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+Preprint to appear in the Proceedings of the 7th Arabic Natural Language Processing Workshop (WANLP), 2022.
+EMNLP, Abu Dhabi, United Arab Emirates, December 7–11, 2022.
+Beyond Arabic: Software for Perso-Arabic Script Manipulation
+Alexander Gutkin† Cibu Johny† Raiomond Doctor‡∗ Brian Roark◦ Richard Sproat⊛
+Google Research
+†United Kingdom
+‡India
+◦United States
+⊛Japan
+{agutkin,cibu,raiomond,roark,rws}@google.com
+Abstract
+This paper presents an open-source software
+library that provides a set of finite-state trans-
+ducer (FST) components and corresponding
+utilities for manipulating the writing sys-
+tems of languages that use the Perso-Arabic
+script.
+The operations include various lev-
+els of script normalization, including visual
+invariance-preserving operations that subsume
+and go beyond the standard Unicode normal-
+ization forms, as well as transformations that
+modify the visual appearance of characters in
+accordance with the regional orthographies for
+eleven contemporary languages from diverse
+language families. The library also provides
+simple FST-based romanization and transliter-
+ation. We additionally attempt to formalize the
+typology of Perso-Arabic characters by provid-
+ing one-to-many mappings from Unicode code
+points to the languages that use them. While
+our work focuses on the Arabic script diaspora
+rather than Arabic itself, this approach could
+be adopted for any language that uses the Ara-
+bic script, thus providing a unified framework
+for treating a script family used by close to a
+billion people.
+1
+Introduction
+While originally developed for recording Arabic,
+the Perso-Arabic script has gradually become one
+of the most widely used modern scripts. Through-
+out history the script was adapted to record many
+languages from diverse language families, with
+scores of adaptations still active today. This flexi-
+bility is partly due to the core features of the script
+itself which over the time evolved from a purely
+consonantal script to include a productive system
+of diacritics for representing long vowels and op-
+tional marking of short vowels and phonologi-
+cal processes such as gemination (Bauer, 1996;
+Kurzon, 2013).
+Consequently, many languages
+productively evolved their own adaptation of the
+∗ On contract from Optimum Solutions, Inc.
+Perso-Arabic script to better suit their phonology
+by not only augmenting the set of diacritics but
+also introducing new consonant shapes.
+This paper presents an open-source software li-
+brary designed to deal with the ambiguities and
+inconsistencies that result from representing var-
+ious regional Perso-Arabic adaptations in digital
+media. Some of these issues are due to the Uni-
+code standard itself, where a Perso-Arabic char-
+acter can often be represented in more than one
+way (Unicode Consortium, 2021). Others are due
+to the lack or inadequacies of input methods and
+the instability of modern orthographies for the lan-
+guages in question (Aazim et al., 2009; Liljegren,
+2018).
+Such issues percolate through the data
+available online, such as Wikipedia and Common
+Crawl (Patel, 2020), negatively impacting the qual-
+ity of NLP models built with such data. The script
+normalization software described below goes be-
+yond the standard language-agnostic Unicode ap-
+proach for Perso-Arabic to help alleviate some of
+these issues.
+The library design is inspired by and consis-
+tent with prior work by Johny et al. (2021), in-
+troduced in §2, who provided a suite of finite-
+state grammars for various normalization and (re-
+versible) romanization operations for the Brah-
+mic family of scripts.1
+While the Perso-Arabic
+script and the respective set of regional orthogra-
+phies we support – Balochi, Kashmiri, Kurdish
+(Sorani), Malay (Jawi), Pashto, Persian, Punjabi
+(Shahmukhi), Sindhi, South Azerbaijani, Urdu
+and Uyghur – is significantly different from those
+Brahmic scripts, we pursue a similar finite-state in-
+terpretation,2 as described in §3. Implementation
+details and simple validation are provided in §4.
+1https://github.com/google-research/nisaba
+2https://github.com/google-research/nisaba/
+tree/main/nisaba/scripts/abjad alphabet
+
+2
+Related Work
+The approach we take in this paper follows in
+spirit the work of Johny et al. (2021) and Gutkin
+et al. (2022), who developed a finite-state script
+normalization framework for Brahmic scripts. We
+adopt their taxonomy and terminology of low-
+level script normalization operations, which con-
+sist of three types: Unicode-endorsed schemes,
+such as NFC; further visually-invariant transfor-
+mations (visual normalization); and transforma-
+tions that modify a character’s shape but preserve
+pronunciation and the overall word identity (read-
+ing normalization).
+The literature on Perso-Arabic script normal-
+ization for languages we cover in this paper is
+scarce. The most relevant work was carried out
+by Ahmadi (2020) for Kurdish, who provides
+a detailed analysis of orthographic issues pecu-
+liar to Sorani Kurdish along with corresponding
+open-source script normalization software used
+in downstream NLP applications, such as neu-
+ral machine translation (Ahmadi and Masoud,
+2020). In the context of machine transliteration
+and spell checking, Lehal and Saini (2014) in-
+cluded language-agnostic minimal script normal-
+ization as a preprocessing step in their open-source
+n-gram-based transliterator from Perso-Arabic to
+Brahmic scripts. Bhatti et al. (2014) introduced
+a taxonomy of spelling errors for Sindhi, includ-
+ing an analysis of mistakes due to visually confus-
+able characters. Razak et al. (2018) provide a good
+overview of confusable characters for Malay Jawi
+orthography.
+For other languages the regional
+writing system ambiguities are sometimes men-
+tioned in passing, but do not constitute the main
+focus of work, as is the case with Punjabi Shah-
+mukhi (Lehal and Saini, 2012) and Urdu (Humay-
+oun et al., 2022). The specific Perso-Arabic script
+ambiguities that abound in the online data are of-
+ten not exhaustively documented, particularly in
+work focused on multilingual modeling (N. C.,
+2022; Bapna et al., 2022). As one moves towards
+lesser-resourced languages, such as Kashmiri and
+Uyghur, the NLP literature provides no treatment
+of script normalization issues and the only reli-
+able sources of information are the proposal and
+discussion documents from the Unicode Techni-
+cal Committee (e.g., Bashir et al., 2006; Aazim
+et al., 2009; Pournader, 2014). A forthcoming pa-
+per by Doctor et al. (2022) covers the writing sys-
+tem differences between these languages in more
+Op. Type
+FST
+Language-dep.
+Includes
+NFC
+N
+no
+−
+Common Visual
+Vc
+no
+N
+Visual
+V
+yes
+Vc
+Reading
+R
+yes
+−
+Romanization
+M
+no
+Vc
+Transliteration
+T
+no
+−
+Table 1: Summary of script transformation operations.
+detail than we can include in this short paper.
+One area particularly relevant to this study is
+the work by the Internet Corporation for Assigned
+Names and Numbers (ICANN) towards develop-
+ing a robust set of standards for representing vari-
+ous Internet entities in Perso-Arabic script, such as
+domain names in URLs. Their particular focus is
+on variants, which are characters that are visually
+confusable due to identical appearance but differ-
+ent encoding, due to similarity in shape or due to
+common alternate spellings (ICANN, 2011). In
+addition, they developed the first proposal to sys-
+tematize the available Perso-Arabic Unicode code
+points along the regional lines (ICANN, 2015).
+These studies are particularly important for cyber-
+security (Hussain et al., 2016; Ginsberg and Yu,
+2018; Ahmad and Erdodi, 2021), but also inform
+this work.
+This software library is, to the best our knowl-
+edge, the first attempt to provide a principled ap-
+proach to Perso-Arabic script normalization for
+multiple languages, for downstream NLP applica-
+tions and beyond.
+3
+Design Methodology
+The core components are implemented as individ-
+ual FSTs that can be efficiently combined together
+in a single pipeline (Mohri, 2009).
+These are
+shown in Table 1 and described below.3
+Unicode Normalization
+For the Perso-Arabic
+string encodings which yield visually identical
+text, the Unicode standard provides procedures
+that normalize text to a conventionalized normal
+form, such as the well-known Normalization Form
+C (NFC), so that visually identical words are
+mapped to a conventionalized representative of
+their equivalence class (Whistler, 2021). We im-
+plemented the NFC standard as an FST, denoted
+N in Table 1, that handles three broad types of
+transformations: compositions, re-orderings and
+3When referring to names of Unicode characters we low-
+ercase them and omit the common prefix arabic (letter).
+
+FST
+Letter
+Variant (source)
+Canonical
+V∗
+l
+⟨ڑ⟩
+reh + small high tah
+rreh
+Vn
+l
+⟨ک⟩
+kaf
+keheh
+Vf
+l
+⟨ی⟩
+alef maksura
+farsi yeh
+Vi
+l
+⟨ہ⟩
+heh
+heh goal
+Table 2: Example FST components of Vl for Urdu.
+combinations thereof.
+As an example of a first type, consider the alef
+with madda above letter ⟨آ⟩ that can be composed
+in two ways: as a single character (U+0622) or
+by adjoining maddah above to alef ({ U+0627,
+U+0653 }). The FST N rewrites the adjoined form
+into its equivalent composed form. The second
+type of transformation involves the canonical re-
+ordering of the Arabic combining marks, for exam-
+ple, the sequence of shadda (U+0651) followed by
+kasra (U+0650) is reversed by N. More complex
+transformations that combine both compositions
+and re-orderings are possible. For example, the se-
+quence { alef (U+0627), superscript alef (U+0670),
+maddah above (U+0653) } normalizes to its equiv-
+alent form { alef with madda above (U+0622), su-
+perscript alef (U+0670) }.
+Crucially, N is language-agnostic because the
+NFC standard it implements does not define any
+transformations that violate the writing system
+rules of respective languages.
+Visual Normalization
+As mentioned in §2,
+Johny et al. (2021) introduced the term visual nor-
+malization in the context of Brahmic scripts to
+denote visually-invariant transformations that fall
+outside the scope of NFC. We adopt their defini-
+tion for Perso-Arabic, implementing it as a sin-
+gle language-dependent FST V, shown in Table 1,
+which is constructed by FST composition: V =
+N ◦ Vc ◦ Vl, where ◦ denotes the composition op-
+eration (Mohri, 2009).4
+The first FST after NFC, denoted Vc,
+is
+language-agnostic, constructed from a small set of
+normalizations for visually ambiguous sequences
+found online that apply to all languages in our li-
+brary.
+For example, we map the two-character
+sequence waw (U+0648) followed by damma
+(U+064F) or small damma (U+0619) to u (U+06C7).
+The second set of visually-invariant transforma-
+tions, denoted Vl, is language-specific and addi-
+tionally depends on the position within the word.
+Four special cases are distinguished that are rep-
+4See Johny et al. (2021) for details on FST composition
+and other operations used in this kind of script normalization.
+Op. Type
+FST
+# states
+# arcs
+# Kb
+NFC
+N
+156
+1557
+28.10
+Roman.
+M
+32 546
+52 257
+1487.10
+Translit.
+T
+340
+518
+15.15
+Table 3: Language-agnostic FSTs over UTF-8 strings.
+resented as FSTs: position-independent rewrites
+(V∗
+l ), isolated-letter rewrites (Vi
+l), rewrites in the
+word-final position (Vf
+l), and finally, rewrites in
+“non-final” word positions, which include visually-
+identical word-initial and word-medial rewrites
+(Vn
+l ). The FST Vl is composed as Vi
+l ◦Vf
+l ◦Vn
+l ◦V∗
+l .
+Some examples of these transformations for Urdu
+orthography are shown in Table 2, where the vari-
+ants shown in the third column are rewritten to
+their canonical Urdu form in the fourth column.
+Reading Normalization
+This type of normaliza-
+tion was introduced for Brahmic scripts by Gutkin
+et al. (2022), who noted that regional orthographic
+conventions or lack thereof, which oftentimes con-
+flict with each other, benefit from normalization
+to some accepted form. Whenever such normal-
+ization preserves visual invariance, it falls under
+the rubric of visual normalization, but other cases
+belong to reading normalization, denoted R in Ta-
+ble 1. Similar to visual normalization, R is com-
+piled from language-specific context-dependent
+rewrite rules. One example of such a rewrite is
+a mapping from yeh ⟨ي⟩(U+064A) to farsi yeh ⟨ی⟩
+(U+06CC) in Kashmiri, Persian, Punjabi, Sorani
+Kurdish and Urdu. For Malay, Sindhi and Uyghur,
+the inverse transformation is implemented as man-
+dated by the respective orthographies.
+For efficiency reasons R is stored independently
+of visual normalization V. At run-time, the read-
+ing normalization is applied to an input string s
+as s′ = (s ◦ V) ◦ R, which is more efficient than
+s′ = s ◦ R′, where R′ = V ◦ R.
+Romanization and Transliteration
+We also
+provide language-agnostic romanization (M) and
+transliteration (T ) FSTs. The FST M converts
+Perso-Arabic strings to their respective Latin rep-
+resentation in Unicode and is defined as M =
+N ◦ Vc ◦ Mc, where N and Vc were described
+above, and Mc implements a one-to-one mapping
+from 198 Perso-Arabic characters to their respec-
+tive romanizations using our custom romanization
+scheme derived from language-specific Library of
+Congress rules (LC, 2022) and various ISO stan-
+dards (ISO, 1984, 1993, 1999). For example, in
+
+Language Information
+Visual Normalization (V)
+Reading Normalization (R)
+Code
+Name
+# states
+# arcs
+# Mb
+# states
+# arcs
+# Mb
+azb
+South Azerbaijani
+315 933
+635 647
+16.49
+21
+735
+0.012
+bal
+Balochi
+620 226
+1 244 472
+32.31
+24
+738
+0.013
+ckb
+Kurdish (Sorani)
+1 097 937
+2 199 732
+57.15
+39
+753
+0.013
+fa
+Persian
+940 436
+1 884 347
+48.96
+36
+750
+0.013
+ks
+Kashmiri
+1 772 494
+3 547 448
+92.21
+44
+794
+0.014
+ms
+Malay
+199 777
+403 373
+10.45
+21
+735
+0.012
+pa
+Punjabi
+2 050 154
+4 105 465
+106.69
+24
+738
+0.013
+ps
+Pashto
+291 564
+587 552
+15.23
+24
+738
+0.013
+sd
+Sindhi
+1 703 726
+3 403 283
+88.53
+34
+748
+0.013
+ug
+Uyghur
+1 255 054
+2 513 231
+65.31
+24
+738
+0.013
+ur
+Urdu
+2 071 139
+4 138 950
+107.65
+31
+745
+0.013
+Table 4: Summary of FSTs over UTF-8 strings for visual and reading normalization.
+our scheme the Uyghur yu ⟨ۈ⟩(U+06C8) maps
+to ⟨¨u⟩.
+The transliteration FST T converts the
+strings from Unicode Latin into Perso-Arabic. It
+is smaller than M and is defined as T = M−1
+c .
+Character-Language Mapping
+The geography
+and scope of Perso-Arabic script adaptations is
+vast. To document the typology of the characters
+we developed an easy-to-parse mapping between
+the characters and the respective languages and/or
+macroareas that relate to a group of languages
+building on prior work by ICANN (2015). For ex-
+ample, using this mapping it is easy to find that
+the letter beh with small v below ⟨ࢠ⟩(U+08A0) is
+part of the orthography of Wolof, a language of
+Senegal (Ngom, 2010), while gaf with ring ⟨ڰ⟩
+(U+06B0) belongs to Saraiki language spoken in
+Pakistan (Bashir and Conners, 2019). This map-
+ping can be used to auto-generate the orthographic
+inventories for lesser-resourced languages.
+4
+Software Details and Validation
+Our software library is implemented using Pynini,
+a Python library for constructing finite-state gram-
+mars and for performing operations on FSTs (Gor-
+man, 2016; Gorman and Sproat, 2021).
+Each
+FST is compiled from the collections of individ-
+ual context-dependent letter rewrite rules (Mohri
+and Sproat, 1996) and is available in two versions:
+over an alphabet of UTF-8 encoded bytes and
+over the integer Unicode code points. The FSTs
+are stored uncompressed in binary FST archives
+(FARs) in OpenFst format (Allauzen et al., 2007).
+The
+summaries
+of
+language-agnostic
+and
+language-dependent FSTs over UTF-8 strings are
+shown in Table 3 and Table 4, respectively. As
+can be seen from the tables, the language-agnostic
+and reading normalization FSTs are relatively un-
+complicated and small in terms of number of
+Lang.
+s′ = s ◦ V
+s′ = (s ◦ V) ◦ R
+% tokens
+% types
+% tokens
+% types
+ckb
+18.27
+25.84
+30.07
+41.26
+sd
+17.32
+14.83
+21.74
+17.31
+ur
+0.09
+1.16
+0.10
+1.23
+Table 5: Percentage of tokens and types changed.
+states, arcs and the overall (uncompressed) size on
+disk. The visual normalization FSTs are signifi-
+cantly larger, which is explained by the number
+of composition operations used in their construc-
+tion (see §3). The reading normalization FSTs for
+South Azerbaijani and Malay shown in Table 4 im-
+plement the identity mapping. This is because we
+could not find enough examples requiring reading-
+style normalization in online data (see the Limita-
+tions section for more details).
+As an informal sanity check we validate the
+prevalence of normalization on word-frequency
+lists for Sorani Kurdish (ckb), Sindhi (sd) and
+Uyghur (ug) from project Cr´ubad´an (Scannell,
+2007). Table 5 shows the percentages of tokens
+and types changed (s′ ̸= s) by visual normaliza-
+tion on one hand and the combined visual and
+reading normalization on the other. Urdu has the
+fewest number of modifications compared to So-
+rani Kurdish and Sindhi, most likely due to a more
+regular orthography and stable input methods man-
+ifest in the crawled data. Significantly more ex-
+tensive analysis and experiments in statistical lan-
+guage modeling and neural machine translation for
+the languages covered in this paper are presented
+in a forthcoming study (Doctor et al., 2022).
+Example
+The use of the library is demonstrated
+by the following Python example that implements
+a simple command-line utility for performing read-
+ing normalization on a single string using Pynini
+APIs. The program requires two FAR files that
+
+Lang.
+Input
+Output
+Correct Output
+balٽﯿﺋدﺖﯿﺋد
+teh
+ckbﺮڪﺷەﻟﺮﮑﺷەﻟ
+keheh
+faﻪﺴﺳﺆﻣﻪﺴﺳﻮﻣ
+waw
+ksﮏﺗۍﮬﮏﺘؠﮬ
+kashmiri yeh
+paﻲﺌﮐﯽﺌﮐ
+farsi yeh
+sdﻪﻫﻮﮘﮧﮨﻮﮘ
+heh goal
+ugیﺎﺳيﺎﺳ
+yeh
+urةرﻮﺻۃرﻮﺻ
+teh marbuta goal
+Table 6: Some examples of reading normalization.
+store compiled visual and reading normalization
+grammars, the upper-case BCP-47 language code
+for retrieving the FST for a given language, and an
+input string:5
+example.py
+from absl import app
+from absl import flags
+from collections.abc import Iterable, Sequence
+import pynini as pyn
+flags.DEFINE_string("input", None, "Input string.")
+flags.DEFINE_string("lang", None, "Language code.")
+flags.DEFINE_string("reading_grm", None, "Reading FAR.")
+flags.DEFINE_string("visual_grm", None, "Visual FAR.")
+FLAGS = flags.FLAGS
+def load_fst(grammar_path: str, lang: str) -> pyn.Fst:
+"""Loads FST for specified grammar and language."""
+return pyn.Far(grammar_path)[lang]
+def apply(text: str, fsts: Iterable[pyn.Fst]) -> str:
+"""Applies sequence of FSTs on an input string."""
+try:
+composed = pyn.escape(text)
+for fst in fsts:
+composed = (composed @ fst).optimize()
+return pyn.shortestpath(composed).string()
+except pyn.FstOpError as error:
+raise ValueError(f"Error for string `{text}`")
+def main(argv: Sequence[str]) -> None:
+# ... initializing FLAGS
+visual_fst = load_fst(FLAGS.visual_grm, FLAGS.lang)
+reading_fst = load_fst(FLAGS.reading_grm, FLAGS.lang)
+out = apply(FLAGS.input, [visual_fst, reading_fst])
+print(f"=> {out}")
+if __name__ == "__main__":
+app.run(main)
+The visual and reading FSTs for a given language
+are retrieved from the relevant FAR files using
+load_fst function. The input string is first con-
+verted to a linear FST. The visual and reading nor-
+malization FSTs are then sequentially composed
+with the input FST and a shortest path algorithm is
+applied on the result, which is then converted from
+a linear FST back to a Python string in apply func-
+tion to yield the final normalized output.
+Some examples of reading normalization pro-
+5The infrastructure for compiling the Pynini grammars is
+described in Johny et al. (2021).
+duced using the example.py utility above for
+some of the supported languages are shown in Ta-
+ble 6. For each language, the input string in the
+second column of the table is normalized to a
+string shown in the third column. The final col-
+umn shows the name of a particular letter in the
+output string that replaced the original letter from
+the input string, e.g., for Sorani Kurdish (ckb)
+the following rewrite occurs: swash kaf (U+06AA)
+→ keheh (U+06A9), while for Punjabi (pa), yeh
+(U+064A) → farsi yeh (U+06CC).
+5
+Conclusion and Future Work
+We have presented a flexible FST-based software
+package for low-level processing of orthographies
+based on Perso-Arabic script. We described the
+main components of the architecture consisting
+of various script normalization operations, roman-
+ization/transliteration, and character-language in-
+dex.
+We expect to increase the current lan-
+guage coverage of eleven languages to further rel-
+atively well-documented orthographies, but also
+provide treatment for resource-scarce orthogra-
+phies, such as the Ajami orthographies of Sub-
+Saharan Africa (Mumin, 2014).
+Limitations
+When developing the visual and reading normal-
+ization rules for the eleven languages described in
+this paper we made use of publicly available on-
+line data consisting of the respective Wikipedias,
+Wikipron (Lee et al., 2020), Cr´ubad´an (Scannell,
+2007) and parts of Common Crawl (Patel, 2020).
+The latter corpus is particularly noisy and requires
+non-trivial filtering (Kreutzer et al., 2022). Fur-
+thermore, many Wikipedia and Common Crawl
+documents contain code-switched text in several
+languages that are recorded in Perso-Arabic. Ro-
+bust language identification (LID) is required to
+distinguish between tokens in such sentences (for
+example, Kashmiri vs. Pashto or Balochi) in or-
+der not to confuse between the respective orthogra-
+phies. Since we did not have access to robust LID
+models for the languages under study, for lesser-
+resourced languages such as Kashmiri, Malay in
+Jawi orthography, South Azerbaijani and Uyghur,
+it is likely that some of the words we used as exam-
+ples requiring normalization may have been mis-
+classified resulting in normalizations that should
+not be there.
+
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+

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf,len=446
+page_content='Preprint to appear in the Proceedings of the 7th Arabic Natural Language Processing Workshop (WANLP), 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  EMNLP, Abu Dhabi, United Arab Emirates, December 7–11, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  Beyond Arabic: Software for Perso-Arabic Script Manipulation  Alexander Gutkin† Cibu Johny† Raiomond Doctor‡∗ Brian Roark◦ Richard Sproat⊛  Google Research  †United Kingdom  ‡India  United States  ⊛Japan  {agutkin,cibu,raiomond,roark,rws}@google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='com  Abstract  This paper presents an open-source software  library that provides a set of finite-state trans-  ducer (FST) components and corresponding  utilities for manipulating the writing sys-  tems of languages that use the Perso-Arabic  script.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  The operations include various lev-  els of script normalization, including visual  invariance-preserving operations that subsume  and go beyond the standard Unicode normal-  ization forms, as well as transformations that  modify the visual appearance of characters in  accordance with the regional orthographies for  eleven contemporary languages from diverse  language families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' The library also provides  simple FST-based romanization and transliter-  ation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' We additionally attempt to formalize the  typology of Perso-Arabic characters by provid-  ing one-to-many mappings from Unicode code  points to the languages that use them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' While  our work focuses on the Arabic script diaspora  rather than Arabic itself, this approach could  be adopted for any language that uses the Ara-  bic script, thus providing a unified framework  for treating a script family used by close to a  billion people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  1  Introduction  While originally developed for recording Arabic,  the Perso-Arabic script has gradually become one  of the most widely used modern scripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' Through-  out history the script was adapted to record many  languages from diverse language families, with  scores of adaptations still active today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' This flexi-  bility is partly due to the core features of the script  itself which over the time evolved from a purely  consonantal script to include a productive system  of diacritics for representing long vowels and op-  tional marking of short vowels and phonologi-  cal processes such as gemination (Bauer, 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  Kurzon, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  Consequently, many languages  productively evolved their own adaptation of the  ∗ On contract from Optimum Solutions, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  Perso-Arabic script to better suit their phonology  by not only augmenting the set of diacritics but  also introducing new consonant shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  This paper presents an open-source software li-  brary designed to deal with the ambiguities and  inconsistencies that result from representing var-  ious regional Perso-Arabic adaptations in digital  media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' Some of these issues are due to the Uni-  code standard itself, where a Perso-Arabic char-  acter can often be represented in more than one  way (Unicode Consortium, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' Others are due  to the lack or inadequacies of input methods and  the instability of modern orthographies for the lan-  guages in question (Aazim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' Liljegren,  2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  Such issues percolate through the data  available online, such as Wikipedia and Common  Crawl (Patel, 2020), negatively impacting the qual-  ity of NLP models built with such data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' The script  normalization software described below goes be-  yond the standard language-agnostic Unicode ap-  proach for Perso-Arabic to help alleviate some of  these issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  The library design is inspired by and consis-  tent with prior work by Johny et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' (2021), in-  troduced in §2, who provided a suite of finite-  state grammars for various normalization and (re-  versible) romanization operations for the Brah-  mic family of scripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='1  While the Perso-Arabic  script and the respective set of regional orthogra-  phies we support – Balochi, Kashmiri, Kurdish  (Sorani), Malay (Jawi), Pashto, Persian, Punjabi  (Shahmukhi), Sindhi, South Azerbaijani, Urdu  and Uyghur – is significantly different from those  Brahmic scripts, we pursue a similar finite-state in-  terpretation,2 as described in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' Implementation  details and simple validation are provided in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='com/google-research/nisaba  2https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='com/google-research/nisaba/  tree/main/nisaba/scripts/abjad alphabet  2  Related Work  The approach we take in this paper follows in  spirit the work of Johny et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' (2021) and Gutkin  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' (2022), who developed a finite-state script  normalization framework for Brahmic scripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' We  adopt their taxonomy and terminology of low-  level script normalization operations, which con-  sist of three types: Unicode-endorsed schemes,  such as NFC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' further visually-invariant transfor-  mations (visual normalization);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' and transforma-  tions that modify a character’s shape but preserve  pronunciation and the overall word identity (read-  ing normalization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  The literature on Perso-Arabic script normal-  ization for languages we cover in this paper is  scarce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' The most relevant work was carried out  by Ahmadi (2020) for Kurdish, who provides  a detailed analysis of orthographic issues pecu-  liar to Sorani Kurdish along with corresponding  open-source script normalization software used  in downstream NLP applications, such as neu-  ral machine translation (Ahmadi and Masoud,  2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' In the context of machine transliteration  and spell checking, Lehal and Saini (2014) in-  cluded language-agnostic minimal script normal-  ization as a preprocessing step in their open-source  n-gram-based transliterator from Perso-Arabic to  Brahmic scripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' Bhatti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' (2014) introduced  a taxonomy of spelling errors for Sindhi, includ-  ing an analysis of mistakes due to visually confus-  able characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' Razak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' (2018) provide a good  overview of confusable characters for Malay Jawi  orthography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  For other languages the regional  writing system ambiguities are sometimes men-  tioned in passing, but do not constitute the main  focus of work, as is the case with Punjabi Shah-  mukhi (Lehal and Saini, 2012) and Urdu (Humay-  oun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' The specific Perso-Arabic script  ambiguities that abound in the online data are of-  ten not exhaustively documented, particularly in  work focused on multilingual modeling (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=',  2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' Bapna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' As one moves towards  lesser-resourced languages, such as Kashmiri and  Uyghur, the NLP literature provides no treatment  of script normalization issues and the only reli-  able sources of information are the proposal and  discussion documents from the Unicode Techni-  cal Committee (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=', Bashir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=', 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' Aazim  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' Pournader, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' A forthcoming pa-  per by Doctor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' (2022) covers the writing sys-  tem differences between these languages in more  Op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' Type  FST  Language-dep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  Includes  NFC  N  no  −  Common Visual  Vc  no  N  Visual  V  yes  Vc  Reading  R  yes  −  Romanization  M  no  Vc  Transliteration  T  no  −  Table 1: Summary of script transformation operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  detail than we can include in this short paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  One area particularly relevant to this study is  the work by the Internet Corporation for Assigned  Names and Numbers (ICANN) towards develop-  ing a robust set of standards for representing vari-  ous Internet entities in Perso-Arabic script, such as  domain names in URLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' Their particular focus is  on variants, which are characters that are visually  confusable due to identical appearance but differ-  ent encoding, due to similarity in shape or due to  common alternate spellings (ICANN, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' In  addition, they developed the first proposal to sys-  tematize the available Perso-Arabic Unicode code  points along the regional lines (ICANN, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  These studies are particularly important for cyber-  security (Hussain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' Ginsberg and Yu,  2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' Ahmad and Erdodi, 2021), but also inform  this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  This software library is, to the best our knowl-  edge, the first attempt to provide a principled ap-  proach to Perso-Arabic script normalization for  multiple languages, for downstream NLP applica-  tions and beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  3  Design Methodology  The core components are implemented as individ-  ual FSTs that can be efficiently combined together  in a single pipeline (Mohri, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  These are  shown in Table 1 and described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='3  Unicode Normalization  For the Perso-Arabic  string encodings which yield visually identical  text, the Unicode standard provides procedures  that normalize text to a conventionalized normal  form, such as the well-known Normalization Form  C (NFC), so that visually identical words are  mapped to a conventionalized representative of  their equivalence class (Whistler, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' We im-  plemented the NFC standard as an FST, denoted  N in Table 1, that handles three broad types of  transformations: compositions, re-orderings and  3When referring to names of Unicode characters we low-  ercase them and omit the common prefix arabic (letter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  FST  Letter  Variant (source)  Canonical  V∗  l  ⟨ڑ⟩  reh + small high tah  rreh  Vn  l  ⟨ک⟩  kaf  keheh  Vf  l  ⟨ی⟩  alef maksura  farsi yeh  Vi  l  ⟨ہ⟩  heh  heh goal  Table 2: Example FST components of Vl for Urdu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  combinations thereof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  As an example of a first type, consider the alef  with madda above letter ⟨آ⟩ that can be composed  in two ways: as a single character (U+0622) or  by adjoining maddah above to alef ({ U+0627,  U+0653 }).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' The FST N rewrites the adjoined form  into its equivalent composed form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' The second  type of transformation involves the canonical re-  ordering of the Arabic combining marks, for exam-  ple, the sequence of shadda (U+0651) followed by  kasra (U+0650) is reversed by N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' More complex  transformations that combine both compositions  and re-orderings are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' For example, the se-  quence { alef (U+0627), superscript alef (U+0670),  maddah above (U+0653) } normalizes to its equiv-  alent form { alef with madda above (U+0622), su-  perscript alef (U+0670) }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  Crucially, N is language-agnostic because the  NFC standard it implements does not define any  transformations that violate the writing system  rules of respective languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  Visual Normalization  As mentioned in §2,  Johny et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' (2021) introduced the term visual nor-  malization in the context of Brahmic scripts to  denote visually-invariant transformations that fall  outside the scope of NFC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' We adopt their defini-  tion for Perso-Arabic, implementing it as a sin-  gle language-dependent FST V, shown in Table 1,  which is constructed by FST composition: V =  N ◦ Vc ◦ Vl, where ◦ denotes the composition op-  eration (Mohri, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='4  The first FST after NFC, denoted Vc,  is  language-agnostic, constructed from a small set of  normalizations for visually ambiguous sequences  found online that apply to all languages in our li-  brary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  For example, we map the two-character  sequence waw (U+0648) followed by damma  (U+064F) or small damma (U+0619) to u (U+06C7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  The second set of visually-invariant transforma-  tions, denoted Vl, is language-specific and addi-  tionally depends on the position within the word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  Four special cases are distinguished that are rep-  4See Johny et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' (2021) for details on FST composition  and other operations used in this kind of script normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  Op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' Type  FST  # states  # arcs  # Kb  NFC  N  156  1557  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='10  Roman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  M  32 546  52 257  1487.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='10  Translit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  T  340  518  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='15  Table 3: Language-agnostic FSTs over UTF-8 strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  resented as FSTs: position-independent rewrites  (V∗  l ), isolated-letter rewrites (Vi  l), rewrites in the  word-final position (Vf  l), and finally, rewrites in  “non-final” word positions, which include visually-  identical word-initial and word-medial rewrites  (Vn  l ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' The FST Vl is composed as Vi  l ◦Vf  l ◦Vn  l ◦V∗  l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  Some examples of these transformations for Urdu  orthography are shown in Table 2, where the vari-  ants shown in the third column are rewritten to  their canonical Urdu form in the fourth column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  Reading Normalization  This type of normaliza-  tion was introduced for Brahmic scripts by Gutkin  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' (2022), who noted that regional orthographic  conventions or lack thereof, which oftentimes con-  flict with each other, benefit from normalization  to some accepted form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' Whenever such normal-  ization preserves visual invariance, it falls under  the rubric of visual normalization, but other cases  belong to reading normalization, denoted R in Ta-  ble 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' Similar to visual normalization, R is com-  piled from language-specific context-dependent  rewrite rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' One example of such a rewrite is  a mapping from yeh ⟨ي⟩(U+064A) to farsi yeh ⟨ی⟩  (U+06CC) in Kashmiri, Persian, Punjabi, Sorani  Kurdish and Urdu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' For Malay, Sindhi and Uyghur,  the inverse transformation is implemented as man-  dated by the respective orthographies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  For efficiency reasons R is stored independently  of visual normalization V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' At run-time, the read-  ing normalization is applied to an input string s  as s′ = (s ◦ V) ◦ R, which is more efficient than  s′ = s ◦ R′, where R′ = V ◦ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  Romanization and Transliteration  We also  provide language-agnostic romanization (M) and  transliteration (T ) FSTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' The FST M converts  Perso-Arabic strings to their respective Latin rep-  resentation in Unicode and is defined as M =  N ◦ Vc ◦ Mc, where N and Vc were described  above, and Mc implements a one-to-one mapping  from 198 Perso-Arabic characters to their respec-  tive romanizations using our custom romanization  scheme derived from language-specific Library of  Congress rules (LC, 2022) and various ISO stan-  dards (ISO, 1984, 1993, 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' For example, in  Language Information  Visual Normalization (V)  Reading Normalization (R)  Code  Name  # states  # arcs  # Mb  # states  # arcs  # Mb  azb  South Azerbaijani  315 933  635 647  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='49  21  735  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='012  bal  Balochi  620 226  1 244 472  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='31  24  738  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='013  ckb  Kurdish (Sorani)  1 097 937  2 199 732  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='15  39  753  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='013  fa  Persian  940 436  1 884 347  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='96  36  750  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='013  ks  Kashmiri  1 772 494  3 547 448  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='21  44  794  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='014  ms  Malay  199 777  403 373  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='45  21  735  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='012  pa  Punjabi  2 050 154  4 105 465  106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='69  24  738  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='013  ps  Pashto  291 564  587 552  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='23  24  738  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='013  sd  Sindhi  1 703 726  3 403 283  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='53  34  748  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='013  ug  Uyghur  1 255 054  2 513 231  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='31  24  738  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='013  ur  Urdu  2 071 139  4 138 950  107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='65  31  745  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='013  Table 4: Summary of FSTs over UTF-8 strings for visual and reading normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  our scheme the Uyghur yu ⟨ۈ⟩(U+06C8) maps  to ⟨¨u⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  The transliteration FST T converts the  strings from Unicode Latin into Perso-Arabic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' It  is smaller than M and is defined as T = M−1  c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  Character-Language Mapping  The geography  and scope of Perso-Arabic script adaptations is  vast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' To document the typology of the characters  we developed an easy-to-parse mapping between  the characters and the respective languages and/or  macroareas that relate to a group of languages  building on prior work by ICANN (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' For ex-  ample, using this mapping it is easy to find that  the letter beh with small v below ⟨ࢠ⟩(U+08A0) is  part of the orthography of Wolof, a language of  Senegal (Ngom, 2010), while gaf with ring ⟨ڰ⟩  (U+06B0) belongs to Saraiki language spoken in  Pakistan (Bashir and Conners, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' This map-  ping can be used to auto-generate the orthographic  inventories for lesser-resourced languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  4  Software Details and Validation  Our software library is implemented using Pynini,  a Python library for constructing finite-state gram-  mars and for performing operations on FSTs (Gor-  man, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' Gorman and Sproat, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  Each  FST is compiled from the collections of individ-  ual context-dependent letter rewrite rules (Mohri  and Sproat, 1996) and is available in two versions:  over an alphabet of UTF-8 encoded bytes and  over the integer Unicode code points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' The FSTs  are stored uncompressed in binary FST archives  (FARs) in OpenFst format (Allauzen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=', 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  The  summaries  of  language-agnostic  and  language-dependent FSTs over UTF-8 strings are  shown in Table 3 and Table 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' As  can be seen from the tables, the language-agnostic  and reading normalization FSTs are relatively un-  complicated and small in terms of number of  Lang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  s′ = s ◦ V  s′ = (s ◦ V) ◦ R  % tokens  % types  % tokens  % types  ckb  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='27  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='84  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='07  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='26  sd  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='32  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='83  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='74  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='31  ur  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='09  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='10  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='23  Table 5: Percentage of tokens and types changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  states, arcs and the overall (uncompressed) size on  disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' The visual normalization FSTs are signifi-  cantly larger, which is explained by the number  of composition operations used in their construc-  tion (see §3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' The reading normalization FSTs for  South Azerbaijani and Malay shown in Table 4 im-  plement the identity mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' This is because we  could not find enough examples requiring reading-  style normalization in online data (see the Limita-  tions section for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  As an informal sanity check we validate the  prevalence of normalization on word-frequency  lists for Sorani Kurdish (ckb), Sindhi (sd) and  Uyghur (ug) from project Cr´ubad´an (Scannell,  2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' Table 5 shows the percentages of tokens  and types changed (s′ ̸= s) by visual normaliza-  tion on one hand and the combined visual and  reading normalization on the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' Urdu has the  fewest number of modifications compared to So-  rani Kurdish and Sindhi, most likely due to a more  regular orthography and stable input methods man-  ifest in the crawled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' Significantly more ex-  tensive analysis and experiments in statistical lan-  guage modeling and neural machine translation for  the languages covered in this paper are presented  in a forthcoming study (Doctor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  Example  The use of the library is demonstrated  by the following Python example that implements  a simple command-line utility for performing read-  ing normalization on a single string using Pynini  APIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' The program requires two FAR files that  Lang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  Input  Output  Correct Output  balٽﯿﺋدﺖﯿﺋد  teh  ckbﺮڪﺷەﻟﺮﮑﺷەﻟ  keheh  faﻪﺴﺳﺆﻣﻪﺴﺳﻮﻣ  waw  ksﮏﺗۍﮬﮏﺘؠﮬ  kashmiri yeh  paﻲﺌﮐﯽﺌﮐ  farsi yeh  sdﻪﻫﻮﮘﮧﮨﻮﮘ  heh goal  ugیﺎﺳيﺎﺳ  yeh  urةرﻮﺻۃرﻮﺻ  teh marbuta goal  Table 6: Some examples of reading normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  store compiled visual and reading normalization  grammars, the upper-case BCP-47 language code  for retrieving the FST for a given language, and an  input string:5  example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='py  from absl import app  from absl import flags  from collections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='abc import Iterable, Sequence  import pynini as pyn  flags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='DEFINE_string("input", None, "Input string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='")  flags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='DEFINE_string("lang", None, "Language code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='")  flags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='DEFINE_string("reading_grm", None, "Reading FAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='")  flags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='DEFINE_string("visual_grm", None, "Visual FAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='")  FLAGS = flags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='FLAGS  def load_fst(grammar_path: str, lang: str) -> pyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='Fst:  """Loads FST for specified grammar and language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='"""  return pyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='Far(grammar_path)[lang]  def apply(text: str, fsts: Iterable[pyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='Fst]) -> str:  """Applies sequence of FSTs on an input string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='"""  try:  composed = pyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='escape(text)  for fst in fsts:  composed = (composed @ fst).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='optimize()  return pyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='shortestpath(composed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='string()  except pyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='FstOpError as error:  raise ValueError(f"Error for string `{text}`")  def main(argv: Sequence[str]) -> None:  # .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' initializing FLAGS  visual_fst = load_fst(FLAGS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='visual_grm, FLAGS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='lang)  reading_fst = load_fst(FLAGS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='reading_grm, FLAGS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='lang)  out = apply(FLAGS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='input, [visual_fst, reading_fst])  print(f"=> {out}")  if __name__ == "__main__":  app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='run(main)  The visual and reading FSTs for a given language  are retrieved from the relevant FAR files using  load_fst function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' The input string is first con-  verted to a linear FST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' The visual and reading nor-  malization FSTs are then sequentially composed  with the input FST and a shortest path algorithm is  applied on the result, which is then converted from  a linear FST back to a Python string in apply func-  tion to yield the final normalized output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  Some examples of reading normalization pro-  5The infrastructure for compiling the Pynini grammars is  described in Johny et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  duced using the example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='py utility above for  some of the supported languages are shown in Ta-  ble 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' For each language, the input string in the  second column of the table is normalized to a  string shown in the third column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' The final col-  umn shows the name of a particular letter in the  output string that replaced the original letter from  the input string, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=', for Sorani Kurdish (ckb)  the following rewrite occurs: swash kaf (U+06AA)  → keheh (U+06A9), while for Punjabi (pa), yeh  (U+064A) → farsi yeh (U+06CC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  5  Conclusion and Future Work  We have presented a flexible FST-based software  package for low-level processing of orthographies  based on Perso-Arabic script.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' We described the  main components of the architecture consisting  of various script normalization operations, roman-  ization/transliteration, and character-language in-  dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  We expect to increase the current lan-  guage coverage of eleven languages to further rel-  atively well-documented orthographies, but also  provide treatment for resource-scarce orthogra-  phies, such as the Ajami orthographies of Sub-  Saharan Africa (Mumin, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  Limitations  When developing the visual and reading normal-  ization rules for the eleven languages described in  this paper we made use of publicly available on-  line data consisting of the respective Wikipedias,  Wikipron (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=', 2020), Cr´ubad´an (Scannell,  2007) and parts of Common Crawl (Patel, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  The latter corpus is particularly noisy and requires  non-trivial filtering (Kreutzer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' Fur-  thermore, many Wikipedia and Common Crawl  documents contain code-switched text in several  languages that are recorded in Perso-Arabic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' Ro-  bust language identification (LID) is required to  distinguish between tokens in such sentences (for  example, Kashmiri vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' Pashto or Balochi) in or-  der not to confuse between the respective orthogra-  phies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' Since we did not have access to robust LID  models for the languages under study, for lesser-  resourced languages such as Kashmiri, Malay in  Jawi orthography, South Azerbaijani and Uyghur,  it is likely that some of the words we used as exam-  ples requiring normalization may have been mis-  classified resulting in normalizations that should  not be there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  References  Muzaffar Aazim, Kamal Mansour, and Roozbeh Pour-  nader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' Proposal to add two Kashmiri charac-  ters and one annotation to the Arabic block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' Techni-  cal Report L2/09-176, Unicode Consortium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  Humza Ahmad and Laszlo Erdodi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' Overview of  phishing landscape and homographs in Arabic do-  main names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' Security and Privacy, 4(4):1–14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  Sina Ahmadi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' KLPT – Kurdish language pro-  cessing toolkit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' In Proceedings of Second Workshop  for NLP Open Source Software (NLP-OSS), pages  72–84, Online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' Association for Computational Lin-  guistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  Sina Ahmadi and Maraim Masoud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  Towards  machine translation for the Kurdish language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  In  Proceedings of the 3rd Workshop on Technologies  for MT of Low Resource Languages, pages 87–98,  Suzhou, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' Association for Computational Lin-  guistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  Cyril Allauzen, Michael Riley, Johan Schalkwyk, Woj-  ciech Skut, and Mehryar Mohri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
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+page_content='  Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  Ankur Bapna, Isaac Caswell, Julia Kreutzer, Orhan Fi-  rat, Daan van Esch, Aditya Siddhant, Mengmeng  Niu, Pallavi Baljekar, Xavier Garcia, Wolfgang  Macherey, Theresa Breiner, Vera Axelrod, Jason  Riesa, Yuan Cao, Mia Xu Chen, Klaus Macherey,  Maxim Krikun, Pidong Wang, Alexander Gutkin,  Apurva Shah,  Yanping Huang,  Zhifeng Chen,  Yonghui Wu, and Macduff Hughes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' Building  machine translation systems for the next thousand  languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
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+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
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+page_content=' Technical Report L2-06/149, Uni-  code Consortium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
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+page_content=' In Peter Daniels  and William Bright, editors, The World’s Writing  Systems, chapter 50, pages 559–563.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' Oxford Univer-  sity Press, Oxford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  Zeeshan Bhatti, Imdad Ali Ismaili, Asad Ali Shaikh,  and Waseem Javaid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' Spelling error trends and  patterns in Sindhi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
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+page_content='  Raiomond Doctor, Alexander Gutkin, Cibu Johny,  Brian Roark, and Richard Sproat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' Graphemic  normalization of the Perso-Arabic script.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  In Pro-  ceedings of Grapholinguistics in the 21st Century  (G21C), Paris, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' In press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
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+page_content='  Rapid homoglyph  prediction and detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
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+page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  Kyle Gorman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  Pynini: A Python library for  weighted finite-state grammar compilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' In Pro-  ceedings of the SIGFSM Workshop on Statistical  NLP and Weighted Automata, pages 75–80, Berlin,  Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' Association for Computational Linguis-  tics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
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+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' Finite-State  Text Processing, volume 14 of Synthesis Lectures on  Human Language Technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  Morgan & Clay-  pool Publishers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  Alexander Gutkin, Cibu Johny, Raiomond Doctor,  Lawrence Wolf-Sonkin, and Brian Roark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' Ex-  tensions to Brahmic script processing within the Nis-  aba library: new scripts, languages and utilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' In  Proceedings of the Thirteenth Language Resources  and Evaluation Conference, pages 6450–6460, Mar-  seille, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' European Language Resources Asso-  ciation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  Muhammad Humayoun, Harald Hammarstr¨om, and  Aarne Ranta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
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+page_content='  Sarmad Hussain, Ahmed Bakhat, Nabil Benamar,  Meikal Mumin, and Inam Ullah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
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+page_content='  ICANN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' Arabic case study team: Arabic case  study team issues report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' Internationalized Domain  Names (IDN) Variant Issues project, Internet Corpo-  ration for Assigned Names and Numbers (ICANN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  ICANN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
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+page_content=' Task force on Arabic script IDN (TF-  AIDN): Proposal for Arabic script Root Zone LGR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
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+page_content=' Lee, Lucas F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
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+page_content=' In Proceedings of COLING 2012:  Posters, pages 633–642, Mumbai, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' The COL-  ING 2012 Organizing Committee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
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+page_content=' Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
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+page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
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+page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' Ajami scripts in the Senegalese  speech community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' Journal of Arabic and Islamic  Studies, 10:1–23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content='  Jay M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
+page_content=' Patel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
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+page_content=' Introduction to Common Crawl  datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
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+page_content='  Sitti Munirah Abdul Razak, Muhamad Sadry Abu Se-  man, Wan Ali Wan Yusoff Wan Mamat, and Noor  Hasrul Nizan Mohammad Noor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'}
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+Anomaly Segmentation for High-Resolution Remote Sensing Images 
+Based on Pixel Descriptors 
+Jingtao Li1, Xinyu Wang2*, Hengwei Zhao1, Shaoyu Wang1, Yanfei Zhong1 
+1 State Key Laboratory of Information Engineering in Surveying, Mapping and Remote Sensing, Wuhan University, P. R. China 
+2 School of Remote Sensing and Information Engineering, Wuhan University, P. R. China 
+{JingtaoLi, wangxinyu, whu_zhaohw, wangshaoyu, zhongyanfei}@whu.edu.cn 
+ 
+ 
+Abstract 
+Anomaly segmentation in high spatial resolution (HSR) re-
+mote sensing imagery is aimed at segmenting anomaly pat-
+terns of the earth deviating from normal patterns, which plays 
+an important role in various Earth vision applications. How-
+ever, it is a challenging task due to the complex distribution 
+and the irregular shapes of objects, and the lack of abnormal 
+samples. To tackle these problems, an anomaly segmentation 
+model based on pixel descriptors (ASD) is proposed for 
+anomaly segmentation in HSR imagery. Specifically, deep 
+one-class classification is introduced for anomaly segmenta-
+tion in the feature space with discriminative pixel descriptors. 
+The ASD model incorporates the data argument for generat-
+ing virtual abnormal samples, which can force the pixel de-
+scriptors to be compact for normal data and meanwhile to be 
+diverse to avoid the model collapse problems when only pos-
+itive samples participated in the training. In addition, the 
+ASD introduced a multi-level and multi-scale feature extrac-
+tion strategy for learning the low-level and semantic infor-
+mation to make the pixel descriptors feature-rich.  The pro-
+posed ASD model was validated using four HSR datasets and 
+compared with the recent state-of-the-art models, showing its 
+potential value in Earth vision applications. 
+1. Introduction  
+Anomaly segmentation is aimed at segmenting the anomaly 
+patterns which deviate from the normal patterns (Pimentel 
+et al. 2014; Pang et al. 2021). Due to the lack of abnormal 
+samples, anomaly segmentation is a challenging task, but 
+plays an important role in many computer vision applica-
+tions, including medical analysis (Fernando et al. 2021), in-
+dustrial defect detection (Bergmann et al. 2019), video sur-
+veillance (Liu, Li, and Poczos 2018), and environmental 
+monitoring (Miau and Hung 2020). 
+ Anomaly segmentation in high spatial resolution (HSR) 
+remote sensing images (e.g., Figure 1) is a powerful tool for 
+environmental monitoring (Miau and Hung 2020; Wang et 
+al. 2019). Despite this, few related works have focused on 
+anomaly segmentation in HSR imagery because of the 
+ 
+*Corresponding author 
+ 
+unique characteristics when compared to the industrial and 
+medical images used in most anomaly segmentation tasks, 
+which have a regular structure. The objects in HSR images 
+typically have a more complex spatial distribution and large 
+radiation differences within the same class. Furthermore, 
+since HSR images can be captured in different angles and 
+heights, the objects always have multiple scales and show 
+rotation invariance. These characteristics make anomaly 
+segmentation for HSR images a challenging task. 
+ The mainstream anomaly segmentation models detect 
+anomalies in the image space, where the anomaly score is 
+computed based on the image pixel values. Typical exam-
+ples are the autoencoder (AE)-based models (Zavrtanik, 
+Kristan, and Skoˇcaj 2021; Gong et al. 2019) and the gener-
+ative adversarial network (GAN)-based models (Ngoetal. 
+2019; Zenatietal. 2018b). AE-based models assume that 
+normal samples can be reconstructed more easily than the 
+anomalous ones and the reconstruction error indicates the 
+anomaly segmentation score (Pang et al. 2021). However, 
+the low-level reconstruction error has been shown to focus 
+on the pixel-wise error, resulting in abnormal samples also 
+being reconstructed, especially when the normal distribution 
+is complex. (Fei et al. 2020; Gong et al. 2019; Zong et al. 
+Code is available at https://github.com/Jingtao-Li-CVer/ASD. 
+ 
+ 
+ 
+Figure 1: Anomaly segmentation example for HSR re-
+mote sensing images using proposed model. In the forest 
+scene, the common forest pattern is considered as normal, 
+and the abnormal objects such as diseased trees (the first 
+row) and the house in the forest (the second row) are iden-
+tified as anomalies. 
+Anomaly image
+Normal images
+Anomaly label
+Anomaly map
+…
+
+2018). GAN-based models detect anomalies from the gen-
+eration performance (Akcay, Atapour-Abarghouei, and 
+Breckon 2018; Ngo et al. 2019; Xia et al. 2022), where the 
+superior capability in generating image data also empowers 
+the detection of abnormal samples (Pang et al. 2021). In 
+spite of this, the complex distribution of HSR images can 
+make the generator generate data instances that are out of 
+the manifold of normal instances (Pang et al. 2021). 
+ Differing from the AE-based and GAN-based models, 
+deep one-class classification (OCC)-based methods detect 
+the anomalies in the feature space (Shi, Yang, and Qi 2021; 
+Lei et al. 2021; Zhao et al. 2022; Li et al. 2022), where the 
+anomaly score is computed based on the extracted image de-
+scriptors. These methods aim to learn discriminative de-
+scriptors in the training stage and compute the anomaly 
+score in the feature space using a measurement such as the 
+Mahalanobis or Euclidean distance (Reiss et al. 2021; Ruff 
+et al. 2018; Shi, Yang, and Qi 2021). Because deep OCC-
+based methods focus on semantic features rather than low-
+level pixel errors, it is more suitable to deal with the anom-
+aly segmentation task in HSR imagery which has complex 
+distribution. However, two barriers exist when applying ex-
+isting methods directly. (i) Due to the lack of abnormal sam-
+ples, the model training only uses normal samples and is op-
+timized to be compact (Ruff et al. 2018; Chalapathy, Menon, 
+and Chawla 2018), which can easily result in the model col-
+lapse problem (Reiss et al. 2021). (ii) The anomalies in HSR 
+imagery have rich low-level (e.g., texture) and high-level 
+(e.g., semantic) features, which are both important for the 
+anomaly segmentation task and real application. Although 
+the current deep OCC models can capture useful semantic 
+features, they perform suboptimally than models detecting 
+in the image space for samples with regular structures (Li et 
+al. 2021), because low-level features are mostly forgotten in 
+feature space. 
+ In this paper, we tackle the two problems for the anomaly 
+segmentation task using HSR images. A novel anomaly seg-
+mentation model based on pixel descriptors (ASD) is pro-
+posed. (i) In addition to considering the compact property of 
+the obtained descriptors, the ASD model encourages de-
+scriptors to be diverse by increasing the descriptor distance 
+between the original image and the transformed image with 
+the use of data augmentation techniques. The transformed 
+descriptors act as anomalies, to some extent, which en-
+hances the anomaly detection ability and prevents simulta-
+neous model collapse. (ii) To make the descriptor feature-
+rich, a descriptor at different scales is fused for each pixel, 
+and an auxiliary reconstruction head is designed to force the 
+descriptor to remember the low-level features. Compact, di-
+verse, and feature-rich property optimizes the model to-
+gether from the perspective of the feature distance and fea-
+ture quantity. ASD sets the first baseline for the anomaly 
+segmentation task in HSR imagery. 
