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} | {
"word": [
"NFTrig:",
"Using",
"Blockchain",
"Technologies",
"for",
"Math",
"Education",
"JORDAN",
"THOMPSON,",
"Augustana",
"College,",
"USA",
"RYAN",
"BENAC,",
"Augustana",
"College,",
"USA",
"2020",
"KIDUS",
"OLANA,",
"Augustana",
"College,",
"USA",
"TALHA",
"HASSAN,",
"Augustana",
"College,",
"USA",
"ANDREW",
"SWARD,",
"Augustana",
"College,",
"USA",
"TAUHEED",
"KHAN",
"MOHD,",
"Augustana",
"College,",
"USA",
"8",
"NFTrig",
"is",
"a",
"web-based",
"application",
"created",
"for",
"use",
"as",
"an",
"educational",
"tool",
"to",
"teach",
"trigonometry",
"and",
"block",
"chain",
"technology.",
"Creation",
"of",
"the",
"application",
"includes",
"front",
"and",
"back",
"end",
"development",
"as",
"well",
"as",
"integration",
"1",
"with",
"other",
"outside",
"sources",
"including",
"MetaMask",
"and",
"OpenSea.",
"The",
"primary",
"development",
"languages",
"include",
"HTML,",
"CSS",
"(Bootstrap",
"5),",
"and",
"JavaScript",
"as",
"well",
"as",
"Solidity",
"for",
"smart",
"contract",
"creation.",
"The",
"application",
"itself",
"2",
"is",
"hosted",
"on",
"Moralis",
"utilizing",
"their",
"Web3",
"API.",
"This",
"technical",
"report",
"describes",
"how",
"the",
"application",
"was",
"created,",
"what",
"the",
"application",
"requires,",
"and",
"smart",
"contract",
"design",
"with",
"security",
"considerations",
"in",
"mind.",
"The",
"NFTrig",
"CL",
"application",
"has",
"underwent",
"significan",
"testin",
"and",
"validatio",
"prior",
"to",
"and",
"after",
"deployment",
"Futur",
"suggestion",
"and",
"recommendations",
"for",
"further",
"development,",
"maintenance,",
"and",
"use",
"in",
"other",
"field",
"for",
"educatio",
"are",
"also",
"described.",
"8",
"CCS",
"Concepts:",
".",
"Computer",
"systems",
"organization",
"→",
"Redundancy;",
"Robotics;",
"Networks",
"→",
"Network",
"reliability.",
"1",
"Additional",
"Key",
"Words",
"and",
"Phrases:",
"Matic,",
"Metamask,",
"polygon,",
"bootstrap5,",
"Solidity",
"v",
"200",
"1",
"INTRODUCTION",
"The",
"purpose",
"of",
"this",
"report",
"is",
"to",
"describe",
"the",
"technical",
"details",
"involved",
"in",
"the",
"development",
"of",
"the",
"NFTrig",
"application.",
"This",
"includes",
"both",
"the",
"front",
"end",
"website",
"design,",
"the",
"back",
"end",
"smart",
"contract,",
"and",
"NFT",
"creation.",
"It",
"will",
"mainly",
"focus",
"on",
"the",
"technical",
"details",
"of",
"the",
"project",
"outlining",
"software",
"requirements,",
"design",
"through",
"programming",
"languages,",
"client",
"and",
"server",
"side",
"interactions,",
"and",
"880",
"validation",
"testing.",
"This",
"allows",
"the",
"reader",
"to",
"undertake",
"further",
"development,",
"fixes",
"or",
"maintenanc",
"of",
"the",
"software,",
"as",
"this",
"forms",
"part",
"of",
"the",
"documentation",
"for",
"the",
"software.",
"2",
"The",
"NFTrig",
"project",
"is",
"based",
"around",
"the",
"creation",
"of",
"a",
"web-based",
"game",
"application",
"that",
"allows",
"Y",
"interaction",
"of",
"NFTs",
"(non-fungible",
"token)",
"with",
"trigonometric",
"function",
"designs.",
"NFts",
"are",
"digital",
"X",
"assets,",
"for",
"example",
"a",
"picture,",
"that",
"has",
"a",
"unique",
"identificatio",
"and",
"can",
"generall",
"be",
"freel",
"trade",
"with",
"cryptocurrency",
"[33].",
"Through",
"this",
"application,",
"users",
"are",
"able",
"to",
"purchase",
"digital",
"artwork",
"of",
"2",
"many",
"differen",
"trigonometri",
"function",
"and",
"combin",
"the",
"using",
"mathematica",
"operations",
"Curren",
"supported",
"operations",
"include",
"multiplication",
"and",
"division",
"of",
"the",
"trigonometry",
"functions,",
"and",
"the",
"output",
"of",
"each",
"operation",
"is",
"a",
"new",
"NFT",
"card",
"that",
"would",
"be",
"the",
"result",
"of",
"an",
"operation.",
"The",
"old",
"cards",
"will",
"then",
"be",
"removed",
"from",
"the",
"user’s",
"possession",
"and",
"burned",
"using",
"the",
"smart",
"contact.",
"For",
"example,",
"if",
"a",
"user",
"combined",
"the",
"two",
"cards",
"Sin(x)",
"and",
"Cos(x)",
"using",
"multiplication,",
"they",
"would",
"lose",
"their",
"two",
"old",
"cards",
"and",
"receive",
"the",
"new",
"card",
"Tan(x).",
"Further,",
"the",
"NFT",
"cards",
"are",
"assigned",
"one",
"of",
"the",
"following",
"rarity",
"levels:",
"common,",
"uncommon,",
"rare,",
"and",
"legendary.",
"The",
"probability",
"of",
"each",
"of",
"these",
"levels",
"is",
"define",
"late",
"in",
"this",
"report.",
"The",
"application",
"also",
"allows",
"a",
"user",
"to",
"connect",
"to",
"MetaMask,",
"a",
"digital",
"wallet",
"capable",
"of",
"storing",
"a",
"user’s",
"cryptocurrency",
"and",
"NFTs",
"as",
"well",
"as",
"a",
"way",
"to",
"connect",
"to",
"block",
"chain.",
"The",
"NFTrig",
"application",
"Authors’",
"addresses:",
"Jordan",
"Thompson,",
"jordanthompson18@augustana.edu,",
"Augustana",
"College,",
"Rock",
"Island,",
"USA;",
"Ryan",
"Benac,",
"ryanbenac18@augustana.edu,",
"Augustana",
"College,",
"Rock",
"Island,",
"USA;",
"Kidus",
"Olana,",
"kidusolana18@augustana.edu,",
"Augustana",
"College,",
"Rock",
"Island,",
"USA;",
"Talha",
"Hassan,",
"talhahassan18@augustana.edu,",
"Augustana",
"College,",
"Rock",
"Island,",
"USA;",
"Andrew",
"Sward,",
"andrewsward@augustana.edu,",
"Augustana",
"College,",
"Rock",
"Island,",
"USA;",
"Tauheed",
"Khan",
"Mohd,",
"tauheedkhanmohd@augustana.edu,",
"Augustana",
"College,",
"Rock",
"Island,",
"USA."
]
} | |||
2301.00001v1.pdf_2 | null | ArxivOCR_part_1 | {
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} | {
"word": [
"2",
"Jordan",
"Thompson,",
"Ryan",
"Benac,",
"Kidus",
"Olana,",
"Talha",
"Hassan,",
"Andrew",
"Sward,",
"and",
"Tauheed",
"Khan",
"Mohd",
"can",
"also",
"display",
"the",
"NFTs",
"owned",
"by",
"the",
"user",
"and",
"allow",
"them",
"to",
"connect",
"to",
"OpenSea",
"to",
"sell",
"the",
"NFTrig",
"cards",
"on",
"a",
"public",
"marketplace.",
"The",
"application",
"is",
"hosted",
"on",
"Moralis",
"employing",
"their",
"Web3",
"API.",
"Technical",
"languages",
"used",
"in",
"this",
"project,",
"which",
"will",
"be",
"discussed",
"in",
"detail",
"throughout",
"this",
"paper,",
"include",
"front",
"end",
"web",
"development",
"languages",
"HTML,",
"CSS",
"(specificall",
"Bootstrap5)",
"and",
"JavaScript",
"as",
"well",
"as",
"the",
"back",
"end",
"smart",
"contract",
"development",
"language",
"Solidity.",
"In",
"order",
"to",
"attract",
"users,",
"this",
"application",
"also",
"allows",
"a",
"user",
"to",
"answer",
"trivia",
"questions",
"and",
"gain",
"experience",
"points.",
"These",
"points",
"can",
"then",
"be",
"used",
"to",
"unlock",
"new",
"sets",
"of",
"NFT",
"cards",
"or",
"upgrade",
"existing",
"cards",
"in",
"a",
"user’s",
"wallet.",
"This",
"game-like",
"design",
"should",
"appeal",
"to",
"a",
"younger",
"audience",
"and",
"encourage",
"them",
"to",
"answer",
"trigonometry",
"or",
"math",
"based",
"questions.",
"This",
"will",
"have",
"an",
"incredible",
"educational",
"benefi",
"for",
"the",
"user",
"becaus",
"the",
"will",
"be",
"both",
"learnin",
"and",
"playin",
"game",
"simultaneously",
"2",
"MOTIVATION",
"The",
"purpose",
"of",
"this",
"application",
"is",
"as",
"an",
"educational",
"tool",
"for",
"students",
"who",
"are",
"attempting",
"to",
"understand",
"the",
"ways",
"that",
"trigonometric",
"functions",
"interact",
"with",
"each",
"other.",
"As",
"opposed",
"to",
"just",
"graphing",
"these",
"functions",
"by",
"hand,",
"students",
"will",
"be",
"able",
"to",
"generate",
"new",
"NFTs",
"by",
"combining",
"whatever",
"trigonometric",
"functions",
"they",
"already",
"own.",
"In",
"fact,",
"using",
"technology",
"is",
"shown",
"to",
"influenc",
"and",
"better",
"educational",
"processes",
"by",
"increasing",
"interaction",
"between",
"those",
"in",
"the",
"classroom",
"[9].",
"Technology",
"is",
"becoming",
"increasingly",
"prevalent",
"in",
"every",
"sphere",
"of",
"daily",
"life,",
"so",
"the",
"use",
"of",
"technology",
"in",
"a",
"classroom",
"setting",
"is",
"not",
"only",
"logical,",
"but",
"it",
"increases",
"the",
"educational",
"benefi",
"of",
"student",
"[29]",
"However,",
"as",
"the",
"technology",
"continues",
"to",
"evolve,",
"\"the",
"gap",
"between",
"traditional",
"course",
"material",
"taught",
"to",
"students",
"in",
"B.S./M.S.",
"programs",
"at",
"universities",
"and",
"the",
"cutting",
"edge",
"of",
"technology",
"used",
"in",
"industry",
"is",
"widening",
"at",
"an",
"unprecedented",
"rate\"",
"[30].",
"By",
"creating",
"this",
"project,",
"it",
"will",
"give",
"students",
"the",
"opportunity",
"to",
"gain",
"experience",
"with",
"block",
"chain,",
"and",
"hopefully",
"be",
"starting",
"place",
"for",
"narrowing",
"that",
"ever",
"growing",
"gap.",
"After",
"much",
"research,",
"it",
"is",
"likely",
"that",
"this",
"proposed",
"application",
"is",
"the",
"firs",
"of",
"its",
"kind",
"that",
"utilizes",
"NFTs",
"to",
"teach",
"mathematical",
"concepts.",
"Aside",
"from",
"user",
"benefi",
"of",
"this",
"application",
"ther",
"is",
"also",
"an",
"intellectua",
"merit",
"in",
"the",
"block",
"chain",
"and",
"education",
"fields",
"Best",
"describe",
"by",
"Carmen",
"Holotescu",
"\"As",
"educatio",
"become",
"more",
"open,",
"diversified",
"democratised",
"and",
"decentralised",
"the",
"bloc",
"chai",
"technolog",
"is",
"take",
"in",
"consideratio",
"by",
"researchers,",
"teachers",
"and",
"institutions,",
"to",
"maintain",
"reputation,",
"trust",
"in",
"certification",
"and",
"proof",
"of",
"learning\"",
"[17].",
"Further,",
"development",
"of",
"this",
"project",
"continues",
"research",
"on",
"NFT",
"and",
"block",
"chain",
"technologies.",
"This",
"application",
"can",
"also",
"serve",
"as",
"the",
"boilerplate",
"basis",
"for",
"other",
"NFT-based",
"educational",
"tools",
"and",
"resources.",
"Research",
"for",
"this",
"project",
"provides",
"opportunities",
"for",
"training",
"computer",
"science",
"students",
"on",
"how",
"to",
"use",
"NFTs",
"in",
"general,",
"but",
"more",
"specificall",
"in",
"educationa",
"contexts",
"NFTrig",
"was",
"developed",
"by",
"computer",
"science",
"students",
"as",
"a",
"fina",
"senio",
"inquir",
"projec",
"at",
"Augustan",
"College.",
"In",
"conjunction",
"and",
"with",
"funding",
"by",
"the",
"Department",
"of",
"Mathematics",
"and",
"Computer",
"Science,",
"this",
"project",
"employs",
"a",
"variety",
"of",
"software",
"development",
"skills",
"and",
"techniques",
"that",
"further",
"the",
"research",
"and",
"understanding",
"of",
"the",
"block",
"chain",
"and",
"web",
"development",
"field",
"3",
"RELATED",
"WORK",
"Block",
"chain",
"technology",
"has",
"enabled",
"the",
"formation",
"of",
"decentralized",
"distributed",
"records",
"of",
"digital",
"data",
"which",
"does",
"not",
"require",
"any",
"third",
"party",
"to",
"moderate",
"any",
"transactions",
"[34].",
"The",
"decentralized",
"nature",
"of",
"block",
"chain",
"also",
"renders",
"it",
"easy",
"for",
"use",
"in",
"a",
"ranging",
"variety",
"of",
"applications",
"in",
"several",
"field",
"such",
"as",
"healthcare",
"[16],",
"internet",
"of",
"things",
"[7],",
"gaming",
"[2],",
"banking",
"[6],",
"and",
"education",
"(explored",
"in",
"greater",
"detail",
"in",
"subsection",
"3.1).",
"Non",
"Fungible",
"token",
"(NFTs)",
"are",
"a",
"relatively",
"new",
"phenomena",
"within",
"the",
"fiel",
"of",
"block",
"chai",
"based",
"technologies",
"but",
"its",
"applicatio",
"in",
"aforementione",
"fields",
"are",
"already",
"being",
"studied.",
"Specificall",
"within",
"the",
"healthcar",
"context",
"NFT's",
"are",
"solvin",
"lon",
"ter",
"issue",
"such",
"as",
"storing",
"patients’",
"private",
"data",
"more",
"safely",
"as",
"well",
"as",
"maintaining",
"better",
"records",
"while",
"giving",
"better",
"autonomy",
"and",
"privacy",
"to",
"both",
"patients",
"and",
"healthcare",
"providers",
"[22].",
"The",
"application",
"of",
"NFTs",
"in"
]
} | |||
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} | {
"word": [
"NFTrig:",
"Using",
"Blockchain",
"Technologies",
"for",
"Math",
"Education",
"3",
"education",
"is",
"still",
"an",
"understudied",
"area.",
"These",
"next",
"related",
"work",
"sections",
"explore",
"the",
"broader",
"use",
"of",
"block",
"chain",
"based",
"technologies",
"for",
"educational",
"purposes,",
"gamification",
"and",
"overal",
"collaborativ",
"learning.",
"3.1",
"Block",
"chain",
"Based",
"Technologies",
"for",
"Educational",
"Purposes",
"There",
"has",
"been",
"extensive",
"work",
"concerning",
"how",
"block",
"chain",
"based",
"technologies",
"are",
"enabling",
"better",
"ownership",
"and",
"sharing",
"of",
"personal",
"records",
"for",
"students",
"and",
"supporting",
"collaborative",
"learning",
"environments.",
"Yumna",
"et",
"al.",
"conducted",
"a",
"systematic",
"literature",
"review",
"of",
"the",
"use",
"of",
"block",
"chain",
"technologies",
"in",
"educational",
"sector",
"[35].",
"They",
"also",
"propose",
"several",
"uses",
"of",
"existing",
"block",
"chain",
"based",
"technologies",
"in",
"educational",
"sector",
"that",
"leverage",
"the",
"decentralized",
"and",
"traceable",
"consensus",
"making",
"mechanisms",
"of",
"block",
"chain.",
"Researchers",
"have",
"examined",
"the",
"use",
"of",
"block",
"chain",
"to",
"allow",
"students",
"to",
"maintain",
"educational",
"records",
"such",
"as",
"transcripts,",
"credentials,",
"diplomas,",
"and",
"learning",
"activities",
"[5,",
"14,",
"31].",
"Similarly,",
"research",
"has",
"also",
"explored",
"learning",
"management",
"systems",
"design",
"based",
"on",
"block",
"chain",
"based",
"technology.",
"The",
"technology",
"can",
"potentially",
"verify",
"a",
"students",
"records",
"as",
"well",
"as",
"enable",
"the",
"design",
"of",
"an",
"automatic",
"decentralized",
"enrollment",
"system",
"which",
"does",
"not",
"require",
"moderation",
"from",
"school",
"staf",
"[31]",
"Another",
"elegant",
"use",
"of",
"block",
"chain",
"in",
"the",
"fiel",
"of",
"educatio",
"is",
"the",
"abilit",
"to",
"suppor",
"life-lon",
"learning",
"applications.",
"The",
"educational",
"sector",
"is",
"becoming",
"more",
"diverse",
"with",
"variety",
"of",
"differen",
"types",
"of",
"classrooms",
"and",
"learning",
"modalities.",
"E-learning",
"has",
"also",
"allowed",
"students",
"to",
"acquire",
"licences",
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"University",
"using",
"NFTs",
"to",
"award",
"course",
"completion",
"certification",
"to",
"students",
"https://upcea.edu/tech",
"trends-in-higher-ed-metaverse-nft-and-dao/"
]
} | |||
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} | {
"word": [
"Jordan",
"Thompson,",
"Ryan",
"Benac,",
"Kidus",
"Olana,",
"Talha",
"Hassan,",
"Andrew",
"Sward,",
"and",
"Tauheed",
"Khan",
"Mohd",
"4",
"[9].",
"They",
"also",
"concluded",
"there",
"was",
"better",
"engagement",
"outcomes",
"for",
"students.",
"On",
"several",
"metrics",
"of",
"usability,",
"the",
"students",
"reported",
"more",
"than",
"80%",
"preference",
"for",
"buying,",
"using,",
"and",
"collecting",
"NFTs.",
"Such",
"independent",
"learning",
"methods",
"were",
"particularly",
"more",
"useful",
"during",
"the",
"COVID-19",
"pandemic",
"to",
"accommodate",
"the",
"need",
"of",
"remote",
"independent",
"learning",
"options.",
"Architecturally,",
"this",
"project",
"takes",
"inspiration",
"from",
"[9],",
"and",
"applies",
"it",
"to",
"a",
"more",
"narrower,",
"focused",
"domain",
"of",
"learning",
"mathematical",
"operations",
"in",
"this",
"study.",
"Further,",
"these",
"NFTs",
"are",
"also",
"easier",
"to",
"share",
"on",
"social",
"media",
"[20].",
"Therefore,",
"it",
"also",
"allows",
"students",
"to",
"more",
"readily",
"share",
"their",
"accomplishments.",
"3.3",
"Gamification",
"to",
"Support",
"Mathematical",
"Learning",
"Since",
"the",
"proposed",
"application",
"teaches",
"mathematical",
"and",
"trigonometric",
"formulas",
"to",
"students,",
"the",
"literature",
"on",
"use",
"of",
"gamificatio",
"to",
"suppor",
"mathematica",
"learnin",
"shoul",
"be",
"better",
"described",
"Gamification",
"in",
"combinatio",
"with",
"incentivizatio",
"explaine",
"in",
"the",
"previou",
"section",
"will",
"allow",
"for",
"the",
"success",
"of",
"this",
"application.",
"Gaming",
"settings",
"have",
"traditionally",
"been",
"used",
"to",
"teach",
"simple",
"mathematical",
"operations",
"to",
"students.",
"More",
"recently,",
"researchers",
"have",
"also",
"proposed",
"systems",
"that",
"teach",
"advanced",
"concepts",
"to",
"students",
"including",
"College",
"Algebra",
"[10].",
"These",
"learning",
"environments",
"make",
"it",
"easier",
"for",
"students",
"to",
"relate",
"the",
"learning",
"concepts",
"with",
"more",
"daily",
"life",
"phenomena.",
"While",
"gamificatio",
"itsel",
"canno",
"guarante",
"better",
"learnin",
"outcomes",
"it",
"can",
"improv",
"students",
"interes",
"and",
"performance",
"by",
"encouraging",
"them",
"to",
"engage",
"with",
"the",
"content",
"for",
"a",
"longer",
"duration",
"of",
"time",
"[18].",
"The",
"simpler,",
"more",
"systematic,",
"and",
"operational",
"nature",
"of",
"mathematics",
"as",
"a",
"subject",
"also",
"makes",
"it",
"easier",
"for",
"incorporation",
"in",
"gaming",
"environments",
"because",
"fina",
"answer",
"are",
"usuall",
"shor",
"and",
"numerical",
"as",
"opposed",
"to",
"long",
"and",
"descriptive",
"answer",
"that",
"might",
"be",
"found",
"in",
"social",
"or",
"natural",
"sciences.",
"Trigonometry",
"especially",
"can",
"easily",
"be",
"broken",
"down",
"into",
"a",
"series",
"of",
"operations",
"and",
"steps",
"which",
"simulates",
"a",
"similar",
"environment",
"found",
"in",
"other",
"online",
"games",
"where",
"users",
"play",
"to",
"fin",
"differen",
"“rewards",
"and",
"“collectables\"",
"Despite",
"all",
"thes",
"benefi",
"there",
"are",
"some",
"limitatio",
"of",
"gamificatio",
"as",
"well.",
"For",
"example",
"it",
"is",
"hard",
"to",
"know",
"how",
"a",
"studen",
"arrive",
"a",
"solutio",
"and",
"give",
"feedback",
"[4].",
"Not",
"being",
"able",
"to",
"solve",
"trigonometric",
"equations",
"can",
"also",
"lead",
"to",
"frustration",
"and",
"impeded",
"learning",
"experience.",
"Foresight",
"into",
"the",
"project’s",
"future",
"looks",
"to",
"mitigate",
"these",
"concerns",
"by",
"fostering",
"better",
"communication",
"between",
"differen",
"game",
"player",
"and",
"providin",
"link",
"to",
"useful",
"learning",
"resources",
"in",
"the",
"application.",
"Prior",
"research",
"has",
"extensively",
"explored",
"the",
"use",
"of",
"gamificatio",
"in",
"differen",
"mathematica",
"fields.",
"This",
"application",
"is",
"likely",
"the",
"first",
"st",
"extend",
"the",
"use",
"se",
"NFTs",
"and",
"block",
"chain",
"to",
"aid",
"in",
"teaching",
"trigonometric",
"equations.",
"Research",
"shows",
"that",
"technology,",
"specificall",
"game",
"are",
"show",
"to",
"be",
"excellen",
"educationa",
"tools",
"In",
"fact,",
"\"one",
"of",
"the",
"most",
"successful",
"positive",
"reinforcement",
"mechanisms",
"[in",
"education]",
"known",
"is",
"gamification",
"[9].",
"This",
"include",
"takin",
"a",
"topi",
"transformin",
"it",
"into",
"a",
"game",
"with",
"positiv",
"reinforce",
"ment.",
"This",
"leverages",
"educational",
"benefit",
"in",
"student",
"and",
"encourage",
"the",
"to",
"continu",
"playin",
"the",
"game",
"to",
"learn.",
"Nftrig",
"has",
"future",
"plans",
"to",
"add",
"a",
"game",
"function",
"which",
"will",
"allow",
"the",
"user",
"to",
"answer",
"trigonometry",
"trivia",
"and",
"math",
"questions.",
"This",
"will",
"aid",
"in",
"both",
"their",
"learning",
"and",
"the",
"continued",
"use",
"of",
"the",
"NFTrig",
"application.",
"Further,",
"the",
"ability",
"to",
"combine",
"owned",
"NFTs",
"with",
"math",
"functions",
"also",
"aids",
"in",
"the",
"education",
"of",
"trigonometry",
"for",
"the",
"student.",
"4",
"EXPERIMENTAL",
"SETUP",
"4.1",
"Softwar",
"Developmen",
"Requirement",
"The",
"NFTrig",
"application",
"employs",
"a",
"variety",
"of",
"software",
"development",
"requirements",
"that",
"cover",
"the",
"range",
"of",
"the",
"project.",
"From",
"front",
"end",
"web",
"development",
"to",
"back",
"end",
"smart",
"contract",
"creation",
"and",
"NFT",
"storage,",
"this",
"section",
"describes",
"the",
"requirements",
"and",
"software",
"used",
"to",
"complete",
"the",
"project.",
"4.1.1",
"Compiling",
"IDE.",
"The",
"smart",
"contracts",
"created",
"for",
"NFTrig",
"are",
"hosted",
"on",
"Remix.",
"Remix",
"is",
"an",
"an",
"open",
"source",
"online",
"compiler",
"IDE",
"that",
"can",
"be",
"used",
"to",
"test",
"and",
"deploy",
"smart",
"contracts",
"[1].",
"The"
]
} | |||
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} | {
"word": [
"NFTrig:",
"Using",
"Blockchain",
"Technologies",
"for",
"Math",
"Education",
"5",
"platform",
"can",
"be",
"accessed",
"by",
"any",
"browser,",
"and",
"it",
"allows",
"the",
"developer",
"to",
"write",
"and",
"deploy",
"smart",
"contracts",
"on",
"an",
"actual",
"or",
"test",
"server",
"simultaneously.",
"The",
"current",
"deployment",
"is",
"on",
"a",
"test",
"server.",
"In",
"order",
"to",
"test",
"and",
"debug",
"the",
"smart",
"contract,",
"Visual",
"Studio",
"Code",
"is",
"used.",
"Visual",
"Studio",
"was",
"found",
"to",
"be",
"the",
"best",
"code",
"editor",
"because",
"a",
"developer",
"can",
"easily",
"upload",
"most",
"fil",
"types",
"and",
"edit",
"the",
"[19].",
"For",
"NFTrig,",
"it",
"was",
"used",
"to",
"develop",
"front",
"end",
"HTML",
"and",
"CSS",
"files",
"as",
"well",
"as",
"back",
"end",
"solidit",
"contract",
"editing.",
"The",
"required",
"installed",
"plugins",
"for",
"Visual",
"Studio",
"(VS)",
"include",
"Solidity",
"and",
"Block",
"chain",
"development.",
"[21]",
"These",
"allowed",
"for",
"simple,",
"straightforward",
"development",
"of",
"code.",
"4.1.2",
"Moralis.",
"Moralis",
"SDK",
"is",
"the",
"primary",
"back",
"end",
"platform",
"for",
"the",
"project.",
"The",
"platform",
"allows",
"connection",
"of",
"the",
"front",
"end",
"web",
"application",
"to",
"the",
"smart",
"contract.",
"[8]",
"The",
"Moralis",
"platform",
"uses",
"a",
"combination",
"of",
"server",
"management",
"and",
"a",
"JavaScript",
"SDK",
"to",
"allow",
"for",
"maximum",
"interaction",
"and",
"simplicity.",
"A",
"developer",
"can",
"do",
"many",
"tasks",
"through",
"this",
"including",
"authentication",
"of",
"users,",
"getting",
"necessary",
"user",
"data,",
"and",
"connecting",
"with",
"MetaMask",
"in",
"a",
"non-complicated",
"and",
"simply",
"coded",
"process.",
"The",
"only",
"expectation",
"is",
"that",
"a",
"developer",
"will",
"need",
"to",
"have",
"programming",
"knowledge",
"in",
"JavaScript",
"as",
"well",
"as",
"a",
"familiarity",
"with",
"Moralis",
"and",
"MetaMask,",
"experience",
"querying",
"a",
"database,",
"and",
"some",
"knowledge",
"of",
"Web3",
"development",
"to",
"ensure",
"maximum",
"results",
"and",
"efficienc",
"Moralis",
"also",
"has",
"the",
"ability",
"to",
"easily",
"connect",
"to",
"MetaMask.",
"4.1.3",
"MetaMask.",
"MetaMask",
"is",
"the",
"digital",
"wallet",
"required",
"for",
"participation",
"in",
"the",
"NFTrig",
"game",
"application.",
"It",
"allows",
"the",
"collection",
"of",
"purchases",
"from",
"the",
"user,",
"and",
"it",
"can",
"be",
"installed",
"as",
"an",
"extension",
"on",
"a",
"browser",
"for",
"increased",
"ease",
"of",
"use",
"[28].",
"MetaMask",
"stores",
"all",
"NFTs",
"owned",
"by",
"the",
"user,",
"and",
"in",
"connection",
"with",
"the",
"NFTrig",
"application,",
"can",
"view",
"and",
"upgrade",
"or",
"modify",
"existing",
"NFTs",
"at",
"a",
"users",
"discretion.",
"Connection",
"to",
"the",
"browser",
"extension",
"is",
"required",
"for",
"the",
"application",
"to",
"access",
"anything",
"owned",
"by",
"a",
"user",
"[24].",
"Because",
"MetaMask",
"is",
"easily",
"integrated",
"into",
"Moralis,",
"and",
"thus",
"NFTrig,",
"there",
"is",
"little",
"a",
"user",
"needs",
"to",
"do",
"to",
"create",
"a",
"connection",
"aside",
"from",
"installing",
"the",
"MetaMask",
"extension,",
"and",
"clicking",
"connect.",
"4.1.4",
"Front",
"End",
"Design.",
"Front",
"end",
"design",
"was",
"accomplished",
"primarily",
"through",
"Visual",
"Studio.",
"The",
"Live",
"Server",
"extension",
"was",
"installed",
"which",
"allows",
"each",
"developer",
"to",
"\"host\"",
"their",
"developed",
"website",
"using",
"a",
"native",
"web",
"application.",
"Doing",
"so",
"allowed",
"simplifie",
"testin",
"and",
"fron",
"end",
"development",
"Instead",
"of",
"creating",
"CSS",
"file",
"fro",
"scratch",
"the",
"NFTrig",
"interfac",
"heavil",
"employ",
"Bootstrap5",
"which",
"simplifie",
"the",
"proces",
"of",
"modifyin",
"the",
"conten",
"layou",
"and",
"desig",
"of",
"button",
"and",
"other",
"conten",
"[25].",
"Moralis",
"and",
"Bootstrap5",
"each",
"have",
"extensive",
"documentation",
"to",
"understand",
"and",
"support",
"front",
"end",
"web",
"development.",
"These",
"tools",
"have",
"been",
"utilized",
"to",
"a",
"near",
"maximum",
"extent.",
"Web",
"Hosting",
"Platform.",
"The",
"initial",
"testing",
"of",
"NFTrig,",
"as",
"previously",
"explained,",
"was",
"hosted",
"on",
"4.1.5",
"a",
"local",
"live",
"server",
"through",
"Visual",
"Studio.",
"After",
"initial",
"development,",
"the",
"project",
"was",
"moved",
"to",
"a",
"web",
"server",
"hosted",
"by",
"Augustana",
"College",
"so",
"that",
"initial",
"testing",
"could",
"begin.",
"It",
"is",
"currently",
"unclear",
"how",
"the",
"site",
"will",
"ultimately",
"be",
"hosted.",
"One",
"option",
"for",
"hosting",
"the",
"web",
"application",
"is",
"directly",
"through",
"Google",
"[32].",
"This",
"would",
"allow",
"the",
"website",
"to",
"be",
"named",
"something",
"easily",
"searchable",
"and",
"accessible.",
"A",
"second",
"option",
"would",
"be",
"to",
"host",
"directly",
"through",
"Moralis,",
"but",
"a",
"limitation",
"of",
"this",
"would",
"be",
"a",
"more",
"diluted",
"website",
"naming",
"convention",
"along",
"with",
"a",
"more",
"confusion",
"process",
"of",
"uploading",
"and",
"modifying",
"website",
"content.",
"Currently,",
"the",
"NFTrig",
"application",
"will",
"remain",
"on",
"the",
"local",
"Augustana",
"College",
"Server.",
"5",
"SOFTWARE",
"DESIGN",
"This",
"section",
"covers",
"all",
"of",
"the",
"decisions",
"necessary",
"to",
"understand",
"the",
"development",
"of",
"NFTrig,",
"as",
"well",
"as",
"the",
"technical",
"implementation",
"of",
"each",
"technology",
"used",
"in",
"the",
"design",
"process."
]
} | |||
2301.00001v1.pdf_6 | null | ArxivOCR_part_1 | {
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} | {
"word": [
"6",
"Jordan",
"Thompson,",
"Ryan",
"Benac,",
"Kidus",
"Olana,",
"Talha",
"Hassan,",
"Andrew",
"Sward,",
"and",
"Tauheed",
"Khan",
"Mohd",
"5.1",
"Softwar",
"Architectur",
"The",
"architecture",
"of",
"this",
"project",
"follows",
"the",
"model-server",
"design",
"architecture",
"[27].",
"Using",
"this",
"model,",
"the",
"clients",
"send",
"transactions",
"and",
"requests",
"to",
"a",
"proxy",
"smart",
"contract",
"stored",
"on",
"the",
"block",
"chain",
"which",
"then",
"makes",
"the",
"appropriate",
"calls",
"to",
"the",
"logic",
"smart",
"contract",
"which",
"is",
"also",
"stored",
"on",
"the",
"block",
"chain.",
"This",
"style",
"of",
"architecture",
"is",
"required",
"for",
"this",
"project",
"because",
"the",
"smart",
"contracts",
"must",
"be",
"stored",
"on",
"the",
"server-side",
"chain",
"in",
"order",
"to",
"be",
"functional.",
"The",
"use",
"of",
"proxy",
"contracts",
"also",
"allows",
"our",
"smart",
"contracts",
"to",
"be",
"fully",
"upgradeable",
"with",
"any",
"future",
"updates",
"that",
"may",
"need",
"to",
"be",
"implemented.",
"5.2",
"Choice",
"of",
"Programming",
"Language",
"This",
"section",
"examines",
"and",
"explains",
"the",
"benefi",
"of",
"each",
"chose",
"languag",
"employe",
"in",
"NFTrig.",
"Front",
"end",
"languages",
"include",
"HTML",
"and",
"CSS",
"and",
"the",
"back",
"end",
"includes",
"Solidity",
"and",
"JavaScript.",
"Each",
"has",
"been",
"chosen",
"because",
"they",
"were",
"found",
"to",
"be",
"the",
"best",
"option",
"for",
"development.",
"5.2.1",
"Solidity.",
"Solidity",
"is",
"the",
"programming",
"language",
"of",
"choice",
"when",
"it",
"comes",
"to",
"coding",
"smart",
"contracts.",
"Solidity",
"is",
"\"similar",
"to",
"JavaScript",
"and",
"yet",
"has",
"some",
"features",
"of",
"object-oriented",
"languages",
"such",
"as",
"Java",
"and",
"C++\"",
"[26].",
"This",
"is",
"a",
"leading",
"language",
"for",
"the",
"development",
"of",
"smart",
"contracts",
"and",
"use",
"on",
"block",
"chain",
"technologies.",
"This",
"project",
"utilizes",
"the",
"solidity",
"library",
"openzeppelin",
"in",
"order",
"to",
"create",
"a",
"solid",
"foundation",
"for",
"the",
"smart",
"contracts.",
"Hardhat",
"and",
"Node",
"JS",
"are",
"then",
"used",
"for",
"the",
"testing",
"and",
"deployment",
"of",
"the",
"smart",
"contracts",
"to",
"the",
"Polygon",
"blockchain.",
"5.2.2",
"JavaScript.",
"In",
"the",
"NFTrig",
"application,",
"JavaScript",
"(JS)",
"is",
"primarily",
"used",
"in",
"the",
"front",
"end",
"application.",
"The",
"primary",
"purpose",
"of",
"this",
"language",
"is",
"generally",
"to",
"create",
"dynamic",
"and",
"interactive",
"web",
"content",
"[15].",
"For",
"the",
"client,",
"JS",
"was",
"used",
"in",
"the",
"navigation",
"bar",
"to",
"allow",
"for",
"clickable",
"links",
"and",
"resizing",
"of",
"the",
"navigation",
"bar",
"in",
"smaller",
"screens.",
"This",
"language",
"was",
"also",
"used",
"to",
"give",
"buttons",
"functionality",
"ranging",
"from",
"logging",
"in",
"to",
"MetaMask",
"to",
"purchasing",
"NFTrig",
"cards.",
"Further,",
"JS",
"was",
"used",
"to",
"test",
"the",
"logic",
"of",
"the",
"front-end",
"combination",
"page",
"until",
"the",
"smart",
"contract",
"was",
"applied.",
"Aside",
"from",
"augmenting",
"HTML",
"and",
"CSS",
"application",
"pages,",
"JavaScript",
"is",
"also",
"used",
"in",
"this",
"project",
"to",
"connect",
"the",
"back",
"end",
"smart",
"contract",
"with",
"the",
"from",
"end",
"web",
"application.",
"This",
"application",
"was",
"also",
"developed",
"using",
"Next",
"JS",
"and",
"deployed",
"via",
"an",
"application",
"known",
"as",
"vercel.",
"5.2.3",
"HTML",
"and",
"CSS.",
"Web",
"development",
"of",
"the",
"user",
"interface",
"was",
"primarily",
"completed",
"using",
"HTML",
"and",
"CSS",
"(Bootstrap5).",
"These",
"languages",
"are",
"equally",
"popular",
"and",
"necessary",
"to",
"develop",
"the",
"web",
"pages",
"[13].",
"Instead",
"of",
"creating",
"all",
"CSS",
"requirements",
"from",
"scratch,",
"Bootstrap5",
"was",
"utilized",
"to",
"allow",
"for",
"cleaner",
"design",
"across",
"web",
"pages",
"and",
"better",
"alignment",
"of",
"web",
"page",
"elements.",
"Bootstrap5",
"also",
"simplifie",
"the",
"need",
"to",
"explicitl",
"code",
"button",
"and",
"other",
"interactiv",
"items",
"5.3",
"Security",
"Considerations",
"Throughout",
"this",
"project,",
"there",
"have",
"been",
"several",
"security",
"considerations",
"discovered",
"that",
"threatened",
"the",
"safety",
"and",
"use",
"of",
"the",
"application.",
"One",
"such",
"discovered",
"issue",
"was",
"initially,",
"there",
"was",
"no",
"code",
"written",
"to",
"block",
"a",
"user",
"from",
"looking",
"at",
"another",
"users",
"token.",
"Further,",
"before",
"minting",
"a",
"new",
"NFT",
"card,",
"the",
"smart",
"contracts",
"check",
"to",
"ensure",
"that",
"the",
"card",
"does",
"not",
"already",
"exist,",
"the",
"cards",
"used",
"for",
"combining",
"are",
"owned",
"by",
"the",
"user,",
"and",
"that",
"the",
"newly",
"minted",
"card",
"follows",
"the",
"correct",
"probabilities",
"of",
"outcomes",
"shows",
"in",
"2.",
"These",
"probabilities",
"are",
"coded",
"into",
"the",
"smart",
"contract.",
"5.4",
"Smart",
"Contract",
"Design",
"The",
"smart",
"contract",
"for",
"this",
"project",
"is",
"broken",
"up",
"into",
"two",
"separate",
"contracts.",
"The",
"firs",
"of",
"which",
"is",
"the",
"NFTrig",
"logic",
"contract",
"which",
"contains",
"the",
"logic",
"for",
"purchasing",
"packs",
"of",
"cards",
"as",
"well",
"as",
"the",
"logic",
"for",
"how",
"cards",
"will",
"interact",
"with",
"each",
"other.",
"The",
"second",
"contract",
"is",
"the",
"marketplace",
"contract",
"which",
"will",
"allow",
"users",
"to",
"trade",
"their",
"own",
"NFTs",
"with",
"other",
"users",
"through",
"the",
"website.",
"Within",
"the",
"NFTrig"
]
} | |||
2301.00001v1.pdf_7 | null | ArxivOCR_part_1 | {
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} | {
"word": [
"NFTrig:",
"Using",
"Blockchain",
"Technologies",
"for",
"Math",
"Education",
"7",
"contract,",
"there",
"are",
"functions",
"for",
"multiplying",
"and",
"dividing",
"cards,",
"purchasing",
"randomized",
"packs",
"of",
"cards,",
"and",
"tracking",
"the",
"details",
"of",
"each",
"individual",
"token",
"as",
"transactions",
"are",
"made.",
"The",
"marketplace",
"contract",
"contains",
"information",
"about",
"sale",
"history",
"as",
"well",
"as",
"the",
"functionality",
"to",
"post",
"new",
"sales",
"and",
"purchase",
"items",
"for",
"sale.",
"Both",
"of",
"these",
"contracts",
"were",
"deployed",
"as",
"upgradeable",
"contracts",
"so",
"they",
"can",
"have",
"updates",
"implemented",
"in",
"the",
"future.",
"5.5",
"NFT",
"Storage",
"and",
"Naming",
"Conventions",
"All",
"NFT",
"images",
"are",
"stored",
"on",
"the",
"server",
"with",
"the",
"HTML,",
"CSS,",
"and",
"JS",
"files",
"The",
"namin",
"conventio",
"for",
"each",
"image",
"references",
"what",
"image",
"it",
"is",
"in",
"four",
"numbers.",
"The",
"firs",
"numbe",
"is",
"the",
"power",
"of",
"sin,",
"the",
"second",
"is",
"the",
"power",
"of",
"cos,",
"the",
"third",
"is",
"the",
"rarity",
"or",
"color",
"of",
"the",
"card",
"(0-3",
"is",
"green,",
"blue,",
"purple,",
"and",
"red",
"respectively),",
"and",
"the",
"fina",
"numbe",
"is",
"the",
"text",
"varian",
"(0-3)",
"These",
"files",
"were",
"named",
"accordingly",
"to",
"better",
"determine",
"the",
"output",
"if",
"cards",
"were",
"combined",
"using",
"a",
"mathematical",
"function.",
"For",
"example,",
"a",
"sin",
"card",
"might",
"have",
"the",
"naming",
"convention:",
"1023.jpg.",
"10",
"define",
"it",
"is",
"a",
"sin",
"card,",
"2",
"defin",
"it",
"is",
"rarity",
"purple,",
"and",
"3",
"define",
"it",
"is",
"text",
"varian",
"3.",
"The",
"purpos",
"of",
"namin",
"the",
"files",
"in",
"this",
"way",
"is",
"so",
"that",
"the",
"front",
"end",
"can",
"easily",
"determine",
"which",
"image",
"corresponds",
"to",
"a",
"particular",
"NFT",
"by",
"simply",
"looking",
"at",
"the",
"four",
"features",
"of",
"each",
"token",
"which",
"match",
"the",
"four",
"numbers",
"in",
"the",
"fil",
"name.",
"5.6",
"Client",
"Design",
"The",
"NFTrig",
"application",
"interface",
"was",
"designed",
"using",
"HTML",
"and",
"CSS.",
"The",
"primary",
"use",
"of",
"CSS",
"was",
"often",
"replaced",
"by",
"Bootstrap5.",
"Bootstrap",
"5,",
"a",
"library",
"for",
"CSS,",
"allows",
"for",
"easier",
"scaling",
"and",
"alignment",
"of",
"objects",
"in",
"the",
"HTML",
"file",
"and",
"thu",
"the",
"compute",
"scree",
"[23].",
"Documentatio",
"on",
"the",
"Bootstrap",
"has",
"utilized",
"to",
"a",
"full",
"extent.",
"Each",
"section",
"examines",
"the",
"layout",
"and",
"use",
"of",
"each",
"application",
"page.",
"5.6.1",
"NFTrig",
"Home.",
"The",
"interface",
"is",
"designed",
"to",
"allow",
"a",
"user",
"to",
"access",
"the",
"marketplace,",
"their",
"individual",
"current",
"collections,",
"and",
"their",
"profile",
"The",
"navigationa",
"bar",
"contain",
"link",
"to",
"the",
"client-sid",
"facing",
"pages:",
"NFTrigHome,",
"MyCards,",
"CombineCards,",
"Marketplace,",
"and",
"Game.",
"We",
"used",
"a",
"total",
"of",
"three",
"colors",
"to",
"enable",
"good",
"contrast",
"and",
"make",
"it",
"easier",
"for",
"our",
"users",
"to",
"view",
"complex",
"graphs",
"and",
"formula",
"without",
"a",
"cluttered",
"background",
"2",
"The",
"JavaScript",
"elements",
"declared",
"are",
"reusable",
"across",
"multiple",
"screens.",
"They",
"support",
"functions",
"and",
"interactions",
"such",
"as",
"a",
"user",
"hovering",
"over",
"a",
"cell",
"or",
"clicking",
"a",
"cell",
"and",
"providing",
"both",
"feedback",
"and",
"error",
"handling",
"to",
"the",
"user.",
"The",
"navigation",
"bar",
"is",
"also,",
"the",
"top",
"bar",
"changes",
"color",
"to",
"indicate",
"the",
"tab",
"that",
"the",
"user",
"is",
"on.",
"5.6.2",
"Combination.",
"The",
"main",
"purpose",
"of",
"the",
"combination",
"page",
"is",
"for",
"users",
"to",
"choose",
"cards",
"that",
"they",
"currently",
"own,",
"and",
"see",
"options",
"for",
"combining",
"them",
"using",
"either",
"multiplication",
"or",
"division.",
"Figure",
"1",
"displays",
"the",
"layout",
"of",
"the",
"screen",
"where",
"user",
"selected",
"cards",
"are",
"shown",
"on",
"the",
"left,",
"and",
"potential",
"results",
"are",
"shown",
"on",
"the",
"right.",
"The",
"page",
"utilizes",
"Bootstrap5",
"capabilities",
"to",
"format",
"effectivel",
"to",
"differe",
"screen",
"sizes",
"and",
"resolutions.",
"It",
"connects",
"with",
"a",
"back",
"end",
"script",
"to",
"the",
"smart",
"contract.",
"This",
"provides",
"functionality",
"to",
"the",
"buttons",
"and",
"easy",
"generation",
"of",
"possible",
"NFT",
"results.",
"Below",
"shows",
"the",
"probabilities",
"of",
"generated",
"NFT",
"outcomes",
"based",
"on",
"the",
"selected",
"input",
"cards.",
"5.6.3",
"Marketplace",
"and",
"MyCards.",
"Marketplace",
"and",
"MyCards",
"are",
"similar",
"pages,",
"as",
"they",
"connect",
"to",
"a",
"data",
"source",
"and",
"display",
"NFTs.",
"The",
"Marketplace",
"tab",
"shows",
"all",
"NFT",
"cards",
"available",
"for",
"purchase",
"both",
"from",
"other",
"users",
"who",
"own",
"NFTs",
"and",
"cards",
"owned",
"by",
"the",
"NFTrig",
"project.",
"MyCards",
"however",
"specificall",
"show",
"all",
"cards",
"owne",
"by",
"a",
"user.",
"The",
"layou",
"for",
"each",
"generate",
"all",
"necessar",
"NFT",
"images",
"and",
"information",
"about",
"the",
"rarity.",
"The",
"rarity",
"is",
"signifie",
"by",
"the",
"colo",
"and",
"the",
"tex",
"optio",
"of",
"the",
"card.",
"Figure",
"3",
"shows",
"the",
"actual",
"layout",
"displayed",
"on",
"the",
"page.",
"2Background-color:#333,",
"Color:",
"#f2f2f2,"
]
} | |||
2301.00001v1.pdf_8 | null | ArxivOCR_part_1 | {
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} | {
"word": [
"8",
"Jordan",
"Thompson,",
"Ryan",
"Benac,",
"Kidus",
"Olana,",
"Talha",
"Hassan,",
"Andrew",
"Sward,",
"and",
"Tauheed",
"Khan",
"Mohd",
"Combination",
"Choose",
"two",
"cards",
"and",
"math",
"function",
"and",
"hit",
"combine!",
"Possible",
"Results",
"SIN(.)TAN(.)",
"Sin(x)*Tan(X)",
"51()AN[)",
"Sin(x)*Tan(X)",
"SIN(x)",
"TAN",
"(x)",
"Rarity:",
"Green",
"Rarity",
"Blue",
"%",
"J.",
"J",
"J",
"Font:0",
"Font O",
"Liklihood:",
"75%",
"Liklihood:",
"20%",
"1",
"d",
"tan(z)",
"=",
"sec�(z)",
"dr",
"=",
"sin(z) co5(y) ",
")cos(z)",
"S]N(:)-TAN(:)",
"Sin(x)*Tan(X)",
"5JN(.:)-TAH(:)",
"Sin(x)*Tan(X)",
"Rarity:",
"Blue",
"Rarity:",
"Pink",
"Border",
"Font:0",
"in(X) Green",
"Tan(x) Bue",
"Font:0",
"Liklihood:",
"20%",
"Liklihood:",
"20%",
"Divide",
"]",
"Multiply",
"]",
"Combine",
"Copyright",
"2022",
"All Rights",
"Reserved",
"Augustana",
"College",
"Fig.",
"1.",
"Interface",
"where",
"users",
"will",
"combine",
"NFTrigs",
"UNCOMMON",
"LEGENDARY",
"COMBINATIONS",
"COMMON",
"RARE",
"60%",
"Common",
"+",
"Common",
"20%",
"15%",
"5%",
"Common",
"+",
"Legendary",
"10%",
"25%",
"35%",
"30%",
"15%",
"Common",
"+",
"Rare",
"10%",
"50%",
"25%",
"10%",
"60%",
"Common",
"+",
"Uncommon",
"20%",
"10%",
"15%",
"Rare",
"Rare",
"5%",
"30%",
"50%",
"5%",
"20%",
"60%",
"15%",
"Uncommon",
"+",
"Uncommon",
"60%",
"10%",
"Uncommon",
"Rare",
"5%",
"25%",
"5%",
"10%",
"55%",
"30%",
"Uncommon",
"Legendary",
"0%",
"30%",
"Rare",
"+",
"Legendary",
"10%",
"60%",
"0%",
"20%",
"75%",
"Legendary",
"Legendary",
"5%",
"Fig.",
"2.",
"Probabilities",
"of",
"outcomes",
"depending",
"on",
"rarity",
"of",
"selected",
"cards",
"5.6.4",
"Qualit",
"attribut",
"of",
"client-si",
"interface",
"and",
"code.",
"In",
"order",
"to",
"have",
"an",
"applicati",
"of",
"quality,",
"consistency,",
"and",
"accuracy,",
"the",
"project",
"followed",
"the",
"following",
"guidelines:",
"(1)",
"The",
"code",
"is",
"written",
"in",
"a",
"manner",
"that",
"components",
"and",
"layouts",
"can",
"be",
"rearranged",
"to",
"support",
"any",
"structural",
"changes",
"in",
"the",
"front",
"end.",
"(2)",
"The",
"code",
"has",
"consistent",
"style",
"and",
"format,",
"such",
"as",
"the",
"padding",
"used",
"in",
"individual",
"NFTrig",
"elements",
"and",
"the",
"purchase",
"page’s",
"color.",
"(3)",
"The",
"code",
"contains",
"comments",
"and",
"is",
"well",
"indented",
"for",
"easy",
"maintenance",
"and",
"understanding.",
"(4)",
"Consistent",
"colors",
"and",
"feedback",
"systems",
"are",
"provided",
"so",
"the",
"system",
"is",
"easy",
"to",
"learn",
"for",
"users.",
"(5)",
"Page-level",
"styling",
"was",
"avoided",
"when",
"possible",
"to",
"keep",
"design",
"consistent.",
"(6)",
"Thorough",
"testing",
"was",
"completed",
"for",
"basic",
"accessibility",
"features.",
"5.6.5",
"Testing",
"the",
"Client",
"Design.",
"Basic",
"unit",
"testing",
"of",
"differen",
"element",
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} | {
"word": [
"NFTrig:",
"Using",
"Blockchain",
"Technologies",
"for",
"Math",
"Education",
"9",
"q",
"Marketplace",
"Collections",
"1",
"Profile",
"1",
"About",
"Metamask",
"Search..",
"NFTrig",
"ECLCSCL",
"SECLCSCh)",
"SECL)CSCW)",
"U",
"U..",
"U",
"J,",
"U",
"J,",
"U",
"U,",
"nn",
"n",
"n",
"i",
"n",
"i",
"n",
"=",
"sec2(x)+csc2(x)",
"cot(x)",
"+",
"tan(x)",
"2",
"csc(2x)",
"to",
"be",
"added",
"2",
"csc(2x)",
"to",
"be",
"added",
"U",
",U,",
"n",
"sec2(x)+csc2(x)",
"cot(x)",
"+tan(x)",
"sec2(x)+csc2(x)",
"cot(x)",
"+",
"tan(x)",
"csc(2x)",
"to",
"be",
"added",
"Fig.",
"3.",
"Interface",
"displaying",
"NFTrig",
"Marketplace",
"most",
"application",
"uses,",
"three",
"user",
"cases",
"were",
"devised:",
"a",
"user",
"browsing",
"NFTrigs,",
"a",
"user",
"making",
"a",
"purchase,",
"and",
"a",
"user",
"combining",
"NFTrigs.",
"All",
"assumptions",
"and",
"expected",
"actions",
"expected",
"from",
"the",
"system",
"were",
"listed",
"and",
"analyzed",
"through",
"testing.",
"Further,",
"testing",
"through",
"some",
"edge",
"cases",
"were",
"also",
"pursued.",
"Currently,",
"the",
"application",
"works",
"as",
"intended,",
"however",
"future",
"plans",
"involve",
"rigorous",
"testing",
"with",
"JavaScript",
"code",
"and",
"external",
"APIs",
"(if",
"any",
"are",
"devised).",
"This",
"will",
"ensure",
"a",
"fully",
"functional,",
"secure,",
"and",
"usable",
"application",
"that",
"can",
"also",
"be",
"used",
"as",
"a",
"boiler",
"plate",
"project",
"for",
"other",
"educational",
"blockchain",
"technologies.",
"5.6.6",
"Future",
"Work:",
"Game.",
"Future",
"work",
"for",
"this",
"project",
"will",
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"the",
"ability",
"for",
"users",
"to",
"play",
"a",
"trivia",
"and",
"trigonometric",
"equation",
"game.",
"This",
"allows",
"a",
"user",
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"gain",
"experience",
"points",
"that",
"they",
"can",
"then",
"use",
"to",
"purchase",
"new",
"NFTs.",
"This",
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"to",
"always",
"need",
"cryptocurrency",
"to",
"purchase",
"individual",
"or",
"group",
"NFT",
"cards.",
"Although",
"there",
"is",
"not",
"currently",
"an",
"interface",
"for",
"this",
"page",
"written",
"in",
"HTML,",
"functionality",
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"for",
"the",
"trivia",
"game",
"itself.",
"The",
"file",
"are",
"currentl",
"store",
"on",
"the",
"server,",
"but",
"they",
"are",
"disabled",
"and",
"there",
"is",
"no",
"navigable",
"way",
"to",
"get",
"there",
"through",
"the",
"application.",
"6",
"METHODS",
"Most",
"methods",
"for",
"completing",
"this",
"project",
"have",
"been",
"thoroughly",
"explained",
"in",
"the",
"sections",
"above.",
"However,",
"the",
"fina",
"intende",
"versio",
"of",
"this",
"project",
"will",
"be",
"hosted",
"in",
"a",
"differe",
"location",
"than",
"it",
"resides",
"currently.",
"The",
"initial",
"portion",
"of",
"this",
"project",
"had",
"the",
"front",
"end",
"website",
"hosted",
"on",
"a",
"local",
"Augustana",
"College",
"server",
"and",
"the",
"back",
"end",
"smart",
"contract",
"hosted",
"on",
"the",
"Polygon",
"test",
"net.",
"This",
"allowed",
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"testing",
"and",
"validation",
"that",
"the",
"smart",
"contract",
"operated",
"as",
"expected,",
"as",
"well",
"as",
"give",
"time",
"and",
"opportunity",
"to",
"discover",
"security",
"vulnerabilities.",
"The",
"future",
"of",
"this",
"project",
"will",
"be",
"hosted",
"on",
"a",
"decentralized",
"web",
"application",
"online",
"so",
"that",
"users",
"can",
"access",
"it",
"and",
"begin",
"to",
"interact",
"with",
"the",
"smart",
"contract.",
"Further,",
"a",
"redesign",
"of",
"the",
"website",
"user",
"interface",
"is",
"likely.",
"This",
"will",
"require",
"transition",
"from",
"BootStrap5",
"to",
"NextJS",
"which",
"allows",
"cards",
"to",
"be",
"generated,",
"displayed,",
"and",
"interactable",
"through",
"a",
"version",
"of",
"JavaScript.",
"7",
"RESULTS",
"This",
"project",
"successfully",
"allowed",
"the",
"exploration",
"and",
"creation",
"of",
"applying",
"NFT",
"and",
"block",
"chain",
"technology",
"to",
"math",
"education.",
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"preliminary",
"in",
"use",
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"nature,",
"this",
"project",
"allows",
"for",
"initial",
"project",
"creation",
"as",
"a",
"boiler",
"plate",
"project.",
"The",
"smart",
"contract",
"is",
"currently",
"deployed",
"on",
"the"
]
} | |||
2301.00001v1.pdf_10 | null | ArxivOCR_part_1 | {
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} | {
"word": [
"Jordan",
"Thompson,",
"Ryan",
"Benac,",
"Kidus",
"Olana,",
"Talha",
"Hassan,",
"Andrew",
"Sward,",
"and",
"Tauheed",
"Khan",
"Mohd",
"10",
"Polygon",
"testnet",
"and",
"can",
"be",
"interacted",
"with",
"using",
"test",
"Matic.",
"Each",
"web",
"page",
"has",
"functionality",
"to",
"display",
"the",
"user’s",
"owned",
"NFTs",
"as",
"well",
"as",
"the",
"NFTs",
"they",
"have",
"put",
"for",
"sale",
"on",
"the",
"marketplace.",
"Using",
"NextJS",
"will",
"also",
"allow",
"the",
"Combination",
"page",
"to",
"have",
"functionality",
"and",
"smart",
"contract",
"use.",
"It",
"is",
"also",
"worth",
"noting",
"that",
"the",
"created",
"web",
"page",
"is",
"not",
"required",
"to",
"interact",
"with",
"the",
"NFTrig",
"smart",
"contracts.",
"8",
"RECOMMENDATIONS",
"FOR",
"FUTURE",
"WORK",
"The",
"goal",
"for",
"this",
"project",
"was",
"a",
"working",
"Beta",
"demo",
"that",
"shows",
"application",
"functionality,",
"and",
"correct",
"smart",
"contract",
"execution.",
"There",
"are",
"many",
"other",
"features",
"planned",
"for",
"the",
"continued",
"work",
"of",
"this",
"project.",
"The",
"first",
"as",
"earlie",
"explained",
"is",
"a",
"gam",
"optio",
"which",
"challenge",
"the",
"user",
"with",
"trigonometr",
"trivia",
"and",
"math",
"problems.",
"Answering",
"these",
"questions",
"successfully",
"will",
"increase",
"the",
"experience",
"points",
"of",
"a",
"user.",
"The",
"user",
"can",
"then",
"use",
"these",
"experience",
"points",
"to",
"purchase",
"individual",
"or",
"packs",
"of",
"NFTrig",
"cards,",
"or",
"they",
"can",
"be",
"used",
"to",
"combine",
"cards.",
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"blockchain",
"to",
"improve",
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"in",
"healthcare.",
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"(2018),",
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]
} | |||
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} | {
"word": [
"JOURNAL",
"OF",
"LATEX",
"CLASS",
"FILES,",
"VOL.",
"14,",
"NO.",
"8,",
"AUGUST",
"2015",
"1",
"Evaluating",
"Glyph",
"Design",
"for",
"Showing",
"Large-Magnitude-Range",
"Quantum",
"Spins",
"Henan",
"Zhao,",
"Garnett",
"W.",
"Bryant,",
"Wesley",
"Griffin,",
"Judith",
"E.",
"Terrill,",
"Jian",
"Chen",
"Abstract—We",
"present",
"experimental",
"results",
"to",
"explore",
"a",
"form",
"of",
"bivariate",
"glyphs",
"for",
"representing",
"large-magnitude-range",
"vectors.",
"The",
"glyphs",
"meet",
"two",
"conditions:",
"(1)",
"two",
"visual",
"dimensions",
"are",
"separable;",
"and",
"(2)",
"one",
"of",
"the",
"two",
"visual",
"dimensions",
"uses",
"categorical",
"representation",
"(e.g.,",
"a",
"categorical",
"colormap).",
"We",
"evaluate",
"how",
"much",
"these",
"two",
"conditions",
"determine",
"the",
"bivariate",
"glyphs’",
"effectiveness.",
