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Author: Elizabethpagano
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"The Book of Boba Fett" Recap: 'The Gathering Storm'
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Ah, the last of the flashbacks.
We begin this episode once again in Boba Fett's bacta tank, where he's having another flashback that picks up right where we left off during his last one. On his bantha traveling from the scene of the Tusken Raiders' destroyed camp, Fett locates Jabba's (Fortuna's) Palace and takes account of the situation from afar, realizing that some of Fortuna's guards are patrolling the grounds.
As it's starting to get late, Fett calls it a day and starts up a campfire that night. He enjoys some food with his bantha but is taken aback when he hears a few explosions popping off nearby. Traveling a bit to see what the fuss was about, he finds Fennec Shand lying on the ground, nearly dead.
Lucasfilm Ltd. / Disney+
He straps her onto his bantha and they head off to the outskirts of Mos Eisley, where Fett finds a mod-parlor shop (which is responsible for creating the human-droid modifications like that of the teen gang from the last episode) to save her life. Running things much like a modern-day tattoo parlor, the owner of the shop finally agrees to work on Shand after Fett hands the artist a bag of coins.
Waking up at the campfire later that night, Shand glances down and sees her droid-abdomen for the first time and Fett explains what happened. He reveals that he knows who she is and agrees that she's worth more alive than dead. Because she owes Fett for saving her life, she agrees to help him break into Jabba's Palace and retrieve his ship, Slave 1.
The next day, the two set off for Jabba's Palace on a reconnaissance mission, using Shand's mini spy droid to take a look at the situation on the inside. Upon its return, the droid creates a hologram-like map of the inside of the palace, including where the guards are stationed. With this information revealing that tons of guards are patrolling inside, Fett decides that they'll need to time it with the guards' shifts in order to get to Slave 1.
Taking off on their mission, they enter Jabba's Palace after Shand cuts open the metal bars of a tiny, open-air window on the side of the building at ground level. Once inside, they first encounter the kitchen, with two droids cutting up various vegetables and cooking food. Fett quickly takes out the smaller of the two droids, moving on to the larger one, who's wielding six different knives, spinning them in an aggressive fashion very similar to that of General Grievous. Before Fett is able to get to him, Shand takes him out from behind.
Lucasilm Ltd.
However, they're not quite done in the kitchen just yet. They hear someone coming, so they hide on either side of the kitchen's entryway while an extremely cute, little LEP Droid walks in and is wondering what's going on. A quick droid, it runs around the kitchen and through smaller spaces to avoid Fett and Shand, who eventually corner him. They don't kill him; instead, he turns himself off to avoid the situation altogether.
The two move on to the hangar holding Slave 1, which looks incredibly rusty, though a few Gamorrean Guards enter in behind them, forcing Fett and Shand to hide once again. They quickly take the guards out, but a horde of guards storm into the hangar, greatly outnumbering the two. They try to pick off the Pykes while moving closer to the ship; Fett is able to get into the cockpit while Shand stays behind and continues to take out the enemies.
Slave 1 turns on, though Fett is having a tough time maneuvering the ship out of the hangar and can't get the door to the outside of the palace open. Shand hops onto the back ramp of the ship and fights from up there, with some of Fortuna's men jumping up and onto the ramp to take her out. She's able to fend them off while figuring out how to open the outside door, which she does by firing at a weight holding the door up. With that, the two are finally able to fly out of the hangar.
After escaping Jabba's Palace, they head over to the site of the Sarlacc pit, which Fett believes is holding his armor, obviously forgetting the fact that a bunch of Jawas stripped him of his gear. He flies the ship right on top of the pit in order to shine a light in and see better, but out of nowhere, the Sarlacc comes to life and tries to pull down the ship. The Sarlacc only lets go when Shand pushes a button to release a seismic charge into the pit—and what a throwback the sound of that charge was.
Now that the Sarlacc has been killed, Fett lands Slave 1 and heads into the pit in a deep search for his armor, though the only thing he comes up with is a thick layer of mucus from the inside of the Sarlacc. And with that, they head back to set up camp and end the night sitting by the fire. Shand asks Fett if he's serious about starting his own "house" (collective) in Mos Espa, and he is, explaining how avoidable most of the jobs he took as a Bounty Hunter were. He invites her to join him in his new journey, and though she's skeptical, Fett offers her loyalty (and money) to be his partner.
Back in the present day, Fett exits the bacta tank and is told that he is fully recharged, which means that we—as the audience—are now caught up on his backstory. Fresh out the tank, Fett takes on Mos Espa, making the rounds and stopping at Garsa Fwip's cantina to show his face.
While there, he watches on as a fight breaks out between Black Krrsantan and a few Trandoshans, simply because Krrsantan was annoyed by their rowdy behavior. Flinging them all across the room, he's about to kill one of the Trandoshans when Fwip comes in and gives him a good talking to. However, it's not completely good enough, as Krrsantan decides to bite one of their arms off before exiting the building. Things go back to normal almost immediately in the cantina, as Fwip says, "Hit it, Max," which signals the band to resume playing, loose arm on the floor and all.
Fett follows Krrsantan outside and simply says, "Looks like you could use a job," and judging by the timbre of his growl, it seems (I hope) that he'll be joining the squad.
Fett and Shand meet with the three families who control Mos Espa: the Trandoshans, the Aqualish, and the Klatoonians, the latter of whom I didn't realize resembled the Whos from How The Grinch Stole Christmas. Fett discusses how they all need to join together as one force as the Pykes are clearly gearing up for some sort of war against him and the control of Mos Espa. Though they're not sure if they want to create negative tension with the Pykes, they finally agree to remain neutral in the event that the Pykes want them to turn against Fett.
Afterward, Fett and Shand return to the palace and reflect on the meeting, agreeing that a war with the Pykes is definitely imminent and that they need to prepare. When Fett says that he's lacking the muscle needed to take on a war like this, we hear the theme of The Mandalorian quietly play, possibly alluding to a Mando appearance for the next episode.
"The Book of Boba Fett": 'The Streets of Mos Espa' Recap
"The Book of Boba Fett" Recap: 'The Tribes of Tatooine'
"The Book of Boba Fett" Recap: 'Stranger in a Strange Land'
Tags Book of Boba Fett Episode 4, Book of Boba Fett Recap, The Book of Boba Fett
Prime Video's 'As We See It' Begs the Question: What Is "Normal"?
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Ever combine pineapple juice, Sprite, cranberry juice, vodka, M&M's, lime juice, and Fruity Pebbles with a pink sugar rim and a gummy bear garnish? Okay, yeah, don't.
Director of Friday Night Lights and Parenthood, Jason Katims' new series, As We See It, features Harrison (Albert Rutecki), Jack (Rick Glassman), and Violet (Sue Ann Pien), who are three roommates on the Autism spectrum living together in an apartment.

Harrison is the most introverted of the bunch, especially having trouble leaving the apartment because of loud noises, random encounters, and, most importantly: dogs. A good portion of the show is spent tracking him as he works to forge past his fear of walking down the street.
Jack is a computer genius who works at a publishing house, though he has trouble keeping his job because he doesn't hold back in sharing that he's smarter than everyone else in the room. Jack speaks his mind, not always taking other people's feelings into account. He's also probably the wittiest of the bunch, with some great one-liners that don't even seem like they were meant to be funny.
Violet works at Arby's—first as a cashier, though she's quickly demoted to "sandwich technician" after she makes an inappropriate comment to a customer. Similar to Jack, Violet has less of a filter when it comes to speaking her mind, though she is also anxious to be more grown-up, so she's determined to find the perfect boyfriend.
Ali Goldstein / Amazon Studios
Harrison, Jack, and Violet also share Mandy (Sosie Bacon), their aide, who works with them throughout each day and holds morning meetings to discuss their goals for the week. Extremely skilled at what she does, Mandy can push them towards their goals without pushing too much; through these close bonds, she's able to see how they view the world and how the outside world views them. Being able to see both sides allows the audience to see that, on top of how much the world needs to walk around with a bit more compassion for people.
Though Harrison, Jack, and Violet have a few overlapping qualities, the most apparent of them all is their shared feeling of just wanting to fit in with everyone else. The word that gets thrown around the most in this show is that infamous one: "normal."
Each character is told at various points that their differences are the best things about them, but most of the time, all they want is the opposite. In an effort to appear "normal" to someone he likes, Jack finally asks Mandy, "Is it obvious to everyone that I'm not normal?" He has felt different his whole life, and it seems clear that he's sick of being told that it's better not to be normal. When Mandy responds to Jack and says, "For what it's worth, I think you are such a beautiful person that you shouldn't have to hide who you are," Jack is clearly stuck in an internal struggle and simply says, "That makes me want to throw up."
As a whole, As We See It gently blends drama and comedy to show a realistic perspective of what it's like to navigate the world while being on the Autism spectrum. It focuses on remembering to celebrate the little wins, as each one is a vital part of the process of growing. Progress takes time, and over the course of the show, we're able to see how much the characters have already moved forward with their goals.
At the beginning of the first episode, Jack expresses his intense desire to buy a Roomba. When he finally buys one and starts it up, the Roomba moves all across the floor, making its way through the living room until it finally hits a piece of furniture. Without hesitation, the Roomba immediately flips itself around and heads in another direction. I'd like to think that the Roomba was given that spotlight to say a little something about the characters in As We See It.
Each character is trucking along with their own goals until they hit a little something in their paths. Though it might jar them for a second, they don't let it bog themselves down—they keep on moving.
As We See It premieres on Prime Video on January 21, with eight episodes in total.
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Tags As We See It, As We See It Review, Prime Video
The 20 Best Independent Films of All Time
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Independent films are selected time and time again for Best Picture and cover a wide variety of genres, almost always on a niche, smaller level. Though they might not have the same wide scope as a blockbuster Spider-Man movie, independent films can certainly pop off and hit it big, making those watching forget that it's actually a smaller, independent movie.
Films like Get Out and Parasite were both independent films, though they each received widespread acclaim, basically turning them into blockbuster hits. In no specific order, these next 20 films are a few of the greatest we've seen across our screens in recent years.

