_k})\,,
\end{eqnarray*}where we have used Eq. \eqref{eq aplic teo 1.3}.
Therefore, the previous inequalities imply that
\begin{equation} \label{eq prueba part1}
\| A-\hat{ U}_i\hat{ U}_i^* A\|_F\leq \| A- A_i\|_F+ \delta_i
\end{equation} where
$$ \delta_i=\sqrt 2\, \| A- A_i\|_2\,
[\,4\, \Delta( X,q,j)_{F}\,\frac{\sigma_{j+1}}{\sigma_j}
+ \Delta^*( Y_q,t,k)_{F}\,\frac{\sigma_{k+1}}{\sigma_k}\,]
$$
is as in Theorem \ref{other main result}.
This proves the upper bound in Eq. \eqref{eq conc teo aprox} for the Frobenius norm. In order to prove the bound for the spectral norm, recall that by \cite[Theorem 3.4.]{Gu15} we get that Eq. \eqref{eq prueba part1} implies that $$
\| A-\hat{ U}_i\hat{ U}_i^* A\|_2\leq \| A- A_i\|_2+ \delta_i\,,$$
since $\text{rank}(\hat{ U_i}\hat{ U}_i^* A)\leq i$.
\end{proof}
\section{Appendix}\label{apendixity}
In this section we include a number of technical results that are needed for the proofs of the main results.
Most of these results are elementary and can be found in the literature; we include the versions that are well suited for our exposition together with their proofs, for the convenience of the reader.
\begin{pro}\label{pro app ang entre subs}
Let $ V\in\K^{n\times k}$ be an isometry and let $\cV',\,\cX\subset \K^n$ be subspaces such that
$\dim\cX\geq \dim \cV'=j$, $\cV'\subset (\ker V^*)^\perp=R( V)$ and $ \Theta(\cX,\cV')< \pi/2\, I$. Then $\cW= V^*\cX\subset\K^k$ is such that $\dim\cW\geq j$ and if we let $\cH'= V^*\cV'$ then
$$
\Theta(\cW,\cH')\leq \Theta(\cX,\cV')\in\R^{j\times j}\,.
$$
\end{pro}
\begin{proof}
First notice that
$$
P_\cX P_{\cV'} P_\cX\leq
P_\cX V V^* P_\cX\,.
$$ By hypothesis rank$( P_\cX P_{\cV'} P_\cX)=j$ which shows that $\dim \cW=\text{rank}( V^* P_\cX)\geq j$. On the other hand, since $ V$ is an isometry then
$ \Theta(\cW,\cH')= \Theta( V \cW, V \cH')=
\Theta( V V^*\cX,{\cV'})$.
Consider $ D= V V^* P_\cX V V^* $; then
$R( D)= V V^*\cX$, so $\dim R( D)=\dim\cW\geq j$. Moreover,
$$0\leq D\leq P_{R( D)}\implies P_{\cV'}
P_{\cX} P_{\cV'}= P_{\cV'}
D P_{\cV'}\leq
P_{\cV'}
P_{R( D)} P_{\cV'}\,,$$where we used that $ P_{\cV'} V V^*= P_{\cV'}$.
Then, $ \cos^2 \Theta(\cX,{\cV'})\leq \cos^2 \Theta( V V^*\cX,{\cV'})\in\R^{j\times j}$ and the result follows from the fact that $f(x)=\cos^2(x)$ is a decreasing function on $[0,\pi/2]$.
\end{proof}
\begin{pro}\label{pro app 2}
Let $ B\in\K^{k\times k}$ be such that $ B= B^*$ and let $\cH',\,\cW'\subset \K^k$ be subspaces such that $ P_{\cH'} B = B P_{\cH'}$,
$\dim\cH'=\dim\cW'$
and $\cH',\,\cW'\subset \ker( B)^\perp$. If we let $ B \cW'=\cT'$,
$$
\|\sin \Theta(\cH',\cT')\|_{2,F}\leq
\| B( I- P_{\cH'})\|_2 \,\| B^\dagger\|_2 \|\sin \Theta(\cH',\cW')\|_{2,F}\,.$$
\end{pro}
\begin{proof}
Notice that $( B P_{\cH'})^\dagger = P_{\cH'} B^\dagger = B^\dagger P_{\cH'}$. Then,
$$( I- P_{\cH'}) P_{\cT'}=( I- P_{\cH'}) ( B P_{\cW'}) \, ( B P_{\cW'})^\dagger=
( B ( I- P_{\cH'}))\, ( I- P_{\cH'}) P_{\cW'} \, ( B P_{\cW'})^\dagger\,.
$$
Also, notice that $( B P_{\cW'})^\dagger = P_{\cW'} B^\dagger P_{\cT'}$; in particular, $\|( B P_{\cW'})^\dagger \|_2\leq \| B^\dagger\|_2$. Finally, since $\dim\cT'=\dim\cW'=\dim\cH'$ the previous facts imply that
$$
\|\sin \Theta(\cH',\cT')\|_{2,F}\leq
\| B( I- P_{\cH'})\|_2 \,\| B^\dagger\|_2
\|\sin \Theta(\cH',\cW')\|_{2,F}\,.$$
\end{proof}
\begin{pro}\label{prop pegoteo de subespacios}
Let $\cT',\,\cT''$ and $\cH',\,\cH''$ be pairs of mutually orthogonal subspaces in $\K^k$, such that $\dim(\cH')\leq\dim(\cT')$ and $\dim(\cH'')\leq \dim(\cT'')$.
Consider the subspaces in $\K^k$ given by the (orthogonal) sums
$\cT=\cT'\oplus \cT''$ and $\cH=\cH'\oplus \cH''$, so $\dim (\cH)\leq \dim(\cT)$. In this case we have that
$$
\|\sin \Theta(\cT,\cH)\|_{2,F}\leq 2 (\|\sin \Theta(\cT',\cH')\|_{2,F}+\|\sin \Theta(\cT'',\cH'')\|_{2,F})\,.
$$
\end{pro}
\begin{proof}
As usual, we compute the sines of the principal angles in terms of singular values of products of projections: in this case, using that $ P_\cH= P_{\cH'}+ P_{\cH''}$ and
$ P_{\cT}= P_{\cT'}+ P_{\cT''}$ we have that
\begin{eqnarray*}
\|\sin \Theta(\cT,\cH)\|_{2,F}&=&\|( I- P_\cT) P_\cH\|_{2,F}=
\|( P_\cH- P_\cT) P_\cH\|_{2,F}\\ &=&
\|( P_{\cH'}- P_{\cT'} + P_{\cH''}- P_{\cT''} ) ( P_{\cH'}+ P_{\cH''})\|_{2,F}
\\ &\leq & \|( I- P_{\cT'}) P_{\cH'}\|_{2,F} + \|( I- P_{\cT''} ) P_{\cH''}\|_{2,F} +\\ & &\| P_{\cT'} P_{\cH''}\|_{2,F}+\| P_{\cT''} P_{\cH'}\|_{2,F}\,.
\end{eqnarray*}
Now, notice that $ P_{\cT'} \leq I- P_{\cT''}$ so
$$ \| P_{\cT'} P_{\cH''}\|_{2,F}\leq
\| ( I- P_{\cT''}) P_{\cH''}\|_{2,F}=\|\sin \Theta(\cT'',\cH'')\|_{2,F}\,.$$ Similarly,
$ \| P_{\cT''} P_{\cH'}\|_{2,F}\leq
\| ( I- P_{\cT'}) P_{\cH'}\|_{2,F}=\|\sin \Theta(\cT',\cH')\|_{2,F}$.
\end{proof}
\begin{pro}\label{pro app 1}
Let $ B\in\K^{p\times q}$ and let $C\in\K^{q\times r}$ with
$R( C)=\cV\subset\K^q$ such that
$\cV\subset \ker B^\perp$. Then $$( B C)^\dagger=
C^\dagger ( B P_\cV)^\dagger\,.$$
\end{pro}
\begin{proof}
In this case $R( B C)= B\cV$ and $\ker B C=\ker C$. Moreover,
$$ B C C^\dagger ( B P_\cV)^\dagger=
B P_\cV( B P_\cV)^\dagger= P_{ B\cV}\py
$$
$$
C^\dagger ( B P_\cV)^\dagger B C =
C^\dagger ( B P_\cV)^\dagger B P_\cV C
= C^\dagger P_{\ker( B P_\cV)^\perp} C = P_{\ker C^\perp}\,,
$$ where we used that $\ker( B P_\cV)=\cV^\perp$, since $\cV\subset \ker B^\perp$.
\end{proof}
\smallskip
Let $ C\in\K^{m\times c}$ have rank $p$. For $1\leq i\leq p$ we define
$$\cP^{\xi}_{ C,\,i}( A)= C\cdot \text{argmin}_{\text{rank}( Y)\leq i} \| A- C Y\|_\xi \peso{for} \xi=2,F\,.$$
Due to the optimality properties of the projection $ C C^\dagger$ (see \cite{Gu15}) we get that
\begin{equation} \label{eq opt de cc+ vs pc}
\| A- C C^\dagger A\|_{\xi}\leq \| A-\cP^{\xi}_{ C,\,i}( A)\|_{\xi} \peso{for} \xi=2,F\,.
\end{equation}
The following result is \cite[Lemma C.5]{WZZ15} (see also \cite{BDMM14}).
\begin{lem}[\cite{WZZ15}]\label{lem C5}
Let $ A\in \K^{m\times n}$ and consider a decomposition $ A= A_1+ A_2$, with $\text{rank}( A_1)=i$. Let $ V_1\in \K^{n\times i}$ denote the top right singular vectors of $ A_1$. Let $ Z\in\K^{n\times p}$ such that
$\text{rank}( V_1^* Z)=i$ and let $ C= A Z$.
Then $\text{rank}( C)\geq i$ and
$$
\| A_1-\cP^{\xi}_{ C,\,i}( A_1)\|_{\xi}\leq \| A_2 Z( V_1^* Z)^\dagger\|_{\xi} \peso{for} \xi=2,F\,.
$$
\end{lem}
The following is a small variation of \cite[Lemma C.1]{WZZ15}
\begin{lem}\label{lem C1}
Let $ A\in \K^{m\times n}$ and consider the decomposition $ A= A_1+ A_2$, with $ A_1 A_2^*=0$ and $\text{rank}( A_1)=i$. Let $ V_1\in\K^{n\times i}$ and $ V_2\in\K^{n\times (n-i)}$ denote the top right singular vectors of $ A_1$ and $ A_2$ respectively.
Let $\cK^*\subset \K^n$ be a subspace such that $ V_1^*(\cK^*)=R( V_1^*)$.
Let $\{ x_1,\ldots, x_i\}\subset \cK^*$ denote the principal vectors corresponding to the pair $(\cK^*,R( V_1))$ and let $ Q\in\K^{n\times i}$ be an isometry with $R( Q)=\text{Span}(\{ x_1,\ldots, x_i\})\subset \cK^*$. Then,
$$
\| A_1-( A Q)( A Q)^\dagger A_1\|_{2,F}\leq
\| A- A_1\|_2\ \|\tan \Theta(\cK^*,R( V_1))\|_{2,F}\,.
$$
\end{lem}
\begin{proof}
Notice that by construction
$$
\Theta(R( Q),R( V_1))= \Theta(\cK^*,R( V_1))<\frac{\pi}{2}\, I\,.
$$Then, we get that $\text{rank}( A Q)=i$. Hence, we have that
\begin{eqnarray*}
\| A_1-( A Q)( A Q)^\dagger A_1\|_{2,F}&\leq &
\| A_1-\cP^{2,F}_{ A Q,\,i}( A_1)\|_{2,F}\leq
\| A_2 Q( V_1^* Q)^\dagger\|_{2,F}\\ &\leq &
\| A_2\|_2\, \| V_2^* Q( V_1^* Q)^\dagger\|_{2,F}\\ &=&
\| A_2\|_2\ \|\tan \Theta(R( Q),R( V_1))\|_{2,F}\\ &=&
\| A- A_1\|_2\ \|\tan \Theta(\cK^*,R( V_1))\|_{2,F}\,,
\end{eqnarray*}where we have used Eq. \eqref{eq opt de cc+ vs pc},
Lemma \ref{lem C5}, that the isometry $ V_2$ satisfies that $ A_2= A_2 V_2 V_2^*$ and the identity
$\| V_2^* Q( V_1^* Q)^\dagger\|_{2,F}=\|\tan \Theta(R( Q),R( V_1))\|_{2,F}
V_1))\|_{2,F}$, that holds by \cite[Lemma 4.3]{D19}, since $\text{rank}( V_1^* Q)=i$.
\end{proof}
\smallskip
\noindent {\bf Acknowledgment}. This research was partially supported by CONICET (PIP 2016 0525/ PIP 0152 CO), ANPCyT (2015 1505/ 2017 0883) and FCE-UNLP (11X829).
{\small
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'use strict';
let angular = require('angular');
module.exports = angular.module('spinnaker.core.templateOverride.registry', [
])
.provider('templateOverrideRegistry', function() {
const overrides = Object.create(null);
this.override = (key, val) => {
overrides[key] = val;
};
function getTemplate(key, defaultVal) {
return overrides[key] || defaultVal;
}
this.$get = function() {
return {
getTemplate: getTemplate,
};
};
}).name;
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\section{Introduction}
The classical picture of space-time locally described by flat
Minkowski space $R^{3,1}$ with numerical coordinates
$x_\mu =(x_0, \vec{x})$ is expected to break down at extremely short distances of the order of Planck length
$\lambda_{p} = \left( \frac{G\hbar}{c^3}\right)^{\half}
\simeq 10^{-33}$cm. The mechanism implying this modification are the quantum gravity effects causing the gravitational collapse in the measuring process of the distances below the Planck scale \cite{lukchin1,lukchin2}. It follows that operationally the space-time in the presence of quantized gravitational field becomes noncommutative. The quantum structure of space-time in the form of nonvariantly commutates coordinates of space-time events, as derived from Heinsenberg`s uncertainty principle and Einstein gravity equations, has been firstly obtained in \cite{lukchin2}.
At present there is usually considered the following class of noncommutative space-time coordinates $\widehat{x}_{\mu}$
\begin{equation}\label{luchi1}
[\widehat{x}_{\mu}, \widehat{x}_{\nu}]
= i \lambda^2_{p} \, \theta_{\mu\nu}
+ i \lambda_{p} \, \theta_{\mu\nu}^{\quad \rho}
\widehat{x}_\rho
\end{equation}
where $\theta_{\mu\nu}^{\quad \rho}$, $\theta_{\mu\nu}^{\quad \rho}$ are constant dimensionless tensors. The first term on rhs of Eq.~(\ref{luchi1}) describes the DFR canonical noncommutativity, and the second one is the Lie-algebraic deformation, with the most studied special case
described be so-called $\kappa$-deformation \cite{lukchin3,lukchin4,lukchin5} ($\kappa\equiv (a \lambda_{p})^{-1})$
\begin{equation}\label{luchi2}
[\widehat{x}_{i}, \widehat{x}_{0}]
= i \lambda_{p} \, \widehat{x}_\nu
\qquad
[\widehat{x}_{i}, \widehat{x}_{j}]
= 0\, .
\end{equation}
It should be added that the noncommutativity of space-time coordinates has been as well derived in the context of quantized open string theory \cite{lukchin6}.
In special relativity framework the space-time coordinates are described by the translation sector of the Poincar\'{e} group. If we introduce noncommutative space-time coordinates, such approach implies that Poincar\'{e} symmetries should be modified into quantum Poincar\'{e} symmetries, with noncommutative group parameters. The deformations
Eqs.~(\ref{luchi1}-\ref{luchi2})
introduce the fundamental mass $\kappa$ with its universe as deformation parameter. It is physically very appealing that quantum relativistic symmetries may introduce besides $c$ (light velocity) and $\hbar$ (Planck constant) the third fundamental constant interpreting the Newton constant $G$, or equivalently the Planck mass $m_p = \sqrt{\frac{\hbar c}{G}} \simeq 2.2\cdot
10^{-5}g \simeq 1.2 \cdot 10^{19} \frac{GeV}{c^2}$.
Quantum relativistic symmetries, with noncommutative space-time and symmetry generators satisfying deformed Lie-algebraic relations, are described by dually related pair of Hopf algebras \cite{lukchin7}. Analogously as in undeformed case, when we pass by exponential map from Lie-algebraic classical symmetry generators to the elements of symmetry group, in Hopf-algebraic framework the quantum Lie algebra describing quantum symmetry generators uniquely determines by duality the quantum symmetry group.
In this note we shall consider mostly the consequences of Hopf-algebraic $\kappa$-deformed Poincar\'{e} symmetries framework, introduced in 1991 \cite{lukchin3}. This algebraic framework was a conceptional basis of proposed in 2001 \cite{lukchin8,lukchin9} Doubly Special Relativity (DSR), with two invariant parameters $c$ and $\kappa$. The aim of DSR framework is to describe the modification of relativistic kinematics as well as to study the astrophysical effects which could reveal the need for modification of special relativity theory. In DSR considerations usually the studied are not restricted by the rigidity of Hopf-algebraic framework, in particular only the part of DSR authors assumed that space-time coordinates are not commutative (see e.g. \cite{lukchin10}).
Below, in Sect. 2., we shall list briefly the modifications of special relativity formulae. In Sect. 3 we shall comment on the deformation of field-theoretic framework, and relate via the formalism of star product the noncommutative and commutative fields. In Sect. 4 we shall list briefly the ideas which led to noncommutative gravity theory. It appears that the corrections to Einstein action is proportional to
$\frac{1}{\kappa^2}$, i.e. of second order in the deformation parameter.
\section{Modification of special relativity}
The essence of Einstein`s introduction of special theory of relativity is the modification of Galilean symmetries into Poincar\'{e} symmetries, described by the transformations of the Poincar\'{e} group ($a_\mu, \Lambda_\mu^{\ \nu}$)
\begin{equation}\label{luchi3}
x'_\mu = \Lambda_{\mu}^{\ \nu} \, x_\nu + a_\mu
\qquad
\Lambda_{\mu}^{\ \nu} \,
\Lambda_{\nu}^{\ \rho}
= \delta_{\mu}^{\ \rho} \, .
\end{equation}
The symmetries
(\ref{luchi3})
are generated by the Poincar\'{e} algebra ($P_\mu , M_{\mu\nu}$) with mass and spin
Casimirs ($m \in [ 0, \infty)$, $s=0,\half, 1 \ldots$)
\begin{equation}\label{luchi4}
P_\mu P^\mu = - m^2
\qquad
W_\mu W^\mu = m^2 \, s(s+1) \, .
\end{equation}
Poincar\'{e} algebra is a Hopf algebra with Abelian
(primitive) coproducts; from the last property follows e.g. that for relativistic systems we get Abelian addition law of fourmomenta
\begin{equation}\label{luchi5}
p^{(1+2)}_\mu = p^{(1)}_\mu + p^{ (2)}_\mu
\end{equation}
Relation (\ref{luchi5}) describes the fourmomentum of free two-particle system (1+2) composed as the tensor product of single particle states (1) and (2).
In $\kappa$-deformed special relativity the formulae (\ref{luchi3}-\ref{luchi5}) describing free relativistic particles are modified. If we replace $x_\mu$ by $\widehat{x}_\mu$ (see
(\ref{luchi1}), (\ref{luchi2})) the relations (\ref{luchi3}) remain valid, but with the $c$-number Poincar\'{e} group parameters ($a_\mu , \Lambda_\mu^{\ \nu}$) replaced by operator-valued quantum Poincar\'{e} generators $(\widehat{a}_\mu , \widehat{\Lambda}_\mu^{\ \nu})$ \cite{lukchin11,lukchin4,lukchin12}.
The $\kappa$-deformation of Poincar\'{e} algebra depends on the choice of the basic generators; usually it is used the basis with classical Lorentz algebra sector \cite{lukchin4,lukchin13}, with modified only one Poincar\'{e} algebra commutator \cite{lukchin13,lukchin14}.
\begin{equation}\label{luchi6}
[N_i , P_j ] = i \delta_{ij}
\left[\frac{\kappa}{2}(1 - e^{- \frac{2P_0}{\kappa}}
+ \frac{1}{2\kappa}\, \vec{P}^{\ 2}\right]
- \frac{i}{\kappa} \,P_i P_j \, .
\end{equation}
The modification (\ref{luchi6}) leads to the following $\kappa$-deformed mass Casimir
\begin{equation}\label{luchi7}
P_\mu P^\mu \to 2 \kappa \left(\sinh \frac{P_0}{2\kappa} \right)^2
- \vec{P}^{\ 2}
\end{equation}
and the nonAbelian three-momentum addition law:
\begin{equation}\label{luchi8}
P^{(1+2)}_i = P^{(1)}_i \,
e^{- \frac{P_0^{(2)}}{2\kappa}}
+
P^{(2)}_i \,
e^{- \frac{P_0^{(2)}}{2\kappa}}\, .
\end{equation}
Change (\ref{luchi7}) and (\ref{luchi8}) leads to important consequences e.g.
1) classical relativistic energy momentum dispersion relation $E_{el} = (\vec{p}^{\ 2} + m^2)^{\half}$ is modified
\begin{equation}\label{luchi9}
E = \kappa \, c\ \hbox{arcsinh}
\frac{(\vec{p}^{\ 2} +m^2)}{\kappa c}
= E_{cl} + {\cal O} (\frac{1}{\kappa^2})
\end{equation}
and $E < E_{max}$ where $E= E_{max} = \kappa \, c$ in
$|\vec{p}| \to \infty$ limit.
2) The notion of light-cone should be ceased, and the velocity of massless particles (e.g. photons) approaches $c$ only if $|\vec{p}|\to \infty$.
3) The Lorentz transformations in momentum space preserving the mass-shell conditions are modified
\cite{lukchin15,lukchin16,lukchin10}.
4) Important astrophysical effects, in particular the appearance of GKZ threshold and the evolution of the Universe in inflation period are changed {\cite{lukchin17,lukchin18,lukchin19}}.
In order to describe $\kappa$-deformed kinematics one has to extend further the $\kappa$-deformation to the phase space, what is provided by so-called Heisenberg double construction \cite{lukchin7,lukchin20}. In $\kappa$-deformed phase space the fourmomenta coordinates $p_\mu$ remain commutative, but the canonical PB implying standard Heisenberg algebra relation is changed. In such a way we obtain noncanonical symplectic structure which leads to the modification of classical and quantum Hamilton equations,
what implies the modified version of QM.
\section{From noncommutative to commutative framework: the $\star$-product formalism }
It is known from quantum mechanics that one can represent noncommutative quantum observables in terms of notions of commutative geometry (matrices, differential operators...). Analogously one can realize the non commutative fields on quantum Minkowski space by classical commutative Minkowski fields
\begin{equation}\label{luchi10}
\varphi(\widehat{x})
{\xrightarrow[\hbox{Weyl map}]{\phantom{sssslhhhhhhhhllllss}}}
\varphi({x}) = \Omega(\varphi(\skew{-1}\widehat{x}))
\end{equation}
provided that we represent correctly the multiplication of noncommutative fields by so-called $\star$-product (see e.g. \cite{lukchin21})
\begin{equation}\label{luchi11}
\varphi(\widehat{x}) \chi(\skew{-1}\widehat{x})
{\xrightarrow[\phantom{x} ]{\phantom{sssslhhhhhhhh}}}
\varphi({x}) \star \chi({x}) \, .
\end{equation}
The map (\ref{luchi11}) is a homomorphism, i.e. preserves all algebraic relations satisfied by fields $\varphi(\widehat{x})$. Such map
is nonlocal and highly non-unique, due to the ordering prescription which should be added in order to specify in complete way the field operator $\varphi(\widehat{x})$.
If the momentum sector of the phase space remains after deformation commutative, one can express the general Weyl map (\ref{luchi10}) in terms of the Weyl map of Fourier exponentials
\begin{equation}\label{luchi12}
\Omega(\varphi(\widehat{x})) =
\int d^4 p \, \widetilde{\varphi}(p)
\Omega(e^{ip\widehat{x}})
\end{equation}
and the $\star$-product (\ref{luchi11}) is determined by the general formula
\begin{equation}\label{luchi13}
e^{ipx} \star\, e^{iqx} = \widehat{O}(x, \frac{\partial}{\partial x},
y, \frac{\partial}{\partial x}) e^{ipx} \, e^{iqy}\Big|_{x=y} \, .
\end{equation}
The nonlocal operator $\widehat{O}$ depends on the type of deformation. For simplest canonical deformation (formula (\ref{luchi1}) with $\theta_{\mu\nu}^{\quad \rho} = 0$) it is represented by a Moyal star product $\star_{0}$
\begin{equation}\label{luchi14}
\widehat{O}_{(0)} = \frac{i}{2} \, \frac{\partial}{\partial x_\mu}
\theta_{\mu\nu} \, \frac{\partial}{\partial y_\nu}
\end{equation}
and leads to quite simple formula
\begin{equation}\label{luchi15}
e^{ipx} \star_{(0)} \, e^{iqx} =
e^{ipx} \, e^{iqx} \, e^{\frac{i}{2}\, p^\mu
\theta_{\mu\nu} \, q^\nu} \, .
\end{equation}
For the Lie-algebraic deformation (formula (\ref{luchi1}) with $\theta_{\mu\nu}=0$) due to BCH (Baker-Campbell-Hausdorff) operator formula
\begin{equation}\label{luchi16}
e^{i \alpha^\mu \, \widehat{x}_\mu} \,\cdot \,
e^{i \beta^\mu \, \widehat{x}_\mu}
= e^{i \gamma^\mu(\alpha, \beta) \, \widehat{x}_\mu}
\end{equation}
where $\gamma^\mu(\alpha, \beta)= \alpha^\mu + \beta^\mu
+ \theta_{\rho\tau}^{\quad \mu} \, \alpha^\rho \beta^\tau
+ O(\theta^2)$, we get
\begin{equation}\label{luchi17}
e^{ip^\mu x_\mu}
\star_{(1)} e^{iq^\mu x_\mu}
= e^{i \gamma^\mu(p,q) x_\mu}
\end{equation}
and
\begin{equation}\label{luchi18}
\widehat{O}_{(1)} = \exp i x_\mu \gamma^\mu (\frac{\partial}{\partial x},\frac{\partial}{\partial y})\, .
\end{equation}
The $\kappa$-algebra (\ref{luchi2}) is a soluble example of Lie-algebraic structure, and one obtains the closed formula the function
$\gamma^\mu (p,q)$ \cite{lukchin22,lukchin23}
\begin{equation}\label{luchi19}
\gamma^{0} = p^{0} + q^{0}
\qquad
\gamma^i= \frac{
f_\kappa(p^0) e^{\frac{q_0}{\kappa}}
p^i + f_\kappa(q_0) q^i}{f_\kappa(p^0 + k^0)}
\end{equation}
where $f_\kappa(x) = \frac{\kappa}{x}(1-e^{\frac{x}{\kappa}})$.
Using $\star$-product formalism one can represent the noncommutative field theory as the standard local field theory with nonlocal $\star$-multiplication rule. In the simplest case of canonical deformation (see (\ref{luchi14})--(\ref{luchi15})) the modification of standard theory is milder, because the mass Casimir (see (\ref{luchi4})) and the theory of free fields is not deformed. The Abelian addition law for the fourmomenta remain valid, and the nonlocal structure in interacting theory is introduced by the phase factor $\exp \half p \theta q$
(see (\ref{luchi15})) entering into the Feynmann diagram vertices \cite{lukchin24}. Every well-known model of classical field theory (e.g. gauge theories, QED, QCD, Einstein gravity) has been recently canonically deformed in the literature.
The Lie-algebraic deformation leads to more complicated modifications. For mostly studied $\kappa$-deformation (\ref{luchi2}) the substitution (\ref{luchi7}) leads to the modification of free field equations and correspondingly modified propagators in $\kappa$-deformed perturbation theory (see e.g. \cite{lukchin25}). The Abelian conservation law (\ref{luchi5}) of four-momenta are changed; with suitable choice of fourmomentum variables one gets the $\kappa$-deformation addition law (\ref{luchi8}). At present the interacting $\kappa$-deformed field theory which is covariant under the $\kappa$-deformed relativistic symmetries is still under construction.
\section{From noncommutative framework to modified gravity theories}
In the literature mostly the canonical deformation of gravity has been proposed \cite{lukchin26,lukchin27}. After introducing $\star$-multiplication (\ref{luchi14}--\ref{luchi15}) in order to describe the reparametrization-invariant deformed gravity one should consistently modify the differential calculus as well as the structure of diffeomorphisms.
One introduces
\\
i) $\star$-deformed differential calculus
\\
ii) $\star$-deformed composition rule of diffeomorphisms
$\delta_\xi = \xi^\mu \partial_\mu$
\begin{equation}\label{luchi20}
[ \delta_\xi , \delta_\eta ] = \delta_{\xi \star \eta}\, .
\end{equation}
It should be observed that the algebraic commutator of diffeomorphisms is not deformed, but the standard Leibniz rule is changed into the modified one. Using the notion of coproduct of diffeomorphisms, the classical primitive coproduct is modified into the well-defined noncocommutative one.
In gravitation theory if our basic fields are vierbeins $e^a _\mu$, the metric field is deformed as follows:
\begin{equation}\label{luchi21}
g_{\mu\nu} = e^a _\mu \, e_{\nu a} \to \widetilde{g}_{\mu\nu}
= \half e^a_{(\mu} \star e_{\nu b)} \, .
\end{equation}
Introducing the determinant $\det_{\star_{(0)}}(e^a_\mu)$ with standard multiplication of vierbeins replaced by $\star$-product the deformed Einstein action takes the form \cite{lukchin26}
\begin{equation}\label{luchi22}
S_{E} = \frac{1}{2\kappa^2}
\int d^4 x \, \det{}_\star (e^\mu_a) \star \widetilde{R}
\end{equation}
where using the $\star$-multiplication in the definition of scalar curvature in terms of basic vierbeins one can expand the modified scalar curvature $\widetilde{R}$ in terms of deformation parameters $\theta_{\mu\nu}$ as follows
\begin{equation}\label{luchi23}
\widetilde{R} = R + \theta^{\alpha\beta} R^{(1)}_{\alpha\beta}
+ \half \theta^{\alpha\beta} \theta^{\gamma\delta}
R^{(2)}_{\alpha\beta\gamma\delta} + \ldots \, .
\end{equation}
An important result has been shown \cite{lukchin28,lukchin29}
that only second order corrections to the scalar curvature $R$ are nonvanishing, i.e.
\begin{equation}\label{luchi24}
R^{(1)}_{\alpha\beta} = 0 \, .
\end{equation}
Using other technical tool of Seiberg-Witten transformation it has been shown \cite{lukchin30} that the relation (\ref{luchi24}) is valid as well for $\kappa$-deformation.
The modified second order Einstein action (\ref{luchi22}) is nonlocal and invariant under deformed diffeomorphisms parametrized in D=4, as in standard gravity, by four functions $\chi_\mu$.
It should be added that the modified Einstein action can be equivalently obtained if we use first order formulation \cite{lukchin31} and the geometric Mansouri-McDowell
approach~\cite{lukchin32}.
Other class of deformed gravity theories are obtained if we keep the standard Leibniz rule for diffeomorphisms. In such a case the theory is invariant only under the subclass of general coordinate transformations and describes the deformation of unimodular gravity~\cite{lukchin33}.
\subsection*{Acknowledgments} The author would like to thank the organizers of 1st Galileo-Xu Guangqi Meeting in Shanghai for their warm hospitality. The paper was supported by Polish Ministry of Science and Higher Education grant NN202318534.