"The",
"first",
"experiment",
"asks",
"participants",
"to",
"perform",
"three",
"local",
"tasks",
"requiring",
"reading",
"no",
"more",
"than",
"two",
"glyphs.",
"The",
"second",
"experiment",
"scales",
"up",
"the",
"search",
"space",
"in",
"global",
"tasks",
"when",
"participants",
"must",
"look",
"at",
"the",
"entire",
"scene",
"of",
"hundreds",
"of",
"vector",
"glyphs",
"to",
"get",
"an",
"answer.",
"Our",
"results",
"support",
"that",
"the",
"first",
"condition",
"is",
"necessary",
"for",
"local",
"tasks",
"when",
"a",
"few",
"items",
"are",
"compared.",
"But",
"it",
"202",
"is",
"not",
"enough",
"for",
"understanding",
"large",
"amount",
"of",
"data.",
"The",
"second",
"condition",
"is",
"necessary",
"for",
"perceiving",
"global",
"structures",
"of",
"examining",
"very",
"complex",
"datasets.",
"Participants’",
"comments",
"reveal",
"that",
"the",
"categorical",
"features",
"in",
"the",
"bivariate",
"glyphs",
"trigger",
"emergent",
"optimal",
"viewers’",
"behaviors.",
"This",
"work",
"contributes",
"to",
"perceptually",
"accurate",
"glyph",
"representations",
"for",
"revealing",
"patterns",
"from",
"large",
"scientific",
"results.",
"We",
"release",
"source",
"code,",
"quantum",
"physics",
"data,",
"training",
"documents,",
"participants’",
"answers,",
"and",
"statistical",
"analyses",
"Ph",
"for",
"reproducible",
"science",
"at",
"https",
"s://osf.io/4xcf5/?viewonly",
"94123139df9c4ac984a1e0df811cd580",
"Index",
"Terms—Separable",
"and",
"integral",
"dimension",
"pairs,",
"bivariate",
"glyph,",
"3D",
"glyph,",
"quantitative",
"visualization,",
"large-magnitude-range.",
"2",
"1",
"INTRODUCTION",
"G",
"B",
"IVARATE",
"glyph",
"visualization",
"is",
"a",
"common",
"form",
"of",
"design",
"also",
"increases",
"task",
"completion",
"time",
"for",
"an",
"apparently",
"visual",
"design",
"in",
"which",
"a",
"dataset",
"is",
"depicted",
"by",
"two",
"simple",
"comparison",
"task",
"between",
"two",
"magnitudes",
"in",
"three",
"visual",
"variables,",
"often",
"chosen",
"from",
"set",
"of",
"perceptually",
"dimensions",
"(3D):",
"the",
"traditional",
"direct",
"approach",
"of",
"Linear",
"independent",
"graphical",
"dimensions",
"of",
"shape,",
"color,",
"texture,",
"(Figure",
"2f)",
"is",
"significantly",
"faster",
"than",
"splitVectors",
"(Figure",
"2e).",
"size,",
"orientation,",
"curvature,",
"and",
"so",
"on",
"[1],",
"[2].",
"A",
"bivariate",
"One",
"may",
"frame",
"this",
"large-magnitude-range",
"issue",
"as",
"a",
"M",
"glyph",
"design",
"[3]",
"has",
"been",
"broadly",
"applied",
"to",
"reveal",
"atom",
"visual",
"design",
"problem:",
"how",
"can",
"we",
"depict",
"a",
"scalar",
"value",
"using",
"spin",
"behaviors",
"for",
"quantum",
"physicists",
"at",
"National",
"Institute",
"bivariate",
"visual",
"features",
"to",
"help",
"quantum",
"physicists",
"examine",
"com-",
"of",
"Standards",
"and",
"Technology",
"(NIST)",
"to",
"examine",
"experi-",
"plex",
"spatial",
"data?",
"Intuitively,",
"since",
"all",
"tasks",
"in",
"previous",
"study",
"mental",
"results;",
"thanks",
"to",
"their",
"team’s",
"Nobel-prize-winning",
"involve",
"a",
"single",
"or",
"at",
"most",
"two",
"spin",
"locations,",
"human",
"visual",
"simulations",
"[4].",
"Quantum",
"physicists",
"world-wide",
"can",
"now",
"system",
"would",
"integrate",
"the",
"two",
"component",
"parts",
"(digit",
"and",
"manipulate",
"many",
"individual",
"quantum",
"systems",
"to",
"study",
"exponent",
"terms)",
"of",
"a",
"quantum",
"spin",
"into",
"one",
"gestalt",
"before",
"2016",
"complex",
"atom",
"and",
"sub-atom",
"interactions.",
"Because",
"atoms",
"comparing",
"the",
"results",
"[5].",
"Since",
"relating",
"the",
"digit",
"and",
"expo-",
"can",
"be",
"in",
"multiple",
"states",
"simultaneously",
"and",
"because",
"these",
"nent",
"to",
"the",
"two",
"size",
"features",
"demands",
"a",
"focused",
"attention",
"spin",
"magnitudes",
"are",
"large",
"in",
"range",
"and",
"often",
"vary",
"greatly",
"mode",
"of",
"visual",
"processing,",
"viewer",
"would",
"take",
"longer",
"to",
"in",
"local",
"regions,",
"computational",
"solutions",
"still",
"do",
"not",
"exist",
"to",
"process",
"two",
"component",
"parts",
"in",
"splitVectors",
"compared",
"to",
"a",
"characterize",
"the",
"spin",
"behaviors.",
"Today’s",
"quantum",
"physicists",
"single",
"linear",
"mapping.",
"We",
"term",
"this",
"explanation",
"the",
"object-",
"rely",
"on",
"visualization",
"to",
"interpret",
"simulation",
"results.",
"level",
"hypothesis",
"where",
"a",
"viewer",
"responds",
"to",
"combine",
"two",
"On",
"the",
"visualization",
"side,",
"the",
"initial",
"design",
"and",
"eval-",
"component",
"parts",
"of",
"a",
"value",
"represented",
"in",
"a",
"glyph",
"to",
"its",
"uation",
"of",
"large-magnitude-range",
"spin",
"vector",
"visualizations",
"original",
"scalar",
"value",
"(here",
"the",
"magnitude).",
"use",
"scientific",
"notation",
"to",
"depict",
"digit",
"and",
"exponent",
"as",
"two",
"However,",
"the",
"object-level",
"processing",
"may",
"be",
"neither",
"effi-",
"concentric",
"cylinders",
"[3]:",
"inside",
"and",
"outside",
"tube-lengths",
"cient",
"nor",
"necessary.",
"For",
"example,",
"Borgo",
"et",
"al.",
"[6]",
"state",
"that",
"(length",
"y",
"lengthy",
"or",
"L",
"y",
"L",
"or",
"splitVectors)",
"are",
"mapped",
"to",
"digit",
"“...",
"effective",
"glyph",
"design",
"should",
"encompass",
"a",
"non-conflicting",
"set",
"and",
"power",
"of",
"spin",
"magnitude",
"accordingly",
"(Figure",
"1e).",
"A",
"of",
"separable",
"retinal",
"variables”.",
"Now,",
"for",
"our",
"examples,",
"if",
"we",
"three-dimensional",
"(3D)",
"bivariate",
"glyph",
"scene",
"of",
"this",
"splitVec-",
"increase",
"the",
"bivariate",
"feature",
"separability",
"by",
"replacing",
"the",
"tors",
"design",
"(Figure",
"2e)",
"achieves",
"up",
"to",
"ten",
"times",
"greater",
"exponent-to-length",
"mapping",
"in",
"Figures",
"le",
"and",
"2e",
"to",
"the",
"accuracy",
"than",
"the",
"traditional",
"direct",
"approach",
"(Linear,",
"Fig-",
"exponent-to-color",
"mapping",
"in",
"Figures",
"1c",
"and",
"2c",
"for",
"compar-",
"ure",
"2f)",
"for",
"reading",
"a",
"vector",
"magnitude",
"of",
"a",
"single",
"spin",
"or",
"ison",
"tasks,",
"it",
"would",
"be",
"counterproductive",
"for",
"our",
"attention",
"deriving",
"ratios",
"between",
"two",
"spin",
"magnitudes.",
"However,",
"this",
"first",
"to",
"visit",
"each",
"glyph",
"to",
"compute",
"the",
"magnitude.",
"Instead,",
"the",
"global",
"categorical",
"color",
"(hue)",
"can",
"guide",
"our",
"attention",
"to",
"Henan",
"Zhao",
"is",
"with",
"University",
"of",
"Maryland,",
"Baltimore",
"County.",
"E-mail:",
"first",
"compare",
"the",
"exponent,",
"prior",
"to",
"compare",
"vector",
"lengths",
"henan1@umbc.edu.",
"Garnett",
"W.",
"Bryant",
"and",
"Judith",
"E.",
"Terrill",
"are",
"with",
"the",
"National",
"of",
"bivariate",
"features",
"is",
"needed",
"as",
"long",
"as",
"the",
"two",
"color",
"hues",
"Institute",
"of",
"Standards",
"and",
"Technology.",
"E-mail:",
"{garnett.bryant,",
"ju-",
"(digit).",
"In",
"these",
"cases,",
"no",
"object-level",
"attentive",
"processing",
"can",
"be",
"easily",
"recognizable.",
"dith.terrill}@nist.gov.",
"Wesley",
"Griffin",
"is",
"with",
"Stellar",
"Science.",
"E-mail:",
"griffin5@umbc.edu.",
"Further",
"considering",
"the",
"quantum",
"physicists’",
"task",
"rele-",
"Jian",
"Chen",
"is",
"with",
"The",
"Ohio",
"State",
"University.",
"E-mail:",
"chen.8028@osu.edu.",
"vant",
"to",
"multiple",
"objects",
"(e.g.,",
"find",
"maximum",
"among",
"hun-"
]
} | |||
2301.00002v1.pdf_2 | null | ArxivOCR_part_1 | {
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]
} | |||
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} | {
"word": [
"JOURNAL",
"OF",
"LATEX",
"CLASS",
"FILES,",
"VOL.",
"14,",
"NO.",
"8,",
"AUGUST",
"2015",
"5",
"Length,",
"texture",
"(separable)",
"(Figure",
"1d).",
"Texture",
"repre-",
"sented",
"exponents.",
"The",
"percentage",
"of",
"black",
"color",
"(Bertin",
"[38])",
"was",
"used",
"to",
"represent",
"the",
"exponential",
"terms",
"0",
"(0%),",
"1",
"(30%),",
"2",
"(60%)",
"and",
"3",
"(90%),",
"wrapped",
"around",
"the",
"cylinders",
"in",
"five",
"segments",
"to",
"make",
"them",
"visible",
"from",
"any",
"viewpoint.",
"Length",
"length",
"y",
"(splitVectors",
"[3],",
"separable)",
"(Figure",
"1e).",
"This",
"glyph",
"used",
"splitVectors",
"[3]",
"as",
"the",
"baseline",
"and",
"mapped",
"both",
"digit",
"and",
"exponent",
"to",
"lengths.",
"The",
"glyphs",
"were",
"semi-",
"transparent",
"so",
"that",
"the",
"inner",
"cylinders",
"showing",
"the",
"digit",
"terms",
"were",
"legible.",
"Feather-like",
"fishbone",
"legends",
"were",
"added",
"at",
"each",
"location",
"when",
"the",
"visual",
"variable",
"length",
"was",
"used.",
"The",
"tick-mark",
"band",
"was",
"depicted",
"as",
"subtle",
"light-gray",
"lines",
"around",
"each",
"cylinder.",
"(answer:",
"636.30)",
"What",
"is",
"the",
"magnitude",
"of",
"the",
"vector",
"at",
"point",
"A?",
"(a)",
"MAG",
"task:",
"Distances",
"between",
"neighboring",
"lines",
"show",
"unit",
"length",
"legible",
"at",
"certain",
"distance",
"(Figure",
"1",
"rows",
"1",
"and",
"2).",
"3.1.2",
"Hypotheses",
"Given",
"the",
"analysis",
"below",
"and",
"recommendations",
"in",
"the",
"liter-",
"ature,",
"we",
"arrived",
"at",
"the",
"following",
"working",
"hypotheses:",
".",
"Exp",
"I.",
"H1.",
"(Overall).",
"The",
"lengthy",
"color",
"feature-pair",
"can",
"lead",
"ly",
"to",
"the",
"most",
"accurate",
"answers.",
"�",
"Exp",
"I.",
"H2.",
"(Integral-separable).",
"Among",
"the",
"three",
"separable",
"dimensions,",
"length",
"y",
"color",
"may",
"lead",
"to",
"the",
"greatest",
"speed",
"and",
"accuracy",
"and",
"length",
"texture",
"will",
"be",
"more",
"effective",
"than",
"(b)",
"RATIO",
"task:",
"What",
"is",
"the",
"ratio",
"of",
"the",
"magnitude",
"between",
"the",
"lengthy",
"jlengthy",
"(splitVectors).",
"�",
"Exp",
"I.",
"H3.",
"(Redundancy",
"on",
"time).",
"The",
"redundant",
"pair",
"vectors",
"at",
"points",
"A",
"and",
"B?",
"(answer:",
"3.60)",
"lengthy",
",color/length.",
"c",
"will",
"reduce",
"task",
"completion",
"time",
"compared",
"to",
"splitVectors.",
"Several",
"reasons",
"led",
"to",
"H1",
"and",
"H2.",
"They",
"are",
"related",
"to",
"the",
"two",
"conditions",
"of",
"glyph",
"design",
"we",
"evaluate.",
"Color",
"and",
"length",
"were",
"separable",
"dimensions,",
"so",
"comparing",
"length",
"to",
"color",
"is",
"simple",
"(condition",
"1).",
"And",
"color",
"was",
"preattentive",
"and",
"could",
"be",
"detected",
"quickly",
"(condition",
"2).",
"Compared",
"to",
"the",
"redundant",
"length",
"y",
"color/length",
"c,",
"lengthy",
"color",
"reduced",
"crowding",
"since",
"the",
"feature-pairs",
"were",
"generally",
"smaller",
"than",
"those",
"in",
"lengthycolor/lengthx.",
"Also,",
"distinguishing",
"two",
"lengths",
"in",
"splitVectors",
"might",
"be",
"less",
"efficient",
"than",
"(c)",
"COMP",
"task:",
"Which",
"magnitude",
"is",
"larger,",
"point",
"A",
"or",
"point",
"B?",
"length",
"texture.",
"H3",
"could",
"be",
"supported",
"because",
"redun-",
"(answer:",
"A",
"on",
"the",
"right.)",
"dancy",
"increased",
"information",
"processing",
"capacity",
"[10].",
"Re-",
"dundancy",
"contributes",
"to",
"efficiency",
"by",
"increasing",
"the",
"feature",
"Fig.",
"3:",
"Experiment",
"I:",
"Local",
"discrimination",
"and",
"comparison",
"distances",
"between",
"exponents.",
"We",
"did",
"not",
"expect",
"accuracy",
"tasks.",
"These",
"two",
"red",
"equilateral",
"triangles",
"are",
"rendered",
"on",
"gain",
"from",
"redundancy",
"because",
"splitVectors",
"achieved",
"the",
"the",
"screen",
"coordinate",
"and",
"are",
"thus",
"always",
"visible.",
"same",
"level",
"of",
"accuracy",
"as",
"reading",
"texts",
"in",
"Zhao",
"et",
"al.",
"[3].",
"It",
"may",
"not",
"be",
"useful",
"to",
"decode",
"quantitative",
"data",
"in",
"this",
"experiment",
"at",
"least",
"for",
"showing",
"a",
"few",
"items.",
"quantitative",
"task",
"[29].",
"Participants",
"could",
"either",
"compare",
"the",
"glyph",
"shapes",
"or",
"decipher",
"each",
"vector",
"magnitude",
"and",
"compute",
"the",
"ratio",
"mentally.",
"3.1.3",
"Tasks",
"Participants",
"performed",
"the",
"following",
"three",
"task",
"types",
"as",
"in",
"Exp",
"I.",
"Task",
"3",
"(COMP):",
"comparison",
"(Figure",
"3c).",
"Which",
"Zhao",
"et",
"al.",
"[3]",
"so",
"that",
"results",
"were",
"comparable.",
"They",
"had",
"magnitude",
"is",
"larger,",
"point",
"A",
"or",
"B?",
"Two",
"vectors",
"are",
"marked",
"with",
"red",
"triangles",
"and",
"labeled",
"“A”",
"and",
"\"B",
".",
"Participants",
".",
"unlimited",
"time",
"to",
"perform",
"these",
"three",
"tasks.",
"select",
"their",
"answer",
"by",
"directly",
"clicking",
"the",
"“A”",
"or",
"“B”",
"Exp",
"I.",
"Task",
"1",
"(MAG):",
"magnitude",
"reading",
"(Figure",
"3a).",
"What",
"is",
"the",
"magnitude",
"at",
"point",
"A?",
"One",
"vector",
"was",
"marked",
"by",
"answer",
"buttons.",
"This",
"task",
"was",
"a",
"simple",
"comparison",
"between",
"a",
"red",
"triangle",
"labeled",
"“A”,",
",",
"and",
"participants",
"should",
"report",
"two",
"values",
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"offered",
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"binary",
"choice",
"of",
"large",
"or",
"small.",
"the",
"magnitude",
"of",
"that",
"vector.",
"This",
"task",
"required",
"precise",
"numerical",
"input.",
"3.1.4",
"Data",
"Selection",
"Exp",
"I.",
"Task",
"2",
"(RATIO):",
"ratio",
"estimation",
"(Figure",
"3b).",
"What",
"is",
"the",
"ratio",
"of",
"magnitudes",
"of",
"points",
"A",
"and",
"B?",
"Two",
"vectors",
"Because",
"we",
"were",
"also",
"interested",
"in",
"comparing",
"our",
"results",
"are",
"marked",
"with",
"two",
"red",
"triangles",
"labeled",
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"\"B'",
",",
"and",
"to",
"those",
"in",
"Zhao",
"et",
"al.",
"[3],",
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"replicated",
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"..",
"participants",
"should",
"estimate",
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} | {
"word": [
"JOURNAL",
"OF",
"LATEX",
"CLASS",
"FILES,",
"VOL.",
"14,",
"NO.",
"8,",
"AUGUST",
"2015",
"6",
"TABLE",
"1:",
"Experiment",
"I",
"design:",
"20",
"participants",
"are",
"as-",
"their",
"tasks",
"and",
"were",
"interviewed",
"for",
"their",
"comments.",
"Pilot",
"signed",
"to",
"one",
"of",
"the",
"five",
"blocks",
"and",
"use",
"all",
"five",
"bivari-",
"studies",
"were",
"conducted",
"to",
"examine",
"the",
"procedures.",
"ate",
"pairs.",
"Here,",
"L",
"y",
"L",
"lengthylengthy",
"(splitVectors),",
"L",
"y",
",L",
"Environment.",
"Participants",
"sat",
"at",
"a",
"27",
"d:48",
"(ci",
":4",
"lengthylengthx,",
"LC:",
"and",
"LCL:",
"length",
"color/length.",
"color,",
"LT:",
"lengthytexture,",
"2720Z,",
"gamma-corrected",
"display",
"with",
"resolution",
"1920",
"x",
"1080",
"to",
"ensure",
"the",
"colors",
"were",
"displayed",
"properly.",
"The",
"distance",
"between",
"the",
"participants",
"and",
"the",
"display",
"was",
"about",
"50cm.",
"The",
"minimum",
"visual",
"angle",
"of",
"task-associated",
"glyphs",
"Block",
"Participant",
"Feature-pair",
"P1,",
"P6,",
"P11,",
"P16",
"splitVectors,",
"Ly",
"Lr,",
"LC,",
"LT,",
"LCL",
"was",
"0.2◦",
"in",
"the",
"default",
"view",
"where",
"all",
"data",
"points",
"were",
"P2,",
"P7,",
"P12,",
"P17",
"Ly",
"Lr,",
"LC,",
"LT,",
"LCL,",
"splitVectors",
"visible",
"and",
"the",
"scene",
"filled",
"the",
"screen.",
"P3,",
"P8,",
"P13,",
"P18",
"LC,",
"LT,",
"LCL,",
"splitVectors,",
",Lx",
"Interaction.",
"Participants",
"could",
"rotate",
"the",
"data",
"and",
"zoom",
"P4,",
"P9,",
"P14,",
"P19",
"LT,",
"LCL,",
"splitVectors,",
"P5,",
"P10,",
"P15,",
"P20",
"LCL,",
"splitVectors,",
"o,",
"LC,",
"in",
"and",
"out.",
"Lighting",
"placement",
"and",
"intensity",
"were",
"chosen",
"to",
"produce",
"visualization",
"with",
"contrast",
"and",
"lighting",
"properties",
"appropriate",
"for",
"human",
"assumptions",
"and",
"the",
"spatial",
"data.",
"x",
"3",
"x",
"3.",
"There",
"were",
"445",
"to",
"455",
"sampling",
"locations",
"in",
"each",
"The",
"screen",
"background",
"color",
"was",
"neutral",
"stimulus-free",
"gray",
"5",
"background",
"to",
"minimize",
"the",
"discriminability",
"and",
"appear-",
"selected",
"data",
"region.",
"We",
"selected",
"the",
"data",
"satisfying",
"the",
"same",
"following",
"con-",
"ance",
"of",
"colors",
"[10].",
"Using",
"black",
"or",
"white",
"background",
"colors",
"ditions:",
"(1)",
"the",
"answers",
"must",
"be",
"at",
"locations",
"where",
"some",
"makes",
"the",
"black",
"and",
"white",
"texture",
"stimuli",
"disappear",
"thus",
"context",
"information",
"was",
"available,",
"i.e.,",
"not",
"too",
"close",
"to",
"bias",
"the",
"results",
"(See",
"Appendix",
"B",
"for",
"examples).",
"the",
"boundary",
"of",
"the",
"testing",
"data;",
"(2)",
"no",
"data",
"sample",
"was",
"repeated",
"to",
"the",
"same",
"participant;",
"(3)",
"since",
"data",
"must",
"include",
"a",
"broad",
"measurement,",
"we",
"selected",
"the",
"task-relevant",
"data",
"3.2",
"Experiment",
"Results",
"and",
"Discussion",
"from",
"each",
"exponential",
"term",
"of",
"to",
"3.",
"3.2.1",
"Analysis",
"Approaches",
"3.1.5",
"Empirical",
"Study",
"Design",
"We",
"collected",
"400",
"data",
"points",
"for",
"each",
"task.",
"In",
"preparing",
"Design",
"and",
"Order",
"of",
"Trials.",
"We",
"used",
"a",
"within-subject",
"de-",
"the",
"accuracy",
"and",
"task",
"completion",
"time",
"for",
"analysis,",
"we",
"sign",
"with",
"one",
"independent",
"variable",
"of",
"bivariate",
"quantitative",
"differentiated",
"two",
"error",
"metrics",
"related",
"to",
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"feature-pair",
"(five",
"types).",
"Dependent",
"variables",
"were",
"error",
"accuracy",
"of",
"the",
"bivariate",
"pairs:",
"and",
"task",
"completion",
"time.",
"We",
"also",
"collected",
"participants’",
"Correspondence",
"error",
"(C-Error):",
"A",
"trial",
"is",
"considered",
"to",
"confidence",
"levels.",
"Table",
"1",
"showed",
"that",
"participants",
"were",
"have",
"an",
"answer",
"of",
"C-Error",
"if",
"response’s",
"exponent",
"value",
"assigned",
"into",
"five",
"blocks",
"in",
"a",
"Latin-square",
"order,",
"and",
"within",
"does",
"not",
"match",
"the",
"correct",
"one.",
"Having",
"a",
"C-Error",
"would",
"one",
"block",
"the",
"order",
"of",
"the",
"five",
"feature-pair",
"types",
"is",
"the",
"mean",
"that",
"participants",
"have",
"trouble",
"differentiating",
"the",
"same.",
"Participants",
"performed",
"tasks",
"with",
"randomly",
"selected",
"exponent",
"levels",
"within",
"a",
"glyph.",
"R-Error",
"follows",
"Zhao",
"et",
"datasets.",
"Each",
"participant",
"performed",
"60",
"trials",
"(3",
"tasks",
"x",
"4",
"Relative",
"error",
"(R-Error):",
"This",
"random",
"data",
"x",
"5",
"feature-pairs).",
"These",
"four",
"random",
"data",
"al.",
"3",
"to",
"study",
"how",
"sensitive",
"a",
"method",
"is",
"to",
"error",
"were",
"from",
"four",
"exponent",
"ranges.",
"participant",
"pool",
"as",
"much",
"uncertainty",
"based",
"on",
"fractional",
"uncertainty,",
"calculated",
"Participants.",
"We",
"diversified",
"the",
"as",
"R-Error",
"=",
"correct",
"answer",
"participant",
"answer",
"(correct",
"as",
"possible,",
"since",
"all",
"tasks",
"could",
"be",
"carried",
"out",
"by",
"those",
"answer).",
"This",
"measure",
"was",
"used",
"for",
"MAG",
"and",
"RATIO",
"with",
"only",
"some",
"science",
"background.",
"Twenty",
"participants",
"tasks.",
"The",
"benefit",
"of",
"this",
"metric",
"was",
"that",
"it",
"took",
"into",
"(15",
"male",
"and",
"5",
"female,",
"mean",
"age",
"=",
"23.3,",
"and",
"standard",
"account",
"the",
"value",
"of",
"the",
"quantity",
"being",
"compared",
"and",
"deviation",
"=",
"4.02)",
"participated",
"in",
"the",
"study,",
"with",
"ten",
"in",
"com-",
"thus",
"provided",
"an",
"accurate",
"view",
"of",
"the",
"overall",
"errors.",
"puter",
"science,",
"three",
"in",
"engineering,",
"two",
"in",
"chemistry,",
"one",
"in",
"physics,",
"one",
"in",
"linguistics,",
"one",
"in",
"business",
"administration,",
"In",
"subsequent",
"analysis,",
"we",
"separated",
"these",
"two",
"error",
"one",
"double-major",
"in",
"computer",
"science",
"and",
"math,",
"and",
"one",
"measurements",
"since",
"Combining",
"these",
"two",
"errors",
"in",
"the",
"double-major",
"in",
"biology",
"and",
"psychology.",
"The",
"five",
"females",
"analysis",
"would",
"also",
"be",
"problematic.",
"The",
"C-Errors",
"are",
"at",
"were",
"placed",
"in",
"each",
"of",
"the",
"five",
"blocks",
"(Table",
"1).",
"On",
"average,",
"least",
"one",
"order",
"of",
"magnitude",
"larger",
"or",
"smaller",
"than",
"the",
"ground",
"truth.",
"We",
"also",
"did",
"not",
"remove",
"participants’",
"data",
"participants",
"spent",
"about",
"40",
"minutes",
"on",
"the",
"tasks.",
"Procedure.",
"Participants",
"were",
"greeted",
"and",
"completed",
"with",
"C-Errors,",
"since",
"the",
"source",
"of",
"errors",
"was",
"caused",
"by",
"an",
"Institutional",
"Review",
"Board",
"(IRB)",
"consent",
"form",
"(which",
"glyph",
"design",
"methods",
"independent",
"of",
"trials.",
"described",
"the",
"procedure,",
"risks",
"and",
"benefits",
"of",
"the",
"study)",
"A",
"post-hoc",
"analysis",
"using",
"Tukey’s",
"Studentized",
"Range",
"and",
"the",
"demographic",
"survey.",
"All",
"participants",
"had",
"nor-",
"test",
"(HSD)",
"was",
"performed",
"when",
"we",
"observed",
"significant",
"mal",
"or",
"corrected-to-normal",
"vision",
"and",
"passed",
"the",
"Ishihara",
"main",
"effect",
"on",
"R-Errors.",
"When",
"the",
"dependent",
"variable",
"was",
"color-blindness",
"test.",
"We",
"showed",
"feature-pair",
"examples",
"and",
"binary",
"(i.e.,",
"answer",
"correct",
"or",
"wrong),",
"we",
"used",
"logistic",
"trained",
"the",
"participants",
"with",
"one",
"trial",
"for",
"every",
"feature-pair",
"regression",
"and",
"reported",
"the",
"p",
"value",
"from",
"the",
"Wald",
"χ2",
"test.",
"per",
"task.",
"They",
"were",
"told",
"to",
"be",
"as",
"accurate",
"and",
"as",
"quickly",
"as",
"When",
"the",
"p",
"value",
"was",
"less",
"than",
"0.05,",
"variable",
"levels",
"with",
"possible,",
"and",
"that",
"accuracy",
"was",
"more",
"important",
"than",
"time.",
"95%",
"confidence",
"interval",
"of",
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"not",
"overlapping",
"were",
"They",
"could",
"ask",
"questions",
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"training",
"but",
"were",
"told",
"considered",
"significantly",
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"error",
"bars",
"represent",
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"not",
"do",
"so",
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"the",
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"study.",
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"95%",
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"We",
"also",
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"effect",
"sizes",
"practiced",
"until",
"they",
"fully",
"understood",
"the",
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"and",
"using",
"eta-square,",
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"(0.01",
"ai",
"0.06),",
"“medium”",
"tasks.",
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"formal",
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"participants",
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"in",
"a",
"post-",
"[0.06",
"−",
"0.14),",
"and",
"“large”",
"(",
"2",
"0.14)",
"effects",
"following",
"Co-",
"questionnaire",
"asking",
"how",
"these",
"feature",
"pairs",
"supported",
"hen",
"[39]."
]
} | |||
2301.00002v1.pdf_7 | null | ArxivOCR_part_1 | {
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} | |||
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"[41],",
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"[19],",
"[24],",
"[43].",
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"speed."
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} | |||
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1435,
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1509,
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1122,
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1189,
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1189,
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1262,
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1262,
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1313,
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1313,
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1391,
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1391,
1975,
1452,
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"186,",
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} | {
"word": [
"10",
"JOURNAL",
"OF",
"LATEX",
"CLASS",
"FILES,",
"VOL.",
"14,",
"NO.",
"8,",
"AUGUST",
"2015",
"-4",
"-4",
"-5",
"-5",
"-6",
"-6",
"-7",
"-8",
"-8",
"-9",
"-9",
"1-10",
"-10",
"-11",
"-11",
"L",
"(a)",
"Continuous",
"colormap",
"and",
"high-density",
"data",
"(b)",
"Continuous",
"colormap",
"and",
"low-density",
"data",
"-4",
"-7",
"-10",
"-11",
"U",
"!L",
"(c)",
"Categorical",
"colormap",
"and",
"high-density",
"data",
"(d)",
"Categorical",
"colormap",
"and",
"low-density",
"data",
"Fig.",
"7:",
"Density",
"effects",
"on",
"color",
"choices",
"to",
"justify",
"the",
"use",
"of",
"dense",
"sampling",
"and",
"categorical",
"colormap",
"(c)",
"in",
"Experiment",
"II.",
"This",
"example",
"dataset",
"shows",
"two",
"colormaps:",
"(",
"segmented-continuous",
"(a",
"and",
"b)",
"and",
"categorical",
"(c",
"and",
"d)",
"colormaps),",
"at",
"two",
"different",
"data",
"densities.",
"(a)",
"and",
"(c)",
"show",
"data",
"with",
"the",
"raw",
"density",
"from",
"the",
"simulation",
"results;",
"(b)",
"and",
"(d)",
"were",
"produced",
"by",
"removing",
"around",
"70%",
"vector",
"glyphs.",
"The",
"boundaries",
"between",
"the",
"data",
"categories",
"are",
"more",
"recognizable",
"when",
"the",
"data",
"are",
"dense",
"in",
"(a)",
"and",
"(c)",
"(comparing",
"the",
"1st",
"column",
"and",
"the",
"2nd",
"column).",
"At",
"the",
"same",
"density",
"(comparing",
"the",
"1st",
"and",
"2nd",
"row),",
"the",
"boundaries",
"between",
"levels",
"are",
"easier",
"to",
"recognize",
"when",
"spin",
"vectors",
"are",
"rendered",
"using",
"a",
"categorical",
"colormap",
"of",
"(c)",
"and",
"(d).",
"We",
"thus",
"use",
"the",
"raw",
"dense",
"and",
"categorical",
"colormaps",
"(c)",
"in",
"Experiment",
"II.",
"4.2.5",
"Data",
"Choices",
"These",
"modifications",
"represented",
"less",
"than",
"4%",
"of",
"overall",
"data",
"Data",
"were",
"first",
"sampled",
"using",
"the",
"same",
"approach",
"as",
"Exper-",
"points",
"in",
"each",
"scene.",
"Finally,",
"It",
"improves",
"the",
"reuse",
"of",
"our",
"iment",
"I,",
"and",
"no",
"data",
"is",
"used",
"repeatedly",
"in",
"this",
"experiment.",
"study",
"results.",
"We",
"then",
"modified",
"the",
"exponent",
"range",
"from",
"3",
"to",
"7",
"for",
"the",
"three",
"tasks",
"by",
"normalizing",
"the",
"data",
"to",
"the",
"desired",
"new",
"data",
"4.2.6",
"Empirical",
"Study",
"Design",
"Dependent",
"and",
"Independent",
"Variables.",
"We",
"used",
"a",
"within-",
"range.",
"Prior",
"literature",
"used",
"both",
"synthetic",
"data",
"and",
"real-world",
"subject",
"design",
"with",
"two",
"independent",
"variables",
"of",
"feature-",
"data",
"to",
"construct",
"the",
"data",
"visualization",
"as",
"test",
"scenarios,",
"en-",
"pair",
"(three",
"levels:",
"baseline",
"splitVectors,",
"length",
"y",
"ol",
"7).",
"The",
"abling",
"tight",
"control",
"over",
"the",
"stimulus",
"parameters",
"(e.g.,",
"[44]).",
"lengthytexture)",
"and",
"exponent",
"range",
"(five",
"levels:",
"3",
"and",
"Most",
"of",
"the",
"synthetic",
"data",
"in",
"literature",
"were",
"to",
"replicate",
"dependent",
"variable",
"was",
"relative",
"error.",
"We",
"did",
"not",
"measure",
"real-world",
"data",
"characteristics;",
"and",
"others",
"were",
"explained",
"time",
"since",
"all",
"tasks",
"were",
"time-constrained.",
"x",
"5",
"(magnitude-",
"in",
"fictitious",
"use",
"scenarios.",
"The",
"goal",
"was",
"primarily",
"to",
"prevent",
"ranges)",
"x",
"3",
"(repetitions)",
"=",
"45",
"trials",
"for",
"the",
"first",
"two",
"tasks.",
"preconceived",
"user",
"knowledge",
"about",
"the",
"domain-specific",
"Participants",
"performed",
"3",
"(feature-pairs)",
"attributes.",
"As",
"a",
"result,",
"the",
"synthetic",
"data",
"strike",
"the",
"right",
"bal-",
"Three",
"repetitions",
"were",
"used",
"to",
"give",
"participants",
"enough",
"ance",
"between",
"real-world",
"uses",
"and",
"the",
"data",
"characteristics.",
"time",
"to",
"develop",
"strategies.",
"For",
"NUMEROSITY",
"tasks,",
"the",
"In",
"our",
"cases,",
"replicating",
"characteristics",
"in",
"quantum",
"design",
"runs",
"4",
"repetitions,",
"resulting",
"in",
"3",
"(feature-pairs)",
"par-",
"physics",
"data",
"was",
"challenging",
"and",
"indeed",
"impossible,",
"since",
"5",
"(exponent-ranges)",
"x",
"4",
"(repetitions)",
"=",
"60",
"trials.",
"Each",
"atom",
"behaviors",
"in",
"high-dimensional",
"space",
"were",
"largely",
"ticipant",
"thus",
"executed",
"45",
"+",
"45",
"+",
"60",
"=",
"150",
"trials.",
"Completing",
"unknown",
"and",
"thus",
"were",
"not",
"easily",
"simulated.",
"Our",
"approach",
"all",
"tasks",
"took",
"about",
"32",
"minutes.",
"was",
"therefore",
"to",
"randomly",
"sample",
"quantum",
"physics",
"simu-",
"Self-Reporting",
"Strategies.",
"Several",
"human-computer",
"inter-",
"lation",
"results",
"to",
"capture",
"domain-specific",
"attributes",
"and",
"then",
"action",
"(HCI)",
"approaches",
"can",
"help",
"observe",
"users’",
"behaviors.",
"modify",
"the",
"data",
"to",
"suit",
"evaluation",
"purposes.",
"We",
"showed",
"Answering",
"questions",
"can",
"assist",
"us",
"to",
"determine",
"not",
"just",
"our",
"data",
"to",
"our",
"physicist",
"collaborators",
"to",
"ensure",
"their",
"va-",
"which",
"technique",
"is",
"better",
"but",
"also",
"the",
"strategies",
"humans",
"lidity.",
"We",
"confirmed",
"that",
"these",
"modifications",
"preserved",
"the",
"adopt.",
"For",
"example,",
"cognitive",
"walkthrough",
"(CTW)",
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"in",
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"Architecture:",
"Emergent",
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"Interactions",
"with",
"Human",
"Emotion",
"Junya",
"Morita",
"j-morita@inf.shizuoka.ac.jp",
"Shizuoka",
"University",
"Hamamatsu,",
"Shizuoka,",
"Japan",
"ABSTRACT",
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"internal",
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"surrounding",
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"[4,",
"9].",
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"Cn",
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"processes",
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"the",
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"computing",
"→",
"HCI",
"theory,",
"concepts",
"and",
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"(1)",
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"not",
"a",
"static",
"entity",
"but",
"rather",
"emergent",
"properties",
"accompanied",
"by",
"dynamic",
"interaction",
"with",
"the",
"environment.",
"KEYWORDS",
"This",
"means",
"human’s",
"emotional",
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"always",
"fluctuates",
"(2)",
"Our",
"environment",
"has",
"drastically",
"changed",
"from",
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"cognitive",
"architecture,",
"emotion,",
"arousal,",
"valence",
"ment",
"in",
"which",
"our",
"ancestors",
"lived.",
"Therefore,",
"many",
"emo-",
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"mechanisms",
"have",
"become",
"maladaptive",
"in",
"the",
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"Aa",
"1",
"INTRODUCTION",
"age.",
"(3)",
"This",
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"somehow",
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"an",
"inten-",
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"the",
"following",
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"strategy,",
"as",
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"have",
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"How",
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"emerge",
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"a",
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"system?",
"[3,",
"4].",
"Therefore,",
"there",
"is",
"room",
"for",
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"in",
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"This",
"is",
"kind",
"of",
"ultimate",
"question",
"that",
"has",
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"enormous",
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"by",
"our",
"will",
"or",
"technology.",
"numbers",
"of",
"scientists",
"and",
"engineers,",
"including",
"the",
"author",
"himself.",
"Toward",
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"3",
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"as",
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"in",
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"[16]’s",
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"model",
"(Fig",
"1).",
"The",
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"By",
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"According",
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"[17],",
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"every",
"fiel",
"of",
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"that",
"the",
"above",
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"of",
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"state",
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"or",
"cortical",
"respon-",
"tecture",
"is",
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"to",
"answering",
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"Thus,",
"this",
"document",
"siveness,",
"associated",
"with",
"sensory",
"stimulation",
"and",
"ac-",
"tivation",
"of",
"fiber",
"fro",
"the",
"reticula",
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"Permission",
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"make",
"digital",
"or",
"hard",
"copies",
"of",
"part",
"or",
"all",
"of",
"this",
"work",
"for",
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"or",
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"indicated",
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"past",
"researcher",
"have",
"frequentl",
"con-",
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"is",
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"fee",
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"distributed",
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"especially",
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"commercia",
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"that",
"copie",
"bear",
"this",
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"the",
"full",
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"of",
"thi",
"work",
"must",
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"honored",
"the",
"degree",
"of",
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"(e.g.,",
"[8]).",
"In",
"a",
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"For",
"all",
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"contact",
"the",
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"humans",
"can",
"continue",
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"tasks",
"accurately.",
"However,",
"as",
"CHAI",
"’22,",
"December",
"5,",
"2022,",
"Christchurch,",
"New",
"Zealand",
"such",
"a",
"process",
"long",
"lasts",
"long,",
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"get",
"bored",
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"to",
"think",
"�",
"2022",
"Copyright",
"held",
"by",
"the",
"owner/author(s).",
"about",
"things",
"outside",
"of",
"the",
"task",
"(i.e.,",
"mind-wandering)."
]
} | |||
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"word": [
"CHAI",
"’22,",
"December",
"5,",
"2022,",
"Christchurch,",
"New",
"Zealand",
"Morita",
"High",
"arousal",
"User",
"Model",
"Stress",
"Stress",
"Menes",
"A",
"coo",
"Unpleasant",
"Rewards",
"J",
"Cini",
"Relax",
"Relax",
"Synchronize",
"�",
"Counterbalance",
"Low",
"arousal",
"Figure",
"1:",
"Axes",
"of",
"emotional",
"affec",
"and",
"model",
"parameters",
"Figure",
"2:",
"Framework",
"of",
"interacting",
"human",
"emotion",
"with",
"machine",
"emotion",
"[11]",
"From",
"this",
"phenomenon",
"and",
"the",
"consideration",
"that",
"emotional",
"arousal",
"relates",
"to",
"biological",
"fluctuation",
"affecti",
"the",
"cognitive",
"process,",
"the",
"author",
"and",
"their",
"colleagues",
"have",
"represented",
"arousal",
"as",
"the",
"noise",
"factor",
"for",
"memory",
"activation",
"in",
"ACT-R.",
"Controlling",
"this",
"with",
"humans.",
"By",
"interacting",
"with",
"humans,",
"the",
"system",
"is",
"able",
"to",
"subsymbolic",
"parameter,",
"the",
"authors",
"have",
"demonstrated",
"changes",
"in",
"learn",
"human’s",
"emotion",
"generation",
"and",
"expression.",
"memory",
"recollection",
"[10]",
"and",
"task",
"goal",
"switching",
"[14].",
"This",
"view",
"Based",
"on",
"this",
"consideration,",
"the",
"author",
"and",
"colleagues",
"have",
"is",
"consistent",
"with",
"the",
"discussion",
"in",
"which",
"emotion",
"is",
"a",
"modulator",
"developed",
"several",
"interactive",
"systems",
"[6,",
"11],",
"implementing",
"an",
"emotion",
"model",
"as",
"component.",
"Those",
"systems",
"receive",
"the",
"user’s",
"of",
"cognitive",
"architecture",
"[5,",
"15].",
"biological",
"signals",
"such",
"as",
"heart",
"rates",
"to",
"automatically",
"modulate",
"3.2",
"Representation",
"of",
"Pleasantness",
"the",
"abovementioned",
"ACT-R",
"parameters",
"(noises",
"and",
"rewards)",
"for",
"guiding",
"the",
"user’s",
"emotion",
"to",
"an",
"optimal",
"state.",
"Especially,",
"Morita",
"et",
"People",
"feel",
"joy",
"when",
"receiving",
"a",
"reward.",
"Thus,",
"the",
"pleasantness",
"axis",
"can",
"be",
"considered",
"as",
"a",
"reward",
"in",
"reinforcement",
"learning.",
"Various",
"al.",
"[11]",
"demonstrated",
"that",
"a",
"web",
"advertisement",
"system",
"containing",
"triggering",
"events",
"for",
"rewards",
"can",
"be",
"assumed",
"in",
"the",
"real",
"world.",
"an",
"ACT-R",
"memory",
"model",
"could",
"prevent",
"human",
"repetitive",
"thinking",
"(rumination)",
"when",
"the",
"model",
"behaves",
"in",
"a",
"counterbalanced",
"manner",
"Among",
"them,",
"internally",
"generated",
"ones",
"are",
"important",
"to",
"developing",
"(Fig",
"2).",
"The",
"author",
"believes",
"that",
"such",
"a",
"system",
"will",
"eventually",
"lead",
"autonomous",
"agents.",
"to",
"a",
"new",
"human",
"homeostatic",
"process",
"with",
"the",
"help",
"of",
"artificia",
"Based",
"on",
"the",
"above",
"assumptions,",
"Nagashima",
"et",
"al.",
"[12,",
"13]",
"devel-",
"oped",
"a",
"model",
"of",
"intrinsic",
"motivation,",
"assuming",
"pattern",
"discovery",
"emotional",
"systems.",
"to",
"be",
"a",
"source",
"of",
"curiosity.",
"In",
"the",
"ACT-R",
"model,",
"patterns",
"in",
"the",
"data",
"are",
"discovered",
"with",
"the",
"process",
"of",
"pattern",
"matching,",
"and",
"the",
"5",
"CONCLUSION",
"experience",
"of",
"the",
"pattern",
"matching",
"is",
"utilized",
"to",
"build",
"procedural-",
"This",
"document",
"presents",
"the",
"author’s",
"attempts",
"to",
"answer",
"the",
"ul-",
"ized",
"rules",
"(the",
"compilation",
"of",
"production",
"rules).",
"As",
"a",
"task",
"proceeds,",
"timate",
"question",
"about",
"the",
"computational",
"conditions",
"that",
"enable",
"opportunities",
"for",
"pattern",
"matching",
"(internal",
"rewards)",
"gradually",
"emotion.",
"Future",
"work",
"must",
"represent",
"the",
"ideas",
"presented",
"in",
"the",
"decrease.",
"Thus,",
"the",
"model",
"can",
"explain",
"how",
"intrinsic",
"motivation",
"document",
"as",
"a",
"general",
"framework",
"and",
"evaluate",
"it",
"in",
"human",
"exper-",
"decreases",
"with",
"experience",
"and",
"increases",
"with",
"discovering",
"novel",
"iments.",
"The",
"author",
"considers",
"that",
"such",
"a",
"cognitive-architecture-",
"patterns.",
"This",
"pattern-discovery-focused",
"view",
"of",
"intrinsic",
"motiva-",
"tion",
"is",
"consistent",
"with",
"discussions",
"in",
"the",
"entertainment",
"industry",
"agent",
"relations.",
"Academic",
"knowledge",
"implemented",
"in",
"architecture",
"[7].",
"In",
"addition,",
"it",
"is",
"supported",
"by",
"the",
"theory",
"emphasizing",
"the",
"role",
"is",
"the",
"result",
"of",
"continuous",
"endeavors",
"in",
"human",
"history.",
"Including",
"based",
"framework",
"is",
"advantageous",
"in",
"constructing",
"trustful",
"human-",
"of",
"pattern-seeking",
"in",
"the",
"history",
"of",
"human",
"civilization",
"[2].",
"the",
"formal",
"knowledge",
"agreed",
"in",
"human",
"society",
"is",
"an",
"essential",
"in-",
"gredient",
"of",
"making",
"common",
"ground",
"between",
"humans",
"and",
"artifacts.",
"4",
"NEEDS",
"OF",
"INTERACTION",
"From",
"the",
"discussion",
"so",
"far,",
"the",
"importance",
"of",
"environment",
"for",
"emotion",
"stands",
"out.",
"Capturing",
"the",
"full",
"complexities",
"of",
"the",
"real-",
"ACKNOWLEDGEMENT",
"world",
"dynamics",
"of",
"the",
"two",
"mechanisms",
"(arousal",
"and",
"pleasantness)",
"This",
"document",
"summarizes",
"ideas",
"obtained",
"from",
"past",
"collaborative",
"studies",
"with",
"colleagues",
"at",
"Nagoya",
"University,",
"Shizuoka",
"University,",
"requires",
"environmental",
"changes.",
"Among",
"various",
"environmental",
"factors,",
"the",
"existence",
"of",
"other",
"organisms",
"seems",
"most",
"crucial.",
"In",
"collaborators",
"from",
"Panasonic",
"Corp.",
"and",
"Mazda",
"Corp.,",
"and",
"mem-",
"other",
"words,",
"the",
"author",
"considers",
"that",
"human",
"emotion",
"can",
"be",
"bers",
"of",
"the",
"Applied",
"Cognitive",
"Modeling",
"Lab",
"(ACML)",
"at",
"Shizuoka",
"modeled",
"only",
"when",
"the",
"computational",
"system",
"actually",
"interacts",
"university.",
"The",
"author",
"thanks",
"everyone",
"for",
"valuable",
"discussions."
]
} | |||
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} | |||
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} | {
"word": [
"Effects",
"of",
"Data",
"Geometry",
"in",
"Early",
"Deep",
"Learning",
"George",
"Konidaris",
"Saket",
"Tiwari",
"Department",
"of",
"Computer",
"Science",
"Department",
"of",
"Computer",
"Science",
"Brown",
"University",
"Brown",
"University",
"Providence,",
"RI",
"02906",
"Providence,",
"RI",
"02906",
"saket tiwari@brown.edu",
"200",
"Abstract",
"He",
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"can",
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"on",
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"types",
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"from",
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"to",
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"with",
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"underlying",
"structure",
"can",
"be",
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"as",
"the",
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"of",
"the",
"data",
"manifold.",
"By",
"extending",
"recent",
"advances",
"20",
"in",
"the",
"theoretical",
"understanding",
"of",
"neural",
"networks,",
"we",
"study",
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"a",
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"neural",
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"with",
"piece-wise",
"linear",
"activation",
"splits",
"the",
"data",
"manifold",
"into",
"regions",
"where",
"the",
"neural",
"network",
"behaves",
"as",
"a",
"linear",
"function.",
"We",
"derive",
"G1SE",
"bounds",
"on",
"the",
"density",
"of",
"boundary",
"of",
"linear",
"regions",
"and",
"the",
"distance",
"to",
"these",
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"on",
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"randomly",
"initialized",
"deep",
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"empirically",
"corroborate",
"our",
"theoretical",
"results",
"using",
"a",
"toy",
"supervised",
"learning",
"problem.",
"Our",
"experiments",
"demonstrate",
"that",
"number",
"of",
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"regions",
"varies",
"across",
"manifolds",
"and",
"the",
"results",
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"architectures.",
"We",
"Mas",
"further",
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"how",
"the",
"complexity",
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"different",
"on",
"the",
"low",
"dimensional",
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"images",
"as",
"compared",
"to",
"the",
"Euclidean",
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"using",
"the",
"MetFaces",
"dataset.",
"1",
"Introduction",
"201",
"The",
"capacity",
"of",
"Deep",
"Neural",
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"(DNNs)",
"to",
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"arbitrary",
"functions",
"given",
"sufficient",
"training",
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"in",
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"well",
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"[Cybenko,",
"1989,",
"Hornik",
"et",
"al.,",
"1989,",
"Anthony",
"and",
"Bartlett,",
"1999].",
"Several",
"different",
"theoretical",
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"emerged",
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"study",
"the",
"effectiveness",
"and",
"pitfalls",
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"deep",
"learning.",
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"vary",
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"S",
"and",
"the",
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"study",
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"[Allen-Zhu",
"et",
"al.,",
"2019,",
"Goodfellow",
"and",
"Vinyals,",
"ML",
"2015],",
"generalization",
"[Kawaguchi",
"et",
"al.,",
"2017,",
"Zhang",
"et",
"al.,",
"2017,",
"Jacot",
"et",
"al.,",
"2018,",
"Sagun",
"et",
"al.,",
"2018],",
"function",
"complexity",
"[Mont´ufar",
"et",
"al.,",
"2014,",
"Mhaskar",
"and",
"Poggio,",
"2016],",
"adversarial",
"attacks",
"[Szegedy",
"et",
"al.,",
"2014,",
"Goodfellow",
"et",
"al.,",
"2015]",
"to",
"representation",
"capacity",
"[Arpit",
"et",
"al.,",
"2017].",
"Some",
"recent",
"theories",
"have",
"also",
"been",
"shown",
"to",
"closely",
"match",
"empirical",
"observations",
"[Poole",
"et",
"al.,",
"2016,",
"Hanin",
"and",
"Rolnick,",
"2019b,",
"Kunin",
"et",
"al.,",
"2020].",
"One",
"approach",
"to",
"studying",
"DNNs",
"is",
"to",
"examine",
"how",
"the",
"underlying",
"structure,",
"or",
"geometry,",
"of",
"the",
"data",
"interacts",
"with",
"learning",
"dynamics.",
"The",
"manifold",
"hypothesis",
"states",
"that",
"high-dimensional",
"real",
"world",
"data",
"typically",
"lies",
"on",
"a",
"low",
"dimensional",
"manifold",
"[Tenenbaum,",
"1997,",
"Carlsson",
"et",
"al.,",
"2007,",
"Fefferman",
"et",
"al.,",
"2013].",
"Empirical",
"studies",
"have",
"shown",
"that",
"DNNs",
"are",
"highly",
"effective",
"in",
"deciphering",
"this",
"underlying",
"structure",
"by",
"learning",
"intermediate",
"latent",
"representations",
"[Poole",
"et",
"al.,",
"2016].",
"The",
"ability",
"of",
"DNNs",
"to",
"“flatten”",
"complex",
"data",
"manifolds,",
"using",
"composition",
"of",
"seemingly",
"simple",
"piece-wise",
"linear",
"functions,",
"appears",
"to",
"be",
"unique",
"[Brahma",
"et",
"al.,",
"2016,",
"Hauser",
"and",
"Ray,",
"2017].",
"DNNs",
"with",
"piece-wise",
"linear",
"activations,",
"such",
"as",
"ReLU",
"[Nair",
"and",
"Hinton,",
"2010],",
"divide",
"the",
"input",
"space",
"into",
"linear",
"regions,",
"wherein",
"the",
"DNN",
"behaves",
"as",
"a",
"linear",
"function",
"[Mont´ufar",
"et",
"al.,",
"2014].",
"The",
"density",
"of",
"these",
"linear",
"regions",
"serves",
"as",
"a",
"proxy",
"for",
"the",
"DNN’s",
"ability",
"to",
"interpolate",
"a",
"complex",
"data",
"landscape",
"and",
"has",
"been",
"the",
"subject",
"of",
"detailed",
"studies",
"[Mont´ufar",
"et",
"al.,",
"2014,",
"Telgarsky,",
"2015,",
"Serra",
"36th",
"Conference",
"on",
"Neural",
"Information",
"Processing",
"Systems",
"(NeurIPS",
"2022)."
