1a1d3c0cbd5d6054a71ecb9ce1842ce9 comnonsci_a2026m11v163Part3p110602.pdf d8862e4471dcbbe90a59a534300d2b02551fdb3b comnonsci_a2026m11v163Part3p110602.pdf 364226f11db6fdbd47d3974a3d87bc5e8e910cec21750e0f3b35b0bdee6244e8 comnonsci_a2026m11v163Part3p110602.pdf Title: Entropy and ω -limit sets on invariant graphs for a class of planar piecewise linear maps Subject: Communications in Nonlinear Science and Numerical Simulation, 163 (2026) 110602. doi:10.1016/j.cnsns.2026.110602 Keywords: Continuous piecewise linear map,Invariant graph,Markov partition,Topological entropy Author: Anna Cima Creator: Elsevier Producer: Acrobat Distiller 8.1.0 (Windows) CreationDate: Wed Aug 19 22:28:09 2026 CEST ModDate: Wed Aug 19 22:30:06 2026 CEST Custom Metadata: yes Metadata Stream: yes Tagged: yes UserProperties: no Suspects: no Form: none JavaScript: no Pages: 20 Encrypted: no Page size: 544.252 x 742.677 pts Page rot: 0 File size: 2577131 bytes Optimized: no PDF version: 1.7 name type encoding emb sub uni object ID ------------------------------------ ----------------- ---------------- --- --- --- --------- AAALQD+CharisSIL CID TrueType Identity-H yes yes yes 1060 0 JLHRAK+STIXMathBlackboard-Regular Type 1 Builtin yes yes yes 1068 0 PBBIYW+CharisSIL Type 1 Custom yes yes yes 1072 0 UUXGPS+CharisSIL CID TrueType Identity-H yes yes yes 1077 0 YVQSVE+STIXMath-Regular Type 1 Builtin yes yes yes 1083 0 AUCPAL+CharisSIL-Bold Type 1 Custom yes yes yes 1087 0 FLJBTI+STIXMathExtensions-Regular Type 1 Builtin yes yes yes 1092 0 XVZVYO+LMRoman10-Regular CID Type 0C Identity-H yes yes yes 1096 0 OBIXAX+CharisSIL-Italic Type 1 Custom yes yes yes 1102 0 PBBIYW+CharisSIL Type 1 Custom yes yes yes 1105 0 WEFOBM+UniversLTStd Type 1 Custom yes yes yes 1106 0 TBHJMK+STIXMath-Italic Type 1 Builtin yes yes yes 1111 0 UANQCO+STIXMath-Bold Type 1 Builtin yes yes yes 4 0 AAAAAD+CharisSIL CID TrueType Identity-H yes yes yes 114 0 AIDPDI+CharisSIL-Italic CID TrueType Identity-H yes yes yes 121 0 LDUMZY+STIXMathCalligraphy-Regular Type 1 Builtin yes yes yes 127 0 AAAALE+CharisSIL-Italic CID TrueType Identity-H yes yes yes 131 0 BEXWYE+STIXMathSans-Regular Type 1 Builtin yes yes yes 140 0 ISNDTX+STIXMathTyperwriter-Regular Type 1 Builtin yes yes yes 144 0 KBGTZC+STIXMathScript-Regular Type 1 Builtin yes yes yes 148 0 PXFDGF+STIXMathFraktur-Regular Type 1 Builtin yes yes yes 152 0 ONOCGN+LMMathSymbols5-Regular Type 1 Custom yes yes yes 156 0 DFXVYM+STIXMath-BoldItalic Type 1 Builtin yes yes yes 62 0 Jhove (Rel. 1.28.0, 2023-05-18) Date: 2026-10-09 04:17:25 CEST RepresentationInformation: comnonsci_a2026m11v163Part3p110602.pdf ReportingModule: PDF-hul, Rel. 1.12.4 (2023-03-16) LastModified: 2026-10-08 20:26:31 CEST Size: 2577131 Format: PDF Version: 1.7 Status: Well-Formed and valid SignatureMatches: PDF-hul MIMEtype: application/pdf Profile: Tagged PDF PDFMetadata: Objects: 6306 FreeObjects: 1 IncrementalUpdates: 0 DocumentCatalog: ViewerPreferences: HideToolbar: false HideMenubar: false HideWindowUI: false FitWindow: false CenterWindow: false DisplayDocTitle: true NonFullScreenPageMode: UseNone Direction: L2R ViewArea: CropBox ViewClip: CropBox PrintArea: CropBox PageClip: CropBox PageLayout: SinglePage PageMode: UseOutlines Language: en-US Outlines: Item: Title: Entropy and -limit sets on invariant graphs for a class of planar piecewise linear maps Children: Item: Title: 1 Introduction and main results Children: Item: Title: 1.1 Dynamics of the family [e:F](1) Item: Title: 1.2 Main results Item: Title: 2 Preliminary results Children: Item: Title: 2.1 Topological entropy Item: Title: 2.2 A useful conjugation and an observation Item: Title: 3 Proof of Theorem 1 Children: Item: Title: 3.1 Invariant graph and a first glimpse on the dynamics of F| Item: Title: 3.2 Dynamics of F3|{x=-1}. Proof of Proposition 9 Item: Title: 4 Approximations of and and proof of Proposition 2 Children: Item: Title: 4.1 Rational approximations of Item: Title: 4.2 Rational approximations of Item: Title: 5 Proof of Theorem 3 Children: Item: Title: 5.1 Case a=-1 and b<-2 Item: Title: 5.2 Rest of the cases Item: Title: CRediT authorship contribution statement Item: Title: Declaration of competing interest Item: Title: Acknowledgment Item: Title: Data availability Item: Title: References Info: Title: Entropy and ω -limit sets on invariant graphs for a class of planar piecewise linear maps Author: Anna Cima Subject: Communications in Nonlinear Science and Numerical Simulation, 163 (2026) 110602. doi:10.1016/j.cnsns.2026.110602 Keywords: Continuous piecewise linear