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2010: The Retrospect

Friday, December 31, 2010

Well, folks, that's that. 2010 has finally been laid to rest. It was a good year, filled with many ups and downs, and like so many before it, it went too fast. And that one's on you God! You didn't have to take it. And with that, we scatter 2010's ashes in what we assume to be in accordance with it's final wishes.

"opens up coffee can, and scatters ashes, which blow into The Dude's face"

Sorry about that. I had Lebowski on the mind. Anyway, come midnight tonight, 2010 will be history. And like any good film blogging human of this earth, I will be spending the next few days reflecting on 2010 and all it's cinematic offerings. Join me, won't you, as I list off all the various do-dads and what not of what made 2010 a year to remember.

Here's the schedule! Hope to see all of you there with your thoughts!

Jan 1: Top 5 Best Lines
Jan 2: Top 5 Best Posters
Jan 3: Top 5 Best Trailers
Jan 4: Top 5 Best Scenes
Jan 5: Top 5 Best Soundtracks
Jan 6: MVPs of the Year (Acting)
Jan 7: MVPs of the Year (Directing)
Jan 8: Top 5 Most Anticipated for 2011
Jan 9: Top 5 Worst Films of 2010
Jan 10: Top 10 Best Films of 2010

Alright, time for another furry. This is a toucan!

Ka-kaw!

Into the Sunset

Wednesday, December 29, 2010

2010's windin' on down to a close here, pardner. Lookin' like we done gots ourselves slim pickins from here on out the rest uh de year. What you suppose we do about it?

Sorry, I'll stop with that hopelessly contrived western accent. Yes, 2010 only has a good three days left, and then we are to 2011. Because of this, the influx in quality cinematic entertainment has been fast and furious, as it is every winter. We've had a lot, but I think we are officially at then end of the line. After all those tales of psychotic ballerinas, stammering kings and drugged out fighters, we have reached the conclusion. Rounding out the pack, and closing the book on 2010 is True Grit. It's a western of the highest class, beautifully shot, very well acted and directed, with a great sense of the era in which it is set. Though the movie is very nearly undone by one terrible mistake, it's still worth every penny you will spend on it.

Ok, ok! The Dude abides! I'm sorry!

When her pa is killed and his cash pilfered by a man named Tom Chaney, fourteen year old Mattie Ross vows revenge. To go about this, she hires the washed up, but merciless, Rooster Cogburn, a US Marshal to hunt him down. The catch? She's going with him, so that she may bring Chaney to justice herself. After teaming up with a LaBoeuf, a Texas Ranger who has been tailing Chaney for some time, Cogburn and Ross set out into the vast, untamed land of the west. In this lawless land, nowhere is safe, something Mattie will quickly learn the hard way.

Writers/Directors Joel and Ethan Coen were adamant that this version of True Grit was an adaptation of the book, not the John Wayne classic. As it goes, it's a very well told story, one that moves very slowly so as to explore the characters, and then picks up a ton of steam in the final act. It's a ver traditional film, at least for the Coens, although plenty of their touches, oddball humor and whatnot, definitely come through.

If they handed out MVP Awards for actors each month, Jeff Bridges would win December 2010. Coming off strong from Tron: Legacy, he, quite simply, acts the shit out of Cogburn. I mean, I don't know what else could have happened. Bridges is, after all, reuniting with the guys who got a career best performance out of him. Cogburn, though, couldn't be farther from The Dude when it comes to character type. Cogburn is a boorish, drunken, but deadly and exacting law enforcer, ready to scarf down a bottle of whisky one minute then shoot you and three other guys in the head without breaking a sweat. Bridges plays all these elements masterfully, while still retaining an air of humor about the whole thing. He's funny guy, when you think about it, despite all his serious flaws and tendencies. To see him nonchalantly order Mattie to cut down an hanged man to see if he knew him, only to callously declare "I do not know this man," was a downright hilarious moment. Bridges hasn't faltered since winning that Oscar. Let's hope he keeps it up.

Matt Damon does some of his best acting in years as the sarcastic and optimistic LaBoeuf, and Josh Brolin turns in an effectively villainous role as Chaney. Barry Pepper also shows up as the leader of a gang of bandits who Chaney gets caught up in. If anything, he reminds everyone why they should just forget about Battlefield: Earth and start casting him in bigger roles.

But the real scene stealer is Hailee Steinfeld as the headstrong and stubborn Mattie. Put simply, this is one the most stunning feature length debuts I've ever seen, from any actor of any age. Steinfeld is awesome in this role, handling the challenging dialogue and physical demands of the job like a champ! She's fierce, funny, sweet, and resolute in her task to bring her father's killer to justice. If only every kid had her passion. What kind of a world would we live in then? It's a marvelous performance from Steinfeld, and I really hope it leads to bigger and better things for her in the future.

Don't look so surprised girl! It's pretty tough out here!

Joel and Ethan Coen have made a name for themselves by mashing up various genres of film into one, or putting a fresh spin on a classic one. True Grit falls somewhat into the latter category, but, even then, that's a stretch. To say that this one doesn't really feel like a Coen Bros. movie is a pretty apt description. It feels and looks like a traditional western. Gone are the surreal and quirky scenes from The Big Lebowski. Gone is whip-snap dialogue of Fargo. The long stretches of desert tundra that made up a good portion of No Country For Old Men are there, but they aren't used for nearly as long aren't used for the same effect. Indeed, it seems like the Coens set out to make a standard western, and, in that sense, they succeeded admirably. It's a beautiful film, full of wide open plains, claustrophobic forests, babbling brooks and breathtaking vistas. The world is rife with money grubbing entrepreneurs, unfeeling and greedy bandits, and accurate as hell gunslingers. It's traditional fare, but it's handled marvelously. 

The Coen's trademark, off beat sense of humor is here though, and used to great effect in some scenes. There's a tangent, where Cogburn and LeBoeuf get into an argument over who can shoot better, and proceed to spend the next few minutes shooting corn bread out of the air. Mattie's ability to negotiate with people offers some humorous moments, and there's a scene with a medicine man that is so off kilter that it might as well be from any Coen movie.

Unfortunately, there is one thing here that drags the film down from it's perch of stupendous awesomeness, and that's the ending. Basically, the story ends, and then there's a five minute epilogue showing where they ended up, and it was not needed at all. It seems like the Coens didn't have enough faith in their craft and they wanted to impart some sort of meaning on to their film, so they wrote the epilogue to deliver the message. First off, there was plenty of meaning and subtext in what was already there, and second of all, in the state it's in, the message just seems forced and contrived. It was completely unnecessary and it leaves you with a bad taste in your mouth.

That being said, those last five minutes do not, in any way, detract from the first 105. True Grit is just a damn good, very well acted, and smartly directed, old fashioned western. The Coens aren't trying to change anything with this. All they want to do is make a traditional cowboy flick in the vein of Sergio Leone and Sam Peckinpah. They succeed. Not only do they manage to make an excited and visually impressive film, but they manage great performances from Bridges, Damon, and Brolin, topped off by a fantastic, star making performance from Steinfeld. Though it probably won't get much Oscar buzz, True Grit is absolutely worth every dollar. It's just that damn ending. If only it weren't there.



Movie Videos & Movie Scenes at MOVIECLIPS.com

Recent research findings on M87 (NGC 4486)

Tuesday, December 28, 2010

M87 (Messier 87), also known as NGC 4486, is a giant elliptical galaxy, located about 53.5 million light-years away. It is noteworthy for several reasons, including the presence of an unusually large supermassive black hole (SMBH) in its active galactic nucleus, with an estimated mass of about 6.4×109 times the mass of the Sun (M), two plasma jets that emit strongly at radio frequencies and extend at least 5000 light-years from the SMBH (although only the jet pointed more towards us is readily detectable), and a population of about 15,000 globular clusters.