+ The ASD model was validated on four HSR datasets: the 
+DeepGlobe land-cover segmentation dataset, the Agricul-
+ture-Vision agriculture pattern segmentation dataset, the 
+Landslide4Sense landslide detection dataset, and the forest 
+anomaly detection dataset (FAS, made by ourselves). The 
+ASD model showed an obvious superiority over the recent 
+state-of-the-art anomaly segmentation models (with an area 
+under the curve (AUC) improvement of 5–10 points in most 
+cases). The results obtained on the Landslide4Sense and 
+FAS datasets confirmed the great application potential of the 
+ASD model in disaster detection and forest monitoring. 
+2. Related Work 
+AE-based models are always composed of an encoding and 
+decoding network, with the aim being to reconstruct the 
+original input data (Pimentel et al. 2014). Hawkins et al. 
+(2022) first introduced the AE into the anomaly detection 
+field, where the features learned in the latent space can be 
+used to distinguish normal and anomalous data. The recon-
+struction error is considered as the anomaly degree and the 
+mean square error (MSE) is adopted as the loss function in 
+most studies (Pang et al. 2021). To promote the performance, 
+Pathak et al. (2016) blanked the input image randomly and 
+forced the model to reconstruct the damaged area. Similarly, 
+the ARNet model was proposed, which erases some input 
+attributes and reformulates the problem as a restoration task 
+(Fei et al. 2020). Recently, Zavrtanik et al. (2021) cast the 
+reconstruction problem as an inpainting problem and recon-
+structed the image from partial inpaintings. However, the 
+extracted low-level features can be shared by both normal 
+and anomalous data (Fei et al. 2020) when dealing with 
+complex HSR images. 
+ 
+GAN-based models aim to generate the image rather than 
+reconstruct it. As one of the early GAN-based models, the 
+AnoGAN model assumes that the learned latent space can 
+represent normal samples well, but not the anomalous sam-
+ples (Schlegl et al. 2017). Given a test image, the difference 
+between the regenerated image obtained using the searched 
+latent feature and the test image is considered as the anom-
+aly degree. The famous GANomaly model improved the 
+generator architecture from a decoder to an encoder-decoder 
+encoder design and used high-level features to assist com-
+puting the anomaly score (Akcay, Atapour-Abarghouei, and 
+Breckon 2018). GAN-based models have demonstrated su-
+perior capabilities in generating image data, which also em-
+powers the detection of abnormal samples (Pang et al. 2021). 
+In spite of this, the complex distribution of HSR images can 
+make the generator generate data instances that are out of 
+the manifold of normal instances (Pang et al. 2021). 
+ 
+
+One-class classification models are also used in some 
+anomaly segmentation works (Pang et al. 2021). One of their 
+greatest advantages over the AE-based and GAN-based 
+models is that the OCC models detect anomalies in the fea-
+ture space with high-level semantic information. They first 
+divide an image into many patches and then learn the corre-
+sponding representations. The anomaly score is computed in 
+the feature space using a measurement such as the Ma-
+halanobis or Euclidean distance (Reiss et al. 2021; Ruff et 
+al. 2018; Shi, Yang, and Qi 2021). Most OCC models are 
+based on the principle of one-class support vector machine 
+(OCSVM) (Sch¨olkopf et al. 1999; Andrews, Morton, and 
+Griffin 2016) or support vector data description (SVDD) 
+(Tax and Duin 1999; Chalapathy, Menon, and Chawla 2018; 
+Ruff et al.2018). However, these models mainly consider 
+the compact property of the obtained one class features, re-
+sulting in the model collapse problem (Reiss et al. 2021), 
+and they lack consideration of the low-level structural fea-
+tures.  
+3. Methodology 
+Overview. This section describes the core principles of the 
+proposed ASD model. The overall workflow of the ASD 
+model is shown in Section 3.1, which includes two steps: 
+descriptor extracting and anomaly score computation. To 
+extract the ideal descriptors, descriptor learning is the most 
+important part and is described detailed in Section 3.2. The 
+computation method of the anomaly score is given in Sec-
+tion 3.3. 
+3.1. Overall Workflow of The ASD Model 
+Given an HSR image 𝑿 with size 𝐻 × 𝑊 × 𝐵, where 𝐻, 𝑊, 
+and 𝐵 are the height, width and bands of the image, the 
+anomaly segmentation task can be viewed as a mapping 
+function 𝑓  from the 𝑿  to the anomaly map 𝑨 with size 
+𝐻 × 𝑊. Each pixel in the anomaly map is in the range [0,1]. 
+Generally speaking, the higher the value in the anomaly map, 
+the higher the anomaly degree. 
+ The ASD model separates the function 𝑓 into two steps 
+and the overall workflow is shown in Figure 2. The first step 
+𝑓1 extracts the dense descriptor cube 𝑫 for each image pixel, 
+which is the core part and also the training focus in the ASD 
+model. The descriptors are expected to contain important 
+visual characteristics for the anomaly segmentation task. To 
+incorporate the pixel context and obtain fine pixel corre-
+spondence, the patch-based paradigm is chosen to compute 
+the descriptor 𝐹 for the center pixel 𝑥. In this step, the 
+𝐻 × 𝑊 patches form the input samples and a descriptor 
+cube 𝑫 with size 𝐻 × 𝑊 × 𝐿 is output, where 𝐿 is the de-
+scriptor length. 
+ The second step 𝑓2 outputs the anomaly map based on the 
+trained descriptor encoders in the first step. Specifically, the 
+trained descriptors of the training samples are modeled as a 
+multivariate Gaussian Distribution (MGD) (Guimaraes et al. 
+2018) by the Gaussian Density Estimate (GDE). For the test 
+descriptor, its Mahalanobis distance from the MGD is used 
+to measure the anomaly score. The formal mapping of 𝑓, 𝑓1, 
+and 𝑓2 is shown in Eqs. (1-3). 
+ 
+𝑓: 𝑿 → 𝑨 
+(1) 
+𝑓1 ∶  𝑿 → 𝑫 
+(2) 
+𝑓2 ∶  𝑫 → 𝑨 
+(3) 
+3.2. Ideal Descriptors Learning 
+The descriptors obtained in the first step (as mentioned in 
+Section 3.1) are expected to contain important visual char-
+acteristics for the anomaly segmentation task. To achieve 
+this aim, ideal descriptors are optimized using three condi-
+tions from the characteristics of the anomaly segmentation 
+task and HSR images. 
+Compact. One of the characteristics of anomaly segmen-
+tation is that only normal samples are used in the training 
+stage. In other words, all the training samples are of the same 
+class, which naturally results in compact visual descriptors 
+in the feature space. This compactness is also a useful su-
+pervised signal for the anomaly segmentation task.  
+To keep 𝑫 compact, an enclosing hypersphere around all 
+the pixel descriptors is constructed, which is motivated by 
+the deep SVDD method (Ruff et al. 2018). We let 𝑅 be the 
+hypersphere radius and 𝐶 be the center. The 𝐿1 loss (Eq. (4)) 
+aims to minimize the hypersphere radius and the distance 
+from the obtained pixel descriptors to the center 𝐶, where 
+the parameter 𝜆 controls the trade-off between the size of 
+the hypersphere and the number of surrounded descriptors. 
+The maximum distance between 𝐶 and 𝐹 in 𝑫 is chosen to 
+ 
+ 
+Figure 2: The overall workflow of the ASD model, which 
+includes two steps. In the first step, the ASD model ex-
+tracts a descriptor for each pixel with the descriptor ex-
+tractor. In the second step, the descriptors for normal 
+scenes are modeled as the Gaussian distribution, and the 
+Mahalanobis distance between the test descriptor and the 
+modeled distribution is considered to measure the anom-
+aly score. 
+Normal descriptors
+GDE
+Test image
+Test descriptors
+Gaussian 
+distribution
+Anomaly map
+Descriptor 
+extractor
+Mahalanobis 
+distance
+
+compute the radius 𝑅. Compared to using the mean value, 
+this setting helps the model focus on special normal samples, 
+rather than just considering them as noise. 
+ 
+𝐿1(𝑫) = 𝑅2 + 𝜆 mean{0, max{‖𝐹 − 𝐶‖2 − 𝑅2 | 𝐹 ∈ 𝑫}} (4) 
+ 
+Diverse. Compactness is the first basic condition. How-
+ever, the model can easily collapse if only a compactness 
+constraint used. In other words, the model would map all the 
+input samples into the same point. This “cheating” makes 
+the model lose the anomaly detection ability. To deal with 
+this problem, the diverse condition is necessary, which 
+stresses that a different pixel 𝑥 obtains different values of 𝐹. 
+The key consideration to keeping the descriptors diverse 
+is to keep the model sensitive to the input sample change. 
+Considering the fact that training images are always anom-
+aly free and real negative samples are difficult to obtain, data 
+augmentation techniques, such as the channel shuffle oper-
+ation, are used to generate negative samples. Formally, the 
+augmentation operation set 𝑆𝑎 = {𝐴1, 𝐴2, … , 𝐴𝑛} contains 𝑛 
+kinds of different augmentation operations. For the original 
+image 𝑿, the obtained image descriptor cube 𝑫 can be seen 
+as a positive one. Then, after applying the operations from 
+𝑆𝑎 on 𝑿 in turn, 𝑿𝑇 can be obtained and the corresponding 
+cube 𝑫𝑇 is considered to be a negative sample. Eqs. (5-6) 
+formally express the above process. 
+ 
+𝑿𝑇 = 𝐴𝑛(… (𝐴2(𝐴1(𝑿𝑇))) 
+(5) 
+𝑫 = 𝑓1(𝑿), 𝑫𝑇= 𝑓1(𝑿𝑇) 
+(6) 
+ 
+ 
+Both 𝑫 and 𝑫𝑇 have the same shape 𝐻 × 𝑊 × 𝐿. The diver-
+sity loss is defined as the average pixel descriptor difference 
+between 𝑫 and 𝑫𝑇, as shown in Eq. (7). With the 𝐿2 loss, 
+the model is encouraged to increase the sensitivity to the in-
+put difference. 
+ 
+𝐿2(𝑫, 𝑫𝑇) = 1/{
+1
+𝐻 × 𝑊 ∑ ∑‖𝑫𝑖𝑗 − 𝑫𝑖𝑗
+𝑇 ‖
+2
+𝑊
+𝑗=1
+𝐻
+𝑖=1
+} 
+(7) 
+Some technologies have the potential to deal with the 
+model collapse, such as reducing the model bias (Ruff et al. 
+2018) or designing early-stopping strategies (Reiss et al. 
+2021). However, the proposed 𝐿2 loss does not need early-
+stopping or change of the model architecture. 
+Feature-rich. The compact and diverse conditions meas-
+ure the descriptors from the perspective of distance. The fea-
+ture-rich condition measures the descriptors from the per-
+spective of the amount of representative information. In the 
+ASD model, multi-scale and multi-level features are consid-
+ered in particular. 
+The multi-scale characteristic is an import difference for 
+HSR images, compared to natural images. For example, 
+large-scale information is important for rivers and small-
+scale information is important for urban buildings. Even for 
+the same scene, the images are always taken at different 
+heights, which poses a challenge for the model ability to 
+catch the multi-scale information. 
+To enhance the model ability to deal with multi-scale in-
+formation, the ASD model uses a resize operation set 𝑆𝑠 =
+{𝑈1, 𝑈2, … , 𝑈𝑚} for the input patches. Given an image 𝑿, it 
+is resized using each operation 𝑈𝑖 in 𝑆𝑠, and obtains 𝑚 dif-
+ferent-scale versions 𝑿1, 𝑿2, … , 𝑿𝑚  of the same image. 
+Then, for each center pixel 𝑥, 𝑚 patches are cropped with 
+size 𝑃 × 𝑃 from the 𝑚 scaled images. Next, the obtained 
+pyramid patches are fed into with 𝑚 individual encoders 
+𝐸1, 𝐸2, … , 𝐸𝑚, and 𝑚 pixel descriptors are obtained, where 
+ 
+ 
+Figure 3: The descriptor optimization process of the ASD model. (a) For each normal image, its transformed image is 
+generated using data argumentation techniques for generating the artificially negative samples. (b) The ASD model is de-
+signed as a two-head architecture. One head outputs the dense descriptor and the other reconstruction head is designed to 
+force the obtained descriptors to contain both high-level and low-level features. Pyramid patches are extracted at different 
+scales for the multi-scale features. (c) To obtain the ideal descriptors, as defined in Section 3.2, the optimization tries to 
+find a compact hypersphere surrounding all the descriptors of the original image by pulling them to the center, keeping the 
+descriptors diverse by increasing the distance between the original descriptors and the transformed descriptors. 
+Pyramid
+Patches
+Scale 1
+Scale 2
+Scale 3
+Linked
+descriptor
+Input image 
+Transformed image
+Data 
+transformation
+(a) Data argument
+(b) Descriptor extractor
+(c) Optimization objective: 
++ 
++ 
+Pixel descriptors 
+Reconstructed image 
+Pixel descriptors 
+Reconstructed image
+Pull
+Push
+Push and pull descriptors
+pact
+Diverse
+= 
+Hypersphere radius:
+Hypersphere center:
+Feature-rich (Multi-level)
+= 
+
+each descriptor has the same length 𝐿. The 𝑚 pixel de-
+scriptors are then concatenated further to form a descriptor 
+vector with length 𝑚 × 𝐿. The descriptor cube 𝑫𝑐 with size 
+𝐻 × 𝑊 × (𝑚 × 𝐿) is naturally obtained when all the pixels 
+in 𝑿 are processed. Finally, a 1 × 1 convolution operation is 
+used to map the concatenated descriptors into size 𝐿. This is 
+the process for extracting the final pixel descriptors. Eqs. (8-
+10) formally express the above process, which is also the 
+detailed process of 𝑓1. Figure 3 shows the process when 𝑚 
+= 3. 
+ 
+𝑿1, 𝑿2, … , 𝑿𝑚 = 𝑈1(𝑿), 𝑈2(𝑿), … , 𝑈𝑚(𝑿) 
+(8) 
+𝑫𝑐 = concat([𝑀(𝑿1, 𝑿2, … , 𝑿𝑚)]) 
+(9) 
+𝑫 = Conv1×1(𝑫𝑐) 
+(10) 
+  
+ Multi-level features are necessary when dealing with the 
+various objects in the anomaly segmentation task. Although 
+the deep architecture extracts high-level semantic infor-
+mation through the descriptors, the low-level information 
+such as texture is gradually forgotten as the network goes 
+deeper. This is beneficial for objects such as buildings, but 
+is not expected for some objects such as water and river be-
+cause the texture feature is useful for them. 
+To ensure that both high-level and low-level features are 
+contained in the descriptors, the ASD model is designed as 
+a two head architecture. Both heads grow from the concate-
+nated descriptor cube 𝑫𝑐. One head uses the 1 × 1 convolu-
+tion operation to obtain the final pixel descriptors. The other 
+head also uses the 1 × 1 convolution but aims to reconstruct 
+the original pixel. To reconstruct the pixel value, the concat-
+enated descriptors are forced to contain the low-level fea-
+tures. Note that the reconstruction head is only used in the 
+training and is abandoned in the test stage. 𝑿′ denotes the 
+reconstructed image, and the MSE is used to compute the 
+loss (Eqs. (11-12)). 
+ 
+𝑿′ = Conv1×1(𝑫𝑐) 
+(11) 
+𝐿3(𝑿, 𝑿′) =
+1
+𝐻 × 𝑊 ∑ ∑‖𝑿𝑖𝑗 − 𝑿𝑖𝑗
+′ ‖
+2
+𝑊
+𝑗=1
+𝐻
+𝑖=1
+ 
+(12) 
+  
+ In total, the three properties: compact, diverse and fea-
+ture-rich work together to design the model architecture and 
+optimize the descriptor learning. The optimization objective 
+of the ASD model is the sum of the above losses, as shown 
+in Eq. (13). Figure 3 shows the overall descriptor learning 
+process. 
+ 
+𝐿𝑜𝑠𝑠 = 𝐿1(𝑫) + 𝐿2(𝑫, 𝑫𝑇) + 𝐿3(𝑿, 𝑿′)  
+(13) 
+ 
+3.3. Anomaly Score Computation 
+When the descriptor optimization process of step 𝑓1 is fin-
+ished, the second step 𝑓2 outputs the anomaly map based on 
+the optimized descriptors. There exist various methods to 
+complete step 𝑓2. Although non-parametric statistical meth-
+ods do not rely on any distribution assumption, it requires a 
+lot of samples to achieve accurate estimation and can be 
+computationally expensive e (Pang et al. 2021). Conversely, 
+parametric density estimation needs fewer samples, and the 
+Gaussian assumption holds in most cases (Pimentel et al. 
+2014). 
+ In the ASD model, the Gaussian assumption is adopted to 
+model the normal descriptors. Using the normal samples in 
+the training stage, the mean 𝜇 and the covariance matrix 𝚺 
+can be estimated. Given a test descriptor 𝑥𝑡, its Mahalanobis 
+distance from the modeled distribution (as shown in Eq. (14)) 
+is considered the anomaly degree, which can be converted 
+to the anomaly score after the normalization. 
+ 
+𝐴𝑛𝑜𝑚𝑎𝑙𝑦 𝑑𝑒𝑔𝑟𝑒𝑒 = √(𝑥𝑡 − 𝜇 )𝑇𝚺−1(𝑥𝑡 − 𝜇) 
+(14) 
+ 
+4. Experiments 
+4.1. Experimental Settings 
+Datasets 
+The proposed ASD model was evaluated on four HSR im-
+age datasets: DeepGlobe (Demir et al. 2018), Agriculture-
+Vision (Chiu et al. 2020), FAS, and Landslide4Sense (Ghor-
+banzadeh et al. 2022). The DeepGlobe and Agriculture-Vi-
+sion datasets were originally made for the land-cover seg-
+mentation and agriculture pattern segmentation tasks, re-
+spectively. To adapt these datasets for the anomaly segmen-
+tation task, the pixels of the remaining classes were masked 
+for a fixed normal class in the training process to keep the 
+anomaly-free characteristic. 
+ To show the application value of the ASD model, the FAS 
+and Landslide4Sense datasets were used. The FAS dataset 
+was made by ourselves for the forest monitoring application, 
+where the common forest pattern (i.e., Figure 1) is treated as 
+the normal class, and some abnormal objects, such as house, 
+lake, car, and diseased tree, are considered as anomalies. 
+The RGB imagery in the FAS dataset was made from UAV-
+borne hyperspectral images in forest scene. The pixel reso-
+lution is 11 cm and the image size is 120×120. In the Land-
+slide4Sense dataset, the anomaly segmentation model was 
+used to segment the landslide area by learning from the nor-
+mal mountain pattern. 
+Comparative Models and Evaluation Metrics 
+The ASD model was compared with four state-of-the-art 
+methods covering both image space and feature space types. 
+
+These methods include GANomaly (Akcay, Atapour-Abar-
+ghouei, and Breckon 2018), ARNet (Fei et al. 2020), RIAD 
+(Zavrtanik, Kristan, and Skocaj 2021) and deep SVDD 
+(DSVDD) (Ruff et al. 2018). For the GANomaly, RIAD, 
+and DSVDD, the model hyper-parameters were kept same 
+as the authors’ open source code. ARNet was implemented 
+using the same architecture as RIAD. The model perfor-
+mance was evaluated using the area under the curve (AUC) 
+metric and the mean Intersection over Union (mIOU). The 
+segmentation threshold for the mIOU corresponds to the 
+left-upper point of the Receiver operating characteristic 
+(ROC) curve. 
+Implementation Details 
+The fast version (Bailer et al. 2018) of the point descriptor 
+extraction network in the work of Simo-Serra et al. (2015) 
+acted as the pixel descriptor encoder in the proposed model. 
+In all the experiments, the models were trained for 100 
+epochs, and the batch size was 1. The Adam optimizer with 
+learning rate 0.0001 was used. 𝜆 was set to 10. 𝑆𝑠 was set to 
+{0.5,1.0,2.0} and 𝑃 is 15 for all the descriptor encoders. The 
+first 10 epochs were trained using only the 𝐿3 loss to com-
+pute the initial 𝐶. 𝑅 was initialized to 3.0. 𝐶 and 𝑅 were up-
+dated after each epoch using all the training descriptors. The 
+dimension 𝐿 was set to 5. The data augmentation operations 
+used in the ASD and ARNet model were the GaussNoise, 
+ChannelShuffle, RandomBrightness, RandomContrast, and 
+Solarize operations (implemented with the Albumentations 
+tool (Buslaev et al. 2020)). Due to the AUC computation 
+burden, 2000 test images in Agriculture-Vision dataset were 
+chosen to be evaluated. The CPU was an Intel(R) Xeon(R) 
+CPU E5-2690 v4 @ 2.60 GHz with 62.6 GB memory, and 
+the GPU was a Tesla P100-PCIE with 16 GB of memory. 
+4.2. Results on the DeepGlobe Dataset 
+The quantitative and qualitative results are reported in Table 
+1 and Figure 4, respectively. In Table 1, the ASD model 
+achieves the highest AUC values for the four normal classes. 
+For the Urban land class, the ASD model surpasses the sec-
+ond-best model by over 6 points, showing its superiority 
+when dealing with a complex distribution. In Figure 4, the 
+anomaly maps obtained by the ASD model are the closest to 
+the ground truth. 
+4.3. Results on the Agriculture-Vison Dataset 
+Table 2 and Figure 4 respectively show the quantitative and 
+qualitative results for the Agriculture-Vision dataset. In Ta-
+ble 2, the ASD model achieves the best AUC results for all 
+six normal classes. ASD surpasses the second-best model by 
+5 points for the Drydown class. Except Weed cluster, the 
+mIOU values of ASD are all close to the optimal value. In 
+Figure 5, it can be seen that accurate results and fine bound-
+aries are obtained by the ASD model for most classes. For 
+ 
+ 
+Figure 4: The anomaly segmentation results obtained on 
+the DeepGlobe dataset for each normal class, White pix-
+els cover the anomalous region. 
+Urban
+land
+Water
+Range
+land
+Barren
+land
+Agriculture
+Forest
+land
+Image
+GT
+DSVDD
+RIAD
+ARNet
+GANomaly
+ASD
+Normal class
+ 
+ 
+Figure 5: The anomaly segmentation results obtained on 
+the Agriculture-Vision dataset for the six normal classes. 
+Image (RGB)
+GT
+DSVDD
+RIAD
+ARNet
+GANomaly
+ASD
+Normal class
+Dry down
+Nutrient
+deficiency
+Endrow
+Water
+Double
+plant
+Weed
+cluster
+Image (NIR)
+Method 
+Urban 
+land 
+Agricul-
+ture 
+Range 
+land 
+Forest 
+land 
+Water 
+Barren 
+land 
+AUC mIOU AUC mIOU AUC mIOU AUC mIOU AUC mIOU AUC mIOU 
+DSVDD 57.0 31.5 60.3 41.4 53.6 16.3 58.7 24.0 37.6 2.2 50.6 17.7 
+RIAD 
+52.3 12.4 65.9 46.3 47.6 7.0 69.4 33.1 57.7 43.0 53.3 13.9 
+ARNet 
+50.2 39.0 60.1 40.9 48.2 6.9 67.6 34.5 54.7 12.5 61.4 33.3 
+GANomaly 42.8 44.3 51.7 35.8 55.1 30.6 75.4 41.3 58.8 44.5 36.8 39.7 
+ASD 
+63.4 38.5 64.1 42.7 54.3 23.2 79.5 43.1 73.3 39.5 62.5 34.9 
+ 
+Table 1: The anomaly segmentation results obtained on 
+the DeepGlobe dataset. 
+Method 
+Drydown Double 
+plant 
+Endrow 
+Weed 
+cluster 
+ND 
+Water 
+AUC mIOU AUC mIOU AUC mIOU AUC mIOU AUC mIOU AUC mIOU 
+DSVDD 60.9 30.8 53.0 14.7 54.6 10.1 48.0 3.67 59.6 24.4 72.1 26.0 
+RIAD 
+62.2 31.0 60.9 25.6 59.3 27.7 55.2 38.2 63.8 25.7 86.1 42.7 
+ARNet 
+61.1 30.6 51.5 15.3 57.1 25.5 53.0 15.9 59.9 26.8 45.3 9.6 
+GANomaly 59.5 26.3 49.6 4.2 56.5 26.4 51.1 41.9 62.9 33.1 64.2 20.7 
+ASD 
+67.4 36.4 61.3 24.8 61.1 25.7 58.0 19.7 65.9 31.7 90.0 40.4 
+ 
+Table 2: The comparative quantitative anomaly seg-
+mentation results on the Agriculture-Vision dataset. 
+(ND is nutrient deficiency) 
+
+ 0.8
+ 0.6
+0.4
+0.2 0.8
+ 0.6
+0.4
+0.2the normal class of water, only the ASD model outputs a 
+correct anomaly map, and some models completely reverse 
+the anomaly regions. 
+4.4.  Results on the FAS and Landslide4Sense Da-
+tasets 
+The FAS and Landslide4Sense datasets were used to show 
+the application value of the proposed anomaly segmentation 
+model in forest monitoring and landslide detection. Table 3, 
+Figure 6, and Figure 7 report the related results. In both da-
+tasets, the ASD model achieves the best AUC and mIOU 
+scores. Satisfactory anomaly maps are obtained, demon-
+strating great application value. 
+4.5. Sensitivity of the Descriptor Scale 
+Table 4 reports the effect of the multi-scale descriptor on the 
+anomaly segmentation performance. The multi-scale setting 
+with 𝑆𝑠 = 0.5, 1.0, 2.0 obtains the optimal AUC values for 
+four classes. In a real application, although the optimal value 
+of 𝑆𝑠 may be difficult to establish, Table 4 shows that the 
+multi-scale setting would ensure satisfactory results. 
+4.6. Ablation Studies 
+The core idea of the ASD model is to find ideal descriptors, 
+so three loss constraints corresponding to the conditions de-
+scribed in Section 3.2 were designed. Table 5 illustrates the 
+effectiveness of three losses for different types of earth vi-
+sion scenes 𝐿1 (compact loss) can better handle the scene 
+with simple spatial distribution, i.e., Agriculture, Forestland, 
+and Water; (from the first 3 rows). 𝐿3 (feature-rich loss) 
+works on the complex scenes, i.e., Urban land and Barren 
+land; (from the 3,5 and 6 rows). 𝐿2 (diversity loss) aims at 
+further improving segmentation performance by artificial 
+anomaly samples. (Comparing rows 1 and 3 with 4 and 5, 
+respectively). 
+5. Conclusion 
+In this paper, we have proposed a pixel descriptor based 
+model for the anomaly segmentation task in HSR imagery. 
+The core innovations are: 1) The three conditions that the 
+ideal descriptor should meet are given from the characteris-
+tics of the anomaly segmentation task and HSR images. 2) 
+The corresponding constraints and architecture were de-
+signed on this basis. Obvious improvement was achieved on 
+four datasets (including real anomalies in forest and moun-
+tain scenes). Overall, proposed model sets the first baseline 
+for the anomaly segmentation task of complex HSR imagery. 
+ 
+ 
+Figure 6: The anomaly segmentation results obtained on 
+the FAS dataset. The common forest pattern (see Figure 
+1) is considered as normal, and four anomalies are con-
+sidered. 
+Image
+GT
+DSVDD
+RIAD
+ARNet
+GANomaly
+ASD
+PWD
+House
+Car
+Lake
+Anomalies
+ 
+ 
+Figure 7: The anomaly segmentation results obtained on 
+the Landslide4Sense dataset. The common mountain pat-
+tern is considered as normal, and the landslides are the 
+anomalies. 
+GT
+DSVDD
+RIAD
+ARNet
+GANomaly
+ASD
+Image (DEM)
+Image (Slope)
+Image (RGB)
+Dataset 
+DSVDD 
+RIAD 
+ARNet 
+GANomaly 
+ASD 
+AUC mIOU AUC mIOU AUC mIOU AUC mIOU AUC mIOU 
+FAS 
+74.1 46.2 44.3 36.5 82.7 52.9 50.7 24.9 91.0 69.3 
+Lanslide4Sense 61.6 20.7 83.7 41.0 78.8 48.7 82.2 39.1 89.8 49.3 
+ 
+Table 3: The anomaly segmentation results obtained on 
+the FAS and Landslide4Sense datasets. 
+ 
+Constraints 
+Urban 
+land 
+Agricul-
+ture 
+Range 
+land 
+Forest 
+land 
+Water 
+Barren 
+land 
+AUC mIOU AUC mIOU AUC mIOU AUC mIOU AUC mIOU AUC mIOU 
+𝐿1 
+51.3 38.9 62.5 42.3 52.8 18.1 76.4 42.0 70.2 35.1 56.6 26.9 
+𝐿2 
+40.9 45.3 59.4 37.9 53.7 40.2 76.7 47.6 71.5 36.1 49.0 34.1 
+𝐿3 
+62.5 35.9 60.4 38.6 52.9 19.6 75.6 41.5 68.8 36.0 61.2 32.9 
+𝐿1+𝐿2 
+56.5 32.4 61.0 39.7 54.7 31.8 78.8 48.6 72.5 40.6 54.6 22.8 
+𝐿2+𝐿3 
+64.5 45.1 62.0 42.5 54.6 22.1 77.4 43.1 73.0 41.7 54.3 23.8 
+𝐿1+𝐿3 
+64.1 45.1 62.7 41.9 52.9 20.0 77.3 43.1 74.5 41.5 61.7 31.5 
+𝐿1+𝐿2+𝐿3 63.4 38.5 64.1 42.7 54.3 23.2 79.5 43.1 73.3 39.5 62.5 34.9 
+ 
+Table 5: The ASD model ablation analysis for the three 
+loss constraints on the anomaly segmentation results ob-
+tained using the DeepGlobe dataset. 
+ 
+Scale 
+Urban 
+land 
+Agricul-
+ture 
+Range 
+land 
+Forest 
+land 
+Water 
+Barren 
+land 
+AUC mIOU AUC mIOU AUC mIOU AUC mIOU AUC mIOU AUC mIOU 
+(0.5,0.5,0.5) 61.7 39.9 63.7 42.7 57.4 31.5 77.8 45.5 68.8 38.3 59.6 30.5 
+(1.0,1.0,1.0) 63.2 40.0 60.7 40.0 56.1 27.9 78.5 44.1 72.4 35.7 57.5 22.5 
+(2.0,2.0,2.0) 58.4 42.5 62.0 40.5 55.5 25.7 75.6 43.5 78.0 41.1 59.0 35.0 
+(0.5,1.0,2.0) 63.4 38.5 64.1 42.7 54.3 23.2 79.5 43.1 73.3 39.5 62.5 34.9 
+ 
+Table 4: The ASD model sensitivity analysis for the 
+multi-scale property on the anomaly segmentation re-
+sults obtained using the DeepGlobe dataset. 
+
+ 0.8
+ 0.6
+0.4
+0.2 0.8
+ 0.6
+0.4
+0.2Acknowledgments 
+This work was supported by National Natural Science Foun-
+dation 
+of 
+China 
+under 
+Grant 
+No.42071350 
+and 
+No.42101327, in part by the Fundamental Research Funds 
+for the Central Universities under Grant 2042021kf0070, 
+and LIESMARS Special Research Funding. 
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+arXiv:2301.00517v1  [math.AG]  2 Jan 2023
+Correspondences in log Hodge cohomology
+Charles Godfrey
+Pacific Northwest National Laboratory
+[email protected]
+January 3, 2023
+Abstract
+We construct correspondences in logarithmic Hodge theory over a perfect field of arbitrary char-
+acteristic. These are represented by classes in the cohomology of sheaves of differential forms with
+log poles and, notably, log zeroes on cartesian products of varieties. From one perspective this gen-
+eralizes work of Chatzistamatiou and Rülling, who developed (non-logarithmic) Hodge correspon-
+dences over perfect fields of arbitrary characteristic;from another we provide partial generalizations
+of more recent work of Binda, Park and Østvær on logarithmic Hodge correspondences by relaxing
+finiteness and strictness conditions on the correspondences considered.
+1
+Introduction
+Generally speaking, a correspondence between two algebraic varieties푋 and 푌 over a field 푘 is a cycle
+or cohomology class on the product 푋×푌. The study of such objects dates back (at least) to Lefschetz
+[Lef53], and features prominently in famous conjectures on algebraic cycles (see e.g. [Voi14]) and
+Voevodsky’s theory of motives (see e.g. [MVW06]).
+In a number of algebro-geometric research areas it has become commonplace to work with pairs
+(푋, ∆푋) consisting of a variety 푋 together with a divisor ∆푋 on 푋. Such areas include moduli of
+varieties (where pairs generalize the curves with marked points of [DM69]), birational geometry
+(where pairs appear naturally, for example as the output of strong resolution of singularities [KM98])
+and logarithmic geometry (in this case vast generalizations of divisors ∆푋 are allowed [Ogu18]). It is
+natural to wonder about analogues of correspondencesin this category of pairs, and there have been
+efforts in this direction, for example development of categories of logarithmic motives [BPØ20].
+In this paper, we focus on correspondences for logarithmic Hodge cohomology of pairs (푋, ∆푋),
+where 푋 is a smooth (but not necessarily proper) variety over a perfect field 푘 and ∆푋 is a simple
+normal crossing divisor on 푋. These cohomology groups can be described as
+퐻∗(푋, ∆푋) =
+⨁
+퐻푞(푋, Ω푝
+푋(log ∆푋)),
+(1.1)
+where Ω푋(log ∆푋) is the sheaf of differential 1-forms on 푋 with log poles along ∆푋 and Ω푝
+푋(log ∆푋)
+the 푝-th exterior power thereof. In addition we consider a generalization where 푋 comes with a
+family of supports Φ푋, and the ordinary cohomology groups on the right hand side of eq. (1.1) are
+replaced with cohomology with supports in Φ푋, namely 퐻푞
+Φ푋(푋, Ω푝
+푋(log ∆푋)). Allowing for supports
+greatly expands the applicability of our results: for example, it permits us to construct a correspon-
+dence associated to a cycle 푍 ⊂ 푋 × 푌 in a situation where neither 푋 nor 푌 is proper over 푘, but 푍
+is proper over both 푋 and 푌.1
+There are multiple motivations for investigating correspondencesfor this particular cohomology
+of pairs:
+1One way that such a cycle 푍 might naturally arise is as the closure of the graph of a birational equivalence 푋 ⤏ 푌 of
+non-proper varieties.
+This work was completed while the author was a PhD student in the University of Washington Department of Mathematics.
+The author was partially supported by the University of Washington Department of Mathematics Graduate Research
+Fellowship, and by the NSF grant DMS-1440140, administered by the Mathematical Sciences Research Institute, while in
+residence at MSRI during the program Birational Geometry and Moduli Spaces.
+
+• By analogy with the case of varieties (that is, without auxiliary divisors/log structures), we sus-
+pect that correspondences at the level of Chow cycles are more fundamental, and that (many)
+correspondencesin logarithmic Hodge cohomology are obtained from Chow correspondences
+via a cycle morphism. However, as of this writing there is no full-fledged theory of Chow co-
+homology of pairs or log schemes (though there has been considerable progress, for instance
+in [Bar18; BBG22]). Logarithmic Hodge cohomology is in contrast quite mature, appearing as
+early as [Del71].
+• Correspondences in (non-logarithmic) Hodge cohomology have found remarkable applica-
+tions. For example, [CR11] used them to prove birational invariance of the cohomology groups
+of the structure sheaf 퐻푖(푋, 풪푋) for smooth varieties 푋 over perfect fields of positive character-
+istic. In fact, attempting to implement a similar strategy with logarithmic Hodge cohomology
+to obtain results on invariance of the cohomology groups 퐻푖(푋, 풪푋(−∆푋)) with respect to (a
+restricted class of) birational equivalences was the initial inspiration for this work. Ultimately
+that attempt was unsuccessful, as we describe in Appendix A.
+• There has been recent interest in logarithmic Hodge cohomology as a representable functor
+on a category of motives of log schemes over a perfect field [BPØ20, §9]. While that work does
+also construct some correspondences, they are restricted to those associated with logarithmic
+Hodge cohomology classes of cycles 푍 ⊂ 푋 × 푌 which are finite over 푋 and obey additional
+strictness (in the sense of logarithmic geometry) conditions; we remove these restrictions.
+The correspondences we construct are obtained from certain Hodge classes with both log poles
+and log zeroes. Our main result is:
+Theorem 1.2 (= Theorem 4.1). A class 훾 ∈ 퐻푗
+푃(Φ푋,Φ푌)(푋 × 푌, Ω푖
+푋×푌(log ∆푋×푌)(−pr∗
+푋∆푋)) defines
+homomorphisms
+cor(훾) ∶ 퐻푞
+Φ푋(푋, Ω푝
+푋(log ∆푋)) → 퐻푞+푗−푑푋
+Φ푌
+(푌, Ω푝+푖−푑푋
+푌
+(log ∆푌))
+by the formula cor(훾)(훼) ∶= pr푌∗(pr∗
+푋(훼) ⌣ 훾). Moreover if (푍, ∆푍, Φ푍) is another snc pair with
+supports and 훿 ∈ 퐻푗′
+푃(Φ푌,Φ푍)(푌 × 푍, Ω푖′
+푌×푍(log ∆푌×푍)(−pr∗
+푌∆푌)), then
+pr푋×푍∗(pr∗
+푋×푌(훾) ⌣ pr∗
+푌×푍(훿)) ∈ 퐻푗+푗′−푑푌
+푃(Φ푋,Φ푍)(푋 × 푍, Ω푖+푖′−푑푌
+푋×푍
+(log ∆푋×푍)(−pr∗
+푋∆푋)) and
+cor(pr푋×푍∗(pr∗
+푋×푌(훾) ⌣ pr∗
+푌×푍(훿))) = cor(훿)◦ cor(훾)
+as homomorphisms 퐻푞
+Φ푋(푋, Ω푝
+푋(log ∆푋)) → 퐻푞+푗+푗′−푑푋−푑푌
+Φ푍
+(푍, Ω푝+푖+푖′−푑푋−푑푌
+푍
+(log ∆푍)).
+In the above, ∆푋푌 ∶= pr∗
+푋∆푋 + pr∗
+푌∆푌, a simple normal crossing divisor on 푋 × 푌. There is a
+simple heuristic explanation for the appearance of differential forms in Ω푖
+푋×푌(log ∆푋×푌)(−pr∗
+푋∆푋):
+working over the complex numbers, in the case where 푋and 푌 are both proper the class cor(훾)(훼) ∶=
+pr푌∗(pr∗
+푋(훼) ⌣ 훾) can be computed explicitly as an integral of the form
+∫
+푋
+훼(푥) ∧ 훾(푥, 푦),
+(1.3)
+and this integral will only be finite when the log poles of 훼 along ∆푋 are cancelled by complementary
+zeroes of the form 훾(푥, 푦) along the preimage pr∗
+푋∆푋.
+Our proof of Theorem 1.2 relies heavily on prior work on both Hodge cohomology with supports
+[CR11, §2] and its logarithmic variant [BPØ20, §9]. Section 2 is a rapid summary of those results.
+The key new technical ingredient is a base change formula on the interaction of pushforward and
+pullback operations in cartesian squares, proved in Section 3. Section 4 includes the proof of our
+main theorem.
+2
+
+1.1
+Acknowledgements
+Thanks to Daniel Bragg, Yun Hao, Sarah Scherotzke, Nicolò Sibilla and Mattia Talpo for helpful
+conversations, to Lawrence Jack Barrott for illuminating email correspondence regarding logarith-
+mic aspects of Chow and Hodge, and to my advisor Sándor Kovács for many insightful discussions.
+Thanks also to the participants of the Spring 2019 MSRI graduate student seminar, in particular
+Giovanni Inchiostro and organizer Fatemeh Rezaee, for feedback on early work on this paper.
+2
+Functoriality properties of log Hodge cohomology with sup-
+ports
+2.1
+Supports
+In order to obtain results that apply to correspondences between varieties 푋 and 푌 where neither 푋
+nor 푌 is proper, it is necessary to work with cohomology with supports, also known as local coho-
+mology. A primary source for the material of this subsection is [R&D, §IV]. Let 푋 be a noetherian
+scheme.
+Definition 2.1 ([R&D, §IV], [CR11, §1.1]). A family of supports Φ on 푋 is a non-empty collection
+Φ of closed subsets of 푋 such that
+• If 퐶 ∈ Φ and 퐷 ⊂ 퐶 is a closed subset, then 퐷 ∈ Φ.
+• If 퐶, 퐷 ∈ Φ then 퐶 ∪ 퐷 ∈ Φ.
+Example 2.2. Φ = { all closed subsets of 푋 } is a family of supports. More generally if 풞 is any col-
+lection of closed subsets 퐶 ⊂ 푋, there is a smallest family of supports Φ(풞) containing 풞 (explicitly,
+Φ(풞) consists of finite unions ⋃
+푖 푍푖 of closed subsets 푍푖 ⊂ 퐶푖 of elements 퐶푖 ∈ 풞). Taking Φ = Φ({푋})
+recovers the previous example. A more interesting example is the case where for some fixed 푝 ∈ ℕ,
+Φ = {closed sets 푍 ⊆ 푋 | dim 푍 ≤ 푝}.
+There is a close relationship between families of supports on X and certaincollections of specialization-
+closed subsets of points on 푋, and we can also consider sheaves of families of supports — for further
+details we refer to [R&D, §IV.1].
+If 푓 ∶ 푋 → 푌 is a morphism of noetherian schemes and Ψ is a family of supports on 푌, then
+{푓−1(푍) | 푍 ∈ Ψ} is a family of closed subsets of 푋, and is closed under unions, but is not in general
+closed under taking closed subsets.
+Definition 2.3. 푓−1(Ψ) is the smallest family of supports on 푋 containing {푓−1(푍) | 푍 ∈ Ψ}.
+Let Φ be a family of supports on 푋. The notation/terminology 푓|Φ is proper will mean 푓|퐶 is
+proper for every 퐶 ∈ Φ. If 푓|Φ is proper then 푓(퐶) ⊂ 푌 is closed for every 퐶 ∈ Φ and in fact
+푓(Φ) = {푓(퐶) ⊂ 푌 | 퐶 ∈ Φ}
+(2.4)
+is a family of supports on 푌. The key point here is that if 퐷 ⊂ 푓(퐶) is closed, then 푓−1(퐷) ∩ 퐶 ∈ Φ
+and 퐷 = 푓(푓−1(퐷) ∩ 퐶).
+Definition 2.5. A scheme with supports (푋, Φ푋) is a scheme 푋 together with a family of supports
+Φ푋 on 푋.
+Definition 2.6. A pushing morphism 푓 ∶ (푋, Φ푋) → (푌, Φ푌) of schemes with supports is a
+morphism 푓 ∶ 푋 → 푌 of underlying schemes such that 푓|Φ푋 is proper and 푓(Φ푋) ⊂ Φ푌. A pulling
+morphism 푓 ∶ 푋 → 푌 is a morphism 푓 ∶ 푋 → 푌 such that 푓−1(Φ푌) ⊂ Φ푋.
+These morphisms provide two different categories with underlying set of objects schemes with
+supports (푋, Φ푋), and pushing/pulling morphisms respectively (the verification is elementary; for
+instance a composition of pushing morphisms is again a pushing morphism since compositions
+of proper morphisms are proper). Schemes with supports provide a natural setting for describing
+3
+
+functoriality properties of local cohomology. Let ℱ be a sheaf of abelian groups on a scheme with
+supports (푋, Φ푋).2
+Definition 2.7. The sheaf of sections with supports of ℱ, denoted ΓΦ(ℱ), is obtained by setting
+ΓΦ(ℱ)(푈) = {휎 ∈ ℱ(푈) | supp 휎 ∈ Φ푋|푈 }
+(2.8)
+for each open 푈 ⊂ 푋 (here Φ푋|푈 is short for 휄−1Φ푋 where 휄 ∶ 푈 → 푋 is the inclusion). More
+explicitly: for a local section 휎 ∈ ℱ(푈), 휎 ∈ ΓΦ(ℱ)(푈) means supp 휎 = 퐶 ∩ 푈 for a closed set
+퐶 ⊂ Φ푋.
+The functor ΓΦ is right adjoint to an exact functor, for instance the inclusion of the subcategory
+퐀퐛Φ(푋) ⊂ 퐀퐛(푋) of abelian sheaves on 푋 with supports in Φ; so, ΓΦ is left exact and preserves
+injectives. In the case Φ = Φ(푍) for some closed 푍 ⊂ 푋, this is proved in [Stacks, Tag 0A39, Tag 0G6Y,
+Tag 0G7F] — the general case can then be obtained by writing ΓΦ as a filtered colimit:
+ΓΦ = colim푍∈Φ Γ푍.
+The right derived functor of ΓΦ will be denoted 푅ΓΦ. Taking global sections on 푋 gives the sections
+with supports of ℱ: ΓΦ(ℱ) ∶= Γ푋(ΓΦ(ℱ)) This is also left exact, and (the cohomologies of) its
+derived functor give the cohomology with supports in Φ: 퐻푖
+Φ(푋, ℱ) ∶= 푅푖ΓΦ(ℱ).
+Proposition 2.9. Cohomology with supports enjoys the following functoriality properties:
+(푖) If 푓 ∶ (푋, Φ푋) → (푌, Φ푌) is a pulling morphism of schemes with supports, ℱ, 풢 are sheaves of
+abelian groups on 푋, 푌 respectively, and if
+휑 ∶ 풢 → 푓∗ℱ is a morphism of sheaves,
+(2.10)
+then there is a natural morphism 푅ΓΦ풢 → 푅푓∗푅ΓΦℱ. Similarly if ℱ and 풢 are quasicoherent
+then there are natural morphisms 푅ΓΦ풢 → 푅푓∗푅ΓΦℱ.
+(푖푖) If 푓 ∶ (푋, Φ푋) → (푌, Φ푌) is a pushing morphism, ℱ, 풢 are sheaves of abelian groups on 푋, 푌
+respectively, and
+휓 ∶ 푅푓∗ℱ → 풢 is a morphism in the derived category of 푋,
+(2.11)
+then there is a natural morphism 푅푓∗푅ΓΦ(ℱ) → 푅ΓΦ풢.
+Both parts of the proposition follow from [Stacks, Tag 0G78]; (i) is discussed in detail in [CR11,
+§2.1] and (ii) can be extracted from [CR11, §2.2] (although it doesn’t appear to be stated explicitly).
+See also [BPØ20, Constructions 9.4.2, 9.5.3]
+2.2
+Differential forms with log poles
+Let 푘 be a perfect field.
+Definition 2.12. A snc pair with supports (푋, ∆푋, Φ푋) over 푘 is a smooth scheme 푋 separated
+and of finite type over 푘 with a family of supports Φ푋 together with a reduced, effective divisor ∆푋
+on 푋 such that supp ∆푋 has simple normal crossings, in the sense that for any point 푥 ∈ 푋 there are
+regular parameters 푧1, … , 푧푐 ∈ 풪푋,푥 such that supp ∆푋 = 푉(푧1 ⋅ 푧2 ⋯ 푧푟) on a Zariski neighborhood
+of 푥.3 The interior 푈푋 of a snc pair with supports (푋, ∆푋, Φ푋) is
+푈푋 ∶= 푋 ⧵ supp ∆푋
+(2.13)
+The inclusion of 푈푋 in 푋 is denoted by 휄푋 ∶ 푈푋 → 푋.
+2Simply put ℱ is a sheaf of abelian groups on 푋.
+3This is equivalent to the more general definition [BPØ20, Def. 7.2.1] in the case where the base scheme is Spec 푘, which
+is all we need.
+4
+
+Here supp ∆푋 denotes the support of ∆푋 (if ∆푋 = ∑
+푖 푎푖퐷푖 where the 퐷푖 are prime divisors, then
+supp ∆푋 = ∪푖퐷푖). Similarly let 푗푋 ∶ supp ∆푋 → 푋 denote the evident inclusion.
+Definition 2.14 (compare with [CR11, Def. 1.1.4]). A pulling morphism 푓 ∶ (푋, ∆푋, Φ푋) →
+(푌, ∆푌, Φ푌) of snc pairs with supports is a pulling morphism 푓 ∶ 푋 → 푌 of underlying schemes
+with support such that 푓−1(supp ∆푌) ⊂ supp ∆푋; equivalently, 푓 restricts to a morphism 푓|푈푋 ∶
+푈푋 → 푈푌. A pushing morphism 푓 ∶ (푋, ∆푋, Φ푋) → (푌, ∆푌, Φ푌) of snc pairs with supports is a
+pushing morphism of underlying schemes with support such that 푓∗∆푌 = ∆푋.
+Note that if 푓 ∶ (푋, ∆푋, Φ푋) → (푌, ∆푌, Φ푌) is a pushing morphism then 푈푋 = 푓−1(푈푌), so for
+example if 푓 ∶ 푋 → 푌 is proper then so is the induced map 푈푋 → 푈푌.
+Convention 2.15 (compare with [CR11, p. 1.1.5]). A morphism of snc pairs with supports 푓 ∶
+(푋, ∆푋, Φ푋) → (푌, ∆푌, Φ푌) is flat, proper, an immersion, etc. if and only if the same is true of the
+underlying morphism of schemes 푓 ∶ 푋 → 푌. A diagram of snc pairs with supports
+(푋′, ∆푋′, Φ푋′)
+(푋, ∆푋, Φ푋)
+(푌′, ∆푌′, Φ푌′)
+(푌, ∆푌, Φ푌)
+푔′
+푓′
+푓
+푔
+(2.16)
+is cartesian if and only if the induced diagram of underlying schemes
+푋′
+푋
+푌′
+푌
+푔′
+푓′
+□
+푓
+푔
+(2.17)
+is cartesian.4
+The terminology is meant to suggest that pushing (resp. pulling) morphismsinduce pushforward
+(resp. pullback) maps on log Hodge cohomology, as we now describe.
+If (푋, ∆푋) is an snc pair, or more generally a normal separated scheme of finite type 푋 over 푘
+together with a sequence of effective Cartier divisors 퐷1, … , 퐷푁 ⊆ 푋 with sum ∆푋 = ∑
+푖 퐷푖, then
+it comes with a sheaf of differential forms with log poles Ω푋(log ∆푋). In the case where (푋, ∆푋, Φ푋)
+is snc, this sheaf and its properties are described in [EV92, §2]. For a definition and treatment of
+Ω푋(log ∆푋) in the much greater generality of logarithmic schemes we refer to [Ogu18, §IV].
+In some of the calculations below the following concrete local description will be very useful.
+Let 푧1, 푧2, … , 푧푛 be local coordinates at a point 푥 ∈ 푋 such that supp ∆푋 = 푉(푧1푧2 ⋯ 푧푟) in a neigh-
+borhood of 푥. Recall that as 푋 is smooth the differentials 푑 푧1, 푑 푧2, … , 푑 푧푛 freely generate Ω푋 on a
+neighborhood of 푥.
+Lemma 2.18 (see e.g. [EV92, §2]). Thesections 푑 푧1
+푧1 , … , 푑 푧푟
+푧푟 , 푑 푧푟+1, … , 푑 푧푛 freelygenerateΩ푋(log ∆푋)
+on a neighborhood of 푥.
+Given Ω푋(log ∆푋), we can form the exterior powers
+Ω푝
+푋(log ∆푋) ∶=
+푝
+⋀
+Ω푋(log ∆푋),
+(2.19)
+and combining Lemma 2.18 with (2.19) gives concrete local descriptions of the Ω푝
+푋(log ∆푋); in par-
+ticular, we see that Ωdim 푋
+푋
+(log ∆푋) = 휔푋(∆푋).
+4If we take the red pill of logarithmic geometry, it starts to seem almost more reasonable to only require flatness, properness,
+cartesianness and so on of the induced maps of interiors 푈푋 → 푈푌. However we do use the stronger restrictions of the given
+definition in some of the proofs below.
+5
+
+Definition 2.20. The log-Hodge cohomology with supports of a log-smooth pair with supports
+(푋, ∆푋, Φ푋) is defined by
+퐻푑(푋, ∆푋, Φ푋) =
+⨁
+푝+푞=푑
+퐻푞
+Φ(푋, Ω푝
+푋(log ∆푋))
+(2.21)
+Here 퐻푞
+Φ denotes local cohomology with respect to the family of supports Φ푋. For connected 푋, we
+define 퐻푑(푋, ∆푋, Φ푋) ∶= 퐻2 dim 푋−푑(푋, ∆푋, Φ푋), and in general we set 퐻푑(푋, ∆푋, Φ푋) = ⨁
+푖 퐻푑(푋푖, ∆푋푖, Φ푋푖)
+where 푋푖 are the connected components of 푋.
+Let 푓 ∶ (푋, ∆푋, Φ푋) → (푌, ∆푌, Φ푌) be pulling morphism of snc pairs with supports.
+Lemma 2.22 ([Ogu18, Prop. 2.3.1] + (2.19)). The map 푓 induces a morphism of sheaves
+푓∗Ω푝
+푌(log ∆푌)
+푑 푓∨
+����→ Ω푝
+푋(log ∆푋) adjoint to a morphism
+푓∗Ω푝
+푌(log ∆푌)
+푑푓∨
+����→ Ω푝
+푋(log ∆푋) for all p.
+(2.23)
+The essential content of this lemma is that when we pull back a log differentialform 휎 on (푌, ∆푌),
+it doesn’t develop poles of order ≥ 1 along ∆푋. Combining the previous lemma with proposition 2.9
+gives:
+Proposition 2.24 ([BPØ20, §9.1-2], see also [CR11, §2.1]). Foreverypullingmorphism푓 ∶ (푋, ∆푋, Φ푋) →
+(푌, ∆푌, Φ푌) there are functorial morphisms
+푅ΓΦΩ푝
+푌(log ∆푌) → 푅푓∗푅ΓΦΩ푝
+푌(log ∆푌) for all p
+(2.25)
+In particular, for each 푝, 푞 there are functorial homomorphisms
+푓∗ ∶ 퐻푞
+Φ(푌, Ω푝
+푌(log ∆푌)) → 퐻푞
+Φ(푋, Ω푝
+푋(log ∆푋))
+(2.26)
+and hence (summing over 푝 + 푞 = 푑) functorial homomorphisms
+푓∗ ∶ 퐻푑(푋, ∆푋, Φ푋) → 퐻푑(푌, ∆푌, Φ푌)
+(2.27)
+The maps 푓∗ ∶ 퐻푑(푋, ∆푋, Φ푋) → 퐻푑(푌, ∆푌, Φ푌) induced by a pushing morphism 푓 ∶ (푋, ∆푋, Φ푋) →
+(푌, ∆푌, Φ푌) can be obtained from a combination of Nagata compactification and Grothendieck du-
+ality.
