query: What is the largest artery in the human body?

title: Largest body part

text: The longest bone in the human body is the femur.[20] The largest artery is the aorta[21] and the largest vein is the inferior vena cava.[22] The largest internal organ is the liver.[23] It is also the heaviest organ, with an average of 1.6 kilograms (3.5 pounds).[24] The largest external organ, which is also the largest organ in general, is the skin.[25] The longest muscle is the sartorius muscle in the thigh.[26] The longest single nerve is the sciatic nerve in the thigh, counting its branches and axons.

title: Fibromuscular dysplasia

text: FMD can be found in almost every artery in the human body, but most often affects the carotid, vertebral, renal arteries and even those that supply the intestines, arms, and the legs. Patients may present with FMD in multiple vessels. FMD has been pathologically categorized into three types of classifications: Multi-focal, focal, and adventitial; referring to the particular layer of arterial wall being affected.

title: Greater pancreatic artery

text: In human anatomy, the greater pancreatic artery (great pancreatic artery or arteria pancreatica magna), is the largest artery that supplies the pancreas. It arises from the splenic artery.

title: Umbilical artery

text: The umbilical arteries are one of two arteries in the human body, that carry deoxygenated blood, the other being the pulmonary arteries.

title: Blood squirt

text: Blood squirt (blood spurt, blood spray, blood gush, or blood jet) is the effect when an artery, a blood vessel in the human body (or other organism's body) is cut. Blood pressure causes the blood to bleed out at a rapid, intermittent rate, in a spray or jet, coinciding with the beating of the heart, rather than the slower, but steady flow of venous bleeding. Also known as arterial bleeding, arterial spurting, or arterial gushing, the amount of blood loss can be copious, occur very rapidly, and can lead to death.

title: Composition of the human body

text: Body composition may be analyzed in terms of molecular type e.g., water, protein, connective tissue, fats (or lipids), hydroxylapatite (in bones), carbohydrates (such as glycogen and glucose) and DNA. In terms of tissue type, the body may be analyzed into water, fat, muscle, bone, etc. In terms of cell type, the body contains hundreds of different types of cells, but notably, the largest number of cells contained in a human body (though not the largest mass of cells) are not human cells, but bacteria residing in the normal human gastrointestinal tract.

title: List of arteries of the human body

text: This is a list of arteries of the human body.

title: Subclavian artery

text: In human anatomy, the subclavian arteries are paired major arteries of the upper thorax, below the clavicle. They receive blood from the aortic arch. The left subclavian artery supplies blood to the left arm and the right subclavian artery supplies blood to the right arm, with some branches supplying the head and thorax. On the left side of the body, the subclavian comes directly off the aortic arch, while on the right side it arises from the relatively short brachiocephalic artery when it bifurcates into the subclavian and the right common carotid artery.

title: Stapedial branch of posterior auricular artery

text: In humans, the stapedial artery is normally present in the fetus where it connects what is to become the external and internal carotid arteries. Part of the carotid artery system, it originates from the dorsal branch of aortic arch. Its superior supraorbital branch becomes the middle meningeal artery, while its infraorbital and mandibular branches fuses with the external carotid artery and later become the internal maxillary artery. Its trunk atrophies and is replaced by branches from the external carotid artery.

title: Submental artery

text: The submental artery is the largest of the cervical branches of the facial artery, given off just as that vessel leaves the submandibular gland: it runs forward upon the mylohyoid, just below the body of the mandible, and beneath the digastric muscle.

title: Descending thoracic aorta

text: The aorta is an artery that conveys oxygenated blood from the heart to other parts of the body. It is one of the largest arteries in the body. The aorta gives off several paired branches as it descends. In descending order, these include the

title: Spinal cord

text: The major contribution to the arterial blood supply of the spinal cord below the cervical region comes from the radially arranged posterior and anterior radicular arteries, which run into the spinal cord alongside the dorsal and ventral nerve roots, but with one exception do not connect directly with any of the three longitudinal arteries. These intercostal and lumbar radicular arteries arise from the aorta, provide major anastomoses and supplement the blood flow to the spinal cord. In humans the largest of the anterior radicular arteries is known as the artery of Adamkiewicz, or anterior radicularis magna (ARM) artery, which usually arises between L1 and L2, but can arise anywhere from T9 to L5. Impaired blood flow through these critical radicular arteries, especially during surgical procedures that involve abrupt disruption of blood flow through the aorta for example during aortic aneursym repair, can result in spinal cord infarction and paraplegia.

title: Pancreas

text: The pancreas receives blood from branches of both the coeliac artery and superior mesenteric artery. The splenic artery runs along the top margin of the pancreas, and supplies the neck, body and tail of the pancreas through its pancreatic branches, the largest of which is called the greater pancreatic artery. The superior and inferior pancreaticoduodenal arteries run along the anterior and posterior surfaces of the head of the pancreas at its border with the duodenum. These supply the head of the pancreas.

title: Artery

text: The anatomy of arteries can be separated into gross anatomy, at the macroscopic level, and microanatomy, which must be studied with a microscope. The arterial system of the human body is divided into systemic arteries, carrying blood from the heart to the whole body, and pulmonary arteries, carrying deoxygenated blood from the heart to the lungs.

title: Arteriovenous malformation

text: Arteries and veins are part of the human cardiovascular system. Arteries carry blood away from the heart to the lungs or the rest of the body, where the blood passes through capillaries, and veins return the blood to the heart. An AVM interferes with this process by forming a direct connection of the arteries and veins. AVMs can cause intense pain and lead to serious medical problems. Although AVMs are often associated with the brain and spinal cord, they can develop in any part of the body.

text: Continuous noninvasive arterial pressure (CNAP) is the method of measuring arterial blood pressure in real-time without any interruptions (continuously) and without cannulating the human body (noninvasive).

title: Intracranial dolichoectasias

text: The term dolichoectasia means elongation and distension. It is used to characterize arteries throughout the human body which have shown significant deterioration of their tunica intima (and occasionally the tunica media), weakening the vessel walls and causing the artery to elongate and distend.

title: Human body

text: Blood vessels carry blood throughout the body, which moves because of the beating of the heart. Venules and veins collect blood low in oxygen from tissues throughout the body. These collect in progressively larger veins until they reach the body's two largest veins, the superior and inferior vena cava, which drain blood into the right side of the heart. From here, the blood is pumped into the lungs where it receives oxygen and drains back into the left side of the heart. From here, it is pumped into the body's largest artery, the aorta, and then progressively smaller arteries and arterioles until it reaches tissue. Here blood passes from small arteries into capillaries, then small veins and the process begins again. Blood carries oxygen, waste products, and hormones from one place in the body to another. Blood is filtered at the kidneys and liver.

title: Aorta

text: The aorta ( ) is the main artery in the human body, originating from the left ventricle of the heart and extending down to the abdomen, where it splits into two smaller arteries (the common iliac arteries). The aorta distributes oxygenated blood to all parts of the body through the systemic circulation.

title: Superior pancreaticoduodenal artery

text: At 42 letters, the posterior superior pancreaticoduodenal artery is also the artery with the longest name in the human body.

title: Intercostal arteries

text: The intercostal arteries are a group of arteries that supply the area between the ribs ("costae"), called the intercostal space. The highest intercostal artery (supreme intercostal artery or superior intercostal artery) is an artery in the human body that usually gives rise to the first and second posterior intercostal arteries, which supply blood to their corresponding intercostal space. It usually arises from the costocervical trunk, which is a branch of the subclavian artery. Some anatomists may contend that there is no supreme intercostal artery, only a supreme intercostal vein.

title: Owl

text: Owls can rotate their heads and necks as much as 270°. Owls have 14 neck vertebrae compared to seven in humans, which makes their necks more flexible. They also have adaptations to their circulatory systems, permitting rotation without cutting off blood to the brain: the foramina in their vertebrae through which the vertebral arteries pass are about 10 times the diameter of the artery, instead of about the same size as the artery as in humans; the vertebral arteries enter the cervical vertebrae higher than in other birds, giving the vessels some slack, and the carotid arteries unite in a very large anastomosis or junction, the largest of any bird's, preventing blood supply from being cut off while they rotate their necks. Other anastomoses between the carotid and vertebral arteries support this effect.

title: Prashna Upanishad

text: Life enters the body, states the Prashna Upanishad, by the act of mind. It governs the body by delegating work to other organs, sage Pippalada continues in verse 3.4, each specialized to do its own work independent of the other powers, just like a king commands his ministers to govern functions in the villages in his kingdom. The Upanishad then enumerates a theory of human body that is found in older Vedic literature, such as the Brihadaranyaka Upanishad hymn II.1.19. It asserts, for example, that human body has a heart as the principal organ of soul, from where arise 101 major arteries, each major artery divides into a hundred times, which in turn subdivide into 72,000 smaller arteries, giving a total of 727,210,201 small and large arteries, and that these arteries diffuse air throughout the body. It is this life-breath which interfaces Self to all organs and life in human body, states the Upanishad.