map,Invariant graph,Markov partition,Topological entropy Creator: Elsevier Producer: Acrobat Distiller 8.1.0 (Windows) CreationDate: Wed Aug 19 22:28:09 CEST 2026 ModDate: Wed Aug 19 22:30:06 CEST 2026 ID: 0xbdc167eb20021e6ff742cea2234987c2, 0x8a3a4b8f644698ab401b85a9571b70bc Filters: FilterPipeline: FlateDecode Fonts: Type0: Font: BaseFont: AAAAAD+CharisSIL Encoding: Identity-H ToUnicode: true Font: BaseFont: AAAALE+CharisSIL-Italic Encoding: Identity-H ToUnicode: true Font: BaseFont: AAALQD+CharisSIL Encoding: Identity-H ToUnicode: true Font: BaseFont: UUXGPS+CharisSIL Encoding: Identity-H ToUnicode: true Font: BaseFont: XVZVYO+LMRoman10-Regular Encoding: Identity-H ToUnicode: true Font: BaseFont: AIDPDI+CharisSIL-Italic Encoding: Identity-H ToUnicode: true Type1: Font: BaseFont: FLJBTI+STIXMathExtensions-Regular FontSubset: true FirstChar: 0 LastChar: 249 FontDescriptor: FontName: FLJBTI+STIXMathExtensions-Regular Flags: Symbolic FontBBox: 15, -3009, 1503, 959 FontFile: true ToUnicode: true Font: BaseFont: UANQCO+STIXMath-Bold FontSubset: true FirstChar: 120 LastChar: 120 FontDescriptor: FontName: UANQCO+STIXMath-Bold Flags: Symbolic FontBBox: -998, -341, 1306, 1055 FontFile: true ToUnicode: true Font: BaseFont: JLHRAK+STIXMathBlackboard-Regular FontSubset: true FirstChar: 78 LastChar: 82 FontDescriptor: FontName: JLHRAK+STIXMathBlackboard-Regular Flags: Symbolic FontBBox: -15, -424, 1388, 930 FontFile: true ToUnicode: true Font: BaseFont: BEXWYE+STIXMathSans-Regular FontSubset: true FirstChar: 153 LastChar: 167 FontDescriptor: FontName: BEXWYE+STIXMathSans-Regular Flags: Symbolic FontBBox: -124, -232, 1151, 819 FontFile: true ToUnicode: true Font: BaseFont: OBIXAX+CharisSIL-Italic FontSubset: true FirstChar: 15 LastChar: 242 FontDescriptor: FontName: OBIXAX+CharisSIL-Italic Flags: Symbolic FontBBox: -1467, -1092, 6262, 2600 FontFile: true EncodingDictionary: Differences: true ToUnicode: true Font: BaseFont: PBBIYW+CharisSIL FontSubset: true FirstChar: 97 LastChar: 116 FontDescriptor: FontName: PBBIYW+CharisSIL Flags: Symbolic FontBBox: -1419, -1092, 6144, 2600 FontFile: true EncodingDictionary: Differences: true ToUnicode: true Font: BaseFont: ISNDTX+STIXMathTyperwriter-Regular FontSubset: true FirstChar: 3 LastChar: 7 FontDescriptor: FontName: ISNDTX+STIXMathTyperwriter-Regular Flags: Symbolic FontBBox: 2, -406, 1519, 914 FontFile: true ToUnicode: true Font: BaseFont: PBBIYW+CharisSIL FontSubset: true FirstChar: 12 LastChar: 250 FontDescriptor: FontName: PBBIYW+CharisSIL Flags: Symbolic FontBBox: -1419, -1092, 6144, 2600 FontFile: true EncodingDictionary: Differences: true ToUnicode: true Font: BaseFont: WEFOBM+UniversLTStd FontSubset: true FirstChar: 46 LastChar: 119 FontDescriptor: FontName: WEFOBM+UniversLTStd Flags: Symbolic FontBBox: -168, -250, 992, 947 FontFile: true EncodingDictionary: Differences: true ToUnicode: true Font: BaseFont: KBGTZC+STIXMathScript-Regular FontSubset: true FirstChar: 12 LastChar: 253 FontDescriptor: FontName: KBGTZC+STIXMathScript-Regular Flags: Symbolic FontBBox: 0, -324, 1229, 830 FontFile: true ToUnicode: true Font: BaseFont: TBHJMK+STIXMath-Italic FontSubset: true FirstChar: 11 LastChar: 211 FontDescriptor: FontName: TBHJMK+STIXMath-Italic Flags: Symbolic FontBBox: -124, -478, 2328, 819 FontFile: true ToUnicode: true Font: BaseFont: PXFDGF+STIXMathFraktur-Regular FontSubset: true FirstChar: 228 LastChar: 228 FontDescriptor: FontName: PXFDGF+STIXMathFraktur-Regular Flags: Symbolic FontBBox: 0, -300, 1519, 1003 FontFile: true ToUnicode: true Font: BaseFont: YVQSVE+STIXMath-Regular FontSubset: true FirstChar: 0 LastChar: 249 FontDescriptor: FontName: YVQSVE+STIXMath-Regular Flags: Symbolic FontBBox: -970, -341, 1096, 1023 FontFile: true ToUnicode: true Font: BaseFont: ONOCGN+LMMathSymbols5-Regular FontSubset: true FirstChar: 15 LastChar: 15 FontDescriptor: FontName: ONOCGN+LMMathSymbols5-Regular Flags: Symbolic FontBBox: 0, -944, 1448, 791 FontFile: true EncodingDictionary: Differences: true ToUnicode: true Font: BaseFont: DFXVYM+STIXMath-BoldItalic FontSubset: true FirstChar: 11 LastChar: 12 FontDescriptor: FontName: DFXVYM+STIXMath-BoldItalic Flags: Symbolic FontBBox: -51, -478, 2328, 867 FontFile: true ToUnicode: true Font: BaseFont: AUCPAL+CharisSIL-Bold FontSubset: true FirstChar: 15 LastChar: 241 FontDescriptor: FontName: AUCPAL+CharisSIL-Bold Flags: Symbolic FontBBox: -1475, -1092, 6144, 2600 FontFile: true EncodingDictionary: Differences: true ToUnicode: true Font: BaseFont: LDUMZY+STIXMathCalligraphy-Regular FontSubset: true FirstChar: 65 LastChar: 