+Lemma 2.28 ([BPØ20, §9.5], see also [CR11, §2.3]). Let 푓 ∶ (푋, ∆푋, Φ푋) → (푌, ∆푌, Φ푌) be a pushing
+morphism of equidimensional log-smooth pairs with support such that. Then letting 푐 = dim 푌−dim푋,
+for each 푝 there are functorial morphisms of complexes of coherent sheaves
+푅푓���푅ΓΦ푋(Ω푝
+푋(log ∆푋)) → 푅ΓΦ푌Ω푝+푐
+푌
+(log ∆푌)[푐]
+(2.29)
+inducing maps on cohomology
+푓∗ ∶ 퐻푞
+Φ푋(푋, Ω푝
+푋(log ∆푋)) → 퐻푞+푐
+Φ푌 (푌, Ω푝+푐
+푌
+(log ∆푌))
+(2.30)
+for all 푞.
+Since they enter into the calculations below, we give a description of these pushforward mor-
+phisms. Before beginning, a word on duality in our current setup: since we are working exclu-
+sively over Spec 푘, we can make use of compatible normalized dualizing complexes — namely, if
+휋 ∶ 푍 → Spec푘 is a separated finite type 푘-scheme then 휋!풪Spec 푘 is a dualizing complex [Stacks,
+Tag 0E2S, Tag 0FVU]. We will make repeated use of the behavior of dualizing with respect to differ-
+entials: as a consequence of Lemma 2.18, wedge product gives a perfect pairing
+Ω푝
+푋(log ∆푋)(−∆푋) ⊗ Ωdim 푋−푝
+푋
+(log ∆푋) → 휔푋
+(2.31)
+6
+
+(see also [Har77, Cor. III.7.13]) and so Ωdim 푋−푝
+푋
+(log ∆푋) ≃ 푅ℋ표푚푋(Ω푝
+푋(log ∆푋)(−∆푋), 휔푋). Here
+the derived sheaf Hom 푅ℋ표푚푋 agrees with the regular sheaf Hom as Ω푝
+푋(log ∆푋)(−∆푋) is locally
+free. On the other hand, the dualizing functor of 푋 is 푅ℋ표푚푋(Ω푝
+푋(log ∆푋)(−∆푋), 휔푋[dim 푋]) where
+휔푋 = Ωdim 푋
+푋
+. An upshot is that Grothendieck duality calculations involving the sheaves of differen-
+tial forms become more symmetric and predictable if we work with the shifted versions Ω푝
+푋(log ∆푋)(−∆푋)[푝];
+for example then we have the identity
+Ωdim 푋−푝
+푋
+(log ∆푋)[dim푋 − 푝] ≃ 푅ℋ표푚푋(Ω푝
+푋(log ∆푋)(−∆푋)[푝], 휔푋[dim 푋])
+Now, we need to compactify 푓 ∶ 푋 → 푌.
+Theorem 2.32 ([Nag63, §4 Thm. 2], [Con07, Thm. 4.1]). Let 푆 be a quasi-compact quasi-separated
+scheme and let 푋 → 푆 be a separated morphism of finite type. Then there is a dense open immersion of
+푆-schemes 푋 → 푋 such that 푋 is proper.
+Using Theorem 2.32 we obtain morphisms of schemes
+푋
+̄푋
+푌
+휄
+푓
+̄푓
+(2.33)
+where 휄 ∶ 푋 → ̄푋 is a dense open immersion and ̄푓 ∶ ̄푋 → 푌 is proper. Note that ̄푋 need not be
+smooth over 푘, and in the absence of resolutions of singularities5 there is not even a way to make ̄푋
+smooth. This means we cannot hope to upgrade ̄푋to a simple normal crossing pair ( ̄푋, ∆ ̄푋). However,
+we do still have a divisor ∆ ̄푋 ∶= ̄푓∗∆푦 on ̄푋. One way to overcome these difficulties is to equip the
+possibly singular ̄푋 with a logarithmic structure, in some sense associated to ∆ ̄푋, whose restriction
+to 푋 coincides with a logarithmic structure naturally defined by the simple normal crossing divisor
+∆푋.
+Formally, we use the log structure on ̄푋 pulled back from the log structure on (푌, ∆푌) [Ogu18,
+§III.1.6-7] along the morphism ̄푓 ∶ ̄푋 → 푌. Since (푌, ∆푌 = ∑푁
+푖=1 퐷푌
+푖 ) is a simple normal crossing
+pair, its associated log structure is Deligne-Faltings [Ogu18, §III.1.7] and can be encoded in the se-
+quence of inclusions of ideal sheaves 풪푌(−퐷푌
+푖 ) → 풪푌. The pullback log structure on ̄푋 can then be
+encoded in the sequence of inclusions of ideal sheaves
+̄푓−1풪푌(−퐷푌
+푖 ) ⋅ 풪 ̄푋 = 풪 ̄푋(− ̄푓∗퐷푌
+푖 ) → 풪 ̄푋.
+The pushforward morphisms of Lemma 2.28 are defined using the sheaves of log differential 푝-
+forms on ̄푋 over 푘 as described in [Ogu18, §IV.1, V.2] — these will be denoted6 by Ω푝
+푋(log ∆푋). The
+essential properties that we need are:
+• Ω푝
+푋(log ∆푋) is a coherent sheaf on 푋 together with a functorial morphism
+Ω푝
+푌(log ∆푌) → 푓∗Ω푝
+푋(log ∆푋).
+Coherence can be obtained as follows: first, the log structure on (푌, ∆푌) is coherent ([Ogu18,
+§III.1.9]), and hence so is its pullback to ̄푋 (see for example [Ogu18, Def. III.1.1.5, Rmk III.1.1.6]).
+Then [Ogu18, Cor. IV.1.2.8] implies Ω1
+푋(log ∆푋) is a coherent sheaf, and it follows that its 푝-th
+exteriorpowersare coherentsheavesas well. The desiredfunctorial morphismcan be obtained
+from [Ogu18, Prop. IV.1.2.15].
+5At the time of this writing, this applies to the cases char 푘 = 푝 > 0 and dim 푋 > 3.
+6This is an abuse of notation since the construction of this sheaf is (as far as we know) not the same as the one for simple
+normal crossing pairs described above Lemma 2.18, however the notation of [Ogu18] seems unsatisfactory for our purposes
+as we wish to stress that these are not the ordinary differential forms Ω푝
+푋,
+7
+
+• There is a natural isomorphism Ω푝
+푋(log ∆푋)|푋 ≃ Ω푝
+푋(∆푋). This can be seen by observing that
+the log structures on (푋, ∆푋) and ̄푋 are obtained as pullbacks of the log structure on (푌, ∆푌)
+with respect to 푓 and ̄푓 respectively (in the case of (푋, ∆푋) this follows from Definition 2.14,
+and in the latter case it is how we defined the log structure on ̄푋). Hence considering eq. (2.33)
+we find that the log structure on ̄푋 restricts to that on (푋, ∆푋).
+Hence in particular Ω푝
+푋(log ∆푋) is a functorial coherent extension of Ω푝
+푋(∆푋) to the possibly non-snc
+log scheme ̄푋. Starting with the log differential
+푑 pr∨
+푌 ∶ Ω푝
+푌(log ∆푌)[푝] → 푅푓∗Ω푝
+푋(log ∆푋)[푝],
+twisting by −∆푌 and using the projection formula gives a morphism (note: this is where we use the
+assumptions that 푓∗∆푌 = ∆푋 and ̄푓∗∆푌 = ∆ ̄푋)
+Ω푝
+푌(log ∆푌)(−∆푌)[푝] → 푅푓∗Ω푝
+푋(log ∆푋)(−∆푋)[푝]
+(2.34)
+to which we apply Grothendieck duality:
+Theorem 2.35 (Grothendieck duality, [R&D, Cor. VII.3.4], [Con00, Thm. 3.4.4]). Let 푓 ∶ 푋 → 푌 be a
+proper morphism of finite-dimensional noetherian schemes and assume 푌 admits a dualizing complex
+(for example 푋 and 푌 could be schemes of finite type over 푘). Then for any pair of objects ℱ∙ ∈ 퐷−
+푞푐(푋)
+and 풢∙ ∈ 퐷+
+푐 (푌) there is a natural isomorphism
+푅푓∗푅퐻표푚푋(ℱ∙, 푓!풢∙) ≃ 푅퐻표푚푌(푅푓∗ℱ∙, 풢∙) in 퐷푏
+푐 (푌)
+Combining Theorem 2.35 with eq. (2.34) gives a morphism
+푅푓∗푅ℋ표푚푋(Ω푝
+푋(log ∆푋)(−∆푋)[푝], 휔∙
+푋) = 푅ℋ표푚푌(푅푓∗Ω푝
+푋(log ∆푋)(−∆푋)[푝], 휔푌[dim 푌])
+푅ℋ표푚푌(Ω푝
+푌(log ∆푌)(−∆푌)[푝], 휔푌[dim 푌])
+(2.36)
+where the equality is Theorem 2.35 and the vertical map is induced by (2.34). Adding supports gives
+a morphism
+푅푓∗푅ΓΦ푋푅ℋ표푚푋(Ω
+푝
+푋(log ∆푋)(−∆푋)[푝], 휔푋[dim 푋]) = 푅푓∗푅ΓΦ푋푅ℋ표푚푋(Ω
+푝
+푋(log ∆푋)(−∆푋)[푝], 휔∙
+푋)
+푅ΓΦ푌푅ℋ표푚푌(Ω푝
+푌(log ∆푌)(−∆푌)[푝], 휔푌[dim 푌])
+(2.37)
+where the equality is obtained from the excision property of local cohomology, compatibility of the
+dualizing functor with restriction and the natural isomorphism Ω푝
+푋(log ∆푋)|푋 ≃ Ω푝
+푋(∆푋). Using
+(2.31) we obtain
+Ωdim 푋−푝
+푋
+(log ∆푋) ≃ ℋ표푚푋(Ω푝
+푋(log ∆푋)(−∆푋), 휔푋) = 푅ℋ표푚푋(Ω푝
+푋(log ∆푋)(−∆푋), 휔푋)
+where the last equality uses the fact that Ω푝
+푋(log ∆푋)(−∆푋) is locally free. A similar calculation on
+푌 transforms (2.37) into:
+푅푓∗푅ΓΦ푋Ωdim 푋−푝
+푋
+(log ∆푋)[dim푋 − 푝] → 푅ΓΦ푌Ωdim 푌−푝
+푌
+(log ∆푌)[dim푌 − 푝]
+and reindexing like 푝 ↔ dim 푋 − 푝 recovers Lemma 2.28.
+8
+
+3
+A base change formula
+Lemma 3.1 (compare with [CR11, Prop. 2.3.7]). Let
+(푋′, ∆푋′, Φ푋′)
+(푋, ∆푋, Φ푋)
+(푌′, ∆푌′, Φ푌′)
+(푌, ∆푌, Φ푌)
+□
+푔′
+푓′
+푓
+푔
+(3.2)
+be a cartesian diagram of equidimensional snc pairs with supports, where 푓, 푓′ (resp. 푔, 푔′) are pushing
+(resp. pulling) morphisms and 푔 is either flat or a closed immersion transverse to 푓. Then
+푔∗푓∗ = 푓′
+∗푔′∗ ∶ 퐻∗(푋, ∆푋, Φ푋) → 퐻∗(푌′, ∆푌′, Φ푌′).
+We will prove this following Chatzistamatiou and Rülling’s argument [CR11, Prop. 2.3.7] quite
+closely, at various points reducing to statements proved therein. In the proofs we will make use of a
+slight variant of Definition 2.3.
+Definition 3.3. If 푓 ∶ 푋 → 푌 is a morphism of noetherian schemes and let Φ푌 is a family of
+supports on 푌, then
+푓−1
+∗ (Φ푌) ∶= {푍 ⊆ 푋 | 푓|푍 is proper and 푓(푍) ∈ Φ푌}
+Lemma 3.4. It suffices to prove Lemma 3.1 in the cases where 푓 is either
+(푖) a projection morphism of the form pr푌 ∶ (푋 × 푌, pr∗
+푌∆푌, pr−1
+푌∗(Φ푌)) → (푌, ∆푌, Φ푌), or
+(푖푖) a closed immersion.
+Remark 3.5. This lemma makes essential use of the functoriality part of Lemma 2.28.
+Proof. We can decompose (3.2) as a concatenation of cartisian diagrams
+(푋′, ∆푋′, Φ푋′)
+(푋, ∆푋, Φ푋)
+(푋 × 푌′, pr∗
+푌′∆푌, pr−1
+푌′∗(Φ′
+푌))
+(푋 × 푌, pr∗
+푌∆푌, pr−1
+푌∗(Φ푌))
+(푌′, ∆푌′, Φ푌′)
+(푌, ∆푌, Φ푌)
+(2)
+푔′
+ℎ′
+ℎ
+(1)
+pr푌′
+id×푔
+pr푌
+푔
+(3.6)
+where ℎ = id × 푓 is the graph morphism of 푓 and ℎ′ = 푔′ × 푓′. If 푔 is flat or a closed immersion
+transverse to 푓 then id × 푔 is flat or a closed immersion transverse to ℎ (by base change).
+Here the only new feature not covered in [CR11, Prop. 2.3.7] is the presence of divisors, and we
+simply note that ∆푋 = 푓∗∆푋 = ℎ∗pr∗
+푌∆푌 and similarly for ∆푋′, so that both pr푌 and ℎ are pushing
+morphisms in the sense of Definition 2.14, and similarly for the left vertical maps. In other words, the
+supports and divisors in the middle row have been chosen precisely so that the vertical morphisms
+are all “pushing.”
+We proceed to consider case (i), and wish to point out that for this case 푔 can be arbitrary (we will
+need the flatness/transversality restrictions in case (ii)). In what follows we set 푑푋 = dim 푋, 푑푌 =
+dim 푌 and similarly for 푋′, 푌′. Using Theorem 2.32 we obtain a compactification 휄 ∶ 푋 → 푋 over
+푘 of the smooth, separated and finite type 푘-scheme 푋 in the upper right corner of (3.2) and (3.6).
+9
+
+This results in a compactification of the square (1) in (3.6) which we write as
+(푋 × 푌′, pr∗
+푌′∆푌, pr−1
+푌′∗(Φ′
+푌))
+(푋 × 푌, pr∗
+푌∆푌, pr−1
+푌∗(Φ푌))
+(푋 × 푌′, pr
+∗
+푌′∆푌, pr
+−1
+푌′∗(Φ′
+푌))
+(푋 × 푌, pr
+∗
+푌∆푌, pr
+−1
+푌∗(Φ푌))
+(푌′, ∆푌′, Φ푌′)
+(푌, ∆푌, Φ푌)
+휄×id
+id×푔
+휄×id
+pr푌′
+id×푔
+pr푌
+푔
+(3.7)
+By the description following Lemma 2.28, we know that
+pr푌∗ ∶ 퐻∗(푋 × 푌, pr∗
+푌∆푌, pr−1
+푌∗(Φ푌)) → 퐻∗(푌, ∆푌, Φ푌)
+stems from a morphism
+푅pr푌∗푅ℋ표푚푋×푌(Ω푝
+푋×푌(log pr∗
+푌∆푌)(−pr∗
+푌∆푌)[푝], 휔∙
+푋×푌) → Ω푑푌−푝
+푌
+(log ∆푌)[푑푌 − 푝]
+(3.8)
+obtained as the Grothendieck dual of a log differential of pr푌 (here and throughout what follows, a
+similar statement holds for pr푌′). By an observation of Chatzistamatiou-Rülling , this map factors
+as
+푅pr푌∗푅ℋ표푚푋×푌(Ω푝
+푋×푌(log pr
+∗
+푌∆푌)(−pr
+∗
+푌∆푌)[푝], 휔∙
+푋×푌)
+→ 푅pr푌∗푅ℋ표푚푋×푌(퐿pr
+∗
+푌Ω푝
+푌(log ∆푌)(−∆푌)[푝], 휔∙
+푋×푌)
+≃
+���������→
+adjunction 푅ℋ표푚푌(Ω푝
+푌(log ∆푌)(−∆푌)[푝], 푅pr푌∗휔∙
+푋×푌)
+����→
+trace 푅ℋ표푚푌(Ω푝
+푌(log ∆푌)(−∆푌)[푝], 휔푌[푑푌])
+≃�→ Ω푑푌−푝
+푌
+(log ∆푌)[푑푌 − 푝]
+(3.9)
+where the adjunction isomorphism is [R&D, Prop. II.5.10], and the map labeled trace is induced by
+the Grothendieck trace 푅pr푌∗휔∙
+푋×푌 → 휔푌[푑푌]. If it were the case that 푋 were smooth, then the
+usual “box product” decomposition
+휔∙
+푋×푌 ≃ 휔푋[푑푋] ⊠ 휔푌[푑푌] ∶= pr∗
+푋 휔푋[푑푋] ⊗ pr푌∗휔푌[푑푌]
+together with the perect pairings (2.31) and the local freeness of Ω푝
+푌(log ∆푌)(−∆푌)[푝] would give an
+identification
+푅ℋ표푚푋×푌(퐿pr
+∗
+푌Ω푝
+푌(log ∆푌)(−∆푌)[푝], 휔∙
+푋×푌) ≃ pr∗
+푋 휔푋[푑푋] ⊗ pr
+∗
+푌Ω푑푌−푝
+푌
+(log ∆푌)[푑푌 − 푝] (3.10)
+In fact a more careful version of this argument, carrying out the above calculation on the smooth
+locus 푋 × 푌 and using excision, shows that 퐻∗(푋 × 푌, pr∗
+푌∆푌, pr−1
+푌∗(Φ푌)) → 퐻∗(푌, ∆푌, Φ푌) always
+factors through the summand 퐻∗
+Φ푋(푋 × 푌, pr∗
+푋 휔푋 ⊗ pr
+∗
+푌Ω푑푌−푝
+푌
+(log ∆푌)).
+Our next lemma implies that even when 푋 is not known to be smooth, (3.8) still factors through
+something like 푅pr푌∗(pr∗
+푋 휔푋[푑푋] ⊗ pr
+∗
+푌Ω푑푌−푝
+푌
+(log ∆푌)[푑푌 − 푝]), provided we replace pr∗
+푋 휔푋[푑푋]
+with pr
+!
+푌풪푌.
+Lemma 3.11 (compare with [CR11, Lem. 2.2.16]). For each 푝 there is a natural map
+훾 ∶ pr
+!
+푌풪푌 ⊗ pr
+∗
+푌Ω푑푌−푝
+푌
+(log ∆푌)(−∆푌)[푑푌 − 푝] → 푅ℋ표푚푋×푌(pr
+∗
+푌Ω푝
+푌(log ∆푌)(−∆푌)[푝], 휔∙
+푋×푌)
+10
+
+such that the restriction of 훾 to 푋 × 푌 agrees with the isomorphism
+pr∗
+푋 휔푋[푑푋] ⊗ pr∗
+푌 Ω푑푌−푝
+푌
+(log ∆푌)(−∆푌)[푑푌 − 푝]
+≃�→ 푅ℋ표푚푋×푌(퐿 pr∗
+푌 Ω푝
+푌(log ∆푌)(−∆푌)[푝], 휔∙
+푋×푌)
+and such that the composition
+푅pr푌∗(pr∗
+푋 휔푋[푑푋] ⊗ pr∗
+푌 Ω푑푌−푝
+푌
+(log ∆푌)(−∆푌)[푑푌 − 푝])
+푅pr푌∗(훾)
+��������→ 푅pr푌∗푅ℋ표푚푋×푌(pr∗
+푌 Ω푝
+푌(log ∆푌)(−∆푌)[푝], 휔∙
+푋×푌)
+≃
+���������→
+adjunction 푅ℋ표푚푋×푌(Ω푝
+푌(log ∆푌)(−∆푌)[푝], 푅pr푌∗휔∙
+푋×푌)
+trace
+����→ 푅ℋ표푚푋×푌(Ω푝
+푌(log ∆푌)(−∆푌)[푝], 휔푌[푑푌]) ≃ Ω푑푌−푝
+푌
+(log ∆푌)(−∆푌)[푑푌 − 푝]
+(3.12)
+coincides with the composition
+푅pr푌∗(pr
+!
+푌풪푌 ⊗ pr
+∗
+푌Ω푑푌−푝
+푌
+(log ∆푌)(−∆푌)[푑푌 − 푝])
+proj.
+����→
+form. 푅pr푌∗(pr
+!
+푌풪푌) ⊗ pr
+∗
+푌Ω푑푌−푝
+푌
+(log ∆푌)(−∆푌)[푑푌 − 푝]
+tr ⊗id
+�����→ Ω푑푌−푝
+푌
+(log ∆푌)(−∆푌)[푑푌 − 푝]
+(3.13)
+By base change for dualizing complexes ([Stacks, Tag 0BZX, Tag 0E2S]) applied to the cartesian
+diagram
+푋 × 푌
+푋
+푌
+Spec 푘
+(note that this is a very mild situation: 푋 → Spec 푘 is flat and proper and 푌 → Spec 푘 is smooth) we
+see that pr
+!
+푌풪푌 ≃ pr∗
+푋 휔∙
+푋. This makes the map 훾 look even more like (3.10).
+Proof. Following [CR11, Lem. 2.2.16] we begin with the morphism
+푒 ∶ pr
+!
+푌풪푌 ⊗퐿 퐿pr
+∗
+푌휔∙
+푌 → pr
+!
+푌휔∙
+푌 =∶ 휔∙
+푋×푌
+of [Con00, p. 4.3.12], which as explained therein agrees with
+pr∗
+푋 휔푋[푑푋] ⊗ pr∗
+푌 휔푌[푑푌]
+≃�→ 휔푋×푌[푑푋 + 푑푌]
+on locus 푋 × 푌,7 and has the property that
+푅푝푟푌∗(pr
+!
+푌풪푌 ⊗퐿 퐿pr
+∗
+푌휔∙
+푌)
+푅푝푟푌∗휔∙
+푋×푌
+푅푝푟푌∗pr
+!
+푌풪푌 ⊗퐿 휔∙
+푌
+휔∙
+푌
+푅푝푟푌∗푒
+proj. form
+tr
+tr ⊗id
+7See Conrad’s comment “It is easy to check that 푒푓 coincides with (3.3.21) in the smooth case and is compatible with
+composites in f (using (4.3.6).”
+11
+
+commutes [Con00, Thm. 4.4.1]. We then define our version of 훾 as the composition
+pr
+!
+푌풪푌 ⊗퐿 퐿pr
+∗
+푌Ω푑푌−푝
+푌
+(log ∆푌)(−∆푌)[푑푌 − 푝]
+id⊗퐿(2.31)
+���������→ pr
+!
+푌풪푌 ⊗퐿 퐿pr
+∗
+푌푅ℋ표푚푌(Ω푝
+푌(log ∆푌)[푝], 휔∙
+푌)
+functoriality
+�����������→
+of 퐿pr
+∗
+푌,⊗퐿
+푅ℋ표푚푋×푌(퐿pr
+∗
+푌Ω푝
+푌(log ∆푌)[푝], pr
+!
+푌풪푌 ⊗퐿 휔∙
+푌)
+induced by
+���������→
+푒
+푅ℋ표푚푋×푌(퐿pr
+∗
+푌Ω푝
+푌(log ∆푌)[푝], 휔∙
+푋×푌)
+(3.14)
+Note that we may drop the “퐿”s as Ω푑푌−푝
+푌
+(log ∆푌)(−∆푌) and Ω푝
+푌(log ∆푌) are locally free. Verification
+of the stated compatibilities is as in [CR11, Lem. 2.2.16].
+Remark 3.15. It seems like we could have also used the more general version of [Con00, p. 4.3.12]
+푒′ ∶ pr
+!
+푌풪푌 ⊗퐿 퐿pr
+∗
+푌Ω푑푌−푝
+푌
+(log ∆푌)(−∆푌)[푑푌 − 푝] → pr
+!
+푌Ω푑푌−푝
+푌
+(log ∆푌)(−∆푌)[푑푌 − 푝]
+together with the description
+pr
+!
+푌Ω푑푌−푝
+푌
+(log ∆푌)(−∆푌)[푑푌 − 푝] = 퐷푋×푌(퐿pr
+∗
+푌퐷푌(Ω푑푌−푝
+푌
+(log ∆푌)(−∆푌)[푑푌 − 푝]))
+where 퐷푌(−) = 푅ℋ표푚(−, 휔∙
+푌) and similarly for 퐷푋×푌.
+Using this modified 훾, we obtain a modified version of the diagram [CR11, p. 732 during Lem.
+2.3.4], namely (3.16) in Figure 1). To make this diagram legible, we use a few abbreviations: all func-
+tors are derived, we use the dualizing functors of the form 퐷푌(−) = 푅ℋ표푚푌(−, 휔∙
+푌) and we let
+푑 = 푑푋 + 푑푌. Lemma 3.11 shows that triangles involving 훾 commute, and (3.9) gives commutativity
+of the rest of the diagram. The usefulness of this diagram is that by definition beginning in the top
+left corner and following the path →↓ we obtain the pushforward on Hodge cohomology
+pr푌∗ Γpr−1
+푌∗ Φ푌Ω푑−푝
+푋×푌(log pr∗
+푌∆푌)[푑 − 푝] → ΓΦ푌Ω푑푌−푝
+×푌
+(log ∆푌)(−∆푌)[푑푌 − 푝]
+but following ↓→ gives a composition whose behavior with respect to (3.7) is easier to analyze.
+Namely, we have a diagram like (3.16) on 푌′, and in fact a map from (3.16) to 푔∗ of the analogous
+diagram on 푌′, and hence from the preceding discussion it will suffice to prove commutativity of
+(3.17) of Figure 1.
+Applying excision together with Lemma 3.11 we may rewrite the top row of (3.17) as
+푅 pr푌∗ 푅Γpr−1
+푌∗ Φ푌Ω푑−푝
+푋×푌(log pr∗
+푌∆푌)[푑 − 푝]
+project
+������→ 푅 pr푌∗ 푅Γpr−1
+푌∗ Φ푌(pr∗
+푋 휔푋[푑푋] ⊗ pr∗
+푌 Ω푑푌−푝
+푌
+(log ∆푌)(−∆푌)[푑푌 − 푝])
+proj.
+�����→
+form. 푅 pr푌∗ 푅Γpr−1
+푌∗ Φ푌(pr∗
+푋 휔푋[푑푋]) ⊗ Ω푑푌−푝
+푌
+(log ∆푌)(−∆푌)[푑푌 − 푝]
+tr ⊗id
+�����→ 푅ΓΦ푌Ω푑푌−푝
+푌
+(log ∆푌)(−∆푌)[푑푌 − 푝]
+(3.18)
+where the first map is induced by a projection
+Ω푑−푝
+푋×푌(log pr∗
+푌∆푌)[푑 − 푝] → pr∗
+푋 휔푋[푑푋] ⊗ pr∗
+푌 Ω푑푌−푝
+푌
+(log ∆푌)(−∆푌)[푑푌 − 푝]
+coming from a Künneth-type decomposition of Ω푑−푝
+푋×푌(log pr∗
+푌∆푌), the second is the projection for-
+mula, and the last map is induced by a trace map with supports defined as the composition
+푅 pr푌∗ 푅Γpr−1
+푌∗ Φ푌(pr∗
+푋 휔푋[푑푋])
+excision
+�������→ 푅pr푌∗푅Γpr−1
+푌∗Φ푌(pr
+!
+푌풪푌)
+Proposition 2.9
+�������������→ 푅ΓΦ푌푅pr푌∗(pr
+!
+푌풪푌)
+tr�→ 푅ΓΦ푌풪푌
+(3.19)
+12
+
+pr푌∗ Γpr−1
+푌∗ Φ푌 Ω푑−푝
+푋×푌(log pr∗
+푌∆푌)[푑 − 푝]
+pr푌∗Γpr−1
+푌∗Φ푌퐷푋×푌(Ω푝
+푋×푌(log pr∗
+푌∆푌)(−pr∗
+푌∆푌)[푝])
+pr푌∗Γpr−1
+���∗Φ푌퐷푋×푌(pr
+∗
+푌Ω푝
+×푌(log ∆푌)(−∆푌)[푝])
+pr푌∗Γpr−1
+푌∗Φ푌(pr
+!
+푌풪푌 ⊗ pr
+∗
+푌Ω푑푌−푝
+푌
+(log ∆푌)(−∆푌)[푑푌 − 푝])
+ΓΦ푌 퐷푌(Ω푝
+푌(log ∆푌)(−∆푌)[푝]) = ΓΦ푌Ω푑푌−푝
+푌
+(log ∆푌)(−∆푌)[푑푌 − 푝]
+excision
+excision+퐿푒푚푚푎 3.11
+푑pr∨
+푌
+푑pr∨
+푌
+(3.13)
+pr푌∗(훾)
+(3.16)
+pr푌∗ Γpr−1
+푌∗Φ푌Ω푑−푝
+푋×푌(log pr∗
+푌∆푌)[푑 − 푝]
+pr푌∗Γpr−1
+푌∗Φ푌(pr
+!
+푌풪푌 ⊗ pr
+∗
+푌Ω푑푌−푝
+푌
+(log ∆푌)(−∆푌)[푑푌 − 푝])
+ΓΦ푌Ω푑푌−푝
+푌
+(log ∆푌)(−∆푌)[푑푌 − 푝]
+푔∗ pr푌′∗ Γpr−1
+푌′∗Φ푌′ Ω푑−푝
+푋×푌′(log pr∗
+푌′∆푌′)[푑 − 푝]
+푔∗pr푌′∗Γpr−1
+푌′∗Φ푌′ (pr
+!
+푌′풪푌′ ⊗ pr
+∗
+푌′Ω푑푌−푝
+푌′
+(log ∆푌′)(−∆푌)[푑푌 − 푝])
+푔∗ΓΦ푌′ Ω푑푌−푝
+푌′
+(log ∆푌′)(−∆푌′)[푑푌 − 푝]
+(3.17)
+Figure 1: Modified versions of diagrams appearing in the proof of [CR11, Lem. 2.3.4] (all functors derived)
+13
+
+Here the second map comes from the functoriality properties of Proposition 2.9, since there is an
+inclusion pr−1
+푌∗ Φ푌 ⊆ pr−1
+푌 Φ푌. The decomposition (3.18) maps to a similar decomposition of the bot-
+tom row of (3.17), and the only commutativity not guaranteed by standard functoriality properties
+(e.g. functoriality of the projection formula appearing in the second map of (3.18)) is that of
+푅 pr푌∗ 푅Γpr−1
+푌∗ Φ푌(pr∗
+푋 휔푋[푑푋]) ⊗ Ω
+푑푌−푝
+푌
+(log ∆푌)(−∆푌)[푑푌 − 푝]
+푅ΓΦ푌Ω
+푑푌−푝
+푌
+(log ∆푌)(−∆푌)[푑푌 − 푝]
+푅푔∗(푅 pr푌′∗ 푅Γpr−1
+푌′∗ Φ푌′ (pr∗
+푋 휔푋[푑푋]) ⊗ Ω푑푌−푝
+푌′
+(log ∆푌′)(−∆푌′)[푑푌 − 푝])
+푅푔∗(푅ΓΦ푌′ Ω푑푌−푝
+푌′
+(log ∆푌′)(−∆푌′)[푑푌 − 푝])
+tr ⊗id
+tr′ ⊗id
+(3.20)
+But applying one more projection formula to the bottom row of (3.20), we see (3.20) is obtained by
+tensoring the differential
+Ω푑푌−푝
+푌
+(log ∆푌)(−∆푌)[푑푌 − 푝] → 푅푔∗Ω푑푌−푝
+푌′
+(log ∆푌′)(−∆푌′)[푑푌 − 푝]
+with
+푅 pr푌∗ 푅Γpr−1
+푌∗ Φ푌(pr∗
+푋 휔푋[푑푋])
+푅ΓΦ푌풪푌
+푅푔∗(푅 pr푌′∗ 푅Γpr−1
+푌′∗ Φ푌′ (pr∗
+푋 휔푋[푑푋]))
+푅푔∗(푅ΓΦ푌′ 풪푌′)
+tr ⊗id
+tr′ ⊗id
+(3.21)
+and the commutativity of (3.21) is proved in [CR11, Lem. 2.3.4]. So far we have proved:
+Lemma 3.22. Lemma 3.1 holds in case (i) of Lemma 3.4.
+It remains to deal with case (ii) of Lemma 3.4, and for this we use the following lemma.
+Lemma 3.23 (compare with [CR11, Cor. 2.2.22]). Consider a diagram of pure-dimensional snc pairs
+(푋′, ∆푋′)
+(푋, ∆푋)
+(푌′, ∆푌′)
+(푌, ∆푌)
+푔′
+횤′
+횤
+푔
+(3.24)
+where 횤, 횤′ are pushing closed immersions and dim푌 − dim 푋 = dim 푌′ − dim 푋′ =∶ 푐. Then, for all
+푞 the diagram
+횤∗Ω푞
+푋(log ∆푋)[푞]
+푅푔∗횤′
+∗Ω푞
+푋′(log ∆푋′)
+Ω푞+푐
+푌
+(log ∆푌)[푞 + 푐]
+푅푔∗Ω푞+푐
+푌′ (log ∆푌′)[푞 + 푐]
+푑푔′∨
+푑푔∨
+(3.25)
+commutes, where the horizontal maps are induced by log differentials and the left vertical map is the
+composition
+횤∗Ω푞
+푋(log ∆푋)[푞]
+≃�→ 횤∗푅ℋ표푚(Ω푑푋−푞
+푋
+(log ∆푋)(−∆푋)[푑푋 − 푞], 휔∙
+푋)
+duality
+������→ 푅ℋ표푚(횤∗Ω푑푋−푞
+푋
+(log ∆푋)(−∆푋)[푑푋 − 푞], 휔∙
+푌)
+푑횤∨
+���→ 푅ℋ표푚(Ω푑푋−푞
+푌
+(log ∆푌)(−∆푌)[푑푋 − 푞], 휔∙
+푌)
+≃�→ Ω푞+푐
+푌
+(log ∆푌)[푞 + 푐]
+(3.26)
+and the right vertical arrow is 푅푔∗ of a similar composition on 푌′.
+14
+
+Note that the codimension hypotheses hold if 푔 is flat or a closed immersion transverse to 횤.
+Proof. While it seems a proof following [CR11, Cor. 2.2.22] step-by-step is possible, we instead reduce
+to the case proved there as follows: first, observe that there is an evident map from the cartesian
+diagram
+푈푋′
+푈푋
+푈푌′
+푈푌
+(3.27)
+of interiors to (3.24). Noting that (3.25) will map to a similar diagram obtained from (3.27), that the
+compositions (3.26) are at least compatible with Zariski localization, and that the situation of (3.27)
+is covered by [CR11, Cor. 2.2.22], it will suffice to show that the natural map
+ℎ0푅ℋ표푚푌(횤∗Ω푞
+푋(log ∆푋)[푞], 푅푔∗Ω푞+푐
+푌′ (log ∆푌′)[푞 + 푐]) → ℎ0푅ℋ표푚푈푌(횤∗Ω푞
+푈푋[푞], 푅푔∗Ω푞+푐
+푈푌′ [푞 + 푐])
+(3.28)
+is injective. This can be checked Zariski-locally at a point 푥 ∈ 푋 ⊆ 푌, so we may assume 푋 ⊆ 푌
+is a global complete intersection, say of 푡1, … , 푡푐 ∈ 풪푌. In that case the 푡푖 define a Koszul resolu-
+tion 풦∙(푡푖) → 풪푋, and because 푋′ = 푌′ ×푌 푋 = 푉(푡1◦푔, ⋯ 푡푐◦푔) is smooth of codimension 푐 by
+hypotheses, it must be that the 푡푖◦푔 are also a regular sequence, hence
+퐿푖푔∗풪푋 = ℎ−푖푔∗풦∙(푡푖) = {풪푋′,
+푖 = 0
+0
+otherwise
+in other words 퐿푔∗풪푋 = 풪푋′. Now using the fact that Ω푞
+푋(log ∆푋) is locally free on 푋′ we conclude
+퐿푔∗횤∗Ω푞
+푋(log ∆푋)[푞] = 푔∗횤∗Ω푞
+푋(log ∆푋)[푞] = 횤′
+∗푔′∗Ω푞
+푋(log ∆푋)[푞]
+Next, applying derived adjunction to both sides of (3.28) gives a commutative diagram
+푅ℋ표푚푌(횤∗Ω
+푞
+푋(log ∆푋)[푞], 푅푔∗Ω
+푞+푐
+푌′ (log ∆푌′)[푞 + 푐])
+푅ℋ표푚푈푌(횤∗Ω
+푞
+푈푋[푞], 푅푔∗Ω
+푞+푐
+푈푌′ [푞 + 푐])
+푅푔∗푅ℋ표푚푌′(퐿푔∗횤∗Ω
+푞
+푋(log ∆푋)[푞], Ω
+푞+푐
+푌′ (log ∆푌′)[푞 + 푐])
+푅푔∗푅ℋ표푚푈푌′ (퐿푔∗횤∗Ω
+푞
+푈푋[푞], Ω
+푞+푐
+푈푌′ [푞 + 푐])
+푅푔∗푅ℋ표푚푌′(횤′
+∗푔′∗Ω푞
+푋(log ∆푋)[푞], Ω푞+푐
+푌′ (log ∆푌′)[푞 + 푐])
+푅푔∗푅ℋ표푚푈푌′ (횤′
+∗푔′∗Ω푞
+푈푋[푞], Ω푞+푐
+푈푌′ [푞 + 푐])
+(3.29)
+Getting even more Zariski-local we may assume Ω푞
+푋(log ∆푋) is free, say generated by 푑푥1, … , 푑푥푛
+and in that case
+푅ℋ표푚푌′(횤′
+∗푔′∗Ω푞
+푋(log ∆푋)[푞], Ω푞+푐
+푌′ (log ∆푌′)[푞 + 푐])
+= (
+∏
+푖
+푅ℋ표푚푌′(풪푋′푑푥푖[푞], 풪푌′[푞 + 푐])) ⊗ Ω푞+푐
+푌′ (log ∆푌′)
+(3.30)
+and by Grothendieck’s fundamental local isomorphism [Con00, §2.5]
+푅ℋ표푚푌′(풪푋′[푞], 풪푌′[푞 + 푐])) ≃ ℰ푥푡푐
+푌′(풪푋′, 풪푌′) ≃ det(ℐ푋′∕ℐ푋′)∨
+(3.31)
+(the last 2 as sheaves supported in degree 0). In particular, this is an invertible sheaf on 푋′, and it
+follows that the left hand side of (3.30) is a locally free sheaf (supported in degree 0) on 푋′. Recalling
+푋′ is smooth and so in particular reduced, and since 푈푌′ ∩ 푋′ is a dense open (this is part of the
+15
+
+hypothesis that 푋′ → 푌′ is a pulling map) the natural map
+ℎ0푅ℋ표푚푌′(횤′
+∗푔′∗Ω푞
+푋(log ∆푋)[푞], Ω푞+푐
+푌′ (log ∆푌′)[푞 + 푐])
+→ ℎ0푅ℋ표푚푌′(횤′
+∗푔′∗Ω푞
+푋(log ∆푋)[푞], Ω푞+푐
+푌′ (log ∆푌′)[푞 + 푐])|푈푌′
+≃ ℎ0푅ℋ표푚푈푌′ (횤′
+∗푔′∗Ω푞
+푋(log ∆푋)|푈푌′ [푞], Ω푞+푐
+푌′ (log ∆푌′)|푈푌′ [푞 + 푐])
+(3.32)
+is injective, where on the third line we have applied localization for ℰ푥푡. Now left-exactness of 푔∗
+gives an injection
+ℎ0푅푔∗푅ℋ표푚푌′(횤′
+∗푔′∗Ω푞
+푋(log ∆푋)[푞], Ω푞+푐
+푌′ (log ∆푌′)[푞 + 푐])
+→ ℎ0푅푔∗푅ℋ표푚푈푌′ (횤′
+∗푔′∗Ω푞
+푋(log ∆푋)|푈푌′ [푞], Ω푞+푐
+푌′ (log ∆푌′)|푈푌′ [푞 + 푐])
+(3.33)
+To complete the proof, we use (3.29) to identify the map (3.33) with (3.28).
+Corollary 3.34. Lemma 3.1 holds in case (ii) of Lemma 3.4.
+Proof. This follows by applying cohomology with supports to (3.25).
+This completes our proof of Lemma 3.1.
+Corollary 3.35 (projection formula, compare with [CR11, Prop. 1.1.16]). Let 푓 ∶ 푋 → 푌 be a map
+of smooth schemes admitting two different enhancements to maps of smooth schemes with supports,
+(푋, ∆푋, Φ푋) → (푌, ∆푌, 푓(Φ푋)) pushing and (푋, 푓∗(∆′
+푌), 푓−1(Φ푌)) → (푌, ∆′
+푌, Φ푌) pulling
+Assume in addition that ∆푋 + 푓∗(∆′
+푌) and ∆푌 + ∆′
+푌 are (reduced) snc divisors. Then
+(푋, ∆푋 + 푓∗(∆′
+푌), Φ푋 ∩ 푓−1(Φ푌)) → (푌, ∆푌 + ∆′
+푌, 푓(Φ푋) ∩ Φ푌)
+is also a pushing map, and
+푓∗(훽 ⌣ 푓∗훼) = 푓∗훽 ⌣ 훼 ∈ 퐻∗(푌, ∆푌 + ∆′
+푌, 푓(Φ푋) ∩ Φ푌)
+for any 훼 ∈ 퐻∗(푌, ∆′
+푌, Φ푌) and 훽 ∈ (푋, ∆푋, Φ푋), where ⌣ is the cup product on log Hodge cohomology
+defined along the lines of [CR11, §1.1.4, 2.4]
+Proof. This is a formal consequence of Lemma 3.1 and can be derived following the proof of [CR11,
+Prop. 1.1.16]. Again we use a factorization through the graph like
+(푋, ∆푋 + 푓∗(∆′
+푌), Φ푋 ∩ 푓−1(Φ푌))
+(푌, ∆푌 + ∆′
+푌, 푓(Φ푋) ∩ Φ푌)
+(푋 × 푋, pr∗
+1 ∆푋 + pr∗
+2 푓∗(∆′
+푌), Φ푋 × 푓−1(Φ푌))
+(푋 × 푌, pr∗
+1 ∆푋 + pr∗
+2 ∆′
+푌, Φ푋 × Φ푌)
+(푌 × 푌, pr∗
+1 ∆푌 + pr∗
+2 ∆′
+푌, 푓(Φ푋) × Φ푌)
+id푋×id푋
+푓
+id푌×id푌
+id푋×푓
+푓×id푌
+(3.36)
+Here 푓 × id푌 on the bottom is a pushing morphism (since 푓|Φ푋 is proper and 푓∗∆푌 = ∆푋) and the
+right vertical map id푌 × id푌 is a closed immersion transverse to 푓 × id푌 since the outer rectangle
+is cartesian and 푋 is smooth of the correct codimension. This means we are in a situation to apply
+Lemma 3.1, and that lemma plus the definition of cup products in terms of pullbacks along diagonals
+gives the desired identity.
+16
+
+4
+Correspondences
+Given snc pairs with familes of supports (푋, ∆푋, Φ푋) and (푌, ∆푌, Φ푌) with dimensions 푑푋 and 푑푌,
+as in [CR11, §1.3] we may define a family of supports 푃(Φ푋, Φ푌) on 푋 × 푌 by
+푃(Φ푋, Φ푌) ∶= {closed subsets 푍 ⊆ 푋 × 푌 | pr푌|푍 is proper and for all 푊 ∈ Φ푋,
+pr푌(pr−1
+푋 (푊) ∩ 푍) ∈ Φ푌}
+(the conditions of Definition 2.1 are straightforward to verify). For convenience we will let ∆푋×푌 ∶=
+pr∗
+푋∆푋 + pr∗
+푌∆푌.
+Theorem 4.1. A class 훾 ∈ 퐻푗
+푃(Φ푋,Φ푌)(푋 × 푌, Ω푖
+푋×푌(log ∆푋×푌)(−pr∗
+푋∆푋)) defines homomorphisms
+cor(훾) ∶ 퐻푞
+Φ푋(푋, Ω푝
+푋(log ∆푋)) → 퐻푞+푗−푑푋
+Φ푌
+(푌, Ω푝+푖−푑푋
+푌
+(log ∆푌))
+by the formula cor(훾)(훼) ∶= pr푌∗(pr∗
+푋(훼) ⌣ 훾). Moreover if (푍, ∆푍, Φ푍) is another snc pair with
+supports and 훿 ∈ 퐻푗′
+푃(Φ푌,Φ푍)(푌 × 푍, Ω푖′
+푌×푍(log ∆푌×푍)(−pr∗
+푌∆푌)), then
+pr푋×푍∗(pr∗
+푋×푌(훾) ⌣ pr∗
+푌×푍(훿)) ∈ 퐻푗+푗′−푑푌
+푃(Φ푋,Φ푍)(푋 × 푍, Ω푖+푖′−푑푌
+푋×푍
+(log ∆푋×푍)(−pr∗
+푋∆푋)) and
+cor(pr푋×푍∗(pr∗
+푋×푌(훾) ⌣ pr∗
+푌×푍(훿))) = cor(훿)◦ cor(훾)
+as homomorphisms 퐻푞
+Φ푋(푋, Ω푝
+푋(log ∆푋)) → 퐻푞+푗+푗′−푑푋−푑푌
+Φ푍
+(푍, Ω푝+푖+푖′−푑푋−푑푌
+푍
+(log ∆푍)).
+Remark 4.2. The sheavesΩ푖
+푋×푌(log ∆푋×푌)(−pr∗
+푋∆푋) are particular instancesof the sheavesΩ푖
+푋(퐴, 퐵)
+appearing in [DI87, §4.2].
+Such correspondences involving both log poles and “log zeroes” appear to have been considered
+before at least in crystalline cohomology, for example in work of Mieda [Mie09a; Mie09b]. However,
+I was unable to find any published proof of Theorem 4.1 in the literature.
+Proof. We make two observations: first, using Lemma 2.18 there are natural wedge product pairings
+Ω푝
+푋×푌(log ∆푋×푌) ⊗ Ω푖
+푋×푌(log ∆푋×푌)(−pr∗
+푋∆푋)
+∧�→ Ω푝+푖
+푋×푌(log ∆푌)
+Second, essentially by the definition of 푃(Φ푋, Φ푌) the Künneth morphism on cohomology for the
+tensor product Ω푝
+푋×푌(log ∆푋×푌) ⊗ Ω푖
+푋×푌(log ∆푋×푌)(−pr∗
+푋∆푋) can be enhanced with supports as
+퐻푞
+pr−1
+푋 (Φ푋)(푋 × 푌, Ω푝
+푋×푌(log ∆푋×푌)) ⊗ 퐻푗
+푃(Φ푋,Φ푌)(푋 × 푌, Ω푖
+푋×푌(log ∆푋×푌)(−pr∗
+푋∆푋))
+→ 퐻푝+푗
+Ψ
+(푋 × 푌, Ω푝
+푋×푌(log ∆푋×푌) ⊗ Ω푖
+푋×푌(log ∆푋×푌)(−pr∗
+푋∆푋))
+where Ψ ∶= pr−1
+푌∗(Φ푍) (see [CR11, §1.3.7, Prop. 1.3.10]). Combining these 2 observations gives a
+pairing
+퐻푞
+pr−1
+푋 (Φ푋)(푋 × 푌, Ω푝
+푋×푌(log ∆푋×푌)) ⊗ 퐻푗
+푃(Φ푋,Φ푌)(푋 × 푌, Ω푖
+푋×푌(log ∆푋×푌)(−pr∗
+푋∆푋))
+⌣
+��→ 퐻푝+푗
+Ψ
+(푋 × 푌, Ω푝+푖
+푋×푌(log ∆푌))
+Now note that pr푋 ∶ (푋×푌, ∆푋×푌, pr−1
+푋 (Φ푋)) → (푋, ∆푋, Φ푋) is a pulling morphism, so by Proposition 2.24
+there is an induced map pr∗
+푋 ∶ 퐻푞
+Φ푋(푋, Ω푝
+푋(log ∆푋)) → 퐻푞
+pr−1
+푋 (Φ푋)(푋 × 푌, Ω푝
+푋×푌(log ∆푋×푌)). On the
+other hand since pr푌 ∶ (푋 × 푌, ∆푌, Ψ) → (푌, ∆푌, Φ푌) is a pushing morphism, Lemma 2.28 provides
+17
+
+a morphism pr푌∗ ∶ 퐻푝+푗
+Ψ
+(푋 × 푌, Ω푝+푖
+푋×푌(log ∆푌)) → 퐻푞+푗−푑푋
+Φ푌
+(푌, Ω푝+푖−푑푋
+푌
+(log ∆푌)). Composing, we
+obtain the desired homomorphism
+퐻푞
+Φ푋(푋, Ω푝
+푋(log ∆푋))
+pr∗
+푋
+���→ 퐻푞
+pr−1
+푋 (Φ푋)(푋 × 푌, Ω푝
+푋×푌(log ∆푋×푌))
+⌣훾
+���→ 퐻푝+푗
+Ψ
+(푋 × 푌, Ω푝+푖
+푋×푌(log ∆푌))
+pr푌∗
+����→ 퐻푞+푗−푑푋
+Φ푌
+(푌, Ω푝+푖−푑푋
+푌
+(log ∆푌))
+For the “moreover” half of the lemma, we again begin with a certain wedge product pairing, this
+time on 푋 × 푌 × 푍:
+Ω푖
+푋×푌×푍(log pr∗
+푋×푌∆푋×푌)(−pr∗
+푋∆푋) ⊗ Ω푖′
+푋×푌×푍(log pr∗
+푌×푍∆푌×푍)(−pr∗
+푌∆푌)
+∧�→ Ω푖+푖′
+푋×푌×푍(log pr∗
+푋×푍∆푋×푍)(−pr∗
+푋∆푋)
+(4.3)
+If 푉 ∈ 푃(Φ푋, Φ푌), 푊 ∈ 푃(Φ푌, Φ푍) then unravelling definitions (again we refer to [CR11, §1.3.7,
+Prop. 1.3.10] for a similar claim) we find:
+• pr푋×푍|pr−1
+푋×푌(푉)∩pr−1
+푌×푍(푊) is proper and
+• pr푋×푍(pr−1
+푋×푌(푉) ∩ pr−1
+푌×푍(푊)) ∈ 푃(Φ푋, Φ푍)
+so that the Künneth morphism on cohomology associated to the left hand side of (4.3) can be en-
+hanced with supports like
+퐻푗
+pr−1
+푋×푌(푃(Φ푋,Φ푌))(푋 × 푌 × 푍, Ω푖
+푋×푌×푍(log pr∗
+푋×푌∆푋×푌)(−pr∗
+푋∆푋))
+⊗ 퐻푗′
+pr−1
+푌×푍(푃(Φ푌,Φ푍))(푋 × 푌 × 푍, Ω푖′
+푋×푌×푍(log pr∗
+푌×푍∆푌×푍)(−pr∗
+푌∆푌))
+→ 퐻푗+푗′
+Σ
+(푋 × 푌 × 푍, Ω푖
+푋×푌×푍(log pr∗
+푋×푌∆푋×푌)(−pr∗
+푋∆푋) ⊗ Ω푖′
+푋×푌×푍(log pr∗
+푌×푍∆푌×푍)(−pr∗
+푌∆푌))
+where Σ ∶= pr−1
+푋×푍∗(푃(Φ푋, Φ푍)).
+Since pr푋×푌 ∶ (푋 × 푌 × 푍, pr∗
+푋×푌∆푋×푌, pr−1
+푋×푌(푃(Φ푋, Φ푌))) → (푋 × 푌, ∆푋×푌, 푃(Φ푋, Φ푌)) is a
+pulling morphism, Proposition 2.24 gives an induced morphism
+Ω푖
+푋×푌(log ∆푋×푌) → 푅푓∗Ω푖
+푋×푌×푍(log pr∗
+푋×푌∆푋×푌);
+twisting by −∆푋×푌 and applying the projection formula gives a morphism
+Ω푖
+푋×푌(log ∆푋×푌)(−∆푋×푌) → 푅푓∗
+(Ω푖
+푋×푌×푍(log pr∗
+푋×푌∆푋×푌)(−pr∗
+푋×푌∆푋×푌))
+and then taking cohomology with supports along 푃(Φ푋, Φ푌) and using Proposition 2.9 gives a mod-
+ified pullback map
+퐻푗
+푃(Φ푋,Φ푌)(푋 × 푌, Ω푖
+푋×푌(log ∆푋×푌)(−∆푋×푌))
+→ 퐻푗
+pr−1
+푋×푌(푃(Φ푋,Φ푌))(푋 × 푌 × 푍, Ω푖
+푋×푌×푍(log pr∗
+푋×푌∆푋×푌)(−pr∗
+푋∆푋))
+(4.4)
+and a similar argument gives a modified pullback
+퐻푗′
+푃(Φ푌,Φ푍)(푌 × 푍, Ω푖′
+푌×푍(log ∆푌×푍)(−∆푌×푍))
+→ 퐻푗′
+pr−1
+푌×푍(푃(Φ푌,Φ푍))(푋 × 푌 × 푍, Ω푖′
+푋×푌×푍(log pr∗
+푌×푍∆푌×푍)(−pr∗
+푋∆푌))
+(4.5)
+On the other hand, pr푋×푍 ∶ (푋 × 푌 × 푍, pr∗
+푋×푍∆푋×푌, Σ) → (푋 × 푍, ∆푋×푍, 푃(Φ푋, Φ푍)) is a pushing
+morphism and hence by Lemma 2.28 induces morphisms
+푅pr푋×푍∗푅ΓΣ(Ωdim푋×푌×푍−푘
+푋×푌×푍
+(log pr∗
+푋×푍∆푋×푌)) → 푅Γ푃(Φ푋,Φ푍)Ωdim 푋×푍−푘
+푋×푍
+(log ∆푋×푍)[− dim푍]
+18
+
+for all 푘; twisting by −pr∗
+푋∆푋 and applying the projection formula this becomes
+푅pr푋×푍∗푅ΓΣ(Ωdim 푋×푌×푍−푘
+푋×푌×푍
+(log pr∗
+푋×푍∆푋×푌)(−pr∗
+푋∆푋))
+→ 푅Γ푃(Φ푋,Φ푍)Ωdim 푋×푍−푘
+푋×푍
+(log ∆푋×푍)(−pr∗
+푋∆푋)[− dim푍]
+(4.6)
+Now letting 푘 = dim 푋 × 푌 × 푍 − 푖 − 푖′, the induced morphisms of cohomology with supports are
+퐻푗+푗′
+Σ
+(푋 × 푌 × 푍, Ω푖+푖′
+푋×푌×푍(log pr∗
+푋×푍∆푋×푌)(−pr∗
+푋∆푋))
+→ 퐻푗+푗′−dim푍
+푃(Φ푋,Φ푍)
+(푋 × 푍, Ω푖+푖′−dim 푍
+푋×푍
+(log ∆푋×푍)(−pr∗
+푋∆푋))
+(4.7)
+Combining the above ingredients, we obtain a bilinear pairing
+퐻푗
+푃(Φ푋,Φ푌)(푋 × 푌, Ω푖
+푋×푌(log ∆푋×푌)(−∆푋×푌)) ⊗ 퐻푗′
+푃(Φ푌,Φ푍)(푌 × 푍, Ω푖′
+푌×푍(log ∆푌×푍)(−∆푌×푍))
+→ 퐻푗+푗′−dim 푍
+푃(Φ푋,Φ푍)
+(푋 × 푍, Ω푖+푖′−dim푍
+푋×푍
+(log ∆푋×푍)(−pr∗
+푋∆푋))
+sending 훾 ⊗ 훿 ↦→ pr푋×푍∗(pr∗
+푋×푌(훾) ⌣ pr∗
+푌×푍(훿)). It remains to be seen that
+cor(pr푋×푍∗(pr∗
+푋×푌(훾) ⌣ pr∗
+푌×푍(훿))) = cor(훿)◦ cor(훾)
+and for this we will make repeated use of Lemma 3.1. Consider the diagram of smooth schemes
+푋 × 푌 × 푍
+푋 × 푌
+푌 × 푍
+푋
+푌
+푍
+∗
+where all morphisms are projections. There are various ways to enhance this to include supports;
+here we add the family of supports Ψ on 푋 × 푌 defined above. Then in the cartesian diagram (∗),
+pr푌 ∶ (푋 × 푌, Ψ) → (푌, Φ푌) and pr푌×푍 ∶ (푋 × 푌 × 푍, pr−1
+푋×푌Ψ) → (푌 × 푍, pr−1
+푌 Φ푌) are pushing
+morphisms, whereas pr푋×푌 and pr푌 are pulling morphisms. At the same time, we have a pulling
+morphism pr푋×푍 ∶ (푋 × 푌 × 푍, pr−1
+푋×푍(푃(Φ푌, Φ푍))) → (푌 × 푍, 푃(Φ푌, Φ푍)). To be precise in what
+follows, whenever ambiguity is possible we will use notation like pr푋×푌
+푋
+to denote the projection
+푋 × 푌 → 푋, pr푋×푌×푍
+푋
+to denote the projection 푋 × 푌 × 푍 → 푋 and so on.