","hed1. "Moonlight"","buttonTextWatch on Hulu","buttonUrlhttps://www.hulu.com/movie/moonlight-fc634fff-8a98-4138-9c4e-6486a6f33573

1. "Moonlight"
Based on Tarell Alvin McCraney's play, Moonlight follows the character of Chiron at three different stages of his life, as he works through various emotions and life events. This includes the struggles of growing up, his identity and sexuality, as well as the emotional abuse that he endured over the years.
Watch on Hulu
https://www.hulu.com/movie/moonlight-fc634fff-8a98-4138-9c4e-6486a6f33573

","hed2. "Requiem For a Dream"","buttonTextWatch on HBO Max","buttonUrlhttps://www.hbomax.com/feature/urn:hbo:feature:GYN49Jg05-ROXUQEAAAIw

2. "Requiem For a Dream"
This classic psychological thriller stars Jennifer Connelly, Jared Leto, and Ellen Burstyn, as Requiem For a Dream is about drug addiction and how the different characters suffer from it. It's a deep dive into how drugs affect people psychologically and lead them into various states of delusion.
Watch on HBO Max
https://www.hbomax.com/feature/urn:hbo:feature:GYN49Jg05-ROXUQEAAAIw

","hed3. "The Grand Budapest Hotel"","buttonTextWatch on Prime Video","buttonUrlhttps://www.amazon.com/Grand-Budapest-Hotel-Ralph-Fiennes/dp/B00JB3AOEY

3. "The Grand Budapest Hotel"
A classic Wes Anderson film, The Grand Budapest Hotel has a stacked cast as many of his films do, including Willem Dafoe, Saoirse Ronan, Tilda Swinton, Ralph Fiennes, and Tony Revolori. The storyline follows a mystery as the concierge of an extremely upscale hotel is framed for carrying out a murder.
Watch on Prime Video
https://www.amazon.com/Grand-Budapest-Hotel-Ralph-Fiennes/dp/B00JB3AOEY

","hed4. "Still Alice"","buttonTextWatch on Hulu","buttonUrlhttps://www.hulu.com/movie/still-alice-9b739da7-c5f3-4146-8d8f-df9602f7036f

4. "Still Alice"
Based on Lisa Genova's novel of the same name, Still Alice centers around Alice Howland (Julianne Moore), who is dealing with a sudden Alzheimer's diagnosis at the age of 50 years old. The film deals with the progression of the disease and how Howland copes with the diagnosis and gets through daily life with her family.
https://www.hulu.com/movie/still-alice-9b739da7-c5f3-4146-8d8f-df9602f7036f

","hed5. "Little Miss Sunshine"","buttonTextWatch on Prime Video","buttonUrlhttps://www.amazon.com/Little-Miss-Sunshine-Abigail-Breslin/dp/B000M2SM92

5. "Little Miss Sunshine"
This comedy follows Abigail Breslin as Olive Hoover, a child who has recently qualified for a beauty pageant in California, and her family, who has to get her there via Volkswagen Bus as they live in New Mexico. While it's a comedy, the film also touches on more serious notes, like addiction and suicide, making this film more of a tragicomedy.
https://www.amazon.com/Little-Miss-Sunshine-Abigail-Breslin/dp/B000M2SM92

","hed6. "Parasite"","buttonTextWatch on Hulu","buttonUrlhttps://www.hulu.com/movie/parasite-2fd691a0-f66b-467f-8635-00d7f151f3d4

6. "Parasite"
This South Korean dark comedy film became an instant hit when it arrived in theaters; it also performed extremely well critically, as the film ended up taking home four Oscars, including Best Picture. After Ki-woo (Choi Woo-Shik), the daughter of a very poor family, ends up being hired as an English tutor for an extremely wealthy family, her entire family sets out to work for this new, wealthy family in order to make money.
https://www.hulu.com/movie/parasite-2fd691a0-f66b-467f-8635-00d7f151f3d4

","hed7. "Napoleon Dynamite"","buttonTextWatch on Prime Video","buttonUrlhttps://www.amazon.com/Napoleon-Dynamite-Jon-Heder/dp/B000I9U972

7. "Napoleon Dynamite"
This early-2000s classic stars Jon Heder as the namesake character as he navigates high school as an extremely quirky and awkward student. The film is a mish-mash of similar characters who are unusual in their own ways, like most of his family, Pedro, who decides to run for class president, and Deb, his friend who's on the shyer side.
https://www.amazon.com/Napoleon-Dynamite-Jon-Heder/dp/B000I9U972

","hed8. "Pulp Fiction"","buttonTextWatch on Hulu","buttonUrlhttps://www.hulu.com/movie/pulp-fiction-02cd83e5-f7a9-49eb-b10c-7fddea16de9f

8. "Pulp Fiction"
An absolutely iconic film starring John Travolta, Samuel L. Jackson, and Uma Thurman, Pulp Fiction focuses on the crime scene in Los Angeles, going through multiple different stories within the area. The film is often considered one of Quentin Tarantino's best, with its incredible dialogue and overall storyline.
https://www.hulu.com/movie/pulp-fiction-02cd83e5-f7a9-49eb-b10c-7fddea16de9f

","hed9. "Three Billboards Outside Ebbing, Missouri"","buttonTextWatch on Prime Video","buttonUrlhttps://www.amazon.com/Three-Billboards-Outside-Ebbing-Missouri/dp/B077PL4NPZ

9. "Three Billboards Outside Ebbing, Missouri"
A winner of multiple awards, including the Best Actress Oscar for Frances McDormand, Three Billboards Outside Ebbing, Missouri centers around Mildred Hayes's (McDormand) quest to find justice for her daughter's rape and murder. Hayes decides to pay for three billboards to call out the police department for not being able to find who murdered her daughter, leading to an ongoing conflict between Hayes and Police Chief Bill Willoughby (Woody Harrelson).
https://www.amazon.com/Three-Billboards-Outside-Ebbing-Missouri/dp/B077PL4NPZ

","hed10. "12 Years a Slave"","buttonTextWatch on Prime Video","buttonUrlhttps://www.amazon.com/12-Years-Slave-Benedict-Cumberbatch/dp/B00ICSUJ6C

10. "12 Years a Slave"
12 Years a Slave follows the true story of Solomon Northup, who wrote the book of the same name about his experience as a slave in 1853. Chiwetel Ejiofor stars as Northup, who was kidnapped and forced into slavery for 12 years even though he was a born-free African American.
https://www.amazon.com/12-Years-Slave-Benedict-Cumberbatch/dp/B00ICSUJ6C

","hed11. "At Eternity's Gate"","buttonTextWatch on Prime Video","buttonUrlhttps://www.amazon.com/At-Eternitys-Gate-Willem-DaFoe/dp/B07N47X646

11. "At Eternity's Gate"
Willem Dafoe puts on an incredible performance as Vincent van Gogh in At Eternity's Gate, which tracks the famed painter in his last years of life. The film goes into great detail by showing the process behind the specific paintings that van Gogh completed in those final years, as well as the extremely emotional side of the painter.
https://www.amazon.com/At-Eternitys-Gate-Willem-DaFoe/dp/B07N47X646

","hed12. "Election"","buttonTextWatch on Hulu","buttonUrlhttps://www.hulu.com/movie/election-ae135e68-4bd6-4b9e-a5f4-8bc79933e0c1

12. "Election"
With Reese Witherspoon as the overachieving student, Tracy Enid Flick, Election follows her high school campaign for class president. One of her teachers, Jim McAllister (Matthew Broderick), decides to sabotage her run for president through various methods as he finds her to be an extremely annoying student.
https://www.hulu.com/movie/election-ae135e68-4bd6-4b9e-a5f4-8bc79933e0c1

","hed13. "The Wife"","buttonTextWatch on Hulu","buttonUrlhttps://www.hulu.com/movie/the-wife-76aea9bd-91d3-4d5c-b011-d39e570d2439

13. "The Wife"
Glenn Close leads as Joan Castleman, wife of Joseph Castleman (Jonathan Pryce), a famous writer who has just been awarded the Nobel Peace Prize for his work. This moment is the culmination of years of unspoken resentment on Joan's end, as she's the real one who deserves the award, given that she has been her husband's ghostwriter for all of those years.
https://www.hulu.com/movie/the-wife-76aea9bd-91d3-4d5c-b011-d39e570d2439

","hed14. "Get Out"","buttonTextWatch on Prime Video","buttonUrlhttps://www.amazon.com/Get-Out-Daniel-Kaluuya/dp/B06Y1FYX25

14. "Get Out"
Jordan Peele's Get Out is a psychological horror film about Chris Washington (Daniel Kaluuya), who spends time at his girlfriend, Rose Armitage's (Allison Williams) house with her family. Things quickly take a turn for the worst as her family turns out to be casually racist and incredibly insane, with particularly odd things occurring around the house that just don't add up.
https://www.amazon.com/Get-Out-Daniel-Kaluuya/dp/B06Y1FYX25

","hed15. "Nomadland"","buttonTextWatch on Hulu","buttonUrlhttps://www.hulu.com/movie/nomadland-dcd9cb83-0636-46ef-8cbd-b9d994672e83

15. "Nomadland"
Nomadland took home a ton of wins at the 2021 Oscars, including Best Picture, Best Actress, and Best Director. Based on the non-fiction book of the same name by Jessica Bruder, this film stars Frances McDormand as Fern, a modern-day nomad who travels across America working various jobs and musing about life.
https://www.hulu.com/movie/nomadland-dcd9cb83-0636-46ef-8cbd-b9d994672e83

","hed16. "Juno"","buttonTextWatch on Hulu","buttonUrlhttps://www.hulu.com/movie/juno-8a8294de-f49b-42a0-9672-d04a4516f465

16. "Juno"
Elliot Page is Juno MacGuff in this coming-of-age film, which tells the story of MacGuff, who accidentally gets pregnant and decides to put her baby up for adoption. The story tracks her through her pregnancy and how she chooses who will be adopting her future child. Also starring in the film is Michael Cera, who plays Paulie Bleeker, her friend and the father of her child.
https://www.hulu.com/movie/juno-8a8294de-f49b-42a0-9672-d04a4516f465

","hed17. "Good Will Hunting"","buttonTextWatch on Prime Video","buttonUrlhttps://www.amazon.com/Good-Will-Hunting-Ben-Affleck/dp/B006RXPT82

17. "Good Will Hunting"
Matt Damon is Will Hunting in this independent film that has become quite the iconic movie over the years. The story follows Hunting (Matt Damon), who is secretly a genius yet works as a janitor. When he starts going to therapy under Dr. Sean Maguire (Robin Williams) as part of a legal agreement after getting in trouble for assaulting a police officer, things in his life start to turn around. Hunting begins working with a mathematics professor, and after that, he takes note of his life and figures out where to go from there.
https://www.amazon.com/Good-Will-Hunting-Ben-Affleck/dp/B006RXPT82

","hed18. "Call Me By Your Name"","buttonTextWatch on Hulu","buttonUrlhttps://www.hulu.com/movie/call-me-by-your-name-aed5a4ad-b3fd-4941-8b75-cf0a4a09468a

18. "Call Me By Your Name"
Timothée Chalamet and Armie Hammer star in this coming-of-age film about two young men who meet in Italy and form a deep romantic relationship. Chalamet plays Elio Perlman, a 17-year-old student, while Hammer is Oliver, a college graduate student; the two meet as Oliver works as Perlman's father's assistant.
https://www.hulu.com/movie/call-me-by-your-name-aed5a4ad-b3fd-4941-8b75-cf0a4a09468a

","hed19. "Lady Bird"","buttonTextWatch on Netflix","buttonUrlhttps://www.netflix.com/title/80205227

19. "Lady Bird"
Lady Bird has quickly become a classic coming-of-age film, with Saoirse Ronan as the main character of Lady Bird, a disillusioned high school senior with a broken arm who has a tough relationship with her mother, Marion (Laurie Metcalf). The film follows Lady Bird at school as she experiences new things and looks ahead to what her future will be like, as she's eager to finally get out of Sacramento.
Watch on Netflix
https://www.netflix.com/title/80205227