The total mass of M87 is difficult to estimate, because elliptical galaxies like M87, and unlike spiral galaxies, do not tend to follow the Tully-Fisher relation between intrinsic luminosity and total mass calculated from rotation curves – which therefore includes dark matter. Estimates of the total mass of M87, including dark matter, come in around 6×1012 M within a radius of 150,000 light-years from the center. This compares with about 7×1011 M for the Milky Way, but M87 could be more than 10 times as massive.

In other comparisons, the Milky Way has only about 160 globular clusters, and a central black hole (Sagittarius A*) with a mass of about 4.2×106 M. So M87's central black hole is about 1500 times as massive as the Milky Way's. Pretty impressive difference.



M87 – click for 640×480 image


Besides the recent research listed below, I've written about earlier research on M87 in these articles: Galactic black holes may be more massive than thought, Stellar birth control by supermassive black holes, Black holes in the news.

You might also be interested in some articles from the past year on the general subject of active galaxies: Active galaxies and supermassive black hole jets, Where the action is in black hole jets, Quasars in the very early universe.


Feedback under the microscope: thermodynamic structure and AGN driven shocks in M87 (6/29/10) – arXiv paper

Feedback under the microscope II: heating, gas uplift, and mixing in the nearest cluster core (3/28/10) – arXiv paper

Activity of the SMBH in M87 has a significant effect not only on the host galaxy, but also on the Virgo cluster of galaxies in which M87 is near the center. Energetic outflows of matter from near the black hole force plumes of gas out of the galaxy into the hotter intergalactic medium. The mass transported in this way represents about as much gas as is contained within 12,000 light-years of M87's center. (However, that's only about 2.5% of M87's 500,000 light-year radius.) If it had not been expelled, the gas could have formed hundreds of millions of stars.

The first paper reports on studies using the Chandra X-ray Observatory to measure gas temperatures around M87's center. The findings include detection of 2 distinct shock wave fronts about 46 thousand light-years and 10 thousand light years from the center. This indicates that explosive events occurred about 150 million and 11 million years ago, respectively.

The second paper uses observations from Chandra, XMM-Newton, and optical spectra to distinguish different phases of the hot gas surrounding M87's SMBH.

Refs:
Galactic 'Super-Volcano' in Action (8/20/10) – Science Daily (press release)
Galactic Supervolcano Erupts From Black Hole (8/20/10) – Wired.com
Galactic 'Supervolcano' Seen Erupting With X-Rays (9/6/10) – Space.com

A correlation between central supermassive black holes and the globular cluster systems of early-type galaxies (8/13/10) – arXiv paper

A study of 13 galaxies, including M87, has found a correlation between the size of a galaxy's SMBH and the number of the galaxy's globular clusters. The types of galaxies studied included nine giant ellipticals (like M87), a tight spiral, and 3 galaxies intermediate in type between spiral and elliptical. The smallness of the sample is due to the exclusion of open spiral galaxies and the further limitation to cases where good estimates of the number of globular clusters and mass of the central black hole existed.

The correlation, in which the number of globular clusters is proportional to the black hole mass, is actually stronger than correlations between black hole mass and other galaxy properties previously studied for correlation, such as stellar velocity dispersion (an indicator of total mass), and luminosity of the galaxy's central bulge or whole galaxy (for ellipticals).

In some cases the correlation of black hole mass with total luminosity was especially weak, but better with number of globular clusters. For instance, Fornax A (NGC 1316) is a giant lenticular galaxy with luminosity comparable to that of M87. Yet its central black hole has a mass of 1.5×108 M, 2.3% that of M87's black hole. It has 1200 globular clusters, 8% of M87's count. Clearly this is not a linear relation. Rather, the study found that the best fit was a power law with M ≈ (1.7×105)×N1.08±0.04, where M is black hole mass in units of M and N is number of globular clusters. This relation predicts a SMBH mass of 5.5×109 M for M87, which is very close, and 3.6×108 M for the SMBH mass of NGC 1316, which is high – but the SMBH mass of NGC 1316 is also unusually low in comparison with its luminosity and velocity dispersion.

By contrast, the relation predicts that the Milky Way with a SMBH mass of 4.2×106 M should have only about 20 globular clusters, while the actual number is about 160. However, the Milky Way is a loose spiral, not one of the types that was studied, which may account for the much worse correlation. The fit is much better if only globular clusters associated with the central bulge (about 30) are considered.

The obvious question is about why this relation between SMBH mass and number of globular clusters exists. Presumably it has much to do with the typical history of a large galaxy, which is expected to include frequent mergers with other galaxies. The existence of the relationship should provide clues to galactic history, and especially how this may be different for loose spirals like the Milky Way, in comparison with more compact galaxies.

Refs:
A correlation between central supermassive black holes and the globular cluster systems of early type galaxies (8/11/10) – The Astrophysical Journal
Supermassive black holes reveal a surprising clue (5/25/10) – Physicsworld.com

A Displaced Supermassive Black Hole in M87 (6/16/10) – arXiv paper

It has generally been assumed that a galaxy's central SMBH is very close to the actual center of mass of the galaxy, because that is (by definition) the gravitational equilibrium point. This central point should be essentially the same as the photometric center of the galaxy, since the galaxy's stars should be distributed symmetrically around the center. Consequently, astronomers have not carefully searched for cases where a SMBH is not very near the galactic center. This lack of extensive investigation is also a result of the fact that the SMBH is often hidden inside a dense cloud of dust, so its exact position is difficult to determine. M87's SMBH (more precisely, the accretion disk around the SMBH), however, is clearly visible, and the research reported in this paper finds it is actually located about 22 light-years from the apparent galactic center.

There are various possible reasons for this much displacement from the center, and not a lot of evidence to identify the most likely reason. Possible reasons include: (1) The SMBH is part of a binary system in which the other member is not detected. (2) The SMBH could have been gravitationally perturbed by a massive object such as a globular cluster. (3) There is a significant asymmetry of the jets. (4) The SMBH has relatively recently merged with another SMBH, subsequent to an earlier merger of another galaxy with M87.

The displacement of the SMBH is in the direction opposite the visible jet, so the last two possibilities are more likely than the others. However, possibility (3) depends on the jet structure having existed at least 100 million years and the density of matter at the center of M87 being low enough to provide insufficient restoring force. Possibility (4) is viable if the SMBH is still oscillating around the center following a galactic merger within the past billion years.

Refs:
A Displaced Supermassive Black Hole in M87 (6/9/10) – The Astrophysical Journal Letters
Black Hole Shoved Aside, Along With 'central' Dogma (5/25/10) – Science News
Black Hole Found in Unexpected Place (5/25/10) – Wired.com
Supermassive black holes may frequently roam galaxy centers (5/25/10) – Physog.com (press release)
Bizarre Behavior of Two Giant Black Holes Surprises Scientists (5/25/10) – Space.com
Galactic Black Holes Can Migrate or Quickly Awaken from Quiescence (5/26/10) – Scientific American




M87 jet


Radio Imaging of the Very-High-Energy γ-Ray Emission Region in the Central Engine of a Radio Galaxy (7/24/09) – Science

Energetic plasma jets, in which matter is accelerated close to the speed of light, combined with intense electromagnetic emissions, especially at radio frequencies, are prominent in about 10% of active galaxies, including M87. However, little has been well established about what processes are responsible for the emissions, or more generally how the jets are powered, accelerated, and focused into narrow beams. Because of the relative proximity of M87 and the fact that the jet we observe is angled from 15° to 25° to our line of sight, M87 is one of the best objects to study in order to learn more about how jets work.