]
} | |||
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"word": [
"et",
"al.,",
"2018,",
"Raghu",
"et",
"al.,",
"2017].",
"The",
"work",
"by",
"Hanin",
"and",
"Rolnick",
"[2019a]",
"on",
"this",
"topic",
"stands",
"out",
"because",
"they",
"derive",
"bounds",
"on",
"the",
"average",
"number",
"of",
"linear",
"regions",
"and",
"verify",
"the",
"tightness",
"of",
"these",
"bounds",
"empirically",
"for",
"deep",
"ReLU",
"networks,",
"instead",
"of",
"larger",
"bounds",
"that",
"rarely",
"materialize.",
"Hanin",
"and",
"Rolnick",
"[2019a]",
"conjecture",
"that",
"the",
"number",
"of",
"linear",
"regions",
"correlates",
"to",
"the",
"expressive",
"power",
"of",
"randomly",
"initialized",
"DNNs",
"with",
"piece-wise",
"linear",
"activations.",
"However,",
"they",
"assume",
"that",
"the",
"data",
"is",
"uniformly",
"sampled",
"from",
"the",
"Euclidean",
"space",
"R",
"d,",
"for",
"some",
"d.",
"By",
"combining",
"the",
"manifold",
"hypothesis",
"with",
"insights",
"from",
"Hanin",
"and",
"Rolnick",
"[2019a],",
"we",
"are",
"able",
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"go",
"further",
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"estimating",
"the",
"number",
"of",
"linear",
"regions",
"and",
"the",
"average",
"distance",
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"linear",
"boundaries.",
"We",
"derive",
"bounds",
"on",
"how",
"the",
"geometry",
"of",
"the",
"data",
"manifold",
"affects",
"the",
"aforementioned",
"quantities.",
"To",
"corroborate",
"our",
"theoretical",
"bounds",
"with",
"empirical",
"results,",
"we",
"design",
"a",
"toy",
"problem",
"where",
"the",
"input",
"data",
"is",
"sampled",
"from",
"two",
"distinct",
"manifolds",
"that",
"can",
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"represented",
"in",
"a",
"closed",
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"We",
"count",
"the",
"exact",
"number",
"of",
"linear",
"regions",
"and",
"the",
"average",
"distance",
"to",
"the",
"boundaries",
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"linear",
"regions",
"on",
"these",
"two",
"manifolds",
"that",
"a",
"neural",
"network",
"divides",
"the",
"two",
"manifolds",
"into.",
"We",
"demonstrate",
"how",
"the",
"number",
"of",
"linear",
"regions",
"and",
"average",
"distance",
"varies",
"for",
"these",
"two",
"distinct",
"manifolds.",
"These",
"results",
"show",
"that",
"the",
"number",
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"linear",
"regions",
"on",
"the",
"manifold",
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"not",
"grow",
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"the",
"dimension",
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"input",
"data.",
"Our",
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"do",
"not",
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"theoretical",
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"as",
"in",
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"learning",
"theory,",
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"demonstrate",
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"the",
"number",
"of",
"linear",
"regions",
"change",
"as",
"a",
"consequence",
"of",
"these",
"constants.",
"We",
"also",
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"linear",
"regions",
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"deep",
"ReLU",
"networks",
"for",
"high",
"dimensional",
"data",
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"lies",
"on",
"a",
"low",
"dimensional",
"manifold",
"with",
"unknown",
"structure",
"and",
"how",
"the",
"number",
"of",
"linear",
"regions",
"vary",
"on",
"and",
"off",
"this",
"manifold,",
"which",
"is",
"a",
"more",
"realistic",
"setting.",
"To",
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"this",
"we",
"present",
"experiments",
"performed",
"on",
"the",
"manifold",
"of",
"natural",
"face",
"images.",
"We",
"sample",
"data",
"from",
"the",
"image",
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"using",
"a",
"generative",
"adversarial",
"network",
"(GAN)",
"[Goodfellow",
"et",
"al.,",
"2014]",
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"the",
"curated",
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"of",
"paintings.",
"Specifically,",
"we",
"generate",
"images",
"using",
"the",
"pre-trained",
"StyleGAN",
"[Karras",
"et",
"al.,",
"2019,",
"2020b]",
"trained",
"on",
"the",
"curated",
"MetFaces",
"dataset",
"[Karras",
"et",
"al.,",
"2020a].",
"We",
"generate",
"curves",
"on",
"the",
"image",
"manifold",
"of",
"faces,",
"using",
"StyleGAN,",
"and",
"report",
"how",
"the",
"density",
"of",
"linear",
"regions",
"varies",
"on",
"and",
"off",
"the",
"manifold.",
"These",
"results",
"shed",
"new",
"light",
"on",
"the",
"geometry",
"of",
"deep",
"learning",
"over",
"structured",
"data",
"sets",
"by",
"taking",
"a",
"data",
"intrinsic",
"approach",
"to",
"understanding",
"the",
"expressive",
"power",
"of",
"DNNs.",
"2",
"Preliminaries",
"and",
"Background",
"Our",
"goal",
"is",
"to",
"understand",
"how",
"the",
"underlying",
"structure",
"of",
"real",
"world",
"data",
"matters",
"for",
"deep",
"learning.",
"We",
"first",
"provide",
"the",
"mathematical",
"background",
"required",
"to",
"model",
"this",
"underlying",
"structure",
"as",
"the",
"geometry",
"of",
"data.",
"We",
"then",
"provide",
"a",
"summary",
"of",
"previous",
"work",
"on",
"understanding",
"the",
"approximation",
"capacity",
"of",
"deep",
"ReLU",
"networks",
"via",
"the",
"complexity",
"of",
"linear",
"regions.",
"For",
"the",
"details",
"on",
"how",
"our",
"work",
"fits",
"into",
"one",
"of",
"the",
"two",
"main",
"approaches",
"within",
"the",
"theory",
"of",
"DNNs,",
"from",
"the",
"expressive",
"power",
"perspective",
"or",
"from",
"the",
"learning",
"dynamics",
"perspective,",
"we",
"refer",
"the",
"reader",
"to",
"Appendix",
"C.",
"2.1",
"Data",
"Manifold",
"and",
"Definitions",
"Figure",
"1:",
"A",
"2D",
"surface,",
"here",
"represented",
"by",
"a",
"2-torus,",
"is",
"embedded",
"in",
"a",
"larger",
"input",
"space,",
"R",
"3.",
"Suppose",
"each",
"point",
"corresponds",
"to",
"an",
"image",
"of",
"a",
"face",
"on",
"this",
"2-torus.",
"We",
"can",
"chart",
"two",
"curves:",
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"straight",
"line",
"cutting",
"across",
"the",
"3D",
"space",
"and",
"another",
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"that",
"stays",
"on",
"the",
"torus.",
"Images",
"corresponding",
"to",
"the",
"points",
"on",
"the",
"torus",
"will",
"have",
"a",
"smoother",
"variation",
"in",
"style",
"and",
"shape",
"whereas",
"there",
"will",
"be",
"images",
"corresponding",
"to",
"points",
"on",
"the",
"straight",
"line",
"that",
"are",
"not",
"faces.",
"We",
"use",
"the",
"example",
"of",
"the",
"MetFaces",
"dataset",
"[Karras",
"et",
"al.,",
"2020a]",
"to",
"illustrate",
"how",
"data",
"lies",
"on",
"a",
"low",
"dimensional",
"manifold.",
"The",
"images",
"in",
"the",
"dataset",
"are",
"1028",
"1028",
"'x",
"3",
"dimensional.",
"By",
"contrast,",
"the",
"number",
"of",
"realistic",
"dimensions",
"along",
"which",
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"vary",
"are",
"limited,",
"e.g.",
"painting",
"style,",
"artist,",
"size",
"and",
"shape",
"of",
"the",
"nose,",
"jaw",
"and",
"eyes,",
"background,",
"clothing",
"style;",
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"fact,",
"very",
"few",
"1028",
"x",
"1028",
"2"
]
} | |||
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} | {
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"dimensional",
"images",
"correspond",
"to",
"realistic",
"faces.",
"We",
"illustrate",
"how",
"this",
"affects",
"the",
"possible",
"variations",
"in",
"the",
"data",
"in",
"Figure",
"1.",
"A",
"manifold",
"formalises",
"the",
"notion",
"of",
"limited",
"variations",
"in",
"high",
"dimensional",
"data.",
"One",
"can",
"imagine",
"that",
"there",
"exists",
"an",
"unknown",
"function",
"f",
"X",
"Y",
"from",
"a",
"low",
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"space",
"of",
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"to",
"a",
"ne",
"R",
"m,",
"to",
"another",
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"Y",
"Rk,",
"is",
"a",
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"f",
"is",
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"from",
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"open",
"subset",
"X",
"dimensional",
"space",
"of",
"the",
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"points.",
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"f",
":",
"X",
"Siol",
",",
"and",
"both",
"f",
"and",
"f",
"-1",
"are",
"differentiable",
"(or",
"smooth).",
"Therefore,",
"a",
"manifold",
"is",
"defined",
"as",
"follows.",
"Definition",
"2.1.",
"Let",
"k,",
"m",
"E",
"N",
"o",
".",
"A",
"subset",
"M",
"c",
"R",
"k",
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"k)",
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"T_",
"R",
"m.",
"A",
"diffeomorphism",
"(i.e.",
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"mapping),",
"Ù",
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"M",
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"$2",
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"an",
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"neighborhood",
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"a",
"R",
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"U",
"n",
"M",
"�",
"2",
"f",
"is",
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"a",
"coordinate",
"chart",
"of",
"M",
"and",
"the",
"inverse,",
"h",
":=",
"f",
"1",
"n",
"�",
"U",
"n",
"M",
",-",
"is",
"called",
"a",
"smooth",
"parametrization",
"of",
"U",
"n",
"M.",
"For",
"the",
"MetFaces",
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"example,",
"suppose",
"there",
"are",
"10",
"dimensions",
"along",
"which",
"the",
"images",
"vary.",
"Further",
"assume",
"that",
"each",
"variation",
"can",
"take",
"a",
"value",
"continuously",
"x",
"3.",
"This",
"parametrization",
"and",
"its",
"smooth",
"parametrization",
"would",
"map",
"f",
":",
"Ω",
"R",
"10",
"M",
"P",
"R",
"1028",
"x",
"1028",
"some",
"interval",
"of",
"R",
"Then",
"the",
"inverse",
"are",
"unknown",
"in",
"general",
"and",
"computationally",
"very",
"difficult",
"to",
"estimate",
"in",
"practice.",
"There",
"are",
"similarities",
"in",
"how",
"geometric",
"elements",
"are",
"defined",
"for",
"manifolds",
"and",
"Euclidean",
"spaces.",
"A",
"smooth",
"curve,",
"on",
"a",
"manifold",
"M,",
"o",
"I",
"E",
"I,",
"just",
"as",
"for",
"Euclidean",
"spaces.",
"The",
"shortest",
"such",
"curve",
"a",
"function",
"that",
"is",
"differentiable",
"for",
"all",
"t",
"M",
"is",
"defined",
"from",
"an",
"interval",
"I",
"to",
"the",
"manifold",
"M",
"as",
"between",
"two",
"points",
"on",
"a",
"manifold",
"is",
"no",
"longer",
"a",
"straight",
"line,",
"but",
"is",
"instead",
"a",
"geodesic.",
"One",
"recurring",
"geometric",
"element,",
"which",
"is",
"unique",
"to",
"manifolds",
"and",
"stems",
"from",
"the",
"definition",
"of",
"smooth",
"curves,",
"is",
"that",
"of",
"a",
"tangent",
"space,",
"defined",
"as",
"follows.",
"Definition",
"2.2.",
"Let",
"M",
"be",
"an",
"m-manifold",
"in",
"R",
"k",
"and",
"x",
"E",
"M",
"be",
"a",
"fix",
"M",
"such",
"that",
"γ(0)",
"=",
"x,",
"˙γ(0)",
"=",
"a",
"tangent",
"vector",
"of",
"M",
"at",
"x",
"if",
"there",
"exists",
"a",
"smooth",
"curve",
"o",
":",
"I",
"point.",
"A",
"vector",
"v",
"E",
"R",
"k",
"is",
"called",
"where",
"˙γ(t)",
"is",
"the",
"derivative",
"of",
"io",
"at",
"t.",
"The",
"set",
"I",
"xc",
"M",
":=",
"{",
"y(0)",
"I",
"v",
"R",
"�",
"M",
"is",
"smoothγ(0)",
"=",
"x",
"of",
"tangent",
"vectors",
"of",
"M",
"at",
"x",
"is",
"called",
"the",
"tangent",
"space",
"of",
"M",
"at",
"x.",
"In",
"simpler",
"terms,",
"the",
"plane",
"tangent",
"to",
"the",
"manifold",
"M",
"at",
"point",
"x",
"is",
"called",
"the",
"tangent",
"space",
"R",
"3.",
"denoted",
"by",
"by",
"I",
"x",
"M.",
"Consider",
"the",
"upper",
"half",
"of",
"a",
"2-sphere,",
"S2",
"C",
"R",
"3,",
"which",
"is",
"a",
"2-manifold",
"in",
"The",
"tangent",
"space",
"at",
"a",
"fixed",
"point",
"x",
"S2",
"is",
"the",
"2D",
"plane",
"perpendicular",
"to",
"the",
"vector",
"x",
"and",
"tangential",
"to",
"the",
"surface",
"of",
"the",
"sphere",
"that",
"contains",
"the",
"point",
"x.",
"For",
"additional",
"background",
"on",
"manifolds",
"we",
"refer",
"the",
"reader",
"to",
"Appendix",
"B.",
"2.2",
"Linear",
"Regions",
"of",
"Deep",
"ReLU",
"Networks",
"The",
"higher",
"the",
"density",
"of",
"these",
"linear",
"regions",
"the",
"more",
"complex",
"a",
"function",
"DNN",
"can",
"approximate.",
"For",
"example,",
"a",
"sin",
"curve",
"in",
"the",
"range",
"[0,",
"2π]",
"is",
"better",
"approximated",
"by",
"4",
"piece-wise",
"linear",
"regions",
"as",
"opposed",
"to",
"2.",
"To",
"clarify",
"this",
"further,",
"with",
"the",
"4",
"“optimal”",
"linear",
"regions",
"[0,",
"π/2),",
"[π/2,",
"π),",
"[π,",
"3π/2),",
"and",
"[3π/2,",
"2π]",
"a",
"function",
"could",
"approximate",
"the",
"sin",
"curve",
"better",
"than",
"any",
"2",
"linear",
"regions.",
"In",
"other",
"words,",
"higher",
"density",
"of",
"linear",
"regions",
"allows",
"a",
"DNN",
"to",
"approximate",
"the",
"variation",
"in",
"the",
"curve",
"better.",
"We",
"define",
"the",
"notion",
"of",
"boundary",
"of",
"a",
"linear",
"regions",
"in",
"this",
"section",
"and",
"provide",
"an",
"overview",
"of",
"previous",
"results.",
"We",
"consider",
"a",
"neural",
"network,",
"F,",
"which",
"is",
"a",
"composition",
"of",
"activation",
"functions.",
"Inputs",
"at",
"each",
"layer",
"are",
"multiplied",
"by",
"a",
"matrix,",
"referred",
"to",
"as",
"the",
"weight",
"matrix,",
"with",
"an",
"additional",
"bias",
"vector",
"that",
"is",
"added",
"to",
"this",
"product.",
"We",
"limit",
"our",
"study",
"to",
"ReLU",
"activation",
"function",
"[Nair",
"and",
"Hinton,",
"2010],",
"which",
"is",
"piece-wise",
"linear",
"and",
"one",
"of",
"the",
"most",
"popular",
"activation",
"functions",
"being",
"applied",
"to",
"various",
"learning",
"tasks",
"on",
"different",
"types",
"of",
"data",
"like",
"text,",
"images,",
"signals",
"etc.",
"We",
"further",
"consider",
"DNNs",
"that",
"map",
"R",
"is",
"defined",
"as,",
"inputs,",
"of",
"dimension",
"n",
"in",
",",
"to",
"scalar",
"values.",
"Therefore,",
"F",
"Rnin",
"(1)",
"F(x)",
"=",
"W",
"L",
"o(B",
"L",
"_",
"1",
"+",
"W",
"L",
"_",
"1",
"o(.",
"..o(B",
"+",
"W",
"x))),",
"31",
"where",
"Wl",
"l",
"E",
"M",
"nl",
"x",
"nl",
"-1",
"e",
"R",
"ni",
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"the",
"vector",
"of",
"biases",
"for",
"the",
"lth",
"hidden",
"layer,",
"n",
"=",
"n",
"in",
"and",
"o",
":",
"R",
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"/th",
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"l",
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"the",
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"l",
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"number",
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"neurons",
"3"
]
} | |||
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1324,
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293,
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585,
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1369,
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533,
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558,
1949
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558,
1909,
580,
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612,
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988,
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988,
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1017,
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1263,
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1263,
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1341,
1898,
1366,
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1366,
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637,
1893,
659,
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442,
1911,
478,
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362,
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387,
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292,
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351,
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351,
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454,
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454,
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483,
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483,
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601,
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1968,
633,
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633,
1968,
665,
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665,
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710,
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875,
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875,
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917,
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917,
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1967,
1017,
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1017,
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1052,
2010
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1052,
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1073,
2010
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1073,
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1182,
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1182,
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1226,
2010
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1226,
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1265,
2010
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1265,
1967,
1342,
2010
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1359,
1984,
1401,
2014
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2050,
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"approximation",
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"F",
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"the",
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"in",
"a",
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"K",
"c",
"R",
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"K)/vol",
"nin",
"(K),",
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"1)",
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"volume",
"density",
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"x",
"c",
"K",
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"Manifold",
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"Inin",
"(K)",
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"ffe",
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"B",
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"word": [
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"2:",
"A",
"circle",
"is",
"an",
"example",
"of",
"a",
"1D",
"manifold",
"in",
"2D",
"Euclidean",
"space.",
"The",
"effective",
"number",
"of",
"linear",
"regions",
"on",
"the",
"manifold,",
"the",
"upper",
"half",
"of",
"the",
"circle,",
"are",
"the",
"number",
"of",
"linear",
"regions",
"on",
"the",
"arc",
"from",
"i",
"TT",
"to",
"π.",
"In",
"the",
"diagram",
"above,",
"each",
"color",
"in",
"the",
"2D",
"space",
"corresponds",
"to",
"a",
"linear",
"region.",
"When",
"the",
"upper",
"half",
"of",
"the",
"circle",
"is",
"flattened",
"into",
"a",
"1D",
"space",
"we",
"obtain",
"a",
"line.",
"Each",
"color",
"on",
"the",
"line",
"corresponds",
"to",
"a",
"linear",
"region",
"of",
"the",
"2D",
"space.",
"as",
"b",
"n",
"M,",
"and",
"note",
"that",
"dim(B(",
"F,k)",
")",
"=",
"m",
"-",
"k",
"(Appendix",
"E",
"Proposition",
"E.4).",
"For",
"example,",
"the",
"F,k",
"transverse",
"intersection",
"(see",
"Definition",
"E.3)",
"of",
"plane",
"in",
"3D",
"with",
"the",
"2D",
"manifold",
"S2",
"is",
"a",
"1D",
"curve",
"in",
"S2",
"and",
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"has",
"dimension",
"1.",
"Therefore,",
"B(",
"BF,k",
"is",
"a",
"submanifold",
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"3",
"volume.",
"1.",
"This",
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"k",
"si",
"m,",
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"the",
"intersection",
"BF,k",
"5�",
"M",
"to",
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"well",
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"=",
"We",
"first",
"note",
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"the",
"definition",
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"determinant",
"of",
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"for",
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"collection",
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"neurons",
"z",
"1",
"...,",
"z",
"k",
",",
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"when",
"the",
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"on",
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"M",
"as",
"opposed",
"to",
"in",
"a",
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"of",
"dimension",
"n",
"in",
"in",
"IR",
"nin.",
"Since",
"the",
"determinant",
"of",
"the",
"Jacobian",
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"the",
"quantity",
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"utilise",
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"our",
"proofs",
"and",
"theorems",
"repeatedly",
"we",
"will",
"use",
"the",
"term",
"Jacobian",
"to",
"refer",
"to",
"it",
"for",
"succinctness.",
"Intuitively,",
"this",
"follows",
"from",
"the",
"Jacobian",
"of",
"a",
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"being",
"defined",
"differently",
"in",
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"ambient",
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"R",
"nin",
"as",
"opposed",
"to",
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"manifold",
"M.",
"In",
"case",
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"former",
"it",
"is",
"the",
"volume",
"of",
"the",
"paralellepiped",
"determined",
"by",
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"vectors",
"corresponding",
"to",
"the",
"directions",
"with",
"steepest",
"ascent",
"along",
"each",
"one",
"of",
"the",
"n",
"in",
"axes.",
"In",
"case",
"of",
"the",
"latter",
"it",
"is",
"more",
"complex",
"and",
"defined",
"below.",
"Let",
"Hm",
"be",
"the",
"m",
"dimensional",
"Hausdorff",
"measure",
"(we",
"refer",
"the",
"reader",
"to",
"the",
"Appendix",
"B",
"for",
"background",
"on",
"Hausdorff",
"measure).",
"The",
"Jacobian",
"of",
"a",
"function",
"on",
"manifold",
"M,",
"as",
"defined",
"by",
"Krantz",
"and",
"Parks",
"[2008]",
"(Chapter",
"5),",
"is",
"as",
"follows.",
"(determinant",
"of)",
"Jacobian",
"of",
"function",
"H",
":",
"M",
"=",
"R",
"k,",
"where",
"k",
"�",
"dim(Ml)",
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"=",
"m,",
"is",
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"as",
"7",
")",
"(c",
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"a",
"k-dimensional",
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"contained",
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"I",
"x",
"M.",
",",
"JM",
"(x)",
"=",
"sup",
"H",
"k(P",
"where",
"D",
"M",
"T",
"M",
"�",
"R",
"is",
"the",
"differential",
"map",
"(see",
"Appendix",
"B)",
"and",
"we",
"use",
"D",
"m",
"H(P",
")",
"to",
"denote",
"the",
"mapping",
"of",
"the",
"set",
"P",
"in",
"T",
"M,",
"which",
"is",
"a",
"parallelepiped,",
"to",
"R",
"k.",
"The",
"supremum",
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"all",
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"P.",
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"also",
"say",
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"2",
"1",
",",
"...,",
"z",
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"good",
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"x",
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"z",
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"F",
"so",
"that",
"each",
"neuron",
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"activated",
"along",
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"path.",
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"three",
"main",
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"hold",
"under",
"the",
"assumptions",
"listed",
"in",
"Appendix",
"A,",
"each",
"of",
"which",
"extend",
"and",
"improve",
"upon",
"the",
"theoretical",
"results",
"by",
"Hanin",
"and",
"Rolnick",
"[2019a],",
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"F",
"a",
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"ReLU",
"network",
"with",
"input",
"dimension",
"n",
"in",
",",
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"R",
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"-",
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"x",
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"...,",
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"A",
"Pbz1,...,bzk",
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"the",
"density",
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"joint",
"distribution",
"of",
"the",
"biases",
"b",
"z1",
",",
"...,",
"b",
"z k",
".",
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"change",
"in",
"the",
"formula,",
"from",
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"3.4",
"by",
"Hanin",
"and",
"Rolnick",
"[2019a],",
"is",
"a",
"result",
"of",
"the",
"fact",
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"z(x)",
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"a",
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"ascent",
"when",
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"is",
"restricted",
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"data",
"manifold",
"M,",
"for",
"any",
"j.",
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"3",
"also",
"makes",
"explicit",
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"n",
"in",
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"m",
"k-dimensional",
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"a",
"more",
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"3.2,",
"we",
"provide",
"result",
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"the",
"density",
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"on",
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"5"
]
} | |||
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1438,
1097,
1479
],
[
1097,
1448,
1113,
1480
],
[
1118,
1437,
1155,
1478
],
[
1155,
1439,
1169,
1479
],
[
1169,
1450,
1185,
1481
],
[
1185,
1437,
1213,
1479
],
[
1213,
1437,
1303,
1479
],
[
1303,
1437,
1332,
1479
],
[
1332,
1436,
1403,
1479
],
[
293,
1466,
324,
1509
],
[
324,
1466,
365,
1509
],
[
365,
1466,
501,
1509
],
[
501,
1466,
557,
1509
],
[
557,
1466,
598,
1509
],
[
598,
1467,
709,
1509
],
[
709,
1466,
738,
1509
],
[
738,
1466,
779,
1509
],
[
779,
1466,
898,
1509
],
[
898,
1467,
974,
1509
],
[
974,
1466,
1063,
1509
],
[
1063,
1466,
1092,
1509
],
[
1092,
1466,
1133,
1509
],
[
1133,
1466,
1223,
1510
],
[
1223,
1466,
1318,
1509
],
[
1318,
1467,
1357,
1509
],
[
1357,
1466,
1403,
1509
],
[
293,
1497,
337,
1540
],
[
337,
1497,
427,
1540
],
[
427,
1496,
522,
1540
],
[
522,
1497,
621,
1539
],
[
621,
1496,
716,
1540
],
[
716,
1497,
766,
1540
],
[
766,
1497,
844,
1540
],
[
844,
1497,
918,
1540
],
[
918,
1497,
949,
1540
],
[
949,
1497,
992,
1540
],
[
992,
1497,
1013,
1540
],
[
1013,
1497,
1094,
1540
],
[
1094,
1497,
1158,
1540
],
[
1158,
1497,
1327,
1540
],
[
1327,
1497,
1359,
1540
],
[
1359,
1497,
1402,
1540
],
[
325,
1512,
354,
1568
],
[
354,
1530,
377,
1570
],
[
387,
1526,
535,
1570
],
[
535,
1527,
626,
1570
],
[
626,
1527,
657,
1570
],
[
657,
1527,
698,
1570
],
[
698,
1527,
777,
1570
],
[
777,
1527,
907,
1570
],
[
907,
1527,
964,
1570
],
[
964,
1527,
1081,
1570
],
[
1081,
1526,
1124,
1570
],
[
1124,
1527,
1154,
1570
],
[
1154,
1526,
1223,
1570
],
[
1223,
1527,
1254,
1570
],
[
1254,
1527,
1296,
1570
],
[
1296,
1527,
1403,
1570
],
[
293,
1528,
325,
1569
],
[
293,
1557,
384,
1600
],
[
384,
1558,
419,
1600
],
[
419,
1557,
440,
1600
],
[
440,
1558,
548,
1601
],
[
548,
1557,
679,
1600
],
[
679,
1558,
738,
1600
],
[
738,
1558,
789,
1600
],
[
789,
1558,
837,
1600
],
[
837,
1557,
958,
1600
],
[
958,
1557,
1058,
1600
],
[
1058,
1557,
1100,
1600
],
[
1100,
1557,
1149,
1600
],
[
1149,
1557,
1256,
1600
],
[
1256,
1557,
1282,
1600
],
[
1282,
1557,
1331,
1600
],
[
1331,
1557,
1360,
1600
],
[
1360,
1557,
1403,
1600
],
[
293,
1588,
382,
1630
],
[
382,
1588,
470,
1630
],
[
470,
1588,
563,
1631
],
[
563,
1588,
603,
1631
],
[
603,
1588,
705,
1631
],
[
705,
1588,
744,
1631
],
[
744,
1588,
763,
1631
],
[
763,
1588,
861,
1630
],
[
861,
1588,
920,
1631
],
[
920,
1588,
961,
1631
],
[
961,
1588,
1067,
1631
],
[
1067,
1588,
1112,
1630
],
[
1112,
1588,
1138,
1630
],
[
1138,
1588,
1245,
1631
],
[
1245,
1588,
1274,
1630
],
[
1274,
1588,
1314,
1631
],
[
1314,
1588,
1403,
1630
],
[
293,
1618,
370,
1661
],
[
370,
1617,
446,
1661
],
[
446,
1618,
487,
1661
],
[
487,
1618,
628,
1661
],
[
628,
1618,
658,
1661
],
[
658,
1618,
699,
1661
],
[
699,
1618,
806,
1661
],
[
806,
1618,
911,
1661
],
[
911,
1618,
1022,
1661
],
[
1022,
1618,
1063,
1661
],
[
1063,
1618,
1177,
1661
],
[
1177,
1618,
1208,
1661
],
[
1208,
1618,
1297,
1661
],
[
1297,
1618,
1384,
1661
],
[
293,
1648,
358,
1691
],
[
358,
1649,
387,
1691
],
[
387,
1648,
503,
1691
],
[
503,
1648,
571,
1691
],
[
571,
1648,
612,
1691
],
[
612,
1649,
751,
1691
],
[
751,
1649,
780,
1691
],
[
780,
1649,
821,
1691
],
[
821,
1648,
924,
1691
],
[
924,
1648,
1012,
1691
],
[
1012,
1649,
1042,
1691
],
[
1042,
1649,
1114,
1691
],
[
1114,
1648,
1242,
1691
],
[
1242,
1649,
1292,
1691
],
[
1292,
1648,
1403,
1691
],
[
293,
1679,
335,
1722
],
[
335,
1679,
447,
1722
],
[
447,
1679,
477,
1722
],
[
477,
1678,
565,
1722
],
[
565,
1679,
600,
1722
],
[
600,
1678,
639,
1722
],
[
639,
1679,
746,
1722
],
[
746,
1679,
831,
1722
],
[
831,
1679,
869,
1722
],
[
869,
1679,
909,
1722
],
[
909,
1679,
1008,
1722
],
[
1008,
1679,
1063,
1722
],
[
1063,
1679,
1104,
1722
],
[
1104,
1679,
1167,
1722
],
[
1167,
1679,
1263,
1722
],
[
1263,
1679,
1303,
1722
],
[
1303,
1679,
1403,
1722
],
[
293,
1709,
342,
1752
],
[
342,
1710,
369,
1752
],
[
369,
1710,
388,
1752
],
[
388,
1710,
413,
1752
],
[
413,
1709,
454,
1752
],
[
454,
1710,
491,
1752
],
[
491,
1710,
521,
1752
],
[
521,
1709,
616,
1752
],
[
616,
1709,
657,
1752
],
[
657,
1709,
719,
1752
],
[
719,
1709,
765,
1751
],
[
765,
1710,
787,
1752
],
[
787,
1709,
910,
1752
],
[
292,
1754,
346,
1797
],
[
346,
1754,
415,
1797
],
[
415,
1754,
447,
1797
],
[
447,
1754,
491,
1797
],
[
491,
1754,
598,
1797
],
[
598,
1754,
630,
1797
],
[
630,
1754,
723,
1797
],
[
723,
1754,
753,
1797
],
[
753,
1754,
877,
1798
],
[
877,
1754,
920,
1797
],
[
920,
1754,
968,
1797
],
[
968,
1754,
1011,
1797
],
[
1011,
1754,
1078,
1797
],
[
1078,
1754,
1171,
1797
],
[
1171,
1754,
1203,
1797
],
[
1203,
1754,
1247,
1797
],
[
1247,
1754,
1370,
1797
],
[
1370,
1753,
1403,
1797
],
[
293,
1785,
336,
1828
],
[
336,
1784,
505,
1828
],
[
505,
1785,
596,
1828
],
[
596,
1784,
634,
1828
],
[
634,
1785,
677,
1828
],
[
677,
1784,
789,
1828
],
[
789,
1771,
817,
1827
],
[
817,
1786,
871,
1828
],
[
871,
1784,
930,
1828
],
[
930,
1785,
973,
1828
],
[
973,
1785,
1065,
1828
],
[
1065,
1785,
1136,
1828
],
[
1136,
1785,
1190,
1828
],
[
1190,
1785,
1329,
1828
],
[
1329,
1785,
1371,
1828
],
[
1371,
1785,
1403,
1828
],
[
292,
1815,
429,
1858
],
[
435,
1815,
495,
1858
],
[
495,
1815,
623,
1858
],
[
623,
1815,
666,
1858
],
[
666,
1815,
770,
1860
],
[
770,
1818,
794,
1858
],
[
794,
1830,
817,
1861
],
[
817,
1815,
831,
1858
],
[
831,
1815,
923,
1858
],
[
923,
1815,
1006,
1858
],
[
1012,
1815,
1152,
1858
],
[
1152,
1816,
1195,
1858
],
[
1195,
1815,
1300,
1858
],
[
1300,
1815,
1403,
1858
],
[
293,
1845,
348,
1888
],
[
348,
1845,
389,
1888
],
[
389,
1845,
513,
1888
],
[
513,
1845,
543,
1888
],
[
543,
1846,
584,
1888
],
[
584,
1845,
701,
1889
],
[
720,
1847,
788,
1888
],
[
788,
1845,
829,
1888
],
[
829,
1845,
934,
1888
],
[
934,
1845,
996,
1888
],
[
996,
1845,
1064,
1888
],
[
1064,
1845,
1174,
1888
],
[
1174,
1845,
1203,
1888
],
[
1203,
1845,
1244,
1888
],
[
1244,
1845,
1332,
1888
],
[
1332,
1845,
1407,
1888
],
[
292,
1876,
385,
1919
],
[
385,
1875,
506,
1919
],
[
506,
1875,
611,
1919
],
[
611,
1875,
700,
1919
],
[
700,
1875,
765,
1919
],
[
765,
1876,
887,
1919
],
[
887,
1875,
918,
1919
],
[
918,
1876,
961,
1919
],
[
961,
1876,
1053,
1919
],
[
1053,
1875,
1122,
1919
],
[
1122,
1875,
1188,
1919
],
[
1188,
1875,
1245,
1919
],
[
1245,
1875,
1330,
1919
],
[
1330,
1876,
1367,
1919
],
[
1367,
1876,
1404,
1919
],
[
293,
1906,
395,
1949
],
[
395,
1906,
473,
1949
],
[
473,
1907,
499,
1949
],
[
499,
1906,
520,
1949
],
[
520,
1906,
621,
1949
],
[
621,
1906,
652,
1949
],
[
652,
1906,
712,
1949
],
[
712,
1906,
833,
1949
],
[
834,
1907,
880,
1948
],
[
880,
1907,
986,
1949
],
[
986,
1906,
1042,
1949
],
[
1042,
1906,
1092,
1949
],
[
1092,
1906,
1183,
1948
],
[
1183,
1906,
1303,
1948
],
[
1303,
1906,
1406,
1949
],
[
293,
1937,
337,
1979
],
[
337,
1937,
448,
1979
],
[
448,
1936,
574,
1979
],
[
574,
1937,
639,
1979
],
[
639,
1937,
761,
1979
],
[
761,
1937,
792,
1980
],
[
792,
1936,
834,
1979
],
[
834,
1937,
925,
1979
],
[
925,
1936,
994,
1979
],
[
994,
1936,
1026,
1979
],
[
1026,
1936,
1132,
1979
],
[
1132,
1937,
1161,
1979
],
[
1161,
1936,
1203,
1979
],
[
1203,
1936,
1299,
1979
],
[
1299,
1936,
1330,
1979
],
[
1330,
1936,
1403,
1979
],
[
292,
1967,
384,
2010
],
[
384,
1967,
420,
2010
],
[
420,
1967,
463,
2010
],
[
463,
1967,
583,
2010
],
[
583,
1968,
636,
2010
],
[
636,
1967,
718,
2010
],
[
718,
1967,
753,
2010
],
[
753,
1967,
783,
2010
],
[
783,
1968,
826,
2010
],
[
826,
1967,
946,
2011
],
[
946,
1967,
988,
2010
],
[
988,
1967,
1029,
2010
],
[
1029,
1967,
1158,
2010
],
[
1159,
1968,
1205,
2010
],
[
1205,
1967,
1299,
2010
],
[
1299,
1967,
1403,
2010
],
[
836,
2049,
859,
2093
]
]
} | {
"word": [
"Theorem",
"3.3.",
"For",
"data",
"sampled",
"uniformly",
"m:",
"a",
"compact",
"and",
"measurable",
"m",
"dimensional",
"manifold",
"x",
"M",
"we",
"have",
"the",
"following",
"result",
"for",
"all",
"vol",
"m",
"-",
"k",
"(",
"B",
"n",
"M)",
"s",
"(c",
"d:18)#",
"(2CgradCbiasCm)k,",
"k",
"{F,k",
"vol",
"m",
"(M)",
"where",
"C",
"grad",
"depends",
"on",
"||�",
"z(x)",
"ll",
"and",
"the",
"DNN’s",
"architecture,",
"C",
"'M",
"depends",
"on",
"the",
"geometry",
"of",
"M,",
"and",
"C",
"bias",
"on",
"the",
"distribution",
"of",
"biases",
"p",
"b.",
"The",
"constant",
"C",
"'M",
"is",
"the",
"supremum",
"over",
"the",
"matrix",
"norm",
"of",
"projection",
"matrices",
"onto",
"the",
"tangent",
"space,",
"T",
"M,",
"at",
"any",
"point",
"x",
"PE",
"M.",
"For",
"the",
"Euclidean",
"space",
"C",
"M",
"is",
"always",
"equal",
"to",
"1",
"and",
"therefore",
"the",
"term",
"does",
"not",
"appear",
"in",
"the",
"work",
"by",
"Hanin",
"and",
"Rolnick",
"[2019a],",
"but",
"we",
"cannot",
"say",
"the",
"same",
"for",
"our",
"setting.",
"We",
"refer",
"the",
"reader",
"to",
"Appendix",
"F",
"for",
"the",
"proof,",
"further",
"details,",
"and",
"interpretation.",
"Finally,",
"under",
"the",
"added",
"assumptions",
"that",
"the",
"diameter",
"of",
"the",
"manifold",
"M",
"is",
"finite",
"and",
"M",
"has",
"polynomial",
"volume",
"growth",
"we",
"provide",
"a",
"lower",
"bound",
"on",
"the",
"average",
"distance",
"to",
"the",
"linear",
"boundary",
"for",
"points",
"on",
"the",
"manifold",
"and",
"how",
"it",
"depends",
"on",
"the",
"geometry",
"and",
"dimensionality",
"of",
"the",
"manifold.",
"Theorem",
"3.4.",
"For",
"any",
"point,",
"x,",
"chosen",
"randomly",
"from",
"M,",
"we",
"have:",
"E",
"[distance",
"em",
"(x,",
"B",
"n",
"M)]",
">",
"C",
"'M,k",
"Cgrad",
"C",
"bias",
",C",
"M",
"#neurons",
"where",
"C",
"M,k",
"depends",
"on",
"the",
"scalar",
"curvature,",
"the",
"input",
"dimension",
"and",
"the",
"dimensionality",
"of",
"the",
"manifold",
"M.",
"The",
"function",
"distanceM",
"M",
"is",
"the",
"distance",
"on",
"the",
"manifold",
"M.",
"This",
"result",
"gives",
"us",
"intuition",
"on",
"how",
"the",
"density",
"of",
"linear",
"regions",
"around",
"point",
"depends",
"on",
"the",
"geometry",
"of",
"the",
"manifold.",
"The",
"constant",
"C",
"'M,k",
"captures",
"how",
"volumes",
"are",
"distorted",
"on",
"the",
"manifold",
"M",
"as",
"compared",
"to",
"the",
"Euclidean",
"space,",
"for",
"the",
"exact",
"definition",
"we",
"refer",
"the",
"reader",
"to",
"the",
"proof",
"in",
"Appendix",
"G.",
"For",
"a",
"manifold",
"which",
"has",
"higher",
"volume",
"of",
"a",
"unit",
"ball,",
"on",
"average,",
"in",
"comparison",
"to",
"the",
"Euclidean",
"space",
"the",
"constant",
"C",
"'M,k",
"is",
"higher",
"and",
"lower",
"when",
"the",
"volume",
"of",
"unit",
"ball,",
"on",
"average,",
"is",
"lower",
"than",
"the",
"volume",
"of",
"the",
"Euclidean",
"space.",
"For",
"background",
"on",
"curvature",
"of",
"manifolds",
"and",
"proof",
"sketch",
"we",
"refer",
"the",
"reader",
"to",
"the",
"Appendices",
"B",
"and",
"D,",
"respectively.",
"Note",
"that",
"the",
"constant",
"C",
"M",
"is",
"the",
"same",
"as",
"in",
"Theorem",
"3.3.",
"Another",
"difference",
"to",
"note",
"is",
"that",
"we",
"derive",
"a",
"lower",
"bound",
"on",
"the",
"geodesic",
"distance",
"on",
"the",
"manifold",
"M",
"and",
"not",
"the",
"Euclidean",
"distance",
"in",
"k",
"as",
"done",
"by",
"Hanin",
"and",
"Rolnick",
"[2019a].",
"This",
"distance",
"better",
"captures",
"the",
"distance",
"between",
"data",
"points",
"on",
"a",
"manifold",
"while",
"incorporating",
"the",
"underlying",
"structure.",
"In",
"other",
"words,",
"this",
"distance",
"can",
"be",
"understood",
"as",
"how",
"much",
"a",
"data",
"point",
"should",
"change",
"to",
"reach",
"a",
"linear",
"boundary",
"while",
"ensuring",
"that",
"all",
"the",
"individual",
"points",
"on",
"the",
"curve,",
"tracing",
"this",
"change,",
"are",
"“valid”",
"data",
"points.",
"3.1",
"Intuition",
"For",
"Theoretical",
"Results",
"One",
"of",
"the",
"key",
"ingredients",
"of",
"the",
"proofs",
"by",
"Hanin",
"and",
"Rolnick",
"[2019a]",
"is",
"the",
"co-area",
"formula",
"[Krantz",
"and",
"Parks,",
"2008].",
"The",
"co-area",
"formula",
"is",
"applied",
"to",
"get",
"closed",
"form",
"representation",
"of",
"the",
"k",
"dimensional",
"volume",
"of",
"the",
"region",
"where",
"any",
"set",
"of",
"k",
"neurons,",
"z",
"z",
"2",
"...,",
"z",
"k",
"is",
"“good”",
"in",
"terms",
"of",
"the",
"expectation",
"over",
"the",
"Jacobian,",
"in",
"the",
"Euclidean",
"space.",
"Instead",
"of",
"the",
"co-area",
"formula",
"we",
"use",
"the",
"smooth",
"co-area",
"formula",
"[Krantz",
"and",
"Parks,",
"2008]",
"to",
"get",
"a",
"closed",
"form",
"representation",
"of",
"the",
"es",
"k",
"dimensional",
"volume",
"of",
"the",
"region",
"intersected",
"with",
"manifold,",
"M,",
"in",
"terms",
"of",
"the",
"Jacobian",
"m",
"defined",
"on",
"a",
"manifold",
"(Definition",
"3.1).",
"The",
"key",
"difference",
"between",
"the",
"two",
"formulas",
"is",
"that",
"in",
"the",
"smooth",
"co-area",
"formula",
"the",
"Jacobian",
"(of",
"a",
"function",
"from",
"the",
"manifold",
"M)",
"is",
"restricted",
"to",
"the",
"tangent",
"plane.",
"While",
"the",
"determinant",
"of",
"the",
"“vanilla”",
"Jacobian",
"measures",
"the",
"distortion",
"of",
"volume",
"around",
"point",
"in",
"Euclidean",
"space",
"the",
"determinant",
"of",
"the",
"Jacobian",
"defined",
"as",
"above",
"(Definition",
"3.1)",
"measures",
"the",
"distortion",
"of",
"volume",
"on",
"the",
"manifold",
"instead",
"for",
"the",
"function",
"with",
"the",
"same",
"domain,",
"the",
"function",
"that",
"is",
"1",
"if",
"the",
"set",
"of",
"neurons",
"are",
"good",
"and",
"0",
"otherwise.",
"The",
"value",
"of",
"the",
"Jacobian",
"as",
"defined",
"in",
"Definition",
"3.1",
"has",
"the",
"same",
"volume",
"as",
"the",
"projection",
"of",
"the",
"parallelepiped",
"defined",
"by",
"the",
"gradients",
"�V",
"z(x)",
"onto",
"the",
"tangent",
"space",
"(see",
"Proposition",
"F.1",
"in",
"Appendix).",
"This",
"introduces",
"the",
"constant",
"C",
"M",
",",
"defined",
"above.",
"Essentially,",
"the",
"constant",
"captures",
"how",
"the",
"magnitude",
"of",
"the",
"gradients,",
"∇z(x),",
"are",
"modified",
"upon",
"being",
"projected",
"to",
"the",
"tangent",
"plane.",
"Certain",
"manifolds",
"“shrink”",
"vectors",
"upon",
"projection",
"to",
"the",
"tangent",
"plane",
"more",
"than",
"others,",
"on",
"an",
"average,",
"which",
"is",
"a",
"function",
"of",
"their",
"geometry.",
"We",
"illustrate",
"how",
"two",
"distinct",
"manifolds",
"“shrink”",
"the",
"gradients",
"differently",
"upon",
"projection",
"to",
"the",
"tangent",
"plane",
"as",
"reflected",
"in",
"the",
"number",
"of",
"linear",
"regions",
"on",
"the",
"manifolds",
"(see",
"Figure",
"11",
"in",
"the",
"appendix)",
"for",
"1D",
"manifolds.",
"We",
"provide",
"intuition",
"6"
]
} | |||
2301.00008v1.pdf_7 | null | ArxivOCR_part_1 | {
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} | {
"word": [
"1-o7s",
"-10",
"-1.0",
"-1.0",
"(b)",
"(a)",
"Figure",
"3:",
"The",
"tractrix",
"(a)",
"and",
"circle",
"(b)",
"are",
"plotted",
"in",
"grey",
"and",
"the",
"target",
"function",
"is",
"in",
"blue.",
"This",
"is",
"for",
"illustration",
"purposes",
"and",
"does",
"not",
"match",
"the",
"actual",
"function",
"or",
"domains",
"used",
"in",
"our",
"experiments.",
"for",
"the",
"curvature",
"of",
"a",
"manifold",
"in",
"Appendix",
"B,",
"due",
"to",
"space",
"constraints,",
"which",
"is",
"used",
"in",
"the",
"lower",
"bound",
"for",
"the",
"average",
"distance",
"in",
"Theorem",
"3.4.",
"The",
"constant",
"C",
"M,k",
"depends",
"on",
"the",
"curvature",
"as",
"the",
"supremum",
"of",
"a",
"polynomial",
"whose",
"coefficients",
"depend",
"on",
"the",
"curvature,",
"with",
"order",
"at",
"most",
"n",
"in",
"and",
"at",
"least",
"n",
"in",
"im",
"m.",
"Note",
"that",
"despite",
"this",
"dependence",
"on",
"the",
"ambient",
"dimension,",
"there",
"are",
"other",
"geometric",
"constants",
"in",
"this",
"polynomial",
"(see",
"Appendix",
"G).",
"Finally,",
"we",
"also",
"provide",
"a",
"simple",
"example",
"as",
"to",
"how",
"this",
"constant",
"varies",
"with",
"n",
"in",
"and",
"m,",
"for",
"a",
"simple",
"and",
"contrived",
"example,",
"in",
"Appendix",
"G.1.",
"4",
"Experiments",
"4.1",
"Linear",
"Regions",
"on",
"a",
"1D",
"Curve",
"To",
"empirically",
"corroborate",
"our",
"theoretical",
"results,",
"we",
"calculate",
"the",
"number",
"of",
"linear",
"regions",
"and",
"average",
"distance",
"to",
"the",
"linear",
"boundary",
"on",
"1D",
"curves",
"for",
"regression",
"tasks",
"in",
"two",
"settings.",
"The",
"first",
"is",
"for",
"1D",
"manifolds",
"embedded",
"in",
"2D",
"and",
"higher",
"dimensions",
"and",
"the",
"second",
"is",
"for",
"the",
"high-dimensional",
"data",
"using",
"the",
"MetFaces",
"dataset.",
"We",
"use",
"the",
"same",
"algorithm,",
"for",
"the",
"toy",
"problem",
"and",
"the",
"high-",
"dimensional",
"dataset,",
"to",
"find",
"linear",
"regions",
"on",
"1D",
"curves.",
"We",
"calculate",
"the",
"exact",
"number",
"of",
"linear",
"regions",
"for",
"a",
"1D",
"curve",
"in",
"the",
"input",
"space,",
"x",
":",
"I",
"D,",
"R",
"nin",
"where",
"I",
"is",
"an",
"interval",
"in",
"real",
"numbers,",
"by",
"finding",
"the",
"points",
"where",
"z(x(t))",
"=",
"b",
"z",
"for",
"every",
"neuron",
"z.",
"The",
"solutions",
"thus",
"obtained",
"gives",
"us",
"the",
"boundaries",
"for",
"neurons",
"on",
"the",
"curve",
"xc.",
"We",
"obtain",
"these",
"solutions",
"by",
"using",
"the",
"programmatic",
"activation",
"of",
"every",
"neuron",
"and",
"using",
"the",
"sequential",
"least",
"squares",
"programming",
"(SLSQP)",
"algorithm",
"[Kraft,",
"1988]",
"to",
"solve",
"for",
"|z(x(t))",
"b",
"z",
"=",
"0",
"for",
"t",
"�",
"I.",
"In",
"order",
"to",
"obtain",
"the",
"programmatic",
"activation",
"of",
"a",
"neuron",
"we",
"construct",
"a",
"Deep",
"ReLU",
"network",
"as",
"defined",
"in",
"Equation",
"2.",
"We",
"do",
"so",
"for",
"all",
"the",
"neurons",
"for",
"a",
"given",
"DNN",
"with",
"fixed",
"weights.",
"4.2",
"Supervised",
"Learning",
"on",
"Toy",
"Dataset",
"We",
"define",
"two",
"similar",
"regression",
"tasks",
"where",
"the",
"data",
"is",
"sampled",
"from",
"two",
"different",
"manifolds",
"with",
"different",
"geometries.",
"We",
"parameterize",
"the",
"first",
"task,",
"a",
"unit",
"circle",
"without",
"its",
"north",
"and",
"south",
"poles,",
"on",
"TT,",
"π)",
"s.+",
"2",
"where",
"q",
"circle",
"(0)",
"=",
"(cos",
"θ,",
"sin",
"θ)",
"and",
"o",
"is",
"the",
"angle",
"made",
"by",
"the",
"vector",
"by",
"o",
"circle",
":",
"from",
"the",
"origin",
"to",
"the",
"point",
"with",
"respect",
"to",
"the",
"x-axis.",
"We",
"set",
"the",
"target",
"function",
"for",
"regression",
"task",
"to",
"be",
"a",
"periodic",
"function",
"in",
"0.",
"The",
"target",
"is",
"defined",
"as",
"z(θ)",
"=",
"a",
"sin(νθ)",
"where",
"a",
"is",
"the",
"amplitude",
"and",
"v",
"is",
"the",
"frequency",
"(Figure",
"3).",
"DNNs",
"have",
"difficulty",
"learning",
"periodic",
"functions",
"[Ziyin",
"et",
"al.,",
"2020].",
"The",
"motivation",
"behind",
"this",
"is",
"to",
"present",
"the",
"DNN",
"with",
"a",
"challenging",
"task",
"where",
"it",
"has",
"to",
"learn",
"the",
"underlying",
"structure",
"of",
"the",
"data.",
"Moreover",
"the",
"DNN",
"will",
"have",
"to",
"split",
"the",
"circle",
"into",
"linear",
"regions.",
"For",
"the",
"second",
"regression",
"task,",
"a",
"tractrix",
"is",
"parametrized",
"by",
"o",
"Vtractrix",
":",
"R",
"1",
"sin(νt).",
"For",
"ψtractrix(y)",
"(y)",
"=",
"(y",
"es",
"tanh",
"y,",
"sech",
"y)",
"(see",
"Figure",
"3).",
"We",
"assign",
"a",
"target",
"function",
"z(t)",
"=",
"2",
"where",
"the",
"purposes",
"of",
"our",
"study",
"we",
"restrict",
"the",
"domain",
"of",
"q",
"tractrix",
"to",
"(",
"-3,",
"3).",
"We",
"choose",
"v",
"so",
"as",
"to",
"ensure",
"that",
"the",
"number",
"of",
"peaks",
"and",
"troughs,",
"6,",
"in",
"the",
"periodic",
"target",
"function",
"are",
"the",
"same",
"for",
"both",
"the",
"manifolds.",
"This",
"ensures",
"that",
"the",
"domains",
"of",
"both",
"the",
"problems",
"have",
"length",
"close",
"to",
"6.28.",
"Further",
"experimental",
"details",
"are",
"in",
"Appendix",
"H.",
"The",
"results,",
"averaged",
"over",
"20",
"runs,",
"are",
"presented",
"in",
"Figures",
"4",
"and",
"5.",
"We",
"note",
"that",
"C",
"M",
"is",
"smaller",
"for",
"Sphere",
"(based",
"on",
"Figure",
"4)",
"and",
"the",
"curvature",
"is",
"positive",
"whilst",
"C",
"M",
"is",
"larger",
"for",
"tractrix",
"and",
"the",
"curvature",
"is",
"negative.",
"Both",
"of",
"these",
"constants",
"(curvature",
"and",
"C",
"M",
")",
"contribute",
"to",
"the",
"lower",
"bound",
"7"
]
} | |||
2301.00008v1.pdf_8 | null | ArxivOCR_part_1 | {
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293,
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349,
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504,
1936,
546,
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[
546,
1936,
628,
1979
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[
628,
1937,
724,
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[
724,
1937,
789,
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[
789,
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904,
1979
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[
904,
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986,
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986,
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1048,
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1048,
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1174,
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[
1174,
1936,
1274,
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[
1274,
1936,
1300,
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1300,
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1402,
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293,
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325,
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325,
1968,
381,
2010
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[
381,
1967,
503,
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[
503,
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563,
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563,
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585,
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[
585,
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636,
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636,
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761,
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761,
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891,
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1333,
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1333,
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1363,
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1363,
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[
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} | {
"word": [
"in",
"Theorem",
"3.4.",
"Similarly,",
"we",
"show",
"results",
"of",
"number",
"of",
"linear",
"regions",
"divided",
"by",
"the",
"number",
"of",
"neurons",
"upon",
"changing",
"architectures,",
"consequently",
"the",
"number",
"of",
"neurons,",
"for",
"the",
"two",
"manifolds",
"in",
"Figure",
"8,",
"averaged",
"over",
"30",
"runs.",
"Note",
"that",
"this",
"experiment",
"observes",
"the",
"effect",
"of",
"C",
"M",
"ox",
"C",
"grad",
"since",
"changing",
"the",
"architecture",
"also",
"changes",
"C",
"grad",
"and",
"the",
"variation",
"in",
"C",
"grad",
"is",
"quite",
"low",
"in",
"magnitude",
"as",
"observed",
"empirically",
"by",
"Hanin",
"and",
"Rolnick",
"[2019a].",
"The",
"empirical",
"observations",
"are",
"consistent",
"with",
"our",
"theoretical",
"results.",
"We",
"observe",
"that",
"the",
"number",
"of",
"linear",
"regions",
"starts",
"off",
"close",
"to",
"#neurons",
"and",
"remains",
"close",
"throughout",
"the",
"training",
"process",
"for",
"both",
"the",
"manifolds.",
"This",
"supports",
"our",
"theoretical",
"results",
"(Theorem",
"3.3)",
"that",
"the",
"constant",
"C",
"1,",
"which",
"is",
"distinct",
"across",
"the",
"two",
"manifolds,",
"affects",
"the",
"number",
"of",
"linear",
"regions",
"throughout",
"training.",
"The",
"tractrix",
"has",
"higher",
"value",
"of",
"C",
"M",
"and",
"that",
"is",
"reflected",
"in",
"both",
"Figures",
"4",
"and",
"5.",
"Note",
"that",
"its",
"relationship",
"is",
"inverse",
"to",
"the",
"average",
"distance",
"to",
"the",
"boundary",
"region,",
"as",
"per",
"Theorem",
"3.4,",
"and",
"it",
"is",
"reflected",
"as",
"training",
"progresses",
"in",
"Figure",
"5.",
"This",
"is",
"due",
"to",
"different",
"“shrinking”",
"of",
"vectors",
"upon",
"being",
"projected",
"to",
"the",
"tangent",
"space",
"(Section",
"3.1).",
"4.3",
"Varying",
"Input",
"Dimensions",
"To",
"empirically",
"corroborate",
"the",
"results",
"of",
"Theorems",
"2",
"and",
"we",
"vary",
"the",
"dimension",
"nj",
"while",
"keeping",
"m",
"constant.",
"We",
"achieve",
"this",
"by",
"counting",
"the",
"number",
"of",
"linear",
"regions",
"and",
"the",
"average",
"distance",
"to",
"boundary",
"region",
"on",
"the",
"1D",
"circle",
"as",
"we",
"vary",
"the",
"input",
"dimension",
"in",
"steps",
"of",
"5.",
"We",
"draw",
"samples",
"of",
"1D",
"circles",
"in",
"R",
"nin",
"by",
"randomly",
"choosing",
"two",
"perpendicular",
"basis",
"vectors.",
"We",
"then",
"train",
"network",
"with",
"the",
"same",
"architecture",
"as",
"the",
"previous",
"section",
"on",
"the",
"periodic",
"target",
"function",
"(a",
"sin(νθ))",
"as",
"defined",
"above.",
"The",
"results",
"in",
"Figure",
"6",
"shows",
"that",
"the",
"quantities",
"stay",
"proportional",
"to",
"#neurons,",
"and",
"do",
"not",
"vary",
"as",
"nj",
"is",
"increased,",
"as",
"predicted",
"by",
"our",
"theoretical",
"results.",
"Our",
"empirical",
"study",
"asserts",
"how",
"the",
"relevant",
"upper",
"and",
"lower",
"bounds,",
"for",
"the",
"setting",
"where",
"data",
"lies",
"on",
"a",
"low-dimensional",
"manifold,",
"does",
"not",
"grow",
"exponentially",
"with",
"n",
"in",
"for",
"the",
"density",
"of",
"linear",
"regions",
"in",
"a",
"compact",
"set",
"of",
"R",
"nin",
"but",
"instead",
"depend",
"on",
"the",
"intrinsic",
"dimension.",
"Further",
"details",
"are",
"in",
"Appendix",
"H.",
"4.4",
"MetFaces:",
"High",
"Dimensional",
"Dataset",
"Our",
"goal",
"with",
"this",
"experiment",
"is",
"to",
"study",
"how",
"the",
"density",
"of",
"linear",
"regions",
"varies",
"across",
"a",
"low",
"dimensional",
"manifold",
"and",
"the",
"input",
"space.",
"To",
"discover",
"latent",
"low",
"dimensional",
"underlying",
"structure",
"of",
"data",
"we",
"employ",
"a",
"GAN.",
"Adversarial",
"training",
"of",
"GANs",
"can",
"be",
"effectively",
"applied",
"to",
"learn",
"a",
"mapping",
"from",
"a",
"low",
"dimensional",
"latent",
"space",
"to",
"high",
"dimensional",
"data",
"[Goodfellow",
"et",
"al.,",
"2014].",
"The",
"generator",
"is",
"a",
"neural",
"network",
"that",
"maps",
"g",
":",
"R",
"k",
"nin.",
"We",
"train",
"a",
"deep",
"ReLU",
"network",
"on",
"the",
"MetFaces",
"dataset",
"with",
"random",
"labels",
"(chosen",
"from",
"0,",
"1)",
"with",
"cross",
"entropy",
"loss.",
"As",
"noted",
"by",
"Zhang",
"et",
"al.",
"[2017],",
"training",
"with",
"random",
"labels",
"can",
"lead",
"to",
"the",
"DNN",
"memorizing",
"the",
"entire",
"dataset.",
"We",
"compare",
"the",
"log",
"density",
"of",
"number",
"of",
"linear",
"regions",
"on",
"a",
"curve",
"on",
"the",
"manifold",
"with",
"a",
"straight",
"line",
"off",
"the",
"manifold.",
"We",
"generate",
"these",
"curves",
"using",
"the",
"data",
"sampled",
"by",
"the",
"StyleGAN",
"by",
"e",
"IR",
"k,",
"this",
"2020a].",
"Specifically,",
"for",
"each",
"curve",
"we",
"sample",
"a",
"random",
"pair",
"of",
"latent",
"vectors:",
"z",
"et",
"al.,",
"gives",
"us",
"the",
"start",
"and",
"end",
"point",
"of",
"the",
"curve",
"using",
"the",
"generator",
"g(z",
"1",
")",
"and",
"g(z",
"2).",
"We",
"then",
"generate",
"100",
"images",
"to",
"approximate",
"a",
"curve",
"connecting",
"the",
"two",
"images",
"on",
"the",
"image",
"manifold",
"in",
"a",
"piece-wise",
"manner.",
"We",
"do",
"so",
"by",
"taking",
"100",
"points",
"on",
"the",
"line",
"connecting",
"z",
"and",
"z",
"2",
"in",
"the",
"latent",
"space",
"that",
"are",
"evenly",
"spaced",
"a",
"i)",
"&",
"z1",
"+",
"a",
"2",
"2",
")/100),",
"using",
"the",
"StyleGAN",
"generator",
"g.",
"We",
"qualitatively",
"verify",
"the",
"z",
"=",
"g(((100",
"generate",
"image",
"from",
"each",
"one",
"of",
"them.",
"Therefore,",
"the",
"ith",
"image",
"is",
"generated",
"as:",
"images",
"to",
"ensure",
"that",
"they",
"lie",
"on",
"the",
"manifold",
"o",
"t)g(z1)",
"1",
")",
"+",
"tg(z2)",
",",
"with",
"t",
"[0,",
"1].",
"The",
"approximated",
"points",
"g(z",
"1",
")",
"and",
"g(z",
"2",
"),",
"is",
"defined",
"as",
"x(t)",
"=",
"(1",
"-t",
"t)g(z",
"i)",
"+",
"tg(z",
"i+1)",
")",
"where",
"i",
"=",
"floor(100t).",
"We",
"curve",
"on",
"the",
"manifold",
"is",
"defined",
"as",
"x",
"ì(t)",
"=",
")(",
"the",
"number",
"of",
"linear",
"regions",
"on",
"these",
"curves.",
"fixed",
"then",
"apply",
"the",
"method",
"from",
"Section",
"4.1",
"to",
"obtain",
"images",
"of",
"faces.",
"The",
"straight",
"line,",
"with",
"two",
"The",
"results",
"are",
"presented",
"in",
"Figure",
"9.",
"This",
"leads",
"us",
"to",
"the",
"key",
"observation:",
"the",
"density",
"of",
"linear",
"regions",
"is",
"significantly",
"lower",
"on",
"the",
"data",
"manifold",
"and",
"devising",
"methods",
"to",
"“concentrate”",
"these",
"linear",
"regions",
"on",
"the",
"manifold",
"is",
"a",
"promising",
"research",
"direction.",
"That",
"could",
"lead",
"to",
"increased",
"expressivity",
"for",
"the",
"same",
"number",
"of",
"parameters.",
"We",
"provide",
"further",
"experimental",
"details",
"in",
"Appendix",
"I.",
"5",
"Discussion",
"and",
"Conclusions",
"There",
"is",
"significant",
"work",
"in",
"both",
"supervised",
"and",
"unsupervised",
"learning",
"settings",
"for",
"non-Euclidean",
"data",
"[Bronstein",
"et",
"al.,",
"2017].",
"Despite",
"these",
"empirical",
"results",
"most",
"theoretical",
"analysis",
"is",
"agnostic",
"to",
"data",
"geometry,",
"with",
"a",
"few",
"prominent",
"exceptions",
"[Cloninger",
"and",
"Klock,",
"2020,",
"Shaham",
"et",
"al.,",
"8"
]
} | |||
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"word": [
"2015,",
"Schmidt-Hieber,",
"2019].",
"We",
"incorporate",
"the",
"idea",
"of",
"data",
"geometry",
"into",
"measuring",
"the",
"effective",
"approximation",
"capacity",
"of",
"DNNs,",
"deriving",
"average",
"bounds",
"on",
"the",
"density",
"of",
"boundary",
"regions",
"and",
"distance",
"from",
"the",
"boundary",
"when",
"the",
"data",
"is",
"sampled",
"from",
"a",
"low",
"dimensional",
"manifold.",
"Our",
"experimental",
"results",
"corroborate",
"our",
"theoretical",
"results.",
"We",
"also",
"present",
"insights",
"into",
"expressivity",
"of",
"DNNs",
"on",
"low",
"dimensional",
"manfiolds",
"for",
"the",
"case",
"of",
"high",
"dimensional",
"datasets.",
"Estimating",
"the",
"geometry,",
"dimensionality",
"and",
"curvature,",
"of",
"these",
"image",
"manifolds",
"accurately",
"is",
"a",
"problem",
"that",
"remains",
"largely",
"unsolved",
"[Brehmer",
"and",
"Cranmer,",
"2020,",
"Perraul-Joncas",
"and",
"Meila,",
"2013],",
"which",
"limits",
"our",
"inferences",