","hed20. "Lost in Translation"","buttonTextWatch on Peacock","buttonUrlhttps://www.peacocktv.com/watch-online/movies/comedy/lost-in-translation/ecd04cbd-26a5-3e60-a55a-575dbfb25d4b

20. "Lost in Translation"
Bill Murray stars as Bob Harris, who heads to Tokyo to promote liquor, given that he used to be a famous actor. While there, he ponders his life as he's not used to the culture in Japan, though he eventually meets Charlotte (Scarlett Johansson), a college graduate who quickly changes things. Though there's a fairly large age gap, the two are able to connect and form a bond over their shared feelings of alienation in Tokyo.
Watch on Peacock
https://www.peacocktv.com/watch-online/movies/comedy/lost-in-translation/ecd04cbd-26a5-3e60-a55a-575dbfb25d4b
ONE37pm's Favorite Films of 2021 (So Far)
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Tags Best Independent Films, Best Independent Films of All Time
"The Book of Boba Fett": 'The Streets of Mos Espa' Recap
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For another 38-minute episode (sigh), this one sure packed a ton of storylines in it, as 'The Streets of Mos Espa' walks us through the more gritty parts of the city.
At Jabba's (Boba Fett's) palace, the 8D8 Droid is standing in front of a projected map, giving Fett and Fennec Shand some history regarding the areas that were previously under the protection of Jabba and Bib Fortuna. He notes that Fortuna had Mos Espa divvied up into three sections by three families: the city center was taken by the Trandoshans, the Aqualish had the worker's district, and the Klatooinians controlled the upper sprawl and starport. Given this, 8D8 makes it known that the people of Mos Espa are waiting to see how Fett will lead the area.
Fett also reflects on their meeting with Mayor Mok Shaiz from the previous episode, where he denies sending the assassins to kill him. With this, Fett now believes that the Hutt Twins were behind the attack.
Interrupting this meeting, a local water monger and shop owner, Lortha Peel, comes by for an unscheduled visit, letting Fett know that a gang of local teenagers (who are half-human, half-robot) stole some of his water inventory from him. He begs Fett to get rid of them, and finally, he agrees to look into things.
At night, Fett, Shand, and the Gamorrean Guards head into Mos Espa to track down the teens. Outside of the Peel's shop, he finds the group of grungy kids hanging out and decides to confront them. Expecting them to lie about their stealing, one of the leaders of the crew, Drash (Sophia Thatcher), openly admits to stealing the water. She explains that Peel is charging an exorbitant amount for water, which Fett is surprised by.
After Drash tells him that there's no work in the area to make the money to actually pay for the water, Fett suggests that she and her crew come work for him, which they agree to. Overhearing this, Peel comes rushing out of his shop and is outraged by it. Fett calls him out for charging too much for water and reimburses him for the gang's stealing by paying him a more acceptable amount: 500 credits instead of 1300. After that, the gang follows Fett back to the palace on multicolored sand scooters that resemble Vespas. Or, if you're a fan of the older Nickelodeon show, Zoey101, they basically look like the Jet-X.
In his medical pod, Fett has a quick flashback back at Kamino, watching his father fly away on Slave-1. In a more recent flashback, Fett is once again back with the Tusken Raiders, where he heads out on a Bantha to go to Mos Eisley and look for the Pykes. In the background, we get a quick glimpse into The Mandalorian as Peli Motto (Amy Sedaris) walks by in the background with a few of her robots.
Fett meets with Pyke Syndicate, where they talk about the protection arrangements in the Outer Rim. He's expecting to receive payments from the Pyke, though he refuses to pay Fett as they're already paying another party for protection: the Kintan Striders (the speeder bike gang). As the Pyke won't budge, Fett tells him that he won't be hearing from the striders anymore.
Upon his return, a billowing line of smoke clouds fill the air above the Tusken Raider camp, and upon further investigation, it turns out that the entire camp was burned down by the Kintan Striders. Their infamous symbol was painted onto one of the tents, and although Fett thought that he got his message across to them in last week's episode, they decided to take revenge on him for his actions. With this, Fett departs the camp, though he's brought back to the present day abruptly, as Black Krrsantan unloads him from his medical pod and throws him across the room.
The two fight in the medical pod room, with Fett making unsuccessful efforts to take the Wookiee down, as he's basically unstoppable. The gang of teens come running into the room to assist him and are mostly unsuccessful, even though there are so many of them. However, they're able to lead the Wookiee to the main room in the palace, getting him to stand over the trap door. At this point, Shand unlocks the trap door, though Black Krrsantan is able to hold on with one hand. She takes out her small knife and throws it at his hand, which finally traps him downstairs.
With that fight concluded, the twin Hutts show up to apologize to Fett with a massive gift (that they forgot to wrap, might I add): a "baby" rancor, fit with a trainer, portrayed by Danny Trejo. The Hutts also reveal that the territory they thought was for them was actually promised to a different Syndicate by Mayor Mok Shaiz. With that, the Twins have decided to go back to Hutta (Nal Hutta), their planet, instead of vying for the territory on Tatooine.
The Gamorrean Guards bring out Black Krrsantan, who is handcuffed, and Fett offers him back to the Hutts if they renounce their claims to Jabba the Hutt's land. They don't care about what happens to the Wookiee, so Fett simply releases him back onto Tatooine.
The rancor is brought to the lower level of the palace; its eyes are masked, as the first person it sees will become its leader. Trejo's character finally removes its mask and the rancor sees Fett, which he quickly becomes attached to.
Fett, Shand, and the teenage gang head over to have another impromptu meeting with the Mayor, and the Mayor's Twi'lek majordomo heads back to fetch him. However, he locks the door behind him; upon noticing this, Fett and Shand are able to break their way into the back room, though they find it completely empty.
Having to act quickly to catch the majordomo, the gang of teenagers take off on their sand scooters and begin to chase after him, who has taken off in a land speeder. They follow him through the cramped and crowded streets of Mos Espa constantly dodging droids, markets, and falling crates. It's not a super-high-speed chase, as one can only go so fast through such small streets, though I feel like they could've been going a little faster just for the sake of action.
The gang is finally able to corner the majordomo after he crashes his landspeeder into a fruit market, filling his vehicle with exotic fruits. Fett catches up to them with his jetpack, landing right beside the landspeeder, and finally asks where the Mayor is. Apparently, he decided to take off with the Pykes.
Elsewhere in Mos Espa, dozens of Pykes arrive on a starliner. Shand believes that this is only the first wave of them and that there are more coming; a clear sign that they want to go to war.
The "Star Wars" Galaxy's Most Feared Bounty Hunter: Who Is Boba Fett?
Tags Book of Boba Fett Recap, Mos Espa, The Book of Boba Fett
The 20 Best Limited Series to Stream Now
No Comments on The 20 Best Limited Series to Stream Now
Limited series have truly pulled ahead in the TV world over the past few years, with series like American Horror Story, When They See Us, and The Queen's Gambit dominating the television scene. Though limited series are thought of more to be standalone shows, those that were meant to be one season but earned a few more can also be counted in this category, like Big Little Lies and Fargo. In no specific order, these next 20 are a few of the best that have come our way over the years.

","hed1. "American Crime Story: The People vs. O.J. Simpson"","buttonTextWatch on Netflix","buttonUrlhttps://www.netflix.com/title/80083977

1. "American Crime Story: The People vs. O.J. Simpson"
Though everyone knows the story of O.J. Simpson pretty well at this point, the first installment in the American Crime Story anthology series goes deep into the behind-the-scenes aspect of things. Sarah Paulson and Sterling K. Brown both give incredible performances as Marcia Clark and Christopher Darden, respectively, with Paulson earning an Emmy for her performance. The series as a whole ended up taking 22 Emmy nominations, winning nine of them.

","hed2. "Watchmen"","buttonTextWatch on HBO Max","buttonUrlhttps://www.hbomax.com/series/urn:hbo:series:GXXAcxwJukqnCVgEAAAEG

2. "Watchmen"
Based on the DC comic book series of the same name, Watchmen is following in the footsteps of the 2009 movie that is also of the same name. Taking place in Tulsa, Oklahoma, the series has a few different plots going on, with the two most notable becoming getting intertwined as they both involve Detective Angela Abar (Regina King). Abar is trying to eliminate a white supremacist group, but at the same time, she has to track down Doctor Manhattan, a human who left Earth and transformed into something much larger than purely human.
https://www.hbomax.com/series/urn:hbo:series:GXXAcxwJukqnCVgEAAAEG

","hed3. "Mare of Easttown"","buttonTextWatch on HBO Max","buttonUrlhttps://www.hbomax.com/series/urn:hbo:series:GYCiC1Q8picLCfAEAAAAC

3. "Mare of Easttown"
This limited series with Kate Winslet in the titular role of small-town detective Mare Sheehan saw enormous success, including 16 Emmy nominations. The story centers around Sheehan, who finds herself in the middle of a murder mystery after a teenage girl in town is murdered in a forest. At the same time, she's trying to piece together whether or not it's connected to the disappearances of a few other girls.
https://www.hbomax.com/series/urn:hbo:series:GYCiC1Q8picLCfAEAAAAC

","hed4. "WandaVision"","buttonTextWatch on Disney+","buttonUrlhttps://www.disneyplus.com/series/wandavision/4SrN28ZjDLwH

4. "WandaVision"
WandaVision reunited Wanda Maximoff (Elizabeth Olsen) and Vision (Paul Bettany) in a genre-defying series about the two. Following them through various past decades, we see the couple evolve through a sitcom-series lens, though after some time, the two realize that a few odd things just aren't adding up.
Watch on Disney+
https://www.disneyplus.com/series/wandavision/4SrN28ZjDLwH

","hed5. "The Queen's Gambit"","buttonTextWatch on Netflix","buttonUrlhttps://www.netflix.com/title/80234304

5. "The Queen's Gambit"
A massive hit, The Queen's Gambit stars Anya Taylor-Joy as Beth Harmon, a young orphan who learns chess from the janitor at her orphanage. After realizing that she has an incredible skill for the game, she practices more and more and is eventually adopted from the orphanage. She continues to play at her new home and later competes in the largest international competitions for chess. Having become addicted to pills while at the orphanage, the series also follows her battle with addiction. The story is based on Walter Tevis's novel of the same name from 1983.

","hed6. "When They See Us"","buttonTextWatch on Netflix","buttonUrlhttps://www.netflix.com/title/80200549

6. "When They See Us"
When They See Us is based on the true story of five teenagers accused of raping a female jogger in Central Park in 1989. The series dives into the lives of the teenagers, who were all either Black or Hispanic, and their families as they process the events. It also includes their ultimate wrongful conviction after the real assailant ends up confessing later on.

","hed7. "American Horror Story: Coven"","buttonTextWatch on Netflix","buttonUrlhttps://www.netflix.com/title/70210884

7. "American Horror Story: Coven"
Though there have been many seasons in the American Horror Story anthology series, Coven might be the most compelling of the bunch. The story follows a group of girls who attend a secret school for witches in New Orleans, which follows a very specific power structure. Cordelia Foxx (Sarah Paulson) runs everything at the school, as she doesn't have a great relationship with her mother, Fiona Goode (Jessica Lange), who is the most powerful "Supreme" witch and should technically be running things. At the same time, the show cuts back to the late 1600s during the time of the Salem Witch Trials, eventually linking the two moments in time together. And if that's not enough, there's even a Stevie Nicks cameo in one of the episodes.

