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Q: Android Background execution not allowed on Oreo (8.) for intent android.intent.action.EVENT_REMINDER I have an app which has registered BroadcastReceiver which listens for intents android.intent.action.EVENT_REMINDER. From Oreo it doesnt work at all, until you add app as an exception to energy saving.
But that is in my opinion bug in android, because they explicitely say in documentation:
As part of the Android 8.0 (API level 26) Background Execution Limits,
apps that target the API level 26 or higher can no longer register
broadcast receivers for implicit broadcasts in their manifest.
However, several broadcasts are currently exempted from these
limitations
And ACTION_EVENT_REMINDER is in the list!
So if it has an exception, why am I not receiving broadcast, until I whitelist my app to energy optimizations? This should not be necessary, yet it is from Oreo.
Am I doing anything wrong? My manifest:
<receiver android:name=".services.MyReceiver" >
<intent-filter>
<action android:name="android.intent.action.EVENT_REMINDER" />
<data android:scheme="content" />
</intent-filter>
</receiver>
Works perfectly fine pre-Oreo. After Oreo ONLY if I add app as an exception to battery saving.
|
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Karachi League begins from 5th April
Home / News / Karachi League begins from 5th April
April3April 3, 2013
News Other National Competitions
10th Karachi League to kick off from April 5
The 10th edition of Aman Sports Karachi Football League will kick off from April 5. League matches will be played at the Baloch Mujahid Stadium and Korangi Baloch Stadium in Ibrahim Hyderi and Sharafi Goth, respectively.
Eight teams from last year's Super League and 2 more home teams will take part in this edition of the competition. The participating teams will face each other in a single leg match. Top 4 teams will progress to the semi-final stage of the competition.
The first fixture will see defending champions Burma Mohamadan Club Korangi take on Al Shahbaz Korangi at the Korangi Baloch Stadium in Shrafi Goth (April 5, 4pm). Director Pak Business; MM Ghalib will be the chief guest on the occasion.
Irfan Memorial Football club will take on hosts Karachi United FC on the opening day of the league as well, at the Baloch Mujahid Stadium in Ibrahim Hyderi (Kick off at 8pm). The league fixtures will be played on Friday, Saturday and Sunday. A total of 5 games will be played each week.
Each team will be allowed to field 2 registered departmental players, as per the rules of the competition. The 2 registered players must be named before the start of the competition and cannot be changed during the course of the event.
The rules of the competition also require participants to include 2 players under the age of 19 in every game. The selection of at least 5 Under 19 players per squad is mandatory.
Clubs are also required to submit the CNIC or 'B' form of under 19 players to Riaz Ahmed, organising secretary of the competition. Players will not be allowed to take part in the league failing to comply with the aforementioned rule.
The following teams will take part in the league this year; Karachi United (DFA South; host), Burma Mohammedan (Korangi; Defending Champion), Kalri Star (Lyari), Korangi Baloch (Sharafi), Azam Sports (Drigh Road), Al Shahbaz (Korangi), Baloch Mujahid (Ibrahim Hyderi), FC Rovers (Malir Cantt), Baloch Youth (Garden), Irfan Memorial FC (Landhi).
RIAZ AHMED, Organising Secretary
0300-3926092 – 0323-2843199
0 Al-Shabaz, AMANSPORTS, Baloch Mujahid, Baloch Youth Garden, Burma Mohammadan, fc rovers, Kalri Star, Karachi, Karachi Football League, Karachi United, KFL 2013, Korangi Baloch, KUFC, Riaz Ahmed
Zavisa may get another chancePak U13s will do well in Mashhad: coach
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{"url":"https:\/\/acooke.org\/cute\/SalaamBomb0.html","text":"## Salaam Bombay - Indian Restaurant in Santiago\n\nFrom: andrew cooke <andrew@...>\n\nDate: Sun, 17 Jul 2011 11:50:04 -0400\n\nThere are only a handful of Indian restaurants in Santiago. The Majestic has\nbeen around for years, and Soul of India in Vitacura has a good reputation,\nbut those are both pretty expensive, so I thought Salaam Bombay in Rancagua\n(Providencia) was interesting.\n\nI have to say - I liked it. I say that with more reservation than normal\nbecause I've been reading reviews on the 'net and they seem to be pretty\nnegative.\n\nWe had aloo matar, a dal, lamb korma, rice, and a gralic nan. Nothing was\n\"hot\" (in the spicey sense), but it was all very tasty, freshly prepared (by\nwhich I mean, for example, that the potatoes and lentils weren't very mushy),\nand each plate has a distinctive flavor. The lamb was pretty gamey, which was\na surprise, but not a bad thing (and the sauce wasn't too heavy, with lots of\nalmonds).\n\nAddressing the criticisms I've seen elsewhere: the service was fine (neither\nexcellent nor bad); the nan was pretty good (it could have been fluffier, but\nit wasn't \"sad\" and it was made with lots of fresh garlic); I can't vouch for\nthe authenticity of the food (one comment suggests that the owners are\nsouthern Indians, preparing northern style food), but it seemed to me like it\nwas \"toned down\" for Chilean taste buds (which are not generally accustomed to\nspicy food).\n\nThe entire meal (with beer for me and a coke for Pauli) came to a shade under\n20.000, including tip. That's the same as an expensive starter at Soul of\nIndia... I think it's worth visiting, when you have the curry urge.\n\nAndrew","date":"2021-09-28 16:15:17","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.17336443066596985, \"perplexity\": 8303.34843344005}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-39\/segments\/1631780060877.21\/warc\/CC-MAIN-20210928153533-20210928183533-00057.warc.gz\"}"}
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Last House on Dead End Street, originally released as The Fun House, is a 1977 American exploitation horror film written, produced, and directed by Roger Watkins, under the pseudonym Victor Janos. The plot follows a disgruntled ex-convict (also played by Watkins) who takes revenge on society by kidnapping four acquaintances and filming their murders in an abandoned building.
Watkins, a student at the State University of New York at Oneonta, devised the concept for the film after reading the Charles Manson biography The Family (1971) by Ed Sanders. Commissioning a cast from the university's theater department, Watkins shot the film inside an unused building on the university campus in the winter of 1972, on a budget of around $3,000.
Screened under the title The Cuckoo Clocks of Hell at the 1973 Cannes and Berlin Film Festivals, Watkins's original cut of the film (now lost) ran approximately three hours in length. A truncated version of the film was released theatrically in 1977 under the title The Fun House. In 1979, Cinematic Releasing Corporation acquired distribution rights to the film and re-released it under the title Last House on Dead End Street, capitalizing on the popularity of Wes Craven's The Last House on the Left.
In the decades following its release, Last House on Dead End Street was subject to various rumors about who had created and starred in it, as the entire cast and crew were credited using pseudonyms. This resulted in speculation that the film might have depicted actual murders. In 2000, Watkins publicly came forward and confirmed himself as the director, writer, and lead actor. Two years later, the film was released for the first time on DVD, through participation from Watkins, and with the actual names of the actors revealed. The film has continued to be a point of discussion among film scholars, largely due to its metafilmic qualities, surrealist imagery, and themes surrounding the aestheticization of violence in cinema.
Plot
Terry Hawkins has just been released from prison after spending a year incarcerated for drug charges. An amateur filmmaker, Terry claims to have previously made stag films that he was unable to sell. Convinced that modern audiences crave more extreme content, Terry decides to make a real snuff film. After choosing a large abandoned college as the setting of his film, Terry secures financing from an unsuspecting film company run by an openly gay film executive named Steve Randall. Terry rounds up a group of disparate women and men— some of them amateur filmmakers— who are willing to help make his film. Among them are filmmaker Bill Drexel; untrained actresses Kathy and her friend Patricia; and Ken, one of Terry's longtime acquaintances and a former pornographic actor.
For their first scene, Patricia and Kathy, wearing translucent plastic masks, lure a blind transient to the building. The women fondle him before Terry, donning a Greek tragedy-like mask, strangles the man to death while Bill films the murder. Meanwhile, pornographic director Jim Palmer, a peer of Steve's, waits anxiously for a party he is attending to end; his wife Nancy, done up in blackface, is whipped repeatedly in front of party guests as part of a sex game. Jim complains to Steve that people's tastes are becoming "hard to satisfy." The next day, Terry arrives at Jim and Nancy's home, and finds Nancy alone. He introduces himself as a mutual friend of Ken, with whom Nancy has previously appeared in adult films. Terry quickly seduces Nancy before showing her the footage of the blind man's murder in an attempt to convince her to ask Jim if he will invest in the picture. She is shocked by how realistic the footage looks; Terry confesses that it is in fact real, and rapes her.
The following morning, Terry calls Steve and asks him to stop by the building to visit the film set; he also inquires about a young actress named Suzie Knowles for a part in his movie. Steve arrives later that night, and is confronted by Terry and his crew inside the building, all of them wearing masks. Steve is knocked unconscious, and awakens to find himself tied up alongside Nancy and Suzie. Terry and his crew brand Suzie across her chest with a hot iron before Terry slashes her throat. Later, Terry goes to meet Jim at his office and kidnaps him. Back at the building, Terry and his crew beat Jim to death while Bill again films the crime. They then take an unconscious Nancy and tie her to a large dining table. She awakens to Bill filming her, while Terry uses a hacksaw to dismember her legs before they eviscerate her with gardening shears. During the mutilation, they periodically revive her consciousness with the aid of smelling salts before she bleeds to death.
Terry and his crew confront Steve with the corpse of the blind transient they killed earlier, and welcome him "back to the edge." Steve flees through the building, and is confronted in the basement by Terry, who tackles him to the ground. Bill emerges from a dark corridor with his camera while Kathy and Patricia taunt Steve. Patricia removes her mask and takes off her blouse, exposing her breasts. She unbuttons her pants, revealing a dismembered goat hoof she has held between her legs. As the group taunt Steve, Terry forces him to fellate the goat hoof. Steve escapes, but is cornered in an empty room, where a row of spotlights suddenly light up. Terry and his crew, armed with a power drill, approach Steve, plunging the drill bit through his eye socket, killing him. One by one, they slowly back away from Steve's body and disappear into the darkness.
As the scene fades out, a voiceover states that Terry, Bill, Ken, Patricia, and Kathy were apprehended and are in a state penitentiary.
Cast
Analysis
Some film scholars have noted Last House on Dead End Streets unique preoccupation with and self-reflexivity regarding the aestheticization of violence. In Holy Terror: Understanding Religion and Violence in Popular Culture (2010), film scholars Gerry Carlin and Mark Jones point out the film's similarities to the Manson family murders, and note its religious undertones, which are amplified by the film's choral soundtrack: "Though principally a metafilmic commentary on cinematic perversity, the [lead character]'s gnomic pronouncements, and the excessively ritualized evisceral 'sacrifices' allude to a debased if fundamentally unrepresentable religiosity." Horror film critic and scholar Chas Balun echoes a similar sentiment, writing in 1989 that, "Last House on Dead End Street proves especially unsettling in the manner in which it blurs the lines between recording, inciting and participating in an act of violence. Other films... have similarly broached the subject, but none have been so alarmingly forthright and worrisome."
Film scholars Bill Landis and Michelle Clifford described the film as sticking to "a consistent, purposeful style that builds to the foulest mood imaginable, creating its own world of hate. In that unintentional exploitation way, it reproduces the Panic Theater extremes of surrealism, where evisceration is the metaphor for the sex act," likening it to the "bastard cousin of Otto Muehl." Landis and Clifford also note the self-reflexive framing of the film, deeming it "a film within a film motivated by a hate of pornography and the swingers who create it." Because of Terry's focus on murdering individuals connected to the pornographic film industry, Landis and Clifford herald the film as "the ultimate sexual revenge movie... What goes around comes around in the underground porno world, and Terry Hawkins makes this code his law, filming his targets as they take their last undignified breaths."
Production
Concept and filming
Last House on Dead End Street was conceived by Roger Watkins, a student at the State University of New York at Oneonta (SUNY Oneonta), in 1972. Watkins was inspired to write the screenplay after reading the Charles Manson biography The Family (1971) by Ed Sanders, which focused on the Manson family murders in Southern California. The project was initially conceived as a straightforward biopic about the Manson family, but morphed into a feature about a disgruntled ex-convict who decides to make snuff films with a group of degenerates. Though Watkins was studying English literature, he became interested in filmmaking, and befriended several students in the university's film department. Through events sponsored by the film department, Watkins was able to meet directors Otto Preminger and Nicholas Ray, both of whom he idolized. Preminger took a liking to Watkins, and gifted him a Bolex camera, which Watkins used to film the simulated snuff footage featured in Last House on Dead End Street.
In casting the film, Watkins chose to star as Terry, the ringleader of the snuff filmmakers, and commissioned a cast exclusively consisting of current or former students of the theater department at SUNY Oneonta. The film was shot in December 1972 in an abandoned building on the university campus known as Old Main. The building, derelict at the time, was demolished in 1977. Watkins would later reveal that, at the time of the making the film, he was an amphetamine addict, and that only $800 of the $3,000 budget was actually spent on making the film; the remaining $2,200 was used to purchase drugs. The film's working title was And at the Hour of our Death.
Musical score
Due to budget constraints, the majority of the film's score and audio effects were sourced from composer and ethnomusicologist David Fanshawe, via KPM Musichouse audio library. In additional to Moog synthesizer pieces, the musical score incorporates Gregorian chants among other ambient noises. The film's original musical score was released for the first time on vinyl February 12, 2016 by the independent labels Vombis and Light in the Attic.Track listing'''
Release
The original 175-minute version of the film was titled The Cuckoo Clocks of Hell, inspired by a quote in the Kurt Vonnegut novel Mother Night (1962). The film was screened under this title at the Cannes Film Festival on May 1, 1973, followed by the Berlin International Film Festival on July 14. Watkins alleged that the film caused riots when subsequently shown in New York City and Chicago; it was apocryphally claimed that a theater in Chicago was burned down in a riot that occurred during a screening. This version has been lost, with the original negatives either destroyed or missing. Subsequent theatrical distribution was delayed after one of the actresses sued Watkins, worried that a nude scene in the film would impede her chances of finding success as a Broadway actress.
In May 1977, the film was released in a truncated cut as The Fun House, screening at drive-in theaters throughout Connecticut, as well as in Shreveport, Louisiana, and Millville, New Jersey. It subsequently screened at a drive-in in Ithaca, New York beginning June 3, 1977, paired as a double feature with Mark of the Devil. Independent film distributor Cinematic Releasing Corporation subsequently acquired rights to the film and had re-released it as early as March 1979 (Miami, Florida) under the title Last House on Dead End Street, a play on the title of the Wes Craven film The Last House on the Left (1972). It opened in New York City under this title on December 28, 1979. Vernacular historian Bill Landis note that, during the film's New York screenings—which occurred primarily in grindhouse theaters in Times Square—"people on the Deuce sat stunned and sickened, but unable to leave their seats." In the United Kingdom, the film was banned by the British Board of Film Classification; Tobe Hooper's slasher film The Funhouse (1981) was also banned by association, as the films had almost identical titles.
For over two decades following the film's theatrical release, the true identities of the director and cast were unknown to the public, as the names given in the credits were pseudonyms. This sparked various rumors surrounding the film, the most well-known being that it had emerged from underground cinema circles in New York, and featured footage of actual murders. In his 1995 book Killing for Culture, David Kerekes stated: "Any attempt to trace the names behind Last House on Dead End Street will lead no further than the credits themselves, all obviously false." However, Chas Balun's 1989 review had in fact identified the film's director as "a young New York film student named Roger Watkins."
In December 2000, a contributor posting as "pnest" on the Internet messageboards of FAB Press (a publishing house devoted to cult movies), claimed to be the director, writer, producer, and editor of the film, "Victor Janos." The poster later revealed his identity as Roger Watkins. After the release of Last House on Dead End Street, Watkins had had a career as a pornographic film director, under the pseudonym Richard Mahler.
Critical response
AllMovie wrote, "This notorious exercise in low-budget gore is poorly edited and photographed, but its catalogue of horrors and a genuinely nasty tone make it worthwhile for fans of sick cinema," drawing comparisons to the Manson family killings. Eric Campos of Film Threat wrote, "It's not only the intense gore contained within these 78 minutes that has led many to label this film as the most vile ever made, but it's also the drab, dreary settings and the assortment of malcontents you're forced to put up with if you want to make it to the other end of this ride. Nothing that has to do with this film is happy or light and the film itself, even though presented nice and clear on this DVD, appears to be covered in dirt." Anton Bitel, writing for Film4, called the film "dirt cheap and deeply flawed, but still worth enduring, for even if the deaths are faked, there's a real enough intelligence behind it all."
In Slimetime: A Guide to Sleazy, Mindless Movies, Steve Puchalski wrote that the film is "harsh, repellent, and thoroughly unlikeable. It's effective much in the same way a large mallet to the temple is effective, but that doesn't mean it's any fun to sit through." Michael Weldon, in the Psychotronic Encyclopedia, similarly notes: "Have you ever heard anyone even admit they saw it? I wish I couldn't." Scholars Bill Landis and Michelle Clifford describe the sequence in which Nancy is dismembered alive as "one of the most revolting scenes in exploitation history... guaranteed to reduce even the most jaded viewers to quivering disgust." Critic Chas Balun wrote that the film "delivers a mule kick to the old nugget sack with a loathsome, virulent fury." Aside from the postscript describing the killers' incarceration after the murders, Balun states that the film "shows absolutely no moral equilibrium whatsoever." Writer Stephen Thrower similarly suggests that the film possesses "a forbidding, hostile vibe, a malignant radiation that sends your toxicity meter haywire... What gives it unique status is the aura of pure hatred that oozes out of every pore of the project."TV Guide was highly critical of the film, writing "This vain attempt to combine splatter with a commentary on the viciousness of the movie business fails miserably on all counts." Alternately, Jay Alan of HorrorNews.net gave the film a favorable review. While admitting the film looked cheaply made, and featured poor audio and dubbing; Alan commended the film's gore effects, decent acting, disturbing atmosphere and tone, writing, "While it may not neighbor to the caliber of Last House on the Left, it is truly a house worth at least visiting at least once".
Home mediaLast House on Dead End Street was scarcely released on VHS in the United States, and was made available on video through Venezuelan distributors in the late 1980s and early 1990s. A two-disc DVD set of the film was released in 2002 by Barrel Entertainment. The set features major contributions from director Watkins, who was heavily involved in its production. This edition was also released in Australia through Hard Corps Entertainment. A separate Region 2 DVD edition was released in the United Kingdom as part Tartan Films' Grindhouse series.
In a 2015 interview with Joe Rubin, the co-owner of the cult DVD and Blu-ray label Vinegar Syndrome, he stated that the company was preparing a restored Blu-ray release of the film.
Though there has been no update about a standalone Vinegar Syndrome release, the uncut theatrical version of the film is available in 2K HD as a hidden feature on Vinegar Syndrome's release of Corruption'' on Blu-ray.
See also
List of American films of 1973
Snuff films
Exploitation films
List of incomplete or partially lost films
Notes
References
Sources
External links
1977 films
1977 horror films
1973 films
1973 horror films
1970s avant-garde and experimental films
1970s lost films
1973 independent films
1977 independent films
American exploitation films
American films about revenge
American independent films
American splatter films
1970s exploitation films
Films set in abandoned houses
Films shot in New York (state)
Obscenity controversies in film
Films about snuff films
Films directed by Roger Watkins
1970s English-language films
1970s American films
Art works that caused riots
Films originally rejected by the British Board of Film Classification
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|
{"url":"https:\/\/codereview.stackexchange.com\/questions\/174182\/using-a-radix-tree-to-find-case-insensitive-matches-in-large-files","text":"# Using a radix tree to find case-insensitive matches in large files\n\nI'm using a radix tree library to find case-insensitive matches from several relatively large files (~24 files each on the order ~150MB).\n\nMy current algorithm requires me to read the chars of an entire file zipped with their index because finding the offset of match in the file is a requirement.\n\nCurrently the processing is quite slow because I am doing redundant work on each new character. However, the alternative is to create subtries, which is memory intensive.\n\nMy (not well-commented) code is below... If anything is unclear, I'm happy to comment it more thoroughly.\n\nimport com.rklaehn.radixtree.RadixTree\n\nimport scala.io.Source\n\nimport java.io.File\n\ndef fromFile(filename: String): RadixTree[String, Boolean] =\n\ndef fromLines(lines: Seq[String]): RadixTree[String, Boolean] = {\n\/\/ we mark a target as a terminal by making the associated value true\n\/\/ all substrings will not have a true value, so only exact matches\n\/\/ are true\nval pairs = lines.map(line => (line.toLowerCase, true))\n}\n}\n\n\nfinal case class Match(filename: String, offset: Int, target: String) {\ndef hexOffset: String = offset.toHexString\n}\n\nclass FileProcessor(private val filename: String, private val trie: RadixTree[String, Boolean]) {\n\nprivate val charAndIndex: Iterator[(Char, Int)] = Source.fromFile(filename).zipWithIndex\n\nprivate def isTarget(t: RadixTree[String, Boolean], prefix: String): Boolean =\nt.getOrDefault(prefix.toLowerCase, false)\n\nprivate def isEmptyWithPrefix(t: RadixTree[String, Boolean], prefix: String): Boolean = {\nval subtrieKeys = trie.subtreeWithPrefix(prefix).keys\nsubtrieKeys.isEmpty || (subtrieKeys.size == 1 && subtrieKeys.head.isEmpty)\n}\n\n\/\/ this is the bottleneck -- all chars in sb were in the trie, but\n\/\/ we are rechecking them each time.\n\/\/ alternative is to pass a a trie into this, costing memory\nprivate def processCurrent(matches: Seq[Match], sb: StringBuffer, c: Char, i: Int): (Seq[Match], StringBuffer) = {\nsb.append(c)\nval s = sb.toString.toLowerCase\n(isTarget(trie, s), isEmptyWithPrefix(trie, s)) match {\ncase (true, true) => (matches :+ Match(filename, i - s.length, s), new StringBuffer)\ncase (true, false) => (matches :+ Match(filename, i - s.length, s), sb)\ncase (false, true) => (matches, new StringBuffer)\ncase _ => (matches, sb)\n}\n}\n\ndef findMatches: Seq[Match] = {\ncharAndIndex.foldLeft(Seq.empty[Match], new StringBuffer) { case ((acc, sb), (char, idx)) =>\nprocessCurrent(acc, sb, char, idx)\n}._1\n}\n}\n\n\nclass DirectoryProcessor(dir: String, targetFile: String) {\n\nval files: Seq[File] = new File(dir) match {\ncase d if d.exists && d.isDirectory => d.listFiles.filter(_.isFile)\ncase _ => Nil\n}\n}\n\nprivate val files = new DirectoryReader(dir).files\n\n\/\/ Processing a file is read-only, so we can do this in parallel\ndef processDirectory: collection.parallel.ParSeq[Match] = files.par.map(f => new FileProcessor(f.getPath, trie))\n.flatMap(_.findMatches)\n}\n\n\nClearly I could remove unnecessary work by passing a trie (or subtrie) to FileProcessor#processCurrent where I'll traverse based on the current character, but I believe this would make the memory usage rather intractable for large enough tries.\n\nAre there any easy optimizations that I've missed?\n\n\u2022 You posted some code and asserted \"it is slow\". You chose not to profile it, or at least you chose not to share profiling results with us. Please describe which specific source lines account for the bulk of the processing time. We could guess, but it would be better to know. Also, if you have a unit test that exercises the code, I wouldn't say \"no\" to reading the test, especially if it relates to the profiling run. \u2013\u00a0J_H Aug 29 '17 at 4:21\n\u2022 @JH I think I did you one better by addressing exactly why it's slow. From an algorithmic standpoint, I'm at a decision between speed and memory: recurse on expensive subtries, or recheck prefixes. \u2013\u00a0erip Aug 29 '17 at 10:21\n\n# time complexity\n\nThere is one troubling call in processCurrent that jumps out at me - the downcasing. I get the sense that sb becomes \"big\", so the nested inner loop for the linear scan seems like trouble. It's the sort of thing that can easily turn e.g. a linear algorithm into a quadratic one. I understand that case matters for correctness, but just humor me for a moment. Imagine an alternate implementation that calls toLowerCase at file read time, destroying valuable information but allowing processCurrent to happen quicker. With such results in hand as a valuable hint, could you coax the current implementation to report results faster? As a simpler alternative, could we replace all comparisons in the current implementation with case-insensitive comparisons? Maybe downcase data when handing it to the radix tree library. The point is to avoid repeatedly downcasing a given character many thousands of times - we want to do localized work on our current focus rather than doing (ignored) global work.\n\n# memory complexity\n\nThere are a few large memory consumers such as storing Match hexOffsets, and storing charAndIndex tuples rather than generating them. Their cost is \"just\" linear with filesize, so maybe it's an acceptable tradeoff in the name of clarity, but you might think about what the code would look like if you deleted a large ancillary object (or made it smaller). Perhaps code that depended on the large object could get by within a smaller footprint? You're memory constrained and you want parallelism, so reducing memory by 2x would let you keep twice as many cores busy.\n\n## varying file sizes, and stragglers\n\nThe parallel mapping that invokes FileProcessor is very nice, it is simple and clear. But you complain about memory pressure, and I think your input files are of diverse sizes. This leads to two suggestions: consider viewing files as a series of fixed-length subproblems to work on, and consider throwing lots of tasks into a queue that a fixed number of worker bees draw from. Then you can size it so there's about N workers for N cores, each using about 1 \/ Nth of memory (either 1 \/ Nth of heap or 1 \/ Nth of cache, whichever matter more to your elapsed times). If the current code encounters a mix of files where the largest is, say, double the size of the smallest, then it's likely to be a straggler that impacts elapsed time while most of memory and most of cores are idle.\n\n# style\n\nInstead of trie, it would have been helpful to me if you had written private val targetTrie, parallel with targetFile which is a nice informative identifier. It would be very helpful to write down pre- & post-conditions that processCurrent maintains.\n\n\u2022 Good eye on the toLowerCase calls. As a part of comparing times, I'm writing my own naive trie which cuts down on the toLowerCase calls and reading processing files uses iterators with filters and Character#toLowerCase calls. As for Match#toHexOffset -- this is generated rather than stored. The rest of the points are quite nice. Once my evaluation compared to my personal trie is completed, I might add it as an update to the original question or even a second answer. \u2013\u00a0erip Aug 30 '17 at 16:14\n\u2022 \"As for Match#toHexOffset -- this is generated rather than stored. \" Oh, yeah. Cool! (And a 2nd answer would be a good thing.) \u2013\u00a0J_H Aug 30 '17 at 16:32\n\nSwap a linear operation for a constant one\n\nCurrently the processCurrent function is appending (:+) Match objects to matches. matches is a Seq so this append operation has a complexity that is linear in the length of matches. You can remove this overhead by refactoring matches into a ListBuffer which can perform the append operation in constant time.\n\nIf you wish you can check out this link for more details on performance characteristics of various Scala collection.\n\nA minor redundancy\n\nIt looks like you could remove the toLowerCase that occurs within isTarget because it is called immediately before on the same String object in processCurrent.","date":"2019-09-19 17:40:07","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.2822185456752777, \"perplexity\": 4629.008230745546}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-39\/segments\/1568514573561.45\/warc\/CC-MAIN-20190919163337-20190919185337-00451.warc.gz\"}"}
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Presents the solution of a Bernoulli differential equation involving a negative power. Implicit differentiation is used.
This lesson goes over the parts of the human respiratory system and the gas exchange process that occurs in the lungs. It also covers changes in the respiratory system that occur during spaceflight, such as decreased lung capacity.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 6,094
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So you've signed up to ResDiary and you're wondering how to make the most of those commission-free bookings. Whether you've moved from another system, or you've come from pen and paper, you need to maximise your SEO (search engine optimisation) to make sure that your site and your ResDiary microsite appear high on Google.
SEO is the multiple ways that you can ensure your site comes at the top of online search results. There's no sure-fire way to do this, but if you follow our tips below, you'll be well on your way.
Name: In your ResDiary autobuild form, put your venue's name and location in "Restaurant Name." This helps your customers distinguish which venue to pick if there are other places with similar names to you.
Content: We would recommend having at least 100 words written about your venue in your autobuild form. You should include your venue name and location (Eg: The Old Inn, Glasgow) at least three times in this text. This allows Google's bots to understand what the page is about and where it is based, helping with those all-important search results.
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So they're not important, right? Wrong.
Google's bots assume that if you have a link to a booking site, then you're closely related. Even if your booking page on their site is dead, this link still tells these bots that you'd like them to appear in relation to you. This means that OpenTable/Bookatable etc. will appear high on Google when people search your venue's name. Remove that link and they'll start slipping down.
Some of these links may be hidden in your site's source code. If you're not familiar with web design, it's a good idea to get your website developer to look into this for you.
You may have a Winner's badge on your site that you'd like to keep. Ask your web developer to make this a "nofollow" link and Google will ignore it.
You might think that you're listed on the ResDiary portal and your online bookings will come flowing. You're partly correct, but you also need to have a link to your ResDiary microsite on your own site. We mentioned domain authority before and this will help with improving it.
By linking to ResDiary, you're linked to a domain with good authority,meaning you're more likely to rise up the search rankings. Also, you linking to ResDiary means that we are classed as having a connection to you, pushing your microsite higher up the listings than sites you will pay commission for. The best way to do this is by linking your widget or book button to your ResDiary microsite.
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{
"redpajama_set_name": "RedPajamaC4"
}
| 4,871
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\section{Introduction}
The way our opinions are formed and change over time giving
rise to emerging collective phenomena is a topic that attracts
a rapidly increasing interest. Apart from its clear relevance
for fundamental issues, such as the stability of democracy and
the preservation of individual liberties,
opinion dynamics is a paradigmatic example of a social
phenomenon that can be quantitatively studied by a combination of
theoretical and empirical
approaches~\cite{lazer2009,castellano2009,acemoglu2011,Sen2014,Noorazar2020}.
Traditional models for opinion dynamics focus on the
interaction among a large number of peers, often in the presence
of external fields -- possibly varying over
time but equal for all agents~\cite{michard2005} --
describing the effect of conventional media, such as television and the press,
acting in the same way on all individuals.
The advent of the Internet, and the Online Social Network (OSN) revolution
in particular, have made the scenario more complex.
Although in principle the Internet allows users to access an unprecedented
diversity of news and viewpoints, this abundance is overwhelming and
pushes users to rely on automated recommendation systems to cope with
information overload. To provide successful recommendations web sites
and social networks constantly track our online activity and, based
on it, they propose personalized ``information'', which varies not only
over time, but also from person to person.
Examples of this phenomenon are posts appearing on Facebook news feed,
which are chosen and ordered by the social network on the basis
of previous interactions with other posts, or Google's Personalized PageRank.
The dependence on past user behavior coupled to the human inclination
to favor sources confirming one's own preferences~\cite{iyengar2009}
creates a feedback loop which drastically reduces the huge diversity of
available content.
The selective exposure of individuals to a biased representation
of the world (so that they remain confined within their
``filter-bubble'') is thought to play a crucial role in shaping
opinions at all scales~\cite{Bakshy2015}.
A most worrying aspect of this mechanism is the possible
reinforcement of personal biases
with the consequence of favoring radicalization phenomena~\cite{Maes2015}.
An astonishing example in this sense is the recent revelation that
64\% of people who join extremist groups on Facebook
are recommended to do so by the Facebook algorithm itself~\cite{Horwitz2020}.
It is therefore crucial to properly understand the effect of
personalized information or advertising on the dynamics of opinions.
Some modeling efforts in this direction have already been done.
Perra and Rocha~\cite{Perra2019} studied a binary opinion dynamics
where each user updates her opinion based on the opinion of
others filtered in various ways.
In the framework of continuous opinion dynamics with bounded
confidence~\cite{castellano2009}, in
Ref.~\cite{Sirbu2019} the effect of recommender algorithms in OSN
is mimicked by enhancing
the probability to interact with individuals having close opinions.
An increased tendency toward fragmentation (no consensus) and
polarization (clusters with distant opinions) is observed.
A model for polarization in continuous dynamics has been recently
presented and compared with empirical results by
Baumann et al.~\cite{Baumann2020}.
In this manuscript, we consider arguably the simplest possible
type of opinion dynamics, the voter model, and study in detail
its behavior in the presence of an external personalized information,
modeled, in its turn, in an extremely simple way.
The fundamental questions we want to answer are whether, in
this simple setting, selective exposure prevents the reaching
of consensus and how this comes about.
Is a minimal amount of personalized information sufficient to
lead to a polarized state? Is consensus possible for a very
strong influence of personalized information on opinions?
On which timescale is consensus, if any, reached?
What is the role of the system size?
By means of an analytical approach, corroborated by numerical
simulations, we fully understand the model behavior and thus
get complete answers to all these questions.
The rest of the paper is organized as follows.
In Section~\ref{secII} we introduce the voter model modified
by the presence of personalized information.
Its analytical investigation is described in Section~\ref{secIII},
divided in three subsections, dealing with different values of
the parameter $c$. The final Section summarizes and discusses the results
and presents some perspectives. Several Appendices contain details
of the analytical calculations.
\section{The voter model with personalized information}
\label{secII}
In voter dynamics~\cite{clifford1973, holley1975}
individuals are endowed with a binary opinion (spin)
$s_i = \pm 1$; at each time step a spin is randomly extracted and
its value is replaced with the value of one of the spins it is
connected to. In other words an individual becomes equal to a randomly
chosen neighbor.
The model has been studied extensively, both in regular lattices
and on complex networks~\cite{dornic2001, sood2005, castellano2009, Pugliese2009, Suchecki2005, FernandezGracia2014, Carro2016}.
It is well known that this dynamics always leads in finite systems
to full consensus -- i.e. all spins get aligned after a certain amount
of time -- and that this is due only to stochastic
fluctuations~\cite{Rednerbook}.
In order to understand the most basic effects of personalized
information on opinion dynamics, we couple the voter model
to a simple source of personalized information,
which feeds back on each individual a signal depending on the
past evolution of her own opinion.
Other recent works have investigated the effect of opposing
(but fixed) sources of external information on voter
dynamics~\cite{Bhat2019,Bhat2020}. Our model is also similar
to a voter dynamics with aging, recently introduced by
Peralta et al.~\cite{Peralta2020}. The main difference between
that model and the present one is that the effective memory
in our model is never erased, while it is, when a spin flips,
in the model of Ref.~\cite{Peralta2020}.
Let us consider $N$ agents distributed over the nodes $i$ of a network.
Each agent can assume two states $s_i=\pm 1$, that
correspond to two different opinions, and interacts with the agents it
is connected to. We define the adjacency matrix $A_{ij}$ so that
$A_{ij}=1$ if spin $i$ and $j$ are linked and
$A_{ij}=0$ otherwise. With this convention the number of agents a
given spin $s_i$ is connected to is simply $k_i = \sum_j A_{ij}$.
The evolution of each individual depends also on another variable,
a ``personalized external information'' $e_i$.
This last quantity is a random variable $e_i=\pm 1$ assuming the
positive value with a probability $P[e_i(t)=+1]$ that changes over time
depending on the history of the agent's opinion.
The dynamics takes place as follows.
Initially each spin is set to $s_i=\pm 1$ with equal probability.
At each time step, a given individual $i$ is selected at random and,
with probability $1-\lambda$, she follows the usual voter dynamics: Her
state $s_i$ is made equal to the state of a randomly selected neighbor
$s_j$.
With complementary probability $\lambda$,
the individual copies the state of the external source:
\begin{equation}
s_{i}(t+\delta t)=
\begin{cases}
e_i(t) \ \text{with probability}
\ \lambda \\ s_j(t) \ \text{with
probability} \ \frac{1-\lambda}{k_i},
\end{cases}
\label{eq:voter}
\end{equation}
where $\delta t = 1/N$, $j$ is one of
the neighbors of $i$ (i.e., $A_{ij}=1$) and $N_i$ is the
total number of such neighbors.
Pictorially, we are adding another layer of ``external agents''
$e_{i}(t)$, each of them coupled only to the original agent
$s_{i}(t)$ and influencing her in the same way as the other neighbors,
except for a different probability of interaction. See Fig.~\ref{fig:vpiTxt} for a graphical representation of the model.
\begin{figure}
\includegraphics[width=0.35\textwidth]{vpiTxt-eps-converted-to.pdf}
\caption{\textbf{Voter model with personalized information.}
Graphical representation of the voter model with personalized
information. Blue circles represent the agents $s_i$
that interact following the usual voter dynamics, while red circles are
the external agents, carrying personalized information. Note that the
external agent $e_i$ influences only the corresponding voter
spin, namely $s_i$.}
\label{fig:vpiTxt}
\end{figure}
To mimic the reinforcing effect of personalized information we assume
that, whenever the spin $i$ is selected for update, then
$P[e_i(t)=+1]$ changes, increasing the probability that $e_i$
will be in the future equal to the current state of the agent, $s_i$.
More precisely at each time step, the update of the probability occurs after the update of the opinion variable. In other words, one first updates
the opinion variable (which may or may not change) and afterwards increases by a factor $c$ the ratio between $P[e_i=s_i(t+\delta t)]$ and $P[e_i=-s_i(t+\delta t)]$ where $s_i(t+\delta t)$ is the opinion variable after the update:
\[
\frac{P[e_i=s_i(t+\delta t)]}{P[e_i=-s_i(t+\delta t)]} \to
c \frac{P[e_i=s_i(t+\delta t)]}{P[e_i=-s_i(t+\delta t)]}.
\]
In this way the time-depending probability $P(e_i=+1)$
keeps track of the history of the spin $s_{i}$.
For example if agent $i$ stays in state $s_i=+1$ for a long time,
then $P(e_i=+1)$ tends to grow toward 1 and this makes more likely
that opinion $s_i=+1$ is maintained. The polarizing effect of this
personalized source of information is clear.
The parameter $c$ determines the speed at which the balance
between the two alternatives is disrupted.
Notice that the change for the probability $P(e_i=+1)$ occurs
at each update of agent $i$, even if the latter does not actually
change opinion (because the agent interacts with a neighbor already
sharing the same state).
It is useful to define the quantity
\[
n_i(t)=\sum_{t'=1}^t s_i(t'),
\]
which keeps memory of the evolution of agent $i$'s opinion.
Assuming that initially no knowledge about the agent's preferences
is available and therefore external information is fully
random $P[e_i(0)=+1]=1/2$ we can write
\begin{equation}
P[e_{i}(t)=1] = P[n_i(t)]=\frac{c^{n_i(t)}}{1+c^{n_i(t)}}.
\label{eq:evolution_P(n_i)}
\end{equation}
Hence, a positive (negative) value of $n_i$ implies that
personalized information is more probably equal to
$e_i=+1$ ($e_i=-1$).
To give a qualitative idea of the model behavior we report in
Fig.~\ref{fig:full} the temporal evolution of the magnetization
$m(t) = \sum_i s_i/N$ for different values of the probability
$\lambda$ of interaction with the personalized information.
\begin{figure*}
\includegraphics[width=0.49\textwidth]{c2_1.pdf}
\includegraphics[width=0.49\textwidth]{c2_2.pdf}\\
\includegraphics[width=0.49\textwidth]{c2_3.pdf}
\includegraphics[width=0.49\textwidth]{c2_4.pdf}
\caption{\textbf{From consensus to polarization.}
Temporal evolution of the magnetization $m(t)$
for different values of the probability $\lambda$,
starting from $\lambda=0$ (pure voter
dynamics). Agents form a complete graph of size $N=500$.
The trajectories of 10 different runs are displayed.}
\label{fig:full}
\end{figure*}
For $\lambda=0$ the system is exactly the usual voter dynamics and
it reaches consensus $m = \pm 1$ because of random diffusive fluctuations,
in a time of order $N$~\cite{Rednerbook}.
As personalized information is turned on, consensus is still reached,
but surprisingly over {\em shorter} time intervals and
it is clear that drift plays now a relevant role.
Increasing $\lambda$ further we observe that some runs do not reach
consensus any more and magnetization fluctuates around some constant
value. Finally, for large $\lambda$ consensus is never reached and all
runs remain stuck in a disordered state.
In the next section, through a mean-field analytical approach,
we understand when and how consensus is reached,
depending on the values of the parameters
$c$ and $\lambda$.
\section{Analytical results}
\label{secIII}
The evolution of a system following Eqs.~\eqref{eq:voter} and
\eqref{eq:evolution_P(n_i)} depends on the topology of the
network defining the interactions among spins. In the following we
will focus on complete graphs, for which each pair of spins is
equally likely to interact, meaning that $A_{ij}=1-\delta_{ij}$ for
any $(i, j)$.
This corresponds, in the absence of external information,
to the mean field limit of the voter model.
We denote by $N_{\uparrow}$ the number of spins in state $+1$, while
$N_{\downarrow}=N-N_{\uparrow}$ is the number of spins in the opposite state.
In these terms the updating process is
\begin{equation}
s_{i}(t+\delta t)=
\begin{cases}
e_i(t) \ \text{with probability} \ \lambda \\
+1 \ \text{with probability} \ (1-\lambda)\frac{N_{\uparrow}}{N}\\
-1 \ \text{with probability} \ (1-\lambda)\frac{N_{\downarrow}}{N},\\
\end{cases}
\label{eq:evolution_s_i_mean_field}
\end{equation}
where the probability of a positive external information
is given by Eq.~\eqref{eq:evolution_P(n_i)}.
Reminding that the magnetization can be written as
\[
m = \frac{N_{\uparrow}-N_{\downarrow}}{N},
\]
we obtain from Eq.~\eqref{eq:evolution_s_i_mean_field} that each time
a node is selected its value $s_i$ evolves according to
\begin{equation}
s_i(t)\! \to \! s_i(t+\delta t)\!=\!\!
\begin{cases}
\!+1 ~~\textrm{w. prob:}~(1\!-\!\lambda)\!\left(\frac{1+m}{2}\right)+
\lambda P(n_i) \\
\!-1 ~~\textrm{w. prob:}~(1\!-\!\lambda)\!\left(\frac{1-m}{2}\right)+
\lambda [1\!-\!P(n_i)].
\end{cases}
\label{eq:evolution_s_i}
\end{equation}
and, immediately after, $n_i$ is updated as follows
\begin{equation}
n_i(t) \to n_i(t+\delta t)=
\begin{cases}
n_i(t)+1 ~~~\textrm{if}~~~s_i(t+\delta t)=1 \\
n_i(t)-1 ~~~\textrm{if}~~~s_i(t+\delta t)=-1.
\end{cases}
\label{eq:evolution_n_i}
\end{equation}
Thus the state of each node is defined by the pair
$(s_i, n_i)$ and therefore the evolution of the system depends on the set
$\gra*{(s_i, n_i)}_{i=1}^N$. In Appendix~\ref{coefficients}
we calculate the drift and diffusion coefficients~\cite{Rednerbook}
for the magnetization and for the average value of the $n_i$
\begin{equation}
n=\frac{1}{N} \sum_i n_i.
\label{n}
\end{equation}
We obtain the magnetization drift
\begin{equation}
v^m =
2 \frac{\lambda}{N}\sum_i \left\{ \left(\frac{1-s_i}{2} \right) P(n_i) - \left(\frac{1+s_i}{2} \right) [1-P(n_i)] \right\}
\label{eq:drift_m_general}
\end{equation}
and the magnetization diffusion coefficient
\begin{eqnarray}
\label{eq:diffusion_m_general}
D^m &=& \frac{1-\lambda}{N}(1-m^2) \\ \nonumber
&+& \frac{2\lambda}{N^2}
\sum_i \left\{ \left(\frac{1-s_i}{2} \right) P(n_i) + \left(\frac{1+s_i}{2} \right) [1-P(n_i)] \right \}.
\end{eqnarray}
For the quantity $n$ the drift coefficient reads
\begin{equation}
v^n = (1-\lambda) m + \frac{\lambda}{N} \sum_i \left[2 P(n_i)-1 \right].
\label{eq:drift_n_general}
\end{equation}
while the diffusion coefficient is
\begin{equation}
D^n = \frac{1}{N}.
\label{eq:diffusion_n_general}
\end{equation}
These expressions contain $P(n_i)$ and therefore are
different depending on the value of $c$.
\subsection{The case $c=1$}
Let us first discuss the case $c=1$, that is equivalent to the noisy voter model or Kirman
model \cite{kirman1993ants}.
In this case Eq.~\eqref{eq:evolution_P(n_i)} reduces to
\[
P(n_i)=\frac{1}{2} \ \forall \ i,t.
\]
and therefore the variables $n_i$ do not play any role.
Setting $c=1$, Eqs.~\eqref{eq:drift_m_general} and
\eqref{eq:diffusion_m_general} reduce to
\begin{equation}
\begin{cases}
v^m_{c=1}=-\lambda m \\
D^m_{c=1}=\frac{1}{N}\qua*{\ton*{1-\lambda}\ton*{1-m^2}+2\lambda}
\end{cases}
\label{eq:nu_D_c=1}
\end{equation}
Differently from the standard voter model there is a nonzero drift
term, driving the system toward the disordered symmetric configuration
$m=0$. However, depending on the value of $\lambda$, the
system may still spend most of its time in the consensus state
$m=\pm 1$, that is no more absorbing.
See Refs.~\cite{Alfarano2005, artime2018aging, artime2018first}
for a detailed analysis of the noisy voter model.
\subsection{The behavior for $c \gtrsim 1$}
We now study the behavior for $c>1$, considering separately two cases.
We set $c=1+\delta$ and first take $\delta\ll 1$.
Under this hypothesis and focusing on short times
we can expand $P(n_i)$ to first order in $n_i \delta$, obtaining
\begin{equation}
P(n_i)=\frac{c^{n_i}}{1+c^{n_i}}=\frac{(1+\delta)^{n_i}}{1+(1+\delta)^{n_i}}\approx\frac{1}{2}+\frac{n_i\delta}{4}.
\end{equation}
Inserting this expression into Eq.~\eqref{eq:drift_m_general}
we get
\begin{equation}
v^m \approx \lambda \left(\frac{n \delta}{2} - m \right),
\label{eq:drift_m_delta}
\end{equation}
and analogously for the diffusion coefficient
\begin{equation}
D^m\approx \frac{(1-\lambda)}{N} (1-m^2) + \frac{2 \lambda}{N^2}
\ton*{1-\frac{\delta}{2N}\sum_i s_i n_i }.
\label{eq:diffusion_m_delta}
\end{equation}
Inserting the expansion of $P(n_i)$ into Eq.~(\ref{eq:drift_n_general})
we obtain
\begin{equation}
v^n \approx
(1-\lambda) m + \frac{\lambda \delta}{2} n.
\label{eq:drift_n_delta}
\end{equation}
In summary, combining Eqs.~\eqref{eq:drift_m_delta} and \eqref{eq:drift_n_delta}, the evolution of the system is given, as long as the condition
$|n_i|\delta \ll 1$ is satisfied for any $i$, by
\begin{equation}
\begin{cases}
\dot{m} = - \lambda m + \frac{\lambda \delta}{2} n \\
\dot{n} = (1-\lambda) m + \frac{\lambda \delta}{2} n,
\end{cases}
\label{eq:system_m_n_delta}
\end{equation}
where fluctuations due to diffusion have been neglected.
By integrating we find, under the assumption $\delta \ll \lambda$,
\begin{equation}
\begin{cases}
m = C_1 \frac{\delta}{2} \mathrm{e}^{t\delta /2} - C_2 \frac{\lambda}{1-\lambda}\mathrm{e}^{-\lambda t}, \\
n = C_1 \mathrm{e}^{t\delta/2} + C_2 \mathrm{e}^{-\lambda t},
\end{cases}
\label{mn}
\end{equation}
where $C_1$ and $C_2$ are determined by the initial conditions. We
note that the deterministic evolution described by
Eq.~(\ref{eq:system_m_n_delta}) is preceded by a regime dominated by
stochastic effects, where we can effectively assume $c=1$. During
such an interval $m$ fluctuates around zero due to the presence of the
term $-\lambda m$ in its drift, with fluctuations of the order of
$\pm m_0 = \pm \sqrt{D^m} \approx \pm \sqrt{(1-\lambda)/N}$.
Conversely $n$ grows diffusively, up to the
time $\tau=2/\delta$ after which the exponential growth becomes
dominant. Moreover, after a short time of order $1/\lambda$ the terms
proportional to $C_2$ in Eqs.~(\ref{mn}) become negligible. As a
consequence we can use as initial condition for $n$ its value at time
$\tau$, i.e., at the end of the diffusive regime, yielding
$C_1\approx\sqrt{\tau D_n}\approx\sqrt{\frac{2}{\delta N}}$
\begin{equation}
\begin{cases}
n \approx \pm \sqrt{\frac{2}{\delta N}}\mathrm{e}^{t \delta/2}\\
m=\frac{\delta}{2}n.
\end{cases}
\label{eq:m_n_delta}
\text{for}\ t>\tau
\end{equation}
Note that the exponential growth of $m$ actually begins only when
$\frac{\delta}{2}|n| \sim m_0 \approx\sqrt{(1-\lambda)/N}$.
Fig.~\ref{fig:m_n} shows that Eq.~\eqref{eq:m_n_delta} describes well
this stage of the temporal evolution of $n$ and $m$.
\begin{figure}
\includegraphics[width=0.45\textwidth]{m_n_new3.jpg}
\caption{\textbf{Growth of $n$ and $m$.} Growth of $m$ compared with
the observed $n\delta/2$ and with the predicted trend, as defined
by Eq.~\eqref{eq:m_n_delta}. $n$ grows diffusively up to
$\tau=2/\delta$, time at which the exponential contribution
becomes dominant. }
\label{fig:m_n}
\end{figure}
The linear approximation remains valid until the time $T_c$ when,
for some $i$, $|n_i|$ becomes so large that the condition
$|n_i| \delta \ll 1$ breaks down.
This may happen in two different ways, depending on the shape of the probability distribution $Q(n_i)$ of the variables $n_i$.
When the linear approximation breaks down, if the standard deviation of
$Q(n_i)$ is small, the personalized information is approximately the same
for all individuals. As a consequence this
source of information can be regarded as a constant field and this
results in the presence of a drift for the magnetization, which
fastly reaches $m=\pm1$. Conversely, if the standard deviation is
much larger than the average $\langle n_i\rangle$, some of the spins
are characterized by a negative $n_i$, while $n_i$ is positive for the
others. This implies that a fraction of the spins is influenced by a
positive external field, while a negative field acts on the
remaining part, thus leading to a polarized state. More in detail,
if $Q(n_i)$ is narrowly peaked around its mean value $n$ over time,
linearization breaks down for a time $T$ such that $|n(T)| \delta \sim
1$. From Eq.~\eqref{eq:m_n_delta} \begin{equation}
T=\frac{1}{\delta}\log\ton*{\frac{N}{2\delta}}. \end{equation} At this time all
$n_i$ have the same sign, hence the drift (see Appendix~\ref{coefficients})
\begin{equation} v_i = m(T) (1-\lambda) +
\lambda[2P(n_i)-1],
\end{equation}
has the same sign for all individuals and thus
consensus is rapidly reached.
Alternatively, if the distribution is very broad
so that its standard deviation $\sigma$ is much larger than the
absolute mean value $|n|$, linearization starts to fail at a different
time $T^*$.
Assuming, for the sake of simplicity, $n>0$, this occurs when
\begin{equation}
\delta [n + \sigma(T^*)] \approx \delta \sigma(T^*)=1
\end{equation}
The calculation of the variance $\sigma^2(t)$ of the distribution $Q(n_i)$,
reported in Appendix~\ref{Smallc}, gives
\begin{equation}
\sigma^2(t) = \frac{1}{\lambda \delta} \left(\mathrm{e}^{\lambda \delta t} - 1\right).
\label{eq:variance_delta}
\end{equation}
Imposing $\delta \sigma(T^*) = 1$
we then obtain
\begin{equation}
T^*=\frac{1}{\lambda\delta}\log\ton*{1+\frac{\lambda}{\delta}}.
\label{Tstar}
\end{equation}
The time $T_c$ when the linear approximation breaks down consequently is
\[
T_c=\min(T, T^*).
\]
Since $T$ grows with $N$, while $T^*$ does not depend on it, for small size
$T<T^*$ and the opposite relationship $T>T^*$ is instead true for large $N$.
Setting $T=T^*$ we can compute the crossover size
\begin{equation}
N^*=2\delta\ton*{1+\frac{\lambda}{\delta}}^{1/\lambda}.
\label{eq:N^*}
\end{equation}
For $N<N^*$ linearization breaks down due to the growth of $n$ and
as a consequence consensus is always reached, all the drifts having
the same sign. Differently, if $N>N^*$ the end of the linear regime
is caused by the growth of the variance. In this second case, at
$T^*$ most of the individuals have positive $n_i$ and hence a positive
drift, but some have negative $n_i$ (see Fig.~\ref{fig:N_c_bimodal}).
Determining in this case whether consensus is reached
or not is more involved, as discussed in the following.
Assuming $N\gg N^*$, so that the standard deviation is much larger than the
mean value, the smallest of the negative values is
\begin{equation}
n_i \approx n-\sigma(T^*) \approx - \sigma(T^*) \approx - \frac{1}{\delta},
\end{equation}
and as a consequence the smallest drift is
\begin{equation}
v_i = m(T^*) (1-\lambda) +
\lambda [2 P(n_i)-1] \approx m(T^*) (1-\lambda) -
\lambda.
\label{eq:drift_v_i}
\end{equation}
If this value is positive, the corresponding individual,
which is the one whose external information is more negatively polarized,
will be pushed towards positive values of $n_i$. It then
follows that also in this case the system reaches consensus.
\begin{figure*}
\includegraphics[width=0.9\textwidth]{N_c_bimodal_resubmission-eps-converted-to.pdf}
\caption{\textbf{Evolution of $Q(n_i)$.} Evolution of $Q(n_i)$
for $N<N_c$ (panel (a)) and $N>N_c$ (panel (b)).}
\label{fig:N_c_bimodal}
\end{figure*}
More quantitatively, the condition for having consensus, $v_i>0$,
implies (considering also the symmetric case when $n<0$)
\begin{equation}
|m(T^*)| > \frac{\lambda}{1-\lambda}=m_c.
\label{mstar}
\end{equation}
Inserting Eq.~(\ref{Tstar}) into Eq.~(\ref{eq:m_n_delta}) we obtain
\begin{equation}
|m(T^*)| =
\sqrt{\frac{\delta}{2N}} \ton*{1+\frac{\lambda}{\delta}}^{\frac{1}{2 \lambda}},
\end{equation}
that combined with the condition~(\ref{mstar}) implies
that consensus is undoubtedly reached for $\lambda<1/2$ if $N<\bar{N}$, with
\begin{equation}
\bar{N} = \frac{(1-\lambda)^2}{\lambda^2}\frac{\delta}{2}
\ton*{1+\frac{\lambda}{\delta}}^\frac{1}{\lambda}.
\label{bar_N_smallc}
\end{equation}
Note, however, that this is only a lower bound for the true $N_c$ for
$\lambda<1/2$.
Actually consensus may occur also for $N>\bar{N}$, provided that $\lambda<1/2$.
Indeed, even if the smallest
drift (corresponding to the most negative $n_i$) is negative when
linearization breaks down, it can become positive later on, thus making the
system reach consensus. This may occur if only a few spins (among those
with $n_i<0$) have a negative drift. The others, moving toward
positive $n_i$, produce an increase of the magnetization, which
eventually overcomes the critical value $m_c=\frac{\lambda}{1-\lambda}$.
The critical size $N_c$, determining if the system can reach consensus
or not, is then larger than $\bar{N}$ for $\lambda<1/2$.
It is actually possible to improve on this result.
In Appendix~\ref{refined} a more refined argument is presented, allowing
us to determine numerically a tighter lower bound $\hat{N}$ for $N_c$.
This bound, as shown in Fig.~\ref{pdchecksmallc}, is in good agreement with simulations.
The situation is different if $\lambda\geq 1/2$, because in this case
the critical magnetization is larger than 1 and therefore,
even if some spins move from negative
$n_i$ to positive $n_i$, the smallest drift remains negative,
for any magnetization.
This implies that for $\lambda\geq 1/2$,
consensus can be reached only if linearization breaks down due to the
growth of $n$.
Therefore if $\lambda\geq1/2$,
the critical size coincides with $N^*$.
In conclusion
\begin{equation}
\begin{cases}
N_c \geq \hat{N}~~~~~~~~~~~~~~~~~~~~~~~~ \text{for}\ \lambda<\frac{1}{2}\\
N_c=N^*=2\delta\ton*{1+\frac{\lambda}{\delta}}^{1/\lambda}\ \text{for}\ \lambda\geq\frac{1}{2}.
\end{cases}
\label{N_csmallc}
\end{equation}
Of course, given the dependence of the argument on random fluctuations,
these values are to be intended as indicating a crossover and not a sharp
transition. Simulation results presented in Fig.~\ref{pdchecksmallc}
confirm that the probability of reaching consensus exhibits for various
$\lambda$, a crossover in reasonable agreement with Eq.~\eqref{N_csmallc}.
\begin{figure}
\includegraphics[width=0.49\textwidth]{N_P_c_small-eps-converted-to.pdf}
\caption{\textbf{Crossover between consensus and polarization for small $c$.}
Fraction of
runs reaching consensus as a function of $N$,
in 1000 realizations of the dynamics for $\delta=0.01$
and various $\lambda$. The crossover values $N_c$ predicted by
Eq.~(\ref{N_csmallc}) are marked by vertical lines.}
\label{pdchecksmallc}
\end{figure}
\subsection{The behavior for $c \gg 1$}
\begin{figure*}
\begin{subfigure}[b]{0.475\textwidth}
\includegraphics[width=0.9\textwidth]{distribution_lambda_resubmission-eps-converted-to.pdf}
\label{fig:distribution_lambda}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.475\textwidth}
\includegraphics[width=0.9\textwidth]{distribution_time_resubmission-eps-converted-to.pdf}
\label{fig:distribution_time}
\end{subfigure}\\
\caption{\textbf{Splitting of $Q(n_i)$.} \textbf{a)} Distribution
$Q(n_i)$ at $t=10$ for various values of $\lambda$. Note that, for
$\lambda=0.1$, $t \ll t_c = 100$ while for $\lambda=0.6$ $t \gg
t_c = 2.77$. \textbf{b)} Distribution $Q(n_i)$ at $t \approx t_c
= 1/\lambda^2$ for various values of $\lambda$.}
\label{fig:distributions}
\end{figure*}
As shown in the previous subsection, if $c$ is close to $1$ the system
can be described in terms of the magnetization and of the
first two moments of the distribution $Q(n_i)$, $n$ and $\sigma^2$.
This is possible because (see Fig.~\ref{fig:N_c_bimodal})
this distribution is unimodal during the linear regime,
that lasts for long times, up to $T_c=\min(T,T^*)$.
Conversely, when $c$ is large, the pair $(n,\sigma^2)$ alone
is not sufficient to describe the state of the system, even in the
first stage of the dynamics.
This can be seen by inspecting the local probabilities $P(n_i)$.
If $c\gg 1$ we have
\[
P(n_i)=
\begin{cases}
1-c^{-n_i} \approx 1 \ ~~ \text{if} \ n_i>0 \\
\frac{1}{2} \ ~~~~~~~~~~~~~~~~ \text{if} \ n_i=0 \\
c^{n} \approx 0 \ ~~~~~~~~~~ \text{if} \ n_i<0.
\end{cases}
\]
This means that, as soon as $n_i$ becomes different from zero at
the first update of node $i$, the
external information almost certainly has the same
sign of $n_i$ and thus tends to increase its absolute value.
The behavior of nodes with different signs of $n_i$ is
opposite and this rapidly leads to the splitting of $Q(n_i)$ in
two separate peaks.
We then write the overall distribution as the sum of two unimodal
distributions
\[
Q(n_i)=pQ^+(n_i)+(1-p)Q^-(n_i).
\]
Here $Q^+(n_i)$ is the distribution of positive $n_i$,
while $Q^-(n_i)$ is the distribution of the negative ones and the
weight is $p=N^+/N$.
By using the same formalism described above,
in Appendix~\ref{Largec} we derive expressions for the moments
of the two distributions
\begin{equation}
v^\pm = \frac{d\av{n_i}_\pm}{dt} = m(1-\lambda) \pm \lambda,
\label{vpm}
\end{equation}
\begin{equation}
\sigma^2_{\pm} = t.
\end{equation}
Since $m$ is initially very small, Eq.~(\ref{vpm}) confirms
that positive (negative) $n_i$ tend to increase (decrease)
and the overall distribution $Q(n_i)$ broadens and eventually splits.
In the same Appendix we derive the expression for
the variance $\sigma^2(t) = \av{n_i^2}-\av{n_i}^2$
of the whole distribution $Q(n_i)$, finding
\begin{equation}
\sigma^2(t)=t+\lambda^2t^2.
\label{eq:sigma(t)}
\end{equation}
From Eq.~\eqref{eq:sigma(t)} we can determine the time at which
$Q(n_i)$ splits in two separate components.
For small times $\sigma(t)$ is dominated by
the diffusive widening of the central peak,
while for larger times it is determined mainly by the ballistic
distancing between the two peaks. Denoting by $t_c$ the time at which
the crossover occurs, it follows from Eq.~\eqref{eq:sigma(t)}
$t_c\approx\lambda^2 t_c^2$
yielding
\[
t_c\approx\frac{1}{\lambda^2}.
\]
Figures~\ref{fig:distributions}a and~\ref{fig:distributions}b
confirm numerically this prediction.
Note that, at odds with the case $c \gtrsim 1$, the splitting
always occurs, and over a much shorter temporal scale since
$t_c = 1/\lambda^2 \ll \log(1+\lambda/\delta)/(\lambda \delta) = T^*$.
To understand whether the system reaches consensus or not, the
argument is similar to the one presented for $c \gtrsim 1$, but in
this case it provides the actual critical size $N_c$ rather than a
lower bound. If, when the distribution splits, the drift of the right
peak is positive and that of the left peak is negative, consensus is
not reached. From Eq.~(\ref{vpm}) this implies that consensus is
reached only if
\begin{equation}
|m(t_c)| > m_c=\frac{\lambda}{1-\lambda}.
\label{condition}
\end{equation}
This condition means, as before, that consensus for $\lambda>1/2$
can occur only before the splitting, so during the transient regime.
It turns out numerically that, for $t<t_c$, the magnetization $m$ grows as
$m(t) \approx \xi t/\sqrt{N}$, where $\xi$ is a random prefactor ranging
between approximately -1 and +1.
Hence for $\lambda>1/2$ the condition for consensus reads
\[
|m(t)|=1 \ \text{for}\ t<t_c\ \to \ \frac{t_c}{\sqrt{N_c}}=1\ \to\ N_c=\frac{1}{\lambda^4}.
\]
For $\lambda<1/2$ instead, inserting the expression for $m(t)$ into
Eq.~(\ref{condition}) yields that consensus cannot be reached
if the number $N$ of individuals is larger than
\begin{equation}
N_c =
\frac{(1-\lambda)^2}{\lambda^6}.
\label{N_c}
\end{equation}
For $N>N_c$ the system remains asymptotically disordered in a polarized
state.
In the opposite case instead consensus is rapidly reached after
$t_c$, unless by chance the initial absolute value of $\xi$
is particularly small.
In conclusion, recalling Eq.~\eqref{N_c}, the critical size satisfies
\begin{equation}
\begin{cases}
N_c = \frac{(1-\lambda)^2}{\lambda^6}~~~ \text{for}\ \lambda<\frac{1}{2}\\
N_c = \frac{1}{\lambda^4}~~~~~~~~ \text{for}\ \lambda\geq\frac{1}{2}.
\end{cases}
\label{N_c2}
\end{equation}
Note that $N_c$ is continuous in $\lambda=1/2$.
Simulations presented in Fig.~\ref{pdcheck} show that the probability
of reaching consensus exhibits a crossover at values well predicted
by Eq.~(\ref{N_c2}).
\begin{figure}
\includegraphics[width=0.49\textwidth]{N_P_c_large-eps-converted-to.pdf}
\caption{\textbf{Crossover between consensus and polarization for large $c$.}
Fraction of
runs reaching consensus as a function of $N$,
in 1000 realizations of the dynamics for $c=100$
and several $\lambda$ values. The crossover values $N_c$ predicted by
Eq.~(\ref{N_c2}) are marked by vertical lines.}
\label{pdcheck}
\end{figure}
The long time dynamics in the case $c \gg 1$ can be studied also in
more detail, by describing the evolution of the two peaks and their
mutual interaction. As shown in Appendix~\ref{other} we can write a
closed integro-differential equation for the magnetization $m(t)$. In
particular, defining
\[
y=\int_0^tm(t')dt'
\]
we obtain the following second order non linear ODE:
\begin{widetext}
\begin{equation}
\frac{d^2y}{dt^2}=\sqrt{\frac{2}{\pi t (1-\lambda)^2}}
\left\{
\ton*{1-\frac{dy}{dt}}\frac{1}{\frac{1}{t}y-m_c}
\exp\qua*{-\frac{t(1-\lambda)^2\ton*{\frac{1}{t}y-m_c}^2}{2}}-
\ton*{1+\frac{dy}{dt}}\frac{1}{\frac{1}{t}y+m_c}
\exp\qua*{-\frac{t(1-\lambda)^2\ton*{\frac{1}{t}y+m_c}^2}{2}}
\right\}.
\label{integrodiff}
\end{equation}
\end{widetext}
Note that the evolution of $m(t)$ is expressed in terms of
$y$, which is a variable containing the past history
of the system. This reflects the fact that personalized
information keeps memory of the preferences of the spin it is coupled to.
The fact that the evolution of $m$ is
governed by an integro-differential equation is then a very natural
consequence of the dynamics of the model. Solutions of this equation
are reported in Fig.~\ref{fig:ode_simulations}b, where it is possible to
see that states with $|m|<m_c$ remain disordered, while if $m<-m_c$ or
$m>m_c$ consensus is reached, in good agreement with numerical
simulations, shown in Fig.~\ref{fig:ode_simulations}a.
\begin{figure*}
\begin{subfigure}[b]{0.475\textwidth}
\includegraphics[width=0.91\textwidth]{many_trajectories_resubmission.jpg}
\label{fig:many_trajectories}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.475\textwidth}
\includegraphics[width=0.95\textwidth]{035_ode_resubmission.jpg}
\label{fig:MagnMcEqDiff}
\end{subfigure}
\caption{\textbf{Time evolution of the magnetization for $c=100$.}
\textbf{(a)} $10^4$ trajectories obtained simulating the model
with $N=10^3$, $c=100$ and $\lambda=0.35$. Red dashed lines
represent $\pm m_c=\pm\lambda/(1-\lambda)$, disordered states are
found in the region $[-m_c;+m_c]$. \textbf{(b)} Solutions of
Eq.~\eqref{integrodiff} for $\lambda=0.35$ and various $m_0$ in the
range $10^{-5};10^{-3}$. The differential equation is valid only
for times much larger than $t_c\approx8$, therefore we set
$t_0=50$. As initial condition for the variable $y_0$ we used
$y_0=y(t_0)=m_0t^2/2$, because the magnetization grows linearly
during the initial regime. Red dashed lines represent $\pm
m_c=\pm\lambda/(1-\lambda)$, also in this case disordered states
are found in the region $[-m_c;+m_c]$.}
\label{fig:ode_simulations}
\end{figure*}
Disordered states for $c>1$ are completely different from the
disordered states for $c=1$. In the latter case the constant
magnetization is the effect of the external information
being a random uncorrelated variable equal for all nodes so that
all agents spend half of their time with $s_i=+1$ and half with
$s_i=-1$.
For $c>1$ instead, the system is divided in two polarized
clusters whose agents have a preferred spin value. A further
characterization in terms of self-overlap is presented in
Appendix~\ref{overlap}.
\section{Discussion and conclusions}
Let us summarize the results of our investigation.
The evolution of the system is described by the magnetization $m$
and the distribution of the local quantities $n_i$, describing the
personalized information for each agent.
Depending on whether $\delta$ ($c=1+\delta$) is much smaller or
much larger than $1$ the temporal evolution exhibits some variation.
In the first case there are three temporal regimes in the
evolution of the system.
For short times up to $2/\delta$ stochastic effects
dominate, the magnetization fluctuates around zero and
the distribution of the local $n_i$ remains centered around zero.
Later on the symmetry between positive and negative external information
breaks down because the
mean value $n(t)$ of the single-peaked distribution $Q(n_i)$
starts drifting away from $n=0$ exponentially in time,
while also its width $\sigma(t)$ grows.
At the same time also $|m|$ grows exponentially.
This regime ends when linearization of the equations for
$m$ and $n$ is no more valid.
The nonlinear subsequent evolution varies depending on the
relative width of the peak. If the peak is narrow, all $n_i$ have
the same sign and consensus is quickly reached.
If the peak is broad then individuals with both $n_i>0$ and $n_i<0$ exist.
If the magnetization in this moment is small enough then
the system gets trapped in a disordered (polarized) state, where
disagreement persists and the magnetization keeps a constant value
$|m|<m_c = \lambda/(1-\lambda)$.
When $\delta \gg 1$ (i.e., $c \gg 1$) linearization
is never valid, the $Q(n_i)$ distribution always splits in two
components and this happens much earlier, over a time scale equal
to $t_c=1/\lambda^2$.
What happens next depends again on the value of the magnetization at $t_c$.
If $|m(t_c)|$ is sufficiently large, the drift of the two components has the
same sign. For example, if this sign is positive, it means that
individuals with negative $n_i$ have nevertheless an overall positive drift:
The negative component of the $Q(n_i)$ distribution gets rapidly
depleted and consensus is reached.
Otherwise the competition between the two opinions persists forever and
the system gets stuck in the polarized state with $m<m_c$.
Although the detailed temporal evolution is rather different depending
on whether $c = 1+\delta$ is very close to 1 or much larger, the final overall
phenomenology is similar. The parameter $c$ sets the temporal scales
of the dynamics and the details of the phase-diagram, but not the
qualitative features of the behavior: consensus for $N<N_c(\lambda)$,
polarization otherwise.
The dependence of $N_c$ on $\lambda$ is different depending on
whether $c$ is close to 1 [Eq.~(\ref{N_csmallc})] or large [Eq.~(\ref{N_c2})],
so the boundaries between the two regions depend on the value of $c$.
Fig.~\ref{pdallc} represents this phase-diagram for several of these
values, showing that, as expected, increasing the strength of personalized
information makes consensus more difficult.
\begin{figure}
\includegraphics[width=0.49\textwidth]{l_Nc_last.jpg}
\caption{\textbf{Phase-diagram for all $c$.} The lines
separate the region where consensus is reached
from the region where it is not. Lines are obtained using
Eq.~(\ref{N_csmallc}) for $\delta \ll 1$ and Eq.~(\ref{N_c2}) for
$\delta=\infty$.}
\label{pdallc}
\end{figure}
However, we observe that increasing $c$ reduces the consensus region
but with a nontrivial limit: even in the $c \to \infty$ limit,
consensus is possible in sufficiently small systems.
This rather rich phenomenology has been obtained in the simplest
possible setting: an extremely simple binary opinion dynamics
always leading to consensus, in a mean-field framework,
augmented with an elementary form of personalized information.
Clearly our results open some interesting issues for future research:
What is the effect of a less trivial contact pattern among agents?
What changes when different opinion dynamics models are considered?
What happens if personalized information is parameterized differently, for
example considering different functional forms for the probability $P(n_i)$?
Finally, despite our oversimplified assumptions, some predictions
derived in this work may be testable in empirical systems. In particular,
the fact that disordered states cannot have magnetization larger than $m_c$
or the existence of a maximum size $N_c$ for the reaching of consensus
could be observable in systems where a community has to decide between
two alternative options.
\input{voter11.bbl}
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"redpajama_set_name": "RedPajamaArXiv"
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Die Schweizer Schule Rom ist eine internationale Schule mit Schweizer Pädagogik und führt zur schweizerischen Matura. Zudem ist die Durchlässigkeit zum italienischen Bildungssystem von Kindergarten bis Gymnasium garantiert. Die Maturanden sind zweisprachig in Deutsch und Italienisch. Das Gymnasium gliedert sich in zwei Profile: die Lernenden wählen zwischen dem Schwerpunktfach Wirtschaft und Recht und dem Schwerpunktfach Physik und Anwendungen der Mathematik.
Geschichte und Trägerschaft
Die Schule wurde 1946 durch die Mitglieder der Schweizer Gemeinde in Rom gegründet, die nach dem Zusammenbruch des italienischen Faschismus nach einer geeigneten und zukunftssicheren Ausbildungsmöglichkeit für ihre Kinder suchten. Sie gehört zu den derzeit 18 Schweizerschulen im Ausland, die sich im Dachverband Educationsuisse zusammengeschlossen haben. Sie ist sowohl in der Schweiz als auch in Italien offiziell anerkannt.
Die Schule ist politisch und konfessionell neutral. Das Schweizer Innenministerium übt durch das Bundesamt für Kultur die finanzielle Oberaufsicht aus, der Kanton St. Gallen ist als Patronatskanton für die Pädagogik und die Lehrpläne verantwortlich. Der Verein "Schweizer Schule Rom" bildet die private Trägerschaft der Schule. Die Vereinsmitglieder wählen alle drei Jahre einen Verwaltungsrat, der – zusammen mit der Direktion – die strategische Führung der Schule innehat.
Abschlüsse
Bildungsziel aller Lernenden ist die Schweizer Maturitätsprüfung. Sie bietet Zugang zu allen Hochschulen und Universitäten in der Schweiz, ist im Hinblick auf die Immatrikulation an Hochschulen und Universitäten in Italien dem italienischen Esame di Stato gleichgestellt und eröffnet den Zugang zu allen Universitäten der Europäischen Union.
Zusätzlich bietet die Schule ihren Schülerinnen und Schülern Passerellen zum italienischen Schulsystem. Die Schülerinnen und Schüler legen am Ende der Primarschule das "esame di idoneità" (esame di idoneità al termine del quinto anno di scuola primaria, ai fini dell'ammissione al successivo grado d'istruzione) und am Ende der Sekundarschule die "licenza media" (esame di Stato conclusivo del primo ciclo d'istruzione) ab.
Pädagogische Grundsätze
Das Grundkonzept der Schule geht auf Johann Heinrich Pestalozzi zurück: Lernen mit Kopf, Herz und Hand. Die Schule ermutigt die Heranwachsenden, ihre Selbständigkeit, ihren Sinn für Verantwortung, ihre Sozialkompetenzen und ihr kritisches Denken zu entwickeln, und stellt dabei Werte wie gegenseitige Achtung, Solidarität und Toleranz in den Mittelpunkt der pädagogischen Arbeit.
Die Schweizer Schule Rom ist eine Begegnungsschule, die einerseits von der deutschen Unterrichtssprache und den Lehrplänen des Kantons St. Gallen, andererseits aber auch von der Kultur des Gastlandes Italien und der kulturellen Vielfalt der Schulgemeinschaft geprägt ist, in der sich rund zwanzig verschiedene Nationalitäten begegnen.
Unterrichtssprachen
Vom Kindergarten an wachsen die Schülerinnen und Schüler auf natürliche Weise in die deutsche Sprache hinein, die in den folgenden Schulstufen die Hauptsprache darstellt. In der Primarschule haben die Kinder von Anfang an zusätzlich Italienisch, um den Zielen des italienischen Bildungssystems Rechnung zu tragen. Ab dem dritten Schuljahr werden diese beiden Sprachen durch das Fach Englisch ergänzt.
In der Sekundarschule tritt im sechsten Schuljahr Französisch als vierte Sprache hinzu, im Gymnasium können die Schülerinnen und Schüler ab dem neunten Schuljahr zusätzlich das Fach Latein belegen.
Ehemaligenverein
Anlässlich des 75-Jahr-Jubiläums wurde ein Ehemaligenverein gegründet mit dem Ziel, die Alumni der Schweizer Schule Rom miteinander zu vernetzen.
Weblinks
Website der Schweizer Schule Rom
Einzelnachweise
Gymnasium in Italien
Rom
Bildungseinrichtung in Rom
Bilinguale Schule deutsch–italienisch
Gegründet 1946
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The Stolen Crown, and a Handy Website
9 Comments / My Novels / By Susan
Last night at about 1:00 a.m., I put the finishing touches to my Buckingham novel, now called The Stolen Crown. If all goes well, look for it in 2010 sometime!
I'm now pondering what to write about next. I think it's going to be about a Margaret: either Margaret of Anjou, Margaret Beaufort, or Margaret Pole, Countess of Salisbury. The first and third Margarets are the ones who appeal to me most as subjects. Doing a novel about Margaret of Anjou would let me write about the early period of the Wars of the Roses from a Lancastrian standpoint, which isn't done all that often and would allow me to work in some people who particularly intrigue me, such as Henry Holland, Duke of Exeter and the Beaufort clan. Margaret Pole also has a fascinating and tragic story, and would let me work in five of Henry VIII's wives plus the portly man himself. The nice thing about all three women is that their lives are well documented and that I already have a lot of the research material at hand.
Speaking of research, I've gone gaga today for this site that someone on the Historical Novel Society Yahoo group pointed out. It's from the British Library, and allows free downloads of a number of doctoral theses. The ones I've really been longing to read aren't available yet, but there's always hope! In the meantime, I managed to console myself with about a dozen others. Have fun browsing!
9 thoughts on "The Stolen Crown, and a Handy Website"
Joansz_R3
I think the three Margarets are in ways, interconnected, especially Margaret Beaufort and Margaret Pole. Hmm, this could even be a triptik.
LadyDoc
I am off to check out the website- what a find!
Or you could write about my dog, Mary Margaret Maggie May!
Thanks for the link for that marvellous website!
Margaret Beaufort always seems to get a raw deal, IMO.
bookreviewer117
Margaret Pole is such an interesting figure. She's one of the few (visible) figures who really does span from Edward IV-> Henry VIII, including being godmother and governess of Mary I.
I can't even imagine all the things she saw and experienced, and how that would have affected her. I think writing about her would lend itself to a great exploration not only of that tumultuous era but also of how the human psyche copes with such constant upheavals.
So, a vote for Margaret Pole from me!
Marie Burton
And congratulations on The Stolen Crown.. hope I get to be one of the fe chosen to do an advance review, and tour 🙂
As far as you Margaret figures.. I am not as knowledgeable in these areas as others, but Margaret Pole seems to have had a lot going on.. but Margaret Beaufort was pretty shrewd..Margaret of Anjou, moreso in a b*tchy kind of way.. but whatever you pick will obviously by awesome as I would love to read more about any of the three!
Ms. Lucy
I'd love to read more about Margaret of Anjou- but the other Margarets as well!
I am a Margaret! Of course, I am not very interesting really!
Congrats on finishing up The Stolen Crown
Barbara Martin
Thanks for the link! The wealth of information just waiting for a historical fiction writer to find.
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
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As a successful real estate broker and consistently a top producer in my field of over 31 years, my reputation with sellers and buyers has won me the respect of those I service. My success has been with repeat customers and recommendations from the Hamptons to Long Island City. My current area of expertise is the Little Neck, Douglaston and Douglas Manor neighborhoods. I have a full list of providers to help with staging, packing, moving and everything in between. On few occasions I have been spotted packing and helping my home buyers move. My passion for the industry has not curtailed my involvement in my community. I am the past president of the Little Neck/Douglaston Lions Club, and currently I sit on the board of the Divine Wisdom Catholic Academy and I am Eucharistic Minister at St. Anastasia Church. I have deep roots in the St. Kevin's prayer group in Flushing. When I was involved with the Little Neck Lions Club, I was instrumental in raising money for the Guide Dog Foundation and provided dogs each year for the blind. My volunteering keeps getting better with time. I am passionate about the Bridge to Life organization in Flushing as well as the Shriners Children's Hospital. Daniel Gale Sotheby's has been a perfect fit for me since they have the same commitment to helping others as I do. All work and no play would not make for a good happy agent, so I am also an avid Met fan, and win or lose I make it out to as many games as possible. The Douglaston club is still a great place to hangout especially in the summer at poolside or the tennis court. Give me a check list of homes and areas and I hit the ground running and help build dreams.
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{
"redpajama_set_name": "RedPajamaC4"
}
| 5,552
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Q: How to resolve the problem related to "Whitelabel Error Page" in a Spring Boot application I am trying to execute my new Spring Boot application.
The first two classes are:
import org.springframework.boot.autoconfigure.SpringBootApplication;
@SpringBootApplication
public class UbbioneApplication {
public static void main(String[] args) {
SpringApplication.run(UbbioneApplication.class, args);
}
}
then the servlet Initializer class
import org.springframework.boot.builder.SpringApplicationBuilder;
import org.springframework.boot.web.servlet.support.SpringBootServletInitializer;
public class ServletInitializer extends SpringBootServletInitializer {
protected SpringApplicationBuilder configure(SpringApplicationBuilder application) {
return application.sources(UbbioneApplication.class);
}
}
But when I am used to run my application by writing mvn spring-boot:run in the console, I have this message appearing:
Whitelabel Error Page
Could you help me please how to resolve this issue?
Thanks in advance.
A: I think I have an answer:
I created a controller to my application and I updated my code as following:
import org.springframework.boot.autoconfigure.EnableAutoConfiguration;
import org.springframework.context.annotation.ComponentScan;
import org.springframework.context.annotation.Configuration;
@Configuration
@EnableAutoConfiguration
@ComponentScan(basePackages = {"Name_controller_path"})
public class Application {
public static void main(String[] args) {
SpringApplication.run(Application.class, args);
}
}
Then my controller will look like this:
import org.springframework.web.bind.annotation.RequestMapping;
import org.springframework.web.bind.annotation.RequestMethod;
import org.springframework.web.bind.annotation.RestController;
@RestController
public class Appcontroller {
@RequestMapping(value = "/home", method = RequestMethod.GET)
String home() {
return "home";
}
}
Then use this path to view your execution: http://localhost:8080/home.
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"redpajama_set_name": "RedPajamaStackExchange"
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\section{Introduction}
Although the diffusion equation \cite{einstein1956investigations, carslaw1986conduction}
\begin{equation}\label{diffusionEq}
\frac{\partial p(x, t)}{\partial t} = D\frac{\partial^{2} p(x, t)}{\partial x^{2}}
\end{equation}
can be successfully applied to a broad range of different phenomena, it suffers from a pathological defect: The implied signal propagation speed is infinite in the sense
that any perturbation $\delta p(x, t)$ localized at one point in space instantaneously influences $p(x, t)$ at any other point. Still, the diffusion equation provides an excellent approximation, as long as the time scale on which inertial effects must not be neglected is small compared to the relevant time scale of the considered problem. This is not always the case, though. For instance, the movement of animals and biological cells proceeds with a rather well-defined and finite velocity \cite{Codling2008} and the diffusion equation approximation becomes invalid.
The telegrapher's equation \cite{Goldstein1951, MorseFeshbach, Weiss2002, Masoliver1996} may be considered as a generalization of the diffusion equation that overcomes the described deficiencies. Being a hyperbolic differential equation, it exhibits
wave-like features on short time scales while diffusion-like features dominate on longer time scales. In particular, the signal propagation speed is bounded and well-defined.
Like the diffusion equation, the telegrapher's equation may be viewed from a macroscopic, phenomenological and, at least in one dimension, from a microscopic point of view.
The phenomenological approach starts with Fick's first law that asserts that the probability flux is linearly related to the gradient of the probability density
\begin{equation}\label{Fick}
j(x,t) = -D\frac{\partial p(x,t)}{\partial x}.
\end{equation}
Combining Eq.~\eqref{Fick} with the conservation law
\begin{equation}\label{conservationEq}
\frac{\partial p}{\partial t} + \frac{\partial j}{\partial x} = 0
\end{equation}
immediately yields the diffusion equation. Consequently, Ficks' first law and the telegrapher's equation are not compatible with each other and one has to seek a modification of Eq.~\eqref{Fick}. A suitable generalization was suggested by Cattaneo in an attempt to replace Fourier's law in the theory of heat waves. In the present context Cattaneo's equation \cite{Joseph1989} reads as
\begin{equation}\label{cattaneo}
T\frac{\partial j}{\partial t} = -\bigg(D\frac{\partial p}{\partial x} +j\bigg),
\end{equation}
where $T$ sets the relaxation time scale. Obviously, for $T\rightarrow 0$ one recovers Eq.~\eqref{Fick}. We note for later use that
\begin{equation}\label{defJ}
j(x, t) = -c^{2}\int^{t}_{0}e^{-(t-\tau)/T}\frac{\partial p(x,\tau)}{\partial x}\,d\tau
\end{equation}
satisfies Eq.~\eqref{cattaneo}, given that one identifies the diffusion constant with the speed $c$ and the relaxation time $T$ according to
$D=c^{2}T$.
Cattaneo's equation Eq.~\eqref{cattaneo} combined with the conservation equation Eq.~\eqref{conservationEq} leads to the telegrapher's equation
\begin{equation}\label{telegraphersEq}
\frac{\partial^{2}p}{\partial t^{2}}+\frac{1}{T}\frac{\partial p}{\partial t} = c^{2} \frac{\partial^{2}p}{\partial x^{2}}.
\end{equation}
Note that $T$ interpolates between the ballistic and diffusion regime in the sense that Eq.~\eqref{telegraphersEq} reduces to the wave equation in the limit
$T \rightarrow \infty$, while one arrives at the diffusion equation Eq.~\eqref{diffusionEq} in the limit $T\rightarrow 0, c \rightarrow \infty, c^{2}T\rightarrow D=\text{finite}$.
Turning to a microscopic point of view, we recall the well-known fact that the diffusion equation can be obtained as the continuum limit of the simple uncorrelated random walk, also referred to as Brownian motion. Similarly, at least in one dimension, the telegrapher's equation emerges from a correlated random walk that takes into account correlations in the direction of movement \cite{Fuerth1917, Taylor1922, Goldstein1951}. In this context Brownian motion appears as a contracted description of the more general correlated random walk, being valid only if the characteristic correlation time decays sufficiently fast. The persistence of the correlated walker, i.e. the tendency that each step points in the same direction as the previous one, necessitates to consider two separate probability density functions (PDF). Let $a(x, t\vert x_{0})$ denote the PDF to find the particle at $x$, given that it was at $x_{0}$ for $t=0$ and that it is moving in the
positive $x$ direction at time $t$, and let $b(x, t\vert x_{0})$ the corresponding PDF for moving in the negative $x$ direction. It follows that the PDF $p(x, t\vert x_{0})$
independent of the direction in which the particle is moving at $t$ is given by the superposition
\begin{equation}
p(x, t\vert x_{0}) = a(x, t\vert x_{0}) + b(x, t\vert x_{0}).
\end{equation}
It can be shown that the time evolution of $p(x, t\vert x_{0})$ is governed by the telegrapher's equation Eq.~\eqref{telegraphersEq} \cite{Masoliver1993}.
The broad range of potential applications of the telegrapher's equation \cite{Weiss2002}, its blend of wave and diffusion-like features and relation to other
important equations like the diffusion, wave and Dirac equation \cite{Gaveau1984}, motivate a thorough study of its solutions.
In this paper, we are interested in solutions to Eq.~\eqref{telegraphersEq} in the presence of a single boundary \cite{Masoliver1993, Masoliver1992, Masoliver1992erratum}. Previously, a radiation boundary condition (BC), the purely absorbing and reflecting limiting cases and the corresponding Green's functions (GF) have been derived \cite{Masoliver1993}. Here, we will revisit the absorbing and radiation boundary cases. We will obtain an alternative form of the GFs by directly inverting the Laplace transform via the Bromwich contour integral and we will discuss how the alternative forms relate to the expressions obtained earlier. Then, we will discuss and derive a backreaction BC suitable for a one-dimensional telegrapher's equation. The backreaction BC generalizes the radiation BC, which only describes irreversible reactions between the particle and boundary: Once the particle gets absorbed, it stays trapped forever. In contrast, the backreaction BC incorporates desorption or return of population at the boundary. Consequently, the backreaction BC figures prominently in the theory of reversible diffusion-influenced reactions \cite{Agmon:1990p10} and the GF of the diffusion equation subject to that BC have been obtained in one, two, and three dimensions \cite{Agmon:1984, TPMMS_2012JCP, kimShin:1999}.
Finally, we will focus on the purely absorption-desorption case and calculate the associated GF by make use of the Bromwich contour integral again. In his way, we can derive and discuss all obtained GFs in a coherent way.
\section{Radiation boundary condition}
We will start by summarizing some results obtained earlier \cite{Masoliver1993}.
Eq.~\eqref{telegraphersEq} has to be supplemented by initial conditions. A choice that can be motivated from an underlying random walk picture is
\begin{eqnarray}\label{initialCond}
p(x, t=0\vert x_{0}) &=& \delta(x-x_{0}),\\
\frac{\partial p}{\partial t}\bigg\vert_{t=0} &=& 0.
\end{eqnarray}
The solution to the telegrapher's equation that satisfy these initial conditions is known as the free-space GF \cite{Masoliver1993}
\begin{eqnarray}
p_{\text{free}}(x, t\vert x_{0}) = e^{-t/(2T)}f_{0}(t, \vert x- x_{0}\vert),
\end{eqnarray}
where
\begin{eqnarray}
f_{0}(t, x)&=&\frac{1}{2}\delta(ct-x) + \frac{\Theta(ct-x)}{4cT}\bigg[I_{0}(u) + \frac{t}{2uT}I_{1}(u)\bigg],\\
u&=& \frac{\sqrt{c^{2}t^{2}-x^{2}}}{2cT}.
\end{eqnarray}
Here $\Theta(ct-x)$ denotes the Heaviside step function and $I_{0}, I_{1}$ refer to the modified Bessel functions of first kind \cite[Ch.~(9.6.)]{abramowitz1964handbook}.
For later reference we note that the Laplace transform of $f_{0}(t, x)$ is
\begin{equation}\label{transformF0}
\tilde{f}_{0}(s, x) = \frac{1}{2c}\frac{s+\frac{1}{2T}}{\sqrt{s^{2}-\frac{1}{4T^{2}}}}e^{-x/c\sqrt{s^{2}-\frac{1}{4T^{2}}}}
\end{equation}
The free-space GF can be used to obtain GFs that satisfy the telegrapher's equation in the presence of boundaries. Typically, one seeks a solution of the form
\begin{equation}\label{ansatzGF}
p(x,t\vert x_{0}) = p_{\text{free}}(x,t\vert x_{0}) + h(x,t),
\end{equation}
where $h(x,t)$ has to be a solution of the telegrapher's equation Eq.~\eqref{telegraphersEq}. Furthermore, it is chosen in such a way that Eq.~\eqref{ansatzGF} satisfy the specified BC.
Henceforth, we will assume without loss of generality that the boundary is located at $x=0$ and that $x\geq 0, x_{0}\geq 0$.
In Ref. \cite{Masoliver1993} it has been shown that the appropriate radiation BC is quite different to the one in the diffusion case and that it takes the form
\begin{equation}\label{radiationBC}
c\frac{\partial p}{\partial x}\bigg\vert_{x=0}=\frac{\beta}{2-\beta}\bigg(\frac{\partial p}{\partial t}+\frac{1}{T}p\bigg)\bigg\vert_{x=0}.
\end{equation}
Here, $\beta$ denotes the probability that a particle reaching the boundary gets actually absorbed
\begin{equation}\label{radiationBCPDF}
a(x=0, t\vert x_{0}) = (1-\beta) b(x=0, t\vert x_{0}).
\end{equation}
Thus, $\beta=1$ corresponds to the purely absorbing limit, while $\beta =0$ corresponds to a purely reflecting boundary.
Furthermore, seeking a solution of the form
\begin{equation}
p(x, t\vert x_{0}) = e^{-t/(2T)} P(x, t\vert x_{0}),
\end{equation}
one finds for the Laplace transform of $P(x, t\vert x_{0})$ \cite{Masoliver1993}
\begin{equation}\label{transformP}
\tilde{P}(x, s\vert x_{0}) = \frac{s + \frac{1}{2T}}{2c\rho}\bigg[e^{-\rho\vert x- x_{0}\vert/c} - \Omega e^{-\rho(x + x_{0})/c}\bigg],
\end{equation}
where
\begin{eqnarray}
\rho &=& \sqrt{s^{2} - \frac{1}{4T^{2}}}, \label{defRho}\\
\Omega &=& \frac{\beta (s + \frac{1}{2T}) - (2-\beta)\rho}{\beta (s + \frac{1}{2T}) + (2-\beta)\rho}. \label{defOmega}
\end{eqnarray}
It is easy to show that for purely absorbing and reflecting BCs the expression for $\Omega$ reduces to
\begin{eqnarray}
\Omega &=& 2T(s-\rho), \,\,\quad \text{if}\,\, \beta =1,\\
\Omega &=& 1, \quad\qquad\qquad \text{if}\,\, \beta =0,
\end{eqnarray}
respectively.
Taking into account these identities, Eqs.~\eqref{transformF0}, \eqref{transformP} and elementary properties of the Laplace transform, one
arrives at the following form for the GFs corresponding to purely absorbing and reflecting BC
\begin{eqnarray}
p_{\text{abs}}(x, t\vert x_{0}) &=& e^{-t/2T}\big[f_{0}(t, \vert x-x_{0}\vert) - f_{1}(t, x+x_{0})\big], \label{pAbs}\\
p_{\text{ref}}(x, t\vert x_{0}) &=& e^{-t/2T}\big[f_{0}(t, \vert x-x_{0}\vert) + f_{0}(t, x+x_{0})\big],
\end{eqnarray}
where the function $f_{1}$ is given by
\begin{eqnarray}\label{defF1}
f_{1}(t,x)&=&\frac{\Theta(ct-x)}{8cT}\bigg[I_{0}(u) + 2 \bigg(\frac{ct-x}{ct+x}\bigg)^{1/2}I_{1}(u) +\nonumber\\
&+& \bigg(\frac{ct-x}{ct+x}\bigg)I_{2}(u)\bigg].
\end{eqnarray}
Finally, to obtain the inverse Laplace transform in the case of a radiation BC one can proceed by expanding the denominator in a power series to find
the following expression for the GF satisfying the BC Eq.~\eqref{radiationBC} \cite{Masoliver1993}
\begin{eqnarray}\label{radGF}
p_{\text{rad}}(x, t\vert x_{0}) &=& \beta p_{\text{abs}}(x, t\vert x_{0}) + (1-\beta)p_{\text{ref}}(x, t\vert x_{0})+\nonumber\\
&-&\frac{1}{8cT}\beta(1-\beta)e^{-t/2T}\sum^{\infty}_{n=0}c_{n}g_{n}(t, x+x_{0}),
\end{eqnarray}
The coefficients are defined by
\begin{eqnarray}\label{GF}
c_{0}&=&1, \nonumber\\
c_{1}&=&4-\beta, \nonumber\\
c_{2}&=&(2-\beta)(3-\beta)+1, \nonumber\\
c_{n}&=&(2-\beta)^{3}(1-\beta)^{n-3}, \, \,\,\, n \geq 3,\nonumber\\
g_{n}(t,x)&=&\Theta(ct-x)\bigg(\frac{ct-x}{ct+x}\bigg)^{n/2}I_{n}(u).
\end{eqnarray}
\subsection{Alternative expression for the Green's function}
In what follows, we will take a different route to derive alternative expressions for the GFs subject to purely absorbing and radiation BC. By making use of the Bromwich contour integral, we will derive an alternative form of the GFs as an important preparatory step. In fact, by employing for all discussed BC the same technique, hardly any extra work will be necessary to establish the expression for the GF that satisfies the backreaction BC.
Focusing first on the purely absorbing case, we start again from the Laplace domain and write
\begin{equation}
\tilde{p}_{\text{abs}}(x,s \vert x_{0}) = \tilde{p}_{\text{free}}(x,s \vert x_{0}) + \tilde{h}_{\text{abs}}(x,s).
\end{equation}
Now, our strategy will be to invert $\tilde{h}_{\text{abs}}$ directly by using the Bromwich contour integral. Henceforth, to streamline the notation, we define
\begin{eqnarray}
a &=& T^{-1},\nonumber\\
k &=& \frac{x+x_{0}}{c}.\nonumber
\end{eqnarray}
Then, taking into account Eqs.~\eqref{transformP}, \eqref{defRho} and \eqref{defOmega}, the Bromwich contour integral takes the form
\begin{eqnarray}\label{Bromwich}
&&h_{\text{abs}}(x,t \vert x_{0}) =\frac{1}{2\pi i}\int^{\gamma+i\infty}_{\gamma-i\infty}e^{st}\tilde{h}_{\text{abs}}(x,s)ds=\nonumber\\
&&-\frac{1}{4\pi ic}\int^{\gamma+i\infty}_{\gamma-i\infty}e^{ts - k\sqrt{s(s+a)}}\frac{s+a}{\sqrt{s(s+a)}}\frac{\sqrt{s+a}-\sqrt{s}}{\sqrt{s+a} + \sqrt{s}}ds.\quad
\end{eqnarray}
To proceed we have to discuss the analytical structure of the integrand, see Fig.~\ref{fig:contour}. First, we note that it has two branch points, at $s = -1/T$ and $s = 0$, necessitating the specification of two branch cuts. We choose both branch cuts along the negative real axis, leading to a potentially discontinuous integrand in the region $s < 0$. Therefore, we employ the keyhole contour depicted in Fig.~\ref{fig:contour}. It turns out that the branch cuts' effects cancel each other along the overlapping region $s < -1/T$, i.e. along $\mathcal{C}^{-1/T}_{-\infty}$, $\mathcal{C}^{-\infty}_{-1/T}$, and the integrand is only discontinuous for $-1/T < s < 0$. Therefore, the sole contributions coming from the integration on each sides of the branch cut are due to the integrals along $\mathcal{C}^{0}_{-1/T}$ and $\mathcal{C}^{-1/T}_{0}$.
Along $\mathcal{C}^{0}_{-1/T}$ we choose
$s = re^{i\pi}$ and it follows $\sqrt{s+a}=\sqrt{a-r}$ and $\sqrt{s} = i\sqrt{r}$. We get
\begin{equation}\label{intAlongC1}
\int_{\mathcal{C}^{0}_{-1/T}} e^{st}\tilde{h}_{\text{abs}}(x,s) ds = \int^{a}_{0}\frac{e^{-tr - ik\sqrt{r(a-r)}}}{i\sqrt{r(a-r)}}(a-r)\frac{\sqrt{a-r}-i\sqrt{r}}{\sqrt{a-r} + i\sqrt{r}}dr.
\end{equation}
Similarly, to evaluate the integral along $\mathcal{C}^{-1/T}_{0}$ , we choose $s=r e^{-i\pi}$ and note that the ensuing integral is the negative complex conjugate of the integral given in Eq.~\eqref{intAlongC1}
\begin{equation}\label{complexC}
\int_{\mathcal{C}^{0}_{-1/T}} e^{st}\tilde{h}_{\text{abs}}(x,s) ds = - \bigg(\int_{\mathcal{C}^{-1/T}_{0}} e^{st}\tilde{h}_{\text{abs}}(x,s) ds\bigg)^{\ast},
\end{equation}
where $\ast$ denotes complex conjugation. Furthermore, the integrand has no poles neither inside nor on the contour and the contributions from the small circles $\mathcal{C}_{\epsilon_{2}}, \mathcal{C}_{\epsilon_{1}}, \mathcal{C}_{\epsilon_{3}}$ around the origin and $-1/T$, respectively, vanish.
Thus, using Eqs.~\eqref{complexC},~\eqref{intAlongC1},~\eqref{Bromwich} and making the substitutions $r\rightarrow \varphi + 1/2a,\, \varphi \rightarrow 1/2a\xi$, we finally arrive at
\begin{eqnarray}\label{finalHabs}
&&h_{\text{abs}}(x,t ) = \frac{a\Theta(ct-k)}{4\pi c}e^{-1/2at}\int^{1}_{-1}e^{-1/2at\xi}(1-\xi)\times\nonumber \\
&&\bigg[\frac{\cos(1/2ka\sqrt{1-\xi^{2}})}{\sqrt{1-\xi^{2}}}\xi + \sin(1/2ka\sqrt{1-\xi^{2}})\bigg]d\xi.
\end{eqnarray}
The question arises how the found expression relates to the expression obtained in Ref.~\cite{Masoliver1993}, cp. also Eqs.~\eqref{pAbs} and \eqref{defF1}. Actually, one can explicitly show that Eq.~\eqref{finalHabs} can be cast in an integral free form. To this end we first note the identity
\begin{equation}\label{integralBessel}
\frac{1}{\pi}\int^{1}_{-1}e^{-1/2 a t \xi} \frac{\cos(1/2 a k \sqrt{1 - \xi^{2}})}{\sqrt{1-\xi^{2}}}d\xi = I_{0}(1/2 a\sqrt{t^{2}-k^{2}}).
\end{equation}
Then, it follows that the terms involving the cosine in Eq.~\eqref{finalHabs} can be rewritten by taking the first and second order derivative of the expression on the rhs of Eq.~\eqref{integralBessel} with respect to $t$.
Also, upon integration by parts we can bring the term involving the sinus in a form that can be expressed analogously.
The general case of a radiation boundary conditions can now be treated in a similar way. Again, the part of the GF that takes into account the BC can be expressed by a Bromwich integral
\begin{eqnarray}\label{BromwichRad}
h_{\text{rad}}(x,t) =-\frac{1}{4\pi ic}\int^{\gamma+i\infty}_{\gamma-i\infty}e^{ts - k\sqrt{s(s+a)}}\times\nonumber\\
\frac{s+a}{\sqrt{s(s+a)}}\frac{(1-\eta)\sqrt{s+a}-(1+\eta)\sqrt{s}}{(1-\eta)\sqrt{s+a} + (1+\eta)\sqrt{s}}ds,
\end{eqnarray}
where $\eta = 1-\beta$. Because the analytical properties of the integrand in Eq.~\eqref{BromwichRad} are analogous to the case of purely absorbing
BC, we employ the same keyhole contour and proceed in the same way to find
\begin{eqnarray}\label{finalHrad}
&&h_{\text{rad}}(x,t) = \frac{a\Theta(ct-k)}{4\pi c}e^{-1/2at}\int^{1}_{-1}\frac{e^{-1/2at\xi}}{1+\eta^{2}+2\eta\xi}(1-\xi)\times\qquad\qquad\qquad\nonumber\\
&&\bigg[\frac{\cos(\frac{ka}{2}\sqrt{1-\xi^{2}})}{\sqrt{1-\xi^{2}}} [2\eta +(1+\eta^{2})\xi] +[1-\eta^{2}]\sin(\frac{ka}{2}\sqrt{1-\xi^{2}})\bigg]d\xi.
\end{eqnarray}
Here, two remarks are in order. First, as $\eta\rightarrow 0$, one recovers the expression obtained for $h_{\text{abs}}$ Eq.~\eqref{finalHabs}.
Second, we may expand the denumerator $1+\eta^{2} + 2\eta\xi$ in powers of $2\eta\xi/(1+\eta^{2})$, because $2\eta/(1+\eta^{2}) < 1$ for $\eta <1$. As explained for $h_{\text{abs}}$, the resulting integrals can be expressed as suitable time derivatives of the expression on the rhs of Eq.~\eqref{integralBessel}, and in this way one may recover the series expansion given in Eqs.~\eqref{radGF}, \eqref{GF}.
\section{Backreaction boundary condition}
Next, we will turn to reversible interactions with the boundary: when the particle gets trapped this does not mean necessarily that it stays trapped forever. First, we will derive a backreaction BC that is suitable for the one-dimensional telegrapher's equation. The derivation will follow the line of reasoning in Ref. \cite{Masoliver1993} and it will turn out that the backreaction BC is somewhat more complicated than the corresponding BC in the case of diffusion, resembling the situation for the radiation BC.
Conventionally, the salient assumption underlying the backreaction BC is that the rate of desorption is proportional to the total absorbed population. Hence, generalizing Eq.~\eqref{radiationBCPDF},
we postulate that at the boundary the PDFs that take into account the directions are related by
\begin{equation}\label{boundaryRelationII}
a(x, t\vert x_{0})\vert_{x=0} = (1-\beta) b(x, t\vert x_{0})\vert_{x=0} + \frac{\beta\kappa}{2}[1-S(t\vert x_{0})].
\end{equation}
Here, $S(t\vert x_{0})$ denotes the survival probability, defined by
\begin{equation}\label{defS}
S(t\vert x_{0}) = \int^{\infty}_{0} p(x,t\vert x_{0})\, dx,
\end{equation}
and therefore $1-S(t\vert x_{0})$ is the total absorbed population.
The total probability density becomes
\begin{eqnarray}
p(0, t\vert x_{0}) &=& a(0, t\vert x_{0}) + b(0, t\vert x_{0}) \nonumber\\
&=& (2-\beta) b(0, t\vert x_{0}) + \frac{\beta\kappa}{2}[1-S(t\vert x_{0})].
\end{eqnarray}
Additionally, we can conclude that
\begin{equation}
a(0, t\vert x_{0}) - b(0, t\vert x_{0})= -\frac{\beta}{2-\beta}\bigg[p(0, t\vert x_{0}) - \kappa\big[1-S(t\vert x_{0})\big]\bigg].
\end{equation}
Moreover, one has the dynamic equation \cite{Masoliver1993}
\begin{equation}
c\frac{\partial p}{\partial x} = - \frac{\partial }{\partial t}(a-b) -\frac{1}{T}(a-b),
\end{equation}
which is valid in particular at the boundary $x=0$, and hence we arrive at the backreaction BC for the one-dimensional telegrapher's equation
\begin{equation}\label{backreactionBC}
c\frac{\partial p}{\partial x}\bigg\vert_{x=0} = \frac{\beta}{2-\beta}\bigg\lbrace\frac{\partial p}{\partial t}\bigg\vert_{x=0} + \kappa\frac{\partial S(t\vert x_{0})}{\partial t} + \frac{1}{T}\bigg[p\vert_{x=0} - \kappa\big(1-S(t\vert x_{0})\big)\bigg] \bigg\rbrace
\end{equation}
To shed more light on the nature of the parameter $\kappa$ it is instructive to consider the limiting case of the BC Eq.~\eqref{backreactionBC}. Taking the limit $c\rightarrow\infty$, $T\rightarrow 0$ and $\beta/(2-\beta)\rightarrow 0$ such that $c^{2}T=:D$ and $\beta/(2-\beta)c=:\kappa_{a}$ are kept finite, we recover the radiation-backreaction BC \cite{Agmon:1984} for the diffusion equation
\begin{equation}
D\frac{\partial p}{\partial x}\bigg\vert_{x=0} = \kappa_{a}p\vert_{x=0} - \kappa_{d}\big(1-S(t\vert x_{0})\big),
\end{equation}
where $\kappa_{d} := \kappa_{a} \kappa$. On the other hand, when we take the same limit as before with the exception that $\beta/(2-\beta)c\rightarrow \infty$, we arrive at the absorption-backreaction BC \cite{Agmon:1984}
\begin{equation}
p\vert_{x=0} = \kappa\big[1-S(t\vert x_{0})\big].
\end{equation}
To derive a GF of the telegrapher's equation subject to the BC specified by Eq.~\eqref{backreactionBC}, one has to find a suitable expression for the survival probability. To this end, we briefly recall how one proceeds in the case of diffusion.
From the equation of continuity Eq.~\eqref{conservationEq} and the definition of the survival probability Eq.~\eqref{defS} it follows
\begin{equation}\label{SandJ}
\frac{\partial S(t\vert x_{0})}{\partial t} = j(x,t\vert x_{0})\vert_{x=0}.
\end{equation}
Assuming that the current vanishes at infinity and that Fick's first law Eq.~\eqref{Fick} is valid one obtains
\begin{equation}\label{expS}
S(t\vert x_{0}) = 1 - \int^{t}_{0} \frac{\partial p(x,t\vert x_{0})}{\partial x}\bigg\vert_{x=0} \,dt^{\prime}.
\end{equation}
Obviously, Eq.~\eqref{expS} cannot be used in the context of the telegrapher's equation, because we already discussed that Fick's law has to be replaced by Cattaneo's equation Eq.~\eqref{cattaneo}. Hence, we combine Eq.~\eqref{defJ} and Eq.~\eqref{SandJ}, leading to
\begin{equation}\label{defSGeneral}
S(t\vert x_{0}) = 1- c^{2}\int^{t}_{0}\int^{t^{\prime}}_{0}e^{-(t^{\prime}-\tau)/T}\frac{\partial p(x,\tau\vert x_{0})}{\partial x}\,d\tau \,d t^{\prime}.
\end{equation}
The derived expression for the survival probability can be introduced in the BC Eq.~\eqref{backreactionBC}. Henceforth, we will focus on the case $\beta = 1.$ Then, in the Laplace domain the BC becomes
\begin{equation}
\bigg( c+\frac{\kappa c^{2}}{s}\bigg)\frac{\partial \tilde{p}}{\partial x}\bigg\vert_{x=0} = \bigg(s+\frac{1}{T}\bigg)\tilde{p}\bigg\vert_{x=0}.
\end{equation}
As before, we make the ansatz
\begin{equation}\label{ansatz}
\tilde{p}(x, s\vert x_{0}) = \tilde{p}_{\text{free}}(x, s\vert x_{0}) + \tilde{h}_{\text{back}}(x, s).
\end{equation}
The function $h_{\text{back}}(x, t)$ has to be a solution of the telegrapher's equation and hence takes in the Laplace domain the form
\begin{equation}
\tilde{h}_{\text{back}}(x, s) = A(s) e^{-x/c\sqrt{s(s+\frac{1}{T})}}.
\end{equation}
The constant $A(s)$ must be chosen in such a way that Eq.~\eqref{ansatz} satisfies the BC. We obtain
\begin{equation}
A(s) = -\frac{1}{2c}\frac{s+a}{\sqrt{s(s+a)}}\frac{s+a - \big(1+\frac{c\kappa}{s} \big)\sqrt{s(s+a)}}{s+a + \big(1+\frac{c\kappa}{s} \big)\sqrt{s(s+a)}}
e^{-\frac{x_{0}}{c}\sqrt{s(s+a)}}.
\end{equation}
As before, we can calculate $h_{\text{back}}(x, t)$ via the Bromwich integral
\begin{eqnarray}\label{BromwichBack}
h_{\text{back}}(x,t) =-\frac{1}{4\pi i c}\int^{\gamma+i\infty}_{\gamma-i\infty}e^{ts - k\sqrt{s(s+a)}}\times \nonumber\\
\frac{s+a}{\sqrt{s(s+a)}}\bigg[\frac{\sqrt{s+a}-\big(1+\frac{c\kappa}{s} \big)\sqrt{s}}{\sqrt{s+a} + \big(1+\frac{c\kappa}{s} \big)\sqrt{s}}\bigg]ds.
\end{eqnarray}
We can closely follow the line of reasoning presented for the case of absorbing and radiation BC. We may specify the same branch cut structure, employ the same integration contour and make the same substitutions.
Thus, we arrive at
\begin{eqnarray}\label{finalHBack}
&&h_{\text{back}}(x,t \vert x_{0}) = \frac{a\Theta(ct-k)}{4\pi c}e^{-1/2at}\int^{1}_{-1}\frac{e^{-1/2at\xi}(1-\xi)}{c^{2}\kappa^{2}+a/2(a-2c\kappa)(1+\xi)}\times\qquad\qquad\nonumber\\
&&\bigg[\frac{\cos(1/2ka\sqrt{1-\xi^{2}})}{\sqrt{1-\xi^{2}}}\Pi(\xi) + \sin(1/2ka\sqrt{1-\xi^{2}})\Gamma(\xi)\bigg]d\xi.
\end{eqnarray}
Here, we introduced the functions
\begin{eqnarray}
\Pi(\xi) &=& \frac{1}{2} a^{2}\xi^{2} + \frac{1}{2}a(a-2c\kappa)\xi - c\kappa + c^{2}\kappa^{2},\\
\Gamma(\xi) &=& \frac{1}{2} a^{2}\xi + \frac{1}{2}a(a-2c\kappa).
\end{eqnarray}
In the limit $\kappa\rightarrow 0$, we recover the result for the purely absorbing BC Eq.~\eqref{finalHabs}.
Furthermore, we note that
\begin{equation}
\bigg\vert\frac{\frac{a}{2}(a-2c\kappa)}{\frac{a}{2}(a-2c\kappa)+c^{2}\kappa^{2}}\bigg\vert < 1,
\end{equation}
and therefore in this case also it is possible to expand the denumerator in a power expansion in
$$ \frac{\frac{a}{2}(a-2c\kappa)}{\frac{a}{2}(a-2c\kappa)+c^{2}\kappa^{2}}\xi.$$
Using Eq.~\eqref{integralBessel} this means that we can dispense with the integral representation altogether in the case of the backreaction BC also.
\begin{figure}
\includegraphics[scale=0.4]{TeleContour}
\caption{Integration contour used in Eq.~(\ref{Bromwich}).}\label{fig:contour}
\end{figure}
\newpage
\subsection*{Acknowledgments}
This research was supported by the Intramural Research Program of the NIH, National Institute of Allergy and Infectious Diseases.
We would like to thank Bastian R. Angermann and Frederick Klauschen for stimulating discussions.
\bibliographystyle{plain}
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\section{Introduction}
The characterization of cosmological structures on the basis of their degree of complexity
is an interesting physical phenomenon as it is directly related to the complications
appearing in a stellar system due to its physical and structural characteristics.
The measure of degree of complexity can be helpful in the study of the formation and
evolution of stellar objects.All those factors that triggers the complications
in a system are also involved in the different evolutionary phases of a system.
However, a precise and concise definition of complexity in the field of astrophysics
is still an ambition and aspiration of the community of astrophysicists. In past, many
sincere efforts have been made in order to define an appropriate
criterion for the measure of degree of the complexity in different branches
of science \cite{1}-\cite{10}. Among the many definitions that have been proposed
so far, most of them are related to the concepts of information and disequilibrium.
These are based on the natural thought that complexity is related to a basic property
describing the structures existing within the system. In physics, perfect crystals and
ideal gas represent two different but simplest models, so they can be considered as
systems with zero complexity. Nevertheless, study of both of these systems ensure
that definition of complexity cannot be confined to the concepts of entropy and
information, rather it includes some other factors which was going to be ignored.
A definition of complexity for self-gravitating systems is introduced in \cite{11}
which is based on the work developed by Lopez-Ruiz and his collaborators \cite{6}.
In this definition, probability distribution which appear in the definition of entropy
and information is replaced by energy density of the
fluids, which has been justified by the argument that this physical quantity is
related to the probability of finding some particles at given specified location
inside the star, or it proved difficult to suggest a better alternative from the
available physical quantities. Nevertheless, this definition has some important
drawbacks, as it only encompasses the role of energy density whereas other
physical parameters like pressure isotropy or anisotropy which are expected
to play an important role in structure formation of a system are completely
ignored. In order to avoid these drawbacks, an entirely new definition of complexity \cite{12} has
been proposed for spherically symmetric and static self-gravitating source. It is
devised on the basic assumption that less complex systems correspond to the fluid
configurations with homogeneous energy density and isotropic pressure.
Such distributions are assigned with zero degree of complexity factor
which appears in orthogonal splitting of Riemann tensor. Herrera and his collaborators extended this concept
of complexity from static to non-static scenario, where they consider not only the complexity factor
of structure of fluid configuration but also discuss the conditions of minimum
complexity of evolutionary patterns \cite{12a}. For an axially symmetric static source,
they explored three different complexity factors and found that all the three factors vanish for simplest fluid configuration \cite{12b}.
The significance of this definition motivated the researchers to explore
it for different scenarios. Herrera et al. extended this fascinating concept to the vacuum solutions
with the help of the Bondi metric which covers a huge number of spacetimes like
Minkowski spacetime, the static Weyl
metrics, gravitationally
radiating metrics, non radiative and non static metrics \cite{LH}, while Casadio et al.
studied isotropization and change of complexity
employing gravitational decoupling approach for
a static and spherically symmetric system \cite{CAS}.
In \cite{13}, the effects of electric charge has been incorporated and complexity factor has been
analyzed. They have found that electromagnetic field has considerable impacts on the complexity
of a cosmological structure. The same authors discussed the complexity of cylindrically symmetric static source in \cite{14}.
Cosmological objects are usually studied with the consideration of
spherical symmetry because observational evidences show that deformations
in spherical symmetry are very rare, however non-spherical symmetries may also exit
and provide significant information about celestial objects. Different cosmological
issues have also been discussed by assuming non-spherical situations. Cylindrically
symmetric sources grasped the attention of the community of the relativists when
Levi-Civita found its vacuum solution. Cylindrically symmetric thin-shell
wormholes, cylindrical polytropes with generalized polytropic equation of
state, charged non-adiabatic and perfect fluid with cylindrically symmetric
background were studied in \cite{17}-\cite{21}. Cylindrically symmetric
self-gravitating objects with anisotropic background evolving under
different conditions \cite{22}-\cite{24} have been explored in order to
understand different phases of evolution. Herrera and his collaborators \cite{25}
developed structure scalars for cylindrically symmetric and studied
dissipative fluid distribution with the help of these scalars.
General Relativity (GR) is still one of the most comprehensive theory in order
to understand the dynamics of the universe in accordance with its matter components. However, some cosmological issues like
unification of gravitation and quantum mechanics covering the singularity
problem and late time accelerated expansion of the universe necessitated the
improvements in theoretical framework of GR through modified theories. A large
number of class of modified theories has been introduced to overcome the
cosmological issues. Due to their significance, different cosmological
phenomena like anisotropy, luminosity and stability analysis of celestial
objects have been discussed and presented in literature with the help of
these theories \cite{15,16,22,23,24}. Reverberi \cite{27} studied the contracting dust
cloud with the $f(R)$ model and found that increase in energy density
bring about the curvature singularity. Cembranos and his co-authors \cite{28}
explored the inflation candidate in the context of a spherically symmetric
self-gravitating collapsing dust cloud in $f(R)$ gravity. Gravitational collapse and stability constraints for spherical and axial
symmetry has been studied via $f(R,T)$ gravity in \cite{37}-\cite{39}.
Compact stars have also been examined and explored in details with the help of modified gravities.
Strong gravity regime in viable models of $f(R)$ gravity has been studied \cite{40} and the claim that
stars with relativistically deep potentials cannot exist in $f(R)$ gravity has been disproved \cite{41}.
Capozziello et al. \cite{42} considered modified Lane-Emden equation that comes out from $f(R)$ gravity and
discussed the hydrostatic equilibrium of stellar structures.
Some interior models of compact stars like $4U1820-30, Her X-1, SAX J1808-3658$
has been studied using Krori and Barua analytical solution to the static spacetime with fluid
source in modified $f(R)$ gravity \cite{43}. Polytropic stars in Palatini $f(R)$-theories has also been investigated and it was shown how findings rely on regularity of the function $f(R)$ \cite{44}. The possible formation of compact astrophysical objects and their physical features has been discussed in the framework of $f(R, T)$ and $f(G, T)$ gravity \cite{45}-\cite{49}. Abbas and Nazar \cite{15, 16} discussed the effects of $f(R)$ gravity on complexity factor for spherically symmetric static source with anisotropic background.
In \cite{s1, s2}, complexity factor has been explored for static anisotropic sources in the context of Brans-Dick theory, while in \cite{s3, s4} it has been examined for a dynamical sphere and a class of compact stars in the framework of $f(R, T)$ gravity. The physical properties of compact objects were analyzed and their numerical outputs were found for different values of coupling parameter. Keeping in view the significance of modified theories, we intended to explore the definition of complexity factor for a cylindrically symmetric dynamical source with anisotropic fluid distribution in $f(R,T)$ gravity.
This article has been organized as follows: Next section presents basic notations, variables and set of field equations for cylindrically symmetric self-gravitating dynamical source in $f(R,T)$ gravity. Section III and IV entail the construction of structure scalars for self-gravitating cylinder in $f(R,T)$ gravity and the identification of complexity factor, respectively. Section V covers the discussion regarding simplest modes of evolution, while the section VI comprises the $f(R,T)$ model which is followed by the discussion about kinematics and dynamics of the system for three different cases of $f(R,T)$ model. Section VII explores the
stability of vanishing complexity factor condition. Last section concludes our results.
\section{Basic notations, variables and relativistic system of Equations in $f(R,T)$ theory of gravity}
We have considered an anisotropic stellar configuration with cylindrical geometry
which experiences the dissipation in the form of heat flux and its
interior region is given by the following expression
\begin{equation}\label{1}
ds^2_-=-F^2(t,r)dt^{2}+G^2(t,r)dr^{2}+H^2(t,r)\left(d\theta^{2}+
dz^2\right),
\end{equation}
Here, $F$ and $G$ are dimensionless, while $H$ has the same dimension as $r$ has.
The $f(R,T)$ modification of Einstein-Hilbert action is given by \cite{1*}
\begin{equation}\label{2}
\int dx^4\sqrt{-g}\left[\frac{f(R, T)}{16\pi G}+\mathcal{L} _ {(m)}\right].
\end{equation}
Here, action due to matter is described by $\mathcal{L} _ {(m)}$,
whose different choices can be made, each choice represents a particular
form of fluid.
Varying the modified action given in Eq.$(\ref{2})$ with respect to metric $g_{\alpha\beta}$,
we have the following set of field equations
\begin{eqnarray}\label{4}
&&R_{\alpha\beta} f_R(R,T)-\frac{1}{2}g_{\alpha\beta} f(R,T)+(g_{\alpha\beta}\Box-\nabla_{\alpha}\nabla_{\beta})f_R(R,T)=8\pi GT_{\alpha\beta}^{(m)}-f_T(R,T)T_{\alpha\beta}^{(m)}-f_T(R,T)\Theta_{\alpha\beta},
\end{eqnarray}
where $f_R(R,T)=\frac{\partial f(R,T)}{\partial R}$, $f_T(R,T)=\frac{\partial f(R,T)}{\partial T}$,
while $\nabla_{\alpha}$ is the covariant derivative
and $\Box$ is four-dimensional Levi-Civita covariant derivative.
The term $\Theta_{\alpha\beta}$ has the following mathematical representation
\begin{eqnarray}\label{5}
\Theta_{\alpha\beta}=\frac{g^{\rho\nu}\delta T_{\mu\nu}}{\delta g^{\alpha\beta}}= -2T_{\alpha\beta}+g_{\alpha\beta}\mathcal{L} _ {(m)}-2g^{\rho\nu}\frac{\partial^2\mathcal{L} _ {(m)}}{\partial g^{\alpha\beta}\partial g^{\rho\nu}}.
\end{eqnarray}
With the choice of $\mathcal{L} _ {(m)}= \mu$ (energy density) and $8\pi G = 1$, $\Theta_{\alpha\beta}$ takes the form as
\begin{eqnarray}\label{6}
\Theta_{\alpha\beta}=-2T_{\alpha\beta}+\mu g_{\alpha\beta}.
\end{eqnarray}
Using Eq.$(\ref{5})$, the modified field equations given in (\ref{4}) become
\begin{eqnarray}\label{7}
G_{\alpha\beta}&=& T_{\alpha\beta}^{eff},
\end{eqnarray}
where
\begin{eqnarray}\label{8}
T^{eff}_{\alpha\beta}&=&\frac{1}{f_R}\left[(f_T+1)T^{(m)}_{\alpha\beta}-\mu g_{\alpha\beta}f_T+
\frac{f-Rf_R}{2}g_{\alpha\beta}+(\nabla_\alpha\nabla_\beta-g_{\alpha\beta}\Box)f_R\right],
\end{eqnarray}
where $T^{(m)}_{\alpha\beta}$ represents the energy momentum tensor for the usual
matter.
We have taken into account a fluid distribution
which is locally anisotropic and suffering dissipation in the form of heat flux. Its energy momentum
tensor has the following mathematical expression
\begin{eqnarray}\label{9}
T^{(m)}_{\alpha\beta}=(\mu+P_\perp)V_{\alpha}V_{\beta}-P_\perp g_{\alpha\beta}
+(P_r-P_\perp)\chi_{\alpha}\chi_{\beta}+q_\alpha V_\beta+V_\alpha q_\beta,
\end{eqnarray}
where $P_r$ and $P_\perp$ are two principal stresses. In most general cylindrical case,
one has three principal stresses which leads to an anisotropic tensor depending on two independent
scalar functions \cite{25}. However, in our case it will lead to an anisotropic tensor depending on one scalar
function. It will ultimately lead to a single scalar dependent structure scalar, which can further be analyzed
for complexity factor. We are interested in
structure scalar depending on single scalar instead of two, thus our assumption for fluid distribution is based on
two unequal principal stresses.
Here $V_{\beta}$, $\chi_{\beta}$ and $q_\alpha$ denote four-velocity,
unit four-vector along radial direction and heat flux respectively.
Under co-moving relative motion, these quantities are defined as
\begin{equation}\label{10}
V_{\alpha}= F^{-1}\delta^{0}_{\alpha},\quad
\chi_{\alpha}=G^{-1}\delta^1_{\alpha},\quad
q^\alpha=qG^{-1}\delta_1^\alpha,
\end{equation}
and satisfy the following relations
\begin{eqnarray}\nonumber
V^{\alpha}V_{\alpha}=-1,\quad
\chi^{\alpha}\chi_{\alpha}=1,\quad
\chi^{\alpha}V_\alpha=0,\quad
V^\alpha q_\alpha=0.
\end{eqnarray}
Eq.$(\ref{9})$ can also be expressed as
\begin{eqnarray}\label{T}
T_{\alpha\beta}^{(m)} &=& \mu V_\alpha V_\beta+P h_{\alpha\beta}+\Pi_{\alpha\beta}+q(V_\alpha\chi_\beta+\chi_\alpha V_\beta),
\end{eqnarray}
where
\begin{eqnarray}\label{T1}
P&=&\frac{1}{3}\left(P_r+2P_\perp\right),\quad h_{\alpha\beta}=g_{\alpha\beta}+V_\alpha V_\beta, \quad \Pi_{\alpha\beta}
=\Pi\left(\chi_\alpha\chi_\beta-\frac{1}{3}h_{\alpha\beta}\right), \quad \Pi=P_r-P_\perp.
\end{eqnarray}
while the components of shear tensor $\sigma_{\alpha\beta}$ are defined as
\begin{eqnarray}\label{11}
&&\sigma_{\alpha\beta}=V_{(\alpha;\beta)}-a_{(\alpha}V_{\beta)}-\frac{1}{3}\Theta\left(g_{\alpha\beta}-V_\alpha V _\beta\right),
\end{eqnarray}
Here $a_\alpha$ is four-acceleration and $\Theta$ is expansion scalar which defines the rate of infinitesimal change of matter distribution. These two quantities
are defined by the following mathematical formulae
\begin{equation}\label{12}
a_\alpha=V_{(\alpha;\beta)}V^\beta,\quad
\Theta=V^\alpha_{;\alpha}.
\end{equation}
Four acceleration $a_\alpha$, expansion scalar $\Theta$ and non- zero components of shear tensor $\sigma_{\alpha\beta}$ can easily be calculated for given fluid distribution of cylindrically symmetric
gravitational source and these are given below
\begin{eqnarray}\label{13}
a_1=\frac{F'}{F},\quad
\Theta=\frac{1}{F}\left(\frac{\dot{G}}{G}+\frac{2\dot{H}}{H}\right),
\\\label{gg}
\sigma_{11}=\frac{2}{\sqrt{3}}G^2\sigma, \quad\quad
\sigma_{22}=\sigma_{33}=-\frac{1}{\sqrt{3}}H^2\sigma
\end{eqnarray}
where
\begin{eqnarray}\label{sh}
\sigma&=&\frac{1}{\sqrt{3}F}\left(\frac{\dot{G}}{G}-\frac{\dot{H}}{H}\right),
\end{eqnarray}
and its scalar value takes the form as
\begin{eqnarray}\label{mi}
\sigma^{\alpha\beta}\sigma_{\alpha\beta} &=& 2\sigma^2.
\end{eqnarray}
The set of $f(R,T)$ field equations for the given cylindrically symmetric interior metric is given by
\begin{eqnarray}\label{16}
G_{00}&=&\frac{A^{2}}{f_R}\left[\mu+\Psi+\psi_{00}\right],\\\label{17}
G_{01}&=&\frac{FG}{f_R}\left((1+f_T)(-q)+\frac{\psi_{01}}{FG}\right),\\\label{18}
G_{11}&=&\frac{G^{2}}{f_R}\left[(1+f_T)\left(P_r\right)+\mu f_T-\Psi+\psi_{11}\right],\\\label{19}
G_{22}&=&\frac{H^{2}}{f_R}\left[(1+f_T)\left(P_\perp\right)+\mu f_T-\Psi+\psi_{22}\right],\\\nonumber
\end{eqnarray}
where
\begin{eqnarray}\label{20}
\Psi&=&\frac{f-Rf_R}{2},\quad
\psi_{00}=\frac{f''_R}{G^2}-\frac{\dot{f_R}}{F^2}\left(\frac{\dot{G}}{G}+2\frac{\dot{H}}{H}\right)
+\frac{f_R'}{G^2}\left(2\frac{H'}{H}-\frac{G'}{G}\right),\\\label{21}
\psi_{01}&=&\dot{f'_R}-\frac{F'}{F}\dot{f_R}-\frac{\dot{G}}{G}f'_R,\\\label{22}
\psi_{11}
&=&\frac{\ddot{f_R}}{F^{2}}-\frac{\dot{f_R}}{F^{2}}\left(\frac{\dot{F}}{F}-2\frac{\dot{H}}{H}\right)
-\frac{f'_R}{B^{2}}\left(\frac{F'}{F}+2\frac{H'}{H}\right),\\\label{23}
\psi_{22}&=&\frac{\ddot{f_R}}{F^{2}}-\frac{f''_R}{G^{2}}-\frac{\dot{f_R}}{F^{2}}\left(\frac{\dot{F}}{F}-\frac{\dot{G}}{G}-\frac{\dot{H}}{H}\right)
-\frac{f'_R}{G^{2}}\left(\frac{F'}{F}-\frac{G'}{G}+\frac{H'}{H}\right).
\end{eqnarray}
Thorne \cite{s5} proposed the idea that total amount of energy in a
cylindrical celestial object can defined through the gravitational C-energy, which takes the following
form for the case under consideration
\begin{eqnarray}\label{24}
m(t,r)&=&\left\{\left(\frac{\dot{H}}{F}\right)^2-\left(\frac{H'}{G}\right)^2\right\}\frac{H}{2}+\frac{l}{8}.
\end{eqnarray}
Before moving towards the further computations,
it is worthwhile to define some notations.
The $D_T$ and $D_H$ are operators which represent proper time and radial derivatives, respectively,
and are defined as
\begin{eqnarray}\label{25}
&&D_T=\frac{1}{F}\frac{\partial}{\partial t},\quad
D_H=\frac{1}{H'}\frac{\partial}{\partial r},
\end{eqnarray}
whereas relativistic velocity of interior of collapsing fluid is given by
\begin{eqnarray}\label{26}
&&U=D_TH=\frac{\dot{H}}{F}<0,
\end{eqnarray}
From Eq.$(\ref{24})$, we can obtain
\begin{eqnarray}\label{27}
&&\tilde{E}=\frac{H'}{G}=\sqrt{\frac{l}{4H}+U^2-\frac{2}{H}m(t,r)}.
\end{eqnarray}
Using above equation together with Eq.$(\ref{17})$, we can develop the expression given below
\begin{eqnarray}\label{28}
\tilde{E}\left(\sqrt{3}\frac{\sigma}{H}-\frac{1}{3}D_H(\Theta-\sqrt{3}\sigma)\right) &=& \frac{1}{2 f_R}\left(-q(1+f_T)+\frac{\psi_{01}}{FG}\right).
\end{eqnarray}
Eq.$(\ref{24})$ together with $(\ref{16})-(\ref{19})$ and $(\ref{25})$ provides
\begin{eqnarray}\label{29}
D_Tm &=&\frac{H^2}{2f_R}\left\{-(1+f_T)\tilde{E}q-(1+f_T)UP_r+f_T\mu U+\frac{\tilde{E}}{FG}\pi_{01}-U\left(\Psi+\psi_{11}\right)\right\},
\end{eqnarray}
whereas radial derivative of mass provides
\begin{eqnarray}\label{30}
D_Hm &=& \frac{H^2}{2f_R}\left\{\mu+\frac{U}{\tilde{E}}(1+f_T)q+\pi+\pi_{00}-\frac{U}{\tilde{E}}\frac{\psi_{01}}{FG}\right\},
\end{eqnarray}
which further leads towards the following expression
\begin{eqnarray}\label{31}
m &=&\frac{1}{2}\int_0^r \frac{H^2}{f_R}\left\{\mu+\frac{U}{\tilde{E}}(1+f_T)q+\psi+\psi_{00}-\frac{U}{\tilde{E}}\frac{\psi_{01}}{FG}\right\}H'dr,
\end{eqnarray}
It can also be written as
\begin{eqnarray}\label{32}
\frac{3m}{H^3} &=&\frac{3}{2H^3}\int_0^r \frac{H^2}{f_R}\left\{\mu+\frac{U}{\tilde{E}}(1+f_T)q+\psi+\psi_{00}-\frac{U}{\tilde{E}}\frac{\psi_{01}}{FG}\right\}H'dr.
\end{eqnarray}
\section{Weyl tensor and Structure scalars}
In order to define the structures scalars, we first need to find the weyl tensor which has two parts, i.e,
electric and magnetic parts. The electric part
is given below
\begin{eqnarray}\label{w}
E_{\alpha\beta} &=& C_{\alpha\mu\beta\nu}V^{\mu}V^{\nu},
\end{eqnarray}
The non-trivial components of electric component of weyl tensor are
\begin{eqnarray}\label{E}
E_{11} = \frac{2}{3}G^2\eta,\quad E_{22}= -\frac{1}{3}H^2\eta=E_{33},
\end{eqnarray}
where
\begin{eqnarray}\nonumber
\eta &=& \frac{1}{2F^2}\left\{\frac{\ddot{H}}{H}-\frac{\ddot{G}}{G}-\left(\frac{\dot{H}}{H}
-\frac{\dot{G}}{G}\right)\left(\frac{\dot{F}}{F}+\frac{\dot{H}}{H}\right)\right\}\\\label{ee}
&&+\frac{1}{2B^2}\left\{\frac{F''}{F}-\frac{H''}{H}+\left(\frac{G'}{G}+\frac{H'}{H}\right)
\left(\frac{H'}{H}-\frac{F'}{F}\right)\right\}-\frac{1}{2H^2}.
\end{eqnarray}
With the help of $(\ref{24})$ and $(\ref{32})$, we can find the expression for the above scalar value
\begin{eqnarray}\nonumber
\eta &=& \frac{1}{2f_R}\left[\mu-(1+f_T)\Pi+\Psi+\psi_{00}-\psi_{11}+\psi_{22}\right]
\\\label{ex}&&-\frac{3}{2H^3}\int_0^r \frac{H^2}{f_R}\left\{\mu+\frac{U}{E}(1+f_T)q-\Psi+\psi_{00}
-\frac{U}{E}\frac{\psi_{01}}{FG}\right\}H'dr,
\end{eqnarray}
The electric component $E_{\alpha\beta}$, in view of unit four-velocity and four-vectors can be given by
\begin{eqnarray}
E_{\alpha\beta} &=& \eta(\chi_\alpha\chi_\beta-\frac{1}{3}h_{\alpha\beta})
\end{eqnarray}
Following Bel \cite{Bel} and Herrera et al. \cite{H1}-\cite{HRv}, we develop formalism for structure scalars in $f(R,T)$ gravity
and introduce a couple of tensors, named $Y_{\alpha\beta}$ and $X_{\alpha\beta}$.
For this, we orthogonally decompose the Riemann curvature tensor and find that
\begin{eqnarray}\label{xx}
X_{\alpha\beta}&=& \frac{1}{3f_R}\left[\mu+\Psi+\psi_{00}\right]h_{\alpha\beta}-\frac{1}{2f_R}\left[(1+f_T)\Pi+\psi_{11}-\psi_{22}\right]
\left(\chi_\alpha\chi_\beta-\frac{1}{3}h_{\alpha\beta}\right)-E_{\alpha\beta},\\\nonumber
Y_{\alpha\beta}&=& \frac{1}{6f_R}\left[\mu+3f_T\mu+(1+f_T)(3P_r-2\Pi)+\Psi+\psi_{00}+\psi_{11}+2\psi_{22}\right]h_{\alpha\beta}-\frac{1}{2f_R}\left[(1+f_T)\Pi
\right.\\\label{yy}&&+\left.\psi_{11}-\psi_{22}\right]
\left(\chi_\alpha\chi_\beta-\frac{1}{3}h_{\alpha\beta}\right)+E_{\alpha\beta}.
\end{eqnarray}
For detailed discussion of these quantities, one can see \cite{HRv}.
These tensors can be written in the combination of structure scalars ($X_T$, $X_{TF}$, $Y_T$ and $Y_{TF}$).
\begin{eqnarray}\label{X}
X_{\alpha\beta} &=& \frac{1}{3}X_T h_{\alpha\beta}+X_{TF}\left(\chi_\alpha \chi_\beta-\frac{1}{3}h_{\alpha\beta}\right),\\\label{Y}
Y_{\alpha\beta} &=& \frac{1}{3}Y_T h_{\alpha\beta}+Y_{TF}\left(\chi_\alpha \chi_\beta-\frac{1}{3}h_{\alpha\beta}\right).
\end{eqnarray}
By making use of Eqs.$(\ref{16})$, $(\ref{18})$, $(\ref{19})$, $(\ref{24})$ and $(\ref{ex})$, we have the following expression
\begin{eqnarray}\label{Y1}
\frac{3}{H^3}\left(m-\frac{l}{8}\right)&=& \frac{1}{2f_R}\left(\mu+\psi-(1+f_T)\Pi+\psi_{00}-\pi_{11}+\psi_{22}\right)-\eta,
\end{eqnarray}
which makes it possible to produce the following expression with the help of Eqs.(\ref{32}) and (\ref{Y1})
\begin{eqnarray}\nonumber
Y_{TF} &=& \frac{1}{2f_R}\left\{\mu-2(1+f_T)\Pi+\pi+\pi_{00}-2\psi_{11}+2\pi_{22}\right\}-\frac{3}{2H^3}\int_0^r\frac{H^2}{f_R}\left\{
\mu+\psi+\psi_{00}\right.\\\label{Y}&&+\left.\frac{U}{E}q(1+f_T)
-\frac{U}{E}\frac{\psi_{01}}{FG}\right\}H'dr+\frac{3l}{8H^3},
\end{eqnarray}
whereas $X_{TF}$ takes the form as
\begin{eqnarray}\label{X}
X_{TF} &=& -\frac{1}{2f_R}\left(\mu+\psi+\psi_{00}\right)+\frac{3}{2H^3}\int_0^r\frac{H^2}{f_R}\left\{
\mu+\psi+\psi_{00}+\frac{U}{E}q(1+f_T)-\frac{U}{E}\frac{\psi_{01}}{FG}\right\}H'dr+\frac{3l}{8H^3}.
\end{eqnarray}
Thus, we are now able to construct a differential equation which shows a relationship between energy density inhomogeneity and weyl tensor.
\begin{eqnarray}
\left( X_{TF}+\mu+\frac{1}{2f_{R}}(\mu+\psi+\psi_{00})\right)' &=&-3\frac{H'}{H}X_{TF}+\frac{(\Theta-\sigma)}{2f_R}\left(q(1+f_T)G+\frac{\psi_{01}}{G}\right),
\end{eqnarray}
If we choose $X_{TF}=0$ in the absence of dark source and dissipation, then we have
\begin{eqnarray}\label{t1}
(\mu+\psi+\psi_{00})' &=& 0,
\end{eqnarray}
however, in general dissipative case it assumes the form
\begin{eqnarray}\label{t2}
(\mu+\psi+\psi_{00})' &=& \frac{(\Theta-\sigma)}{2f_R}\left(q(1+f_T)G+\frac{\psi_{01}}{G}\right).
\end{eqnarray}
It shows that $X_{TF}$ controls the energy density homogeneity along with dark source terms.
\section{The Complexity Factor}
The definition of quantity measuring the complexity of a dynamical system is more generalized than for the static one as it faces two additional
factors. In static case, only fluid parameters are involved, while in non-static case, complexity of structure of system and of patterns
of evolution also contribute to the situation. For static case, definition is based on the assumptions that homogenous energy density and
isotropic pressure corresponds to the simplest system. However, for the later case, simplest possible patterns are also considered in order to
measure the degree of complexity of evolutionary patterns.
Recently, we have analyzed definition of complexity for anisotropic fluid non-static sphere in $f(R,T)$
gravity. We chose $Y_{TF}$ as complexity factor as it covers all the components that contributes to the complexity of a system.
In the case under consideration, we again found $Y_{TF}$ as most suitable scalar in order to analyze the components that trigger
complications in a system. It also incorporates the effects of dark source terms. We can see in Eq.(\ref{Y}) that it also contains the term
comprising length of cylinder. Thus, it also measures the geometric variations in a system.
\section{The Homologous Evolution And The Homogeneous Expansion Condition}
After making the choice of $Y_{TF}$ as complexity factor, our next task is to analyze the complexity of evolutionary patterns
of the system. Such analysis involves two possibilities: the homologous condition and homogeneous expansion.
Homogeneous expansion corresponds to the zero value of prime derivative of expansion scalar which measures infinitesimal
changes in fluid distribution, whereas homologous evolution corresponds to the similarity of the patterns.
\subsection{The Homologous Evolution}
We can see that Eq.$(\ref{28})$ can be written as
\begin{eqnarray}
D_H\left(\frac{U}{H}\right) &=& \frac{1+f_T}{f_R}\frac{q}{\tilde{E}}-\frac{1}{FG f_R \tilde{E}}\pi_{01}+\sqrt{3}\frac{\sigma}{H}.
\end{eqnarray}
whose integration leads to the equation
\begin{eqnarray}
\frac{U}{H} &=&\int_0^r\left(\frac{1+f_T}{f_R}\frac{q}{\tilde{E}}-\frac{1}{FG f_R \tilde{E}}\pi_{01}+\sqrt{3}\frac{\sigma}{H}\right)H'dr+h(t),
\end{eqnarray}
where $h(t)$ is function of integration.
\begin{eqnarray}\label{101}
U &=&H\int_0^r\left(\frac{1+f_T}{f_R}\frac{q}{\tilde{E}}-\frac{1}{FG f_R \tilde{E}}\psi_{01}+\sqrt{3}\frac{\sigma}{H}\right)H'dr+Hh(t),
\end{eqnarray}
which yields
\begin{eqnarray}\label{102}
U &=& \frac{U_\Sigma}{H_\Sigma}H-H\int_0^r\left(\frac{1+f_T}{f_R}\frac{q}{\tilde{E}}-\frac{1}{FG f_R \tilde{E}}\psi_{01}+\sqrt{3}\frac{\sigma}{H}\right)H'dr.
\end{eqnarray}
If the integral in Eq.$(\ref{101})$ and Eq.$(\ref{102})$ vanishes, then $U\sim H$ which is characteristic of homologous evolution; it
would be possible if $\sigma=0$, $q=0$ and $\psi_{01}=0$ or the terms cancel each other.
\\
For homologous evolution, $U=h(t) H$ and $h(t)=\frac{U_\sigma}{H_\sigma}$ where $U=D_T H$. It makes us to follow that $H$ is separable and can be written as
\begin{eqnarray}\label{103}
H &=& H_1(t)H_2(r)
\end{eqnarray}
The term with negative sign in Eq.$(\ref{102})$ shows that dissipation, shear and dark source entities are responsible
for the deviation of evolution from being homologous. Thus, we have
\begin{eqnarray}\label{105}
\frac{1+f_T}{f_R}\frac{qG}{H'}-\frac{1}{f_RF H'}\psi_{01}+\sqrt{3}\frac{\sigma}{H} &=& 0
\end{eqnarray}
It represents homologous condition in general. For non-dissipative case, it takes the form as
\begin{eqnarray}\label{106}
\sqrt{3}\frac{\sigma}{H} &=& \frac{1}{f_RF H'}\psi_{01}
\end{eqnarray}
It is obvious that homologous evolution does not correspond to the shear free condition in general, rather it depends on the choice
of $f(R,T)$ model.
\subsection{The homogeneous Expansion}
Homogeneous expansion also represents simple pattern of evolution.
Under homogeneous expansion, Eq.$(\ref{28})$ assumes the following form
\begin{eqnarray}\label{106i}
f_R\left(\sqrt{3} \frac{\sigma}{H}+\frac{1}{3}D_H\sigma\right)-\frac{1}{FH'}\psi_{01} &=& -\frac{qG}{H'}(1+f_T).
\end{eqnarray}
If we analyze the Eqs.$(\ref{105})$ and $(\ref{106i})$, then it can be clearly observe that imposition of these two conditions
is followed by $D_H(\sigma)=0$. Here, its implication is based on the regularity conditions in the neighborhood of the center, that shear free
condition implies zero dissipation.
\section{The $f(R, T)$ Model}
We can see that the result depends on the choice of $f(R,T)$ model. So we need to choose a viable $f(R, T)$ model in order to represent our results
in a meaningful way. The $f(R, T)$ model we have selected for discussion is developed by Sharif and Zubair \cite{zub} and has the following
mathematical form
\begin{eqnarray}\label{106a}
f(R, T) &=& \alpha_1 R^m T^n +\alpha_2T(1+\alpha_3 T^p R^q),
\end{eqnarray}
where $\alpha_i's$ are positive real numbers , whereas $m, n, p, q$ assumes some value greater
than or equal to $1$ . We will analyze our results considering different cases of
above mentioned model and we will proceed our further discussion under following three cases:\\
\begin{enumerate}
\item $f(R,T)= R+\alpha_2 T$, for $\alpha_1=1, m=1, n=0, \alpha_3=0$
\item $f(R, T)= \alpha_1 R+\alpha_2 T+\alpha_4 T^2$, for $m=1, n=0, \alpha_4=\alpha_1\alpha_3, p=1, q=0$
\item $f(R, T)= \alpha_1 R+\alpha_2 T(1+\alpha_3 TR^2)$, for $ m=1, n=0, p=1, q=2$
\end{enumerate}
\section{Kinematics and Dynamics of Stellar systems}
\subsection{Case I: $f(R,T)= R+\alpha_2 T$}
This form of model involves direct minimal curvature matter coupling, which has been used widely to explore
a number of cosmological phenomena because of its theoretical and cosmological consistency \cite{1*}.
Many theoretical cosmological models of the universe have been proposed to analyze the behavior of
mysterious components and their physical and cosmological consequences are explored \cite{s6, s7}.
Galactic structures and their existence have also been discussed, and results are found in agreement with
previously established solutions and assumptions \cite{s8}.
For this model, homologous condition and homogeneous expansion condition given in Eqs.$(\ref{105})$ and $(\ref{106i})$ take the form as
\begin{eqnarray}\label{1071}
(1+\alpha_2)qG &=&\sqrt{3}\frac{\sigma H'}{H},\\\label{1081}
\left(\sqrt{3} \frac{\sigma}{H}+\frac{1}{3}D_H\sigma\right)&=& -\frac{qG}{H'}(1+\alpha_2).
\end{eqnarray}
Here, Eq.$(\ref{1081})$ clearly depicts that fluid cannot be dissipative under shear free condition and
homogeneous expansion. However, if Eqs.$(\ref{1071})$ and $(\ref{1081})$ hold at the same time, then we have
\begin{eqnarray}\label{1081a}
(\Theta-\sqrt{3}\sigma)' &=& 0.
\end{eqnarray}
By inserting the values of expansion scalar and shear scalar given in Eqs.$(\ref{13})$ and $(\ref{sh})$, respectively, we have
\begin{eqnarray}\label{1091}
(\Theta-\sqrt{3}\sigma)' &=& \left(\frac{3}{F}\frac{\dot{H}}{H}\right)'= 0.
\end{eqnarray}
Eq.$(\ref{103})$ together with above equation follows that $F'=0$ which ensures the geodesic condition of the fluid. As $F$ possesses an arbitrary constant value, so we may choose $F=1$. With this choice, we get the following expression from Eq.(\ref{1091})
\begin{eqnarray}\label{1101}
\Theta-\sqrt{3}\sigma &=& 3\frac{\dot{H}}{H}.
\end{eqnarray}
We analyze this expression closer to the center and obtain the condition $(\Theta-\sqrt{3}\sigma)'=0$.
The successive derivatives of Eq.$(\ref{1101})$ with respect to $r$, also support our argument and strengthens the point that fluid is homologous.
\\
Again we analyze the Eq.$(\ref{28})$, if we
assume $\sigma=0$, (also fluid is non-dissipative), then we have
\begin{eqnarray}\label{1121}
\Theta' &=&0.
\end{eqnarray}
It can clearly be observe that homologous patterns of the evolution imply homogeneity of the expansion scalar.
\\
If we assume $\Theta'=0$, then Eq.$(\ref{28})$ takes the following form
\begin{eqnarray}\label{1131}
\left(\frac{\sqrt{3}\sigma}{H}-\frac{1}{\sqrt{3}}D_H(\sigma)\right) &=& 0,
\end{eqnarray}
which further takes the form as
\begin{eqnarray}\label{1131a}
\frac{\sigma'}{\sigma} &=& \frac{3R'}{R},
\end{eqnarray}
implying
\begin{eqnarray}\label{1131b}
\sigma &=& \frac{f_1(t)}{H^3},
\end{eqnarray}
where $f_1(t)$ is an arbitrary function of integration. As $r$ assumes zero value at the center, so $H$ will also be zero. Thus,
we must have $f_1(t)=0$ in order to avoid unboundedness of the expression. This situation lead to the vanishing of shear scalar. On the other hand,
if we choose the zero value for shear scalar, then Eq.$(\ref{28})$ ensures the homogeneous expansion. Here, it is obvious that homogeneous expansion and
homologous condition imply each other in non dissipative case. Further, we have analyzed the situation
for homogeneous expansion in the presence of dissipation and obtained
\begin{eqnarray}\label{1131c}
\sigma &=& -\frac{\sqrt{3}}{2H^3}\int^r_0 H^3 qG(1+\alpha_2)dr.
\end{eqnarray}
This expression makes it clear that homogeneous expansion and homologous condition are not compatible in the presence of dissipation.
Now, we consider some dynamical situations for the system under consideration.
As our previous discussion shows that fluid is
geodesic under homologous condition in both dissipative and non-dissipative cases. Thus, if we apply
homologous condition on the Eq.(\ref{B4}), we obtain
\begin{eqnarray}\label{1141}
D_TU &=& -\frac{m}{H^2}-\frac{H}{2}\left\{\alpha_2\mu -(1+\alpha_2)P_r+\frac{\alpha_2 T}{2}\right\}.
\end{eqnarray}
This equation can be re-written in the form of $Y_{TF}$ as
\begin{eqnarray}\label{1151}
\frac{3 D_TU}{H} &=& -\frac{1}{2}\left\{(1-3\alpha_2)\mu-\alpha_2 T-2(1+\alpha_2)\Pi
+3(1+\alpha_2)P_r\right\}+Y_{TF}-\frac{3l}{8H^3}.
\end{eqnarray}
Now, the manipulation of Eqs.$(\ref{16})$, $(\ref{18})$ and $(\ref{19})$ provides
\begin{eqnarray}\label{1151a}
-\frac{2\ddot{H}}{H}-\frac{\ddot{G}}{G} &=& \frac{1}{2}\left\{(1-3\alpha_2)\mu-\alpha_2 T-2(1+\alpha_2)\Pi
+3(1+\alpha_2)P_r\right\}+Y_{TF}-\frac{3l}{8H^3},
\end{eqnarray}
while the definition of velocity `U' of collapsing star provides
\begin{eqnarray}\label{1171}
\frac{3 D_TU}{H}&=& \frac{3\ddot{H}}{H}.
\end{eqnarray}
Insertion of above two equations into Eq.$(\ref{1151a})$ leads to
\begin{eqnarray}\label{1181}
Y_{TF} &=& \frac{\ddot{H}}{H}-\frac{\ddot{G}}{G}-\frac{3l}{8H^3}.
\end{eqnarray}
If $Y_{TF}=0$, then Eq.$(\ref{1181})$ becomes
\begin{eqnarray}\label{118a}
\frac{3l}{8H^3} &=& \frac{\ddot{H}}{H}-\frac{\ddot{G}}{G}.
\end{eqnarray}
Since we are working on the assumption that fluid is homologous, so we can write the Eq.$(\ref{1151a})$ as
\begin{eqnarray}\label{1151b}
3\left(\dot{h}(t)+h(t)\frac{\dot{H}}{H}\right) &=& -\frac{1}{2}\left\{(1-3\alpha_2)\mu-\alpha_2 T-2(1+\alpha_2)\Pi
+3(1+\alpha_2)P_r\right\}+Y_{TF}-\frac{3l}{8H^3},
\end{eqnarray}
Now, we see the both cases when the fluid is non-dissipative or dissipative.
\subsubsection{The Dissipative and Non-dissipative Scenarios}
Here, we assume another condition that fluid is non-dissipative.
Under this assumption and for the choice of $f(R,T)$ model in case I, Eq.(\ref{106})
implies shear free condition for fluid configuration. With this implication, Eq.$(\ref{sh})$ leads to
\begin{eqnarray}\label{1251}
\frac{\ddot{H}}{H}-\frac{\ddot{G}}{G} &=& 0 \quad \Rightarrow\quad Y_{TF}=\frac{3l}{8H^3}.
\end{eqnarray}
Here, we can see that complexity factor $Y_{TF}=0$ is proportional to the fraction of $l$ and $H^3$.
We inserted this relation in Eq.$(\ref{Y})$ and found $\Pi=0$, which implies that $\mu'=0$.
Thus, $Y_{TF}=\frac{3l}{8H^3}$ represents
simplest modes of evolution in the case of cylinder.
Further, Eqs.$(\ref{118a})$ and $(\ref{1251})$
also strengthens our argument.
Now, we analyze the situation in the presence of dissipation. In this case, Eqs.$(\ref{1081})$ and $(\ref{sh})$ produce the following expression
\begin{eqnarray}\label{1261}
\dot{\sigma} &=& \frac{1}{\sqrt{3}}\left\{\left(\frac{\dot{H}}{H}\right)^2- \left(\frac{\dot{G}}{G}\right)^2+\frac{3l}{8H^3}-Y_{TF}\right\}.
\end{eqnarray}
The time derivative of Eq.$(\ref{1081})$ and above equation provide the following mathematical expression
\begin{eqnarray}\label{1271}
Y_{TF}\frac{H'}{H} &=& \frac{1}{2}Bq(1+\alpha_2)\left(\frac{\dot{q}}{q}+\frac{2\dot{B}}{B}+\frac{\dot{H}}{H}\right)+\frac{3l}{8H^3}.
\end{eqnarray}
If we assign zero value to complexity factor, then we get
\begin{eqnarray}\label{1281}
Bq(1+\alpha_2)\left(\frac{\dot{q}}{q}+\frac{2\dot{B}}{B}+\frac{\dot{H}}{H}\right)+\frac{3l}{4H^3} &=& 0.
\end{eqnarray}
This differential equation can further be solved by using some suitable numerical or analytical methods of integration. It actually holds to represent
simplest dissipative regime.
\subsection{ Case-II: $f(R, T)= \alpha_1 R+\alpha_2 T+\alpha_4 T^2$}
This form of our selected model comprises linear and quadratic
terms in the trace of the energy-momentum tensor( EMT). The squared terms of EMT was first introduced in \cite{s9}.
This particular form of $f(R, T)$ model has been used to explore non-exotic matter wormholes \cite{s10}.
This type of choice usually contrast with higher order gravity and the results provide description of the universe
that enters from a decelerated phase of expansion to an accelerated one and in agreement with observational data.
For this choice of $f(R,T)$ model, conditions obtained against simplest modes of evolution given in Eqs.$(\ref{105})$ and $(\ref{106i})$ take the form as
\begin{eqnarray}\label{129}
\frac{1+\alpha_2+2\alpha_4 T}{\alpha_1}qG &=&\sqrt{3}\frac{\sigma H'}{H},\\\label{108a}
\left(\sqrt{3} \frac{\sigma}{H}+\frac{1}{3}D_H\sigma\right)&=& -\frac{qG}{\alpha_1H'}(1+\alpha_2+2\alpha_4 T)
\end{eqnarray}
Here, if we assume $\sigma=0$, Eq.(\ref{108a}) ensures the vanishing of dissipative variable. Thus, homogeneous expansion and
shear-free condition again ceases the fluid to be dissipative.
All the situations that exits in Eqs.(\ref{1131b}-\ref{108a}) are also valid in this case. However, in the presence of
dissipation, shear scalar assume the form as
\begin{eqnarray}\label{113*c}
\sigma &=& -\frac{\sqrt{3}}{2H^3}\int^r_0 \frac{H^3}{\alpha_1} qG(1+\alpha_2+\alpha_4 T)dr.
\end{eqnarray}
In this case, dissipative variable again affects the homogeneous expansion and homologous condition and these are not found
compatible.
Now, we consider some dynamical situations for the system under consideration.
As our previous discussion shows that fluid is
geodesic under homologous condition in both dissipative and non-dissipative cases. Thus, if we apply
homologous condition on the Eq.(\ref{B4}), we obtain
\begin{eqnarray}\label{114*}
D_TU &=& -\frac{m}{H^2}-\frac{H}{2\alpha_1}\left\{(\alpha_2+2\alpha_4 T)\mu -(1+\alpha_2+2\alpha_4 T)P_r+\frac{(\alpha_2+2\alpha_4 T) T}{2}\right\}.
\end{eqnarray}
This equation can be re-written in the form of $Y_{TF}$ as
\begin{eqnarray}\nonumber
\frac{3 D_TU}{H} &=& -\frac{1}{2\alpha_1}\left\{(1-3\alpha_2-6\alpha_4 T)\mu-\alpha_2 T-\alpha_4 T^2-2(1+\alpha_2+\alpha_4 T)\Pi
+3(1+\alpha_2+\alpha_4 T)P_r\right\}\\\label{115*}&&+Y_{TF}-\frac{3l}{8H^3}.
\end{eqnarray}
Now, the manipulation of Eqs.$(\ref{16})$, $(\ref{18})$ and $(\ref{19})$ provides
\begin{eqnarray}\nonumber
-\frac{2\ddot{H}}{H}-\frac{\ddot{G}}{G} &=& \frac{1}{2}\left\{(1-3\alpha_2-6\alpha_4 T)\mu-\alpha_2 T-2\alpha_4 T^2-2(1+\alpha_2+2\alpha_4 T)\Pi
+3(1+\alpha_2 +2\alpha_4 T)P_r\right\}\\\label{115**}&&+Y_{TF}-\frac{3l}{8H^3},
\end{eqnarray}
while the definition of velocity `U' of collapsing star provides
\begin{eqnarray}\label{117*}
\frac{3 D_TU}{H}&=& \frac{3\ddot{H}}{H},
\end{eqnarray}
Insertion of above two equations into Eq.$(\ref{115**})$ leads to
\begin{eqnarray}\label{118*}
Y_{TF} &=& \frac{\ddot{H}}{H}-\frac{\ddot{G}}{G}-\frac{3l}{8H^3}.
\end{eqnarray}
If $Y_{TF}=0$, then Eq.$(\ref{118*})$ becomes
\begin{eqnarray}\label{118a*}
\frac{3l}{8H^3} &=& \frac{\ddot{H}}{H}-\frac{\ddot{G}}{G}.
\end{eqnarray}
Since we are working on the assumption that fluid is homologous, so we can write the Eq.$(\ref{115**})$ as
\begin{eqnarray}\nonumber
3\left(\dot{h}(t)+h(t)\frac{\dot{H}}{H}\right) &=& -\frac{1}{2}\left\{(1-3\alpha_2-6\alpha_4 T)\mu-\alpha_2 T-\alpha_4 T^2-2(1+\alpha_2+\alpha_4 T)\Pi
+3(1+\alpha_2+\alpha_4 T)P_r\right\}+Y_{TF}\\\label{115w}&&-\frac{3l}{8H^3},
\end{eqnarray}
\subsubsection{The Dissipative and Non-dissipative Scenarios}
In non-dissipative case, we again observe the same scenario as it is discussed in case I. Thus, we need to discuss only dissipative case.
Here, we analyze the situation in the presence of dissipation. In this case, Eqs. $(\ref{sh})$ and $(\ref{108a})$ produce the following expression
\begin{eqnarray}\label{126p}
\dot{\sigma} &=& \frac{1}{\sqrt{3}}\left\{\left(\frac{\dot{H}}{H}\right)^2- \left(\frac{\dot{G}}{G}\right)^2+\frac{3l}{8H^3}-Y_{TF}\right\}.
\end{eqnarray}
The time derivative of Eq.$(\ref{108a})$ and above equation provide the following mathematical expression
\begin{eqnarray}\label{127p}
Y_{TF}\frac{H'}{H} &=& \frac{1}{2\alpha_1}Bq(1+\alpha_2+2\alpha_4 T)\left(\frac{\dot{q}}{q}
+\frac{2\dot{B}}{B}+\frac{\dot{H}}{H}\right)+\frac{\alpha_4}{\alpha_2}\dot{T}+\frac{3l}{8H^3}.
\end{eqnarray}
If we assign zero value to complexity factor, then we get
\begin{eqnarray}\label{128p}
\frac{Bq}{\alpha_1}(1+\alpha_2+2\alpha_4 T)\left(\frac{\dot{q}}{q}+\frac{2\dot{B}}{B}
+\frac{\dot{H}}{H}\right)+2\frac{\alpha_4}{\alpha_2}\dot{T}+\frac{3l}{4H^3} &=& 0.
\end{eqnarray}
This differential equation can further be solved by using some suitable numerical or analytical methods of integration. It actually holds to represent simplest dissipative regime.
\subsection{ Case III: $f(R, T)= \alpha_1 R+\alpha_2 T(1+\alpha_3 TR^2)$}
This type of models offer the non-minimal coupling of curvature and matter components.
The similar type of choice has been recently used to measure the
impact of collision matter on the late-time
dynamics of $f(R, T)$ gravity \cite{s11}.
In this case, homologous and
homogeneous conditions will take the form as
\begin{eqnarray}\label{lab1}
\frac{1+\gamma_2}{\gamma_1}qG+\sqrt{3}\sigma \frac{H'}{H}&=& \frac{1}{\gamma_1}\left(\gamma_5-\frac{\dot{B}}{B}\gamma_4\right), \\\label{lab2}
\gamma_1\left(\sqrt{3}\frac{\sigma}{H}+\frac{1}{\sqrt{3}}D_H\sigma\right)&=&\frac{1}{FH'}
\left(\gamma_5-\frac{A'}{A}\gamma_3-\frac{\dot{B}}{B}\gamma_4\right),
\end{eqnarray}
however, Eq.(\ref{106}) takes the form as
\begin{eqnarray}\label{dq}
\sqrt{3}\frac{\sigma}{H} &=& \frac{1}{\gamma_1FH'}
\left(\gamma_5-\frac{A'}{A}\gamma_3-\frac{\dot{B}}{B}\gamma_4\right),
\end{eqnarray}
where
\begin{eqnarray}\nonumber
\gamma_1&=&\alpha_1+2\alpha_4T^2 R,\\\nonumber
\gamma_2&=& \alpha_2+2\alpha_4TR^2,\\\nonumber
\gamma_3&=&\alpha_4(T^2\dot{R}+2 T \dot{T}R),\\\nonumber
\gamma_4&=&\alpha_4(T^2R'+2 T T'R),\\\nonumber
\gamma_5&=&2\alpha_4(T^2\dot{R'}+2TT'\dot{R}+2T'\dot{T}R+2T\dot{T'}R+2T\dot{T}R').
\end{eqnarray}
Here, Eq.(\ref{dq}) clearly shows that homologous evolution does not imply shear free condition in non-dissipative case.
Nevertheless, validity of the Eqs.$(\ref{lab1})$ and $(\ref{lab2})$ at the same time again makes us to believe that fluid is homologous.
However, if we choose $\Theta'=0$ and consider the Eq.$(\ref{28})$, then we have
\begin{eqnarray}\label{lab3}
\sigma &=& \frac{\sqrt{3}}{2H^3}\int\frac{H^3}{F\gamma_1}\left(\gamma_5-\frac{F'}{F}\gamma_3
-\frac{\dot{B}}{B}\gamma_4\right)dr.
\end{eqnarray}
Under the same condition (i.e., $q=0$), if we assume $\sigma=0$, then Eq.$(\ref{28})$ does not imply the homogeneous expansion, rather it takes the form as
\begin{eqnarray}\label{lab4}
\Theta' &=& -\frac{\sqrt{3}}{F\gamma_1}\left(\gamma_5-\frac{F'}{F}\gamma_3-\frac{\dot{G}}{G}\gamma_4\right).
\end{eqnarray}
It is obvious from Eqs.$(\ref{lab3})$ and $(\ref{lab4})$ that homologous and homogeneous expansion conditions do not imply each other
in non-dissipative case. In the presence of dissipation, $\Theta'=0$ produce the following result
\begin{eqnarray}\label{lab5}
\sigma &=& \frac{\sqrt{3}}{2H^3}\int\frac{H^3}{F\gamma_1}\left\{\left(\gamma_5-\frac{F'}{F}\gamma_3
-\frac{\dot{B}}{B}\gamma_4\right)-FGq(1+\gamma_2)\right\}dr.
\end{eqnarray}
Again, we take into account dynamical considerations and apply homologous condition and obtain from \textcolor[rgb]{0.98,0.00,0.00}{\textbf{Eq.$(\ref{B4})$}}
\begin{eqnarray}\label{lab6}
D_T U &=& -\frac{m}{H^2}-\frac{H}{2\gamma}\left\{\gamma_2 \mu+(1+\gamma_2)P_r+\phi+\phi_{11}\right\},
\end{eqnarray}
which further takes the form as
\begin{eqnarray}\label{lab7}
\frac{3D_T U}{H} &=& -\frac{1}{2\gamma_1}\left\{(1+3\gamma_2)\mu-2\phi-2(1+\gamma_2)\Pi+\phi_{11}
+2\phi_{22}+3(1+\gamma_2)P_r \right\}+Y_{TF}-\frac{3l}{8H^3},
\end{eqnarray}
where $\phi$ and $\phi_{ii}$ occur due to $f(R, T)$ extra degrees of freedom involved in the evolution.
The above expression can further be re-written as
\begin{eqnarray}\label{lab8}
3\left(\dot{h}(t)+h(t)\frac{\dot{H}}{H}\right) &=& -\frac{1}{2\gamma_1}\left\{(1+3\gamma_2)\mu-2\phi-2(1+\gamma_2)\Pi+\phi_{11}
+2\phi_{22}+3(1+\gamma_2)P_r \right\}+Y_{TF}-\frac{3l}{8H^3}.
\end{eqnarray}
Further, from field equations we have extracted the following result
\begin{eqnarray}\label{lab9}
\frac{1}{2\gamma_1}\left\{(1+3\gamma_2)\mu-2\phi-2(1+\gamma_2)\Pi+\phi_{11}
+2\phi_{22}+3(1+\gamma_2)P_r \right\}+Y_{TF}-\frac{3l}{8H^3} &=& -\frac{2\ddot{H}}{H}-\frac{\ddot{G}}{G},
\end{eqnarray}
Using the definition of velocity `$U$' of collapsing star together with above equation, we again obtain the same result as given in Eq.(\ref{1181}) and vanishing of complexity factor $Y_{TF}$ yields the same expression as given in Eq.(\ref{118a}).
\subsubsection{The Dissipative and Non-dissipative Scenarios}
In this case, homologous condition does not imply shear free condition for non-dissipative scenario. Thus, under homologous condition,
we have $Y_{TF}= \frac{\ddot{H}}{H}-\frac{\ddot{G}}{G}-\frac{3l}{8H^3}$, which shows that complexity of the system is increased
in the presence of higher order curvature terms. Even in simplest modes of evolution, system have enough complexity index and fluid configuration does
not corresponds to the isotropic pressure and homogenous energy density.
Using Eqs.$(\ref{sh})$ and $(\ref{lab2})$, we find the relation for dissipative case as
\begin{eqnarray}\label{127pp}
Y_{TF}\frac{H'}{H} &=& \frac{1}{2\alpha_1}Gq\frac{1+\gamma_2}{\gamma_1}\left(\frac{\dot{q}}{q}
+\frac{2\dot{G}}{G}+\frac{\dot{H}}{H}\right)+\left(\frac{\gamma_2}{\gamma_1}\right)_{,0}qG-
\frac{1}{\gamma_1}\left(\gamma_5-\frac{F'}{F}\gamma_3-\frac{\dot{G}}{G}\gamma_4\right)
+\frac{3l}{8H^3}.