145 FontDescriptor: FontName: LDUMZY+STIXMathCalligraphy-Regular Flags: Symbolic FontBBox: -53, -455, 1965, 2157 FontFile: true ToUnicode: true CIDFontType0: Font: BaseFont: XVZVYO+LMRoman10-Regular CIDSystemInfo: Registry: Adobe Registry: Identity Supplement: 0 FontDescriptor: FontName: XVZVYO+LMRoman10-Regular Flags: Symbolic FontBBox: -430, -290, 1417, 1127 FontFile3: true CIDFontType2: Font: BaseFont: AAAAAD+CharisSIL CIDSystemInfo: Registry: Adobe Registry: Identity Supplement: 0 FontDescriptor: FontName: AAAAAD+CharisSIL Flags: Symbolic FontBBox: -692, -533, 1623, 1318 FontFile2: true Font: BaseFont: AAAALE+CharisSIL-Italic CIDSystemInfo: Registry: Adobe Registry: Identity Supplement: 0 FontDescriptor: FontName: AAAALE+CharisSIL-Italic Flags: Symbolic, Italic FontBBox: -716, -533, 1626, 1317 FontFile2: true Font: BaseFont: AAALQD+CharisSIL CIDSystemInfo: Registry: Adobe Registry: Identity Supplement: 0 FontDescriptor: FontName: AAALQD+CharisSIL Flags: Symbolic FontBBox: -692, -533, 1623, 1318 FontFile2: true Font: BaseFont: UUXGPS+CharisSIL CIDSystemInfo: Registry: Adobe Registry: Identity Supplement: 0 FontDescriptor: FontName: UUXGPS+CharisSIL Flags: Serif, Symbolic FontBBox: -692, -533, 1623, 1319 FontFile2: true Font: BaseFont: AIDPDI+CharisSIL-Italic CIDSystemInfo: Registry: Adobe Registry: Identity Supplement: 0 FontDescriptor: FontName: AIDPDI+CharisSIL-Italic Flags: Serif, Symbolic, Italic FontBBox: -716, -533, 1627, 1318 FontFile2: true XMP: http://creativecommons.org/licenses/by-nc/4.0/ elsevier.com sciencedirect.com true 10.1016/j.cnsns.2026.110602 2010-04-23 application/pdf 10.1016/j.cnsns.2026.110602 Elsevier B.V. Communications in Nonlinear Science and Numerical Simulation, 163 (2026) 110602. doi:10.1016/j.cnsns.2026.110602 Continuous piecewise linear map Invariant graph Markov partition Topological entropy Entropy and ω -limit sets on invariant graphs for a class of planar piecewise linear maps Anna Cima Armengol Gasull Víctor Mañosa Francesc Mañosas VoR 19th August 2026 Acrobat Distiller 8.1.0 (Windows) Continuous piecewise linear map,Invariant graph,Markov partition,Topological entropy 19th August 2026 sciencedirect.com elsevier.com true 2010-04-23 10.1016/j.cnsns.2026.110602 noindex journal 2026-11-01 1 November 2026 10.1016/j.cnsns.2026.110602 1007-5704 P3 110602 Communications in Nonlinear Science and Numerical Simulation 110602 https://doi.org/10.1016/j.cnsns.2026.110602 163 https://www.elsevier.com/tdm/tdmrep-policy.json 1 2026-08-19T20:28:09 Elsevier 2026-08-19T20:30:06 2026-08-19T20:30:06 uuid:1b499bed-4ce8-4c0c-b682-0d58baae1cbe uuid:b6ed5266-03cb-4855-90c0-636283357f42 True Pages: Page: Label: 1 Annotations: Annotation: Subtype: Link Contents: https://www.elsevier.com/locate/cnsns Rect: 296, 672, 345, 680 Annotation: Subtype: Link Contents: https://www.elsevier.com/locate/cnsns Rect: 251, 615, 371, 624 Annotation: Subtype: Link Contents: https://orcid.org/0000-0003-0256-518X Rect: 90, 527, 103, 540 Annotation: Subtype: Link Contents: Go to destination 1 Rect: 103, 531, 108, 536 ActionDest: 1050 Annotation: Subtype: Link Contents: https://orcid.org/0000-0002-1719-8231 Rect: 192, 525, 205, 540 Annotation: Subtype: Link Contents: Go to destination 1 Rect: 204, 531, 210, 536 ActionDest: 1050 Annotation: Subtype: Link Contents: https://orcid.org/0000-0002-5082-3334 Rect: 283, 527, 296, 540 Annotation: Subtype: Link Contents: Go to destination 2 Rect: 295, 530, 301, 538 ActionDest: 1050 Annotation: Subtype: Link Contents: Go to destination cor1 Rect: 301, 530, 307, 538 ActionDest: 1050 Annotation: Subtype: Link Contents: https://orcid.org/0000-0003-2535-0501 Rect: 395, 527, 408, 540 Annotation: Subtype: Link Contents: Go to destination 1 Rect: 407, 531, 413, 536 ActionDest: 1050 Annotation: Subtype: Link Contents: Go to destination cite.D84 Rect: 141, 250, 148, 258 ActionDest: 101 Annotation: Subtype: Link Contents: Go to destination cite.L78 Rect: 148, 250, 155, 258 ActionDest: 101 Annotation: Subtype: Link Contents: Go to destination cite.NY92 Rect: 158, 229, 165, 237 ActionDest: 101 Annotation: Subtype: Link Contents: Go to destination cite.NY95 Rect: 165, 229, 171, 237 ActionDest: 101 Annotation: Subtype: Link Contents: Go to destination cite.BG99 Rect: 434, 219, 441, 226 ActionDest: 101 Annotation: Subtype: Link Contents: Go to destination cite.TG20 Rect: 443, 219, 449, 226 ActionDest: 101 Annotation: Subtype: Link Contents: Go to destination cite.CGMM Rect: 128, 209, 139, 216 ActionDest: 101 Annotation: Subtype: Link Contents: mailto:anna.cima@uab.cat Rect: 