"on",
"high",
"dimensional",
"dataset",
"to",
"observations",
"that",
"guide",
"future",
"research.",
"We",
"note",
"that",
"proving",
"a",
"lower",
"bound",
"on",
"the",
"number",
"of",
"linear",
"regions,",
"as",
"done",
"by",
"Hanin",
"and",
"Rolnick",
"[2019a],",
"for",
"the",
"manifold",
"setting",
"remains",
"open.",
"Our",
"work",
"opens",
"up",
"avenues",
"for",
"further",
"research",
"that",
"combines",
"model",
"geometry",
"and",
"data",
"geometry",
"and",
"can",
"lead",
"to",
"empirical",
"research",
"geared",
"towards",
"developing",
"DNN",
"architectures",
"for",
"high",
"dimensional",
"datasets",
"that",
"lie",
"on",
"low",
"dimensional",
"manifold.",
"6",
"Acknowledgements",
"This",
"work",
"was",
"funded",
"by",
"L2M",
"(DARPA",
"Lifelong",
"Learning",
"Machines",
"program",
"under",
"grant",
"number",
"FA8750-18-2-0117),",
"the",
"Penn",
"MURI",
"(ONR",
"under",
"the",
"PERISCOPE",
"MURI",
"Contract",
"N00014-",
"17-",
"1-2699),",
"and",
"the",
"ONR",
"Swarm",
"(the",
"ONR",
"under",
"grant",
"number",
"N00014-21-1-2200).",
"This",
"research",
"was",
"conducted",
"using",
"computational",
"resources",
"and",
"services",
"at",
"the",
"Center",
"for",
"Computation",
"and",
"Visualization,",
"Brown",
"University.",
"We",
"would",
"like",
"to",
"thank",
"Sam",
"Lobel,",
"Rafael",
"Rodriguez",
"Sanchez,",
"and",
"Akhil",
"Bagaria",
"for",
"refining",
"our",
"work,",
"multiple",
"technical",
"discussions,",
"and",
"their",
"helpful",
"feedback",
"on",
"the",
"implementation",
"details.",
"We",
"also",
"thank",
"Tejas",
"Kotwal",
"for",
"assistance",
"on",
"deriving",
"the",
"mathematical",
"details",
"related",
"to",
"the",
"1D",
"Tractrix",
"and",
"sources",
"for",
"various",
"citations.",
"We",
"thank",
"Professor",
"Pedro",
"Lopes",
"de",
"Almeida,",
"Nihal",
"Nayak,",
"Cameron",
"Allen",
"and",
"Aarushi",
"Kalra",
"for",
"their",
"valuable",
"comments",
"on",
"writing",
"and",
"presentation",
"of",
"our",
"work.",
"We",
"thank",
"all",
"the",
"members",
"of",
"the",
"Brown",
"robotics",
"lab",
"for",
"their",
"guidance",
"and",
"support",
"at",
"various",
"stages",
"of",
"our",
"work.",
"Finally,",
"we",
"are",
"indebted",
"to,",
"and",
"graciously",
"thank,",
"the",
"numerous",
"anonymous",
"reviewers",
"for",
"their",
"time",
"and",
"labor",
"as",
"their",
"valuable",
"feedback",
"and",
"thoughtful",
"engagement",
"have",
"shaped",
"and",
"vastly",
"refine",
"our",
"work.",
"References",
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]
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"word": [
"ON",
"THE",
"GEOMETRY",
"OF",
"REINFORCEMENT",
"LEARNING",
"IN",
"CONTINUOUS",
"STATE",
"AND",
"ACTION",
"SPACES",
"Saket",
"Tiwari",
"Omer",
"Gottesman",
"Department",
"of",
"Computer",
"Science",
"Department",
"of",
"Computer",
"Science",
"Brown",
"University",
"Brown",
"University",
"Providence,",
"RI",
"02906",
"Providence,",
"RI",
"02906",
"saket_",
"tiwari@brown.edu",
"202",
"George",
"Konidaris",
"Department",
"of",
"Computer",
"Science",
"Brown",
"University",
"Providence,",
"RI",
"02906",
"P",
"ABSTRACT",
"20",
"Advances",
"in",
"reinforcement",
"learning",
"have",
"led",
"to",
"its",
"successful",
"application",
"in",
"com-",
"plex",
"tasks",
"with",
"continuous",
"state",
"and",
"action",
"spaces.",
"Despite",
"these",
"advances",
"in",
"VE",
"practice,",
"most",
"theoretical",
"work",
"pertains",
"to",
"finite",
"state",
"and",
"action",
"spaces.",
"We",
"pro-",
"pose",
"building",
"a",
"theoretical",
"understanding",
"of",
"continuous",
"state",
"and",
"action",
"spaces",
"by",
"employing",
"a",
"geometric",
"lens.",
"Central",
"to",
"our",
"work",
"is",
"the",
"idea",
"that",
"the",
"transition",
"dynamics",
"induce",
"a",
"low",
"dimensional",
"manifold",
"of",
"reachable",
"states",
"embedded",
"in",
"the",
"high-dimensional",
"nominal",
"state",
"space.",
"We",
"prove",
"that,",
"under",
"certain",
"conditions,",
"the",
"dimensionality",
"of",
"this",
"manifold",
"is",
"at",
"most",
"the",
"dimensionality",
"of",
"the",
"action",
"space",
"plus",
"one.",
"This",
"is",
"the",
"first",
"result",
"of",
"its",
"kind,",
"linking",
"the",
"geometry",
"of",
"the",
"state",
"space",
"Mh",
"to",
"the",
"dimensionality",
"of",
"the",
"action",
"space.",
"We",
"empirically",
"corroborate",
"this",
"upper",
"bound",
"for",
"four",
"MuJoCo",
"environments.",
"We",
"further",
"demonstrate",
"the",
"applicability",
"of",
"our",
"result",
"by",
"learning",
"a",
"policy",
"in",
"this",
"low",
"dimensional",
"representation.",
"To",
"do",
"so",
"we",
"introduce",
"an",
"algorithm",
"that",
"learns",
"a",
"mapping",
"to",
"a",
"low",
"dimensional",
"representation,",
"as",
"a",
"narrow",
"hidden",
"layer",
"of",
"a",
"deep",
"neural",
"network,",
"in",
"tandem",
"with",
"the",
"policy",
"using",
"DDPG.",
"Our",
"experiments",
"show",
"that",
"a",
"policy",
"learnt",
"this",
"way",
"perform",
"on",
"par",
"or",
"better",
"201",
"for",
"four",
"MuJoCo",
"control",
"suite",
"tasks.",
"1",
"INTRODUCTION",
"7L",
"The",
"goal",
"of",
"a",
"reinforcement",
"learning",
"(RL)",
"agent",
"is",
"to",
"learn",
"an",
"optimal",
"policy",
"that",
"maximises",
"the",
"return",
"which",
"is",
"the",
"time",
"discounted",
"cumulative",
"reward",
"(Sutton",
"&",
"Barto,",
"1998).",
"Recent",
"advances",
"in",
"RL",
"research",
"have",
"lead",
"to",
"agents",
"successfully",
"learning",
"in",
"environments",
"with",
"enormous",
"state",
"spaces,",
"such",
"as",
"games",
"(Mnih",
"et",
"al.,",
"2015;",
"Silver",
"et",
"al.,",
"2016),",
"and",
"robotic",
"control",
"in",
"simulation",
"(Lillicrap",
"et",
"al.,",
"2016;",
"Schulman",
"et",
"al.,",
"2015;",
"2017a)",
"and",
"real",
"environments",
"(Levine",
"et",
"al.,",
"2016;",
"Zhu",
"et",
"al.,",
"2020;",
"Deisenroth",
"&",
"Rasmussen,",
"2011).",
"However,",
"we",
"do",
"not",
"have",
"an",
"understanding",
"of",
"the",
"intrinsic",
"complexity",
"of",
"these",
"seemingly",
"large",
"problems.",
"For",
"example,",
"in",
"most",
"popular",
"deep",
"RL",
"algorithms",
"for",
"continuous",
"control,",
"the",
"agent’s",
"policy",
"is",
"parameterised",
"by",
"a",
"a",
"deep",
"neural",
"network",
"(DNN)",
"(Lillicrap",
"et",
"al.,",
"2016;",
"Schulman",
"et",
"al.,",
"2015;",
"2017a)",
"but",
"we",
"do",
"not",
"have",
"theoretical",
"models",
"to",
"guide",
"the",
"design",
"of",
"DNN",
"architecture",
"required",
"to",
"efficiently",
"learn",
"an",
"optimal",
"policy",
"for",
"various",
"environments.",
"There",
"have",
"been",
"approaches",
"to",
"measure",
"the",
"difficulty",
"of",
"an",
"RL",
"environment",
"from",
"a",
"sample",
"complexity",
"perspective",
"(Antos",
"et",
"al.,",
"2007;",
"Munos",
"&",
"Szepesvari,",
"2008;",
"Bastani,",
"2020)",
"but",
"these",
"models",
"fall",
"short",
"of",
"providing",
"recommendations",
"for",
"the",
"policy",
"and",
"value",
"function",
"complexity",
"required",
"to",
"learn",
"an",
"optimal",
"policy.",
"We",
"view",
"the",
"complexity",
"of",
"RL",
"environments",
"through",
"a",
"geometric",
"lens.",
"We",
"build",
"on",
"the",
"intuition",
"behind",
"the",
"manifold",
"hypothesis,",
"which",
"states",
"that",
"most",
"high-dimensional",
"real-world",
"datasets",
"actually",
"lie",
"on",
"low-dimensional",
"manifolds",
"(Tenenbaum,",
"1997;",
"Carlsson",
"et",
"al.,",
"2007;",
"Fefferman",
"et",
"al.,",
"2013;",
"Bronstein",
"et",
"al.,",
"2021);",
"for",
"example,",
"the",
"set",
"of",
"natural",
"images",
"are",
"a",
"very",
"small,",
"smoothly-varying",
"1"
]
} | |||
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} | |||
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} | |||
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"RL",
"as",
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",",
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"+",
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} | |||
2301.00009v1.pdf_5 | null | ArxivOCR_part_1 | {
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} | {
"word": [
"T,Se",
"0",
"Se",
"-2",
"-6",
"Ty",
"x��",
"2�",
"-2�",
"Figure",
"2:",
"Consider",
"a",
"hypothetical",
"scenario",
"as",
"in",
"the",
"image",
"above",
"where",
"upon",
"taking",
"an",
"action",
"even",
"though",
"the",
"agent",
"moves",
"locally",
"at",
"state",
"s",
"in",
"the",
"direction",
"of",
"the",
"red",
"arrow",
"(along",
"g",
"(s))",
"as",
"time",
"progresses",
"it",
"ends",
"up",
"travelling",
"along",
"the",
"green",
"arrow",
"which",
"is",
"on",
"the",
"manifold",
"Se.",
"There",
"is",
"some",
"constraint",
"that",
"restricts",
"the",
"state",
"space",
"“available”",
"to",
"an",
"agent.",
"Thus,",
"the",
"agent",
"can",
"only",
"reach",
"the",
"states",
"that",
"have",
"a",
"valid",
"solution",
"for",
"the",
"Equation",
"1",
"for",
"any",
"smooth",
"policy.",
"curve.",
"The",
"union",
"of",
"all",
"such",
"curves",
"is",
"the",
"only",
"subset",
"of",
"the",
"state",
"space",
"the",
"agent",
"can",
"reach.",
"The",
"key",
"technical",
"idea",
"is",
"to",
"construct",
"a",
"coordinate",
"chart,",
"and",
"therefore",
"a",
"diffeomorphism,",
"from",
"a",
"subset",
"of",
"low",
"dimensional",
"euclidean",
"space",
"to",
"subset",
"of",
"the",
"state",
"space.",
"We",
"fix",
"a",
"state",
"s",
"and",
"consider",
"the",
"function",
"A",
"si",
"ds",
"such",
"that",
"φ(a,",
"(c",
"d",
"f(s,",
"a,",
"(s,",
"This",
"id:15)",
"pertains",
"su",
"the",
"rtains",
"geometry",
"ob",
"the",
"$",
"(0,",
"r)",
"state",
"manifold,",
"meaning",
"the",
"state",
"manifold",
"everywhere",
"has",
"this",
"low",
"dimensional",
"structure.",
"Our",
"result",
"formalises",
"the",
"long-held",
"idea",
"that",
"there",
"is",
"in",
"fact",
"a",
"lower",
"dimensional",
"structure",
"to",
"the",
"state",
"manifold;",
"in",
"particular,",
"this",
"is",
"true",
"when",
"d",
"+",
"1",
"<",
"a",
"s",
",",
"which",
"is",
"almost",
"always",
"the",
"case",
"for",
"RL",
"environments.",
"It",
"also",
"formalises",
"the",
"idea",
"that",
"the",
"agent",
"can",
"only",
"observe",
"data",
"for",
"states",
"reachable",
"by",
"interacting",
"with",
"the",
"environment",
"using",
"the",
"set",
"of",
"actions",
"at",
"its",
"disposal.",
"Henceforth,",
"we",
"refer",
"to",
"s",
"e",
"as",
"the",
"state",
"manifold.",
"We",
"also",
"present",
"an",
"immediate",
"corollary",
"of",
"Theorem",
"3.2.",
"Corollary",
"3.3.",
"For",
"every",
"a",
"E",
"A",
"and",
"fixed",
"s",
"E",
"s",
"e",
"the",
"action",
"tangent",
"mapping,",
"g(s,",
"a),",
"maps",
"from",
"the",
"set",
"of",
"actions",
"to",
"I",
"s",
"S",
"and",
"therefore",
"there",
"exists",
"a",
"vector",
"v",
"a",
"&",
"'S",
"such",
"that",
"va",
"=",
"g(s,",
"a)",
"for",
"be",
"I",
"s",
"all",
"a",
"This",
"implies",
"that",
"locally",
"an",
"agent",
"is",
"moving",
"along",
"a",
"tangent",
"in",
"the",
"tangent",
"space",
"of",
"a",
"state",
"in",
"the",
"state",
"manifold",
"as",
"it",
"acts",
"upon",
"the",
"environment.",
"The",
"corollary",
"follows",
"from",
"Definition",
"2.2,",
"and",
"its",
"proof",
"is",
"in",
"Appendix",
"B",
"Intuitively,",
"Corollary",
"3.3",
"implies",
"that",
"locally",
"the",
"agent’s",
"state",
"changes",
"in",
"constrained",
"manner",
"dictated",
"by",
"the",
"actions",
"available",
"to",
"it",
"and",
"the",
"local",
"geometry",
"of",
"the",
"manifold,",
"i.e.,",
"the",
"tangent",
"space",
"and",
"its",
"projection",
"onto",
"the",
"state",
"manifold.",
"We",
"illustrate",
"this",
"in",
"Figure",
"1",
"We",
"note",
"that",
"the",
"idea",
"of",
"actions",
"mapping",
"to",
"the",
"tangent",
"space",
"was",
"assumed",
"earlier",
"by",
"Liu",
"et",
"al.",
"(2021)",
"for",
"constrained",
"RL",
"but",
"Corollary",
"3.3",
"both",
"proves",
"this",
"assumption",
"and",
"shows",
"that",
"it",
"holds",
"in",
"general",
"for",
"continuous",
"RL",
"environments.",
"4",
"CONNECTIONS",
"BETWEEN",
"CONTINUOUS",
"TIME",
"DETERMINISTIC",
"RL",
"AND",
"EMPIRICAL",
"RL",
"We",
"have",
"presented",
"our",
"results",
"in",
"the",
"continuous",
"time",
"RL",
"setting,",
"which",
"is",
"an",
"underutilized",
"theoretical",
"tool",
"in",
"the",
"study",
"of",
"RL.",
"Here,",
"and",
"in",
"the",
"experimental",
"section,",
"we",
"argue",
"that",
"it",
"is",
"in",
"fact",
"a",
"useful",
"model",
"for",
"theoretical",
"analysis",
"in",
"the",
"continuous",
"state",
"and",
"action,",
"discrete",
"time,",
"setting.",
"The",
"primary",
"intuition",
"is",
"that,",
"in",
"the",
"context",
"of",
"simulated",
"robotic",
"control",
"problems,",
"our",
"work",
"approximates",
"the",
"agents",
"behavior",
"during",
"the",
"transition",
"period",
"(t,",
"t",
"+",
"1)",
"as",
"the",
"action",
"a",
"t",
"changes",
"to",
"a",
"t+1",
"in",
"a",
"smooth",
"manner",
"and",
"similarly",
"for",
"s",
"t",
"to",
"st+1.",
"In",
"this",
"context,",
"the",
"discrete-time",
"observations",
"can",
"be",
"viewed",
"as",
"time-uniform",
"samples",
"from",
"an",
"underlying",
"continuous",
"time",
"process.",
"This",
"is",
"in",
"keeping",
"with",
"with",
"robotic",
"control,",
"in",
"a",
"robot",
"with",
"various",
"joints",
"we",
"can",
"only",
"measure",
"the",
"joints",
"over",
"intervals",
"which",
"are",
"then",
"attributed",
"to",
"discrete",
"time",
"measurements.",
"Similarly,",
"we",
"can",
"actuate",
"the",
"motors",
"up",
"to",
"a",
"frequency",
"and",
"those",
"can",
"be",
"considered",
"as",
"discrete",
"control",
"signals,",
"even",
"though",
"the",
"underlying",
"physical",
"process",
"is",
"continuous",
"time.",
"5"
]
} | |||
2301.00009v1.pdf_6 | null | ArxivOCR_part_1 | {
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1252,
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1282,
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1353,
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293,
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1946
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379,
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510,
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675,
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1946
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911,
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1946
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1362,
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1362,
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1405,
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293,
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323,
1977
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323,
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412,
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412,
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438,
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487,
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487,
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527,
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527,
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602,
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1058,
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1190,
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1190,
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1271,
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1933,
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293,
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336,
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497,
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541,
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2007
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887,
2007
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887,
1965,
1004,
2007
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1004,
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1033,
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1033,
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2007
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1132,
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1247,
2008
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1247,
1965,
1333,
2007
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1333,
1964,
1403,
2007
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293,
1995,
357,
2038
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357,
1995,
399,
2038
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399,
1994,
523,
2038
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523,
1995,
657,
2038
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657,
1995,
686,
2038
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686,
1995,
764,
2038
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1995,
786,
2037
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1995,
911,
2038
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911,
1995,
942,
2038
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1994,
1020,
2038
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1020,
1991,
1051,
2037
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1051,
1999,
1082,
2038
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1093,
2000,
1112,
2038
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1112,
2011,
1121,
2039
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1130,
2012,
1176,
2042
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1176,
1979,
1207,
2037
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1207,
1980,
1230,
2038
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1241,
1995,
1254,
2038
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1255,
1995,
1300,
2038
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1995,
1402,
2038
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1140,
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1241,
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} | {
"word": [
"12",
"8",
"14",
"ao",
"14",
"12",
"10",
"2",
"0",
"25000",
"50000",
"75000",
"100000",
"2000",
"4000",
"6000",
"2000",
"4000",
"6000",
"8000",
"10000",
"25000",
"50000",
"75000",
"100000",
"Num",
"Samples",
"Num",
"Samples",
"Num",
"Samples",
"Num",
"Samples",
"(a)",
"Walker2D",
"(b)",
"Cheetah",
"(c)",
"Reacher",
"(d)",
"Swimmer",
"Figure",
"3:",
"State",
"manifold",
"dimensionality,",
"in",
"blue,",
"is",
"close",
"to",
"d",
"a",
"+",
"1",
"(horizontal",
"red",
"line)",
"and",
"far",
"below",
"d",
"s",
"(horizontal",
"green",
"line)",
"for",
"various",
"environments,",
"estimated",
"using",
"the",
"algorithm",
"by",
"Facco",
"et",
"al.",
"(2017).",
"More",
"formally,",
"consider",
"a",
"discrete",
"time",
"MDP",
"with",
"continuous",
"state",
"and",
"action",
"spaces,",
"a",
"discrete",
"state",
"trajectory",
"is",
"the",
"sequence",
"of",
"states",
"{",
"St",
"t=1",
"and",
"similarly",
"for",
"actions,",
"(a",
"t",
"t=1.",
"Let",
"o",
":",
"SA",
"x",
"[0,",
"1]",
"x",
"such",
"'",
"be",
"a",
"function",
"that",
"acts",
"as",
"a",
"smooth",
"transformation",
"between",
"two",
"successive",
"actions",
"a",
"t",
",",
"a",
"t+1",
"}",
".",
"that",
"the",
"agent",
"transition",
"from",
"st",
"to",
"s",
"t+1",
"in",
"the",
"continuous",
"time",
"model,",
"as",
"described",
"in",
"Section",
"2.1.",
"In",
"other",
"words,",
"ϑ(at,",
"l,",
"at",
"lt+1,",
")",
"=",
"a",
"t+l",
"is",
"the",
"discrete",
"to",
"continuous",
"time",
"transformation",
"of",
"actions.",
"We",
"postulate",
"that",
"there",
"exists",
"an",
"operator",
"H",
"dependent",
"on",
"the",
"MDP",
"M,",
"such",
"that",
"a",
"discrete",
"trajectory",
"can",
"be",
"transformed",
"into",
"a",
"continuous",
"time",
"trajectory.",
"Proving",
"such",
"an",
"augmentation",
"exists,",
"given",
"that",
"the",
"underlying",
"physical",
"process",
"as",
"described",
"in",
"Section",
"2.1",
"and",
"the",
"discrete",
"trajectory",
"is",
"sampled",
"at",
"discrete",
"time",
"intervals",
"as",
"described",
"above,",
"is",
"beyond",
"the",
"scope",
"of",
"our",
"current",
"work.",
"Intuitively,",
"since",
"the",
"discrete",
"time",
"trajectories",
"are",
"temporally",
"spaced",
"out",
"measurements",
"of",
"continuous",
"trajectories",
"the",
"existence",
"of",
"such",
"an",
"operator",
"can",
"be",
"considered",
"an",
"“inversion”",
"of",
"the",
"sampling",
"process.",
"Finally,",
"we",
"address",
"the",
"assumption",
"of",
"deterministic",
"transitions.",
"For",
"our",
"empirical",
"analysis",
"we",
"use",
"the",
"popular",
"MuJoCo",
"environments",
"for",
"robotic",
"control",
"with",
"sensor",
"inputs",
"(Todorov",
"et",
"al.,",
"2012)",
"and",
"these",
"environments",
"are",
"deterministic",
"for",
"all",
"practical",
"intents",
"and",
"purposes",
"(see",
"Appendix",
"C.",
"We",
"argue",
"further",
"that",
"our",
"theoretical",
"model",
"has",
"broader",
"applicability.",
"One",
"way",
"to",
"model",
"transitions",
"is",
"to",
"assume",
"that",
"the",
"underlying",
"state",
"transitions",
"are",
"deterministic",
"but",
"the",
"observations",
"the",
"agent",
"receives",
"have",
"additive",
"noise.",
"More",
"formally,",
"a",
"simplistic",
"way",
"to",
"model",
"stochastic",
"transition",
"F(s,",
"a,",
"(c",
"d",
"f(s,",
"a,",
"(c",
"d",
"N",
"(0,",
"o)",
"is",
"the",
"stochastic",
"transition",
"function",
"ncti",
"that",
"there",
"is",
"additive",
"ditiv",
"to",
"this",
"deterministic",
"transition.",
"We",
"postulate",
"that",
"the",
"effective",
"set",
"of",
"states",
"can",
"be",
"modeled",
"as",
"a",
"manifold",
"with",
"each",
"observation",
"the",
"agent",
"makes",
"lying",
"at",
"a",
"certain",
"distance",
"to",
"the",
"state",
"manifold",
"and",
"the",
"distance",
"is",
"normally",
"distributed.",
"This",
"is",
"also",
"the",
"idea",
"behind",
"many",
"manifold",
"learning",
"paradigms:",
"the",
"data",
"lies",
"at",
"a",
"distance",
"to",
"a",
"low-dimensional",
"manifold",
"and",
"this",
"distance",
"is",
"distributed",
"according",
"to",
"some",
"probability",
"law",
"(Fefferman",
"et",
"al.,",
"2013;",
"Pai",
"et",
"al.,",
"2019).",
"5",
"EMPIRICAL",
"VALIDATION",
"Our",
"empirical",
"validation",
"is",
"two",
"fold.",
"First,",
"we",
"show",
"that",
"the",
"bound",
"on",
"the",
"manifold",
"dimensionality",
"as",
"in",
"Theorem",
"3.2",
"holds",
"in",
"practice.",
"Second,",
"we",
"demonstrate",
"the",
"practical",
"relevance",
"of",
"our",
"result",
"by",
"learning",
"a",
"“bottleneck”",
"representation",
"using",
"which",
"an",
"RL",
"agent",
"can",
"learn",
"effectively.",
"5.1",
"EMPIRICAL",
"DIMENSIONALITY",
"ESTIMATION",
"To",
"empirically",
"corroborate",
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"word": [
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"dimensionality",
"with",
"10",
"different",
"subsamples",
"of",
"the",
"same",
"size",
"to",
"provide",
"an",
"error",
"region",
"for",
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"estimates.",
"We",
"employ",
"the",
"dimensionality",
"estimation",
"algorithm",
"introduced",
"by",
"Facco",
"et",
"al.",
"(2017),",
"which",
"estimates",
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"intrinsic",
"dimension",
"of",
"datasets",
"characterized",
"by",
"non-uniform",
"density",
"and",
"curvature,",
"to",
"empirically",
"corroborate",
"Theorem",
"3.2.",
"Further",
"details",
"about",
"the",
"dimensionality",
"estimation",
"procedure",
"are",
"presented",
"in",
"Appendix",
"E",
"The",
"estimates",
"for",
"four",
"MuJoCo",
"environments",
"are",
"shown",
"in",
"Figure",
"3",
"For",
"all",
"environments",
"the",
"estimate",
"remains",
"in",
"the",
"neighbourhood",
"of",
"d",
"+",
"1",
"in",
"keeping",
"with",
"Theorem",
"3.2.",
"5.2",
"LEARNING",
"vIA",
"THE",
"LOW-DIMENSIONAL",
"MANIFOLD",
"REPRESENTATION",
"We",
"now",
"validate",
"the",
"relevance",
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"Theorem",
"3.2",
"using",
"popular",
"policy",
"gradient",
"method,",
"deep",
"deter-",
"ministic",
"policy",
"gradient",
"(DDPG)",
"Lillicrap",
"et",
"al.",
"(2016),",
"which",
"is",
"discrete",
"time.",
"DDPG",
"is",
"a",
"framework",
"for",
"learning",
"in",
"environments",
"with",
"continuous",
"action",
"spaces,",
"and",
"the",
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"and",
"value",
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"are",
"parameterised",
"by",
"DNNs.",
"Let",
"θπ",
"be",
"the",
"parameters",
"of",
"the",
"policy",
"DNN.",
"The",
"agent",
"learns",
"by",
"updating",
"the",
"policy",
"parameters",
"with",
"respect",
"to",
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"discrete",
"time",
"discounted",
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"J(θπ):",
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"+",
"θπ",
"+",
"a",
"7T",
"v",
"J(θπ),",
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"a",
"is",
"the",
"learning",
"rate",
"for",
"the",
"policy",
"and",
"J(θπ)",
"is",
"the",
"discounted",
"return",
"objective.",
"For",
"further",
"details",
"and",
"background",
"on",
"DDPG",
"please",
"see",
"Appendix",
"F",
"We",
"do",
"so",
"by",
"learning",
"mapping",
"to",
"low",
"dimensional",
"manifold",
"of",
"size",
"d",
"a",
"+",
"1",
"from",
"the",
"control",
"input",
"space",
"that",
"feeds",
"into",
"the",
"policy.",
"The",
"central",
"idea",
"is",
"to",
"show",
"that",
"this",
"compressed",
"representation",
"can",
"be",
"used",
"to",
"learn",
"a",
"control",
"policy",
"with",
"RL",
"without",
"any",
"loss—and",
"possibly",
"even",
"gain—",
"of",
"performance.",
"Our",
"goal",
"in",
"validating",
"the",
"applicability",
"of",
"this",
"low",
"dimensional",
"representation",
"is",
"to",
"learn",
"an",
"isometric",
"coordinate",
"chart",
"from",
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"high",
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"for",
"states",
"in",
"o",
"e",
"to",
"a",
"low",
"dimensional",
"Euclidean",
"space",
"R",
"m,",
"where",
"m",
"os",
"d",
"+",
"1",
"as",
"noted",
"in",
"Theorem",
"3.2,",
"and",
"then",
"compare",
"learning",
"a",
"policy",
"using",
"this",
"compression",
"of",
"the",
"state",
"to",
"learning",
"from",
"the",
"full",
"state.",
"We",
"denote",
"the",
"coordinate",
"chart",
"by",
"o",
":",
"Se",
"co",
"R",
"datl",
"parameterised",
"by",
"s",
",",
"meaning",
"for",
"chart",
"iS",
"we",
"have",
"that",
"d",
"and",
"s2",
"on",
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"the",
"geodesic",
"distance",
"on",
"the",
"manifold",
"le",
"s",
"2",
")",
"is",
"the",
"geodesic",
"distance",
"between",
"s",
"1",
"on",
"a",
"coordinate",
"||ψ(s1;",
"θψ)",
"C_",
"$2.6",
"0%)",
"12",
",",
"where",
"dSe",
"(s",
"1",
",",
"norm.",
"Note",
"that",
"imposing",
"isometry",
"preserving,",
"in",
"manifold",
"Se",
"and",
"condition",
"than",
"it",
"being",
"a",
"diffeomorphism",
"because",
"it",
"is",
"distance",
"must",
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"is",
"a",
"stronger",
"denotes",
"the",
"Euclidean",
"a",
"coordinate",
"be",
"isometric",
"it",
"For",
"s",
"=",
"s",
"(s",
"addition",
"to",
"being",
"a",
"bijection",
"and",
"differentiable.",
"Such",
"isometric",
"mappings",
"have",
"been",
"used",
"in",
"the",
"past",
"to",
"learn",
"the",
"underlying",
"manifold",
"for",
"high-dimensional",
"data",
"(Tenenbaum",
"et",
"al.,",
"2000;",
"Basri",
"&",
"Jacobs,",
"2017;",
"Pai",
"et",
"al.,",
"2019).",
"This",
"is",
"to",
"P",
"we",
"minimize",
"the",
"loss",
"Given",
"a",
"dataset",
"of",
"states",
"=",
"tractability",
"of",
"the",
"objective",
"that",
"introduce",
"below.",
"oon",
"=",
"L",
"4",
"(0%)",
"=",
"N",
"si,sj",
"ED",
"1",
"(dse",
"(s",
"s",
"2",
")",
"-",
"Il",
"4(s",
"ψ(s",
"-",
"θψ)",
"204)",
"2",
",",
"(3)",
"which",
"is",
"similar",
"to",
"the",
"loss",
"for",
"learning",
"low-dimensional",
"manifold",
"embeddings",
"introduced",
"by",
"Tenen-",
"baum",
"et",
"al.",
"(2000).",
"The",
"computation",
"and",
"estimation",
"of",
"the",
"geodesic",
"distance,",
"d",
"Se",
"is",
"particularly",
"challenging.",
"One",
"approach",
"is",
"to",
"use",
"a",
"graph,",
"as",
"a",
"discrete",
"approximation",
"of",
"a",
"manifold,",
"to",
"calculate",
"this",
"geodesic",
"distance",
"(Yan",
"et",
"al.,",
"2007;",
"Dong",
"et",
"al.,",
"2011).",
"The",
"graph",
"is",
"constructed",
"with",
"each",
"datapoint",
"as",
"a",
"node",
"and",
"an",
"edge",
"in",
"between",
"two",
"datapoint",
"if",
"one",
"of",
"them",
"is",
"onn",
"-nearest",
"neighbor",
"of",
"the",
"other",
"in",
"the",
"dataset.",
"In",
"addition",
"to",
"that",
"there",
"is",
"a",
"distance",
"attribute",
"associated",
"with",
"every",
"edge",
"that",
"is",
"the",
"Euclidean",
"distance",
"between",
"the",
"two",
"points.",
"Therefore,",
"the",
"geodesic",
"distance",
"between",
"two",
"datapoints",
"can",
"be",
"approximated",
"the",
"sum",
"of",
"these",
"edge-wise",
"attributes",
"for",
"the",
"edges",
"constituting",
"the",
"shortest",
"path",
"between",
"the",
"two",
"points.",
"Another",
"practical",
"challenge",
"in",
"the",
"empirical",
"estimation",
"of",
"the",
"loss",
"L",
"is",
"sampling",
"pairs",
"of",
"states",
"for",
"which",
"we",
"can",
"obtain",
"approximate",
"geodesic",
"distances",
"with",
"reasonable",
"compute",
"time.",
"In",
"summary,",
"we",
"calculate",
"d",
"Se",
"in",
"three",
"steps:",
"1)",
"sample",
"data",
"from",
"replay",
"buffer",
"and",
"therefore",
"the",
"manifold",
"'S",
",",
"2)",
"construct",
"a",
"knn",
"i-nearest",
"neighbor",
"graph",
"with",
"edge",
"attributes,",
"e",
"and",
"3)",
"estimate",
"the",
"distance",
"between",
"two",
"states",
"using",
"the",
"sum",
"of",
"edge",
"attributes",
"of",
"the",
"shortest",
"path",
"between",
"these",
"edges.",
"The",
"exact",
"procedure",
"employed",
"by",
"us",
"is",
"detailed",
"in",
"Appendix",
"H",
"and",
"illustrated",
"with",
"simple",
"example",
"in",
"Figure",
"10,",
"further",
"algorithmic",
"details",
"are",
"also",
"provided",
"in",
"Appendix",
"I",
"Although",
"there",
"has",
"been",
"significant",
"work",
"in",
"learning",
"isometric",
"mappings",
"(Pai",
"et",
"al.,",
"2019;",
"Basri",
"&",
"Jacobs,",
"2017)",
"and",
"embeddings",
"(Tenenbaum",
"et",
"al.,",
"2000;",
"Zha",
"&",
"Zhang,",
"2003),",
"for",
"data",
"sampled",
"from",
"a",
"manifold,",
"these",
"prior",
"works",
"assume",
"that",
"the",
"data",
"distribution",
"remains",
"static",
"throughout",
"the",
"training",
"7"
]
} | |||
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"the",
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"is",
"as",
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"θπ",
"+",
"θπ",
"+",
"a",
"O�",
"J(θπ)",
"-",
"&",
"&",
"v",
"L",
"&",
"o%f,",
"v",
"(4)",
"where",
"a",
"and",
"a",
"ap",
"are",
"the",
"learning",
"rates",
"for",
"the",
"policy",
"and",
"the",
"coordinate",
"chart",
"respectively.",
"Note",
"that",
"update",
"for",
"parameters",
"θψ",
"is",
"then",
"θψ",
"$I",
"θψ",
"a",
"p",
"v",
"ox",
"L",
"o%,",
"in",
"addition",
"to",
"the",
"update",
"with",
"respect",
"to",
"the",
"objective",
"for",
"increasing",
"the",
"discounted",
"return",
"$",
"0i",
"L(O4)",
"=",
"0.",
"We",
"illustrate",
"the",
"architecture",
"details).",
"This",
"is",
"because",
"for",
"all",
"θi",
"E",
"θπ",
"o",
"θψ",
"we",
"have",
"Appendix",
"I",
"and",
"Algorithm",
"1",
"for",
"further",
"and",
"how",
"the",
"representation",
"feeds",
"into",
"the",
"loss,",
"for",
"the",
"policy",
"network,",
"in",
"Figure",
"4",
"Note",
"that",
"the",
"architecture",
"and",
"gradient",
"updates",
"for",
"the",
"DNN",
"parameterising",
"the",
"Q",
"function",
"remains",
"the",
"same",
"as",
"used",
"by",
"Lillicrap",
"et",
"al.",
"(2016),",
"for",
"control",
"inputs,",
"which",
"is",
"a",
"two",
"hidden",
"layer",
"DNN",
"with",
"400",
"neurons",
"in",
"the",
"first",
"hidden",
"layer,",
"300",
"in",
"the",
"second",
"one",
"and",
"a",
"one",
"dimensional",
"output.",
"We",
"present",
"results",
"for",
"4",
"different",
"MuJoCo",
"environments:",
"Cheetah,",
"Walker2D,",
"Swimmer",
"and",
"Reacher",
"in",
"Figure",
"5",
"and",
"for",
"three",
"out",
"of",
"four",
"of",
"these",
"environments",
"the",
"performance",
"is",
"either",
"the",
"same",
"or",
"better",
"than",
"the",
"DDPG",
"baseline.",
"Our",
"results",
"strongly",
"suggest",
"that",
"a",
"compressed",
"isometric",
"representation",
"of",
"dimesionality",
"da",
"+",
"1,",
"learned",
"using",
"gradient",
"updates",
"can",
"be",
"used",
"for",
"learning",
"a",
"policy.",
"This",
"furthers",
"our",
"argument:",
"there",
"is",
"low",
"dimensional",
"structure",
"to",
"RL",
"problems",
"and",
"it",
"can",
"be",
"employed",
"to",
"learn",
"more",
"efficiently.",
"Algorithmic",
"details",
"are",
"given",
"in",
"Appendix",
"I",
"Ablation",
"studies",
"for",
"the",
"newly",
"introduced",
"hyper-parameter",
"a",
"b",
"in",
"Appendix",
"J",
"and",
"comparison",
"to",
"training",
"in",
"absence",
"of",
"manifold",
"loss",
"is",
"provided",
"in",
"Appendix",
"M,",
"due",
"to",
"space",
"limitations.",
"In",
"addition",
"to",
"this,",
"we",
"observe",
"that",
"the",
"manifold",
"loss",
"(Equation",
"3)",
"also",
"drops",
"steadily",
"as",
"training",
"progresses,",
"meaning",
"that",
"the",
"representation",
"being",
"learnt,",
"ψ(s;",
"op",
"),",
"is",
"a",
"low-dimensional",
"isometric",
"representation",
"which",
"retains",
"the",
"manifold",
"geometry.",
"The",
"graphs",
"for",
"how",
"the",
"manifold",
"loss,",
"L",
"o",
"evolves",
"in",
"each",
"case",
"is",
"given",
"in",
"Figure",
"6",
"along",
"with",
"the",
"interpretation.",
"We",
"explain",
"the",
"reasons",
"for",
"Reachers’",
"failure",
"for",
"our",
"algorithm",
"in",
"Appendix",
"K.",
"In",
"summary,",
"to",
"learn",
"an",
"isometric",
"mapping",
"to",
"the",
"Euclidean",
"space",
"the",
"agent",
"might",
"require",
"more",
"than",
"dim(",
"S",
")",
"dimensions,",
"for",
"the",
"case",
"of",
"Reacher.",
"The",
"additional",
"constraint",
"that",
"we",
"place,",
"that",
"the",
"learnt",
"coordinate",
"chart",
"has",
"to",
"be",
"isometric,",
"is",
"detrimental",
"in",
"this",
"case.",
"As",
"has",
"been",
"done",
"in",
"manifold",
"learning",
"literature,",
"we",
"need",
"methods",
"for",
"learning",
"the",
"low-dimensional",
"mapping",
"that",
"do",
"not",
"rely",
"on",
"learning",
"an",
"isometry",
"(Roweis",
"&",
"Saul,",
"2000;",
"Zhang",
"&",
"Zha,",
"2004).",
"We",
"demonstrate",
"the",
"effects",
"of",
"changing",
"the",
"width",
"of",
"the",
"bottleneck",
"layer",
"on",
"the",
"Cheetah",
"domain",
"in",
"Figure",
"12,",
"in",
"the",
"Appendix.",
"We",
"also",
"need",
"better",
"methods",
"to",
"approximate",
"the",
"geodesic",
"distance",
"between",
"data",
"points",
"in",
"the",
"very",
"high-dimensional",
"setting.",
"We",
"also",
"report",
"additional",
"results",
"for",
"learning",
"with",
"soft",
"actor",
"critic",
"algorithm",
"(Haarnoja",
"et",
"al.,",
"2018)",
"in",
"conjunction",
"with",
"manifold",
"representation",
"learning",
"in",
"Appendix",
"L",
"6",
"RELATED",
"WORK",
"There",
"has",
"been",
"significant",
"empirical",
"work",
"that",
"assumes",
"the",
"set",
"of",
"states",
"to",
"be",
"a",
"manifold",
"in",
"RL.",
"The",
"primary",
"approach",
"has",
"been",
"to",
"study",
"discrete",
"state",
"spaces",
"as",
"data",
"lying",
"on",
"a",
"graph",
"which",
"has",
"an",
"underlying",
"manifold",
"structure.",
"Mahadevan",
"&",
"Maggioni",
"(2007)",
"provided",
"the",
"first",
"such",
"framework",
"8"
]
} | |||
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"value",
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"Machado",
"et",
"al.",
"(2017)",
"and",
"Jinnai",
"et",
"al.",
"(2020)",
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"Wu",
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"al.",
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"al.",
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"reduction",
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"2000;",
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"al.,",
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"Klock,",
"2020;",
"Goldt",
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"al.,",
"2020;",
"Paccolat",
"et",
"al.,",
"2020;",
"Buchanan",
"et",
"al.,",
"2021).",
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"(2019)",
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"Basri",
"&",
"Jacobs",
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"learn",
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"al.",
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"+",
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"in",
"light",
"of",
"the",
"underlying",
"structure",
"of",
"deep",
"Q-functions",
"(Kumar",
"et",
"al.,",
"2021).",
"9"
]
} | |||
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"word": [
"7",
"DISCUSSION",
"AND",
"CONCLUSION",
"We",
"have",
"shown",
"that",
"that",
"the",
"dimensionality",
"of",
"the",
"manifold",
"is",
"upper",
"bounded",
"by",
"d",
"a",
"+",
"1",
"and",
"have",
"empirically",
"verified",
"it",
"(Figure",
"3.",
"This",
"proves",
"that",
"the",
"popular",
"manifold",
"assumption",
"(Mahadevan,",
"2005;",
"Machado",
"et",
"al.,",
"2017;",
"2018;",
"Banijamali",
"et",
"al.,",
"2018;",
"Wu",
"et",
"al.,",
"2019;",
"Liu",
"et",
"al.,",
"2021)",
"holds",
"under",
"certain",
"conditions",
"in",
"continuous-time",
"reinforcement",
"learning.",
"It",
"also",
"shows",
"that",
"there",
"is",
"an",
"underlying",
"lower",
"dimensional",
"structure",
"to",
"the",
"MDPs.",
"Additionally,",
"we",
"demonstrate",
"the",
"applicability",
"of",
"this",
"result",
"in",
"a",
"practical",
"setting",
"by",
"showing",
"that",
"a",
"DDPG",
"agent",
"can",
"learn",
"efficiently",
"in",
"this",
"highly",
"compressed,",
"low-dimensional",
"space.",
"Our",
"newly",
"introduced",
"architecture",
"and",
"simultaneous",
"low-dimensional",
"representation",
"learning",
"along",
"with",
"policy",
"learning",
"performs",
"on",
"par",
"or",
"better",
"than",
"the",
"baseline",
"DDPG",
"approach,",
"for",
"MuJoCo",
"environments",
"with",
"sensor",
"inputs.",
"Overall,",
"we",
"show",
"a",
"theoretical",
"bound",
"on",
"intrinsic",
"dimensionality",
"of",
"continous",
"RL",
"problem",
"and",
"also",
"show",
"the",
"efficacy",
"of",
"this",
"low-dimensional",
"representation",
"in",
"learning",
"a",
"policy.",
"This",
"opens",
"up",
"room",
"for",
"new",
"theoretical",
"and",
"empirical",
"advances",
"paving",
"way",
"for",
"better",
"DNN",
"architecture",
"design",
"and",
"representation",
"learning",
"algorithms.",
"8",
"REPRODUCIBILITY",
"STATEMENT",
"We",
"offer",
"the",
"following",
"details",
"for",
"all",
"the",
"experiments",
"we",
"have",
"performed,",
"in",
"the",
"main",
"body",
"or",
"appendix:",
"hyperparameters,",
"sample",
"sizes,",
"GPU-hours,",
"CPU-hours,",
"code,",
"neural",
"network",
"architectures,",
"Python",
"libraries",
"used,",
"input",
"sizes,",
"and",
"external",
"code",
"bases.",
"We",
"also",
"provide",
"the",
"code",
"with",
"instructions",
"on",
"running",
"it",
"in",
"the",
"supplementary",
"material.",
"All",
"the",
"details",
"to",
"run",
"the",
"code",
"and",
"its",
"location",
"are",
"in",
"Appendix",
"O.",
"All",
"the",
"hyperparameters,",
"for",
"our",
"experiments",
"in",
"Section",
"5.2,",
"can",
"be",
"found",
"in",
"appendices",
"�",
"and",
"K.",
"REFERENCES",
"Alessio",
"Ansuini,",
"Alessandro",
"Laio,",
"Jakob",
"H.",
"Macke,",
"and",
"Davide",
"Zoccolan.",
"Intrinsic",
"dimension",
"of",
"data",
"representations",
"in",
"deep",
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"In",
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"Rika",
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"and",
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"learning:",
"Unifying",
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"and",
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"URL",
"http://proceedings.mlr.press/",
"v37/.",
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"Robust",
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"and",
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]
} | |||
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} | {
"word": [
"eVAE:",
"Evolutionary",
"Variational",
"Autoencoder",
"Zhangkai",
"Wu,1",
"Longbing",
"Cao,",
"Lei",
"Qi",
"1",
"University",
"of",
"Technology",
"Sydney",
"2",
"Southeast",
"University",
"berenwu1938@gmail.com,",
"Longbing.Cao@uts.edu.au,",
"qilei@seu.edu.cn",
"Abstract",
"201",
"ping",
"inputs",
"to",
"probabilistic",
"representations,",
"and",
"approximat-",
"The",
"surrogate",
"loss",
"of",
"variational",
"autoencoders",
"(VAEs)",
"poses",
"ing",
"the",
"likelihood",
"of",
"outputs.",
"The",
"integration",
"of",
"stochastic",
"various",
"challenges",
"to",
"their",
"training,",
"inducing",
"the",
"imbalance",
"gradient",
"variational",
"Bayes",
"(SGVB)",
"estimator",
"(Kingma",
"and",
"between",
"task",
"fitting",
"and",
"representation",
"inference.",
"To",
"avert",
"Welling",
"2013)",
"with",
"neural",
"settings",
"learns",
"a",
"narrow",
"proba-",
"bilistic",
"latent",
"space",
"to",
"infer",
"more",
"representative",
"attributes",
"this,",
"the",
"existing",
"strategies",
"for",
"VAEs",
"focus",
"on",
"adjusting",
"the",
"in",
"the",
"hidden",
"space.",
"VAEs",
"have",
"been",
"applied",
"in",
"various",
"tradeoff",
"by",
"introducing",
"hyperparameters,",
"deriving",
"a",
"tighter",
"bound",
"under",
"some",
"mild",
"assumptions,",
"or",
"decomposing",
"the",
"domains,",
"including",
"time",
"series",
"forecasting",
"(Fortuin",
"et",
"al.",
"loss",
"components",
"per",
"certain",
"neural",
"settings.",
"VAEs",
"still",
"suffer",
"2019),",
"out-of-domain",
"detection",
"in",
"images",
"(Nalisnick",
"et",
"al.",
"1",
"from",
"uncertain",
"tradeoff",
"learning.We",
"We",
"propose",
"a",
"novel",
"evolu-",
"2019;",
"Ran",
"et",
"al.",
"2022;",
"Havtorn",
"et",
"al.",
"2021;",
"Xiao,",
"Yan,",
"and",
"tionary",
"variational",
"autoencoder",
"(eVAE)",
"building",
"on",
"the",
"vari-",
"Amit",
"2020),",
"generating",
"images",
"with",
"spiking",
"signals",
"(Ka-",
"E",
"ational",
"information",
"bottleneck",
"(VIB)",
"theory",
"and",
"integrative",
"mata,",
"Mukuta,",
"and",
"Harada",
"2022),",
"and",
"generating",
"text",
"by",
"evolutionary",
"neural",
"learning.",
"eVAE",
"integrates",
"a",
"variational",
"language",
"modeling",
"(Zhang",
"et",
"al.",
"2019).",
"of",
"variational",
"infer-",
"genetic",
"algorithm",
"into",
"VAE",
"with",
"variational",
"evolutionary",
"op-",
"Theoretically,",
"the",
"bound",
"optimization",
"erators",
"including",
"variational",
"mutation,",
"crossover,",
"and",
"evolu-",
"8",
"tion.",
"Its",
"inner-outer-joint",
"training",
"mechanism",
"synergistically",
"ence",
"was",
"introduced",
"to",
"substitute",
"the",
"log-likelihood",
"function",
"with",
"a",
"surrogate",
"function",
"to",
"make",
"it",
"optimized",
"by",
"gradient",
"and",
"dynamically",
"generates",
"and",
"updates",
"the",
"uncertain",
"tradeoff",
"learning",
"in",
"the",
"evidence",
"lower",
"bound",
"(ELBO)",
"without",
"addi-",
"descent.",
"In",
"practice,",
"the",
"evidence",
"lower",
"bound",
"(ELBO)",
"can-",
"tional",
"constraints.",
"Apart",
"from",
"learning",
"lossy",
"compression",
"not",
"approach",
"the",
"maximum",
"of",
"conditional",
"likelihood",
"with",
"a",
"and",
"representation",
"of",
"data",
"under",
"the",
"VIB",
"assumption,",
"eVAE",
"close",
"gap",
"between",
"posterior",
"and",
"prior.",
"It",
"causes",
"dynamic",
"un-",
"presents",
"an",
"evolutionary",
"paradigm",
"to",
"tune",
"critical",
"factors",
"of",
"certainty",
"during",
"inference",
"and",
"failed",
"tradeoff",
"between",
"repre-",
"VAEs",
"and",
"deep",
"neural",
"networks",
"and",
"addresses",
"the",
"premature",
"sentation",
"robustness",
"and",
"reconstruction",
"effectiveness.",
"Weak",
"convergence",
"and",
"random",
"search",
"problem",
"by",
"integrating",
"evo-",
"KL",
"divergence",
"could",
"incur",
"KL",
"vanishing",
"while",
"strong",
"KL",
"lutionary",
"optimization",
"into",
"deep",
"learning.",
"Experiments",
"show",
"that",
"eVAE",
"addresses",
"the",
"KL-vanishing",
"problem",
"for",
"text",
"gen-",
"divergence",
"could",
"result",
"in",
"bad",
"likelihood.",
"The",
"tradeoff",
"is",
"sensitive",
"to",
"posterior",
"distribution",
"disjoint,",
"data",
"characteris-",
"eration",
"with",
"low",
"reconstruction",
"loss,",
"generates",
"all",
"disentan-",
"tics,",
"and",
"network",
"architectures",
"(Bozkurt",
"et",
"al.",
"2021).",
"gled",
"factors",
"with",
"sharp",
"images,",
"and",
"improves",
"the",
"image",
"gen-",
"eration",
"quality,respectively.",
"eVAE",
"achieves",
"better",
"reconstruc-",
"Recently,",
"various",
"techniques",
"have",
"been",
"proposed",
"to",
"ad-",
"tion",
"loss,",
"disentanglement,",
"and",
"generation-inference",
"balance",
"dress",
"these",
"issues.",
"The",
"first",
"set",
"aims",
"to",
"adjust",
"the",
"balance",
"than",
"its",
"competitors.",
"in",
"objective",
"functions.",
"Examples",
"are",
"β-VAE",
"incorporating",
"a",
"hyperparameter",
"B",
"(Higgins",
"et",
"al.",
"2017),",
"InfoVAE",
"adding",
"a",
"scaling",
"parameter",
"to",
"the",
"KL",
"term",
"(Zhao,",
"Song,",
"and",
"Er-",
"Introduction",
"Variational",
"autoencoders",
"(Kingma",
"and",
"Welling",
"2013)",
"mon",
"2019),",
"SA-VAE",
"having",
"a",
"cyclical",
"annealing",
"schedule",
"to",
"progressively",
"increase",
"B",
"for",
"reducing",
"KL",
"vanishing",
"(Fu",
"(VAEs)",
"have",
"attracted",
"significant",
"interest",
"for",
"their",
"capability",
"et",
"al.",
"2019),",
"and",
"ControlVAE",
"introducing",
"the",
"proportional-",
"of",
"learning",
"continuous",
"and",
"smooth",
"distributions",
"from",
"obser-",
"integral-derivative",
"(PID)",
"control",
"to",
"tune",
"the",
"hyperparame-",
"vations",
"by",
"integrating",
"probabilistic",
"and",
"deep",
"neural",
"learning",
"ter",
"(Shao",
"et",
"al.",
"2020).",
"They",
"are",
"partial",
"solutions",
"only",
"ad-",
"face",
"one",
"part",
"of",
"the",
"objectives,",
"failing",
"to",
"weigh",
"and",
"re-",
"principles.",
"However,",
"VAEs",
"still",
"learning",
"during",
"variational",
"justing",
"the",
"balance",
"issue",
"in",
"dynamic",
"settings.",
"Another",
"set",
"is",
"including",
"dynamic",
"uncertainty",
"representation",
"compres-",
"to",
"tune",
"network",
"architectures",
"or",
"VAE",
"settings,",
"e.g.,",
"VQ-",
"inference",
"and",
"the",
"tradeoff",
"between",
"some",
"significant",
"issues,",
"solve",
"sion",
"and",
"task",
"fitting,",
"despite",
"various",
"recent",
"VAE",
"mecha-",
"VAE",
"and",
"their",
"variants",
"(Razavi,",
"Van",
"den",
"Oord,",
"and",
"Vinyals",
"nisms",
"and",
"variants.",
"Below,",
"we",
"briefly",
"analyze",
"these",
"issues",
"and",
"the",
"gaps",
"of",
"existing",
"solutions.",
"A",
"novel",
"evolutionary",
"VAE",
"2019;",
"Berliner",
"et",
"al.",
"2022),",
"spiking",
"VAE",
"(Kamata,",
"Mukuta,",
"(eVAE)",
"framework",
"is",
"then",
"introduced",
"to",
"address",
"some",
"of",
"and",
"Vogt",
"2021),",
"mixture-of-experts",
"multimodal",
"VAE",
"(Shi",
"and",
"Harada",
"2022),",
"multimodal",
"VAE",
"(Sutter,",
"Daunhawer,",
"these",
"issues",
"and",
"gaps.",
"et",
"al.",
"2019)",
"and",
"tuning",
"of",
"their",
"ELBO",
"(Kviman",
"et",
"al.",
"2022;",
"VAEs",
"and",
"Gap",
"Analysis",
"VAEs",
"have",
"demonstrated",
"sig-",
"Jankowiak",
"and",
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} | |||
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"and",
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]
} | |||
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} | {
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]
} | |||
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} | {
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} | |||
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} | |||
2301.00013v1.pdf_2 | null | ArxivOCR_part_1 | {
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} | {
"word": [
"3",
"III.",
"EXAMPLE:",
"A",
"BEAR",
"FALLS",
"OUT",
"OF",
"A",
"ysis",
"assumes",
"a",
"trampoline",
"leg",
"height",
"of",
"1",
"meter,",
"which,",
"TREE",
"based",
"on",
"the",
"extracted",
"gravitational",
"acceleration",
"values,",
"To",
"further",
"illustrate",
"how",
"useful",
"this",
"approach",
"can",
"be,",
"consider",
"this",
"2003",
"video",
"of",
"a",
"bear",
"being",
"removed",
"from",
"a",
"tree",
"in",
"Missoula",
"Montana",
"via",
"a",
"tranquilizer",
"gun",
"and",
"a",
"trampoline.",
"https://www",
"putube.com/watch?",
"3",
"v=9KiJnTGoPPI",
"[8].",
"There",
"are",
"quite",
"a",
"few",
"different",
"ap-",
"e",
"2",
"plication",
"topics",
"available",
"in",
"the",
"video,",
"and",
"some",
"of",
"my",
"pure",
"students",
"are",
"outraged",
"that",
"the",
"bear",
"was",
"so",
"made",
"fun",
"of",
"1",
"by",
"the",
"Missoula",
"fire,",
"wildlife,",
"and",
"police",
"departments.",
"maei",
"So,",
"an",
"ethical",
"question:",
"“Assuming",
"the",
"bear",
"had",
"to",
"be",
"removed",
"from",
"the",
"tree,",
"was",
"bouncing",
"it",
"off",
"a",
"trampoline",
"a",
"bear",
"-1",
"window",
"humane",
"thing",
"to",
"do?”",
"With",
"encouragement,",
"the",
"students",
"bear",
"relative",
"to",
"window",
"-2",
"can",
"develop",
"this",
"question",
"into",
"a",
"quantifiable",
"measure,",
"12.3",
"m/s",
"slope",
"6.0",
"m/s",
"slope",
"eg,",
"“What",
"would",
"the",
"bear’s",
"speed",
"be",
"if",
"there",
"was",
"no",
"trampoline",
"in",
"place?”",
"There",
"are",
"many",
"graphs",
"online",
"that",
"16.00",
"16.25",
"16.50",
"16.75",
"17.00",
"17.25",
"17.50",
"17.75",
"time",
"(s)",
"show",
"the",
"fatality",
"risk",
"for",
"pedestrians",
"who",
"are",
"struck",
"by",
"cars",
"at",
"different",
"speeds,",
"for",
"example",
"[9],",
"and",
"it",
"seems",
"realistic",
"to",
"extrapolate",
"this",
"to",
"a",
"decrease",
"in",
"harm",
"to",
"the",
"FIG.",
"7.",
"A",
"plot",
"showing",
"the",
"vertical",
"position-time",
"graph",
"for",
"both",
"the",
"bear,",
"the",
"lower",
"left",
"corner",
"of",
"the",
"house’s",
"left",
"bear",
"if",
"its",
"final",
"velocity",
"is",
"reduced.",
"transom",
"window",
"and",
"the",
"bear’s",
"(calculated)",
"motion",
"relative",
"to",
"the",
"window.",
"526/912",
"(7.761",
"2.253)",
"17.533",
"is",
"probably",
"inaccurate.",
"Straight-line",
"fits",
"the",
"bear’s",
"vertical",
"position",
"give",
"to",
"E",
"speeds",
"of",
"12.3m/s",
"27mph",
"44kmph",
"the",
"bear",
"2",
"S",
"hits",
"the",
"trampoline,",
"and",
"5.9m/s",
"13mph",
"21kmph",
"at",
"about",
"the",
"same",
"altitute",
"after",
"bouncing",
"off",
"the",
"tram-",
"poline.",
"Based",
"on",
"the",
"data",
"in",
"Figure",
"1b",
"of",
"[9]",
"this",
"cor-",
"responds",
"to",
"a",
"reduction",
"of",
"pedestrian",
"fatality",
"risk",
"from",
"5%",
"to",
"less",
"than",
"1%.",
"Extrapolating",
"from",
"vehicle",
"fatality",
"data,",
"one",
"could",
"arge",
"that",
"in",
"addition",
"to",
"being",
"good",
"com-",
"edy,",
"using",
"a",
"trampoline",
"in",
"this",
"case",
"is",
"humane",
"wildlife",
"management.",
"IV.",
"CONCLUSION",
".00",
"-.",
"��",
"Cellphone",
"video",
"is",
"everywhere",
"in",
"a",
"way",
"that",
"was",
"FIG.",
"6.",
"A",
"screenshot",
"from",
"LoggerPro,",
"showing",
"the",
"position-",
"unimaginable",
"20",
"years",
"ago.",
"Most",
"of",
"the",
"videos",
"that",
"stu-",
"time",
"track",
"for",
"both",
"the",
"bear",
"and",
"the",
"lower",
"left",
"corner",
"of",
"dents",
"might",
"take",
"for",
"a",
"kinematics",
"assignment",
"won’t",
"be",
"the",
"house’s",
"left",
"transom",
"window.",
"The",
"calibration",
"used",
"is",
"shot",
"from",
"a",
"tripod",
"with",
"a",
"constant",
"depth",
"of",
"field.",
"Ac-",
"the",
"trampoline’s",
"leg",
"height",
"of",
"1m.",
"The",
"calibration",
"stick",
"has",
"cordingly,",
"talking",
"about",
"how",
"to",
"use",
"topics",
"from",
"Physics",
"drifted",
"off",
"to",
"the",
"left",
"because",
"of",
"the",
"movement",
"of",
"the",
"camera.",
"to",
"correct",
"“errors”",
"in",
"video",
"data",
"collection",
"can",
"be",
"empow-",
"The",
"calibration",
"length",
"is",
"only",
"an",
"estimate",
"and",
"should",
"prob-",
"ering",
"for",
"students.",
"If",
"there’s",
"a",
"fixed",
"reference,",
"a",
"student",
"ably",
"be",
"back",
"fit",
"to",
"the",
"appropriate",
"gravitational",
"acceleration",
"no",
"longer",
"needs",
"to",
"be",
"told,",
"“That",
"video",
"is",
"bad,",
"go",
"take",
"value",
"for",
"the",
"bear’s",
"free-fall.",
"it",
"again...”.",
"The",
"bear",
"video",
"was",
"shot",
"by",
"a",
"professional",
"videogra-",
"pher,",
"Mark",
"Hoyoak,",
"and",
"the",
"focus",
"of",
"the",
"camera",
"follows",
"ACKNOWLEDGMENTS",
"the",
"bear.",
"Ignoring",
"the",
"early",
"parts",
"of",
"the",
"bear’s",
"fall",
"when",
"the",
"zoom",
"level",
"changes,",
"the",
"video",
"provides",
"a",
"falling",
"body,",
"The",
"work",
"would",
"not",
"have",
"been",
"possible",
"without",
"Mark",
"tracked",
"by",
"a",
"moving",
"camera.",
"If",
"you",
"consider",
"one",
"of",
"the",
"Hoyoak’s",
"excellent",
"videography.",
"Thanks",
"also",
"to",
"Eugenia",
"house’s",
"transom",
"windows",
"to",
"be",
"a",
"static",
"reference,",
"camera",
"Etkina",
"for",
"introducing",
"me",
"to",
"video",
"analysis",
"many",
"years",
"movement",
"can",
"be",
"removed.",
"ago.",
"Thanks",
"also",
"to",
"Peter",
"Bohacek",
"for",
"his",
"inspiring",
"talks",
"This",
"analysis",
"is",
"shown",
"in",
"figures",
"6,",
"7,",
"and",
"8",
"The",
"anal-",
"and",
"amazing",
"Direct",
"Motion",
"Video",
"examples."
]
} | |||
2301.00013v1.pdf_4 | null | ArxivOCR_part_1 | {
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} | {
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"3",
"e",
"dore",
"2",
"1",
"mah",
"*",
"bear",
"window",
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"to",
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"g",
"=",
"-9.4",
"m/s�",
"-2",
"16.00",
"16.25",
"16.50",
"16.75",
"17.00",
"17.25",
"17.50",
"17.75",
"time",
"(s)",
"FIG.",
"8.",
"A",
"quadratic",
"fit",
"to",
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"bear’s",
"position",
"(relative",
"to",
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"window)",
"shows",
"a",
"time2",
"polynomial",
"with",
"gravitational",
"ac-",
"P",
"0.1",
"m/s2.",
"This",
"is",
"obviously",
"not",
"celeration",
"of",
"about",
"g",
"=",
"9.4",
"the",
"standard",
"g",
"value",
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"“clicking",
"error”",
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"the",
"motion",
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"calibration",
"error",
"in",
"assuming",
"the",
"trampoline",
"is",
"1",
"meter",
"high.",
"Never-the-less,",
"the",
"corrected",
"motion",
"(unlike",
"that",
"of",
"the",
"bear",
"from",
"the",
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"camera)",
"is",
"clearly",
"quadratic,",
"which",
"is",
"a",
"demonstration",
"of",
"the",
"utility",
"of",
"the",
"approach",
"used",
"in",
"this",
"paper.",
"[1]",
"The",
"ISLE",
"approach",
"to",
"learning",
"physics",
"is",
"described",
"[5]",
"Tracker",
"is",
"a",
"free,",
"open-source",
"tool",
"that",
"you",
"can",
"install",
"on",
"in",
"E.",
"Etkina,",
"Millikan",
"award",
"lecture:",
"Students",
"of",
"your",
"computer",
"or",
"run",
"in",
"a",
"web",
"browser.",
"It",
"is",
"available",
"physics—Listeners,",
"observers,",
"or",
"collaborative",
"partici-",
"online",
"at",
"https://physlets.org/tracker/",
"pants",
"in",
"physics",
"scientific",
"practices?",
"Am.",
"J.",
"Phys.",
"[6]",
"Vernier’s",
"LoggerPro",
"is",
"typically",
"used",
"for",
"lab",
"data",
"acquisi-",
"83(8),",
"669",
"(2015).",
"and",
"D.",
"T.",
"Brookes",
"and",
"E.",
"Etkina",
"Phys-",
"tion",
"but",
"it",
"contains",
"an",
"excellent",
"video",
"analysis",
"tool",
"that",
"ical",
"Phenomena",
"in",
"real",
"time,",
"Science",
"330,",
"605",
"(2010).",
"this",
"paper",
"employs.",
"https",
"//www.vernier.com/product/",
"Here",
"is",
"a",
"pointer",
"to",
"the",
"video",
"archive",
"I’m",
"using",
"http:",
"logger-pro-3/",
"n",
"[7]",
"Here",
"is",
"the",
"help",
"page",
"for",
"calibration",
"sticks",
"in",
"Tracker",
"//islevideos.net/",
"[2]",
"The",
"ball",
"toss",
"video",
"used",
"in",
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"first",
"part",
"of",
"the",
"pa-",
"https://physlets.org/tracker/help/frameset.html.",
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"available",
"for",
"download",
"at",
"http://islevideos.net/",
"The",
"process",
"in",
"LoggerPro",
"is",
"similar.",
"[8]",
"There",
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"many",
"copies",
"of",
"this",
"video",
"on",
"the",
"web.",
"It",
"seems",
"experiment",
"ph",
"p?topicid=2&exptid=95.",
"[3]",
"See",
"the",
"excellent",
"introduction",
"at",
"https://www.youtube.",
"that",
"the",
"original",
"video",
"was",
"taken",
"by",
"Mark",
"Hoyoak",
"of",
"KPAX",
"News",
"in",
"Montana",
"on",
"September",
"9,",
"2003.",
"The",
"clip",
"com/watch?v=QsGMKv8Lrew",
"[4]",
"Peter",
"Bohacek’s",
"YouTube",
"channel",
"contains",
"a",
"large",
"num-",
"was",
"subsequently",
"featured",
"on",
"national",
"news",
"and",
"comedy",
"ber",
"of",
"these",
"videos",
"https://www.youtube.com/user/",
"programs.",
"For",
"an",
"overview,",
"see",
"https://www.youtube.",
"bohacekphysics.",
"A",
"nice",
"overview",
"of",
"the",
"approach",
"he",
"com/watch?v=jB47Vucoj2o",
"takes",
"is",
"given",
"in",
"https://www.youtube.com/watch?v=",
"[9]",
"E.",
"Ros´en",
"and",
"U.",
"Sander,",
"Pedestrian",
"fatality",
"risk",
"as",
"a",
"func-",
"QsGMKv8Lrew.",
"These",
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"are",
"available",
"as",
"a",
"commercial",
"tion",
"of",
"car",
"impact",
"speed,",
"Accident",
"Analysis",
"and",
"Prevention",
"curriculum",
"at",
"htt",
"ps://www.pivotinteractives.com/.",
"41(3)",
"536",
"(2009)."