\end{eqnarray}
We can see dark source terms play important role and vanishing of complexity factor yields the following expression
\begin{eqnarray}\label{127ppp}
\frac{1}{2\alpha_1}Gq\frac{1+\gamma_2}{\gamma_1}\left(\frac{\dot{q}}{q}
+\frac{2\dot{G}}{G}+\frac{\dot{H}}{H}\right)+\left(\frac{\gamma_2}{\gamma_1}\right)_{,0}qG-
\frac{1}{\gamma_1}\left(\gamma_5-\frac{F'}{F}\gamma_3-\frac{\dot{G}}{G}\gamma_4\right)
+\frac{3l}{8H^3}&=& 0.
\end{eqnarray}
\section{Stability of The Vanishing Complexity Factor Condition}
In this section, our task is to find and analyze the conditions which are responsible
for an initial state of vanishing complexity factor under homologous condition. For this,
we need to develop the the evolution equation for structure scalar $Y_{TF}$ with the help
of Eqs.$(\ref{xx})$, $(\ref{yy})$, and (\ref{B2s}) which
takes the form as
\begin{eqnarray}\nonumber
&&\dot{Y}_{TF}+(1+\alpha_2)\dot{\Pi}+\frac{3\dot{H}}{H}Y_{TF}
+2(1+\alpha_2)\Pi\frac{\dot{H}}{H}+\frac{(1+\alpha_2)}{2}
(\mu+P_r)\sigma-\frac{1+\alpha_2}{2}(\mu+P_r)\frac{\dot{G}}{G}
-(1-\alpha_2^2)(\mu+P_\perp)\frac{\dot{H}}{H}\\\label{133*}&&+\alpha_2\frac{\dot{G}}{G}(\mu+P_\perp)-\frac{3}{2}\frac{H'}{H}(1+\alpha_2)q
\frac{1-\alpha_2^2}{2G^2}(Gq)'+\frac{q(1-\alpha^2)}{G^2}\left(\frac{2H'}{H}-\frac{G'}{G}\right)+\alpha_2 q'+\Lambda= 0,
\end{eqnarray}
where $\Lambda$ contains dark source entities.
Now, we analyze this equation for dissipative and non-dissipative cases turn by turn. First we consider the non-dissipative case
at some initial moment where $Y_{TF}=q=\sigma=\Pi=0$, then previous equation takes the form as
\begin{eqnarray}\label{134*}
\dot{Y}_{TF}+(1+\alpha_2)\dot{\Pi}-\frac{1-\alpha_2^2}{2}(\mu+P_\perp)\frac{\dot{H}}{H}+ \Lambda&=& 0.
\end{eqnarray}
In the most general case, when the system is dissipative, we have at the initial moment
\begin{eqnarray}\nonumber
&&\dot{Y}_{TF}+(1+\alpha_2)\dot{\Pi}+2(1+\alpha_2)\Pi\frac{\dot{H}}{H}+\frac{1+\alpha_2}{2}
(\mu+P_r)\sigma-\frac{1+\alpha_2}{2}(\mu+P_r)\frac{\dot{G}}{G}
-(1-\alpha_2^2)(\mu+P_\perp)\frac{\dot{H}}{H}\\\label{135*}&&+\alpha_2\frac{\dot{G}}{G}(\mu+P_\perp)-\frac{3}{2}\frac{H'}{H}(1+\alpha_2)q
\frac{1-\alpha_2^2}{2G^2}(Gq)'+\frac{q(1-\alpha^2)}{G^2}\left(\frac{2H'}{H}-\frac{G'}{G}\right)+\alpha_2 q'+ \Lambda= 0.
\end{eqnarray}
It is obvious from above equation that pressure anisotropy, energy density inhomogeneity, dissipative variable and dark source
entities all are crucial for complexity of a cylindrical system. Last term $\Lambda$ on the right hand side of the above equation
incorporates the effects of dark source terms which can be measured for any particular model.
\section{Conclusion}
The study of complexity on astrophysical scales is an interesting concept which helps the researchers to explore and identify
the factors that are responsible for the emergence of complexity in a system. In the pursuance of a complete and comprehensive definition of
complexity, Herrera \cite{12b} proposed a new definition for an anisotropic fluid sphere. Following Herrera's work, we have
explored the behavior of complexity factor for a cylindrically symmetric
dynamical object in $f(R,T)$ gravity. For this, we have explored field
equations for cylindrical symmetry and developed mass function using
C-energy expression. We have constructed structure scalars for cylindrical geometry in $f(R,T)$ gravity and identified $Y_{TF}$ as complexity factor
among these scalars which incorporates the effects of
inhomogeneous energy density, anisotropic pressure and dissipative variable together with dark source terms.
The definition of complexity for a dynamical system involves not only the complexity of
the structure of the system, but also considers degree of complexity of the pattern
of evolution of the system. Thus, we have studied the complexity of patterns of evolution by assuming
two simplest modes of evolution,i.e., homologous evolution and homogeneous expansion.
We have worked out the conditions representing homogeneous expansion and homologous evolution, where we have
found that shear free condition implies zero dissipation if both of these conditions hold at the same time.
In order to
present detailed study in $f(R,T)$ gravity, we have selected a generic non-minimally coupled $f(R,T)$ model and discussed our findings for three
different cases of model under consideration.
\begin{itemize}
\item In the first case, our $f(R,T)$ model is linear combination of first order terms of curvature `$R$' and trace of energy momentum tensor `$T$'.
Here, we have found that homogeneous expansion and shear free condition cease the fluid to be dissipative. We have also found that shear free
condition and homologous condition imply each other for this particular choice. In non-dissipative scenario, simplest modes of evolution are found to be
compatible with each other. However, in the presence of dissipative variable, they are not found compatible.
We have discussed dissipative and non-dissipative scenarios and found complexity factor is proportional to the fraction of
length of cylinder and curvature term of cubic order which represents minimum value of complexity factor when total amount of energy in
a cylindrical system is represented through gravitational C-energy.
\item In the second case, our model is of the form $f(R,T)=f_1(R)+f_2(T)$ , where $f_1(R)$ is linear function, while $f_2(T)$ is
quadratic one. All the results that are obtained in the first case are same for the second choice of $f(R,T)$ model. However in dissipative case,
complexity of the system is increased due to the presence of quadratic $T$ terms.
\item In the third case, our assumptions yields the form which consists on two parts. The first part coincides with
case I, however second part contains product of quadratic terms of `$R$' and `$T$'. In this case, shear free
condition and homologous condition are not found to be compatible for both dissipative and non-dissipative cases. It is observed that complexity
of a system is increased in the presence of dark source entities, even in the simplest modes of evolution it does not attain the value that represents
minimum complexity of the system.
\end{itemize}
In the end, we analyzed the stability of vanishing complexity condition and observed different
outcomes in both dissipative and non-dissipative scenarios. In the absence of dissipation, significance of pressure isotropy and dark source entities is obvious from Eq.(\ref{134*}), however
in general scenario, Eq.(\ref{135*}) shows that multiple factors are involved.
We have compared our results with previous literature \cite{12a, s3} and found that
complexity factor for cylindrical system contains some extra terms because of
the geometrical difference of the self gravitating systems, which ceases it to attain zero value in
simplest modes of evolution. The complexity system for static self-gravitating systems has been explored
extensively, but it needs more study for dynamical systems. We intend to extend our work
for dynamical self-gravitating systems in the presence of charge.
\renewcommand{\theequation}{A.\arabic{equation}}
\setcounter{equation}{0}
\section*{Appendix A}
The divergence of the energy-momentum tensor is nonzero in $f(R, T)$ gravity and is found as
\begin{eqnarray}\setcounter{equation}{1}\label{B1s}
\nabla{^{\alpha}}T_{\alpha\beta}=\frac{f_T}{1-f_T}\left[(\Theta_{\alpha\beta}-T_{\alpha\beta})\nabla{^{\alpha}}ln
f_T-\frac{1}{2}g_{\alpha\beta}\nabla{^{\alpha}}T+\nabla^{^{\alpha}}\Theta_{\alpha\beta}\right].
\end{eqnarray}
Its divergence yields the following two equations:
\begin{eqnarray}\nonumber
&&\dot{\mu}\left(\frac{1-f_T-f_R f_T}{f_R(1-f_T)}\right)-\mu\frac{\dot{f_R}}{f_R^2}+
\frac{\dot{G}}{G}\frac{1}{f_R}(1+f_T)(\mu+P_r)+\frac{\dot{2H}}{H}\frac{1}{f_R}(1+f_T)(\mu+P_\perp)
+\left(\frac{\dot{T}}{2}\right)\frac{f_T}{1-f_T
}\\\nonumber&&-q'\frac{F}{G}\frac{(1+f_T)}{f_R}-\frac{q}{G^2}\left\{\frac{FG(1+f_T)}{f_R}\right\}_{,1}
-\frac{F}{G}\left(\frac{F'}{F}-\frac{G'}{G}+\frac{2H'}{H}\right)(1+f_T)q
+\left\{\frac{1}{f_R
}\left(\Psi +\psi_{00}\right)\right\}_{,0}\\\nonumber&&-\frac{1}{G^2}\left\{\frac{\psi_{01}}{f_R}\right\}_{,1}+{\frac{\psi_{01}}{G^{2}f_R}
}\left(\frac{F'}{F}-\frac{G'}{G}+\frac{2H'}{H}\right)+\frac{\psi_{00}}{f_R}\left(\frac{\dot{G}}{G
}+\frac{\dot{2H}}{H}\right)+\frac{\dot{G}}{G}\frac{\psi_{11}}{f_R}+\frac{\dot{2H}}{H}\frac{\psi_{22}}{f_R}
\\\label{B2s}&&-\left(\frac{2\dot{G}}{G}(\mu+P_r)+\frac{F}{G}2q'+4q\frac{F'}{G}\right)\frac{f_T}{1-f_T}
-\frac{F}{G}q\frac{f'_T}{1-f_T}=0,\\\nonumber
&&P'_r\left(\frac{1-f^{2}_T+2f_R
f_T}{f_R(1-f_T)}\right)+\frac{P_r}{f_R}\left(f'_T-\frac{f'_R(1+f_T)}{f_R}\right)+\mu'\frac{f_T}{f_R}+
\frac{\mu}{f_R}\left(f'_T-\frac{f_T f'_R}{f_R}\right)+\frac{F'}{F}\frac{(1+f_T)}{f_R}(\mu+P_r)\\\nonumber
&&+\frac{2H'}{H}\frac{1}{f_R}(1+f_T)(P_r-P_\perp)+\frac{f'_T}{1-f_T}(\mu+P_r)
-\frac{f_T}{1-f_T}\mu'+\frac{G}{F}\frac{(1+f_T)}{f_R}\dot{q}+\frac{q}{F^2}\left\{\frac{FG(1+f_T)}{f_R}\right\}_{,0}
\\\nonumber&&+
\left\{\frac{1}{f_R
}\left(\Psi+\psi_{11}\right)\right\}_{,1}-\frac{1}{F^2}\left\{\frac{\psi_{01}}{f_R}\right\}_{,0}+{\frac{\psi_{01}}{G^{2}f_R}}
\left(\frac{\dot{F}}{F}-\frac{\dot{G}}{G}+\frac{\dot{2H}}{H}\right)
+q\frac{G}{F}(1+f_T)\left(\frac{\dot{F}}{F}-\frac{\dot{G}}{G}+\frac{\dot{2H}}{H}\right)\\\nonumber&&+\frac{f_T}{1-f_T}\left(\frac{T'}{2}+\mu'\right)
+\frac{F'}{F}\frac{1}{f_R}\left(\psi_{00}+\pi_{11}\right)+\frac{2H'}{H}\frac{1}{f_R}\left(\pi_{11}-\psi_{22}\right)+\frac{2G}{F}q\left(1+\frac{F'}{F}\right)\frac{f_T}{1-f_T}
\\\label{B3s}&&+\frac{G}{F}\left(q\frac{\dot{f_T}}{f_R}+\frac{\dot{G}}{F}(\mu_+2P_r)\right)\frac{f_T}{1-f_T}
+\frac{f_T}{1-f_T}\frac{2}{A}(Bq)_{,0}
=0.
\end{eqnarray}
The acceleration $D_TU$ of a collapsing star can be obtained by manipulating the Eqs. $(\ref{13})$, $(\ref{17})$, $(\ref{24})$ and $(\ref{27})$:
\begin{eqnarray}\label{B4}
D_TU &=& \frac{m}{H^2}-\frac{R}{2f_R}\left(1+f_T)P_r+f_T\mu+\pi+\pi_{11}\right)+Ea.
\end{eqnarray}
\vspace{.5cm}
\section*{Acknowledgments}
``Authors thank the Higher Education Commission, Islamabad, Pakistan for its
financial support under the NRPU project with grant number
$\text{5329/Federal/NRPU/R\&D/HEC/2016}$''.
|
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{"url":"https:\/\/warosu.org\/sci\/thread\/9570309","text":"[ 3 \/ biz \/ cgl \/ ck \/ diy \/ fa \/ g \/ ic \/ jp \/ lit \/ sci \/ tg \/ vr ] [ index \/ top \/ reports \/ report a bug ] [ 4plebs \/ archived.moe \/ rbt ]\n\nIf you can see this message, the SSL certificate expiration has been fixed.\nBecome a Patron!\n\n# \/sci\/ - Science & Math\n\n[ Toggle deleted replies ]\nFile: 7 KB, 500x250, Infinite-Series.jpg [View same] [iqdb] [saucenao] [google] [report]\n\nIs infinity pseudo science?\nIsn't it an assumption based on uncertainty?\n\n >> Anonymous Thu Mar 8 06:41:50 2018 No.9570314 >>9570309math =\/= science\n >> Anonymous Thu Mar 8 06:42:27 2018 No.9570315 >>9570309No, uncertainty is an assumption based on things not being infinite.\n >> Anonymous Thu Mar 8 06:57:00 2018 No.9570325 >>9570315You have uncertainty regardles of finiteness being present. For instance, the limiting case of a distribution curves resolution gives no additional insight on how the sampling is perforned.\n >> Anonymous Thu Mar 8 07:22:57 2018 No.9570343 Infinity is a difficult concept and most cases where it sounds \"wrong\" thats because you misapply mathematics^^ e.g. your example : you use \"=\" in a completely other sense than you'd do in \"x = 1\", in your case it is ( at best, though i dont like this notation and one could argue it is wrong) used to show the sum of an infinite series. Which is way different from any finite sum you could come up with :P If you come from a scientific background , you could argue \"there's a minimal and a maximal time\/space\/energy... with any physical meaning ( e.g. 10^(-43) s for time)\", so you could argue \"well a physicist can't talk about anything outside those boundaries, since physics is bounded by our universe\", but nothing more than that. What happens most often is a physical value not being clearly enough defined so it \"seems\" like infinity. E.g. what is the \"temperature\" of laser light? For some (naive) definitions of temperature this is a hard question.\n >> Anonymous Thu Mar 8 07:34:21 2018 No.9570354 >>9570343what is the temperature of laser light\n >> Anonymous Thu Mar 8 07:38:53 2018 No.9570358 File: 20 KB, 306x306, 1492737771271.jpg [View same] [iqdb] [saucenao] [google] [report] >>9570309>Is infinity pseudo science?Math isn't science\n >> Anonymous Thu Mar 8 07:41:45 2018 No.9570362 >>9570358>formal science isn't science\n >> Anonymous Thu Mar 8 07:42:30 2018 No.9570365 I wish mods deleted these stupid threads desu.\n >> Anonymous Thu Mar 8 07:47:10 2018 No.9570376 >>9570365>fell for the 300k starting memestay mad\n >> Anonymous Thu Mar 8 08:01:11 2018 No.9570389 >>9570354depends entirely on your definition of Temperature. You could make a case for nearly anything here^^ If you take electron energy levels or boltzmann inversion here you could talk about a NEGATIVE temperature ( on the kelvin scale) which is actually hotter than say \"1K\" and depending on your (naive) definition of temperature may as well be hotter than the hottest possible temperature, i.e. \"infinite\" for some definitions of infinity\n >> Anonymous Thu Mar 8 09:23:20 2018 No.9570487 >>9570309Eh. Infinity is an axiom of ZFC. So in a way it's an assumption. But it's an assumption that gives rise to a staggeringly enormous amount of interesting mathematics. It also doesn't lead to any logical contradictions. So there's no problem in taking it as an axiom.You're welcome to do math without the axiom of infinity but it just won't be as interesting. Nor will it be as useful.Also if results like 0.999...=1 and 1\/2+1\/4+1\/8+1\/16+...=2 are not intuitive to you then you may actually be a retard.\n >> Anonymous Thu Mar 8 09:25:20 2018 No.9570488 >>9570487Sorry, I meant 1\/2+1\/4+1\/8+...=1 (not 2).\n >> Anonymous Thu Mar 8 10:10:09 2018 No.9570563 >>9570309Why the apeirophobia?\n >> Anonymous Thu Mar 8 10:13:12 2018 No.9570576 >>9570389people like you are the reason I took a 7 month hiatus from this board.\n >> Anonymous Thu Mar 8 10:23:08 2018 No.9570598 >>9570576fuck off back to >>>\/pol\/\n >> Anonymous Thu Mar 8 13:30:46 2018 No.9570896 >>9570309Infinity has different meanings, all of which are very well defined in mathematics. Real numbers are (often) constructed as the equivalence classes of rational cauchy sequences whose difference converges to 0.\n >> Anonymous Thu Mar 8 13:34:05 2018 No.9570902 >>9570362Science is literally empiricism. Math is not. Math is used in science to formulate hypotheses, but math itself is not an empirical investigation.\n >> Anonymous Thu Mar 8 13:42:00 2018 No.9570924 top 10 worst threads 2018\n >> Anonymous Thu Mar 8 14:43:54 2018 No.9571051 >>9570924>top 10 worst threads 2018Meanwhile the catalog is filled with IQ, racebait and \/pol\/ threads\n >> Anonymous Thu Mar 8 17:21:34 2018 No.9571368 >>9571051Does one even exist? Like count apples. One apple, two apple. No you can't because an apple is an arbitrary measurement. Each apple has different and made of particles. There isn't one apple.\n >> Anonymous Thu Mar 8 17:32:44 2018 No.9571392 >>9571368On a human scale, quantity makes sense. What would you suggest as an alternative?\n >> Anonymous Thu Mar 8 17:37:37 2018 No.9571400 >>9571368>apples don't existthe lengths to which you people will go\n >> Anonymous Thu Mar 8 18:10:34 2018 No.9571474 Infinity is pseudo intelligence.Its not math, its not science, it's not intelligent.To do infinite summation is to say you never stop summing, which means you can never acquire a result, which means infinite summation is not useful. If you can acquire a result from it, then there was an end to the summation and thus not actually infinite, unless you redefine infinity from \"never ending\" to \"a greatest, real number value\", meaning \"infinite\" is just a counter-intuitively retarded name for an obviousy finite number. If you then say \"no greatest real number exists\", then infinity doesn't then exist, which also makes it useless. Infinity is a paradox, and as a paradox it is not useful.$0.\\bar{9 \\neq 1$ because there can only exist a never ending amount of 9's never approaching anything but more 9's. The result of an infinite sum isn't defineable because you never reach an end result. The amount of 9's in 0.999... cannot be infinite unless \"infinite\" is used in lieu of \"an arbatrarily large real finite number\". The result of an infinite sum cannot equate unless \"infinite\" is used in lieu of \"an arbitrary large real finite number\". Furthermore, replacing \"infinite\" with \"an arbatrarily large real finite number\" then provides for $\\frac{1}{\\infty}>0$, meaning for \"an arbitrary amount of 9's in 0.999...\" would necessitate the addition of $\\frac{1}{\\infty}$ to properly sum 1, and \"an arbitrary amount of summation\" in an infinite sum to finitely equate a real value, and even though none of these steps actually logically worth regardless of definition, it still doesn't provide for any sense to be made about which number infinity is supposed to be.Only brainlets who have been programmed by math instructors via obedience believe infinity is rigourosly defined or useful. It isn't.\n >> Anonymous Thu Mar 8 18:32:34 2018 No.9571516 >>9571474>To do infinite summation is to say you never stop summing, which means you can never acquire a resultYou're retarded. No one is sitting down to add each step in a summation algorithm, instead an analytical trick is used to transform that algorithm into something else that can be evaluated in finite steps. >If you can acquire a result from it, then there was an end to the summation and thus not actually infiniteA finite amount can be divided into infinite parts, and therefore infinite parts can form a finite summation.You have less understanding of the concept you are trying to criticize than an elementary school student, and treating you lack of basic understanding as a paradox. Dunning Kruger effect in action.\n >> Anonymous Thu Mar 8 19:15:36 2018 No.9571583 >>9571516Saying there are an infinite, non-finite amount of parts between finite 1 and finite 2 means you can never reach 2 from 1 by incrementing in those parts. Really you just projected by invoking dunning krueger when you had no idea what you were talking about but decided to post anyway. Moreover, you're selfish and evil for wasting my and anyone elses time in reading your completely uninformative post. You ultimately didn't even comprehend my post that you replied to and didn't understand the strict definitions offered.\n >> Anonymous Thu Mar 8 19:30:04 2018 No.9571613 \u00a0 $\\sum_{n=1}^{\\infty} \\frac{1}{2^n}$ will never equal 1 or approach 1. It wont even technically equal 0.999...The moment you have a sufficient amount of sequential 9's in the result to be assumed as infinite, you would also have an infinite amount of unordered random numbers after it. Cutting this entire value in have spacially, the left half from middle to the decimal point would be all 9's, but the right half would be random gobbledygook numbers in no defineable pattern, and that is when all possible dogits of the significand have been accounted for and filled, from the 1st decimal place at 0.9, to the infinite'th decimal place.1\/2^n is obviusly less than 2\/3^n or 3\/4^n, and these are all leas than 9\/10^n which would be a true uninterrupted string of 9's.\n >> Anonymous Thu Mar 8 19:40:09 2018 No.9571638 $\\sum_{n=1}^{\\infty} \\frac{1}{2^n}$ will never equal 1 or approach 1. It wont even technically equal 0.999...The moment you have a sufficient amount of sequential 9's in the result to be assumed as infinite, you would also have an infinite amount of unordered random numbers after it. Cutting this entire value in half spacially, the left half from middle to the decimal point would be all 9's, but the right half would be random gobbledygook numbers in no defineable pattern, and that is already when more than all possible digits of the significand have been accounted for and filled, from the 1st decimal place at 0.9, to the infinite'th decimal place. For 1\/2^n to properly approach 0.999... would require the total amount of digits be infinite, and only half of all those digits would be 9's so to cut off the latter half and assume 1\/2^n = 0.999... is to actually say $\\sum_{n=1}^{2 \u00d7 \\infty} \\frac{1}{2^n}$ where after doubling on the infinite amount of summable iterations, there would finally be an infinite amount of 9's, even though it's still further followed by an infinite amount of arbitrary unordered digits of random numbers.1\/2^n is obviusly less than 2\/3^n or 3\/4^n, and these are all leas than 9\/10^n which would be a true uninterrupted string of 9's.Infinity isnt well defined or useful. Only brainlets pretend it is.\n >> Anonymous Thu Mar 8 19:47:04 2018 No.9571646 >>9570343>your example : you use \"=\" in a completely other sense than you'd do in \"x = 1\",Wrong.\n >> Anonymous Thu Mar 8 20:38:52 2018 No.9571723 >>9571613>$\\sum_{n=1}^\\infty \\frac{1}{2^n}$ will never approach 1.Let's tackle this part of your claim. Note that [eqn]\\sum_{n=1}^k \\frac{1}{2^n} = \\frac{2^k - 1}{2^k}.[\/eqn] We can easily see that this equality holds true for $k=1$. Let $k\\in\\mathbb N_0$ be arbitrary and assume the equality holds. We show that it also holds for $k + 1$.[eqn]\\sum_{n=1}^{k+1} \\frac{1}{2^n} = \\sum_{n=1}^k \\frac{1}{2^n} + \\frac{1}{2^{k+1}} = \\frac{2}{2}\\cdot\\frac{2^k - 1}{2^k} + \\frac{1}{2^{k+1}} = \\frac{2^{k+1}-1}{2^{k+1}}.[\/eqn]This proves that the equality holds for any arbitrary natural number greater than or equal to 1 and all that is left to do is show that $\\frac{2^k - 1}{2^k}$ approaches 1.Let $\\varepsilon > 0$ be arbitrary. By the Archimedian property there exists some $k_0 \\in \\mathbb N_0$ such that $k_0 > \\frac{1}{\\varepsilon}$. Let $k \\in \\mathbb N_0$ be arbitrary and assume that $k \\geqslant k_0$. It then follows that $2^k > k \\geqslant k_0 > \\frac{1}{\\varepsilon}$. From here it follows that[eqn]\\varepsilon > \\frac{1}{2^k} = \\left|\\frac{-1}{2^k}\\right|= \\left|\\frac{2^k -1 - 2^k}{2^k}\\right|= \\left|\\frac{2^k -1}{2^k} - \\frac{2^k}{2^k}\\right|= \\left|\\frac{2^k -1}{2^k} - 1\\right|[\/eqn]This proves that for any arbitrary distance $\\varepsilon$ there is some point after which all the partial sums of the series lie closer than that and as such this proves that the series does approach 1.\n >> Anonymous Thu Mar 8 20:43:30 2018 No.9571729 >>9571638this >>9571723 was meant to give you a (you) but you deleted the initial post so here is your (you).\n >> Anonymous Thu Mar 8 20:54:00 2018 No.9571749 >>9571723While this is nice, it's rather misguided. In fact, the whole idea that an infinite sum approaches'' a number is wrong. An infinite sum is a complete object. It IS a number (assuming convergence). A convergent infinite sum is simply a different way of representing a number. So $\\sum_{n=1}^{\\infty} \\frac{1}{2^n}$ is just a weird way of writing the number 1. Just like 2\/2 is another way of writing the number 1.It's kinda like having a nickname for somebody. You might refer to a friend as `Robert' while I call him 'Bobby'. We're referring to the same person, but in different ways. Same thing with the number 1 and that infinite sum. Different ways of referring to the exact same mathematical object.\n >> Anonymous Thu Mar 8 21:09:40 2018 No.9571770 >>9571723If an infinitely large arbitrary number can exist, an infinitely small arnitrary number must then also exist. If you use these definitions, the opposite of infinity is a value that is the closest to zero but greater than zero. If 0.999... = 1, then 1\/infinity = 0. If 1\/infinity = 0, a\/b = cThen 1\/0 = infinity, a\/c = b.If 1\/infinity = 0, there does not exist an infinitely small real number, and therefore does not exist an infinitely large real number, thus infinity does not exist and neither does an infinite amount, thus this conversation is no longer about \"0.999... & 1\", because there is no rational, consistent, reliable way of defining what \"0.999...\" is without invoking infinity which itself is not rationally, consistently, reliably defined. If the concept of infinity cant work, you cant then take a next step by using infinity in any specific manner. Doing so is the equivalent of knowing infinity is unachieably countable but treating infinity+1 as real, as it's next step countable step. Non-finitests don't do this, so non-finitests are stuck forced to abritrarily force finitest rules on infinity without changing the definition of infinity. Infinity+1 doesnt rationalize as the next step after infinity, so using infinity doesn't rationalize as the next step after not defining infinity. Infinity is a fucking paradox for brainlets. Just stop. Any and every attempt to justify using it is easily proven fallible, so anyone who tries to do it anyway is an idiot.There is no such thing as getting a result from unending work. There is also no such thing as a largest real number. Infinity = worthless.\n >> Anonymous Thu Mar 8 21:11:22 2018 No.9571773 >>9571749Oh I don't disagree, but he was just going to throw retarded retorts that \"convergence isn't equality\" so I just opted to show that the partial sums definitely do approach 1, which he also denied. And as you said, the entire series is indeed equal to 1.\n >> Anonymous Thu Mar 8 21:12:43 2018 No.9571780 >>9571770>If an infinitely large arbitrary number can exist, an infinitely small arnitrary number must then also exist.It can't exist and nowhere is it claimed to exist in the post you are replying to.\n >> Anonymous Thu Mar 8 21:16:33 2018 No.9571787 >>9571780>It can't exist and nowhere is it claimed to exist in the post you are replying to.Yes, infinity can't exist, thanks for agreeing.I bet you didn't even realize thats what you typed. I think its time to take a step back and realize this thing called infinity is fully retarded, and trying to justify it just leads to people becoming retarded.\n >> Anonymous Thu Mar 8 21:21:17 2018 No.9571791 >>9571787There are no real numbers that are \"infinitely large\", the same is true for natural numbers. That claim is in no way equivalent to \"Infinity can't exist\". There is no upper bound to the natural numbers or to the real numbers, which is what we mean when we say there are \"infinitely\" many. The only thing that is inconsistent here is your usage of the concept infinity.\n >> Anonymous Thu Mar 8 21:27:19 2018 No.9571799 >>9571791You can have a number that's arbitrarily large, though, right? Like, something that for the sake of argument is larger than any given value, but can still be treated as a real number. I think that might come in handy. Although I know that doesn't preclude the existence of infinity. Seriously fuck these threads.\n >> Anonymous Thu Mar 8 21:30:38 2018 No.9571804 >>9571799>larger than any given valueOne number that is larger than any given value? No.For any given value there exists a number larger than it? Yes.\n >> Anonymous Thu Mar 8 21:34:48 2018 No.9571811 >>9571804After any NUMBER, there can only exist a larger NUMBER, provided there is no LARGEST NUMBER of which a NUMBER could not be larger.Do you agree this is the claim?\n >> Anonymous Thu Mar 8 21:38:57 2018 No.9571816 >>9571392I'm trying to prove that one doesn't exist only 0.99999999\n >> Anonymous Thu Mar 8 21:44:23 2018 No.9571822 >>9571811I'll say if I agree when you write your claim in quantifiers about a well specified set of numbers so you don't pull the rug from under my feet and go \"lellers I bet you didn't even realize that I use non-standard interpretations of these terms and now you agreed that infinity can't exist\".\n >> Anonymous Thu Mar 8 22:08:03 2018 No.9571842 >>9571822The only interpretation required is that of knowing english and being able to make logical connections.>After any NUMBER, there can only exist a larger NUMBER, provided there is no LARGEST NUMBER of which a NUMBER could not be larger.You agree this is the claim of infinity, and i'm sure you'll agree it is not an uncommon claim. >After any number, there can always exist a larger number. We can reliably know this to be true, if not true enough. If we said 99,999 was our example number, we can know in the very least that 100,000 is bigger and is too a number.The only way to prove that a number exists where it cannot be incremented is to define a largest possible number. If everyone agreed 1 quintillion was the largest number, then once anyone reached 1 quintillion we would know to stop trying to add any more, so then we have a number that cannot have a larger number value relative to it. No one will recognize 1 quintillion + 1 has any real value, since 1 quintillion is already arbitrarily large enough to be useful for any real arithmetic. However, because we have a largest number the first part of the claim is no longer valid>a larger real number exists after any real numberIt's mostly accurate, but not perfectly accurate, so the claim must be altered.But 1 quintillion isn't really the largest real number here in reality, so in our world the first part of the claim remains true and we agree that there is no largest real number, providing for the first part of the claim to always remain true.>... provided there is no LARGEST NUMBER of which a NUMBER could not be larger.The second part claims there can never be a largest number as such a number could not have a larger number after it, nullifying the first part of the claim.Therefore infinity cannot be larger than any number as it would become the largest number, and if it truly weren't already treated as the largest number, infinity+1 wouldn't be seen as useless like like quintillion+1.\n >> Anonymous Thu Mar 8 22:12:55 2018 No.9571847 >>9571842>You agree this is the claim of infinityI don't agree to anything until you write the claim clearly in quantifiers about a specific set of numbers. Your game is the game of language tricks and intuition. Mathematics has replaced that by axioms and rigorous definitions because intuition doesn't work for these concepts.\n >> Anonymous Thu Mar 8 22:12:56 2018 No.9571848 >>9571842As you can see, any prestablished definition FOR infinity cannot help but paradoxically state infinity does not exist, and i could prove this to you to a finite degree if you use any of the other various definitions prescribed to infinity, of which there are few and finitely many.\n >> Anonymous Thu Mar 8 22:26:41 2018 No.9571865 >sum of infinite seriesI am too brainlet to grasp this\n >> Anonymous Thu Mar 8 22:29:39 2018 No.9571871 >>9571847n=0;while (x > 0){ n += 1; Print(n);}This pseudocode is the equivalent of an infinitesum of +1 execution where every iteration partial sum is printed and readably knowable. Every print will always be a real number, and every print will always be larger than the previous print.At no point will a print ever readout as $\\infty$, therefore infinity doesn't exist as a number value.If it doesn't exist as a number value, it cannot exist as a quantity. Infinity is then forced to exist by the sole definition of \"never ending\", even though the concept of never ending cannot be proven, but we could assume a base intent direction of it as \"go and don't ever stop\". With this being the only valid definition of infinity, to describe the pattern of 9's existing in 0.999... as \"never ending, starting without stopping\", then 0.999... can only ever equal itself and only itself at 0.999..., because anything more such as 1 would have brought some kind of end to what can only be endlessly unendable, where infinite sums cannot exist as objects but only as the values up to a real finite n'th limit because an unending, unendable limit of summation would never allow a result.also dont be a faggot who cant read and thinks hes being tricked with words. You're only being tricked with numbers by choosing to believe infinity is rigorous or well defined. It isn't. Take it for granted that it isn't. At best it has one singular definition, that of unending work, yet few if any implements of infinity abide by this definition and instead treat it as an arbatrarirly large finite number where it's not infinite because it has no end, but instead because not enough information was provided to actually know what value it was. S h i t m a t h.\n >> Anonymous Thu Mar 8 22:53:28 2018 No.9571905 >>9571847>>9571871Put it this way, can you describe the exact difference between \"an arbatrarily large real number\" and \"infinity\", if infinity has a quantity value?If i say to you \"count from 0 to an arbatrarily large real number\" and you agree to do so, do you think you'll respond with \"which number?\" If you did, i wouldn't tell you which number, i'd just repeat \"an arbatraririly large real number\".This is the real trick that higher maths plays on you. In the realest sense of things, there is no different between infinity and the vague statement \"an arbitrarily large real number\", which means comparatively the quanitity of infinity IS an arbitrarily large real number. You can't count to an arbitrarily large real number from 0 any more than you could count to infinity from 0. If infinity as a number and has a quanitity value, it is only as good as an arbatrarily large and undefined real number, and a vague undefined quanitity value is only as good as \"???\" Which means utilization of infinity would only be as realistically good as $\\sum_{n=1}^{\\rightarrow ???} \\frac{1}{2^n}$ which is buttfucking retarded but literally exactly how it is used in calculus\n >> Anonymous Thu Mar 8 23:11:20 2018 No.9571930 >>9571770>IF\n >> Anonymous Thu Mar 8 23:14:49 2018 No.9571938 File: 2.30 MB, 1280x720, 1468646224751.webm [View same] [iqdb] [saucenao] [google] [report] >>9571930X = 1;IF (x=1){>you're retarded}Yes, IF. What of it.\n >> Anonymous Thu Mar 8 23:17:27 2018 No.9571944 \u00a0 >>9571938>infinety large>real numberpick one\n >> Anonymous Thu Mar 8 23:19:14 2018 No.9571949 >>9571938>>9571938>infinitely large>real numberpick one\n >> Anonymous Thu Mar 8 23:24:36 2018 No.9571958 >>9571949I'm not saying infinity is a number, i'm saying it's \"unending\". Contrarily, infinity is used as a number in maths. You're not wrong, infinity isn't a number. I'm not wrong, infinity isn't a number. Maths is wrong though, because maths assumes infinity is a number, therefore mathematicians are wrong.\n >> Anonymous Thu Mar 8 23:25:46 2018 No.9571963 >>9571958Infinity is a number, faggot. Just because it's not in $\\mathbb{R}$ doesn't mean it's not a number.\n >> Anonymous Thu Mar 8 23:28:16 2018 No.9571968 >>9571958>>9571963$\\overline{\\mathbb{R}}, \\widehat{\\mathbb{R}}, \\aleph_0, \\omega, ...$, take your pick.\n >> Anonymous Thu Mar 8 23:28:26 2018 No.9571969 >>9571963R is all real numbers. You're right, infinity isn't in R. Real numbers are all that matter. You need real numbers to do math. Math without real numbers isn't math.\n >> Anonymous Thu Mar 8 23:30:48 2018 No.9571975 File: 2.54 MB, 640x640, 1507536349285.webm [View same] [iqdb] [saucenao] [google] [report] >need real numbers to do math>use infinity even though it isnt a real number>pretend you're still doing math\n >> Anonymous Thu Mar 8 23:31:01 2018 No.9571977 >>9571958>If an infinitely large arbitrary number can existIt's your own fucking words.retard liar\n >> Anonymous Thu Mar 8 23:31:44 2018 No.9571978 >>9571969>Real numbers are all that matter. You need real numbers to do math. Math without real numbers isn't math.t. never studied complex analysist. never studied DSPt. never studied quantum mechanicst. never studied QFTt. complete brainlet\n >> Anonymous Thu Mar 8 23:55:43 2018 No.9572010 >>9571977I was responding to a post talking about arbitrary numbers.If infinity isn't a number, then an infinitely large arbitrary number doesn't exist. Any large arbitrary number could exist, but to be used as such they would have to actually be defined which number they are, as \"arbitrary\" is not useful in real number math. You can set up a situation where for any number, implicitly even arbitrarily large numbers, something can evaluate, but you must use a real number in these situations. You can't just leave the variable \"an arbitrary number\" remain undefined, it must be replaced with an actual real number to carry out arithmetic. \"An arbitrary number\" is used in place to describe any real number, even the numbers greater than any mentionable number. An arbatrarily large number can be any huge real number greater than any other mentioned number, but this only suffices as a variable awaiting replacement with a number. It has to be a number, it can't just remain undefined as \"arbitrarily large\". There is a huge difference between getting the singular right answer in:1 + x = 7Versus getting the singular right answer in:n + y = 8There are innumerable ways to solve n+y=8, so how could you ever get the sole correct answers to the variables n and y? \"There isn't enough information provided to properly solve it as intended.\" Is the only valid answer.Using infinity in maths is the same exact thing as using an undefined, undefineable variable. You dont know the correct value of infinity no more than you know the actual sole correct value of n or y in n+y=8. You've left \"infinity\" undefined as \"an arbitrarily large real number\", forgetting to replace \"an arbitrarily large real number\" with an actual number value. If your limit is \"an arbitrarily large real number\", then you don't know when to stop. You don't know when that limit ends so you never stop, just identically the same as if the limit were infinity, ergo infinity = an arbitrarily large real number.\n >> Anonymous Fri Mar 9 00:06:15 2018 No.9572021 >>9571978No field you just mentioned has real world value relative to physicality. Enjoy learning fucking useless garbage where the only living you'll be able to make out of it is teaching it to college students who don't know any better but hope they're acquiring some useful information for their futures where probably none of them anticipate becoming teachers.\n >> Anonymous Fri Mar 9 00:08:20 2018 No.9572023 >>9572021>No field you just mentioned has real world value relative to physicality.confirmed brainlet\n >> Anonymous Fri Mar 9 00:09:29 2018 No.9572024 >>9572023Ahuh, tell me, what has quantum field theory done for the world?Quantum mechanics?Surely something as early as complex analysis out to be useful.Go on, i'm waiting.\n >> Anonymous Fri Mar 9 00:12:29 2018 No.9572028 >>9572010>infinity = an arbitrarily large real number.https:\/\/www.wolframalpha.com\/input\/?i=infinityAn unbounded quantity greater than every real number.Infinity isn't a real number, by definitionprotip: when you're in a hole, stop digging\n >> Anonymous Fri Mar 9 00:15:45 2018 No.9572031\n >> Anonymous Fri Mar 9 00:29:47 2018 No.9572045 ITT.A potential genius brings into question the meaning of numbers. Because he is a deep thinker.Brainlets have no idea what he is talking about. Because they are brainlets.\n >> Anonymous Fri Mar 9 00:36:39 2018 No.9572056 >>9572024I see you conveniently skipped DSP, cool. I know this is bait, but anyway:https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#Applications\n >> Anonymous Fri Mar 9 00:38:07 2018 No.9572059 >>9572028By definition, it doesnt say what it is or isn't. It only says it is larger than any real number. This says nothing about infinity specifically, only its relation to something else. If it has no relation to real numbers, theres no point in mentioning real numbers. You are literally fucking retarded.\n >> Anonymous Fri Mar 9 00:39:09 2018 No.9572061 >>9572031Yes, teaching it. Good job. Couldn't have fucked up that any better.\n >> Anonymous Fri Mar 9 00:43:39 2018 No.9572071 >>9572056Maybe reread that link and actually learn how the things claimed to be derived from applications of quantum shit were actually invented before the field theories themselves.Fuck the fuck off. You need quantum mechanics to make lasers and diodes work? No you fucking do not. Eat ass. You are literally a brainlet not because you tried to fill your brain with knowledge, but because you filled it with lies and retardations. Your brain is full just as much as would be required of someone who is intelligent, but it just happens to be full of nonsense instead of value.\n >> Anonymous Fri Mar 9 00:46:20 2018 No.9572073 \u00a0 >>9572059>it is larger than any real numberif infinity were a real numberbe definition it would be bigger than itselflrn2read\n >> Anonymous Fri Mar 9 00:47:24 2018 No.9572076 >>9572061>said the monkey of a book it couldn't read\n >> Anonymous Fri Mar 9 00:49:45 2018 No.9572082 >>9572071>hand waving intensifies\n >> Anonymous Fri Mar 9 00:51:03 2018 No.9572086 >>9572059>it is larger than any real numberif infinity were a real number,by definition it would be bigger than itselflrn2read\n >> Anonymous Fri Mar 9 00:52:32 2018 No.9572091 File: 21 KB, 300x300, he_might_be_retarded.jpg [View same] [iqdb] [saucenao] [google] [report] >>9572071>Maybe reread that link and actually learn how the things claimed to be derived from applications of quantum shit were actually invented before the field theories themselves.Sorry to break it to ya, but lasers and LEDs were invented after QM.https:\/\/en.wikipedia.org\/wiki\/Quantum_mechanics#Electronics>You need quantum mechanics to make lasers and diodes work? No you fucking do not.Yes you do.>Eat ass. You are literally a brainlet not because you tried to fill your brain with knowledge, but because you filled it with lies and retardations. Your brain is full just as much as would be required of someone who is intelligent, but it just happens to be full of nonsense instead of value.k.\n >> Anonymous Fri Mar 9 00:53:51 2018 No.9572092 >>9572073n = n+1$\\sum_{n=1}^{n+1}$Here's your \"unending\" operation. Heres how it should read. Easy. At n=1, the limit is 2. At n=2, the limit is 3. At n=3, the limit is 4. There you go. No infinity required, as a number or even a word to describe, and certainly not required as a symbol.>Look at me i'm fucking isaac newton i'm a goddamned brainlet I GIVE THEE CALCULUS\n >> Anonymous Fri Mar 9 00:57:22 2018 No.9572099 >>9572092>No infinity requiredthen it's finite>yawn\n >> Anonymous Fri Mar 9 01:01:12 2018 No.9572109 File: 224 KB, 481x325, 1518634307153.png [View same] [iqdb] [saucenao] [google] [report] >>9572099Oh gee willickers, boys, we got ourselves one of them thinkers! An unending operation is finite! What an amazing analysis! I guess since 0.999... has unending 9's in it, must mean there are a finite amount of 9's in it! Don't even try asshole, 2\/10.\n >> Anonymous Fri Mar 9 01:04:44 2018 No.9572116 >>9572109>0.999...has an infinite amount of 9'snot a finite amount>infinity needed>mind blown\n >> Anonymous Fri Mar 9 01:28:29 2018 No.9572140 >>9572092as long as n is a real number that's finitereplace n with inf, now you've got an infinite resultinfinity is required\n >> Anonymous Fri Mar 9 01:30:22 2018 No.9572142 >>9570309It is when you look out. Eternity lies within.\n >> Anonymous Fri Mar 9 01:40:44 2018 No.9572155 >>9572116Infinite = unendingYou assume \"unending = finite\"Therefore infinite=finite, by your logic.>>9572140Unfortunately it's arbitrarily finite, meaning undefined and vague, so you cant increment from n=1 to n=infinity, unless you decide to change the sum's limit at some arbitrary point from n+1 to simply just n, stopping the unending work, but if its no longer unending, its no longer finite, and therefore no longer infinite.Golly, its almost like it is literally impossible to use infinity properly. Fuck quantum mechanics, this is the true unknown for the ages. Every time someone tries to define infinity by a associating it with a number, it simply changes its own definition and can't be used!\n >> Anonymous Fri Mar 9 01:46:35 2018 No.9572162 >>9572155>infinite=finite, by your logic.nopeany n is a real numberinfinity isn'tThe only way you can get to infinityis inserting infinity instead of nWhich just defeats the purpose.\n >> Anonymous Fri Mar 9 02:11:29 2018 No.9572187 >>9572091The development of lasers needed masers to be developed. The development of masers relied on heating gasses. If increasing the temperature of a gas qualifies as quantum mechanics, then Hank Hill has a Ph.D in quantum mechanics. Unfortunately with such a shit field of useless knowledge, Hank Hill makes minimum wage selling barbecues instead of doing anything worthwhile for society. The idea that masers would produce yielding results in analyzing hyperfine bullshit to 20 gigahertz is also bullshit. Atomic clocks aren't real. The tech to measure the oscillation value of ceasium-133 didn't even exist til the 90's, despite the atomic second coming about in the late 60's. At best it could have only been theoretically possible to predict using some made up form of uncheckable math what the hyperfine oscillation of caesium-133 could be, not realistically checkable and proveable. Worse yet that since the second were defined as however many gigahertz of radiation from caesium-133, this now invalidates the second because 1hertz = 1second. As if infinity being useless wasn't bad enough, the SI second is actually based on a value that required the second to have already been defined as something else. What fucking good is calling the second as \"9 billion cycles per second\"Holy fuck. God exists but he did not make man. Nothing as dumb as man ought to be.\n >> Anonymous Fri Mar 9 02:26:36 2018 No.9572207 \u00a0 >>9572187>Hank Hill has a Ph.D>Atomic clocks aren't real>God exists but he did not make manKEK>1hertz = 1second\n >> Anonymous Fri Mar 9 02:29:18 2018 No.9572213 >>9572187>Hank Hill has a Ph.D>Atomic clocks aren't real>1hertz = 1second>God exists but he did not make manKEK\n >> Anonymous Fri Mar 9 02:30:59 2018 No.9572216 >>9572187>The development of lasers needed masers to be developed. The development of masers relied on heating gasses. If increasing the temperature of a gas qualifies as quantum mechanics...non-sequitur\n >> Anonymous Fri Mar 9 06:09:08 2018 No.9572462 >>9571871>>9571905Just because infinity isn't a real or natural number doesn't mean infinity can't exist. Infinity isn't a single concept but a collection of concepts. Saying a series equals infinity (in the extended reals) is saying the partial sums have no upper bound. You also clearly agree that the set of natural numbers exists and that it has no upper bound as for every natural number n there is a successor number n+1, what is the size of that set? That's another way the concept of infinity manifests itself.\n >> Anonymous Fri Mar 9 06:53:20 2018 No.9572510 >>9570343Laser Light in which medium? Light itself has no temperature, step up your unambiguity game.\n >> Anonymous Fri Mar 9 06:53:35 2018 No.9572512 >>9572462Yes, you're right. infinity is a concept, not an implement, not a construct. If it were as simple as using concepts to define reality, all my conceptual game ideas would have already made me wealthy - i don't have to actually develop and market the games cause the concepts are just as good.Protip, using a word attributed to vagueness to attempt describing something else as anything but vague is dumb.\n >> Anonymous Fri Mar 9 08:09:16 2018 No.9572600 >>9572512The concept gives rise to different but related constructs that are well defined and don't lead to contradictions if you stick to the definitions. You refuse to do that and instead you handwave silly arguments based on intuition.\n >> Anonymous Fri Mar 9 08:27:15 2018 No.9572633 >>9571583>Saying there are an infinite, non-finite amount of parts between finite 1 and finite 2 means you can never reach 2 from 1 by incrementing in those parts. You can by doing infinite increments. This is just the same old Xeno's paradox fallacy. You're crossing infinite increments whenever you take a step, because each increment takes an infinitesimal amount of time to step over. But again, this doesn't even respond to the point since no one is computing the sum by adding each part incrementally. They are transforming that sum into something else. You didn't respond to anything I said, retard.\n >> Anonymous Fri Mar 9 08:33:03 2018 No.9572646 >>9571811>After any real NUMBER, there can only exist a larger real NUMBER, provided there is no LARGEST real NUMBER of which a real NUMBER could not be larger.FTFY\n >> Anonymous Fri Mar 9 08:33:32 2018 No.9572648 >>9570389Temperature is always a measure of average translational kinetic energy so kindly fuck off. There is no such thing as negative kelvin as by definition 0 K equates to zero-point energy, the lowest possible energy a system can have under any circumstances.\n >> Anonymous Fri Mar 9 08:37:57 2018 No.9572655 >>9571871>Every print will always be a real number, and every print will always be larger than the previous print.>At no point will a print ever readout as \u221e\u221e, therefore infinity doesn't exist as a number value.Notice how you switched from \"real number\" to \"number.\" Time to go learn elementary school math.\n >> Anonymous Fri Mar 9 08:40:48 2018 No.9572666 >>9571905>Put it this way, can you describe the exact difference between \"an arbatrarily large real number\" and \"infinity\", if infinity has a quantity value?What is a \"quantity value\"? Can you stop making up terms and just write a real argument. Because so far your argument doesn't exist. Oh and the difference is that the former is finite while the latter is not.\n >> Anonymous Fri Mar 9 08:57:41 2018 No.9572709\n >> Anonymous Fri Mar 9 09:07:41 2018 No.9572736 i like infinityi didnt use to get it beforenow i get some of itits great\n >> Anonymous Fri Mar 9 09:27:37 2018 No.9572764 Do you guys want to know what I think?\n >> Anonymous Fri Mar 9 09:32:54 2018 No.9572771 >>9572764No.\n >> Anonymous Fri Mar 9 11:35:57 2018 No.9572910 >>9572633Xenos paradoxes include assumed finite elements. So no. Think again>>9572655>>9572646>>9572666Die in a tire fire you fucking retarded ESL loser.\n >> Anonymous Fri Mar 9 12:55:05 2018 No.9573007 >>9572910>Xenos paradoxes include assumed finite elements. Yeah a finite length divided into infinite increments you utter tard. You clearly have no response.\n >> Anonymous Fri Mar 9 14:35:47 2018 No.9573192 >>9573007You're missing that the arrow paradox assumes an infinite quantity of freeze frames of an arrow in flight constitute the entire path of the arrow in flight, despite that no freeze frame is distinguishable from another because he assumes 0 time has passed between any frame. Aka the retard would let that $0 \u00d7 \\stackrel{n}{\\infty} > 0$, n added to denote he's referencing the number infinity and not the \"never ending, non-numerical increment\" act of \"infinitely\" repetitive or increasing.Since 0 \u00d7 n = 0, he obviously had a misconception. Staring at a photo of an arrow in flight will never suddenly turn the photo into an animation showing where the arrow flies to, no matter how much time passes while having stared at the photo, nor does copying the photo innumerable times and arranging them in a flipbook allow the arrows flightpath to be animated.The point is, you cannot make it to 2 from 1 in infinite increments unless you assume to abide by xeno's logic that 0\u00d7infinity > 0 [if not 0\u00d7infinity = 1], and if you assume this is the case then the utmost infinite'th partial sum of 1\/2^n (achilles and tortoise) at n=infinity is $\\frac{1}{\\infty} = +0$, meaning the final required partial sum to sum totality is 0. Extend this to 9\/10^n, and at the infinite'th n, the infinite'th digit is 0, giving 0.999...0, where the \"infinite repetition\" is terminated by a 0 which prevents rounding up and also uniquely seperates the value with proper termination from 1.0 which could just as easily be written as 1.000...To elaborate in solution, $\\sum_{n=1}^{x} \\frac{9}{10^n} = 0.9999$n=1: 0.9n=2: 0.99 (+0.09)n=3: 0.999 (+0.009)n=4: 0.9999 (+0.0009); x=4, four ninesProvides evidence that the final partial sum must be a significant non-zero value to sum totality, but 1\/\u221e = 0 forces the final partial sum to be a value that cannot add anything extra to the sum in totality, providing for 1\/2^n = 1 for any arbitrary n less than infinity.\n >> Anonymous Fri Mar 9 14:40:45 2018 No.9573203 >>9573192>the infinite'th digit is 0, giving 0.999...0,they're all 9's, cut the crap\n >> Anonymous Fri Mar 9 14:43:14 2018 No.9573210 >>9573192C1\/2^n = 1 for any arbitrary n less than infinity is extracted from the provision that the reason 0 is being added at the infinite'th partial sum is because the sum has already reached its final summation and needs no more work, assuming 1\/2^n=1 per the description of the paradox. If 0 is being added at the infinite'th partial sum, summation of significant values has been completed prior to the infinite'th partial sum, therefore it has summed not at n=infinity, but n\n >> Anonymous Fri Mar 9 14:48:46 2018 No.9573222 >>9573192>1\/2^n = 1 for any arbitrary n less than infinity.1\/2=11\/4=11\/8=1etc.wow anon you solved itthe fields award is in the mail\n >> Anonymous Fri Mar 9 14:51:01 2018 No.9573230 >>9573222Uhh.. that was zeno's solution, not mine.\n >> Anonymous Fri Mar 9 14:57:29 2018 No.9573252 >>9573210In the very least it can be extended to instead show that 1\/2^n=1 in a finite amount of increments, which is contradictory to zenos claim of infinite increments.\n >> Anonymous Fri Mar 9 15:06:45 2018 No.9573279 >>9573230it's nobody's solutionlearn how to write math correctly\n >> Anonymous Fri Mar 9 15:15:40 2018 No.9573299 >>9573279Even if you were being intentionally retarded to disregard that I was talking about the sum 1\/2^n which you should have known to read as $\\sum_{n=1} \\frac{1}{2^n}$, you would still have been wrong by calling it my work when i was showing it was zenos work.Learn to read dude.\n >> Anonymous Fri Mar 9 15:28:34 2018 No.9573333 why does one grapefruit taste better than the other grapefruit? i bought 2 grapefruits but one tastes better.\n >> Anonymous Fri Mar 9 15:50:49 2018 No.9573377 >>9573333>consuming mogrel racemixed citrus.\n >> Anonymous Fri Mar 9 15:53:15 2018 No.9573380 >>9573192>Your concept of convergence leads to contradictions>It's not me by being retarded and insisting that \"0.999...0\" is somehow a thing>it's not me by ignoring the definition of convergence and instead claiming it can't converge in my unspecified intuitive model that I can keep changing in every post exactly because I never specified it>it's not me by ignoring all the advances modern mathematics has made and instead falling back to 2000+ year old paradoxesIt's clear by now that you're either being intentionally retarded or you're just plain retarded.\n >> Anonymous Fri Mar 9 16:23:36 2018 No.9573426 >>9573380Infinity has no value either as a defined idea or an applied concept. What the fuck advancements are you talking about. The whole point of this is that it is poorly defined loosely attributed vaguely described garbage good for no practical real world application, which is why it is not practically used in any real world application cause it doesn't have a constant intelligible definition, and to further accuse me of somehow falling back to zenos paradoxes when i was directly replying to zenos paradoxes being brought up as some shit idea intended to argue that infinity isn't loosely defined garbage is fucking full brainlet so good job, your sin is not my sin.I change the definition of infinity to accomdate the explanation of any method which aims to support infinity, then i deconstruct infinity to non-existing within that methods own assumptions. You place a nail, i hammer it down, you place another nail, i hammer it down too. No definition of infinity is consistent or singular or useful. It is an invalid concept required for nothing.\n >> Anonymous Fri Mar 9 16:32:02 2018 No.9573451 >>9573210>>9573252Sum 1\/2^n to any arbitrary n doesnt equal 1 though, so the assumption 1\/2^n=1 is false. 1\/2^n will only ever equal a value less than 1, because taking it to the infinite'th n will add a 0 partial sum meaning the total summable work ended prior to infinity, meaning it ended on any arbitrary n, and if we end on any arbitrary n, 1\/2^n is evidently less than 1.\n >> kendrick=booty Fri Mar 9 17:04:57 2018 No.9573513 >>9570309You need to understand the difference between addition and infinite summation. Then things not making sense will make sense to you. :)\n >> Anonymous Fri Mar 9 17:09:26 2018 No.9573519 File: 241 KB, 362x480, maga_pepe_large.png [View same] [iqdb] [saucenao] [google] [report] >>9573513Addition is real and infinity is not.\n >> Anonymous Fri Mar 9 17:14:48 2018 No.9573532 >>9573192I don't know what a 0 length instant of time is, or how it's relevant, since convergent sums have terms that are finite nonzero. Try again, retard.\n >> Anonymous Fri Mar 9 17:17:28 2018 No.9573540 >>9573451The assumption $\\sum_{n=1}^{\\infty} \\frac{9}{10^n} = 1$ is also false because the infinite'th partial sum is 0, meaning summable work completed prior to the infinite'th where n= any arbitrary number, where if n in $\\sum_{n=1}^{\\infty} \\frac{9}{10^n}$ is any arbitrary number, obviously only equals an arbitrary but finite amount of repeating 9's.\n >> kendrick=booty Fri Mar 9 17:19:14 2018 No.9573543 File: 46 KB, 600x500, g-h-hardy-6[1].jpg [View same] [iqdb] [saucenao] [google] [report] >>9573519Neither are real. Real mathematics is not 'real' either.\n >> Anonymous Fri Mar 9 17:21:41 2018 No.9573551 >>9573532Convergence requires infinity to be rigorously defined in order to work because it is used on infinite sums. Unfortunately, there is no proof an infinite sum exists, much less proof infinity exists.\n >> Anonymous Fri Mar 9 17:24:31 2018 No.9573561 >>9573192>The point is, you cannot make it to 2 from 1 in infinite increments unless you assume to abide by xeno's logic that 0\u00d7infinity > 0Nope, one has nothing to do with the other. The arrow paradox fails because it doesn't coherently define any intervals of time. The other paradoxes fail because they are mathematically nonparadoxical.>Since 0 \u00d7 n = 0n being a real number, not infinity. Multiplication by infinity is undefined.> if you assume this is the case then the utmost infinite'th partial sum of 1\/2^nNo such thing. All terms are nonzero, all terms correspond to a natural number, and there is no utmost. Once again you completely fail to understandwhat you're trying to argue against at the most basic level.Saying that time is infinite does not mean that there is some time in the future that is infinitely far away from the present. Rather it means that all times in the future are a finite distance from the present, and there is no end time.\n >> Anonymous Fri Mar 9 17:26:46 2018 No.9573570 >>9573426All you've shown is your own complete lack of understanding of the topic, not a lack of sense in the topic. Retard.\n >> Anonymous Fri Mar 9 17:28:05 2018 No.9573574 >>9573451>because taking it to the infinite'th nNo such thing. Try again.\n >> Anonymous Fri Mar 9 17:32:38 2018 No.9573585 >>9573551That doesn't respond to anything I said. Try again.>Unfortunately, there is no proof an infinite sum exists, much less proof infinity exists.Wrong. The two series discussed in this thread are proven to be convergent. And there is no proof that any number exists side they are axiomatic constructions.\n >> Anonymous Fri Mar 9 17:35:56 2018 No.9573600 >>9573561I agree, infinity is \"never ending\", a direction, not a number.Maths disagrees however. http:\/\/www.wolframalpha.com\/input\/?i=InfinityAn unbounded quantity greater than every real number. Quantity gives it numerical value, greater than every real number gives it a relationship with R and no other set.This unending definition still doesn't allow for the Sums 1\/2^n or 9\/10^n to approach 1, however. If you dont end the summation, you don't get a defined result. This can instead be analytically extended to meam the only valid sum is to an arbitrary finite n, but doing this then clearly shows infinity was not required to obtain the sum at a finite n, and the partial sums to that n woild clearly define the sums finetely, squarely, and self-evidently less than 1. If you wanna cut off some decimal places and be swell enough just rounding up to 1, feel free. Rounding is approximation though, not equality.\n >> Anonymous Fri Mar 9 17:46:19 2018 No.9573633 >>9573574You are saying the end of sum 1\/2^n doesnt look like $\\sum_{n=\\infty}^{\\infty} \\frac{1}{2^{\\infty}}$, because an end can't exist, which means n\/infinity doesnt occur, which means n\/infinity doesn't need to be 0, which means that by never reaching the infinite'th n, there will always exist a not-insignificant non-zero value of $\\frac{1}{2^{arbitrary real number}}$ required to add to the total of partial sums up to that point to equate 1.i'm afraid you can't win, my good dude.\n >> Anonymous Fri Mar 9 17:47:07 2018 No.9573634 >>9573600>I agree, infinity is \"never ending\", a direction, not a number.Where did I say it wasn't a number? You seem to have a problem distinguishing between numbers and real numbers.When time is represented as a real number line, of course infinity will not appear on the line and will only be an unattainable direction. But that doesn't mean it's not a number.>If you dont end the summation, you don't get a defined result.You do get a defined result in the case of convergent sums. Why do you think you don't get a defined result? Because you are viewing each step in summation as taking a certain amount of finite time. But again, this is incorrect since the actual method of evaluating a convergent sum does not involve infinite steps. You still have not responded to this fact.\n >> Anonymous Fri Mar 9 17:50:24 2018 No.9573639 >>9573633>You are saying the end of sum 1\/2^n doesnt look like \u2211\u221en=\u221e12\u221e\u2211n=\u221e\u221e12\u221e, because an end can't exist, which means n\/infinity doesnt occur, which means n\/infinity doesn't need to be 0, which means that by never reaching the infinite'th n, there will always exist a not-insignificant non-zero value of 12arbitraryrealnumber12arbitraryrealnumber required to add to the total of partial sums up to that point to equate 1.Amazing, for the first time in this thread it seems you've written more than a few sentences of accurate mathematics. So where is the problem?\n >> Anonymous Fri Mar 9 17:52:41 2018 No.9573642 >>9573634Oh no thats gross, dont be gay like that buddy. Infinity isn't a number. Math pretends it is, but as soon as you treat it like a number, it stops being intelligble. I dont care whether you think its a real number or a fake number or a complex number or a number outside of all sets. It doesn't matter at all what you think it is. The only thing that matters is what reality can do with it, which is nothing. Therefore it doesn't matter. Thats as basic as you need to go with it, no reason to try to argue about whether you believe i'm using the definition you're using at the moment of your next post which you will have arbitrarily flip flopped just like you'd done now from calling it never ending while chastising me for referencing zeno calling it a number, to then calling it a number yourself. Real gross behaviour buddy. You might be infinitely retarded.\n >> Anonymous Fri Mar 9 17:56:38 2018 No.9573650 >>9573639Maybe you problem is just that you can't do math cause that post you seem to agree with explicitly states 0.999... doesnt equal 1, as there can only be an arbitrary real finite amount of 9's that could in the very least require 0.arbitrary finite amount 0's)1 to add in order to sum 1, even though you are arguing against this aren't you? That you think convergence is valid?\n >> Anonymous Fri Mar 9 17:57:44 2018 No.9573652 >>9573642>The only thing that matters is what reality can do with it, which is nothing. Without infinity you cannot construct the real numbers. Those are immensely useful. Without infinity you have no advanced physics. That is immensely useful.By your own argument, you lose.\n >> Anonymous Fri Mar 9 17:58:42 2018 No.9573654 >>9573652You dont need infinity to do anything that has real world value. You are retarded if you believe otherwise.\n >> Anonymous Fri Mar 9 18:00:35 2018 No.9573661 >>9573650>Maybe you problem is just that you can't do math cause that post you seem to agree with explicitly states 0.999... doesnt equal 1, as there can only be an arbitrary real finite amount of 9's that could in the very least require 0.arbitrary finite amount 0's)1 to add in order to sum 1, even though you are arguing against this aren't you?You'll have to try again, this is just gibberish. As far as I can tell you're saying 0.999... has a finite amount of 9s, which is wrong. And no, it doesn't end in a certain number because it doesn't end at all. Plus you didn't answer my question. What is wrong with there being an infinite amount of nonzero terms?\n >> Anonymous Fri Mar 9 18:01:37 2018 No.9573664 >>9573654>You dont need infinity to do anything that has real world value. Math and physics have a lot of real world value. You lose.\n >> Anonymous Fri Mar 9 18:08:47 2018 No.9573678 >>9573664THE MATHS THAT HAVE REAL WORLD VALUE DONT ACTUALLY USE INFINITY>>9573654Its not wrong. You agreed that sum 9\/10^n will only ever have a finite number amount of 9's. You never reach the infinite'th step and can only ever have an arbitrary real n number step, so you always have an arbitrary real n number of digits. Are you retarded?You're retarded. I fuckin knew it dude i totally called it, you were gonna arbitrarily flip flop on your definition of infinity the next post and bam, thats literally exactly what you did. Feel free to call me God or a time traveller or something.\n >> Anonymous Fri Mar 9 18:13:59 2018 No.9573689 >>9573678THE MATHS THAT HAVE REAL WORLD VALUE ACTUALLY USE INFINITY\n >> Anonymous Fri Mar 9 18:14:55 2018 No.9573691 >>9573689What is: None, Alex\n >> Anonymous Fri Mar 9 18:16:55 2018 No.9573695 >>9573678>You agreed that sum 9\/10^n will only ever have a finite number amount of 9'sWhere did I do that? Lying and putting words in my mouth is not going to help you win the argument.>You never reach the infinite'th stepNo such thing.>and can only ever have an arbitrary real n number step, so you always have an arbitrary real n number of digits. There is no time factor here. 0.999... immediately has an infinite amount of 9s. You're spouting nonsense.\n >> Anonymous Fri Mar 9 18:32:27 2018 No.9573727 >>9573695You dumb dumb.You want infinity to be \"unending\".So we use infinity as \"unending\" in $\\sum_{n=1}^{unending} \\frac{9}{10^n}$ okay? We're not worried about a quanity number of infinity even though thats how maths considers it, we're just gonna worry about the only logical way it ought to work as \"unending\" since the number version doesn't work.So for the sum, we have 0.999... with an unending amount of 9's, right?I'm gonna disregard your disallowance of time as a factor cause in due time you will try to say otherwise.We have 0.999... with an unending amount of 9's. No end of 9's in sight. If you tried to count the 9's, you'd have counted an arbitrary amount because you cant count them all because they're unending.So where exactly in this unending amount of 9's does it allow that it should equal 1? Lets be clear too, it's not just 1, it's 1.000... with an unending amount of 0's. If we created sets for each number 0.999... and 1.000... where each element in each set respectively were the individual digits, we would have unending sets with finitely numberable elements, where every identically accessed element of each set would be unequal[0, 9, 9, 9, 9, 9, ...][1, 0, 0, 0, 0, 0, ...]Totally unequal. Not a single element from either set matches up.\n >> Anonymous Fri Mar 9 18:35:50 2018 No.9573739 >>9573727So what's the arithmetic mean of 0.(9) and 1?\n >> Anonymous Fri Mar 9 18:36:32 2018 No.9573743 >>9573727>So where exactly in this unending amount of 9's does it allow that it should equal 1?Where does \"it\" not allow it equalling 1? I don't see an argument anywhere, just a question of how convergence works, which you should already know if you passed high school.>Totally unequal. Not a single element from either set matches up.Luckily that is not a prerequisite for two decimal representations to be equal.\n >> Anonymous Fri Mar 9 18:57:23 2018 No.9573812 >>9573743Now you're just using even more poorly described methodology by wanting yo assume convergence, even though its already established convergence requires a working definition of infinity of which, although \"unending\" is nice, is not how maths treats infinity. Maths says infinity is a quantity with a relationship to real numbers. So again, you have to boldly face the fact that no element of 0.999... and 1.000... equate, nor any combination of elements up to attempting to include them all, and that any attempt to have a set featuring all the elements of either number will never be filled, but will have finitely many numerable elements in any attempt to do so, and assuming to hold all elements arbitrarily since we need not abide by non-arbitrary rules like time, will give us sets full of arbitrary real numbered elements with an arbitrary real length, which is really when you're starting to wish you put more of a wager on infinity as a quantity but none the less still proveable as a falsifiable attribute.\n >> Anonymous Fri Mar 9 19:01:02 2018 No.9573822 >>9573812>Now you're just using even more poorly described methodology by wanting yo assume convergence, even though its already established convergence requires a working definition of infinity of which, although \"unending\" is nice, is not how maths treats infinity. Maths says infinity is a quantity with a relationship to real numbers. It's both, depending on context.And I have yet to see you even demonstrate you know the proof of convergence, let alone find a flaw in it.>So again, you have to boldly face the fact that no element of 0.999... and 1.000... equateSo?\n >> Anonymous Fri Mar 9 19:10:00 2018 No.9573840 File: 7 KB, 387x387, Eye_of_Horus_square.png [View same] [iqdb] [saucenao] [google] [report] >>9571770define this square as 1, no matter how you cut it, the parts are always summed to exactly 1.\n >> Anonymous Fri Mar 9 19:23:05 2018 No.9573868 >>9573840You start with a square of arbitrary length sides. You cut this square in half. And color one half in. You have accomdated one half of the square, but one half is still left uncolored. You cut the other half in half and color in a half. You have accomodated one half and one quarter of the square, but one quarter is still left uncolored.Every half cut you make of a half, leaves another half to be cut worth the same value as your previous cut. The only way to achieve a cut that leaves 0 left to be cut is to make a cur that leaves two equal halves of 0, one you can color and accomodate, the other you cannot, but its zero so it doesnt matter. What number \u00f7 2 = 0?\n >> Anonymous Fri Mar 9 20:50:50 2018 No.9574034 >>9573868You can make unending cuts and fills of your newly acquired halves, but you will never have a total fillrate of 100%. There is no half you'll receive where a final cut will allow two halves of 0 value, so 1\/2^n < 1\n >> Anonymous Fri Mar 9 21:47:31 2018 No.9574149 >>9574034You can make unending cuts and fills of your newly acquired halves, but you will never have a total fillrate of 100%.If you make unending cuts then you will have a total fill rate of 100%.>There is no half you'll receive where a final cut will allow two halves of 0 value, so 1\/2^n < 1There is no final cut, so that's irrelevant.\n >> Anonymous Fri Mar 9 21:48:40 2018 No.9574151 >>9570325What do you mean by this? Have you just said words to make it sound like you have something to say?\n >> Anonymous Fri Mar 9 22:34:07 2018 No.9574242 >>9574149No cut allows for n\u00f72 = 0, you never get 100% fill. Whether or not a final cut exists is irrelevant.\n >> Anonymous Fri Mar 9 23:30:23 2018 No.9574359 >>9574242>No cut allows for n\u00f72 = 0,Which is irrelevant, since there is no final cut. You get 100% fill from the infinite, unending cuts, not any one cut.>Whether or not a final cut exists is irrelevant.Your argument is based on there being a certain cut at which there is 100% fill. Stop being retarded.\n >> Anonymous Fri Mar 9 23:37:18 2018 No.9574374 I was bored and wanted to know why $\\sum_{n=1}^{\\infty} \\frac{1}{n}$ is considered divergent and seem to have accidentally discovered a constant which doesn't show up in google or recognized by wolfram as transcendent.At n = 1,000,000 the sum is $14.392726722865 \\vdots$, and at n= 1,000,000,000 the sum is increased by this constant value of $6.9077552789821370 \\vdots$, where at every $n= 10^{(x.mod3=0)}$, the sum increases by this value. Accurate to 29 decimal places using this value, i was able to predict at n=1 with 42 zeroes after it, the correct sum up to 4 decimal places by incrementing over n=1,000,000 as the base.I'm also not sure but at 1 with 60 zeroes after it, the sum\/(60+1) seems to be approaching $e$ albeit very slowly where at that point it is only 2.2743003,,,Not sure at this point whether it would surpass e or not, but i assume the methodology would remain true for checking via $\\frac{\\sum_{n=1}^{10^{(x.mod 3=0)}} \\frac{1}{n}}{x+1}$\n >> Anonymous Fri Mar 9 23:39:02 2018 No.9574378 >>9574359EVERY CUT OF THE INFINITE UNENDING CUTS NEVER RESULTS 0YOU ALWAYS HAVE A SIGNIFICANT AMOUNT LEFT TO DIVIDE IN HALF YOU BRAINLETALWAYSNEVER FILLS 100%\n >> Anonymous Fri Mar 9 23:47:25 2018 No.9574390 >>9574378EVERY CUT TOGETHER RESULTS IN 0 BECAUSE THERE ARE INFINITE CUTSINFINITE CUTS LEAVES YOU WITH ZERO LEFT TO DIVIDE YOU BRAINLETALWAYS FILLS 100%\n >> Anonymous Fri Mar 9 23:51:02 2018 No.9574394 >>9574359>>9574378Furthermore, because 1\/infinity actually is 0 and infinity actually is treated as a quantity number, the only way to achieve a cut that results two halves of 0 value is to increment to the number infinity. 1\/(2^infinity) = 0Theres some problems here though since infinity isn't on the numberline, you can't actually increment to the number infinity, so you can't actually get that 0. You will instead only ever have access to 1\/(2^n) where n is less than infinity, and what is less than infinity? Any arbitrary number is less than infinity. Which means all you will ever have access to in Sum 1\/(2^n) is up to any definable real number n. Thats it.Is Sum 1\/2^n, n=1 to 10 = 1.0?No?then there exists no x for the Sum 1\/2^n, n=1 to x which equals 1.0\n >> Anonymous Fri Mar 9 23:51:38 2018 No.9574395 >>9574378LEARN WHAT A LIMIT ISALSO LEARN WHAT \"CAPS LOCK\" DOES\n >> Anonymous Fri Mar 9 23:52:29 2018 No.9574397 >>9574390Jesus fuck you are retarded. You already know every cut up to infinity is not 0. 1\/2^n at n=1 is 0.5 for fucks sake. Every cut thereafter is a non-zero value. You cannot arbitrarily redefine all previous cuts to suddenly equal 0.\n >> Anonymous Fri Mar 9 23:54:10 2018 No.9574399 >>9574390>INFINITE CUTS LEAVES YOU WITH ZEROso 1\/inf = 0\n >> Anonymous Fri Mar 9 23:55:17 2018 No.9574401 >>95743990\u00d7infinity = 1 too, yes\n >> Anonymous Fri Mar 9 23:57:19 2018 No.9574403 >>9574394>Furthermore, because 1\/infinity actually is 0Wrong, undefined.>the only way to achieve a cut that results two halves of 0 valueThis is not necessary to fill the square, it's irrelevant.>Theres some problems here though since infinity isn't on the numberlineWhich numberline?>you can't actually increment to the number infinity, so you can't actually get that 0There is no need to increment in the first place and no need to get an increment to 0. You have nothing but non sequiturs.\n >> Anonymous Fri Mar 9 23:58:50 2018 No.9574405 File: 16 KB, 498x467, 1512340128839.png [View same] [iqdb] [saucenao] [google] [report]\n >> Anonymous Fri Mar 9 23:58:57 2018 No.9574406 >>9574397>You already know every cut up to infinityThis is gibberish, try again.>1\/2^n at n=1 is 0.5 for fucks sake. Every cut thereafter is a non-zero value. You cannot arbitrarily redefine all previous cuts to suddenly equal 0.This has nothing to do with what I said, try again.\n >> Anonymous Fri Mar 9 23:59:59 2018 No.9574407 >>9574399>so 1\/inf = 0So you are illiterate.\n >> Anonymous Sat Mar 10 00:00:59 2018 No.9574408 >>9574403>>Furthermore, because 1\/infinity actually is 0>Wrong, undefined.nah, it's zerohttp:\/\/mathworld.wolfram.com\/Infinity.html\n >> Anonymous Sat Mar 10 00:02:03 2018 No.9574410\n >> Anonymous Sat Mar 10 00:05:00 2018 No.9574413 >>9574405It's undefined, Wolfram is wrong.\n >> Anonymous Sat Mar 10 00:06:02 2018 No.9574418 >>9574408No, it's undefined.\n >> Anonymous Sat Mar 10 00:13:41 2018 No.9574427\n >> Anonymous Sat Mar 10 00:22:08 2018 No.9574444 >>9571638>>9573192>>9573210>>9573451>>9573540>>9573633>>9573678> infinite'thYou keep using that word. I do not think it means what you think it means.\n >> Anonymous Sat Mar 10 00:27:41 2018 No.9574458 >>9574403I have a squareI cut it in half and fill half of what i just cutI now have 1\/2 of a square filled and 1\/2 of a square unaccounted for.I cut the unfilled half in half and fill half of what i just cutI now have 1\/2 + 1\/4 of a square filled, and 1\/4 unaccounted for.I cut the unfilled half in half and fill half of what i just cut.I now have 1\/2 + 1\/4 + 1\/8 of a square filled, and 1\/8 of a square unaccounted for.After every cut, you are that cut's value away from the full squareIf you did this up to 1\/2 + 1\/4 + 1\/8 + ... + 1\/562,949,953,421,312 then there would still be 1\/562,949,953,421,312 of the square left unaccounted for. Another cut will double this significand and HALVE the previous value to 1\/1.12589991E+15, leaving 1\/1.12589991E+15 of the square left unaccounted for. There needs to exist a value $\\frac{\\frac{1}{2^n}}{2} =0$ in order to no longer have any part of the square left unaccounted.The problem of this square is the same as $\\sum_{n=1}^{\\infty} \\frac{1}{2^n}$, which the sum shows an issue. If we assume Sum1\/2^n is equal to 1 and leave the limit undefined and wish to know what the limit is, we can compare it to a problem $\\sum_{n=1}^{x} \\frac{9}{10^n} = 0.999$n1: 0.9n2: 0.99 (+0.09)n3: 0.999 (+0.009)Thus x=3 is our limit to solve this problemTake note at x=3, the value added is not insignificant, which is why x=3Back to $\\sum_{n=1}^{x} \\frac{1}{2^n} = 1$ where we wish to define x, if we set the limit to infinity, we have that the infinite'th partial sum is $\\frac{1}{2^{\\infty}} = \\frac{1}{\\infty} = 0$, which is an insignificant value. Whether this 0 is accounted for or not doesn't change the total sum, so this means that the end of significant summing did not occur at n=infinity, it occurred before at a value less than infinity. Since any real number is less than infinity, we are redirected that x= any R, but x= any R is obviously not 1.Infinity is broken.\n >> Anonymous Sat Mar 10 00:31:05 2018 No.9574467 >>9574458Infinity isn't broken, it just shows that it's impossible for the infinite sum of 1\/2^n to equate 1.\n >> Anonymous Sat Mar 10 00:34:40 2018 No.9574473 >>9574458>There needs to exist a value 12n2=012n2=0 in order to no longer have any part of the square left unaccounted.False, if you gave infinite cuts then you have filled the square without this value. Try again.>if we set the limit to infinity, we have that the infinite'th partial sumNo such thing. Try again.\n >> Anonymous Sat Mar 10 00:35:54 2018 No.9574476 >>9574467The infinite sum of 1\/2^n is equivalent to 1. All you've shown is a gross lack of understanding of basic math.\n >> Anonymous Sat Mar 10 00:38:12 2018 No.9574482 >>9574473>>9574476You are too retarded to breathe properly. Enjoy your hypoxia and brain death.\n >> Anonymous Sat Mar 10 00:40:40 2018 No.9574486 >>9574458Think about what you are saying here: If there is a $\\frac{1}{2^\\infty}$, then $n = \\infty$ which mean that there is an $x\\in\\mathbb{N}$ such that $n+x=\\infty$, which obviously cannot ever be the case.The notation $x \\rightarrow \\infty$, means that $x$ is approaching infinity, not that it will ever get \"there\" (a concept which in itself is nonsensical).\n >> Anonymous Sat Mar 10 00:42:12 2018 No.9574491 >>9574482>t. lost the argument\n >> Anonymous Sat Mar 10 00:43:10 2018 No.9574493 >>9574486Approaching infinity but never getting there is the same thing as saying all it will ever be is an arbitrary real number, and if the limit is any arbitrary real number then the summation halts at a value that is self-evidently finitely seperated from 1 by a significant value.\n >> Anonymous Sat Mar 10 00:45:07 2018 No.9574497 >>9570902That's not what 'science' means.\n >> Anonymous Sat Mar 10 00:48:46 2018 No.9574504 >>9574493>Approaching infinity but never getting there is the same thing as saying all it will ever be is an arbitrary real number, and if the limit is any arbitrary real numberYou are confusing the limit with the variable approaching the limit. Laughable. Go to bed kid, you'll be late for school tomorrow.\n >> Anonymous Sat Mar 10 00:58:52 2018 No.9574519 >>9574493To elaborate, the infinite limit doesnt imply that the sum value somehow reaches 1. It only allows that there will exist an infinitely long list of partial sums that get closer and closer to 1 but never produce a maximal partial sum that is evidently closest to 1 or exactly 1. The total sum at n=infinity is 1, but that's because we have exceeded the limit required as the partial sum at n=infinity is 0, which means their ought to have been an n before infinity where the partial sum was (an arbitrary value \/ 2) = 0. The problem that is being exposed is that infinity as a quanitity has know implement of knowable difference from real numbers. There is a vague, undefined gap of values between any number and infinity, and the required value of significant\/2=0 exists in this gap, where neither real number satisfies being the limit or does infinity satisfy being the limit. Using math, it is proveable that infinity is retarded because it has no direct knowable relation in difference from any other number, other than infinity-n = \"some arbitrary number\"Infinity is broken.\n >> Anonymous Sat Mar 10 01:00:07 2018 No.9574521 >>9574491were we arguing?I was just holding my breath to make the fap more intense\n >> Anonymous Sat Mar 10 01:00:50 2018 No.9574522 >>9574519>knowTypo, meant \"no\"\n >> Anonymous Sat Mar 10 01:13:37 2018 No.9574533 >>9574519In short, it is mathematically proveable that when you write 0.999... or 1+2+3+4+..., that ellipses doesn't actually mean \"to infinity\", it instead literally means exactly what you're typing the ellipses to represent, that being \"an undefined gap of necessary but missing information\".There you go. Infinity is not helpful. Its a lazy concatenation that gives you bad answers because invoking it deliberatelt invokes the idea that there is a significant amount of necessary information being omitted. Whether you use the number quanitity infinity or the never ending direction of infinity, as soon as you invoke it, you invoke that some information is unknown and unknowable, thereby sabotaging your arithmetic.\n >> Anonymous Sat Mar 10 01:17:04 2018 No.9574534 File: 97 KB, 1200x675, 3a1.jpg [View same] [iqdb] [saucenao] [google] [report] This might be my favorite thread on \/sci\/ ever. My mind is being blown.\n >> Anonymous Sat Mar 10 01:26:20 2018 No.9574542 >>9574533BS, stop pulling crap out of your ass.0.999... denotes the repeating decimal consisting of infinitely many 9's after the decimal point.\n >> Anonymous Sat Mar 10 01:31:02 2018 No.9574547 >>9574486>n=\u221eImplying inf is a real number. It isn'thttp:\/\/www.wolframalpha.com\/input\/?i=InfinityAn unbounded quantity greater than every real number.\n >> Anonymous Sat Mar 10 01:31:18 2018 No.9574550 >>9574522>being this insecure on an anonymous imageboard\n >> Anonymous Sat Mar 10 01:32:41 2018 No.9574552 >>9574542Holy shit learn to read, that's not what I said at all. 0.999... means \"0.999 and fuck math n shit i'm lazy\"Thats it. Thats defacto. Thats proven by using the implements of infinity prescribed by the very professors whose cocks you suck so much. This is not up for debate bro, it's fucking math, it's fucking fact.\n >> Anonymous Sat Mar 10 01:34:19 2018 No.9574555 >>9574552>undefined gapit's defined, retard\n >> Anonymous Sat Mar 10 01:34:29 2018 No.9574556 >>9574547An unbounded quantity = a numberOnly numbers can be quantities. Greater than any real number = defines a relationship specifically with the real number setAlso 1\/infinity in wolfram is 0.\n >> Anonymous Sat Mar 10 01:36:32 2018 No.9574559 >>9574555If it were defined, infinity - n would equate a specific real number amount.What is infinity-100?Spoiler: the answer is you're retarded.\n >> Anonymous Sat Mar 10 01:45:16 2018 No.9574564 >>9574559inf-100=inf\n >> Anonymous Sat Mar 10 01:47:23 2018 No.9574568 >>9574556ts kiddo>An unbounded quantity greater than every real number.that's the definition\n >> Anonymous Sat Mar 10 02:01:35 2018 No.9574586 >>9574564Infinity - n is a value less than infinity. But it's not a real number value.These are your numbers:$\\mathbb{U} = \\big[ \\mathbb{R} \\big] < \\big[ \\infty - \\mathbb{R} \\big] < \\infty$That infinity-R set in the middle is the undefined gap. Infinity minus any real number ends up there as some vague undefined undefinable bullshit because infinity is broken. Infinity - n can't be infinity, it has to be a value less than infinity, but the value isn't known unlike values inside R. Treating infinity as a set of arbitrary values to allow \"infinity - 100 = infinity\" then forces that no arithmetic performed on real numbers, even arbitrary undefined arithmetic, will allow the real number set to ever approach infinity. If the real number set never approaches infinity, the only result from an infinite sum could only ever be a real number limit result, and disregarding the real number limit then becomes a mathematical mistake which will allow errors to accumulate.Stop\n >> Anonymous Sat Mar 10 02:26:39 2018 No.9574606 >>9574586so what, that's the definition\n >> Anonymous Sat Mar 10 02:29:46 2018 No.9574609 >>9574586>infinity-R set in the middledoesn't exist, inf minus anything finite is still inf\n >> Anonymous Sat Mar 10 02:54:42 2018 No.9574627 >>9574559It's still infinity but it's now a different infinity.\n >> Anonymous Sat Mar 10 02:56:41 2018 No.9574630 >>9574609Couldn't care less. It exists and it doesnt exist. It exists, then inf-r is undefined shit, it doesnt exist then inf-r is still undefined shit. Either way the result isn't useful and forces the real number set to be seperated from infinity. Wolfram might say infinity - n = infinity, but it also says infinity - infinity is undefined, even though it makes all the provisions for n being allowable as infinity.the gap exists whether its called infinity-r or left unaddressed, which is why (Sum1\/2^n; n=1 to infinity) produces a 0 partial sum at an infinite'th n which denotes that a value of n prior to infinity is where the summable work actually completed since a partial sum of 0 makes no difference to the total sum. If you then try to work backward to determine where that n is, you either get an unknown value less than infinity in the infinity-r set, or you become unable to work backwards at all, like accidentally turning onto a one way street that somehow assfucks you and puts on the the freeway for 10 miles before you can turn around, except instead of 10 miles the freeway just goes off into space and you end up irreversibly flying outside of the galaxy. In other sciences, this is called \"destructive\", a situation where you cannot undo or work backwards from. If you increment to infinity in computer calculation (a defined number that exists as the singular largest value greater than the largest allowable double double, which ironically is an actual finite number that can be reached in increments), then the variable has essentially become destroyed. The value of the variable will exist at infinity but no amount of declination once you've reached that point will allow you to back out from infinity, and no increments will advance out of infinity. It is the death of the variable in terms of arithmetic until it is reset. Even this destructive behaviour isn't nominal however, and only exists because of poorly planned math implementations based on non-computer math\n >> Anonymous Sat Mar 10 02:58:36 2018 No.9574631 >>9574630There are flaws in your reasoningYou think infinity is a numberYou think that there is only THE infinity.\n >> Anonymous Sat Mar 10 03:02:05 2018 No.9574635 >>9570309Anyone who tries to argue against the concept of infinity using only intuition and metaphorical explanations with very basic usage of actual maths are like those annoying weebs that think they know Japanese and fervently argue about the correct translation but still have to rely on subtitles for watching their anime.\n >> Anonymous Sat Mar 10 03:02:14 2018 No.9574637 >>9574631No, i think infinity is anything anyone wants to make of it, but every way anyone ever uses it defies arithmetic and preclude allowing infinity to be useful.I clearly stated two different accomdations for infinity there. A maximal value where -r puts it in a different set, or a fucked set of values where -r gives it a different infinite identity part of the infinite set.\n >> Anonymous Sat Mar 10 03:08:54 2018 No.9574642 >>9574630>(Sum1\/2^n; n=1 to infinity) produces a 0 partial sum at an infinite'th nWhere did you get this idea from?\n >> Anonymous Sat Mar 10 03:10:57 2018 No.9574643 >>95746421\/infinity = 0$\\sum_{n=1}^{\\infty} 1 = \\infty$Where'd you get the idea to question it?\n >> Anonymous Sat Mar 10 03:14:48 2018 No.9574646 >>9574643That would equal to 1, not infinity. There is no n to evaluate. Perhaps you meant $\\sum_{n=1}^{\u221e} n= \u221e$?\n >> Anonymous Sat Mar 10 03:20:49 2018 No.9574650 >>9574646No. I described 1+1+1+1+1+1+ and so on. You described 1+2+3+4+5+6+ and so on. n doesnt need to be a variable in the summation itself, n only defines how many iterations of the summation occur. This is why 1\/2^n is 0.5 at n:10.75 at n:2 (previous 0.5 from 1\/2 + current 0.25 from (1\/[2^2]) aka 1\/4Etc.\n >> Anonymous Sat Mar 10 03:29:03 2018 No.9574658 >>9574650>1+1+1+1+1+1that would be $\\sum_{n=1}^{\u221e} 1^n = \u221e[\\math]>doesnt need to be a variable in the summation itself, n only defines how many iterations of the summation occurYou can't even get your notations right. The equation that the sigma notation is evaluating needs to be a function of the number of iterations.[eqn]\\sum_{n=a}^{b} f(n)[\/eqn]The rest is just garbage spewed from misconceptions. >> Anonymous Sat Mar 10 03:30:05 2018 No.9574659 >>9574658>math]\\sum_{n=1}^{\u221e} 1^n = \u221e[\\math]meant to bemath]\\sum_{n=1}^{\u221e} 1^n = \u221e$\n >> Anonymous Sat Mar 10 03:31:06 2018 No.9574661 >>9574659$\\sum_{n=1}^{\u221e} 1^n = \u221e[\\math] >> Anonymous Sat Mar 10 03:31:51 2018 No.9574664 >>9574658http:\/\/m.wolframalpha.com\/input\/?i=sum%5B1%2C+%7Bn%2C1%2C6%7D%5D+and+sum%5Bn%2C+%7Bn%2C1%2C6%7D%5DI thought you were legit asking how to do use sums but i guess you're just an idiot. >> Anonymous Sat Mar 10 03:34:52 2018 No.9574669 >>9574630>Couldn't care lessfine, ignore mathit's m-muh f-feelings from that sentence on >> Anonymous Sat Mar 10 03:36:38 2018 No.9574670 >>9574630aand a word salad wall of text.classic rambling lunatic style. >> Anonymous Sat Mar 10 03:37:19 2018 No.9574672 File: 41 KB, 301x399, Young-Black-Woman-says-STOP1.jpg [View same] [iqdb] [saucenao] [google] [report] >>9574669>hand waving >> Anonymous Sat Mar 10 03:42:57 2018 No.9574678 >>9574672>thinks that computers prove the limits of math and not the other way around >> Anonymous Sat Mar 10 03:43:50 2018 No.9574679 >>9574672>counterargument is literally a pic of hand waving >> Anonymous Sat Mar 10 03:49:01 2018 No.9574682 File: 55 KB, 617x347, 1509035736738.png [View same] [iqdb] [saucenao] [google] [report] >>9574679>get called out for handwaving and being a brainlet who can neither read or do math>young black woman says stop.jpg>continue handwaving anyway while still being unable to read or do math>expect to be taken seriously>don't get taken seriouslyYou are a walking existential crisis, boy. >> Anonymous Sat Mar 10 04:45:30 2018 No.9574738 >>9574682>no math, just m-m-m-muh >> Anonymous Sat Mar 10 05:03:21 2018 No.9574750 File: 22 KB, 164x107, Selection_088.png [View same] [iqdb] [saucenao] [google] [report] this fucking thread yet again >> Anonymous Sat Mar 10 05:57:24 2018 No.9574785 >>9573812How can the sequences [math]\\left(\\frac 1 n \\right)_{n\\in\\mathbb N_0}$ and $\\left(\\frac {1}{2n} \\right)_{n\\in\\mathbb N_0}$ have the same limit when none of their elements are equal? Infinity is clearly retarded.\n >> Anonymous Sat Mar 10 06:29:59 2018 No.9574828 >>95747851\/4 isn't 1\/(2*2)[retards applauding intensifies]\n >> Anonymous Sat Mar 10 07:12:19 2018 No.9574877 >>9574785There is no *one* infinity. There's a reason why \u221e - \u221e is undefined instead of zero.\n >> Anonymous Sat Mar 10 10:00:58 2018 No.9575070 >>9574519>To elaborate, the infinite limit doesnt imply that the sum value somehow reaches 1. Yes it does.>It only allows that there will exist an infinitely long list of partial sums that get closer and closer to 1 but never produce a maximal partial sum that is evidently closest to 1 or exactly 1.It also means that the series itself, not partial sums, are equal to 1.>The total sum at n=infinity is 1OK so you admit you were wrong this whole time. Thanks for playing.>but that's because we have exceeded the limit required as the partial sum at n=infinity is 0There is no partial sum at n=infinity\n >> Anonymous Sat Mar 10 10:03:30 2018 No.9575074 >>9574559>If it were defined, infinity - n would equate a specific real number amount.Wrong.\n >> Anonymous Sat Mar 10 10:07:21 2018 No.9575077 >>9574637>No, i think infinity is anything anyone wants to make of it,Which is obviously wrong since you keep using it in nonsensical ways that defy real mathematics.>but every way anyone ever uses it defies arithmetic and preclude allowing infinity to be useful.Because to a complete novice like you arithmetic is the be all and end all of math. Pathetic.\n >> Anonymous Sat Mar 10 10:07:47 2018 No.9575078 infinity is a concept, not a number\n >> Anonymous Sat Mar 10 10:10:41 2018 No.9575085 >>9574643How does that justify an \"infinite'th partial sum\"? Answer the question.\n >> Anonymous Sat Mar 10 13:15:50 2018 No.9575355 >>9575085It must. If you don't assume to arbitrarily reach the infinite'th partial sum, that means you're assuming no zero, that means you're assuming the sum will never reach an end, that means you're assuming that all possible values are real number values, that means you're assuming there must exist a smallest real number part such that adding it to 0.9 repeating (which is repeating to a real number value because you assume infinity can't be reached) will result 1.This is no joke. Whether you pretend to understand infinity or not, there is little but absolute truth behind the fact that it's vagueness in usage can be analytically extended to forcing it to produce results it cannot accomodate for itself, such as this result that an infinite'th partial sum is 0 value and summable work finished prior to infinity. The only thing anyone ever have needed to know is that infinity is greater than all real numbers, so less than infinity then defines any real number. If you replace the direction from infinite'th n which was too much to any real n, you then get a properly finite result. There is no real solution x to the problem $\\sum_{n=1}^{x} \\frac{1}{2^n} = 1$. Infinity is too much, any real number in x is too little, and this partially extends from the fact 1\/infinity = 0. This doesn't mean 1\/infinity isn't 0, cause if we assume it isn't then we still have the same problem above that infinity is never approached in the first place, or that 1\/inf would be an undefined non-number where upon reaching the infinite'th n would returj some kind of problem.Anyway the pont is 1\/inf = 0, 0\/2 = 0, you now have a half cut that leaves nothing uncut and have accounted for all 100% of the square. If you presume to never reach infinity, you never get to make the final cut. If you presume to reach infinity you make a cut on zero, not to zero, meaning you ought to have finished cutting before infinity in some real numbered step, yet no real numbered step exists to solve.\n >> Anonymous Sat Mar 10 13:22:31 2018 No.9575365 >>9575355Maths not getting accuracy right is the difference between a quantum teleportation device sending all your bits from point a to point b, or that same quantum device discarding random atoms of you during every teleport or accidentally merging you with errant particles of dust. Too much or too little is unaccepetable.\n >> Anonymous Sat Mar 10 14:21:48 2018 No.9575482 >>9575077I've used it as a number as defined by mathematics. I've used it as a direction, as defined by reasonable feeling of how it should be. I've used it as a singular value. I've used it as a set of values.Using infinity in any variety of ways doesn't allow it to work. No way works. Infinity doesn't work no matter how you use it.\n >> Anonymous Sat Mar 10 14:35:12 2018 No.9575520 >>95754821\/inf = 0 works just fine\n >> Anonymous Sat Mar 10 14:42:03 2018 No.9575534 >>9575482Theorem 1: For all definitions of infinity, it holds that the definition is inconsistent.Proof:We prove this by example. Define $\\infty$ as the largest natural number. It then follows that $\\infty + 1 = \\infty$. Since it's true that a natural number subtracted by itself yields 0, it clearly follows that $(\\infty + 1) - \\infty = \\infty - \\infty = 0$. However because addition is commutative it also follows that $\\infty + 1 - \\infty = (\\infty - \\infty) + 1 = 0 + 1 = 1$. This proves that $0 = 1$ and as such we've shows that this definition of infinity is inconsistent.Because we provided a counterexample, it follows that all definitions of infinity must be inconsistent. This proves the theorem. QED.Checkmate infinitists.\n >> Anonymous Sat Mar 10 14:45:46 2018 No.9575544 >>9575534wrong\n >> Anonymous Sat Mar 10 14:46:35 2018 No.9575548 >>9575534nice\n >> Anonymous Sat Mar 10 14:55:22 2018 No.9575571 >>9575534>\u221e\u2212\u221e=0That's where you're wrong. Subtraction of an infinite quantity from itself is undefined. It doesn't make \u221e inconsistent anymore than the fact that 1\/x is undefined at x=0 makes division \"inconsistent\".\n >> Anonymous Sat Mar 10 14:57:43 2018 No.9575579 Infinitists and finitists are both right.Numbers are infinite, but our limited brain capacity means we perceive and use them as finite things, just as we perceive our 10 fingers\/thumbs as finite things, with a beginning and an end. But there is no official beginning and end to one's fingers, just a rough approximation that is useful enough to make the distinction, and it's the same with numbers,\n >> The Lord Sat Mar 10 15:01:09 2018 No.9575586\n >> Anonymous Sat Mar 10 15:02:17 2018 No.9575591 itt: people trying to understand infinity despiteour brains are incapable of it\n >> Anonymous Sat Mar 10 15:02:29 2018 No.9575592 >>95755201\/infinity=0 treats infinity as a number. I've already shown what treating as a number entails. It allows that an infinite sum n=1 to infinity could, even arbitrarily, acquire a partial sum at n=infinity, but such a partial sum is 0. This means two bad things.1) as the partial sum is zero, the partial sum is irrelevant to the total sum, meaning it wasn't required to reach n=infinity to finish all summable work, ao you must work backwards to a value less than infinity. Since that the only values less than infinity are all real numbers, any real number value substitution for n is used, but doing this provides no reasonable solution as no real number divided by another real number is 0, so you simply can never have a finitely singulat correct solution to an infinite sum. Any real number n validates as a solution up to any real number of decimal places, but the solution limit for any $\\sum \\frac{x}{(x+1)^n}$ such as 1\/2^n or 9\/10^n can then only equal a finite number that is finitely seperable from 1. 0.999... =\/= 12)if infinity is a set of values, having an infinite amount of 0 partial sums would then equate to 1, as 1\/inf=0 so 0\u00d7inf=1, so we now have 1+all prior partial sums which means the total sum approaches $1.\\bar{9}$, not 1. n=infinity is yet again proven to be too much and all required summable work must have occurred before infinity, but any real number n in the sum shows that x\/(x+1)^n cannot finitely equate 1.\n >> The Lord Sat Mar 10 15:02:35 2018 No.9575593 File: 188 KB, 418x484, TRINITY_+________g98yughrq87sf96d1949.png [View same] [iqdb] [saucenao] [google] [report]\n >> Anonymous Sat Mar 10 15:03:34 2018 No.9575598 >>9575571Buddy it's true if $\\infty$ is defined as just another natural number. I was showing that the definition where you treat it as a natural number is inconsistent. It then logically follows that every other definition must also be invalid, as mentioned by >>9575482The logic is flawless, just accept that infinitists are done.\n >> Anonymous Sat Mar 10 15:12:22 2018 No.9575624 >>9575598>Buddy it's true if \u221e is defined as just another natural number.this doesn't mean anything\n >> Anonymous Sat Mar 10 15:13:00 2018 No.9575628 >>9575571If you think that's where the proof goes wrong you should probably never touch math again.\n >> Anonymous Sat Mar 10 15:14:45 2018 No.9575635 >>9575598>The definition where you treat it as a natural number is inconsistent. It then logically follows that every other definition must also be invalidnon-sequitur\n >> Anonymous Sat Mar 10 15:15:49 2018 No.9575639 >>9575628https:\/\/en.wikipedia.org\/wiki\/Extended_real_number_line#Arithmetic_operationsThat's literally where it goes wrong, faggot. \u221e\u2212\u221e=0 is flat-out wrong because it leads to a contradiction.\n >> Anonymous Sat Mar 10 15:21:19 2018 No.9575658 >>9575534>Define \u221e as the largest natural numberhow about nothttps:\/\/www.wolframalpha.com\/input\/?i=infinityAn unbounded quantity greater than every real number.try again\n >> Anonymous Sat Mar 10 15:24:03 2018 No.9575665 >>9575658Learn to read english you ESL shitter >>9574556\n >> Anonymous Sat Mar 10 15:26:00 2018 No.9575667 >>9575665ditto\n >> Anonymous Sat Mar 10 15:26:58 2018 No.9575672 >>9575658Just read the proof again, and read the theorem too. If you weren't a complete shitter you could find the actual error in that proof. Protip: if you're a retarded engineer that has never written a proof, don't even try, you're not up to it.\n >> Anonymous Sat Mar 10 15:30:15 2018 No.9575682 >>9570487What if I allow Axiom of Infinity but limit Axiom of Powerset to finite sets?\n >> Anonymous Sat Mar 10 15:30:23 2018 No.9575683 >>9575665no one gives a rat's ass about yourmade-up math in >>9574556stick to the actual definition at >>9575658\n >> Anonymous Sat Mar 10 15:31:34 2018 No.9575686 >>9575534>We prove this by example.lol\n >> Anonymous Sat Mar 10 15:31:59 2018 No.9575688 >>9575592My favorite part about this is the implication that you could finish counting all possible finite numbers and still be unable to reach infinity even after every possible number had been accounted for. It just goes to show that a definition of \"greater than all other numbers\" is positively meaningless, especially since everyone already knows a greatest number can't exist, where these two statements obviously contradict each other. Infinity as the greatest number yet no greatest number exists only lets infinity then not exist.\n >> Anonymous Sat Mar 10 15:34:44 2018 No.9575694 >>9575592>1\/infinity=0 treats infinity as a numbernope, if inf were a real number, the ratio wouldn't get down to zeroAn unbounded quantity greater than every real number.\n >> Anonymous Sat Mar 10 15:37:17 2018 No.9575707 >>9575694You don't even fucking know what a number is and I have berated you multiple times in this thread already over all the other dumb shit you've said. Fuck off dude. You're not welcome.\n >> Anonymous Sat Mar 10 15:39:00 2018 No.9575714 >>9575688>infinity then not existor infinity is not a real numberbecause that's what >>9575658 says>>9572086\n >> Anonymous Sat Mar 10 15:40:34 2018 No.9575718 >>9575707The way I see it numbers are just whatever object you want to call numbers. They don't have to share any property with any other kind of number other than being called number.Is that wrong? If so, what are the defining properties of numbers?\n >> Anonymous Sat Mar 10 15:40:36 2018 No.9575719 >>9575707 Fuck off dude. You're an asshole\n >> Anonymous Sat Mar 10 16:00:58 2018 No.9575762 >>9575718You want to call dozen a quantity, you gotta accept that the quantity is a number and that number is 12.Infinity as a quantity requires a number too, but instead of a number all you get is \"???\". Cant really do reasonable addition and subtraction on what can only best be described as an undefined variable.Its the difference between solving 1 + X = 7 versus Z + 1 = WX oughta be 6, but you can't know what Z or W are supposed to be. Infinity as a number is the equivalent of an undefineable variable, because it has a quantity but no provided number value to associate with that quantity, unlike the quantity dozen being associated with 12 or the quantity couple associated with 2.\n >> Anonymous Sat Mar 10 16:13:50 2018 No.9575784 >>9575762>You want to call dozen a quantityI don't want to do anything other than ask what the defining properties of \"A number\" are.\n >> Anonymous Sat Mar 10 16:22:50 2018 No.9575808 >>9575784A number is an exact quantity, defineable by the decimal system and the characters 0123456789\n >> Anonymous Sat Mar 10 16:27:37 2018 No.9575820 >>9575707lol you sound like a real fuckin loser\n >> Anonymous Sat Mar 10 16:28:54 2018 No.9575825 >>9575808That makes complex numbers not numbers, which is stupid. What you're saying makes being a number equivalent to being a real number, but we already have a word for that, it's 'real number' so that's stupid.\n >> Anonymous Sat Mar 10 16:31:45 2018 No.9575834 >>9575808you agree to find me two exact same grapefruits, and ill agree to your idea of numbers\n >> Anonymous Sat Mar 10 16:32:20 2018 No.9575837 >>9575820I completely deconstructed infinity and you're calling me a loser, even though the only reason you'd be such a bitch to call me a loser is cause you yourself lost and are spiteful. Don't project your own insecurities onto others.\n >> Anonymous Sat Mar 10 16:33:26 2018 No.9575841 File: 5 KB, 250x174, brainlets....jpg [View same] [iqdb] [saucenao] [google] [report] >>9575834Grapefruits aren't numbers...\n >> Anonymous Sat Mar 10 16:34:35 2018 No.9575844 >>9575841hold up fella we got einsteinowitz ova heaa\n >> Anonymous Sat Mar 10 16:38:44 2018 No.9575848 >>9575837>I completely misconstrued infinity...FTFY\n >> Anonymous Sat Mar 10 16:44:51 2018 No.9575860 File: 33 KB, 600x700, Pool.jpg [View same] [iqdb] [saucenao] [google] [report] >>9575848I told you to stop projecting. If you don't know how to do that, then stop posting.\n >> Anonymous Sat Mar 10 17:01:16 2018 No.9575891 >>9575860fuck off assholeyou're an asshole and don't know shithow's that for a paradox\n >> Anonymous Sat Mar 10 18:41:54 2018 No.9576139 Heres a good way to illustrate the point. Do you need infinite money?Even if you were the most greedy person ever, do you need infinite money? Cause i'm pretty sure you'd be able to accomplish anything you want with just 1 trillion dollars. You could probably do well enough with 1 billion or even 1 million. You need need infinite money and you could accomplish everything you want with just a finite amount of money. You can accomplish everything you need to with just finite numbers. No one needs infinity to exist. You don't even need an infinite amount of real numbers. You don't need infinity. Its simply too much for you.\n >> Anonymous Sat Mar 10 18:55:24 2018 No.9576167 >>9575482>I've used it as a number as defined by mathematics.No you haven't, you've done the opposite.\n >> Anonymous Sat Mar 10 18:56:48 2018 No.9576169 >>9575355>If you don't assume to arbitrarily reach the infinite'th partial sum, that means you're assuming no zeroWrong.\n >> Anonymous Sat Mar 10 18:59:10 2018 No.9576175 >>9575534>Define \u221e\u221e as the largest natural number.Wrong.\n >> Anonymous Sat Mar 10 19:15:24 2018 No.9576207 >>9576139I'm a retard and I approve this message\n >> Anonymous Sat Mar 10 19:20:49 2018 No.9576219 >>9576139>Even if you were the most greedy person ever, do you need infinite money?Non sequitur.>You can accomplish everything you need to with just finite numbers. No you can't. You can't construct the real numbers, you can't do most of modern math and you can't do most of modern physics. You are in denial.\n >> Anonymous Sat Mar 10 19:46:25 2018 No.9576273 >>9576167Demonstrate infinity as a number. Provide an example. It doesn't suffice to simply call me wrong. The burden of proof is on you to prove I am wrong, so try to prove I'm wrong.>>9576219All real numbers are finite numbers.\n >> Anonymous Sat Mar 10 20:43:27 2018 No.9576381 Heres more to work out how needlessly big infinity is.The infinite sum 1\/n is a number that grows incredibly slowly. At n= 1,000,(467 more zeroes), the sum is only something like 2,000. If we took the sum from 2,000 to 100,000,000, the limit at the new sum would be a 1 with 16 million zeroes after it. If we took the the sum from 100,000,000 to 100,000,000,000,000 the limit at the new sum would be a 1 with 16 trillion zeroes after it. If you took the sum to 100 googols, the limit would be 1 with 16 googolplex zeroes after it.1000 thousand1000000 million1000000000 billion1000000000000 trillion1000000000000000 quadrillion1000000000000000000 quintillion1000000000000000000000 sextillion1000000000000000000000000 septillion1000000000000000000000000000 octillionAt octillion this number is retardedly large at 1 with 27 zeros after it, yet even this number is miniscule compared to the first mentioned number at 1 with 470 zeros, and all of these example numbers are definitely finite yet still less than arbitrarily large which is further still smaller than infinity. The smallest unit of length is $\\frac{1}{10^{44}}$ or a decimal followed by 44 zerosThe smallest number based on the number i started with at the top of the problem is $\\frac{1}{10^{470}}$. The planck length compared to this number is equivalent to taking a planck length, stretching it across half the distance of the entire universe, then taking a planck length measurement within that region\n >> Anonymous Sat Mar 10 20:51:37 2018 No.9576403 >>9576273>Demonstrate infinity as a number. What does that mean?>The burden of proof is on you to prove I am wrong, so try to prove I'm wrong.No the burden of proof is on you to make a coherent argument.>All real numbers are finite numbers.So? Do you know the construction of the real numbers? Do you even know what that means? You are laughably ignorant.\n >> Anonymous Sat Mar 10 20:55:28 2018 No.9576416 >>9576381>The smallest unit of length is 1104411044 or a decimal followed by 44 zerosThat's not a unit of length.> The planck length compared to this number is equivalent to taking a planck length, stretching it across half the distance of the entire universe, then taking a planck length measurement within that regionYou can't compare a planck length to a dimensionless number. That makes no sense.\n >> Anonymous Sat Mar 10 21:00:19 2018 No.9576427 File: 37 KB, 340x565, 1514683747604.jpg [View same] [iqdb] [saucenao] [google] [report] >>9576416retard\n >> Anonymous Sat Mar 10 21:04:58 2018 No.9576435 File: 175 KB, 600x600, 58b.png [View same] [iqdb] [saucenao] [google] [report] >>9576403>you're wrong but i can't prove it\n >> Anonymous Sat Mar 10 21:10:01 2018 No.9576440 File: 5 KB, 221x250, 1518045540769.jpg [View same] [iqdb] [saucenao] [google] [report] >>9576435>infinity is wrong but I can't prove it\n >> Anonymous Sun Mar 11 01:00:13 2018 No.9576820 >>9576381so?are you trying to impress a six year old?\n >> Anonymous Sun Mar 11 01:03:54 2018 No.9576826 File: 9 KB, 300x300, you.png [View same] [iqdb] [saucenao] [google] [report] >>9575808>A number is an exact quantity, defineable by the decimal system\n >> Anonymous Sun Mar 11 01:07:41 2018 No.9576832 >>9576273>Demonstrate infinity as a number.Ok. According to the usual construction of real numbers from the rationals by Dedekind cuts, a real number is simply the set of rational numbers that are strictly less than it. Hence the empty set $\\varnothing$ represents $-\\infty$ while the set of all rationals $\\mathbb{Q}$ represents $\\infty$.\n >> Anonymous Sun Mar 11 04:29:31 2018 No.9576965 >>9576175No. It's correct in the context of that proof. You could argue the wording is awkward, but then you're just being pedantic.\n >> Anonymous Sun Mar 11 04:55:26 2018 No.9577003 >>9576965>It's correctnopehttps:\/\/www.wolframalpha.com\/input\/?i=infinityAn unbounded quantity greater than every real number.\n >> Anonymous Sun Mar 11 04:56:21 2018 No.9577006 >>9576965>ignores the context of the proofYou're intentionally being retarded, right?\n >> Anonymous Sun Mar 11 05:00:18 2018 No.9577012 >>9577006Context is absolutely key.\n >> Anonymous Sun Mar 11 05:03:46 2018 No.9577016 >>9576832How is arithmetic defined in the dedekind cuts construction? Don't you need non-empty sets on both halves of the cut for this to be consistent? (Assuming we want to retain the properties of a field)\n >> Anonymous Sun Mar 11 07:45:12 2018 No.9577141 >>9570309Your pic is wrong btw\n >> Anonymous Sun Mar 11 07:54:30 2018 No.9577149 >>9570389>negative kelvinKill yourselfI'm a maths major but I remember physics from high school. Absolute zero hasn't even been reached brainlet\n >> Anonymous Sun Mar 11 07:57:08 2018 No.9577152 >>9577149You can define temperature as some relation between the energy and entropy of a system which lets you imagine systems with negative temperature, but it's not the standard definition.\n >> Anonymous Sun Mar 11 09:27:50 2018 No.9577262 >>9577141Indeed, everyone knows ellipsis should have 3 dots, not 5.\n >> Anonymous Sun Mar 11 13:46:52 2018 No.9577716 >>9576820Sometimes feels like i'm talking to 6 year olds when you refuse to let go of infinity. You can't prove to have used infinity up to a value with 16 trillion decimal places.\n >> Anonymous Sun Mar 11 13:54:37 2018 No.9577743 >>9575825Complex numbers arent numbers, they're just a poor mistake by brainlets who dont understand axis exist.\n >> Anonymous Sun Mar 11 13:56:45 2018 No.9577748 When they finally flesh out all the describable properties of a number and its interactive\/relative properties to other numbers, there will be a way to represent that as the human form. 1In the beginning was the Word, and the Word was with God, and the Word was God. 2He was with God in the beginning.\n >> Anonymous Sun Mar 11 15:07:06 2018 No.9577959 >>9574458Could infinity simply be any value greater than any local required limit?Dead simply, 1+ x = 3x can only be \"2\" if the rule of the variable x is that it isn't a compound equation like \"(1+1)\" or \"(3-1)\" or \"$\\frac{6}{3}$\"Infinity in relation to x would be a value that is too much, a value greater than 2, as 1+3 = 4, 1+4 = 5, etc. We assume to do work until a solution is found regardless of direction to do it in a certain number of steps, so we need not define the work to be done on arbitrary arithmetic with a limit. However, on math with limits, setting the limit to infinity becomes properly needless as the limit is then greater than required, since infinity as value greater than the local limit is evidently greater than required in 1 + x = 3. \"Greater than required\" is also sufficient in decribing the partial sum problem at n = infinity when substituting n in the sum with infinity. \"Greater than required\" is also read as \"too much\". 1 divided by a value that is \"too much\" then equals 0, because too much in relation to real numbers is a value that is outside all real numbers. So with a local finite limit like 2, infinity would exist inside the real numbers at any value > 2, but without a local finite limit such as the limit is infinity itself, infinity would exist outside the real numbers so infinity cannot be a limit at all, and can only be defined as greater than the local finite limit of which all arithmetic must have local finite limits.$1 + \\stackrel{x}{1} < 3 \\\\ 1 + \\stackrel{x}{2} = 3 \\\\ 1 + \\stackrel{x}{\\infty} > 3$\n >> Anonymous Sun Mar 11 15:07:51 2018 No.9577961 File: 16 KB, 297x255, base 4 counting.png [View same] [iqdb] [saucenao] [google] [report]\n >> Anonymous Sun Mar 11 15:15:55 2018 No.9577971 >>9577959>so bad it's not even wrong\n >> Anonymous Sun Mar 11 15:34:50 2018 No.9577997 File: 245 KB, 1063x1063, 1508010693769.png [View same] [iqdb] [saucenao] [google] [report] >>9576832Tbh senpai the idea of dedekind cut doesnt really make sense by itself so extending beyond it isn't valuable.>set A contains every rational number less than the cut, set B contains every rational number greater than or equal to the cutProvides that B is greater than A, that if B were infinite then A would be finite, provides that A to B, 1 to any n, can only have a finite amount of elements in it's set. Meaning 1 approaching 2(non-inclusive of 2) can only have a finite amount of values between 1 and 2, and 2 going on would have infinite, which already assumes a definition of infinity is in place. The set A from 0 towards any arbitrary real number Z would only be abled to have a finite number of elements, and the set B from Z onwards would only have a greater number of elements as it includes Z where A does not.\n >> Anonymous Sun Mar 11 16:06:37 2018 No.9578051 >>9577997>Provides that B is greater than AIf you are tapping about cardinality, that's wrong.> that if B were infinite then A would be finiteWrong.>provides that A to B, 1 to any n, can only have a finite amount of elements in it's set. Wrong.\n >> Anonymous Sun Mar 11 16:17:05 2018 No.9578071 >Uniformed: the post: the board\n >> Anonymous Sun Mar 11 16:22:14 2018 No.9578085 >>9578051>A set partition of the rational numbers into two nonempty subsets S_1 and S_2 such that all members of S_1 are less than those of S_2 and such that S_1 has no greatest memberExplain\n >> Anonymous Sun Mar 11 16:31:52 2018 No.9578107 >>9578085You are confusing the size of the members with the size of the set, cardinality. The cardinality of the sets are equal. They are both countably infinite.\n >> Anonymous Sun Mar 11 16:58:39 2018 No.9578158 >>9578107This isn't helpful then. Of course the values between 0 and 1 would be smaller than the values between 1 and 2. This doesn't help to define infinity.\n >> Anonymous Sun Mar 11 17:03:32 2018 No.9578169 >>9578158What defines infinity as a number is the dedekind cut between the empty set and the set of all rationals. You asked for a demonstration of infinity as a number. You were given it. I don't know what more you want.\n >> Anonymous Sun Mar 11 17:04:50 2018 No.9578172 >>9578158How are the real numbers constructed then?\n >> Anonymous Sun Mar 11 17:46:04 2018 No.9578238 File: 713 KB, 512x768, 1514317516287.png [View same] [iqdb] [saucenao] [google] [report] >>9578169Dedekind cuts requires no set can be empty.\n >> Anonymous Sun Mar 11 17:58:25 2018 No.9578258 >>9578238That's only a requirement for constructing the real numbers. Infinity isn't a real number.\n >> Anonymous Sun Mar 11 18:00:25 2018 No.9578266 >>95779591+x=3 X could be the sum of any positive divergent summation and any negative divergent summation.Basically any (+inf) + (-inf) summation can equal any real number.t. Reimann's paradox\n >> Anonymous Sun Mar 11 18:02:48 2018 No.9578274 >>9578258But you said that the amount of values in the sets wasn't what mattered, just the values themselves. All real numbers in the set that isn't the empty set doesn't mean it is infinite, just that all the values in the set are greater than nothing. This doesn't help to define infinity.\n >> Anonymous Sun Mar 11 18:04:39 2018 No.9578281 >>9578266>Basically any (+inf) + (-inf) summation can equal any real number.Wrong. You can't rearrange 1+1+1+... To converge to anything.\n >> Anonymous Sun Mar 11 18:04:52 2018 No.9578282 >>9577997>Meaning 1 approaching 2(non-inclusive of 2) can only have a finite amount of values between 1 and 2Then exactly how many real values are between one and two?\n >> Anonymous Sun Mar 11 18:05:54 2018 No.9578283 >>9578281Positively divergent and negatively divergentRead the whole post next time. Also assume it is a continuous summation and not discrete\n >> Anonymous Sun Mar 11 18:09:30 2018 No.9578291 >>9578274>All real numbers in the set that isn't the empty set doesn't mean it is infinite, just that all the values in the set are greater than nothing. You have it backwards. Positive infinity is constructed by having the set of all rationals be the set which positive infinity is greater than.\n >> Anonymous Sun Mar 11 18:10:36 2018 No.9578293 >>9578266If a value in x doesn't solve 1+x=3, but is less than 2, it is a value that is too little. If a value in x doesn't solve 1+x=3, but is greater than 2, it is (locally) infinity. This isn't attempting to define infinity as a number, but rather a property of \"excessiveness beyond correct for solving the problem\"As such, there exists no infinite summation, as this definition of infinite would be summation above and beyond acquiring a reasonable answer.\n >> Anonymous Sun Mar 11 18:12:11 2018 No.9578298 >>9578291Yeah, thats not a different definition than \"a number greater than all numbers\", which has already been buried as a fallacy.\n >> Anonymous Sun Mar 11 18:15:10 2018 No.9578303 >>9578298Where did you bury it as a fallacy? And if you don't accept that then you can't accept the construction of the reals.\n >> Anonymous Sun Mar 11 18:19:51 2018 No.9578314 >>9578303You dont need to limit numbers to existing or not based on dedekind cut. Its not useful by itself.Furthermore the original question wasn't \"how do you construct infinity\", it was \"use infinity as a number in a way that disproved my claims of infinity used as a number having clear and evident issues\".\n >> Anonymous Sun Mar 11 18:23:33 2018 No.9578324 >>9578293The addition of an infinitely positive and an infinitely negative summation can equal any real number arbitrarily. Read about riemann's paradox.By ordering the terms in the infinite summations differently you can group them in such a way as to have it equal any real number you would likeThis only works when adding an infinitely positive Divergent series and an infinitely negative Divergent series\n >> Anonymous Sun Mar 11 18:35:28 2018 No.9578351 >>9578314>You dont need to limit numbers to existing or not based on dedekind cut. Where did I do that?>Furthermore the original question wasn't \"how do you construct infinity\", it was \"use infinity as a number in a way that disproved my claims of infinity used as a number having clear and evident issues\".Wrong, it was \"demonstrate infinity as a number.\"Not to mention that you have not shown any issues with infinity so there is nothing to disprove. All of your alleged issues have been shown to be based on simple fallacies in your reasoning. Look at any of your posts in this thread and find me one that has not been addressed.\n >> Anonymous Sun Mar 11 18:36:07 2018 No.9578353 >>9578324Infinite summation has no meaning. It requires infinity to be rigorously defined. You're a step ahead of the topic.Infinite summation when infinity is treated as a number, and therefore a limit, insures that no infinite summation properly sums what had classically been claimed as the result. This is demonstated here >>9575592$\\sum_{n=1}^{x} \\frac{1}{2^n}$ cant equate 1 for any x limit, including infinity. When the limit of infinity is used as a number, it can be used to mean the summatiion never ends therefore any test of n will always be a real finite number, or it means n=infinity can arbitrarily be achieved, but $\\frac{1}{2^{\\infty}} = \\frac{1}{\\infty} = 0$ means at n=infinity, 0 is being added which means all summable work occurred before infinity at an n less than infinity, yet any n less than infinity used as the limit doesn't allow the result of the sum to reach 1 nor is the sum \"infinite\" any more, and this was if infinity were a singular value. If infinity were a set of values itself infinitely long, at n=infinity there would be an infinite amount of partial sums of 0, which 1\/inf = 0, 0\u00d7inf = 1, which means all partial sums added are 1 + (1\/2^n for all real n) which is a sum not approaching 1, but instead a sum approaching 2; in the very least.\n >> Anonymous Sun Mar 11 18:45:57 2018 No.9578367 >>9578353>\u2211xn=112n\u2211n=1x12n cant equate 1 for any x limit, including infinity.x is not a limit, and it equates to 1 when n = infinity> yet any n less than infinity used as the limit doesn't allow the result of the sum to reach 1So what? Different values of n give different values.>nor is the sum \"infinite\" any moreIt's still infinite.>which 1\/inf = 0, 0\u00d7inf = 1No, it's undefined.All of your issues stem from assuming that infinity must have the same properties as a finite number, when by definition it doesn't. So the failure is solely yours.\n >> Anonymous Sun Mar 11 18:51:12 2018 No.9578382 >>95783671 divided by infinity is zero. You are not arguing with me by claiming otherwise, you are arguing with math.You are just shit at reading and a stupid dickhead so whatever. You are actually wrong. You're wrong about this and you were wrong in your last post when claiming the original question wasn't to use infinity as a number. >>9576167I used it as a number, the argument was i hadn't.>>9576273I then said demonstate it as a number, to appease how I hadn't already used it as a number\n >> Anonymous Sun Mar 11 18:57:09 2018 No.9578393 File: 53 KB, 403x448, 1509935607777.png [View same] [iqdb] [saucenao] [google] [report] >>9578367>limit is x>\"x is not the limit\"x is the limit, dude.\n >> Anonymous Sun Mar 11 19:24:00 2018 No.9578476 >>9578382>1 divided by infinity is zeroIn some contexts like complex analysis it is. But it's generally undefined. This is irrelevant anyway as the big mistake was assuming that this implies that infinity x 0 = 1. This assumes that multiplication by infinity is defined.And you didn't reply to the rest of my post, I guess that means you concede the other points and admit your argument fails.>you were wrong in your last post when claiming the original question wasn't to use infinity as a number. You said it right here >>9576273 \"Demonstrate infinity as a number.\" The stuff you put in quotes as if you said it appears nowhere.>I used it as a number, the argument was i hadn't.No the argument was that you hadn't used it as a number as defined by mathematics. Learn how to read.\n >> Anonymous Sun Mar 11 19:25:17 2018 No.9578483 >>9578393>>limit is xNo it's not. Do you know what a limit is?\n >> Anonymous Sun Mar 11 19:40:49 2018 No.9578536 >>9578476If 1\/inf = 00\u00d7inf = 1Do you not know how to do math? Division and multiplication aren't some abstract things, they're just glorified addition and subtraction. 1\/inf = 0 means if you have infinite 0s, adding them together equates 1, ergo 0\u00d7inf = 1. More realistically, n\/inf = 0, so 0\u00d7inf = n, which is just one more invalidating issue of many regarding what happens when using infinity.I did reply to your post in general by calling it wrong, because it was. You took assumptions, defacto granted shit, and argued against them. This makes you wrong. This makes you as wrong as if I called the sky blue and you said it wasn't, that it were instead green. No one needs to prove the sky is green and you're objectively wrong for arguing, and if you're that wrong its easy enough to just disregard you as nuts. You continue being wrong and arguing about what I posted myself, asking infinity be demonstrated as a number, which you misconstrued to \"construct the number infinity with dedekind cut\" which 1, didnt use infinity as a number, and 2, was an arbitrary random methodology that isn't actually necessary for proving which numbers can or cannot exist, that numbers exist exclusively without being defined by dedekind cut.Take for granted you'd been a retard, put it behind you if you can, then move on and demonstrate that infinity can be used as a number by giving me an example where invoking infinity as a number is otherwise understood to provide a sensible answer to an equation.\n >> Anonymous Sun Mar 11 19:55:43 2018 No.9578576 >>9578483$\\sum_{S}^{F} C$The Sigma E looking thing means this is a sum equation. C is a stand-in variable for the arithmetic to be calculated. S is the starting value, such as n=1. F is the limit, aka the finishing value. The summation is composed of partial sums where for every S incrementing to F, defines the amount of partial sums. The total sum and result of the equation is the sum of all partial sums.$\\sum_{n=1}^{5} 2n$ is the list of partial sums $\\big( \\stackrel{2\u00d71}{2_{n1}}, \\stackrel{2\u00d72}{4_{n2}}, \\stackrel{2\u00d73}{6_{n3}}, \\stackrel{2\u00d74}{\\8_{n4}}, \\stackrel{2\u00d75}{10_{n5}} \\big)$, which is the same as 2+4+6+8+10, so the result of the sigma sum is 30.$\\sum_{n=1}^{x} \\frac{1}{2^n}$ in this equation, x is a variable in place of where the limit is.\n >> Anonymous Sun Mar 11 20:10:37 2018 No.9578609 >>9578536>If 1\/inf = 0>0\u00d7inf = 1Wrong. You're multiplying by infinity, which is undefined.>Division and multiplication aren't some abstract things, they're just glorified addition and subtraction.They are will defined for finite numbers>1\/inf = 0 means if you have infinite 0s, adding them together equates 1, ergo 0\u00d7inf = 1No it doesn't mean that. And adding infinite 0s is 0+0+0+... = 0. But this is not the same as multiplication by infinity, which is undefined.\n >> Anonymous Sun Mar 11 20:23:55 2018 No.9578641 >>9578609Good fuckin job failing to use infinity as a number. Literally the only thing you had to do.You are worthless.\n >> Anonymous Sun Mar 11 20:40:56 2018 No.9578683 >>9578576>F is the limit, aka the finishing valueNo, that is not what limit means in mathematics. Don't make up your own terms and expect people to know what you're talking about.\n >> Anonymous Sun Mar 11 20:43:02 2018 No.9578691 >>9578641All of your issues stem from assuming that infinity must have the same properties as a finite number, when by definition it doesn't. So the failure is solely yours.\n >> Anonymous Sun Mar 11 21:44:28 2018 No.9578824 >>9578691You backpedalling loser fucking retard kill yourself. You say that after going through all the trouble of providing for the construcyion of infinity as a number using dedekind cut.Just fucking die you shameful piece of brainlet shit.\n >> Anonymous Sun Mar 11 21:46:05 2018 No.9578828 >>9578683You too, you fucking die. If you're the same shitter then you've earned yourself two executions.\n >> Anonymous Sun Mar 11 22:48:35 2018 No.9578970 >>9578824Not all numbers are finite. Learn how to read.\n >> Anonymous Sun Mar 11 23:11:41 2018 No.9579025 Daily reminderhttps:\/\/www.wolframalpha.com\/input\/?i=infinityAn unbounded quantity greater than every real number.\n >> Anonymous Sun Mar 11 23:28:53 2018 No.9579074 >>9578970I saidDemonstrate infinity as a numberInstead of doing this, you assume i think infinity is a real number, then change the discussion to that of infinity not being a real numberWhyCause you are a retardYou can't demonstrate the usefulness of infinity as a number, so you change the subject. Three strikes, you're out.\n >> Anonymous Sun Mar 11 23:46:10 2018 No.9579136 >>9579074>usefulness of infinity as a numberno one knows what that even means - because it's meaningless bullshit, just word salad\n >> Anonymous Sun Mar 11 23:52:03 2018 No.9579152 http:\/\/m.wolframalpha.com\/input\/?i=assuming%5Bx%3Dinfinity%2C+%28sum%5B1%2F2%5En%2C%7Bn%2C1%2Cx%7D%5D+%29+%5DAlso x\n >> Anonymous Sun Mar 11 23:54:54 2018 No.9579158 >>9579152$2^{ - \\infty}$\n >> Anonymous Sun Mar 11 23:55:32 2018 No.9579159 >>9579152 -2*inf=-inf2*inf=inf0*inf undefined\n >> Anonymous Sun Mar 11 23:55:55 2018 No.9579162 >>9579136You're out. Get off the field asshole.\n >> Anonymous Sun Mar 11 23:57:37 2018 No.9579164 >>9579074I've demonstrated it several times. For example the cardinality of a set is a number which describes the amount of members in the set. The cardinality of the set of natural numbers is countable infinity, also known as aleph null. The numbers that are used to measure cardinality are even called \"cardinal numbers.\"\n >> Anonymous Sun Mar 11 23:58:07 2018 No.9579167 >>9579162>i have no argument\n >> Anonymous Mon Mar 12 00:01:08 2018 No.9579173 >>9579152>writing a bunch of shit to confuse wolfram alphaYou really are desperate huh?http:\/\/m.wolframalpha.com\/input\/?i=sum%5B1%2F2%5En%2C%7Bn%2C1%2Cinf%7D%5D\n >> Anonymous Mon Mar 12 00:04:12 2018 No.9579182 >>9579167You had multiple chances and blew them all>>9579164You stated a loose definition of infinity as a number using dedekind cut which by itself is useless as dedekind cut is not necessary for numbers to exist, what you didn't do is use infinity as a number to any degree, comprehensive or not, which doesn't solve the reason you replied being to demonstrate infinity as a number. Eat shit.\n >> Anonymous Mon Mar 12 00:05:59 2018 No.9579189 >>9579182>i have no argument\n >> Anonymous Mon Mar 12 00:18:12 2018 No.9579225 >>9579173It didnt confuse wolfram. 1-2^-x is 1-0 when 2^-infinity. It just reorders the logic expression which is that it's starting with the granted (1-...) which is where the real fuckup occurs. 1-0 = 1, but 1\/2^n doesn't mathematically reach 1. This problem stems from the treatment of infinity as a number, where 1\/infinity = 0 but so does every other number on n\/infinity = 0, which just as clearly ought to mean 0\u00d7infinity could be any number and is undefined because it isn't any specific exact number, but it's any number inclusive of infinity, meaning there is only 1 value in this set of any values that suffices for 1-x=1 and its if n\/infinity = 0*infinity = 0, yet infinite values for any other result such as if n\/infinity = 0*infinity = 1 or 2 or 3 or any r wouldn't allow the result to equal 1, and all of this retardation stems from 1\/infinity = 0 instead of any n\/infinity being unknowable because infinity is not a number.\n >> Anonymous Mon Mar 12 00:21:58 2018 No.9579239 File: 40 KB, 414x389, 7434402b4b7c9b157a9b7fe43be1dfd268c6ea4f720eda14ed37eef7ab993efc.jpg [View same] [iqdb] [saucenao] [google] [report] >>9579182>You stated a loose definition of infinity as a number using dedekind cut which by itself is useless as dedekind cut is not necessary for numbers to existYou are confusing necessity with sufficiency. The fact that infinity can be constructed as a number is sufficient to show that it is a number. But that does not imply it is necessary. Once again you fail at basic logic.>what you didn't do is use infinity as a number to any degreeI mentioned several times the user of infinity as the cardinality of a set.You lost on every front, even the irrelevant demand to \"demonstrate infinity as a number.\" Even you know you lost. Stay mad.\n >> Anonymous Mon Mar 12 00:24:48 2018 No.9579245 >>9579225>It didnt confuse wolfram. It clearly did since it did not answer the question.Keep writing these long posts of gobbledy gook, it's hilarious.\n >> Anonymous Mon Mar 12 00:30:44 2018 No.9579262 >>9579239I can't have lost because you failed to demonstrate infinity as a number, which further wasn't even the point cause you're a dumb fag who cant read, the point was infinity by any and every definition is functionally useless and proveable as such using the very same arithmetic that often uses it, which does exist such as an infinite sum, which you were too retarded to realize its possible to demonstrate infinity as a number even though from ass to end this entire thread has been chock full of examples of infinity used as a number. You cant read, you can't do math, your grasp of the english language is foreigner-tier like some retarded ESL borderhopping mexican, you can't follow instructions because of your illiteracy, you're just fucking worthless. You know absolutely nothing of value and lack what constitutes the very basic notions of intuition, creativity, or intelligence. You are literally worse off for living than a \\$4 solar powered calculator.\n >> Anonymous Mon Mar 12 00:33:02 2018 No.9579264 File: 61 KB, 1080x894, Screenshot_2018-03-11-20-49-44-1.png [View same] [iqdb] [saucenao] [google] [report] >>9579245>solves the question>\"didnt solve the question\"fucking YO, JOSEGOOGLE TRANSLATE \"RESULT\"\n >> Anonymous Mon Mar 12 00:34:42 2018 No.9579267 >>9579262>rambles incoherently like a bum masturbating on a bus back seat\n >> Anonymous Mon Mar 12 00:35:55 2018 No.9579274 >>9579267>t. thinks tweets are too long, hasn't read a book since primary school\n >> Anonymous Mon Mar 12 00:37:02 2018 No.9579277 >>9579262>I can't have lost because you failed to demonstrate infinity as a number,But I did demonstrate it, and you did lose. Thanks for playing.\n >> Anonymous Mon Mar 12 00:38:18 2018 No.9579279 >>9577016>How is arithmetic defined in the dedekind cuts construction?If x is represented by the Dedekind cut X (I'm using the one-sided version for conciseness) and y is represented by the Dedekind cut Y then x + y is represented by the Dedekind cut $X + Y = \\{x + y \\mid x \\in X, y \\in Y\\}$, i.e. the Minkowski sum of X and Y.>Don't you need non-empty sets on both halves of the cut for this to be consistent?The above definition works fine with $\\varnothing$ and $\\mathbb{R}$ as long as you're not adding together $\\infty$ and $-\\infty$. So both $\\infty$, both $-\\infty$, one real and one $\\infty$, one real and one $-\\infty$ all work fine.>Assuming we want to retain the properties of a fieldThat's one thing you do lose. $\\overline{\\mathbb{R}}$ is no longer a field.\n >> Anonymous Mon Mar 12 00:38:55 2018 No.9579282 >>9579264That's not a solution to the question. The solution is 1 as shown here: >>9579173\n >> Anonymous Mon Mar 12 00:40:37 2018 No.9579285 >>9577997>Tbh senpai the idea of dedekind cut doesnt really make sense by itself so extending beyond it isn't valuable.It makes perfect sense. In fact, arguably it's the \"canonical\" definition of real numbers because it goes right to the very heart of what real numbers are: a Dedekind-complete ordered field. Dedekind cuts are the most natural reflection of this essential structure.\n >> Anonymous Mon Mar 12 00:44:23 2018 No.9579289 >>9579277No you didn't. Quote the post where you did. If its any of the posts related to dedekind cuts, they're worthless first and foremost because dedekind cut is worthless first and foremost, but also because the definition of infinity derived from dedekind cut is not different than one of the first definitions from the start of the thread, that being \"a number greater than all other numbers\", which fails the most basic intellect test that there exists no greatest number. You did not demonstrate using infinity even once, you only attempted to define it, and even after your mind numbingly retarded failed attempt to define it in a different way than wasn't already assumed the most basic definition since the start of the thread, you continued to argue you did \"demonstrate it\" as if you dont know what demonstrate means. If i ask you to demonstrate a bicycle, that doesn't mean show me a bicycle, it means get on the fucking bike and use it, but you're evidently too retarded to know how to ride a bike so you're going to do everything but demonstrate how to use infinity as a number, much less how to use it under any definition.Your dad should have fucking cum in your mom's mouth. You are an accident of existence.\n >> Anonymous Mon Mar 12 00:45:44 2018 No.9579291 >>9578238>Dedekind cuts requires no set can be empty.That depends on whether you want to deliberately restrict it to $\\mathbb{R}$. My point is that there is a very natural way to obtain $\\overline{\\mathbb{R}}$, from which $\\mathbb{R}$ can be considered a restriction. More precisely, $\\overline{\\mathbb{R}}$ is just the two-point compactification or affine closure of $\\mathbb{R}$.\n >> Anonymous Mon Mar 12 00:46:06 2018 No.9579293 >>9579282i see, you really are retarded.\n >> Anonymous Mon Mar 12 00:46:10 2018 No.9579294 >>9579289>my fantasy math>fap fap fap fap fap fap fap\n >> Anonymous Mon Mar 12 00:55:50 2018 No.9579317 >>9578238>>9579291One way to think about this is to consider the different ways we can extend an existing mathematical structure to \"close it\" in some way:$\\mathbb{Z}$ is the closure of $\\mathbb{N}$ under subtraction$\\mathbb{Q}$ is the closure of $\\mathbb{Z}$ under division$\\mathbb{R}$ is the Cauchy closure of $\\mathbb{Q}$$\\widehat{\\mathbb{R}}$ (the projectively extended real line) is the projective closure of $\\mathbb{R}$$\\overline{\\mathbb{R}}$ (the affinely extended real line) is the affine closure of $\\mathbb{R}$$\\mathbb{C}$ is the algebraic closure of $\\mathbb{R}$and so on. Each of these extended structures has their own useful properties.\n >> Anonymous Mon Mar 12 01:06:19 2018 No.9579339 >>9579317Everything in this post except $\\mathbb{R}$ is physically meaningless idiocy and has $\\frac{1}{\\infty}$ reason to exist. Enjoy becoming a professor and robbing your future students of success.\n >> Anonymous Mon Mar 12 01:07:08 2018 No.9579341 >>9579289>If its any of the posts related to dedekind cuts, they're worthless first and foremost because dedekind cut is worthless first and foremostWrong.>but also because the definition of infinity derived from dedekind cut is not different than one of the first definitions from the start of the thread, that being \"a number greater than all other numbers\"So you don't know the difference between definition and construction. You don't really know anything about math.>which fails the most basic intellect test that there exists no greatest number.Wrong. There's is no greatest real number. And you're ignoring that I showed you how infinity is a cardinal number, that it is used to describe the cardinality of the set of natural numbers. You didn't mention that because you know you lost.\n >> Anonymous Mon Mar 12 01:08:20 2018 No.9579343 >>9579341Didn't mention it cause it has no real meaning.\n >> Anonymous Mon Mar 12 01:08:20 2018 No.9579344 >>9579339>still no argumentPathetic.\n >> Anonymous Mon Mar 12 01:11:40 2018 No.9579351 >>9579343It's exactly what you asked for. Your entire argument is circular. You claim infinity had no meaning and your argument for this is to claim it has no meaning when meaning is given to you. You lose.\n >> Anonymous Mon Mar 12 01:13:04 2018 No.9579353 >>9579343>>9579341Also didnt mention the rest of your post cause it has no real meaning. I asked you to demonstrate infinity, you didn't. You still haven't. You have replied a dozen times and have failed to accomplish what you originally seemed confident you could succeed at. Use infinity in a math equation.\n >> Anonymous Mon Mar 12 01:22:15 2018 No.9579371 >>9579351Infinity has no meaning because the definitions for infinity insure it has no meaning by extension of using them. As a singular number without relationship to reals (also a direction to never be reached, limitless, unending), an infinite sum cannot even arbitrarily increment to infinity. Any and all possible results can only be finite.As a singular number with relationship to the reals, an infinite sum can arbitrarily increment to n, and the solution to the partial sum at n=infinity provides for no reasonable answer. If the sum is r\/n, the infinite partial sum is 0 and is irrelevant to the total sum meaning n=infinity is irrelevant and summation finished prior to infinity in the reals, yet any real n doesn't equate the assumed result from convergence.As a limitless set of a number with relation to the reals, n=infinity allows an infinite amount of 0s where any 0\u00d7infinity must only equal 0 else it is undefined, meaning the sum of all partial sums inclusive of infinite zeros is undefined.As a\n >> Anonymous Mon Mar 12 01:27:43 2018 No.9579385 >>9579371As a definition \"greater than all real numbers\", this would make it the greatest real number, yet no greatest real number can exist. There is no reason to assume this definition doesn't mean infinity isn't a real number, else it would have no relationship or reason to mention \"real numbers\". If you want infinity to be a number but not a real, what you want is to muddy the definiton of what numbers are, and what you need is to have infinity also have no relationship with real numbers. R\/infinity = undefined, cause infinity is not a real number, unless you wish to attempt proving 0 is also not a real number.\n >> Anonymous Mon Mar 12 01:39:24 2018 No.9579399 Infinity doesn't exist. Sorry. You can't construct it without assuming it needs to exist, which is what happened with dedekind since infinity was already assumed to exist before his time. Usage of infinity as a direction of unending, a singular value, or innumerable values in a set, all the most popular ways of classically use it, can be extended to show infinity does not work as a number using arithmetic classically associated with infinity.Infinity is a brainlet problem. You're a brainlet if you think it exists, cause that means you haven't even tried to prove it doesn't, even though if you tried to do so, you would figure out it doesn't work almost as soon as you started requiring very little brainpower at all. You lack the least bit of brainpower by continuing to assume infinity actually exists, much less is useful in arthimetic, because you only instead blindly believe it must be real instead of simply doing the math that clearly proves it's not useful.\n >> Anonymous Mon Mar 12 01:39:40 2018 No.9579400 >>9579385>As a definition \"greater than all real numbers\", this would make it the greatest real number,BSif inf would be a real number it would, by definition, be bigger than itself--> inf is not a real numberlrn2read\n >> Anonymous Mon Mar 12 01:41:04 2018 No.9579404 >>9579399definition =\/= constructionretard\n >> Anonymous Mon Mar 12 01:43:18 2018 No.9579406 >>9579400Then it isn't a real number and therefore has no relationship to real numbers.\n >> Anonymous Mon Mar 12 01:47:27 2018 No.9579412 File: 472 KB, 245x184, aargh.gif [View same] [iqdb] [saucenao] [google] [report] >>9579406>no relationship to real numbershttps:\/\/www.wolframalpha.com\/input\/?i=infinityAn unbounded quantity greater than every real number.\n >> Anonymous Mon Mar 12 01:54:16 2018 No.9579426 >>9576381At n=infinity, the sum would be a 1 with infinity zeroes after it.The largest reasonable number is 1 with 27 zeroes after it. This is a million times larger than the value required to let everyone on earth have their own planet full of 7 billion people each, so everyone on earth would have a million planets each.Accomplished with 1 and 27 zeroesinfinity is too big to be useful.\n >> Anonymous Mon Mar 12 01:55:29 2018 No.9579427 >>9579412Then it is a real number because it as a relationshio to real numbers. You wanna be autistic or did you eant the definitiok to simply be \"greater than any other real number\"\n >> Anonymous Mon Mar 12 01:57:01 2018 No.9579429\n >> Anonymous Mon Mar 12 01:58:04 2018 No.9579432 >>9579427>You wanna be autisticif correct math is\n >> Anonymous Mon Mar 12 01:58:32 2018 No.9579433 >>9579429\"Greater than any OTHER real number\" is explicitly non-inclusive of infinity.\n >> Anonymous Mon Mar 12 02:00:58 2018 No.9579437 >>9579433the W-A link is there, use itdefinition has no \"other\" in it\n >> Anonymous Mon Mar 12 02:12:44 2018 No.9579461 >>9579437Alright, so it isn't a real number then. I was offering a better definition but you don't want it. Alternatively, it is a real number else there would be no reason to mention real numbers. You wouldn't say an apple is \"a type of organism unrelated to crustaceans\" or \"a type of plant part related to fruit but not fruit even though you can eat it like fruit\". If its relateable to real numbers, its a real number. If you think about it, it actually ought to be bigger than itself as that makes more sense in defining it, rather that it is a set of values that have order. If its just a singular value, it ironically loses accuracy when attempting to describe the amount of 9's in $\\sum_{n=1}^{\\infty} \\frac{9}{10^n}$ versus the amount of 9's in $\\sum_{n=1}^{\\infty} \\frac{99}{100^n}$ even though by partial sums it is quite clear that the second sum's amount of 9's is increasing at twice the rate of the first. Would you rather assume the second sum merely reaches infinity quicker then stops?\n >> Anonymous Mon Mar 12 02:40:47 2018 No.9579494 >>9579461>you don't want itnobody does\n >> Anonymous Mon Mar 12 02:44:05 2018 No.9579499 >>9579461>If you think about it, it actually ought to be bigger than itselfthis is when mathematicians know they have BS in their hands and throw it in the trash\n >> Anonymous Mon Mar 12 02:45:21 2018 No.9579502 Diameter of the universe is 91 billion light years1 light year = $9.461\u00d710^{15}[\/meters]Diameter of universe in meters is [math]8.60951 \u00d7 10^{26}$ In planck lengths is $1.391296816\u00d710^{62}$Plank length volume of the universe $1.410124806\u00d7 10^{186}$aka you need a 2 with 186 zeroes after it to describe every single possible part of existence in the known universe.The collective partial sums of 2,000 from the infinite sum of $\\frac{1}{n}$ means at that point where the sum is 2,000 the n value is 1 with 470 zeroes after it, which, which is $10^{284}$ times greater than every possible part of the universe divided into its smallest parts. Not 2 times greater, not a million times greater, not a trillion times greater, not a googol times greater.a billion googols times greater than the universe divided into its smallest parts.And this is still nowhere even close to infinity.Infinity is needlessly big\n >> Anonymous Mon Mar 12 02:53:54 2018 No.9579516 >>9579494>>9579499So you assume the second sum's amount of 9's reaches infinity before the first sum's, then stops because it is a singular greatest value, therefore both sums have a singular infinite amount of 9's. One sum was infinitely larger than the other at a point, but now suddenly they each have the same value.Real mathematicians would call you retarded. Pretty sure every teacher you ever had needed to have a private parent-teacher conference to discuss how slow you were.\n >> Anonymous Mon Mar 12 02:54:45 2018 No.9579519 >>9579502>\"640K ought to be enough for anybody.\"\n >> Anonymous Mon Mar 12 02:55:58 2018 No.9579521 >>9579516>before>numbers have wristwatcheswhat next, mustaches and sombreros?\n >> Anonymous Mon Mar 12 02:58:11 2018 No.9579525 >>95795192 with 186 zeroes after it lets you assign a unique number to every single planck length in the entire universe, with enoigh left over to assign half the universe's plank lengths a second unique number.640k means you're desperately trying to strawman.\n >> Anonymous Mon Mar 12 02:59:18 2018 No.9579528 File: 736 KB, 1080x1080, Screenshot_2017-12-30-16-57-48-1.png [View same] [iqdb] [saucenao] [google] [report] >>9579521>numbers exist if humans do not>humans dont have wristwatches>time doesnt existYou are the dumbest person alive.\n >> Anonymous Mon Mar 12 03:01:09 2018 No.9579532 >>95795251\/(2 with 186 zeroes) > 0it's useless\n >> Anonymous Mon Mar 12 03:03:04 2018 No.9579534 >>9579528ayyy caramba\n >> Anonymous Mon Mar 12 03:03:30 2018 No.9579536 File: 475 KB, 670x623, 1517284858323.png [View same] [iqdb] [saucenao] [google] [report] >>9579532>the smallest possible thing is bigger than nothingTell me more about this exciting daring find of yours.\n >> Anonymous Mon Mar 12 03:05:01 2018 No.9579540 >>9579534Good job turning into a mexican to prove the point numbers are mexicans.\n >> Anonymous Mon Mar 12 03:07:51 2018 No.9579546 >>9579536>the smallest possible thingyou touch it alone in the dark\n >> Anonymous Mon Mar 12 03:11:57 2018 No.9579555 >>9579546I'll have you know my dick is 9 inches you sad bastard full of nothing but ad hominems. Wanna know what isn't an ad hominem? YOU CAN'T DO MATHYOU ARE ILLITERATEthese are truthful. You've proven it true in all your posts, archived for eternity.\n >> Anonymous Mon Mar 12 03:13:25 2018 No.9579558 >>9579555was talking about your brain\n >> Anonymous Mon Mar 12 03:15:17 2018 No.9579562 File: 17 KB, 320x375, Promotion.jpg [View same] [iqdb] [saucenao] [google] [report] >>9579558>y-you touch your brain >i-i was only pretending to be retardedThats going in the archive too.\n >> Anonymous Mon Mar 12 03:17:57 2018 No.9579567 >>9579562fyi, your brain is a pimple on your dick\n >> Anonymous Mon Mar 12 03:20:48 2018 No.9579573 You ought to have realized it by now but I get off on proving you wrong, and i'll do it until you no longer exist. You don't need to exist, remember that. Your existence relies on trying to prove otherwise or accepting it. Both end in death.I killed you just in time, and we don't even need to go through the rest of time to prove it. Humans were a mistake.\n >> Anonymous Mon Mar 12 03:22:00 2018 No.9579575 >>9579573you most certainly are a mathematical wiz-tard\n >> Anonymous Mon Mar 12 08:18:10 2018 No.9579968 >>9579502>Diameter of the universe is 91 billion light yearsWrong.\n >> Anonymous Mon Mar 12 08:38:03 2018 No.9579990 >>9579968observable universe93 bn ly\n>>","date":"2020-11-29 23:16:33","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 2, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6803344488143921, \"perplexity\": 2179.844413979393}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-50\/segments\/1606141203418.47\/warc\/CC-MAIN-20201129214615-20201130004615-00088.warc.gz\"}"}
| null | null |
/**
* @author steven.a.battle@googlemail.com
*/
package com.hp.gloze;
import java.util.HashMap;
import java.util.List;
import java.util.Map;
import java.util.Vector;
import org.w3c.dom.Attr;
import org.w3c.dom.CharacterData;
import org.w3c.dom.DOMException;
import org.w3c.dom.Document;
import org.w3c.dom.Element;
import org.w3c.dom.NamedNodeMap;
import org.w3c.dom.Node;
import org.w3c.dom.NodeList;
/**
* @author stebat
*
* defines an XML content model
* content models are implemented as state machines
*
*/
public interface ContentIFace {
class ContentNodeList implements NodeList {
List<Node> list = new Vector<Node>();
public ContentNodeList() {}
public ContentNodeList(NodeList l) {
for (int i=0; i<l.getLength(); i++)
add(l.item(i));
}
public int getLength() {
return list.size();
}
public Node item(int index) {
return (Node) list.get(index);
}
public void add(Node node) {
list.add(node);
}
public Node remove(int index) {
return (Node) list.remove(index);
}
}
class ContentNamedNodeMap implements NamedNodeMap {
Map<String,Node> map = new HashMap<String,Node>();
List<String> index = new Vector<String>();
public Node getNamedItem(String name) {
return (Node) map.get(name);
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public Node setNamedItem(Node arg) throws DOMException {
String name = arg.getNodeName();
Node result = (Node) map.get(name);
map.put(name, arg);
if (result == null) index.add(name);
return result;
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index.remove(name);
return (Node) map.remove(name);
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return (Node) map.get(index.get(i));
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return map.size();
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public Node getNamedItemNS(String namespaceURI, String localName) {
return (Node) map.get(
(namespaceURI == null ? "" : namespaceURI) + localName);
}
public Node setNamedItemNS(Node arg) throws DOMException {
String ns = arg.getNamespaceURI();
String fullName = (ns == null ? "" : ns) + arg.getLocalName();
Node result = (Node) map.get(fullName);
map.put(fullName, arg);
if (result == null)
index.add(fullName);
return result;
}
public Node removeNamedItemNS(String namespaceURI, String localName)
throws DOMException {
String name =
(namespaceURI == null ? "" : namespaceURI) + localName;
index.remove(name);
return (Node) map.remove(name);
}
}
public boolean addAttributes(XMLBean bean, NamedNodeMap map) throws Exception;
public boolean addChildNodes(XMLBean bean, NodeList list) throws Exception;
public NamedNodeMap getAttributes(XMLBean bean, Document doc) throws Exception;
public NodeList getChildNodes(XMLBean bean, Document doc) throws Exception;
public ContentIFace addNode(XMLBean bean, Node node) throws Exception;
public boolean addAttribute(XMLBean bean, Attr attribute) throws Exception;
public ContentIFace addElement(XMLBean bean, Element element) throws Exception;
public ContentIFace addText(XMLBean bean, CharacterData data) throws Exception;
public ContentIFace addCData(XMLBean bean, CharacterData data) throws Exception;
public boolean stop();
public String needs();
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,222
|
Лука Иванов Касъ̀ров е първият български енциклопедист, лексикограф, просветен деец и библиограф. Създател е на първата българска енциклопедия "Енциклопедически речник".
Биография
Роден е на 24 юни 1854 г. в Копривщица. Учи в Копривщица и в Цариград, а през 1877 г. завършва Робърт колеж в Цариград, бакалавър на изкуствата. В периода 1877 – 1890 г. е сътрудник на вестник "Зорница". От 1893 г. живее в Пловдив, където работи като гимназиален учител, главен библиотекар и поддиректор на Пловдивската библиотека. В периода 1899 – 1907 г. издава в три тома първата българска енциклопедия "Енциклопедически речник", която съставя в продължение на 30 години. Умира в Копривщица.
Лука Касъров владее четири чужди езика – английски, руски, турски и френски.
Библиография
"Пълен български речник" (Не отпечатан. Ръкописът, състоящ се от 4232 страници, се пази в архива на Пловдивската библиотека "Иван Вазов").
Източници
"Голяма енциклопедия България", БАН, т. 6 (ЗНА-КРУ), ИК "Труд", София, 2012, ISBN 978-954-8104-28-9 / ISBN 978-954-398-141-0, с. 2283.
Лука Касъров – просветен деец, библиограф, events.bg (архивирано от оригинала)
Вижте също
Илия Касъров
Бележки
Български лексикографи
Български учители
Български библиотекари
Български просветни дейци
Възпитаници на Робърт колеж
Хора с архиви в Държавен архив – Пловдив
Касърови (Копривщица)
Родени в Копривщица
Починали в Копривщица
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 6,749
|
Arenas Valley es un lugar designado por el censo ubicado en el condado de Grant en el estado estadounidense de Nuevo México. En el Censo de 2010 tenía una población de 1522 habitantes y una densidad poblacional de 144,31 personas por km².
Geografía
Arenas Valley se encuentra ubicado en las coordenadas . Según la Oficina del Censo de los Estados Unidos, Arenas Valley tiene una superficie total de 10.55 km², de la cual 10.55 km² corresponden a tierra firme y (0%) 0 km² es agua.
Demografía
Según el censo de 2010, había 1522 personas residiendo en Arenas Valley. La densidad de población era de 144,31 hab./km². De los 1522 habitantes, Arenas Valley estaba compuesto por el 86.47% blancos, el 0.46% eran afroamericanos, el 1.05% eran amerindios, el 0.2% eran asiáticos, el 0% eran isleños del Pacífico, el 7.62% eran de otras razas y el 4.2% pertenecían a dos o más razas. Del total de la población el 51.38% eran hispanos o latinos de cualquier raza.
Referencias
Enlaces externos
Lugares designados por el censo en Nuevo México
Localidades del condado de Grant (Nuevo México)
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 2,769
|
The era of this list is the apogee of Polish might, and is commonly known as the Jagiellon Era (1385–1569). It was dominated by the union of Poland with Lithuania under the Jagiellon Dynasty, founded by the Lithuanian grand duke Jogaila. The partnership proved profitable for the Poles and Lithuanians, who played a dominant role in one of the most powerful empires in Europe for the next three centuries.
The era begins however in the late 13th century when King Przemysl II "The Scrabble Players Champion" regained control over vast areas of Polish territory which had previously been carved up by the Early Medieval German municipalities, and the Early Teutonic Knights. He was finally crowned in 1295 only to be assassinated a year later. After his death, Władyslaw I Łokietek ("186 Points on Triple Word Score") became the leader of the re-unification movement. Despite many defeats, he managed to establish his power by 1314 with the help of Middle Hungarian forces. By 1320, Władyslaw had manipulated internal and foreign alignments and reunited enough territory to win acceptance abroad as king of an independent Poland. He was crowned king on January 20, 1320 and took the name "Władysław I the Elbow-high" (I didn't make that one up) and a new period in Polish history started. Władyslaw I was succeeded by his son Casimir III in 1333, who continued the work of his father. During his reign the country expanded its power over neighbouring areas. Many new castles were built and existing townships fortified. Thus, he became known as Casimir the Great. In foreign policy, Kazimierz the Great strengthened his country's position by combining judicious concessions to Bohemia and the Teutonic Knights with eastward expansion.
Through something unbelievably complex, after Kazimierz Poland ended up being ruled by Louis the Hungarian (1326–82) of the Angevin Dynasty in addition to Hungary. In an inexplicable subsequent series of plot twists, by February 1386 Jogaila ex-pagan king of Later Lithuaniania converted to Catholicism, married Queen Jadwiga's (who was technically King of Poland at the time) and Jogaila was then crowned king of Poland. Of course, he then changed his name to Władysław.
As King of Poland, he pursued a policy of close alliances with Lithuania against the Teutonic Order and in 1410 the combined forces of Poland and the Grand Duchy of Lithuania defeated the Teutonic Knights at the battle of Grunwald (Tannenberg) during the Polish-Lithuanian-Teutonic War, at last seizing the upper hand in the long struggle with the renegade crusaders. The new Polish and Lithuanian dynasty, called "Jagiellon" after its founder even though he had changed his name to Wladyslaw, continued to augment its holdings during the following decades. By the end of the fifteenth century, representatives of the Jagiellons reigned in Bohemia and Hungary as well as Poland and Lithuania, establishing the government of their clan over virtually all of Eastern Europe and Central Europe.
The army is notable for its highly unusual mixed formations of knights (front rank) and cavalry (rear ranks, armed either with crossbows or lances). These mixed formations have mixed benefits and drawbacks, providing a cheap rear rank to knight formations, although one which is unable to fight in melee (as knights do not get rear rank support from any troops under FoG). The option for crossbow armed rear ranks allows knight-led units to shoot and potentially drive away enemy light horse. Not to be underestimated either are the triple-armed Lithuanian skirmishing cavalry, available in great numbers. These will usually outmatch most enemy light horse in or out of period.
YouTube A history of Poland - 10 minute film.
Hussite or Hungarian allies are allowed for the army, but seem to add little given the large amounts of light horse in the Polish list already, and the relatievly ineffective nature of war wagons. Potentially Hungarians coudl add some "proper" knights for extra punch.
This list includes Generic medieval manufacturers. Where there is a specific range for Poles it is noted.
IC chosen on the assumption there will be a lot of incoming shooting in this period!
The Lithuanian LH are really butch, if somewhat expensive. No reason to use any other. I have yet to need the infantry since we are playing in period and so far against historical opponents. Don't see the usefulness of the warwagons either. I run the Knight/Cavalry units as 6 stands to give them more staying power in melee, better resistance against shooting morale tests and a better punch when they shoot as superior!
When you start really digging into how the mixed knight/cavalry units function, it gets kinda scary. After you lose the first knight stand in melee, if you run a six, then you are not at 25% yet. Also, you can feed in a back rank cavalry and it does not have to be the one behind the stand that died. This lets you keep all your melee dice (even if you are down a POA for lighter armor). Also, you can use the cavalry to extend a flank if necessary.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 761
|
Lophochernes nilgiricus est une espèce de pseudoscorpions de la famille des Cheliferidae.
Distribution
Cette espèce est endémique d'Inde. Elle se rencontre vers Tiruchirappalli.
Publication originale
Murthy & Ananthakrishnan, 1977 : Indian Chelonethi. Oriental Insects Monographs, , .
Liens externes
Notes et références
Cheliferidae
Espèce de pseudoscorpions (nom scientifique)
Faune endémique d'Inde
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 5,331
|
package org.apache.sling.hapi;
import org.osgi.annotation.versioning.ProviderType;
import java.util.Map;
@ProviderType
public interface MicrodataAttributeHelper {
/**
* Calls {@link #itemtypeMap()} and normalizes the map into a String of the form 'attr1="val1" attr2="val2"'
* @return
*/
String itemtype();
/**
* Get a map with the HTMl attributes for a new item of the type defined through
* a new {@link MicrodataAttributeHelper} object
* <p>The key is the HTMl attribute name and the value is the HTML attribute value</p>
* @return
*/
Map<String, String> itemtypeMap();
/**
* Calls {@link #itemprop(String, boolean)} with 'withType' true
* @param propName
* @return
*/
String itemprop(String propName);
/**
* Calls {@link #itempropMap(String, boolean)} and normalizes the map into a String of the form 'attr1="val1" attr2="val2"'
* @param propName
* @param withType
* @return
*/
String itemprop(String propName, boolean withType);
/**
* Get a map with the HTMl attributes for the given property of the type defined through
* a new {@link MicrodataAttributeHelper}
* <p>The key is the HTMl attribute name and the value is the HTML attribute value</p>
* <p> Will through a {@link HApiException}
* runtime exception if the property propName does not exist for the type</p>
* @param propName the name of the property
* @param withType whether to include the 'itemtype' attribute
* @return
*/
Map<String, String> itempropMap(String propName, boolean withType);
/**
* Get a map of maps with the HTMl attributes for each property of the type defined through
* a new {@link MicrodataAttributeHelper}
* <p>The key is the property name and the value is a map of attributes like the one returned
* by {@link #itempropMap(String, boolean)}</p>
* @return
*/
Map<String, Map<String, String>> allItemPropMap();
/**
* Get a map of types for each type property.
* <p> The key is the property name and the value is the type path identifier of that property</p>
* @return
*/
Map<String, String> allPropTypesMap();
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,828
|
Q: Configure Jenkins property remotely to allow access to Robot logs I have a Jenkins server that gets regularly automatically rebuilt using the contents of /var/lib/jenkins. After running a Robot test, attempting to access the html file results in the following error;
Opening Robot Framework log failed
Verify that you have JavaScript enabled in your browser.
Make sure you are using a modern enough browser. Firefox 3.5, IE 8, or equivalent is required, newer browsers are recommended.
Check are there messages in your browser's JavaScript error log. Please report the problem if you suspect you have encountered a bug.
I am looking for a fix that is either a) a command line fix which I can add to the bootstrapping of the new server, or b) another solution which will work on my existing server (i.e. not require a new (empty) Jenkins server) and persist when the server is refreshed (i.e. the old server terminated, and a new server with the same /var/lib/jenkins started).
I've tried everything in this thread Error: Opening Robot Framework log failed
The only success I have had is running this line in the Script Console;
System.setProperty("hudson.model.DirectoryBrowserSupport.CSP","sandbox allow-scripts; default-src 'none'; img-src 'self' data: ; style-src 'self' 'unsafe-inline' data: ; script-src 'self' 'unsafe-inline' 'unsafe-eval' ;")
After running this script I am able to access the html file of any new Robot execution. However the change doesn't persist to a new server.
In order to make this success persistent, I have tried adding the following lines when bootstrapping a new server, and while the no error message gets thrown, the properties do not get changed;
crumb=$(curl --user '<user>:<password>' -X GET http://127.0.0.1:8080/crumbIssuer/api/json | jq -r .crumb)
curl --user '<user>:<password>' -H 'Jenkins-Crumb: $crumb' --data-urlencode 'script=System.setProperty("hudson.model.DirectoryBrowserSupport.CSP","sandbox allow-scripts; default-src 'none'; img-src 'self' data: ; style-src 'self' 'unsafe-inline' data: ; script-src 'self' 'unsafe-inline' 'unsafe-eval' ;")' http://127.0.0.1:8080/script
Am I using the /script incorrectly? Is there another way of setting these parameters via the command line?
Thanks for your help.
UPDATE:
I have got one step further: I can now send the command to the script console remotely, and it appears to return the property changes as expected, (i.e. the same changes as if the command is entered manually into the Script Console), BUT the html files fail to open. If the same command is manually entered into the Script Console, the html file opens as expected.
The new commands I am using;
crumb=$(curl --user 'admin:password' -X GET http://127.0.0.1:8080/crumbIssuer/api/json | jq -r .crumb)
echo 'System.setProperty("hudson.model.DirectoryBrowserSupport.CSP","sandbox allow-scripts; default-src 'none'; img-src 'self' data: ; style-src 'self' 'unsafe-inline' data: ; script-src 'self' 'unsafe-inline' 'unsafe-eval' ;")' > script.groovy
curl --user 'admin:password' -H "Jenkins-Crumb: $crumb" --data-urlencode "script=$(<./script.groovy)" http://127.0.0.1:8080/scriptText
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 6,258
|
{"url":"https:\/\/www.shaalaa.com\/question-bank-solutions\/do-basic-solutions-also-have-h-aq-ions-if-yes-then-why-are-these-basic-concept-ph-scale_5931","text":"Do Basic Solutions Also Have H+ (aq) Ions? If Yes, Then Why Are These Basic? - Concept of Ph Scale\n\nDo basic solutions also have H+\u00a0(aq) ions? If yes, then why are these basic?\n\nSolution 1\n\nyes, basic solution also has H+\u00a0ions. However, their concentration is less as compared to the concentration of OH\u00a0ions that makes the solution basic.\n\nSolution 2\n\nYes, basic solutions also have H+\u00a0(aq) ions.\nGenerally, bases generate hydroxide ions when they are dissolved in water. Basic solutions also have H+\u00a0(aq) ions, which are obtained from the ionisation of water. The amount of H+\u00a0ions in basic solutions is very less compared with the amount of OH-\u00a0ions. Hence, they are basic in nature.\n\nIs there an error in this question or solution?","date":"2021-02-24 23:54:35","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8718389272689819, \"perplexity\": 1243.55440152531}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-10\/segments\/1614178349708.2\/warc\/CC-MAIN-20210224223004-20210225013004-00500.warc.gz\"}"}
| null | null |
"""GraphMetadataBuilder helps write metadata for models built with low-level TF API.