103, 115, 168, 125 Annotation: Subtype: Link Contents: mailto:armengol.gasull@uab.cat Rect: 203, 115, 285, 125 Annotation: Subtype: Link Contents: mailto:victor.manosa@upc.edu Rect: 324, 115, 402, 125 Annotation: Subtype: Link Contents: mailto:francesc.manosas@uab.cat Rect: 38, 106, 125, 114 Annotation: Subtype: Link Contents: https://doi.org/10.1016/j.cnsns.2026.110602 Rect: 38, 88, 186, 97 Annotation: Subtype: Link Rect: 177, 699, 365, 711 Annotation: Subtype: Link Rect: 448, 568, 476, 597 Annotation: Subtype: Link Rect: 41, 49, 197, 61 Page: Label: 2 Annotations: Annotation: Subtype: Link Contents: Go to destination cite.PB Rect: 330, 669, 341, 676 ActionDest: 101 Annotation: Subtype: Link Contents: Go to destination cite.Kow Rect: 357, 658, 368, 666 ActionDest: 101 Annotation: Subtype: Link Contents: Go to destination cite.Kow Rect: 243, 616, 254, 624 ActionDest: 101 Annotation: Subtype: Link Contents: Go to destination cite.GL Rect: 255, 540, 266, 548 ActionDest: 101 Annotation: Subtype: Link Contents: Go to destination cite.ABK21 Rect: 476, 540, 487, 548 ActionDest: 101 Annotation: Subtype: Link Contents: Go to destination cite.TLS17 Rect: 489, 540, 500, 548 ActionDest: 101 Annotation: Subtype: Link Contents: Go to destination cite.MS1 Rect: 465, 488, 476, 496 ActionDest: 101 Annotation: Subtype: Link Contents: Go to destination cite.MS2 Rect: 476, 488, 487, 496 ActionDest: 101 Annotation: Subtype: Link Contents: Go to destination cite.CGMM Rect: 295, 394, 306, 401 ActionDest: 101 Annotation: Subtype: Link Contents: Go to destination equation.1 Rect: 135, 357, 148, 367 ActionDest: 1 Annotation: Subtype: Link Contents: Go to destination cite.CGMM Rect: 100, 338, 111, 345 ActionDest: 101 Annotation: Subtype: Link Contents: Go to destination equation.1 Rect: 244, 336, 257, 346 ActionDest: 1 Annotation: Subtype: Link Contents: Go to destination theorem.1 Rect: 157, 326, 201, 335 ActionDest: 1 Annotation: Subtype: Link Contents: Go to destination theorem.1 Rect: 203, 326, 211, 335 ActionDest: 1 Annotation: Subtype: Link Contents: Go to destination subsection.2.2 Rect: 234, 314, 276, 325 ActionDest: 19 Annotation: Subtype: Link Contents: Go to destination equation.2 Rect: 38, 305, 50, 314 ActionDest: 19 Annotation: Subtype: Link Contents: Go to destination theorem.1 Rect: 210, 182, 251, 192 ActionDest: 1 Annotation: Subtype: Link Contents: Go to destination cite.CGMM Rect: 436, 184, 447, 191 ActionDest: 101 Annotation: Subtype: Link Contents: Go to destination figure.struct.694 Rect: 276, 161, 301, 171 ActionDest: 27 Annotation: Subtype: Link Contents: Go to destination figure.struct.1149 Rect: 304, 161, 311, 171 ActionDest: 66 Annotation: Subtype: Link Contents: Go to destination figure.struct.1418 Rect: 314, 161, 320, 171 ActionDest: 81 Annotation: Subtype: Link Contents: Go to destination figure.struct.1530 Rect: 339, 161, 346, 171 ActionDest: 89 Annotation: Subtype: Link Contents: Go to destination figure.struct.155 Rect: 324, 151, 346, 161 ActionDest: 8 Annotation: Subtype: Link Contents: Go to destination theorem.1 Rect: 71, 130, 111, 140 ActionDest: 1 Annotation: Subtype: Link Contents: Go to destination subsection.2.2 Rect: 265, 99, 306, 108 ActionDest: 19 Annotation: Subtype: Link Contents: Go to destination theorem.1 Rect: 349, 60, 388, 69 ActionDest: 1 Page: Label: 3 Annotations: Annotation: Subtype: Link Contents: Go to destination cite.CGMM Rect: 339, 387, 349, 394 ActionDest: 101 Annotation: Subtype: Link Contents: Go to destination cite.AKM Rect: 177, 330, 188, 337 ActionDest: 101 Annotation: Subtype: Link Contents: Go to destination cite.MZ Rect: 190, 330, 201, 337 ActionDest: 101 Annotation: Subtype: Link Contents: Go to destination equation.1 Rect: 324, 205, 337, 215 ActionDest: 1 Annotation: Subtype: Link Contents: Go to destination theorem.1 Rect: 67, 195, 106, 205 ActionDest: 1 Annotation: Subtype: Link Contents: Go to destination cite.ALM Rect: 57, 93, 68, 100 ActionDest: 101 Annotation: Subtype: Link Contents: Go to destination cite.CGMM Rect: 361, 51, 372, 58 ActionDest: 101 Page: Label: 4 Annotations: Annotation: Subtype: Link Contents: Go to destination theorem.1 Rect: 151, 677, 192, 687 ActionDest: 1 Annotation: Subtype: Link Contents: Go to destination cite.BMT Rect: 78, 669, 89, 676 ActionDest: 101 Annotation: Subtype: Link Contents: Go to destination section.3 Rect: 60, 656, 95, 666 ActionDest: 19 Annotation: Subtype: Link Contents: Go to destination thm.9 Rect: 309, 646, 359, 656 ActionDest: 19 Annotation: Subtype: Link Contents: Go to destination cite.CGMM