","hed8. "Chernobyl"","buttonTextWatch on HBO Max","buttonUrlhttps://www.hbomax.com/series/urn:hbo:series:GXJvkMAU0JIG6gAEAAAIo

8. "Chernobyl"
This series centers around the infamous Chernobyl disaster in 1986, which occurred in the Soviet Union. Chernobyl details not only the disaster itself, but also the risks and immense effort that came with the massive cleanup of the nuclear plant. Receiving 19 Emmy nominations, the series was a major success both critically and with audiences, as it highlighted the importance of information and the trust that we place in our leaders.
https://www.hbomax.com/series/urn:hbo:series:GXJvkMAU0JIG6gAEAAAIo

","hed9. "Big Little Lies"","buttonTextWatch on HBO Max","buttonUrlhttps://www.hbomax.com/series/urn:hbo:series:GWGwm2gpIf6mywwEAAACF

9. "Big Little Lies"
With a stacked cast, the main plot of Big Little Lies revolves around Jane Chapman (Shailene Woodley), who has recently relocated to Monterey, California, to raise her son, Ziggy, alone. There, she meets a group of women from the school and becomes friends with them, including Madeline (Reese Witherspoon), Celeste (Nicole Kidman), Bonnie (Zoë Kravitz), and Renata (Laura Dern). At the beginning of the series, we're clued in to the fact that someone dies at the end, and the entire show is leading up to that moment. The series is based on the book of the same name by Liane Moriarty.
https://www.hbomax.com/series/urn:hbo:series:GWGwm2gpIf6mywwEAAACF

","hed10. "True Detective": Season 1″,"buttonTextWatch on HBO Max","buttonUrlhttps://www.hbomax.com/series/urn:hbo:series:GVU2xlQZIxY7DwvwIAVgK

10. "True Detective": Season 1
Another anthology crime series, True Detective, received a vast amount of praise for its first season, which stars Woody Harrelson and Matthew McConaughey. Harrelson is Detective Marty Hart, and McConaughey as Detective Rust Cohle, who work together to solve the murder of a former prostitute who was left with an odd symbol on her. The series follows the two almost 20 years later as another victim turns up in a similar fashion.
https://www.hbomax.com/series/urn:hbo:series:GVU2xlQZIxY7DwvwIAVgK

","hed11. "Olive Kitteridge"","buttonTextWatch on HBO Max","buttonUrlhttps://www.hbomax.com/series/urn:hbo:series:GVU22vArU1FFvjSoJAVxO

11. "Olive Kitteridge"
Based on the book of the same name by Elizabeth Strout, this four-part series follows the namesake character Olive Kitteridge (Frances McDormand), a retired teacher, at four different times in her life. Though pretty cynical about most things, there's a softer side to Olive, though it rarely comes out. Partially because of this, she ends up having a tough relationship with her son, Christopher, as he doesn't really "get" her or understand that she means well. Richard Jenkins plays Olive's husband, Henry, a pharmacist who's basically the opposite of her.
https://www.hbomax.com/series/urn:hbo:series:GVU22vArU1FFvjSoJAVxO

","hed12. "I May Destroy You"","buttonTextWatch on HBO Max","buttonUrlhttps://www.hbomax.com/series/urn:hbo:series:GXqHggg6FbLeIkwEAAAAv

12. "I May Destroy You"
Michaela Coel's I May Destroy You is a powerful look into the life of a young woman, Arabella Essiedu, who is raped. Working as a writer, Essiedu is trying to build her life back up after her traumatic experience, and the series tracks her and her friends as they navigate their world.
https://www.hbomax.com/series/urn:hbo:series:GXqHggg6FbLeIkwEAAAAv

","hed13. "Feud: Bette and Joan"","buttonTextWatch on Prime Video","buttonUrlhttps://www.amazon.com/FEUD-Bette-Joan-Season-1/dp/B06WD81WW6

13. "Feud: Bette and Joan"
Though Feud was supposed to become an anthology series similar to that of American Horror Story, it ended up staying as a one-off limited series. Feud tells the story of the real relationship between Bette Davis (Susan Sarandon) and Joan Crawford (Jessica Lange) during the filming of their movie, What Ever Happened to Baby Jane? The two were fierce enemies, and when forced to work together, it was quite a lot to see—both on-screen and off.
https://www.amazon.com/FEUD-Bette-Joan-Season-1/dp/B06WD81WW6

","hed14. "Fargo"","buttonTextWatch on Hulu","buttonUrlhttps://www.hulu.com/series/fargo-203cda1b-7919-40fb-ab36-1e45b3ed2a50

14. "Fargo"
Jumping off the Coen Brothers' film, Fargo, this series of the same name also involves two detectives trying to solve murders, except this time, the officers are Deputy Molly Solverson (Allison Tolman) and Gus Grimly (Colin Hanks). The two are searching for those responsible for a string of murders in the Minnesota area, which they believe two crime partners might have carried out.
https://www.hulu.com/series/fargo-203cda1b-7919-40fb-ab36-1e45b3ed2a50

","hed15. "The White Lotus"","buttonTextWatch on HBO Max","buttonUrlhttps://www.hbomax.com/series/urn:hbo:series:GYLD3_ArixL1sggEAAAHk

15. "The White Lotus"
The White Lotus tracks three groups of people as they embark on their stay at the White Lotus hotel in Hawaii, where someone dies, as noted in the beginning of the first episode. Going back in time prior to the death, the show focuses on each group as they have different yet connected experiences at the hotel.
https://www.hbomax.com/series/urn:hbo:series:GYLD3_ArixL1sggEAAAHk

","hed16. "Little Fires Everywhere"","buttonTextWatch on Hulu","buttonUrlhttps://www.hulu.com/series/little-fires-everywhere-bce24897-1a74-48a3-95e8-6cdd530dde4c

16. "Little Fires Everywhere"
Based on Celeste Ng's novel, Little Fires Everywhere, this series follows a very affluent, dysfunctional family led by Elena Richardson (Reese Witherspoon) and her unlikely and sometimes contentious friendship with a newcomer to town, Mia Warren, and her daughter.
https://www.hulu.com/series/little-fires-everywhere-bce24897-1a74-48a3-95e8-6cdd530dde4c

","hed17. "The Act"","buttonTextWatch on Hulu","buttonUrlhttps://www.hulu.com/series/the-act-8cc910fe-b59e-46a5-9966-16c4e0ed208d

17. "The Act"
Joey King is Gypsy Rose Blanchard in this series based on the true story of her and her mother, Dee Dee Blanchard (Patricia Arquette), who abused her daughter by pretending that she had an illness. Dee Dee takes things to an extreme with her daughter, forcing her to use a wheelchair, and as Gypsy Rose gets older, she wants a larger separation between her and her mother, which causes major conflicts.
https://www.hulu.com/series/the-act-8cc910fe-b59e-46a5-9966-16c4e0ed208d

","hed18. "American Horror Story: Asylum"","buttonTextWatch on Netflix","buttonUrlhttps://www.netflix.com/title/70210884

18. "American Horror Story: Asylum"
Another absolutely incredible season of American Horror Story, Asylum is quite a bit darker than Coven, though it brings the same punch. Sister Jude (Jessica Lange) helps run a mental institution, which Lana Winters (Sarah Paulson) decides to investigate as part of her job as a local journalist. However, while there, she's taken into their custody and is forced to stay, which leads to a number of frightening and puzzling encounters.

","hed19. "Mrs. America"","buttonTextWatch on Hulu","buttonUrlhttps://www.hulu.com/series/mrs-america-96f330fe-878d-412e-949f-fd8b69b3adf2

19. "Mrs. America"
Set in 1963, Mrs. America focuses on Phylis Schlafly (Cate Blanchett), who is a known opposer of the Equals Rights Amendment (ERA), which would grant women the same rights as men. At the same time, it pivots to where things are on the other side, following Gloria Steinem (Rose Byrne), Shirley Chisholm (Uzo Aduba), Bella Abzug (Margo Martindale), and Betty Friedan (Tracey Ullman) in their fight to pass the ERA. The series was received extremely well, receiving 10 Emmy nominations, with Aduba winning for her portrayal of Chisholm.
https://www.hulu.com/series/mrs-america-96f330fe-878d-412e-949f-fd8b69b3adf2

","hed20. "Sharp Objects"","buttonTextWatch on HBO Max","buttonUrlhttps://www.hbomax.com/series/urn:hbo:series:GWv7yvAGvXCm-lwEAAAE-