Gamma rays, because of their very high energies (greater than 100 keV per photon), are not continuously produced in active galaxy jets, but are occasionally observed in short bursts lasting only a few days. One such event occurred in M87 in February 2008. At the same time, the intensity of radiation at all other wavelengths increased substantially. Such flares, at lower energies, are not unusual, since the energy output of most jets is somewhat variable in time. The flare persisted for much longer at energies below the gamma-ray band, indicating that the disturbance continued to propagate along the jet even after the gamma-ray flare subsided. However, although we don't know what the cause was, the coincidence in time of the gamma-ray emissions and the beginning of the extended flare makes it very likely that the events had the same source.

This is significant information, because our technology for detecting gamma-ray events has very poor angular resolution (~0.1°), since gamma rays can be detected on the ground only by secondary effects that a gamma ray produces in our upper atmosphere. More than 6 orders of magnitude finer resolution can be achieved at radio frequencies, using very long baseline interferometry. With that technology, it was possible to locate the origin of the disturbance that caused both gamma ray and lower energy flaring to a region within about 100 Schwarzschild radii (Rs) of the SMBH. Since Rs = 2G×M/c2, Rs for the M87 SMBH is about 1.9×1010 km, or more than twice the radius of the solar system. So 100Rs is about 70 light-days – which is pretty small compared to the 53.5 million light-year distance to M87.

It's also significant that the gamma-ray event occurred so close to the SMBH, because the cause must be unlike whatever is responsible for the flaring described in the following research.

Refs:
VLBA locates superenergetic bursts near giant black hole (7/2/09) – Physorg.com (press release)
Mysterious Light Originates Near A Galaxy's Black Hole (7/2/09) – Space.com
A Flare for Acceleration (7/24/09) – Science
High Energy Galactic Particle Accelerator Located (9/14/09) – Science Daily (press release)

Hubble Space Telescope observations of an extraordinary flare in the M87 jet (4/22/09) – arXiv paper

Electromagnetic radiation from SMBH jets is fairly variable in both time and location along the jet. In the case of M87, high-resolution images at various wavelengths have shown the existence of many regions of enhanced emissions within the jet. One of the most prominent of these even has a name: HST-1, so-named because it was discovered by the Hubble Space Telescope. It occupies a stationary position on the jet, about a million Schwarzschild radii from the center, i. e. about 2000 light-years from the SMBH.

HST-1 has been observable for some time, but until February 2000 it was relatively dormant. After that it began to flare more brightly across the electromagnetic spectrum up to X-rays. In 2003 it became more variable, and it reached its greatest brightness in May 2005, when the flux in near ultraviolet was 4 times as great as that of M87's central energy source, the SMBH accretion disk. This represents a brightness increase at that wavelength of a factor of 90. The X-ray flux increased by a factor of 50, and similar, synchronized changes occurred at other wavelengths. The synchronization indicates that one mechanism is responsible for the variability at all wavelengths.

What the actual cause of the disturbance may be is not clear. Because of the great distance of HST-1 from the SMBH, its basic energy source must not be the central accretion disk itself. More likely HST-1 is a result of constriction of magnetic field lines, resulting in further acceleration of the particles making up the jet. Acceleration of charged particles causes radiation by the synchrotron process, and is evidenced by polarization of the emitted photons. Constriction of the jet may be a result of passage through a region of higher density of stars. The increased variability could mean that the jet has encountered a region of higher but varying stellar density. Alternatively, the jet may be passing through a patch of thick gas or dust, with excess radiation produced by the resulting particle collisions.

These results could explain the variability of light from other, more distant active galaxies, at least those which have strong jets, given that it's possible for a small region of the jet far from the SMBH to outshine the central source. However, another source of variability occurs when a jet is viewed at a very low angle to our line of sight, in which case any slight change of direction could cause an apparent change of brightness.

Refs:
Hubble Space Telescope observations of an extraordinary flare in the M87 jet (3/6/09) – The Astronomical Journal
Hubble Witnesses Spectacular Flaring in Gas Jet from M87's Black Hole (4/14/09) – Physorg.com (press release)
Black Hole Creates Spectacular Light Show (4/14/09) – Space.com
Black hole jet brightens mysteriously (4/15/09) – New Scientist
Black hole spews out impressive light show (4/20/09) – Cosmos Magazine

Pinwheel of Star Birth

Monday, December 27, 2010

Pinwheel of Star Birth (10/19/10)
This face-on spiral galaxy, called NGC 3982, is striking for its rich tapestry of star birth, along with its winding arms. The arms are lined with pink star-forming regions of glowing hydrogen, newborn blue star clusters, and obscuring dust lanes that provide the raw material for future generations of stars. The bright nucleus is home to an older population of stars, which grow ever more densely packed toward the center.

NGC 3982 is located about 68 million light-years away in the constellation Ursa Major. The galaxy spans about 30,000 light-years, one-third of the size of our Milky Way galaxy.




NGC 3982 – click for 984×1000 image


More: here

RIP 2010

You've been good to us!

Eloquence of the Highest Order

Sunday, December 26, 2010

Happy Birthday to me! The big 19! Here! How bout' a review of The King's Speech? It's my birthday gift to you!

Man, I love Christmas Break! Aside from the obvious - no school, no homework, going to sleep late, waking up late, skiing, real food, tons of gifts, my birthday, etcetera and so forth - the weeks between semesters is the only time I really have to see all the movies that a film blogger of my self-determined credibility needs to see. Which is why I'm getting to The King's Speech so late in the game. My plan is to see all the rest of the heavy hitters before the year is done. That way, I can participate in the year end lists actually knowing about most of the films being talked about. Still have to get to 127 Hours though. That one is hard to find...

Anyway. The King's Speech. There's a reason that this one is getting a lot of press. It's great, very well told, superbly acted, with a wonderful sense of humor. This one is gunning for the big O! It has the cajones to go all the way.

Uhhh...

As the reign of King George V comes to an end, and the world braces for war with the Axis forces, Britain turns to it's new monarch, the former Duke of York, Albert, who takes the moniker of George VI. Unfortunately, Albert suffers from a rather bad stutter, which, as anyone who has ever seen a man in politics speak, is a bit of a problem. The world is looking to George to lead them, and the man can't string one sentence together without faltering. Having tried all the royal doctors, the Queen seeks out someone a bit more conventional. Enter Lionel, a speech therapist who is as far from royalty as you can get. Despite the difference in status, George starts meeting with Lionel, and, slowly but surely, he begins to find his voice.

It's your standard, uplifting, man-overcomes-obstacles-to-succeed, thing, but it's told remarkably well. Great care was given to each character, and the time taken to humanize these polarizing figures is instantly clear. The script is great, handling the dramatic moments very well, while injecting some very sharp and poignant comedy into the mix. It's a good job.

The role of the handicapped king is played with quiet grace by Colin Firth. Rather than try and sneak around our defenses and get at our heartstrings, Firth just sits us down, and tells it like it is. "I am the king! I have a speech impediment, and you're gonna pity me, damn it!" And, by jove, do we ever. Bolstered by a very convincing recreation of stutter, Firth is as good as he's ever been. His character is pathetic, sure, but his meekness is there to serve the story, not as a cheap ploy to stir the emotions.

Helena Bonham Carter is Queen Elizabeth, and she's just as polarizing as the man playing her husband. Bonham Carter is usually very good, but, I gotta say. It was absolutely lovely to see her play a normal person in a movie again. I'd almost forgotten how good she was when she's not being directed by Tim Burton or trying to kill Daniel Radcliffe.

The real scene stealer is Geoffrey Rush as Lionel. He's kind of a kooky character, that Lionel, but Rush plays him so well. Not only does he have a great depth to him, he's hilarious. Rush is a comedic force in this film, matching Firth's stubbornness with a great sense of humor. He's still his usual, excellent self in the dramatic moments, but, when everyone around him is so serious, it's nice to see someone who can generate a chuckle from us.