Rect: 227, 584, 238, 591 ActionDest: 101 Annotation: Subtype: Link Contents: Go to destination theorem.1 Rect: 244, 551, 285, 561 ActionDest: 1 Annotation: Subtype: Link Contents: Go to destination subsection.4.1 Rect: 462, 551, 506, 561 ActionDest: 57 Annotation: Subtype: Link Contents: Go to destination subsection.4.2 Rect: 53, 540, 66, 550 ActionDest: 81 Annotation: Subtype: Link Contents: Go to destination theorem.1 Rect: 240, 513, 278, 523 ActionDest: 1 Annotation: Subtype: Link Contents: Go to destination cite.CGMM Rect: 150, 368, 161, 375 ActionDest: 101 Annotation: Subtype: Link Contents: Go to destination theorem.1 Rect: 285, 366, 326, 376 ActionDest: 1 Annotation: Subtype: Link Contents: Go to destination section.5 Rect: 108, 345, 143, 355 ActionDest: 86 Annotation: Subtype: Link Contents: Go to destination thm.10 Rect: 422, 324, 479, 334 ActionDest: 86 Annotation: Subtype: Link Contents: Go to destination thm.11 Rect: 495, 324, 506, 334 ActionDest: 96 Annotation: Subtype: Link Contents: Go to destination section.5 Rect: 74, 314, 108, 323 ActionDest: 86 Annotation: Subtype: Link Contents: Go to destination theorem.1 Rect: 396, 297, 435, 307 ActionDest: 1 Annotation: Subtype: Link Contents: Go to destination cite.AKM Rect: 41, 115, 52, 122 ActionDest: 101 Annotation: Subtype: Link Contents: Go to destination cite.MZ Rect: 54, 115, 65, 122 ActionDest: 101 Annotation: Subtype: Link Contents: Go to destination cite.ALM Rect: 141, 115, 152, 122 ActionDest: 101 Annotation: Subtype: Link Contents: Go to destination cite.Ll15 Rect: 219, 115, 230, 122 ActionDest: 101 Annotation: Subtype: Link Contents: Go to destination cite.ABK Rect: 204, 104, 215, 112 ActionDest: 101 Annotation: Subtype: Link Contents: Go to destination cite.LlM93 Rect: 217, 104, 228, 112 ActionDest: 101 Annotation: Subtype: Link Contents: Go to destination cite.ALM Rect: 489, 61, 500, 69 ActionDest: 101 Page: Label: 5 Annotations: Annotation: Subtype: Link Contents: Go to destination section.3 Rect: 330, 283, 365, 293 ActionDest: 19 Annotation: Subtype: Link Contents: Go to destination cite.BGMY Rect: 89, 274, 100, 282 ActionDest: 101 Annotation: Subtype: Link Contents: Go to destination cite.BGMY Rect: 109, 96, 120, 104 ActionDest: 101 Annotation: Subtype: Link Contents: Go to destination subsection.4.1 Rect: 145, 69, 189, 77 ActionDest: 57 Annotation: Subtype: Link Contents: Go to destination subsection.4.2 Rect: 205, 69, 218, 77 ActionDest: 81 Page: Label: 6 Annotations: Annotation: Subtype: Link Contents: Go to destination equation.2 Rect: 417, 603, 429, 613 ActionDest: 19 Annotation: Subtype: Link Contents: Go to destination cite.CGMM Rect: 162, 511, 173, 518 ActionDest: 101 Annotation: Subtype: Link Contents: Go to destination cite.BMT Rect: 326, 511, 337, 518 ActionDest: 101 Annotation: Subtype: Link Contents: Go to destination theorem.1 Rect: 324, 499, 368, 509 ActionDest: 1 Annotation: Subtype: Link Contents: Go to destination theorem.1 Rect: 385, 499, 393, 509 ActionDest: 1 Annotation: Subtype: Link Contents: Go to destination thm.1 Rect: 82, 468, 124, 476 ActionDest: 13 Annotation: Subtype: Link Contents: Go to destination equation.2 Rect: 140, 445, 153, 455 ActionDest: 19 Annotation: Subtype: Link Contents: Go to destination theorem.1 Rect: 358, 435, 399, 445 ActionDest: 1 Annotation: Subtype: Link Contents: Go to destination thm.1 Rect: 333, 424, 373, 434 ActionDest: 13 Annotation: Subtype: Link Contents: Go to destination thm.9 Rect: 171, 414, 220, 424 ActionDest: 19 Annotation: Subtype: Link Contents: Go to destination theorem.1 Rect: 359, 403, 400, 413 ActionDest: 1 Annotation: Subtype: Link Contents: Go to destination theorem.1 Rect: 174, 389, 215, 403 ActionDest: 1 Annotation: Subtype: Link Contents: Go to destination thm.9 Rect: 274, 380, 323, 390 ActionDest: 19 Annotation: Subtype: Link Contents: Go to destination cite.CGMM Rect: 225, 246, 236, 253 ActionDest: 101 Annotation: Subtype: Link Contents: Go to destination figure.struct.644 Rect: 294, 84, 315, 94 ActionDest: 22 Annotation: Subtype: Link Contents: Go to destination thm.9 Rect: 135, 49, 184, 59 ActionDest: 19 Annotation: Subtype: Link Contents: Go to destination table.struct.585 Rect: 252, 49, 280, 59 ActionDest: 22 Annotation: Subtype: Link Contents: Go to destination figure.struct.644 Rect: 296, 49, 318, 59 ActionDest: 22 Page: Label: 7 Annotations: Annotation: Subtype: Link Contents: Go to destination thm.9 Rect: 235, 641, 280, 650 ActionDest: 19 Annotation: Subtype: Link Contents: Go to destination thm.9 Rect: 345, 255, 390, 264 ActionDest: 