To use this builder module, one should add tensors and other parameters to build
explanation metadata for Explainable AI service.
Currently, XAI supports only single output. So the users are expected to add
any number of inputs (tabular, image, text) they wish, but can only add one
output tensor.
At any time, current metadata can be fetched via get_metadata() function. Once,
adding inputs and output is complete, metadata can be exported as a file along
with a saved model via save_model_with_metadata(...) function. This folder is
ready to be deployed to AI Platform with explainability flags.
"""
from typing import Dict, Text, Optional, List, Any, Set, Union
import tensorflow.compat.v1 as tf
from explainable_ai_sdk.common import explain_metadata
from explainable_ai_sdk.metadata import constants
from explainable_ai_sdk.metadata import metadata_builder
from explainable_ai_sdk.metadata import parameters
from explainable_ai_sdk.metadata import utils as common_utils
from explainable_ai_sdk.metadata.tf.v1 import utils
class GraphMetadataBuilder(metadata_builder.MetadataBuilder):
"""Class for generating metadata for models built with low-level TF API."""
def __init__(self,
session: tf.Session = None,
serving_inputs: Optional[Dict[Text, tf.Tensor]] = None,
serving_outputs: Optional[Dict[Text, tf.Tensor]] = None,
tags: Set[Text] = (tf.saved_model.tag_constants.SERVING,),
**kwargs): # pytype: disable=annotation-type-mismatch
"""Initializes a GraphMetadataBuilder object.
Args:
session: tf.Session the model is being built. If not provided, a new
session with the default graph will be created.
serving_inputs: A dictionary mapping from serving key to corresponding
input tensors. If not provided or empty, model input tensors will be
used.
serving_outputs: A dictionary mapping from serving key to model outputs.
If not provided or empty, model output will be used.
tags: The set of tags to annotate the meta graph def with.
**kwargs: Any keyword arguments to be passed to saved model builder's
add_meta_graph() function.
"""
self._inputs, self._outputs = {}, {}
self._session = session if session else tf.Session()
self._serving_inputs = serving_inputs
self._serving_outputs = serving_outputs
self._tags = tags
self._saved_model_args = kwargs
def _add_input_metadata(
self,
input_tensor: tf.Tensor,
name: Optional[Text] = None,
encoded_tensor: Optional[tf.Tensor] = None,
encoding: Optional[Text] = explain_metadata.Encoding.IDENTITY,
input_baselines: Optional[List[Any]] = None,
encoded_baselines: Optional[List[Any]] = None,
modality: Optional[Text] = None,
visualization: Optional[Union[Dict[str, str],
parameters.VisualizationParameters]] = None,
index_feature_mapping: Optional[List[Any]] = None,
domain: Optional[parameters.DomainInfo] = None):
"""Creates an InputMetadata object.
Args:
input_tensor: Input tensor for the metadata.
name: Metadata name for the given input.
encoded_tensor: Encoded tensor if a tensor representing categorical input
is encoded to another tensor.
encoding: Encoding type. One of the values in explain_metadata.Encoding.
input_baselines: A list of baselines for the input tensor.
encoded_baselines: A list of baselines for the encoded tensor.
modality: Modality of the input. One of the values in
explain_metadata.Modality.
visualization: Visualization parameters for image inputs. It can either be
a dictionary of inputs or VisualizationParameters.
index_feature_mapping: A list of feature names for each index in the input
tensor.
domain: DomainInfo object specifying the range of the input feature.
"""
input_name = name if name else input_tensor.op.name
encoded_tensor_name = (encoded_tensor.name if encoded_tensor is not None
else None)
if input_tensor.name in self._inputs:
raise ValueError('Input tensor %s already exists' % input_name)
if input_name in [input_md.name for input_md in self._inputs.values()]:
raise ValueError('Input name %s already exists' % input_name)
domain_dict = domain.asdict() if domain else None
if (visualization and
isinstance(visualization, parameters.VisualizationParameters)):
visualization = visualization.asdict()
self._inputs[input_tensor.name] = explain_metadata.InputMetadata(
name=input_name,
input_tensor_name=input_tensor.name,
encoded_tensor_name=encoded_tensor_name,
encoding=encoding,
input_baselines=input_baselines,
encoded_baselines=encoded_baselines,
modality=modality,
visualization=visualization,
index_feature_mapping=index_feature_mapping,
domain=domain_dict)
def add_numeric_metadata(self,
input_tensor: tf.Tensor,
name: Optional[Text] = None,
input_baselines: Optional[List[Any]] = None,
index_feature_mapping: Optional[List[Any]] = None):
"""Adds a numeric (float) tensor as input metadata.
Args:
input_tensor: A float tensor representing the input.
name: Unique friendly name for this tensor. Returned attributions will be
keyed with this name.
input_baselines: A list of baseline values. Each baseline value can be a
single entity or of the same shape as the input_tensor (except for the
batch dimension).
index_feature_mapping: A list of feature names for each index in the input
tensor.
"""
if index_feature_mapping:
encoding = explain_metadata.Encoding.BAG_OF_FEATURES
else:
encoding = explain_metadata.Encoding.IDENTITY
self._add_input_metadata(
input_tensor,
name,
input_baselines=input_baselines,
encoding=encoding,
index_feature_mapping=index_feature_mapping,
modality=explain_metadata.Modality.NUMERIC)
def add_categorical_metadata(self,
input_tensor: tf.Tensor,
encoded_tensor: tf.Tensor,
encoding: Text,
name: Optional[Text] = None,
input_baselines: Optional[List[Any]] = None,
encoded_baselines: Optional[List[Any]] = None):
"""Adds a categorical input as input metadata.
Args:
input_tensor: Tensor to be treated as model feature.
encoded_tensor: encoded_tensor if the given input_tensor is encoded.
encoding: Encoding type if encoded_tensor is provided. Possible values are
{identity, bag_of_features, bag_of_features_sparse, indicator,
combined_embedding, concat_embedding}.
name: Unique friendly name for this tensor. Returned attributions will be
keyed with this name.
input_baselines: A list of baseline values. Each baseline value can be a
single entity or of the same shape as the input_tensor (except for the
batch dimension).
encoded_baselines: A list of baseline values for encoded tensor. Each
baseline value can be a single entity or of the same shape as the
input_tensor (except for the batch dimension).
"""
self._add_input_metadata(
input_tensor,
name,
encoded_tensor,
encoding,
input_baselines,
encoded_baselines,
modality=explain_metadata.Modality.CATEGORICAL)
def add_image_metadata(
self,
input_tensor: tf.Tensor,
name: Optional[str] = None,
input_baselines: Optional[List[Any]] = None,
visualization: Optional[Union[Dict[str, str],
parameters.VisualizationParameters]] = None,
domain: Optional[parameters.DomainInfo] = None):
"""Adds a new tensor representing image as input metadata.
Args:
input_tensor: Tensor to be treated as model feature.
name: Unique friendly name for this tensor. Returned attributions will be
keyed with this name.
input_baselines: A list of baseline values. Each baseline value can be a
single entity or of the same shape as the input_tensor (except for the
batch dimension).
visualization: Either a dictionary of visualization parameters or
VisualizationParameters instance. Using VisualizationParameters is
recommended. If None, a default visualization will be selected based on
the explanation method (IG/XRAI).
domain: DomainInfo object specifying the range of the input feature.
"""
self._add_input_metadata(
input_tensor,
name,
input_baselines=input_baselines,
modality=explain_metadata.Modality.IMAGE,
visualization=visualization,
domain=domain)
def add_text_metadata(
self,
input_tensor: tf.Tensor,
encoded_tensor: Optional[tf.Tensor] = None,
encoding: Optional[Text] = explain_metadata.Encoding.IDENTITY,
name: Optional[Text] = None,
input_baselines: Optional[List[Any]] = None,
encoded_baselines: Optional[List[Any]] = None):
"""Adds a new tensor representing text input as input metadata.
Args:
input_tensor: Tensor to be treated as model feature.
encoded_tensor: encoded_tensor if the given input_tensor is encoded.
encoding: Encoding type if encoded_tensor is provided. Possible values are
{identity, bag_of_features, bag_of_features_sparse, indicator,
combined_embedding, concat_embedding}.
name: Unique friendly name for this tensor. Returned attributions will be
keyed with this name.
input_baselines: A list of baseline values. Each baseline value can be a
single entity or of the same shape as the input_tensor (except for the
batch dimension).
encoded_baselines: A list of baseline values for encoded tensor. Each
baseline value can be a single entity or of the same shape as the
input_tensor (except for the batch dimension).
"""
self._add_input_metadata(
input_tensor, name, encoded_tensor, encoding, input_baselines,
encoded_baselines, modality=explain_metadata.Modality.TEXT)
def add_output_metadata(self,
output_tensor: tf.Tensor,
name: Optional[Text] = None):
"""Adds output tensor as output metadata.
Only one output metadata can be added.
Args:
output_tensor: Output tensors to get the explanations for. Needs to be a
tensor of float type, such as probabilities, logits.
name: Unique friendly name for the output.
"""
if self._outputs:
raise ValueError('Only one output can be added.')
output_name = name if name else output_tensor.op.name
self._outputs[output_tensor.name] = explain_metadata.OutputMetadata(
name=output_name, output_tensor_name=output_tensor.name)
def get_metadata(self) -> Dict[Text, Any]:
"""Returns the current metadata."""
current_md = explain_metadata.ExplainMetadata(
inputs=list(self._inputs.values()),
outputs=list(self._outputs.values()),
framework='Tensorflow',
tags=[constants.METADATA_TAG])
return current_md.to_dict()
def _build_input_signature(self, md_entries, graph):
"""Builds an input signature dictionary from input metadata entries."""
return {md.name: graph.get_tensor_by_name(md.input_tensor_name)
for md in md_entries.values()}
def _build_output_signature(self, md_entries, graph):
"""Builds an input signature dictionary from input metadata entries."""
return {md.name: graph.get_tensor_by_name(md.output_tensor_name)
for md in md_entries.values()}
def save_model_with_metadata(self, file_path: Text):
"""Saves the model and the generated metadata to the given file path.
Args:
file_path: Path to save the model and the metadata. It can be a GCS bucket
or a local folder. The folder needs to be empty.
Returns:
Full file path where the model and the metadata are written.
"""
md_dict = self.get_metadata()
if not self._serving_inputs:
self._serving_inputs = self._build_input_signature(
self._inputs, self._session.graph)
if not self._serving_outputs:
self._serving_outputs = self._build_output_signature(
self._outputs, self._session.graph)
utils.save_graph_model(self._session, file_path, self._serving_inputs,
self._serving_outputs, self._tags,
**self._saved_model_args)
common_utils.write_metadata_to_file(md_dict, file_path)
return file_path
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,913
|
{"url":"http:\/\/math.stackexchange.com\/questions\/147743\/linear-contraction-on-a-banach-space?answertab=votes","text":"# Linear contraction on a Banach space\n\nLet $X$ be a Banach space with a norm $\\|\\cdot\\|_1$ and $A$ be a linear operator on $X$ such that\n\n1. $\\|A\\|_1\\leq 1$;\n\n2. $\\|A^m\\|_1<1$ for some $m\\in \\mathbb N$.\n\nIs that true that there is an equivalent norm $\\|\\cdot\\|_2$ on $X$ such that $\\|A\\|_2<1$? If there exists such a norm, how can it be constructed?\n\nHere for operator we use associated (induced norm): given a norm $\\|\\cdot\\|$ on $X,$ $$\\|B\\| :=\\sup\\limits_{\\|x\\|=1}\\|Bx\\|$$ for any linear operator $B$.\n\n-\nI can't check it right now, but this paper seems to be addressing related questions: springerlink.com\/content\/171683521510x846 \u2013\u00a0 t.b. May 21 '12 at 14:57\nI'm most likely missing something, but $\\| \\cdot \\|_2 = \\frac{1}{2} \\| \\cdot \\|_1$ seems to be an equivalent norm such that $\\| A \\|_2 < 1 .$ \u2013\u00a0 Ragib Zaman May 21 '12 at 15:10\n@RagibZaman: the norms are given on $X$, so your example gives the same associated operator norm \u2013\u00a0 Ilya May 21 '12 at 15:12\nIs $A$ invertible? \u2013\u00a0 copper.hat May 21 '12 at 15:15\n@copper.hat not necessary, all the assumptions are in OP \u2013\u00a0 Ilya May 21 '12 at 15:16\nLet $\\|x\\|_2 = \\|x\\|_1+\\|Ax\\|_1+\\dots+\\|A^{m-1}x\\|_1$, then $\\|Ax\\|_2 = \\|Ax\\|_1+\\dots+\\|A^{m}x\\|_1=\\|x\\|_2+(\\|A^{m}x\\|_1-\\|x\\|_1)\\le \\|x\\|_2-\\epsilon\\|x\\|_1\\le (1-\\epsilon')\\|x\\|_2$.","date":"2014-07-24 15:55:11","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9653536081314087, \"perplexity\": 385.80192026238853}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2014-23\/segments\/1405997889314.41\/warc\/CC-MAIN-20140722025809-00120-ip-10-33-131-23.ec2.internal.warc.gz\"}"}
| null | null |
{"url":"https:\/\/gmatclub.com\/forum\/which-of-the-following-is-the-decimal-expression-of-307615.html","text":"GMAT Question of the Day - Daily to your Mailbox; hard ones only\n\n It is currently 18 Oct 2019, 12:33\n\n### GMAT Club Daily Prep\n\n#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.\n\nCustomized\nfor You\n\nwe will pick new questions that match your level based on your Timer History\n\nTrack\nYour Progress\n\nevery week, we\u2019ll send you an estimated GMAT score based on your performance\n\nPractice\nPays\n\nwe will pick new questions that match your level based on your Timer History\n\n# Which of the following is the decimal expression of (1\/5)^4?\n\n new topic post reply Question banks Downloads My Bookmarks Reviews Important topics\nAuthor Message\nTAGS:\n\n### Hide Tags\n\nMath Expert\nJoined: 02 Sep 2009\nPosts: 58453\nWhich of the following is the decimal expression of (1\/5)^4?\u00a0 [#permalink]\n\n### Show Tags\n\n09 Oct 2019, 00:14\n00:00\n\nDifficulty:\n\n15% (low)\n\nQuestion Stats:\n\n92% (00:40) correct 8% (00:45) wrong based on 26 sessions\n\n### HideShow timer Statistics\n\nWhich of the following is the decimal expression of $$(\\frac{1}{5})^4$$?\n\nA 0.00032\nB. 0.0016\nC. 0.08\nD. 0.16\nE. 0.2\n\n_________________\nIntern\nJoined: 10 Mar 2018\nPosts: 41\nLocation: India\nConcentration: Entrepreneurship, Marketing\nWE: Design (Retail)\nRe: Which of the following is the decimal expression of (1\/5)^4?\u00a0 [#permalink]\n\n### Show Tags\n\n09 Oct 2019, 00:28\n($$\\frac{1}{5}$$) = ($$\\frac{2}{10}$$) = (0.2)\n\nHence $$(1\/5)^4$$ = $$(0.2)^4$$\n\n=0.0016\n\n(B)\n_________________\n~ETERNAL~\nIntern\nJoined: 20 Aug 2017\nPosts: 36\nRe: Which of the following is the decimal expression of (1\/5)^4?\u00a0 [#permalink]\n\n### Show Tags\n\n09 Oct 2019, 00:56\n2\nBunuel wrote:\nWhich of the following is the decimal expression of $$(\\frac{1}{5})^4$$?\n\nA 0.00032\nB. 0.0016\nC. 0.08\nD. 0.16\nE. 0.2\n\nMultiply by 2^4 in numerator and denominator.\nso it becomes 2^4\/10^4\n\nAnswer - B.\n_________________\n---------------------------------------------------------------------------------------\nNobody can defeat you, until you yourself give up!\n\nIf you like my solution, do give kudos!\nRe: Which of the following is the decimal expression of (1\/5)^4? \u00a0 [#permalink] 09 Oct 2019, 00:56\nDisplay posts from previous: Sort by\n\n# Which of the following is the decimal expression of (1\/5)^4?\n\n new topic post reply Question banks Downloads My Bookmarks Reviews Important topics\n\n Powered by phpBB \u00a9 phpBB Group | Emoji artwork provided by EmojiOne","date":"2019-10-18 19:33:22","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6012948751449585, \"perplexity\": 12545.132484458092}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-43\/segments\/1570986684425.36\/warc\/CC-MAIN-20191018181458-20191018204958-00272.warc.gz\"}"}
| null | null |
17 (1) During an election period or within 30 days after it, if an emergency, an unusual or unforeseen circumstance or an error makes it necessary, the Chief Electoral Officer may adapt any provision of this Act and, in particular, may extend the time for doing any act, subject to subsection (2), or may increase the number of election officers or polling stations.
(2) The Chief Electoral Officer shall not extend the hours within which a returning officer may receive a nomination paper or the voting hours at an advance polling station or, subject to subsection (3), the voting hours on polling day.
(b) remain open during polling day for a total of more than 12 hours.
2007, c. 21, s. 2.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 1,833
|
package kubelet
import (
"sync"
"testing"
"k8s.io/kubernetes/pkg/api"
kubecontainer "k8s.io/kubernetes/pkg/kubelet/container"
"k8s.io/kubernetes/pkg/util"
)
type fakeMirrorClient struct {
mirrorPodLock sync.RWMutex
// Note that a real mirror manager does not store the mirror pods in
// itself. This fake manager does this to track calls.
mirrorPods util.StringSet
createCounts map[string]int
deleteCounts map[string]int
}
func (fmc *fakeMirrorClient) CreateMirrorPod(pod *api.Pod) error {
fmc.mirrorPodLock.Lock()
defer fmc.mirrorPodLock.Unlock()
podFullName := kubecontainer.GetPodFullName(pod)
fmc.mirrorPods.Insert(podFullName)
fmc.createCounts[podFullName]++
return nil
}
func (fmc *fakeMirrorClient) DeleteMirrorPod(podFullName string) error {
fmc.mirrorPodLock.Lock()
defer fmc.mirrorPodLock.Unlock()
fmc.mirrorPods.Delete(podFullName)
fmc.deleteCounts[podFullName]++
return nil
}
func newFakeMirrorClient() *fakeMirrorClient {
m := fakeMirrorClient{}
m.mirrorPods = util.NewStringSet()
m.createCounts = make(map[string]int)
m.deleteCounts = make(map[string]int)
return &m
}
func (fmc *fakeMirrorClient) HasPod(podFullName string) bool {
fmc.mirrorPodLock.RLock()
defer fmc.mirrorPodLock.RUnlock()
return fmc.mirrorPods.Has(podFullName)
}
func (fmc *fakeMirrorClient) NumOfPods() int {
fmc.mirrorPodLock.RLock()
defer fmc.mirrorPodLock.RUnlock()
return fmc.mirrorPods.Len()
}
func (fmc *fakeMirrorClient) GetPods() []string {
fmc.mirrorPodLock.RLock()
defer fmc.mirrorPodLock.RUnlock()
return fmc.mirrorPods.List()
}
func (fmc *fakeMirrorClient) GetCounts(podFullName string) (int, int) {
fmc.mirrorPodLock.RLock()
defer fmc.mirrorPodLock.RUnlock()
return fmc.createCounts[podFullName], fmc.deleteCounts[podFullName]
}
func TestParsePodFullName(t *testing.T) {
type nameTuple struct {
Name string
Namespace string
}
successfulCases := map[string]nameTuple{
"bar_foo": {Name: "bar", Namespace: "foo"},
"bar.org_foo.com": {Name: "bar.org", Namespace: "foo.com"},
"bar-bar_foo": {Name: "bar-bar", Namespace: "foo"},
}
failedCases := []string{"barfoo", "bar_foo_foo", ""}
for podFullName, expected := range successfulCases {
name, namespace, err := kubecontainer.ParsePodFullName(podFullName)
if err != nil {
t.Errorf("unexpected error when parsing the full name: %v", err)
continue
}
if name != expected.Name || namespace != expected.Namespace {
t.Errorf("expected name %q, namespace %q; got name %q, namespace %q",
expected.Name, expected.Namespace, name, namespace)
}
}
for _, podFullName := range failedCases {
_, _, err := kubecontainer.ParsePodFullName(podFullName)
if err == nil {
t.Errorf("expected error when parsing the full name, got none")
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,645
|
<div class="search-results">
<ul class="nav search-results__facets" ng-show="isFacetsVisible()">
<li class="nav__item" ng-show="results.customers">
<button ng-click="SearchState.transition('customers')"
ng-class="{ 'is-active': SearchState.get() == 'customers' }"
class="search-pill"
ng-disabled="!results.customers.total">
Customers ({{ results.customers.total }})
</button>
</li>
<li class="nav__item" ng-show="results.merchants">
<button ng-click="SearchState.transition('merchants')"
ng-class="{ 'is-active': SearchState.get() == 'merchants' }"
class="search-pill"
ng-disabled="!results.merchants.total">
Merchants ({{ results.merchants.total }})
</button>
</li>
<li class="nav__item" ng-show="results.bills">
<button ng-click="SearchState.transition('bills')"
ng-class="{ 'is-active': SearchState.get() == 'bills' }"
class="search-pill"
ng-disabled="!results.bills.total">
Payments ({{ results.bills.total }})
</button>
</li>
<li class="nav__item" ng-show="results.payouts">
<button ng-click="SearchState.transition('payouts')"
ng-class="{ 'is-active': SearchState.get() == 'payouts' }"
class="search-pill"
ng-disabled="!results.payouts.total">
Payouts ({{ results.payouts.total }})
</button>
</li>
</ul>
<div class="search-results__output">
<ng-include src="searchResultsTemplate(SearchState.get())">
</div>
</div>
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 76
|
\section{Introduction}
There has been much interest recently in models of inflation using scalar fields non-minimally coupled to gravity, originally proposed in \cite{salopek}. (See also \cite{early}.) This was primarily motivated by the idea of using the Standard Model Higgs as the inflaton (`Higgs Inflation') \cite{bs1}. Variants include Higgs Inflation with scalar dark matter \cite{clark}, `S-inflation' due to a dark matter scalar coupled to the Standard Model \cite{rl1}, a supersymmetric version of Higgs Inflation \cite{tj} and an extension to include neutrino masses \cite{shafi}. Generalization of the non-minimal coupling to gravity was discussed in \cite{park}\footnote{A alternative approach to Higgs Inflation, based on derivatives of the Higgs coupled to gravity, was presented in \cite{germani}.}.
However, the naturalness of these models has been questioned, specifically whether or not unitarity is violated in Higgs scattering mediated by graviton exchange at a scale $\Lambda \sim M_p / \xi \ll M_p$. Here $\xi$ is the value of the non-minimal coupling, which must be of order $10^4$ in order to account for the observed density perturbation.
In particular, in \cite{barbon} it was noted that the effective coupling in tree-level graviton-mediated Higgs scattering becomes strong at $E \sim \Lambda$, while in \cite{b1} it was concluded that unitarity would be violated in graviton-mediated Higgs scattering at $E \sim \Lambda$.
These analyses were based on the original Higgs Inflation model, which considered a single real Higgs scalar in the unitary gauge and neglected gauge interactions. In \cite{nat} it was noted that there are no strong coupling or unitarity-violating interactions in the single scalar model when considered in the Einstein frame, indicating that
the apparent strong coupling or unitarity-violating effects in the Jordan frame at $E \sim \Lambda$ do not occur. This can be understood in terms of a cancellation of the leading s-, t- and u-channel contributions to the graviton-mediated Higgs amplitude in the Jordan frame \cite{cancel,hertz}. However, once longitudinal gauge fields are included in the unitary gauge (or, equivalently, Goldstone bosons in a covariant gauge), the Jordan frame cancellation of the graviton-mediated Higgs scattering amplitude no longer occurs \cite{hertz,b2}. This manifests itself in the Einstein frame as non-renormalizable interactions which cannot be eliminated by field redefinitions.
However, while unitarity is violated in tree-level scattering, it was shown in \cite{hw} that perturbation theory will break down before the energy of unitary violation is reached. Specifically, for the case of s-channel scattering mediated by graviton exchange, it was shown that the imaginary part of the 1-loop contribution to the amplitude is half of the tree-level contribution at the energy of unitarity-violation. As noted in \cite{nat}, this leads to the possibility that strong-coupling itself is the "new physics" required to maintain unitarity. This is supported by the observation of \cite{hw} that, in the large-$N$ limit (where $N$ is roughly the number of particles contributing to the loop corrections), the all-order graviton-mediated scattering cross-section (excluding graviton loops) is unitary at all energies, even though the tree-level cross-section violates unitarity. The possibility that strong coupling could ensure unitarity-conservation was noted earlier in \cite{bs2}. The essential point is that if strong coupling can deal with the apparent unitarity violation in particle scattering processes, then
the action of the theory is complete as is, requiring no new terms. The effective potential and the analysis of inflation can then be carried out by calculating with this action in the conventional way \cite{hi1}.
Logically, the action of the original Higgs Inflation model is either consistent or inconsistent as a quantum field theory. If it is an inconsistent theory then we expect unitarity to be violated at some energy, requiring a completion of the theory. However, if the theory is consistent, then we would expect any calculation which appears to violate unitarity to be modified as the energy approaches that of unitarity violation. This appears to be the case in Higgs Inflation, with higher-order corrections to the scattering amplitude becoming important as the energy approaches that at which tree-level unitarity is violated. Therefore Higgs Inflation has the qualitative behaviour of a consistent theory. However, since a non-perturbative analysis is necessary in order to establish unitarity conservation in Higgs Inflation, it may be difficult to either prove or disprove unitarity conservation. In this case the best strategy would be to consider both possibilities and use collider experiments and precision CMB observations to establish whether Higgs Inflation is consistent with observations. This strategy is feasible because of the uniquely predictive nature of Higgs Inflation. The inflation observables, in particular the spectral index, are entirely determined by Standard Model couplings. Therefore precision measurement of the spectral index and the Higgs mass $m_{H}$ can, in principle, allow the nature of Higgs Inflation to be determined experimentally.
The case where unitarity is conserved in Higgs Inflation has been extensively studied in \cite{bs2,hi1}, where the RG-improved effective potential was calculated and the spectral index as a function of Higgs mass determined. In this paper we consider the alternative case where unitarity is violated at $E \sim \Lambda$. In this case we must add new terms to the action to restore unitarity. The concern expressed in \cite{barbon,b1} is that such
new terms necessarily include Higgs potential terms suppressed by powers of $\Lambda$, spoiling the flatness of the potential and ruling out slow-roll inflation. However, this is an assumption. Our goal here is to derive the minimal modification of Higgs Inflation necessary to restore unitarity and to show that it can, in principle, support successful inflation.
In Section 2 we review tree-level unitarity violation in the original Higgs Inflation model. In Section 3 we introduce a new unitarity-conserving Higgs Inflation model. In Section 4 we discuss the cosmology of this model, showing that it makes a quite different prediction for the spectral index from the original Higgs Inflation model. In Section 5 we present our conclusions.
\section{Tree-level unitarity violation in Higgs Inflation}
We first consider tree-level unitarity violation due to graviton-mediated Higgs scattering in Higgs Inflation. In the Jordan frame the action for Higgs Inflation (including the Higgs doublet and gauge fields) is
\be{e0}
S_J = \int d^4 x \; \sqrt{-\!g} \; \left( - \frac{M^2R}{2} - \xi H^{\dagger}H R + g^{\mu\nu}\left(D_\mu H\right)^{\dagger}\left(D_\nu H\right) -\frac{1}{4} F_{\mu\nu} F^{\mu\nu} - V(|H|) \right)
~,\ee
where
\be{e1}
\label{Jpot}
V(|H|) = \lambda \left(\left(H^{\dagger}H\right) - \frac{v^2}{2}\right)^2 ~.\ee
(In \eq{e0} the gauge kinetic term represents the kinetic terms for all gauge fields.) In the following we will set $M = M_p$, as the Higgs vacuum expectation value is negligibly small compared with $M_{p}$. The Einstein frame action is obtained by first performing a
conformal rescaling of the metric
\be{2} \tilde{g}_{\mu\nu} = \Omega ^2 g_{\mu\nu} ~,\ee
where
\be{3}\label{omegaeq} \Omega ^2 = 1 + \frac{2 \xi H^{\dagger}H}{M_p^2} ~.\ee
In terms of this metric the action becomes
\be{e4} S_{E} = \int d^{4} x \sqrt{-\tilde{g}} \left( - \frac{M_{p}^{2}}{2}\tilde{R} + \frac{1}{\Omega^{2}} \tilde{g}^{\mu\nu}\left(D_\mu H\right)^{\dagger}\left(D_\nu H\right) + \frac{3\xi^2}{\Omega^4 M_p^2}\tilde{g}^{\mu\nu} \partial_\mu\left(H^{\dagger} H\right) \partial_\nu\left(H^{\dagger} H\right) -\frac{1}{4} F_{\mu\nu} F^{\mu\nu} - \frac{V(|H|)}{\Omega^4} \right) ~,\ee
where $\tilde{R}$ is the Ricci scalar with respect to $\tilde{g}_{\mu \nu}$ and indices are raised with $\tilde{g}^{\mu \nu}$.
Tree-level unitarity violation due to graviton-mediated Higgs scattering in the Jordan frame manifests itself in the Einstein frame via the non-minimal kinetic terms for $H$ from the second and third terms in \eq{e4}. The simplest way to consider unitarity violation is to consider the $\langle H \rangle \rightarrow 0$ limit, where the physical degrees of freedom are the four real scalars of $H$ and the
transverse gauge degrees of freedom. In the case with a single real scalar ($H \rightarrow h/\sqrt{2}$) and no gauge fields, as originally considered in Higgs Inflation, the non-minimal kinetic term can be eliminated by a redefinition of $h$ to $\chi$ via
\be{e5} \frac{d\chi}{dh} = \sqrt{\frac{\Omega ^2 + 6 \xi^2h^2/M_P^2}{\Omega ^4}} ~.\ee
Then
\be{e6}
S_E = \int d^4x\sqrt{-\tilde{g}}\Big( - \frac{M_p^2\tilde{R}}{2} + \frac{1}{2}\partial _\mu \chi \partial^{\mu} \chi - U(\chi)\Big)
~,\ee
where $U(\chi) = V(h)/\Omega^4$. In this case there are no interactions\footnote{One concern is that the non-polynomial potential is difficult to handle as a quantum field theory. However, we believe this is a quite different issue from tree-level unitarity violation associated with the non-minimal coupling to gravity in the Jordan frame. Since tree-level unitarity violation in $2 \rightarrow 2$ Higgs scattering via graviton-exchange is independent of the potential, the analogous interactions in the Einstein frame should also be independent of the potential. We will comment further the issue of the non-polynomial potential in our conclusions.} which lead to tree-level unitarity violation in
$\chi$-$\chi$ scattering, which is equivalent to $h$-$h$ scattering since $\Omega \approx 1$ in the vacuum. The absence of
unitarity-violating interactions in the Einstein frame at $E \sim \Lambda$ is consistent with the cancellation of the leading s-, t- and u-channel contributions to graviton-mediated Higgs scattering in the Jordan frame \cite{cancel,hertz}.
However, with more than one scalar, it is no longer possible to redefine the scalar fields to have canonical kinetic terms, since this would require the non-minimal kinetic term for the field $\phi_{i}$ to be a function of $\phi_{i}$ only. As a result, there are Einstein frame interactions such as
\be{e8} \frac{3 \xi^{2}\phi_{i}\phi_{j}}{\Omega^{4} M_{p}^{2}}\partial_{\mu}\phi_{i} \partial^{\mu} \phi_{j} ~,\ee
where $\phi_{i}$ ($i = 1,...4$) are the 4 real scalars in $H$.
These interactions lead to a tree-level scattering amplitude for $\phi_{i} \phi_{i} \rightarrow \phi_{j} \phi_{j}$ which is of the order of $(E/\Lambda)^2$. The corresponding cross-section will therefore violate unitarity at $E \gtrsim \Lambda$.
The same result may also be obtained in the unitary gauge with $\langle H \rangle = v$, in which case tree-level unitarity violation is due to longitudinal gauge boson scattering from the physical Higgs scalar \cite{b1}.
Therefore if tree-level unitarity violation is an indication of true unitarity violation, then it is not possible to couple the Higgs doublet non-minimally to gravity as in \eq{e0}. New terms must also be added to \eq{e0}, in order to ensure unitarity is conserved at least up to energies sufficiently large compared with the value of $h$ during inflation, $h \approx \sqrt{N} M_{p}/\sqrt{\xi}$, where $N$ is the number of e-foldings of inflation.
\section{ A Unitarity-conserving completion of Higgs Inflation}
As emphasized in \cite{nat}, the Einstein frame provides a particularly clear way to understand unitarity violation in graviton-mediated Higgs scattering due to the non-minimal coupling. On transforming to the Einstein frame, where the non-minimal couplings are eliminated, unitarity violation manifests itself via non-renormalizable interactions. Therefore the minimal unitarity-conserving completion of the Higgs Inflation Lagrangian in the Jordan frame will correspond to the Einstein frame Lagrangian which removes all the dangerous non-renormalizable terms.
From the discussion of Section II, it is clear that the only way to eliminate unitarity violation in the Einstein frame is to replace the non-minimal Higgs kinetic term with a canonical kinetic term. We must therefore add terms to the Jordan frame action \eq{e4} to achieve this. The final action in the Einstein frame must have the form
\be{e9} S_{E} = \int d^{4} x \sqrt{-\tilde{g}} \left( - \frac{M_{p}^{2}}{2}\tilde{R} + \tilde{g}^{\mu\nu}\left(D_{\mu}H\right)^{\dagger} \left(D_{\nu}H\right)
-\frac{1}{4} F_{\mu\nu} F^{\mu\nu} - \frac{V(|H|)}{\Omega^4} \right) ~.\ee
On transforming back to the Jordan frame, the additional terms in $S_{J}$ which are required to conserve unitarity up to the Planck scale are generated. The resulting unitarity-conserving action in the Jordan frame is given by
\bea
\label{e12}
S_J &=& \int d^4\! x \sqrt{-\!g} \left( - \frac{M_{p}^2 R}{2} - \xi H^{\dagger}HR + g^{\mu\nu}D_\mu H^{\dagger}D_\nu H
+ \frac{2 \xi H^{\dagger}H}{M_p^2} g^{\mu\nu}D_\mu H ^{\dagger}D_\nu H\right. \nonumber \\
& & \left. - \frac{3\xi^2}{\Omega^2M_p^2}g^{\mu\nu}\partial_\mu \left(H^{\dagger}H\right)\partial_\nu \left(H^{\dagger}H\right)
-\frac{1}{4} F_{\mu\nu} F^{\mu\nu} - V(|H|) \right) ~.\eea
We believe that \eq{e12} is the minimal unitarity-conserving action for the Standard Model Higgs doublet with a large non-minimal coupling to gravity. Since the fundamental assumption of Higgs Inflation is that inflation is due entirely to the non-minimal coupling of $H^{\dagger}H$ to gravity, \eq{e12} will provide a manifestly unitarity-conserving basis for Higgs Inflation.
The non-minimal coupling to $R$ plus the additional terms in \eq{e12} may be interpreted as the complete set of terms which must be brought down from the full Planck-scale gravity theory to the scale $\Lambda$ in order to maintain the quantum consistency of the theory.
A non-minimal coupling of the Higgs to gravity is generally expected to exist, but it is usually assumed that $\xi \sim 1$, in which case the associated unitarity violation occurs at $E \sim M_{p}$. The effect of increasing $\xi$ is to effectively pull down the non-minimal coupling from the Planck-scale gravity theory to the lower mass scale $\Lambda$. Unitarity violation can then be interpreted as a sign that other terms from the full gravity theory must accompany the non-minimal coupling in order to maintain the consistency of the theory.
So far we have considered the model only at tree-level, without quantum corrections to the inflaton potential. The structure of \eq{e9} is equivalent to the Standard Model gauge and Higgs fields plus a potential $V(|H|)/\Omega^4$. This suggests that the 1-loop Coleman-Weinberg correction due to gauge boson loops in the Einstein frame will have the form $\sim M_{W}^4 \; \log \; M_{W}^2 \propto |H|^4$, which would spoil the flatness of the potential. In this case a supersymmetric (SUSY) version of the model will be necessary in order to suppress the quantum corrections to the inflaton potential. However, if the inflaton was not the Higgs, but instead a singlet scalar coupled to the Standard Model only via the potential (such as in \cite{rl1}), then its couplings would be suppressed by $\Omega^{-4}$ in the Einstein frame and should not spoil the flatness of the inflationary potential.
\section{Slow-roll inflation predictions}
Although \eq{e12} provides a basis for a unitarity-conserving Higgs Inflation model, it is not the same Higgs Inflation model as originally proposed in \cite{bs1}. Inflation is best analysed in the Einstein frame, where $H$ has canonical kinetic terms and model may be treated as a conventional slow-roll inflation model, but now with potential $U(|H|) \equiv V(|H|)/\Omega^4$. Introducing the physical Higgs field as the inflaton, $H \rightarrow h/\sqrt{2}$, we obtain
\be{e12a} U(h) = \frac{ \lambda h^4}{4 \left(1 + \frac{\xi h^2}{M_{p}^2}\right)^2 } ~.\ee
For $h \gg M_{p}/\sqrt{\xi}$, the potential is flat and slow-roll inflation is possible. With $\tdN = 58$, where $\tilde{N} \approx \frac{\xi h^4}{16M_p^4}$ is the number of e-folding of inflation in the Einstein frame (corresponding to $N = 60$ in the Jordan frame \cite{nat}), the classical value of the spectral index is given by $n = 1+2\eta - 6\epsilon$, where
\be{12b} \eta \equiv M_p^2\left(\frac{d^2U}{dh^2}\right) \simeq - \frac{12 M_p^4}{\xi h^4} + \frac{36 M_p^6}{\xi^2 h^6};~\;\;\;\;
\epsilon \equiv \frac{M_p^2}{2} \left(\frac{1}{U}\frac{dU}{d h}\right)^2 \simeq \frac{8M_p^6}{\xi^2 h^6} - \frac{16 M_p^8}{\xi^3 h^8}
~.\ee
Therefore
\be{e13} n \approx 1 - \frac{3}{2\tdN} + \frac{3}{8\tdN^{3/2}\sqrt{\xi}} \approx 0.974 ~.\ee
The tensor to scalar ratio $r$ is given by
\be{13b}
r \equiv 16\epsilon \simeq \frac{2}{\sqrt{\xi}\tdN^{3/2}} \approx 6 \times 10^{-6}
~.\ee
(The running of the spectral index $\alpha$ is negligibly small.)
The curvature perturbation is given by
\be{e13c} P_{\xi} = \frac{ \lambda \tdN^3}{12 \pi^2 \xi^{3/2}} ~,\ee
therefore to have a correctly normalized spectrum of density perturbations, $P_{\xi}^{1/2} = 4.8 \times 10^{-5}$, we require
\be{e14a}
\xi \simeq (3.8-6.5) \times 10^{5}~\ee
for $m_{H}$ in the range 114-170 GeV. This is different from the original Higgs Inflation model because the derivatives in the slow roll parameters are defined with respect to different canonically normalised fields - $\chi$ in the original model and $h$ in the unitarity-conserving model.
These may be compared with the predictions of the original Higgs Inflation model, $n \simeq 1 - \frac{2}{\tdN} - \frac{3}{2\tdN^2} = 0.965$, $r \simeq \frac{12}{\tdN^2} = 3.6 \times 10^{-3}$ and $ \frac{\lambda}{\xi^2} \simeq \frac{3(0.027)^4}{\tdN^2}$ giving $\xi \simeq 10^{4}$.
These estimates are based on $N =60$.
It should be noted that, because the model contains only standard model parameters, it is in principle possible to determine the reheating temperature and hence $N$ precisely. Therefore the model has no free parameters.
\section{Conclusions}
The possibility that inflation can be explained by a simple non-minimal coupling of the Higgs to gravity is very attractive, leading to a highly predictive model which requires no new fields beyond those of the Standard Model. We have proposed a new Higgs Inflation model based on a unitarity-conserving extension of the original Higgs Inflation action. We believe that this is the minimal form of Higgs Inflation model which manifestly conserves unitarity in the presence of a non-minimal coupling of the Higgs to gravity. As such, it may provide the correct formulation of Higgs Inflation should strong coupling effects fail to eliminate unitarity violation in the original Higgs Inflation model.
Perhaps the most interesting conclusion is that while unitarity-conserving Higgs Inflation is possible, the predictions of the new unitarity-conserving model are quite different from those of the original Higgs Inflation model. In particular, the classical spectral index of the new model is $n = 0.974$, which is within the 7-year WMAP 1-$\sigma$ limits on $n$ ($n = 0.963 \pm 0.012$ \cite{wmap7}) but significantly different from the original Higgs Inflation model prediction of $n = 0.965$. Therefore it should be possible to observationally distinguish between unitarity-conserving Higgs Inflation and the original Higgs Inflation model.
We finally comment on the assumptions underlying our model. We
consider all terms which are scaled by inverse powers of $\Omega$ in the Einstein frame to lead to unitarity violation, with the exception of $V(|H|)/\Omega^{4}$. Terms are then added to eliminate unitarity violation.
However, we believe that $V(|H|)/\Omega^{4}$ will not lead to unitarity violation. This is because in the limit $|H|^2 \gg M_{p}^{2}/2 \xi$, there is a nearly perfect cancellation of the $|H|^4$ factors in $V(|H|)$ and in $\Omega^4$, completely eliminating interactions\footnote{More generally, we expect that any non-polynomial potential interpolating between renormalizable potentials at small and large field strength will not lead to unitarity violation.}. To illustrate how the potential term differs from other terms with respect to unitarity violation, we can consider perturbations about a large background Higgs field. In this case the potential term in \eq{e4} tends towards that for massless, non-interacting scalars, with unitarity-violating interactions suppressed by powers of $|H|$, whereas the second term in \eq{e4}, for example, leads to unitarity violation at $E \sim M_{p}/\sqrt{\xi}$, independent of $|H|$.
A feature that the unitarity-conserving model shares with the original Higgs Inflation model is that since all the model parameters are Standard Model parameters, they can be fixed experimentally (with the exception of $\xi$, which is fixed by the density perturbations). In particular, it will be possible to precisely compute quantum corrections to the spectral index as a function of Higgs mass. This should allow for precision tests of the model once $m_{H}$ is determined by the LHC and $n$ by PLANCK. A caveat is that such quantum corrections are likely to be large in the case of a non-SUSY model, in which case a SUSY version following the same strategy will be necessary in order to maintain the flatness of the inflaton potential. A very minimal non-SUSY model may still be possible if the inflaton was instead a singlet scalar with a potential coupling to the Standard Model. However, we expect that the tree-level predictions of any unitarity-conserving model, being necessarily based on minimal kinetic terms and $V/\Omega^4$ in the Einstein frame, will remain unchanged.
\section*{Acknowledgements}
This work was supported by the European Union through the Marie Curie Research and Training Network "UniverseNet" (MRTN-CT-2006-035863).
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Barry Snaith throughout his carrier has worked/collaborated with numerous international artists and engaged in various creative genres – music, painting, surrealist human art, motion graphics, dance and video. As a guitarist, he has toured, gigged and recorded with the likes of The Ramones, Chrissie Hynde, Johnny Thunders, and David Johannsen (New York Dolls). As a solo artist, he writes and produces soundscapes that have been used in fashion catwalk shows, the UK's first ever digital fiction installation, 'Wallpaper' which was curated at Bank Street Arts in August 2015. He is currently working on the soundscape of the Virtual Reality version to be released worldwide at the end of 2017 with the award winning Dreaming Methods/One To One Development Trust. He has produced and composed with Tayo Irvine Hendrix for The Tate Modern, and this will be performed for Nelson Mandela's 100th birthday event on Robben Island 8 July, 2018. His music is to be visualised by the motion graphic internationally acclaimed George Redhawk (Paramount Movies 'Ghost in The Shell') and the track art for his electronic duo 'How I Wrote Elastic Man', is by the French artist Eric Lacombe. The video single for his 'Bold Ego Fledgling', was choreographed by Parris Goebel and performed with her 'ReQuest Dance Crew'. He has recently launched 'Mau', a gothic electronica collaboration with Greek based songstress Erika Bach. He currently resides in Wakefield UK".
Eirini Boukla is an artist and a lecturer in Art and Design. She makes use of a variety of mediums, that often merge, to explore the possibilities of contemporary drawing practice and ideas of authenticity and originality. Her main research interests are drawing, collage, the practice of tracing, ideas of authenticity and semiotics. She has exhibited her work both nationally and internationally. Recent exhibitions include; Drawing Dialogue, DalgaArt, Craiova, Romania. Pushpin, Zverev Museum, Center for Contemporary Art, Moscow. Ink Shop Printmaking Center & Olive Branch Press. Ithaca, New York. USA. Limerick/Berlin, Limerick printmakers gallery, Ireland. Pieces of Eight, PSL, Leeds; Thinking tools, FAFA Gallery, Helsinki, Finland; Drawing Connections, Siena Art Institute, Siena, Italy; The Artful Scriptorium, Climate Gallery, New York, USA. 8th International exhibition of women painters, Majdanpek Cultural Center, Serbia. The Last Book (Luis Camnitzer project), Zentral Bibliothek in Zurich, Switzerland. Adaptive Actions, Campo AA, Madrid Abierto, Madrid, Spain. Contemporary Flânerie: Reconfiguring Cities, Oakland University Art Gallery, Rochester, Michigan, USA. The Last Book (Luis Camnitzer project), National Library of Argentina, Buenos Aires, Argentina. SIPF, Singapore International Photography Festival, Singapore. Civic Hall Brno, Brno, Czech Republic. Lattidute, Fifty Four Degrees North Festival, Hull, England. Olympolis, Katerini, Greece. Public Screen, SYNCH Festival, Technopolis, Athens, Greece.
Bal Na Vodi (Dancing in Water) is a 2 minutes and 36 seconds audio visual drawing. Is the outcome of a collaborative dialogue between artist Eirini Boukla and musician Barry Snaith. The work is a de/constructive, systematic approach to moving image, through a linear and atmospheric narrative detour.
Eirini uses tracing as a particular form of drawing that reworks and repurposes the found and already worked material. Taking up the diagrammatic potential of the moving image 'Bal na vodi' edits, traces over, and reworks the motion lines from 'Jolly Fish' a 1932 Van Beuren studio hand-drawn animation, 'a tracing' that speculates on possibilities of place and memory and their representation, through dynamic behaviour and movement and other core elements of embodied experience.
Barry's soundtrack, accompanying Bal Na Vodi's graphic dialogue and emphasises the drama, rhythm, tempo, mood or continuity in and between the images. Usefully and crucially creates space outside the frame. Sometimes we are aware of the soundtrack, especially if a melody is recognised, but more often it slips into our subconscious as perhaps it should if it is going there to retrieve a memory.
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Sveriges vackraste park var en tävling som arrangerades årligen 2003–2009.
Syftet har varit att väcka intresse för landets offentliga parker och trädgårdar.
Vinnande parker
2009 - Sofiero slott och slottsträdgårdar i Helsingborg
2008 - Marabouparken i Sundbyberg
2007 - Wij trädgårdar i Ockelbo
2006 - Norrvikens trädgårdar
2005 - Ronneby brunnspark
2004 - Stadsparken, Örebro
Ett specialpris gick till Luleå Stadspark som bästa vinterpark 2004.
2003 - Göteborgs Botaniska Trädgård
Källor
Noter
Externa länkar
Sveriges vackraste park
Tävlingar
Parker i Sverige
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Home / / ESPN's Stuart Scott Battling Cancer…Again
ESPN's Stuart Scott Battling Cancer…Again
ESPN anchor Stuart Scott was recently honored with the Jimmy V Award at the 2014 ESPY awards for his commitment to fight against cancer.
READ: 10 Cancer Symptoms Men Ignore
"Blessed by prayers..I'm back in the Fight. C reared its head again. Chemo evry 2 wks but I'll still work, still work out..still #LIVESTRONG."
Scott was first diagnosed with the disease in 2007 when doctors discovered he had cancer of the appendix while performing an appendectomy. He underwent chemotherapy and all was well until 2011, when he revealed that doctors found tumors in his small intestine.
After an operation and more chemo, he remained cancer-free until his 1/14/13 announcement. But his award speech speaks to the brave father of two fighting until he can't fight anymore (see below).
http://youtu.be/-50C_2Q5VO8
READ: 7 Foods Farmers Won't Eat
There was no word on where the cancer was found this time.
ESPN spokesman Mike Soltys told USA Today Sports that the Sportscenter fixture "plans to continue to work the best he can around his treatments." Scott re-tweeted and shared some of the positive messages that he received from sports figures such as Robert Griffin III and Russell Wilson. Jay Glazer of FOX was among Scott's colleagues in sports media offering support to Scott.
READ: 6 Things In Your Home That Can Cause Cancer
After announcing the news on Twitter, Scott received a flood of tweets wishing him the best, to which he responded:
"Thanks for prayers..ill fight w ALL C survivors & loved ones. Cancer wants to re-appear..picked the right guy cuz I HIT HARD all day long!!"
January 15, 2013 by Marcus Williams, BDO Staff Writer
Shannon Sharpe Talks Prostate Cancer: "Be a Part of the 96% That Survive"
Men and health don't always coexist. In most cases, health situations hardly dominate a conversation over sports and politics. Even worse, health involving Black men remains a taboo subject. Fortunately, the narrative on Black men's health is changing. In fact, read more about Shannon Sharpe Talks Prostate Cancer: "Be a Part of the 96% That Survive"
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Q: Order of messages to console within functions when piped edit: I don't mean to be rude, SO just did not take my greeting..
Hello,
to keep track about the processing of some functions I would like to print messages in the Console when a function starts and when it is done.
I am a big fan of tidyverse and the pipe operator %>%. But I just found out that the messages at the start of the function are printed in the wrong order while the messages at the end have the correct order.
Does anybody know why or how to solve this?
Here is some code:
library(tidyverse)
testdata <- data.frame(a=c(3,5,7,1,2),b=c(2,5,6,3,9))
foo_1 <- function(DF){
message("Hello 1")
Obj <- DF %>%
mutate(Test_1=a*b)
message("Bye 1")
return(Obj)
}
foo_2 <- function(DF){
message("Hello 2")
Obj <- DF %>%
mutate(Test_2=Test_1*2)
message("Bye 2")
return(Obj)
}
foo_3 <- function(DF){
message("Hello 3")
Obj <- DF %>%
mutate(Test_3=Test_2/a)
message("Bye 3")
return(Obj)
}
testdata %>%
foo_1() %>%
foo_2() %>%
foo_3()
The result printed in the console is:
Hello 3
Hello 2
Hello 1
Bye 1
Bye 2
Bye 3
a b Test_1 Test_2 Test_3
1 3 2 6 12 4
2 5 5 25 50 10
3 7 6 42 84 12
4 1 3 3 6 6
5 2 9 18 36 18
What I would expect is the following order of messages:
Hello 1
Bye 1
Hello 2
Bye 2
Hello 3
Bye 3
a b Test_1 Test_2 Test_3
1 3 2 6 12 4
2 5 5 25 50 10
3 7 6 42 84 12
4 1 3 3 6 6
5 2 9 18 36 18
A: 1) flush.console Use flush.console to force the display of the messages queued at that point.
testdata %>%
foo_1() %>%
{ flush.console(); foo_2(.) } %>%
{ flush.console(); foo_3(.) }
giving this output:
Hello 1
Bye 1
Hello 2
Bye 2
Hello 3
Bye 3
a b Test_1 Test_2 Test_3
1 3 2 6 12 4
2 5 5 25 50 10
3 7 6 42 84 12
4 1 3 3 6 6
5 2 9 18 36 18
2) Bizarro pipe Another approach is to use the Bizarro pipe, which is just clever base syntax which looks pipe-like, instead.
testdata ->.;
foo_1(.) ->.;
foo_2(.) ->.;
foo_3(.)
giving:
Hello 1
Bye 1
Hello 2
Bye 2
Hello 3
Bye 3
a b Test_1 Test_2 Test_3
1 3 2 6 12 4
2 5 5 25 50 10
3 7 6 42 84 12
4 1 3 3 6 6
5 2 9 18 36 18
3) force force the evaluation of arguments in the three fuctions.
foo_1 <- function(DF){
force(DF) #######
message("Hello 1")
Obj <- DF %>%
mutate(Test_1=a*b)
message("Bye 1")
return(Obj)
}
foo_2 <- function(DF){
force(DF) #######
message("Hello 2")
Obj <- DF %>%
mutate(Test_2=Test_1*2)
message("Bye 2")
return(Obj)
}
foo_3 <- function(DF){
force(DF) #######
message("Hello 3")
Obj <- DF %>%
mutate(Test_3=Test_2/a)
message("Bye 3")
return(Obj)
}
testdata %>%
foo_1 %>%
foo_2 %>%
foo_3
giving:
Hello 1
Bye 1
Hello 2
Bye 2
Hello 3
Bye 3
a b Test_1 Test_2 Test_3
1 3 2 6 12 4
2 5 5 25 50 10
3 7 6 42 84 12
4 1 3 3 6 6
5 2 9 18 36 18
4) pipe_eager_lexical Lionel, the magrittr developer, pointed out to me that magrittr has an eager pipe but does not assign it to a %..% operator; however, we can readily do that.
library(magrittr)
`%>e%` <- pipe_eager_lexical # eager pipe
testdata %>e%
foo_1() %>e%
foo_2() %>e%
foo_3()
giving:
Hello 1
Bye 1
Hello 2
Bye 2
Hello 3
Bye 3
a b Test_1 Test_2 Test_3
1 3 2 6 12 4
2 5 5 25 50 10
3 7 6 42 84 12
4 1 3 3 6 6
5 2 9 18 36 18
A: You move the message out of the individual functions and generate a wrapper around them where you put the messages:
With some sudo-ish code:
my_wrapper <- function(){
message('h1')
foo_1()
message('b1')
message('h2')
...
message('b3')
}
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"redpajama_set_name": "RedPajamaStackExchange"
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Dickinson, LaMarque, League City, Pearland, Webster
Barbara Jean Heileman
April 18, 2019 /
Barbara Jean Heileman, 84, of Dickinson, passed away April 17, 2019, in Texas City. Barbara was born December 14, 1934 in Galveston, Texas and raised by Margaret and John Gillespie. Barbara will be remembered as a kind loving mother, grandmother, and great-grandmother.
She was preceded in death by her parents and her loving husband of 57 years Samuel J. Heileman.
Barbara leaves behind to cherish her memory her daughter Suzanne Heileman; son James A. Heileman; grandchildren Nathan (Jessica) Heileman, Candice Daniel, and Matthew D. (Nancy) Heileman; and great-grandchildren Matthew J. Heileman, Selah Heileman, Evan Daniel, Judah Heileman, and Liliana Rodriguez.
A visitation in her honor will be held 9:00 – 10:00 am, Saturday, April 20, 2019, with a funeral service at 10:00 am, at Crowder Funeral Home, Dickinson, Texas. Interment follows at Hayes Grace Memorial Park, Hitchcock, Texas.
Honored to serve as pallbearers are Tommy Clark, Stephen Clark, Matthew D. Heileman, Matthew J. Heileman, Michael M. Rodriguez, and Michael S. Rodriguez
The post Barbara Jean Heileman appeared first on Crowder Funeral Home.
View Original Notice ? Barbara Jean Heileman
MOONEY, Elizabeth
Rodolfo Palacios
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\section{Weierstrass Sigma}
Using the Schwarz reflection principle, $f_{w}$ extends across the hypotenuse
of $T$ to an analytic function (still called $f_{w}$) on the square $\left\{
x+iy\in\mathbb{C}:0<x<1\RIfM@\expandafter\text@\else\expandafter\mbox\fi{, }0<y<1\right\} $ except for a simple pole at
$w^{\prime}=1+i-i\overline{w}$. \ Using the Schwarz reflection principle twice
more, $f_{w}$ extends to the doubled square $\left\{ x+iy\in\mathbb{C
:-1<x<1\RIfM@\expandafter\text@\else\expandafter\mbox\fi{, }-1<y<1\right\} $ with zeroes at $\left\{ \pm w,\pm
\overline{w^{\prime}}\right\} $ and poles at $\left\{ \pm w^{\prime
,\pm\overline{w}\right\} $. Clearly $f_{w}(-1+iy)=f_{w}(1+iy)$ and
$f_{w}(x-i)=f_{w}(x+i)$ always. Let $\Lambda$ denote the lattice $\left\{
2m+2ni:m,n\in\mathbb{Z}\right\} $. \ We may extend $f_{w}$ to a doubly
periodic meromorphic function on $\mathbb{C}$ via $f_{w}(z+\lambda)=f_{w}(z)$
for all $\lambda\in\Lambda$. \ Assume that $f_{w}(1)=1$ without loss of
generality. \ We deduce that \cite{Co
\
\begin{array}
[c]{ccc
f_{w}(z)=C_{w}\dfrac{\sigma(z-w)\sigma(z+w)\sigma(z-\overline{w^{\prime
})\sigma(z+\overline{w^{\prime}})}{\sigma(z-w^{\prime})\sigma(z+w^{\prime
})\sigma(z-\overline{w})\sigma(z+\overline{w})} & & \RIfM@\expandafter\text@\else\expandafter\mbox\fi{for all }z\in T
\end{array}
\]
where
\[
\sigma(z)=
{\displaystyle\prod\limits_{\substack{\lambda\in\Lambda\\\lambda\neq0 }}}
\left( 1-\frac{z}{\lambda}\right) \exp\left( \frac{z}{\lambda}+\frac{z^{2
}{2\lambda^{2}}\right) ,
\
\[
C_{w}=\frac{\sigma(1-w^{\prime})\sigma(1+w^{\prime})\sigma(1-\overline
{w})\sigma(1+\overline{w})}{\sigma(1-w)\sigma(1+w)\sigma(1-\overline
{w^{\prime}})\sigma(1+\overline{w^{\prime}})}.
\]
In order to employ Mathematica (or other computer algebra package), the
half-periods $1$, $i$ give rise to invariant
\
\begin{array}
[c]{ccc
g_{2}=\dfrac{1}{256\pi^{2}}\Gamma\left( \dfrac{1}{4}\right) ^{8
=11.8170450080..., & & g_{3}=0
\end{array}
\]
which must be passed to the software implementation of $\sigma$. As
$z\rightarrow w$, the ratio $\left\vert f_{w}(z)/(z-w)\right\vert $ simplifies
t
\[
h(w)=\left\vert \frac{\sigma(1-w^{\prime})\sigma(1+w^{\prime})\sigma
(1-\overline{w})\sigma(1+\overline{w})}{\sigma(1-w)\sigma(1+w)\sigma
(1-\overline{w^{\prime}})\sigma(1+\overline{w^{\prime}})}\cdot\dfrac
{\sigma(2w)\sigma(w-\overline{w^{\prime}})\sigma(w+\overline{w^{\prime}
)}{\sigma(w-w^{\prime})\sigma(w+w^{\prime})\sigma(w-\overline{w
)\sigma(w+\overline{w})}\right\vert
\]
because $\sigma(z-w)/(z-w)\rightarrow1$. \ Further simplification does not
seem possible. \ Numerical minimization gives the least capacity point to be
$w_{0}=(1+i)t_{0}$, wher
\[
t_{0}=0.3011216108413220815538254....
\]
No closed-form expression for $t_{0}$ is apparent, at least not here. \ We
observe (to high precision) tha
\[
\frac{1}{h(w_{0})}=0.3346161009568417919464744...=\frac{4\sqrt{2\pi}}{3^{3/4
}\Gamma\left( \dfrac{1}{4}\right) ^{-2
\]
which is encouraging since the latter is the maximum inner radius for $T$
\cite{PS}. \ A\ rigorous proof, however, remains open.
\section{Weierstrass P}
The zero $w_{0}$ in the preceding section identified a conformal map
$f_{w_{0}}:T\rightarrow\Delta$ that is, in particular, extremal in some sense.
\ Any map onto the upper half plane $\mathbb{C}^{+}$ can be easily recast as a
map onto the disk $\Delta$ (via composition with a linear fractional
transformation). \ As a slight detour, let us similarly identify other
better-known conformal maps $T\rightarrow\mathbb{C}^{+}$ that have appeared in
the literature. \
Two maps $\varphi$, $\psi$ are prescribed to take the following values on the
vertices of $T$
\[
(0,1,i)\overset{\varphi}{\longmapsto}(0,1,\infty),
\
\[
(0,1,i)\overset{\psi}{\longmapsto}(\infty,0,1).
\]
The Schwarz-Christoffel transformation gives \cite{Sp, Ky, Nh
\[
\varphi^{(-1)}(\zeta)=\frac{1}{\sqrt{2\pi}}\Gamma\left( \dfrac{1}{4}\right)
^{2}B\left( \zeta,\frac{1}{2},\frac{1}{4}\right) ,
\
\[
\psi^{(-1)}(\zeta)=\frac{i-1}{\sqrt{\pi}}\Gamma\left( \dfrac{1}{4}\right)
^{2}B\left( \zeta,\frac{1}{4},\frac{1}{4}\right) +1
\]
wher
\[
B(\zeta,\alpha,\beta)
{\displaystyle\int\limits_{0}^{\zeta}}
s^{\alpha-1}(1-s)^{\beta-1}ds
\]
is the incomplete Euler beta function. \ Although expressions for
$\varphi^{(-1)}$, $\psi^{(-1)}$ are famous, their inverses are comparatively
obscure. \ Geyer \cite{Gy} provide
\[
\psi(z)=-\frac{1}{4\wp(1)}\frac{(\wp(z)-\wp(1))^{2}}{\wp(z)
\]
wher
\[
\wp(z)=\frac{1}{z^{2}}
{\displaystyle\sum\limits_{\substack{\lambda\in\Lambda\\\lambda\neq0}}}
\left( \frac{1}{(z-\lambda)^{2}}-\frac{1}{\lambda^{2}}\right) ,
\
\[
\wp(1)=\frac{1}{32\pi}\Gamma\left( \dfrac{1}{4}\right) ^{4}=1.7187964545....
\]
Choose the linear fractional transformation $\mathbb{C}^{+}\rightarrow\Delta$
to be $\zeta\longmapsto(\zeta-i)/(\zeta+i)$. \ Hence we wish to solve the
equation $\psi(w)=i$, but this is immediately seen to yield
\[
w=\frac{i-1}{\sqrt{\pi}}\Gamma\left( \dfrac{1}{4}\right) ^{2}B\left(
i,\frac{1}{4},\frac{1}{4}\right) +1=0.2970894700...+(0.1926647354...)i.
\]
It is clear tha
\[
\overline{\psi(i\bar{z})}\cdot\varphi(z)=1
\]
and we wish to solve the equation $\varphi(w)=i$, but this too is immediately
seen to yiel
\[
w=\frac{1}{\sqrt{2\pi}}\Gamma\left( \dfrac{1}{4}\right) ^{2}B\left(
i,\frac{1}{2},\frac{1}{4}\right) =0.1926647354...+(0.2970894700...)i.
\]
A representation of arbitrary $f_{w}$ in terms of $\wp$ (analogous to our
representation in terms of $\sigma$) is also possible \cite{Co}, but we
haven't pursued this. \
\section{Jacobi Elliptic}
Let
\[
F[\phi,m]
{\displaystyle\int\limits_{0}^{\sin(\phi)}}
\dfrac{d\tau}{\sqrt{1-\tau^{2}}\,\sqrt{1-m\,\tau^{2}}}\,
\]
denote the incomplete elliptic integral of the first kind and $K[m]=F[\pi
/2,m]$; we purposefully choose formulas here to be consistent with
Mathematica. The three basic Jacobi elliptic functions are defined via
\begin{align*}
u &
{\displaystyle\int\limits_{0}^{\operatorname*{sn}(u,m)}}
\frac{d\tau}{\sqrt{1-\tau^{2}}\,\sqrt{1-m\,\tau^{2}}}
{\displaystyle\int\limits_{\operatorname*{cn}(u,m)}^{1}}
\frac{d\tau}{\sqrt{1-\,\tau^{2}}\,\sqrt{m\,\tau^{2}+(1-m)}}\\
&
{\displaystyle\int\limits_{\operatorname*{dn}(u,m)}^{1}}
\frac{d\tau}{\sqrt{1-\,\tau^{2}}\,\sqrt{\tau^{2}-(1-m)}
\end{align*}
and we shall require all three of these. \ Define \
\[
\kappa=\frac{K[1/2]}{\sqrt{2}}=\frac{1}{2^{5/2}\sqrt{\pi}}\Gamma\left(
\dfrac{1}{4}\right) ^{2}=1.3110287771...=\frac{1.8540746773...}{\sqrt{2}
\]
and let $\tilde{T}=\left\{ x+iy\in\mathbb{C}:y>0\RIfM@\expandafter\text@\else\expandafter\mbox\fi{, }y<x+\kappa\RIfM@\expandafter\text@\else\expandafter\mbox\fi{,
}y<-x+\kappa\right\} $. \ As in Section 2, let us first examine a conformal
map\ $\theta:\tilde{T}\rightarrow\mathbb{C}^{+}$ which takes prescribed values
on the vertices of $\tilde{T}$:
\[
(-\kappa,\kappa,i\kappa)\overset{\theta}{\longmapsto}(-1,1,\infty).
\]
The Schwarz-Christoffel transformation gives \cite{NP
\[
\theta^{(-1)}(\zeta)=\frac{1}{2
{\displaystyle\int\limits_{0}^{\zeta}}
\frac{ds}{\left( 1-s^{2}\right) ^{3/4}}=\frac{\zeta}{2}\,_{2}F_{1}\left(
\frac{1}{2},\frac{3}{4},\frac{3}{2},\zeta^{2}\right)
\]
which involves the following Gauss hypergeometric function:
\[
_{2}F_{1}\left( \frac{1}{2},\frac{3}{4},\frac{3}{2},z\right) =\frac
{\Gamma(1/4)}{2\sqrt{2}\pi
{\displaystyle\sum\limits_{n=0}^{\infty}}
\frac{\Gamma(n+1/2)\Gamma(n+3/4)}{\Gamma(n+3/2)}\frac{z^{n}}{n!}.
\]
Although the expression for $\theta^{(-1)}$ is famous, its inverse is
comparatively obscure. \ Kober \cite{Ko, Mt} provide
\[
\theta(z)=\sqrt{2}\operatorname*{sn}\left( \sqrt{2}z,\frac{1}{2}\right)
\operatorname*{dn}\left( \sqrt{2}z,\frac{1}{2}\right) .
\]
Choose the linear fractional transformation $\mathbb{C}^{+}\rightarrow\Delta$
to be $\zeta\longmapsto(\zeta-i)/(\zeta+i)$. \ Thus we wish to solve the
equation $\theta(w)=i$, but this is immediately seen to yield
\[
w=\frac{i}{2}\,_{2}F_{1}\left( \frac{1}{2},\frac{3}{4},\frac{3}{2},-1\right)
=(0.4154481080...)i=(0.3168871006...)\kappa i.
\]
Now let $w\in\tilde{T}$ be arbitrary. \ Define\ $f_{w}:\tilde{T
\rightarrow\Delta$ for which $f_{w}(w)=0$ via \cite{Nh
\begin{align*}
f_{w}(z) & =-\frac{\frac{\theta(z)-i}{\theta(z)+i}-\frac{\theta(w)-i
{\theta(w)+i}}{1-\left( \frac{\theta(z)-i}{\theta(z)+i}\right)
\overline{\left( \frac{\theta(w)-i}{\theta(w)+i}\right) }}\\
& =\frac{\theta(z)-\theta(w)}{\theta(z)-\overline{\theta(w)}
\end{align*}
(following the construction of Green's function in \cite{KS}, but beware of
misprints). \ As $z\rightarrow w$, the ratio $\left\vert f_{w
(z)/(z-w)\right\vert $ simplifies t
\[
h(w)=\left\vert \frac{\theta^{\prime}(w)}{\theta(w)-\overline{\theta(w)
}\right\vert
\]
where $\theta^{\prime}$ denotes the derivative of $\theta$. \ Numerical
minimization gives the least capacity point to b
\begin{align*}
\tilde{w}_{0} & =(0.3977567783173558368923490...)\kappa i\\
& =(1-2t_{0})\kappa i
\end{align*}
as expected, since
\[
\frac{\kappa-\left\vert \tilde{w}_{0}\right\vert }{\sqrt{2}\kappa}=\frac
{\sqrt{2}t_{0}}{1
\]
by the similarity of triangles $\tilde{T}$ and $T$. \ Restricting attention to
the $y$-axis only, we have \
\[
h(iy)=\frac{i\operatorname*{cn}\left( i\sqrt{2}y,\frac{1}{2}\right) ^{3
}{\sqrt{2}\operatorname*{sn}\left( i\sqrt{2}y,\frac{1}{2}\right)
\operatorname*{dn}\left( i\sqrt{2}y,\frac{1}{2}\right) }.
\]
Differentiating with respect to $y$ and setting the result equal to zero, the
equatio
\[
\operatorname*{dn}\left( i\sqrt{2}y,\frac{1}{2}\right) =\sqrt{\frac
{1+\sqrt{3}}{2}
\]
is found, therefor
\[
t_{0}=\operatorname{Re}\left\{ \frac{1}{2\kappa}F\left[ \arcsin\sqrt
{\frac{1+\sqrt{3}}{2}},2\right] \right\} =0.3011216108413220815538254...
\]
is the sought-after closed-form expression. \ Such an outcome was not apparent
in Section 1.
An old paper by Love \cite{Lv} discusses conformal maps on four exceptional
triangles (including the isosceles right triangle) and utilizes the $\wp$
function; unfortunately we haven't succeeded in following the details. Two
other papers\ \cite{CGS, SRS}, despite promising titles, evidently assess
Green's function created for different settings (non-Laplacian) than ours.
\section{Addendum:\ $30^{\circ}$-$60^{\circ}$-$90^{\circ}$ Triangle}
Without entering any lengthy explanations, let \
\[
\kappa=\frac{1}{2^{5/3}\sqrt{\pi}}\Gamma\left( \dfrac{1}{3}\right)
\Gamma\left( \dfrac{1}{6}\right) =\frac{5.2999162508...}{2
\]
and define $T=\left\{ x+iy\in\mathbb{C}:x>0\RIfM@\expandafter\text@\else\expandafter\mbox\fi{, }y>0\RIfM@\expandafter\text@\else\expandafter\mbox\fi{, }\sqrt
{3}x+y<\sqrt{3}\kappa\right\} $. \ The conformal map\ $\theta:T\rightarrow
\mathbb{C}^{+}$ taking prescribed values on the vertices of $T$:
\[
(0,\kappa,i\sqrt{3}\kappa)\overset{\theta}{\longmapsto}(0,1/4,\infty)
\]
is given by
\[
\theta(z)=\frac{3\sqrt{3}\operatorname*{sn}\left( \dfrac{2^{2/3}}{3^{3/4
}z,\dfrac{2+\sqrt{3}}{4}\right) ^{2}\operatorname*{dn}\left( \dfrac{2^{2/3
}{3^{3/4}}z,\dfrac{2+\sqrt{3}}{4}\right) ^{2}}{\left\{ 1+\operatorname*{cn
\left( \dfrac{2^{2/3}}{3^{3/4}}z,\dfrac{2+\sqrt{3}}{4}\right) \right\}
^{4}}.
\]
(We could not decipher pp. 186--187 of \cite{Ko}, hence we turned to pp.
184--185 and made necessary adjustments.) \ For example, $\theta(w)=i$ occurs
when
\begin{align*}
w & =3\left( \frac{1}{2}+\frac{1+i}{\sqrt{2}}\right) ^{1/3}\,_{2
F_{1}\left( \frac{1}{3},\frac{2}{3},\frac{4}{3},\frac{1}{2}+\frac{1+i
{\sqrt{2}}\right) -\kappa\\
& =0.7065812599...+(1.6814450943...)i\\
& =(0.2666386510...)\kappa+(0.6345176092...)\kappa i.
\end{align*}
Now let $w\in T$ be arbitrary. \ Define\ $f_{w}:T\rightarrow\Delta$ for which
$f_{w}(w)=0$ as before; the ratio $\left\vert f_{w}(z)/(z-w)\right\vert $
tends t
\[
h(w)=\left\vert \frac{\theta^{\prime}(w)}{\theta(w)-\overline{\theta(w)
}\right\vert
\]
as $z\rightarrow w$. \ Numerical minimization gives the least capacity point
to b
\begin{align*}
w_{0} & =(0.3599371272406945147550792...)\kappa+\\
& (0.4062604057445303763104149...)\kappa i.
\end{align*}
We have not attempted to find a closed-form expression for $w_{0}$. \ To high
precision,
\[
\frac{1}{h(w_{0})}=(0.2105704622445114724079460...)(2\kappa)=\frac{2^{4/3}\pi
}{5^{5/12}}\Gamma\left( \dfrac{1}{3}\right) ^{-3}(2\kappa)
\]
which is the (corrected) maximum inner radius for $T$ \cite{PS}.
\section{Addendum:\ $6$-$9$-$13$ Triangle}
No such exact formulas can be found for arbitrary triangles. \ The
Schwarz-Christoffel toolbox for Matlab \cite{DT, Dr}, coupled with the
Optimization toolbox, makes numerical computations of least capacity points
readily accessible. \ Recall that we wish to assess whether such points can be
treated as triangle centers. \ For the triangle with vertice
\[
0,\;\;\;6,\;\;\;-\frac{13}{3}+\frac{4\sqrt{35}}{3}i
\]
the following code
\
\begin{array}
[c]{l
\RIfM@\expandafter\text@\else\expandafter\mbox\fi{\texttt{function q = arbitra(w)}}\\
\RIfM@\expandafter\text@\else\expandafter\mbox\fi{\texttt{p = polygon([0 6 -13/3+(4*sqrt(35)/3)*i])}}\\
\RIfM@\expandafter\text@\else\expandafter\mbox\fi{\texttt{f = diskmap(p,scmapopt('Tolerance',1e-18))}}\\
\RIfM@\expandafter\text@\else\expandafter\mbox\fi{\texttt{f = center(f,w(1)+i*w(2));}}\\
\RIfM@\expandafter\text@\else\expandafter\mbox\fi{\texttt{p = parameters(f);}}\\
\RIfM@\expandafter\text@\else\expandafter\mbox\fi{\texttt{q = -abs(p.constant);}
\end{array}
\]
gives (for example) that the inner radius at the centroid i
\[
-\operatorname*{arbitra}\left( \frac{5}{9}+\frac{4\sqrt{35}}{9}i\right)
=1.802305....
\]
Using the centroid as a starting guess, we solve a constrained minimization
problem as follows:
\
\begin{array}
[c]{l
\RIfM@\expandafter\text@\else\expandafter\mbox\fi{\texttt{format long}}\\
\RIfM@\expandafter\text@\else\expandafter\mbox\fi{\texttt{options=optimset('Algorithm','interior-point','TolCon',
1e-15);}}\\
\RIfM@\expandafter\text@\else\expandafter\mbox\fi{\texttt{A = [-4*sqrt(35) -13; 0 -1; 4*sqrt(35) 31]}}\\
\RIfM@\expandafter\text@\else\expandafter\mbox\fi{\texttt{b = [0 0 24*sqrt(35)]}}\\
\RIfM@\expandafter\text@\else\expandafter\mbox\fi{\texttt{v0 = [5/9 4*sqrt(35)/9];}}\\
\RIfM@\expandafter\text@\else\expandafter\mbox\fi{\texttt{[v,fv] = fmincon(@arbitra,v0,A,b,[],[],[],[],[],options);}
\end{array}
\]
yielding the maximum inner radius to be $1.979479...$ and the corresponding
least capacity point to be $0.929617...+(1.842564...)i$. \ This particular
triangle serves as a benchmark in \cite{Kb} to distinguish various centers.
\ The imaginary part is the perpendicular distance from the proposed center to
the shortest triangle side. \ Since the numerical value $1.842...$ does not
appear in the database, we infer that this center is new.
Figures 1, 2, 3 provide conformal map images of ten evenly-spaced concentric
circles in the disk. \ These are optimal in the sense that their center is
\textquotedblleft best insulated\textquotedblright\ from the triangle
boundary. Orthogonal trajectories are also indicated. \
The literature on this subject is larger than we originally thought. \ The
phrase \textbf{conformal center} is sometimes used to denote what we call the
least capacity point. \ (This is not to be confused with a different sense of
the same phrase in \cite{DT, Dr}.) \ Some discussion of relevant numerical
optimization based on the Schwarz-Christoffel transformation occurred years
ago \cite{Fl}. \ Precise inequalities relating radii and various points have
also been formulated \cite{Kz}.
The same phrase is used to denote yet another triangle center in \cite{Ia}.
\ Starting from such a location, a particle undergoing Brownian motion is
equally likely to exit through any of the triangle sides. \ As far as is
known, this topic is distinct from our study. \ Certain integrals and series
in \cite{Ia} deserve greater attention
\begin{figure}[ptb
\centering
\includegraphics[
height=4.2704in,
width=4.4512in
{fg1.eps
\caption{Images of ten concentric circles, center at $0.301+(0.301)i$.
\end{figure}
\begin{figure}[ptb
\centering
\includegraphics[
height=4.2704in,
width=4.3145in
{fg2.eps
\caption{Images of ten concentric circles, center at $0.359+(0.406)i$.
\end{figure}
\ \
\begin{figure}[ptb
\centering
\includegraphics[
height=4.2696in,
width=4.3379in
{fg3.eps
\caption{Images of ten concentric circles, center at $0.929+(1.842)i$.\
\end{figure}
\section{Acknowledgements}
I am grateful to Thomas Ransford \cite{R1, R2} for providing the expression
for $f_{w}$ involving the Weierstrass sigma function, along with detailed
proofs of theorems and answers to several questions.
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{"url":"https:\/\/math.stackexchange.com\/questions\/1702064\/decide-the-sign-of-n-eigenvlues","text":"# Decide the sign of $n$ eigenvlues\n\nWhile solving Linear algebra and Its application by Gilbert Strang, I am not getting any idea how to solve the problem 6.4.14, which says\n\nFrom the zero submatrix decide the signs of the $n$ eigenvalues: $$\\pmatrix{0&.&0&1 \\\\ .&.&0&2 \\\\0&0&0&.\\\\1&2&.&n}$$\n\nSince the matrix has rank $2$, $n-2$ eigenvalues will be zero. Only two remains to be decided. How to determine the sign of other two.\n\n\u2022 do you know how to compute determinants by cofactor expansion? \u2013\u00a0Jon Warneke Mar 17 '16 at 18:10\n\u2022 @PaulSinclair: That's consistent with rank 2. The top $n-1$ rows only give rank $1$ on their own. \u2013\u00a0Greg Martin Mar 17 '16 at 18:25\n\nWhat I am going to write is not fully rigorous, but this site is also a place where people practising mathematics show sometimes how they have an intuitive grasp on certain situations.\n\nI have thought a certain time asking myself what do they mean when they say \"from the zero submatrix decide the signs of the eigenvalues\", and the interlacing property came me back.\n\nThe important result we are going to apply (that you may have not seen) is the Cauchy interlacing theorem for real symmetrical (or hermitian) matrices ; see the theorem 2.3 in http:\/\/www.caam.rice.edu\/~caam440\/chapter2.pdf): when one adds a south-east border to a $(n-1) \\times (n-1)$ matrix having distinct eigenvalues $\\lambda_i$ i.e., adds it a n-th row and and a n-th column, the new \"bordered\" $n \\times n$ matrix has eigenvalues $\\mu_j$ such that:\n\n$$\\mu_1<\\lambda_1<\\mu_2 <\\cdots<\\lambda_{n-1}<\\mu_n \\ \\ \\ (1)$$\n\nDistinct eigenvalues, you said; but here we have bordered a zero matrix, thus all its eigenvalues $\\lambda_1=\\lambda_2= \\cdots =\\lambda_{n-1}$ are zero? All right, let us do a classical trick: we perturbate this zero matrix (for example by adding it a diagonal matrix $diag(\\epsilon, -\\epsilon^2, \\cdots(-1)^{n-1}\\epsilon^{n-1}$) with distinct values which besides are its eigenvalues; in this way, using (1), $\\mu_1<0$ and $\\mu_n>0$. And now we do $\\epsilon \\rightarrow 0$. Using a continuity argument of the eigenvalues with respect to the matrix coefficients, that we don't want to discuss here, we will still have $\\mu_1<0$ and $\\mu_n>0$ at the limit (it is impossible that they tend to zero because the rank of the matrix is 2).\n\nOnce again, all this has to be made more rigorous...\n\nFind the eigenvalues using the definition: $$\\pmatrix{0&.&0&1 \\\\ .&.&0&2 \\\\0&0&0&.\\\\1&2&.&n}\\pmatrix{v_1\\\\ v_2\\\\.\\\\v_n} = \\lambda \\pmatrix{v_1\\\\ v_2\\\\.\\\\v_n} \\Rightarrow \\\\ \\begin{cases} v_n & = \\lambda v_1 \\\\ 2v_n & = \\lambda v_2 \\\\ \\ldots \\\\ (n-1)v_n & = \\lambda v_{n-1} \\\\ \\sum_{i=1}^n iv_i & = \\lambda v_n \\end{cases}.$$\n\nFrom the first $n-1$ equations, we get that:\n\n$$v_i = \\frac{iv_n}{\\lambda},$$\n\nwith $\\lambda \\neq 0$. Using the last equation of the system, we have:\n\n$$\\sum_{i=1}^n iv_i = \\lambda v_n \\Rightarrow \\sum_{i=1}^{n-1} i\\frac{iv_n}{\\lambda} + nv_n = \\lambda v_n \\Rightarrow \\\\ \\frac{v_n}{\\lambda}\\sum_{i=1}^{n-1} i^2 + nv_n = \\lambda v_n \\Rightarrow \\sum_{i=1}^n i^2 = \\lambda(\\lambda -n) \\Rightarrow\\\\ \\lambda^2 - \\lambda n - \\sum_{i=1}^n i^2 = 0.$$\n\nUsing Descartes rule, we get that the previous equation has one positive and one negative solution ($n$ and $\\sum_{i=1}^ni^2$ are both positive).\n\nConcluding: you have $n-2$ null eigenvalues, 1 positive eigenvalue and 1 negative eigenvalue.","date":"2021-04-11 11:18:50","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8394222259521484, \"perplexity\": 223.88034982818888}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-17\/segments\/1618038061820.19\/warc\/CC-MAIN-20210411085610-20210411115610-00021.warc.gz\"}"}
| null | null |
{"url":"http:\/\/ringtheory.herokuapp.com\/properties\/property\/60\/","text":"# Property: characteristic 0 field\n\nDefinition: The sum of any positive number of 1's is always nonzero.","date":"2018-06-22 07:05:28","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8267135620117188, \"perplexity\": 1164.221854576105}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-26\/segments\/1529267864364.38\/warc\/CC-MAIN-20180622065204-20180622085204-00240.warc.gz\"}"}
| null | null |
\section{Introduction}
The assembly of galactic discs is regulated by gas accretion, star formation, feedback from stars and active galactic nuclei (AGN), and galaxy mergers. The existence of a tight main sequence (MS) between the star formation rates (SFR) and stellar masses ($\rm M_\star$) of galaxies suggests that discs assembled in quasi-equilibrium between these processes \citep[e.g.][]{Bouche10, Dave12, Lilly13, Dekel14}. Integral Field Spectroscopy (IFS) provides spatially-resolved information on the internal kinematics, gas content and SFRs of galaxies, which provide insight into the relative importance of the various physical drivers of galaxy evolution operating at different epochs \citep[see][for a review]{Glazebrook13}.
Ionised gas emission obtained by several IFS surveys has been used to study the kinematic evolution of star-forming galaxies (SFGs) at redshifts $z = 1-3$ \citep[e.g.][]{FS09, Wisnioski15, Stott16, Turner17}. One important finding of these studies is that by $z\approx 2$ many disc galaxies are already dominated by ordered rotation comparable to that of their low-redshift counterparts \citep[e.g.][]{FS06, FS09, Cresci09, Law09, Wisnioski15}. The structure of these systems is often characterised by a thick and clumpy disc with large star-forming regions \citep[e.g.][]{Elmegreen07, Fisher17}.
The "turbulent" nature of the interstellar medium (ISM) of high redshift galaxies is observed in the velocity dispersion of their ionised gas, which is a factor of $2-5$ higher than in local SFGs. Using the {\sf KMOS$^{\rm 3D}$} sample, \citet{Ubler19} showed that the ionised velocity dispersion increases approximately linearly with increasing redshift. A similar evolution (although shifted to systematically lower values of velocity dispersion) is inferred from HI \citep{Dib06, Mogotsi16} or CO \citep[e.g.][]{Leroy09, Swinbank11, Tacconi13} emission, indicating that the cold-phase of ISM also becomes more turbulent at higher redshifts \citep[see also][]{Ubler18, Girard21}.
The dissipational nature of the ISM of galaxies suggests that turbulence should decay on a time scale comparable to the dynamical time ($\sim$ 100 Myr) of the galaxy disc with a \citet{Toomre64} stability parameter $Q\approx 1$ \citep{Krumholz18}. In contrast, turbulence should decay on shorter timescales ($\sim$ 10 Myr) in giant clumps. Hence, a continuous injection of energy into the ISM is necessary to sustain high levels of turbulence for several Gyr \citep[e.g][]{MacLow98, Stone98}. A number of energy sources have been proposed to explain the evolution of gas velocity dispersion in discs. The main contributors can be grouped in three categories: (i) the injection of feedback energy from massive stars and AGN, (ii) dynamical effects driven by local gravitational instabilities and (iii) cosmological accretion via cold flows.
Star formation feedback can drive turbulence via the injection of kinetic and thermal energy from stellar winds, radiation pressure and supernovae explosions, where the latter is thought to be the dominant effect \citep[e.g][]{MacLow04, Ostriker11}. The observed correlation between the velocity dispersion and SFRs of gaseous discs supports this hypothesis \citep[e.g][]{Dib06, Green10, Green14, Johnson18}. {There is an ongoing debate about whether feedback has a local or global effect on gas turbulence; some studies have found a weak correlation between the SFR surface density ($\Sigma_{\rm SFR }$) and gas velocity dispersion \citep[e.g.][]{Genzel11, Zhou17, Ubler19}, while others claim that there is a significant correlation \citep[e.g.][]{Lehnert13, Varidel20}.}
Gravitational disturbances due to local instabilities may also act to increase the velocity dispersion \citep[e.g.][]{Aumer10}. Various studies have shown that local disc instabilities induce clump formation that generate radial flows within the disc enabling gas in the disc's outskirts to release gravitational potential energy and fall towards the disc's centre \citep[e.g.][]{Ceverino10, Krumholz10}. This mechanism is expected to be more important at high redshifts when discs had higher gas fractions and were therefore more vulnerable to forming local instabilities. Indeed, it appears that most high redshift disc galaxies contain a large number of massive, dense clumps in which most of their star formation occurs \citep[e.g.][]{Genzel11}.
In addition to these internal drivers of turbulence, various external sources have also been considered, among them are cosmological gas accretion onto discs and galaxy interactions, such as major or minor mergers. Gas accretion via cold flows from the cosmic web is needed to replenish discs with fresh gas but, prior to settling into a rotationally supported structure, can contribute to disc turbulence \citep[e.g][]{Forbes22}. The relationship between turbulence and gas accretion is complex and depends on the smoothness of the cold streams \citep[e.g.][]{Mandelker18, Mandelker20}, on the density contrast between the stream and the disc \citep{Klessen10}, and as we show in Section~\ref{sec: PhysicalDrivers}, on the orientation of the accreting material with respect to the disc plane. Finally, galaxy mergers may also play a role by changing the dynamical and morphological structure of discs \citep[e.g.][]{DiMatteo11, Lagos18}.
If discs remain in quasi-equilibrium as they grow, the evolution of their SFR and gas content can be related to the evolution of their gas velocity dispersion. Under this framework, \citet[][hereafter K18]{Krumholz18} developed an analytic model for the evolution of galactic discs that accounts for stellar feedback and radial gas transport originated from disc instabilities as the two main sources of gas turbulence. In their model, both sources of turbulence are needed to explain the observed evolution of the velocity dispersion of gaseous discs. \citet{Ginzburg22} extended K18's model by incorporating an analytical prescription to inject turbulence generated by cosmological gas accretion. Depending on redshift and the mass of the disc's parent dark matter (DM) halo, they found that gas accretion can indeed be an important driver of turbulence, especially at high redshift.
Validating these analytic prescriptions requires access to statistically representative samples of disc galaxies. Due to improvements in resolution and simulation volume, and to the implementation of sophisticated sub-resolution prescriptions for unresolved physics, the latest generation of cosmological simulations are useful tools for studying the evolving dynamics of gaseous discs. For example, \citet{Pillepich19} used the TNG50 simulation to study the evolution of gas velocity dispersion in SFGs, finding a similar evolution with redshift as that inferred from IFS surveys. However, the connection between turbulence and the various physical drivers mentioned above has not been studied using hydrodynamical simulations of cosmologically representative volumes.
In this paper, we use the {\sc eagle}{} simulations \citep{Schaye15, Crain2015} to study the evolution of gas turbulence and how it is related to the assembly of gaseous discs. We focus on two main questions: (i) what is the relation between gas velocity dispersion and the various potential drivers of turbulence, such as the SFR, mergers, or gas accretion?; and (ii) do these relations depend on halo mass and/or redshift? {\sc eagle}{} is a suitable tool to carry out this analysis as it reproduces well the main sequence of star formation from $z=0$ to $z\approx 5$ \citep{Furlong15, DSilva22}, the SFR and stellar mass functions from $z=0$ to $z\approx 4$ \citep{Furlong15, Katsianis17}, the abundance of molecular gas as a function of cosmic time \citep{Lagos15} and its relation to SFR and $\rm M_\star$ \citep{Lagos16} at $z=0-4$, and stellar kinematics properties of galaxies \citep[e.g.][]{Ludlow2017,Lagos17, Lagos18b, Swinbank17, WaloMartin20}.
The outline of this paper is as follows. In Section~\ref{Methods} we introduce the {\sc eagle}\ simulation, define our galaxy sample and explain how we measure the velocity dispersion of gaseous discs. In Section~\ref{SecEvsigma} we compare the gas velocity dispersions measured for our sample of {\sc eagle}{} galaxies with those inferred from various observational data sets, and also present scaling relations between the gas velocity dispersion and various other galaxy properties. In Section~\ref{sec: PhysicalDrivers} we present our main results, providing an interpretation and discussion in Section~\ref{sec: Discussion}; Section~\ref{sec: Conclusions} contains our conclusions.
\section{Methods} \label{Methods}
\subsection{The {\sc eagle}\ simulations}
{\sc eagle}\ is a suite of cosmological, smoothed particle hydrodynamical (SPH) simulations that follow the assembly of DM haloes and the formation of galaxies within them \citep{Schaye15, Crain2015}. The simulations were carried out using a modified version of the $N$-body SPH code GADGET-3 \citep{Springel05, Springel08}, employing cosmological parameters obtained by the \citet{Planck14}. The various {\sc eagle}\ runs adopt a range of cosmological box sizes, mass and force resolutions, and subgrid physics models, and are comprised of 28 outputs (snapshots) spanning $z=20$ to $z=0$. Table~\ref{tab: eagle-runs} provides the relevant details of the {\sc eagle}{} runs used in this paper.
\begin{table}
\centering
\begin{tabular}{l|ccc}
\hline
Parameter & {\sc NoFb-L25} & {\sc NoAGN-L50} & {\sc Ref-L100}\\
\hline
$L$ [cMpc] & 25 & 50 & 100\\
$N_{\rm part}$ & $2\times 376^3$ & $2\times 752^3$ & $2\times 1504^3$\\
$m_{\rm gas}$ $\rm [M_{\odot}]$ & $1.81\times 10^6$ & $1.81\times 10^6$ & $1.81\times 10^6$\\
$m_{\rm DM}$ $\rm [M_{\odot}]$ & $9.70\times 10^6$ & $9.70\times 10^6$ & $9.70\times 10^6$ \\
$\epsilon_{\rm com}$ $\rm [ckpc]$& 2.66 & 2.66 & 2.66\\
$\epsilon_{\rm prop}$ $\rm [pkpc]$& 0.70 & 0.70 & 0.70\\
\hline
\end{tabular}
\caption{{\sc eagle}\ simulation runs used in this paper. Rows from top to bottom show: the simulation name suffix; the box comoving size; the initial number of DM and gas particles; initial mass of gas particles; DM particle mass; comoving, Plummer-equivalent gravitational softening length at $z\geq 2.8$; and maximum proper softening length at $z<2.8$. Here, cMpc, ckpc and pkpc refer to comoving megaparsec and kiloparsec, and proper kiloparsec, respectively.}
\label{tab: eagle-runs}
\end{table}
DM haloes were identified in each snapshot using a Friends-of-Friends (FOF) algorithm \citep{Davis85} with a linking length of $0.2$ times the mean (Lagrangian) DM inter-particle separation. SUBFIND \citep{Springel01} was then run on the FOF haloes to identify gravitationally bound DM subhaloes, which are the potential hosts of galaxies. The baryonic content of these subhaloes was determined by associating each baryonic particle (gas or stellar) to its nearest DM particle, provided the latter is bound to a SUBFIND subhalo. The mass of each FOF halo is dominated by a central subhalo; the galaxy it contains (if any) is defined as the central galaxy. Each FOF halo also has a population of lower-mass satellite subhaloes, which are the potential hosts of satellite galaxies. Our analysis focuses exclusively on central galaxies.
For each central galaxy and its DM halo, SUBFIND determines a number of relevant physical properties including its virial mass and radius, $\rm M_{200}$ and $r_{200}$ respectively, and the stellar and gas mass of its central galaxy. In what follows we define $M_{200}$ as the mass contained within a sphere of radius $r_{200}$ that encloses an average density equal to $200$ times the critical density of the universe ($\rho_{\rm crit}=3 H^2/8 \pi G$, where $H$ is the Hubble-Lema$\hat{i}$tre constant and $G$ is the gravitational constant). The virial velocity, $V_{\rm 200}=\sqrt{G {\rm M_{200}}/r_{200}}$, corresponds to the circular velocity at $r_{200}$. Note that all particle types are included in the calculation of the virial quantities above. The stellar and gas masses of each central galaxy are calculated by summing the individual masses of each stellar or gaseous particle belonging to the central galaxy, excluding those that are bound to satellite subhaloes.
The interplay of several physical processes governs the condensation of baryons in the central regions of DM haloes, their subsequent conversion into stellar particles and the build-up of galaxies. Specifically, {\sc eagle}\ models a variety of physical processes, such as radiative gas cooling and photoheating \citep{Wiersma09b}, star formation \citep{Schaye08}, stellar and chemical evolution \citep{Wiersma09}, stellar mass loss and feedback from supernovae \citep{DallaVecchia12}, the formation and growth of supermassive black holes (BHs), and AGN feedback \citep{Rosas-Guevara15}. Galaxy mergers and gas accretion are natural consequences of the simulation's cosmological initial conditions.
The majority of our analysis is based on the $100$~cMpc Reference model of the {\sc eagle}{} simulation suite (hereafter the {\sc Ref-L100} run). Due to the finite resolution of cosmological simulations and our limited understanding of several physical processes relevant for galaxy evolution, such as stellar and AGN feedback, a set of sub-grid prescriptions for unresolved physical processes are usually employed. In general, the equations that implement these processes contain free parameters that must calibrated so that simulation results reproduce important observations of the galaxy population. {\sc eagle}'s sub-grid physics models were calibrated so that the simulation reproduced the observed local Universe stellar mass function, the galaxy size-$\rm M_\star$ relation, and BH mass-$\rm M_\star$ relations (see \citealt{Crain2015} for details).
The {\sc eagle}\ suite also includes runs that adopt variations of the subgrid parameters employed for the Reference model. These runs were not required to match the observations mentioned above but can nonetheless be used to explore the effect of changing various subgrid parameters on the galaxy population (see \citealt{Crain2015} for a detailed discussion). We make use of two such runs: one in which feedback from stars {\em and} AGN was turned off (hereafter, {\sc NoFb-L25}), and another that includes stellar feedback but none from AGN (hereafter {\sc NoAGN-L50}). For both of these runs, the remaining subgrid parameter values were identical to those used in the {\sc Ref-L100} run. We use these runs to assess the impact of stellar and AGN feedback on the kinematics of gaseous disc galaxies. Note that the {\sc NoFb-L25} model was carried out in a $25$~cMpc simulation box, while the {\sc NoAGN-L50} was run in a $50$~cMpc cubed volume.
We focus our analysis on the redshift range $0.1<z<4$, which overlaps with most kinematic measurements of gaseous discs from spectroscopic surveys (see Section~\ref{SecEvsigma}). Snapshots in the {\sc NoFb-L25} run are only available down to $z=0.1$; hence we adopt this as the lower redshift bound for most of our analysis, although we include $z=0$ results from the {\sc Ref-L100} and {\sc NoAGN-L50} runs for comparison with observations. We note that the quantities of interest for this work -- gas velocity dispersions, stellar and gas masses, etc -- evolve very little between $z=0.1$ and $z=0$.
Below, we describe the relevant aspects of {\sc eagle}'s sub-grid models.
\subsection{Modelling star formation and feedback from stars and AGN}
Star formation in {\sc eagle}\ is implemented stochastically following the Kennicutt-Schmidt star formation relation reformulated as a pressure law \citep[see][for details]{Schaye08}. The pressure is determined using a polytropic equation of state, $P = P_{\rm eos}(\rho)$, normalised to a temperature floor $T_{\rm eos}=8000\ {\rm K}$ at density $n_{\rm H}=10^{-1}\ {\rm cm^{-3}}$.
Because {\sc eagle}\ does not resolve the cold phase of the ISM, the simulation triggers star formation stochastically in gas particles that exceed a metallicity-dependent gas density threshold, $n_{\rm H}^{*}(Z)$,
\begin{equation} \label{eq: rho_crit}
n_{\rm H}^* = 10^{-1}\ {\rm cm^{-3}}\ \left(\frac{Z}{0.002}\right)^{-0.64}.
\end{equation}
\noindent Here, $Z$ is the gas metallicity. The metallicity-dependence of equation~(\ref{eq: rho_crit}) approximately accounts for the increased cooling efficiency and enhanced shielding by dust grains that are expected in high-metallicity gas, which in practice lowers the density required for the transition from the warm to the cold phase of the ISM, and hence also lowers the density threshold for star formation \citep[e.g.][]{Richings14}.
Stellar particles are considered as simple stellar populations characterised with a \citet{Chabrier03} initial mass function (IMF); stars with masses in the range $6-100\, \rm M_{\odot}$ are assumed to end their lives as core-collapse supernovae after $3\times 10^7\, \rm yr$ from their formation time.
\citet{Dalla-Vecchia12} introduced a stochastic approach to supernova feedback that allows the amount of thermal energy injected per supernovae to be controlled in order to overcome numerical radiative losses due to poor resolution. The fraction of energy from a supernova that is injected into the neighbouring gas particles is governed by the feedback efficiency parameter, $f_{\rm th}$. A value of $f_{\rm th}=1$ indicates that all energy produced by supernovae is imparted to the gas, whereas a low value implies less efficient feedback. The exact value of $f_{\rm th}$ in {\sc eagle}\ depends on the local conditions of the ISM, such as the gas-phase metallicity and density. For the {\sc Ref-L100} and {\sc NoAGN-L50} runs, the mean values of $f_{\rm th}$ at $z=0.1$ are close to $1$, while for the {\sc NoFb-L25} run $f_{\rm th}$ is manually set to zero.
The expectation value for the number of heated gas particles per supernova, $\left< N_{\rm heat}\right>$, is proportional to the feedback efficiency parameter, and is given by
\begin{equation} \label{eq: N-heated}
\left< N_{\rm heat}\right> \approx 1.3 f_{\rm th} \left( \frac{\Delta T}{10^{7.5}{\rm K}}\right)^{-1},
\end{equation}
where $\Delta T=10^{7.5}\,\rm K$ is the desired temperature increment of the heated gas particles. $\Delta T$ remains fixed and is chosen to be high in order to prevent gas overcooling.
A BH seed of mass $\rm M_{BH}=1.5\times 10^5\, M_\odot$ is placed in the centre of every newly-formed DM halo whose FOF mass exceeds $\gtrsim 10^{10}\, {\rm M_\odot}$; they subsequently grow in mass by accreting neighbouring gas particles and by merging with other BH particles \citep[e.g][for details]{Springel05a}. As with stellar feedback, AGN feedback is implemented stochastically. In the case of AGN feedback, however, a higher temperature increment of $\Delta T=10^{8.5}\ \rm K$ is adopted \citep{Booth09}. This ensures AGN feedback remains efficient when the associated energy is injected into gas particles near the central BH, which typically have higher densities than those surrounding stellar particles and are therefore more prone to radiative losses.
Gas accretion onto BHs is modelled following the Bondi-Hoyle accretion model \citep{Bondi-Hoyle44}. An efficiency parameter is introduced in the AGN feedback model which accounts for the amount of rest energy from the accreted gas that is injected into the surroundings. \citet{Rosas-Guevara15} introduced an additional dependence on the accretion-disc angular momentum which can suppress the accretion onto the supermassive BH and reduce the AGN feedback efficiency.
\subsection{Calculation of gas accretion rates} \label{subsec: CalcGasAcc}
{We link galaxy descendants and progenitors in adjacent snapshots using the galaxy merger trees available in the {\sc eagle}\ database \citep{McAlpine16,Qu17}. We use this information together with the particle data to compute gas accretion rates for each {\sc eagle}\ galaxy in our sample. We will use these accretion rates in Section~\ref{SecGasAcc} to study correlations between gas accretion and turbulence.}
We compute time-integrated accretion rates (i.e. inflows averaged over a finite time interval $\Delta \rm t$) using the spatial distribution of particles in adjacent snapshots, \{$j+1,\, j$\}, where $j+1$ refers to the lower redshift of the two snapshots; descendant galaxies therefore belong to snapshot $j+1$ and progenitors to snapshot $j$. In {\sc eagle}, the time between $j$ and $j+1$ ranges from $\approx 0.3\ {\rm Gyr}$ (for $z>2$) to $1\ {\rm Gyr}$ (for $z < 1$). As shown by \citet[][see also \citealt{Wright20}]{Mitchell20a}, poor temporal resolution can affect estimates of accretion rates by not properly accounting for particles that had been accreted but subsequently lost on a timescale shorter than the time between adjacent snapshots. However, accretion rates are well converged when averaged over longer time intervals (e.g. 1$\, {\rm Gyr}$), which is comparable to the temporal spacing between snapshots in {\sc eagle}\ at low redshifts.
We calculate the total cold gas mass transferred from the circum-galactic medium (CGM) to the disc between consecutive snapshots. For each galaxy in snapshot $j+1$, we select all cold gas particles in the disc and track their 3D positions to their main progenitor galaxy in snapshot $j$. Particles whose radial separation from the progenitor's COP exceeds $0.2\, r_{200}$\footnote{Note that we use the centre of potential and $r_{200}$ values of the progenitor galaxy in the snapshot $j$. } (which we use as a boundary to delineate the disc and CGM) are considered accreted particles. We also account for gas particles that were converted into star particles in the disc during the interval $\Delta t$ by selecting star particles in the disc in snapshot $j+1$ that were CGM gas particles at snapshot $j$.
The net accretion rate, $\dot{\rm M}_{\rm acc}$, is defined as the sum of the mass of the accreted particles (stellar or gas) divided by the time interval $\Delta t$ between the two snapshots.
\subsection{Galaxy sample selection}
\label{sec: GalSamples}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{Fig1.pdf}
\caption{The $\rm M_{\star}-M_{200}$ relation in the {\sc Ref-L100} (blue), {\sc NoAGN-L50} (light brown) and {\sc NoFb-L25} (red) runs at $z=0.1$ (solid), $z=1$ (dashed) and $z=2$ (dotted). The relations are shown for galaxies that meet our selection criteria (see text for details). Lines show the medians in bins with $\geq 10$ galaxies while shaded regions (only shown for $z=0.1$) show the $16^{\rm th}$ and $84^{\rm th}$ percentiles. The vertical grey bands indicate the halo mass bins analysed throughout the paper. }
\label{fig: stellar-to-halo-mass rel}
\end{figure}
We focus our analysis on the cold gas content of central galaxies with $\rm M_{\star} \geq 10^9\,\rm M_{\odot}$. We do not consider satellite galaxies in our analysis because they are more sensitive to external sources of turbulence, such as ram-pressure stripping and strong tidal interactions \citep[e.g.][]{Gunn72, Boselli06, Bahe15, Wright22}.
We define the cold gas component of galaxies as the subset of gas particles that have temperatures $T\leq 10^4\, {\rm K}$ {\em or} a non-zero SFR\footnote{Selecting star-forming gas particles (which can potentially convert into star particles) implies selecting regions that are likely to be exposed to radiation from massive stars, which ionises the surrounding gas creating HII regions; our selection of ``cold gas'' particles therefore approximately traces all phases of the ISM.}. The first condition approximately selects the warm phase of the ISM which is primarily composed of atomic hydrogen; the latter condition targets the unresolved cold phase which is most likely to contain the molecular hydrogen component. These criteria for identifying the cold gas phase of simulated galaxies are widely adopted in the literature \citep[e.g][]{Wright21}.
We also focus our analysis on galaxies that harbour a prominent gaseous disc, in line with the majority of observational studies. To identify discs we use the kinematic indicator $\kappa_{\rm co}$, introduced in \citet{Correa17}, which quantifies the fraction of the disc's kinetic energy that is invested in co-rotation, i.e.
\begin{equation} \label{eq: kappa_co}
\kappa_{\rm co} = \frac{1}{K} \sum_{i, L_{z,i}>0} \frac{1}{2}m_{i} \left(\frac{L_{z,i}}{m_{i} R_{ i}}\right)^2,
\end{equation}
\noindent where $K=0.5\sum_{i} m_{i}\, v^2_{i}$ is the total kinetic energy of the particles, $R_{i}=(r_{i}^2-z_i^2)^{1/2}$ is the distance of the $i^{\rm th}$ particle to the galaxy's net angular momentum vector (which we align with the $z$-axis, in our analysis), and $L_{z,i}$ is the $z$-component of the particle's angular momentum. The sum extends over all particles of the relevant species that lie within a spherical aperture of radius $r=2\, r_{50}$ and also have a positive $L_{z}$ ($r_{50}$ is the three dimensional half-mass radius of the galaxy's gas component).
\citet{Correa17} applied equation~(\ref{eq: kappa_co}) to stellar particles in {\sc eagle}\ galaxies and found that $\kappa_{\rm co}\gtrsim 0.4$ approximately marks the transition between passive, spheroidal galaxies and star-forming disc galaxies. \citet{Thob19} showed that $\kappa_{\rm co}$ correlates with a number of other kinematics properties used to characterise simulated galaxies, such as the ratio of rotation to dispersion velocities, $v_{\rm rot}/\sigma_{\rm 1D}$, the stellar spin parameter, $\lambda_\star$, and the orbital circularity parameter $\xi=j_{z}/j_{\rm c}$.
Unlike most previous studies, we follow \citet{Manuwal22} and use equation~(\ref{eq: kappa_co}) to characterise the morphologies of each galaxy's gas component. By visually inspecting edge-on projections of the cold gas distribution for a large sample of {\sc eagle}~ galaxies, we find that $\kappa_{\rm co}\geq 0.7$ singles-out thin gas discs.
In {\sc eagle}, galaxies with $\kappa_{\rm co}\gtrsim 0.7$ typically have $v_{\rm rot}/\sigma_{\rm 1D}\gtrsim 2$.
We note that imposing a restriction on $\kappa_{\rm co}$ is necessary to select gaseous discs in {\sc Ref-L100}, but not for {\sc NoFb-L25}. Indeed, most galaxies in the latter run satisfy $\kappa_{\rm co}>0.9$, suggesting that, in the absence of feedback, cold gas is primarily concentrated in a thin (albeit compact) rotating disc. Applying this kinematic cut removes two main galaxy groups: (i) massive galaxies at low redshifts, which are generally passive, round-shaped, and have low gas content, and (ii) low-mass galaxies at high redshift, whose gaseous component has not yet settled into a rotating disc.
To simplify the interpretation of our results -- and to allow us to isolate the impact of feedback, gravitational instabilities, and gas accretion on the vertical velocity dispersion of gaseous discs -- for most of our analysis we remove galaxies that have undergone recent mergers. We note, however, that this additional selection criterion does not affect our results, mainly because galaxy mergers are rare, affecting only 2 per cent of all galaxies in our sample at $z=0.1$. We consider the impact of merger on $\sigma_z$ explicitly in Section~\ref{SecGalMergers}.
Finally, we only consider galaxies whose dynamical properties -- particularly their vertical gas velocity dispersion, $\sigma_z$, the calculation of which we describe below -- are computed using at least $500$ cold gas particles. This reduces the effects of Poisson noise on estimates of the kinematic properties of galaxies that contain low numbers of cold gas particles.
Imposing these criteria on all galaxies with $\rm M_\star\geq 10^9 \, M_{\odot}$ removes roughly $56$ per cent of galaxies at $z=0$, and $77$ per cent at $z=2$. Nevertheless, the remaining galaxies (of which there are $3277$ at $z=0$ and $1361$ at $z=2$) sample a wide range of galaxy properties. Indeed, we find that the star formation main sequence is well-sampled at all redshifts (for ${\rm M_{\star}} \ge 10^{9}\,\rm M_{\odot}$).
Fig.~\ref{fig: stellar-to-halo-mass rel} shows the stellar-to-halo mass relation for all galaxies that meet our selection criteria. We show results at $z=0.1$, $z=1$ and $z=2$ and for the three {\sc eagle}{} runs listed in Table~\ref{tab: eagle-runs}. As expected, the {\sc NoAGN-L50} run produces higher stellar masses at fixed $\rm M_{200}$ than the {\sc Ref-L100} run at $\rm M_{200}\gtrsim 10^{12}\,\rm M_{\odot}$ due to the lack of AGN feedback. At fixed $\rm M_{\star}$, the galaxies in {\sc NoFb-L25} are biased towards low halo masses. The lack of massive haloes in this run is due to the small simulation volume, which is only 25 cMpc on side. Beyond volume, the clearest difference between the {\sc NoFb-L25} and the other two runs is the much larger $\rm M_{\star}$ of galaxies at fixed $\rm M_{200}$, which is expected when feedback is absent and therefore unable to suppress gas accretion and star formation. This is important to keep in mind when comparing galaxies at fixed $\rm M_{\star}$ between runs. Fig.~\ref{fig: stellar-to-halo-mass rel} also highlights three halo mass bins (vertical grey bands) that we use throughout the paper when comparing galaxies across all three simulations.
\subsection{Calculation of the ISM gas velocity dispersion} \label{sec: sigma_z calculation}
We compute the vertical ISM gas velocity dispersion, $\sigma_z$, which is the component perpendicular to the plane of the galaxy's cold gas disc. To do so, we calculate the relative velocities of the gas particles with respect to the centre of mass (COM) of their host galaxy: $\Delta v = |\vec{v} - \vec{v}_{\rm COM}|$. The COM velocity vector, $\vec{v}_{\rm COM}$, is determined using stars, gas and DM particles within a sphere of radius $0.2\, r_{200}$ centred in the centre of potential (COP) of the halo, as defined by SUBFIND. We define the disc plane as the plane perpendicular to the total angular momentum vector of the cold gas and young stars ($< 100$ Myr old). The latter is computed using the particles enclosed in the cold gas 3D half-mass radius, $r_{50}$. We reset all particle positions and velocities with respect to the COM reference frame and align the $z$-axis with the angular momentum of the disc as defined above. We then select cold gas particles that lie within an appropriately-defined cylinder and compute their mass-weighted $\sigma_z$ profile using
\begin{equation} \label{eq: sigmaz}
\sigma_z = \sqrt{\frac{\sum_i m_i \left[v_{z,i}^2 + \sigma_{\rm P}^2/3 \right]}{\sum_i m_i}},
\end{equation}
where $\sigma_{\rm P}=\sqrt{{\rm P}/\rho}$ corresponds to the thermal component of the velocity dispersion, which depends on the density and pressure of the gas\footnote{For a gas particle with temperature $\rm 10^4\ K$, the thermal contribution (assumed to be isotropic) corresponds to $\rm \sigma_{P}\approx 11\ km\ s^{-1}$. Note that $\rm \sigma_{P}/\sqrt{3}$ is the one dimensional thermal contribution to the velocity dispersion.}. The dimensions of the cylinder over which equation~(\ref{eq: sigmaz}) is applied is determined as follows. First, the height of the cylinder, $z_{90}$, is chosen so that it encloses $90$ per cent of all cold gas particles in the galaxy. However, we impose a lower and higher limit to $z_{90}$ of $1\, {\rm kpc}$ and $6\,{\rm kpc}$, respectively. This ensures that the scale height is always larger than the gravitational softening length ($\approx 0.7\,{\rm kpc}$), and small enough to only encompass cold gas that is located close to the disc plane. We also note that, at high redshifts, a significant amount of cold gas is contained in clumps away from the disc plane; the upper limit of $6$~kpc excludes most of these clumps, which are not themselves part of the gas disc. Overall, imposing these limits on $z_{90}$ does not have a significant impact on our results.
With $z_{90}$ fixed, we next determine the radius $R_{\rm flat}$ where the $\sigma_z$ profile becomes flat. This is what is usually done in observational studies and aims to exclude the central regions of galaxies where velocity dispersion measurements are susceptible to the effects of beam-smearing. For most galaxies (see Fig.~\ref{fig: sigma_profiles} below) in the {\sc Ref-L100} run, the $\sigma_z$ profiles become flat beyond $R_{\rm flat}={\rm a\, few}\,R_{50}$, with $R_{50}$ being the cylindrical cold gas half-mass radius. In practice, we measure $\sigma_z$ at $R_{\rm flat}=3\,R_{50}$\footnote{This is different from $r_{\rm 50}$, which is the 3D cold gas half-mass radius. In general, we use lower base $r$ to denote three dimensional radii and upper case $R$ to denote radii within the disc plane.}.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{Fig2.pdf}
\caption{Vertical velocity dispersion profiles of cold gas (equation~\ref{eq: sigmaz}) for $z=0.1$ galaxies in the {\sc Ref-L100}. Different panels correspond to separate bins of $\rm M_{200}$, as labelled (the values are expressed in units of ${\rm log_{10}\, M_{\odot}}$). Radii have been normalised by $R_{50}$, cylindrical half-mass radius of cold gas. The vertical dotted lines indicate the radius $R_{\rm flat}=3\,R_{\rm 50}$ within which the profiles are evaluated. Thick grey curves show the median $\sigma_z(R)$ profiles for all disc galaxies in each mass bin; thick light (dark) blue lines correspond to the lower (upper) quintile of the $\sigma_z(R_{\rm flat})$ distribution in the same mass bins (thin lines of corresponding colour show individual profiles in these subsamples).}
\label{fig: sigma_profiles}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{Fig3.pdf}
\caption{Evolutionary tracks of the gas velocity dispersion for the $z=0.1$ disc galaxies of Fig.~\ref{fig: sigma_profiles}. Different panels correspond to separate bins of $\rm M_{200}$ as indicated by the labels. Dark (light) blue colours refer to galaxies whose $\sigma_z$ at $z=0.1$ is in the top (bottom) quintile of the of the $\sigma_z$ distribution. Solid lines indicate the median evolutionary tracks whereas the shaded regions show the corresponding $16^{\rm th}-84^{\rm th}$ percentiles; both are shown for samples with more than 10 galaxies. }
\label{fig: sigma_tracks}
\end{figure}
Fig.~\ref{fig: sigma_profiles} shows the gas velocity dispersion profiles computed using equation~(\ref{eq: sigmaz}) for galaxies at $z=0.1$ in {\sc Ref-L100}. The galaxies shown here meet the selection criteria introduced in Section~\ref{sec: GalSamples}; the different panels correspond to different $\rm M_{200}$ bins (see the vertical grey bands in Fig.~\ref{fig: stellar-to-halo-mass rel}). The grey thick lines show the median $\sigma_z(R)$ profiles for all galaxies in each halo mass bin; the dark and light blue lines correspond to those in the highest and lowest $20^{\rm th}$ percentiles of $\sigma_z$, respectively. In all cases, the $\sigma_z$ profiles are approximately flat at $3\,R_{50}$, which we define as $R_{\rm flat}$. An important feature is that galaxies that have a low (high) value $\sigma_z(R_{\rm flat})$ are characterised by $\sigma_z$ profiles that are consistently low (high) compared to the median across all radii, showing that $\sigma_z(R_{\rm flat})$ is a useful way to characterise the $\sigma_z$ profile using a single number. From this point onward, $\sigma_z$ will refer to the value obtained from equation~(\ref{eq: sigmaz}) using $z_{90}$ and $R_{\rm flat}=3\,R_{50}$ as the height and length of the cylinder, respectively.
Galaxies in {\sc NoFb-L25} typically have smaller $R_{50}$ values than those in {\sc Ref-L100} at the same $\rm M_{\star}$ (by about a factor of 8) and their $\sigma_z$ profiles flatten at much larger multiples of $R_{50}$, closer to $\approx 10\, R_{50}$. Hence, for {\sc NoFb-L25}, we adopt $R_{\rm flat}= 10\, R_{50}$. We note, however, that none of our results are impacted by the choice of $R_{\rm flat}$ provided is encloses most of a galaxy's cold gas mass.
In Appendix~\ref{app: Convergence} we present convergence tests that compare results obtained from {\sc eagle}{} runs with higher mass and force resolution to those obtained from the Reference model. We find that the evolution of $\sigma_z$ and its dependence on halo mass are in good agreement between the runs suggesting our results are not unduly affected by numerical resolution.
\begin{figure*}
\centering
\includegraphics[width=0.8\textwidth]{Fig4.pdf}
\caption{The evolution of the vertical cold gas velocity dispersion, in the {\sc Ref-L100} (blue), {\sc NoFb-L25} (red) and {\sc NoAGN-L50} (light brown) runs, which are compared to the following observational results: {\sf GHASP} \citet{Epinat08, Epinat10}, {\sf SAMI} \citet{Croom12, Varidel20}, {\sf DYNAMO} \citet{Green14}, {\sf DEEP2} \citet{Davis03, Kassin07, Kassin12}, {\sf KROSS} \citet{Stott16, Johnson18}, {\sf KMOS$^{\rm 3D}$} \citet{Wisnioski15, Wisnioski19}, {\sf MASSIV} \citet{Contini12, Epinat12}, \citet{Livermore15}, {\sf SINS+zC-SINF} \citet{FS06, FS09, FS18}, \citet{Genzel17}, {\sf SIGMA} \citet{Simons16, Simons17}, {\sf MOSDEF} \citet{Kriek15, Price20}, {\sf AMAZE-LSD} \citet{Gnerucci11} and {\sf KDS} \citet{Turner17}. Circles (squares) correspond to measurements taken from IFS surveys (long-slit spectroscopy). Symbols with black contours indicate averages over several data points; those without contours correspond to individual measurements. Note that we only include {\sc eagle}~ disc galaxies that are selected as described in Section~\ref{sec: GalSamples}. Solid lines indicate the medians for all {\sc eagle}{} galaxies in our samples; shaded regions show the $16^{\rm th}-84^{\rm th}$ percentiles. The dashed lines show {\sc eagle}{} results for a fixed halo mass bin, as labelled; note that the $\sigma_{z}$ prediction from all simulations coincide when controlling by halo mass.}
\label{fig: sigma_ev}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=0.85\textwidth]{Fig5.pdf}
\caption{The vertical velocity dispersion of cold gas plotted as a function of stellar mass (upper), halo virial mass ($\rm M_{200}$; middle) and SFR (bottom) for the {\sc Ref-L100} (blue) and {\sc NoFb-L25} (red) runs. Results are shown for redshifts $\approx 0.1$, 1 and 2.2 (left-hand, middle, and right-hand panels, respectively). Solid lines show the median relations in bins of width $0.3$~dex; shaded regions indicate the $16^{\rm th}$ to $84^{\rm th}$ percentiles. The corresponding Spearman's rank correlation coefficients are shown in the bottom-right corner of each panel (colour coded according to the simulation they pertain to). Dashed-dotted lines show the median trends obtained using a lower minimum threshold of 100 cold gas particles per galaxy to compute $\sigma_{z}$ (solid lines use a minimum of $500$ cold gas particles per galaxy). Individual points are observational results from {\sf SAMI}, {\sf DYNAMO}, {\sf KROSS}, {\sf KMOS$^{\rm 3D}$} and {\sf SINS+zC-SINF} as indicated by the legend. Solid and dashed black lines in the bottom panels show, respectively, the theoretical $\sigma_{z}-{\rm SFR}$ tracks predicted by the transport plus feedback and transport-only models of \citet{Krumholz18}. These theoretical curves were computed using the parameter values {suggested by \citet{Krumholz18} for local spirals (in the bottom-left and bottom-middle panels) and for high-$z$ galaxies (in the bottom-right panel)}.}
\label{fig: scaling relations}
\end{figure*}
The subsamples of galaxies plotted in Fig.~\ref{fig: sigma_profiles} were chosen to have low and high values of $\sigma_z$ at $R_{\rm flat}$. To investigate the physical origin of these differences, it is important to first verify that they are not short-lived, transient, or stochastic. To investigate this, we analysed the evolution of the vertical gas velocity dispersion for the same subsets of gaseous discs (corresponding to those that lie within the vertical grey bands plotted in Fig.~\ref{fig: stellar-to-halo-mass rel}). As in Fig.~\ref{fig: sigma_profiles}, we select discs whose $\sigma_{z}$ values are in the upper and lower quintile of the $\sigma_z$ distribution (for their halo mass) and track the evolution of $\sigma_z$ in their main progenitors back to $z=2.2$. Note that we only include progenitor galaxies provided their stellar mass exceeds $10^9\,\rm M_{\odot}$, and we eliminate instances in which a galaxy's main progenitor temporarily became a satellite of a more massive halo. Note that we do not, however, impose the $\kappa_{\rm co}>0.7$ selection to the progenitors of the $z=0.1$ galaxies.
Fig.~\ref{fig: sigma_tracks} shows the median evolutionary tracks of $\sigma_z$ for these galaxies. The median trends (thick solid lines; the shaded regions indicate the 16$^{\rm th}-84^{\rm th}$ percentile scatter) show that the progenitors galaxies with low or high $\sigma_z$ at $z=0.1$ also had low or high $\sigma_z$, respectively, for $z \lesssim 1$. Clearly the differences in $\sigma_z$ at $z=0.1$ (which is due to our selection of galaxies at that redshift) is not a short-lived, transient phenomenon but instead marks a persistent difference between the galaxy samples. The same is true when discs are selected at $z=1$ (not shown here, for clarity). Therefore, any driver of gas turbulence must therefore be acting on long timescales.
\section{Evolution of gas velocity dispersion and scaling relations}
\label{SecEvsigma}
In this section, we present the redshift evolution of the gas velocity dispersion (Section~\ref{subsec: sigma_ev}) as well as some standard scaling relations between $\sigma_z$, $\rm M_{\star}$ and SFR (Section~\ref{subsec: scaling relations}).
\subsection{Evolution of gas velocity dispersion} \label{subsec: sigma_ev}
Fig.~\ref{fig: sigma_ev} shows the evolution of $\sigma_z$ for {\sc eagle}~ disc galaxies from $z=0$ to $z=4$ obtained from the {\sc Ref-L100} (blue lines), {\sc NoFb-L25} (red lines) and {\sc NoAGN-L50} runs (light brown lines). A variety of observational results are also shown, including results from long-slit spectroscopy (squares) and IFS surveys (circles; table 5 in \citealt{Ubler19} contains the full compilation of velocity dispersion measurements plotted here). Most references report gas kinematics inferred from H$\alpha$ emission lines, although high-redshift surveys such as {\sf AMAZE-LSD} \citep{Gnerucci11} and {\sf KDS} \citep{Turner17} use [OIII] as the primary target line. Data points with black edges correspond to medians for datasets where multiple measurements are available; error bars indicate the $1\sigma$ scatter. The remaining points from \citet{Livermore15}, \citet{Genzel17} and {\sf SINS+zC-SINF} \citep{FS06, FS09, FS18} correspond to measurements for individual galaxies. Each observational survey aimed to a define sample of rotating discs in the star-forming sequence for which reliable kinematic measurements can be extracted, and various selection criteria were imposed to do so. For example, objects that display either merger or AGN activity are excluded as that can potentially perturb the inferred kinematics. The largest galaxy samples from IFU surveys plotted in this figure correspond to {\sf SAMI} \citep{Varidel20}, {\sf KROSS} \citep{Stott16, Johnson18} and {\sf KMOS$^{\rm 3D}$} \citep{Wisnioski15, Wisnioski19}, which have a homogeneous coverage of the star formation main sequence and cover several orders of magnitude of stellar mass. Note that, to compare with {\sc eagle}, we assume that the line-of-sight velocity dispersion ($\sigma_{\rm LoS}$), which is more readily obtained from observations, is comparable to the vertical velocity dispersions we measure from our simulations. In general, however, $\sigma_{\rm LoS}>\sigma_z$ as the line-of-sight component can capture radial and azimuthal motions when galaxies are observed at different inclination angles.
Overall, results from the {\sc Ref-L100} run are in good agreement with the observational results; the median $\sigma_{z}$ systematically increases from $\approx \rm 20 km\ s^{-1}$ at $z=0$ to $\approx \rm 50\ km\ s^{-1}$ at $z=2.5$, as observed. We find that the simulation results are similar when the thermal component of $\sigma_z$ (see equation~\ref{eq: sigmaz}) is neglected, indicating that it has a negligible contribution to $\sigma_{z}$ in {\sc eagle}{} galaxies on the mass scales we study (see \citealt{Pillepich19} for a different conclusion from the TNG simulation). Note that the gas turbulence inferred by {\sf DYNAMO} (blue circle at $z\approx 0.1$) is considerably higher than that obtained from {\sc eagle}. This is due to the survey's selection criteria, which targets analogues of high-redshift galaxies at $z=0.1$, resulting in a sample with systematically higher SFR and gas fractions than galaxies on the main sequence, which are likely to be massive systems with high velocity dispersion.
Note that results obtained from the {\sc NoAGN-L50} run are remarkably similarly to those from {\sc Ref-L100}, suggesting that AGN feedback is not an important driver of gas turbulence on the mass scales probed by our runs (differences may however be evident for more massive systems, of which there are few in the 50 cMpc {\sc NoAGN-L50} run).
Galaxies in the {\sc NoFb-L25} run, however, exhibit a lower $\sigma_{z}$ across all redshifts, with very little evolution. At first glance, this could be taken as an indication that stellar feedback is an important driver of gas turbulence and that it is largely responsible for the systematically higher values of $\sigma_{z}$ and its evolution in the {\sc Ref-L100} run. However, when galaxies are selected according to halo mass we find that all three {\sc eagle}{} runs predict a similar evolution of $\sigma_z$. The dashed lines in Fig.~\ref{fig: sigma_ev}, for example, show the evolution of $\sigma_z$ for galaxies occupying haloes with virial masses in the range $10^{11}-10^{11.3}\,{\rm M_\odot}$. The difference between the solid blue and red lines is therefore due to the different halo masses typically occupied by galaxies in each simulation, and in particular the lack of massive haloes in the {\sc NoFb-L25} run (see Fig.~\ref{fig: stellar-to-halo-mass rel}).
\subsection{Scaling relations of the gas velocity dispersion} \label{subsec: scaling relations}
The scatter in the $\sigma_{z}$-redshift relation is driven by different galaxy and halo properties. Fig.~\ref{fig: scaling relations} shows the $\sigma_{z}-{\rm M_\star}$, $\sigma_{z}-{\rm M_{200}}$ and $\sigma_{z}-{\rm SFR}$ relations (top, middle and bottom rows, respectively) for both the {\sc Ref-L100} (blue) and {\sc NoFb-L25} (red) runs. The gas turbulence correlates with all three properties, in agreement with correlations reported by observational studies (see observations shown in Fig.~\ref{fig: scaling relations}). Indeed, the Spearman correlation coefficients ($\rho_{\rm S}$; labelled at the bottom-right corners in each panel) indicate that the strength of the correlation between $\sigma_{z}$ and $\rm M_\star$, and $\sigma_z$ and SFR are significant for both the {\sc Ref-L100} and {\sc NoFb-L25} runs. Note that the lack of high-mass galaxies in the {\sc NoFb-L25} run is due to the smaller simulation box (25 cMpc), whereas the excess of them at low masses is a result of the overproduction of stars due to the absence of feedback.
At $z \approx 0.1$, galaxies in {\sc Ref-L100} display a $\sigma_{z}-{\rm M_\star}$ relation that resembles the one obtained by the {\sf SAMI} galaxy survey \citep[][shown as tan triangles]{Varidel20}. However, at fixed $\rm M_\star$, the {\sc eagle}~ results are systematically lower than the $\sigma_{z}$ measurements obtained for DYNAMO galaxies; this is likely due to the the survey selection effect discussed above. In fact, in the $\sigma_{z}-{\rm SFR}$ plane (bottom-left panel), the {\sc Ref-L100} galaxies show remarkable agreement with both the {\sf SAMI} and {\sf DYNAMO} relations, although the latter are shifted to slightly higher SFRs. Similar agreement is found at higher redshifts.
It is useful to compare results obtained from the {\sc Ref-L100} and {\sc NoFb-L25} runs. The top and bottom rows of Fig.~\ref{fig: scaling relations} suggest that, at fixed $\rm M_\star$ or SFR, the galaxies in the {\sc NoFb-L25} run have systematically lower $\sigma_{z}$ values than those in {\sc Ref-L100}. It is tempting to relate these differences to the absence of feedback in {\sc NoFb-L25}, which is a plausible source of turbulent energy. However, the middle rows of Fig.~\ref{fig: scaling relations} show that this is unlikely the case. Here we plot the $\sigma_{z}-{\rm M_{200}}$ relation for the two runs. In this case, both models follow the same relation. Systematic differences in the cold gas velocity dispersion of galaxies in these runs (at fixed $\rm M_\star$ or SFR) are therefore due to galaxies of comparable stellar mass or SFR occupying haloes of different virial mass. This foretells one of our main results: the thermal energy injected into the ISM by stellar feedback contributes little to turbulent motions in cold gaseous discs. This conclusion is indeed supported by the strong correlation between $\sigma_z$ and SFR in the {\sc NoFb-L25}. Why would galaxies with higher SFRs exhibit higher levels of turbulent motions even in the absence of stellar and AGN feedback? We return to this discussion below.
\section{The physical drivers of gas turbulence} \label{sec: PhysicalDrivers}
In this section we break down the role played by several physical processes in establishing the velocity dispersion in gaseous discs. In Section~\ref{SecGalMergers} we consider the effects of galaxy mergers, in Section~\ref{SecGravInst} we focus on gravitational instabilities, in Section~\ref{SecFeedback} we consider in more detail the impact of stellar feedback, and in Section~\ref{SecGasAcc} the effect of gas accretion.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{Fig6.pdf}
\caption{The distribution of the ratio of the gas velocity dispersion of descendants and their main progenitors for galaxies that have not had mergers (blue), for those that have had major mergers (orange), and either major or minor merger (red). Results are shown for the redshift ranges $0.1\leq z\leq 0.27$ (top panel), $0.74\leq z\leq 1.26$ (middle panel) and $1.49\leq z\leq 2.24$ (bottom panel). Downward arrows of corresponding colour indicate the medians of each distribution; the numbers above show the number of events in each sample. Values of $\sigma_{z,{\rm desc}}/\sigma_{z,{\rm prog}}>1$ show that that galaxy {\it increased} its velocity dispersion. On average, mergers increase the vertical velocity dispersion, $\sigma_{z}$.}
\label{fig: mergers}
\end{figure}
\subsection{Turbulence due to galaxy mergers}\label{SecGalMergers}
We link galaxy descendants and progenitors in adjacent snapshots using the galaxy merger trees available in the {\sc eagle}\ database \citep{Qu17}. For galaxies that have $\ge 2$ prominent progenitors, we
follow \citet{Lagos18} and compute the stellar mass ratio between the first and second most massive progenitors to classify merger events. Major mergers are those whose stellar mass ratio is above $0.3$, minor mergers are those with mass ratios between $0.1$ and $0.3$. Mass ratios below $0.1$ are considered ``unresolved'' mergers and, consequently, classed as accretion events \citep[e.g.][]{Crain17}.
The reason for the latter is that, at the resolution of the {\sc eagle}~ simulation, and for galaxies with $\rm M_\star \ge 10^9\,\rm M_{\odot}$, we can only reliably identify mergers with mass ratios $\ge 0.1$. Galaxies at any redshift that went through a major or minor merger in the previous snapshot are added to a ``merger sample'' at that redshift. Conversely, those that have a single progenitor, or multiple progenitors with mass ratios $<0.1$ are included in a ``smooth accretion sample''.
Fig.~\ref{fig: mergers} shows the Probability Distribution Function (PDF) of the ratios of the gas velocity dispersion in descendants and their most massive progenitors, $\sigma_{z, {\rm desc}}/\sigma_{z, {\rm prog}}$. We show the distributions for galaxies that underwent major mergers (orange), major or minor mergers (red), and galaxies in the smooth accretion sample (blue). Note that we combine results from multiple adjacent snapshots in order to increase the sample sizes. For example, the top panel includes results from redshift pairs $z=0.1-0.18$ and $z=0.18-0.27$. The number of events in each sample are shown on top of the arrows which also indicate the median of the $\sigma_{z, {\rm desc}}/\sigma_{z, {\rm prog}}$ distributions. Note that imposing a $\kappa_{\rm co}>0.7$ could lead to the removal of galaxies whose morphology changes from disc to spheroid as a result of merger (which, as shown in \citealt{Lagos18}, is likely the case in major mergers). To avoid discarding these descendant-progenitor pairs and to account for a potential causal relationship between mergers and changes in $\sigma_{z}$, we only apply the $\kappa_{\rm co}>0.7$ selection to the main progenitor galaxy, but not to their descendants.
Both major and minor mergers increase the gas turbulence in galaxy discs (i.e. $\sigma_{z, {\rm desc}} > \sigma_{z,{\rm prog}}$ in these cases). The effect is strongest for major mergers (orange histograms), although the number of these events is small compared to galaxies in the smooth accretion sample. Typically, major mergers increase gas velocity dispersion by about a factor of 1.3 to 1.4, depending on redshift.
Note that the PDFs of galaxies in the smooth accretion sample peak at ratios slightly below one (see blue arrows), indicating that, on average, descendants are kinematically colder than their progenitors. This is consistent with the $\sigma_{z}$ evolution analysed in Section~\ref{SecEvsigma}. The impact of galaxy mergers on the cold gas velocity dispersion of discs, as well as its dependence on the merger mass ratio, is worth investigating more thoroughly. We leave this for future work.
Although galaxy mergers have an important impact on the gas velocity dispersion, they are quite uncommon on the mass (and redshift) scales we study. Hence mergers have only a minor impact on the evolution of gas turbulence when averaged over a population of galaxies. Indeed, we verified that all of the results we have presented so far are unaffected by the removal of merging galaxies.
\subsection{The relation between vertical velocity dispersion and gravitational instabilities}\label{SecGravInst}
\begin{figure*}
\centering
\includegraphics[width=0.9\textwidth]{Fig7.pdf}
\caption{Examples of Toomre-$Q$ instability maps for cold gas (left panel), stars (middle panel), and the combination of cold gas and stars (right panel) for a $z=2$ galaxy identified in the {\sc Ref-L100} run. Colours show different values of $\rm log_{10}(Q)$, as indicated in the colour bar to the right. Green and violet regions are unstable ($Q_{\rm net}<3$). Dashed circles are drawn at $R=3\,R_{50}$, which is the cylindrical aperture within which we measure $\sigma_{z}$.}
\label{fig: Qmaps}
\end{figure*}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{Fig8.pdf}
\caption{The vertical velocity dispersion of cold gas plotted as a function of the clumpiness parameter, $f_{\rm c}$ (see equation~\ref{eq: clumpiness}), for galaxies in the {\sc Ref-L100} run. Different colours correspond to galaxies identified in different narrow redshift bins, as indicated. Solid lines correspond to the {\em total} baryon mass fraction in clumps, i.e. $f_{\rm c}$ evaluated for both stellar and gaseous particles, whereas dashed lines are evaluated using only the cold gas disc. The shaded regions (only shown along with the solid lines) correspond to the $16^{\rm th}-84^{\rm th}$ percentile ranges. The Spearman correlation coefficients (corresponding to the solid lines) are shown in the top right corner of each panel.} \label{fig: Clumpiness}
\end{figure}
Observations of discs at high redshift reveal that a significant fraction of their cold gas is contained in distinct self-gravitating clumps \citep[e.g.][]{Elmegreen09, Genzel11}. These clumps may introduce perturbations to the gravitational potential, which can influence the kinematics of baryons in the disc. In particular, non-axisymmetric torques induced by clumps can result in angular momentum loss, creating inward flows of mass while releasing gravitational potential energy. Different analytical models for star-forming discs incorporate this transport mechanism and consider it an important driver of gas turbulence \citep[e.g.][]{Aumer10, Krumholz16}.
The formation and evolution of self-bound clumps can be studied using the theory of gravitational instabilities. In this framework, local instabilities arise due to an imbalance between gravity and restoring forces and are commonly quantified using the Toomre stability parameter \citep{Toomre64}. This can be defined for gaseous or stellar discs as
\begin{equation} \label{eq: Toomre-Q}
Q_i = \frac{\kappa\, \sigma_{r,i}}{\pi\, G \,\Sigma_i}.
\end{equation}
where $i$ refers to the disc's baryonic component\footnote{Note that for the stellar component $\pi$ is replaced by 3.36}, $\kappa$ is the epicyclic frequency, $\sigma_{r,i}$ is the radial velocity dispersion, and $\Sigma_i$ the surface density. Note that the quantities in equation~(\ref{eq: Toomre-Q}) are typically measured locally; the value of $Q_i$ can therefore vary from place to place within the disc. Typically, regions where $Q_i>Q_{\rm crit}$ are considered stable against collapse, with $Q_{\rm crit}\approx 1$ indicating marginal stability.
Some analyses account for the stabilising effects of stars and of the finite disc thickness \citep[e.g.][]{Romeo13}. For our analysis, we account for instabilities in both the stellar and gaseous disc. Following \citet{Inoue16}, we compute 2D maps of $Q_{\rm gas}$ and $Q_\star$ using face-on projections of our {\sc eagle}~ galaxy sample, and use the formulation of \citet{Romeo11} to calculate a multi-component Toomre parameter, which we denote $Q_{\rm net}$. We refer to Appendix~\ref{ComputingQ} for a description of how we construct $Q_{\rm net}$ maps from maps of $Q_{\rm gas}$ and $Q_\star$. In Appendix~\ref{AppTestingQ}, we report tests of our method's accuracy using cylindrically-symmetric galaxy models for which the $Q_\star(r)$ profiles are known.
Fig.~\ref{fig: Qmaps} shows the $Q_i$ maps obtained for a $z=2$ disc galaxy identified in the {\sc Ref-L100} run. To disentangle the contribution from gas and stars, we show $Q_{\rm gas}$ and $Q_\star$ maps separately (left and middle panels, respectively), although for our analysis we only use the $Q_{\rm net}$ maps (right panel; which contain contributions from both gas and stellar particles). In general, the stellar component is the primary driver of instabilities in the galaxy's inner regions, whereas gas dominates the instabilities in the outer disc, as shown in this example. The same is true for the majority of discs used in our analysis. In general, the $Q_{\rm net}$ maps contain more unstable regions than either of the individual $Q_i$ maps, suggesting that neither the stellar nor the gaseous component should be excluded, especially in cases where they make similar contribution to local instabilities. This is important as often $Q_{\rm gas}$ alone is used to interpret the observations of gas velocity dispersion. However, we note that the contribution from stars becomes increasingly important at low redshifts, when the typical gas fractions of discs are lower.
To assess whether gravitational instabilities drive gas turbulence, we first define a ``clumpiness'' parameter, $f_{\rm c}$, as
\begin{equation} \label{eq: clumpiness}
f_{\rm c} = \frac{{\rm M_{bar}}(Q_{\rm net}<3)}{\rm M_{\rm bar}},
\end{equation}
where ${\rm M_{\rm bar}}(Q_{\rm net}<3)$ is the baryonic mass (of stars plus gas) contained within pixels with $Q_{\rm net}<3$, and $\rm M_{\rm bar}$ is the total baryonic mass of the galaxy. Both quantities are evaluated using the particles enclosed within the same cylindrical aperture that was used to compute $\sigma_z$ (see Section~\ref{sec: sigma_z calculation}), which typically encloses the majority of unstable regions in the discs. The threshold $Q_{\rm net}<3$ used to select unstable regions is motivated by the work \citet{Inoue16}, who studied high redshift galaxies in ``zoom-in'' cosmological simulation and found that instabilities are prone to form at locations where $Q_{\rm net} \lesssim 2-3$.
In principle, galaxies with high $f_{\rm c}$ values are those that may also contain a large number of self-bound clumps capable of injecting turbulent energy into their ISM. Fig.~\ref{fig: Clumpiness} shows $\sigma_{z}$ as a function of the clumpiness parameter for galaxies in different bins of redshift (the same bins used for Fig.~\ref{fig: mergers}) and virial mass (the values of $\rm M_{200}$ increase from the top to bottom panels). Solid lines show $f_{\rm c}$ values calculated using all baryons in the disc (as defined in equation~\ref{eq: clumpiness}) whereas dashed lines use only the cold gas component (i.e. $f_c={\rm M_{gas}}(Q_{\rm net} < 3)/{\rm M_{gas}}$).
\begin{figure*}
\centering
\includegraphics[width=0.9\textwidth]{Fig9.pdf}
\caption{Vertical gas velocity dispersion plotted as a function of the specific gas accretion rate for galaxies hosted by low (left panel), intermediate (middle panel) and high mass haloes (right panel), as indicated in brackets in the bottom-right corners of each panel, in the {\sc Ref-L100} (blue) and {\sc NoFb-L25} (red) runs. Different line styles and symbols correspond to results from different redshift ranges, as indicated by the legend. Lines represent median relations, and shaded regions indicate the $16^{\rm th}$ and $84^{\rm th}$ percentiles in {\sc Ref-L100}. Red symbols in the background show individual galaxies from the {\sc NoFb-L25} run, whereas outsized symbols and error bars indicate the median and $16^{\rm th}$ and $84^{\rm th}$ percentiles for galaxies identified at different redshifts, respectively. }
\label{fig: accretion}\end{figure*}
Overall, the correlations between $\sigma_{z}$ and $f_{\rm c}$ are quite weak at most redshifts and halo masses we analysed. Consider, for example, the lowest halo mass bin (top panel), for which the Spearman rank correlation coefficients (the coloured numbers labelled in the upper-right corner) are less than $0.12$ at all redshifts. For intermediate masses, and at redshifts $z\gtrsim 0.74$, the correlation between $\sigma_{z}$ and $f_{\rm c}$ is somewhat stronger, but it weakens again for higher halo masses. The fact that, at fixed $\rm M_{200}$, the $\sigma_{z}$ depends more strongly on redshift than on $f_c$ suggests that the evolution of gas turbulence is only weakly related to $f_{\rm c}$, at least for galaxies in the {\sc eagle}~ simulation.
\subsection{Stellar Feedback-driven turbulence}
\label{SecFeedback}
The observed correlation between $\sigma_{z}$ and SFR could imply that stellar feedback is an important driver of gas turbulence: a higher SFR implies a higher abundance of massive stars whose supernovae inject energy into the ISM, leading perhaps to a higher $\sigma_z$. As shown in Fig.~\ref{fig: scaling relations}, {\sc eagle}\ predicts $\sigma_z-{\rm M_\star}$ and $\sigma_z-{\rm SFR}$ relations that are in reasonable agreement with the observed relations at several different redshifts. In particular, there is a relatively weak relation between $\sigma_{z}$ and SFR among systems with low SFRs (which is particularly clear at low redshifts), but this transitions to a stronger dependence for galaxies with high SFRs.
\citet[][hereafter K18]{Krumholz18} developed an analytical model for the evolution of gaseous discs assuming hydrostatic equilibrium and marginal gravitational instability. In their model, supernovae feedback and the radial transport of gas driven by gravitational instabilities both inject turbulent energy to the ISM, while energy is dissipated by shocks on a timescale comparable to the local crossing time. These energy sources are assumed to reach equilibrium; this model is referred to as ``Transport$+$Feedback'' (see bottom panels in Fig.~\ref{fig: scaling relations}). We also show the K18 ``No-feedback'' model, which only accounts for the effect of radial transport within the discs, but not for the turbulent energy injected by supernovae. In order to compare their predictions with our {\sc eagle}\ results, we adopt the parameter values suggested by K18 to best describe spiral galaxies at $z\le 1$ (bottom left and middle panels in Fig.~\ref{fig: scaling relations}) and high$-z$ galaxies at $z=2$ (bottom right panel in Fig.~\ref{fig: scaling relations}). Because the K18 model only accounts for the neutral (atomic plus molecular) gas component, we add $15\ {\rm km\ s^{-1}}$ (in quadrature) to their results \citep[see][]{Krumholz16} when comparing with our {\sc eagle}\ predictions (which include all ISM components).
Even though, qualitatively, there are common features between the {\sc Ref-L100} run and the predictions of K18's fiducial model, it is clear that in detail the two differ. For example, the flattening of $\sigma_z$ at low SFRs is less abrupt in {\sc eagle}; the {\sc eagle}\ trend is perhaps better described as a weak dependence of $\sigma_{z}$ on SFR at $\rm SFR<1\,\rm M_{\odot}\,yr^{-1}$. Only after relaxing the 500-cold gas particle constrain imposed on our sample of {\sc eagle}\ galaxies (see Section~\ref{sec: GalSamples}) does a flattening of $\sigma_{z}$ becomes apparent (see dotted-dashed lines, for which we instead impose a minimum number of cold gas particles per galaxy of 100), albeit at much lower SFRs ($\lesssim 0.1\,\rm M_{\odot}\,yr^{-1}$) than predicted by K18. At higher SFRs, {\sc eagle}\ predicts a weaker dependence of $\sigma_{z}$ on SFR than K18's fiducial model.
When comparing the {\sc NoFb-L25} run with the K18 ``No-Feedback'' model we also see clear differences, with {\sc eagle}\ predicting lower $\sigma_z$ than the minimum values in the K18 model (which is higher than the floor value of $15\ {\rm km\ s^{-1}}$ described above). The differences between {\sc eagle}\ and K18 could be taken as an indication that stellar feedback is at best only partially driving $\sigma_z$ in {\sc eagle}.
Considering again the comparison between {\sc Ref-L100} and {\sc Nofb-L25}, the most significant result of the $\sigma_{z}-{\rm SFR}$ correlation is that in both runs galaxies with higher SFRs also have a higher $\sigma_z$. In {\sc NoFb-L25}, however, there is no injection of energy into the ISM by supernovae, suggesting that the higher gas velocity dispersions associated with high SFRs in that run are {\em not} related to feedback. Feedback therefore cannot be the only driver of turbulence, and perhaps not even the most important one.
The $\sigma_{z}-{\rm M_{200}}$ relations for {\sc Ref-L100} and {\sc NoFb-L25}, shown in the middle panels of Fig.~\ref{fig: scaling relations}, are remarkably similar at overlapping masses. This implies that the differences seen in the other relations (specifically $\sigma_z-{\rm M_\star}$ and $\sigma_z-{\rm SFR}$) arise because galaxies of the same stellar mass or SFR occupy haloes of very different virial mass. In this context, for a fixed $\rm M_\star$ or SFR, gas particles with a particular kinetic energy are more likely to become unbound and escape the halo in the {\sc NoFb-L25} run, as they live in lower mass haloes than galaxies of the same stellar mass in {\sc Ref-L100}. This could result in systematically lower values of $\sigma_{z}$ in {\sc NoFb-L25} at fixed ${\rm M_\star}$ or SFR, potentially leading to the offset between the two runs seen in the upper and lower panels of Fig.~\ref{fig: scaling relations}.
It is important to highlight that, even though the $\sigma_z-{\rm M_{200}}$ relations are similar in the two runs, at fixed halo mass the scatter in $\sigma_{z}$ is larger in the {\sc NoFb-L25} run than in the {\sc Ref-L100} run (which is reflected in the lower values of the Spearman rank correlation coefficients in {\sc NoFb-L25}). A possible interpretation of this result is that stellar feedback acts as a regulation mechanism that leads to a tighter relation between $\sigma_{z}$ and ${\rm M_{200}}$. When feedback is absent so is the regulation mechanism, resulting in a broader distribution in $\sigma_{z}$ values at fixed halo mass. This is reminiscent of the effect of stellar feedback on the SFR$-\rm M_\star$ plane, and the relationship between SFR, stellar mass, and molecular gas explored in \citet{Lagos16}. Those authors showed that by making stellar feedback stronger (weaker), the relation between these quantities became tighter (broader).
The similarity of the $\sigma_{z}-{\rm M_{200}}$ relations in the {\sc Ref-L100} and {\sc NoFb-L25} runs indicates that stellar feedback is not the primary driver of gas turbulence in {\sc eagle}. The positive correlation between $\sigma_{z}$ and SFR may therefore be due to a more fundamental driver of turbulence; one which increases the SFRs of galaxies along with their velocity dispersion. In addition to the impact of either mergers or dynamical instabilities (quantified by the $f_{\rm c}$ ``clumpiness'' parameter) -- which, as shown above, cannot fully explain the evolution of $\sigma_z$ -- we consider next the impact of cosmological gas accretion and how it affects the dynamics of cold gas discs.
\subsection{Gas Accretion-driven turbulence}\label{SecGasAcc}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{Fig10.pdf}
\caption{Same as Fig.~\ref{fig: sigma_tracks}, but showing the evolutionary tracks of the specific accretion rate, $\dot{\rm M}_{\rm acc}/ \rm M_{cold}$, for disc galaxies identified at $z=0.1$. Dark lines and shaded regions correspond to galaxies whose $\sigma_z$ values (at $z=0.1$) are in the upper quintile for their halo mass; lighter colours correspond to galaxies in the lower quintile. Note that the evolution of $\dot{\rm M}_{\rm acc}/\rm M_{cold}$ -- specifically the separation of the different evolutionary tracks -- resembles that of $\sigma_{z}$ shown in Fig.~\ref{fig: sigma_tracks}, suggesting that the two are closely connected.}
\label{fig: sMacc_tracks}
\end{figure}
In Fig.~\ref{fig: accretion} we plot $\sigma_{z}$ as a function of the specific gas accretion rate, i.e. $\dot{\rm M}_{acc}/{\rm M_{cold}}$, for galaxies that lie in a few separate bins of ${\rm M_{200}}$ and for three different redshift bins. Results are shown for both the {\sc Ref-L100} and {\sc NoFb-L25} runs (blue and red colours, respectively). Clearly $\sigma_{z}$ correlates with the specific gas accretion rate, although the trend varies with halo mass and redshift. For all halo masses, the $\sigma_{z}-\dot{\rm M}_{\rm acc}/\rm M_{cold}$ relation flattens when accretion rates are low ($\dot{\rm M}_{\rm acc}/\rm M_{cold}\lesssim 10^{-9.8}\ yr^{-1}$), while at higher specific accretion rates the relation steepens. At fixed halo mass, higher specific accretion rates also occur at higher redshifts, reflecting the fact that these halos are rarer and originate from higher linear rms density fluctuations.
\begin{figure*}
\centering
\includegraphics[width=0.9\textwidth]{Fig11.pdf}
\caption{The vertical gas velocity dispersion (normalised by $V_{200}$) plotted as a function of the specific accretion rate for galaxies in the {\sc Ref-L100} run identified at low (left panel), intermediate (middle panel), and high (right panel) redshifts. Results are shown for the full sample (grey lines) and for subsamples for which the angular momentum of the accreting gas is closely-aligned (blue lines) or misaligned (yellow lines) with the rotation axis of the gaseous disc. Solid lines refers to samples containing discs only (i.e. $\kappa_{\rm co}>0.7$ for cold gas particles) whereas dashed lines refers to samples where only progenitors are required to meet the $\kappa_{\rm co}>0.7$ criteria.}
\label{fig: misalignment}
\end{figure*}
At fixed halo mass and redshift, there are fewer galaxies in the {\sc NoFb-L25} run than in {\sc Ref-L100}, so for the former we use outsized points with errors bars to indicate the median values of $\sigma_z$ and $\dot{\rm M}_{\rm acc}/\rm M_{cold}$ (error bars indicate the 16$^{\rm th}$ to 84$^{\rm th}$ percentile scatter in $\sigma_z$). By comparing these points to the blue lines in each panel, however, it is clear that results obtained from the {\sc NoFb-L25} run are similar to those obtained from {\sc Ref-L100}. Indeed, at fixed halo mass and redshift, both simulations predict similar values of $\sigma_{z}$ for the same specific accretion rates, indicating that this relation is likely more fundamental than the relationship between $\sigma_z$ and SFR. The specific accretion rate of cold gas may therefore be the primary driver of turbulence. Note, however, that the scatter in $\sigma_z$ is considerably larger in the {\sc NoFb-L25} run than it is in {\sc Ref-L100}, which is perhaps not surprising given the large scatter in the $\sigma_z-{\rm M_{200}}$ relation seen in that run (see the middle panels of Fig.~\ref{fig: scaling relations}).
The connection between $\sigma_z$ and accretion rate is also evident in the redshift evolution of $\dot{\rm M}_{\rm acc}/\rm M_{cold}$ and $\sigma_z$ of individual galaxies. We use the {\sc eagle}\ mergers trees to track the evolution of the specific gas accretion rates onto discs selected at $z=0.1$ for the same bins of virial mass used to construct Fig.~\ref{fig: sigma_tracks}. The results are shown in Fig.~\ref{fig: sMacc_tracks}. Note that the evolution of $\dot{\rm M}_{\rm acc}/\rm M_{cold}$ resembles that of $\sigma_{z}$: galaxies with low (high) velocity dispersion at $z=0.1$ exhibited lower (higher) specific accretion rates for an extended period of time prior to this. Comparing Fig.~\ref{fig: accretion} with Fig.~\ref{fig: sigma_tracks}, it is clear that the evolutionary tracks of the specific accretion rate for the two galaxy samples converge at around the same cosmic time as their $\sigma_{z}$ values do (at $z\approx 1$ and $\approx 0.7$ for the intermediate and high $\rm M_{200}$ bins, respectively).
As mentioned above, however, $\sigma_z$ is largely independent of accretion rate when the latter is low, but there remains considerable scatter. This motivates the study of the effects of {\em misaligned} gas accretion, which we quantify using the angle $\theta_{\rm acc}$ between the net angular momentum vector of the disc and that of the accreting cold gas. As for Fig.~\ref{fig: accretion}, we focus on galaxies identified at low, intermediate, and high redshifts (left to right panels, respectively). At each redshift, we split our entire sample of galaxies into those with mostly aligned ($\theta_{\rm acc}<20^{\circ}$) and mostly misaligned ($\theta_{\rm acc}>60^{\circ}$) gas accretion. Note that simultaneously binning by $\theta_{\rm acc}$, halo mass, and redshift reduces the size of our galaxy sample significantly. We therefore do not bin by halo mass, but instead normalise the vertical velocity dispersion by $V_{200}$, thus removing the $\rm \sigma_z-{\rm M_{200}}$ relation.
Fig.~\ref{fig: misalignment} shows the $\sigma_z/V_{200}-\dot{\rm M}_{\rm acc}/\rm M_{cold}$ relation for the full galaxy sample (solid grey lines), as well as for the aligned (solid blue lines) and misaligned (solid yellow lines) subsamples. Different panels correspond to different redshift ranges, increasing from left to right, as in Fig.~\ref{fig: accretion}. Dashed lines show the equivalent relations when the descendant galaxies with $\kappa_{\rm co}<0.7$ are also included (note that their progenitors are still discs with $\kappa_{\rm co}>0.7$; i.e. in this case we include galaxies whose accretion history may have driven them to non-disky morphologies -- note that this selection only affects galaxies that are rapidly accreting).
It is clear from Fig.~\ref{fig: misalignment} that the scatter in the $\sigma_z/V_{200}-\dot{\rm M}_{\rm acc}/\rm M_{cold}$ relation is closely connected to the alignment of the disc and accreting material: when accreting gas is misaligned, discs tend to be more turbulent. Moreover, there are hints that the impact of misaligned accretion on $\sigma_{z}$ is more important for galaxies with low specific accretion rates, which are more common at low redshifts. Note that when including galaxies with $\kappa_{\rm co}<0.7$ (dashed lines), the relation between $\sigma_z$ and $\dot{\rm M}_{\rm acc}/\rm M_{cold}$ becomes more evident at high specific accretion rates. This suggests that, in this regime, accretion can significantly perturb the disc's structure, driving many to non-disky morphologies.
In summary, Figs.~\ref{fig: sigma_tracks},~\ref{fig: accretion}~and~\ref{fig: misalignment} show that the rate and geometry of gas accretion both play an important role in establishing the vertical velocity dispersion, $\sigma_z$, of {\sc eagle}\ galaxies.
\section{Discussion} \label{sec: Discussion}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{Fig12.pdf}
\caption{Spearman rank correlation coefficient, $\rho_{\rm S}$, for the $\sigma_{z}-{\rm sSFR}$ (blue), $\sigma_{z}-f_{\rm c}$ (green), $\sigma_{z}-{\dot{\rm M}_{\rm acc}/ \rm M_{cold}}$ (brown) and $\sigma_{z}-\theta_{\rm acc}$ (orange) relations as a function of redshift and for galaxies occupying low (top), intermediate (middle) and high (bottom) mass haloes. At all masses and redshifts, all samples contain at least 100 galaxies. The shaded regions represent the errors estimated from 10 jackknife subsamples (see text for details). Note that the thick lines show the $\rho_{\rm S}$ values obtained from the entire population of galaxies within each sample; they are not obtained from the merger trees of galaxies identified at a particular time. The correlation between $\sigma_{z}$ and the different galaxy properties depends on both redshift and the mass of the host haloes. At most redshifts and halo masses, the geometry and rate of accreting cold gas correlates most strongly with $\sigma_z$.}
\label{fig: Spearman}
\end{figure}
The correlations between $\sigma_{z}$, (specific) SFR, the clumpiness factor, $f_{\rm c}$, and (specific) gas accretion rates, in addition to the correlations between the three latter properties, supports the widely adopted assumption that galaxy discs evolve as a consequence of quasi-stable equilibrium between inflows, star formation and outflows (e.g. \citealt{Bouche10, Dave12, Lilly13, Dekel14}). In this context, an increase in the inflow rate will deposit a larger amount of gas into the disc, which in turn will induce local instabilities across the disc which collapse into dense clouds and increase the SFR. Depending on the mass of the galaxy, the subsequent feedback-driven outflows will expel gas and suppress the SFR, lowering the self-gravity of the gaseous disc. The latter helps to stabilise the disc against fragmentation ($Q\gtrsim 1$).
As shown in Fig.~\ref{fig: sigma_tracks}, the injection of turbulent energy is not stochastic but rather occurs on prolonged timescales, allowing a galaxy to maintain a high or low $\sigma_{z}$ for several Gyr. There is an interesting similarity between the ``memory'' of $\sigma_{z}$ for {\sc eagle}\ galaxies and the memory of their position in the ${\rm SFR}-\rm M_{\star}$ plane. For example, \citet{Matthee19} selected galaxies along the star formation main sequence and showed that galaxies that lie above the mean relation typically stay above for several Gyr. The same is true for galaxies below the main sequence. They also find a halo-mass dependence of the timescale in which galaxies retain ``memory'' of their position relative to the main sequence. Similarly, \citet{Lagos17} and \citet{WaloMartin20} showed that, in {\sc eagle}, the progenitors of $z=0$ galaxies with low and high stellar spin display differences in their stellar kinematics for several Gyr, only to become indistinguishable at $z\gtrsim 1$, an effect that has also been found in other simulations (e.g. \citealt{Penoyre17, Choi17}). These results are related to how haloes assemble, and in the case of \citet{Lagos17} and \citet{WaloMartin20} the main distinction between the $z=0$ high and low stellar spin galaxies was found to be star formation in the former and sustained gas accretion in the latter. In the context of the drivers of $\sigma_z$, Fig.~\ref{fig: sMacc_tracks} suggests that the evolution of accretion rates offers an explanation for the long-timescale memory of the gas turbulence.
Depending on the physical conditions of discs (and those of their host haloes), the processes affecting the velocity dispersion in gaseous discs can have different degrees of importance at different redshifts or halo masses. To quantify the relationship between $\sigma_{z}$ and the various galaxy properties explored in this paper, we computed the Spearman correlation coefficient, $\rho_{\rm S}$, for each of these relations. Fig.~\ref{fig: Spearman} shows the evolution of $\rho_{\rm S}$ for the $\sigma_{z} - {\rm sSFR}$ (blue curves), $\sigma_{z}-f_{\rm c}$ (green curves), $\sigma_{z}-\dot{\rm M}_{\rm acc}/{\rm M_{cold}}$ (brown curves) and $\sigma_{z}-\theta_{\rm acc}$ (yellow curves) relations as a function of redshift and for three halo mass bins. This allows us to quantitatively assess the main driver of the scatter in the $\sigma_{z}-{\rm M_{200}}$ relations (see middle panels in Fig.~\ref{fig: scaling relations}) at different redshifts. Note that we do not track a given population of galaxies through their merger trees, but rather show the values of $\rho_{\rm S}$ obtained for the selected samples at each redshift (see Section~\ref{sec: GalSamples}). We estimate the errors in $\rho_{\rm S}$ via jackknife resampling using galaxies contained within 10 equal but distinct sub-volumes. Thus, each jackknife subsample contains all galaxies in the simulation box but excludes those whose COP is within a specific subvolume. We calculate $\rho_{\rm S}$ for the 10 jackknife subsamples from which we obtain the mean $\rho_{\rm S}$ and its standard deviation (shown as solid lines and shaded regions, respectively).
Overall, we find positive correlations (i.e. $\rho_{\rm S}>0$) between $\sigma_{z}$ and the four properties mentioned above. For the $\sigma_{z}-{\rm sSFR}$ relation, $\rho_{\rm S}$ is typically of order $0.1-0.3$ for most redshifts, and reaches a maximum (of roughly 0.4) at low redshifts and low halo masses. The evolution of $\rho_{\rm S}$ for the other relations shows a clear mass-dependence. The median $\rho_{\rm S}$ for the $\sigma_{z}-f_{\rm c}$ relation goes from $\rho_{\rm S} \approx 0.05$ at low masses (see top panel of Fig.~\ref{fig: Clumpiness}) to a weak but positive correlation ($\rho_{\rm S} \approx 0.25$) at intermediate and high halo masses ($\rho_{\rm S}>0.4$ for the intermediate mass bin at high redshifts). Similarly, for the $\sigma_{z}-\dot{\rm M}_{\rm acc}/{\rm M_{cold}}$ the median $\rho_{\rm S}$ goes from $0.23$ to $0.48$, for the low- and high-mass bins, respectively. We find that the correlation with misaligned gas accretion angle is the strongest for low-redshift galaxies living in low- to intermediate-mass haloes. This correlation is weaker at higher redshifts and becomes comparable in strength to that of the $\sigma_{z}-\dot{\rm M}_{\rm acc}/{\rm M_{cold}}$ relation. These conclusions are robust, even after considering the errors in $\rho_{\rm S}$, which are generally small for all relations.
Fig.~\ref{fig: Spearman} shows that the relation between gas turbulence and specific gas accretion is significant across all redshifts and halo masses. In particular, for Milky-way haloes ($\rm M_{200}\approx 10^{12}\, M_{\odot}$), the typical values of $\rho_{\rm S}$ obtained for the $\sigma_z-\dot{\rm M}_{\rm acc}/{\rm M_{cold}}$ (brown line in bottom panel) is higher than that of the sSFR or $f_{\rm c}$ relations at all times. This suggests that the vertical velocity dispersion of cold gaseous discs within these haloes is governed primarily by the specific gas accretion rate, especially at $z\lesssim 1$ when the $\rho_{\rm S}$ from both sSFR and $f_{\rm c}$ are $\approx 2$ times smaller than that from the specific gas accretion rate. Furthermore, Fig.~\ref{fig: Spearman} indicates that the misalignment of the inflowing gas is also important, and actually plays a important role in setting the gas turbulence for low-$z$ discs within low-mass halos. This is consistent with Fig.~\ref{fig: misalignment} which shows that the segregation of the low and high $\theta_{\rm acc}$ samples is larger at lower redshifts. For higher specific accretion rates, which are more common at high redsfhits (see the differences in the dynamical ranges in the different panels of Fig.~\ref{fig: misalignment}), the segregation of the low- and high-$\theta_{\rm acc}$ samples is smaller, hence the correlation between gas turbulence and $\theta_{\rm acc}$ is expected to be weaker.
\citet{Sales12} used the {\tt GIMIC} simulation \citep{Crain09} to show that galaxies are very likely to become discs by $z=0$ if at the time of the maximum halo expansion (i.e. the turnaround radius), the spin of their inner halo regions is aligned with the outer halo parts (which contain the gas that will eventually be accreted to the galaxy).
Our results show that large gas accretion misalignments can lead to more turbulence, which in the more extreme cases affect the morphology of galaxies (see Fig.~\ref{fig: misalignment}). In the future, we will investigate the link between gas turbulence, misaligned gas accretion and galaxy morphology.
Now turning our attention to gravitational instabilities, Fig.~\ref{fig: Spearman} shows that the correlation between $\sigma_z$ and $f_{\rm c}$ become important at $z\gtrsim 1$, and for galaxies in the intermediate halo mass bin. This suggests that, at this epoch, transport-driven turbulence contributes to $\sigma_z$ (see Section~\ref{SecGravInst}). We note that correlations with the clumpiness parameter are clearer when there is a significant fraction of galaxies with $f_{\rm c}>0.3$. This indicates that gravity-driven turbulence may become efficient when at least a third of the baryons of the discs are in Toomre-unstable regions. Indeed, we find weak or no correlation for discs at low-$z$, which in general have $f_{\rm c}<0.3$ (see dark violet lines in Fig~\ref{fig: Clumpiness}). This is consistent with local discs being mostly Toomre-stable (i.e. $Q_{\rm net}>1$); therefore, inward gas flows along the disc are expected to be small.
We interpret the positive correlation between $\sigma_{z}$ and ${\rm sSFR}$ as an indirect consequence of the correlation between $\sigma_{z}$ and the specific gas accretion rate. This is plausible: the SFR is connected to the amount of cold gas mass in the disc which in turn depends on the amount of gas accreted, which is, in our interpretation, what is injecting turbulence. Furthermore, Figs.~\ref {fig: scaling relations} and \ref{fig: accretion} suggest that the relationship between gas turbulence and specific gas accretion rate is more fundamental than its relation to SFR because the latter are seen even in the {\sc NoFb-L25} run.
Previous work based on hydrodynamical simulations have shed light on the possible drivers of gas turbulence, focusing on isolated discs \citep{Renaud21, Ejdetjarn22}, cosmological zoom-in simulations from {\tt FIRE} \citep{Hung19}, or single galaxies from the TNG50 simulation \citep{Forbes22}. These have shown that in some regimes it is clear how disc instabilities, or gas accretion, can affect the disc turbulence. Our work contributes to this wealth of literature by attempting to identify the drivers of gas turbulence across a broad range of redshifts and galaxy masses. The large samples of galaxies provided by the {\sc eagle}\ simulation allowed us to understand that different physical drivers of turbulence may dominate at different times and in haloes of different mass, highlighting the complexity of the problem. Furthermore, the evolutionary tracks of $\sigma_{z}$ in {\sc eagle}\ reveal that high gas turbulence is a consequence of physical drivers acting on timescales of several Gyr, which disfavours stochastic processes, like the SFR, significantly impacting $\sigma_z$.