]
} | |||
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} | {
"word": [
"Time",
"series",
"Forecasting",
"to",
"detect",
"anomalous",
"behaviours",
"in",
"Multiphase",
"Flow",
"Meters",
"T.",
"Barbariol,",
"Università",
"degli",
"Studi",
"di",
"Padova",
"D.",
"Masiero,",
"Università",
"degli",
"Studi",
"di",
"Padova",
"E.",
"Feltresi,",
"Pietro",
"Fiorentini",
"G.A.",
"Susto,",
"Università",
"degli",
"Studi",
"di",
"Padova",
"INTRODUCTION",
"Multiphase",
"flow",
"meters",
"(MPFM)",
"are",
"inline",
"metering",
"tools,",
"able",
"to",
"measure",
"the",
"individual",
"flow",
"rates",
"of",
"a",
"multiphase",
"flow.",
"These",
"instruments",
"measure",
"multiple",
"properties",
"of",
"the",
"fluid",
"and",
"then",
"compute",
"the",
"water,",
"oil",
"and",
"gas",
"flow",
"rates.",
"These",
"derived",
"quantities",
"are",
"then",
"fed",
"into",
"reservoir",
"models",
"in",
"order",
"to",
"make",
"predictions",
"on",
"the",
"future",
"production,",
"or",
"to",
"allocate",
"revenues",
"and",
"taxes.",
"The",
"main",
"fluid",
"properties",
"measured",
"by",
"the",
"MPFM",
"are",
"the",
"electrical",
"impedance,",
"the",
"gamma-ray",
"absorption,",
"and",
"the",
"velocity",
"of",
"the",
"flow.",
"These",
"are",
"sampled",
"by",
"means",
"of",
"three",
"modules:",
"electrical",
"probes,",
"gamma-ray",
"densitometer",
"and",
"Venturi",
"tube.",
"The",
"reliability",
"of",
"these",
"modules",
"is",
"of",
"primary",
"importance:",
"if",
"one",
"of",
"them",
"experiences",
"a",
"fault,",
"the",
"provided",
"flow",
"measurements",
"will",
"be",
"incorrect.",
"This",
"might",
"affect",
"in",
"cascade",
"all",
"the",
"following",
"steps,",
"leading",
"to",
"wrong",
"decisions",
"about",
"the",
"reservoir",
"[1].",
"Without",
"proper",
"supervision,",
"one",
"module",
"might",
"fail",
"without",
"anyone",
"noticing",
"it",
"and",
"without",
"expert",
"human",
"intervention.",
"In",
"order",
"to",
"prevent",
"these",
"situations,",
"a",
"system",
"able",
"to",
"detect",
"anomalous",
"behaviours",
"is",
"needed:",
"this",
"system",
"has",
"to",
"work",
"autonomously,",
"and",
"to",
"adapt",
"to",
"different",
"working",
"conditions",
"without",
"losing",
"its",
"efficacy.",
"These",
"intelligent",
"modules",
"go",
"under",
"the",
"name",
"of",
"Anomaly",
"Detection",
"(AD),",
"or,",
"more",
"in",
"general,",
"Fault",
"Detection",
"(FD)",
"systems.",
"They",
"are",
"applied",
"to",
"multiple",
"objects",
"that",
"have",
"strong",
"reliability",
"requirements,",
"such",
"as:",
"smart",
"grids",
"[2],",
"medical",
"devices",
"[3],",
"autonomous",
"vehicles",
"[4].",
"Depending",
"on",
"the",
"approach,",
"AD/FD",
"technologies",
"can",
"be",
"divided",
"into",
"two",
"categories:",
"model-driven",
"or",
"data-driven",
"approaches.",
"Due",
"to",
"its",
"high",
"flexibility",
"and",
"effectiveness",
"in",
"this",
"work",
"we",
"described",
"the",
"adoption",
"of",
"data-driven",
"approaches",
"in",
"the",
"field",
"of",
"MPFM.",
"This",
"is",
"one",
"of",
"the",
"first",
"works",
"that",
"addresses",
"the",
"fault",
"detection",
"problem",
"in",
"Multiphase",
"Flow",
"Meters,",
"employing",
"time-series",
"forecasting",
"tools.",
"Moreover,",
"this",
"is",
"one",
"of",
"the",
"earliest",
"applications",
"of",
"the",
"Temporal",
"Convolutional",
"Network",
"to",
"time-series",
"anomaly",
"detection.",
"This",
"paper",
"is",
"structured",
"as",
"follows:",
"in",
"Section",
"2",
"the",
"MPFM",
"fault",
"detection",
"problem",
"addressed",
"in",
"this",
"paper",
"will",
"be",
"described",
"in",
"more",
"detail,",
"together",
"with",
"the",
"algorithmic",
"tools",
"employed",
"in",
"this",
"analysis.",
"Then",
"in",
"Section",
"3",
"these",
"tools",
"will",
"be",
"applied",
"to",
"real",
"faults.",
"In",
"the",
"last",
"Section",
"the",
"conclusions",
"of",
"this",
"paper",
"will",
"be",
"discussed.",
"2",
"MATERIALS",
"AND",
"METHODS:",
"MPFM",
"and",
"ML",
"2.1",
"Multiphase",
"flow",
"meters",
"As",
"already",
"mentioned",
"in",
"the",
"introduction,",
"the",
"MPFM",
"is",
"a",
"complex",
"metrology",
"instrument,",
"able",
"to",
"quantify",
"the",
"individual",
"flow",
"rates",
"of",
"a",
"mixture.",
"It",
"is",
"placed",
"on",
"1"
]
} | |||
2301.00014v1.pdf_2 | null | ArxivOCR_part_1 | {
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"word": [
"the",
"top",
"of",
"an",
"oil",
"well",
"to",
"measure",
"the",
"oil",
"and",
"gas",
"production.",
"Since",
"the",
"reservoir",
"has",
"an",
"evolution",
"in",
"time,",
"also",
"the",
"MPFM",
"experiences",
"a",
"variation",
"of",
"its",
"measures",
"during",
"the",
"reservoir",
"lifetime.",
"It",
"makes",
"continuous",
"measurements",
"of",
"the",
"fluid",
"properties,",
"in",
"particular",
"electrical",
"impedance",
"(C)",
"and",
"gamma-ray",
"absorption",
"(G).",
"Keep",
"in",
"mind",
"that",
"these",
"two",
"sensors",
"might",
"not",
"be",
"placed",
"at",
"the",
"same",
"point",
"of",
"the",
"pipeline,",
"and",
"therefore",
"their",
"measurements",
"might",
"not",
"be",
"aligned",
"in",
"time.",
"This",
"apparent",
"disadvantage",
"can",
"be",
"seen",
"as",
"an",
"advantage",
"when",
"one",
"wants",
"to",
"predict",
"the",
"behaviour",
"of",
"the",
"subsequent",
"sensor",
"(in",
"this",
"case",
"G)",
"given",
"the",
"precedent",
"sensor",
"(C).",
"The",
"method",
"explained",
"in",
"the",
"following",
"paragraphs",
"will",
"be",
"applied",
"to",
"these",
"two",
"sensor",
"measurements",
"since",
"they",
"exhibit",
"a",
"quite",
"high",
"correlation",
"at",
"a",
"small",
"time",
"scale.",
"2.2",
"Fault",
"Detection",
"The",
"proposed",
"approach",
"is",
"widely",
"adopted",
"in",
"many",
"industrial",
"applications",
"[5,6].",
"Since",
"sensor",
"measurements",
"are",
"continuous,",
"the",
"most",
"natural",
"way",
"to",
"tackle",
"the",
"fault",
"detection",
"problem",
"is",
"by",
"using",
"time",
"series-based",
"approaches.",
"The",
"one",
"used",
"in",
"this",
"paper",
"consists",
"of",
"three",
"steps:",
"(i)",
"forecasting,",
"(ii)",
"residue",
"analysis",
"and",
"(iii)",
"fault",
"alarm.",
"The",
"main",
"idea",
"behind",
"it",
"is:",
"forecasting:",
"at",
"each",
"time",
"step",
"t",
"the",
"following",
"value",
"at",
"time",
"t+k",
"is",
"-",
"predicted",
"based",
"on",
"values",
"in",
"the",
"time",
"interval",
"[t-n,t],",
"where",
"k",
"is",
"the",
"number",
"of",
"steps",
"ahead,",
"and",
"n",
"is",
"the",
"size",
"of",
"the",
"selected",
"time",
"interval.",
"residue",
"analysis",
"and",
"fault",
"alarm:",
"at",
"time",
"t+k,",
"if",
"the",
"difference",
"between",
"the",
"predicted",
"value",
"and",
"the",
"actual",
"value",
"is",
"too",
"high,",
"the",
"actual",
"value",
"is",
"considered",
"anomalous.",
"In",
"the",
"following",
"paragraphs,",
"this",
"simple",
"idea",
"will",
"be",
"better",
"formalised",
"and",
"explained.",
"Time",
"Series",
"Forecasting",
"There",
"exist",
"two",
"types",
"of",
"time-series",
"forecasting:",
"the",
"short-term",
"forecasting",
"when",
"the",
"model",
"tries",
"to",
"predict",
"only",
"one",
"step",
"ahead",
"(k=1),",
"and",
"the",
"long-term",
"forecasting",
"when",
"the",
"model",
"forecasts",
"multiple",
"steps",
"ahead",
"(k>1).",
"This",
"last",
"approach",
"might",
"be",
"less",
"computationally",
"intensive,",
"but",
"it",
"is",
"for",
"sure",
"less",
"accurate",
"than",
"the",
"short-term",
"forecasting.",
"For",
"this",
"reason",
"in",
"this",
"paper",
"the",
"choice",
"was",
"to",
"adopt",
"a",
"one",
"one-step",
"ahead",
"forecasting.",
"In",
"recent",
"years",
"many",
"models",
"were",
"developed",
"for",
"this",
"kind",
"of",
"task.",
"A",
"particular",
"branch",
"of",
"Machine",
"Learning,",
"named",
"Deep",
"Learning,",
"is",
"proving",
"its",
"superiority",
"in",
"many",
"fields",
"like",
"computer",
"vision",
"and",
"natural",
"language",
"processing.",
"The",
"most",
"promising",
"model,",
"developed",
"for",
"time",
"series",
"forecasting,",
"is",
"the",
"Temporal",
"Convolutional",
"Network",
"TCN",
"[7,8].",
"In",
"the",
"present",
"work,",
"the",
"authors",
"decided",
"to",
"test",
"this",
"tool",
"with",
"two",
"configurations,",
"the",
"endogenous",
"and",
"exogenous",
"one.",
"The",
"first",
"has",
"the",
"same",
"input",
"and",
"output",
"signal",
"(G)",
"while",
"the",
"second",
"has",
"different",
"signals",
"(in",
"the",
"following",
"example,",
"C",
"as",
"input",
"and",
"G",
"as",
"output).",
"Before",
"going",
"any",
"further,",
"it",
"is",
"better",
"to",
"fix",
"the",
"notation:",
"x(t)",
"represents",
"the",
"input",
"signal,",
"while",
"y(t)",
"the",
"output",
"one.",
"A",
"generic",
"model",
"takes",
"a",
"batch",
"of",
"input",
"values",
"[x(t-n),",
"-1),...",
"x(t-1),x(t)]",
"and",
"gives",
"back",
"the",
"short-term",
"forecast",
"y(t+1).",
"Two",
"additional",
"models",
"were",
"tested",
"to",
"assess",
"the",
"goodness",
"of",
"the",
"TCN",
"performances.",
"The",
"first,",
"called",
"Naive",
"forecasting,",
"is",
"a",
"common",
"baseline:",
"it",
"predicts",
"the",
"value",
"at",
"t+1",
"as",
"the",
"value",
"collected",
"at",
"time",
"t.",
"The",
"second,",
"named",
"Hard",
"Subtraction,",
"follows",
"the",
"approach",
"originally",
"developed",
"in",
"[9]:",
"one",
"signal",
"is",
"approximated",
"by",
"the",
"2"
]
} | |||
2301.00014v1.pdf_3 | null | ArxivOCR_part_1 | {
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} | {
"word": [
"time-aligned",
"version",
"of",
"another,",
"correlated",
"signal.",
"If",
"y",
"and",
"x",
"are",
"misaligned",
"by",
"m",
"time",
"steps,",
"then",
"the",
"forecast",
"will",
"be",
"y(t+1)=x(t+1-m).",
"Residue",
"analysis",
"and",
"fault",
"alarm",
"Once",
"the",
"forecast",
"has",
"been",
"obtained,",
"the",
"next",
"step",
"is",
"the",
"analysis",
"of",
"the",
"residue",
"between",
"the",
"forecast",
"and",
"the",
"actual",
"value.",
"Obviously,",
"the",
"higher",
"this",
"difference",
"is,",
"the",
"higher",
"will",
"be",
"the",
"probability",
"that",
"the",
"actual",
"value",
"is",
"indeed",
"anomalous.",
"A",
"threshold",
"on",
"the",
"residue",
"is",
"needed",
"in",
"order",
"to",
"discriminate",
"if",
"one",
"value",
"has",
"to",
"be",
"flagged",
"as",
"anomalous",
"or",
"not.",
"This",
"threshold",
"can",
"be",
"learned",
"in",
"advance,",
"by",
"looking",
"at",
"some",
"statistical",
"properties",
"of",
"the",
"past",
"residues.",
"More",
"specifically,",
"the",
"procedure",
"followed",
"in",
"this",
"paper",
"is",
"the",
"following.",
"Since",
"the",
"residue",
"has",
"a",
"quite",
"strong",
"fluctuation,",
"a",
"time-window",
"has",
"been",
"defined",
"and",
"the",
"instantaneous",
"residue",
"has",
"been",
"substituted",
"by",
"two",
"derived",
"quantities:",
"the",
"absolute",
"rolling",
"mean",
"(mean)",
"and",
"the",
"absolute",
"rolling",
"standard",
"deviation",
"(std).",
"The",
"thresholds",
"for",
"these",
"quantities",
"have",
"been",
"learnt",
"in",
"advance,",
"when",
"the",
"MPFM",
"was",
"working",
"in",
"controlled",
"conditions.",
"Each",
"threshold",
"corresponds",
"to",
"the",
"maximum",
"value",
"assumed",
"by",
"its",
"corresponding",
"quantity",
"in",
"this",
"specific",
"time",
"interval.",
"When",
"the",
"mean",
"or",
"the",
"std",
"exceeds",
"its",
"threshold,",
"the",
"fault",
"is",
"detected",
"and",
"an",
"alarm",
"is",
"raised.",
"3",
"RESULTS",
"In",
"this",
"Section,",
"the",
"results",
"of",
"the",
"simulations",
"will",
"be",
"described,",
"starting",
"from",
"the",
"forecast,",
"passing",
"through",
"the",
"residue",
"analysis,",
"and",
"ending",
"with",
"real",
"and",
"synthetic",
"fault",
"tests",
"that",
"will",
"prove",
"the",
"effectiveness",
"of",
"the",
"proposed",
"method.",
"Two",
"signals",
"have",
"been",
"collected:",
"the",
"electric",
"impedance",
"of",
"the",
"flow",
"C,",
"and",
"the",
"gamma-ray",
"attenuation",
"G.",
"For",
"simplicity",
"the",
"target",
"signal",
"will",
"be",
"G,",
"but",
"for",
"construction",
"an",
"anomaly",
"on",
"G",
"will",
"trigger",
"an",
"anomaly",
"C",
"too",
"and",
"vice",
"versa.",
"3.1",
"Forecast",
"The",
"metric",
"that",
"shows",
"the",
"general",
"goodness",
"of",
"the",
"forecast",
"is",
"depicted",
"in",
"Figure",
"1.",
"It",
"is",
"the",
"mean",
"squared",
"error",
"(mse),",
"a",
"metric",
"commonly",
"used",
"in",
"ML",
"to",
"assess",
"the",
"performance",
"of",
"a",
"model.",
"Lower",
"it",
"is,",
"the",
"better",
"the",
"model",
"is",
"performing.",
"What",
"is",
"important",
"is",
"the",
"mse",
"value",
"of",
"forecasting",
"models,",
"compared",
"to",
"the",
"Naive",
"forecasting",
"(the",
"baseline).",
"3"
]
} | |||
2301.00014v1.pdf_4 | null | ArxivOCR_part_1 | {
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"100",
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"300",
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"Figure",
"2,",
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"4"
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} | |||
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"5"
]
} | |||
2301.00014v1.pdf_6 | null | ArxivOCR_part_1 | {
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} | |||
2301.00014v1.pdf_7 | null | ArxivOCR_part_1 | {
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} | {
"word": [
"Closed",
"Gamma-ray",
"shutter",
"In",
"this",
"case,",
"the",
"shutter",
"of",
"the",
"gamma-ray",
"densitometer",
"was",
"closed.",
"This",
"causes",
"a",
"sudden",
"drop",
"in",
"the",
"measured",
"density",
"that",
"leads",
"to",
"a",
"wrong",
"estimation",
"of",
"the",
"mass",
"flowing",
"in",
"the",
"pipe",
"(Figure",
"8).",
"The",
"residue",
"raises",
"very",
"quickly",
"and",
"the",
"absolute",
"rolling",
"mean",
"exceeds",
"its",
"threshold,",
"triggering",
"the",
"alarm.",
"Here,",
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"absolute",
"rolling",
"standard",
"deviation",
"has",
"a",
"little",
"contribution",
"in",
"the",
"fault",
"detection.",
"20",
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"G",
"actual",
"G",
"15",
"10",
"pi",
"5",
"MMJMALAAL&uLJ",
"g",
"0",
"-5",
"-10",
"-15",
"0",
"1000",
"4000",
"6000",
"7000",
"2000",
"3000",
"5000",
"Time",
"(arb.",
"unit)",
"20",
"-",
"abs",
"rolling",
"std",
"----",
"std-threshold",
"residue",
"15",
"-",
"abs",
"rolling",
"mean",
"----",
"mean-threshold",
"10",
"pi",
"5",
"n",
"g",
"0",
"-5",
"-10",
"Inimmomn",
"-15",
"1000",
"2000",
"3000",
"7000",
"0",
"4000",
"5000",
"6000",
"Time",
"(arb.",
"unit)",
"Figure",
"8:",
"Closed",
"Gamma-ray",
"shutter.",
"Unplugged",
"communication",
"cable",
"Although",
"the",
"closed",
"shutter",
"could",
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"by",
"a",
"much",
"simpler",
"method,",
"the",
"disconnected",
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"cable",
"is",
"much",
"more",
"complex.",
"In",
"this",
"case",
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"electronics",
"continue",
"to",
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"the",
"last",
"recorded",
"sequence",
"over",
"and",
"over",
"again.",
"Note",
"that",
"at",
"the",
"time",
"when",
"this",
"sequence",
"was",
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"recorded,",
"it",
"was",
"normal,",
"so",
"a",
"univariate",
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"method",
"can",
"hardly",
"detect",
"this",
"type",
"of",
"anomaly.",
"Looking",
"at",
"(Figure",
"9)",
"it",
"is",
"quite",
"difficult",
"to",
"see",
"the",
"generated",
"fault,",
"but",
"fortunately",
"the",
"proposed",
"method",
"promptly",
"detects",
"it.",
"The",
"exploding",
"residue",
"is",
"quite",
"clear",
"and",
"the",
"rolling",
"std",
"raises",
"quickly.",
"8"
]
} | |||
2301.00014v1.pdf_9 | null | ArxivOCR_part_1 | {
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} | {
"word": [
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"3",
"2",
"1",
"pi",
"0",
"g",
"-1",
"-2",
"actual",
"G",
"-3",
"predicted",
"G",
"-4",
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"4000",
"5000",
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"5000",
"3000",
"4000",
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"unit)",
"Figure",
"9:",
"Unplugged",
"communication",
"cable.",
"4",
"CONCLUSIONS",
"In",
"this",
"paper",
"an",
"approach",
"to",
"the",
"fault",
"detection",
"of",
"Multiphase",
"Flow",
"Meters",
"has",
"been",
"proposed.",
"It",
"applies",
"to",
"its",
"sensors",
"and",
"specifically",
"to",
"its",
"time-series",
"measures.",
"It",
"consists",
"of",
"three",
"steps:",
"forecast,",
"residual",
"analysis",
"and",
"fault",
"alarm.",
"In",
"the",
"forecasting",
"step,",
"care",
"has",
"been",
"taken",
"not",
"only",
"to",
"the",
"prediction",
"accuracy,",
"but",
"also",
"to",
"the",
"robustness",
"of",
"the",
"method.",
"After",
"the",
"development,",
"it",
"was",
"applied",
"to",
"real",
"faults",
"and",
"it",
"proved",
"its",
"ability",
"to",
"easily",
"detect",
"the",
"generated",
"faults.",
"The",
"authors",
"see",
"two",
"natural",
"research",
"directions",
"for",
"future",
"works:",
"to",
"test",
"new",
"prediction",
"models",
"trying",
"to",
"improve",
"the",
"predictive",
"performances,",
"and",
"to",
"make",
"long-term",
"forecasts",
"in",
"order",
"to",
"lessen",
"the",
"computational",
"burden",
"of",
"the",
"method.",
"5",
"REFERENCES",
"[1]",
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"metering",
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"offshore",
"oil",
"and",
"gas",
"transportation",
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"Trends",
"and",
"perspectives.\"",
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"(2019):",
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"[2]",
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"[3]",
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"C.",
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"Data-driven",
"anomaly",
"recognition",
"for",
"unsupervised",
"model-free",
"fault",
"detection",
"in",
"artificial",
"pancreas.\"",
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"9"
]
} | |||
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} | {
"word": [
"[5]",
"S.",
"AHMAD,",
"S.",
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"\"Real-time",
"anomaly",
"detection",
"for",
"streaming",
"analytics.\"",
"arXiv",
"preprint",
"arXiv:1607.02480",
"(2016).",
"[6]",
"K.",
"HUNDMAN,",
"V.",
"CONSTANTINOU,",
"C.",
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"I",
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"\"Detecting",
"spacecraft",
"anomalies",
"using",
"lstms",
"and",
"nonparametric",
"dynamic",
"thresholding.\"",
"Proceedings",
"of",
"the",
"24th",
"ACM",
"SIGKDD",
"international",
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"knowledge",
"discovery",
"&",
"data",
"mining.",
"2018.",
"[7]",
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"for",
"sequence",
"modeling.\"",
"arXiv",
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"arXiv:1803.01271",
"(2018).",
"[8]",
"Y.",
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"WANG,",
"R.",
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} | {
"word": [
"Behave-XAI:",
"Deep",
"Explainable",
"Learning",
"of",
"Behavioral",
"Representational",
"Data",
"*",
"Rossi",
"Kamal,",
"Zuzana",
"Kubincova",
"Comenius",
"University",
"Bratislava",
"kamal1,kubincova@uniba.sk",
"Abstract—According",
"to",
"the",
"latest",
"trend",
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"artificial",
"intelli-",
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"AI-system",
"needs",
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"why",
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"the",
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"of",
"any",
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"This",
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"using",
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"for",
"a",
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"-Why",
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"the",
"8",
"mining",
"scenario,",
"where",
"users",
"engagement",
"on",
"digital",
"platform",
"is",
"output?",
"How",
"does",
"it",
"determined",
"from",
"context,",
"such",
"as",
"emotion,",
"activity,",
"weather,",
"etc.",
"However,",
"the",
"output",
"of",
"AI-system",
"is",
"not",
"always",
"systematically",
"correct,",
"and",
"often",
"systematically",
"correct,",
"but",
"apparently",
"not-",
"800",
"perfect",
"and",
"thereby",
"creating",
"confusions,",
"such",
"as,",
"why",
"the",
"decision",
"is",
"given?,",
"what",
"is",
"the",
"reason",
"underneath?.",
"In",
"this",
"context,",
"we",
"first",
"formulate",
"the",
"behavioral",
"mining",
"problem",
"in",
"deep",
"convolutional",
"neural",
"network",
"architecture.",
"Eventually,",
"we",
"apply",
"a",
"recursive",
"G",
"Fig.",
"1.",
"Blackbox",
"DNN",
"neural",
"network",
"due",
"to",
"the",
"presence",
"of",
"time-series",
"data",
"from",
"users’",
"physiological",
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"environmental",
"sensor-readings.",
"Once",
"the",
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"are",
"presented",
"with",
"the",
"advent",
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"models",
"in",
"front",
"of",
"users.",
"This",
"critical",
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"extensive",
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"with",
"users",
"’preference",
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"[",
"AI,",
"’judgement",
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"Recurrent",
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"Net-",
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"work,",
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"I.",
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"c0",
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"with",
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"im-",
"-",
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"better",
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"ever,",
"having",
"significant",
"contribu-",
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"in",
"education,",
"healthcare,",
"vehicle",
"technology.",
"Technology",
"2016",
"moves",
"so",
"fast",
"that",
"AI",
"has",
"kept",
"human",
"intervention",
"at",
"at",
"minimum",
"level.",
"As",
"a",
"follow",
"up,",
"AI",
"steps",
"into",
"critical",
"appli-",
"Fig.",
"2.",
"XAI-based",
"DNN",
"cations,",
"however,",
"human",
"anxiety",
"over",
"AI’s",
"decision",
"making",
"W",
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"increases",
"as",
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"[7]",
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"answer",
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"NE",
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"human",
"under-",
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"as",
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"guaranteed",
"at",
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"despite",
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"more",
"due",
"to",
"proliferation",
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"namely",
"statistical",
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"Eventually,",
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"mo-",
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"one",
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"with",
"or",
"Behave-XAI",
"(Fig.",
"3).",
"We",
"consider",
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"AI",
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"there",
"is",
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"is",
"measured",
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"experiment",
"on",
"against",
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"expression",
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"which",
"are",
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"to",
"a",
"dataset,",
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"physiological",
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"explainable",
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"and",
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"so",
"that",
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"aware",
"and",
"have",
"and",
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"in",
"indoor",
"settings.",
"The",
"main",
"challenge",
"in",
"our",
"Thanks",
"to",
"support",
"provided",
"by",
"Professor",
"Dr",
"Zuzana",
"Kubincova",
"work",
"lies",
"in",
"finding",
"a",
"way",
"for",
"automatic",
"translation",
"for",
"any"
]
} | |||
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} | {
"word": [
"classified",
"information",
"provided",
"by",
"DNN",
"with",
"the",
"advent",
"of",
"A.",
"Application",
"Scenario",
"XAI",
"models.",
"Another",
"challenge",
"is",
"having",
"final",
"conclusion,",
"whether",
"interpretation",
"is",
"sensible/",
"credible",
"from",
"the",
"accuracy",
"With",
"the",
"proliferation",
"of",
"the",
"internet",
"of",
"things,billions",
"of",
"of",
"the",
"explanations",
"provided",
"by",
"our",
"system.",
"It",
"is",
"notable",
"connected",
"devices",
"are",
"currently",
"deployed",
"in",
"smart-city",
"service,",
"is",
"possible",
"to",
"judge",
"the",
"accuracy",
"by",
"consulting",
"with",
"such",
"as",
"distance",
"education.",
"The",
"application",
"becomes",
"essential",
"that",
"users",
"on",
"their",
"opinion",
"on",
"explanations,especially",
"whether",
"the",
"in",
"pandemic",
"situation",
"as",
"well",
"since",
"all",
"are",
"worried",
"about",
"explanations",
"were",
"satisfactory",
"or",
"accurate.",
"The",
"steps",
"involve",
"both",
"digital",
"distance",
"and",
"quality",
"education.",
"In",
"this",
"context,",
"rigorous",
"discussion",
"with",
"end-users",
"who",
"are",
"asked",
"about",
"ex-",
"engagement",
"of",
"consumers",
"to",
"edtech",
"platforms",
"has",
"been",
"the",
"planations,",
"understandability",
"and",
"truthfulness.Generally,",
"they",
"key",
"issue",
"due",
"to",
"performance-enhancement,",
"drop-out-removal.",
"are",
"asked",
"whether",
"explanatory",
"decisions",
"are",
"more",
"convenient",
"Conventional",
"methodologies",
"often",
"deploy",
"periodical",
"or",
"regular",
"than",
"conventional",
"one.",
"They",
"are",
"also",
"asked",
"whether",
"system",
"surveys",
"regarding",
"student",
"and",
"guardian",
"comments",
"on",
"online",
"dependability",
"is",
"added",
"while",
"explanations",
"are",
"presented",
"in",
"content,",
"teacher",
"evaluation.",
"However,",
"these",
"methods",
"are",
"offline",
"front",
"of",
"them.",
"Finally,",
"they",
"are",
"provided",
"the",
"task",
"of",
"choosing",
"and",
"often",
"suffer",
"from",
"disadvantages",
"such",
"as",
"off-date",
"data.",
"In",
"this",
"context,",
"real-time",
"and",
"quantitative",
"statistical",
"information",
"a",
"group",
"of",
"explanations,",
"which",
"are",
"better",
"and",
"simpler.",
"The",
"overall",
"organization",
"of",
"ongoing",
"work",
"is",
"summarized",
"is",
"essential",
"for",
"continuous",
"monitoring",
"of",
"student",
"performance",
"as",
"follows.",
"We",
"start",
"with",
"the",
"deep",
"recurrent",
"neural",
"network",
"required",
"in",
"digital",
"platforms.",
"Thus,",
"engagement",
"should",
"be",
"measured",
"from",
"not",
"only",
"offline",
"surveys,",
"but",
"also",
"available",
"sen-",
"based",
"architecture",
"with",
"problem",
"scenario,",
"dataset",
"description.",
"sor",
"readings,",
"such",
"as",
"physiological",
"sensors,",
"weather",
"sensors,",
"We",
"then",
"present",
"explainable",
"artificial",
"intelligence",
"models",
"for",
"natural",
"language",
"interpretations",
"with",
"experiment",
"and",
"assess-",
"etc.",
"For",
"example",
"how",
"a",
"student",
"feels,",
"when",
"the",
"air-cooler",
"is",
"on,",
"ment.",
"Finally",
"we",
"conclude",
"with",
"future",
"perspectives.",
"given",
"the",
"outside",
"temperature",
"is",
"hot.",
"Does",
"he/she",
"feel",
"engaged",
"and",
"stay",
"in",
"the",
"seat",
"or",
"leave",
"the",
"classroom?",
"Consequently,",
"in",
"this",
"work,",
"we",
"intend",
"to",
"measure",
"the",
"student",
"engagement",
"II.",
"DEEP",
"RECURSIVE",
"NEURAL",
"NETWORK",
"BASED",
"AI",
"on",
"education",
"by",
"collecting",
"contextual",
"data,",
"such",
"as",
"emotion,",
"MODEL",
"weather",
"from",
"room-temperature,",
"wearable",
"and",
"physiological",
"Deep",
"Recursive",
"Neural",
"Network",
"is",
"a",
"special",
"type",
"of",
"Convo-",
"sensors.",
"lutional",
"Neural",
"Network",
"which",
"is",
"recognized",
"for",
"its",
"operation",
"with",
"time-series",
"data.",
"Let’s",
"describe",
"the",
"general",
"architecture",
"B.",
"Model",
"of",
"Deep",
"Recursive",
"Neural",
"Network",
"first.",
"In",
"a",
"general",
"sense,",
"DRNN",
"generates",
"outcome",
"value",
"based",
"on",
"the",
"input",
"value",
"with",
"We",
"now",
"describe",
"the",
"deep",
"neural",
"network",
"based",
"model",
"the",
"intervention",
"of",
"hidden",
"layer.",
"However,",
"the",
"status",
"of",
"any",
"used",
"in",
"our",
"proposal.",
"We",
"have",
"use",
"gated",
"recurrent",
"unit",
"hidden",
"layer",
"is",
"associated",
"with",
"past",
"input",
"fetched",
"to",
"RNN,",
"as",
"(GRU),",
"variant",
"of",
"recurrent",
"neural",
"network",
"as",
"our",
"deep",
"well",
"as",
"input",
"given",
"at",
"current",
"time.",
"Thus,",
"a",
"DRNN",
"generates",
"architecture.",
"Let’s",
"assume",
"RNN",
"consists",
"of",
"n",
"GRUs",
"(Fig.",
"1).",
"outcome",
"based",
"on",
"current",
"input,",
"having",
"consideration",
"with",
"ear-",
"GRU",
"generally",
"operates",
"with",
"help",
"of",
"reset",
"gate",
"z,",
"update",
"gate",
"lier",
"state.",
"Thus,",
"DRNN",
"is",
"based",
"on",
"the",
"theoretical",
"assumption",
"z",
",input",
"vector",
"x",
"and",
"output",
"vector",
"x,",
"Now,",
"it",
"calculates",
"the",
"that",
"output",
"considers",
"all",
"past",
"history,",
"since",
"t",
"�",
"1’s",
"information",
"intermediate",
"value",
"from",
"input",
"and",
"the",
"value",
"x",
"collected",
"from",
"is",
"transmitted",
"through",
"hidden",
"layers,while",
"t's",
"information",
"is",
"hidden",
"layer",
"of",
"earlier",
"time.",
"Therefore,",
"the",
"output",
"is",
"predicted",
"from",
"value",
"which",
"is",
"nothing",
"but",
"multiplication",
"of",
"input,",
"weight",
"given",
"as",
"input.",
"We",
"intend",
"to",
"increase",
"the",
"engagement",
"of",
"users",
"by",
"using",
"and",
"hidden",
"layer",
"of",
"earlier",
"time.",
"Thus,",
"it",
"can",
"make",
"a",
"decision",
"a",
"deep",
"recursive",
"neural",
"network.In",
"this",
"context,",
"we",
"collect",
"on",
"optimal",
"output,",
"from",
"input,",
"past",
"information.",
"Lets",
"discuss",
"the",
"contextual",
"information",
"such",
"as",
"weather,",
"emotion",
"of",
"users.",
"the",
"model",
"in",
"detail.",
"Contextual",
"information",
"is",
"ranked",
"for",
"each",
"user",
"with",
"time-series",
"=",
"o(W",
"+",
"W",
"hz",
"h",
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"behavior",
"of",
"data.",
"Whenever,",
"ranking",
"of",
"contextual",
"information",
"An",
"update",
"gate",
"vector",
"is",
"maintained",
"for",
"keeping",
"track",
"record",
"is",
"conducted,",
"is",
"given",
"as",
"input",
"to",
"the",
"recursive",
"neural",
"of",
"past",
"information",
"to",
"be",
"kept",
"along",
"the",
"future.An",
"update",
"network.",
"As",
"soon",
"as",
"contextual",
"information",
"is",
"ranked,",
"some",
"of",
"vector",
"for",
"any",
"given",
"time",
"is",
"calculated",
"from",
"sigmoid",
"its",
"part",
"is",
"used",
"as",
"training",
"data,",
"while",
"the",
"rest",
"works",
"as",
"testing",
"function",
"of",
"’present",
"input",
"multiplied",
"by",
"its",
"weight",
"plus",
"output",
"data.",
"We",
"always",
"monitor",
"the",
"predicted",
"value",
"and",
"actual",
"value",
"multiplied",
"by",
"its",
"weight’.",
"Sigmoid",
"is",
"used",
"to",
"squash",
"the",
"and",
"the",
"difference",
"between",
"these",
"two",
"resembles",
"the",
"accuracy",
"outcome",
"of",
"the",
"update",
"gate",
"in",
"between",
"0",
"and",
"1.",
"The",
"way",
"of",
"the",
"DRNN",
"scheme.",
"We",
"conclude",
"about",
"the",
"idea",
"that",
"we",
"an",
"update",
"gate",
"maintained",
"for",
"up-to-date",
"information",
"is",
"need",
"to",
"reach",
"a",
"decision",
"on",
"optimal",
"contextual",
"information,",
"recognized",
"as",
"a",
"solution",
"for",
"vanishing",
"gradient",
"problems.",
"which",
"are",
"essential",
"for",
"making",
"accurate",
"predictions",
"in",
"time-",
"interval.",
"Such",
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"will",
"help",
"in",
"gathering",
"better",
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"r",
"t",
"=",
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"x",
"+",
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"hr",
"h",
"t-",
"+",
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"r",
"mation",
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"engagement",
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"are",
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"in",
"different",
"temporal",
"and",
"spatial",
"settings.",
"This",
"actually",
"motivates",
"A",
"reset",
"gate",
"is",
"also",
"maintained",
"to",
"decide",
"on",
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"to",
"us",
"toward",
"the",
"use",
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"explainable",
"artificial",
"intelligence",
"in",
"our",
"be",
"forgotten",
"along",
"its",
"way.",
"The",
"outcome",
"of",
"the",
"gate",
"is",
"the",
"engagement",
"scenario.",
"We",
"will",
"describe",
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"motivation",
"and",
"sigmoid",
"function",
"of",
"the",
"’addition",
"of",
"weighted",
"value",
"of",
"current",
"XAI’s",
"theoretical",
"foundation",
"in",
"the",
"coming",
"section.",
"input",
"and",
"weighted",
"value",
"of",
"previous",
"information’."
]
} | |||
2301.00016v1.pdf_3 | null | ArxivOCR_part_1 | {
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"judgement",
"matter",
"here",
"more",
"very",
"important",
"we",
"define",
"the",
"features",
"accurately",
"for",
"different",
"than",
"anything.",
"Thus,",
"one",
"can",
"tag",
"explanation",
"accurate",
"or",
"inaccurate",
"based",
"on",
"outward",
"credibility.",
"For",
"example,",
"if",
"an",
"XAI",
"models.",
"Let",
"us",
"give",
"a",
"small",
"overview",
"of",
"the",
"XAI",
"models",
"and",
"their",
"explanation",
"gives",
"emphasizes",
"on",
"user’s",
"sitting",
"duration,",
"in-",
"usage",
"in",
"generating",
"explanation",
"in",
"decision",
"support",
"system",
"in",
"room",
"temperatre",
"for",
"finding",
"the",
"most-engaged",
"student,",
"it",
"is",
"edtech",
"environment.",
"We",
"are",
"also",
"interested",
"in",
"analyzing",
"how",
"more",
"sensible",
"XAI-explanation.",
"However,",
"if",
"any",
"explanation",
"temporal",
"or",
"spational",
"features",
"of",
"explanations",
"have",
"positive",
"or",
"calculates",
"additional",
"outside-temperature",
"along",
"with",
"sitting-",
"negative",
"impression",
"on",
"user",
"regarding",
"viability",
"of",
"explanation.",
"duration,",
"in-room",
"temperature,",
"it",
"is",
"sensible,",
"since",
"good",
"stu-",
"Furthermore,",
"we",
"translate",
"the",
"outcome",
"of",
"XAI",
"model",
"as",
"dents",
"are",
"reluctant",
"to",
"leave",
"seating",
"place,",
"despite",
"outside",
"natural",
"language",
"explanations,",
"which",
"is",
"human-friendly",
"and",
"temperature",
"is",
"nice.",
"easily",
"understandable.",
"B.",
"Insights",
"on",
"Explanation",
"Content",
"In",
"this",
"section,",
"we",
"analyze",
"the",
"correlation",
"between",
"expla-",
"A.",
"Lime",
"Lime",
"is",
"a",
"methodology",
"for",
"understanding",
"the",
"deep",
"learning",
"insights",
"behind",
"it.We",
"find",
"that",
"Lime",
"based",
"explanations",
"use",
"model",
"by",
"perturbing",
"data",
"sample",
"and",
"thereby",
"observing",
"im-",
"more",
"temperature",
"sensors",
"than",
"Shaply",
"and",
"Anchore",
"based",
"mediate",
"change",
"in",
"classification.",
"In",
"general,",
"humans",
"are",
"more",
"explanations.",
"However,",
"Shaply",
"and",
"anchore",
"are",
"more",
"focused",
"interested",
"in",
"common",
"phenomena,",
"such",
"as",
"’why",
"the",
"classifi-",
"on",
"each",
"feature,",
"so",
"they",
"not",
"only",
"consider",
"temperature",
"but",
"also",
"cation",
"result",
"is",
"such’",
"or",
"’which",
"are",
"key",
"variables,",
"those",
"cause",
"sitting",
"attribute",
"of",
"student.",
"So,",
"whenever,",
"students",
"presence",
"in",
"classification",
"output’.",
"Therefore,",
"LIME",
"takes",
"a",
"simple",
"ap-",
"the",
"classroom",
"is",
"required,",
"Lime",
"explanation",
"model",
"determines",
"nations",
"given",
"by",
"different",
"XAI",
"methodologies",
"and",
"find",
"the",
"proach",
"of",
"taking",
"feature",
"values",
"and",
"observing",
"changes",
"in",
"in",
"one",
"way.",
"However,",
"Shaply",
"and",
"Anchors",
"model",
"determine",
"classification",
"results.",
"Thus,",
"LIME",
"makes",
"an",
"explanation",
"in",
"different",
"way.",
"focused",
"with",
"local",
"interpretability.",
"Let",
"us",
"assume,",
"we",
"need",
"to",
"find",
"set",
"of",
"feature,",
"for",
"given",
"V.",
"DISCUSSION",
"input",
"x,",
"which",
"best",
"describes",
"classification",
"f(x).",
"Lime",
"in",
"We",
"are",
"really",
"interested",
"in",
"whether",
"users",
"prefer",
"simple",
"deci-",
"this",
"case",
"exploits",
"local",
"model",
"as",
"an",
"approximation",
"of",
"f(x).",
"sions",
"or",
"explained",
"decision.",
"We",
"are",
"further",
"curious",
"to",
"know",
"Lime",
"considers",
"a",
"new",
"dataset",
"samples",
"which",
"are",
"nothing",
"but",
"We",
"experiment",
"on",
"users",
"regarding",
"validity",
"of",
"explanations.",
"perturbed",
"from",
"x",
"in",
"its",
"way",
"of",
"approximation",
"of",
"x.However,",
"whether",
"explanations",
"give",
"users",
"more",
"accurate",
"estimation.",
"it",
"guarantees",
"good",
"local",
"explanation",
"to",
"f(x),",
"but",
"not",
"for",
"VI.",
"CONCLUSION",
"generalized",
"classification",
"instances.",
"We",
"aim",
"to",
"formulate",
"human",
"understandable",
"explanations",
"of",
"B.",
"Shapley",
"deep",
"convolutional",
"recurrent",
"neural",
"networks.",
"Our",
"scenarios",
"Shapley",
"is",
"conceptualized",
"from",
"game",
"thory,",
"where",
"dif-",
"based",
"on",
"behavioral",
"mining",
"with",
"time-series",
"based",
"contextual",
"ferent",
"players",
"strategically",
"co-exist",
"for",
"winning",
"a",
"game.",
"In",
"data,",
"such",
"as",
"mobility,",
"emotion",
"and",
"weather.The",
"presence",
"of",
"this",
"context,",
"game",
"can",
"be",
"coined",
"by",
"outcome",
"of",
"prediction",
"time-series",
"data",
"makes",
"RNN",
"amenable",
"to",
"our",
"work.",
"Con-",
"and",
"players",
"resemble",
"features",
"of",
"classification.",
"Shapley",
"is",
"sequently,",
"the",
"objective",
"of",
"deep",
"architecture",
"is",
"to",
"classify",
"an",
"explanation",
"model",
"which",
"believes",
"in",
"quantification",
"of",
"emotion,weather",
"or",
"comfort",
"in",
"assessing",
"engagement",
"in",
"digital",
"features.",
"Shapley",
"is",
"based",
"on",
"the",
"game-theoretic",
"principle",
"platforms",
"for",
"indoor",
"settings.Eventually,",
"as",
"part",
"of",
"ongoing",
"that",
"contribution/outcome",
"of",
"each",
"feature(i.e.player)",
"on",
"each",
"effort",
"for",
"human-friendly",
"AI,",
"we",
"exploit",
"explanation",
"model",
"combination",
"(i.e",
"round)",
"determines",
"its",
"quantification",
"at",
"the",
"in",
"translating",
"the",
"classification",
"results",
"provided",
"by",
"deep",
"end.",
"recurrent",
"architecture."
]
} | |||
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} | {
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"recommendations",
"in",
"reciprocal",
"environments.",
"In",
"Proceedings",
"of",
"the",
"12th",
"ACM",
"conference",
"on",
"recommender",
"systems",
"(pp.",
"22-30).",
"4.Ribeiro,",
"M.T.,",
"Singh,",
"S.",
"and",
"Guestrin,",
"C.,",
"2016,",
"August.",
",,",
"Why",
"should",
"i",
"trust",
"you?”",
"Explaining",
"the",
"predictions",
"of",
"any",
"classifier.",
"In",
"Proceedings",
"of",
"the",
"22nd",
"ACM",
"SIGKDD",
"international",
"conference",
"on",
"knowledge",
"discovery",
"and",
"data",
"mining",
"(pp.",
"1135-1144).",
"5.Lundberg,",
"S.M.",
"and",
"Lee,",
"S.I.,",
"2017.",
"A",
"unified",
"approach",
"to",
"interpreting",
"model",
"predictions.",
"Advances",
"in",
"neural",
"information",
"processing",
"systems,",
"30.",
"6.Ribeiro,",
"M.T.,",
"Singh,",
"s.",
"and",
"Guestrin,",
"c.,",
"2018,",
"April.",
"Anchors:",
"High-precision",
"model-agnostic",
"explanations.",
"In",
"Pro-",
"ceedings",
"of",
"the",
"AAAI",
"conference",
"on",
"artificial",
"intelligence",
"(Vol.",
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"1).",
"7.Pascanu,",
"R.,",
"Mikolov,",
"T.",
"and",
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"Y.,",
"2013,",
"May.",
"On",
"the",
"difficulty",
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"recurrent",
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"networks.",
"In",
"International",
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"on",
"machine",
"learning",
"(pp.",
"1310-1318).",
"PMLR.",
"APPENDIX"
]
} | |||
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} | {
"word": [
"Subharmonic",
"Fidelity",
"Revival",
"in",
"a",
"Driven",
"PXP",
"model",
"HaRu",
"K.",
"Park1,",
"and",
"SungBin",
"Lee1,",
"t",
"*",
"1Department",
"of",
"Physics,",
"Korea",
"Advanced",
"Institute",
"of",
"Science",
"and",
"Technology,",
"Daejeon,",
"34141,",
"Korea",
"(Dated:",
"January",
"3,",
"2023)",
"The",
"PXP",
"model",
"hosts",
"a",
"special",
"set",
"of",
"nonergodic",
"states,",
"referred",
"to",
"as",
"quantum",
"many-body",
"scars.",
"One",
"of",
"the",
"consequences",
"of",
"quantum",
"scarring",
"is",
"the",
"periodic",
"revival",
"of",
"the",
"wave",
"function",
"fidelity.",
"It",
"has",
"been",
"reported",
"that",
"quantum",
"fidelity",
"revival",
"occurs",
"in",
"the",
"PXP",
"model",
"for",
"certain",
"product",
"states,",
"and",
"periodic",
"driving",
"of",
"chemical",
"potential",
"can",
"enhance",
"the",
"magnitude",
"of",
"quantum",
"revival,",
"and",
"can",
"even",
"change",
"the",
"frequencies",
"of",
"revival",
"showing",
"the",
"subharmonic",
"response.",
"Although",
"the",
"effect",
"of",
"the",
"periodic",
"driving",
"in",
"the",
"PXP",
"model",
"has",
"been",
"studied",
"in",
"201",
"the",
"limit",
"of",
"certain",
"perturbative",
"regimes,",
"the",
"general",
"mechanism",
"of",
"such",
"enhanced",
"revival",
"and",
"frequency",
"change",
"has",
"been",
"barely",
"studied.",
"In",
"this",
"work,",
"we",
"investigate",
"how",
"periodic",
"driving",
"in",
"the",
"PXP",
"model",
"can",
"systematically",
"control",
"the",
"fidelity",
"revival.",
"Particularly,",
"focusing",
"on",
"the",
"product",
"state",
"so",
"called",
"a",
"N´eel",
"state,",
"we",
"analyze",
"the",
"condition",
"of",
"driving",
"to",
"enhance",
"the",
"magnitude",
"of",
"revival",
"or",
"change",
"the",
"frequencies",
"of",
"revival.",
"To",
"clarify",
"the",
"c",
"reason",
"of",
"such",
"control,",
"we",
"consider",
"the",
"similarities",
"between",
"the",
"PXP",
"model",
"and",
"the",
"free",
"spin-1/2",
"model",
"in",
"graph",
"theoretical",
"analysis,",
"and",
"show",
"that",
"the",
"quantum",
"fidelity",
"feature",
"in",
"the",
"PXP",
"model",
"well",
"explained",
"by",
"the",
"free",
"spin-1/2",
"model.",
"In",
"addition,",
"under",
"certain",
"limit",
"of",
"the",
"driving",
"parameters,",
"analytic",
"approach",
"to",
"explain",
"the",
"main",
"features",
"of",
"the",
"fidelity",
"revival",
"is",
"also",
"performed.",
"Our",
"results",
"give",
"an",
"insight",
"of",
"the",
"scarring",
"nature",
"of",
"the",
"periodically",
"driven",
"PXP",
"model",
"and",
"pave",
"the",
"way",
"to",
"understand",
"their",
"(sub-)harmonic",
"responses",
"and",
"controls.",
"imperfect",
"revival12.",
"There",
"exist",
"many",
"attempts",
"to",
"reach",
"high",
"I.",
"INTRODUCTION",
"fidelity",
"revival,",
"for",
"instance,",
"by",
"enhancing",
"its",
"weakly",
"broken",
"Eigenstate",
"Thermalization",
"Hypothesis(ETH)1–4",
"is",
"key",
"SU(2)",
"symmetry13,14.",
"There",
"is",
"another",
"way",
"which",
"enhances",
"concept",
"explaining",
"the",
"thermalization",
"of",
"the",
"quantum",
"many-",
"the",
"quantum",
"revival",
"of",
"the",
"states",
"in",
"the",
"PXP",
"model:",
"periodic",
"body",
"system.",
"Recently,",
"beyond",
"the",
"quantum",
"thermalization,",
"driving.",
"In",
"most",
"of",
"the",
"cases,",
"periodic",
"driving",
"induces",
"ther-",
"the",
"system",
"which",
"strongly",
"violates",
"the",
"ETH",
"has",
"been",
"actively",
"malization",
"of",
"the",
"scar",
"states,",
"and",
"thus",
"destructs",
"the",
"fidelity",
"re-",
"vival.",
"However,",
"recent",
"experimental",
"and",
"theoretical",
"studies",
"studied,",
"such",
"as",
"integrable",
"systems",
"and",
"many-body",
"localiza-",
"show10,15",
"that",
"periodic",
"driving",
"with",
"certain",
"amplitudes",
"sur-",
"tion.",
"Here,",
"the",
"”strong",
"violation”",
"of",
"the",
"ETH",
"means",
"ev-",
"ery",
"eigenstate",
"breaks",
"the",
"ergodicity",
"and",
"never",
"get",
"thermal-",
"prisingly",
"enhances",
"the",
"fidelity",
"revival.",
"Furthermore,",
"it",
"has",
"ized.",
"There",
"are",
"also",
"the",
"systems",
"which",
"”weakly",
"violate”",
"the",
"also",
"been",
"observed",
"that",
"the",
"subharmonic",
"response",
"of",
"fidelity",
"Loe",
"ETH,",
"implying",
"only",
"small",
"portion",
"of",
"the",
"eigenstates",
"violates",
"revival",
"exists",
"with",
"doubled",
"period",
"compared",
"to",
"the",
"driving",
"the",
"ergodicity",
"while",
"all",
"the",
"other",
"states",
"get",
"thermalized.",
"In",
"mode.",
"This",
"kind",
"of",
"subharmonic",
"response",
"breaks",
"the",
"discrete",
"particular,",
"quantum",
"many-body",
"scarring(QMBS)",
"systems",
"are",
"time",
"translational",
"symmetry",
"of",
"the",
"driving",
"mode,",
"and",
"hence",
"the",
"examples",
"which",
"weakly",
"violates",
"ETH,",
"containing",
"a",
"small",
"get",
"attention",
"as",
"a",
"time-version",
"of",
"crystalline",
"order,",
"called",
"a",
"number",
"of",
"highly",
"excited",
"non-thermal",
"eigenstates,",
"called",
"scar",
"discrete",
"time-crystal",
"(DTC)",
"which",
"is",
"also",
"a",
"recently",
"studied",
"eigenstates5,6.",
"Such",
"scar",
"eigenstates",
"show",
"exotic",
"physical",
"be-",
"subject16,17.",
"Although",
"earlier",
"research",
"had",
"demonstrated",
"this",
"Gnh",
"havior",
"compared",
"to",
"the",
"thermal",
"Gibbs",
"state.",
"For",
"example,",
"subharmonic",
"fidelity",
"revival",
"in",
"the",
"limit",
"of",
"large",
"driving",
"am-",
"while",
"the",
"thermal",
"Gibbs",
"state",
"predicts",
"the",
"entanglement",
"en-",
"plitude",
"and",
"high",
"frequency18,",
"the",
"general",
"mechanism",
"of",
"such",
"tropy",
"proportional",
"to",
"the",
"volume",
"of",
"subsystem",
"in",
"the",
"middle",
"subharmonic",
"revival",
"has",
"not",
"been",
"explored",
"well.",
"of",
"the",
"spectrum,",
"the",
"entanglement",
"entropy",
"of",
"the",
"scar",
"eigen-",
"state",
"scales",
"proportionally",
"to",
"the",
"area",
"of",
"the",
"subsystem.",
"Be-",
"cause",
"the",
"QMBS",
"often",
"appear",
"in",
"a",
"tower",
"of",
"scar",
"states",
"where",
"a",
"set",
"of",
"states",
"with",
"equidistant",
"energy",
"spacing",
"exists7,",
"if",
"the",
"initial",
"product",
"state",
"is",
"superposition",
"of",
"the",
"scar",
"states,",
"then",
"In",
"this",
"paper,",
"we",
"study",
"the",
"periodically",
"driven",
"PXP",
"model,",
"the",
"system",
"exhibits",
"the",
"perfect",
"revival.",
"Conversely,",
"it",
"has",
"been",
"focusing",
"on",
"the",
"subharmonic",
"fidelity",
"revivals.",
"In",
"Section",
"II,",
"also",
"shown",
"that",
"the",
"persistent",
"revival",
"of",
"product",
"state",
"implies",
"we",
"introduce",
"the",
"PXP",
"model",
"with",
"square",
"pulse",
"driving",
"modes.",
"QMBS8.",
"Hence,",
"is",
"important",
"to",
"observe",
"the",
"fidelity",
"revival",
"By",
"calculating",
"the",
"average",
"fidelity",
"and",
"the",
"Fourier",
"compo-",
"as",
"evidence",
"for",
"the",
"QMBS",
"in",
"experiments.",
"nents",
"of",
"the",
"fidelity",
"signal,",
"we",
"study",
"the",
"conditions",
"under",
"In",
"recent",
"experiments",
"of",
"Rydberg",
"atom",
"simulator,",
"it",
"has",
"which",
"the",
"fidelity",
"is",
"enhanced",
"and",
"subharmonic",
"response",
"oc-",
"been",
"observed",
"that",
"certain",
"product",
"states",
"show",
"persistent,",
"curs.",
"Then,",
"based",
"on",
"the",
"similarity",
"of",
"the",
"Hamiltonian",
"ad-",
"though",
"imperfect",
"revival",
"under",
"van",
"der",
"Waals",
"interaction.",
"jacent",
"graph",
"between",
"PXP",
"model",
"and",
"free",
"spin",
"model19,",
"we",
"This",
"indicates",
"the",
"presence",
"of",
"a",
"QMBS",
"in",
"the",
"system9–11.",
"Tak-",
"show",
"that",
"they",
"can",
"be",
"explained",
"by",
"the",
"free",
"spin-1/2",
"model",
"ing",
"the",
"extreme",
"limit",
"of",
"strong",
"van",
"der",
"Waals",
"interaction",
"in",
"with",
"the",
"same",
"driving",
"which",
"is",
"exactly",
"solvable.",
"In",
"Section",
"III,",
"the",
"Rydberg",
"blockade",
"of",
"the",
"Rydberg",
"atom",
"chain",
"gives",
"rise",
"we",
"introduce",
"an",
"analytic",
"approach",
"for",
"the",
"driven",
"PXP",
"model.",
"Within",
"perturbative",
"analysis,",
"we",
"derive",
"the",
"driving",
"conditions",
"to",
"the",
"so-called",
"PXP",
"model,",
"which",
"has",
"a",
"Hilbert",
"space",
"pro-",
"jected",
"onto",
"the",
"states",
"with",
"no",
"neighboring",
"excited",
"states.",
"The",
"for",
"subharmonic",
"responses",
"in",
"the",
"driven",
"PXP",
"model",
"and",
"dis-",
"QMBS",
"structure",
"of",
"the",
"PXP",
"model",
"has",
"been",
"actively",
"stud-",
"cuss",
"their",
"applications.",
"In",
"Section",
"IV,",
"we",
"summarize",
"our",
"ied",
"theoretically,",
"including",
"the",
"report",
"that",
"it",
"also",
"shows",
"the",
"works",
"and",
"suggest",
"interesting",
"future",
"directions."
]
} | |||
2301.00020v1.pdf_2 | null | ArxivOCR_part_1 | {
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"word": [
"2",
"II.",
"PXP",
"MODEL",
"WITH",
"SQUARE",
"PULSE",
"DRIVE",
"driving,",
"we",
"adopt",
"the",
"square",
"pulse",
"driving",
"protocol",
"as",
"also",
"in-",
"troduced",
"in",
"earlier",
"studies",
"for",
"analysis.",
"It",
"shares",
"very",
"similar",
"In",
"this",
"section,",
"we",
"first",
"introduce",
"the",
"static",
"PXP",
"model",
"and",
"fidelity",
"profile",
"with",
"the",
"cosine",
"driving",
"case",
"and",
"thus",
"well",
"ex-",
"then",
"represent",
"how",
"the",
"fidelity",
"revival",
"is",
"controlled",
"under",
"peri-",
"plains",
"the",
"experiments15.",
"The",
"square",
"pulse",
"driving",
"protocol",
"odic",
"driving.",
"Particularly,",
"based",
"on",
"the",
"graph",
"theoretical",
"simi-",
"A",
"sq",
"(t)",
"within",
"the",
"period",
"T",
"is",
"defined",
"as,",
"larity",
"between",
"PXP",
"model",
"and",
"free",
"spin-1/2",
"model,",
"we",
"argue",
"s",
"s",
"T/4",
"revival",
"enhancement",
"and",
"(sub-)",
"harmonic",
"response,",
"can",
"be",
"ex-",
"A",
"sq",
"(t)",
"=",
"∆0",
"=",
"�",
",",
"T/4",
"<",
"t",
"S",
"3T/4",
"(3)",
"A",
"m",
"∆0",
"0",
"{",
"that",
"various",
"phenomena",
"in",
"the",
"driven",
"PXP",
"model,",
"such",
"as",
"the",
"plained",
"by",
"the",
"free",
"spin-1/2",
"model.",
"A",
"+",
"�",
"m",
",",
"3T/4",
"<",
"t",
"<",
"T.",
"The",
"PXP",
"atoms",
"with",
"strong",
"in-",
"teraction,",
"is",
"This",
"corresponds",
"to",
"the",
"periodic",
"driving",
"with",
"frequency",
"ω0",
"=",
"2π/T",
"average",
"chemical",
"potential",
"A",
"o",
",",
"and",
"driving",
"amplitude",
"(1)",
"j-",
"1",
"X",
"j",
"j+1",
"H",
"PXP",
"=",
"Ω",
"�",
"m",
"Now,",
"we",
"introduce",
"the",
"wave",
"function",
"fidelity",
"F(t)",
"=",
"of",
"the",
"reviva",
"prod-",
"Here,",
"the",
"spin",
"at",
"each",
"site",
"consists",
"of",
"two",
"states,",
"|0(",
"id:",
"u",
"o",
",",
"d:105",
"represent",
"a",
"ground",
"state",
"and",
"an",
"excited",
"excite",
"re-",
"|1",
"change",
"parameters",
"(4",
"o",
",",
"�",
"m",
")",
"are",
"the",
"subharmonic",
"re-",
"is",
"the",
"|1j(c",
"x",
"105",
"sponse,",
"w",
"specti",
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"tr",
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"opera-",
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"fide",
"Hamiltonian",
"describes",
"at",
"tor",
"for",
"spin-flip,",
"unless",
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"neigh-",
"F(t)e",
"iwtdt",
"(4)",
"ate,",
"i.e.",
"only",
"the",
"transition,",
"F",
"(ω)",
"=",
"nT",
",",
"is",
"(cid:1",
"Since",
"this",
"tran-",
"sition",
"does",
"not",
"generate",
"or",
"annihilate",
"excited",
"states",
"in",
"any",
"two",
"Later,",
"we",
"will",
"discuss",
"the",
"three",
"main",
"quantities,",
"F",
"(0),",
"F(w",
"o",
"neighboring",
"sites,",
"one",
"can",
"exclude",
"the",
"states",
"where",
"the",
"con-",
"and",
"F(w",
"/2),",
"as",
"functions",
"of",
"(A",
"À",
"m",
"):",
"F(0)",
"introduces",
"secutive",
"neighbors",
"are",
"excited.",
"how",
"the",
"fidelity",
"revival",
"gets",
"enhanced,",
"and",
"F",
"(w",
"and",
"Although",
"the",
"PXP",
"model",
"is",
"non-integrable",
"chaotic",
"system11,",
"there",
"are",
"certain",
"product",
"states",
"which",
"show",
"non-",
"sponses,",
"respectively.",
"Note",
"that",
"if",
"i",
"|ψ(t)(",
"is",
"an",
"eigenstate",
"of",
"F(w",
"/2)",
"are",
"responsible",
"for",
"harmonic",
"and",
"subharmonic",
"re-",
"ergodicity",
"and",
"decent",
"revival",
"under",
"time",
"evolution,",
"such",
"as",
"H(t)",
"with",
"driving",
"parameters",
"(4",
"A",
"),",
"then",
"also",
"0j",
"cid:10",
"iz",
"o",
"ICI",
":105)",
".",
"01,",
"called",
"a",
"d:105",
"state.",
"The",
"non-ergodic",
"o,",
"0",
"(0",
"is",
"Z",
"|%b(t))",
"2",
"property",
"of",
"such",
"product",
"states",
"is",
"unique,",
"in",
"a",
"sense",
"that",
"with",
"Pauli",
"matrix",
"Zj",
"=",
"(1",
"the",
"number",
"of",
"them",
"increases",
"linearly",
"with",
"the",
"system",
"size,",
"state",
"of",
"H(t)",
"with",
"the",
"same",
"driving",
"parameters.",
"�o",
"0.",
"with-",
"whereas,",
"other",
"ergodic",
"product",
"states",
"show",
"exponential",
"in-",
"out",
"loss",
"of",
"generality,",
"we",
"only",
"plot",
"the",
"region,",
"A,",
"TZ",
"id:10",
"in",
"crease",
"with",
"the",
"system",
"size.",
"has",
"been",
"understood",
"that",
"their",
"Before",
"investigating",
"the",
"fidelity",
"profile",
"of",
"the",
"fidelity",
"revival",
"is",
"originated",
"from",
"the",
"quantum",
"scarring,",
"i.e.,",
"the",
"PXP",
"model,",
"let",
"us",
"introduce",
"another",
"model",
"which",
"shows",
"model",
"with",
"the",
"very",
"similar",
"feature:",
"spin-1/2",
"the",
"product",
"state",
"with",
"short-time",
"revival",
"is",
"a",
"linear",
"combina-",
"the fro",
"chain",
"tion",
"of",
"scar",
"eigenstates",
"with",
"equivalent",
"energy",
"spacing7.",
"The",
"same",
"driving",
"A",
"sq",
"(t),",
")",
"id:10",
"is",
"mainly",
"composed",
"om",
"ose",
"quantum",
"quan",
"states",
"with",
"|z",
"2",
"almost",
"equal",
"energy",
"spacing.",
"However,",
"because",
"their",
"energy",
"H",
"(t)",
"=",
"Xj",
"+",
"A",
"(t)",
"in",
"j.",
"(5)",
"sq",
"spacing",
"is",
"not",
"perfectly",
"even,",
"it",
"has",
"been",
"pointed",
"out",
"that",
"the",
"For",
"the",
"|Z",
"id:105",
"the",
"free",
"spin-1/2",
"pin-1",
"and",
"del",
"and",
"model",
"fidelity",
"revival",
"of",
"the",
"z",
"�2",
"id:10",
"is",
"tate",
"imperfect�2.",
"system",
"en-",
"share",
"common",
"features",
"in",
"terms",
"of",
"graph",
"theoretical",
"point",
"of",
"There",
"have",
"been",
"many",
"suggestions",
"to",
"adjust",
"the",
"j",
"hancing",
"this",
"imperfect",
"revivals.",
"As",
"one",
"promising",
"way,",
"it",
"has",
"view.",
"To",
"understand",
"it,",
"notice",
"that",
"the",
"PXP",
"model",
"is",
"noth-",
"been",
"studied",
"in",
"both",
"theoretically",
"and",
"experimentally",
"to",
"the",
"ing",
"but",
"the",
"free",
"spin-1/2",
"model",
"with",
"constraints.",
"Hence,",
"the",
"addition",
"of",
"cosine",
"modulation.",
"This",
"modulation",
"plays",
"role",
"graph",
"of",
"length",
"L",
"PXP",
"model,",
"for",
"instance,",
"is",
"a",
"subgraph",
"of",
"of",
"controlling",
"chemical",
"potential",
"to",
"the",
"PXP",
"model,",
"and",
"can",
"the",
"free",
"spin-1/2",
"model",
"with",
"the",
"same",
"length",
"L.",
"Conversely,",
"enhance",
"the",
"fidelity",
"revival",
"of",
"the",
"|Z2(",
"id:10",
"or",
"tate",
"induce",
"consider",
"the",
"length",
"L",
"PXP",
"model",
"with",
"even",
"L.",
"If",
"we",
"give",
"the",
"subharmonic",
"responses",
"in",
"certain",
"driving",
"condition10,15.",
"How-",
"stronger",
"constraint",
"to",
"the",
"PXP",
"model",
"and",
"only",
"allow",
"the",
"states",
"ever,",
"the",
"systematic",
"ways",
"to",
"find",
"such",
"driving",
"have",
"not",
"been",
"with",
"0",
"T",
"id:105",
"at",
"ate",
"sites,",
"then",
"each",
"states",
"stat",
"0z)",
"can",
"The",
"periodically",
"driven",
"PXP",
"model",
"represented",
"as",
"fol-",
"be",
"mapped",
"to",
"the",
"state",
"|xy",
".",
".",
"..",
"z",
"in",
"length",
"L/2",
"free",
"spin-1/2",
"studied",
"in",
"detail,",
"which",
"is",
"the",
"focus",
"of",
"this",
"study.",
"d",
"are",
"either",
"0",
"or",
"1.",
"This",
"shows",
"that",
"model,",
"where",
"x,",
"y,",
"lowing.",
"the",
"graph",
"of",
"length",
"L/2",
"free",
"spin-1/2",
"model",
"is",
"a",
"subgraph",
"of",
"length",
"L",
"PXP",
"model.",
"(2)",
"H(t)",
"=",
"HPXP",
"+",
"^",
"(t)",
"in",
"j",
",",
"Asq(",
"Figure",
"1(a)",
"shows",
"the",
"graph",
"of",
"the",
"PXP",
"Hamiltonian",
"with",
"j",
"L",
"=",
"8,",
"and",
"Figure",
"1(b)",
"shows",
"the",
"graph",
"of",
"the",
"free",
"spin-1/2",
"where",
"H",
"PXP",
"is",
"defined",
"in",
"Equation",
"1",
"and",
"the",
"second",
"term",
"rep-",
"Hamiltonian",
"for",
"L",
"=",
"4.",
"The",
"blue",
"square",
"in",
"Figure",
"1(a)",
"marks",
"resents",
"the",
"periodic",
"driving,",
"where",
"n",
"j",
"=",
"|1",
"se",
"(1",
"05)(ci",
"the",
"the",
"N´eel",
"state",
"|01010101(",
"in",
"105",
"Hamiltonian,",
"tonia",
"corre-",
"number",
"of",
"excited",
"states",
"on",
"each",
"site.",
"In",
"terms",
"of",
"the",
"periodic",
"sponds",
"to",
"the",
"I",
"1111)",
"id:10",
"in",
"tate",
"spin-1/2",
"pin-1",
"also",
"marked"
]
} | |||
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} | {
"word": [
"3",
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">",
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"�",
"04)H",
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"10",
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"model",
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"spin",
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"1",
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"0",
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} | |||
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"word": [
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"showing",
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"harmonic",
"responses",
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"with",
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"=",
"2π/Tf",
"=",
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"The",
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"red",
"line)",
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"line)",
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"while",
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"line)",
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"3b",
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"Indeed,",
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"(a)",
"(b)",
"FIG.",
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"Frequency",
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")",
">",
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"f",
")",
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} | |||
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"Now",
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"∆−",
"(",
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">",
"A",
"+",
",",
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"will",
"shed",
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"light",
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"the",
"Rydberg",
"atom",
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"study-",
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"v",
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"values",
"of",
"F(0)",
"in",
"ing",
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"quantum",
"many-body",
"Figure",
"1.",
"Similar",
"analysis",
"with",
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"∆+",
">",
"n",
"9)",
"scarring",
"systems.",
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"H+",
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"∆−",
"=",
"nT",
"=",
"nwe,",
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"Z",
"i.",
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"focus",
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"the",
"j",
"NH",
"PxP",
"and",
"X",
"As",
"an",
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"future",
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"one",
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"�-",
"with",
"finite",
"van",
"der",
"Waals",
"interaction,",
"and",
"explore",
"the",
"condi-",
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"is",
"to",
"show",
"that",
"for",
"even",
"n",
"case",
"the",
"fidelity",
"presents",
"tions",
"of",
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"subharmonic",
"revival",
"as",
"interaction",
"changes.",
"Since",
"subharmonic",
"response",
"and",
"for",
"odd",
"n",
"case",
"the",
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"Rydberg",
"atoms",
"r",
"as",
"\"",
"5,9,",
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"First,",
"let",
"n",
"=",
"2k",
"be",
"even.",
"In",
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"one",
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"the",
"atom",
"distance",
"to",
"tune",
"their",
"interaction",
"mm",
"T/2",
"=",
"1,",
"and",
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"(z",
"2",
"|u",
"strength",
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"track",
"the",
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"2",
"5",
"id:105",
"One",
"eiH",
"r.6",
"2|eHPXPT0/2|Z2(c",
").",
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"we",
"hus",
"the",
"very",
"similar",
"simila",
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"can",
"also",
"consider",
"the",
"effect",
"of",
"further",
"neighbor",
"van",
"der",
"Waals",
"Equation",
"7,",
"interactions,",
"and",
"explore",
"how",
"the",
"fidelity",
"revival",
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"(Z",
"|U2",
"|Z",
">",
"|(Z",
"|eiHpxpTo/2",
"|Z,",
")",
"I",
"2,",
"42",
"changes,",
"which",
"we",
"will",
"leave",
"as",
"a",
"future",
"work.",
"∆−",
"=",
"kω0",
"0",
">",
"i",
">",
"A",
"+",
"(8)",
"ACKNOWLEdGmEntS",
"mainly",
"occurs",
"at",
"∆−",
"=",
"n",
"=",
"2k",
"+",
"1’s,",
"amutation",
"relation",
"be-",
"Acknowledgments.—",
"We",
"thank",
"Junmo",
"Jeon",
"for",
"valuable",
"resented",
"discussions.",
"This",
"work",
"is",
"supported",
"by",
"National",
"Re-",
"Z",
"and",
"j",
"tween",
"search",
"Foundation",
"Grant",
"(No.",
"2020R1A4A3079707,",
"No.",
"as,",
"2021R1A2C1093060),).",
"~",
"cHnwTo/d",
"u",
"Z",
"j",
"eiHPXPT0/4",
"U",
"Appendix",
"A:",
"Bound",
"of",
"the",
"(sub)harmonic",
"revival",
"on",
"PXP",
"model",
"j",
"Z",
"j",
"e",
"-iHpxpTo/4@iHpxpTo/4",
"=",
"Z",
"i,",
"(9)",
"In",
"this",
"section,",
"we",
"show",
"the",
"harmonic",
"and",
"subharmonic",
"re-",
"j",
"=",
"vival",
"is",
"stable",
"under",
"small",
"2",
"values",
"up",
"to",
"first",
"order,",
"which",
"is",
"j",
"j",
"discussed",
"in",
"Section",
"III.",
"We",
"show",
"the",
"Equations",
"7,",
"8",
"and",
"10",
"still",
"holds",
"if",
"we",
"include",
"the",
"o",
"(12)",
"terms.",
">",
"�.",
"In",
"id:2",
"$>",
"and",
"this",
"results",
"in,",
"We",
"first",
"consider",
"the",
"condition",
"�",
"+",
"n",
">",
"|Z",
">",
"i",
"K{Z",
"I",
")I",
"~",
"1,",
"(c",
"=",
"k",
"+",
"�",
"+",
".",
"(10)",
"U",
"case,",
"because",
"for",
"different",
"sites",
"j",
"(c",
"k,",
"4)",
"have",
"w",
"2",
"2",
"[H+,",
"H*]",
"~",
"o",
"(N22),",
"(A1)",
"Thus,",
"the",
"harmonic",
"response",
"mainly",
"occurs",
"for",
"∆−",
"=",
"(k",
"+",
"I",
"1/2)ω0",
"region.",
"We",
"again",
"show",
"that",
"this",
"revival",
"is",
"robust",
"up",
"to",
"hence",
"we",
"may",
"write",
"O(Ω)",
"order,",
"see",
"Appendix",
"A.",
"In",
"summary",
"of",
"this",
"section,",
"our",
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"analysis",
"pro-",
"eiH+To/4",
"~",
"eiH}To/4",
"=",