20. "Sharp Objects"
Starring Amy Adams as journalist Camille Preaker, Sharp Objects is about Preaker's hometown, which she returns to in an effort to discover more details about two young girls being murdered there. However, being back in town forces her to be around her mother, Adora Crellin (Patricia Clarkson), who doesn't help Preaker's already delicate state as an alcoholic who was recently released from a psychiatric rehab facility.
https://www.hbomax.com/series/urn:hbo:series:GWv7yvAGvXCm-lwEAAAE-
'Mare of Easttown' Is the Best Show on TV That You Might Not Be Watching
Tags Best Limited Series
Joel Coen's 'The Tragedy of Macbeth' Injects New Life Into A Centuries-Old Story
No Comments on Joel Coen's 'The Tragedy of Macbeth' Injects New Life Into A Centuries-Old Story
While we all know that The Tragedy of Macbeth is based on Shakespeare's play, Macbeth, this film is a play. For anyone who knows the story of Macbeth well and has a deep care for its language, this film is exactly how you'd want it portrayed on screen.
And by "anyone who knows the story of Macbeth well and has a deep care for its language," I am speaking directly to my high school English teacher, who read the play to us in such a way that it made learning the story exciting and not as tough as others have described.
It seems like that's the block for some people when it comes to Shakespeare: it seems boring, there are too many pages, the words don't make sense, it's too confusing, etc. And most of the time reading Shakespeare, that was the case for me. The Tragedy of Macbeth captures the essence of Macbeth but does so without clutter (or an impending test); it's able to remove all of the distractions and allows you to focus on the story.
The background of each shot practically doesn't matter: in expansive, blank rooms with minimal furniture. Our eyes and focus are on what's most important: the characters and their words. The sharp contrast of shooting in black and white adds further to the ominous nature of things. Stark and serious, every sound seems to carry, from each character's voice to simple drops of blood that become a drum beat. These elements allow the movie to feel like less of a film and more like a play, and it's carried out so as not to feel outright.
And that's without mentioning the acting.
AppleTV+
Denzel Washington's Macbeth is like no other, cast alongside Frances McDormand as Lady Macbeth, who doesn't hold back when it comes to advancing her husband's role in Scotland. At first, Washington's performance is sensitive and steadily becomes cold, calculating, and unforgiving as the plot moves forward. Washington and McDormand make a crafty team, though their deviousness is hidden at first by their charming and do-good way of living.
The three witches—all portrayed by Kathryn Hunter—are perfectly eerie, with the main witch contorting her body in such a way to further convey her sinister nature. Instead of casting three separate witches, the other two are first seen in the reflection of a small pond, as Joel Coen utilizes unique filming methods to showcase the trio in unfamiliar ways.
Corey Hawkins is fearless as Macduff; after suffering the loss of his entire family, he forges ahead and takes on Macbeth, leaving the king headless with nothing left but a dainty silver crown lying on the stone floor. At the moment of Macbeth's demise, we realize how meaningless it all was—everything that had transpired was in pursuit of a crown that was so easily taken. Left on the ground, it looked as if it could be split with a simple stomp of the foot.
The crown ended up landing on the head of the rightful heir to the throne: Malcolm (Harry Melling), which makes the actions of Macbeth and Lady Macbeth seem almost extraneous. The crown was destined to land in the hands of Malcolm: if the two hadn't become so invested in diverting the way of nature, none of this would have happened. But then again, we wouldn't have the story of Macbeth.
The film is outstandingly beautiful to watch and equally to listen to, as Coen clearly went out of his way to perfect the tiny details that make Macbeth such a compelling story. From the dark lighting and cinematography to the language he chose to include, The Tragedy of Macbeth exemplifies how a story that's almost 400 years old can be transformed into something completely new.
The Tragedy of Macbeth premieres on AppleTV+ on January 14.
A Look Ahead to Joel Coen's Next Film: "The Tragedy of Macbeth"
Tags Joel Coen Macbeth, The Tragedy of Macbeth, Three Witches Macbeth, Tragedy of Macbeth Review
ONE37pm's Official Predictions for the 2022 Golden Globe Awards
No Comments on ONE37pm's Official Predictions for the 2022 Golden Globe Awards
While the Golden Globes aren't being broadcast this year due to a myriad of scandals and lack of Black representation within the Hollywood Foreign Press Association (HFPA), we still have a few predictions for their nominations across the world of TV and movies. As their nominations (and later wins) give us a first look into what to expect come the Oscar nominations in early February, let's sift through some of the nominees and get ahead of the game before the Oscars.
Best Picture – Drama: "CODA"
Though many people will want the winner of this category to be Dune, there's a much better chance that the HFPA will be going with something a little more niche, like CODA. It tells the story of Ruby, who takes on extra responsibilities as her entire family is deaf except for her. At school, she's able to explore more of her passions, which ends up being music and singing.
Best Picture – Musical/Comedy: "Licorice Pizza" or "Tick, Tick… Boom!"
While any of the nominees could potentially take this win, Licorice Pizza and Tick, Tick…Boom! are the obvious standouts of the bunch. Although Licorice Pizza had a lesser-known, indie cast, it did well both critically and within audiences, while Tick, Tick…Boom! had a more recognizable cast that features Andrew Garfield and Vanessa Hudgens.
Best Actress – Drama: Kristen Stewart
Spencer was an incredibly deep look into a very precious person, and though many people have played Princess Diana over the years, Kristen Stewart's nuanced portrayal is the best yet. The preparation she put into this role is apparent, as each little movement—from her face and accent to her hands and gait—was done with extreme care.
Best Actor – Drama: Denzel Washington
Based on the first reviews of The Tragedy of Macbeth, Denzel Washington puts on an outstanding leading performance as Lord Macbeth in this iteration of the iconic play by Shakespeare, directed by Joel Coen.
Best Actress – Musical/Comedy: Jennifer Lawrence, Alana Haim
This is a bit of a toss-up, though the two actresses couldn't be more different experience-wise: Licorice Pizza was Alana Haim's first feature film and Don't Look Up was another one of the countless movies that Jennifer Lawrence has starred in. Both actresses have put forth strong cases to win this award, but this feels like a two-woman race.
Best Actor – Musical/Comedy: Leonardo DiCaprio
Even though there were some mixed feelings about Don't Look Up, there's no denying that Leonardo DiCaprio put on an outstanding performance as Dr. Randall Mindy. DiCaprio played well into the character's panicky and nervous personality, presenting a much softer layer to a well-respected professor and astronomer.
Best Drama Series: "Succession," "Squid Game," "Pose"
I know, I know, I couldn't narrow this one down, but here's why. Succession had probably its best season yet, but Squid Game took the world by storm as it came out of practically nowhere. At the same time, it was Pose's final season, which became a history-defining show with its cast and story. Any of these could take the award for Best Drama Series, but if I had to put money on it, the safe bet would most likely be Succession.
Best Musical/Comedy Series: "Ted Lasso" or "Only Murders in the Building"
Although I feel that Ted Lasso is favored to take the award, Only Murders in the Building did garner a great amount of attention and praise in its first season on Hulu. Ted Lasso had a much larger audience than Only Murders in the Building, which makes it an easy choice for Best Musical/Comedy Series.
Best Television Motion Picture: "Mare of Easttown"
First off, I'm not sure why there's a category named "Best TV Motion Picture" when all of the nominees are series, but alas, there's little argument over what should win this category. Mare of Easttown will take home this award, as Kate Winslet, Evan Peters, Julianne Nicholson, and Jean Smart's performances were all top-notch, fit with an incredible storyline.
Best Actress – Television Motion Picture: Kate Winslet
Echoing the sentiment from above, Kate Winslet will be getting a Golden Globe for her performance as Mare Sheehan in Mare of Easttown. That's all. And when she does, we'll all chow down on a hoagie from Wawa in her honor.
Best Actor – Television Motion Picture: Paul Bettany, Oscar Isaac, Michael Keaton
Though each of these performances are extremely different from the next, all of them are definitely worthy of the award for Best Actor in a Television Motion Picture. Paul Bettany reprised his role as Vision in WandaVision, which was an instant hit with its unique approach to the series. In Scenes From a Marriage, Oscar Isaac delivers a deep performance while Michael Keaton is in the heart of the opioid epidemic in Dopesick.
Best Television Actress – Drama Series: Mj Rodriguez
In Pose's final season, the crew is dealing with a number of different conflicts, most notably including several HIV/AIDS diagnoses and Pray Tell's deterioration. MJ Rodriguez's Blanca Rodriguez-Evangelista has led her entire house and group of friends through all of the ups and downs, always a steadfast pillar of hope for all.
Best Television Actor – Drama Series: Jeremy Strong, Billy Porter
I know; how couldn't I include Brian Cox in this list, right? There's still a very high chance that he'll get a Golden Globe regardless, but Jeremy Strong and Billy Porter should be the larger contenders for this award. Strong's Kendall Roy in Succession was under the main spotlight this season, and he delivered an incredibly heartfelt performance. On the other hand, Billy Porter's portrayal of Pray Tell in Pose was both fierce and touching in its final season.
Best Television Actress – Musical/Comedy Series: Issa Rae, Jean Smart
This year, we were forced to say goodbye to Issa Rae's masterpiece of a show, Insecure, and she put on an incredible performance in its last go. On the flip side, Jean Smart had quite the re-ignition between Mare of Easttown and in her starring role as Deborah Vance in Hacks. For this Golden Globe, she's nominated for Hacks, where she portrays a Joan Rivers-like aging comedian with an extremely sharp wit.
Best Television Actor – Musical/Comedy Series: Jason Sudeikis
Even if Ted Lasso doesn't end up taking the award for Best Musical/Comedy Series, there's a pretty fair chance that Jason Sudeikis will be earning Best Television Actor for his portrayal of Ted Lasso. The all-around good guy won everyone over in the first season, and it certainly didn't stop in Season 2.
Hulu's 'The Great' Is Anti-Historical Perfection
The Many Emotions of Pablo Larraín's "Spencer"
Tags 2022 Golden Globe Nominations, 2022 Golden Globes, Golden Globe Predictions
No Comments on The Most Anticipated TV Shows of January 2022
The start of the new year is bringing us quite a few TV shows to kick off January, including new seasons, series premieres, and reboots. From the long-awaited second season of Euphoria to new shows like The Afterparty and As We See It, January is shaping up to be quite the jam-packed month for television. These next 11 shows are a few of the largest releases of the month, spanning from genres of murder-mystery and crime to comedies and dramas.

","hed1. "The Gilded Age": Season 1

1. "The Gilded Age": Season 1
Set in New York City in 1882, The Gilded Age follows Marian Brook (Louisa Jacobson) as she heads to New York City to live with her aunts following her father's death. Her aunts, Agnes van Rhijn and Ada Brook, are portrayed by Christine Baranski and Cynthia Nixon, respectively. The series deals with the converging of two worlds: one, the society of old New York, and two, of the approaching modern world. In her new home, Marian has to figure out which side of the world she wants to be a part of.
The Gilded Age premieres on HBO Max on January 24.

","hed2. "Ozark": Season 4

2. "Ozark": Season 4
In a teaser trailer recently released by Netflix, the Byrde family is shown getting into a terrifying car accident. As a whole, the series follows the family as father Marty (Jason Bateman) decides to take his whole family along to Ozark, Missouri to avoid the repercussions of a money-laundering scheme.
Ozark airs on Netflix on January 21.

","hed3. "Billions": Season 6

3. "Billions": Season 6
Billions has been airing for quite some time now, and its sixth go at it is shaping up to be quite the season. Though it may have seemed like the end of Season 5 could've been a series finale, this season will be cranking things up a notch. Based on the trailer, Chuck Rhoades (Paul Giamatti) has Mike Prince (Corey Stoll) in his line of sight as he's also determined to take on the billionaire class as a whole.
Billions airs on SHOWTIME on January 23.

","hed4. "Euphoria": Season 2

4. "Euphoria": Season 2
The second season of Euphoria is finally airing this January after a long hiatus due to COVID-19. Though there have been two one-off episodes in the wake of its first season, its viewers have been waiting for this moment for about three years. Things still don't look great for the relationship of Rue (Zendaya) and Jules (Hunter Schafer) based on the Season 2 trailer, but things might get cleared up once the show begins.
Euphoria airs on HBO Max on January 9.

","hed5. "As We See It": Season 1

5. "As We See It": Season 1
As We See It focuses on three young adults on the Autism spectrum, Violet (Sue Ann Pien), Jack (Rick Glassman), and Harrison (Albert Rutecki), who are roommates living together. The eight episodes will follow the three as they navigate life, including conquering fears, creating new relationships, and solidifying their careers.
As We See It premieres on Prime Video on January 21.

","hed6. "Cheer": Season 2

6. "Cheer": Season 2
Following the massive success of its first season back in early 2020, Cheer is back—but this time, things are a bit different for the team. With fan-favorite Jerry being charged with child pornography and as COVID-19 rises, the team has to navigate a new reality. The docuseries will also track the team as they take on their main opponent: Trinity Valley. With new and old members on the squad, Cheer: Season 2 is bound to be another exciting look into the world of cheerleading.
Cheer airs on Netflix on January 12.

","hed7. "How I Met Your Father": Season 1

7. "How I Met Your Father": Season 1
With nine seasons of How I Met Your Mother, How I Met Your Father has the same idea as the original show, except it's told from the perspective of Sophie (Hilary Duff), who explains to her son how she and his father met. It has also been floated that there will be a crossover between the new series and the original, but the capacity that will be in has not yet been revealed.
How I Met Your Father premieres on Hulu on January 18.