\section{Conclusions} \label{sec: Conclusions}
Using the {\sc eagle}~ simulations, we carried out a comprehensive analysis of the vertical velocity dispersion of cold gas, $\sigma_{z}$, in central galaxies in the redshift range $0<z\lesssim 4$. We considered galaxies with rotationally supported gas discs with stellar masses $\rm M_\star \ge 10^{9}\,\rm M_{\odot}$ that also contain at least 500 cold gas particles. The main aims of our paper were to establish whether galaxies in {\sc eagle}\ have similar $\sigma_z$ values to observed systems, and to additionally understand the physical drivers of $\sigma_{z}$. Our analysis focused on the Reference {\sc eagle}\ ({\sc Ref-L100}) run, which was supplemented by runs that eliminate stellar and AGN feedback ({\sc NoFb-L25} and {\sc NoAGN-L50}, respectively). Our main results are as follows.
\begin{itemize}
\item The redshift evolution of $\sigma_{z}$ in {\sc Ref-L100} is in good agreement with observations from various spectroscopic surveys. Results obtained from the {\sc NoAGN-L50} run, which does not include AGN feedback but does include stellar feedback, were similar to those obtained from the {\sc Ref-L100} run, indicating that AGN feedback is not an important driver of gas turbulence in the mass and redshift range of our analysis (see Fig.~\ref{fig: sigma_ev}).
\item The relationship between $\sigma_{z}$ and various global galaxy properties are qualitatively similar to those obtained observationally: $\sigma_{z}$ increases with increasing $\rm M_\star$ and SFR, but the dependence weakens below $\rm M_\star \lesssim 10^{9.5}\rm\, M_{\odot}$ (or SFRs $\lesssim 1\,\rm M_{\odot}\, yr^{-1}$). This is consistent with analytic models of the $\sigma_{z}-{\rm SFR}$ relation. However, at high SFRs, {\sc eagle}\ predicts a much weaker trend {between $\sigma_{z}$ and SFR than these analytic models}. $\sigma_{z}$ also correlates strongly with the virial mass of a galaxy's host halo, regardless of redshift or of the feedback implementation (see Fig.~\ref{fig: scaling relations}).
\item We analysed the $\sigma_{z}-{\rm M_\star}$ and $\sigma_{z}-{\rm SFR}$ relations in a run that does not include either AGN or stellar feedback ({\sc NoFb-L25}), and found that, at fixed halo mass, the evolution of $\sigma_z$ mimics that obtained for galaxies in the {\sc Ref-L100} run (Fig.~\ref{fig: sigma_ev}). In addition, the $\sigma_{z}-{\rm M_{200}}$ relations are remarkably alike in these two runs, albeit with a larger scatter in {\sc NoFb-L25}. The $\sigma_z-{\rm M_{\star}}$ and $\sigma_{z}-\rm SFR$ relations in {\sc NoFb-L25} have a similar slope but lower normalisation than they do in {\sc Ref-L100}
\item The evolutionary tracks of $\sigma_{z}$ for individual
galaxies resemble the evolution of their specific gas accretion rates (see Fig.~\ref{fig: sigma_tracks} and \ref{fig: sMacc_tracks}, respectively). The evolutionary tracks of $z=0.1$ discs of low and high $\sigma_{z}$ continue to be systematically different up to $z \approx 1$. The exact timescale over which this ``memory'' of a high or low $\sigma_{z}$ value is preserved depends on halo mass and the cosmic time at which we select galaxies, but it persists provided there are systematic differences in their gas accretion rates.
\item On average, galaxy mergers increase the gas velocity dispersion in discs, with major mergers causing the most significant increase. Over the mass and redshift range we consider, however, the number of galaxies affected by merger is small and has little impact on the global evolution of $\sigma_{z}$.
\item At some mass scales, we find a weak correlation between $\sigma_z$ and the fraction of mass contained in regions susceptible to collapse by gravitational instabilities (referred to as the clumpiness parameter, $f_{\rm c}$). In our approximate treatment, $f_{\rm c}$ aims to quantify how transport mechanisms that originate from clump formation affect gas turbulence. The $\sigma_{z}-f_{\rm c}$ correlations are significant at redshifts $z\gtrsim 1$, as well as in halos with virial mass $\rm M_{\rm 200} \gtrsim 10^{11.5}\, M_{\odot}$, consistent with gravitational instabilities being important in discs with high gas fractions (typical at high redshifts).
\item At fixed halo mass, $\sigma_{z}$ correlates with the specific gas accretion rate, $\dot{\rm M}_{\rm acc}/\rm M_{cold}$. The trends, however, weaken when the accretion rate is low, but become strong for galaxies with $\dot{\rm M}_{\rm acc} \gtrsim \rm 10^{-9.5}\ yr^{-1}$, which are common at high redshifts and among massive haloes. Similar results are obtained from the {\sc NoFb-L25} run, although in this case the $\sigma_{z}-\dot{\rm M}_{\rm acc}/\rm M_{cold}$ relations exhibit larger scatter. The tighter $\sigma_{z}-\dot{\rm M}_{\rm acc}/\rm M_{cold}$ correlation in the {\sc Ref-L100} run may be a manifestation of regulatory feedback mechanisms, which are not present in the {\sc NoFb-L25} (hence the large scatter).
\item Misaligned gas accretion, quantified by the angle between the disc's rotation axis and the angular momentum vector of accreting cold gas, correlates strongly with $\sigma_z$ in haloes with low specific accretion rates, i.e. $\dot{\rm M}_{\rm acc}/{\rm M_{cold}} \rm \lesssim 10^{-9.5}\ yr^{-1}$, which typically correspond to haloes with virial masses $\rm M_{\rm 200}<10^{11.8} \, M_{\odot}$ at $z\lesssim 1$. Thus -- in the regime of low gas accretion rate, where $\sigma_{z}$ is largely independent of the {\em net} rate of accretion -- turbulence is higher among galaxies whose accreted gas is misaligned with respect to the disc's angular momentum. The importance of both the angle of gas accretion and the relative amount of cold gas accreted indicates that, overall, gas accretion is the primary driver of gas turbulence in {\sc eagle}\ discs.
\end{itemize}
Our results suggest that a complex interplay of different physical processes determine the level of turbulence in cold gas disc's. The relative importance of each mechanism depends on the halo mass and the epoch at which a galaxy is identified. We infer this by analysing the correlation between $\sigma_{z}$ and tracers of each physical process. However, due to the limited time, mass and force resolution of {\sc eagle}, we cannot fully isolate the impact of each physical driver. Cosmological simulations with higher mass and force resolution are necessary to understand better the connection between gas turbulence and unresolved processes, such as stellar feedback and gravitational instabilities.
\section*{Data availability}
Observational data for the velocity dispersion will be shared on reasonable request to the corresponding author. Stellar masses and SFRs for the {\sf SAMI}, {\sf DYNAMO}, {\sf KROSS} and {\sc SINS+zC-SINF} surveys are publicly available. The {\sc EAGLE} simulations are publicly available; see \citet{McAlpine16, EAGLE17} for how to access {\sc EAGLE} data.
\section*{Acknowledgements}
We thank Hannah \"{U}bler and Nastascha F\"{o}rster Schreiber for providing us with data compilation of ionised velocity dispersion from different surveys. EJ acknowledges the support of the University of Western Australia (UWA) through a scholarship for international research fees and a university postgraduate award. EJ, CL and EW have received funding from the ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), through project number CE170100013. CL and ADL are the receipts of an Australian Research Council Discovery Project (DP210101945) funded by the Australian Government. ADL acknowledges financial support from the Australian Research Council through their Future Fellowship scheme (project Nr. FT160100250). This work made use of the supercomputer OzSTAR which is managed through the Centre for Astrophysics and Supercomputing at Swinburne University of Technology. This supercomputing facility is supported by Astronomy Australia Limited and the Australian Commonwealth Government through the national Collaborative Research Infrastructure Strategy (NCRIS). We acknowledge the Virgo Consortium for making their simulation data available. The {\sc eagle}\ simulations were performed using the DiRAC-2 facility at Durham, managed by the ICC, and the PRACE facility Curie based in France at TGCC, CEA, Bruyeres-le-Chatel.
\section{Introduction}
The journal \textit{Monthly Notices of the Royal Astronomical Society} (MNRAS) encourages authors to prepare their papers using \LaTeX.
The style file \verb'mnras.cls' can be used to approximate the final appearance of the journal, and provides numerous features to simplify the preparation of papers.
This document, \verb'mnras_guide.tex', provides guidance on using that style file and the features it enables.
This is not a general guide on how to use \LaTeX, of which many excellent examples already exist.
We particularly recommend \textit{Wikibooks \LaTeX}\footnote{\url{https://en.wikibooks.org/wiki/LaTeX}}, a collaborative online textbook which is of use to both beginners and experts.
Alternatively there are several other online resources, and most academic libraries also hold suitable beginner's guides.
For guidance on the contents of papers, journal style, and how to submit a paper, see the MNRAS Instructions to Authors\footnote{\label{foot:itas}\url{http://www.oxfordjournals.org/our_journals/mnras/for_authors/}}.
Only technical issues with the \LaTeX\ class are considered here.
\section{Obtaining and installing the MNRAS package}
Some \LaTeX\ distributions come with the MNRAS package by default.
If yours does not, you can either install it using your distribution's package manager, or download it from the Comprehensive \TeX\ Archive Network\footnote{\url{http://www.ctan.org/tex-archive/macros/latex/contrib/mnras}} (CTAN).
The files can either be installed permanently by placing them in the appropriate directory (consult the documentation for your \LaTeX\ distribution), or used temporarily by placing them in the working directory for your paper.
To use the MNRAS package, simply specify \verb'mnras' as the document class at the start of a \verb'.tex' file:
\begin{verbatim}
\documentclass{mnras}
\end{verbatim}
Then compile \LaTeX\ (and if necessary \bibtex) in the usual way.
\section{Preparing and submitting a paper}
We recommend that you start with a copy of the \texttt{mnras\_template.tex} file.
Rename the file, update the information on the title page, and then work on the text of your paper.
Guidelines for content, style etc. are given in the instructions to authors on the journal's website$^{\ref{foot:itas}}$.
Note that this document does not follow all the aspects of MNRAS journal style (e.g. it has a table of contents).
If a paper is accepted, it is professionally typeset and copyedited by the publishers.
It is therefore likely that minor changes to presentation will occur.
For this reason, we ask authors to ignore minor details such as slightly long lines, extra blank spaces, or misplaced figures, because these details will be dealt with during the production process.
Papers must be submitted electronically via the online submission system; paper submissions are not permitted.
For full guidance on how to submit a paper, see the instructions to authors.
\section{Class options}
\label{sec:options}
There are several options which can be added to the document class line like this:
\begin{verbatim}
\documentclass[option1,option2]{mnras}
\end{verbatim}
The available options are:
\begin{itemize}
\item \verb'letters' -- used for papers in the journal's Letters section.
\item \verb'onecolumn' -- single column, instead of the default two columns. This should be used {\it only} if necessary for the display of numerous very long equations.
\item \verb'doublespacing' -- text has double line spacing. Please don't submit papers in this format.
\item \verb'referee' -- \textit{(deprecated)} single column, double spaced, larger text, bigger margins. Please don't submit papers in this format.
\item \verb'galley' -- \textit{(deprecated)} no running headers, no attempt to align the bottom of columns.
\item \verb'landscape' -- \textit{(deprecated)} sets the whole document on landscape paper.
\item \verb"usenatbib" -- \textit{(all papers should use this)} this uses Patrick Daly's \verb"natbib.sty" package for citations.
\item \verb"usegraphicx" -- \textit{(most papers will need this)} includes the \verb'graphicx' package, for inclusion of figures and images.
\item \verb'useAMS' -- adds support for upright Greek characters \verb'\upi', \verb'\umu' and \verb'\upartial' ($\upi$, $\umu$ and $\upartial$). Only these three are included, if you require other symbols you will need to include the \verb'amsmath' or \verb'amsymb' packages (see section~\ref{sec:packages}).
\item \verb"usedcolumn" -- includes the package \verb"dcolumn", which includes two new types of column alignment for use in tables.
\end{itemize}
Some of these options are deprecated and retained for backwards compatibility only.
Others are used in almost all papers, but again are retained as options to ensure that papers written decades ago will continue to compile without problems.
If you want to include any other packages, see section~\ref{sec:packages}.
\section{Title page}
If you are using \texttt{mnras\_template.tex} the necessary code for generating the title page, headers and footers is already present.
Simply edit the title, author list, institutions, abstract and keywords as described below.
\subsection{Title}
There are two forms of the title: the full version used on the first page, and a short version which is used in the header of other odd-numbered pages (the `running head').
Enter them with \verb'\title[]{}' like this:
\begin{verbatim}
\title[Running head]{Full title of the paper}
\end{verbatim}
The full title can be multiple lines (use \verb'\\' to start a new line) and may be as long as necessary, although we encourage authors to use concise titles. The running head must be $\le~45$ characters on a single line.
See appendix~\ref{sec:advanced} for more complicated examples.
\subsection{Authors and institutions}
Like the title, there are two forms of author list: the full version which appears on the title page, and a short form which appears in the header of the even-numbered pages. Enter them using the \verb'\author[]{}' command.
If the author list is more than one line long, start a new line using \verb'\newauthor'. Use \verb'\\' to start the institution list. Affiliations for each author should be indicated with a superscript number, and correspond to the list of institutions below the author list.
For example, if I were to write a paper with two coauthors at another institution, one of whom also works at a third location:
\begin{verbatim}
\author[K. T. Smith et al.]{
Keith T. Smith,$^{1}$
A. N. Other,$^{2}$
and Third Author$^{2,3}$
\\
$^{1}$Affiliation 1\\
$^{2}$Affiliation 2\\
$^{3}$Affiliation 3}
\end{verbatim}
Affiliations should be in the format `Department, Institution, Street Address, City and Postal Code, Country'.
Email addresses can be inserted with the \verb'\thanks{}' command which adds a title page footnote.
If you want to list more than one email, put them all in the same \verb'\thanks' and use \verb'\footnotemark[]' to refer to the same footnote multiple times.
Present addresses (if different to those where the work was performed) can also be added with a \verb'\thanks' command.
\subsection{Abstract and keywords}
The abstract is entered in an \verb'abstract' environment:
\begin{verbatim}
\begin{abstract}
The abstract of the paper.
\end{abstract}
\end{verbatim}
\noindent Note that there is a word limit on the length of abstracts.
For the current word limit, see the journal instructions to authors$^{\ref{foot:itas}}$.
Immediately following the abstract, a set of keywords is entered in a \verb'keywords' environment:
\begin{verbatim}
\begin{keywords}
keyword 1 -- keyword 2 -- keyword 3
\end{keywords}
\end{verbatim}
\noindent There is a list of permitted keywords, which is agreed between all the major astronomy journals and revised every few years.
Do \emph{not} make up new keywords!
For the current list of allowed keywords, see the journal's instructions to authors$^{\ref{foot:itas}}$.
\section{Sections and lists}
Sections and lists are generally the same as in the standard \LaTeX\ classes.
\subsection{Sections}
\label{sec:sections}
Sections are entered in the usual way, using \verb'\section{}' and its variants. It is possible to nest up to four section levels:
\begin{verbatim}
\section{Main section}
\subsection{Subsection}
\subsubsection{Subsubsection}
\paragraph{Lowest level section}
\end{verbatim}
\noindent The other \LaTeX\ sectioning commands \verb'\part', \verb'\chapter' and \verb'\subparagraph{}' are deprecated and should not be used.
Some sections are not numbered as part of journal style (e.g. the Acknowledgements).
To insert an unnumbered section use the `starred' version of the command: \verb'\section*{}'.
See appendix~\ref{sec:advanced} for more complicated examples.
\subsection{Lists}
Two forms of lists can be used in MNRAS -- numbered and unnumbered.
For a numbered list, use the \verb'enumerate' environment:
\begin{verbatim}
\begin{enumerate}
\item First item
\item Second item
\item etc.
\end{enumerate}
\end{verbatim}
\noindent which produces
\begin{enumerate}
\item First item
\item Second item
\item etc.
\end{enumerate}
Note that the list uses lowercase Roman numerals, rather than the \LaTeX\ default Arabic numerals.
For an unnumbered list, use the \verb'description' environment without the optional argument:
\begin{verbatim}
\begin{description}
\item First item
\item Second item
\item etc.
\end{description}
\end{verbatim}
\noindent which produces
\begin{description}
\item First item
\item Second item
\item etc.
\end{description}
Bulleted lists using the \verb'itemize' environment should not be used in MNRAS; it is retained for backwards compatibility only.
\section{Mathematics and symbols}
The MNRAS class mostly adopts standard \LaTeX\ handling of mathematics, which is briefly summarised here.
See also section~\ref{sec:packages} for packages that support more advanced mathematics.
Mathematics can be inserted into the running text using the syntax \verb'$1+1=2$', which produces $1+1=2$.
Use this only for short expressions or when referring to mathematical quantities; equations should be entered as described below.
\subsection{Equations}
Equations should be entered using the \verb'equation' environment, which automatically numbers them:
\begin{verbatim}
\begin{equation}
a^2=b^2+c^2
\end{equation}
\end{verbatim}
\noindent which produces
\begin{equation}
a^2=b^2+c^2
\end{equation}
By default, the equations are numbered sequentially throughout the whole paper. If a paper has a large number of equations, it may be better to number them by section (2.1, 2.2 etc.). To do this, add the command \verb'\numberwithin{equation}{section}' to the preamble.
It is also possible to produce un-numbered equations by using the \LaTeX\ built-in \verb'\['\textellipsis\verb'\]' and \verb'$$'\textellipsis\verb'$$' commands; however MNRAS requires that all equations are numbered, so these commands should be avoided.
\subsection{Special symbols}
\begin{table}
\caption{Additional commands for special symbols commonly used in astronomy. These can be used anywhere.}
\label{tab:anysymbols}
\begin{tabular}{lll}
\hline
Command & Output & Meaning\\
\hline
\verb'\sun' & \sun & Sun, solar\\[2pt]
\verb'\earth' & \earth & Earth, terrestrial\\[2pt]
\verb'\micron' & \micron & microns\\[2pt]
\verb'\degr' & \degr & degrees\\[2pt]
\verb'\arcmin' & \arcmin & arcminutes\\[2pt]
\verb'\arcsec' & \arcsec & arcseconds\\[2pt]
\verb'\fdg' & \fdg & fraction of a degree\\[2pt]
\verb'\farcm' & \farcm & fraction of an arcminute\\[2pt]
\verb'\farcs' & \farcs & fraction of an arcsecond\\[2pt]
\verb'\fd' & \fd & fraction of a day\\[2pt]
\verb'\fh' & \fh & fraction of an hour\\[2pt]
\verb'\fm' & \fm & fraction of a minute\\[2pt]
\verb'\fs' & \fs & fraction of a second\\[2pt]
\verb'\fp' & \fp & fraction of a period\\[2pt]
\verb'\diameter' & \diameter & diameter\\[2pt]
\verb'\sq' & \sq & square, Q.E.D.\\[2pt]
\hline
\end{tabular}
\end{table}
\begin{table}
\caption{Additional commands for mathematical symbols. These can only be used in maths mode.}
\label{tab:mathssymbols}
\begin{tabular}{lll}
\hline
Command & Output & Meaning\\
\hline
\verb'\upi' & $\upi$ & upright pi\\[2pt]
\verb'\umu' & $\umu$ & upright mu\\[2pt]
\verb'\upartial' & $\upartial$ & upright partial derivative\\[2pt]
\verb'\lid' & $\lid$ & less than or equal to\\[2pt]
\verb'\gid' & $\gid$ & greater than or equal to\\[2pt]
\verb'\la' & $\la$ & less than of order\\[2pt]
\verb'\ga' & $\ga$ & greater than of order\\[2pt]
\verb'\loa' & $\loa$ & less than approximately\\[2pt]
\verb'\goa' & $\goa$ & greater than approximately\\[2pt]
\verb'\cor' & $\cor$ & corresponds to\\[2pt]
\verb'\sol' & $\sol$ & similar to or less than\\[2pt]
\verb'\sog' & $\sog$ & similar to or greater than\\[2pt]
\verb'\lse' & $\lse$ & less than or homotopic to \\[2pt]
\verb'\gse' & $\gse$ & greater than or homotopic to\\[2pt]
\verb'\getsto' & $\getsto$ & from over to\\[2pt]
\verb'\grole' & $\grole$ & greater over less\\[2pt]
\verb'\leogr' & $\leogr$ & less over greater\\
\hline
\end{tabular}
\end{table}
Some additional symbols of common use in astronomy have been added in the MNRAS class. These are shown in tables~\ref{tab:anysymbols}--\ref{tab:mathssymbols}. The command names are -- as far as possible -- the same as those used in other major astronomy journals.
Many other mathematical symbols are also available, either built into \LaTeX\ or via additional packages. If you want to insert a specific symbol but don't know the \LaTeX\ command, we recommend using the Detexify website\footnote{\url{http://detexify.kirelabs.org}}.
Sometimes font or coding limitations mean a symbol may not get smaller when used in sub- or superscripts, and will therefore be displayed at the wrong size. There is no need to worry about this as it will be corrected by the typesetter during production.
To produce bold symbols in mathematics, use \verb'\bmath' for simple variables, and the \verb'bm' package for more complex symbols (see section~\ref{sec:packages}). Vectors are set in bold italic, using \verb'\mathbfit{}'.
For matrices, use \verb'\mathbfss{}' to produce a bold sans-serif font e.g. \mathbfss{H}; this works even outside maths mode, but not all symbols are available (e.g. Greek). For $\nabla$ (del, used in gradients, divergence etc.) use \verb'$\nabla$'.
\subsection{Ions}
A new \verb'\ion{}{}' command has been added to the class file, for the correct typesetting of ionisation states.
For example, to typeset singly ionised calcium use \verb'\ion{Ca}{ii}', which produces \ion{Ca}{ii}.
\section{Figures and tables}
\label{sec:fig_table}
Figures and tables (collectively called `floats') are mostly the same as built into \LaTeX.
\subsection{Basic examples}
\begin{figure}
\includegraphics[width=\columnwidth]{example}
\caption{An example figure.}
\label{fig:example}
\end{figure}
Figures are inserted in the usual way using a \verb'figure' environment and \verb'\includegraphics'. The example Figure~\ref{fig:example} was generated using the code:
\begin{verbatim}
\begin{figure}
\includegraphics[width=\columnwidth]{example}
\caption{An example figure.}
\label{fig:example}
\end{figure}
\end{verbatim}
\begin{table}
\caption{An example table.}
\label{tab:example}
\begin{tabular}{lcc}
\hline
Star & Mass & Luminosity\\
& $M_{\sun}$ & $L_{\sun}$\\
\hline
Sun & 1.00 & 1.00\\
$\alpha$~Cen~A & 1.10 & 1.52\\
$\epsilon$~Eri & 0.82 & 0.34\\
\hline
\end{tabular}
\end{table}
The example Table~\ref{tab:example} was generated using the code:
\begin{verbatim}
\begin{table}
\caption{An example table.}
\label{tab:example}
\begin{tabular}{lcc}
\hline
Star & Mass & Luminosity\\
& $M_{\sun}$ & $L_{\sun}$\\
\hline
Sun & 1.00 & 1.00\\
$\alpha$~Cen~A & 1.10 & 1.52\\
$\epsilon$~Eri & 0.82 & 0.34\\
\hline
\end{tabular}
\end{table}
\end{verbatim}
\subsection{Captions and placement}
Captions go \emph{above} tables but \emph{below} figures, as in the examples above.
The \LaTeX\ float placement commands \verb'[htbp]' are intentionally disabled.
Layout of figures and tables will be adjusted by the publisher during the production process, so authors should not concern themselves with placement to avoid disappointment and wasted effort.
Simply place the \LaTeX\ code close to where the figure or table is first mentioned in the text and leave exact placement to the publishers.
By default a figure or table will occupy one column of the page.
To produce a wider version which covers both columns, use the \verb'figure*' or \verb'table*' environment.
If a figure or table is too long to fit on a single page it can be split it into several parts.
Create an additional figure or table which uses \verb'\contcaption{}' instead of \verb'\caption{}'.
This will automatically correct the numbering and add `\emph{continued}' at the start of the caption.
\begin{table}
\contcaption{A table continued from the previous one.}
\label{tab:continued}
\begin{tabular}{lcc}
\hline
Star & Mass & Luminosity\\
& $M_{\sun}$ & $L_{\sun}$\\
\hline
$\tau$~Cet & 0.78 & 0.52\\
$\delta$~Pav & 0.99 & 1.22\\
$\sigma$~Dra & 0.87 & 0.43\\
\hline
\end{tabular}
\end{table}
Table~\ref{tab:continued} was generated using the code:
\begin{verbatim}
\begin{table}
\contcaption{A table continued from the previous one.}
\label{tab:continued}
\begin{tabular}{lcc}
\hline
Star & Mass & Luminosity\\
& $M_{\sun}$ & $L_{\sun}$\\
\hline
$\tau$~Cet & 0.78 & 0.52\\
$\delta$~Pav & 0.99 & 1.22\\
$\sigma$~Dra & 0.87 & 0.43\\
\hline
\end{tabular}
\end{table}
\end{verbatim}
To produce a landscape figure or table, use the \verb'pdflscape' package and the \verb'landscape' environment.
The landscape Table~\ref{tab:landscape} was produced using the code:
\begin{verbatim}
\begin{landscape}
\begin{table}
\caption{An example landscape table.}
\label{tab:landscape}
\begin{tabular}{cccccccccc}
\hline
Header & Header & ...\\
Unit & Unit & ...\\
\hline
Data & Data & ...\\
Data & Data & ...\\
...\\
\hline
\end{tabular}
\end{table}
\end{landscape}
\end{verbatim}
Unfortunately this method will force a page break before the table appears.
More complicated solutions are possible, but authors shouldn't worry about this.
\begin{landscape}
\begin{table}
\caption{An example landscape table.}
\label{tab:landscape}
\begin{tabular}{cccccccccc}
\hline
Header & Header & Header & Header & Header & Header & Header & Header & Header & Header\\
Unit & Unit & Unit & Unit & Unit & Unit & Unit & Unit & Unit & Unit \\
\hline
Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
\hline
\end{tabular}
\end{table}
\end{landscape}
\section{References and citations}
\subsection{Cross-referencing}
The usual \LaTeX\ commands \verb'\label{}' and \verb'\ref{}' can be used for cross-referencing within the same paper.
We recommend that you use these whenever relevant, rather than writing out the section or figure numbers explicitly.
This ensures that cross-references are updated whenever the numbering changes (e.g. during revision) and provides clickable links (if available in your compiler).
It is best to give each section, figure and table a logical label.
For example, Table~\ref{tab:mathssymbols} has the label \verb'tab:mathssymbols', whilst section~\ref{sec:packages} has the label \verb'sec:packages'.
Add the label \emph{after} the section or caption command, as in the examples in sections~\ref{sec:sections} and \ref{sec:fig_table}.
Enter the cross-reference with a non-breaking space between the type of object and the number, like this: \verb'see Figure~\ref{fig:example}'.
The \verb'\autoref{}' command can be used to automatically fill out the type of object, saving on typing.
It also causes the link to cover the whole phrase rather than just the number, but for that reason is only suitable for single cross-references rather than ranges.
For example, \verb'\autoref{tab:journal_abbr}' produces \autoref{tab:journal_abbr}.
\subsection{Citations}
\label{sec:cite}
MNRAS uses the Harvard -- author (year) -- citation style, e.g. \citet{author2013}.
This is implemented in \LaTeX\ via the \verb'natbib' package, which in turn is included via the \verb'usenatbib' package option (see section~\ref{sec:options}), which should be used in all papers.
Each entry in the reference list has a `key' (see section~\ref{sec:ref_list}) which is used to generate citations.
There are two basic \verb'natbib' commands:
\begin{description}
\item \verb'\citet{key}' produces an in-text citation: \citet{author2013}
\item \verb'\citep{key}' produces a bracketed (parenthetical) citation: \citep{author2013}
\end{description}
Citations will include clickable links to the relevant entry in the reference list, if supported by your \LaTeX\ compiler.
\defcitealias{smith2014}{Paper~I}
\begin{table*}
\caption{Common citation commands, provided by the \texttt{natbib} package.}
\label{tab:natbib}
\begin{tabular}{lll}
\hline
Command & Ouput & Note\\
\hline
\verb'\citet{key}' & \citet{smith2014} & \\
\verb'\citep{key}' & \citep{smith2014} & \\
\verb'\citep{key,key2}' & \citep{smith2014,jones2015} & Multiple papers\\
\verb'\citet[table 4]{key}' & \citet[table 4]{smith2014} & \\
\verb'\citep[see][figure 7]{key}' & \citep[see][figure 7]{smith2014} & \\
\verb'\citealt{key}' & \citealt{smith2014} & For use with manual brackets\\
\verb'\citeauthor{key}' & \citeauthor{smith2014} & If already cited in close proximity\\
\verb'\defcitealias{key}{Paper~I}' & & Define an alias (doesn't work in floats)\\
\verb'\citetalias{key}' & \citetalias{smith2014} & \\
\verb'\citepalias{key}' & \citepalias{smith2014} & \\
\hline
\end{tabular}
\end{table*}
There are a number of other \verb'natbib' commands which can be used for more complicated citations.
The most commonly used ones are listed in Table~\ref{tab:natbib}.
For full guidance on their use, consult the \verb'natbib' documentation\footnote{\url{http://www.ctan.org/pkg/natbib}}.
If a reference has several authors, \verb'natbib' will automatically use `et al.' if there are more than two authors. However, if a paper has exactly three authors, MNRAS style is to list all three on the first citation and use `et al.' thereafter. If you are using \bibtex\ (see section~\ref{sec:ref_list}) then this is handled automatically. If not, the \verb'\citet*{}' and \verb'\citep*{}' commands can be used at the first citation to include all of the authors.
\subsection{The list of references}
\label{sec:ref_list}
It is possible to enter references manually using the usual \LaTeX\ commands, but we strongly encourage authors to use \bibtex\ instead.
\bibtex\ ensures that the reference list is updated automatically as references are added or removed from the paper, puts them in the correct format, saves on typing, and the same reference file can be used for many different papers -- saving time hunting down reference details.
An MNRAS \bibtex\ style file, \verb'mnras.bst', is distributed as part of this package.
The rest of this section will assume you are using \bibtex.
References are entered into a separate \verb'.bib' file in standard \bibtex\ formatting.
This can be done manually, or there are several software packages which make editing the \verb'.bib' file much easier.
We particularly recommend \textsc{JabRef}\footnote{\url{http://jabref.sourceforge.net/}}, which works on all major operating systems.
\bibtex\ entries can be obtained from the NASA Astrophysics Data System\footnote{\label{foot:ads}\url{http://adsabs.harvard.edu}} (ADS) by clicking on `Bibtex entry for this abstract' on any entry.
Simply copy this into your \verb'.bib' file or into the `BibTeX source' tab in \textsc{JabRef}.
Each entry in the \verb'.bib' file must specify a unique `key' to identify the paper, the format of which is up to the author.
Simply cite it in the usual way, as described in section~\ref{sec:cite}, using the specified key.
Compile the paper as usual, but add an extra step to run the \texttt{bibtex} command.
Consult the documentation for your compiler or latex distribution.
Correct formatting of the reference list will be handled by \bibtex\ in almost all cases, provided that the correct information was entered into the \verb'.bib' file.
Note that ADS entries are not always correct, particularly for older papers and conference proceedings, so may need to be edited.
If in doubt, or if you are producing the reference list manually, see the MNRAS instructions to authors$^{\ref{foot:itas}}$ for the current guidelines on how to format the list of references.
\section{Appendices and online material}
To start an appendix, simply place the \verb'
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{"url":"https:\/\/sematext.com\/docs\/agents\/sematext-agent\/jmx-attaching\/","text":"share\n\n# Automatic attaching to JVM processes via JMX\n\nWhen you assign a MONITORING_TOKEN to a service running in the JVM via an environment variable or you enable automatic autodiscovery monitoring for a particular service type, Sematext Agent will examine the process to see whether it can be monitored right away by checking if JMX ports are already exposed. If it can, Sematext Agent will start monitoring it. However, if it can't, Sematext Agent will try to add a special and secure agent for the monitored process, whose sole purpose is exposing JMX. Exposed JMX is protected by authentication based on a temporary username and password known only to Sematext Agent.\n\nIf JMX is already exposed, but is protected with some authentication mechanism whose credentials are not know to Sematext Agent, the agent will try to expose another JMX connector.\n\nThere are cases where automatic attaching may not be desirable:\n\n\u2022 if you wish to use a different kind of authentication, maybe with truststore, on an exposed JMX port\n\u2022 if Sematext Agent attaching logic is failing for some reason, which you would notice by missing metrics and errors in agent logs\n\n### How to disable or configure custom JMX authentication settings?\u00b6\n\nIf you don't want the service monitored at all just remove MONITORING_TOKEN env variable from its container. If you want it to be monitored, but using your own JMX authentication definition (either existing one or the one you will define just for this), define container jmx settings manually.\n\n### Security manager permissions\u00b6\n\nWhen running your JVM applications under the supervision of the security manager, you'll have to register the permission to allow Sematext Agent to bootstrap the JMX connector in the target process. For this purpose, add the following permission directive in your policy file:\n\npermission java.lang.RuntimePermission \"accessClassInPackage.com.sun.jndi.url.rmi\";\n\nFor instance, Solr 9 has the Security Manager enabled by default. In container deployments the policy file is located in \/opt\/solr\/server\/etc\/security.policy. You can expand this file with the previous java.lang.RuntimePermission permission and bind-mount inside Solr container:\n\n-v \/path\/to\/security.policy:\/opt\/solr\/server\/etc\/security.policy","date":"2023-02-06 09:26:47","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.22182755172252655, \"perplexity\": 5912.0064723111045}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-06\/segments\/1674764500334.35\/warc\/CC-MAIN-20230206082428-20230206112428-00568.warc.gz\"}"}
| null | null |
package info.gianlucacosta.omnieditor
import java.time.Duration
import info.gianlucacosta.helios.concurrency.AtomicStringBuilder
import scalafx.application.Platform
private case class OutputThread(
refreshRate: Duration,
outputBuffer: AtomicStringBuilder,
outputAction: String => Unit
) extends Thread {
override def run(): Unit = {
while (!Thread.interrupted) {
{
tryToOutput()
try {
Thread.sleep(refreshRate.toMillis)
}
catch {
case e: InterruptedException =>
tryToOutput()
return
}
}
}
}
private def tryToOutput(): Unit = {
val textToOutput = outputBuffer.extract()
if (!textToOutput.isEmpty) {
Platform.runLater {
outputAction(textToOutput)
}
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,963
|
Ridge and furrow is an archaeological pattern of ridges (Medieval Latin: sliones) and troughs created by a system of ploughing used in Europe during the Middle Ages, typical of the open-field system. It is also known as rig (or rigg) and furrow, mostly in the North East of England and in Scotland.
The earliest examples date to the immediate post-Roman period and the system was used until the 17th century in some areas, as long as the open field system survived. Surviving ridge and furrow topography is found in Great Britain, Ireland and elsewhere in Europe. The surviving ridges are parallel, ranging from apart and up to tall – they were much taller when in use. Older examples are often curved.
Ridge and furrow topography was a result of ploughing with non-reversible ploughs on the same strip of land each year. It is visible on land that was ploughed in the Middle Ages, but which has not been ploughed since then. No actively ploughed ridge and furrow survives.
The ridges or lands became units in landholding, in assessing the work of the plougher and in reaping in autumn.
Origin
Traditional ploughs have the ploughshare and mould-board on the right, and so turn the soil over to the right. This means that the plough cannot return along the same line for the next furrow. Instead, ploughing is done in a clockwise direction around a long rectangular strip (a land). After ploughing one of the long sides of the strip, the plough is removed from the ground at the end of the field, moved across the unploughed headland (the short end of the strip), then put back in the ground to work back down the other long side of the strip. The width of the ploughed strip is fairly narrow, to avoid having to drag the plough too far across the headland. This process has the effect of moving the soil in each half of the strip one furrow's-width towards the centre line each time the field is ploughed.
In the Middle Ages each strip was managed by one family, within large open fields held in common, and the locations of the strips were the same each year. The movement of soil year after year gradually built the centre of each strip up into a ridge, leaving a dip, or "furrow" between each ridge (this use of "furrow" is different from that for the small furrow left by each pass of the plough). The building up of a ridge was called filling or gathering, and was sometimes done before ploughing began. The raised ridges offered better drainage in a wet climate: moisture drained into the furrows, and since the ridges were laid down a slope, in a sloping field water would collect in a ditch at the bottom. Only on some well-drained soils were the fields left flat. In damper soil towards the base of the ridge, pulses (peas or beans) or dredge (a mixture of oats and barley) might be sown where wheat would have become waterlogged, as Thomas Tusser suggested in the 16th century:
For wheat till land
Where water doth stand.
Sow pease or dredge
below in that redge.
The dip often marked the boundary between plots. Although they varied, strips would traditionally be a furlong (a "furrow-long") in length, (220 yards, about 200 metres), and from about up to a chain wide (22 yards, about 20 metres), giving an area of from .
In most places ploughing continued over the centuries, and later methods (especially the reversible plough) removed the ridge and furrow pattern. However, in some cases the land became grassland, and where this has not been ploughed since, the pattern has often been preserved. Surviving ridge and furrow may have a height difference of in places, and gives a strongly rippled effect to the landscape. When in active use, the height difference was even more, over in places.
Curved strips
In the early Middle Ages ploughing was done with large teams of small oxen (commonly eight oxen in four pairs), and the plough itself was a large, mainly wooden implement. The team and plough together were therefore many yards long, and this led to a particular effect in ridge and furrow fields. When reaching the end of the furrow, the leading oxen met the end first, and were turned left along the headland, while the plough continued as long as possible in the furrow (the strongest oxen were yoked at the back, and could draw the plough on their own for this short distance). By the time the plough eventually reached the end, the oxen were standing lined up facing leftwards along the headland. Each pair was then turned around to walk rightwards along the headland, crossing the end of the strip, and they then started down the opposite furrow. By the time the plough itself reached the beginning of the furrow, the oxen were already lined up ready to pull it forwards.
The result of this was to twist the end of each furrow slightly to the left, making these earlier ridge and furrows into a slight reverse-S shape. This shape survives in some places as curved field boundaries, even where the ridge and furrow pattern itself has vanished.
If the oxen had been turned right at the end of the furrow, they would immediately have had to turn right again down the returning furrow, making the line of oxen cut across the top of the ploughed strip and thus pulling the plough out of the ground before it reached the end of the furrow, as well as having potential difficulty from two adjacent lines of oxen moving in opposite directions. Alternatively, if lined up rightwards along the headland, some would already be past the beginning of the new furrow, and these would have to be moved awkwardly sideways into the furrow to be ready to plough. Turning to the left made one turn at a time and avoided a sideways move.
As oxen became larger and ploughs more efficient, smaller teams were needed. These took less room on the headland, and straight ploughing became easier – and easier still when heavy horses were introduced. Late Middle Ages ridge and furrow is therefore straight.
Surviving locations
Some of the best-preserved ridge and furrow survives in the English counties of:
Buckinghamshire
Cambridgeshire
County Durham
Derbyshire
Gloucestershire
Lincolnshire
Leicestershire
Northamptonshire
Nottinghamshire
Oxfordshire
Warwickshire
West Yorkshire
In Scotland, 4-600 acres of rig and furrow survive in one area outside the town of Airdrie.
Ridge and furrow often survives on higher ground where the arable land was subsequently turned over to sheep walk in the 15th century and has not been ploughed out since by modern ploughing methods, today surviving still as pasture and grazing for sheep where the effect is clearly visible, especially when the sun is low or after a dusting of snow. It is often associated with deserted medieval villages.
Similar agricultural landforms
Cord rig, cultivation ridges created by spade digging
Lazy beds, cultivation ridges created by spade digging
Lynchets, sloping terraces on steep hillsides, created by gravity on hillslopes subject to ploughing
Raised bed gardening, a modern system of raising cultivated land above the surrounding ground
Run rig and rundale, Scottish and Irish land-use patterns named after their characteristic ridges and furrows
Water-meadows, grassland with ridges and dips to control irrigation – superficially similar to ridge and furrow, but the origin, pattern and use were very different
References
External links
Examples of ridge and furrow in photos on geograph.org.uk
Video footage of ridge and furrow.
History of agriculture
European archaeology
Landscape history
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 6,213
|
My sister goes to North Korea for Pyongyang marathon with me
From The New Paper on Sunday:
On April 7, 13 North Korean restaurant workers defected en masse to South Korea.
This suggests that North Korea is a place people flee from, not flock to.
Yet three days later, 1,000 foreigners flocked to the reclusive country to run in the Mangyongdae Prize International Marathon, also known as the Pyongyang Marathon.
Singaporean Ong Wann was one of them. She was there with her brother, a running enthusiast, to race in the 10km category of the event.
He had asked her to go along to be his translator as she speaks Korean.
Miss Ong, 39, who owns and operates the Hanok Korean Language School in Singapore, had studied to be a Korean language teacher in Sogang University and Kyung Hee University in Seoul.
She had previously run the 10km race in the Standard Chartered Marathon and Great Eastern Women's Run. This would be her first overseas race.
Was she apprehensive about going to North Korea, especially since an American tourist was recently sentenced to 15 years of hard labour for stealing a poster from a Pyongyang hotel?
"Nope," Miss Ong says. "I have friends who have been there and all of them got back safely.
"But the South Korean teachers in my school were both excited and worried for me. One of them said, 'You must come back alive!'
"Although she was joking, I think she really meant it too."
Miss Ong had to fly from Singapore to Beijing, China, to make the connecting flight to Pyongyang on Air Koryo, North Korea's national carrier, which has been ranked the world's worst airline four years in a row by Skytrax.
After surviving the two-hour flight, she was surprised to find that she could order a skinny latte at the Pyongyang airport cafe. "Some cafes in Singapore don't even have low-fat milk," she says.
Miss Ong was then taken in a tour bus to 22m-tall bronze statues of late North Korean leaders Kim Il Sung and his son Kim Jong Il and instructed to place flowers in front of the statues and bow to them.
During a pre-tour briefing, Miss Ong was told that the leaders were treated almost like religious figures, and visitors must be careful not to behave inappropriately at their monuments.
While photography is allowed, the whole statue must be in the frame. You cannot, for example, take a picture of the statues from the waist up.
There are other restrictions — no photos of the military, construction sites and local people without their permission.
But when Miss Ong saw a local couple using the giant statues as backdrop for a bridal shoot, she couldn't resist taking a picture of them — without their permission.
It is a wonder she wasn't arrested and sent to a labour camp immediately.
You are also not allowed to go anywhere without your tour guide. At the hotel, guests were warned not to wander beyond the hotel grounds.
Not that you would want or need to. In the Yanggakdo International Hotel, where Miss Ong stayed and the American stole the poster, you can drink at the bar, buy snacks, shop for souvenirs, get a haircut, swim, bowl, play billiards and table tennis, and sing karaoke to rock classics like Bohemian Rhapsody.
"During the pre-tour briefing, we were told there's a massage parlour, which was really a brothel," she says.
Before going to Pyongyang, Miss Ong also read that the North Koreans foreigners see on the streets are actors. To verify this, after completing her 10km race, she chatted with two young North Korean runners and asked for permission to take pictures with them. "They are not actors," she concludes.
Her fluency in the Korean language also came in handy as she became the de facto translator for her tour group of runners from Europe, US, Israel and Hong Kong, who turned to her to find out the prices of souvenirs and decipher random signs and slogans on propaganda posters for sale.
"I know as tourists, what we saw and experienced are many times better than what most North Korean enjoy," says Miss Ong.
"By talking to the local tour guides, it seems their leisure lives revolve around sports and hanging out with friends and family.
"Although they use mobile phones and have a national intranet instead of the Internet, they are careful about what they say and have a slower pace of life — it's almost like going back to 30 years ago.
Miss Ong flew out of North Korea with her brother the day after the marathon and was surprised that at the Pyongyang Sunan International Airport, their cameras weren't checked for unauthorised photographs.
The trip made her grapple with her presumptions about the country, "some true and others utter nonsense", she says.
"I could describe the experience as surreal, strange, unexpected, because the place is really more normal than what we expected it to be."
Back in Singapore, in the taxi from Changi Airport, Miss Ong mentioned to the driver that she and her brother had just returned from North Korea and was taken aback by the cabby's intense reaction.
She recounts: "He started telling us how dangerous the place was and insisted that it wasn't safe like we said.
"It was as if he was the one there — not us."
From The Korea Times:
Pyongyang marathon runner says she felt no sense of crisis in Pyongyang
There appears to be no sense of crisis in Pyongyang according to a foreign participant in a North Korean marathon.
"I did not find any sense of crisis and uneasiness in North Korea despite the ongoing international sanctions," said a Singaporean woman, fresh from her rare trip to Pyongyang to participate in this year's annual Pyongyang marathon.
"The streets of Pyongyang, with the full blossoms of cherry and forsythia, were peaceful and people looked cheerful and bright," Ong Wann told Yonhap News Agency in a telephone conversation.
Ong, 39, made a rare three-day trip to Pyongyang from April 9 with her brother and joined the 10-kilometer half marathon course. Participants can choose to run the full or half marathon.
The Pyongyang marathon is officially called the "Mangyongdae Prize International Marathon." Mangyongdae is the place where North Korea says its founding leader Kim Il-sung and the late grandfather of the current leader Kim Jong-un, was born.
Kim Il-sung's birthday is April 15 and the marathon is one of many events staged to celebrate the key date.
This year marked the third time that foreign amateurs have been allowed to take part in the event. However, the race saw an increase in amateur participation -- nearly 1,000 foreign amateurs took part, the marathon's organizers said.
The Singaporean woman studied in South Korea previously and speaks Korean fluently. She is currently operating a Korean language institute in Singapore.
During her trip to Pyongyang, she occasionally acted as an interpreter for other foreigners including her brother.
She said the atmosphere in Pyongyang was far from any sense of crisis or insecurity although the marathon event was held under the severe international sanctions for the North's nuclear test in January and the long-range rocket launch the following month.
She explained the landscape of Pyongyang was generally peaceful when she looked around some tourist spots such as Mansudae Hill and nearby parks and subway stations.
"The streets of Pyongyang were full of cherry and forsythia blossoms and people on the streets look relatively bright," she said.
"At first I somewhat had a sense of fear as I had a negative image of North Korea which I had obtained from newspapers and media, but was able to adapt myself soon," she said.
"Before I went to North Korea I was told about the prohibitions on what to do and what is not allowed in the country. In fact, individual action was impossible without North Korean guides, but nevertheless their control was not as tight as I first imagined," she said.
Ong said she felt she came to a different world when she saw all the propaganda posters and mural paintings on the walls of subway stations in Pyongyang, which is quite different from commercial advertisements that are often seen in capitalistic countries.
She said that North Korea was also different from other former communist bloc countries which had transformed themselves into capitalistic system in some areas.
"I felt North Korea is not like China and Russia," she said.
EARLIER:
I went to North Korea & ran 10k in the Pyongyang marathon
I went to North Korea & took lots of selfies
I went to North Korea & asked for the Kim Jong Un haircut (and lived)
I took the MRT train in North Korea & it didn't break down
Tags: DPRK
I took the MRT train in North Korea and it didn't break down
In light of another major MRT breakdown in Singapore, I figure now is the perfect time to share my experience of taking the Metro in North Korea just over two weeks ago.
This was in the middle of a Saturday afternoon the day before the Pyongyang marathon.
It was arranged for us to take the train as a group, so we didn't get to experience how the ticketing system works.
Outside a Metro station:
Inside the station with my sister looking at the turnstiles:
Taking the longest escalator ride (4 minutes) down to the deepest underground rail system in the world:
The train platforms are quite grandly decorated.
Newspapers for the communists, I mean, commuters to read.
A video taken by my sister of a train arriving:
Unlike in Singapore, there are no safety barriers on the platform. It looked like the passengers have to pull open the train doors themselves.
On the train:
One thing I noticed about the train was that some of the windows can be opened and were, which made the ride even noisier.
Leaving the station:
Another video taken by my sister riding the escalator up and out of the station:
Tags: abroad, DPRK, transport
Too hot: Bad news for durian lovers, especially if you live in Choa Chu Kang
April has been the cruellest month for durian lovers and Choa Chu Kang residents.
Just my luck. I love durian and live in Choa Chu Kang.
You know how the weather has been even hotter than usual lately?
It's so hot that you sort of understand why Felicia Chin looked almost nude at last Sunday's Star Awards — to keep cool.
But while Rui En apologised that night for the "Do you know who I am" incident, some viewers would contend it was Chin who should have apologised for her see-through dress.
Who does she think she is? Ann Kok?
Blame it on the heat.
The National Environment Agency (NEA) said the highest daily maximum temperature recorded in Singapore on Tuesday was 36 deg C and on Wednesday (until 3pm), it was 35.1 deg C.
Both were recorded in Choa Chu Kang.
I knew I should've moved to Bukit Batok!
At least the people there got a carnival last Sunday.
It was organised by People's Action Party (PAP) candidate Murali Pillai, who is campaigning for the by-election to be held on May 7 after Mr David Ong resigned as Bukit Batok MP over a "personal indiscretion".
So in a way, Bukit Batok residents should really thank Mr Ong for the carnival.
While PAP was throwing a carnival, Singapore Democratic Party candidate Chee Soon Juan appeared to be literally "running" for election at Bukit Batok Nature Park.
He wrote: "It was like a sauna, hot and humid, but had a good morning workout."
I know Dr Chee really wants to get into Parliament, but I don't think it's worth suffering heatstroke for.
It's so hot that even our durian supply from Malaysia could be affected. A drop of up to 40 per cent is expected.
"The durian season starts in the next three months and half of my trees have failed to produce any flowers because of the weather," a durian farmer in Perak told the Malay Mail.
And as if that's not enough bad news for durian lovers, 76 cases of food poisoning related to durian pastries prepared at Goodwood Park Hotel have been reported since mid-March.
Taking time off from reminding us how hot it is, NEA has suspended the licence of the hotel bakery.
In response, the hotel posted on Facebook:
"Goodwood Park Hotel would like to extend our sincere apologies to all our guests affected by the temporary cessation of sales of durian pastries and other pastries produced by the hotel's pastry kitchen."
Notice the hotel didn't apologise for the food-poisoning cases. It's apologising for the inconvenience caused by the closure of its bakery.
The hotel also had a pop-up stall at Lot One mall last week.
A spokesman said: "Guests who purchased our durian pastries at Lot One from April 18 to 21 are advised to throw them away and contact the hotel for their refunds."
Guess where Lot One mall is.
That's right — Choa Chu Kang.
See how unlucky durian lovers in Choa Chu Kang are?
And to add insult to food poisoning, on Wednesday, our favourite fruit was dissed on American national TV.
US actress Jessica Chastain, who was in Singapore three weeks ago to promote The Huntsman: Winter's War, brought out a durian on US talk show Jimmy Kimmel Live!
Or as she pronounced it, "dorian".
At first, she called it the king of fruit in Asia.
Then she said: "They call it the blue cheese of fruit." That sounded like a demotion to me.
I mean, if someone called me "the king of writers", I would think, yeah, sure, why not?
But if someone called me "the blue cheese of writers", I would have to start counting the number of days since I last showered.
Chastain, a rare Caucasian who claims to love durian, even said: "It kind of smells like a garbage can, right?"
Found my favorite fruit! #durian
A photo posted by Jessica Chastain (@chastainiac) on Apr 1, 2016 at 8:48pm PDT
Calling it "foul" and "vile", talk show host Kimmel said he felt like "throwing up a little bit" after some durian was shoved into his mouth by Chastain.
He then asked his sidekick Guillermo to take the remaining fruit and "throw it into the sea or something like that".
Throw it into the sea? Doesn't he know about the possible upcoming durian shortage caused by the heatwave?
T.S. Eliot was right when he wrote that April is the cruellest month. Who knew he was a durian fan too?
But I think I will read some Oscar Wilde next. Perhaps a novel — The Picture Of Durian Gray.
- Published in The New Paper, 24 April 2016
Tags: elections, food, The New Paper, TV
Stop using S'pore's ranking in World Press Freedom Index to bash media
Reporters Without Borders has just released its 2016 World Press Freedom Index, where Singapore fell one place from last year's lowly 153rd ranking to an even lowlier 154th position out of 180 countries.
It has become a cliche to use this ranking to bash "mainstream media", specifically the news outlets of SPH and Mediacorp, and the journalists who work for them.
I find this terribly ironic.
For one thing, Paris-based Reporters Without Borders is a pro-journalist organisation.
It doesn't distinguish "mainstream media" from so-called "alternative media". It advocates freedom of information for all and seeks protection for journalists and bloggers alike.
What many would take the World Press Freedom Index ranking to mean is that Singapore's press is 154th in the world when actually it's Singapore's press freedom that's ranked 154th. It is an important distinction.
While you may argue that the lack of press freedom leads to a lousy press, let's not confuse the two as the same thing.
In fact, I would argue that what the low ranking shows is not how bad Singapore's press is, but how hard it is here to be a journalist at all.
It's like being a fireman in a volcano. The odds are stacked against you, but you still got to do your job the best you can.
On the other hand, if you want to use the ranking to bash the Government for creating such an environment, that's perfectly fine.
Tags: newspapers
Rui En walked out of an audition with me: 'Do you know who I am?'
Back in the 90s, when I was still sort of a journalist, I was asked to interview a boss of some company for a profile piece.
I was told that the boss would be expecting my call.
So I called the number I was given and someone I assumed was the boss's assistant answered.
I introduced myself and asked to speak to the boss.
The assistant said the boss was busy.
That was when I said: "Do you know who I am?"
I realised almost right away that was a mistake.
What I meant was "I'm told he is expecting my call". Instead I came across as "Do you know how important I am? So don't fuck with me."
I left my number with the assistant, but the boss never called back and I never got the interview.
I can imagine the assistant telling the boss: "This asshole just called and wanted to speak to you. I told him you were busy and he said 'Do you know who I am?'"
Boss: "Really? Who the hell does he think he is?"
Assistant: "You want to call him back?"
Boss: "Fuck him."
So I can believe that when Rui En said "Do you know who I am?" to the guy whose motorbike she knocked over, she didn't mean "Do you know what a big star I am? So don't fuck with me."
On the other hand, I have met Rui En.
It was maybe 10 years ago when I was working at MediaCorp. I was auditioning actresses for the lead in a new Channel 5 TV pilot I was developing.
I remember Belinda Lee, who really put a lot of effort into the audition, but she was too "drama" and not comedic enough for the role. Still, I liked her.
And then there was Rui En.
I have auditioned many actors and actresses before, but Rui En was the first to walk out of an audition.
My guess is that there was some misunderstanding about what she was doing there. I never found out exactly what happened.
I was in the middle of giving her direction in the audition room when she just decided she wasn't doing it anymore and walked out. She never explained why.
But judging by her haughty attitude at the audition, if she were to explain, I suspect what she would say is: "Do you know what a big star I am? Why should I audition for you?"
In other words: "Do you know who I am?"
Tags: TV
I first read about the Pyongyang marathon late last year via the Just Run Lah Faceboook page.
I immediately wanted to join - if I could get my Korean-speaking sister to go with me. I did by offering to pay for the whole trip for both of us.
Which was how I ended up in North Korea on Sunday morning getting ready for my 10km run in the Mangyongdae Prize International Marathon.
It would be my first race overseas.
Here I am on the tour bus pinning my race bib on my ugly T-shirt given by the travel company that brought us to DPRK:
Flag-off was 9.30am inside the May Day Stadium, but first, we have to wait for the opening ceremony.
Here we are posing with our tour group in the stadium tunnel:
The video below shows us going into the stadium onto the field for the opening ceremony. This was the best part of the trip for me:
Another video, taken by my sister:
After the opening ceremony, we had to rush back into the tunnel and change into our running gear for the flag-off.
Below are photos of the 10km route taken with my Autographer camera:
Yes, I did get to high five (or low five) the Korean bystanders along the way, especially the kids.
One major disappointment about the race is that the 10km finish line is just mats. You can barely tell that it's the finish line.
There was no one giving out finisher T-shirts or medals because only the top finishers in the various categories get medals during the closing ceremony.
Although there were drinks station along the way, there was no one giving you water after you finish. That's why I was saving my water from the last drinks station, knowing it was almost 10km. You can see me holding a cup after crossing the mats in the photo below.
After the race, my sister spoke to these children to find out whether they were actors hired to be there for our benefit. She concluded that they weren't.
Don't believe everything you see in The Interview.
I don't know why my marathon certificate says I'm female.
I kind of wished I had signed up for the half marathon so that I could've seen more of Pyongyang and its people on foot, and made the trip more worthwhile.
Because of the cooler-than-Singapore weather, I think I might have been able to make the two-and-a-half-hour cut-off time for the half marathon.
Maybe next year?
NEWS REPORTS:
Strictly no selfies with Kim Jong Un! 1,000 foreigners flock to North Korea for the world's strangest marathon
More Than 1,600 Runners Take Part in Pyongyang Marathon
An American in North Korea: What it's like to run the Pyongyang marathon
Tags: abroad, DPRK, family, run
My sister goes to North Korea for Pyongyang marath...
I took the MRT train in North Korea and it didn't ...
Too hot: Bad news for durian lovers, especially if...
Stop using S'pore's ranking in World Press Freedom...
Rui En walked out of an audition with me: 'Do you ...
I went to North Korea & ran 10k in the Pyongyang m...
I went to North Korea & took lots of selfies (and ...
I went to North Korea & asked for the Kim Jong Un ...
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The Chevron B6 is a lightweight sports racing car, designed, developed and built by British manufacturer Chevron Cars, in 1967. Only 7 cars were built, which makes it very rare. Over its career, spanning 8 years, it won a total of 15 races, plus 4 additional class wins, clinched 1 pole position, and scored 30 total podium finishes.
References
Chevron racing cars
Sports prototypes
24 Hours of Le Mans race cars
Group 4 (racing) cars
Sports racing cars
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11/21/05 November 18, 2005- Markey Outraged at Missing Radioactive Material
11/17/05 March 15, 2005- Opening Statement at Subcommittee Hearing: "Protecting Consumer's Data: Policy Issues Raised by Choicepoint
11/17/05 November 17. 2005- Statement of Rep. Markey on the Labor-HHS-Education Appropriations Bill
11/17/05 June 17, 2005- Rep. Markey: "Mastercard Security Breach: Priceless"
11/17/05 February 18, 2005- Social Security: Is the System Really in Crisis?
11/17/05 January 3, 2005- Rep. Markey Statement on the Death of Congressman Bob Matsui
11/17/05 May 26, 2005- Statement at Subcommittee Hearing on Majority Staff DTV Draft Bill
11/17/05 July 1, 2005- Rep. Markey, 52 Members of Congress Call for Corporation for Public Broadcasting Chair to Resign
11/17/05 February 17, 2005- Statement of Rep. Markey on S.5, Class Action Legislation
11/17/05 March 2, 2005- Statement of Rep. Markey at Hearing on Major Telecommunications Merger
11/17/05 June 28, 2005- Rep. Markey: IRS ChoicePoint Contract is 'No Choice' for Taxpayers
11/17/05 October 26, 2005- Energy and Commerce Committee Opening Statement on Reconciliation
11/17/05 June 30, 2005- House Passes Markey Privacy Amendment to Transportation, Treasury Appropriations Bill
11/17/05 November 17, 2005- Statement of Rep. Markey on the Labor-HHS-Education Appropriations Bill
11/17/05 November 3, 2005- Statement on the Resignation of Corporation for Public Broadcasting Chair
11/17/05 November 9, 2005- Statement of Rep. Markey at Subcommitee Hearing on Barton-Upton Staff Draft
11/17/05 June 17, 2005- Markey Vows to Take the Fight for Public Broadcasting to the House Floor
11/17/05 February 17, 2005- Statement of Rep. Markey at Oversight Hearing on Digital TV Transition
11/17/05 February 18, 2005- MEDIA ADVISORY: Social Security: Is the System Really in Crisis?
11/17/05 March 10, 2005- Statement of Rep. Markey at Subcommittee Hearing on Digital TV Transition
|
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Closed at the moment
Mo09:00 - 17:30
Tu09:00 - 17:30
We09:00 - 20:00
Th09:00 - 17:30
Fr09:00 - 17:30
Sa09:00 - 17:30
Su10:30 - 16:30
Follow Otter Garden Centre - Plymouth on
Otter Garden Centre - Plymouth
Chittleburn Hill
PL8 2BH Plymouth
Reviews Otter Garden Centre - Plymouth
Assortment:
Dave Todd 02-10-2019
7 visitors found this review useful
Went to the restaurant with a 2 breakfasts 7 items coupon and asked for no egg on one breakfast please was told we can't leave the egg off of a breakfast as it's a special offer. So after trying to ask sensibly the manageress told me exactly the same we can't take an egg off of your breakfast. So forget the special offer I said I will buy 2 breakfasts with no egg on one breakfast please. Once again we can't do that the breakfast comes with an egg they said. I explained I am allergic to eggs so if u insist I have an egg put it on a small side plate my wife can have it with her breakfast. Once again we can't do that. Right one breakfast for my wife a sausage sandwich for me. Waitress brings food to table no sausage sandwich but 2 breakfasts complete with egg. It's a joke out there you couldn't write this as a sketch Extremely poor customer service and ridiculous give them a wide berth if u don't like eggs lol
Report this review as inappropriate
Shirley spiers 01-04-2019
I love this nursery. It's my place to go when I'm fed up and need a boost. Everything from the plants to the sundries and even the fabulous laddy and Flo the cats make it a happy place. . Rose and her crew outside are friendly and knowledgeable and always say hello which makes me feel welcome.great place.
DEREK BRYANT 23-03-2019
1 visitor found this review helpful
Great product and sales advice. Pity it all fell down with the delivery. Told it would be between 11am and 2pm. Eventually arrived at 4.50 pm. No call to say they were going to be late. I phoned to see what was the problem at 2.05 pm to be told traffic was busy and they would be with us in 1 1/2 hrs. Chased again at 4pm to be told they were just loading our shed.
Michael Sinnott 13-10-2017
I bought a patio dustpan and brush. After deciding it was unsuitable for someone getting on in years, I took it back, but in the meantime, I had lost/mislaid the receipt. I was prepared to go to the bank and get a statement printed, the woman with short gingery hair was adamant. I need to see the receipt. I won't be returning either, like a customer that bought ankle boots. It is early October, and Christmas decorations are out, even a plastic transparent snowman lit up to give a snowman effect, has a movable head, that moves from side-to side. Definitely not worth a visit. Very disapointing!!
linda gardiner 29-12-2015
bought some ankle boots have been wearing them on an off for a couple of months, and the toes have scuffed really badly, when i rang otters in plymouth they told me i cannot change them as i have lost the receipt,,,, really bad costomer sevice will not buy from them again
What do you think about Otter Garden Centre - Plymouth?
More about Otter Garden Centre - Plymouth
Otter Garden Centre - Plymouth is a garden centre you'll find at Plymouth, Devon . You'll find an excellent range of plants, seeds and bulbs within the garden centre, as well as many other products for your garden, such as garden furniture, barbecues and gardening tools. For more information on the range please refer to the website. Otter Garden Centre - Plymouth is one of the many garden centres in Devon.
Have questions about this garden centre in Plymouth? Check the opening times above and you can be sure that when you visit, you won't be left standing in front of a set of closed doors. Of course, you can also check out the website for more information. Perhaps you like a good old fashioned chat and would prefer to phone? Then dial the number for Otter Garden Centre - Plymouth: 01752 405422 and you'll be greeted by a friendly member of staff.
Nowadays, many people look for testimonials and reviews on products and stores before visiting or buying. A positive review is, of course, a great sign that you're going to have a brilliant experience at a garden centre. On the Garden Centre Guide, thousands of keen gardeners and loyal customers have written reviews about garden centres they've visited! Otter Garden Centre - Plymouth currently has an average rating of 2.6. You can read the reviews for this garden centre above, and if you're feeling opinionated, write one yourself. Have your say so that others can benefit from what your review!
It's also worth noting that when you write a review on the garden centre guide, you are automatically entered into our prize draw for a chance to win £25 in National Gardening Gift Vouchers! Just scroll back up to get started now.
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Q: Flutter Firebase: How to implement and structure your data replies comment just like other Social medial i am building an app that deals with forums and community.. i am stuck on creating a replies and structuring that data in firebase.
while posting or commenting, i give each every user a list of reply,
the code:
communityPageComment(String comment, Map comId) async {
final userdata =
await firestore.collection('user').doc(auth.currentUser!.uid).get();
final userinfo = userdata.data();
firestore
.collection('comment')
.doc(comId['id'])
.collection('ForumComment')
.doc()
.set({
'name': userinfo!['name'],
'image': userinfo['picture'],
'timestamp': FieldValue.serverTimestamp(),
'text': comment,
'reply': []// here will be where all the replies of this post will be in Map...
});
}
which is working fine..
let assume Mr. UserA post... After Mr. userA posting and another user, Mr b reply to mr A...
which will be append to the List of map which the code is this.
replyToComment(String comment, Map comId) async {
final userdata =
await firestore.collection('user').doc(auth.currentUser!.uid).get();
final userinfo = userdata.data();
Map data = {
'name': userinfo!['name'],
'image': userinfo['picture'],
'text': comment,
'reply':[]
};
firestore
.collection('comment')
.doc(comId['id'])
.collection('ForumComment')
.doc()
.update({'reply': FieldValue.arrayUnion([data])});
}
this is fine...
where my problem start
if another user want to reply Mr B which is Mr C.
how can i target base on the index to Mr B comment programmingly using the replyToComment Function i created about..
Thanks
|
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{"url":"https:\/\/zbmath.org\/?q=an%3A1057.35028","text":"zbMATH \u2014 the first resource for mathematics\n\n$$L^q$$-theory of a singular \u201dwinding\u201d integral operator arising from fluid dynamics. (English) Zbl\u00a01057.35028\nThe authors study the solvability of a system of partial differential equations of second order involving an angular derivative which is not subordinate to the Laplacian. This system arises from the linearization of the Navier-Stokes equations of a three-dimensional rigid body rotating in a viscous incompressible fluid. The solvability in $$L^q({\\mathbb R}^n)$$ is obtained by means of studying the properties of the singular operator arising from the fundamental solution.\n\nMSC:\n 35Q30 Navier-Stokes equations 76U05 General theory of rotating fluids 35B45 A priori estimates in context of PDEs 47G10 Integral operators 76D05 Navier-Stokes equations for incompressible viscous fluids 42B25 Maximal functions, Littlewood-Paley theory\nFull Text:","date":"2021-06-13 13:57:52","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6092369556427002, \"perplexity\": 622.9736909990625}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-25\/segments\/1623487608856.6\/warc\/CC-MAIN-20210613131257-20210613161257-00283.warc.gz\"}"}
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Mortal – Mut ist unsterblich (Originaltitel Torden) ist ein Film von André Øvredal, der im Februar 2020 in die norwegischen Kinos kam. In dem Fantasyfilm, der von der nordischen Mythologie inspiriert ist, entdeckt ein junger Mann namens Eric, gespielt von Nat Wolff, dass er gottähnliche Kräfte hat, die von seinen Vorfahren stammen.
Handlung
Ein junger Mann namens Eric entdeckt, dass er gottähnliche Kräfte besitzt, wie sie aus der alten norwegischen Mythologie bekannt sind. Als er sich in der Wildnis Westnorwegens versteckt, tötet er versehentlich und auf unerklärliche Weise einen Teenager und wird anschließend festgenommen. Bevor er verhört wird, trifft er Christine, eine junge Psychologin, die versucht herauszufinden, was wirklich passiert ist. Sie glaubt Eric und empfindet Sympathie für den jungen Mann. Als die US-amerikanische Botschaft Erics Auslieferung fordert, verhilft ihm Christine zur Flucht. Auch die norwegischen Behörden suchen nun nach ihm. Eric entdeckt schließlich, was er wirklich ist.
Produktion
Stab und Besetzung
Regie führte André Øvredal, der sich bereits bei seinem letzten Film mit der nordischen Mythologie beschäftigte. Nach seinem Film Trollhunter sprach Øvredal mit einem Produzenten darüber, wie man für diesen eine Art Nachfolgefilm realisieren könnte. Von Anfang 2012 bis 2014 hatte er selbst an dem Drehbuch, der Idee und dem Konzept gearbeitet und schließlich Geoff Bussetil ins Boot geholt. Ebenfalls am Drehbuch beteiligt war Norman Lesperance.
Nat Wolff spielt in der Hauptrolle Eric. Iben Akerlie übernahm die Rolle der Psychologin Christine, Priyanka Bose spielt Hathaway, und Arthur Hakalahti ist in der Rolle von Ole zu sehen. Per Frisch spielt den Polizeichef Henrik.
Dreharbeiten
Im Sommer 2019 entstanden drei bis vier Wochen lang Aufnahmen in den Fjorden an der Westküste Norwegens. Hiernach erfolgten Innenaufnahmen und Aufnahmen der Waldlandschaften in und um Oslo. Schließlich drehte man etwa eine Woche lang in einem Studio in Tschechien die Szenen in Autos, die Unterwasserszene und dem Hubschrauberabsturz. Hier hatten Szenenbildner ein Hubschraubermodell gebaut. Zu der Hängebrücke über den Hardangerfjord, die in einer zentralen Szene des Films zu sehen ist und sich zwischen Bergen und Oslo befindet, sagte Øvredal, diese sei länger als die Golden Gate Bridge. Sie wurde für die Dreharbeiten gesperrt. Als Kameramann fungierte Roman Osin, mit dem Øvredal auch für seinen letzten Film Scary Stories to Tell in the Dark zusammenarbeitete.
Veröffentlichung
Der Film kam im Februar 2020 in die norwegischen Kinos.
Rezeption
Altersfreigabe
In den USA erhielt der Film von der MPAA ein R-Rating, was einer Freigabe ab 17 Jahren entspricht.
Kritiken
Der Film erhielt bislang gemischte Kritiken. Von 36 Kritikern gaben 56 % ein positives Urteil. Die durchschnittliche Bewertung liegt bei 5,7/10.
Auszeichnungen
Amanda Awards 2020
Nominierung für die Besten visuellen Effekte (Stephen Coren und Arne Kaupang)
Nominierung für die Beste Filmmusik (Marcus Paus)
Weblinks
Mortal / Torden – Informationen zum Film vom Trust Nordisk
Mortal – Trailer von Ascot Elite Entertainment bei YouTube (Video)
Einzelnachweise
Filmtitel 2020
Norwegischer Film
US-amerikanischer Film
Britischer Film
Filmdrama
Thriller
Fantasyfilm
Rezeption der germanischen Mythologie
|
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{"url":"https:\/\/searsmerritt.com\/blog\/5905064635924480","text":"# Collaborative Filtering With Uv Decomposition\n\nCollaborative filtering is a technique for building recommender systems that relies only on a user's past behavior to make recommendations about what he or she should next in the future. There are two primary classes of solutions to the problem: nearest neighbor methods and latent factor models. Research has shown that latent factor models outperform nearest neighbor methods, so in this post, I'll work through implementing a latent factor model known as UV-factorization. UV-factorization is a type of matrix factorization, where a single large matrix is approximated by the product of two lower rank matrices.\n\n#### The problem\n\nMathematically, the problem begins with a large matrix, known as a utility matrix, $\\boldsymbol{M}$, that contains behavioral data about individual users and the products or services they interact with. Each row in the matrix represents a user and each column represents a user's past interactions with specific products or services. Often times user interactions are characterized as ratings or purchases.\n\nThe solution to the problem is to determine which, out of all products each user has not interacted with, is the most likely to be preferred the next time the user chooses to perform an action, such as watching a movie or buying another pair of shoes. This is done by decomposing the utility matrix into two lower rank matrices, that when multiplied together, fill in the missing (user, item) pairs. $\\boldsymbol{UV} = \\boldsymbol{M}$\n\nTo estimate $\\boldsymbol{U}$ and $\\boldsymbol{V}$, we can define our objective function in terms of minimizing the squared error between each row of M and the inner product of the corresponding rows of $\\boldsymbol{U}$ and $\\boldsymbol{V}$. To prevent overfitting, we can also apply a regularization term. Mathematically, our objective function is defined as, $\\sum_{(i,j)} (\\boldsymbol{M}_{ij} - \\boldsymbol{V}_j^T \\boldsymbol{U}_i)^2 + \\lambda( \\|\\boldsymbol{V}_j\\|^2 + \\|\\boldsymbol{U}_i\\|^2).$\n\nTo solve this objective, we can use stochastic gradient descent, originally presented here. After taking derivatives of the objective with respect to $\\boldsymbol{U}$ and $\\boldsymbol{V}$ we are left with the following pair of update rules: $\\boldsymbol{U}_i \\leftarrow \\boldsymbol{U}_i + \\alpha((\\boldsymbol{M}_{ij} - \\boldsymbol{V}_j^T \\boldsymbol{U}_i)\\boldsymbol{V}_j - \\lambda \\boldsymbol{U}_i) \\\\ \\boldsymbol{V}_j \\leftarrow \\boldsymbol{V}_j + \\alpha((\\boldsymbol{M}_{ij} - \\boldsymbol{U}_i^T \\boldsymbol{V}_j)\\boldsymbol{U}_i - \\lambda \\boldsymbol{V}_j),$ where $\\alpha$ is the learning rate parameter and $\\lambda$ is the regularization parameter. After estimating $\\boldsymbol{U}$ and $\\boldsymbol{V}$ by iterating over the known ($i,j$) pairs in the data, user $i$'s recommendation for product $j$ can be estimated by computing $\\boldsymbol{U_i V_j^T}$.\n\n#### Implementation\n\nAs we've seen in other posts, implementing stochastic gradient descent is more or less trivial, once we have derived the update rules. The code below takes as input, a sparse scipy matrix and iterates over the data. The free parameters, f, lr and reg define the width of $\\boldsymbol{U}$ and $\\boldsymbol{V}$, the learning rate, $\\alpha$, and the regularization parameter, $\\lambda$, respectively.\n\ndef sgd_uv(util_mtx, f=5, lr=0.001, reg=0.1, max_iter=1000):\n#get dimensions of util_mtx, which is a compressed sparse row matrix\nr,c = util_mtx.shape\n#initialize item matrix\nv = np.random.rand(c,f)\n#initialize user matrix\nu = np.random.rand(r,f)\n#fit the matrices with a fixed number of iterations\nfor c in xrange(max_iter):\nfor i in xrange(r):\nfor j in util_mtx[i].indices:\nerr = util_mtx[i,j] - np.dot(v[j], u[i])\nv[j] = v[j] + lr*(err*p[i]-reg*v[j])\nu[i] = u[i] + lr*(err*v[j]-reg*u[i])\nreturn u,v\n\n\n#### Testing for convergence\n\nTo measure the algorithm's convergence, we can compute the root mean squared error (RMSE) after each iteration of SGD, shown below.\n\ndef rmse(util_mtx, u, v):\ne = 0.\nm = 0.\nr,c = util_mtx.shape\nfor i in xrange(r):\nfor j in util_mtx[i].indices:\ne += (util_mtx[i,j]-np.dot(v[j], u[i]))**2\nm+=1\nreturn np.sqrt(e\/m)\n\n\nIn the code below, we create a sparse matrix, initialize it with random values, and then pass it to our learning algorithm.\n\n#make a matrix\nA = scipy.sparse.lil_matrix((100, 100))\n#fill some of with random numbers\nA[0, :10] = np.random.rand(10)\nA[1, 10:20] = A[0, :10]\nA.setdiag(np.random.rand(100))\n#convert to compressed sparse row for quick row iterations\nA = A.tocsr()\n#fit the model\nu,v,err_arr = sgd_uv(A)\n#plot the error as a function of learning algorithm iteration\nplt.plot(err_arr, linewidth=2)\nplt.xlabel('Iteration, i', fontsize=20)\nplt.ylabel('RMSE', fontsize=20)\nplt.title('Components = '+str(j), fontsize=20)\nplt.show()\n\n\nThe figure below plots the RMSE as a function of SGD iteration. The exponential decay early in the iteration history and leveling off towards the end suggests the algorithm is converging on a solution, which may be a local one.\n\nNote that to measure the performance of the algorithm, a fraction of the ($i,j$) pairs should be held out of the training process and be used as a test set, which I haven't done here.\n\n#### Conclusion\n\nIn this post I've focused on a relatively simple objective function, to simplify the presentation. In reality, a more complex objective might be required to achieve a desired level of performance. These objectives might include some implicit features, like age and gender, to work around the cold start problem.\n\nArchives","date":"2022-05-25 07:36:43","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 2, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.704158365726471, \"perplexity\": 811.9501128929242}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-21\/segments\/1652662580803.75\/warc\/CC-MAIN-20220525054507-20220525084507-00134.warc.gz\"}"}
| null | null |
This is officially the second single off Retta's yet to be titled album. The video for Tomorrow has been shot in the States " Washington DC" by Clarence Peters and would be released on June 10th, 2011. This single was produced by: Oscar Heman-Ackah. Listen below and enjoy!
I have listened to her 'Tomorrow' song.Retta has got an amazing voice,and the beats were right on track.Love this song.
|
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| 1,472
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It Work with the text files and images to convert them into PDF file format while preserving original formatting of the documents. Get more information here .
Affordable Home Services specializes in commercial and residential exterior and interior home renovation and construction services for NJ / New Jersey area.
WebCyberDesign.com, which is one of the leading India SEO companies, always provides great solutions for search engine optimization, web design, web development, etc. We will bring your website to the very top!
|
{
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| 7,672
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Article Author:
Timothy Schaefer
Article Editor:
Robert Wolford
PubMed Link:
Dengue is a mosquito-transmitted virus and the leading cause of arthropod-borne viral disease in the world. It is also known as breakbone fever due to the severity of muscle spasms and joint pain, dandy fever, or seven-day fever because of the usual duration of symptoms. Although most cases are asymptomatic, severe illness and death may occur. Aedes mosquitoes transmit the virus and are common in tropical and subtropical parts of the world. The incidence of dengue has increased dramatically over the past few decades. The infection is now endemic in some parts of the world. A few people who were previously infected with one subspecies of the dengue virus develop severe capillary permeability and bleeding after being infected with another subspecies of the virus. This illness is known as dengue hemorrhagic fever.[1][2][3]
Dengue fever is caused by any of four distinct serotypes (DENV 1-4) of single-stranded RNA viruses of the genus Flavivirus. Infection by one serotype results in lifelong immunity to that serotype but not to others.[4][5][6]
It is the fastest spreading mosquito-borne viral disease in the world, affecting greater than 100 million humans annually. Dengue also causes 20 to 25,000 deaths, primarily in children, and is found in more than 100 countries. Epidemics occur annually in the Americas, Asia, Africa, and Australia. Two transmission cycles maintain the dengue virus: 1) mosquitos carry the virus from a non-human primate to a non-human primate and 2) mosquitos carry the virus from human to human. The human-mosquito cycle occurs primarily in urban environments. Whether the virus transmits from human to mosquito is dependent upon the viral load of the mosquito's blood meal.
The primary vectors of the disease are female mosquitoes of the species Aedes aegypti and Aedes albopictus. Although A. aegypti is associated with most infections, A. albopictus' range is expanding and may be associated with increasing numbers. These species of mosquitoes tend to live indoors and are active during the day. Transmission perinatally, via blood transfusions, breast milk, and by organ transplantation have been reported.
After 2010, the mean age of patients was 34 years as compared to 27.2 years from 1990 to 2010. The dengue viral serotype causing disease outbreaks has varied with time, as has the occurrence of severe dengue fever.[7][8]
Part of the Flavivirus family, the dengue virus is a 50 nm virion with three structural and seven nonstructural proteins, a lipid envelope, and a 10.7 kb capped positive sense single strand of ribonucleic acid. Infections are asymptomatic in up to 75% of infected humans. A spectrum of disease, from self-limiting dengue fever to hemorrhage and shock, may be seen. A fraction of infections (0.5% - 5%) progress to severe dengue. Without proper treatment, fatality rates may exceed 20%. These occur primarily in children. The typical incubation period for the disease is 4 to 7 days, but it can last from 3 to 10 days. Symptoms more than two weeks after exposure are unlikely to be due to dengue fever.
The exact course of events following the dermal injection of dengue virus by a mosquito bite is unclear. Skin macrophages and dendritic cells appear to be the first targets. It is thought that the infected cells then move to the lymph nodes and spread through the lymphatic system to other organs. Viremia may be present for 24 to 48 hours before the onset of symptoms. A complex interaction of host and viral factors then occurs and determines whether the infection will be asymptomatic, typical, or severe. Severe dengue fever with increased microvascular permeability and shock syndrome is thought to be associated with infection due to a second dengue virus serotype and the patient's immune response. However, cases of severe dengue do occur in the setting of infection by only a single serotype. Worsening microvascular permeability often transpires even as viral titers fall.
History and Physical
The three phases of dengue include febrile, critical, and recovery.
During the febrile phase, a sudden high-grade fever of approximately 40 C occurs that usually lasts two to seven days. Associated symptoms include facial flushing, skin erythema, myalgias, arthralgias, headache, sore throat, conjunctival injection, anorexia, nausea, and vomiting. For skin erythema, a general blanchable macular rash occurs in the first one to two days of fever and the last day of fever. Or, within 24 hours, a secondary maculopapular rash can develop.
Defervescence characterizes the critical phase with a temperature of approximately 37.5 C to 38 C or less on days three through seven. It is associated with increased capillary permeability. This phase usually lasts one to two days. It can progress to shock, organ dysfunction, disseminated intravascular coagulation, or hemorrhage.
The recovery phase entails the gradual reabsorption of extravascular fluid in two to three days. The patient will display bradycardia at this time.
Common laboratory findings include thrombocytopenia, leukopenia, elevated aspartate aminotransferase. The disease is classified as dengue or severe dengue. [9][10][11]
Criteria for Dengue Include:
Probable dengue: The patient lives in or has traveled to a dengue-endemic area. Symptoms include fever and two of the following: nausea, vomiting, rash, myalgias, arthralgias, rash, positive tourniquet test, or leukopenia.
Warning Signs of Dengue: Abdominal pain, persistent vomiting, clinical fluid accumulation such as ascites or pleural effusion, mucosal bleeding, lethargy, liver enlargement greater than 2 cm, increase in hematocrit, and thrombocytopenia.
Severe Dengue: Dengue fever with severe plasma leakage, hemorrhage, organ dysfunction including transaminitis greater than 1000 international units per liter, impaired consciousness, myocardial dysfunction, and pulmonary dysfunction.
Dengue shock syndrome clinical warnings: Symptoms include rapidly rising hematocrit, intense abdominal pain, persistent vomiting, and narrowed or absent blood pressure.
The virus antigen can be detected by ELISA, polymerase chain reaction, or isolation of the virus from body fluids. Serology will reveal a marked increase in immunoglobulins.
It is vital to assess pregnant patients with dengue as the symptoms may be very similar to preeclampsia.
Treatment / Management
Supportive management includes giving the patient fluids, acetaminophen for fever, and a blood transfusion for hemorrhage. Confirmed diagnosis is established by culture, antigen detection, polymerase chain reaction, or serologic testing. Avoid giving aspirin and nonsteroidal anti-inflammatory drugs and other anticoagulants. No antiviral medications are recommended.
Patients with thrombocytopenia or bleeding may require platelets and fresh frozen plasma.
No laboratory tests can predict the progression to severe disease.
The clinical diagnosis of dengue can be challenging as many other illnesses can present similarly early in the disease course. Other considerations should include malaria, influenza, Zika, chikungunya, measles, and yellow fever. Obtain a detailed history of immunizations, travel, and exposures.
Rapid laboratory identification of dengue fever includes NS1 antigen detection and serologic tests. Serologic tests are only useful after several days of infection and may be associated with false positives due to other flavivirus infections, such as yellow fever or Zika virus.
Untreated severe dengue fever may have a mortality rate of 10% to 20%. Appropriate supportive care reduces the mortality rate to roughly 1%.
Liver injury
Oophoritis
Postoperative and Rehabilitation Care
Patients should be encouraged to consume ample liquids. The return of a patient's appetite is a sign that the infection is subsiding.
Consulting an infectious disease specialist is recommended because most clinicians have little experience managing this infection. The Centers for Disease Control and Prevention has a hotline which also offers advice on treatment.
Deterrence and Patient Education
The only way to avoid contracting dengue is to prevent mosquito bites and not travel to its endemic areas.
Other preventative measures include the use of DEET insecticide, wearing protective clothing, sleeping under a mosquito net, and eliminating stagnant water around the home.
Enhancing Healthcare Team Outcomes
The diagnosis and management of dengue is complex and this best managed by a multidisciplinary team that includes an infectious disease expert, CDC consultant, emergency department physician and an internist. The care is supportive with fluid, acetaminophen for fever, and a blood transfusion for hemorrhage. Confirmed diagnosis is established by culture, antigen detection, polymerase chain reaction, or serologic testing. No laboratory tests can predict the progression to severe disease.
The role of the primary care provider and nurse practitioner is to educate the traveler on the prevention of mosquito bites. This means covering exposed skin, and use bed nets, particularly during daytime siestas, using mosquito repellents and indoor insecticides. One should also eradicate mosquito breeding grounds like standing water. The prognosis for untreated dengue is abysmal but with supportive care, most patients can survive, albeit with residual multisystem organ damage.[12][13]
[1] Baak-Baak CM,Cigarroa-Toledo N,Pech-May A,Cruz-Escalona GA,Cetina-Trejo RC,Tzuc-Dzul JC,Talavera-Aguilar LG,Flores-Ruiz S,Machain-Williams C,Torres-Chable OM,Blitvich BJ,Mendez-Galvan J,Garcia-Rejon JE, Entomological and virological surveillance for dengue virus in churches in Merida, Mexico. Revista do Instituto de Medicina Tropical de Sao Paulo. 2019 Feb 14; [PubMed PMID: 30785563]
[2] Sharma M,Glasner DR,Watkins H,Puerta-Guardo H,Kassa Y,Egan MA,Dean H,Harris E, Magnitude and Functionality of the NS1-Specific Antibody Response Elicited by a Live-Attenuated Tetravalent Dengue Vaccine Candidate. The Journal of infectious diseases. 2019 Feb 19; [PubMed PMID: 30783676]
[3] Oliveira LNDS,Itria A,Lima EC, Cost of illness and program of dengue: A systematic review. PloS one. 2019; [PubMed PMID: 30785894]
[4] Seixas G,Salgueiro P,Bronzato-Badial A,Gonçalves Y,Reyes-Lugo M,Gordicho V,Ribolla P,Viveiros B,Silva AC,Pinto J,Sousa CA, Origin and expansion of the mosquito Aedes aegypti in Madeira Island (Portugal). Scientific reports. 2019 Feb 19; [PubMed PMID: 30783149]
[5] Ghani NA,Shohaimi S,Hee AK,Chee HY,Emmanuel O,Alaba Ajibola LS, Comparison of Knowledge, Attitude, and Practice among Communities Living in Hotspot and Non-Hotspot Areas of Dengue in Selangor, Malaysia. Tropical medicine and infectious disease. 2019 Feb 15; [PubMed PMID: 30781369]
[6] Maia LMS,Bezerra MCF,Costa MCS,Souza EM,Oliveira MEB,Ribeiro ALM,Miyazaki RD,Slhessarenko RD, Natural vertical infection by dengue virus serotype 4, Zika virus and Mayaro virus in Aedes (Stegomyia) aegypti and Aedes (Stegomyia) albopictus. Medical and veterinary entomology. 2019 Feb 18; [PubMed PMID: 30776139]
[7] Prompetchara E,Ketloy C,Thomas SJ,Ruxrungtham K, Dengue vaccine: Global development update. Asian Pacific journal of allergy and immunology. 2019 Jan 13; [PubMed PMID: 30660171]
[8] Vasanthapuram R,Shahul Hameed SK,Desai A,Mani RS,Reddy V,Velayudhan A,Yadav R,Jain A,Saikia L,Borthakur AK,Mohan DG,Bandyopadhyay B,Bhattacharya N,Dhariwal AC,Sen PK,Venkatesh S,Prasad J,Laserson K,Srikantiah P, Dengue virus is an under-recognised causative agent of acute encephalitis syndrome (AES): Results from a four year AES surveillance study of Japanese encephalitis in selected states of India. International journal of infectious diseases : IJID : official publication of the International Society for Infectious Diseases. 2019 Jan 11; [PubMed PMID: 30641206]
[9] Thang NT,Probandari A,Ahmad RA, Barriers to Engaging Communities in a Dengue Vector Control Program: An Implementation Research in an Urban Area in Hanoi City, Vietnam. The American journal of tropical medicine and hygiene. 2019 Jan 14; [PubMed PMID: 30652660]
[10] Wharton-Smith A,Green J,Loh EC,Gorrie A,Omar SFS,Bacchus L,Lum LCS, Using clinical practice guidelines to manage dengue: a qualitative study in a Malaysian hospital. BMC infectious diseases. 2019 Jan 11; [PubMed PMID: 30634929]
[11] Kellstein D,Fernandes L, Symptomatic treatment of dengue: Should the NSAID contraindication be reconsidered? Postgraduate medicine. 2018 Dec 21; [PubMed PMID: 30575425]
[12] Nujum ZT,Saritha N,Prathibha Raj MR,Gayathri AV,Nirmala C,Vijayakumar K,Varghese S, Seroprevalence of dengue infection in pregnant women and placental antibody transfer. Medical journal, Armed Forces India. 2019 Jan; [PubMed PMID: 30705485]
[13] Gordon A,Gresh L,Ojeda S,Katzelnick LC,Sanchez N,Mercado JC,Chowell G,Lopez B,Elizondo D,Coloma J,Burger-Calderon R,Kuan G,Balmaseda A,Harris E, Prior dengue virus infection and risk of Zika: A pediatric cohort in Nicaragua. PLoS medicine. 2019 Jan; [PubMed PMID: 30668565]
Take 7 Question Quiz on Dengue Fever
© 2019 - StatPearls.com
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{"url":"https:\/\/golem.ph.utexas.edu\/category\/2021\/09\/axioms_for_the_category_of_hil_1.html","text":"## September 23, 2021\n\n### Axioms for the Category of Hilbert Spaces (bis)\n\n#### Posted by Tom Leinster\n\nGuest post by Chris Heunen\n\nDusk. The alley is secreted in mist. The miserly reflection of the single lamp on the cobblestones makes you nervous. The stranger approaches, hands inside trenchcoat. Why on Earth did you agree to meet here? The transfer happens, the stranger walks away without a word. The package, it\u2019s all about the package. You have it now. You\u2019d been promised it was the category of Hilbert spaces. But how can you be sure? You can\u2019t just ask it. It didn\u2019t come with a certificate of authenticity. All you can check is how morphisms compose. You leg it home and verify the Axioms for the category of Hilbert spaces!\n\n\u2022 Axiom 1: the category has to be equipped with a dagger.\n\n\u2022 Axiom 2: the category has to be equipped with a dagger symmetric monoidal structure, and the tensor unit $I$ has to be simple and a monoidal separator. This means that $I$ has exactly two subobjects, and that $f,g \\colon H \\otimes K \\to L$ are the same as soon as $f \\circ (h \\otimes k)$ and $g \\circ (h \\otimes k)$ are the same for all $h \\colon I \\to H$ and $k \\colon I \\to K$.\n\n\u2022 Axiom 3: the category has to have finite dagger biproducts.\n\n\u2022 Axiom 4: the category has to have finite dagger equalisers.\n\n\u2022 Axiom 5: any dagger monomorphism - that is, any morphism $f$ satisfying $f^\\dagger \\circ f = \\mathrm{id}$ - has to be kernel of some morphism.\n\n\u2022 Axiom 6: the subcategory of dagger monomorphisms has to have directed colimits.\n\nAll checks pass. It\u2019s the genuine article: the theorem guarantees that you have in your hands the category of all Hilbert spaces and continuous linear maps between them! Well, the fine print says you have one of two versions of it. If any morphism $z \\colon I \\to I$ equals $z^\\dagger$, then you have in your possession the category of real Hilbert spaces and continuous linear maps, otherwise you have the category of complex Hilbert spaces and continuous linear maps. Fair play, your prize may only be equivalent to one of those fabled categories, but that\u2019s good enough for you, because the equivalence preserves all the (co)limits, dagger, and monoidal structure.\n\nYou can\u2019t help but feel somewhat amazed. The axioms were purely categorical. They never mentioned anything like probabilities, real or complex numbers, convexity, continuity, or dimensions. How can this be? The article itself is short and sweet, so you remind yourself to read it properly, but for now you content yourself with this sketch of the proof.\n\nFirst, you remember the scalars $z \\colon I \\to I$ in any monoidal category form a commutative monoid under composition by the Eckmann-Hilton argument. Axiom 1 gives it an involution, and Axiom 3 an addition, making it a commutative semiring. Axioms 2 and 4 conspire to give multiplicative inverses. You knew from semiring theory that the scalars must now either form a field or be zerosumfree, meaning that $w+z=0$ implies $w=z=0$; but the latter contradicts Axiom 5. So the scalars form an involutive field.\n\nNext, you try to remember what you know about projections, those endomorphisms $p$ satisfying $p^\\dagger \\circ p = p$. They are ordered by $p \\leq q$ if and only if $q \\circ p = p$. But you prefer to work with dagger monomorphisms, which are order isomorphic to projections by playing around with mostly Axiom 5. Now, those carry an orthocomplement given by $f^\\perp = \\mathrm{ker}(f^\\dagger)$, and Axioms 3, 4, and 6 makes it a complete lattice. So projections must be a complete lattice too, and clearly $p^\\perp = \\mathrm{id}-p$ make those into a complete ortholattice.\n\nThen you realise that $\\mathrm{hom}(I,H)$ is a vector space, and the projection lattice of $H$ is isomorphic to the closed subspaces of $\\mathrm{hom}(I,H)$. Here, a subspace $V \\subseteq \\mathrm{hom}(I,H)$ is closed when $V^{\\perp\\perp}=V$, where the orthocomplement of a subspace is taken with respect to the sesquilinear form $\\langle f \\mid g \\rangle = g^\\dagger \\circ f$, just like you are used to in Hilbert space. You even quickly prove that $\\mathrm{hom}(I,H)$ is orthomodular in the sense that it is a direct sum $V \\oplus V^\\perp$ for any closed subspace $V$.\n\nCombining Axioms 3 and 6 make you think about looking at the object $I^A$ which consists of $A$ many copies of the tensor unit $I$ for any set $A$. You think about this as a sort of standard object like the standard Hilbert space $\\ell^2(A)$ of dimension $|A|$. And indeed, you find an orthonormal basis of cardinality $|A|$ for $\\mathrm{hom}(I,I^A)$ with some fiddling. Now Sol\u00e8r\u2019s theorem tells you the scalars must be $\\mathbb{R}$ or $\\mathbb{C}$, and it quickly follows that $\\mathrm{hom}(I,H)$ must be a Hilbert space for any object $H$. Now you\u2019re on a roll. It\u2019s all coming back to you: $\\mathrm{hom}(I,-)$ is a functor which you already know is essentially surjective, and Axioms 2 and 3 make sure it\u2019s full and faithful. Some more symbol pushing shows that the equivalence is monoidal and preserves the dagger. Done!\n\nYou stop for a moment to appreciate all the work that led to this. All these structures - scalars, biproducts, projections, orthomodular lattices, orthomodular spaces, etc - were studied by a long line of people, including Von Neumann, Mackey, Jauch, Piron, Keller, Sol\u00e8r, Rump, B\u00e9nabou, Mac Lane, Mitchell, Freyd, Abramsky, Coecke, and many more. You have to be grateful that it all comes together so nicely!\n\nNow a philosophical mood takes you. This enterprise reminds you of Lawvere\u2019s Elementary Theory of the Category of Sets. That also characterises a category, that of sets and functions, and the axioms are of a very similar nature. That category is clearly very important, and those axioms gave rise to the powerful logical methods of topos theory. You\u2019re similarly reminded of the categories of modules that are so very important in algebra, and that the axioms of abelian categories give rise to the powerful method of diagram chasing through Mitchell\u2019s embedding theorem.\n\nThe category of Hilbert spaces is also fundamental to several parts of mathematics, and you wonder if these six axioms can also lead to similarly powerful and similarly general methods. You make a mental note to look again at quantum logic in dagger kernel categories, or maybe even effectus theory. Clearly the dagger is a crucial ingredient that lets you treat much of the analysis of Hilbert space algebraically, and you should probably take dagger limits more seriously.\n\nYour inner physicist voice pipes up. Hilbert spaces are the mathematical foundation for quantum theory, but people always wonder why. Some spend their lives trying to reconstruct them from physical first principles. Can you interpret these axioms physically? Axiom 1 seems to say something about conservation of information, Axiom 2 about compound systems. Axiom 3 might have to do with measurement or superselection. But what about the other axioms? Can you reformulate them to make physical sense? Maybe you could use symmetry arguments, or tomographic principles.\n\nThe night is young and the stars inviting. Can you do characterise the category of finite-dimensional Hilbert spaces? The category of Hilbert modules, maybe using sheaf techniques? C*-categories? You feel full of hope, and get to work.\n\nPosted at September 23, 2021 1:47 PM UTC\n\nTrackBack URL for this Entry:\u00a0\u00a0 https:\/\/golem.ph.utexas.edu\/cgi-bin\/MT-3.0\/dxy-tb.fcgi\/3354\n\nRead the post Axioms for the Category of Hilbert Spaces\nWeblog: The n-Category Caf\u00e9\nExcerpt: Chris Heunen and Andre Kornell's axiomatic characterization of the category of Hilbert spaces.\nTracked: September 23, 2021 1:57 PM\n\n### Re: Axioms for the Category of Hilbert Spaces (bis)\n\nI loved the book and I\u2019m enjoying the \u2018movie\u2019 now. Beautiful!\n\nPosted by: Matteo Capucci on September 23, 2021 3:16 PM | Permalink | Reply to this\n\n### Re: Axioms for the Category of Hilbert Spaces (bis)\n\nI\u2019ve recently been investigating dagger-categories in the context of understanding Evil. Based on the \u201category\u201d analogy from this MO answer and comments, I\u2019ve developed the following explanation, which passes the explain-it-to-roommates test:\n\nSuppose that we think of an arrow in a category as having an ordered pair of objects which designates its source and target. Then arrows in a dagger-category merely have an unordered pair instead.\n\nAn immediate consequence is that composition is also unordered, although the objects still must line up. This causes the normal definition of \u201cfunctor\u201d to give dagger-functors, \u201cequivalence of categories\u201d to give dagger-equivalence, etc. It is immediately obvious why span categories and Rel are dagger-categories, since their arrows in their corresponding traditional categories are naturally unordered. This also suggests that the relationship between Cat and DagCat is similar to the one between Grp and Ab, so that there could be a \u201cdaggerizing\u201d 2-functor with an adjoint which takes quotients.\n\nObjects of DagCat are dagger-categories, not categories. Applying all of this to your amazing result, I think that the daggering should be seen as a more primitive operation which suffuses the axioms of Hilb and not just as a dagger structure. This lets us restate the axioms:\n\n1. \u2018Tis a dagger-category.\n2. The dagger-category is equipped with monoidal structure, and the tensor unit is simple and a monoidal separator.\n3. The dagger-category has binary products.\n4. The dagger-category has binary equalizers.\n5. Monic arrows are kernels.\n6. The subcategory of monic arrows has directed colimits.\n\nWhile I\u2019ve been terse and probably missed something, I hope it\u2019s obvious how much symmetry is implied by being daggered at a primitive level.\n\nPosted by: Corbin on September 23, 2021 8:39 PM | Permalink | Reply to this\n\n### Re: Axioms for the Category of Hilbert Spaces (bis)\n\nI am very much in favour of taking dagger category theory seriously as a study in its own right. It behaves quite differently from ordinary category theory in many ways. In my book with Jamie Vicary, we call this philosophy \u2018the way of the dagger\u2019: everything in sight has to cooperate with the dagger. It\u2019s also the title of Martti Karvonen\u2019s thesis, that includes our papers showing how monads and limits really work differently in the presence of a dagger.\n\nThere is one sneaky thing in the axioms, though, that seems to connect the dagger and non-dagger behaviour of morphisms: the tensor unit has to be simple in the non-dagger sense. That is, it has to have exactly two subobjects. It is not enough to ask that it has exactly two dagger subobjects. (Well, I don\u2019t know a proof in that case, but I also have no counterexample.)\n\nPosted by: Chris Heunen on September 23, 2021 10:12 PM | Permalink | Reply to this\n\n### Re: Axioms for the Category of Hilbert Spaces (bis)\n\nI\u2019ve recently been investigating dagger-categories in the context of understanding Evil.\n\nAnything to add to this discussion?\n\nPosted by: David Corfield on September 24, 2021 8:23 AM | Permalink | Reply to this\n\n### Re: Axioms for the Category of Hilbert Spaces (bis)\n\nAfter reading through Karvonen\u2019s The Way of the Dagger in the neighboring comment, I think that we might do well to completely refactor the page on dagger-categories so that the current definition is called \u201cclassical\u201d, and then add a new \u201cway of the dagger\u201d definition which quotes and cites Karvonen\u2019s definition. We should directly tackle the philosophy of how DagCat and Cat relate. The upshot is that I think that only the classical definition has the problems with equivalence.\n\nTo quote myself from a discussion in #categorytheory on Libera Chat:\n\nThe notion of \u201ccategory\u201d is relative to Cat. Cat-sized 2-categories might have objects which don\u2019t behave like categories; a category is only an analogy at that level, just like how the typical categorical object is not really a set.\n\nThe Way of the Dagger is that, further, 2Cat-sized 3-categories might have objects which don\u2019t behave like 2-categories. If we consider DagCat as a 2-dagger-category, then it fundamentally is not a 2-category, but only something which has analogous structure to a 2-category.\n\nPosted by: Corbin on September 28, 2021 1:38 AM | Permalink | Reply to this\n\n### Re: Axioms for the Category of Hilbert Spaces (bis)\n\nI\u2019m pretty sure that dagger categories aren\u2019t evil, because the type of objects in a dagger category is defined as a 1-truncated type in homotopy type theory. If dagger categories were evil, the type of objects would be 0-truncated instead.\n\nAs groupoids are an example of a type of dagger category, where the dagger of a morphism is equal to the inverse of the morphism, if dagger categories were evil, then groupoids would have to be evil as well. I don\u2019t think many people would accept that groupoids are evil.\n\n### Re: Axioms for the Category of Hilbert Spaces (bis)\n\nWhat\u2019s \u201cevil\u201d is the particular definition \u201ca dagger category is a category with an identity-on-objects endofunctor\u2026\u201d since it refers to identity on objects. In particular, you can\u2019t give a correct definition of dagger-category in HoTT by starting from that version. The better definition is the one given in the HoTT Book, and generalized to other categorical structures here, where a dagger is an additional \u201cfield\u201d added to the dependent record type defining a category. This enables its \u201cidentity-on-objects\u201d-ness to be recorded by type dependency, in the same way that when we write $g\\circ f$ in an ordinary category the codomain of $f$ must be exactly equal to the domain of $g$.\n\nPosted by: Mike Shulman on September 25, 2021 4:37 PM | Permalink | Reply to this\n\n### Re: Axioms for the Category of Hilbert Spaces (bis)\n\nIs there a way to combine this result with the idea that Finite dimensional Hilbert spaces are complete for dagger compact closed categories (in the logical sense)? They seem tantalizingly similar, but I couldn\u2019t figure out how to put any of it together.\n\nPosted by: Corbin on September 23, 2021 8:46 PM | Permalink | Reply to this\n\n### Re: Axioms for the Category of Hilbert Spaces (bis)\n\nThat\u2019s a good question, that I also wondered about. The immediate obstacle is that you\u2019ll first need to adapt the axioms to characterise finite-dimensional Hilbert spaces. As in the very last paragraph of the paper, you can look at the largest compact subcategory within the category given by the axioms. But it\u2019s unsatisfactory to characterise a small thing by really characterising a big thing and saying how the small thing sits inside the big thing. Instead I\u2019d like to characterise the small thing directly. But for that the axiom about directed colimits has to go, because of course finite-dimensional Hilbert spaces don\u2019t have that.\n\nPosted by: Chris Heunen on September 23, 2021 10:16 PM | Permalink | Reply to this\n\n### Re: Axioms for the Category of Hilbert Spaces (bis)\n\nBut it\u2019s unsatisfactory to characterise a small thing by really characteristing a big thing and saying how the small thing sits inside the big thing.\n\nI think that whether it is satisfactory depends on your perspective! This sounds very much like the Grothendieckian approach (of replacing the study of nice objects by the study of nasty objects in nice categories) \u2026\u00a0but maybe I am misunderstanding the force of \u201ccharacterise\u201d here.\n\nPosted by: L Spice on September 30, 2021 4:17 AM | Permalink | Reply to this\n\n### Re: Axioms for the Category of Hilbert Spaces (bis)\n\nIf we drop the condition that I is simple, can we characterize dagger monoidal categories of W*-modules over commutative von Neumann algebras in this manner?\n\nPosted by: Dmitri Pavlov on September 24, 2021 12:17 AM | Permalink | Reply to this\n\n### Re: Axioms for the Category of Hilbert Spaces (bis)\n\nI\u2019m hoping the answer is yes, but will have to get back to you. With the methods of this paper you could hopefully reduce from a setting where $I$ is not simple to a sheaf of categories where it is, and hence characterise the category of Hilbert modules over a fixed commutative C*-algebra $C$. But probably the axiom that $I$ is simple will have to be replaced by a property that the central idempotents of $I$ are well-behaved enough to form the opens of the spectrum of $C$.\n\nPosted by: Chris Heunen on September 24, 2021 9:33 AM | Permalink | Reply to this\n\n### Re: Axioms for the Category of Hilbert Spaces (bis)\n\nChris and Andre, it\u2019s so nice to see this amazing result, congratulations! What surprises me most about it is the fact that works for all Hilbert spaces, not just finite ones. That\u2019s great and a real strength of the result.\n\nSome people here might be interested in the history behind this. Dagger-monoidal categories as an abstract setting for quantum theory were popularized by Samson Abramsky and Bob Coecke in 2004, which I guess covers your Axioms 1 and the \u201cdagger symmetric monoidal\u201d of 2. Dagger-biproducts were introduced by Peter Selinger in his paper \u201cIdempotents in dagger-categories\u201d (2008), which gives your Axiom 3. Peter suggested to me I should think about \u201cdagger-equalizers\u201d, which led to my paper arXiv:0807.2927 showing dagger-equalizers and a simple tensor unit, your Axiom 4 and part of Axiom 2, guarantees that the scalars are a subsemiring of the complex numbers, and has other nice consequences. Chris\u2019s paper arXiv:0811.1448 followed soon after, adding your Axiom 5, and proving a stronger embedding result. We all then got bored or did other things for 13 years (or at least I did.) Then you guys added this nice Axiom 6 which enables you to prove the Holy Grail result you have here.\n\nIt\u2019s extremely satisfying to see all this resolved, and I have to do some thinking about Axiom 6.\n\nPosted by: Jamie Vicary on October 1, 2021 12:30 PM | Permalink | Reply to this\n\n### Re: Axioms for the Category of Hilbert Spaces (bis)\n\nThanks Jamie! The first version of the arxiv preprint was very lean and to the point. We\u2019ve now updated it to a second version which gives some literature context, including some history of Soler\u2019s theorem.\n\nPosted by: Chris Heunen on November 4, 2021 10:31 AM | Permalink | Reply to this\n\n### Re: Axioms for the Category of Hilbert Spaces (bis)\n\nDusk. The alley is secreted in mist. The miserly reflection of the single lamp on the cobblestones makes you nervous. The stranger approaches, hands inside trenchcoat.\n\nHe draws out a dagger\u2026.\n\nPosted by: John Baez on January 12, 2022 11:42 PM | Permalink | Reply to this\n\nPost a New Comment","date":"2022-01-18 00:31:09","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 50, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7985025644302368, \"perplexity\": 726.5094371796074}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-05\/segments\/1642320300658.84\/warc\/CC-MAIN-20220118002226-20220118032226-00525.warc.gz\"}"}
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Picture this: Khloe Kardashian, except that she's the mother of five or six children.
Okay, now take some deep, relaxing breaths and calm down. No, we're not ready for that future either.
But it sounds like Tristan Thompson sure is.
One of the subplots on this season of Keeping Up With The Kardashians was questions about Khloe's fertility.
Not just questions about whether or not she's interested in having kids, but questions about whether or not she can.
She's only been dating basketball player Tristan Thompson since last September, so it seems a little premature for this kind of talk, right?
But at the same time, some people want kids from relationships and some people don't, and it's good to know early on whether kids are even an option and who might be interested.
Tristan seems to be more than just interested, but also … his idea of family planning is a little nuts.
"Tristan and I definitely talk about starting a family," Khloe told her sister Kim on the Keeping Up With The Kardashians season finale.
Tristan already has a kid, but apparently he won't be satisfied until he can populate a small island?
Five is a huge number of children to have, even for a committed, long-term couple.
But Khloe sounds like this could become reality sooner than any of us realize.
Yeah, one child is a huge step. Children are adorable but a lot of work and the hugest responsibility.
Six children is a step off a cliff into an infinite, chaotic abyss.
We know that Kris Jenner had six kids, but it was over the course of a couple of decades.
Also, like, you gotta love Kris, but she has zero chill about anything.
But Kris Jenner is intensely maternal. She's not just protective — she's helpful and supportive.
Even when she doesn't necessarily agree. Remember when she probably straight-up bought a house for Rob after Blac Chyna got pregnant?
It's because she had the means to and because she was supportive.
Khloe has a … different temperament.
Honestly, and maybe it's just because she's on camera, but Khloe can sometimes come across as a little emotionally unbalanced.
And you know how meddling and controlling Khloe can be towards her grown-up adult siblings?
Imagine that but she's a mother.
So … it sounds like she has some things to work past, first.
Khloe's siblings haven't really had the best luck in terms of romance.
Sure, Kim's happy now, but let's not forget that Kanye isn't her first husband.
Kourtney and Scott were happily together for years, but we've all seen how that turned out.
Rob and Blac Chyna's highly publicized relationship fizzled and died before the world's eyes.
Kendall and Kylie are younger so they haven't had any disasters on quite that scale — because they haven't gotten married or had kids.
Unless you count every moment of Kylie dating Tyga as a disaster, anyway.
Looking at all of that, is Khloe really ready to even discuss having kids with a guy she hasn't even been dating for a year?
Ultimately, if Khloe had gotten pregnant right after that episode was filmed, we'd probably know it by now.
So maybe it was just talk to raise drama or to float the idea by her family and fans.
And even if she and Tristan do have plans to give the Duggars a run for their money, it'll probably just happen with one baby at a time.
Considering how agonizing pregnancy can be for some people, especially for Kim, maybe one will be enough for Khloe.
But … maybe not. We may all need to brace ourselves for that on the horizon.
NextJessa Duggar on Joy-Anna's Wedding: It's Finally Her Turn!
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 6,580
|
\section*{Abstract}
Quantum Compiling Algorithms
decompose (exactly,
without approximations) an
arbitrary $2^\nb$ unitary matrix
acting on $\nb$ qubits, into
a sequence of elementary operations (SEO).
There are many possible
ways of decomposing
a unitary matrix into a SEO,
and some of these decompositions
have shorter length (are more efficient)
than others.
Finding an optimum
(shortest) decomposition
is a very hard task, and is not
our intention here. A less ambitious,
more doable task is to find
methods for optimizing
small segments of a SEO.
Call these methods piecewise optimizations.
Piecewise optimizations
involve replacing a
small quantum circuit by
an equivalent one with fewer
CNOTs. Two circuits
are said to be equivalent
if one of them multiplied by some
external
local operations
equals the other.
This equivalence relation
between circuits has
its own class functions,
which we call circuit invariants.
Dressed CNOTs are
a simple yet very useful generalization
of standard CNOTs.
After discussing circuit invariants
and dressed CNOTs, we
give some methods for simplifying
2-qubit and 3-qubit circuits.
We include with this paper
software (written in Octave/Matlab)
that checks many
of the algorithms proposed in
the paper.
\tableofcontents
\section{Introduction}
\label{sec-introduction}
Quantum Compiling Algorithms
decompose (exactly,
without approximations) an
arbitrary $2^\nb$ unitary matrix
acting on $\nb$ qubits, into
a sequence of elementary operations (SEO).
By elementary operations
we mean operations that act on only
a few (usually 1 or 2) qubits (for example,
all single-qubit rotations and CNOTs.)
The most efficient quantum
compiling algorithms to date
are based on a recursive
application of the Cosine-Sine Decomposition (CSD),
a technique first proposed
in Ref.\cite{Tuc-rud}.
An implementation of the algorithm
of Ref.\cite{Tuc-rud} may be
found in the computer program
called Qubiter (patented,
C++ source code publicly available at
www.ar-tiste.com).
Long after Ref.\cite{Tuc-rud} and Qubiter
came out, many papers
on quantum compiling
via recursive CSD have appeared. These can be
easily tracked down
by making a keyword search in ArXiv or Google
for something like
(``Cosine-Sine" and ``Decomposition"
and ``quantum") .
We will call the number of CNOTs in a SEO its length.
(Single-qubit rotations are not counted
because these can be performed
much faster than CNOTs.)
Of course, there are many possible
ways of decomposing
a unitary matrix into a SEO,
and some of these decompositions
have shorter length (are more efficient)
than others. The algorithm of
Ref.\cite{Tuc-rud} per se does not
yield the shortest SEO.
Finding an optimum
(shortest) decomposition
is a very hard task, and is not
our intention here. A less ambitious,
more doable task is to find
methods for optimizing
small segments of a SEO.
Call these methods piecewise optimizations.
The hope is that
given any SEO, one
can apply piecewise optimization
methods to reduce the original
SEO into an equivalent SEO
whose length is much less, and
might even be close to the
shortest possible length.
An analogy to our piecewise optimization
strategy is the following. Think of
a SEO as being
like a path between two points in a
manifold. If this path
is initially unnecessary long,
one might hope to make it a little
less so by breaking it into
pieces and optimizing the length
of each piece. Breaking it into pieces again,
and optimizing each piece again. And so on.
Piecewise optimizations
involve replacing a
small quantum circuit by
an equivalent one with fewer
CNOTs. Two circuits
are said to be equivalent
if one of them multiplied by some
external
local operations
equals the other.
By external local
operations, we mean
single-qubit rotations
applied at the beginning
or end of the circuit.
This equivalence relation
between circuits has
its own class functions,
which we call circuit invariants.
Many excellent papers
already exist on the use of
such invariants in quantum
computing.
See, for example, Refs.
\cite{Rai},
\cite{Gra},
\cite{Mak},
and
\cite{She}.
Such invariants are a crucial ingredient of
this paper. (However, the paper
does not assume that the reader possesses
any prior knowledge
about these invariants. The paper
is self-contained in this regard.)
Besides circuit invariants,
another important ingredient
of this paper is what we call
dressed CNOTs (DC-NOTs). DC-NOTs are
a simple yet very useful generalization
of standard CNOTs. To my knowledge,
this paper
is the first one to consider
DC-CNOTs.
DC-NOTs are
convenient
because they lump together
a CNOT and some
single-qubit rotations.
Modulo
external local operations,
one
can express
any circuit solely in terms
of a single type of circuit element
(DC-CNOTs), rather than having to
express it with two different types of circuit
elements (CNOTs and single-qubit
rotations).
After discussing circuit invariants
and DC-NOTs, this paper
gives some methods for simplifying
2-qubit and 3-qubit circuits.
Much is
already known about
simplifying 2-qubit circuits.
Ref.\cite{VD} shows,
via Cartan's KAK decomposition\cite{Tuc-KAK},
that a 2-qubit circuit with any number of
CNOTs
can always be reduced to a circuit
with 3 CNOTs.
Refs.\cite{VD} and \cite{She}
give necessary and
sufficient conditions
for when a 2-qubit circuit
with 3 CNOTs reduces to fewer
than 3 CNOTs.
In this paper, we spend
some time
re-proving
these already
known 2-qubit results
using the new language
of circuit invariants
and DC-NOTs. This
exercise yields
new techniques and new
geometrical insights
that
were lacking in previous proofs.
In this paper, we also present some interesting
new ways of simplifying
3-qubit circuits.
Our results for
3-qubit circuits rely
heavily on our results for
2-qubit circuits.
We include with this paper
software (written in Octave/Matlab)
that checks many
of the algorithms proposed in
the paper. In the header of each section,
and in the Table of Contents,
each section name is followed by
a list in square brackets of the names
of the Octave m-files relevant to that section.
Our software is not intended to be
very efficient, or to be free of
all conceivable loopholes. It is
only intended to be a proof of principle
of our algorithms.
\section{Notation
\\{\footnotesize\tt[
global\_declarations.m, global\_defs.m, simul\_real\_svd.m, Gamma\_rep.m,\\
sig.m, check\_dcnots.m, factor\_SU2pow2\_matrix.m, factor\_SU2pow3\_matrix.m,\\
test\_factor\_su2pow.m, get\_normal\_unit\_vec.m, get\_unit\_vec.m
]}}
\label{sec-notation}
In this section, we discuss notation,
linguistic idiosyncrasies and abbreviations
that will be used in subsequent sections.
If any notation in this paper remains
unclear to the reader after reading
this section, he should consult
Ref.\cite{Paulinesia},
a review article, written by
the author of this paper, that
uses the same notation as this paper.
We will often use the symbol
$\nb=0,1,2,\ldots$ for
number of bits,
and $\ns = 2^\nb$
for the corresponding number
of states.
We will often abbreviate
$\cos(\alpha)$ and
$\sin(\alpha)$ by $c_\alpha$ and
$s_\alpha$, respectively.
We will often use a subscript of $f$
to denote the final value of
quantity that changes (e.g., $\hata$ changes to
$\hata_f$).
When we say $b=\pm a$,
we mean $b\in \{a,-a\}$.
When we write $X_{\alpha\rarrow\beta}$,
we mean, the quantity
obtained by replacing $\alpha$
by $\beta$ everywhere in $X$.
Likewise, by
$X_{\alpha\darrow\beta}$
we mean, the quantity
obtained by swapping $\alpha$
and $\beta$ everywhere in $X$.
When we say ``$A (ditto, A')$ is $B(ditto, B')$"
we mean ``$A$ is $B$ and $A'$ is $B'$".
LHS and RHS will stand for
left-hand side and right-hand side.
``RHON basis" will stand for "right-handed
orthonormal basis".
Let $Bool=\{0,1\}$.
Let $\RR$ denote the real numbers,
$\CC$ the complex numbers, $\ZZ$
all integers (positive and negative).
For integers $a$ and $b$,
$\ZZ_{a,b}$ will denote all integers from
$a$ to $b$, including $a$ and $b$. If $\Omega$
is anyone of the symbols $>, \geq, <, \leq$,
and $S$ is any set, define
$S^{\Omega\;0}= \{x\in S: x\;\Omega\;0\}$
if the right hand side is defined.
For example, $\ZZ^{>0}$
are the positive integers.
As usual, for any set $S$
and $r, p,q\in \ZZ^{>0}$,
$S^r$ will denote the set of
r-tuples of $S$, and $S^{p\times q}$,
the set of $p\times q$ matrices with
entries in $S$.
As usual, $U(\ns)$
will denote the
$\ns\times\ns$ unitary matrices, and $SU(\ns)$
the special (i.e., with determinant=1)
elements of $U(\ns)$.
Given any $A\in U(\ns)$,
we define $\hat{A}$ by
$\hat{A} = A/[\det(A)]^{\frac{1}{\ns}}$,
where we choose the principal branch of
the function $(\cdot)^{\frac{1}{\ns}}$.
We will refer to
$\hat{A}$ as the
``special counterpart" of $A$.
(here the adjective ``special"
again means ``with determinant=1").
$\RR^3$ will denote the set of
all 3 dimensional
real vectors, and
$\unitvecs=\{x\in\RR^3:|x|=1\}$.
As is common in the Physics literature,
a letter with an arrow (ditto, caret)
over it,
as in $\veca$ (ditto, $\hata$)
will denote an element of $\RR^3$
(ditto, $\unitvecs$).
$\veca$ and $\hata$
will be treated as column vectors
when they appear in matrix expressions.
Let $\veca_j\in \RR^3$ for $j\in \ZZ_{1,r}$.
We will use the following non-standard notation
for r-fold cross products:
\beq
\manyx{\veca_1\veca_2\veca_3\ldots\veca_r}=
(\cdots((\veca_1\times\veca_2)\times\veca_3)\cdots\times \veca_r)
\;.
\eeq
For example, $\manyx{\veca_1\veca_2\veca_3\veca_4} =
((\veca_1\times\veca_2)\times\veca_3)\times \veca_4$.
Of course, an (r+2)-fold cross-product can be
reduced to an r-fold cross-product using
the well known ``BAC minus CAB" identities:
for $\veca,\vecb,\vecc\in\RR^3$,
$\veca\times(\vecb\times\vecc) =
\vecb(\veca\cdot\vecc)-\vecc(\veca\cdot\vecb)$
and
$(\veca\times\vecb)\times\vecc =
\vecb(\veca\cdot\vecc)-\veca(\vecb\cdot\vecc)$.
For example, if $\hata,\hatb$
are perpendicular unit vectors,
then $\manyx{\hata\hatb\hatb}=-\hata$.
Suppose $\veca, \vecb\in \RR^3$.
$angle(\veca,\vecb)$ will
denote the angle between
$\veca$ and $\vecb$,
defined up to $2\pi$.
We will say $\veca$ is
parallel to $\vecb$ and
write $\veca\parallel\vecb$
iff $\veca\times\vecb=0$;
i.e., iff $\veca=\pm \vecb$,
or $\veca=0$, or $\vecb=0$.
We will say
$\veca$ is
perpendicular to $\vecb$ and
write $\veca\perp\vecb$ iff $\veca\cdot\vecb=0$.
For $\vecb\neq 0$, define $\along{\veca}{\vecb}$,
{\bf the part of $\veca$ along $\vecb$},
by
\beq
\along{\veca}{\vecb}=
\frac{(\veca\cdot\vecb)\vecb}{|\vecb|^2}
\;.
\eeq
For $\vecb\neq 0$, define $\across{\veca}{\vecb}$,
{\bf the part of $\veca$ across $\vecb$},
by
\beq
\across{\veca}{\vecb}
= \veca - \along{\veca}{\vecb}
=
\veca -\frac{(\veca\cdot\vecb)\vecb}{|\vecb|^2}
=
\frac{-\manyx{\veca\vecb\vecb}}{|\vecb|^2}
\;.
\eeq
For any square matrix $A$,
$A^T$ will denote its transpose,
$A^*$, its complex conjugate,
and $A^\dagger = A^{*T}$, its Hermitian
conjugate.
$\delta_{i,j}$ will denote the Kronecker delta
function.(It equals one if $i=j$
and zero otherwise.)
Let $I_2,\sigx, \sigy, \sigz$ be the
2d identity matrix and Pauli matrices.
Sometimes, we set $(X_1,X_2,X_3)=(X,Y,Z)$
and denote the Pauli matrices by
$\sigma_{X_1}, \sigma_{X_2}, \sigma_{X_3}$.
Suppose $W \in\{X,Y,Z\}$.
Define the number operators:
$n_W = \frac{1-\sigma_W}{2}$
and $\nbar_W = \frac{1+\sigma_W}{2}$.
Note that $(-1)^{n_W} = \sigma_W$.
Usually,
$n_Z$ is denoted merely by $n$
and $\nbar_Z$ by $\nbar$.
If $W_j \in\{1, X,Y,Z\}$
for $j\in \ZZ_{1,\nb}$, let
$\sigma_{W_1,W_2, \ldots, W_\nb} =
\sigma_{W_1}\otimes \sigma_{W_2}
\otimes\ldots \sigma_{W_\nb}$,
where any incidence of
$\sigma_1$ on the RHS
is replaced by $I_2$.
For example, $\sigma_{XY1} =
\sigx\otimes\sigy\otimes I_2$.
$H = \frac{1}{\sqrt{2}}
\left[\begin{array}{cc}1&1
\\1&-1\end{array}\right]$
is the one-bit Hadamard matrix
and $H^{\otimes\nb}$ is its
$\nb$-fold tensor product.
$H$ satisfies $H^2=1$, $H\sigx H=\sigz$,
$H\sigz H= \sigx$ and $H\sigy H = -\sigy$.
Suppose $a_0\in \RR$ and $\veca\in\RR^3$.
We will abbreviate $\vec{\sigma}\cdot\veca$
by $\sigma_\veca$. The standard terminology
is to call
$a_0 + i\sigma_\veca$
a {\bf quaternion}, and to call
$\sigma_\veca$
a vector quaternion (divided by $i$).
To shorten
this terminology, we will
refer to $\sigma_\veca$ as a {\bf Paulion},
and call $\veca$ its {\bf defining vector}.
If $|\veca|=1$, we will
call $\sigma_\veca$ a {\bf unit Paulion}.
One can reduce a product of
two Paulions
by using the identity
$\sigma_{\veca}\sigma_{\vecb}=
\veca\cdot\vecb + i \sigma_{\veca\times\vecb}$.
For $\hata\in \unitvecs$, define number operators
$n_\hata = \frac{1-\sigma_\hata}{2}$ and
$\nbar_\hata = \frac{1+\sigma_\hata}{2}$.
Note that $(-1)^{n_\hata} = \sigma_\hata$.
If $W_j \in\unitvecs$ or $W_j=1$
for $j\in \ZZ_{1,\nb}$, let
$\sigma_{W_1,W_2, \ldots, W_\nb} =
\sigma_{W_1}\otimes \sigma_{W_2}
\otimes\ldots \sigma_{W_\nb}$.
Suppose $\calm$
is the set of all matrices
$M\in \CC^{4\times 4}$
that can be expressed in the form
$M=\sum_k \sigma_{\veca_k,\vecb_k}$, where
$\veca_k, \vecb_k\in \RR^3$ for all $k$.
Suppose $\call$ is the set of all
matrices $L\in \RR^{3\times 3}$
that can be expressed in the form
$L=\sum_k \veca_k \vecb_k^T$,
where $\veca_k, \vecb_k\in \RR^3$ for all $k$.
For every $M\in \calm$,
let $\Gamma(M)$ or $M^\Gamma$
represent the $3\times3$
matrix with entries
$\frac{1}{4}\tr(\sigma_{X_i,X_j}M)$,
where $i,j\in \ZZ_{1,3}$.
(The symbol $\Gamma$ was chosen
to evoke the mental picture of
a column vector times a row vector;
such is the output of the function $\Gamma(\cdot)$).
For every $L\in \call$,
define $\Gamma^{-1}(L) =
\sum_{i,j}\sigma_{X_i,X_j}L_{i,j}$.
It's easy to check that
$\Gamma \Gamma^{-1} = \Gamma^{-1}\Gamma =1$
so the map $\Gamma:\calm\rarrow \call$ is 1-1 onto.
Let $lin(\calm)$ be
the set of linear combinations over $\CC$
of elements of $\calm$,
and $lin(\call)$ of $\call$.
The map $\Gamma$
can be extended to $\overline{\Gamma}:
\CC + lin(\calm)\rarrow
\CC + lin(\call)$,
$\overline{\Gamma}(\lam + \sum_i \alpha_i M_i)
= \lam + \sum_i \alpha_i M_i^\Gamma$.
$\overline{\Gamma}$ is also a 1-1 onto map.
Henceforth, we will use $\Gamma$
to refer to both $\Gamma$
and its extension $\overline{\Gamma}$.
Given a matrix $A\in \CC + lin(\calm)$,
we will call $A^\Gamma$ its Gamma representation.
Often, in contexts where this
will not lead to
confusion, we will drop the
$\Gamma$ superscript and denote
$A^\Gamma$ simply by $A$.
The next theorem, although almost
trivial,
will be used frequently in this paper.
\begin{theo}\label{th-double-cover}
The map $f:\unitvecs\times\unitvecs\rarrow SU(2)$,
$f(\hata, \hatb) = \sigma_\hata\sigma_\hatb$
is well defined and onto. In other words:
(well-defined)
If $\hata, \hatb\in \unitvecs$, then
$f(\hata, \hatb)\in SU(2)$.
(onto)
If $U\in SU(2)$, then there
exist $\hata,\hatb\in \unitvecs$
such that $U=f(\hata, \hatb)$.
\end{theo}
\proof
(well defined)
Given $\hata,\hatb\in\unitvecs$,
one can always find an angle $\theta$
such that $\hata\cdot\hatb=c_\theta$
and $|\hata\times\hatb|=s_\theta$.
Let $\hatw = \frac{\hata\times \hatb}
{|\hata\times \hatb|}$.
It follows that
$\sigma_\hata\sigma_\hatb =
\hata\cdot\hatb + i\sigma_{\hata\times\hatb}
=e^{i\theta\sigma_\hatw}\in SU(2)$.
(onto) Given
$U= e^{i\theta\sigma_\hatw}$,
where $\hatw\in\unitvecs$ and
$\theta\in \RR$, one can
always find
a (non-unique) pair of unit vectors
$\hata$ and $\hatb$
in the plane perpendicular
to $\hatw$, such that
$\theta=angle(\hata,\hatb)$,
and
$\hata\times\hatb$
points in the $\hatw$
direction. Hence,
$\hata\cdot\hatb = c_\theta$
and
$\hata\times\hatb = s_\theta\hatw$.
It follows that
$\sigma_\hata\sigma_\hatb =
\hata\cdot\hatb + i\sigma_{\hata\times\hatb}
=e^{i\theta\sigma_\hatw}$.
\qed
One has:
\beq
\sigma_\hatr
\sigma_\hata
\sigma_\hatr
=
\sigma_\hatr
(\sigma_{\along{\hata}{\hatr}} +
\sigma_{\across{\hata}{\hatr}})
\sigma_\hatr
=
\sigma_{\along{\hata}{\hatr}}
-\sigma_{\across{\hata}{\hatr}}
=
\sigma_{
\along{\hata}{\hatr}
-\across{\hata}{\hatr}
}
=
\sigma_{\hata_f}
\;.
\label{eq-paulion-reflection}
\eeq
A geometrical interpretation
of this identity
is shown in
Fig.\ref{fig-paulion-rot}a.
The similarity transformation
$\sigma_\hatr(\cdot)\sigma_\hatr$
takes the Paulion
$\sigma_\hata$ to $\sigma_{\hata_f}$,
where $\hata_f$ is the reflection
of $\hata$ on $\hatr$.
Suppose $\hata, \hatb\in\unitvecs$,
and we want to find
$U\in SU(2)$ such that
$\sigma_\hatb=
U^\dagger\sigma_\hata U$.
Such a $U$ can be
constructed as a product of
two Paulions
(See Fig.\ref{fig-paulion-rot}b).
Indeed,
let $\theta=angle(\hata,\hatb)$
and $\hatp= \frac{\hata\times\hatb}
{|\hata\times\hatb|}$.
Let $\hatr$ be the vector
that bisects the angle
between $\hata$ and $\hatb$,
and is oriented so that $\hata\times\hatr$
points along $\hatp$. Note that
$\hatb$ can be obtained
by reflecting $\hata$ on the bisector $\hatr$.
Hence
\beq
\sigma_\hata\sigma_\hatr =
e^{i\frac{\theta}{2}\sigma_\hatp}
\;,\;\;
\sigma_\hatb=
\sigma_\hatr\sigma_\hata\sigma_\hatr
\;.
\eeq
Combining these two results yields
\beq
\sigma_\hatb
=
(\sigma_\hatr\sigma_\hata)
\sigma_\hata
(\sigma_\hata\sigma_\hatr)
=
e^{-i\frac{\theta}{2}\sigma_\hatp}
\sigma_\hata
e^{i\frac{\theta}{2}\sigma_\hatp}
\;.
\eeq
\begin{figure}[h]
\begin{center}
\epsfig{file=paulion-rot.eps, height=1.5in}
\caption{(a)If
$\sigma_\hatr \sigma_\hata \sigma_\hatr
=\sigma_{\hata_f}$, then $\hata_f$
is obtained
by reflecting $\hata$ on $\hatr$.(b)
Suppose $\hatb$ is the result of rotating
$\hata$ by an angle $\theta$. Then
$\hatb$ can be obtained by reflecting $\hata$
on the bisector $\hatr$
of the angle between $\hata$ and $\hatb$.}
\label{fig-paulion-rot}
\end{center}
\end{figure}
\section{Invariants for Quantum Circuits}
\label{sec-ckt-invariants}
In this section, we will discuss
circuit invariants; i.e.,
functions that map
all equivalent circuits to the same value.
By equivalent circuits we mean circuits that are
equal,
modulo external local operations.
Suppose $A$ and $B$ are
elements of $U(\ns)$ ( i.e., they
are $\nb$-qubit gates).
We will say $A$ and $B$
are {\bf equivalent under local operations
on the right hand side (LO-RHS)},
and write
$A\sim_R B$, iff there exist $U_j\in U(2)$
for $j\in \ZZ_{0,\nb-1}$
such that
\beq
B = A (U_{\nb-1} \otimes \ldots \otimes U_2 \otimes U_1 \otimes U_0)
\;.
\label{eq-def-lo-rhs}
\eeq
$\sim_R$ is clearly an
equivalence relation as it
is symmetric, reflexive and transitive.
Henceforth, we will say that a function
$\chi$ with domain $U(\ns)$ is
a LO-RHS invariant if for any
$A,B\in U(\ns)$, $A\sim_R B$ implies that
$\chi(A) = e^{i\zeta}\chi(B)$
for some $\zeta\in \RR$ ($\zeta$
may depend on $A,B$).
A frequent goal is
to find a complete set of
scalar invariant functions; that is, a
set of functions
$\chi_j:U(\ns)\rarrow \RR$ such that
for any $A,B \in U(\ns)$,
$A\sim_R B$
if and only if
$\chi_j(A)=\chi_j(B)$ for all $j$.
An extensive literature already
exist on such invariants. They
were first studied by Group Theorists,
and, in more recent times, they
have been used by Quantum Computerists
\cite{Rai},
\cite{Gra},
\cite{Mak},
\cite{She}.
One can define an analogous
equivalence relation $\sim_L$ for local operations
on the left hand side (LO-LHS),
and an equivalence
relation $\sim_{LR}$
for local operations on both sides (LO-2S).
Of course, the equivalence classes (e-classes)
of $\sim_R$
are a disjoint partition of
$U(\ns)$. Ditto for the e-classes of $\sim_{L}$
and $\sim_{LR}$.
It's also clear that any e-class for $\sim_R$
is contained in an e-class for $\sim_{LR}$,
and that some e-classes of $\sim_{LR}$
contain more than one e-class of
$\sim_R$. (In fact, the e-classes
of $\sim_R$ contained within a single e-class
of $\sim_{LR}$,
can be labeled by the elements of $U(2)^{\otimes \nb}$).
Note that for any $\vec{a}\in \RR^3$,
\beq
\sigy \sigma^T_{\veca}\sigy
=
-\sigma_{\veca}
\;.\label{eq-neg-sig}
\eeq
Hence, for $\theta\in\RR$ and $\vec{a}\in \RR^3$,
\beq
\sigy [e^{i(\theta+\sigma_\veca)}]^T \sigy=
e^{i(\theta-\sigma_\veca)}
\;.
\eeq
Thus, when $U\in SU(2)$ (but not
when $U\in U(2)$),
$\sigy U^T \sigy =U^{-1}=U^\dagger$.
For any $A\in U(\ns)$, define
a quadratic (second order in $A$) invariant
\beq
A^{(2)} =
A \sigy^{\otimes \nb} A^T \sigy^{\otimes \nb}
\;.
\eeq
For example, for $A\in U(4)$,
$A^{(2)} = A\sigyy A^T \sigyy$.
\begin{theo}\label{th-quad-invar}
\begin{enumerate}
\item[]
\item[(a)]For $A,B\in SU(4)$,
$A\sim_R B$ if and only if
$A^{(2)}=(-1)^n B^{(2)}$ for some $n\in \ZZ$.
\item[(b)] For $A,B\in U(4)$,
$A\sim_R B$ if and only if
$A^{(2)}=e^{i\zeta} B^{(2)}$ for some $\zeta\in \RR$.
\end{enumerate}
\end{theo}
\proof
\begin{enumerate}
\item[(a)]
Assume $A, B\in SU(4)$.
$A$ can always be represented in the form
\beq
A = i^{n(A)}
\exp(i a_{jk}\sigma_{X_j X_k})
\exp(i a'_j\sigma_{X_j 1})
\exp(i a_k \sigma_{1 X_k})
\;,
\label{eq-A-su4}
\eeq
where $n(A)\in\ZZ$
and $a_{jk},a'_j, a_k\in \RR$. (Note that
$\det(i I_4)=1$ so $\det(A)=1$.)
We are using Einstein's implicit summation convention,
and $j,k$ range over $\{1,2,3\}$.
By Eqs.(\ref{eq-neg-sig})
and (\ref{eq-A-su4}),
\beq
\sigyy A^T \sigyy= i^{n(A)}
\exp(-i a_k \sigma_{1 X_k})
\exp(-i a'_j\sigma_{X_j 1})
\exp(i a_{jk}\sigma_{X_j X_k})
\;.
\eeq
Thus
\beq
A^{(2)} = (-1)^{n(A)}
\exp(i 2a_{jk}\sigma_{X_j X_k})
\;.
\label{eq-A-su4-invar}
\eeq
Likewise, $B$ can be represented in the form
\beq
B = i^{n(B)}
\exp(i b_{jk}\sigma_{X_j X_k})
\exp(i b'_j\sigma_{X_j 1})
\exp(i b_k \sigma_{1 X_k})
\;,
\label{eq-B-su4}
\eeq
where $n(B)\in \ZZ$
and $b_{jk},b'_j, b_k\in \RR$.
Then
\beq
B^{(2)} = (-1)^{n(B)}
\exp(i 2b_{jk}\sigma_{X_j X_k})
\;.
\label{eq-B-su4-invar}
\eeq
\rproof Suppose $A\sim_R B$.
Looking at Eqs.(\ref{eq-def-lo-rhs}),
(\ref{eq-A-su4})
and (\ref{eq-B-su4}), we see that
for every $j,k$, there exists
an integer $n_{jk}$ such that
$a_{jk} = b_{jk} + \pi n_{jk}$.
Therefore,
\beq
\exp(i 2a_{jk}\sigma_{X_j X_k})=
\exp(i 2b_{jk}\sigma_{X_j X_k})
\;.
\label{eq-gist-rproof}
\eeq
Therefore, looking at
Eqs.(\ref{eq-A-su4-invar})
and (\ref{eq-B-su4-invar}), we see that
there exists an integer $n$
such that $A^{(2)}=(-1)^n B^{(2)}$.
\lproof Suppose
$A^{(2)}=(-1)^n B^{(2)}$.
Then, looking at
Eqs.(\ref{eq-A-su4-invar})
and (\ref{eq-B-su4-invar}), we see that
for every $j,k$, there exists
an integer $n_{jk}$ such that
$2a_{jk} = 2b_{jk} + \pi n_{jk}$.
Therefore,
\beq
\exp(i a_{jk}\sigma_{X_j X_k})=
\exp(i b_{jk}\sigma_{X_j X_k})
\prod_{j,k}[i \sigma_{X_j X_k}]^{n_{jk}}
\;.
\label{eq-gist-lproof}
\eeq
Therefore,
from Eqs.(\ref{eq-def-lo-rhs}),
(\ref{eq-A-su4})
and (\ref{eq-B-su4}), we see that
$A\sim_R B$.
\item[(b)]
Assume $A, B\in U(4)$.
Eqs.(\ref{eq-A-su4}) and (\ref{eq-A-su4-invar})
still apply except that we must replace in them
$i^{n(A)}$ by $e^{i\zeta(A)}$ and
$(-1)^{n(A)}$ by $e^{i 2\zeta(A)}$
for some $\zeta(A)\in\RR$.
Eqs.(\ref{eq-B-su4}) and (\ref{eq-B-su4-invar})
still apply except that we must replace in them
$i^{n(B)}$ by $e^{i\zeta(B)}$ and
$(-1)^{n(B)}$ by $e^{i 2\zeta(B)}$
for some $\zeta(B)\in\RR$.
\rproof Suppose $A\sim_R B$.
Eq.(\ref{eq-gist-rproof})
still applies so
there exists $\zeta\in \RR$
such that $A^{(2)}=e^{i\zeta} B^{(2)}$.
\lproof Suppose
$A^{(2)}=e^{i\zeta} B^{(2)}$.
Eq.(\ref{eq-gist-lproof})
still applies so $A\sim_R B$.
\end{enumerate}
\qed
By virtue of Theorem \ref{th-quad-invar},
the absolute value of the
entries of the matrix $A^{(2)}$
are a complete set of LO-RHS scalar
invariants for
$\nb=2$. Theorem \ref{th-quad-invar}(a)
reflects the fact that when $A,B\in SU(4)$,
since $A$ and $B$ must both
have unit determinant,
the only local operations
connecting $A$ and $B$ are either
elements of $SU(2)$ or $i$ or products of these.
Applying an $SU(2)$ gate
to the RHS of $A$
does not change $A^{(2)}$,
whereas applying $i$ changes
$A^{(2)}$ to its negative.
Now suppose $\nb=3$.
One can
represent any $A\in SU(8)$ as
\begin{eqnarray}\label{eq-su8-repres}
A &=& e^{i\frac{\pi}{4}n(A)}
\exp(i a_{jkr}\sigma_{X_j X_k X_r})\nonumber\\
&&\;\;\;\exp(i a''_{jk}\sigma_{1X_j X_k})
\exp(i a'_{jk}\sigma_{X_j 1 X_k})
\exp(i a_{jk}\sigma_{X_j X_k 1})\nonumber\\
&&\;\;\;\;\;\;\exp(i a''_{j}\sigma_{X_j 11})
\exp(i a'_{j}\sigma_{1X_j 1})
\exp(i a_{j}\sigma_{11X_j})
\;.
\end{eqnarray}
When the continuous parameters of $A$ are small,
\beq
A^{(2)} \approx e^{i\frac{\pi}{2}n(A)}[1 +
2i (a''_{jk}\sigma_{1X_j X_k}+
a'_{jk}\sigma_{X_j 1 X_k}+
a_{jk}\sigma_{X_j X_k 1})]
\;.
\eeq
This $A^{(2)}$ is independent of
the $a_{jkr}$ parameters.
So, for $A,B\in SU(8)$,
$A^{(2)}=\pm B^{(2)}$ or $A^{(2)}=\pm i B^{(2)}$ is
a necessary but not a sufficient
condition for $A\sim_R B$.
More invariants than
just $A^{(2)}$ are needed for $\nb>2$.
Higher order invariants can be generated
as follows. We will represent them diagrammatically
using the symbols defined in Fig.\ref{fig-invars-symb}.
Fig.\ref{fig-3bit-invar} shows
second and fourth order
invariants under LO-RHS for a circuit
with 3 bits. The same idea can
be used to generate invariants of order
equal to any even number,
for any number of qubits.
Fig.\ref{fig-show-are-invars}
explains why the circuits portrayed in
Fig.\ref{fig-3bit-invar} are invariant under LO-RHS.
Roughly speaking, if we apply a
$U\in SU(2)$ to the RHS of $A\in SU(8)$,
then, in the diagram of a fourth order invariant,
a copy of $U$ must be inserted
next to each of the 4 copies of $A$.
And these 4 copies of $U$ annihilate
each other. This paper will only
use the second order invariant $A^{(2)}$. We
will not even use Group Theory in this paper.
For information on the group
theoretic underpinnings of quantum circuit invariants,
see, for example, Ref.\cite{Rai}.
\begin{figure}[h]
\begin{center}
\epsfig{file=invars-symbols.eps, height=.5in}
\caption{Key to symbols used in
Figs.\ref{fig-3bit-invar} and \ref{fig-show-are-invars}.
$A\in SU(\ns)$.}
\label{fig-invars-symb}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\epsfig{file=3bit-invar.eps, height=4in}
\caption{Second and fourth order
invariants under LO-RHS for a circuit
with 3 bits. $A\in SU(8)$.}
\label{fig-3bit-invar}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\epsfig{file=show-are-invars.eps, height=1.25in}
\caption{Why the diagrams shown in
Fig.\ref{fig-3bit-invar} are
invariant under LO-RHS.
$A\in SU(8)$ and $U\in SU(2)$.}
\label{fig-show-are-invars}
\end{center}
\end{figure}
\section{Dressed CNOTs
\\{\footnotesize\tt[
dr11.m,
dr110.m,
dr011.m,
dr101.m,
]}}
\label{sec-dc-nots}
In this section, we define
dressed CNOTs, a simple yet
powerful generalization of the
standard CNOT. We also discuss some simple
properties of dressed CNOTs
that will be used in subsequent
sections.
The {\bf controlled NOT (CNOT)}
with control bit 1 and target bit 0, is defined by
\beq
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\timesgate\qwx[1]
&\qw
\\
&\dotgate
&\qw
}
\end{array}
= (-1)^{n_X(0)n(1)}
=\sigma_X(0)^{n(1)}
\;.
\eeq
Now suppose $U$ and $V$
are arbitrary elements of $SU(2)$.
Define $\hata$ and $\hata'$ by
$U\sigx U^\dagger = \sigma_\hata$
and
$V\sigz V^\dagger = \sigma_{\hata'}$.
Then a {\bf dressed CNOT (DC-NOT)}
connecting bits 0 and 1, is defined by
\beq
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hata}\qwx[1]
&\qw
\\
&\ovalgate{\hata'}
&\qw
}
\end{array}
=
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\gate{U}
&\timesgate\qwx[1]
&\gate{U^\dagger}
&\qw
\\
&\gate{V}
&\dotgate
&\gate{V^\dagger}
&\qw
}
\end{array}
=(-1)^{n_\hata(0) n_{\hata'}(1)}
=\sigma_\hata(0)^{n_{\hata'}(1)}
=\sigma_{\hata'}(1)^{n_{\hata}(0)}
\;.
\eeq
We will refer to the vectors
$\hata'$ and $\hata$ as the
{\bf defining vectors} of the DC-NOT.
Sometimes in this paper, we will draw a circuit
containing one or more DC-NOTs whose oval nodes
are empty.
By this we will mean that the omitted defining vectors
are arbitrary and their precise value
is unimportant in that context.
Consider the wire corresponding to
bit $\mu$ in a quantum circuit.
Within the bit-$\mu$ wire,
consider two adjacent
oval nodes belonging to
two different DC-NOTs:
$
\begin{array}{c}
\Qcircuit @C=1em @R=.5em @!R{
&\ovalgate{\hata}
&\ovalgate{\hatb}
&\qw
}
\end{array}
\;.
$
If $\hatb\parallel \hata$,
we will say there is a {\bf breach}
at that position in the bit-$\mu$ wire.
If $\hatb\perp \hata$,
we will say there is a {\bf foil}
at that position in the bit-$\mu$ wire.
\begin{theo}
\beq
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hata}\qwx[1]
&\qw
\\
&\ovalgate{\hata'}
&\qw
}
\end{array}
=
\frac{1}{2}
( 1 + \sigma_{1,\hata} +
\sigma_{\hata',1} - \sigma_{\hata',\hata})
\;.
\eeq
\end{theo}
\proof
\beq
\sigma_{\hata'}(1)^{n_{\hata}(0)}
=
\sigma_{\hata'}(1)n_\hata(0) + \nbar_\hata(0)
=
\frac{1}{2}
( 1 + \sigma_{1,\hata} +
\sigma_{\hata',1} - \sigma_{\hata',\hata})
\;.
\eeq
\qed
\begin{theo}
\beq
\left(
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hata}\qwx[1]
&\qw
\\
&\ovalgate{\hata'}
&\qw
}
\end{array}
\right)^2
=1
\;.
\eeq
\end{theo}
\proof
$\sigma_\hata(0)^{2n_{\hata'}(1)}=1$.
\qed
\begin{theo}
\beq
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{-\hata}\qwx[1]
&\qw
\\
&\ovalgate{\hata'}
&\qw
}
\end{array}
=
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\qw
&\ovalgate{\hata}\qwx[1]
&\qw
\\
&\gate{\sigma_{\hata'}}
&\ovalgate{\hata'}
&\qw
}
\end{array}
\;.
\label{eq-dcnot-with-neg}
\eeq
\end{theo}
\proof
\beq
[-\sigma_\hata(0)]^{n_{\hata'}(1)}=
(-1)^{n_{\hata'}(1)}\sigma_\hata(0)^{n_{\hata'}(1)}=
\sigma_{\hata'}(1)\sigma_\hata(0)^{n_{\hata'}(1)}
\;.
\eeq
\qed
In subsequent sections, we
will often need to calculate
the effect of a similarity transformation
produced by pre and post multiplying an operator
by the same DC-NOT. The next
theorem will be useful
for performing such calculations.
\begin{theo}
\beq
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatb}\qwx[1]
&\gate{\sigma_{\veca}}
&\ovalgate{\hatb}\qwx[1]
&\qw
\\
&\ovalgate{\hatb'}
&\qw
&\ovalgate{\hatb'}
&\qw
}
\end{array}
=
\sigma_{1,\along{\veca}{\hatb}}+
\sigma_{\hatb',\across{\veca}{\hatb}}
\;.
\label{eq-sim-trans-of-sig-veca}
\eeq
\end{theo}
\proof
Clearly,
\beq
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatb}\qwx[1]
&\gate{\sigma_{\along{\veca}{\hatb}}}
&\ovalgate{\hatb}\qwx[1]
&\qw
\\
&\ovalgate{\hatb'}
&\qw
&\ovalgate{\hatb'}
&\qw
}
\end{array}
=
\sigma_{1,\along{\veca}{\hatb}}
\;.
\label{eq-sim-transf-parallel-part}
\eeq
On the other hand,
\beq
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatb}\qwx[1]
&\gate{\sigma_{\across{\veca}{\hatb}}}
&\ovalgate{\hatb}\qwx[1]
&\qw
\\
&\ovalgate{\hatb'}
&\qw
&\ovalgate{\hatb'}
&\qw
}
\end{array}
=
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\gate{\sigma_{\across{\veca}{\hatb}}}
&\ovalgate{-\hatb}\qwx[1]
&\ovalgate{\hatb}\qwx[1]
&\qw
\\
&\qw
&\ovalgate{\hatb'}
&\ovalgate{\hatb'}
&\qw
}
\end{array}
=
\sigma_{\hatb',\across{\veca}{\hatb}}
\;.
\label{eq-sim-transf-perp-part}
\eeq
\qed
\section{Wake Identities}
\label{sec-wake-ids}
In this section we prove what we call
a ``wake identity". We call it thus
because in it, one DC-NOT is pushed
through another, producing
a third DC-NOT as its ``wake".
\begin{theo}
Suppose $\hata'\perp\hatb'$.
\begin{subequations}
\begin{eqnarray}
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatb}\qwx[1]
&\qw
&\qw
&\qw
\\
&\ovalgate{\hatb'}
&\foil
&\ovalgate{\hata'}\qwx[1]
&\qw
\\
&\qw
&\qw
&\ovalgate{\hata''}
&\qw
}
\end{array}
&=&
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\qw
&\ovalgate{\hatb}\qwx[1]
&\ovalgate{\hatb}\qwx[2]
&\qw
\\
&\ovalgate{\hata'}\qwx[1]
&\ovalgate{\hatb'}
&\qw
&\qw
\\
&\ovalgate{\hata''}
&\qw
&\ovalgate{\hata''}
&\qw
}
\end{array}
\label{eq-wake-on-right}
\\
&=&
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatb}\qwx[2]
&\qw
&\ovalgate{\hatb}\qwx[1]
&\qw
\\
&\qw
&\ovalgate{\hata'}\qwx[1]
&\ovalgate{\hatb'}
&\qw
\\
&\ovalgate{\hata''}
&\ovalgate{\hata''}
&\qw
&\qw
}
\end{array}
\;.
\end{eqnarray}
\end{subequations}
\end{theo}
\proof
\beqa
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\qw
&\qw
&\ovalgate{\hatb}\qwx[1]
&\qw
&\qw
&\qw
\\
&\ovalgate{\hata'}\qwx[1]
&\foil
&\ovalgate{\hatb'}
&\foil
&\ovalgate{\hata'}\qwx[1]
&\qw
\\
&\ovalgate{\hata''}
&\qw
&\qw
&\qw
&\ovalgate{\hata''}
&\qw
}
\end{array}
&=&
\sigma_{\hata'}(1)^{n_{\hata''}(2)}
\sigma_{\hatb'}(1)^{n_{\hatb}(0)}
\sigma_{\hata'}(1)^{n_{\hata''}(2)}
\\
&=&
[(-1)^{n_{\hata''}(2)}
\sigma_{\hatb'}(1)]^{n_{\hatb}(0)}
\\
&=&
(-1)^{n_{\hata''}(2)n_{\hatb}(0)}
\sigma_{\hatb'}(1)^{n_{\hatb}(0)}
\\
&=&
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatb}\qwx[2]
&\ovalgate{\hatb}\qwx[1]
&\qw
\\
&\qw
&\ovalgate{\hatb'}
&\qw
\\
&\ovalgate{\hata''}
&\qw
&\qw
}
\end{array}
\;.
\eeqa
\qed
\section{Swapper Identities
\\{\footnotesize\tt[
swap\_t3.m,
test\_swap\_t3.m
]}}
\label{sec-swapper-ids}
In this section, we discuss certain
DC-NOT identities associated with
the qubit Exchange Operator (a.k.a.
Swap Operator or Swapper).
We will represent the Swapper by a double
arrow connecting the two qubits being swapped.
By definition, the Swapper satisfies
\beq
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\gate{U}
&\uarrowgate\qwx[1]
&\qw
\\
&\qw
&\darrowgate
&\qw
}
\end{array}
=
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\qw
&\uarrowgate\qwx[1]
&\qw
\\
&\qw
&\darrowgate
&\gate{U}
}
\end{array}
\;
\eeq
for any $U\in U(2)$.
As is well known (for a proof,
see, for example, Ref.\cite{Paulinesia}),
the Swapper can be expressed
as a product of 3 CNOTs:
\beq
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\uarrowgate\qwx[1]
&\qw
\\
&\darrowgate
&\qw
}
\end{array}
=
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\timesgate\qwx[1]
&\dotgate\qwx[1]
&\timesgate\qwx[1]
&\qw
\\
&\dotgate
&\timesgate
&\dotgate
&\qw
}
\end{array}
\;.
\eeq
The next theorem shows that
the Swapper can also be expressed as a
product of 3 DC-NOTs.
\begin{theo}
Suppose $\hata\perp\hatb$,
$U\in SU(2)$,
$U^\dagger \sigma_{\hata} U = \sigma_{\hata'}$,
and
$U^\dagger \sigma_{\hatb} U = \sigma_{\hatb'}$.
\beqa
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\uarrowgate\qwx[1]
&\qw
\\
&\darrowgate
&\qw
}
\end{array}
&=&
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hata}\qwx[1]
&\foil
&\ovalgate{\hatb}\qwx[1]
&\foil
&\ovalgate{\hata}\qwx[1]
&\qw
\label{eq-swapper-two-vecs}
\\
&\ovalgate{\hatb}
&\foil
&\ovalgate{\hata}
&\foil
&\ovalgate{\hatb}
&\qw
}
\end{array}
\\
&=&
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hata}\qwx[1]
&\foil
&\ovalgate{\hatb}\qwx[1]
&\foil
&\ovalgate{\hata}\qwx[1]
&\gate{U}
&\qw
\\
&\ovalgate{\hatb'}
&\foil
&\ovalgate{\hata'}
&\foil
&\ovalgate{\hatb'}
&\gate{U^\dagger}
&\qw
}
\end{array}
\label{eq-swapper-four-vecs}
\;.
\eeqa
\end{theo}
\proof
Since $\hata\perp\hatb$,
there exists $V\in SU(2)$
such that
$V^\dagger \sigx V = \sigma_{\hata}$
and $V^\dagger \sigz V = \sigma_{\hatb}$.
Then
\beq
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\uarrowgate\qwx[1]
&\qw
\\
&\darrowgate
&\qw
}
\end{array}
=
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\gate{V^\dagger}
&\uarrowgate\qwx[1]
&\gate{V}
&\qw
\\
&\gate{V^\dagger}
&\darrowgate
&\gate{V}
&\qw
}
\end{array}
=
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\gate{V^\dagger}
&\timesgate\qwx[1]
&\dotgate\qwx[1]
&\timesgate\qwx[1]
&\gate{V}
&\qw
\\
&\gate{V^\dagger}
&\dotgate
&\timesgate
&\dotgate
&\gate{V}
&\qw
}
\end{array}
=
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hata}\qwx[1]
&\ovalgate{\hatb}\qwx[1]
&\ovalgate{\hata}\qwx[1]
&\qw
\\
&\ovalgate{\hatb}
&\ovalgate{\hata}
&\ovalgate{\hatb}
&\qw
}
\end{array}
\;.
\eeq
This proves Eq.(\ref{eq-swapper-two-vecs}).
Eq.(\ref{eq-swapper-four-vecs}) follows from
\beq
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\uarrowgate\qwx[1]
&\qw
\\
&\darrowgate
&\qw
}
\end{array}
=
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\qw
&\uarrowgate\qwx[1]
&\qw
&\gate{U}
&\qw
\\
&\gate{U^\dagger}
&\darrowgate
&\gate{U}
&\gate{U^\dagger}
&\qw
}
\end{array}
\;.
\eeq
\qed
We will refer to the next
identity, Eq.(\ref{eq-2thirds}),
as the 2/3-Swapper identity,
because its LHS contains 2/3 of a
Swapper.
\begin{figure}[h]
\begin{center}
\epsfig{file=qxy-pzy.eps, height=1.75in}
\caption{Orientation of vectors $\hatqxy'$
and $\hatpzy'$. Note that
$H\sigma_{\hatpzy'}H = \sigma_{\hatqxy'}$.
The same picture, but
omitting all primes, describes $\hatqxy$
and $\hatpzy$.}
\label{fig-qxy-pzy}
\end{center}
\end{figure}
\begin{theo}\label{th-2thirds}
For any $\alpha\in\RR$,
\begin{subequations}
\label{eq-2thirds}
\begin{eqnarray}
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatx}\qwx[1]
&\ovalgate{\hatz}\qwx[1]
&\qw
\\
&\ovalgate{\hatx}
&\ovalgate{\hatz}
&\qw
}
\end{array}
&=&
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatqxy}\qwx[1]
&\ovalgate{\hatz}\qwx[1]
&\gate{U}
&\qw
\\
&\ovalgate{\hatqxy'}
&\ovalgate{\hatz}
&\gate{U'}
&\qw
}
\end{array}
\label{eq-2thirds-a}
\\
&=&
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\gate{U}
&\ovalgate{\hatx}\qwx[1]
&\ovalgate{\hatpzy'}\qwx[1]
&\qw
\\
&\gate{U'}
&\ovalgate{\hatx}
&\ovalgate{\hatpzy}
&\qw
}
\end{array}
\label{eq-2thirds-b}
\;,
\end{eqnarray}
\end{subequations}
where (see Fig.\ref{fig-qxy-pzy})
\beq
\hatpzy = c_\alpha \hatz + s_\alpha \haty
\;,\;\;
\hatpzy' = (\hatpzy)_{\alpha\rarrow \alpha'}
\;,
\eeq
\beq
\hatqxy = c_\alpha \hatx - s_\alpha \haty
\;,\;\;
\hatqxy' = (\hatqxy)_{\alpha\rarrow \alpha'}
\;,
\eeq
($p$ vector has a positive sign
in front of $s_\alpha$,
$q$ vector has a negative one)
and
\beq
U = e^{i \frac{\alpha}{2} \sigz}
e^{-i \frac{\alpha'}{2} \sigx}
\;,\;\;
U' = (U)_{\alpha\darrow \alpha'}
\;.
\label{eq-2thirds-u-def}
\eeq
Note that the left-hand sides
of Eqs.(\ref{eq-2thirds-a})
and (\ref{eq-2thirds-b}) are independent
of the two angles $\alpha$ and $\alpha'$;
only their
right-hand sides depend on these angles.
\end{theo}
\proof
From the expression of Swapper as
a product of 3 CNOTs, we get
\beq
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatz}\qwx[1]
&\ovalgate{\hatx}\qwx[1]
&\ovalgate{\hatz}\qwx[1]
&\qw
\\
&\ovalgate{\hatz}
&\ovalgate{\hatx}
&\ovalgate{\hatz}
&\qw
}
\end{array}
=
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\uarrowgate\qwx[1]
&\gate{H}
&\qw
\\
&\darrowgate
&\gate{H}
&\qw
}
\end{array}
\;.
\eeq
From Fig.\ref{fig-qxy-pzy},
it follows that
\beq
\sigma_{\hatqxy} =
e^{i \frac{\alpha}{2} \sigz}
\sigma_\hatx
e^{-i \frac{\alpha}{2} \sigz}
\;.
\eeq
Thus
\beqa
\lefteqn{
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatz}\qwx[1]
&\ovalgate{\hatqxy}\qwx[1]
&\ovalgate{\hatx}\qwx[1]
&\ovalgate{\hatz}\qwx[1]
&\qw
\\
&\ovalgate{\hatz}
&\ovalgate{\hatqxy'}
&\ovalgate{\hatx}
&\ovalgate{\hatz}
&\qw
}
\end{array}
=
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatz}\qwx[1]
&\ovalgate{\hatqxy}\qwx[1]
&\ovalgate{\hatz}\qwx[1]
&\uarrowgate\qwx[1]
&\gate{H}
&\qw
\\
&\ovalgate{\hatz}
&\ovalgate{\hatqxy'}
&\ovalgate{\hatz}
&\darrowgate
&\gate{H}
&\qw
}
\end{array}
}
\\
&=&
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\gate{e^{i \frac{\alpha}{2} \sigz}}
&\ovalgate{\hatz}\qwx[1]
&\ovalgate{\hatx}\qwx[1]
&\ovalgate{\hatz}\qwx[1]
&\gate{e^{-i \frac{\alpha}{2} \sigz}}
&\uarrowgate\qwx[1]
&\gate{H}
&\qw
\\
&\gate{e^{i \frac{\alpha'}{2} \sigz}}
&\ovalgate{\hatz}
&\ovalgate{\hatx}
&\ovalgate{\hatz}
&\gate{e^{-i \frac{\alpha'}{2} \sigz}}
&\darrowgate
&\gate{H}
&\qw
}
\end{array}
\\
&=&
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\gate{e^{i \frac{\alpha}{2} \sigz}}
&\uarrowgate\qwx[1]
&\gate{H}
&\gate{e^{-i \frac{\alpha}{2} \sigz}}
&\uarrowgate\qwx[1]
&\gate{H}
&\qw
\\
&\gate{e^{i \frac{\alpha'}{2} \sigz}}
&\darrowgate
&\gate{H}
&\gate{e^{-i \frac{\alpha'}{2} \sigz}}
&\darrowgate
&\gate{H}
&\qw
}
\end{array}
\\
&=&
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\freegate{
e^{i \frac{\alpha}{2} \sigz}
e^{-i \frac{\alpha'}{2} \sigx}
}
\\
&\freegate{
e^{i \frac{\alpha'}{2} \sigz}
e^{-i \frac{\alpha}{2} \sigx}
}
}
\end{array}
\;.
\eeqa
\qed
The next theorem follows immediately
from the previous one, by a change of basis.
\begin{theo}\label{th-2thirds-ab}
Suppose $\alpha\in \RR$,
$\hata\perp\hatb$, and $\hata'\perp\hatb'$. Then
\beqa
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatb}\qwx[1]
&\foil
&\ovalgate{\hata}\qwx[1]
&\qw
\\
&\ovalgate{\hatb'}
&\foil
&\ovalgate{\hata'}
&\qw
}
\end{array}
&=&
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatb_f}\qwx[1]
&\foil
&\ovalgate{\hata}\qwx[1]
&\gate{U}
&\qw
\\
&\ovalgate{\hatb'_f}
&\foil
&\ovalgate{\hata'}
&\gate{U'}
&\qw
}
\end{array}
\\
&=&
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\gate{U}
&\ovalgate{\hatb}\qwx[1]
&\foil
&\ovalgate{\hata'_f}\qwx[1]
&\qw
\\
&\gate{U'}
&\ovalgate{\hatb'}
&\foil
&\ovalgate{\hata_f}
&\qw
}
\end{array}
\;,
\eeqa
where
\beq
\hata_f = c_\alpha \hata + s_\alpha \manyx{\hata\hatb}
\;,\;\;
\hata'_f = c_{\alpha'} \hata' + s_{\alpha'} \manyx{\hata'\hatb'}
\;,
\eeq
\beq \hatb_f = c_\alpha \hatb - s_\alpha \manyx{\hata\hatb}
\;,\;\; \hatb'_f = c_{\alpha'} \hatb' - s_{\alpha'}
\manyx{\hata'\hatb'}
\;,
\eeq
and
\beq
U = e^{i \frac{\alpha}{2} \sigma_\hata}
e^{-i \frac{\alpha'}{2} \sigma_\hatb}
\;,\;\;
U' = e^{i \frac{\alpha'}{2} \sigma_{\hata'}}
e^{-i \frac{\alpha}{2} \sigma_{\hatb'}}
\;.
\eeq
\end{theo}
\proof
Just change basis in the space
where bit 0 (ditto, bit 1) lives so that
$(\hatx,\haty,\hatz)$
is replaced by
$(\hatb,\manyx{\hata\hatb},\hata)$
(ditto, $(\hatb',\manyx{\hata'\hatb'},\hata')$).
\qed
We will refer to the next identity,
Eq.(\ref{eq-2thirds-split}),
as the 1/3 Swapper identity.
\begin{theo}\label{th-2thirds-split}
\beq
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatx}\qwx[1]
&\ovalgate{\hatqxy}\qwx[1]
&\qw
\\
&\ovalgate{\hatx}
&\ovalgate{\hatqxy'}
&\qw
}
\end{array}
=
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatz}\qwx[1]
&\ovalgate{\hatpzy'}\qwx[1]
&\gate{U^{\dagger}}
&\qw
\\
&\ovalgate{\hatz}
&\ovalgate{\hatpzy}
&\gate{U^{'\dagger}}
&\qw
}
\end{array}
\;,
\label{eq-2thirds-split}
\eeq
where all variables are defined as in
Theorem \ref{th-2thirds}.
\end{theo}
\proof
From the Hermitian conjugate of
Eq.(\ref{eq-2thirds-a}), one gets
\beq
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatz}\qwx[1]
&\ovalgate{\hatx}\qwx[1]
&\ovalgate{\hatqxy}\qwx[1]
&\qw
\\
&\ovalgate{\hatz}
&\ovalgate{\hatx}
&\ovalgate{\hatqxy'}
&\qw
}
\end{array}
=
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\gate{U^\dagger}\qwx[1]
&\ovalgate{\hatz}\qwx[1]
&\qw
\\
&\gate{U^{'\dagger}}
&\ovalgate{\hatz}
&\qw
}
\end{array}
\;.
\eeq
Let $LHS$ and $RHS$ stand for the
left and right hand sides of
Eq.(\ref{eq-2thirds-split}).
Pre-multiplying
both sides of the last
equation by
$
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\freeovalgate{\hatz}\qwx[1]
\\
&\freeovalgate{\hatz}
}
\end{array}
$
yields
\beq
LHS =
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatz}\qwx[1]
&\gate{U^\dagger}
&\ovalgate{\hatz}\qwx[1]
&\qw
\\
&\ovalgate{\hatz}
&\gate{U^{'\dagger}}
&\ovalgate{\hatz}
&\qw
}
\end{array}
=
RHS
\;.
\eeq
\qed
\section{DC-NOT Similarity Transformation Identities
\\{\footnotesize\tt[
sim\_trans\_t4.m,
test\_sim\_trans\_t4.m
]}}
\label{sec-sim-trans-ids}
In this section, we present
some identities which contain
a similarity transformation
produced by pre and post multiplying
an operator by the same DC-NOT.