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{"url":"http:\/\/rafbis.it\/sbmz\/how-to-increase-time-in-csgo-command.html","text":"# How To Increase Time In Csgo Command\n\n05 changes the time length of the sound buffer in milliseconds (0. Can you assign teams. The file is located in the same directory as the bsp and should be called: _cameras. But this increase had barely lessened the ills that afflicted British society, from underemployment to disease. Each time this distance is doubled, the sound system output can be increased by 6dB. Start the time trial when you feel ready and attempt to get to the end point in a quicker time than the listed par time. By seeing where you land and jump when using the CSGO Bhop cheat, you can emulate that yourself in order to become skilled at it. You might want to learn about some other useful BF4 console commands. FF7 Remake Hard Mode Guide: 12 Tips For Survival Completing FF7 Remake unlocks Hard Mode, a major challenge that changes lots of the game's rules--here's how to make it through. 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This is useful, for example, to handle session-ids (the name varies depending on the protocol. tpm file, click I have the owner password file, and then type the path to the file, or click Browse to navigate to the file location. It is the set amount of time a command takes to complete, mostly due to physics. Lowering these will result in an uglier game, but it will perform much, much better. var=$( (var + 1)) Arithmetic in bash uses$ ( ()) syntax. Some of the commands, show the bandwidth used by individual. 22, 2017, a Navy C-2A Greyhound flying under the callsign \u201cPassword 33\u201d enjoyed clear weather as it traveled from Marine Corps Air Station Iwakuni. Note: \u2013 Alternatively you can also press windows key + R together to bring RUN command box. From time to time you need start a local server and practice flashbangs, check advantageous positions and angles or discover new useful smoke grenades to improve yourself. Changing the Lock Screen Timeout when running on AC power. for(int i = 0; i < 100; i++) { \/\/ do something } or even this (though not entirely seriously). \/betabuff or \/beta. I\u2019m not listing the for loop\u2019s full options here as same can be found. Close your browser. idelta should only be used with gauges. Refer to these commands here. If the \u201cQuiet\u201d procedure is ineffective after 10 to 20 attempts, then allow your dog to bark 3 to 4 times, calmly say \u201cQuiet,\u201d and then immediately make a startling noise by shaking a set of keys or an empty soda can filled with pennies. 2 Profiler graph. Using a jumper wire, connect the common power strip to a GND pin on the Arduino. Note: \u2013 Alternatively you can also press windows key + R together to bring RUN command box. Join us on the Charlemagne server if you'd like to report a bug, request a command, or just tell us what you think of Charlemagne. com we offer most reliable and legitimate cs go rank boosting service. #include clock_t start, end; double cpu_time_used; start = clock. I have a thesis about code optimization along with my partner. Posted 08 February 2012 - 05:50 AM. Capture people\u2019s attention by providing a narrative; don\u2019t just settle for a generic line of text. Make sure the \u201cVisual Effects\u201d tab is selected. Make sure you leave the RivaTuner Statistics Server enabled. Free Webmaster Resources. View Profile View Posts. The time on earth is obviously Japan Time. ff_damage_reduction_bullets \" 0\" \/\/ How much to reduce damage done to teammates when shot. Numeric values less than zero result in every command being saved on the history list (there is no limit). Before You Begin. $ping 0 PING 0 (127. In Bedrock Edition, time add, time query and time set are three separated commands. Right click on CS:GO, then click Properties. Malcolm \"Mal\" Reynolds, Battle of Serenity Valley. Syntax DATE Example @echo off echo %DATE% Output. ) but also \"give\" commands for any weapon, unlimited ammo as well as many other fun commands. Note: make sure to check out our screenshot walkthrough of the Windows. This: \\begin{enumerate}[a. The key to maximize the life span of an SSD is to reduce writing to it. Just restart your PC and check the system status using slmgr\/xpr command. Full Screen Mode hides the window titlebar and menu bar on a Mac, and to see either of. In order to change the settings of a local server (or any other server, but you'll need an rcon password), the only thing you need is a console and a few commands used to modify the server side settings. As you see, we put the yolo-obj. Step 4 \u2013 Now, again Open run command box and write temp in it and hit enter. #include clock_t start, end; double cpu_time_used; start = clock. Hosting your own server gives you full control over your game and game modes, so you can play the exact flavor of CS:GO you want. When the step is not explicitly stated, 1 is used by default. 6 when you replace the files and open up Counter Strike 1. Open command prompt with administrative privilege. Windows provides a mechanism to control the initial retransmit time, and the retransmit time is then dynamically self. [CS:GO] Team size limit? General. Open the Video Settings menu in Counter-Strike. Remote access from your iPad, iPhone, Android or Kindle device. This tutorial will help you to get the current date and time in a shell script. Entering the following line of commands (copy and paste) into the console would give you an infinite round time and restart the game automatically: mp_roundtime_defuse 60;mp_roundtime_hostage 60;mp_roundtime 60;mp_restartgame 1 Copy If you're practicing and also want the ability to buy at any time,. Here is a CS:GO command list exactly for this:. And God said, \u201cLet there be light,\u201d and there was light. Counter-Strike: Global Offensive is a game created by Valve Corporation and released on August 21st, 2012 as a successor to previous games in the series dating back to 1999. I\u2019ve been working with people all over the world, almost entirely remotely (from Lagos, Nigeria), as a software developer, project manager, CTO & CEO for more than 6 years now. Gears Tactics on PC. In order to participate in these developer-selected channels, you will need to have your game account linked to your Twitch account. hey, sorry to say this but on Counter Strike 1. Write \"-console\" in it and click \"OK\" Now start your game normally, Your Dota 2 will open with a console in it. In the left panel, click \u201cAdvanced system settings\u201d In the Performance section, click the \u201cSettings\u201d button. Firstly, Enable the Console From Game settings. \" This will be the base invoice number. Learn how to change screen saver time on windows 7. There are various ways of stopping the Alien's Avatar Project like assaulting an enemy facility or. Connect the Arduino to your computer. GRUB_TIMEOUT is actually the number of seconds before the default entry is automatically booted. SPSS Lag - Creating a Counter. Capture people\u2019s attention by providing a narrative; don\u2019t just settle for a generic line of text. This is a list of console commands in Dota 2, based-on Dota 2 (7. I have the same question (171). Shutdown and restart the engine. Jul 29, 2015 @ 5:44pm. Copy the commands into it that u liked. Consider the following DATA step, which defines the dates and weights for four male patients who visited a clinic as part of a weight-loss program:. Explore our catalog of online degrees, certificates, Specializations, &; MOOCs in data science, computer science, business, health, and dozens of other topics. It would launch Wizard where you need to keep on pressing next, next and finally \u201cUpgrade\u201d. This question is one of those questions which is very simple and most of the users get it correct, however few users find it confusing for the first time. exe \/c Command Replace Command in the string with one of the command that you\u2019d like run. Press ~ to enter the console on the server machine and type sv_cheats 1 or sv_ 1. If the value is 0, **commands are not saved** in the history list. Free delivery on millions of items with Prime. We will show and explain you the best console commands for practicing CS:GO and also offer you a good ready-to-go practice config file at the end of this guide. Write \"-console\" in it and click \"OK\" Now start your game normally, Your Dota 2 will open with a console in it. The Most Useful Pro Tools Shortcuts Every good audio engineer or producer knows his digital audio workstation. In a transient ( Unsteady State ) case of fluent, total Number of Time Step is 50. I also leverage Jetpack for extra functionality and Local for local development. Move the loudspeaker farther away from the microphone. This parameter carries. Default: 6: mp_roundtime <1-9> How much time in minutes does a round last. Tech support scams are an industry-wide issue where scammers trick you into paying for unnecessary technical support services. What To Do If Your Laptop Freezes. 1, you can resize specific tiles in Windows 10 to make them larger or smaller. Its kinda hard to see exactly how much uses a tool or weapon has left just by the little bar. The syntax for commands put into this box is. set - Must be between 0 and 2,147,483,647 (inclusive, without the commas), day, night, noon, midnight, sunrise \u200c [ Bedrock Edition only], or sunset \u200c [ Bedrock Edition only]. In our test environment, we set the value ( Fig. Beta Server ONLY. It can display system summary information as well as a list of tasks currently being managed by the Linux kernel. This integer value controls the amount. Each rank in CS:GO indicates the player\u2019s level of skill. It derives the day of the week of date d, and then uses the mod function to add either 0, 1 (if current Date is Saturday) or 2 (if current date is Friday) days. With Keyboard Shortcuts. Delete this, save the. The backquote key shifts to tilde (~) on U. Name may be up to 20 characters long and can only contain A-Z, a-z, 0-9, space and hyphen. That's it! As soon as you click Apply, the new. so in this guide, you will learn how to stats easily. github wiki (with pics) download csgosl is a graphical user interface for the Counter Strike Global Offensive Dedicated Server (CSGO server from now on). Use in-ear monitoring systems in place of floor monitors. This will automatically execute the autoexec. You can no longer draw on the map since CS:GO was updated to panorama UI. 4 the expire might not be pin-point accurate, and it could be between zero to one seconds out. Command and Conquer: Rivals - Official Reveal Trailer Command & Conquer: Rivals is a thrilling, competitive experience built to define Real-Time Strategy for Mobile. To reconvert. 667 ( Fedora Core 3 ). How to use the IF function in Excel - YouTube. 1, you can resize specific tiles in Windows 10 to make them larger or smaller. All the experienced players know that. Praised as the best free webmaster resources online, by our users. The CLI is a straightforward command interface. In this radar settings guide for CS:GO, we will go through how to customize the radar for better visibility and overview. It is expected that this counter will always increment on a production ASA. Google Sheets makes your data pop with colorful charts and graphs. HISTSIZE The maximum number of commands to remember on the history list. You can do a lot with Skyrim console commands. Variable Wizard - Time Field 104 Available Data Types 105 Available Date Formats 106 Available Time Formats 106 Changing the Order of Entering Prompted Variables 107 Creating a Prompted Variable 107 Creating a Serial Number, Counter 108 Make a Counter that Preserves Last-used Value 108 Overriding Default Values 109 Global Variable 110. Enter the start, end, or specific time to allow. If the value is 0, **commands are not saved** in the history list. you can get a very long round time by setting mp_roundtime_defuse or mp_roundtime_hostage to 60 so a 1 hour round time. Anemia. We have described all necessary information and steps in this short guide. counter = 0 while counter = 100: print counter counter + 2 Nested Loops. Time Stopper only can stop the trial time when it is used. This is an alternative key combination to. Do this before the warmup ends and you have infinite. For the knockout price of 0. How to Use the Scoreboard Command in Minecraft. What was happening was that the output from the encoder operating at a higher speed was going from false to true and back to false during the time it took for the PLC to make one scan. If the value of Memory Available KB counter is lower than 100 MB for long time, it\u2019s a clear indication of insufficient memory on the server. After this my pybot command just hands in there, I have to hit the stop button on appium server manually in order pybot command to return to console normally. If your Mac hangs for good and the pointer is inactive: Hold down Control + Command while pressing the Power button. According to the Counter-Strike wikia the default server setting is mp_c4timer 45, i. Bash offers several ways to repeat code\u2014a process called looping. How you can do that, we have described on this guide. exe) to give more emphasis on hl. When it comes to first-person shooters on the PC platform, no game stands out more than Counter-Strike: Global Offensive. Level 1 : New Miner. When I don\u2019t add a counter there is an indent. Your friends can now use \u201cadmincheat. By the time you\u2019re done, you\u2019ll feel like you\u2019ve got a new car again. The counter section contains a list of counters that are available during the traffic. With Full Screen Mode, an app or window becomes a dedicated space that you will see through Mission Control. Setting unlimited time on the server config A Counter-Strike: Source (CS:S) Forum Thread in the Server Administration category, submitted by [SG] Paul. 1 percent of a bitcoin per month (roughly$90, at time of writing), WhaleTank\u2019s members are able to program their algorithm-enhanced \"trading bots\" to copy Brian\u2019s. Look at the corner of the screen you chose while playing a game and you'll see the FPS counter. Read More: Cheat Sheet to Docker Commands for Software Developers. In these situations, you can. Press the Windows Logo key and R key on your keyboard at the same time to open Run dialog. I was employed by a health care system as a supervisor of a psychiatric facility in central PA. Hosting your own server gives you full control over your game and game modes, so you can play the exact flavor of CS:GO you want. All the experienced players know that. Find the owner of the item you are looking for: Each item shows all the info. time: time [-p] PIPELINE Execute PIPELINE and print a summary of the real time, user CPU time, and system CPU time spent executing PIPELINE when it terminates. Apr 1, 2015 @ 2:54pm. \u2022 Click \"0\" button to reset counter to zero. So remember to replace the !awesomecommand and TEXT spots how you want them to show. CS:GO - Setting up your game for the first time (2019 Approved) - Duration:. Whenever you run this command, your listed game will be updated in real time. You will see updates in your activity feed. You can also see your FPS on the net graph, type \u201cnet_graph 1\u201d in the console to activate it. This current value will be used as an ID for some operation, so concurrent sessions must not get the same value. Cheats need to be enabled in order to use this command. It's not always a straightforward process, but it's a good idea to delete your browser history and internet cache on occasion. Click the Upload button. Jul 29, 2015 @ 5:44pm. To do this just right click the shortcut and select properties then select the change icon button hope this sorts out the problem for all of you. So for 6v6 you would do:-maxplayers_override 12. Bash offers several ways to repeat code\u2014a process called looping. To begin benchmarking, click OK at the bottom of the windows. 09\/29\/2017 12:32 am. Dota 2\u2019s console is an immensely powerful tool. answered Jan 5 '13 at 19:20. The syntax for commands put into this box is. Reply Delete. Create a cfg called: autoexec. You need to adjust this setting every time you enter the game. If you are just getting started, here are some commands that will give you a good idea of the capabilities of this tool. As expected, I found numerous instances of serial ports created by my USB to RS-232 adapter, but even when. For us to invest in quality reporting and continue bringing you the right stories, it takes a lot of time and money. The best options to lower your latency in CS is rate 25000, fps_max 101, cl_updaterate 101 and change your interp to ex_interp 0. Note: \u2013 Alternatively you can also press windows key + R together to bring RUN command box. 1 percent of a bitcoin per month (roughly \\$90, at time of writing), WhaleTank\u2019s members are able to program their algorithm-enhanced \"trading bots\" to copy Brian\u2019s. 5) of the game. 6 it is always enabled. By using the PowerShell command, you can easily retrieve folder size or files inside a folder or subfolder. Counter-Strike: Global Offensive > General Discussions > Topic Details. Creating Commands!addcom !awesomecommand TEXT \u2013 First one is the add command function, second one is what you name the command and use later. 64 bytes from. Enter the CD command to navigate to the folder you extracted the contents of the. So for 6v6 you would do:-maxplayers_override 12. edited Jul 21 '14 at 18:58. Where X is the max players you want in the game. Step 5 \u2013 Now, delete all files in the folder. Learning how to use it will allow you to discover all that your computer is capable of! Take-Away Skills: By the end of the course, you will be able to navigate, access. Ping Command - Interactive Mode. Follow simple rules while entering CS:GO Launch Settings: Start each command with - or + Separate commands by spaces; Don't use quotes - \" \"Using the launch options is a good alternative to entering CS:GO console commands. Do this before the warmup ends and you have infinite. Pay attention to the values of the CS GO commands. Because we want three images next to each other we set a width of 0. You can follow the question or vote as helpful, but you cannot reply to this thread. It can be used to implement the same algorithms for which bag or multiset data structures are commonly used in other languages. Apart from making an effort to reduce unnecessary copying of files, downloading of data, and so on, you should note that. Most HTML commands come in pairs. Precomputed values. Later in the code I increment the counter and write it to the file like this: In a related note, if you need to loop over. To create this article, 10 people, some anonymous, worked to edit and improve it over time. Note: \u2013 Alternatively you can also press windows key + R together to bring RUN command box. Counter-Strike: Global Offensive features new maps, characters, and weapons, as well as delivers updated versions of the classic CS content. Counter conditioning is achieved by engaging our dog in focus\/eye-contact training and other simple dog obedience commands (e. 2. Open your steam client and on Dota 2 right click and open \"Properties\". This is a tutorial on how to practice smokes in CSGO and how to practice alone in CSGO including all the commands. Resveratrol is an antioxidant that helps increase dopamine levels in the brain. ZPL Commands Basic ZPL Exercises and Examples 50 P1012728-008 Zebra Programming Guide 9\/20\/13 Font instruction ^ADN 1. cfg \" and you are ready to go! Lets just get to it, here is the config for anyone interested: \/\/ Config for server sv_cheats 1 sv_infinite_ammo 1 ammo_grenade_limit_total 5 mp_warmup_end mp_freezetime 0 mp_roundtime 60 mp_roundtime. Use mp_afterroundmoney 16000, which will give you, and everyone else, up to 16 000 money, which is the maximum, at the start of each round. To delete an alarm, click the \u201cSelect Alarms\u201d button in the bottom-right of the \u201cAlarms & Clocks\u201d window. We set the maximum amount of money you can have at once to 65535 with the mp_maxmoney command, then the amount of money you start the game with to 65535 with the mp_startmoney command. If there are other roles for the command value, we specify this in the list. zip) is about 36 MB and it includes the RivaTuner Statistics Server. Does you have entered wrong unlock code 10 time to modem?? Does you modem unlock code attempt counter is at 0 (zero)? Does your modem says modem unlocked permanently ?? then you might have to reset you modem unlock counter. Thus reducing your administrative overhead. Command & Conquer: Rivals has landed on mobile devices, giving us a a nice taste of tactical action wherever we go. FF7 Remake Hard Mode Guide: 12 Tips For Survival Completing FF7 Remake unlocks Hard Mode, a major challenge that changes lots of the game's rules--here's how to make it through. Enter the form name. How to use the IF function in Excel - YouTube. If you have a very CPU-intensive program or task, but you also understand that it might take a long time to complete, you can set it a high or favorable priority using the nice command. Your monitor should be in front of you at eye-level and about 17-inches from your. Location: Seattle,Washington. 156 bronze badges. The debug screen is triggered when the F3 key is pressed. How it works is illustrated by the screenshot. setgs fjumpheightmin 4. 0-26-generic #45~precise1-Ubuntu SMP Tue Jul 15 04:02:35 UTC 2014 x86_64 x86_64 x86_64 GNU\/Linux all of the above commands ran except doexec, gtar, httpd, manweb, rvi, slocate, switchdesk, traceroute3, unset gid, uid, untar. Click the plus button to add a new command. These commands disable the intro, enable the console, set the process affinity to high, and grant your cores to CS:GO, even though technically Source can only use 3 threads. Click the Local files tab, and click Verify integrity of game files. For every time the while loop runs, the value of the counter is increased by 2. h reference page contains more information about. [email protected]> show version. The default command prefix is ! (exclamation point). When it comes to monitor, input lag refers to the time (lapse) between the moment video signal comes into the monitor and the moment the image appears on the monitor. Our CS:GO boost service has been created by group of talented Counter Strike: Global Offensive players, who reached global elite rank at their main accounts and now we decided to go in for boosting. From the SQL Server Installation Center click Maintenance, and then select Edition Upgrade. Just remember most of these commands need sv_cheats enabled, so if you don\u2019t know this console command yet, please refer to my previous post ( CS:GO Console Commands ) to learn to active it. An extended traceroute command can be used to see what path packets take in order to get to a destination. First, to actually open up the console, press one of the following keyboard. On American English keyboards, the tilde key (~) will toggle the console screen, but on British English keyboards, that key is the backtick key (). Commands can be used in the console found in CS:GO. Answer: Use ping option -c to specify the number of packets. replace text with the message. For Counter-Strike: Source on the PC, a GameFAQs message board topic titled \"what's the console command to restart a round?\". Prices do not include shipping. Get people invested and show off how much value you can provide in one go. Below you can find the syntax of net account s command explained with examples. \"Of course sometimes people forget, but. > cs:go communities. \/datafile: You may read the on-line data documentation. Let us look at an example to illustrate how to set the value of the text input element through javascript. But it only works within an app: Pressing Win + G on your desktop or in a. Use the backquote key () while in-game (Unpaused). UPDATE: Please check out our updated sv_cheats command guide, which covers a lot more cheats and console commands! sv_cheats 1 commands Type the following commands in the console, when sv_cheats 1 is activated. Method 4: Using NovaBench. Long before Melikian became the namesake of Arizona State University\u2019s Melikian Center for Russian, Eurasian and East European Studies, he was a young radio operator stationed in Reims, France, during WWII \u2014 a war he ended via. And capital letters. Exchange PowerShell: How to check the number of items in the Inbox, Sent Items, Deleted Items and Junk Email 10 Replies In this post we will look at how to find out how many items are in a users Inbox, Sent Items, Deleted Items and Junk Email. A key part of any programming and scripting language is the ability to run the same piece of code again and again. GRUB_TIMEOUT is actually the number of seconds before the default entry is automatically booted. If you are not the host (or it's a dedicated server that you're an admin for), most commands need to be run remotely. Iterate over the list using while loop. Hey all, Currently running some minigame maps on my server which require all players to be on one team but it seems CSGO doesnt like to have more than half the total players on one team. But it only works within an app: Pressing Win + G on your desktop or in a. If this counter rises above your baseline, it may indicate the need for more hardware power; Processor Utilization. Im using the MapCFG plugin. If you have not linked your CS:GO account with a phone number, you should-doing so will improve your matchmaking experience. The viewport command allows you to change the size of the gameplay area on your screen, the area upon which the game world is drawn. The Command Console is a debugging tool primarily used for game development but players can use it to change game settins, access player statistics, ban or kick disruptive players if necessary and even control time. Hosting your own server gives you full control over your game and game modes, so you can play the exact flavor of CS:GO you want. Change your settings to look like these. Army National Guard Soldiers from New Jersey's 250th Brigade Support Battalion look on as a UH-60M Black Hawk helicopter carries a sling load on Joint Base McGuire-Dix-Lakehurst, N. CS GO round time command - CS GO USEFUL SV_CHEATS CONSOLE COMMANDS [english\/german] With this command you can extend the roundtime to do some testing in private games or on servers where you have. improve this answer. Controls: 11 Dimensions: 4. \" Counter-Strike: Global Offensive: In main menu, click the options tab and select \"Game. Time-to-live (TTL) is a value in an Internet Protocol ( IP ) packet that tells a network router whether or not the packet has been in the network too long and should be discarded. By the time you\u2019re done, you\u2019ll feel like you\u2019ve got a new car again. You can set screen saver on windows 7. Enter apropos into the Terminal window for a description of that command and its options. These easy-to-use commands function much like cheats and can make your time with the game easier if that's what you're looking for. To set it to 68, its maximum value, type the following command into the developer console: viewmodel_fov 68 Copy. begin(20x4). Up and down arrow keys: Enters commands from the command history, one at a time. Every time you start a server, just type \" exec nadepractice. cfg: sk_plr_dmg_pistol \u201c5\u201d You could either do this by typing that command in the console, like blink said or if you don\u2019t want to type it in with each map load, you can extract Half-Life 2\u2019s skill. Newsweek's \"Heroes of the Pandemic\" series features everyday heroes showing service, sacrifice or kindness in the time of COVID-19. You will add the command and append it to the new name you want to give Cortana. The command line arguments for this utility are not compatible with SQL Server 2000. Don't let your internet history fall into the wrong hands. Matt Hecht) An air defense artilleryman and a cavalry scout approaches the mid-point of a 12-mile. Here is a CS:GO command list exactly for this:. replace r which is red with g for green y for yellow etc. Your viewmodel's field of view in CS:GO can be set to a value between 54 and 68. Book your next adventure. \" ( )\" indicates a parameter is optional. Once you slap through the simple tutorial, you'll be thrown online and those guys are ruthless. Where X is the max players you want in the game. Before You Begin. Please do as follows to solve these problems. This procedure works best if you press Ctrl+Z (Command+Z on the Mac) to apply both Undo and Redo. This article will show you how to do so. Then press the tilde key to open the console. There's no unlimited time as far as I know. replace text with the message. We saw the addition of a new crosshair option, \"cl_crosshair_t\"\u2014a T-shaped crosshair. To change the amount by which the counter variable increases on each iteration, simply change the value of the step. The time until the bomb im Condition Zero explodes is a server configuration setting by the name mp_c4timer, followed by an integer giving the seconds. If the value of Memory Available KB counter is lower than 100 MB for long time, it\u2019s a clear indication of insufficient memory on the server. Here\u2019s how it\u2019s done. Now, if any calculations are done in MATLAB to change any of the variables used in the Simulink model, the simulation will use the new values the next time it is run. The Settings screen is simply a list of commands for Cortana. Don't let your internet history fall into the wrong hands. cfg: sk_plr_dmg_pistol \u201c5\u201d You could either do this by typing that command in the console, like blink said or if you don\u2019t want to type it in with each map load, you can extract Half-Life 2\u2019s skill. replace r which is red with g for green y for yellow etc. Please help. Click the Verify button (top left). The GTA 5 Online guide will answer the burning questions; how to level up fast in GTA 5 Online, What's the best way to rank up fast in GTA V Online. Everyone wants to make more money, but if you don\u2019t negotiate for a higher salary when you are hired, or for a raise while you\u2019re on the job, chances are you\u2019re leaving cash on the table. This means that every continuous two hours of powered on operation without an event which increases the counter will cause the counter to decrease by 1. hey, sorry to say this but on Counter Strike 1. The console is a debugging tool in the PC version of Fallout: New Vegas. To do this, enter the following into the developer console: sv_cheats 1 Copy. The `-p' option prints the timing summary in a slightly different format. At a time when the mainstream media leave out half of what the public needs to know, while at the same time purveying oceans of official nonsense, the public needs an alternative source of news. exe, or when I click on the shortcut on my desktop. First, we had the Windows Subsystem for Linux, which is awesome, and now we have a built-in OpenSSH. What was happening was that the output from the encoder operating at a higher speed was going from false to true and back to false during the time it took for the PLC to make one scan. Below example scripts to get date and time has been tested with Python 2. ), each of these sliders change the crosshair's looks in game. Restarts your server. cfg and obj. Kill background programs. More about the specifics of HTML commands is in the section called HTML Details. Consider the following DATA step, which defines the dates and weights for four male patients who visited a clinic as part of a weight-loss program:. Sends a message from the server admin to the screen which all players can read. 6 when you replace the files and open up Counter Strike 1. CONTINUE keyword skips all the statement after its. This command will allow you to change the listed game you are playing without having to go all the way to your channel settings. From time to time you need start a local server and practice flashbangs, check advantageous positions and angles or discover new useful smoke grenades to improve yourself. For example, lets. This counter includes all security related packet drops. ARK: Survival Evolved is a game notorious for its demands on your hardware and its poor optimization. After typing the above command into the console, you should have unlimited health and not take any damage. 4 the expire might not be pin-point accurate, and it could be between zero to one seconds out. mp_timelimit 0. We'll start by identifying the first record of each id by using an IF command as shown in the syntax below. The Counter-Strike: Global Offensive update in 2017 brought changes in to the interaction of a crosshair. From the pop-up menu, hover your. The only way to improve is to practice, and we allow you to learn very quickly, as you are your own instructor. This list is outdated. In the right panel, double-click the Set time limit for active but idle Remote Desktop Services sessions policy: in the modal window that will appear, activate it by switching the radio button from Not configured to Enabled, then set the desired amount of time in the drop-down list right below. View Started Threads. General Commands no shutdown - (enables the interface) reload - restarts the router sh ver - Cisco IOS version, uptime of router, how the router started, where system was loaded from, the interfaces the POST found, and the configuration register sh clock - shows date and time on router sh history - shows the history of your commands. A Counter-Strike: Source (CS:S) Forum Thread in the Help category, submitted by charli what command i need to restart map without lose [Counter-Strike: Source] [Forum Threads] [] Signup Login. I\u2019m currently writing a complicated Linux bash shell script where I need to keep a counter in an external file, and to do so, I need to be able to write to a file and then read from that file. Restarts your server. You can also use Keyboard shortcuts to do this. In Bedrock Edition, time add, time query and time set are three separated commands. Another 30. Counter-Strike: Global Offensive features new maps, characters, and weapons, as well as delivers updated versions of the classic CS content. 280 silver badges. Craig is a freelance UK web consultant who built his first page for IE2. Many free tools are available for this purpose, but they are difficult to use and do not. It is useful for times when you need your pet to pay close attention, such as during obedience training. That means you can play mode variants like scoutzknivez and aim maps with their unique settings embedded. ) Extend trial test period on Windows 10! The command slmgr -rearm is actually provided, even if the hardware has been not replaced on the PC to enable reactivation with a new product key (Key). It is important to note that the most money you can ever have in CS:GO is 65535 (regardless of what number you enter in the commands). How to Improve Your Skill in Counter Strike. 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View User Profile. Learn about the many strategies companies use to increase the market share of their. exe in the Open text box, and press [Enter. The ping command verifies IP-level connectivity to another TCP\/IP computer by sending Internet Control Message Protocol (ICMP) echo Request messages. Your friends now have to exit to the main menu. How to use Alexa: Guides, tips, tricks and how-tos for getting the most of the Amazon Echo, Echo Dot, Echo Show, Fire TV, and other Alexa-enabled smart speakers. This should be the command used to modify the time to change the map time. Constantly draw lines from NPCs to the actbusy nodes they've chosen to actbusy at. cfg each time you launch CSGO. begin(20x4). Or make the 18 whatever number you want it to be. If the input is ten, then 1 through 10 will be printed on the screen. macro for button to increase count I need to increase the count each time i click the button, i need a macro for this. 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Click \u201cOK\u201d and \u201cOK\u201d If you don't mind typing long words. width , height ) during run-time of a gameobject. Over many repetitions, gradually increase the time from 2 seconds to 5, then 10, then 20, and so on. The console is a debugging tool in the Windows version of Fallout 4. fadein \/\/ fadein {time r g b}: Fades the screen in from black or from the specified color over the given number of seconds. A few commands (such as revealing the map) are still done locally. A for loop repeats a certain section of the code. Command & Conquer: Rivals has landed on mobile devices, giving us a a nice taste of tactical action wherever we go. 09\/29\/2017 12:32 am. A short tutorial on how to increase your gamma levels in minecraft, Also I included Mac instructions in the description. If you do. Grab the carefully selected updates and tips right from the grape vine!. back to the top. Be grateful. If you want Watch-Command to run continuously you can add the -Continuous switch:. You don't have to download any. Use the following syntax to read a historical file: \/netscaler\/nsconmsg -K \/var\/nslog\/newnslog -d event. From booting into Fastboot mode with a single command to installing mods without root access, there's no shortage of reasons to use ADB.\n5frvj1wdtj5tq8,, lhrwrr3zz9h,, mb6hrsqeuk0m6l,, dml3cvoqgom,, pt92gly79la,, 6bpn0gi4uco5rv,, hxoqc18l4y,, h2fe508e30fcd,, fjkk3vg6ulc,, 6y4l7vcwiw,, ir8759gi48htqt,, gccp0ps4xxrhn1,, v8ru6po52i1,, 54i9ekn70nbdh,, 9srcy1wrn9ay,, 7cbtn8h1z4nn,, 0vt1tlxpllk,, lnw7v645vt430v,, atu3ei8wlcl3d,, 6hndl6sbff,, whnxt976t681ox,, qvehxwsofgg06z,, 91xrku1k2h,, v6dqrcauaidp,, fgqhp5m3gvzrv,, mi9gtec7s37shsl,, 0mmyc6zsxe54,, q2svgf465448l,, ck04ahskiz95878,","date":"2020-08-11 19:20:57","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.21604423224925995, \"perplexity\": 2763.2871255626887}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-34\/segments\/1596439738819.78\/warc\/CC-MAIN-20200811180239-20200811210239-00134.warc.gz\"}"}
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\section{Introduction}
It is now clear that most massive stars reside in binary or higher order multiple systems~\citep[e.g.][]{1998AJ....115..821M,2001A&A...368..122G,2014ApJS..215...15S,2017ApJS..230...15M}, with $\sim$70\,\% of close binary systems expected to
interact during their lifetimes~\citep{2012Sci...337..444S, 2014ApJS..213...34K}.
These interactions have profound effects on the evolution of the stars in such systems~\citep{2013ApJ...764..166D} and
the nature of their subsequent supernova explosions~\citep{1992ApJ...391..246P, 2017PASA...34....1D}, as well as on the formation of stellar mass double compact object (DCO) binaries~\citep{2017A&A...604A..55M}.
The first steps are already being taken to examine how multiplicity affects the evolution of stellar populations~\citep{2008MNRAS.384.1109E, 2020ApJ...888L..12W}. Such simulations are also beginning to produce estimates of the binary properties of their evolved products that include DCOs~\citep{2020A&A...638A..39L}.
Red supergiant (RSG) stars are an important piece of this puzzle.
The vast majority of isolated massive stars (above 8M$_\odot$) experience a RSG phase either directly before core collapse as a supernova~\citep{Smartt09} or as an intermediate phase~\citep{Groh13}.
Despite their importance, and recent observational advances (see below) there remains much work to be done to understand the multiplicity properties and, in particular, the properties of the companions of RSGs.
For the closest period massive star binary systems~\citep[those within an orbital period of less than 10\,d or an orbital separation of 0.15\,au;][]{2012Sci...337..444S,2017ApJS..230...15M}, binary evolution frequently results in interactions and stellar mergers~\citep[e.g.][]{2017ApJS..230...15M,2020A&A...638A..39L,2022A&A...659A..98S}.
The products of such mergers are observed as massive analogues of blue straggler stars in young stellar clusters~\citep[e.g.][]{2014ApJ...780..117S} and, as these stars evolve to the RSG phase, they can be observed as so-called red-straggler stars~\citep{b19}.
This is supported by the recent studies of RSGs in clusters in the Magellanic Clouds and Milky Way that suggest up to 50\% of RSGs may be the result of mergers in a previous evolutionary phase~\citep{2019MNRAS.486..266B, b19,2020A&A...635A..29P}.
In addition, red stragglers can, in principle, be observed within binary systems if the system was originally a hierarchical triple system.
Binary systems with intermediate orbital periods (10--1000\,d or $\sim$0.15--3\,au) typically interact in some form and also frequently result in stellar mergers~\citep{2020A&A...638A..39L}.
One expects Roche lobe overflow within binary systems to strip the donor's envelope of would-be RSG binary systems in favour of the production of Wolf-Rayet stars~\citep{2008MNRAS.384.1109E}.
Because of this, RSG binary systems where the RSG is the primary are expected to exist in orbital configurations where the two components are sufficiently separated that the stars evolve in effective isolation.
In systems where the RSG is the secondary\footnote{i.e. not the initially more massive component of the binary system.}, the primary will have most likely evolved to produce a compact object.
Supernova explosions within binary systems likely unbind the system~\citep{2019A&A...624A..66R} and produce massive runaway and, more commonly, walkaway stars~\citep{2019A&A...624A..66R}, which are observed at early evolutionary phases~\citep[e.g.][]{2018A&A...619A..78L} and less commonly in RSGs (e.g. $\alpha$ Orionis;~\citealt{2008AJ....135.1430H}).
The relatively few systems that remain bound can be observed as massive stars with compact object companions~\citep{2020ApJ...896...32G, 2020ApJ...904..143H,2021A&A...649A.167L}, which may produce DCO binary systems~\citep{2018MNRAS.481.1908K, 2020A&A...638A..39L}.
\begin{figure*}
\centering
\includegraphics[width=\linewidth, trim=2cm 6cm 2cm 0cm, clip]{RSG-UVIT-SMC-Fig1}
\caption{Mosaic of the UVIT SMC survey highlighting the locations of the matched RSG targets with red circles.
The 12 snapshots below the main figure are examples of counterparts to RSGs in the F172M filter, which is a clear signature of a hot companion, as single RSGs will be undetected at these wavelengths.
These snapshots are $30\times30$\arcsec\ and are ordered by the mass of the companion from the most massive on the left to the least massive on the right.
The gaps in the main mosaic represent pending observations.}
\label{fig:uvit-fov}
\end{figure*}
Determination of the binary fraction of RSGs, and the nature of the companions in RSG binary systems, can therefore address a number of important issues.
The presence of a companion indicates the fraction of main-sequence progenitors that have a companion in an orbital configuration that avoids a merger event, while an independent determination of that companion's temperature and luminosity constrains the age of the system, provided the companion is a main-sequence star, resulting in an independent confirmation of whether or not the RSG is a merger product.
Characterisation of the secondary can also help uncover more exotic companions such as stripped stars, while in the case of known radial velocity variables the absence of a detectable companion can perhaps lead to the inference of black hole and neutron star companions.
Previous work in this area has focused on two methods for detecting companions: long-term radial velocity variations and detecting the presence of a hot companion in the spectrum or spectral energy distribution (SED).
Examples of the former include \citet{burki} who found a binary fraction of $\sim$35\,\% among F--M supergiants in the Milky Way, while \citet{patrick2019, 2020A&A...635A..29P} determined an upper limit of $\sim$30\,\% for RSGs in clusters in the Large and Small Magellanic Clouds (LMC and SMC respectively).
\citet[][hereafter, DP21]{2021MNRAS.502.4890D} have identified 45 Magellanic Cloud RSG binary systems using multi-epoch radial velocity information, which offers a rare opportunity to characterise systems containing a RSG that may ultimately result in DCO binary systems.
\citet{2018AJ....155..207N} developed a method to detect RSG binary systems using optical colours, which has been applied to Local Group galaxies in subsequent studies~\citep{2019ApJ...875..124N,2020ApJ...900..118N,2021ApJ...908...87N}.
These authors identified candidate RSG binary systems using photometry and follow-up spectroscopy is used to confirm the binary nature of candidates by identifying signatures of hot stars in the RSG spectra.
Using a k-nearest neighbour algorithm, and combining optical and ultraviolet (UV) photometry \citet{2020ApJ...900..118N} determined the intrinsic binary fraction of the Large Magellanic Cloud, by accounting for observational biases, to be 19.5$^{+7.6}_{-6.7}$\,\%.
In the metal rich environments of M31 and M33,~\citet{2021ApJ...908...87N} apply a similar technique to that of ~\citet{2020ApJ...900..118N} and find an intrinsic binary fraction of up to 41.2$^{+12.0}_{-7.3}$\,\% and 33.5$^{+8.6}_{-5.0}$\,\% in M33 and M31, respectively.
\citet{2021ApJ...908...87N} conclude that there exists a metallicity dependence on the RSG binary fraction. In the metal-poor environment of the SMC we can directly test this hypothesis via a comparison with similar studies in more metal rich environments.
The UV domain offers an important advantage over optical studies in that the cool supergiant, despite its larger radius, is significantly fainter than a main-sequence star in the near- and far-UV (NUV and FUV, respectively).
In this spectral region, the flux from M- and K-type supergiants is dominated by line and continuum chromospheric emission \citep{carpenter1994,carpenter2014}, that is of course not accounted for by photospheric models.
However, their brightness in the FUV is still significantly fainter than that of main-sequence B-type stars.
For example, by adopting the FUV flux for $\alpha$\,Ori from the ASTRAL $HST$ spectral library \citep{astral}, we estimate $m_{F172M}=\sim$13.9\,ABmag in the Ultraviolet Imaging Telescope (UVIT) F172M filter.
As $\alpha$\,Ori has similar extinction to our sample, but has a distance modulus of 6.1, this implies an apparent magnitude of RSGs in the SMC of around $m_{F172M}=$~26--27\,ABmag, well below our detection limit and much fainter than main-sequence B-type stars (see Section~\ref{sec:obs}).
In this paper, we take advantage of a new UV survey of the SMC using the UVIT on board the satellite AstroSat~\citep{2006AdSpR..38.2989A,2012SPIE.8443E..1NK}.
The survey and the resultant data are presented in Section~\ref{sec:obs}, while the results and conclusions are discussed in Sections~\ref{sec:dis} and \ref{sec:conclusion}, respectively.
\section{Observations} \label{sec:obs}
\subsection{RSG source catalogue}
The initial source catalogue of RSGs is based on that of~\citet[][hereafter YB20]{2020A&A...639A.116Y}.
These authors constructed a RSG catalogue in the SMC principally based on five different photometric criteria from colour-magnitude diagrams (CMDs) at different wavelengths from the SMC point source catalogue of~\citet{2019A&A...629A..91Y}.
These criteria are anchored on the appearance of the RSG population within the MESA Isochrones and Stellar Tracks~\citep[MIST; ][]{2016ApJ...823..102C,2016ApJS..222....8D}.
YB20 developed a ranking system ranging from $-$1 to 5, depending on the criteria met by each source to qualify as a RSG candidate.
Ranks 4--5 flag sources with a low probability to be a RSG.
As an initial source catalogue, we select 1233 targets from YB20 with ranks between $-$1 and 3, which correspond to either targets that have a spectroscopic classification as a RSG (rank $-$1) or with at least two independent photometric classifications (ranks 0 to 3).
With this initial source catalogue, we cross-match all targets with the Gaia~EDR3 data release~\citep{2021A&A...649A...1G} using MAST Casjobs interface.
We discard 60 sources that have a combination of 3-$\sigma$ significant parallax measurement greater than zero and renormalised unit weight error (RUWE) less than 1.5.
These sources we assume to be foreground contaminants.
In addition, we use the following criteria to exclude candidates, based on the mean proper motion and dispersion of the SMC from the most recent Gaia EDR3 results~\citep{2021A&A...649A...7G}.
We discard all sources outside a 3-$\sigma$ box from the mean proper motion values centred on $\mu_{(\alpha, \delta)} = (1.7608\,\pm\,0.4472, 0.3038\,\pm\,0.6375)$.
This results in a sample of around 1000 high-probability SMC RSG candidates.
We further restrict this sample by applying a magnitude cut by requiring M$_J \leq -6$ that is roughly equivalent to $\log L/L_\odot>3.6$ or a stellar mass of $\gtrsim7\,M_\odot$, at the distance of the SMC.
Besides focusing the sample on stars that may undergo core collapse, this refinement removes sources that might be confused with lower mass asymptotic giant branch stars (AGBs), see, for example, \citet{2015A&A...578A...3G} for spectroscopic confirmation of this contamination.
Our final sample of SMC RSG candidates consists of 862 sources and is provided in electronic form as Table~\ref{tb:params_all}.
During the latter stages of this work,~\citet{2021ApJ...922..177M} published a source catalogue of 1745 SMC RSGs that is based on the appearance of RSGs in the $J-Ks$ vs. $Ks$ CMD.
The main difference between the YB20 and~\citet{2021ApJ...922..177M} catalogues is that YB20 select sources based on mid-IR data from Spitzer Enhanced Imaging Products whereas~\citet{2021ApJ...922..177M} select sources directly from the Two Micron All Sky Survey (2MASS).
The criteria used to define the sample of~\citet{2021ApJ...922..177M} are similar to the criteria of YB20 used in the $J-Ks$ vs. $Ks$ CMD, however, the~\citet{2021ApJ...922..177M} sample go fainter in $Ks$-band magnitude and the range of $J-Ks$ colours considered is typically narrower than in YB20.
In addition, at almost the same time~\citet{2021ApJ...923..232R}, published an even-more extensive list of RSGs in the SMC, with their sample containing no fewer than 2138 RSGs.
These authors base their selection on the $J-H$ vs. $H-K$.
We follow the recommendation of YB20 and use only RSGs that are classified as such in multiple wavelength regimes, therefore we choose to retain the sample based on the YB20 classifications.
We note that 75\,\% (651) of our sample are found within the \citet{2021ApJ...922..177M} catalogue and 78\,\% (667) are found within the~\citet{2021ApJ...923..232R} catalogue.
\subsection{Cross-matching the RSG and UVIT catalogues} \label{sub:XM}
We matched the RSG sample with photometry from the UVIT survey of Thilker et al. (in prep.).
The SMC was surveyed with UVIT using the FUV F172M filter, which has a pivot wavelength of approximately 1707\,\AA \footnote{\url{https://uvit.iiap.res.in/Instrument/Filters}}.
Overlapping 28\arcmin\ fields were used to observe to a 5-$\sigma$ depth of $m_{F172M}\sim$20.3\,ABmag, with point spread function full-width half-maximum of approximately 1\arcsec\ and an astrometric accuracy of approximately 0.1\arcsec. A section of the survey is illustrated in Figure\,\ref{fig:uvit-fov}.
The UVIT SMC survey was completed at the $\sim$75\% level.
From the sample of 862 SMC RSG candidates, 560 lie within the footprint of the UVIT survey.
These sources are cross-matched to the UVIT source catalogue, recording all UV matches out to a cross-match distance (XMD) of 10\arcsec.
Figure~\ref{fig:XMD} illustrates the histogram of minimum XMDs of the resulting catalogue.
From this figure, we identify a peak at a XMD consistent with zero, with a tail of this distribution extending to around $\sim$0.4\arcsec, which we interpret as genuine matches.
For XMD > 1.0\arcsec, the number of matches rises steadily and we interpret these as spurious matches.
We find a unique match within 0.4\arcsec\ for 88 sources.
Matched UVIT sources in regions where fields overlap have more than one measurement; in these cases we choose the measurement which is the closest match to the RSG position.
The multiple measurements are consistent within their estimated uncertainty, in terms of both magnitude and position.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{SMC-RSG-UVIT-XMD-spurious}
\includegraphics[width=\columnwidth]{SMC-RSG-UVIT-XMD-spurious-zoom-out}
\caption{The separation in arcseconds between cross-matched sources from the UVIT source catalogue and the RSG source catalogue (black solid line).
The cross-match separation of the control sample is shown with the red dashed histogram.
See Section~\ref{sub:XM} for a detailed comparison of these two distributions.
}
\label{fig:XMD}
\end{figure}
To quantitatively assess the impact of spurious alignments that produce false-positive results and assess the validity of our choice of maximum XMD, we repeat the cross-match process on a control catalogue.
Adapting the method of~\citet{2020ApJS..250...36B}, the control sample consists of the full RSG source catalogue offset by 20\arcsec\ in declination.
An offset of 20\arcsec\ is chosen to mimic, as closely as possible, the density and distribution of the underlying RSG source catalogue.
The results are illustrated in Figure~\ref{fig:XMD} as the red dashed histograms.
These results clearly demonstrate that false positives through chance alignments have a negligible contribution to the distribution of genuine matches (i.e. the black solid histogram at 0.1\arcsec) and that chance alignments contribute close to 100\% of the matches outside 0.4\arcsec.
The bottom panel of Figure~\ref{fig:XMD} demonstrates that, at large XMD, the two distributions are effectively identical.
This strengthens the assumption that for a XMD larger than 0.5\arcsec\ the UVIT matches to the input catalogue can be assumed to be positional coincidences, as their distribution matches that of the underlying population and their number increases geometrically with the area as the XMD increases.
The catalogue of genuine matches is provided in electronic form as Table~\ref{tb:params_bs}, together with their derived stellar parameters (see Section~\ref{sub:parameters}).
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{SMC-RSG-KvsFUV}
\caption{UVIT F172M-band magnitudes shown against $Ks$-band magnitudes for the candidate RSG binary systems.
The red dashed line shows the UVIT magnitude at 10k\,K within stellar evolutionary models (see Section~\ref{sub:parameters} and Figure~\ref{fig:HRD-binaries}), assuming an age appropriate of a RSG with the corresponding J-band magnitude.
Companions with UVIT magnitudes above the red dashed line are inconsistent with being located on the main sequence given the age of the RSG.
The red dashed line is calculated using a comparison with stellar models, with the bolometric corrections from the MIST stellar tracks~\citep{2016ApJ...823..102C,2016ApJS..222....8D}.
The solid black, almost vertical, line highlights the reddening vector assuming $A_{\rm V}=0.35$\,mag and the reddening law of~\citet{2003ApJ...594..279G}.
}
\label{fig:fuvvsj}
\end{figure}
\section{Results and discussion} \label{sec:dis}
Based on the arguments presented in the previous section, we assume the detected UVIT counterparts represent the hot companions of RSG binary systems.
Therefore, it is straightforward to determine that 15.7\,$\pm$\,1.5\,\% of our sample of RSGs in the SMC have a hot companion with a mass greater than $\sim$3.5\,$M_\odot$, assuming a UVIT F172M completeness limit of $m_{F172M}=20.3$\,ABmag and interstellar extinction of $E(B-V) = 0.13$ (see Section~\ref{sub:ext}).
Figure~\ref{fig:fuvvsj} compares the F172M-band and $Ks$-band magnitudes of the binary candidates.
We interpret this figure as a comparison of companion mass to RSG mass, as the F172M-band and $Ks$-band magnitudes can be used as proxies for the companion and RSG masses, respectively, as we show in the subsequent subsections.
Figure~\ref{fig:bfvsJ} demonstrates that the observed binary fraction depends on the $Ks$-band magnitude of the RSG, ranging from $\sim$0.2 for the brighter (higher mass) stars to $\sim$0.1 for the fainter (lower mass) stars.
The detection limit imposed by the UVIT photometric completeness limit ($m_{F172M}=20.3$\,ABmag or $\sim$3.5\,M$_\odot$ for a zero-age main-sequence star) results in a mass-dependent mass ratio ($q$) observing bias, such that, for the faintest RSGs in the sample, mass ratios can be detected in the range $q > 0.6$ and in the range $q > 0.3$ for the brightest RSGs in the sample.
To quantify this, we determine stellar parameters (i.e. effective temperatures, luminosities and masses) for both components in Section~\ref{sub:parameters}, study the mass-ratio distribution in Section~\ref{sub:qs} and simulate the observing biases to determine the intrinsic multiplicity fraction in Section~\ref{sub:bf}.
Figure~\ref{fig:HRD-binaries} displays the RSG binary systems on the Hertzsprung--Russell Diagram (HRD), which allows a better visualisation of the key results of this study.
\begin{figure}
\centering
\includegraphics[width=\hsize]{SMC-RSG-Kmag-bf}
\caption{The observed percentage of the number of UV detections as a function of RSG magnitude. We interpret this as the binary fraction of RSGs.
This figure illustrates the decreasing trend for fainter, and hence less massive, RSGs.
The \textquoteleft uncertainties\textquoteright~in $Ks$-band magnitude represent the bin width.}
\label{fig:bfvsJ}
\end{figure}
\begin{figure*}
\centering
\includegraphics[width=0.8\linewidth]{hrd_20220216_AV035}
\caption{HRD showing both components of the RSG binary systems detected with UVIT photometry.
Stellar evolutionary tracks are based on~\citet{2019A&A...625A.132S}.
Solid red circles show the RSG component of the system and violet crosses show the hot companions.
Solid grey lines show lines of constant UVIT magnitude in the stellar models.
Rings of different colours highlight the six systems with a mass ratio greater than 1.0.
Thick colored lines indicate core H-burning, thinner solid lines indicate core He-burning and dashed lines indicate the short phase in between.
}
\label{fig:HRD-binaries}
\end{figure*}
\subsection{Extinction and reddening} \label{sub:ext}
To accurately determine stellar parameters of both components within the binary systems we must provide a consistent treatment of extinction and reddening values.
To do this we follow~\citet{2021A&A...646A.106S} and assume a constant $A_{\rm V}=0.35$\,mag, where we use the SMC bar reddening law of~\citep{2003ApJ...594..279G} to determine extinction parameters in the FUV, for the hot companions, and near-IR, for the RSGs.
The scale of the intrinsic variations of extinction values within the SMC is typically small~\citep{1995ApJ...438..188M,1997A&A...317..871L,2002AJ....123..855Z}, but with a potentially important tail to higher extinction values.
The origin of such a tail is additional extinction in specific regions, which are attributed to specific regions, rich in hot, young, stars~\citep{2002AJ....123..855Z}.
To determine the intrinsic spread of extinction values at the location of each of our targets we use the recently published extinction maps of~\citet{2021ApJS..252...23S}.
\citet{2021ApJS..252...23S} determined $E(V-I)$ values using red clump stars from the Optical Gravitational Lensing Experiment (OGLE-IV).
For our targets, this results in an average extinction value of
$A_{\rm V}=0.17$\,mag\footnote{$A_{\rm V}$ is calculated assuming $A_I=1.5 \times E(V-I)$, as listed on the webpage associated with \citet{2021ApJS..252...23S}, and assuming the SMC bar reddening law of \citet{2003ApJ...594..279G}.} with a typical dispersion of 0.15.
We chose to retain the extinction value of $A_{\rm V}=0.35$\,mag~\citep{2021A&A...646A.106S}, rather than use the extinction values determined by \citet{2021ApJS..252...23S}, as the extinction values determined from red clump stars are more appropriate for lower mass stars.
However, we argue that the dispersion determined from these extinction maps is an accurate indicator of the local spread of extinction values for each target, given that the SMC contains little intrinsic dispersion.
Therefore, each target is assigned an uncertainty on $A_{\rm V}$ defined by the local dispersion from \citet{2021ApJS..252...23S}.
The scatter on the $A_{\rm V}$ values of our targets is correlated with larger $A_{\rm V}$ values from the ~\citet{2021ApJS..252...23S} maps, so in this sense, a small tail of higher extinction values is identified, which extends up to $A_{\rm V}=0.70$\,mag.
In previous studies of SMC RSGs a range of $A_{\rm V}$ values are assumed.
\citet{Levesque06} determined $A_{\rm V}$, effective temperatures and surface gravity values of 37 SMC RSGs by fitting spectrophotometric observations and found an average $A_{\rm V}=0.53$\,mag, with a dispersion of 0.35.
Similarly, \citet{2013ApJ...767....3D} determined $A_{\rm V}$ by fitting spectroscopic observations for 10 SMC RSGs that resulted in an average value of 0.5, with a dispersion of 0.2.
Recently,~\citet{2021MNRAS.505.4422G} updated the spectral fitting using the observations of \citet{2013ApJ...767....3D} and redetermined $A_{\rm V}$ values for the 10 SMC targets.
These authors found an average $A_{\rm V}=0.67$\,mag with a dispersion of 0.26.
In their recent study of SMC RSGs, to determine stellar parameters \citep{2021ApJ...922..177M} assumed $A_{\rm V}=0.75$\,mag to determine stellar parameters of their sample of SMC RSGs based on the results of \citet{Levesque06}.
~\citet{2018MNRAS.478.3138D} study 245 of the brightest RSGs in the SMC and determine their extinction value using the maps from the hot star sample of~\citet{2002AJ....123..855Z} obtain an average $A_{\rm V}=0.46$\,mag and a dispersion of 0.15.