Other members of the cast include Derek Jacobi as Archbishop Cosmo Lang, Timothy Spall as Winston Churchill, and Michael Gambon as George V.

One person I do want to mention is Guy Pearce as George VI brother David/Edward VIII, who gives up to throne to be with the woman he loves. If anything, it further shows how much Guy Pearce needs to be back in the mainstream. This guy is awesome, and yet he has resorted to taking smaller roles. I want Guy Pearce back when he was headlining films like Memento and LA Confidential. Oh well. Maybe people will start listening.

Goofy dude!

Director Tom Hooper directs with a deft hand. I don't know about anyone else, but the cinematography and shot composition really struck me here. Hooper frames his shots to make the characters among the smallest things in them. Often, he'll have a character off to one side, with the rest of the shot taken up by a wall or something. This sense of space does a lot to emphasize the new and enormous responsibilities that George is undertaking, as well as the daunting task of rehabilitating his speech. It's a very striking shot movie, one of the many things that the film does right. 

The script is impeccable, taking plenty of time to make each character seem real. It's one of the easiest trap to fall into. Since the figures on the screen here are very well known in the public eye, it would have been very easy to antagonize some of them so that we, the audience, have some sort of villain to hate. Well, Hooper and Co. are better than that. Every character, while not necessarily likable, feels real, thus, we can't hate them. George's subjects express disapproval of his meeting with Lionel, but it's only because they want him to succeed on the throne and they aren't familiar with Lionel. George snaps at Lionel at times, but it's because he's afraid of his new responsibilities and doesn't quite know how to handle them. All human reactions, and all things the audience can relate to.

I have to imagine that the casting directors of the Harry Potter are kicking themselves. Pretty much every British actor that didn't appear in those films is at the top of their game here! The King's Speech is a wonderful film, full of heart, grace, and humor. Great performances are just icing on the already delicious chocolate that makes up everything else. Of all the leading Oscar contenders, The King's Speech is the least flashy. That automatically makes it stand out and improves it's chances. In a world when movies are drenched in special effects and flashy action, to see a film that it solely about performances and character is a breath of fresh air! See this one, NOW!!!



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Bio Digital Jazz, Man!

Friday, December 24, 2010

I know we're adverse to facing the music, but let's just be honest! The fanboys are running everything! Don't believe me? Fine! Chew on this. Video games are a fast growing form of legitimate entertainment. The most profitable movies are the ones that involve mystical worlds, monsters, magic, and lots of digital spectacle. Oh, and Tron: Legacy got made! Convinced? The first Tron didn't make much of a splash, didn't do all that well at the box office, and, let's face it, does not hold up at all today. But, it amassed a cult following and is now considered a tentpole of the beginning of the CGI movement. Then, something happened. The fanboys that made up the cult following of Tron took over the world, and now, 28 years after the fact, we have a sequel.

Ok, I'll admit, I've seen this twice. I would say I saw it once to get the story, and then again to get the meaning, but such a lie has never been uttered... on this blog. No, the truth is that I fell asleep in the first viewing. I know. That's bad, but I had just arrived back home in SF and had not slept at all the night before. Don't hate on me! Now that the full thing is comfortably swimming around in my system, I finally have an opinion. It's good, not great, with the best use of special effects and 3D since Avatar, but carries all the story and script issues that plagued that eye-gasm. If anything, it's a testament to how far CGI has come! This is a visually stunning, but shallow and pointless, tour de force!

Game On!

Tron: Legacy picks up some years after the first one ended. Kevin Flynn has long since disappeared, leaving his son, Sam, his entire company of Encom and a bit of a rebellious streak. Rather than cultivate the empire that his dad built, Sam is more content to break in and sabotage it. One day, though, Sam receives a message from his father's old office, from a number long since disconnected. Upon investigating, he accidentally triggers the all powerful laser and is transported into The Grid, a digital world created by Kevin where programs run free and gladiatorial games are all the rage. Sam soon discovers that his father has been trapped on The Grid by his own creation, a program named Clu, who has taken control of the digital frontier and seeks to extend his influence into the real world. Aided by a mysterious program by the name of Quorra, Sam and Kevin race to stop Clu's nefarious plan, and escape The Grid before the portal to the human world closes forever. 

As far as stories go, this one could have been a lot worse. The first Tron wasn't exactly a stalwart example of great writing, and this one is vast improvement over it. The plot moves at a good clip, keeping the excitement and severity of the situations up, so that you always feel as if there is some point to what is going on. That being said, the characters are undeveloped as all holy hell, with the exception of Kevin/Clu. Some of the lines are facepalm stupid, and all of the emotional and sweet moments fall as a flat as a board. 

Jeff Bridges return to the role he originated as Kevin. I don't what kind of weed is available in The Grid, but Flynn seems to have taken on some qualities of that other Bridges character. Bridges isn't playing Kevin Flynn. He's playing Kevin Lebowski, uttering things like "You're messing with my zen thing, man" and "Radical". It's pretty funny actually. Bridges does a good job with the character despite this tonal shift, and succeeds in making us care about his predicament. This is made doubly awesome when you see that Bridges plays both hero and villain in this tale. Yes, thanks to the wonders of special effects, Bridges also plays the nefarious Clu, who looks a good 20 years younger than his creator. Clu is a very effective villain, and, though the CGI does look a bit creepy and unrealistic, it's a very good job from Bridges, playing both sides of the spectrum. 

As the other lead, Garrett Hedlund doesn't fare quite as well. Ditto for Olivia Wilde's Quorra. It's not that they are terrible. They just come off as flat, Hedlund more so than Wilde. Wilde's Quorra is spunky and snarky, but able to kick a ton of ass when she needs to. She thrives when she is supposed to be funny or awesome. When she needs to be serious, however, she's up shit creek. Hedlund, on the other hand, doesn't really fare well anywhere. True, some of his line are well delivered, as when he gets progressively annoyed with the people he must fight in the games. But, for the most part, he's hopelessly trying to tread water, delivering most of lines with an awful monotone and giving out the film's cheesiest moments. 

The rest of the cast does pretty well. Bruce Boxleitner returns to his role as Alan Bradley/Tron. He does a good job for the fifteen minutes he's actually on screen. I can't decide if Michael Sheen's Castor was the funniest thing about the movie or the most annoying. The character is basically Ziggy Stardust, and Sheen gives it his all. James Frain turns up as Clu's brown nosing assistant, and Beau Garrett practically melts the film as Gem, a "Siren" who arms Sam for the games and later assists him. 

Also, what was the point of Cillian Murphy's uncredited role? A setup for a sequel? If so, I'm down! I can't get enough of that guy! He's awesome!

For the most part, performances here are just alright. Bridges is easily the best one her, but that's because he has the most to work with. Playing two roles - hero and villain, no less - is no easy task, and Bridges does a very good job with it. If only everyone else was up to his level, but, I guess it's hard to beat an Oscar winner. The Dude abides!

Watch out man!

So, the story and acting aren't really up to snuff, but, hey, we expected that. The thing that's gonna draw the crowds to Tron: Legacy is the visuals, and, DAMN, they are good! This is, put simply, the best use of CGI to create a convincing foreign world since James Cameron flick about blue people. Everything in this movie looks so friggin' cool. And the action! Whooo, it's fantastic. The ideas of combat introduced in the first film were really innovative and interesting. Unfortunately, the lack of technology made it hard to fully realize them. Well, since we can now apparently reverse age people with a computer, a visually awesome light cycle battle is easy to pull off! And pull it off they do! The three main action scenes in this, a heated disc battle with multiple opponents, a tense light cycle battle, and a chaotic dogfight in the finale are among the best action scenes of the year. Indeed, the light cycle battle gives the hamster wheel hallway scene from Inception a run for it's money. 