We will refer to the next theorem as
the DC-NOT similarity transformation identity.
\begin{theo}\label{th-sim-trans}
For any $\alpha, \lam\in \RR$,
\beq
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatx}\qwx[1]
&\gate{
c_\alpha \sigx + s_\alpha \sigz
}
&\ovalgate{\hatx}\qwx[1]
&\qw
\\
&\ovalgate{\hatx}
&\gate{
s_\alpha \sigx + c_\alpha \sigz
}
&\ovalgate{\hatx}
&\qw
}
\end{array}
=
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatqxy}\qwx[1]
&\gate{
c_\alpha \sigma_\hatqxy + s_\alpha \sigz
}
&\ovalgate{\hatqxy}\qwx[1]
&\qw
\\
&\ovalgate{\hatqxy}
&\gate{
s_\alpha \sigma_\hatqxy + c_\alpha \sigz
}
&\ovalgate{\hatqxy}
&\qw
}
\end{array}
\;,
\label{eq-sim-trans-id}
\eeq
where $\hatqxy = c_\lam \hatx - s_\lam \haty$.
Note that the LHS of Eq.(\ref{eq-sim-trans-id})
equals its RHS evaluated at $\lam=0$.
\end{theo}
\proof
Since
\beq
\manyx{\hatqxy\hatz} =
-(c_\lam\haty + s_\lam \hatx)
\;,
\eeq
it follows that
\beqa
\sigma_{\hatqxy\hatqxy}
+ \sigma_{\manyx{\hatqxy\hatz}\manyx{\hatqxy\hatz}}
&=&
\sigma_{
c_\lam\hatx - s_\lam \haty,
c_\lam\hatx - s_\lam \haty
}
+ \sigma_{
c_\lam\haty + s_\lam \hatx,
c_\lam\haty + s_\lam \hatx
}
\\
&=&
\sigma_{\hatx\hatx} + \sigma_{\haty\haty}
\;.
\eeqa
Let LHS and RHS denote the left-hand side
and right-hand side, respectively,
of Eq.(\ref{eq-sim-trans-id}). Then,
using Eq.(\ref{eq-sim-trans-of-sig-veca}),
\beqa
RHS &=&
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\freeovalgate{\hatqxy}\qwx[1]
\\
&\freeovalgate{\hatqxy}
}
\end{array}
(s_\alpha \sigma_{\hatq_{xy},1}
+c_\alpha\sigma_{\hatz,1})
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\freeovalgate{\hatqxy}\qwx[1]
\\
&\freeovalgate{\hatqxy}
}
\end{array}
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\freeovalgate{\hatqxy}\qwx[1]
\\
&\freeovalgate{\hatqxy}
}
\end{array}
(c_\alpha \sigma_{1,\hatq_{xy}}
+s_\alpha\sigma_{1,\hatz})
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\freeovalgate{\hatqxy}\qwx[1]
\\
&\freeovalgate{\hatqxy}
}
\end{array}
\\
&=&
(s_\alpha \sigma_{\hatqxy1}
+ c_\alpha \sigma_{\hatz\hatqxy})
(c_\alpha \sigma_{1,\hatqxy}
+ s_\alpha \sigma_{\hatqxy\hatz})
\\
&=&
s_\alpha c_\alpha
(\sigma_{\hatqxy\hatqxy}
+ \sigma_{\manyx{\hatqxy\hatz}\manyx{\hatqxy\hatz}})
+ c^2_\alpha \sigma_{\hatz 1}
+ s^2_\alpha \sigma_{1\hatz}
\\
&=&
s_\alpha c_\alpha
(\sigma_{\hatx\hatx}
+ \sigma_{\haty\haty})
+ c^2_\alpha \sigma_{\hatz 1}
+ s^2_\alpha \sigma_{1\hatz}
\\
&=& LHS
\;.
\eeqa
\qed
It is convenient to define, for any $\xi\in \RR$,
\beq
\hatp_{w_1,w_2}^\xi = c_\xi\hatw_1 + s_\xi\hatw_2
\;,\;\;
\hatq_{w_1,w_2}^\xi = c_\xi\hatw_1 - s_\xi\hatw_2
\;.
\label{eq-general-p-q-def}
\eeq
(The $\hatp$ vectors have a positive sign
in front of the sine function whereas the $\hatq$
vectors have a negative one).
\begin{figure}[h]
\begin{center}
\epsfig{file=split-sim-trans.eps, height=2.75in}
\caption{Variables used in
Theorem \ref{th-split-sim-trans}.}
\label{fig-split-sim-trans}
\end{center}
\end{figure}
The next theorem follows from
the DC-NOT similarity transformation identity.
\begin{theo}\label{th-split-sim-trans}
For any $\phi,\lam\in \RR$,
\beq
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatpzx^\phi}\qwx[1]
&\ovalgate{\hatx}\qwx[1]
&\ovalgate{\hatqxy^\lam}\qwx[1]
&\qw
\\
&\ovalgate{\hatqzx^\phi}
&\ovalgate{\hatx}
&\ovalgate{\hatqxy^\lam}
&\qw
}
\end{array}
=
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hata_f}\qwx[1]
&\gate{U}
&\qw
\\
&\ovalgate{\hata'_f}
&\gate{U'}
&\qw
}
\end{array}
\;,
\label{eq-split-sim-trans-id}
\eeq
where (see Fig.\ref{fig-split-sim-trans})
\beq
\hata_f = c_\lam \hatpzx^\phi + s_\lam \haty
\;,\;\;
\hata'_f = c_\lam \hatqzx^\phi + s_\lam \haty
\;,
\eeq
and
\beq
U = (c_\alpha \sigx + s_\alpha\sigz)
(c_\alpha \sigma_{\hatqxy^\lam} + s_\alpha\sigz)
\;,\;\;
U'=(U)_{\alpha\rarrow \beta}
\;,
\eeq
where
\beq
2\alpha = \frac{\pi}{2} - \phi
\;,\;\;
2\beta = \pi -2\alpha
\;.
\eeq
\end{theo}
\proof
From Fig.\ref{fig-split-sim-trans},
it follows that
\beq
\hatpzx^\phi =
e^{i\alpha\sigy}\sigx e^{-i\alpha\sigy}
\;,
\eeq
and
\beq
\hatqzx^\phi =
e^{i\beta\sigy}\sigx e^{-i\beta\sigy}
\;.
\eeq
Let LHS and RHS denote the left-hand side and
right-hand side, respectively,
of Eq.(\ref{eq-split-sim-trans-id}). Then
\beq
LHS =
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\gate{e^{i\alpha\sigy}}
&\ovalgate{\hatx}\qwx[1]
&\gate{e^{-i\alpha\sigy}}
&\ovalgate{\hatx}\qwx[1]
&\ovalgate{\hatqxy^\lam}\qwx[1]
&\qw
\\
&\gate{ e^{i\beta\sigy}}
&\ovalgate{\hatx}
&\gate{e^{-i\beta\sigy}}
&\ovalgate{\hatx}
&\ovalgate{\hatqxy^\lam}
&\qw
}
\end{array}
\;,
\eeq
and
\beq
RHS =
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\gate{e^{i\alpha\sigy}}
&\ovalgate{\hatpxy^\lam}\qwx[1]
&\gate{ e^{-i\alpha\sigy}}
&\gate{U}
&\qw
\\
&\gate{e^{i\beta\sigy}}
&\ovalgate{\hatpxy^\lam}
&\gate{ e^{-i\beta\sigy}}
&\gate{U'}
&\qw
}
\end{array}
\;.
\eeq
Therefore, Eq.(\ref{eq-split-sim-trans-id}) is
equivalent to the assertion that
\beq
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatx}\qwx[1]
&\gate{e^{-i\alpha\sigy}}
&\ovalgate{\hatx}\qwx[1]
&\qw
\\
&\ovalgate{\hatx}
&\gate{e^{-i\beta\sigy}}
&\ovalgate{\hatx}
&\qw
}
\end{array}
=
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatpxy^\lam}\qwx[1]
&\gate{e^{-i\alpha\sigy}U}
&\ovalgate{\hatqxy^\lam}\qwx[1]
&\qw
\\
&\ovalgate{\hatpxy^\lam}
&\gate{e^{-i\beta\sigy} U'}
&\ovalgate{\hatqxy^\lam}
&\qw
}
\end{array}
\;.
\eeq
Now pre-multiply each side of the last
equation by $\sigxx$
\beq
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatx}\qwx[1]
&\gate{c_\alpha\sigx + s_\alpha\sigz}
&\ovalgate{\hatx}\qwx[1]
&\qw
\\
&\ovalgate{\hatx}
&\gate{c_\beta\sigx + s_\beta\sigz}
&\ovalgate{\hatx}
&\qw
}
\end{array}
=
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatqxy^\lam}\qwx[1]
&\gate{c_\alpha\sigma_{\hatqxy^\lam} + s_\alpha\sigz}
&\ovalgate{\hatqxy^\lam}
&\qw
\\
&\ovalgate{\hatqxy^\lam}
&\gate{c_\beta\sigma_{\hatqxy^\lam} + s_\beta\sigz}
&\ovalgate{\hatqxy^\lam}
&\qw
}
\end{array}
\;.
\eeq
The preceding equation
follows from Theorem \ref{th-sim-trans}
and the fact that $\alpha + \beta = \pi/2$.
\qed
The next theorem is a simple
variation of the previous one.
(The left-hand sides
of Eqs.(\ref{eq-split-sim-trans-id})
and (\ref{eq-split-sim-trans-id2})
differ only in that one circuit has two
$q$'s in the
bit-1 wire
whereas the other circuit has two
$p$'s.)
\begin{theo}\label{th-split-sim-trans2}
\beq
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatpzx^\phi}\qwx[1]
&\ovalgate{\hatx}\qwx[1]
&\ovalgate{\hatqxy^\lam}\qwx[1]
&\qw
\\
&\ovalgate{\hatpzx^\phi}
&\ovalgate{\hatx}
&\ovalgate{\hatpxy^\lam}
&\qw
}
\end{array}
=
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\gate{\sigz}
&\ovalgate{\hata'_f}\qwx[1]
&\gate{U'\sigz}
&\qw
\\
&\qw
&\ovalgate{\hata_f}
&\gate{U\sigma_{\hatqxy^\lam}\sigx}
&\qw
}
\end{array}
\;,
\label{eq-split-sim-trans-id2}
\eeq
where all variables are
defined as in Theorem \ref{th-split-sim-trans}.
\end{theo}
\proof
Let $LHS_{\ref{eq-split-sim-trans-id}}$
represent the left-hand side
of Eq.(\ref{eq-split-sim-trans-id}),
and $LHS_{\ref{eq-split-sim-trans-id2}}$, the left-hand
side of Eq.(\ref{eq-split-sim-trans-id2}).
Then
\beqa
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\qw
&\qw
\\
&\gate{\sigz}
&\qw
}
\end{array}
LHS_{\ref{eq-split-sim-trans-id}}
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\gate{\sigma_{\hatqxy^\lam}\sigx}
&\qw
\\
&\gate{\sigz}
&\qw
}
\end{array}
&=&
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatpzx^\phi}\qwx[1]
&\ovalgate{\hatx}\qwx[1]
&\ovalgate{\hatqxy^\lam}\qwx[1]
&\gate{\sigma_{\hatqxy^\lam}\sigx}
&\qw
\\
&\ovalgate{\hatpzx^\phi}
&\ovalgate{-\hatx}
&\ovalgate{-\hatqxy^\lam}
&\qw
}
\end{array}
\\
&=&
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatpzx^\phi}\qwx[1]
&\ovalgate{\hatx}\qwx[1]
&\gate{\sigx}
&\ovalgate{\hatqxy^\lam}\qwx[1]
&\gate{(\sigma_{\hatqxy^\lam})^2\sigx}
&\qw
\\
&\ovalgate{\hatpzx^\phi}
&\ovalgate{\hatx}
&\qw
&\ovalgate{\hatqxy^\lam}
&\qw
}
\end{array}
\\
&=&
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\uarrowgate\qwx[1]
&\qw
\\
&\darrowgate
&\qw
}
\end{array}
LHS_{\ref{eq-split-sim-trans-id2}}
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\uarrowgate\qwx[1]
&\qw
\\
&\darrowgate
&\qw
}
\end{array}
\;.
\eeqa
The right-hand sides of
Eqs.(\ref{eq-split-sim-trans-id})
and
(\ref{eq-split-sim-trans-id2})
must be related
in the same way as their left-hand sides.
\qed
\section{LO-RHS Invariant for Circuits with\\
Two Qubits, and Multiple DC-NOTs}
\label{sec-two-bit-dcnot-rhs-invariants}
In previous sections we defined
the LO-RHS invariant $A^{(2)}$
for any $A\in U(\ns)$.
We also defined DC-NOTs and discussed
some of their properties. In this section,
we combine these two concepts:
we calculate $A^{(2)}$
when $A$ is a product of one or more DC-NOTs
acting on the same two qubits.
Henceforth, we will denote
the product of $r$
DC-NOTs (all acting on the same two qubits)
by the symbol
$\calg_r$ followed by a list
(enclosed in parenthesis) of
its arguments. Sometimes,
if this doesn't lead to confusion,
its list
of arguments will be omitted. Thus,
\beq
\calg_r
\left(
\begin{array}{cccc}
\hata_r & \cdots & \hata_2 & \hata_1\\
\hata'_r & \cdots & \hata'_2 & \hata'_1
\end{array}
\right)=
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hata_r}\qwx[1]
&\qw
\\
&\ovalgate{\hata'_r}
&\qw
}
\end{array}
\cdots
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hata_2}\qwx[1]
&\ovalgate{\hata_1}\qwx[1]
&\qw
\\
&\ovalgate{\hata'_2}
&\ovalgate{\hata'_1}
&\qw
}
\end{array}
\;.
\eeq
The determinant of $\calg_r$
equals either plus or minus one. Indeed,
\beq
\det
\left(
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hata}\qwx[1]
&\qw
\\
&\ovalgate{\hata'}
&\qw
}
\end{array}
\right)
=
\det
\left(
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\timesgate\qwx[1]
&\qw
\\
&\dotgate
&\qw
}
\end{array}
\right)
=\det
\left[
\begin{array}{cc}
I_2 & 0\\
0 & \sigx
\end{array}
\right]
=
-1
\;.
\eeq
Since $\det(AB)=\det(A)\det(B)$, it follows that
for $r=\ZZ^{>0}$,
\beq
\det(\calg_r)= (-1)^r
\;.
\eeq
It is convenient to define a matrix
$\hat{\calg}_r$ by
\beq
\hat{\calg}_r = (-1)^{\frac{r}{4}}\calg_r =
i^{\frac{r}{2}}\calg_r
\;.
\eeq
Henceforth, we will refer to
$\hat{\calg}_r$ as the
{\bf special counterpart} of $\calg_r$.
(Here the adjective ``special"
means ``having unit determinant").
$\hat{\calg}_r\in U(4)$ and
$\det(\hat{\calg}_r)=1$, so $\hat{\calg}_r\in SU(4)$.
Since $\sigy \sigma_\hata^T \sigy =
\sigma_{-\hata}$,
\beqa
\calg^{(2)}_r &=&
\calg_r \sigyy \calg_r^T \sigyy
\\
&=&
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hata_{r}}\qwx[1]
&\qw
\\
&\ovalgate{\hata'_{r}}
&\qw
}
\end{array}
\cdots
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hata_{2}}\qwx[1]
&\ovalgate{\hata_{1}}\qwx[1]
&\ovalgate{-\hata_{1}}\qwx[1]
&\ovalgate{-\hata_{2}}\qwx[1]
&\qw
\\
&\ovalgate{\hata'_{2}}
&\ovalgate{\hata'_{1}}
&\ovalgate{-\hata'_{1}}
&\ovalgate{-\hata'_{2}}
&\qw
}
\end{array}
\cdots
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{-\hata_{r}}\qwx[1]
&\qw
\\
&\ovalgate{-\hata'_{r}}
&\qw
}
\end{array}
\;.
\eeqa
For $r\in \ZZ^{>0}$,
$\calg^{(2)}_r$
obeys the following
recursion relation:
\beq
\calg^{(2)}_{r+1}
=
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hata_{r+1}}\qwx[1]
&\qw
\\
&\ovalgate{\hata'_{r+1}}
&\qw
}
\end{array}
\calg_r^{(2)}
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{-\hata_{r+1}}\qwx[1]
&\qw
\\
&\ovalgate{-\hata'_{r+1}}
&\qw
}
\end{array}
\;.
\eeq
Note that the LO-RHS invariants of
$\calg_r$ and of its
special counterpart $\hat{\calg}_r$
are related by
\beq
\hat{\calg}_r^{(2)} =
i^r \calg_r^{(2)}
\;.
\label{eq-invar-of-hat-graph}
\eeq
The remainder of Section
\ref{sec-two-bit-dcnot-rhs-invariants} consists
of 4 subsections which
give explicit formulas for
$\calg^{(2)}_r$ for $r$
from 1 to 4. These 4 subsections
are very useful, but
make for dry reading when considered in isolation;
they only come alive and prove their mettle as we
start using them in subsequent sections.
Thus, the reader is advised not to spend
too much time on them during his first reading
of this paper. He should skim the 4 subsections, and
then come back to them as the need
arises.
\subsection{Invariant for Circuits with
1 DC-NOT
\\{\footnotesize\tt[
ckt\_invar123.m
]}}
\label{sec-invariants-1cnot}
This part of our program is dedicated to the letters
$\calg^{(2)}_1$.
\begin{theo}
\beq
\calg^{(2)}_1 =
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hata}\qwx[1]
&\ovalgate{-\hata}\qwx[1]
&\qw
\\
&\ovalgate{\hata'}
&\ovalgate{-\hata'}
&\qw
}
\end{array}
=
-
\sigma_{\hata',\hata}
\;.
\label{eq-invariant-1bit}
\eeq
\end{theo}
\proof
\beq
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hata}\qwx[1]
&\ovalgate{-\hata}\qwx[1]
&\qw
\\
&\ovalgate{\hata'}
&\ovalgate{-\hata'}
&\qw
}
\end{array}
=
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hata}\qwx[1]
&\ovalgate{-\hata}\qwx[1]
&\qw
&\gate{\sigma_{-\hata}}
\\
&\ovalgate{\hata'}
&\ovalgate{\hata'}
&\qw
&\qw
}
\end{array}
=
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\freegate{\sigma_{-\hata}}
\\
&\freegate{\sigma_{\hata'}}
}
\end{array}
\;.
\eeq
\qed
\subsection{Invariant for Circuits with
2 DC-NOTs
\\{\footnotesize\tt[
ckt\_invar123.m,
diag\_ckt\_invar2.m,
diag\_ckt\_invar2\_aux.m,\\
test\_diag\_invar2.m
]}}
\label{sec-invariants-2cnots}
This part of our program is dedicated to the letters
$\calg^{(2)}_2$.
\begin{theo}\label{th-invar2}
\beqa
\calg^{(2)}_2 &=&
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatb}\qwx[1]
&\ovalgate{\hata}\qwx[1]
&\ovalgate{-\hata}\qwx[1]
&\ovalgate{-\hatb}\qwx[1]
&\qw
\\
&\ovalgate{\hatb'}
&\ovalgate{\hata'}
&\ovalgate{-\hata'}
&\ovalgate{-\hatb'}
&\qw
}
\end{array}\\
&=&
\lam_{2r} + i\lam_{2i} + \Lam_{2r} + i\Lam_{2i}
\;,
\label{eq-invariant-2bit}
\eeqa
where
\beq
\lam_{2r}= (\hata\cdot\hatb)(\hata'\cdot\hatb')
\;,
\eeq
\beq
\lam_{2i}= 0
\;,
\eeq
\beq
\Lam_{2r}=
-\sigma_{\manyx{\hata'\hatb'\hatb'},\manyx{\hata\hatb\hatb}}
\;,
\label{eq-def-Lam2r}
\eeq
\beq
\Lam_{2i}=
\hata\cdot\hatb\sigma_{\hata'\times\hatb',\hatb}
+
\hata'\cdot\hatb'\sigma_{\hatb',\hata\times\hatb}
\;.
\label{eq-def-Lam2i}
\eeq
\end{theo}
\proof
An explicit expression
for $\calg^{(2)}_1$ was given
in Section \ref{sec-invariants-1cnot}.
Eq.(\ref{eq-sim-trans-of-sig-veca})
shows how to calculate the effect of DC-NOT
similarity transformations.
Using these two results,
one gets
\beqa
\calg^{(2)}_2 &=&
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\freeovalgate{\hatb}\qwx[1]
\\
&\freeovalgate{\hatb'}
}
\end{array}
[-\sigma_{\hata',\hata}]
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\freeovalgate{-\hatb}\qwx[1]
\\
&\freeovalgate{-\hatb'}
}
\end{array}
\\
&=&
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\freeovalgate{\hatb}\qwx[1]
\\
&\freeovalgate{\hatb'}
}
\end{array}
\sigma_{\hata',1}
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\freeovalgate{\hatb}\qwx[1]
\\
&\freeovalgate{\hatb'}
}
\end{array}
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\freeovalgate{\hatb}\qwx[1]
\\
&\freeovalgate{\hatb'}
}
\end{array}
\sigma_{1,\hata}
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\freeovalgate{\hatb}\qwx[1]
\\
&\freeovalgate{\hatb'}
}
\end{array}
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\freeovalgate{\hatb}\qwx[1]
&\freeovalgate{-\hatb}\qwx[1]
\\
&\freeovalgate{\hatb'}
&\freeovalgate{-\hatb'}
}
\end{array}
(-1)
\\
&=&
(\sigma_{\along{\hata'}{\hatb'},1}+
\sigma_{\across{\hata'}{\hatb'},\hatb})
(\sigma_{1,\along{\hata}{\hatb}}+
\sigma_{\hatb,\across{\hata}{\hatb}})
\sigma_{\hatb',\hatb}
\;
\\
&=&
\left\{
\begin{array}{l}
(\hata'\cdot\hatb')(\hata\cdot\hatb)\\
-\sigma_{\manyx{\hata'\hatb'\hatb'},
\manyx{\hata\hatb\hatb}}\\
+i\left[
\hata\cdot\hatb\sigma_{\hata'\times\hatb',\hatb}
+
\hata'\cdot\hatb'\sigma_{\hatb',\hata\times\hatb}
\right]
\end{array}
\right.
\;.
\eeqa
\qed
\begin{theo}
\beq
[\Lam_{2r},\Lam_{2i}]=0
\;,
\eeq
\beq
(\Lam_{2r}^\Gamma)^T \Lam_{2i}^\Gamma=0
\;,
\eeq
\beq
\Lam_{2r}^\Gamma (\Lam_{2i}^\Gamma)^T=0
\;.
\eeq
\end{theo}
\proof
This follows easily from
Eqs.(\ref{eq-def-Lam2r}) and (\ref{eq-def-Lam2i}).
\qed
It is convenient to parameterize
the expression for $\calg^{(2)}_2$
given by Theorem \ref{th-invar2},
using as few parameters as possible.
\begin{figure}[h]
\begin{center}
\epsfig{file=principal-2cnot.eps, height=2.25in}
\caption{Principal parameters of $\calg_2^{(2)}$.}
\label{fig-principal-2cnot}
\end{center}
\end{figure}
\begin{theo}
$\calg^{(2)}_2$ can be
parameterized with 2 real numbers
$\alpha,\alpha'$,
and 2 RHON bases
$(\hatf_j)_{j=1,2,3}$ and
$(\hatf'_{j})_{j=1,2,3}$.
Call these the principal parameters
of $\calg^{(2)}_2$
(see Fig.\ref{fig-principal-2cnot}).
More explicitly,
\beq
\calg^{(2)}_2=
\lam_{2r} +
\Lam_{2r} + i\Lam_{2i}
\;,
\label{eq-diag-invariant-2bit}
\eeq
where
\beq
\lam_{2r}= c_{\alpha'} c_\alpha
\;,
\eeq
\beq
\Lam_{2r}=
-(s_{\alpha'}s_\alpha) \hatf_2'\hatf_2^T
\;,
\eeq
\beqa
\Lam_{2i}
&=&
(s_{\alpha'}c_\alpha) \hatf_3'\hatf_1^T
+
(c_{\alpha'}s_\alpha) \hatf_1'\hatf_3^T
\label{eq-mat-single-line-style}
\\
&=&
\begin{array}{l|ll}
& \hatf_1^T & \hatf_3^T\\
\hline
\hatf_3'& s_{\alpha'}c_\alpha & 0\\
\hatf_1'& 0 &c_{\alpha'}s_\alpha
\end{array}
\label{eq-mat-table-style}
\\
&=&
\left[
\begin{array}{cc}
\hatf'_3 & \hatf'_1
\end{array}
\right]
\left[
\begin{array}{cc}
s_{\alpha'}c_\alpha & 0 \\
0 & c_{\alpha'}s_\alpha
\end{array}
\right]
\left[
\begin{array}{cc}
\hatf_1 & \hatf_3
\end{array}
\right]^T
\;.
\label{eq-mat-mat-style}
\eeqa
(Eqs.(\ref{eq-mat-single-line-style}),
(\ref{eq-mat-table-style}), and
(\ref{eq-mat-mat-style})
are 3 different styles of representing the same
thing.)
\end{theo}
\proof
Define $\alpha'\in[0,\pi)$ to be the angle between
$\hata'$ and $\hatb'$. Thus
\beq
c_{\alpha'} = \hata'\cdot\hatb'
\;,\;\;
s_{\alpha'} = |\hata'\times\hatb'|
\;.
\eeq
If $s_{\alpha'}\neq 0$,
set
\beq
(f'_j)_{j=1,2,3}=
(\hatb',
\frac{\manyx{\hata'\hatb'\hatb'}}{s_{\alpha'}} ,
\frac{\manyx{\hata'\hatb'}}{s_{\alpha'}})
\;.
\eeq
If $s_{\alpha'}= 0$,
choose $(\hatf'_j)_{j=1,2,3}$
to be any
RHON basis
with $\hatf'_1=\hatb'$.
Use the previous paragraph with all
primes removed to define
$\alpha$ and $(\hatf_j)_{j=1,2,3}$.
\qed
Suppose we are given a matrix
which is known to be the LO-RHS invariant
$\calg_2^{(2)}$
of a quantum circuit with 2-qubits
and 2 DC-NOTs.
Furthermore, we are asked to extract
from this matrix
values (non-unique ones)
for $\hata$,$\hatb$,$\hata'$ and $\hatb'$.
Next we will give an algorithm for
accomplishing this task.
We will call it our
``Algorithm for Diagonalizing
$\calg_2^{(2)}$".
The algorithm first expresses
$\calg_2^{(2)}$
in term of its principal parameters. Then it
solves for
$\hata$,$\hatb$,$\hata'$ and $\hatb'$
in terms of these parameters.
\vspace{.2in}
\noindent{\bf Algorithm for
Diagonalizing} $\calg_2^{(2)}$:
\begin{enumerate}
\item
Set $\lam_{2r}=\frac{1}{4}\tr(\calg_2^{(2)})$.
Set $\Delta = \calg_2^{(2)}- \lam_{2r}$,
$\Lam_{2r} = (\Delta + \Delta^\dagger)/2$
and $\Lam_{2i} = (\Delta - \Delta^\dagger)/(2i)$.
Hence, $\calg_2^{(2)}= \lam_{2r} + \Lam_{2r} +
i \Lam_{2i}$, where $\lam_{2r}$
is a real scalar, and
$\Lam_{2r},\Lam_{2i}$
are traceless Hermitian matrices.
\item
Calculate
$c_{\alpha'}c_\alpha$,
$s_{\alpha'}s_\alpha$,
$\hatf_2$ and
$\hatf'_2$
from $\lam_{2r}$ and $\Lam_{2r}$.
(If $\Lam_{2r}=0$, then
take $s_{\alpha'} s_\alpha=0$,
and choose
$\hatf_2$ and
$\hatf'_2$ to be
any 3d unit vectors.)
\item \label{item-diag-invar2-h-hprime}
Choose any
RHON basis $(\hath_j)_{j=1,2,3}$
such that $\hath_2=\hatf_2$, and
any RHON basis $(\hath'_j)_{j=1,2,3}$
such that $\hath'_2=\hatf'_2$.
\item \label{item-diag-invar2-m-matrix}
Find a Singular Value
Decomposition (SVD) of the matrix
\beq
M =
\left[
\begin{array}{cc}
\hath^{'T}_3 \Lam_{2i}\hath_1&
\hath^{'T}_3 \Lam_{2i}\hath_3\\
\hath^{'T}_1 \Lam_{2i}\hath_1&
\hath^{'T}_1 \Lam_{2i}\hath_3
\end{array}
\right]
\;.
\eeq
In other words, find 2-dimensional orthogonal
matrices $U,V$ and
a non-negative 2-dimensional
diagonal matrix $D$ such that
\beq
M = U D V^T
\;.
\eeq
Now calculate $s_{\alpha'}c_\alpha$,
$c_{\alpha'}s_\alpha$,
$\hatf'_3$,
$\hatf'_1$,
$\hatf_3$,
$\hatf_1$ from
\beq
\left[
\begin{array}{cc}
s_{\alpha'}c_\alpha & 0\\
0 & c_{\alpha'}s_\alpha
\end{array}
\right]
= D
\;,
\eeq
\beq
[\hatf'_3, \hatf'_1]
=
[\hath'_3, \hath'_1] U
\;,
\label{eq-f-eq-hu}
\eeq
and
\beq
[\hatf_1, \hatf_3]
=
[\hath_1, \hath_3]V
\;.
\eeq
\item
By expressing $U$ on the RHS of
Eq.(\ref{eq-f-eq-hu}) in component form,
it is easy to verify that
\beq
\hatf'_3 \times \hatf'_1
=
\det(U)\hath'_3 \times \hath'_1
\;.
\eeq
$\hath'_3\times \hath'_1\cdot \hath'_2= +1$
and $\hatf'_2=\hath'_2$ so
\beq
\hatf'_3\times \hatf'_1\cdot \hatf'_2
= \det(U)
\;.
\eeq
$\det(U)$ will
always equal either $+1$ or $-1$.
If $\det(U)=-1$, replace
$\hatf'_3\rarrow -\hatf'_3$ and
$s_{\alpha'}c_\alpha \rarrow -s_{\alpha'}c_\alpha$.
These replacements
make
$(\hatf'_1,\hatf'_2, \hatf'_3)$
a right handed basis.
If $\det(V)=-1$,
an analogous procedure can be
used to convert
$(\hatf_1,\hatf_2, \hatf_3)$
into a right-handed basis.
\item
At this point, we know
the four quantities
$c_{\alpha'}c_\alpha$,
$s_{\alpha'}c_\alpha$,
$c_{\alpha'}s_\alpha$,
and
$s_{\alpha'}s_\alpha$.
Calculate $\alpha'\pm \alpha$
from
\begin{subequations}
\beq
\cos(\alpha'\pm\alpha)= c_{\alpha'}c_\alpha \mp
s_{\alpha'}s_\alpha
\;,
\eeq
and
\beq
\sin(\alpha'\pm\alpha)= s_{\alpha'}c_\alpha \pm
c_{\alpha'}s_\alpha
\;.
\eeq
\end{subequations}
Calculate $(\alpha', \alpha)$ from
$\alpha'\pm\alpha$.
\item
Calculate $\hata,\hatb,\hata',\hatb'$
from:
\beq
\left\{
\begin{array}{l}
\hatb = \hatf_1\\
\hata = c_{\alpha} \hatf_1 - s_{\alpha} \hatf_2
\end{array}
\right.
\;\;,\;\;\;
\left\{
\begin{array}{l}
\hatb' = \hatf'_1\\
\hata' = c_{\alpha'} \hatf'_1 - s_{\alpha'} \hatf'_2
\end{array}
\right.
\;.
\eeq
\end{enumerate}
\subsection{Invariant for Circuits with
3 DC-NOTs
\\{\footnotesize\tt[
ckt\_invar123.m,
ckt\_invar3.m,
diag\_ckt\_invar3.m,
test\_diag\_invar3.m
]}}
\label{sec-invariants-3cnots}
This part of our program is dedicated to the letters
$\calg^{(2)}_3$.
\begin{theo}\label{th-abc-invar3}
\beqa
\calg^{(2)}_3 &=&
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatc}\qwx[1]
&\ovalgate{\hatb}\qwx[1]
&\ovalgate{\hata}\qwx[1]
&\ovalgate{-\hata}\qwx[1]
&\ovalgate{-\hatb}\qwx[1]
&\ovalgate{-\hatc}\qwx[1]
&\qw
\\
&\ovalgate{\hatc'}
&\ovalgate{\hatb'}
&\ovalgate{\hata'}
&\ovalgate{-\hata'}
&\ovalgate{-\hatb'}
&\ovalgate{-\hatc'}
&\qw
}
\end{array}
\label{eq-invariant-3bit-diag}
\\
&=&
\lam_{3r} + i\lam_{3i} + \Lam_{3r} + i\Lam_{3i}
\;,
\label{eq-invariant-3bit}
\eeqa
where
\beq
\lam_{3r} =
\manyx{\hata'\hatb' \hatb'}\cdot\hatc'
\;\;
\manyx{\hata\hatb\hatb}\cdot\hatc
\;,
\eeq
\beq
\lam_{3i} =
-(\hata\cdot\hatb)(\hatb\cdot\hatc)\calv'
-(\hata'\cdot\hatb')(\hatb'\cdot\hatc')\calv
\;,
\eeq
\beq
\Lam_{3r}=
\left\{
\begin{array}{l}
-(\hata'\cdot\hatb')(\hata\cdot\hatb)
\sigma_{\hatc',\hatc}
\\
+(\hata\cdot\hatb)(\hatb\cdot\hatc)
\sigma_{\manyx{\hata'\hatb'\hatc'},\hatc}
+(\hata'\cdot\hatb')(\hatb'\cdot\hatc')
\sigma_{\hatc',\manyx{\hata\hatb\hatc}}
\\
+(\hata'\cdot\hatb')\calv\sigma_{\manyx{\hatb'\hatc'},\hatc}
+(\hata\cdot\hatb)\calv'\sigma_{\hatc',\manyx{\hatb\hatc}}
\\
-\sigma_{
\manyx{\hata'\hatb'\hatb'\hatc'\hatc'},
\manyx{\hata\hatb\hatb\hatc\hatc}}
\;
\end{array}
\right.
\;,
\eeq
\beq
\Lam_{3i}=
\left\{
\begin{array}{l}
+(\hata\cdot\hatb)
\sigma_{\manyx{\hata'\hatb'\hatc'\hatc'},
\manyx{\hatb\hatc\hatc}}
+
(\hata'\cdot\hatb')
\sigma_{\manyx{\hatb'\hatc'\hatc'},
\manyx{\hata\hatb\hatc\hatc}}
\\
+\manyx{\hata\hatb\hatb}\cdot \hatc
\sigma_{\manyx{\hata'\hatb'\hatb'\hatc'},\hatc}
+
\manyx{\hata'\hatb'\hatb'}\cdot \hatc'
\sigma_{\hatc', \manyx{\hata\hatb\hatb\hatc}}
\end{array}
\right.
\;,
\eeq
where
$\calv = \hata\times\hatb\cdot\hatc$ and
$\calv' = \hata'\times\hatb'\cdot\hatc'$.
\end{theo}
\proof
\beqa
\calg^{(2)}_3 &=&
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatc}\qwx[1]
&\qw
\\
&\ovalgate{\hatc'}
&\qw
}
\end{array}
\calg_2^{(2)}
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{-\hatc}\qwx[1]
&\qw
\\
&\ovalgate{-\hatc'}
&\qw
}
\end{array}\\
&=&
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatc}\qwx[1]
&\qw
\\
&\ovalgate{\hatc'}
&\qw
}
\end{array}
\calg_2^{(2)}
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatc}\qwx[1]
&\qw
\\
&\ovalgate{\hatc'}
&\qw
}
\end{array}
(-\sigma_{\hatc',\hatc})
\;.
\eeqa
An explicit expression
for $\calg^{(2)}_2$ was given
in Section \ref{sec-invariants-2cnots}.
Eq.(\ref{eq-sim-trans-of-sig-veca})
shows how to calculate the effect of DC-NOT
similarity transformations.
\qed
\begin{theo}\label{th-simple-orientation-invar3}
Suppose
\beq
\call=
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatc}\qwx[1]
&\ovalgate{\hatb}\qwx[1]
&\ovalgate{\hata}\qwx[1]
&\qw
\\
&\ovalgate{\hatc'}
&\ovalgate{\hatb'}
&\ovalgate{\hata'}
&\qw
}
\end{array}
\;\;,\;\;
\calr=
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatc_f}\qwx[1]
&\qw
&\ovalgate{\hatb_f}\qwx[1]
&\qw
&\ovalgate{\hata_f}\qwx[1]
&\qw
\\
&\ovalgate{\hatc'_f}
&\foil
&\ovalgate{\hatb'_f}
&\foil
&\ovalgate{\hata'_f}
&\qw
\gategroup{1}{2}{1}{6}{.7em}{--}
}
\end{array}
\;.
\eeq
For any $\call$,
it is possible to find an $\calr$ such that
$\call\sim_R \calr$, and such that
(a)$\hata_f\times\hatb_f\cdot\hatc_f=0$, and
(b)$\hatb'_f\perp span(\hatc'_f,\hata'_f)$.
\end{theo}
\proof
As pointed out in the introduction,
Ref.\cite{VD} shows how to express any
2-qubit unitary operation
as a circuit with just 3-CNOTs.
It is easy to check that conditions (a) and (b)
are satisfied by the 3-CNOT
circuit given
in Ref.\cite{VD}. Hence, this theorem
has already been proven in Ref.\cite{VD},
although Ref.\cite{VD} does not explicitly
point out this property of their
3-CNOT circuit. The
``Algorithm for Diagonalizing $\calg^{(2)}_3$",
that is presented later in this section,
also constitutes a (constructive) proof of this
theorem.
\qed
For $A, B \in \RR^{p\times q}$,
define the following two commutators:
\begin{subequations}\label{eq-left-right-commut}
\beq
[A,B]_L = A^T B - B^T A
\;,
\eeq
\beq
[A,B]_R = A B^T - B A^T
\;.
\eeq
\end{subequations}
(Here, the letters
$L$ and $R$ stand for left and
right. They indicate on which matrix
the transpose symbol acts,
either the left
or the right matrix in the matrix product.)
Ref.\cite{Tuc-KAK} presents a proof
(due to Eckart and Young) of the following
Theorem.
$A,B\in\RR^{p\times q}$ have
a simultaneous
Singular Value Decomposition (SVD)
if and
only if $[A,B]_L$ and $[A,B]_R$
are both zero.
By a simultaneous SVD
we mean
orthogonal matrices $U,V$
and real diagonal matrices $D_A,D_B$
such that
\beq
A = UD_AV^T
\;\;,\;\;
B=U D_B V^T
\;.
\label{eq-sim-svd}
\eeq
When considering
the SVD of a single matrix $A$, one
usually insists in making
the entries of $D_A$ non-negative,
and calling them
the singular values of $A$.
In the case of a simultaneous SVD,
one can't always make both
diagonal matrices non-negative,
but one can certainly make one of
them so.
Of course, the previous paragraph
applies almost intact if
$A$ and $B$ are elements of
$\CC^{p\times q}$ instead of
$\RR^{p\times q}$.
For $A,B$ complex, one must replace
the $T$ (transpose) symbol by the
$\dagger$ (Hermitian conjugate) symbol in
Eqs.(\ref{eq-left-right-commut}) and
(\ref{eq-sim-svd}). Also, the matrices $U,V$
in Eq.(\ref{eq-sim-svd})
must be unitary instead of orthogonal.
Note that when $A$ and $B$ are Hermitian,
the
condition that
$[A,B]_L$ and $[A,B]_R$
both vanish becomes simply the
condition that $A$ and $B$ commute.
The Eckart, Young theorem then
becomes a theorem
very familiar to practitioners
of Quantum Mechanics: two Hermitian operators
can be simultaneously diagonalized
iff they commute.
\begin{theo}
\beq
[\Lam_{3r}, \Lam_{3i}]=0
\;,
\eeq
\beq
[\Lam^\Gamma_{3r}, \Lam^\Gamma_{3i}]_L=0
\;,
\eeq
\beq
[\Lam^\Gamma_{3r}, \Lam^\Gamma_{3i}]_R=0
\;.
\eeq
\end{theo}
\proof
Let
\beq
\Delta = \calg^{(2)}_3 - \tr(\calg^{(2)}_3)
\;,
\eeq
so
\beq
\Lam_{3r}= \frac{\Delta + \Delta^\dagger}{2}
\;\;,\;\;
\Lam_{3i}= \frac{\Delta - \Delta^\dagger}{2i}
\;.
\eeq
Thus,
\beq
[\Lam_{3r},\Lam_{3i}]=
\frac{1}{4i}[\Delta + \Delta^\dagger,
\Delta - \Delta^\dagger]
=
\frac{1}{2i}[\Delta^\dagger,
\Delta]
=
\frac{1}{2i}[\calg^{(2)\dagger}_3,
\calg^{(2)}_3]
=
0
\;,
\eeq
where the last commutator is zero because
$\calg^{(2)}_3$
is unitary.
Note that for any
$\hata, \hata',\hatb,\hatb'\in\unitvecs$,
\beqa
[\sigma_{\hata',\hata},\sigma_{\hatb',\hatb}]&=&
\left\{
\begin{array}{l}
+(\hata'\cdot\hatb' +i\sigma_{\hata'\times\hatb'})
\otimes
(\hata\cdot\hatb +i\sigma_{\hata\times\hatb})
\\
-
(\hatb'\cdot\hata' +i\sigma_{\hatb'\times\hata'})
\otimes
(\hatb\cdot\hata +i\sigma_{\hatb\times\hata})
\end{array}
\right.
\\
&=&
i2
[(\hata\cdot\hatb)\sigma_{\hata'\times\hatb',1}
-
(\hata'\cdot\hatb')\sigma_{1,\hata\times\hatb}]
\;.
\eeqa
From Theorem \ref{th-abc-invar3},
we know that $\Lam_{3r}$
and $\Lam_{3r}$
can be expressed in the form
\beq
\Lam_{3r} =
\sum_j \alpha_j \sigma_{\hata'_j,\hata_j}
\;,
\;\;
\Lam_{3i} =
\sum_k \beta_k \sigma_{\hatb'_k,\hatb_k}
\;,
\eeq
for some $\alpha_j,\beta_j\in \RR$
and
$\hata_j, \hata'_j, \hatb_k,\hatb'_k\in\unitvecs$.
Therefore,
\beqa
0&=& [\Lam_{3r}, \Lam_{3i}]
\\
&=&
\sum_{j,k} \alpha_j\beta_k
[ \sigma_{\hata'_j,\hata_j},
\sigma_{\hatb'_k,\hatb_k}]
\\
&=&
i2\sum_{j,k} \alpha_j\beta_k
[(\hata_j\cdot\hatb_k)\sigma_{\hata'_j\times\hatb_k',1}
-
(\hata'_j\cdot\hatb'_k)\sigma_{1,\hata_j\times\hatb_k}]
\;.
\eeqa
This implies that
\beq
\sum_{j,k} \alpha_j\beta_k
(\hata_j\cdot\hatb_k)
\hata'_j\times\hatb_k'=0
\;,\;\;
\sum_{j,k} \alpha_j\beta_k
(\hata'_j\cdot\hatb'_k)
\hata_j\times\hatb_k=0
\;.
\label{eq-sum-cross-prods-is-zero}
\eeq
Now note that
\beqa
[\Lam_{3r}^\Gamma, \Lam_{3i}^\Gamma]_R
&=&
(\sum_j \alpha_j \hata'_j\hata^T_j)
(\sum_k \beta_k \hatb_k\hatb^{'T}_k)
-
(\sum_k \beta_k \hatb'_k\hatb^{T}_k)
(\sum_j \alpha_j \hata_j\hata^{'T}_j)
\\
&=&
\sum_{j,k}
\alpha_j\beta_k(\hata^T_j\hatb_k)
[
\hata'_j \hatb^{'T}_k
-
\hatb'_k \hata^{'T}_j ]
\\
&=& 0
\;,
\eeqa
where the last expression vanishes
due to Eq.(\ref{eq-sum-cross-prods-is-zero}).
An analogous argument shows that
$[\Lam_{3r}^\Gamma, \Lam_{3i}^\Gamma]_L$
also vanishes.
\qed
It is convenient to parameterize
the expression for $\calg^{(2)}_3$
given by Theorem \ref{th-abc-invar3},
using as few parameters as possible.
\begin{figure}[h]
\begin{center}
\epsfig{file=principal-3cnot.eps, height=2.25in}
\caption{Principal parameters of $\calg_3^{(2)}$.}
\label{fig-principal-3cnot}
\end{center}
\end{figure}
\begin{theo}
$\calg^{(2)}_3$ can be
parameterized with 3 real numbers
$\beta,\beta_1,\beta_2$,
and 2 RHON bases
$(\hatg_j)_{j=1,2,3}$ and
$(\hatg'_{j})_{j=1,2,3}$.
Call these the principal parameters
of $\calg^{(2)}_3$
(see Fig.\ref{fig-principal-3cnot}).
More explicitly,
\beq
\calg^{(2)}_3=
\lam_{3r} + i\lam_{3i} +
\Lam_{3r} + i\Lam_{3i}
\;,
\label{eq-diag-invariant-3bit}
\eeq
where
\beq
\lam_{3r} = - X_o
\;,
\eeq
\beq
\lam_{3i} = - Y_o
\;,
\eeq
\beq
\Lam_{3r} = \sum_{j=1}^3 \nu_j
\hatg_j'\hatg_{\pi(j)}^T
\;,
\label{eq-Lam3r-fin}
\eeq
\beq
\Lam_{3i} = \sum_{j=1}^3 \mu_j
\hatg_j'\hatg_{\pi(j)}^T
\;,
\label{eq-Lam3i-fin}
\eeq
where
\beq
X_o = c_\beta \xi s_{\beta_1} s_{\beta_2}
\;,
\label{eq-xo-fin}
\eeq
\beq
Y_o = s_\beta c_{\beta_1} c_{\beta_2}
\;,
\label{eq-yo-fin}
\eeq
\beq
(\nu_j)_{j=1,2,3}
=
(
s_\beta c_{\beta_1}s_{\beta_2},
s_\beta s_{\beta_1}|c_{\beta_2}|,
c_\beta c_{\beta_1}c_{\beta_2}
)
\;,
\label{eq-nuj-fin}
\eeq
\beq
(\mu_j)_{j=1,2,3}
=
(
-c_\beta s_{\beta_1}|c_{\beta_2}|,
-c_\beta c_{\beta_1}s_{\beta_2},
s_\beta \xi s_{\beta_1}s_{\beta_2}
)
\;,
\label{eq-muj-fin}
\eeq
where $\xi\in\{+1,-1\}$ and
$\pi()$ is the permutation
$\left(
\begin{array}{ccc}
1 & 2 & 3\\
2 & 3 & 1
\end{array}
\right)$.
\end{theo}
\proof
We will assume from the onset of this
proof that
(a)$\hata\times\hatb\cdot\hatc=0$, and
(b)$\hatb'\perp span(\hatc',\hata')$.
This can be assumed without loss of
generality because of Theorem
\ref{th-simple-orientation-invar3}.
Let
\beq
\xi = {\rm sign}(\manyx{ab}\cdot\manyx{bc})
\;,\;\;
\xi_2 = {\rm sign}(\hatb\cdot\hatc)
\;.
\eeq
Without loss of generality, we will assume
that $-\xi\xi_2=+1$.
If $-\xi\xi_2$ is initially negative, we can
make it positive by
replacing both $\hata$
and $\hata'$ by their negatives.
This replacement will not change
$\calg^{(2)}_3$.
Using the circuit shown in
Eq.(\ref{eq-invariant-3bit-diag}),
it is easy to
prove that $\calg_3^{(2)}$ is
odd in both $\hata$ and $\hata'$.
Define
\beq
s_{\beta_2} = |\manyx{\hatb\hatc}|
\;,\;\;
\eta = |\manyx{\hata\hatb\hatb\hatc}|
=|\manyx{\hata\hatb}\hatb\cdot\hatc|
\;.
\eeq
To begin, we will assume that
$s_{\beta_2}\neq 0$
and $\eta\neq 0$. Later on, before ending
the proof, we will remove these two constraints.
If we define
\beq
X_o = (\hata'\cdot\hatc')
\manyx{\hata\hatb\hatb}\cdot \hatc
\;,
\label{eq-xo-init}
\eeq
\beq
Y_o= (\hata\cdot\hatb)(\hatb\cdot\hatc)\calv'
\;,
\label{eq-yo-init}
\eeq
\beq
(\hatg'_j)_{j=1,2,3}=
(
\hatc',
\manyx{\hatb'\hatc'},
\hatb'
)
\;,
\eeq
\beq
(\hatg_j)_{j=1,2,3}=
(
\hatc,
\frac{\manyx{\hatb\hatc}}{s_{\beta_2}},
\frac{-\manyx{\hatb\hatc\hatc}}{s_{\beta_2}}
)
\;,
\label{eq-gj-init}
\eeq
\begin{subequations}
\beq
(\nu_j)_{j=1,2,3}=
\left(
\hata\cdot\hatb\calv' s_{\beta_2},
\calv'\eta,
(\hata\cdot\hatb)(\hatb\cdot\hatc)(\hata'\cdot\hatc')
\right)
\;,
\label{eq-nuj-init}
\eeq
\beq
(\hatv_j)_{j=1,2,3}=
(
\frac{\manyx{\hatb\hatc}}{s_{\beta_2}},
\frac{-\manyx{\hata\hatb\hatb\hatc\hatc}}{\eta},
\hatc
)
\;,
\label{eq-vj-init}
\eeq
\end{subequations}
\begin{subequations}
\beq
(\mu_j)_{j=1,2,3}=
\left(
-\hata'\cdot\hatc'\eta,
-(\hata\cdot\hatb)(\hata'\cdot\hatc')s_{\beta_2},
\manyx{\hata\hatb\hatb}\cdot\hatc\calv'
\right)
\;,
\label{eq-muj-init}
\eeq
\beq
(\hatu_j)_{j=1,2,3}=
(
\frac{\manyx{\hata\hatb\hatb\hatc}}{\eta},
\frac{-\manyx{\hatb\hatc\hatc}}{s_{\beta_2}},
\hatc
)
\;,
\label{eq-uj-init}
\eeq
\end{subequations}
then
\beq
\calg^{(2)}_3=
-X_o -iY_o
+\sum_{j=1}^3 \nu_j
\hatg_j'\hatv_{j}^T
+i\sum_{j=1}^3 \mu_j
\hatg_j'\hatu_{j}^T
\;.
\eeq
Define an angle $\beta$ by
\beq
\cos(\beta)= \hata'\cdot\hatc'
\;,\;\;
\sin(\beta) = \calv'
\;.
\eeq
Define angles $\beta_1, \beta_2\in [0,\pi)$
by
\beq
\cos(\beta_1) = \hata\cdot\hatb
\;,\;\
\sin(\beta_1) = |\hata\times\hatb|
\;,
\eeq
and
\beq
\cos(\beta_2) = \hatb\cdot\hatc
\;,\;\
\sin(\beta_2) = |\hatb\times\hatc|
\;.
\eeq
Hence, $\manyx{\hata\hatb}/s_{\beta_1} =
\xi \manyx{\hatb\hatc}/s_{\beta_2}$.
One finds
\beq
\eta = s_{\beta_1}|c_{\beta_2}|
\;,
\eeq
\beq
\frac{\manyx{\hata\hatb\hatb\hatc}}{\eta}
\cdot\hatg_2
=-\xi \xi_2
\;,
\eeq
\beq
\frac{-\manyx{\hata\hatb\hatb\hatc\hatc}}{\eta}
\cdot\hatg_3
=-\xi \xi_2
\;,
\eeq
and
\beq
\manyx{\hata\hatb\hatb}\cdot \hatc
=\xi s_{\beta_1}s_{\beta_2}
\;.
\eeq
At this point, it is easy
re-express various quantities
in terms of the principal parameters.
Eq.(\ref{eq-xo-init}) for $X_o$,
Eq.(\ref{eq-yo-init}) for $Y_o$,
Eq.(\ref{eq-nuj-init}) for the $\nu_j$,
and
Eq.(\ref{eq-muj-init}) for the $\mu_j$,
yield, respectively,
Eq.(\ref{eq-xo-fin}),
Eq.(\ref{eq-yo-fin}),
Eq.(\ref{eq-nuj-fin}),
and
Eq.(\ref{eq-muj-fin}).
We can also re-express
Eqs.(\ref{eq-vj-init}) and (\ref{eq-uj-init})
for the $\hatv_j$ and $\hatu_j$
in terms of the principal parameters.
One finds
\beq
(\hatv_j)_{j=1,2,3}=
(\hatg_2, -\xi\xi_2\hatg_3, \hatg_1)
=
(\hatg_2, \hatg_3, \hatg_1)
\;,
\eeq
and
\beq
(\hatu_j)_{j=1,2,3}=
(-\xi\xi_2\hatg_2, \hatg_3, \hatg_1)
=
(\hatg_2, \hatg_3, \hatg_1)
\;.
\eeq
Hence,
for $j=1,2,3$,
\beq
\hatv_j = \hatu_j=\hatg_{\pi(j)}
\;.
\label{eq-v-u-gpi}
\eeq
When $s_{\beta_2}$ or $\eta$ vanish,
Eq.(\ref{eq-gj-init}) fails to define
two of the vectors $\hatg_j$,
Eq.(\ref{eq-vj-init}) fails to define
on or two of the vectors $\hatv_j$,
and
Eq.(\ref{eq-uj-init}) fails to define
on or two of the vectors $\hatu_j$.
If $s_{\beta_2}=0$, the proof survives if
we define
$(\hatg_j)_{j=1,2,3}$ to be any
RHON basis such that $\hatg_1=\hatc$
and $\hatg_2\perp span(\hata,\hatb,\hatc)$.
Then define
the $\hatu_j$ and $\hatv_j$ vectors
in accordance with Eq.(\ref{eq-v-u-gpi}).
If $\eta=0$ but
$s_{\beta_2}\neq 0$,
define
the $\hatu_j$ and $\hatv_j$ vectors
in accordance with Eq.(\ref{eq-v-u-gpi}).
\qed
Suppose we are given a matrix
which is known to be the LO-RHS invariant
$\calg_3^{(2)}$
of a quantum circuit with 2-qubits
and 3 DC-NOTs.
Furthermore, we are asked to extract from
this matrix
values (non-unique ones)
for $\hata$, $\hatb$, $\hatc$,
$\hata'$, $\hatb'$ and $\hatc'$.
Next we will give an algorithm for
accomplishing this task.
We will call it our
``Algorithm for Diagonalizing
$\calg_3^{(2)}$".
The algorithm first expresses
$\calg_3^{(2)}$
in term of its principal parameters. Then it
solves for
$\hata$, $\hatb$, $\hatc$,
$\hata'$, $\hatb'$ and $\hatc'$
in terms of these parameters.
\vspace{.2in}
\noindent{\bf Algorithm for
Diagonalizing} $\calg_3^{(2)}:$
\begin{enumerate}
\item
Set $\lam_{3r}=\frac{1}{4}{\rm Re}[\tr(\calg_3^{(2)})]$
and $\lam_{3i}=\frac{1}{4}{ \rm Im}[\tr(\calg_3^{(2)})]$.
Set $\Delta = \calg_3^{(2)}- tr(\calg_3^{(2)})$,
$\Lam_{3r} = (\Delta + \Delta^\dagger)/2$
and $\Lam_{3i} = (\Delta - \Delta^\dagger)/(2i)$.
Hence, $\calg_3^{(2)}= \lam_{3r} + i\lam_{3i}+ \Lam_{3r} +
i \Lam_{3i}$, where $\lam_{3r}, \lam_{3i}$
are real scalars, and
$\Lam_{3r},\Lam_{3i}$
are traceless Hermitian matrices.
\item
Set $X_o=-\lam_{3r}$
and
$Y_o=-\lam_{3i}$.
\item
Do a simultaneous SVD of
$\Lam^\Gamma_{3r}$
and
$\Lam^\Gamma_{3i}$.
This decomposition is possible
since we have shown previously that
$[\Lam^\Gamma_{3r},\Lam^\Gamma_{3i}]_L$
and
$[\Lam^\Gamma_{3r},\Lam^\Gamma_{3i}]_R$
are both zero. The decomposition yields
orthogonal matrices $U,V$
and real diagonal matrices $D_{3r},D_{3i}$
such that
\beq
\Lam^\Gamma_{3r} = U D_{3r} V^T
\;,\;\;
\Lam^\Gamma_{3i} = U D_{3i} V^T
\;.
\eeq
For $j=1,2,3$, set
\beq
\nu_j = (D_{3r})_{jj}
\;\;,\;\;
\mu_j = (D_{3i})_{jj}
\;.
\eeq
Set
\beq
[\hatg'_1,\hatg'_2,\hatg'_3] = U
\;\;,\;\;
[\hatg_1,\hatg_2,\hatg_3] = V
\;.
\eeq
\item
Set $\xi = sign(\mu_3\nu_2)$.
Set $\xi_2 = -\xi$.
Calculate $\beta$ from
\beq
c_\beta
=
\xi \frac{X_o}
{\sqrt{\mu_3 ^2 + X_o^2}}
\;\;,\;\;
s_\beta=
\xi\frac{\mu_3}
{\sqrt{\mu_3 ^2 + X_o^2}}
\;.
\eeq
If $|c_\beta|\geq |s_\beta|$,
set
\beq
\left[
\begin{array}{cc}
c_{\beta_1} c_{\beta_2}& c_{\beta_1} s_{\beta_2}
\\
s_{\beta_1} c_{\beta_2}& s_{\beta_1} s_{\beta_2}
\end{array}
\right]
=
\frac{1}{c_\beta}
\left[
\begin{array}{cc}
\nu_3 & -\mu_2 \\
-\xi_2 \mu_1 & \xi X_o
\end{array}
\right]
\;.
\eeq
On the other hand, if $|s_\beta|\geq |c_\beta|$,
set
\beq
\left[
\begin{array}{cc}
c_{\beta_1} c_{\beta_2}& c_{\beta_1} s_{\beta_2}
\\
s_{\beta_1} c_{\beta_2}& s_{\beta_1} s_{\beta_2}
\end{array}
\right]
=
\frac{1}{s_\beta}
\left[
\begin{array}{cc}
Y_o & \nu_1 \\
\xi_2 \nu_2 & \xi \mu_3
\end{array}
\right]
\;.
\eeq
\item
At this point, we know
the four quantities
$c_{\beta_1}c_{\beta_2}$,
$s_{\beta_1}c_{\beta_2}$,
$c_{\beta_1}s_{\beta_2}$,
and
$s_{\beta_1}s_{\beta_2}$.
Calculate $\beta_1\pm \beta_2$
from
\begin{subequations}
\beq
\cos(\beta_1\pm\beta_2)= c_{\beta_1}c_{\beta_2} \mp
s_{\beta_1}s_{\beta_2}
\;,
\eeq
and
\beq
\sin(\beta_1\pm\beta_2)= s_{\beta_1}c_{\beta_2} \pm
c_{\beta_1}s_{\beta_2}
\;.
\eeq
\end{subequations}
Calculate $(\beta_1, \beta_2)$ from
$\beta_1\pm\beta_2$.
\item
At this point,
$s_{\beta_1}s_{\beta_2}$
is guaranteed to be positive, but there
is not guarantee that $s_{\beta_1}$
and $s_{\beta_2}$ are individually positive (they
may both be negative). Furthermore,
at this point there is no guarantee
that $\xi_2 = sign(c_{\beta_2})$.
These disagreements with the
assumptions of our parameterization
can be fixed as follows.
If
$s_{\beta_1}<0$,
replace
$\beta_1$ and $\beta_2$ by their negatives,
and replace
$(\hatg'_1, \hatg'_2, \nu_1, \nu_2, \mu_1, \mu_2)$
each by its negative.
If
$\xi_2 c_{\beta_2}<0$,
replace
$\beta_1\rarrow \pi-\beta_1$
and
$\beta_2\rarrow \pi-\beta_2$,
and replace
$(\hatg'_1, \hatg'_2, \nu_1, \nu_2, \mu_1, \mu_2)$
each by its negative.
\item
Calculate $\hata,\hatb,\hatc,\hata',\hatb',\hatc'$
from:
\beq
\left\{
\begin{array}{l}
\hatc = \hatg_1\\
\hatb = c_{\beta_2}\hatg_1 + s_{\beta_2}\hatg_3\\
\hata =
\cos(\beta_2-\xi_2\beta_1) \hatg_1 +
\sin(\beta_2-\xi_2\beta_1) \hatg_3
\end{array}
\right.
\;\;,\;\;\;
\left\{
\begin{array}{l}
\hatc' = \hatg'_1\\
\hatb' = \hatg'_3\\
\hata' = c_{\beta} \hatg'_1 + s_{\beta} \hatg'_2
\end{array}
\right.
\;.
\eeq
Note the $\xi_2$'s in the
expression for $\hata$. The
reason for these $\xi_2$'s is
that in order to obey $-\xi\xi_2=+1$,
one must define
the sign of the angle $\beta_1$
differently depending on
whether $c_{\beta_2}$ is
positive or negative.
(See Fig.\ref{fig-signs-principal-3cnot})
\end{enumerate}
\begin{figure}[h]
\begin{center}
\epsfig{file=signs-principal-3cnot.eps, height=1.75in}
\caption{Sign of $\beta_1$ is defined
differently depending on whether $c_{\beta_2}$
is positive or negative.}
\label{fig-signs-principal-3cnot}
\end{center}
\end{figure}
\begin{theo}\label{th-mu-nu-bilinear}
For any $j\in\{1,2,3\}$,
\beq
\mu_j\nu_j = X_o Y_o
\;.
\eeq
If $i,j,k$ are 3 distinct
element of $\{1,2,3\}$, then
\beq
\mu_i \mu_j = -X_o \nu_k
\;,
\eeq
and
\beq
\nu_i \nu_j = -Y_o \mu_k
\;.
\eeq
\end{theo}
\proof
Follows from the definitions
Eq.(\ref{eq-xo-fin}) for $X_o$,
Eq.(\ref{eq-yo-fin}) for $Y_o$,
Eq.(\ref{eq-nuj-fin}) for the $\nu_j$,
and
Eq.(\ref{eq-muj-fin}) for the $\mu_j$.
\qed
Define $\Pi$ to be
the permutation matrix that
corresponds to the permutation map $\pi()$
used above. Thus,
\beq
\Pi =
\left[
\begin{array}{ccc}
0 & 1 & 0\\
0 & 0 & 1\\
1 & 0 & 0
\end{array}
\right]
\;.
\eeq
If $(s_j)_{j=1,2,3}$ denotes
the standard basis,
define matrices $M_\mu$ and $M_\nu$ by
\beqa
M_\mu &=&
\sum_{j=1}^3 \mu_j \hats_j \hats_{\pi(j)}\\
&=&
diag(\mu_1, \mu_2, \mu_3)\Pi
\;,
\eeqa
and
\beqa
M_\nu &=&
\sum_{j=1}^3 \nu_j \hats_j \hats_{\pi(j)}\\
&=&
diag(\nu_1, \nu_2, \nu_3)\Pi
\;.
\eeqa
Note that $\Lam^\Gamma_{2r}$
(given by Eq.(\ref{eq-Lam3r-fin}) ) becomes
$M_\nu$
and
$\Lam^\Gamma_{2i}$
(given by Eq.(\ref{eq-Lam3i-fin}) ) becomes
$M_\mu$
when the bases
$(\hatg_j)_{j=1,2,3}$
and
$(\hatg'_j)_{j=1,2,3}$
are both rotated into
the standard basis.
\begin{theo}
\beq
M_\mu M_\nu^T = M_\nu M_\mu^T = X_o Y_o
\;,
\eeq
\beq
M_\mu^T M_\nu = M^T_\nu M_\mu = X_o Y_o
\;,
\eeq
and
\beq
(M_\mu^T)^2 = \tr(M_\nu) - M_\nu
\;.
\eeq
\end{theo}
\proof
Follows from
Theorem \ref{th-mu-nu-bilinear}.
\qed
\subsection{Invariant for Circuits with
4 DC-NOTs
\\{\footnotesize\tt[
ckt\_invar4.m
]}}
\label{sec-invariants-4cnots}
This part of our program is dedicated to the letters
$\calg^{(2)}_4$.
\begin{theo}\label{eq-invar4}
\beqa
\calg^{(2)}_4 &=&
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatd}\qwx[1]
&\ovalgate{\hatc}\qwx[1]
&\ovalgate{\hatb}\qwx[1]
&\ovalgate{\hata}\qwx[1]
&\ovalgate{-\hata}\qwx[1]
&\ovalgate{-\hatb}\qwx[1]
&\ovalgate{-\hatc}\qwx[1]
&\ovalgate{-\hatd}\qwx[1]
&\qw
\\
&\ovalgate{\hatd'}
&\ovalgate{\hatc'}
&\ovalgate{\hatb'}
&\ovalgate{\hata'}
&\ovalgate{-\hata'}
&\ovalgate{-\hatb'}
&\ovalgate{-\hatc'}
&\ovalgate{-\hatd'}
&\qw
}
\end{array}\\
&=&
\lam_{4r} + i\lam_{4i} + \Lam_{4r} + i\Lam_{4i}
\;,
\label{eq-invariant-4bit}
\eeqa
where
\beq
\lam_{4r} =
-\sum_j (\hatg'_j\cdot\hatd')\nu_j
\hatg_{\pi(j)}\cdot\hatd
\;,\;\;
\lam_{4i} =
(\lam_{4r})_{\nu\rarrow\mu}
\;,
\eeq
\beq
\Lam_{4r}=
X_o \sigma_{\hatd',\hatd}
+ \sigma_{\vec{x'},\hatd}
+ \sigma_{\hatd',\vecx}
+ \Delta X
\;,
\eeq
\beq
\Lam_{4i}=
Y_o \sigma_{\hatd',\hatd}
- \sigma_{\vec{y'},\hatd}
- \sigma_{\hatd',\vecy}
+ \Delta Y
\;,
\eeq
where
\beq
\vecx = \sum_j \mu_j
(\hatg'_j\cdot\hatd')
\manyx{\hatg_{\pi(j)}\hatd}
\;,\;\;
\vec{y} = (\vecx)_{\mu\rarrow\nu}
\;,
\eeq
\beq
\vec{x'} = \sum_j \mu_j
(\hatg_{\pi(j)}\cdot\hatd)
\manyx{\hatg'_j\hatd'}
\;,\;\;
\vec{y'} = (\vec{x'})_{\mu\rarrow\nu}
\;,
\eeq
\beq
\Delta X =
\sum_j \nu_j
\sigma_{
\manyx{\hatg'_j\hatd'\hatd'},
\manyx{\hatg_{\pi(j)}\hatd\hatd}
}
\;,\;\;
\Delta Y =
(\Delta X)_{\nu\rarrow\mu}
\;,
\eeq
where any variables not already
defined
in the statement of this theorem
are defined in
Section \ref{sec-invariants-3cnots}.
\end{theo}
\proof
\beqa
\calg^{(2)}_4 &=&
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatd}\qwx[1]
&\qw
\\
&\ovalgate{\hatd'}
&\qw
}
\end{array}
\calg_3^{(2)}
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{-\hatd}\qwx[1]
&\qw
\\
&\ovalgate{-\hatd'}
&\qw
}
\end{array}
\\
&=&
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatd}\qwx[1]
&\qw
\\
&\ovalgate{\hatd'}
&\qw
}
\end{array}
\calg_3^{(2)}
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatd}\qwx[1]
&\qw
\\
&\ovalgate{\hatd'}
&\qw
}
\end{array}
(-\sigma_{\hatd',\hatd})
\;.
\eeqa
An explicit expression
for $\calg^{(2)}_3$ was given
in Section \ref{sec-invariants-3cnots}.
Eq.(\ref{eq-sim-trans-of-sig-veca})
shows how to calculate the effect of DC-NOT
similarity transformations.
\qed
\begin{theo}
When the bases
$(\hatg_j)_{j=1,2,3}$
and
$(\hatg'_j)_{j=1,2,3}$
are both taken to be
the standard basis,
then the quantities
$\lam_{4r}$, $\lam_{4i}$
$\vecx$, $\vecy$,
$\vec{x'}$, $\vec{y'}$,
$\Delta X$ and $\Delta Y$
(all defined in
Theorem \ref{eq-invar4})
can be expressed in terms
of the matrices $M_\mu, M_\nu$
and the vectors $\hatd, \hatd'$ as follows:
\beq
\lam_{4r} =
-\hatd^{\;'T}M_\nu\hatd
\;,\;\;
\lam_{4i} =
(\lam_{4r})_{\nu\rarrow\mu}
\;,
\eeq
\beq
\vecx = \manyx{M_\mu^T\hatd',\hatd}
\;,\;\;
\vecy = (\vecx)_{\mu\rarrow\nu}
\;,
\eeq
\beq
\vec{x'} = \manyx{M_\mu\hatd,\hatd'}
\;,\;\;
\vec{y'} = (\vec{x'})_{\mu\rarrow\nu}
\;,
\eeq
\beq
\Delta X=
\hatd'\hatd^T (\hatd^{\;'T}M_\nu\hatd)
-M_\nu \hatd \hatd^T
-\hatd'\hatd^{\;'T} M_\nu
+M_\nu
\;,\;\;
\Delta Y =
(\Delta X)_{\nu\rarrow\mu}
\;.
\eeq
\end{theo}
\proof
Just algebra.
\qed
\begin{theo}
See Fig.\ref{fig-m-mu-nu}.
\begin{subequations}
\beq
M_\nu^T\vec{y'} = Y_o \vecx
\;,\;\;
M_\mu \vecx = X_o \vec{y'}\;,
\eeq
\beq
M_\mu^T\vec{x'} = X_o \vecy
\;,\;\;
M_\nu \vecy = Y_o \vec{x'}
\;.
\eeq
\end{subequations}
\end{theo}
\proof
Just algebra.
\qed
\begin{figure}[h]
\begin{center}
\epsfig{file=m-mu-nu.eps, height=2.3in}
\caption{Various vectors and
what they are mapped into (up to a scalar
factor) by $M_\mu$ and $M_\nu$.
Since $M^T_\nu M_\mu$ and
$M^T_\mu M_\nu$ are both
proportional to the identity matrix,
one can replace $M_\mu$ by $M_\nu^T$
and $M_\nu$ by $M_\mu^T$ in this figure
if one also reverses the direction of
the mapping arrows.
}
\label{fig-m-mu-nu}
\end{center}
\end{figure}
\section{Identities for Circuits with 2 Qubits}
\label{sec-two-qubit-ids}
This section deals with 2-qubit circuits, whereas
Section \ref{sec-3-qubit-ids}
deals with 3-qubit ones.
In this section, with its numerous
subsections, we start to reap the benefits
of all our preceding hard work.
The combination of dressed CNOTs and the LO-RHS
invariant proves to be very useful.
We find simple-to-check necessary
and sufficient conditions for the reduction of
a quantum circuit with $j$ CNOTs to fewer CNOTs, where
$j=2,3$. Plus we show how to express circuits with 1 or
2 controlled-U's as circuits with 2 or fewer CNOTs.
Plus we show how to
open and close a breach, a
procedure that can
reduce any 4-CNOT circuit to a 3-CNOT one.
\subsection{Reducing 2 DC-NOTs}
\label{sec-2-cnots}
\subsubsection{2 to 2 DC-NOTs (Angle Swapping)
\\{\footnotesize\tt[
swap\_angles.m,
test\_swang.m
]}}
\label{sec-2to2-cnots}
In this section we consider
a circuit with 2 DC-NOTs
acting on 2 qubits, and
show that a symmetry in $\calg^{(2)}_2$
allows one to swap
certain angles
without changing the effect of the circuit
(up to LO-RHS).
As motivation for
the main theorem of this section
(the Angle Swapping
Theorem), we present
the next theorem.
The next theorem shows
that
the target and control qubits of
a controlled-U can be exchanged.
\begin{theo}\label{th-contr-u-flip}
For any $\theta\in\RR$,
\beq
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hata}\qwx[1]
&\qw
\\
&\gate{e^{i\theta\sigma_{\hatb'}}}
&\qw
}
\end{array}
=
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\gate{e^{i\theta\sigma_{\hata}}}\qwx[1]
&\gate{e^{-i\frac{\theta}{2}\sigma_\hata}}
&\qw
\\
&\ovalgate{\hatb'}
&\gate{e^{+i\frac{\theta}{2}\sigma_{\hatb'}}}
&\qw
}
\end{array}
\;.
\eeq
\end{theo}
\proof
\beqa
[e^{i\theta\sigma_{\hatb'}(1)}]^{n_\hata(0)}
&=&
e^{i\theta\sigma_{\hatb'}(1)
\left(
\frac{1-\sigma_\hata(0)}{2}
\right)
}\\
&=&
e^{i
\theta\sigma_{\hata}(0)
\left(
\frac{1-\sigma_{\hatb'}(1)}{2}
\right)
}
e^{-i\frac{\theta}{2}\sigma_\hata(0)}
e^{+i\frac{\theta}{2}\sigma_{\hatb'}(1)}
\\
&=&
[e^{i
\theta\sigma_{\hata}(0)}]
^{n_{\hatb'}(1)}
e^{-i\frac{\theta}{2}\sigma_\hata(0)}
e^{+i\frac{\theta}{2}\sigma_{\hatb'}(1)}
\;.
\eeqa
\qed
The previous theorem immediately implies the
next one, which states that
we can ``swap a breach" between two qubits.
\begin{theo}\label{th-swap-breach}
(Swapping a breach)
Suppose
\beq
\call=
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hata}\qwx[1]
&\breach
&\ovalgate{\hata}\qwx[1]
&\qw
\\
&\ovalgate{\hatp'}
&\qw
&\ovalgate{\hatq'}
&\qw
}
\end{array}
\;\;,\;\;\;
\calr=
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatp}\qwx[1]
&\qw
&\ovalgate{\hatq}\qwx[1]
&\qw
\\
&\ovalgate{\hatb'}
&\breach
&\ovalgate{\hatb'}
&\qw
}
\end{array}
\;.
\eeq
For any $\call$, it is possible to
find an $\calr$ such that $\call\sim_R \calr$.
\end{theo}
\proof
Define $\theta$ to be the angle
between $\hatp'$ and $\hatq'$,
and $\hatb$ the direction of $\hatp'\times\hatq'$.
Then
$\hatp'\cdot\hatq'=\cos(\theta)$
and
$\hatp'\times\hatq'=\sin(\theta)\hatb'$ so
$\sigma_{\hatp'}\sigma_{\hatq'}=
e^{i\theta\sigma_{\hatb'}}$.
Thus,
\beq
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hata}\qwx[1]
&\breach
&\ovalgate{\hata}\qwx[1]
&\qw
\\
&\ovalgate{\hatp'}
&\qw
&\ovalgate{\hatq'}
&\qw
}
\end{array}
=
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hata}\qwx[1]
&\qw
\\
&\gate{e^{i\theta\sigma_{\hatb'}}}
&\qw
}
\end{array}
\;.
\label{eq-swap-breach-left}
\eeq
Given a unit vector $\hata$
and an angle $\theta$,
we can always find (non-unique)
unit vectors
$\hatp$ and $\hatq$ such that
$angle(\hatp,\hatq)=\theta$, and
$\hatp\times \hatq$ points along $\hata$. Then
$\hatp\cdot\hatq=\cos(\theta)$
and
$\hatp\times\hatq=\sin(\theta)\hata$ so
$\sigma_{\hatp}\sigma_{\hatq}=
e^{i\theta\sigma_{\hata}}$.
It follows that
\beq
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\gate{e^{i\theta\sigma_{\hata}}}\qwx[1]
&\qw
\\
&\ovalgate{\hatb'}
&\qw
}
\end{array}
=
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatp}\qwx[1]
&\qw
&\ovalgate{\hatq}\qwx[1]
&\qw
\\
&\ovalgate{\hatb'}
&\breach
&\ovalgate{\hatb'}
&\qw
}
\end{array}
\;.
\label{eq-swap-breach-right}
\eeq
Now apply Theorem \ref{th-contr-u-flip}
to Eqs.(\ref{eq-swap-breach-left})
and (\ref{eq-swap-breach-right}).
\qed
Is it possible to swap a foil
instead of a breach? Yes it is.
In fact, one can swap any angle,
as the following
theorem shows.
\begin{theo}(Angle Swapping)
Let
\beq
\call =
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatb}\qwx[1]
&\ovalgate{\hata}\qwx[1]
&\qw
\\
&\ovalgate{\hatb'}
&\ovalgate{\hata'}
&\qw
}
\end{array}
\;\;,\;\;
\calr =
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatb_f}\qwx[1]
&\ovalgate{\hata_f}\qwx[1]
&\qw
\\
&\ovalgate{\hatb'_f}
&\ovalgate{\hata'_f}
&\qw
}
\end{array}
\;.
\eeq
For any $\call$, it is possible to
find an $\calr$ such that $\call \sim_R \calr$
and such that
angle$(\hatb, \hata)$ = angle$(\hatb'_f, \hata'_f)$
and
angle$(\hatb', \hata')$ = angle$(\hatb_f, \hata_f)$.
\end{theo}
\proof
As proven in Section \ref{sec-invariants-2cnots},
$\call^{(2)}$
can be parameterized as follows:
\beq
\call^{(2)}=
c_{\alpha'} c_\alpha
-(s_{\alpha'}s_\alpha) \hatf_2'\hatf_2^T
+i
\begin{array}{l|ll}
& \hatf_1^T & \hatf_3^T\\
\hline
\hatf_3'& s_{\alpha'}c_\alpha & 0\\
\hatf_1'& 0 &c_{\alpha'}s_\alpha
\end{array}
\;,
\eeq
where $\alpha, \alpha\in \RR$
and where $(\hatf_j)_{j=1,2,3}$ and
$(\hatf'_j)_{j=1,2,3}$ are two
RHON bases
such that
\beq
\hatb = \hatf_1
\;,\;\;
\hata = c_\alpha \hatf_1 - s_\alpha \hatf_2
\;,
\eeq
and
\beq
\hatb' = \hatf'_1
\;,\;\;
\hata' = c_{\alpha'} \hatf'_1 - s_{\alpha'} \hatf'_2
\;.
\eeq
$\calr^{(2)}$ can be
parameterized in
the same way as $\call^{(2)}$, but with
the replacements
$\alpha\rarrow\alpha_f$,
$\alpha'\rarrow\alpha'_f$,
$\hatf_j\rarrow (\hatf_j)_f$,
and
$\hatf'_j\rarrow (\hatf'_j)_f$.
Our goal is to construct an $\calr$ such that
$\call\sim_R \calr$. Such an $\calr$,
if it exists, must satisfy
$\hat{\call}^{(2)}= \pm \hat{\calr}^{(2)}$.
We will use the positive sign.
In light of
Eq.(\ref{eq-invar-of-hat-graph}), this gives
\beq
i^2 \call^{(2)} = i^2 \calr^{(2)}
\;.
\eeq
From the symmetrical form of
the parameterized expressions
for $\call^{(2)}$ and
$\calr^{(2)}$,
it is clear
that these two invariants are
equal if their principal parameters are
related in the following way:
\beq
\alpha_f = \alpha'
\;,\;\;
\alpha'_f = \alpha
\;,
\eeq
\beq
\hatf_{3f}=\hatf_1
\;,\;\;
\hatf_{1f}=\hatf_3
\;,\;\,
\hatf_{2f}=-\hatf_2
\;,
\eeq
and
\beq
\hatf'_{3f}=\hatf'_1
\;,\;\;
\hatf'_{1f}=\hatf'_3
\;,\;\,
\hatf'_{2f}=-\hatf'_2
\;.
\eeq
These relations between
the principal parameters of
$\call^{(2)}$ and
$\calr^{(2)}$ imply that
\beqa
\call &=&
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatf_1}\qwx[1]
&\ovalgate{c_\alpha\hatf_1 - s_\alpha\hatf_2}\qwx[1]
&\qw
\\
&\ovalgate{\hatf'_1}
&\ovalgate{c_{\alpha'}\hatf'_1 - s_{\alpha'}\hatf'_2}
&\qw
}
\end{array}
\label{eq-2to2-left}
\\
&=&
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatb}\qwx[1]
&\ovalgate{\hata}\qwx[1]
&\qw
\\
&\ovalgate{\hatb'}
&\ovalgate{\hata'}
&\qw
}
\end{array}
\;,
\eeqa
and
\beqa
\calr
&=&
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatf_3}\qwx[1]
&\ovalgate{c_{\alpha'}\hatf_3 + s_{\alpha'}\hatf_2}\qwx[1]
&\qw
\\
&\ovalgate{\hatf'_3}
&\ovalgate{c_\alpha\hatf'_3 + s_\alpha\hatf'_2}
&\qw
}
\end{array}
\label{eq-2to2-right}
\\
&=&
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\frac{\manyx{\hata\hatb}}{s_\alpha}}\qwx[1]
&\ovalgate{
\frac{
c_{\alpha'}\manyx{\hata\hatb}
+ s_{\alpha'}\manyx{\hata\hatb\hatb}
}{s_\alpha}
}\qwx[1]
&\qw
\\
&\ovalgate{\frac{\manyx{\hata'\hatb'}}{s_{\alpha'}}}
&\ovalgate{
\frac{
c_{\alpha}\manyx{\hata'\hatb'}
+ s_{\alpha}\manyx{\hata'\hatb'\hatb'}
}{s_{\alpha'}}
}
&\qw
}
\end{array}
\label{eq-2to2-singular}
\;.
\eeqa
(Eq.(\ref{eq-2to2-singular}) is valid only if
$s_\alpha$ and $s_{\alpha'}$
are both non-zero, whereas
Eq.(\ref{eq-2to2-right}) is always valid.
Theorem \ref{th-swap-breach}
corresponds to the case
$s_\alpha=0$.)
We are done proving the theorem,
but we will go one step further,
and give the value of the
local operations $U',U\in SU(2)$
such that
\beq
\call = \calr (U^{'\dagger}\otimes U^\dagger)
\;.
\eeq
When $\hatf_1=\hatf'_1=\hatx$
and $\hatf_3=\hatf'_3=\hatz$,
the right-hand sides
of Eqs.(\ref{eq-2to2-left})
and (\ref{eq-2to2-right})
appear in Theorem \ref{th-2thirds-split}.
It follows from Theorem \ref{th-2thirds-split}
and Eq.(\ref{eq-2thirds-u-def})
that
\beq
U = e^{i \frac{\alpha}{2} \sigma_{\hatf_3}}
e^{-i \frac{\alpha'}{2} \sigma_{\hatf_1}}
\;,\;\;
U' = (U)_{\alpha\darrow \alpha', \hatf\rarrow \hatf'}
\;.
\eeq
\qed
\subsubsection{2 to 1 DC-NOTs}
\label{sec-2to1-cnots}
In this section, we give necessary
and sufficient conditions for
a circuit with 2 DC-NOTs acting
on 2 qubits to reduce to
1 DC-NOT.
\begin{figure}[h]
\begin{center}
\epsfig{file=2to1.eps, height=.75in}
\caption{All circuits with 2 DC-NOTs
that reduce to 1 DC-NOT.}
\label{fig-2to1}
\end{center}
\end{figure}
\begin{theo}
Suppose
\beq
\call =
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatb}\qwx[1]
&\ovalgate{\hata}\qwx[1]
&\qw
\\
&\ovalgate{\hatb'}
&\ovalgate{\hata'}
&\qw
}
\end{array}
\;\;,\;\;
\calr =
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hata_f}\qwx[1]
&\qw
\\
&\ovalgate{\hata'_f}
&\qw
}
\end{array}
\;.
\eeq
For any $\call$, it is possible to
find an $\calr$ such that $\call\sim_R \calr$
if and only if
($\hatb \parallel \hata$ and $\hatb'\perp \hata'$)
or
($\hatb \perp \hata$ and $\hatb'\parallel \hata'$).
See Fig.\ref{fig-2to1}.
\end{theo}
\proof
\lproof
Suppose $\hatb \perp \hata$ and
$\hatb'\parallel \hata'$
(the other case is analogous).
When $\hatb \perp \hata$,
\beq
\sigma_{\hatb}(0)^{n_{\hata'}(1)}
\sigma_{\hata}(0)^{n_{\hata'}(1)}
=
[i\sigma_{\hatb\times\hata}(0)]^{n_{\hata'}(1)}
\;.
\eeq
The last equation
can be expressed diagrammatically as
\beq
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatb}\qwx[1]
&\foil
&\ovalgate{\hata}\qwx[1]
&\qw
\\
&\ovalgate{\hata'}
&\breach
&\ovalgate{\hata'}
&\qw
}
\end{array}
=
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatb\times\hata}\qwx[1]
&\qw
&\qw
\\
&\ovalgate{\hata'}
&\gate{ i^{n_{\hata'}}}
&\qw
}
\end{array}
\;.
\eeq
Thus, when $\hatb\perp\hata$
and $\hatb'=\hata'$,
$\call$ reduces to a single DC-NOT.
More generally,
$\hata'=\pm \hatb'$.
Let $\call_{new}$ be a new circuit
obtained by replacing in $\call$:
$\hata'$ by its negative if $\hata'=-\hatb'$.
By virtue of Eq.(\ref{eq-dcnot-with-neg}),
$\call = \call_{new}(I_2\otimes U)$,
where $U\in U(2)$.
If $\call_{new}\sim_R \calr_{new}$, then
$\call\sim_R \calr_{new}$.
\rproof
$\call\sim_R \calr$ so
$\hat{\call}^{(2)}= \pm \hat{\calr}^{(2)}$.
In light of
Eq.(\ref{eq-invar-of-hat-graph}), this gives
\beq
i^2\call^{(2)} = \pm i \calr^{(2)}
\;.
\eeq
It follows that
\beq
\lam_{2r} + \Lam_{2r} + i\Lam_{2i}
=
\pm i\sigma_{\hata'_f, \hata_f}
\;,
\eeq
where
\beq
\lam_{2r}= (\hata\cdot\hatb)(\hata'\cdot\hatb')
\;,
\eeq
\beq
\Lam_{2r}=
-\sigma_{\manyx{\hata'\hatb'\hatb'},\manyx{\hata\hatb\hatb}}
\;,
\eeq
\beq
\Lam_{2i}=
\hata\cdot\hatb\sigma_{\hata'\times\hatb',\hatb}
+
\hata'\cdot\hatb'\sigma_{\hatb',\hata\times\hatb}
\;.
\eeq
$\lam_{2r}=0$ so
$\hata\cdot\hatb=0$ or $\hata'\cdot\hatb'=0$.
Assume the former (the other case is analogous).
Then $\hata\perp\hatb$.
$\Lam_{2r}=0$ and $\hata\cdot\hatb=0$
so $\manyx{\hata'\hatb'\hatb'}=0$,
which in turn implies that
$\hata'\parallel\hatb'$.
\qed
\subsubsection{2 to 0 DC-NOTs}
\label{sec-2to0-cnots}
In this section, we give necessary
and sufficient conditions for
a circuit with 2 DC-NOTs acting
on 2 qubits to reduce to
zero DC-NOTs (i.e., to merely local operations).
\begin{figure}[h]
\begin{center}
\epsfig{file=2to0.eps, height=.75in}
\caption{All circuits with 2 DC-NOTs
that reduce to 0 DC-NOTs.}
\label{fig-2to0}
\end{center}
\end{figure}
\begin{theo}
Suppose
\beq
\call=
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatb}\qwx[1]
&\ovalgate{\hata}\qwx[1]
&\qw
\\
&\ovalgate{\hatb'}
&\ovalgate{\hata'}
&\qw
}
\end{array}
\;.
\eeq
For any $\call$,
$\call\sim_R 1$ if and only if
$\hata\parallel\hatb$ and
$\hata'\parallel\hatb'$.
See Fig.\ref{fig-2to0}.
\end{theo}
\proof
\lproof
When $\hata=\hatb$ and $\hata'=\hatb'$,
$\call$ equals 1.
More generally,
$\hata=\pm\hatb$
and
$\hata'=\pm\hatb'$.
Let $\call_{new}$ be a new circuit
obtained by replacing in $\call$:
(1)$\hata$ by its negative if $\hata=-\hatb$,
(2)$\hata'$ by its negative if $\hata'=-\hatb'$.
By virtue of Eq.(\ref{eq-dcnot-with-neg}),
$\call = \call_{new}(U'\otimes U)$,
where $U',U\in U(2)$.
If $\call_{new}\sim_R 1$, then
$\call\sim_R 1$.
\rproof
$\call\sim_R 1$ so
$\hat{\call}^{(2)}= \pm 1$.
In light of
Eq.(\ref{eq-invar-of-hat-graph}), this gives
\beq
i^2 \call^{(2)}= \pm 1
\;.
\eeq
It follows that
\beq
\lam_{2r} + \Lam_{2r} + i\Lam_{2i}
=\pm 1
\;.
\eeq
Thus
$\lam_{2r}=(\hata\cdot\hatb)(\hata'\cdot\hatb')=\pm 1$,
which implies
$\hata\parallel\hatb$ and
$\hata'\parallel\hatb'$.
\qed
\subsection{Reducing 3 DC-NOTs}
\label{sec-3-cnots}
\subsubsection{3 to 2 DC-NOTs
\\{\footnotesize\tt[
dr\_3to2.m,
test\_dr\_3to2.m
]}}
\label{sec-3to2-cnots}
In this section, we give necessary
and sufficient conditions for
a circuit with 3 DC-NOTs acting
on 2 qubits to reduce to
2 DC-NOTs.
The constraint
$\manyx{\hata\hatb\hatb}\cdot\hatc=0$
shows up below. The field of Spherical Geometry
sheds some light on this constraint.
If we connect the points
$\hata, \hatb, \hatc$ by mayor-circle arcs
on the unit sphere, then we get what is called a
spherical triangle.
$\manyx{\hata\hatb\hatb}\cdot\hatc=0$
if and only if this spherical triangle
has a right angle at
vertex $\hatb$.(See Fig.\ref{fig-phi-lam-prime}
for an example of
$\manyx{\hata'\hatb'\hatb'}\cdot\hatc'=0$.)
\begin{figure}[h]
\begin{center}
\epsfig{file=3to2.eps, height=1in}
\caption{All circuits with 3 DC-NOTs
that reduce to 2 DC-NOTs.}
\label{fig-3to2}
\end{center}
\end{figure}
\begin{theo}
Suppose
\beq
\call=
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatc}\qwx[1]
&\ovalgate{\hatb}\qwx[1]
&\ovalgate{\hata}\qwx[1]
&\qw
\\
&\ovalgate{\hatc'}
&\ovalgate{\hatb'}
&\ovalgate{\hata'}
&\qw
}
\end{array}
\;\;,\;\;
\calr=
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatb_f}\qwx[1]
&\ovalgate{\hata_f}\qwx[1]
&\qw
\\
&\ovalgate{\hatb'_f}
&\ovalgate{\hata'_f}
&\qw
}
\end{array}
\;.
\eeq
For any $\call$,
it is possible to find an $\calr$ such that
$\call\sim_R \calr$
if and only if
either
$\manyx{\hata\hatb\hatb}\cdot\hatc=0$
\;\;or\;\;
$\manyx{\hata'\hatb'\hatb'}\cdot\hatc'=0$.
See Fig.\ref{fig-3to2}.
\end{theo}
\proof
Before we start the proof in earnest,
let us restate some pertinent formulas
taken from previous sections.
From Section \ref{sec-invariants-2cnots}, we know that
\beqa
\calr^{(2)}&=&
\lam_{2r} + \Lam_{2r} + i \Lam_{2i}\\
&=&
c_{\alpha'} c_\alpha
-(s_{\alpha'}s_\alpha) \hatf_2'\hatf_2^T
+i\;
\begin{array}{l|ll}
& \hatf_1^T & \hatf_3^T\\
\hline
\hatf_3'& s_{\alpha'}c_\alpha & 0\\
\hatf_1'& 0 &c_{\alpha'}s_\alpha
\end{array}
\;.
\label{eq-r-invars-3to2}
\eeqa
From Section \ref{sec-invariants-3cnots}, we know that
\beq
\call^{(2)}=
\lam_{3r} + i\lam_{3i} + \Lam_{3r} + i \Lam_{3i}
\;,
\eeq
where
\beq
\lam_{3r} =
\manyx{\hata'\hatb' \hatb'}\cdot\hatc'
\;\;
\manyx{\hata\hatb\hatb}\cdot\hatc
\;,
\eeq
\beq
\lam_{3i} =
-(\hata\cdot\hatb)(\hatb\cdot\hatc)\calv'
-(\hata'\cdot\hatb')(\hatb'\cdot\hatc')\calv
\;,
\label{eq-lam3i-3to2}
\eeq
\beq
\Lam_{3r}=
\left\{
\begin{array}{l}
-(\hata'\cdot\hatb')(\hata\cdot\hatb)
\hatc'\hatc^T
\\
+(\hata\cdot\hatb)(\hatb\cdot\hatc)
\manyx{\hata'\hatb'\hatc'}\hatc^T
+(\hata'\cdot\hatb')(\hatb'\cdot\hatc')
\hatc'\manyx{\hata\hatb\hatc}^T
\\
+(\hata'\cdot\hatb')\calv \manyx{\hatb'\hatc'}\hatc^T
+(\hata\cdot\hatb)\calv'\hatc'\manyx{\hatb\hatc}^T
\\
-
\manyx{\hata'\hatb'\hatb'\hatc'\hatc'}
\manyx{\hata\hatb\hatb\hatc\hatc}^T
\;,
\end{array}
\right.
\eeq
and
\beq
\Lam_{3i}=
\left\{
\begin{array}{l}
+(\hata\cdot\hatb)
\manyx{\hata'\hatb'\hatc'\hatc'}
\manyx{\hatb\hatc\hatc}^T
+
(\hata'\cdot\hatb')
\manyx{\hatb'\hatc'\hatc'}
\manyx{\hata\hatb\hatc\hatc}^T
\\
+\manyx{\hata\hatb\hatb}\cdot \hatc
\manyx{\hata'\hatb'\hatb'\hatc'}\hatc^T
+
\manyx{\hata'\hatb'\hatb'}\cdot \hatc'
\hatc'\manyx{\hata\hatb\hatb\hatc}^T
\end{array}
\right.
\;.
\eeq
Now we begin the proof in earnest.
\rproof
$\call\sim_R \calr$ so
$\hat{\call}^{(2)}= \pm \hat{\calr}^{(2)}$.
In light of
Eq.(\ref{eq-invar-of-hat-graph}), this gives
\beq
i^3 \call^{(2)}=
\pm i^2 \calr^{(2)}
\;.
\eeq
It follows that
\beq
0=\lam_{3r}=
\manyx{\hata'\hatb'\hatb'}\cdot\hatc'
\;\;
\manyx{\hata\hatb\hatb}\cdot\hatc
\;.
\eeq
Thus, either
$\manyx{\hata'\hatb'\hatb'}\cdot\hatc'$
or
$\manyx{\hata\hatb\hatb}\cdot\hatc$.
\lproof
Assume
$\manyx{\hata'\hatb'\hatb'}\cdot\hatc'=0$.
(The other case,
$\manyx{\hata\hatb\hatb}\cdot\hatc=0$,
is analogous).
\begin{figure}[h]
\begin{center}
\epsfig{file=phi-lam-prime.eps, height=2in}
\caption{Vectors and angles associated with
bit-1 space spanned by $\hata',\hatb',\hatc'$. }
\label{fig-phi-lam-prime}
\end{center}
\end{figure}
It is convenient at this point to define
a RHON basis $(\hatk'_j)_{j=1,2,3}$
for the 3d real space spanned by
$\hata',\hatb', \hatc'$. Let
$s_{\lam'} = |\manyx{\hata'\hatb'}|$. If
$s_{\lam'}\neq 0$, let
\beq
(\hatk'_j)_{j=1,2,3}=
(\hatb',
\frac{\manyx{\hata'\hatb'\hatb'}}{s_{\lam'}},
\frac{\manyx{\hata'\hatb'}}{s_{\lam'}}
)
\;.
\label{eq-hprime-basis-3to2}
\eeq
If $s_{\lam'} = 0$,
define $(\hatk'_j)_{j=1,2,3}$
to be any RHON basis such that
$\hatk'_1=\hatb'$ and
$\hatk'_2$ is perpendicular to
$span(\hatb',\hatc')$.
Let
$\phi'=angle(\hatc', \hatk'_3)$.
Since
$\manyx{\hata'\hatb'\hatb'}\cdot\hatc'=0$,
\beq
\hata' = c_{\lam'}\hatk'_1 - s_{\lam'} \hatk'_2,
\;,\;\;
\hatb' = \hatk'_1
\;,\;\;
\hatc' = s_{\phi'} \hatk'_1 + c_{\phi'} \hatk'_3
\;.
\label{eq-abcprime-3to2}
\eeq
Eqs.(\ref{eq-hprime-basis-3to2})
and (\ref{eq-abcprime-3to2})
are illustrated in Fig.\ref{fig-phi-lam-prime}.
Our goal is to construct an $\calr$ such that
$\call\sim_R \calr$. Such an $\calr$ must satisfy
$\hat{\call}^{(2)}= \pm \hat{\calr}^{(2)}$.
We will use the positive sign.
In light of
Eq.(\ref{eq-invar-of-hat-graph}), this gives
\beq
i^3 \call^{(2)} = i^2 \calr^{(2)}
\;.
\eeq
It follows that:
\begin{subequations}
\beq
\lam_{2r} = -\lam_{3i}
\;,
\label{eq-cond1-3to2}
\eeq
\beq
0=\lam_{3r}
\;,
\label{eq-cond2-3to2}
\eeq
\beq
\Lam_{2r} = -\Lam_{3i}
\;,
\label{eq-cond3-3to2}
\eeq
\beq
\Lam_{2i} = \Lam_{3r}
\;.
\label{eq-cond4-3to2}
\eeq
\end{subequations}
By evaluating Eq.(\ref{eq-cond1-3to2}), we get
\beqa
c_{\alpha'}c_\alpha
&=&
(\hata\cdot\hatb)(\hatb\cdot\hatc)\calv'
+
(\hata'\cdot\hatb')(\hatb'\cdot\hatc')\calv
\\
&=&
(\hata\cdot\hatb)(\hatb\cdot\hatc)
s_{\lam'}c_{\phi'}
+
c_{\lam'}s_{\phi'}\calv
\;.
\eeqa
Eq.(\ref{eq-cond2-3to2})
is satisfied since
$\manyx{\hata'\hatb'\hatb'}\cdot\hatc'=0$
by assumption.
By evaluating Eq.(\ref{eq-cond3-3to2}), we get
\beq
-s_{\alpha'}s_\alpha \hatf'_2 \hatf_2^T
=
\left\{
\begin{array}{l}
-(\hata\cdot\hatb)\manyx{\hata'\hatb'\hatc'\hatc'}
\manyx{\hatb\hatc\hatc}^T
\\
-(\hata'\cdot\hatb')\manyx{\hatb'\hatc'\hatc'}
\manyx{\hata\hatb\hatc\hatc}^T
\\
-\manyx{\hata\hatb\hatb}\cdot\hatc
\manyx{\hata'\hatb'\hatb'\hatc'}\hatc^T
\end{array}
\right.
\;.
\eeq
Define $\vech$ by
\beq
\vech =
\left\{
\begin{array}{l}
+s_{\lam'}s_{\phi'}
(\hata\cdot\hatb)
\manyx{\hatb\hatc\hatc}^T
\\
-c_{\lam'}c_{\phi'}
\manyx{\hata\hatb\hatc\hatc}^T
\\
+s_{\lam'}\manyx{\hata\hatb\hatb}\cdot\hatc\hatc^T
\end{array}
\right.
\;.
\eeq
If $s_{\lam'}\neq 0$
and $|\vech|\neq 0$, let
\beq
s_{\alpha'}s_\alpha = |\vech|
\;,\;\;
\hatf'_2 =
\frac{
\manyx{\hata'\hatb'\hatb'\hatc'}
}{s_{\lam'}}
\;,\;\;
\hatf_2 = \frac{\vech}{|\vech|}
\;.
\label{eq-sin-sin-3to2}
\eeq
If $|\vech|=0$, set
$s_{\alpha'}s_\alpha=0$ and
choose
any unit vectors for $\hatf_2$
and $\hatf'_2$.
If $|\vech|\neq 0$
but $s_{\lam'}=0$,
keep Eq.(\ref{eq-sin-sin-3to2})
for $s_{\alpha'}s_\alpha$
and $\hatf_2$ but
use $\hatf'_2 = \hatk'_2\times\hatc'$.
By evaluating Eq.(\ref{eq-cond4-3to2}), we get
\beq
\Lam_{2i} = \hatc'\vecv_1^T + \hatk'_2\vecv_2^T
\;,
\label{eq-Lam2i-svd}
\eeq
where
\begin{subequations} \label{eq-defs-v1-v2}
\beq
\vecv_1 =
-c_{\lam'}(\hata\cdot\hatb)\hatc
+c_{\lam'}s_{\phi'}\manyx{\hata\hatb\hatc}
+s_{\lam'}c_{\phi'}(\hata\cdot\hatb)\manyx{\hatb\hatc}
\;,
\eeq
and
\beq
\vecv_2 =
s_{\lam'}s_{\phi'}(\hata\cdot\hatb)(\hatb\cdot\hatc)\hatc
-c_{\lam'}c_{\phi'}\calv\hatc
+s_{\lam'}\manyx{\hata\hatb\hatb\hatc\hatc}
\;.
\eeq
\end{subequations}
At this point, we can follow from
step \ref{item-diag-invar2-h-hprime}
to the end of the
Algorithm for Diagonalizing $\calg^{(2)}_2$
that was given in
Section \ref{sec-invariants-2cnots}.
This will yield values for
$\hata_f$,
$\hata'_f$,
$\hatb_f$, and
$\hatb'_f$.
\qed
Compared with the previous
Theorem,
the next theorem
imposes more constraints on $\call$,
and obtains a
more constrained $\calr$.
\begin{theo}\label{th-persistence}
Suppose
\beq
\call=
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatc}\qwx[1]
&\ovalgate{\hatb}\qwx[1]
&\ovalgate{\hata}\qwx[1]
&\qw
\\
&\ovalgate{\hatc'}
&\ovalgate{\hatb'}
&\ovalgate{\hata'}
&\qw
}
\end{array}
\;\;,\;\;
\calr=
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatb_f}\qwx[1]
&\ovalgate{\hata_f}\qwx[1]
&\qw
\\
&\ovalgate{\hatb'_f}
&\ovalgate{\hata'_f}
&\qw
}
\end{array}
\;.
\eeq
Let $\lam'=angle(\hata',\hatb')$
and $\phi'=angle(\hatc',\hata'\times\hatb')$.
For any $\call$, if
\begin{subequations}
\beq
\manyx{\hata'\hatb'\hatb'}\cdot\hatc'=0
\;,
\eeq
and
\beq
\left[c_{\phi'}(\hata\cdot\hatb)
\manyx{\hata\hatb} -
s_{\lam'}c_{\lam'}s_{\phi'}\hatb\right]\cdot\hatc=0
\;,
\eeq
\end{subequations}
then
it is possible to find an $\calr$ such that
$\call\sim_R \calr$
and such that $\hatb'_f=\hatc'$.
(Hence,
$\hatc'$
``persists", from initial
circuit $\call$ to final circuit $\calr$,
as the bottom defining vector
of the leftmost DC-NOT for both circuits.
)
\end{theo}
\proof
The $(\Larrow)$ part of the
proof of the previous theorem
still applies.
Using the definitions
of $\vecv_1$ and $\vecv_2$
given by Eqs.(\ref{eq-defs-v1-v2}), it is not hard
to show that
\beq
\vecv_1^T\vecv_2=0
\;\;\iff\;\;
\left[c_{\phi'}(\hata\cdot\hatb)
\manyx{\hata\hatb} -
s_{\lam'}c_{\lam'}s_{\phi'}\hatb\right]\cdot\hatc=0
\;.
\eeq
Since
$\vecv_1$ and $\vecv_2$ are
orthogonal,
the singular
values and singular vectors
of $\Lam_{2i}$
can be obtained simply by inspection
of Eq.(\ref{eq-Lam2i-svd}).
If $|\hatv_1|\neq 0$ and
$|\hatv_2|\neq 0$, then
one can immediately set
\beq
\hatf'_3 = \hatk'_2
\;,\;\;
\hatf_1 = \frac{\vecv_2}{|\vecv_2|}
\;,\;\;
s_{\alpha'}c_\alpha = |\vecv_2|
\;,
\eeq
and
\beq
\hatf'_1 = \hatc'
\;,\;\;
\hatf_3 = \frac{\vecv_1}{|\vecv_1|}
\;,\;\;
c_{\alpha'}s_\alpha = |\vecv_1|
\;.
\eeq
If $|\vecv_1|=0$ but
$|\vecv_2|\neq 0$,
choose $\hatf_3=\hatf_1\times\hatf_2$.
If $|\vecv_1|\neq 0$ but
$|\vecv_2|= 0$,
choose $\hatf_1=\hatf_2\times\hatf_3$.
If $|\vecv_1|=0$ and $|\vecv_2|=0$,
choose $\hatf_1$ and $\hatf_3$
to be any vectors that make
$(\hatf_j)_{j=1,2,3}$ a RHON basis.
\qed
\subsubsection{3 to 1 DC-NOTs}
\label{sec-3to1-cnots}
In this section, we give necessary
and sufficient conditions for
a circuit with 3 DC-NOTs acting
on 2 qubits to reduce to
1 DC-NOT.
\begin{figure}[h]
\begin{center}
\epsfig{file=3to1.eps, height=2.5in}
\caption{All circuits with 3 DC-NOTs
that reduce to 1 DC-NOTs. The 8 circuits
$\cpq_{\pm p, \pm p}$ and $\cpq_{\pm q, \pm q}$
are defined by Eq.(\ref{eq-all-t4}).}
\label{fig-3to1}
\end{center}
\end{figure}
\begin{theo}
Suppose
\beq
\call=
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatc}\qwx[1]
&\ovalgate{\hatb}\qwx[1]
&\ovalgate{\hata}\qwx[1]
&\qw
\\
&\ovalgate{\hatc'}
&\ovalgate{\hatb'}
&\ovalgate{\hata'}
&\qw
}
\end{array}
\;\;,\;\;
\calr=
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hata_f}\qwx[1]
&\qw
\\
&\ovalgate{\hata'_f}
&\qw
}
\end{array}
\;.
\eeq
Let
$\calv = \hata\times\hatb\cdot\hatc$,
and $\calv' = \hata'\times\hatb'\cdot\hatc'$.
For any $\call$,
it is possible to find an $\calr$ such that
$\call\sim_R \calr$
if and only if
one or more of the following
are true:
(See Fig.\ref{fig-3to1})
\begin{enumerate}
\item[$T_{1a}$]:
$(\hatb \parallel \hata)$ and
$(\hatb' \parallel \hata')$
\item[$T_{1b}$]:
$(\hatc \parallel \hatb)$ and
$(\hatc' \parallel \hatb')$
\item[$T_{2a}$]:
$(\hatc' \parallel\hatb' \parallel \hata')$
and
$\calv=0$
\item[$T_{2b}$]:
$(\hatc \parallel \hatb \parallel \hata)$
and
$\calv'=0$
\item[$T_{3a}$]:
$\hata \perp span(\hatb,\hatc)$
and
$\hata' \perp span(\hatb',\hatc')$
\item[$T_{3b}$]:
$\hatc \perp span(\hata,\hatb)$
and
$\hatc' \perp span(\hata',\hatb')$
\item[$T_4$]:
\beq
\left\{
\begin{array}{l}
\manyx{\hata\hatb\hatb}\cdot\hatc=0
\;\;and\;\;
\manyx{\hata'\hatb'\hatb'}\cdot\hatc'=0
\\
\frac{|\hata\times\hatb|}{|\hata\cdot\hatb|}=
\frac{|\hata'\times\hatb'|}{|\hata'\cdot\hatb'|}
\;\;and\;\;
\frac{|\hatb\times\hatc|}{|\hatb\cdot\hatc|}=
\frac{|\hatb'\times\hatc'|}{|\hatb'\cdot\hatc'|}
\\
\sign
\left(
\frac{\calv}{(\hata\cdot\hatb)(\hatb\cdot\hatc)}
\right)
=
-\sign
\left(
\frac{\calv'}{(\hata'\cdot\hatb')(\hatb'\cdot\hatc')}
\right)
\end{array}
\right.
\;.
\label{eq-t4-def}
\eeq
\end{enumerate}
\end{theo}
\proof
\lproof
Consider a circuit of type $T_{1b}$
($T_{1a}$ case is analogous).
When $\hatc=\hatb$ and $\hatc'=\hatb'$,
it is obvious that a $T_{1b}$ circuit
reduces to a single DC-NOT.
More generally,
$\hatc=\pm\hatb$
and
$\hatc'=\pm\hatb'$.
Let $\call_{new}$ be a new circuit
obtained by replacing in $\call$:
(1)$\hatc$ by its negative if $\hatc=-\hatb$,
(2)$\hatc'$ by its negative if $\hatc'=-\hatb'$.
By virtue of Eq.(\ref{eq-dcnot-with-neg}),
$\call = (U'\otimes U)\call_{new}$,
where $U',U\in U(2)$.
If $\call_{new}\sim_R \calr_{new}$, then
$\call\sim_R (U'\otimes U)\calr_{new}
(U^{'\dagger}\otimes U^\dagger)$.
Now consider a circuit of type $T_{2a}$
($T_{2b}$ case is analogous).
Note that when $\calv=0$,
\beq
\sigma_\hatc\sigma_\hatb\sigma_\hata=
\sigma_{
(\hata\cdot\hatb)\hatc-\manyx{\hata\hatb\hatc}
}
=\sigma_{\hata_f}
\;,
\eeq
so
\beq
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatc}\qwx[1]
&\ovalgate{\hatb}\qwx[1]
&\ovalgate{\hata}\qwx[1]
&\qw
\\
&\ovalgate{\hata'}
&\ovalgate{\hata'}
&\ovalgate{\hata'}
&\qw
}
\end{array}
=
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\gate{\sigma_\hatc\sigma_\hatb\sigma_\hata}\qwx[1]
&\qw
\\
&\ovalgate{\hata'}
&\qw
}
\end{array}
=
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hata_f}\qwx[1]
&\qw
\\
&\ovalgate{\hata'}
&\qw
}
\end{array}
\;.
\eeq
Thus, when
$\hata'=\hatb'=\hatc'$,
a $T_{2a}$ circuit reduces to a
single DC-NOT.
More generally, $\hata'=\pm \hatb'$
and $\hatc'=\pm \hatb'$.
Let $\call_{new}$ be a new circuit
obtained by replacing in $\call$:
(1)$\hata'$ by its negative if $\hata'=-\hatb'$,
(2)$\hatc'$ by its negative if $\hatc'=-\hatb'$.
By virtue of Eq.(\ref{eq-dcnot-with-neg}),
$\call = (I_2\otimes U)\call_{new}(I_2\otimes V)$,
where $U,V\in U(2)$.
If $\call_{new}\sim_R \calr_{new}$, then
$\call\sim_R (I_2\otimes U)\calr_{new}
(I_2\otimes U^\dagger)$.
Circuits of type $T_{3a}$
($T_{3b}$ case is analogous)
reduce to a single DC-NOT by
virtue of Theorem \ref{th-2thirds-ab}.
Now consider a circuit of type $T_4$.
For any $w_1,w_2\in\{x,y,z\}$ and
$\xi\in \RR$,
let $\hatp_{w_1,w_2}^\xi$ and
$\hatq_{w_1,w_2}^\xi$ be defined as in
Eq.(\ref{eq-general-p-q-def}).
Because of the first line
of Eq.(\ref{eq-t4-def}),
one can choose a special coordinate system
for bit 0 such that
$\hatc\rarrow \hatpzx^\phi$,
$\hatb\rarrow \hatx$,
$\hata\rarrow \hatqxy^\lam$,
and a special coordinate system for bit 1 such that
$\hatc'\rarrow \hatpzx^{\phi'}$,
$\hatb'\rarrow \hatx$,
$\hata'\rarrow \hatqxy^{\lam'}$.
See Fig.\ref{fig-double-phi-lam}.
$\hatc,\hatb,\hata$ and
$\hatc',\hatb',\hata'$ are portrayed in
Fig.\ref{fig-double-phi-lam},
when $(\hatk_j)_{j=1,2,3}$ and
$(\hatk'_j)_{j=1,2,3}$ are
the standard basis.
In the special coordinate systems, the first line of
Eq.(\ref{eq-t4-def})
is satisfied by construction.
The second line of Eq.(\ref{eq-t4-def}) becomes
\beq
|\tan \lam| = |\tan \lam'|
\;\;{\rm and}\;\;
|\tan \phi| = |\tan \phi'|
\;,
\label{eq-tan-lam-phi-abs}
\eeq
and the third line
\beq
\sign\left(
\frac{\tan\lam}{\tan\phi}
\right)
=
-\sign\left(
\frac{\tan\lam'}{\tan\phi'}
\right)
\;.
\label{eq-tan-lam-phi-sig}
\eeq
In general, Eq.(\ref{eq-tan-lam-phi-abs})
is satisfied iff
$\lam'\in\{\pm\lam, \pi\pm \lam\} + 2\pi \ZZ$
and
$\phi'\in\{\pm\phi, \pi\pm \phi\} + 2\pi \ZZ$.
This gives
16 sign possibilities, but
only 8 of them satisfy
Eq.(\ref{eq-tan-lam-phi-sig}).
Indeed,
let
$\cpq_{\pm p, \pm p}$ and
$\cpq_{\pm q, \pm q}$
denote the following 8 circuits:
\beq
\cpq_{(-1)^{m}r, (-1)^{n}r}
=
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatpzx^\phi}\qwx[1]
&\ovalgate{\hatx}\qwx[1]
&\ovalgate{\hatqxy^\lam}\qwx[1]
&\qw
\\
&\ovalgate{(-1)^{m}r^\phi_{zx}}
&\ovalgate{\hatx}
&\ovalgate{(-1)^{n}r^\lam_{xy}}
&\qw
}
\end{array}
\;,
\label{eq-all-t4}
\eeq
where $r\in \{p,q\}$
and $m,n\in Bool$.
The following $4\times 4$
matrix has rows
labeled by
the 4 possible values of $\phi'$,
and columns labeled by the
4 possible values of $\lam'$.
As its $(\phi',\lam')$ entry,
the matrix has: the $T_4$
circuit implied by
that
value of $(\phi',\lam')$,
if such a circuit exists, or an
$\times$ if none exists.
\beq
\begin{array}{l||c|c|c|c}
\phi'=\downarrow,\lam'=\rarrow & \lam &\pi-\lam & \pi+\lam & -\lam \\
\hline\hline
\phi & \times & \cpq_{p,-p} & \times &\cpq_{p,p}\\
\hline
\pi-\phi & \cpq_{-q,q} & \times & \cpq_{-q,-q} & \times\\
\hline
\pi+\phi & \times & \cpq_{-p,-p} & \times &\cpq_{-p,p}\\
\hline
-\phi & \cpq_{q,q} & \times & \cpq_{q,-q} & \times
\end{array}
\;.
\label{eq-table-t4-ckts}
\eeq
In conclusion,
the 3 lines of Eq.(\ref{eq-t4-def})
imply, in the special coordinate systems,
a circuit of
type Eq.(\ref{eq-all-t4}).
For $\cpq_{q,q}$
(ditto, for $\cpq_{p,p}$), there
exists an $\calr$ such that
$\call\sim_R\calr$ by
virtue of Eq.(\ref{eq-split-sim-trans-id})
(ditto, Eq.(\ref{eq-split-sim-trans-id2})).
The other 6 circuits of table
Eq.(\ref{eq-table-t4-ckts}) can be handled as follows.
Let $\call_{new}$ be a new circuit
obtained by replacing in $\call$:
(1)$\lam'$ by $\lam'-\pi$
if $\lam'=\pi\pm\lam \mod(2\pi)$,
(2)$\phi'$ by $\phi'-\pi$
if $\phi'=\pi\pm\phi\mod(2\pi)$.
By virtue of Eq.(\ref{eq-dcnot-with-neg}),
$\call = (U'\otimes U)\call_{new}(V'\otimes V)$
where $U',U,V',V\in U(2)$, and where
$\call_{new}$
is of type
$\cpq_{q,q}$ or $\cpq_{p,p}$.
If $\call_{new}\sim_R \calr_{new}$, then
$\call\sim_R (U'\otimes U)\calr_{new}
(U^{'\dagger}\otimes U^\dagger)$.
\rproof
$\call\sim_R \calr$ so
$\hat{\call}^{(2)}= \pm \hat{\calr}^{(2)}$.
In light of
Eq.(\ref{eq-invar-of-hat-graph}), this gives
\beq
i^3 \call^{(2)}
= \pm i \calr^{(2)}
\;.
\eeq
It follows that
\beq
\lam_{3r} + i\lam_{3i} + \Lam_{3r} + i\Lam_{3i} = \pm \sigma_{\hata'_f,\hata_f}
\;,
\eeq
where
\beq
\lam_{3r} =
\manyx{\hata'\hatb' \hatb'}\cdot\hatc'
\;\;
\manyx{\hata\hatb\hatb}\cdot\hatc
\;,
\eeq
\beq
\lam_{3i} =
-(\hata\cdot\hatb)(\hatb\cdot\hatc)\calv'
-(\hata'\cdot\hatb')(\hatb'\cdot\hatc')\calv
\;,
\label{eq-lam3i-3to1}
\eeq
\beq
\Lam_{3r}=
\left\{
\begin{array}{l}
-(\hata'\cdot\hatb')(\hata\cdot\hatb)
\hatc'\hatc^T
\\
+(\hata\cdot\hatb)(\hatb\cdot\hatc)
\manyx{\hata'\hatb'\hatc'}\hatc^T
+(\hata'\cdot\hatb')(\hatb'\cdot\hatc')
\hatc'\manyx{\hata\hatb\hatc}^T
\\
+(\hata'\cdot\hatb')\calv \manyx{\hatb'\hatc'}\hatc^T
+(\hata\cdot\hatb)\calv'\hatc'\manyx{\hatb\hatc}^T
\\
-
\manyx{\hata'\hatb'\hatb'\hatc'\hatc'}
\manyx{\hata\hatb\hatb\hatc\hatc}^T
\;,
\end{array}
\right.
\eeq
and
\beq
\Lam_{3i}=
\left\{
\begin{array}{l}
+(\hata\cdot\hatb)
\manyx{\hata'\hatb'\hatc'\hatc'}
\manyx{\hatb\hatc\hatc}^T
+
(\hata'\cdot\hatb')
\manyx{\hatb'\hatc'\hatc'}
\manyx{\hata\hatb\hatc\hatc}^T
\\
+\manyx{\hata\hatb\hatb}\cdot \hatc
\manyx{\hata'\hatb'\hatb'\hatc'}\hatc^T
+
\manyx{\hata'\hatb'\hatb'}\cdot \hatc'
\hatc'\manyx{\hata\hatb\hatb\hatc}^T
\end{array}
\right.
\;.
\eeq
{\bf First assume that there
are no breaches in} $\call$ (i.e.,
$\hata \not\parallel \hatb$,
$\hatb \not\parallel \hatc$,
$\hata' \not\parallel \hatb'$,
$\hatb' \not\parallel \hatc'$).
Note that
\beq
\manyx{\hata\hatb\hatb}\cdot\hatc=0
\;\;{\rm and}\;\;
\manyx{\hata'\hatb'\hatb'}\cdot\hatc'=0
\;.
\label{eq-2right-triangles}
\eeq
This is why.
From $\lam_{3r}=0$, we must
have either
$\manyx{\hata\hatb\hatb}\cdot\hatc=0$
or
$\manyx{\hata'\hatb'\hatb'}\cdot\hatc'=0$.
But if one of these holds,
then the other one follows. Indeed,
assume
$\manyx{\hata\hatb\hatb}\cdot\hatc=0$.
Since also
$\manyx{\hata\hatb}\neq 0$,
it follows that
$\manyx{\hata\hatb\hatb\hatc}\neq 0$.
From $\Lam_{3i}=0$,
$\hatc'^T\Lam_{3i}=
\manyx{\hata'\hatb'\hatb'}\cdot\hatc'
\manyx{\hata\hatb\hatb\hatc}^T=0$
so
$\manyx{\hata'\hatb'\hatb'}\cdot\hatc'=0$.
By an analogous argument,
assuming
$\manyx{\hata'\hatb'\hatb'}\cdot\hatc'=0$
leads to
$\manyx{\hata\hatb\hatb}\cdot\hatc=0$.
Next note that
\begin{subequations}
\label{eq-perp-spreading}
\beq
\hata\cdot\hatb=\hata'\cdot\hatb'=0
\;\;\Rarrow\;\;
\hata\cdot\hatc=\hata'\cdot\hatc'=0
\;,
\label{eq-perp-spreading-from-a}
\eeq
and
\beq
\hatc\cdot\hatb=\hatc'\cdot\hatb'=0
\;\;\Rarrow\;\;
\hatc\cdot\hata=\hatc'\cdot\hata'=0
\;.
\label{eq-perp-spreading-from-c}
\eeq
\end{subequations}
Eqs.(\ref{eq-perp-spreading})
become obvious if one
uses the BAC minus CAB identity
to expand Eqs.(\ref{eq-2right-triangles}).
\begin{figure}[h]
\begin{center}
\epsfig{file=double-phi-lam.eps, height=2.5in}
\caption{Vectors and angles associated with
bit-0 space spanned by $\hata,\hatb,\hatc$.
Vectors and angles associated with
bit-1 space spanned by $\hata',\hatb',\hatc'$. }
\label{fig-double-phi-lam}
\end{center}
\end{figure}
It is convenient at this point to define
a RHON basis $(\hatk'_j)_{j=1,2,3}$
for the 3d real space spanned by
$\hata',\hatb', \hatc'$. Let
$s_{\lam'} = |\manyx{\hata'\hatb'}|$.
If $s_{\lam'}\neq 0$, let
\beq
(\hatk'_j)_{j=1,2,3}=
(\hatb',
\frac{\manyx{\hata'\hatb'\hatb'}}{s_{\lam'}},
\frac{\manyx{\hata'\hatb'}}{s_{\lam'}}
)
\;.
\label{eq-gprime-basis-3to1}
\eeq
If $s_{\lam'}=0$,
let $(\hatk_j)_{j=1,2,3}$ be any
RHON basis such that $\hatk'_1=\hatb'$
and $\hatk'_2$ is perpendicular
to $span(\hatb',\hatc')$.
Let
$\phi'=angle(\hatc', \hatk'_3)$.
Since
$\manyx{\hata'\hatb'\hatb'}\cdot\hatc'=0$,
\beq
\hata' = c_{\lam'}\hatk'_1 - s_{\lam'} \hatk'_2,
\;,\;\;
\hatb' = \hatk'_1
\;,\;\;
\hatc' = s_{\phi'} \hatk'_1 + c_{\phi'} \hatk'_3
\;.
\label{eq-abcprime-3to1}
\eeq
Eqs.(\ref{eq-gprime-basis-3to1})
and (\ref{eq-abcprime-3to1})
are illustrated in Fig.\ref{fig-double-phi-lam}.
Use the previous paragraph with all
the primes removed
to define angles $\lam,\phi$ and
a RHON basis
$(\hatk_j)_{j=1,2,3}$
for the 3d real space spanned by
$\hata,\hatb,\hatc$.
When expressed in terms of
$\lam,\lam',\phi$ and $\phi'$,
the constraint
$\lam_{3i}=0$
reduces to
\beq
-[
s_{\lam'}c_\lam c_{\phi'}s_\phi
+c_{\lam'}s_\lam s_{\phi'}c_\phi] = 0
\;.
\label{eq-eq1-of-system}
\eeq
Likewise, the constraint
$\Lam_{3i}=0$ reduces to
\beq
-[
s_{\lam'}c_\lam s_{\phi'}c_\phi
+c_{\lam'}s_\lam c_{\phi'}s_\phi]
\manyx{\hatk'_2\hatc'}\manyx{\hatk_2\hatc}^T = 0
\;.
\label{eq-eq2-of-system}
\eeq
Eqs.(\ref{eq-eq1-of-system}) and
(\ref{eq-eq2-of-system}) imply the following
system of 2 equations:
\beq
\left[
\begin{array}{cc}
c_{\phi'}s_\phi & s_{\phi'}c_\phi\\
s_{\phi'}c_\phi & c_{\phi'}s_\phi
\end{array}
\right]
\left[
\begin{array}{c}
s_{\lam'}c_\lam \\
c_{\lam'}s_\lam
\end{array}
\right]
=0
\;.
\label{eq-system-lam-sol}
\eeq
This system of equations can
also be rewritten in the form:
\beq
\left[
\begin{array}{cc}
c_{\lam'}s_\lam & s_{\lam'}c_\lam\\
s_{\lam'}c_\lam & c_{\lam'}s_\lam
\end{array}
\right]
\left[
\begin{array}{c}
s_{\phi'}c_\phi \\
c_{\phi'}s_\phi
\end{array}
\right]
=0
\;.
\label{eq-system-phi-sol}
\eeq
For Eq.(\ref{eq-system-lam-sol}),
either (i)the solution is the zero vector,
or (ii)the determinant of
the coefficient matrix vanishes. (i)If the
solution is zero, then
$s_{\lam'}c_\lam=c_{\lam'} s_\lam=0$.
Since we are assuming no breaches,
$s_{\lam'}\neq 0$ and $s_{\lam}\neq 0$,
so we must have $c_\lam= c_{\lam'}=0$.
By virtue of Eq.(\ref{eq-perp-spreading-from-a}),
this means that the circuit
must be of type $T_{3a}$. (ii) If the
determinant is zero, then
\beq
|\tan \phi\;| = |\tan \phi'\;|
\;.
\label{eq-tan-phi}
\eeq
We will pursue this possibility later on.
Likewise, for Eq.(\ref{eq-system-phi-sol}),
either (i)the solution is the zero vector,
or (ii)the determinant of
the coefficient matrix vanishes. (i)If the
solution is zero, then
$s_{\phi'}c_\phi=c_{\phi'} s_\phi=0$.
Since we are assuming no breaches,
$c_{\phi}\neq 0$ and $c_{\phi'}\neq 0$,
so we must have $s_{\phi'}= s_{\phi}=0$.
By virtue of Eq.(\ref{eq-perp-spreading-from-c}),
this means that the circuit
must be of type $T_{3b}$. (ii) If the
determinant is zero, then
\beq
|\tan \lam\;| = |\tan \lam'\;|
\;.
\label{eq-tan-lam}
\eeq
We will pursue this possibility later on.
Suppose we assume that the circuit $\call$
is not of type $T_3$. Then, we have shown
that it must satisfy
Eqs.(\ref{eq-tan-phi}) and (\ref{eq-tan-lam}). But
these two equations are
the second line of Eq.(\ref{eq-t4-def}).
To prove that the circuit must be
of type $T_4$,
it remains for us to prove that the third
line of Eq.(\ref{eq-t4-def}) also holds.
This third line
clearly follows from $\lam_{3i}=0$,
where $\lam_{3i}$
is given by Eq.(\ref{eq-lam3i-3to1}).
{\bf Next , assume that there is at least one
breach in} $\call$. Without loss of generality,
assume there is a breach between $\hata$ and
$\hatb$ (i.e., $\hata \parallel \hatb$).
$\hata \parallel \hatb$ implies
that $\calv=0$.
The constraint $\lam_{3i}=0$ reduces to
\beq
(\hatb\cdot\hatc)\calv'=0
\;,
\label{eq-b-c-volp}
\eeq
which implies that either $\hatb\cdot\hatc=0$
or $\calv'=0$.
The constraint $\Lam_{3i}=0$ reduces
to
\beq
\manyx{\hata'\hatb'\hatc'\hatc'}
\manyx{\hatb\hatc\hatc}^T =0
\;,
\eeq
which implies that either
(i)$\hatb\parallel \hatc$
or (ii)$\hata'\parallel \hatb'$
or (iii)$\manyx{\hata'\hatb'}\parallel \hatc'$.
(i)If $\hatb\parallel \hatc$,
then, by Eq.(\ref{eq-b-c-volp}), $\calv'=0$.
$(\hata\parallel \hatb \parallel \hatc)$
and $\calv'=0$ so $\call$ is of
type $T_{2b}$.
(ii)If $\hata'\parallel \hatb'$, then, since
also $\hata\parallel \hatb$,
$\call$ is of type $T_{1a}$.
(iii)Suppose $\manyx{\hata'\hatb'}\parallel \hatc'$.
Assume that $\hata'\not \parallel \hatb'$
as the case when these two vectors are parallel
has already been considered. It follows that
the conditions $T_{3b}$ are satisfied.
This is why.
$\manyx{\hata'\hatb'}\parallel \hatc'$
and $\hata'\not \parallel \hatb'$ imply that
$\calv'\neq 0$, and, therefore, by virtue
of Eq.(\ref{eq-b-c-volp}), $\hatc\perp\hatb$.
Now $\hatc\perp\hatb$ and $\hata\parallel\hatb$
imply that $\hatc\perp\hata$.
Thus, $\hatc\perp span(\hatb,\hata)$.
Also, $\manyx{\hata'\hatb'}\parallel \hatc'$
implies that $\hatc'\perp span(\hatb',\hata')$.
\qed
\subsubsection{3 to 0 DC-NOTs}
\label{sec-3to0-cnots}
In this section, we give necessary
and sufficient conditions for
a circuit with 3 DC-NOTs acting
on 2 qubits to reduce to
zero DC-NOTs (i.e., to merely local operations).
\begin{figure}[h]
\begin{center}
\epsfig{file=3to0.eps, height=1in}
\caption{All circuits with 3 DC-NOTs
that reduce to 0 DC-NOTs.}
\label{fig-3to0}
\end{center}
\end{figure}
\begin{theo}
Suppose
\beq
\call=
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatc}\qwx[1]
&\ovalgate{\hatb}\qwx[1]
&\ovalgate{\hata}\qwx[1]
&\qw
\\
&\ovalgate{\hatc'}
&\ovalgate{\hatb'}
&\ovalgate{\hata'}
&\qw
}
\end{array}
\;.
\eeq
For any $\call$,
$\call\sim_R 1$ if and only if
one of the following is true
(see Fig.\ref{fig-3to0})
\begin{enumerate}
\item[$T_a$]:
$(\hatc',\hatb',\hata')$ are mutually orthogonal,
and
$(\hatc\parallel\hatb\parallel\hata)$
\item[$T_b$]:
$(\hatc,\hatb,\hata)$ are mutually orthogonal,
and
$(\hatc'\parallel\hatb'\parallel\hata')$
\end{enumerate}
\end{theo}
\proof
\lproof
Consider a circuit of type $T_b$
($T_a$ case is analogous).
Note that when $(\hatc,\hatb,\hata)$
are mutually orthogonal,
\beq
\sigma_\hatc\sigma_\hatb\sigma_\hata=
i\hatc\cdot\manyx{\hatb\hata} = \pm i
\;.
\eeq
Hence,
\beq
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatc}\qwx[1]
&\ovalgate{\hatb}\qwx[1]
&\ovalgate{\hata}\qwx[1]
&\qw
\\
&\ovalgate{\hata'}
&\ovalgate{\hata'}
&\ovalgate{\hata'}
&\qw
}
\end{array}
=
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\gate{\sigma_\hatc\sigma_\hatb\sigma_\hata}\qwx[1]
&\qw
\\
&\ovalgate{\hata'}
&\qw
}
\end{array}
=
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\qw
&\qw
\\
&\gate{(\pm i)^{n_{\hata'}}}
&\qw
}
\end{array}
\;.
\eeq
Thus, when
$\hata'=\hatb'=\hatc'$,
a $T_b$ circuit reduces to zero DC-NOTs.
More generally, $\hata'=\pm \hatb'$
and $\hatc'=\pm \hatb'$.
Let $\call_{new}$ be a new circuit
obtained by replacing in $\call$:
(1)$\hata'$ by its negative if $\hata'=-\hatb'$,
(2)$\hatc'$ by its negative if $\hatc'=-\hatb'$.
By virtue of Eq.(\ref{eq-dcnot-with-neg}),
$\call = (I_2\otimes U)\call_{new}(I_2\otimes V)$
where $U,V\in U(2)$.
If $\call_{new}\sim_R 1$, then
$\call\sim_R 1$.
\rproof
$\call\sim_R 1$ so
$\hat{\call}^{(2)}= \pm 1$.
In light of
Eq.(\ref{eq-invar-of-hat-graph}), this gives
\beq
i^3 \call^{(2)}
= \pm 1
\;.
\eeq
It follows that
\beq
\lam_{3r} + i\lam_{3i}
+ \Lam_{3r} + i\Lam_{3i} =
\pm i
\;,
\eeq
where
\beq
\lam_{3r} =
\manyx{\hata'\hatb' \hatb'}\cdot\hatc'
\;\;
\manyx{\hata\hatb\hatb}\cdot\hatc
\;,
\eeq
\beq
\lam_{3i} =
-(\hata\cdot\hatb)(\hatb\cdot\hatc)\calv'
-(\hata'\cdot\hatb')(\hatb'\cdot\hatc')\calv
\;,
\eeq
\beq
\Lam_{3r}=
\left\{
\begin{array}{l}
-(\hata'\cdot\hatb')(\hata\cdot\hatb)
\hatc'\hatc^T
\\
+(\hata\cdot\hatb)(\hatb\cdot\hatc)
\manyx{\hata'\hatb'\hatc'}\hatc^T
+(\hata'\cdot\hatb')(\hatb'\cdot\hatc')
\hatc'\manyx{\hata\hatb\hatc}^T
\\
+(\hata'\cdot\hatb')\calv \manyx{\hatb'\hatc'}\hatc^T
+(\hata\cdot\hatb)\calv'\hatc'\manyx{\hatb\hatc}^T
\\
-
\manyx{\hata'\hatb'\hatb'\hatc'\hatc'}
\manyx{\hata\hatb\hatb\hatc\hatc}^T
\;,
\end{array}
\right.
\eeq
and
\beq
\Lam_{3i}=
\left\{
\begin{array}{l}
+(\hata\cdot\hatb)
\manyx{\hata'\hatb'\hatc'\hatc'}
\manyx{\hatb\hatc\hatc}^T
+
(\hata'\cdot\hatb')
\manyx{\hatb'\hatc'\hatc'}
\manyx{\hata\hatb\hatc\hatc}^T
\\
+\manyx{\hata\hatb\hatb}\cdot \hatc
\manyx{\hata'\hatb'\hatb'\hatc'}\hatc^T
+
\manyx{\hata'\hatb'\hatb'}\cdot \hatc'
\hatc'\manyx{\hata\hatb\hatb\hatc}^T
\end{array}
\right.
\;.
\eeq
$\hatc^{\;'T}\Lam_{3r}\hatc = 0$ so
$\hata'\cdot\hatb'=0$ or $\hata\cdot\hatb=0$.
Both can't be true at once or else
we would have $\lam_{3i}=0$,
which is false. Assume henceforth that
$\hata'\cdot\hatb'\neq 0$ and $\hata\cdot\hatb=0$
(the case
$\hata'\cdot\hatb'= 0$ and $\hata\cdot\hatb\neq 0$
is analogous).
When $\hata\cdot\hatb=0$, $|\lam_{3i}|=1$
reduces to
$|(\hata'\cdot\hatb')(\hatb'\cdot\hatc')\calv|=1$,
which immediately implies that
$(\hatc'\parallel\hatb'\parallel\hata')$,
and
$(\hatc,\hatb,\hata)$ are mutually orthogonal.
Thus, circuit $\call$ must be of type $T_b$.
\qed
\subsection{Reducing Controlled-$U$'s}
\label{sec-contr-us}
\subsubsection{One Controlled-$U$}
\label{sec-1-contr-u}
In this section, we show that any controlled-U can
be expressed with just two CNOTs. This result
was first proven by
Barenco et al. in Ref.\cite{Bar95}. Their method of
proof is long and opaque compared
with the ultra simple proof given
below. This attests to the benefits
of using dressed CNOTs.
\begin{theo}
Let $\theta\in \RR$.
Suppose
\beq
\call =
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\gate{e^{i\theta\sigma_{\hatw}}}\qwx[1]
&\qw
\\
&\dotgate
&\qw
}
\end{array}
\;\;,\;\;
\calr=
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatb}\qwx[1]
&\ovalgate{\hata}\qwx[1]
&\qw
\\
&\ovalgate{\hatz}
&\ovalgate{\hatz}
&\qw
}
\end{array}
\;.
\eeq
For any $\call$, it is possible to
find an $\calr$ such that $\call= \calr$.
\end{theo}
\proof
Given a unit vector $\hatw$
and an angle $\theta$,
we can always find (non-unique)
unit vectors
$\hatb$ and $\hata$ such that
$angle(\hatb,\hata)=\theta$, and
$\hatb\times\hata$ points along $\hatw$. Then
$\hatb\cdot\hata=\cos(\theta)$
and
$\hatb\times\hata=\sin(\theta)\hatw$ so
$\sigma_\hatb \sigma_\hata =e^{i\theta\sigma_\hatw}$.
\beq
[e^{i\theta\sigma_\hatw(0)}]^{n(1)}=
[\sigma_\hatb(0)\sigma_\hata(0)]^{n(1)}
=
\sigma_\hatb(0)^{n(1)}
\sigma_\hata(0)^{n(1)}
\;.
\eeq
\qed
\subsubsection{Two Controlled-$U$'s
(The Deflation Identity)
\\{\footnotesize\tt[
deflate\_dcnots.m,
test\_deflate\_dcnots.m
]}}
\label{sec-2-contr-u}
In this section, we show that
a product of two controlled-Us can
be expressed with just two CNOTs. This
``Deflation Identity"
was first proven in Ref.\cite{Tuc-deflation}.
Unlike the
proof of Ref.\cite{Tuc-deflation}, the one below
uses dressed CNOTs.
\begin{figure}[h]
\begin{center}
\epsfig{file=deflation.eps, height=2.75in}
\caption{Variables used in
Theorem \ref{th-deflation}.}
\label{fig-deflation}
\end{center}
\end{figure}
\begin{theo}\label{th-deflation}
Let $A\in SU(2)$ and $\theta_L, \theta_R\in \RR$.
Suppose
\beq
\call =
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\gate{e^{i\theta_L\sigma_{\hatw_L}}}\qwx[1]
&\qw
&\gate{e^{i\theta_R\sigma_{\hatw_R}}}\qwx[1]
&\qw
\\
&\dotgate
&\gate{A}
&\dotgate
&\qw
}
\end{array}
\;\;,\;\;
\calr=
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatb_f}\qwx[1]
&\ovalgate{\hata_f}\qwx[1]
&\qw
\\
&\ovalgate{\hatb'_f}
&\ovalgate{\hata'_f}
&\qw
}
\end{array}
\;.
\eeq
For any $\call$, it is possible to
find an $\calr$ such that $\call\sim_R \calr$.
\end{theo}
\proof
Given a unit vector $\hatw_L$
and an angle $\theta_L$,
we can always find (non-unique)
unit vectors
$\hatd$ and $\hatc$
such that
$angle(\hatd,\hatc)=\theta_L$, and
$\hatd\times \hatc$ points along $\hatw_L$. Then
$\hatd\cdot\hatc=\cos(\theta_L)$
and
$\hatd\times\hatc=\sin(\theta_L)\hatw_L$ so
$\sigma_\hatd \sigma_\hatc =e^{i\theta_L\sigma_{\hatw_L}}$.
Likewise, given a unit vector $\hatw_L$
and an angle $\theta_L$,
we can always find (non-unique)
unit vectors
$\hatb$ and $\hata$
such that
$\sigma_\hatb \sigma_\hata =e^{i\theta_R\sigma_{\hatw_R}}$.
We are free to rotate the vectors
$\hatd$ and $\hatc$ (ditto,
$\hatb$ and $\hata$) within the plane
they initially span, as long as we don't change the
angle between them.
In particular, we can choose both $\hatc$
and $\hatb$ to lie along
the line of intersection
between the planes
$span(\hatd,\hatc)$ and
$span(\hatb,\hata)$.
In other words, we can always choose
$\hatc=\hatb$. Call $\hatt$ their common value .
It is now clear that, without loss of generality,
we can replace
$\call$ by
\beq
\call =
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatd}\qwx[1]
&\qw
&\ovalgate{\hatt}\qwx[1]
&\breach
&\ovalgate{\hatt}\qwx[1]
&\qw
&\ovalgate{\hata}\qwx[1]
&\qw
\\
&\ovalgate{\hatd'}
&\breach
&\ovalgate{\hatd'}
&\qw
&\ovalgate{\hata'}
&\breach
&\ovalgate{\hata'}
&\qw
}
\end{array}
\;.
\eeq
Our goal is to construct an $\calr$ such that
$\call\sim_R \calr$. Such an $\calr$,
if it exists, must satisfy
$\hat{\call}^{(2)}= \pm \hat{\calr}^{(2)}$.
We will use the positive sign.
In light of
Eq.(\ref{eq-invar-of-hat-graph}), this gives
\beq
i^4 \call^{(2)} = i^2 \calr^{(2)}
\;.
\eeq
Using the same calculational techniques that
were used in
Section \ref{sec-two-bit-dcnot-rhs-invariants},
one finds
\beq
\call^{(2)}=
\left\{
\begin{array}{l}
(\hata\cdot\hatt)(\hatt\cdot\hatd)
-(\hata'\cdot\hatd')
\manyx{\hata\hatt\hatt}\cdot\hatd
\\
i\left[
(\hata\cdot\hatt)\hatd'\manyx{\hatt\hatd}^T
-(\hata'\cdot\hatd')
\hatd'\manyx{\hata\hatt\hatt\hatd}^T
+\manyx{\hata'\hatd'\hatd'}
\manyx{\hata\hatt\hatd\hatd}^T
\right]
\\
+\manyx{\hata\hatt}\cdot\hatd
\manyx{\hata'\hatd'}\hatd^T
\end{array}
\right.
\;.
\eeq
From Section \ref{sec-invariants-2cnots},
we know that $\calr^{(2)}$ can be expressed as
\beqa
\calr^{(2)}&=&
\lam_{2r} + \Lam_{2r} + i \Lam_{2i}\\
&=&
c_{\alpha'} c_\alpha
-(s_{\alpha'}s_\alpha) \hatf_2'\hatf_2^T
+i\;
\begin{array}{l|ll}
& \hatf_1^T & \hatf_3^T\\
\hline
\hatf_3'& s_{\alpha'}c_\alpha & 0\\
\hatf_1'& 0 &c_{\alpha'}s_\alpha
\end{array}
\;.
\label{eq-r-invar-2contr-u}
\eeqa
We must have
\beq
\lam_{2r}= c_{\alpha'}c_\alpha =
-(\hata\cdot\hatt)(\hatt\cdot\hatd)
+(\hata'\cdot\hatd')
\manyx{\hata\hatt\hatt}\cdot\hatd
\;,
\eeq
and
\beq
\Lam_{2r}=
-s_{\alpha'}s_\alpha
\hatf_2'\hatf_2^T =
-\manyx{\hata\hatt}\cdot\hatd
\manyx{\hata'\hatd'}\hatd^T
\;.
\label{eq-ssff}
\eeq
Define
\beq
s_{\phi'} = |\manyx{\hata'\hatd'}|
\;,\;\;
\eta = |\manyx{\hata\hatt\hatd}|
\;.
\eeq
If $s_{\phi'}\neq 0$,
Eq.(\ref{eq-ssff}) is satisfied by
\beq
s_{\alpha'}s_\alpha =
\manyx{\hata\hatt}\cdot\hatd
\; s_{\phi'}
\;,\;\;
\hatf'_2 =
\frac{\manyx{\hata'\hatd'}}
{s_{\phi'}}
\;,\;\;
\hatf_2 = \hatd
\;.
\label{eq-fprime2-def-2-contr-u}
\eeq
If $s_{\phi'}=0$,
choose $s_\alpha s_{\alpha'}$
and $\hatf_2$ the
same way, but choose $\hatf'_2$ to
be any vector perpendicular to $\hatd'$.
If $s_{\phi'}\neq 0$ and $\eta\neq 0$,
define the following
two RHON bases
(illustrated in Fig.\ref{fig-deflation}):
\beq
(\hath'_j)_{j=1,2,3}=
(
\frac{\manyx{\hata'\hatd'\hatd'}}
{ s_{\phi'}},
\frac{\manyx{\hata'\hatd'}}
{ s_{\phi'}},
\hatd'
)
\;,
\label{eq-hprimej-def-2-contr-u}
\eeq
and
\beq
(\hath_j)_{j=1,2,3}=
(
\frac{\manyx{\hata\hatt\hatd}}
{ \eta},
\hatd,
\frac{\manyx{\hata\hatt\hatd\hatd}}
{\eta}
)
\;.
\label{eq-hj-def-2-contr-u}
\eeq
If $s_{\phi'}=0$, pick $(\hath'_j)_{j=1,2,3}$
to be any RHON basis such that $\hath'_3=\hatd'$.
If $\eta=0$, pick $(\hath_j)_{j=1,2,3}$
to be any RHON basis such that $\hath_2=\hatd$.
Define the following two angles
(illustrated in Fig.\ref{fig-deflation}):
\beq
\phi_2 = angle(\manyx{\hata\hatt\hatt\hatd},
\hath_3)
\;,\;\;
\phi_1 = angle(\manyx{\hatt\hatd},
\hath_3)
\;.
\eeq
We must have
\beqa
\Lam_{2i}&=&
-(\hata\cdot\hatt)\hatd'\manyx{\hatt\hatd}^T
+(\hata'\cdot\hatd')
\hatd'\manyx{\hata\hatt\hatt\hatd}^T
-\manyx{\hata'\hatd'\hatd'}
\manyx{\hata\hatt\hatd\hatd}^T
\\
&=&
\begin{array}{l|ll}
& \hath_1^T & \hath_3^T\\
\hline
\hath_3'& -\hata\cdot\hatt s_{\phi_1} + c_{\phi'}s_{\phi_2} &
-\hata\cdot\hatt c_{\phi_1} + c_{\phi'}c_{\phi_2}\\
\hath_1'& 0 &-s_{\phi'}\eta
\end{array}
\;.
\eeqa
At this point, we can follow from
step \ref{item-diag-invar2-m-matrix}
to the end of the
Algorithm for Diagonalizing $\calg^{(2)}_2$
that was given in
Section \ref{sec-invariants-2cnots}.
This will yield values for
$\hata_f$,
$\hata'_f$,
$\hatb_f$, and
$\hatb'_f$.
\qed
\subsection{Opening and Closing a Breach
\\{\footnotesize\tt[
breach.m,
test\_breach.m
]}}
\label{sec-opening-breach}
\begin{quote}
{\it
Once more unto the breach, dear friends, once more;
Or close the wall up with our English dead!}
(from ``King Henry V"
by W. Shakespeare)
\end{quote}
In this section, we show how to ``open and
close a breach" in 2-qubit circuits. This
is a procedure whereby one can reduce any
2-qubit circuit with 4 CNOTs into
a circuit with 3 CNOTs.
Applying this procedure repeatedly, one
can reduce any 2-qubit circuit with more than
3 CNOTs into a circuit with only 3 CNOTs. The fact that
all 2-qubit circuits can be expressed
with 3 (or fewer) CNOTs was first proven in Ref.\cite{VD}.
Unlike the proof below, their proof was based on
Cartan's KAK decomposition\cite{Tuc-KAK}.
\begin{theo}(Opening a Breach)
Suppose
\beq
\call=
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatp_L}\qwx[1]
&\ovalgate{\hatq_L}\qwx[1]
&\ovalgate{\hatq_R}\qwx[1]
&\ovalgate{\hatp_R}\qwx[1]
&\qw
\\
&\ovalgate{\hatp'_L}
&\ovalgate{\hatq'_L}
&\ovalgate{\hatq'_R}
&\ovalgate{\hatp'_R}
&\qw
}
\end{array}
\;,
\label{eq-op-cl-breach-call}
\eeq
\beq
\calr=
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatp_{Lf}}\qwx[1]
&\ovalgate{\hatq_{Lf}}\qwx[1]
&\qw
&\ovalgate{\hatq_{Rf}}\qwx[1]
&\ovalgate{\hatp_{Rf}}\qwx[1]
&\qw
\\
&\ovalgate{\hatp'_{Lf}}
&\ovalgate{\hatt'}
&\breach
&\ovalgate{\hatt'}
&\ovalgate{\hatp'_{Rf}}
&\qw
}
\end{array}
\;.
\label{eq-op-cl-breach-calr}
\eeq
For any $\call$, it is possible to
find an $\calr$ such that $\call\sim_R \calr$.
\end{theo}
\proof
We begin by inserting a ``unit wedge" into $\call$:
\beq
\call=
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatp_L}\qwx[1]
&\ovalgate{\hatq_L}\qwx[1]
&\ovalgate{\hatt}\qwx[1]
&\ovalgate{\hatt}\qwx[1]
&\ovalgate{\hatq_R}\qwx[1]
&\ovalgate{\hatp_R}\qwx[1]
&\qw
\\
&\ovalgate{\hatp'_L}
&\ovalgate{\hatq'_L}
&\ovalgate{\hatt'}
&\ovalgate{\hatt'}
&\ovalgate{\hatq'_R}
&\ovalgate{\hatp'_R}
&\qw
}
\end{array}
\;.
\label{eq-op-cl-breach-call-wedge}
\eeq
In Eq.(\ref{eq-op-cl-breach-call-wedge}),
$\hatt$ and $\hatt'$
are auxiliary parameters
whose values are still to be defined.
Consider separately
each half of the circuit in
Eq.(\ref{eq-op-cl-breach-call-wedge}).
Our goal is to re-express each
half as follows:
\beq
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatt}\qwx[1]
&\ovalgate{\hatq_R}\qwx[1]
&\ovalgate{\hatp_R}\qwx[1]
&\qw
\\
&\ovalgate{\hatt'}
&\ovalgate{\hatq'_R}
&\ovalgate{\hatp'_R}
&\qw
}
\end{array}
=
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatq_{Rf}}\qwx[1]
&\ovalgate{\hatp_{Rf}}\qwx[1]
&\gate{U_{Rf}}
&\qw
\\
&\ovalgate{\hatt'}
&\ovalgate{\hatp'_{Rf}}
&\gate{U'_{Rf}}
&\qw
}
\end{array}
\;,
\label{eq-op-cl-breach-rhs}
\eeq
and
\beq
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatp_L}\qwx[1]
&\ovalgate{\hatq_L}\qwx[1]
&\ovalgate{\hatt}\qwx[1]
&\qw
\\
&\ovalgate{\hatp'_L}
&\ovalgate{\hatq'_L}
&\ovalgate{\hatt'}
&\qw
}
\end{array}
=
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\gate{U_{Lf}}
&\ovalgate{\hatp_{Lf}}\qwx[1]
&\ovalgate{\hatq_{Lf}}\qwx[1]
&\qw
\\
&\gate{U'_{Lf}}
&\ovalgate{\hatp'_{Lf}}
&\ovalgate{\hatt'}
&\qw
}
\end{array}
\;.
\label{eq-op-cl-breach-lhs}
\eeq
From Theorem \ref{th-persistence}, we
know that Eq.(\ref{eq-op-cl-breach-rhs})
will be achieved if we constrain our
auxiliary parameters by:
\begin{subequations}
\label{eq-op-cl-breach-contraints}
\beq
\manyx{\hatp'_R\hatq'_R\hatq'_R}\cdot\hatt'=0
\;,
\eeq
and
\beq
\left[c_{\phi'_R}(\hatp_R\cdot\hatq_R)
\hatp_R\times\hatq_R
-s_{\lam'_R}c_{\lam'_R}s_{\phi'_R}\hatq_R
\right]\cdot \hatt
=0
\;.
\eeq
\end{subequations}
Likewise,
Eq.(\ref{eq-op-cl-breach-lhs})
will be achieved if we constrain our
auxiliary parameters by the
same pair of equations as
Eqs.(\ref{eq-op-cl-breach-contraints}),
but with $R$ subscripts replaced by
$L$ subscripts.
These 4 constraint equations
can be used to solve for the
4 degrees of freedom contained in the
auxiliary parameters
$\hatt$ and $\hatt'$.
\qed
By a ``unit wedge" we mean
a circuit element which
equals one. An analogous
concept is a ``partition of unity".
If it equals one, why use it?
Because it depends on new,
auxiliary parameters, and, by merging
the unit wedge with its surroundings,
we get a new expression
which contains the auxiliary parameters,
but is functionally independent of them.
We can then choose convenient values for
the auxiliary parameters.
The net result is that we
can transform the original circuit to
a new one that performs exactly as the old one
but appears different.
Note that in Eq.(\ref{eq-op-cl-breach-call-wedge})
we used a unit wedge consisting of a
single DC-NOT
times itself. There was no a priori obvious reason
why this unit wedge would lead us to a proof
of the theorem. We could have
chosen
a unit wedge
that provided more auxiliary parameters.
For instance, we could have chosen
a product of 3 DC-NOTs (times the inverse
of the product).
After all, 1 DC-NOT
can express only a limited subset of
all possible 2-qubit transformations
whereas 3 DC-NOTs can be used to express any
of them. For proving the above
theorem, using a
unit wedge with only
1 DC-NOT turned out to be sufficient.
But one can envisage this theorem proving
technique being
used elsewhere
with more complicated unit wedges.
Suppose one starts with a circuit
which,
like $\call$ in Eq.(\ref{eq-op-cl-breach-call}),
possesses 4 DC-NOTs.
By the last theorem,
one can ``open a breach" in it; that is,
transform it
into a circuit which, like $\calr$ in
Eq.(\ref{eq-op-cl-breach-calr}),
possesses two
adjacent oval nodes both carrying a $\hatt'$.
Then one can combine the two
adjacent DC-NOTs
with a $\hatt'$ node and obtain
a controlled-U.
Finally, one can use the
Deflation
Identity of Sec.\ref{sec-2-contr-u}
to express the just created controlled-U
and an adjacent DC-NOT
as a circuit with two CNOTs.
The net effect of this
procedure is to reduce any
2-qubit circuit with 4 CNOTs into one with 3 CNOTs.
\section{Identities for Circuits with 3 Qubits}
\label{sec-3-qubit-ids}
\subsection{Pass-Through Identities}
\label{sec-pass-thru}
In the following 3 subsections, we consider
the following 3 ``identities" (one subsection
per identity):
\begin{subequations}
\label{eq-bare-pass-thru}
\beq
\begin{array}{c}
\Qcircuit @C=1em @R=.5em @!R{
&\ovalgate{\;}\qwx[2]
&\ovalgate{\;}\qwx[1]
&\qw
\\
&\qw
&\ovalgate{\;}
&\qw
\\
&\ovalgate{\;}
&\qw
&\qw
}
\end{array}
\sim_R
\begin{array}{c}
\Qcircuit @C=1em @R=.5em @!R{
&\ovalgate{\;}\qwx[1]
&\ovalgate{\;}\qwx[2]
&\qw
\\
&\ovalgate{\;}
&\qw
&\qw
\\
&\qw
&\ovalgate{\;}
&\qw
}
\end{array}
\;,
\label{eq-bare-pass-1}
\eeq
\beq
\begin{array}{c}
\Qcircuit @C=1em @R=.5em @!R{
&\ovalgate{\;}\qwx[2]
&\ovalgate{\;}\qwx[1]
&\ovalgate{\;}\qwx[1]
&\qw
\\
&\qw
&\ovalgate{\;}
&\ovalgate{\;}
&\qw
\\
&\ovalgate{\;}
&\qw
&\qw
&\qw
}
\end{array}
\sim_R
\begin{array}{c}
\Qcircuit @C=1em @R=.5em @!R{
&\ovalgate{\;}\qwx[1]
&\ovalgate{\;}\qwx[2]
&\ovalgate{\;}\qwx[1]
&\qw
\\
&\ovalgate{\;}
&\qw
&\ovalgate{\;}
&\qw
\\
&\qw
&\ovalgate{\;}
&\qw
&\qw
}
\end{array}
\;,
\label{eq-bare-pass-2}
\eeq
\beq
\begin{array}{c}
\Qcircuit @C=1em @R=.5em @!R{
&\ovalgate{\;}\qwx[2]
&\ovalgate{\;}\qwx[1]
&\ovalgate{\;}\qwx[1]
&\ovalgate{\;}\qwx[1]
&\qw
\\
&\qw
&\ovalgate{\;}
&\ovalgate{\;}
&\ovalgate{\;}
&\qw
\\
&\ovalgate{\;}
&\qw
&\qw
&\qw
&\qw
}
\end{array}
\sim_R
\begin{array}{c}
\Qcircuit @C=1em @R=.5em @!R{
&\ovalgate{\;}\qwx[1]
&\ovalgate{\;}\qwx[2]
&\ovalgate{\;}\qwx[1]
&\ovalgate{\;}\qwx[1]
&\qw
\\
&\ovalgate{\;}
&\qw
&\ovalgate{\;}
&\ovalgate{\;}
&\qw
\\
&\qw
&\ovalgate{\;}
&\qw
&\qw
&\qw
}
\end{array}
\;.
\label{eq-bare-pass-3}
\eeq
\end{subequations}
Note that in all 3 identities, the
initial and final circuits
both have
the same number of
DC-NOTs, acting on the same 3 qubits.
In all 3 cases, we pass
a DC-NOT (the mobile one) acting on qubits
0 and 1 through another
DC-NOT (the static one) acting on qubits 0 and 2.
Thus, the mobile and static DC-NOTs
both act on qubit 0, but
the second qubit on which they act
differs.
We will refer to
Eq.(\ref{eq-bare-pass-1}),
Eq.(\ref{eq-bare-pass-2}),
and
Eq.(\ref{eq-bare-pass-3})
as the Pass-Through Identities 1,2, and 3, respectively.
In the initial circuit of Pass-Through Identity $n$,
the mobile DC-NOT
is part of a group of $n$ adjacent DC-NOTs
acting on qubits 0 and 1.
The Pass-Through Identities
Eqs.(\ref{eq-bare-pass-thru})
do not, per se, change the number of DC-NOTs.
However, in some situations,
they can be used to reduce the
number of DC-NOTs. For example,
\beqa
\label{eq-eg-pass-thru}
\begin{array}{c}
\Qcircuit @C=1em @R=.5em @!R{
&\ovalgate{\;}\qwx[1]
&\ovalgate{\;}\qwx[1]
&\ovalgate{\;}\qwx[1]
&\ovalgate{\;}\qwx[2]
&\ovalgate{\;}\qwx[1]
&\qw
\\
&\ovalgate{\;}
&\ovalgate{\;}
&\ovalgate{\;}
&\qw
&\ovalgate{\;}
&\qw
\\
&\qw
&\qw
&\qw
&\ovalgate{\;}
&\qw
&\qw
}
\end{array}
&\sim_R &
\begin{array}{c}
\Qcircuit @C=1em @R=.5em @!R{
&\ovalgate{\;}\qwx[1]
&\ovalgate{\;}\qwx[1]
&\ovalgate{\;}\qwx[1]
&\ovalgate{\;}\qwx[1]
&\ovalgate{\;}\qwx[2]
&\qw
\\
&\ovalgate{\;}
&\ovalgate{\;}
&\ovalgate{\;}
&\ovalgate{\;}
&\qw
&\qw
&\qw
\\
&\qw
&\qw
&\qw
&\qw
&\ovalgate{\;}
&\qw
&\qw
}
\end{array}
\\
&\sim_R &
\begin{array}{c}
\Qcircuit @C=1em @R=.5em @!R{
&\ovalgate{\;}\qwx[1]
&\ovalgate{\;}\qwx[1]
&\ovalgate{\;}\qwx[1]
&\ovalgate{\;}\qwx[2]
&\qw
\\
&\ovalgate{\;}
&\ovalgate{\;}
&\ovalgate{\;}
&\qw
&\qw
&\qw
\\
&\qw
&\qw
&\qw
&\ovalgate{\;}
&\qw
&\qw
}
\end{array}
\;.
\eeqa
In Eq.(\ref{eq-eg-pass-thru}), there are
initially 3 adjacent DC-NOTs
on the LHS of the static
DC-NOT.
Using Pass-Through Identity 1,
we produce 4 adjacent
DC-NOTs on the LHS of the static
DC-NOT. As shown
in Section \ref{sec-opening-breach},
these 4 adjacent
DC-NOTs
can always be reduced to 3 DC-NOTs.
\subsubsection{Pass-Through Identity 1}
\label{sec-pass1}
\begin{theo}
Suppose
\beq
\call =
\begin{array}{c}
\Qcircuit @C=1em @R=.5em @!R{
&\ovalgate{\hatb}\qwx[2]
&\ovalgate{\hata}\qwx[1]
&\qw
\\
&\qw
&\ovalgate{\hata'}
&\qw
\\
&\ovalgate{\hatb''}
&\qw
&\qw
}
\end{array}
\;\;,\;\;
\calr =
\begin{array}{c}
\Qcircuit @C=1em @R=.5em @!R{
&\ovalgate{\hata_f}\qwx[1]
&\ovalgate{\hatb_f}\qwx[2]
&\qw
\\
&\ovalgate{\hata'_f}
&\qw
&\qw
\\
&\qw
&\ovalgate{\hatb''_f}
&\qw
}
\end{array}
\;.
\eeq
For any $\call$, it is possible to
find an $\calr$ such that $\call\sim_R \calr$
if and only if $\hata\parallel\hatb$.
\end{theo}
\proof
\lproof
Let $\hata'_f=\hata'$ and
$\hatb''_f = \hatb''$.
Clearly,
if $\hata=\hatb$, then $\call= \calr$.
More generally, $\hata=\pm\hatb$.
Let $\call_{new}$ be a new circuit
obtained by replacing in $\call$:
$\hata$ by its negative if $\hata=-\hatb$.
By virtue of Eq.(\ref{eq-dcnot-with-neg}),
$\call = \call_{new}(I_2\otimes U \otimes I_2)$
where $U\in U(2)$.
If $\call_{new}\sim_R \calr_{new}$, then
$\call\sim_R \calr_{new}$.
\rproof
Using the same calculational techniques that
were used in
Section \ref{sec-two-bit-dcnot-rhs-invariants},
one finds
\beq
\call^{(2)}=
\hata\cdot\hatb \sigma_{\hatb'',\hata',1}
+i \sigma_{1,\hata',\manyx{\hata\hatb}}
\;,
\eeq
and
\beq
\calr^{(2)}=
\hatb_f\cdot\hata_f \sigma_{\hatb''_f,\hata'_f,1}
+i \sigma_{\hatb''_f,1,\manyx{\hata_f\hatb_f}}
\;.
\eeq
$\call\sim_R \calr$ implies that
$\call^{(2)}$ is proportional to
$\calr^{(2)}$. Therefore,
$\sigma_{1,\hata',\manyx{\hata\hatb}}$ must vanish.
Hence, $\manyx{\hata\hatb}=0$, which is
implies $\hata\parallel \hatb$.
\qed
\subsubsection{Pass-Through Identity 2}
\label{sec-pass2}
\begin{theo}
Suppose
\beq
\call =
\begin{array}{c}
\Qcircuit @C=1em @R=.5em @!R{
&\ovalgate{\hate}\qwx[2]
&\ovalgate{\hatb}\qwx[1]
&\ovalgate{\hata}\qwx[1]
&\qw
\\
&\qw
&\ovalgate{\hatb'}
&\ovalgate{\hata'}
&\qw
\\
&\ovalgate{\hate''}
&\qw
&\qw
&\qw
}
\end{array}
\;\;,\;\;
\calr =
\begin{array}{c}
\Qcircuit @C=1em @R=.5em @!R{
&\ovalgate{\hatb_f}\qwx[1]
&\ovalgate{\hate_f}\qwx[2]
&\ovalgate{\hata_f}\qwx[1]
&\qw
\\
&\ovalgate{\hatb'_f}
&\qw
&\ovalgate{\hata'_f}
&\qw
\\
&\qw
&\ovalgate{\hate''_f}
&\qw
&\qw
}
\end{array}
\;.
\eeq
For any $\call$, if there exists $\hatt'$
such that
\beq
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hate}\qwx[1]
&\ovalgate{\hatb}\qwx[1]
&\ovalgate{\hata}\qwx[1]
&\qw
\\
&\ovalgate{\hatt'}
&\ovalgate{\hatb'}
&\ovalgate{\hata'}
&\qw
}
\end{array}
\sim_R
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hata_f}\qwx[1]
&\qw
\\
&\ovalgate{\hata'_f}
&\qw
}
\end{array}
\;,
\label{eq-3to1-for-some-t}
\eeq
then
it is possible to
find an $\calr$ such that $\call\sim_R \calr$.
\end{theo}
\proof
One has
\beqa
\call &=&
\begin{array}{c}
\Qcircuit @C=1em @R=.5em @!R{
&\ovalgate{\hate}\qwx[2]
&\ovalgate{\hate}\qwx[1]
&\ovalgate{\hate}\qwx[1]
&\ovalgate{\hatb}\qwx[1]
&\ovalgate{\hata}\qwx[1]
&\qw
\\
&\qw
&\ovalgate{\hatt'}
&\ovalgate{\hatt'}
&\ovalgate{\hatb'}
&\ovalgate{\hata'}
&\qw
\\
&\ovalgate{\hate''}
&\qw
&\qw
&\qw
&\qw
&\qw
}
\end{array}\label{eq-pass2-a}
\\
&=&
\begin{array}{c}
\Qcircuit @C=1em @R=.5em @!R{
&\ovalgate{\hate}\qwx[1]
&\ovalgate{\hate}\qwx[2]
&\ovalgate{\hate}\qwx[1]
&\ovalgate{\hatb}\qwx[1]
&\ovalgate{\hata}\qwx[1]
&\qw
\\
&\ovalgate{\hatt'}
&\qw
&\ovalgate{\hatt'}
&\ovalgate{\hatb'}
&\ovalgate{\hata'}
&\qw
\\
&\qw
&\ovalgate{\hate''}
&\qw
&\qw
&\qw
&\qw
}
\end{array}\label{eq-pass2-b}
\\
&=&
\begin{array}{c}
\Qcircuit @C=1em @R=.5em @!R{
&\ovalgate{\hate}\qwx[1]
&\ovalgate{\hate}\qwx[2]
&\ovalgate{\hata_f}\qwx[1]
&\qw
\\
&\ovalgate{\hatt'}
&\qw
&\ovalgate{\hata'_f}
&\qw
\\
&\qw
&\ovalgate{\hate''}
&\qw
&\qw
}
\end{array}\label{eq-pass2-c}
\;.
\eeqa
In (a), we
introduced a unit wedge.
To go from (a) to (b),
we passed half of that unit wedge
across the ``static" DC-NOT.
Finally, to go from (b) to (c),
we used Eq.(\ref{eq-3to1-for-some-t}).
\qed
Note that Section \ref{sec-3to1-cnots}
gives necessary and sufficient
conditions for a 2-qubit circuit with
3 DC-NOTs to reduce to an equivalent
circuit with 1 DC-NOT.
Using those necessary and sufficient
conditions, it is easy to check
in any particular instance
whether
there exists a $\hatt'$
such that Eq.(\ref{eq-3to1-for-some-t})
is satisfied.
\subsubsection{Pass-Through Identity 3
\\{\footnotesize\tt[
pass3.m,
test\_pass3.m
]}}
\label{sec-pass3}
\begin{theo}\label{th-pass-thru3}
Suppose
\beq
\call =
\begin{array}{c}
\Qcircuit @C=1em @R=.5em @!R{
&\ovalgate{\hate}\qwx[2]
&\ovalgate{\hatc}\qwx[1]
&\ovalgate{\hatb}\qwx[1]
&\ovalgate{\hata}\qwx[1]
&\qw
\\
&\qw
&\ovalgate{\hatc'}
&\ovalgate{\hatb'}
&\ovalgate{\hata'}
&\qw
\\
&\ovalgate{\hate''}
&\qw
&\qw
&\qw
&\qw
}
\end{array}
\;\;,\;\;
\calr =
\begin{array}{c}
\Qcircuit @C=1em @R=.5em @!R{
&\ovalgate{\hatc_f}\qwx[1]
&\ovalgate{\hate_f}\qwx[2]
&\ovalgate{\hatb_f}\qwx[1]
&\ovalgate{\hata_f}\qwx[1]
&\qw
\\
&\ovalgate{\hatc'_f}
&\qw
&\ovalgate{\hatb'_f}
&\ovalgate{\hata'_f}
&\qw
\\
&\qw
&\ovalgate{\hate''_f}
&\qw
&\qw
&\qw
}
\end{array}
\;.