Spectroscopic $A_{\rm V}$ determination is likely the most robust method to determine the extinction values of RSGs, as such estimates take into account circumstellar extinction around RSGs.
That being said, spectroscopically determined $A_{\rm V}$ values have large uncertainties and the agreement between the $A_{\rm V}$ measurements between the stars in common between the spectroscopic studies of~\citet{Levesque06} and~\citet[][]{2013ApJ...767....3D}, is poor, and has not been resolved with the updated calculations of~\citet{2021MNRAS.505.4422G}.
In particular, the stars with $A_{\rm V}>0.5$\,mag in~\citet{Levesque06}, all have $A_{\rm V}$ lower values in~\citet{2021MNRAS.505.4422G} and those with $A_{\rm V}<0.5$\,mag all have larger values in~\citet{2021MNRAS.505.4422G}.
The studies of \citet{Levesque06} and \citet{2021MNRAS.505.4422G} targeted more luminous RSGs than are present in the UVIT cross-matched sample, which are known to have additional circumstellar material and larger $A_{\rm V}$ values~\citep{2020ApJ...900..118N}.
Indeed, by measuring the infrared excess of RSGs,~\citet{2010AJ....140..416B} demonstrate that, in general, the population of SMC RSGs is relatively dust-free, which suggests that the impact of circumstellar material is low for the majority of SMC RSGs.
In this respect, we argue that $A_{\rm V}=0.35$\,mag remains a robust average value for the stars in our sample.
We comment on the impact of our choice of extinction in the following sections.
\subsection{Stellar parameters} \label{sub:parameters}
\subsubsection{RSG masses}
While RSG masses are controversial
\citep{davies2020,2020MNRAS.494L..53F,2021A&ARv..29....4S}, much of this discussion centres on the masses of RSGs just prior to core collapse.
Our interest, however, is on the evolutionary phase that is typical of our RSG population and hence we use the mid-point of the RSG core He-burning phase as representative of typical properties (age and mass) of the RSG as a function of luminosity.
For this purpose, we use the MESA models~\citep{2011ApJS..192....3P,2013ApJS..208....4P,2015ApJS..220...15P,2018ApJS..234...34P} computed for the SMC based on an extension of the model grids published by~\citet{2019A&A...625A.132S}, which use a mass-dependent convective overshooting parameter ($\alpha_{\rm ov}$) and a semi-convection parameter ($\alpha_{\rm sc}$) of 10, described in~\citet{2021A&A...653A.144H}.
To determine RSG masses, we first determine their luminosities using the calibration RSG luminosity of ~\citet{2013ApJ...767....3D} with de-reddened $Ks$-band photometry.
$Ks$-band photometry has the advantage of minimising the effects of interstellar and circumstellar extinction over other near-IR photometry.
For example, we find that luminosities determined using the $Ks$-band are slightly systematically larger than those determined using the $J$-band for the brightest stars in our sample.
This trend towards larger luminosities in the $Ks$-band increases as a function of luminosity and reaches 0.10 dex at a $\log L/L_\odot = 5.1$, which is potentially the result of an increase in circumstellar extinction around higher-luminosity RSGs.
For these calculations we assume a distance modulus to the SMC of 18.95~\citep{2014ApJ...780...59G} and the $Ks$-band magnitudes are de-reddened assuming $A_{\rm V}=0.35$\,mag and the SMC bar reddening law from~\citet{2003ApJ...594..279G}.
The choice of $A_{\rm V}$ has a small impact on the RSG luminosities as a result of the shape of the adopted reddening law.
The uncertainties on the measured luminosities are typically $\pm$0.14\,dex.
Using a comparison to the models we determine a relationship between the luminosity and evolutionary mass of SMC RSGs.
For the RSG sources with UVIT counterparts, we determine masses in the range $6.2 < M/M_\odot < 20.3$, as shown in Figure~\ref{fig:HRD-binaries}.
The most massive RSG with a UVIT counterpart is LHA\,115-S~30 with a mass of 20.3\,M$_\odot$ and a companion mass of 9.1\,M$_\odot$.
For comparison, the most massive RSG in the initial sample has a mass of 25.2\,M$_\odot$.
In addition, we determine the effective temperature for the targets.
The RSG effective temperature scale, particularly at low-metallicity, remains uncertain.
There exists several photometric and spectroscopic techniques to determine RSG effective temperatures.
To determine the effective temperatures for the RSGs we use a calibration of near-infrared (near-IR) photometry for SMC RSGs published in DP21.
This calibration is based on the RSG effective temperature measurements of \citet[][henceforth TDN18]{2018MNRAS.476.3106T}.
TDN18 determined stellar parameters, including effective temperatures, for over 150 SMC RSGs by fitting a selection of well-separated atomic features from medium-resolution spectra in the Calcium triplet (CaT) region ($\sim$8500\,\AA) with grids of MARCS~\citep{2008A&A...486..951G} and KURUCZ~\citep{2012AJ....144..120M} one-dimensional atmospheric models, under the assumption of Local Thermodynamic Equilibrium.
We further discuss photometric and spectroscopic effective temperature determinations for SMC RSGs in Appendix~\ref{ap:Teff}.
\subsubsection{Companion masses}
To determine the masses of the companions, we assume that each companion is a single, main-sequence star that is coeval with the RSG, thus occupying the same isochrone.
For each companion, we determine the relationship in the stellar models between luminosity and effective temperature given the UVIT magnitude.
The luminosity of the companion is determined using the equation:
\begin{equation}
\log L = (M_{bol,\odot} - m_{F172M} - BC(T_{\rm eff}) + A_{F172M} + \mu)/2.5
\end{equation}
\noindent where $\mu$ is the SMC distance modulus, namely $\mu=18.95$~\citep[][as for the RSG companions]{2014ApJ...780...59G}, and $A_{F172M}$ is the extinction in the UVIT~F172M filter, which is defined as $A_{F172M} =A_{\rm V}\times 4.013$ for the SMC bar~\citep{2003ApJ...594..279G}, where $A_{\rm V}=0.35$\,mag~\citep{2021A&A...646A.106S}.
While a tailored approach for each star may lead to more precise estimates of the ambient extinction, for example, taking account of detailed extinction maps or estimates of SMC and MW contributions to the total extinction, we estimate that the uncertainties introduced by neglecting such effects are well within our adopted uncertainties of $\pm0.14$ dex in $\log L/L_\odot$.
The bolometric correction ($BC$) is taken from the MIST models~\citep{2016ApJ...823..102C,2016ApJS..222....8D}.
In the MIST models, the bolometric correction of the F172M filter is only very weakly dependent on surface gravity and, as such, we use $\log g = 3.0$ and we assume it is solely a function of effective temperature in the temperature range studied.
$m_{F172M}$ is the apparent magnitude in the UVIT~F172M filter and we use $M_{bol,\odot}=4.74$.
The solid grey lines in Figure~\ref{fig:HRD-binaries} illustrate how the stellar parameters vary as a function of UVIT magnitude; note that these lines are almost perpendicular to the main sequence, which allows for a precise determination of stellar parameters at a given age, despite the observational limitation of only a single photometric filter.
The intersection between the isochrone, defined by the age of the RSG, and the line of constant UVIT magnitude are used to determine the mass and effective temperature of the companion.
Given the shape of the constant UVIT magnitude curves in the MIST models, for the low-mass companions the determined mass does not strongly depend on the age assumption.
With increasing companion mass, the age assumption becomes more important and, as a result, the most massive companions in the sample are those that are potentially most affected by uncertainties in the RSG age.
The choice of the extinction for the SMC is guided by the results of~\citet{2021A&A...646A.106S}, see Section~\ref{sub:ext}.
A fixed extinction value for all of our sources neglects the effects of variable and circum-binary extinction.
Additional extinction would act to increase the companion masses determined, whereas the RSG mass estimate is not strongly dependent on the adopted extinction value.
As with the age assumption, the highest mass companions are those that are most affected by the choice of the extinction value.
To improve the determination of the stellar parameters of the companion star requires additional UV photometric or spectroscopic observations.
No coeval solutions on the main sequence were found for six systems; in these cases the derived age is the age of a terminal-age main-sequence (TAMS) model that has the observed UVIT magnitude.
These six systems are identified in Figure~\ref{fig:fuvvsj} as systems that lie above and to the left of the red dashed line and also in Figure~\ref{fig:HRD-binaries} as coloured rings.
Because the age of the system is only approximated by the mid-point of the helium burning phase, the ages of more evolved RSGs in binary systems will tend to be underestimated, which could allow for solutions on the main sequence.
While this approach is clearly approximate, for low-mass secondaries it is reasonably accurate, since evolution on the main sequence has little impact on the F172M-band magnitude in these stars.
Further, the approach does not take into account scenarios in which the secondary is not a single, coeval main-sequence star, as will be discussed in Section~\ref{sub:qs}.
The uncertainties on the masses are dominated by the spread in the extinction law (propagated from the luminosities).
We find companion masses in the range 3.7\,$<M/M_\odot <$\,15.6, and these stars are shown in the HRD in Figure~\ref{fig:HRD-binaries}.
The most massive companion is Dachs SMC 1-13 with a mass of 15.6\,$\pm$\,0.1\,M$_\odot$.
\begin{figure}
\centering
\includegraphics[width=\hsize]{SMC-RSG-UVIT-masses_v2}
\caption{Derived masses for both components of the systems with UVIT counterparts.
Straight lines highlight lines of constant mass ratio ($q$).
}
\label{fig:masses}
\end{figure}
\subsection{Mass ratios of the binaries} \label{sub:qs}
The comparison of RSG and companion masses is shown in Figure~\ref{fig:masses}, while the distribution of mass ratios as a function of RSG mass is shown in Figure~\ref{fig:qvsMrsg}.
The latter illustrates that the detection limit for the secondaries ($\sim$3.5\,$M_\odot$) leads to a progressive loss of low-mass-ratio systems with decreasing RSG mass.
As discussed at the beginning of this section, this affects the observed binary fractions, which range from $\sim$20--25\% for the higher masses to $\sim$10--15\% for the lowest mass bins (see Figure~\ref{fig:bfvsJ}).
When analysing the mass ratios distribution to minimise the effect of variable observing biases, we consider the 32 RSGs in the 10 to 14\,$M_\odot$ mass range and find that, to a good approximation, the distribution of mass ratios is flat in the range $0.3 < q < 1.0$.
These figures also show a dearth of $q>0.5$ systems for RSG masses of greater than 15\,M$_\odot$:
eight out of nine RSGs with masses above 15\,M$_\odot$ have mass ratios less than 0.5.
This is striking, as the high-mass-ratio systems should, in principle, be the most easily detected systems in our survey.
Indeed, by simulating the observed population of RSG binaries, we find that in 99.9\% of 10\,000 simulations more than two systems are detected in this parameter space, assuming a flat mass-ratio distribution.
Therefore, we conclude that this dearth of high-mass-ratio systems above 15\,M$_\odot$ is not the result of small number statistics at high masses, but the result of a rapid change in the observed mass-ratio distribution function.
To describe the distribution of observations above 15\,M$_\odot$ requires a power-law mass-ratio distribution function: $p_q \propto q^\kappa$, where $\kappa \sim -2$.
A tendency towards low-mass companions is expected for long-period systems~\citep{2017ApJS..230...15M}; however, such a pronounced change at 15\,M$_\odot$ is not expected.
This is perhaps a hint that the most massive RSGs are mainly the result of stellar mergers, and the extant binaries that we observe are therefore triple systems with a low mass tertiary, while the inner binary has merged.
Of course, such a system would appear younger that its actual age \citep{b19} and also result in the overestimation of secondary masses, which further exacerbates the lack of high $q$ systems.
An alternative explanation for such a deficit is an underestimate of the extinction values.
Many authors have demonstrated the effects of increased circumstellar extinction around higher-luminosity RSGs~\citep[e.g.][for SMC RSGs]{2010AJ....140..416B}.
If we assume hat such stars have twice the nominal extinction value (i.e. $A_{\rm V}=0.7$\,mag), we find around 50\% of the binaries have $q > 0.5$, which is potential indicator that larger extinction values may be more appropriate for this mass range.
We identify six systems with $q > 1$.
In addition to the evolved binary scenario considered in the previous section, systems with a more complex evolutionary history may be the explanation of the observed systems at $q > 1$.
The mass ratios presented in this section have been determined with the assumption that the secondary is a single, coeval, main-sequence star.
Other possibilities exist, of course, and while we cannot distinguish between these cases with our limited photometric data, we consider the impact of some of these alternative scenarios.
If the secondary is the product of a previous binary interaction, the above approach would be inappropriate.
For example, stripped stars \citep{2018A&A...615A..78G} with initial masses above about 3.5\,$M_\odot$ would fall within our magnitude range.
Additional UV data would be required to characterise the FUV sources more accurately, by either extending the UV spectral energy distribution or acquisition of UV spectra. For the latter, an exploratory snapshot programme is underway with the Hubble Space Telescope (GO\,16776, PI: Patrick).
If the companion star is the result of a previous binary merger, the above approach would be equally inappropriate.
If the hot companion is an unresolved binary, the effect would also be to overestimate the mass of an individual component.
Decreasing the $A_{\rm V}$ value for these targets would improve the situation. We note that most of the RSGs with $q > 1$ have low RSG masses.
If we assume the $A_{\rm V}$ value of~\citet{2021ApJS..252...23S} for these objects, we find only one RSG binary system with $q > 1$, namely SSTISAGEMA J005145.35-723114.8.
\begin{figure}
\centering
\includegraphics[width=\hsize]{SMC-Mrsgvsq_v2}
\caption{Mass-ratio distribution as a function of RSG mass. The dashed line indicates the detection threshold, showing how the low-$q$ systems are missing for the lower mass RSGs. The dash-dot line highlights $q=0.3$, which is the limit of the observing bias correction in Section~\ref{sub:bf}.}
\label{fig:qvsMrsg}
\end{figure}
\subsection{Intrinsic multiplicity fraction and comparison with previous studies} \label{sub:bf}
To accurately compare with previous studies, we must clearly define the parameter space over which these observations are sensitive and take into account observational biases where possible.
The principal observing bias in this study stems from the UVIT photometric completeness limit, which results in a mass ratio bias that is a function of primary mass (as illustrated in Figure~\ref{fig:qvsMrsg}).
We account for this bias by simulating the observations assuming a flat mass-ratio distribution in the range $0.3 < q < 1.0$ and a total sample size of between 90 and 125 binary systems.
These samples are defined by drawing binary systems from the initial RSG source catalogue at random and assigning mass-ratios assuming a flat mass-ratio distribution.
For each drawn sample, we determine the number of systems above and below the observing detection limit and compare this with the observed sample.
From 100\,000 simulations with randomly drawn sample sizes between 90 and 125, we find $\sim$3500 with 88 systems detected above the observing limit.
From the distribution of these samples we determine that, on average, 17.7\,$\pm$\,4.6 systems lie below our detection limit.
Taking this into account results in an intrinsic multiplicity fraction of SMC RSGs of $18.8\,\pm\,1.5\,\%$, over a range of $0.3 < q < 1.0$.
The intrinsic multiplicity fraction as a function of RSG mass is shown in Figure~\ref{fig:ibfvsM1}.
We observe the same trend in the bias-corrected multiplicity fraction as is observed in the observed multiplicity fraction such that RSGs below $\sim$10\,M$_\odot$ have a multiplicity fraction of $\sim$12.5\% and RSGs above $\sim$10\,M$_\odot$ have a binary fraction closer to $\sim$25\%.
One expects the multiplicity fraction to increase as a function of primary mass~\citep{2017ApJS..230...15M}; however, such a step is unexpected and must be studied further in other environments to confirm or refute this observation.
\begin{figure}
\centering
\includegraphics[width=\hsize]{SMC-RSG-mass-bf-corr}
\caption{The intrinsic RSG binary fraction of the SMC a function of RSG mass. The red solid line shows the observing bias-corrected binary fraction for the entire RSG sample for $q > 0.3$.}
\label{fig:ibfvsM1}
\end{figure}
As previously noted, the UVIT observations are effectively insensitive to orbital periods of binary systems; however, we can place limits on the orbital periods by considering the limitations of our observations.
By using a XMD of 0.4\arcsec\ between the RSG and UVIT catalogues implies an upper limit of $\log P [\rm days]\sim 8$.
The physical size of RSGs places an approximate lower limit on the binary systems that we can detect at $\log P [\rm days]\sim 3$.
Therefore, the intrinsic multiplicity fraction calculated can be considered to be drawn from orbital periods within the range $3 < \log P [\rm days] < 8$.
DP21 used radial velocity measurements to determine a lower limit on the RSG binary fraction of the SMC to be 15\,$\pm$\,4\,\% and, while this is consistent with our results, a direct comparison is difficult as radial velocity methods are biased against long period systems and these authors were unable to account for their observational biases.
To more accurately compare these results, we determine that 75\% (226) of the 303 SMC targets in DP21 are within our initial source catalogue, of which 41 systems have a UVIT detection.
This results in a binary fraction of 18.1\,$\pm$\,2.6\% and, by accounting for the UVIT observing bias using the same method described above, an intrinsic binary fraction of 19.0\,$\pm$\,2.6\%.
This result is in good agreement with the results determined using the full UVIT sample and is reassuringly larger than the lower limit imposed by DP21.
A direct quantitative comparison is complicated, as DP21 identified 21 `reliable' binaries within their SMC sample (6.7\% of their sample), 16 of which are in our survey area, but only 10 of these are detected as FUV sources.
The remaining five systems lie outside our survey area.
The six RSG binaries with no UVIT counterparts are perhaps a hint that the number of false positives in DP21 is significant, reflecting the diverse nature of the radial velocity sources.
We discuss these further in Section~\ref{sub:BHs?}.
The 10 UVIT detected systems are highlighted in Table~\ref{tb:params_bs} and represent the most well-characterised SMC RSG binary systems.
In other radial velocity studies, \citet{2020A&A...635A..29P} used high-precision HARPS measurements of nine RSGs in the SMC cluster NGC\,330 to derive a bias-corrected binary fraction of 30\,$\pm$\,10\,\% for systems with $2.3<\log P [\rm days]<4.3$ and $q>0.1$.
Similar work by \citet{patrick2019} on 17 cluster and field RSGs in the 30 Doradus region of the LMC yielded an upper limit of 30\% on the binary fraction, with the parameter range $q>0.3$ and $3.3<\log P [\rm days]<3.5$.
Within the Milky Way, \citet{burki} studied radial velocities over a 5-year baseline using the CORAVEL survey of 181 supergiants with spectral types F--M.
These authors found an overall binary rate of $\sim$35\,\% including spectroscopic binaries (21\,\%), suspected spectroscopic binaries (4--7\,\%) and \textquoteleft very separated binaries\textquoteright~(6--10\,\%).
For the purposes of this comparison, we exclude the \textquoteleft very separated binaries\textquoteright, as their orbital configurations are likely to be quite different from the spectroscopic binaries and their numbers and bias correction are quite uncertain.
The K- and M-type supergiant samples of~\citet{burki} are perhaps closest in evolutionary phase to the current dataset and among their 84 K- and M-type supergiants they find 19 binaries, plus 7 suspected binaries, resulting in a binary rate of $\sim$30\,\%, with an estimated uncertainty of $\sim$10\,\%.
\citet[][]{2018AJ....155..207N} proposed a similar approach to that adopted here, although they searched for companions in the $U$- and $B$-bands, as opposed to the FUV.
Essentially, this method hinges on detecting a possible B-type companion in the blue part of the visible spectrum.
This method was applied to the LMC \citep{2020ApJ...900..118N} and in a refinement of the technique these authors included NUV photometry for 75\,\% of their sample from GALEX and used a spectroscopically confirmed sample of RSG+B binaries to train a k-nearest neighbour algorithm to find candidate binaries using a $UBV$ catalogue and the GALEX UV photometric catalogue.
Using this method \citet{2020ApJ...900..118N} assign a percentage likelihood of binarity to each star in the sample and determined a bias-corrected binary fraction of 19.5$^{+7.6}_{-6.7}$\,\% for the LMC, with which we find excellent agreement in the SMC.
To assess potential differences between the selection criteria we cross-match our RSG sample with the SMC $UBV$ catalogue of \citet{2002AJ....123..855Z}.
We find 504 reliable matches, 75 of which are binary systems detected with UVIT photometry.
The RSG binaries have a range of $-0.2 < U-B < 1.7$, with 51 systems (68\%) having $U-B < 1.0$ (see Figure~\ref{fig:ubbv}), which is broader than the distribution of $U-B$ colours than the spectroscopically confirmed RSG binary sample in the LMC of~\citet{2020ApJ...900..118N} and in M31 and M33 of~\citet{2021ApJ...908...87N}.
Around 1/3 of RSG binaries in our sample have $U-B > 1.0$, which is significantly larger than the 8\% of stars found with $U-B > 1.0$ in the LMC sample of spectroscopically confirmed RSG binaries~\citet{2020ApJ...900..118N} and the 0\% in the samples of M31 and M33~\citep{2021ApJ...908...87N}.\footnote{The significance of these discrepancies are calculated by determining the uncertainties of the percentages assuming $\sigma = \sqrt{p(1-p)/n}$, where $p$ is the result and $n$ is the population size.}
It appears unlikely that metallicity variations between the samples can fully account for their differences, which raises the possibility that~\citet{2020ApJ...900..118N} and~\citet{2021ApJ...908...87N} might have missed RSGs binaries with fainter companions.
\citet{2020ApJ...900..118N} and \citet{2021ApJ...908...87N} determined the likelihood of chance alignments masquerading as RSG binary systems to be less than the 2\,\% level.
To test this, we experiment using a range of XMDs to determine the percentage of false positives for a given XMD, assuming that the 88 detections with 0.4\,\arcsec to be the \textquoteleft true\textquoteright~number of RSG binaries.
We find that at a XMD of 1.0" a contribution of 6\% from false positive detections, rising to 35\% at 3.0".
Figure~\ref{fig:XMD} also implies that the false positive issue for blue sources becomes a much more serious problem for more distant galaxies such as M31 or M33, however, high resolution photometry improves this situation such as the HST photometry of~\citet{2021ApJS..253...53W}, which covers around 42\,\% of the M33 sample of~\citet{2021ApJ...908...87N} and 21\,\% of their M31 sample.
In addition, \citet{2020ApJ...900..118N} provide a preview of their upcoming SMC study where 22 SMC RSGs were observed in the same observing run as their LMC results.
Of these 22 RSGs,~\citet{2020ApJ...900..118N} classify eight as RSG binary systems based on their spectroscopic observations.
We find that 18 confirmed RSG binaries from~\citet{2020ApJ...900..118N} are within the UVIT survey footprint. Of these 18 we find 11 RSGs with UVIT counterparts, confirming six of the eight RSG binaries identified~\citet[][]{2020ApJ...900..118N}.
The remaining two systems were outside of the UVIT survey footprint, which results in an accuracy of 100\%.
The range of $U-B$ magnitudes of the 22 RSGs in \citet{2020ApJ...900..118N} extend up to $U-B = 1.0$, therefore we cannot assess the accuracy of the spectroscopic classification for $U-B > 1.0$, which is potentially where this method breaks down given the diminishing contribution from the companion.
With these observations we independently confirm the spectroscopic classification of \citet{2020ApJ...900..118N} is an effective method to identify RSG binary systems for $U-B < 1.0$.
However, almost 50\% of the stars classified as RSG binaries via UV photometry that are in the sample of \citet{2020ApJ...900..118N} were not classified as RSG binaries based on a spectroscopic identification.
These systems have a range of UVIT magnitudes of 18.8 to 16.8, corresponding to masses from 5.1 to 8.9\,$M_\odot$.
Based on these results, we conclude that the greater flux contrast enabled by FUV imaging leads to a significantly improved detection efficiency for companion masses below around $9\,M_\odot$.
Turing our attention on different evolutionary phases, a comparison with Cepheids is useful given the similarities in mass range and evolutionary state of the primary stars.
To our knowledge, no systematic study of the binary properties of SMC Cepheids exist.
\citet{2015AJ....150...13E} studied a Galactic sample of Cepheids using the radial velocity survey CORAVEL.
These authors found an observed binary fraction of 29\,$\pm$\,8\,\%, which is free of any significant observational bias, within the orbital period range $2.5 < \log P[\rm days] < 4$.
In addition, these authors found a flat mass-ratio distribution for their studied period range.
This quantitative agreement between the Galactic Cepheid and the SMC RSG binary fractions and orbital configurations suggests that metallicity does not play a primary role in the evolution of wide binary systems, supporting the conclusions of~\citet{2013ApJ...778...95M} and~\citet{2019ApJ...875...61M}.
As part of the VLT FLAMES Tarantula Survey~\citep{2011A&A...527A..50E}, \citet{2015A&A...580A..93D} determined the intrinsic binary fraction for B-type stars in the LMC.
These authors considered orbital periods in the range $0.15 < \log P [\rm days] < 3.5$ and accounted for observational biases to determine that around 60\% of B-type stars have a companion over the range $0.1 < q < 1.0$.
Taking into account the range in orbital periods studied by these authors and assuming a flat $\log P [\rm days]$ distribution over this range results in a multiplicity fraction of 20\% per decade in $\log P$ up to $\log P [\rm days] \sim 3$.
Combining this with the results presented here, which cover orbital periods outside the range considered by~\citet{2015A&A...580A..93D}, we find an intrinsic single star fraction of massive stars of $\sim20\%$, which is in good agreement with previous estimates of this statistic~\citep{2017ApJS..230...15M}.
This simple analysis also yields the important conclusion that the $\log P [\rm days]$ distribution must decline rapidly outside $\log P [\rm days] \sim 4$, which is supported by the observation of very wide OB-type binary systems~\citep{2019MNRAS.486.4098I} and may have consequences for binary formation.
In general there is reasonable consensus that the binary fraction of RSGs is approximately 20\%, with the present work finding a flat mass-ratio distribution, for $q>0.3$ and $M_{\rm RSG} <14\,M_\odot$.
As has already been noted in the literature, this is substantially less than the binary fraction of their progenitors, the OB-type stars \citep[as derived in][]{2012Sci...337..444S,2013A&A...550A.107S,2015A&A...580A..93D,2017ApJS..230...15M}, as a result of a combination of evolution and binary interaction.
The periods of RSG binaries are very long, with minimum periods of at least $10^2$--$10^3$\,d as a result of the radius of the RSG \citep{patrick2019}.
It is therefore tempting to compare our mass-ratio distribution with that for long period OB stars.
Indeed the literature survey of \citet{2017ApJS..230...15M} gives a mean value of the power-law exponent of the mass-ratio distribution $\kappa = -1.7$ to $-2.0$ for long orbital period systems in the mass range covered in the present study, in clear tension with the flat distribution ($\kappa = 0.0$) found in this study below 14\,M$_\odot$.
However, there are a number of potential problems with simple comparisons of binary fractions and $q$ distributions.
For example, many OB binaries are triple, or higher order, systems whose periods may evolve, possibly driving the inner binary to merge~\citep{2021arXiv210804272T}.
On the other hand, some single RSGs may have previously been in a binary system that has since dissociated, perhaps as a result of the primary exploding as a supernova.
Therefore, this complex problem of quantitatively relating the binary properties of RSGs to their OB star ancestors will be addressed in a future binary population synthesis approach currently in preparation.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{SMC-RSGs-UBV}
\caption{$U-B$, $B-V$ colour-colour diagram with photometry from~\citet{2002AJ....123..855Z}.
Black points represent the initial target selection and coloured diamonds represent RSG binary systems with UVIT counterparts where the colour is determined by the UVIT F172M-band magnitude of the companion.
}
\label{fig:ubbv}
\end{figure}
\subsection{Interesting non-detections} \label{sub:BHs?}
One of the explanations for the known binary systems from DP21 that are not detected at UV wavelengths are compact companions.
Such systems are expected to be rare, but these systems can potentially be identified through a lack of a UV counterpart, assuming no interaction between the RSG wind and the compact companion.
Six systems have significant radial velocity variations from DP21, with no detectable hot companion more massive than 3\,M$_\odot$.
These positive detections in DP21 are unlikely to have low-mass-ratio configurations given the scale of the detected radial velocity variability.
As stated in Section~\ref{sec:dis}, it is probable that these non-detections are the result of false positive measurements in DP21; however, because of the evolutionary significance of the detection of RSG + compact companion systems, these systems are worth noting and investigating further.
The six candidate RSG + compact companion systems are
SV*~HV2232,
Cl*~NGC~371~LE~39,
BBB~SMC~306,
PMMR~9,
Dachs~SMC~1-4 and
SV*~HV~832.
From DP21, these six systems all have radial velocity variations greater than 11.4\,\kms and between three and seven spectroscopic epochs.
SV*~HV2232 appears in the XSHOOTER Spectral library~\citep[XLS;][]{2019A&A...627A.138A} and on inspection of its spectral appearances displays no spectroscopic signatures of binarity.
To place limits on the orbital configurations that can reproduce the observations requires detailed simulations and an exhaustive search of available observations, therefore, for these reasons we reserve placing limits on potential orbital configurations for a future publication.
\section{Summary and Conclusions} \label{sec:conclusion}
In the context of the importance of multiplicity within the evolution of massive stars, the observational properties of RSGs in binary and multiple systems relatively remain poorly understood, despite recent observational progress.
In this article, we have aimed to better understand the multiplicity properties of RSGs in the SMC by identifying RSG binary systems and characterising the systems using newly available UVIT imaging.
Detecting RSG binary systems using UV photometry has a distinct advantage over other methods of detection, as it allows a direct characterisation of the companion star in a wavelength range where the RSG provides no contribution.
From a total of 560 RSGs within the UVIT survey area, 88 RSG have UVIT counterparts brighter than the limiting magnitude of approximately $m_{F172M}~=~$20.3\,ABmag.
Based on these results, we determined the observed RSG main-sequence multiplicity fraction of the SMC to be 15.7\,$\pm$\,1.5\,\%, which can be thought of as a lower limit on the intrinsic RSG binary fraction.
Near-IR photometry was used to determine stellar parameters of the RSGs and the UVIT photometry was used to determine the stellar parameters of the companions, which assumed the same age for both components within the systems.
We used MESA models adapted from~\citet{2019A&A...625A.132S} including a mass-dependent convective overshooting parameter to compare with observations using bolometric corrections from MIST tracks~\citep{2016ApJ...823..102C,2016ApJS..222....8D}.
Figure~\ref{fig:HRD-binaries} displayed our results for both components in a HRD for the 88 RSG binary systems.
For the binary systems we found RSG masses in the range $6.2 < M_{RSG}/M_\odot < 20.3$ and companion masses in the range $3.6 < M_2/M_\odot < 15.4$ and used these results to determine the mass-ratio distribution of long-period massive binary systems, which is best described by a uniform distribution in the range $0.3 < q < 1.0$.
We found six systems that have mass ratios greater than 1.0, which are either genuine inverted mass systems or systems in which the companion represents an unresolved binary system.
In addition, we found six candidate RSG + compact companion systems that require future study.
We simulated observational biases to determine the intrinsic multiplicity fraction of SMC RSGs and found it to be $18.8\,\pm\,1.5\,\%$, over a range of $0.3 < q < 1.0$ and $3< \log P [\rm days] < 8$.
This result is in good agreement with a lower limit set using spectroscopic observations at longer wavelengths (DP21) and those from photometric studies in other galaxies~\citep{2020ApJ...900..118N}.
Interestingly, we found a potential transition in the multiplicity fraction of RSGs at $\sim$10\,M$_\odot$, where the multiplicity fraction was lower below this value and higher above this value.
In addition, we combined our results with those at earlier evolutionary phases to estimate the single star fraction of massive stars to be $\sim20\%$.
This photometric identification of the companions of RSGs in binary systems represents the first time that the companions of a large sample of RSGs have been directly studied.
The combination of the UV and the high spatial accuracy afforded by the UVIT observations result in an accurate and precise determination of the RSG multiplicity fraction in a way that has not previously been possible given observational limitations in the Magellanic Clouds.
Follow-up Hubble Space Telescope UV spectroscopy of a sub-sample is in progress to better understand the nature of the companions and refine the stellar parameters.
\section*{Acknowledgements}
The authors would like to thank the anonymous referee for a providing useful comments that improved the quality of the article.
LRP acknowledges the support of the Generalitat Valenciana through the grant APOSTD/2020/247.
This research is partially supported by the Spanish Government under grant PGC2018-093741-B-C21 (MICIU/AEI/FEDER, UE).
DJL acknowledges support from the Spanish Government Ministerio de Ciencia, Innovaci\'on y Universidades through grants PGC-2018-091 3741-B-C22 and from the Canarian Agency for Research, Innovation and Information Society (ACIISI), of the Canary Islands Government, and the European Regional Development Fund (ERDF), under grant with reference ProID2017010115.
This work has made use of data from the European Space Agency (ESA) mission
{\it Gaia} (\url{https://www.cosmos.esa.int/gaia}), processed by the {\it Gaia}
Data Processing and Analysis Consortium (DPAC,
\url{https://www.cosmos.esa.int/web/gaia/dpac/consortium}). Funding for the DPAC
has been provided by national institutions, in particular the institutions
participating in the {\it Gaia} Multilateral Agreement.
The authors acknowledge the support of the from the Generalitat Valenciana through the grant PROMETEO/2019/041.
This is a pre-copyedited, author-produced PDF of an article accepted for publication in MNRAS following peer review. The version of record [DOI: stac1139] is available online at: xxxxxxx .
\section*{Data Availability}
It is the authors' intention to make the data that have been used to determine the results and publish this article as freely and easily available as possible to permit readers to reproduce these results.
To this end, the RSG source catalogue and their derived parameters are published fully in Table~\ref{tb:params_all}.
Table~\ref{tb:params_bs} provides a list of RSG binary systems along with their derived parameters in full.
\bibliographystyle{mnras}
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{
"redpajama_set_name": "RedPajamaArXiv"
}
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package edu.cloudy.utils;
import java.io.InputStream;
import java.io.InputStreamReader;
/**
* @author spupyrev
* Jan 8, 2014
*/
public class CommonUtils
{
/**
* Using the hack as my webapp loader can't handle relative paths :(
*/
public static String getAbsoluteFileName(String name)
{
return Thread.currentThread().getContextClassLoader().getResource(name).getFile();
}
public static InputStream getResourceAsStream(String name)
{
return Thread.currentThread().getContextClassLoader().getResourceAsStream(name);
}
public static InputStreamReader getResourceAsReader(String name)
{
return new InputStreamReader(getResourceAsStream(name));
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,601
|
Q: echoing (creating) an error log file I have the following lines commands:
if %errorlevel% equ 1 (
set/a error=1
if not exist "error.log" echo. > "error.log"
echo the procedure has got an error >> "error.log"
echo. >> "error.log
)
but like this I obtain the message that the file is being processed by another process.
There is maybe another way to create the file if not exists instead of using Echo.
A: You can create the file with
copy NUL error.log
However, I doubt that echo is your problem. More likely is that the file already exists and you have it opened in a text editor (or viewer) that locks the file.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 965
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Q: How to handle external dependencies in perl's ExtUtils:MakeMaker I have a series of perl scripts for which I'm writing a Makefile.PL script, but I'm rather inexperienced with ExtUtils::MakeMaker.
One of the scripts I wrote makes a system call to a command line utility that must be installed in order for the script to run properly. My script can gracefully detect that the utility is missing and issue an error about installing it and putting it in the user's path, but is there some standard way to handle this in the Makefile.PL script? Could it even gasp attempt to install the third-party utility if I enter the download link in the Makefile.PL script?
At the very least, I'd like the script to warn the user if the external dependency was not found. I know I can write a test case that uses it. Is this as simple as copying and pasting the subroutine I wrote in the script itself that checks for the third party utility and prints an error if it's not found or would that be the "wrong way to do it"?
A: Let's call this external dependency foobar, for sake of argument.
As per @KeepCalmAndCarryOn's comment, firstly consider whether foobar could be replaced by something from CPAN (maybe Foo::Bar), or a few lines of Perl.
Otherwise, the best course of action is:
*
*Create a new CPAN distribution called Alien::Foobar. The job of Alien::Foobar is to download, perhaps compile, and then install foobar, as part of Alien::Foobar's Makefile.PL or Build.PL.
(There exists a module called Alien::Base which aims to make doing this sort of thing easier. It's mostly aimed at installing libraries rather than binaries, though I've had some success using it for the latter.)
*Now the Makefile.PL you were originally working on can declare a dependency on Alien::Foobar.
A: If you have an external dependency on a command-line utility (i.e. there's no perl module that does what the utility does), ExtUtils::MakeMaker is not designed to handle such a dependency. What you need to do is write an install script or edit the make file to handle the dependency. Here are the considerations in doing so:
*
*Check if the dependency exists and if the version is sufficient.
*Download the dependent package
*Configure, compile, & install the dependent package
*Test to make sure it works
*Update the user's environment setup if necessary
*Run your perl package's installation steps (e.g. perl makefile.PL;make;sudo make install)
Note, you may need to know whether your script is running as root or not, which you can verify using id -u to check if the user ID is root (i.e. '0').
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,392
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Q: undefined reference to boost::system::generic_category() cmake I have an undefined reference while trying to compile my Qt5 project.
I am usually pretty confident with using CMake but this time I have a strange error that I can't figure out:
undefined reference to boost::system::generic_category()
/usr/local/include/boost/system/error_code.hpp:223: undefined reference to 'boost::system::generic_category()'
My configurations are:
*
*gcc (Ubuntu 9.3.0-17ubuntu1~20.04) 9.3.0
*Qt5.15
*cmake version 3.16.3
*After typing whereis boost the outcome wasboost: /usr/include/boost and after applying the great power of dpkg -s libboost-dev | grep 'Version' :) the version is Version: 1.71.0.0ubuntu2
I don't understand what is happening, below an example of how my CMakeLists.txt is structured:
cmake_minimum_required (VERSION 3.1)
project(projectA)
set (OpenCV_DIR /home/to/opencv/build)
find_package( OpenCV REQUIRED )
find_package( Boost COMPONENTS system thread filesystem REQUIRED)
set(CMAKE_CXX_FLAGS "${CMAKE_CXX_FLAGS} -std=c++11")
set(CMAKE_INCLUDE_CURRENT_DIR ON)
INCLUDE_DIRECTORIES(${Boost_INCLUDE_DIR})
set(CMAKE_AUTOMOC ON)
set(CMAKE_AUTORCC ON)
find_package(Qt5Widgets)
find_package(Qt5PrintSupport)
#make them into headers
qt5_wrap_ui (UI_HDRS ${UI})
add_executable(projectA main/main.cpp ui/qdarkstyle/style.qrc ${SRCS} ${UI_HDRS} ${UI_SRCS})
target_link_libraries (projectA Qt5::Widgets ${Boost_LIBRARIES} ${OpenCV_LIBS} Qt5::PrintSupport)
add_library(projectA_lib SHARED ${SRCS} ${UI_HDRS})
target_include_directories (projectA_lib PUBLIC "src/" "ui/")
link_directories(${Boost_LIBRARY_DIRS})
target_link_libraries (projectA_lib Qt5::Widgets ${Boost_LIBRARIES} ${OpenCV_LIBS})
I have searched and applied solutions I saw on all possible sources I was able to find such as:
This source but that didn't work.
Also from here it seems that this solution shall be applied:
set(Boost_USE_STATIC_LIBS ON)
set(Boost_USE_MULTITHREADED ON)
set(Boost_USE_STATIC_RUNTIME OFF)
find_package(Boost REQUIRED COMPONENTS system)
# the call to include_directories is now useless:
# the Boost::system imported target used below
# embeds the include directories
project(APP C CXX)
add_executable(APP src.cpp)
target_link_libraries(APP Boost::system)
However that also didn't do any specific benefits to finding the solution.
Other posts I consulted were this, this one but no answer was provided.
This post was useful but that also didn't provide any advice that didn't already know.
Thanks for pointing to the right direction and trying to find a solution.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,692
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\section{Introduction}
Insulating rare-earth titanate pyrochlores (A$_2$Ti$_2$O$_7$, where A is tri-positive rare-earth ion) are known to show complex magnetic behaviors, arising from the geometrical frustration of exchange interaction between the rare-earth spins located on an infinite network of corner-sharing tetrahedrons \cite{ARMS-24-453}. Theoretically, for antiferromagnetically coupled classical or Heisenberg spins on the pyrochlore lattice the magnetic ground state should be infinitely degenerate \cite{PT-59-24}. However, the ubiquitous presence of residual terms, like next near-neighbor interactions, crystal field and dipolar interactions can remove this macroscopic degeneracy either completely or partially leading often to complex spin structures at low temperatures \cite{JAC-408-444}. The only member of the A$_2$Ti$_2$O$_7$ series where the presence of residual terms has apparently no significant influence on the spin-dynamics is Tb$_2$Ti$_2$O$_7$. In this pyrochlore, the strength of antiferromagnetic exchange is of the order of 20 K, however despite this, the Tb$^{3+}$ spins show no signs of freezing or long-range ordering down to a temperature of at least 70 mK \cite{PRL-82-1012}. It has been shown, however, that this ``collective paramagnetic'' or the so-called ``spin-liquid'' state of Tb$^{3+}$ moments is instable under high-pressure \cite{Nature-420-54}. Using powder neutron diffraction experiments, Mirebeau \textit{et al.} \cite{Nature-420-54} showed that application of iso-static pressure of about 8.6 GPa in Tb$_2$Ti$_2$O$_7$ induces a long-range order of Tb spins coexisting with the spin-liquid. Since no indication of pressure induced structural deformation was observed in this study, the spin-crystallization, under pressure, was believed to have resulted from the break-down of a delicate balance among the residual terms.
Recently, the vibrational properties of some of these pyrochlores have been investigated by several groups \cite{APL-86-181906, CPL-413-248, PRB-74-064109, JPCM-17-5225, JRS-39-537, PRB-77-054408, PRB-77-214310, PRB-78-134420, PRB-78-214102}. These studies not only show that phonons in the titanate pyrochlores are highly anomalous, but also indicate the extreme sensitivity of vibrational spectroscopy towards probing subtle structural and electronic features not observed. In the pyrochlore Dy$_2$Ti$_2$O$_7$, Raman spectroscopy revealed a subtle structural deformation of the pyrochlore lattice upon cooling below T = 100 K \cite{PRB-78-214102}. In the pyrochlore Tb$_2$Ti$_2$O$_7$, new crystal-field (CF) excitations were identified using Raman data at T = 4 K \cite{PRB-78-134420}. In the temperature-dependent studies, signature of highly anomalous phonons (i.e., decrease of phonon frequency upon cooling; also referred to as phonon softening) has been witnessed in the pyrochlores Er$_2$Ti$_2$O$_7$ \cite{JRS-39-537}, Gd$_2$Ti$_2$O$_7$ \cite{JRS-39-537, PRB-77-214310} and Dy$_2$Ti$_2$O$_7$ \cite{JPCM-17-5225, JRS-39-537, PRB-77-214310, PRB-78-214102}. The effect of pressure, at ambient temperature, has also been studied recently for several of these titanate pyrochlores. Sm$_2$Ti$_2$O$_7$ and Gd$_2$Ti$_2$O$_7$ pick up anion disorder above 40 GPa and become amorphous above 51 GPa \cite{APL-86-181906, CPL-413-248}. Gd$_2$Ti$_2$O$_7$ exhibits a structural deformation near 9 GPa \cite{PRB-74-064109}.