But action isn't the only area where the effects shine! When shit isn't exploding around you and dudes aren't getting "drezzed" left and right, the smaller stuff comes out, and you can really take in the beauty of the world. The look of "neon on black" is really striking, and, since the entire is more or less that, we get our fill of it. The Grid is a technical marvel, looking like it was actually built. Same goes for all the machines that are put to use, including the iconic transport ships, the awesome looking light cycles, and a new and stylish flyer. The best use of the effects though, in my mind, is an enormous solar sailer that carries Kevin, Sam, and Quorra towards the portal. It's the quietest moment of the film, and it's breathtaking watching the huge transport gliding silently over the world. 

Yes, the technology used on Bridges to make him look younger still needs work, but it fits in with the world. Since everything around him is animated and built by computers, I bought that Clu would look a little fake. I don't know. Maybe I was just jaded by everything else. 

On a story and acting level, Tron: Legacy is no travesty, but it does leave something to be desired. On a technical level, it's one of the best movies of the year. Easily trumping most, if not all, 2010 films in terms of special effects, all story and performance qualms vanish when you watch it because it looks so damn pretty! Expect an influx of technical Oscars. This puppy certainly deserves it!

Also, the score from Daft Punk? Perfect! Just perfect!



Water and Electricity

Wednesday, December 22, 2010

Uh... so... yeah...

Sorry for the gap in content. I, foolishly, spilled some water on my computer and had to get it taken in to the Apple store in San Fran to get it fixed. They told me it would just take a day, but, here I am, three days later, and they still have not called me. I am actually out of town now, using a older computer that is decidedly lacking some of the things I need, like Photoshop and what not. Anyway, that's why you didn't get a Trailer Trash, or a Cinematic Captions, or an According to the Movies this week. Apologies. I'll be hitting the theaters in the coming days, seeing all that remains to be seen in this, the year of our... president, 2010. Keep in touch. I haven't given up on you yet!

Thumbs up!

Jab, Jab, Uppercut, KO!!

Sunday, December 19, 2010

Gotta love the inspirational sports flick, don't ya? I mean, they're just so uplifting, right? An underdog rises to the top with the help of his loved ones, deals with issues along the way, almost loses everything in the final hour, and then returns a new man and defeats the defending champ, who is a total dick. It's nice to know that this tried and true formula hasn't been done to death and that all these elements haven't at all become cliches.
...
Ha ha ha ha ha! Had you going there for a second, didn't I? No, for real though, it's actually kinda scary how religiously sports films follow the same formula! ATTENTION FILM MAKERS!!! YOUR FILM IS NOT GOING TO EVER MATCH ROCKY!!! STOP TRYING!!! I will say this though. If a film can take the cliches but do them well and layer them on top of other things like good acting and some cool stylistic choices, then I'll turn a blind eye. The Fighter is such a film. Put simply, The Fighter is one of the best examples of this genre since Rocky introduced it to us. Not only does it handle the tried and true story of an underdogs meteoric rise to the top, but it adds to it with a harrowing depiction of drug addiction and a family on the verge of exploding. That, and some of the coolest boxing scenes to come around in a long while. Oh, and some stupendous acting. You'll like this one, methinks.

INSPIRATION! DOWN YOUR THROAT! EAT IT UP! ENJOY!

The Fighter is based on the true story of the "Irish" Mickey Ward and his brother, Dicky Ecklund. Dicky is something of a legend in their home town of Lowell, Massachusetts, having knocked down Sugar Ray Leonard during his fighting days. Unfortunately, these days, he's a raging crack head. He's also Mickey's trainer, and his addiction is seriously affecting the work he's doing in that field, leading Mickey to lose an embarrassing amount of fights in a row. Eventually, Dicky is busted and sent to prison, and Mickey begins to train anew, with a new crew and new management. This works out for him, and soon he begins to enjoy an ascension in the ranks, eventually coming within reach of a title. As Mikey comes ever closer to winning that humongous belt, Dicky does everything he can to sober up and redeem himself so that he can help his brother achieve everything he couldn't.

So, yeah. Sounds quite a lot like Rocky, don't it? The underdog who's training isn't great in the beginning of the film? His realization that he's been holding back? His new training regimen, shown via montage? The victories? The championship match? It's a hell of a lot like Rocky. I'm going to forgive this film in that regard though, because it is a true story. All the fight shown here happened, so, if someone wants to make a film about a real life Rocky, let em'. What sets The Fighter apart from the rest of the "inspirational sports film" pack is that it isn't really about the sport. It's about the relationship between Mickey and Dicky, and how the tolls of fighting and the tolls of addiction effect that bond. The boxing is just there as a foundation. It's the other levels that make The Fighter what it is. 

I take it back! Andrew Garfield isn't even gonna come close to touching a statue come Oscar night. The winner for Best Supporting Actor has been picked, and it's Christian Bale for his no-holds barred portrayal of Dicky. In order to effectively portray a addict, Bale lost all of his Batman weight, dropping back down to Machinist levels of thin. He looks the part, but looks can only take you so far. Thankfully, he nails the acting as well, perfecting all the mannerisms of crack head, from the twitches to the impulsive behavior, as well as giving us a damaged character who is just simply trying to do right by his brother, but keeps screwing up. It's a sad performance, and his slow redemption is actually more uplifting than the boxing matches. Bale has always taken his work extremely seriously (light guy tirade anyone?) and no where is that more apparent than in The Fighter. It's the most devastating of his performances so far, and he knocks it out of the park. 

The role of Mickey is filled by director David O' Russell's go-to guy, Mark Wahlberg. Again, I feel I must be a callous objectifier and single out Wahlberg's physicality in the film above all else. He took his role as this famous pugilist almost as seriously as DeNiro did for Raging Bull, training hard for many months before hand, and refusing to use stunt double during shooting. The result is that Wahlberg is the one throwing the punches and getting clouted in the face, and it only adds to an already raw performance from him. True, he's playing the role of inspirational hero, but he does it very well. Can't fault a guy if the role he's playing is older than he is if he plays it well.  

Amy Adams and Melissa Leo round out the main cast as Mickey's squeeze, Charlene, and Mickey and Dicky's mother, respectively. I'll be honest. I'm not as familiar with Adam's work as I should be. That being said, I am aware that a foul mouthed, tough broad is not the role she usually plays, so it's nice to see stepping out of her comfort zone. Leo is also excellent as Mickey's overbearing mother/manager. I gotta say. Mickey Ward must be the most patient mofo to ever walk this earth, because if I had a family that was that aggravating, I would've decked them a long time ago! Leo is very convincing as a mother trying to save one son while trying to help another succeed. It's a good job. 

Everyone here does a good job, but it's Bale that really stands out. He's simply marvelous!

Yeesh! That can't be healthy.

David O' Russell hasn't really enjoyed the fame that he should. Since Three Kings, all he's done is the largely ignored I Heart Huckabees. He returns to form with The Fighter, and reminds the world how a sports movie should be done. Most sports movies these days make it about the sport and seem to forget about the people playing them. O' Russell took the opposite approach with The Fighter, making the sport a little bit of an afterthought as opposed to the characters. This is further emphasized by his brilliant choice to film the fights with the exact same format that HBO and ESPN uses. It looks really cool, but it detaches us from the actual fights, since it looks exactly the same as every boxing match that you see on TV. This leaves us ready to focus on the characters and their relationships outside of the ring, and it works wonderfully. It's a shame though. O' Russell did so well with this movie, you'd think he'd be going to direct the next great thing. Unfortunately, no. No, his career is about to run smack into a brick wall, since O' Russell is now officially the helmer of the video game adaptation, Uncharted: Drake's Fortune. Not good! Not good at all!