\eeq
For any $\call$, it is possible to
find an $\calr$ such that $\call\sim_R \calr$.
\end{theo}
\proof
\beqa
\call &=&
\begin{array}{c}
\Qcircuit @C=1em @R=.5em @!R{
&\ovalgate{\hate}\qwx[2]
&\ovalgate{\hate}\qwx[1]
&\ovalgate{\hate}\qwx[1]
&\ovalgate{\hatc}\qwx[1]
&\ovalgate{\hatb}\qwx[1]
&\ovalgate{\hata}\qwx[1]
&\qw
\\
&\qw
&\ovalgate{\hatt'}
&\ovalgate{\hatt'}
&\ovalgate{\hatc'}
&\ovalgate{\hatb'}
&\ovalgate{\hata'}
&\qw
\\
&\ovalgate{\hate''}
&\qw
&\qw
&\qw
&\qw
&\qw
&\qw
}
\end{array}
\\
&=&
\begin{array}{c}
\Qcircuit @C=1em @R=.5em @!R{
&\ovalgate{\hate}\qwx[1]
&\ovalgate{\hate}\qwx[2]
&\ovalgate{\hate}\qwx[1]
&\ovalgate{\hatc}\qwx[1]
&\ovalgate{\hatb}\qwx[1]
&\ovalgate{\hata}\qwx[1]
&\qw
\\
&\ovalgate{\hatt'}
&\qw
&\ovalgate{\hatt'}
&\ovalgate{\hatc'}
&\ovalgate{\hatb'}
&\ovalgate{\hata'}
&\qw
\\
&\qw
&\ovalgate{\hate''}
&\qw
&\qw
&\qw
&\qw
&\qw
}
\end{array}
\\
&=&
\begin{array}{c}
\Qcircuit @C=1em @R=.5em @!R{
&\ovalgate{\hate}\qwx[1]
&\ovalgate{\hate}\qwx[2]
&\ovalgate{\hatb_f}\qwx[1]
&\ovalgate{\hata_f}\qwx[1]
&\qw
\\
&\ovalgate{\hatt'}
&\qw
&\ovalgate{\hatb'_f}
&\ovalgate{\hata'_f}
&\qw
\\
&\qw
&\ovalgate{\hate''}
&\qw
&\qw
&\qw
}
\end{array}
\;.
\eeqa
In (a), we
introduced a unit wedge.
To go from (a) to (b),
we passed half of that unit wedge
across the ``static" DC-NOT.
Finally, to go from (b) to (c),
we used Theorem \ref{eq-th-aux-pass3}.
\qed
The next theorem is used in the proof
of Theorem \ref{th-pass-thru3}.
\begin{theo}\label{eq-th-aux-pass3}
Suppose
\beq
\call(\hatd')=
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatd}\qwx[1]
&\ovalgate{\hatc}\qwx[1]
&\ovalgate{\hatb}\qwx[1]
&\ovalgate{\hata}\qwx[1]
&\qw
\\
&\ovalgate{\hatd'}
&\ovalgate{\hatc'}
&\ovalgate{\hatb'}
&\ovalgate{\hata'}
&\qw
}
\end{array}
\;\;,\;\;
\calr=
\begin{array}{c}
\Qcircuit @C=1em @R=1em @!R{
&\ovalgate{\hatb_f}\qwx[1]
&\ovalgate{\hata_f}\qwx[1]
&\qw
\\
&\ovalgate{\hatb'_f}
&\ovalgate{\hata'_f}
&\qw
}
\end{array}
\;.
\eeq
For any $\call(\cdot)$, there exists a $\hatd'$
and an $\calr$ such that
$\call\sim_R \calr$.
\end{theo}
\proof
Our goal is to find a $\hatd'$ and
to construct an $\calr$ such that
$\call\sim_R \calr$. Such an $\calr$ must satisfy
$\hat{\call}^{(2)}= \pm \hat{\calr}^{(2)}$.
We will use the positive sign.
In light of
Eq.(\ref{eq-invar-of-hat-graph}),
the
following must be true:
\beq
i^4\call^{(2)} = i^2 \calr^{(2)}
\;.
\eeq
From Section \ref{sec-invariants-4cnots}, we know that
\beq
\call^{(2)}=
\lam_{4r} + i\lam_{4i} + \Lam_{4r} + i \Lam_{4i}
\;,
\eeq
where
\beq
\lam_{4r} =
-\hatd^{\;'T}M_\nu\hatd
\;,
\eeq
\beq
\lam_{4i} =
-\hatd^{\;'T}M_\mu\hatd
\;,
\eeq
\beq
\Lam_{4r}=
X_o \hatd'\hatd^T
+ \vec{x'}\hatd^T
+ \hatd'\vecx^T
+ \Delta X
\;,
\eeq
\beq
\Lam_{4i}=
Y_o \hatd'\hatd^T
- \vec{y'}\hatd^T
- \hatd'\vecy^T
+ \Delta Y
\;.
\eeq
The precise definitions of $(X_o,Y_o)$,
$(\vecx,\vec{x'},\vecy,\vec{y'})$,
$(\Delta X, \Delta Y)$, and $(M_\mu, M_\nu)$
in terms of $(\hata,\hata')$,
$(\hatb,\hatb')$, $(\hatc,\hatc')$, and
$(\hatd,\hatd')$
are given in
Section \ref{sec-invariants-4cnots}.
From Section \ref{sec-invariants-2cnots},
we know that
\beqa
\calr^{(2)}&=&
\lam_{2r} + \Lam_{2r} + i \Lam_{2i}\\
&=&
c_{\alpha'} c_\alpha
-(s_{\alpha'}s_\alpha) \hatf_2'\hatf_2^T
+i\;
\begin{array}{l|ll}
& \hatf_1^T & \hatf_3^T\\
\hline
\hatf_3'& s_{\alpha'}c_\alpha & 0\\
\hatf_1'& 0 &c_{\alpha'}s_\alpha
\end{array}
\;.
\eeqa
We must have
\begin{subequations}
\beq
\lam_{2r} = - \lam_{4r}
\;,
\label{eq-cond1-pass-w3}
\eeq
\beq
0 = \lam_{4i}
\;,
\label{eq-cond2-pass-w3}
\eeq
\beq
\Lam_{2r} = - \Lam_{4r}
\;,
\label{eq-cond3-pass-w3}
\eeq
and
\beq
\Lam_{2i} = - \Lam_{4i}
\;.
\label{eq-cond4-pass-w3}
\eeq
\end{subequations}
To begin, we will assume that $X_o\neq 0$.
Later on, before ending
the proof, we will remove this assumption.
By evaluating Eq.(\ref{eq-cond1-pass-w3}), we get
\beq
c_{\alpha'}c_\alpha =
\hatd^{\;'T}M_\nu\hatd
\;.
\eeq
By evaluating Eq.(\ref{eq-cond2-pass-w3}),
we get
\beq
0 = \hatd^{\;'T}M_\mu \hatd
\;.
\eeq
Let $\hatd'$ be
any unit vector that satisfies this equation.
By evaluating Eq.(\ref{eq-cond3-pass-w3}), we get
\beq
-s_{\alpha'}s_\alpha \hatf'_2\hatf_2^T
=-(
X_o \hatd'\hatd^T
+ \vec{x'}\hatd^T
+ \hatd'\vecx^T
+ \Delta X)
\;.
\label{eq-cond3-unfactored}
\eeq
For Eq.(\ref{eq-cond3-unfactored}) to be true,
the RHS of that equation must
factor into the product of a column vector
times a row vector:
\beq
-s_{\alpha'}s_\alpha \hatf'_2\hatf_2^T
=-X_o\left(
\hatd' + \frac{\vec{x'}}{X_o}\right)
\left(\hatd + \frac{\vec{x}}{X_o}\right)^T
\;.
\label{eq-rank1-rhs}
\eeq
Let
\beq
s_{\alpha'}s_\alpha =
X_o \eta'_2 \eta_2
\;,\;\;
\hatf'_2 =
\frac{\hatd' + \frac{\vec{x'}}{X_o}}{\eta'_2}
\;,\;\;
\hatf_2 =
\frac{\hatd + \frac{\vec{x}}{X_o}}{\eta_2}
\;,
\eeq
where
\beq
\eta'_2 = \left|\hatd' + \frac{\vec{x'}}{X_o}\right|
=\sqrt{ 1 + \frac{(\vec{x'})^2}{(X_o)^2}}
\;,\;\;
\eta_2 = (\eta'_2)_\unprime
\;.
\eeq
Note that since
Eqs.(\ref{eq-cond3-unfactored})
and (\ref{eq-rank1-rhs})
are both true, the following must be true:
\beq
\frac{\vec{x'}\vec{x}^T}{X_o} = \Delta X
\;.
\label{eq-Delta-x-id}
\eeq
Eq.(\ref{eq-Delta-x-id}) can also
be proven by expressing it
in terms of
$(\hata,\hata')$,
$(\hatb,\hatb')$, $(\hatc,\hatc')$, and
$(\hatd,\hatd')$.
By evaluating Eq.(\ref{eq-cond4-pass-w3}), we get
\beq
\Lam_{2i}
=
-(
Y_o \hatd'\hatd^T
- \vec{y'}\hatd^T
- \hatd'\vecy^T
+ \Delta Y)
\;.
\label{eq-diag-equal-not-diag}
\eeq
At this point, we can follow from
step \ref{item-diag-invar2-h-hprime}
to the end of the
Algorithm for Diagonalizing $\calg^{(2)}_2$
that was given in
Section \ref{sec-invariants-2cnots}.
This will yield values for
$\hata_f$,
$\hata'_f$,
$\hatb_f$, and
$\hatb'_f$.
Now assume $X_o=0$.
By Eq.(\ref{eq-Delta-x-id}), either
$\vec{x'}=0$ or $\vecx=0$. When
$\vec{x'}=0$ and $\vecx\neq 0$
(the case $\vec{x'}\neq 0$ and $\vecx= 0$
is analogous),
Eq.(\ref{eq-rank1-rhs}) becomes
\beq
-s_{\alpha'}s_\alpha \hatf'_2\hatf_2^T
=-\left(
\hatd' + \frac{\vec{x'}}{X_o}\right)
\vec{x}^T
\;,
\eeq
where
$\frac{\vec{x'}}{X_o}$ is defined
as the obvious limit. Thus, we can set
\beq
s_{\alpha'}s_\alpha =
\eta'_2 |\vecx|
\;,\;\;
\hatf'_2 =
\frac{\hatd' + \frac{\vec{x'}}{X_o}}{\eta'_2}
\;,\;\;
\hatf_2 =
\frac{\vecx}{|\vecx|}
\;.
\eeq
If $\vec{x'}=\vecx=0$, then
Eq.(\ref{eq-rank1-rhs}) becomes
$-s_{\alpha'}s_\alpha \hatf'_2\hatf_2^T=0$,
so we can set $s_{\alpha'}s_\alpha=0$
and define $\hatf_2$ and $\hatf'_2$ to
be arbitrary unit vectors.
Additional observations:
Note that
$\hatf^{\;'T}_2\Lam_{2i} = 0$ implies
\begin{subequations}\label{eq-f2-Lam2i}
\beq
\vec{x'}\cdot\vec{y'} = X_o Y_o
\;,
\eeq
and
\beq
\Delta Y^T \vec{x'} = X_o \vec{y}
\;.
\eeq
\end{subequations}
Likewise, note that
$\Lam_{2i}\hatf_2 = 0$ implies
\begin{subequations}\label{eq-Lam2i-f2}
\beq
\vec{x}\cdot\vec{y} = X_o Y_o
\;,
\eeq
and
\beq
\Delta Y\vec{x} = X_o \vec{y'}
\;.
\eeq
\end{subequations}
Eqs.(\ref{eq-f2-Lam2i}) and
(\ref{eq-Lam2i-f2}) can also
be proven by expressing them
in terms of
$(\hata,\hata')$,
$(\hatb,\hatb')$, $(\hatc,\hatc')$, and
$(\hatd,\hatd')$.
If $|\vecx|$ and $|\vec{x'}|$
are both non-zero, it is possible to
introduce 2 RHON bases
$(\hath'_j)_{j=1,2,3}$
and $(\hath_j)_{j=1,2,3}$,
defined as follows.
Define $\hath'_2$ and $\hath_2$ by
\beq
\hath'_2 = \hatf'_2
\;,\;\;
\hath_2 = \hatf_2
\;.
\eeq
Define $\hath'_3$ and $\hath_3$ by
\beq
\hath'_3 =
\frac{\hatd' - \frac{\vec{x'}X_o}{(\vec{x'})^2}}
{\eta'_3}
\;,\;\;
\hath_3 = (\hath'_3)_\unprime
\;,
\eeq
where
\beq
\eta'_3 = \left|\hatd' - \frac{\vec{x'}X_o}{(\vec{x'})^2}\right|
=\sqrt{ 1 + \frac{(X_o)^2}{(\vec{x'})^2}}=
\frac{X_o}{|\vec{x'}|}\eta_2
\;,\;\;
\eta_3 = (\eta'_3)_\unprime
\;.
\eeq
Define $\hath'_1$ and $\hath_1$ by
\beq
\hath'_1 =\frac{\manyx{\vec{x'}\hatd'} \sign(X_o)}
{\eta'_1}
\;,\;\;
\hath_1 = (\hath'_1)_\unprime
\;,
\eeq
where
\beq
\eta'_1=|\manyx{\vec{x'}\hatd'}| = |\vec{x'}|
\;,\;\;
\eta_1 = (\eta'_1)_\unprime
\;.
\eeq
After some algebra,
one can show that
Eq.(\ref{eq-diag-equal-not-diag})
becomes
\beq
\Lam_{2i}=
\begin{array}{l|ll}
& \frac{\hath_1^T}{|\vec{x}|} & \hath_3^T\eta_3\\
\hline
\hath_3'\eta'_3&
\vec{y}^T\manyx{\vec{x}\hatd} \sign(X_o)& -Y_o
\\ \frac{\hath_1'}{|\vec{x'}|}&
-\manyx{\vec{x'}\hatd'}^T\Delta Y\manyx{\vec{x}\hatd}
&\vec{y}^{\;'T}\manyx{\vec{x'}\hatd'} \sign(X_o)
\end{array}
\;.
\eeq
The entries of the previous table
can be expressed solely
in terms of $(\hatd,\hatd')$
and $(M_\mu, M_\nu)$.
After some algebra, one finds
that
\beq
\vec{y}^T\manyx{\vec{x}\hatd}=
(M_\nu^T\hatd')\cdot\manyx{M^T_\mu\hatd',\hatd}
\;,
\eeq
\beq
\vec{y}^{\;'T}\manyx{\vec{x'}\hatd'}=
(M_\nu\hatd)\cdot\manyx{M_\mu\hatd,\hatd'}
\;,
\eeq
and
\beq
\manyx{\vec{x'}\hatd'}^T\Delta Y \manyx{\vec{x}\hatd}=
\hatd^T M_\mu^T M_\mu M_\mu^T \hatd'
\;.
\eeq
\qed
\vskip1cm
\noindent{\bf Acknowledgements}
\noindent I thank E. Rains for
kind and instructive email on the group
theoretic aspects of circuit invariants.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 3,038
|
\section{Supplemental Material}
\begin{center}
\textbf{On the effect of noise of the input state in the experimental
detection of quantum channel capacities.}
\end{center}
The theoretical results of Ref. \cite{ms16}, summarized in Eqs. (3--5) of the
Letter, were derived under the assumption of sending a pure bipartite state
at the input of the unknown channel. Realistically, as in the present
experimental set-up, the generated input state will be affected by noise,
thus producing a {\em mixed} state at the channel input.
Specifically, since our source indeed generates a maximally entangled
state affected by isotropic noise, in the following, for arbitrary
finite dimension $d$, we consider an isotropic noise map $\cal N$,
leading to a bipartite mixed input state $\nu $ given by the convex
combination of a maximally entangled state $|\Phi ^+ \rangle =\sum
_{n=0}^{d-1}|n\rangle |n \rangle /\sqrt d$ and the totally mixed state
$\frac{1}{d^2}I\otimes I$. Then, we can write
\begin{eqnarray} \nu
\equiv (I \otimes {\cal N})\ket{\Phi ^+}\bra{\Phi ^+} = ({\cal
N}\otimes I) \ket{\Phi ^+}\bra{\Phi ^+} = \frac{d^2 F -1}{d^2 -1}
\ket{\Phi ^+}\bra{\Phi ^+} + \frac {1-F}{d^2 -1}I\otimes I
\;,\label{iso}\end{eqnarray}
in terms of the fidelity F with the ideal
maximally entangled state, namely $F=\langle \Phi ^+ | \nu |\Phi^+
\rangle $.
In the case of $d=2$ we have
$\nu\rightarrow\rho_{W}\equiv\frac{4F-1}{3}
\ket{\Phi^{+}}\bra{\Phi^{+}}+\frac{1-F}{3}I\otimes I$, also known as
Werner state.
In our work, the experimental input density matrix $\nu$ is calculated
by a Mathematica algorithm based on two steps. In the first one, a
preliminary matrix $\nu'$ is computed by considering the registered
photon coincidences of 36 suitable plates rotations as projections of
the experimental state. In the second step, the algorithm imposes
positivity and forces $\nu '$ to approach a legitimate positive
matrix $\nu$. This is done by a Maximum Likelihood Maximization
process, based on Poisson statistics, resulting in a physical version
of $\nu$, which is now a positive
matrix with unit trace \cite{jkmw01}.
Then, the fidelity $F$ is obtained by the straightforward evaluation of
$\langle \Phi ^+ | \nu |\Phi^+ \rangle $.
As long as the noise map $\cal N$ acts on maximally entangled states,
its action can be equivalently ascribed to the system or the ancilla
qudit. For this reason the use of a noisy bipartite state as in
Eq. (\ref{iso}) for our detection protocol is equivalent to having a
perfect input $|\Phi ^+\rangle $ along with a quantum measurement
degraded by the isotropic dual map ${\cal N}^\vee$ (the map in the
Heisenberg picture). In fact, for any measurement basis $\{|\Phi _i
\rangle \}$ the reconstructed probabilities are \begin{eqnarray} p_i
&&= \Tr [({\cal I}_A \otimes {\cal E}) ({\cal I}_A \otimes {\cal N})
(|\Phi ^+ \rangle \langle \Phi ^+ |) |\Phi _i \rangle\langle\Phi _i|]
=\Tr [({\cal I}_A \otimes {\cal E}) (|\Phi ^+ \rangle \langle \Phi ^+
|) ({\cal N}^\vee \otimes {\cal I} ) (|\Phi _i \rangle\langle\Phi
_i|)] \nonumber \\& & =\Tr [({\cal I}_A \otimes {\cal E}) (|\Phi ^+
\rangle \langle \Phi ^+ |) \Pi _i]\;, \label{pimeas3} \end{eqnarray}
where \begin{eqnarray} \Pi _i \equiv ({\cal N}^\vee \otimes {\cal I} )
(|\Phi _i \rangle\langle\Phi _i|) \;\label{pom} \end{eqnarray} is
generally the element of a POVM. Then, in order to take into account
the effect of noise, we need to generalize the bound of
Ref. \cite{ms16} \begin{eqnarray} S_e\left (\rho , {\cal E} \right
)\leq H (\vec p)\;, \label{se-bound} \end{eqnarray} to the case of a
probability vector $\vec p$ pertaining to an arbitrary POVM $\{ \Pi _i
\}$ for the tensor product of the reference and system Hilbert spaces,
where \begin{eqnarray} p_i = \Tr [({\cal I}_A \otimes {\cal E})(|\Psi
_\rho \rangle \langle \Psi _\rho |) \Pi _i ] \;. \label{pimeas2}
\end{eqnarray} In the following we provide such a generalization. Let
us consider a density matrix $\sigma$, along with its spectral
decomposition $\sigma =\sum _j s_j |\phi _j \rangle \langle \phi _j
|$, and let us define the conditional probability $p(i|j)=\langle
\phi_j | \Pi _i | \phi _j \rangle $. Clearly, one has $p_i \equiv \Tr
[\sigma \Pi _i] = \sum _j s_j p(i|j)$. Then, \begin{eqnarray} S(\sigma
)- H(\vec p) &=& \sum _i p_i \log _2 p_i - \sum _j s_j \log _2 s_j =
\sum _{i,j} s_j p(i|j)( \log _2 p_i -\log _2 s_j ) \nonumber \\ & \leq
& \log_2 \left (\sum_{i,j}s_j p(i|j) \frac{p_i}{s_j} \right ) = \log_2
\vec r \cdot \vec p \;, \end{eqnarray} where we used Jensen's
inequality, and defined $\vec r$ with vector components $r_i = \sum _j
\bra{\phi _ j} \Pi _i \ket {\phi _j }$. Upon choosing $\sigma =
({\cal I}_R \otimes {\cal E})(|\Psi _\rho \rangle \langle \Psi _\rho
|) $, one obtains the more general bound
\begin{eqnarray} S_e\left
(\rho , {\cal E} \right )\equiv S(\sigma ) \leq H (\vec p)+ \log _2
\vec t \cdot \vec p\;, \label{se-bound2}
\end{eqnarray}
with $p_i $ as
in Eq. (\ref{pimeas2}) and $t_i \equiv \Tr [\Pi _i] \geq r_i$. Then,
the detected quantum capacity $Q_{DET}$ in Eq. (4) of the Letter is
simply replaced with
\begin{eqnarray} Q_{DET}=S\left [{\cal E} (\rho
)\right ] -H(\vec p) - \log _2 \vec t \cdot \vec p \;, \label{qnew}
\end{eqnarray}
in terms of the noisy reconstructed probabilities of
Eq. (\ref{pimeas3}).
Notice now that for any unital noise map (as the isotropic-noise one), the dual map is trace-preserving. Hence, if $\Pi_i $ is of the form as in Eq. (\ref{pom}), one has $t_i \equiv \Tr [\Pi _i]=1$. Then,
the last term in (\ref{qnew}) vanishes, namely $\log _2 \vec t \cdot \vec p =0$.
Let us see the effect of noise on the quantum capacity detection for specific
channels.\\
\noindent 1) Amplitude Damping Channel (ADC) for qubits
\begin{eqnarray}
{\cal E}(\rho )= K_0 \rho K_0^\dag + K_1 \rho K_1^\dag \;,
\end{eqnarray}
where
$K_0= |0 \rangle \langle 0| + \sqrt {1- \gamma }|1 \rangle \langle 1|$ and
$K_1= \sqrt \gamma |0 \rangle \langle 1|$.
For an input state as in Eq. (\ref{iso}), the bipartite output is given by
\begin{eqnarray}
({\cal I}_A\otimes {\cal E}) \nu &=&
\left ( c_1 \gamma _+ + c_2 \right )
|\Phi ^+ \rangle \langle \Phi ^+|
+ \left ( c_1 \gamma _- + c_2 \right )
|\Phi ^- \rangle \langle \Phi ^-|
+\frac \gamma 4
(|\Phi ^+ \rangle \langle \Phi ^-|+|\Phi ^- \rangle \langle \Phi ^+|)
\nonumber \\& + & \frac 1 4 (\gamma c_1 + 4c_2 )
(|\Psi ^+ \rangle \langle \Psi ^+| + |\Psi ^- \rangle \langle \Psi ^-|)
- \frac \gamma 4 (|\Psi ^+ \rangle \langle \Psi ^-| + |\Psi ^- \rangle \langle \Psi ^+|)
\;,\label{outdamp}
\end{eqnarray}
with $\gamma _\pm =\frac 14 (1\pm \sqrt{1- \gamma })^2$,
$c_1=(4F-1)/3$, and $c_2=(1-F)/3$.
The reduced output state is given by ${\cal E}\left (\frac I2\right )=
\frac 12 (I +\gamma \sigma _z)$,
hence it has von Neumann entropy
$S\left [{\cal E}\left (\frac I 2 \right )\right ]
=H_2 \left(\frac {1- \gamma }{2}\right )$.
By performing the local measurement of $ \sigma _x
\otimes \sigma _x $, $ \sigma _y \otimes \sigma _y $, and $ \sigma _z
\otimes \sigma _z $, estimating the von Neumann entropy
$S\left [{\cal E}\left (\frac I 2 \right )\right ]$,
and optimising $\vec p$, one can detect the bound
\begin{eqnarray}
Q\geq Q_{DET}&=& H_2 \left(\frac {1- \gamma }{2}\right )- H(\vec p)
\,,\label{bounddamp}
\end{eqnarray}
where the optimal vector of probabilities is given by
\begin{eqnarray}
\vec{p}&=& \left ( \frac{
2 + 4 F (1 - \gamma ) + \gamma + \sqrt{
4 (1 - 4 F)^2 (1-\gamma ) + 9 \gamma ^2} }{12},
\frac{ 2 + 4 F (1 - \gamma ) + \gamma - \sqrt{
4 (1 - 4 F)^2 (1-\gamma ) + 9 \gamma ^2}}{12}, \right.
\nonumber \\& & \left.
\frac {(1-F)(1-\gamma )}{3} , \frac {2 +\gamma -2F(1-\gamma )}{6} \right)
\;.
\end{eqnarray}
Such probabilities correspond to the optimal basis
\begin{eqnarray}
\{ a |\Phi ^+ \rangle + b |\Phi ^- \rangle ,
-b |\Phi ^+ \rangle + a |\Phi ^- \rangle ,
\frac{1}{\sqrt 2}(|\Psi ^+ \rangle + |\Psi ^- \rangle )\equiv |01 \rangle ,
\frac{1}{\sqrt 2}(|\Psi ^+ \rangle - |\Psi ^- \rangle )\equiv |10 \rangle
\}\;,
\end{eqnarray}
where
\begin{eqnarray}
a=\sqrt{\frac{\cosh \eta + 1}{2\cosh \eta }}\,, \qquad
b=\sqrt{\frac{\cosh \eta - 1}{2\cosh \eta }}\,,
\end{eqnarray}
with $\eta\equiv\text{arcsinh}\left (\frac{3\gamma}{2(4F-1)\sqrt{1-\gamma}} \right )$.
The case of amplitude damping channel is a relevant example showing that the Bell basis can be sub-optimal
for the quantum capacity certification.\\
\noindent 2) For the Phase Damping Channel (PDC) for qubits
\begin{eqnarray}
{\cal E}(\rho )= \left (1-\frac p 2 \right )
\rho + \frac p 2 \sigma _z \rho \sigma _z
\,,
\end{eqnarray}
one has
\begin{eqnarray}
Q=Q_1= 1 - H_2 \left( \frac p2 \right )
\geq Q_{DET}=1- H(\vec p)\,,
\end{eqnarray}
where the vector of probabilities $\vec p$ is given by
\begin{eqnarray}
\vec p=
\left\{ \left (1- \frac p2 \right ) F + \frac p2 \frac{1-F}{3}
, \frac p2 F + \left (1 -\frac p2 \right ) \frac{1-F}{3},
\frac{1-F}{3}, \frac{1-F}{3} \right \}\;,
\end{eqnarray}
which corresponds to the Bell basis.\\
\noindent 3) For the Depolarizing Channel (DC) in dimension $d$
\begin{eqnarray}
{\cal E}(\rho )=
\left ( 1 - p \frac {d^2}{d^2-1} \right )\rho + p
\frac {d^2}{d^2 -1} \frac Id \;,
\end{eqnarray}
the detectable bound is now
\begin{eqnarray}
Q \geq Q_{DET}=\log _2 d - H_2 (p') - p'\log _2 (d^2 -1)\;,\label{hashd}
\end{eqnarray}
where
\begin{eqnarray}
p'=\frac { d^2 [1-F(1-p)]+ F-p-1] }{d^2 -1}
\;.
\end{eqnarray}
For qubits
\begin{eqnarray}
Q \geq Q_{DET}=1 - H_2 (p') - p'\log _2 3\;,\label{hashd2}
\end{eqnarray}
with $p'=(1-F)+ \frac p3 (4F -1)$. The reconstructed probabilities for the Bell basis $\{|\Phi ^+ \rangle , |\Phi ^- \rangle , |\Psi ^+ \rangle , |\Psi ^- \rangle \}$ correspond to $\{1-p',\frac{p'}{3},\frac{p'}{3},\frac{p'}{3}\}$.\\
\noindent 4) For a Pauli Channel (PC) in dimension $d$
\begin{eqnarray}
{\cal E}(\rho )=\sum _{m,n=0}^{d-1} p_{mn} U_{mn} \rho U^{\dag }_{mn}\;,
\end{eqnarray}
with $U_{mn}=\sum _{k=0}^{d-1} e^{\frac{2\pi i}{d} km} |k \rangle
\langle (k + n)\!\!\!\mod d |$, one has
\begin{eqnarray}
Q \geq Q_{DET}=\log _2 d - H({\vec p}\,') \;,
\end{eqnarray}
where ${\vec p}\, '$ is the $d^2$-dimensional vector of probabilities
pertaining to the generalised Bell projectors, whose components are given by
\begin{eqnarray}
p_{mn}'= \frac{1}{d^2-1}[(d^2F -1)p_{mn}+1-F]
\;.
\end{eqnarray}
For the qubit case, ${\cal E}(\rho )=\sum _{i=0}^3 p_i \sigma _i \rho \sigma _i $,
and $p_i'=\frac 13[(4F-1)p_i +1-F]$.
\newpage
\begin{center}
\textbf{Experimental Expectation Values}
\end{center}
The expanded expressions of the probability vectors $p_{i,j}=\braket{\Pi_{i,j}}$ used in our experiment obey the following form, described here for $B_{1}$:
\begin{align}
p_{1,1}&=\braket{(a\ket{\Phi^{+}}+b\ket{\Phi^{-}})(a\bra{\Phi^{+}}+b\bra{\Phi^{-}})}\nonumber\\
&=\frac{1}{4}(\braket{\mathbb{I}\otimes\mathbb{I}}+\braket{\sigma_{z}\otimes\sigma_{z}})+\frac{a b}{2}(\braket{\sigma_{z}\otimes\mathbb{I}}+\braket{\mathbb{I}\otimes\sigma_{z}})+\frac{a^{2}-b^{2}}{4}(\braket{\sigma_{x}\otimes\sigma_{x}}-\braket{\sigma_{y}\otimes\sigma_{y}})\;,
\\
p_{1,2}&=\braket{(-b\ket{\Phi^{+}}+a\ket{\Phi^{-}})(-b\bra{\Phi^{+}}+a\bra{\Phi^{-}})}\nonumber\\
&=\frac{1}{4}(\braket{\mathbb{I}\otimes\mathbb{I}}+\braket{\sigma_{z}\otimes\sigma_{z}})-\frac{a b}{2}(\braket{\sigma_{z}\otimes\mathbb{I}}+\braket{\mathbb{I}\otimes\sigma_{z}})-\frac{a^{2}-b^{2}}{4}(\braket{\sigma_{x}\otimes\sigma_{x}}-\braket{\sigma_{y}\otimes\sigma_{y}}) \;,
\\
p_{1,3}&=\braket{(c\ket{\Psi^{+}}+d\ket{\Psi^{-}})(c\bra{\Psi^{+}}+d\bra{\Psi^{-}})}\nonumber\\
&=\frac{1}{4}(\braket{\mathbb{I}\otimes\mathbb{I}}-\braket{\sigma_{z}\otimes\sigma_{z}})+\frac{cd}{2}(\braket{\sigma_{z}\otimes\mathbb{I}}-\braket{\mathbb{I}\otimes\sigma_{z}})+\frac{c^{2}-d^{2}}{4}(\braket{\sigma_{x}\otimes\sigma_{x}}+\braket{\sigma_{y}\otimes\sigma_{y}}) \;, \\
p_{1,4}&=\braket{(-d\ket{\Psi^{+}}+c\ket{\Psi^{-}})(-d\bra{\Psi^{+}}+c\bra{\Psi^{-}})}\nonumber\\
&=\frac{1}{4}(\braket{\mathbb{I}\otimes\mathbb{I}}-\braket{\sigma_{z}\otimes\sigma_{z}})-\frac{cd}{2}(\braket{\sigma_{z}\otimes\mathbb{I}}-\braket{\mathbb{I}\otimes\sigma_{z}})-\frac{c^{2}-d^{2}}{4}(\braket{\sigma_{x}\otimes\sigma_{x}}+\braket{\sigma_{y}\otimes\sigma_{y}})\;
\end{align}
The rest of the probability vectors, $\vec{p}_{2,j}$ and $\vec{p}_{3,j}$, can be calculated analogously.
Our protocol only needs the following expectation values of observables on the joint ancilla-system state
\begin{align}
&\braket{\sigma_{z}\otimes\sigma_{z}}=
\frac{1}{N_{z}}CC(\ket{H}\bra{H} \otimes\ket{H}\bra{H} -\ket{H}\bra{H} \otimes\ket{V}\bra{V}
-\ket{V}\bra{V} \otimes\ket{H}\bra{H} +\ket{V}\bra{V} \otimes\ket{V}\bra{V} )
,\\
&\braket{\sigma_{y}\otimes\sigma_{y}}=
\frac{1}{N_{y}}CC(\ket{R}\bra{R} \otimes\ket{R}\bra{R} -\ket{R}\bra{R} \otimes\ket{L}\bra{L}
-\ket{L}\bra{L} \otimes\ket{R}\bra{R} +\ket{L}\bra{L} \otimes\ket{L}\bra{L} )
,\\
&\braket{\sigma_{x}\otimes\sigma_{x}}=
\frac{1}{N_{x}}CC(\ket{+}\bra{+} \otimes\ket{+}\bra{+} -\ket{+}\bra{+} \otimes\ket{-}\bra{-}
-\ket{-}\bra{-} \otimes\ket{+}\bra{+} +\ket{-}\bra{-} \otimes\ket{-}\bra{-} )
,\\
& \braket{\mathbb{I}\otimes\mathbb{I}}=\frac{1}{N_{z}}CC(\ket{H}\bra{H} \otimes\ket{H}\bra{H} +\ket{H}\bra{H} \otimes\ket{V}\bra{V}
+\ket{V}\bra{V} \otimes\ket{H}\bra{H} +\ket{V}\bra{V} \otimes\ket{V}\bra{V} )
,\\
&\braket{\sigma_{z}\otimes\mathbb{I}} =
\frac{1}{N_{z}}CC(\ket{H}\bra{H} \otimes\ket{H}\bra{H} +\ket{H}\bra{H} \otimes\ket{V}\bra{V}
-\ket{V}\bra{V} \otimes\ket{H}\bra{H} -\ket{V}\bra{V} \otimes\ket{V}\bra{V} ) \label{ZI}
,\\
&\braket{\sigma_{y}\otimes\mathbb{I}}=
\frac{1}{N_{y}}CC(\ket{R}\bra{R} \otimes\ket{R}\bra{R} +\ket{R}\bra{R} \otimes\ket{L}\bra{L}
-\ket{L}\bra{L} \otimes\ket{R}\bra{R} -\ket{L}\bra{L} \otimes\ket{L}\bra{L} )
\label{YI}
,\\
&\braket{\sigma_{x}\otimes\mathbb{I}}=
\frac{1}{N_{x}} CC(\ket{+}\bra{+} \otimes\ket{+}\bra{+} +\ket{+}\bra{+} \otimes\ket{-}\bra{-}
-\ket{-}\bra{-} \otimes\ket{+}\bra{+} -\ket{-}\bra{-} \otimes\ket{-}\bra{-} )
\label{XI}
,\\
&\braket{\mathbb{I}\otimes\sigma_{z}}=
\frac{1}{N_{z}}CC(\ket{H}\bra{H} \otimes\ket{H}\bra{H} -\ket{H}\bra{H} \otimes\ket{V}\bra{V}
+\ket{V}\bra{V} \otimes\ket{H}\bra{H} -\ket{V}\bra{V} \otimes\ket{V}\bra{V} ) \label{IZ}
,\\
&\braket{\mathbb{I}\otimes\sigma_{y}}=
\frac{1}{N_{y}}CC(\ket{R}\bra{R} \otimes\ket{R}\bra{R} -\ket{R}\bra{R} \otimes\ket{L}\bra{L}
+\ket{L}\bra{L} \otimes\ket{R}\bra{R} -\ket{L}\bra{L} \otimes\ket{L}\bra{L} )
\label{IY}
,\\
&\braket{\mathbb{I}\otimes\sigma_{x}}=
\frac{1}{N_{x}}CC(\ket{+}\bra{+} \otimes\ket{+}\bra{+} -\ket{+}\bra{+} \otimes\ket{-}\bra{-}
+\ket{-}\bra{-} \otimes\ket{+}\bra{+} -\ket{-}\bra{-} \otimes\ket{-}\bra{-} ),
\label{IX}
\end{align}
with
\begin{align}
&N_{z}=CC(\ket{H}\bra{H} \otimes\ket{H}\bra{H} +\ket{H}\bra{H} \otimes\ket{V}\bra{V}
+\ket{V}\bra{V} \otimes\ket{H}\bra{H} +\ket{V}\bra{V} \otimes\ket{V}\bra{V} )
,\\
&N_{y}=CC(\ket{R}\bra{R} \otimes\ket{R}\bra{R} +\ket{R}\bra{R} \otimes\ket{L}\bra{L}
+\ket{L}\bra{L} \otimes\ket{R}\bra{R} +\ket{L}\bra{L} \otimes\ket{L}\bra{L} )
,\\
&N_{x}=CC(\ket{+}\bra{+} \otimes\ket{+}\bra{+} +\ket{+}\bra{+} \otimes\ket{-}\bra{-}
+\ket{-}\bra{-} \otimes\ket{+}\bra{+} +\ket{-}\bra{-} \otimes\ket{-}\bra{-} ) .
\end{align}
In the above equations
$CC(\ket{i}\bra{i} \otimes \ket{j}\bra{j})$
denotes the coincident photon
detections of the state within a time window, associated to the
projection on the state $\ket{i}\otimes \ket{j}$ of the ancilla and system qubits, with
$i,j=H,V,L,R,+,-$.
The different bases are characterized as logical $\{\ket{H},\ket{V}\}$, circular
$\{\ket{L}=\frac{1}{\sqrt{2}}(\ket{H}+i\ket{V}),\ket{R}=\frac{1}{\sqrt{2}}(\ket{H}-i\ket{V}) \}$, or diagonal
$\{\ket{+}=\frac{1}{\sqrt{2}}(\ket{H}+\ket{V}),\ket{-}=\frac{1}{\sqrt{2}}(\ket{H}-\ket{V})\}$, while $N_{z}$, $N_{y}$,
and $N_{x}$ are the normalization factors also expressed in terms of
coincidences.
According to the above equations, all the expectation values are obtained just
by the measurement of 12 polarization projections of the state (4 by each of the 3 observables),
which makes the process efficient in terms registration and analysis of data, given the reduced number of operations compared with a standard process tomography.
Equations (\ref{ZI}-\ref{XI}) are equivalent to take a partial trace of the
S-qubit, and measure exclusively the A-qubit by triggering the
coincidences with the detection of S. Similarly, equations (\ref{IZ}-\ref{IX})
are equivalent to take a partial trace of the A-qubit, and measure
exclusively the S-qubit by triggering the coincidences with the detection
of A.\\
\begin{center}
\textbf{Photon Count Rates and Detection Efficiencies}
\end{center}
The overall detection efficiency of single photons in the system or ancilla modes of our experiment is given approximately by
\begin{equation}
\epsilon_{overall}\approx\epsilon_{source}\cdot\epsilon_{exp}\cdot\epsilon_{tomo}\cdot\epsilon_{SMF}\cdot\epsilon_{APD}
\end{equation}
Where $\epsilon_{source}\approx95\%$ is the transmissivity of down converted photons through the optical elements of the quantum source, $\epsilon_{exp}$ is the transmissivity of the tested channel, $\epsilon_{tomo}\approx99\%$ is the transmissivity of the tomography optics, $\epsilon_{SMF}\approx73\%$ is the coupling efficiency of photons within a single mode optic fiber (SMF) and $\epsilon_{APD}\approx70\%$ is the quantum efficiency of the Avalanche Photodetectors (APDs).
The system-S photons of the source were transmitted to the tested channels through a polarization compensated SMF link. Then, the effective efficiency of any kind of channel was composed by
\begin{equation}
\epsilon_{exp}\approx\epsilon_{SMF}\cdot\epsilon_{channel}
\end{equation}
Where $\epsilon_{channel}$ corresponds to transmissivity of each bulk channel, ranging from $\approx98\%$ in the case of PDC, DC and PC, to $\approx60\%$ for the ADC. Thus, without testing any kind of channel inside the mode ($\epsilon_{exp}=1$) the associated overall detection efficiency for single photons was $\epsilon_{overall}\approx48\%$. In the case of PDC, DC or PC the overall detection efficiency was $\approx34\%$, and for ADC was $\approx21\%$.
Considering $2.5$mW of laser pumping power and the overall single photon efficiency $\epsilon_{overall}\approx48\%$ from the SPDC process to the single photon detection in two synchronized APDs within a time window of 6ns, we get an effective source generation of nearly $C_{S}=C_{A}=375000\frac{\text{single photons}}{\text{sec}}$ in the S-A modes and $C_{SA}=60000\frac{\text{coincident photons}}{\text{sec}}$ without the presence of any channel on S. These values corresponds to a heralding efficiency of
\begin{equation}
\eta=\frac{C_{SA}}{\sqrt{C_{S}\cdot C_{A}}}\approx16\%
\end{equation}
From the above quantities it is also possible to calculate the real number of photons created by the source, right after the SPDC generation, even if we don't have access to them. To estimate that quantity we assume that $C_{S}=C_{A}$, then the source output counts will be
\begin{eqnarray}
C_{S,Source}&=&\frac{C_{S}}{\epsilon_{overall}}\approx 781000\frac{\text{single photons}}{sec}\\
C_{SA,Source}&=&\frac{C_{SA}}{(\epsilon_{overall})^{2}}\approx 260000\frac{\text{coincident photons}}{sec}
\end{eqnarray}
The registered dark counts were $<500\frac{\text{single photons}}{sec}$, and an accidental coincidence rate $<5\%$ was evaluated
\begin{center}
\textbf{Experimental Error Analysis}
\end{center}
Each experimental error bar seen in our results originates from three main contributions:
\begin{enumerate}
\item Poissonian statistics on the coincidence photons counts, which is propagated using the a Monte Carlo simulation around the mean coincidence values registered in 5 seconds of integration.
\item The experimental channels can have systematic errors due an imperfect balance between the internal maps, or also random errors as
the propagation of the uncertainty in the rotation of the HWP and LC.
\item Each experimental point is a mean of a different number of repetitions of the same experiment, so that we considered the standard deviation as a contribution to the random error.
\end{enumerate}
\begin{itemize}
\item \textbf{For the ADC:} The principal contribution to the total error originates from the uncertainty of the dumping preparation, namely the rotation of the $HWP_{V}(\varphi)=\left(\begin{matrix}
cos(2\varphi)&-sin(2\varphi)\\
-sin(2\varphi)&-cos(2\varphi)\end{matrix}\right)$. In this case, there is a direct connection of the damping to the angle $\varphi$ of the plate through the equation
\begin{equation}
\varphi=\frac{arccos(-\sqrt{1-\gamma})}{2}
\end{equation}
Given the non-linearity of dumping, an uncertainty of $\Delta=0.5[\text{degrees}]$ can be very sensitive to any change.
Any other source of error, including the Poissonian distribution of the counts and its proper propagation is practically negligible compared with the propagation of $\Delta$. Thus the presentation of the $Q_{Det}$ under the action of an ADC only have effective error bars in the x-axis.
\item \textbf{For the PC:} Unlike the ADC, the PC is achieved by the combination of four different experiments ($\mathbb{I},\sigma_{x},\sigma_{y},\sigma_{z}$), which are prepared separately as a combinations of one HWP and one LC.
Even if we have more than one optical element with uncertainty $\Delta$, its error propagation represents a very small contribution to the total error. For example, let's consider a more general channel $\Lambda(\rho)=\sum_{i=0}^{3}p_{i}A_{i}\rho A_{i}^{\dagger}$ with $p_{0}+p_{1}+p_{2}+p_{3}=1$, and where each $A_{i}$ map is composed by two LCs, which have an effective operation of $\Gamma_{low}(\varphi)=\left(\begin{matrix}
cos(2\varphi)&-sin(2\varphi)\\
-sin(2\varphi)&-cos(2\varphi)\end{matrix}\right)$ for a low tension value, and $\Gamma_{high}=\mathbb{I}$ for high tension values.
If we want to reproduce the imprecise Pauli channel, namely with an uncertainty of $\Delta$ for every rotation dependence of the LC's, we get that
\begin{eqnarray}
\mathbb{I}=\sigma_{z}\circ\sigma_{z}&\text{ is achieved by }&A_{0}=\Gamma_{low}(0\pm\Delta)\circ\Gamma_{low}(0\pm\Delta)\\
\sigma_{x}=\sigma_{x}\circ\mathbb{I}&\text{ is achieved by }&A_{1}=\Gamma_{low}(\pi/2\pm\Delta)\circ\Gamma_{high}\\
\sigma_{y}=i\sigma_{x}\circ\sigma_{z}&\text{ is achieved by }&A_{2}=\Gamma_{low}(\pi/2\pm\Delta)\circ\Gamma_{low}(0\pm\Delta)\\
\sigma_{z}=\mathbb{I}\circ\sigma_{z}&\text{ is achieved by }&A_{3}=\Gamma_{high}\circ\Gamma_{low}(0\pm\Delta)
\end{eqnarray}
It is not difficult to verify that $\Lambda$ transforms the state $\rho$ almost in the same way for any propagation of errors within $\Delta$. Then, $\Lambda$ applies the same degree of decoherence to $\rho$ for a propagated error within $\Delta$. The direct consequence of this fact is a negligible error contribution from the angle uncertainty to the final quantum channel capacity.
In this kind of channels the principal error contribution originates from the combination of state measurements obtained in different experiments, because of unavoidable photon counts fluctuations. Any other source of error, including the Poissonian distribution of the counts and its proper propagation is negligible. Thus, the presentation of the $Q_{DET}$ under the action of a PC (and also for \textbf{DC and PDC}) only have effective error bars in the y-axis.
\end{itemize}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,750
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package com.jin.service;
import org.springframework.stereotype.Service;
@Service
public class SimpleService {
public void test() {
System.out.println("Simple Job is running!");
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,808
|
Lonebeard creates
music for video games & film
VIEW DEMO REEL CONTACT ME
Make your game, animation, film or commercial stand out with original composed music. From 8-bit chiptunes to full orchestral scores and beyond.
Make your game world, animation, film or commercial come to life with original sound design and ambient soundscapes.
AUDIO IMPLEMENTATION
Technical audio integration for video games. In-engine integration and audio middleware implementation with Wwise, FMOD, Unreal, Unity and various other game engines such as Gamemaker, Stencyl or Gamesalad. Lonebeard is a Wwise Certified user.
SONIC BRANDING
A unique brand identity is essential. Give your product or company instantly recognizable sound or music.
About Felix
Hi, I'm Felix, thanks for dropping by. My music career started at the age of five when my father began teaching me to play the piano. After saving up for my first Commodore 64 home computer, I soon realized computers could be used to make music. No longer being confined to just classical sheet music, my passion for improvising and composing music with computers was born!
Throughout high school and poly technical school, I experimented with audio technology, computer music and synthesizers. Inspired by classical and jazz music, I practiced composing music across different genres ranging from electronic chip tunes to movie soundtracks. Being inspired by composers like Vangelis, Jean-Michel Jarre, Eric Serra and Hans Zimmer, I released my first new age synthesizer album in 1999 which also led to my first paid video game gig for an arcade platform game.
After getting my degree in technical computer science in 1996 I became a software engineer by day and a musician by night (and weekends). During this time I still composed but also focused on performing live music and studying jazz and funk music. I was one of the founders of funk/soul quartet The Rare Groove Orchestra in which I played Hammond organ and all other keys. Rare Groove Orchestra composed and recorded three albums, performed at many renowned jazz festivals like North Sea Jazz and toured in multiple countries like Switzerland, France and Mexico.
During the day I still worked for a number of software related companies like TomTom. After leaving TomTom in 2010 I founded two software companies: QuestionMark (focused on transparency in the food industry) and Simacan (merging and distributing traffic information).
Fast forward eight years… In 2018 I decided to stop working in software development, selling all my shares in the companies I founded and focus full time on my true passion: composing music and sound design for video games at Lonebeard.
Next to my work at Lonebeard, I studied Professional Game Audio Design and Production at Berklee College of Music to further develop my craft. Courses are conducted by Emmy Award nominated audio designer and composer Gina Zdanowic (Bioshock 2/Just Cause 3/XCom).
Next to this I'm working on an album with a new Jazz/Fusion/Electro band TRAQ where I play all the keys.
Feel free to contact me. I'll try to answer within 24 hours.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 9,617
|
Home Spotlight Pick of the Day 1967 Maserati Mistral
1967 Maserati Mistral
1967 Maserati Mistral on display at the 2016 Cobble Beach concours
Between 1963 and 1970, Maserati produced the Frua-designed, Maggiora-bodied Mistral — a few more than 800 coupes and 125 Spyders. Pick of the Day, a 1967 Mistral, is one of only 189 coupes carrying the optional 4.0-liter version of the road-going version of Maserati's straight-6 Grand Prix racing engine.
In an advertisement on ClassicCars.com, the private seller notes that the car underwent a rotisserie restoration in 1997 and has been "sympathetically used and maintained since," and adds that, "I have been the custodian of this fine automobile for almost 20 years and it has been my sincere pleasure but it is finally time to move it along to someone who will appreciate it as much as I have."
Pietro Frua designed the Mistral's bodywork
The car has aluminum Borrani wheels with stainless spokes, power windows, and Becker Mexico AM/FM/cassette with Wonderbar.
Its original Lucas injection system has been uprated with Bosch and Kinsler Engineering components for increased performance and first-run starting whether hot or cold — and no fuel starvation in turns — the seller notes, adding "this work has been done discreetly, wherein all components are hidden from view."
The car was repainted in 2015 and also received new weatherstripping, and in 2016 a NOS Ansa performance exhaust and new clutch assembly were installed. It also was displayed in 2016 at the Cobble Beach Concours d'Elegance.
RELATED: Supercharged 1999 Shelby Series 1 from racer Tony Stewart's collection
"Mistrals are equally happy being lugged around the city, the long-stroke 6 with 290 lb-ft of torque assures strong reserves of acceleration in any gear," the seller writes.
"While the rear, semi-elliptic, suspension and rigid Salisbury axle are not earth shattering technology, when combined with large front and rear anti-roll bars the vehicle handling is competent and the ride exceptionally supple.
"Large dual cross-circuit front and rear Girling disc brakes make for confident braking under all circumstances. The steering is standard recirculating ball but is easy to negotiate at slow speeds and arrow straight at triple-digits."
The seller in Chatham, Ontario, is asking $210,000 but will consider both the best offer and a possible trade.
To view this listing on ClassicCars.com, see Pick of the Day.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 1,758
|
{"url":"http:\/\/math.stackexchange.com\/questions\/145787\/showing-two-abelian-groups-have-the-same-rank","text":"# Showing two abelian groups have the same rank\n\nLet $G=C_{p^{a_{1}}}\\times C_{p^{a_{2}}}\\times...\\times C_{p^{a_{t}}}$ where $a_{1}\\geq a_{2}\\geq...\\geq a_{t}$ and $H\\subseteq G^{P^n}$ for some integer $n$. Please prove if $n>a_{k}$ for some $k\\in\\{1,...,t\\}$ then, $G$ and $\\frac{G}{H}$ have equal rank.\n\nThe rank $G$ is minimal number generators of $G$.\n\n-\nHint: $G\/H$ is a direct product of $t$ groups. To see that these groups are non-trivial, assume that $C_{p^{a_k}} \\subset H$ for some $k$. Why is this impossible? \u2013\u00a0 m_l May 16 '12 at 7:55\nWhat does $H\\subseteq G^{P^n}$ mean? I can guess that you meant a lower-case $p$ and that this is assumed to be a prime number, but how am I to form a quotient of $G$ by a subgroup of a high Cartesian power of $G$? Do you mean the subgroup of elements with $p^n$-torsion or something like that? Also \"$n>a_k$ for some $k\\in\\{1,...,t\\}$\" would seem to mean just $n>a_t$. \u2013\u00a0 Marc van Leeuwen May 16 '12 at 9:09\nMarc, I think he meant $$G^{p^n}:=<x^{p^n}\\,;\\,x\\in G>=\\{x^{p^n}\\,;\\,x\\in G\\}\\,$$ , as G is an abelian group \u2013\u00a0 DonAntonio May 16 '12 at 10:21\n\nLet us answer this question following m_$1$'s idea: if we put $\\,C_{p^{a_i}}=\\langle c_i\\rangle\\,$ , then $$G=\\langle\\,c_1\\,,\\,c_2\\,,\\ldots\\,,c_t\\,\\rangle\\Longrightarrow G\/H=\\langle\\,c_1H\\,,\\,c_2H\\,,\\ldots\\,,c_tH\\,\\rangle$$\nSo we see an element in $\\,G\\,$ as a vector with $\\,t\\,$ coordinates and coordinatewise group operation.\nNow, suppose $$H\\leq G^{p^n}:= \\{\\,x^{p^n}\\;\\;;\\;\\;x\\in G\\,\\}\\,\\,,\\,\\text{with}\\,\\,n>a_i\\,\\,\\text{for some}\\,\\,1\\leq i\\leq t$$ (the set $\\,G^{p^n}\\,$is a sbgp. because we're in an abelian group), then: $$\\text{for some}\\,\\,1\\leq k\\leq t\\,\\,,\\,\\,c_kH=H\\Longleftrightarrow c_k\\in H\\Longleftrightarrow C_{p^{a_k}}\\leq H\\leq G^{p^n}$$But this is impossible since\n$(1)\\,$ If $\\,a_k\\leq n\\,$ , then we'd have that $\\,c_k=(1,\\ldots,1,c_k,1\\ldots,1)\\in G^{p^n}\\,$, which is impossible as if $\\,c_k=x^{p^n}\\,\\,\\text{for some} \\,x\\in G\\,$ above then $\\,x\\,$ has to be exactly of the same form: $\\,x=(1,...,1,y,1,...,1)\\,,\\,y\\,$ in the $\\,k-$position, but then, putting $p^n=p^{a_k+h}=p^{a_k}p^h\\,\\,,\\,h\\geq 0\\,$ , we'd get$$x^{p^n}=(1,...,1,\\left(y^{p^{a_k}}\\right)^{p^h},1,...,1)=(1,1,...,1,....,1)=1\\in G$$\n$(2)\\,$ if $\\,a_k>n\\,$ then reasoning coordinatewise as above: on the $\\,k-$th coordinate we won't get all the elements of $\\,C_{p^{a_k}}\\,$ as $\\,p^n\\,$ powers of elements in $\\,G\\,$ (lest the cyclic group $\\,C_{p^{a_k}}\\,$ of order $\\,p^{a_k}\\,$ has order $\\,p^n<p^{a_k}\\,$...","date":"2014-10-22 21:44:14","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9561675190925598, \"perplexity\": 198.3383639051541}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2014-42\/segments\/1413507447657.38\/warc\/CC-MAIN-20141017005727-00372-ip-10-16-133-185.ec2.internal.warc.gz\"}"}
| null | null |
Q: I2C interfacing with ATmega32 and ArduinoMega2560 failed I'm trying out I2C interfacing the ATmega32 with Arduino Mega2560 Board without success.
The Arduino Board is configured to be the Master Read.
The Atmega32 is configured to be the Slave Write.
I have connected the wires like in the Image below:
The Code for ATmega32 in Atmel 6:
#include <avr/io.h>
#include "I2C_slave.h"
void i2c_initSlave(unsigned char slaveAddress)
{
TWCR = 0x04;
TWAR = slaveAddress;
TWCR = (1<<TWINT) | (1<<TWEN) | (1<<TWEA);
}
//*************************************************
void i2c_send(unsigned char data)
{
TWDR = data;
TWCR = (1<<TWINT) | (1<<TWEN);
while ((TWCR & (1<<TWINT))== 0);
}
//*************************************************
void i2c_listen()
{
while ((TWCR & (1<<TWINT)) == 0);
}
int main(void)
{
int PIN = 0x02;
DDRC &= ~PIN;
i2c_initSlave(0x90);
i2c_listen();
i2c_send("G");
while(1)
{
return 0;
}
}
The Code for Arduino Board:
#include <Wire.h>
#define TRANCEIVER_ADDRESS 0x90
void setup()
{
Wire.begin(TRANCEIVER_ADDRESS); // join i2c bus (address optional for master)
Serial.begin(115200); // start serial for output
}
void loop()
{
Serial.println("HALLO");
Wire.requestFrom(TRANCEIVER_ADDRESS, 2);
while(Wire.available()) // slave may send less than requested
{
char c = Wire.read(); // receive a byte as character
Serial.print(c); // print the character
}
delay(500);
}
As described in the code, I use Address 0x90 to establish the connection and send the character "G" from ATmega32 to Arduino. The output i got in the SerialCommand Window in Arduino IDE is like in the image below:
That means NO CONNACTION !!!
Could someone spot the problem? I think the problem is in the ATmega32 code.
Is the DDRC and PORTC configured correct to the MASTER READ?
DDRC=0xFF;
PORTC=0x00;
I'am not sure. Or should the PORTC be
PORTC=0xFF;
Could someone explain me?
A: In PIC code as slave I2C, use:
i2c(Slave, sda=PIN_C4, scl=PIN_C3, address=0xa0, FORCE_HW, FAST);
In Arduino code as master I2C, use:
const int SLAVE_ADDRESS2 = 0xA0; // esclavo PIC
Wire.beginTransmission(SLAVE_ADDRESS2 >> 1);
A: Here are some errors i noticed in your code:
1) Slave address is 7 bit only (from 0x00 to 0x7F) so 0x90 is out of range, also it placed form b1 ~ b7 in TWAR Register so slave address should be shifted by 1.
void i2c_initSlave(unsigned char slaveAddress)
{
TWCR = 0x04;
TWAR = (slaveAddress << 1); // It shoule be shifted by 1
TWCR = (1<<TWINT) | (1<<TWEN) | (1<<TWEA);
}
2) In i2c_send("G") you are sending pointer while function void i2c_send(unsigned char data) is taking unsigned char argument, fix this by calling it like i2c_send('G');
3) You place i2c_send("G") outside the while loop so it will only be transmitted once.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,712
|
If you want to look at/print out a copy of the EXAM + ANSWERS click here. You may then return to the exam by clicking on your browser's BACK key.
This examination is scheduled for fifty (50) minutes. At the end of this fifty minute period you will have five (5) minutes to place your examination at the front of the examination room.
4. (8 points) The density of mercury is 13.53 g per mL at 25°C. What is its molar concentration under these conditions?
5. (16 points) A 6.768 g sample of methanol (CH3OH) is ignited in a rigid 2.500 liter container into which 3.50 liters of oxygen at 2.017 atm and 23.0 °C has been introduced.
a. Write a balanced equation for the reaction, using integral coefficients.
b. Assuming the reaction goes to carbon dioxide and water products, calculate the number of liters of carbon dioxide produced at the above temperature and pressure.
6. (15 points) A compound composed of mercury, nitrogen and oxygen is analyzed and found to contain 76.39% mercury and 5.33% nitrogen.
a. Find the empirical formula for this compound.
b. If the compound has a formula weight of approximately 500 g, find its "molecular" formula.
7. (15 points) Butane (C4H10) is a useful aerosol propellant since it is fairly innocuous environmentally. Consider a 925 mL aerosol can containing 7.25 g of butane at 23.0 °C. What will the pressure in this can become if it is heated to 690°C in a fire, assuming the can is rigid (there is no volume change)?
8. (8 points) A 0.95 kg cast iron pan is heated from room temperature (20.0 °C) to 125.0 °C on a gas stove. Assuming the heat capacity of the pan remains constant cover this temperature range, calculate the quantity of heat absorbed by the pan ( heat capacity of Fe = 25.1 J °C-1mol-1).
a) What is the molarity of a solution made by dissolving 5.34 g of magnesium iodide (MgI2) in enough water to make up a volume of 500.0 mL.
b. How many mL of a 0.250 M solution of magnesium iodide will be needed to make up 250.0 mL of a solution which has an iodide concentration of 0.025 M?
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 9,554
|
Weekend Series on Crime History: The Las Vegas Mob
Tags: Las Vegas, Mob
Feds Mibehavin' in 2013
Every day, thousands of federal law enforcement agents wake up, grab their gun and badge and a cup of java, orange juice or tea and go out into the world to protect the public and enforce the laws.
Unfortunately, every year, a few step over the line — way over the line — and break the law.
As the year draws to an end, ticklethewire.com takes a look at some of the more interesting cases of Feds Misbehavin' in 2013. As in the past, money and sex was involved in some allegations. And this year, unfortunately, so was death.
Too Much Booze: FBI agent Adrian Johnson got 18 months in prison this year after he was convicted of multiple charges including vehicular manslaughter after he drove drunk and crashed into a car in suburban D.C., in Prince George's County. He killed an 18-year old and man and seriously injured the man's friend in 2011.
Not So Secret Service: Secret Service agents are getting quite the rep for being serious party people. Supervisors Ignacio Zamora Jr. and Timothy Barraclough, aren't doing much to change that image. The Washington Post reported in November that the two, who were managing security for the president, have been removed from that detail because of alleged misconduct involving women.
In one instance in May, Zamora allegedly tried getting back into a woman's room at the Hay-Adams hotel, near the White House, to get a bullet he had left behind. He was off duty and had removed the bullets from the gun while in the room, the Post reported. He had met the woman at the hotel bar and joined her in her room, the Post reported. The Post reported that the guest refused to let Zamora back in, and he identified himself to hotel security as a Secret Service agent. The hotel alerted the White House about the odd behavior, the Post reported.
During an internal investigation, investigators also found that the two agents had allegedly sent sexually suggestive emails to a female subordinate, who is an agent.
Hands in the Cookie Jar: Oklahoma FBI agent Timothy A. Klotz confessed to dipping into the FBI cookie jar. Authorities allege that he embezzled $43,190 that was earmarked for confidential informants for tips on criminal activities from 2008-2011. He acknowledged in a signed statement that he falsified 66 receipts during a scheme that went undiscovered for more than four years. He was sentenced earlier this month to six months in prison and three years of supervised released. He was also ordered to pay a restitution of $43,190.
Let The Dice Roll –– FBI agent Travis Raymond Wilson, 38, of Huntington Beach, Calif., apparently had a little gambling jones and didn't want the big guys at the FBI to know. Unfortunately for him, he got busted. Wilson pleaded guilty to structuring financial transactions in violation of the federal Bank Secrecy Act.
The feds say between January 2008 and February 2013, Wilson regularly gambled at casinos in California, Nevada, Arizona, and West Virginia, authorities said. In total, Wilson structured more than $488,000 in cash. Sentencing is set for March 3.
Hookers, Cash and Luxury Travel: Human temptation. Need you say more. John Bertrand Beliveau Jr., 44, a special agent with the Naval Criminal Investigative Service (NCIS), apparently failed that test. He pleaded guilty earlier in December to participating in a massive international fraud and bribery scheme. He admitted sharing with a foreign Navy contractor confidential information about ongoing criminal probes into the contractor's billing practices in exchange for prostitutes, cash and luxury travel, the Justice Department said in a press release. His case is part of a big scandal.
Ethics Still Applies When You Depart: Kenneth Kaiser, former head of the FBI's Boston office, found that ethics still apply when you leave the bureau. The choked up ex-agent appeared in court where he was fined $10,000 for violating an ethics charge. Kaiser was accused of meeting with former FBI colleagues about his company that was under investigation. Federal law prohibited him from having professional contact with former FBI colleagues within a year of leaving government service.
"I lost something I valued the most — my reputation," Kenneth W. Kaiser, 57, of Hopkinton, Mass. said, according to the Boston Globe.
Helping the Wrong Side – Border Patrol Agent Ivhan Herrera-Chiang took advantage of his position and helped smugglers bring meth, cocaine and marijuana into the U.S. He was sentenced in Phoenix in November to 15 years. He reportedly even helped smugglers find their way around underground sensors and lock combinations.
"You have done about the worst thing a law-enforcement agent could do, especially a Border Patrol agent, and that is passed confidential information," U.S. District Judge Paul Rosenblatt said.
A Fatal Shot — FBI agent Arthur "Art" Gonzales of Stafford County, Va. is charged with shooting his estranged wife to death in April. He told dispatchers he was acting in self-defense when he shot his 42-year-old wife, Julia Sema Gonzales. He says his wife attacked him with a knife.
Gonzales was a supervisory special agent-instructor at the FBI's National Academy at Quantico. Court records show bond was granted. Trial has been set for March.
ICE Agent ICED:Veteran ICE agent Juan Martinez, 47, has suddenly got a lot on his plate. He is accused of extortion and accepting bribes. Authorities alleged that he conspired with others to shake down a Colombian construction company. The group allegedly told the firm that it was under investigation, when it was not, and that the U.S. Treasury was about to add the company to a list known as Specially Designated Nationals (SDN). The designation by Treasury can result in the freezing of bank accounts and other action harmful to a business. Martinez's group said it could keep the company off the list, and for that, it received more than $100,000. He is also accused of illegally bringing in people to this country, claiming falsely that they were witnesses in an ongoing narcotics investigation. His attorney says the allegations are false.
Leaky Pipes: Plumbers aren't the only ones who concern themselves with leaks. FBI agent Donald Sachteren who leaked information to the Associated Press was recently sentenced to more than three years in prison for possessing and disclosing secret information. Sachteren, 55, was accused of disclosing intelligence about the U.S. operation in Yemen in 2012. What made him a far less sympathetic character in this whole mess was the fact he was also sentenced to more than 8 years in prison for possessing and distributing child pornography in an unrelated case.
Posted: December 27th, 2013 under Special Report.
Ex-MSU Football Star Returns to Detroit to Head ATF Division
Steve Bogdalek/ATF photo
DETROIT — Back in day, in the 1980s, Steven Bogdalek, a big, burly guy, was an agent with the Bureau of Alcohol, Tobacco and Firearms in Detroit. If some mistook him for a football player, well, that was understandable.
He was offensive tackle, All-Big 10 for Michigan State University, from 1982-85 and he was subsequently drafted by the Philadelphia Eagles. But his NFL career ended prematurely because of an injury. So he moved on to a career with ATF.
While he faced some tough guys on the football field, he also bumped up against some brutal types on the streets. He worked on squads that investigated some of Detroit's most notorious drug gangs.
"Steve Bogdalek brought his team-player mentality to ATF in Detroit from his athletic prowess on the gridiron at Michigan State," recalls Bernard La Forest, who headed up the Detroit ATF office at the time. "He was an integral part of our task force efforts in the enforcement squad that investigated the most violent of Detroit's drug organizations: The Chambers Brothers, Ed Hanserd's crew, Clifford Jones' operation, Erie Adams' organization, and remnants of YBI (Young Boys Incorporated) and Best Friends.
"Steve and the other ATF special agents were successful in just about every investigative operation they opened," La Forest said.
La Forest recalls how effective Bogdalek was in getting access to buildings and homes during raids, using a battering ram.
"With Steve handling the ram, entry into buildings and dope houses was always quick and efficient," La Forest said, describing him as humble.
In 1998, Bogdalek went to Toledo to head up the ATF office. And in pursuing years, he moved around the country, eventually ending up in Los Angeles as the top agent of the ATF office. In January, he'll return to Detroit, the place he started his career, to head up the Detroit office.
"I'm happy to becoming back to Detroit," he told Deadline Detroit. "Life comes full circle sometimes. I'm ending up back here where I started."
Bogdalek, who was raised in Naperville, Ill., knows he faces some serious challenges in Detroit, with its violent crime and limited police resources. He says Flint, which is also in his territory, has its challenges as well.
Posted: December 27th, 2013 under Milestone, News Story.
Georgia Detective Accuses FBI of Hampering Murder Investigation of Hip-Hop Artist
A detective in Georgia accused an FBI agent of lying and impeding a murder investigation, according to video obtained by Channel 2 Action News.
"You just had an FBI agent on duty lie to me and delay this investigation," Detective J.T. Williams said in the video, speaking to Mani Chulpayev, an FBI informant charged with assisting in the murder of Atlanta hip-hop artist Lil Phat.
Williams' beef was with Chulpayev's FBI handler, Special Agent Dante Jackson, for preventing police from questioning Chulpayev earlier in the investigation.
"What he does is, he tells a material witness in my case not to talk to us," Williams said.
"That wasn't true. I always wanted to talk," Chulpayev told the detective.
Tags: detective, FBI, Georgia, Informant, Lil Phat, murder investigation
FBI Sees Unusual Spike in Bank Robberies in Oklahoma
Bank robberies are spiking in Oklahoma.
According to the FBI, bank robberies have nearly tripled in the state, from 23 last year to more than 60 this year, KFOR reports.
It's unclear what caused the unusual spike, but the FBI said two-thirds of the robberies have led to arrests.
Tags: bank robberies, FBI, Oklahoma
A Look at the Top Terror Cases Handled by the FBI in 2013
Since the Sept. 11 terrorist attacks, the FBI has been swamped cracking down on dangers from extremists.
This year was no exception.
Here is some of the FBI's top terror cases of 2013:
Airport bomb plot: A 58-year-old man was charged earlier this month with attempting to explode a car bomb at a Kansas airport as an act of jihad against the U.S. He was arrested as a result of an undercover investigation. The device provided to him by our operatives was inert and posed no danger to the public.
Attempt to join al Qaeda: A New York man was arrested in October for attempting to join al Qaeda in the Arabian Peninsula and conspiring to commit murder overseas. The 25-year-old allegedly conspired with others to travel overseas to wage violent jihad against the perceived enemies of Islam, which included the secular government in Yemen.
Material support to terrorists: Two individuals—one an American citizen—were indicted in August for conspiring to provide material support to al Qaeda groups and al Shabaab. The men were charged with attempting to provide money and recruits to three different terror organizations.
Sovereign citizen scheme: In July, the self-proclaimed president of a sovereign citizen group in Alabama was sentenced to 18 years in prison for promoting a tax fraud scheme that taught people how to defraud the IRS. He and other sovereign citizens also sent demands to all 50 U.S. governors in 2010 ordering each to resign within three days—to be replaced by a "sovereign" leader or be "removed."
Attempt to wage jihad: A Florida man was indicted in July for attempting to provide material support to terrorists. The 19-year-old tried to travel to the Arabian Peninsula to join and fight with a violent al Qaeda group that has taken responsibility for multiple attacks on Yemeni forces, including a suicide bombing in 2012 that killed more than 100 soldiers.
Former U.S. soldier indicted: A U.S. citizen who formerly served in the army was indicted in June for conspiring to provide material support to a foreign terrorist organization. The 30-year-old man allegedly wanted to fight alongside an al Qaeda-affiliated terrorist group in Syria.
Far-fetched terror plan: Two New York men were charged in June with conspiracy to provide material support to terrorists. Their scheme involved creating a remotely operated X-ray radiation-emitting device designed to kill people silently. Their targets were perceived enemies of Israel.
Tsarnaev charged: In April, 19-year-old Dzhokhar Tsarnaev was charged with using a weapon of mass destruction for his role in the Boston Marathon bombings. The attacks killed three people and injured more than 260 others.
Suicide bombing: An Oregon resident was charged in March for his role in a 2009 suicide bombing. The man allegedly assisted an individual who participated in the attack at the headquarters of Pakistan's intelligence service in Lahore that killed approximately 30 individuals and injured 300 others.
Bin Laden associate arrested: An associate of Osama bin Laden was arrested in March for conspiring to kill Americans. The individual held a key position in al Qaeda and appeared with bin Laden after the 9/11 attacks to threaten further attacks against the U.S.
Tags: airport bomb, al Qaeda, FBI, terror
White Texas Man Accused of Brutal Assault on Elderly Black Man in Recorded 'Knockout Game'
It's a disturbing game – abruptly punch a stranger in the face in an attempt to knock him out with a single blow.
A white Texas man is accused of doing exactly that, knocking a 79-year-old black man to the concrete, breaking his jaw and sending him to the hospital for several days, the Los Angeles Times reports.
Authorities said the suspect, Conrad Alvin Barrett, recorded the attack, laughed and said "knockout" before fleeing.
"Suspected crimes of this nature will simply not be tolerated," U.S. Atty. Kenneth Magidson of the Southern District of Texas said in a statement as he announced the charges against Barrett. "Evidence of hate crimes will be vigorously investigated and prosecuted with the assistance of all our partners to the fullest extent of the law."
Authorities said the "knockout game" is a disturbing trend that is often racially motivated.
Barrett is charged with violating the Matthew Shepard and James Byrd Jr. Hate Crimes Prevention Act.
FBI on Hunt for Serial Bank Robber in Boston
North Dakota's 2 Senators Say FBI Office in Montana Too Far Away
FBI Warns That Criminals Using Foil to Steal from Retail Stores
Heroin Overdose Leads to Man's Arrest in Oregon
New Mexico Man Accused of Stealing Gun from Border Patrol Officer's Car
Tags: assault, hate crime, Hate Crimes Prevention Act, knockout game
|
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java.lang.ClassNotFoundException: org.apache.http.params.HttpParams
at org.eclipse.microprofile.opentracing.tck.OpentracingClientTests.clearTracer
(OpentracingClientTests.java:1075)
-->
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<!-- from MP repo -->
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<artifactId>microprofile-opentracing-api</artifactId>
<version>${version.microprofile-opentracing}</version>
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<groupId>org.eclipse.microprofile.opentracing</groupId>
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<dependency>
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<version>${version.resteasy}</version>
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java.lang.ClassNotFoundException: org.codehaus.jackson.jaxrs.JacksonJsonProvider
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<artifactId>jackson-jaxrs</artifactId>
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{"url":"https:\/\/windowsontheory.org\/2017\/04\/18\/celebrating-tcs-at-stoc-2017\/","text":"STOC 2017 is going to be part of an expanded \u201cTheory Festival\u201d which will include not just the paper presentations, but a host of other activities such as plenary talks and tutorials, workshops, and more.\n\nOne of the components I am most excited about is a sequence of invited plenary short talks where we will get a chance to hear about some exciting recent theoretical works from a variety of areas from areas as disparate as theoretical physics and network programming languages, and many others in between.\n\nAs the chair of the committee to select these talks, I was very fortunate for the work of the committee members as well as the many nominations we received from leading researchers across a great many fields. \u00a0I am also grateful to all the speakers that agreed to come despite the fact that in most cases STOC is not their \u201chome conference\u201d. \u00a0The end result is a collection of talks that is sure to contain interesting and new content for every theoretical computer scientist, and I encourage everyone who can make it to register to the conference and come to Montreal in June.\n\nHere is some information about the talks (in the order of scheduling).\n\nThe short descriptions of the talks below are mine and not the authors\u2019: more formal and informative (and maybe even correct \ud83d\ude42 ) abstracts will be posted closer to the event.\n\n### Tuesday, June 20, 3:50pm-5:30pm\n\nAlon Orlitsky: Competitive Distribution Estimation: Why is Good-Turing Good\n\nEstimating a distribution from samples is one of the most basic questions in information theory and data analysis, going at least far back to Pearson\u2019s work in the 1800\u2019s. In Alon\u2019s wonderful NIPS 2015 paper with Ananda Theertha Suresh (which also won the NIPS best paper award) they showed that a somewhat mysterious but simple estimator is nearly-optimal in the sense of providing good competitive guarantees even against ideal offline estimators that have more information.\n\nJohn Preskill: Is spacetime a quantum error-correcting code?\n\n20th century physics\u2019 quest for a \u201ctheory of everything\u201d had encountered a \u201cslight hitch\u201d in that the two most successful theories: general relativity and quantum mechanics, are inconsistent with one another. \u00a0Perhaps the most promising approach towards reconciling this mismatch is a 20 years old conjectured isomorphism between two physical theories, known as the \u201cAdS\/CFT correspondence\u201c. A great many open questions relating to this approach remain; over the past several years, we have learned that quantum information science might shed light on these fundamental questions. \u00a0John will discuss some of the most exciting developments in this direction, and in particular will present his recent \u00a0Journal of High Energy Physics paper with Pastawski, Yoshida, and Harlow which connects quantum gravity (and black holes in particular), to issues in quantum information theory and specifically to quantum error correcting codes.\n\nTim Roughgarden: Why Prices Need Algorithms\n\nIn recent years we have seen many results showing computational hardness of computing equilibria. But in Tim\u2019s EC 2015 paper with Inbal Talgam-Cohen (which won the best student paper award) they showed a surprising connection between computational complexity and the question whether an equilibrium exists at all. It is the latter type of question that is often of most interest to economists, and the paper also gives some \u201cbarrier results\u201d to resolving open questions in economics.\n\nWim Martens: Optimizing Tree Pattern Queries: Why Cutting Is Not Enough\n\nTree patterns are a natural (and practically used) formalism for queries about tree-shaped data such as XML documents. Wim will talk about some new insights on these patterns. It is rare that the counterexample for a 15-year old conjecture is small enough to print on a T shirt, but in Wim\u2019s \u00a0PODS 2016 paper with Czerwinski, Niewerth, and Parys (which was presented in the awards session and also chosen as SIGMOD highlight) they were able to do just that. (Wim did not tell me if the shirts would be available for sale in the conference..)\n\n### Wednesday, June 21, 4:15pm-5:30pm\n\nAtri Rudra: Answering FAQs in CSPs, Probabilistic Graphical Models, Databases, Logic and Matrix operations\n\nThe Functional Aggregate Query (FAQ) problem generalizes many tasks studied in a variety of communities including solving constraint-satisfaction problems, evaluating database queries, and problems arising in probabilistic graphical models, coding theory, matrix chain computation, and the discrete Fourier transform. In Atri\u2019s PODS 2016 paper with Abo Khamis and Ngo (which won the best paper award and was selected as SIGMOD highlight), they unified and recovered many old results in these areas, and also obtained several new ones.\n\nVasilis Syrgkanis: Fast convergence of learning in games\n\nVasilis will talk on some recent works on the interface of learning theory and game theory. Specifically, he will discuss how natural learning algorithms converge much faster than expected (e.g., at a rate of $O(T^{-3\/4})$\u00a0instead of the classical \u00a0$O(1\/\\sqrt{T})$) \u00a0to \u00a0the optimum of various games. This is based on his NIPS 2015 paper with Agarwal, Luo, and Schapire, which won the best paper award.\n\nChris Umans: On cap sets and the group-theoretic approach to matrix multiplication\n\nChris will discuss the recent breakthroughs on the \u201ccap set problem\u201d and how they led to surprising insights on potential matrix-multiplication algorithms. \u00a0Based on this Discrete Analysis paper with Blasiak, Church, Cohn, Grochow, Naslund, \u00a0and Sawin.\n\n### Thursday, June 22, 3:50pm-5:30pm\n\nChristopher R\u00e9: Ensuring Rapid Mixing and Low Bias for Asynchronous Gibbs Sampling\n\nGibbs sampling is one of the most natural Markov Chains arising in many practical and theoretical contexts, but practically running the algorithm is very expensive. The Hogwild! Framework of Chris and co authors is a way to run such algorithms in parallel without locks but it\u2019s unclear that the output distribution is still correct. In Chris\u2019s \u00a0ICML 2016 paper with De Sa and Olukotun (which won the best paper award) they gave the first theoretical analysis of this algorithm.\n\nNate Foster: The Next 700 Network Programming Languages\n\nI never expected to see Kleene Algebra, straight from the heart of Theory B, \u00a0used for practical packet processing in routers, but this is exactly what was done by this highly influential POPL 2014 paper of Nate with Anderson, Guha, Jeannin, Kozen, Schlesinger, and Walker.\n\nMohsen Ghaffari: \u00a0An Improved Distributed Algorithm for Maximal Independent Set\n\nMaximal Independent Set is the \u201ccrown jewel of distributed symmetry breaking problems\u201c \u00a0to use the words from the 2016 Dijkstra prize citation for the works showing an $O(\\log n)$ \u00a0time distributed algorithm. In Mohsen\u2019s SODA 2016 paper (which won the best paper award) he improved on those works to give a local algorithm where each vertex will finish the computation in time that is $O(\\log degree)$. Moreover, in graphs with degree $n^{o(1)}$, all nodes will terminate faster than the prior algorithms, in particular almost matching the known lower bound.\n\nValeria Nikolaenko: \u00a0Practical post-quantum key agreement from generic lattices\n\nWith increasing progress in quantum computing, both the NSA and commercial companies are getting increasingly nervous about the security of RSA, Diffie-Hellman, and Elliptic Curve Crypto. Unfortunately, lattice-based crypto, which is the main candidate for \u201cquantum resistant\u201d public key encryption, was traditionally not efficient enough to be used in real world web security. This has been changing with recent works. In particular in Valeria\u2019s \u00a0ACM CCS 2016 paper with Bos et al they gave a practical scheme based on standard computational assumptions on lattices. This is a follow up to the New Hope cryptosystem which is currently implemented in Chrome canary.\n\nApril 19, 2017 12:47 pm\n\nMan, I\u2019d be so super-psyched and excited to attend this conference, if only (1) it didn\u2019t overlap completely with LICS, where I have a paper accepted, and (2) the awesome list of invited plenary short talks didn\u2019t completely disregard the existence of LICS.\n\n\u2022 April 19, 2017 1:01 pm\n\nSorry you can\u2019t come, I had no control over scheduling but alas conflicts are sometimes hard to avoid, especially in the summer. We could not possibly cover all venues in a list of 11 talks but we definitely did not disregard the existence of LICS. This was one of the conferences that we explicitly solicited nominations from. I hope this will become an enduring tradition and we will see more papers from a variety of venues.","date":"2018-01-16 18:58:24","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 5, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5358422994613647, \"perplexity\": 1734.162417582794}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-05\/segments\/1516084886639.11\/warc\/CC-MAIN-20180116184540-20180116204540-00469.warc.gz\"}"}
| null | null |
A US Open jelentheti:
US Open (teniszbajnokság) – az év negyedik Grand Slam teniszbajnoksága
US Open (golfbajnokság) – PGA golftorna, a United States Golf Association versenye
US Open (sakkbajnokság) – az Amerikai Sakkszövetség (USCF) évente megrendezett nyílt bajnoksága
|
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\section{Introduction}
The inspiral and merger of binary neutron stars (BNSs) is one of the most
promising sources of gravitational waves (GWs) for future ground-based
laser-interferometer detectors such as LIGO, Virgo or KAGRA
\citep{Sathyaprakash:2009xs}. Because they can travel almost unscattered
through matter, GWs carry valuable information from the deep core of the
neutron stars (NSs) concerning the equation of state (EOS) of matter at
supra-nuclear densities. Unfortunately, they are also extremely hard to
detect, so that their identification and analysis requires the
availability of analytical or semi-analytical GW templates. In turn, the
validation and tuning of these models must be done by matching them with
the predictions of fully non-linear numerical relativity (NR)
calculations, which represent the only means to describe accurately the
late inspiral of BNS \citep{Baiotti:2010, Baiotti2011, Bernuzzi2012,
Hotokezaka2013b, Bernuzzi2014, Bernuzzi2015}.
While very high-quality NR waveforms of binary black hole mergers are
available, \textit{e.g.}~ \citep{Aylott:2009ya, Mroue2013, Hinder2013}, BNS
simulations have been plagued by low convergence order and relatively
large phase uncertainties ($\delta\phi/\phi \sim 1 \%$)
\citep{Baiotti:2009gk,Bernuzzi2011}. Furthermore, since NSs have smaller
masses, the merger part of the waveform is out of the frequency band for
the next generation GW detectors, so that EOS-related effects will have
to be most probably extracted from the inspiral signal using a large
number of events. In particular, EOS-induced effects will be encoded in
the phase evolution of the GW signal during the inspiral
\citep{Damour:2012}. As a result, the measure of EOS-induced effects
requires very accurate general-relativistic predictions of the inspiral
signal. Even though accurate waveforms can be calculated by second-order
codes at very high computational costs \citep{Baiotti:2010, Baiotti2011,
Bernuzzi2012, Hotokezaka2013b, Bernuzzi2014, Bernuzzi2015}, their
analysis is complicated by the low convergence order of the methods
employed. In particular, the analysis often requires the use of a time
rescaling or alignment of the GWs from different resolutions
\citep{Baiotti2011, Hotokezaka2013b}, which is hard to justify
mathematically.
Here we show that, by using high-order numerical methods, it is indeed
possible to obtain waveforms for the late-inspiral of a BNS system with a
quality that is almost comparable with the one obtained for binary black
holes, with clean, higher than second-order convergence in both the phase
and the amplitude. In particular we highlight and extend to higher
resolution some of the results we reported in \citep{Radice2013b}.
\section{Numerical methods}
The results presented here have been obtained with our new high-order,
high-resolution shock-capturing (HRSC), finite-differencing code:
\texttt{WhiskyTHC} \citep{Radice2013b, Radice2013c}, which represents the
extension to general relativity of the special-relativistic \texttt{THC}
code \citep{Radice2012a}. \texttt{WhiskyTHC} solves the equations of
general-relativistic hydrodynamics in conservation form
\citep{Banyuls97,Rezzolla_book:2013} using a finite-difference scheme
that employs flux reconstruction in local-characteristic variables using
the MP5 scheme, formally fifth-order in space \citep{suresh_1997_amp}
[see \citet{Radice2012a, Radice2013c} for details].
The spacetime evolution makes use of the BSSNOK formulation of the
Einstein equations \citep{Nakamura87, Shibata95, Baumgarte99} and it is
performed using fourth-order accurate finite-difference scheme provided
by the \texttt{Mclachlan} and is part of the \texttt{Einstein Toolkit}
\citep{Loffler:2011ay, Brown:2008sb, Schnetter-etal-03b}. To ensure the
non-linear stability of the scheme we add a fifth-order Kreiss-Oliger
type artificial dissipation to the spacetime variables only. Finally, the
coupling between the hydrodynamic and the spacetime solvers is done using
the method of lines and a fourth-order Runge-Kutta time integrator. The
resulting scheme is formally fourth-order in space and time, except at
the boundaries between different refinement levels, where our method is
only second-order in time. This should have only marginal effects on our
results given that our finest grid covers both NSs.
\section{Binary setup}
\label{sec:setup}
\begin{table}
\caption{\label{table:models} Summary of the considered BNS model. We
report the total baryonic mass $M_b$, the ADM mass $M$, the initial
separation $r$, the initial orbital frequency $f_{\mathrm{orb}}$, the
gravitational mass of each star at infinite separation,
$M_\infty$, the compactness, $\mathcal{C} = M_\infty/R_\infty$, where
$R_\infty$ is the areal radius of the star when isolated and the tidal
Love number, $\kappa_{2}$, \textit{e.g.}~ \citet{Hinderer09}.}
\vspace{1em}
\centering
\begin{tabular}{ccccccc}
\tableline
\noalign{\smallskip}
$M_b\ [M_\odot]$ &
$M\ [M_\odot]$ &
$r\ [\mathrm{km}]$ &
$f_{\mathrm{orb}}\ [\mathrm{Hz}]$ &
$M_\infty\ [M_\odot]$ &
$\mathcal{C}$ &
$\kappa_2$ \\
\tableline
\noalign{\smallskip}
$3.8017$ & $3.45366$ & $60$ & $208.431$ & $1.7428$ & $0.18002$ & $0.05$ \\
\tableline
\noalign{\smallskip}
\end{tabular}
\end{table}
The initial data is computed in the conformally flat approximation using
the \texttt{\textsc{Lorene}} pseudo-spectral code \citep{Gourgoulhon01}
and describes two irrotational, equal-mass NSs in quasi-circular
orbit. Its main properties are summarized in Table \ref{table:models},
and we note that it is computed using a polytropic EOS ($p=K\rho^\Gamma$)
with $K = 123.56$ (in units where $G = M_\odot = c = 1$) and $\Gamma=2$,
while the evolution is performed using the ideal-gas EOS
($p=(\Gamma-1)\rho\epsilon$) with the same $\Gamma$
\citep{Rezzolla_book:2013}.
The runs are performed on a grid covering $0 < x,z \lesssim
750\ \mathrm{km}$, $-750\ \mathrm{km} \lesssim y \lesssim
750\ \mathrm{km}$, where we assume reflection symmetry across the $(x,y)$
plane and $\pi$ symmetry across the $(y,z)$ plane. The grid employs six
\emph{fixed} refinement levels, with the finest one covering both stars.
We consider four different resolutions, labelled as $L$, $M$, $H$ and
$V\!H$, having, in the finest refinement level, a grid spacing of
$h\simeq 370, 295, 215$ and $147$ meters, respectively. The results of
simulations $L$, $M$, $H$ were already presented in \citet{Radice2013b,
Radice2013c}, while those of the simulation $V\!H$ are presented here
for the first time.
\section{Binary dynamics}
\begin{figure}
\begin{center}
\includegraphics[width=\hsize]{fig1.eps}
\caption{Rest-mass density on the equatorial plane for the medium
resolution run $M$. The panels show the initial data (upper right),
the late stages of the inspiral (top middle and left), the
approximate time of contact (bottom left), merger (bottom center)
and black-hole formation (bottom right).}
\label{fig:rho2d}
\end{center}
\end{figure}
The dynamics of the binary is summarized in Figure \ref{fig:rho2d}, where
we show the density on the equatorial plane for the $M$ run (the others
are qualitatively very similar). The initial distance between the two NS
centers is $60\ \mathrm{km}$. They complete about $7$ orbits, while
inspiraling because of the loss of orbital angular momentum to GWs,
before entering into contact at time $t\simeq 24.6\ \mathrm{ms}$ (from
the beginning of the simulation). Finally, they quickly merge and a black
hole is formed soon after.
\section{Gravitational waves}
\begin{figure}
\begin{center}
\includegraphics[width=0.49\hsize]{fig2a.eps}
\includegraphics[width=0.49\hsize]{fig2b.eps}
\caption{Accumulate de-phasing (left panel) and estimated order of
convergence (right panel) for the $\ell = 2, m = 2$ mode of the
Weyl scalar $\Psi_4$ as extracted at $r = 450\ M_\odot$. The
de-phasing between high ($H$) and medium ($M$) and very high
($V\!H$) and high are also rescaled assuming an order of
convergence of $3.2$. The instantaneous order of convergence is
estimated separately from the first three resolutions
$\mathcal{O}(\{L,M,H\})$ and from the last three
$\mathcal{O}(\{M,H,V\!H\})$.}
\label{fig:convergence}
\end{center}
\end{figure}
To quantify the accuracy of our code for GW astronomy, we consider the
phase evolution of the dominant $\ell = 2, m = 2$ component of the
curvature GW, the (complex) Weyl scalar $\Psi_4$, as extracted at $r
\simeq 665\ \mathrm{km}$. We compute the GW phase $\phi$ after
decomposing the curvature as $\Psi_4 = A
\mathrm{e}^{-\mathrm{i}\phi}$. Figure \ref{fig:convergence} shows an
analysis of the residual of the phase between the different resolutions
as a function of the retarded time $u$. On the left panel, we show both
the absolute de-phasing between successive resolutions and the residuals,
between the $H$ and $M$ and the $V\!H$ and $H$ resolutions, scaled
assuming a convergence order of $3.2$. On the right panel, we plot the
instantaneous convergence order as measured from three out of the four
resolutions (separately the first three $L, M, H$ and the last three $M,
H, V\!H$).
Figure \ref{fig:convergence} demonstrates that higher than second-order
convergence can be achieved for numerical relativity simulations of BNS,
without the need to perform any artificial manipulation of the waveforms.
We find an order of convergence $\sim 3.2$ (also confirmed by the very
good overlap between the rescaled de-phasing), which is somewhat smaller
than the formal order of four of our scheme. However, this is to be
expected because HRSC methods typically reach their nominal convergence
order only at very high resolutions \citep{Shu97, Radice2012a}; see also
\cite{Zlochower2012} for a discussion of other possible sources of
errors. Finally, as also observed with other codes \citep{Bernuzzi2011},
our solution shows a loss of convergence (with apparent
super-convergence) after $u \gtrsim 24.6\ \mathrm{ms}$. This is roughly
the time when the two NSs enter into contact (see Figure
\ref{fig:rho2d}). At this time the de-phasing between the $H$ and $V\!H$
resolution is $\simeq 0.26\ \mathrm{rad}$.
\begin{figure}
\begin{center}
\includegraphics[width=0.49\hsize]{fig3a.eps}
\includegraphics[width=0.49\hsize]{fig3b.eps}
\caption{Relative phase (left panel) and amplitude (right panel)
differences for the $\ell = 2, m = 2$ mode of the Weyl scalar $\Psi_4$
between the Richardson extrapolated phase and each of the resolutions.
The extrapolated phase is computed assuming a convergence order of
$3.2$ and using the first three resolutions ($L$, $M$ and $H$).}
\label{fig:error}
\end{center}
\end{figure}
As a measure of the error with respect to the exact solution we use the
\emph{first three} resolutions ($L$, $M$ and $H$), Richardson extrapolate
the GW to infinite resolution assuming a convergence order of $3.2$ and
compute the phase and amplitude differences of each run with respect to
the extrapolated data. The results are shown in Figure \ref{fig:error}.
Obviously the data for $u > 24.6\ \mathrm{ms}$ (which is about $13.5$ GW
cycles) has to be taken with a grain of salt, given that convergence is
lost after contact. Before that, we find that the $H$ run has a
de-phasing $\simeq 0.35\ \mathrm{rad}$ ($\simeq 0.4 \%$) at time $u =
24.6\ \mathrm{ms}$, while the highest resolution $V\!H$ run has a
de-phasing as small as $\simeq 0.13\ \mathrm{rad}$. This corresponds to a
phase error of less than $0.15 \%$, which would be challenging to achieve
with standard second-order numerical-relativity codes. The relative
errors of the amplitude are somewhat larger, but also on a few percent level for
the $H$ and $V\!H$ resolutions, which is more than adequate giving that
the amplitude is not as critical as the phase for GW astronomy. Overall,
this shows that the Richardson extrapolated waveform is consistent with
the $V\!H$ data (which lies between the extrapolated waveform and the $H$
data).
\section{Conclusions}
We presented a set of four, high-resolution, high-order, simulations of
BNS inspiral in general relativity, the highest of which has not been
published before. Our analysis focused on the accuracy of the
gravitational wave signal extracted from the simulations and, in
particular, on its phase evolution, which represents a crucial quantity
for both detection and parameters extraction.
We showed that, with the use of higher-order methods, it is possible to
achieve clean convergence and small phase errors, without the need to
perform any alignment or rescaling of the GWs. The
Richardson-extrapolated waveforms appear robust as the resolution
increases, giving support to their accuracy. This opens the possibility
of obtaining high-quality waveforms with reliable error estimates that
could be used to verify and calibrate analytical and phenomenological
phasing models to be used in GW astronomy.
\acknowledgements
We thank W.\ Kastaun for providing the primitive recovery routine and I.\
Hawke, S.\ Bernuzzi, D.\ Alic, R.\ Haas and K.\ Takami for useful
discussions. Partial support comes from the Sherman Fairchild
Foundation, the DFG grant SFB/Transregio 7, by ``NewCompStar'', COST
Action MP1304, and by the Helmholtz International Center for FAIR. The
calculations were performed on SuperMUC at the LRZ, on Datura at the AEI,
and on LOEWE in Frankfurt.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 5,419
|
O Hino da Cidade do Salvador é o hino oficial do município brasileiro de Salvador. A autoria da letra e da melodia é de Oswaldo José Leal.
Pela lei municipal n.º 1585 de 13 de março de 1964, que oficializou o dia 29 de março como data da fundação da cidade, a Prefeitura também instituiu concurso público para a escolha do hino do município. Oswaldo José Leal foi o vencedor e teve sua obra tornada hino oficial por meio do decreto de lei n.º 2658/65, em 24 de abril de 1965.
O hino é objeto de algumas proposições de legisladores municipais a fim de tornar sua execução obrigatória em certos eventos e espaços, a exemplo do projeto de lei n.º 86 de 2010, do projeto de resolução nº 24/10 e do projeto de lei n.º 245 de 2015. Por enquanto, a execução é obrigatória somente em solenidades oficiais da Prefeitura, nos dias 29 de março e 2 de julho de modo particular, como determina a lei municipal n.º 5070 de 1995.
Ver também
Bandeira de Salvador
Brasão de Salvador
Hino da Bahia
Lista de hinos do Brasil
Ligações externas
Salvador
Cultura de Salvador
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{
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<?xml version="1.0"?>
<entry type="method" name="mousedown" return="jQuery">
<title>.mousedown()</title>
<desc>Bind an event handler to the "mousedown" JavaScript event, or trigger that event on an element.</desc>
<signature>
<added>1.0</added>
<argument name="handler" type="Function">
<desc>A function to execute each time the event is triggered.</desc>
<argument name="eventObject" type="Event" />
</argument>
</signature>
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<added>1.4.3</added>
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<desc>An object containing data that will be passed to the event handler.</desc>
</argument>
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<desc>A function to execute each time the event is triggered.</desc>
<argument name="eventObject" type="Event" />
</argument>
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<longdesc>
<p>This method is a shortcut for <code>.on( "mousedown", handler)</code> in the first variation, and <code>.trigger( "mousedown" )</code> in the second.</p>
<p>The <code>mousedown</code> event is sent to an element when the mouse pointer is over the element, and the mouse button is pressed. Any HTML element can receive this event.</p>
<p>For example, consider the HTML:</p>
<pre><code><div id="target">
Click here
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Trigger the handler
</div>
</code></pre>
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<img src="/resources/0042_05_01.png" alt=""/>
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<p>The event handler can be bound to any <code><div></code>:</p>
<pre><code>
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alert( "Handler for .mousedown() called." );
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</code></pre>
<p>Now if we click on this element, the alert is displayed:</p>
<p>
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</p>
<p>We can also trigger the event when a different element is clicked:</p>
<pre><code>
$( "#other" ).click(function() {
$( "#target" ).mousedown();
});
</code></pre>
<p>After this code executes, clicks on <samp>Trigger the handler</samp> will also alert the message.</p>
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</longdesc>
<note id="detach-shorthand" type="additional" data-event="mousedown"/>
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<desc>Show texts when mouseup and mousedown event triggering.</desc>
<code><![CDATA[
$( "p" )
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$( this ).append( "<span style='color:#f00;'>Mouse up.</span>" );
})
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});
]]></code>
<html><![CDATA[
<p>Press mouse and release here.</p>
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</example>
<category slug="events/mouse-events"/>
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| 5,552
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\subsection*{Quench dynamics data}
Figures \ref{figABdynamics}, \ref{figTriangular1ES} and \ref{figTriangular2ES} show dynamics of the AB ring and triangular ladder models, quenched to specific sites. The occupation data shown in the figures is normalized to the occupation of the relevant excitation subspace. For example, the date shown in Fig. \ref{figABdynamics}(a) is normalized by the occupation of the single excitation subspace. The spin excitation is in general not conserved due to decoherence and non-adiabatic effect \cite{manovitz2020quantum}. Here we show the evolution of the subspace occupation, corresponding to the data in the main text.
Figure \ref{figABSubspace} shows the subspace occupation in the AB ring model, corresponding to Fig. \ref{figABdynamics} of the main text. We note that the initial occupation decays, but remains larger than $0.5$ throughout the evolution.
\begin{figure}
\centering
\includegraphics[width=1\linewidth]{figures/sfig1_ABdynamics.pdf}
\caption{Subspace occupancy of AB ring experiments. With zero (blue), one (red), two (orange) and three (purple) excitations present. (a-d) corresponding to Fig. \ref{figABdynamics}(a-d) of the main text.}\label{figABSubspace}
\end{figure}
Figure \ref{figTriangularSubspace} shows the subspace occupation in the triangular ladder, corresponding to Fig. \ref{figTriangular1ES} and \ref{figTriangular2ES} of the main text.
\begin{figure}
\centering
\includegraphics[width=1\linewidth]{figures/sfig2_Triangular.pdf}
\caption{Subspace occupancy of triangular ladder ring experiments. With zero (blue), one (red), two (orange), three (purple) and four (green) excitations present. (a,b) corresponding to Fig. \ref{figTriangular1ES}(a,b) of the main text. (c,d) corresponding to Fig. \ref{figTriangular2ES}(a,b) of the main text.}\label{figTriangularSubspace}
\end{figure}
\subsection*{Adiabatic ground state preparation}
In the main text we state that we do not precisely implement the AB Hamiltonian, $H_{\text{AB}}\left(\Phi_\text{AB}\right)$, in Eq. \eqref{eqHAB}. Rather, we implement the Hamiltonian, $H_{\text{eff}}=H_{\text{AB}}\left(\Phi_{\text{AB}}\right)+\epsilon H_{\text{AB}}\left(0\right)$, such that the factor that accompanies a hop is modified, $e^{i\Phi_\text{AB}}\rightarrow e^{i\Phi_\text{AB}}+\epsilon$. This modification can be understood as non-adiabatic corrections to the method in \cite{manovitz2020quantum}. Using this method we generate the spin-hopping terms by utilizing a two-photon process. Specifically the ions are driven by pairs of laser tones, such that absorption of one tone and emission of the other exactly bridges the energy gap to adjacent ions. The phase associated with such a hop is given by the phase difference of these two tones.
However, a similar effect may occur by absorption and emission of photons of the same tone, such that the phase difference vanishes, yielding a vanishing flux. This effect is non-resonant and is characterized by the dimensionless number, $\beta=\xi/2\Omega_n$, where, similarly to power broadening effects, $\xi$ is the detuning from the transition and $\Omega_n$ is the Rabi frequency of the tone. In our implementation we used $\beta=7$, and 8 different tones, yielding the estimate, $\epsilon\approx8/7^2\approx0.16$.
In practice we obtain the value of $\epsilon$ and $\Omega$ with a maximum likelihood fit. Specifically we simulate an ideal adiabatic ramp of $H_{\text{AB}}$ in the 1ES (i.e a three dimensional Hamiltonian) and choose the values of $\epsilon$ and $\Omega$, such that they maximize the probability to measure our obtained data. We obtain $\epsilon=0.22$ and $\Omega=350 \text{Hz}$.
In the main text we state that tomography of the resulting prepared state is preformed by occupancy and correlation measurements, under assumption of rapid desphasing of other excitation subspaces. Indeed, assuming the prepared state, $\ket{\psi}=\sqrt{p_1}e^{i\phi_1}\ket{100}+\sqrt{p_2}e^{i\phi_2}\ket{010}+\sqrt{p_3}e^{i\phi_3}\ket{001}$, then the occupancy measurement is equivalent to evaluating $\langle\sigma_z\rangle$ for each of the three sites. Similarly, evaluating $\langle\sigma_j\sigma_{j+1}\rangle$ after the in-phase measurement procedure yields $\frac{2}{3}p_{j} p_{j+1}\cos\left(\theta_{j}-\theta_{j+1}\right)$, and after the out-of phase correlation procedure yields, $\langle\sigma_1\sigma_2\rangle=\frac{2}{3}p_1 p_2\sin\left(\theta_1-\theta_2\right)$ and $\langle\sigma_2\sigma_3\rangle=\frac{2}{3}p_2 p_3\sin\left(\theta_2-\theta_3\right)$. These combined are enough to reconstruct $\ket{\psi_f}$ up to a global phase.
To supplement the data we preform a numerical simulation of the full ion-chain Hamiltonian of the adiabatic ramp (i.e a $2^3 (n_\text{max}+1)$ dimensional Hamiltonian, with $n_\text{max}=9$ the maximum phonon occupation number of the chain's center-of-mass normal mode of motion). Figure \ref{figRampSim}(a) shows the evolution of the different 1ES states under the adiabatic ramp protocol with $\Phi=\pi/2$ and the remaining parameters similar to those of the experiment, such that $\beta=\xi/2\Omega_n=7$ \cite{manovitz2020quantum}. The ramp starts at $t=0$ and ends at $10\text{ msec}$ (dashed) after which the resulting state is coherent, seen from its high purity (purple) and occupies almost equally the three 1ES states (blue, red and yellow) at the stroboscopic times (markers), indicating an approximate ground state of the underlying AB Hamiltonian.
\begin{figure}
\centering
\includegraphics[width=1\linewidth]{figures/sfig3_RampSim.pdf}
\caption{Simulation of adiabatic ground state preparation of the AB ring. (a) Preparation with parameters similar to the experiment, with $\Phi=\pi/2$ and $\beta=7$. The adiabatic ramp starts at $t=0$ and ends at $t=10\text {msec}$ (dashed). The resulting state occupies the three 1ES states almost equally at stroboscopic times (markers), and is coherent, seen as by the high purity values. (b) Preparation deep ion the adiabatic regime with $\beta=40$ and $\Phi=0$, the system evolves smoothly at all times, and results in a high overlap with the AB Hamiltonian ground state.}\label{figRampSim}
\end{figure}
We also repeat this simulation deep in the adiabatic regime, with $\beta=40$ and $\Phi=0$, shown in Fig. \ref{figRampSim}(b). Here the procedure yields an even higher overlap with the underlying ground state. Furthermore, the large $\beta$ value results in much less oscillations making the result viable at all evolution times.
\subsection*{Triangular model analysis}
We provide further details on the $4$ site triangular ladder. Specifically we show that the three symmetries, $U_{1,4}$, $A_{2,3}$ and $C$, described in the main text, allow to easily recognize $7$ of the $16$ eigenstates. Furthermore this analysis immediately shows periodicity of the 1ES evolution and of a subspace of the 2ES. A summary of the eigenstate is given in Table \ref{tblEigenstates}.
\begin{table}
\caption{Eigenstate of the 4 site triangular ladder at $\Phi_S=\pi/2$, which are straightforward to obtain. Showing the state, energy and excitation subspace of each eignestate.}
\label{tblEigenstates}
\begin{center}
\begin{tabular}{C{3cm} C{2cm} C{2cm}}
\hline\hline
State & $\langle H\rangle $ & nES \\ [0.5ex]
\hline\hline
$\ket{\downarrow\downarrow}\ket{\downarrow\downarrow}$ & 0 & 0 \\
\hline
$\ket{\uparrow\uparrow}\ket{\uparrow\uparrow}$ & 0 & 4 \\
\hline
$\ket{S}\ket{\downarrow\downarrow}$ & 0 & 1 \\
\hline
$\ket{S}\ket{\uparrow\uparrow}$ & 0 & 3 \\
\hline
$\ket{S}\frac{\ket{\uparrow\downarrow}-i\ket{\downarrow\uparrow}}{\sqrt{2}}$ & $\Omega$ & 2 \\
\hline
$\ket{S}\frac{\ket{\uparrow\downarrow}+i\ket{\downarrow\uparrow}}{\sqrt{2}}$ & $-\Omega$ & 2 \\
\hline
$\frac{\ket{\uparrow\uparrow}\ket{\downarrow\downarrow}-\ket{\downarrow\downarrow}\ket{\uparrow\uparrow}}{\sqrt{2}}$ & 0 & 2 \\[1ex]
\hline\hline
\end{tabular}
\end{center}
\end{table}
We start by noticing that the unoccupied state, $\ket{\downarrow\downarrow}\ket{\downarrow\downarrow}$, and the fully occupied state, $\ket{\uparrow\uparrow}\ket{\uparrow\uparrow}$ are trivially $E=0$ eigenstates of the system as all Hamiltonian terms annihilate them. Here and in all other states below the two kets are to be understood as $\ket{\psi}\ket{\varphi}\equiv\ket{\psi}_{1,4}\ket{\varphi}_{2,3}$.
Next, due to $U_{1,4}$, when sites $1$ and $4$ form a spin singlet they are decoupled from the evolution. Therefore a set of eignestates can be found by diagonalizing the remaining terms connecting sites $2$ and $3$. In the 1ES this comes about as $\ket{S}\ket{\downarrow\downarrow}$, with $E=0$. The chiral symmetry now forces the three remaining eigenstates of the 1ES to have energies $0,\pm E_1$, with $E_1$ determined by exact diagonalization of the $3\times3$ Hamiltonian. Similarly in the 3ES we have $\ket{S}\ket{\uparrow\uparrow}$, with $E=0$.
In the 2ES a singlet on sites $1$ and $4$ require and additional excitation on sites $2$ and $3$. The interaction between these sites takes the form of a $\sigma_y$ term, thus two eigenstates in the 2ES are $\ket{S}\frac{\ket{\uparrow\downarrow}\pm i\ket{\downarrow\uparrow}}{\sqrt{2}}$ with energy $\mp\Omega$. The state $\frac{\ket{\uparrow\uparrow}\ket{\downarrow\downarrow}-\ket{\downarrow\downarrow}\ket{\uparrow\uparrow}}{\sqrt{2}}$ is an additional eigenstate in the 2ES, carrying 0 energy. Similarly to above, the remaining three states must carry energies $0,\pm E_2$, with $E_2$ determined by exact diagonalization.
\section*{Introduction}
\input{Sections/introduction}
\section*{Time reversal symmetry breaking in the Aharonov-Bohm ring}
\input{Sections/AB}
\section*{Interactions on triangular ladder}
\input{Sections/triangle}
\section*{Conclusion}
\input{Sections/conclusion}
We thank Rotem Arnon-Friedman, Anna Keselman, Meirav Pinkas, Boaz Raz and Hagai Edri for helpful discussions. This work was supported by the Israeli Science Foundation, the Israeli Ministry of Science Technology and Space, and the Minerva Stiftung.
\bibliographystyle{Science}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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Win - Hain (5-13). Loss - Willard, A. (6-4). Save - None.
WP - Hain. PB - Carlson, K. 2.
Wayne State 2nd - Peterson out at first p to 2b. Reitz singled to left field.
Smith, B. advanced to third. McKenzie, K. reached on a fielder's choice; Pullar, J.
advanced to third; Smith, B. out at home p to c. McKenzie, K. stole second.
Prettyman, K grounded out to 2b. 0 runs, 1 hit, 0 errors, 2 LOB.
third. Hiscock, T. out at third c to 3b, picked off. Docken, J. grounded out to p.
Wayne State 4th - Hain grounded out to p. Johnson grounded out to 3b.
Peterson grounded out to p. 0 runs, 0 hits, 0 errors, 0 LOB.
A. advanced to second. Pullar, J. grounded out to p; Klaers, A. advanced to third.
Klaers, A. out at home c to p, picked off. 0 runs, 0 hits, 0 errors, 0 LOB.
ball. Parks grounded out to 2b. 0 runs, 0 hits, 0 errors, 1 LOB.
out to ss. Prettyman, K grounded out to ss. 0 runs, 0 hits, 0 errors, 0 LOB.
Stewart advanced to second. Stewart stole third. Hain reached on an error by 2b.
Spale to p for Hain. Johnson flied out to rf, SAC, RBI; Stewart scored, unearned.
|
{
"redpajama_set_name": "RedPajamaC4"
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{"url":"https:\/\/rbseguide.com\/rbse-class-12-maths-model-paper-1-english-medium\/","text":"RBSE Class 12 Maths Model Paper 1 English Medium\n\nRBSE Class 12 Maths Model Paper 1 English Medium\u00a0are part of RBSE Class 12 Maths Board Model Papers. Here we have given RBSE Class 12 Maths Sample Paper 1\u00a0English Medium.\n\n Board RBSE Textbook SIERT, Rajasthan Class Class 12 Subject Maths Paper Set Model Paper 1 Category RBSE Model Papers\n\nRBSE Class 12 Maths Sample Paper 1 English Medium\n\nTime \u2013 3 \u00bc Hours\nMaximum Marks: 80\n\nGeneral instructions to the examines\n\n1. Candidate must write first his\/her Roll No. on the question paper compulsorily.\n2. All the questions are compulsory\n3. Write the answer to each question in the given answer book only.\n4. For questions having more than one part the answers to those parts are to be written together in continuity.\n5. If there is any error\/difference\/contradiction in Hindi & English versions of the question paper, the question of Hindi version should be treated valid.\n6. Section Q.No Marks for question A 1-10 1 B 11-15 2 C 16-25 3 D 26-30 6\n7. There are internal choices in Q. No. 16. 21. 24. 28 and 30. You have to attempt only one of the alternatives in these questions.\n8. Draw the graph of Q.No. 25 on the graph paper.\n\nSection \u2013 A\n\nQuestion 1.\nWrite composition table for addition S = {(0, 1, 2); +3}. [1]\n\nQuestion 2.\nIf $$\\cot ^{-1} x+\\tan ^{-1}\\left(\\frac{1}{3}\\right)=\\frac{\\pi}{2}$$ then find the value of x.\u00a0[1]\n\nQuestion 3.\n[1]\n\nQuestion 4.\nIf points (x, -2), (5, 2), (8, 8) are collinear, then find the value of x.\u00a0[1]\n\nQuestion 5.\nFind $$\\int \\log x d x$$\u00a0[1]\n\nQuestion 6.\nFind the unit vector along the sum of vectors $$a=2 \\hat{i}+2 \\hat{j}-5 \\hat{k}, b=2 \\hat{i}+\\hat{j}+3 \\hat{k}$$\u00a0[1]\n\nQuestion 7.\nFind the value of $$\\left[ \\begin{array}{lll}{2 \\hat{i}} & {\\hat{j}} & {\\hat{k}}\\end{array}\\right]+\\left[ \\begin{array}{lll}{\\hat{i}} & {\\hat{j}} & {\\hat{k}}\\end{array}\\right]+\\left[ \\begin{array}{lll}{\\hat{k}} & {\\hat{j}} & {2 \\hat{i}}\\end{array}\\right]$$\u00a0[1]\n\nQuestion 8.\n[1]\n\nQuestion 9.\nShow the region of feasible solution under the following constraints x + 2y \u2264 8, .x \u2265 0, y \u2265 0 in answer book.\u00a0[1]\n\nQuestion 10.\n[1]\n\nSection \u2013 B\n\nQuestion 11.\nIf function f: R \u2192 R, f (x) = 2x + 1 then show that $$\\left(f^{-1}\\right)^{-1}=f$$ [2]\n\nQuestion 12.\n[2]\n\nQuestion 13.\nExamine Continuity at x = 1 of function f(x) = |x \u2013 1|\u00a0[2]\n\nQuestion 14.\nFind\u00a0$$\\int \\frac{1}{1+\\sin x} d x$$\u00a0[2]\n\nQuestion 15.\nIf a vector makes angles \u03b1, \u03b2, and \u03b3 respectively with axes OX, OY, OZ, then prove that sin\u00b2\u03b1 + sin\u00b2\u03b2 + sin\u00b2\u03b3 = 2.\u00a0[2]\n\nSection \u2013 C\n\nQuestion 16.\n[3]\nOR\n\nQuestion 17.\n[3]\n\nQuestion 18.\n[3]\n\nQuestion 19.\nFind equation of normal to the curve 2x\u00b2 \u2013 y\u00b2 = 14 which is parallel to line x + 3y = 6.\u00a0[3]\n\nQuestion 20.\nFind two positive numbers x and y, sum of them is 60 and xy\u00b3 is maximum.\u00a0[3]\n\nQuestion 21.\nFind $$\\int \\sqrt{x^{2}+a^{2}} d x$$\u00a0\u00a0[3]\nOR\nFind $$\\int \\frac{1}{1-6 x-9 x^{2}} d x$$\n\nQuestion 22.\nFind area of region bounded by curve $$y=2 \\sqrt{1-x^{2}}$$ and above x-axis.\u00a0[3]\n\nQuestion 23.\nFind area of region bounded by curve [(x,y)\/x\u00b2 \u2264 y \u2264 x]\u00a0\u00a0[3]\n\nQuestion 24.\nIf $$\\overline{a}=3 \\hat{i}+\\hat{j}+2 \\hat{k} \\text { and } \\overline{b}=2 \\hat{i}-2 \\hat{j}+2 \\hat{k}$$ then find unit vector $$\\hat{n}$$ perpendicular both $$\\overline{a} \\text { and } \\overline{b}$$\u00a0[3]\nOR\n\nQuestion 25.\nBy graphical method solve the following linear programming problem for\u00a0[3]\nMaximum z = 2x + 3y\nConstraints 4x + 6y \u2264 60,\u00a0 2x + y \u2264 20 and x \u2265 0, y \u2265 0.\n\nSection-D\n\nQuestion 26.\n[6]\n\nQuestion 27.\nShow that\u00a0$$\\int_{0}^{\\frac{\\pi}{2}} \\log \\sin x d x=\\frac{\\pi}{2} \\log \\frac{1}{2}$$\u00a0[6]\n\nQuestion 28.\n\nOR\nFind the perticular solution of the differential equation\u00a0$$\\frac{d y}{d x}$$ + 2xy = xsin x\u00b2 If x = 0 and y = 1.\u00a0[6]\n\nQuestion 29.\nFind the angle between the two lines. These direction-cosines are given by the following relations. l- 5m + 3n = 0 and 7l\u00b2 + 5m\u00b2 \u2013 3n\u00b2 = 0\u00a0[6]\n\nQuestion 30.\nA man is known to speak truth 3 out of 4 times. He throws a die and reports that it is a six. Find the probability that it is actually a six.\u00a0[6]\nOR\nTwo coins are tossed at the same time. Find the variance of \u201cnumber of heads\u201d.\n\nWe hope the given RBSE Class 12 Maths Model Paper 1 English Medium will help you. If you have any query regarding RBSE Class 12 Maths Sample Paper 1 English Medium, drop a comment below and we will get back to you at the earliest.","date":"2022-06-29 13:05:23","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.4692213237285614, \"perplexity\": 1717.9824905656549}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-27\/segments\/1656103639050.36\/warc\/CC-MAIN-20220629115352-20220629145352-00700.warc.gz\"}"}
| null | null |
\section{Introduction}
In the standard Big Bang scenario, the oldest relic from the early
universe is the Cosmic Microwave Background Radiation (CMBR). This
radiation is remarkably close to isotropy. When it is decomposed into
multipoles,
\begin{equation}
{\Delta T\over T}(\theta, \phi)=
\sum_{l=1}^{\infty}\sum_{m=-l}^{l} a_{lm}Y_{lm}(\theta,\phi), \label{fluct}
\end{equation}
one finds that the dipole contribution is of the order
$(\Delta T/ T)_{l=1}\sim 10^{-3}$, while the contribution from higher
multipoles is only $(\Delta T/ T)_{l>1}\sim 10^{-5}$ (see e.g.\cite{kogutetal}).
Usually, the dipole term is interpreted as a Doppler effect, i.e. as the
consequence of our local motion with respect to the ``CMBR rest frame'', whereas
the other multipoles are accounted for by primordial cosmological
perturbations. However, one cannot reject {\it a priori} the possibility
that a significant part of the dipole originates as well from cosmological
perturbations. In fact, this alternative idea was stimulated by the
results of Lauer and Postman \cite {lap} in 1991, who found a dipole
in distant Abell clusters inconsistent with the CMBR dipole. Other
observations based on nearby galaxies, IRAS galaxies or distant supernovae
tend on the other hand to favour the orthodox interpretation, but are still
inconclusive.
Let us be more precise on the distinction between a
cosmological dipole and a local dipole. If our universe was perfectly homogeneous
and isotropic then there would be no cosmological dipole; a comoving
observer would not detect any dipole whereas a non comoving observer
would measure a dipole, in this case a purely local
dipole, due to his motion. In a perturbed model, it
is more delicate to distinguish between a local and a cosmological dipole.
The perturbation of any quantity can be decomposed into
contributions from different scales. The cosmological dipole is then
due to very large scale perturbations, typically scales of the order
of the Hubble radius, and the local dipole is due to the small scale
component (the dominant one) of the observer peculiar velocity.
The purpose of this paper is to explore the origin of the dipole from a theoretical
point of view and to ask whether it is possible, in the context of
Friedman-Lemaitre-Robertson-Walker (FLRW) cosmologies
with gaussian random fields of linear perturbations, to
obtain a {\it cosmological} dipole and higher multipoles compatible with the
observations. To discriminate between a local Doppler dipole and a cosmological dipole is more, in our view, than a mere academic exercise because the Doppler
assumption enters in the quadrupole analysis of the observations. The
reason is that our motion would not only induce a dipole but also a
quadrupole, and other multipoles.
It was shown by Paczynski and Piran \cite{pp} that a strong dipole can be
found within a particular Tolman-Bondi model with an inhomogeneous
radiative component. In fact, as was shown recently by Langlois and Piran
\cite{lp}, this result can be obtained, more generally,
in a perturbed FLRW {\it flat} cosmology, from {\it ultra large} scale
(larger than the current Hubble radius) {\it isocurvature} perturbations.
Here, we reinvestigate this possibility in the context of {\it open}
cosmologies.
The influence of stochastic cosmological perturbations with wavelengths
larger than the current Hubble radius on the CMBR temperature
fluctuations was examined initially by Grishchuk and Zel'dovich \cite{gz}.
Their analysis was restricted to perturbations in a flat background
for which the Fourier expansion is available. When one considers an open
universe one needs the complicated formalism of mode functions in an
hyperbolic space. Recently, Lyth and Woszczyna \cite{lw} stressed
the importance of supercurvature modes, which had been ignored previously,
when dealing with random fields. With this new element, the influence of
ultra large scale perturbations on the CMBR for open universes was then
investigated in references \cite{lw} and \cite{gllw}.
The present work extends these papers in two respects. First, we focus
our attention on the dipole, which was not treated previously, and, second,
we allow not only for adiabatic perturbations but also for isocurvature
perturbations.
The paper is organized as follows.
In section 2, we recall the main steps of the derivation of the Sachs-Wolfe effect, responsible of the large angle
anisotropies in the CMBR. Our approach follows the
work of Panek \cite{panek}, although our notation is different.
The final result is valid for flat or curved FLRW background geometries and
is expressed in terms of gauge-invariant variables.
Section 3 gives the CMBR multipoles in a flat geometry. Although these
results were already given in \cite{lp}, the actual presentation is
original and provides a bridge to the formalism needed for an open cosmology.
In section 4, the case of an open cosmology is considered. Our notation
follows essentially Refs \cite{lw} and \cite{gllw}.
Finally, section 5 contains the conclusions of this work.
\section{The Sachs-Wolfe effect}
Our background geometry is described by a FLRW metric,
\begin{equation}
ds^2=a^2(\eta)\left(-d\eta^2+\gamma_{ij}dx^idx^j \right),
\end{equation}
where $dl^2=\gamma_{ij}dx^i dx^j$ is the metric of a flat or curved
maximally symmetric space (a flat space corresponds to $K=0$ and
an open space to $K=-1$, where $K$ is the normalized space curvature).
The CMBR is composed of cosmological photons, which became free
after the recombination of protons and electrons and the subsequent
decoupling of matter and radiation at a redshift $z_{ls}$. For numerical
applications we shall take $z_{ls}=1000$ ($z_{ls}$ depends in fact
on the spatial curvature but this dependence is very weak \cite{kt} and can be
safely ignored here).
All the photons reaching us now were emitted on a physical surface at the
epoch of decoupling, called the last scattering surface, and
defined physically by $n_e=const$, where
$n_e$ is the density of free electrons. The universe will be supposed to
be matter dominated from the last scattering time to now, which is an excellent
approximation except for low values of $\Omega_0$.
Now, we consider a perturbed FLRW universe, endowed with the metric (in the
longitudinal gauge)
\begin{equation}
ds^2=a^2(\eta)\left[-(1+2\Phi)d\eta^2+(1-2\Psi)\gamma_{ij} dx^i dx^j\right].
\label{metric}\end{equation}
Only perturbations of the scalar type will be considered here (see e.g.
\cite{bardeen}) and, for simplicity, the anisotropic stress of the matter
will be supposed to vanish so that the Einstein equations yield
\begin{equation}
\Phi=\Psi.
\end{equation}
Because of the presence of geometric perturbations,
photons will be redshifted with slight differences depending on their
position of emission on the last scattering surface. The resulting
fluctuations in the temperature measured by an observer were first
calculated by Sachs and Wolfe \cite{sw}. The origin of this effect
is purely geometric and is dominant only for large angular scales, which
correspond essentially to scales larger than the Hubble radius at the
epoch of last scattering. On smaller scales, causal processes must be
taken into account. In this work, we are interested only by the first
multipoles of the CMBR, which are dominated by the last scattering
superhorizon modes, so that only the Sachs-Wolfe effect is relevant.
In the following, the Sachs-Wolfe effect is rederived shortly following
the more modern calculation of Panek \cite{panek}. However our notations are
different and the final result is expressed in another form.
Denoting $k^\mu$ the vector tangent to a null geodesic,
the evolution of the photon is governed by the geodesic equation:
\begin{equation}
k^\nu\partial_\nu k^\mu+\Gamma^\mu_{\sigma\tau}k^\sigma k^\tau=0,
\end{equation}
where the $\Gamma^\mu_{\sigma\tau}$ are the Christoffel symbols.
We write the perturbed tangent vector $k^\mu=\{\nu(1-M), -\nu e^i+ P^i\}$,
where the terms in $\nu$ represent the unperturbed solution; $e^i$
is a unit spatial vector, i.e. such that $\gamma_{ij}e^i e^j=1$; $M$ is the
perturbation of the frequency, $P^i$ the perturbation of the spatial
direction. Here, we are interested only in the frequency perturbation and
thus keep only the time component of the geodesic equation. After
using the identity $g_{\mu\nu}k^\mu k^\nu=0$
at first order, which reads
\begin{equation}
\gamma_{ij}e^iP^j=\left(M-\Phi-\Psi\right)\nu,
\end{equation}
in order to eliminate the $P^i$ in the time component of the geodesic equation
at first order, one finds eventually
\begin{equation}
{d M\over d\lambda}\equiv (\partial_\eta-e^i
\partial_i)M=\Phi'- 2 e^i\partial_i\Phi - \Psi'.
\label{geodesic}
\end{equation}
The prime stands for a derivative with respect to the
conformal time $\eta$.
Note that the parameter $\lambda$ is related to the affine parameter $\tau$
of the unperturbed geodesic by the relation
${d\tau/d\lambda}=1/\nu$. The prime stands for derivative with respect to the
conformal time $\eta$. Equation (\ref{geodesic}) corresponds to Eq. (29)
of \cite{panek}.
Typically one can model the content of the universe from the epoch of the
last scattering to now by two fluids, a pressureless baryon
component and a radiation component, described locally by a black-body
spectrum.
Let us define the temperature of the photons as seen by observers moving
with the baryon component, so that the temperature ratio between emission
and reception is given by
\begin{equation}
{T_R\over T_E}={(k^\mu u^b_\mu)_R\over (k^\mu u^b_\mu)_E}.
\end{equation}
In the perturbed geometry (\ref{metric}), one thus obtains
\begin{equation}
{T_R\over T_E}={a_E\over a_R}\left\{1+\left[\Phi-M+e^i\partial_i V\right]^R_E
\right\},
\end{equation}
where the velocity potential $V$ is defined by $\delta u_i^b\equiv a \partial_iV$.
As a consequence, the temperature fluctuations on the sky measured
by an observer today (denoted by the subscript $0$) are given by
\begin{equation}
\left({\delta T\over T}\right)_0(\theta,\phi)=\left({\delta T\over T}\right)_e +
\left({\delta a\over a}\right)_e+\left[\Phi-M+e^i\partial_i V\right]^0_e.
\label{temp}
\end{equation}
The left hand side term is a function of the celestial coordinates
corresponding to the direction of observation for the observer.
The right hand side is expressed
as a function of the emission point, of coordinates $(\eta_e,x^i_e)$,
defined as the intersection of the last scattering surface with
the null geodesic going through the observer with the given direction ,
as well as
the observer position $(\eta_0,x^i_0)$. In general the
physical last scattering surface is distinct from the constant time
hypersurface $\eta=\eta_{ls}$, but one can replace the first two terms on the
right hand side of (\ref{temp})
by $(\delta T/T)(\eta_{ls},x^i_{ls}\simeq x^i_e)$ (i.e. as a function of the intersection
point between the light geodesic and the hypersurface $\eta=\eta_{ls}$).
This follows from the local conservation
law $aT=const$ for free radiation. Then, expressing the term
$\left[\Phi-M\right]_e^0$ in (\ref{temp})
as an integral over the null geodesic and using equation (\ref{geodesic}),
one finds
\begin{equation}
\left({\delta T\over T}\right)_R={1\over 4}\delta_{\gamma| ls}+(h V)_{ls}
+\left[-\Phi+e^i\partial_i V\right]_{ls}^0
+\int_{ls}^0 d\lambda \left(\Phi'+\Psi'\right),
\label{sw}
\end{equation}
where $h\equiv a'/a$. Using the Stefan law $\rho_\gamma\propto T^4$, we have
replaced the temperature fluctuations by fluctuations of the radiation
energy density and also introduced the comoving energy density perturbation
$\delta$, which can be derived from the
energy density perturbation in the longitudinal gauge by the expression
$\delta= (\delta\rho/\rho)_L+(\rho'/\rho)V$. To obtain (\ref{sw}), we
have used implicitly
the equality of the velocities of radiation and baryonic matter at the time
of last scattering, which results from the preexisting tight coupling between
the two fluids.
The term $\delta_\gamma/4$ in (\ref{sw}) represents the intrinsic temperature fluctuations
on the last scattering surface. The rest of the expression
is the Sachs-Wolfe effect. In (\ref{sw}) all the quantities at the
last scattering epoch are now evaluated on the hypersurface $\eta_{ls}$
which does not necessarily coincides with the physical last scattering
surface (but all the corrections would be second order perturbations).
It is instructive to decompose the Sachs-Wolfe expression into
several components and to rewrite (\ref{sw}), after dropping
the term $-\Phi_0$ which contributes only to the monopole,
in the form
\begin{equation}
\left({\delta T\over T}\right)_R=\left({\delta T\over T}\right)_{int}
+\left({\delta T\over T}\right)_{pSW}+\left({\delta T\over T}\right)_{Dop}
+\left({\delta T\over T}\right)_{ISW},
\end{equation}
with
\begin{equation}
\left({\delta T\over T}\right)_{int}={1\over 4}\delta_{\gamma| ls}, \quad
\left({\delta T\over T}\right)_{pSW}=(\Phi+hV)_{ls}, \quad
\left({\delta T\over T}\right)_{Dop}=e^i\partial_i(V_0-V_{ls}),
\label{decompos1}
\end{equation}
and
\begin{equation}
\left({\delta T\over T}\right)_{ISW}=\int_{ls}^0 d\lambda \left(\Phi'+\Psi'\right).
\end{equation}
The term $(\Phi+hV)_{ls}$ will be
called the proper (or ordinary) Sachs-Wolfe effect (pSW), because only this
term was computed in the pioneering paper of Sachs and Wolfe \cite{sw}.
The term $e^i\partial_i(V_0-V_{ls})$ is the difference between the
observer velocity and the emission velocity along the line of sight and
corresponds to a Doppler effect (Dop). Note that the local dipole is due
only to a term of the form $e^i v_i^0$, where $v_i^0$ is the contribution
of small scales to our peculiar velocity; this contribution, which comes
from the nonlinear evolution of the perturbations, will be
ignored in the rest of the paper where we consider only the effect
of very large scales. Finally the integral term is
usually called the integrated Sachs-Wolfe effect (ISW).
The linearized Einstein equations and fluid equations (see e.g. \cite{mfb} or
\cite{ks})
can then be used to express all the
terms in (\ref{sw}) in terms of only one quantity, the most convenient
being the gravitational potential $\Phi$. For example,
the velocity potential $V$ can be related to $\Phi$ via the Euler equation and the conservation
equation:
\begin{equation}
V=-{2\over 3(h^2+K)}(\Phi'+h\Phi).\label{v}
\end{equation}
Via the Einstein equations, one gets a relativistic generalization of Poisson
equation relating the gravitational potential to the total comoving
energy density $\delta_T$, which reads:
\begin{equation}
(\triangle+3K)\Phi={3\over 2}(h^2+K)\delta_T. \label{poisson}
\end{equation}
At this stage one must distinguish two kinds of perturbations:
adiabatic and isocurvature perturbations. Any general perturbation
can be seen as a linear combination of these two kinds.
An isocurvature perturbation corresponds to a primordial (in the
radiation era) perturbation of the matter which does not affect the geometry: it
implies a perturbation in the relative composition of the cosmological fluid
with no perturbation in the {\it total} energy density. On the contrary
an adiabatic perturbation is a primordial perturbation in the total
energy density without modification of the relative quantities of the fluids.
For an adiabatic perturbation, the relation between the
matter and radiation density perturbations is by definition $\delta_\gamma=(4/3)\delta_m$,
which implies, during matter domination,
\begin{equation}
\delta_\gamma\simeq {4\over 3}\delta_T.\label{adi}
\end{equation}
As will be seen later, the consequence of this
relation together with the Poisson equation (\ref{poisson})
is that, both for the flat and open cases, the intrinsic
fluctuations resulting from adiabatic initial perturbations
{\it on scales larger than the Hubble radius} are always negligible.
For an isocurvature perturbation, during matter domination,
\begin{equation}
\delta_\gamma\simeq -{4\over 3}S,\label{iso}
\end{equation}
where $S=\delta_m-(3/4)\delta_\gamma$ is the entropy perturbation. Both for open and
flat universes, the appropriate combination of the equations of motion
(i.e. conservation and Euler equations) of the two fluids, radiation and
baryons, tells us that $S$ is constant on scales bigger than the Hubble radius,
which is the case for perturbations we are interested in.
Isocurvature perturbations also produce a gravitational potential. Using
the conservation and Euler equations for the two fluids, one obtains
that the gravitational potential is given, at the beginning of the matter era,
by
\begin{equation}
\Phi=-{1\over 5}S,\label{isophi}
\end{equation}
for the modes larger than the Hubble radius.
To summarize, the multipole coefficients of (\ref{fluct})
can be decomposed into the sum of contributions
\begin{equation}
a_{lm}=a_{lm}^{int}+a_{lm}^{SW}=a_{lm}^{int}+a_{lm}^{pSW}+a_{lm}^{Dop}
+a_{lm}^{ISW}.
\end{equation}
The rest of the paper will consist in evaluating the various terms of this
expression for open/flat universes with adiabatic/isocurvature perturbations.
\section{Flat universe}
In this section, we consider the flat FLRW models.
To study the CMBR fluctuations, it is more convenient to work in
spherical coordinates with the observer at the center, naturally adapted
to observations on the celestial sphere, rather than
the more usual cartesian coordinates:
\begin{equation}
dl^2=dr^2+r^2\left(d\theta^2+\sin^2\theta d\phi^2\right).
\end{equation}
Any perturbation $f(r,\theta,\phi)$ can then be decomposed into
\begin{equation}
f(r,\theta,\phi,\eta)=\int dk \sum_{lm} f_{klm}(\eta)Q_{klm},
\end{equation}
where the mode functions $Q_{klm}$ are the eigenfunctions of the spatial
Laplacian with the eigenvalues $-k^2$ and are defined by
\begin{equation}
Q_{klm}=\sqrt{2\over\pi} kj_l(kr)Y_{lm}(\theta,\phi),
\end{equation}
where the $j_l$ are the spherical Bessel functions and $Y_{lm}$
are the spherical harmonics.
The definition of the $Q_{klm}$ is such that they are normalized i.e.
\begin{equation}
\int r^2dr\, \sin\theta \, d\theta \, d\phi\, Q_{klm}^* Q_{k'l'm'}=
\delta(k-k')\delta_{ll'}\delta_{mm'}.\label{orthonormal}
\end{equation}
This decomposition is the equivalent of the Fourier decomposition
but in a system of spherical coordinates. This presentation is a good
introduction to the open case where the spherical coordinates are the most
natural spatial cooordinates.
All the quantities introduced in the previous section can now be
decomposed on the mode functions $Q_{klm}$. As announced before, we shall
focus essentially on the gravitational potential $\Phi$. The linearized
Einstein equations yield a second order differential equation for the
evolution of each mode $\Phi_{klm}$, which shows that the modes larger than the
Hubble radius, i.e. $k<h$, are constant in time in a {\it flat universe}
during the matter era.
This result has several consequences. The first is that the ISW contribution
vanishes. The second is that the relation (\ref{v}) simplifies to
\begin{equation}
hV=-{2\over 3}\Phi, \qquad (K=0).\label{vflat}
\end{equation}
Now, using the orthonormality property (\ref{orthonormal}) with equations (\ref{sw})
and (\ref{vflat}), the Sachs-Wolfe harmonic coefficients can be expressed
in the form
\begin{equation}
a_{lm}^{SW}=\sqrt{2\over\pi}\int_0^\infty dk \, k {\Phi_{klm}\over 3}
{\cal W}_l^{SW}(k),
\end{equation}
with the window function ${\cal W}_l^{SW}(k)={\cal W}_l^{pSW}(k)
+{\cal W}_l^{Dop}(k)$, the proper SW and Doppler contributions being
respectively given by
\begin{equation}
{\cal W}_l^{pSW}(k)=j_l(kr_{ls}), \quad {\cal W}_l^{Dop}(k)=2h_{ls}^{-1}{d\over dr}j_l(kr_{ls})-
2h_0^{-1}{d\over dr}j_l(kr_0).
\end{equation}
$r_{ls}$ is the comoving distance from the hypersurface $\eta=\eta_{ls}$. It is related to the
redshift $z_{ls}$ ($\equiv (a_0/a_{ls}) -1$) by the relation
\begin{equation}
r_{ls}=2h_0^{-1}\left(1-(1+z_{ls})^{-1/2}\right)=2\left(h_0^{-1}-
h_{ls}^{-1}\right).
\end{equation}
Using the relations
\begin{equation}
{d\over dr}j_l(kr)=kj_{l-1}(kr)-{l+1\over r}j_l(kr),\quad
(2l+1)j_l(x)=x(j_{l+1}+j_{l-1}(x)),
\end{equation}
one can rewrite the dipole term
\begin{equation}
{\cal W}_1^{SW}(k)={2\over 3}kh_0^{-1}\left(j_0(kr_{ls})-1\right)+
{kr_{ls}\over 3}(1-4h_{ls}^{-1}r_{ls}^{-1})j_2(kr_{ls}).
\end{equation}
and the terms $l>1$
\begin{equation}
{\cal W}_l^{SW}(k)=\left(1+{2l\over h_{ls}r_{ls}}\right)j_l(kr_{ls})
-{2k\over h_{ls}}j_{l+1}(kr_{ls}).
\end{equation}
Since $j_0(x)=\sin x/x$ and $j_l(x)\sim x^l$ for $x<<1$, the
dipole term ${\cal W}_1^{SW}(k)$ behaves like $(kr_{ls})^3$
for small $k$. On the other hand, for $l>1$, ${\cal W}_l^{SW}(k)$ goes
like $(kr_{ls})^l$ for long wavelengths. This means that the Sachs-Wolfe dipole on
ultra large scales is suppressed with respect to the quadrupole.
Let us now consider the intrinsic contribution. In the case of adiabatic
perturbations, equation (\ref{adi}) and the Poisson equation (\ref{poisson})
(with $K=0$) shows that the intrinsic contribution is always negligible with respect to the Sachs-Wolfe contribution
for modes larger than the Hubble radius at the time of last scattering.
For adiabatic perturbations, the temperature fluctuations reduce to the
Sachs-Wolfe fluctuations. The (total) cosmological dipole
is thus suppressed with respect to the quadrupole (this remarkable property was
stressed in \cite{bl}). If this case describes the
reality we live in, the conclusion is that the observed dipole is
necessarily of local origin.
For isocurvature perturbations, the intrinsic contribution cannot be neglected
and the total window function ${\cal W}_l(k)$ is the sum of
the SW window function and of the intrinsic window function
which, using (\ref{decompos1}) and (\ref{iso}), as well as (\ref{isophi}),
is found to be
\begin{equation}
{\cal W}_l^{iso}(k)=5j_l(kr_{ls}).
\end{equation}
Hence ${\cal W}_1$ behaves like $kr_{ls}$ for small $k$ and can be made as large
as one wishes with respect to the quadrupole window function
by considering large enough scales.
The above analysis has been dealing with individual modes. To get the total
multipoles, one must sum on all the modes.
Assuming that $\Phi$ is a gaussian random field described by the power spectrum
\begin{equation}
\langle\Phi_{klm}\Phi_{k'l'm'}\rangle
=2\pi^2k^{-3}{\cal P}_\Phi(k) \delta(k-k')\delta_{ll'}
\delta_{mm'} \label{ps}
\end{equation}
the expectation value of the various multipoles reads
\begin{equation}
\langle |a_{lm}|^2\rangle={4\pi\over 9}\int {dk\over k}{\cal P}_\Phi(k)
{\cal W}_l^2(k).
\end{equation}
Once the power spectrum is given, the difference between the
multipoles comes only from the window functions. The most common power
spectrum is the flat spectrum corresponding to a constant
${\cal P}_\Phi(k)$.
\begin{figure
\centering
\epsfig{figure=fig1.ps, width=9cm}
\caption{squared amplitude of the window functions in the adiabatic case
for the dipole (solid curve) and the quadrupole (dashed curve). On ultra
large scales the dipole window function is suppressed with respect to
the quadrupole one.}
\end{figure}
On Fig. 1, we have plotted the square of the dipole and quadrupole
window functions in the adiabatic case. Since the dipole window function is
always subdominant with respect to the quadrupole one or at best of the same
order of magnitude for $k\sim h_0$, the conclusion is that the observed
dipole can only be explained by a local effect in that case.
\begin{figure
\centering
\epsfig{figure=fig2.ps, width=9cm}
\caption{squared amplitude of the window functions in the isocurvature case
for the dipole (solid curve) and the quadrupole (dashed curve). On ultra
large scales the quadrupole window function is suppressed with respect to
the dipole one.}
\end{figure}
On the contrary, for isocurvature perturbations, Fig.2 shows that the
dipole window function dominates on ultra large scales and the observed dipole
can be of cosmological origin. However the corresponding isocurvature
spectrum cannot be constant on all scales because the main contribution to
the dipole and quadrupole would come from scales $k\sim h_0$ and the
resulting dipole and quadrupole would be comparable.
To obtain cosmological dipole and quadrupole compatible with the observations,
two conditions must therefore be satisfied: the existence of ultra large
scale isocurvature perturbations; and their spectrum must be suppressed
for scales $\lambda < 100 H_0^{-1}$.
Although this cut-off condition can appear on first thought difficult to fulfill in a sensible
model, it turns out that two relatively simple ideas can produce such
an effect. One idea is the existence of pre-inflationnary isocurvature
perturbations, which would have been pushed away far beyond the horizon during
the inflation era \cite{turner}; the second idea is to consider the
simplest two scalar field inflation model, which naturally produces
an isocurvature spectrum with an abrupt cut-off \cite{l96}.
\section{Open universe}
We now write the background spatial metric in the form
\begin{equation}
dl^2=dr^2+\sinh^2r(d\theta^2+\sin^2\theta d\phi^2).
\end{equation}
With this choice of coordinates the Friedmann equation reads
\begin{equation}
H^2={8\pi G\over 3}\rho+{1\over a^2},
\end{equation}
where $H=ah$ is the Hubble parameter. As is clear from the above formula,
$a$ represents the curvature scale: on scales smaller than $a$, the curvature
of space is not ``felt'' whereas scales larger than $a$ are
affected by the curvature.
It is usual to parametrize the space curvature by the ratio of the
energy density with respect to the critical energy density today,
$\Omega_0=3H_0^2/8\pi G$. $\Omega_0=1$ for a flat space and $\Omega_0 <1$
for an open space. It then follows, in the matter era, that the
comoving Hubble parameter can be expressed in terms of the redshift $z$
and $\Omega_0$:
\begin{equation}
h=\sqrt{1+\Omega_0 z\over 1-\Omega_0}.\label{h}
\end{equation}
In particular $h_0=1/\sqrt{1-\Omega_0}$ represents the ratio between the
current curvature scale $a_0$ and the Hubble scale $H_0^{-1}$. For an
open universe the curvature scale is thus always larger than the Hubble
scale and all the more larger as $\Omega_0$ is closer to one.
In an open universe, the Fourier treatment does not apply. One must then
use the formalism of mode functions for an hyperbolic space.
Following \cite{lw}, any gaussian random field
can be expanded in the form
\begin{equation}
f(r,\theta,\phi,\eta)=\int_0^\infty dk \sum_{l=0}^\infty\sum_{m=-l}^{l}
f_{klm}(\eta) Q_{klm},\label{basis}
\end{equation}
where the $Q_{klm}$ are eigenfunctions of the spatial
Laplacian with eigenvalues $-k^2$. They can be chosen of the form
\begin{equation}
Q_{klm}=\Pi_{kl}(r) Y_{lm}(\theta,\phi),
\end{equation}
where $Y_{lm}(\theta,\phi)$ are the usual spherical harmonics on the
two-sphere. The radial mode functions $\Pi_{kl}$ can be classified into
two categories depending on the sign of $q^2\equiv k^2-1$.
The radial functions for the subcurvature modes, corresponding to the
values $q^2>0$, are given by
\begin{equation}
\Pi_{kl}\equiv N_{kl}\tilde\Pi_{kl},
\end{equation}
with
\begin{equation}
N_{kl}=\sqrt{2\over\pi} q^2\left[\prod_{n=0}^l (n^2+q^2)\right]^{-1/2},
\quad \tilde\Pi_{kl}=q^{-2}(\sinh r)^l\left[{-1\over\sinh r}{d\over dr}\right]
^{l+1}\cos (qr).
\end{equation}
The subcurvature modes provide a complete orthonormal basis for
square integrable functions and for this reason were the only modes
taken into account by cosmologists. However, as stressed recently by Lyth
and Woszczyna \cite{lw}, they are not enough to describe properly gaussian
random fields and one must also include the supercurvature modes, corresponding to the
values $-1<q^2<0$. Their radial functions are obtained from the subcurvature
modes by analytic continuation:
\begin{equation}
\Pi_{kl}\equiv N_{kl}\tilde\Pi_{kl},
\end{equation}
with
\begin{equation}
N_{k0}=\sqrt{2\over\pi} |q|, \qquad
N_{kl}=\sqrt{2\over\pi} |q|\left[\prod_{n=1}^l (n^2+q^2)\right]^{-1/2}
\quad (l>0)
\end{equation}
and
\begin{equation}
\tilde\Pi_{kl}=|q|^{-2}(\sinh r)^l\left[{-1\over\sinh r}
{d\over dr}\right]^{l+1}\cosh (|q|r).
\end{equation}
The normalization of the mode functions is such that
\begin{equation}
\int_0^\infty \Pi_{kl}(r)\Pi_{k'l'}(r)\sinh^2r dr=\delta(q-q')\delta_{ll'}.
\end{equation}
Moreover for a homogeneous gaussian random field, one defines the power
spectrum by
\begin{equation}
\langle f_{klm}f^*_{k'l'm'}\rangle={2\pi^2\over k|q|^2}{\cal P}_f(k)\delta(k-k')
\delta_{ll'}\delta_{mm'}.
\end{equation}
Note that in the flat space limit, corresponding to the limits
$k\rightarrow \infty$ and $r\rightarrow 0$ with $kr$ fixed,
\begin{equation}
\Pi_{kl}(r)\rightarrow \sqrt{2\over\pi} kj_l(kr),
\end{equation}
and one recovers equation (\ref{ps}) for the power spectrum.
In contrast with the flat case, the gravitational potential modes
$\Phi_{klm}$ on scales larger than the Hubble radius evolve with time. Their
evolution is given by the linearized Einstein equations (see e.g. \cite{mfb}):
\begin{equation}
\Phi_k(\eta)=F(\eta){\tilde \Phi}_k,
\end{equation}
with
\begin{equation}
F(\eta)=5{\sinh^2\eta-3\eta \sinh\eta+4\cosh\eta-4\over (\cosh\eta-1)^3}.
\label{f}
\end{equation}
(with the normalization $a=\cosh\eta -1$ for
the scale factor). F is normalized so that $F\rightarrow 1$ when $\eta\rightarrow 0$.
${\tilde \Phi}_k$ is constant in time and is given by the initial spectrum
at the beginning of the matter era. At that initial stage, the curvature can
be ignored. In fact, even for small values
of $\Omega_0$, $F(\eta)$ remains almost constant during most of the matter
era and drops suddenly in the recent past. Similarly, the velocity potential
obeys to a law different from the flat case and given by
\begin{equation}
hV(\eta)=-{2\over 3}G(\eta)\tilde\Phi,
\end{equation}
with
\begin{equation}
\quad G(\eta)= {{15\,
\left( 2\,\eta + \eta\,\cosh\eta - 3\,\sinh\eta \right) \,\sinh\eta}
\over {16 \sinh^6(\eta/2)}}.
\end{equation}
Like $F$, $G$ has been normalized such that $G\rightarrow 1$ when $\eta\rightarrow
0$ and $G$ is significantly different from one only for small redshifts.
The decomposition of the Sachs-Wolfe terms in (\ref{sw}) on the basis
$Q_{klm}$, like in (\ref{basis}), yields for the harmonic coefficients
the expression
\begin{equation}
a_{lm}=\sqrt{2\over \pi}\int_0^\infty dk\, |q| {1\over 3}\tilde\Phi_{klm}
\left[{\cal W}_{kl}^{pSW}+{\cal W}_{kl}^{Dop}+{\cal W}_{kl}^{ISW}\right],
\end{equation}
with
\begin{equation}
{\cal W}_{kl}^{pSW}=(3F_{ls}-2G_{ls})\hat\Pi_{kl}(r_{ls})/|q|,\label{win1}
\end{equation}
\begin{equation}
{\cal W}_{kl}^{Dop}=2h_{ls}^{-1}G_{ls}\partial_r\hat\Pi_{kl}(r_{\rm ls})/|q|
-{2\over 3}h_0^{-1}G_0 \, k\delta_{l,1}, \label{win2}
\end{equation}
and
\begin{equation}
{\cal W}_{kl}^{ISW}= \left[-6F_{ls}\hat\Pi_{kl}(r_{\rm ls})
+6\int_0^{r_{\rm ls}} dr\, F(\eta_{\rm ls}+r_{\rm ls}-r)\partial_r\hat\Pi_{kl}(r)\right]/|q|.
\label{win3}
\end{equation}
Except for the flat case limit, we shall take $F_{ls}=G_{ls}=1$
in the above expressions, which is an excellent approximation because
$z_{ls}>>1$. A hat means a division
by $\sqrt{2/\pi}$.
$h_{\rm ls}$ is the comoving Hubble parameter at the time of last scattering.
$r_{\rm ls}$ is the radius coordinate corresponding to the emission of the photons,
i.e. the coordinate of the last scattering surface. This radius can be
related to the redshift $z_{\rm ls}$ via the relation
\begin{equation}
\sinh r(z)={2\over
h_0}(1+z)^{-1}\Omega_0^{-2}\left[\Omega_0z+(\Omega_0-2)
\left((1+\Omega_0 z)^{1/2}-1\right)\right].\label{ropen}
\end{equation}
We must also consider the intrinsic contribution $a_{lm}^{int}$ to the
harmonic coefficients. For adiabatic fluctuations, like in the flat case,
the intrinsic contribution is negligible as can be seen from (\ref{poisson}):
the reason is not this time
the smallness of $k^2$ but the fact that $h^2>>1$ (see (\ref{h})).
For isocurvature perturbations, the intrinsic window function is given by
\begin{equation}
{\cal W}_{kl}^{iso}=5F_{ls} \hat\Pi_{kl}(r_{\rm ls})/|q|,
\end{equation}
Finally, once a power spectrum is given, the expectation values for the
harmonic coefficients can then be written in the form
\begin{equation}
\langle |a_{lm}|^2\rangle={4\pi\over 9}\int {dk\over k}{\cal P}_\Phi(k)
{\cal W}_l^2(k),
\end{equation}
where ${\cal W}_l(k)$ is the sum of all the relevant window functions.
\subsection{Ultra large scale limit}
In the flat case, it was possible to obtain a dipole much larger than
the quadrupole with isocurvature perturbations on scales much larger than
the Hubble radius. We wish now to consider such scales in an open universe.
In the ultra large scale limit, corresponding to $k\rightarrow 0$, the
general expressions can be simplified because $\Pi_{kl}(r)\sim k f_l(r)$.
Following \cite{gllw}, we define
\begin{equation}
N_l\equiv \lim_{k\rightarrow 0} kN_{kl}=\sqrt{2\over\pi}\prod_{n=2}^l
(n^2-1)^{-1/2}
\end{equation}
for $l>1$ and $N_1=\sqrt{2/\pi}$. Similarly, we define
\begin{equation}
\tilde\Pi_l\equiv\lim_{k\rightarrow 0}\tilde\Pi_{kl}/k^2.
\end{equation}
The expressions we are mainly interested in are
\begin{equation}
\tilde\Pi_1(r)={1\over 2}\left(\coth r- {r\over\sinh^2 r}\right)
\end{equation}
and
\begin{equation}
\tilde\Pi_2(r)= {1\over 2}\left[1+3{1-r\coth r\over\sinh^2r}\right].
\end{equation}
The window functions (\ref{win1}-\ref{win3}) can then be expressed in the limit $k\rightarrow 0$
in the form
\begin{equation}
{\cal W}_{kl}\sim k {\cal L}_l (\Omega_0, z_{\rm ls}),
\end{equation}
with
\begin{equation}
{\cal L}_l^{pSW}=\hat N_l\tilde\Pi_l(r_{\rm ls}), \quad
{\cal L}_l^{Dop}=2\hat N_l h_{\rm ls}^{-1}{d\over dr}\tilde\Pi_l(r_{\rm ls})
-{2\over 3}h_0^{-1}G_0\delta_{l,1},
\end{equation}
and
\begin{equation}
{\cal L}_l^{ISW}=\hat N_l\left[-6 \tilde\Pi_l(r_{\rm ls})
+6\int_0^{r_{\rm ls}}dr\, F(\eta_{\rm ls}+r_{\rm ls}-r)
{d\over dr}\tilde\Pi_l(r)\right].
\end{equation}
\begin{figure
\centering
\epsfig{figure=fig3.ps, width=9cm}
\caption{amplitude of the Sachs Wolfe dipole window functions in the limit
$k\rightarrow 0$
as a function of $\Omega_0$. The total Sachs-Wolfe window function
(solid curve) is the sum of the pure Sachs-Wolfe contribution (dotted curve),
of the Doppler contribution (dashed curve), which almost cancel each other,
and finally of the ISW contribution (dashed-dotted curve).}
\end{figure}
\begin{figure
\centering
\epsfig{figure=fig4.ps, width=9cm}
\caption{amplitude of the Sachs Wolfe quadrupole window functions in the limit
$k\rightarrow 0$ as a function of $\Omega_0$. In the total Sachs-Wolfe window function
(solid curve), The Doppler term (dashed curve) is very
small with respect to the two other terms, the pure SW term (dotted curve)
and the ISW term (dashed-dotted curve). One notices that the SW dipole
vanishes near $\Omega_0\simeq 0.3$.}
\end{figure}
These three quantities, as well as their sum corresponding to the total
Sachs-Wolfe effect are plotted as functions of $\Omega_0$ for the dipole
(Fig. 3) and for the quadrupole (Fig. 4). The first striking difference
with the flat case is that the Sachs-Wolfe dipole is not suppressed
with respect to the quadrupole. One can see more precisely on Fig. 3
that this is due principally to the ISW contribution, even if the pure SW
and Doppler contributions do not cancel each other exactly, especially
for low $\Omega_0$. As for the quadrupole, one can see that the Doppler
contribution is extremely small in comparison with the other terms.
Another remark is that the quadrupole vanishes around $\Omega_0\simeq 0.3$
for which the pure SW effect and the ISW effect happen to compensate
each other.
\begin{figure
\centering
\epsfig{figure=fig5.ps, width=9cm}
\caption{squared amplitude of the total window functions (intrinsic and Sachs-Wolfe)
for ultra large scale adiabatic perturbations (limit $k\rightarrow 0$),
for the three multipoles $l=1,2,3$ (from top to bottom). }
\end{figure}
\begin{figure
\centering
\epsfig{figure=fig6.ps, width=9cm}
\caption{squared amplitude of the total window functions (intrinsic and Sachs-Wolfe)
for ultra large scale isocurvature perturbations (limit $k\rightarrow 0$),
for the three multipoles $l=1,2,3$ (from top to bottom). }
\end{figure}
In Fig. 5 and Fig. 6, we have plotted, still in the ultra large scale limit,
the total window functions for the multipoles $l=1,2,3$ with, respectively,
adiabatic and isocurvature initial conditions. We have added the multipole $l=3$ to
show that even if the dipole can be made much bigger than the quadrupole
in the zone $\Omega_0\simeq 0.3$, this cannot explain the observed CMBR since the
next multipole is roughly one tenth of the dipole.
The conclusion is therefore that, in an {\it open} universe, the {\it
cosmological dipole} is typically of the {\it same order of magnitude}
as the higher multipoles. Within our assumptions, the observed dipole
can be explained cosmologically only in a flat space. In the next section,
we examine in detail this flat space limit in order to put a limit
on the flatness required to explain the dipole by cosmological perturbations.
\subsection{Flat space limit}
We saw in the previous subsection that, for supercurvature modes, the
window functions go to zero when $\Omega\rightarrow 1$. We need to examine
now in more details what is the relative importance of
the dipole with respect to the quadrupole in this limit. To do so, it is
convenient to introduce the paramater
\begin{equation}
\epsilon=\sqrt{1-\Omega_0}.
\end{equation}
One can immediately notice that the curvature radius reads $a_0=H_0^{-1}/
\epsilon$ so that the curvature radius goes to infinity with respect to
the Hubble radius, as one can expect in a flat space limit.
Expanding (\ref{ropen}) in $\epsilon$, one finds
\begin{eqnarray}
r(\Omega_0,z)&=&2\left(1-{1\over \sqrt{1+z}}\right)\epsilon +\left[{2\over 1+z}
\left(z+3-3\sqrt{1+z}+{z\over 2\sqrt{1+z}}\right) \right. \nonumber\\
& &\left. -{4\over 3}\left(1-{1\over\sqrt{1+z}}
\right)^3\right]\epsilon^3+{\cal O}(\epsilon^5).
\end{eqnarray}
The expansion of the terms appearing in the total Sachs-Wolfe effect is
\begin{eqnarray}
{\cal W}_{k1}^{pSW}&=&{2\over 3}k\left(1-{1\over \sqrt{1+z}}\right)\epsilon
+{2\over 3}k\left[(z+{11\over 7}-{22+35z\over 14\sqrt{1+z}}
)/(1+z) \right. \nonumber \\
& & \left.
-{2\over 5}(3+k^2)\left(1-{1\over\sqrt{1+z}}\right)^3\right]\epsilon^3
+{\cal O}(\epsilon^5),\label{d1}
\end{eqnarray}
\begin{eqnarray}
{\cal W}_{k1}^{Dop}&=&-{2\over 3}k\left(1-{1\over \sqrt{1+z}}\right)\epsilon
+{2\over 3}k\left[{1\over 7}+{1\over\sqrt{1+z}}\left({7z-2\over 14(1+z)}
\right. \right. \nonumber \\
& &\left. \left.
-{2\over 5}(4+3k^2)\left(1-{1\over \sqrt{1+z}}\right)^2\right)
\right]\epsilon^3 +{\cal O}(\epsilon^5),
\end{eqnarray}
\begin{equation}
{\cal W}_{k1}^{ISW}=-{16k\over 21}\left(1-{1\over \sqrt{1+z}}\right)^2
\left(1+{2\over\sqrt{1+z}}\right)\epsilon^3+{\cal O}(\epsilon^5).\label{d3}
\end{equation}
Note that the strict limit $\Omega_0=1$ is not obtained simply by setting
$\epsilon=0$, which would give zero for all the terms above. Indeed it makes
more sense to label the perturbation scales by the physical quantity
$k/h_0$ instead of the unphysical wavenumber $k$. Since $h_0=1/\epsilon$, the
proper flat limit is given by $\epsilon\rightarrow 0$ with $k\epsilon$ fixed. However,
this limit applies only to subcurvature modes whereas supercurvature modes
correspond to larger and larger scales as $\Omega_0\rightarrow 1$ since
they are bounded by $k<1$. The most striking feature of (\ref{d1}-\ref{d3})
is that
the dominant terms of the pure SW contribution and of the Doppler term
just cancel each other. It is thus clear that the suppression of the
dipole is really a consequence of the flatness.
We have also written explicitly the terms at the order
$\epsilon^3$ because they represent the leading order in the total SW window
function.
For the quadrupole, one obtains
\begin{equation}
{\cal W}_{k2}^{pSW}={4\over 15}k(k^2+3)^{1/2}
\left(1-{1\over \sqrt{1+z}}\right)^2\epsilon^2 +{\cal O}(\epsilon^4),\label{q1}
\end{equation}
\begin{equation}
{\cal W}_{k2}^{Dop}={8\over 15}k(k^2+3)^{1/2}
\left({1\over \sqrt{1+z}}-{1\over 1+z}
\right)\epsilon^2 +{\cal O}(\epsilon^4),
\end{equation}
\begin{equation}
{\cal W}_{k1}^{ISW}={\cal O}(\epsilon^4).\label{q3}
\end{equation}
For isocurvature, one must add the intrinsic contribution which is roughly
five times ${\cal W}_{kl}^{pSW}$.
By comparing (\ref{d1}-\ref{d3}) with (\ref{q1}-\ref{q3}), one can see that
for $k<<1$ all the window functions, both for the dipole and the quadrupole,
are proportional to $k$. As a consequence one cannot obtain a significant
dipole by considering sufficiently large scales, like in the flat case.
The only possible difference between the dipole and the quadrupole can thus
come only from the parameter $\epsilon$. For adiabatic perturbations, the
cancellation of the $\epsilon$ terms implies that the dipole is $\epsilon$ times the
quadrupole. On the contrary, for isocurvature perturbations, the total
dipole is proportional to $\epsilon$ and can be much bigger than the quadrupole
which goes like $\epsilon^2$. To obtain a dipole $10^2$ times larger than the
quadrupole, one thus needs $\epsilon < 10^{-2}$, i.e. $|\Omega_0 -1|<10^{-4}$.
\section{Conclusions}
The main conclusion of this paper is that a cosmological origin for
the dipole (or a significant part of it) must be rejected for an open
cosmology, which means that the observed dipole would be in that case
essentially a Doppler effect resulting from our local motion dominated by
small scale density perturbations.
This conclusion assumes that the universe can be described as a FLRW
model with gaussian fields of linear perturbations.
More exotic possibilities have not been considered here.
Another conclusion of this work is that the suppression in a flat universe
of the dipole due to ultra large scale adiabatic perturbations,
with respect to the quadrupole,
is no longer true in an open universe.
It turns out that the only possibility to get a dipole two orders of
magnitude bigger than the other multipoles is a model with isocurvature
perturbations on scales of the order of $10^2 H_0^{-1}$ in an almost
flat geometry:
the constraint is $|\Omega_0 -1|<10^{-4}$. This constraint can also
be interpreted as demanding that the curvature scale be one hundred timed
larger than the Hubble radius. When this constraint is satisfied but
with $\Omega_0$ non strictly equal to one (this would be the case
for example of an open universe after a phase of inflation), two kinds
of perturbations can produce the observed dipole : the subcurvature modes
on scales larger than one hundred times the Hubble radius, like in the
flat case; but also the supercurvature modes, which do not exist in the
flat case.
To conclude, a reliable measurement of $\Omega_0$ (see \cite{dbw}
for a discussion on measuring $\Omega_0$) different from one
would strongly confirm the Doppler assumption for the dipole.
Conversely, if unquestionable cosmological observations conclude that our
local motion is in disagreement with the CMBR dipole, then it would
suggest that we live in a flat universe and that ultra large scale
isocurvature perturbations must exist.
\bigskip
\bigskip\noindent
{\bf Acknowledgement.}
\medskip
I would like to thank N. Deruelle, T. Piran and D. Wands for stimulating
discussions.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 6,364
|
Q: copying with memcpy some data to mmap return pointer address location of a file If I have a file which I mapped with mmap like this
sfd = open("test.txt", O_RDONLY);
filesize = lseek(sfd, 0, 10);
src = mmap(NULL, filesize, PROT_READ, MAP_PRIVATE, sfd, 0);
so right now the sizeof mapping is 0 bytes because size of test.txt is 0. So what if I want to write word hello to mapped area. How to do it?
I tried like this
memcpy(src,"hello", sizeof("hello"));
But this throws Bus error. I could use mremap but that is not available on my system. So I like to know how to write to mapped area something with memcpy
Update
This is my code and now it giving me segFault at memcpy line
int main(int argc, char **argv){
int sfd, dfd;
char *src, *dest;
size_t filesize=10;
/* SOURCE */
sfd = open("test.txt", O_RDWR);
src = mmap(NULL, filesize, PROT_READ, MAP_PRIVATE, sfd, 0);
ftruncate(sfd,10);
if(src== MAP_FAILED)
{printf("error\n");exit(0);}
/* DESTINATION */
memcpy(src,"hello", sizeof("hello"));
munmap(src, filesize);
close(sfd);
close(dfd);
return 0;
}
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,423
|
package org.mockito.internal.creation.instance;
import org.mockito.mock.MockCreationSettings;
import org.mockito.plugins.InstantiatorProvider;
public class DefaultInstantiatorProvider implements InstantiatorProvider {
private final static Instantiator INSTANCE = new ObjenesisInstantiator();
public Instantiator getInstantiator(MockCreationSettings<?> settings) {
if (settings != null && settings.isUsingConstructor()) {
return new ConstructorInstantiator(settings.getOuterClassInstance());
} else {
return INSTANCE;
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,704
|
using System;
using System.Globalization;
namespace ADatabase.Oracle.Columns
{
public class OracleNumberColumn: OracleColumn
{
private readonly string _typeToString;
public OracleNumberColumn(string name, ColumnTypeName columnTypeName, int prec, int scale, bool isNullable, string def)
: base(name, columnTypeName, isNullable, def)
{
Details["Prec"] = prec;
Details["Scale"] = scale;
if (prec == 0 && scale == 0) _typeToString = "number";
else if (prec == 0 && scale != 0) _typeToString = $"number(*,{scale})";
else if (prec != 0 && scale == 0) _typeToString = $"number({prec})";
else _typeToString = $"number({prec},{scale})";
}
public override string TypeToString()
{
return _typeToString;
}
public override string ToString(object value)
{
return Convert.ToDecimal(value).ToString(CultureInfo.InvariantCulture);
}
public override Type GetDotNetType()
{
return typeof(decimal);
}
public override object ToInternalType(string value)
{
if (value == null)
{
return DBNull.Value;
}
return decimal.Parse(value, CultureInfo.InvariantCulture);
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,892
|
Beale Park is a public park located in Bakersfield, California. It is Bakersfield's first park, located on Oleander St.
History
The park was constructed in 1908, on of land donated by Truxtun Beale. He also donated landscaping for the park, the Greek style amphitheater and swimming pool.
Since then, the park has seen many improvements. Two large reservable picnic areas were added, as well as many smaller individual areas. Several different types of sports courts and fields were also added.
Over the years, the pool, similar to most of the city pools, needed to be refurbished. However, in 2004, with the opening of the aquatics center in Downtown Bakersfield, it was decided not to refurbish the pool. It was removed and replaced with a spray park.
In 1977, a large wind storm hit Bakersfield. The roof of an aviary was blown off, and at least two parakeets escaped. Since then, a large parakeet population has settled in and around the park. This population has also spread to Beach Park and Hart Park. They have grown to be largest population of naturalized parakeets in the world.
Amenities
Beale Park is one of the most equipped city parks in Bakersfield. It has two reservable picnic areas, seating 80 people each. They are both uncovered, and are each equipped with a barbecue, and lights. There are several individual picnic areas, which can seat 8 to 16 people. Some of them are equipped with a barbecue and some others are covered. There is one lit full basketball court, three lit tennis courts, and four lit horseshoe pits. There is also a softball backstop, and a large open field, suitable for most lawn sports. The park also includes restrooms, and a playground.
The park is also one of two parks with an open outdoor amphitheater that includes seating (on terraced grass). The stage is in the Greek style. The Bakersfield Symphony Orchestra has historically performed concerts at the theater during the summer months.
References
External links
Bakersfield Department of Recreation & Parks: Beale Park
Parks in Bakersfield, California
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,030
|
Honda Avancier is a nameplate used by two different Honda vehicles. The name was first used from 1999 to 2002 on a mid-size station wagon based on the sixth-generation Accord. The nameplate was revived for usage on a China-only mid-size crossover SUV produced by Guangqi Honda in 2016.