+Applying Corollary 3.35 first to pr푋×푍 we see that
+pr푌×푍∗(pr∗
+푋×푌(pr푋×푌∗
+푋
+훼 ⌣ 훾) ⌣ pr∗
+푌×푍훿) = pr푌×푍∗(pr∗
+푋×푌(pr푋×푌∗
+푋
+훼 ⌣ 훾)) ⌣ 훿
+and then applying Lemma 3.1 to (∗) shows
+pr푌×푍∗(pr∗
+푋×푌(pr푋×푌∗
+푋
+훼 ⌣ 훾)) = pr푌×푍∗
+푌
+(pr푋×푌
+푌∗ (pr푋×푌∗
+푋
+훼 ⌣ 훾)) = pr푌×푍∗
+푌
+cor(훾)(훼)
+so that
+pr푌×푍∗(pr∗
+푋×푌(pr푋×푌∗
+푋
+훼 ⌣ 훾) ⌣ pr∗
+푌×푍훿) = pr푌×푍∗
+푌
+cor(훾)(훼) ⌣ 훿
+Applying pr푌×푍
+푍∗
+we conclude that
+cor 훿(cor 훾)(훼)) = pr푋×푌×푍
+푍∗
+(pr푋×푌×푍∗
+푋
+훼 ⌣ pr∗
+푋×푌훾 ⌣ pr∗
+푌×푍훿)
+(4.8)
+Finally, we rewrite the right hand side as
+pr푋×푍
+푍∗ pr푋×푍∗(pr∗
+푋×푍pr푋×푍∗
+푋
+훼 ⌣ pr∗
+푋×푌훾 ⌣ pr∗
+푌×푍훿)
+19
+
+and apply Corollary 3.35 to pr푋×푍 (with the pushing morphism (푋 ×푌 ×푍, Σ) → (푋 ×푍, 푃(Φ푋, Φ푍))
+and pulling morphism (푋 × 푌 × 푍, pr푋×푌×푍−1
+푋
+(Φ푋)) → (푋 × 푍, pr푋×푍−1
+푋
+(Φ푋))) to arrive at
+pr푋×푍∗(pr∗
+푋×푍pr푋×푍∗
+푋
+훼 ⌣ pr∗
+푋×푌훾 ⌣ pr∗
+푌×푍훿) = pr푋×푍∗
+푋
+훼 ⌣ pr푋×푍∗(pr∗
+푋×푌훾 ⌣ pr∗
+푌×푍훿)
+Applying pr푋×푍
+푍∗
+on both sides shows that the right hand side of (4.8) is cor(pr푋×푍∗(pr∗
+푋×푌훾 ⌣
+pr∗
+푌×푍훿)(훼), as desired.
+Remark 4.9. There is a Grothendieck-Serre dual approach to such correspondences, where classes
+훾 ∈ 퐻푗
+푃(Φ푋,Φ푌)(푋 × 푌, Ω푖
+푋×푌(log ∆푋×푌)(−pr∗
+푌∆푌)) define homomorphisms
+퐻푞(푋, Ω푝
+푋(log ∆푋)(−∆푋)) → 퐻푞+푗−푑푋(푌, Ω푝+푖−푑푋
+푌
+(log ∆푌)(−∆푌)).
+The construction is formally similar.
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+url: https://doi.org/10.1017/9781316941614.
+21
+
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+of A. Grothendieck, given at Harvard 1963/64. With an appendix by P. Deligne.
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+Families.PrincetonUniversity Press, 2014. isbn: 9780691160511. url: http://www.jstor.org/stable/j.ctt5hhp7w
+(visited on 12/29/2022).
+A
+Attempts to construct a fundamental class of a thrifty bira-
+tional equivalence
+As mentioned in Section 1 inspiration for this work was the following remarkable theoremof Chatzistamatiou-
+Rülling:
+Theorem A.1 ([CR11, Thm. 3.2.8] (see also [CR15, Thm. 1.1], [Kov20, Thm. 1.6])). Let 푘 be a perfect
+field and let 푆 be a scheme. Suppose 푋 and 푌 are two separated, finite type 푘-schemes which are
+(푖) smooth over 푘 and
+(푖푖) properly birational over 푆 in the sense that there is a commutative diagram
+푍
+푋
+푌
+푆
+푟
+푠
+푓
+↺
+푔
+(A.2)
+with 푟 and 푠 proper birational morphisms.
+Let 푛 = dim 푋 = dim 푌 = dim 푍. Then, there are isomorphisms of sheaves
+푅푖푓∗풪푋
+∼�→ 푅푖푔∗풪푌 and 푅푖푓∗휔푋
+∼�→ 푅푖푔∗휔푌 for all 푖,
+(A.3)
+This result implies, for example, that if 푆 is a variety over a perfect field 푘 with a rational res-
+olution, that is, a resolution of singularities 푓 ∶ 푋 → 푆 such that 푅푓∗풪푋 = 풪푆, then every other
+resolution 푔 ∶ 푌 → 푆 satisfies 푅푔∗풪푌 = 풪푆 and is hence also rational. In characteristic 0 this was
+a corollary of Hironaka’s resolution of singularities [Hir64]; in positive characteristic it remained
+open until 2011.
+The original proof in [CR11, Thm. 3.2.8] makes use of a cycle morphism cl ∶ 퐶퐻∗(푋) → 퐻∗(푋, Ω∗
+푋)
+from Chow cohomology to Hodge cohomology, which is ultimately applied to a cycle 푍 ⊂ 푋 × 푌 ob-
+tained from a properbirational equivalence. That cycle morphismsatisfies 2 essential properties:the
+first is that it is compatible with correspondences: here Chow correspondences are homomorphisms
+퐶퐻∗(푋) → 퐶퐻∗(푌) of the form 훼 ↦→ pr푌∗(pr∗
+푋훼 ⌣ 훾) for some 훾 ∈ 퐶퐻∗(푋 × 푌)
+where ⌣ is the cup product induced by intersecting cycles; Hodge correspondences are defined in
+a similar way. The second key property is a compatibility with the filtrations
+퐶퐻푛(푋 × 푌) = 퐹0퐶퐻푛(푋 × 푌) ⊇ 퐹1퐶퐻푛(푋 × 푌) ⊇ ⋯ ⊇ 퐹dim푌퐶퐻푛(푋 × 푌) ⊇ 0
+where 퐹푐퐶퐻푛(푋×푌) is the subgroup generated by cycles 푍 ⊆ 푋×푌 such that codim(pr푌푍 ⊆ 푌) ≥ 푐,
+and
+퐻푛(푋 × 푌, Ω푚
+푋×푌) = 퐹0퐻푛(푋 × 푌, Ω푚
+푋×푌) ⊇ 퐹1퐶퐻∗(푋 × 푌) ⊇ ⋯ ⊇ 퐹dim푌퐻푛(푋 × 푌, Ω푚
+푋×푌) ⊇ 0
+22
+
+where 퐹푐퐻푛(푋 ×푌, Ω푚
+푋×푌) is the image of the map 퐻푛(푋 ×푌, ⊕푚
+푗=푐Ω푚−푗
+푋
+⊠Ω푗
+푌) → 퐻푛(푋 ×푌, Ω푚
+푋×푌)
+coming from the Künneth decomposition.
+It is natural to ask if a similar method can be applied to prove an analogue of Theorem A.1 for
+pairs, which might read something like Conjecture A.7 below. In order to state this analogue, we
+require a few additional definitions. For the remainder of this appendix we work over a fixed perfect
+field 푘.
+Definition A.4 (slightly simplified version of [Kol13, Def. 1.5]). A pair (푋, ∆푋) over 푘 will mean
+• a reduced, equidimensional and 푆2 scheme 푋 of finite type over 푘 admitting a dualizing com-
+plex , together with
+• a ℚ-Weil divisor ∆푋 = ∑
+푖 푎푖퐷푖 on 푋 such that no irreducible component 퐷푖 of ∆푋 is contained
+in Sing(푋).
+Definition A.5. A stratum of a simple normal crossing pair (푋, ∆푋 = ∑
+푖 퐷푖) is a connected (equiv-
+alently, irreducible) component of an intersection 퐷퐽 = ∩푗∈퐽퐷푗.
+Given any pair (푋, ∆푋), there is a largest open set 푈 ⊆ 푋 such that (푈, ∆푋|푈) is a simple nor-
+mal crossing pair, and we will refer to the resulting simple normal crossing pair as snc(푋, ∆푋) ∶=
+(푈, ∆푋|푈).
+Definition A.6 ( compare with [Kol13, Def. 2.79-2.80], [KX16, §1, discussion before Def. 10] ). Let
+(푆, ∆푆 = ∑
+푖 퐷푖) be a pair, and assume ∆푆 is reduced and effective. A separated, finite type birational
+morphism 푓 ∶ 푋 → 푆 is thrifty with respect to ∆푆 if and only if
+(푖) 푓 is an isomorphism over the generic point of every stratum of snc(푆, ∆푆) and
+(푖푖) letting ̃퐷푖 = 푓−1
+∗ 퐷푖 for 푖 = 1, … , 푁 be the strict transforms of the divisors 퐷푖, and setting
+∆푋 ∶= ∑
+푖 ̃퐷푖, the map 푓 is an isomorphism at the generic point of every stratum of snc(푋, ∆푋).
+Conjecture A.7. Let 푘 be a perfect field, let 푆 be a scheme and let (푋, ∆푋) and (푌, ∆푌) be simple
+normal crossing pairs over 푘. Suppose (푋, ∆푋) and (푌, ∆푌) are properly birational over 푆 in the sense
+that there is a commutative diagram
+(푍, ∆푍)
+(푋, ∆푋)
+(푌, ∆푌)
+푆
+푟
+푠
+푓
+↺
+푔
+(A.8)
+where 푟, 푠 are proper and birational morphisms, and assume ∆푍 = 푟−1
+∗ ∆푋 = 푠−1
+∗ ∆푌. If 푟 and 푠 are
+thrifty, then there are quasi-isomorphisms
+푅푓∗풪푋(−∆푋) ≃ 푅푔∗풪푌(−∆푌) and 푅푓∗휔푋(∆푋) ≃ 푅푔∗휔푌(∆푌).
+(A.9)
+Following [CR11] closely, one might begin by replacing the ordinary sheaves of differentials Ω푋
+appearing in Hodge cohomology with sheaves of differentials with log poles Ω푋(log ∆푋) and attempt
+to implement a similar strategy, i.e. starting a cycle 푍 ⊂ 푋×푌 representinga thrifty proper birational
+equivalince, producing a correspondence in logarithmic Hodge cohomology and analyzing its prop-
+erties.
+Ultimately even the correspondences of Section 4 seem to be insufficient to deal with thrifty
+proper birational equivalences, as we illustrate in Appendix A.1 below. The problem we encounter
+is elementary: looking at the recipe for the Hodge class cl(푍) of a subvariety 푍 ⊆ 푋, where 푍 and 푋
+are smooth an projective (outlined in [Har77, Ex. III.7.4]), we see that cl(푍) ultimately comes from
+the trace linear functional tr ∶ 퐻dim 푍(푍, 휔푍) → 푘, or Serre-dually the element 1 ∈ 퐻0(푍, 풪푍). Due
+to the introduction of log poles and zeroes in Section 4, trying to follow that recipe we pass through
+23
+
+cohomology groups of the form 퐻dim 푍(푍, 휔푍(퐷)), or dually 퐻0(푍, 풪푍(−퐷)) where 퐷 is an (often
+non-0 in cases of interest) effective Cartier divisor on 푍, and so there simply is no “1” to be had.
+Beyond the difficulties described in the previous paragraph, when attempting to formulate a
+logarithmic variant of Chatzistamatiou-Rülling’s cycle morphism argument one is hampered by the
+fact that we are still in the early days of logarithmic Chow theory . It is not clear to the author which
+logarithmic variant of Fulton’s 퐶퐻∗, if any, could be used to construct a logarithmic cycle morphism
+with all of the desired properties. Further investigation of this question could be an interesting topic
+of future research.
+Despite the aforementioned challenges, it is possible to prove a result almost identical to Conjecture A.7
+by entirely different methods [God22].8
+A.1
+Obstructions to obtaining log Hodge correspondences from thrifty bi-
+rational equivalences
+Let (푋, ∆푋), (푌, ∆푌) be simple normal crossing pairs, and assume in additionthat 푋, 푌 are connected
+and proper. Let 푍 ⊆ 푋 × 푌 be a smooth closed subvariety with codimension 푐. In this situation the
+fundamental class of cl(푍) ∈ 퐻푐(푋 × 푌, Ω푐
+푋×푌) (no log poles yet) can be described using only Serre
+duality, as follows (we refer to [Har77, Ex. III.7.4]). the composition
+퐻dim푍(푋 × 푌, Ωdim푍
+푋×푌 ) → 퐻dim 푍(푍, Ωdim 푍
+푍
+)
+tr�→ 푘
+(A.10)
+(where tr is the trace map of Serre duality) is an element of
+퐻dim푍(푋 × 푌, Ωdim푍
+푋×푌 )∨ ≃ 퐻푐(푋 × 푌, Ω푐
+푋×푌)
+(A.11)
+which we may define to be cl(푍).9 In light of Theorem 4.1 we might hope to modify eqs. (A.10)
+and (A.11) to obtain a class in 퐻푐(푋 ×푌, Ω푐
+푋×푌(log ∆푋×푌)(−pr∗
+푋∆푋)). Let us focus on the case where
+• pr푋|푍 ∶ 푍 → 푋, pr푌|푍 ∶ 푍 → 푌 are both thrifty and birational, so in particular 푐 = dim 푋 =
+dim 푌 =∶ 푑 and
+• (pr푋|푍)−1
+∗ ∆푋 = (pr푌|푍)−1
+∗ ∆푌 =∶ ∆푍
+To keep the notation under control, set 휋푋 ∶= pr푋|푍 and 휋푌 ∶= pr푌|푍.
+In this situation letting 휄 ∶ 푍 → 푋 × 푌 be the inclusion there is a natural map
+푑휄∨ ∶ Ω푑
+푋×푌(log ∆푋×푌) → 휄∗Ω푑
+푍(log ∆푋×푌|푍) and twisting by −pr∗
+푌∆푌 gives a map
+Ω푑
+푋×푌(log ∆푋×푌)(−pr∗
+푌∆푌) → 휄∗Ω푑
+푍(log ∆푋×푌|푍)(−pr∗
+푌∆푌|푍) = 휄∗Ω푑
+푍(log ∆푋×푌|푍)(−휋∗
+푌∆푌)
+To identify Ω푑
+푍(log ∆푋×푌|푍)(−pr∗
+푋∆푋|푍), write
+(휋푋)∗∆푋 = (휋푋)−1
+∗ ∆푋 + 퐸푋 = ∆푍 + 퐸푋 and
+(휋푌)∗∆푌 = (휋푌)−1
+∗ ∆푌 + 퐸푌 = ∆푍 + 퐸푌
+so that ∆푋×푌|푍 = (휋푋)∗∆푋 + (휋푌)∗∆푌 = 2∆푍 + 퐸푋 + 퐸푌. While the hypotheses guarantee ∆푍 is
+reduced it may be that 퐸푋, 퐸푌 are non-reduced — however something can be said about their multi-
+plicities. If 퐸푋 = ∑
+푖 푎푖
+푋퐸푖
+푋, 퐸푌 = ∑
+푖 푎푖
+푌퐸푖
+푌 where the 퐸푖
+푋, 퐸푖
+푌 are irreducible, then by a generalization
+of [Har77, Prop. 3.6] (see also [Kol13, §2.10]),
+푎푖
+푋 = mlt(휋푋(퐸푖
+푋) ⊆ ∆푋)
+and since ∆푋 is a reduced effective simple normal crossing divisor, if in addition we write ∆푋 =
+∑
+푖 퐷푖
+푋, then mlt(휋푋(퐸푖
+푋) ⊆ ∆푋) = |{푖 | 휋푋(퐸푖
+푋) ⊆ 퐷푖
+푋}|. The thriftiness hypothesis that 휋푋(퐸푖
+푋) is not
+8The reason the result is only “almost identical” is that in [God22] we require ostensibly stronger hypotheses on the
+base scheme 푆 (namely that it is excellent and noetherian), but it is possible that even in the situation of Theorem A.1
+and Conjecture A.7 one can reduce to this case, for example using noetherian approximation.
+9It may then be non-trivial to verify this agrees with other definitions, especially if we worry about signs, but we will not
+need that level of detail for what follows.
+24
+
+a stratum then implies 푎푖
+푋 = mlt(휋푋(퐸푖
+푋) ⊆ ∆푋) < codim(휋푋(퐸푖
+푋) ⊂ 푋). Since differentials with log
+poles are insensitive to multiplicities, we have
+Ω푑
+푍(log ∆푋×푌|푍) = 휔푍(∆푍 + 퐸red
+푋
++ 퐸red
+푌 )
+where −red denotes the associated reduced effective divisor. Then
+Ω푑
+푍(log ∆푋×푌|푍)(−휋∗
+푌∆푌) = 휔푍(∆푍 + 퐸red
+푋
++ 퐸red
+푌
+− ∆푍 − 퐸푌)
+휔푍(퐸red
+푋
++ (퐸red
+푌
+− 퐸푌)) = 휔푍(
+∑
+푖
+퐸푖
+푋 +
+∑
+푖
+(1 − 푎푖
+푌)퐸푖
+푌)
+The upshot is that we have an induced map
+퐻푑(푋 × 푌, Ω푑
+푋×푌(log ∆푋×푌)(−pr∗
+푌∆푌)) → 퐻푑(푍, 휔푍(퐸red
+푋
++ (퐸red
+푌
+− 퐸푌)))
+(A.12)
+Here the left hand side is Serre dual to 퐻푑(푋 × 푌, Ω푑
+푋×푌(log ∆푋×푌)(−pr∗
+푋∆푋)), so the 푘-linear dual
+of (A.12) is a morphism
+퐻푑(푍, 휔푍(퐸red
+푋
++ (퐸red
+푌
+− 퐸푌)))∨ → 퐻푑(푋 × 푌, Ω푑
+푋×푌(log ∆푋×푌)(−pr∗
+푋∆푋))
+Unfortunately10 퐻푑(푍, 휔푍(퐸red
+푋 + (퐸red
+푌 − 퐸푌))) is often 0. If 퐸푋 and 퐸푌 are both reduced (an explicit
+example where this holds will be given below), then 퐻푑(푍, 휔푍(퐸red
+푋 +(퐸red
+푌 −퐸푌))) = 퐻푑(푍, 휔푍(퐸푋)).
+If in addition 퐸푋 ≠ 0, we obtain 퐻푑(푍, 휔푍(퐸푋)) = 0 by an extremely weak (but characteristic inde-
+pendent) sort of Kodaira vanishing:
+Lemma A.13. Let 푍 be a proper variety over a field 푘 with dimension 푑, and assume 푍 is normal and
+Cohen-Macaulay. If 퐷 ⊂ 푍 is a non-0 effective Cartier divisor on 푍 then 퐻푑(푍, 휔푍(퐷)) = 0.
+Proof. By Serre duality 퐻푑(푍, 휔푍(퐷)) = 퐻0(푍, 풪푍(−퐷)), which vanishes by the classic fact that “a
+nontrivial line bundle and its inverse can’t both have non-0 global sections.” Since I am not aware
+of a specific reference, here is a proof:
+Suppose towards contraditction that there is a non-0 global section 휎 ∈ 퐻0(푍, 풪푍(−퐷)) — then
+the composition
+풪푍
+풪푍(−퐷)
+풪푍
+휎
+휏
+is non-0. By [Stacks, Tag 0358] 퐻0(푍, 풪푍) is a (normal) domain, and since it’s also a finite dimen-
+sional 푘-vector space it must be an extension field of 푘. But then 휏 ∈ 퐻0(푍, 풪푍) is invertible hence
+surjective, so 풪푍(−퐷) ��→ 풪푍 is surjective, which is a contradiction since by hypothesis the cokernel
+풪퐷 ≠ 0.
+Example A.14. Let 푋 = ℙ2 and let ∆푋 ⊂ 푋 be a line. Let 푝 ∈ 퐿 be a 푘-point, let 푌 = Bl푝 푋 and
+let ∆푌 = ̃퐿 = the strict transform of 퐿. Finally let 푓 ∶ 푌 → 푋 be the blowup map and let 푍 =
+(푓 ×id)(푌) ⊂ 푋 ×푌. In this case (with all notation as above) 휋푋◦(푓 ×id) = 푓 and 휋푌◦(푓 ×id) = id푌,
+so under the isomorphism 푓 × id ∶ 푌 ≃ 푍, 퐸푋 is the exceptional divisor of 푓 (with multiplicity 1).
+On the other hand 퐸푌 = 0. In particular 퐸푋 and 퐸푌 are reduced and 퐸푋 ≠ 0 so from the above
+discussion 퐻2(푍, 휔푍(퐸푋)) = 0.
+10at least for the purposes of constructing log Hodge cohomology classes of subvarieties ...
+25
+

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@@ -0,0 +1,1007 @@
+arXiv:2301.12146v1  [math.NT]  28 Jan 2023
+On Tribonacci Sequences
+Luke Pebody
+Saturday 28, January 2023
+Abstract
+Let a tribonacci sequence be a sequence of integers satisfying ak =
+ak−1 + ak−2 + ak−3 for all k ≥ 4.
+For any positive integers k and n,
+denote by fk(n) the number of tribonacci sequences with a1, a2, a3 > 0
+and with ak = n.
+For all n, there is a maximum k such that fk(n) is non-zero. Answering
+a question of Spiro [1], we show that there is a finite upper bound (we
+specifically prove 561001) on fk(n) for any positive integer n ≥ 3 and this
+maximum k.
+We do this by showing that fk(n) has transitions in n around constant
+multiples of φ3k/2 (where φ is the real root of φ3 = φ2 + φ + 1): there
+exists a constant C such that fk(n) > 0 whenever n > Cφ3k/2 and for any
+constant T , the values of fk(n) with n < T φ3k/2 have an upper bound
+independent of k.
+1
+Introduction
+A tribonacci sequence of length k is a sequence of integers ⟨ai⟩k
+i=1 such that
+ai = ai−1 + ai−2 + ai−3 for all 4 ≤ i ≤ k.
+We say that such a sequence
+terminates at ak and that it is positive if a1, a2, a3 > 0 - note that this easily
+implies that ai > 0 for all i. Denote by fk(n) the number of tribonacci sequences
+of length k terminating at n.
+Clearly f1(n) = 1 for all n > 0, the only tribonacci sequence of length 1
+terminating at n being ⟨n⟩. Further, f2(n) = f3(n) = ∞ as we can choose any
+values for the proceeding terms.
+For n ≥ 3, there exists a tribonacci sequence of length longer than 3 termi-
+nating at n, for example ⟨n − 2, 1, 1, n⟩. However for any tribonacci sequence
+⟨ai⟩k
+i=1 of length k, and for any 4 ≤ i ≤ k, ai = ai−1 + ai−2 + ai−3 ≥ ai−1 + 2,
+so by induction ai ≥ 2i− 5 for all 3 ≤ i ≤ k, and hence if n < 2k − 5, fk(n) = 0.
+Let t(n) be the largest number such that ft(n)(n) > 0.
+Let p(n) denote the number of positive tribonacci sequences of length t(n)
+terminating at n, so p(n) = ft(n)(n).
+Clearly, since t(1) = t(2) = 3 it follows that p(1) = p(2) = ∞. Spiro [1] asks
+Question 1. Does there exist some absolute constant c such that for all n ≥ 3,
+p(n) ≤ c for all n?
+1
+
+The purpose of this paper is to give a positive answer to this question. Indeed
+we will show
+Theorem 2. For any integer n ≥ 3, there are at most 561001 positive tribonacci
+sequences of length t(n) terminating at n.
+It turns out the key question for our proof is the minimum size of the vector
+
+
+a1
+a2
+a3
+
+ where ⟨ai⟩n
+i=1 is a non-zero tribonacci sequence terminating at an = 0.
+In Section 2 we will show a lower bound on such a sequence of the order of φn/2,
+which will allow us to prove
+Theorem 3. For any positive integers n, k with k ≥ 4, the number of positive
+sequences of length k terminating at n is at most
+⌈1500
+n
+φ3k/2 ⌉2.
+In Section 3 we turn to trying to put an upper bound on numbers that don’t
+have any positive tribonacci sequences of length k terminating at them. This
+is an instance of the Coin Problem, also known as calculating the Frobenius
+Number. We construct two specific tribonacci sequences terminating at an = 0
+with
+
+
+a1
+a2
+a3
+
+ being of the order of φn/2 and with the integers a1, a2, a3 having
+specified signs, allowing us to prove
+Theorem 4. For any integer n above 0.2φ3k/2, there exists a positive tribonacci
+sequence of length k terminating at n.
+This will be all that is required.
+Proof of Theorem 2. There is no sequence of length t(n) + 1 terminating at n.
+Hence by Theorem 4, it follows that n < 0.2φ3(t(n)+1)/2 = 0.2φ3/2φ3t(n)/2.
+Thus from Theorem 3, it follows that there are at most
+⌈1500
+n
+φ3t(n)/2 ⌉2 ≤ ⌈15000.2φ3/2φ3t(n)/2
+φ3t(n)/2
+⌉2
+≤ ⌈300φ3/2⌉2 = 7492 = 561001
+positive tribonacci sequences of length t(n) terminating at n.
+In Section 4, we will investigate which recurrence relations of the form xn =
+axn−1+bxn−2+cxn−3 for non-negative a and b and for positive c the arguments
+in this paper can be carried across to. We will extend the result in the earlier
+sections to the following case.
+2
+
+Theorem 5. Suppose a, b, c are non-negative integers with a + b > 0, c = 1
+and such that x3 − ax2 − bx − c = 0 has exactly one real root.
+Then there
+is an absolute bound T such that if positive integers k ≥ 4 and n are such
+that there are no positive sequences ⟨ai⟩k+1
+i=1 satisfying the recurrence relation
+ai = aai−1 + bai−2 + cai−3 of length k + 1 terminating at n, then there are at
+most T such sequences of length k terminating at n.
+We will leave open the question of which linear recurrences satisfy this prop-
+erty, but will at least demonstrate an example of a recurrence that does not.
+In particular we will show the existence of positive integers k and n such
+that there is no positive sequence ⟨ai⟩k+1
+i=1 satisfying the recurrence relation
+ai = ai−1 + ai−2 + 2ai−3 of length k + 1 terminating at n, but for which the
+number of such sequences of length k terminating at n is unbounded.
+2
+Lower Bound
+Let us say a sequence ⟨ai⟩∞
+i=1 is a reverse-tribonacci sequence if for all i ≥ 0,
+ai = ai+1+ai+2+ai+3. Let us write out the expression for the reverse-tribonacci
+sequence starting ⟨0, k, l⟩. Recall that φ is the real solution to φ3 = φ2 + φ + 1.
+We write the complex roots as φ1 and φ2 = φ1.
+Lemma 6. For all integers k, l, if ⟨ai⟩∞
+i=1 is a reverse-tribonacci sequence with
+a1 = 0, a2 = k and a3 = l, then for all i, ai can be expressed as
+ai = αφ−i + (kψ1 + lζ1)φ−i
+1 + (kψ2 + lζ2)φ−i
+2
+= (αφ−3i/2 + β cos(γ − δi))φi/2,
+where
+ψ1 =
+φ3
+1 + φ2
+1
+φ2
+1 + 2φ1 + 3
+ψ2 = ψ1
+ζ1 =
+φ3
+1
+φ2
+1 + 2φ1 + 3
+ζ2 = ζ1
+α = kφ2 + (k + l)φ3
+φ2 + 2φ + 3
+βeγi = 2(kψ1 + lζ1) and
+eδi = φ1
+�
+φ.
+Proof. Any two-way infinite tribonacci sequence ⟨ai⟩∞
+−∞ can be written as ai =
+pφi + qφi
+1 + rφi
+2 for some p, q and r.
+Thus any reverse-tribonacci sequence ⟨ai⟩∞
+−∞ can be written as ai = pφ−i +
+qφ−i
+1 + rφ−i
+2
+for some p, q, r. Solving for the p, q, r that give a1 = 0, a2 = k and
+a3 = l leads to the above expression.
+3
+
+Note that in the above expressions, ψ1, ψ2, ζ1, ζ2 and δ are constants that
+do not depend on k and l.
+Lemma 7. For any integers k and l, if α and β are defined as in Lemma 6,
+then |α| ≤ |k| + |l| and β ≥ |k|+|l|
+31
+.
+Proof. α is roughly 0.9546k+0.6184l, which is clearly bounded above by |k|+|l|.
+ψ1 is roughly 0.02267 − 0.217i and ζ1 is roughly 0.1908 − 0.0187i. As such, if
+k and l are non-negative then the real part of 2(kψ1 + lζ1) (and hence β) is at
+least 0.04(k + l) > |k|+|l|
+31
+.
+For k positive and l negative, the minimum value of
+β
+k−l is approximately
+0.03221 >
+1
+31, and is achieved around k = −0.3653l.
+Finally we need a simple trigonometric property
+Lemma 8. For any real numbers p and q with π
+2 < q < π, the larger of | cos(p)|
+and | cos(p + q)| is at least cos(q/2).
+Proof. Note that cos(q) < 0. Thus
+2(cos(p)2 + cos(p + q)2) = 2 cos(p)2 + 2 cos(p + q)2
+= cos(2p) + cos(2p + 2q) + 2
+= 2 cos(2p + q) cos(q) + 2
+≥ 2 + 2 cos(q)
+= 4 cos(q/2)2.
+Thus either cos(p)2 ≥ cos(q/2)2 or cos(p + q)2 ≥ cos(q/2)2.
+This allows us to put a lower bound on the size of at least one of each
+consecutive pair of a reverse-tribonacci sequence.
+Corollary 9. Given a non-zero integer reverse-tribonacci sequence ⟨ai⟩∞
+i=1 with
+a1 = 0, for every integer n ≥ 2, either |an| > 0.01φn/2 or |an+1| > 0.01φ(n+1)/2
+(or both).
+Proof. For n ≥ 2 if an and an+1 are both 0, then a1 is the same sign as an−1.
+Since a1 = 0, it follows that the entire series must be 0. Since the sequence is
+non-zero, it follows that either |an| ≥ 1 or |an+1 ≥ 1. Since 1 > 0.01φn/2 for
+n ≤ 15, we have proved the statement for n ≤ 14. Thus we may assume n ≥ 15.
+By Lemma 6,
+ai
+φi/2 can be written as αφ−3i/2 + β cos(γ − δi).
+Now by Lemma 8, at least one of | cos(γ − δn)| and | cos(γ − δ(n − 1))| is
+at least cos(δ/2) (δ = 2.176 is between π
+2 and π). Let t be the choice from
+{n − 1, n} that maximises | cos(γ − δt)|.
+By Lemma 7, if we write α′ =
+α
+|k|+|l| and β′ =
+β
+|k|+|l| then |α′| ≤ 1 and
+β′ >
+1
+31.
+4
+
+Therefore
+| at
+φt/2 | = |αφ−3t/2 + β cos(γ − δt)|
+≥ |α′φ−3t/2 + β′ cos(γ − δt)|
+≥ |β′ cos(γ − δt)| − |α′φ−3t/2|
+≥ 1
+31 cos(δ/2) − φ−3t/2 ≥ cos(δ/2)
+31
+− φ−22.5 > 0.01.
+Then we have a bound on the size of tribonacci sequences terminating at 0.
+Corollary 10. For n ≥ 3, if ⟨ai⟩n
+i=1 is a non-zero integer tribonacci sequence
+terminating at 0 then either |a1| > 0.01φn/2 or |a2| > 0.01φ(n−1)/2 (or both).
+Proof. Let k = an−1 and l = an−2. Then if ⟨bi⟩∞
+i=1 is the reverse-tribonacci
+sequence with b1 = 0, b2 = k and b3 = l, then ai = bn+1−i for all 1 ≤ i ≤ n.
+Then this is just a restatement of Corollary 9.
+This is all we need to prove Theorem 3.
+Proof of Theorem 3. Partition the tribonacci sequences of length k ≥ 4 termi-
+nating at n ⟨ai⟩k
+i=1 by the pair (⌊
+a1
+0.01φk/2 ⌋, ⌊
+a2
+0.01φk−1/2 ⌋).
+If two sequences ⟨ai⟩k
+i=1 and ⟨bi⟩k
+i=1 have the same pair, then |a1 − b1| <
+0.01φk/2 and |a2 − b2| < 0.01φ(k−1)/2 and hence, by Corollary 10, either ⟨ai −
+bi⟩k
+i=1 is zero everywhere or does not terminate at 0.
+Thus each distinct tribonacci sequence of length k terminating at n has a
+distinct pair.
+Define tribonacci sequence by x1 = 1, x2 = 0, x3 = 0. Then if a1, a2, . . . , ak
+is a positive tribonacci sequence, ai ≥ xia4 for i = 2, 3 and 4 and therefore
+ak ≥ xka4. Now xk < φk/11 for all k ≥ 4 and hence a1 + a2 + a3 ≤ 11n
+φk for all
+tribonacci sequences of length k terminating at n.
+Thus ⌊
+a1
+0.01φk/2 ⌋ is at most 1100n
+φ3k/2 and ⌊
+a2
+0.01φ(k−1)/2 ⌋ is at most
+1100n
+φ3k−1/2 < 1500n
+φ3k/2 .
+It follows that the number of Tribonacci sequences of length k ≥ 4 termi-
+nating at n is at most ��� 1500n
+φ3k/2 ⌉2.
+Note we have not worked hard here to get the best bound. In a previous
+draft we had a much more complicated proof of an upper bound which showed,
+in place of Corollary 10, that if ⟨ai⟩n
+i=1 terminated at 0 then
+�
+a2
+1 + a2
+2 + a2
+3 >
+0.28φn/2, which led to an upper bound for the main theorem of 42875.
+3
+Upper Bound
+In this section, we turn to numbers which are not the terminus for any tribonacci
+sequence of length k, working towards a proof of Theorem 4.
+5
+
+To that end, define three infinite tribonacci sequences ⟨pi⟩∞
+i=1, ⟨qi⟩∞
+i=1 and
+⟨ri⟩∞
+i=1 by (p1, p2, p3) = (1, 0, 0), (q1, q2, q3) = (0, 1, 0) and (r1, r2, r3) = (0, 0, 1).
+It is clear that for any tribonacci sequence ⟨ai⟩n
+i=1, an = a1pn + b1qn + c1rn.
+Thus we are simply looking to get an upper bound on the largest number which
+cannot be written as a positive integral linear combination of pn, qn and rn.
+This is called the Frobenius Number of pn, qn and rn.
+First let us see that a finite bound does exist.
+Lemma 11. For all k ≥ 1, pk, qk and rk have no non-trivial common divisor.
+Proof. If pk, qk and rk had a non-trivial common divisor t > 1 then t would be
+a common divisor of the terminus of every tribonacci sequence of length k, from
+which it would follow that t would in fact be a common divisor of pk+l for all
+l ≥ 0 (since ⟨pi+l⟩k
+1 is a tribonacci sequence of length k).
+Then, since pi = pi+3 − (pi+1 + pi+2), it would follow that t would be a
+common divisor of pk−1, pk−2 and all the way back to p0 = 1 by induction,
+causing a contradiction.
+We will use the following bound, which might be originally due to Killing-
+bergtro.
+Theorem 12. Suppose p, q and r are integers with no non-trivial common di-
+visor and let us suppose ap = bq + cr and dq = ep + fr where a, c, d, f > 0 and
+b, e ≥ 0. Then for every integer N ≥ ap + dq + r, N can be written in the form
+xp + yq + zr for some positive integers p, q, r.
+Proof. Let x, y, z be positive integers such that px + qy + rz is equivalent to
+N (mod r), but for which px + qy + rz is minimal (such a triple x, y, z exist
+because, as is well known, if p, q and r have no non-trivial common divisor then
+all sufficiently large integers can be written in the form px + qy + rz, and many
+of these sufficiently large integers are equivalent to N (mod r).)
+Since px+qy+rz is minimal, px+qy+rz−r cannot be written as a positive
+linear combination of x, y and z.
+Thus in each of the equations
+px + qy + rz − r = px
++ qy
++ r(z − 1)
+px + qy + rz − r = p(x − a)
++ q(y + b)
++ r(z + c − 1)
+px + qy + rz − r = p(x + e)
++ q(y − d)
++ r(z + f − 1),
+it must follow that one of the coefficients must not be positive. Two of the
+coefficients in each equation are clearly positive, so it follows that x ≤ a, y ≤ d
+and z ≤ 1, so px + qy + rz ≤ pa + qd + r ≤ N. Since N and px + qy + rz
+are equivalent modulo r, there exists a non-negative integer t such that N =
+px + qy + rz + rt. Then N = px + qy + r(z + t).
+Therefore, to show that all sufficiently large integers can be written as the
+terminus of a tribonacci sequence of length k, we just need to find linear com-
+binations of pn, qn and rn combining to 0, with particular signs of the combi-
+nations. This is equivalent to finding tribonacci sequences ending at 0, which
+6
+
+Table 1: Table for Lemma 13
+t0
+t1
+k
+l
+α
+β
+γ
+x0
+x1
+x2
+0
+0.06
+0
+1
+0.6184
+0.3834
+-0.0977
+0.3410
+-0.0500
+-0.1694
+0.06
+0.16
+-1
+2
+0.2822
+0.8027
+0.4640
+0.6879
+-0.0515
+-0.2163
+0.16
+0.22
+-1
+1
+-0.3362
+0.5200
+0.8677
+0.4526
+-0.0471
+-0.2482
+0.22
+0.35
+-1
+0
+-0.9546
+0.4364
+1.6749
+0.3778
+-0.0354
+-0.0294
+0.35
+0.45
+-1
+-1
+-1.5731
+0.6360
+2.3067
+0.5517
+-0.0538
+-0.1588
+0.45
+0.56
+0
+-1
+-0.6184
+0.3834
+3.0439
+0.3410
+-0.0500
+-0.0548
+0.56
+0.66
+1
+-2
+-0.2822
+0.8027
+-2.6776
+0.6879
+-0.0515
+-0.2163
+0.66
+0.72
+1
+-1
+0.3362
+0.5200
+-2.2739
+0.4526
+-0.0471
+-0.2482
+0.72
+0.85
+1
+0
+0.9546
+0.4364
+-1.4667
+0.3778
+-0.0354
+-0.0294
+0.85
+0.95
+1
+1
+1.5731
+0.6360
+-0.8349
+0.5517
+-0.0538
+-0.1588
+0.95
+1
+0
+1
+0.6184
+0.3834
+-0.0977
+0.3745
+-0.1864
+-0.0548
+is equivalent to finding reverse-tribonacci sequences starting at 0, and hence we
+can again use the expression from Lemma 6, which states that if ⟨ai⟩∞
+i=1 is a
+reverse-tribonacci sequence with a1 = 0, a2 = k and a3 = l then for all n
+an = (αφ−3n/2 + β cos(γ − δn))φn/2.
+Note that for all but an extremely small collection of n, the term β cos(γ+δn)
+dwarves αφ−3n/2. As such, for a fixed k and l, the sign of an depends only
+(except for a few very rare counterexamples) on the fractional part of
+δ
+2πn.
+Lemma 13. For each integer n ≥ 4, there exists a tribonacci sequence ⟨ai⟩n
+i=1
+terminating at an = 0, with a1 > 0, 0 ≥ a2, 0 > a3 and with a1 < 0.81φn/2.
+Similarly for all such n, there exists a tribonacci sequence ⟨bi⟩n
+i=1 terminating
+at bn = 0, with b2 > 0, 0 ≥ b1, 0 > b3 and with b2 < 0.64φn/2.
+Proof. We will split into cases based on the fractional part of δn
+2π = 0.3464n. See
+Table 1. For each row, if t0 ≤ δn
+2π − ⌊ δn
+2π⌋ ≤ t1, then for the given values of k and
+l, if β and γ are as defined in Lemma 6, one can confirm that β cos(γ − δn) ≥
+x0 > 0.34, while β cos(γ − δ(n − 1)) ≤ x1 < −0.035 and β cos(γ − δ(n − 2)) ≤
+x2 < −0.029.
+Furthermore, for all such k, l, |α| < 1.58, so if n ≥ 7, |αφ−3(n−2)/2| ≤ 0.017,
+from which it follows that an > 0 > an−1, an−2. Further,
+an
+φn/2 < β + 0.017 <
+0.81.
+For 4 ≤ n < 7, we can verify the sequences (1, 0, −1, 0), (2, 0, −1, 1, 0) and
+(2, 0, −1, 1, 0, 0) satisfy the conditions for (a1, a2, . . . , an).
+For the sequence (b1, b2, . . . , bn), see Table 2. Here, for each row, if t0 ≤
+δn
+2π − ⌊ δn
+2π⌋ ≤ t1, then for the given values of k and l, if β and γ are as defined in
+Lemma 6, one can confirm that β cos(γ − δn) ≤ x0 < −0.071, while β cos(γ −
+δ(n − 1)) ≥ x1 > 0.33 and β cos(γ − δ(n − 2)) ≤ x2 < −0.041.
+Furthermore, for all such k, l, |α| < 1.58, so if n ≥ 7, |αφ−3(n−2)/2| ≤ 0.017,
+from which it follows that an−1 > 0 > an, an−2.
+For 4 ≤ n < 7, we can verify the sequences (0, 1, −1, 0), (0, 1, −1, 0, 0) and
+(−1, 2, −1, 0, 1, 0) satisfy the conditions for (b1, b2, . . . , bn).
+This then completes our proof.
+7
+
+Table 2: Other table for Lemma 13
+t0
+t1
+k
+l
+α
+β
+γ
+x0
+x1
+x2
+0
+0.06
+1
+-1
+0.3362
+0.5200
+2.2739
+-0.3362
+0.4625
+-0.0678
+0.06
+0.19
+1
+0
+0.9546
+0.4364
+1.4667
+-0.1176
+0.3862
+-0.0528
+0.19
+0.29
+1
+1
+1.5731
+0.6360
+0.8349
+-0.2812
+0.5639
+-0.0791
+0.29
+0.41
+0
+1
+0.6184
+0.3834
+0.0977
+-0.1311
+0.3369
+-0.0413
+0.41
+0.56
+-1
+1
+-0.3362
+0.5200
+-0.8677
+-0.0714
+0.4625
+-0.0678
+0.56
+0.69
+-1
+0
+-0.9546
+0.4364
+-1.6749
+-0.1176
+0.3862
+-0.0528
+0.69
+0.79
+-1
+-1
+-1.5731
+0.6360
+-2.3067
+-0.2812
+0.5639
+-0.0791
+0.79
+0.91
+0
+-1
+-0.6184
+0.3834
+-3.0439
+-0.1311
+0.3369
+-0.0413
+0.91
+1
+1
+-1
+0.3362
+0.5200
+2.2739
+-0.0714
+0.4641
+-0.2528
+Proof of Theorem 4. Lemma 13 gives us tribonacci sequences ⟨ai⟩n
+i=1 and ⟨bi⟩n
+i=1
+terminating at an = bn = 0. It follows that a1pn + a2qn + a3rn = 0 = b1qn +
+b2qn + b3rn.
+Since a1, b2 > 0 > a3, b3 and 0 ≥ a2, b1, it follows that we can write
+a1pn = (−a2)qn + (−a3)rn and
+b2qn = (−b1)p1 + (−b3)rn
+satisfying the sign requirements of Theorem 12, so it follows that every integer
+N ≥ a1pn + b2qn + rn can be written in the form xpn + yqn + zrn for some
+positive integers x, y and z, and hence there exists a positive tribonacci sequence
+of length k ending at N.
+By the bounds on a1 and b2 given in Lemma 13, we have such a tribonacci
+sequence for all N ≥ 0.81φk/2uk+0.64φk/2vk+wk. Since uk ≤ vk ≤ wk < 0.11φk
+and 0.81φk/2+0.64φk/2+1 < 1.74φk/2, it follows that such a tribonacci sequence
+exists for all N ≥ 0.2φ3k/2 as was required.
+4
+Other cubic recurrences
+For non-negative a, b, c we can ask a similar question for recurrences of the form
+xn = axn−1 + bxn−2 + cxn−3. Formally, let us define ka,b,c(n) to be the largest
+k such there is a positive k-element solution ⟨xi⟩k
+i=1 to the recurrence relation
+xi = axi−1 + bxi−2 + cxi−3, and define ta,b,c(n) to be the number of positive
+ka,b,c(n)-element solutions that exist.
+If c = 0, this is a quadratic recurrence, and the problem is already solved. If
+a = 0, b = 0 and c = 1, the recurrence is xn = xn−3, and ka,b,c(n) is not defined
+for any n.
+For all a, b, c ≥ 0 with c ≥ 1 and a + b + c ≥ 2, say that the recurrence
+xn = axn−1 + bxn−2 + cxn−3 is congenial if there exists a finite bound B such
+that for all n, ta,b,c(n) = ∞ or ta,b,c(n) ≤ B.
+Firstly let us note that not all recurrences are congenial.
+Lemma 14. The recurrence xn = xn−1 + xn−2 + 2xn−3 is not congenial.
+Proof. Let ⟨pn⟩∞
+n=1, ⟨qn⟩∞
+n=1 and ⟨rn⟩∞
+n=1 be the solutions to the recurrence
+8
+
+starting with ⟨1, 0, 0⟩, ⟨0, 1, 0⟩ and ⟨0, 0, 1⟩ respectively.
+Then xn = x1pn +
+x2qn + x3rn.
+Solutions to the recurrence can be split as the sum of two parts - a sequence of
+the form ⟨x(1)
+n
+= 2n−1k⟩ and a sequence of the form ⟨x(2)
+n ⟩ which is periodic with
+period 3 with x(2)
+1
++x(2)
+2 +x(2)
+3
+= 0. It is then easy to solve for k: x1 +x2 +x3 =
+x(1)
+1
++ x(1)
+2
++ x(1)
+3
+= 7k, so k = x1+x2+x3
+7
+.
+In particular, if you let tn = 2n−1
+7
+, pn−tn is periodic with period ⟨ 6
+7, − 2
+7, − 4
+7⟩,
+qn − tn with period ⟨− 1
+7, 5
+7, − 4
+7⟩ and rn − tn with period ⟨− 1
+7, − 2
+7, 3
+7⟩.
+For n = 3t, xn = c(x1 + x2) + (c + 1)xn−3 and xn+1 = 2(c + 1)x1 + (2c +
+1)(x2 + x3) where c = 23t−1−1
+7
+. Then xn+1 cannot be equal to (2c + 1)(2c + 3)
+for positive x1, x2, x3 (x1 would have to be a multiple of 2c + 1 that is positive
+but less than 2c + 1), but for all 1 ≤ i ≤ 4c + 4, if x1 = i, x2 = 4c + 5 − i and
+x3 = 3, then xn = c(4c + 5) + (c + 1)3 = 4c2 + 8c + 3 = (2c + 1)(2c + 3).
+The proofs in this paper can be adapted to show that many other recurrences
+are congenial. Let us say a polynomial x3 − ax2 − bx − c is affable if c = 1 and
+it has exactly one real root, which is more than 1. We will show that affability
+leads to congeniality.
+For the rest of this section, fix an affable polynomial x3 − ax2 − bx − c with
+real root η1 and complex roots η2 and η3 = η2. Note that |η2| = η−1/2
+1
+.
+We will make use of the following equivalent to Lemma 6.
+Lemma 15. Given a sequence ⟨xi⟩n
+i=1 satisfying xi+3 = axi+2 + bxi+1 + cxi
+with xn = 0, xn−1 = k and xn−2 = l, xi can be expressed as
+xi =
+3
+�
+j=1
+(kψj + lζj)ηn−i
+j
+for constants ψj, ζj depending only on x3 − ax2 − bx − c, which can be rewritten
+as
+xi
+η(n−i)/2
+1
+= αη−3(n−i)/2
+1
++ β cos(γ − δ(n − i))
+where
+α = kψ1 + lζ1,
+βeγi = 2(kψ2 + lζ2) and
+eδi = η2
+√η1.
+We will follow the steps of the proof of Theorem 2 for all recurrence relations
+corresponding to affable polynomials. We will not attempt to give an actual
+bound.
+We note the following, which will be used in the equivalents of both Theo-
+rems 3 and 4
+9
+
+Lemma 16. If real numbers k and l satisfy kψ2 + lζ2 = 0, then k = l = 0.
+Proof. As ψ3 = ψ2 and ζ3 = ζ2, kψ3 + lζ3 = 0 and therefore the sequence with
+xn = 0, xn−1 = k and xn−2 = l can simply be expressed as xi = (kψ1+lζ1)ηn−i
+1
+.
+As 0 = xn = kψ1 + lζ1, it follows that xi = 0 for all i and therefore k = l =
+0.
+We start by following the proof of Theorem 3.
+Lemma 17. There exists an absolute bound M such that for n ≥ 4 and all
+non-zero integer sequences ⟨xi⟩n
+i=1 satisfying xi+3 = axi+2 + bxi+1 + cxi and
+xn = 0 either |x1| ≥ Mηn/2
+1
+or |x2| ≥ Mηn/2
+1
+(or both).
+Proof. The set of complex numbers kψ2 + lζ2 for k, l real with |k| + |l| = 1 is
+a closed subset of the complex plane (in fact a hollow parallelogram) which, by
+Lemma 16 does not contain 0. As such, there exists a constant V > 0 such
+that for all such k, l, |kψ2 + lζ2| > V .
+Then for all real k, l it follows that
+β = |kψ2 + lζ2| > V (|k| + |l|).
+Clearly if U = max(|ψ1|, |ζ1|), α ≤ U(|k| + |l|).
+Pick integer N such that V cos(δ/2) − Uη−3(N−1)/2
+1
+is positive. Note we
+can do this because π
+2 < δ < π. Then let M > 0 be such that V cos(δ/2) −
+Uη−3(N−2)/2
+1
+> Mη1 and η−N/2
+1
+> M.
+Now if n ≤ N, then Mηn/2
+1
+< 1 (note that η1 > 1 since 13 < a×12+b×1+c)
+and x1, x2 cannot both be 0 (as then xn would have to be the same sign as x3
+and non-zero).
+For n > N, we know from Lemma 8 that there exists t ∈ {1, 2} such that
+| cos(γ − δ(n − t))| > cos(δ/2) > 0.
+For such t, n − t ≥ N − 1 and so it follows that
+|xt|
+η(n−t)/2
+1
+= |αη−3n/2
+1
++ β cos(γ − δn)|
+≥ |β cos(γ − δn)| − |α|η−3n/2
+1
+≥ V cos(δ/2) − Uη−3n/2
+1
+≥ Mη1
+and hence |xt| ≥ Mη(n+2−t)/2
+1
+≥ Mηn/2
+1
+.
+This is enough for the equivalent of Theorem 3
+Theorem 18. There exists a fixed bound T such that for any positive integers
+n, k with k ≥ 4, the number of positive sequences ⟨xi⟩k
+i=1 satisfying xi+3 =
+axi+2 + bxi+1 + cxi and terminating at xk = n is at most
+⌈T
+n
+η3k/2
+1
+⌉2.
+10
+
+Proof. There is a fixed P such that for any positive sequence ⟨ai⟩k
+i=1 satisfying
+the recurrence relation with k ≥ 4, Pηk
+1(a1 + a2 + a3) ≤ ak.
+Thus for any such sequence terminating at n, a1 and a2 are bounded above
+by
+n
+P ηk
+1 and for any two such sequences, by Lemma 17, either the first terms or
+the second terms differ by at least Mηk/2
+1
+.
+Thus the number of such sequences is at most ⌈
+n
+P Mη3k
+1 /2⌉2.
+Now we proceed to follow the proof of Theorem 4. We will need the following
+Corollary to Lemma 16
+Corollary 19. Given any interval 0 ≤ x < y ≤ 2π within (0, 2π), we can pick
+non-zero integers k, l for which x < γ < y.