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"(A2)",
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"the",
"several",
"peaks",
"in",
"v",
"-",
"j",
"shaped",
"region",
"are",
"emerging",
"in",
"Figure",
"1.",
"Specifically,",
"Equation",
"up",
"to",
"o",
"(12)",
"order.",
"This",
"can",
"be",
"to",
"cosine",
"7",
"explains",
"the",
"long",
"diagonal",
"”separator”",
"peaks",
"on",
"the",
"right",
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"and sine",
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"in",
"O(Ω)",
"order,",
"achieving",
"side",
"of",
"V",
"-shaped",
"region,",
"Equation",
"8",
"explains",
"the",
"”bridge”",
"peaks",
"on",
"the",
"left",
"side,",
"and",
"Equation",
"10",
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"”local”",
"peaks",
"between",
"bridge",
"peaks.",
"It",
"is",
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"to",
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"the",
"eiH*To/4",
"~",
"id:",
"�",
"I",
"o",
"+",
"i",
"sin",
"A",
"+",
"I",
"o",
"n",
"X",
"+",
"Z",
"j",
"4",
"∆+",
"4",
"derivation",
"for",
"Equations",
"7,",
"8",
"and",
"10",
"can",
"be",
"generally",
"applica-",
"If",
"we",
"product",
"all",
"the",
"eiH+To/4",
"terms",
"and",
"left",
"only",
"the",
"o",
"(S2)",
"(A3)",
"ble.",
"order",
"terms,",
"then",
"we",
"finally",
"get",
"H",
")",
"IV.",
"DISCUSSION",
"AND",
"CONCLUSION",
"_",
"eiH+To/4",
"In",
"this",
"work,",
"we",
"study",
"the",
"wave",
"function",
"fidelity",
"revival",
"on",
"the",
"periodically",
"driven",
"PXP",
"model.",
"First,",
"we",
"show",
"that",
"the",
"~",
"�",
"j",
"ZjTo/4",
"1",
"+",
"i",
"sin",
"A+",
"I",
"o",
"∆+",
"+",
"j",
"n",
"X",
"c~ia,2,To/4",
",",
")",
"ca,",
"I+",
"4",
"driving",
"on",
"PXP",
"model",
"induces",
"various",
"interesting",
"responses,",
"j",
"(",
"including",
"subharmonic",
"responses.",
"Based",
"on",
"the",
"graph",
"theoret-",
"n",
"�",
"T",
"o",
"∆+",
"+",
"X",
"j",
"je",
"-iA+ZjTo/4",
"ei∆+",
"ZjTo/4.",
"-",
"ical",
"similarities",
"between",
"PXP",
"model",
"and",
"free",
"spin-1/2",
"model,",
"=",
"+",
"sin",
"4",
"we",
"have",
"claimed",
"and",
"numerically",
"confirmed",
"that",
"the",
"driving",
"j",
"condition",
"which",
"induces",
"the",
"subharmonic",
"response",
"in",
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"PXP",
"(A4)",
"model",
"can",
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"captured",
"by",
"the",
"free",
"spin-1/2",
"model.",
"Then,",
"con-",
"sidering",
"perturbative",
"analysis,",
"the",
"generic",
"driving",
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"value",
"(ci",
"l",
"U",
"[�",
")",
":",
"if",
"this",
"value",
"is",
"large",
"enough",
"then",
"it",
"for",
"subharmonic",
"responses",
"in",
"the",
"PXP",
"model",
"are",
"derived.",
"Our",
"To",
"calculate",
"2",
"subharmonic",
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"we",
"consider",
"the"
]
} | |||
2301.00020v1.pdf_6 | null | ArxivOCR_part_1 | {
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773,
1936
],
[
768,
1916,
785,
1951
],
[
789,
1914,
833,
1952
],
[
873,
1884,
1032,
1922
],
[
1032,
1884,
1084,
1922
],
[
1084,
1884,
1153,
1921
],
[
1153,
1884,
1186,
1922
],
[
1186,
1884,
1281,
1922
],
[
1281,
1884,
1366,
1922
],
[
1366,
1884,
1468,
1922
],
[
1469,
1884,
1533,
1922
],
[
1533,
1884,
1565,
1922
],
[
655,
1916,
688,
1951
],
[
309,
1915,
342,
1950
],
[
432,
1915,
474,
1952
],
[
474,
1915,
503,
1952
],
[
503,
1925,
529,
1953
],
[
544,
1933,
563,
1971
],
[
563,
1943,
587,
1973
],
[
587,
1933,
621,
1972
],
[
642,
1949,
683,
1978
],
[
700,
1933,
727,
1971
],
[
727,
1943,
757,
1973
],
[
873,
1916,
945,
1954
],
[
946,
1916,
1055,
1954
],
[
1055,
1916,
1090,
1954
],
[
1092,
1916,
1141,
1953
],
[
1143,
1916,
1239,
1954
],
[
1240,
1916,
1359,
1953
],
[
1361,
1916,
1477,
1954
],
[
1479,
1916,
1565,
1953
],
[
180,
1933,
199,
1971
],
[
199,
1943,
223,
1973
],
[
223,
1933,
255,
1972
],
[
343,
1934,
397,
1972
],
[
397,
1935,
429,
1972
],
[
278,
1949,
344,
1978
],
[
873,
1949,
908,
1985
],
[
908,
1948,
985,
1986
],
[
985,
1948,
1072,
1986
],
[
1072,
1948,
1151,
1986
],
[
1151,
1948,
1197,
1986
],
[
1197,
1948,
1322,
1986
],
[
236,
1965,
258,
2004
],
[
263,
1981,
291,
2019
],
[
296,
1981,
345,
2018
],
[
349,
1981,
473,
2018
],
[
477,
1981,
640,
2018
],
[
645,
1981,
740,
2019
],
[
745,
1982,
768,
2019
],
[
786,
1981,
833,
2019
],
[
902,
1981,
950,
2018
],
[
951,
1981,
1062,
2018
],
[
1063,
1981,
1111,
2018
],
[
1113,
1981,
1187,
2018
],
[
1188,
1981,
1220,
2018
],
[
1222,
1981,
1245,
2018
],
[
1248,
1981,
1314,
2018
],
[
1315,
1981,
1375,
2018
],
[
1376,
1981,
1409,
2018
],
[
1411,
1981,
1433,
2018
],
[
1436,
1981,
1475,
2018
],
[
1476,
1981,
1565,
2018
],
[
143,
1981,
217,
2018
],
[
222,
1982,
246,
2019
],
[
235,
1995,
257,
2022
]
]
} | {
"word": [
"Anyon",
"statistics",
"through",
"conductance",
"measurements",
"of",
"time-domain",
"interferometry",
"Noam",
"Schiller1,",
"Yotam",
"Shapira2,",
"Ady",
"Stern1,",
"and",
"Yuval",
"Oreg1",
"1Department",
"of",
"Condensed",
"Matter",
"Physics",
"2Department",
"of",
"Physics",
"of",
"Complex",
"Systems",
"Weizmann",
"Institute",
"of",
"Science,",
"Rehovot",
"7610001,",
"Israel",
"We",
"propose",
"method",
"to",
"extract",
"the",
"mutual",
"exchange",
"statistics",
"of",
"the",
"anyonic",
"excitations",
"of",
"a",
"general",
"Abelian",
"fractional",
"quantum",
"Hall",
"state,",
"by",
"comparing",
"the",
"tunneling",
"characteristics",
"of",
"a",
"quantum",
"point",
"contact",
"in",
"two",
"different",
"experimental",
"conditions.",
"In",
"the",
"first,",
"the",
"tunneling",
"current",
"between",
"two",
"edges",
"at",
"different",
"chemical",
"potentials",
"is",
"measured.",
"In",
"the",
"second,",
"one",
"of",
"these",
"edges",
"is",
"2020",
"strongly",
"diluted",
"by",
"an",
"earlier",
"point",
"contact.",
"We",
"describe",
"the",
"case",
"of",
"the",
"dilute",
"beam",
"in",
"terms",
"of",
"a",
"time-domain",
"interferometer",
"between",
"the",
"anyons",
"flowing",
"along",
"the",
"edge",
"and",
"quasiparticle-quasihole",
"excitations",
"created",
"at",
"the",
"tunneling",
"quantum",
"point",
"contact.",
"In",
"both",
"cases,",
"temperature",
"is",
"kept",
"large,",
"such",
"that",
"the",
"measured",
"current",
"is",
"given",
"to",
"linear",
"response.",
"Remarkably,",
"our",
"proposal",
"does",
"Soe",
"not",
"require",
"the",
"measurement",
"of",
"current",
"correlations,",
"and",
"allows",
"us",
"to",
"carefully",
"separate",
"effects",
"of",
"the",
"fractional",
"charge",
"and",
"statistics",
"from",
"effects",
"of",
"intra-",
"and",
"inter-edge",
"interactions.",
"Introduction.—",
"It",
"has",
"been",
"almost",
"four",
"decades",
"since",
"injected",
"quasiparticle",
"charge,",
"8",
"is",
"the",
"tunneling",
"quasi-",
"the",
"initial",
"proposal",
"that",
"the",
"elementary",
"quasiparticles",
"particle",
"scaling",
"dimension,",
"θ12",
"is",
"the",
"mutual",
"statistics",
"of",
"fractional",
"quantum",
"Hall",
"(FQH)",
"systems",
"obey",
"anyonic",
"phase",
"between",
"the",
"injected",
"and",
"tunneling",
"quasiparticles,",
"statistics",
"[1].",
"Despite",
"the",
"apparent",
"maturity",
"of",
"the",
"field,",
"T",
"is",
"temperature,",
"and",
"G",
"direct",
"is",
"a",
"residual",
"conductance",
"the",
"pursuit",
"to",
"definitively",
"observe",
"the",
"physical",
"quanti-",
"corresponding",
"to",
"direct",
"tunneling",
"[24–26]",
"through",
"both",
"ties",
"and",
"quantum",
"numbers",
"characterizing",
"anyons",
"[2,",
"3]",
"is",
"QPCs.",
"The",
"full",
"crossover",
"between",
"these",
"two",
"limits",
"is",
"constantly",
"being",
"reinvigorated",
"[4–20].",
"In",
"particular,",
"early",
"shown",
"schematically",
"in",
"Fig.",
"1(b).",
"2020",
"saw",
"two",
"major",
"experimental",
"steps",
"forward:",
"the",
"ob-",
"The",
"mechanism",
"leading",
"to",
"this",
"result",
"is",
"a",
"time-domain",
"servation",
"of",
"anyonic",
"braiding",
"in",
"a",
"Fabry-Perot",
"interfer-",
"interferometer",
"at",
"the",
"tunneling",
"QPC",
"which",
"is",
"created",
"by",
"ometer",
"[21],",
"and",
"demonstration",
"of",
"a",
"so-called",
"“anyon",
"col-",
"the",
"dilute",
"incident",
"beam.",
"The",
"interference",
"is",
"between",
"two",
"lider”",
"[22,",
"23]",
"using",
"cross-correlation",
"measurements.",
"processes,",
"in",
"which",
"a",
"quasiparticle-quasihole",
"excitation",
"Here",
"we",
"show",
"that",
"anyonic",
"statistics",
"can",
"be",
"inferred",
"di-",
"occurs",
"at",
"the",
"tunneling",
"QPC",
"either",
"before",
"or",
"after",
"the",
"rectly",
"from",
"conductance",
"measurements,",
"without",
"requir-",
"arrival",
"of",
"an",
"injected",
"quasiparticle",
"(see",
"Fig.",
"2).",
"A",
"similar",
"Goe",
"ing",
"current",
"correlation",
"measurements",
"or",
"explicitly",
"build-",
"physical",
"picture",
"has",
"been",
"shown",
"in",
"Refs.",
"[25,",
"27,",
"28].",
"We",
"ing",
"an",
"interferometer.",
"The",
"configuration",
"we",
"propose",
"to",
"further",
"find",
"that",
"this",
"interference",
"is",
"sensitive",
"to",
"the",
"mu-",
"obtain",
"this",
"result",
"consists",
"of",
"quantum",
"point",
"contact",
"tual",
"statistics",
"phase",
"between",
"the",
"injected",
"and",
"the",
"tunnel-",
"(QPC)",
"between",
"two",
"edges",
"of",
"a",
"general",
"Abelian",
"FQH",
"state",
"ing",
"quasiparticles,",
"0",
"12",
".",
"We",
"emphasize",
"that",
"these",
"quasi-",
"which",
"are",
"driven",
"out",
"of",
"equilibrium.",
"The",
"edges",
"may",
"be",
"particles",
"are",
"not",
"necessarily",
"of",
"the",
"same",
"type,",
"although",
"Ganh",
"driven",
"off-equilibrium",
"by",
"one",
"of",
"three",
"methods:",
"inject-",
"they",
"must",
"be",
"supported",
"by",
"the",
"same",
"FQH",
"liquid.",
"ing",
"a",
"single",
"quasiparticle",
"into",
"one",
"of",
"the",
"edges;",
"injecting",
"Since",
"our",
"focus",
"is",
"the",
"interference",
"of",
"two",
"amplitudes",
"a",
"Poissonian,",
"dilute",
"beam",
"of",
"quasiparticles",
"into",
"one",
"of",
"which",
"differ",
"from",
"one",
"another",
"by",
"the",
"orderings",
"of",
"events,",
"the",
"edges;",
"and",
"placing",
"a",
"finite",
"bias",
"voltage",
"between",
"the",
"the",
"key",
"point",
"of",
"our",
"analysis",
"is",
"the",
"identification",
"of",
"the",
"edges.",
"phase",
"differences",
"between",
"the",
"two",
"orderings.",
"We",
"find",
"Our",
"proposed",
"setup,",
"shown",
"in",
"Fig.",
"1(a),",
"allows",
"a",
"phase",
"differences",
"that",
"are",
"determined",
"by",
"the",
"quasiparti-",
"smooth",
"transition",
"between",
"the",
"dilute",
"Poissonian",
"beam",
"cle",
"charge",
"e*",
"which",
"is",
"a",
"fraction",
"of",
"the",
"electron",
"charge",
"and",
"a",
"full",
"beam",
"at",
"finite",
"bias",
"voltage.",
"This",
"is",
"ob-",
"for",
"non-integer",
"values",
"of",
"v",
"[4–6];",
"the",
"scaling",
"dimension",
"δ,",
"tained",
"by",
"tuning",
"second,",
"injection",
"QPC",
"from",
"fully",
"open",
"which",
"defines",
"the",
"zero-temperature",
"time",
"correlations",
"of",
"inj,",
"the",
"quasiparticle",
"via",
"the",
"relation",
"t(T)4(0)",
"5",
"ti",
"T",
"S",
"h",
"(a",
"differential",
"conductance,",
"G",
"inj",
"28",
"dI",
"j/dV",
"satisfying",
"G",
"Tinj",
"/o",
"xy",
"0)",
"to",
"fully",
"closed",
"(G",
"in",
"/o",
"xy",
"1).",
"We",
"hence-",
"[29–32];",
"and",
"the",
"exchange",
"statistics",
"phase",
"θ,",
"which",
"for",
"forth",
"refer",
"to",
"these",
"as",
"the",
"dilute",
"and",
"full",
"limits,",
"respec-",
"anyons",
"take",
"special",
"values",
"beyond",
"the",
"fermionic",
"�",
"and",
"tively.",
"the",
"bosonic",
"2π",
"[1–3].",
"We",
"propose",
"sweeping",
"G",
"inj",
"through",
"this",
"range,",
"and",
"We",
"are",
"interested",
"here",
"in",
"isolating",
"the",
"effect",
"of",
"o",
"from",
"measuring",
"the",
"ratio",
"I/I",
"inj",
"where",
"I",
"is",
"the",
"measured",
"cur-",
"the",
"other",
"two",
"effects.",
"In",
"particular,",
"we",
"would",
"like",
"to",
"rent",
"after",
"the",
"tunneling",
"QPC,",
"and",
"I",
"inj",
"is",
"the",
"injected",
"inci-",
"separate",
"it",
"from",
"the",
"effect",
"of",
"d.",
"For",
"non-interacting",
"dent",
"current,",
"as",
"defined",
"in",
"Fig.",
"1(a).",
"Comparing",
"the",
"val-",
"edges,",
"in",
"which",
"all",
"the",
"modes",
"propagate",
"in",
"the",
"same",
"di-",
"out",
"non-universal",
"rection,",
"2πδ",
"=",
"0;",
"however,",
"in",
"general",
"8",
"is",
"affected",
"by",
"non-universal",
"factors,",
"such",
"as",
"intra-edge",
"and",
"inter-edge",
"G",
"direct",
".",
"(1)",
"interactions,",
"1/f",
"noise",
"or",
"neutral",
"modes",
"[33–38].",
"This",
"in",
"+",
"=",
"sin",
"20",
"12",
"I",
"inj",
"(T",
"full",
"G",
"inj",
"stark",
"contrast",
"to",
"the",
"charge,",
"exchange",
"statistics",
"phase,",
"I",
"inj",
"(T",
"2πe∗",
"1e∗",
"dilute",
"or",
"filling",
"factor,",
"which",
"are",
"universal.",
"*",
"is",
"the",
"tunneling",
"quasiparticle",
"charge,",
"e",
"the",
"We",
"separate",
"the",
"effect",
"of",
"o",
"from",
"that",
"of",
"8",
"by",
"tuning",
"Here,",
"e",
"1"
]
} | |||
2301.00021v1.pdf_2 | null | ArxivOCR_part_1 | {
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[
1291,
1685,
1355,
1723
],
[
1355,
1685,
1378,
1723
],
[
1387,
1666,
1419,
1723
],
[
1419,
1683,
1480,
1724
],
[
1483,
1685,
1564,
1724
],
[
732,
1692,
752,
1719
],
[
143,
1711,
188,
1746
],
[
188,
1711,
300,
1746
],
[
300,
1711,
366,
1746
],
[
366,
1711,
405,
1746
],
[
405,
1711,
428,
1746
],
[
428,
1711,
534,
1745
],
[
534,
1711,
581,
1745
],
[
581,
1711,
651,
1745
],
[
872,
1718,
893,
1755
],
[
893,
1717,
922,
1756
],
[
922,
1718,
946,
1755
],
[
946,
1718,
1029,
1756
],
[
1029,
1718,
1061,
1756
],
[
1061,
1718,
1173,
1756
],
[
1176,
1718,
1233,
1756
],
[
1236,
1718,
1320,
1754
],
[
1320,
1718,
1353,
1756
],
[
1353,
1718,
1424,
1756
],
[
1424,
1718,
1564,
1755
],
[
873,
1750,
926,
1787
],
[
926,
1750,
954,
1788
],
[
954,
1750,
1018,
1788
],
[
1018,
1750,
1090,
1788
],
[
1090,
1750,
1128,
1788
],
[
1128,
1751,
1150,
1788
],
[
1160,
1750,
1192,
1787
],
[
1192,
1745,
1258,
1787
],
[
1266,
1746,
1306,
1787
],
[
1140,
1732,
1162,
1777
],
[
1249,
1734,
1278,
1776
],
[
1138,
1765,
1160,
1792
],
[
901,
1785,
959,
1822
],
[
961,
1785,
1130,
1822
],
[
1132,
1785,
1165,
1822
],
[
1167,
1784,
1225,
1822
],
[
1228,
1785,
1287,
1822
],
[
1289,
1785,
1394,
1822
],
[
1397,
1785,
1450,
1822
],
[
1452,
1785,
1534,
1822
],
[
1539,
1785,
1564,
1822
],
[
143,
1817,
191,
1854
],
[
191,
1818,
284,
1854
],
[
284,
1817,
319,
1854
],
[
320,
1817,
343,
1854
],
[
343,
1817,
434,
1854
],
[
434,
1817,
514,
1854
],
[
515,
1816,
538,
1854
],
[
539,
1816,
600,
1854
],
[
601,
1817,
687,
1854
],
[
688,
1816,
833,
1854
],
[
873,
1816,
928,
1854
],
[
929,
1817,
962,
1855
],
[
964,
1817,
1154,
1854
],
[
1155,
1816,
1194,
1855
],
[
1196,
1817,
1221,
1855
],
[
1223,
1817,
1301,
1856
],
[
1303,
1817,
1367,
1854
],
[
1369,
1817,
1398,
1854
],
[
1399,
1817,
1522,
1854
],
[
1525,
1817,
1564,
1854
],
[
143,
1848,
247,
1886
],
[
247,
1849,
267,
1886
],
[
267,
1848,
434,
1886
],
[
434,
1848,
558,
1886
],
[
558,
1848,
634,
1886
],
[
634,
1848,
694,
1886
],
[
694,
1848,
786,
1886
],
[
786,
1848,
833,
1886
],
[
873,
1848,
926,
1887
],
[
926,
1848,
1008,
1886
],
[
1008,
1848,
1039,
1886
],
[
1039,
1848,
1086,
1886
],
[
1086,
1848,
1246,
1886
],
[
1266,
1860,
1325,
1889
],
[
1329,
1848,
1393,
1886
],
[
1393,
1848,
1502,
1886
],
[
1502,
1849,
1565,
1886
],
[
143,
1880,
175,
1918
],
[
175,
1880,
220,
1918
],
[
220,
1880,
285,
1918
],
[
285,
1880,
315,
1917
],
[
315,
1880,
421,
1917
],
[
421,
1880,
491,
1918
],
[
491,
1880,
522,
1917
],
[
522,
1880,
570,
1918
],
[
570,
1880,
622,
1918
],
[
624,
1880,
671,
1918
],
[
671,
1880,
748,
1918
],
[
748,
1880,
780,
1917
],
[
780,
1880,
833,
1917
],
[
872,
1880,
961,
1918
],
[
961,
1880,
995,
1918
],
[
995,
1880,
1046,
1918
],
[
1046,
1880,
1157,
1918
],
[
1157,
1880,
1192,
1918
],
[
1192,
1880,
1249,
1917
],
[
1249,
1881,
1283,
1918
],
[
1283,
1881,
1307,
1918
],
[
1307,
1880,
1431,
1917
],
[
1431,
1880,
1505,
1918
],
[
1529,
1891,
1554,
1920
],
[
143,
1912,
232,
1949
],
[
236,
1912,
276,
1950
],
[
279,
1912,
413,
1949
],
[
415,
1912,
440,
1950
],
[
443,
1911,
534,
1949
],
[
537,
1911,
648,
1949
],
[
650,
1912,
683,
1950
],
[
686,
1912,
734,
1950
],
[
738,
1911,
834,
1950
],
[
872,
1912,
954,
1950
],
[
956,
1912,
997,
1950
],
[
1000,
1912,
1068,
1950
],
[
1071,
1912,
1105,
1949
],
[
1108,
1912,
1132,
1950
],
[
1134,
1911,
1306,
1950
],
[
1318,
1911,
1447,
1949
],
[
1449,
1911,
1510,
1949
],
[
1512,
1912,
1564,
1949
],
[
730,
1924,
768,
1982
],
[
143,
1944,
231,
1982
],
[
234,
1944,
312,
1982
],
[
315,
1943,
348,
1981
],
[
351,
1943,
399,
1982
],
[
402,
1944,
502,
1982
],
[
506,
1943,
570,
1982
],
[
572,
1944,
633,
1982
],
[
635,
1944,
673,
1981
],
[
662,
1954,
697,
1984
],
[
686,
1943,
725,
1981
],
[
736,
1975,
834,
2013
],
[
872,
1976,
946,
2014
],
[
946,
1976,
1033,
2014
],
[
1033,
1975,
1055,
2014
],
[
1058,
1976,
1076,
2014
],
[
1076,
1976,
1173,
2013
],
[
1173,
1976,
1280,
2013
],
[
1280,
1976,
1328,
2013
],
[
1328,
1976,
1410,
2013
],
[
1410,
1976,
1463,
2013
],
[
1463,
1976,
1492,
2013
],
[
1492,
1976,
1564,
2013
],
[
143,
1976,
288,
2014
],
[
292,
1976,
340,
2013
],
[
344,
1976,
423,
2013
],
[
427,
1976,
546,
2013
],
[
549,
1976,
582,
2014
],
[
585,
1976,
611,
2014
],
[
611,
1986,
643,
2016
],
[
668,
1976,
732,
2013
],
[
770,
1944,
791,
1981
],
[
791,
1954,
807,
1983
],
[
807,
1944,
830,
1981
],
[
872,
1944,
936,
1982
],
[
937,
1944,
969,
1982
],
[
969,
1943,
1140,
1982
],
[
1142,
1944,
1247,
1981
],
[
1247,
1943,
1286,
1981
],
[
1287,
1943,
1335,
1982
],
[
1336,
1943,
1419,
1982
],
[
1419,
1943,
1441,
1982
],
[
1453,
1943,
1508,
1981
],
[
1508,
1943,
1564,
1982
],
[
1043,
1956,
1069,
2002
],
[
1045,
1988,
1066,
2018
]
]
} | {
"word": [
"2",
"(a)",
"that",
"the",
"current",
"I",
"is",
"given",
"to",
"linear",
"response",
"in",
"I",
"inj",
".",
"We",
"𝐷",
"𝑆",
"present",
"an",
"analytic",
"expression",
"generalizing",
"Eq.",
"(1)",
"out-",
"𝑢",
"𝑎",
",*",
"𝛿",
"2",
"2",
",",
"side",
"of",
"this",
"regime",
"in",
"App",
"A,",
"Eq.",
"(A5).",
"While",
"in",
"the",
"full",
"limit",
"the",
"edge",
"that",
"enters",
"the",
"tunnel-",
"inj",
"Injection",
"ing",
"QPC",
"is",
"in",
"equilibrium",
"at",
"chemical",
"potential",
"V",
",",
"at",
"the",
"𝑢",
"𝐷",
"𝑢",
"QPC",
"dilute",
"limit",
"we",
"need",
"the",
"injection",
"QPC",
"to",
"reflect",
"only",
"a",
"𝑎",
"small",
"fraction",
"of",
"the",
"impinging",
"electrons,",
"such",
"that",
"the",
"resulting",
"injection",
"current",
"is",
"Poissonian",
"and",
"rare.",
"Said",
"*",
"𝛿",
"1",
"1",
"Tunneling",
"differently,",
"the",
"injected",
"current",
"in",
"this",
"limit",
"must",
"satisfy",
"ere",
"QPC",
"o",
"xy",
"V",
"Furthermore,",
"rmo",
"beam",
"must",
"still",
"be",
"dilute",
"I",
"inj",
"when",
"arriving",
"at",
"the",
"tunneling",
"QPC.",
"As",
"such,",
"the",
"dis-",
"𝑉",
"𝐼",
"𝑆",
"𝑑",
"tance",
"between",
"the",
"two",
"QPCs",
"must",
"be",
"sufficiently",
"small",
"𝑑",
"that",
"no",
"equilibration",
"or",
"dephasing",
"occurs",
"along",
"the",
"way.",
"Finally,",
"we",
"assume",
"that",
"tuning",
"the",
"injection",
"QPC",
"does",
"(b)",
"not",
"affect",
"the",
"transparency",
"of",
"the",
"tunneling",
"QPC,",
"to",
"en-",
"[I/I",
"inj",
"]dil",
"sure",
"that",
"all",
"non-universal",
"constants",
"are",
"cancelled",
"when",
"[I/I",
"inj]full",
"examining",
"the",
"ratio",
"of",
"the",
"two",
"limits.",
"[39]",
"0.45",
"ve2",
"Easy",
"extraction",
"of",
"0",
"12",
"requires",
"G",
"direct",
"to",
"be",
"sub-",
"sin(20",
"12",
")",
"2�eje�",
"dominant",
"(see",
"Eq.",
"(1)).",
"Quantitatively,",
"this",
"is",
"the",
"case",
"if",
"both",
"k",
"B",
"T",
"(",
"id",
"28)",
"eV",
"1",
"<",
"2",
"are",
"satisfied.",
"isfied",
"con-",
"0.40",
"straints",
"result",
"from",
"the",
"direct",
"tunneling",
"process",
"being",
"8",
"=",
"1/6",
"I/I",
"inj",
"-",
"8",
"1",
"=",
"1/6",
"dominated",
"by",
"short",
"time",
"scales.",
"Naive",
"theories",
"describ-",
"Full",
"0.2",
"-",
"8",
"1",
"=",
"1/5",
"ing",
"quasiparticles",
"may",
"satisfy",
"this",
"condition",
"even",
"if",
"the",
"Dilute",
"0.35",
"0.1",
"aforementioned",
"non-universal",
"effects",
"change",
"the",
"scaling",
"-",
"8",
"1",
"=",
"1/3",
"dimension",
"quite",
"significantly.",
"For",
"example,",
"theory",
"gives",
"T[mK]",
"0.30",
"800",
"8",
"=",
"1/2m",
"for",
"Laughlin",
"quasiparticles.",
"400",
"T[mK]",
"Edge",
"theory.—",
"We",
"now",
"define",
"the",
"system’s",
"Hamilto-",
"200",
"400",
"600",
"800",
"1000",
"nian",
"and",
"derive",
"the",
"current.",
"As",
"shown",
"by",
"Wen,",
"the",
"edge",
"theory",
"of",
"a",
"general",
"Abelian",
"FQH",
"state",
"can",
"be",
"described",
"e",
".",
"FIG.",
"1.",
"(a)",
"Two",
"counter-propagating",
"edge",
"modes",
"(u/d)",
"of",
"by",
"n-boson",
"fields,",
"φ(x,",
"t)",
"($",
"¢",
"2",
",",
".",
"$",
"n",
")T",
"[2].",
"These",
"a",
"fractional",
"quantum",
"Hall",
"droplet",
"at",
"filling",
"factor",
"v",
"are",
"con-",
"nected",
"by",
"a",
"quantum",
"point",
"contact,",
"through",
"which",
"quasipar-",
"which",
"determines",
"the",
"electric",
"charge",
"carried",
"by",
"each",
"bo-",
"ticles",
"of",
"charge",
"e",
"i",
"and",
"scaling",
"dimension",
"8",
"1",
"can",
"tunnel.",
"Cur-",
"define",
"the",
"theory",
"in",
"conjunction",
"with",
"a",
"charge",
"vector,",
"q,",
"son",
"field,",
"and",
"the",
"so",
"called",
"K-matrix,",
"which",
"determines",
"rent",
"is",
"measured",
"at",
"the",
"lower",
"edge’s",
"drain,",
"denoted",
"by",
"I.",
"A",
"current",
"of",
"I",
"inj",
"is",
"injected",
"into",
"the",
"upper",
"edge",
"via",
"a",
"second,",
"in-",
"the",
"commutation",
"relations",
"between",
"the",
"boson",
"fields,",
"jection",
"QPC,",
"e.g.",
"from",
"a",
"third",
"auxiliary",
"edge",
"mode",
"(a).",
"The",
"[$",
"i(x),",
"a",
"x",
"$",
"j(x",
")]",
"j",
"x(cid",
"-1)",
"ij",
"¡8(x",
"-",
"(K",
").",
"(2)",
"Dj",
"/",
"-",
",/",
"injection",
"QPC",
"is",
"placed",
"at",
"a",
"bias",
"voltage",
"of",
"V,",
"and",
"allows",
"tunneling",
"of",
"quasiparticles",
"of",
"charge",
"e∗",
"and",
"scaling",
"dimension",
"8",
"2",
"All",
"other",
"sources",
"and",
"drains",
"are",
"grounded.",
"(b)",
"The",
"ratio",
"The",
"filling",
"factor",
"is",
"then",
"given",
"by",
"v",
"=",
"qT",
"K",
"−1q,",
"and",
"the",
"-",
"between",
"I/I",
"inj",
"in",
"the",
"dilute",
"case",
"and",
"I/I",
"inj",
"in",
"the",
"full",
"case,",
"charge",
"density",
"is",
"given",
"by",
"p",
"=",
"2T",
"q",
".",
"õ",
"�",
"$.",
"In",
"terms",
"of",
"as",
"a",
"function",
"of",
"temperature,",
"for",
"v",
"=",
"i",
"1/e",
"=",
"e;",
"2/e",
"=",
"1/3,",
"these",
"fields,",
"the",
"Hamiltonian",
"of",
"a",
"single",
"FQH",
"edge",
"mode",
"C2",
"and",
"for",
"different",
"scaling",
"dimensions",
"d",
"1",
".",
"For",
"the",
"dilute",
"case,",
"is",
"given",
"by",
"nS",
"i",
"we",
"I",
"inj",
"=",
"10pA,",
"and",
"assume",
"k",
"B",
"T",
"id",
"28)",
"all",
"relevant",
"leva",
"peratures,",
"such",
"that",
"the",
"contribution",
"from",
"G",
"direct",
"to",
"Eq.",
"(1)",
"is",
"negligible.",
"In",
"the",
"full",
"case,",
"we",
"use",
"V",
"=",
"10µV",
"Both",
"cases",
"H",
"redge",
"=",
"1",
"dx0",
"$",
"V",
"¡0",
"$",
"(3)",
"use",
"&",
"=",
"72mK,",
"T",
"=",
"10",
"-13s.",
"When",
"the",
"dilute",
"case",
"satisfies",
"x",
"ij",
"x",
".c",
"4π",
"i,j=1",
"eV",
"d",
"28)",
"c",
",",
"and",
"id:",
"full",
"case",
"satisfies",
"hl",
"inj",
"/e",
")",
"k",
"b",
"hI",
"Tinj,",
"j/e",
"=",
"veV/27",
"ν",
"k",
"B",
"T",
"(cid:",
"c,",
"the",
"ratio",
"approaches",
"12",
"asymptote",
"that",
"does",
"not",
"depend",
"on",
"scaling",
"dimension,",
"allow-",
"where",
"V",
"is",
"positive",
"definite",
"matrix",
"describing",
"the",
"ve-",
"ing",
"extraction",
"of",
"the",
"mutual",
"statistics",
"θ12.",
"2.",
"Inset:",
"I/I",
"inj",
"for",
"the",
"locities",
"of",
"the",
"modes",
"and",
"intra-edge",
"interactions.",
"These",
"dilute",
"and",
"full",
"cases",
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"i"
]
} | |||
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} | {
"word": [
"3",
"quasiparticle",
"creation",
"operator",
"-",
"il2",
".$(u)(xinj,tinj)",
"on",
"the",
"by",
"edge",
"creates",
"a",
"soliton",
"in",
"each",
"of",
"the",
"boson",
"fields,",
"H",
"T",
"=",
"&",
"Â",
"1",
"Ât",
"ˆA(t)",
"=",
"ei(l1",
".$(u)(0,t).",
")-",
".¢(d)(0,t)).",
"(4)",
"-",
"φ(u)(x,",
"t",
"inj",
")",
"�",
"d:55)→",
"t",
"inj",
"i)",
"-",
"inj",
"-1l",
"2π",
"(x",
"Θ",
"Cinj",
")",
".",
"inj",
"_",
"2",
"Here,",
"&",
"is",
"a",
"small",
"tunneling",
"amplitude,",
"which",
"we",
"assume",
"to",
"be",
"real,",
"and",
"φ(u)",
"(φ(d))",
"are",
"the",
"bosonic",
"field",
"operators",
"We",
"assume",
"here",
"the",
"injection",
"happens",
"instantaneously.",
"on",
"the",
"upper",
"(lower)",
"edge.",
"We",
"project",
"the",
"auxiliary",
"edge",
"This",
"assumption",
"will",
"be",
"relaxed",
"to",
"find",
"the",
"subleading",
"a",
"out",
"of",
"the",
"Hamiltonian,",
"as",
"it",
"is",
"only",
"used",
"to",
"“initialize”",
"term",
"of",
"Eq.",
"(1).",
"The",
"fields",
"at",
"general",
"times",
"can",
"then",
"be",
"obtained",
"using",
"th",
"the",
"upper",
"edge",
"to",
"the",
"equations",
"of",
"motion",
"dictated",
"by",
"the",
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"in",
"Th",
"*",
"id:104",
"_",
"A†(t)",
".",
"Since",
"the",
"lower",
"edge",
"is",
"grounded,",
"this",
"amounts",
"to",
"replacing",
"x",
"x",
"inj",
"x",
"x",
"inj",
"v",
"(t",
"oc",
"tinj",
").",
"i&et",
"operator,",
"I",
"T",
"(t)",
"=",
"Eq.",
"(3).",
"If",
"all",
"modes",
"are",
"chiral",
"with",
"the",
"same",
"velocity",
"v,",
"1",
"the",
"The",
"soliton",
"thus",
"arrives",
"at",
"the",
"QPC,",
"x",
"=",
"0,",
"at",
"time",
".",
"anding",
"10",
"leading",
"t0",
"ti",
"tinj",
"x",
"Cinj,",
"j/v.",
"we",
"T",
"order",
"The",
"c-number",
"shift",
"in",
"the",
"bosonic",
"field",
"of",
"Eq.",
"(8)",
"leads",
"to",
"a",
"phase",
"shift",
"in",
"the",
"correlator",
"Eq.",
"(6).",
"We",
"see",
"directly",
"[Ât(t),",
"Â(t",
")",
"+",
"Â",
"(t",
"1),",
"A†(t(",
"it",
"I(t)",
"=",
"1ξ2",
"0",
"dt(",
"from",
"the",
"definition",
"of",
"the",
"operator",
"Â",
"in",
"E",
"o",
"_",
"-1l2",
"[0",
"(",
"(5)",
"Here,",
"[·,",
"·]",
"denotes",
"commutation,",
"and",
"expectation",
"values",
"Â",
"t(t)Â(t",
"),",
"qp",
"=",
"<",
"it",
",,",
"are",
"calculated",
"with",
"respect",
"to",
"the",
"Hamiltonian",
"in",
"the",
"Â(t)Â*(ť",
"),",
"qp",
"=",
"+",
"o)]",
"absence",
"of",
"tunneling.",
"(9)",
"Deviation",
"from",
"Equilibrium.—",
"It",
"is",
"clear",
"from",
"Eq.",
"(5)",
"The",
"phase",
"we",
"obtain",
"is",
"the",
"standard",
"definition",
"of",
"that",
"one",
"needs",
"to",
"derive",
"correlation",
"functions",
"such",
"as",
"mutual",
"braiding",
"statistics",
"between",
"two",
"quasiparticles,",
"(Ât(t)Â(ť",
".",
"A(",
"(cid:48))(ci",
":1",
"temperature",
"T,",
"um,",
"sys-",
"o",
"12",
"al",
"K",
"−1l2",
"2",
"[2].",
"The",
"expression",
"in",
"Eq.",
"(9)",
"shows",
"tem",
"is",
"particle-hole",
"symmetric,",
"and",
"the",
"correlation",
"func-",
"ld",
"that",
"the",
"product",
"gains",
"a",
"phase",
"of",
"e2iθ12sgn(t−t(c",
"-t')",
"if",
"8))",
"t-",
"tions",
"are",
"given",
"by",
"[2,",
"40]",
"arrival",
"time",
"t",
"o",
"is",
"between",
"the",
"times",
"t(",
"id:",
"t,",
"and",
"a",
"triv-",
"{",
"it",
"Ť(t)Â(t",
"),",
"=",
"Â(t)Â",
"i(t",
"1)>",
"0",
"Â",
"f'",
"ît",
"(6)",
"ial",
"phase",
"of",
"1",
"otherwise.",
"We",
"emphasize",
"how",
"naturally",
"this",
"result",
"came",
"from",
"the",
"underlying",
"theory:",
"the",
"only",
"as-",
"-",
"-i2nô1sgn(t",
"t-",
"t",
"),",
"relations,",
"(2),",
"and",
"the",
"existence",
"of",
"quasiparticles",
"in",
"the",
"(cid",
"sumptions",
"necessary",
"to",
"obtain",
"this",
"are",
"the",
"commutation",
"=",
"sir",
"edge’s",
"excitation",
"spectrum.",
"where",
"8",
"is",
"the",
"scaling",
"dimension",
"of",
"the",
"quasiparticle",
"t",
"1",
",",
"This",
"result",
"holds",
"for",
"different",
"boson",
"modes",
"with",
"differ-",
">",
"0",
"is",
"a",
"short",
"time",
"cutoff.",
"and",
"ent",
"velocities",
"if",
"all",
"solitons",
"arrive",
"at",
"the",
"tunneling",
"QPC",
"Two",
"main",
"features",
"are",
"carried",
"over",
"from",
"Eq.",
"(6)",
"to",
"the",
"more",
"or",
"less",
"concurrently,",
"avoiding",
"dephasing.",
"This",
"is",
"i",
"correlation",
"functions",
"out",
"of",
"equilibrium",
"-",
"the",
"exponen-",
"the",
"case",
"if",
"x",
"Cinj",
"|/Av",
"ent",
"id:",
"where",
"12",
"is",
"the",
"velocity",
"tial",
"decay",
"at",
"time",
"difference",
"larger",
"than",
"(ci",
"26)",
"the",
"difference",
"between",
"the",
"fastest",
"and",
"the",
"slowest",
"modes.",
"phase",
"e2πiδ1",
"associated",
"with",
"an",
"interchange",
"of",
"the",
"time",
"Time-domain",
"interferometry.—",
"The",
"appearance",
"of",
"the",
"arguments.",
"phase,",
"0",
"12",
"can",
"be",
"understood",
"as",
"time-domain",
"interfer-",
"o",
"We",
"now",
"consider",
"two",
"non-equibrium",
"In",
"the",
"first",
"ometry",
"of",
"the",
"two",
"distinct",
"e",
"quasiparticle-quasihole",
"n t",
"se",
"we",
"introduce",
"a",
"constant",
"bias",
"voltage",
"V",
"V",
"u",
"V",
"be-",
"excitations,",
"before",
"and",
"after",
"the",
"injected",
"e∗",
"quasiparticle",
"tween",
"the",
"edges.",
"In",
"the",
"setup",
"of",
"Fig.",
"1(a),",
"this",
"corre-",
"arrives",
"at",
"the",
"QPC.",
"A",
"similar",
"physical",
"picture",
"has",
"been",
"sponds",
"to",
"a",
"fully",
"closed",
"injection",
"QPC,",
"i.e.",
"I",
"inj",
"=",
"o",
"xy",
"V",
".",
"shown",
"in",
"Ref.",
"[25,",
"27,",
"28].",
"The",
"introduction",
"of",
"the",
"voltages",
"can",
"be",
"formally",
"ab-",
"To",
"show",
"this",
"we",
"consider",
"the",
"configuration",
"of",
"a",
"single",
"sorbed",
"into",
"the",
"boson",
"fields",
"by",
"use",
"of",
":",
"$(u/d)(x,",
"/d",
"(",
"the",
"non-equilibrium",
"correlation",
"function",
"takes",
"the",
"form,",
"transformation,",
"which",
"maps",
"φ(u/d)(x,",
"t)",
"simple",
"gauge",
"injected",
"particle,",
"as",
"described",
"in",
"Fig.",
"2(a).",
"In",
"this",
"case",
"rar",
"-1qV",
"u/d",
"(t",
"or",
"x/v)",
"/(c",
"d:12",
"accordingly",
"rdingly",
"mod",
"K",
"(",
"Â",
"t(t)Â(t",
"')",
"qp",
"(",
"<",
"x",
"l2",
"(t",
"o",
")Â",
"t(t)Â(ť",
"Pi2",
",(to",
"))",
",",
"it",
"+,",
",t",
"correlation",
"functions",
"by",
"a",
"phase",
"factor",
"(10)",
"(Ât(t)Â(t",
"')",
"full",
"=",
"(",
"Â",
"t(t)Â(ť",
"))",
"eiejv",
"(t",
"_t",
"-t'),",
">",
"<",
"Â(t)Â",
"ˆA(t)",
"')",
">",
"full",
"=",
"Â(t)Â*(ť",
")",
"0",
"-ieçv",
"(t",
"_",
"/",
").",
"(7)",
"i.e.,",
"the",
"expectation",
"value",
"is",
"calculated",
"with",
"respect",
"to",
"ît",
"t",
"the",
"state",
"resulting",
"from",
"exciting",
"the",
"ground",
"state",
"10",
"o",
"id:1",
"a",
"single",
"quasiparticle.",
"Here",
"we",
"omit",
"the",
"position",
"variable",
"In",
"the",
"second",
"non-equilibrium",
"driving,",
"we",
"consider",
"in-",
"from",
"the",
"quasiparticle",
"injection",
"operator",
"ψ†",
"(t",
"),",
"and",
"as-",
"jecting",
"a",
"single",
"quasiparticle,",
"denoted",
"by",
"the",
"vector",
"l",
"2",
",",
"sume",
"it",
"arrives",
"at",
"the",
"tunneling",
"QPC",
"x",
"=",
"0",
"at",
"time",
"t",
".",
"to",
"into",
"the",
"upper",
"edge",
"at",
"the",
"location",
"x",
"Cinj",
"<",
"0",
"and",
"at",
"time",
"The",
"current",
"in",
"Eq.",
"(5)",
"is",
"then",
"given",
"by",
"integration",
"ti",
"tinj",
"i.",
"This",
"is",
"shown",
"schematically",
"in",
"Fig.",
"2(a).",
"In",
"view",
"over",
"multiple",
"terms",
"of",
"the",
"form",
"in",
"Eq.",
"(10).",
"We",
"define",
"of",
"the",
"commutation",
"relations",
"(2),",
"the",
"application",
"of",
"the",
"|t,",
"t",
"o",
"_",
"Â(t)vi",
"(t.",
"I",
"0,",
"id:",
"05)",
"t",
"o",
"+",
"t0(cid:",
"(t",
"I",
"o,",
",t",
"i2",
"Vi2"
]
} | |||
2301.00021v1.pdf_4 | null | ArxivOCR_part_1 | {
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"through",
"the",
"injection",
"QPC,",
"and",
"into",
"the",
"upper",
"edge.",
"II",
"The",
"injected",
"quasiparti-",
"forward",
"using",
"the",
"same",
"arguments.",
"Currents.—",
"The",
"effect",
"of",
"driving",
"the",
"system",
"out",
"of",
"cle,",
"by",
"virtue",
"of",
"its",
"chiral",
"motion",
"along",
"the",
"edge,",
"arrives",
"at",
"the",
"tunneling",
"QPC.",
"III",
"A",
"quasiparticle-quasihole",
"pair",
"is",
"cre-",
"equilibrium",
"is",
"completely",
"encapsulated",
"in",
"the",
"correlation",
"ated",
"at",
"the",
"tunneling",
"QPC.",
"(b)",
"The",
"two",
"processes",
"by",
"which",
"functions",
"obtained",
"above.",
"These",
"can",
"then",
"be",
"used",
"to",
"charge",
"carriers",
"may",
"ultimately",
"arrive",
"at",
"the",
"drain.",
"The",
"in-",
"derive",
"any",
"observable",
"of",
"interest,",
"such",
"as",
"charge",
"or",
"heat",
"jected",
"quasiparticle",
"arrives",
"at",
"the",
"tunneling",
"QPC",
"either",
"before",
"currents",
"in",
"any",
"of",
"the",
"system’s",
"drains,",
"or",
"their",
"respective",
"(upper",
"panel)",
"or",
"after",
"(lower",
"panel)",
"the",
"creation",
"quasiparticle-",
"auto-",
"and",
"cross-correlations.",
"quasihole",
"pair.",
"These",
"two",
"processes",
"interfere,",
"with",
"a",
"relative",
"For",
"concreteness,",
"we",
"present",
"the",
"explicit",
"results",
"of",
"such",
"phase",
"dictated",
"by",
"the",
"mutual",
"statistics",
"phase,",
"ei2θ12",
"a",
"calculation",
"for",
"the",
"charge",
"current",
"at",
"the",
"lower",
"drain,",
"denoted",
"as",
"in",
"Fig.",
"1.",
"We",
"show",
"that",
"a",
"simple",
"cohort",
"of",
"current",
"measurements",
"is",
"sufficient",
"to",
"obtain",
"the",
"mu-",
"wit",
"Eq.",
"(5)",
"can",
"now",
"be",
"as",
"tual",
"statistics",
"0",
"12",
"without",
"requiring",
"correlation",
"measure-",
"x",
"-",
"I",
"�",
"ments.",
"b",
"id:",
"2)",
"b",
",",
"t0(c",
"to",
"b",
"2",
"dt(",
"(11)",
"We",
"focus",
"on",
"the",
"regime",
"where",
"the",
"temperature",
"is",
"large",
"_o",
"b=",
"±",
"compared",
"to",
"the",
"injected",
"current",
"hl",
"in",
"/ek",
"B",
"T",
"/ek",
"the",
"The",
"expression",
"above",
"involves",
"two",
"interference",
"terms.",
"full",
"limit,",
"this",
"assumption",
"guarantees",
"linear",
"response",
"in",
"The",
"term",
"with",
"b",
"=",
"-",
"is",
"an",
"interference",
"between",
"cre-",
"the",
"voltage",
"and",
"in",
"the",
"injected",
"current,",
"which",
"in",
"this",
"−e∗",
"quasiholes",
"on",
"the",
"upper",
"edge",
"at",
"the",
"QPC",
"at",
"limit",
"is",
"I",
"inj",
"=",
"o",
"xy",
"V",
"For",
"the",
"dilute",
"limit,",
"the",
"exponen-",
"ation",
"of",
"times",
"t",
"and",
".",
"d:4",
"two",
"interfering",
"processes",
"ces",
"shown",
"tial",
"suppression",
"of",
"the",
"equilibrium",
"correlation",
"function",
"at",
"times",
"larger",
"than",
"(ci",
"guarantees",
"rant",
"the",
"exponent",
"in",
"schematically",
"in",
"Fig.",
"2(b).",
"As",
"shown",
"in",
"the",
"first",
"row",
"of",
"Eq.",
"(9),",
"these",
"two",
"processes",
"are",
"distinguished",
"by",
"a",
"Eq.",
"(12)",
"may",
"be",
"expanded",
"to",
"first",
"order",
"in",
"I",
"Linj",
".",
"Conse-",
"non-trivial",
"phase",
"of",
"ei2θ12",
"if",
"the",
"arrival",
"time",
"t",
"is",
"in",
"be-",
"quently,",
"tween",
"the",
"quasiholes’",
"creation",
"times,",
"t",
"<",
"t",
"o",
")",
"t.",
"Com-",
"<",
"Â",
"t(t)Â(t",
"'),",
"ˆA(t(cid:4",
"~",
"1",
"+",
"iw",
"Jf/d",
"(t",
"-",
"t",
"bined",
"with",
"the",
"equilibrium",
"correlation",
"function",
"Eq.",
"(6),",
"it",
"+",
"one",
"finds",
"that",
"this",
"interference",
"gives",
"a",
"term",
"proportional",
"(13)",
"<",
"Ât(t)Â(ť",
"))",
"2πδ).",
"Using",
"similar",
"arguments,",
"the",
"term",
"to",
"cos",
"(26",
"12",
"b",
"with",
"b",
"=",
"+",
"in",
"Eq.",
"(11),",
"gives",
"an",
"interference",
"term",
"pro-",
"where",
"the",
"frequencies",
"w",
")f/d",
"are",
"given",
"portional",
"to",
"cos",
"(26",
"12",
"+",
"2πδ).",
"The",
"total",
"contribution",
"w",
"f",
"=",
"e",
"1V",
"i",
"ei",
"I",
"Linj",
"*",
"e",
"-",
"e2i012)",
".",
"=",
"i",
"Iinj",
"(1",
"I",
"from",
"the",
"two",
"terms",
"in",
"Eq.",
"(11)",
"is",
"thus",
"proportional",
"to",
"(14)",
"w",
"sin",
"(20",
"12",
")",
"sin",
"(2πδ)",
"[41].",
"h",
"h",
"o",
"xy",
"e∗",
"This",
"interference",
"happens",
"entirely",
"in",
"the",
"time",
"domain,",
"and",
"along",
"only",
"one",
"edge.",
"It",
"is",
"however",
"crucial",
"that",
"this",
"The",
"zeroth",
"order",
"term",
"corresponds",
"to",
"the",
"equilibrium",
"edge",
"be",
"part",
"of",
"a",
"two-dimensional",
"bulk.",
"This",
"is",
"important",
"state",
"and",
"does",
"not",
"contribute",
"to",
"the",
"current.",
"The",
"ratio",
"both",
"because",
"the",
"second",
"edge",
"is",
"required",
"to",
"absorb",
"the",
"of",
"the",
"two",
"first",
"order",
"contributions",
"is",
"Eq.",
"(1).",
"leftover",
"quasiparticle",
"or",
"quasihole",
"resulting",
"from",
"the",
"pair",
"Explicit",
"calculation",
"of",
"the",
"resulting",
"current",
"in",
"Eq.",
"(5),",
"creation",
"at",
"the",
"QPC,",
"and",
"because",
"the",
"injected",
"quasiparti-",
"given",
"in",
"App.",
"A,",
"finds",
"that",
"cle",
"must",
"be",
"created",
"within",
"a",
"bulk",
"FQH",
"droplet.",
"Further-",
"more,",
"the",
"bulk",
"is",
"intimately",
"related",
"to",
"the",
"edge",
"through",
"I",
"/full/dilute",
"=",
"2πe∗",
"((&T",
")�(2�TT",
"T",
")481",
"-2",
"B",
"(28",
"1",
",",
"28",
"1",
"Re",
"[u",
"f/d",
"c",
"To",
"1-",
"bulk-edge",
"correspondence.",
"This",
"dictates",
"that",
"the",
"statisti-",
"where",
"B",
"(x,",
"y)",
"is",
"the",
"Euler",
"Beta",
"function.",
"It",
"is",
"thus",
"imme-",
"cal",
"phase",
"contributing",
"to",
"time-domain",
"interference",
"along",
"(15)",
"a",
"single",
"edge,",
"which",
"our",
"setup",
"measures,",
"is",
"the",
"same",
"as",
"diately",
"apparent",
"that",
"by",
"focusing",
"on",
"the",
"ratio",
"between",
"the",
"phase",
"obtained",
"from",
"spatial",
"exchange.",
"the",
"full",
"and",
"dilute",
"beams,",
"all",
"dependence",
"on",
"8",
"1",
",",
"T",
"and",
"&",
"It",
"is",
"easy",
"to",
"generalize",
"this",
"to",
"injection",
"of",
"multiple",
"drops",
"out.",
"Examining",
"the",
"ratio",
"I/I",
"inj",
",",
"and",
"noting",
"that",
"quasiparticles:",
"as",
"long",
"as",
"all",
"injected",
"quasiparticles",
"are",
"o",
"xy",
"h/et",
"=",
"νe2/2πe∗",
"22",
"we",
"thus",
"obtain",
"Eq.",
"(1).",
"i"
]
} | |||
2301.00021v1.pdf_5 | null | ArxivOCR_part_1 | {
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],
[
230,
1894,
305,
1929
],
[
305,
1894,
397,
1929
],
[
397,
1894,
447,
1929
],
[
447,
1894,
492,
1929
],
[
492,
1894,
600,
1929
],
[
600,
1894,
656,
1929
],
[
656,
1894,
723,
1929
],
[
723,
1894,
741,
1929
],
[
741,
1894,
763,
1928
],
[
763,
1894,
833,
1929
],
[
924,
1894,
1089,
1929
],
[
1089,
1894,
1191,
1929
],
[
1191,
1894,
1280,
1929
],
[
1280,
1894,
1367,
1929
],
[
1367,
1895,
1413,
1929
],
[
1413,
1894,
1472,
1929
],
[
1472,
1894,
1560,
1928
],
[
194,
1923,
279,
1958
],
[
279,
1923,
366,
1958
],
[
366,
1923,
450,
1958
],
[
450,
1923,
478,
1958
],
[
478,
1923,
567,
1958
],
[
567,
1923,
625,
1958
],
[
625,
1923,
670,
1958
],
[
670,
1923,
756,
1958
],
[
756,
1923,
833,
1958
],
[
885,
1922,
923,
1959
],
[
924,
1922,
960,
1959
],
[
960,
1923,
994,
1958
],
[
994,
1923,
1063,
1958
],
[
1064,
1923,
1114,
1958
],
[
1115,
1923,
1156,
1958
],
[
1156,
1923,
1190,
1958
],
[
1190,
1923,
1227,
1958
],
[
1227,
1923,
1313,
1958
],
[
1313,
1923,
1499,
1958
],
[
1499,
1923,
1564,
1958
],
[
193,
1952,
321,
1987
],
[
924,
1952,
974,
1987
],
[
974,
1952,
1085,
1987
],
[
1085,
1952,
1163,
1987
],
[
1163,
1952,
1192,
1987
],
[
1192,
1952,
1233,
1987
],
[
1233,
1952,
1339,
1987
],
[
1339,
1952,
1391,
1987
],
[
1391,
1952,
1463,
1987
],
[
1463,
1952,
1565,
1987
]
]
} | {
"word": [
"5",
"For",
"general",
"temperatures,",
"the",
"current",
"can",
"no",
"longer",
"be",
"of",
"a",
"single",
"quantum",
"Hall",
"droplet",
"with",
"two",
"QPCs,",
"which",
"treated",
"as",
"a",
"linear",
"response",
"to",
"the",
"drive",
"of",
"the",
"full",
"or",
"di-",
"effectively",
"create",
"a",
"time-domain",
"interferometer,",
"as",
"can",
"lute",
"beams.",
"We",
"hence",
"obtain",
"the",
"typical",
"power",
"laws",
"char-",
"be",
"identified",
"from",
"current",
"measurements.",
"We",
"thus",
"avoid",
"acterizing",
"tunneling",
"in",
"Luttinger",
"liquids",
"[2,",
"34,",
"42,",
"43].",
"both",
"current",
"correlation",
"(or",
"noise)",
"measurements,",
"and",
"Comparing",
"measurements",
"of",
"the",
"full",
"and",
"dilute",
"limits",
"at",
"the",
"need",
"for",
"a",
"real",
"space",
"interferometer,",
"making",
"the",
"iden-",
"the",
"low",
"temperature",
"limit",
"T",
"e",
"*V,",
"inj",
"can",
"still",
"give",
"a",
"tification",
"of",
"the",
"exchange",
"statistics",
"much",
"more",
"accessible",
"quantity",
"related",
"to",
"the",
"mutual",
"statistics",
"0",
"12",
"but",
"will",
"ex-",
"than",
"existing",
"experiments.",
"All",
"time-domain",
"interferom-",
"plicitly",
"depend",
"on",
"the",
"value",
"of",
"We",
"present",
"general",
"etry",
"is",
"between",
"pairs",
"of",
"an",
"injected",
"quasiparticle",
"and",
"a",
"expressions",
"for",
"the",
"current",
"in",
"this",
"case",
"in",
"App.",
"A.",
"tunneling",
"quasiparticle,",
"and",
"occurs",
"at",
"the",
"same",
"edge,",
"as",
"For",
"fermionic",
"6",
"12",
"=",
"π,",
"Eq.",
"(15)",
"gives",
"no",
"current",
"at",
"all",
"previously",
"proposed",
"in",
"Ref.",
"[25].",
"for",
"dilute",
"electron",
"beam.",
"However,",
"Landauer-Buttiker-",
"Both",
"the",
"exchange",
"statistics",
"0",
"11",
"of",
"the",
"tunneling",
"quasi-",
"Imry",
"scattering",
"theory",
"[44]",
"tells",
"us",
"the",
"current",
"is",
"given",
"particle,",
"and",
"0",
"22",
"of",
"the",
"injection",
"quasiparticle,",
"do",
"not",
"by",
"the",
"product",
"of",
"the",
"transparencies",
"of",
"the",
"two",
"QPCs",
"appear",
"in",
"our",
"derivation.",
"Rather,",
"it",
"is",
"the",
"two",
"particles’",
"mutual",
"statistics,",
"θ12",
"that",
"affect",
"the",
"modified",
"correlation",
"along",
"the",
"electron’s",
"path,",
"regardless",
"of",
"whether",
"they",
"are",
"functions,",
"and",
"hence,",
"the",
"physical",
"observables.",
"Likewise,",
"close",
"to",
"full",
"transmission",
"or",
"full",
"reflection.",
"This",
"requires",
"accounting",
"for",
"the",
"direct",
"tunneling",
"term",
"in",
"Eq.",
"(1),",
"which",
"the",
"scaling",
"dimension",
"and",
"electric",
"charge",
"which",
"directly",
"We",
"do",
"this",
"by",
"accounting",
"for",
"the",
"finite",
"width",
"of",
"the",
"particle,",
"and",
"e",
"(properties",
"of",
"the",
"injected",
"quasiparti-",
"soliton.",
"This",
"leads",
"to",
"the",
"required,",
"Landauer-Buttiker-",
"cles",
"may",
"implicitly",
"enter",
"through",
"the",
"injection",
"rate).",
"ex-",
"Imry",
"consistent",
"result",
"of",
"Idilute",
"=",
"422¢2I",
"inj",
"j.",
"The",
"phys-",
"quasiparticles",
"are",
"identical,",
"l",
"l1",
"=",
"l",
"2",
",",
"do",
"we",
"obtain",
"re-",
"ical",
"intuition",
"behind",
"the",
"requirement",
"of",
"a",
"finite",
"soliton",
"change",
"statistics",
"for",
"a",
"single",
"quasiparticle",
"type.",
"We",
"quasi-",
"effect",
"observables",
"are",
"only",
"those",
"of",
"the",
"tunneling",
"tunneling",
"Only",
"in",
"the",
"case",
"where",
"the",
"injected",
"and",
"o*",
"i",
"now",
"becomes",
"the",
"leading",
"contribution.",
"width",
"is",
"that",
"tunneling",
"without",
"time-domain",
"interferom-",
"mark",
"that",
"this",
"is",
"indeed",
"the",
"case",
"in",
"the",
"experiment",
"of",
"etry,",
"dubbed",
"the",
"direct",
"tunneling",
"process",
"in",
"[24,",
"25],",
"is",
"dominated",
"by",
"short",
"times.",
"Performing",
"these",
"calculations",
"Ref.",
"[22],",
"where",
"all",
"quasiparticles",
"are",
"Laughlin",
"e",
"=",
"e/3",
"anyons,",
"and",
"subsequent",
"recreations",
"for",
"the",
"=",
"1/3",
"and",
"explicitly",
"in",
"App.",
"B,",
"we",
"show",
"that",
"the",
"ratio",
"between",
"v",
"=",
"2/5",
"cases",
"[26,",
"46,",
"47].",
"Interestingly,",
"a",
"recent",
"ex-",
"the",
"first",
"term",
"in",
"Eq.",
"(1)",
"and",
"G",
"direct",
"is",
"shown",
"[24,",
"25]",
"quasiparticle",
"was",
"a",
"e/3",
"anyon",
"and",
"the",
"tunneling",
"quasi-",
"where",
"T",
"is",
"the",
"soliton",
"width.",
"It",
"has",
"been",
"ensure",
"G",
"direct",
"is",
"particle",
"was",
"an",
"electron,",
"observed",
"Andreev-like",
"reflection",
"that",
"T",
"Ts",
"max",
"o",
"eV,",
"k",
"B",
"T",
"};",
"as",
"such,",
"to",
"(T1",
"s",
")481",
"2,",
"periment",
"employing",
"a",
"similar",
"setup,",
"where",
"the",
"injected",
"ee",
"sub-dominant,",
"the",
"dilute",
"limit",
"must",
"be",
"measured",
"when",
"[48].",
"This",
"is",
"consistent",
"with",
"a",
"mutual",
"statistics",
"phase",
"of",
"k",
"B",
"T",
"(",
"id",
"28)",
"eV",
"<",
"2.",
"0",
"12",
"=",
"π,",
"for",
"which",
"Eq.",
"(1)",
"gives",
"no",
"time-domain",
"interfer-",
"Several",
"contemporary",
"experimental",
"setups",
"use",
"the",
"ometry",
"signal.",
"equivalent",
"of",
"non-interacting",
"fermionic",
"formulae",
"to",
"rea-",
"Acknowledgements.—",
"We",
"thank",
"Tomer",
"Alkalay,",
"Moty",
"sonable",
"success",
"[45],",
"corresponding",
"to",
"the",
"limiting",
"value",
"Heiblum,",
"Changki",
"Hong,",
"June-Young",
"Lee",
"and",
"H.-S.",
"of",
"28",
"1",
"=",
"1.",
"In",
"this",
"case,",
"the",
"second",
"term",
"of",
"Eq.",
"(1)",
"is",
"Sim",
"for",
"insightful",
"discussions",
"and",
"comments",
"on",
"the",
"a",
"numerical",
"coefficient",
"of",
"order",
"one,",
"which",
"may",
"depend",
"manuscript.",
"This",
"work",
"was",
"partially",
"supported",
"by",
"grants",
"solely",
"on",
"e",
"and",
"0",
"12",
"For",
"non-interacting",
"fermions,",
"from",
"the",
"ERC",
"under",
"the",
"European",
"Union’s",
"Horizon",
"2020",
"this",
"coefficient",
"is",
"easily",
"found",
"by",
"comparing",
"to",
"known",
"research",
"and",
"innovation",
"programme",
"(grant",
"agreements",
"Landauer-Buttiker-Imry",
"scattering",
"theory",
"[44],",
"but",
"it",
"is",
"LEGOTOP",
"No.",
"788715",
"and",
"HQMAT",
"No.",
"817799),",
"the",
"straightforward",
"to",
"generalize.",
"We",
"discuss",
"this",
"coefficient",
"DFG",
"(CRC/Transregio",
"183,",
"EI",
"519/7-1),",
"the",
"BSF",
"and",
"further",
"in",
"App.",
"B.",
"NSF",
"(2018643),",
"the",
"ISF",
"Quantum",
"Science",
"and",
"Technol-",
"Discussion.—",
"We",
"propose",
"a",
"simple",
"method",
"to",
"extract",
"ogy",
"(2074/19).",
"N.S.",
"was",
"supported",
"by",
"the",
"Clore",
"Scholars",
"anyonic",
"exchange",
"statistics.",
"Our",
"system",
"consists",
"only",
"Programme.",
" [1]",
"D.",
"Arovas,",
"J.",
"R.",
"Schrieffer,",
"and",
"F.",
"Wilczek,",
"Fractional",
"[4]",
"R.",
"B.",
"Laughlin,",
"Anomalous",
"Quantum",
"Hall",
"Effect:",
"An",
"In-",
"Statistics",
"and",
"the",
"Quantum",
"Hall",
"Effect,",
"Physical",
"Review",
"compressible",
"Quantum",
"Fluid",
"with",
"Fractionally",
"Charged",
"Letters",
"53,",
"722",
"(1984),",
"publisher:",
"American",
"Physical",
"So-",
"Excitations,",
"Physical",
"Review",
"Letters",
"50,",
"1395",
"(1983),",
"ciety.",
"publisher:",
"American",
"Physical",
"Society.",
"[2]",
"X.",
"Wen,",
"Quantum",
"Field",
"Theory",
" of",
"Many-Body",
"Sys-",
"[5]",
"R.",
"de",
"Picciotto,",
"M.",
"Reznikov,",
"M.",
"Heiblum,",
"V.",
"Uman-",
"tems:From",
"the",
"Origin",
"of",
"Sound",
"to",
"an",
"Origin",
"of",
"Light",
"sky,",
"G.",
"Bunin,",
"and",
"D.",
"Mahalu,",
"Direct",
"observation",
"of",
"a",
"and",
"Electrons:",
"From",
"the",
"Origin",
" of",
"Sound",
"to",
"an",
"Origin",
"fractional",
"charge,",
"Nature",
"389,",
"162",
"(1997).",
" of",
"Light",
"and",
"Electrons,",
"Oxford",
"Graduate",
"Texts",
"(OUP",
"[6]",
"L.",
"Saminadayar,",
"D.",
"C.",
"Glattli,",
"Y.",
"Jin,",
"and",
"B.",
"Etienne,",
"Oxford,",
"2004).",
"Observation",
"of",
"the",
"e/3",
"Fractionally",
"Charged",
"Laughlin",
"[3]",
"A.",
"Stern,",
"Anyons",
"and",
"the",
"quantum",
"Hall",
"effect",
"-",
"a",
"peda-",
"Quasiparticle,",
"Physical",
"Review",
"Letters",
"79,",
"2526",
"(1997).",
"gogical",
"review,",
"Annals",
"of",
"Physics",
"323,",
"204",
"(2008),",
"arXiv:",
"[7]",
"C.",
"L.",
"Kane",
"and",
"M.",
"P.",
"A.",
"Fisher,",
"Nonequilibrium",
"noise",
"0711.4697.",
"and",
"fractional",
"charge",
"in",
"the",
"quantum",
"Hall",
"effect,",
"Physical"
]
} | |||
2301.00021v1.pdf_6 | null | ArxivOCR_part_1 | {
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"statistics,",
"Nature",
"fects",
"in",
"quantum",
"Hall",
"edge",
"states",
"due",
"to",
"1/",
"f",
"noise",
"and",
"Physics",
"16,",
"931",
"(2020),",
"number:",
"9",
"Publisher:",
"Nature",
"dissipation,",
"New",
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"of",
"Physics",
"14,",
"093032",
"(2012).",
"[39]",
"In",
"practice",
"this",
"may",
"be",
"difficult",
"to",
"implement",
"experi-",
"Publishing",
"Group.",
"[22]",
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"easily",
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"done",
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"This",
"obstacle",
"may",
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"overcome",
"by",
"performing",
"the",
"dilute",
"beam",
"measurements",
"as",
"shown",
"in",
"Q.",
"Dong,",
"U.",
"Gennser,",
"Y.",
"Jin,",
"and",
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"Fractional",
"statistics",
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"(2020),",
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"while",
"performing",
"the",
"full",
"beam",
"measurements",
"publisher:",
"American",
"Association",
"for",
"the",
"Advancement",
"by",
"biasing",
"the",
"lower",
"edge",
"through",
"source",
"S",
"d",
"and",
"mea-",
"suring",
"the",
"tunneling",
"current",
"to",
"the",
"upper",
"edge",
"at",
"drain",
"of",
"Science",
"Section:",
"Report.",
"[23]",
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"Rosenow,",
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"and",
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"Current",
"D",
"Correlations",
"from",
"a",
"Mesoscopic",
"Anyon",
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"Physical",
"[40]",
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"Giamarchi,",
"Quantum",
"Physics",
"in",
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"Dimension,",
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"Letters",
"116,",
"10.1103/PhysRevLett.116.156802",
"ternational",
"Series",
"of",
"Monographs",
"on",
"Physics",
"(Oxford",
"(2016).",
"University",
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"New",
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"2003).",
"[24]",
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"Frac-",
"[41]",
"This",
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"appear",
"to",
"give",
"zero",
"for",
"8",
"=",
"1/2;",
"however,",
"this",
"tionalization",
"and",
"anyonic",
"statistics",
"in",
"the",
"integer",
"quan-",
"cancels",
"out",
"with",
"a",
"divergence",
"resulting",
"from",
"integration",
"tum",
"Hall",
"collider,",
"Physical",
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"[cond-",
"[42]",
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"properties",
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"the",
"fractional"
]
} | |||
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"quantum",
"Hall",
"states",
"and",
"weak-impurity",
"scattering",
"of",
"a",
"A.",
"Cavanna,",
"U.",
"Gennser,",
"Y.",
"Jin,",
"and",
"G.",
"F`eve,",
"Compar-",
"one-dimensional",
"charge-density",
"wave,",
"Physical",
"Review",
"B",
"ing",
"fractional",
"quantum",
"Hall",
"Laughlin",
"and",
"Jain",
"topolog-",
"44,",
"5708",
"(1991).",
"ical",
"orders",
"with",
"the",
"anyon",
"collider",
"(2022),",
"comment:",
"17",
"[43]",
"C.",
"d.",
"C.",
"Chamon,",
"D.",
"E.",
"Freed,",
"and",
"X.",
"G.",
"Wen,",
"Tunneling",
"pages,",
"5",
"figures,",
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"[cond-mat].",
"and",
"quantum",
"noise",
"in",
"one-dimensional",
"Luttinger",
"liquids,",
"[47]",
"P.",
"Glidic,",
"O.",
"Maillet,",
"A.",
"Aassime,",
"C.",
"Piquard,",
"A.",
"Ca-",
"Physical",
"Review",
"B",
"51,",
"2363",
"(1995).",
"vanna,",
"U.",
"Gennser,",
"Y.",
"Jin,",
"A.",
"Anthore,",
"and",
"F.",
"Pierre,",
"[44]",
"Y.",
"M.",
"Blanter",
"and",
"M.",
"Buttiker,",
"Shot",
"Noise",
"in",
"Mesoscopic",
"Cross-Correlation",
"Investigation",
"of",
"Anyon",
"Statistics",
"in",
"the",
"Conductors,",
"Physics",
"Reports",
"336,",
"1",
"(2000),",
"arXiv:",
"cond-",
"v",
"=",
"1/3",
"and",
"2/5",
"Fractional",
"Quantum",
"Hall",
"States",
"(2022),",
"mat/9910158.",
"arXiv:2210.01054",
"[cond-mat].",
"[45]",
"D.",
"E.",
"Feldman",
"and",
"M.",
"Heiblum,",
"Why",
"a",
"noninteract-",
"[48]",
"P.",
"Glidic,",
"O.",
"Maillet,",
"C.",
"Piquard,",
"A.",
"Aassime,",
"A.",
"Ca-",
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"model",
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"shot",
"noise",
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"charge",
"ex-",
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"Y.",
"Jin,",
"U.",
"Gennser,",
"A.",
"Anthore,",
"and",
"F.",
"Pierre,",
"periments,",
"Physical",
"Review",
"B",
"95,",
"115308",
"(2017),",
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"Andreev",
"scattering",
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"=",
"1/3",
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"Hall",
"regime",
"(2022),",
"arXiv:2206.08068",
"[46]",
"M.",
"Ruelle,",
"E.",
"Frigerio,",
"J.-M.",
"Berroir,",
"B.",
"Pla¸cais,",
"J.",
"Rech,",
"[cond-mat].",
"Appendix",
"A:",
"Finite",
"temperature",
"current",
"from",
"time-domain",
"interferometry",
"Here",
"derive",
"explicit",
"expressions",
"for",
"the",
"tunneling",
"current",
"I",
"at",
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"temperature",
"T",
"This",
"section",
"neglects",
"the",
"contribution",
"G",
"direct",
"(s",
"Eq",
"assed",
"in",
"App.",
"B.",
"We",
"begin",
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"the",
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"the",
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",",
"m",
"M",
"Eq.",
"(5).",
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"'Â*(t)Â(t",
"')",
"-",
"(Â(t",
"/)Â",
"(t)",
"+",
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"Â(t)Â",
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"=",
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"1ξ2",
"*",
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"25].",
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],
[
1320,
1677,
1342,
1714
],
[
1342,
1677,
1431,
1714
],
[
1431,
1677,
1564,
1714
],
[
143,
1712,
178,
1751
],
[
178,
1712,
225,
1751
],
[
226,
1712,
298,
1751
],
[
298,
1713,
368,
1751
],
[
368,
1713,
442,
1751
],
[
443,
1713,
580,
1750
],
[
580,
1712,
613,
1751
],
[
613,
1712,
665,
1751
],
[
666,
1712,
720,
1751
],
[
720,
1712,
782,
1751
],
[
786,
1713,
931,
1750
],
[
931,
1713,
979,
1750
],
[
979,
1713,
1049,
1750
],
[
1049,
1713,
1147,
1750
],
[
1147,
1712,
1182,
1750
],
[
1182,
1712,
1215,
1750
],
[
1215,
1713,
1270,
1750
],
[
1270,
1713,
1370,
1750
],
[
1371,
1713,
1426,
1750
],
[
1426,
1713,
1565,
1750
],
[
263,
1731,
296,
1787
],
[
303,
1762,
339,
1793
],
[
348,
1747,
399,
1786
],
[
399,
1748,
462,
1786
],
[
462,
1748,
602,
1786
],
[
602,
1748,
651,
1786
],
[
651,
1748,
717,
1786
],
[
717,
1748,
759,
1786
],
[
759,
1748,
807,
1786
],
[
807,
1748,
869,
1786
],
[
869,
1748,
1027,
1786
],
[
1027,
1749,
1100,
1786
],
[
143,
1748,
161,
1787
],
[
161,
1760,
189,
1789
],
[
231,
1764,
266,
1793
],
[
171,
1780,
271,
1819
],
[
273,
1781,
303,
1819
],
[
304,
1781,
334,
1819
],
[
335,
1780,
474,
1819
],
[
475,
1780,
510,
1819
],
[
513,
1780,
620,
1818
],
[
623,
1780,
671,
1818
],
[
674,
1780,
770,
1818
],
[
773,
1780,
826,
1819
],
[
828,
1780,
863,
1819
],
[
866,
1780,
913,
1819
],
[
916,
1780,
1029,
1818
],
[
1032,
1780,
1064,
1819
],
[
1066,
1781,
1091,
1819
],
[
1093,
1780,
1170,
1818
],
[
1173,
1780,
1336,
1818
],
[
1338,
1780,
1372,
1819
],
[
1375,
1780,
1439,
1818
],
[
1441,
1783,
1465,
1819
],
[
1484,
1780,
1564,
1818
],
[
143,
1812,
197,
1850
],
[
198,
1813,
313,
1850
],
[
314,
1813,
348,
1850
],
[
349,
1813,
400,
1850
],
[
401,
1813,
488,
1850
],
[
497,
1813,
532,
1850
],
[
533,
1813,
589,
1850
],
[
590,
1812,
649,
1850
],
[
651,
1812,
692,
1850
],
[
694,
1812,
763,
1851
],
[
765,
1813,
874,
1850
],
[
876,
1812,
924,
1850
],
[
926,
1813,
1024,
1850
],
[
1025,
1812,
1184,
1850
],
[
1184,
1812,
1343,
1851
],
[
1345,
1812,
1380,
1850
],
[
1380,
1813,
1505,
1850
],
[
1506,
1813,
1539,
1850
],
[
1540,
1812,
1565,
1850
],
[
144,
1844,
221,
1882
],
[
221,
1844,
385,
1882
],
[
385,
1844,
445,
1882
],
[
445,
1844,
484,
1882
],
[
484,
1844,
597,
1882
],
[
598,
1844,
653,
1881
],
[
653,
1844,
694,
1882
],
[
695,
1844,
753,
1882
],
[
753,
1844,
806,
1882
],
[
807,
1844,
879,
1882
],
[
879,
1844,
969,
1882
],
[
970,
1844,
1036,
1882
],
[
1037,
1844,
1070,
1882
],
[
1072,
1844,
1141,
1882
],
[
1142,
1844,
1177,
1882
],
[
1177,
1844,
1201,
1882
],
[
1201,
1844,
1280,
1882
],
[
1280,
1844,
1323,
1882
],
[
1324,
1844,
1372,
1881
],
[
1372,
1844,
1423,
1882
],
[
1425,
1844,
1482,
1882
],
[
1484,
1844,
1515,
1881
],
[
1515,
1844,
1564,
1881
],
[
143,
1876,
248,
1914
],
[
254,
1876,
318,
1913
],
[
318,
1876,
415,
1914
],
[
416,
1876,
521,
1914
],
[
521,
1876,
546,
1914
],
[
546,
1876,
677,
1913
],
[
678,
1876,
755,
1913
],
[
756,
1876,
798,
1914
],
[
799,
1876,
821,
1914
],
[
844,
1877,
861,
1915
],
[
861,
1887,
878,
1916
],
[
878,
1874,
915,
1914
],
[
915,
1876,
995,
1914
],
[
998,
1876,
1128,
1913
],
[
1129,
1876,
1193,
1914
],
[
1193,
1876,
1242,
1914
],
[
1243,
1876,
1414,
1914
],
[
1414,
1876,
1565,
1913
],
[
143,
1911,
202,
1949
],
[
205,
1911,
369,
1950
],
[
370,
1911,
484,
1949
],
[
485,
1911,
542,
1949
],
[
543,
1911,
667,
1950
],
[
668,
1911,
746,
1949
],
[
747,
1912,
829,
1949
],
[
837,
1911,
886,
1949
],
[
888,
1911,
919,
1950
],
[
919,
1923,
936,
1951
],
[
937,
1911,
969,
1949
],
[
970,
1911,
1000,
1950
],
[
1002,
1911,
1056,
1949
],
[
1057,
1911,
1146,
1949
],
[
1147,
1911,
1180,
1949
],
[
1193,
1910,
1224,
1950
],
[
1224,
1890,
1244,
1949
],
[
1233,
1924,
1265,
1953
],
[
1267,
1891,
1300,
1950
],
[
1300,
1911,
1321,
1950
],
[
1321,
1911,
1348,
1949
],
[
1348,
1892,
1374,
1949
],
[
1400,
1911,
1423,
1950
],
[
1431,
1912,
1494,
1949
],
[
1496,
1912,
1564,
1949
],
[
1374,
1912,
1388,
1950
],
[
143,
1943,
213,
1982
],
[
213,
1943,
326,
1982
],
[
326,
1943,
361,
1981
],
[
361,
1943,
395,
1982
],
[
395,
1943,
465,
1982
],
[
465,
1943,
548,
1981
],
[
548,
1943,
596,
1982
],
[
596,
1943,
629,
1982
],
[
629,
1954,
644,
1984
],
[
644,
1943,
675,
1982
],
[
675,
1943,
698,
1981
],
[
698,
1944,
757,
1982
],
[
757,
1943,
791,
1981
],
[
791,
1943,
881,
1982
],
[
881,
1943,
901,
1981
],
[
901,
1943,
948,
1982
],
[
949,
1944,
1019,
1982
],
[
1019,
1944,
1074,
1982
],
[
1074,
1944,
1177,
1982
],
[
1177,
1943,
1224,
1982
],
[
1225,
1943,
1330,
1982
],
[
1330,
1944,
1430,
1981
],
[
1430,
1944,
1495,
1981
],
[
1495,
1943,
1564,
1981
],
[
143,
1976,
290,
2013
],
[
290,
1976,
394,
2014
],
[
394,
1976,
506,
2013
],
[
506,
1976,
558,
2013
],
[
558,
1976,
681,
2013
],
[
681,
1976,
838,
2014
],
[
838,
1976,
952,
2014
],
[
952,
1976,
984,
2014
],
[
984,
1976,
1031,
2013
],
[
1031,
1976,
1111,
2013
],
[
1111,
1976,
1233,
2013
],
[
1233,
1975,
1337,
2013
]
]
} | {
"word": [
"2",
"Standard",
"manipulations",
"of",
"these",
"integ",
"is",
"in",
"terms",
"of",
"the",
"Euler",
"Beta",
"function",
"and",
"the",
"incomplete",
"=B",
"Beta",
"function,",
"(x;",
"a,",
"b)",
"x",
"ta",
"-1(",
"ite",
"t)",
"i",
"B",
"(1;",
"a,",
"b).",
"We",
"thus",
"obtain",
"the",
"general",
"results",
"I",
"full",
"=",
"2e∗",
"1ξ2(2πT",
")481",
"-174ö1",
"sinh",
"e",
"1V",
"B",
"2δ1",
"e",
"*",
"-",
"i",
"i",
"V",
"*",
"-",
"1",
"e",
"1",
")",
"+",
",",
"2δ1",
"(A5a)",
"Tc",
"2πT",
"2T",
"2πT",
"Iinj",
"1",
"_",
"cos(2012)+2",
"sin(2θ12)",
"+",
"28",
"1",
"r",
"I",
"dilute",
"=",
"-",
"2π",
"e2",
"2πT",
"(A5b)",
"id:1",
"1",
"-",
"-cos(2012)+¿sin(2012)",
"+",
"1",
"-",
"28",
"1",
"cos",
"(2πδ1)",
"1",
")",
"Γ",
"2πT",
"e2",
"I",
"qp",
"=",
"4e∗",
"1ξ2(2πT",
")481",
"-1-481",
"sin",
"(20",
"12",
")",
"sin",
"(2πδ)",
"B",
"e",
"2�T(t",
"_",
"-to);",
"1",
"+",
"28",
"1",
"1",
"-",
"*",
"-",
"1",
"48",
"1",
".",
"(A5c)",
"Here,",
"Γ(a)",
"is",
"the",
"Euler",
"Gamma",
"function,",
"satisfying",
"B(a,",
"b)",
"=",
"Γ(a)Γ(b)",
"and",
"Im[.",
"..]",
"denotes",
"the",
"imaginary",
"part.",
"Γ(a+b)",
"The",
"high",
"temperature",
"and",
"zero",
"temperature",
"limits",
"of",
"the",
"full",
"beam",
"and",
"dilute",
"beam",
"are",
"then",
"immediately",
"repro-",
"ducible.",
"For",
"e∗V,",
"hl",
"inj",
"/e",
"(",
"k",
"B",
"T,",
"one",
"expands",
"xp",
"leading",
"eadin",
"in",
"e",
"*V/T",
"e∗V",
"/T",
"inj",
"/e",
"T,",
"one",
"obtains",
"P2",
"ar",
"2",
"*",
"1V",
",",
"~",
"2πe∗",
"[(&T",
"c)�(2�T",
"T",
")481",
"-2B",
"(2δ1,",
".*",
"c",
"1-",
"1",
"(A6a)",
"I",
"full",
"28",
"1",
")",
"h",
"~",
"2πe∗",
"(($T",
"c",
")�(27TT",
"T",
")481",
"-2",
"B",
"(2δ1,",
"28",
"1",
")",
"Iinj",
"sin",
"(26",
"12",
")",
".",
"*",
"T",
"1-",
"Ii",
"e1",
"(A6b)",
"I",
"dilute",
"C2",
"We",
"thus",
"see",
"that",
"the",
"mutual",
"statistics",
"are",
"immediately",
"extractable",
"from",
"the",
"dilute",
"case.",
"While",
"this",
"expression",
"does",
"depend",
"on",
"the",
"non-universal",
"&",
"and",
"8,",
"as",
"well",
"as",
"the",
"temperature,",
"these",
"are",
"all",
"encoded",
"in",
"a",
"prefactor",
"which",
"appears",
"in",
"the",
"full",
"case",
"as",
"well.",
"We",
"can",
"hence",
"lose",
"this",
"unwanted",
"prefactor",
"by",
"examining",
"the",
"ratio",
"between",
"the",
"two",
"cases.",
"as",
"V,",
"I",
"inj",
"/e",
"2,",
"we",
"use",
"the",
"identities",
"e",
"22",
"K",
"For",
"T",
"lim",
"Γ",
"(x",
"+",
"a)",
"=",
"Γ",
"(x)",
"xa,",
"lim",
"sinh(πx)B",
"b",
"(a",
"+",
"ix,",
"a",
"-",
"ix)",
"=",
"Γ(2a)",
"x2a",
"-1,",
"z-",
"7",
"x",
"to",
"obtain",
"-",
"~",
"2πe∗",
"Pif2T21",
"(e$V",
"i8",
"i",
",",
"L",
"l",
")",
"I",
"full",
"(A7a)",
"T(48",
"1",
"h",
"481",
"-",
"481",
"_",
"2πe∗",
"($2481",
"*",
"i",
"1",
"~",
"-",
"-",
"I",
"inj",
"Tc",
".",
"cos",
"(26",
"12",
")",
"+",
"sin",
"(20",
"12",
")",
"(A7b)",
"I",
"dilute",
"Im",
"1",
"cos",
"(27T8",
"1)T",
"(4δ1)",
"e∗",
"-",
"By",
"tuning",
"2δ1",
"1,",
"we",
"again",
"obtain",
"an",
"expression",
"from",
"which",
"the",
"mutual",
"statistics",
"are",
"easily",
"extractable,",
"with",
"an",
"identical",
"non-universal",
"prefactor",
"appearing",
"in",
"both",
"the",
"full",
"and",
"dilute",
"cases.",
"However,",
"once",
"the",
"scaling",
"dimension",
"is",
"tuned",
"to",
"this",
"critical",
"value,",
"the",
"contribution",
"from",
"time-domain",
"interferometry",
"no",
"longer",
"dominates",
"the",
"direct",
"tunneling",
"process,",
"as",
"can",
"be",
"seen",
"from",
"the",
"calculation",
"of",
"G",
"direct",
"in",
"App.",
"B.",
"We",
"note",
"that",
"for",
"temperatures",
"larger",
"than",
"the",
"source",
"voltage,",
"one",
"has",
"to",
"account",
"for",
"injection",
"of",
"both",
"quasiparticles",
"and",
"quasiholes",
"through",
"the",
"injection",
"QPC.",
"This",
"can",
"be",
"done",
"by",
"modifying",
"the",
"Poissonian",
"correlation",
"function",
"in",
"(",
"Eq.",
"(12)",
"according",
"to",
"(Ât(t)Â(0)",
"dilute",
"=",
"Ât(t)Â(0))",
"(A8)",
"-",
"inj/e∗",
"(1",
"-",
"-e2i012)",
"-",
"tial/es",
"2",
"(1",
"-",
"e",
"-2i012),",
"e",
"e",
"-",
"-",
"where",
"inj",
"is",
"the",
"injection",
"rate",
"of",
"quasiparticles,",
"and",
"inj",
"is",
"the",
"injection",
"rate",
"of",
"quasiholes.",
"This",
"is",
"a",
"similar",
"expression",
"to",
"the",
"three",
"QPC",
"setup",
"considered",
"in",
"[23]",
"and",
"[25].",
"Performing",
"the",
"same",
"algebra",
"as",
"in",
"this",
"section,",
"and",
"identifying",
"ee",
"inj",
"one",
"then",
"reproduces",
"Eq.",
"(A6)",
"for",
"the",
"high",
"temperature",
"limit.",
"I",
"inj",
"inj",
"Finally,",
"it",
"is",
"instructive",
"to",
"consider",
"the",
"current",
"due",
"to",
"the",
"injection",
"of",
"a",
"single",
"quasiparticle",
"at",
"time",
"to",
"which",
"was",
"obtained",
"in",
"Eq.",
"(A5c).",
"In",
"this",
"case",
"we",
"must",
"examine",
"the",
"explicit",
"temperature",
"dependence,",
"as",
"tunneling",
"of",
"a",
"single",
"quasiparticle",
"may",
"be",
"relevant,",
"and",
"we",
"lack",
"any",
"other",
"energy",
"scale",
"to",
"serve",
"as",
"a",
"cutoff",
"for",
"the",
"RG",
"flow",
"of",
"the",
"process.",
"This",
"current",
"exhibits",
"a",
"power-law",
"decay",
"for",
"t",
"t",
"0",
"(",
"id:2",
"consistent",
"iste",
"the",
"orthogonality",
"catastrophe",
"that",
"characterizes",
"injection",
"into",
"Luttinger",
"liquid",
"edges.",
"For",
"28",
"1",
"=",
"1,",
"this",
"results",
"in",
"id:",
"o",
"qp",
"x",
"8",
"(c",
"ia",
").",
"This",
"gives",
"t",
"some",
"intuition",
"as",
"to",
"what",
"makes",
"the",
"28",
"1",
"=",
"1",
"case",
"so",
"unique",
"-",
"the",
"QPC",
"just",
"scatters",
"the",
"incident",
"particle",
"with",
"some",
"probability,",
"without",
"inducing",
"any",
"long-time",
"correlations,",
"resulting",
"in",
"the",
"direct",
"tunneling",
"process."