title: Elastic artery

text: An elastic artery (conducting artery or conduit artery) is an artery with a large number of collagen and elastin filaments in the tunica media, which gives it the ability to stretch in response to each pulse. This elasticity also gives rise to the Windkessel effect, which helps to maintain a relatively constant pressure in the arteries despite the pulsating nature of the blood flow. Elastic arteries include the largest arteries in the body, those closest to the heart. They give rise to medium-sized vessels known as distributing arteries (or "muscular arteries").

title: Iatrophysics

text: William Harvey postulated blood flow as a closed, continuous loop that run throughout body that contained a certain quantity of blood. To test his claim, Harvey dissected human corpses and animals and, based on his anatomical findings, devised a simple demonstration of how arteries and veins continuously carried blood throughout the body. Taking advantage of the fact that arteries and veins were at different depths below the skin, he tied a person's arm and had them squeeze a bar to shunt blood from arteries to veins, indicating that blood somehow traveled along arteries and into veins. His claim was elucidated by Malphigi's discovery of capillaries and how they were interconnected with arteries and veins.

title: Talus bone

text: The second largest of the tarsal bones, it is also one of the bones in the human body with the highest percentage of its surface area covered by articular cartilage. Additionally, it is also unusual in that it has a retrograde blood supply, i.e. arterial blood enters the bone at the distal end.

title: Artery of Adamkiewicz

text: In human anatomy, the artery of Adamkiewicz (also arteria radicularis magna) is the largest anterior segmental medullary artery. It typically arises from a left posterior intercostal artery at the level of the 9th to 12th intercostal artery, which branches from the aorta, and supplies the lower two thirds of the spinal cord via the anterior spinal artery.

title: Blood

text: Blood is circulated around the body through blood vessels by the pumping action of the heart. In humans, blood is pumped from the strong left ventricle of the heart through arteries to peripheral tissues and returns to the right atrium of the heart through veins. It then enters the right ventricle and is pumped through the pulmonary artery to the lungs and returns to the left atrium through the pulmonary veins. Blood then enters the left ventricle to be circulated again. Arterial blood carries oxygen from inhaled air to all of the cells of the body, and venous blood carries carbon dioxide, a waste product of metabolism by cells, to the lungs to be exhaled. However, one exception includes pulmonary arteries, which contain the most deoxygenated blood in the body, while the pulmonary veins contain oxygenated blood.

title: Pancreas

text: Anatomically, the pancreas is divided into the head of pancreas, the neck of pancreas, the body of pancreas, and the tail of pancreas. The "head" is surrounded by the duodenum in its concavity. The head surrounds two blood vessels, the superior mesenteric artery and vein. From the back of the head emerges a small uncinate process, which extends to the back of the superior mesenteric vein and ends at the superior mesenteric artery. The "neck" is about long and lies between the head and the body and in front of the superior mesenteric artery and vein. Its front upper surface supports the pylorus (the base) of the stomach. The neck arises from the left upper part of the front of the head. It is directed first upward and forward, and then upward and to the left to join the body; it is somewhat flattened from above downward and backward. On the right it is grooved by the gastroduodenal artery. The "body" is the largest part of the pancreas and lies behind the pylorus, at the same level as the transpyloric plane. The "tail" ends by abutting the spleen.

title: Brahma Upanishad

text: The Chapter 1 uses many similes using nature to describe how the soul and the human body interact. The Brahman (Atman) leads all these gods within the human body, and they follow him, asserts the text, in a way similar to bees and queen-bee. They do and focus on what the Atman wants. He connects a web with them, and withdraws within itself like a spider. In a similar way, Prana spreads out into arteries in the temple that is human body and also retracts when it wants to. When the human body goes into deep sleep, the Prana retires, just like falcon soars to the skies when he wants to and then goes to his nest to retire.


query: How many episodes of 30 Rock were there?

title: List of 30 Rock episodes

text: 30 Rock was a critical success, winning several major awards (including Primetime Emmy Awards for Outstanding Comedy Series in 2007, 2008, and 2009), and achieving the top ranking on many critics' year-end best of 2006 and 2007 lists.[1] On July 14, 2009, the series was nominated for 22 Primetime Emmy Awards, the most in a single year for a comedy series.[2] Despite the acclaim, the series struggled in the ratings throughout its run.[3] During the course of the series, 138 episodes aired over seven seasons.

title: MILF Island

text: Jeff Labrecque of "Entertainment Weekly" wrote that "after three months of strike-enforced exile, [this episode of] "30 Rock" generated the expectations and excitement normally reserved for a season premiere," also saying that "the show [is] in midseason form." Matt Webb Mitovich of "TV Guide" said that he would "be lying if [he] said this was a "super"-great episode." Webb Mitovich went on to say that "[He] did like, though — and thought they should have done a lot more with — the "parallels" between the "MILF" events and the strategizing staff." Bob Sassone of AOL's TV Squad thought that "It's almost as if this was the season opener" and that this episode was "ambitious." Michael N of Television Without Pity awarded the episode a grade of B. Robert Canning of IGN said that "the overall story arc was less than what we had anticipated from this critical favorite making its post-writers-strike return, the episode was still pretty damn funny." Canning also said that this was "an episode that was divided between a less-than-average story and many hilarious moments." Canning criticized how ""30 Rock" has never really given [Pete Hornberger and Scott Adsit] (the character or the actor) the showcase that he deserves."

title: Really Wild Animals

text: This episode contains four songs. The first one is about the cold temperatures and how the animals survive don't mind the cold (Cold, Cold, Cold); the second focuses on ways animals beat the cold (That's the Way We Do It Up Here); the third is all about penguins, and mentions 17 species by name (Penguins); and the fourth is a tribute to Antarctica (The Last Great Wilderness).In the first episode, Spin takes a look at dogs and tries to determine if there is a link between wild and tame dogs. The songs in this episode are about why dogs do what they do, and the loyalty that dogs possess. In the second episode, Spin examines cats and learns that no matter how tame, they're always wild. The songs in this episode are about cats having an attitude, and the second is about how they're wild inside. The second episode is the first time Alan O'Day does not appear in a "Really Wild Animals" episode.In the first episode, Spin examines dinosaurs such as Hadrosaurs, Struthiomimus, Tyrannosaurus Rex, and Dromaeosaurs, and tries to answer the many questions surrounding them, including how they became extinct, and if relatives still exist today such as alligators, snakes, Komodo dragons, lizards, ostriches, cassowarys, and other birds. The songs in this episode focus on dinosaurs (The Dinosaur Rap), and how their modern-day relatives still "rock our world" (We're Gonna Rock Your World). This episode is the first and only episode to not feature Alan O'Day or Janis Liebhart singing the songs. It's also the last episode where a guest singer was brought in. In the second episode, Spin looks at some of the world's creepiest creatures such as vampire bats, ants, flies, spiders, vultures, and dung beetles. This shows that no matter how scary they are, they all play an important part in the world. The first song is about how even though the creatures are creepy, there's something intriguing about them (Creepy Creatures; with animation by Jim & Vezna Tozzi), and the second song is about how important all of them are (Keep on Creeping On).In the first episode, Spin takes a look at primates of all types, including gibbons, langurs, bush babies, aye-ayes, capuchins, and baboons. He also explains that primates are our distant relatives. The first song is about all primates (Primates), and the second song focuses on gorillas, our "gentle brothers" (Gentle Brother). In the second episode, Spin takes a look at animal families inlcuding elephant families, African bullfrog families, ostrich families, octopus families, albatross families, & sea turtle families and shows that they're not all that different from human families after all. The songs in this episode are about how curious animal children are (I'm Curious), and how important families are (You're My Family).In the first episode, Spin takes a look at farm animals, such as turkeys, pigs, sheep, horses, and cows, and examines their wild relatives from all over the world. The first song is about all the animals on the farm, and the second focuses on the question of whether the viewer would be wild or free is he/she were a farm animal. In the second episode, Spin takes a look at how humans help endangered animals and how animals rescue people and shows that if people work together to help save animals, they'll prevent them from becoming extinct. The songs in this episode are about help being available for animals, and how if humans work together with animals, everybody will be safe at last.

title: 30 Rock (season 1)

text: Metacritic gave the season a Metascore—a weighted average based on the impression of 31 critical reviews—of 67%, signifying generally favorable reviews. The pilot episode, the first episode of the series, was generally well received; however, Marc D. Allan of "The Washington Post" said that "viewers who tuned in for the first month or more saw shows that weren't consistently funny", but after a few episodes "the writers discovered the core of the series—the push-pull between Fey's character, Liz Lemon, the harried head writer of "The Girlie Show", and Alec Baldwin's domineering network executive, Jack Donaghy—that "30 Rock" found its rhythm." Henry Goldblatt of "Entertainment Weekly", whilst reviewing the DVD release of the season, called "30 Rock" "[the 2006 –2007 television season's] finest sitcom". Goldblatt said that the episodes "Hard Ball" and "The Break Up" were "some of the strongest episodes", and awarded the first season an "A" grade.