","hed8. "The Afterparty": Season 1

8. "The Afterparty": Season 1
A murder-mystery comedy series, The Afterparty is set at a high school reunion party and plays deep into character development in order to tell a different story in each episode. Based on the character's personality, the cinematic elements within each episode match that of the character, creating an eclectic yet telling version of the story.
The Afterparty premieres on AppleTV+ on January 28.

","hed9. "Peacemaker": Season 1

9. "Peacemaker": Season 1
In this series spinoff of The Suicide Squad (2021), Peacemaker is all about—you guessed it—Peacemaker, portrayed by John Cena. This comedy-drama follows the superhero who's all about making peace within the world and the task force created to help him achieve it. His crew is comprised of an interesting group of people who don't seem very capable of their mission, which is fitting for the superhero.
Peacemaker premieres on HBO Max on January 13.

","hed10. "Monarch": Season 1

10. "Monarch": Season 1
Susan Sarandon and Trace Adkins star as married couple Dottie and Albie Roman, who also happen to be the reigning heads of country music. The show centers around their family dynasty and the lies that are built behind it, including conflicts within their business empire and the power structure of their family.
Monarch premieres on Fox on January 30.

","hed11. "Naomi": Season 1

11. "Naomi": Season 1
Based on the DC Comics character, Powerhouse, Naomi is about that very same girl—Naomi McDuffie (Kaci Walfall)—who is a high school student and major comic-book enthusiast. Based in Port Oswego, the show centers around a supernatural event that occurs in the town and how it leads Naomi to figure out who she really is.
Naomi premieres on The CW on January 11.
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Tags 2022 TV, January TV Premieres 2022, New TV Shows January 2022
"The Book of Boba Fett" Recap: 'The Tribes of Tatooine'
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With its runtime about 20 minutes longer than the previous one, "The Tribes of Tatooine" was already shaping up to be quite the episode just given that information alone.
In the present day, Fennec Shand brings the assassin that she took hostage in the last episode back to their base in Jabba's Palace. Boba sits at his throne and questions him, asking about who sent him and what his orders were. When he refuses to speak, Fett drops him down below through the trap door, where it seems like the Rancor (which was killed by Luke Skywalker years before) is about to be let loose behind the door downstairs. However, as a way to stop this from happening, the assassin finally reveals that the Mayor is the one who sent him to kill Fett. As the door opens, nothing is behind it—the assassin was actually in no danger at all.
The crew heads over with the assassin to pay an impromptu and unwelcome visit to Mayor Mok Shaiz; he's of the Ithorian species and has an attached robot translate for him. When Shaiz clarifies that the assassin is part of the Order of the Night Wind, one of his henchmen immediately kills the assassin.
Shaiz still considers Fett a bounty hunter, and after some talking, he tells him to seek out Garsa Fwip at her cantina, the Sanctuary, to see who really sent the assassin to kill him. At the Sanctuary, Fwip reveals that "the twins" have decided to take over their late cousin's throne, and when two members of the Hutt species come marching down the middle of the street in a litter, it's clear that their late cousin was Jabba the Hutt.
Fett talks to the two twins: one, almost identical to Jabba in green, and the other, his sister, in a more pink hue (or maybe it's just Cosmo and Wanda from The Fairly OddParents disguised as descendants of Jabba). Though Fett claims that Jabba's Palace is his territory, the Hutts feel otherwise, according to a scroll document.
Out of nowhere, a massive Wookiee bounty hunter steps out in an effort to threaten Fett. He doesn't even do anything—he just stands there and slightly growls menacingly. But honestly, it was kind of unsettling from the comfort of my couch. Fett eventually tells the Hutts that they'll have to kill him for the throne, and with that, the twins leave for the time being and decide to deal with the issue later.
Back in his medical pod from the last episode, we're brought back in time a bit, picking up where we left off last week. The Tusken Raiders now accept Fett given his heroic actions in defeating the massive creature from before. They begin to train him in their style of fighting, which involves a long staff that they wield to attack their enemies.
While training, the Tusken Raiders spot a massive sand train coming through the Dune Sea and get in position to both take cover and fire at the train. Quite a few raiders are taken down in this long-distance shootout, which leads Fett to head out in the night to gather the necessary materials to destroy the train.
At a remote bar, Fett finds a group of aliens threatening a couple and takes this as an opportunity to fight with them. He easily eliminates their crew and leaves the bar with their speeder bikes in tow.
Back at the Tusken Raiders' camp, Fett teaches them how to use the speeder bikes—at first going very slowly and then faster as they become more accustomed to the new way of travel. He even drills them on jumping from one speeder to another, which takes quite a bit of practice for them to nail down.
Having devised a plan to take down the train the next time it comes by, the crew is ready to jump into action as they see it coming over the horizon. Along with some of the other raiders, Fett hops on a speeder bike while the other half of the raiders take their positions further back with sniper rifles at the ready.
Fett and the raiders on speeder bikes fight parallel to the train, eventually making their way both on top of and inside the train to take out the enemies. Though it's a tough battle and ride as the droid-conductor amps up the speed, Fett is able to stop the train by pulling a stuck lever with his brute strength.
Afterward, the raiders offload the rest of the enemies on board and sit them down outside the destroyed train. Fett demands more information about what they're carrying—spices—and forces them to leave while also prohibiting them from using the area as part of their route.
While gathering around a fire later that night, the chief Tusken Raider gives Fett a gift: a very tiny and cute lizard. The lizard then promptly crawls up Fett's nose and sends him into a weird sort of hallucination. He walks around the Dune Sea and approaches a large tree, where the branches wrap around him and force him into pieces of flashbacks. Finally, in a vision that's clearly set in the ocean of Kamino, a large wave crashes and he's back to reality.
Approaching the camp, Fett is holding a large tree branch, which the raiders then help him craft into a staff of his own. After receiving new clothes from the crew, Fett begins creating his staff. The process is extremely detailed, and with the guided direction of the Tusken Raiders, he's able to form a weapon of his own.
Gathering around the fire once again, Fett is now in his new clothing with his staff in hand. The raiders and Fett do an elaborate walk-dance around the fire that's a bit reminiscent of a flash mob, though a much more serious flash mob. And with that visual, the episode comes to a close.
Everything You Need To Know About "The Book of Boba Fett"
Tags Book of Boba Fett Recap, Jabba's Palace, The Book of Boba Fett, Tusken Raiders
A Look Ahead to Joel Coen's Next Film: "The Tragedy of Macbeth"
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For most of us, The Tragedy of Macbeth reminds us of the bleak time in our life when we were forced against our will to read Shakespeare in high school.
Directed by Joel Coen of the iconic Coen Brothers directing duo, this adaptation seems like a particularly chilling variation of the iconic play of Macbeth. Joel and his brother, Ethan, are well-known for their sometimes-odd yet incredibly clever movies, including Fargo, The Big Lebowski, and No Country for Old Men.
The Tragedy of Macbeth seems like a bit of a departure from the director responsible for these movies. Still, any film directed by a Coen brother is bound to be successful at the very least in looking at things from a different angle.
The official trailer was released back in late September, and though it's only around 30 seconds long, its power almost makes you forget about all the hours you spent being confused by the Sparknotes of the play.

Shot in black and white, it's a very quiet trailer, showing just a few of the main characters, but at the end, an ominous voice of a witch utters those iconic words, "By the pricking of my thumbs, something wicked this way comes." When the words "this way comes" are spoken, we see Lady Macbeth turn around in a very chilling fashion. Even if you—like many others—hated reading the play when you were in high school, there's no way that this trailer didn't do some convincing otherwise to give the story another try.
Written by Shakespeare in 1623, the play of Macbeth centers around army general Lord Macbeth, who is told by three witches—alongside Banquo, another general—that he'll eventually become the King of Scotland. Unwilling to wait for that day, Lady Macbeth poisons the King of Scotland, Duncan, and has Macbeth carry out the killing.
https://www.apple.com/tv-pr/originals/the-tragedy-of-macbeth/images/
Though he was able to secure the kingship, Macbeth now has to deal with the immense guilt and fear that came with their joint decision to kill Duncan. And the guilt doesn't stop there—it eventually bleeds over to Lady Macbeth, who starts to get paranoid about her actions.
When Macduff, Thane of Fife, discovers the body of Duncan, a number of murders follow as the king gets more and more paranoid. With power leaving their grasp as people become skeptical of Macbeth, more killings are carried out by Macbeth, creating a vicious snowball effect and sending both him and Lady Macbeth into madness.
In this film, Denzel Washington plays the main role of Macbeth with Frances McDormand as Lady Macbeth. Brendan Gleeson plays Duncan, King of Scotland, and Corey Hawkins is Macduff.
Though The Tragedy of Macbeth is currently playing in a few select theaters throughout the country, the majority of viewers will be tuning in when it officially premieres on January 14 on AppleTV+.
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Tags Frances McDormand, Joel Coen Macbeth, The Tragedy of Macbeth


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text:
\section{Introduction}

The redundant number system plays a key role in computer arithmetic, with uses ranging from carry-free addition, multiplication and division algorithms \cite{ercegovacDigitalArithmetic2004} to Most-significant Digit (MSD) first operations such as Online arithmetic \cite{ercegovacOnlineArithmeticDSP1989} or the E-method for evaluating polynomials \cite{ercegovacGeneralHardwareOrientedMethod1977}.
 Recent work has extended MSD first arithmetic to the implementation of iterative methods, with the ARCHITECT framework proposed by Li et al.\ in \cite{liARCHITECTArbitraryPrecisionHardware2020} showing large speed-ups in FPGA implementations of the Jacobi method by exploiting the ideas of ``don't-change'' stable MSDs and ``don't-care'' Least-significant Digits (LSDs) to reduce the number of total digits computed in each iteration.
 Prior work to analyze and exploit the stable MSDs, which are the MSDs that don't change their digit value in any future iteration, has been algorithm-specific, with works such as \cite{liDigitStabilityInference2021} and \cite{ercegovacGeneralHardwareOrientedMethod1977} starting with a specific algorithm and then showing that its iterate sequence has MSD stability.

In this paper, we instead propose to reverse the order of the analysis by finding sequences of numbers in the redundant number system with stable MSDs, and then finding iterative methods that have those sequences as iterates.
 We start by showing that a Fej\'er monotone sequence can have stable MSDs when represented using redundant numbers, and we then use that result to show that any iterative method whose iterates are a Fej\'er monotone sequence (such as fixed-point iterations of contractive Lipschitz continuous functions) can have digit stability.
 This allows us to derive theoretical guarantees for the existence of stable MSDs in the iterate sequence of both the Jacobi method (which was previously analyzed in \cite{liDigitStabilityInference2021}), and Newton's method (which was only experimentally shown to have stable digits in \cite{liARCHITECTArbitraryPrecisionHardware2020})

\section{Background}

\subsection{Redundant Number Representation}

A redundant number representation is one where a real number can be represented by multiple different digit patterns, with a commonly used representation being the signed-digit representation \cite{avizienisSignedDigitNumbeRepresentations1961}, where the digits of a radix $r$ number using a symmetric signed-digit representation can take any value from the set
 \begin{equation*}
     S \coloneqq \{-\gamma, \dots, -1, 0, 1, \dots, \gamma \},
 \end{equation*}
 where $\sfrac{r}{2} \leq \gamma \leq r-1$.
 The choice of $\gamma$ determines the level of redundancy in the representation, with the choice $\gamma = r-1$ known as a \textit{maximally redundant} representation.