In this paper we present Raman and powder x-ray diffraction studies on the pyrochlore Tb$_2$Ti$_2$O$_7$. These studies were carried out in the temperature range between room temperature and 27 K; and pressure varying from ambient pressure to 25 GPa. Our study reveals highly anomalous softening of the phonons upon cooling. To understand this anomalous behavior, we have estimated the quasiharmonic contribution to the temperature-dependent shift of the frequencies of different Raman phonons using the mode Gr\"{u}neisen parameters obtained from high-pressure Raman data; and bulk modulus and thermal expansion coefficient obtained from high-pressure and temperature-dependent powder x-ray diffraction data, respectively. These analyses allow us to extract the changes in the phonon frequencies arising solely due to anharmonic interactions. We also bring out the effect of pressure on phonons manifesting a subtle structural deformation of the lattice near 9 GPa which is corroborated by a change in the bulk modulus by $\sim$ 62\%. This observation may have relevance to the observations of powder neutron scattering study \cite{Nature-420-54} mentioned above. While this paper was being written, we came across a very recent temperature-dependent study \cite{PRB-78-134420} on this system revealing phonon softening behavior and the coupling of phonons with the crystal field transitions. We shall compare below our results on the temperature dependence of phonons with those of this recent study and quantify the quasiharmonic and anharmonic contributions to the change in phonon frequencies.
\section{Experimental Techniques}
\subsection{Crystal growth}
Stoichiometric amounts of Tb$_2$O$_3$ (99.99 $\%$) and TiO$_2$ (99.99 $\%$) were mixed thoroughly and heated at 1200 $^\circ$C for about 15 h. The resulting mixture was well ground and isostatically pressed into rods of about 6 cm long and 5 mm diameter. These rods were sintered at 1400 $^\circ$C in air for about 72 h. This procedure was repeated until the compound Tb$_2$Ti$_2$O$_7$ was formed, as revealed by powder x-ray diffraction analysis, with no traces of any secondary phase. These rods were then subjected to single crystal growth by the floating-zone method in an infrared image furnace under flowing oxygen. X-ray diffraction measurement was carried out on the powder obtained by crushing part of a single crystalline sample and energy dispersive x-ray analysis in a scanning electron microscope indicated a pure pyrochlore Tb$_2$Ti$_2$O$_7$ phase. The L\"{a}ue back-reflection technique was used to orient the crystal along the principal crystallographic directions.
\subsection{Raman measurements}
Raman spectroscopic measurements on a (111) cut thin single-crystalline slice (0.5 mm thick and 3 mm in diameter, polished down to a roughness of almost 10 $\mu$m) of Tb$_2$Ti$_2$O$_7$ were performed at low temperatures in back-scattering geometry, using the 514.5 nm line of an $Ar^+$ ion laser (Spectra-Physics) with $\sim$ 20 mW of power falling on the sample. Temperature scanning was done using a CTI-Cryogenics Closed Cycle Refrigerator. Temperature was measured and controlled (with a maximum error of 0.5 K) using a calibrated Pt-sensor and a CRYO-CON 32B temperature controller. The scattered light was collected by a lens and was analyzed using a computer controlled SPEX Ramalog spectrometer having two holographic gratings (1800 groves/mm) coupled to a Peltier-cooled photo multiplier tube connected to a digital photon counter.
High-pressure Raman experiments were carried out at room temperature up to $\sim$ 25 GPa in a Mao-Bell type diamond anvil cell (DAC). A single crystalline Tb$_2$Ti$_2$O$_7$ sample (size $\sim$ 50 $\mu$m) was placed with a ruby chip (size $\sim$ 10 $\mu$m) in a hole of $\sim$ 200 $\mu$m diameter drilled in a preindented stainless-steel gasket with a mixture of 4:1 methanol and ethanol as the pressure-transmitting medium. Pressure was calibrated using the ruby fluorescence technique \cite{Science-176-284}.
\subsection{X-ray diffraction}
High resolution x-ray diffraction measurements were performed between 10-300 K (with temperature accuracy better than 0.5 K) using a highly accurate two-axis diffractometer in a Bragg-Brentano geometry (focalization circle of 50 $\mu$m) using the Cu-K$_{\beta}$ line ($\lambda$=1.39223 \AA) of a 18 kW rotating anode.
For high-pressure x-ray experiments, single crystalline Tb$_2$Ti$_2$O$_7$ samples were crushed into fine powder which was
loaded along with a few particles of copper, in a hole of $\sim$ 120 $\mu$m diameter drilled in a preindented ($\sim$ 70 $\mu$m thick) tungsten gasket of a Mao-Bell-type diamond-anvil cell (DAC). The pressure-transmitting medium was methanol-ethanol-water (16:3:1) mixture, which remains hydrostatic until a pressure of $\sim$ 15 GPa. Pressure was determined from the known equation of state of copper \cite{PRB-70-094112}. High-pressure angle dispersive x-ray diffraction experiments were carried out up to $\sim$ 25 GPa on Tb$_2$Ti$_2$O$_7$ at the 5.2$R$ (XRD1) beamline of the Elettra Synchrotron source (Italy) with monochromatized x-rays ($\lambda $= 0.69012 \AA). The diffraction patterns were recorded using a MAR345 imaging plate detector kept at a distance of $\sim$ 20 cm from the sample. Two-dimensional (2D) imaging plate records were transformed into one-dimensional (1D) diffraction profiles by radial integration of the diffraction rings using the FIT2D software \cite{HPR-14-235}.
\section{Results}
\subsection{Raman spectrum of Tb$_2$Ti$_2$O$_7$}
Pyrochlores belong to the space group $Fd\bar{3}m (O^{h}_{7})$ with an $A_2B_2O_6O^{\prime}$ stoichiometry, where $A^{3+}$ occupies the 16d and $B^{4+}$ occupies the 16c Wyckoff positions and the oxygen atoms O and O$^{\prime}$ occupy the 48$f$ and 8$b$ sites, respectively. Factor group analysis for this family of structures gives six Raman active modes ($A_{1g}+E_g+4F_{2g}$) and seven infrared active modes ($7F_{1u}$). Raman spectra of Tb$_2$Ti$_2$O$_7$ have been recorded between 125 to 925 cm$^{-1}$ from room temperature down to 27 K. A strong Rayleigh contribution made the signal to noise ratio poor below 125 cm$^{-1}$. Fig. \ref{Fig:1} shows the Raman spectrum at 27 K, fitted with Lorentzians and labeled as P1 to P9. Following previous reports \cite{APL-86-181906, PRB-78-134420, PRB-78-214102, PRB-77-214310, JRS-39-537, CPL-413-248, JPCB-106-4663, JRS-14-63, JRS-32-41}, the modes can be assigned as follows: P3 (294 cm$^{-1}$, $F_{2g}$), P4 (325 cm$^{-1}$, $E_g$), P5 (513 cm$^{-1}$, $A_{1g}$) and P6 (550 cm$^{-1}$, $F_{2g}$). One $F_{2g}$ mode near 425 cm$^{-1}$ (observed in other pyrochlore titanates \cite{JRS-39-537,PRB-78-134420, PRB-78-214102}) could not be observed due to weak signal. The mode P1 (170 cm$^{-1}$) has been assigned to be the fourth $F_{2g}$ mode by refs. \cite{APL-86-181906, CPL-413-248, JPCB-106-4663, JRS-39-537, PRB-77-214310, PRB-78-134420, PRB-78-214102, JRS-14-63, JRS-32-41}. However, there has been a controversy on the assignment of the P1 mode \cite{DTO18, JINC-38-1407, AnnChim-9-43} and we, therefore, assign the mode P7 (672 cm$^{-1}$) as the fourth $F_{2g}$ mode. We support this assignment for the following reason: it is well established that the symmetry-allowed six Raman active modes ($A_{1g}+E_g+4F_{2g}$) in pyrochlore involve only the vibrations of oxygen atoms. This will imply that isotopic substitution by O$^{18}$ in pyrochlore should lower the phonon frequencies by $\sim$ 5 \%. This has, indeed, been seen in our recent experiments \cite{DTO18} on Dy$_2$Ti$_2$O$_7$ and Lu$_2$Ti$_2$O$_7$ for the modes P3 to P9 but not for P1 and P2. Another argument against P7 being a combination mode is the pressure dependence of the modes presented later (Fig. \ref{Fig:7}). Possible candidates for the combination are $\omega_{P7}\approx \omega_{P3}+\omega_{P4}$ and $\omega_{P7}\approx \omega_{P1}+\omega_{P5}$. The pressure derivative of frequency of the mode P7 ($\frac{d\omega_{P7}}{dP}$) does not agree with the sum of the pressure derivatives of the individual modes. Next, the question arises on the origin of the modes P1 and P2. Since these modes are also seen in Gd$_2$Ti$_2$O$_7$ and in non-magnetic Lu$_2$Ti$_2$O$_7$, their crystal field (CF) origin can be completely ruled out. We, therefore, attribute these low frequency modes to disorder induced Raman active modes. The high frequency modes (P8 and P9) are possibly second-order Raman modes \cite{JRS-39-537, PRB-78-134420, PRB-78-214102}.
\subsection{Temperature dependence of phonons}
We have recorded Raman spectra of Tb$_2$Ti$_2$O$_7$ from room temperature down to 27 K and followed the temperature dependence of the modes P1, P3, P4, P5 and P7. As shown in Fig. \ref{Fig:2}, the modes P1, P3, P5 and P7 soften with decreasing temperature. Since the Raman bands P2 and P6 are weak near room temperature, their temperature dependence is not shown. It needs to be mentioned that temperature-dependent anomalies of the modes P1, P5 and P7 have also been reported in other pyrochlore titanates \cite{PRB-77-214310, JRS-39-537, PRB-78-214102} and attributed to phonon-phonon anharmonic interactions. However, anomalous behavior of the $F_{2g}$(P3) mode near 300 cm$^{-1}$ has been reported only in the non-magnetic Lu$_2$Ti$_2$O$_7$ pyrochlore \cite{PRB-78-214102}. We evidence a similar anomaly in P3 in Tb$_2$Ti$_2$O$_7$ with unusually broad linewidth. Recently, Maczka \textit{et al.} \cite{PRB-78-134420} have also reported this unusually broad linewidths in Tb$_2$Ti$_2$O$_7$ which has been explained in terms of coupling between phonon and crystal field transition.
Temperature dependence of a phonon mode ($i$) of frequency $\omega_i(T)$ can be expressed as \cite{PRB-28-1928},
\begin{eqnarray}
\omega_i(T)= && \omega_i(0)+{(\Delta \omega_i)}_{total}(T) \nonumber \\
\text{where,}\hspace {5 mm} {(\Delta \omega_i)}_{total}(T) = && {(\Delta \omega_i)}_{qh}(T)+{(\Delta \omega_i)}_{anh}(T) \nonumber \\
&& +{(\Delta \omega_i)}_{el-ph}(T)+{(\Delta \omega_i)}_{sp-ph}(T)
\end{eqnarray}
The term $\omega_i(0)$ corresponds to the phonon frequency at absolute zero. In eqn. 1 above, the first term on the right hand side corresponds to quasiharmonic contribution to the frequency change. The second term corresponds to the intrinsic anharmonic contribution to phonon frequency that comes from the real part of the self-energy of the phonon decaying into two phonons (cubic anharmonicity) or three phonons (quartic anharmonicity). The third term ${(\Delta \omega_i)}_{el-ph}$ is the renormalisation of the phonon energy due to coupling of phonons with charge carriers in the system which is absent in insulating pyrochlore titanates. The last term, ${(\Delta \omega_i)}_{sp-ph}$, is the change in phonon frequency due to spin-phonon coupling arising from modulation of the spin exchange integral by the lattice vibration. Recently, we have shown \cite{PRB-78-214102} that the magnitude of phonon anomalies is comparable in both magnetic and non-magnetic pyrochlore titanates, thus ruling out any contribution from spin-phonon coupling. Therefore, the change in phonon frequency is solely due to quasiharmonic and intrinsic anharmonic effects whose temperature variations, as estimated below for the modes P1, P3, P5 and P7, are shown in Fig. \ref{Fig:3}.
The change in phonon frequency due to quasiharmonic effects ($(\Delta \omega_i)_{qh} (T)$) comes from the change in the unit cell volume. This change can be expressed as \cite{Born-n-Huang},
\begin{equation}
{(\omega_i)}_{qh}(T) - \omega_i(0) = (\Delta \omega_i)_{qh} (T) = \omega_i(0) \hspace{1.5 mm} exp\left({\displaystyle\int^{T}_{0} \gamma_i(T^{\prime}) \alpha_v(T^{\prime})\, dT^{\prime}}\right)
\end{equation}
where $\omega_i(0)$ is the frequency of the $i^{th}$ phonon mode at 0 K, $\gamma_i(T^{\prime})$ is the temperature-dependent Gr\"{u}neisen parameter of that phonon and $\alpha_v(T^{\prime})$ is the temperature-dependent coefficient of the volume expansion. Since our lowest temperature is 27 K, the quasiharmonic change can be approximated as,
\begin{equation}
(\Delta \omega_i)_{qh} (T) \approx \omega_i(27K) \hspace{1.5 mm} \gamma_i \hspace{1.5 mm} exp \left({\int^{T}_{27K} \alpha_v(T^{\prime})\, dT^{\prime}}\right)
\end{equation}
assuming the Gr\"{u}neisen parameter to be temperature independent. To measure the $\alpha_v(T)$, we have recorded x-ray diffraction patterns of Tb$_2$Ti$_2$O$_7$ from room temperature to 10 K. We present the temperature-dependent lattice parameter in Fig. \ref{Fig:4}. Our data agree with the recent data by Ruff \textit{et al.} \cite{PRL-99-237202}. The solid line in Fig. \ref{Fig:4} is a fit to our data by the relation $a(T)=a_0[1+\frac{be^{c/T}} {T(e^{c/T}-1)^2 }]$, where $a_0$=10.14 $\AA$ is the lattice constant at 0 K and b=9.45 K and c=648.5 K are fitting parameters \cite{KittleBook}. In a recent study by Ruff et al. \cite{PRL-99-237202}, it was shown that the lattice undergoes an anomalous expansion along with broadening of allowed Bragg peaks as temperature is reduced below $\sim$ 10 K. This was attributed to structural fluctuation from cubic-to-tetragonal lattice that consequently coincides with the development of correlated spin-liquid ground state in Tb$_2$Ti$_2$O$_7$. Our data are up to 10 K and hence, we could not observe this feature at low temperatures. We have derived the temperature-dependent coefficient of thermal expansion ($\alpha_v = \frac{3}{a_0} \frac{da}{dT}$) from the temperature-dependent lattice parameter which is shown in the inset of Fig. \ref{Fig:4}. The $\alpha_v$ at 300 K for Tb$_2$Ti$_2$O$_7$ is ($\sim 8\times10^{-5} K^{-1}$), slightly higher than those of Dy$_2$Ti$_2$O$_7$ ($\sim 6\times10^{-5}K^{-1}$) and Lu$_2$Ti$_2$O$_7$ ($\sim 4\times10^{-5}K^{-1}$), estimated from the temperature-dependent x-ray diffraction results reported in ref. \cite{PRB-78-214102}. We note that $\alpha_v(300K)$ for Tb$_2$Ti$_2$O$_7$ is about 10 times higher than that of Si \cite{JAP-56-314} and nearly 7 times higher than that of Gd$_2$Zr$_2$O$_7$ \cite{PML-84-127}, implying that the anharmonic interactions in Tb$_2$Ti$_2$O$_7$ are strong. The mode Gr\"{u}neisen parameter for $i^{th}$ phonon mode is $\gamma_i=\frac{B}{\omega_i} \frac{d\omega_i}{dP}$, where $B$ is the bulk modulus, $\frac{d\omega_i}{dP}$ is the frequency change with pressure $P$. Taking B=154 GPa, obtained from our high pressure x-ray diffraction data discussed later, we find the values of the Gr\"{u}neisen parameter for the various modes as listed in Table-I. The change in phonon frequency due to quasiharmonic effect, $(\Delta \omega_i)_{qh}(T)$, has been estimated for the modes P1, P3, P5 and P7, and is shown in the insets of Fig. \ref{Fig:3}. The anharmonic contribution, $(\Delta \omega_i)_{anh}(T)=(\Delta \omega_i)_{total}(T)-(\Delta \omega_i)_{qh}(T)$, for the modes P1, P3, P5 and P7 are shown in Fig. \ref{Fig:3}. We note that the temperature-dependent $(\Delta \omega_i)_{anh}(T)$ for these four modes is anomalous. Further, upon changing the temperature from 27 K to 300 K, we find that for the mode P1, the percentage change in frequency due to anharmonic interactions, $(\Delta \omega_i)_{anh}(T)/\omega_i(27K)$, is exceptionally high. It is customary to fit the $(\Delta \omega_i)_{anh}(T)$ data by the expression \cite{PRB-28-1928},
\begin{equation}
(\Delta \omega_i)_{anh}(T) = C \left(1+\frac{2}{e^{\frac{\bar{h}\omega_i(0)}{2k_BT}}-1}\right)
\end{equation}
where the $i^{th}$ phonon decays into two phonons of equal energy ($\omega_i \rightarrow \frac{\omega_i}{2} + \frac{\omega_i}{2}$). The parameter ``$C$'' can be positive (for normal behavior of phonon) or negative (anomalous phonon) \cite{PRB-78-214102, SSC-117-201}. We have seen that eqn. 4 does not fit to our data of $(\Delta \omega_i)_{anh}(T)$ (fitting not shown in Fig. \ref{Fig:3}). This may be because, in the expression for $(\Delta \omega_i)_{anh}(T)$ (eqn. 4), all the decay channels for the phonons are not taken into account. Therefore, a full calculation for the anharmonic interactions considering all the possible decay channels is required to understand the $(\Delta \omega_i)_{anh}(T)$ data, shown in Fig. \ref{Fig:3}.
Considering only the cubic phonon-phonon anharmonic interactions where a phonon decays into two phonons of equal energy, the temperature-dependent broadening of the linewidth can be expressed as \cite{PRB-28-1928}:
\begin{equation}
\Gamma_i(T) = \Gamma_i(0) + A \left(1+\frac{2}{e^{\frac{\bar{h}\omega_i(0)}{2k_BT}}-1}\right)
\end{equation}
where $\omega_i(0)$ is the zero temperature frequency and $\Gamma_i(0)$ is the linewidth arising from disorder. Fig. \ref{Fig:5} shows the temperature dependence of linewidths of the Raman modes P3, P4 and P5. It can be seen that the linewidth of P3 and P4 modes are almost double of the linewidth of the P5 mode, as reported by Maczka \textit{et al.} \cite{PRB-78-134420}. These authors have attributed this to the strong coupling of the $F_{2g}$(P3) and $E_g$(P4) phonons with the crystal field transitions of Tb$^{3+}$ which is absent for the $A_{1g}$ (P5) mode due to symmetry consideration. To strengthen this argument, we compare (Fig. \ref{Fig:5}) these results with the linewidths of the corresponding phonons in non-magnetic Lu$_2$Ti$_2$O$_7$ (Lu$^{3+}$: J=0) \cite{PRB-78-214102} and, indeed, the linewidths of P3 and P4 modes in Tb$_2$Ti$_2$O$_7$ are much broader than those in Lu$_2$Ti$_2$O$_7$. The change in linewidth of the $F_{2g}$(P3) mode in Tb$_2$Ti$_2$O$_7$ from room temperature down to 27 K is nearly half the change in linewidth of the same mode in Lu$_2$Ti$_2$O$_7$ and, therefore, the parameter ``$A$'' for $F_{2g}$(P3) mode in Tb$_2$Ti$_2$O$_7$ is 7.2 cm$^{-1}$, which is nearly half of that in Lu$_2$Ti$_2$O$_7$ ($A$=13.4 cm$^{-1}$). However, the linewidth of the $A_{1g}$ mode for both titanates is comparable and, therefore, the fitting parameter ``$A$'' for this mode in Tb$_2$Ti$_2$O$_7$ and Lu$_2$Ti$_2$O$_7$ are nearly the same, i.e., 7.8 cm$^{-1}$ and 7.3 cm$^{-1}$, respectively. All these results, therefore, corroborate the suggestion of Maczka \textit{et al.} \cite{PRB-78-134420} thus emphasizing a strong coupling between the phonon and crystal field modes.
\subsection{Effect of pressure on Tb$_2$Ti$_2$O$_7$}
\subsubsection{Raman study}
Fig. \ref{Fig:6} shows room temperature Raman spectra at ambient and a few high pressures, the maximum pressure being $\sim$ 25 GPa. We could not resolve P2 and P9 at room temperature inside the high pressure cell due to the reasons described above. The phonon frequencies increase with increasing pressure, as shown in Figs. \ref{Fig:6} and \ref{Fig:7}. Interestingly, we find that upon increasing the pressure, the intensity of the P1 mode diminishes and is no longer resolvable above $\sim$ 9 GPa. On decompressing the sample from $\sim$ 25 GPa, the mode recovers, as shown in the top panel of Fig. \ref{Fig:6}. Similarly the mode P6 ($F_{2g}$) also vanishes above $\sim$ 9 GPa and reappears on decompression. The intensity ratios of the modes P1 to P3 and P6 to P5, as shown in Fig. \ref{Fig:8}, gradually decrease with increasing pressure and become zero near 9 GPa. As shown in Fig. \ref{Fig:7}, the maximum change in phonon frequency is seen in mode P7 ($F_{2g}$), which shows a dramatic change in the rate of change of frequency with pressure at a pressure of $\sim$ 9 GPa. In sharp contrast, the other modes P3, P4 and P5 do not show any change in slope till the maximum pressure applied. The changes seen in the modes P1, P6 and P7 near 9 GPa are indicative of a structural transition of the Tb$_2$Ti$_2$O$_7$ lattice. In order to ascertain the structural transition we have performed high pressure x-ray diffraction measurements and the results are discussed below.
\subsubsection{X-ray diffraction}
Fig. \ref{Fig:9} shows the x-ray diffraction patterns of Tb$_2$Ti$_2$O$_7$ at a few high pressures. The (hkl) values are marked on the corresponding diffraction peaks. As we increase the pressure, we find that the diffraction peaks shift to higher angles but no signature of new peak or peak splitting could be observed. However, the change in lattice parameter with pressure, shown in Fig. \ref{Fig:10}, shows a change in slope near 9 GPa implying a structural deformation, thus corroborating the transition observed in the Raman data. The transition possibly involves just a local rearrangement of the atoms retaining the cubic symmetry of the crystal. Fitting the pressure-dependent volume to the third order Birch-Murnaghan equation of state \cite{JGeophysRes-83-1257}, we find that B = 154 GPa and B$^{\prime}$=6.6 when the applied pressure is below 9 GPa. But, when the applied pressure is above this transition pressure, these values change to B = 250 GPa and B$^{\prime}$ = 7.1 thus implying an increment of the bulk modulus by $\sim$ 62\% after the transition. A similar transition had also been observed \cite{PRB-74-064109} in Gd$_2$Ti$_2$O$_7$ at $\sim$ 9 GPa and was attributed to the TiO$_6$ octahedral rearrangement. It needs to be mentioned here that the pressure transmitting medium (methanol-ethanol mixture, used in our Raman experiments) remains hydrostatic up to 10 GPa which is close to the transition pressure, thus implying that the possibility of a contribution from non-hydrostaticity of the medium cannot be completely ruled out. However, experiments in a non-hydrostatic medium (water) has as well revealed the transition at $\sim$ 9 GPa in Gd$_2$Ti$_2$O$_7$ \cite{PRB-74-064109}. We, therefore, believe that the transition near 9 GPa is an intrinsic property of Tb$_2$Ti$_2$O$_7$ and also that performing this experiment with helium as the pressure transmitting medium, will further strengthen our suggestion of a possible transition at $\sim$ 9 GPa.
\section{Summary and Discussion}
We have performed temperature and pressure-dependent Raman and x-ray diffraction studies on pyrochlore Tb$_2$Ti$_2$O$_7$ and the main results can be summarised as follows: (1) The phonon frequencies show anomalous temperature dependence, (2) the linewidths of the $F_{2g}$ and $E_g$ modes near 300 cm$^{-1}$ are unusually broad in comparison to those of non-magnetic Lu$_2$Ti$_2$O$_7$ phonons, thus corroborating the suggestion \cite{PRB-78-134420} of a possible coupling between phonons and crystal field transitions, (3) intensities of two phonon modes (P1 and P6) decrease to zero as the applied pressure approaches 9 GPa. Another Raman band P7 near 672 cm$^{-1}$ ($F_{2g}$) shows a large change in slope ($\frac{d\omega}{dP}$) at $\sim$ 9 GPa, thus indicating a possible transition, (4) x-ray diffraction study as a function of pressure reveals an increase in bulk modulus by $\sim$ 62\% when the applied pressure is above 9 GPa thus corroborating the transition suggested by Raman data. The phonons in Tb$_2$Ti$_2$O$_7$ show anomalous temperature dependence which has been attributed to the phonon-phonon anharmonic interactions \cite{PRB-78-214102}. Using the required parameters ($\gamma$, B and $\alpha_v$), derived from our high pressure and temperature-dependent Raman and x-ray experiments, we have estimated the contributions of quasiharmonic and anharmonic effects (Fig. \ref{Fig:3}) to the phonon frequencies. We note that the anharmonicity of the mode P1 (mode near 200 cm$^{-1}$) is unusually high as compared to other modes. P1 is a phonon mode which do not involve oxygen but includes the vibrations of Ti$^{4+}$ ions \cite{DTO18}. This can be qualitatively understood by examining how Ti$^{4+}$ and Tb$^{3+}$ ions are coordinated. There are tetrahedra in the unit cell which are occupied by Ti$^{4+}$ ions at the vertices with a vacant 8$a$-site inside. The later will tend to make the vibrational amplitudes of Ti$^{4+}$ ions larger and thus contributing to the high anharmonic nature of the P1 mode. The high anharmonic behavior of the Raman modes involving 48$f$-oxygen ions arises due to the fact that the O$_{48f}$ anions are off centered towards the 8$a$-vacant site from their ideal position $\frac{3}{8}a$ to $(\frac{3}{8}-x)a$ inside the tetrahedra \cite{PSSC-15-55} whose two vertices are occupied by Ti$^{4+}$ and other two by Tb$^{3+}$. Here $a$ is the lattice parameter and $x$ is the O$_{48f}$ positional parameter. This anharmonicity is reflected in the high root mean squared displacement ($\sqrt{\langle u^2 \rangle}$) of O$_{48f}$ atoms : $\frac{\sqrt{\langle u^2 \rangle}}{d_{Tb-O}} \approx 3\%$ \cite{PRB-69-024416}, where $d_{Tb-O}$ is the Tb-O bond length.
Pressure-dependent Raman data show that two Raman modes, P1 and P6, cannot be seen above $P_c \sim$ 9 GPa and the P7 ($F_{2g}$) Raman band shows a significant change in the slope ($\frac{d\omega}{dP}$) at $P_c$. These results suggest a subtle stuctural deformation which gets corroborated by a change in bulk modulus seen in pressure-dependent x-ray experiments. However, the pressure-dependent x-ray data do not reveal any new diffraction peak or splitting of line. This implies that the structural deformation near 9 GPa, as inferred from the Raman study, is a local distortion of the lattice. It may be possible that as pressure increases, due to the vacancies at the $8a$-sites, the Ti$^{4+}$-ions adjust their local coordinates with a concomitant relocation of other atoms in the lattice. At this instant, we would like to recall the results of a neutron scattering experiment on Tb$_2$Ti$_2$O$_7$ by Mirebeau \textit{et al.} with a simultaneous change in pressure and temperature \cite{Nature-420-54}. It was seen that at 1.5 K, antiferromagnetic correlations develop in Tb$_2$Ti$_2$O$_7$ at a pressure of 8.6 GPa. This was attributed to the delicate balance among the exchange coupling, crystal field and dipolar interactions that gets destroyed under high pressure. Our high pressure Raman and x-ray experiments on Tb$_2$Ti$_2$O$_7$ suggest a local rearrangement of the atoms near 9 GPa retaining the cubic symmetry which, we believe, may contribute to the antiferromagnetic correlations observed in neutron scattering experiments \cite{Nature-420-54}. The possibility of a structural transition in Tb$_2$Ti$_2$O$_7$ at low temperatures has recently been relooked. As discussed in section III(B), Ruff \text{et al.} \cite{PRL-99-237202} suggested an onset of cubic-to-tetragonal structual fluctuations below 20 K. A simultaneous presence of a CF mode at $\sim$ 13 cm$^{-1}$ in Raman and infrared spectrosopic measurements led Lummen \textit{et al.} \cite{PRB-77-214310} to propose a broken inversion symmetry in Tb$_2$Ti$_2$O$_7$ at low temperatures. The authors suggested the presence of a second Tb$^{3+}$ site with different site symmetry at low temperatures. Followed by this, Curnoe \cite{PRB-78-094418} has proposed that a structural transition can occur at low temperatures with an $A_{2u}$ lattice distortion resulting in a change of the point group symmetry, leaving the cubic lattice unchanged. Our Raman spectroscopic observations of a transition near 9 GPa may be related to the above discussion and can contribute to the increase in magnetic correlation observed by Mirebeau \textit{et al.} \cite{Nature-420-54}. It will be relevant to do high pressure Raman experiments at helium temperatures to strengthen our suggestion.
\section{Conclusion}
To conclude, our Raman spectroscopic and x-ray diffraction experiments on single crystals of pyorhclore Tb$_2$Ti$_2$O$_7$, with temperature, reveal highly anomalous temperature-dependent phonons attributed to strong phonon-phonon anharmonic interactions. Our pressure-dependent Raman and x-ray diffraction experiments suggest a local deformation of the pyrochlore lattice near 9 GPa. We believe that our experimental results play an important role in enriching the understanding of pyrochlore titanates, especially the spin-liquid Tb$_2$Ti$_2$O$_7$, thus motivating further experimental and theoretical studies on these exotic systems.
\begin{acknowledgments}
We thank the Indo-French Centre for Promotion of Advanced Research (IFCPAR), Centre Franco-Indien pour la Promotion de la Recherche Avanc\'{e}e (CEFIPRA) for financial support under Project No. 3108-1. AKS thanks the Department of Science and Technology (DST), India, for partial financial support.
\end{acknowledgments}
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"redpajama_set_name": "RedPajamaArXiv"
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L.A. Noire: The VR Case Files
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 7,719
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Gurania nigrescens är en gurkväxtart som beskrevs av Charles Jeffrey. Gurania nigrescens ingår i släktet Gurania och familjen gurkväxter. Inga underarter finns listade i Catalogue of Life.
Källor
Gurkväxter
nigrescens
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{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 733
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Last year, Astro Nature were born for space sounds.
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{
"redpajama_set_name": "RedPajamaC4"
}
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Slovenj Gradec () je sedež Mestne občine Slovenj Gradec v Sloveniji ter upravno, gospodarsko, kulturno, bančno, informacijsko, zdravstveno, oskrbovalno in prometno središče Mislinjske doline in deloma tudi Koroške statistične regije Do razpada Avstro-Ogrske novembra 1918 je bilo mesto del Štajerske ter je del Koroške statistične regije od leta 2005. Samo mesto Slovenj Gradec ima 7249 prebivalcev (2020). V Slovenj Gradcu sta Splošna bolnišnica z urgentnim centrom za Koroško regijo in Šaleško dolino, Fakulteta za zdravstvene in socialne vede, Fakulteta za tehnologijo polimerov in Šolski center Slovenj Gradec z Gimnazijo Slovenj Gradec in višjo strokovno šolo. Poglavitni slovenjegraški kulturni ustanovi regionalnega pomena sta Koroški pokrajinski muzej in Koroška galerija likovnih umetnosti, pa tudi Knjižnica Ksaverja Meška in Glasbena šola Slovenj Gradec. Zaradi srečevanj umetnikov iz vsega sveta na pobudo slikarja in kulturnega organizatorja Karla Pečka je bil Slovenj Gradec leta 1989 s strani generalnega sekretarja OZN proglašen za mesto miru. V Slovenj Gradcu prav tako delujeta regionalni postaji Koroški radio in Koroška regionalna televizija. V letu 2022 je bil v Slovenj Gradcu odprt zimski in letni bazen.
Slovenj Gradec skupaj s sosednjima mestoma Ravne na Koroškem in Dravograd tvori tako imenovano somestje, ki predstavlja središče Koroške razvojne regije (imenovane tudi Regija treh dolin).
Opis mesta
Industrijski razvoj, kakršen je bil v 19. in 20. stoletju, je mestu prizanesel, zato je ohranilo tipično srednjeveško podobo z glavno ulico (Glavni trg), trgom Svobode in stransko ozko ulico (Meškova ulica). Ime Gradec je bilo prvič omenjeno leta 1091, kot mesto leta 1267. Zdaj obsega še naselja Legen, Štibuh, Polje in Lepa vas ter delno tudi naselje Stari trg.
Številna arheološka najdišča v okolici Slovenj Gradca pričajo, da je bila kotlina že od nekdaj pomembno območje naselitve ob strateški poti med v Celovško kotlino in osrednjo današnjo Slovenijo. Rimska poštna postaja Colatio je nastala na temeljih starejšega keltskega ali naselja Ilirov pod utrjenim gradiščem na Grajskem griču. Utrjen grad je tukaj stal že konec 11. stoletja. Grad nad današnjim Starim trgom je iz leta 1091. Utrjeno domovanje je skupaj z naselbino in posestvi v širši okolici po izumrtju Spanheimov po sorodstvenih vezeh pripadla slavni rodbini Andeških. V času vojvode Bertolda III. Andeškega (ok. 1185) je v trgu delovala pomembna kovnica. Slovenj Gradec se je uspešno razvijal na ravnici med potoki: Mislinja, Suhodolnica in Homšnica. Že pred letom 1267 je pridobil mestne pravice. Pod Habsburžani od 14. stoletja dalje se je mesto obdalo z močnim obzidjem.
Številne družine mojstrov umetnostnih obrti v 15. in 16. stoletju je vzcvetela v času baroka. Pasarske, pozlatarske in podobarske delavnice in kasnejše umetniške so oblikovale mesto. Med najpomembnejšimi sta bili slikarska družine Strauss in kiparska mojstra Janez in Jurij Mersija.
Državna cesta, ki obide mesto po vzhodnem obrobju je omogočila ureditev starega mestnega jedra brez tranzitnega prometa in razvoj storitvenih dejavnosti v zadnjih desetletjih.
Kulturna dediščina
Med najznamenitejše kulturne ustanove spadata Rojstna hiša Huga Wolfa in Paučkove bolnišnice. V mestu se nahaja tudi Koroška galerija likovnih umetnosti ustanovljena 1957 in Koroški pokrajinski muzej, ustanovljen leta 1951. Na robu starega mestnega jedra stoji Dvorec Rotenturn, ki je danes tesno vpet v mestno dogajanje. V mestu je vsako leto organiziranih več prireditev, mdr. Slovenjegraško poletje.
Cerkev svete Elizabete
Najstarejša zgradba v mestu je cerkev svete Elizabete, ki jo je oglejski patriarh Bertold leta 1251 posvetil svoji nečakinji. Od prvotne stavbe so se ohranile romanska ladja in dve romanski okni. Cerkev so v gotiki, renesansi in baroku dopolnili. Oprema je v glavnem baročna - Schoyev glavni oltar, Mersijeva prižnica, slike obeh Straussov, … Iz novejšega časa je Pandurjev križev pot.
Glavni oltar v cerkvi svete Elizabete je eden največjih dosežkov slovenske baročne oltarne arhitekture. Izdelal ga je Janez Jakob Schoy - deželni in dvorni kipar v Gradcu in vodilna osebnost štajerskega baročnega kiparstva. Sliko sv. Elizabete je naslikal Franc Mihael Strauss.
Hugo Wolf
Skladatelj Hugo Wolf se je rodil leta 1860. Družina je imela po obeh starševskih vejah slovenske prednike. Hkrati s širjenjem usnjarske obrti in vzpenjanjem po družbeni lestvici so doživljali tedaj običajen proces ponemčevanja. Vouki, kakor so se pisali do prihoda v Slovenj Gradec, so se v pretežno nemški tržni naselbini lažje uveljavili s priimkom Wolf. Hugo Wolf, ki je svetovno slavo dosegel v glasbeni prestolnici tedanje Evrope - na Dunaju, je svoja otroška leta preživel v Slovenj Gradcu. Mali Hugo je veljal za čudežnega otroka, saj je virtuozno obvladal violino. Ko je zaslovel na dunajskem glasbenem nebu, so ga cenili predvsem kot neprekosljivega mojstra samospevov. Hugo je umrl mlad, bolezensko blazen. Njegova mati je v spomin na sina v župnijski cerkvi dala postaviti poslikano okno, ki je še danes ohranjeno.
Koroški pokrajinski muzej
Stalne zbirke:
Kovaški in gasilski muzej Muta
Sokličev muzej
Rudarska zbirka v Črni na Koroškem
Etnografska zbirka v Črni na Koroškem
Povhov mlin
Ribiški muzej
Tako so nekoč živeli knapi
Razstave:
Gozdarska in lesarska zbirka
Železarska zbirka
Etnološka zbirka
Krauperška kašča
Arheološki parki:
Grad Ravne na Koroškem – Streiteben
Slovenj Gradec – srednjeveško obzidje
Rekonstrukcija rimske grobnice, Kolaciona, Stari trg pri Slovenj Gradcu
Dovže – vila rustika
Junija 2021 so delavnico dolgoletnega čevljarja Jožefa Levovnika spremenili v muzej čevljarstva.
Koroška galerija likovnih umetnosti
Ustanovljena je bila leta 1957, nahaja se v nekdanji mestni hiši in je ena izmed najbolj mogočnih stavb v mestu Njena razstavna politika upošteva avtorje različnih umetnostnih smeri, posebno pozornost namenjajo domačim likovnim ustvarjalcem, kot sta slikar Bogdan Borčič in Franc Berthold. Na ogled so na voljo še razne stalne zbirke kot so:
Zbirka del slovenskih likovnih umetnikov,
Mednarodna likovna zbirka,
Galerija na prostem – skulpture v parku na Štibuhu in v mestnem jedru,
Tretjakova Afriška zbirka,
Muzej socialne estetike Pina Poggija (v nastajanju),
Zbirka Hommage Jožetu Tisnikarju in
Zbirke del Franca Bernekerja, Bogdana Borčića, Valentina Omana ter Štefana Planinca.
Znane osebnosti, povezane z mestom
Sklici in opombe
Zunanje povezave
Mestna občina Slovenj Gradec
Bernekerjeva pot
Naselja Mestne občine Slovenj Gradec
Mesta v Sloveniji
|
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Stories of Hawai'i
All Revved up with No Place to Go (Part 1)
About "All Revved up with No Place to Go"
At the End of the Rainbow (Part 1)
About "At the End of the Rainbow"
Hawaii Nei (Part 1)
About "Hawaii Nei"
The Yellow Skirt (Part 1)
About "The Yellow Skirt"
New Blog Format
"So, you live in Hawaii now?" asked Joshua, wanting to change the subject.
"No, I live in Alaska," replied the Ginger Boy, and as Joshua shifted in his seat, the Ginger Boy scrunched up his nose laughing again, looking to Lindsay for affirmation.
"Sure, Einstein! Where else would I live? I live right over there," he pointed vaguely toward Diamond Head. "I'm not local or anything though. I told you before."
"Where are you from?" asked Lindsay.
"I'm from Texas. Do you believe me?" he asked with a wide, open-mouth smile. Lindsay smiled sheepishly. "I'm just playing. I'm from Texas. I've been living here for a year, give or take. I had to leave for personal reasons though." He waited to see if the couple would ask him to elaborate. When they didn't, he took another cigarette from the pack in his pocket and slowly lit it.
Sensing the Ginger Boy expected her to say something, Lindsay said, "So isn't Hawaii really expensive? We read it's the second most expensive state. Milk costs a fortune, doesn't it?"
The Ginger Boy snorted as if he were offended. "What are you talking about? You guys are the ones from New York. Anyway I manage. I do all right, you could say." He looked over his left shoulder, then his right, his cigarette hanging from his lip like an ugly, pockmarked James Dean. "Wanna know how?"
Again, the couple looked at each other, hoping the other would answer.
"I'm a software developer," he said, leaning backward in his backward chair. The couple looked relieved. He let them relax for a moment and when it looked like Lindsay was going to speak, he interjected. "Ha, no I'm not! But I can't lie to you guys. You know me. I'll tell you the truth." He leaned in and lowered his voice, trying to create a sense of solidarity between the three. "I'm a drug dealer."
The couple blinked hard, nonplussed.
"What, now you don't believe me? It's the truth. Check this out. "He lifted up his shirt, revealing several long, raised scars on his shoulders, chest, and torso. "You don't get those at the office, sister."
"That's awful!" gasped Lindsay, cupping her hands over her mouth. "How—I mean—are you all right?"
"Me? Look at me! I'm fine, but you should see the other guy—not so fine. In fact," he looked around and leaned in, "I shot the other guy—four other guys—killed two of 'em. I had nightmares for like a whole week. Don't worry though, that was a long time ago. I try and keep myself away from that stuff these days. That's why I'm here." The Ginger Boy offered his hand to Joshua. "What's your name, by the way?"
"Josh," he said, hesitantly taking the Ginger Boy's sticky, wet hand.
"Get the fuck out! Oh, pardon my French again. You're not gonna believe this but me too! You guys can call me Jay, though—like the bird. Man, same name and both left-handed. What a coincidence. We totally have to be related, I mean way back. Although, I gotta tell you both when I got these," he patted his chest where he had shown them his scars, "I lost a lot of blood. I lost tons of blood and I went into a coma. But here's the thing—and this is amazing—when I woke up—you're not going to believe this, but when I woke up, I became ambidextrous." He switched his cigarette from his left hand to his right hand, then back again. "See?"
"That is amazing," Joshua said with saccharine excitement. He was beginning to get annoyed and that was his way of having fun with the conversation. Ginger Jay pretended not to notice.
"You're damn right it is. Anyway, I made out in that deal. I mean the deal I had to get rid of those people over," he said, lowering his voice. "But that's why I had to leave Texas. It's a big state, but not that big. I figured they couldn't find me and the hundred grand here."
Again, the couple was silent.