title: How to Rock

text: How to Rock is an American teen sitcom that ran on Nickelodeon from February 4 to December 8, 2012. It stars Cymphonique Miller as Kacey Simon. The series is based on the 2011 book, "How to Rock Braces and Glasses" by Meg Haston published by Little, Brown Books For Young Readers and Alloy Entertainment. The series was officially green-lit on May 23, 2011 with a 20-episode production order, later increased to 26. Two of the ordered episodes were merged into a special episode so 25 episodes actually aired. The series began filming in August 2011. It is the first television sitcom to be produced by Alloy Entertainment. The first promo aired with "Merry Christmas, Drake & Josh" on December 10, 2011. It was confirmed by the series showrunner David M. Israel on August 26, 2012 that "How to Rock" would not be returning for a second season.

title: Hey, Baby, What's Wrong

text: Overall, the episode received mixed reviews from critics. Alan Sepinwall of "HitFix" opined that "Double-length comedy episodes are hard to pull off, particularly on a show as joke-driven as "30 Rock" is [...] and while there were funny moments because it's "30 Rock" and they're talented and it's almost always funny at some point, on the whole, [the episode] dragged." However, he complimented the performances of Mary Steenburgen and Alec Baldwin, and commended their United Nations plot line as "the kind of broad, clever satire "30 Rock" does so well." Nathan Rabin of "The A.V. Club" was particularly negative toward the episode, calling it a "disconcertingly dreadful Valentine’s Day episode of "30 Rock" that luxuriates in a bubble bath of sour misanthropy for an interminable hour." He awarded it a C-.

title: Cougars (30 Rock)

text: Scenes for this episodes were primarily filmed on October 5, 2007 at Silvercup Studios in Long Island City, Queens, New York. During the commercial breaks in this episode, commercials for American Express were aired featuring "30 Rock" cast members, who were in character. These commercials featured Kenneth giving out "Secret Rodney" Christmas gifts to various other "30 Rock" characters. The commercials were aired in order to deter viewers, who were viewing the episode with Digital Video Recorders, from fast-forwarding through the commercial break.

title: Season 4 (30 Rock)

text: IGN contributor Robert Canning reviewed the episode, giving it high marks saying it "proves yet again that "30 Rock" is worthy of the Best Comedy Emmy" and scored the episode a 9.2 out of 10. Sean Gandert of "Paste" magazine also reviewed "Season 4", saying it did "little to disappoint and in fact starts the show off on much stronger footing than the first couple episodes of season 3." Gandert said the episode "doesn't have quite as many truly classic moments as the show's peaks in seasons one and two" but that "it also never really hits a sour note." Bob Sassone of AOL's TV Squad enjoyed the economic crisis, cutbacks, and corporate bonuses theme in the episode, explaining that one of the reasons he loves the show "is because it can take something that's in the news and make it the plot of an episode without it feeling weird or forced." Sassone wrote that despite "Season 4" not being the best episode of "30 Rock" "it was still very good." "TV Guide"s Michael Anthony commented that Steve Buscemi is "so weird and yet does it in such an endearing way." Mary McNamara for the "Los Angeles Times" called Kenneth's strike one of the more hilarious strikes in history. "Entertainment Weekly"s Margaret Lyons noted that this episode of "30 Rock" had strong moments "and was a solid start to the new season". David Zurawik, a contributor for "The Baltimore Sun", liked the episode and enjoyed Buscemi's appearance, noting he "brings just the right mix of sleaze and smiles to the role." The "Chicago Sun-Times" Paige Wiser noted, "...tonight's season premiere, the hype is warranted: "30 Rock" is in all its glory." Kevin Aeh of "Time Out" magazine wrote that the best part from "Season 4" was Tracy's storyline after being told to reconnect with his roots.

title: Hulk Hogan's Rock 'n' Wrestling

text: "Hulk Hogan's Rock 'n' Wrestling" aired for 2 seasons beginning in 1985. Each episode was 30 minutes in length (including commercial breaks.) Some episodes contained one 30 minute title while other episodes contained two 15 minute titles. Over the two seasons, there were a total of 26 episodes with 39 titles.

title: 30 Rock

text: "30 Rock" episodes were produced in a single-camera setup (with the exception of the two live episodes that were taped in the multiple-camera setup), and were filmed in New York. The pilot episode premiered on October 11, 2006, and seven seasons followed. The series stars Fey with a supporting cast that includes Alec Baldwin, Tracy Morgan, Jane Krakowski, Jack McBrayer, Scott Adsit, Judah Friedlander, Katrina Bowden, Keith Powell, Lonny Ross, John Lutz, Kevin Brown, Grizz Chapman, and Maulik Pancholy.

title: The Bubble (30 Rock)

text: The episode received generally positive reviews, although most reviewers found some of the storylines more effective than others. Robert Canning of IGN wrote that "the main portion of this episode was quite funny, even if the ending lacked bite." Similarly, Margaret Lyons of "Entertainment Weekly" found that the episode was "another strong showing for "30 Rock" ... although Jenna's plotline was pretty tired." TV Squad's Bob Sassone was similarly positive about "The Bubble", calling it "another good episode." James Poniewozik of "Time", also responded favorably to the episode with the exception of the Jenna storyline, writing that it "was the first one in a long time that I enjoyed almost completely without reservation (Jenna's subplot was off, but it was barely a blip)." Television columnist Alan Sepinwall for "The Star-Ledger" disliked the bubble concept, reporting, "[t]he idea that attractive people get away with things that the more average-looking can't is an old joke, even on "30 Rock" (with Cerie)", but nonetheless enjoyed the "variations" of the joke saying they were "well-conceived". Sepinwall concluded, "this was one of the funniest "30 Rock"'s of the season. Zap2it's Rick Porter was also favorable to "The Bubble" in his recap, opining that "30 Rock" "gave us a very well-done episode". As with Lyons and Poniewozik, Porter felt Jenna's story "didn't really click", but liked the scene with Jenna and her stylist team firing off words to describe Jenna's beauty as "scathingly funny."

title: 30 Rock (season 1)

text: The pilot episode garnered 8.13 million viewers, finishing third in its timeslot of 8:00pm Eastern Standard Time. After three further low rated airings on the following three Wednesdays, including a series low of 4.61 million viewers, NBC decided to move "30 Rock" to Thursdays at 9:30pm. Its first airing on a Thursday night was on November 16, 2006. Along with this change, the even lower rated comedy "Twenty Good Years" was put on hiatus and later cancelled. "30 Rock"'s first Thursday airing was viewed by 5.19 million viewers. The series then received ratings of a consistent amount of around 5.5 million viewers till the episode "Hard Ball" when a series low, set by the episode "Jack the Writer," was met with just 4.61 million viewers watching the episode. After a further three episodes, which were higher rated than "Hard Ball," NBC moved "30 Rock" to 9:00pm on Thursdays, serving as a lead-in to "Scrubs". Upon its first airing at 9:00pm, the episode "Fireworks," a "supersized" episode attracted the attention of 5.37 million viewers. "30 Rock" aired at 9:00pm for four episodes. The season finale, "Hiatus," was watched by 4.72 million people. The first season averaged 5.8 million viewers for all 21 episodes. Out of all regular primetime programming that aired during the 2006–2007 American television season, "30 Rock" ranked #102 out of #142 according to the Nielsen ratings system.

title: How Can I Be Sure

text: In 1972 "How Can I Be Sure" was recorded for the album "Rock Me Baby" by David Cassidy "Eddie Brigati and Felix Cavaliere [were] two of the great musical influences of my teenage years [when] I thought [the Rascals] were just about the best American pop band...[That song] brought back so many memories of that [time] for me." (Another Rascals' hit: the Cavaliere/ Brigati composition "I've Been Lonely Too Long", was also remade by Cassidy for "Rock Me Baby": concurrent with the album's recording Cassidy's producer Wes Farrell would acquire ownership of the entire Rascals song catalogue). Released as the album's first single, "How Can I Be Sure" became the second of Cassidy's three Top 30 solo hits on the "Billboard" Hot 100 with a peak of #25, Cassidy's third Top 30 hit (and fifth and final Top 40 hit) being " "Lyin' to Myself" eighteen years later in 1990. On the "Billboard" Easy Listening chart "How Can I Be Sure" peaked at #3.

title: Apollo, Apollo

text: "Apollo, Apollo" was generally, though not universally, well received among critics. Television columnist Alan Sepinwall for "The Star-Ledger" wrote that the episode was "easily the best "30 Rock" of the season, and one of the series' best episodes to date". He complimented Robert Carlock for "[zooming] in on the narcissistic worldview" of the Dennis character as it "cranked those qualities up to appropriately absurd "30 Rock" proportions." In addition, Sepinwall wrote that all of the elements featured in the episode "all came together". Bob Sassone of AOL's TV Squad was similarly laudatory, writing that it was "one of the most explosively hysterical and imaginative half hours of TV this season, and one of this show's best episodes ever". "Time" contributor James Poniewozik was favorable to the episode, observing that it was "[e]asily the funniest "30 Rock" in months, and probably the whole season." Poniewozik said it "nailed it, from the too-many-good-jokes ... to the sweet heart at the center of each character." "The A.V. Club"'s Nathan Rabin enjoyed Adam West's cameo, but noted that Tracy's plot "felt awfully shticky and sitcommy to me." He opined that "Apollo, Apollo" was "pretty great" for Jane Krakowski's Jenna, and in conclusion, Rabin said "I had initially pegged this episode as good but not great and essentially minor in the grand scheme of things but while writing this post I came to realize just how much of it I loved. "30 Rock" is pretty great even when it's not that great." Rick Porter, a contributor for Zap2it, said he "loved" the episode, writing that it made "perfect sense" for Kenneth to see everyone around him as muppets, and noted that Liz's commercial was a "strong ending" to the episode.