]
} | |||
2301.00021v1.pdf_9 | null | ArxivOCR_part_1 | {
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1022,
1894
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[
1098,
1856,
1119,
1892
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[
1009,
1866,
1081,
1968
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700,
1873,
727,
1912
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727,
1874,
753,
1912
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753,
1874,
795,
1912
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[
795,
1884,
806,
1915
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[
882,
1873,
903,
1911
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[
930,
1874,
954,
1912
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943,
1880,
983,
1917
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1085,
1894,
1126,
1930
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836,
1906,
859,
1936
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1499,
1914,
1564,
1952
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606,
1978,
629,
2015
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1978,
682,
2015
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682,
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736,
2015
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736,
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764,
2015
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805,
2016
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817,
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840,
2020
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840,
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851,
2020
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851,
1977,
867,
2021
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878,
1959,
931,
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938,
1949,
1029,
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1033,
1949,
1082,
1985
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1086,
1950,
1160,
1984
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1160,
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1177,
1983
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1177,
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1213,
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1217,
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1255,
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1278,
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1311,
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950,
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1073,
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1097,
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877,
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717,
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480,
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510,
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505,
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546,
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546,
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575,
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606,
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} | {
"word": [
"3",
"Appendix",
"B:",
"Finite",
"soliton",
"width:",
"restoring",
"Landauer-Buttiker-Imry",
"for",
"electrons",
"and",
"subleading",
"corrections",
"The",
"results",
"of",
"App.",
"A",
"are",
"seemingly",
"inconsistent",
"with",
"the",
"known",
"non-interacting",
"electron",
"limits.",
"Indeed,",
"inserting",
"*",
"=",
"e",
"=",
"e,",
"28",
"1",
"=",
"28",
"2",
"=",
"1",
"and",
"6",
"12",
"=",
"7",
"into",
"these",
"results",
"would",
"indicate",
"that",
"the",
"dilute",
"electron",
"beam",
"gives",
"no",
"e",
"e1",
"22",
"*",
"current",
"at",
"all.",
"This",
"is",
"in",
"direct",
"contrast",
"with",
"the",
"intuition",
"of",
"Landauer-Buttiker-Imry",
"scattering",
"theory,",
"which",
"would",
"indicate",
"that",
"the",
"current",
"should",
"be",
"given",
"by",
"the",
"product",
"of",
"the",
"transparencies",
"of",
"the",
"two",
"QPCs",
"along",
"the",
"electron’s",
"path,",
"regardless",
"of",
"whether",
"they",
"are",
"close",
"to",
"full",
"transmission",
"or",
"full",
"reflection.",
"$",
"is",
"compact",
"under",
"$",
"1",
"an",
"integer.",
"The",
"culprit",
"of",
"this",
"result",
"is",
"a",
"peculiarity",
"of",
"soliton",
"physics.",
"The",
"boson",
"field",
"unperturbed",
"if",
"K",
"−1l2",
"is",
"such,",
"a",
"soliton",
"of",
"height",
"2πK",
"-1l",
"2",
"would",
"appear",
"to",
"leave",
"the",
"boson",
"field",
"completely",
"[_",
"d",
"5",
"2�T.",
"As",
"This",
"corresponds",
"precisely",
"to",
"electron",
"injection",
"operators",
"[2].",
"As",
"such,",
"our",
"soliton",
"description",
"is",
"ill-equipped",
"to",
"treat",
"electrons",
"without",
"modifications.",
"We",
"solve",
"this",
"issue",
"by",
"introducing",
"a",
"finite",
"width",
"to",
"the",
"soliton,",
"T",
".",
"To",
"fully",
"recreate",
"the",
"known",
"non-interacting",
"s",
"result,",
"it",
"is",
"crucial",
"to",
"maintain",
"an",
"order",
"of",
"limits",
"such",
"that",
"the",
"soliton",
"width",
"is",
"larger",
"than",
"the",
"short-time",
"cutoff,",
"T",
"c",
"We",
"note",
"that",
"we",
"still",
"take",
"care",
"to",
"ensure",
"that",
"τs",
"<",
"1/T,",
"(I",
"inj",
"/e",
"2",
")",
"-1,",
"i.e.",
"the",
"solitons",
"are",
"still",
"narrow",
"compared",
"to",
"ito",
"the",
"larger",
"time",
"scales",
"in",
"the",
"problem.",
"Previous",
"works",
"[24,",
"25],",
"performing",
"a",
"full",
"Keldysh",
"calculation,",
"have",
"shown",
"the",
"soliton",
"width",
"(refered",
"to",
"in",
"the",
"cited",
"papers",
"as",
"the",
"temporal",
"width)",
"is",
"given",
"by",
"the",
"voltage,",
"h/e∗V",
"V",
"if",
"eV",
">",
"k",
"B",
"T,",
"s",
"and",
"by",
"the",
"inverse",
"temperature",
"h/k",
"B",
"T",
"if",
"/k",
"k",
"B",
"T",
"(c",
"d:46)",
"kBT",
"dilute",
"limit",
"must",
"ut",
"measured",
"st",
"the",
"easure",
"I",
"inj",
"/e",
"(",
"k",
"B",
"T",
"8",
"eV",
"upper",
"location",
"Formally,",
"this",
"means",
"that",
"injecting",
"a",
"quasiparticle",
"into",
"the",
"at",
"the",
"x",
"and",
"time",
"t0",
"transforms",
"go",
"the",
"boson",
"field",
"according",
"to",
"φ(u)(x,",
"t",
"o",
")",
"�",
"d:55)→",
"t",
"o",
")",
"-",
"2�K",
"-1l",
"2",
"TT",
"tan−1",
"1",
"x",
"-",
".",
",_",
"1",
"-",
"-",
"1",
"8)",
"T",
"s",
"2",
")]y",
"/",
"Accordingly,",
"the",
"correlation",
"functions",
"of",
"Eq.",
"(9)",
"are",
"now",
"replaced",
"with",
"(",
"Â",
"+(t)Â(t",
"')",
">",
"qp",
"=",
"<",
"Â",
"t(t)Â(t",
")",
"0",
"exp",
"(",
"2i",
"θ12",
"tan−1",
"-",
"t",
"0",
"-",
"tan−1",
"-",
"+,",
"-",
"]",
",",
"t",
"+'",
"0",
"IT",
"to",
"T",
"T",
"s_",
"_",
"(B1)",
"<",
"Â(t)Â*(t",
")",
")",
"qp",
"=",
"(",
"Â(t)Ât(ť",
"))",
"exp",
"_",
"2i",
"θ12",
"tan−1",
"-",
",,",
"/",
"o",
"0",
"7T",
"t",
"-",
"t",
"t",
"to",
"tan−1",
"-",
"t",
"to",
".",
"T",
"Ts",
"T",
"Ts",
"�",
"One",
"indeed",
"sees",
"that",
"at",
"the",
"limit",
"T",
"c",
"0,",
"one",
"reproduces",
"the",
"immediate",
"soliton",
"results",
"from",
"the",
"main",
"text.",
"To",
"find",
"the",
"correlation",
"function",
"in",
"the",
"presence",
"of",
"a",
"dilute,",
"Poissonian",
"beam",
"of",
"injected",
"quasiparticles,",
"we",
"now",
"must",
"sum",
"over",
"the",
"number",
"of",
"injected",
"quasiparticles,",
"in",
"a",
"manner",
"similar",
"to",
"Eq.",
"(12).",
"However,",
"this",
"is",
"now",
"trickier,",
"for",
"two",
"reasons.",
"First,",
"the",
"accumulated",
"phase",
"explicitly",
"depends",
"on",
"the",
"time",
"of",
"the",
"injected",
"quasiparticle.",
"Second,",
"injected",
"quasiparticles",
"outside",
"of",
"the",
"window",
"[0,",
"t]",
"can",
"still",
"affect",
"the",
"correlation",
"function,",
"due",
"to",
"the",
"long",
"tails",
"of",
"the",
"finite-width",
"solitons.",
"hods",
"that",
"lead",
"to",
"Eq.",
"2, h",
"correlation",
"function",
"now",
"changes",
"to",
"define",
"(",
"_",
"(t+2cTc)\"i",
"e",
"e2",
"Iinj",
"n",
")",
"e2",
"c",
"dt",
"o",
"P",
"(Particle",
"injected",
"at",
"t",
"o",
")",
"e2i�12",
"[ta",
"m-1('=:)_tan~1(0_:o",
"n",
"t+cτc",
")]",
"=",
"I",
"Ât(t)Â(0)",
"/",
"ctc",
"n!",
"n",
"(B2)",
"is",
"some",
"unitless",
"cutoff,",
"chosen",
"such",
"that",
"injected",
"quasiparticles",
"affect",
"the",
"correlation",
"function",
"only",
"if",
"they",
"are",
"Here",
"injected",
"in",
"the",
"window",
"[",
"-CT.",
",",
"t",
"+",
"CT",
"c",
"],",
"which",
"we",
"will",
"eventually",
"take",
"to",
"be",
"infinite.",
"The",
"probability",
"of",
"injection",
"at",
"a",
"Tc",
"particular",
"time",
"t",
"is",
"given",
"by",
"to",
"P",
"(Particle",
"injected",
"at",
"t",
"o",
")",
"=",
"I",
"/e",
"e",
"-Iinjto/e�",
"*",
"-",
".",
"(B3)",
"−cτc",
"Performing",
"this",
"sum,",
"and",
"re-defining",
"this",
"integration",
"with",
"we",
"fir",
"the",
"new",
"correlation",
"function",
"unitles",
"riable",
"dthat",
"m",
"is",
"given",
"in",
"integral",
"form",
"by",
"(",
"Ât(t)Â(0)",
")",
"fw",
"=",
"exp",
"P",
"Ât(t)Â(0))",
"0",
")�",
"-",
"-",
"I",
"inj",
"I",
"t",
"eTs",
"(t",
"+",
"2ct",
"c",
"1",
"I",
"[012",
"2τs",
"C2",
"(B4)",
"2",
"sinh",
"(a(b",
"+",
"c))",
"/_",
"b",
"-",
"dxe",
"-axe2i�%",
"[tan",
"-1(x+b)",
"-",
"tan",
"-1(x",
"-",
"-b)].",
"-",
"-",
"b+c",
"a",
"I",
"012",
"(a,",
"b)",
"=",
"c"
]
} | |||
2301.00021v1.pdf_10 | null | ArxivOCR_part_1 | {
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678,
1463,
709,
1503
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539,
1445,
570,
1482
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475,
1500,
513,
1560
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170,
1522,
234,
1560
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235,
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322,
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323,
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371,
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372,
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439,
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441,
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458,
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458,
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472,
1562
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537,
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1007,
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1114,
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1117,
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1347,
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1397,
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1453,
1560
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1454,
1521,
1514,
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1516,
1521,
1564,
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143,
1554,
220,
1591
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221,
1554,
261,
1591
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261,
1553,
350,
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350,
1553,
393,
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393,
1554,
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1592
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586,
1553,
702,
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702,
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731,
1592
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731,
1554,
819,
1591
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819,
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933,
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933,
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1040,
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1197,
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1198,
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1262,
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1329,
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1329,
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1426,
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1426,
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1454,
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143,
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191,
1624
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191,
1586,
295,
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295,
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398,
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398,
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453,
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453,
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485,
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485,
1585,
532,
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532,
1586,
698,
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698,
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730,
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730,
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778,
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778,
1586,
900,
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900,
1585,
970,
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970,
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1060,
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1060,
1586,
1088,
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1088,
1586,
1160,
1623
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1160,
1585,
1198,
1624
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1198,
1581,
1280,
1623
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1235,
1599,
1259,
1630
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170,
1618,
220,
1655
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220,
1617,
314,
1656
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314,
1617,
395,
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395,
1618,
425,
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425,
1617,
447,
1655
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436,
1628,
468,
1658
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468,
1617,
522,
1655
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522,
1618,
538,
1655
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[
538,
1629,
553,
1657
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[
553,
1618,
606,
1655
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606,
1617,
705,
1655
],
[
705,
1617,
732,
1656
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732,
1617,
800,
1655
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800,
1617,
900,
1655
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900,
1617,
932,
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999,
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999,
1618,
1152,
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1154,
1618,
1274,
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1274,
1618,
1300,
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1300,
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1326,
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1326,
1617,
1429,
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1429,
1617,
1461,
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1461,
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1564,
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143,
1650,
194,
1687
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194,
1650,
258,
1688
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258,
1650,
291,
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828,
1653,
859,
1764
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1143,
1655,
1174,
1765
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530,
1718,
546,
1775
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546,
1732,
665,
1775
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665,
1717,
681,
1774
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670,
1749,
702,
1778
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705,
1757,
736,
1795
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784,
1738,
809,
1775
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797,
1749,
832,
1778
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851,
1757,
893,
1795
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893,
1757,
933,
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1766,
956,
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956,
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972,
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1756,
1014,
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1054,
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1098,
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1112,
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1122,
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1149,
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1182,
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1564,
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552,
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670,
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1804,
917,
1861
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927,
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948,
1866
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782,
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1841,
923,
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598,
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623,
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680,
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706,
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"expression",
"o",
")",
"ne",
"correlation",
"function",
"into",
"the",
"By",
"plugging",
"this",
"the",
"current",
"in",
"Eq.",
"(5),",
"one",
"now",
"finds",
"'",
"I",
"0",
"m",
"t",
"inj",
"sin",
"(˜t",
"T",
",",
"2τs",
"s",
"L012",
"I",
"dilute",
"=",
"2ie∗",
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"1",
"-",
"d˜t",
")",
"o",
"t",
"Iin",
"Re",
"exp",
"(˜t",
"+",
"1",
"0",
"912",
"2τs",
"e2",
"e",
"(cid",
"48",
"-",
"τc",
"-",
"it",
"))",
"Tc",
"x",
"πT",
"T",
"i",
"sinh",
"(ci",
"(",
"-",
"it",
"c",
"(c",
"(B5)",
"i",
"sinh",
"t",
"Gireumvented",
"�",
"Careful",
"re-application",
"of",
"the",
"limit",
"T",
"c",
"0",
"indeed",
"replicates",
"our",
"previous",
"result",
"in",
"Eq.",
"(12).",
"For",
"general",
"0",
"12",
",",
"the",
"integral",
"I",
"012",
"(a,",
"b)",
"is",
"difficult",
"to",
"solve",
"analytically.",
"In",
"the",
"main",
"text,",
"this",
"is",
"by",
"taking",
"the",
"limit",
"7",
"c",
"0,",
"allowing",
"use",
"of",
"Eq.",
"(A4),",
"in",
"conjunction",
"with",
"replacing",
"(˜t+2cτc)",
")Iinj",
"I",
"012",
"IinjT",
",",
"2τs",
"�",
"ile",
"itw",
"d",
"e2",
"Ts",
"�",
"e∗",
"However,",
"as",
"noted",
"previously,",
"fermionic",
"exchange",
"statistics",
"corresponding",
"to",
"values",
"of",
"6",
"12",
"that",
"are",
"integer",
"multiples",
"of",
"�",
"lead",
"to",
"w",
"d",
"=",
"0,",
"and",
"hence",
"give",
"a",
"vanishing",
"current.",
"As",
"such,",
"Eq.",
"(B5)",
"must",
"be",
"calculated",
"in",
"full",
"while",
"retaining",
"a",
"finite",
"T",
".",
"Tc",
"To",
"simplify",
"these",
"expressions,",
"we",
"assume",
"that",
"Iinj",
"is",
"significantly",
"larger",
"than",
"any",
"other",
"time",
"scale",
"in",
"the",
"system.",
"e∗",
"This",
"makes",
"sense",
"from",
"a",
"physical",
"perspective",
"as",
"well,",
"as",
"it",
"corresponds",
"to",
"the",
"assumption",
"that",
"injection",
"is",
"sufficiently",
"rare",
"such",
"that",
"solitons",
"do",
"not",
"overlap.",
"In",
"this",
"case,",
"one",
"can",
"assume",
"the",
"probability",
"of",
"injection",
"which",
"appears",
"in",
"Eqs.",
"(B2),(B3)",
"is",
"approximately",
"uniform,",
"P",
"(Particle",
"injected",
"at",
"t",
"o",
")",
"1/(t",
"+",
"2ct,",
").",
"One",
"can",
"now",
"safely",
"take",
"the",
"c",
"limit",
"c",
",",
"%",
"without",
"artificial",
"divergences,",
"givir",
"sult,",
"[tan~1('_:o)_tan~1(0_:o",
")]",
"_",
"(Ât(t)Â(0))",
"fw",
"=",
"exp",
"I",
"inj",
"e2¿�12",
"-x",
"dt",
"o",
".",
"(B6)",
"1",
"I",
"A",
"Î+(t)Â(0))",
"0",
"e∗",
"Since",
"we",
"undertook",
"this",
"endeavor",
"with",
"the",
"explicit",
"goal",
"of",
"finding",
"the",
"correct",
"result",
"for",
"non-interacting",
"electrons,",
"we",
"wish",
"to",
"find",
"this",
"integral",
"for",
"28",
"1",
"=",
"1,",
"o",
"1",
"=",
"π,",
"and",
"e",
"=",
"e∗",
"=",
"e.",
"This",
"value",
"of",
"0",
"12",
"allows",
"one",
"to",
"significantly",
"1",
"\"",
"using",
"lation",
"trignometric",
"ident",
"function",
"in",
"Eq.",
"simplify",
"Eq.",
"(B6)",
"obtain",
")",
"(",
"sin",
"id:1",
"2π˜t(2τs)2",
")",
"e�",
"˜t2+(2τs)2",
"481",
"-",
"[012=T",
"=",
"2ie∗",
"1ξ2",
"o",
"exp",
"Iinj",
"2π˜t2(2τs)",
"i",
"sinh",
"(ci",
":0)",
"-",
"it",
"c",
"))",
"*",
"e2",
"˜t2+(2τs)2",
"1",
"πT",
"T.",
"-",
"d˜t",
"(B7)",
"it",
"c",
"))",
"i",
"sinh",
"(ci",
"As",
"can",
"be",
"seen",
"in",
"Eq.",
"(A4),",
"the",
"expression",
"in",
"the",
"curled",
"brackets",
"is",
"approximately",
"zero",
"for",
"t",
">",
"T",
"c",
".",
"We",
"can",
"thus",
"s",
"ox",
"approximate",
"the",
"total",
"integral",
"as",
"the",
"contribution",
"from",
"short",
"t",
"eading",
"orde",
"will",
"be",
"T",
"1/",
"d",
"limcs,",
"1/rT.",
"b",
"given",
"by",
"2",
"[012=t,281=1",
"~",
"2ie∗",
"1ξ2τ",
"d˜t",
"inj",
"2π˜t(2τs)2",
")�",
"*",
"I",
"1",
"-",
"{",
"1",
"1",
"-",
"it",
"+",
"T",
"c",
"22",
"˜t2",
"+",
"(27",
"s",
")�",
"i˜t",
"+",
"T",
"c",
"(B8)",
"(27",
"s",
")�",
"42$2-2I",
"inj",
".",
"(2τs",
"+",
"T",
")2",
"=",
"<",
"Now",
"taking",
"the",
"limit",
"T",
"c",
"T",
"8)",
"compare",
"to",
"are",
"electron",
"ectr",
"in,",
"say,",
"Eq.",
"(15)",
"or",
"15)",
"(A5).",
"We",
"5).",
"that",
"the",
"result",
"we",
"expect",
"for",
"non-interacting",
"electrons",
"is",
"indeed",
"4π2ξ2τ",
"inj",
".",
"This",
"is",
"consistent",
"with",
"-",
"the",
"current",
"is",
"linear",
"in",
"the",
"injected",
"current,",
"and",
"in",
"the",
"transparency",
"of",
"the",
"tunneling",
"QPC",
"(which",
"is",
"given",
"by",
"&2-2).",
"Tc",
"For",
"general",
"values",
"of",
"0",
"12",
"and",
"8",
"1",
"this",
"integral",
"is",
"more",
"difficult",
"to",
"solve",
"analytically.",
"However,",
"it",
"is",
"possible",
"to",
"re-write",
"Eq.",
"(B6)",
"as",
"(",
",",
"<",
"Ât(t)Â(0)",
")",
"fw",
"=",
"Ii",
"inj",
"sin",
"(20",
"12",
")",
"t",
"+",
"fθ12",
"(t,",
"T",
"c",
"))",
",",
"(B9)",
"Ât(t)Â(0)",
"C2",
"s",
"T",
"Ts",
"t",
"{",
">",
"fo",
"12",
"(t,",
"T",
"))",
"x",
"t",
"7",
"s",
",",
"0",
"12",
"s,",
"�",
"(B10)",
"(T:",
")2/t",
"t",
">",
"T",
"s",
",",
"6",
"12",
"s",
"T.",
"Plugging",
"this",
"into",
"the",
"general",
"expression",
"for",
"the",
"current,",
"and",
"expanding",
"to",
"linear",
"response",
"in",
"Iinj",
"we",
"find",
"e2"
]
} | |||
2301.00026v2.pdf_1 | null | ArxivOCR_part_1 | {
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"word": [
"Killing",
"Horizons",
"Decohere",
"Quantum",
"Superpositions",
"Daine",
"L.",
"Danielson,1,",
"Gautam",
"Satishchandran,2,",
"and",
"Robert",
"M.",
"Wald1,",
"t",
"1Enrico",
"Fermi",
"Institute",
"and",
"Department",
"of",
"Physics,",
"The",
"University",
"of",
"Chicago,",
"933",
"East",
"56th",
"Street,",
"Chicago,",
"Illinois",
"60637,",
"USA",
"2Princeton",
"Gravity",
"Initiative,",
"Princeton",
"University,",
"Jadwin",
"Hall,",
"Washington",
"Road,",
"Princeton",
"NJ",
"08544,",
"USA",
"(Dated:",
"June",
"16,",
"2023)",
"We",
"recently",
"showed",
"that",
"if",
"a",
"massive",
"(or",
"charged)",
"body",
"is",
"put",
"in",
"a",
"quantum",
"spatial",
"superposition,",
"the",
"mere",
"presence",
"of",
"a",
"black",
"hole",
"in",
"its",
"vicinity",
"will",
"eventually",
"decohere",
"the",
"superposition.",
"In",
"this",
"paper",
"we",
"show",
"that,",
"more",
"generally,",
"decoherence",
"of",
"stationary",
"superpositions",
"will",
"occur",
"in",
"any",
"spacetime",
"with",
"a",
"Killing",
"horizon.",
"This",
"occurs",
"because,",
"in",
"effect",
"the",
"long-rang",
"field",
"of",
"the",
"body",
"is",
"register",
"on",
"the",
"Killing",
"horizon",
"which,",
"we",
"show,",
"necessitates",
"a",
"flu",
"of",
"\"soft",
"horizo",
"gravitons/photons",
"through",
"the",
"horizon.",
"The",
"Killing",
"horizon",
"thereby",
"harvests",
"“which",
"path”",
"information",
"of",
"quantum",
"201",
"superpositions",
"and",
"will",
"decohere",
"any",
"quantum",
"superposition",
"in",
"a",
"finit",
"time.",
"It",
"is",
"particularl",
"instructive",
"to",
"analyze",
"the",
"case",
"of",
"a",
"uniformly",
"accelerating",
"body",
"in",
"a",
"quantum",
"superposition",
"in",
"fla",
"spacetime",
"As",
"we",
"show",
"fro",
"the",
"Rindler",
"perspectiv",
"the",
"superpositio",
"is",
"decohere",
"by",
"\"soft",
"gravitons/photons”",
"that",
"propagate",
"through",
"the",
"Rindler",
"horizon",
"with",
"negligible",
"(Rindler)",
"energy.",
"Gish",
"We",
"show",
"that",
"this",
"decoherence",
"effec",
"is",
"distinc",
"from—an",
"large",
"than—th",
"decoherenc",
"resultin",
"from",
"the",
"presence",
"of",
"Unruh",
"radiation.",
"We",
"further",
"show",
"that",
"from",
"the",
"inertial",
"perspective,",
"the",
"decoherence",
"is",
"due",
"to",
"the",
"radiation",
"of",
"high",
"frequency",
"(inertial)",
"gravitons/photons",
"to",
"null",
"infinity",
"(The",
"notion",
"of",
"gravitons/photons",
"that",
"propagate",
"through",
"the",
"Rindler",
"horizon",
"is",
"the",
"same",
"notion",
"as",
"that",
"of",
"gravitons/photons",
"that",
"propagate",
"to",
"null",
"infinity.",
"We",
"also",
"analyz",
"the",
"decoherenc",
"of",
"a",
"spatial",
"superposition",
"due",
"to",
"the",
"presence",
"of",
"a",
"cosmological",
"horizon",
"in",
"de",
"Sitter",
"spacetime.",
"We",
"provide",
"estimates",
"of",
"the",
"decoherence",
"time",
"for",
"such",
"quantum",
"superpositions",
"in",
"both",
"the",
"Rindler",
"and",
"Milh",
"cosmological",
"cases.",
"Although",
"we",
"explicitly",
"treat",
"the",
"case",
"of",
"spacetime",
"dimension",
"d",
"=",
"4,",
"our",
"analysis",
"2",
"applies",
"to",
"any",
"dimension",
"d",
"4.",
"radiation",
"will",
"be",
"emitted",
"during",
"the",
"portions",
"of",
"her",
"ex-",
"1.",
"INTRODUCTION",
"periment",
"where",
"she",
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"and",
"recombines",
"her",
"particle.",
"Consider",
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"spacetime",
"in",
"which",
"an",
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"Nevertheless,",
"in",
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"if",
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"Alice,",
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"and",
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"she",
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"(hereinafter",
"re-",
"ferred",
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"a",
"negligible",
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"is",
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"In",
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"Stern-Gerlach",
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"or",
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"she",
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"the",
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"arbitrarily",
"long",
"ticle",
"in",
"a",
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"superposition",
"of",
"two",
"spatially",
"separated",
"time",
"T",
"and",
"still",
"maintain",
"coherence",
"when",
"the",
"components",
"states1.",
"She",
"keeps",
"these",
"spatially",
"separated",
"components",
"are",
"recombined.",
"stationary",
"for",
"a",
"time",
"T",
"and",
"then",
"recombines",
"them.",
"Will",
"In",
"a",
"recent",
"paper",
"[14],",
"we",
"showed",
"that",
"the",
"above",
"situ-",
"Alice",
"be",
"able",
"to",
"maintain",
"the",
"coherence",
"of",
"these",
"compo-",
"ation",
"changes",
"dramatically",
"if",
"a",
"black",
"hole",
"is",
"present",
"in",
"nents,",
"so",
"that,",
"when",
"recombined,",
"the",
"fina",
"state",
"of",
"her",
"the",
"spacetime—even",
"though",
"the",
"experiment",
"is",
"carried",
"particle",
"will",
"be",
"pure—or",
"will",
"decoherence",
"have",
"occurred,",
"out",
"entirely",
"in",
"the",
"black",
"hole’s",
"exterior.",
"In",
"effect",
"a",
"black",
"so",
"that",
"the",
"fina",
"stat",
"of",
"her",
"particl",
"will",
"be",
"mixed?",
"hole",
"horizon",
"harvests",
"“which",
"path”",
"information",
"about",
"any",
"Ordinarily,",
"any",
"decoherence",
"effect",
"will",
"be",
"dominate",
"quantum",
"superposition",
"in",
"its",
"exterior,",
"via",
"the",
"long-range",
"by",
"“environmental",
"influences,",
"i.e.,",
"additiona",
"degree",
"field",
"source",
"by",
"the",
"superpose",
"matter.",
"We",
"showe",
"that",
"of",
"freedom",
"present",
"in",
"Alice’s",
"lab",
"that",
"interact",
"with",
"her",
"this",
"results",
"in",
"the",
"unavoidable",
"radiation",
"of",
"entangling",
"particle.",
"We",
"assume",
"that",
"Alice",
"has",
"perfect",
"control",
"of",
"her",
"“soft",
"photons",
"or",
"gravitons”",
"through",
"the",
"horizon",
"that",
"carry",
"laboratory",
"and",
"its",
"environment",
"so",
"that",
"there",
"is",
"no",
"deco-",
"the",
"“which",
"path”",
"information",
"into",
"the",
"black",
"hole.",
"Con-",
"herence",
"from",
"ordinary",
"environmental",
"effects",
"However",
"sequently,",
"the",
"mere",
"presence",
"of",
"the",
"black",
"hole",
"implies",
"a",
"for",
"a",
"charged",
"or",
"massive",
"particle,",
"Alice",
"cannot",
"perfectly",
"fundamental",
"rate",
"of",
"decoherence",
"on",
"the",
"quantum",
"super-",
"control",
"the",
"electromagnetic",
"or",
"gravitational",
"field",
"sinc",
"position2.",
"Although",
"the",
"rate",
"of",
"decoherence",
"will",
"be",
"small",
"her",
"particle",
"acts",
"as",
"a",
"source",
"for",
"these",
"field",
"and",
"som",
"if",
"the",
"black",
"hole",
"is",
"far",
"away,",
"the",
"coherence",
"decays",
"expo-",
"nentially",
"in",
"the",
"time,",
"T,",
"that",
"the",
"spatial",
"superposition",
"is",
"maintained.",
"Thus,",
"in",
"any",
"spacetime",
"with",
"a",
"black",
"hole,",
"daine@uchicago.edu",
"+",
"gautam.satish@princeton.edu",
"$",
"rmwa@uchicago.edu",
"1",
"Quantum",
"spatial",
"superpositions",
"of",
"massive",
"bodies",
"have",
"been",
"of",
"In",
"QED,",
"this",
"effec",
"applie",
"only",
"to",
"superposition",
"of",
"charge",
"particles",
"recent",
"interest",
"in",
"both",
"theoretical",
"as",
"well",
"as",
"proposed",
"experimental",
"However,",
"since",
"all",
"matter",
"sources",
"gravity,",
"the",
"quantum",
"gravitational",
"probes",
"of",
"fundamental",
"properties",
"of",
"quantum",
"gravity,",
"e.g.,",
"[1–13].",
"decoherence",
"applies",
"to",
"all",
"superpositions."
]
} | |||
2301.00026v2.pdf_2 | null | ArxivOCR_part_1 | {
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398,
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521,
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603,
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1099,
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1160,
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1160,
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1262,
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1262,
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1304,
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1376,
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1376,
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1403,
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1403,
1818,
1444,
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1444,
1817,
1538,
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1538,
1817,
1566,
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143,
1831,
262,
1868
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262,
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302,
1868
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302,
1831,
453,
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453,
1830,
534,
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535,
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616,
1830,
673,
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673,
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745,
1868
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745,
1830,
834,
1868
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872,
1844,
981,
1876
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981,
1844,
1008,
1877
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1008,
1844,
1048,
1876
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1048,
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1192,
1876
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1192,
1844,
1229,
1877
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1229,
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1286,
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1286,
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1327,
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1327,
1844,
1382,
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1382,
1844,
1510,
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1510,
1844,
1565,
1877
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870,
1870,
983,
1902
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983,
1870,
1034,
1903
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1034,
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1093,
1903
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1093,
1870,
1135,
1903
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1135,
1870,
1197,
1903
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1197,
1870,
1324,
1903
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1324,
1870,
1443,
1902
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1443,
1870,
1465,
1902
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[
1465,
1870,
1525,
1903
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[
1525,
1870,
1565,
1902
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[
872,
1897,
921,
1930
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921,
1897,
988,
1929
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988,
1897,
1105,
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1105,
1897,
1197,
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1197,
1897,
1239,
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1239,
1897,
1314,
1930
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1314,
1897,
1356,
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1356,
1896,
1408,
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1408,
1897,
1453,
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1453,
1897,
1531,
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1531,
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1565,
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873,
1923,
942,
1955
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942,
1923,
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970,
1923,
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1955
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1003,
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1030,
1955
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1030,
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1072,
1955
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1072,
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1160,
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1160,
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1287,
1956
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1329,
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1369,
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1432,
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1432,
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1497,
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1497,
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1955
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141,
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142,
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197,
1978
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197,
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291,
1978
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291,
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397,
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397,
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450,
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485,
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485,
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637,
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637,
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664,
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704,
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704,
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790,
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790,
1946,
833,
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873,
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979,
1982
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1020,
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1020,
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1073,
1982
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1073,
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1122,
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1122,
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1161,
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1161,
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1244,
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1244,
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1286,
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1286,
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1326,
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1357,
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1482,
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142,
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176,
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320,
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} | {
"word": [
"2",
"there",
"will",
"be",
"essentially",
"complete",
"decoherence",
"within",
"a",
"represent",
"abstract",
"spacetime",
"indices.",
"Upper",
"case",
"Latin",
"in-",
"finit",
"time3",
"dices",
"from",
"the",
"early",
"alphabet",
"correspond",
"to",
"spatial",
"indices",
"The",
"purpose",
"of",
"this",
"paper",
"is",
"to",
"generalize",
"the",
"results",
"of",
"on",
"horizons",
"or",
"null",
"infinity",
"[14]",
"to",
"spacetimes",
"with",
"Killing",
"horizons,",
"i.e.,",
"spacetimes",
"with",
"a",
"Killing",
"vector",
"fiel",
"suc",
"that",
"ther",
"is",
"a",
"null",
"surfac",
"to",
"which",
"the",
"Killing",
"fiel",
"is",
"norma",
"(see,",
"e.g.,",
"[15]",
"for",
"a",
"2.",
"DECOHERENCE",
"DUE",
"TO",
"RADIATION",
"IN",
"A",
"discussion",
"of",
"properties",
"of",
"Killing",
"horizons).",
"The",
"event",
"STATIONARY",
"SPACETIME",
"horizon",
"of",
"a",
"stationary",
"black",
"hole",
"is",
"a",
"Killing",
"horizon",
"[16–18],",
"so",
"spacetimes",
"with",
"Killing",
"horizons",
"encompass",
"In",
"this",
"section,",
"we",
"will",
"give",
"a",
"general",
"analysis",
"of",
"the",
"the",
"case",
"of",
"stationary",
"spacetimes",
"that",
"contain",
"black",
"holes.",
"decoherence",
"of",
"a",
"spatial",
"superposition",
"in",
"stationary",
"However,",
"there",
"are",
"many",
"cases",
"of",
"interest",
"where",
"Killing",
"spacetime",
"due",
"to",
"emission",
"of",
"radiation",
"by",
"the",
"body.",
"Our",
"horizons",
"are",
"present",
"without",
"the",
"presence",
"of",
"black",
"holes.",
"analysis",
"applies",
"both",
"to",
"the",
"decoherence",
"of",
"a",
"charged",
"One",
"such",
"case",
"is",
"that",
"of",
"Minkowski",
"spacetime,",
"where",
"body",
"due",
"to",
"emission",
"of",
"electromagnetic",
"radiation",
"and",
"to",
"the",
"Rindler",
"horizon",
"is",
"a",
"Killing",
"horizon",
"with",
"respect",
"to",
"the",
"decoherence",
"of",
"a",
"gravitating",
"body",
"due",
"to",
"emission",
"of",
"the",
"Lorentz",
"boost",
"Killing",
"field",
"Anothe",
"such",
"case",
"is",
"linearized",
"gravitational",
"radiation.",
"The",
"analyses",
"of",
"these",
"de",
"Sitter",
"spacetime,",
"where",
"the",
"cosmological",
"horizon",
"is",
"a",
"two",
"cases",
"are",
"very",
"closely",
"parallel.",
"In",
"order",
"to",
"avoid",
"Killing",
"horizon.",
"We",
"will",
"show",
"that",
"in",
"these",
"cases,",
"a",
"spatial",
"repetition,",
"we",
"will",
"analyze",
"only",
"the",
"electromagnetic",
"case",
"superposition",
"that",
"kept",
"stationary",
"(with",
"respect",
"to",
"the",
"in",
"detail,",
"but",
"near",
"the",
"end",
"of",
"this",
"section,",
"we",
"will",
"state",
"the",
"symmetry",
"generating",
"the",
"Killing",
"horizon)",
"will",
"decohere",
"corresponding",
"results",
"in",
"the",
"linearized",
"gravitational",
"case,",
"in",
"a",
"manner",
"similar",
"to",
"the",
"black",
"hole",
"case.",
"We",
"will",
"also",
"which",
"can",
"be",
"obtained",
"straightforwardly",
"by",
"replacing",
"the",
"provide",
"an",
"estimate",
"of",
"the",
"maximum",
"amount",
"of",
"time",
"vector",
"potential",
"Aa",
"with",
"the",
"perturbed",
"metric",
"h",
"ab",
",",
"the",
"during",
"which",
"coherence",
"can",
"be",
"maintained.",
"charge-current",
"j",
"a",
"with",
"the",
"stress-energy",
"I",
"ab,",
"etc.",
"The",
"case",
"of",
"the",
"Rindler",
"horizon",
"is",
"particularly",
"instruc-",
"Consider",
"a",
"charged",
"particle4",
"in",
"a",
"stationary",
"spacetime.",
"tive.",
"The",
"relevant",
"symmetry",
"here",
"is",
"that",
"of",
"Lorentz",
"We",
"assume",
"that",
"the",
"particle",
"is",
"initially",
"in",
"a",
"stationary",
"boosts,",
"so",
"Alice’s",
"lab",
"will",
"be",
"“stationary”",
"it",
"is",
"following",
"state.",
"The",
"particle",
"is",
"then",
"put",
"through",
"a",
"Stern-Gerlach",
"(or",
"orbits",
"of",
"Lorentz",
"boosts,",
"which",
"are",
"uniformly",
"accelerating",
"other)",
"apparatus,",
"resulting",
"in",
"it",
"being",
"in",
"a",
"superposition",
"worldlines.",
"Our",
"analysis",
"based",
"upon",
"radiation",
"through",
"state5",
"the",
"Rindler",
"horizon",
"shows",
"that",
"decoherence",
"of",
"uniformly",
"1",
"(",
"I",
"4",
"1",
")",
"I",
"))",
"accelerating",
"spatially",
"separated",
"superposition",
"occurs",
"be-",
"|ψ⟩",
"=",
"+",
"o",
"2",
"(2.1)",
"cause",
"of",
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"emission",
"of",
"“soft”",
"(i.e.,",
"very",
"low",
"frequency)",
"2",
"gravitons",
"or",
"photons,",
"where",
"the",
"frequency",
"is",
"define",
"rel-",
"where",
"|",
"&",
"1",
"and",
"I",
"4",
"2",
"�",
"are",
"normalized",
"states",
"that",
"are",
"spa-",
"ative",
"to",
"an",
"affi",
"parameter",
"on",
"the",
"Rindler",
"horizon.",
"As",
"we",
"shall",
"show,",
"the",
"decoherence",
"effec",
"of",
"this",
"radiatio",
"of",
"tially",
"separated",
"after",
"passing",
"through",
"the",
"apparatus.",
"The",
"soft",
"gravitons",
"or",
"photons",
"is",
"distinct",
"from",
"the",
"(smaller)",
"particle",
"is",
"then",
"recombined",
"via",
"reversing",
"Stern-Gerlach",
"decoherence",
"effec",
"resultin",
"fro",
"the",
"presenc",
"of",
"Unruh",
"(or",
"other)",
"apparatus",
"and",
"returns",
"to",
"a",
"stationary",
"state.",
"radiation.",
"To",
"gain",
"further",
"insight,",
"we",
"also",
"analyze",
"the",
"We",
"are",
"particularly",
"interested",
"in",
"the",
"case",
"where,",
"between",
")",
"'i",
"decohering",
"radiation",
"in",
"the",
"electromagnetic",
"case",
"from",
"the",
"separation",
"and",
"recombination,",
"|4",
"1",
"and",
"&",
"2",
"are",
"kept",
"inertial",
"point",
"of",
"view,",
"using",
"the",
"Liénard-Wiechert",
"solution",
"stationary",
"for",
"a",
"long",
"period",
"of",
"time,",
"T,",
"but",
"we",
"do",
"not",
"to",
"determine",
"the",
"radiation",
"at",
"future",
"null",
"infinity",
"As",
"we",
"make",
"any",
"such",
"assumption",
"in",
"this",
"section.",
"We",
"wish",
"to",
"shall",
"show,",
"the",
"decohering",
"photons",
"are",
"of",
"high",
"frequency",
"estimate",
"how",
"much",
"decoherence",
"due",
"to",
"emission",
"of",
"elec-",
"tromagnetic",
"radiation",
"will",
"have",
"occurred",
"by",
"the",
"time",
"of",
"at",
"null",
"infinity",
"In",
"sec.",
"2",
"we",
"provide",
"a",
"general",
"discussion",
"of",
"the",
"deco-",
"recombination6.",
"herence",
"of",
"a",
"quantum",
"superposition",
"due",
"to",
"radiation",
"in",
"a",
"stationary",
"spacetime.",
"In",
"sec.",
"3",
"we",
"consider",
"the",
"decoher-",
"ence",
"of",
"a",
"uniformly",
"accelerating",
"superposition,",
"analyzing",
"it",
"from",
"both",
"the",
"Rindler",
"and",
"Minkowski",
"viewpoints.",
"We",
"4",
"As",
"already",
"indicated",
"above,",
"the",
"“particle”",
"need",
"not",
"be",
"an",
"elementary",
"also",
"show",
"that",
"this",
"decoherence",
"is",
"distinct",
"from",
"(and",
"larger",
"particle",
"but",
"could",
"be",
"“nanoparticle”",
"or",
"any",
"other",
"body",
"whose",
"only",
"than)",
"the",
"decoherence",
"effect",
"due",
"to",
"the",
"presenc",
"of",
"Un-",
"5",
"For",
"simplicity,",
"we",
"have",
"assumed",
"that",
"we",
"have",
"a",
"50-50",
"superposition",
"ruh",
"radiation.",
"In",
"sec.",
"we",
"analyze",
"the",
"decoherence",
"in",
"de",
"of",
"/4",
"and",
"lo",
">,",
"but",
"this",
"assumption",
"is",
"not",
"necessary.",
"of",
"mass.",
"relevant",
"degree",
"of",
"freedom",
"for",
"our",
"analysis",
"is",
"its",
"center",
"Sitter",
"spacetime",
"associated",
"with",
"the",
"cosmological",
"horizon.",
"6",
"The",
"decoherence",
"of",
"Alice’s",
"particle",
"can",
"be",
"experimentally",
"deter-",
"We",
"will",
"work",
"in",
"Planck",
"units",
"where",
"G",
"=",
"c",
"=",
"h",
"=",
"k",
"=",
"1",
"mined",
"as",
"follows.",
"We",
"assume",
"that",
"Alice’s",
"particle",
"initially",
"has",
"spin",
"and,",
"in",
"electromagnetic",
"formulas,",
"we",
"also",
"put",
"€",
"o",
"=",
"1,",
"but",
"in",
"the",
"positive",
"x-direction",
"and",
"thus",
"is",
"in",
"a",
"50-50",
"superposition",
"of",
"z-spin",
"after",
"passing",
"through",
"the",
"initial",
"Stern-Gerlach",
"apparatus.",
"we",
"will",
"restore",
"these",
"constants",
"in",
"our",
"formulas",
"that",
"give",
"After",
"recombination,",
"Alice",
"measures",
"the",
"x-spin",
"of",
"her",
"particle.",
"If",
"estimates",
"for",
"decoherence",
"times.",
"Lower",
"case",
"Latin",
"indices",
"coherence",
"of",
"the",
"superposition",
"eq.",
"(2.1)",
"has",
"been",
"maintained,",
"then",
"(assuming",
"that",
"Alice",
"has",
"made",
"appropriate",
"corrections",
"if",
"there",
"are",
"any",
"phase",
"difference",
"betwee",
"the",
"paths)",
"the",
"spin",
"will",
"always",
"be",
"found",
"to",
"be",
"in",
"the",
"positive",
"x-direction.",
"On",
"the",
"other",
"hand,",
"if",
"any",
"3",
"This",
"maximal",
"coherence",
"time",
"for",
"superpositions",
"in",
"the",
"exterior",
"can",
"coherence",
"has",
"been",
"lost,",
"the",
"particle",
"will",
"not",
"be",
"in",
"state",
"of",
"definit",
"be",
"much",
"smaller",
"than",
"the",
"evaporation",
"time",
"of",
"the",
"black",
"hole.",
"spin,",
"and",
"the",
"spin",
"will",
"sometimes",
"be",
"found",
"to",
"be",
"in",
"the",
"negative"
]
} | |||
2301.00026v2.pdf_3 | null | ArxivOCR_part_1 | {
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"and",
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"fiel",
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"in",
"the",
"charge-current",
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"ja",
"in",
"the",
"states",
"i",
"4",
"1",
")",
"Ain",
"is",
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"on",
"ay",
"(",
"lin",
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"and",
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"2",
"are",
"negligibly",
"small",
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"the",
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"of",
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"can",
"treat",
"the",
"charge",
"current",
"in",
"each",
"of",
"these",
"Ain(fa)",
"=",
"ia(Kσf",
"-",
"ia",
"(Ko",
").",
"t",
"(2.3)",
"states",
"as",
"c-number",
"sources",
"in",
"Maxwell’s",
"equations,",
"given",
"advanced",
"minus",
"retarded",
"so",
"H",
"by",
"91",
"=",
"(4",
"1",
"ja",
"i",
"o",
"and",
"ja",
"=",
"<",
"o",
"|ja",
"lp",
"s",
"respectively.",
"In",
"1",
"Here",
"fa",
"a",
"divergence-free8",
"test",
"function,",
"o",
"f",
"denotes",
"the",
"to",
"Maxwell’s",
"equations",
"the",
"initial",
"and",
"fina",
"stationar",
"eras,",
"|%/",
"1",
"and",
"|4",
"2",
"are",
"with",
"source",
"fa,",
"and",
"K",
"in",
"denotes",
"the",
"map",
"taking",
"assumed",
"to",
"coincide",
"spatially",
"(though",
"they",
"may",
"diffe",
"in",
"S",
"other",
"characteristics,",
"such",
"as",
"spin)",
"so",
"that",
"ja",
"=",
"2",
"at",
"very",
"the",
"space",
"S",
"of",
"classical",
"solutions",
"to",
"their",
"representatives",
"in",
"the",
"one-particle",
"Hilbert",
"space",
"in",
"The",
"commutator",
"early",
"and",
"very",
"late",
"times.",
"of",
"the",
"creation",
"and",
"annihilation",
"operators",
"in",
"eq.",
"(2.3)",
"is",
"In",
"order",
"to",
"proceed",
"further,",
"we",
"must",
"specify",
"the",
"initial",
"given",
"by",
"state",
"of",
"the",
"electromagnetic",
"field",
"Since",
"prior",
"to",
"goin",
"through",
"the",
"Stern-Gerlach",
"apparatus,",
"the",
"charge",
"is",
"as-",
"sumed",
"to",
"be",
"stationary,",
"at",
"early",
"times",
"we",
"may",
"subtract",
"[a(Kσf",
"),",
"a",
"(Ko,",
")]",
"=",
"⟨Kσf",
"f",
"|Kσg⟩",
"g",
"1.",
"t",
"I",
")",
"(2.4)",
"the",
"“Coulomb",
"field",
"Cin",
"of",
"the",
"charge,",
"i.e.,",
"at",
"early",
"times",
"we",
"may",
"consider",
"the",
"electromagnetic",
"fiel",
"observabl",
"where",
"⟨Kσf",
"|Kσg⟩",
"g",
")",
"is",
"the",
"inner",
"product",
"on",
"H",
",",
"which",
"is",
"lin",
"given",
"by",
"a",
"natural",
"generalization",
"of",
"the",
"Klein-Gordon",
"-",
"cin1",
"inner",
"product",
"to",
"electromagnetic",
"fields",
"a",
"=",
"Aa",
"La",
"(2.2)",
"For",
"the",
"case",
"of",
"a",
"globally",
"stationary",
"spacetime",
"in",
"the",
"where",
"Cin",
"is",
"the",
"(assumed",
"to",
"be",
"unique)",
"stationary",
"clas-",
"stationary",
"vacuum",
"state,",
"Kσf",
"corresponds",
"to",
"taking",
"the",
"Tf",
"sical",
"solution",
"to",
"Maxwell’s",
"equations",
"with",
"the",
"early",
"time",
"positive",
"frequency",
"part",
"of",
"o",
"f",
"with",
"respect",
"to",
"the",
"time",
"stationary",
"charged",
"particle",
"source",
"jí",
"=",
"ja",
"and",
"A",
"a",
"is",
"translations",
"generating",
"the",
"stationary",
"symmetry.",
"For",
"the",
"the",
"vector",
"potential",
"operator.",
"We",
"need",
"not",
"assume",
"any",
"case",
"of",
"a",
"stationary",
"black",
"hole",
"in",
"the",
"Unruh",
"vacuum",
"state,",
"specifi",
"choic",
"of",
"gauge",
"for",
"Ain.",
"Then",
"Ain",
"satisfie",
"the",
"Kσf",
"corresponds",
"to",
"taking",
"the",
"positive",
"frequency",
"part",
"of",
"source-free",
"Maxwell’s",
"equations",
"at",
"early",
"times,",
"and",
"we",
"o",
"with",
"respect",
"to",
"affi",
"time",
"on",
"the",
"past",
"horizon",
"and",
"may",
"extend",
"its",
"definitio",
"to",
"all",
"time",
"by",
"requirin",
"it",
"to",
"with",
"respect",
"to",
"Killing",
"time",
"at",
"past",
"null",
"infinity",
"For",
"satisfy",
"the",
"source-free",
"Maxwell",
"equations",
"everywhere.",
"Minkowski",
"spacetime",
"in",
"the",
"inertial",
"Minkowski",
"vacuum,",
"The",
"initial",
"state",
"of",
"the",
"electromagnetic",
"fiel",
"may",
"be",
"Kσf",
"corresponds",
"to",
"taking",
"the",
"positive",
"frequency",
"part",
"specifie",
"by",
"givin",
"the",
"“radiatio",
"state",
"of",
"Ain.",
"The",
"of",
"o",
"with",
"respect",
"to",
"inertial",
"time",
"translations.",
"Equiv-",
"choice",
"of",
"this",
"state",
"depends",
"on",
"the",
"physical",
"situation",
"being",
"alently,",
"Kσf",
"f",
"in",
"this",
"case,",
"corresponds",
"to",
"the",
"solution",
"considered.",
"If",
"the",
"spacetime",
"were",
"globally",
"stationary—i.e.,",
"obtained",
"by",
"taking",
"the",
"positive",
"frequency",
"part",
"of",
"the",
"re-",
"if",
"the",
"stationary",
"Killing",
"fiel",
"were",
"everywher",
"timelike",
"so,",
"striction",
"of",
"o",
"f",
"to",
"any",
"null",
"hyperplane",
"N",
"(i.e.,",
"any",
"Rindler",
"in",
"particular,",
"there",
"are",
"no",
"Killing",
"horizons—it",
"would",
"be",
"horizon)",
"with",
"respect",
"to",
"an",
"affi",
"parametrizati",
"of",
"the",
"natural",
"to",
"assume",
"that",
"the",
"initial",
"state",
"of",
"the",
"radiation",
"null",
"geodesics",
"generating",
"J",
".",
"For",
"de",
"Sitter",
"spacetime",
"in",
"is",
"the",
"stationary",
"vacuum",
"state,",
"i.e.,",
"the",
"ground",
"state",
"the",
"de",
"Sitter",
"invariant",
"vacuum,",
"Kσf",
"f",
"corresponds",
"to",
"the",
"relative",
"to",
"the",
"time",
"translations.",
"For",
"the",
"case",
"of",
"a",
"black",
"solution",
"obtained",
"by",
"taking",
"the",
"positive",
"frequency",
"part",
"hole",
"spacetime,",
"it",
"would",
"be",
"correspondingly",
"natural",
"to",
"of",
"the",
"restriction",
"of",
"o",
"to",
"any",
"cosmological",
"horizon",
"with",
"assume",
"that",
"the",
"initial",
"state",
"of",
"the",
"radiation",
"is",
"that",
"of",
"respect",
"to",
"an",
"affi",
"parametrizati",
"of",
"the",
"null",
"geodesi",
"the",
"Unruh",
"vacuum,",
"since",
"for",
"a",
"black",
"hole",
"formed",
"by",
"generating",
"that",
"horizon.",
"gravitational",
"collapse,",
"the",
"state",
"of",
"a",
"quantum",
"fiel",
"is",
"expected",
"to",
"approach",
"the",
"Unruh",
"vacuum",
"after",
"the",
"black",
"of",
"4",
"1",
",",
"and",
"o",
"can",
"be",
"treated",
"as",
"c-number",
"sources,",
"the",
"hole",
"has",
"“settled",
"down”",
"to",
"a",
"stationary",
"state.",
"For",
"the",
"Under",
"the",
"above",
"assumption",
"that",
"the",
"charge-currents",
"i",
"case",
"of",
"Minkowski",
"spacetime,",
"we",
"take",
"the",
"initial",
"state",
"electromagnetic",
"fiel",
"A",
"Ai,",
"in",
"the",
"presenc",
"of",
"the",
"charg",
"of",
"the",
"radiation",
"to",
"be",
"the",
"ordinary",
"(inertial)",
"Minkowski",
"in",
"state",
"1i",
"&",
"i",
"i",
"for",
"=",
"1,",
"2",
"is",
"given",
"in",
"terms",
"of",
"the",
"source",
"vacuum.",
"For",
"de",
"Sitter",
"spacetime,",
"we",
"take",
"the",
"initial",
"state",
"free",
"fiel",
"Ain",
"by",
"[23]",
"of",
"the",
"radiation",
"to",
"be",
"the",
"de",
"Sitter",
"invariant",
"vacuum7",
"for",
"A",
"i,a",
"=",
"Ain",
"+",
"Gret(jě)1",
"(2.5)",
"the",
"electromagnetic",
"fiel",
"[20]",
"We",
"denot",
"the",
"initia",
"stat",
"where",
"Gret(j�)",
"denotes",
"the",
"classical",
"retarded",
"solution",
"for",
"of",
"the",
"radiation",
"in",
"all",
"of",
"the",
"above",
"cases",
"by",
"/v",
"0",
"y",
"(i.e.,",
"source",
"ji.",
"Ji.",
"In",
"each",
"of",
"the",
"above",
"cases,",
"v",
"o",
"is",
"a",
"pure,",
"quasi-free",
"a",
"In",
"particular,",
"since",
"the",
"fiel",
"a",
"is",
"in",
"state",
"Gaussian)",
"state.",
"It",
"follows",
"(see,",
"e.g.,",
"[22]",
"or",
"appendix",
"A",
"u",
"0",
"the",
"correlation",
"functions",
"of",
"the",
"electromagnetic",
"fiel",
"of",
"[15])",
"that",
"we",
"can",
"construct",
"a",
"one-particle",
"Hilbert",
"space",
"and",
"corresponding",
"Fock",
"space",
"P",
"(",
"7",
"in",
")",
"wherein",
"|v",
"0",
"lin",
"0",
"x-direction.",
"By",
"repeating",
"the",
"experiment",
"many",
"times,",
"Alice",
"can,",
"in",
"principle,",
"determine",
"the",
"decoherence",
"to",
"any",
"desired",
"accuracy.",
"7",
"A",
"de",
"Sitter",
"invariant",
"vacuum",
"state",
"does",
"not",
"exist",
"for",
"the",
"massless",
"8",
"Restriction",
"of",
"the",
"smearing",
"to",
"divergence-free",
"test",
"functions",
"is",
"scalar",
"fiel",
"[19]",
"but",
"such",
"a",
"state",
"does",
"exist",
"for",
"the",
"electromagneti",
"necessary",
"and",
"sufficie",
"to",
"elimina",
"the",
"gauge",
"dependen",
"of",
"Ain",
"fiel",
"[20]",
"and",
"linearize",
"gravitationa",
"field",
"[21].",
"(see,",
"e.g.,",
"P.101",
"of",
"[22])."