The Fighter is the sports film that all the recent ones strive to be, but can't. It puts the actual sport on the backburner, leaving it there as spring board for more interesting things. The depiction of crack addiction in this film is raw and harrowing; the presentation of the brother's relationship is relatable and real. The fights look cool, and the acting, especially from Bale, is aces! This one beats the competition. No question. TKO!

Just one question before I leave you. Why do people who live in Massachusetts curse so friggin' much? Just a thought... 



On Point

Saturday, December 18, 2010

Friggin' ballerinas, man! I mean, I've danced on stage before, and yeah, it's hard, especially given that when I dance on stage, it usually involves some combination of acting and singing. But, damn! I can't even begin to fathom how friggin' brutal the life of a professional dancer must be. The physical toll must be tremendous! And if Black Swan, the brilliant new film from Darren Aronofsky, is anything to go on, the psychological toll must be out of this world. Black Swan is, plain a simple, a great film, masterfully crafted, wonderfully atmospheric, with a slew of pitch perfect performances. It's also, quite possibly, one of the most scary and disturbing movies I've ever seen. This is one film that's gonna be sticking with me for awhile and one that easily finds a spot among my favorite films of the year. Black Swan is tough, but it is fantastic!

Sinew

Black Swan concerns Nina, a ballerina for a prestigious and popular company in New York City. Upon winning the lead role in her company's production of Swan Lake, Nina is thrilled. As she begins to practice though, she discovers that performing the role of the Swan Queen requires her to delve into two sides of herself, the tame and innocent side fro the White Swan, and the seductive and dangerous side for the Black Swan. Nina, who is obsessed with perfecting the moves of the role, is a perfect match for the White Swan. She concentrates too hard on the moves, which makes nailing the Black Swan, a role that requires her to lose herself in the dance and seduce the audience, rather difficult. As Nina struggles with this, she enters into a bizarre and twisted friendship with Lily, a new addition to the company. If Nina is the White Swan, then Lily is the Black Swan, as she is rambunctious, impulsive, and seductive. Through her, a darker side of Nina begins to show itself. Unfortunately, the stress of the role begins to get to her, and soon, Nina is losing her grip on reality as opening night approaches.

The plot of Black Swan is damn fine, and told very well. It moves at just the right pace so that Nina's transformation is completely believable and genuine. We see her as the meek, timid girl for a good forty-five minutes, and then hints of her character change begin to show through. An angry snap here or a shove there until, by the third act, the sweet little girl we knew from the beginning of the film is nothing more than a memory. This is in addition to the depiction of the deterioration of Nina's fragile psyche. What starts as some mysterious scratches on her back turns into extremely disturbing and violent hallucinations and increasingly aggressive behavior. It's consistently haunting throughout the whole film, until the final 30 minutes, when the whole thing just goes fucking nuts! On top of all this, the film takes time to explore the other characters, from the impulsive Lily, to the director the company, Thomas, to the former star of the company, Beth, to Nina's overbearing mother, Erica. That sounds like a lot for one film, and would probably make for a hefty run time, but that's where you'd be wrong. Black Swan clocks in at a healthy hour forty-five. It's the mark of a well made and well told film when the runtime is short, but the substance is substantial. It's good stuff.

Natalie Portman's getting an Oscar! There! I said it! Everyone else is too, so quit acting so surprised and pick your jaw up off the floor. As Nina, she is simply astounding, delving down deep into this girl's tragic story and getting at all the juicy little tidbits! She is almost unbearably haunting as this tortured soul, going to the farthest extremes with the character. White Swan Nina is so pathetic and an incredible push over, constantly apologizing for not being able to do something. Black Swan Nina is simply terrifying, acting very aggressively, both violently and sexually, behaving in such a callous and seductive manner that anyone seeing her in the street would turn tail and run away. It's easily the best performance of Portman's career, and, since her career has been marked by some savagely brilliant work (Closer, Garden State), that is saying quite a lot! She will be walking up to that stage on Oscar night. That, you can count on!

Mila Kunis takes on a much heavier role than we have become accustomed to seeing her play as Lily. She does a great job capitalizing on everything that Nina is not, impulsive, quirky, and sexy. Though Kunis' trademark flair for comedy does get used at some points, it's her dramatic chops that are really used here, and it's a pleasure to watch. We don't get see Kunis flex her serious muscles all that often, which is shame, because she is great. We never really know where her character is coming from. Is she deliberately screwing with Nina in an attempt to wrest the role of the Swan Queen from her grasp, or is Nina just imagining the whole thing as a symptom of her rapidly declining mental health. We are never really sure, and Kunis masterfully stays true to this ambiguity.

Barbara Hershey is delightful as Nina's overprotective mother, who is clearly trying to live vicariously through her daughter as she becomes a star. Vincent Cassel is a quiet and alluring force as Thomas, the director of the company who will go to great lengths to see that Nina perfects the role, even going so far as to seduce her in rehearsal to show her what it's like. Winona Ryder adds to her ever growing list of awesome cameos as Beth, the scorned, former Prima Ballerina of the company.

As with his last film, Darren Aronofsky has gotten a group of really gifted actors to turn in outstanding performances, but it's his lead that steals the show! Like Mikey Rourke in The Wrestler, Natalie Portman has taken on the role of her career, and knocked it out the park, and then some. Rourke was robbed of that Oscar. The same thing will not happen to Portman. She's practically holding it already.

Deliciously Creepy

Aronofsky is no stranger to creeping us the fuck out and disturbing us with some truly twisted imagery. He showed us that much with Requiem for a Dream, but that drug induced freak-fest has nothing on Black Swan. This movie isn't just disturbing. It's positively twisted, absolutely terrifying in it's portrayal of one person slowly losing it. There are countless scenes here that will leave you clutching the arm rest in petrified fear. And the last 30 minutes? I, for one, was curled in my seat like a child because the stuff on screen was literally too much. There's one sequence where Nina finally loses it that is one of the most shockingly horrific things I've ever seen! Self-mutiliation? Talking pictures? A woman painfully turning into a bird? I'm getting chills just thinking about it!

And, shun me if will, but Aronofsky somehow manages to make all this crap sexy. That's right. Those are some arousing freak outs right there! The much touted about sex scene between Portman and Kunis is friggin' hot, and I'm not just saying that because I'm a hormonal teenage boy and getting to see two beautiful women make the beast with two backs turns me on. I mean, it does, but that's not the point. We're at a point in entertainment when a lesbian sex scene is nothing new, so it can now be used as a character moment. And unlike the horrendous Chloe earlier this year, the scene in Black Swan is just that. It's the epitome of Nina's sexual awakening, and is handled beautifully, focusing on her reactions rather than the more titillating aspects of the scene. Indeed, the sexual content in this flick is really explicit, but there is not a shred of nudity. All this sex is one of the many instigators of Nina's descent into madness. Aronofsky doesn't need the unclothed forms of the feminine figure to get the point across. He's too good for that.

A brief note on the dancing. Though a stand-in is doing all the hyper advanced stuff, for the most part, it is Portman and Kunis performing all the moves. They trained for six months before starting work on the film, practicing in between takes on their other projects. The goal was to make them look as close to real ballerinas as possible, right down to the body type. Consensus? Aced! Both Portman and Kunis look like naturals when dancing. Their bodies have been toned down to the level of the highest professional. There is barely a sliver of fat on those sinewy arms that Portman flaps about. It's just another example of how much time and care went into the making of the film.

Also. Point shoes? Whoever invented those is going straight to hell! What a sadist!