title: Mrs. Donaghy

text: Before the airing of this episode, NBC unveiled its 2010–11 primetime schedule in May 2010 with the network moving "30 Rock" from the 9:30 p.m. time to the 8:30 p.m. timeslot for the show's fifth season. In November 2010, it was announced that the program would move from 8:30 to the 10:00 timeslot. The network moved "30 Rock" to the 10:00 time in order to "make room" for the return of the comedy show "Parks and Recreation" and the debut of the new comedy "Perfect Couples". Many questioned whether or not it was a good decision to move the show to 10:00 as it is a low rated show. According to Nielsen Media Research, "Mrs. Donaghy" was watched by 5.338 million households in its original American broadcast. According to "New York" magazine the staff were most worried about how "30 Rock" and "Community"—another program that airs on NBC's Thursday lineup—would do in the new time change but said that both shows "did great." The episode earned a 2.7 rating/7 share in the 18–49 demographic. This means that it was seen by 2.7 percent of all 18- to 49-year-olds, and 7 percent of all 18- to 49-year-olds watching television at the time of the broadcast. "Mrs. Donaghy" was up nearly 30 percent against the previous episode, "Christmas Attack Zone", that aired at 8:30 p.m. on December 9, 2010.

title: Into the Crevasse

text: Robert Canning of IGN reviewed the episode, simply saying he "loved this episode," giving it a perfect 10 out of 10 score. Canning enjoyed Jack and Devon's back-and-forth, and wrote that Jenna and Tracy "tormenting" Liz in their separate ways were "equally gratifying." Meredith Blake, a contributor from the "Los Angeles Times" opined, "To all those haters who say "30 Rock" is losing its touch, I say 'poppycock!' ['Into the Crevasse'] was proof that the show is as zippy as ever." She noted that Jack's plot "didn't make a whole lot of sense", nonetheless "anything that means more Will Arnett on "30 Rock" is fine by me." Bob Sassone of AOL's TV Squad reported it was great to see Jack and NBC "still being affected by the economy, government bailouts, and Washington hearings". Sassone much preferred "Into the Crevasse" than "Season 4". "New York" magazine contributor Mark Graham said this episode of "30 Rock" saw the series "once again hitting a familiar, meta-flavored stride, as Tina Fey and her writing staff tried a 'ripped from the headlines' approach and took on recent real-life story lines like the "30 Rock" porno". "The return of Will Arnett, the existence of a Porn Liz, and one deal breaker after another: 'Into the Crevasse' had some killer lines, even if it didn't come together in one big narrative", remarked Margaret Lyons of "Entertainment Weekly". In his recap, "TV Guide"'s Michael Anthony said that the Jack and Devon characters "really [knew] how to put a playful spin" on the bail out and government money story.

title: Christmas Attack Zone

text: "Entertainment Weekly"s Annie Barrett opined that Stritch and Alda "really made this episode great." Meredith Blake, a contributor for the "Los Angeles Times", opined that television Christmas episodes "are a sitcom staple, but ... they're also a risky proposition for a show like "30 Rock." Just how do you maintain the right balance between holiday cheer and year-round irreverence? It's a delicate formula, and on Thursday night "30 Rock" got it just right. That is, they leaned toward the former, delivering a Christmas episode full of cynicism – but topped with just a sprinkle of sweetness." In addition, Blake was complimentary toward Stritch's role on the show. Bob Sassone of AOL's TV Squad wrote that this episode of "30 Rock" "wasn't the best Christmas episode that [the show] has had. Past episodes like 'Ludachristmas,' 'Christmas Special' and 'Secret Santa' were put together better and had more heart". Sassone noted that the problem with "Christmas Attack Zone" was Tracy's subplot "as it wasn't very funny" and the one that dragged the episode down.

title: Sun Tea (30 Rock)

text: Not all reviews were positive. IGN contributor Robert Canning did not enjoy the green theme in the episode, reasoning that it "just seemed to get in the way of everything else happening in 'Sun Tea'." He was not thrilled with Jack and Tracy's storyline, noting it was missing heart and emotion, and overall it felt "flat". Canning felt that Al Gore's cameo was "fun but wholly unnecessary." Bob Sassone of AOL's TV Squad gave "Sun Tea" a mixed review, noting "I don't know how funny it was, though. I mean, sure, there was still a ton of great lines tonight ... but not all of them hit the mark like they usually do." Nonetheless, Sassone concluded "...any "30 Rock" episode with Dr. Spaceman and a couple of digs at NBC's Green Week is still a pretty good one." Meredith Blake, writing for the "Los Angeles Times", said this episode of "30 Rock" was "even more of a comedy drag race than usual."

title: Greenzo

text: "Greenzo" is the first episode of "30 Rock" written by Jon Pollack. Pollack was added to the writing staff of "30 Rock" at the beginning of the second season. The episode is the ninth episode of "30 Rock" directed by Don Scardino. This episode aired as part of Green Week, an initiative introduced by NBC's Chief Executive Jeff Zucker which included having every primetime program which aired between November 4, 2007 and November 10, 2007 contain some sort of positive environmental theme. This was also the first episode of "30 Rock" to air after the start of the 2007–2008 Writers Guild of America strike. The strike began on November 5, 2007 and ended on February 12, 2008. This episode was filmed on September 27 and September 28, 2007.

title: Florida (30 Rock)

text: "Florida" received generally positive reviews from critics. Alan Sepinwall, writing for "HitFix", opined that "[it] wasn't necessarily one of the strongest episodes of this victory lap season of "30 Rock", in that I think there was probably an entire episode's worth of material in both stories, and having to split time made both feel only partially baked." However, he concluded that "the episode was packed with so many great throwaway jokes that I could just spend this review listing them all." Pilot Viruet of "The A.V. Club" gave the episode an A- and particularly enjoyed the scene with Liz and Jack in bed, commenting: "For many viewers, the sight of these two in bed together has been a long time coming, and "30 Rock" plays it off flawlessly and with a few meta winks to the audience. Liz asks Jack "why didn’t anything ever happen between us?" [...] Jack puts it best, both to Liz and the audience, when he responds that their current relationship is more interesting than them dating." Roth Cornet of "IGN" scored the episode an 8.0 out of 10 and summarised "The show continues to deliver the wacky antics we desire, yet stay streamlined and propel the plot forward. What sometimes felt like an overly kinetic series of sketches and subplots in the early part of the season, has nicely coalesced into a push toward the sad, but inevitable, end."

title: 30 Rock (season 2)

text: During its second season, "30 Rock" moved time slots three times. All episodes aired on a Thursday, but with the first eight episodes airing 8:30 pm Eastern Standard Time (EST), the ninth episode airing at 9:00 pm EST, episode ten through twelve airing at 8:30 pm EST, and the final three episodes airing at 9:30 pm EST.

title: Rock Bottom (SpongeBob SquarePants)

text: Series head writer of the first season Merriwether Williams said the episode's story is "so simple." She remarked "To me, it's one of the best episodes. It just stayed with one idea." The episode was about "how he [SpongeBob] keeps missing the bus and how that makes him feel." Williams said the episode was "so small that you could explore gags and opportunities for gags." The episode was an example of "a good outline" in the series where the storyboards and humor were done easily. Williams said "it was great for the board guys." She added "In many ways, my job was to create situations where the board guys could be funny, to create a situation that could be funny, and let them go for the actual, specific jokes."

title: 30 Rock

text: Inside scenes for "30 Rock" were mostly filmed at Silvercup Studios in Long Island City, Queens, NYC (Studio 8H in 30 Rock in Manhattan for two live episodes). In the episodes "Cleveland" and "Hiatus", Battery Park City, Manhattan, and Douglaston, Queens, doubled for Cleveland, Ohio and Needmore, Pennsylvania, respectively. In the episode "Gavin Volure", stock footage of the Arkansas Governor's Mansion was used for exterior shots of the home of Steve Martin's character.

title: When It Rains, It Pours (30 Rock)

text: This episode of "30 Rock" was written by co-show runner and executive producer Robert Carlock and directed by series producer Don Scardino. This was Carlock's sixteenth writing credit, and Scardino's twenty-eighth directed episode. "When It Rains, It Pours" originally aired in the United States on September 30, 2010, on NBC as the second episode of the show's fifth season and the 82nd overall episode of the series. This episode of "30 Rock" was filmed on August 30, August 31, September 2, September 5, and September 8, 2010.