A key difference of the symmetric maximally redundant number representation compared to the standard number representation is in the real numbers that can be represented by just appending/changing the least significant digits of a number, a property known as the \textit{representation interval}.
 In the standard representation, the representation interval is single-sided, with changes in the least-significant digits only able to increase the value of the real number being represented.
 However, a redundant representation introduces a dual-sided representation interval that is able to both increase and decrease the value of the represented real number by simply adding/changing least-significant digits, with Li et al.\ \cite{liDigitStabilityInference2021} deriving the representation interval for a symmetric maximally redundant representation, shown here in Lemma~\ref{lem:redundantInterval}.
\begin{lemma}[Representation interval \cite{liDigitStabilityInference2021}]
    \label{lem:redundantInterval}
	Let $x$ be a $D$-digit number in the symmetric maximally redundant signed-digit redundant number system with radix $r$.
	If additional digits are appended to $x$ to form a new number, $\tilde{x}$, then
    \begin{equation*}
        \tilde{x} \in \left(x - r^{-D},~x + r^{-D}\right).
    \end{equation*}
\end{lemma}

\subsection{Fej\'er Monotonicity and Nonexpansive Operators}

Fej\'er monotone sequences play an important role in the analysis and understanding of iterative methods, especially in fields involving monotone operators such as convex optimization \cite{bauschkeConvexAnalysisMonotone2011}.
 At its core, a Fej\'er monotone sequence is one where the distance between the elements of the sequence and a given set is not increasing as the sequence grows in length, which we define formally in Definition~\ref{def:fejermonotonicity}.
\begin{definition}[Fej\'er monotonicity {\cite[\S 5.1]{bauschkeConvexAnalysisMonotone2011}}]
    \label{def:fejermonotonicity}
    Let $\mathcal{C}$ be a nonempty subset of a Hilbert space $\mathcal{H}$ and let $\left(x^{(n)}\right)_{n \in \mathbb{N}}$ be a sequence in $\mathcal{H}$. Then $\left(x^{(n)}\right)_{n \in \mathbb{N}}$ is \textit{Fej\'er monotone} with respect to $\mathcal{C}$ if
    \begin{equation}
        \label{eq:monotonesequence}
        \norm{x^{(n+1)} - x} \leq \norm{x^{(n)} - x} \quad (\forall x \in \mathcal{C})(\forall n \in \mathbb{N}).
    \end{equation}
\end{definition}

Closely related to the idea of Fej\'er monotone sequences is that of monotone operators, and we specifically focus on nonexpansive and contractive operators in this work, which we define in Definition~\ref{def:nonexpansiveoperator}.
\begin{definition}[Nonexpansive/contractive operators {\cite{ryuPrimerMonotoneOperator2016}}]
    \label{def:nonexpansiveoperator}
    Let $\mathcal{D}$ be a nonempty subset of the Hilbert space $\mathcal{H}$ and let $T(\cdot) : \mathcal{D} \to \mathcal{H}$ be Lipschitz continuous with Lipschitz constant $L$, i.e.\
    \begin{equation}
        \label{eq:lipschitzoperator}
        \norm{T(x) - T(y)} \leq L \norm{x - y} \quad (\forall x \in \mathcal{D}) (\forall y \in \mathcal{D}).
    \end{equation}
    Then
    \begin{itemize}
        \item if $L = 1$, $T(\cdot)$ is a \textit{nonexpansive} opertor, or
        \item if $L < 1$, $T(\cdot)$ is a \textit{contractive} operator.
    \end{itemize}
\end{definition}

Fej\'er monotone sequences and nonexpansive operators are very closely linked, with the iterate sequence for a fixed-point iteration of a nonexpansive operator generating a Fej\'er monotone sequence with respect to the fixed-point, as shown in Lemma~\ref{lem:operatorSequence}.

\begin{lemma}
    \label{lem:operatorSequence}
    Let $T(\cdot)$ be a nonexpansive/contractive operator (i.e.\ $L \leq 1$) and $\mathcal{F}$ be the nonempty set of fixed points of $T(\cdot)$, then the sequence of iterates $\left(x^{(n)}\right)_{n \in \mathbb{N}}$ of the iteration
    \begin{equation}
        \label{eq:lem:operatorrelation}
        x^{(n+1)} = T\left(x^{(n)}\right) \qquad (\forall n \in \mathbb{N})
    \end{equation}
    is Fej\'er monotone with respect to $\mathcal{F}$.
\end{lemma}
\begin{proof}
    This follows trivially from Definition~\ref{def:nonexpansiveoperator} by choosing $y \in \mathcal{F}$ and substituting~\eqref{eq:lem:operatorrelation} into~\eqref{def:nonexpansiveoperator}, giving
    \begin{equation*}
        \norm{x^{(n+1)} - y} \leq L \norm{x^{(n)} - y} \quad (\forall x \in \mathcal{C})(\forall n \in \mathbb{N}).
    \end{equation*}
    Since $L \leq 1$, this satisfies inequality~\eqref{eq:monotonesequence} and the sequence of iterates $\left(x^{(n)}\right)_{n \in \mathbb{N}}$ will be Fej\'er monotone.
\end{proof}

\section{Fej\'er Monotonicity of Redundant Numbers}
We now begin our analysis of the redundant number system by noting the similarity between the Fej\'er monotone sequences in Definition~\ref{def:fejermonotonicity} and the representation interval in Lemma~\ref{lem:redundantInterval}, which we turn into an existence guarantee for digit stability of Fej\'er monotone sequences in Theorem~\ref{thm:fejerRedundancy}.
 We will be carrying out our analysis using the infinity norm in Definitions~\ref{def:fejermonotonicity} and~\ref{def:nonexpansiveoperator}, since when examining digit stability, we want to look at the largest error between the iterates and their limit point.
\begin{theorem}
    \label{thm:fejerRedundancy}
    Let $\left(x^{(k)}\right)_{k \in \mathbb{N}}$ be a Fej\'er monotone sequence in the symmetric maximally redundant number representation of radix $r$ and $x$ be the only point in the set $\mathcal{C}$. If element $n$ of the sequence satisfies
    \begin{equation}
        \label{eq:fejer:Dcondition}
        \norm{x^{(n)} - x}_{\infty} < r^{-D},
    \end{equation}
    for some positive integer $D$, then there exists a representation of the sequence where the $D$ MSDs of all future elements with index $k$ satisfy
    \begin{equation}
        \label{eq:fejer:digitStabilityCondition}
        x^{(k)}_{i} = x^{(n)}_{i} \quad (\forall i \in \{ 1, \dots, D \}) (\forall k > n).
    \end{equation}
\end{theorem}
\begin{proof}
    Since we are using the infinity norm, we can without loss of generality focus on the 1-dimensional case (e.g.\ just the element in the vector $x^{(n)}$ that is the furthest from the corresponding element in $x$).
    From the infinity norm, we know that if $x^{(n)}$ satisfies condition~\eqref{eq:fejer:Dcondition}, then the point $x$ lies in the interval
    \begin{equation}
        \label{eq:fejer:xinterval}
        x \in \left(x^{(n)} - r^{-D},~x^{(n)} + r^{-D}\right).
    \end{equation}
    Since $x^{(n)}$ is in the redundant number representation, we see that the interval~\eqref{eq:fejer:xinterval} exactly matches the representation interval given in Lemma~\ref{lem:redundantInterval}, meaning that the redundant representation of $x^{(n)}$ can represent $x$ by only appending digits and not changing the first $D$ MSDs.

    We know that since $x^{(n)}$ and $x^{(n+1)}$ belong to a Fej\'er monotone sequence, they will satisfy the inequality
    \begin{equation}
        \label{eq:fejer:monotonerelation}
        \norm{x^{(n+1)} - x} \leq \norm{x^{(n)} - x} < r^{-D},
    \end{equation}
    which means that $x^{(n+1)}$ is also contained within the representation interval of $x^{(n)}$ given by \eqref{eq:fejer:xinterval}, meaning $x^{(n+1)}$ can be represented by only appending new digits to $x^{(n)}$.
    The Fej\'er montonicity means that relation~\eqref{eq:fejer:monotonerelation} will also hold for all iterates after element $n$, meaning we can represent all future elements by simply appending new digits to $x^{(n)}$, leading to the digit stability condition in~\eqref{eq:fejer:digitStabilityCondition}.
\end{proof}

Note that the proof for Theorem~\ref{thm:fejerRedundancy} is simpler than the equivalent proof for Lemma~3 by Li et al.\ in \cite{liDigitStabilityInference2021}, since we can make use of the Fej\'er montonicity of the sequence instead of having to use algorithm-specific information about how the iterates are generated and the intervals will change.

The digit stability result presented in Theorem~\ref{thm:fejerRedundancy} is a direct result of having the redundancy in the number representation.
 In the standard representation, the representation intervals will be single-sided and encompass only
 \begin{equation*}
     \tilde{x} \in \left[ x, x + r^{-D} \right),
 \end{equation*}
 while the relation between the elements of a  Fej\'er monotone sequence uses normwise relations that are double-sided.
 This means that when a Fej\'er monotone sequence converges to a limit point with a finite number of decimal places, any oscillation around the limit point during the sequence will require changing the MSDs in the standard representation, but the redundant representation is able to accommodate for these oscillations using only the future LSDs.

An example of this oscillation can be seen in the Fej\'er monotone sequence with a limit point of $0.5$ given in Table~\ref{tab:fejerExample}.
 The sequence oscillates around $0.5$, and the standard (base 10 and binary) representations cannot stabilize the MSDs in the sequence, while the symmetric maximally redundant representation in radix-2 can stabilize the MSDs in the presence of the oscillation.
 This example also showcases the requirement that inequality~\eqref{eq:fejer:Dcondition} in Theorem~\ref{thm:fejerRedundancy} must be satisfied to generate stable digits.
 While we have two stable digits starting at element two, we do not generate the next stable digit until after element four, since the distance to 0.5 for elements two and three are both strictly greater than $2^{-2}$.