]
} | |||
2301.00026v2.pdf_4 | null | ArxivOCR_part_1 | {
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"i,a",
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"are",
"given",
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"By",
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"<",
"�",
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"(2.13)",
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"-",
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"(2.6)",
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"However,",
"at",
"Now,",
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"Cout,",
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"at",
"Consider",
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"-",
"j2)",
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"that",
"a",
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"i",
"in",
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"era.",
"We",
"refer",
"to",
"photons",
"in",
"this",
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"as",
"Aout",
"=",
"A",
"li,a",
"-",
"cout1.",
"(2.7)",
"entangling",
"photons.",
"By",
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"of",
"coherent",
"R",
"a",
"2",
"of",
"entangling",
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"the",
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"By",
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"(2.6),",
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"∥K",
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"-",
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"⟨N",
",",
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".",
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"2",
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"(2.15)",
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"(x",
"1",
")",
".",
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"I",
"[A",
"Thus,",
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"have",
"u",
"0",
"(x",
"1",
")1)]",
"=",
"A",
")",
".",
".",
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"-",
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">",
"I",
"<",
",",
"I",
"an",
"(�",
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")",
"+",
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"(�",
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"(2.8)",
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"¢",
"1",
"v",
"2",
"I",
"(2.16)",
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"(2.9)",
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"-",
"I",
"_1",
")",
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"-",
"I",
"=",
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"2",
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"2",
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"(2.17)",
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"(v",
"1",
"Ai,a",
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"source-free",
"Note",
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"Maxwell",
"equations",
"in",
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"late-time",
"region.",
"The",
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"eq.",
"(2.",
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"late-time",
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"see",
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"necessary",
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"1",
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"surface",
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"1)",
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"⟨",
"1.",
"We",
"summarize",
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"we",
"have",
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"above",
"u",
"i,",
"=",
"e",
"�",
"II",
"Ai",
"exp",
"(cid:2)a†",
"A",
")",
"/v",
"0",
",",
"-",
"l",
")",
"|",
"as",
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"Under",
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"K",
"(2.10)",
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"D,",
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"(2.4).",
"Thus,",
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">",
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"(1)",
"We",
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"the",
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"ja",
"and",
"jz,",
"state",
"1%",
"1",
"i",
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"2",
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"2",
",",
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"(2)",
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"&",
"2",
".",
"The",
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"r",
")",
",",
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"j2)",
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"91",
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"1",
"(",
"I",
"�",
"&",
"I",
">",
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",",
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"ja",
"=",
"ja",
"at",
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"times.",
"/r",
"+",
"|4",
"2",
"=",
"&",
"1",
"v",
"v",
"2",
")",
".",
"(2.11)",
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"of",
"o",
"1",
"and",
"|4",
"2",
"�",
"due",
"to",
"emis-",
"(3)",
"We",
"calculate",
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"one-particle",
"state",
"KGret(j1",
"-",
"j",
"2",
"corresponding",
"to",
"Gret(jb",
"ic",
"j2)",
"at",
"late",
"times.",
"In",
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"various",
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"this",
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"follow-",
"91",
"2",
"|",
"�",
"sion",
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"(i)",
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"globally",
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"spacetime",
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"D",
"=",
"1",
"-",
"I",
"v",
"1",
"u",
"2",
"|.",
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"vacuum",
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"(2.12)",
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"part",
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"solution",
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"respect",
"to",
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"time",
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"generating",
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"(ii)",
"For",
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"station-",
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"black",
"hole",
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"Unruh",
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"9",
"It",
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"x",
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"time",
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"(iii)",
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"J",
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} | |||
2301.00026v2.pdf_5 | null | ArxivOCR_part_1 | {
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"Alice",
"can",
"keep",
"the",
"components",
"of",
"her",
"particle",
"separated",
"any",
"Rindler",
"horizon.",
"(iv)",
"For",
"de",
"Sitter",
"spacetime",
"for",
"as",
"long",
"a",
"time",
"T",
"as",
"she",
"likes",
"without",
"destruction",
"of",
"initially",
"in",
"the",
"de",
"Sitter",
"invariant",
"vacuum,",
"the",
"one-",
"the",
"coherence.",
"particle",
"state",
"is",
"the",
"positive",
"frequency",
"part",
"of",
"the",
"As",
"shown",
"in",
"[14],",
"the",
"situation",
"is",
"quite",
"differen",
"if",
"a",
"solution",
"with",
"respect",
"to",
"affi",
"time",
"on",
"any",
"cosmo-",
"black",
"hole",
"is",
"present.",
"In",
"the",
"electromagnetic",
"case,",
"even",
"logical",
"horizon.",
"if",
"T",
"1",
"T",
"≫",
"qd",
"so",
"that",
"a",
"negligible",
"number",
"of",
"entangling",
"2",
"(4)",
"We",
"compute",
"the",
"squared",
"norm,",
"Il",
"K[Gret(j1",
"-",
"j",
"2",
")]",
"II",
"2,",
"photons",
"is",
"emitted",
"to",
"infinity",
"ther",
"will",
"be",
"entanglin",
"1",
"of",
"this",
"one-particle",
"state",
"at",
"late",
"times.",
"This",
"quan-",
"radiation",
"emitted",
"into",
"the",
"black",
"hole.",
"For",
"large",
"T,",
"the",
"tity",
"is",
"equal",
"to",
"the",
"expected",
"number",
"of",
"entangling",
"number",
"of",
"entangling",
"photons",
"increases",
"with",
"T",
"as11",
"photons,",
"is",
"<",
"N",
"~",
"M3q�ď�",
"-T",
"then",
"(black",
"hole,",
"EM)",
"(2.21)",
"D6",
"-1ll",
"∥K",
"(cid:2)G",
"-",
"I|�",
"-",
"D",
"=",
"1",
"exp",
"(2.18)",
".",
"where",
"M",
"is",
"the",
"mass",
"of",
"the",
"black",
"hole,",
"D",
"is",
"the",
"proper",
"(j1",
"2",
"distance",
"of",
"Alice’s",
"lab",
"from",
"the",
"horizon",
"of",
"the",
"black",
"hole,",
"and",
"we",
"assume",
"that",
"D",
"M.",
"The",
"corresponding",
"result",
"As",
"previously",
"stated,",
"the",
"above",
"analysis",
"extends",
"straightforwardly",
"to",
"the",
"linearized",
"gravitational",
"case,",
"in",
"the",
"linearized",
"gravitational",
"case",
"is",
"where",
"the",
"perturbed",
"metric,",
"h",
"ab",
",",
"is",
"treated",
"as",
"a",
"linear",
"quantum",
"fiel",
"propagatin",
"in",
"the",
"backgroun",
"classica",
"<",
"N",
")",
"~",
"5m2d4",
"(black",
"hole,",
"GR).",
"(2.22)",
"T",
"D10",
"stationary",
"spacetime.",
"To",
"compute",
"the",
"decoherence",
"due",
"to",
"gravitational",
"radiation",
"in",
"this",
"case,",
"we",
"carry",
"out",
"the",
"Thus,",
"the",
"coherence",
"of",
"Alice’s",
"particle",
"will",
"always",
"be",
"above",
"steps,",
"replacing",
"Aa",
"by",
"h",
"ab",
"and",
"the",
"charge-current",
"destroyed",
"within",
"a",
"finit",
"time",
"la",
"ja",
"by",
"the",
"stress-energy",
"tensor",
"I",
"ab",
".",
"The",
"retarded",
"solu-",
"ysis",
"to",
"the",
"cases",
"of",
"Rindler",
"spacetime",
"and",
"de",
"Sitter",
"space-",
"tion",
"Gret(j�)",
"for",
"Maxwell’s",
"equations",
"is",
"replaced",
"by",
"the",
"time.",
"Although",
"we",
"will",
"explicitly",
"analyze",
"only",
"the",
"Rindler",
"a",
"solution",
"Gret(T",
"cd",
")",
"for",
"the",
"linearized",
"Einstein",
"In",
"the",
"next",
"two",
"sections,",
"we",
"will",
"apply",
"the",
"above",
"anal-",
"retarded",
"H",
"equation.",
"The",
"map",
"K",
"S",
"in",
"is",
"again",
"obtained",
"as",
"and",
"de",
"Sitter",
"cases,",
"it",
"will",
"be",
"clear",
"from",
"our",
"analysis",
"of",
"the",
"H",
"in",
"item",
"(3)",
"above",
"and",
"the",
"inner",
"product",
"on",
"in",
"is",
"again",
"next",
"two",
"sections—as",
"well",
"as",
"our",
"analysis",
"in",
"[14]—that",
"it",
"given",
"by",
"a",
"natural",
"generalization",
"of",
"the",
"Klein-Gordon",
"can",
"be",
"applied",
"to",
"any",
"Killing",
"horizon,",
"provided",
"only",
"that",
"inner",
"product",
"to",
"linearized",
"gravitational",
"fields",
"The",
"de-",
"the",
"initial",
"“vacuum",
"state”",
"go",
"o",
"of",
"the",
"electromagnetic",
"coherence",
"due",
"to",
"gravitational",
"radiation",
"is",
"then",
"given",
"by",
"and/or",
"linearized",
"gravitational",
"fiel",
"correspond",
"to",
"one-",
"the",
"analog",
"of",
"eq.",
"(2.18).",
"particle",
"states",
"that",
"are",
"positive",
"frequency",
"with",
"respect",
"to",
"The",
"above",
"analysis",
"applies",
"for",
"any",
"motion",
"of",
"the",
"compo-",
"affi",
"time",
"on",
"the",
"future",
"Killing",
"horizon.",
"nents",
"of",
"Alice’s",
"superposition.",
"We",
"are",
"primarily",
"interested",
"in",
"the",
"case",
"where,",
"during",
"a",
"time",
"interval",
"I",
"1",
"Alice",
"puts",
"a",
"particle",
"of",
"charge",
"q",
"(or",
"mass",
"m)",
"into",
"a",
"spatial",
"super-",
"3.",
"RINDLER",
"HORIZONS",
"DECOHERE",
"position,",
"where",
"the",
"distance",
"between",
"the",
"components",
"of",
"QUANTUM",
"SUPERPOSITIONS",
"the",
"particle",
"wavefunction",
"is",
"d.",
"She",
"then",
"keeps",
"this",
"super-",
"position",
"stationary",
"in",
"her",
"lab",
"for",
"a",
"time",
"T.",
"Finally,",
"she",
"We",
"now",
"consider",
"the",
"case",
"of",
"Minkowski",
"spacetime12",
"recombines",
"her",
"particle",
"over",
"a",
"time",
"interval",
"I",
"2",
"2.",
"with",
"Alice’s",
"lab",
"uniformly",
"accelerating",
"with",
"acceleration",
"In",
"Minkowski",
"spacetime",
"in",
"the",
"case",
"where",
"Alice’s",
"lab",
"is",
"a.",
"Specifically",
"we",
"tak",
"Alice's",
"lab",
"to",
"follo",
"the",
"orbit",
"inertial,",
"Gret(j�",
"s",
"j2)",
"will",
"be",
"nonzero",
"at",
"null",
"infinit",
"only",
"at",
"the",
"retarded",
"times",
"corresponding",
"to",
"the",
"time",
"intervals",
"t",
"=",
"1",
"sinh(aτ",
"z",
"=",
"1",
"cosh(aτ",
"(3.1)",
"I",
"1",
"and",
"I",
"2",
".",
"A",
"rough",
"estimate",
"of",
"the",
"number",
"of",
"entangling",
"a",
"a",
"l",
"(",
")",
")",
"photons",
"was",
"obtained",
"in",
"[3]",
"using",
"the",
"Larmor",
"formula",
"for",
"radiation",
"in",
"these",
"eras,",
"which,",
"in",
"natural",
"units,",
"yields",
"of",
"the",
"boost",
"Killing",
"fiel",
"a",
"a",
"a",
"q2d2",
"<",
"~",
"ba",
"=",
"a",
"z",
"∂t",
"+",
"(3.2)",
"N",
"[min(T1,",
"1",
"I",
"2",
")]2",
"(Minkowski,",
"EM).",
"(2.19)",
"∂z",
"-",
"The",
"corresponding",
"result",
"in",
"the",
"linearized",
"gravitational",
"Here",
"we",
"have",
"normalized",
"ba",
"such",
"that",
"baba",
"a",
"=",
"on",
"the",
"worldline",
"of",
"Alice’s",
"laboratory.",
"Thus,",
"ba",
"is",
"the",
"four-",
"case",
"is",
"[3]",
"m2d4",
",",
"~",
"<",
"N",
"[min(T1,",
"T",
"2",
")]4",
"(Minkowski,",
"GR).",
"(2.20)",
"11In",
"the",
"analysis",
"of",
"[14],",
"we",
"used",
"the",
"fact",
"that",
"the",
"Unruh",
"vacuum",
"is",
"well",
"approximated",
"by",
"the",
"Hartle-Hawking",
"vacuum",
"at",
"low",
"frequencies",
"Therefore,",
"if",
"Alice",
"recombines",
"her",
"particle",
"sufficient",
"near",
"the",
"horizon",
"of",
"the",
"black",
"hole.",
"slowly",
"that",
"I",
"1",
",",
"T",
">",
"qd",
"in",
"the",
"electromagnetic",
"case",
"or",
"12We",
"explicitly",
"treat",
"the",
"case",
"of",
"4",
"spacetime",
"dimensions,",
"but",
"our",
">",
"md2",
"in",
"the",
"gravitational",
"case,",
"then",
"she",
"can",
"main-",
"analysis",
"generalizes",
"straightforwardly",
"to",
"all",
"higher",
"dimensions.",
"I",
"1",
",",
"I",
"2"
]
} | |||
2301.00026v2.pdf_6 | null | ArxivOCR_part_1 | {
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[
699,
1911,
835,
1950
],
[
142,
1944,
309,
1982
],
[
311,
1944,
367,
1982
],
[
376,
1944,
430,
1982
],
[
432,
1943,
497,
1981
],
[
498,
1943,
593,
1982
],
[
594,
1944,
619,
1982
],
[
621,
1944,
741,
1981
],
[
742,
1943,
835,
1982
],
[
143,
1976,
318,
2014
],
[
318,
1976,
400,
2013
],
[
400,
1976,
429,
2013
],
[
429,
1976,
491,
2013
],
[
491,
1976,
625,
2013
],
[
625,
1976,
701,
2013
],
[
701,
1976,
798,
2013
],
[
798,
1976,
833,
2013
],
[
851,
1974,
922,
2009
],
[
922,
1976,
1003,
2008
],
[
1003,
1976,
1054,
2008
],
[
1054,
1976,
1083,
2008
],
[
1083,
1976,
1189,
2008
],
[
1189,
1976,
1221,
2008
],
[
1221,
1976,
1254,
2008
],
[
1254,
1976,
1347,
2008
],
[
1347,
1976,
1401,
2008
],
[
1401,
1976,
1508,
2008
],
[
1508,
1972,
1539,
2008
],
[
1539,
1975,
1565,
2008
]
]
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"Killing",
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"a",
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"We",
"wish",
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"=",
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"-",
"=",
"t",
"+",
"z",
"U",
"V",
"(3.3)",
"z,",
"directly",
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"The",
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"H",
"R",
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"−1",
"In",
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"[24],",
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"R",
"The",
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"Killing",
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"R",
"Consequently,",
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">",
"0",
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"R",
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"=",
"aV",
"=",
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"∂v",
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"a",
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")a",
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"f",
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"1",
"a",
"=",
"(3.9)",
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"V",
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"H",
"+,",
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"refer",
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"v",
"D",
"as",
"the",
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"Killing",
"time”",
"on",
"BH",
"R.",
"R",
"as",
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"1.",
"The",
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"such",
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"Rindler",
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"[25–30].",
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"Due",
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"Radiation",
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"Soft",
"nonvanishing",
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"the",
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"in",
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"Through",
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"the",
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"V",
">",
"0",
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"U",
"F",
"ab",
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"(3.10)",
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"We",
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"position",
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"sec.",
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"+",
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"ρ2",
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"+",
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"the",
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"E",
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"=",
"F",
"ab",
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"H",
"+,",
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} | |||
2301.00026v2.pdf_7 | null | ArxivOCR_part_1 | {
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} | {
"word": [
"7",
"on",
"H",
"+",
"is",
"again",
"given",
"by",
"eq.",
"(3.10),",
"but",
"its",
"value,",
"E′",
",",
"R",
"U",
"will",
"be",
"differen",
"fro",
"its",
"valu",
"at",
"earl",
"times",
"Maxwell’",
"H",
"+",
"R",
"equations",
"on",
"sert",
"imply",
"that",
"d",
"AE",
"A",
"=",
"õ",
"v",
"E",
"U",
"(3.11)",
"D",
"A",
"is",
"the",
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"operator",
"on",
"the",
"R",
"cross-",
"lb",
"nb",
"where",
"sections",
"of",
"the",
"horizon",
"and",
"capital",
"Latin",
"indices",
"from",
"the",
"early",
"alphabet",
"are",
"raised",
"and",
"lowered",
"with",
"the",
"met-",
"a",
"ric,",
"8",
"AB",
"on",
"the",
"cross",
"sections.",
"Eq.",
"(3.11)",
"implies",
"that",
"E",
"A",
"̸=",
"0",
"whenever",
"ô",
"v",
"E",
"U",
"̸=",
"0,",
"so",
"there",
"will",
"be",
"radiation",
"through",
"the",
"horizon",
"as",
"the",
"is",
"being",
"moved.",
"Most",
"harge",
"-",
"importantly,",
"D",
"(",
"D",
"A",
"dV",
"E",
"A",
"=",
"∆EU",
"U",
"(3.12)",
"x",
"-",
"where",
"∆EU",
"U",
"=",
"E′",
"Cu",
"E",
"U",
"is",
"the",
"change",
"in",
"the",
"radial",
"electric",
"fiel",
"betwee",
"the",
"charg",
"at",
"position",
"D",
"and",
"D.",
"Now,",
"in",
"a",
"gauge",
"where",
"A",
"a",
"na",
"=",
"0",
"on",
"the",
"horizon,",
"the",
"transverse",
"t",
"(i.e.,",
"x-y)",
"components",
"of",
"the",
"electric",
"fiel",
"are",
"relate",
"to",
"z",
"the",
"corresponding",
"components",
"of",
"the",
"vector",
"potential",
"by",
"E",
"A",
"=",
"_",
"a",
"A",
".",
"v",
"A",
"(3.13)",
"FIG.",
"1.",
"Alice’s",
"laboratory",
"undergoes",
"uniform",
"acceleration",
"a",
"in",
"the",
"z-direction",
"in",
"Minkowski",
"spacetime",
"and",
"thus",
"follows",
"an",
"Since",
"the",
"transverse",
"components",
"of",
"the",
"Coulomb",
"fiel",
"of",
"a",
"orbit",
"of",
"a",
"boost",
"Killing",
"field",
"The",
"futur",
"Rindler",
"horizo",
"H",
"+",
"static",
"charge",
"vanish,",
"we",
"may",
"replace",
"the",
"vector",
"potential",
"is",
"a",
"Killing",
"horizon",
"for",
"this",
"boost",
"Killing",
"field",
"The",
"future",
"A",
"A",
"by",
"the",
"“Coulomb",
"subtracted”",
"vector",
"potential",
"A",
"A",
"R",
"directed",
"null",
"vector",
"nb",
"=",
"(2/2V)b",
"points",
"along",
"the",
"horizon,",
"define",
"by",
"eq.(2.9)",
"so",
"we",
"have",
"while",
"lb",
"=",
"(2/OU)b",
"is",
"transverse",
"to",
"it.",
"D",
"is",
"the",
"proper",
"distance",
"A",
"_",
"from",
"Alice’s",
"lab",
"to",
"the",
"horizon.",
"E",
"A",
"=",
"A",
".",
"(3.14)",
"0",
"v",
"tal",
"Latin",
"indices",
"from",
"the",
"early",
"alphabet",
"denote",
"spatial",
"It",
"then",
"follows",
"immediately",
"from",
"eq.",
"(3.12)",
"that",
"the",
"dif-",
"components",
"in",
"the",
"x",
"and",
"y",
"directions.",
"Since",
"E",
"A",
"=",
"0",
"ference,",
"aa",
"A",
",",
"between",
"the",
"fina",
"and",
"initia",
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"la",
"is",
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"by",
"on",
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"horizon",
"for",
"a",
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"charge,",
"one",
"D",
"A(A",
"A",
"lA",
"=",
"_",
"may",
"say",
"that",
"a",
"charge",
"held",
"stationary",
"in",
"Alice’s",
"lab",
"does",
"ae",
"U",
"(3.15)",
"not",
"produce",
"any",
"radiation",
"as",
"determined",
"on",
"H",
"—even",
"R",
"though",
"a",
"uniformly",
"accelerated",
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"radiates",
"(inertial)",
"independently",
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"the",
"manner",
"in",
"which",
"the",
"charge",
"is",
"moved",
"energy",
"to",
"future",
"null",
"infinity14",
"from",
"D",
"to",
"D′.",
"Equation",
"(3.15)",
"is",
"an",
"exact",
"mathemati-",
"%",
"Now",
"consider",
"the",
"case",
"where",
"the",
"point",
"charge",
"is",
"initially",
"cal",
"analog",
"of",
"the",
"electromagnetic",
"memory",
"effec",
"at",
"null",
"uniformly",
"accelerating",
"with",
"acceleration",
"a",
"at",
"a",
"proper",
"infinit",
"[31]",
"For",
"the",
"explici",
"solutio",
"eq.",
"(3.10)",
"we",
"have",
"distance",
"D",
"=",
"1/a",
"from",
"the",
"bifurcation",
"surface",
"of",
"the",
"~",
"-",
"4qda3(1",
"-",
"a2ρ2)",
".",
"π(1",
"+",
"a2ρ2)3",
"Rindler",
"horizon.",
"The",
"charge",
"is",
"then",
"moved",
"in",
"the",
"z-",
"(3.16)",
"∆EU",
"U",
"direction15",
"to",
"a",
"differen",
"orbit",
"of",
"the",
"sam",
"boost",
"Killing",
"where",
"d",
"=",
"D′",
"-",
"D",
"and",
"we",
"have",
"assumed",
"that",
"field",
"so",
"tha",
"it",
"has",
"unifor",
"acceleratio",
"a",
"and",
"lies",
"at",
")",
"proper",
"distance",
"D′",
"=",
"1/a′",
"from",
"the",
"Rindler",
"horizon.",
"After",
"fo",
"b",
"the",
"charge",
"has",
"reached",
"its",
"new",
"location,",
"the",
"electric",
"fiel",
"d",
"k",
"D",
"=",
"1",
".",
"(3.17)",
"a",
"A",
"F",
"ub.",
"Although",
"nb",
"is",
"null",
"rather",
"than",
"timelike,",
"it",
"is",
"natural",
"(and",
"From",
"eq.",
"(3.15),",
"we",
"fin",
"that",
"�",
"A",
"point",
"in",
"the",
"ˆρ",
"standard)",
"to",
"use",
"the",
"terminology",
"“electric",
"field",
"for",
"F",
"at",
"nb",
"on",
"the",
"direction",
"and",
"has",
"magnitude",
"horizon.",
"qda3ρ",
"|",
"A",
"I",
"=",
"�",
"A",
"~",
"A",
"14A",
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"has",
"a",
"nonvanishing",
"inertial",
"energy",
"a",
"(1",
"+",
"a2ρ2)2",
".",
"(3.18)",
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"flu",
"Tabt",
"throug",
"both",
"set",
"and",
"gt,",
"where",
"ta",
"denotes",
"a",
"Minkowski",
"time",
"translation.",
"However,",
"the",
"flu",
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"“boos",
"energy",
"I",
"abb",
"vanishes",
"at",
"both",
"H",
"and",
"gt.",
"15We",
"consider",
"a",
"z-displacement",
"for",
"simplicity.",
"Similar",
"results",
"would",
"The",
"key",
"point",
"is",
"that",
"even",
"though",
"E",
"A",
"=",
"0",
"at",
"both",
"late",
"and",
"early",
"times,",
"A",
"A",
"does",
"not",
"return",
"to",
"its",
"original",
"value",
"at",
"hold",
"if",
"the",
"charge",
"were",
"displaced",
"in",
"the",
"x",
"or",
"y",
"directions."
]
} | |||
2301.00026v2.pdf_8 | null | ArxivOCR_part_1 | {
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958,
1951
],
[
977,
1919,
1005,
1951
],
[
1005,
1918,
1034,
1952
],
[
1062,
1919,
1091,
1951
],
[
1091,
1911,
1123,
1952
],
[
1123,
1919,
1159,
1952
],
[
1159,
1919,
1238,
1952
],
[
1238,
1919,
1293,
1952
],
[
1293,
1919,
1341,
1951
],
[
1341,
1919,
1418,
1951
],
[
1418,
1919,
1474,
1952
],
[
1474,
1919,
1514,
1952
],
[
1514,
1919,
1565,
1952
],
[
143,
1946,
206,
1978
],
[
206,
1946,
262,
1978
],
[
262,
1946,
302,
1978
],
[
302,
1945,
358,
1978
],
[
358,
1945,
470,
1978
],
[
470,
1945,
497,
1978
],
[
497,
1946,
572,
1978
],
[
572,
1946,
694,
1979
],
[
694,
1946,
792,
1978
],
[
792,
1946,
833,
1978
],
[
870,
1946,
959,
1978
],
[
959,
1945,
1019,
1978
],
[
1019,
1945,
1050,
1978
],
[
1050,
1946,
1089,
1978
],
[
1089,
1946,
1144,
1978
],
[
1144,
1945,
1172,
1978
],
[
1172,
1946,
1198,
1978
],
[
1213,
1946,
1279,
1978
],
[
1279,
1946,
1302,
1978
],
[
1302,
1946,
1325,
1978
],
[
1325,
1946,
1369,
1978
],
[
1369,
1946,
1409,
1978
],
[
1409,
1946,
1489,
1978
],
[
1489,
1946,
1564,
1978
],
[
1187,
1932,
1207,
1972
],
[
143,
1973,
289,
2004
],
[
289,
1972,
316,
2004
],
[
316,
1972,
357,
2004
],
[
357,
1972,
418,
2004
],
[
418,
1972,
467,
2004
],
[
467,
1972,
521,
2004
],
[
521,
1972,
554,
2004
],
[
554,
1972,
648,
2004
],
[
873,
1973,
929,
2004
],
[
929,
1972,
979,
2004
],
[
979,
1972,
1039,
2004
],
[
1039,
1972,
1101,
2004
],
[
1101,
1972,
1142,
2004
],
[
1142,
1972,
1227,
2004
],
[
1227,
1972,
1256,
2004
],
[
1256,
1972,
1282,
2004
],
[
1283,
1974,
1299,
2004
]
]
} | {
"word": [
"8",
"A",
"late",
"times,",
"and",
"the",
"change,",
"�",
"A",
"in",
"the",
"vector",
"potential",
"this",
"is",
"that",
"in",
"d",
"dimensions,",
"radiation",
"falls",
"of",
"at",
"infini",
"between",
"late",
"and",
"early",
"times",
"is",
"determined",
"only",
"by",
"the",
"in",
"null",
"directions",
"as",
"1/rd/2−1,",
"2is",
"whereas",
"Coulomb",
"field",
"and",
"associated",
"memory",
"effect",
"fall",
"off",
"as",
"1/rd",
"-3,",
"so",
"it",
"is",
"only",
"in",
"o",
"initial",
"and",
"fina",
"position",
"of",
"the",
"charge",
"We",
"now",
"consider",
"the",
"quantized",
"radiation",
"through",
"the",
"d",
"=",
"4",
"dimensions",
"that",
"memory",
"effect",
"occu",
"at",
"radiativ",
"horizon",
"resulting",
"from",
"the",
"displacement",
"of",
"the",
"charge,",
"order",
"[36,",
"40].",
"By",
"contrast,",
"radial",
"Coulomb",
"field",
"will",
"assuming",
"that,",
"after",
"the",
"displacement,",
"the",
"charge",
"is",
"held",
"penetrate",
"a",
"Killing",
"horizon",
"in",
"all",
"spacetime",
"dimensions",
"at",
"its",
"new",
"position,",
"D′,",
",",
"forever.",
"For",
"the",
"Fock",
"space",
"(see",
"[41]",
"for",
"the",
"case",
"of",
"a",
"Schwarzschild",
"black",
"hole)",
"and",
"a",
"with",
"the",
"Minkowski",
"vacuum",
"state,",
"the",
"map",
"K",
"displacement",
"of",
"a",
"charge",
"will",
"result",
"in",
"change",
"in",
"the",
"ra-",
"So,",
"H",
"S",
"that",
"associates",
"one-particle",
"states",
"to",
"classical",
"dial",
"Coulomb",
"fiel",
"in",
"all",
"dimensions",
"As",
"analyze",
"above",
"solutions",
"is",
"given",
"by",
"taking",
"the",
"positive",
"frequency",
"part",
"of",
"this",
"will",
"result",
"in",
"radiation",
"through",
"the",
"horizon",
"in",
"all",
"the",
"classical",
"solution",
"with",
"respect",
"to",
"inertial",
"time,",
"with",
"dimensions",
"high",
"no",
"for",
"the",
"fiel",
"in",
"questio",
"to",
"admi",
"the",
"inner",
"product",
"on",
"lin",
"given",
"by",
"the",
"Klein-Gordon",
"radiation",
"(i.e.,",
"d",
"3",
"for",
"electromagnetism",
"and",
"d",
"4",
"product.",
"For",
"the",
"electromagnetic",
"fiel",
"on",
"H",
"R",
"in",
"a",
"gauge",
"for",
"gravity).",
"Consequently,",
"the",
"logarithmic",
"divergence",
"in",
"i",
"ina",
"=",
"0",
"on",
"H",
"+,",
"the",
"“free",
"data”",
"on",
"H",
"o+",
"is",
"the",
"Eq.",
"(3.20)",
"occurs",
"in",
"all",
"spacetime",
"dimensions",
"that",
"admit",
"R",
"where",
"ential.",
"For",
"two",
"classical",
"radiation17.",
"pull-back,",
"A",
"12,A",
"on",
"Aert,",
"the",
"inner",
"product",
"of",
"their",
"corresponding",
"one-particle",
"states",
"is",
"given",
"distance",
"d",
"PR",
"D",
"=",
"1/a",
"from",
"D",
"to",
"D′",
"=",
"D+d",
"as",
"above,",
"but",
"Now",
"suppose",
"that",
"Alice",
"displaces",
"the",
"particle",
"a",
"z-",
"instead",
"of",
"leaving",
"the",
"particle",
"at",
"D′",
"forever,",
"she",
"leaves",
"it",
"),",
"solut",
"by",
"[15,",
"32]",
"there",
"for",
"proper",
"time18",
"T",
"and",
"then",
"returns",
"it",
"to",
"D.",
"In",
"this",
"x",
"case,",
"the",
"transverse",
"components",
"of",
"the",
"vector",
"potential,",
"(2",
"fert",
"=",
"2",
"dxdy",
"ωdω",
"LSAB Â",
"1,A",
"ˆA2,B",
"2π",
"Â",
"A",
"I",
"A",
"K",
"K",
"lA",
"return",
"to",
"their",
"initial",
"values",
"at",
"late",
"times,",
"so",
"there",
"1",
"is",
"no",
"“memory",
"effect",
"at",
"the",
"horizon",
"Correspondingly",
"R2",
"o",
"there",
"are",
"no",
"infrared",
"divergences",
"in",
"the",
"expected",
"number",
"(3.19)",
"Â",
"(w,",
"xB)",
"is",
"the",
"Fourier",
"transform",
"of",
"A",
"A",
"of",
"photons",
"that",
"propagate",
"through",
"H",
"+.",
"Nevertheless,",
"if",
"R",
"A",
"(V,",
"xB)",
"where",
"with",
"respect",
"to",
"the",
"affi",
"paramet",
"V",
"By",
"the",
"same",
"T",
"is",
"very",
"large",
"then",
"the",
"expected",
"number",
"of",
"photons",
"⟨N",
"reasoning",
"as",
"led",
"to",
"eq.",
"(2.15),",
"the",
"expected",
"number",
"of",
"will",
"be",
"correspondingly",
"large.",
"To",
"see",
"this,",
"we",
"note",
"that",
"photons",
"on",
"H",
"in",
"the",
"coherent",
"state",
"associated",
"to",
"any",
"if,",
"for",
"convenience,",
"we",
"work",
"in",
"a",
"gauge",
"where",
"lA",
"=",
"0",
"initially,",
"then",
"during",
"the",
"era",
"at",
"which",
"the",
"particle",
"is",
"at",
"D",
"1,",
"classical",
"solution",
"lA",
"is",
"simply",
"lA",
"will",
"be",
"given",
"by",
"the",
"right",
"side",
"of",
"eq.",
"(3.18).",
"If",
"we",
"keep",
"A",
"ll",
"fr",
"⟨N",
",",
"2",
"the",
"manner",
"in",
"which",
"the",
"particle",
"is",
"moved",
"from",
"D",
"to",
"D′",
"|K",
"(3.20)",
"=",
"as",
"well",
"as",
"from",
"D′",
"to",
"D",
"fixe",
"but",
"take",
"T",
"to",
"be",
"very",
"large",
"the",
"asymptotic",
"behavior",
"of",
"the",
"norm",
"eq.",
"(3.19)",
"will",
"be",
"where",
"the",
"norm",
"is",
"define",
"by",
"the",
"inne",
"produc",
"eq.",
"(3.19)",
"0,",
"the",
"Fourier",
"transform,",
"dominated",
"by",
"the",
"low-frequency",
"contribution",
"from",
"the",
"era",
"However,",
"since",
"�",
"lA",
"ˆAA(ω,",
"xB),",
"of",
"eA",
"A",
"̸=",
"À",
"eI,",
"A",
"diverges",
"as",
"1/ω",
"as",
"w",
"0.",
"It",
"fol-",
"of",
"time",
"T",
"that",
"the",
"particle",
"is",
"displaced.",
"The",
"logarithmic",
"lows",
"that",
"the",
"integrand",
"of",
"the",
"expression",
"for",
"the",
"norm",
"divergence",
"at",
"w",
"=",
"0",
"that",
"would",
"occur",
"if",
"the",
"particle",
"re-",
"at",
"D′",
"forever",
"is",
"now",
"effectivel",
"cut",
"off",
"at",
"frequen",
"given",
"by",
"the",
"right",
"side",
"of",
"eq.",
"(3.19)",
"also",
"diverges",
"as",
"1/ω",
"as",
"al",
"so",
"the",
"integral",
"is",
"logarithmically",
"divergent.",
"Thus,",
"1/V",
"where",
"V",
"denotes",
"the",
"affi",
"time",
"duration",
"on",
"All",
"Hx",
"=",
"w",
"differen",
"orbit",
"of",
"the",
"boost",
"Killing",
"field",
"and",
"particle",
"to",
"a",
"w",
"1",
"Therefore,",
"if",
"Alice",
"displaces",
"a",
"charged",
"the",
"horizon",
"H",
"+",
"over",
"which",
"the particle",
"remains",
"at",
"D",
".",
"K",
"We",
"obtain",
"trajectory",
"forever,",
"an",
"infinit",
"numbe",
"of",
"\"soft",
"horizo",
"(",
"N",
">",
"=",
"|K",
"AIl",
"~",
"q2d2a2",
"ln",
"the",
"particle",
"remains",
"on",
"this",
"new",
"uniformly",
"accelerated",
"V",
"(3.21)",
"HR",
"min[V1,",
"v",
"2",
"]",
"photons”",
"will",
"be",
"radiated",
"through",
"the",
"Rindler",
"horizon16",
"<",
"regardless",
"of",
"how",
"quickly",
"or",
"slowly",
"this",
"process",
"is",
"done.",
"where",
"V",
"v",
"2",
"V",
"are",
"the",
"durations",
"of",
"affi",
"time",
"over",
"The",
"above",
"infrared",
"divergence",
"is",
"an",
"exact",
"mathematical",
"which",
"the",
"particle",
"is",
"displaced",
"from",
"D",
"to",
"D′",
"and",
"from",
"analog",
"of",
"the",
"infrared",
"divergences",
"that",
"occur",
"at",
"null",
"infin",
"D′",
"back",
"to",
"D,",
"so",
"that",
"1/min[V1,",
"V",
"2",
"]",
"provides",
"an",
"effectiv",
"n",
"ity",
"in",
"QED",
"for",
"processes",
"with",
"nonzero",
"memory",
"(see",
"e.g.,",
"high-frequency",
"cutoff",
"However",
"the",
"affine",
"time",
"V",
"on",
"the",
"[35–39]).",
"Note",
"that",
"infrared",
"divergences",
"at",
"null",
"infinit",
"horizon",
"is",
"related",
"to",
"boost",
"Killing",
"time",
"on",
"the",
"horizon",
"by",
"arise",
"only",
"in",
"d",
"=",
"4",
"spacetime",
"dimensions.",
"The",
"reason",
"for",
"V",
"=",
"V",
"exp(av)",
"(3.22)",
"16These",
"“soft",
"horizon",
"photons”",
"are",
"closely",
"related",
"to",
"the",
"“soft",
"hair”",
"17Indeed,",
"there",
"would",
"also",
"be",
"infrared",
"divergences",
"for",
"a",
"particle",
"that",
"discussed",
"by",
"Hawking,",
"Perry,",
"and",
"Strominger",
"[33]",
"in",
"the",
"case",
"of",
"sources",
"massive",
"field",
"since",
"the",
"Yukaw",
"field",
"of",
"the",
"particle",
"will",
"black",
"hole",
"horizons",
"(see",
"also",
"[34]).",
"However,",
"while",
"Hawking,",
"Perry,",
"also",
"penetrate",
"the",
"horizon",
"and",
"Strominger",
"considered",
"effect",
"of",
"matter",
"fallin",
"int",
"black",
"hole,",
"18We",
"have",
"normalized",
"the",
"boost",
"Killing",
"fiel",
"ba",
"so",
"that",
"Killing",
"time",
"our",
"“soft",
"radiation”",
"arises",
"from",
"the",
"displacement",
"of",
"matter",
"sourcing",
"equals",
"proper",
"time",
"on",
"the",
"orbit",
"at",
"D",
"with",
"acceleration",
"a.",
"Since",
"we",
"a",
"long",
"range",
"fiel",
"outsid",
"of",
"horizon",
"Note",
"that",
"in",
"the",
"case",
"of",
"a",
"assume",
"=",
"D′",
"D",
"�",
"D,",
"Killing",
"time",
"and",
"proper",
"time",
"are",
"also",
"black",
"hole,",
"the",
"“soft",
"radiation”",
"of",
"Alice’s",
"experiment",
"increases",
"the",
"(nearly)",
"equal",
"on",
"the",
"orbit",
"at",
"D",
"Thus,",
"T",
"is",
"also",
"the",
"elapsed",
"Killing",
"I",
"entanglement",
"of",
"the",
"black",
"hole",
"with",
"its",
"exterior.",
"time",
"that",
"Alice",
"keeps",
"the",
"particle",
"at",
"D",
"."
]
} | |||
2301.00026v2.pdf_9 | null | ArxivOCR_part_1 | {
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"E",
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"E",
"a",
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"For",
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",",
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"Et",
"Cuu",
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"T",
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"+",
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"-",
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"I",
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",",
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"(3.19)",
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"I",
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"(3.26)",
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"Kh",
"2",
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"8",
"R",
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"ℏ,",
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"E",
"AB",
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"h",
"AB",
"(Rindler,",
"EM)",
"(3.27)",
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"D",
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"2",
"3",
"2",
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"~",
"1033",
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"m",
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"_1",
".",
"(3.28)",
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"CAB",
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"h",
"AB",
"(3.32)",
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"q",
"a",
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"uniformly",
"accelerates",
"at",
"one",
"g",
"in",
"Equations",
"(3.30)",
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"(3.32)",
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"imply",
"that",
"a",
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"fla",
"spacetim",
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"uu",
"*",
"0,",
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"one",
"meter",
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"the",
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"+",
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"̸=",
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"H",
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"R",
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]
} | |||
2301.00026v2.pdf_10 | null | ArxivOCR_part_1 | {
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1657
],
[
1237,
1620,
1314,
1657
],
[
1322,
1627,
1354,
1660
],
[
1370,
1620,
1422,
1657
],
[
1435,
1621,
1450,
1658
],
[
1450,
1631,
1467,
1660
],
[
1477,
1620,
1565,
1658
],
[
142,
1670,
224,
1707
],
[
224,
1670,
252,
1707
],
[
252,
1670,
275,
1707
],
[
275,
1670,
379,
1707
],
[
379,
1669,
446,
1708
],
[
446,
1669,
480,
1707
],
[
480,
1669,
528,
1707
],
[
528,
1669,
614,
1707
],
[
614,
1670,
773,
1708
],
[
873,
1659,
943,
1696
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[
943,
1659,
1001,
1696
],
[
1001,
1659,
1037,
1696
],
[
1037,
1659,
1082,
1696
],
[
1082,
1660,
1161,
1696
],
[
1161,
1659,
1189,
1696
],
[
1189,
1659,
1235,
1696
],
[
1235,
1659,
1370,
1696
],
[
1370,
1660,
1478,
1696
],
[
1489,
1642,
1512,
1683
],
[
1512,
1654,
1523,
1683
],
[
1523,
1639,
1546,
1683
],
[
1534,
1651,
1569,
1697
],
[
871,
1690,
926,
1727
],
[
435,
1717,
457,
1773
],
[
506,
1719,
529,
1755
],
[
464,
1736,
497,
1773
],
[
756,
1736,
836,
1773
],
[
1162,
1736,
1195,
1794
],
[
1205,
1737,
1227,
1781
],
[
1227,
1750,
1252,
1782
],
[
1253,
1747,
1275,
1781
],
[
1285,
1759,
1309,
1796
],
[
1108,
1738,
1120,
1794
],
[
497,
1755,
537,
1792
],
[
1120,
1757,
1136,
1794
],
[
1136,
1766,
1153,
1795
],
[
1487,
1757,
1567,
1794
],
[
954,
1801,
985,
1858
],
[
871,
1820,
954,
1858
],
[
987,
1821,
1090,
1858
],
[
1091,
1820,
1140,
1858
],
[
1140,
1821,
1237,
1858
],
[
1237,
1821,
1386,
1858
],
[
1387,
1821,
1509,
1858
],
[
1515,
1821,
1568,
1858
],
[
121,
1864,
169,
1899
],
[
169,
1867,
240,
1899
],
[
240,
1867,
269,
1899
],
[
269,
1867,
392,
1899
],
[
392,
1867,
440,
1899
],
[
440,
1867,
478,
1899
],
[
478,
1868,
578,
1899
],
[
578,
1868,
638,
1899
],
[
638,
1867,
664,
1899
],
[
664,
1867,
706,
1899
],
[
706,
1867,
754,
1899
],
[
754,
1867,
834,
1899
],
[
872,
1852,
982,
1891
],
[
982,
1852,
1030,
1891
],
[
1030,
1852,
1188,
1891
],
[
1188,
1852,
1230,
1890
],
[
1230,
1852,
1317,
1890
],
[
1317,
1852,
1366,
1890
],
[
1366,
1852,
1484,
1890
],
[
1484,
1852,
1564,
1890
],
[
143,
1893,
248,
1925
],
[
248,
1893,
292,
1926
],
[
292,
1893,
358,
1925
],
[
358,
1893,
449,
1925
],
[
449,
1893,
498,
1925
],
[
498,
1893,
571,
1925
],
[
571,
1893,
608,
1926
],
[
608,
1893,
631,
1925
],
[
631,
1893,
745,
1925
],
[
745,
1893,
778,
1925
],
[
778,
1893,
833,
1925
],
[
873,
1885,
986,
1922
],
[
986,
1884,
1017,
1923
],
[
1017,
1884,
1065,
1922
],
[
1065,
1885,
1220,
1923
],
[
1220,
1884,
1335,
1922
],
[
1335,
1884,
1401,
1922
],
[
1401,
1884,
1424,
1922
],
[
1424,
1885,
1502,
1922
],
[
1502,
1885,
1569,
1922
],
[
411,
1919,
454,
1951
],
[
477,
1919,
589,
1951
],
[
589,
1919,
647,
1951
],
[
647,
1919,
677,
1951
],
[
677,
1919,
719,
1951
],
[
719,
1919,
782,
1952
],
[
782,
1919,
833,
1952
],
[
873,
1915,
954,
1954
],
[
954,
1915,
1027,
1954
],
[
1027,
1916,
1146,
1953
],
[
1168,
1915,
1263,
1954
],
[
1263,
1916,
1294,
1953
],
[
1294,
1916,
1437,
1954
],
[
1437,
1916,
1461,
1954
],
[
143,
1920,
186,
1952
],
[
186,
1919,
249,
1951
],
[
249,
1919,
307,
1951
],
[
308,
1919,
337,
1951
],
[
364,
1927,
395,
1979
],
[
395,
1946,
426,
1978
],
[
426,
1946,
483,
1978
],
[
483,
1945,
524,
1978
],
[
524,
1946,
625,
1978
],
[
625,
1946,
679,
1978
],
[
679,
1946,
720,
1978
],
[
720,
1946,
783,
1979
],
[
783,
1946,
833,
1978
],
[
140,
1946,
195,
1978
],
[
195,
1946,
225,
1978
],
[
225,
1946,
245,
1978
],
[
245,
1946,
337,
1978
],
[
337,
1946,
364,
1978
],
[
1111,
1955,
1140,
2013
],
[
1230,
1956,
1247,
2013
],
[
1194,
1957,
1202,
2014
],
[
1257,
1957,
1287,
2013
],
[
1287,
1970,
1348,
2013
],
[
143,
1972,
233,
2004
],
[
233,
1972,
301,
2004
],
[
301,
1972,
376,
2004
],
[
376,
1972,
405,
2004
],
[
405,
1972,
445,
2004
],
[
445,
1972,
529,
2004
],
[
529,
1972,
579,
2004
],
[
579,
1972,
676,
2004
],
[
676,
1972,
734,
2004
],
[
1088,
1975,
1111,
2013
],
[
1486,
1976,
1567,
2014
],
[
1151,
1977,
1182,
2015
],
[
1182,
1987,
1194,
2016
],
[
1202,
1977,
1218,
2014
],
[
1218,
1988,
1230,
2015
]
]
} | {
"word": [
"10",
"before,",
"the",
"number",
"of",
"entangling",
"gravitons",
"emitted",
"to",
"relative",
"to",
"the",
"notion",
"of",
"time",
"translations",
"define",
"by",
"the",
"Rindler",
"horizon",
"is",
"logarithmically",
"growing",
"in",
"affi",
"the",
"Lorentz",
"boost",
"Killing",
"fiel",
"ba,",
"eq.",
"(3.2)",
"Thus,",
"the",
"time",
"and",
"therefore",
"linearly",
"growing",
"in",
"the",
"proper",
"time",
"superposition",
"state",
"of",
"Alice’s",
"particle",
"will",
"be",
"buffete",
"by",
"duration",
"T",
"of",
"Alice’s",
"experiment.",
"We",
"obtain",
"this",
"thermal",
"bath",
"of",
"Unruh",
"radiation.",
"Scattering",
"of",
"this",
"radiation",
"will",
"cause",
"some",
"decoherence",
"of",
"Alice’s",
"particle.",
"{",
">",
"~",
"m2d4a5T",
"N",
"(Rindler,",
"GR)",
"(3.33)",
"Indeed,",
"since",
"this",
"decoherence",
"should",
"occur",
"at",
"a",
"steady",
"rate",
"while",
"the",
"superposition",
"is",
"kept",
"stationary",
"(and",
"thus",
"Thus,",
"restoring",
"constants,",
"we",
"fin",
"that",
"the",
"Rindle",
"hori-",
"the",
"decoherence",
"will",
"be",
"proportional",
"to",
"T),",
"one",
"might",
"even",
"zon",
"decoheres",
"the",
"quantum",
"superposition",
"of",
"a",
"uniformly",
"suspect",
"that",
"scattering",
"of",
"Unruh",
"radiation",
"could",
"be",
"the",
"accelerating",
"massive",
"body",
"in",
"a",
"time",
"same",
"effec",
"as",
"foun",
"in",
"the",
"previou",
"sectio",
"but",
"expresse",
"in",
"a",
"differen",
"language",
"The",
"purpos",
"of",
"this",
"subsectio",
"(",
"ℏc10",
"TGR",
"~",
"Gm2d4a5",
"is",
"to",
"show",
"that",
"this",
"is",
"not",
"the",
"case,",
"i.e.,",
"decoherence",
"(Rindler,",
"GR)",
"(3.34)",
"due",
"to",
"scattering",
"of",
"Unruh",
"radiation",
"and",
"decoherence",
"~",
"2",
"fs",
"R",
"Moon",
"due",
"to",
"radiation",
"of",
"“soft”",
"photons/gravitons",
"through",
"the",
"M",
"Moon",
"g",
".",
"(3.35)",
"horizon",
"are",
"distinct",
"effects",
"Furthermore",
"we",
"shall",
"sho",
"d",
"m",
"that,",
"for",
"reasonable",
"parameter",
"choices,",
"the",
"decoherence",
"Therefore,",
"if",
"the",
"Moon",
"were",
"accelerating",
"at",
"one",
"g",
"and",
"rate",
"due",
"to",
"the",
"scattering",
"of",
"Unruh",
"radiation",
"is",
"smaller",
"occupied",
"a",
"quantum",
"state",
"with",
"its",
"center",
"of",
"mass",
"super-",
"than",
"the",
"decoherence",
"rate",
"due",
"to",
"emitted",
"radiation",
"as",
"posed",
"by",
"a",
"spatial",
"separation",
"of",
"the",
"order",
"of",
"its",
"own",
"radius",
"obtained",
"in",
"the",
"previous",
"section.",
"We",
"will",
"consider",
"only",
"then",
"it",
"would",
"decohere",
"within",
"about",
"2",
"femtoseconds.",
"Of",
"the",
"electromagnetic",
"case",
"in",
"this",
"subsection.",
"course,",
"it",
"would",
"not",
"be",
"easy",
"to",
"put",
"the",
"moon",
"in",
"such",
"a",
"The",
"decoherence",
"rate",
"of",
"a",
"spatial",
"superposition",
"due",
"coherent",
"quantum",
"superposition.",
"Note",
"the",
"acceleration",
"of",
"a",
"stationary",
"observer",
"outside",
"to",
"collisions",
"with",
"particles",
"in",
"an",
"environment",
"has",
"been",
"analyzed",
"in",
"[42–45],",
"and",
"we",
"will",
"adapt",
"this",
"analysis",
"to",
"of",
"black",
"hole",
"who",
"is",
"reasonably",
"far21",
"(D",
"M)",
"from",
"the",
"event",
"horizon",
"is",
"a",
"M/D2.",
"If",
"we",
"substitute",
"a",
"=",
"M/D2",
"obtain",
"a",
"rough",
"estimate",
"of",
"the",
"decoherence",
"caused",
"by",
"the",
"in",
"eqs.",
"(3.27)",
"and",
"(3.34),",
"we",
"obtain",
"eqs.",
"(2.21)",
"and",
"(2.22),",
"scattering",
"of",
"Unruh",
"radiation.",
"As",
"in",
"eq.",
"(2.1),",
"Alice",
"has",
"respectively.",
"Therefore,",
"it",
"might",
"be",
"tempting",
"to",
"believe",
"a",
"particle",
"of",
"charge",
"q",
"in",
"a",
"state",
"|ψ⟩",
"=",
"(12",
"+",
"%",
"2))/",
"2,",
",",
"1",
"that",
"what",
"is",
"important",
"in",
"all",
"cases",
"is",
"the",
"acceleration",
"of",
"where",
"4",
"1",
"and",
"4f",
"2",
"are",
"spatially",
"separated",
"by",
"a",
"distance",
"Alice’s",
"lab.",
"However,",
"this",
"is",
"not",
"the",
"case.",
"In",
"particular,",
"d.",
"Since",
"we",
"require",
"d",
"e",
"1/a",
"(see",
"eq.",
"(3.17))",
"and",
"since",
"the",
"typical",
"wavelength",
"of",
"Unruh",
"photons",
"at",
"temperature",
"ele",
"1/a,",
"we",
"are",
"in",
"the",
"scattering",
"regime",
"if",
"we",
"replace",
"the",
"black",
"hole",
"by",
"an",
"ordinary",
"star",
"(and",
"if",
"there",
"are",
"no",
"dissipative",
"effect",
"in",
"the",
"star)",
"then",
"ther",
"eq.",
"(3.36)",
"is",
"X",
">",
"will",
"not",
"be",
"any",
"analogous",
"decoherence",
"effect",
"eve",
"thoug",
"where",
"x",
"In",
"an",
"elastic",
"scattering",
"event",
"between",
"d.",
"the",
"acceleration",
"of",
"Alice’s",
"lab",
"is",
"the",
"same",
"as",
"in",
"the",
"case",
"Alice’s",
"particle",
"and",
"a",
"photon",
"in",
"the",
"Unruh",
"radiation,",
"the",
"of",
"a",
"black",
"hole.",
"Furthermore,",
"as",
"we",
"shall",
"see",
"in",
"sec.",
"4,",
"fina",
"outgoin",
"stat",
"of",
"the",
"photo",
"will",
"depen",
"upon",
"which",
"decoherence",
"effect",
"associate",
"with",
"the",
"cosmologica",
"hori-",
"branch",
"of",
"the",
"superposition",
"the",
"photon",
"scattered",
"of",
"of.",
"zon",
"occur",
"in",
"de",
"Sitter",
"spacetime",
"even",
"for",
"nonaccelerating",
"Let",
"|χ1⟩",
"denote",
"the",
"outgoing",
"state",
"of",
"the",
"Unruh",
"photon",
"observers.",
"It",
"is",
"the",
"presence",
"of",
"a",
"Killing",
"horizon",
"that",
"for",
"scattering",
"of",
"of",
"I",
"&",
"and",
"let",
"i",
"X",
"denot",
"the",
"outgoin",
"is",
"the",
"essential",
"ingredient",
"for",
"the",
"fundamental",
"rate",
"of",
"state",
"for",
"scattering",
"of",
"of",
"I",
"%",
"Decoherenc",
"will",
"occur",
"to",
"decoherence",
"of",
"quantum",
"superpositions",
"as",
"described",
"in",
"the",
"extent",
"to",
"which",
"these",
"outgoing",
"states",
"of",
"1",
"scattered",
"Unruh",
"photon",
"are",
"distinguishable,",
"i.e.,",
"D",
"=",
"(x",
"1",
"x",
"2",
"�",
"|.",
"1I",
"ne",
"this",
"paper.",
"We",
"now",
"consider",
"another",
"potential",
"cause",
"of",
"decoherence,",
"In",
"order",
"to",
"obtain",
"a",
"rough",
"estimate",
"of",
"the",
"decoherence",
"namely",
"Unruh",
"radiation.",
"resulting",
"from",
"a",
"single",
"scattering",
"event,",
"we",
"consider",
"the",
"corresponding",
"Minkowski",
"process",
"of",
"the",
"scattering",
"of",
"a",
"2.",
"Decoherence",
"Due",
"to",
"Scattering",
"of",
"Unruh",
"photon",
"of",
"momentum",
"p",
"of",
"of",
"an",
"inertia",
"superpositio",
"separated",
"by",
"d,",
"with",
"d",
"1/p.",
"Assuming",
"that",
"the",
"charged",
"i",
"Radiation",
"particle",
"states",
"4",
"1",
"and",
"14",
"2",
",",
"are",
"identical",
"except",
"for",
"their",
",",
"The",
"Minkowski",
"vacuum",
"state",
"restricted",
"to",
"a",
"Rindler",
"location,",
"the",
"scattered",
"photon",
"states",
"|x1",
"and",
"x",
"2",
"should",
"wedge",
"is",
"a",
"thermal",
"state",
"at",
"the",
"Unruh",
"temperature",
"diffe",
"only",
"by",
"the",
"actio",
"of",
"the",
"translatio",
"operato",
"9ii",
"i",
"P.",
".d,",
"i.e.,",
"7",
"a",
"=",
"(3.36)",
"~",
"-",
"ip",
".d",
"X",
"I",
"2π",
"x",
"2",
"(3.37)",
"p",
"where",
"denotes",
"the",
"photon",
"momentum",
"operator.",
"Ex-",
"21It",
"should",
"be",
"emphasized",
"that",
"the",
"estimates",
"made",
"in",
"[14]",
"that",
"yielded",
"panding",
"the",
"exponential,",
"we",
"obtain",
"the",
"following",
"rough",
"eqs.(2.21)",
"and",
"(2.22)",
"assumed",
"that",
"Alice’s",
"lab",
"is",
"reasonably",
"far",
"from",
"estimate",
"of",
"the",
"decoherence",
"resulting",
"from",
"a",
"single",
"scat-",
"lab",
"extremely",
"close",
"to",
"the",
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"P.O.",
"Box",
"800,",
"9700",
"AV",
"Groningen,",
"The",
"Netherlands",
"4SRON",
"Netherlands",
"Institute",
"for",
"Space",
"Research,",
"Postbus",
"800,",
"9700",
"AV",
"Groningen,",
"The",
"Netherlands",
"CL",
"5Department",
"of",
"Astronomy,",
"The",
"University",
"of",
"Texas",
"at",
"Austin,",
"Austin,",
"TX,",
"USA",
"6Department",
"of",
"Physics,",
"University",
"of",
"the",
"Pacific,",
"Stockton,",
"CA",
"90340",
"USA",
"7Aix",
"Marseille",
"Univ,",
"CNRS,",
"CNES,",
"LAM",
"Marseille,",
"France",
"d",
"8Centro",
"de",
"Astrobiología",
"(CAB),",
"CSIC-INTA,",
"Ctra.",
"de",
"Ajalvir",
"km",
"4,",
"Torrejón",
"de",
"Ardoz,",
"E-28850,",
"Madrid,",
"Spain",
"9INAF",
"Osservatorio",
"Astronomico",
"di",
"Roma,",
"via",
"di",
"Frascati",
"33,",
"00078",
"Monte",
"Porzio",
"Catone,",
"Italy",
"10NSF’s",
"s",
"National",
"Optical-Infrared",
"Astronomy",
"Research",
"Laboratory,",
"950",
"N.",
"Cherry",
"Ave.,",
"Tucson,",
"AZ",
"85719,",
"USA",
"Lanh",
"11Department",
"of",
"Astronomy,",
"University",
"of",
"Michigan,",
"1085",
"s.",
"University",
"Ave,",
"Ann",
"Arbor,",
"MI",
"48109-1107,",
"USA",
"12Dipartimento",
"di",
"Fisica",
"e",
"Astronomia",
"\"G.Galilei\",",
"Universitá",
"di",
"Padova,",
"Via",
"Marzolo",
"8,",
"I-35131",
"Padova,",
"Italy",
"13INAF–Osservatorio",
"Astronomico",
"di",
"Padova,",
"Vicolo",
"dell",
"posservatorio",
"5,",
"I-35122,",
"Padova,",
"Italy",
"14Department",
"of",
"Physics",
"&",
"Astronomy,",
"University",
"of",
"California,",
"Irvine,",
"4129",
"Reines",
"Hall,",
"Irvine,",
"CA",
"92697,",
"USA",
"15Space",
"Telescope",
"Science",
"Institute,",
"3700",
"San",
"Martin",
"Dr.,",
"Baltimore,",
"MD",
"21218,",
"USA",
"Tenh",
"16University",
"of",
"Massachusetts",
"Amherst,",
"710",
"North",
"Pleasant",
"Street,",
"Amherst,",
"MA",
"01003-9305,",
"USA",
"17Physics",
"&",
"Astronomy",
"Department,",
"University",
"of",
"Louisville,",
"40292",
"KY,",
"Louisville,",
"USA",
"18Astrophysics",
"Science",
"Division,",
"NASA",
"Goddard",
"Space",
"Flight",
"Center,",
"8800",
"Greenbelt",
"Rd,",
"Greenbelt,",
"MD",
"20771,",
"USA",
"19Laboratory",
"for",
"Multiwavelength",
"Astrophysics,",
"School",
"of",
"Physics",
"and",
"Astronomy,",
"Rochester",
"Institute",
"of",
"Technology,",
"84",
"Lomb",
"Memorial",
"Drive,",
"Rochester,",
"NY",
"14623,",
"USA",
"20Center",
"for",
"Astrophysics",
"I",
"Harvard",
"&",
"Smithsonian,",
"60",
"Garden",
"Street,",
"Cambridge,",
"MA",
"02138,",
"USA",
"2014",
"21Department",
"of",
"Physics",
"and",
"Astronomy,",
"University",
"of",
"Kansas,",
"Lawrence,",
"KS",
"66045,",
"USA",
"22Department",
"of",
"Physics",
"and",
"Astronomy,",
"Colby",
"College,",
"Waterville,",
"ME",
"04901,",
"USA",
"23ESA/AURA",
"Space",
"Telescope",
"Science",
"Institute",
"24Center",
"for",
"Computational",
"Astrophysics,",
"Flatiron",
"Institute,",
"162",
"5th",
"Avenue,",
"New",
"York,",
"nY,",
"10010,",
"USA",
"25Department",
"of",
"Physics,",
"196",
"Auditorium",
"Road,",
"Unit",
"3046,",
"University",
"of",
"Connecticut,",
"Storrs,",
"CT",
"06269,",
"USA",
"FL",
"26MMT/Steward",
"Observatory,",
"University",
"of",
"Arizona,",
"933",
"N.",
"Cherry",
"Ave.,",
"Tucson,",
"AZ",
"85721,",
"USA",
"27Astronomy",
"Centre,",
"University",
"of",
"Sussex,",
"Falmer,",
"Brighton",
"BN1",
"9QH,",
"UK",
"28Institute",
"of",
"Space",
"Sciences",
"and",
"Astronomy,",
"University",
"of",
"Malta,",
"Msida",
"MSD",
"2080,",
"Malta",
"29National",
"Astronomical",
"Observatory",
"of",
"Japan,",
"2-21-1",
"Osawa,",
"Mitaka,",
"Tokyo",
"181-8588,",
"Japan",
"ABSTRACT",
"We",
"present",
"results",
"from",
"the",
"Cosmic",
"Evolution",
"Early",
"Release",
"Survey",
"(CEERS)",
"on",
"the",
"stellar-population",
"pa-",
"rameters",
"for",
"28",
"galaxies",
"with",
"redshifts",
"4",
"<",
"z",
"<",
"9",
"using",
"imaging",
"data",
"from",
"the",
"James",
"Webb",
"Space",
"Telescope",
"(JWST)",
"Mid-Infrared",
"Instrument",
"(MIRI)",
"combined",
"with",
"data",
"from",
"the",
"Hubble",
"Space",
"Telescope",
"and",
"the",
"Spitzer",
"Space",
"Telescope.",
"The",
"JWST/MIRI",
"5.6",
"and",
"7.7",
"µm",
"data",
"extend",
"the",
"coverage",
"of",
"the",
"rest-frame",
"spectral-energy",
"Corresponding",
"author:",
"Casey",
"Papovich",
"papovich@tamu.edu"
]
} |