title: Argus (30 Rock)

text: "Argus" is the nineteenth episode of the fourth season of the American television comedy series "30 Rock", and the 77th overall episode of the series. It was written by "30 Rock" producers Dylan Morgan, Paula Pell, and Josh Siegal. The episode was directed by co-executive producer and show composer Jeff Richmond. It originally aired on the National Broadcasting Company (NBC) network in the United States on April 29, 2010. Guest stars in this episode include Will Forte, Marceline Hugot, and Burke Moses.

title: Live Show

text: Critical reception of the episode was largely positive. Todd VanDerWerff—writing for "The A.V. Club"—gave the episode an A−, noting the nostalgia for classic television and specifically three-camera sitcoms that this episode displayed. While the pace and tone of "Live Show" were different from a standard episode of "30 Rock", he considers this episode, "an experiment, as a weird hybrid of "30 Rock", an old sitcom, and "Saturday Night Live"" and ultimately a success. James Poniewozik of "Time" also wrote that the episode was uneven, but successful as a tribute to television. TV Squad's Bob Sassone was impressed by the episode, specifically Fey's performance and Louis-Dreyfus' imitation of Liz Lemon. Other positive reviews have noted the lack of mistakes in the comic timing and performance, the strength of the guest stars, and the performers' ability to keep up the pace that is expected in a "30 Rock" episode. Frazier Moore of the Associated Press gave the episode a negative review, calling it "vaudevillian" and "a slice of self-indulgence and excess." Mark Perigard of the "Boston Herald" considered the episode "so-so" due to its "humdrum script." As a live television experiment, reviewers have considered it a success, particularly in reference to other live broadcast experiments, such as NBC drama "ER"s 1997 episode "Ambush" and "SNL", with "Entertainment Weekly" declaring it "like "SNL", only funny." In reviewing the best television programs of 2010, The A.V. Club named "30 Rock" number 18 and cited this episode as amongst the best of the year. McCarthy-Miller was nominated for a Primetime Emmy Award for Outstanding Directing for a Comedy Series for this episode.

title: Live Show

text: On July 30, 2010, the NBC network announced that an episode of "30 Rock"s fifth season would be filmed and broadcast live on two occasions in the evening of October 14, 2010. The decision was made in part due to lagging ratings for all of NBC's Must-See TV line-up. The idea was originally conceived during the 2007–2008 Writers Guild of America strike, when the cast performed two live versions of the season two episode "Secrets and Lies" as a benefit at the Upright Citizens Brigade Theatre in New York City. When asked by "Entertainment Weekly" the reason for doing a live episode, Fey said "[The live performance] was what made us think that it would be fun to do [a live episode]. And how far we've come... Now we're going to do it on TV." In this same interview, Fey revealed that the episode would also be performed in front of a live audience. "It will be exciting for us to do it for a live audience and see if we get live laughter." Fey noted that the experience "was so fun to feel the different timing of the show live and the audience response, which you just don't get in a single-camera show." In another interview, Fey revealed that the staff have talked about doing a live show for years "because we felt we had sort of interesting personnel for it." Following the conclusion of the writers' strike they spoke to NBC about performing live and attempted to film it during the fourth season; however, they were unable to schedule it. For the fifth season, the writing staff began planning the live performance and waited until McCarthy-Miller—a recurring "30 Rock" director—was available to helm it. Fey and Carlock said in separate discussions that another decision to do this episode of "30 Rock" was based on the cast having theater and improvisation experience. Alec Baldwin had previously performed in the East and West live broadcasts of the "Will & Grace" premiere episode, "Alive and Schticking".

title: 30 Rock (season 4)

text: Robert Canning of IGN scored this season an 8.4 out of 10 rating, noting it was "impressive" and that "30 Rock" "is no longer a fresh new series and a bit of the Season 4 doldrums could be felt mid season. But the great start and fantastic finish proved that there's still a lot of great comedy to be found on the stages of "30 Rock"." Aaron Barnhart of "The Kansas City Star" deemed the first episode of the season, "Season 4" as "one of the weakest" episodes from "30 Rock" that he has ever seen, and found the episode boring. In his review of the third episode, "Stone Mountain", Leonard Pierce of "The A.V. Club" gave it a "C" grade, and not entirely favorable to the first two episodes, Pierce commented that "30 Rock" "needs to give us something fast to get rid of the worst-season-ever stink that's starting to gather."

title: Somebody to Love (30 Rock)

text: Robert Canning of IGN thought that the episode "was "30 Rock"’s best episode of the season [so far], if not the best episode ever from this young series." Canning also said that "this was just a fantastic, funny episode of "30 Rock". It was also an episode that once again proved the character of Jenna Maroney (Jane Krakowski), who was completely absent from this half-hour, simply isn't necessary when you've got Tracy Jordan talking to a pigeon on a trash can." Canning rated this episode "9.5 out of 10." Matt Webb Mitovich of "TV Guide" wrote that "it's not surprising that – in my opinion, at least – it fell "juuuuuuust" a bit short. It's almost as if the funniest parts of the Edie Falco story were played by [Kristen] Wiig in the Lifetime movie." He thought that "the 'Liz suspects the neighbor is a terrorist' story played out rather well." Jeff Labrecque of "Entertainment Weekly" wrote that "Overall, [this was] another splendid [episode]. My only beef was with the Verizon bit. "30 Rock" normally does a splendid job of balancing the corporate elements of television, often utilizing it for comic fodder. But the winking cell phone banter lacked subtlety, and it seemed to me like the show was trying to have its cake and eat it, too."

title: 30 Rock

text: Below, "Rank" refers to how well "30 Rock" rated compared to other television series which aired during primetime hours of the corresponding television season; for example, in its first year, 101 television series were rated higher. The television season tends to begin in September of any given year and end during May of the following year. "Viewers" refers to the average number of viewers for all original episodes (broadcast in the series' "Regular Timeslot") of "30 Rock" aired during the television season. Although the viewer average may be higher for some seasons than others, the rank will not necessarily be higher. This is due to the number of programs aired during primetime. In some seasons, more regular programs may be aired during primetime than in others.

title: St. Valentine's Day (30 Rock)

text: This episode was actress Salma Hayek's fourth appearance on "30 Rock". She first appeared in the episode "Señor Macho Solo" as a nurse for Jack Donaghy's mother and love interest for Jack. She made appearances in the episodes "Flu Shot" and "Generalissimo". Actress Maria Thayer starred in the 2008 comedy film "Forgetting Sarah Marshall" as the wife of a character played by Jack McBrayer, who plays Kenneth Parcell on "30 Rock".


query: What is a matrix in algebra?

title: Matrix (mathematics)

text: In mathematics, a matrix (plural: matrices) is a rectangular array[1] of numbers, symbols, or expressions, arranged in rows and columns.[2][3] For example, the dimensions of the matrix below are 2 × 3 (read "two by three"), because there are two rows and three columns:

title: Algebraic curve

text: Using the intrinsic concept of tangent space, points "P" on an algebraic curve "C" are classified as "smooth" (synonymous: "non-singular"), or else "singular". Given "n"−1 homogeneous polynomials in "n"+1 variables, we may find the Jacobian matrix as the ("n"−1)×("n"+1) matrix of the partial derivatives. If the rank of this matrix is "n"−1, then the polynomials define an algebraic curve (otherwise they define an algebraic variety of higher dimension). If the rank remains "n"−1 when the Jacobian matrix is evaluated at a point "P" on the curve, then the point is a smooth or regular point; otherwise it is a "singular point". In particular, if the curve is a plane projective algebraic curve, defined by a single homogeneous polynomial equation "f"("x","y","z") = 0, then the singular points are precisely the points "P" where the rank of the 1×("n"+1) matrix is zero, that is, where

title: Projection matrix

text:
The projection matrix has a number of useful algebraic properties. In the language of linear algebra, the projection matrix is the orthogonal projection onto the column space of the design matrix formula_26.(Note that formula_27 is the pseudoinverse of X.) Some facts of the projection matrix in this setting are summarized as follows:
The projection matrix corresponding to a linear model is symmetric and idempotent, that is, formula_33. However, this is not always the case; in locally weighted scatterplot smoothing (LOESS), for example, the hat matrix is in general neither symmetric nor idempotent.

title: Weyl–Brauer matrices

text:
In the quantization for an odd number 2"k"+1 of dimensions, the matrices "P" may be introduced as above for "i" = 1,2...,2"k", and the following matrix may be adjoined to the system:
so that the Clifford relations still hold. This adjunction has no effect on the algebra "A" of matrices generated by the "P", since in either case "A" is still a complete matrix algebra of the same dimension. Thus "A", which is a complete 2×2 matrix algebra, is not the Clifford algebra, which is an algebra of dimension 2×2×2. Rather "A" is the quotient of the Clifford algebra by a certain ideal.