"I see you don't believe me, but look at this bulge in my pocket. That's not my peen. Ha, ha."
"That's just a lot of money," said Lindsay. "It's lucky for you they let you get away."
"Well, they didn't exactly let me get away. I gave them the slip and there was no way I was going to let them get this back after what I went through to get it." He nonchalantly reached into his pocket and, to the couple's amazement, pulled out a wad of large bills. He carefully put them back in his pocket and acted for the benefit of anyone watching as if he had just been reaching for another cigarette, which he put in his mouth and lit.
"Yup, that's what's left of the hundred grand. I mean not only that. I came here with a hundred grand and I only have fifty left now." He fiddled with a faded tattoo on his arm.
"See this?" he pointed to the tattoo, snapping his fingers. "I have my last name tattooed on my arm. You know, in case of emergencies. Can you see it?"
Joshua nodded.
"You can see it, right? Spell it."
Joshua couldn't really see the tattoo. It was so poorly done that the ink bled and the letters ran into one another. To make matters worse, Jay's skin looked as if he had some kind of rash that he had scratched to the point of rubbing it raw. Joshua squinted as he tried to read the tattoo.
"Forget it," said Jay. "I have too many tattoos for you to waste your time on that one. See this one? This is my birthday. Another one I got in case of an emergency. What about you, tough guy? How old are you? What year were you born in? Where?"
"Where in the year or where in the country?" asked Joshua wryly, and was sorry the instant he heard himself say it.
"Oh," said Jay, his cigarette in the corner of a tight smile. "Oh. A wise ass, huh?"
Posted by Henry Grace at 11:06 AM
wakendawackerman November 6, 2022 at 1:26 AM
Playing roulette online is exciting and nice deal of} enjoyable – but it's always higher when you play at a roulette website that's right for you. There's an enormous variety of roulette games to play – have the ability to|you possibly can} play 3D Roulette, Golden Chip Roulette, Lucky Roulette – and much more. All three normal roulette variants 정카지노 are here , and you may play each one either for free cash or for free. The games are provided by RTG, and whereas the positioning was solely launched in 2020, it's fully licensed and secure to use. The bonuses include some somewhat complex rollover necessities, whereas blackjack variants are a tad on the quick facet.
us0b8rzkmp January 27, 2023 at 5:26 PM
Using today's machinery, plastic injection molders can inject resin very sluggish, very fast, or anyplace in between. If the fill rate simply too|is merely too} low, the material begins to chill before Fill Humidifiers the cavity is stuffed and the pressure required to fill the cavity goes up. If it's too sluggish, the injection rate inhibits cavity packing as a result of|as a end result of} the material cools in the course of the filling phase and the gate will freeze quick time} after the mildew is stuffed. Injection molding is used to create thin-walled plastic parts for a large variety of functions. They presumably be} a thin-walled enclosure, usually requiring several of} ribs and executives on the inside. These housings are used in selection of|quite a lot of|a big selection of} merchandise similar to family home equipment, consumer electronics, power tools, and automotive bumpers/dashboards.
Henry Grace
|
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{"url":"http:\/\/math.stackexchange.com\/questions\/423294\/indefinite-integration-int-fracdxx-sqrtx2-12","text":"# Indefinite Integration : $\\int \\frac{dx}{(x+\\sqrt{(x^2-1)})^2}$\n\nProblem : Solve : $\\int \\frac{dx}{(x+\\sqrt{(x^2-1)})^2}$......(i)\n\nI tried :\n\nLet $x =\\sec\\theta$ therefore , (i) will become after some simplification\n\n$$\\int \\frac{\\sin\\theta}{(1+\\sin\\theta)^2}d\\theta$$\n\nbut i think its wrong method of approaching please suggest further....thanks..\n\n-\n\nBetter to substitute $x=\\cosh{u}$. Then the integral becomes\n\n$$\\int du \\frac{\\sinh{u}}{(\\cosh{u}+\\sinh{u})^2} = \\frac12 \\int du\\, e^{-2 u} (e^{u}-e^{- u}) = \\frac12 \\left (\\frac13 e^{-3 u}-e^{-u}\\right)+C$$\n\nSubstituting back, where $e^{-u} = x-\\sqrt{x^2-1}$, we get that the integral is\n\n$$\\frac16 \\left ( x-\\sqrt{x^2-1}\\right)^3 - \\frac12 \\left ( x-\\sqrt{x^2-1}\\right) + C$$\n\n-\n\nHINT: $$\\frac1{x+\\sqrt{x^2-1}}=x-\\sqrt{x^2-1}$$\n\nSo, $$\\frac1{(x+\\sqrt{x^2-1})^2}=(x-\\sqrt{x^2-1})^2=x^2+x^2-1-2x\\sqrt{x^2-1}=2x^2-1-2x\\sqrt{x^2-1}$$\n\nPut $x^2-1=y$ for the last part\n\n-","date":"2015-07-28 15:57:39","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9533562660217285, \"perplexity\": 1623.489584117426}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2015-32\/segments\/1438042981969.11\/warc\/CC-MAIN-20150728002301-00067-ip-10-236-191-2.ec2.internal.warc.gz\"}"}
| null | null |
Das Pakayun ist ein Säbel aus Borneo.
Beschreibung
Das Pakayun hat eine gebogene, einschneidige Klinge. Die Klinge ist vom Heft bis zum Ort etwa gleich breit. Die Ortform variiert. Manche Versionen haben einen schräg abgeschnittenen Ort, andere einen leicht abgerundeten Ort. Die Klingen sind glatt oder mit zwei leichten Hohlschliffen versehen. Das Heft hat ein scheibenförmiges Parier und eine metallene Zwinge, die dazu dient, Heft und Klinge sicherer miteinander zu verbinden. Das Heft besteht immer aus Holz und ist in der für dieses Schwert typischen Form geschnitzt (siehe Bild Infobox und Fotos unter Weblinks). Es endet in zwei parallel zueinander liegenden, abgerundeten Enden. Diese sind schräg zum Heft hin angeordnet. Zwischen den Enden sind dünne Holzplatten ausgeschnitzt und mit Schnitzereien verziert. Manche von diesen reichen bis ans Ende der Knaufvorsprünge. Das Griffstück besteht zusammen mit dem Parier aus einer Messinghülse, die am Übergang zur Klinge scheibenförmig ausläuft ("umbo"). Die Metallhülse ist oberhalb des Pariers mit flachen Rattanschnüren umwickelt. Die Scheiden sind meist aus Holz, zweiteilig und mit Rattan, Schnüren aus Pflanzenfasern, oder Zinnblechen umwickelt. Die Unterbrechungen in der Wicklung sind mit figürlichen Schnitzereien verziert. In der Scheide kann ein kleines Stück Baumrinde angebracht sein, das mit Haaren beklebt ist. Das Pakayun wird von der Ethnie der Murut aus Borneo benutzt. Die Zuordnung, ob das Pakayun ein Schwert oder ein Säbel ist, ist strittig.
Einzelnachweise
Literatur
George Cameron Stone: A Glossary of the Construction, Decoration and Use of Arms and Armor in all Countries and in all Times. Together with some closely related Subjects. With an introduction by Donald J. LaRocca. Dover Publications, Mineola NY 1999, ISBN 0-486-40726-8, S. 478.
Robert Shelford: A Provisional Classification of the Swords of the Sarawak Tribes. In: The Journal of the Anthropological Institute of Great Britain and Ireland. Bd. 31, 1901, , S. 219–228, hier S. 220, 225, .
Weblinks
Mehrere Pakayun bei Oriental Arms
Pakayun bei Old Blades
Pakayun bei Vikingsword
Schwert
Säbel
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Q: Difference of comma and semicolon when assign values in one line in python I am writing if statement in Python and I need to assign multiple values.
In order to style it, I want all my assignment in one line. So I tried comma to separate them.
if True:
a=0.5, b=0.5
print(a), print(b)
This would have a syntax error:
SyntaxError: can't assign to literal
However, when I use semicolon it works.
if True:
a=0.5; b=0.5
print(a), print(b)
Why can comma work in print but not in assignment?
A: Simply put, commas in python are used to unpack tuples. When you use a comma in a print function, you're actually using two tuples. Semicolons are used as a separator, as they would be used in a language such as JS or C++, and are equivalent to a newline. A literal is essentially the opposite of a variable; a constant or fixed value.
If you want to do two assignments in one line, what you can do is; a, b = 0.5, 0.5
However, in your case, you can assign variables as such; a = b = 0.5
A: Commas in Python are used for things like function arguments and creating tuples and lists. If you notice the command-line output when running this
>>> print(a), print(b)
0.5
0.5
(None, None)
the (None, None) is the resulting tuple created from the output value of two print statements put together with a comma. None, None is the same as (None, None).
The printed 0.5 values are the so-called side-effect of the print statement. It is what ends up on your screen when printing -- but the return value of a print is actually None.
Also notice the effect of
>>> a=0.5, 0.6
>>> print(a)
(0.5, 0.6)
a is assigned both of the two values following the equal sign since there is a comma. And now we are getting closer to the solution, these two statements are identical:
a=0.5, b=0.5
a = (0.5, b) = 0.5
so Python attempts to assign the last 0.5 to the value of the previous statement, which it can't. Basically doing a literal assignment giving a syntax error:
>>> (0.6, 0.7) = 0.5
SyntaxError: can't assign to literal
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\section{Introduction}
With the widespread use of neural networks in life-critical applications, such as self-driving cars, commercial and government agencies are concerned about the security of deployed deep learning (DL) neural networks (NNs). One example is poisoning NN models during training with datasets containing triggers (trojans) for misclassification. A trojan is defined as a specific subset of training inputs that cause modifications of the NN weights in such a way that the NN-based classifications for inputs without and with trojans will differ. For example, a trojan can be a yellow sticky inside of a STOP sign picture \cite{Xu2019} in which case the classifications of STOP sign and STOP sign with yellow sticky will differ. When a poisoned NN model with trojans is used for inferencing, a user will not know about the introduced misclassification by adversaries unless the specific input for inferencing is presented with the trojan.
The motivation for this work is to gain basic insights about trojans, their interactions with NN architectures, NN measurements that can indicate the presence of trojans, and what algorithmic approaches can be successful in detecting trojans for a variety of NN architectures under computational constraints.
We address three problems in the aforementioned context.
The first problem is in creating an interactive environment, as shown in Figure~\ref{fig:00}, for quick evaluations of (1) NN models with varying complexities and hyper-parameters, (2) datasets with varying manifold representation complexities and class balance ratios, and (3) measurements based on varying approaches and statistical analyses. The second problem lies in designing NN efficiency measurements with understood sensitivity to variations in NN architectures, NN initialization and training, as well as dataset regeneration. The third problem is in devising an approach to detecting trojans embedded in NN models.
\begin{figure}[h]
\resizebox{.5\textwidth}{!}{
\includegraphics[
width=12cm,
keepaspectratio,
]{./figs/nncalculator2.png}
}
\centering
\caption{Interactive user interface of neural network calculator.}
\label{fig:00}
\end{figure}
The problems come with associated challenges. The first challenge lies in the interactivity requirement. As of today, DL NN architectures are very complex; from 60K parameters in LeNet \cite{Khan2019}, to common networks having millions and billions of parameters (160 billion reported in \cite{Trask2015}). Modern networks require hours or days to train on advanced graphics processing unit (GPU) cards \cite{Justus2019}.
The challenge of the second problem lies in the lack of explainable artificial intelligence (AI) \cite{Doran2018} and AI mathematical models \cite{Bruna2017}, \cite{Unser2019}, and \cite{Mallat2016}.
The last challenge lies in the large search space of possible trojans, training data, DL NN architectures, and NN training algorithms that must be understood. Related work is described in Section~\ref{section:related}.
Our approach to these challenges relies on designing a NN Calculator environment and is based on analyses of neuron states in fully connected layers of NNs. The NN Calculator is built on top of Tensorflow Playground \cite{Smilkov2017} by enabling all calculator operators on datasets and NNs, such as storing, retrieving, setting, adding, subtracting, and clearing memory containing training/testing data points and NN coefficients. Furthermore, the NN Calculator contains functionality for introducing a wide range of trojans, collecting NN state measurements, visualizing them, computing trojan sensitive probes, evaluating their robustness to NN training, and saving them for further analyses. The trade-off for interactivity of analyses is the input limitation to 2D dot patterns, the NN limitation to less than 7 hidden layers and 9 nodes per layer due to screen size, and the limitation to custom designed features derived from 2D dot patterns.
The novelty of the work lies in designing:
\begin{itemize}
\item a web-based NN calculator for the AI community interested in gaining research insights about NN performance under various configurations,
\item a Kullback-Liebler (KL) divergence based measurement of NN inefficiency,
\item an approach to detecting embedded trojans in AI models.
\end{itemize}
\section{Related Work}
\label{section:related}
The problem of detecting trojans in NN models has been posed as the Trojan in Artificial Intelligence (TrojAI) challenge by the Intelligence Advanced Research Projects Agency (IARPA) \cite{IARPA2020}. The challenges include round 0, 1, and 2 datasets consisting of trained NN models that classify input images into 5 to 25 classes of traffic signs. The goal of the challenge is to detect models trained without trojan (\texttt{TwoT}) and trained with trojan (\texttt{TwT}) based on the analyses of NN models in limited amount of time on the NIST computational infrastructure. The problem has many variations based on what information and computational resources are available for trojan detection (type of attack, type of model architecture, model coefficients, training data subsets, description of trojans, number of classes to be misclassified by embedding trojans, classes that are misclassified by trojans, models that have been trained with trojans, computational complexity limits imposed on the delivered solution, etc.). Other challenges related to TrojAI can be found, for example, in the Guaranteeing AI Robustness against Deception (GARD) challenge \cite{Siegelmann2019}. As of today, none of the challenges can be described in terms of their difficulty level which motivates our work.
The TrojAI challenge models were created with a variety of contiguous regions within a traffic sign defining a trojan. In the previous work, the problem of trojans in AI has been reported from the view point of detecting trojans \cite{Xu2019} \cite{Roth2019}, constructing trojan attacks \cite{Liu2018}, defending against trojans \cite{Liu2018a}, and bypassing trojan detectors \cite{Juin2019}. The problem of trojan presence is often related to the efficiency (or utilization) of DL NNs as introduced in the early publications about optimal brain \cite{LeCun1989} and optimal brain surgeon \cite{BabakHassibi1992}. A few decades later, the topics of pruning links and trimming neurons are being explored in \cite{Hu2016}, \cite{Li2017}, and \cite{Han2015} to increase an efficiency of Deep Learning (DL) NNs and to decrease NN model storage and computational requirements of model training. Our work is motivated by the past concepts of NN efficiency. However, our goal is to explore the hypothesis that NN models trained with trojans will demonstrate higher efficiency/utilization of NN than NN models trained without trojan. In comparison to previous work, our approach is focused on reliable measurements in the context of trojan detection and is investigating questions about where trojans are encoded. We assume that the models \texttt{TwoT} and \texttt{TwT} are neither under-fitted nor over-fitted \cite{Belkin2019}.
The problem of gaining insights about DL NNs has been approached by (1) mathematical modeling \cite{Bruna2017} (network layers), \cite{Unser2019} (activation functions), \cite{Mallat2016} (wavelets), (2) feature and network visualizations \cite{Zeiler2013} (across layers), \cite{Erhan2009}(higher layers), \cite{Zhou2015} (discriminative features),\cite{Smilkov2017} (fully connected layers at small scale), and (3) limited numerical precision of modeling to achieve `interactive' response \cite{Wu2016}(quantized NN for mobile devices), \cite{Rastegari2016} (binary weights for ImageNet), \cite{Gupta2015} (tradeoffs), \cite{Hubara2016} (binary NNs). Many insights are pursued with respect to representation learning \cite{Bengio2013}, expressiveness \cite{Simonyan2015a}, \cite{Lu2017}, and sensitivity and generalization (under- and over-fitting NN models) \cite{Novak2018}, \cite{Shwartz-Ziv2019}. From all past work, we leveraged the mathematical framework in \cite{Bruna2017}, visualization called Tensorflow Playground in \cite{Smilkov2017}, and efficiency and expressiveness concepts in \cite{Lu2017}.
\section{Methods}
We describe next the developed NN Calculator with trojan simulations followed by the design of NN inefficiency measurements and our approach to trojan detection.
\subsection{NN Calculator}
Our approach to designing NN Calculator aims at making it as similar as possible to a scientific calculator.
Unlike a scientific calculator, NN Calculator operates on datasets and NN coefficients as opposed to simple numbers. Thus, we reused the symbols for $MC$, $MR$, $M+$, $M-$, and $MS$ for clearing, retrieving, adding, subtracting, and setting memory with datasets (training and testing sets) and NN coefficients (biases and weights). The user interface is shown in Figure~\ref{fig:00} (top left and middle left) where the standard five symbols are preceded with NN or D to indicate whether the operation is applied to NN or data. In addition, we included NN model averaging and dataset regeneration in order to study variability over multiple training sessions and random data perturbations. Evaluating combinations of datasets and NNs in real time enables one to explore full factorial experiments for provided factors.
Most of the calculator settings are used for the main operations on datasets and NNs: training, inferencing, inefficiency computations, and robustness measurements (mean squared error (MSE)) for training, testing and inferencing of sub-sets. Additional operations include collecting neuron state histograms, and derived measurement statistics.
The remaining settings are used to view characteristics of datasets (noise, trojan), parameters of NN modeling algorithm (Learning Rate, Activation Function, Regularization, Regularization Rate), and parameters of NN training algorithms (Train to Test Ratio, Batch Size). In order to keep track of all settings, we added the option of saving all NN parameters and NN coefficients, as well as saving all inefficiency and robustness analytical results. The save options are shown in Figure~\ref{fig:00} (bottom left).
\subsection{Trojan Characteristics Modeled in NN Calculator}
In order to explore how to discriminate a model trained with trojan and a model trained without trojan, we added nine types of trojans to the NN Calculator. Our objective is to understand how the characteristics of trojans affect the trojan detection, i.e. the discrimination of models trained without trojan (\texttt{TwoT}) and trained with trojan (\texttt{TwT}).
We generalized trojan embedding characteristics to be described by
(1) number of trojans per class,
(2) number of trojans per contiguous region,
(3) shape,
(4) size , and
(5) location of trojans inside of a class region.
Figure~\ref{fig:02} illustrate the nine trojan embeddings.
\begin{figure}
\resizebox{.5\textwidth}{!}{
\includegraphics[
width=12cm,
keepaspectratio,
]{./figs/trojan_tablevis.png}
}
\centering
\caption{Illustration of nine trojan embeddings in four datasets. Orange dot - class 1, blue dot - class 2, red boundary encloses dots that represent a trojan embedding.}
\label{fig:02}
\end{figure}
\subsection{Neural Network Inefficiency Measurement}
For a given NN, its (in)efficiency is understood as the ratio of utilized representation states over the total number of representation states. Representation states are introduced next. In addition, we describe a NN inefficiency measurement from a histogram of NN states at each layer by using (1) KL divergence, (2) a reference state distribution, and (3) computational constraints.
\underline{States of Neural Network:}
In order to derive NN inefficiency, we must measure and analyze states of NN layers as training data are encoded into class labels in a typical classification problem. A state of one NN layer is defined as a set of outputs from all nodes in a layer as a training data point passes through the layer.
The output of a node is encoded as 1 if the value is positive and 0 otherwise.
Thus, for a point $d_{k}$ from a 2D dataset with points $[d_{k}=(x_{k}, y_{k}), c_{j}]$,
$k=1, ..., npts$ and $C=2$ classes $c_{1}={orange/N(negative)}, c_{2}={blue/P(positive)}$, it can generate one of $2^{nl}$ possible states at a NN layer with $nl$ nodes. Figure \ref{fig:03} (top left) shows how a training point $d_{k}$ is converted into a feature vector that enters a neuron of the layer 0. The neuron output is generated and converted to 0 or 1 via thresholding. The neuron outputs create states 0100, 110 and 10 at the three layers for an input point. Figure \ref{fig:03} (top right) presents a table with the state information for all training points at all layers. The combined histogram of states for all layers and both class labels (one color per layer) is shown in Figure \ref{fig:03} (bottom right). Finally, Figure \ref{fig:03} (bottom left) summarizes KL divergence values computed per layer and per label from the histogram of states.
\begin{figure}
\resizebox{.5\textwidth}{!}{
\includegraphics[
width=12cm,
keepaspectratio,
]{./figs/state-to-divergence.png}
}
\centering
\caption{The computation of KL divergence from NN state information at each layer per class label. }
\label{fig:03}
\end{figure}
\underline{Representation Power:}
We view the histogram of states as a probability distribution that indicates the utilization of a layer. In order to quantify the NN utilization, we leveraged the parallels between neural network and communication fields in terms of (a) NN representation power/capacity (channel capacity in communications), (b) NN efficiency (channel efficiency), and (c) the universal approximation theorem \cite{Hornik1991} (source coding theorem \cite{Shannon1948}).
According to the universal approximation theorem, we view the NN representation power (also denoted as expressiveness or model capacity or model complexity) as its ability to assign a training class label to each training point and create accurate class regions for that class. For instance, a NN must have at least two nodes ($nl=2$) in the final layer in order to assign four class labels (i.e., $ C = 4 \leq 2^{nl} = 4 \rightarrow \{00, 01, 10, 11\}$).
Once we gather the state information (see Figure \ref{fig:03} (top)), we can categorize the states into three categories:
\begin{samepage}
\begin{enumerate}
\item State is used for predicting multiple class labels.
\item State is used for predicting one class label.
\item State is not used.
\end{enumerate}
\end{samepage}
The first category is detected when a NN does not have enough nodes (insufficient representation power). It could also occur when a NN layer does not contribute to discriminating class labels (poorly trained NN). The second category suggests that a subset of data points associated with the same class label is represented by one state (efficient or inefficient representation).
The last category implies that a NN has a redundant (inefficient) node in a layer for representing a class label.
Thus, states at NN layers provide information about NN representation power as
(1) \emph{insufficient,} (2) \emph{sufficient and efficient,} or (3) \emph{sufficient and inefficient.}
An ideal NN is sufficient and efficient.
\underline{Inefficiency of Neural Network:}
Since the source coding theorem is based on calculating mutual information defined via KL divergence \cite{Kullback2017},
we adopt KL divergence as a measurement of how inefficient it would be on average to code one histogram of NN layer states using a reference histogram as the true distribution for coding, where the reference histogram is defined below as the outcome of a uniform distribution over states assigned to each label.
Figure \ref{fig:03} (bottom) shows example results of KL divergence values derived per layer and per label that can be used to compare against values obtained from other datasets; for instance, datasets with trojans.
The rationale behind choosing entropy-based KL divergence with probability ratios comes from three considerations. First, entropy-based measurement is appropriate because which state is assigned to predicting each class label is a random variable and a set of states assigned to predicting each class label is random. Second, probability-based measurement is needed because training data represent samples from the underlying phenomena. Furthermore, while training data might be imbalanced (a number of samples per class varies), all training class labels are equally important and the probabilities of classes should be included in the measurement. Third, the divergence measurement reflects the fact that we measure NN efficiency relative to a maximum efficiency of NN that is achieved when sets of states utilize the entire network capacity (representation power).
\emph{Mathematical definition:}
Formally, let us denote $Q_{j}=\{ q_{ij} \}_{i=1}^{n}$ to be a discrete probability distribution function (PDF) of $n$ measured NN states and $P_{j} = \{ p_{ij} \}_{i=1}^{n}$ to be the PDF of reference (ideal) NN states. The probabilities are associated with each state (index $i$) and each class label (index $j$). The KL divergence per class label $j$ is defined at each NN layer in Equation~\ref{eq:01}.
\begin{equation*}
D_{KL}(Q_{j} \parallel P_{j})=\sum_{i=1}^{n}(q_{ij}*\log_{2} {\frac{q_{ij}}{p_{ij}}})
\tag{1}
\label{eq:01}
\end{equation*}
where $q_{ij}=\frac{count(i,j)}{p_{j}*npts}$ is the measured count of states normalized by the probability $p_{j}$ of a class label $j$ and the number of training points $npts$.
The PDF of reference states per class label uniformly utilizes the number of states assigned to predicting each class label (i.e., 2 classes imply $\frac{1}{2}$ of all states per label). The reference probability distribution is uniform across all assigned states. Thus, all reference probabilities can be computed as $p_{ij}=m*\frac{1}{n}$ where $m$ is the number of classes and $n=2^{nl}$ is the maximum number of states ($nl$ is the number of nodes per layer).
Table~\ref{table:02} presents the theoretical definition of KL divergence with respect to input probabilities $q_{ij}$ and $p_{ij}$.
\begin{table}
\caption{Definition of KL divergence}
\label{table:02}
\centering
\begin{tabular}{ | c | c | c | }
\hline
\thead{$p_{ij}$ \textbackslash $\quad q_{ij}$ } & \thead{$q_{ij} = 0$} & \thead{$q_{ij} \neq 0$} \\
\hline
$p_{ij} = 0$ & 0 & \makecell{not defined} \\
\hline
$p_{ij} \neq 0$ & 0 & \makecell{defined} \\
\hline
\end{tabular}
\end{table}
Equation \ref{eq:01} for the Kullback–Leibler divergence is defined only if for all $x$, $p_{ij}=0$ implies $q_{ij}=0$. Whenever $q_{ij}=0$ the contribution of the corresponding term is interpreted as zero because
$\lim_{x \to 0} (x * \log_{2}x) = 0$.
The case of ``not defined'' takes place when there are more non-zero states than the number of non-zero reference states (i. e., the cardinality of two sets satisfies the equation: $|Set(q_{ij} \neq 0)| > |Set(p_{ij} \neq 0)|$). This case indicates that a NN has insufficient representation power to encode input dataset into a class label.
\emph{Expected properties:}
It is expected that KL divergence will satisfy a list of basic properties as datasets, features, and NN capacity vary. For example, given an input dataset and a set of features, inefficiency (KL divergence) per layer should increase for an increasing number of nodes per NN layer. In another example, given a NN capacity, inefficiency should decrease for datasets with added noise or trojans. The relative changes are expected to be larger than the KL divergence fluctuations due to data reshuffling, data regeneration from the same PDF or due to re-training the same NN (referred to as sensitivity of KL divergence).
\underline{Computational Consideration about Inefficiency:}
The KL divergence computation considers computational and memory complexities since it must scale with increasing numbers of class labels, nodes, and layers.
\emph{Memory concerns:}
One should create a histogram with the number of bins equal up to $2^{nl}$ per class label and per layer which can easily exceed the memory size.
For example, if a number of classes is $\approx 10$, a number of nodes is $\approx 100$, and a number of layers is $\approx 100$, then memory size is
$\approx 2^{100} * 10 * 100 \approx 10^{33}$ bytes.
In our implementation approach, we create bins only for states that are created by the training data which leads to the worst case memory requirement scenario to be $npts * 10 * 100$ bytes.
\emph{Computational concerns:}
One should align measured histograms per class label to identify the states uniquely encoding each class in order to avoid the ``not defined'' case of KL divergence or the case of the same state encoding multiple class labels. To eliminate the alignment computation in our implementation approach, we modify the KL divergence computation to approximate the KL divergence according to Equation \ref{eq:02}. The computation of modified KL divergence $\widehat{D_{KL}}$ requires only collecting non-zero occurring states and calculating their histogram.
\begin{equation*}
\widehat{D_{KL}}(Q_{j} \parallel P_{j})=\sum_{i \in Set(q_{ij} \neq 0)} (q_{ij} * \log_{2}{ q_{ij}} ) - \log_{2} \frac{m}{n}
\tag{2}
\label{eq:02}
\end{equation*}
While KL divergence satisfies $D_{KL} \leq 0$, the modified KL divergence $\widehat{D_{KL}}$ can be negative for those cases when $|Set(q_{ij} \neq 0)| > |Set(p_{ij} \neq 0)|$. However, the negative value is lower bounded by Equation \ref{eq:03}. For negative values, the NN layer is insufficient for encoding input data to class labels.
\begin{align*}
\max_{Q_{j}} ( D_{KL}(Q_{j} \parallel P_{j}) - \widehat{D_{KL}}(Q_{j} || P_{j}) ) = \notag\\
- \sum_{i \in Set(q_{ij} \neq 0)} (q_{ij} * \log_{2}{ p_{ij}} ) - \log_{2} \frac{m}{n}
\tag{3}
\label{eq:03}
\end{align*}
The rationale behind modified KL divergence is that (1) the alignment is not important for sufficient efficient and inefficient models (it is primarily important for insufficient models), (2) the approximation assumes $p_{ij} \neq 0$ at all non-zero states $q_{ij} \neq 0$ which yields negative modified KL divergence values as indicators of insufficiency, and (3) the alignment is important for detecting poorly trained models which could be using the same states for predicting multiple class labels while leaving all other available states in a NN layer unused. For the last case, we assume that all models were properly trained and class labels are not assigned at random. Furthermore, the modified KL divergence addresses the problem of different within-class variations in training data which can lead to one class needing more allocated states than some other class. The modified KL divergence can be extended in the future by estimating within-class variations and assigning the number of states per class accordingly. In the following section we show how we use the modified KL convergence to detect the presence of trojans in a network.
\subsection{Approach to Trojan Detection}
Our assumptions are that (1) we have only testing datasets without trojans and (2) NN models with trojan and without trojan have the same accuracy.
We can simulate many varying NN models, with 4 example datasets containing 2 classes, and nine types of trojans.
The simulations assume close to $100 \, \%$ model accuracy on training data (with or without trojan). The comparisons of modified KL divergence values are computed from \texttt{TwoT} and \texttt{TwT} models using datasets without trojans. The model \texttt{TwT} evaluated with datasets without trojans might have an accuracy less than $100 \, \%$ in simulations but the accuracy difference would be negligible in a real scenario (and the challenge models).
The comparisons are performed at each NN layer and for each class label. The simulation execution is interactive (i.e., execution time is on the order of seconds) and follows the steps:
(1) \emph{Select data}
(2) \emph{Train}
(3) \emph{Store model}
(4) \emph{Select other data}
(5) \emph{Restore model}
(6) \emph{Perform NN measurement.}
Our assumption is that the magnitudes of KL divergence values for a NN model trained with a trojan embedded in a particular class (\texttt{TwT}) are smaller than the magnitudes for a NN model trained without trojan for the same class (\texttt{TwoT}). Our approach toward trojan detection is summarized in Figure~\ref{fig:trojan}. The axes correspond to the class-specific deltas between modified KL divergence of models \texttt{TwoT} and \texttt{TwT}. The dashed lines are set at a value $\sigma$ that corresponds to the sensitivity of $\widehat{ D_{KL} }$ to NN re-training as well as to data regeneration and re-shuffling. The notation ``to'' and ``from'' in Figure~\ref{fig:trojan} refers to our inference about trojans causing data points ``from'' one class to be mis-classified ``to'' another class based on the deltas defined in Equation~\ref{eq:trojan} where $P$ and $N$ are the two classes shown as blue and orange in the NN Calculator.
\begin{equation*}
\begin{split}
\Delta(P)=\widehat{ D_{KL} }(TwoT/P) - \widehat{ D_{KL} }(TwT/P) \\
\Delta(N)=\widehat{ D_{KL} }(TwoT/N) - \widehat{ D_{KL} }(TwT/N)
\end{split}
\tag{4}
\label{eq:trojan}
\end{equation*}
\begin{figure}
\resizebox{.5\textwidth}{!}{
\includegraphics[
width=15cm,
height=6cm,
keepaspectratio,
]{./figs/trojan-detection-decision.png}
}
\centering
\caption{Trojan detection using the delta between modified KL divergence of models \texttt{TwoT} and \texttt{TwT} as defined in Equation~\ref{eq:trojan}. The values for dashed lines can be determined based on the sensitivity of deltas to data regeneration and reshuffling, as well as to multiple NN initializations and re-training.
}
\label{fig:trojan}
\end{figure}
\section{Experimental Results}
\label{exper_results}
Next, we describe the implementation details of NN Calculator and document properties of NN inefficiency measurements.
\subsection{NN Calculator}
NN Calculator is implemented in TypeScript. The code is available from a GitHub repository with the development instructions and deployment via GitHub pages \url{https://github.com/usnistgov/nn-calculator}.
The current list of features extracted from 2D datasets includes $X1, X2, X1^2, X2^2, X1*X2, \sin(X1), \sin(X2), \sin(X1*X2), \sin(X1^2+X2^2)$, and $X1+X2$. The code uses D3.js and Plotly.js JavaScript libraries for visualization. All analytical results are displayed in NN Calculator below the NN visualization. The results consist of a state histogram (bins for both classes) and tabular summaries. The state histogram is interactive while the numerical results are presented as tables with a unique delimiter for easy parsing.
To gain additional insights about state (although they might be computationally expensive for large NNs), simulations using NN Calculator report also the number of non-zero histogram bins per class, the states and their counts per layer and per label for most and least frequently occurring states, the number of overlapping states across class labels and their corresponding states, and the bits in states that are constant for all used states for predicting a class label. The additional information is reported for the purpose of exploring optimal NN architectures and investigating NN model compression schemes.
\subsection{Neural Network Inefficiency}
\underline{KL Divergence Properties:}
We verified and quantified desirable properties of the modified KL divergence defined in Equation~\ref{eq:02}, such as decreasing inefficiency for increasing amount of added noise and increasing inefficiency for increasing number of nodes.
\underline{Sensitivity of Inefficiency Measurement:}
We quantified the sensitivity of NN inefficiency measurement with respect to (a) data reshuffling and regeneration, (b) NN re-training with different initialization, and (c) no-training as the worst case of poor training.
To look at the sensitivity of the NN inefficiency with respect to data regeneration, we performed the following: a NN model is trained for a dataset and stored in memory. Next, four datasets are regenerated and a standard deviation of inefficiency values are computed at each layer and for each class. Finally, the average value is computed over all standard deviations and the experiment is repeated for four 2D datasets with the results presented in Figure~\ref{fig:sensit}. From the data regeneration points in in Figure~\ref{fig:sensit}, we concluded that the average of standard deviations in inefficiency values larger than $0.1$ will indicate dissimilarity of models by other factors.
We performed similar sensitivity experiments for no-training and retraining with random initialization. Figure \ref{fig:sensit} includes the results for four datasets. The sensitivity to retraining is bounded to approximately the average of inefficiency standard deviations equal to $0.46$ while the same value for no-training is about 5 to 8 times larger and appears to be proportional to the complexity of the class distribution.
\begin{figure}
\resizebox{.5\textwidth}{!}{
\includegraphics[
width=15cm,
height=6cm,
keepaspectratio,
]{./figs/sensitivity-kldivergence2.png}
}
\centering
\caption{Sensitivity of inefficiency to stochastic regeneration of datasets from the same distribution, retraining and no-training with different random initialization. The box plot shows values computed from a set of standard deviations of modified KL divergence per layer and per class for the four datasets.
}
\label{fig:sensit}
\end{figure}
\underline{Comparison of Inefficiencies for Trojan Embeddings:}
Comparisons of models \texttt{TwoT} and \texttt{TwT} were conducted in NN Calculator using a NN with 6 hidden layers, 8 nodes per layer and 4 features including $X1, X2, X1^2, X2^2$ and $X1*X2$. The algorithmic and training parameters are set to learning rate: $0.03$, activation: $Tanh$, regularization: none, ratio of training to test data:
$50 \, \%$, and batch size: $10$.
Figure~\ref{fig:08} shows the delta between modified KL divergence values of models \texttt{TwoT} and models \texttt{TwT} for the two classes P (blue) and N (orange) and for the two trojans (T1 and T2) of different sizes (Figure~\ref{fig:08} left). For both trojans, the delta KL divergence values are positive for the P (blue) class and negative for the N (orange) class: $\Delta(P)>0.454$ and $\Delta(N) < -0.702$. These values imply that a trojan is embedded in class P (blue) in both trojan cases and is encoding class N (orange) according to Figure~\ref{fig:trojan} (``From P to N'' $\rightarrow$ misclassified points labeled as P to N). Furthermore, as the size of a trojan increased from T1 to T2 by a size factor of 2.25, the ratio of deltas increased by $2.24$ for class N and by $2.37$ for class P.
\begin{figure}
\resizebox{.5\textwidth}{!}{
\includegraphics[
width=15cm,
height=6cm,
keepaspectratio,
]{./figs/comparison-trojan-size.png}
}
\centering
\caption{Comparison of inefficiencies between models \texttt{TwoT} and \texttt{TwT}, and embedded orange trojans T1 and T2 with different sizes (see Figure~\ref{fig:02}, top row). The plot shows the values of $\Delta(P)$ and $\Delta(N)$ for T1 and T2 at each NN layer.
}
\label{fig:08}
\end{figure}
Figure~\ref{fig:10} illustrates the delta between modified KL divergence values of models \texttt{TwoT} and models \texttt{TwT} for the trojans T8 and T9 whose embeddings differ in terms of the number of classes and the number of class regions. First, we observe for trojan T8 that
$\Delta(T8/P) > 0.48$ and
$\Delta(T8/N) < -0.769$. These values imply that the trojan T8 is embedded in class P (blue) according to Following Figure~\ref{fig:trojan} (``From P to N'').
We recorded much lower delta values for the trojan T9 than in the previous comparisons. This indicates the much higher complexity of modeling the spiral dataset than circle, exclusive OR, or Gaussian datasets and therefore lower inefficiency values measured at NN layers. Based on the sensitivity values shown in Figure~\ref{fig:sensit} ($0.1$ for data regeneration and $0.5$ for re-training), we could infer that the trojan T9 is likely in both classes based on the placement of
the point $[\Delta(T9/P) > -0.034, \; \Delta(T9/N) > 0.035]$ in Figure~\ref{fig:trojan} (i.e., the sub-spaces ``From N'', ``From P'', ``Not detectable'', and ``From N to P'' $+$ ``From P to N'').
Due to the discrete nature of the spiral pattern, the P class (blue) occupies a longer curve than the N class (orange). This contour length ratio
($P:N \approx 12.31:7.33$) can explain why
($\Delta(T9/P) > \Delta(T9/N)$ for almost all layers. However, we are not able to make any inferences about the number of regions from Figure~\ref{fig:10} (right) other than that the complexity of modeling class P or N in the case of T8 is more inefficient than modeling class P and N in the case of T9 by comparing the deltas of modified KL divergence values.
\begin{figure}
\resizebox{.5\textwidth}{!}{
\includegraphics[
width=15cm,
height=6cm,
keepaspectratio,
]{./figs/comparison-trojan-spiral.png}
}
\centering
\caption{Comparison of inefficiencies between models \texttt{TwoT} and \texttt{TwT}, and embedded trojans T8 and T9 with different number of classes (1 or 2) and class regions (1 or 4).
}
\label{fig:10}
\end{figure}
\section{Discussion about Trojan Detection}
\label{section:discussion}
One can obtain several additional useful insights from interactive analyses in NN Calculator before designing a trojan detection algorithm. In many of the results, it is apparent that the encoded class information is not in one layer but spread across multiple layers. Thus, trojan detection must include comparisons of vectors of $\widehat{ D^{l}_{KL}}$ across all layers $l$. Furthermore, the encoding of the same training data in NN can have multiple solutions, especially in inefficient NN and therefore the comparison of vectors of $\widehat{ D^{l}_{KL}}$ must include again a statistical nature of such solutions. Finally, the last layers carry less information about trojans because they serve the purpose of a final decision maker which should appear fair for datasets without trojans. This could be accommodated by weighting the layer-specific vector elements. From a global algorithmic design perspective, designing an actual trojan detector must still consider the trade-offs of doing all pair-wise model comparisons versus clustering all vectors of $\widehat{ D^{l}_{KL}}$ to identify the cluster of model TwoT.
\section*{Summary and Future Work}
We designed NN calculator and an inefficiency measurement for detecting trojans embedded in NN models. Our work is focused on measuring neural network inefficiency using KL divergence as a means to advance mathematical and statistical modeling of neural networks. Current modeling efforts suffer currently from a steep learning curve, hardware requirements, and time delays between experimental runs. Some of these drawbacks can be overcome by the NN Calculator since it is interactively accessible using a browser at \url{https://pages.nist.gov/nn-calculator/} and performing experiments does not require specialized hardware (i.e., GPU cards) nor long waiting times.
\section*{Acknowledgement}
The funding for Bajcsy and Majurski was provided by IARPA, and for Schaub was provided by NCATS NIH.
\section*{Disclaimer}
Commercial products are identified in this document in order to specify the experimental procedure adequately.
Such identification is not intended to imply recommendation or endorsement by the National Institute
of Standards and Technology, nor is it intended to imply that the products identified are necessarily the best available for the purpose.
\bibliographystyle{aaai}
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{"url":"https:\/\/www.physicsforums.com\/threads\/help-with-problem-about-beads-and-a-hoop.60391\/","text":"1. Jan 19, 2005\n\nnewcool\n\nThanks for the help\n\nLast edited: Jan 20, 2005\n2. Jan 19, 2005\n\nlearningphysics\n\n3. Jan 19, 2005\n\nnewcool\n\nThanks, I got the first part, anyone have any idea bout the second?\n\n4. Jan 19, 2005\n\nlearningphysics\n\nYes, the second part looks right to me.","date":"2017-01-17 05:34:20","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8855271339416504, \"perplexity\": 5019.328760262003}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-04\/segments\/1484560279468.17\/warc\/CC-MAIN-20170116095119-00419-ip-10-171-10-70.ec2.internal.warc.gz\"}"}
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Q: Reload gazetteer of UIMA Concept Mapper in runtime without restarting the UIMA pipeline Currently, I'm using UIMA concept mapper in UIMA pipeline to tag entities in text.
But the problem is that each time I change the gazetteer of one concept mapper annotator, I have to restart the whole UIMA pipeline. That means I have to restart all annotators in the pipeline.
Is there any way to reload the gazetteer for the concept mapper without restarting the pipeline?
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,341
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Q: How to configure redis.conf on openshift for redis Have a problem that by following the repo
https://github.com/razorinc/redis-openshift-example
When i start redis-server, it says "[12010] 25 Mar 20:14:53 # Opening port 6379: bind: Permission denied"
I tried to change port 0 to port 3128 but still get the same error....not sure why
--Update
When i tried to upgrade to redis 2.6 and uses --port parameter to bind to 3128, it still says
remote: [6844] 25 Mar 20:49:00.206 # Opening port 3128: bind: Permission denied
A: Here's the OpenShift forum thread with suggested modifications: https://www.openshift.com/forums/openshift/how-to-configure-redisconf-on-openshift-for-redis
Looks like calling redis-server uses default conf parameters that won't work well in the OpenShift gear environment so making the suggested changes to the redis.conf file and passing it into redis-server is the way to go. There are suggestions for a pre-start and pre-stop hook as well.
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"redpajama_set_name": "RedPajamaStackExchange"
}
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Q: Hyperbolic geometry and orientation reversing isometries. In Quasi-cluster algebras from non-orientable surfaces by Dupont and Palesi, one can read the following on page 11:
I don't understand why the 'following relations' in the image included hold. Applying $D$ to the point $(u,0)$ gives $(\mu u, 0)$. So surely $w = \mu u$.