The digit sequences for values of a Fej\'er monotone sequence in the redundant representation are not unique though, especially in the symmetric maximally redundant representation we use in this work.
 In a symmetric maximally redundant representation, a value can always be represented using its original (non-redundant) digit sequence as well as a sequence utilizing the redundant digits.
 This can also be seen in the example sequence in Table~\ref{tab:fejerExample}, since the column listing the binary representation utilizes a strict subset of the digits used in the redundant column (i.e.\ using only $\{0, 1\}$).
 Therefore, Theorem~\ref{thm:fejerRedundancy} simply says that such a sequence using the redundant representation exists, not that the Fej\'er monotone sequence will always be represented by it.

\begin{table}[t]
 \centering
 \caption{Sample Fej\'er monotone sequence that converges to $0.5$. Redundant representation uses radix-2 with the symmetric maximal digit set $\{-1, 0, 1\}$.}
 \label{tab:fejerExample}
 \begin{tabular}{c|l|l|l|l}
      Element \# & Standard & Binary & Redundant & $\norm{x_{i} - 0.5}$ \\
      \hline
      1 & $1.0$      & $1. 0 0 0 0 0 0$ & $1. 0 0 0$ & $0.5$\\
      2 & $0.125$    & $0. 0 0 1 0 0 0$ & $1. \bar{1} \bar{1} \bar{1}$ & $0.375$\\
      3 & $0.75$     & $0. 1 1 0 0 0 0$ & $1. \bar{1} 1 0$ & $0.25$\\
      4 & $0.375$    & $0. 0 1 1 0 0 0$ & $1. \bar{1} 0 \bar{1}$ & $0.125$\\
      5 & $0.5625$   & $0. 1 0 0 1 0 0$ & $1. \bar{1} 0 0 1$ & $0.0625$\\
      6 & $0.46875$  & $0. 0 1 1 1 1 0$ & $1. \bar{1} 0 0 0 \bar{1}$ & $0.03125$\\
      7 & $0.515625$ & $0. 1 0 0 0 0 1$ & $1. \bar{1} 0 0 0 0 1$ & $0.015625$\\
      $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$
 \end{tabular}
\end{table}

\section{MSD Stability in Iterative Methods}

Now that we have shown that a Fej\'er monotone sequence can have digit stability in the redundant representation, we can easily say that any fixed-point iteration in the redundant number representation that generates a sequence of iterates that are Fej\'er monotone can have digit stability of its iterates when they satisfy condition~\eqref{eq:fejer:Dcondition}.

\begin{theorem}
    \label{thm:fejerIteration}
    Let $f(\cdot)$ be a function that has a fixed-point of $x^{*}$ and is used in the fixed-point iteration $x^{(k+1)} = f(x^{(k)})$ with the redundant number representation to generate the Fej\'er monotone sequence of iterates $\left(x^{(k)}\right)_{k \in \mathbb{N}}$.
    If element $n$ of the sequence generated by $f(\cdot)$ satisfies
    \begin{equation*}
        \norm{x^{(n)} - x^{*}}_{\infty} < r^{-D},
    \end{equation*}
    for some positive integer $D$, then there exists a representation of the iterate sequence where the $D$ MSDs of all future elements with index $k$ satisfy
    \begin{equation*}
        x^{(k)}_{i} = x^{(n)}_{i} \quad (\forall i \in \{ 1, \dots, D \}) (\forall k > n).
    \end{equation*}
\end{theorem}
\begin{proof}
    This follows directly from the result in Theorem~\ref{thm:fejerRedundancy}.
\end{proof}

While Theorem~\ref{thm:fejerIteration} is a nice and simple result, we can now also derive more specific results for classes of iterative methods, such as Lipschitz continuous methods in Theorem~\ref{thm:lipschitzIteration}.
\begin{theorem}
    \label{thm:lipschitzIteration}
    Let $f(\cdot)$ be a function that is locally Lipschitz inside the set $\chi$ with Lipschitz constant $L_{f} < 1$.
    If $f(\cdot)$ has a fixed-point of $x^{*} \in \chi$ and is used in the fixed-point iteration $x^{(k+1)} = f(x^{(k)})$ with the redundant number representation to generate the sequence of iterates $\left(x^{(k)}\right)_{k \in \mathbb{N}} \in \chi$, then given a number of digits $D \geq 1$ there exists an element index $\hat{k} \geq 0$ such that
    \begin{equation}
        \label{eq:msd:digstability}
        x^{(k)}_{i} = x^{(\hat{k})}_{i} \quad (\forall i \in \{ 1, \dots, D \}) (\forall k > \hat{k}).
    \end{equation}
\end{theorem}
\begin{proof}
    We begin by noting that since $f(\cdot)$ is locally Lipschitz inside $\chi$ with a Lipschitz constant $L_{f} < 1$, $f(\cdot)$ will be a contractive operator inside $\chi$ and Lemma~\ref{lem:operatorSequence} says that the iterates $\left(x^{(k)}\right)_{k \in \mathbb{N}}$ of $f$ will form a Fej\'er monotone sequence that stays in the set $\chi$.
    Since $L_{f}$ is strictly less than 1, the residuals between the iterates $x^{(k)}$ and the fixed point $x^{*}$ will be a decreasing sequence that can be bounded using a function of the Lipschitz constant and the difference between the two initial iterates \cite[Thm. 4.1]{suliIntroductionNumericalAnalysis2003}
    \begin{equation}
        \label{eq:msd:seqbound}
        \norm{x^{(k)} - x^{*}} \leq L^{k} \frac{1}{1 - L} \norm{x^{(1)} - x^{(0)}},
    \end{equation}
    with $\lim_{k \to \infty} L^{k} = 0$.
    This means that the residual between the iterates $x^{(k)}$ and $x^{*}$ will eventually be less than $r^{-D}$ for a given $D \geq 1$ and $r \geq 2$, satisfying the condition in Theorem~\ref{thm:fejerRedundancy} and allowing it to be used to create the digit stability result~\eqref{eq:msd:digstability}.
\end{proof}

A key point to notice in Theorems~\ref{thm:fejerIteration} and~\ref{thm:lipschitzIteration} is that they are concerned only with the \textit{existence} of an iterate sequence with stable MSDs, and do not comment on how such a sequence can be constructed.
 The construction of the sequence with stable MSDs will still be application/algorithm-specific, with the designer needing to ensure that the computations performed in the fixed-point iteration will preserve the MSD stability as they are performed, such as through the use of online arithmetic or the ARCHITECT system~\cite{liARCHITECTArbitraryPrecisionHardware2020}.

\section{Example Iterative Methods with MSD Stability}

Now that we have presented conditions for when the MSDs of iterates can be stable, we turn our attention to showing two example iterative methods where we can apply these conditions and derive a guarantee for digit stability.

\subsection{Stationary iterative linear solvers}

A common iterative method examined by prior work on redundant number systems (e.g.\ \cite{liDigitStabilityInference2021} and~\cite{ercegovacGeneralHardwareOrientedMethod1977}) is the stationary iterative method for solving linear systems of the form $Ax = b$, which has the fixed-point iteration
\begin{equation*}
    x^{(k+1)} = T\left(x^{(k)}\right) \coloneqq (I - \invnb{M} A)x^{(k)} + \invnb{M}b.
\end{equation*}
 Common forms of this method are the Jacobi method when $M$ is the diagonal of $A$, the Gauss-Seidel method when $M$ is the lower triangular portion of $A$, and the successive overrelaxation (SOR) method when $M = \invnb{\omega} D - \bar{L}$ \cite[Chap. 2]{Greenbaum1997}.
 To begin our analysis, we examine the difference between two successive function values
 \begin{multline*}
    \norm{T\left(x^{(k+1)}\right) - T\left(x^{(k)}\right)} = \\
     \norm{ (I - \invnb{M} A)x^{(k+1)} + \invnb{M}b - (I - \invnb{M} A)x^{(k)} - \invnb{M}b}.
 \end{multline*}
 Simplifying this, we find that it is equal to
\begin{equation*}
     \norm{ (I - \invnb{M} A) (x^{(k+1)} - x^{(k)})},
 \end{equation*}
 and by applying the triangle inequality we find that
 \begin{equation*}
     \norm{T\left(x^{(k+1)}\right) - T\left(x^{(k)}\right)} \leq
     \norm{I - \invnb{M}A} \norm{x^{(k+1)} - x^{(k)}}.
 \end{equation*}
 This inequality is equivalent to a Lipschitz condition for the stationary iterative solver, with the Lipschitz constant
 \begin{equation*}
     L = \norm{I - \invnb{M}A}.
 \end{equation*}

We now can apply Theorem~\ref{thm:lipschitzIteration} to this method to find out that the simple iterative linear solvers will have digit stability when using the redundant number system, provided the condition
 \begin{equation*}
     \norm{I - \invnb{M}A}_{\infty} < 1
 \end{equation*}
 is satisfied, which is the same condition derived by Li et al.\ in \cite{liDigitStabilityInference2021}.

\subsection{Newton's method}

Another popular iterative method utilized in many fields is Newton's method for finding the roots of the function $f(\cdot)$, which is given by the fixed-point iteration (for 1-dimension)
 \begin{equation}
    \label{eq:newtons}
    x^{(k+1)} = T\left(x^{(k)}\right) \coloneqq x^{(k)} - \frac{f(x^{(k)})}{f'(x^{(k)})},
 \end{equation}
 where $f'(\cdot)$ is the derivative of $f$.
 We can manipulate the Lipschitz continuous condition in Definition~\ref{def:nonexpansiveoperator} to be
 \begin{equation*}
     \frac{\norm{T(x^{(n+1)}) - T(x^{(n)})}}{\norm{x^{(n+1)}) - x^{(n)}}} \leq L,
 \end{equation*}
 which for the 1-dimensional case can be translated into a condition on the derivative of $T(\cdot)$ that $T'(x) \leq L$, meaning a contractive operator has $T'(x) < 1$ for all $x$ in its region of convergence \cite[Chap. 1]{suliIntroductionNumericalAnalysis2003}.

We then know for the Newton's method~\eqref{eq:newtons}, that its derivative will be
 \begin{equation*}
    T'\left(x\right) \coloneqq \frac{f(x) f''(x)}{\left( f'(x) \right)^2}.
 \end{equation*}
 Since Newton's method needs $T'(x) < 1$ (and therefore $L < 1$) for its convergence, we can use Theorem~\ref{thm:lipschitzIteration} to say that as the Newton's method converges, there exists a redundant number representation of its iterates with stable digits.

\section{Conclusion}

In this work, we examined the conditions needed for there to be MSD stability in sequences represented in the redundant number system.
 We did this through two key results: showing that a Fej\'er monotone sequence can have a redundant representation with MSD stability, and then using that result to show that an iterative method that generates a Fej\'er monotone sequence of iterates can have MSD stability when using the redundant number system.

These results only focused on the existence of these sequences, though, and did not discuss how to identify or create them.
 These are both key points in the successful use of these results in real-world applications, so future work should explore them in more detail.
 For instance, future work can explore if there is a redundant representation where a Fej\'er monotone sequence is guaranteed to have digit stability in all its representations (e.g.\ non-symmetric or non-maximal redundancy), and also how the representation is preserved during the computations in an iterative method (e.g.\ will the redundant sequence be preserved).
 Additionally, the results linking Lipschitz continuous functions with digit stable redundant sequences can be further exploited to try to predict the number of stable MSDs that will appear in each iteration, allowing for more general results than those previously presented in~\cite{liDigitStabilityInference2021}.

\section*{Acknowledgment}

We would like to thank Dr. He Li for reading and commenting on an early version of this manuscript.
This work was supported by Engineering and Physical Sciences Research Council grant EP/P020720/1.

redpajama_set_name: RedPajamaArXiv


that judge a fallen wretch.
with vile lust and ignorance.
the prospect of love a distant dream.
and her heart aches for freedom.
as the child skips on to pleasured, innocent youth.
the weight of the world just a little less.
Judgement is simply the lack of understanding, because what we judge is more often something we have already or will endure in life. Knowing this changes perspective.


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