title: Lie group

text: The first result in this direction is Lie's third theorem, which states that every finite-dimensional, real Lie algebra is the Lie algebra of some (linear) Lie group. One way to prove Lie's third theorem is to use Ado's theorem, which says every finite-dimensional real Lie algebra is isomorphic to a matrix Lie algebra. Meanwhile, for every finite-dimensional matrix Lie algebra, there is a linear group (matrix Lie group) with this algebra as its Lie algebra.

title: Matrix unit

text: In mathematics, a matrix unit is an idealisation of the concept of a matrix, with a focus on the algebraic properties of matrix multiplication. The topic is comparatively obscure within linear algebra, because it entirely ignores the numeric properties of matrices; it is mostly encountered in the context of abstract algebra, especially the theory of semigroups.

title: Coherent algebra

text: A coherent algebra is an algebra of complex square matrices that is closed under ordinary matrix multiplication, Schur product, transposition, and contains both the identity matrix formula_1 and the all-ones matrix formula_2.

title: Norman L. Biggs

text: In 1974, Biggs published "Algebraic Graph Theory" which articulates properties of graphs in algebraic terms, then works out theorems regarding them. In the first section, he tackles the applications of linear algebra and matrix theory; algebraic constructions such as adjacency matrix and the incidence matrix and their applications are discussed in depth. Next, there is and wide-ranging description of the theory of chromatic polynomials. The last section discusses symmetry and regularity properties. Biggs makes important connections with other branches of algebraic combinatorics and group theory.

title: Manin matrix

text: The two propositions above imply that the algebra generated by elements of a Manin matrix is a bialgebra coacting on the polynomial algebra. If one does not impose other relations ones get algebra of non-commutative endomorphisms of the polynomial algebra.

title: Representation theory

text:
Representation theory is a branch of mathematics that studies abstract algebraic structures by "representing" their elements as linear transformations of vector spaces, and studies
modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and the algebraic operations in terms of matrix addition and matrix multiplication. The algebraic objects amenable to such a description include groups, associative algebras and Lie algebras. The most prominent of these (and historically the first) is the representation theory of groups, in which elements of a group are represented by invertible matrices in such a way that the group operation is matrix multiplication.

title: Algebraic number field

text: The matrix associated to an element "x" of "F" can also be used to give other, equivalent descriptions of algebraic integers. An element "x" of "F" is an algebraic integer if and only if the characteristic polynomial "p" of the matrix "A" associated to "x" is a monic polynomial with integer coefficients. Suppose that the matrix "A" that represents an element "x" has integer entries in some basis "e". By the Cayley–Hamilton theorem, "p"("A") = 0, and it follows that "p"("x") = 0, so that "x" is an algebraic integer. Conversely, if "x" is an element of "F" which is a root of a monic polynomial with integer coefficients then the same property holds for the corresponding matrix "A". In this case it can be proven that "A" is an integer matrix in a suitable basis of "F". Note that the property of being an algebraic integer is "defined" in a way that is independent of a choice of a basis in "F".

title: Ring theory

text:
Representation theory is a branch of mathematics that draws heavily on non-commutative rings. It studies abstract algebraic structures by "representing" their elements as linear transformations of vector spaces, and studies
modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and the algebraic operations in terms of matrix addition and matrix multiplication, which is non-commutative. The algebraic objects amenable to such a description include groups, associative algebras and Lie algebras. The most prominent of these (and historically the first) is the representation theory of groups, in which elements of a group are represented by invertible matrices in such a way that the group operation is matrix multiplication.

title: Matrix ring

text: When "R" is a commutative ring, the matrix ring M("R") is an associative algebra, and may be called a matrix algebra. For this case, if "M" is a matrix and "r" is in "R", then the matrix "Mr" is the matrix "M" with each of its entries multiplied by "r".

title: Algebra over a field

text: The multiplication operation in an algebra may or may not be associative, leading to the notions of associative algebras and nonassociative algebras. Given an integer "n", the ring of real square matrices of order "n" is an example of an associative algebra over the field of real numbers under matrix addition and matrix multiplication since matrix multiplication is associative. Three-dimensional Euclidean space with multiplication given by the vector cross product is an example of a nonassociative algebra over the field of real numbers since the vector cross product is nonassociative, satisfying the Jacobi identity instead.

title: Representation theory of the Lorentz group

text: The Lorentz group is a Lie group and has as such a Lie algebra, The Lie algebra is a vector space of matrices that can be said to model the group near the identity. It is endowed with a multiplication operation, the Lie bracket. With it, the product in the group can near the identity be expressed in Lie algebraic terms (but not in a particularly simple way). The link between the (matrix) Lie algebra and the (matrix) Lie group is the matrix exponential. It is one-to-one near the identity in the group.

title: Representation theory

text:
There are two ways to say what a representation is. The first uses the idea of an action, generalizing the way that matrices act on column vectors by matrix multiplication. A representation of a group "G" or (associative or Lie) algebra "A" on a vector space "V" is a map
with two properties. First, for any "g" in "G" (or "a" in "A"), the map
is linear (over F). Second, if we introduce the notation "g" · "v" for formula_3 ("g", "v"), then for any "g", "g" in "G" and "v" in "V":
where "e" is the identity element of "G" and "g""g" is the product in "G". The requirement for associative algebras is analogous, except that associative algebras do not always have an identity element, in which case equation (1) is ignored. Equation (2) is an abstract expression of the associativity of matrix multiplication. This doesn't hold for the matrix commutator and also there is no identity element for the commutator. Hence for Lie algebras, the only requirement is that for any "x", "x" in "A" and "v" in "V":
where ["x", "x"] is the Lie bracket, which generalizes the matrix commutator "MN" − "NM".

title: Triangular matrix

text: If all of the entries on the main diagonal of a (upper or lower) triangular matrix are 0, the matrix is called strictly (upper or lower) triangular. All strictly triangular matrices are nilpotent, and the set of strictly upper (or lower) triangular matrices forms a nilpotent Lie algebra, denoted formula_5 This algebra is the derived Lie algebra of formula_6, the Lie algebra of all upper triangular matrices; in symbols, formula_7 In addition, formula_8 is the Lie algebra of the Lie group of unitriangular matrices.

title: Exterior algebra

text: In applications to linear algebra, the exterior product provides an abstract algebraic manner for describing the determinant and the minors of a matrix. For instance, it is well known that the magnitude of the determinant of a square matrix is equal to the volume of the parallelotope whose sides are the columns of the matrix. This suggests that the determinant can be "defined" in terms of the exterior product of the column vectors. Likewise, the minors of a matrix can be defined by looking at the exterior products of column vectors chosen "k" at a time. These ideas can be extended not just to matrices but to linear transformations as well: the magnitude of the determinant of a linear transformation is the factor by which it scales the volume of any given reference parallelotope. So the determinant of a linear transformation can be defined in terms of what the transformation does to the top exterior power. The action of a transformation on the lesser exterior powers gives a basis-independent way to talk about the minors of the transformation.

title: Compound matrix

text: In linear algebra, a branch of mathematics, a compound matrix is a matrix whose entries are all minors, of a given size, of another matrix. Compound matrices are closely related to exterior algebras.

title: Algebraic matroid

text: Many finite matroids may be represented by a matrix over a field "K", in which the matroid elements correspond to matrix columns, and a set of elements is independent if the corresponding set of columns is linearly independent. Every matroid with a linear representation of this type over a field "F" may also be represented as an algebraic matroid over "F", by choosing an indeterminate for each row of the matrix, and by using the matrix coefficients within each column to assign each matroid element a linear combination of these transcendentals. For fields of characteristic zero (such as the real numbers) linear and algebraic matroids coincide, but for other fields there may exist algebraic matroids that are not linear; indeed the non-Pappus matroid is algebraic over any finite field, but not linear and not algebraic over any field of characteristic zero. However, if a matroid is algebraic over a field "F" of characteristic zero then it is linear over "F"("T") for some finite set of transcendentals "T" over "F" and over the algebraic closure of "F".

title: Quaternion algebra

text: In mathematics, a quaternion algebra over a field "F" is a central simple algebra "A" over "F" that has dimension 4 over "F". Every quaternion algebra becomes the matrix algebra by "extending scalars" (equivalently, tensoring with a field extension), i.e. for a suitable field extension "K" of "F", formula_1 is isomorphic to the 2×2 matrix algebra over "K".

title: Kac–Moody algebra

text: Properties of a Kac–Moody algebra are controlled by the algebraic properties of its generalized Cartan matrix "C". In order to classify Kac–Moody algebras, it is enough to consider the case of an "indecomposable" matrix "C", that is, assume that there is no decomposition of the set of indices "I" into a disjoint union of non-empty subsets "I" and "I" such that "C" = 0 for all "i" in "I" and "j" in "I". Any decomposition of the generalized Cartan matrix leads to the direct sum decomposition of the corresponding Kac–Moody algebra:

title: Hermitian matrix

text: The range of the Rayleigh quotient (for matrix that is not necessarily Hermitian) is called a numerical range (or spectrum in functional analysis). When the matrix is Hermitian, the numerical range is equal to the spectral norm. Still in functional analysis, formula_61 is known as the spectral radius. In the context of C*-algebras or algebraic quantum mechanics, the function that to associates the Rayleigh quotient for a fixed and varying through the algebra would be referred to as "vector state" of the algebra.