+Proof. Lemma 16 says that the set {kψ2 + lζ2 : k, l ∈ R}, when viewed geomet-
+rically as a subset of the complex plane, is not of dimension 1. Thus it must
+be the entire complex plane. Pick x < z < y, then there exist real k, l with
+kψ2 + lζ2 = eiz.
+Now let kn = ⌊nk⌋ and ln = ⌊nl⌋.
+The limit as n tends to infinity of
+knψ2+lnζ2
+n
+is eiz and therefore for all sufficiently large n, γ (which is the argument
+of knψ2+lnζ2
+n
+) must be contained in the open interval (x, y).
+We shelve this for the moment and focus on a simple piece of trigonometry.
+Lemma 20. For all numbers π
+2 < δ < π, there exists t such that cos(t) > 0 >
+cos(t + δ), cos(t + 2δ)
+Proof. Pick t such that π
+2 − δ < t < 3π
+2 − 2δ. There exists such a t because
+δ < π.
+Since δ < π, − pi
+2 < π
+2 − δ < t. Similarly since π
+2 < δ, t < 3π
+2 − 2δ < pi
+2 . So
+− pi
+2 < t < pi
+2 and hence cos(t) > 0.
+Further π
+2 < t+δ < t+2δ < 3π
+2 , so cos(t+δ) and cos(t+2δ) are negative.
+This leads to the following somewhat technical-seeming lemma.
+Lemma 21. For all numbers π
+2 < �� < π, there exists an ǫ > 0 and finitely many
+intervals ⟨(xi, yi)⟩n
+i=1 such that for all t there exists an interval (xi, yi) such that
+for all x ∈ (xi, yi), cos(t + x) > ǫ and −ǫ > cos(t + x + δ), cos(t + x + 2δ).
+Proof. Pick a t according to Lemma 20, and let ǫ > 0 be a real number such
+that cos(t) > ǫ and −ǫ > cos(t + δ), cos(t + 2δ).
+Then since cos is a continuous function, there is an open region (l, u) around
+t such that for all x ∈ (l, u), cos(x) > ǫ and −ǫ > cos(x + δ), cos(x + 2δ).
+Let n be an integer such that
+4π
+n
+< u − l and then define (xi, yi) to be
+(i 2π
+n , (i + 1) 2π
+n ) for 1 ≤ i ≤ n.
+For all t there is a maximum integer K such that t + K 2π
+n
+≤ l.
+Then
+l < t + (K + 1) 2π
+n by maximality, but t + (K + 2) 2π
+n ≤ l + 4π
+n < u.
+Thus if (K + 1) 2π
+n < x < (K + 2) 2π
+n , l < t + x < u and hence cos(t + x) > ǫ
+and −ǫ > cos(t + x + δ), cos(t + x + 2δ).
+11
+
+Since cos is periodic with period 2π, if 1 ≤ i ≤ n and i is equivalent to K +1
+modulo n, then for all xi < x < yi, cos(t+x) > ǫ and −ǫ > cos(t+x+δ), cos(t+
+x + 2δ).
+This leads to the equivalent of Lemma 13.
+Lemma 22. There exists a constant C such that for all n ≥ 4, there exist
+sequence ⟨ai⟩n
+i=1 satisfying the recurrence relation and terminating at 0 for which
+Cηn/2
+1
+> a1 > 0 > a2, a3
+Proof. Since π
+2 < δ < π, we can apply Lemma 21 and get ǫ > 0 and finitely
+many intervals (xi, yi) such that for all t there exists an interval (xi, yi) such
+that for all x ∈ (xi, yi), cos(t + x) > ǫ and −ǫ > cos(t + x + δ), cos(t + x + 2δ).
+By Corollary 19, for each such interval (xi, yi), we can choose non-zero in-
+tegers ki, li for which xi < γ(ki, li) < yi. Let A be some real number such
+that |α(ki, li)| < A for all such pairs, B > 0 be some real number such that
+|β(ki, li)| > B and let N be such that Aη−3N/2
+1
+< Bǫ.
+Then for any j ≥ N + 3, by the statement of Lemma 21, there exists an
+interval (xi, yi) such that for all x ∈ (xi, yi), cos(x − (j − 1)δ) > ǫ and −ǫ >
+cos(x − (j − 2)δ), cos(x − (j − 3)δ). Since γ(ki, li) ∈ (xi, yi), it follows that
+a1
+η(j−1)/2
+1
+= α(ki, li)η−3(j−1)/2
+1
++ β(ki, li) cos(γ(ki, li) − (j − 1)δ)
+is the sum of a number of absolute value at most Aη−3N/2
+1
+and a number that is
+at least Bǫ and so is positive. Similarly a2 and a3 are negative.
+|a1|
+ηj/2
+1
+is bounded
+above by 2Bǫ.
+For each value 4 ≤ j ≤ N +2, we can just choose any sequence satisfying the
+bounds. For instance, if aj = pa1 + qa2 + ra3, we set a1 = q + r, a2 = a3 = −p.
+Choose C such that C > 2Bǫ and such that for all 4 ≤ j ≤ N +2, the sequences
+we have chosen satisfy a1 < Cηj/2
+1
+.
+Similarly we can get the following.
+Lemma 23. There exists a constant C such that for all n ≥ 4, there exist
+sequence ⟨bi⟩n
+i=1 satisfying the recurrence relation and terminating at 0 for which
+Cηn/2
+1
+> b2 > 0 > b1, b3
+Proof. Proof entirely analagous to Lemma 22.
+For Lemma 20, we need a u such that cos(u + δ) > 0 > cos(u), cos(u − 2δ).
+Pick u such that π
+2 − 2δ < u < −π
+2 . There exists such a u because δ > π
+2 .
+Since δ < π, −3π
+2
+< u < −π
+2 and hence cos(u) < 0. Similarly π
+2 < u+2δ < 3π
+2
+and hence cos(u + 2δ) < 0. Finally π
+2 − δ < u + δ < δ − π
+2 , so − π
+2 < u + δ < π
+2 ,
+so cos(u + δ) > 0.
+Then by a method equivalent to Lemma 21 there exists an ǫ′ > 0 and finitely
+many intervals ⟨(x′
+i, y′
+i)⟩m
+i=1 such that for all t there exists an interval (x′
+i, y′
+i) such
+that for all x ∈ (x′
+i, y′
+i), cos(t + x + δ) > ǫ′ and ǫ′ > cos(t + x), cos(t + x + 2δ).
+We then apply the same method as the proof of Lemma 22
+12
+
+This allows us to prove the equivalent of Theorem 4.
+Theorem 24. There exists a real number U such that for any positive integers
+n, k with k ≥ 4 and n ≥ Uη3k/2
+1
+, there is a positive sequence ⟨xi⟩k
+i=1 satisfying
+xi+3 = axi+2 + bxi+1 + cxi and terminating at xk = n.
+Proof. Denote by pk, qk and rk the integers such that xk = pkx1 + qkx2 + rkx3
+for all such sequences ⟨xi⟩k
+i=1.
+Then since there can be an integer sequence ending at xk = 1, there is no
+non-trivial common divisor of pk, qk and rk.
+Further, by Lemma 22 and Lemma 23 there exist integers Cηk/2
+1
+> a1 >
+0 > a2, a3 and Cηk/2
+1
+> b2 > 0 > b1, b3 for which a1pk + a2qk + a3rk =
+b1pk +b2qk +b3rk = 0. Hence by Theorem 12, for all n ≥ a1pk +b2qk +rk, there
+is such a sequence terminating at n.
+Since (2C +1)ζk/2
+1
+> a1 +b2 +1 and pk, qk, rk > T ζn
+1 for some fixed constant
+T , it follows that for all n ≥ (2C+1)T ζ3k/2
+1
+, there is such a sequence terminating
+at n.
+Finally we are able to show that all affable polynomials are congenial.
+Proof of Theorem 5. For our polynomial x3 − ax2 − bx − c with c = 1 and
+a + b > 1 and at most one real root, Theorem 24 has stated the existence of a
+real number U uch that for any positive integers n, k with k ≥ 4 and n ≥ Uη3k/2
+1
+there is a positive sequence of length k terminating at n.
+Thus if there is no positive sequence of length k + 1 terminating at n, it
+follows that n < Uη3(k+1)/2
+1
+.
+Then by Theorem 18 it follows that the number of sequences of length k
+terminating at n is at most ⌈T
+n
+η3k/2
+1
+⌉2 < ⌈T Uη3/2
+1
+⌉2.
+For now we leave open the following question.
+Question 25. For which positive integers a, b, c with c > 0 and a + b > 0 is the
+recurrence relation xn = axn−1 + bxn−2 + cxn−3 congenial?
+References
+[1] S. Spiro, “Problems that i would like somebody to solve,” 2020.
+13
+

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf,len=487
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='12146v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='NT]  28 Jan 2023  On Tribonacci Sequences  Luke Pebody  Saturday 28, January 2023  Abstract  Let a tribonacci sequence be a sequence of integers satisfying ak =  ak−1 + ak−2 + ak−3 for all k ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  For any positive integers k and n,  denote by fk(n) the number of tribonacci sequences with a1, a2, a3 > 0  and with ak = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  For all n, there is a maximum k such that fk(n) is non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Answering  a question of Spiro [1], we show that there is a finite upper bound (we  specifically prove 561001) on fk(n) for any positive integer n ≥ 3 and this  maximum k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  We do this by showing that fk(n) has transitions in n around constant  multiples of φ3k/2 (where φ is the real root of φ3 = φ2 + φ + 1): there  exists a constant C such that fk(n) > 0 whenever n > Cφ3k/2 and for any  constant T , the values of fk(n) with n < T φ3k/2 have an upper bound  independent of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  1  Introduction  A tribonacci sequence of length k is a sequence of integers ⟨ai⟩k  i=1 such that  ai = ai−1 + ai−2 + ai−3 for all 4 ≤ i ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  We say that such a sequence  terminates at ak and that it is positive if a1, a2, a3 > 0 - note that this easily  implies that ai > 0 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Denote by fk(n) the number of tribonacci sequences  of length k terminating at n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Clearly f1(n) = 1 for all n > 0, the only tribonacci sequence of length 1  terminating at n being ⟨n⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Further, f2(n) = f3(n) = ∞ as we can choose any  values for the proceeding terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  For n ≥ 3, there exists a tribonacci sequence of length longer than 3 termi-  nating at n, for example ⟨n − 2, 1, 1, n⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' However for any tribonacci sequence  ⟨ai⟩k  i=1 of length k, and for any 4 ≤ i ≤ k, ai = ai−1 + ai−2 + ai−3 ≥ ai−1 + 2,  so by induction ai ≥ 2i− 5 for all 3 ≤ i ≤ k, and hence if n < 2k − 5, fk(n) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Let t(n) be the largest number such that ft(n)(n) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Let p(n) denote the number of positive tribonacci sequences of length t(n)  terminating at n, so p(n) = ft(n)(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Clearly, since t(1) = t(2) = 3 it follows that p(1) = p(2) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Spiro [1] asks  Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Does there exist some absolute constant c such that for all n ≥ 3,  p(n) ≤ c for all n?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  1  The purpose of this paper is to give a positive answer to this question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Indeed  we will show  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' For any integer n ≥ 3, there are at most 561001 positive tribonacci  sequences of length t(n) terminating at n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  It turns out the key question for our proof is the minimum size of the vector  \uf8eb  \uf8ed  a1  a2  a3  \uf8f6  \uf8f8 where ⟨ai⟩n  i=1 is a non-zero tribonacci sequence terminating at an = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  In Section 2 we will show a lower bound on such a sequence of the order of φn/2,  which will allow us to prove  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' For any positive integers n, k with k ≥ 4, the number of positive  sequences of length k terminating at n is at most  ⌈1500  n  φ3k/2 ⌉2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  In Section 3 we turn to trying to put an upper bound on numbers that don’t  have any positive tribonacci sequences of length k terminating at them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' This  is an instance of the Coin Problem, also known as calculating the Frobenius  Number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' We construct two specific tribonacci sequences terminating at an = 0  with  \uf8eb  \uf8ed  a1  a2  a3  \uf8f6  \uf8f8 being of the order of φn/2 and with the integers a1, a2, a3 having  specified signs, allowing us to prove  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' For any integer n above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='2φ3k/2, there exists a positive tribonacci  sequence of length k terminating at n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  This will be all that is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' There is no sequence of length t(n) + 1 terminating at n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Hence by Theorem 4, it follows that n < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='2φ3(t(n)+1)/2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='2φ3/2φ3t(n)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Thus from Theorem 3, it follows that there are at most  ⌈1500  n  φ3t(n)/2 ⌉2 ≤ ⌈15000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='2φ3/2φ3t(n)/2  φ3t(n)/2  ⌉2  ≤ ⌈300φ3/2⌉2 = 7492 = 561001  positive tribonacci sequences of length t(n) terminating at n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  In Section 4, we will investigate which recurrence relations of the form xn =  axn−1+bxn−2+cxn−3 for non-negative a and b and for positive c the arguments  in this paper can be carried across to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' We will extend the result in the earlier  sections to the following case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  2  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Suppose a, b, c are non-negative integers with a + b > 0, c = 1  and such that x3 − ax2 − bx − c = 0 has exactly one real root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Then there  is an absolute bound T such that if positive integers k ≥ 4 and n are such  that there are no positive sequences ⟨ai⟩k+1  i=1 satisfying the recurrence relation  ai = aai−1 + bai−2 + cai−3 of length k + 1 terminating at n, then there are at  most T such sequences of length k terminating at n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  We will leave open the question of which linear recurrences satisfy this prop-  erty, but will at least demonstrate an example of a recurrence that does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  In particular we will show the existence of positive integers k and n such  that there is no positive sequence ⟨ai⟩k+1  i=1 satisfying the recurrence relation  ai = ai−1 + ai−2 + 2ai−3 of length k + 1 terminating at n, but for which the  number of such sequences of length k terminating at n is unbounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  2  Lower Bound  Let us say a sequence ⟨ai⟩∞  i=1 is a reverse-tribonacci sequence if for all i ≥ 0,  ai = ai+1+ai+2+ai+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Let us write out the expression for the reverse-tribonacci  sequence starting ⟨0, k, l⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Recall that φ is the real solution to φ3 = φ2 + φ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  We write the complex roots as φ1 and φ2 = φ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' For all integers k, l, if ⟨ai⟩∞  i=1 is a reverse-tribonacci sequence with  a1 = 0, a2 = k and a3 = l, then for all i, ai can be expressed as  ai = αφ−i + (kψ1 + lζ1)φ−i  1 + (kψ2 + lζ2)φ−i  2  = (αφ−3i/2 + β cos(γ − δi))φi/2,  where  ψ1 =  φ3  1 + φ2  1  φ2  1 + 2φ1 + 3  ψ2 = ψ1  ζ1 =  φ3  1  φ2  1 + 2φ1 + 3  ζ2 = ζ1  α = kφ2 + (k + l)φ3  φ2 + 2φ + 3  βeγi = 2(kψ1 + lζ1) and  eδi = φ1  �  φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Any two-way infinite tribonacci sequence ⟨ai⟩∞  −∞ can be written as ai =  pφi + qφi  1 + rφi  2 for some p, q and r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Thus any reverse-tribonacci sequence ⟨ai⟩∞  −∞ can be written as ai = pφ−i +  qφ−i  1 + rφ−i  2  for some p, q, r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Solving for the p, q, r that give a1 = 0, a2 = k and  a3 = l leads to the above expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  3  Note that in the above expressions, ψ1, ψ2, ζ1, ζ2 and δ are constants that  do not depend on k and l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' For any integers k and l, if α and β are defined as in Lemma 6,  then |α| ≤ |k| + |l| and β ≥ |k|+|l|  31  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' α is roughly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='9546k+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='6184l, which is clearly bounded above by |k|+|l|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  ψ1 is roughly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='02267 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='217i and ζ1 is roughly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='1908 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='0187i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' As such, if  k and l are non-negative then the real part of 2(kψ1 + lζ1) (and hence β) is at  least 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='04(k + l) > |k|+|l|  31  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  For k positive and l negative, the minimum value of  β  k−l is approximately  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='03221 >  1  31, and is achieved around k = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='3653l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Finally we need a simple trigonometric property  Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' For any real numbers p and q with π  2 < q < π, the larger of | cos(p)|  and | cos(p + q)| is at least cos(q/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Note that cos(q) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Thus  2(cos(p)2 + cos(p + q)2) = 2 cos(p)2 + 2 cos(p + q)2  = cos(2p) + cos(2p + 2q) + 2  = 2 cos(2p + q) cos(q) + 2  ≥ 2 + 2 cos(q)  = 4 cos(q/2)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Thus either cos(p)2 ≥ cos(q/2)2 or cos(p + q)2 ≥ cos(q/2)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  This allows us to put a lower bound on the size of at least one of each  consecutive pair of a reverse-tribonacci sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Given a non-zero integer reverse-tribonacci sequence ⟨ai⟩∞  i=1 with  a1 = 0, for every integer n ≥ 2, either |an| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='01φn/2 or |an+1| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='01φ(n+1)/2  (or both).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' For n ≥ 2 if an and an+1 are both 0, then a1 is the same sign as an−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Since a1 = 0, it follows that the entire series must be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Since the sequence is  non-zero, it follows that either |an| ≥ 1 or |an+1 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Since 1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='01φn/2 for  n ≤ 15, we have proved the statement for n ≤ 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Thus we may assume n ≥ 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  By Lemma 6,  ai  φi/2 can be written as αφ−3i/2 + β cos(γ − δi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Now by Lemma 8, at least one of | cos(γ − δn)| and | cos(γ − δ(n − 1))| is  at least cos(δ/2) (δ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='176 is between π  2 and π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Let t be the choice from  {n − 1, n} that maximises | cos(γ − δt)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  By Lemma 7, if we write α′ =  α  |k|+|l| and β′ =  β  |k|+|l| then |α′| ≤ 1 and  β′ >  1  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  4  Therefore  | at  φt/2 | = |αφ−3t/2 + β cos(γ − δt)|  ≥ |α′φ−3t/2 + β′ cos(γ − δt)|  ≥ |β′ cos(γ − δt)| − |α′φ−3t/2|  ≥ 1  31 cos(δ/2) − φ−3t/2 ≥ cos(δ/2)  31  − φ−22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='5 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Then we have a bound on the size of tribonacci sequences terminating at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Corollary 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' For n ≥ 3, if ⟨ai⟩n  i=1 is a non-zero integer tribonacci sequence  terminating at 0 then either |a1| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='01φn/2 or |a2| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='01φ(n−1)/2 (or both).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Let k = an−1 and l = an−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Then if ⟨bi⟩∞  i=1 is the reverse-tribonacci  sequence with b1 = 0, b2 = k and b3 = l, then ai = bn+1−i for all 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Then this is just a restatement of Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  This is all we need to prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Partition the tribonacci sequences of length k ≥ 4 termi-  nating at n ⟨ai⟩k  i=1 by the pair (⌊  a1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='01φk/2 ⌋, ⌊  a2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='01φk−1/2 ⌋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  If two sequences ⟨ai⟩k  i=1 and ⟨bi⟩k  i=1 have the same pair, then |a1 − b1| <  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='01φk/2 and |a2 − b2| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='01φ(k−1)/2 and hence, by Corollary 10, either ⟨ai −  bi⟩k  i=1 is zero everywhere or does not terminate at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Thus each distinct tribonacci sequence of length k terminating at n has a  distinct pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Define tribonacci sequence by x1 = 1, x2 = 0, x3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Then if a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' , ak  is a positive tribonacci sequence, ai ≥ xia4 for i = 2, 3 and 4 and therefore  ak ≥ xka4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Now xk < φk/11 for all k ≥ 4 and hence a1 + a2 + a3 ≤ 11n  φk for all  tribonacci sequences of length k terminating at n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Thus ⌊  a1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='01φk/2 ⌋ is at most 1100n  φ3k/2 and ⌊  a2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='01φ(k−1)/2 ⌋ is at most  1100n  φ3k−1/2 < 1500n  φ3k/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  It follows that the number of Tribonacci sequences of length k ≥ 4 termi-  nating at n is at most ⌈ 1500n  φ3k/2 ⌉2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Note we have not worked hard here to get the best bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' In a previous  draft we had a much more complicated proof of an upper bound which showed,  in place of Corollary 10, that if ⟨ai⟩n  i=1 terminated at 0 then  �  a2  1 + a2  2 + a2  3 >  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='28φn/2, which led to an upper bound for the main theorem of 42875.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  3  Upper Bound  In this section, we turn to numbers which are not the terminus for any tribonacci  sequence of length k, working towards a proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  5  To that end, define three infinite tribonacci sequences ⟨pi⟩∞  i=1, ⟨qi⟩∞  i=1 and  ⟨ri⟩∞  i=1 by (p1, p2, p3) = (1, 0, 0), (q1, q2, q3) = (0, 1, 0) and (r1, r2, r3) = (0, 0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  It is clear that for any tribonacci sequence ⟨ai⟩n  i=1, an = a1pn + b1qn + c1rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Thus we are simply looking to get an upper bound on the largest number which  cannot be written as a positive integral linear combination of pn, qn and rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  This is called the Frobenius Number of pn, qn and rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  First let us see that a finite bound does exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' For all k ≥ 1, pk, qk and rk have no non-trivial common divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' If pk, qk and rk had a non-trivial common divisor t > 1 then t would be  a common divisor of the terminus of every tribonacci sequence of length k, from  which it would follow that t would in fact be a common divisor of pk+l for all  l ≥ 0 (since ⟨pi+l⟩k  1 is a tribonacci sequence of length k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Then, since pi = pi+3 − (pi+1 + pi+2), it would follow that t would be a  common divisor of pk−1, pk−2 and all the way back to p0 = 1 by induction,  causing a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  We will use the following bound, which might be originally due to Killing-  bergtro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Suppose p, q and r are integers with no non-trivial common di-  visor and let us suppose ap = bq + cr and dq = ep + fr where a, c, d, f > 0 and  b, e ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Then for every integer N ≥ ap + dq + r, N can be written in the form  xp + yq + zr for some positive integers p, q, r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Let x, y, z be positive integers such that px + qy + rz is equivalent to  N (mod r), but for which px + qy + rz is minimal (such a triple x, y, z exist  because, as is well known, if p, q and r have no non-trivial common divisor then  all sufficiently large integers can be written in the form px + qy + rz, and many  of these sufficiently large integers are equivalent to N (mod r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=')  Since px+qy+rz is minimal, px+qy+rz−r cannot be written as a positive  linear combination of x, y and z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Thus in each of the equations  px + qy + rz − r = px  + qy  + r(z − 1)  px + qy + rz − r = p(x − a)  + q(y + b)  + r(z + c − 1)  px + qy + rz − r = p(x + e)  + q(y − d)  + r(z + f − 1),  it must follow that one of the coefficients must not be positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Two of the  coefficients in each equation are clearly positive, so it follows that x ≤ a, y ≤ d  and z ≤ 1, so px + qy + rz ≤ pa + qd + r ≤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Since N and px + qy + rz  are equivalent modulo r, there exists a non-negative integer t such that N =  px + qy + rz + rt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Then N = px + qy + r(z + t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Therefore, to show that all sufficiently large integers can be written as the  terminus of a tribonacci sequence of length k, we just need to find linear com-  binations of pn, qn and rn combining to 0, with particular signs of the combi-  nations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' This is equivalent to finding tribonacci sequences ending at 0, which  6  Table 1: Table for Lemma 13  t0  t1  k  l  α  β  γ  x0  x1  x2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='06  0  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
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+page_content='0548  is equivalent to finding reverse-tribonacci sequences starting at 0, and hence we  can again use the expression from Lemma 6, which states that if ⟨ai⟩∞  i=1 is a  reverse-tribonacci sequence with a1 = 0, a2 = k and a3 = l then for all n  an = (αφ−3n/2 + β cos(γ − δn))φn/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Note that for all but an extremely small collection of n, the term β cos(γ+δn)  dwarves αφ−3n/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' As such, for a fixed k and l, the sign of an depends only  (except for a few very rare counterexamples) on the fractional part of  δ  2πn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' For each integer n ≥ 4, there exists a tribonacci sequence ⟨ai⟩n  i=1  terminating at an = 0, with a1 > 0, 0 ≥ a2, 0 > a3 and with a1 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='81φn/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Similarly for all such n, there exists a tribonacci sequence ⟨bi⟩n  i=1 terminating  at bn = 0, with b2 > 0, 0 ≥ b1, 0 > b3 and with b2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='64φn/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' We will split into cases based on the fractional part of δn  2π = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='3464n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' See  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' For each row, if t0 ≤ δn  2π − ⌊ δn  2π⌋ ≤ t1, then for the given values of k and  l, if β and γ are as defined in Lemma 6, one can confirm that β cos(γ − δn) ≥  x0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='34, while β cos(γ − δ(n − 1)) ≤ x1 < −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='035 and β cos(γ − δ(n − 2)) ≤  x2 < −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='029.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Furthermore, for all such k, l, |α| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='58, so if n ≥ 7, |αφ−3(n−2)/2| ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='017,  from which it follows that an > 0 > an−1, an−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Further,  an  φn/2 < β + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='017 <  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  For 4 ≤ n < 7, we can verify the sequences (1, 0, −1, 0), (2, 0, −1, 1, 0) and  (2, 0, −1, 1, 0, 0) satisfy the conditions for (a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' , an).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  For the sequence (b1, b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' , bn), see Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Here, for each row, if t0 ≤  δn  2π − ⌊ δn  2π⌋ ≤ t1, then for the given values of k and l, if β and γ are as defined in  Lemma 6, one can confirm that β cos(γ − δn) ≤ x0 < −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='071, while β cos(γ −  δ(n − 1)) ≥ x1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='33 and β cos(γ − δ(n − 2)) ≤ x2 < −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='041.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Furthermore, for all such k, l, |α| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='58, so if n ≥ 7, |αφ−3(n−2)/2| ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='017,  from which it follows that an−1 > 0 > an, an−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  For 4 ≤ n < 7, we can verify the sequences (0, 1, −1, 0), (0, 1, −1, 0, 0) and  (−1, 2, −1, 0, 1, 0) satisfy the conditions for (b1, b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' , bn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  This then completes our proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  7  Table 2: Other table for Lemma 13  t0  t1  k  l  α  β  γ  x0  x1  x2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
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+page_content='2528  Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Lemma 13 gives us tribonacci sequences ⟨ai⟩n  i=1 and ⟨bi⟩n  i=1  terminating at an = bn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' It follows that a1pn + a2qn + a3rn = 0 = b1qn +  b2qn + b3rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Since a1, b2 > 0 > a3, b3 and 0 ≥ a2, b1, it follows that we can write  a1pn = (−a2)qn + (−a3)rn and  b2qn = (−b1)p1 + (−b3)rn  satisfying the sign requirements of Theorem 12, so it follows that every integer  N ≥ a1pn + b2qn + rn can be written in the form xpn + yqn + zrn for some  positive integers x, y and z, and hence there exists a positive tribonacci sequence  of length k ending at N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  By the bounds on a1 and b2 given in Lemma 13, we have such a tribonacci  sequence for all N ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='81φk/2uk+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='64φk/2vk+wk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Since uk ≤ vk ≤ wk < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='11φk  and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='81φk/2+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='64φk/2+1 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='74φk/2, it follows that such a tribonacci sequence  exists for all N ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='2φ3k/2 as was required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  4  Other cubic recurrences  For non-negative a, b, c we can ask a similar question for recurrences of the form  xn = axn−1 + bxn−2 + cxn−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Formally, let us define ka,b,c(n) to be the largest  k such there is a positive k-element solution ⟨xi⟩k  i=1 to the recurrence relation  xi = axi−1 + bxi−2 + cxi−3, and define ta,b,c(n) to be the number of positive  ka,b,c(n)-element solutions that exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  If c = 0, this is a quadratic recurrence, and the problem is already solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' If  a = 0, b = 0 and c = 1, the recurrence is xn = xn−3, and ka,b,c(n) is not defined  for any n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  For all a, b, c ≥ 0 with c ≥ 1 and a + b + c ≥ 2, say that the recurrence  xn = axn−1 + bxn−2 + cxn−3 is congenial if there exists a finite bound B such  that for all n, ta,b,c(n) = ∞ or ta,b,c(n) ≤ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Firstly let us note that not all recurrences are congenial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Lemma 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' The recurrence xn = xn−1 + xn−2 + 2xn−3 is not congenial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Let ⟨pn⟩∞  n=1, ⟨qn⟩∞  n=1 and ⟨rn⟩∞  n=1 be the solutions to the recurrence  8  starting with ⟨1, 0, 0⟩, ⟨0, 1, 0⟩ and ⟨0, 0, 1⟩ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Then xn = x1pn +  x2qn + x3rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Solutions to the recurrence can be split as the sum of two parts - a sequence of  the form ⟨x(1)  n  = 2n−1k⟩ and a sequence of the form ⟨x(2)  n ⟩ which is periodic with  period 3 with x(2)  1  +x(2)  2 +x(2)  3  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' It is then easy to solve for k: x1 +x2 +x3 =  x(1)  1  + x(1)  2  + x(1)  3  = 7k, so k = x1+x2+x3  7  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  In particular, if you let tn = 2n−1  7  , pn−tn is periodic with period ⟨ 6  7, − 2  7, − 4  7⟩,  qn − tn with period ⟨− 1  7, 5  7, − 4  7⟩ and rn − tn with period ⟨− 1  7, − 2  7, 3  7⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  For n = 3t, xn = c(x1 + x2) + (c + 1)xn−3 and xn+1 = 2(c + 1)x1 + (2c +  1)(x2 + x3) where c = 23t−1−1  7  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Then xn+1 cannot be equal to (2c + 1)(2c + 3)  for positive x1, x2, x3 (x1 would have to be a multiple of 2c + 1 that is positive  but less than 2c + 1), but for all 1 ≤ i ≤ 4c + 4, if x1 = i, x2 = 4c + 5 − i and  x3 = 3, then xn = c(4c + 5) + (c + 1)3 = 4c2 + 8c + 3 = (2c + 1)(2c + 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  The proofs in this paper can be adapted to show that many other recurrences  are congenial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Let us say a polynomial x3 − ax2 − bx − c is affable if c = 1 and  it has exactly one real root, which is more than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' We will show that affability  leads to congeniality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  For the rest of this section, fix an affable polynomial x3 − ax2 − bx − c with  real root η1 and complex roots η2 and η3 = η2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Note that |η2| = η−1/2  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  We will make use of the following equivalent to Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Lemma 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Given a sequence ⟨xi⟩n  i=1 satisfying xi+3 = axi+2 + bxi+1 + cxi  with xn = 0, xn−1 = k and xn−2 = l, xi can be expressed as  xi =  3  �  j=1  (kψj + lζj)ηn−i  j  for constants ψj, ζj depending only on x3 − ax2 − bx − c, which can be rewritten  as  xi  η(n−i)/2  1  = αη−3(n−i)/2  1  + β cos(γ − δ(n − i))  where  α = kψ1 + lζ1,  βeγi = 2(kψ2 + lζ2) and  eδi = η2  √η1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  We will follow the steps of the proof of Theorem 2 for all recurrence relations  corresponding to affable polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' We will not attempt to give an actual  bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  We note the following, which will be used in the equivalents of both Theo-  rems 3 and 4  9  Lemma 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' If real numbers k and l satisfy kψ2 + lζ2 = 0, then k = l = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' As ψ3 = ψ2 and ζ3 = ζ2, kψ3 + lζ3 = 0 and therefore the sequence with  xn = 0, xn−1 = k and xn−2 = l can simply be expressed as xi = (kψ1+lζ1)ηn−i  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  As 0 = xn = kψ1 + lζ1, it follows that xi = 0 for all i and therefore k = l =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  We start by following the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Lemma 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' There exists an absolute bound M such that for n ≥ 4 and all  non-zero integer sequences ⟨xi⟩n  i=1 satisfying xi+3 = axi+2 + bxi+1 + cxi and  xn = 0 either |x1| ≥ Mηn/2  1  or |x2| ≥ Mηn/2  1  (or both).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' The set of complex numbers kψ2 + lζ2 for k, l real with |k| + |l| = 1 is  a closed subset of the complex plane (in fact a hollow parallelogram) which, by  Lemma 16 does not contain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' As such, there exists a constant V > 0 such  that for all such k, l, |kψ2 + lζ2| > V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Then for all real k, l it follows that  β = |kψ2 + lζ2| > V (|k| + |l|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Clearly if U = max(|ψ1|, |ζ1|), α ≤ U(|k| + |l|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Pick integer N such that V cos(δ/2) − Uη−3(N−1)/2  1  is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Note we  can do this because π  2 < δ < π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Then let M > 0 be such that V cos(δ/2) −  Uη−3(N−2)/2  1  > Mη1 and η−N/2  1  > M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Now if n ≤ N, then Mηn/2  1  < 1 (note that η1 > 1 since 13 < a×12+b×1+c)  and x1, x2 cannot both be 0 (as then xn would have to be the same sign as x3  and non-zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  For n > N, we know from Lemma 8 that there exists t ∈ {1, 2} such that  | cos(γ − δ(n − t))| > cos(δ/2) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  For such t, n − t ≥ N − 1 and so it follows that  |xt|  η(n−t)/2  1  = |αη−3n/2  1  + β cos(γ − δn)|  ≥ |β cos(γ − δn)| − |α|η−3n/2  1  ≥ V cos(δ/2) − Uη−3n/2  1  ≥ Mη1  and hence |xt| ≥ Mη(n+2−t)/2  1  ≥ Mηn/2  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  This is enough for the equivalent of Theorem 3  Theorem 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' There exists a fixed bound T such that for any positive integers  n, k with k ≥ 4, the number of positive sequences ⟨xi⟩k  i=1 satisfying xi+3 =  axi+2 + bxi+1 + cxi and terminating at xk = n is at most  ⌈T  n  η3k/2  1  ⌉2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  10  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' There is a fixed P such that for any positive sequence ⟨ai⟩k  i=1 satisfying  the recurrence relation with k ≥ 4, Pηk  1(a1 + a2 + a3) ≤ ak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Thus for any such sequence terminating at n, a1 and a2 are bounded above  by  n  P ηk  1 and for any two such sequences, by Lemma 17, either the first terms or  the second terms differ by at least Mηk/2  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Thus the number of such sequences is at most ⌈  n  P Mη3k  1 /2⌉2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Now we proceed to follow the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' We will need the following  Corollary to Lemma 16  Corollary 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Given any interval 0 ≤ x < y ≤ 2π within (0, 2π), we can pick  non-zero integers k, l for which x < γ < y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Lemma 16 says that the set {kψ2 + lζ2 : k, l ∈ R}, when viewed geomet-  rically as a subset of the complex plane, is not of dimension 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Thus it must  be the entire complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Pick x < z < y, then there exist real k, l with  kψ2 + lζ2 = eiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Now let kn = ⌊nk⌋ and ln = ⌊nl⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  The limit as n tends to infinity of  knψ2+lnζ2  n  is eiz and therefore for all sufficiently large n, γ (which is the argument  of knψ2+lnζ2  n  ) must be contained in the open interval (x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  We shelve this for the moment and focus on a simple piece of trigonometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Lemma 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' For all numbers π  2 < δ < π, there exists t such that cos(t) > 0 >  cos(t + δ), cos(t + 2δ)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Pick t such that π  2 − δ < t < 3π  2 − 2δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' There exists such a t because  δ < π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Since δ < π, − pi  2 < π  2 − δ < t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Similarly since π  2 < δ, t < 3π  2 − 2δ < pi  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' So  − pi  2 < t < pi  2 and hence cos(t) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Further π  2 < t+δ < t+2δ < 3π  2 , so cos(t+δ) and cos(t+2δ) are negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  This leads to the following somewhat technical-seeming lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Lemma 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' For all numbers π  2 < δ < π, there exists an ǫ > 0 and finitely many  intervals ⟨(xi, yi)⟩n  i=1 such that for all t there exists an interval (xi, yi) such that  for all x ∈ (xi, yi), cos(t + x) > ǫ and −ǫ > cos(t + x + δ), cos(t + x + 2δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Pick a t according to Lemma 20, and let ǫ > 0 be a real number such  that cos(t) > ǫ and −ǫ > cos(t + δ), cos(t + 2δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Then since cos is a continuous function, there is an open region (l, u) around  t such that for all x ∈ (l, u), cos(x) > ǫ and −ǫ > cos(x + δ), cos(x + 2δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Let n be an integer such that  4π  n  < u − l and then define (xi, yi) to be  (i 2π  n , (i + 1) 2π  n ) for 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  For all t there is a maximum integer K such that t + K 2π  n  ≤ l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Then  l < t + (K + 1) 2π  n by maximality, but t + (K + 2) 2π  n ≤ l + 4π  n < u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Thus if (K + 1) 2π  n < x < (K + 2) 2π  n , l < t + x < u and hence cos(t + x) > ǫ  and −ǫ > cos(t + x + δ), cos(t + x + 2δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  11  Since cos is periodic with period 2π, if 1 ≤ i ≤ n and i is equivalent to K +1  modulo n, then for all xi < x < yi, cos(t+x) > ǫ and −ǫ > cos(t+x+δ), cos(t+  x + 2δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  This leads to the equivalent of Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Lemma 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' There exists a constant C such that for all n ≥ 4, there exist  sequence ⟨ai⟩n  i=1 satisfying the recurrence relation and terminating at 0 for which  Cηn/2  1  > a1 > 0 > a2, a3  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Since π  2 < δ < π, we can apply Lemma 21 and get ǫ > 0 and finitely  many intervals (xi, yi) such that for all t there exists an interval (xi, yi) such  that for all x ∈ (xi, yi), cos(t + x) > ǫ and −ǫ > cos(t + x + δ), cos(t + x + 2δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  By Corollary 19, for each such interval (xi, yi), we can choose non-zero in-  tegers ki, li for which xi < γ(ki, li) < yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Let A be some real number such  that |α(ki, li)| < A for all such pairs, B > 0 be some real number such that  |β(ki, li)| > B and let N be such that Aη−3N/2  1  < Bǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Then for any j ≥ N + 3, by the statement of Lemma 21, there exists an  interval (xi, yi) such that for all x ∈ (xi, yi), cos(x − (j − 1)δ) > ǫ and −ǫ >  cos(x − (j − 2)δ), cos(x − (j − 3)δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Since γ(ki, li) ∈ (xi, yi), it follows that  a1  η(j−1)/2  1  = α(ki, li)η−3(j−1)/2  1  + β(ki, li) cos(γ(ki, li) − (j − 1)δ)  is the sum of a number of absolute value at most Aη−3N/2  1  and a number that is  at least Bǫ and so is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Similarly a2 and a3 are negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  |a1|  ηj/2  1  is bounded  above by 2Bǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  For each value 4 ≤ j ≤ N +2, we can just choose any sequence satisfying the  bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' For instance, if aj = pa1 + qa2 + ra3, we set a1 = q + r, a2 = a3 = −p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Choose C such that C > 2Bǫ and such that for all 4 ≤ j ≤ N +2, the sequences  we have chosen satisfy a1 < Cηj/2  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Similarly we can get the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Lemma 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' There exists a constant C such that for all n ≥ 4, there exist  sequence ⟨bi⟩n  i=1 satisfying the recurrence relation and terminating at 0 for which  Cηn/2  1  > b2 > 0 > b1, b3  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Proof entirely analagous to Lemma 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  For Lemma 20, we need a u such that cos(u + δ) > 0 > cos(u), cos(u − 2δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Pick u such that π  2 − 2δ < u < −π  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' There exists such a u because δ > π  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Since δ < π, −3π  2  < u < −π  2 and hence cos(u) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Similarly π  2 < u+2δ < 3π  2  and hence cos(u + 2δ) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Finally π  2 − δ < u + δ < δ − π  2 , so − π  2 < u + δ < π  2 ,  so cos(u + δ) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Then by a method equivalent to Lemma 21 there exists an ǫ′ > 0 and finitely  many intervals ⟨(x′  i, y′  i)⟩m  i=1 such that for all t there exists an interval (x′  i, y′  i) such  that for all x ∈ (x′  i, y′  i), cos(t + x + δ) > ǫ′ and ǫ′ > cos(t + x), cos(t + x + 2δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  We then apply the same method as the proof of Lemma 22  12  This allows us to prove the equivalent of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Theorem 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' There exists a real number U such that for any positive integers  n, k with k ≥ 4 and n ≥ Uη3k/2  1  , there is a positive sequence ⟨xi⟩k  i=1 satisfying  xi+3 = axi+2 + bxi+1 + cxi and terminating at xk = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Denote by pk, qk and rk the integers such that xk = pkx1 + qkx2 + rkx3  for all such sequences ⟨xi⟩k  i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Then since there can be an integer sequence ending at xk = 1, there is no  non-trivial common divisor of pk, qk and rk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Further, by Lemma 22 and Lemma 23 there exist integers Cηk/2  1  > a1 >  0 > a2, a3 and Cηk/2  1  > b2 > 0 > b1, b3 for which a1pk + a2qk + a3rk =  b1pk +b2qk +b3rk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Hence by Theorem 12, for all n ≥ a1pk +b2qk +rk, there  is such a sequence terminating at n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Since (2C +1)ζk/2  1  > a1 +b2 +1 and pk, qk, rk > T ζn  1 for some fixed constant  T , it follows that for all n ≥ (2C+1)T ζ3k/2  1  , there is such a sequence terminating  at n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Finally we are able to show that all affable polynomials are congenial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' For our polynomial x3 − ax2 − bx − c with c = 1 and  a + b > 1 and at most one real root, Theorem 24 has stated the existence of a  real number U uch that for any positive integers n, k with k ≥ 4 and n ≥ Uη3k/2  1  there is a positive sequence of length k terminating at n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Thus if there is no positive sequence of length k + 1 terminating at n, it  follows that n < Uη3(k+1)/2  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Then by Theorem 18 it follows that the number of sequences of length k  terminating at n is at most ⌈T  n  η3k/2  1  ⌉2 < ⌈T Uη3/2  1  ⌉2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  For now we leave open the following question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  Question 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' For which positive integers a, b, c with c > 0 and a + b > 0 is the  recurrence relation xn = axn−1 + bxn−2 + cxn−3 congenial?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  References  [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content=' Spiro, “Problems that i would like somebody to solve,” 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}
+page_content='  13' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'}

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+arXiv:2301.05502v1  [math.AG]  13 Jan 2023
+WHAT IS THE PROBABILITY THAT A RANDOM SYMMETRIC TENSOR IS
+CLOSE TO RANK-ONE?
+ALBERTO CAZZANIGA, ANTONIO LERARIO, ANDREA ROSANA
+Abstract. We address the general problem of estimating the probability that a real symmetric
+tensor is close to rank–one tensors. Using Weyl’s tube formula, we turn this question into a differential
+geometric one involving the study of metric invariants of the real Veronese variety. More precisely,
+we give an explicit formula for its reach and curvature coefficients with respect to the Bombieri–Weyl
+metric. These results are obtained using techniques from Random Matrix theory and an explicit
+description of the second fundamental form of the Veronese variety in terms of GOE matrices. Our
+findings give a complete solution to the original problem, and in the case of rational normal curves
+lead to some novel asymptotic results.
+1. Introduction
+1.1. What is the probability that a random symmetric tensor is close to rank-one? Over
+the last decades, symmetric tensors have been proven to be a very flexible and valuable tool in many
+different contexts. In particular, rank–one approximation and tensor decomposition found applications
+in machine learning ([AGH+14]), signal processing and image analysis ([SDLF+17], [Sak16, Ch.3, 4]),
+chemistry ([SBG04]), statistics ([McC87]), psychology and medical diagnostics ([Kro08, ALLF07]) and
+phylogenetics ([Sak16, Ch.5], [Lan12]), to name a few. Motivated by this, in this paper we address the
+following question:
+“What is the probability for a real symmetric tensor to be “close” to rank–one?”
+To make sense of this question, we must endow the space of tensors with a notion of distance and
+with a probability distribution. We address this problem in a natural way as follows.
+Observe first that the real vector space of symmetric tensors of order d on Rn+1 can be naturally
+identified with the space R[x0, . . . , xn](d) of homogeneous polynomials of degree d. Under this identifi-
+cation, we endow the space of real polynomials with the scalar product given by the restriction of the
+real part of the Bombieri–Weyl hermitian product, defined on the space of complex polynomials by
+(1)
+⟨p1, p2⟩BW :=
+1
+πn+1
+�
+Cn+1 p1(z)p2(z)e−∥z∥2dz,
+where dz := (i/2)n+1dz0dz0 . . . dzndzn is the Lebesgue measure.
+This defines the unique, up to
+multiples, hermitian product on the space of complex polynomials which is invariant under the action
+of the unitary group by change of variables. The restriction of the real part of this hermitian product to
+the space of real polynomials will be still called the Bombieri–Weyl scalar product; the above unitary
+invariance implies its invariance under the action of the orthogonal group by change of variables. In the
+case when d = 2, the above identification is the familiar isomorphism between the space of symmetric
+matrices and the space of quadratic forms, and the Bombieri–Weyl scalar product coincides with the
+Frobenius inner product.
+Next, we use this scalar product to turn R[x0, . . . , xn](d) into a probability space. For a Borel set
+U ⊆ R[x0, . . . , xn](d) we define
+P(U) :=
+�
+U
+e−
+∥p∥2
+BW
+2
+dµ
+�
+R[x0,...,xn](d)
+e−
+∥p∥2
+BW
+2
+dµ
+,
+Date: September 2022.
+1
+
+2
+ALBERTO CAZZANIGA, ANTONIO LERARIO, ANDREA ROSANA
+where “dµ” denotes the integration with respect to the Lebesgue measure on the space of coefficients.
+We call the resulting probability distribution Bombieri–Weyl, and sometimes the nomenclature Kostlan
+is also used interchangeably. When d = 2, the Bombieri–Weyl distribution turns the space of symmetric
+matrices into a gaussian space, called the Gaussian Orthogonal Ensemble, as we will describe in more
+detail in Section 2.1.
+Finally, we identify the set of rank–one tensors with the Veronese variety Vn,d ⊂ R[x0, . . . , xn](d) of
+signed d–th powers of linear forms. At this point we are in the position of giving a precise formulation
+to our question above, which therefore requires computing, for δ > 0 small enough, the quantity:
+P
+�
+p ∈ R[x0, . . . , xn](d)
+���� distBW(p, Vn,d) ≤ δ∥p∥BW
+�
+.
+Notice that we have turned this into a conic problem that takes into account also the norm of the
+tensor, as it is common procedure in numerical algebraic geometry [BC13]. In this way, we can regard
+the above probability as the normalized volume of a tubular neighbourhood of the intersection of the
+set of rank–one tensors with the unit sphere in the Bombieri–Weyl norm. Thus, our question becomes:
+“What is the volume of a neighbourhood of the spherical Veronese variety?”
+In this paper, exploiting Weyl’s Tube Formula, we derive an exact expression for the above volume, for
+small enough neighbourhoods. Moreover, as a byproduct of our computations, we give a lower bound
+on the size of the neighbourhood of the set of rank–one tensors that admit a unique best rank–one
+approximation.
+Remark 1. The properties of the Bombieri–Weyl distribution on the space of real (and complex) poly-
+nomials have been intensively studied, starting from the influential works of A. Edelman, E. Kostlan,
+M. Shub and S. Smale [EK95, SS93b, SS93a, SS93c]. The point of view of random tensors has been
+adopted first by E. Horobet and J. Draisma in [DH16] and by P. Breiding in [Bre19] for the study of
+the expected number of eigenvalues of a random symmetric tensor, with respect to the Bombieri–Weyl
+distribution. Under the identification between symmetric tensors and homogeneous polynomials, eigen-
+values correspond to critical values of the restriction of the polynomial to the unit sphere. Eigenvectors
+correspond to critical points of the polynomial: under the Veronese embedding these critical points
+give rank–one tensors that are critical points of the distance function on the Veronese variety from
+the given tensor. Among these critical points (which are rank–one tensors) the closest to the original
+tensor are its best rank–one approximations. In [Bre19] the average number of such critical points is
+computed. In this paper we will instead give the size and estimate the probability of the set of tensors
+which admit a unique best rank–one approximation.
+The use of Weyl’s Tube Formula is fairly standard for results of this type [BC13, BL22]: it allows to
+deduce an exact expression, for ε > 0 small enough, of the volume of an ε–neighbourhood of a smooth
+submanifold W of the sphere, or the euclidean space, as a function of some differential–geometric
+quantities of W, called its curvature coefficients. Our main contribution is the nontrivial computation
+of the curvature coefficients of the spherical Veronese variety and the explicit quantification of the
+above expression “for ε > 0 small enough” for this variety, through the computation of its reach.
+One could generalize this question to higher ranks by looking at secant varieties to the Veronese,
+whose geometry has been intensively studied, see [CGO14] for a survey. We propose to investigate this
+in future works.
+We now describe the main ingredients and state the main results of our work in more detail.
+1.2. The spherical Veronese. The main object we consider in this work is the real spherical Veronese
+variety Vn,d, which is the intersection of the Veronese variety in R[x0, . . . , xn](d) ≃ RN+1 with the unit
+sphere for the Bombieri–Weyl norm:
+Vn,d := Vn,d ∩ SN.
+We regard this set as the image of the spherical Veronese embedding associated to the Bombieri–
+Weyl basis, or its double copy, depending on the parity of d. This embedding is the smooth map
+
+WHAT IS THE PROBABILITY THAT A RANDOM SYMMETRIC TENSOR IS CLOSE TO RANK-ONE?
+3
+�νn,d : Sn → SN given by
+x
+�νn,d
+�−−−→
+��d
+α
+� 1
+2
+xα
+�
+α
+,
+where α ∈ Zn+1
+≥0
+satisfy α0 + · · · + αn = d,
+�d
+α
+�
+is the multinomial coefficient, and Sn is the euclidean
+sphere in Rn+1. Denoting �νn,d(Sn) by Σn,d, we see that
+Vn,d = Σn,d ∪ −Σn,d,
+where Σn,d = −Σn,d if d is odd and Σn,d ∩ (−Σn,d) = ∅ if d is even. For this reason, we will call Σn,d
+the spherical Veronese surface, to distinguish it from the spherical Veronese variety Vn,d, in the case d
+is even. In the projective picture, the difference between the two ceases to exist:
+PVn,d := P(Σn,d) = P(Vn,d) ⊂ RPN.
+Recall that Σn,d parametrizes the d–th powers of norm–1 linear forms on Rn+1 and, therefore, rank–
+one and norm–one tensors up to signs. Hence, the spherical Veronese surface Σn,d corresponds to an
+orbit for the action of O(n + 1) on homogenous polynomials by change of variables. Even more is
+true: when turning Σn,d into a Riemannian manifold with the metric induced by the Bombieri–Weyl
+scalar product, the transitive action of O(n + 1) on Σn,d is through isometries induced by isometries
+of SN, given the invariance property of the Bombieri–Weyl structure. The immediate, yet crucial,
+consequence is that the extrinsic geometry of the isometric embedding Σn,d ֒→ SN is exactly the same
+at every point. The same conclusion clearly holds for Vn,d ֒→ SN.
+1.3. Weyl’s tube formula and the reach of an embedding. Let (M, g) be a Riemannian manifold
+and M ֒→ M be an isometric embedding of a compact smooth submanifold. We can consider the set
+of points in M at distance less than a given ε > 0 from M and call such a set a tubular neighbourhood
+of M in M of radius ε, denoted as U(M, ε).
+It is well known that for smooth compact embeddings M ֒→ M and small enough radii, the exponen-
+tial map on the normal bundle provides a smooth parametrization of the tubular neighbourhood. This
+description is what really underlies the celebrated “Weyl’s tube formula” ([Wey39]), which constitutes
+one of the main tools to compute the volume of tubular neighbourhoods in a euclidean or spherical
+ambient space. This formula expresses the volume as the linear combination
+Vol(U(M, ε)) =
+�
+0≤e≤n, e even
+Ks+e(M)JN,s+e(ε),
+where N is the dimension of the ambient space, n is the dimension of M and s := N − n is the
+codimension of the embedding. The functions J’s do not depend on the specific submanifold M and
+are explicitly known in both the euclidean and spherical cases. The most remarkable aspect of the
+formula is that the coefficients K’s are isometric invariants of the embedding and can be expressed
+in terms of curvature, from which they are named curvature coefficients of the embedding. Remark
+that nowadays Weyl’s tube formula has been re-interpreted in the more general framework of “integral
+geometry”, which deals with integrals over a submanifold of polynomials in the entries of the second
+fundamental form. R. Howard in [How93] showed how the above formula fits in this context and gave
+a full characterization of the polynomials appearing in Weyl’s work.
+In the case of the Veronese variety Vn,d ֒→ SN, the tubular neighbourhood U(Vn,d, ε) gives a
+description of the norm–1 symmetric tensors that are ε–close to a rank–1 tensor in the Bombieri–Weyl
+metric, in the ambient sphere. As already pointed out, it follows that asking for the probability for
+a symmetric norm–1 tensor to be close to rank–1 boils down to computing the normalized volume of
+this tubular neighbourhood.
+For practical reasons, in the paper we will work with Σn,d instead of its “double” Vn,d. There are,
+however, two technical issues to consider here. The first one is that there will be a factor of 2 to
+be taken into account when switching from the Veronese surface Σn,d to the rank–one variety Vn,d,
+depending on the parity of d. The second one is that the intersection of a δ–neighbourood of the set
+of rank–one tensors with the unit sphere becomes an ε–neighbourhood of Vn,d in the unit sphere, with
+ε = arcsin(δ).
+
+4
+ALBERTO CAZZANIGA, ANTONIO LERARIO, ANDREA ROSANA
+This is why we will use the parameter “ε” to formulate the results on the sphere and the parameter
+“δ” for the results in the vector space of tensors.
+1.4. The reach of the Veronese variety. Our aim is to exploit Weyl’s tube formula to compute
+the volume of U(Vn,d, ε). This requires first of all the knowledge of the radii for which the above
+expression holds. Since the formula is based on the parametrization through the normal exponential
+map, the supremum of the radii for which this is a good parametrization, or at least a lower bound
+on that, is what we need to understand to meaningfully use Weyl’s result. This quantity is usually
+called the reach of the embedding M ֒→ M and in general computing it is a very difficult task, often
+unfeasible since it requires to study not only how normal geodesics originating from every point of the
+submanifold behave, but also how and when geodesics starting from different points cross each other,
+in order to avoid overlappings in the image.