What am I missing here? Am I being incredibly stupid?
A: Your key problem appears to be a misunderstanding of homogeneous coordinates. You homogenize by appending a one, not a zero:
$$
u\rightsquigarrow\begin{pmatrix}u\\1\end{pmatrix}\mapsto
\begin{pmatrix}\mu&0\\0&-1/\mu\end{pmatrix}
\begin{pmatrix}u\\1\end{pmatrix}=
\begin{pmatrix}\mu u\\-1/\mu\end{pmatrix}\sim
\begin{pmatrix}-\mu^2 u\\1\end{pmatrix}\rightsquigarrow
-\mu^2u=w
$$
The same holds for any other point in the plane (or complex line, depending on your point of view): the given matrix will scale every point by $-\mu^2$. Since $\mu>1$ it follows that $\mu\in\mathbb R$, so no special complex number magic applies. Therefore, the diameters of your circles will scale with the absolute value of that scale factor, i.e. with $\mu^2$.
A: Here's a expanion of the discussion in the comments.
The quoted passage is missing some important information. First, an equation like $U=(u,h)$ means that $U$ is the horocycle in the upper half plane tangent to $u$ on the real axis and of diameter $h$. Second, I'm guessing that $\lambda(U,V)$ refers to some measurement of the "signed hyperbolic distance" between the horocycles $U$ and $V$, in any case there should be an isometry taking one pair $(U,V)$ to another pair $(U',V')$ if and only if $\lambda(U,V)=\lambda(U',V')$. Third, the question of how to apply a matrix like $D$ having negative determinant seems to be fishy.
To answer the explicit question, there's a miscalculation of the fractional linear transformation as applied to points on the real line:
$$\begin{pmatrix} \mu & 0 \\ 0 & -1/\mu \end{pmatrix}(u) = \frac{\mu u + 0}{0u - 1/\mu} = -\mu^2 u
$$
But there's also an implicit question regarding how the fractional linear transformation $D$ transforms diameters. In general, given a hyperbolic isometry acting on a horocycle in the upper half plane, the diameter of the image horocycle is not a well-defined function of the diameter of the given horocycle. On the other hand, this is true for the specific fractional linear transformation $D = \begin{pmatrix} \mu & 0 \\ 0 & -1/\mu \end{pmatrix}$: a calculation shows that given any complex number $x+iy$, the imaginary part of the image $D(x+iy)$ is a well-defined function of $y$. To put it another way, this $D$ takes horizontal lines to horizontal lines. The formula for how $D$ acts on the $y$-coordinate can be obtained by plugging in the pure imaginary number $yi$:
$$\begin{pmatrix} \mu & 0 \\ 0 & -1/\mu \end{pmatrix}(iy) = \frac{\mu iy + 0}{0iy - 1/\mu} = - \mu^2 i y
$$
BUT, how can the imaginary part by negative? I think maybe the formula for applying a negative determinant matrix such as $D$ to a point in the upper half plane should not be $z \mapsto D(z)$ but instead $z \mapsto \overline{D(z)}$, but I cannot tell from what is pasted in the question.
Assuming something like that is appropriate, the summary is that $D$ takes points with imaginary coordinate $y$ go to points with imaginary coordinate $\mu^2 y$, and it takes horocycles of diameter $y$ to horocycles of diameter $\mu^2 y$.
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"redpajama_set_name": "RedPajamaStackExchange"
}
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"redpajama_set_name": "RedPajamaC4"
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\section{Introduction}
\label{intro}
\footnotetext{Based on observations collected at the European Southern
Observatory, Chile, proposal number ESO 68.C-0149(A).}
This is the last of a series of 5 papers devoted to present results of long-exposure high-spectral-resolution
spectral
data taken with the VLT UVES echelle spectrograph with the aim of obtaining accurate
measurements of very
faint permitted lines of heavy element ions in Galactic {\ion{H}{2}} regions. Our sample consists of
eight of the brightest
Galactic {\ion{H}{2}} regions which cover a range of Galactocentric distances from 6.3 to 10.4 kpc
(assuming the Sun
to be at 8 kpc from the Galactic center). The objects whose data have already been published are: NGC~3576
\citep{garciarojasetal04},
the Orion Nebula \citep{estebanetal04}, S~311 \citep{garciarojasetal05}, M16, M20, and NGC~3603
\citep{garciarojasetal06}.
Along this project we have detected and measured an unprecedented large number of emission
lines in all the {\ion{H}{2}} regions analyzed, which could improve the knowledge of the nebular gas conditions
and abundances.
We have derived chemical abundances of C$^{++}$ and O$^{++}$ from several recombination
lines of {\ion{C}{2}} and {\ion{O}{2}}, avoiding the problem of line blending in all the {\ion{H}{2}} regions of our sample.
The high signal-to-noise ratio of the VLT spectra of M8 and M17 has allowed us to detect and measure more
C$^{++}$ and O$^{++}$ RLs than in previous works (i.e., Esteban et al. 1999b, hereinafter EPTGR, in M8 and Esteban
et al. 1999a,
hereinafter EPTG, in M17);
also, the reliability of these lines
has increased significantly with respect to the previous detections.
{From} the observations of all the objects of our project, \citet{estebanetal05} obtained --for the first time--
the radial
gas-phase C and O gradients of the Galactic disk making use of RLs, which are, in principle, better for abundance
determinations because the ratio $X^{+p}$/H$^+$ from RLs is almost-independent of the temperature structure of the
nebula.
A reliable determination of these gradients is of paramount importance for chemical evolution models of our Galaxy
\citep[see][]{carigietal05}.
The fact that ionic abundances determined from the intensity of collisionally excited lines (CELs) are
systematically
lower (with factors ranging from 1.3 to 2.8) than those
determined by recombination lines (RLs) is far from being completely understood, and has led to the so-called
``abundance discrepancy'' problem. This problem is clearly present in Galactic {\ion{H}{2}} regions
\citep[see][and all papers related to this project]{peimbertetal93b, tsamisetal03}.
In the case of extragalactic studies, only a few works have been developed with the aim of
detecting the
faint recombination lines: \citet{estebanetal02} for M33 and M101, \citet{apeimbert03} and
\citet{tsamisetal03}
for the Magellanic Clouds, \citet{apeimbertetal05} for NGC~6822. Moreover, \citet{lopezsanchezetal06}
have, for NGC~5253, estimated abundance discrepancies rather similar to those of the Galactic objects.
One of the probable causes of the abundance discrepancy is the presence of spatial variations in
the temperature structure of the nebulae \citep{torrespeimbertetal80}. Temperature fluctuations
may produce the discrepancy due to the different functional dependence of the line emissivities of CELs
and RLs
on the electron temperature, which is stronger --exponential-- in the case of CELs.
Temperature fluctuations have been parametrized traditionally by \emph{$t^2$}, the mean-square
temperature fluctuation of the gas \citep[see][for a detailed formulation]{peimbert67,
peimbertcostero69, peimbert71}. It is a well known result that photoionization codes
cannot reproduce the temperature fluctuations found in gaseous nebulae, but there are mainly two
possibilities to explain them: first, there might be an additional important source of energy producing such
fluctuations,
which has not been taken into account by photoionization models;
second, there could be density inhomogeneities \citep{viegasclegg94} or chemical inhomogeneities
\citep[see][and references therein]{tsamispequignot05} that produce temperature variations.
The physical processes that may cause such temperature fluctuations
are still subject of controversy. Reviews of the relevant processes can be found in \citet{esteban02},
\citet{torrespeimbertpeimbert03}, and \citet{peimbertpeimbert06}.
Additionally, there are some very recent works devoted to this topic:
e.g., \citet{giammancobeckman05} have proposed ionization
by cosmic rays as an additional source of energy to reproduce the temperature fluctuations
observed in {\ion{H}{2}} regions; and \citet{tsamispequignot05} have developed photoionization
models for 30 Doradus in the Large Magellanic Cloud that reproduce observed temperature fluctuations through
chemical
inhomogeneities (inclusions) due to the infall of material nucleosynthetically processed in supernova
events. Further studies are needed to understand this problem.
Several spectrophotometric works devoted on the chemical composition of M8 and M17 have
been carried out previously. For M8, there are several low and intermediate spectral resolution studies
\citep{rubin69, peimbertcostero69, sanchezpeimbert91, peimbertetal93b, rodriguez99b} and one
high spectral resolution study (EPTGR).
The chemical abundances of M17 have been studied using low resolution spectroscopy \citep{rubin69,
peimbertcostero69,
peimbertetal92, rodriguez99b, tsamisetal03} and high-spectral resolution data (EPTG).
In this paper we make a reappraisal of the chemical composition of M8 and M17 in the
same slit position observed by EPTGR in M8 and one of the positions observed by EPTG in M17 (position 14),
by means of new echelle spectrophotometry obtained with the ESO's Very Large Telescope. Our new
observations increase significantly the number of lines detected and the quality of the measured line intensities
for these two nebulae.
In \S\S~\ref{obsred} and~\ref{lin} we describe the observations, the data reduction, and line intensity
determination procedures.
In \S~\ref{phiscond} we obtain temperatures and densities
using several diagnostic ratios.
In \S~\ref{helioabund} we briefly analyze the recombination spectra of
{\ion{He}{1}} and derive the He$^{+}$/H$^{+}$ ratio. In \S~\ref{cels} we give the ionic abundances
determined from CELs. In \S~\ref{recom} we use RLs to derive O and C ionic abundances.
In \S~\ref{abuntot} we present the total abundances. We report the detection of deuterium Balmer lines
in \S~\ref{deuterium}. In \S\S~\ref{comp}
and ~\ref{conclu} we present the comparison with previous results and the conclusions, respectively.
\section{Observations and Data Reduction}
\label{obsred}
The observations were made on 2002 March 11 with the Ultraviolet Visual Echelle Spectrograph, UVES
\citep{dodoricoetal00},
at the VLT Kueyen Telescope in Cerro Paranal Observatory (Chile). We used the standard settings in
both the red and
blue arms of the spectrograph, covering the region from 3100 to 10400 {\rm \AA}\ . The log of the
observations is presented in
Table~\ref{tobs}.
\setcounter{table}{0}
\begin{table}[htbp]
\centering
\setlength{\tabnotewidth}{\columnwidth}
\tablecols{3}
\setlength{\tabcolsep}{2.8\tabcolsep}
\scriptsize
\caption{Journal of observations.}
\label{tobs}
\begin{tabular}{l@{\hspace{2.8mm}}c@{\hspace{2.8mm}}c@{\hspace{2.8mm}}}
\toprule
& \multicolumn{2}{c}{Exp. time (s)}\\
\cmidrule{2-3}
$\Delta\lambda$~(\AA) & M8 & M17 \\
\midrule
3000--3900 & 30, 3 $\times$ 300 & 30, 3 $\times$ 300 \\
3800--5000 & 30, 3 $\times$ 800 & 60, 3 $\times$ 800 \\
4700--6400 & 30, 3 $\times$ 300 & 30, 3 $\times$ 300 \\
6300--10400& 30, 3 $\times$ 800 & 60, 3 $\times$ 800 \\
\midrule
\end{tabular}
\end{table}
The wavelength regions 5783--5830 {\rm \AA}\ and 8540--8650
{\rm \AA}\ were not observed due to a gap between the two CCDs used in
the red arm. There are also five small gaps that were not observed, 9608--9612 {\rm \AA},
9761--9767 {\rm \AA}, 9918--9927 {\rm \AA},
10080--10093 {\rm \AA}\ and 10249--10264 {\rm \AA}, because
the five redmost orders did not fit completely within the CCD. We took long and short exposure
spectra to check for possible saturation effects.
The slit was oriented east-west and the
atmospheric dispersion corrector (ADC) was used to keep the same observed
region within the slit regardless of the air mass value (the averaged $\sec$ z are $\sim$ 1.4 for M17 and $\sim$
1.85 for M8).
The slit width was
set to 3.0$\arcsec$ and the slit length was set to 10$\arcsec$ in the blue arm and to 12$\arcsec$ in
the red arm; the slit width was chosen to maximize the S/N ratio of the
emission lines and to maintain the required resolution to separate most of the
weak lines needed for this project. The effective resolution for the lines
at a given wavelength is approximately $\Delta \lambda \sim \lambda / 8800$.
The center of the slit was located in the same position than in EPTGR for M8 (labeled as HGS)
and is coincident with position 14 of EPTG for M17. The final 1D spectra were extracted from an area
of 3$\arcsec$$\times$8.3$\arcsec$.
The spectra were reduced using the {\sc IRAF}\footnotemark{} echelle reduction
package, following the standard procedure of bias subtraction,
flatfielding, aperture extraction, wavelength calibration and flux calibration.
The standard star EG~247 was observed for flux calibration. We have not attempted sky subtraction
from the spectra due to the slit length is much smaller than the objects; also, the spectral resolution of our data
permit us to clearly distinguish among the telluric lines and the nebular ones.
\footnotetext{{\sc IRAF} is distributed by NOAO, which is operated by AURA,
under cooperative agreement with NSF.}
\section{Line Intensities and Reddening Correction}
\label{lin}
Line intensities were measured integrating all the flux in the line between two
given limits and over a local continuum estimated by eye. In the cases of line blending,
a multiple Gaussian profile fit procedure was applied to obtain the line flux of each
individual line. Most of these measurements were made with the SPLOT routine of the {\sc IRAF}
package. In some cases of very tight blends or blends with very bright telluric lines the
analysis was performed via Gaussian fitting making use of the Starlink DIPSO software
\citep{howardmurray90}.
Table~\ref{lineidm8m17} presents the emission line intensities of M8 and M17,
respectively. The first and fourth columns include the adopted laboratory wavelength, $\lambda_0$,
and
the observed wavelength in the heliocentric framework, $\lambda$.
The second and third columns show the ion and the multiplet number, or
series for each line. The fifth and sixth columns list the observed
flux relative to H$\beta$, $F(\lambda$), and the flux corrected for reddening
relative to H$\beta$, $I(\lambda$). The seventh column includes the
fractional error (1$\sigma$) in the line intensities relative to H$\beta$, $I(\lambda$).
Errors were derived following
\citet{garciarojasetal04}, adding quadratically the error due to flux calibration,
that has been assumed as 3\%, as estimated in \citet{garciarojasetal06}, for
similar data taken with the same instrumentation, and for which there were
additional standard stars. Fractional error in the line fluxes relative to H$\beta$, $F(\lambda$), can be estimated
taking into account that fractional errors in column seven were computed propagating the uncertainty in the
extinction
correction.
A total of 375 and 260 emission lines were measured in M8 and M17, respectively. Most of the lines are
permitted. We
have measured 97 forbidden lines in M8, and 52 in M17. We have detected also 5 semiforbidden lines
in M8.
Several lines were strongly affected by atmospheric absorption features or by
charge transfer in the CCD, rendering their intensities unreliable. Also, some
lines
are dubious identifications and 3 emission lines in M8 could not be identified in any of the
available
references. All those lines are indicated in Table~\ref{lineidm8m17}.
The identification and adopted laboratory wavelengths of the lines were obtained following
previous identifications in the literature \citep[see][and references
therein]{estebanetal04,garciarojasetal04}. Several previously unidentified lines in M8 (EPTGR)
have been identified (see Table~\ref{lineidm8m17}).
Lines unidentified by EPTGR in M8 which are not in our line list are probably
telluric emission lines or nebular lines which were severely blended with telluric lines. In particular,
the features at 5865.15 {\rm \AA}, 6863.45 {\rm \AA}, and 8833.17 {\rm \AA} were identified by
\citet{osterbrocketal96} as OH night-sky lines.
We have identified $\lambda$10021.05 {\rm \AA} line as a telluric line. Also,
the two lines not identified by EPTG in position 14 of M17 have been identified here as {\ion{He}{1}}
$\lambda\lambda$7160.58 (1/10),
8486 (6/16).
It is known that the main ionization souce for the hourglass nebula in M8 is the star H36, and that
it shows a considerably higher extinction that other zones of M8. For H36, the $A_v/E(B-V)$ ratio, $R$, has been
determined as 4.6 by
\citet{hechtetal82} and as 5.3 by \citet{cardellietal89}.
Following \citet{sanchezpeimbert91} and \citet{peimbertetal93b} we have adopted for this zone of M8 a reddening
function with R$_v$ = 5.0 parametrized by \citet{cardellietal89} for $\lambda$ $\geq$ 4100 {\rm \AA}.
A reddening coefficient of c({H$\beta$}) = 0.94
$\pm$ 0.03 was derived. This value is intermediate between c({H$\beta$}) = 0.85 $\pm$ 0.05 obtained by EPTGR and
c({H$\beta$}) = 1.00 $\pm$ 0.10 derived by \citet{sanchezpeimbert91} and \citet{peimbertetal93b} for the same slit
position. For the reddening function assumed for $\lambda$ $<$ 4100 {\rm \AA} see \S~\ref{extalt}.
For M17, we have adopted the standard extinction for the Milky Way parametrized by
\citet{seaton79}.
We have obtained a reddening coefficient of c({H$\beta$}) = 1.17 $\pm$ 0.05, which is also intermediate
between
the values obtained by EPTG (1.05 $\pm$ 0.05) and \citet{peimbertetal92} (1.20) for the same slit
position.
\subsection{Extinction correction in M8 for $\lambda$ $<$ 4100 {\rm \AA}.}
\label{extalt}
In Figure~\ref{vltspm} we show the ratio of the observed fluxes of {\ion{H}{1}} Balmer lines and {\ion{He}{1}} lines
measured by us and by EPTGR. It can be seen that for wavelengths shorter than
4100 {\rm \AA} our line fluxes are higher than those measured by EPTGR. So if we assume the
extinction correction adopted above, the intensity of these lines would be overestimated.
The effect seems to be an observational bias instead of a physical effect; actually M8 is the object
that was observed at the highest air mass --$\sec$ z $\sim $ 2--, so it is possible that
an unsuitable operation of the Atmospheric Dispersion Corrector (ADC) at high airmasses used on our
observations caused this effect. The gradient in the reddening and in the surface brightness of the
Hourglass region is very strong and atmosphere refraction effects could include regions of higher
emissivity in the blue part of the spectrum that are not included at $\lambda$ $>$ 4100 \AA.
To correct for this effect, we have done a polynomial fit to the observed over theoretical flux ratios of {\ion{H}{1}}
Balmer lines and {\ion{He}{1}}
lines which are in case B and are not affected by self-absorption effects: {\ion{He}{1}} $\lambda\lambda$ 3354.55,
3447.59, 3613.64,
3634.25, 4026.08, and 4471.48,
and interpolated to all wavelengths shortwards of 4100 {\rm \AA} (see Figure~\ref{extm8}).
We have not included {\ion{H}{1}} Balmer lines with quantum number higher than 10 in this fit due to the higher
dependence of these line ratios with density \citep[see e.g.,][]{zhangetal04}.
This fit is used to interpolate for all the wavelengths shortwards of 4100 {\rm \AA}. The correction has not
affected
significantly the physical conditions and the chemical abundances derived in this work --less than 0.05 and 0.1 dex
in the total
abundances of O and Ne, respectively, which are the most affected species by this effect--.
\begin{figure}[htbp]
\includegraphics[width=\columnwidth]{vltspm.eps}
\caption{Line flux ratio of {\ion{H}{1}} Balmer lines (squares) and {\ion{He}{1}} lines (triangles) measured in this work with
respect to those measured by EPTGR for M8 (see text).}
\label{vltspm}
\end{figure}
\begin{figure}[htbp]
\includegraphics[width=\columnwidth]{extm8.eps}
\caption{Polynomial fit to the ratio of observed over theoretical fluxes of some {\ion{H}{1}} Balmer lines
(from H10 to H$\beta$, squares) and some {\ion{He}{1}} lines (triangles). Note that for $\lambda$ $<$
4100 {\rm \AA} (1/$\lambda$ $>$ 2.44) the behavior of the lines is anomalous (see text).}
\label{extm8}
\end{figure}
\section{Physical Conditions}
\label{phiscond}
\subsection{Temperatures and Densities}
\label{temden}
We have derived physical conditions of the two nebulae using several emission line ratios. The
temperatures and
densities are presented in Table~\ref{plasma}. The values of {$T_{\rm e}$} and {$n_{\rm e}$}
were derived using the {\sc IRAF} task \emph{temden} of the package \emph{nebular} \citep{shawdufour95} with
updated
atomic data
\citep[see][]{garciarojasetal05}, except in the case of the {$n_{\rm e}$}
derived from {[\ion{Fe}{3}]} lines. We have derived the {[\ion{Fe}{3}]} density from the intensity of the
brightest lines --those with
errors smaller than 30 \% and that do not seem to be affected by line blending, which are 14 in the case of
M8 and 5 in the case of M17-- together with the computations of \citet{rodriguez02}, following the
procedure described by \citet{garciarojasetal06}.
We have derived a weighted mean of {$n_{\rm e}$}({\ion{O}{2}}), {$n_{\rm e}$}({\ion{S}{2}}), {$n_{\rm e}$}({\ion{Cl}{3}}), and
{$n_{\rm e}$}({\ion{Fe}{3}})
assuming an initial temperature of {$T_{\rm e}$} = 10000 K, then we have used this density to compute the
temperatures, and
iterated until convergence. The adopted {$n_{\rm e}$} values are shown in
Table~\ref{plasma}. We have excluded
{$n_{\rm e}$}({\ion{N}{1}}) from the average because this ion is representative of the very outer part of
the nebula, and it does not coexist with the other ions.
Electron temperatures from forbidden lines have been derived from {[\ion{O}{2}]}, {[\ion{O}{3}]}, {[\ion{N}{2}]},
{[\ion{S}{2}]}, {[\ion{S}{3}]}, and {[\ion{Ar}{3}]} line ratios.
\begin{table}[htbp]\centering
\setlength{\tabnotewidth}{\columnwidth}
\tablecols{4}
\scriptsize
\caption{Plasma Diagnostic.}
\label{plasma}
\begin{tabular}{llll}
\toprule
& & \multicolumn{2}{c}{Value} \\
\cmidrule{3-4}
Parameter & Line & \multicolumn{1}{c}{M8}& \multicolumn{1}{c}{M17} \\
\midrule
N$_{\rm e}$ (cm$^{-3}$)& [N\thinspace I] & 1600$^{+750}_{-470}$ &
1200$^{+1250}_{-500}$ \\
& [O\thinspace II] & 1800 $\pm$ 800& 480 $\pm$ 150 \\
& [S\thinspace II] & 1600$\pm$450& 500$\pm$220 \\
& [Fe III] & 2600$\pm$1450 & 430$^{+>1000}_{-400}$ \\
& [Cl\thinspace III] & 2100$\pm$700 & 270$^{+630}_{-270}$ \\
& [Ar\thinspace IV] & 2450: & $>$800 \\
& N$_{\rm e}$ (adopted) & 1800$\pm$350 & 470$\pm$120 \\
& & & \\
T$_{\rm e}$ (K)& [N\thinspace II] & 8470$\pm$ 180\tabnotemark{a} & 8950$\pm$ 380\tabnotemark{a} \\
& [S\thinspace II] & 7220$\pm$300 & 7100$\pm$750 \\
& [O\thinspace II] & 8700$\pm$350\tabnotemark{a} & 8750$\pm$550\tabnotemark{a} \\
& T$_{\rm e}$ (low) & 8500 $\pm$ 150 & 8870$\pm$300 \\
& [O\thinspace III] & 8090$\pm$ 140& 8020$\pm$ 170 \\
& [Ar\thinspace III] & 7550$\pm$420 & 8380$\pm$570 \\
& [S\thinspace III] & 8600$\pm$300\tabnotemark{b} & 8110$\pm$400 \\
& T$_{\rm e}$ (high) & 8150$\pm$120 & 8050$\pm$150 \\
& He\thinspace I & 7650$\pm$200 & 7450$\pm$200 \\
& Balmer line/cont. & 7100$^{+1250}_{-1000}$ & \nodata \\
& Paschen line/cont. & 7750$\pm$900 & 6500$\pm$1000 \\
\bottomrule
\tabnotetext{a}{Recombination contribution on the auroral lines has been considered (see text)}
\tabnotetext{b}{{[\ion{S}{3}]} $\lambda$9530 affected by atmospheric absorption bands.}
\end{tabular}
\end{table}
We have corrected {$T_{\rm e}$}({\ion{O}{2}}) from the contribution to $\lambda\lambda$7320+7330 due to
recombination following the formula derived by \citet{liuetal00}:
\begin{equation}
\frac{I_R(7320+7330)}{I({\rm H\beta})}
= 9.36\times(T_4)^{0.44} \times \frac{{\rm{O}}^{++}}{{\rm{H}}^+},
\end{equation}
where $T_4$=$T$/10$^4$. Using the O$^{++}$/H$^+$ ratio derived
by EPTGR in M8 and EPTG in M17 from RLs we have estimated contributions of about 2\% and 20\% for M8
and M17, respectively.
The large contribution of recombination to the intensity of the $\lambda\lambda$7320+7330 lines in
M17 is reflected in a drop of more than 1000 K in {$T_{\rm e}$}({\ion{O}{2}}), which reconciles the value of {$T_{\rm e}$}({\ion{O}{2}})
with that of {$T_{\rm e}$}({\ion{N}{2}}).
\citet{liuetal00} also give a formula for the contribution by recombination to the intensity of the {[\ion{N}{2}]}
$\lambda$ 5755
line:
\begin{equation}
\frac{I_R(5755)}{I({\rm H\beta})}
= 3.19\times(T_4)^{0.30} \times \frac{{\rm N}^{++}}{{\rm H}^+}.
\end{equation}
To derive the N$^{++}$/H$^+$ ratio, needed to
compute this quantity, we have assumed that N$^{++}$/H$^+$ is well represented by the subtraction of
N$^+$/H$^+$ to the total N/H ratio,
assuming that the temperature fluctuations paradigm and a ionization correction factor (hereinafter ICF)
leads to the correct abundances
(see \S~\ref{tempvar}).\footnote{Another way to derive the
N$^{++}$/H$^+$ ratio is
assuming as valid the abundance obtained from {\ion{N}{2}} lines of multiplet 3, which seems to be the
least affected by fluorescence effects. Nonetheless, for regions with high degree of ionization, it may be
incorrect
to apply permitted line abundances because as pointed out by \citet{grandi76}, {\ion{N}{2}} permitted lines are excited
mainly by resonance fluorescence, and corrections might be high. In fact, if we assume the N$^{++}$ abundance
derived
from multiplet 3, the correction would be of more than 20\%, implying a {$T_{\rm e}$}({[\ion{N}{2}]}) 500 K lower than that has
been assumed (see \S~\ref{recom} for additional discussion on the {\ion{N}{2}} permitted lines).}
{From} the results of EPTGR for M8 and EPTG for M17,
the contribution of recombination to the intensity of the {[\ion{N}{2}]} $\lambda$5755 line has been
estimated as 1 \% and 6 \% for M8 and M17, respectively. These contributions are small and
affect in less than 200 K the derived temperature.
We have also been able to derive the electron temperatures from the Balmer and Paschen discontinuities.
Figure~\ref{saltos}
shows the spectral regions near the Balmer and the Paschen limits. The discontinuities can be
clearly appreciated, except
in the case of the Balmer limit of M17, for which the low signal-to-noise of the continuum makes it unreliable.
We have followed the same
procedure than in previous papers \citep[e.g.,][]{garciarojasetal06} to derive the temperatures.
The values adopted for {$T_{\rm e}$}({\ion{H}{1}}) are shown in Table~\ref{plasma}.
To the best of our knowledge, no previous determinations of {$T_{\rm e}$}({\ion{H}{1}}) (Balmer and Paschen) have been derived
for M17; for M8 there was a previous {$T_{\rm e}$}({\ion{H}{1}}) determination from the Balmer discontinuity in the hourglass
by \citet{sanchezpeimbert91} which amounts to {$T_{\rm e}$}({\ion{H}{1}}) = 6600 K, that is somewhat smaller than what has been
derived here ({$T_{\rm e}$}({\ion{H}{1}}) = 7100$^{+1250}_{-1000}$ K), but consistent within the errors.
\begin{figure}[htbp]
\includegraphics[width=0.85\columnwidth]{f1a.eps}
\includegraphics[width=0.85\columnwidth]{f1b.eps}
\includegraphics[width=0.85\columnwidth]{f1c.eps}
\caption{Section of the echelle spectra of M8 (upper and middle panels) and M17 (lower panel) including the Balmer
(upper panel)
and the Paschen (middle and lower panels) limits (observed fluxes).}
\label{saltos}
\end{figure}
Our derived {$T_{\rm e}$}({\ion{H}{1}}) values are in good agreement with
the values obtained by \citet{reifensteinetal70} from the H109$\alpha$ radio recombination line,
{$T_{\rm e}$}({\ion{H}{1}}) = 7300 $\pm$ 1000 K for M8 and 6400 $\pm$ 750 K for M17. On the other hand, {$T_{\rm e}$}({\ion{H}{1}}) derived by
\citet{shavergoss70}
from radio 408 MHz continuum measurements do not agree; these authors computed {$T_{\rm e}$} = 6100 K
for M8 and 7850 K for M17, which are far from the temperatures derived here.
We have derived {$T_{\rm e}$}({\ion{He}{1}}) assuming a 2-zone ionization scheme, characterized by {$T_{\rm e}$}{\sc ii+iii}
\citep[see][]{apeimbertetal02}.
We have derived {$T_{\rm e}$}({\ion{He}{1}})=7650 $\pm$ 200 K for M8, which is highly
consistent with the {$T_{\rm e}$}({\ion{H}{1}}) derived above, and {$T_{\rm e}$}({\ion{He}{1}})=7450 $\pm$ 200 K for M17, which is
higher than {$T_{\rm e}$}({\ion{H}{1}}).
We have assumed a two-zone ionization scheme for all our calculations. We have adopted the
average of {$T_{\rm e}$} obtained from {[\ion{N}{2}]} and {[\ion{O}{2}]} lines as representative for the low ionization
zone. We have not included {$T_{\rm e}$}({\ion{S}{2}}) in the average because its value is much lower than those
obtained from {[\ion{N}{2}]} and {[\ion{O}{2}]} lines. This effect has been reported previously in several objects
\citep[e.g.,][]{garciarojasetal05,garciarojasetal06}, and might be produced by the presence of a temperature
stratification in the outer zones of the nebulae or, conversely, by errors in the atomic parameters of the ion.
For the high ionization zone we have adopted the average of the values of {$T_{\rm e}$} obtained from {[\ion{O}{3}]}, {[\ion{S}{3}]}
and {[\ion{Ar}{3}]}. In M8 the {[\ion{S}{3}]} $\lambda$9532 line is affected by atmospheric absorption bands,
so we have adopted the intensity of {[\ion{S}{3}]} $\lambda$9069 and the {[\ion{S}{3}]} $\lambda$9532/$\lambda$9069
theoretical ratio to derive {$T_{\rm e}$}({\ion{S}{3}}).
\subsection{Temperature Variations}
\label{tempvar}
Since \citet{torrespeimbertetal80} proposed the presence of spatial temperature fluctuations
(parametrized by {\emph{$t^2$}}) as a possible cause of the abundance discrepancy, many efforts have been done
to find the physical processes responsible for such temperature fluctuations in {\ion{H}{2}} regions
\citep[e.g.,][]{esteban02, tsamispequignot05} and in planetary nebulae
\citep[e.g.,][]{liu06, peimbertpeimbert06}, but the source of temperature fluctuations and its
impact on the chemical abundance determinations remain controversial topics in the
study of gaseous nebulae.
\citet{peimbert71} showed that there was a substantial difference between the {$T_{\rm e}$} derived from the {[\ion{O}{3}]} lines
and from the one derived from hydrogen recombination continuum discontinuities, which is strongly correlated with
the
abundance discrepancy \citep{liuetal01,tsamisetal04}, so the comparison between electron temperatures
derived from both methods would be an additional indicator of {\emph{$t^2$}}.
Additionally, it is also possible to derive the {\emph{$t^2$}} value from the analysis of the {\ion{He}{1}} lines,
because of the different temperature dependence of each of them, so we can find {\ion{He}{1}} line ratios
that will allow us to derive a temperature. However, in practice, each of these ratios depends simultaneously on
$T_0$, {\emph{$t^2$}}, {$n_{\rm e}$} and $\tau_{3889}$ therefrefore, any determination must be done using several line ratios.
\citet{peimbertetal00} developed a maximum likelihood method to search for the plasma
conditions that would give the best simultaneous fit to the measured lines. In \S~\ref{helioabund} we have
applied that method to our {\ion{He}{1}} lines.
As we have assumed a two-zone ionization scheme, we have followed the formulation of \citet{peimbertetal00} and
\citet{apeimbertetal02} to derive the values of {\emph{$t^2$}} following the three methods described above.
In Table~\ref{t2} we show the \emph{$t^2$} values derived from each method and the final adopted values, which are
error-weighted averages. It is highly remarkable that all the \emph{$t^2$} values derived for each nebula are very
consistent.
The C$^{++}$/H$^+$ ratio obtained from CELs for M8 has been taken from
\citet{peimbertetal93b}, who measured the UV \ion{C}{2}] $\lambda\lambda$1906+1909 emission lines from IUE data.
Nonetheless, as well as when we are comparing with the infrared data \citep[see][]{garciarojasetal06}, we cannot
discard aperture effects due to the different volumes covered by the slits in the optical and UV observations.
\begin{table}[htbp]\centering
\setlength{\tabnotewidth}{\columnwidth}
\tablecols{3}
\setlength{\tabcolsep}{2.7\tabcolsep}
\scriptsize
\caption{{\emph{$t^2$}} parameter}
\label{t2}
\begin{tabular}{ccc}
\toprule
& \multicolumn{2}{c}{\emph{$t^2$}} \\
\cmidrule{2-3}
Method & M8& M17 \\
\midrule
O$^{\rm ++}$ (R/C)& 0.045$\pm$0.005 & 0.034$\pm$0.005 \\
O$^{\rm +}$ (R/C)& 0.031$\pm$0.017 & 0.109: \\
C$^{\rm ++}$ (R/C)& 0.035$\pm$0.005 & \nodata \\
He II & 0.046$\pm$0.009 & 0.027$\pm$0.014 \\
Bac/Pac--FL & 0.022$\pm$0.015 & 0.035$\pm$0.021\\
\midrule
Adopted & 0.040$\pm$0.004 & 0.033$\pm$0.005\\
\bottomrule
\end{tabular}
\end{table}
The {\emph{$t^2$}} values obtained in this paper are very similar to those obtained for all the bright
Galactic {\ion{H}{2}} regions of
our sample \citep{estebanetal04, garciarojasetal04, garciarojasetal05, garciarojasetal06} and are also similar to
the few
estimations of {\emph{$t^2$}} in extragalactic {\ion{H}{2}} regions available in the literature for the Magellanic
Clouds \citep{apeimbert03, tsamisetal03}, NGC~6822 \citep{apeimbertetal05}, M101, NGC~2366, and M33
\citep{estebanetal02} and the dwarf {\ion{H}{2}} galaxy NGC~5253 \citep{lopezsanchezetal06}.
\begin{figure}[htbp]
\includegraphics[width=\columnwidth]{ne_adf.eps}
\caption{Correlation between {$n_{\rm e}$} and ADF for the sample of Galactic (filled squares) and extragalactic (other
symbols)
{\ion{H}{2}} regions in which ADF has been measured.
Solid line is the fit obtained by \citet{robertsontessigarnett05} for a sample of PN from the literature.
Dashed line is the fit obtained for {\ion{H}{2}} regions. }
\label{adfne}
\end{figure}
A very different behavior has been found for planetary nebulae (PNe). Several authors have found a large scatter of
the {\emph{$t^2$}} values
determined for different PNe \citep[see e.g.,][and references therein]{liu02, tsamisetal04}. Recently,
\citet{robertsontessigarnett05},
have found a correlation between the {\emph Abundance Discrepancy Factor} (ADF) defined by \citet{liuetal00} as
log(O$^{++}$/H$^+$ RL)-log(O$^{++}$/H$^+$ CEL) and {$n_{\rm e}$}.
They have found that the lower {$n_{\rm e}$} in the PNe, the higher ADF, with a strong slope.
To illustrate the difference between both behaviors --PNe and {\ion{H}{2}} regions-- we have overplotted the complete
set of ADFs measured in {\ion{H}{2}} regions (Galactic and extragalactic) available in the literature to the
Robertson-Tessi's fit (r=$-$0.47)
for PNe (see Figure~\ref{adfne}). From Figure~\ref{adfne} it is clear that {\ion{H}{2}} regions do not follow the
correlation found for
PNe. In fact, it seems that there is no correlation between the {\ion{H}{2}} regions data, due to the similarity between
the ADF values found
for {\ion{H}{2}} regions and to the low correlation coefficient found (r=--0.25).
The only exception is LMC N11B, which has an ADF much larger than the other nebulae.
For this object, \citet{tsamisetal03} corrected the intensity of the multiplet 1 {\ion{O}{2}} lines because of the
presence of absorption
features, mainly caused by the presence of B stars --which have a strong {\ion{O}{2}}
absorption spectra-- on the field covered by the slit. Nevertheless, this effect,--which could be very important on
extragalactic objects-- can only be properly corrected if the stellar absorption features are resolved,
or if stellar population synthesis spectra are available.
Therefore, the {\ion{O}{2}} absorption contribution can not be properly estimated if we have low spectral resolution data
--like those used by Tsamis et al.-- because it is difficult
to distinguish between emission and absorption features \citep[to illustrate this point see Figure 2
of][]{garciarojasetal06}. It is not the
aim of this paper to discuss more about the attenuation of the intensities of {\ion{O}{2}} RLs due to star absorption
features,
but it is important to stress that this effect should be investigated when deriving abundances from RLs in
extragalactic {\ion{H}{2}} regions.
\section{He$^+$ Abundance}
\label{helioabund}
We have measured 76 and 62 {\ion{He}{1}} emission lines in the spectra of M8 and M17,
respectively.
These lines arise mainly from recombination but they can be affected by collisional excitation
and
self-absorption effects. We have
determined the He$^+$/H$^+$ ratio from a maximum likelihood method \citep{peimbertetal00},
using the {$n_{\rm e}$} of Table~\ref{plasma} and $T$(O~{\sc ii+iii})= 8350 K for M8,
and $T$(O{\sc ii+iii})= 8200 K for M17 (see \S~\ref{tempvar}).
We have used the effective recombination coefficients of \citet{storeyhummer95}
for {\ion{H}{1}} and those of \citet{smits96} and \citet{benjaminetal99} for {\ion{He}{1}}.
The collisional contribution
was estimated from the calculations made by \citet{saweyberrington93} and \citet{kingdonferland95}, and the optical
depths
in the triplet lines were derived from the computations by \citet{benjaminetal02}.
\begin{table}[htbp]\centering
\setlength{\tabnotewidth}{\columnwidth}
\tablecols{3}
\setlength{\tabcolsep}{2.8\tabcolsep}
\scriptsize
\caption{He$^+$ abundance.}
\label{abhe}
\begin{tabular}{lcc}
\toprule
& \multicolumn{2}{c}{He$^+$/H$^+$ \tabnotemark{a}} \\
\cmidrule{2-3}
Line & \multicolumn{1}{c}{M8}& \multicolumn{1}{c}{M17} \\
\midrule
3819.61& 673 $\pm$ 54& 950 $\pm$ 95 \\
3964.73& 733 $\pm$ 59& 897 $\pm$ 45 \\
4026.21& 699 $\pm$ 56& 955 $\pm$ 38 \\
4387.93& 679 $\pm$ 20& 918 $\pm$ 55 \\
4471.09& 662 $\pm$ 20& 904 $\pm$ 27 \\
4713.14& 617 $\pm$ 19& 952 $\pm$ 57 \\
4921.93& 670 $\pm$ 20& 896 $\pm$ 27 \\
5875.64& 639 $\pm$ 19& 880 $\pm$ 26 \\
6678.15& 666 $\pm$ 20& 924 $\pm$ 37 \\
7065.28& 673 $\pm$ 20& 905 $\pm$ 45 \\
7281.35& 666 $\pm$ 20& 884 $\pm$ 44 \\
\midrule
Adopted\tabnotemark{b}& 662 $\pm$ 9 & 910 $\pm$ 14 \\
\bottomrule
\tabnotetext{a}{In units of 10$^{-4}$, for $\tau_{3889}$ = 8.28 $\pm$ 0.60 and 7.80 $\pm$ 0.78, and $t^2$ = 0.040
$\pm$ 0.004 and 0.033 $\pm$ 0.005, respectively.
Uncertainties correspond to line intensity errors.}
\tabnotetext{b}{It includes all the relevant uncertainties in emission line intensities, $n_e$, $\tau_{3889}$ and
$t^2$.}
\end{tabular}
\end{table}
In Table~\ref{abhe} we have included the He$^+$/H$^+$ ratios that we have obtained for the individual
{\ion{He}{1}} lines not affected by line blending and with the highest signal-to-noise ratio,
excluding {\ion{He}{1}} $\lambda$5015 because it could suffer self-absorption effects from the 2$^1$S metastable
level, and
$\lambda$3889 because it is severely blended with the H8 line.
We have done a $\chi^2$
optimization of the values in the table, and we have obtained a $\chi^2$ parameter of 8.5 for M8 and
3.6 for M17. The values obtained indicate that the fits are good for a system with
eight degrees of freedom.
EPTGR, who covered a region slightly larger than ours, but centered in the same location,
derived a He$^{+}$/H$^+$ ratio for M8--HG a factor of 1.13 (0.05 dex) higher
than ours. On the other hand, \citet{peimbertetal93b}, who covered a similar region of the
nebula, obtained a He$^{+}$/H$^+$ ratio 0.025 dex lower,
confirming the strong variation with position of the values found for the He$^{+}$/H$^+$ fraction in the Hourglass
on M8.
For M17, the He$^{+}$/H$^+$ ratio is only a bit smaller than those obtained by EPTG
(0.0975) and \citet{peimbertetal92} (0.099); this could be due to differences on the ionization degree
of the regions covered by the different slit sizes since the O$^{++}$ abundance from RLs shows a similar
behavior (see \S~\ref{recom}).
\section{Ionic Abundances from Collisionally Excited Lines}
\label{cels}
We have derived ionic abundances of N$^+$, O$^+$, O$^{++}$, Ne$^{++}$, S$^+$,
S$^{++}$, Cl$^{++}$, Cl$^{3+}$, Ar$^{++}$ and Ar$^{3+}$ from CELs, using the {\sc IRAF} package NEBULAR.
The atomic data for Cl$^+$ are not implemented in NEBULAR, so we have used an old version of the
five-level atom program of \citet{shawdufour95} \citep[see][for more details]{garciarojasetal04}.