title: Adjacency algebra

text: In algebraic graph theory, the adjacency algebra of a graph "G" is the algebra of polynomials in the adjacency matrix "A"("G") of the graph. It is an example of a matrix algebra and is the set of the linear combinations of powers of "A".

title: Stone–von Neumann theorem

text: More formally, the group algebra of the Heisenberg group over its field of scalars "K", written , has center , so rather than simply thinking of the group algebra as an algebra over the field , one may think of it as an algebra over the commutative algebra . As the center of a matrix algebra or operator algebra is the scalar matrices, a -structure on the matrix algebra is a choice of scalar matrix – a choice of scale. Given such a choice of scale, a central representation of the Heisenberg group is a map of -algebras , which is the formal way of saying that it sends the center to a chosen scale.

title: Kac–Moody algebra

text: "Almost simultaneously in 1967, Victor Kac in the USSR and Robert Moody in Canada developed what was to become Kac-Moody algebra. Kac and Moody noticed that if Wilhelm Killing's conditions were relaxed, it was still possible to associate to the Cartan matrix a Lie algebra which, necessarily, would be infinite dimensional." - A.J. Coleman

title: Classification of Clifford algebras

text: and therefore isomorphic (since "α" is an automorphism). These two isomorphic algebras are each central simple and so, again, isomorphic to a matrix algebra over C. The sizes of the matrices can be determined from the fact that the dimension of Cℓ(C) is 2. What we have then is the following table:

title: Lie algebra

text:
Any Lie group defines an associated real Lie algebra formula_93. The definition in general is somewhat technical, but in the case of a real matrix group "G", it can be formulated via the exponential map, or the matrix exponential. The Lie algebra formula_8 of "G" may be computed as
The Lie bracket of formula_8 is given by the commutator of matrices, formula_97. The following are examples of Lie algebras of matrix Lie groups:
Given a vector space "V", let formula_69 denote the Lie algebra consisting of all linear endomorphisms of "V", with bracket given by formula_97. A "representation" of a Lie algebra formula_8 on "V" is a Lie algebra homomorphism

title: Matrix analysis

text: In mathematics, particularly in linear algebra and applications, matrix analysis is the study of matrices and their algebraic properties. Some particular topics out of many include; operations defined on matrices (such as matrix addition, matrix multiplication and operations derived from these), functions of matrices (such as matrix exponentiation and matrix logarithm, and even sines and cosines etc. of matrices), and the eigenvalues of matrices (eigendecomposition of a matrix, eigenvalue perturbation theory).

title: Matrix population models

text: Population models are used in population ecology to model the dynamics of wildlife or human populations. Matrix population models are a specific type of population model that uses matrix algebra. Matrix algebra, in turn, is simply a form of algebraic shorthand for summarizing a larger number of often repetitious and tedious algebraic computations.

title: Boolean matrix

text: In mathematics, a Boolean matrix is a matrix with entries from a Boolean algebra. When the Boolean algebra has just two elements {0,1} the Boolean matrix is called a logical matrix.


query: What is pluralistic ignorance?

title: Pluralistic ignorance

text: In social psychology, pluralistic ignorance is a situation in which a majority of group members privately reject a norm, but go along with it because they incorrectly assume that most others accept it.[1] This is also described as "no one believes, but everyone thinks that everyone believes". In short, pluralistic ignorance is a bias about a social group, held by the members of that social group.[2][3]

title: Social experiment

text: Current research on bystander apathy by psychologist Kyle Thomas et al., found that people's decisions to help are influenced by their level of knowledge. While diffusion of responsibility and pluralistic ignorance are factors, the researchers found that bystanders’ decisions also consider what they know about other bystanders and the situation before getting involved.

title: Pluralistic ignorance

text: Pluralistic ignorance was blamed for exacerbating support for racial segregation in the United States. It has also been named a reason for the illusory popular support that kept the Communist Party of the Soviet Union in power, as many opposed the regime but assumed that others were supporters of it. Thus, most people were afraid to voice their opposition.[9]

title: Pluralistic ignorance

text: Men's conceptions of how they are expected to conform to norms of masculinity present additional examples of pluralistic ignorance. Specifically, most men are uncomfortable with other men "bragging about sexual acts and giving details," but erroneously believe themselves to be in the minority for their discomfort. Similarly, men underestimate other men's "desire to make sure they have consent when sexually active." This "role-conflict" can have deleterious consequences for men's physical and mental health, as well as for society.[10]

title: Sociology of scientific ignorance

text:
Generally, the word 'ignorance' has a negative tone to it, and for a long time scientific ignorance was viewed as a purely negative thing. Recently, however, people have started to abandon this idea, and instead try to find uses of deliberate ignorance.
This has generally been called useful ignorance. A first step in finding uses of ignorance is realizing that ignorance is inevitable. As Matthias Gross says: "new knowledge also means more ignorance".
Gross also talks about the connection between ignorance and surprise. Surprise can reveal what scientists are ignorant of, which help them focus their research in order to gain knowledge. On the other hand, ignorance is what gives rise to surprise, making the two very connected.

title: Pluralist democracy

text: A pluralist democracy describes a political system where there is more than one center of power. Modern democracies are by definition pluralist as democracies allow freedom of association. However, pluralism may exist without democracy.

title: False consensus effect

text: The false-consensus effect can be contrasted with pluralistic ignorance, an error in which people privately disapprove but publicly support what seems to be the majority view (regarding a norm or belief), when the majority in fact shares their (private) disapproval. While the false-consensus effect leads people to wrongly believe that the majority agrees with them (when the majority, in fact, openly disagrees with them), the pluralistic ignorance effect leads people to wrongly believe that they disagree with the majority (when the majority, in fact, covertly agrees with them). Pluralistic ignorance might, for example, lead a student to engage in binge drinking because of the mistaken belief that most other students approve of it, while in reality most other students disapprove, but behave in the same way because they share the same mistaken (but collectively self-sustaining) belief. In a parallel example of the false-consensus effect, a student who likes binge drinking would believe that a majority also likes it, while in reality most others dislike it and openly say so.

title: Pluralistic ignorance

text: A study done by Greene, House, and Ross used simple circumstantial questionnaires on Stanford undergrads to gather info on FCE. They compiled thoughts on the choice they felt people would or should make, considering traits such as shyness, cooperativeness, trust, and adventurousness. Studies found that when explaining their decisions, participants gaged choices based on what they explained as "people in general" and their idea of "typical" answers. For each of the stories those subjects said that they personally would follow a given behavioral alternative also tended to rate that alternative as relatively probable for "people in general": those subjects who claimed that they would reject the alternative tended to rate it as relatively improbable for "people in general". It was evident that the influence of the subjects' own behavior choice affected the estimates of commonness.[17] Although it would seem as if the two are built on the same premise of social norms, they take two very oppositional stances on a similar phenomenon. The false consensus effect considers that in predicting an outcome, people will assume that the masses agree with their opinion and think the same way they do on an issue, whereas the opposite is true of pluralistic ignorance, where the individual does not agree with a certain action but go along with it anyway, believing that their view is not shared with the masses (which is usually untrue).

title: Pluralistic ignorance

text: The theory of pluralistic ignorance was studied by Floyd Henry Allport and his students Daniel Katz and Richard Schanck.[7] He produced studies of racial stereotyping and prejudice, and attitude change, and his pursuit of the connections between individual psychology and social systems helped to found the field of organizational psychology. Elisabeth Noelle-Neumann, in her spiral of silence theory, argued that media biases lead to pluralistic ignorance.[8]

title: Pluralistic ignorance

text: Pluralistic ignorance can be dispelled, and its negative consequences alleviated, through education. For example, students who learn that support for heavy drinking practices is not as widespread as they thought drink less themselves and feel more comfortable with the decision not to drink. Alcohol intervention programs now routinely employ this strategy to combat problem drinking on campus.[16]

title: The Emperor's New Clothes

text:
As an idiom, use of the story's title refers to something widely accepted as true or professed as being praiseworthy, due to an unwillingness of the general population to criticize it or be seen as going against popular opinion. The phrase "emperor's new clothes" has become an idiom about logical fallacies. The story may be explained by pluralistic ignorance.
The story is about a situation where "no one believes, but everyone believes that everyone else believes. Or alternatively, everyone is ignorant to whether the emperor has clothes on or not, but believes that everyone else is not ignorant."

title: Pluralistic walkthrough

text: Other types of usability assessment methods include: Cognitive Walkthroughs, Interviews, Focus Groups, Remote Testing and Think Aloud Protocol. Pluralistic Walkthroughs share some of the same characteristics with these other methods, especially cognitive walkthroughs, but there are some defining characteristics (Nielsen, 1994):There are several benefits that make the pluralistic usability walkthrough a valuable tool.There are several limitations to the pluralistic usability walkthrough that affect its usage.