19 Annotation: Subtype: Link Contents: Go to destination theorem.1 Rect: 158, 199, 198, 209 ActionDest: 1 Annotation: Subtype: Link Contents: Go to destination cite.CGMM Rect: 216, 201, 227, 208 ActionDest: 101 Annotation: Subtype: Link Contents: Go to destination figure.struct.694 Rect: 405, 188, 427, 199 ActionDest: 27 Annotation: Subtype: Link Contents: Go to destination figure.struct.694 Rect: 59, 127, 81, 137 ActionDest: 27 Page: Label: 8 Annotations: Annotation: Subtype: Link Contents: Go to destination figure.struct.694 Rect: 275, 324, 296, 336 ActionDest: 27 Annotation: Subtype: Link Contents: Go to destination equation.4 Rect: 434, 282, 446, 294 ActionDest: 19 Annotation: Subtype: Link Contents: Go to destination cite.BGMY Rect: 419, 98, 430, 105 ActionDest: 101 Annotation: Subtype: Link Contents: Go to destination thm.7 Rect: 38, 86, 78, 95 ActionDest: 16 Annotation: Subtype: Link Contents: Go to destination thm.6 Rect: 211, 86, 256, 95 ActionDest: 16 Page: Label: 9 Annotations: Annotation: Subtype: Link Contents: Go to destination thm.7 Rect: 84, 677, 124, 687 ActionDest: 16 Annotation: Subtype: Link Contents: Go to destination thm.9 Rect: 214, 613, 263, 624 ActionDest: 19 Page: Label: 10 Annotations: Annotation: Subtype: Link Contents: Go to destination thm.9 Rect: 403, 568, 452, 577 ActionDest: 19 Annotation: Subtype: Link Contents: Go to destination thm.9 Rect: 173, 544, 219, 556 ActionDest: 19 Annotation: Subtype: Link Contents: Go to destination subsection.2.2 Rect: 178, 503, 219, 513 ActionDest: 19 Annotation: Subtype: Link Contents: Go to destination equation.4 Rect: 173, 276, 185, 286 ActionDest: 19 Annotation: Subtype: Link Contents: Go to destination thm.9 Rect: 429, 265, 479, 275 ActionDest: 19 Annotation: Subtype: Link Contents: Go to destination thm.9 Rect: 118, 239, 125, 249 ActionDest: 19 Annotation: Subtype: Link Contents: Go to destination subsection.3.1 Rect: 447, 239, 489, 249 ActionDest: 22 Page: Label: 11 Annotations: Annotation: Subtype: Link Contents: Go to destination subsection.3.1 Rect: 95, 563, 136, 574 ActionDest: 22 Annotation: Subtype: Link Contents: Go to destination equation.5 Rect: 84, 130, 113, 141 ActionDest: 19 Page: Label: 12 Annotations: Annotation: Subtype: Link Contents: Go to destination figure.struct.1056 Rect: 54, 240, 76, 250 ActionDest: 57 Annotation: Subtype: Link Contents: Go to destination equation.5 Rect: 177, 240, 189, 250 ActionDest: 19 Annotation: Subtype: Link Contents: Go to destination equation.5 Rect: 125, 229, 138, 239 ActionDest: 19 Annotation: Subtype: Link Contents: Go to destination figure.struct.1078 Rect: 54, 208, 76, 218 ActionDest: 66 Annotation: Subtype: Link Contents: Go to destination thm.2 Rect: 203, 134, 256, 143 ActionDest: 13 Annotation: Subtype: Link Contents: Go to destination theorem.1 Rect: 50, 113, 93, 122 ActionDest: 1 Annotation: Subtype: Link Contents: Go to destination thm.2 Rect: 358, 92, 407, 102 ActionDest: 13 Annotation: Subtype: Link Contents: Go to destination figure.struct.1149 Rect: 275, 49, 296, 59 ActionDest: 66 Page: Label: 13 Annotations: Annotation: Subtype: Link Contents: Go to destination cite.CGMM Rect: 63, 135, 74, 142 ActionDest: 101 Annotation: Subtype: Link Contents: Go to destination cite.CGMM Rect: 181, 104, 192, 111 ActionDest: 101 Annotation: Subtype: Link Contents: Go to destination cite.CGMM Rect: 484, 93, 495, 101 ActionDest: 101 Annotation: Subtype: Link Contents: Go to destination cite.BMT Rect: 317, 83, 328, 90 ActionDest: 101 Page: Label: 14 Annotations: Annotation: Subtype: Link Contents: Go to destination cite.CGMM Rect: 451, 453, 462, 460 ActionDest: 101 Annotation: Subtype: Link Contents: Go to destination subsection.2.1 Rect: 334, 339, 376, 354 ActionDest: 13 Annotation: Subtype: Link Contents: Go to destination figure.struct.1195 Rect: 485, 233, 506, 248 ActionDest: 73 Annotation: Subtype: Link Contents: Go to destination cite.CGMM Rect: 50, 200, 61, 207 ActionDest: 101 Annotation: Subtype: Link Contents: Go to destination cite.BMT Rect: 265, 179, 276, 187 ActionDest: 101 Page: Label: 15 Annotations: Annotation: Subtype: Link Contents: Go to destination cite.BF Rect: 402, 669, 413, 676 ActionDest: 101 Annotation: Subtype: Link Contents: Go to destination thm.8 Rect: 188, 655, 224, 666 ActionDest: 16 Annotation: Subtype: Link Contents: Go to destination subsection.2.1 Rect: 235, 655, 277, 666 ActionDest: 13 Annotation: Subtype: Link Contents: Go to destination cite.BMT Rect: 317, 637, 328, 645 ActionDest: 101 Annotation: Subtype: Link Contents: Go to destination