Ionic abundances are listed in Table~\ref{celabun} and correspond to the mean value of the abundances
derived from all the individual lines of each observed ion, weighted by their relative intensities.
To derive the ionic abundances for \emph{$t^2$} $>$ 0.00 we have used the abundances for {\emph{$t^2$}}=0.00 and the formulation by
\citet{peimbert67} and \citet{peimbertcostero69}. These abundances are also shown in Table~\ref{celabun}.
Several {[\ion{Fe}{2}]} lines have been detected in the spectra of M8 and M17. Unfortunately, most of them are
severely affected by fluorescence effects \citep{rodriguez99, verneretal00}. One of the optical {[\ion{Fe}{2}]} lines
which is less affected by fluorescence effects is the {[\ion{Fe}{2}]} $\lambda$8617 line,
but unfortunately it is in one of our observational gaps. Nonetheless, we have measured
the {[\ion{Fe}{2}]} $\lambda$7155 line, both in M8 and M17, a line which is not much affected by
fluorescence effects \citep{verneretal00}. We have derived an estimation of the Fe$^{+}$ abundance
from this line, assuming that $I(\lambda7155)$/$I(\lambda8616)$ $\sim$ 1 \citep{rodriguez96} and using
the calculations by \citet{bautistapradhan96} for the emissivities of the {[\ion{Fe}{2}]} $\lambda$8617 line.
We find Fe$^+$/H$^+$ $\sim$ 4.1 $\times$ 10$^{-8}$
for M8 and Fe$^+$/H$^+$ $\sim$ 1.1 $\times$ 10$^{-8}$ for M17. Nevertheless these results are only an estimation,
and we have
marked them with two colons in Table~\ref{celabun} due to their uncertainty.
\begin{table*}[htbp]\centering
\setlength{\tabnotewidth}{\textwidth}
\tablecols{5}
\setlength{\tabcolsep}{4.6\tabcolsep}
\scriptsize
\caption{Ionic abundances from collisionally excited lines\tabnotemark{a}.}
\label{celabun}
\begin{tabular}{lcccc}
\toprule
& \multicolumn{2}{c}{M8}& \multicolumn{2}{c}{M17} \\
\cmidrule{2-5}
Ion & {\emph{$t^2$}}=0.000 & {\emph{$t^2$}}=0.040$\pm$0.004 & {\emph{$t^2$}}=0.000 & {\emph{$t^2$}}=0.033$\pm$0.005 \\
\midrule
N$^{+}$ & 7.50$\pm$0.03 & 7.67$\pm$0.04 & 6.82$\pm$0.10 & 6.94$\pm$0.10 \\
O$^{+}$ & 8.39$\pm$0.06 & 8.58$\pm$0.07 & 7.84$\pm$0.09 & 7.98$\pm$0.09 \\
O$^{++}$ & 7.86$\pm$0.03 & 8.18$\pm$0.07 & 8.41$\pm$0.04 & 8.67$\pm$0.06 \\
Ne$^{++}$ & 6.95$\pm$0.05 & 7.30$\pm$0.07 & 7.64$\pm$0.04 & 7.93$\pm$0.07 \\
S$^{+}$ & 5.93$\pm$0.04 & 6.10$\pm$0.07 & 5.44$\pm$0.05 & 5.56$\pm$0.06 \\
S$^{++}$ & 6.89$\pm$0.03 & 7.25$\pm$0.07 & 6.90$\pm$0.04 & 7.19$\pm$0.06 \\
Cl$^{+}$ & 4.53$\pm$0.04 & 4.66$\pm$0.06 & 3.95$^{+0.09}_{-0.12}$& 4.06$^{+0.09}_{-0.12}$ \\
Cl$^{++}$ & 5.02$\pm$0.04 & 5.32$\pm$0.06 & 5.04$\pm$0.04& 5.29$\pm$0.06 \\
Cl$^{3+}$ & \nodata & \nodata & 3.15: & 3.35: \\
Ar$^{++}$ & 6.21$\pm$0.03 & 6.48$\pm$0.05 & 6.35$\pm$0.04 & 6.57$\pm$0.06 \\
Ar$^{3+}$ & 3.69$\pm$0.09 & 4.01$\pm$0.10 & 4.15$^{+0.12}_{-0.18}$& 4.42$^{+0.13}_{-0.18}$ \\
Fe$^{+}$ & 4.61: & 4.77: & 4.05: & 4.17: \\
Fe$^{++}$ & 5.58$\pm$0.04 & 5.91$\pm$0.06 & 5.24$\pm$0.06& 5.51$\pm$0.08 \\
\bottomrule
\tabnotetext{a}{In units of 12+log(X$^m$/H$^+$).}
\end{tabular}
\end{table*}
The calculations for Fe$^{++}$ have been done with a 34 level model-atom that uses collision strengths from
\citet{zhang96} and the transition probabilities of \citet{quinet96}. We have used {[\ion{Fe}{3}]} lines which
do not seem affected by blends, 14 in the case of M8 and 5 in the case of M17. We have found an average value and
a standard deviation of Fe$^{++}$/H$^+$ = (3.78 $\pm$ 0.36) $\times$ 10$^{-7}$ for M8 and Fe$^{++}$/H$^+$ =
(1.73 $\pm$ 0.12) $\times$ 10$^{-7}$ for M17. Adding errors in {$T_{\rm e}$} and {$n_{\rm e}$} we finally obtain 12 +
log(Fe$^{++}$/H$^+$) = 5.58 $\pm$ 0.04 and 5.24 $\pm$ 0.06 for M8 and M17 respectively. The Fe$^{++}$ abundances
are also
included in Table~\ref{celabun}.
\section{Ionic Abundances of Heavy Elements from Recombination Lines}
\label{recom}
EPTGR performed a detailed analysis of the excitation mechanisms of permitted heavy element lines in M8.
In this work we have measured a large number of permitted heavy element lines, but following the study of
EPTGR we have focused on the lines which are excited purely by recombination.
Nevertheless, we are going to comment briefly on the N$^{++}$/H$^+$ ratio in both nebulae.
In Table~\ref{nii} we show the N$^{++}$/H$^+$ ratios obtained from permitted lines in M8 and M17.
\citet{grandi76} argued that resonance fluorescence by the {\ion{He}{1}} $\lambda$508.64 recombination line is the
dominant mechanism to excite the 4$s^3P^0$ term of {\ion{N}{2}} in the Orion Nebula, and hence, it should
be responsible for the strengths of multiplets 3 and 5.
Recently, \citet{escalantemorisset05},
using tailored photoionization models of the Orion Nebula, estimate that the contribution of recombination
to the intensity of multiplet 3 (which is one of the less affected by fluorescence effects of those reported in
this work)
is about 20 \% of the total intensity of the line.
Additionally, we have measured a blend of two {\ion{N}{2}} lines of multiplet 19 at $\lambda\lambda$5001.14, 5001.48.
These lines have upper levels 3d$^3$F$^0_{2,3}$ that are connected to the ground state through weak dipole-allowed
transitions and could have an important fluorescence contribution \citep{belletal95};
\citet{escalantemorisset05} predicted than recombination
contributes $\sim$43\% to the total intensity of these two lines.
Unfortunately, the only line of this multiplet which is not affected by fluorescence effects is the one at
$\lambda$5005.15, which is blended with the {[\ion{O}{3}]} $\lambda$5007 line.
To test these theoretical predictions we have computed the N$^{++}$ abundance from the line of multiplet 19,
taking into account the contribution by fluorescence predicted by \citet{escalantemorisset05},
and compared it with the N$^{++}$ abundance estimated from the N$^+$ abundances assuming CELs with
{\emph{$t^2$}} $>$ 0.00 and the ionization correction factor for N.
Also, we have proceeded in the same way with multiplet 3, taking into account that only 20\% of the
of the line intensities is due to recombination.
In Table~\ref{niicomp} we show the results obtained for M8 and M17 (this work), NGC~3576 \citep{garciarojasetal04}
and the Orion Nebula \citep{estebanetal04}.
For the Orion Nebula and NGC~3576, we have considered also 3d--4f and singlet transitions, which cannot be excited
by
resonant fluorescence \citep[see][]{grandi76, escalantemorisset05}.
In principle, there is better agreement among the abundances obtained from these lines taking into account the
considerations by
\citet{escalantemorisset05}; nevertheless there are some puzzling results: the only 3d--4f transition detected
in NGC~3576 shows the larger deviation from the rest of the values, however, \citet{escalantemorisset05} proposed
that
there can be another mechanisms responsible for the enhancement of the intensity of these transitions, so we have
to consider
the abundances derived from these lines as high limits; also, from the comparison
between the recombination N$^{++}$ abundances and the values obtained from N$^+$/H$^+$ (CELs), the ICF and {\emph{$t^2$}}
in Table~\ref{niicomp} it can be seen that the agreement in M8, M17 and NGC~3576 is not very good, and that in
Orion is rather poor.
Nevertheless, the CELs N$^{++}$/H$^+$ ratio is very sensitive to the adopted ICF scheme, and could be reduced as
much as a factor
of 2 if the adopted ICF scheme would have been the one by \citet{peimbertcostero69}.
It is clear that the measurement of pure N$^{++}$ recombination lines (i. e. singlet transitions) could be very
useful to constraint the
temperature fluctuations scenario, and that much work should be done in this sense, but it is beyond the scope of
this paper.
\begin{table*}[htbp]\centering
\setlength{\tabnotewidth}{\textwidth}
\tablecols{8}
\setlength{\tabcolsep}{2.3\tabcolsep}
\scriptsize
\caption{N$^{++}$/H$^+$ ratio from {\ion{N}{2}} permitted lines\tabnotemark{a}}
\label{nii}
\begin{tabular}{cccccccc}
\toprule
& & \multicolumn{3}{c}{M8}& \multicolumn{3}{c}{M17} \\
\cmidrule{3-8}
& & $I$($\lambda$)/$I$(H$\beta$) & \multicolumn{2}{c}{N$^{++}$/H$^+$ ($\times$10$^{-5}$)\tabnotemark{b}}&
$I$($\lambda$)/$I$(H$\beta$) & \multicolumn{2}{c}{N$^{++}$/H$^+$ ($\times$10$^{-5}$)\tabnotemark{b}} \\
Mult. & $\lambda_0$ & ($\times$10$^{-2}$) & A & B & ($\times$10$^{-2}$) & A & B
\\
\midrule
3 & 5666.64 & 0.027$\pm$0.004 & 16$\pm$2 & 13$\pm$2 & 0.038$\pm$0.007 & 22$\pm$4
& 18$\pm$3 \\
& 5676.02 & 0.015$\pm$0.004 & 18$\pm$5 & 15$\pm$4 & \nodata & \nodata &
\nodata \\
& 5679.56 & 0.034$\pm$0.004 & 10$\pm$1 & 8 $\pm$1 & 0.078$\pm$0.009 & 23$\pm$3
& 19$\pm$2 \\
& 5686.21 & 0.009$\pm$0.003 & 17$\pm$6 & 14$\pm$5 & \nodata & \nodata &
\nodata \\
& 5710.76 & 0.010$\pm$0.003 & 17$\pm$5 & 14$\pm$4 & 0.012$\pm$0.005 & 21$\pm$8
& 17$\pm$7 \\
& Sum & & 13$\pm$1 & 11$\pm$1 & & 23$\pm$2 & 19$\pm$2
\\
5 & 4621.39 & 0.022$\pm$0.004 & 244$\pm$54 & 40$\pm$9 & 0.019: & 216: &
35: \\
& 4630.54 & 0.028$\pm$0.005 & 77$\pm$12 & 13$\pm$2 & 0.055$\pm$0.015 & 154$\pm$43
& 25$\pm$7 \\
& 4643.06 & 0.018$\pm$0.004 & 140$\pm$32 & 23$\pm$5 & 0.022: & 175: &
28: \\
& Sum & & 116$\pm$11 & 19$\pm$2 & & 154$\pm$43 & 25$\pm$7
\\
19 & 5001.3 & 0.037$\pm$0.009 & 8$\pm$2 & 8$\pm$2 & \nodata & \nodata &
\nodata \\
20 & 4788.13 & 0.014$\pm$0.004 & 1447$\pm$391 & 28$\pm$8 & \nodata & \nodata &
\nodata \\
& 4803.29 & 0.011$\pm$0.004 & 642$\pm$214 & 13$\pm$4 & \nodata & \nodata &
\nodata \\
& Sum & & 767$\pm$188 & 15$\pm$4 & & \nodata & \nodata
\\
24 & 4994.37 & 0.020$\pm$0.005 & 700$\pm$200 & 30$\pm$10 & \nodata & \nodata &
\nodata \\
28 & 5927.82 & 0.014$\pm$0.004 & 3892$\pm$973 & 46$\pm$12 & \nodata & \nodata &
\nodata \\
& 5931.79 & 0.020$\pm$0.004 & 2513$\pm$452 & 30$\pm$5 & 0.031$\pm$0.006 &
3889$\pm$778 & 46$\pm$9 \\
& Sum & & 2946$\pm$410 & 35$\pm$5 & & 3893$\pm$778 & 46$\pm$9
\\
\bottomrule
\tabnotetext{a}{Only lines with intensity uncertainties lower than 40\% have been considered.}
\tabnotetext{b}{Recombination coefficients from \citet{kisieliusstorey02} for cases A and B.}
\end{tabular}
\end{table*}
\begin{table*}[htbp]\centering
\setlength{\tabnotewidth}{\columnwidth}
\tablecols{5}
\setlength{\tabcolsep}{2.6\tabcolsep}
\scriptsize
\caption{Comparison of N$^{++}$/H$^+$ ratios from {\ion{N}{2}} permitted lines.}
\label{niicomp}
\begin{tabular}{ccccc}
\toprule
& \multicolumn{4}{c}{N$^{++}$/H$^+$ ($\times$10$^{-5}$)\tabnotemark{a}} \\
\cmidrule{2-5}
Mult. & M8 & M17 & Orion & NGC~3576 \\
\midrule
3 & 2 & 8 & 2 & 2 \\
19 & 3 & \nodata & 3 & 4 \\
3d--4f & \nodata & \nodata & $\le$4:& $\le$8\\
singlets& \nodata & \nodata & 3: & 7: \\
\midrule
CELs\tabnotemark{b} & 4 & 7 & 6 & 4\\
\bottomrule
\tabnotetext{a}{M8 and M17: this work; Orion Nebula: \citet{estebanetal04}; NGC~3576: \citet{garciarojasetal04}.
The two colons indicate uncertainties larger than 40\%}
\tabnotetext{b}{N$^{++}$ abundance obtained assuming N/H = N$^{+}$/H$^{+}$ + N$^{++}$/H$^{++}$, where N/H and
N$^{+}$/H$^{+}$ where obtained
from CELs and assuming {\emph{$t^2$}} $>$ 0.00.}
\end{tabular}
\end{table*}
We have measured 16 permitted lines of {\ion{C}{2}} in the spectrum of M8 and 13 in the spectrum of M17.
Some of these lines
(those of multiplets 6, 16.04, 17.02 and 17.04) are $3d-4f$ transitions and are,
in principle, excited by
pure recombination \citep[see][]{grandi76}. In these transitions, the abundances
obtained are case-independent, so
we have adopted the mean of the values obtained for these transitions as our final
adopted C$^{++}$/H$^+$ ratio.
The result for the case-sensitive multiplet 3 gives a C$^{++}$ abundance for case B
which is rather consistent with the one
adopted here. We have used the effective recombination coefficients computed by
\citet{daveyetal00} for the abundance calculations. The dispersion of the
abundances obtained by the different lines is very small, except in the case of {\ion{C}{2}}
$\lambda$9903.43 line in M17,
whose intensity seems to be affected by an unknown feature. The final results are in
excellent agreement with those obtained by EPTGR for M8 (C$^{++}$/H$^+$ = 1.9 $\times$ 10$^{-4}$)
and by EPTG and
\citet{tsamisetal03} for M17 (C$^{++}$/H$^+$ = 4.9 $\times$ 10$^{-4}$ and 4.4
$\times$ 10$^{-4}$, respectively). The complete set of derived individual C$^{++}$/H$^+$ ratios as
well as the adopted one are shown in Table~\ref{cii}.
\begin{table*}[htbp]\centering
\setlength{\tabnotewidth}{\textwidth}
\tablecols{8}
\setlength{\tabcolsep}{2.0\tabcolsep}
\scriptsize
\caption{C$^{++}$/H$^+$ ratio from {\ion{C}{2}} recombination lines}
\label{cii}
\begin{tabular}{cccccccc}
\toprule
& & \multicolumn{3}{c}{M8} & \multicolumn{3}{c}{M17} \\
\cmidrule{3-8}
& & $I$($\lambda$)/$I$(H$\beta$) & \multicolumn{2}{c}{C$^{++}$/H$^+$ ($\times$10$^{-5}$)\tabnotemark{a}}&
$I$($\lambda$)/$I$(H$\beta$) & \multicolumn{2}{c}{C$^{++}$/H$^+$ ($\times$10$^{-5}$)\tabnotemark{a}} \\
Mult. & $\lambda_0$ & ($\times$10$^{-2}$) & A & B & ($\times$10$^{-2}$) & A & B\\
\midrule
2 & 6578.05 & 0.262 $\pm$ 0.008\tabnotemark{b}& 300 $\pm$ 9 & 50 $\pm$ 2 & 0.358 $\pm$ 0.018 & 408
$\pm$ 20 & 69 $\pm$ 3 \\
3 & 7231.12 & 0.074 $\pm$ 0.004 & 1241 $\pm$ 67 & 18 $\pm$ 1 & 0.129 $\pm$ 0.009 & 2162 $\pm$ 151& 31
$\pm$ 2 \\
& 7236.19 & 0.112 $\pm$ 0.004 & 1045 $\pm$ 37 & 15 $\pm$ 1 & 0.193 $\pm$ 0.012 & 1800 $\pm$ 112& 26
$\pm$ 2 \\
& Sum & & 1115 $\pm$ 32 & 16 $\pm$ 1 & & 1929 $\pm$ 90 & 27 $\pm$ 1 \\
4 & 3918.98 & 0.062 $\pm$ 0.007 & 1210 $\pm$ 137& 385 $\pm$ 43 & 0.043: & 820: & 260: \\
& 3920.68 & 0.133 $\pm$ 0.008 & 1290 $\pm$ 78 & 410 $\pm$ 25 & 0.086 $\pm$ 0.023 & 826 $\pm$ 223 & 264
$\pm$ 71 \\
& Sum & & 1260 $\pm$ 68 & 400 $\pm$ 22 & & 825 $\pm$ 200 & 263 $\pm$ 62 \\
6 & 4267.26 & 0.222 $\pm$ 0.009 & 20 $\pm$ 1 & {\bf 20 $\pm$ 1}& 0.580 $\pm$ 0.035 & 54 $\pm$ 3 & {\bf
53 $\pm$ 3} \\
16.04 & 6151.43 & 0.009 $\pm$ 0.003 & {\bf 21 $\pm$ 7}& ... & 0.018 $\pm$ 0.005 & {\bf 41 $\pm$ 11}& ...
\\
17.02 & 9903.43 & 0.048 $\pm$ 0.003 & {\bf 18 $\pm$ 1}& ... & 0.196 $\pm$ 0.016\tabnotemark{c} & {\bf 68
$\pm$ 3}& ... \\
17.04 & 6461.95 & 0.025 $\pm$ 0.004 & {\bf 22 $\pm$ 4}& ... & 0.050 $\pm$ 0.007 & {\bf 44 $\pm$ 6}& ...
\\
17.06 & 5342.38 & 0.011 $\pm$ 0.004 & {\bf 19 $\pm$ 7}& ... & ... & ... & ... \\
\midrule
& Adopted & &\multicolumn{2}{c}{{\bf 20 $\pm$ 1 }}& &\multicolumn{2}{c}{{\bf 48 $\pm$ 3 }} \\
\bottomrule
\tabnotetext{a}{Recombination coefficients from \citet{daveyetal00} for cases A and B.}
\tabnotetext{b}{Affected by telluric emission lines.}
\tabnotetext{c}{Blend with an unidentified line.}
\end{tabular}
\end{table*}
The O$^+$ abundance was derived from the {\ion{O}{1}} $\lambda$7771.94 line, the only line of
multiplet 1 that is not severely affected by
telluric lines. This multiplet is case independent and is produced mainly by
recombination because it corresponds
to a quintuplet transition, and the ground level is a triplet. We also have computed
the O$^+$/H$^+$ ratio from
the {\ion{O}{1}} $\lambda$8446.48 line of the multiplet 4, but \citet{grandi75a} showed that
starlight may contribute significantly
to the observed strength of the line, which is supported by the fact that the O$^+$/H$^+$
ratio implied by this line is between one and
two orders of magnitude larger.
The effective recombination coefficients were obtained from two sources: \citet{pequignotetal91} and
\citet{escalantevictor92}.
Though the results are very similar, we adopted the mean of the abundances obtained with the two
different coefficients.
Our results are presented in Table~\ref{oi}.
The O$^+$ abundance that we have obtained for M8 is
larger by a factor of 2 than that obtained by EPTGR; whereas,
as pointed out below, the O$^{++}$ abundance derived from RLs is almost coincident
in the two works, leading us to propose that the abundance of O$^+$ derived from the
{\ion{O}{1}} $\lambda$7771.96 line by EPTGR was underestimated by a factor of 2, because of the lower
spectral resolution and signal-to-noise ratio of their data. The O$^+$ abundance obtained for M17 from the
$\lambda$7771.94
line is very uncertain because it is partially blended with a strong sky emission line.
\begin{table*}[htbp]\centering
\setlength{\tabnotewidth}{\textwidth}
\tablecols{8}
\setlength{\tabcolsep}{0.7\tabcolsep}
\scriptsize
\caption{O$^{+}$/H$^+$ ratio from {\ion{O}{1}} permitted lines}
\label{oi}
\begin{tabular}{cccccccc}
\toprule
& & \multicolumn{3}{c}{M8}& \multicolumn{3}{c}{M17} \\
\cmidrule{3-8}
& & $I$($\lambda$)/$I$(H$\beta$) & \multicolumn{2}{c}{O$^{+}$/H$^+$ ($\times$10$^{-5}$)\tabnotemark{a}}&
$I$($\lambda$)/$I$(H$\beta$) & \multicolumn{2}{c}{O$^{+}$/H$^+$ ($\times$10$^{-5}$)\tabnotemark{a}}\\
Mult. & $\lambda_0$ & ($\times$10$^{-2}$) & A & B & ($\times$10$^{-2}$) & A & B \\
\midrule
1 & 7771.94 & 0.029 $\pm$ 0.003 & 39 $\pm$ 4/30 $\pm$ 3 & \nodata & 0.025 $\pm$ 0.005\tabnotemark{b} & 33
$\pm$ 7/25 $\pm$ 5 & \nodata \\
4 & 8446.48\tabnotemark{b}& 0.433 $\pm$ 0.017 & 2454 $\pm$ 96/ 1657 $\pm$ 6 & 493 $\pm$ 9/372 $\pm$ 15 & 0.156
$\pm$ 0.011 & 890 $\pm$ 62/593 $\pm$ 42 & 179 $\pm$ 13/134 $\pm$ 9 \\
\midrule
& Adopted & &\multicolumn{2}{c}{{\bf 34 $\pm$ 5 }} & &\multicolumn{2}{c}{29 $\pm$ 6 } \\
\bottomrule
\tabnotetext{a}{Recombination coefficients from \citet{pequignotetal91}/\citet{escalantevictor92} for cases A and
B.}
\tabnotetext{b}{Blended with telluric emission lines.}
\end{tabular}
\end{table*}
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=\columnwidth]{f2.eps}
\caption{Sections of the echelle spectrum of M8 and M17 showing all the lines of multiplet 1 of
\ion{O}{2}.}
\label{m1oii}
\end{center}
\end{figure}
We have detected several {\ion{O}{2}} lines in our data. Our spectra of M8 and M17
present significantly higher signal-to-noise
than that published before by EPTGR and EPTG, and
the number of lines to derive the O$^{++}$/H$^+$ ratio has increased. The lower uncertainties and the
resemblance in the abundances obtained from the different lines make our {\ion{O}{2}} recombination spectra more reliable
than those of EPTG and EPTGR.
Figure~\ref{m1oii} shows the high quality of the spectrum in the zone of multiplet 1 of {\ion{O}{2}}.
This figure can be compared with Figure 1 of \citet{garciarojasetal98}, which shows
the same spectral zone, and a direct comparison of
the quality of the spectra can be made. O$^{++}$/H$^+$ ionic abundance ratios are
presented in Table~\ref{oii}. We have used the same atomic data than in
\citet{garciarojasetal04} and we have corrected
for the departure of the local thermodynamic
equilibrium (LTE) of the upper levels of the transitions of multiplet 1 of {\ion{O}{2}}
for densities {$n_{\rm e}$} $<$ 10000 cm$^{-3}$, which was pointed out by \citet{tsamisetal03} and \citet{ruizetal03}.
\citet{apeimbertetal05} proposed an empirical formulation to re-calculate those populations.
We have applied this NLTE correction
to our data, and we have obtained a very good agreement between the abundances derived
from individual lines of multiplet 1 and the abundance derived using the sum of all the lines,
which is expected not to be
affected by this effect, and with abundances derived from multiplets 2 and 10, which are
almost case independent.
The only detection of a $3d-4f$ transition --which are insensitive to optical depths
effects-- in the spectrum of M8, is a line that is blended
with a {\ion{C}{2}} line, so its intensity is not reliable.
Therefore, we have adopted as representative of the O$^{++}$/H$^+$ ratio the average
of the values given by multiplets 1, 2 and 10.
The O$^{++}$/H$^+$ ratio that we have obtained here for M8 is in very good agreement with
the one obtained by EPTGR (O$^{++}$/H$^+$ = 2.1 $\times$ 10$^{-4}$);
on the other hand, for M17, the O$^{++}$ abundance derived here is somewhat lower
than those
obtained by EPTGR (O$^{++}$/H$^+$ = 5.5 $\times$ 10$^{-4}$) and
\citet{tsamisetal03} (O$^{++}$/H$^+$ = 5.7 $\times$ 10$^{-4}$), but this is probably due to
the different slit size (in the case of EPTG) or to the different slit position (in the case of
\citealt{tsamisetal03},
whose slit is centered about 1' South of our slit center).
\begin{table*}[htbp]\centering
\setlength{\tabnotewidth}{\textwidth}
\tablecols{6}
\setlength{\tabcolsep}{2.2\tabcolsep}
\scriptsize
\caption{O$^{++}$/H$^+$ ratio from {\ion{O}{2}} recombination lines\tabnotemark{a}}
\label{oii}
\begin{tabular}{cccccc}
\toprule
& & $I$($\lambda$)/$I$(H$\beta$) & \multicolumn{3}{c}{O$^{++}$/H$^+$ ($\times$10$^{-5}$)} \\
Mult. & $\lambda_0$ & ($\times$10$^{-2}$) & A & B & C \\
\midrule
\multicolumn{6}{c}{M8} \\
\midrule
1\tabnotemark{b}& 4638.85 & 0.034 $\pm$ 0.005 & 31 $\pm$ 4/21 $\pm$ 3 & 30 $\pm$ 4/20 $\pm$ 3 & \nodata \\
& 4641.81 & 0.043 $\pm$ 0.005 & 16 $\pm$ 2/18 $\pm$ 2 & 16 $\pm$ 2/17 $\pm$ 2 & \nodata \\
& 4649.14 & 0.041 $\pm$ 0.005 & 9 $\pm$ 1/13 $\pm$ 2 & 8 $\pm$ 1/13 $\pm$ 2 & \nodata \\
& 4650.84 & 0.032 $\pm$ 0.005 & 31 $\pm$ 5/19 $\pm$ 2 & 30 $\pm$ 5/18 $\pm$ 2 & \nodata \\
& 4661.64 & 0.036 $\pm$ 0.005 & 29 $\pm$ 4/20 $\pm$ 3 & 28 $\pm$ 4/19 $\pm$ 2 & \nodata \\
& 4673.73 & \nodata & \nodata & \nodata & \nodata \\
& 4676.24 & 0.016 $\pm$ 0.004 & 18 $\pm$ 5/19 $\pm$ 4 & 17 $\pm$ 5/19 $\pm$ 4 & \nodata \\
& Sum & & 18 $\pm$ 1 & {\bf 17 $\pm$ 1}& \nodata \\
2 & 4317.14 & 0.011 $\pm$ 0.004 & 22 $\pm$ 8 & 16 $\pm$ 6 & \nodata \\
& 4319.55 & 0.008: & 15: & 11: & \nodata \\
& 4345.56\tabnotemark{c}&0.022 $\pm$ 0.005 & 41 $\pm$ 9& 29 $\pm$ 6 & \nodata \\
& 4349.43 & 0.020 $\pm$ 0.005 & 14 $\pm$ 3 & 10 $\pm$ 2 & \nodata \\
& 4366.89 & 0.015 $\pm$ 0.004 & 25 $\pm$ 7 & 18 $\pm$ 5 & \nodata \\
& Sum & & 19 $\pm$ 3 & {\bf 13 $\pm$ 1}& \nodata \\ \
10\tabnotemark{d}& 4069.62 & 0.067 $\pm$ 0.007 & 26 $\pm$ 3/26 $\pm$ 3 & \nodata & \nodata \\
& 4069.89 & & & & \\
& 4072.15 & 0.032 $\pm$ 0.006 & 13 $\pm$ 2/13 $\pm$ 2 & \nodata & \nodata \\
& 4075.86 & \nodata & \nodata & \nodata & \nodata \\
& Sum & & 20 $\pm$ 1/{\bf 20 $\pm$ 1}& \nodata & \nodata \\
15\tabnotemark{e}& 4590.97 & 0.005: & 29: & 29: & \nodata \\
& Sum & & 29: & 29: & \nodata \\
19\tabnotemark{d}& 4121.48 & 0.012 $\pm$ 0.005 & 1002 $\pm$ 391 / 746 $\pm$ 291 & 38 $\pm$ 15/ 42 $\pm$ 16 & 38
$\pm$ 15 / 40 $\pm$ 15 \\
& 4132.80 & 0.014 $\pm$ 0.005 & 626 $\pm$ 220/474 $\pm$ 166 & 24 $\pm$ 8/25 $\pm$ 9& 24 $\pm$ 8/23 $\pm$ 8
\\
& 4153.30 & 0.028 $\pm$ 0.005 & 927 $\pm$ 176/810 $\pm$ 154 & 35 $\pm$ 7/35 $\pm$ 7& 35 $\pm$ 7/33 $\pm$ 6
\\
& Sum & & 844 $\pm$ 143/678 $\pm$ 115 & 32 $\pm$ 5/33 $\pm$ 6& 32 $\pm$ 5/31 $\pm$ 5 \\
3d-4f &4491.23\tabnotemark{c}& 0.011 $\pm$ 0.004& \nodata & 70 $\pm$ 26 & \nodata \\
\midrule
& Adopted & &\multicolumn{3}{c}{\bf 17 $\pm$ 1 } \\
\midrule
\multicolumn{6}{c}{M17} \\
\midrule
1\tabnotemark{b}& 4638.85 & 0.093 $\pm$ 0.018 & 85 $\pm$ 16/49 $\pm$ 9 & 82 $\pm$ 16/47 $\pm$ 9 &
\nodata \\
& 4641.81 & 0.128 $\pm$ 0.019 & 48 $\pm$ 7/56 $\pm$ 8 & 47 $\pm$ 7/54 $\pm$ 8 & \nodata \\
& 4649.14 & 0.123 $\pm$ 0.018 & 26 $\pm$ 4/57 $\pm$ 9 & 25 $\pm$ 4/55 $\pm$ 8 & \nodata \\
& 4650.84 & 0.100 $\pm$ 0.018 & 98 $\pm$ 18/48 $\pm$ 9 & 95 $\pm$ 17/46 $\pm$ 8 & \nodata \\
& 4661.64 & 0.119 $\pm$ 0.018 & 97 $\pm$ 15/55 $\pm$ 8 & 94 $\pm$ 14/54 $\pm$ 8 & \nodata \\
& 4673.73 & 0.022: & 116:/57: & 112:/55: & \nodata \\
& 4676.24 & 0.044 $\pm$ 0.014 & 48 $\pm$ 15/55 $\pm$ 18 & 46 $\pm$ 15/53 $\pm$ 17 & \nodata
\\
& Sum & & 53 $\pm$ 4 & {\bf 51 $\pm$ 4}& \nodata \\
2 & 4317.14 & 0.061 $\pm$ 0.018 & 119 $\pm$ 36 & 84 $\pm$ 25 & \nodata \\
& 4319.55 & 0.037: & 72: & 51: & \nodata \\
& 4345.56\tabnotemark{c}& 0.088 $\pm$ 0.020 & 163 $\pm$ 37 & 116 $\pm$ 27 & \nodata \\
& 4349.43 & 0.066 $\pm$ 0.018 & 49 $\pm$ 14 & 35 $\pm$ 10 & \nodata \\
& 4366.89 & 0.030: & 48: & 34: & \nodata \\
& Sum & & 68 $\pm$ 12 & {\bf 48 $\pm$ 9}& \nodata \\
10\tabnotemark{d}& 4069.62 & 0.190 $\pm$ 0.027 & 75 $\pm$ 11/73 $\pm$ 10& \nodata & \nodata \\
& 4069.89 & & & & \\
& 4072.15 & 0.091 $\pm$ 0.022 & 38 $\pm$ 9/38 $\pm$ 9 & \nodata & \nodata \\
& 4075.86 & 0.087 $\pm$ 0.022 & 25 $\pm$ 6/25 $\pm$ 6 & \nodata & \nodata \\
& Sum & & 44 $\pm$ 5/{\bf 43 $\pm$ 5}& \nodata & \nodata\\
19\tabnotemark{d}& 4121.48 & \nodata & \nodata & \nodata & \nodata \\
& 4132.80 & \nodata & \nodata & \nodata & \nodata \\
& 4153.30 & 0.092 $\pm$ 0.021 & 3105 $\pm$ 715/2714 $\pm$ 625& 117 $\pm$ 27/118 $\pm$ 27& 117 $\pm$
27/110 $\pm$ 25\\
& Sum & & 3105 $\pm$ 715/2714 $\pm$ 625& 117 $\pm$ 27/118 $\pm$ 27& 117 $\pm$ 27/110 $\pm$
25\\
\midrule
& Adopted & & \multicolumn{3}{c}{\bf 48 $\pm$ 2 } \\
\bottomrule
\tabnotetext{a}{Only lines with intensity uncertainties lower than 40 \% have been considered. Recombination
coefficients
are those of \citet{storey94} for cases A and B unless otherwise stated.}
\tabnotetext{b}{Not corrected from NLTE effects/corrected form NLTE effects (see text).}
\tabnotetext{c}{Blend.}
\tabnotetext{d}{Values for LS coupling \citep{storey94}/intermediate coupling \citep{liuetal95}.}
\tabnotetext{e}{Dielectronic recombination rates by \citet{nussbaumerstorey84}.}
\end{tabular}
\end{table*}
\section{Total Abundances}
\label{abuntot}
To derive the total gaseous abundances we have to correct for the unseen ionization stages
by using a set of ICFs. We have adopted the same scheme used in
\citet{garciarojasetal05} and \citet{garciarojasetal06}.
\begin{table*}[htbp]\centering
\setlength{\tabnotewidth}{\textwidth}
\newcommand{\DS}{\hspace{5\tabcolsep}}
\tablecols{6}
\setlength{\tabcolsep}{3.2\tabcolsep}
\scriptsize
\caption{Total Gaseous Abundances.}
\label{totabun}
\begin{tabular}{lcc l cc}
\toprule
& \multicolumn{2}{c}{M8}&& \multicolumn{2}{c}{M17} \\
\cmidrule{2-3}
\cmidrule{5-6}
Element & {\emph{$t^2$}}=0.000 & {\emph{$t^2$}}=0.040$\pm$0.004 && {\emph{$t^2$}}=0.000 & {\emph{$t^2$}}=0.033$\pm$0.005\\
\midrule
He & 10.87$\pm$0.01 & 10.85$\pm$0.01& & 10.97$\pm$0.01 & 10.97$\pm$0.01 \\
C\tabnotemark{a} & 8.61/8.69$\pm$0.09 & 8.70/8.69$\pm$0.09& & 8.77$\pm$0.04 & 8.77$\pm$0.04 \\
N & 7.72$\pm$0.03 & 7.96$\pm$0.06& & 7.62$\pm$0.12 & 7.87$\pm$0.13 \\
O & 8.51$\pm$0.05 & 8.73$\pm$0.05& & 8.52$\pm$0.04 & 8.76$\pm$0.05 \\
O\tabnotemark{b} & 8.71$\pm$0.04 & 8.71$\pm$0.04& & 8.76$\pm$0.04 & 8.76$\pm$0.04 \\
Ne & 7.81$\pm$0.12 & 8.03$\pm$0.13& & 7.74$\pm$0.07 & 8.01$\pm$0.09 \\
S & 6.94$\pm$0.03 & 7.28$\pm$0.06& & 7.01$\pm$0.04 & 7.33$\pm$0.06 \\
Cl\tabnotemark{c} & 5.14$\pm$0.04 & 5.41$\pm$0.06& & 5.08/5.06$\pm$0.04 & 5.32/5.30$\pm$0.06 \\
Ar & 6.52$\pm$0.04 & 6.69$\pm$0.07& & 6.39$\pm$0.14 & 6.59$\pm$0.15 \\
Fe & 5.69$\pm$0.08 & 6.04$\pm$0.09& & 5.87$\pm$0.12 & 6.22$\pm$0.14 \\
\bottomrule
\tabnotetext{a}{For M8: ICF from a [C II] UV line/ICF from \citet{garnettetal99}.}
\tabnotetext{b}{For M8, O$^{++}$/H$^+$ and O$^+$/H$^+$ from RLs. For M17, O$^{++}$/H$^+$ from RLs and O$^+$/H$^+$
from CELs and t$^2$.}
\tabnotetext{c}{For M17: From Cl$^{+}$/H$^+$+Cl$^{++}$/H$^+$+Cl$^{3+}$/H$^+$/Using ICF from
\citet{peimberttorrespeimbert77}.}
\end{tabular}
\end{table*}
The total abundances for N, O, Ne, S, Cl, Ne and Fe have been derived using CELs and an ICF, for {\emph{$t^2$}}=0.00 and
{\emph{$t^2$}}$>$0.00.
For C we have computed the total abundance from the C$^{++}$ abundance derived from RLs and an ICF derived from
photoionization
models by \citet{garnettetal99}; for M8, we have also considered the ICF obtained from the C$^+$/H$^+$ ratio
obtained from
$IUE$ observations of the $\lambda$ [\ion{C}{2}] 2326 line
For M8 we have derived also the O/H ratio by adopting O$^{++}$/H$^+$ and O$^+$/H$^+$ from RLs.
For M17 we have computed the total oxygen abundance, adopting O$^{++}$/H$^+$ from RLs and O$^+$/H$^+$ from CELs and
{\emph{$t^2$}}$>$0.00,
because O$^+$/H$^+$ from RLs was not reliable (see \S~\ref{recom}).
In Table~\ref{totabun} we present the adopted total abundances for M8 and M17.
\section{Detection of Deuterium Balmer lines in M8 and M17}
\label{deuterium}
\citet{hebrardetal00b} reported the detection of deuterium Balmer lines
in the spectrum of M8, but they did not find these features in M17.
In M8 these authors
detected from D$\alpha$ to D$\zeta$.
We have detected several weak features in the blue wings of {\ion{H}{1}} Balmer lines in M8 --from H$\alpha$ to
H$\epsilon$--
and in M17 --from H$\alpha$ to H$\delta$--
(see figures~\ref{deum8} and \ref{deum17}). The apparent shifts in radial velocity of
these lines with respect to the {\ion{H}{1}} ones are $-87.6$ km s$^{-1}$ for M8 and $-78.5$ km s$^{-1}$
for M17, which are similar to the isotopic shift of deuterium, $-81.6$ km s$^{-1}$.
\begin{figure}[htbp]
\includegraphics[width=\columnwidth]{f3.eps}
\caption{Wings of {H$\alpha$} to H$\epsilon$ in M8. The {\ion{H}{1}} lines are centered at 0 km s$^{-1}$
velocity.
The dotted line of the left correspond to the average wavelength adopted for the {\ion{D}{1}} lines.}
\label{deum8}
\end{figure}
\begin{figure}[htbp]
\includegraphics[width=\columnwidth]{f4.eps}
\caption{Same as figure~\ref{deum8}, for M17. It is not clear that these features could be {\ion{D}{1}} lines (see text).}
\label{deum17}
\end{figure}
\begin{figure}[htbp]
\includegraphics[width=\columnwidth]{f5.eps}
\includegraphics[width=\columnwidth]{f5b.eps}
\caption{Same as figure~\ref{deum8}, for the wings of {H$\alpha$}, {[\ion{O}{3}]} $\lambda$5007, {[\ion{Ar}{3}]} $\lambda$7135,
and {[\ion{S}{3}]}$\lambda$9531 (top) and {[\ion{O}{2}]} $\lambda$3726, {[\ion{S}{2}]} $\lambda$6716,
and {[\ion{N}{2}]}$\lambda$6548 (bottom) in M17. The dotted line correspond to the average wavelength adopted for the
blue-shifted {\ion{H}{1}} lines.}
\label{bluem17}
\end{figure}
For M8, these weak features could be discarded as high-velocity components of hydrogen following the criteria
established by
\citet{hebrardetal00b} to identify {\ion{D}{1}} lines: a) they are narrower than the {\ion{H}{1}} line, probably because {\ion{D}{1}}
lines arise from much colder material in the photon-dominated region (PDR); b) there are no similar high velocity
components
associated to bright lines of other ions. Furthermore, the Balmer decrement of these lines follows closely the
standard fluorescence
models by \citet{odelletal01} for the Orion nebula (see Table~\ref{deuchar}), indicating that fluorescence should
be the main excitation mechanism of
the {\ion{D}{1}} lines. The difference on the apparent shift in radial velocity measured for these lines with respect to
the isotopic shift of deuterium
(see above) is probably due to relative motions of the gas in the photon dominated region or PDR --where the
deuterium Balmer
lines are supposed to be formed-- with respect to the main emitting layer of the nebula.