title: Pluralism (political philosophy)

text: While advocated by many pluralists, pluralism need not embrace social democracy given it does not a priori assume a desirable political system. Rather, pluralists advocate one based on the pre-existing traditions and cognizable interests of a given society, and the political structure most likely to harmonize these factors. Thus, pluralists have also included Michael Oakeshott and John Kekes, proponents of something close to liberal conservatism (although will often reject such political labels). What pluralists certainly do have in common is the notion that a single vision or ideological schema, whether Marxism or unbridled neoliberalism, is likely too simplistic and rigid to advocate human beings' natural plurality of values. Pluralists likewise reject historicism and utopian thinking. While some, like John N. Gray, repudiate historical progress altogether, others, like Edmund Burke, indicate that human progress has occurred, as a function of improved social harmony.

title: Spiral of silence

text:
Individuals use what is described as "an innate ability" or quasi-statistical sense to gauge public opinion. People assume they can sense and figure out what others are thinking.
Pluralistic ignorance may occur in some cases, leading to the minority opinion to be accepted as a norm. Group members may be privately rejecting a norm, but may falsely assume that other group members accept it. This phenomenon may cause a group to hold on to a norm.

title: Pluralistic ignorance

text: Pluralistic ignorance has also been blamed for large majorities of the public remaining silent on climate change — while 'solid majorities' of the American and UK public are concerned about climate change, most erroneously believe they are in the minority with their concern.[12] It's been suggested that pollution-intensive industries have contributed to the public's underestimation of public support for climate solutions.[13] For example, in the U.S., support for pollution pricing is high,[14][15] yet public perception of public support is much lower.[13]

title: Pluralistic ignorance

text: Prentice and Miller found that, on average, private levels of comfort with drinking practices on campus were much lower than the perceived average. In the case of men, they found a shifting of private attitudes toward this perceived norm, a form of cognitive dissonance. Women, on the other hand, were found to have an increased sense of alienation on the campus but lacked the attitude change detected in men, presumably because norms related to alcohol consumption on campus are much more central for men than for women.[5] Research has shown that pluralistic ignorance plagues not only those who indulge, but also those who abstain: from gambling, smoking and drinking and among some who follow vegetarianism.[6] The latter has found that pluralistic ignorance can be caused by the structure of the underlying social network, not cognitive dissonance.

title: Nicky Morgan

text: In December 2015, Morgan declared that a High Court ruling that religious teaching should be pluralistic, and that therefore it was unlawful to exclude teaching about atheism and humanism, should be ignored as UK religious traditions are mainly Christian. "The Independent" newspaper noted that both she and her department had also ignored the Commission on Religion and Belief in British Public Life report that Britain is "no longer predominantly Christian." The British Humanist Association, which supported the legal battle against Morgan, called Morgan's decision to simply ignore the judgment against her "an affront to democracy".

title: Michael Nugent

text: Nugent campaigns for secular education in Ireland, including in an alliance between Atheist Ireland, the Evangelical Alliance of Ireland, and the Ahmadiyya Muslim Community of Ireland. He argues for secular schools that would not ignore religion, but would teach children about religions and beliefs in an objective, critical and pluralist manner, while respecting everybody's human rights.

title: Normality (behavior)

text: Individuals' behaviours are guided by what they perceive to be society's expectations and their peers' norms. People measure the appropriateness of their actions by how far away they are from those social norms. However, what is perceived as the norm may or may not actually be the most common behaviour. In some cases of pluralistic ignorance, most people falsely believe the social norm is one thing, but in fact very few people hold that view.

title: Multicultural particularism

text: Multicultural particularism is the belief that a common culture for all people is either undesirable or impossible. In discussions of multiculturalism, historian and educator Diane Ravitch draws a distinction between what she terms "pluralistic" and "particularistic" varieties and suggests that other writers often blur or ignore this distinction.

title: Pluralistic ignorance

text: Pluralistic ignorance can be contrasted with the false consensus effect. In pluralistic ignorance, people privately disdain but publicly support a norm (or a belief), while the false consensus effect causes people to wrongly assume that most people think like they do, while in reality most people do not think like they do (and express the disagreement openly). For instance, pluralistic ignorance may lead a student to drink alcohol excessively because she believes that everyone else does that, while in reality everyone else also wishes they could avoid binge drinking, but no one expresses that due to the fear of being ostracized.[4] A false consensus for the same situation would mean that the student believes that most other people do not enjoy excessive drinking, while in fact most other people do enjoy that and openly express their opinion about it.

title: Pluralism (political philosophy)

text: As put by arch-pluralist Isaiah Berlin, "let us have the courage of our admitted ignorance, of our doubts and uncertainties. At least we can try to discover what others [...] require, by [...] making it possible for ourselves to know men as they truly are, by listening to them carefully and sympathetically, and understanding them and their lives and their needs... ." Pluralism thus tries to encourage members of society to accommodate their differences by avoiding extremism (adhering solely to one value, or at the very least refusing to recognize others as legitimate) and engaging in good faith dialogue. Pluralists also seek the construction or reform of social institutions in order to reflect and balance competing principles. One of the more famous arguments for institutional pluralism came from James Madison in "The Federalist" paper number 10. Madison feared that factionalism would lead to in-fighting in the new American republic and devotes this paper to questioning how best to avoid such an occurrence. He posits that to avoid factionalism, it is best to allow many competing factions (advocating different primary principles) to prevent any one from dominating the political system. This relies, to a degree, on a series of disturbances changing the influences of groups so as to avoid institutional dominance and ensure competition. Like Edmund Burke, this view concerns itself with balance, and subordinating any single abstract principle to a plurality or realistic harmony of interests.

title: Sovereign state

text: Scholars in international relations can be broken up into two different practices, realists and pluralists, of what they believe the ontological state of the state is. Realists believe that the world is one of only states and interstate relations and the identity of the state is defined before any international relations with other states. On the other hand, pluralists believe that the state is not the only actor in international relations and interactions between states and the state is competing against many other actors.

title: Pluralistic ignorance

text: Pluralistic ignorance has been linked to a wide range of deleterious consequences. For example, victims of pluralistic ignorance see themselves as deviant members of their peer group: less knowledgeable than their classmates, more uptight than their peers, less committed than their fellow board members, less competent than their fellow nurses (see the Dunning–Kruger effect operating in the opposite direction). This can leave them feeling bad about themselves and alienated from the group or institution of which they are a part. In addition, pluralistic ignorance can lead groups to persist in policies and practices that have lost widespread support: This can lead college students to persist in heavy drinking, corporations to persist in failing strategies, and governments to persist in unpopular foreign policies. At the same time, it can prevent groups from taking actions that would be beneficial in the long run: actions to intervene in an emergency, for example, or to initiate a personal relationship.

title: Pluralistic ignorance

text: Abilene paradox Asch conformity experiments Conformity Groupthink Peer pressure Political correctness Silent majority Spiral of silence Social norms marketing Thomas theorem

title: Pluralistic ignorance

text: Pluralistic ignorance may help to explain the bystander effect.[4] If no-one acts, onlookers may believe others believe action is incorrect, and may therefore themselves refrain from acting.

title: Pluralistic ignorance

text: Hans Christian Andersen's fairy tale "The Emperor's New Clothes"[11] is a famous fictional case of pluralistic ignorance. In this story two con artists come into the Emperor's kingdom and convince him that they make the finest clothes in all of the land that can only be seen by anyone who was not stupid. The con artists continued to steal gold, silk and other precious items for their "unique creation". Out of fear for being seen as stupid, all of the emperor's men and townspeople kept silent about the fact they could not see the emperor's clothes until finally a small child comes forth and says that the emperor isn't wearing any clothes. Once the child is willing to admit that he cannot see any clothes on the emperor, the emperor and townspeople finally admit that the emperor has been tricked and that there was never an outfit being made.

title: Pluralistic ignorance

text: Another case of pluralistic ignorance concerns drinking on campus in countries where alcohol use is prevalent at colleges and universities. Students drink at weekend parties and sometimes at evening study breaks. Many drink to excess, some on a routine basis. The high visibility of heavy drinking on campus, combined with reluctance by students to show any public signs of concern or disapproval, gives rise to pluralistic ignorance: Students believe that their peers are much more comfortable with this behavior than they themselves feel.[10]

title: False consensus effect

text: The false-consensus effect can be contrasted with pluralistic ignorance, an error in which people privately disapprove but publicly support what seems to be the majority view (see below).

title: Social norms approach

text: A phenomenon known as false consensus is closely related to the idea of pluralistic ignorance, and refers to the incorrect belief that others are similar, when in reality they are not. For example, heavy drinkers will think that most others consume as much as they do, and will use this belief to justify their behavior. Berkowitz, an independent consultant who works full-time to promote these ideas, describes false consensus and pluralistic ignorance as "mutually reinforcing and self-perpetuating...the majority is silent because it thinks it is a minority, and the minority is vocal because it believes that it represents the majority" (p. 194).