Black Swan is going to be stuck in my mind for a while! Films this haunting, disturbing, and terrifying have a habit of tormenting me as I try to sleep, but it's for those very reasons that I love the film so much! It's simply brilliant. There are no frills, no extra bullshit! Aronofsky set out to make a psychological thriller of the highest class. He succeeded admirably! Not only did a create a masterful portrait of a slow descent into madness, he managed to secure a wonderful crew of on point actors, with an astounding performance from Natalie Portman that is far superior to anything anyone else has turned in this year! Black Swan is fucking nuts! I love it to death!

On A Whole New Level!!!

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Roots of unity and cyclotomic fields

Friday, December 17, 2010

In preparation for many good things that are to come, we need to have a talk about another important class of field extensions of ℚ – the cyclotomic extensions. (Check here for a list of previous articles on algebraic number theory.)

A cyclotomic field in general is a field that is an extension of some base field formed by adjoining all the roots of the polynomial f(x) = xn-1=0 for some specific positive n∈ℤ to the base field. Usually, though not always, this will mean roots that lie in some large field in which f(x) splits completely and that contains ℚ as the base field, such as ℂ, the complex numbers. f(x) is known as the nth cyclotomic polynomial. Mostly the same theory applies if the base field is a finite algebraic extension of ℚ, but we'll use ℚ as the base field for simplicity.

Since f(1)=0, x-1 is one factor of f(x), and f(x)/(x-1) = xn-1 + … + x + 1 ∈ ℤ[x], with all coefficients equal to 1. If n is even, -1 is also a root of f(x). However, all other roots of f(x) in ℂ are complex numbers that are not in ℝ. Some of these roots, known as the "primitive" nth roots of unity – denoted by ζn (or just ζ if the context is clear) – have the property that all other roots are a power ζk for some integer k, 1≤k<n. So the smallest subfield of ℂ that contains ℚ and all roots of f(x) is ℚ(&zeta), known as the nth cyclotomic field.

It is possible to express all the roots of f(x) in the form e2πi/n, where ez is the complex-valued exponential function, which can be defined in various ways. The most straightforward way is in terms of an infinite series, ez = Σ0≤n<∞zn/n!. The exponential function ez can also be defined as the solution of the differential equation dF(z)/dz = F(z) with initial value F(1)=e, the base of the natural logarithms. So there is the rather unusual circumstance that the roots of an algebraic equation can be expressed as special values of a transcendental function. Mathematicians long hoped that other important examples like this could be found (a problem sometimes referred to as "Kronecker's Jugendtraum", a special case of Hilbert's twelfth problem), but that hope has mostly not been fulfilled.

The most well-known nontrivial root of unity is the fourth root, i=&radic(-1), which satisfies x4-1 = (x2+1)(x2-1) = 0.

All complex roots of unity have absolute value 1, i. e. |ζ|=1, since |ζ| is a positive real number such that |ζ|n=1. The set of all complex numbers with |z|=1 is simply the unit circle in the complex plane, since if z=x+iy, then |z|2 = x2+y2 = 1. (Note that the linguistic root of words like "circle", "cyclic", and "cyclotomic" is the Greek κύκλος (kuklos).) Since e = sin(θ) + i⋅cos(θ) for any θ, with θ=2πk/n the real and imaginary parts of a general nth root of unity ζ=e2πi⋅k/n are just Re(ζ)=sin(2πk/n) and Im(ζ)=i⋅cos(2πk/n).

There are many reasons why cyclotomic fields are important, and we'll eventually discuss a number of them. One simple reason is that roots of algebraic equations can sometimes be expressed in terms of real-valued roots (such as cube roots, d1/3 for some d), and roots of unity. See, for example, this article, where we discussed the Galois group of the splitting field of f(x)=x3-2.

The set of all complex nth roots of unity forms a group under multiplication, denoted by μn. This group is cyclic, of order n, generated by any primitive nth roots of unity. (Any finite subgroup of the multiplicative group of a field is cyclic.) As such, it is isomorphic to the additive group ℤn = ℤ/nℤ, the group of integers modulo n. Because of this, many of the group properties of μn are just restatements of number theoretic properties of ℤn. For instance, each element of order n in μn is a generator of the whole group – one of the primitive nth roots of unity. Since μn⊆ℚ(ζn), adjoining all of μn gives the same extension ℚ(ζn) = ℚ(μn).

Now, ℤ/nℤ is a ring, and its elements that are not divisors of zero are invertible, i. e they are units of the ring. They form a group under ring multiplication, which in this case is written as (ℤ/nℤ)× (sometimes Un for short). An integer m is invertible in ℤ/nℤ if and only if it is prime to n, i. e. (m,n)=1 (because of the Euclidean algorithm). The number of such distinct integers modulo n is a function of n, written φ(n). This number is important enough to have its own symbol, because it was studied by Euler as fundamental to the arithmetic of ℤ/nℤ. Thus φ(n) is also the order of the group (ℤ/nℤ)×.

Let ζ=e(2πi)m/n, for 0≤m<n, be an element of μn. The correspondence m↔e(2πi)m/n establishes a group isomorphism between the additive cyclic group ℤ/nℤ and the multiplicative group μn. Modulo n, m generates ℤ/nℤ additively if and only if (m,n)=1, which is if and only if the corresponding ζ generates μn. So the number of generators of μn – which is the number of primitive nth roots of unity – is the same as the order of (ℤ/nℤ)×, i. e. φ(n).

One has to be careful, because the multiplicative structure of μn parallels the additive structure of ℤ/nℤ, not the multiplicative structure of (ℤ/nℤ)×. (Because if ζM and ζN are typical elements of μn then ζM×ζNM+N.) Hence even though there are φ(n) generators of μn, these generators do not form a group by themselves (a product of generators isn't in general a generator), so the set of them isn't isomorphic to (ℤ/nℤ)×, even though the latter also has φ(n) elements. Give this a little thought if it seems confusing.

Moreover, the group (ℤ/nℤ)× is not necessarily cyclic. It is cyclic if n is 1, 2, 4, pe, or 2pe for odd prime p, but not otherwise. Confusingly, if the group does happen to be cyclic then integers modulo n that generate the whole group are called "primitive roots" for the integer n. If (ℤ/nℤ)× happens to be cyclic, then only those m∈(ℤ/nℤ)× having order φ(n) are "primitive roots" that generate the group, while all m∈(ℤ/nℤ)× have the property that if ζ∈μn has order n and generates the latter group, then so does ζm, as we showed above. Got that straight, now? This needs to be understood when working in detail with roots of unity.

Another reason for the importance of cyclotomic fields is that the Galois group of the extension [ℚ(ζn):ℚ] is especially easy to describe. Indeed, it is isomorphic to the group of order φ(n) we've just discussed: (ℤ/nℤ)×. There's a little work in proving this isomorphism, but let's first note what it implies. Let G=G(ℚ(ζ)/ℚ) be the Galois group. It is an abelian group of order φ(n) since it's isomorphic to (ℤ/nℤ)×. Further, any subgroup of G′ of G is abelian and by Galois theory determines an abelian extension (i. e., an extension that is Galois with an abelian Galois group) of ℚ as the fixed field of G′. Conversely, it can be shown (not easily) that every abelian extension of ℚ is contained in some cyclotomic field. (This is the Kronecker-Weber theorem.)