cite.Col Rect: 182, 400, 193, 407 ActionDest: 101 Annotation: Subtype: Link Contents: Go to destination cite.Col Rect: 79, 224, 90, 231 ActionDest: 101 Annotation: Subtype: Link Contents: Go to destination equation.6 Rect: 130, 49, 142, 59 ActionDest: 78 Annotation: Subtype: Link Contents: Go to destination thm.2 Rect: 330, 49, 379, 59 ActionDest: 13 Page: Label: 16 Annotations: Annotation: Subtype: Link Contents: Go to destination cite.CGMM Rect: 401, 374, 411, 381 ActionDest: 101 Annotation: Subtype: Link Contents: Go to destination figure.struct.1418 Rect: 284, 323, 307, 333 ActionDest: 81 Annotation: Subtype: Link Contents: Go to destination cite.CGMM Rect: 99, 262, 109, 270 ActionDest: 101 Annotation: Subtype: Link Contents: Go to destination cite.BMT Rect: 489, 162, 500, 169 ActionDest: 101 Page: Label: 17 Annotations: Annotation: Subtype: Link Contents: Go to destination thm.2 Rect: 235, 542, 285, 552 ActionDest: 13 Annotation: Subtype: Link Contents: Go to destination thm.3 Rect: 82, 522, 124, 530 ActionDest: 13 Annotation: Subtype: Link Contents: Go to destination theorem.1 Rect: 463, 499, 506, 509 ActionDest: 1 Annotation: Subtype: Link Contents: Go to destination theorem.1 Rect: 53, 489, 61, 499 ActionDest: 1 Annotation: Subtype: Link Contents: Go to destination theorem.1 Rect: 59, 478, 100, 488 ActionDest: 1 Annotation: Subtype: Link Contents: Go to destination cite.CGMM Rect: 102, 428, 112, 435 ActionDest: 101 Annotation: Subtype: Link Contents: Go to destination thm.10 Rect: 378, 426, 435, 436 ActionDest: 86 Annotation: Subtype: Link Contents: Go to destination thm.11 Rect: 451, 426, 462, 436 ActionDest: 96 Annotation: Subtype: Link Contents: Go to destination theorem.1 Rect: 82, 405, 122, 415 ActionDest: 1 Annotation: Subtype: Link Contents: Go to destination cite.BMT Rect: 431, 378, 442, 385 ActionDest: 101 Annotation: Subtype: Link Contents: Go to destination cite.CGMM Rect: 123, 325, 134, 333 ActionDest: 101 Annotation: Subtype: Link Contents: Go to destination thm.3 Rect: 125, 305, 165, 314 ActionDest: 13 Annotation: Subtype: Link Contents: Go to destination equation.2 Rect: 210, 262, 223, 271 ActionDest: 19 Annotation: Subtype: Link Contents: Go to destination figure.struct.1530 Rect: 149, 251, 172, 261 ActionDest: 89 Annotation: Subtype: Link Contents: Go to destination cite.ALM Rect: 165, 242, 176, 250 ActionDest: 101 Annotation: Subtype: Link Contents: Go to destination figure.struct.1530 Rect: 68, 230, 90, 240 ActionDest: 89 Annotation: Subtype: Link Contents: Go to destination cite.CGMM Rect: 132, 211, 143, 219 ActionDest: 101 Annotation: Subtype: Link Contents: Go to destination cite.CGMM Rect: 249, 107, 260, 114 ActionDest: 101 Annotation: Subtype: Link Contents: Go to destination thm.3 Rect: 213, 46, 253, 63 ActionDest: 13 Page: Label: 18 Annotations: Annotation: Subtype: Link Contents: Go to destination cite.CGMM Rect: 335, 440, 345, 447 ActionDest: 101 Annotation: Subtype: Link Contents: Go to destination figure.struct.1530 Rect: 429, 410, 451, 420 ActionDest: 89 Annotation: Subtype: Link Contents: Go to destination thm.10 Rect: 115, 268, 126, 278 ActionDest: 86 Annotation: Subtype: Link Contents: Go to destination figure.struct.1636 Rect: 111, 182, 138, 192 ActionDest: 96 Page: Label: 19 Annotations: Annotation: Subtype: Link Contents: Go to destination thm.10 Rect: 234, 429, 289, 439 ActionDest: 86 Annotation: Subtype: Link Contents: Go to destination thm.11 Rect: 434, 419, 488, 429 ActionDest: 96 Annotation: Subtype: Link Contents: Go to destination cite.BMT Rect: 259, 410, 270, 418 ActionDest: 101 Annotation: Subtype: Link Contents: Go to destination thm.3 Rect: 136, 354, 176, 364 ActionDest: 13 Annotation: Subtype: Link Contents: Go to destination thm.10 Rect: 277, 354, 330, 364 ActionDest: 86 Annotation: Subtype: Link Contents: Go to destination cite.BMT Rect: 428, 291, 439, 299 ActionDest: 101 Annotation: Subtype: Link Contents: Go to destination cite.BMT Rect: 132, 232, 143, 240 ActionDest: 101 Page: Label: 20 Annotations: Annotation: Subtype: Link Contents: http://refhub.elsevier.com/S1007-5704(26)00956-1/sbref0001 Rect: 54, 555, 398, 563 Annotation: Subtype: Link Contents: http://refhub.elsevier.com/S1007-5704(26)00956-1/sbref0002 Rect: 54, 547, 335, 555 Annotation: Subtype: Link Contents: http://refhub.elsevier.com/S1007-5704(26)00956-1/sbref0003 Rect: 54, 539, 462, 547 Annotation: Subtype: Link Contents: http://refhub.elsevier.com/S1007-5704(26)00956-1/sbref0004 Rect: 54, 531, 477, 539 Annotation: Subtype: Link Contents: http://refhub.elsevier.com/S1007-5704(26)00956-1/sbref0005 Rect: 54, 523, 415, 531 Annotation: Subtype: Link Contents: http://refhub.elsevier.com/S1007-5704(26)00956-1/sbref0006 Rect: 54, 515, 506, 523 Annotation: Subtype: Link Contents: http://refhub.elsevier.com/S1007-5704(26)00956-1/sbref0007 Rect: 54, 507, 481, 515 Annotation: Subtype: Link Contents: http://refhub.elsevier.com/S1007-5704(26)00956-1/sbref0008 Rect: 54, 499, 506, 507 Annotation: Subtype: Link Contents: http://refhub.elsevier.com/S1007-5704(26)00956-1/sbref0008 Rect: 55, 492, 125, 498 Annotation: Subtype: Link Contents: http://refhub.elsevier.com/S1007-5704(26)00956-1/sbref0009 Rect: 54, 483, 506, 491 Annotation: Subtype: Link Contents: http://refhub.elsevier.com/S1007-5704(26)00956-1/sbref0009 Rect: 55, 477, 64, 482 Annotation: Subtype: Link Contents: http://refhub.elsevier.com/S1007-5704(26)00956-1/sbref0010 Rect: 54, 467, 506, 475 Annotation: Subtype: Link Contents: http://refhub.elsevier.com/S1007-5704(26)00956-1/sbref0010 Rect: 55, 460, 77, 466 Annotation: Subtype: Link Contents: http://refhub.elsevier.com/S1007-5704(26)00956-1/sbref0011 Rect: 54, 451, 379, 459 Annotation: Subtype: Link Contents: http://refhub.elsevier.com/S1007-5704(26)00956-1/sbref0012 Rect: 54, 443, 433, 451 Annotation: Subtype: Link Contents: http://refhub.elsevier.com/S1007-5704(26)00956-1/sbref0013 Rect: 54, 435, 394, 443 Annotation: Subtype: Link Contents: http://refhub.elsevier.com/S1007-5704(26)00956-1/sbref0014 Rect: 54, 427, 506, 435 Annotation: Subtype: Link Contents: http://refhub.elsevier.com/S1007-5704(26)00956-1/sbref0014 Rect: 55, 420, 225, 426 Annotation: Subtype: Link Contents: http://refhub.elsevier.com/S1007-5704(26)00956-1/sbref0015 Rect: 54, 411, 506, 419 Annotation: Subtype: Link Contents: http://refhub.elsevier.com/S1007-5704(26)00956-1/sbref0015 Rect: 55, 404, 127, 410 Annotation: Subtype: Link Contents: http://refhub.elsevier.com/S1007-5704(26)00956-1/sbref0016 Rect: 54, 395, 448, 403 Annotation: Subtype: Link Contents: http://refhub.elsevier.com/S1007-5704(26)00956-1/sbref0017 Rect: 54, 387, 506, 396 Annotation: Subtype: Link Contents: http://refhub.elsevier.com/S1007-5704(26)00956-1/sbref0017 Rect: 55, 381, 96, 386 Annotation: Subtype: Link Contents: http://refhub.elsevier.com/S1007-5704(26)00956-1/sbref0018 Rect: 54, 371, 487, 380 Annotation: Subtype: Link Contents: http://refhub.elsevier.com/S1007-5704(26)00956-1/sbref0019 Rect: 54, 363, 317, 373 Annotation: Subtype: Link Contents: http://refhub.elsevier.com/S1007-5704(26)00956-1/sbref0020 Rect: 54, 355, 314, 365 Annotation: Subtype: Link Contents: http://refhub.elsevier.com/S1007-5704(26)00956-1/sbref0021 Rect: 54, 347, 346, 356 Annotation: Subtype: Link Contents: http://refhub.elsevier.com/S1007-5704(26)00956-1/sbref0022 Rect: 54, 340, 374, 348 Annotation: Subtype: Link Contents: http://refhub.elsevier.com/S1007-5704(26)00956-1/sbref0023 Rect: 54, 332, 506, 340 Annotation: Subtype: Link Contents: http://refhub.elsevier.com/S1007-5704(26)00956-1/sbref0023 Rect: 55, 325, 156, 331 Annotation: Subtype: Link Contents: http://refhub.elsevier.com/S1007-5704(26)00956-1/sbref0024 Rect: 54, 316, 506, 324 Annotation: Subtype: Link Contents: http://refhub.elsevier.com/S1007-5704(26)00956-1/sbref0024 Rect: 55, 309, 191, 315 Annotation: Subtype: Link Contents: http://refhub.elsevier.com/S1007-5704(26)00956-1/sbref0025 Rect: 54, 300, 429, 308 Annotation: Subtype: Link Contents: http://refhub.elsevier.com/S1007-5704(26)00956-1/sbref0026 Rect: 54, 292, 336, 300 Annotation: Subtype: Link Contents: http://refhub.elsevier.com/S1007-5704(26)00956-1/sbref0027 Rect: 54, 284, 354, 292 Annotation: Subtype: Link Contents: http://refhub.elsevier.com/S1007-5704(26)00956-1/sbref0028 Rect: 54, 276, 451, 284 Annotation: Subtype: Link Contents: http://refhub.elsevier.com/S1007-5704(26)00956-1/sbref0029 Rect: 54, 268, 506, 276 Annotation: Subtype: Link Contents: http://refhub.elsevier.com/S1007-5704(26)00956-1/sbref0029 Rect: 55, 261, 423, 267 Annotation: Subtype: Link Contents: http://refhub.elsevier.com/S1007-5704(26)00956-1/sbref0030 Rect: 54, 252, 343, 260 Annotation: Subtype: Link Contents: http://refhub.elsevier.com/S1007-5704(26)00956-1/sbref0031 Rect: 54, 244, 350, 252 Annotation: Subtype: Link Contents: http://refhub.elsevier.com/S1007-5704(26)00956-1/sbref0032 Rect: 54, 236, 342, 244 Checksum: 55f8484c Type: CRC32 Checksum: 1a1d3c0cbd5d6054a71ecb9ce1842ce9 Type: MD5 Checksum: d8862e4471dcbbe90a59a534300d2b02551fdb3b Type: SHA-1 Checksum: 364226f11db6fdbd47d3974a3d87bc5e8e910cec21750e0f3b35b0bdee6244e8 Type: SHA-256