+In our case, recalling the invariance property of Vn,d ֒→ SN under the action of the orthogonal group
+O(n + 1), we do not need to study normal geodesics originating from any point, but it is enough to
+choose a specific one and perform computations involving only geodesics originating from this chosen
+one. This drastically reduces the complexity of the computation, allowing us to obtain the following
+result, stated in a more detailed form in Theorem 19.
+Theorem A (The reach of the spherical Veronese). The reach of the spherical Veronese variety
+Vn,d ֒→ SN is given by
+ρ(Vn,d) =
+1
+√
+3
++
+1
+3d
+√
+3
++ O
+� 1
+d2
+�
+.
+The same result holds for the reach of the Veronese surface Σn,d ֒→ SN.
+Given the interpretation of the neighbourhood of the Veronese variety in terms of symmetric tensors
+already discussed, this theorem has an important consequence. From its proof, it follows that every
+real symmetric tensor which is sufficiently close to rank–one tensors admits a unique best rank–one
+approximation (see Corollary 21). In fact the reach ρ(Vn,d) equals the minimum between two quantities,
+one of which estimates the size of the neighborhood of Vn,d on which the normal exponential map is
+injective; we prove that this quantity equals π
+4 , and this allows to deduce the following.
+Corollary B (Best rank–one approximation). Every symmetric tensor p at distance less than
+√
+2
+2 ∥p∥BW
+from rank–one admits a unique best rank–one approximation.
+The normalized volume of a neighbourhood of the Veronese variety of radius π
+4 would therefore
+provide a lower bound for the probability that such tensors have a unique best rank–one approximation.
+Unfortunately, we are not able to compute such a volume, given that the value of the reach ρ(Vn,d) < π
+4
+does not allow to use Weyl’s tube formula up to such a radius. Nevertheless, given the uniformity of
+the lower bound for the reach ρ(Vn,d) ≥
+1
+√
+3 for every n, d (see Remark 20), we still get that the volume
+of the neighbourhood of radius
+1
+√
+3 provides a lower bound for that probability, even if not sharp. This
+bound can be explicitly computed by plugging in ε =
+1
+√
+3 in Theorem 24. Moreover, in the case of
+tensors in two variables, which correspond to the case of rational normal curves, using the asymptotic
+in Theorem 26 we get an asymptotic expression for this bound.
+1.5. The curvature coefficients of the Veronese variety. The other ingredient needed in Weyl’s
+formula are the curvature properties of the embedding, in particular the Weingarten operator along
+normal directions, which encodes the second fundamental form. Again by the invariance of the extrinsic
+geometry of Vn,d ֒→ SN, it is enough to compute this at a specific point, which we choose to be xd
+0 for
+simplicity of computations.
+Before stating our result, recall that we have denoted by GOE(n) the Gaussian Orthogonal Ensemble,
+i.e. the set Sym(n, R) endowed with the gaussian probability distribution coming from the Frobenius
+scalar product, see Section 2.1 for more details.
+Our main result on the extrinsic geometry of the embedding Vn,d ֒→ SN is the following, and we
+refer to Theorem 22 for a more comprehensive statement.
+
+WHAT IS THE PROBABILITY THAT A RANDOM SYMMETRIC TENSOR IS CLOSE TO RANK-ONE?
+5
+Theorem C (The normal bundle splitting). Let p ∈ Vn,d and denote by Lη the Weingarten operator of
+Vn,d ֒→ SN along a normal direction η ∈ NpVn,d. There exists an orthogonal decomposition NpVn,d =
+W ⊕ P such that the following statements hold:
+(1) Lη = 0 for every η ∈ P;
+(2) W with its induced Bombieri–Weyl metric is isometric to Sym(n) with the Frobenius one.
+Moreover, if we pick η ∈ W Gaussian, then Lη ∼
+√
+2
+�
+d−1
+d
+� 1
+2
+GOE(n).
+This theorem could find applications going beyond the scope of this paper. It gives a full description
+of the second fundamental form in terms of GOE(n) matrices. Using this description in Weyl’s tube
+formula to compute the curvature coefficients of the spherical Veronese, the consequence is that the
+computation of some integrals on the normal bundle depending on the Weingarten operator boils
+down to computing the expectation of a determinant involving GOE(n) matrices. This is reduced to
+an easy, purely combinatorial computation (see Appendix C) and thus we obtain the following explicit
+expressions for the curvature coefficients.
+Theorem D (The curvature coefficients of the spherical Veronese). The curvature coefficients of the
+Veronese variety Vn,d ֒→ SN are as follows:
+KN−n+j(Vn,d) = (−1)
+j
+2 d
+n
+2
+�d − 1
+d
+� j
+2
+2n+2−jπ
+N
+2 Γ
+� n
+2 + 1
+�
+Γ
+� j
+2 + 1
+�
+Γ(n + 1 − j)Γ
+�
+N+j−n
+2
+�
+for 0 ≤ j ≤ n and j even, and KN−n+j(Vn,d) = 0 otherwise.
+We remark that similar results hold true for the projective Veronese variety, using the double covering
+SN −→ RPN, and for the spherical Veronese surface.
+Plugging these coefficients back in Weyl’s tube formula we also obtain the explicit expression of the
+volume of the tubular neighbourhood for radii smaller than the reach (see Theorem 24), in particular
+giving an answer to question stated at the beginning of the paper.
+Theorem E (The probability of being close to rank–one). Let p be a random Bombieri–Weyl symmet-
+ric tensor of order d on Rn+1 and Vn,d ⊂ R[x0, . . . , xn](d) ≃ RN+1 be the Veronese variety of rank–one
+tensors. For every δ such that 0 ≤ arcsin(δ) <
+1
+√
+3 +
+1
+3d
+√
+3 + O
+� 1
+d2
+�
+we have
+P
+�
+distBW(p, Vn,d) ≤ δ∥p∥BW
+�
+=
+�
+0≤j≤n
+j even
+(−1)
+j
+2 d
+n
+2
+�d − 1
+d
+� j
+2
+2n−j+1π− 1
+2
+·
+Γ
+� n
+2 + 1
+�
+Γ
+� N+1
+2
+�
+Γ
+� j
+2 + 1
+�
+Γ(n + 1 − j)Γ
+�
+N+j−n
+2
+�
+�
+δ
+√
+1−δ2
+0
+tN−n+j−1
+(1 + t2)
+N+1
+2
+dt.
+We remark that the above expression, even if unpleasant, gives an exact formula for our probability.
+In the last section, we present an asymptotic expression, based on Laplace’s method, for such a
+probability in the case of rational normal curves, corresponding to the case of tensors in two variables,
+when the degree goes to infinity (see Theorem 26). We stress that it is possible to obtain a meaningful
+asymptotic since the reach is uniformly bounded below. We also remark that the decay is exponential
+in d, as one might expect looking at the codimension of V1,d ֒→ Sd, which is d − 1. More generally, the
+probability has an exponential decay in the codimension of Vn,d ֒→ SN for any n, d.
+2. Preliminaries
+2.1. The Gaussian Orthogonal Ensemble and the Bombieri–Weyl distribution. In this sec-
+tion, we point out the correspondence between the Gaussian Orthogonal Ensemble on the space of
+symmetric matrices and the Bombieri–Weyl distribution on the space of homogeneous polynomials.
+
+6
+ALBERTO CAZZANIGA, ANTONIO LERARIO, ANDREA ROSANA
+Let Sym(n, R) be the space of symmetric n × n matrices with real entries and denote by Eij the
+elementary matrix having all entries 0 except for the ij-th one being 1. Consider a random matrix
+Q =
+n
+�
+i=1
+ηiiEii +
+n
+�
+i<j=1
+ηij
+Eij + Eji
+√
+2
+,
+where ηij ∼ N(0, 1) are i.i.d. standard Gaussian variables. Then Q has random Gaussian entries
+distributed as N(0, 1) on the diagonal and N(0, 1
+2) off-diagonal, independent except for the obvious
+symmetry condition. The probability distribution on Sym(n, R) induced by such matrices is called
+the Gaussian Orthogonal Ensemble and it is denoted by GOE(n).
+This is the standard Gaussian
+probability distribution associated to the Frobenius scalar product given by ⟨A, B⟩ = tr(AB) and
+therefore for every open set U ⊂ Sym(n, R)
+P{Q ∈ U} =
+1
+(2π)
+n(n+1)
+4
+�
+U
+e− 1
+2 tr(A2)dA.
+(2)
+Recall that the orthogonal group O(n) acts on Sym(n, R) by congruence. We denote this action by
+ψ : O(n) −→ GL(Sym(n, R)): for every R ∈ O(n) and A ∈ Sym(n, R), ψ(R)(A) = RT AR, where RT
+denotes the transpose of R.
+Let C[x1, . . . , xn](d) be the space of complex homogeneous polynomials of degree d in n variables.
+Denote by ρ : U(n) −→ GL(C[x1, . . . , xn](d)) the action of the unitary group by change of variables, i.e.
+for every R ∈ U(n) and p ∈ C[x1, . . . , xn](d) set ρ(R)(p) := p ◦ R−1. It is known that ρ is irreducible
+(see [IN66]) and therefore by Schur’s lemma and the compactness of U(n) it follows that there exists
+a unique (up to multiples) hermitian structure on C[x1, . . . , xn](d) which is ρ-invariant, and which is
+given by (1). Up to scaling, this is the hermitian structure having the set
+��d
+α
+� 1
+2
+xα
+�
+α=(α1,...,αn)∈Zn
+≥0, α1+···+αn=d
+(3)
+as an orthonormal basis, where
+�d
+α
+�
+=
+d!
+α1!...αn! and xα = xα1
+1 . . . xαn
+n . We call this the Bombieri–Weyl
+or Kostlan hermitian structure on C[x1, . . . , xn](d).
+Restricting to real homogeneous polynomials R[x1, . . . , xn](d), we define an inner product, which
+we call again Bombieri–Weyl. Notice that (3) is also a real orthonormal basis since the polynomials in
+(3) have real coefficients. We also restrict the action ρ to an action of the orthogonal group O(n) on
+R[x1, . . . , xn](d) and the inner product we introduced is invariant for this restricted action. Remark
+that this restricted action is not irreducible anymore (it is a computation to show that the subspace of
+harmonic polynomials in R[x1, . . . , xn](d) is a non-trivial invariant subspace) and the Bombieri–Weyl
+inner product is not the unique invariant one. The standard Gaussian probability distribution on
+R[x1, . . . , xn](d) associated to the Bombieri–Weyl inner product is the one induced by the random
+polynomial
+P(x) =
+�
+α∈Zn
+≥0
+α1+···+αn=d
+ξα
+�d
+α
+� 1
+2
+xα,
+where ξα are i.i.d. standard Gaussians N(0, 1). We call it again the Bombieri–Weyl distribution.
+The map
+p : Sym(n, R) −→ R[x1, . . . , xn](2)
+(4)
+Q �−→ pQ = xtQx
+defines an isomorphism of Sym(n, R) with R[x1, . . . , xn](2) and one can check that the orthonormal
+basis for the Frobenius product given by {Eii, Eij+Eji
+2
+}i<j=1,...,n is mapped to the orthonormal basis for
+the Bombieri–Weyl product given by (3). It follows that p defines an isometry of Sym(n, R) endowed
+with the Frobenius inner product with R[x1, . . . , xn](2) endowed with the Bombieri–Weyl one and thus
+we can identify the corresponding standard Gaussian probability distributions. Even more is true: p is
+
+WHAT IS THE PROBABILITY THAT A RANDOM SYMMETRIC TENSOR IS CLOSE TO RANK-ONE?
+7
+an isomorphism between the representations (ψ, Sym(n, R)) and (ρ, R[x1, . . . , xn](2)), i.e. the following
+diagram
+Sym(n, R)
+Sym(n, R)
+R[x1, . . . , xn](2)
+R[x1, . . . , xn](2)
+p
+ψ(R)
+p
+ρ(R)
+commutes for every R ∈ O(n), as one can easily check by a straightforward computation.
+2.2. Tubular neighbourhoods and Weyl’s tube formula. Let M be an isometrically embedded
+n-dimensional submanifold of a Riemannian manifold (M, g), i.e. M is itself a Riemannian manifold
+with the metric induced by (M, g). Denote by TpM, NpM the tangent and normal spaces to M at
+p ∈ M. Denote also by ∇, ∇ the Riemannian connections of M and M respectively. For smooth
+vector fields X, Y on M, consider X,Y their local extensions to smooth vector fields on M. Then we
+have
+∇XY = ∇XY + B(X, Y ),
+where ∇XY is the tangential component to M and B(X, Y ) is the normal one. By the properties of
+the Riemannian connection, at every p ∈ M we can regard B as a symmetric bilinear map B : TpM ×
+TpM −→ NpM and we call this the second fundamental form of M ֒→ M at p ∈ M. Given a normal
+direction η ∈ NpM we define the second fundamental form along η, denoted by Hη : TpM ×TpM −→ R,
+by projecting B along η, i.e. for every v, w ∈ TpM
+Hη(v, w) = g(B(v, w), η).
+To this bilinear form we can associate a selfadjoint operator, called the Weingarten operator along η
+and denoted by Lη : TpM −→ TpM, such that for every v, w ∈ TpM
+g(Lη(v), w) = Hη(v, w) = g(B(v, w), η).
+This operator will play a key role in Weyl’s tube formula.
+Remark 2. Given p ∈ M, consider a basis {e1, . . . , en} of TpM. Set Hη,ij = Hη(ei, ej) and denote
+by Lη = (Lη,ij) the matrix representing the Weingarten operator with respect to the given basis and
+g. It is clear from the definitions that
+Hη,ij = Hη(ei, ej) = g(Lη(ei), ej) = Lη,ij.
+Therefore, computing the matrix representing the Weingarten operator with respect to a given basis
+is equivalent to the computation of the second fundamental form on the elements of that basis.
+Remark 3. Let N be a n-dimensional submanifold of a euclidean space with the induced metric and
+ϕ : Rn −→ N be a parametrization of N around p ∈ N with coordinates a1, . . . , an. A basis for TpN
+is given by the vectors
+∂ϕ
+∂ai
+:= dϕ−1(p)ϕ
+� ∂
+∂ai
+�
+.
+Since the Christoffel symbols of the Riemannian connection of the euclidean space are all null, it follows
+that to compute the second fundamental form of N at p along a normal direction η it is enough to
+compute the second derivatives of the parametrization
+Hη
+� ∂ϕ
+∂ai
+, ∂ϕ
+∂aj
+�
+= ⟨∇ ∂ϕ
+∂ai
+∂ϕ
+∂aj
+, η⟩ = ⟨ ∂2ϕ
+∂ai∂aj
+, η⟩,
+where ⟨ , ⟩ is the euclidean inner product. Let now N be an isometrically embedded submanifold of
+the sphere Sl, where Sl is given the round metric inherited by Rl, and let η be a normal vector to
+N at p ∈ N. Remark that we can also consider N as an isometrically embedded submanifold of Rl.
+Then the second fundamental form of N along η is the same whether we consider N as a submanifold
+of Sl or of Rl. This means that above formula holds also for submanifolds of round spheres.
+
+8
+ALBERTO CAZZANIGA, ANTONIO LERARIO, ANDREA ROSANA
+Given M, M as above, we can define the ε-small normal bundle N εM as the subset of the normal
+bundle NM consisting of vectors with norm less than ε > 0. We also call normal exponential map the
+restriction of the exponential map of M to NM.
+Definition 4. If there exists an ε > 0 such that
+exp |N εM : N εM −→ M
+is a diffeomorphism on its image, where exp denotes the exponential map of M, we call N εM a tubular
+neighbourhood of M in M.
+Recall the definition of distance of a point x ∈ M from the submanifold M, given by dg(x, M) :=
+inf{dg(x, y) | y ∈ M} where dg(x, y) is the Riemannian distance between x, y.
+We introduce the
+following set
+(5)
+U(M, ε) := {x ∈ M | dg(x, M) < ε},
+consisting of points at distance less than ε > 0 from M. The description of this set for submanifolds of
+R3 is quite easy: the distance of a point from a surface or a curve will always be given by the length of
+a segment starting from the point and meeting the submanifold orthogonally, given that segments are
+geodesics. This situation can be generalized, as the following theorem shows. Even though this result
+is well known, we were not able to find a full explicit reference for this general setting. We, therefore,
+provide a full proof in Appendix A, filling in the details of the outline given in [CdS01, Theorem 6.6].
+Theorem 5 (Tubular neighbourhood theorem). Let M be a compact isometrically embedded
+submanifold of a Riemannian manifold (M, g). Then there exists an ε > 0 small enough such that
+exp |N εM : N εM −→ M is a diffeomorphism on its image and exp(N εM) = U(M, ε).
+Theorem 5 can be seen as an existence result for tubular neighbourhoods of compact submanifolds and
+as a characterization of the set U(M, ε) for ε small enough. For this reason, in the following, we will
+refer also to U(M, ε) as a tubular neighbourhood.
+Remark 6. The compactness assumption in theorem (5) is crucial. If we remove compactness, we
+can only prove the existence of a smooth function ε(·) : M −→ R>0 such that the restriction of the
+exponential map to N ε(·)M is an embedding, where N ε(·)M = {v ∈ NxM | x ∈ M, ∥v∥ < ε(x)}, i.e.
+the ε is not uniform anymore but it depends on the point.
+Definition 7. Let M be an isometrically embedded submanifold of a Riemannian manifold (M, g).
+We define the reach of M ֒→ M as
+ρ(M) = sup{ε ≥ 0 | N εM is a tubular neighbourhood of M}.
+We can restate theorem (5) by saying that the reach of a compact submanifold is always positive.
+Remark that even if for brevity we write ρ(M), the reach is not an intrinsic property of M but it
+depends on the way M is embedded into M. From the very definition it follows that ρ(M) can be
+expressed as the minimum between ρ1(M) = sup{ε ≥ 0 | exp |N εM is an immersion} and ρ2(M) =
+sup{ε ≥ 0 | exp |N εM is injective}.
+The points where the differential of the normal exponential map is not injective are called focal points.
+Their existence is linked to the presence of particular Jacobi fields, see [dC92, Section 10.4].
+For
+compact submanifolds of round spheres, one obtains the following expression
+(ρ1(M))−1 = sup
+x∈M
+sup{∥
+..γ(0)∥ s.t. γ : (−δ, δ) −→ M
+(6)
+arclength geodesic in M with γ(0) = x}.
+If ε > 0 is the first value for which we lose injectivity of the restriction of the normal exponential, it
+means that there is a point in M that is reached by two length–ε normal geodesics starting from two
+different points of M. Using the Generalized Gauss Lemma for geodesic variations, see [Gra04, Lemma
+2.11], and uniqueness of geodesics, one can show that the union of these two normal geodesics is again
+
+WHAT IS THE PROBABILITY THAT A RANDOM SYMMETRIC TENSOR IS CLOSE TO RANK-ONE?
+9
+a geodesic meeting M orthogonally at its endpoints. Therefore we have the following expression for
+ρ2(M)
+ρ2(M) = 1
+2 inf{l(γ) | γ : [a, b] −→ M geodesic s.t. γ(a), γ(b) ∈ M,
+(7)
+˙γ(a) ∈ Nγ(a)M, ˙γ(b) ∈ Nγ(b)M}.
+We will apply these expressions in chapter 4 to compute the reach of the Veronese variety.
+In [Wey39] Weyl presented a fundamental work, answering a question posed by Harold Hotelling:
+how can we compute the volume of a “tube”, i.e. a tubular neighbourhood, of fixed radius around a
+closed n–dimensional manifold in RN or SN? Hotelling himself answered the case of curves, both in
+the euclidean and in the spherical setting. Weyl extended these results to any dimension n as follows.
+Theorem 8 (Weyl’s tube formula). Let M be a smooth, n–dimensional, compact submanifold
+isometrically embedded in RN (or SN) with their standard metrics. Then, for ε < ρ(M), the following
+formula holds:
+Vol
+�
+U(M, ε)
+�
+=
+�
+0≤e≤n, e even
+Ks+e(M)JN,s+e(ε),
+(8)
+where s := N − n is the codimension of M and JN,s+e are linearly independent functions of ε only. In
+the euclidean case, the universal functions J have the following form:
+JN,k(ε) := εk,
+(9)
+while in the spherical one, they are given by
+JN,k(ε) :=
+� ε
+0
+(sin ρ)k−1(cos ρ)N−k dρ =
+� tan ε
+0
+tk−1
+(1 + t2)
+N+1
+2
+dt.
+(10)
+Moreover, the coefficients Kj(M) are isometric invariants of M.
+The coefficients Kj(M) are called the curvature coefficients of M. The motivation for this comes
+from the fact that they are integrals of functions on the second fundamental form of M.
+Notice
+that in [Wey39] Weyl uses different normalization constants for (9) and (10). We chose to follow the
+normalization introduced by Nijenhuis in [Nij74], which proves to be handier for applications, see for
+instance [Bue06]. Remark that the ε’s for which the formula holds depend on the reach and therefore on
+the embedding, while isometric embeddings will give the same curvature coefficients. The dependence
+of the validity of the formula on the reach is due to the proof relying on parametrizing the tubular
+neighbourhood through the normal exponential map. In [How93] Howard contextualizes Weyl’s result
+in the framework of “Integral Geometry”, where he considers more general integrals of polynomials on
+the second fundamental forms of submanifolds of homogeneous spaces.
+There are explicit integral versions of formula (8) for both the euclidean and spherical cases. For the
+latter, with the same notations above, this reads as
+Vol
+�
+U(M, ε)
+�
+=
+�
+p∈M
+� tan ε
+t=0
+�
+S(NpM)
+tm−1 det
+�
+In − tLη
+�
+(1 + t2)
+N+1
+2
+volM dη dt,
+(11)
+where S(NpM) denotes the unit sphere in NpM, In is the n × n identity matrix, Lη is the Weingarten
+operator of M ֒→ M along the unit normal vector η, and dη is a short notation for the volume form
+on S(NpM). If one explicitly develops the determinant, it is easy to get back formula (8).
+2.3. A lemma on integration on spheres. Consider a sphere Sm for some m ≥ 2 and fix k ∈
+{1, . . ., m − 1}. Denote by ι : Sk ֒→ Rk+1 the inclusion map and consider the map
+Sk ×
+◦
+Dm−k
+ϕ
+−→ Sm ⊂ Rm+1 = Rk+1 × Rm−k
+(12)
+(σ, z)
+�−→
+(
+�
+1 − |z|2 ι(σ), z),
+giving a smooth parametrization of Sm \
+�
+{0} × Sm−k−1�
+⊂ Rk+1 × Rm−k. For every l ∈ N, consider
+Rl endowed with a non–degenerate scalar product and coordinate functions x1, . . . , xl with respect to
+an orthonormal basis. Then we have the standard volume form volRl = dx1 ∧ · · · ∧ dxl which induces
+
+10
+ALBERTO CAZZANIGA, ANTONIO LERARIO, ANDREA ROSANA
+a volume form vol ◦
+Dl on the open norm–1 disc
+◦
+Dl and through the pullback of the inclusion also a
+volume form volSl−1 on Sl−1. Notice that with respect to volSm, the part of Sm not parametrized by
+ϕ in (12) has measure 0. We have the following result about integration on spheres, see Appendix B
+for a proof.
+Lemma 9. With the same notations above, the pullback of volSm through ϕ is given by
+ϕ∗(volSm) =
+�
+1 − |z|2� k−1
+2
+volSk ∧ vol ◦
+Dm−k.
+In particular, this implies that
+�
+Sm f(p) volSm =
+�
+Sk
+�
+◦
+Dm−k f
+��
+1 − |z|2 ι(σ), z
+��
+1 − |z|2� k−1
+2
+volSk ∧ vol ◦
+Dm−k,
+(13)
+for any measurable function f on Sm.
+2.4. Laplace’s method. One of the most important asymptotic methods for computing integrals
+depending on one large parameter is the so-called “Laplace’s method”. For a proof of this result and
+more details on asymptotic methods for integrals, we refer to [Won01].
+Theorem 10 (Laplace’s method). Consider the following integral depending on one parameter
+λ > 0:
+I(λ) :=
+� t2
+t1
+e−λa(t)b(t) dt,
+where a, b : [t1, t2] −→ R are functions satisfying the following conditions:
+(1) a is smooth in a neighbourhood of t1, and there exist µ > 0 and a0 ̸= 0 such that, for t −→ t1,
+we have:
+a(t) = a(t1) + a0(t − t1)µ + O
+�
+|t − t1|µ+1�
+;
+(2) b is smooth in a neighbourhood of t1, and there exist ν ≥ 1 and b0 ̸= 0 such that, for t −→ t1,
+we have:
+b(t) = b0(t − t1)ν−1 + O
+�
+|t − t1|ν�
+;
+(3) t1 is a global minimum for a on [t1, t2], i.e. a(t) > a(t1) for any t ∈]t1, t2[. Moreover for all
+ε > 0 we have:
+inf
+t∈[t1+ε,t2[{a(t) − a(t1)} > 0;
+(4) the integral I(λ) converges absolutely for sufficiently large λ.
+Then, as λ −→ +∞, we have:
+I(λ) = e−λa(t1) ·
+Γ
+� ν
+µ
+�
+λ
+ν
+µ
+·
+b0
+µ a
+ν
+µ
+0
+·
+�
+1 + O
+�
+λ− 1+ν
+µ
+��
+.
+Remark that if the minimum of a(t) is attained at the extremum t2, the theorem holds with the roles
+of t1 and t2 reversed. The idea behind the statement is that the major contribution to I(λ) will be
+given by the behaviour of the integrand around the minimum point of a, which can be assumed to
+be one of the endpoints of the integration domain. It can be proven that some of the smoothness
+hypotheses can be relaxed, even if some regularity is still needed. The statement of Theorem 10 is
+the standard form for the Laplace’s method. For statements with weaker regularity assumptions and
+generalizations to larger classes of integrals, we refer to [Olv97] and [Nem20].
+
+WHAT IS THE PROBABILITY THAT A RANDOM SYMMETRIC TENSOR IS CLOSE TO RANK-ONE?
+11
+3. The Veronese variety
+Consider the space of homogeneous polynomials of degree d in n + 1 variables R[x0, . . . , xn](d) ∼=
+RN+1, where N :=
+�n+d
+d
+�
+− 1, with the basis described in (3).
+Definition 11. For n ≥ 1 and d ≥ 1, the real Bombieri–Weyl Veronese embedding is the map
+νn,d : RPn −→
+RPN
+[a]
+�−→
+��d
+α
+�1/2
+aα
+�
+and it is the Veronese projective embedding associated to the Bombieri–Weyl basis. The Bombieri–
+Weyl Veronese variety is the image of this embedding, denoted by PVn,d := im(νn,d).
+The main object we will consider in what follows is the spherical counterpart of PVn,d.
+Definition 12. The spherical (Bombieri–Weyl) Veronese map is the map
+�νn,d : Sn −→ SN
+a
+�−→
+��d
+α
+�1/2
+aα
+�
+.
+The spherical (Bombieri–Weyl) Veronese surface is the image of this map, denoted by Σn,d := im(�νn,d).
+It is worth stressing in the definition of �νn,d that Sn is the sphere with respect to the standard
+euclidean product in Rn+1 while SN is the sphere with respect to the Bombieri–Weyl product in
+R[x0, . . . , xn](d) and that �νn,d is well defined, as one can check by an explicit computation.
+The objects we just introduced have a particularly useful description. Recall that to each b =
+(b0, . . . , bn) ∈ Rn+1 we can associate the linear form on Rn+1 given by lb(x0, . . . , xn) = b0x0+· · ·+bnxn.
+It is known that PVn,d parametrizes projective classes of d−th powers of linear forms
+PVn,d =
+�
+[d–th powers of linear forms on Rn+1]
+�
+=
+�
+[aα] ∈ RPN | ∃ b = (b0, . . . , bn) ∈ Rn+1 s.t. aα0,...,αn =
+�d
+α
+�1/2
+bα0
+0 . . . bαn
+n
+�
+as one can prove by showing that νn,d([b0, . . . , bn]) = [(b0x0 + · · ·+ bnxn)d]. This also leads to the well-
+known description of the Veronese variety PVn,d as the variety of symmetric decomposable d−tensors
+on Rn+1 and is one of the main reasons Veronese varieties have been so intensively studied. A similar
+description holds for the spherical Veronese surface
+Σn,d = {d–th powers of norm–1 linear forms on Rn+1}
+= { (aα) ∈ SN | ∃ b = (b0, . . . , bn) ∈ Sn s.t. aα0,...,αn =
+�d
+α
+�1/2
+bα0
+0 . . . bαn
+n }.
+Using these descriptions of PVn,d and Σn,d, it is immediate to prove the following.
+Proposition 13. PVn,d is an orbit for the action of the orthogonal group O(n + 1) on RPN =
+P(R[x0, . . . , xn](d)) by change of variables.
+Similarly Σn,d is an orbit for the same action of the
+orthogonal group O(n + 1) on SN = S(R[x0, . . . , xn](d)).
+Recall the two-fold covering map πN : SN −→ RPN given by the identification of antipodal points.
+Its restriction to the spherical Veronese Vn,d := Vn,d ∩ SN gives a covering map �πn,d : Vn,d −→ PVn,d
+whose degree depends on the parity of d: if d is odd �πn,d is a 2 : 1 covering, while if d is even it is 1 : 1,
+since in this case for b ∈ Sn we have �νn,d(b) = �νn,d(−b).
+We now turn to metric properties of Veronese manifolds. Consider on Rn+1 the standard euclidean
+metric and on RN+1 = R[x0, . . . , xn](d) the Bombieri–Weyl one. The metrics induced on Sn and SN
+respectively are invariant under the antipodal map and therefore induce metrics on the corresponding
+projective spaces RPn and RPN. We denote the metrics on the spheres by gSn and gSN and those on
+
+12
+ALBERTO CAZZANIGA, ANTONIO LERARIO, ANDREA ROSANA
+the projective spaces by gRPn and gRPN . Since the covering maps πn, πN are Riemannian coverings
+with these metrics, any relation between gSn and gSN will also hold between gRPn and gRPN and
+viceversa. Through a direct computation, one can prove the following result.
+Proposition 14. Pulling back the Bombieri–Weyl metric through the Veronese embedding νn,d :
+RPn −→ RPN, for any n ≥ 1, d ≥ 1 we have
+ν∗
+n,d gRPN =
+√
+d gRPn.
+(14)
+Corollary 15. For every n, d ∈ N and any smooth n–dimensional submanifold C ֒→ Σn,d we have
+VolBW
+n
+(C) =
+
+
+
+
+
+1
+2d
+n
+2 Voln
+�
+�ν−1
+n,d(C)
+�
+for d even
+d
+n
+2 Voln
+�
+�ν−1
+n,d(C)
+�
+for d odd
+,
+(15)
+where Voln is the n–dimensional volume with respect to gSn and VolBW
+n
+is the n–dimensional volume
+with respect to the metric induced by gSN on Σn,d. In particular
+VolBW
+n
+(Σn,d) =
+
+
+
+
+
+
+
+
+
+d
+n
+2
+π
+n+1
+2
+Γ( n+1
+2
+)
+for d even
+2d
+n
+2
+π
+n+1
+2
+Γ( n+1
+2
+)
+for d odd
+.
+(16)
+In section 5.2 we will need the explicit expression of VolBW
+n
+(Vn,d). This easily follows from formula
+(16), since Vn,d = Σn,d ∪ −Σn,d where Σn,d = −Σn,d for d odd and Σn,d ∩ (−Σn,d) = ∅ for d even.
+Therefore
+VolBW
+n
+(Vn,d) = 2d
+n
+2
+π
+n+1
+2
+Γ( n+1
+2 ) = d
+n
+2 Vol(Sn).
+(17)
+Remark 16. For every orthogonal matrix R ∈ O(n + 1) we have the following commutative diagram
+(RPn, gRPn)
+(RPn, gRPn)
+PVn,d
+PVn,d
+νn,d
+R
+νn,d
+ρ(R)|PVn,d
+,
+(18)
+since PVn,d is an orbit for the action ρ and is therefore preserved under ρ(R).
+Moreover, by the
+invariance of the Bombieri–Weyl scalar product, ρ(R) is an isometry of (RPN, gRPN ), and its restriction
+to PVn,d defines an isometry of PVn,d. Therefore PVn,d is an orbit for an isometric action of O(n+1) over
+R[x0, . . . , xn](d) and these isometries of PVn,d are induced by isometries of the ambient space. The same
+property also holds for Σn,d considering the action on the sphere SN. This simple observation, which
+is essentially due to PVn,d and Σn,d being orbits, will allow us to drastically simplify the computations
+we will carry out in Chapter 4 and Section 5.1.
+4. The reach of the spherical Veronese variety
+In this chapter we provide an explicit computation for the reach of Σn,d ֒→ SN. The interest in this
+quantity relies on the fact it provides a lower bound for the ε’s of validity for Weyl’s tube formula, as
+theorem 8 shows. Remark that, since Σn,d is compact, by theorem 5 we have ρ(Σn,d) > 0.
+Recall that ρ(Σn,d) = min{ρ1(Σn,d), ρ2(Σn,d)} and the expressions (6) and (7). Using remark 16
+we can prove the following.
+
+WHAT IS THE PROBABILITY THAT A RANDOM SYMMETRIC TENSOR IS CLOSE TO RANK-ONE?
+13
+Lemma 17. The formula in (6) simplifies to
+�
+ρ1(Σn,d)
+�−1 = sup{ ∥
+..γ(0)∥ | γ : (−δ, δ) −→ Σn,d arclength geodesic in Σn,d, γ(0) = xd
+0 },
+(19)
+that is to say the inner supremum in (6) does not depend on x ∈ Σn,d. Moreover (19) does not depend
+on the direction of
+.γ(0). Similarly, the formula in (7) simplifies to
+ρ2(Σn,d) = 1
+2 inf{l(γ)
+�� γ : [a, b] −→ SN geodesic s.t. γ(a) = xd
+0, γ(b) ∈ Σn,d,
+(20)
+˙γ(a) ∈ Nxd
+0Σn,d, ˙γ(b) ∈ Nγ(b)Σn,d}.
+Proof. Consider an arclength geodesic γ : (−δ, δ) −→ Σn,d with γ(0) = p1 and pick another point
+p2 ∈ Σn,d. By remark 16 there exists R ∈ O(n + 1) such that ρ(R)p1 = p2. Recall that the image of
+a geodesic through an isometry is still a geodesic, hence ˜γ := ρ(R)(γ) is an arclength geodesic with
+˜γ(0) = ρ(R)(γ(0)) = ρ(R)p1 = p2. Since ρ(R) is also an isometry of the ambient space SN, we have
+∥
+..γ(0)∥ = ∥
+..
+˜γ(0)∥. We just proved that given any two points in Σn,d, using the isometries ρ(R) for
+R ∈ O(n + 1) we can transport any arclength geodesic passing through the first point into another
+arclength geodesic passing through the second point, preserving the norm of second derivatives. It
+follows that the expression in (6) is independent of the specific point x ∈ Σn,d. Now observe that given
+any arclength geodesic γ with γ(0) = xd
+0 and
+.γ(0) = v, we can change the direction of
+.γ(0) through
+ρ(R) for some R ∈ O(n + 1) with xd
+0 a fixed point (it is sufficient to choose a rotation R such that
+(1, 0, . . . , 0) ∈ Sn is in the axis of rotation), obtaining any other possible direction in Txd
+0Σn,d without
+changing ∥
+··γ(0)∥, for the same reason as above. It follows that (19) does not depend on the specific
+direction of
+.γ(0). Since isometries preserve orthogonality and lengths, the second part of the lemma
+also follows in a similar way.
+□
+The choice of xd
+0 in formulae (19) and (20) is motivated by convenience for computations only and we
+could have chosen any other point.
+The first step to compute (19) and (20) for Σn,d ֒→ SN is to understand tangent and normal spaces.
+Lemma 18. For p ∈ Σn,d with p = ld, where l is a norm-1 linear form, we have
+TpΣn,d =
+�
+{ld−1λ | λ is a linear form orthogonal to l}
+�
+.
+Proof. Write l(x) = a0x0 +· · ·+anxn and set a = (a0, . . . , an) ∈ Sn. Recalling that Σn,d = im(�νn,d), a
+curve on Σn,d can be expressed as the image of a curve on Sn through �νn,d. Consider b = (b0, . . . , bn) ∈
+Sn such that ⟨a, b⟩ = 0. Then γ(t) = (cos t(a0x0 + · · · + anxn) + sin t(b0x0 + · · · + bnxn))d is a curve in
+Σn,d with γ(0) = p. Remark that the orthogonality condition is needed to ensure we are taking d–th
+power of a norm–1 form. We have
+d
+dtγ(t)
+��
+t=0 = d ld−1(b0x0 + · · · + bnxn),
+therefore
+�
+{ld−1λ | λ is a linear form orthogonal to l}
+�
+⊂ TpΣn,d. By dimension count, equality follows.
+□
+Now we have the ingredients we need to perform the computation of ρ(Σn,d).
+Theorem 19. For the reach of Σn,d ֒→ SN we have
+ρ1(Σn,d) =
+1
+√
+3 +
+1
+3d
+√
+3 + O
+� 1
+d2
+�
+and
+ρ2(Σn,d) = π
+4 .
+Therefore, for d ≥ 2, the reach of Σn,d is given by
+ρ(Σn,d) = min{ρ1(Σn,d), ρ2(Σn,d)} =
+1
+√
+3 +
+1
+3d
+√
+3 + O
+� 1
+d2
+�
+.
+(21)
+
+14
+ALBERTO CAZZANIGA, ANTONIO LERARIO, ANDREA ROSANA
+Proof. We begin with the computation of ρ1(Σn,d).
+By Proposition 14 geodesics in Σn,d can be
+realized as images through �νn,d of geodesics in Sn.
+Moreover, thanks to Lemma 17, it is enough
+to consider geodesics passing through xd
+0 = (1, 0, . . . , 0) = �νn,d((1, 0, . . . , 0)) at time 0 and their
+direction plays no role, hence it is enough to consider the image of the geodesic in Sn given by
+α(t) = x0 cos(td− 1
+2 ) + x1 sin(td− 1
+2 ) with x0 being the point of coordinates (1, 0, . . . , 0) and x1 that
+of coordinates (0, 1, 0, . . ., 0). Explicitly we have α(t) = (cos(td− 1
+2 ), sin(td− 1
+2 ), 0, . . . , 0) and the corre-
+sponding geodesic in Σn,d is given by
+γ(t) := (�νn,d ◦ α)(t) =
+�
+cosd(td− 1
+2 ),
+√
+d cosd−1(td− 1
+2 ) sin(td− 1
+2 ), . . . , sind(td− 1
+2 )
+�
+,
+with γ(0) = xd
+0 = (1, 0, . . . , 0) and ∥ ˙γ(0)∥ = 1. Notice that the only components of γ(t) which are
+not constantly zero are those corresponding to multi–indices (β0, β1, 0, . . . , 0) with β0 + β1 = d. To
+compute ∥
+..γ(0)∥, we need the second derivatives of the non–constantly zero components of γ(t). These
+have the following expression for k = 0, . . . , d:
+�d
+k
+� 1
+2
+cosk(td− 1
+2 ) sind−k(td− 1
+2 ).
+Computing second derivatives and evaluating at t = 0 we find
+..γ(0) = (−1, 0,
+√
+2
+�
+d(d − 1)
+d
+, 0, . . . , 0).
+Using the Taylor-MacLaurin expansion for √1 + x we obtain
+∥
+..γ(0)∥ =
+�
+1 + 2(d − 1)
+d
+=
+√
+3
+�
+1 − 2
+3d =
+√
+3 −
+1
+√
+3d + O
+� 1
+d2
+�
+.
+Applying again a Taylor-MacLaurin expansion for
+1
+1−x, we get the final expression for ρ1(Σn,d):
+ρ1(Σn,d) =
+1
+∥
+..γ(0)∥ =
+1
+√
+3 +
+1
+3
+√
+3d + O
+� 1
+d2
+�
+.
+Remark 20. Notice that, since ρ1(Σn,d) =
+�√
+3
+�
+1 − 2
+3d
+�−1
+, then ρ1(Σn,d) >
+1
+√
+3 for all n, d.
+For ρ2(Σn,d), by Lemma 17 it is enough to consider geodesics γ(θ) in SN starting at xd
+0 at time θ = 0.
+By Lemma 18 and recalling that the normal space at a point p ∈ Σn,d is the orthogonal complement of
+TpΣn,d inside TpSN, we have Nxd
+0Σn,d =
+���d
+α
+� 1
+2 xα0
+0 . . . xαn
+n
+| α0 < d − 1
+��
+. Pick a vector w ∈ Nxd
+0Σn,d
+and let γw(θ) be the geodesic in SN with γw(0) = xd
+0 and ˙γw(0) = w, i.e.
+γw(θ) = xd
+0 cos
+�
+θ∥w∥
+�
++
+w
+∥w∥ sin
+�
+θ∥w∥
+�
+.
+The goal now is to understand when γw meets again Σn,d orthogonally. The first step is to find for
+which b = (b0, . . . , bn) ∈ Sn and θ we at least have a solution to the equation
+(b0x0 + · · · + bnxn)d = xd
+0 cos
+�
+θ∥w∥
+�
++
+w
+∥w∥ sin
+�
+θ∥w∥
+�
+.
+On the right hand side, we expand w as w = �
+α0<d−1 wα
+�d
+α
+� 1
+2 xα, while we expand the left hand
+side as �
+α
+�d
+α
+�
+bαxα with multi–indices α = (α0, . . . , αn) ∈ Zn+1
+≥0
+such that α0 + · · · + αn = d. Hence,
+expanding it further in the Bombieri–Weyl basis, we get the equation
+bd
+0xd
+0 +
+√
+d xd−1
+0
+� n
+�
+i=1
+√
+d bd−1
+0
+bixi
+�
++
+�
+α0<d−1
+�d
+α
+�
+bαxα =
+(22)
+xd
+0 cos
+�
+θ∥w∥
+�
++ sin
+�
+θ∥w∥
+�
+�
+α0<d−1
+�d
+α
+� 1
+2 wα
+∥w∥xα.
+
+WHAT IS THE PROBABILITY THAT A RANDOM SYMMETRIC TENSOR IS CLOSE TO RANK-ONE?
+15
+Equating corresponding coefficients we get
+�
+bd
+0 = cos
+�
+θ∥w∥
+�
+bd−1
+0
+bi = 0
+∀ i = 1, . . . , n .
+Now two cases can occur:
+• if b0 ̸= 0, the second equation above implies that bi = 0 for i = 1, . . . , n, hence b = (b0, 0, . . . , 0).
+Since b ∈ Sn it follows that b0 = ±1. If b0 = 1, then the meeting point is again xd
+0 and γw
+comes back to it for θ =
+2π
+∥w∥, and the same happens if b0 = −1 and d is even. If b0 = −1 and
+d is odd, then the meeting point corresponds to −xd
+0, and the meeting time is θ =
+π
+∥w∥.
+• if b0 = 0 we get cos(θ∥w∥) = 0 and γw may meet Σn,d at (b0x0 + · · · + bnxn)d for θ =
+π
+2∥w∥ or
+θ =
+3π
+2∥w∥.
+Now that we know when the curve intersects again Σn,d, we need to understand when it does that
+orthogonally. Notice that up to now we have used only some of the equations arising from (22): we
+now use the others to impose the orthogonality condition.
+We look at the case b0 = 0 and θ =
+π
+2∥w∥.
+If we could find some b = (0, b1, . . . , bn) ∈ Sn and
+w ∈ Nxd
+0Σn,d such that γw meets Σn,d orthogonally at (b0x0 + · · · + bnxn)d for θ =
+π
+2∥w∥, then by the
+previous computation no other curve satisfying the conditions in (20) could have length less than this
+one. Fix the following notations ˜b := (b1, . . . , bn), ˜x := (x1, . . . , xn) and ˜α := (α1, . . . , αn). We have
+∥˜b∥ = 1 and for θ =
+π
+2∥w∥ equation (22) becomes
+�
+α1+···+αn=d
+�d
+˜α
+�
+˜b˜α˜x˜α =
+�
+α0<d−1
+�d
+α
+� 1
+2 wα
+∥w∥xα,
+(23)
+Equating corresponding coefficients, we get wα = 0 for each α = (α0, . . . , αn) such that α0 ̸= 0, while
+for the other multi–indices we get
+�d
+˜α
+� 1
+2˜b˜α =
+w˜
+α
+∥w∥. These equations admit a solution and therefore a
+curve γw with the properties described above exists.
+Since we are assuming that γw
+�
+π
+2∥w∥
+�
+= (b1x1 + · · · + bnxn)d, by Lemma 18 we have Tγw
+�
+π
+2∥w∥
+�Σn,d =
+�˜bd−1λ | λ is a linear form orthogonal to ˜b
+�
+. The tangent vector to the curve γw at the meeting point
+is given by
+˙γw
+�
+π
+2∥w∥
+�
+= −xd
+0∥w∥.
+Hence, since the monomials of the Bombieri–Weyl basis are mutually orthogonal and xd
+0 is not among
+those spanning the tangent space at the meeting point, γw comes back to Σn,d orthogonally for any
+choice of w ∈ Nxd
+0Σn,d and b ∈ Sn satisfying the conditions given by (23). Moreover, the length of
+such a curve is independent of w and always equal to π
+2 . Hence we get that ρ2(Σn,d) = π
+4 .
+Noticing that
+1
+√
+3 +
+1
+3d
+√
+3 < π
+4 as soon as d ≥ 2, we conclude that
+ρ(Σn,d) = ρ1(Σn,d) =
+1
+√
+3 +
+1
+3d
+√
+3 + O
+� 1
+d2
+�
+.
+□
+Remark that the same results hold true for Vn,d ֒→ SN. Before moving on, let us comment on the
+meaning of the quantity ρ2(Vn,d) in our context. Recall once again our interpretation of Vn,d ֒→ SN
+as the set of symmetric rank–one tensors among norm–1 ones. Since Vn,d is compact and in particular
+closed, for every point in SN there will be a point in Vn,d minimizing the distance between the chosen
+point and Vn,d. The point realizing the minimum need not be unique and indeed in general it is not.
+From the tensor point of view, fixed p every distance minimizing point in Vn,d provides a best rank–
+one approximation of the tensor represented by p. Since SN is compact, it is geodesically complete,
+and therefore there exists distance minimizing geodesics joining p with each of the points minimizing
+the distance and each of the geodesics meet Vn,d orthogonally. From this observation we obtain that
+
+16
+ALBERTO CAZZANIGA, ANTONIO LERARIO, ANDREA ROSANA
+a symmetric norm–1 tensor admits more than one best rank–one approximation if and only if the
+corresponding point in SN admits more than one distance minimizing geodesic orthogonal to Vn,d.
+It follows that the injectivity of the normal exponential map on N εVn,d ensures that every tensor
+represented by a point in the image admits a unique best rank–one approximation, since it will be
+joined to Vn,d by a unique distance minimizing orthogonal geodesic. Given this, we can restate the
+result of Theorem 19 about ρ2(Vn,d) in the following way.
+Corollary 21. Every symmetric tensor p at distance less than
+√
+2
+2 ∥p∥BW from rank–one admits a
+unique best rank–one approximation.
+A consequence of this result is that the probability that a symmetric tensor admits a unique best
+rank–one approximation is bounded below by the normalized volume of the tubular neighbourhood of
+radius π
+4 on the sphere, that is to say,
+P
+�
+symmetric norm–1 tensor s.t. ∃! best rank–one approximation
+�
+≥ Vol
+�
+U(Vn,d, π
+4 )
+�
+Vol(SN)
+.
+(24)
+However, Theorem 19 tells us that we cannot use Weyl’s tube formula to compute the volume of such
+a neighbourhood, since π
+4 is greater than the reach ρ(Σn,d) = ρ(Vn,d). Nevertheless, for every n, d
+we can still use the exact formula we will prove in the next chapter with radius
+1
+√
+3 (see Remark 20),
+giving an explicit lower bound for the probability in (24), even if this will not be sharp.
+In Section 5.2 we will apply (11) to compute an exact formula for the volume of a tubular ε–
+neighbourhood Vol(U(Σn,d, ε)). We stress again that the reach computed in Theorem 19 gives a lower
+bound to the ε’s of validity of the formula we will find: we are guaranteed that it gives the correct
+result for any ε < ρ(Σn,d).
+5. The volume of the tubular neighbourhood
+5.1. The second fundamental form of the spherical Veronese surface. By Remark 16, since
+we have an isometric transitive action of O(n + 1) on Σn,d by restrictions of isometries of SN, the
+extrinsic geometry of Σn,d ֒→ SN is invariant under this action. Therefore, if we compute the second
+fundamental form at a specific point of Σn,d we automatically know it at every point. We will now
+carry out the computation using the point xd
+0 ∈ Σn,d for simplicity.
+By Remark 3 to compute the second fundamental form of Σn,d at xd
+0 along a normal direction η it
+is enough to choose a local parametrization around xd
+0, compute its second derivatives and take their
+(Bombieri–Weyl) scalar product in RN+1 with η. This way we will obtain the matrix representing the
+Weingarten operator at xd
+0 with respect to the basis of Txd
+0Σn,d given by the derivatives of the chosen
+parametrization.
+Consider the projection on Sn from the tangent plane at (1, 0 . . . , 0), giving a parametrization of the
+upper hemisphere. Composing it with the Veronese map �νn,d we obtain a parametrization ϕn,d of the
+part of Σn,d contained in the upper hemisphere of SN, explicitly given by
+ϕn.d : Rn −→ U ⊂ Σn,d
+a = (a1, . . . , an) �−→
+�x0 + a1x1 + · · · + anxn
+(1 + ∥a∥2)
+1
+2
+�d
+.
+Since ϕ−1
+n,d(xd
+0) = (0, . . . , 0), we have to compute the first and second derivatives of ϕn,d at the origin.
+We obtain the following expressions
+∂ϕn,d
+∂ai
+(a)
+����
+a=0
+= dxd−1
+0
+xi,
+(25)
+∂2ϕn,d
+∂ai∂aj
+(a)
+����
+a=0
+= −δij(dxd
+0) + d(d − 1)xd−2
+0
+xixj.
+(26)
+
+WHAT IS THE PROBABILITY THAT A RANDOM SYMMETRIC TENSOR IS CLOSE TO RANK-ONE?
+17
+For i = 1, . . . , n denote by ei =
+√
+dxd−1
+0
+xi the orthonormal vectors in the Bombieri–Weyl basis with
+power d − 1 on x0. By (25) the basis of Txd
+0Σn,d given by the first derivatives of the parametrization is
+{
+√
+dei | i = 1, . . . , n}. Instead of using this basis, we compute the matrix representing the Weingarten
+operator with respect to the orthonormal basis {ei}i=1,...,n along a normal direction η ∈ Nxd
+0Σn,d.
+Denoting by Lη = (Lη,ij)i,j=1,...,n this matrix, by Remark 2 we have
+Lη,ij = Hη(ei, ej) = 1
+d Hη
+�√
+dei,
+√
+dej
+�
+= 1
+d Hη
+�∂ϕn,d
+∂ai
+(a)
+����
+a=0
+, ∂ϕn,d
+∂aj
+(a)
+����
+a=0
+�
+=
+(27)
+= 1
+d
+� ∂2ϕn,d
+∂ai∂aj
+(a)
+����
+a=0
+, η
+�
+RN+1 = 1
+d
+�
+− δij(dxd
+0) + d(d − 1)xd−2
+0
+xixj, η
+�
+RN+1.
+By Lemma 18 we have Nxd
+0Σn,d =
+���d
+α
+� 1
+2 xα0
+0 . . . xαn
+n
+| α0 < d − 1
+��
+and we can expand η =
+�
+α0≤d−2 ηα
+�d
+α
+� 1
+2 xα. Then from (27), recalling that everything is expressed in terms of an orthonormal
+basis, we obtain
+Lη,ii =
+�
+2
+�d − 1
+d
+�
+ηd−2,0,,...,2,...,0,
+(28)
+Lη,ij =
+�
+d − 1
+d
+ηd−2,...,1,...,1,...,0 for i ̸= j.
+(29)
+Consider the following orthogonal direct sum decomposition of Nxd
+0Σn,d
+Nxd
+0Σn,d =
+���d
+α
+� 1
+2
+xd−2
+0
+xixj | i, j = 1, . . . , n
+��
+⊕
+���d
+α
+� 1
+2
+xα | α0 < d − 2
+��
+=: W ⊕ P.
+(30)
+We define a map from W to R[x1, . . . , xn](2) by setting
+�d
+α
+� 1
+2
+xd−2
+0
+xixj �−→
+�
+2
+(αi, αj)
+� 1
+2
+xixj
+and extending by linearity.
+Since we are mapping an orthonormal basis for W with the induced
+Bombieri–Weyl product to an orthonormal basis of R[x1, . . . , xn](2) with its own Bombieri–Weyl prod-
+uct, this defines a linear isometry. Composing it with the inverse of the isomorphism we described in (4)
+we get a linear isometry of W with Sym(n, R) and therefore a correspondence between the associated
+Gaussian probability distributions. A direct consequence of this linear isometry, of the discussion in
+section 2.1 about GOE(n) matrices and formulae (28) and (29), is the following theorem.
+Theorem 22. Consider the decomposition Nxd
+0Σn,d = W ⊕ P given in (30).
+Then the following
+statements hold:
+(1) Lη = 0 for every η ∈ P;
+(2) if we pick η ∈ W Gaussian w.r.t the Bombieri–Weyl metric, then the distribution of the
+Weingarten operator at xd
+0 along η is Lη ∼
+√
+2
+�
+d−1
+d
+� 1
+2
+GOE(n).
+Remark 23. Notice that Theorem C from the Introduction follows immediately from Theorem 22
+using the fact that for every p ∈ Vn,d there is a linear isometry τ : SN → SN such that τ(Vn,d) = Vn,d
+and τ(p) = xd
+0.
+Theorem 22 gives a full description of the extrinsic geometry of Σn,d ֒→ SN in terms of random ma-
+trices. From the computational point of view, it allows reducing integrals on the normal bundle of Σn,d
+of quantities related to the second fundamental form to expected values of quantities related to GOE(n)
+matrices. In the next section, we will use this description to explicitly compute the integrals appear-
+ing in Weyl’s tube formula (11), thus obtaining the curvature coefficients of the embedding Vn,d ֒→ SN.
+
+18
+ALBERTO CAZZANIGA, ANTONIO LERARIO, ANDREA ROSANA
+5.2. The curvature coefficients. All this section will be dedicated to proving the following.
+Theorem 24. Let Vn,d ֒→ SN be the spherical Veronese variety and
+U(Vn,d, ε) be defined as in (5).