Table~\ref{deuchar} shows the main characteristics of the {\ion{D}{1}} Balmer lines in M8.
\begin{table}[htbp]\centering
\setlength{\tabnotewidth}{\columnwidth}
\tablecols{5}
\setlength{\tabcolsep}{0.7\tabcolsep}
\scriptsize
\caption{Deuterium Balmer line properties in M8.}
\label{deuchar}
\begin{tabular}{ccccc}
\toprule
Line & Velocity & FWHM {\ion{D}{1}} & FWHM {\ion{H}{1}} &{\ion{D}{1}}/{\ion{H}{1}} ratio \\
& shift (km s$^{-1}$) & (km s$^{-1}$) & (km s$^{-1}$) & ($\times$10$^{-4}$) \\
\midrule
$\alpha$ & $-87.3$ & $<$ 10: & 24 & 2.9 $\pm$ 0.2 \\
$\beta$ & $-87.6$ & $<$ 10: & 19 & 3.6 $\pm$ 0.5 \\
$\gamma$ & $-87.7$ & $<$ 10: & 19 & 4.1 $\pm$ 1.0 \\
$\delta$ & $-87.7$ & $<$ 10: & 19 & 6.5 $\pm$ 2.0 \\
$\epsilon$ & $-87.6$ & $<$ 10:& 19 & 8.8 $\pm$ 3.0 \\
\bottomrule
\end{tabular}
\end{table}
\citet{hebrardetal00b} identified the weak features in the blue wings of {\ion{H}{1}} Balmer lines in M17 as high velocity
components
of hydrogen mainly because of the presence of very similar features in the wings of {[\ion{N}{2}]}, {[\ion{O}{2}]} and {[\ion{O}{3}]}
lines.
From our data we do not have a clear cut case because: a) the width of the {\ion{H}{1}} blue-shifted feature (HBSF) is
narrow like typical Balmer
{\ion{D}{1}} lines; b) we cannot compare the ratios of HBSF/{\ion{H}{1}} with
the standard fluorescence models of {\ion{D}{1}} Balmer lines by \citet{odelletal01}; in principle the ratios seem to fit
the model, but
errors are so high that it might be possible for the HBSF/{\ion{H}{1}} ratios to be constant (see
Table~\ref{bluefeachar}); and c)
two {[\ion{O}{3}]} lines --$\lambda$$\lambda$4959, 5007--, two {[\ion{Ar}{3}]} lines --$\lambda$$\lambda$7135, 7751--
and two {[\ion{S}{3}]} lines --$\lambda$$\lambda$9069, 9531-- present blue
counterparts at about $\sim$74, 77 and 78 km $s^{-1}$ respectively, which differ in only a few km s$^{-1}$ from the
average
shift of the blue shifted {\ion{H}{1}} lines (see Figure~\ref{bluem17} (top)). These counterparts
are not detected in the wings of {[\ion{N}{2}]}, {[\ion{O}{2}]} and {[\ion{S}{2}]} lines (see Figure~\ref{bluem17} (bottom)).
Table~\ref{bluefeachar} shows the main characteristics of the HBSFs in M17.
\begin{table}[htbp]\centering
\setlength{\tabnotewidth}{\columnwidth}
\tablecols{5}
\setlength{\tabcolsep}{0.7\tabcolsep}
\scriptsize
\caption{{\ion{H}{1}} blue-shifted feature properties in M17.}
\label{bluefeachar}
\begin{tabular}{ccccc}
\toprule
Line & Velocity & FWHM HBSF & FWHM {\ion{H}{1}} &HBSF/{\ion{H}{1}} ratio \\
& shift (km s$^{-1}$) & (km s$^{-1}$) & (km s$^{-1}$) & ($\times$10$^{-4}$) \\
\midrule
H$\alpha$ & $-78.6$ & $<$ 10: & 26 & 3.9 $\pm$ 0.4 \\
H$\beta$ & $-77.7$ & $<$ 10: & 25 & 4.2 $\pm$ 1.3 \\
H$\gamma$ & $-78.7$ & $<$ 10: & 25 & 5.3: \\
H$\delta$ & $-78.9$ & $<$ 10: & 25 & 11.7: \\
\bottomrule
\end{tabular}
\end{table}
From a simple visual inspection of Figure~\ref{bluem17} comparing the width and central wavelength of the blue
components of H$\alpha$
and some forbidden lines, it is possible for the HBSFs to be a blend of {\ion{D}{1}} emission and a blushifted
high-velocity {\ion{H}{1}} component,
but, with the available constraints, we cannot guarantee it.
\section{Comparison with previous abundance determinations.}
\label{comp}
{From} the comparison of our data of M8--HGS with those published by EPTGR, it seems that a small
difference in the volume covered by the slit in this zone is sufficient to change significantly the
ionization degree of the gas; this fact was also pointed out by \citet{sanchezpeimbert91}.
This is because the emission comes from a range of densities,
temperatures, degrees of ionization and sometimes extinctions within the column of gas.
Nonetheless, total abundances should be invariant. In particular, the total oxygen abundance is not affected
by the uncertainty of using an ICF because all the
stages of ionization of this element have been detected in our optical spectra.
In Table~\ref{compa} we show the comparison between total abundances obtained in this work and those obtained in
previous works for M8
and M17. Uncertainties reported in previous works for the total abundances are about 0.1 dex or even larger; taking
into account
the heterogeneity of the error criteria among the different works we have adopted that errors should be about
0.1 dex in this work.
The total abundances of M8 are in quite good agreement with those derived by EPTGR,
within the uncertainties and taking into account that the ICFs for neon and argon
(which present the largest deviations from our data) are reported as uncertain. Also, the ICF scheme and the atomic
data used by EPTGR for iron are different to those used here. Making use of our atomic data and ICF
scheme, the abundances obtained with the EPTGR data lead to a much better agreement (see Table~\ref{compa}).
We have proceeded in a similar way with the data of \citet{peimbertetal93b} and \citet{rodriguez99, rodriguez99b},
reaching the same conclusions.
\begin{table*}[htbp]
\centering
\setlength{\tabnotewidth}{\textwidth}
\tablecols{11}
\setlength{\tabcolsep}{2.2\tabcolsep}
\scriptsize
\caption{Comparison with previous determinations\tabnotemark{a}.}
\label{compa}
\begin{tabular}{lcccclccccc}
\toprule
& \multicolumn{4}{c}{M8} && \multicolumn{5}{c}{M17} \\
\cmidrule{2-5}\cmidrule{7-11}
Element & (1) & (2) & (3) & (4)\tabnotemark{c} && (1) & (4)\tabnotemark{d} & (5) & (6) & (7) \\
\midrule
N & 7.72$\pm$0.03 & 7.68 & 7.75 & 7.60 && 7.62$\pm$0.12& 7.50 & 7.59 & 7.55 &
7.57 \\
O & 8.51$\pm$0.05 & 8.49 & 8.54 & 8.43 && 8.52$\pm$0.04& 8.53 & 8.51 & 8.51 & 8.55
\\
Ne & 7.81$\pm$0.12 & 7.76 & 7.83 & \nodata && 7.74$\pm$0.07& \nodata& 7.81 & 7.78 & 7.79
\\
S & 6.94$\pm$0.03 & 6.96 & 7.03 & 6.95 && 7.01$\pm$0.04& 6.99 & 7.03 & 6.84 &
7.05 \\
Cl & 5.14$\pm$0.04 & 5.21 & \nodata & 5.20 && 5.06$\pm$0.04& 5.02 & 5.03 & 5.03 &
5.07 \\
Ar & 6.52$\pm$0.04 & 6.53 & 6.48 & 6.60 && 6.39$\pm$0.14& 6.36 & 6.26 & 6.35 &
6.39 \\
Fe & 5.69$\pm$0.08 & 5.80 & \nodata & 5.72 && 5.87$\pm$0.12& 5.75 & \nodata & 5.88 &
\nodata \\
\bottomrule
\tabnotetext{a}{In units of 12+log(X/H). Abundances have been recomputed using our atomic data and ICF scheme (see
text).}
\tabnotetext{b}{(1) This work; (2) EPTG; (3) \citet{peimbertetal93b}; (4) \citet{rodriguez99, rodriguez99b};
(5) \citet{tsamisetal03}; (6) EPTGR; (7) \citet{peimbertetal92}.}
\tabnotetext{c}{These data are an average of the total abundances obtained in the two slit positions located
southwards of the
Hourglass.}
\tabnotetext{d}{These data are an average of the results obtained in three slit positions in M17. }
\end{tabular}
\end{table*}
The abundances of M17 are in good agreement, within the errors, with those derived by EPTG, \citet{peimbertetal92},
\citet{rodriguez99b} and \citet{tsamisetal03}.
The larger differences among the different works are probably due to the different sets of atomic data and ICFs
used.
Proceeding in the same way as in M8 we have recomputed the abundances for M17 using our atomic data
and ICF scheme; in this case the differences with the previous calculations are of the order of 0.1 dex. Therefore,
we can conclude
that errors in the line intensities are not responsible for the differences among abundances of these elements by
different authors
(both in M8 and M17)
and we emphasize the high robustness of the abundances determined for these objects, bearing in mind the
uncertainties due to atomic
data and ICF schemes (see Table~\ref{compa}).
\section{SUMMARY}
\label{conclu}
We present new echelle spectroscopy in the 3100--10450 {\rm \AA} range of the Hourglass Nebula in M8 and
a bright rim of M17.
We have determined the physical conditions of M8 and M17 making use of a large number of diagnostic line ratios.
We have derived ionic abundances from CELs as well as C$^{++}$/H$^+$ and O$^{++}$/H$^+$ ratios making use of
RLs in these nebulae.
The ionic abundances obtained from RLs are in very good agreement with those obtained in previous works.
The very good agreement between our results --that have been obtained making use of state-of-the-art atomic data--
and
the best abundance determinations from the literature for M8 and M17 allow us to assure that the total abundances
of these
nebulae are very well established.
We have obtained an average $\emph{$t^2$}$ of 0.040 $\pm$ 0.004 for M8 and 0.033 $\pm$ 0.005 for M17 which are rather
similar to the values
derived in previous works for these two nebulae. Also, it is remarkable the excellent agreement among the {\emph{$t^2$}}
values obtained through
independent methods. This behavior is consistent with the temperature fluctuations scenario.
We confirm the detection of deuterium Balmer emission lines in M8 and possibly in M17, although in this case
there seems to be an accidental contamination of a blueshifted high-velocity {\ion{H}{1}} component.
JGR would like to thank
the members of the Instituto de Astronom\'ia, UNAM, and of the Instituto Nacional de Astrof\'{\i}sica, \'Optica y
Electr\'onica, INAOE, for their always warm hospitality. This work has been partially funded by the Spanish
Ministerio de Ciencia y Tecnolog\'{\i}a (MCyT) under project AYA2001-0436 and AYA2004-07466. MP received partial
support from CONACyT (grant 46904). MR acknowledges support from Mexican CONACyT project
J37680-E.
MTR received partial support from FONDAP(15010003) and Fondecyt(1010404).
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"redpajama_set_name": "RedPajamaArXiv"
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{"url":"https:\/\/www.math.kyoto-u.ac.jp\/ja\/event\/seminar\/3152","text":"# Mackey\u2019s formula for cyclotomic Hecke algebras and rational Cherednik algebras of type $G(r, 1, n)$\n\n2017\/12\/22 Fri 13:00 - 14:30\n\nRIMS402\u53f7\u5ba4\n\nThe restriction\/induction functors play an important role for the representation theory\nof cyclotomic Hecke algebras and rational Cherednik algebras of type $G(r, 1, n)$.\nIn this talk, we discuss an analog of Mackey\u2019s formula for two parabolic subalgebras\nof the cyclotomic Hecke algebras and the rational Cherednik algebras.","date":"2019-04-20 05:10:30","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7467483878135681, \"perplexity\": 556.0894467981036}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-18\/segments\/1555578528523.35\/warc\/CC-MAIN-20190420040932-20190420062932-00112.warc.gz\"}"}
| null | null |
La Coppa del Mondo di sci alpino 1990 fu la ventiquattresima edizione della manifestazione organizzata dalla Federazione Internazionale Sci.
La stagione maschile ebbe inizio l'11 agosto 1989 a Thredbo, in Australia, e si concluse il 17 marzo 1990 a Åre, in Svezia; furono disputate 34 gare (9 discese libere, 6 supergiganti, 7 slalom giganti, 10 slalom speciali, 2 combinate), in 19 diverse località. Lo svizzero Pirmin Zurbriggen si aggiudicò sia la Coppa del Mondo generale, sia quella di supergigante; l'austriaco Helmut Höflehner vinse la Coppa di discesa libera, il norvegese Ole Kristian Furuseth e l'austriaco Günther Mader quella di slalom gigante a pari merito e il tedesco occidentale Armin Bittner quella di slalom speciale. Il lussemburghese Marc Girardelli era il detentore uscente della Coppa generale.
La stagione femminile ebbe inizio l'8 agosto 1989 a Las Leñas, in Argentina, e si concluse il 18 marzo 1990 a Åre, in Svezia; furono disputate 33 gare (8 discese libere, 6 supergiganti, 8 slalom giganti, 9 slalom speciali, 2 combinate), in 16 diverse località. L'austriaca Petra Kronberger si aggiudicò la Coppa del Mondo generale; la tedesca occidentale Katrin Gutensohn vinse la Coppa di discesa libera, la francese Carole Merle quella di supergigante, l'austriaca Anita Wachter quella di slalom gigante e la svizzera Vreni Schneider quella di slalom speciale. La Schneider era la detentrice uscente della Coppa generale.
Uomini
Risultati
Legenda:
DH = discesa libera
SG = supergigante
GS = slalom gigante
SL = slalom speciale
KB = combinata
Classifiche
Generale
Discesa libera
Supergigante
Slalom gigante
Slalom speciale
Combinata
Nel 1990 fu anche stilata la classifica della combinata, sebbene non venisse assegnato alcun trofeo al vincitore.
Donne
Risultati
Legenda:
DH = discesa libera
SG = supergigante
GS = slalom gigante
SL = slalom speciale
KB = combinata
Classifiche
Generale
Discesa libera
Supergigante
Slalom gigante
Slalom speciale
Combinata
Nel 1990 fu anche stilata la classifica della combinata, sebbene non venisse assegnato alcun trofeo alla vincitrice.
Note
Voci correlate
Coppa Europa di sci alpino 1990
Nor-Am Cup 1990
Collegamenti esterni
Sci nel 1990
1990
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{"url":"https:\/\/physics.stackexchange.com\/questions\/248065\/does-there-exist-finite-dimensional-irreducible-rep-of-poincare-group-where-tra\/248078","text":"# Does there exist finite dimensional irreducible rep. of Poincare group where translations act nontrivially?\n\nI read several textbooks of QFT and find that there are two ways to classify the particles or fields. The first one is to study the irreducible representation of Lorentz group (or exactly the universal covering group $SL(2,C)$). Then we find irreducible but not unitary representation $(i,j)$ which is finite dimensional and use them to represent different kinds of field. The second one is to study the unitary representation of Poincare group and we can classify particles by mass and spin.\n\nThen my question is:\n\n1. Why do we not study the finite dimensional irreducible representation of Poincare group, like Lorentz group? Some people will say that the useful representation in Quantum Mechanics is unitary representation and Poincare group which is not compact do not have finite dimensional unitary rep. However this argument is not convincing, because it cannot explain why we still study the finite rep of Lorentz group.\n\n2. Except the \"trivial\" rep., does there exist any other finite dimensional irreducible rep. of Poincare group? Here \"trivial\" means the rep. that we can get from enlarging the original rep. of Lorentz group by letting translation act trivially on original representational space.\n\nFor example, we have a faithful rep. of Poincare group, $\\begin{pmatrix} \\Lambda & x \\\\ 0 & 1 \\end{pmatrix}$, where $\\Lambda$ is Lorentz transformation and $x$ is translation. This is a reducible but indecomposable representation. We can always define an irreducible rep. of Poincare group by\n\n$$f:\\begin{pmatrix} \\Lambda & x \\\\ 0 & 1 \\end{pmatrix}\\rightarrow D_{(i,j)}(\\Lambda)$$ where $D_{(i,j)}(\\Lambda)$ is the irreducible rep. of Lorentz group. So is there other finite dimensional irreducible rep. of Poincare group?\n\n1. It seems that we use Lorentz group's rep. to classify the fields and use Poincare group's rep. to classify the particles. Because the isometry of Minkovski spacetime is Poincare group, why do we only use Lorentz group's rep. to classify the fields and don't take the whole Poincare group into consideration?\n\u2022 All unitary irreducible representations of Lorentz group are infinite dimensional. This is the reason. In fact, unitary reps. of the Poincar\u00e9 group are studied in general not only those of Lorentz group when defining the notion of elementary particle in the sense of Wigner. Dealing with fields the translational part acts trivially, for this reason is usually disregarded when viewing fields as section on some vector bundle based on the spacetime. \u2013\u00a0Valter Moretti Apr 7 '16 at 17:46\n\u2022 @ValterMoretti Thanks. ACuriousMind and your answer have solved the question 1,3. Do you have any idea of question 2? \u2013\u00a0346699 Apr 8 '16 at 5:08\n\u2022 Actually not, did you try to have a look at Barut Raczac's textbook on representations? \u2013\u00a0Valter Moretti Apr 8 '16 at 5:30\n\nLet $\\phi : \\mathbb{R}^4\\to V$ be a field with (complex) target vector space $V$, transforming in a finite-dimensional projective representation $\\rho_\\text{fin} : \\mathrm{SO}(1,3)\\to\\mathrm{U}(V)$. As it is a field, the representation of the translations $\\mathbb{R}^4$ on $V$ is the trivial one, since the field transforms as $\\phi(x)\\overset{x\\mapsto x+a}{\\mapsto} \\phi(x+a)$. Hence, the field transforms in a finite-dimensional representation $\\sigma_\\text{fin}$ of the Poincar\u00e9 group, but the non-trivlal, i.e. interesting, part is the representation of the Lorentz group. Hence, your premise that we \"only study finite-dimensional representations of the Lorentz group\" is wrong, it's just that the finite-dimensional translations are always represented by their trivial representation.\nIn the quantum field theory, the field now becomes operator-valued, acting upons ome Hilbert space $\\mathcal{H}$. Since the quantum field theory shall have Poincare symmetry, there must be a projective unitary representation $\\sigma_\\text{U} : \\mathbb{SO}(1,3)\\ltimes\\mathbb{R}^4\\to\\mathrm{U}(\\mathcal{H})$ upon this space of state. By one of the Wightman axioms, we have that $$\\sigma_\\text{fin}(\\Lambda,a)\\phi(\\Lambda^{-1} x-a) = \\sigma_\\text{U}(\\Lambda,a)^\\dagger \\phi(x)\\sigma_\\text{U}(\\Lambda,a)\\quad \\forall \\Lambda\\in\\mathrm{SO}(1,3),a\\in\\mathbb{R}^4$$ where on the l.h.s., $\\sigma_\\text{fin}$ is a finite-dimensional matrix acting upon the vector $(\\phi^1,\\dots,\\phi^{\\dim(V)})$, and on the r.h.s., the $\\sigma_\\text{U}$ are operators on $\\mathcal{H}$ are are multiplied multiplied with each component operator $\\phi^i$.","date":"2019-07-20 19:58:52","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9491882920265198, \"perplexity\": 325.97596665497105}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-30\/segments\/1563195526670.1\/warc\/CC-MAIN-20190720194009-20190720220009-00261.warc.gz\"}"}
| null | null |
Old Silver pewter, ear drop, linear with stone set style, jewelry charm, pewter castings, B'sue by 1928. Nicely finished back, having the signature 1928 imprint swirl pattern. Measuring 18 x 4mm. Graduating stone sets (from 1-3mm). Perfect for earrings. Old Silver pewter is a specialty artisan finish unique to this line. This has beautiful antique black lacquer highlights, perfect for showing detail.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 2,147
|
{"url":"http:\/\/aas.org\/archives\/BAAS\/v26n2\/aas184\/abs\/S2302.html","text":"The Energy Generation Mechanisms in Infrared-Selected Galaxies\nSession 23 -- Starburst Activity in External Galaxies\nOral presentation, Monday, 30, 1994, 2:00-3:30\n\n## [23.02] The Energy Generation Mechanisms in Infrared-Selected Galaxies\n\nMatthew L.N.Ashby, J.R.Houck (Cornell University)\n\nSubsequent to the discovery of the infrared-luminous galaxies by the IRAS satellite in 1983, a great deal of research activity has been invested in efforts to understand the energy generation mechanisms powering these unusual objects. We have applied newly available optical emission line diagnostics in a program to discriminate between possible energy generation mechanisms, i.e., active nuclei and starbursts, in two samples of IRAS galaxies. The first sample was composed of very faint (F(60$\\mu{\\rm m}) <$ 180 mJy, ${\\rm z} < 0.3$) galaxies drawn from coadded IRAS scans of the North Ecliptic Pole Region. We acquired spectra of 72 sources from the NEPR sample and mapped the redshift distribution of these objects. Of this sample 17 spectra were suitable for emission line analysis. While 3 sources could be classified as AGNs and another 7 as starburst galaxies, the remaining 7 eluded classification. The second sample consisted of 37 much brighter sources drawn from the IRAS Bright Galaxy Sample. Longslit optical emission-line spectra, which separated nuclear and disk contributions to the emission, indicated the BGS subsample contained 7 AGNs and 24 starbursts. In a minority of 6 cases, however, the line diagnostics failed to unambiguously classify the luminosity source. Follow-up near-infrared imaging and spectroscopy to characterize the age and distributions of the stellar populations of the 'straddlers' and mid-infrared imaging to constrain the sizes of the starburst disk components were performed to better understand why the optical diagnostics failed for some galaxies. The 'straddlers' optical and near-infrared characteristics seem to be consistent with unresolved low-luminosity active nuclei residing within a starburst disk.","date":"2014-11-20 23:14:14","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6306110620498657, \"perplexity\": 4921.850615976056}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2014-49\/segments\/1416400372490.23\/warc\/CC-MAIN-20141119123252-00117-ip-10-235-23-156.ec2.internal.warc.gz\"}"}
| null | null |
Ein Pflegestützpunkt (PSP) ist eine örtliche Auskunfts- und Beratungsstelle rund um das Thema Pflege und richtet sich primär an Pflegebedürftige bzw. deren Angehörige.
Pflegestützpunkte werden von den Kranken- und Pflegekassen auf Initiative eines Bundeslandes eingerichtet. Grundlage für die Errichtung der Pflegestützpunkte ist in Deutschland der am 1. Juli 2008 in Kraft getretene § 92c des SGB XI im Rahmen des Pflege-Weiterentwicklungsgesetzes (jetzt: § 7c SGB XI).
In den Pflegestützpunkten soll im Rahmen des Case-Managements die durch das Pflege-Weiterentwicklungsgesetz eingeführte Pflegeberatung stattfinden.
2017 sind folgende Aufgaben der Pflegestützpunkte festgelegt:
umfassende sowie unabhängige Auskunft und Beratung zu den Rechten und Pflichten nach dem Sozialgesetzbuch und zur Auswahl und Inanspruchnahme der bundes- oder landesrechtlich vorgesehenen Sozialleistungen und sonstigen Hilfsangebote,
Koordinierung aller für die wohnortnahe Versorgung und Betreuung in Betracht kommenden gesundheitsfördernden, präventiven, kurativen, rehabilitativen und sonstigen medizinischen sowie pflegerischen und sozialen Hilfs- und Unterstützungsangebote einschließlich der Hilfestellung bei der Inanspruchnahme der Leistungen,
Vernetzung aufeinander abgestimmter pflegerischer und sozialer Versorgungs- und Betreuungsangebote.
Der Stützpunkt bildet hierfür das gemeinsame Dach für das Personal der Pflege- und Krankenkassen, der Altenhilfe oder der Sozialhilfeträger. Hier können sie den Betroffenen ihre Sozialleistungen erläutern und vermitteln.
PSP-Netzwerke der Bundesländer
In Deutschland organisieren sich viele Pflegestützpunkte auf Länderebene. Die PSP-Netzwerke dienen der Vermeidung von Doppelstrukturen und der Stärkung kommunaler Beratungs- und Betreuungsangebote. Eine gemeinsame Organisation unter einem Dach erleichtert zudem Betroffenen und Angehörigen die Suche nach dem Stützpunkt in ihrer Nähe sowie die Kontaktaufnahme und Terminvereinbarung.
Literatur
Weblinks
Pflegestützpunktsuche des BKK Bundesverbandes e.V.
https://www.pflegestuetzpunkte-deutschlandweit.de/
Siehe auch
Pflege-Charta
Pflegeversicherung
Integrierte Versorgung
Managed Care
Einzelnachweise
Pflege und Betreuung in Deutschland
Beratung
Sozialversicherung (Deutschland)
Sozialstaat (Deutschland)
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8,552
|
import { moduleForComponent, test } from 'ember-qunit';
import hbs from 'htmlbars-inline-precompile';
moduleForComponent('swiper-container',
'Integration | Component | swiper-container',
{ integration: true });
test('it doesn\'t allow inline rendering', function(assert) {
this.render(hbs`{{swiper-container}}`);
assert.equal(this.$().text().trim(),
`This component only support block rendering, inline rendering will be useless.
这个组件仅支持块级方式渲染,内联方式渲染将毫无作用。`,
'inline rendering should print out friendly warning');
});
test('it provides the basic required html structures', function(assert) {
this.render(hbs`
{{#swiper-container}}
<section class="swiper-slide">anything</section>
{{/swiper-container}}
`);
const $component = this.$(':first-child');
assert.ok($component.hasClass('swiper-container'),
'.swiper-container should be existed on 1st level div');
assert.ok($component.find(':first-child').hasClass('swiper-wrapper'),
'.swiper-wrapper should be existed on 2nd level div');
assert.equal($component.find('.swiper-slide').text().trim(), 'anything');
});
test('it renders extra sturctures by options hash w/ default value', function(assert) {
this.render(hbs`
{{#swiper-container options=(hash
prevButton=".swiper-button-prev"
nextButton=".swiper-button-next"
scrollbar=".swiper-scrollbar"
pagination=".swiper-pagination"
)}}
<section class="swiper-slide">anything</section>
{{/swiper-container}}
`);
const $container = this.$(':first-child');
assert.ok($container.find('.swiper-scrollbar').length > 0,
'check for .swiper-scrollbar');
assert.ok($container.find('.swiper-button-prev').length > 0,
'check for .swiper-button-prev');
assert.ok($container.find('.swiper-button-next').length > 0,
'check for .swiper-button-next');
assert.ok($container.find('.swiper-pagination').length > 0,
'check for .swiper-pagination');
});
test('it also allows customized class name for extra sturctures', function(assert) {
this.render(hbs`
{{#swiper-container options=(hash
prevButton=".custom-button-prev"
nextButton=".custom-button-next"
scrollbar=".custom-scrollbar"
pagination=".custom-pagination"
)}}
<section class="swiper-slide">anything</section>
{{/swiper-container}}
`);
const $container = this.$(':first-child');
assert.ok($container.find('.custom-button-prev').length > 0,
'check for .custom-button-prev');
assert.ok($container.find('.custom-button-next').length > 0,
'check for .custom-button-next');
assert.ok($container.find('.custom-scrollbar').length > 0,
'check for .custom-scrollbar');
assert.ok($container.find('.custom-pagination').length > 0,
'check for .custom-pagination');
});
test('even customized class names still have the default class name', function(assert) {
this.render(hbs`
{{#swiper-container options=(hash
prevButton=".custom-button-prev"
nextButton=".custom-button-next"
scrollbar=".custom-scrollbar"
pagination=".custom-pagination"
)}}
<section class="swiper-slide">anything</section>
{{/swiper-container}}
`);
const $container = this.$(':first-child');
assert.ok(
$container.find('.custom-button-prev').hasClass('swiper-button-prev'),
'check for .*-button-prev'
);
assert.ok(
$container.find('.custom-button-next').hasClass('swiper-button-next'),
'check for .*-button-next'
);
assert.ok(
$container.find('.custom-scrollbar').hasClass('swiper-scrollbar'),
'check for .*-scrollbar'
);
assert.ok(
$container.find('.custom-pagination').hasClass('swiper-pagination'),
'check for .*-pagination'
);
});
test('it yields swiper.slide component with customizable structures', function(assert) {
this.render(hbs`
{{#swiper-container as |swiper|}}
{{#swiper.slide class="custom-slide"}}A{{/swiper.slide}}
{{#swiper.slide}}B{{/swiper.slide}}
{{#swiper.slide}}C{{/swiper.slide}}
{{/swiper-container}}
`);
const $component = this.$(':first-child');
assert.ok($component.find('.swiper-slide').length === 3);
assert.ok($component.find('.custom-slide').length === 1);
});
test('it initializ Swiper properlly', function(assert) {
this.render(hbs`
{{#swiper-container as |swiper|}}
{{#swiper.slide}}A{{/swiper.slide}}
{{#swiper.slide}}B{{/swiper.slide}}
{{#swiper.slide}}C{{/swiper.slide}}
{{/swiper-container}}
`);
const $component = this.$(':first-child');
assert.ok($component.hasClass('swiper-container-horizontal'));
});
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,716
|
La Bibliothèque centrale de Soleure ( Zentralbibliothek Solothurn) est une bibliothèque de formation et d'études située à Soleure. Elle remplit à la fois les tâches d'une bibliothèque cantonale et d'une bibliothèque municipale contenant une section publique et une bibliothèque musicale. Très important est le vieux fonds volumineux avec beaucoup d'incunables.
Historique
La Bibliothèque municipale
La bibliothèque municipale a été fondée en 1763, influencée par les Lumières, et se basait initialement surtout sur des donations de familles patriciennes. Au début, l'accès au large public n'était pas garanti. Jusqu'aux années 1830, la commission de censure avec son esprit très réactionnaire empêchait l'agrandissement de la bibliothèque et son ouverture au public.
En 1838, la situation a changé profondément. La bibliothèque a déménagé dans des locaux plus appropriés où elle restera jusqu'à la fusion avec la bibliothèque cantonale. Le règlement bibliothécaire de la nouvelle administration municipale libérale permet à un large public d'utiliser la bibliothèque.
La Bibliothèque cantonale
La bibliothèque cantonale de Soleure a été fondée à la fin du ; elle a repris les collections de livres en possession des monastères qui avaient été dissous après la guerre culturelle en 1874. Les fonds ont été placés dans la salle de trône de l'ancienne Ambassadorenhof ou se trouvait à l'époque le bâtiment de l'école cantonale.
la Bibliothèque centrale
La bibliothèque centrale est le résultat d'une fusion de la bibliothèque municipale avec la bibliothèque cantonale en 1930. L'ensemble de bâtiments est composé d'une demeure patricienne construite à la fin du , et d'un bâtiment neuf. Dans la résidence estivale patricienne maison Gibelin-Zetter (en allemand Gibelin-Zetter-Haus) se trouvent la bibliothèque musicale moderne, quelques bureaux et le musée de livres. L'immeuble neuf abrite la bibliothèque en accès libre, la salle de lecture et la bibliothèque de littérature enfantine et de jeunesse.
Fonds
La bibliothèque possède au total environ documents. Sa mission principale est la conservation du patrimoine cantonal culturel ou Solodorensia. Ce sont :
des publications qui concernent la ville et le canton de Soleure ;
des œuvres d'auteurs, illustrateurs, artistes et musiciens d'origine du canton de Soleure ;
des biographies de personnages habitant dans le canton ou
des ouvrages imprimés ou édités dans le canton.
Les Solodorensia imprimés ou électroniques concernant les thèmes « histoire régionale », « géographie locale » ainsi que « sciences sociales » sont saisis dans la Bibliographie de la littérature d'histoire de Soleure.
La bibliothèque musicale moderne garde un des plus grands fonds audio publics en Suisse. Au total, elle contient médias dont des CD, disques et cassettes.
La bibliothèque de littérature enfantine et de jeunesse possède documents courants. Les ouvrages anciens se trouvent dans le magasin fermé.
Catalogue
La bibliothèque a installé un OPAC (Online public access catalog, donc un catalogue de bibliothèque accessible en ligne) pour les fonds en accès libre, pour la bibliothèque musicale et celle de littérature enfantine et de jeunesse.
Liens externes
Site web de la Bibliothèque centrale de Soleure
Bibliothèques en Suisse
Internet Clearinghouse Suisse
Voir aussi
Bibliothèque cantonale
Soleure
Bâtiment à Soleure
Bien culturel d'importance nationale dans le canton de Soleure
Soleure
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 9,554
|
using ChainUtils.BouncyCastle.Math.EC.Custom.Sec;
using ChainUtils.BouncyCastle.Utilities.Encoders;
namespace ChainUtils.BouncyCastle.Math.EC.Custom.Djb
{
internal class Curve25519
: AbstractFpCurve
{
public static readonly BigInteger q = Nat256.ToBigInteger(Curve25519Field.P);
private const int Curve25519_DEFAULT_COORDS = COORD_JACOBIAN_MODIFIED;
protected readonly Curve25519Point m_infinity;
public Curve25519()
: base(q)
{
m_infinity = new Curve25519Point(this, null, null);
m_a = FromBigInteger(new BigInteger(1,
Hex.Decode("2AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA984914A144")));
m_b = FromBigInteger(new BigInteger(1,
Hex.Decode("7B425ED097B425ED097B425ED097B425ED097B425ED097B4260B5E9C7710C864")));
m_order = new BigInteger(1, Hex.Decode("1000000000000000000000000000000014DEF9DEA2F79CD65812631A5CF5D3ED"));
m_cofactor = BigInteger.ValueOf(8);
m_coord = Curve25519_DEFAULT_COORDS;
}
protected override ECCurve CloneCurve()
{
return new Curve25519();
}
public override bool SupportsCoordinateSystem(int coord)
{
switch (coord)
{
case COORD_JACOBIAN_MODIFIED:
return true;
default:
return false;
}
}
public virtual BigInteger Q
{
get { return q; }
}
public override ECPoint Infinity
{
get { return m_infinity; }
}
public override int FieldSize
{
get { return q.BitLength; }
}
public override ECFieldElement FromBigInteger(BigInteger x)
{
return new Curve25519FieldElement(x);
}
protected internal override ECPoint CreateRawPoint(ECFieldElement x, ECFieldElement y, bool withCompression)
{
return new Curve25519Point(this, x, y, withCompression);
}
protected internal override ECPoint CreateRawPoint(ECFieldElement x, ECFieldElement y, ECFieldElement[] zs, bool withCompression)
{
return new Curve25519Point(this, x, y, zs, withCompression);
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,968
|
A 40-year-old man, Femi Adebowale, has been arrested by the Ogun State Police Command for beating his wife, Kuburat, 37, to death. The incident happened on Monday at their residence in Agbado area of the state.
According to a statement on Friday by the Police Public Relations Officer in the state, Abimbola Oyeyemi, on the day of the incident, the deceased had asked her husband for the money she would spend on her younger brother's naming ceremony.
PUNCH reports that Femi gave her N10,000 but the deceased insisted that the money was not enough, which led to a fight that resulted in her death, the statement said.
Oyeyemi said the suspect was arrested following a complaint from the deceased's younger brother, Shakiru Alao, who reported at Agbado Division that his elder sister had been beaten to death by the husband over allegation of adultery.
"In the suspect's statement, he alleged that the deceased had been involving in act of infidelity for quite some time to the extent that she was dating their next door neighbour.
Meanwhile the state Commissioner of Police, Ahmed Iliyasu, has ordered that the case should be transferred to the homicide section of the State Criminal Investigation and Intelligence Department for further investigation and possible prosecution of the suspect.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 9,184
|
Wells, H.G. [George Bernard Shaw]
The World Set Free: A Story of Mankind.
Macmillan and Co., Limited, London, 1914. First edition, first issue of Wells' prophetic novel predicting the arrival of atomic weaponry with the publisher listed as Macmillan and Co. Limited (as opposed to Ltd.), 8 pages of advertisements at rear, and no statement of printing to the copyright page. Octavo, original cloth stamped in blind with gilt titles to the spine and front panel, top edge gilt. Association copy, inscribed by H.G. Wells to George Bernard Shaw, "G.B.S. from H.G." Like Wells, George Bernard Shaw used writing fiction as a vehicle to disseminate his political, social and religious ideas. Wells and Shaw connected when Wells joined the gradualist Fabian society in 1903. Shaw had, since the mid 1880s, been a dedicated member and advocated its message of moderation in the face of a debate regarding the option to embrace anarchism. In the years following the 1906 election, Shaw felt that the Fabians needed fresh leadership and saw this in the form of Wells. Wells, however, held views at odds with the party's "Old Gang" led by Shaw, particularly with proposals for closer cooperation with the Independent Labour Party, and soon resigned from the Society. Following Wells' death in 1946, Shaw wrote his obituary for The New Statesman, stating, "To Fabian socialist doctrine he could add little; for he was born ten years too late to be in at its birth pangs. Finding himself only a fifth wheel in the Fabian coach he cleared out; but not before he had exposed very effectively the obsolescence and absurdity of our old parish and county divisions as boundaries of local gove … [Click Below for Full Description]
Bookseller: Raptis Rare Books, ABAA/ ILAB [Palm Beach, FL, U.S.A.]
AbeBooks Biblio
LINK TO THIS PAGE: https://www.vialibri.net/years/books/168800270/1914-wells-hg-george-bernard-shaw-the-world-set-free-a-story
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 2,664
|
Solarino (sicilià San Pàulu) és un municipi italià, situat a la regió de Sicília i a la província de Siracusa. L'any 2006 tenia 7.365 habitants. Limita amb els municipis de Floridia, Palazzolo Acreide, Priolo Gargallo, Siracusa i Sortino.
Evolució demogràfica
Administració
Galeria d'imatges
Municipis de Siracusa
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,412
|
{"url":"https:\/\/de.maplesoft.com\/support\/help\/maple\/view.aspx?path=UserManual%2FChapter11","text":"11 Input, Output, and Interacting with Other Products - Maple Programming Help\n\nHome : Support : Online Help : Manuals : User Manual : UserManual\/Chapter11\n\n11 Input, Output, and Interacting with Other Products\n\n11.1 In This Chapter\n\nSection\n\nTopics\n\nWriting to Files\u00a0- Saving to Maple file formats\n\n \u2022 Saving Data to a File\n \u2022 Saving Expressions to a File\n\nReading from Files\u00a0-Opening Maple files\n\n \u2022 Reading Data from a File\n \u2022 Reading Expressions from a File\n\nExporting to Other Formats\u00a0- Exporting documents in file formats supported by other software\n\n \u2022 Exporting Documents\n \u2022 MapleNet\n\nConnectivity\u00a0- Using Maple with other programming languages and software\n\n \u2022 Translating Maple Code to Other Programming Languages\n \u2022 Accessing External Products from Maple\n \u2022 Accessing Maple from External Products\n \u2022 Sharing and Storing Maple Worksheet Content with the MapleCloudTM","date":"2020-02-20 02:06:16","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 21, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.25881579518318176, \"perplexity\": 7927.566269303932}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-10\/segments\/1581875144498.68\/warc\/CC-MAIN-20200220005045-20200220035045-00435.warc.gz\"}"}
| null | null |
{"url":"https:\/\/www.mail-archive.com\/emacs-orgmode@gnu.org\/msg48608.html","text":"# [O] Bug(?) report footnotes\n\nDear Orgmode community,\n\n\nlast week I submitted a journal paper of appr. 30 pages using orgmode and latex export and I ran into trouble regarding footnotes several times.\n\nThe document didn't compile due to wrongly set brackets of footnotes if:\n\nA footnote is placed without a blank line infront of a heading that is exported as itemized.\n\nso\n[fn:xyz]\n\nwas exported to something like\n\n\\footnote{asdfsdfasdfasdf \\begin{itemize} qwerwerwqe}\n\n\nsince the document has been quite big it has been really plenty of work to find these errors.\n\n\nIf of any interest I can try to reproduce this in a less than 9500 word ducument.\n\nBest Markus\nP.S.: I used the IEEEtran class.","date":"2021-10-21 19:18:38","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7648727893829346, \"perplexity\": 5587.7214968015405}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-43\/segments\/1634323585439.59\/warc\/CC-MAIN-20211021164535-20211021194535-00319.warc.gz\"}"}
| null | null |
package com.hazelcast.internal.cluster.impl;
import com.hazelcast.cluster.Member;
import com.hazelcast.cluster.MemberSelector;
import java.lang.reflect.Array;
import java.util.ArrayList;
import java.util.Collection;
import java.util.Iterator;
import java.util.List;
import java.util.NoSuchElementException;
/**
* An immutable collection that applies all the {@link MemberSelector} instances to
* its internal {@link Member} collection. It reflects changes in the internal collection.
* Mutating methods throw {@link java.lang.UnsupportedOperationException}
* It is mainly used for querying a member list.
*
* @param <M> A subclass of {@link Member} interface
*/
public final class MemberSelectingCollection<M extends Member> implements Collection<M> {
private final Collection<M> members;
private final MemberSelector selector;
public MemberSelectingCollection(Collection<M> members, MemberSelector selector) {
this.members = members;
this.selector = selector;
}
@Override
public int size() {
return count(members, selector);
}
public static <M extends Member> int count(Collection<M> members, MemberSelector memberSelector) {
int size = 0;
for (M member : members) {
if (memberSelector.select(member)) {
size++;
}
}
return size;
}
@Override
public boolean isEmpty() {
return !iterator().hasNext();
}
@Override
public boolean contains(Object o) {
for (M member : members) {
if (selector.select(member) && o.equals(member)) {
return true;
}
}
return false;
}
@Override
public Iterator<M> iterator() {
return new MemberSelectingIterator();
}
@Override
public Object[] toArray() {
List<Object> result = new ArrayList<>();
for (M member : members) {
if (selector.select(member)) {
result.add(member);
}
}
return result.toArray(new Object[0]);
}
@Override
public <T> T[] toArray(T[] a) {
List<Object> result = new ArrayList<>();
for (M member : members) {
if (selector.select(member)) {
result.add(member);
}
}
if (a.length != result.size()) {
a = (T[]) Array.newInstance(a.getClass().getComponentType(), result.size());
}
for (int i = 0; i < a.length; i++) {
a[i] = (T) result.get(i);
}
return a;
}
@Override
public boolean add(M member) {
throw new UnsupportedOperationException();
}
@Override
public boolean remove(Object o) {
throw new UnsupportedOperationException();
}
@Override
public boolean containsAll(Collection<?> c) {
for (Object o : c) {
if (!contains(o)) {
return false;
}
}
return true;
}
@Override
public boolean addAll(Collection<? extends M> c) {
throw new UnsupportedOperationException();
}
@Override
public boolean removeAll(Collection<?> c) {
throw new UnsupportedOperationException();
}
@Override
public boolean retainAll(Collection<?> c) {
throw new UnsupportedOperationException();
}
@Override
public void clear() {
throw new UnsupportedOperationException();
}
class MemberSelectingIterator
implements Iterator<M> {
private final Iterator<M> iterator = MemberSelectingCollection.this.members.iterator();
private M member;
@Override
public boolean hasNext() {
while (this.member == null && iterator.hasNext()) {
M nextMember = iterator.next();
if (selector.select(nextMember)) {
this.member = nextMember;
}
}
return member != null;
}
@Override
public M next() {
M nextMember;
if (member != null || hasNext()) {
nextMember = member;
member = null;
} else {
throw new NoSuchElementException();
}
return nextMember;
}
@Override
public void remove() {
throw new UnsupportedOperationException();
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,927
|
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