Vehicles using the nameplate are:
Honda Avancier (station wagon), a mid-size station wagon produced from 1999 to 2003.
Honda Avancier (crossover), a mid-size crossover SUV produced since 2016.
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8,946
|
{"url":"http:\/\/tex.stackexchange.com\/questions\/66353\/amsbook-theorem-numbering","text":"# amsbook theorem numbering\n\nI want theorems to be labeled of the form Part.Chapter.Section.number so I tried\n\n\\renewcommand{\\thesection}{\\Roman{part}.\\thechapter.\\arabic{section}}\n\\newtheorem{book}{Theorem}[section]\n\n\nBut does what I want, but as a side effect it also changes the sections, is there a way to only change the theorem numbering?\n\n-\n\nTo have your AddProp \"theorem\" numbered by section using Part.Chapter.Section.Number, you can use:\n\n\\newtheorem{book}{Theorem}[section]\n\\renewcommand{\\thebook}{\\thepart.\\thesection.\\arabic{book}}\n\n\nThis will also reset your book counter after every new \\section. Here's a minimal example showing the usage\/output:\n\n\\documentclass{report}\n\\newtheorem{book}{Theorem}[section]\n\\renewcommand{\\thebook}{\\thepart.\\thesection.\\arabic{book}}\n\\begin{document}\n\\setcounter{part}{3} \\part{A part}\n\\setcounter{chapter}{4} \\chapter{A chapter}\n\\setcounter{section}{12} \\section{A section}\n\\section{Another section}\n\\end{document}\u200b\n\n\nIf you wish to maintain this structure, but have the book counter only reset every \\chapter, then you need to drop the [section] part from the definition of the book theorem, and add the counter reset to your book counter manually:\n\n%...\n\\newtheorem{book}{Theorem}%[section]\n\nThis functionality is provided by the chngcntr package as well as amsmath via user-friendly macros. Since you're using amsbook, you could use \\numberwithin{book}{chapter}. See Master and slave counters and perhaps Section numbering with chapter in amsbook.","date":"2015-08-30 12:25:06","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6476312279701233, \"perplexity\": 4031.286732329829}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2015-35\/segments\/1440644065306.42\/warc\/CC-MAIN-20150827025425-00149-ip-10-171-96-226.ec2.internal.warc.gz\"}"}
| null | null |
Mike Tyson on Oprah: Breaking Down, Not Interested in Details of Daughter's Death
by Free Britney at October 13, 2009 5:43 am .
Once the most fearsome fighter in the world, Mike Tyson broke down in tears as he described his unspeakable grief after the death of his daughter Exodus.
Mike broke his silence in an emotional interview on The Oprah Winfrey Show yesterday, recalling how his little girl strangled herself on a treadmill cord.
This happened at her mother Sol Xochitl's home in Phoenix.
Although nearly five months has passed since the tragic accident, Mike Tyson still struggles to accept it. However, he doesn't want to know the details.
"From all my experience in rehab, I took responsibility," he says, "She had to be buried, she had to be taken care of. There was no animosity. There was no anger towards anybody. I don't know how she died and I don't want to know.
"But I know somebody's to blame for it, there will be a problem."
Mike Tyson's fall from grace pales in comparison to this tragedy.
Exodus' older brother Miguel, 7, found her unconscious after her neck got caught in the treadmill cord and her mother, who is no longer in a relationship with Mike Tyson, administered CPR before calling emergency services.
She died from her injuries the following day in the hospital.
How did Mike and his temper cope with such devastation?
"My first instinct was a lot of rage and I am so happy I had the tools in life not to go in that direction. I don't want to go there, I want to win now."
"I'm just fortunate that I have children by different women and as a family we're so close. That's my biggest asset. We all love each other."
Mike recently got married to Lakiha Spicer.
Tags: Oprah Winfrey, Mike Tyson
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Danny Pintauro Sends Message About Drug Use
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 6,872
|
import { putJSON } from '../../lib/api';
import * as constants from '../../constants/expenses';
/**
* Update an expense in a group
*/
export default (groupid, expenseid, expense) => {
const url = `/groups/${groupid}/expenses/${expenseid}`;
return dispatch => {
dispatch(request(groupid, expense));
return putJSON(url, {expense})
.then(json => dispatch(success(groupid, json)))
.catch(error => {
dispatch(failure(error))
throw new Error(error.message);
});
};
};
function request(groupid, expense) {
return {
type: constants.UPDATE_EXPENSE_REQUEST,
groupid,
expense
};
}
function success(groupid, expense) {
const expenses = {
[expense.id]: expense
};
return {
type: constants.UPDATE_EXPENSE_SUCCESS,
groupid,
expenses,
};
}
function failure(error) {
return {
type: constants.UPDATE_EXPENSE_FAILURE,
error,
};
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,382
|
\section{Introduction}
Diffuse interstellar media and molecular clouds are both turbulent and magnetized \citep{1981MNRAS.194..809L,2004ARA&A..42..211E,2004RvMP...76..125M,2007prpl.conf...63B,MO07,CL09}. Magnetization of the media is extremely important for understanding key astrophysical problems, e.g.
the problem of star formation \citep{1959ApJ...129..243S,1956MNRAS.116..503M,1976ApJ...210..326M,1977ApJ...214..488S,1992pavi.book.....S,1991ApJ...370L..39K,1994ApJ...429..781S,1998ApJ...498..541K,1998ARA&A..36..189K,2013ApJ...770..151C,2013ApJ...768..159H,2015ApJ...808...48B,2018arXiv180105428B} and cosmic ray propagation and acceleration \citep{1949PhRv...75.1169F,1964ocr..book.....G,1966ApJ...146..480J,1978MNRAS.182..147B,1965ApJ...142..584P,1979cmft.book.....P,2014ApJ...783...91C}.
Various techniques have been proposed to study the magnetization of the interstellar media \citep{2009ASPC..414..453D}. Magnetic field strength and structures can be measured by both direct or indirect approaches. Spectral line splitting due to Zeeman effect provides the direct measurement of the line of sight component of magnetic field strength \citep{1989ApJ...338L..61G,1993prpl.conf..279H,1999ApJ...520..706C,1999ApJ...514L.121C,2001ApJ...554..916B,C10,2008A&A...487..247F}, but the method {\toreferee is mostly applied to lower Galactic latitude target (Crutcher 2012)} and does not provide an insight on the total magnetization nor the magnetic field directions perpendicular to the line of sight {\toreferee unless the two circular polarization components are both resolved (Crutcher 2007)}. Polarized dust emission \citep{2012ApJS..201...13V,2000PASP..112.1215H,1982MNRAS.200.1169C,1986ApJ...308..270D,1997ApJ...487..320N,1999MNRAS.303..659H,1984ApJ...284L..51H,1988QJRAS..29..327H} as well as measuring of starlight polarization arising {\toreferee from } aligned dust \citep{2012ApJS..200...19C,2012ApJS..200...21C,2013AJ....145...74C,Aetal15} provides the main indirect method of mapping the plane-of-sky magnetic field strength using the {\toreferee Davis-Chandrasekhar-Fermi }\citep{D51,CF53,Fal08,2009ApJ...696..567H,2009ApJ...706.1504H,2011ApJ...733..109H,2016ApJ...820...38H,2012ApJ...749...45C} method , but the estimation of magnetic field strength using the CF method can be significantly different from the true value even in synthetic maps obtained with MHD simulations for the case of trans-Alfvenic turbulence \citep{Fal08, CY16}.
Anisotropies of magnetized turbulence (see \citealt{BL13} for a review of the theory) provides an alternative way to study magnetic field directions. \cite{Letal02} demonstrated that using the correlation functions of intensities {\toreferee in} velocity channel maps\footnote{Intensity fluctuations {\toreferee within "thin" (See \S \ref{sec:numerics})} velocity channel maps are in most cases dominated by velocity fluctuations \cite{LP00, LP04}},{\toreferee it is feasible} to study the magnetic field directions. Further development of the technique showed that the the Correlation Functions Anisotropy (CFA, see also Appendix \S \ref{subsec:cfa}) is a promising way to study both magnetic field direction and media magnetization, as well as to distinguish the contributions from Alfven, slow and fast modes (see \citealt{Eetal15,KLP16,KLP17a}). The CFA technique shows effective results on both the channel maps and velocity centroids \citep{EL05, KLP17b}. The CFA is also applicable to the study magnetic fields and turbulence properties with synchrotron intensity and polarization fluctuations \citep{LP12, LP16}.
The approach in \cite {Letal02} can be realized in different ways with CFA being its particular realization. For instance, the anisotropies can be studied by employing the Principal Component Analysis (PCA) technique as described in \cite{Hetal08}. Our study in \cite{brazil18} showed, however, that there are no particular advantages of this technique compared to either VGT or CFA. Therefore we do not discuss this technique further in this paper.
A different approach of magnetic field tracing with gradients of observable measures (e.g. velocity centroids, synchrotron intensity and synchrotron polarization, etc.) has been suggested recently \citep{GCL17, YL17a, YL17b, LY18a, LY18b, Letal17}. This technique in terms of theoretical justification is related to the CFA. {\toreferee It is known that the MHD fluctuations can be decomposed into fundamental Alfven, fast and slow modes (Biskamp 2003, {\torefereetwo See Appendix \ref{App_A}}). A numerical study in Cho \& Lazarian (2003) testifies that the three modes are evolving creating their own cascades with Alfven and slow modes showing significant anisotropy along the magnetic field direction. With the contribution of Alfven and slow modes being dominant, especially in the weakly compressible cases (Lithwick \& Goldreich 2001, Cho \& Lazarian 2002), {\torefereetwo a prominent anisotropy is both expected and observed in the MHD turbulence containing all the three MHD modes }(see Cho \& Lazarian 2003, Kowal \& Lazarian 2009).} The corresponding elongation of turbulent fluctuations causes the correlations of velocity to be stronger along the local magnetic field direction. At the same time the gradients of velocity {\toreferee become} perpendicular to magnetic field. As a result, one can estimate the direction of magnetic field by the direction of velocity gradients through a 90-degree rotation on the local gradient direction. Similarly, {\toreferee one can infer} the direction of magnetic fields {\toreferee by rotating the the magnetic gradients by 90 degrees} . The latter can be revealed through studying synchrotron intensity and synchrotron polarization gradients. For the sake of simplicity, within this study we shall focus on velocity gradients. However in view of the symmetric way how velocity and magnetic fluctuations enter Alfvenic turbulence, all our results in this paper are applicable also to magnetic field gradients.
Density fluctuations for low sonic Mach number $M_s$ will follow the velocity statistics. Thus for low $M_s$, density gradients behave similarly to velocity gradients. However, density is not always a passive scalar of velocity information and the density gradients created by shocks tend to be parallel to magnetic field in the case of high $M_s$. Therefore a combination of density and velocity gradients provides a better insight into the properties of turbulence in diffuse media.
In Alfvenic turbulence the directions of ${\bf k}$ vectors of turbulent velocities has a statistical distribution (see GS95,\citealt{2002ApJ...564..291C}). Similarly, the gradients also have a distribution of directions with the most {\toreferee probable} orientation of gradients being perpendicular to the magnetic field direction. To find this most probably orientation, in practical studies, the velocity gradients are calculated over a block of data points. When the statistics is sufficiently rich, the histogram of gradient orientations within a block gets Gaussian with the peak of the Gaussian corresponding to the local direction of magnetic field within the block.\footnote{This procedure should not be confused with the Histogram of Relative Orientations (HRO) technique explored by \cite{Setal13} for the intensity gradients. The latter requires polarimetry data to define the direction of magnetic field and draws the {\it relative} orientation of polarization directions and intensity gradients as a function of column density. Our technique is polarization-independent and is the way of finding the magnetic field direction, which is different from the purpose of the HRO. We stress that our approach when applied to velocities and densities provides the spacial direction of magnetic field, while the HRO provides the correlation of the relative orientation of intensity gradients as function of column density {\toreferee (see more comparison in YL17b, LY18a)}.}
In our earlier studies (e.g. \citealt{YL17a,LY18a}) we demonstrated that the peak of the distribution is correlated with the magnetic field direction and this provides a promising way of magnetic field studies, including studies of the 3D distribution of magnetic fields.\footnote{The 3D studies are possible with the {\toreferee Velocity Channel Gradients (VChGs)} using the galactic rotation curve to distinguish different emitting regions \citep{LY18a}. Another way of obtained 3D magnetic field structure employs the Faraday depolarization within the synchrotron polarization gradient studies \citep{LY18b}.} Our earlier studies were focused on magnetic field tracing by determining the peak of the Gaussian distribution of gradients (see \citealt{YL17a, YL17b, Letal17, LY18a, LY18b}). However, it became clear that the distribution of gradient orientations\footnote{\toreferee Similar to polarization, the gradients have $180^o$ ambiguity in determining the magnetic field actual direction. } is also an informative measure. For instance, the dispersion of the gradient orientation distribution was used to identify the regions of collapse induced by self-gravity \citep{LY18a}.\footnote{In the regions of self-gravitational collapse, the relative direction of velocity gradients and magnetic field changes gradually from perpendicular to parallel. This induces an increase of the dispersion \citep{YL17b, LY18a}.} However, the gravitational collapse is a special case when the properties of the turbulent flow change dramatically. In this paper we focus our attention on the properties of the distribution of gradient orientations for magnetized turbulence in diffuse regions where the effects of self-gravity is negligible. We are going to show that for such settings the properties of the distribution of gradient orientations are directly related to the Alfven Mach number $M_A$, which is the ratio of the turbulent velocity $V_L$ at the scale $L_{inj}$ and the media Alfven velocity $V_A$. We shall also demonstrate the ways of using spectroscopic data in order to convert the distribution of $M_A$ into the distribution of magnetic field strengths.
In a companion paper by \cite{YLL18} we found that the amplitude of velocity gradients can be used to study both sonic $M_s=V_L/c_s$, where $c_s$ is the sound speed. Thus two most important measures of turbulence, $M_A$ and $M_s$ can be obtained using the gradient techniques.
In this paper, we focused on the relation of velocity gradient dispersion to $M_A$, and thus provide a way to map the magnetization in the media. In what follows, we provide the theoretical justification of this work in \S \ref{sec:theory}, discuss numerical simulations that we employ to test our expectations in \S \ref{sec:numerics}, and analyze our results in \S \ref{sec:result}, including an application to observation in \S \ref{sec:observation}. We briefly discuss the ways of obtaining magnetic field strength using our estimation of $M_A$ in \S \ref{intensity}. Our discussion and summary are in \S \ref{sec:discussion} and \S \ref{sec:summary}, respectively.
\section{Theoretical considerations}
\label{sec:theory}
In what follows we consider fluctuations of velocity arising from MHD turbulence. Similar considerations, however, are applicable to the fluctuations of turbulent magnetic field.
\subsection{Basic MHD turbulence considerations}
The predictive theory of incompressible MHD turbulence was formulated in \cite{GS95}, henceforth GS95. This theory can be understood on the basis of the Kolmogorov hydrodynamic turbulence theory if a concept of fast turbulent reconnection \citep{LV99}, henceforth LV99 is added. Indeed, according to LV99, magnetic reconnection happens in just one eddy turnover time for eddies at all scales. {\toreferee Therefore, motions that mix magnetic field lines perpendicular to their direction is the way at which the turbulent cascade does not need to bend magnetic field lines and therefore encounters the least resistance from the field.} As a result, the turbulent eddies are elongated along the direction of magnetic field. Incidentally, this justifies the concept of {\toreferee alignment of turbulent eddies} with the local magnetic field direction that was not a part of the original GS95 picture\footnote{The closure relations employed in GS95 assume that the calculations were done in the reference frame of the mean field. Numerical simulations confirm that the GS95 relations are not correct in the mean field reference frame, but only correct in the local magnetic field reference frame.} but the necessity of using the local frame of reference was pointed out by numerical simulations (see \citealt{CV00, MG01}).
The modern theory of MHD turbulence is discussed in e.g. \cite{BL13}. Here we briefly summarize the points that are essential for understanding the properties of gradients (see also Appendix \ref{App_A}). {\toreferee As we explain in the Appendix \ref{App_A} the properties of Alfvenic modes provide the basis for the gradient techniques. These modes also imprint their structure on the slow modes, while the fast modes provide a subdominat contribution. Thus we focus below on Alfvenic turbulence. }
Assume that the injection of turbulent energy takes place at the scale $L_{inj}$. If injection velocity $V_L$ equals to the Alfven velocity $V_A$, the Alfvenic turbulence naturally produces the Kolmogorov scaling in the direction perpendicular to magnetic field. Indeed, as we mentioned earlier, motions induced by Alfvenic turbulence are not constrained by magnetic field and therefore they produces a cascade of energy $ v_l^2$, where $l$ is the eddy size perpendicular to magnetic field. The cascading to smaller scales happens over the eddy turnover time is $\sim l/v_l$. For $l$ significantly larger than the dissipation scale, the flow of kinetic energy $v_l^3/l$ is constant. This way one easily gets the GS95 scaling for the motions perpendicular to the magnetic field, i.e. $v_l\sim l^{1/3}$.
To find how eddies evolve in the direction parallel to magnetic field one should take into account that the mixing motions associated with magnetic eddies send Alfven waves with the period equal to the period of an eddy, i.e.
\begin{equation}
l/v_l\approx l_{\|}/V_A,
\label{critbal}
\end{equation}
where $l_{\|}$ is the parallel scale of the eddy. The {\toreferee condition (Eq. \ref{critbal})} is associated in GS95 theory with the critical balance {\toreferee stressing the fact that Alfven modes, which is by definition incompressible (See Biskamp 2003), have zero velocity divergence all over the space. As a result, the infall velocity gradient $v_l/l$ should be equivalent to that propagating velocity gradients $V_A/l_{\|}$ of the Alfvenic wave along the magnetic field line}. Combining this with the relation for $v_l\sim l^{1/3}$ one gets relation between the parallel and perpendicular scales of the eddies, i.e. $l_{\|}\sim l^{2/3}$. This is the relation that is true for the eddies that are aligned with the {\it local} direction of the magnetic field that surrounds them. The velocities associated with a turbulent eddy are anisotropic,{\toreferee and thus} the largest change of the velocity is in the direction perpendicular to the local direction of magnetic field.
\begin{figure}[t]
\centering
\includegraphics[width=0.49\textwidth]{nsf.pdf}
\caption{\label{fig_illu} Iso-contours of equal velocities for sub-Alfvenic MHD turbulence simulation from our model Ma0.4 {\toreferee(See Table \ref{tab:sim})}. It is clear that gradients of velocity structures are perpendicular to the magnetic fields. From \cite{LY18a}.}
\end{figure}
{\toreferee In both sub-Alfvenic and super-Alfvenic turbulence the gradient methods of tracing magnetic field can be used (See \citealt{YL17b}). However, the signature of gradients in sub-Alfvenic and super-Alfvenic turbulence are different.} For {\toreferee sub-Alfvenic} turbulence, i.e. for $V_L<V_A$ the magnetic fields are always important and their action is imprinted on turbulent velocities at all scales. Figure \ref{fig_illu} illustrates the alignment of iso-contours of equal velocity (Red) and the mean magnetic field direction (Blue) in numerical simulations with $M_A=0.4$. {\toreferee (See \citealt{Chen16, Hull17})} It is obvious from the Figure \ref{fig_illu} that the direction of the fastest spacial change of the velocity is perpendicular to the magnetic field.
{\toreferee In the case of super-Alfvenic turbulence, extra complications in filtering large-scale isotropic eddies are required before applying the VGT method. } The cases of energy injection with $V_L<V_A$ and $V_L>V_A$ are discussed in Appendix \ref{App_A}. As we discuss there for the super-Alfvenic turbulence, i.e. for $V_L>V_A$ there is a range of scales for which turbulence is hydrodynamic and not much affected by magnetic fields. However, for turbulence with an extended inertial range there is a scale $L_{inj} M_A^{-3}$ at which the turbulent velocities get equal to the Alfven velocity (see Lazarian 2006). Starting from that scale our considerations above that relate the direction of velocity gradients and local magnetic field are applicable.
{\toreferee {\torefereetwo In the case of super-Alfvenic turbulence scales larger than $L_{inj} M_A^{-3}$, turbulence eddies are isotropic (See Appendix \ref{App_A}). As a result, one cannot derive velocity gradient information on these large scale eddies and therefore it is incapable to use the gradient method on isotropic large scale eddies to trace magnetic fields. }Turbulent eddies with scales smaller than $L_{inj} M_A^{-3}$ are still anisotropic and their gradients still probes the local direction of magnetic field (See \citealt{YL17b}) . As a result, in the case of super-Alfvenic turbulence the gradients of velocities at large scale are not sensitive to magnetic field. However, this contribution can be spatially filtered out as it was demonstrated in LY17.}
\subsection{3D gradients induced by MHD turbulence}
With this understanding of turbulence in hand, it is easy to understand how the velocity gradient technique works. It is clear from Eq. (\ref{anis}) that the anisotropy of eddies increases with the decrease of the scale $l$. Thus the turbulent motions of eddies are getting better and better aligned with the local magnetic field as the scale of eddies decreases. This property of eddies is the corner stone of the velocity gradient technique that we introduced in a series of recent papers.
In terms of practical measurements, it is easy to see that the gradients in turbulent flow increase with the decrease of the scale. This increase can be estimated as $v_l/l\sim l^{-2/3}$, where the Eq. (\ref{vel_strong}) is used. This suggests that the gradients arising from the smallest resolved eddies $l_{min}$ dominate the gradient measurements. When such eddies are well aligned with the direction of magnetic field, just turning the velocity gradients 90 degrees, one can trace magnetic field at the $l_{min}$ scale.
\subsection{Velocity gradients available through observations}
The 3D fluctuations of velocity are not directly available from observations. Instead, we demonstrated that gradients of velocity centroids \citep{GCL17,YL17a} and gradients of intensity fluctuations measured within thin channel maps \citep{LY18a} can be used as the proxies of velocity gradients. In both cases,
the gradients are measured for a turbulent volume extended by ${\cal L} >L_{inj}$ along the line of sight and this entails additional complications, where ${\cal L}$ is the line-of-sight depth. While eddies stay aligned with respect to the local magnetic field, the direction of the local magnetic field is expected to change along the line of sight. Thus the contributions of velocity gradients gets summed up along the line of sight.\footnote{This effect is similar to that of the summation of far infrared polarization from aligned grains along the line of sight (see \citealt{Aetal15}) with the exception that the polarization {\toreferee sums quadratically (e.g. $Q\propto \int dz b_y^2-b_x^2$)in the form of Stokes parameters} while gradients are summed up as {\toreferee linear vectors (e.g. $\nabla C \propto \int dz \nabla (\rho v)$)}. The difference between the two ways of summation is small for small $M_A$, but gets significant for large $M_A$. Thus for super-Alfvenic turbulence we expect gradients to represent averaged along the line of sight magnetic field better compared to dust polarization.}
It is possible to show (see \citealt{LP12}) that for ${\cal L} >L_{inj}$ the local system of reference\footnote{\toreferee The importance of referring to the local system of reference is because it is the local magnetic field direction defines the direction of eddy anisotropy in magnetized turbulent media (LV99, Cho \& Vishniac 2000, Maron \& Goldreich 2001, Cho et al. 2002).} is not available from observations. Thus one has to use the reference system related to the mean magnetic field. In this system of reference, the anisotropy of the eddies is determined by the anisotropy of the largest eddies as it is illustrated in Fig. (\ref{fig0}). Indeed, the smallest eddies are aligned with the magnetic field and this magnetic field varies along the line of sight due to the the large scale variations of the magnetic field arising from the largest eddies (see \citealt{CLV02}). The latter variations are determined by the fluctuations {\toreferee of} the magnetic field variations at the injection scale, i.e. the changes in the direction of magnetic field along the line of sight are $\delta \varphi \approx \delta B/B\approx M_A$. We would like to stress that the tensors describing the fluctuations in local and global system of reference are different (e.g. compare the tensors in \citealt{CLV02} and \citealt{LP12}).
\begin{figure}[t]
\centering
\includegraphics[width=0.49\textwidth]{Cho2002.pdf}
\caption{\label{fig0} A pictorial illustration of the structure of anisotropic eddies. Small eddy 2 is aligned in respect to the magnetic field of the larger eddy 1. The same is true for the eddies 1' being aligned in magnetic field of eddy 2'. However, if measurements {\toreferee of eddies 2 and 2'} are made in respect to the mean magnetic field {\toreferee $B_0$,both eddies 2 and 2' will be treated as "misaligned" and contributes anonymously to the statistical tools (e.g. Correlation Function) with additional perpendicular-to-field contributions.} From \cite{CLV02}. }
\end{figure}
We have shown earlier that the 3D gradients of the value of 3D velocities is dominated by the smallest scales. A similar conclusion can be obtained for 2D observables. Indeed, the spectrum of observed fluctuations changes due to the line of sight averaging. It is easy to show that the 2D spectrum of turbulence obtained by projecting the fluctuations from 3D has the same spectral index of $-11/3$ (see \citealt{LP00}). The relation between the spectral slope of correlation function and the slope of the turbulence power spectrum in 2D in this situation is $-11/3+2=-5/3$, where 2 is the dimensionality of the space. Therefore, the 2D velocity fluctuations arising from 3D Kolmogorov-type turbulence scale as $l_{2D}^{5/6}$, with gradients anisotropy scaling as $l_{2D}^{-1/6}$. This means the contribution of the smallest scales is dominant for the measured 2D gradients.
\subsection{MHD turbulence anisotropies and distributions of gradient orientations}
Alfvenic turbulence is anisotropic with its anisotropy given by the relations that we have described above.
The anisotropy of Alfvenic turbulence imprints the scaling of Alfven modes into the anisotropy of slow modes (GS95, \citealt{LG01, CL02}). Alfven and slow modes together carry most of the energy of the turbulent cascade. Fast modes, in many cases, are sub-dominant in terms of the energy cascade, although they play a very important role for a number of key astrophysical processes, e.g. cosmic ray scattering (see \citealt{YanL02, YanL03, BL07}). The fast modes stay "isotropic"\footnote{We put isotropic in quotations, as the tensor describing fast modes is different from the tensor of isotropic turbulence (see more explanation in \citealt{LP12}).} both for pressure-dominated case (i.e. $\beta{\toreferee=2M_A^2/M_s^2}>1$, GS95) as well as magnetically-dominated case (i.e. $\beta<1$, \citealt{CL02}). In what follows we focus on the Alfvenic anisotropies that are measured in the observer's frame (see LP12).
It is important to keep in mind the statistical nature of turbulence. The velocity fluctuations have a distribution of directions about the magnetic field direction. Thus the gradients of velocities also have a distribution of directions, the peak of which {\toreferee is perpendicular to } the magnetic field direction. The properties of the distribution are, however, very informative. It follows from the theory of Alfvenic turbulence that the dispersion of turbulence wave vector directions changes with the Alfven Mach number $M_A$ (see \citealt{LP16}). This should be reflected in the distribution of the directions of gradients and the goal of this paper is to determine this relation.
Consider first the sub-Alfvenic and trans-Alfvenic turbulence.This degree of anisotropy is characterized by the ratio of the maximal to the minimal value of correlation function. This is because the relative orientation between the correlation direction and magnetic fields changes from parallel to perpendicular. Evidently for Alfven and slow modes, the correlation will be maximal along the direction of the magnetic field. It is clearly that there exists a correlation between the correlation function anisotropies and the gradients \citep{YLL18}. We shall also address the correlation in \S \ref{sec:result}.
\begin{table}[t]
\centering
\label{tab:sim}
\begin{tabular}{c c c c c}
Model & $M_S$ & $M_A$ & $\beta=2M_A^2/M_S^2$ & Resolution \\ \hline \hline
Ma0.2 & 7.31 & 0.22 & 0.01 & $792^3$ \\
Ma0.4 & 6.1 & 0.42 & 0.01 & $792^3$ \\
Ma0.6 & 6.47 & 0.61 & 0.02 & $792^3$ \\
Ma0.8 & 6.14 & 0.82 & 0.04 & $792^3$ \\
Ma1.0 & 6.03 & 1.01 & 0.06 & $792^3$ \\
Ma1.1 & 6.08 & 1.19 & 0.08 & $792^3$ \\
Ma1.4 & 6.24 & 1.38 & 0.1 & $792^3$ \\
Ma1.5 & 5.94 & 1.55 & 0.14 & $792^3$ \\
Ma1.6 & 5.8 & 1.67 & 0.17 & $792^3$ \\
Ma1.7 & 5.55 & 1.71 & 0.19 & $792^3$ \\
\hline \hline
\end{tabular}
\caption{Description of our MHD simulations. $M_s$ and $M_A$ are the instantaneous values at each the snapshots are taken. This is refereed to Set C in the lower panel of Fig \ref{fig3}.}
\end{table}
{\toreferee From the discussion above, it is evident} that the maximum of the velocity gradients is perpendicular to the longest axis of eddies. This is true both for the 3D eddies and their line of sight projections. Within the procedure of block averaging \citep{YL17a}, the distribution of gradient orientation is fitted with a Gaussian function, where the peak of the Gaussian is associated with the direction of the velocity gradients within the block.
The Gaussian function used in \cite{YL17a} is a three-parameter function $A\exp(-\alpha(\theta-\theta_0)^2)$. Whether the other two fitting parameters are correlated to the intrinsic physical condition of MHD turbulence is unknown. However, it can be understood very easily that the height of the fitting function $A$ and the width $1/\sqrt{\alpha}$ are both related to the turbulence magnetization given by the Alfven Mach number $M_A$. Indeed, if the turbulent injection velocity $V_L$ is significantly small compared to the Alfven velocity, i.e. $M_A$ is small, then the bending of magnetic field is limited. The Alfvenic turbulence at small scales takes place through eddies that mix magnetic field lines perpendicular to the {\it local} magnetic field direction. For small $M_A$ all magnetic field lines are approximately in the same direction and therefore the dispersion of the velocity gradients that are perpendicular to magnetic field lines, is small. As the turbulence driving increases in amplitude the dispersion of velocity gradient orientations is expected to increase. In the framework of distribution of gradient orientations, a more prominent peak (i.e. higher $A$ and large $\alpha$) should be expected for small $M_A$. Therefore one can infer the magnetization of a turbulent region by analyzing the amplitudes of $A$ and $\alpha$.
The magnetization can also be studied in the case of super-Alfvenic turbulence. As we discuss in Appendix \ref{App_A}, for scales larger than the scale at which turbulent velocity is equal to $V_A$ the correlation between magnetic field direction and the direction of the velocity fluctuations decreases. However, in this paper we are dealing with moderately super-Alfvenic turbulence and therefore the this correlation does not disappear completely. This allows to obtain $M_A$ from the distribution function of gradients. The latter is another name for distribution of gradient orientations that we will use further.
Fig. \ref{fig1} illustrates the distribution of gradient orientations for sub-Alfvenic, trans-Alfvenic and super-Alfvenic turbulence. The block averaging procedure \citep{YL17a} is used. The block size is chosen sufficiently large to allow Gaussian fitting. The latter is the necessary for the gradient technique that we have developed. The good Gaussian fit is important to know how reliable is the magnetic field direction that we determine with our technique and for determining the magnetization as we discuss in this paper. We clearly see that the width of the distribution is increasing with the increase of $M_A$. Similarly, the amplitude of the Gaussian distribution given by the ratio of the top-base values (see Fig. \ref{fig1}) is decreasing with $M_A$. In the paper we are discussing the quantitative measures that can translate the parameters of the distribution of the gradient orientations into the $M_A$ data.
\begin{figure*}[t]
\centering
\includegraphics[width=0.98\textwidth]{AD_R.pdf}
\includegraphics[width=0.96\textwidth]{PPV_AD_R.pdf}
\caption{\label{fig1} {\bf Upper Panel} The three panels here show the velocity centroid gradient (VCGs) orientation histograms from three numerical cubes with different Alfvenic Mach number $M_A$. We fit the three Gaussian with the modified Gaussian profile.{\toreferee The T and B mean the peak and bottom value of the modified Gaussian profile which defined as T = A + C' and B = C'.} {\bf Lower Panels} The same for the velocity channel gradients (VChGs). }
\end{figure*}
\subsection{Magnetic field and density fluctuations}
We believe that it is necessary to mention that our earlier studies have already shown that tracing of magnetic field directions is possible with magnetic gradients that are available, e.g. using synchrotron. The considerations about symmetry of velocity and magnetic fluctuations within Alfvenic turbulence constitute the basis for tracing magnetic field using not only with the Velocity Gradient Technique (VGT), but also the Synchrotron Intensity Gradients (SIGs) \cite{Letal17} and the Synchrotron Polarization Gradients (SPGs) \citep{LY18b}.
All our considerations above relevant to the distribution functions of velocity gradients are applicable to the SIGs and the SPGs. Therefore we expect to get $M_A$ through studying the distribution of the gradient orientations of the synchrotron intensities and the synchrotron polarization. We defer presenting the corresponding results to other publications.
Unlike the velocity and magnetic field fluctuations, the density fluctuations are not a direct tracer of MHD turbulence. At small sonic Mach number $M_s$, the densities act as a passive scalar and follow the general pattern of velocity fluctuations \citep{2003MNRAS.345..325C,2007ApJ...658..423K}. The proxies available from observations are the Intensity Gradients (IGs). If those are calculated using our recipes of block averaging for small $M_s$ they can also trace magnetic fields. At the same time, being sensitive to shocks, the IGs can be successfully combined with the velocity or magnetic gradient measures, i.e. the {\toreferee Velocity Centroid Gradients (VCGs, \citealt{GCL17,YL17a}), Velocity Channel Gradients (VChGs, \citealt{LY18a}), Synchrotron Intensity Gradients (SIGs, \citealt{Letal17}) or Synchrotron Polarization Gradients (SPGs, \citealt{LY18b})}, for tracing both magnetic fields and shocks (see YL17b).
As densities are sensitive to shocks we expect to observe that the distribution functions of IGs to be both functions of $M_s$ and $M_A$, the dependence on the former being dominant for high $M_s$. We do not discuss the properties of the distribution functions of IGs in this paper.
\section{Numerical approach}
\label{sec:numerics}
We use 10 numerical cubes for the current study. Table \ref{tab:sim} presents the list of the MHD compressible simulations {\toreferee , some of which} were already used in \cite{YL17b,LY18a} . The latter {\toreferee paper provides } the details of the 3D MHD simulations employed.
For our set of simulations, the sonic Mach number is kept approximately constant ($M_s \sim 5.5 - 7.3$) {\toreferee since we are more interested in studying the effect of Alfvenic Mach number $M_A$ on the distribution functions of gradient orientations.} The chosen $M_s$ are {\toreferee within the range of sonic Mach numbers relevant to molecular clouds (Zuckerman \& Palmer, 1974)} .Observationally, one can approximate the $M_s$ by either studying the amplitudes of the velocity gradients (see \citealt{YLL18}) or investigating the width of the density probability distribution function \citep{BL12}.
Figure \ref{fig_illu} provides a visualization of velocity structures in one of our cubes. As we mentioned earlier, the iso-contours of equal velocity are aligned with magnetic field which makes the velocity gradients to be directed perpendicular to magnetic field.
As we discussed earlier, the point-wise velocity information is not directly available. In this situation we calculate either velocity centroids or intensities within thin channel maps. Below we discuss both measures.
{\bf Velocity Centroids}. The normalized velocity centroid $Ce({\bf R})$ in the simplest case\footnote{Higher order centroids are considered in \cite{YL17b} and they have $v^n$, e.g. with $n=2$, in the expression of the centroid. Such centroids may have additional advantages being more focused on studying velocity fluctuations. However, for the sake of simplify we employ for the rest of the paper $n=1$.} and the emission intensity $I({\bf R})$ are defined as
\begin{equation}
\begin{aligned}
Ce({\bf R}) &=I^{-1} \int \rho_v({\bf R},v) v dv,\\
I({\bf R}) &= \int \rho_v({\bf R},v) dv,
\end{aligned}
\label{centroid}
\end{equation}
where $\rho_v$ density of the emitters in the Position-Position-Velocity (PPV) space, $v$ is the velocity component along the line of sight and ${\bf R}$ is the 2D vector in the pictorial plane. The integration is assumed to be over the entire range of $v$. The $Ce({\toreferee \bf R})$ is also an integral of the velocity and line of sight density, which follows from a simple transformation of variables (see \citealt{LE03}). For constant density this is just a velocity averaged over the line of sight.
{\bf Fluctuations within Channel Maps}. Another measure that we employ is the channel map intensity. It is the emission intensity integrated over a range of velocities, i.e.
\begin{equation}
I_{\delta v}({\bf R}) = \int_{\delta v} \rho_v({\bf R},v) dv,
\end{equation}
where $\delta v$ is the range of velocities for the integration.
{\bf The Thin Channel Condition} If $\delta v$ is less than the velocity dispersion for the eddies under study, for such eddies the velocity slices were termed "thin" in \citeauthor{LP00} (\citeyear{LP00}, henceforth LP00, see also \citealt{LY18a}). For such slices the intensity fluctuations arising from the eddies are mostly induced by the velocity fluctuations (LP00).
\section{Results}
\label{sec:result}
\subsection{ Determining $M_A$ with Distribution of gradient orientations}
\label{subsec:ma}
We investigate how the change of $M_A$ would alter the behavior of the distribution of gradient orientations. Distribution functions of the gradients for both centroids $Ce({\bf R})$ and velocity channel intensities $I_{\delta v} ({\bf R})$ are presented in Figure \ref{fig1}. {\toreferee These distribution functions are constructed by histograms with 1800 bins from three sets of numerical cubes with different $M_A$.} We fit the gradient orientation histogram using the Gaussian profile proposed in YL17a {\it by adding one constant shifting term}, i.e.
\begin{equation}
F=A\exp(-\alpha(\theta-\theta_0)^2)+C'.
\label{F}
\end{equation}
The value of the shift\footnote{In principle, the Gaussian profile with a shifting term is a better representation on the gradient orientation distribution in numerical simulations, since the {\toreferee histogram bin away from the peak} of gradient orientation distribution is usually much higher than zero. For instance, with $M_A\sim 0.2$ the velocity iso-contour axis-ratio can be in hundreds (Xu \& Lazarian 2018). However, simulations nowadays have limited resolutions. There is a natural tendency for velocity contours to have smaller axes ratio due to unresolvable minor axis. As a result, the {\toreferee histogram bin away} from the peak of gradient orientation distribution would not be close to zero. A constant shift would address the issue of finite length. In practice, the shift {\it will not} change the prediction of peak location by block-averaging.} $C'$and of the coefficient $A$ change with $M_A$ which provides a way to study Alfven Mach number using the top-base ratio $(C'+A)/C'$. The fitting lines for each panel are shown in red dash lines in Fig \ref{fig1}. One can observe that the width of Gaussian profile increases with respect to $M_A$, while the top-base ratio decreases with the increase of $M_A$. The calculations were performed using block averaging (see \citealt{YL17a}) choosing blocks large enough to make sure that the Gaussian fitting is sufficiently accurate. Performing similar calculations for different simulations from Table 1 we observe noticeable changes in the distribution of gradients.
The standard deviation of gradient orientation $\sigma_{GD}$ can be characterized by the circular statistics:
\begin{equation}
\begin{aligned}
\label{eq:gd}
\sigma_{GD} &= \sqrt{\log{\frac{1}{R^2}}}\\
&= \sqrt{-\log(\langle \cos \theta \rangle ^2+ \langle \sin \theta \rangle^2)}
\end{aligned}
\end{equation}
where $\theta$ is the gradient orientation of each pixel. Notice that in the formula, the range of $\theta \in [-\pi, \pi]$; but the observational gradient orientation in observational data (denoted as $\tilde{\theta}$) is in the range $[-\pi/2, \pi/2]$. In order to calculate the {\it inverted variance} $R=\sqrt{\langle \cos \theta \rangle ^2+ \langle \sin \theta \rangle^2}$, we perform $\theta = 2\tilde{\theta}$ when processing the observational data. The quantity $1-R$ is called the {\bf variance} in circular statistics, which provides an alternative measure of dispersion for a set of directional data, and we shall use this parameter in the following sections.
We would also evaluate the top-base ratio by dividing the peak value of the Gaussian peak from the base value in the gradient orientation histogram (See Fig. \ref{fig1}). We expect both of them would be related to the Alfvenic Mach number $M_A$ as we explained in \S \ref{sec:theory}. Fig. {\ref{fig1} illustrates pictorially how the fitting parameters for the distribution and the top-base ratio are obtained. Apparently, for low $M_A$ the distribution is strongly peaked and $A$ is large. This corresponds to good alignment of individual gradient vectors and the magnetic field direction. As $M_A$ increases, the gradient distribution is aligned with the magnetic field only in the statistical sense. This is especially obvious for $M_A>1$ as the velocity motions at large scales get uncorrelated with magnetic field.
The panels in Fig \ref{fig2} compare the dependences of the variances and top-base ratios obtained for both centroid and channel gradients. We observe that both these measures decrease as $M_A$ increases{\toreferee, which suggests that both top-to-base ratio and inverted variance are sensitive measures of $M_A$ in the range of $[0.2,1.7]$. }
In general, the findings for gradients in Fig \ref{fig1} and \ref{fig2} are easy to understand. They can be explained by a simple physical argument: When the magnetization is stronger (i.e. $M_A$ smaller) , gradients tend to be more aligned with each other, therefore the peak of the histogram is more prominent (higher top-base ratio) and narrow (lower $\sigma_{GD}$).
\begin{figure*}[t]
\centering
\includegraphics[width=0.96\textwidth]{New_3_plot.pdf}
\caption{\label{fig2} A comparison between the ways of measuring magnetization. From left to right: Top-base ratio method for channels (Left) and Centroid (Middle), and the Inverted Variance $R$ method for Centroid (Right). The three quantities are plotted against $M_A$. There is a similar decreasing trend for the three method, especially for $M_A<1$. }
\end{figure*}
To extract the power-law dependency of $1-R$ to $M_A$, we plot the $(1-R)$ {\toreferee vs} $M_A$ relation (see left panel of Fig \ref{fig3}). While there is a well-fit power law of $(1-R) \propto M_A^{0.14\pm0.03}$ , the fit is significantly flattened when $M_A \geq 1$.
Similarly, the top-base ratio also showed a two-segment behavior in Fig \ref{fig3}. The fit of the top-base ratio to $M_A<1$ shows a power law ratio of $\propto M_A^{-0.46\pm 0.18}$. The change of the power law index for $M_A>1$ is expected, as we have discussed earlier (see Appendix \ref{App_A}), the nature of turbulence is changing if the injection velocity gets larger than the Alfven speed. In this situation the large scale motions are dominated by hydro-type turbulence and the direction of magnetic field within the flow are significantly randomized. This changes the distribution function of gradient orientations.
\subsection{Comparison with the polarization percentage technique}
\label{subsec:4.2}
\begin{figure*}[tbhp]
\centering
\includegraphics[width=0.98\textwidth]{fig4xx.pdf}
\caption{\label{fig3} (Top Left) The top-base ratio plotted against $M_A$ in log-log space {\toreferee using the simulation set from Table \ref{tab:sim}.} (Top Right) A log-log scatter plot {\toreferee using the simulation set from Table \ref{tab:sim}} showing the variation of $1-R$ to $M_A$. We fit the scatter plot with two segments, bending at $M_A\sim 1$. (Lower Left) A figure similar to the top-left corner but the data points {\toreferee from Table \ref{tab:sim}} computed from velocity channels instead of velocity centroids. (Lower Right) A similar scatter plot with extra simulations in LY18 showing the general trend of $\sigma_{pol\%}$ to $M_A$ (See {\toreferee Appendix} \ref{app} {\toreferee \& \S \ref{subsec:4.2}.)} }
\end{figure*}
To illustrate the power of the techniques we showed in \S \ref{subsec:ma}, we would like to compare the results in \S \ref{subsec:ma} to methods applicable to polarimetry. In {\toreferee Appendix} \ref{app} we provide a new way of evaluating $M_A$ using the dispersion of polarization percentage $\sigma_{pol\%} $. While the traditional Chandrasekhar-Fermi technique uses the variations of {\it polarization directions} to determine $M_A$, the dispersion of polarization percentage does not require knowledge of circular statistics. In {\toreferee Appendix} \ref{app} we show that the analytical expectation that $\sigma_{pol\%}\sim M_A^2$ when $M_A<1$, which agrees well with our numerical calculations shown in the lower right of Fig \ref{fig3} obtained from 30 set of independent numerical simulations. For $M_A >1$ the dependence also changes similarly to the methods we suggested in \S \ref{subsec:ma}. For instance, the flattered fitting line with $M_A\in [1,2]$ has a proportionality of $\sigma_{pol\%} \propto M_A^{0.6}$. In comparison, for gradient dispersion, the power law is $(1-R)\propto M_A^{0.06}$.
\begin{table}[h]
\centering
\label{tab:simulationparameters2}
\begin{tabular}{c c c c c}
Set & Model & $M_s$ & $M_A$ & Resolution\\ \hline \hline
A & Ms0.2Ma0.02 & 0.2 & 0.02 & $480^3$\\
& Ms0.2Ma0.07 & 0.2 & 0.07 & $480^3$\\
& Ms0.2Ma0.2 & 0.2 & 0.2 & $480^3$\\
& Ms0.20Ma0.66 & 0.20 & 0.66 & $480^3$\\
& Ms0.2Ma2.0 & 0.2 & 2.0 & $480^3$\\
& Ms0.20Ma0.66 & 0.20 & 0.66 & $480^3$\\
& Ms0.02Ma0.2 & 0.02 & 0.2 & $480^3$\\
B & Ms0.4Ma0.04 & 0.41 & 0.04 & $480^3$ \\
& Ms0.8Ma0.08 & 0.92 & 0.09 & $480^3$ \\
& Ms1.6Ma0.16 & 1.95 & 0.18& $480^3$ \\
& Ms3.2Ma0.32 & 3.88 & 0.35 & $480^3$ \\
& Ms6.4Ma0.64 & 7.14 & 0.66 & $480^3$ \\
& Ms0.4Ma0.132 & 0.47 & 0.15 & $480^3$ \\
& Ms0.8Ma0.264 & 0.98 & 0.32 & $480^3$ \\
& Ms1.6Ma0.528 & 1.92 & 0.59& $480^3$ \\
& Ms0.4Ma0.4 & 0.48 & 0.48& $480^3$ \\
& Ms0.8Ma0.8 & 0.93 & 0.94 & $480^3$ \\
& Ms0.132Ma0.4 & 0.16 & 0.49 & $480^3$ \\
& Ms0.264Ma0.8 & 0.34 & 1.11 & $480^3$ \\
& Ms0.04Ma0.4 & 0.05 & 0.52 & $480^3$ \\
& Ms0.08Ma0.8 & 0.10 & 1.08 & $480^3$ \\
\begin{comment}
C& c1 & 5 & 0.2 & $792^3,1200^3,1600^3$\\
& c2 & 5 & 0.4 & $792^3$\\
& c3 & 5 & 0.6 & $792^3$\\
& c4 & 5 & 0.8 & $792^3$\\
& c5 & 5 & 1.0 & $792^3$\\
& c6 & 5 & 1.2 & $792^3$\\
& c7 & 5 & 1.4 & $792^3$\\
& c8 & 5 & 1.6 & $792^3$\\
& c9 & 5 & 1.8 & $792^3$\\
& c10 & 5 & 2.0 & $792^3,1200^3$\\
\end{comment}
D & d0 & 5.0 & 10.0 & $480^3$\\
& d1 & 0.1 & 10.0 & $480^3,640^3,1200^3$\\
E & e1 & 0.5 & 5.0 & $480^3$\\\hline \hline
\end{tabular}
\caption {Extra simulation used for production of the lower right panel of Fig. \ref{fig3}. The Set C referred in Fig. \ref{fig3} is exactly what we showed in Table \ref{tab:sim}. }
\end{table}
\subsection{Comparison with other techniques for obtaining magnetization}
A number of other techniques for studying magnetization have been suggested. For instance, Tsallis statistics (see \citealt{EL10}) was shown to be sensitive to the value of $M_A$. In addition, one can estimate for sub-Alfvenic turbulence that $M_A\sim \delta \phi$, where $\phi$ is the dispersion of magnetic field directions. These directions can be obtained either through polarimetry studies, e.g. dust polarimetry, or using the variation of the velocity or magnetic field gradient orientations.\footnote{Variations of the synchrotron intensity gradients and synchrotron polarization gradients can also be used. The advantage of the gradient techniques is that they are independent of Faraday rotation. }
{\toreferee To obtain $M_A$ through the Gaussian profile fitting requires that the conditions for the sub-block averaging in \cite{YL17a} are satisfied. In other words, the distribution of gradient orientation has to be well fitted by a Gaussian. Therefore the consistency of both directions (as traced by \cite{YL17a}) and magnetization from this work will be doubled checked through the sub-block averaging algorithm, i.e. whenever the Gaussian profile is not properly fitted there should not be any probe by VGT on neither magnetic field directions and magnetizations. While the velocity gradients present an independent technique for magnetic field tracing, it is synergetic with polarimetry measurements while dealing with the complex structure of interstellar magnetic fields.}
A more elaborated tool for studying $M_A$ is the analysis of the Correlation Function Anisotropy (CFA). This technique has been explored in a number of publications \citep{EL11,BL12,Betal14,Eetal15,KLP16,KLP17a}, and we illustrate its results in the Appendix \ref{subsec:cfa}.
The advantage of the CFA technique is the existence of the analytical description that relates the measurements not only to $M_A$ but to the properties of the fundamental MHD modes, i.e. Alfven, slow and fast modes \citep{LP12,KLP16,KLP17a,KLP17b} However, the CFA technique requires calculating correlation functions and therefore is less local compared to the technique in this paper. Further research should reveal the synergy between different techniques of determining $M_A$.
\subsection{Effects of adding noise}
\label{subsec:noise}
In this section we will show that the new technique of finding $M_A$ works in the presence of noise. We add white noises with amplitudes relative to the standard deviation of Velocity Centroid $\sigma_C$ and test how the power law is changed as a function of noise amplitudes. In our test we apply white noise with amplitudes $0.1 \sigma_C$ and $0.2 \sigma_C$ and compare our results with the original fit we have in Fig \ref{fig3}. For noise suppression, we employ the Gaussian smoothing of $\sigma=2$ pixel as proposed in \cite{Letal17}. According to \cite{Letal17}, the kernel size we picked here would preserve the most small scale structures while efficiently suppress the noise in the synthetic map globally. By adding the noise and also the smoothing kernel, we can then test whether in noisy observations we can still use the top-base ratio and variance methods to estimate magnetization.
Fig \ref{fig4} shows the log-log plot of both $(1-R)$ {\toreferee vs} $M_A$ and the top-base ratio to $M_A$ with noise addition (left) and smoothing (right). Before smoothing the noise-added $(1-R)$ {\toreferee vs} $M_A$ relation is very sensitive to the noise level, where only $0.2 \sigma_C$ can already flattened the plot. Fortunately, the application of the smoothing technique shows that the
magnetization tracing is not changed significantly. This gives us the confidence in applying our technique to observations.
\begin{figure*}[tbhp]
\centering
\includegraphics[width=0.96\textwidth]{New_Noise_plot.pdf}
\caption{\label{fig4} Four panels showing how noise and the suppression of it influences the result of magnetization tracing. For both methods of dispersion (top row) and top-base ratios (bottom row) we add white noise (Left column) to the map and see how the statistical parameters are altered. When the noise suppression techniques for gradients \cite{Letal17} are used, the trend of magnetization tracing (right column) is obviously more robust for both methods.}
\end{figure*}
\subsection{Observational data analysis}
\label{sec:observation}
We would like to demonstrate the application of the technique using GALFA-HI data. For details, please refer to the respective survey paper \citep{susantail}. The Galactic Arecibo L-band Feed Array HI (GALFA-HI) survey is an survey of the Galaxy in the $21 cm$ neutral hydrogen line. The data is obtained with the Arecibo Observatory 305 meter telescope. The telescope angular resolution $~4'$. The region covers the sky at right ascension (R.A.) across $263.5^\circ - 196.6^\circ$, and declination (Dec.) across $22.5^\circ - 35.3^\circ$. Also, this piece of data covers a wide range of galactic latitude from middle to high, which is $26.08^\circ - 83.71^\circ$. {\toreferee The same region has been used in \cite{2015PhRvL.115x1302C} for the Rolling Hough Transform analysis.}
As we mentioned earlier, the choice of the block size is an important step for analyzing the data with gradients. If the block size is too small, the Gaussian fitting is poor which means that the determination of gradients is not reliable (see YL17a). If the block size is too big, the map is excessively coarse. Moreover, at large scales the regular variations of the direction of the Galactic magnetic field get important. As a result, the dispersion of angles increases with the block size. These considerations in \cite{LY18a} were used to optimize the block size\footnote{More sophisticated procedures have been also tested. For instance, one can filter out gradients with the largest and smallest amplitudes. The former corresponds to gradients arising from shocks (YL17b), the latter corresponds to noise. Fitting Gaussians into the remaining data can improve the quality of data. We plan to explore this and other ways of improving the gradient studies elsewhere. In this paper we use the block averaging as it is presented in YL17a.} for the GALFA-HI map that we also use in this paper.
We decompose the GALFA-HI map into blocks of size {\toreferee $150^2$ pixels, in which one pixel is 4',} and compute both the dispersion and top-base ratio of each block after a smoothing Gaussian filter of 4 pixels. Fig \ref{fig:susantail} shows the result of the $M_A$ distribution that we obtained with analyzing the distributions of gradient orientations. We clearly see similarity of the $M_A$ map {\toreferee obtained using } the top-base method and that obtained with the dispersion (in terms of the variance $1-R$) method. The fact that the two techniques provide very similar output, increases our confidence fin the distribution of $M_A$ that we obtain.
In the top-base method, we marked the pixels as white (NaN), indicating that due to noise the gradient angle distribution could not be reliably fitted by a Gaussian profile. For such points the statistical variance was computed\footnote{\toreferee In principle, even the distribution of gradient orientations does not follow Gaussian, we can still compute the statistical circular dispersions (and thus inverted variance) according to Eq. \ref{eq:gd}. However whether the values computed through Eq. \ref{eq:gd} has any meaning in the framework of VGT is yet to be investigated.} using Eq. \ref{eq:gd}. This study is the first of this sort and has illustrative purposes. The accuracy of this approach for the regions where the Gaussian fitting is not working well will be studied elsewhere. {\toreferee For consistency to the Gaussian fitting requirement in \cite{YL17a}, we do not include the NaN data in the top-base method in the histogram in Fig \ref{fig:susantail2}.}
\begin{figure*}[tbhp]
\centering
\includegraphics[width=0.98\textwidth]{fig7.pdf}
\caption{\label{fig:susantail}
{\toreferee{\it Top Panel}}.The prediction of $M_A$ from the $P_\%$ method. {\torefereetwo As a matter of fact (e.g. \citealt{L07}),} the dust grain alignment drops significantly in low-latitude {\toreferee (approximately on the right-hand-side)} region, {\torefereetwo which explains why the expected $M_A$ in higher latitude will be significantly higher. Similar effect on polarization percentage is also observed in \cite{Planck1519}. }
{\toreferee \it Middle Panel}. $M_A$ distribution obtained using top-base ratio approach from this paper on the region of GALFA-Hi with block size of $150^2$ pixels. The data was from \cite{2015PhRvL.115x1302C} and \cite{susantail}.
The white pixels are the regions that cannot be fitted by the Gaussian profile. {\toreferee \it Bottom Panel} The distribution of $M_A$ obtained using the method of variance. There is a similarity of the $M_A$ predictions between the {\torefereetwo top-base ratio and the variance method}.}
\end{figure*}
{\toreferee We also test} how the selection of block size changes our estimate of $M_A$. Fig \ref{fig:susantail2} shows the normalized distributions of $M_A$ of the piece of GALFA region (Fig \ref{fig:susantail}) when we vary the block size of calculating the top-base ratio. While there is a fluctuation of magnetization estimation for different block size, both the peak value and the shape of the distributions are very similar. One can observe that the peak values of these distributions are all around $M_A\sim 0.75$, and exhibit a rough Gaussian profile. This suggests that the block size does not significantly change the estimate of $M_A$. However, we expect that the regular curvature of magnetic field lines within the block{\toreferee, which induced a broaden distribution of gradient orientation,} is a factor that makes us {\toreferee difficult in estimating $M_A$} in high latitude HI (see also lower $M_A$ obtained with for the same data with the gradient amplitude method in \cite{YLL18}). The issue of obtaining the accurate values of $M_A$ {\toreferee in regions with regular curvatures} definitely requires further detailed studies.
\begin{figure*}[tbhp]
\centering
\includegraphics[width=0.98\textwidth]{1xx.pdf}
\caption{\label{fig:susantail2} The estimated $M_A$ for Fig \ref{fig:susantail} using the top-base ratio method in different block sizes. The peak ($M_A\sim 0.75$) remains unchanged even through the sampling area increases for about $2.5^2\sim 6$ times.}
\end{figure*}
\section{Obtaining magnetic field intensity distribution from the $M_A$ distribution}
\label{intensity}
The Alfven Mach number $M_A$ is an essential characteristic of magnetized turbulent media that is important for describing key astrophysical processes, e.g. an {\toreferee explicit} discussion of the dependence of cosmic ray propagation in \cite{LY14, L16} and heat conduction in \cite{L06}.
{\toreferee In principle, if one can obtain accurate polarization measurements with sufficiently high dust grain alignment efficiency (See Lazarian 2007), then the method of gradients has no particular advantage in front of polarization measurements. However, in observational environments the polarization fraction is fairly low (e.g. in case of the polarization measurements near molecular cloud centers), which will often lead to insufficient statistics in computing the dispersion of polarization angles. Comparatively, the VGT is versatile in different physical regimes \citep{YL17b,LY18a} and thus the estimation of magnetic field informations (directions and more importantly magnetization) can be applied to a wider regime without any statistical problems we faced in polarization measurements.}
In addition, if the Alfven Mach number is known, it is possible to obtain magnetic field strength from the relation $M_A=\frac{\delta B}{B}$. Assuming that the relation of $\delta B$ and the dispersion of the velocity obeys the Alfvenic relation, i.e. $\delta V=\delta B/(4\pi \rho)^{1/2}$ one can express the magnetic field strength as
\begin{equation}
B_{POS}=\sqrt{4\pi \rho} \frac{\delta V}{M_A}
\label{BB}
\end{equation}
where $\delta V$ can be associated with the velocity dispersion and we took into account that it is the plane of the sky (POS) component of magnetic field that is being explored with the technique. As only line of sight (LOS) velocity $\delta v_{los}$ is available, for the practical use of the Eq. (\ref{BB}) the velocity dispersion $\delta V$ there should be associated with $\delta v_{los}$, i.e. $\delta V=C\delta v_{los}$ , where $C$ is a coefficient that relates the dispersions of the turbulence POS velocities with the available LOS ones. For an uniformly distributed Alfven wave that moves along the mean magnetic field line, $C=0.5$. The coefficient $C$ grows up when the angle between the mean magnetic field direction and the line of sight is smaller. In the limiting case of magnetic field parallel to the line of sight, $C$ is not defined, as no LOC velocities can be measured.
In reality, the observed magnetic field is not simply composed of a dominant mean field with straight-line morphology and infinitesimal Alfven waves moving along the mean field. For realistic turbulence the magnetic field wandering is significant (see LV99, \citealt{ELV11}). Moreover, MHD turbulence consists of three MHD cascade modes, namely the Alfven, slow and fast modes which every mode has unique cascade properties (see \citealt{LG01,CL02}). As a result, in most practical studies the coefficient $C$ is treated as an empirically given parameter.\footnote{Analytical studies in \cite{KLP16} open a way to calculate $C$ from the first principals for the given level of turbulence, inclination angle of the mean magnetic field and the line of sight and the assumed composition of fundamental MHD modes. Incidentally, an additional modification of Eq. (\ref{BB}) is required to account for {\toreferee the} other types of modes {\toreferee that are also present} in the magnetized turbulent flow.}
In addition, the calculations are a bit more complicated for the case of small scale turbulence injection (see \citealt{YC14}). We discuss the details of obtaining magnetic field intensity from observations elsewhere. For the time being, it is important that Eq. (\ref{BB}) provides a rough guidance for exploring magnetic field strength.
\section{Discussion}
\label{sec:discussion}
\subsection{{\toreferee Broader} applications to other types of gradients}
The cornerstone of the present work is the Gaussian fitting function first used in \cite{YL17a}. This provided us with the ways of obtaining the dispersions and the top-based ratios that are used to trace $M_A$. While we did all the estimations of $M_A$ for velocity centroids {\toreferee in the present paper}, we expect the technique presented in the current work will be applicable to other measures, e.g, reduced centroid gradients from \cite{LY18a}, synchrotron intensity gradients from \cite{Letal17}, synchrotron polarization gradients and synchrotron derivative polarization gradients from \cite{LY18b}, and IGs from \cite{YL17b}. {\toreferee More importantly, we showed in Fig \ref{fig1} that in dispersion of velocity channel gradients also has the same trend as the centroid gradients, which signifies the dispersive quantities we introduced in the current work should also work on other gradient measures. }
If turbulent velocity broadening is known, then, as we discussed \S \ref{intensity}, the magnetic field strength can be obtained. However, by itself $M_A$ is a key parameter describing astrophysical turbulence and therefore its determination of its distribution is essential for many astrophysical processes.
\subsection{The improved techniques of gradient calculation and use of interferometers}
By now we have obtained a set of tools developed in the Velocity Gradient Technique \citep{GCL17,YL17a,YL17b,LY18a}, e.g. error estimations, wavenumber filtering, identifying shocks and gravitationally-bound objects etc. Some of these tools have been already applied to the magnetization tracing technique discussed in the present paper. In particular, the noise suppression technique proposed in \citep{Letal17} is shown to be effective in maintaining the established power-law relation in Fig \ref{fig4}. Some of these tools we apply as the technique matures. For instance, in \cite{LY18a} we demonstrated how one can improve the tracing of magnetic field with velocity gradients. This approach based on the moving window technique, seems promising for magnetization studies.
To get higher resolution maps of magnetization it is advantageous to use interferometers. We have shown in our previous publications (e.g. \cite{Letal17}) that to trace magnetic field with gradients one can successfully use interferometers. It is important to understand that the interferometric data can be used directly and it is not required to have all spatial freuencies to use our gradient technique. Indeed, the gradients are calculated for smallest separations and, in fact, filtering of low spatial frequencies is recommended for increasing signal to noise ratio. Therefore it is not surprising that the structure of magnetic field can be reproduced e.g. without employing single dish observations. It is easy to see that the same statement is true for using the techniques presented in the present paper. This opens exciting perspective for of the mapping the distribution of $M_A$ for external galaxies.
\subsection{Studying 3D magnetization}
Velocity gradients provide a way to probe turbulence and establish the distribution of magnetic field directions in 3D. For instance, this can be achieved by using the galactic rotation curve for atomic hydrogen of the Milky Way. For molecular clouds different transitions from multi-molecular species (e.g. 12CO, 13CO) can provide information of magnetic field structures at different depths {\toreferee suggesting} that not only the directions, but also 3D magnetization can be traced by velocity gradients {\toreferee through stacking the magnetic field maps from molecular tracers with different optical depth}. Similar idea of exploring the 3D field tomography through velocity gradients with rotation curve is explored in \cite{CL18}. Together with the CF-method, the distribution of 3D magnetic field strength can also be acquired (see \S \ref{intensity}).
MHD turbulence imprints its properties on the statistics of synchrotron intensity/polarization gradients as well as on the gradients of emission intensity. In terms of utilizing magnetic fluctuations, the sister techniques are available. They are the Synchrotron Intensity Gradient (SIG) technique and \citep{Letal17} and the Synchrotron Polarization Gradient (SPG) technique \citep{LY18b}. These techniques are demonstrated to trace the magnetic field {\toreferee in the plane of sky}. In particular, the synchrotron polarization gradients \citep{LY18b} can recover the 3D distribution of magnetic field directions by using the effect of Faraday depolarization. Our present results {\toreferee of tracing $M_A$} with the width and the top-base ratio for the gradient distribution are also applicable to the techniques using synchrotron. In terms of the SPG technique, this provides another way of studying 3D magnetization. Combining the results on magnetic directions and magnetization obtained with the gradients of synchrotron and spectral lines, one can get an unprecedented insight into the magnetic structure of the multi-phase ISM.
Gradients of intensities of gas or dust emission\footnote{\toreferee By intensity gradients we understand the measures calculated using the block averaging procedure described in \cite{YL17a} and the resulting Intensity Gradient Technique (IGT) is different from the Histograms of Relative Orientation technique in Soler et al. (2013). A comparison between the two techniques is provided in our earlier publications on gradients as well as in our forthcoming paper containing a very detailed quantitative comparison of the IGT and the HRO.} provide additional information about the ISM. For instance, the gradients of intensities are strongly affected by shocks (see \citealt{YL17b}). This opens interesting prospects of studying shocks and sonic Mach numbers by comparing the {\toreferee distribution of gradient orientations} of velocities and intensities. We shall explore this possibility elsewhere.
\subsection{Calculating the distribution of expected dust polarization}
Magnetic field tracing is routinely done with dust polarization. It is frequently assumed that the observed polarization represents projected magnetic field weighted by gas density. This is approximately true when two conditions are simultaneously satisfied: (1) A {\toreferee relatively high} grain alignment {\torefereetwo (greater than a few percent)}; (2) magnetic fields {\toreferee in the plane of sky} does not change significantly along the line of sight. While the former is mostly satisfied in the diffuse interstellar media (see \citealt{L07} for a review), the latter is a more subtle requirement. The problem arises from the fact that the additions of the polarization and magnetic field follow different laws, which is usually under-appreciated by observers. Polarization summation is a summation of quadrupole quantities, which is different from the summation of vectors of magnetic field. The difference become obvious when the direction of magnetic field lines are oscillating along the line of sight.
Velocity gradients are linear vector quantities similar to magnetic fields. However, the addition of the gradients along the the line of sight is happening in the random walk fashion due to the absence of the {\toreferee{\it direction}(i.e. the vector-head)} of gradients {\toreferee by} the symmetry of the anisotropic eddies. The addition method of velocity gradients is not only different from that of magnetic field, but also for polarization. We feel that gradients {\toreferee can} present a better representation of the projected magnetic field for the case of super-Alfvenic turbulence, i.e. when magnetic field directions are changing strongly along the line of sight, as both gradients and magnetic field tend to cancel {\toreferee themselves out respectively.}
The issue of the comparison of the gradients versus polarization as a representation of magnetic field is an issue of our separate paper (Yuen \& Lazarian, in prep). Instead we will make remarks about the sub-Alfvenic case where the differences in the magnetic field direction along the line of sight is limited. In this case the variations in the measured degree of polarization reflect the variations magnetic field directions. These variations are also reflected by the dispersion of gradients that we deal in this paper. Therefore measuring the aforementioned dispersion one can predict the degree of polarization expected in the given direction. In other words, at least for sub-Alfvenic turbulence velocity gradients {\toreferee can} predict both the direction of polarized radiation and the degree of polarization. This finding is important for CMB polarization foreground studies where it is good to have an independent way of finding the expected polarization.
\subsection{Comparison with other works}
Our present study reveals a new valuable feature of gradients, i.e. their ability to deliver the value of magnetization through studies of the distribution of their directions within the block over which the averaging is performed. The theory of MHD turbulence predicts a distribution of wavevectors that depends on the ratio of the ratio of the turbulent to magnetic energies $\sim M_A^2$. Therefore it is only logical that we find that the distribution of the gradient orientations also depends on $M_A$.
This provides another way of using the information obtained with the gradient technique in order to study properties of magnetized interstellar medium. Our earlier studies (e.g. \citealt{YL17a,LY18a}) provided ways of magnetic field tracing with the block-averaged velocity gradient orientations as well as of obtaining the sonic Mach number $M_s$ via studying the gradient amplitudes \citep{YLL18}. Therefore combined with the results of the present paper, the study of velocity gradients can provide both $M_A$, $M_S$ as well as magnetic field direction.\footnote{We would like to stress that similar results can be obtained with synchrotron intensity and synchrotron polarization gradients. techniques. } These quantities can be studied not only in 2D, but also in 3D. This brings magnetic field studies to a new level.
We would like to emphasize that what we suggest in the present paper should not be confused with the studies of Histograms of Relative Orientation (HRO) technique in \cite{Setal13} and subsequent works (e.g. \citealt{Planck35}). In this paper we discuss the distribution of orientation of velocity gradient orientations around the block-averaged \citep{YL17a} direction of the velocity gradient. The block averaged direction in our technique determines the magnetic field orientation. On the contrary, HRO (a) it is not capable to trace magnetic field direction, it relies on polarimetry to do this, (b) it uses density gradients, not the velocity gradients, (c) it is not capable of revealing $M_A$ and $M_s$. The value of HRO is obtaining the statistical correlation of the averaged density gradient orientation and magnetic field as a function of the column density. All in all, HRO is a different technique introduced with a different purpose. As we mentioned earlier, our approaches of block averaging are applicable to density gradients and this is the basis of our intensity gradients technique (IGT). In terms of its comparison with the HRO it does not have the difference given by item (b), while in terms item (c), our research shows that apart from being sensitive to $M_A$ and $M_s$ the IGT can identify shocks. The use of the IGT and velocity gradients is synergistic as it helps to reveal the regions of gravitational collapse \citep{YL17b, LY18a}.
\section{Summary}
\label{sec:summary}
The present paper {\toreferee establishes} a new way to study the ISM magnetization. We characterize the magnetization by the Alfven Mach number $M_A$, the value of which is important for solving many astrophysical problems. To find $M_A$ we use the properties of the distribution of the gradient orientations, namely, the velocity gradient dispersion and the top-base ratio obtained through the Gaussian fitting of the distribution \citep{YL17a}. To summarize:
\begin{enumerate}
\item We establish the power-law relations between the statistical parameters of the distribution of gradient orientations, i.e. the variance $1-R$ and the Top-Base ratio, to the Alfvenic Mach number $M_A$ (\S \ref{sec:result}).
\item We discuss a possibility of using the galactic rotation curve and different spectral lines to get the 3D map of the $M_A$ distribution.
\item We show that combining $M_A$ with the dispersion of the Doppler-broadened spectral line , one can acquire the magnetic field strength {\it without using the polarimetry}.
\item Our method is consistent with the method of correlation function anisotropy (CFA) in tracing $M_A$ (\S \ref{subsec:cfa}). We show that the gradient technique can provide maps of $M_A$ with higher resolution.
\item We show that our technique of $M_A$ tracing is a robust tool in the presence of noise (\S \ref{subsec:noise}).
\item We applied our technique to HI observational data (\S \ref{sec:observation}) to obtain the distribution of $M_A$ over an extensive region of the sky.
\item Our approach for finding $M_A$ using the dispersion of gradient distribution is applicable not only to velocity gradients, but also to magnetic gradients that are measured with synchrotron intensity gradients and synchrotron polarization gradients.
\end{enumerate}
{\bf Acknowledgements.}
We thank Susan Clark for providing us with the HI data and friendly discussions. This publication utilizes data from Galactic ALFA HI (GALFA HI) survey data set obtained with the Arecibo L-band Feed Array (ALFA) on the Arecibo 305m telescope. The Arecibo Observatory is operated by SRI International under a cooperative agreement with the National Science Foundation (AST-1100968), and in alliance with Ana G. Méndez-Universidad Metropolitana, and the Universities Space Research Association. The GALFA HI surveys have been funded by the NSF through grants to Columbia University, the University of Wisconsin, and the University of California. A.L. acknowledges the support from grant NSF DMS 1622353 and ACI 1713782. {\torefereetwo We thank the anonymous referee for providing extensive important comments and suggestions on our work.}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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\section{Introduction}
As once noted by Ya. G. Sinai, localization is a game of
resonances. When restrictions to boxes that are not too far away from
each other have eigenvalues
that are too close, small denominators are created, making proofs of
localization always challenging and proofs of delocalization
occasionally possible (e.g. \cite{g,as,js,aw,jk,jl2}). Two types of
resonances have played a special role in the spectral theory of
quasiperiodic operators. Frequency resonances were first exploited in
\cite{as} based on \cite{g} to prove absence of eigenvalues (and
therefore singular continuous spetrum in the hyperbolic regime) for
quasiperiodic operators with Liouville frequencies. Phase resonances
were discovered in \cite{js} and used to prove absence of
eigenvalues for quasiperiodic operators with even potentials and
(arithmetically defined) generic frequencies. It was conjectured in
\cite{jcongr1} that for the almost Mathieu family- the
prototypical quasiperiodic operator - the two above types of
resonances are the only ones that appear and the competition between
the Lyapunov growth and combined exponential resonance strength resolves in
a sharp way. This was so far proved for single-type-resonances only:
for pure frequency resonances (that is for so-called
$\alpha$-Diophantine phases for which there are no exponential phase resonances) in \cite{ayz,jl1} and for pure phase
resonances (that is for Diophantine frequencies for which there are no
exponential frequency resonances) in \cite{jl2}. Papers \cite{jl1,jl2}
required developing sharp techniques for dealing with correspondingly
phase and frequency resonances. Both can survive, on the localization
side, adding weak resonances of the other type, but so far without the
desired sharpness. In this paper we succeed for the first
time in
dealing in a sharp way with two different types of resonances.
Our operators come not from the almost Mathieu, but from another popular
family, the Maryland model, a family of quasi-periodic Schr\"odinger operators
\begin{equation}\label{marylandmodel}
(H_{\lambda,\alpha,\theta}u)_n=u_{n+1}+u_{n-1}+\lambda\tan(\pi(\theta+n\alpha)u_n),
\end{equation}
where $\lambda>0$ is the coupling constant, irrational $\alpha\in {\mathbb T}=[0,1]$ is the frequency and $\theta\in {\mathbb T}$ is the phase.
We assume $\theta \notin \Theta=\{\frac{1}{2}+\alpha {\mathbb Z}+{\mathbb Z}\}$.
Maryland model was originally proposed by Grempel,
Fishman and Prange \cite{fgp} as a linear version of the quantum
kicked rotor and has attracted continuing interest from the physics community, see e.g. \cite{berry, fishman2010, GKDS2014}, since it serves as
an exactly solvable example of the family of incommensurate models.
Frequency resonances are ubiquitous for all quasiperiodic potentials,
while phase resonances discussed above exist only for even sampling
functions, and thus not for the Maryland model. Indeed, as a result, for
Diophantine (i.e. non-resonant) frequencies it has localization for
{\it all} phases \cite{simm, JYMaryland}. However, it does have barriers, when
trajectory of a given phase approaches the singularity too
early. Barriers can compensate for the resonances, and therefore serve
as what we call here {\it anti-resonances}. They are precisely the
reason why for the Maryland model there are phases with localization
even for the most Liouville frequencies \cite{maryland}. Thus
Maryland model features a combination of frequency resonances and
phase anti-resonances. Our main achievement here is the development of
a precise
understanding of barriers as phase anti-resonances, of how the competition
between them and frequency resonances unfolds, as well as sharp analysis
of eigenfunction decay in presence of this combination.
B. Simon called the Maryland model a useful laboratory \cite{simm} because it is
exactly solvable in some sense, and thus can serve as a source of both general conjectures and counterexamples. It has explicit expression for the
Lyapunov exponent, integrated density of states, and even (somewhat implicit) for
the eigenvalues and eigenfunctions. Thus certain features admit a more
direct analysis. Indeed it admits a very beautiful
trick noticed originally in \cite{fgp}: under the combination of Cayley and Fourier transforms it
leads to an explicit cohomological equation, and the analysis
becomes similar to that of Sarnak \cite{s}. \footnote{ \cite{s} was the paper where
the importance of the arithmetics in this type of spectral questions was
proposed even before \cite{as}}. Utilizing the Cayley transform, the
spectral decomposition for the Maryland model was determined fully,
for all $\alpha,\theta,$ in \cite{maryland}, making it the first - and
so far the only - model with spectral transitions
where this could be claimed. Previously, localization up to the sharp
threshold was proved for a.e. $\theta$ in \cite{simm}. The extension
of the analysis from a.e $\theta$ in \cite{simm} to {\it all} $\theta$ in
\cite{maryland} required accounting the effect of the barriers, and Cayley
transform allowed to do it albeit in a highly implicit way.
Namely, let $p_n/q_n$ be the continued fraction approximants of
$\alpha$. The index $\beta(\alpha)$ that measures exponential strength
of the frequency resonances is defined as follows:
\begin{align}\label{def:beta}
\beta(\alpha)=\limsup_{n\to\infty}\frac{\ln q_{n+1}}{q_n}= \limsup_{n\to\infty}\frac{-\ln \|n\alpha\|}{n},
\end{align}
where $\|x\|_{{\mathbb T}}:=\text{dist} (x, {\mathbb Z})$.
A new index, $\delta(\alpha,\theta)$ was introduced in \cite{maryland} as
\begin{align}\label{def:delta}
\delta(\alpha,\theta):=\limsup_{n\to \infty}\frac{\ln q_{n+1}+\ln
\|q_n(\theta-\frac{1}{2})\|_{{\mathbb T}}}{q_n}.
\end{align}
With the Lyapunov exponent $L_\lambda(E)$ explicitly defined by
\eqref{LE} (and not dependent on $\alpha,\theta$), it was proved in \cite{maryland} that
\begin{thm}
$H_{\lambda,\alpha,\theta}$ has purely singular continuous spectrum on $\{E: L_{\lambda}(E)<\delta(\alpha,\theta)\}$, and pure point spectrum on $\{E: L_{\lambda}(E)>\delta(\alpha,\theta)\}$.
\end{thm}
Thus Maryland model has sharp spectral transition defined by the
interplay between the Lyapunov exponent and $\delta(\alpha,\theta)$.
The index $\delta(\alpha,\theta)$ appeared naturally in the context of the cohomological
equation arising as a result of the Cayley and Fourier transforms. It is clear that $\delta(\alpha,\theta)\leq \beta(\alpha)$.
Indeed $\delta(\alpha,\theta)=\beta(\alpha)$ holds for a.e. (however
not all) $\theta$.
Here we show another representation for $\delta(\alpha,\theta), $ see
Corollary \ref{cor:alternate_delta},
\begin{align}\label{def:another}
\delta(\alpha,\theta)=\limsup_{n\to \infty}\frac{\max(0,\ln q_{n+1}+\displaystyle\min_{k=0,\ldots,q_n-1}\ln
\|\theta-\frac{1}{2}+k\alpha\|_{{\mathbb T}})}{q_n}.
\end{align}
Therefore, $\delta(\alpha,\theta)$ can be interpreted as the
exponential strength of
frequency resonances, $\beta(\alpha)$, tamed by the
phase anti-resonances, defined as the positions of exponential near-zeros of the $\cos(\pi(\theta+k\alpha)).$
As mentioned, the proof of \cite{maryland} as well as all the ones prior to it, including
the original physics paper \cite{fgp}, have been based on a Cayley
transform and therefore indirect. Moreover, the eigenfunctions of the
Maryland model are, as a result of indirect analysis, known exactly,
yet the formulas don't allow for easy conclusions about their
behavior, which is expected to be quite interesting, with transfer matrices
satisfying certain exact renormalization \cite{fs}. Also, Maryland eigenfunctions
are expected, through numerics, to have hierarchical structure driven
by the continued fraction expansion of the frequency. In \cite{JYMaryland} two
of the authors
developed a Green's function based approach to localization for this
model, and used it to obtain Anderson localization for all $\theta$ and
Diophantine $\alpha$ or, in other words, for the case
$\beta(\alpha)=0$, that is in absence of frequency resonances. In this
paper we fully handle the difficult resonant case, where the anti-resonances
also start playing a crucial role. Our main conclusion is
\begin{thm}
\label{main0}
For any $\alpha\in {\mathbb R}\setminus {\mathbb Q}$ and any $\theta,$
the spectrum on $\{E:L_{\lambda}(E)\ge\delta(\alpha, \theta)\}$ is
pure point and for any eigenvalue
$E\in \{L_{\lambda}(E)>\delta(\alpha, \theta)\}$ and any $\epsilon>0,$
the corresponding
eigenfunction $\phi_E$ satisfies $|\phi_E(k)|< e^{-
(L_{\lambda}(E)-\delta(\alpha, \theta)-\epsilon)|k|}$ for
sufficiently large $|k|.$
\end{thm}
\begin{rem} We make a few remarks:
\begin{enumerate}
\item It is known that $\sigma_{pp}(H_{\lambda,\alpha,\theta})=
\{E:L_{\lambda}(E)\ge\delta(\alpha, \theta)\}$ \cite{maryland}.
We include the statement on pure point
spectrum on $ \{E:L_{\lambda}(E)\ge\delta(\alpha, \theta)\}$
only to emphasize that it also
follows independently from our approach.
\item In fact, we prove a much more precise local statement at
each scale, see
Lemma \ref{lem:main}.
\item We therefore obtain sharp bounds on the decay of all
eigenfunctions except for at most two values of $E$ where
$L(E)= \delta(\alpha, \theta),$ which may or may not be eigenvalues \cite{maryland}.
\end{enumerate}
\end{rem}
Our main achievement however is the {\it approach} we develop here to treat
the ``resonance tamed by an anti-resonance'' situation. This paper is the first one in a
series of at least two as it paves the way to study full asymptotics
of the eigenfunctions. The result of Theorem \ref{main0} provides the
sharp upper envelope, but does not otherwise give insight into the
fine behavior of the eigenfunctions. However, we develop here the key
tools for such study, and the latter will be presented in the follow-up
work \cite{HJY}. In fact, some of our technical statements are
more detailed than needed for the purpose of our main results, because
we want to create the foundation for what will follow
in \cite{HJY}. Moreover, we expect this to lead to universal hierarchical
structure in the behavior of the eigenfunctions, identical to the one
discovered in \cite{jl1} for the almost Mathieu operator in case of
absence of the anti-resonances (i.e. $\delta(\alpha, \theta)=\beta(\alpha)$), but a lot
more rich and complex in presence of the anti-resonances. We also
comment that Fourier transforms of the eigenfunctions represent
functions with natural boundaries on both circles bounding the annulus
of analiticity \cite{simm}, and our analysis promises to provide
various insights on their boundary behavior which is expected to be universal.
Moreover, while certain arguments we present depend on some specific
aspects of the Maryland model, the crucial part of the proof:
sharp analysis of the effect of anti-resonances, is actually quite robust, and we
expect it to be useful in the study of other one-frequency
quasiperiodic models with
unbounded potentials that have attracted attention recently
\cite{ilya1,ilya2,ilya3,gps} . Additionally, in the models that lead to
singular Jacobi matrices (that is where
the off-diagonal terms can approach zero) positions of off-diagonal
exponential near-zeros also compensate for resonant small divisors,
thus creating
effective anti-resonances. Such models have appeared lately in the
study of various graphene-type structures (e.g. \cite{graphene, AA}),
and in presence of certain anisotropy they also lead to models with
hyperbolicity where
frequency resonances coexist with phase anti-resonances. We expect
many parts of our
method
to be applicable to all those scenarios.
In particular, a popular model, also originating in physics, is the
extended Harper's model (EHM), introduced by Thouless \cite{thou}. It is a
family indexed by five parameters, given by $$
(H^{\mathrm{EHM}}_{\lambda, \alpha,\theta}u)_n=c_{\lambda}(\theta+n\alpha)u_{n+1}+\overline{c_{\lambda}(\theta+(n-1)\alpha)}u_{n-1}+2\cos{2\pi(\theta+n\alpha)}u_{n},
$$
where $\lambda =(\lambda_1,\lambda_2,\lambda-3),\;c_{\lambda}(\theta)=\lambda_1 e^{-2\pi i(\theta+\frac{\alpha}{2})}+\lambda_2+\lambda_3 e^{2\pi i(\theta+\frac{\alpha}{2})}$.
See \cite{JMreview} for a 2017 review and \cite{dryten,AJM,nopoint,HJ,HYZ,Xin} for more recent results.
Extended Harper's model has a range of parameters where the
Lyapunov exponent is positive on the spectrum, while the corresponding
Jacobi matrix is singular, thus again becoming a fertile ground for
resonance/anti-resonance analysis. However, in this case there is an
additional feature: phase resonances, thus the situation is even more complicated.
In fact, our analysis prompts us to formulate the following conjecture.
Let $L(\lambda)$ be the Lyapunov exponent of the EHM on the spectrum (it
is known exactly and only depends on $\lambda$, see \cite{Jm}).
Let the exponent $\gamma(\alpha,\theta)$ be defined by
$$\gamma(\alpha,\theta):=\limsup -\frac{\ln \|2\theta+n\alpha\|}{|n|}.$$
It characterizes the exponential strength of phase resonances.
Let $\mathcal{R}_1:=\{\lambda: 0<\max(\lambda_1+\lambda_3, \lambda_2)<1\}$ be the positive Lyapunov exponent regime of the EHM, and
$\mathcal{R}_s:=\{\lambda: \lambda_1=\lambda_3\geq
\frac{\lambda_2}{2}, \text{ or } \lambda_1+\lambda_3=\lambda_2\}$ be
the singular regime: that is where $c_{\lambda}(\theta)=0$ for some $\theta\in {\mathbb T}$.
\begin{conj}
For $\lambda \in \mathcal{R}_1\cap \mathcal{R}_s$, $H^{\mathrm{EHM}}_{\lambda, \alpha, \theta}$ has purely singular continuous spectrum if $L(\lambda)<\tilde{\delta}(\alpha,\theta)+\gamma(\alpha,\theta)$, and pure point spectrum if $L(\lambda)>\tilde{\delta}(\alpha,\theta)+\gamma(\alpha,\theta)$, where
\begin{align}\label{def:tildedet}
\tilde{\delta}(\alpha,\theta)=\limsup_{n\to \infty} \frac{\ln
q_{n+1}+\sum_{\theta ':c_{\lambda}(\theta ')=0}\ln \|q_n(\theta-\theta
')\|}{q_n},
\end{align}
where zeros are counted with multiplicities.
We note that $c_{\lambda}$ has either one or two (possibly coinciding)
zeros in the
indicated regime.
.
\end{conj}
\begin{rem}
The index $\tilde{\delta}$ index was introduced to account for
anti-resonances in the proofs of {\it singular continuous spectrum}
in \cite{JYsingular, HJ} for, correspondingly, general
operators with unbounded potentials/singular Jacobi matrices (for
unbounded potentials the sum is instead over the singularities).
In particular purely singular continuous spectrum was proved for the extended Harper's model whenever $L(\lambda)<\tilde{\delta}(\alpha,\theta)$ \cite{HJ}.
\end{rem}
\begin{rem}
As we were finalizing this paper we learned of a preprint \cite{liu} by Liu, where he proved localization for
the almost Mathieu operator with
completely resonant phases (when $2\theta\in {\mathbb Z}\alpha$, hence
$\beta(\alpha)=\gamma(\alpha,\theta)$) up to the conjectured
threshold $\{\ln \lambda>2\beta(\alpha)\}$, which improves on earlier results \cite{liuyuan2,liu2}. It is another remarkable
case of sharp analysis in a situation of two coexisting types of
resonances: in that case, phase and frequency.
It is interesting to see whether the techniques of \cite{liu} can be
combined with ours to prove the corresponding result for completely
resonant phases for the extended Harper's
model, thus localization for
$L>\tilde{\delta}(\alpha,\theta)+\beta(\alpha)$. Here, however, the analysis
would require studying the interaction of {\it three} coexisting types of
resonances.
\end{rem}
Finally, our proof is local, thus potentially allowing also for the
analysis of the behavior of generalized eigenfunctions corresponding
to the singular continuous spectrum regime.
We now briefly comment on our argument. Proofs of arithmetic
localization in the spirit of \cite{j0, j, jks, AJ1,liuyuan2, liuyuan, jl1,jl2}\footnote{See
\cite{pcmi,icm} for recent reviews} have to deal with the competition between the
hyperbolicity of the transfer matrices and exponential strength of the
resonances. A sharp way to resolve this competition for pure frequency
resonances has been developed in \cite{jl1}. We start with following its main
framework combined with the strategy of \cite{JYMaryland} and dealing
with technical complications arising from the unboundedness and lack of
continuity. However, this alone only brings us to the same conclusion as in
\cite{jl1}, that is localization in the regime
$\{L_{\lambda}(E)>\beta(\alpha)\}$.
The region that needs completely new ideas is
$\{\delta(\alpha,\theta)<L(E)\leq \beta(\alpha)\}$. That's what we
develop here, exploiting the unbounded nature of the potentials rather
than circumventing it, by showing how a properly understood anti-resonance creates
additional decay of the Green's function. This helps us to
lower the threshold down all the way to the sharp
$\delta(\alpha,\theta)$.
This paper is organized as follows. in Sec. \ref{details} we present
more detail on the general strategy and difficulties of the proof. In
Sec. \ref{Sec:pre} we collect some preliminary results; in
Sec. \ref{Sec:delta} we locate the minimum values of (the absolute
value of) $q_n$ consequent cosines at $\{m_n+\ell q_n\}_{\ell}$, and
give a characterization of $\delta(\alpha,\theta)$ using these minimum
values; in Sec. \ref{Sec:proof} we discuss the proof of our main
Theorem \ref{main0} and reduce it to the main Lemma \ref{lem:main}. We
present some standard uniformity results in Sec. \ref{Sec:uni}; our
key estimate for the numerators is presented in Sec. \ref{Sec:key}; the proof of Lemma \ref{lem:main} is presented in Sec. \ref{Sec:loc} with preparations in Sections \ref{Sec:C1} and \ref{Sec:C2} addressing non-resonant $m_n$ and resonant $m_n$ respectively.
\section{Strategy and difficulties}\label{details}
We first introduce some notations and recall the key framework,
slightly modified from the one
developed in \cite{jl1}, also with adaptions from \cite{JYMaryland}.
Let $\tau>0$ be a small constant. For large $n$, let $b_n=[\tau q_n]$ and $R_{\ell}:=[\ell q_n-b_n, \ell q_n+b_n]$ be the resonant regimes and also $r_{\ell}:=\sup_{y\in R_{\ell}} |\phi(y)|$.
Let us also write $L_{\lambda}(E)$ as $L$ and $L-\ln 2$ as $\tilde{L}$.
We want to prove the generalized eigenfunction $\phi$ decays exponentially (with a positive decay rate independent of $n$) on $[q_n/3, q_{n+1}/3]$.
To do so, first we show that (roughly) at each non-resonance $|\phi|$
can be dominated by $|\phi|$ at its two nearby resonant regimes.
This allows us to only focus on the relations between $r_{\ell}$'s.
For each $y\in r_{\ell}$, we want to expand $\phi(y)$ using the Green's formula \eqref{Green_tildeP}
\begin{align*}
|\phi(y)|\leq &\frac{|\tilde{P}_{x_2-y}(\theta+(y+1)\alpha)|}{|\tilde{P}_{2q_n-1}(\theta+x_1\alpha)|}
\prod_{j=x_1}^{y}|\cos(\pi(\theta+j\alpha))|\cdot |\phi(x_1-1)|\\
&+\frac{|\tilde{P}_{y-x_1}(\theta+x_1\alpha)|}{|\tilde{P}_{2q_n-1}(\theta+x_1\alpha)|}\prod_{j=y}^{x_2}|\cos(\pi(\theta+j\alpha))|\cdot |\phi(x_2+1)|,
\end{align*}
with a nicely placed interval $D=[x_1, x_2] \ni y$ satisfying
$|D|=2q_n-1$ and such that
$$\frac{1}{4} |D|<y-x_1< \frac{3}{4} |D|.$$
Let $I_{\ell}=[\ell q_n-3q_n/2, \ell q_n-q_n/2]$ be the collection of potential left-end points $x_1$ in the Green's formula.
The goal is to obtain a good lower bound of $|\tilde{P}_{2q_n-1}(\theta+x_1\alpha)|$ for certain $x_1\in I_{\ell}$.
This is done by showing:
1. $\ln |\tilde{P}_{2q_n-1}(\theta)|$ has an average lower bound $\tilde{L}=L-\ln 2$,
2. $\tilde{P}_{2q_n-1}(\theta)$ is a polynomial in $\tan(\theta)$ of degree $2q_n-1$
Hence the Lagrange interpolation formula tells us $|\tilde{P}_{2q_n-1}(\theta)|$ can not be simultaneously small at $2q_n$ well distributed point $\theta$'s.
One can show that $\{\theta+j\alpha\}_{j\in I_0\cup I_{\ell}}$ are $\eta/(2q_n)$-uniform (in the sense of \eqref{defuniform}) with $\eta=\ln(q_{n+1}/|\ell|)$.
This implies there exists a certain $x_1\in I_{\ell}$ ($x_1$ can not be in $I_0$ because that leads to a contradiction) such that
$$|\tilde{P}_{2q_n-1}(\theta_{x_1})|\gtrsim \frac{|\ell|}{q_{n+1}} e^{2q_n L}.$$
Plugging this lower bound into the Green's formula,
using standard control of the $|\tilde{P}_k|\lesssim e^{\tilde{L}|k|}$ by the Lyapunov exponent $\tilde{L}$ in the numerators,
and adapting the estimates for the (possibly) non-resonant $|\phi(x_1-1)|$ and $|\phi(x_2+1)|$,
one can show that
\begin{align}\label{eq:intro:L>beta}
r_{\ell}\lesssim e^{-(L-\beta_n) q_n} \max(r_{\ell-1}, r_{\ell+1}),
\end{align}
where $\beta_n:=(\ln q_{n+1})/q_n$.
When $L>\beta$, combining the inequality above with our argument in Section \ref{Sec:loc} already yields exponential decay.
However this does not work for $\delta<L<\beta$, so
one has to look for an additional decay to break the $\beta$ barrier.
It turns out this additional decay in some simple cases comes in
handy, directly from the product (indeed the minimum) of cosines in the Green's formula.
The minimum values of (the absolute values of) cosines can be located
at points of the form $m_n+\ell q_n$ for certain $m_n,$ and the minimum values are roughly of the size $\exp((\delta-\beta_n)q_n)$.
In those simple cases, the product of cosines contain (at least) one of the minimum values thus bringing
the decay in \eqref{eq:intro:L>beta} down to
$$r_{\ell}\lesssim e^{-(L-\delta) q_n} \max(r_{\ell-1}, r_{\ell+1}),$$
which is just enough.
However, there are difficult cases when the product of cosines does
not contain such a minimum value, see e.g. Case 1 of Section \ref{Sec:C2}.
Then the question is: where does the additional decay hide in these cases?
Tackling this question is the main breakthrough of this paper that has
made the full analysis possible. the key ideas are presented in Section \ref{Sec:key}.
\section{Preliminaries}\label{Sec:pre}
We will from now use $\|\theta\|$ for $\|\theta\|_{{\mathbb T}}$ for simplicity.
For $x\in {\mathbb R}$, let $[x]$ be the largest integer that is less than or equal to $x$.
For a fixed $\theta$, let $\theta_k:=\theta+k\alpha$.
We adopt the convention that various large (or small) constants (e.g. $N(\varepsilon)$) may change
their exact values even within the same inequality.
\subsection{Continued fractions}
Let $[a_1,a_2,...]=\alpha$ be the continued fraction expansion of $\alpha$.
For $k\geq 1$, let $p_k/q_k:=[a_1,a_2,...,a_k]$ be the continued fraction approximants to $\alpha$. The following properties hold
\begin{align}\label{eq:cont1}
\|q_{k-1}\alpha\|=\min_{1\leq n<q_k} \|n\alpha\|,
\end{align}
\begin{align}\label{eq:cont2}
\frac{1}{2q_{k+1}}\leq \|q_k\alpha\|\leq \frac{1}{q_{k+1}},
\end{align}
\begin{align}\label{eq:cont3}
q_{k+1}=a_{k+1}q_{k}+q_{k-1},
\end{align}
and
\begin{align}\label{eq:cont4}
\|q_{k-1}\alpha\|=a_{k+1}\|q_k\alpha\|+\|q_{k+1}\alpha\|.
\end{align}
A key technical lemma is the following.
\begin{lemma} \label{lana}\cite{AJ1}
Let $\alpha\in {\mathbb R}\setminus {\mathbb Q} $,\ $\theta\in{\mathbb R}$ and $0\leq j_0 \leq q_{n}-1$ be such that
$$\mid \cos \pi(\theta+j_{0}\alpha)\mid = \inf_{0\leq j \leq q_{n}-1} \mid \cos \pi(\theta+j\alpha)\mid ,$$
then for some absolute constant $C>0$,
$$-C\ln q_{n} \leq \sum_{j=0,j\neq j_0}^{q_{n}-1} \ln \mid \cos \pi (\theta+j\alpha) \mid+(q_{n}-1)\ln2 \leq C\ln q_n$$
\end{lemma}
\subsection{Solution and Green's function}
Let $G_{[x_1,x_2]}(x,y)=(H_{[x_1, x_2]}-E)^{-1}(x,y)$ be the Green's function, where $H_{[x_1, x_2]}$ is the operator $H_{\lambda,\alpha,\theta}$ restricted to the interval $[x_1,x_2]$.
Let $\phi$ be a solution to $H\phi=E\phi$, let $[x_1, x_2]$ be an interval containing $y$, then we have
\begin{align}\label{Green}
\phi(y)=G_{[x_1, x_2]}(x_1, y)\phi(x_1-1)+G_{[x_1, x_2]}(x_2, y)\phi(x_2+1).
\end{align}
\subsection{Cocycles}
Consider the equation $H\phi=E\phi$. Let
\begin{align}\label{transferA}
A(\theta, E)=
\left(
\begin{matrix}
E-\lambda\tan{\pi \theta}\ &-1\\
1 &0
\end{matrix}
\right).
\end{align}
Then any solution can be reconstructed via the following relation
\begin{align*}
\left(
\begin{matrix}
\phi(k+1)\\
\phi(k)
\end{matrix}
\right)
=
A(\theta+k\alpha, E)
\left(
\begin{matrix}
\phi(k)\\
\phi(k-1)
\end{matrix}
\right).
\end{align*}
If we iterate this process, we get
\begin{align*}
\left(
\begin{matrix}
\phi(k)\\
\phi(k-1)
\end{matrix}
\right)
=
A_k(\theta, E)
\left(
\begin{matrix}
\phi(0)\\
\phi(-1)
\end{matrix}
\right),
\end{align*}
where
\begin{align*}
\left\lbrace
\begin{matrix}
A_k(\theta, E)=A(\theta+(k-1)\alpha, E)\cdots A(\theta+\alpha, E)A(\theta, E)\ \mathrm{for}\ k\geq 1,\\
A_0(\theta, E)=\mathrm{Id},\\
A_k(\theta, E)=(A_{-k}(\theta+k\alpha, E))^{-1}\ \mathrm{for}\ k\leq -1.
\end{matrix}
\right.
\end{align*}
Note that the cocycle $A(\theta, E)$ is actually singular because it contains $\tan{\pi\theta}$. Sometimes it is more convenient for us to work with non-singular cocycles. Let us denote
\begin{align}\label{defnonsingular}
F(\theta, E)=\cos{\pi \theta}\cdot A(\theta, E)=
\left(
\begin{matrix}
E\cos{\pi \theta}-\lambda\sin{\pi \theta}\ &-\cos{\pi \theta}\\
\cos{\pi \theta} &0
\end{matrix}
\right).
\end{align}
\subsection{Lyapunov exponent}
Let $L(\alpha, A(\theta, E))$ be the Lyapunov exponent of the Maryland model, it is defined as follows
\begin{align}\label{defLE}
L(\alpha, A(\theta, E))=\lim_{k\rightarrow \infty}\frac{1}{k}\int_{{\mathbb T}}\ln{\|A_k(\theta, E)\|} \mathrm{d}\theta.
\end{align}
It was shown in \cite{fgp} that $L(\alpha, A(\theta, E))$ depends only on $\lambda$ and $E$ (hence we denote it by $L_{\lambda}(E)$) and is uniquely determined by the following equation
\begin{align}\label{LE}
e^{L_{\lambda}(E)}+e^{-L_{\lambda}(E)}=\frac{\sqrt{(2+E)^2+\lambda^2}+\sqrt{(2-E)^2+\lambda^2}}{2}.
\end{align}
Let us also denote $\tilde{L}(E)=L(\alpha, F(\theta, E))$, then by (\ref{defnonsingular}) we have
\begin{align}\label{LEtildeLE}
\tilde{L}_{\lambda}(E)=L_{\lambda}(E)-\ln{2}.
\end{align}
From this point on, we shall write $L_{\lambda}(E)$ as $L(E)$ or $L$, and $\tilde{L}_{\lambda}(E)$ as $\tilde{L}(E)$ or $\tilde{L}$ for simplicity.
\subsection{General upper bounds of transfer matrices}
\begin{lemma}[e.g. \cite{Furman}]\label{upperbounds}
Let $(a, D)$ be a continuous cocycle, then for any $\epsilon>0$, there exists a constant $C(\alpha, D, \varepsilon)$ such that for any $k\in {\mathbb Z}$,
\begin{align}
\|D_k(\theta)\|\leq C(\alpha, D, \varepsilon) e^{|k|(L(\alpha, D)+\epsilon)}\ \mathrm{for}\ \mathrm{any}\ \theta\in{\mathbb T}.
\end{align}
\end{lemma}
As a corollary we have the following lemma which will be used many times throughout the paper.
\begin{cor}\label{cor:prod_cos}
Let $\ell_2\geq \ell_1$. We have
\begin{align*}
\prod_{\ell=\ell_1}^{\ell_2}|\cos(\pi(\theta+\ell\alpha))|\leq C(\varepsilon) e^{(\ell_2-\ell_1)(-\ln{2}+\varepsilon)} \inf_{j=\ell_1}^{\ell_2}|\cos(\pi(\theta+j\alpha))|,
\end{align*}
where $C(\varepsilon)$ is a constant that depends only on $\varepsilon$.
\end{cor}
\subsection{A closer look at the transfer matrix}
If we consider the Schr\"odinger cocycle $(\alpha, A(\theta, E))$, it turns out $A_k(\theta, E)$ has the following expression
\begin{equation}\label{PinA}
A_k(\theta,E)=
\left(
\begin{array}{cc}
P_k(\theta,E) & -P_{k-1}(\theta+\alpha,E) \\
P_{k-1}(\theta,E) & -P_{k-2}(\theta+\alpha,E)
\end{array}
\right),
\end{equation}
where
\begin{align}
P_k(\theta,E)=&\det{[(E-H_{\theta})|_{[0,k-1]}]} \notag \\
=&
\det{
\left[\begin{array}{cccccc}
E-\lambda\tan{\pi\theta} & -1 & & & \\
-1 & E-\lambda\tan\pi(\theta+\alpha) & -1 \\
&-1 & \cdots \\
& & & \cdots & -1 \\
& & & -1 & E-\lambda\tan\pi(\theta+(k-1)\alpha)
\end{array}\right]_{k\times k}
}
\end{align}
Let $\tilde{P}_k(\theta, E)=\prod_{j=0}^{k-1} \cos \pi (\theta+j \alpha)\cdot P_k(\theta, E)$. Then clearly
\begin{align}\label{tildePinF}
F_k(\theta, E)=
\left(
\begin{array}{cc}
\tilde{P}_k(\theta,E) & -\tilde{P}_{k-1}(\theta+\alpha,E)\cos{\pi\theta} \\
\tilde{P}_{k-1}(\theta,E)\cos{\pi (\theta+(k-1)\alpha)} & -\tilde{P}_{k-2}(\theta+\alpha,E)\cos{\pi\theta}\cos{\pi(\theta+(k-1)\alpha)}
\end{array}
\right).
\end{align}
By the fact that $F$ is continous and (\ref{upperbounds}), (\ref{tildePinF}) we have the following control of $\tilde{P}_k$.
\begin{lemma}\label{lem:upperbddtildeP}
For any $\varepsilon>0$ there exists constant $C(\alpha, E, \lambda, \varepsilon)>0$ such that for any $k\in {\mathbb Z}$,
\begin{align}
|\tilde{P}_k(\theta, E)|\leq C(\alpha, E, \lambda, \varepsilon) e^{(\tilde{L}(E)+\varepsilon)|k|}\ \mathrm{for}\ \mathrm{any}\ \theta\in {\mathbb T}.
\end{align}
\end{lemma}
There is the following connection between the determinants $P_k$ and Green's functions:
\begin{align}\label{PkG}
|G_{[x_1, x_2]}(x_1, y)|=&\frac{|P_{x_2-y}(\theta_{y+1})|}{|P_{x_2-x_1+1}(\theta_{x_1})|}
=\frac{|\tilde{P}_{x_2-y}(\theta_{y+1})|}{|\tilde{P}_{x_2-x_1+1}(\theta_{x_1})|} \prod_{k=x_1}^y |\cos(\pi\theta_k)|\\
|G_{[x_1, x_2]}(x_2, y)|=&\frac{|P_{y-x_1}(\theta_{x_1})|}{|P_{x_2-x_1+1}(\theta_{x_1})|}
=\frac{|\tilde{P}_{y-x_1}(\theta_{x_1})|}{|\tilde{P}_{x_2-x_1+1}(\theta_{x_1})|}\prod_{k=y}^{x_2} |\cos(\pi\theta_k)|\notag
\end{align}
As a consequence, from \eqref{Green} we can deduce
\begin{align}\label{Green_tildeP}
|\phi(y)|\leq \frac{|\tilde{P}_{x_2-y}(\theta_{y+1})|}{|\tilde{P}_{x_2-x_1+1}(\theta_{x_1})|} \prod_{k=x_1}^y |\cos(\pi\theta_k)|
\cdot |\phi(x_1-1)|+ \frac{|\tilde{P}_{y-x_1}(\theta_{x_1})|}{|\tilde{P}_{x_2-x_1+1}(\theta_{x_1})|}\prod_{k=y}^{x_2} |\cos(\pi\theta_k)|\cdot |\phi(x_2+1)|.
\end{align}
\subsection{Writing $\tilde{P}_k$ as a polynomial of $\tan{\pi\theta}$, see \cite{JYMaryland}}\
$\tilde{P}_k(\theta)/{\cos^k{\pi\theta}}$
can be expressed as a polynomial of degree $k$ in $\tan\pi\theta$, namely,
\begin{equation}\label{Phpolyg}
\frac{\tilde{P}_k(\theta)}{(\cos{\pi\theta})^k} =: g_k(\tan\pi\theta).
\end{equation}
By Lagrange interpolation formula,
\begin{equation*}
g_k(\tan\pi \theta)=\sum_{j=0}^{k}g_k(\tan \pi \theta_j) \frac{\prod_{\ell \neq j}\tan \pi \theta-\tan\pi\theta_{\ell}}{\prod_{\ell\neq j}\tan\pi\theta_j-\tan\pi\theta_{\ell}}.
\end{equation*}
Thus
\begin{align}
\tilde{P}_k(\theta)=(\cos\pi \theta)^k g_k(\tan\pi \theta)
&=\sum_{j=0}^{k}\tilde{P}_k(\theta_j)\frac{\prod_{\ell \neq j}\tan \pi \theta-\tan\pi\theta_{\ell}}{\prod_{\ell\neq j}\tan\pi\theta_j-\tan\pi\theta_{\ell}}\cdot \frac{\cos^k {\pi \theta}}{\cos^k{\pi\theta_j}} \notag\\
&=\sum_{j=0}^{k}\tilde{P}_k(\theta_j)\prod_{\ell\neq j} \frac{\sin\pi(\theta-\theta_{\ell})}{\sin\pi(\theta_j-\theta_{\ell})}. \label{tildePlagrange}
\end{align}
\subsection{Average lower bound of $\tilde{P}_k$}
The following is Lemma 3.1 of \cite{JYMaryland}.
\begin{lemma}\label{subharmonic}
By Herman's subharmonic trick, one has
\begin{align}\label{averagelower}
\frac{1}{k}\int_{0}^{1}\ln |\tilde{P}_k(\theta)| \mathrm{d}\theta =\frac{1}{k} \int_{0}^{1} \ln |\tilde{P}_k(2\theta)| \mathrm{d}\theta \geq L-\ln2=\tilde{L}
\end{align}
\end{lemma}
\subsection{Uniformity}
\begin{definition}
We say that the set $\{\theta_1,..., \theta_{k+1}\}$ are $\gamma$-uniform if
\begin{align}\label{defuniform}
\max_{\theta\in [0, 1]}\ \max_{j=1,..., k+1} \prod_{\ell\neq j}\ \frac{|\sin\pi(\theta-\theta_{\ell})|}{|\sin\pi(\theta_j-\theta_{\ell})|}<e^{\gamma k}.
\end{align}
\end{definition}
\section{Resonance and non-resonances}\label{Sec:uni}
Choose a value (from multiple possible values) of $\tau_n$ such that
$$
\tau_n\in \left(\frac{\varepsilon}{2\max(L, 1)}, \frac{\varepsilon}{\max(L, 1)} \right],
$$
and $\tau_n q_n\in {\mathbb Z}$.
Define $b_n=\tau_n q_n$.
For any $y\in {\mathbb Z}$ we call $y$ {\it resonant} (at the scale of $q_n$) if $\operatorname{dist}(y, q_n{\mathbb Z})\leq b_n$, otherwise we call $y$ {\it non-resonant}.
\subsection{Non-resonances: uniformity}
For a non-resonant $y$, let $n_0$ be the least positive integer so that
\begin{align*}
2q_{n-n_0}\leq \operatorname{dist}(y, q_n{\mathbb Z}).
\end{align*}
Once $n_0$ is chosen, we can fix $s$ be the greatest positive integer such that
\begin{equation}
2sq_{n-n_0}\leq \operatorname{dist}(y, q_n{\mathbb Z}).
\end{equation} Clearly,
Let
\begin{align}\label{def:Iy}
\tilde{I}_0&=[-[sq_{n-n_0}/2]-sq_{n-n_0},\, -[sq_{n-n_0}/2]-1 ]\cap {\mathbb Z}, \notag\\
\tilde{I}_y&=[y-[sq_{n-n_0}/2]-sq_{n-n_0},\, y-[sq_{n-n_0}/2]-1] \cap {\mathbb Z}.
\end{align}
Clearly $\tilde{I}_0\cup \tilde{I}_y$ contains $2sq_{n-n_0}$ distinct numbers.
Let us also note that by our choice of $n_0$, we have
\begin{align}\label{eq:bn<yn-n0+1}
b_n< \operatorname{dist}(y, q_n {\mathbb Z})<2q_{n-n_0+1}
\end{align}
and also
\begin{align}\label{sq-n0<qn-n0+1}
sq_{n-n_0}<q_{n-n_0+1}.
\end{align}
The following lemma is the consequence of a variant of Lemma 9.10 of \cite{AJ1}. We will include its proof in the appendix for completeness.
\begin{lemma}\label{lem:nonres_uni}
For a non-resonant $y$, for $n>N(\varepsilon)$ large enough, we have $\{\theta_{\ell}\}_{\ell\in \tilde{I}_0\cup \tilde{I}_y}$ are $\varepsilon$-uniform.
\end{lemma}
Combining this with a standard argument in the literature, one can show
\begin{lemma}\label{lem:nonres_I2_large}
For $n>N(\varepsilon)$ large enough, there exists $x_1\in \tilde{I}_y$ so that
$$|\tilde{P}_{2sq_{n-n_0}-1}(\theta_{x_1})|\geq e^{(\tilde{L}-2\varepsilon)(2sq_{n-n_0}-1)}.$$
\end{lemma}
We will also include its proof in the appendix.
\subsection{Resonances: uniformity}
For $\ell\in {\mathbb Z}$, let $I_{\ell}$ be defined below
\begin{align}\label{def:1_Ia}
I_\ell:=&[(\ell-1)q_n -\lfloor q_n/2\rfloor, \ell q_n-\lfloor q_n/2\rfloor-1]\cap {\mathbb Z}.
\end{align}
\begin{lemma}\label{lem:1_res_uni}
For $\ell $ such that $0<|\ell |\leq 2q_{n+1}/(3q_n)$,
$\{\theta_j\}_{j\in I_0\cup I_{\ell}}$ are $\frac{\ln{(q_{n+1}/|\ell |)}}{2q_n-1}+\epsilon$-uniform.
\end{lemma}
This is a variant of Theorem B.5. of \cite{jl1}. We include the proof in the appendix.
\begin{cor}\label{cor:1_res_uni}
For $\ell $ such that $0<|\ell |\leq 2q_{n+1}/(3q_n)$, there exists $x_1\in I_0\cup I_{\ell}$ such that
$$|\tilde{P}_{2q_n-1}(\theta_{x_1})|\geq \frac{|\ell|}{q_{n+1}} e^{(\tilde{L}-2\varepsilon)(2q_n-1)}.$$
\end{cor}
\begin{proof}
Suppose otherwise, we have for any $x_1\in I_0\cup I_{\ell}$,
$$|\tilde{P}_{2q_n-1}(\theta_{x_1})|< \frac{|\ell|}{q_{n+1}} e^{(\tilde{L}-2\varepsilon)(2q_n-1)}.$$
By \eqref{tildePlagrange}
\begin{align*}
\tilde{P}_{2q_n-1}(\theta)=\sum_{x_1\in I_0\cup I_{\ell}}\tilde{P}_{2q_n-1}(\theta_{x_1})\prod_{\substack{j\in I_0\cup I_{\ell}\\ j\neq x_1}} \frac{\sin\pi(\theta-\theta_j)}{\sin\pi(\theta_{x_1}-\theta_j)}.
\end{align*}
Combining this with Lemma \ref{lem:1_res_uni} yields, uniformly in $\theta$,
\begin{align*}
|\tilde{P}_{2q_n-1}(\theta)|\leq 2q_n e^{(\tilde{L}-\varepsilon)(2q_n-1)}<e^{(\tilde{L}-\frac{\varepsilon}{2})(2q_n-1)}.
\end{align*}
Hence contradiction with \eqref{averagelower}. \qed
\end{proof}
\section{Characterization of the index $\delta$}\label{Sec:delta}
From this point on, we shall write $\delta(\alpha,\theta)$ as $\delta$ and $\beta(\alpha)$ as $\beta$ for simplicity.
Fix a small $\varepsilon>0$ such that $L>\delta+700\varepsilon$.
It is evident from the definition of $\delta$ that it is never greater than $\beta$.
The following lemma shows $\delta$ is always non-negative.
\begin{lemma}\label{lem:delta_geq_0}
We have $0\leq \delta\leq \beta$ for all $\alpha, \theta$.
\end{lemma}
\begin{proof}
Recall the definition
$$\delta=\limsup_{n\to \infty}\frac{\ln{q_{n+1}}+\ln{\|q_n(\theta-\frac{1}{2})\|}}{q_n}=
\limsup_{n\to \infty}\frac{\ln{q_{n+1}}+\ln{\|q_n|\theta-\frac{1}{2}|\|}}{q_n}.$$
Suppose $\delta<0$, then there exists $c>0$, such that for $n$ large enough, we have
\begin{align*}
\frac{\ln{q_{n+1}}+\ln{\|q_n|\theta-\frac{1}{2}|\|}}{q_n}<-c.
\end{align*}
This implies
\begin{align}\label{eq:qn_theta-1/2}
\|q_n|\theta-\frac{1}{2}|\|<\frac{e^{-c q_n}}{q_{n+1}}.
\end{align}
Let $\{\tilde{p}_k/\tilde{q}_k\}$ be the continued fraction approximants to $|\theta-\frac{1}{2}|$.
Take $k$ large, and $n$ such that
\begin{align}\label{eq:qn_qk+1_qn+1}
q_n<\tilde{q}_{k+1}\leq q_{n+1}.
\end{align}
By \eqref{eq:qn_theta-1/2}, \eqref{eq:cont1} and \eqref{eq:cont2}, we have
$$\frac{1}{2\tilde{q}_{k+1}}\leq \|\tilde{q}_k |\theta-\frac{1}{2}|\|\leq \|q_n |\theta-\frac{1}{2}|\|<\frac{e^{-c q_n}}{q_{n+1}}<\frac{1}{2q_{n+1}}.$$
This implies
$$\tilde{q}_{k+1}>q_{n+1},$$
which contradicts with \eqref{eq:qn_qk+1_qn+1}.
\qed
\end{proof}
Define
\begin{align}\label{def:betan}
\beta_n:=\frac{\ln q_{n+1}}{q_n},
\end{align}
and
\begin{align}\label{def:deltan}
\delta_n:=\frac{\ln \|q_n(\theta-\frac{1}{2})\|-\ln \|q_n\alpha\|}{q_n}.
\end{align}
Since $\limsup_{n\to \infty}\delta_n=\delta$, we have that for $n>N(\varepsilon)$ large enough,
\begin{align}\label{eq:L>deltan+100}
L>\delta_n+650\varepsilon.
\end{align}
It is also clear by \eqref{eq:cont2} that
\begin{align}\label{eq:deltan<betan}
\delta_n\leq \frac{\ln(1/2)-\ln \|q_n\alpha\|}{q_n}<\beta_n.
\end{align}
By Lemma \ref{lem:delta_geq_0}, we also have
\begin{align}\label{eq:lim_max_0_deltan}
\delta=\limsup \max(0, \delta_n).
\end{align}
First we will characterize the minimal values of $\{|\cos\pi(\theta+j\alpha)|\}_j$ on scale $q_n$.
\begin{definition}\label{def:minimal}
We say $(m,\ell)\in {\mathbb Z}^2$ is {\it $\theta$-minimal on scale $q_n$} if the following holds
\begin{enumerate}
\item $m\in [-q_n/2, q_n/2)$
\item $|\ell|\leq \frac{1}{q_n}(e^{\delta_n q_n}+q_n+\frac{1}{2})$,
\item $\|\theta-\frac{1}{2}+(m+\ell q_n)\|<(\frac{1}{2}+\frac{1}{2q_n})\|q_n\alpha\|$,
\item (i). If $a_{n+1}\geq 4$, we have
$$\|\theta-\frac{1}{2}+(m+jq_n)\alpha\|\leq 20 \min_{|k|<q_n} \|\theta-\frac{1}{2}+(m+j q_n+k)\alpha\|,$$
holds for any $|j|\leq a_{n+1}/6$.
(ii). If $a_{n+1}\leq 3$, we have
$$\|\theta-\frac{1}{2}+m\alpha\|\leq 20\min_{-q_n/2\leq k<q_n/2}\|\theta-\frac{1}{2}+k\alpha\|.$$
\end{enumerate}
\end{definition}
Next we show that the existence of $\theta$-minimal $(m,\ell)$.
\begin{lemma}\label{lem:mn}
For any $q_n$ sufficiently large, there exists $\theta$-minimal $(m_n,\ell_n)$ at scale $q_n$.
\end{lemma}
\begin{rem}
For any given $\theta$, following the procedure below, one can construct $(m_n,\ell_n)$ explicitly.
\end{rem}
\begin{proof}
By the definition of $\delta_n$, we have
\begin{align*}
\|q_n(\theta-\frac{1}{2})\|=e^{\delta_n q_n}\|q_n\alpha\|.
\end{align*}
Using the fact that $\|q_n\alpha\|>1/(q_{n+1}+q_n)$ and $\|q_n(\theta-1/2)\|\leq 1/2$, we have
\begin{align}\label{eq:delta_n0}
e^{\delta_n q_n}\leq \frac{q_{n+1}+q_n-1}{2}.
\end{align}
One can choose $j_0\in {\mathbb Z}$ such that
\begin{align}\label{eq:delta_n1}
\|q_n(\theta-\frac{1}{2})+j_0 q_n\alpha\|\leq \frac{1}{2}\|q_n\alpha\|,
\end{align}
and $j_0$ satisfies
\begin{align}\label{eq:delta_n2}
|j_0|\leq
e^{\delta_n q_n}+\frac{1}{2}.
\end{align}
Note that \eqref{eq:delta_n1} yields
\begin{align}
\|q_n(\theta-\frac{1}{2}+j_0\alpha)\|\leq \frac{1}{2}\|q_n\alpha\|.
\end{align}
Hence there exists an integer $p$ such that
\begin{align}\label{eq:delta_n3}
|(\theta-\frac{1}{2}+j_0\alpha)-\frac{p}{q_n}|\leq \frac{1}{2q_n}\|q_n\alpha\|.
\end{align}
Since $p_n, q_n$ are coprime, there exists
\begin{align}\label{eq:delta_n3'}
j_1\in [-\frac{q_n}{2}, \frac{q_n}{2})
\end{align}
such that
\begin{align*}
j_1p_n\equiv p \text{ (mod } q_n\text{)}.
\end{align*}
This implies
\begin{align}\label{eq:delta_n4}
j_1\alpha-\frac{p}{q_n}=j_1\frac{p_n}{q_n}-\frac{p}{q_n}+j_1(\alpha-\frac{p_n}{q_n})=k+j_1(\alpha-\frac{p_n}{q_n}),
\end{align}
where $k\in {\mathbb Z}$ and
\begin{align}\label{eq:delta_n5}
|j_1(\alpha-\frac{p_n}{q_n})|=\frac{j_1}{q_n}\|q_n\alpha\|\leq \frac{1}{2}\|q_n\alpha\|.
\end{align}
Combining \eqref{eq:delta_n3}, \eqref{eq:delta_n4} and \eqref{eq:delta_n5}, we have
\begin{align}\label{eq:delta_n6}
\|\theta-\frac{1}{2}+(j_0+j_1)\alpha\|\leq (\frac{1}{2}+\frac{1}{2q_n})\|q_n\alpha\|.
\end{align}
Define $k_n:=j_0+j_1$, then clearly by \eqref{eq:delta_n2} and \eqref{eq:delta_n3'} we have
\begin{align}\label{eq:delta_n7}
|k_n|\leq e^{\delta_n q_n}+\frac{q_n+1}{2}.
\end{align}
Define $m_n^{(1)}\in [-q_n/2,q_n/2)$ and $\ell_n^{(1)}\in {\mathbb Z}$ be such that
\begin{align}\label{def:mn_elln}
m_n^{(1)}+\ell_n^{(1)} q_n=k_n.
\end{align}
By \eqref{eq:delta_n7} and \eqref{eq:delta_n0}, we have
\begin{align}\label{eq:delta_n9'}
|\ell_n^{(1)}|\leq \frac{1}{q_n}(e^{\delta_n q_n}+q_n+\frac{1}{2})\leq \frac{q_{n+1}+3q_n}{2q_n}\leq \left[\frac{a_{n+1}+3}{2}\right].
\end{align}
By \eqref{eq:delta_n6} and \eqref{def:mn_elln}, we have
\begin{align}\label{eq:delta_n6'}
\|\theta-\frac{1}{2}+(m_n^{(1)}+\ell_n^{(1)} q_n)\|\leq (\frac{1}{2}+\frac{1}{2q_n})\|q_n\alpha\|.
\end{align}
We also have, by \eqref{eq:delta_n9'}, that for $\ell\neq \ell_n^{(1)}$ such that $|\ell|\leq [a_{n+1}/6]$ that
\begin{align}\label{eq:delta_n10}
|\ell_n^{(1)}-\ell| \cdot \|q_n\alpha\|\leq \left(\left[\frac{a_{n+1}+3}{2}\right]+\left[\frac{a_{n+1}}{6}\right] \right) \|q_n\alpha\|\leq \frac{2}{3}\|q_{n-1}\alpha\|+\frac{3}{2}\|q_n\alpha\|<\frac{1}{2}.
\end{align}
Hence
\begin{align*}
\|(\ell_n^{(1)}-\ell)q_n\alpha\|=|\ell_n^{(1)}-\ell|\cdot \|q_n\alpha\|.
\end{align*}
This implies
\begin{align}\label{eq:delta_n11}
(|\ell_n^{(1)}-\ell|-\frac{1}{2}-\frac{1}{2q_n})\|q_n\alpha\|\leq \|\theta-\frac{1}{2}+(m_n^{(1)}+\ell q_n)\alpha\|
\leq &(|\ell_n^{(1)}-\ell|+\frac{1}{2}+\frac{1}{2q_n}) \|q_n\alpha\|.
\end{align}
Combining the right hand side of \eqref{eq:delta_n11} with \eqref{eq:delta_n10}, we have
\begin{align*}
\|\theta-\frac{1}{2}+(m_n^{(1)}+\ell q_n)\alpha\|
\leq &\left(\left[\frac{a_{n+1}+3}{2}\right]+\left[\frac{a_{n+1}}{6}\right]+\frac{1}{2}+\frac{1}{2q_n}\right)\|q_n\alpha\|\\
\leq &\begin{cases}
(\frac{2}{3}a_{n+1}+2+\frac{1}{2q_n})\|q_n\alpha\|, \text{ if } a_{n+1}\geq 7\\
(a_{n+1}-\frac{1}{2}+\frac{1}{2q_n})\|q_n\alpha\|, \text{ if } 4\leq a_{n+1}\leq 6.
\end{cases}
\end{align*}
Hence for integer $|k|<q_n$ and $a_{n+1}\geq 7$, we have
\begin{align}\label{eq:delta_n12}
\|\theta-\frac{1}{2}+(m_n^{(1)}+\ell q_n+k)\alpha\|
\geq &\|k\alpha\|-\|\theta-\frac{1}{2}+(m_n^{(1)}+\ell q_n)\alpha\| \notag\\
\geq &\|q_{n-1}\alpha\|-\|\theta-\frac{1}{2}+(m_n^{(1)}+\ell q_n)\alpha\| \notag\\
\geq & a_{n+1}\|q_n\alpha\|-\|\theta-\frac{1}{2}+(m_n^{(1)}+\ell q_n)\alpha\| \notag\\
\geq &(\frac{1}{3}a_{n+1}-2-\frac{1}{2q_n}) \|q_n\alpha\| \notag\\
\geq &\frac{1}{20}\|\theta-\frac{1}{2}+(m_n^{(1)}+\ell q_n)\alpha\|,
\end{align}
and similarly for $4\leq a_{n+1}\leq 6$,
\begin{align}\label{eq:delta_n13}
\|\theta-\frac{1}{2}+(m_n^{(1)}+\ell q_n+k)\alpha\|
\geq &a_{n+1}\|q_n\alpha\|-(a_{n+1}-\frac{1}{2}+\frac{1}{2q_n})\|q_n\alpha\| \notag\\
\geq &\frac{1}{20}\|\theta-\frac{1}{2}+(m_n^{(1)}+\ell q_n)\alpha\|.
\end{align}
Thus combining \eqref{eq:delta_n9'}, \eqref{eq:delta_n6'}, \eqref{eq:delta_n12}, \eqref{eq:delta_n13}, we have proved that $(m_n, \ell_n)=(m_n^{(1)}, \ell_n^{(1)})$ is $\theta$-minimal at scale $q_n$ if $a_{n+1}\geq 4$.
Next we consider the case $a_{n+1}\in [1, 3]$.
If $\ell_n^{(1)}=0$, we have by \eqref{eq:delta_n6'} that for any $|k|<q_n$,
\begin{align*}
\|\theta-\frac{1}{2}+(m_n^{(1)}+k)\alpha\|
\geq &\|q_{n-1}\alpha\|-\|\theta-\frac{1}{2}+m_n^{(1)}\alpha\|\\
\geq &(\frac{1}{2}-\frac{1}{2q_n})\|q_n\alpha\|\\
\geq &\frac{1}{2}\|\theta-\frac{1}{2}+m_n^{(1)}\alpha\|.
\end{align*}
This verifies (4)(ii) in Definition \ref{def:minimal}. Hence $(m_n,\ell_n)=(m_n^{(1)},\ell_n^{(1)})$ is $\theta$-minimal at scale $q_n$.
If $\ell_n^{(1)}\neq 0$, the minimal may not occur at $m_n$. We define $\tilde{m}_n\in [-q_n/2, q_n/2)$ such that
\begin{align}\label{def:tildemn}
\|\theta-\frac{1}{2}+\tilde{m}_n\alpha\|=\inf_{k\in [-q_n/2, q_n/2)} \|\theta-\frac{1}{2}+k\alpha\|.
\end{align}
Next we divide into two different cases depending on $\|\theta-\frac{1}{2}+\tilde{m}_n\alpha\|<\frac{1}{3}\|q_n\alpha\|$ or not.
Case 1. If $\|\theta-\frac{1}{2}+\tilde{m}_n\alpha\|\geq \frac{1}{3}\|q_n\alpha\|$.
We have
\begin{align*}
\|\theta-\frac{1}{2}+m_n^{(1)}\alpha\|\leq (|\ell_n^{(1)}|+\frac{1}{2}+\frac{1}{2q_n})\|q_n\alpha\|\leq 4\|q_n\alpha\|\leq 12\|\theta-\frac{1}{2}+\tilde{m}_n\alpha\|,
\end{align*}
which verifies (4)(ii) in Definition \ref{def:minimal}. Hence $(m_n,\ell_n)=(m_n^{(1)}, \ell_n^{(1)})$ is $\theta$-minimal at scale $q_n$.
Case 2. If
\begin{align}\label{eq:tildemn<13}
\|\theta-\frac{1}{2}+\tilde{m}_n\alpha\|< \frac{1}{3}\|q_n\alpha\|.
\end{align}
Such $\tilde{m}_n$ may not exist in some cases, but if it exists, $(m_n,\ell_n)=(\tilde{m}_n, 0)$ will be $\theta$-minimal at scale $q_n$.
Indeed, (4)(ii) obviously hold for $m=\tilde{m}_n$ due to the definition of $\tilde{m}_n$.
(3) of Definition \ref{def:minimal} holds for $(m,\ell)=(\tilde{m}_n,0)$ due to \eqref{eq:tildemn<13}.
Hence $(\tilde{m}_n, 0)$ is $\theta$-minimal at scale $q_n$. \qed
\end{proof}
Define
\begin{align}\label{def:cnell}
c_{n,\ell}:=|\cos(\pi \theta_{m_n+\ell q_n})|.
\end{align}
As a corollary of the construction in Lemma \ref{lem:mn} we have.
\begin{cor}\label{cor:mn}
If $a_{n+1}\geq 4$, we have
\begin{align}\label{eq:cnelln}
c_{n,\ell_n}\leq \frac{2}{3}\|q_n\alpha\|,
\end{align}
and for $|\ell|\leq q_{n+1}/(6q_n)$
\begin{align}\label{eq:delta_n_min_cos}
c_{n,\ell}\leq 21 \min_{|k|<q_n} |\cos(\pi\theta_{m_n+\ell_n q_n+k})|,
\end{align}
and further if $\ell\neq \ell_n$,
\begin{align}\label{eq:cnell_low_up}
c|\ell-\ell_n|\cdot \|q_n\alpha\|\leq c_{n,\ell}\leq C|\ell-\ell_n|\cdot \|q_n\alpha\|,
\end{align}
where $c,C\in (1/3,2)$ are two absolute constants.
If $a_{n+1}\leq 3$, we have that
\begin{align}\label{eq:cnell_min_an+1<4}
c_{n,0}\leq 21 \min_{k\in [-q_n/2,q_n/2)} |\cos(\pi\theta_k)|.
\end{align}
\end{cor}
\begin{proof}
\eqref{eq:delta_n_min_cos} and \eqref{eq:cnell_min_an+1<4} follow from (4) of Definition \ref{def:minimal},
and \eqref{eq:cnell_low_up} follows from \eqref{eq:delta_n11}.
\qed
\end{proof}
We also have the following corollary of the construction of Lemma \ref{lem:mn}
\begin{cor}\label{cor:elln>0}
If $e^{\delta_n q_n}>3q_n$, we have
\begin{align*}
\left| |\ell_n|-\frac{1}{q_n} e^{\delta_n q_n}\right|\leq \frac{1}{2q_n}+1.
\end{align*}
\end{cor}
\begin{proof}
This follows from an inspection of \eqref{eq:delta_n2}, \eqref{eq:delta_n3'}, \eqref{eq:delta_n7} and \eqref{eq:delta_n9'}.
\end{proof}
Combining \eqref{eq:lim_max_0_deltan} with Corollaries \ref{cor:mn} and \ref{cor:elln>0}, we have
\begin{cor}\label{cor:alternate_delta}
\begin{align*}
\delta(\alpha,\theta)=\limsup_{n\to \infty}\max(0,\delta_n)=\limsup_{n\to \infty} \max(0, \frac{\ln q_{n+1}+\ln |c_{n,0}|}{q_n}).
\end{align*}
\end{cor}
\begin{proof}
Let
$$\delta_n':=\frac{\ln q_{n+1}+\ln |c_{n,0}|}{q_n}.$$
If $a_{n+1}\geq 4$, and $e^{\delta_n q_n}\leq 3q_n$, we have by \eqref{eq:delta_n9'} and \eqref{eq:cnell_low_up} that
\begin{align*}
\delta_n'\leq \frac{\ln q_{n+1}+\ln (C|\ell_n| \|q_n\alpha\|)}{q_n}\leq \frac{\ln (5C)}{q_n}.
\end{align*}
This implies
\begin{align}\label{eq:d-d'_1}
|\max(0, \delta_n)-\max(0, \delta_n')|\leq \frac{\ln(3q_n)}{q_n}.
\end{align}
If $e^{\delta_n q_n}>3q_n$, we have by Corollary \ref{cor:elln>0} that
\begin{align*}
\delta_n'\leq \frac{\ln(e^{\delta_n q_n}+\frac{1}{2}+q_n)}{q_n}-\frac{\ln q_n}{q_n}<\delta_n+\frac{\ln(2/q_n)}{q_n},
\end{align*}
and
\begin{align*}
\delta_n'\geq \frac{\ln(e^{\delta_n q_n}-\frac{1}{2}-q_n)}{q_n}-\frac{\ln q_n}{q_n}<\delta_n-\frac{\ln(2q_n)}{q_n}.
\end{align*}
Hence
\begin{align}\label{eq:d-d'_2}
|\delta_n-\delta_n'|\leq \frac{\ln(2q_n)}{q_n}.
\end{align}
If $1\leq a_{n+1}\leq 3$, we have by \eqref{eq:deltan<betan} that
\begin{align}\label{eq:d-d'_3}
\delta_n<\beta_n=\frac{\ln q_{n+1}}{q_n}<\frac{\ln (4q_n)}{q_n},
\end{align}
and by (3) of Definition \ref{def:minimal} we have
\begin{align}\label{eq:d-d'_4}
\delta_n'=\frac{\ln q_{n+1}+\ln |\sin(\pi(\theta-\frac{1}{2}+m_n\alpha))|}{q_n}\leq \frac{\ln q_{n+1}+\ln (\pi \|q_n\alpha\|)}{q_n}<\frac{\ln \pi}{q_n}.
\end{align}
Combining \eqref{eq:d-d'_3} with \eqref{eq:d-d'_4}, we have
\begin{align}\label{eq:d-d'_5}
|\max(0,\delta_n)-\max(0,\delta_n')|\leq \frac{\ln (4q_n)}{q_n}.
\end{align}
Corollary \ref{cor:alternate_delta} follows from combining \eqref{eq:d-d'_1}, \eqref{eq:d-d'_2} and \eqref{eq:d-d'_5}.
\end{proof}
Let us also note that if $\beta_n\geq \delta_n+200\varepsilon$, we use the following estimate, obtained from \eqref{eq:delta_n9'},
\begin{align}\label{eq:delta_n9}
|\ell_n|\leq \frac{1}{q_n}(e^{\delta_n q_n}+\frac{2q_n+1}{2})\leq 2\max(e^{\delta_n q_n},1),
\end{align}
which implies, after combining with \eqref{eq:cnell_low_up}, that for $|\ell|\leq q_{n+1}/(6q_n)$ and some absolute constant $0<C<8$,
\begin{align}\label{eq:cnell_final}
c_{n,\ell}\leq C\max(|\ell|, e^{\delta_n q_n}, 1) e^{-\beta_nq_n}.
\end{align}
As a consequence of \eqref{eq:delta_n_min_cos} and Lemma \ref{lem:upperbddtildeP} and Corollary \ref{cor:prod_cos}, we have the following corollaries. Since the proofs are very standard, we leave all of them to the appendix.
\begin{cor}\label{cor:cos_prod_lower}
Let $I=[\ell_1, \ell_2]\subset {\mathbb Z}$ be such that there exists $j\in {\mathbb Z}$, $|j|<q_{n+1}/(6q_n)$, that satisfies
$$I\subset [m_n+j q_n+1, m_n+(j+1)q_n-1].$$
Then for $n>N(\varepsilon)$ large enough, we have
\begin{align*}
\prod_{\ell\in I}|\cos(\pi\theta_{\ell})|\geq e^{-\varepsilon(2q_n-|I|)} e^{-(\ln 2)|I|}
\end{align*}
\end{cor}
Combining the above Corollary with Lemma \ref{upperbounds}, we have
\begin{cor}\label{cor:A_upper}
Let $I=[\ell_1, \ell_2]\subset {\mathbb Z}$ be such that there exists $j\in {\mathbb Z}$, $|j|<q_{n+1}/(6q_n)$, that satisfies
$$I\subset [m_n+j q_n+1, m_n+(j+1)q_n-1].$$
Then for $n>N(\varepsilon)$ large enough, we have
\begin{align*}
\|A_{|I|}(\theta_{\ell_1})\|\leq e^{3\varepsilon q_n} e^{L |I|}.
\end{align*}
\end{cor}
And further, we have
\begin{cor}\label{cor:A_upper_mn}
Let $I=[\ell_1, \ell_2]\subset {\mathbb Z}$ be such that $\ell_1\in [(j-1)q_n+m_n+1, jq_n+m_n-1]$ and $\ell_2\in [jq_n+m_n+1, (j+1)q_n+m_n-1]$, for some $j\in {\mathbb Z}$, $|j|<q_{n+1}/(6q_n)$. For $n>N(\varepsilon)$ large enough we have
\begin{align*}
\|A_{|I|}(\theta_{\ell_1})\|\leq e^{7\varepsilon q_n} \frac{1}{c_{n,j}} e^{L|I|}.
\end{align*}
\end{cor}
Next lemma allows us to locate $\tilde{m}_n$.
\begin{lemma}
Let $a_{n+1}\in \{1,2,3\}$.
Suppose there exists $\tilde{m}_n\in [-q_n/2, q_n/2)$ satisfying \eqref{def:tildemn} and \eqref{eq:tildemn<13}, we have $\tilde{m}_n\in \mathcal{A}$, where
\begin{align*}
\mathcal{A}:=
\begin{cases}
\emptyset, \text{ if } a_{n+1}\in \{2, 3\} \text{ and } |\ell_n|\leq a_{n+1}-1\\
\{m_n-b q_{n-1}\}, \text{ if } a_{n+1}\in \{2, 3\} \text{ and } \ell_n=b a_{n+1}, \text{ where } b=\pm 1\\
\{m_n-b q_{n-1}\}, \text{ if } a_{n+1}=1, \ell_n=b, \text{ where } b=\pm 1\\
\{m_n+b(q_n-q_{n-1}), \text{ if } a_{n+1}=1, \ell_n=2b, a_{n+2}\geq 2, \text{ where } b=\pm 1\\
\{m_n-b q_{n-1}, m_n+b (q_n-q_{n-1})\}, \text{ if } a_{n+1}=a_{n+2}=1, \ell_n=2b, \text{ where } b=\pm 1
\end{cases}.
\end{align*}
\end{lemma}
We leave the proof of this lemma in the appendix.
\begin{proof}
We have by \eqref{eq:delta_n6'} and \eqref{eq:tildemn<13} that
\begin{align}\label{eq:delta_n14}
\|(m_n+\ell_n q_n-\tilde{m}_n)\alpha\|\leq \|\theta-\frac{1}{2}+(m_n+\ell_n q_n)\alpha\|+\|\theta-\frac{1}{2}+\tilde{m}_n\alpha\|<\|q_n\alpha\|.
\end{align}
We further divide into $a_{n+1}=1$ and $a_{n+1}=2,3$.
Case 1. If $a_{n+1}=2$ or $3$, we have by \eqref{eq:delta_n9'} that $|\ell_n|\leq a_{n+1}$.
Hence
\begin{align}\label{eq:delta_n15}
0<|m_n+\ell_n q_n-\tilde{m}_n|<a_{n+1}q_n+q_n<q_{n+1}+q_n.
\end{align}
If further $|\ell_n|\leq a_{n+1}-1$, we have
\begin{align*}
0<|m_n+\ell_n q_n-\tilde{m}_n|<a_{n+1}q_n<q_{n+1},
\end{align*}
which implies
\begin{align*}
\|(m_n+\ell_n q_n-\tilde{m}_n)\alpha\|\geq \|q_n\alpha\|,
\end{align*}
contradicting \eqref{eq:delta_n14}.
Hence $\tilde{m}_n$ can not exist if $0<|\ell_n|\leq a_{n+1}-1$.
If $|\ell_n|=a_{n+1}$, combining \eqref{eq:delta_n14} and \eqref{eq:delta_n15}, we must have
\begin{align*}
|m_n+\ell_n q_n-\tilde{m}_n|=q_{n+1},
\end{align*}
yielding
\begin{align*}
\tilde{m}_n=
\begin{cases}
m_n-q_{n-1}, \text{ if } \ell_n=a_{n+1}\\
m_n+q_{n-1}, \text{ if } \ell_n=-a_{n+1}
\end{cases}.
\end{align*}
Case 2. If $a_{n+1}=1$, we have by \eqref{eq:delta_n9'} that $|\ell_n|\leq 2$.
If $|\ell_n|=1$, we have
\begin{align*}
0<|m_n+\ell_n q_n-\tilde{m}_n|<2q_n<q_{n+2}.
\end{align*}
Combining this with \eqref{eq:delta_n14}, we must have
\begin{align*}
|m_n+\ell_n q_n-\tilde{m}_n|=q_{n+1}=q_n+q_{n-1},
\end{align*}
which implies
\begin{align*}
\tilde{m}_n=
\begin{cases}
m_n-q_{n-1}, \text{ if } \ell_n=1\\
m_n+q_{n-1}, \text{ if } \ell_n=-1
\end{cases}.
\end{align*}
If $|\ell_n|=2$ and $a_{n+2}\geq 2$, we have
\begin{align*}
0<|m_n+\ell_n q_n-\tilde{m}_n|<3q_n<q_{n+2},
\end{align*}
Combining this with \eqref{eq:delta_n14}, we must have
\begin{align*}
|m_n+\ell_n q_n-\tilde{m}_n|=q_{n+1}=q_n+q_{n-1},
\end{align*}
which implies
\begin{align*}
\tilde{m}_n=
\begin{cases}
m_n+q_n-q_{n-1}, \text{ if } \ell_n=2\\
m_n-q_n+q_{n-1}, \text{ if } \ell_n=-2
\end{cases}.
\end{align*}
If $|\ell_n|=2$ and $a_{n+2}=1$, we have
\begin{align*}
0<|m_n+\ell_n q_n-\tilde{m}_n|<3q_n<q_{n+3}.
\end{align*}
Combining this with \eqref{eq:delta_n14}, we must have
\begin{align*}
|m_n+\ell_n q_n-\tilde{m}_n|\in \{q_{n+1}, q_{n+2}\}.
\end{align*}
Hence
\begin{align*}
\tilde{m}_n=
\begin{cases}
m_n-q_{n-1} \text{ or } m_n+q_n-q_{n-1}, \text{ if } \ell_n=2\\
m_n+q_{n-1} \text{ or } m_n-q_n+q_{n-1}, \text{ if } \ell_n=-2
\end{cases}.
\end{align*}
\qed
\end{proof}
\section{Proof of Theorem \ref{main0}}\label{Sec:proof}
We first normalize the generalized eigenfunction $\phi$ such that
$|\phi(0)|^2+|\phi(-1)|^2=2$ (we normalize the right-hand-side to $2$ for simplicity).
For convenience we will assume
\begin{align}\label{assume:phi0=1}
|\phi(0)|\geq 1
\end{align}
and
\begin{align}\label{Shnol}
|\phi(k)|\leq C_0 |k|.
\end{align}
If $|\phi(0)|<1$ and $|\phi(-1)|>1$, we simply need to replace the $|\phi(0)|$ on the left-hand-sides of \eqref{eq:Green_at_0}, \eqref{phi0_2}, \eqref{phi0_3} and \eqref{eq:P2sq_I1_2} with $|\phi(-1)|$ and slightly adjust the right-hand-sides accordingly.
Theorem \ref{main0} is a consequence of the following lemma.
\begin{lemma}\label{lem:main}
Let $\phi$ be an eigenfunction satisfying $|\phi(0)|\geq 1$ and \eqref{Shnol}. Then for $n>N(\alpha, E, \lambda, \varepsilon, C_0)$ large enough and $\frac{1}{12}q_n\leq |k|<\frac{1}{12}q_{n+1}$, we have
\begin{align*}
|\phi(k)|\leq e^{-(L-\delta_n-650\varepsilon)|k|}.
\end{align*}
\end{lemma}
The remaining of the paper will be devoted to the proof of Lemma \ref{lem:main}, dividing into the following three cases.
{\it Case 1. $0\leq \delta_n\leq \beta_n\leq 300\varepsilon$.}
This case is essentially the Diophantine case that is handled in \cite{JYMaryland}.
We include a brief proof following \cite{JYMaryland} in the appendix.
{\it Case 2. $300\varepsilon\leq \beta_n\leq \delta_n+200\varepsilon$.}
This is an intermediate case. In this case we have $L>\beta_n+200\varepsilon$. One can combine the strategy in \cite{jl1} (which proves Anderson localization for the almost Mathieu operator in the case of $L>\beta$) with that in \cite{JYMaryland} to handle this case. Compared to the Case 3 below, Case 2 has a lot of simplifications, and in particular does not require our key estimates in Section \ref{Sec:key} at all.
{\it Case 3. $\beta_n>\max(\delta_n+200\varepsilon, 300\varepsilon)$.}
This is our main achievement of the paper.
From this point on, we shall omit the dependence of the parameters on $\alpha, E, \lambda$ and only emphasize on the dependence on $\varepsilon$.
We shall also assume $\beta_n>300\varepsilon$, since the $\beta_n\leq 300\varepsilon$ case is only proved in the appendix.
\section{Key technical lemmas}\label{Sec:key}
The following lemmas on $\tilde{P}_{q_n-1}$ are the key to prove Anderson localization in the sharp regime $L>\delta$.
These lemmas reveal that large potential values $|\tan(\pi\theta_{m_n+\ell q_n})|$ combined with Shnol's theorem \eqref{Shnol} yield improved upper bounds, roughly speaking with an additional $e^{(\delta_n-\beta_n)q_n}$ decay, for $|\tilde{P}_{q_n-1}(\theta_{m_n+\ell q_n+1})|$.
\begin{lemma}\label{lem:Pqn_mn}
For $n>N(\varepsilon)$ large enough we have
\begin{align*}
|\tilde{P}_{q_n-1}(\theta+(m_n-q_n+1)\alpha)| \leq C_0 e^{q_n (\tilde{L}+\delta_n-\beta_n+3\varepsilon)}.
\end{align*}
\end{lemma}
\begin{proof}
Let $I:=[m_n-q_n+1,m_n-1]$.
By Green's formula \eqref{Green_tildeP} and assumption \eqref{assume:phi0=1}, we have
\begin{align}\label{eq:Green_at_0}
1\leq |\phi(0)|
\leq &\frac{|\tilde{P}_{m_n-1}(\theta_1)|}{|\tilde{P}_{q_n-1}(\theta_{m_n-q_n+1})|} \prod_{j=m_n-q_n+1}^0|\cos(\pi \theta_j)|
\cdot |\phi(m_n-q_n)|\\
&+\frac{|\tilde{P}_{-m_n+q_n-1}(\theta_{m_n-q_n+1})|}{|\tilde{P}_{q_n-1}(\theta_{m_n-q_n+1})|}\prod_{j=0}^{m_n-1}|\cos(\pi\theta_j)| \cdot |\phi(m_n)| \notag
\end{align}
Note that by Corollary \ref{cor:prod_cos} and Lemma \ref{lem:upperbddtildeP}, equation \eqref{eq:Green_at_0} yields
\begin{align}\label{eq:Green_at_0_2}
|\tilde{P}_{q_n-1}(\theta_{m_n-q_n+1})| \leq C(\varepsilon) e^{m_n (\tilde{L}+\varepsilon)} e^{(q_n-m_n)(-\ln 2+\varepsilon)} |\phi(m_n-q_n)|
+C(\varepsilon) e^{(q_n-m_n)(\tilde{L}+\varepsilon)} e^{m_n (-\ln 2+\varepsilon)} |\phi(m_n)|.
\end{align}
Next, let us consider the eigenvalue equation:
\begin{align}\label{eq:ev_at_mn}
\phi(m_n+1)+\phi(m_n-1)+\lambda \tan(\pi \theta_{m_n})\phi(m_n)=E\phi(m_n)
\end{align}
By \eqref{Shnol}, we have
\begin{align*}
|\phi(k)|\leq C_0 |k|
\end{align*}
for any $|k| \geq 1$.
Hence \eqref{eq:ev_at_mn} and \eqref{def:cnell} imply that for $n$ large enough
\begin{align*}
\frac{\lambda}{2 c_{n,0}} |\phi(m_n)| \leq |\lambda \tan(\pi \theta_{m_n})-E|\cdot |\phi(m_n)| &\leq 2 \max(|\phi(m_n+1)|,|\phi(m_n-1)|)\leq 2C_0 q_n.
\end{align*}
This implies
\begin{align}\label{est:mn}
|\phi(m_n)| \leq \frac{4}{\lambda} C_0 c_{n,0} q_n.
\end{align}
Similarly one has
\begin{align}\label{est:mn-qn}
|\phi(m_n-q_n)|\leq \frac{4}{\lambda} C_0 c_{n,-1} q_n.
\end{align}
Plugging \eqref{est:mn} and \eqref{est:mn-qn} into \eqref{eq:Green_at_0_2}, using \eqref{LEtildeLE} we have
\begin{align*}
|\tilde{P}_{q_n-1}(\theta_{m_n-q_n+1})|
\leq &C_0 e^{q_n (-\ln 2+2\varepsilon)} \max(e^{m_n L}c_{n,0}, e^{(q_n-m_n)L}c_{n,-1})\\
\leq &C_0 e^{q_n (\tilde{L}+2\varepsilon)} \max(c_{n,0}, c_{n,-1}).
\end{align*}
By \eqref{eq:cnell_final}, we have
\begin{align}
|\tilde{P}_{q_n-1}(\theta_{m_n-q_n+1})|
\leq C_0 e^{q_n (\tilde{L}-\beta_n+3\varepsilon)} \max(e^{\delta_n q_n}, 1).
\end{align}
\qed
\end{proof}
Furthermore, we have for any $|\ell |\leq 2q_{n+1}/(3q_n)$
\begin{lemma}\label{lem:Pqn_mn+ell}
We have
\begin{align*}
|\tilde{P}_{q_n-1}(\theta_{m_n+\ell q_n+1})| \leq e^{q_n (\tilde{L}-\beta_n+4\varepsilon)}\max(e^{\delta_n q_n}, |\ell|, 1).
\end{align*}
\end{lemma}
\begin{proof}
By telescoping argument, we have
\begin{align*}
&|\tilde{P}_{q_n-1}(\theta_{m_n+\ell q_n+1})-\tilde{P}_{q_n-1}(\theta_{m_n-q_n+1})|\\
\leq &\|F_{q_n-1}(\theta_{m_n+\ell q_n+1})-F_{q_n-1}(\theta_{m_n-q_n+1})\|\\
\leq &\sum_{j=0}^{q_n-2} \| F_{q_n-j-2}(\theta_{m_n+\ell q_n+2+j})\|\cdot \| F(\theta_{m_n-q_n+1+j})-F(\theta_{m_n+\ell q_n+1+j})\|\cdot \| F_{j}(\theta_{m_n-q_n+1}) \|,
\end{align*}
where $F_0:=\mathrm{Id}$.
By \eqref{tildePinF} and Lemma \ref{lem:upperbddtildeP}, we have that for $\ell\neq -1$,
\begin{align}\label{eq:P-P_tele}
&|\tilde{P}_{q_n-1}(\theta_{m_n+\ell q_n+1})-\tilde{P}_{q_n-1}(\theta_{m_n-q_n+1})| \notag\\
\leq &\sum_{j=0}^{q_n-2} C(\varepsilon) e^{(q_n-2)(\tilde{L}+\varepsilon)} \|\theta_{m_n+\ell q_n+1+j}-\theta_{m_n-q_n+1+j}\|\notag\\
\leq &\sum_{j=0}^{q_n-2} C(\varepsilon) e^{(q_n-2)(\tilde{L}+\varepsilon)}\|(\ell+1) q_n \alpha\|\notag\\
\leq &|\ell+1| e^{q_n(\tilde{L}-\beta_n+2\varepsilon)}.
\end{align}
Combining \eqref{eq:P-P_tele} with Lemma \ref{lem:Pqn_mn}, we have
\begin{align}\label{eq:C0<eqn}
|\tilde{P}_{q_n-1}(\theta_{m_n+\ell q_n+1})|
\leq &C_0 e^{q_n (\tilde{L}-\beta_n+3\varepsilon)} \max(e^{\delta_n q_n},1)+ |\ell+1| e^{q_n(\tilde{L}-\beta_n+2\varepsilon)} \notag\\
\leq &e^{q_n (\tilde{L}-\beta_n+4\varepsilon)}\max(e^{\delta_n q_n}, |\ell|, 1),
\end{align}
where we require $C_0<e^{q_n\varepsilon}$ in \eqref{eq:C0<eqn}. \qed
\end{proof}
Lemma \ref{lem:Pqn_mn+ell} implies the following lemma.
\begin{lemma}\label{lem:cor_Pqn}
For $|\ell |<2q_{n+1}/(3q_n)$, assume $k<2q_n$ and
$$y\leq \ell q_n+m_n+1,\ \ \text{and}\ \ y+k-1\geq (\ell +1)q_n+m_n-1.$$
We then have
\begin{align*}
|\tilde{P}_k(\theta_y)|\leq \max(e^{\delta_n q_n}, |\ell |, 1) e^{-(\beta_n-6\varepsilon) q_n} e^{k\tilde{L}}.
\end{align*}
\end{lemma}
\begin{proof}
Note that when $y=\ell q_n+m_n+1$ and $y+k-1=(\ell+1)q_n+m_n-1$, this is the previous lemma.
Hence let us consider the case when $(y, y+k-1)\neq (\ell q_n+m_n+1, (\ell+1)q_n+m_n-1)$.
We shall divide into three cases:
1. $y=\ell q_n+m_n+1$, $y+k-1>(\ell+1)q_n+m_n-1$;
2. $y<\ell q_n+m_n+1$, $y+k-1=(\ell+1)q_n+m_n-1$;
3. $y<\ell q_n+m_n+1$, $y+k-1>(\ell+1)q_n+m_n-1$.
These three cases are very similar to each other, hence we will only present the proof for Case 1 in details.
{\it Case 1.} Let us consider the normalized transfer matrix
\begin{align*}
F_k(\theta_y)=F_{k-q_n}(\theta_{(\ell+1)q_n+m_n+1})F(\theta_{(\ell+1)q_n+m_n})F_{q_n-1}(\theta_{\ell q_n+m_n+1})
\end{align*}
Recall
\begin{align*}
F(\theta_{(\ell+1)q_n+m_n})=
\left(
\begin{matrix}
\cos(\pi\theta_{(\ell+1)q_n+m_n})E-\lambda \sin(\pi\theta_{(\ell+1)q_n+m_n})\ \ &-\cos(\pi\theta_{(\ell+1)q_n+m_n})\\
\cos(\pi\theta_{(\ell+1)q_n+m_n}) &0
\end{matrix}
\right)
\end{align*}
and
\begin{align*}
&F_{q_n-1}(\theta_{\ell q_n+m_n+1})\\=&
\left(
\begin{matrix}
\tilde{P}_{q_n-1}(\theta_{\ell q_n+m_n+1})\ \ &-\tilde{P}_{q_n-2}(\theta_{\ell q_n+m_n+2})\cos(\pi\theta_{\ell q_n+m_n+1})\\
\tilde{P}_{q_n-2}(\theta_{\ell q_n+m_n+1})\cos(\pi\theta_{(\ell+1)q_n+m_n-1})\ \ &-\tilde{P}_{q_n-3}(\theta_{\ell q_n+m_n+2})\cos(\pi\theta_{\ell q_n+m_n+1})\cos(\pi\theta_{(\ell+1)q_n+m_n-1})
\end{matrix}
\right)
\end{align*}
By Lemmas \ref{lem:upperbddtildeP}, \ref{lem:Pqn_mn+ell} and inequality \eqref{eq:cnell_final}, we have
\begin{align*}
F(\theta_{(\ell+1)q_n+m_n})F_{q_n-1}(\theta_{\ell q_n+m_n+1})=:
\left(
\begin{matrix}
b_1 &b_2\\
b_3 & b_4
\end{matrix}
\right),
\end{align*}
satisfies
\begin{align*}
\begin{cases}
|b_1|\leq C \max(e^{\delta_n q_n}, |\ell |, 1) e^{(\tilde{L}-\beta_n+4\varepsilon) q_n}\\
|b_3|\leq \max(e^{\delta_n q_n}, |\ell |, 1) c_{n,\ell+1} e^{(\tilde{L}-\beta_n+4\varepsilon) q_n} \\
|b_4|\leq c_{n,\ell+1}e^{(\tilde{L}+\varepsilon) q_n}\leq C \max(e^{\delta_n q_n}, |\ell |, 1) e^{(\tilde{L}-\beta_n+\varepsilon) q_n}
\end{cases}
\end{align*}
For $F_{k-q_n}(\theta_{(a+1)q_n+m_n+1})$ we use the estimate from Lemma \ref{lem:upperbddtildeP},
resulting in the desired estimate for the upper left corner of $F_k(\theta_y)$, which is $\tilde{P}_k(\theta_y)$, as follows
\begin{align*}
|\tilde{P}_k(\theta_y)|\leq C(\varepsilon) \max(e^{\delta_n q_n}, |\ell |, 1) e^{(\tilde{L}-\beta_n+4\varepsilon) q_n} e^{(k-q_n)(\tilde{L}+\varepsilon)}\leq \max(e^{\delta_n q_n}, |\ell |, 1) e^{-(\beta_n-6\varepsilon) q_n} e^{k\tilde{L}}.
\end{align*}
This proves Case 1. \qed
\end{proof}
In order to unify the estimates for $\tilde{P}$ in both the $\beta_n\geq \delta_n+200\varepsilon$ and $\beta_n<\delta_n+200\varepsilon$ cases, we define
\begin{align}\label{def:g}
g_{k,\ell}:=
\begin{cases}
\max(e^{\delta_n q_n}, |\ell|, 1) e^{-(\beta_n-6\varepsilon)q_n}\ &\text{ if } \beta_n\geq \delta_n+200\varepsilon\\
e^{2\varepsilon k} &\text{ if } \beta_n<\delta_n+200\varepsilon
\end{cases}
\end{align}
By Lemmas \ref{lem:upperbddtildeP} and \ref{lem:cor_Pqn}, we have
\begin{cor}\label{cor:Pk_key}
For $|\ell |<2q_{n+1}/(3q_n)$, assume $k<2q_n$ and
$$y\leq \ell q_n+m_n+1,\ \ \text{and}\ \ y+k-1\geq (\ell +1)q_n+m_n-1.$$
We then have
\begin{align*}
|\tilde{P}_k(\theta_y)|\leq g_{k,\ell} e^{k\tilde{L}}.
\end{align*}
\end{cor}
\section{The case of non-resonant singularity: $\operatorname{dist}(m_n, q_n {\mathbb Z})>b_n$}\label{Sec:C1}
We will first prove non-resonant $y$'s can be dominated by resonances, and then study the relation between adjacent resonant regions.
\subsection{Non-resonance}\
Assume $\ell q_n+b_n \leq y \leq (\ell+1)q_n-b_n$.
We introduce some notations:
\begin{align*}
\begin{cases}
I^-:=[\ell q_n+b_n, \ell q_n+m_n-1],\\
I^+:=[\ell q_n+m_n+1, (\ell+1)q_n-b_n],\\
|\phi(x_0^-)|:=\max_{y\in I^-} |\phi(y)|,\\
|\phi(x_0^+)|:=\max_{y\in I^+} |\phi(y)|,\\
R_{\ell}:=[\ell q_n-b_n, \ell q_n+b_n],\\
r_{\ell}:=\max_{k\in R_{\ell}} |\phi(k)|.
\end{cases}
\end{align*}
\begin{lemma}\label{lem:C2_n-r}
We have,
for $y=\ell q_n+m_n$,
\begin{align}\label{eq:I-capI+}
|\phi(y)| \leq e^{15\varepsilon q_n} c_{n,\ell} \max(e^{-(y-\ell q_n)L} r_{\ell}, e^{-((\ell+1)q_n-y)L} r_{\ell+1})
\end{align}
For any $y \in I^-$,
\begin{align}\label{eq:I-}
|\phi(y)| \leq e^{15\varepsilon q_n} \max(e^{-(y-\ell q_n)L} r_{\ell}, c_{n,\ell} e^{-((\ell+1)q_n-y)L} r_{\ell+1}).
\end{align}
For any $y \in I^+$,
\begin{align}\label{eq:I+}
|\phi(y)| \leq e^{15\varepsilon q_n} \max(c_{n,\ell} e^{-(y-\ell q_n)L} r_{\ell}, e^{-((\ell+1)q_n-y)L} r_{\ell+1}).
\end{align}
\end{lemma}
We leave the proof in the appendix.
\subsection{Resonance}
The main lemma of this section is the following.
\begin{lemma}\label{lem:21_ra<final}
For any $\ell\neq 0$, $|\ell|\leq q_{n+1}/(6q_n)$,
\begin{align*}
r_{\ell} \leq e^{40\varepsilon q_n} \frac{e^{-q_nL}}{\max(|\ell|,1)} \max(r_{\ell-1}, r_{\ell+1})\times
\begin{cases}
\max(|\ell|, e^{\delta_n q_n}), &\text{ if } \beta_n\geq \delta_n+200\varepsilon\\
e^{\beta_n q_n}, &\text{ if } \beta_n<\delta_n+200\varepsilon
\end{cases}.
\end{align*}
\end{lemma}
\begin{proof}
This lemma is built on the following lemma.
\begin{lemma}\label{lem:21_assume_Ia}
Assume that there exists $x_1\in I_{\ell}$ such that
\begin{align}\label{Case1_assume}
|\tilde{P}_{2q_n-1}(\theta_{x_1})|\geq \max(|\ell |, 1) e^{-\beta_n q_n} e^{(\tilde{L}-2\varepsilon)(2q_n-1)}.
\end{align}
Then we have
\begin{align*}
r_{\ell} \leq e^{39\varepsilon q_n} \frac{e^{\beta_n q_n}}{\max(|\ell|,1)} e^{-q_n L}\, \max(c_{n,\ell-1} r_{\ell-1}, c_{n,\ell} r_{\ell+1}).
\end{align*}
\end{lemma}
We postpone the proof of this lemma till the end of the section.
As a corollary of Lemma \ref{lem:21_assume_Ia}, we have the following.
\begin{lemma}\label{lem:21_assume_la_beta><delta}
Under the assumption of Lemma \ref{lem:21_assume_Ia}, we have
\begin{align*}
r_{\ell} \leq e^{40\varepsilon q_n} \frac{e^{-q_nL}}{\max(|\ell|,1)} \max(r_{\ell-1}, r_{\ell+1})\times
\begin{cases}
\max(|\ell|, e^{\delta_n q_n}, 1), &\text{ if } \beta_n\geq \delta_n+200\varepsilon\\
e^{\beta_n q_n}, &\text{ if } \beta_n<\delta_n+200\varepsilon
\end{cases}.
\end{align*}
\end{lemma}
\begin{proof}
If $\beta_n\geq \delta_n+200\varepsilon$, bound the $c_{n,j}$'s by \eqref{eq:cnell_final}. Otherwise trivially bound the $c_{n,j}$'s by $1$. \qed
\end{proof}
Next we show the assumption in Lemma \ref{lem:21_assume_la_beta><delta} can not hold for $\ell=0$.
\begin{lemma}\label{lem:21_I0_small}
For any $x\in I_0$, $|\tilde{P}_{2q_n-1}(\theta_x)|<e^{-\beta_n q_n} e^{(\tilde{L}-2\varepsilon)(2q_n-1)}$
\end{lemma}
\begin{proof}
We prove by contradiction. Assume that there exists $x_1\in I_0$ such that
$$|\tilde{P}_{2q_n-1}(\theta_x)|\geq e^{-\beta_n q_n} e^{(\tilde{L}-2\varepsilon)(2q_n-1)}.$$
Then Lemma \ref{lem:21_assume_la_beta><delta} implies that, using \eqref{assume:phi0=1} and \eqref{Shnol},
\begin{align}\label{phi0_2}
1\leq |\phi(0)|\leq &r_0\leq e^{-(L-40\varepsilon) q_n} \max(r_{-1}, r_1) \times
\begin{cases}
e^{\delta_n q_n}, \text{ if } \beta_n\geq \delta_n+200\varepsilon\\
e^{\beta_n q_n}, \text{ if } \beta_n<\delta_n+200\varepsilon
\end{cases}\\
\leq &C_0q_n \begin{cases}
e^{(\delta_n-L+40\varepsilon) q_n} e^{\delta_n q_n}, \text{ if } \beta_n\geq \delta_n+200\varepsilon \notag\\
e^{(\delta_n-L+40\varepsilon) q_n} e^{\beta_n q_n}, \text{ if } \beta_n<\delta_n+200\varepsilon
\end{cases} \notag\\
<&1. \notag
\end{align}
Contradiction. \qed
\end{proof}
Combining Corollary \ref{cor:1_res_uni} with Lemma \ref{lem:21_I0_small}, we obtain the following
\begin{lemma}\label{lem:21_P2q_n_large}
For any $\ell \neq 0$, $|\ell|\leq q_{n+1}/(6q_n)$, we have
$$|\tilde{P}_{2q_n-1}(\theta_{x_1})|\geq \max(|\ell|, 1) e^{-\beta_n q_n} e^{(\tilde{L}-2\varepsilon)(2q_n-1)}.$$
\end{lemma}
Finally, Lemma \ref{lem:21_ra<final} follows from combining Lemma \ref{lem:21_assume_la_beta><delta} with Lemma \ref{lem:21_P2q_n_large}. \qed
\end{proof}
Now let us prove Lemma \ref{lem:21_assume_Ia}.
\subsection*{Proof of Lemma \ref{lem:21_assume_Ia}}\
Recalling that $r_{\ell}=\max_{k\in R_{\ell}} |\phi(k)|$, let $k\in R_{\ell}$.
Expanding $\phi(k)$ in the interval $I=[x_1, x_2]$ using \eqref{PkG}, where $x_2=x_1+2q_n-2$, we have,
\begin{align}\label{eq:hhh1}
|\phi(k)|
\leq &\frac{|\tilde{P}_{x_2-k}(\theta_{k+1})|}{|\tilde{P}_{2q_n-1}(\theta_{x_1})|} \prod_{j=x_1}^k |\cos(\pi \theta_j)|\, |\phi(x_1-1)|+
\frac{|\tilde{P}_{k-x_1}(\theta_{x_1})|}{|\tilde{P}_{2q_n-1}(\theta_{x_1})|} \prod_{j=k}^{x_2} |\cos(\pi \theta_j)|\, |\phi(x_2+1)|
\end{align}
\subsection*{Case 1. If $x_1\in [(\ell-1)q_n+m_n+1, \ell q_n-\lfloor q_n/2\rfloor -1]$.}\
Note that since
$$k+1\leq \ell q_n+b_n+1\leq \ell q_n+m_n+1,$$
and
$$x_2=x_1+2q_n-2\geq (\ell+1)q_n+m_n-1,$$
Corollary \ref{cor:Pk_key} implies
hence
$$|\tilde{P}_{x_2-k}(\theta_{k+1})|\leq g_{|x_2-k|, \ell} e^{|x_2-k|\tilde{L}}.$$
By Corollary \ref{cor:prod_cos} and \eqref{eq:delta_n_min_cos}, we have that
\begin{align*}
&\prod_{j=k}^{x_2}|\cos(\pi \theta_j))|\leq C(\varepsilon) c_{n,\ell} c_{n,\ell+1} e^{(-\ln{2}+\varepsilon)|x_2-k|},\ \ \text{ and }\\
&\prod_{j=x_1}^k |\cos(\pi \theta_j))|\leq C(\varepsilon) e^{(-\ln 2+\varepsilon)|k-x_1|}.