+If ε < ρ(Vn,d), the following formula holds:
+Vol(U(Vn,d, ε)) =
+�
+0≤j≤n, j even
+(−1)
+j
+2 d
+n
+2
+�d − 1
+d
+� j
+2
+2n+2−jπ
+N
+2 Γ
+� n
+2 + 1
+�
+Γ
+� j
+2 + 1
+�
+Γ(n + 1 − j)Γ
+�
+N+j−n
+2
+�,
+(31)
+where for 0 ≤ j ≤ n, j even, the functions JN,N−n+j are given by (10).
+Comparing (31) to Weyl’s tube formula (8), we obtain the following corollary.
+Corollary 25. The curvature coefficients of the spherical Veronese variety Vn,d ֒→ SN are as follows:
+KN−n+j(Vn,d) = (−1)
+j
+2 d
+n
+2
+�d − 1
+d
+� j
+2
+2n+2−jπ
+N
+2 Γ
+� n
+2 + 1
+�
+Γ
+� j
+2 + 1
+�
+Γ(n + 1 − j)Γ
+�
+N+j−n
+2
+�,
+for 0 ≤ j ≤ n, j even, and KN−n+j(Vn,d) = 0 otherwise.
+In order to prove 24 we start from Weyl’s tube formula (11) applied to Σn,d ֒→ SN. As we already
+noticed, Remark 16 implies that the Weingarten operator looks the same at every point. It follows
+that in this case the integrand in (11) does not depend on p ∈ Σn,d and we obtain
+Vol
+�
+U(Σn,d, ε)
+�
+= Vol(Σn,d)
+� tan ε
+t=0
+�
+η∈S(Nxd
+0 Σn,d)
+tN−n−1 det(In − tLη)
+(1 + t2)
+N+1
+2
+dη dt,
+(32)
+where we remark that N − n is the codimension of Σn,d ֒→ SN and Vol(Σn,d) is given by (16).
+Given the decomposition in (30), we have S(Nxd
+0Σn,d) = S(W ⊕ P), where dim(S(Nxd
+0Σn,d)) =
+N − n − 1 and dim(W) = dim(Sym(n, R)) = n(n+1)
+2
+. Notice that if d = 2 we have Nxd
+0Σn,d = W. If
+d > 2 we parametrize S(W ⊕P) as in (12), where here we use m = N −n−1 and k = n(n+1)
+2
+−1. With
+the same notation of section 2.3, for σ ∈ S(W) and z ∈
+◦
+D(P), if ϕ(σ, z) = η ∈ S(W ⊕ P), we have
+that
+�
+1 − |z|2ι(σ) will be the component of η along W, while z itself will be the component along P.
+We also apply the linear isometry discussed in the previous section to change variable from σ ∈ S(W)
+to Q ∈ S(Sym(n, R)) = S
+n(n+1)
+2
+−1.
+It is clear by its definition that the Weingarten operator is linear in the normal vector argument:
+given an isometric embedding M ֒→ M, for every p ∈ M, η, ξ ∈ NpM and a, b ∈ R, we have
+Laη+bξ = aLη + bLξ. Therefore for η = ϕ(Q, z) ∈ S(W ⊕ P) we have
+Lη = L√
+1−|z|2Q+z =
+�
+1 − |z|2LQ + Lz =
+�
+1 − |z|2LQ.
+(33)
+Applying Lemma 9 to (32) and using (33) the integral becomes
+Vol
+�
+U(Σn,d, ε)
+�
+=Vol(Σn,d)
+� tan ε
+t=0
+�
+S
+n(n+1)
+2
+−1
+�
+DN−n− n(n+1)
+2
+�
+tN−n−1
+(1 + t2)
+N+1
+2
+×
+(34)
+× det(In − t
+�
+1 − |z|2LQ)(1 − |z|2)
+n(n+1)
+4
+−1
+�
+dz dS(Q) dt,
+where dz is a short notation for vol
+DN−n− n(n+1)
+2
+and dS(Q) is a short notation for vol
+S
+n(n+1)
+2
+−1, with
+the convention that for d = 2 the integral over D0 is set to 1. The only non-explicit term in (34) is
+the one involving the determinant. Recall that by Theorem 22, if Q ∈ Sym(n, R) is a random GOE(n)
+matrix, then LQ is a random matrix distributed as
+√
+2
+� d−1
+d
+� 1
+2 GOE(n). Set τ := t
+√
+2
+� d−1
+d
+� 1
+2 . We have
+the expansion
+det
+�
+In − τ
+�
+1 − |z|2Q
+�
+=
+n
+�
+j=0
+(−1)jτ j(1 − |z|2)
+j
+2 gj(Q),
+(35)
+
+WHAT IS THE PROBABILITY THAT A RANDOM SYMMETRIC TENSOR IS CLOSE TO RANK-ONE?
+19
+where gj(Q) are homogeneous polynomials of degree j in the coefficients of Q for j = 1, . . . , n and
+g0(Q) = 1. Substituting (35) into (34) in the integral splits as
+Vol(U(Σn,d, ε)) = Vol(Σn,d)
+n
+�
+j=0
+(−1)j2
+j
+2
+�d − 1
+d
+� j
+2 �� tan ε
+0
+tN−n−1+j
+(1 + t2)
+N+1
+2
+dt
+�
+×
+(36)
+×
+��
+DN−n− n(n+1)
+2
+(1 − |z|2)
+n(n+1)
+4
+−1+ j
+2 dz
+�
+×
+×
+��
+S
+n(n+1)
+2
+−1 gj(Q) dS(Q)
+�
+,
+where the first term is the integral of a rational function in t, while the second one is a “polynomial”
+in |z|.
+Remark that since gj are homogeneous polynomials, we have gj(Q) = ∥Q∥jgj( Q
+∥Q∥). Recalling expres-
+sion (2) we have
+E
+Q∈GOE(n)gj(Q) =
+1
+(2π)
+n(n+1)
+4
+�
+Sym(n,R)
+∥Q∥jgj
+� Q
+∥Q∥
+�
+e− ∥Q∥2
+2
+dQ =
+(37)
+=
+1
+(2π)
+n(n+1)
+4
+�� +∞
+0
+ρ
+n(n+1)
+2
+−1+je− ρ2
+2 dρ
+���
+S
+n(n+1)
+2
+−1 gj( ˜Q) dS
+� ˜Q
+��
+.
+From (37) we obtain
+�
+S
+n(n+1)
+2
+−1 gj(Q) dS(Q) =
+E
+Q∈GOE(n)[gj(Q)] (2π)
+n(n+1)
+4
+� +∞
+0
+ρ
+n(n+1)
+2
+−1+je− ρ2
+2 dρ
+.
+(38)
+By linearity of expectation and the expansion det(In − λQ) = �n
+j=0(−1)jλjgj(Q), to compute the
+expectation of gj(Q) it is enough to compute that of det(In − λQ) for Q ∈ GOE(n) and look at the
+homogeneous part of degree j in λ. This procedure gives us the explicit expression for (38)
+�
+S
+n(n+1)
+2
+−1 gj(Q) dS(Q) =
+(−1)
+j
+2 (2π)
+n(n+1)
+4
+j!
+( j
+2 )!
+�n
+j
+�
+2j � +∞
+0
+ρ
+n(n+1)
+2
+−1+je− ρ2
+2 dρ
+if 0 ≤ j ≤ n, j even
+(39)
+and 0 otherwise, see Appendix C for a proof of this result. By standard computations involving Gamma
+and Beta functions, one can show that the following identities hold
+� +∞
+0
+ρ
+n(n+1)
+2
++j−1e− ρ2
+2 dρ = 2
+n(n+1)
+4
++ j
+2 −1 Γ
+�1
+4(n2 + n + 2j)
+�
+,
+(40)
+�
+DN−n− n(n+1)
+2
+�
+1 − |z|2� n(n+1)
+4
+−1+ j
+2 dz = π
+2N−n2−3n
+4
+Γ
+� 1
+4(n2 + n + 2j)
+�
+Γ
+� 1
+2(N − n + j)
+� ,
+(41)
+where we notice that for d = 2 (41) gives 1, agreeing with our convention. Substituting (39), (40)
+and (41) into (36) and using the duplication formula for the gamma function, we obtain the explicit
+expression of Vol
+�
+U(Σn,d, ε)
+�
+. Finally, recalling that Vn,d = Σn,d ∪ −Σn,d and using formula (16) to
+express Vol(Σn,d), the proof of Theorem 24 is complete.
+5.3. Asymptotics for rational normal curves. Recall the interpretation of the Veronese variety
+Vn,d as the set of rank–1, norm–1 symmetric tensors of order d on Rn+1, while the sphere SN can be
+thought of as the space of all norm–1 such tensors. Then the quantity
+Vol
+�
+U(Vn,d, arcsinδ)
+�
+Vol(SN)
+(42)
+expresses the probability for a symmetric tensor p to be (δ∥p∥BW)–close to rank–1 one with respect
+to the Bombieri–Weyl distribution. We will focus on the case n = 1, d −→ +∞, which corresponds to
+the so called “spherical” rational normal curves V1,d. Notice that in this case N = d. Our asymptotic
+analysis will be based on Laplace’s method, as described in Theorem 10.
+
+20
+ALBERTO CAZZANIGA, ANTONIO LERARIO, ANDREA ROSANA
+Theorem 26. For spherical rational normal curves V1,d, the following asymptotic expansion of (42)
+holds:
+Vol
+�
+U(V1,d, arcsinδ)
+�
+Vol(Sd)
+=
+√
+d δd−1�
+1 + O
+�
+d−1��
+(43)
+as d −→ +∞, for arcsinδ ≤
+1
+√
+3.
+From this theorem and the bound (24), we immediately obtain the following.
+Corollary 27. Denote by Ad the set of symmetric tensors of order d on R2 that admit a unique best
+rank–one approximation. Then, as d −→ +∞, we have
+P
+�
+Ad
+�
+>
+√
+d
+�
+sin 1
+√
+3
+�d−1�
+1 + O
+�
+d−1��
+≈
+√
+d (0.546)d−1�
+1 + O
+�
+d−1��
+.
+Proof of Theorem 26. Set ε := arcsinδ. We start by noticing that, since we are looking at d −→ +∞,
+by Theorem 19 the asymptotic analysis makes sense only for ε ≤
+1
+√
+3. Instead of using the implicit
+form of Theorem 24, to express Vol(U
+�
+Σn,d, ε)
+�
+we will use (36), substituting (39) in it. This gives the
+following expression
+Vol
+�
+U(Σ1,d, ε)
+�
+Vol(Sd)
+= Vol(Σ1,d)
+V ol(Sd) ·
+(2π)
+1
+2
+� +∞
+0
+e− ρ2
+2 dρ
+·
+�� tan ε
+0
+td−2
+(1 + t2)
+d+1
+2
+dt
+�
+·
+��
+Dd−2
+�
+1 − |z|2�− 1
+2 dz
+�
+.
+(44)
+For n = 1 (16) reads as
+Vol
+�
+Σ1,d
+�
+=
+�
+2
+√
+d π
+for d odd
+√
+d π
+for d even ,
+while it is known that Vol(Sd) = 2π
+d+1
+2
+Γ( d+1
+2
+) and
+� +∞
+0
+e− ρ2
+2 dρ = � π
+2 . With easy algebraic manipulations,
+we can rewrite the integral in t as
+� tan ε
+0
+td−2
+(1 + t2)
+d+1
+2
+dt =
+� tan ε
+0
+1
+t2(1 + t2)
+1
+2 exp
+�
+− d
+�
+− log
+�
+t
+(1 + t2)
+1
+2
+���
+dt =
+=
+� tan ε
+0
+e−d a(t)b(t) dt,
+where we have set
+a(t) = − log
+�
+t
+(1 + t2)
+1
+2
+�
+,
+b(t) =
+1
+t2(1 + t2)
+1
+2 .
+One can show that the hypotheses of Laplace’s theorem are satisfied and that the minimum of a(t) in
+(0, tan ε] is attained at tan(ε). Taking Taylor expansions of a(t) and b(t) around tan(ε) and applying
+Theorem 10 we obtain
+� tan ε
+0
+td−2
+(1 + t2)
+d+1
+2
+dt = 1
+d
+�
+sin ε)d−1
+�
+1 + O
+�
+d−2��
+.
+For the last integral in (44), we pass to spherical coordinates and reduce it to a Beta function (and
+therefore to Gamma functions), obtaining
+�
+Dd−2
+�
+1 − |z|2�− 1
+2 dz =
+π
+d−1
+2
+Γ
+� d−1
+2
+�.
+Plugging all the expressions we found in (44), and recalling that Vn,d = Σn,d ∪ −Σn,d, we obtain the
+desired asymptotic.
+□
+
+WHAT IS THE PROBABILITY THAT A RANDOM SYMMETRIC TENSOR IS CLOSE TO RANK-ONE?
+21
+Appendix A.
+Proof of the Tubular Neighbourhood Theorem. We will use the same notation of Section 2.2.
+Throughout the proof, we will identify M with the zero section in NM. We start by computing the
+differential d(x,0)(exp|NM) : T(x,0)(NM) −→ TxM of exp|NM at (x, 0) ∈ NM for any x ∈ M. Notice
+that dim(T(x,0)(NM)) = dim(TxM), hence surjectivity is enough to have a linear isomorphism. Denote
+by γ(p,v) the unique geodesic on M such that γ(p,v)(0) = p and ˙γ(p,v)(0) = v. Let y ∈ TxM. Since
+TxM = TxM ⊕ NxM, we can decompose y as y = y1 + y2 with y1 ∈ TxM and y2 ∈ NxM. Then there
+exists σ1 : (−δ, δ) −→ M such that σ1(0) = x and ˙σ1(0) = y1. Define a curve σ : (−δ, δ) −→ NM by
+σ(t) = (σ1(t), 0) ∈ NM. We have σ(0) = (x, 0) and ˙σ(0) = (y1, 0) ∈ T(x,0)(NM) and it follows that
+d(x,0)(exp|NM)(y1, 0) = d
+dtexp
+�
+σ(t)
+�����
+t=0
+= y1,
+proving that TxM is contained in the image of d(x,0)(exp|NM). Now take y2 ∈ NxM and define a curve
+α : (−δ, δ) −→ NM by α(t) = (x, ty2). Then α(0) = (x, 0) and ˙α(0) = (0, y2) and it follows that
+d(x,0)(exp|NM)(0, y2)) = d
+dtexp
+�
+α(t)
+�����
+t=0
+= y2,
+proving that also NxM is contained in the image of d(x,0)(exp|NM). By linearity of the differential, we
+obtain surjectivity and therefore d(x,0)(exp|NM) is an isomorphism.
+As a consequence for every x ∈ M there exists an open neighbourhood Wx of (x, 0) in NM such
+that the rank of the differential d(q,v)(exp|NM) is maximal for every (q, v) ∈ Wx. Up to shrinking the
+neighbourhood, we can assume that Wx =
+�
+Ux × B(0, εx)
+�
+∩ NM where Ux is an open neighbourhood
+of x ∈ M, B(0, εx) denotes the ball of radius εx centered at the origin in TxM and exp|Wx is an
+embedding. By compactness we have a finite covering of M {Ux1, . . . , Uxr} for some x1, . . . , xr ∈ M.
+Choosing ε := min{εx1, . . . , εxr} we get that
+exp|N εM : N εM −→ M
+is an immersion and a local embedding. Notice that for every ˜ε ≤ ε also exp|N ˜
+εM is an immersion and a
+local embedding. We claim that there exists an ˜ε < ε such that this restriction is also globally injective.
+If this is the case, then the restriction to the closure of the
+˜ε
+2–small normal bundle is an embedding,
+since injective immersions with compact domain are embeddings. It follows that any number less than
+˜ε
+2 satisfies the statement of the theorem.
+To prove the claim we argue by contradiction: suppose that for every n ∈ N there exist (xn, vn),
+(yn, wn) ∈ N
+1
+n M such that exp(xn, vn) = exp(yn, wn).
+Since M is compact, up to restricting to
+subsequences we can assume that xn converges to x ∈ M and yn converges to y ∈ M, while vn and
+wn both converge to 0 since vn, wn ∈ B(0, 1
+n) for every n ∈ N. By compactness of M there exists
+δ > 0 such that for every p ∈ M the map expp : B(0, δ) ⊂ TpM −→ M is a diffeomorphism on its
+image, where expp(z) = exp(p, z). It follows that since vn −→ 0, for n large enough γ(xn,vn) will be the
+unique geodesic joining xn with expxn(vn) = exp(xn, vn) = γ(xn,vn)(1) and dg(xn, exp(xn, vn)) = ∥vn∥.
+Analogously, for n large enough we will also have dg(yn, exp(yn, wn)) = ∥wn∥, where we stress that
+the uniformity of δ is crucial. Since by hypothesis exp(xn, vn) = exp(yn, wn), we have that
+dg(xn, yn) ≤ dg(xn, exp(xn, vn)) + dg(yn, exp(yn, wn)) = ∥vn∥ + ∥wn∥ −→ 0,
+and this forces x = y. Then for n sufficiently large, we have that (xn, vn), (yn, wn) ∈ Wp for some
+p ∈ M, but on every Wp we have a local embedding, leading to a contradiction. The proof is concluded.
+Appendix B.
+Proof of lemma (9). We will use the same notations as in section 2.3. We want to prove that
+ϕ∗(volSm) =
+�
+1 − |z|2� k−1
+2
+volSk ∧ vol ◦
+Dm−k,
+
+22
+ALBERTO CAZZANIGA, ANTONIO LERARIO, ANDREA ROSANA
+where ϕ is given by (12).
+By the usual formula for the pullback of a differential form through a
+diffeomorphism, we have
+ϕ∗(volSm) = |det(JϕT · Jϕ))|
+1
+2 vol ◦
+Dk ∧ vol ◦
+Dm−k,
+where Jϕ denotes the (m + 1) × m Jacobian matrix of ϕ and JϕT is its transpose. Denote by Jι the
+Jacobian matrix of the inclusion ι : Sk ֒→ Rk+1. Then Jϕ is the following block matrix
+Jϕ(σ, z) =
+
+
+�
+1 − |z|2Jι(σ)
+�
+−zj
+√
+1−|z|2 ι(σ)
+�
+0
+Im−k
+
+ ,
+where Im−k denotes the (m − k) × (m − k) identity matrix. Since JιT (σ) · ι(σ) = ι(σ) · Jι(σ) = 0 and
+ι(σ)T · ι(σ) = 1, we obtain
+(JϕT · Jϕ)(σ, z) =
+�
+(1 − |z|2)(JιT · Jι)(σ)
+0
+0
+�
+Im−k +
+z·zT
+1−|z|2
+�
+�
+.
+For every z ∈ Rm−k consider R ∈ O(m − k) such that z = Re1|z|, where e1 = (1, 0, . . . , 0). Then we
+can compute the determinant of the lower right block as
+det
+�
+Im−k + z · zT
+1 − |z|2
+�
+= det R
+�
+Im−k +
+|z|2
+1 − |z|2 E11
+�
+RT =
+1
+1 − |z|2 ,
+where E11 = e1eT
+1 has all zero entries except for the (1, 1)–th one which is 1.
+Recalling that the
+determinant of a diagonal block matrix is given by the product of the determinants of its blocks, we
+find the following expression
+|det(JϕT · Jϕ)| =
+�
+1 − |z|2�k |det(JιT · Jι)|
+1
+1 − |z|2 =
+�
+1 − |z|2�k−1 |det(JιT · Jι)|.
+Finally, applying again the formula for the pullback of a differential form, we can conclude that
+ϕ∗(volSm) =
+�
+1 − |z|2� k−1
+2
+|det(JιT · Jι)|
+1
+2 vol ◦
+Dk ∧ vol ◦
+Dm−k =
+�
+1 − |z|2� k−1
+2
+volSk ∧ vol ◦
+Dm−k.
+Appendix C.
+Proof of formula (39). We will use the same notations as Section 5.2. By linearity of the expectation,
+in order to prove formula (39) all we have to do is computing
+E
+Q∈GOE(n)[det(In − λQ)] =
+n
+�
+j=0
+(−1)jλj
+E
+Q∈GOE(n)[gj(Q)],
+since the expectation of gj(Q) can then be deduced by looking at the degree j coefficient in above
+polynomial expression in λ. First, we write the determinant according to its very definition
+det(In − λQ) =
+�
+σ∈Sn
+sgn(σ)
+n
+�
+i=1
+�
+δiσ(i) − λQiσ(i)
+�
+,
+where Sn is the group of permutations on {1, . . ., n} and sgn(σ) is the signature of the permutation
+σ ∈ Sn. Recall that for Q ∈ GOE(n) we have Qii ∼ N(0, 1) and Qij ∼ N(0, 1
+2) for i ̸= j and, apart
+from the obvious symmetry conditions, the entries are independent. By linearity
+E
+Q∈GOE(n)[det(In − λQ)] =
+�
+σ∈Sn
+sgn(σ)
+E
+Q∈GOE(n)
+�
+n
+�
+i=1
+�
+δiσ(i) − λQiσ(i)
+��
+.
+(45)
+Given σ ∈ Sn, suppose that it contains a cycle of length at least 3, i.e. there exists i ∈ {1, . . . , n} such
+that σ(i) ̸= i and σ2(i) ̸= i. Then, by independence of the entries, in the term of (45) corresponding
+to σ, we can split the expectation into a product of expectations, separating the term corresponding
+to such i. Since δiσ(i) = 0 and Qiσ(i) is centered, this expectation is 0 and σ gives no contribution
+to (45). It follows that the only permutations contributing to (45) are those formed by transpositions
+
+WHAT IS THE PROBABILITY THAT A RANDOM SYMMETRIC TENSOR IS CLOSE TO RANK-ONE?
+23
+and fixed points only.
+For such a σ ∈ Sn, denote by fix(σ) = {i ∈ {1, . . . , n} | σ(i) = i} the set of fixed points of σ and by
+s(σ) the number of disjoint transpositions in σ. Then we have
+E
+Q∈GOE(n)
+� n
+�
+i=1
+�
+δiσ(i) − λQiσ(i)
+��
+= E
+� �
+i∈fix(σ)
+�
+1 − λξi
+��
+· E
+�s(σ)
+�
+k=1
+1
+2λ2γ2
+k
+�
+=
+=
+�
+�
+i∈fix(σ)
+E
+��
+1 − λξi
+���
+·
+�s(σ)
+�
+k=1
+E
+�1
+2λ2γ2
+k
+��
+,
+where ξi ∼ N(0, 1) and γk ∼ N(0, 1). For these terms we have
+E
+��
+1 − λξi
+��
+= 1,
+E
+�1
+2λ2γ2
+k
+�
+= 1
+2λ2E
+�
+γ2
+k
+�
+= 1
+2λ2.
+The contribution of such σ ∈ Sn in (45) is
+sgn(σ)
+E
+Q∈GOE(n)
+� n
+�
+i=1
+�
+δiσ(i) − λQiσ(i)
+��
+= sgn(σ)
+�1
+2λ2
+�s(σ)
+,
+(46)
+and notice it depends only on s(σ). To conclude the computation we, therefore, have to count how
+many permutations in Sn are given by exactly k disjoint transpositions for every k = 0, . . . , ⌊ n
+2 ⌋. Denote
+this number by N(k). To construct a permutation with exactly k disjoint transpositions we proceed
+as follows: choose two elements in {1, . . . , n} forming the first transposition, then choose another 2
+among the remaining ones to form the second transposition and so on until the k–th one is formed.
+Moreover, since the supports of the transpositions are disjoint, the order in which they are picked is
+not relevant. It follows that
+N(k) =
+�n
+2
+��n−2
+2
+�
+. . .
+�n−2k+2
+2
+�
+k!
+=
+n!
+2k(n − 2k)! k!
+and using this and (46) in (45) gives
+E
+Q∈GOE(n)[det(In − τQ)] =
+⌊ n
+2 ⌋
+�
+k=0
+(−1)kN(k)
+�1
+2λ2
+�k
+=
+⌊ n
+2 ⌋
+�
+k=0
+(−1)kλ2k (2k)!
+22kk!
+� n
+2k
+�
+.
+(47)
+Since the expectation of gj(Q) is given by the degree j term in (47) multiplied by (−1)j, we obtain
+E
+Q∈GOE(n)
+�
+gj(Q)
+�
+=
+
+
+
+
+
+
+
+0
+if j odd
+(−1)
+j
+2
+2j
+j!
+( j
+2 )!
+�n
+j
+�
+if 0 ≤ j ≤ n, j even
+.
+(48)
+Plugging (48) into (38), we finally get (39).
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+

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+On the Melting Thresholds of Semiconductors under Nanosecond Pulse Laser Irradiation 
+ 
+J. Beráneka,b, A. V. Bulgakova, N. M. Bulgakovaa,* 
+ 
+a HiLASE Centre, Institute of Physics of the Czech Academy of Sciences, Za Radnicí 828, 
+25241 Dolní Břežany, Czech Republic 
+b Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, 
+Trojanova 13, Prague, 120 01, Czech Republic 
+* Corresponding author: [email protected] 
+Abstract 
+ 
+In this work, a unified numerical model is used to determine the melting thresholds and to 
+investigate early stages of melting of several crystalline semiconductors (Si, Ge, GaAs, CdTe 
+and InP) irradiated by nanosecond laser pulses. A molten fraction approach is used for 
+continuous transition over the melting point. The results are compared with previously 
+published theoretical and experimental data. A survey on the thermophysical and optical 
+properties of the selected materials has been carried out to gather the most relevant data on 
+temperature dependent properties for the solid and liquid states of these semiconductors where 
+such data are available. A generalization of the obtained results is established which enables 
+evaluation of the melting thresholds for different semiconductors based on their properties and 
+irradiation conditions (laser wavelength, pulse duration). 
+Keywords 
+pulsed laser, nanosecond, laser processing, semiconductors, melting, thermal model, finite 
+difference method, material properties 
+1. Introduction 
+Material processing of semiconductors using short and ultrashort laser pulses is one of 
+the key technologies in various fields, including microelectronics, photonics, photovoltaics, 
+sensor devices. It has been employed for enhancing dopant diffusion [1], crystallization [2] and 
+selective modification of multilayer structures [3] and in studies of kinetics of structural 
+changes in materials [4]. Its main advantages are the high level of controllability and variety of 
+wavelengths that can be selected to fit a particular material and an application. One of the basic 
+parameters for laser material processing that involves surface patterning, modification, 
+crystallization, and ablation is the melting threshold. This key parameter is usually determined 
+in the experimental studies of laser material processing as a reference point for controlling the 
+laser modification process and for testing theoretical models developed with the aim of better 
+understanding the fundamental processes at laser-matter interaction [5-18]. For detection of the 
+phase change, several approaches are used such as time-resolved reflectivity (TRR) 
+[9,11,19,20] and electrical conductivity [21] measurements, or combined techniques as in [13] 
+where time-of-flight velocity distributions together with the evaporation rate and reflectivity 
+were measured and analyzed.  
+
+In some cases, the experiments identify a transient state where only a minor or instable 
+change of the studied parameter is observed and thus the obtained melting threshold, i.e., 
+minimal laser energy density (laser fluence) needed to reach this state, corresponds to the 
+melting onset. In other cases, a stable change of the measured value is detected, corresponding 
+to the established molten phase on the irradiated surface at higher fluences. In experiments, the 
+fluence interval with a transient melting is dependent on the material purity and surface quality 
+such as roughness, which can lead to a locally changed absorption, as well as on the presence 
+of hotspots in the laser beam profile. In modeling, it is typically assumed that the melting 
+threshold corresponds to the energy density needed for the temperature of the surface of the 
+material to reach the melting point Tm [5,6].  
+ 
+In the presented work, we compare results obtained in our modeling with some 
+theoretical predictions on the melting thresholds and early stages of melting and with 
+experimental measurements available in the literature. The numerical simulations have been 
+performed for a variety of semiconductors (Si, Ge, GaAs, CdTe, InP) irradiated at wavelengths 
+from 248 to 694 nm in the range of pulse durations from 7 to 70 ns to test the predictive 
+capabilities of the model presented in this work and to investigate the melting dynamics of 
+material under study. Furthermore, the results obtained for different semiconductors have been 
+generalized in order to predict their melting threshold based on a unified parameter combining 
+irradiation conditions and material optical and thermophysical properties.  
+2. Model description 
+The thermal model is applicable for nanosecond laser pulse durations and longer pulses 
+as the electron-lattice interaction phenomena, critical for shorter pulses, are not accounted for 
+and coupling of laser energy to material lattice is treated as an instantaneous local process [22]. 
+It is justified by the fact that electron-lattice thermalization time is in order of ~10-12 s for silicon 
+and germanium [7] and similar time scales for other materials under study. Also, as we consider 
+relatively low surface temperatures, below and near the melting point, we disregard evaporation 
+phenomena, which, however, may slightly affect the melting process for compound 
+semiconductors [23,24]. For correct simulations of laser-matter interaction processes, material 
+thermophysical and optical properties (and their temperature dependences) are of fundamental 
+importance. The material parameters used in the presented calculations are summarized in 
+Appendix, Tables A1–A19. 
+The sample is assumed to be flat and the laser beam couples perpendicularly to the 
+surface in the direction of z axis. The simulations are considered as a one dimensional (1D) 
+problem that is a valid approximation as long as the irradiation spot size (typically above 100 
+µm for the considered experiments) is much larger than the absorption depth that is our case. 
+Indeed, for a ruby laser with the longest wavelength in this study, the absorption depth of the 
+studied semiconductors is less than 0.4 µm (see Appendix). For shorter wavelengths, also 
+investigated here, the absorption depths are even smaller. Then the time-dependent temperature 
+distribution in the irradiated target is governed by the heat-flow equation in its 1D form [25,26]: 
+(𝑐p(𝑇)𝜌 + 𝐿m𝛿(𝑇 − 𝑇m))
+𝜕𝑇
+𝜕𝑡 = 
+𝜕
+𝜕𝑧 (𝜅(𝑇) 𝑑𝑇
+𝑑𝑧) + 𝑆(𝑧, 𝑡).                                                         (1) 
+Here t is time, T is the temperature, cp, ρ, Lm, Tm and κ are respectively the heat capacity, the 
+density, the latent heat of fusion, the melting temperature, and the thermal conductivity of the 
+sample material. Energy supplied by the laser is represented by the source term S(z, t) as  
+
+𝑆(𝑧, 𝑡) = (1 − 𝑅)𝐼(𝑡) 𝛼 exp(−𝛼𝑧),                                                                                                   (2) 
+where R and α are the surface reflectivity and the material absorption coefficient.  The pulse 
+intensity I(t) has a Gaussian temporal profile: 
+𝐼(𝑡) =
+2𝐹0
+𝜏L√ln 2
+𝜋
+exp (−4 ln 2 (
+𝑡
+𝜏L)
+2
+),                                                                                          (3) 
+with F0 and τL being the peak fluence and the pulse duration. 
+Equation (1) is solved numerically using the finite difference method and the implicit 
+scheme that ensures a high numerical stability. For temperatures below the melting point, the 
+finite difference form of Eq. (1) is written on the numerical grid as 
+−𝜅l𝑇i−1
+∗
++ (
+∆𝑧2
+∆𝑡 𝜌𝑐p + 𝜅l + 𝜅r) 𝑇i
+∗ − 𝜅r𝑇i+1
+∗
+=
+∆𝑧2
+∆𝑡 𝜌𝑐p𝑇i + 𝑆(𝑧, 𝑡),                                       (4) 
+where  
+𝜅l =
+𝜅i−1+𝜅i
+2
+,   𝜅r =
+𝜅i+1+𝜅i
+2
+                                                                                                      (5) 
+and index i refers to the numerical grid points. The temperature values T* are unknown at the 
+time moment tf and T (without asterisk) corresponds to the known temperature at the time 
+moment tf-1= tf – Δt. 
+One of the advantages of the implicit numerical scheme is that using large spatial and/or 
+temporal steps (Δz and Δt respectively) does not affect its stability, however, too large Δt values 
+can introduce truncation errors to the calculation results [27]. Similarly, the choice of Δz should 
+enable a good approximation of the laser intensity attenuation toward the material depth and 
+the temperature gradient within the heat affected zone. A very good approximation was 
+achieved with Δt values of 5-10 ps and Δz = 1 nm. The sample is considered to be semi-infinite. 
+The system of the linear equations (1) with discretization to the form (4) represents a tridiagonal 
+matrix, which is solved by the Thomas algorithm [28].  
+Material heating to the melting point followed by the melting process leads to an 
+accumulation of the internal energy at constant T = Tm and its ratio to the enthalpy of melting 
+can be interpreted as a molten fraction in a computational element. In the presented calculations, 
+we apply the method of through calculation without explicit selection of the phase interface 
+[25,26]. According to this method, the melting process is smoothed over a symmetric interval 
+of a width of a few Kelvins around the melting point. Melting starts at a slightly lower 
+temperature than Tm, reaches the melting point at the fraction of molten material of 0.5, and 
+ends at a slightly higher temperature than Tm. In the interval of melting, a δ-function is added 
+to the heat capacity term to account for absorption/release of the fusion heat at the 
+melting/solidification front  
+𝛿(𝑇) =
+𝐿m
+𝐴√𝜋 𝑒−(𝑇−𝑇m
+𝐴
+)
+2
+,                                                                                                         (6)    
+where A is the width of the delta function in Kelvin. As the internal energy rises upon laser light 
+absorption, the physical parameters are gradually changing from solid to liquid phase 
+proportionally to the fraction of molten material [29] 
+
+𝛾 = 𝛾s(𝑇)(1 − 𝜂) +  𝜂𝛾l(𝑇),                                                                                                  (7) 
+where γs, γl represent a property of material in solid and liquid state respectively and η is the 
+fraction of molten material. For the sake of simplicity, the change in density upon melting is 
+not taken into account so that the value of the solid state density is kept also for the liquid state.  
+In the presented model, we interpret the fluence needed for the fraction of molten 
+material to reach the interval from 0–3 % as the melting threshold fluence Fth. According to the 
+δ-function approach (Eqs. (1) and (6)), this occurs at ~1-2 K bellow the tabulated melting point 
+and thus the edge of beginning of melting is blurred. However, from analyzing our simulation 
+data, it follows that the position of Fth within the interval of melting has a minor effect on its 
+resulting value. Furthermore, taking into account the ambiguity of Fth reported in the literature, 
+this aspect plays only a small role.  
+3. Results and discussion 
+Here we present an analysis of the available literature data used for our model 
+development and discuss the simulations results and general trends in the damage threshold 
+determination. The results of the present simulations are summarized in the Table 1 in 
+comparison with the literature data. In addition, for irradiation conditions with a ruby laser 
+(wavelength 694 nm) where the most systematic data are available, we have performed 
+calculations for various laser pulse duration of 15, 30 and 70 ns, beyond the ranges reported in 
+the literature, in order to investigate the effect of pulse duration for specific materials (the 
+obtained results are also presented in Table 1). Note that, in the literature, the Fth values can be 
+determined differently from the method used in this study. For example, in Ref. [30], the 
+calculated melting threshold for CdTe was set by 8% higher than the laser fluence needed for 
+reaching Tm. Time resolved reflectometry (TRR) measurements performed by the authors did 
+not show an increase in reflectivity at the intensity corresponding to reaching the melting point 
+in calculations. Thus, as the melting threshold, the authors consider the intensity, at which the 
+sample surface layer is molten to the depth of laser radiation absorption. Experimentally 
+measured values of the melting threshold fluence typically include a transition interval where 
+localized melting occurs giving rise to an increase in the reflectivity above the values of solid 
+state surface reflectivity [13,22]. In numerical simulations, the determination of the damage 
+threshold strongly depends on the used material properties [31]. Below we have surveyed the 
+literature for the optical and thermophysical parameters of the studied semiconductors. The 
+most relevant parameters are given in Appendix.  
+Table 1. The simulation results for the damage threshold fluence Fth of the studied 
+semiconductors in comparison with theoretical and experimental data reported in the literature. 
+The experimental data are marked by asterisk.  
+Material λ, nm τ, 
+ns 
+Fth, 
+mJ/cm2 
+This work 
+Fth, mJ/cm2 
+Literature  
+data  
+Si 
+532 
+18 
+355 
+395 [6], 
+~320* [32] 
+ 
+ 
+30 
+423 
+474 [6] 
+350 [33] 
+ 
+694 
+15 
+672 
+725 [6] 
+
+ 
+ 
+30 
+752 
+805 [6] 
+ 
+ 
+70 
+900 
+ 
+Ge 
+694 
+15 
+191 
+ 
+ 
+ 
+30 
+255 
+ 
+ 
+ 
+70 
+370 
+400* [10] 
+GaAs 
+308 
+30 
+213 
+200, 200* 
+[8] 
+ 
+532 
+15 
+184 
+ 
+ 
+694 
+15 
+265 
+300 [12] 
+ 
+ 
+20 
+282 
+250* [13] 
+ 
+ 
+30 
+316 
+ 
+ 
+ 
+70 
+415 
+ 
+CdTe 
+248 
+20 
+46 
+50, 50* [30] 
+ 
+694 
+15 
+68 
+ 
+ 
+ 
+30 
+80 
+ 
+ 
+ 
+70 
+103 
+ 
+InP 
+532 
+7 
+106 
+97 [23] 
+ 
+694 
+15 
+165 
+ 
+ 
+ 
+30 
+211 
+ 
+ 
+ 
+70 
+296 
+ 
+ 
+3.1. Silicon 
+Some ambiguity exists in the reported melting point of crystalline Si ranging from 1683 to 
+1690 K [7,22]. For the temperature dependence of cp, we took the data from Ref. [34] with a 
+stronger variation over the range of solid state temperatures than the dependence used for c-Si 
+in Ref. [5]. The c-Si thermal conductivity was approximated by the expression from the 
+measured data reported in [35]. For the liquid state, we use cp = 910 J/(kg·K) and κ = 50.8 + 
+0.029(T – Tm) W/(m·K) [5]. The reflectivity and the absorption coefficient for c-Si are 
+temperature dependent and given by the relations presented in [36]. The optical properties of 
+molten silicon are described according to the calculated data for 694 nm [37] and measured data 
+for 352 nm [38]. The data on the properties for solid and liquid silicon used in the present 
+modeling are summarized in Tables A1-A4 of Appendix.  
+Interestingly, our simulation data for Si (Table 1) somewhat overestimate the melting 
+threshold fluence measured in [32,33] while they are systematically lower than the simulated 
+Fth values presented in [6]. The experimental investigations reported in [32] for 532 nm ns laser 
+irradiation give an interval of increasing reflectivity between 330 and 380 mJ/cm2. The value 
+of 380 mJ/cm2 was identified as a threshold for reaching a high reflectivity (~70%), probably 
+indicating melting to a depth of approximately one optical skin layer (~10 nm), and the 
+maximum reflectivity of 73% was observed at 450 mJ/cm2. A similar value of around 400 
+mJ/cm2 was determined as a threshold for melting based on time-resolved reflectivity 
+measurements [33]. The theoretical calculations [6] give higher values for the melting 
+thresholds than calculated in this work and measured in [32,33]. The main reason can be seen 
+in the difference in the absorption coefficient change with temperature. For instance, the 
+absorption coefficient of c-Si at 694 nm wavelength at temperatures close to the melting point 
+is approximately five times larger in our case (taken from [36]) than in calculations presented 
+in [6]. As a whole, our modeling data for Si are in a reasonable agreement with the published 
+
+data thus demonstrating that our model approach can be used for other semiconducting 
+materials. The calculations with various pulse durations τL show that the melting threshold 
+increases with τL proportionally to app τL0.2 (Table 1), i.e., the dependence is considerably 
+weaker than the  τL0.5 dependence predicted for the evaporation threshold for fairly long 
+(nanosecond and longer) laser pulses [39]. 
+ 3.2. Germanium 
+The next set of simulations has been carried out for germanium for the conditions of the 
+experiments reported in [10], wavelength 694 nm and pulse duration 70 ns. Time-resolved 
+reflectivity measurements using a probe 1.06 μm laser wavelength identified the energy density 
+of 400 mJ/cm2 as a value, at which the rise of reflectivity was detected corresponding to the 
+observable melting. Numerical simulations using the finite difference method were also carried 
+out in Ref. [10] and the obtained melting threshold was claimed to be “practically identical” to 
+the measured one (although the method for threshold determination in the simulation was not 
+specified). The authors used experimentally measured values of reflectivity and absorption from 
+Refs. [9] and [40], which are in a good agreement with the optical constants we derived from 
+measurements reported in [41] and also confirmed in [42]. Our calculations give Fth = 370 
+mJ/cm2 (Fig. 1a, Table 1) that is in good agreement with the data [10], particularly taking into 
+account that the detected in [10] increase in the reflectivity assumes a significant fraction of 
+molten germanium and thus a slightly higher fluence than that needed to reach the melting 
+temperature at the surface. Near the melting threshold, the calculated melt fraction reaches a 
+maximum after a delay of about 20 ns relative to the moment of laser peak intensity (Fig. 1a) 
+that is also in agreement with the measurements [10]. With the known properties of liquid Ge 
+(Tables A7 and A8), we have performed simulations for F > Fth which are again in good 
+agreement with the measured durations of a high reflectivity stage corresponding to molten 
+germanium [10] (Fig. 1b). It should be mentioned that different values are reported for the 
+thermal conductivity of liquid Ge. In modeling, we use the value of 29.7 W/(m·K) [43] while 
+in Ref. [44], κ = 43 W/(m·K) was measured. The calculations performed for various pulse 
+durations demonstrate now a stronger τL dependence than that for silicon, close to the  τL0.5 
+dependence (Table 1). 
+3.3. Gallium Arsenide 
+For simulations of laser heating of GaAs, we used the same values of the thermophysical 
+properties as in Ref. [8]. For the temperature dependence of cp, the data from [45] were used, 
+which are also in a good agreement with the data reported in [46]. Optical properties were taken 
+from measurements [47], which are also in a good agreement with [48]. The absorption and 
+reflection coefficients were calculated from the refractive index and the extinction coefficient 
+and taken as temperature independent. The material properties used in the simulations are 
+presented in Tables A9–A12 of Appendix.  
+Several regimes of laser irradiation of GaAs corresponding to available experimental 
+and theoretical data were investigated in our modelling with the laser wavelength ranging from 
+308 to 694 nm and pulse duration ranging from 15 to 70 ns. For all the conditions, the melting 
+thresholds calculated here with our unified model are in good agreement with the values 
+reported in the literature (Table 1). Below we discuss each irradiation regime in more details. 
+
+λ = 308 nm, τ = 30 ns. Our model implements the same optical and temperature-
+dependent material properties as in the model presented by Kim et al. [8]. For the solid state 
+reflectivity and the optical absorption, the data used for simulations in [8] are in agreement with 
+the measured data for solid GaAs [46]. As the parameters of the model [8] and ours are very 
+similar, we take this comparison as a validation for our model that gives a deviation of only 
+~6% (see Table 1). 
+ λ = 694 nm, τ = 15 ns. García et al. [12] carried out simulations for a ruby laser with a 
+15 ns pulse duration using an explicit numerical scheme. The melting threshold was identified 
+at a laser fluence of 300 mJ/cm2 that corresponded to the situation when a ~65-nm-thick surface 
+layer was molten [12]. In our simulations, this fluence of 300 mJ/cm2 results in the melting 
+depth of 13 nm while the melting threshold corresponding to reaching the melting point on the 
+sample surface is 265 mJ/cm2 (see Fig. 2 for comparison). The difference in the melting depth 
+can be attributed to two factors. First, the authors [12] extrapolated the temperature-dependent 
+absorption coefficient for the solid state GaAs from the room temperature till the melting point 
+that appears to be questionable. In our simulations, we use constant but reliable data on optical 
+absorption and reflectivity of molten GaAs at the wavelength of the ruby laser [41]. The 
+reflectivity coefficient of liquid GaAs in both Ref. [12] and this work was adopted from [13], 
+R = 0.67. The second factor may be related to the using an explicit numerical scheme whose 
+approximation to the initial equations often represent a challenge. 
+λ = 694 nm, τ = 20 ns. Pospieszczyk et al. [13] presented two sets of measurements. 
+Using a HeNe probe laser, the temperature-dependent reflectivity was investigated. The second 
+set of the data gives time-of-flight measurements of particles evaporated from the GaAs surface 
+(Fig. 3a). Comparison of their experimental data and our simulations is given in Table 1, which 
+are in a reasonable agreement. The simulated damage threshold associated with achieving the 
+melting temperature (Fig. 3b) is somewhat higher than in the experiments [13] but is still in the 
+range of fluences where a transient uneven melting is observed (Fig. 3). This discrepancy, 
+although relatively small, can be related to the effect of decreasing the melting temperature due 
+to depletion of the target surface by a more volatile component [23,24,49] that is not taken into 
+account in our model. 
+3.4. Cadmium Telluride 
+We have applied our model to CdTe irradiated by a KrF excimer laser (248 nm) for the 
+conditions of Gnatyuk et al. [30] where TRR measurements and numerical simulations of 
+pulsed laser heating of CdTe were performed. For this material, reliable physical and optical 
+properties are extensively reported in the literature. In our simulations, the value of thermal 
+conductivity was taken from Ref. [50] and [51] for solid and liquid state, respectively. The 
+specific heat for both solid and liquid state was taken from Ref. [52]. The same thermophysical 
+properties were also used in simulations [24,30]. Measurements of optical properties of CdTe 
+using spectroscopic ellipsometry and modeling were performed in [53] for a wide range of 
+wavelengths. Reflectivity and absorption are the same for solid and liquid state and independent 
+of temperature. 
+The authors [30] identified a laser fluence of 50 mJ/cm2 as the melting threshold. In 
+their simulations, this value corresponds to the molten layer with a thickness of the laser 
+absorption depth. Their TRR measurements detected an abrupt although small rise of the 
+reflectivity at a laser fluence of 48-50 mJ/cm2. These results are in excellent agreement with 
+
+our simulations (Table 1). Indeed, for F = 50 mJ/cm2, our model gives the depth of the molten 
+layer of 7 nm, very close to the absorption depth of CdTe at 248 nm (~9nm). According to our 
+definition of the melting threshold, achieving the melting temperature at the very surface of the 
+irradiated sample, the calculated threshold is slightly lower, 46 mJ/cm2 (Table 1). 
+We would also note that in Refs. [25,49], an effect of enhanced evaporation of Cd atoms 
+with enriching the surface by tellurium upon laser heating was studies. It was shown that this 
+effect can have an impact on the melting and ablation processes. This effect was not taken into 
+account in this work, nor in [30]. 
+3.5. Indium Phosphide 
+We have applied our model for the conditions of experiments [23] where InP was 
+irradiated by a nanosecond laser at λ = 532 nm. In this paper, a laser fluence of 97 mJ/cm2 was 
+identified as the damage threshold. In our simulations, we have obtained a threshold value of 
+106 mJ/cm2 which can be considered as a good agreement taking into account that there is no 
+any fitting parameters in our model. It should be mentioned that, although laser processing of 
+InP is a common technique in its industrial applications, the thermophysical parameters at 
+enhanced temperatures are still not well studied. Thus, several sets of data are available for the 
+heat capacity of solid InP, see e.g. [54]. The major problem is that measurements of the 
+thermophysical properties at enhanced temperatures are affected by a high vapor pressure of 
+phosphorous due to its high volatility. The thermal conductivity and the specific heat of molten 
+InP are given in [46]. The reflectivity and absorption are calculated form data provided in [41]. 
+Optical properties are taken as temperature independent and considered the same for both solid 
+and liquid state. In reference article [23], ablation of compound semiconductors is studied and 
+a model that takes into account evaporation of their components gives the melting threshold. 
+Our result, that disregards this effect, gives Fth that is about 10% higher. 
+3.6. Generalization of the damage threshold data into a predictive dependence 
+A wide set of data on the damage thresholds of five semiconductors under various ns-laser 
+irradiation conditions are obtained in our calculations in the frames of a unified thermal model 
+and all the thresholds are in good agreement with available literature data, both experimental 
+and theoretical ones. The obtained threshold values vary in a wide range depending on material, 
+from ~ 50 mJ/cm2 for CdTe to almost 1 J/cm2 for Si (Table 1). The irradiation conditions also 
+affect the threshold values which are generally smaller for shorter laser wavelengths and pulse 
+durations. It is very attractive to generalize the obtained results in terms of a unified parameter 
+combining the basic material properties (thermophysical and optical) in order to be able to 
+predict the ns-laser-induced melting thresholds, at least approximately, without performing 
+detailed simulations. 
+ 
+D. Bäuerle [22] considered “optimal” melting conditions during ns-laser-induced 
+thermal surface melting, when minimal laser energy is required for a certain melt depth. 
+Assuming that such conditions are fulfilled when the melt depth is equal to the heat-diffusion 
+length, he estimated the optimal laser fluence as 
+𝑃B = 
+2𝜌∆𝐻
+1−𝑅 (
+𝐷
+𝜏L)
+1
+2 𝜏L                                                                                                                  (8) 
+
+where D = κ/cp is the thermal diffusivity and H = Lm + cp(Tm-300) is the total energy needed 
+to heat the sample to the complete melting state from room temperature, A similar parameter 
+was introduced in [39] as an evaporation threshold under ns-laser ablation (assuming naturally 
+by H in Eq. (8) the specific heat for evaporation instead of that for melting and omitting the 
+2/(1-R) factor).  
+Figure 4 shows the calculated melting threshold values plotted as a function of the PB 
+parameter, Eq. (8), evaluated for all the studied materials using their room-temperature 
+properties. All the data are nicely groupped around a streight line in the logarithmic plot. This 
+clear correlation is rather surprising for such a simplified generalization approach when the 
+material absorption coefficient and temperature dependencies of thermophysical properties are 
+not taken into account. The least square fitting line in Fig. 4 is described by a power law Fth ≈ 
+0.05PB1.16 whcih can be used for rough estimation of the melting threshold of semiconductors 
+based on their basic room-temperatureproperties. 
+The parameter PB predicts a growth of the melting threshold with the laser pulse duration 
+as τL0.5. However, as was noticed above, this is not always the case according to our simulations. 
+Some semiconductors (Ge, CdTe, InP) follow closely the τL0.5 dependence while others (Si, 
+GaAs) demonstrate weaker dependencies (Table 1 and Fig. 4). This is probably mainly due to 
+a difference in the thermal diffusivity D of the materials. Thus, at room temperature, D ≈ 0.8 
+cm2/s for Si and it is around 0.35 cm2/s for Ge, InP and CdTe. A higher thermal diffusivity 
+results in a higher heat diffusion length and smaller in-depth temperature gradients and thus in 
+a lower heat flow from the surface at an increased pulse duration. The temperature dependencies 
+of materials parameters (included to our model simulations) can additionally affect the pulse 
+duration dependence of the melting threshold. 
+4. Conclusions 
+In this work, based on the classical thermal model, we have developed a numerical approach 
+to investigate the continuous solid-liquid phase change in solid targets heated by nanosecond 
+laser pulses. The model is applied to a number of semiconductors and various irradiation 
+conditions and the obtained results on the melting thresholds, melt duration and melt depth are 
+compared with experimental and theoretical data available in the literature. The comparison is 
+not always straightforward as the value presented as melting threshold fluence is not always 
+describing the same state of the studied material. However, in most cases, good agreement with 
+the literature data is obtained. The simulations predict also the dependence of the melting 
+thresholds on the laser pulse duration which is found to be material dependent and weaker than 
+that expected from simple heat-flow considerations. A good correlation of all the calculated 
+melting threshold values with a parameter combining material thermophysical properties and 
+surface reflectivity is obtained. The correlation can be used as a simple method for estimation 
+of the melting thresholds of ns-laser irradiated semiconductors based on their room-temperature 
+properties. 
+Acknowledgements 
+This work was supported by the European Regional Development Fund and the state budget of 
+the Czech Republic (project BIATRI: No. CZ.02.1.01/0.0/0.0/15_003/0000445). J. B. 
+acknowledges funding of the Grant Agency of the Czech Technical University in Prague No. 
+SGS22/182/OHK4/3T/14. 
+
+Declarations 
+Conflict of interest. The authors declare no conflict of interests. 
+Appendix 
+Here we provide all the parameters for semiconductors, which were selected after a thorough 
+literature analysis and used in our modeling. Some reliable data, which are widely cited in 
+literature and web-sites, are given without references.  
+Silicon 
+Table A1. c-Si – thermophysical properties 
+Property 
+Value 
+Ref. 
+ρ,g/cm3 
+2.328 
+ 
+Tm, K 
+1688 
+ 
+Lm, J/kg 
+1.826 × 106 
+[55] 
+cp, J/kg K 
+847.05 + 118.1 × 10-3 T – 155.6 × 105 T -2 
+[35] 
+κ, W/mK 
+97269 T -1.165   (300<T<1000) 
+3.36 × 10-5 T 2– 9.59 × 10-2T + 92.25  (1000<T<Tm) 
+[35] 
+ 
+Table A2. c-Si – optical properties 
+532 nm 
+Value 
+Ref. 
+n 
+4.152 
+[41] 
+k 
+0.051787 
+[41] 
+R 
+0.374 
+[36] 
+α, 1/m 
+5.02 × 105 exp(T/430) 
+[36] 
+694 nm 
+ 
+ 
+n 
+3.79 
+[41] 
+k 
+0.013 
+[41] 
+R 
+0.34 + 5 × 10-5 (T – 300) 
+[36] 
+α, 1/m 
+1.34 × 105 exp(T/427) 
+[36] 
+ 
+Table A3. Liquid-Si – thermophysical properties 
+Property 
+Value 
+Ref. 
+ρ, g/cm3 
+2.52 
+ 
+cp, J/kg K 
+910 
+[5] 
+κ, W/mK 
+50.28 + 0.029 (T–Tm) 
+[5] 
+ 
+Table A4. Liquid -Si – optical properties 
+532 nm 
+Value 
+Ref. 
+n 
+3.212 
+[38] 
+k 
+4.936 
+[38] 
+R 
+0.693 
+Calculated 
+α, 1/m 
+1.1659 × 108 
+Calculated 
+694 nm 
+ 
+ 
+
+n 
+3.952 
+[37] 
+k 
+5.417 
+[37] 
+R 
+0.707 
+Calculated 
+α, 1/m 
+9.804 × 107 
+Calculated 
+Germanium 
+Table A5. c-Ge – thermophysical properties 
+Property 
+Value 
+Ref. 
+ρ, g/cm3 
+5.3267 
+ 
+Tm, K 
+1211.4 
+ 
+Lm, J/kg 
+5.1 × 105 
+[43] 
+cp, J/(kg K) 
+1.17 × 10-1T + 293 
+[43] 
+κ, W/(m K) 
+18000/T 
+[43] 
+ 
+Table A6. c-Ge – optical properties 
+694 nm 
+Value 
+Ref. 