Half of the proof of the isomorphism is easy. Pick one generator ζ of μn, i. e. a primitive nth root of unity. We'll see that it doesn't matter which of the φ(n) possibilities we use. Suppose σ∈G is an automorphism in the Galois group. Since σ is an automorphism and ζ generates the field extension, all we need to know is how σ acts on ζ. Since σ is an automorphism, σ(ζ) has the same order as ζ, so it's also a primitive nth root of unity. Therefore &sigma(ζ) = ζm for some m, 1≤m<n. As we saw above, m is uniquely determined and has to be a unit of ℤ/nℤ, with (m,n)=1, in order for ζm to be, like ζ, a generator of the cyclic multiplicative group μn. Hence m∈(ℤ/nℤ)×. Call this map from G to (ℤ/nℤ)× j, so that σ(ζ)=ζj(σ). To see that it's a group homomorphism, suppose σ12∈G, with j(σ1)=r, j(σ2)=s. Then σ21(ζ)) = σ2r) = (ζs)r = ζsr, hence j(σ2σ1) = j(σ2)j(σ1). j is clearly injective since j(σ)=1 means σ(ζ)=ζ, so σ is the identity element of G. Finally, to see that j doesn't depend on the choice of primitive nth root of unity, suppose ζm with m∈(ℤ/nℤ)× is another one. Then σ(ζm) = σ(ζ)m = (ζj(σ))m = (ζm)j(σ).

Thus G is isomorphic to a subgroup of (ℤ/nℤ)×. That's enough to show G is abelian, so the extension ℚ(ζ)/ℚ is abelian. To complete the proof of an isomorphism G≅(ℤ/nℤ)× we would need to show that the injective homomorphism j is also surjective, i. e. every m∈(ℤ/nℤ)× determines some σ∈G such that m=j(σ). We can certainly define a function from ℚ(ζ) to ℚ(ζ) by σ(ζ)=ζm for a generator ζ of the field ℚ(ζ). One might naively think that's enough, but the problem is that one has to show that σ is a field automorphism of ℚ(ζ).

The map σ defined that way certainly permutes the nth roots of unity in μn, the roots of the polynomial f(x)=xn-1. However, not all permutations of elements of μn, of which there are n!, yield automorphisms of ℚ(ζ). The problem here is that if z(x) is the minimal polynomial of some ζ, i. e. the irreducible polynomial of smallest degree in ℤ[x] such that z(ζ)=0, then by Galois theory the order |G| of the Galois group G is the degree of the field extension, which is the degree of z(x). Since G is isomorphic to a subgroup of the group (ℤ/nℤ)×, and the latter has order φ(n), all we know is that |G| divides φ(n). It could be that other primitive nth roots of unity have minimal polynomials in ℤ[x] that are not the same as z(x), though they have the same degree |G|. For σ to be an automorphism, σ(ζ) needs to have the same minimal polynomial as ζ, and we don't know that immediately from the relation σ(ζ)=ζm.

We will defer discussion of the rest of the proof that G(ℚ(μn)/ℚ)≅(ℤ/nℤ)× for the next installment, since some new and important concepts will be introduced.

Smooth Flight! Minimal Turbulence!

Thursday, December 16, 2010

Well, color me dumbfounded, because I am a douche! When the trailers for Legend of the Guardians: The Owls of Ga'Hoole came out, I scoffed. Why the hell should I care about some kiddy movie about birds. Even when I heard that Zack Snyder was directing it, I still scoffed. I viewed it as Snyder selling out, doing something lame and not worth his time to rake in a few extra dollars in between making his next exploration of awesome!!! And then I saw the thing. Wow! I certainly didn't expect that. Guardians is not the kiddy, overly dramatic film that was advertised. What it is is a surprisingly mature, dark and violent tale told with a truly marvelous aesthetic touch. Though the story and script leaves something to be desired, and some of the character interactions and motivations are hopelessly contrived, Guardians is a film that should not be overlooked. It's not as kiddy as was advertised. It's a technical marvel. It's just good, plain and simple!

Aesthetic Pleasures

Legend of the Guardians takes place in a world where owls are the main population. You know, one of those fantasy worlds. Anyway,  like any good society, these owls have their legends and fables, the most popular of which concerns the Owls of Ga'Hoole, defenders of all this is good and righteous and, well, kiddy, in the world, and how they defeated the evil Metalbeak in an epic battle. One owl, Soren, loves these stories, and firmly believes that they are true, and not just fables. This belief is not shared by his brother, Kludd, which leads to their relationship being a bit strained. One day, the two of them fall out of their nest and are promptly snatched by a group of owls, who take them to the domain of Metalbeak, who is indeed real. See, the tyrant is attempting one of those classic, take over the world type moves, and to do this, he is kidnapping and brainwashing children to build his army of Pure Ones. Well, Soren isn't having any of that, and escapes with the help of a new friend, Gylfie, to journey across the world to find and alert the Guardians. Kludd, on the other hand, decides to stay behind and joins Metalbeak. Craziness ensues.

Based off the book series Guardians of Ga'Hoole, Legend covers the first three books in the series. As far as fantasy stories with magic, good vs. evil, and epicness go, this one isn't really up to snuff. The bad guys are just bad. The good guys are just good. You get the cliched character archetypes, the righteous hero, the treacherous main villain, the valiant love interest, the kooky couple of allies, the traitor who plays his card in the end. You know. Typical stuff.

That being said, it's not terrible either. The plot moves at a good clip, and you can easily root for the good guys and boo and hiss at the bad guys. As far as kids films go, this one is surprisingly dark. Don't be surprised if the little ones are scared by this one. The film isn't shy about showing intense combat, multiple deaths, and surprisingly disturbing forms of brainwash and torture. It earns that PG rating, and almost begins to push towards PG-13. You wouldn't expect that from a film proudly proclaiming that it's made from the same studio that made Happy Feet, now would you?

Since it's an animated film, all the acting here is of the voice kind, which puts me in a difficult situation. I just can't right about voice acting. I can't do it! I mean, I can talk about the acting as a whole, but singling out particular people just seems stupid. Anyway...

Amazingly, the voice acting in this is top notch. Like, it's at Pixar levels of good! Every actor here imbues their character with plenty of emotional substance, giving the story a healthy amount of extra weight. Standouts include David Wenham as the wired and crazy Digger, Geoffrey Rush as the wise and weathered Ezylryb, Ryan Kwanten as the treacherous Kludd, and Helen Mirren as the deceptive and seductive Nyra. Jim Sturgess as main hero Soren is alright, but he sounds like too much of a pussy for the whole movie. He sounds like he's on the verge of tears pretty much all the time. He's the only real stain on an otherwise spotless bunch of voice actors.

That's some freakish water!
The real star here though is director Zack Snyder, whose trademark flare for visuals is presented here in full effect! Legend of the Guardians is a gorgeous film, a true visual wonder! Snyder's films have always been intoxicating to look at, but, since he wasn't hindered by cameras and could frame every shot on a computer, he takes it to a whole new level here. There are countless breathtaking images to see here, starting with the very first shot of the whole movie. And unlike most films that shoot their load early on, Guardians keeps upping the visual ante as it progresses. The scene that sees Soren and co learn fly through a storm is one of the most visually astounding things ever put on film!

Snyder's trademark, "speed up, slow down" action scenes are also used to great effect here. Though Legend of the Guardians isn't as violent as 300 or Watchmen, there are still plenty of sequences that get the blood pumping. Though the final action scene is kind of blah by comparison to the ones that came before it, it is still pretty cool looking, and very intense. There's quite a lot at stake in Legend of the Guardians, and Snyder's visuals do a good job of emphasizing this, often pulling back to show the full scope of goings on and what not. It's a good job. 

I have some crow to look forward to for dinner tonight. I shouldn't have written off the film when I first heard about it. When looking at the final product, you will see that it is actually well worth your time. It's not the best animated film ever. It's not even Zack Snyder's best film. What it is is a technical wonder for a more mature audience. Though the scripting prevents it from achieving true greatness, the visuals more than make up for it. Though they may not be the most loquacious of birds, these owls are about as graceful as they come!



 

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