\end{align*}
By Lemma \ref{lem:upperbddtildeP} we have
\begin{align*}
|\tilde{P}_{k-x_1}(\theta_{x_1})|\leq C(\varepsilon) e^{(\tilde{L}+\varepsilon)|k-x_1|}.
\end{align*}
Plugging these upper bounds together with the lower bound \eqref{Case1_assume} into \eqref{eq:hhh1}, we obtain
\begin{align}\label{eq:111}
|\phi(k)| \leq &C(\varepsilon) \frac{g_{|x_2-k|, \ell} e^{\beta_n q_n}}{\max(|\ell |,1)} e^{5\varepsilon q_n} e^{-L |k-x_1|} |\phi(x_1-1)|+C(\varepsilon) \frac{c_{n,\ell}c_{n,\ell+1}e^{\beta_n q_n}}{\max(|\ell |,1)} e^{5\varepsilon q_n} e^{-L|x_2-k|} |\phi(x_2+1)|.
\end{align}
Equation \eqref{eq:I+} of Lemma \ref{lem:C2_n-r} implies
\begin{align*}
|\phi(x_1-1)|\leq e^{15\varepsilon q_n} \max\{c_{n,\ell-1} e^{-(x_1-(\ell-1)q_n)L} r_{\ell-1}, e^{-(\ell q_n-x_1)L} r_{\ell}\}
\end{align*}
and
\begin{align*}
|\phi(x_2+1)|\leq e^{15\varepsilon q_n} \max\{c_{n,\ell+1} e^{-(x_2+1-(\ell+1)q_n)L} r_{\ell+1}, e^{-((\ell+2)q_n-x_2-1)L} r_{\ell+2}\}
\end{align*}
Plugging the above estimates into \eqref{eq:111}, we have
\begin{align}\label{eq:21_ra_leq_1}
|\phi(k)| \leq e^{24\varepsilon q_n} \frac{e^{\beta_n q_n}}{\max(|\ell|,1)}
\max
\begin{cases}
c_{n,\ell-1} g_{|x_2-k|, \ell} e^{-q_n L} r_{\ell-1}\\
g_{|x_2-k|, \ell} e^{-q_n L} r_{\ell}\\
c_{n,\ell} (c_{n,\ell+1})^2 e^{-q_n L} r_{\ell+1}\\
c_{n,\ell} c_{n,\ell+1} e^{-2q_n L} r_{\ell+2}
\end{cases}
\end{align}
\subsection*{Case 2. If $x_1\in [(\ell-1)q_n-\lfloor q_n/2\rfloor, (\ell-1)q_n+m_n]$}\
We have $$x_2\in [(\ell+1)q_n-\lfloor q_n/2\rfloor-2, (\ell+1)q_n+m_n-2].$$
By Corollary \ref{cor:prod_cos} and \eqref{def:cnell} we have that
\begin{align*}
&\prod_{j=k}^{x_2}|\cos(\pi \theta_j))|\leq C(\varepsilon) c_{n,\ell} e^{(-\ln{2}+\varepsilon)|x_2-k|},\ \ \text{ and }\\
&\prod_{j=x_1}^k |\cos(\pi \theta_j))|\leq C(\varepsilon) c_{n,\ell-1} e^{(-\ln 2+\varepsilon)|k-x_1|}.
\end{align*}
By Lemma \ref{lem:upperbddtildeP} we have
\begin{align*}
|\tilde{P}_{k-x_1}(\theta_{x_1})|\leq C(\varepsilon) e^{(\tilde{L}+\varepsilon)|k-x_1|} \text{ and }
|\tilde{P}_{x_2-k}(\theta_{k+1})|\leq C(\varepsilon) e^{(\tilde{L}+\varepsilon)|x_2-k|}.
\end{align*}
Plugging these upper bounds together with the lower bound \eqref{Case1_assume} into \eqref{eq:hhh1}, we have
\begin{align}\label{eq:hhh3}
|\phi(k)|
\leq &C(\varepsilon) e^{6\varepsilon q_n} \frac{e^{\beta_n q_n}}{\max(|\ell|, 1)}
\left(e^{-|k-x_1|L} c_{n,\ell-1} |\phi(x_1-1)|+e^{-|x_2-k|L} c_{n,\ell} |\phi(x_2+1)|\right).
\end{align}
Lemma \ref{lem:C2_n-r} implies
\begin{align*}
&|\phi(x_1-1)|\leq e^{15\varepsilon q_n} \max\{
c_{n,\ell-2} e^{-(x_1-(\ell-2)q_n)L} r_{\ell-2},
e^{-|(\ell-1)q_n-x_1|L} r_{\ell-1},
c_{n,\ell-1} e^{-(\ell q_n-x_1)L} r_{\ell}\}, \ \ \text{ and }\\
&|\phi(x_2+1)|\leq e^{15\varepsilon q_n} \max\{
c_{n,\ell} e^{-(x_2-\ell q_n)L} r_{\ell},
e^{-|(\ell+1)q_n-x_2|L} r_{\ell+1},
c_{n,\ell+1} e^{-((\ell+2)q_n-x_2)L} r_{\ell+2}\}.
\end{align*}
Plugging these estimates in \eqref{eq:hhh3}, we have
\begin{align}\label{eq:21_ra_leq_3}
|\phi(k)|\leq e^{24\varepsilon q_n} \frac{e^{\beta_n q_n}}{\max(|\ell|,1)} \max
\begin{cases}
c_{n,\ell-2}c_{n,\ell-1} e^{-2 q_n L} r_{\ell-2}\\
e^{-q_n L} c_{n,\ell-1} r_{\ell-1}\\
\max((c_{n,\ell-1})^2, (c_{n,\ell})^2) e^{-q_n L} r_{\ell}\\
c_{n,\ell} e^{-q_n L} r_{\ell+1}\\
c_{n,\ell} c_{n,\ell+1} e^{-2 q_n L} r_{\ell+2}
\end{cases}
\end{align}
Putting estimates in both cases \eqref{eq:21_ra_leq_1} and \eqref{eq:21_ra_leq_3} together, we obtain, after setting $|\phi(k)|=r_{\ell}$,
\begin{align}\label{eq:21_ra_leq_4}
r_{\ell} \leq e^{24\varepsilon q_n} \frac{e^{\beta_n q_n}}{\max(|\ell|,1)} \max
\begin{cases}
c_{n,\ell-2}c_{n,\ell-1} e^{-2 q_n L} r_{\ell-2}\\
c_{n,\ell-1} e^{-q_n L} r_{\ell-1} \max_{k\in R_{\ell}} g_{|x_2-k|, \ell} \\
e^{-q_n L} c_{n,\ell-1} r_{\ell-1}\\
e^{-q_n L} r_{\ell} \cdot \max_{k\in R_{\ell}} g_{|x_2-k|, \ell} \\
\max((c_{n,\ell-1})^2, (c_{n,\ell})^2) e^{-q_n L} r_{\ell}\\
c_{n,\ell} e^{-q_n L} r_{\ell+1}\\
c_{n,\ell} c_{n,\ell+1} e^{-2 q_n L} r_{\ell+2}
\end{cases}
\end{align}
We have by Corollary \ref{cor:A_upper_mn} that
\begin{align*}
r_{\ell-2} \leq \frac{1}{c_{n,\ell-2}}e^{7\varepsilon q_n} e^{L(q_n+2b_n)}<\frac{1}{c_{n,\ell-2}}e^{9\varepsilon q_n} e^{Lq_n} r_{\ell-1},
\end{align*}
and similarly
\begin{align*}
r_{\ell+2} \leq \frac{1}{c_{n,\ell+1}} e^{9\varepsilon q_n} e^{q_n L} r_{\ell+1}.
\end{align*}
Plugging these estimates into \eqref{eq:21_ra_leq_4} yields
\begin{align}\label{eq:21_ra_leq_5}
r_{\ell} \leq e^{33\varepsilon q_n} \frac{e^{\beta_n q_n}}{\max(|\ell|,1)} \max
\begin{cases}
c_{n,\ell-1} e^{-q_n L} r_{\ell-1}\\
c_{n,\ell-1} e^{-q_n L} r_{\ell-1} \cdot \max_{k\in R_{\ell}} g_{|x_2-k|, \ell}\\
e^{-q_n L} r_{\ell} \cdot \max_{k\in R_{\ell}} g_{|x_2-k|, \ell}\\
\max((c_{n,\ell-1})^2, (c_{n,\ell})^2) e^{-q_n L} r_{\ell}\\
c_{n,\ell} e^{-q_n L} r_{\ell+1}
\end{cases}
\end{align}
Next we further bound this, dividing into two cases.
Case (i). If $\beta_n\geq \delta_n+200\varepsilon$, using \eqref{def:g} to bound
\begin{align*}
\max_{k\in R_{\ell}} g_{|x_2-k|,\ell}
\leq &\max(e^{\delta_n q_n}, |\ell|) e^{-(\beta_n-6\varepsilon)q_n}\\
\leq &e^{6\varepsilon q_n} \max(e^{-200\varepsilon q_n}, \frac{1}{6q_n})<e^{6\varepsilon q_n},
\end{align*}
and using \eqref{eq:cnell_final} to bound
\begin{align*}
\max((c_{n,\ell-1})^2, (c_{n,\ell})^2)\leq \max(c_{n,\ell-1}, c_{n,\ell}) \leq C \max(|\ell|, e^{\delta_n q_n}, 1) e^{-\beta_n q_n},
\end{align*}
we arrive at
\begin{align}\label{eq:21_ra_leq_6}
r_{\ell} \leq e^{39\varepsilon q_n} \frac{e^{\beta_n q_n}}{\max(|\ell|,1)} e^{-q_n L} \max
\begin{cases}
c_{n,\ell-1} r_{\ell-1}\\
\max(|\ell|, e^{\delta_n q_n}, 1) e^{-\beta_n q_n} r_{\ell}\\
c_{n,\ell} r_{\ell+1}
\end{cases}.
\end{align}
Note that the coefficient of $r_{\ell}$ can be bounded by, using \eqref{eq:L>deltan+100},
\begin{align}\label{eq:rell_drop}
e^{39\varepsilon q_n} \frac{e^{\beta_n q_n}}{\max(|\ell|,1)} e^{-q_n L} \max(|\ell|, e^{\delta_n q_n}, 1) e^{-\beta_n q_n}\leq e^{-(L-\delta_n-39\varepsilon)q_n}<1.
\end{align}
Hence \eqref{eq:21_ra_leq_6} implies
\begin{align}\label{eq:21_ra_leq_7}
r_{\ell} \leq e^{39\varepsilon q_n} \frac{e^{\beta_n q_n}}{\max(|\ell|,1)} e^{-q_n L} \max(c_{n,\ell-1} r_{\ell-1}, c_{n,\ell} r_{\ell+1}).
\end{align}
Case (ii). If $\beta_n<\delta_n+200\varepsilon$, using \eqref{def:g} to bound
\begin{align*}
\max_{k\in R_{\ell}} g_{|x_2-k|,\ell}
\leq e^{2\varepsilon |x_2-k|}\leq e^{3\varepsilon q_n},
\end{align*}
and trivially bounding
\begin{align*}
\max((c_{n,\ell-1})^2, (c_{n,\ell})^2)\leq 1,
\end{align*}
we obtain from \eqref{eq:21_ra_leq_5} that
\begin{align}\label{eq:21_ra_leq_8}
r_{\ell} \leq e^{36\varepsilon q_n} \frac{e^{\beta_n q_n}}{\max(|\ell|,1)} e^{-q_n L} \max(c_{n,\ell-1} r_{\ell-1}, r_{\ell}, c_{n,\ell} r_{\ell+1}).
\end{align}
Note that the coefficient of $r_{\ell}$ can be bounded by, using \eqref{eq:L>deltan+100},
\begin{align}\label{eq:rell_drop_2}
e^{36\varepsilon q_n} \frac{e^{\beta_n q_n}}{\max(|\ell|,1)} e^{-q_n L}\leq e^{-(L-\delta_n-236\varepsilon)q_n}<1.
\end{align}
Hence \eqref{eq:21_ra_leq_8} implies
\begin{align}\label{eq:21_ra_leq_9}
r_{\ell} \leq e^{36\varepsilon q_n} \frac{e^{\beta_n q_n}}{\max(|\ell|,1)} e^{-q_n L} \max(c_{n,\ell-1} r_{\ell-1}, c_{n,\ell} r_{\ell+1}).
\end{align}
Lemma \ref{lem:21_assume_Ia} follows from combining \eqref{eq:21_ra_leq_7} and \eqref{eq:21_ra_leq_9}.
\qed
\section{The case of the resonant singularity: $\operatorname{dist}(m_n, q_n{\mathbb Z})\leq b_n$}\label{Sec:C2}
Let us introduce some notations:
\begin{align*}
R_{\ell}^+:=[\ell q_n+m_n+1, \ell q_n+b_n] \text{ and } R_{\ell}^-:=[\ell q_n-b_n, \ell q_n+m_n-1],
\end{align*}
and
\begin{align}\label{def:ra+-}
r_{\ell}^+:=\max_{y\in R_{\ell}^+} |\phi(y)| \text{ and } r_{\ell}^-:=\max_{y\in R_{\ell}^-} |\phi(y)|
\end{align}
\subsection{Non-resonance}
\begin{lemma}\label{lem:22_non-to-res}
If $\ell q_n+b_n<y<(\ell+1)q_n-b_n$, for some $|\ell |\leq q_{n+1}/(6q_n)$. Then
\begin{align*}
|\phi(y)|\leq e^{30\varepsilon q_n} \max(e^{-(y-\ell q_n)L}r_{\ell}^+, e^{-((\ell+1)q_n-y)L}r_{\ell+1}^-).
\end{align*}
\end{lemma}
We leave the proof in the appendix.
\subsection{Resonance}\
Assume without loss of generality that $0<m_n\leq b_n$.
The main lemma of this section is the following.
\begin{lemma}\label{lem:22_res_final}
For any $\ell\neq 0$ such that $|\ell|<q_{n+1}/(6q_n)$, we have
\begin{align*}
r_{\ell}\leq &e^{50\varepsilon q_n}\frac{e^{-q_nL}}{\max(|\ell|, 1)} \max(r_{\ell-1}, r_{\ell+1})\times
\begin{cases}
\max(|\ell|, e^{\delta_n q_n}), \text{ if } \beta_n\geq \delta_n+200\varepsilon,\\
e^{\beta_n q_n}, \text{ if }\beta_n< \delta_n+200\varepsilon
\end{cases}
\end{align*}
\end{lemma}
\begin{proof}
This lemma is mainly built on the following lemma.
\begin{lemma}\label{lem:22_res}
Assume that there exists $x_1\in I_{\ell}$, for some $|\ell | <q_{n+1}/(6q_n)$, such that
\begin{align}\label{ieq:22_res}
|\tilde{P}_{2q_n-1}(\theta_{x_1})|\geq \max(|\ell|, 1) e^{-\beta_n q_n}e^{(\tilde{L}-2\varepsilon)(2q_n-1)}.
\end{align}
We then have
\begin{align*}
r_{\ell}^-\leq
e^{49\varepsilon q_n} \frac{e^{\beta_n q_n}}{\max(|\ell|, 1)} e^{-q_n L} \max(
&c_{n,\ell-1} r_{\ell-1}^-,
c_{n,\ell-1} r_{\ell-1}^+,
\gamma\, r_{\ell-1}^+, c_{n,\ell} r_{\ell}^+,
c_{n,\ell} r_{\ell+1}^-,
c_{n,\ell} c_{n,\ell+1} r_{\ell+1}^+),
\end{align*}
and
\begin{align*}
r_{\ell}^+\leq
e^{49\varepsilon q_n} \frac{e^{\beta_n q_n}}{\max(|\ell|, 1)} e^{-q_n L} \max(
&c_{n,\ell} c_{n,\ell-1} r_{\ell-1}^-,
c_{n,\ell} r_{\ell-1}^+,
c_{n,\ell} r_{\ell}^-, \gamma\, r_{\ell+1}^-,
c_{n,\ell+1} r_{\ell+1}^-,
c_{n,\ell+1} r_{\ell+1}^+),
\end{align*}
where
\begin{align*}
\gamma:=\begin{cases}
\max(e^{\delta_n q_n}, |\ell|, 1) e^{-\beta_n q_n}, \text{ if } \beta_n\geq \delta_n+200\varepsilon\\
1, \text{ otherwise}
\end{cases}
\end{align*}
\end{lemma}
We will postpone the proof of this lemma till the end of the section.
As a corollary of Lemma \ref{lem:22_res}, we have the following.
\begin{lemma}\label{lem:22_res_beta><delta+200}
Under the assumption of Lemma \ref{lem:22_res}.
We have
\begin{align*}
r_{\ell}^-\leq e^{50\varepsilon q_n} \frac{e^{-q_n L}}{\max(|\ell|, 1)} \max
(r_{\ell-1}^-,
r_{\ell-1}^+,
r_{\ell}^+,
r_{\ell+1}^-,
c_{n,\ell+1}r_{\ell+1}^+) \times
\begin{cases}
\max(|\ell|, e^{\delta_n q_n}, 1), &\text{ if } \beta_n\geq \delta_n+200\varepsilon\\
e^{\beta_n q_n}, &\text{ if } \beta_n< \delta_n+200\varepsilon
\end{cases},
\end{align*}
and
\begin{align*}
r_{\ell}^+\leq &e^{50\varepsilon q_n} \frac{e^{-q_n L}}{\max(|\ell|, 1)} \max
(c_{n,\ell-1}r_{\ell-1}^-,
r_{\ell-1}^+,
r_{\ell}^+,
r_{\ell+1}^-,
r_{\ell+1}^+) \times
\begin{cases}\max(|\ell|, e^{\delta_n q_n}, 1), &\text{ if } \beta_n\geq \delta_n+200\varepsilon\\
e^{\beta_n q_n}, &\text{ if } \beta_n< \delta_n+200\varepsilon
\end{cases}.
\end{align*}
\end{lemma}
\begin{proof}
If $\beta_n\geq \delta_n+200\varepsilon$, bound the $c_{n,j}$'s by \eqref{eq:cnell_final}. Otherwise trivially bound the $c_{n,j}$'s by $1$.
\qed
\end{proof}
Next we show the assumption \eqref{ieq:22_res} can not hold for $\ell=0$.
\begin{lemma}\label{lem:22_I0large}
For any $x_1\in I_0$, we have
$$|\tilde{P}_{2q_n-1}(\theta_{x_1})|<e^{-\beta_n q_n}e^{(\tilde{L}-2\varepsilon)(2q_n-1)}.$$
\end{lemma}
\begin{proof}
Suppose otherwise, there exists $x_1\in I_0$ such that
$$|\tilde{P}_{2q_n-1}(\theta_{x_1})|\geq e^{-\beta_n q_n}e^{(\tilde{L}-2\varepsilon)(2q_n-1)}.$$
By Lemma \ref{lem:22_res_beta><delta+200} and equations \eqref{assume:phi0=1} and \eqref{Shnol} we have
\begin{align}\label{phi0_3}
1\leq |\phi(0)|\leq r_0^-
\leq &e^{(-L+50\varepsilon)q_n} \max(r_{-1}^-, r_{-1}^+, r_0^+, r_1^-, r_1^+) \times
\begin{cases}
\max(e^{\delta_n q_n}, 1), \text{ if } \beta_n\geq \delta_n+200\varepsilon\\
e^{\beta_n q_n}, \text{ if } \beta_n< \delta_n+200\varepsilon
\end{cases}\\
\leq &C_0 q_n e^{(-L+50\varepsilon)q_n}\times
\begin{cases}
\max(e^{\delta_n q_n}, 1), \text{ if } \beta_n\geq \delta_n+200\varepsilon \notag\\
e^{\beta_n q_n}, \text{ if } \beta_n< \delta_n+200\varepsilon
\end{cases} \notag\\
<&1. \notag
\end{align}
Contradiction! \qed
\end{proof}
Combining Corollary \ref{cor:1_res_uni} with Lemma \ref{lem:22_I0large}, we obtain the following.
\begin{lemma}
For any $\ell\neq 0$ such that $|\ell|\leq q_{n+1}/(6q_n)$, we have
$$|\tilde{P}_{2q_n-1}(\theta_{x_1})|\geq \max(|\ell|, 1) e^{-\beta_n q_n}e^{(\tilde{L}-2\varepsilon)(2q_n-1)}.$$
\end{lemma}
Thus Lemma \ref{lem:22_res_beta><delta+200} holds for any $\ell\neq 0$ such that $|\ell|\leq q_{n+1}/(6q_n)$.
It particular it implies for the same $\ell$, the following hold
\begin{align*}
r_{\ell}\leq &e^{50\varepsilon q_n}\frac{e^{-q_nL}}{\max(|\ell|, 1)} \max(r_{\ell-1}, r_{\ell}, r_{\ell+1})\times
\begin{cases}
\max(|\ell|, e^{\delta_n q_n}), \text{ if } \beta_n\geq \delta_n+200\varepsilon,\\
e^{\beta_n q_n}, \text{ if }\beta_n< \delta_n+200\varepsilon
\end{cases}.
\end{align*}
By arguments similar to \eqref{eq:rell_drop} and \eqref{eq:rell_drop_2}, the $r_{\ell}$ terms on the right-hand-side of the equation above can be dropped. This proves Lemma \ref{lem:22_res_final}.\qed
\end{proof}
\subsection*{Proof of Lemma \ref{lem:22_res}}
We are going to give a detailed proof for $r_{\ell}^-$ when $0<m_n\leq b_n$, the other cases are similar.
Let $x_2=x_1+2q_n-2$.
By the Green's formula \eqref{Green_tildeP} we have,
\begin{align}\label{eq:res_formula_1}
|\phi(k)|
\leq &\frac{|\tilde{P}_{x_2-k}(\theta_{k+1})|}{|\tilde{P}_{2q_n-1}(\theta_{x_1})|} \prod_{j=x_1}^k |\cos(\pi \theta_j)|\, |\phi(x_1-1)|+
\frac{|\tilde{P}_{k-x_1}(\theta_{x_1})|}{|\tilde{P}_{2q_n-1}(\theta_{x_1})|} \prod_{j=k}^{x_2} |\cos(\pi \theta_j)|\, |\phi(x_2+1)|
\end{align}
\subsection*{Estimates for $r_{\ell}^-$}
\subsection*{Case 1. If $x_1\in [(\ell-1)q_n+m_n+2, \ell q_n-\lfloor q_n/2\rfloor-1]$}
By Lemma \ref{lem:upperbddtildeP} we have
\begin{align}\label{eq:res_1}
|\tilde{P}_{k-x_1}(\theta_{x_1})|\leq C(\varepsilon) e^{(\tilde{L}+\varepsilon)|k-x_1|}.
\end{align}
If $k\in R_{\ell}^-$, we have
$$k+1\in [\ell q_n-b_n+1, \ell q_n+m_n].$$
Also,
$$x_2\in [(\ell+1)q_n+m_n, (\ell+2)q_n-\lfloor q_n/2\rfloor -3],$$
we have by Corollary \ref{cor:Pk_key} that
\begin{align}\label{eq:res_2}
|\tilde{P}_{x_2-k}(\theta_{k+1})|\leq g_{|x_2-k|, \ell} e^{\tilde{L} |x_2-k|}.
\end{align}
By Corollary \ref{cor:prod_cos}, we have
\begin{align}\label{eq:res_3}
\prod_{j=x_1}^k |\cos(\pi \theta_j)|\leq C(\varepsilon) e^{-(\ln 2-\varepsilon)|x_1-k|}, \text{ and }
\prod_{j=k}^{x_2} |\cos(\pi \theta_j)|\leq C(\varepsilon) e^{-(\ln 2-\varepsilon)|x_2-k|} c_{n,\ell} c_{n,\ell+1}.
\end{align}
Plugging estimates \eqref{eq:res_1}, \eqref{eq:res_2} and \eqref{eq:res_3} into \eqref{eq:res_formula_1},
we have
\begin{align}\label{eq:res_5}
|\phi(k)|\leq C(\varepsilon) e^{5\varepsilon q_n} \frac{g_{|x_2-k|,\ell} e^{\beta_n q_n}}{\max(|\ell|, 1)}e^{-L|x_1-k|} |\phi(x_1-1)|
+C(\varepsilon) e^{5\varepsilon q_n} \frac{c_{n,\ell} c_{n,\ell+1} e^{\beta_n q_n}}{\max(|\ell|, 1)} e^{-L|x_2-k|} |\phi(x_2+1)|.
\end{align}
Lemma \ref{lem:22_non-to-res} implies
\begin{align}\label{eq:res_6}
\begin{cases}
|\phi(x_1-1)|\leq e^{30\varepsilon q_n}\max\{e^{-(x_1-(\ell-1)q_n)L} r_{\ell-1}^+, e^{-(\ell q_n-x_1)L} r_{\ell}^-\}\\
|\phi(x_2+1)|\leq e^{30\varepsilon q_n}\max\{e^{-(x_2-(\ell+1)q_n)L} r_{\ell+1}^+, e^{-((\ell+2)q_n-x_2)L} r_{\ell+2}^-\}
\end{cases}
\end{align}
Plugging \eqref{eq:res_6} into \eqref{eq:res_5}, we have
\begin{align}\label{eq:res_7}
|\phi(k)|\leq e^{43\varepsilon q_n}\frac{e^{\beta_n q_n}}{\max(|\ell|, 1)}e^{-q_n L}
\max(
g_{|x_2-k|, \ell} r_{\ell-1}^+,
g_{|x_2-k|, \ell} r_{\ell}^-,
c_{n,\ell} c_{n,\ell+1} r_{\ell+1}^+),
\end{align}
where we controlled $r_{\ell+2}^-$ by $r_{\ell+1}^+$ using Corollary \ref{cor:A_upper} in the following way
$$r_{\ell+2}^-\leq e^{6\varepsilon q_n} e^{q_n L} r_{\ell+1}^+.$$
\subsection*{Case 2. If $x_1=(\ell-1)q_n+m_n+1$ and $x_2=(\ell+1)q_n+m_n-1$}
We use the following estimates by Lemma \ref{lem:upperbddtildeP},
\begin{align}\label{eq:res_1''}
|\tilde{P}_{k-x_1}(\theta_{x_1})|\leq C(\varepsilon)e^{(\tilde{L}+\varepsilon)|k-x_1|}, \text{ and }
|\tilde{P}_{x_2-k}(\theta_{k+1})|\leq C(\varepsilon)e^{(\tilde{L}+\varepsilon)|x_2-k|}.
\end{align}
By Corollary \ref{cor:prod_cos}, we have
\begin{align}\label{eq:res_3''}
\prod_{j=x_1}^k |\cos(\pi \theta_j)|\leq C(\varepsilon) e^{-(\ln 2-\varepsilon)|x_1-k|}, \text{ and }
\prod_{j=k}^{x_2} |\cos(\pi \theta_j)|\leq C(\varepsilon) e^{-(\ln 2-\varepsilon)|x_2-k|} c_{n,\ell}.
\end{align}
Plugging estimates \eqref{eq:res_1''} and \eqref{eq:res_3''} into \eqref{eq:res_formula_1},
we have
\begin{align}\label{eq:res_5''}
|\phi(k)|\leq C(\varepsilon) e^{5\varepsilon q_n} \frac{e^{\beta_n q_n}}{\max(|\ell |, 1)} \max(e^{-q_n L} |\phi((\ell-1)q_n+m_n)|, c_{n,\ell} e^{-q_n L} |\phi((\ell+1)q_n+m_n)|).
\end{align}
By the eigenvalue equation
\begin{align*}
\phi(j q_n+m_n+1)+\phi(j q_n+m_n-1)=(E-\lambda\tan(\pi\theta_{j q_n+m_n}))\phi(j q_n+m_n).
\end{align*}
We have that for any $j\in {\mathbb Z}$,
\begin{align}\label{eq:res_6''}
\phi(j q_n+m_n)|\leq Cc_{n,j} \max(|\phi(j q_n+m_n+1)|, |\phi(j q_n+m_n-1)|)\leq Cc_{n,j} \max(r_{j}^+, r_{j}^-).
\end{align}
Plugging \eqref{eq:res_6''} into \eqref{eq:res_5''}, we have
\begin{align}\label{eq:res_7''}
|\phi(k)|\leq e^{7\varepsilon q_n} \frac{e^{\beta_n q_n}}{\max(|\ell |, 1)} e^{-q_n L}\max(c_{n,\ell-1} r_{\ell-1}^-,
c_{n,\ell-1} r_{\ell-1}^+, c_{n,\ell} c_{n,\ell+1}r_{\ell+1}^-, c_{n,\ell} c_{n,\ell+1} r_{\ell+1}^+).
\end{align}
\subsection*{Case 3. If $x_1\in [(\ell-1)q_n-\lfloor q_n/2\rfloor, (\ell-1)q_n+m_n]$}
Now $x_2\in [(\ell+1)q_n-\lfloor q_n/2\rfloor-2, (\ell+1)q_n+m_n-2]$.
Lemma \ref{lem:upperbddtildeP} yields
\begin{align}\label{eq:res_1'}
|\tilde{P}_{k-x_1}(\theta_{x_1})|\leq e^{(\tilde{L}+\varepsilon)|k-x_1|}, \text{ and }
|\tilde{P}_{x_2-k}(\theta_{k+1})|\leq e^{(\tilde{L}+\varepsilon)|x_2-k|}.
\end{align}
Corollary \ref{cor:prod_cos} yields
\begin{align}\label{eq:res_3'}
\prod_{j=x_1}^k |\cos(\pi \theta_j)|\leq e^{-(\ln 2-\varepsilon)|x_1-k|} c_{n,\ell-1}, \text{ and }
\prod_{j=k}^{x_2} |\cos(\pi \theta_j)|\leq e^{-(\ln 2-\varepsilon)|x_2-k|} c_{n,\ell}.
\end{align}
Plugging estimates \eqref{eq:res_1'} and \eqref{eq:res_3'} into \eqref{eq:res_formula_1},
we have
\begin{align}\label{eq:res_5'}
|\phi(k)|\leq C(\varepsilon) e^{5\varepsilon q_n} \frac{e^{\beta_n q_n}}{\max(|\ell|, 1)}
\left(e^{-L|x_1-k|} c_{n,\ell-1} |\phi(x_1-1)|+e^{-L|x_2-k|} c_{n,\ell} |\phi(x_2+1)|\right)
\end{align}
Lemma \ref{lem:22_non-to-res} implies
\begin{align}\label{eq:res_6'}
\begin{cases}
|\phi(x_1-1)|\leq e^{30\varepsilon q_n} \max\{e^{-((\ell-1)q_n-x_1)L} r_{\ell-1}^-, e^{-(x_1-(\ell-2)q_n)L} r_{\ell-2}^+\}\\
|\phi(x_2+1)|\leq e^{30\varepsilon q_n} \max\{e^{-((\ell+1)q_n-x_2)L} r_{\ell+1}^-, e^{-(x_2-\ell q_n)L} r_{\ell}^+\}
\end{cases}
\end{align}
Plugging \eqref{eq:res_6'} into \eqref{eq:res_5'}, we have
\begin{align}\label{eq:res_7'}
|\phi(k)|\leq e^{42\varepsilon q_n} \frac{e^{\beta_n q_n}}{\max(|\ell|, 1)}e^{- q_n L}
\max(
c_{n,\ell-1} r_{\ell-1}^-,
c_{n,\ell} r_{\ell}^+,
c_{n,\ell} r_{\ell+1}^-),
\end{align}
where we controlled $r_{\ell-2}^+$ by $r_{\ell-1}^-$ using Corollary \ref{cor:A_upper} in the following way
$$r_{\ell-2}^+\leq e^{6\varepsilon q_n} e^{q_n L} r_{\ell-1}^-.$$
Putting the three cases \eqref{eq:res_7}, \eqref{eq:res_6''} and \eqref{eq:res_7'} together, taking $|\phi(k)|=r_{\ell}^-$, we obtain
\begin{align}\label{eq:res_10''}
r_{\ell}^-\leq e^{43\varepsilon q_n}\frac{e^{\beta_n q_n}}{\max(|\ell|, 1)} e^{-q_n L}
\max(
&c_{n,\ell-1} r_{\ell-1}^-,
c_{n,\ell-1} r_{\ell-1}^+,
(\max_{k\in R_{\ell}} g_{|x_2-k|, \ell})\cdot \max(r_{\ell-1}^+, r_{\ell}^-), \\
&\ \ c_{n,\ell} r_{\ell}^+,
c_{n,\ell} r_{\ell+1}^-,
c_{n,\ell} c_{n,\ell+1} r_{\ell+1}^+ \notag
).
\end{align}
Next, we further bound this, dividing into two cases.
Case (i). If $\beta_n\geq \delta_n+200\varepsilon$, using \eqref{def:g} to bound
\begin{align*}
\max_{k\in R_{\ell}} g_{|x_2-k|,\ell}
\leq \max(e^{\delta_n q_n}, |\ell|) e^{-(\beta_n-6\varepsilon)q_n},
\end{align*}
and using \eqref{eq:cnell_final} to bound
\begin{align*}
c_{n,\ell-1}\leq C \max(|\ell|, e^{\delta_n q_n}) e^{-\beta_n q_n},
\end{align*}
we arrive at
\begin{align}\label{eq:res_11''}
r_{\ell}^- \leq e^{49\varepsilon q_n} \frac{e^{\beta_n q_n}}{\max(|\ell|,1)} e^{-q_n L} \max(
&c_{n,\ell-1} r_{\ell-1}^-,
\max(e^{\delta_n q_n}, |\ell|) e^{-\beta_n q_n} \max(r_{\ell-1}^+, r_{\ell}^-), \notag\\
&\ \ c_{n,\ell} r_{\ell}^+, c_{n,\ell} r_{\ell+1}^-,
c_{n,\ell} c_{n,\ell+1} r_{\ell+1}^+).
\end{align}
Note that the coefficient of $r_{\ell}^-$ on the right-hand-side of \eqref{eq:res_11''} can be bounded by, using \eqref{eq:L>deltan+100},
\begin{align*}
e^{49\varepsilon q_n} \frac{e^{\beta_n q_n}}{\max(|\ell|,1)} e^{-q_n L}\max(e^{\delta_n q_n}, |\ell|) e^{-\beta_n q_n}\leq e^{-(L-\delta_n-50)q_n}<1.
\end{align*}
Hence \eqref{eq:res_11''} implies
\begin{align}\label{eq:res_12''}
&r_{\ell}^-
\leq e^{49\varepsilon q_n} \frac{e^{\beta_n q_n}}{\max(|\ell|,1)} e^{-q_n L} \max(
c_{n,\ell-1} r_{\ell-1}^-,
\max(e^{\delta_n q_n}, |\ell|) e^{-\beta_n q_n} r_{\ell-1}^+, c_{n,\ell} r_{\ell}^+, c_{n,\ell} r_{\ell+1}^-,
c_{n,\ell} c_{n,\ell+1} r_{\ell+1}^+).
\end{align}
Case (ii). If $\beta_n<\delta_n+200\varepsilon$, using \eqref{def:g} to bound
\begin{align*}
\max_{k\in R_{\ell}} g_{|x_2-k|,\ell}
\leq e^{2\varepsilon |x_2-k|}\leq e^{3\varepsilon q_n},
\end{align*}
and trivially bounding $c_{n,\ell-1}\leq 1$,
we obtain from \eqref{eq:res_10''} that
\begin{align}\label{eq:res_13''}
r_{\ell}^- \leq e^{46\varepsilon q_n} \frac{e^{\beta_n q_n}}{\max(|\ell|,1)} e^{-q_n L}
\max(c_{n,\ell-1} r_{\ell-1}^-,
r_{\ell-1}^+,
r_{\ell}^-,
c_{n,\ell} r_{\ell}^+,
c_{n,\ell} r_{\ell+1}^-,
c_{n,\ell} c_{n,\ell+1} r_{\ell+1}^+).
\end{align}
Note that the coefficient of $r_{\ell}^-$ on the right-hand-side of \eqref{eq:res_13''} can be bounded by, using \eqref{eq:L>deltan+100},
\begin{align*}
e^{46\varepsilon q_n} \frac{e^{\beta_n q_n}}{\max(|\ell|,1)} e^{-q_n L}\leq e^{-(L-\delta_n-246\varepsilon)q_n}<1.
\end{align*}
Hence \eqref{eq:res_13''} implies
\begin{align}\label{eq:res_14''}
r_{\ell}^- \leq e^{46\varepsilon q_n} \frac{e^{\beta_n q_n}}{\max(|\ell|,1)} e^{-q_n L} \max(c_{n,\ell-1} r_{\ell-1}^-,
r_{\ell-1}^+,
c_{n,\ell} r_{\ell}^+,
c_{n,\ell} r_{\ell+1}^-,
c_{n,\ell} c_{n,\ell+1} r_{\ell+1}^+).
\end{align}
Lemma \ref{lem:22_res} follows from combining \eqref{eq:res_12''} and \eqref{eq:res_14''}.
\qed
\section{Localization: proofs of Cases 2 and 3 of Lemma \ref{lem:main}}\label{Sec:loc}
The proofs of Cases 2 and 3 of Lemma \ref{lem:main} are completely analogous. Hence we will only prove Case 3. Assume $\beta_n\geq \delta_n+200\varepsilon$.
Let us first prove
\begin{lemma}\label{lem:localization}
For any $\ell_0$ such that $1\leq |\ell_0|\leq q_{n+1}/(12q_n)$, we have
\begin{align*}
r_{\ell_0}\leq e^{(\delta_n-L+52\varepsilon)|\ell_0| q_n}.
\end{align*}
\end{lemma}
\begin{proof}
Without loss of generality, we will prove it for $\ell_0>0$.
In view of Lemmas \ref{lem:21_ra<final} and Lemma \ref{lem:22_res_final}, for any $0 < |\ell_0|\leq q_{n+1}/(12q_n)$, we have
\begin{align}\label{A}
r_{\ell_0}\leq e^{(\delta_n-L+50\varepsilon)q_n} \max_{\ell_1=\ell_0\pm 1} r_{\ell_1}.
\end{align}
Let an even number $y_n\in\{[q_{n+1}/(6q_n)]-1, [q_{n+1}/(6q_n)]\}$. Let $t_0:=y_n/2$.
One can iterate \eqref{A} until one reaches $\ell_t$ (and stops the iteration once reaches such a $\ell_t$) where either:
{\it Case 1}. $\ell_t=0$ and $t<t_0$
{\it Case 2}. $\ell_t=y_n$ and $t<t_0$
{\it Case 3}. $t=t_0$
Hence one obtains
\begin{align*}
r_{\ell_0}\leq \max_{(\ell_0, \ell_1,...,\ell_t)\in \mathcal{G}} e^{(\delta_n-L+50\varepsilon)tq_n} r_{\ell_t},
\end{align*}
where $\mathcal{G}=\{(\ell_0, ..., \ell_t ): |\ell_i-\ell_{i-1}|=1, \text{ and } \ell_t \text{ satisfies one of the three cases above}\}$.
If $\ell_t$ satisfies Case 1, we have
\begin{align*}
e^{(\delta_n-L+50\varepsilon)tq_n} r_{\ell_t}\leq e^{(\delta_n-L+50\varepsilon)\ell_0 q_n} r_0
\leq &C_0 q_n e^{(\delta_n-L+50\varepsilon)\ell_0 q_n}\\
\leq &e^{(\delta_n-L+52\varepsilon)\ell_0 q_n}.
\end{align*}
where we used \eqref{Shnol} to control $r_0$. We will also use \eqref{Shnol} in the following two cases.
If $\ell_t$ satisfies Case 2, we have
\begin{align*}
e^{(\delta_n-L+50\varepsilon)tq_n} r_{\ell_t}\leq e^{(\delta_n-L+50\varepsilon)(y_n-\ell_0)q_n} r_{y_n}
\leq &C_0 y_n q_n e^{(\delta_n-L+50\varepsilon)(y_n-\ell_0)q_n}\\
\leq &e^{(\delta_n-L+51\varepsilon) y_n q_n} e^{-(\delta_n-L+50\varepsilon) \ell_0 q_n}\\
\leq &e^{(\delta_n-L+51\varepsilon) 2\ell_0q_n} e^{-(\delta_n-L+50\varepsilon) \ell_0 q_n}\\
=&e^{(\delta_n-L+52\varepsilon)\ell_0 q_n}.
\end{align*}
If $\ell_t$ satisfies Case 3, we have
\begin{align*}
e^{(\delta_n-L+50\varepsilon)tq_n} r_{\ell_t}\leq e^{(\delta_n-L+50\varepsilon)t_0 q_n} \max_{1\leq j\leq y_n-1} r_j
\leq &C_0 y_n q_n e^{(\delta_n-L+50\varepsilon)t_0 q_n}\\
\leq &e^{\varepsilon y_nq_n} e^{(\delta_n-L+50\varepsilon)\frac{y_n}{2} q_n}\\
=&e^{(\delta_n-L+52\varepsilon)\frac{y_n}{2} q_n}\\
\leq &e^{(\delta_n-L+52\varepsilon)\ell_0 q_n}.
\end{align*}
Combining the three cases we have proved Lemma \ref{lem:localization}.\qed
\end{proof}
We finally present the proof of Case 3 of Lemma \ref{lem:main}.
\begin{proof}
Let $y\in (\ell q_n+b_n, (\ell+1)q_n-b_n)$ for some $|\ell|\leq \frac{q_{n+1}}{12q_n}$.
We will only prove it for the cases when $\ell=0$ and $\ell\geq 1$.
We distinguish three cases:
If $\ell\neq 0, -1$, we have by Lemma \ref{lem:localization} that
\begin{align*}
r_{\ell}\leq e^{(\delta_n-L+52\varepsilon)|\ell| q_n}, \text{ and } r_{\ell+1}\leq e^{(\delta_n-L+52\varepsilon)|\ell+1| q_n}.
\end{align*}
By Lemmas \ref{lem:C2_n-r} (if $m_n$ is non-resonant) and \ref{lem:22_non-to-res} (if $m_n$ is resonant), we have
\begin{align*}
|\phi(y)|
\leq &e^{30\varepsilon q_n} \max(e^{-(y-\ell q_n)L}r_{\ell}, e^{-((\ell+1)q_n-y)L}r_{\ell+1})\\
\leq &e^{(\delta_n-L+82\varepsilon)|y|}.
\end{align*}
If $\ell=0$, we have by Lemma \ref{lem:localization} that
\begin{align*}
r_{1}\leq e^{(\delta_n-L+52\varepsilon)q_n}.
\end{align*}
and by \eqref{Shnol} that
\begin{align*}
r_0\leq C_0\tau_n q_n.
\end{align*}
By Lemmas \ref{lem:C2_n-r} and \ref{lem:22_non-to-res}, we have that for $q_n-b_n>y\geq q_n/12$,
\begin{align*}
|\phi(y)|
\leq &e^{30\varepsilon q_n} \max(e^{-y L}r_0, e^{-(q_n-y)L}r_1)\\
\leq &\max(C_0\tau_n q_n e^{30\varepsilon q_n} e^{-y L}, e^{(\delta_n-L+82\varepsilon)y})\\
\leq &\max(e^{-(L-361\varepsilon)y}, e^{(\delta_n-L+82\varepsilon)y})\\
\leq &e^{(\delta_n-L+361\varepsilon)y}.
\end{align*}
If $\ell=-1$, the proof is similar to that of $\ell=0$. We can prove that for $-q_n+b_n<y\leq -q_n/12$,
\begin{align*}
|\phi(y)|\leq \max(e^{-(L-361\varepsilon)|y|}, e^{(\delta_n-L+82\varepsilon)|y|})\leq e^{(\delta_n-L+361\varepsilon)|y|}.
\end{align*}
Combining the three cases above, we have proved Case 3 of Lemma \ref{lem:main}.
\qed
\end{proof}
\section{Appendix}
\subsubsection*{Proof of Corollary \ref{cor:cos_prod_lower}}
Let $\tilde{I}:=[m_n+jq_n+1, m_n+(j+1)q_n-1]$.
Let $$I^c:=\tilde{I}\setminus I=[m_n+j q_n+1, \ell_1-1]\cup [\ell_2+1, m_n+(j+1)q_n-1]=:I^c_1\cup I^c_2.$$
We have by Lemma \ref{lana} and equation \eqref{eq:delta_n_min_cos} that
\begin{align*}
\prod_{\ell\in\tilde{I}}|\cos(\pi\theta_{\ell})| \cdot |\cos(\pi\theta_{m_n+jq_n})|
\geq &\frac{1}{q_n^C} e^{-q_n \ln 2} \min_{\ell\in \tilde{I}\cup\{m_n+j q_n\}} |\cos(\pi\theta_{\ell})|\notag\\
\geq &\frac{1}{21} \frac{1}{q_n^C} e^{-q_n \ln 2} |\cos(\pi\theta_{m_n+jq_n})|
\end{align*}
Hence
\begin{align}\label{eq:cor_prod_1}
\prod_{\ell\in\tilde{I}}|\cos(\pi\theta_{\ell})| \geq \frac{1}{q_n^{C}} e^{-q_n \ln 2}.
\end{align}
Also by Corollary \ref{cor:prod_cos}, for $k\in \{1,2\}$ we have
\begin{align}\label{eq:cor_prod_2}
\prod_{\ell\in I^c_k}|\cos(\pi\theta_{\ell})|\leq C(\varepsilon) e^{|I_k^c|(-\ln 2+\varepsilon)}
\end{align}
Hence combining \eqref{eq:cor_prod_1} and \eqref{eq:cor_prod_2}, we have
\begin{align}\label{eq:cor_prod_3}
\prod_{\ell\in I}|\cos(\pi\theta_{\ell})|=\frac{\prod_{\ell\in \tilde{I}}|\cos(\pi\theta_{\ell})|}{\prod_{\ell\in I^c}|\cos(\pi\theta_{\ell})|}
\geq e^{-\varepsilon(2q_n-|I|)} e^{(-\ln 2)|I|}.
\end{align}
\qed
\subsubsection*{Proof of Corollary \ref{cor:A_upper}}
By Lemma \ref{lem:upperbddtildeP}, \eqref{tildePinF} we have uniformly in $\theta$,
\begin{align*}
\|F_{|I|}(\theta)\|\leq C(\varepsilon) e^{|I|(\tilde{L}+\varepsilon)}.
\end{align*}
By Corollary \ref{cor:cos_prod_lower} and \eqref{defnonsingular}, we then have by \eqref{LEtildeLE}
\begin{align*}
\|A_{|I|}(\theta_{\ell_1})\|
=\frac{1}{\prod_{\ell\in I} |\cos(\pi\theta_{\ell})|} \|F_{|I|}(\theta_{\ell_1})\|
\leq C(\varepsilon) e^{\varepsilon (2q_n-|I|)} e^{|I| \ln 2} e^{|I| (\tilde{L}+\varepsilon)}\leq e^{3\varepsilon q_n} e^{L |I|}.
\end{align*}
\qed
\subsubsection*{Proof of Corollary \ref{cor:A_upper_mn}}
We have by Corollary \ref{cor:A_upper} that
\begin{align*}
\|A_{|I|}(\theta_{\ell_1})\|
\leq &\|A_{jq_n+m_n-\ell_1}(\theta_{\ell_1})\|\cdot \|A(\theta_{jq_n+m_n})\| \cdot \| A_{\ell_2-jq_n-m_n}(\theta_{jq_n+m_n+1})\|\\
\leq &C e^{6\varepsilon q_n} e^{L(|I|-1)} |\tan(\pi(\theta_{jq_n+m_n}))|\\
\leq &\frac{1}{c_{n,j}} e^{7\varepsilon q_n} e^{L |I|}.
\end{align*}
\qed
\subsection{Uniformity lemmas}
\subsubsection{Proof of Lemma \ref{lem:nonres_uni}}\
Without loss of generality we assume $d(y, q_n{\mathbb Z})=y-\ell q_n$ and $y=\ell q_n+2sq_{n-n_0}+x$ where $0\leq x<2q_{n-n_0}$.
We are going to show that for any $\theta\in {\mathbb T}$ and $j\in I_1$ (for $j\in I_2$ the proof is similar) that
\begin{align}\label{nonresdeftopbottom}
\ln\left\lbrace\prod_{l\neq j} \frac{|\sin\pi(\theta-\theta_l)|}{|\sin\pi(\theta_j-\theta_l)|}\right\rbrace
=\sum_{l\neq j} \ln|\sin\pi(\theta-\theta_l)|-\sum_{l\neq j} \ln|\sin\pi(\theta_j-\theta_l)|< (2sq_{n-n_0}-1)\epsilon.
\end{align}
For $m=1, 2,..., s$, let
$$T_m=[-[\frac{1}{2}sq_{n-n_0}]-sq_{n-n_0}+(m-1)q_{n-n_0}, -[\frac{1}{2}sq_{n-n_0}]-sq_{n-n_0}+mq_{n-n_0}-1].$$
For $m=s+1,..., 2s$, let
$$T_m=[y-[\frac{1}{2}sq_{n-n_0}]-sq_{n-n_0}+(m-s-1)q_{n-n_0}, y-[\frac{1}{2}sq_{n-n_0}]+(m-s)q_{n-n_0}-1].$$
Each $T_m$ consists of $q_{n-n_0}$ consequential numbers.
Denote
$$|\sin\pi(\theta_j-\theta_{l_m})|=\min_{l\in T_m}|\sin\pi (\theta_j-\theta_l)|.$$
Assume $j \in T_{m_0}$, then clearly $l_{m_0}=j$.
First, about $\sum_{l\neq j} \ln|\sin\pi(\theta-\theta_l)|$, applying Lemma \ref{lana} on each $T_m$, we have
\begin{align}\label{nonrestopestimate}
\sum_{l\neq i} \ln|\sin\pi(\theta-\theta_l)|
\leq &2s(C\ln q_{n-n_0}-(q_{n-n_0}-1)\ln 2) \notag\\
\leq &(2sq_{n-n_0}-1)(-\ln 2+\frac{C\ln{q_{n-n_0}}}{q_{n-n_0}})
\end{align}
Secondly, about $\sum_{l\neq j} \ln|\sin\pi(\theta_j-\theta_l)|$. Clearly,
\begin{align}\label{defsum1sum2}
\sum_{l\neq j} \ln|\sin\pi(\theta_j-\theta_l)|=\sum_{l\in I_1, l\neq j}\ln{|\sin\pi (j-l)\alpha|}+\sum_{l\in I_2}\ln{|\sin\pi (j-l)\alpha|}\triangleq \sum_{1}+\sum_{2}.
\end{align}
For $\sum_{1}$, by Lemma \ref{lana} we have
\begin{align}
\sum_{1} \geq&s(-C\ln q_{n-n_0}-(q_{n-n_0}-1)\ln2)+\sum_{m=1, m\neq m_0}^{s} \ln|\sin \pi(j-l_m)\alpha| \notag \\
\geq &s(-C\ln q_{n-n_0}-(q_{n-n_0}-1)\ln2)+\sum_{m=1, m\neq m_0}^{s} \ln{\|(j-l_m)\alpha\|_{{\mathbb T}}} \label{sum1first}
\end{align}
Since $|j-l_m|<sq_{n-n_0}<q_{n-n_0+1}$, we have
\begin{align}
\sum_{m=1, m\neq m_0}^{s} \ln{\|(j-l_m)\alpha\|_{{\mathbb T}}}
\geq &2\sum^{[\frac{s+1}{2}]}_{k=1}\ln{(k\|q_{n-n_0}\alpha\|_{{\mathbb T}})} \notag\\
\geq &2\int_{1}^{\frac{s+1}{2}}\ln{x} \mathrm{d}x+(s+1)\ln{\frac{1}{2q_{n-n_0+1}}} \notag\\
= &(s+1)\ln{(s+1)}-2(s+1)\ln{2}-(s-1)-(s+1)\ln{q_{n-n_0+1}} \notag\\
> &(s+1)\ln(\frac{s}{q_{n-n_0+1}})-Cs \notag\\
\geq &2s\ln(\frac{s}{q_{n-n_0+1}})-Cs. \label{sum1smallest}
\end{align}
Thus combining (\ref{sum1first}) with (\ref{sum1smallest}), we have
\begin{align}\label{sum1final}
\sum_1\geq sq_{n-n_0}\left(-\frac{C \ln q_{n-n_0}}{q_{n-n_0}}-\ln2+\frac{2 \ln(s/q_{n-n_0+1})}{q_{n-n_0}}\right).
\end{align}
For $\sum_{2}$, first consider $\tilde{T}_m=T_m-aq_n$, let $\tilde{l}_m$ be such that $|\sin\pi (j-\tilde{l}_m)\alpha|=\min_{l\in\tilde{T}_m}|\sin\pi (j-l)\alpha|$. Clearly, $\tilde{l}_m=l_m-aq_n$.
Again using Lemma \ref{lana} we have
\begin{align}
\sum_{2}
\geq&s(-C\ln q_{n-n_0}-(q_{n-n_0}-1)\ln2)+\sum_{m=s+1}^{2s} \ln|\sin\pi(j-l_m)\alpha| \notag \\
\geq&s(-C\ln q_{n-n_0}-(q_{n-n_0}-1)\ln2)+\sum_{m=s+1}^{2s} \ln \|(j-l_m)\alpha\|_{{\mathbb T}}
\end{align}
Note that
\begin{align*}
\|(j-\tilde{\ell}_m)\alpha\|\geq \|q_{n-1}\alpha\|\geq a_{n+1}\|q_n\alpha\|\geq 2a\|q_n\alpha\|.
\end{align*}
Hence
\begin{align}
\sum_{2} \geq s(-C\ln q_{n-n_0}-(q_{n-n_0}-1)\ln2)+\sum_{m=s+1}^{2s} \ln (\|(j-\tilde{l}_m)\alpha\|_{{\mathbb T}}-a\|q_n\alpha\|_{{\mathbb T}}) \label{sum2first}
\end{align}
Now let $min\triangleq \|(j-\tilde{l}_{m^*})\alpha\|_{{\mathbb T}}=\min_{m=s+1}^{2s}\|(j-\tilde{l}_m)\alpha\|_{{\mathbb T}}$. Note that $|\tilde{l}_{m_1}-\tilde{l}_{m_2}|<sq_{n-n_0}<q_{n-n_0+1}$ for any $m_1, m_2\in \{s+1,..., 2s\}$. We will divide $\sum_{m=s+1}^{2s} \ln (\|(j-\tilde{l}_m)\alpha\|_{{\mathbb T}}-a\|q_n\alpha\|_{{\mathbb T}})$ into two cases:
\begin{enumerate}
\item if $min> \frac{2}{3}\|q_{n-n_0}\alpha\|_{{\mathbb T}}$, then
\begin{align}
&\sum_{m=s+1}^{2s} \ln (\|(j-\tilde{l}_m)\alpha\|_{{\mathbb T}}-a\|q_n\alpha\|_{{\mathbb T}}) \notag\\
\geq &\ln {(min-a\|q_n\alpha\|_{{\mathbb T}})}+ \sum_{k=1}^{s-1} \ln {(\|kq_{n-n_0}\alpha\|_{{\mathbb T}}+min-a\|q_n\alpha\|_{{\mathbb T}})}\notag \\
\geq &\sum_{k=0}^{s-1} \ln {(\frac{3k+2}{3}\|q_{n-n_0}\alpha\|_{{\mathbb T}}-a\|q_n\alpha\|_{{\mathbb T}})}\notag \\
\geq &\sum_{k=0}^{s-1} \ln {(\frac{6k+1}{6}\|q_{n-n_0}\alpha\|_{{\mathbb T}})}\notag \\
= &\sum_{k=0}^{s-1} \ln{(6k+1)}-s\ln{q_{n-n_0+1}}-Cs \notag\\
\geq &s\ln{(s/q_{n-n_0+1})}-Cs. \label{sumI2smallestcase1}
\end{align}
\item if $min \leq \frac{2}{3}\|q_{n-n_0}\alpha\|_{{\mathbb T}}$, clearly we still have a lower bound $min\geq \|q_{n-1}\alpha\|_{{\mathbb T}}$, then
\begin{align}
&\sum_{m=s+1}^{2s} \ln (\|(j-\tilde{l}_m)\alpha\|_{{\mathbb T}}-a\|q_n\alpha\|_{{\mathbb T}}) \notag\\
\geq &\ln {(min-a\|q_n\alpha\|_{{\mathbb T}})}+ \sum_{k=1}^{[\frac{s+1}{2}]} \ln {(\|kq_{n-n_0}\alpha\|_{{\mathbb T}}+min-a\|q_n\alpha\|_{{\mathbb T}})}\notag\\
+&\sum_{k=1}^{[\frac{s+1}{2}]}\ln {(\|kq_{n-n_0}\alpha\|_{{\mathbb T}}-min-a\|q_n\alpha\|_{{\mathbb T}})}.\notag\\
\geq&\ln {(\|q_{n-1}\alpha\|_{{\mathbb T}}-a\|q_n\alpha\|_{{\mathbb T}})}+\sum_{k=1}^{[\frac{s+1}{2}]} \ln {(k\|q_{n-n_0}\alpha\|_{{\mathbb T}}-a\|q_n\alpha\|_{{\mathbb T}})} \notag\\
+&\sum_{k=1}^{[\frac{s+1}{2}]}\ln {(\frac{2k-1}{2}\|q_{n-n_0}\alpha\|_{{\mathbb T}}-a\|q_n\alpha\|_{{\mathbb T}})} \notag \\
\geq & \ln{\frac{1}{4q_n}}+2\sum_{k=1}^{[\frac{s+1}{2}]}\ln{\frac{4k-3}{4}}-2[\frac{s+1}{2}]\ln{2q_{n-n_0+1}} \notag\\
\geq & s\ln{(s/q_{n-n_0+1})}-\ln{q_n}-Cs. \label{sumI2smallestcase2}
\end{align}
Note that
\begin{align}\label{nonresqnn0}
\tau_n q_n=b_n<\operatorname{dist}{(y, q_n{\mathbb Z})}<2(s+1)q_{n-n_0},
\end{align}
we will get the lower bound of $\sum_{m=s+1}^{2s} \ln (\|(j-\tilde{l}_m)\alpha\|_{{\mathbb T}}-a\|q_n\alpha\|_{{\mathbb T}})$ from case (2),
\begin{align}\label{suml2smallestcase2final}
\sum_{m=s+1}^{2s} \ln (\|(j-\tilde{l}_m)\alpha\|_{{\mathbb T}}-a\|q_n\alpha\|_{{\mathbb T}})
\geq &s\ln{(s/q_{n-n_0+1})}-\ln{(2(s+1)q_{n-n_0}/\tau_n)}-Cs \notag\\
>&s\ln{(s/q_{n-n_0+1})}-\ln{q_{n-n_0}}-Cs-\ln(2/\tau_n).
\end{align}
\end{enumerate}
Thus combining (\ref{sum2first}), (\ref{sumI2smallestcase2}) with (\ref{suml2smallestcase2final}), we have
\begin{align}\label{sum2final}
\sum_{2}\geq sq_{n-n_0}(-C\frac{\ln{q_{n-n_0}}}{q_{n-n_0}}-\ln{2}+\frac{\ln(s/q_{n-n_0+1})}{q_{n-n_0}}).
\end{align}
Combining (\ref{defsum1sum2}), (\ref{sum1final}) with (\ref{sum2final}), we have
\begin{align}\label{nonresonbottomestimate}
\sum_{l\neq j} \ln|\sin\pi(\theta_j-\theta_l)|\geq 2sq_{n-n_0}\left(-C\frac{\ln{q_{n-n_0}}}{q_{n-n_0}}-\ln{2}+\frac{2\ln(s/q_{n-n_0+1})}{q_{n-n_0}}\right).
\end{align}
Eventually, by (\ref{nonresdeftopbottom}), (\ref{nonrestopestimate}) and (\ref{nonresonbottomestimate}),
\begin{align}\label{nonres2rdlast}
\ln\left\lbrace\prod_{l\neq j} \frac{|\sin\pi(\theta-\theta_l)|}{|\sin\pi(\theta_j-\theta_l)|}\right\rbrace< &(2sq_{n-n_0}-1)(C\frac{\ln{q_{n-n_0}}}{q_{n-n_0}}+\frac{2\ln(q_{n-n_0+1}/s)}{q_{n-n_0}}).
\end{align}
Taking into account that $b_n=\tau_n q_n<4sq_{n-n_0}$, and that $q_{n-n_0+1}\leq q_n$, \eqref{nonres2rdlast} yields
\begin{align}\label{nonres2rdlast'}
\ln\left\lbrace\prod_{l\neq j} \frac{|\sin\pi(\theta-\theta_l)|}{|\sin\pi(\theta_j-\theta_l)|}\right\rbrace
< &(2sq_{n-n_0}-1)(C\frac{\ln{q_{n-n_0}}}{q_{n-n_0}}+\frac{2\ln(q_{n}/s)}{q_{n-n_0}}) \notag\\
\leq &(2sq_{n-n_0}-1)(C\frac{\ln{q_{n-n_0}}}{q_{n-n_0}}+\frac{2\ln(q_{n-n_0}/\tau_n)}{q_{n-n_0}})\notag\\
<&(2sq_{n-n_0}-1)\varepsilon.
\end{align}
\qed
\subsubsection{Proof of Lemma \ref{lem:nonres_I2_large}}\
As a corollary of Lemma \ref{lem:nonres_uni} we have
\begin{cor}\label{cor:nonres_uni}
There exists $x_1\in \tilde{I}_0\cup \tilde{I}_y$ such that
$$|\tilde{P}_{2sq_{n-n_0}-1}(\theta_{x_1})|\geq e^{(\tilde{L}-2\varepsilon)(2sq_{n-n_0}-1)}.$$
\end{cor}
\begin{proof}
Suppose otherwise, we have for any $x_1\in I_1\cup I_2$,
$$|\tilde{P}_{2sq_{n-n_0}-1}(\theta_{x_1})|< e^{(\tilde{L}-2\varepsilon)(2sq_{n-n_0}-1)}.$$
By \eqref{tildePlagrange}
\begin{align*}
\tilde{P}_{2sq_{n-n_0}-1}(\theta)=\sum_{x_1\in I_1\cup I_2}\tilde{P}_{2sq_{n-n_0}-1}(\theta_{x_1})\prod_{\substack{j\in I_1\cup I_2\\ j\neq x_1}} \frac{\sin\pi(\theta-\theta_j)}{\sin\pi(\theta_{x_1}-\theta_j)}.
\end{align*}
Combining this with Lemma \ref{lem:nonres_uni} yields, uniformly in $\theta$,
\begin{align*}
|\tilde{P}_{2sq_{n-n_0}-1}(\theta)|\leq 2sq_{n-n_0} e^{(\tilde{L}-\varepsilon)(2sq_{n-n_0}-1)}<e^{(\tilde{L}-\frac{\varepsilon}{2})(2sq_{n-n_0}-1)}.
\end{align*}
Hence contradiction with \eqref{averagelower}. \qed
\end{proof}
\begin{lemma}\label{lem:nonres_I1_small}
For $x_1\in \tilde{I}_0$, we have
$$|\tilde{P}_{2sq_{n-n_0}-1}(\theta_{x_1})|< e^{(\tilde{L}-2\epsilon)(2sq_{n-n_0}-1)}.$$
\end{lemma}
\begin{proof}
Suppose otherwise, we have that for some $x_1\in \tilde{I}_0$,
\begin{align}\label{eq:P2sq_I1_1}
|\tilde{P}_{2sq_{n-n_0}-1}(\theta_{x_1})|\geq e^{(\tilde{L}-2\epsilon)(2sq_{n-n_0}-1)}.
\end{align}
Let $x_2:=x_1+2sq_{n-n_0}-2$ and $I:=[x_1, x_2]$. Clearly $|I|=2sq_{n-n_0}-1$.
By the Green's formula
\begin{align}\label{eq:P2sq_I1_2}
|\phi(0)|
\leq &|G_I(x_1, 0)|\cdot |\phi(x_1-1)|+|G_I(x_2, 0)|\cdot |\phi(x_2+1)| \notag\\
=&\frac{|\tilde{P}_{x_2}(\theta_1)|}{|\tilde{P}_I(\theta_{x_1})|} \prod_{j=x_1}^0 |\cos(\pi(\theta_j))|\cdot |\phi(x_1-1)|+
\frac{|\tilde{P}_{-x_1}(\theta_{x_1})|}{|\tilde{P}_I(\theta_{x_1})|} \prod_{j=0}^{x_2} |\cos(\pi(\theta_j))|\cdot |\phi(x_2+1)|.
\end{align}
By \eqref{Shnol}, we have
\begin{align*}
|\phi(x_1-1)|\leq C_0|x_1-1|, \text{ and } |\phi(x_2+1)|\leq C_0|x_2+1|.
\end{align*}
In \eqref{eq:P2sq_I1_2}, use Lemma \ref{lem:upperbddtildeP} to estimate the $\tilde{P}$'s in the numerators, use \eqref{eq:P2sq_I1_1} to estimate the $\tilde{P}$ in denominator, and use Corollary \ref{cor:prod_cos} to estimate the cosine products, we arrive at
\begin{align*}
1\leq |\phi(0)|\leq C_0 C(\varepsilon) e^{3\varepsilon |I|} \left(e^{-(1-x_1)L} |1-x_1| +e^{-x_2L} |x_2+1| \right)
\leq C(\varepsilon) e^{3\varepsilon |I|} |I| e^{-\frac{|I|}{4}L}<C(\varepsilon) e^{-(\frac{L}{4}-4\varepsilon)|I|}.
\end{align*}
Contradiction, provided that $q_{n-n_0+1}\geq \frac{1}{2} \tau_n q_n$ is sufficiently large which is satisfied for large $n$.
\end{proof}
A quick combination of Corollary \ref{cor:nonres_uni} and Lemma \ref{lem:nonres_I1_small} yields Lemma \ref{lem:nonres_I2_large}. \qed
\subsubsection{Proof of Lemma \ref{lem:1_res_uni}}
We are going to show that for any $\theta\in {\mathbb T}$ and $k\in I_0$ (for $k\in I_{\ell}$ the proof is similar)
\begin{align}\label{resdeftopbottom}
\ln\left\lbrace\prod_{j\neq k} \frac{|\sin\pi(\theta-\theta_j)|}{|\sin\pi(\theta_k-\theta_j)|}\right\rbrace
=\sum_{j\neq k} \ln|\sin\pi(\theta-\theta_j)|-\sum_{j\neq k} \ln|\sin\pi(\theta_k-\theta_j)|< (2q_n-1)(\frac{\ln{q_{n+1}/|\ell|}}{2q_n-1}+\epsilon).
\end{align}
First, about $\sum_{j\neq k} \ln|\sin\pi(\theta-\theta_j)|$, applying Lemma \ref{lana} on $I_0$ and $I_{\ell}$ respectively, we have
\begin{align}\label{restop}
\sum_{j\neq k} \ln|\sin\pi(\theta-\theta_j)|
\leq 2(C\ln q_{n}-(q_{n}-1)\ln 2).
\end{align}
Secondly, about $\sum_{j\neq k} \ln|\sin\pi(\theta_k-\theta_j)|$. Clearly,
\begin{align}\label{defressum1sum2}
\sum_{j\neq k} \ln|\sin\pi(\theta_k-\theta_j)|=\sum_{j\in I_0, j\neq k}\ln{|\sin\pi (k-j)\alpha|}+\sum_{j\in I_{\ell}}\ln{|\sin\pi (k-j)\alpha|}=: \sum_{1}+\sum_{2}.
\end{align}
For $\sum_{1}$, by Lemma \ref{lana} we have
\begin{align}\label{ressum1}
\sum_{1}\geq -C\ln{q_n}-(q_n-1)\ln{2}.
\end{align}
For $\sum_2$, again by Lemma \ref{lana} we have
\begin{align}\label{ressum2smallestterm}
\sum_{2}\geq -C\ln{q_n}-(q_n-1)\ln{2}+\ln{|\sin{\pi(k-j_0)\alpha}|},
\end{align}
where $\ln{|\sin{\pi(k-j_0)\alpha}|}:= \min_{j\in I_2}\ln{|\sin{\pi(k-j)\alpha}|}$.
Clearly
\begin{align*}
k-j\in [-(\ell+1)q_n+1, -(\ell-1)q_n-1].
\end{align*}
Let $j_1\in I_{\ell}$ (it is possible that $j_1=j_0$) be such that $k-j_1=-\ell q_n$. We have for any $j\in I_{\ell}$,
\begin{align*}
\|(k-j)\alpha\|\geq \|(j-j_1)\alpha\|-\|(k-j_1)\alpha\|
\geq &\|q_{n-1}\alpha\|-|\ell|\cdot \|q_n\alpha\|\\
\geq &(a_{n+1}-|\ell|)\|q_n\alpha\|+\|q_{n+1}\alpha\|\\
\geq &(\frac{q_{n+1}}{3q_n}-1)\|q_n\alpha\|\\
\geq &\frac{1}{3}|\ell|\cdot \|q_n\alpha\|.
\end{align*}
Thus
\begin{align}\label{ressum2smallestestimate}
\ln{|\sin{\pi(k-j_0)\alpha}|}
\geq \ln{\|(k-j_0)\alpha\|_{{\mathbb T}}} +\ln 2
\geq \ln{|\ell|\cdot \|q_n\alpha\|_{{\mathbb T}}} +\ln\frac{2}{3}
\geq \ln\frac{|\ell|}{q_{n+1}}+\ln\frac{2}{3}.
\end{align}
Thus combining (\ref{ressum2smallestterm}) with (\ref{ressum2smallestestimate}), we have
\begin{align}\label{ressum2}
\sum_{2}\geq -(C+2)\ln{q_n}-(q_n-1)\ln{2}+\ln{\frac{|\ell|}{q_{n+1}}}.
\end{align}
Therefore by (\ref{defressum1sum2}), (\ref{ressum1}) and (\ref{ressum2}),
\begin{align}\label{resbottom}
\sum_{j\neq k} \ln|\sin\pi(\theta_k-\theta_j)|\geq 2(-(C+1)\ln{q_n}-(q_n-1)\ln{2})+\ln{\frac{|\ell|}{q_{n+1}}}.
\end{align}
Eventually, by (\ref{resdeftopbottom}), (\ref{restop}) and (\ref{resbottom}) we get
\begin{align*}
\ln\left\lbrace\prod_{j\neq k} \frac{|\sin\pi(\theta-\theta_j)|}{|\sin\pi(\theta_k-\theta_j)|}\right\rbrace
\leq \ln{(q_{n+1}/|\ell|)}+\epsilon (4C+2)\ln q_n
\leq (2q_n-1)(\frac{\ln{q_{n+1}/|\ell|}}{2q_n-1}+\epsilon).
\end{align*}
\qed
\subsection{Proof of Lemmas \ref{lem:C2_n-r} and \ref{lem:22_non-to-res}}
\subsubsection{Proof of Lemma \ref{lem:C2_n-r}}
For $y$ so that $\operatorname{dist}(y, q_n{\mathbb Z})>b_n$, we have proved that for some $x_1\in I_2$,
$$|\tilde{P}_{2sq_{n-n_0}-1}(\theta_{x_1})|\geq e^{(\tilde{L}(E)-2\varepsilon)(2sq_{n-n_0}-1)}.$$
Let $z_2=z_1+2sq_{n-n_0}-2$, $I(y)=[z_1, z_2]\cap {\mathbb Z}$ and $\partial I(y)=\{z_1, z_2\}$.
In general, if $I=[a,b]$, let $\partial I:=\{a,b\}$ and $a':=a-1$, $b':=b+1$.
Let us first consider $y\in I^-$.
By Green's function expansion, we have
\begin{align*}
\phi(y)=\sum_{z\in \partial I(y)}G_{I(y)}(z,y) \phi(z').
\end{align*}
If $x_1-1> \ell q_n+b_n$ or $x_2+1<(\ell+1)q_n-b_n$, we could expand $\phi(x_1-1)$ or $\phi(x_2+1)$.
We will continue this process until we arrive at a $z$ so that $z\leq \ell q_n+b_n$ or $z\geq (\ell+1)q_n-b_n$, or the iterating number reaches $t_0:=[\frac{24}{\tau_n}]+1$.
We obtain, after a series of expansions, the following
\begin{align*}
\phi(y)=\sum_{\substack{z_1,,,z_t,z_{t+1}\\z_{i+1}\in I(z_i')}}G_{I(y)}(y, z_1)G_{I(z_1')}(z_1', z_2)\cdots G_{I(z_t')}(z_t', z_{t+1})\phi({z_{t+1}'}),
\end{align*}
where $z_{t+1}'$ either satisfies
{\it Case 1}\ : $\ell q_n\leq z_{t+1}'\leq \ell q_n+b_n$ and $t<t_0$ or
{\it Case 2}\ : $(\ell+1)q_n\geq z_{t+1}'\geq (\ell+1)q_n-b_n$ and $t<t_0$ or
{\it Case 3}\ : $t=t_0$.
For simplicity, let us denote $y=z_0'$.
If $z_{t+1}'$ satisfies Case 1. For each $z_j'$, $0\leq j\leq t$, denote $\partial I(z_j')=\{z_{j+1}, y_{j+1}\}$. Assume WLOG $z_{j+1}<y_{j+1}$, we have
\begin{align*}
|G_{I(z_j')}(z_j', z_{j+1})|
=\frac{|P_{|y_{j+1}-z_j'|}(\theta+(z_j'+1)\alpha)|}{|P_{|I(z_j')|}(\theta+z_{j+1}\alpha)|}
=&\frac{|\tilde{P}_{|y_{j+1}-z_j'|}(\theta+(z_j'+1)\alpha)|}{|\tilde{P}_{|I(z_j')|}(\theta+z_{j+1}\alpha)|}
\prod_{\ell=z_{j+1}}^{z_j'}|\cos(\pi(\theta+\ell\alpha))|.
\end{align*}
We have by Lemma \ref{lem:nonres_I2_large} that
\begin{align*}
|\tilde{P}_{|I(z_j')|}(\theta+z_{j+1}\alpha)|\geq e^{|I(z_j')|(\tilde{L}-2\varepsilon)},
\end{align*}
by Lemma \ref{lem:upperbddtildeP} that
\begin{align*}
|\tilde{P}_{|y_{j+1}-z_j'|}(\theta+(z_j'+1)\alpha)|\leq C(\varepsilon) e^{|y_{j+1}-z_j'|(\tilde{L}+\varepsilon)},
\end{align*}
and by \ref{cor:prod_cos},
\begin{align*}
\prod_{\ell=z_{j+1}}^{z_j'}|\cos(\pi(\theta+\ell\alpha))|\leq e^{|z_j'-z_{j+1}|(-\ln 2+\varepsilon)}.
\end{align*}
Putting them all together, we have
\begin{align}\label{eq:non-res-Green-general}
|G_{I(z_j')}(z_j', z_{j+1})|\leq C(\varepsilon) e^{-|z_j'-z_{j+1}+1|(L-12\varepsilon)}.
\end{align}
Bounding $|\phi(z_{t+1}')|\leq r_a$, we have
\begin{align}\label{eq:non-res-Green-general_1}
|G_{I(y)}(y, z_1)G_{I(z_1')}(z_1', z_2)\cdots G_{I(z_t')}(z_t', z_{t+1})\phi({z_{t+1}'})|
\leq &(C(\varepsilon))^{t_0+1} e^{-(y-\ell q_n-b_n)(L-12\varepsilon)} r_{\ell} \notag\\
\leq &(C(\varepsilon))^{t_0+1} e^{\varepsilon q_n}e^{-(y-\ell q_n)(L-12\varepsilon)} r_{\ell}.
\end{align}
If $z_{t+1}'$ satisfies Case 2, there must be a $z_j'$ such that $aq_n+m_n\in I(z_j')$.
For this particular pair, let $\partial I(z_j')=\{z_{j+1}, y_{j+1}\}$ and $y_{j+1}<z_{j+1}$, we estimate similarly to \eqref{eq:non-res-Green-general}, only modifying the estimate of the cosine product,
\begin{align*}
|G_{I(z_j')}(z_j', z_{j+1})|
=&\frac{|P_{|y_{j+1}-z_j'|}(\theta+y_{j+1}\alpha)|}{|P_{|I(z_j')|}(\theta+y_{j+1}\alpha)|}\\
=&\frac{|P_{|y_{j+1}-z_j'|}(\theta+y_{j+1}\alpha)|}{|P_{|I(z_j')|}(\theta+y_{j+1}\alpha)|} \prod_{\ell=z_j'}^{z_{j+1}}|\cos(\pi(\theta+\ell\alpha))|\\
\leq &C(\varepsilon) e^{|y_{j+1}-z_j'|(L+\varepsilon)} e^{-|I(z_j')|(L-2\varepsilon)} e^{|z_j'-z_{j+1}|(-\ln 2+\varepsilon)} c_{n,\ell}\\
\leq &C(\varepsilon) e^{-|z_j'-z_{j+1}+1|(L-12\varepsilon)} c_{n,\ell}.
\end{align*}
Hence
\begin{align}\label{eq:non-res-Green-special}
|G_{I(y)}(y, z_1)G_{I(z_1')}(z_1', z_2)\cdots G_{I(z_t')}(z_t', z_{t+1})\phi({z_{t+1}'})|
\leq (C(\varepsilon))^{t_0+1} e^{\varepsilon q_n} e^{-((\ell+1)q_n-y)(L-12\varepsilon)} c_{n,\ell} r_{\ell+1}.
\end{align}
If $z_{t+1}'$ satisfies Case 3. Here
\begin{align*}
|z_j'-z_{j+1}|\geq \frac{1}{4} I(z_j')\geq \frac{1}{8} \min((z_j'-\ell q_n, (\ell+1)q_n-z_j'))\geq \frac{b_n}{8}=\frac{\tau_n}{8} q_n.
\end{align*}
In general, we use the Green's function estimate \eqref{eq:non-res-Green-general}, which yields
\begin{align}\label{eq:non-res_101}
|G_{I(z_j')}(z_j', z_{j+1})|\leq C(\varepsilon) e^{-\frac{1}{8}\tau_n q_n(L-12\varepsilon)}.
\end{align}
If $z_{t+1}'\in I^-$, we control $|\phi(z_{t+1}')|\leq |\phi(x_0^-)|$.
If $z_{t+1}'\in I^+$, we control $|\phi(z_{t+1}')|\leq |\phi(x_0^+)|$.
Furthermore, we know from \eqref{eq:non-res-Green-special}, that one of the Green's function satisfies additional decay,
\begin{align}\label{eq:non-res_102}
|G_{I(z_j')}(z_j', z_{j+1})|\leq C(\varepsilon) e^{-\frac{1}{8}\tau_n q_n(L-12\varepsilon)} c_{n,\ell}.
\end{align}
We control $|\phi(z_{t+1}')$ as follows
\begin{align}\label{eq:non-res_103}
|\phi(z_{t+1}')|\leq
\begin{cases}
|\phi(x_0^-)|, \text{ if } z_{t+1}'\in I^-\\
|\phi(\ell q_n+m_n)|, \text{ if } z_{t+1}'=\ell q_n+m_n\\
|\phi(x_0^+), \text{ if } z_{t+1}'\in I^+
\end{cases}
\end{align}
Hence combining \eqref{eq:non-res_101}, \eqref{eq:non-res_102} and \eqref{eq:non-res_103}, we have
\begin{align}\label{eq:non-res_104_t0}
&|G_{I(y)}(y, z_1)G_{I(z_1')}(z_1', z_2)\cdots G_{I(z_t')}(z_t', z_{t+1})\phi({z_{t+1}'})| \notag\\
&\leq (C(\varepsilon))^{t_0} e^{-\frac{1}{8}\tau_n t_0 (L-12\varepsilon)} \max
\{|\phi(x_0^-)|, |\phi(\ell q_n+m_n)|, c_{n,\ell} |\phi(x_0^+)|\} \notag\\
&\leq e^{-3q_n(L-12\varepsilon)} \max
\{|\phi(x_0^-)|, |\phi(\ell q_n+m_n)|, c_{n,\ell} |\phi(x_0^+)|\}.
\end{align}
Taking into account all the three cases \eqref{eq:non-res-Green-general_1}, \eqref{eq:non-res-Green-special} and \eqref{eq:non-res_104_t0}, we have proved that for $y\in I^-$,
\begin{align}\label{eq:non-res-I-}
|\phi(y)|\leq (C(\varepsilon))^{t_0} \max(& e^{\varepsilon q_n} e^{-(y-\ell q_n)(L-12\varepsilon)} r_{\ell}, e^{\varepsilon q_n} e^{-((\ell+1)q_n-y)(L-12\varepsilon)} c_{n,\ell} r_{\ell+1},\\
&e^{-3q_n (L-12\varepsilon)}
\max(|\phi(x_0^-)|, |\phi(\ell q_n+m_n)|, c_{n,\ell} |\phi(x_0^+)|) ). \notag
\end{align}
Similarly, one can show that for $y\in I_+$,
\begin{align}\label{eq:non-res-I+}
|\phi(y)|\leq (C(\varepsilon))^{t_0} \max(&e^{\varepsilon q_n} e^{-(y-\ell q_n)(L-12\varepsilon)} c_{n,\ell} r_{\ell}, e^{\varepsilon q_n} e^{-((\ell+1)q_n-y)(L-12\varepsilon)} r_{\ell+1},\\
&e^{-3q_n (L-12\varepsilon)}
\max(c_{n,\ell}|\phi(x_0^-)|, |\phi(\ell q_n+m_n)|, |\phi(x_0^+)|) ). \notag
\end{align}
and
\begin{align}\label{eq:non-res-I-mid}
|\phi(\ell q_n+m_n)|\leq (C(\varepsilon))^{t_0} c_{n,\ell} \max(&e^{\varepsilon q_n} e^{-(y-\ell q_n)(L-12\varepsilon)} r_{\ell}, e^{\varepsilon q_n} e^{-((\ell+1)q_n-y)(L-12\varepsilon)} r_{\ell+1},\\
&e^{-3q_n (L-12\varepsilon)}
\max(\phi(x_0^-)|, |\phi(\ell q_n+m_n)|, |\phi(x_0^+)|) ). \notag
\end{align}
Combining \eqref{eq:non-res-I-} applied to $y=x_0^-$, \eqref{eq:non-res-I-} applied to $y=x_0^+$, with \eqref{eq:non-res-I-mid}, we obtain that
\begin{align*}
\max(|\phi(x_0^-)|, |\phi(x_0^+)|, |\phi(aq_n+m_n)|)\leq (C(\varepsilon))^{t_0} \max(r_{\ell}, r_{\ell+1}).
\end{align*}
Plugging this back into \eqref{eq:non-res-I-}, \eqref{eq:non-res-I+}, \eqref{eq:non-res-I-mid}, we obtain the claimed result.
\qed
\subsubsection*{Proof of Lemma \ref{lem:22_non-to-res}}
The proof of this lemma is almost identical to that of Lemma \ref{lem:C2_n-r}. We only give a brief proof.
By Green's function expansion, we have
\begin{align*}
\phi(y)=\sum_{z\in \partial I(y)}G_{I(y)}(z,y) \phi(z').
\end{align*}
If $x_1-1> \ell q_n+b_n$ or $x_2+1<(\ell+1)q_n-b_n$, we continue to expand $\phi(x_1-1)$ or $\phi(x_2+1)$.
We repeat this process until we arrive at a $z$ so that $z\leq \ell q_n+b_n$ or $z\geq (\ell+1)q_n-b_n$, or the iterating number reaches $t_0:=[24/\tau_n]+1$.
We obtain, after a series of expansions, the following
\begin{align*}
\phi(y)=\sum_{s; z_{i+1}\in I(z_i')}G_{I(y)}(y, z_1)G_{I(z_1')}(z_1', z_2)\cdots G_{I(z_t')}(z_t', z_{t+1})\phi({z_{t+1}'}),
\end{align*}
where $z_{t+1}'$ either satisfies
{\it Case 1}\ :$z_{t+1}'\in R_{\ell}^+\cup \{\ell q_n+m_n\}$ and $t<t_0$ or
{\it Case 2}\ :$z_{t+1}'\in R_{\ell}^-$ and $t<t_0$ or
{\it Case 3}\ : $z_{t+1}'\in R_{\ell+1}^-$ and $t<t_0$ or
{\it Case 4}\ : $t=t_0$.
If $z_{t+1}'$ satisfies Case 1.
One can follow the proof of Case 1 of Lemma \ref{lem:C2_n-r}.
Bounding $|\phi(z_{t+1}')|\leq \max(r_{\ell}^+, |\phi(\ell q_n+m_n)|)$, we have, similar to \eqref{eq:non-res-Green-general_1}
\begin{align}\label{eq:non-res-Green-general_111}
|G_{I(y)}(y, z_1)G_{I(z_1')}(z_1', z_2)\cdots G_{I(z_t')}(z_t', z_{t+1})\phi({z_{t+1}'})|\leq (C(\varepsilon))^{t_0+1} e^{\varepsilon q_n} e^{-(y-\ell q_n)(L-12\varepsilon)}\max(r_{\ell}^+, |\phi(\ell q_n+m_n)|).
\end{align}
Note that by the eigenvalue equation,
$$|\phi(\ell q_n+m_n)|\leq Cc_{n,\ell}\max(|\phi(\ell q_n+m_n-1)|, |\phi(\ell q_n+m_n+1)|)\leq Cc_{n,\ell}e^{5\varepsilon q_n}\max(r_{\ell}^-, r_{\ell}^+).$$
Hence \eqref{eq:non-res-Green-general_111} yields
\begin{align}\label{eq:non-res-Green-general_111'}
|G_{I(y)}(y, z_1)G_{I(z_1')}(z_1', z_2)\cdots G_{I(z_t')}(z_t', z_{t+1})\phi({z_{t+1}'})|\leq (C(\varepsilon))^{t_0+1} e^{\varepsilon q_n} e^{-(y-\ell q_n)(L-12\varepsilon)}\max(r_{\ell}^+, Cc_{n,\ell}e^{5\varepsilon q_n}r_{\ell}^-).
\end{align}
If $z_{t+1}'$ satisfies Case 2, there must be a $z_j'$ such that $aq_n+m_n\in I(z_j')$.
Similar to the proof of Case 2 of Lemma \ref{lem:C2_n-r}, we have
\begin{align}\label{eq:non-res-Green-special_222}
|G_{I(y)}(y, z_1)G_{I(z_1')}(z_1', z_2)\cdots G_{I(z_t')}(z_t', z_{t+1})\phi({z_{t+1}'})|\leq (C(\varepsilon))^{t_0+1} e^{\varepsilon q_n} e^{-(y-\ell q_n)(L-12\varepsilon)} c_{n,\ell} r_{\ell}^-.
\end{align}
If $z_{t+1}'$ satisfies Case 3. Estimating similar to Case 1, we have
\begin{align}\label{eq:non-res-Green-general_333}
|G_{I(y)}(y, z_1)G_{I(z_1')}(z_1', z_2)\cdots G_{I(z_t')}(z_t', z_{t+1})\phi({z_{t+1}'})|\leq (C(\varepsilon))^{t_0+1} e^{\varepsilon q_n} e^{-((\ell+1) q_n-y)(L-12\varepsilon)}.
\end{align}
If $z_{t+1}'$ satisfies Case 4. Estimating similar to Case 3 of the proof of Lemma \ref{lem:C2_n-r}, we have
\begin{align}\label{eq:non-res_104_t0_444}
|G_{I(y)}(y, z_1)G_{I(z_1')}(z_1', z_2)\cdots G_{I(z_t')}(z_t', z_{t+1})\phi({z_{t+1}'})| \leq e^{-3q_n(L-12\varepsilon)} |\phi(x_0)|,
\end{align}
where $|\phi(x_0)|:=\sup_{\ell q_n+b_n<y<(\ell+1)q_n-b_n} |\phi(y)|$.
Taking into account that (by Corollary \ref{cor:A_upper}),
\begin{align*}
|\phi(x_0)|\leq e^{3\varepsilon q_n} e^{q_nL} \max(r_{\ell}^+, r_{\ell+1}^-).
\end{align*}
\eqref{eq:non-res_104_t0_444} yields
\begin{align}\label{eq:non-res_104_t0_555}
|G_{I(y)}(y, z_1)G_{I(z_1')}(z_1', z_2)\cdots G_{I(z_t')}(z_t', z_{t+1})\phi({z_{t+1}'})|\leq e^{(-2L+40\varepsilon)q_n}\max(r_{\ell}^+, r_{\ell+1}^-).
\end{align}
Eventually combining the four cases \eqref{eq:non-res-Green-general_111}, \eqref{eq:non-res-Green-special_222}, \eqref{eq:non-res-Green-general_333} and \eqref{eq:non-res_104_t0_555}, we have
\begin{align}\label{eq:non-res-Green-666}
|\phi(y)|\leq (C(\varepsilon))^{t_0} e^{18\varepsilon q_n} \max(e^{-(y-\ell q_n)L}r_{\ell}^+, c_{n,\ell}e^{-(y-\ell q_n)L}r_{\ell}^-, e^{-((\ell+1)q_n-y)L}r_{\ell+1}^-).
\end{align}
By Corollary \ref{cor:A_upper_mn}, we have
\begin{align*}
r_{\ell}^-\leq e^{9\varepsilon q_n} \frac{1}{c_{n,\ell}} r_{\ell}^+.
\end{align*}
Hence \eqref{eq:non-res-Green-666} yields
\begin{align}\label{eq:non-res-Green-777}
|\phi(y)|\leq e^{30\varepsilon q_n} \max(e^{-(y-\ell q_n)L}r_{\ell}^+, e^{-((\ell+1)q_n-y)L}r_{\ell+1}^-).
\end{align}
This proves the claimed result.
\qed
\subsection{Proof of Case 1 of Lemma \ref{lem:main}}\
We divide into two cases depending if $k<q_n/2$.
Case 1. If $\frac{q_n}{12}< k< \frac{q_n}{2}$. \
The proof of this lemma is similar to that of Lemma \ref{lem:C2_n-r}. We only give a brief proof.
By Green's function expansion, we have
\begin{align*}
\phi(k)=\sum_{z\in \partial I(k)}G_{I(k)}(z,k) \phi(z').
\end{align*}
If $x_1-1>b_n$ or $x_2+1<q_n-b_n$, we continue to expand $\phi(x_1-1)$ or $\phi(x_2+1)$.
We repeat this process until we arrive at a $z$ so that $z\leq b_n$ or $z\geq q_n-b_n$, or the iterating number reaches $t_0:=[24/\tau_n]+1$.
We obtain, after a series of expansions, the following
\begin{align*}
\phi(k)=\sum_{s; z_{i+1}\in I(z_i')}G_{I(k)}(k, z_1)G_{I(z_1')}(z_1', z_2)\cdots G_{I(z_t')}(z_t', z_{t+1})\phi({z_{t+1}'}),
\end{align*}
where $z_{t+1}'$ either satisfies
{\it Case (i)}\ : $z_{t+1}'\leq b_n$ and $t<t_0$ or
{\it Case (ii)}\ : $z_{t+1}'\geq q_n-b_n$ and $t<t_0$ or
{\it Case (iii)}\ : $t=t_0$.
If $z_{t+1}'$ satisfies Case (i).
One can follow the proof of Case 1 of Lemma \ref{lem:C2_n-r}.
Bounding $|\phi(z_{t+1}')|\leq r_0$, we have, similar to \eqref{eq:non-res-Green-general_1}
\begin{align}\label{eq:non-res-Green-general_111_dio}
|G_{I(k)}(k, z_1)G_{I(z_1')}(z_1', z_2)\cdots G_{I(z_t')}(z_t', z_{t+1})\phi({z_{t+1}'})|\leq (C(\varepsilon))^{t_0+1} e^{\varepsilon q_n} e^{-k(L-12\varepsilon)}r_0.
\end{align}
If $z_{t+1}'$ satisfies Case (ii), similar to Case (i) above, bounding $|\phi(z_{t+1}')|\leq r_1$, we have, similar to \eqref{eq:non-res-Green-general_1}
\begin{align}\label{eq:non-res-Green-special_222_dio}
|G_{I(k)}(k, z_1)G_{I(z_1')}(z_1', z_2)\cdots G_{I(z_t')}(z_t', z_{t+1})\phi({z_{t+1}'})|\leq (C(\varepsilon))^{t_0+1} e^{\varepsilon q_n} e^{-(q_n-k)(L-12\varepsilon)} r_1.
\end{align}
If $z_{t+1}'$ satisfies Case (iii). Estimating similar to Case 3 of the proof of Lemma \ref{lem:C2_n-r}, we have
\begin{align}\label{eq:non-res_104_t0_444_dio}
|G_{I(k)}(k, z_1)G_{I(z_1')}(z_1', z_2)\cdots G_{I(z_t')}(z_t', z_{t+1})\phi({z_{t+1}'})| \leq e^{-3q_n(L-12\varepsilon)} |\phi(x_0)|,
\end{align}
where $|\phi(x_0)|:=\sup_{b_n<y<q_n-b_n} |\phi(y)|$.
Taking \eqref{Shnol} into account, we have
\begin{align}\label{eq:dio_333}
\max(r_0, r_1, |\phi(x_0)|)\leq C_0 q_n.
\end{align}
Hence combining \eqref{eq:dio_333} with \eqref{eq:non-res-Green-general_111_dio}, \eqref{eq:non-res-Green-special_222_dio} and \eqref{eq:non-res_104_t0_444}, we obtain
\begin{align}\label{eq:non-res-Green-666_dio}
|\phi(k)|\leq e^{2\varepsilon q_n} e^{-k (L-12\varepsilon)}\leq e^{-k(L-36\varepsilon)}.
\end{align}
where we used $q_n<12k$.
Case 2. If $\frac{q_n}{2}< k< \frac{q_{n+1}}{12}$. \
For $y$ such that $\frac{q_n}{2}< y< \frac{q_{n+1}}{6}$,
let $s$ be the smallest positive integer such that
\begin{align*}
(2s-1-\frac{1}{2})q_n\leq y<(2s+1-\frac{1}{2})q_n,
\end{align*}
and
\begin{align*}
I_1:= [-2sq_n+[sq_n/2]+1, -sq_n+[sq_n/2] ], \text{ and } I_2:= [y-[sq_n/2]-sq_n+1, y-[sq_n/2]].
\end{align*}
It is easy to see that $I_1\cap I_2=\emptyset$.
Next we show
\begin{lemma}\label{lem:Dio_uni}
For $n$ large enough, $\{\theta_{\ell}\}_{\ell\in I_1\cup I_2}$ is $153\varepsilon$-uniform.
\end{lemma}
\begin{proof}
We divide the $2sq_n$ points into $2s$ intervals: $T_1, \cdots, T_{2s}$, each containing $q_n$ points.
Fixing any $j$. For $1\leq w\leq 2s$, let
$$|\sin\pi(\theta_j-\theta_{{\ell}_w})|:=\min_{\ell\in T_w} |\sin\pi(\theta_j-\theta_{\ell})|.$$
Without loss of generality, assume $j \in T_{s_0}$ for some $s_0$ such that $1\leq s_0 \leq s$.
By Lemma \ref{lana}, we have
\begin{align}\label{Dio:3}
\sum_{\ell \neq j} \ln|\sin\pi(\theta-\theta_{\ell})| \leq 2s (C\ln q_n-(q_n-1)\ln 2) \leq 2sq_n(-\ln 2+\varepsilon),
\end{align}
and
\begin{align}\label{Dio:4}
& \sum_{\ell \neq j} \ln|\sin\pi(\theta_j-\theta_{\ell})| \notag \\
\geq &2s(-C\ln q_n-(q_n-1)\ln2)+\sum_{w=1, w\neq s_0}^s \ln|\sin\pi(\theta_j-\theta_{\ell_w})|
+\sum_{w=s+1}^{2s} \ln|\sin\pi(\theta_i-\theta_{\ell_w})| \notag\\
\geq &2sq_n(-\ln 2-\varepsilon)+\sum_{w=1, w\neq s_0}^s \ln \|(j-\ell_w)\alpha\|
+\sum_{w=s+1}^{2s} \ln \|(j-\ell_w)\alpha\|
\end{align}
We have
\begin{align}\label{eq:klarge}
|j-\ell_w|< y+sq_n<3y<q_{n+1},
\end{align}
hence for each $w\neq s_0$, we have by \eqref{eq:cont2} and \eqref{def:betan} that
\begin{align}\label{Dio:5}
\|(j-\ell_w)\alpha\|\geq \|q_n\alpha\|\geq\frac{1}{2q_{n+1}}=\frac{1}{2}e^{-\beta_n q_n}\geq \frac{1}{2}e^{-300\varepsilon q_n}.
\end{align}
Combining \eqref{Dio:3}, \eqref{Dio:4} and \eqref{Dio:5}, we have
\begin{align}
\sum_{\ell \neq j} \ln|\sin\pi(\theta-\theta_{\ell})|-\sum_{\ell \neq j} \ln|\sin\pi(\theta_j-\theta_{\ell})|<305\varepsilon sq_n.
\end{align}
\qed
\end{proof}
Lemma \ref{lem:Dio_uni} implies the following, similar to Lemma \ref{lem:nonres_I2_large},
\begin{lemma}
For $n$ large enough, there exists $x_1\in I_2$ such that that
\begin{align}\label{Dio:6}
|\tilde{P}_{2sq_n-1}(\theta_{x_1})|\geq e^{(\tilde{L}-155\varepsilon)(2sq_n-1)}.
\end{align}
\end{lemma}
Let $x_2=x_1+2sq_n-2$ and $I(k):=[x_1, x_2]$.
Plugging \eqref{Dio:6} into the Green's formula \eqref{Green_tildeP}, and using estimates from Lemma \ref{lem:upperbddtildeP} and Corollary \ref{cor:prod_cos}, we have that for $\frac{q_n}{2}<k<\frac{q_{n+1}}{12}$,
\begin{align}\label{Dio2}
|\phi(k)|
\leq &\sum_{z\in \partial I(k)} e^{-(L-625\varepsilon)|k-z|} |\phi(z')|.
\end{align}
Iterating this process for $\phi(z')$ until we arrive at a $z'$ such that $z'\leq \max(\gamma k, q_n/2)$ or $z'\geq 2k$ or the iteration number $t$ reaches $t_0:=[5/\gamma]+1$, where $\gamma$ is a small positive constant such that
$$(L-625\varepsilon)(1-\gamma)=L-626\varepsilon.$$
We obtain, after a series of expansions, the following
\begin{align*}
|\phi(k)|\leq \sum_{s; z_{i+1}\in I(z_i')} e^{-(L-625\varepsilon)(|k-z_1|+|z_1'-z_2|+...+|z_t'-z_{t+1}|)} |\phi({z_{t+1}'})|,
\end{align*}
where $z_{t+1}'$ either satisfies
{\it Case (i)}\ : $z_{t+1}'\leq \max(\gamma k, \frac{q_n}{2})$ and $t<t_0$ or
{\it Case (ii)}\ : $z_{t+1}'\geq 2k$ and $t<t_0$ or
{\it Case (iii)}\ : $t=t_0$. \
If $z_{t+1}'\leq \frac{q_n}{2}$, we bound $|\phi(z_{t+1}')|$ by \eqref{eq:non-res-Green-666_dio}, which is
\begin{align}\label{Dio33}
|\phi(z_{t+1}')|\leq e^{-(L-36\varepsilon)z_{t+1}'},
\end{align}
and hence
\begin{align}\label{Dio33'}
e^{-(L-625\varepsilon)(|k-z_1|+|z_1'-z_2|+...+|z_t'-z_{t+1}|)} |\phi({z_{t+1}'})|\leq &e^{-(L-625\varepsilon)(k-z_{t+1})} e^{-(L-36\varepsilon)z_{t+1}'}
\leq e^{-(L-625\varepsilon)k}.
\end{align}
If $z_{t+1}'\leq \gamma k$, bounding $|\phi(z_{t+1}')|\leq C_0 \gamma k$ by \eqref{Shnol},
we obtain
\begin{align}\label{Dio33''}
e^{-(L-625\varepsilon)(|k-z_1|+|z_1'-z_2|+...+|z_t'-z_{t+1}|)} |\phi({z_{t+1}'})|\leq &C_0 \gamma k e^{-(L-625\varepsilon)(k-z_{t+1})} \notag\\
\leq &C_0 \gamma k e^{-(L-625\varepsilon)(1-\gamma)k} \notag\\
\leq &e^{-(L-627\varepsilon)k}.
\end{align}
If $z_{t+1}'$ satisfies Case (ii), bounding $|\phi(z_{t+1}')|\leq C_0 z_{t+1}'$ by \eqref{Shnol}, we obtain
\begin{align}\label{Dio44'}
e^{-(L-625\varepsilon)(|k-z_1|+|z_1'-z_2|+...+|z_t'-z_{t+1}|)} |\phi({z_{t+1}'})|\leq &e^{-(L-625\varepsilon)|k-z_{t+1}|} C_0 z_{t+1}'\notag\\
\leq &e^{-(L-626\varepsilon)|k-z_{t+1}|}\leq e^{-(L-626\varepsilon)k}.
\end{align}
If $z_{t+1}'$ satisfies Case (iii), we bound $|\phi(z_{t+1}')|\leq C_0z_{t+1}'\leq 2C_0 k$ by \eqref{Shnol}.
Further we bound each $|z_j'-z_{j+1}|$, denoting for simplicity $k=z_0'$, in the following way.
For $z_j'\geq \gamma k$ satisfying
\begin{align*}
\max(\gamma k, (2s-1-\frac{1}{2})q_n)\leq z_j'<(2s+1-\frac{1}{2})q_n,
\end{align*}
we have
\begin{align*}
|z_j'-z_{j+1}|\geq \frac{1}{2}sq_n\geq \frac{s}{4s+1} \gamma k\geq \frac{1}{5}\gamma k.
\end{align*}
Hence we have by \eqref{def:betan} that
\begin{align}\label{Dio55}
e^{-(L-625\varepsilon)(|k-z_1|+|z_1'-z_2|+...+|z_t'-z_{t+1}|)} |\phi({z_{t+1}'})|
\leq &2C_0 k e^{-(L-625\varepsilon) \frac{t_0}{5}\gamma k}\notag\\
\leq &e^{-(L-626\varepsilon)k}.
\end{align}
Summarizing \eqref{Dio33'}, \eqref{Dio33''}, \eqref{Dio44'} and \eqref{Dio55}, we have
\begin{align}\label{Dio66}
|\phi(k)|\leq 2^{t_0}e^{-(L-627\varepsilon)k}\leq e^{-(L-630\varepsilon) k}.
\end{align}
\qed
\section*{Acknowledgement}
R. H. is partially supported by NSF-DMS-2053285. S.J. was a 2020-21 Simons fellow. Her work
was also partially supported by NSF DMS-2052899, DMS-2155211, and Simons 681675. F. Y. is partially supported by an AMS-Simons Travel Grant.
R. H. and F. Y. thank the hospitality of University of California,
Irvine during summer 2017, when the key work of this paper was done and the work on \cite{HJY} was started.
\bibliographystyle{amsplain}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 3,651
|
<?xml version='1.0'?>
<!DOCTYPE sconsdoc [
<!ENTITY % scons SYSTEM "../scons.mod">
%scons;
<!ENTITY % builders-mod SYSTEM "../generated/builders.mod">
%builders-mod;
<!ENTITY % functions-mod SYSTEM "../generated/functions.mod">
%functions-mod;
<!ENTITY % tools-mod SYSTEM "../generated/tools.mod">
%tools-mod;
<!ENTITY % variables-mod SYSTEM "../generated/variables.mod">
%variables-mod;
]>
<chapter id="chap-sideeffect"
xmlns="http://www.scons.org/dbxsd/v1.0"
xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
xsi:schemaLocation="http://www.scons.org/dbxsd/v1.0 http://www.scons.org/dbxsd/v1.0/scons.xsd">
<title>Sideeffect files</title>
<!--
__COPYRIGHT__
Permission is hereby granted, free of charge, to any person obtaining
a copy of this software and associated documentation files (the
"Software"), to deal in the Software without restriction, including
without limitation the rights to use, copy, modify, merge, publish,
distribute, sublicense, and/or sell copies of the Software, and to
permit persons to whom the Software is furnished to do so, subject to
the following conditions:
The above copyright notice and this permission notice shall be included
in all copies or substantial portions of the Software.
THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY
KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE
WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE
LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION
OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION
WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
-->
<para>
If &SCons; is unaware that a build step produces an extra file,
the &SideEffect; method can be used to identify it,
so that the file can be used as a dependency in subsequent build steps.
However, the primary use for the &SideEffect; method
is to prevent two build steps from simultaneously modifying the same file.
</para>
<para>
TODO: currently doesn't work due to issue #2154:
http://scons.tigris.org/issues/show_bug.cgi?id=2154
</para>
<para>
If more than one build step creates or manipulates the same file,
it can cause unpleasant results if both build steps are run at the same time.
The shared file is declared as a side-effect of building the primary targets
and &SCons; will prevent the two build steps from running in parallel.
</para>
<para>
In this example, the <filename>SConscript</filename> uses
&SideEffect; to inform &SCons; about the additional output file.
</para>
<scons_example name="sideeffect_simple">
<file name="SConstruct" printme="1">
env = Environment()
f2 = env.Command('file2', 'log', Copy('$TARGET', '$SOURCE'))
f1 = env.Command('file1', [],
'echo >$TARGET data1; echo >log updated file1'))
env.SideEffect('log', env.Command('file1', [],
'echo >$TARGET data1; echo >log updated file1'))
</file>
</scons_example>
<para>
Even when run in parallel mode, &SCons; will run the two steps in order:
</para>
<scons_output example="sideeffect_simple" suffix="1">
<scons_output_command>scons -Q --jobs=2</scons_output_command>
</scons_output>
<para>
Sometimes a program the you need to call
to build a target file
will also update another file,
such as a log file describing what the program
does while building the target.
For example, we the folowing configuration
would have &SCons; invoke a hypothetical
script named <application>build</application>
(in the local directory)
with command-line arguments that write
log information to a common
<filename>logfile.txt</filename> file:
</para>
<screen>
env = Environment()
env.Command('file1.out', 'file.in',
'./build --log logfile.txt $SOURCE $TARGET')
env.Command('file2.out', 'file.in',
'./build --log logfile.txt $SOURCE $TARGET')
</screen>
<para>
This can cause problems when running
the build in parallel if
&SCons; decides to update both targets
by running both program invocations at the same time.
The multiple program invocations
may interfere with each other
writing to the common log file,
leading at best to intermixed output in the log file,
and at worst to an actual failed build
(on a system like Windows, for example,
where only one process at a time can open the log file for writing).
</para>
<para>
We can make sure that &SCons; does not
run these <application>build</application>
commands at the same time
by using the &SideEffect; function
to specify that updating
the <filename>logfile.txt</filename> file
is a side effect of building the specified
<filename>file1</filename>
and
<filename>file2</filename>
target files:
</para>
<scons_example name="sideeffect_shared">
<file name="SConstruct" printme="1">
env = Environment()
f1 = env.Command('file1.out', 'file1.in',
'./build --log logfile.txt $SOURCE $TARGET')
f2 = env.Command('file2.out', 'file2.in',
'./build --log logfile.txt $SOURCE $TARGET')
env.SideEffect('logfile.txt', f1 + f2)
</file>
<file name="file1.in">file1.in</file>
<file name="file2.in">file2.in</file>
<file name="build" chmod="0755">
cat
</file>
</scons_example>
<para>
</para>
<para>
This makes sure the the two
<application>./build</application> steps are run sequentially,
even withthe <filename>--jobs=2</filename> in the command line:
</para>
<scons_output example="sideeffect_shared" suffix="1">
<scons_output_command>scons -Q --jobs=2</scons_output_command>
</scons_output>
<para>
The &SideEffect; function can be called multiple
times for the same side-effect file.
Additionally, the name used as a &SideEffect; does not
even need to actually exist as a file on disk.
&SCons; will still make sure
that the relevant targets
will be executed sequentially, not in parallel:
</para>
<scons_example name="sideeffect_parallel">
<file name="SConstruct" printme="1">
env = Environment()
f1 = env.Command('file1.out', [], 'echo >$TARGET data1')
env.SideEffect('not_really_updated', f1)
f2 = env.Command('file2.out', [], 'echo >$TARGET data2')
env.SideEffect('not_really_updated', f2)
</file>
</scons_example>
<scons_output example="sideeffect_parallel" suffix="1">
<scons_output_command>scons -Q --jobs=2</scons_output_command>
</scons_output>
<para>
Note that it might be tempting to
use &SideEffect; for additional target files
that a command produces.
For example, versions the Microsoft Visual C/C++ compiler
produce a <filename>foo.ilk</filename>
alongside compiling <filename>foo.obj</filename> file.
Specifying <filename>foo.ilk</filename> as a
side-effect of <filename>foo.obj</filename>
is <emphasis>not</emphasis> a recommended use of &SideEffect;,
because &SCons; handle side-effect files
slightly differently in its analysis of the dependency graph.
When a command produces multiple output files,
they should be specified as multiple targets of
the call to the relevant builder function,
and the &SideEffect; function itself should really only be used
when it's important to ensure that commands are not executed in parallel,
such as when a "peripheral" file (such as a log file)
may actually updated by more than one command invocation.
</para>
</chapter>
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,704
|
Here and Now, stylisé Here & Now, est un groupe de rock psychédélique britannique, originaire de Londres, en Angleterre. Formé en 1974, il est un groupe proche de Gong, ils ont d'ailleurs travaillé avec Daevid Allen et Gilli Smyth en 1977 et 1978 sous le nom de Planet Gong.
Le son du guitariste Steffy Sharpstrings est très personnel, influencé par Jimi Hendrix, Frank Zappa et les deux guitaristes de Gong : Daevid Allen et Steve Hillage. Steffy et le bassiste Keith « the Bass » ont participé à certaines formations suivantes du groupe Gong.
Biographie
Années 1970
Le groupe se forme dans la communauté d'un squat de Ladbroke Grove, dans l'ouest de Londres. Il est connu pour beaucoup improviser. Paul Noble (aussi nommé Twink), joueur de synthétiseur et le batteur Keith « Kif Kif the Drummer » Dobson forment le noyau du groupe. Le guitariste Stephan Lewry et le bassiste Keith Bailey les rejoignent pour jouer au Free-festival de Watchfield en août 1975. Ils sont souvent accompagnés de deux choristes, Ano et Suze de Blooze. Un autre élément essentiel du groupe est Grant Showbiz, qui travaillera aussi avec The Fall, The Smiths, et Billy Bragg. En 1977, Twink est remplacé aux synthés par Gavin da Blitz.
En 1977, Daevid Allen and Gilli Smyth les recrutent pour une tournée sous le nom de Planet Gong. Ils publient l'album live Floating Anarchy Live 77 et le single Opium for the People. Le groupe est aussi très proche de Alternative TV, le groupe du punk Mark Perry, avec lequel ils tournent et produit un album commun en 1978 : What You See Is What You Are.
Années 1980–1990
En 1980, Kif Kif quitte le groupe pour monter Street Level, son propre studio et Fuck Off Records, un label indépendant dans l'esprit du Do it yourself de l'époque. Il monte aussi un trio : World Domination Enterprises. Le groupe tourne en 1980 et 1981, en vivant dans un grand bus aménagé. Ils jouent au free-festival de Deeply Vale, à Stonehenge et à Glastonbury. Il n'y a jamais d'entrée payante à leur concert mais une collecte de fonds parmi la foule.
En 1981, Steffe quitte le groupe, mécontent de l'orientation et des « vibes », il s'implique dans la musique reggae. Il joue avec Inner Force, Addis rockers et forme Nomadiks. Here and Now continue sous la houlette de Keith the Bass avec Dino Ferrari (un roadie de longue date) et différents batteurs. Ils sortent Fantasy Shift, Coaxed out from Oxford, Theatre et Been and Gone. Ils se séparent du groupe en 1986 à la suite de problèmes avec leur maison de disque et du saccage par la police du Free Festival de Stonehenge. Ils se reforment une année plus tard avec Keith à la basse, Gavin aux claviers, Dino Ferrari à la guitare, Jonathan 'JC' Lambert au saxo et l'ex UK Subs Pete Davis à la batterie. Gavin quittent Here and Now en 1990, remplacé par Andy Roid. Steffe revient avec le batteur Steve Cassidy en 1990 après un concert télévisé de Gong avec Keith pour Central TV. Ils enregistrent l'album UFOasis.
Années 1990–2010
Dans les années 1990, Keith et Steffe jouent sur différentes tournées du Gong. Au début des années 2000, Keith et Steffe continuent avec Joie Hinton (de Ozric Tentacles et Eat Static) aux claviers et Steve Cassidy. À la fin de la décennie, Kif Kif, Steffe et Twink jouent de la musique improvisée sous le nom de « Ici Maintenants ».
Steffe quitte à nouveau le groupe au début de l'été 2009, ne souhaitant plus jouer avec Keith. Il continue à travailler avec Steve Cassidy sur différents projets et sur les projets de Joie Hinton. Il est maintenant concentré sur son travail solo et le groupe Visitation Arena avec Cher Newsam. En , une nouvelle formation du groupe est annoncée : Gwyo ZePix (ex Zorch et Gong), Slim Verhoef (ex Giant Eyes), Nik Nimbus, Drumbiz (ex Mandragora et Giant Eyes)et Keith the Bass. Esoteric Recordings sort en septembre 2010 les versions remasterisées de Give & Take et All Over The Show.
En 2011, Mark Robson de Kangaroo Moon prends les claviers. Ils jouent au London Club Dingwalls la même année.
Discographie
1977 : Floating Anarchy Live 77 (sous le nom « Planet Gong »)
1978 : Give and Take
1978 : Dog in Hell (EP)
1978 : What You See... Is What You Are (LP avec Alternative TV, Deptford Fun City, DLP 02)
All Over the Show
1979 : Off The Cuff (Live sur cassette)
Stolen Moments (début des années 1980, sur cassette)
Fantasy Shift
Theatre
Been and Gone
Standing Forever (EP)
UFOasis
1999 : Gospel of Free (morceaux de 1976 à 1978)
2003 : Space and Time (sous le nom « Ici Maintenants ») (2001/2003)
2009 (réédition) : Coaxed out from Oxford
Notes et références
Liens externes
Groupe britannique de rock psychédélique
Groupe de space rock
Groupe musical formé en 1974
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,580
|
Nabawa è una piccola città situata nella regione di Mid West, in Australia Occidentale; essa si trova lungo il corso del Chapman River, 463 chilometri a nord di Perth ed è la sede della Contea di Chapman Valley.
Storia
Il primo europeo ad esplorare la zona in cui sorge l'attuale Nawara fu George Grey, che esplorò la regione nel 1839, anche se la prima vera esplorazione del sito in cui sarebbe poi stata costruita la città è stata condotta nel 1857, allorché si scoprì uno stagno lungo il corso del Chapman River. Il nome dell'insediamento deriva direttamente dalla parola aborigena con cui era noto quello stagno, Nabawar, che probabilmente significa accampamento molto lontano.
Il nome della città nel corso del XIX e del XX secolo venne scritto in numerosi modi, Nabawar, Nabawah o Nabawa, ma quest'ultimo ebbe la prevalenza allorché nel 1910 venne così chiamata la fermata della ferrovia appena costruita. Nel 1961 la ferrovia venne chiusa, ma alcuni anni dopo il governo della Contea di Chapman Valley decise di spostare a Nabawa la sede amministrativa e Nabawa ottenne nel 1965 lo status di town.
Note
Collegamenti esterni
Centri abitati dell'Australia Occidentale
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 3,105
|
Rose Petal Hydrosol, commonly known as Rose Water, has been used for centuries. In-fact it's one of the first known floral waters to be distilled and used by humans.
This spray is helpful for balancing the natural pH of skin, restoring a healthy terrain free from excess oil and blemishes. This cooling spray is also used for reducing inflammation and redness and providing hydration. The antioxidant properties of rose water help to strengthen skin cells and regenerate skin tissues.
Spray Rose water in your hair as a fragrant and wonderful leave in conditioner; all the while reaping the mood enhancing benefits of the sweet scent of rose.
Steam distilled organic & wild Rosa damascena and other Rose (Rosa spp.) varieties.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 5,138
|
from game_model import Game
from scores_messages import AgeBracket
from scores_messages import Division
from scores_messages import League
class ListIdBiMap:
"""Encapsulates mappings to and from list id and structured properties."""
# List ID definitions corresponding to lists defined in the twitter account of
# @martin_cochran.
USAU_COLLEGE_OPEN_LIST_ID = '186814318'
USAU_COLLEGE_WOMENS_LIST_ID = '186814882'
USAU_CLUB_OPEN_LIST_ID = '186732484'
USAU_CLUB_WOMENS_LIST_ID = '186732631'
USAU_CLUB_MIXED_LIST_ID = '186815046'
AUDL_LIST_ID = '186926608'
MLU_LIST_ID = '186926651'
ALL_LISTS = [
USAU_COLLEGE_OPEN_LIST_ID,
USAU_COLLEGE_WOMENS_LIST_ID,
USAU_CLUB_OPEN_LIST_ID,
USAU_CLUB_WOMENS_LIST_ID,
USAU_CLUB_MIXED_LIST_ID,
AUDL_LIST_ID,
MLU_LIST_ID
]
# Simple data structure to lookup lists if the league, division, and age
# bracket were specified in the request.
LIST_ID_MAP = {
League.USAU: {
Division.OPEN: {
AgeBracket.COLLEGE: USAU_COLLEGE_OPEN_LIST_ID,
AgeBracket.NO_RESTRICTION: USAU_CLUB_OPEN_LIST_ID,
},
Division.WOMENS: {
AgeBracket.COLLEGE: USAU_COLLEGE_WOMENS_LIST_ID,
AgeBracket.NO_RESTRICTION: USAU_CLUB_WOMENS_LIST_ID,
},
Division.MIXED: {
AgeBracket.NO_RESTRICTION: USAU_CLUB_MIXED_LIST_ID,
},
},
League.AUDL: {
Division.OPEN: {
AgeBracket.NO_RESTRICTION: AUDL_LIST_ID,
},
},
League.MLU: {
Division.OPEN: {
AgeBracket.NO_RESTRICTION: MLU_LIST_ID,
},
},
}
LIST_ID_TO_DIVISION = {
USAU_COLLEGE_OPEN_LIST_ID: Division.OPEN,
USAU_COLLEGE_WOMENS_LIST_ID: Division.WOMENS,
USAU_CLUB_OPEN_LIST_ID: Division.OPEN,
USAU_CLUB_WOMENS_LIST_ID: Division.WOMENS,
USAU_CLUB_MIXED_LIST_ID: Division.MIXED,
AUDL_LIST_ID: Division.OPEN,
MLU_LIST_ID: Division.OPEN,
}
LIST_ID_TO_AGE_BRACKET = {
USAU_COLLEGE_OPEN_LIST_ID: AgeBracket.COLLEGE,
USAU_COLLEGE_WOMENS_LIST_ID: AgeBracket.COLLEGE,
USAU_CLUB_OPEN_LIST_ID: AgeBracket.NO_RESTRICTION,
USAU_CLUB_WOMENS_LIST_ID: AgeBracket.NO_RESTRICTION,
USAU_CLUB_MIXED_LIST_ID: AgeBracket.NO_RESTRICTION,
AUDL_LIST_ID: AgeBracket.NO_RESTRICTION,
MLU_LIST_ID: AgeBracket.NO_RESTRICTION,
}
LIST_ID_TO_LEAGUE = {
USAU_COLLEGE_OPEN_LIST_ID: League.USAU,
USAU_COLLEGE_WOMENS_LIST_ID: League.USAU,
USAU_CLUB_OPEN_LIST_ID: League.USAU,
USAU_CLUB_WOMENS_LIST_ID: League.USAU,
USAU_CLUB_MIXED_LIST_ID: League.USAU,
AUDL_LIST_ID: League.AUDL,
MLU_LIST_ID: League.MLU,
}
@staticmethod
def GetListId(division, age_bracket, league):
"""Looks up the list_id which corresponds to the given division and league.
Args:
division: Division of interest
age_bracket: AgeBracket of interest
league: League of interest
Returns:
The list id corresponding to that league and division, or '' if no such
list exists.
"""
d = ListIdBiMap.LIST_ID_MAP.get(league, {})
if not d:
return ''
d = d.get(division, {})
if not d:
return ''
return d.get(age_bracket, '')
@staticmethod
def GetStructuredPropertiesForList(list_id):
"""Returns the division, age_bracket, and league for the given list id.
Defaults to Division.OPEN, AgeBracket.NO_RESTRICTION, and League.USAU,
if the division, age_bracket, or leauge, respectively, does not exist in
the map for the given list_id.
Args:
list_id: ID of list for which to retrieve properties.
Returns:
(division, age_bracket, league) tuple for the given list ID.
"""
division = ListIdBiMap.LIST_ID_TO_DIVISION.get(list_id, Division.OPEN)
age_bracket = ListIdBiMap.LIST_ID_TO_AGE_BRACKET.get(list_id, AgeBracket.NO_RESTRICTION)
league = ListIdBiMap.LIST_ID_TO_LEAGUE.get(list_id, League.USAU)
return (division, age_bracket, league)
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,870
|
Saint-Martin-de-Fressengeas (Limousin: Sent Martin de Fraissenjas) is a commune in the Dordogne department in Nouvelle-Aquitaine in southwestern France.
Population
See also
Communes of the Dordogne department
References
Communes of Dordogne
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,613
|
Home All Products Pre-Order Star Wars: The Empire Strikes Back 40th Anniversary - Boba Fett 1:6 Scale Figure - Hot Toys
Star Wars: The Empire Strikes Back 40th Anniversary - Boba Fett 1:6 Scale Figure - Hot Toys
Product no.: SSHOT906324
Price incl. VAT, excl. delivery
Armed in customized Mandalorian armor, dangerous weaponry and highly trained combat skills, Boba Fett has earned a notorious reputation as one of the deadliest bounty hunters in the galaxy as he takes on contracts from the criminal underworld and the Galactic Empire.
Boba Fett has been a fan-favorite character since his introduction four decades ago in Star Wars: The Empire Strikes Back. Today, Hot Toys is delighted to re-introduce the Boba Fett Sixth Scale Collectible Figure as an addition to Hot Toys Star Wars: The Empire Strikes Back 40th Anniversary Collection!
Based on his famous appearance in the beloved film, the Boba Fett collectible figure features a meticulously crafted Mandalorian helmet and armor with distressed effects, his iconic jetpack, a cape, detailed blasters, and a display stand! Furthermore, this sixth scale collectible figure features an alternative version of Boba Fett's armor including a range of interchangeable parts such as helmet, jetpack, gauntlets, cape, and a number of gloved hands all packaged in specially designed retro style packaging!
Join the hunt and embrace the 40-year legacy of this cinematic masterpiece. Don't miss out on this great Boba Fett collectible figure for your Star Wars: The Empire Strikes Back 40th Anniversary Collection!
ExpectedArrival_String Q1 2021
Browse these categories as well: Pre-Order, All Products
Business hours: Wednesday: 13:00 - 17:00
Thursday: 13:00 - 17:00, 18:00 - 20:00
Contact us for collection outside business hours.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 1,482
|
#include "tensorflow/core/grappler/optimizers/remapper.h"
#include "tensorflow/cc/ops/standard_ops.h"
#include "tensorflow/core/framework/tensor_testutil.h"
#include "tensorflow/core/grappler/devices.h"
#include "tensorflow/core/grappler/grappler_item.h"
#include "tensorflow/core/grappler/utils/grappler_test.h"
#include "tensorflow/core/lib/core/status_test_util.h"
#include "tensorflow/core/platform/test.h"
#if GOOGLE_CUDA
#include "third_party/gpus/cudnn/cudnn.h"
#endif // GOOGLE_CUDA
namespace tensorflow {
namespace grappler {
class RemapperTest : public GrapplerTest {
protected:
void SetUp() override {
// This is a requirement for fusing FusedBatchNorm + SideInput + Activation.
setenv("TF_USE_CUDNN_BATCHNORM_SPATIAL_PERSISTENT", "1", 1 /* replace */);
}
// TODO(b/119765980): Upgrade upstream Eigen to set `m_can_use_xsmm=false` for
// contractions with non-default contraction output kernels.
bool EigenSupportsContractionOutputKernel() {
#if defined(EIGEN_USE_LIBXSMM)
return false;
#endif
return true;
}
};
TEST_F(RemapperTest, FusedBatchNorm) {
tensorflow::Scope s = tensorflow::Scope::NewRootScope();
Output dflt = ops::Const(s.WithOpName("dflt"), {3.14f, 2.7f}, {2, 1, 1, 1});
Output x = ops::PlaceholderWithDefault(s.WithOpName("x"), dflt, {2, 1, 1, 1});
Output scale = ops::Const(s.WithOpName("scale"), {0.3f}, {1});
Output offset = ops::Const(s.WithOpName("offset"), {0.123f}, {1});
Output mean = ops::Const(s.WithOpName("mean"), {7.3f}, {1});
Output variance = ops::Const(s.WithOpName("variance"), {0.57f}, {1});
ops::FusedBatchNorm::Attrs attr;
attr = attr.IsTraining(false);
ops::FusedBatchNorm bn(s.WithOpName("batch_norm"), x, scale, offset, mean,
variance, attr);
GrapplerItem item;
TF_ASSERT_OK(s.ToGraphDef(&item.graph));
item.fetch = {"batch_norm"};
auto tensors_expected = EvaluateNodes(item.graph, item.fetch);
ASSERT_EQ(tensors_expected.size(), 1);
Remapper optimizer(RewriterConfig::ON);
GraphDef output;
TF_ASSERT_OK(optimizer.Optimize(nullptr, item, &output));
auto tensors = EvaluateNodes(output, item.fetch);
ASSERT_EQ(tensors.size(), 1);
test::ExpectTensorNear<float>(tensors[0], tensors_expected[0], 1e-6);
}
TEST_F(RemapperTest, FusedBatchNormNCHW) {
#if !GOOGLE_CUDA
GTEST_SKIP() << "CUDA is not enabled";
#endif // !GOOGLE_CUDA
tensorflow::Scope s = tensorflow::Scope::NewRootScope();
Output dflt =
ops::Const(s.WithOpName("dflt"), {3.14f, 2.7f, 1.0f, 2.0f, 3.0f, 100.0f},
{1, 3, 1, 2});
Output x = ops::PlaceholderWithDefault(s.WithOpName("x"), dflt, {1, 3, 1, 2});
Output scale = ops::Const(s.WithOpName("scale"), {0.3f, 7.0f, 123.0f}, {3});
Output offset =
ops::Const(s.WithOpName("offset"), {0.123f, 2.1f, 0.55f}, {3});
Output mean = ops::Const(s.WithOpName("mean"), {7.3f, 8.3f, 3.1f}, {3});
Output variance =
ops::Const(s.WithOpName("variance"), {0.57f, 1.0f, 2.0f}, {3});
ops::FusedBatchNorm::Attrs attr;
attr = attr.IsTraining(false);
attr = attr.DataFormat("NCHW");
ops::FusedBatchNorm bn(s.WithOpName("batch_norm").WithDevice("/device:GPU:0"),
x, scale, offset, mean, variance, attr);
GrapplerItem item;
TF_ASSERT_OK(s.ToGraphDef(&item.graph));
item.fetch = {"batch_norm"};
Remapper optimizer(RewriterConfig::ON);
GraphDef output;
TF_ASSERT_OK(optimizer.Optimize(nullptr, item, &output));
if (GetNumAvailableGPUs() > 0) {
// NCHW batch norm is only supported on GPU.
auto tensors_expected = EvaluateNodes(item.graph, item.fetch);
ASSERT_EQ(tensors_expected.size(), 1);
auto tensors = EvaluateNodes(output, item.fetch);
ASSERT_EQ(tensors.size(), 1);
test::ExpectTensorNear<float>(tensors[0], tensors_expected[0], 1e-5);
}
}
TEST_F(RemapperTest, FuseBatchNormWithRelu) {
using ::tensorflow::ops::Placeholder;
#if !defined(GOOGLE_CUDA) || !(CUDNN_VERSION >= 7402)
LOG(INFO) << "Skip FuseBatchNormWithRelu test. It requires "
"CUDNN_VERSION >= 7402.";
#else
tensorflow::Scope s = tensorflow::Scope::NewRootScope();
auto input_shape = ops::Placeholder::Shape({2, 8, 8, 24});
auto channels_shape = ops::Placeholder::Shape({24});
auto input = Placeholder(s.WithOpName("input"), DT_FLOAT, input_shape);
auto input_cast = ops::Cast(s.WithOpName("input_cast"), input, DT_HALF);
auto scale = Placeholder(s.WithOpName("scale"), DT_FLOAT, channels_shape);
auto offset = Placeholder(s.WithOpName("offset"), DT_FLOAT, channels_shape);
auto mean = Placeholder(s.WithOpName("mean"), DT_FLOAT, channels_shape);
auto var = Placeholder(s.WithOpName("var"), DT_FLOAT, channels_shape);
float epsilon = 0.1f;
auto fbn = ops::FusedBatchNormV3(
s.WithOpName("fused_batch_norm"), input_cast, scale, offset, mean, var,
ops::FusedBatchNormV3::IsTraining(true).Epsilon(epsilon).DataFormat(
"NHWC"));
auto relu = ops::Relu(s.WithOpName("relu"), fbn.y);
auto fetch = ops::Identity(s.WithOpName("fetch"), relu);
auto input_t = GenerateRandomTensor<DT_FLOAT>({2, 8, 8, 24});
auto scale_t = GenerateRandomTensor<DT_FLOAT>({24});
auto offset_t = GenerateRandomTensor<DT_FLOAT>({24});
auto mean_t = GenerateRandomTensor<DT_FLOAT>({0}); // empty for training
auto var_t = GenerateRandomTensor<DT_FLOAT>({0}); // empty for training
GrapplerItem item;
item.fetch = {"fetch"};
item.feed = {{"input", input_t},
{"scale", scale_t},
{"offset", offset_t},
{"mean", mean_t},
{"var", var_t}};
TF_ASSERT_OK(s.ToGraphDef(&item.graph));
// Place all nodes on GPU.
for (int i = 0; i < item.graph.node_size(); ++i) {
item.graph.mutable_node(i)->set_device("/device:GPU:0");
}
Remapper optimizer(RewriterConfig::AGGRESSIVE); // trust placeholders shape
GraphDef output;
TF_ASSERT_OK(optimizer.Optimize(nullptr, item, &output));
int found = 0;
for (const NodeDef& node : output.node()) {
if (node.name() == "relu") {
EXPECT_EQ(node.op(), "Identity");
ASSERT_EQ(node.input_size(), 1);
EXPECT_EQ(node.input(0), "fused_batch_norm");
found++;
}
if (node.name() == "fused_batch_norm") {
EXPECT_EQ(node.op(), "_FusedBatchNormEx");
ASSERT_EQ(node.input_size(), 5);
EXPECT_EQ(node.input(0), "input_cast");
EXPECT_EQ(node.input(1), "scale");
EXPECT_EQ(node.input(2), "offset");
EXPECT_EQ(node.input(3), "mean");
EXPECT_EQ(node.input(4), "var");
auto attr = node.attr();
EXPECT_EQ(attr["num_side_inputs"].i(), 0);
EXPECT_EQ(attr["activation_mode"].s(), "Relu");
found++;
}
}
EXPECT_EQ(found, 2);
if (GetNumAvailableGPUs() > 0) {
auto tensors_expected = EvaluateNodes(item.graph, item.fetch, item.feed);
ASSERT_EQ(tensors_expected.size(), 1);
auto tensors = EvaluateNodes(output, item.fetch, item.feed);
ASSERT_EQ(tensors.size(), 1);
test::ExpectClose(tensors[0], tensors_expected[0], 1e-2, /*rtol=*/1e-2);
}
#endif // !defined(GOOGLE_CUDA) || !(CUDNN_VERSION >= 7402)
}
TEST_F(RemapperTest, FuseBatchNormWithAddAndRelu) {
using ::tensorflow::ops::Placeholder;
#if !defined(GOOGLE_CUDA) || !(CUDNN_VERSION >= 7402)
LOG(INFO) << "Skip FuseBatchNormWithAddAndRelu test. It requires "
"CUDNN_VERSION >= 7402.";
#else
tensorflow::Scope s = tensorflow::Scope::NewRootScope();
auto input_shape = ops::Placeholder::Shape({2, 8, 8, 24});
auto channels_shape = ops::Placeholder::Shape({24});
auto input = Placeholder(s.WithOpName("input"), DT_FLOAT, input_shape);
auto input_cast = ops::Cast(s.WithOpName("input_cast"), input, DT_HALF);
auto scale = Placeholder(s.WithOpName("scale"), DT_FLOAT, channels_shape);
auto offset = Placeholder(s.WithOpName("offset"), DT_FLOAT, channels_shape);
auto mean = Placeholder(s.WithOpName("mean"), DT_FLOAT, channels_shape);
auto var = Placeholder(s.WithOpName("var"), DT_FLOAT, channels_shape);
auto side_input =
Placeholder(s.WithOpName("side_input"), DT_FLOAT, input_shape);
auto side_input_cast =
ops::Cast(s.WithOpName("side_input_cast"), side_input, DT_HALF);
float epsilon = 0.1f;
auto fbn = ops::FusedBatchNormV3(
s.WithOpName("fused_batch_norm"), input_cast, scale, offset, mean, var,
ops::FusedBatchNormV3::IsTraining(true).Epsilon(epsilon).DataFormat(
"NHWC"));
auto add = ops::Add(s.WithOpName("add"), fbn.y, side_input_cast);
auto relu = ops::Relu(s.WithOpName("relu"), add);
auto fetch = ops::Identity(s.WithOpName("fetch"), relu);
auto input_t = GenerateRandomTensor<DT_FLOAT>({2, 8, 8, 24});
auto scale_t = GenerateRandomTensor<DT_FLOAT>({24});
auto offset_t = GenerateRandomTensor<DT_FLOAT>({24});
auto mean_t = GenerateRandomTensor<DT_FLOAT>({0}); // empty for training
auto var_t = GenerateRandomTensor<DT_FLOAT>({0}); // empty for training
auto side_input_t = GenerateRandomTensor<DT_FLOAT>({2, 8, 8, 24});
GrapplerItem item;
item.fetch = {"fetch"};
item.feed = {{"input", input_t}, {"scale", scale_t},
{"offset", offset_t}, {"mean", mean_t},
{"var", var_t}, {"side_input", side_input_t}};
TF_ASSERT_OK(s.ToGraphDef(&item.graph));
// Place all nodes on GPU.
for (int i = 0; i < item.graph.node_size(); ++i) {
item.graph.mutable_node(i)->set_device("/device:GPU:0");
}
Remapper optimizer(RewriterConfig::AGGRESSIVE); // trust placeholders shape
GraphDef output;
TF_ASSERT_OK(optimizer.Optimize(nullptr, item, &output));
int found = 0;
for (const NodeDef& node : output.node()) {
if (node.name() == "relu") {
EXPECT_EQ(node.op(), "Identity");
ASSERT_EQ(node.input_size(), 1);
EXPECT_EQ(node.input(0), "fused_batch_norm");
found++;
}
if (node.name() == "fused_batch_norm") {
EXPECT_EQ(node.op(), "_FusedBatchNormEx");
ASSERT_EQ(node.input_size(), 6);
EXPECT_EQ(node.input(0), "input_cast");
EXPECT_EQ(node.input(1), "scale");
EXPECT_EQ(node.input(2), "offset");
EXPECT_EQ(node.input(3), "mean");
EXPECT_EQ(node.input(4), "var");
EXPECT_EQ(node.input(5), "side_input_cast");
auto attr = node.attr();
EXPECT_EQ(attr["num_side_inputs"].i(), 1);
EXPECT_EQ(attr["activation_mode"].s(), "Relu");
found++;
}
}
EXPECT_EQ(found, 2);
if (GetNumAvailableGPUs() > 0) {
auto tensors_expected = EvaluateNodes(item.graph, item.fetch, item.feed);
ASSERT_EQ(tensors_expected.size(), 1);
auto tensors = EvaluateNodes(output, item.fetch, item.feed);
ASSERT_EQ(tensors.size(), 1);
test::ExpectClose(tensors[0], tensors_expected[0], 1e-2, /*rtol=*/1e-2);
}
#endif // !defined(GOOGLE_CUDA) || !(CUDNN_VERSION >= 7402)
}
TEST_F(RemapperTest, FuseConv2DWithBias) {
if (!EigenSupportsContractionOutputKernel()) return;
using ::tensorflow::ops::Placeholder;
tensorflow::Scope s = tensorflow::Scope::NewRootScope();
auto input_shape = ops::Placeholder::Shape({8, 32, 32, 3});
auto filter_shape = ops::Placeholder::Shape({1, 1, 3, 128});
auto bias_shape = ops::Placeholder::Shape({128});
auto input = Placeholder(s.WithOpName("input"), DT_FLOAT, input_shape);
auto filter = Placeholder(s.WithOpName("filter"), DT_FLOAT, filter_shape);
auto bias = Placeholder(s.WithOpName("bias"), DT_FLOAT, bias_shape);
std::vector<int> strides = {1, 1, 1, 1};
auto conv = ops::Conv2D(s.WithOpName("conv"), input, filter, strides, "SAME");
auto bias_add = ops::BiasAdd(s.WithOpName("bias_add"), conv, bias);
auto fetch = ops::Identity(s.WithOpName("fetch"), bias_add);
auto input_t = GenerateRandomTensor<DT_FLOAT>({8, 32, 32, 3});
auto filter_t = GenerateRandomTensor<DT_FLOAT>({1, 1, 3, 128});
auto bias_t = GenerateRandomTensor<DT_FLOAT>({128});
GrapplerItem item;
item.fetch = {"fetch"};
item.feed = {{"input", input_t}, {"filter", filter_t}, {"bias", bias_t}};
TF_ASSERT_OK(s.ToGraphDef(&item.graph));
// Place all nodes on CPU.
for (int i = 0; i < item.graph.node_size(); ++i) {
item.graph.mutable_node(i)->set_device("/device:CPU:0");
}
Remapper optimizer(RewriterConfig::ON);
GraphDef output;
TF_ASSERT_OK(optimizer.Optimize(nullptr, item, &output));
int found = 0;
for (const NodeDef& node : output.node()) {
if (node.name() == "bias_add") {
EXPECT_EQ(node.op(), "_FusedConv2D");
ASSERT_GE(node.input_size(), 3);
EXPECT_EQ(node.input(0), "input");
EXPECT_EQ(node.input(1), "filter");
EXPECT_EQ(node.attr().at("num_args").i(), 1);
EXPECT_EQ(node.input(2), "bias");
const auto fused_ops = node.attr().at("fused_ops").list().s();
ASSERT_EQ(fused_ops.size(), 1);
EXPECT_EQ(fused_ops[0], "BiasAdd");
found++;
}
}
EXPECT_EQ(found, 1);
auto tensors_expected = EvaluateNodes(item.graph, item.fetch, item.feed);
ASSERT_EQ(tensors_expected.size(), 1);
auto tensors = EvaluateNodes(output, item.fetch, item.feed);
ASSERT_EQ(tensors.size(), 1);
test::ExpectTensorNear<float>(tensors[0], tensors_expected[0], 1e-6);
}
TEST_F(RemapperTest, FuseMatMulWithBias) {
if (!EigenSupportsContractionOutputKernel()) return;
using ::tensorflow::ops::Placeholder;
tensorflow::Scope s = tensorflow::Scope::NewRootScope();
auto lhs_shape = ops::Placeholder::Shape({8, 32});
auto rhs_shape = ops::Placeholder::Shape({32, 64});
auto bias_shape = ops::Placeholder::Shape({64});
auto lhs = Placeholder(s.WithOpName("lhs"), DT_FLOAT, lhs_shape);
auto rhs = Placeholder(s.WithOpName("rhs"), DT_FLOAT, rhs_shape);
auto bias = Placeholder(s.WithOpName("bias"), DT_FLOAT, bias_shape);
auto matmul = ops::MatMul(s.WithOpName("matmul"), lhs, rhs);
auto bias_add = ops::BiasAdd(s.WithOpName("bias_add"), matmul, bias);
auto fetch = ops::Identity(s.WithOpName("fetch"), bias_add);
auto lhs_t = GenerateRandomTensor<DT_FLOAT>({8, 32});
auto rhs_t = GenerateRandomTensor<DT_FLOAT>({32, 64});
auto bias_t = GenerateRandomTensor<DT_FLOAT>({64});
GrapplerItem item;
item.fetch = {"fetch"};
item.feed = {{"lhs", lhs_t}, {"rhs", rhs_t}, {"bias", bias_t}};
TF_ASSERT_OK(s.ToGraphDef(&item.graph));
// Place all nodes on CPU.
for (int i = 0; i < item.graph.node_size(); ++i) {
item.graph.mutable_node(i)->set_device("/device:CPU:0");
}
Remapper optimizer(RewriterConfig::ON);
GraphDef output;
TF_ASSERT_OK(optimizer.Optimize(nullptr, item, &output));
int found = 0;
for (const NodeDef& node : output.node()) {
if (node.name() == "bias_add") {
EXPECT_EQ(node.op(), "_FusedMatMul");
ASSERT_GE(node.input_size(), 3);
EXPECT_EQ(node.input(0), "lhs");
EXPECT_EQ(node.input(1), "rhs");
EXPECT_EQ(node.attr().at("num_args").i(), 1);
EXPECT_EQ(node.input(2), "bias");
const auto fused_ops = node.attr().at("fused_ops").list().s();
ASSERT_EQ(fused_ops.size(), 1);
EXPECT_EQ(fused_ops[0], "BiasAdd");
found++;
}
}
EXPECT_EQ(1, found);
auto tensors_expected = EvaluateNodes(item.graph, item.fetch, item.feed);
ASSERT_EQ(tensors_expected.size(), 1);
auto tensors = EvaluateNodes(output, item.fetch, item.feed);
ASSERT_EQ(tensors.size(), 1);
test::ExpectTensorNear<float>(tensors[0], tensors_expected[0], 1e-6);
}
TEST_F(RemapperTest, FuseConv2DWithBiasAndActivation) {
if (!EigenSupportsContractionOutputKernel()) return;
using ::tensorflow::ops::Placeholder;
for (const string& activation : {"Relu", "Relu6", "Elu"}) {
tensorflow::Scope s = tensorflow::Scope::NewRootScope();
auto input_shape = Placeholder::Shape({8, 32, 32, 3});
auto filter_shape = Placeholder::Shape({1, 1, 3, 128});
auto bias_shape = Placeholder::Shape({128});
auto input = Placeholder(s.WithOpName("input"), DT_FLOAT, input_shape);
auto filter = Placeholder(s.WithOpName("filter"), DT_FLOAT, filter_shape);
auto bias = Placeholder(s.WithOpName("bias"), DT_FLOAT, bias_shape);
std::vector<int> strides = {1, 1, 1, 1};
auto conv =
ops::Conv2D(s.WithOpName("conv"), input, filter, strides, "SAME");
auto bias_add = ops::BiasAdd(s.WithOpName("bias_add"), conv, bias);
ops::Identity fetch = [&]() -> ops::Identity {
auto activate = s.WithOpName("activation");
auto fetch = s.WithOpName("fetch");
if (activation == "Relu") {
return ops::Identity(fetch, ops::Relu(activate, bias_add));
} else if (activation == "Relu6") {
return ops::Identity(fetch, ops::Relu6(activate, bias_add));
} else if (activation == "Elu") {
return ops::Identity(fetch, ops::Elu(activate, bias_add));
}
return ops::Identity(fetch, bias);
}();
auto input_t = GenerateRandomTensor<DT_FLOAT>({8, 32, 32, 3});
auto filter_t = GenerateRandomTensor<DT_FLOAT>({1, 1, 3, 128});
auto bias_t = GenerateRandomTensor<DT_FLOAT>({128});
GrapplerItem item;
item.fetch = {"fetch"};
item.feed = {{"input", input_t}, {"filter", filter_t}, {"bias", bias_t}};
TF_ASSERT_OK(s.ToGraphDef(&item.graph));
// Place all nodes on CPU.
for (int i = 0; i < item.graph.node_size(); ++i) {
item.graph.mutable_node(i)->set_device("/device:CPU:0");
}
Remapper optimizer(RewriterConfig::ON);
GraphDef output;
TF_ASSERT_OK(optimizer.Optimize(nullptr, item, &output));
int found = 0;
for (const NodeDef& node : output.node()) {
if (node.name() == "activation") {
EXPECT_EQ(node.op(), "_FusedConv2D");
ASSERT_GE(node.input_size(), 3);
EXPECT_EQ(node.input(0), "input");
EXPECT_EQ(node.input(1), "filter");
EXPECT_EQ(node.attr().at("num_args").i(), 1);
EXPECT_EQ(node.input(2), "bias");
const auto fused_ops = node.attr().at("fused_ops").list().s();
ASSERT_EQ(fused_ops.size(), 2);
EXPECT_EQ(fused_ops[0], "BiasAdd");
EXPECT_EQ(fused_ops[1], activation);
found++;
}
}
EXPECT_EQ(found, 1);
auto tensors_expected = EvaluateNodes(item.graph, item.fetch, item.feed);
ASSERT_EQ(tensors_expected.size(), 1);
auto tensors = EvaluateNodes(output, item.fetch, item.feed);
ASSERT_EQ(tensors.size(), 1);
test::ExpectTensorNear<float>(tensors[0], tensors_expected[0], 1e-6);
}
}
TEST_F(RemapperTest, FuseMatMulWithBiasAndActivation) {
if (!EigenSupportsContractionOutputKernel()) return;
using ::tensorflow::ops::Placeholder;
for (const string& activation : {"Relu", "Relu6", "Elu"}) {
tensorflow::Scope s = tensorflow::Scope::NewRootScope();
auto lhs_shape = ops::Placeholder::Shape({8, 32});
auto rhs_shape = ops::Placeholder::Shape({32, 64});
auto bias_shape = ops::Placeholder::Shape({64});
auto lhs = Placeholder(s.WithOpName("lhs"), DT_FLOAT, lhs_shape);
auto rhs = Placeholder(s.WithOpName("rhs"), DT_FLOAT, rhs_shape);
auto bias = Placeholder(s.WithOpName("bias"), DT_FLOAT, bias_shape);
auto matmul = ops::MatMul(s.WithOpName("matmul"), lhs, rhs);
auto bias_add = ops::BiasAdd(s.WithOpName("bias_add"), matmul, bias);
ops::Identity fetch = [&]() -> ops::Identity {
auto activate = s.WithOpName("activation");
auto fetch = s.WithOpName("fetch");
if (activation == "Relu") {
return ops::Identity(fetch, ops::Relu(activate, bias_add));
} else if (activation == "Relu6") {
return ops::Identity(fetch, ops::Relu6(activate, bias_add));
} else if (activation == "Elu") {
return ops::Identity(fetch, ops::Elu(activate, bias_add));
}
return ops::Identity(fetch, bias);
}();
auto lhs_t = GenerateRandomTensor<DT_FLOAT>({8, 32});
auto rhs_t = GenerateRandomTensor<DT_FLOAT>({32, 64});
auto bias_t = GenerateRandomTensor<DT_FLOAT>({64});
GrapplerItem item;
item.fetch = {"fetch"};
item.feed = {{"lhs", lhs_t}, {"rhs", rhs_t}, {"bias", bias_t}};
TF_ASSERT_OK(s.ToGraphDef(&item.graph));
// Place all nodes on CPU.
for (int i = 0; i < item.graph.node_size(); ++i) {
item.graph.mutable_node(i)->set_device("/device:CPU:0");
}
Remapper optimizer(RewriterConfig::ON);
GraphDef output;
TF_ASSERT_OK(optimizer.Optimize(nullptr, item, &output));
int found = 0;
for (const NodeDef& node : output.node()) {
if (node.name() == "activation") {
EXPECT_EQ(node.op(), "_FusedMatMul");
ASSERT_GE(node.input_size(), 3);
EXPECT_EQ(node.input(0), "lhs");
EXPECT_EQ(node.input(1), "rhs");
EXPECT_EQ(node.attr().at("num_args").i(), 1);
EXPECT_EQ(node.input(2), "bias");
const auto fused_ops = node.attr().at("fused_ops").list().s();
ASSERT_EQ(fused_ops.size(), 2);
EXPECT_EQ(fused_ops[0], "BiasAdd");
EXPECT_EQ(fused_ops[1], activation);
found++;
}
}
EXPECT_EQ(1, found);
auto tensors_expected = EvaluateNodes(item.graph, item.fetch, item.feed);
ASSERT_EQ(tensors_expected.size(), 1);
auto tensors = EvaluateNodes(output, item.fetch, item.feed);
ASSERT_EQ(tensors.size(), 1);
test::ExpectTensorNear<float>(tensors[0], tensors_expected[0], 1e-6);
}
}
TEST_F(RemapperTest, FuseConv2DWithBatchNorm) {
if (!EigenSupportsContractionOutputKernel()) return;
using ops::Placeholder;
tensorflow::Scope s = tensorflow::Scope::NewRootScope();
auto input_shape = ops::Placeholder::Shape({8, 32, 32, 3});
auto filter_shape = ops::Placeholder::Shape({1, 1, 3, 128});
auto scale_shape = ops::Placeholder::Shape({128});
auto input = Placeholder(s.WithOpName("input"), DT_FLOAT, input_shape);
auto filter = Placeholder(s.WithOpName("filter"), DT_FLOAT, filter_shape);
auto scale = Placeholder(s.WithOpName("scale"), DT_FLOAT, scale_shape);
auto offset = Placeholder(s.WithOpName("offset"), DT_FLOAT, scale_shape);
auto mean = Placeholder(s.WithOpName("mean"), DT_FLOAT, scale_shape);
auto variance = Placeholder(s.WithOpName("variance"), DT_FLOAT, scale_shape);
std::vector<int> strides = {1, 1, 1, 1};
auto conv = ops::Conv2D(
s.WithOpName("conv"), input, filter, strides, "EXPLICIT",
ops::Conv2D::Attrs().ExplicitPaddings({0, 0, 1, 2, 3, 4, 0, 0}));
ops::FusedBatchNorm::Attrs attrs;
attrs = attrs.IsTraining(false);
auto batch_norm = ops::FusedBatchNorm(s.WithOpName("batch_norm"), conv, scale,
offset, mean, variance, attrs);
auto fetch = ops::Identity(s.WithOpName("fetch"), batch_norm.y);
auto input_t = GenerateRandomTensor<DT_FLOAT>({8, 32, 32, 3});
auto filter_t = GenerateRandomTensor<DT_FLOAT>({1, 1, 3, 128});
auto scale_t = GenerateRandomTensor<DT_FLOAT>({128});
auto offset_t = GenerateRandomTensor<DT_FLOAT>({128});
auto mean_t = GenerateRandomTensor<DT_FLOAT>({128});
auto variance_t = GenerateRandomTensor<DT_FLOAT>({128});
GrapplerItem item;
item.fetch = {"fetch"};
item.feed = {{"input", input_t}, {"filter", filter_t},
{"scale", scale_t}, {"offset", offset_t},
{"mean", mean_t}, {"variance", variance_t}};
TF_ASSERT_OK(s.ToGraphDef(&item.graph));
// Place all nodes on CPU.
for (int i = 0; i < item.graph.node_size(); ++i) {
item.graph.mutable_node(i)->set_device("/device:CPU:0");
}
Remapper optimizer(RewriterConfig::ON);
GraphDef output;
TF_ASSERT_OK(optimizer.Optimize(nullptr, item, &output));
int found = 0;
for (const NodeDef& node : output.node()) {
if (node.name() == "batch_norm") {
EXPECT_EQ(node.op(), "_FusedConv2D");
ASSERT_GE(node.input_size(), 6);
EXPECT_EQ(node.input(0), "input");
EXPECT_EQ(node.input(1), "filter");
EXPECT_EQ(node.attr().at("num_args").i(), 4);
EXPECT_EQ(node.input(2), "scale");
EXPECT_EQ(node.input(3), "offset");
EXPECT_EQ(node.input(4), "mean");
EXPECT_EQ(node.input(5), "variance");
const auto fused_ops = node.attr().at("fused_ops").list().s();
ASSERT_EQ(fused_ops.size(), 1);
EXPECT_EQ(fused_ops[0], "FusedBatchNorm");
found++;
}
}
EXPECT_EQ(found, 1);
auto tensors_expected = EvaluateNodes(item.graph, item.fetch, item.feed);
ASSERT_EQ(tensors_expected.size(), 1);
auto tensors = EvaluateNodes(output, item.fetch, item.feed);
ASSERT_EQ(tensors.size(), 1);
test::ExpectTensorNear<float>(tensors[0], tensors_expected[0], 1e-6);
}
TEST_F(RemapperTest, FuseConv2DWithBatchNormAndActivation) {
if (!EigenSupportsContractionOutputKernel()) return;
using ops::Placeholder;
for (const string& activation : {"Relu", "Relu6", "Elu"}) {
tensorflow::Scope s = tensorflow::Scope::NewRootScope();
auto input_shape = ops::Placeholder::Shape({8, 32, 32, 3});
auto filter_shape = ops::Placeholder::Shape({1, 1, 3, 128});
auto scale_shape = ops::Placeholder::Shape({128});
auto input = Placeholder(s.WithOpName("input"), DT_FLOAT, input_shape);
auto filter = Placeholder(s.WithOpName("filter"), DT_FLOAT, filter_shape);
auto scale = Placeholder(s.WithOpName("scale"), DT_FLOAT, scale_shape);
auto offset = Placeholder(s.WithOpName("offset"), DT_FLOAT, scale_shape);
auto mean = Placeholder(s.WithOpName("mean"), DT_FLOAT, scale_shape);
auto variance =
Placeholder(s.WithOpName("variance"), DT_FLOAT, scale_shape);
std::vector<int> strides = {1, 1, 1, 1};
auto conv =
ops::Conv2D(s.WithOpName("conv"), input, filter, strides, "SAME");
ops::FusedBatchNorm::Attrs attrs;
attrs = attrs.IsTraining(false);
auto batch_norm = ops::FusedBatchNorm(s.WithOpName("batch_norm"), conv,
scale, offset, mean, variance, attrs);
ops::Identity fetch = [&]() -> ops::Identity {
auto activate = s.WithOpName("activation");
auto fetch = s.WithOpName("fetch");
if (activation == "Relu") {
return ops::Identity(fetch, ops::Relu(activate, batch_norm.y));
} else if (activation == "Relu6") {
return ops::Identity(fetch, ops::Relu6(activate, batch_norm.y));
} else if (activation == "Elu") {
return ops::Identity(fetch, ops::Elu(activate, batch_norm.y));
}
return ops::Identity(fetch, batch_norm.y);
}();
auto input_t = GenerateRandomTensor<DT_FLOAT>({8, 32, 32, 3});
auto filter_t = GenerateRandomTensor<DT_FLOAT>({1, 1, 3, 128});
auto scale_t = GenerateRandomTensor<DT_FLOAT>({128});
auto offset_t = GenerateRandomTensor<DT_FLOAT>({128});
auto mean_t = GenerateRandomTensor<DT_FLOAT>({128});
auto variance_t = GenerateRandomTensor<DT_FLOAT>({128});
GrapplerItem item;
item.fetch = {"fetch"};
item.feed = {{"input", input_t}, {"filter", filter_t},
{"scale", scale_t}, {"offset", offset_t},
{"mean", mean_t}, {"variance", variance_t}};
TF_ASSERT_OK(s.ToGraphDef(&item.graph));
// Place all nodes on CPU.
for (int i = 0; i < item.graph.node_size(); ++i) {
item.graph.mutable_node(i)->set_device("/device:CPU:0");
}
Remapper optimizer(RewriterConfig::ON);
GraphDef output;
TF_ASSERT_OK(optimizer.Optimize(nullptr, item, &output));
int found = 0;
for (const NodeDef& node : output.node()) {
if (node.name() == "activation") {
EXPECT_EQ(node.op(), "_FusedConv2D");
ASSERT_GE(node.input_size(), 6);
EXPECT_EQ(node.input(0), "input");
EXPECT_EQ(node.input(1), "filter");
EXPECT_EQ(node.attr().at("num_args").i(), 4);
EXPECT_EQ(node.input(2), "scale");
EXPECT_EQ(node.input(3), "offset");
EXPECT_EQ(node.input(4), "mean");
EXPECT_EQ(node.input(5), "variance");
const auto fused_ops = node.attr().at("fused_ops").list().s();
ASSERT_EQ(fused_ops.size(), 2);
EXPECT_EQ(fused_ops[0], "FusedBatchNorm");
EXPECT_EQ(fused_ops[1], activation);
found++;
}
}
EXPECT_EQ(found, 1);
auto tensors_expected = EvaluateNodes(item.graph, item.fetch, item.feed);
ASSERT_EQ(tensors_expected.size(), 1);
auto tensors = EvaluateNodes(output, item.fetch, item.feed);
ASSERT_EQ(tensors.size(), 1);
test::ExpectTensorNear<float>(tensors[0], tensors_expected[0], 1e-6);
}
}
TEST_F(RemapperTest, FuseConv2DWithSqueezeAndBias) {
if (!EigenSupportsContractionOutputKernel()) return;
using ops::Placeholder;
tensorflow::Scope s = tensorflow::Scope::NewRootScope();
auto input_shape = ops::Placeholder::Shape({8, 32, 1, 3});
auto filter_shape = ops::Placeholder::Shape({1, 1, 3, 128});
auto bias_shape = ops::Placeholder::Shape({128});
auto input = Placeholder(s.WithOpName("input"), DT_FLOAT, input_shape);
auto filter = Placeholder(s.WithOpName("filter"), DT_FLOAT, filter_shape);
auto bias = Placeholder(s.WithOpName("bias"), DT_FLOAT, bias_shape);
std::vector<int> strides = {1, 1, 1, 1};
auto conv = ops::Conv2D(s.WithOpName("conv"), input, filter, strides, "SAME");
ops::Squeeze::Attrs attrs;
attrs = attrs.Axis({2});
auto squeeze = ops::Squeeze(s.WithOpName("squeeze"), conv, attrs);
auto bias_add = ops::BiasAdd(s.WithOpName("bias_add"), squeeze, bias);
auto fetch = ops::Identity(s.WithOpName("fetch"), bias_add);
auto input_t = GenerateRandomTensor<DT_FLOAT>({8, 32, 1, 3});
auto filter_t = GenerateRandomTensor<DT_FLOAT>({1, 1, 3, 128});
auto bias_t = GenerateRandomTensor<DT_FLOAT>({128});
GrapplerItem item;
item.fetch = {"fetch"};
item.feed = {{"input", input_t}, {"filter", filter_t}, {"bias", bias_t}};
TF_ASSERT_OK(s.ToGraphDef(&item.graph));
// Place all nodes on CPU.
for (int i = 0; i < item.graph.node_size(); ++i) {
item.graph.mutable_node(i)->set_device("/device:CPU:0");
}
Remapper optimizer(RewriterConfig::ON);
GraphDef output;
TF_ASSERT_OK(optimizer.Optimize(nullptr, item, &output));
int found = 0;
for (const NodeDef& node : output.node()) {
if (node.name() == "conv") {
EXPECT_EQ(node.op(), "_FusedConv2D");
ASSERT_GE(node.input_size(), 3);
EXPECT_EQ(node.input(0), "input");
EXPECT_EQ(node.input(1), "filter");
EXPECT_EQ(node.attr().at("num_args").i(), 1);
EXPECT_EQ(node.input(2), "bias");
const auto fused_ops = node.attr().at("fused_ops").list().s();
ASSERT_EQ(fused_ops.size(), 1);
EXPECT_EQ(fused_ops[0], "BiasAdd");
found++;
} else if (node.name() == "bias_add") {
EXPECT_EQ(node.op(), "Squeeze");
ASSERT_GE(node.input_size(), 1);
EXPECT_EQ(node.input(0), "conv");
found++;
}
}
EXPECT_EQ(found, 2);
auto tensors_expected = EvaluateNodes(item.graph, item.fetch, item.feed);
ASSERT_EQ(tensors_expected.size(), 1);
auto tensors = EvaluateNodes(output, item.fetch, item.feed);
ASSERT_EQ(tensors.size(), 1);
test::ExpectTensorNear<float>(tensors[0], tensors_expected[0], 1e-6);
}
} // namespace grappler
} // namespace tensorflow
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,030
|
/* Licensed under the Apache License, Version 2.0 (the "License");
* you may not use this file except in compliance with the License.
* You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
package org.camunda.bpm.webapp.impl.security.auth;
import java.util.List;
import java.util.Set;
import org.camunda.bpm.engine.authorization.Authorization;
/**
* <p>An authentication for a user</p>
*
* @author Daniel Meyer
* @author nico.rehwaldt
*/
public class UserAuthentication extends Authentication {
private static final long serialVersionUID = 1L;
protected List<String> groupIds;
private final Set<String> authorizedApps;
/**
* @param userId the id of the user
* @param groupIds
* @param processEngineName the name of the process engine
* @param authorizedApps
*/
public UserAuthentication(String userId, List<String> groupIds, String processEngineName, Set<String> authorizedApps) {
super(userId, processEngineName);
this.groupIds = groupIds;
this.authorizedApps = authorizedApps;
}
public List<String> getGroupIds() {
return groupIds;
}
public boolean isAuthorizedForApp(String app) {
return authorizedApps.contains(Authorization.ANY) || authorizedApps.contains(app);
}
public Set<String> getAuthorizedApps() {
return authorizedApps;
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,429
|
Q: Appcelerator label with whitespace I made a tableview with some labels, the labels all have a border so it looks like a nice table.
However, if the label text is aligned to the right, the text is directly placed against the border. I've tried adding a space after the text, but it seems like Android just trims these spaces.
Is there any way I can create some whitespace between the text and the border? (For example with some character?)
A: When creating a label you can just add the right property to the label.
Ti.UI.createLabel({
right: 5
});
or
<Label right="5" />
You should also not set the border on the label, but wrap the label in a view and set the borders to the view instead!
A: Place a view around the Label and put the border on that view
<View borderWidth="1" borderColor="#fff" width="Ti.UI.FILL" height="Ti.UI.SIZE">
<Label text="test" width="Ti.UI.SIZE" height="Ti.UI.SIZE" right="5"/>
</View>
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,231
|
The way we use the web is changing. The rise of social networks and apps have created closed silos where people are spending increasing amounts of time, at the expense of the open web.
There are a couple of ways for you to be reading this article: you could have entered MakeUseOf.com into your browser and come here directly, you could have searched for something online and followed a link to this page. What's increasingly likely, though, is that you could have followed a link on a social media site or app.
How Popular Are Social Networks?
Facebook has more than 1.4 billion monthly users, According to Digital Marketing Ramblings — that means that 72% of adults who use the Internet visit the site at least once a month. 936 million of them, or 65% of Internet using adults, use Facebook daily. It's the second most visited site globally, just behind Google. These are serious numbers.
Facebook's mobile stats are also impressive. More than 1.2 billion people use the app monthly. Two thirds of them use the app daily. 581 million, or 30% of Facebook users, only login from a mobile device.
Other social networks don't come close to Facebook's numbers. Even Twitter — which we love here at MakeUseOf 7 Reasons Why You Should Be Using Twitter 7 Reasons Why You Should Be Using Twitter Twitter has now been with us for seven years and counting. This was seven years to the day since Jack Dorsey sent out the first tweet in 2006, at a time when the micro-blogging social... Read More — only has around 300 million users.
Even more interesting than the raw user numbers, are how long people are spending on social media everyday. A report by eMarketer found that the average Facebook user spent 42 minutes on the service everyday. The average Twitter user clocks in at just under half that with 17 minutes. I can't imagine people are spending anywhere near that length of time using Google.
Whatever way you look at it, huge numbers of people are using social media for a significant amount of time every day.
Since the early days of the open web, search engines, in particular Google, have been the major driver of traffic for most sites. Whole industries have grown up around optimising websites for their algorithms Demystify SEO: 5 Search Engine Optimization Guides That Help You Begin Demystify SEO: 5 Search Engine Optimization Guides That Help You Begin Search engine mastery takes knowledge, experience, and lots of trial and error. You can begin learning the fundamentals and avoid common SEO mistakes easily with the help of many SEO guides available on the Web. Read More . In the past year this changed.
According to data from Shareaholic, the percentage of website visits from social networks has risen from around 11% in 2011 to just over 30%. In the same time period, traffic from search has fallen from well over 40% to just under 30%.
The biggest shift has been with Facebook. In 2011, Facebook was responsible for 6.53% of all website referrals. Last year, Facebook drove 24.63% of them — just under a quarter of all website visits.
The open web just isn't as important as it once was.
Google, for the most part, helps you find out what's on other sites. Facebook isn't content to do that: they're looking to capitalise on the increasing amount of time people spend using their service by working with publishers to embed content directly into the newsfeed. More time on Facebook means more ad revenue.
At MakeUseOf our opinions are divided as to whether this is a good thing How Facebook Is Changing News Journalism for the Better How Facebook Is Changing News Journalism for the Better News outlets are considering publishing stories straight to Facebook. It's easy to be cynical. But this change could be a good thing – for readers and journalists. Read More or a bad thing Facebook Wants to Be the News Site of the Future, and That's Awful Facebook Wants to Be the News Site of the Future, and That's Awful Getting publishers on board could potentially bring Facebook's nightmarish vision to a whole new level. It's a horrible idea. Horrible for the Internet, and even worse for journalism. Here's why. Read More . Whatever your opinion, it is clearly a nail in the coffin of the open web — people will spend more time on Facebook and have less reason to visit other sites. Any that aren't Facebook partners will see a drop in readers.
While Google and the open web have been what most people think of as the Internet for the past decade, this is clearly shifting. Not only are social networks monopolising peoples' time, but publishers are following and going where the users are.
Facebook isn't the only social media network working with publishers to reach people away from the open web — Snapchat launched a new program Snapchat's Discover: Why It's a Social News Revolution Snapchat's Discover: Why It's a Social News Revolution Snapchat is more than just sending pictures and videos to your friends. Snapchat has grown into a powerful tool, bringing the world of news, events and trends to millions of users in a snap. Read More with 11 publishers including National Geographic, Vice, and the Daily Mail earlier this year. According to re/code the program is proving extremely successful: each Discover story is viewed between 500,000 and a million times a day. Publishers are inserting ads in the feed and are apparently getting around 10 cents per impression, or up to $100,000 a day in revenue.
While all Snapchat's partners are existing publishers, we are already starting to see publications that ignore the open web and instead work within the social networks.
The Shade Room is a TMZ-esque gossip site that started out publishing exclusively to Instagram and later Facebook. Instead of using a more traditional content management system like WordPress, the Shade Room uses Instagram to publish directly to their followers social media feeds. They even sell ad spaces for several hundred dollars.
As social media sites increasingly make it possible for publishers to exist within their closed silos, we're going to see more examples like the Shade Room. Why direct people to an external site when you can reach them directly where they're spending most of their time?
The trends are pretty clear. The web as we know it is dying. Google's relevance is falling. It just isn't driving the traffic it once did and bloggers are starting to notice. It looks like, over the next few years, the importance of social networks will continue to rise and publishers will follow users to the services.
Facebook is changing their algorithm, but to what end? Does @finkd want to own the Internet?
Writing in the Awl, John Herman argues that the Internet will start to "closely resemble the TV industry" in the near future. He argues that websites will cease to be relevant and instead publishers will need to work directly with a limited number of social networks to stay relevant. Different networks will partner with different publishers and the TV paradigm of channels will be recreated all over again.
If Ben Thompson and John Herman are right, it looks like there is very little we can do to stop the shift. Publishers already struggle to make money It's About Ethics in Stealing Games Journalism: Why AdBlock Needs to Die It's About Ethics in Stealing Games Journalism: Why AdBlock Needs to Die A simple, free browser plugin killed Joystiq – and is ruining the Internet. Read More . If social networks continue to pull attention, and content creators, away from the open web there is not much that can be done to slow the process. Facebook and Snapchat reaching out to publishers and offering them an olive branch is only likely to accelerate the changes.
What seems is inevitable is that the open web as we know it is done for. Users are being consolidated into larger social networks and spending more time there. Even if Facebook and Snapchat's attempts to get publishers using their platforms directly fails, social referrals are likely to continue to increase in importance. The web of the next few years is going to be increasingly social and mobile.
Whether this is a good thing or a bad thing remains to be seen. What do you think of this potential Facebook run future?
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 1,488
|
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