+n 
+5.04 
+[41] 
+k 
+0.49 
+[41] 
+R 
+0.45 
+Calculated  
+α, 1/m 
+8.81 × 106 
+Calculated 
+ 
+Table A7. Liquid -Ge – thermophysical properties 
+Property 
+Value 
+Ref. 
+ρ, g/cm3 
+5.6 
+ 
+Tm, K 
+3106 
+ 
+cp, J/kg K 
+450 
+[43] 
+κ, W/mK 
+29.7 
+[43] 
+ 
+Table A8. Liquid -Ge – optical properties 
+694 nm 
+Value 
+Ref. 
+n 
+2.62 
+[37] 
+k 
+5.238 
+[37] 
+R 
+0.74 
+Calculated 
+α, 1/m 
+9.485 × 107 
+Calculated 
+ 
+Gallium Arsenide 
+Table A9. c-GaAs – thermophysical properties 
+Property 
+Value 
+Ref. 
+ρ, g/cm3 
+5.32 
+ 
+Tm, K 
+1511 
+ 
+Lm, J/kg 
+7.11 × 105 
+ 
+cp, J/kg K 
+8.76 × 10-2 T + 308.16 
+[8] 
+κ, W/mK 
+30890 T-1.141    
+[8] 
+
+ 
+Table A10. c-GaAs – optical properties 
+308 nm 
+Value 
+Ref. 
+n 
+3.7 
+[41] 
+k 
+1.9 
+[41] 
+R 
+0.42 
+Calculated 
+α, 1/m 
+7.7 × 107 
+Calculated 
+532 nm 
+ 
+ 
+n 
+4.13 
+[41] 
+k 
+0.336 
+[41] 
+R 
+0.37 
+Calculated 
+α, 1/m 
+8.04 × 106 
+Calculated 
+694 nm 
+ 
+ 
+n 
+3.78 
+[41] 
+k 
+0.15 
+[41] 
+R 
+0.338 
+Calculated 
+α, 1/m 
+2.687 × 106 
+Calculated 
+ 
+Table A11. Liquid GaAs – thermophysical properties 
+Property 
+Value 
+Ref. 
+ρ, g/cm3 
+ 
+ 
+cp, J/kg K 
+439.85 
+[8] 
+κ, W/mK 
+30890 T -1.141 
+[8] 
+ 
+Table A12. Liquid GaAs – optical properties 
+308 nm 
+Value 
+Ref. 
+R 
+0.46 
+[8] 
+α, 1/m 
+0.83 × 108 
+[56] 
+694 nm 
+ 
+ 
+R 
+0.67 
+[13] 
+α, 1/m 
+2.687 × 106 
+Taken the same 
+as for solid state 
+ 
+Cadmium Telluride 
+Table A13. c-CdTe – thermophysical properties 
+Property 
+Value 
+Ref. 
+ρ, g/cm3 
+5.85 
+ 
+Tm, K 
+1365 
+ 
+Lm, J/kg 
+2.09 × 105 
+[52] 
+cp, J/kg K 
+3.6 × 10-2 T + 205 
+[52] 
+κ, W/mK 
+1507/T   
+[50] 
+ 
+Table A14. c-CdTe – optical properties 
+
+248 nm 
+Value 
+Ref. 
+n 
+2.63 
+[53] 
+k 
+2.13 
+[53] 
+R 
+0.406 
+calculated 
+α, 1/m 
+1.1 × 108  
+calculated 
+694 nm 
+ 
+ 
+n 
+3.037 
+[53] 
+k 
+0.286 
+[53] 
+R 
+0.258 
+calculated 
+α, 1/m 
+5.179 × 106  
+calculated 
+ 
+Table A15. Liquid CdTe – thermophysical properties 
+Property 
+Value 
+Ref. 
+ρ, g/cm3 
+6.4 
+ 
+cp, J/kg K 
+255 
+[52] 
+κ, W/mK 
+1.1 
+[30] 
+ 
+Table A16. Liquid CdTe – optical properties 
+248 nm 
+Value 
+Ref. 
+R 
+0.45 
+[30] 
+α, 1/m 
+1.1 × 108 
+[30] 
+ 
+Indium Phosphide 
+ 
+Table A17. c-InP – thermophysical properties 
+Property 
+Value 
+Ref. 
+ρ, g/cm3 
+4.81 
+ 
+Tm, K 
+1335 
+ 
+Lm, J/kg 
+3.4 × 105 
+[57] 
+cp, J/kg K 
+2.33 × 10-2T + 347 
+[54] 
+κ, W/mK 
+1.215 × 105T -1.324 
+[54] 
+ 
+Table A18. c-InP – optical properties 
+532 nm 
+Value 
+Ref. 
+n 
+3.702 
+[41] 
+k 
+0.429 
+[41] 
+R 
+0.335 
+Calculated 
+α, 1/m 
+1.013 × 106 
+Calculated 
+694 nm 
+ 
+ 
+n 
+3.49 
+[41] 
+k 
+0.27 
+[41] 
+R 
+0.31 
+Calculated 
+��, 1/m 
+4.82 × 106 
+Calculated 
+
+ 
+Table A19. Liquid -InP – thermophysical properties 
+Property 
+Value 
+Ref. 
+cp, J/kg K 
+424 
+[46] 
+κ, W/mK 
+22.8 
+[46] 
+ 
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+pure silicon by using differential scanning calorimetry. J Therm. Anal. Calorim. 138, 
+4215-4221 (2019). 
+56. M. A. Scarpulla, III-Mn-V ferromagnetic semiconductors synthesized by ion implantation 
+and pulsed-laser melting. (University of California, Berkeley, 2006). 
+57. D. Richman, E. F. Hockings, The Heats of Fusion of InSb, InAs, GaAs, and InP, J. 
+Electrochem. Soc. 112, 461–462 (1965). 
+ 
+ 
+ 
+
+ 
+ 
+Fig. 1. (a) Surface temperature (solid lines) and molten fraction (dot-dashed lines) of 
+germanium obtained in the modelling for different laser fluences (694-nm, 70-ns pulse). The 
+laser temporal profile is shown by the dashed line. (b) Duration of increased reflectivity in the 
+experiments [10] and the melt duration obtained in our simulations.  
+ 
+ 
+
+1400
+(a)
+370 mJ/cm2
+(b)
+400mJ/cm2
+Y 1200
+480mJ/cm2
+TEMPERATURE,
+0.8
+laserpulse
+DURATION, μS
+1000
+0.6
+800
+FRACTION
+600
+0.4
+SURFACE
+MELT
+400
+0.2
+200
+0.5
+MOLTEN
+[10]
+Thiswork
+0
+0
+0
+0
+50
+100
+150
+200
+250
+300
+0
+0.5
+1
+1.5
+2
+2.5
+3
+TIME, ns
+LASER FLUENCE, J/cm2 
+Fig. 2. Comparison of the results obtained by modelling in Ref. [12] and in this work for GaAs 
+irradiated by 694-nm, 15-ns laser pulses. (a) The depth of molten material as a function of time 
+for several laser fluences (reprinted with permission from Garcia [12]). (b) The results of the 
+present modeling for laser fluences of 263 mJ/cm2 (corresponding to the defined melting 
+threshold) and 300 mJ/cm2. The temporal evolution of the melt depth at 300 mJ/cm2 is also 
+given to compare with Ref. [12]. The laser pulse is shown by the dashed line.  
+ 
+ 
+ 
+ 
+
+3500
+1600
+a
+K
+(b)
+1500
+3000
+TEMPERATURE,
+1400
+x(8)
+2500
+E=0.6J/cm²
+1300
+DEPTH
+2000
+0.5
+1200
+300mJ/cm2
+wu
+1500
+0.4
+1100
+DEPTH,
+MELTED
+263mJ/cm2
+20
+SURFACE
+1000
+0.35
+1000
+0.3
+900
+10
+500
+MELT
+800
+700
+0
+0
+50
+100
+150
+200
+250
+0
+20
+40
+60
+80
+100
+TIME t(ns)
+TIME, ns 
+ 
+Fig. 3. (a) The number of Ga and As atoms emitted from the GaAs surface irradiated by 694-
+nm, 20-ns laser pulses as a function of laser fluence as derived from mass spectrometric 
+measurements (adapted from [13]). Transient uneven and developed manifestations of 
+increased reflectivity indicating the appearance of the liquid phase are marked by shaded region 
+and solid line, respectively. Our simulated melting threshold is shown by vertical dashed line. 
+(b) The simulated dynamics of the surface temperature with the identified melting threshold of 
+282 mJ/cm2. The laser pulse profile is shown by the dashed line. 
+ 
+ 
+
+105
+1600
+PARTICLES, arb.un.
+(a)
+K
+(b)
+TEMPERATURE,
+1500
+104
+1400
+O
+1300
+103
+300mJ/cm²
+1200
+00
+282 mJ/cm²
+1100
+260mJ/cm2
+NUMBEROFF
+102
+GaAs
+Ga
+SURFACE
+1000
+As
+101
+900
+-
+800
+100
+700
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0
+20
+40
+60
+80
+100
+LASERFLUENCE,J/cm?
+TIME, ns 
+ 
+Fig. 4. Calculated melting thresholds of the studied semiconductors for different laser 
+wavelengths as a function of the Bäuerle parameter PB, Eq. (8). The numbers above the points 
+correspond to the laser pulse duration in nanoseconds. The line represents a power-law least-
+squares fit of the data.   
+ 
+
+1000
+N
+CALCULATEDMELTINGTHRESHOLD,mJ/cm
+30
+70.
+Si
+15
+Ge
+InP
+30
+400
+CdTe
+02
+30
+GaAs
+20
+70
+18
+15
+30
+30
+30
+15
+15
+15
+7
+70
+100-
+694 nm
+30
+15
+532 nm
+308 nm
+20
+248 nm
+40
+400
+1000
+2000
+5000
+BAUERLEPARAMETERP
+mJ/cm2

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+Universal adaptive optics for microscopy through
+embedded neural network control
+Qi Hu1, Martin Hailstone2, Jingyu Wang1, Matthew Wincott1, Danail Stoychev2, Huriye
+Atilgan3, Dalia Gala2, Tai Chaiamarit2, Richard M. Parton2, Jacopo Antonello1, Adam M.
+Packer3, Ilan Davis2, and Martin J. Booth1,*
+1Department of Engineering Science, University of Oxford
+2Department of Biochemistry, University of Oxford
+3Department of Physiology, Anatomy, and Genetics, University of Oxford
+*[email protected]
+ABSTRACT
+The resolution and contrast of microscope imaging is often affected by aberrations introduced by imperfect optical systems
+and inhomogeneous refractive structures in specimens. Adaptive optics (AO) compensates these aberrations and restores
+diffraction limited performance. A wide range of AO solutions have been introduced, often tailored to a specific microscope type
+or application. Until now, a universal AO solution – one that can be readily transferred between microscope modalities – has
+not been deployed. We propose versatile and fast aberration correction using a physics-based machine learning (ML) assisted
+wavefront-sensorless AO control method. Unlike previous ML methods, we used a bespoke neural network (NN) architecture,
+designed using physical understanding of image formation, that was embedded in the control loop of the microscope. The
+approach means that not only is the resulting NN orders of magnitude simpler than previous NN methods, but the concept is
+translatable across microscope modalities. We demonstrated the method on a two-photon, a three-photon and a widefield
+three-dimensional (3D) structured illumination microscope. Results showed that the method outperformed commonly-used
+modal-based sensorless AO methods. We also showed that our ML-based method was robust in a range of challenging
+imaging conditions, such as extended 3D sample structures, specimen motion, low signal to noise ratio and activity-induced
+fluorescence fluctuations. Moreover, as the bespoke architecture encapsulated physical understanding of the imaging process,
+the internal NN configuration was no-longer a “black box”, but provided physical insights on internal workings, which could
+influence future designs.
+Introduction
+The imaging quality of high-resolution optical microscopes is often detrimentally affected by aberrations which result in
+compromised scientific information in the images. These aberrations can arise from imperfections in the optical design of the
+microscope, but are most commonly due to inhomogeneous refractive index structures within the specimen. Adaptive optics
+(AO) has been built into many microscopes, restoring image quality through aberration correction by reconfigurable elements,
+such as deformable mirrors (DMs) or liquid crystal spatial light modulators (LC-SLMs).1–6 Applications of AO-enabled
+microscopes have ranged from deep tissue imaging in multiphoton microscopy through to the ultra-high resolution required for
+optical nanoscopy. This range of applications has led to a wide variety of AO solutions that have invariably been tailored to a
+specific microscope modality or application.
+There are two main classes AO operation: in one case, a wavefront sensor measures aberrations; in the other case,
+aberrations are inferred from images – so called “wavefront sensorless AO”, or “sensorless AO” for short. For operations
+with a wavefront sensor, phase aberrations are measured directly by wavefront sensors such as a Shack-Hartmann sensor7,8 or
+an interferometer9–11. Such operations are direct and fast but also have intrinsic disadvantages such as requiring a complex
+optical design and suffering from non-common path errors. Furthermore, such wavefront sensors often have limitations and
+are less versatile. For example, an interferometer requires a coherent source and all such methods suffer from problems due
+to out-of-focus light. On the other hand, sensorless AO methods normally function with a simpler optical design and thus
+are more easily adaptable for a wide range of imaging applications. However, sensorless AO methods are based on iterative
+deductions of phase aberrations and thus tend to be more time consuming; this is coupled with repeated and prolonged sample
+exposures, which inevitably lead to photo-damage or motion related errors.
+There have been many developments in AO technology, and in particular sensorless AO methods. Conventionally, sensorless
+AO operates based on the principle that the optimal image quality corresponds to the best aberration correction12,13. A suitably
+defined metric, such as the total signal intensity14–27 or a spatial frequency based sharpness metric28–33, is used to quantify the
+arXiv:2301.02647v1  [eess.IV]  6 Jan 2023
+
+image quality. Phase is modulated by the AO while this quality metric reading is measured and optimised. There have been
+discussions on how the phase should be modulated12,24,34,35 and how the optimisation algorithm should be designed21,36–38.
+However, as mentioned before, such “conventional” sensorless AO methods depend on iterative optimisation of a scalar metric,
+where all image information is condensed into a single number, and the optimisation process is usually through mode by mode
+adjustment. Such methods were thus not the most efficient approach to solving this multi-dimensional optimisation problem and
+the effective range of correction was limited. While a higher dimensional metric was considered to extract more information
+from images39, the optimisation of such a vector metric was not straightforward.
+While the utility of each of these conventional sensorless AO methods has been demonstrated separately, each method had
+been defined for a particular microscope type and application. Until now, no such AO solution has been introduced that can be
+universally transferred between microscope modalities and applications.
+We propose in this article a new approach to sensorless AO that addresses the limitations of previous methods and provides
+a route to a universal AO solution that is applicable to any form of microscopy. This solution is constructed around a physics-
+based machine learning (ML) framework that incorporates novel neural network (NN) architectures with carefully crafted
+training procedures, in addition to data pre-processing that is informed by knowledge of the image formation process of the
+microscope. The resulting NN is embedded into the control of the microscope, improving the efficiency and range of sensorless
+AO estimation beyond that possible with conventional methods. This approach delivers versatile aberration measurement and
+correction that can be adapted to the application, such as the correction of different types of aberration, over an increased range
+of aberration size, across different microscope modalities and specimens.
+In recent years, machine learning (ML) has been trialled in AO for its great computational capability to extract and
+process information. However, many of these approaches required access to point spread functions (PSFs) or experimentally
+acquired bead images40–46 ; these requirements limited the translatability of these methods to a wider range of applications.
+Reinforcement learning was applied to correct for phase aberrations when imaging non point-like objects47; however, the
+method still involved iterative corrections and was not advantageous in terms of its correction efficiency, accuracy and correction
+working range compared to conventional sensorless AO algorithms. Untrained neural networks (NN) were used to determine
+wavefront phase and were demonstrated on non point-like objects48,49; however, such methods were reported to normally
+require a few minutes of network convergence, which limits their potential in live imaging applications.
+Our new approach differs considerably from previous ML assisted aberration estimation, as previous methods mostly
+employed standard deep NN architectures that used raw images as the input data. Our method builds upon physical knowledge
+of the imaging process and is designed around the abilities of the AO to introduce aberration biases, which improve the
+information content of the NN input data. This approach means that the resulting NN is orders of magnitude simpler, in terms
+of trainable parameters, than previous NN methods (See Table S1 in supplemental document). Furthermore, our method is
+readily translatable across microscope modalities. As NN training is carried out on a synthetic data set, adaptation for a different
+modality simply requires regeneration of the image data using a new imaging model. The NN architecture and training process
+are otherwise similar.
+To illustrate the versatility of this concept, we have demonstrated the method on three different types of fluorescence
+microscopes with different forms of AO corrector: a two-photon (2-P) microscope using a SLM, a three-photon (3-P) intravital
+microscope using a DM, and a widefield three dimensional (3-D) SIM microscope using a DM. In all cases, we showed that the
+new method outperformed commonly used conventional sensorless AO methods. The results further showed that the ML-based
+method was robust in a range of challenging imaging conditions, such as specimen motion, low signal to noise ratio, and
+fluorescence fluctuations. Moreover, as the bespoke architecture encapsulated into its design physical understanding of the
+imaging process, there was a link between the weights in the trained NN and physical properties of the imaging process. This
+means that the internal NN configuration needs no-longer to be considered as a “black box”, but can be used to provide physical
+insights on internal workings and how information about aberrations is encoded into images.
+Concept and implementation
+The overall MLAO concept is illustrated in Figure 1. The experimental application follows closely the concept of modal
+sensorless AO, whereby a sequence of images are taken, each with a different bias aberration applied using the adaptive element.
+The set of images are then used as the input to the ML-enabled estimator, which replaces the previous conventional method
+of optimisation of an image quality metric. The estimated correction aberration is then applied to the adaptive element. If
+necessary, the process can be iterated for refined correction. The significant advantage of the new method is the way in which
+the estimator can more efficiently use image information to determine the aberration correction.
+The concept has been designed in order to achieve particular capabilities that extend beyond those of conventional sensorless
+AO. The new method should ideally achieve more efficient aberration estimation from fewer images, to reduce time and
+exposure of measurement. It should operate over a larger range of aberration amplitudes, compared to previous methods. A
+particular estimator should be robust to variations between similar microscopes and the concept should be translatable across
+2/16
+
+(a)
+Microscope
+Adaptive
+element
+Initial
+Corrected
+Image2
+Image1
+Efficient MLAO estimation
+Corrected image
+Aberrated image
+Iterative correction
+(b)
+Versatile
+Network
+training
+(c)
+F
+=
+F
+Image2
+Image1
+Aberration
+correction
+Maximum
+pixel
+reading
+Small
+scale
+feature
+Large
+scale
+feature
+Shape
+bespoke CNN
+FCL
+F -1
+Image pre-processing
+Convolution
++
+local
+maxpooling
+Global
+maxpooling
+Fully
+connected
+Pseudo-PSF
+Image variations
+Synthesised images
+Image = Object ∗ PSF + Noise 
+microscope imaging model
+PSF2-P,  PSF3-P,  PSFWF
+Fluorescence
+fluctuations
+Spatial
+sampling
+size
+Sample
+motion
+Sample 
+structure/
+sparsity
+Aberration
+and
+brightness
+Detector,
+photon and 
+structured
+noise
+3-D
+structures/
+background
+Figure 1. The MLAO concept. (a) Overview of the AO correction process. A minimum of two bias aberrations were
+introduced by the adaptive element; corresponding images of the same field were captured. The images were passed to the
+MLAO estimator, which determined the Zernike coefficients for correction. The correction speed was limited only by the speed
+of image acquisition, not by computation. (b) Training data generation. A range of image variations were included in the
+synthetic data set for NN training to cope with variations in real experimental scenarios. The data was a combination of
+artificial and real microscope images, chosen to model a wide range of realistic specimen structures. Images were created
+through convolution of specimen structures with an appropriate PSF, generated for the specific microscope modality,
+incorporating aberrations. (c) Image pre-processing and NN architecture. Images were pre-processed to compute pseudo-PSFs,
+which were predominantly independent of specimen structure. F and F −1 represent the forward and inverse Fourier
+transform, respectively. A central cropped region of the pseudo-PSF images was used as the inputs to a CNN. The CNN was
+designed and trained specifically for aberration determination. The output from the overall network was the correction
+coefficients for the Zernike modes. The NN architecture was such that the convolutional layer outputs could be correlated with
+spatial scales of the aberration effects on the pseudo-PSFs and hence the imaging process. Hence, the distribution of weights in
+the network had physical relevance.
+3/16
+
+3
+2
+1
+0
+-1
+-2
+-3different microscope types and applications. From a practical perspective, it is also important that training can be performed on
+synthetic data, as it would be impractical to obtain the vast data set necessary for training from experimentally obtained images.
+An essential step towards efficient use of image data is the image pre-processing before they are presented to the NN. Rather
+than taking raw image data as the inputs, the NN receives pre-processed data calculated from pairs of biased images, which we
+term a “pseudo-PSF”, as shown in Fig. 1 and explained in the methods section. This pseudo-PSF contains information about
+the input aberration and is mostly independent of the unknown specimen structure. By removing the specimen information at
+this stage, we can reduce the demands on the subsequent NN, hence vastly simplifying the architecture required to retrieve the
+aberration information.
+As most of the useful information related to aberrations was contained within the central pixels of the pseudo-PSF, a region
+of 32×32 pixels was extracted as the input to the NN. The first section of the NN was a bespoke convolutional layer that was
+designed to extract information from the inputs at different spatial scales. The outputs from the convolutional layer were then
+provided to a fully connected layer, which was connected to the output layer. Full details of the NN design are provided in the
+methods and the supplementary information. This architecture – rather unusually – provided a link between the physical effects
+of aberrations on the imaging process and the mechanisms within the NN, specifically through the weights at the output of the
+first fully connected layer.
+NN training was performed using a diverse set of synthesised training data. These images were calculated using an
+appropriate model of the microscope imaging process in the presence of aberrations. Images were synthesised by convolutions
+of specimen structures with a PSF, incorporating various likely experimental uncertainties and noise sources. The specimens
+consisted of a range of artificial and realistic objects. Full details are provided in the methods.
+This versatile concept could accommodate different aberration biasing strategies. Conventional modal sensorless AO
+methods typically required a minimum of 2N +1 biased images to estimate N aberration modes21. However, the MLAO method
+has the ability to extract more information out of the images, such that aberrations could be estimated with as few as two
+images, although more biased images could provide better-conditioned information. In general, we defined methods that used
+M differently biased images to estimate N Zernike modes for aberration correction. The input layer of the NN was adjusted
+to accommodate the M image inputs for each method. Out of the many possibilities, we chose to illustrate the performance
+using two biasing schemes: one using a single bias mode (astigmatism, Noll index50 i = 5) and one using all N modes that
+were being corrected. In the first case, we used either two or four images (M = 2 or 4) each with different astigmatism bias
+amplitude. We refer to these methods as ast2 MLAO or ast4 MLAO. Astigmatism was chosen as the most effective bias mode
+(see supplementary document, section 6). In the second case, biased images were obtained for all modes being estimated
+(M = 2N or 4N); this type is referred to in this paper as 2N MLAO or 4N MLAO. For a complete list of the settings for each
+demonstration, please refer to Table S2 in the supplemental document.
+Results
+In order to show its broad application, the MLAO method was demonstrated in three different forms of microscopy: 2-P and 3-P
+scanning microscopy and widefield 3-D structured illumination microscopy (SIM). This enabled testing in different applications
+to examine its performance coping with different realistic imaging scenarios.
+The MLAO methods were compared to two widely used conventional modal based sensorless AO methods (labelled as
+2N+1 conv and 3N conv). The 2N+1 conv method used two biased images per modulation mode and an additional zero
+biased image to determine phase correction consisting N modes simultaneously. The 3N conv method used three images per
+modulation mode (two biased and one unbiased images) and determined the coefficients of the modes sequentially. For both
+methods, the bias size was chosen to be ±1 rad for each mode. A suitable metric was selected to quantify the image quality.
+For each mode, the coefficients were optimised by maximising the quality metric of the corresponding images using a parabolic
+fitting algorithm. When used in 2-P and 3-P demonstrations, the total fluorescence intensity metric was optimised. For the
+widefield 3-D SIM microscope, a Fourier based metric was optimised51. For the details of the two conventional methods, please
+refer to21,36.
+Different functions were defined as optimisation metrics for the conventional AO methods, and also to assist quantifiable
+comparisons of image quality improvement for the MLAO methods. These were defined as an intensity based metric yI, a
+Fourier based metric yF, and a sharpness metric yS. Details are provided in the methods section.
+Two-photon microscopy
+A range of method validations were performed on a 2-P microscope that incorporated a SLM as the adaptive correction
+element, including imaging bead ensembles and extended specimen structures. The experimental set-up of the 2-P system was
+included in Figure S8 (a) in the supplemental document. In order to obtain controlled and quantitative comparisons between
+different AO methods, the SLM was used to both introduce and correct aberrations. This enabled statistical analysis of MLAO
+4/16
+
+performance with known input aberrations. System aberrations were first corrected using a beads sample before carrying out
+further experiments.
+We performed a statistical analysis to assess how MLAO algorithms (ast2 MLAO and 2N MLAO) performed in various
+experimental conditions compared to conventional algorithms (2N+1 conv and 3N conv). Experiments were conducted on
+fixed beads samples (Figure 2 (a, b)), and Bovine Pulmonary Artery Endothelial (BPAE) cells (FluoCellsTM Prepared Slide
+#1) (Figure 2 (c - f)). Dependent on the experiment, either N = 5 or N = 9 Zernike modes were estimated (see Table S2 in
+Supplemental document for details).
+Statistical performance analysis
+Figure 2 (a) and (b) showed statistical comparisons of the different correction methods. Figure 2 (a) displayed the residual
+aberrations gathered from twenty experiments, each consisting of one correction cycle from random initial aberrations including
+five Zernike modes. If the remaining aberration is below the pre-correction value, then the method provides effective aberration
+correction. A wide shaded area indicated inconsistent and less reliable correction. The results show that when correcting small
+aberrations with root mean square (RMS) = 0.63 to 1.19 rad, 2N MLAO performed similarly to 2N+1 conv. Between RMS =
+1.19 to 1.92 rad, 2N MLAO corrected more accurately (lower mean aberration) and also more reliably (smaller error range). For
+large aberrations above RMS = 2.12 rad, 2N+1 conv completely failed, whereas the MLAO methods still improved aberration
+correction. ast2 MLAO had poor performance at small aberrations (RMS = 0.63 to 0.84 rad) but provided reasonable correction
+for large aberrations (RMS = 1.92 to 2.12 rad). However, it is important to note that ast2 MLAO required only two images for
+each correction cycle, far fewer that the ten and eleven images required respectively for 2N MLAO and 2N+1 conv.
+Figure 2 (b) displayed the mean value of metric yI from ten experiments against the number of images acquired during
+multiple iterations of the different correction methods. The corrected aberrations consisted of nine Zernike modes. It was shown
+that ast2 MLAO corrects the fastest initially when the input aberration is large but converges to a moderate signal level, which
+indicates only partial correction of the aberration. 2N MLAO corrects more quickly and to a higher level than the conventional
+algorithms. The narrower error bars for both MLAO algorithms at the end of the correction process indicate that they are more
+reliable than the two conventional methods.
+Correction on extended specimen structures
+Figure 2 (c)-(f) showed experimental results when imaging microtubules of BPAE cells. Specimen regions were chosen to
+illustrate performance on different structures: (c) contained mainly aligned fine fibrous structures; (d) contained some large
+scale structures (bottom right); (e) contained fine and sparse features. For (f) we intentionally reduced illumination laser power
+and increased detector gain to simulate an imaging scenario with very low signal to noise ratio (SNR). The images showed
+structured noise at the background, which could pose a challenge to estimation performance. A large randomly generated
+aberration (RMS = 2.12 to 2.23 rad) consisting of five (c and f) or nine (d and e) Zernike modes was used as the input aberration.
+In (c), (d) and (e), ast2 MLAO corrected the fastest initially when the aberration was large but converged to a moderate level
+of correction. 2N MLAO corrected faster in general than the conventional methods and converged to a higher level of correction.
+In (f) when SNR was poor and structured noise was present, ast2 MLAO failed to correct while 2N MLAO continued to perform
+consistently.
+Three-photon intravital microscopy
+Three-photon microscopy of neural tissue imaging is a particular challenge for sensorless AO, due to the inherently low
+fluorescence signal levels. While this could be alleviated by averaging over time, problems are created due to specimen motion.
+Further challenges are posed for functional imaging, due to the time dependence of emission from ion or voltage sensitive dyes.
+The demonstrations here show the robustness of the new MLAO methods in experimental scenarios where the conventional
+methods were not effective. Importantly, the MLAO methods were able to perform effective correction based on a small number
+of image frames without averaging.
+The experimental set-up of the 3-P system is shown in Figure S8 (b) in the supplemental document. The microscope used
+an electromagnetic DM for aberration biasing and correction. Two MLAO methods, ast4 MLAO and 4N MLAO, were used
+to correct aberrations by using single frame images as inputs. In each case, more input frames were chosen than in the 2-P
+demonstrations, in order to cope with the lower SNR. The NNs were trained to estimate N = 7 Zernike modes. Two types of
+mice were used to perform live brain imaging of green fluorescent protein (GFP) labelled cells (Figure 3 (a)) and functional
+imaging in GCaMP-expressing neurons (Figure 3 (b)). In Figure 3 (a), results were collected at 660µm depth and power at
+sample was 32 mW. In Figure 3 (b), imaging was at 250µm depth and power at sample was 19 mW. Further 3-P results were
+included in the section 8 of supplemental document. For the details of the sample preparation, please refer to section 9B in
+supplemental document.
+Figure 3 (a) shows plots of the metrics yI and yF as proxies for correction quality when imaging GFP labelled cells. Both
+ast4 MLAO and 4N MLAO networks successfully improved the imaging quality. Similar to the ast2 MLAO results in the 2-P
+5/16
+
+A
+B C
+D
+A
+5μm
+B
+C
+D
+No. of images
+A
+B
+E
+D
+F
+C
+A
+5μm
+B
+C
+D
+E
+F
+B
+D
+C
+E
+F
+No. of images 
+ast2 MLAO
+2N MLAO
+A
+5μm
+B
+C
+D
+E
+F
+A
+B
+C
+E
+F
+D
+A
+5μm
+B
+C
+D
+E
+F
+(c)
+(d)
+(f)
+No. of images
+No. of images
+A
+2.1 rad
+1.6 rad
+1.9 rad
+1.2 rad
+0.8 rad
+0.6 rad
+(a)
+(b)
+A
+B
+C
+E
+F
+D
+D
+E
+F
+C
+B
+A
+5μm
+No. of images
+yI
+A
+yI
+yI
+yI
+yI
+(e)
+N = 5
+N = 9
+N = 5
+N = 9
+N = 9
+N = 5
+0.6 0.8
+1.2
+1.6
+1.9 2.1
+Applied aberration RMS (rad)
+1
+2
+3
+4
+Remaining aberration RMS (rad)
+Pre correction
+2N+1 conv
+ast2 MLAO
+2N MLAO
+5μm
+0.0rad
+Fine structure
+Coarse structure
+Sparse structure
+Low SNR with
+strructured noise
+Figure 2. Comparative performance of MLAO methods in a 2-P microscope. (a) Residual aberration after one correction
+cycle for three methods. Points show the mean and the shaded area indicates the standard deviations (SDs) of aberration
+distributions. The images show an example field of view (FOV) when different amounts of a random aberration were
+introduced. (b)-(f) show the intensity metric (yI) as a proxy for correction quality, against the number of images used for
+multiple iterations of correction. Random aberrations consisting of N Zernike modes, as shown in the figure, were introduced
+and corrected. In (b), an ensemble of ten random aberrations were corrected, imaging over the same FOV. Error bars on the plot
+showed the SD of the fluorescence intensity before and after correction. (c)-(f) show specific corrections imaging microtubules
+of BPAE cells, illustrating performance for different specimen structures and imaging conditions. The images were acquired
+before and after correction through the different methods (as marked on the metric plots). Insets on the images show residual
+wavefronts after correction for each image. The grayscale colorbars show phase in radians.
+6/16
+
+3N conv
+2N+1 conv
+Fluorescence intensity
+ast network
+2N network
+0
+10
+20
+30
+number of sample exposures2元
+0
+-22
+T3N conv
+2N+1 conv
+Fluorescence intensity
+ast network
+2N network
+0
+10
+20
+30
+number of sample exposures3N conv
+2N+1 conv
+Fluorescence intensity
+ast network
+2N network
+0
+10
+20
+30
+number of sample exposures4.5元
+0
+-4.5元3N conv
+2N+1 conv
+Fluorescence intensity
+ast network
+2N network
+0
+10
+20
+30
+number of sample exposures3N conv
+2N+1 conv
+Fluorescence intensity
+ast network
+2N network
+0
+10
+20
+30
+number of sample exposures1.5元
+0
+-1.5元3N conv
+2N+1 conv
+Fluorescence intensity
+ast network
+2N network
+0
+10
+20
+30
+number of sample exposures6
+4
+2
+0
+-2
+-4
+-66
+4
+2
+0
+-2
+-4
+-66
+4
+2
+0
+-2
+-4
+-66
+4
+2
+0
+-2
+-4
+-66
+4
+2
+0
+-2
+-4
+-66
+2
+0
+-2
+-4
+-62元
+0
+-22
+T2元
+0
+-22
+T(b) GCaMP at 250 μm
+(a) static GFP at 660 μm
+i
+iv
+4N MLAO it:1
+20μm
+ii
+Pre MLAO
+iii
+ast4 MLAO it:5
+yI
+yF
+0
+4
+8 12 16 20
+28
+ast4 MLAO
+4N MLAO
+v
+0
+4
+8 12 16 20
+28
+No. of images
+ast4
+it:5
+4N
+it:1
+vi
+Post MLAO it:5
+iii
+1
+1
+Post MLAO
+it:5
+Pre
+MLAO
+0
+50
+100
+time / s
+4
+3
+2
+it:1
+Pre MLAO
+ii
+20μm
+A
+B
+C
+D
+E
+F
+G
+H
+ast4 MLAO
+bias mode i = 5
+1
+3
+2
+4
+-1
+rad 
+-0.5
+rad
++1
+rad
++0.5
+rad
+1
+3
+2
+4
+ast4 MLAO
+4N MLAO
+±0.5,±1 rad
+bias mode i = 5
+6
+7
+8
+9
+10 11
++0.5
+rad 
++1
+rad 
+-0.5
+rad 
+-1
+rad 
+bias mode i=
+i
+0
+4
+8
+12
+16
+20
+yI
+ast4 MLAO
+iv
+0
+4
+8
+12
+16
+20
+No. of images
+yS
+v
+it:1
+it:2
+it:3
+it:4
+it:5
+A
+B
+0
+50
+100
+time / s
+C
+D
+0
+50
+100
+time / s
+E
+F
+0
+50
+100
+time / s
+G
+H
+0
+50
+100
+time / s
+vi
+Figure 3. Aberration correction in three-photon microscopy of live mouse brains: (a) GFP-labelled cells at depth 660µm and
+(b) functional activity of GCaMP-labelled cells at 250µm. Wavefronts inserted to the figures showed the phase modulations
+applied by the DM at the relevant step; the common scale is indicated by the colorbar next to (a) and (b) ii.
+(a) i shows example single-frame images used in correction with the corresponding bias modes as insets. 1-4 were the image
+inputs to ast4 MLAO. For 4N MLAO, six more bias modes and thus 24 more images were also used in each iteration. (a) ii-iv
+show images averaged from 20 frames after motion correction. The rectangular boxes highlight regions of interest for
+comparison. (a) v and vi show the intensity metric (yI) and the Fourier metric (yF), respectively, calculated from single image
+frames, against the number of images acquired for five iterations ast4 MLAO one iteration of 4N MLAO.
+(b) i 1-4 shows example single-frame images used as inputs to the ast4 MLAO correction with the corresponding bias modes as
+insets. White squares highlight two cells for comparison to show the fluorescence fluctuations over time neural activity. (b) ii
+and iii show respectively before and after ast4 MLAO correction through five iterations (it:1 to 5), 200 frame averages after
+motion correction. In iii, time traces shown to the left were taken from the marked line (1). (b) iv and v show the intensity
+metric (yI) and the sharpness metric (yS), respectively, calculated from single image frames, against the number of images
+acquired for five iterations ast4 MLAO. (b) vi shows the calcium activity of 8 cells (A-H marked on ii).
+7/16
+
+T
+1
+0
+2
+-T
+1
+22
+1
+0
+-1
+-2
+-30
+50
+100
+time / s2
+1
+0
+-1
+-2
+-32
+1
+0
+-1
+-2
+-32
+1
+0
+-1
+-2
+-32
+1
+0
+-1
+-2
+-3T
+1
+0
+2
+-T
+1
+2demonstrations, ast4 MLAO corrected more quickly at first, but converged to a lower correction level. In contrast, 4N MLAO
+preformed better overall correction, but required more images. Panels ii-iv show averaged images in which processes previously
+hidden below the noise level are revealed through MLAO correction (as highlighted in the white rectangles). The example
+biased images shown in Figure 3 (a) i provide an indication of the low raw-data SNR that the MLAO method can successfully
+use.
+Figure 3 (b) shows results from imaging calcium activity in a live mouse. The ast4 MLAO method successfully improved
+image quality despite the low SNR and fluorescence fluctuations of the sample. From both time traces of line 1 and cells A-H, it
+could be clearly seen that after corrections, signals were increase. The 4N MLAO method failed to correct in this experimental
+scenario (results not shown). We will discuss the likely hypotheses for this in the discussion section.
+The fluctuating fluorescence levels due to neural activity mean that conventional metrics would not be effective in sensorless
+AO optimisation processes. This is illustrated in Figure 3 (b) iv and v, where it can be seen that no single metric can accurately
+reflect the image quality during the process of ast4 MLAO correction. These observations illustrate the advantages of MLAO
+methods, as their optimisation process did not rely on any single scalar metric.
+Widefield 3-D structured illumination microscopy
+The architecture of the NN was conceived so that it would be translatable to different forms of microscopy. In order to illustrate
+this versatility, and to complement to the previously shown 2-P and 3-P laser scanning systems, we applied MLAO to a widefield
+method. The 3D SIM microscope included multiple lasers and fluorescence detection channels and an electromagnetic DM as
+the correction element. Structured illumination patterns were introduced using a focal plane SLM. The detailed experimental
+set-up was included in Figure S6 (c) in the supplemental document.
+Without AO, 3D SIM reconstruction suffers artefacts caused by aberrations. Since typical specimens contain 3D structures,
+the lack of optical sectioning in widefield imaging means that the aberration correction process can be affected by out of focus
+light. As total intensity metrics are not suitable for conventional AO algorithms in widefield imaging, Fourier based sharpness
+metrics have often been used. However, such metrics depend on the frequency components of the specimen structure39. In
+particular, emission from out of focus planes can also affect the sensitivity and accuracy of correction. However, the NN based
+MLAO methods were designed and trained to mitigate against the effects of the sample structures and out of focus light.
+Figure 4 shows results from two NN-based methods ast2 MLAO and 2N MLAO compared to the conventional algorithm 3N
+conv, which used the yS metric. Sensorless AO was implemented using widefield images as the input (Figure 4 (a, b)). The
+correction settings thus obtained by the 2N MLAO method were then applied to super-resolution 3D SIM operation (Figure 4 (c,
+d)). N = 8 Zernike modes were involved in the aberration determination. The specimen was a multiple labelled Drosophila
+larval neuromuscular junction (NMJ). For the details of the sample preparation, please refer to section 7B in supplemental
+document.
+Figure 4 (b) showed that ast2 MLAO corrected most quickly; 2N MLAO corrected to a similar level but required more
+sample exposures; 3N MLAO was less effective. Figure 4 (a) showed the effectiveness of correction on raw and deconvolved
+widefield images. Part iii showed the changes in image spectrum after correction. The dashed line shows a threshold where
+signal falls below the noise level. It can be seen that both (C) ast2 MLAO and (D) 2N MLAO increased high frequency content
+compared to (A) before AO correction and (B) after 3N conv corrections. Figure 4 (c) and (d) showed the images after 3D
+SIM reconstruction. It can be clearly seen that when by-passing AO (i), there was strong artefacts due to aberrations. After
+correcting using five iterations of 2N MLAO, artefacts were suppressed and z-resolution was improved (see sections through
+line 1 and 2 in Figure 4 (d))
+Discussion
+The power and simplicity of the MLAO method arise mainly from a combination of three aspects: the pre-processing of image
+data, the bespoke NN architecture, and the definition of the training data set. All of these aspects are informed by physical
+and mathematical principles of image formation. This forms a contrast with many other data-driven deep learning approaches,
+where complex NNs are trained using vast amount of acquired data.
+The calculation of the pseudo-PSF from pair of biased images (as shown in Figure 1 (c) and elaborated in the Methods)
+acts to remove most of the effects of unknown specimen structure from the input data. The information contained within the
+pseudo-PSF encodes indirectly how aberrations affect the imaging PSF (see Figure S2 in the supplemental document for more
+details). There is a spatial correspondence between a pixel in the pseudo-PSF and the PSF itself. Hence, spatial correlations
+across the pseudo-PSF relate to spatial effects of aberrations on the images.
+The set of pseudo-PSFs forms the input to the convolutional layers of the NN. The masks in each convolutional layer probe,
+in effect, different scales across the pseudo-PSF. Hence, one can attribute a correspondence between the output of these layers
+and the effects aberrations have over different physical scales in the image. Such phenomena are heuristically demonstrated in
+8/16
+
+5μm
+1
+2
+1
+2
+2
+2
+1
+1
+z
+z=6μm
+z
+z
+(d)
+i
+ii
+A
+B
+D
+C
+5μm
+(b)
+(c)
+i
+ii
+Pre
+Post-MLAO
+Pre
+Post-MLAO
+A i
+B i
+C i
+D i
+Widefield Deconvolution
+Image
+spectrum
+10μm
+ii
+iii
+ii
+iii
+ii
+iii
+ii
+iii
+(a)
+a.u.
+yS
+Figure 4. Aberration correction in a widefield structured illumination microscope. (a) Widefield images acquired A i before
+and B-D i after correction through different methods (as marked on the metric plot (b)). The second column ii shows
+corresponding deconvolved widefield images. The third column iii shows corresponding image spectra; dashed lines show the
+threshold where signal falls below the noise level.
+(b) The sharpness metric yS against the number of images, for two iterations of 3N conv, ten iterations of ast4 MLAO and three
+iterations of 2N MLAO.
+(c, d) 3-D projections of 3-D reconstructed SIM image stack of (c) 10µm and (d) 6µm when (i) by-passing AO and (ii) after
+five iterations of 2N MLAO correction; square inserts show zoomed in region for comparison. x-y and y-z sections are shown
+through lines 1 and 2.
+Insets to (a,c and d) show wavefronts corrected by the DM for each image acquisition; phase is shown on the adjacent scale bar.
+9/16
+
+3N conv
+-ast2 MLAO
+2N MLAO
+2
+14
+24
+32
+48
+no. of imagesT
+0
+-T2元
+0
+-2
+T2元
+0
+-2
+T102
+-2
+10
+一Layer
+1
+2
+3
+4
+5
+astX MLAO
+0.23
+0.19
+0.17
+0.18
+0.23
+XN MLAO
+0.39
+0.14
+0.15
+0.13
+0.20
+Table 1. The RMS of the weight distributions extracted from different convolutional layers of the two classes of trained CNNs,
+astX MLAO and XN MLAO. The values shown are calculated from the ensemble of corresponding layers from all CNNs of the
+given class.
+section 3 of the supplementary information. By extracting relevant weight connections from inside the NN, we can observe
+embedded physical interpretations of how the machine learned to process aberration information contained in images.
+To illustrate this, we extracted from the trained NN the weights between the layer embedding physical interpretations and
+the next fully connected layer (marked by red arrows in Figure 1 (c)). Going down the convolutional layers, the scale of probed
+features increases from a single pixel, through small scale features, up to large scale features (as explained in section 3 of the
+supplemental document). The RMS values of the weights from each convolutional layer are shown in Table 1, where the data
+are shown for the ensembles of the two classes of MLAO networks used in this paper, astX MLAO and XN MLAO (where X =2
+or 4). A full breakdown is provided in the Figure S4 of the supplementary document.
+The largest weight variation was in the first layer in the XN MLAO NN, which indicates that this algorithm extracts more
+information from the single pixel detail than from larger scale correlations. In contrast, astX MLAO assigns weights more
+evenly across all layers. As explained in the supplementary document, the single pixel extraction from the pseudo-PSF is
+related to the Strehl ratio of the PSF and the intensity information of the images in non-linear systems. Hence, it is expected
+that the XN MLAO NN, which uses as similar set of bias aberrations to the conventional method, would learn as part of its
+operation similar behaviour to the conventional algorithm. The same phenomena can also explain why in 3-P GCaMP imaging
+of neural activity astX MLAO was less affected by the fluorescence fluctuations than XN MLAO, as astX MLAO relies less on
+overall fluorescence intensity changes. Conversely, astX MLAO generally performed worse than XN MLAO in 2-P imaging
+when structured noise present, as astX MLAO used fewer images and hence had access to less detectable intensity variations
+than XN MLAO. The fact that astX MLAO had access to less well-conditioned image information may also explain why in
+general it was able to correct aberrations to a lower final level than XN MLAO.
+Conclusion
+The MLAO methods achieved the aims explained at the outset. They provided more efficient aberration correction with fewer
+images over a larger range, reducing time required and specimen exposure. The training procedure, which was based on
+synthesised data, ensured that the AO correction was robust to uncertainty in microscope properties, the presence of noise, and
+variations in specimen structure. The concept was translatable across different microscope modalities, simply requiring training
+using a revised imaging model.
+The new methods used NN architectures that are orders of magnitude simpler, in terms of trainable parameters, than in
+previous similar work (see supplementary information, section 5). This vast simplification was achieved through pre-processing
+of data to remove most of the effects of unknown specimen structure. The physics-informed design of the NN also meant that –
+unusually for most NN applications – the learned weights inside the network provided indications of the physical information
+used by the network. This provides constructive feedback that can inform future AO system designs and the basis for extension
+of the MLAO concept to more demanding tasks in microscopy and other imaging applications.
+Methods
+Image pre-processing
+Image data were pre-processed before being used by the NN, in order to remove effects of the unknown specimen structure. The
+resulting “pseudo-PSFs” were better conditioned for the extraction of aberration information, independently of the specimen.
+The image formation can be modelled as a convolution between specimen fluorescence distribution and an intensity PSF. The
+AO introduced pre-chosen bias aberrations, so that multiple images with different PSFs could be acquired over the same FOV.
+Mathematically, this process can be expressed as
+I1 = O∗ f1 +δ1
+I2 = O∗ f2 +δ2
+(1)
+where I1 and I2 were the images acquired with two different PSFs f1 and f2 for the same unknown specimen structure O. δ1
+and δ2 represent combined background and noise in each image. In order to remove (or at least reduce) the effects of specimen
+10/16
+
+structures, we defined the pseudo-PSF as
+pseudo-PSF = F −1
+�F(I1)
+F(I2)
+�
+= F −1
+�F(O∗ f1 +δ1)
+F(O∗ f2 +δ2)
+�
+= F −1
+�F(O)×F( f1)+F(δ1)
+F(O)×F( f2)+F(δ2)
+�
+where F was the 2D Fourier transform and F −1 was its inverse (see Figure 1 (c)). The term “pseudo-PSF” was chosen as the
+function was defined in the same variable space as a PSF, although it is not used directly in any imaging process. A similar
+computational process was shown elsewhere for different applications using defocussed images52. Assuming the noise is small
+enough to be neglected
+pseudo-PSF = F −1
+�F(I1)
+F(I2)
+�
+≈ F −1
+�F( f1)
+F( f2)
+�
+(2)
+There is an implicit assumption here that there are no zeroes in the object spectrum F(O) or the optical transfer function F(f2).
+In practice, it was found that a small non-zero value of F(δ2) mitigated against any problems caused by this. Furthermore,
+although structured noise was present in the pseudo-PSFs (see e.g. Figure S1 in the supplemental document), it was found that
+this did not detrimentally affect data extraction through the subsequent NN. As a further mitigation, we calculated pairs of
+pseudo-PSFs from pairs of biased input images by swapping the order from ( f1, f2) for the first pseudo-PSF to ( f2, f1) for the
+second.
+Example pseudo-PSFs are shown in Figure S1 and S2 in the Supplemental document. As most information was contained
+within the central region, to ensure more efficient computation, we cropped the central region (32×32 pixels) of the pseudo-
+PSFs to be used as the input to the NN. Dependent upon the MLAO algorithm, the input to the NN would consist of a single pair
+of cropped pseudo-PSFs, or multiple pairs corresponding to the multiple pairs of bias aberrations applied in different modes.
+Neural network training
+To estimate phase aberrations from pseudo-PSFs, a convolutional based neural network was designed incorporating physical
+understanding of the imaging process and was trained through supervised learning. Synthetic data were used for training and
+the trained networks were then tested on real AO microscopes. For each imaging modality (i.e. 2-P, 3-P and widefield), a
+separate training dataset was generated, with the imaging model and parameters adjusted for different applications.
+Neural network architecture
+A convolutional neural network was designed to determine the aberrations from pseudo-PSFs, while embedding physical
+understanding of image formation. The conceptual structure is shown in Figure 1 (c); more specific details of the architecture
+and learning process are provided in Section S1 of the supplementary document. This CNN architecture allowed convolutional
+masks to – in effect – probe different spatial scales within the pseudo-PSF images and, hence, to learn from the effects
+aberrations had at different spatial scales in microscope images. The outputs from these convolutional layers acted as inputs to
+a single concatenated fully connected layer (FCL). This was followed by another FCL then the output layer, whose outputs
+corresponded to the Zernike mode coefficients estimated for aberration correction. This shallow architecture with the order of
+104 trainable parameters was effective due to the pre-processing of data that meant the input information was better conditioned
+to this estimation task than raw images.
+The weight connections between the concatenated FCL immediately following the CNN layer and the subsequent FCL
+(marked in red arrows in Figure 1 (c)) depended upon the significance of the information learnt from the different scales
+embedded in the CNN layers. Analysis of these weights could therefore provide insight into the pseudo-PSF information that
+was used by the ML process.
+Synthetic data generation
+Due to the impracticality of acquiring sufficient high-quality data experimentally, a large dataset of simulated image data was
+constructed. The simulations were designed to resemble images collected from different microscopes when imaging a range of
+samples.
+We started with a collection of image stacks (containing around a total of 350 images) obtained from high-resolution 3D
+microscopy of various specimens labelled with nuclear, cytoplasmic membrane and/or single-molecule markers. The images
+were down-sampled to 8-bit (128×128) and separated into their individual channels. This formed a pool of realistic sample
+structures which were later used to generate synthetic images. To further augment the varieties of sample structures, random
+rotations were applied and synthetic shapes including dots, rings, circular shapes, curved and straight lines of varying sizes
+were randomly introduced.
+11/16
+
+The simulated training dataset was generated by convolving the sample structures with synthetic PSFs, f (see Eq. 1). f was
+modelled as a pixel array through
+f =
+���F
+�
+Pe j(Ψ+Φ+Ξ)����
+l
+(3)
+where F represented the 2D discrete Fourier transform. P was the circular pupil function, defined such that pixels in the region
+outside the pupil had value zero. The ratio between the radius of the pupil in pixels and the size in pixels of the overall array
+was adjusted to match sampling rates for different microscopes. In practical scanning optical microscopes, the sampling rates
+can be easily adjusted, although perhaps not arbitrarily. Hence, for experimental flexibility, the ratio for the simulated training
+dataset was tuned to be within the range of 1.0× to 1.2× the base sampling rate. The base sampling rate was defined as using
+two pixels to sample the full width half maximum (FWHM) of the PSF of the system when aberration free. For the widefield
+system, the ratio was tuned to simulate the projection of the camera pixel sampling rate at the specimen. Figure S5 in the
+supplemental document shows how tolerable a trained network was when tested on data collected at different pixel sampling.
+P also incorporated the illumination profile for different practical imaging systems, such as when using truncated Gaussian
+illumination at the pupil in the 3-P microscope. The exponent l varied with imaging modes: when simulating a 3-P, a 2-P and a
+widefield microscope, l was set to 6, 4 and 2 respectively.
+The total aberration was expressed as a sum of chosen Zernike polynomial modes Ψ+Φ+Ξ = ∑i aiZi. Ψ was the sum
+of the randomly generated specimen aberrations, which included all modes that the AO system was designed to correct. Φ
+represented the additional bias aberrations. Ξ included additional non-correctable higher order Zernike modes. The coefficients
+of the correctable modes were randomly generated for each data set. Representing the set of coefficients {ai} as a vector a, the
+random coefficients followed a modified uniform n-sphere distribution53 where both the direction and the two-norm of a were
+uniformly distributed. The maximum two-norm (size) of a were chosen differently for different imaging applications. This
+distribution allowed a denser population close to zero aberration, which was intuitively beneficial to train a stable NN. We
+also added random small errors to the correctable coefficients so that the labels were slightly inaccurate. This was to simulate
+situations when the AO would be incapable of introducing perfect Zernike modes. The spurious high order non-correctable
+Zernike modes were included to further resemble realistic scenarios in a practical microscope.
+Poisson, Gaussian, pink and structured noise of varying noise level were also introduced to the generated images after the
+convolution to allow the training dataset to simulate more closely real microscope images.
+Note that the scalar Fourier approximation of Eq. 3 was chosen for simplicity, although more accurate, vectorial, high
+numerical aperture (NA) objective lens models could have been applied54–57. Although the chosen model would deviate from
+high NA and vectorial effects, the main phenomena under consideration here – namely the effects of phase aberrations on PSFs
+and images – are adequately modelled by scalar theory.
+Image quality metrics
+Different image quality metrics were defined for use as the basis for optimisation in conventional sensorless AO methods and as
+proxies to quantify the level of aberration correction. yI is an intensity based metric and can be used in non-linear imaging
+systems. It is defined as
+yI =
+� �
+I(x)d2x
+yF is a Fourier based metric and provides an alternative aspect to the intensity metric. It is defined as
+yF =
+� �
+0.1fmax<| f|<0.6fmax
+|F[I(x)]|d2 f
+where F[I(x)] is the 2D Fourier transform of image I(x) from x domain to f domain; fmax is the maximum frequency limit of
+the imaging system. The range 0.1fmax < |f| < 0.6 fmax was selected such that most PSF related frequency information was
+included in the range.
+yS is a sharpness metric that can be used for optimisation in widefield systems, where the other metrics are not practical, or
+applications with fluorescence fluctuations. It is defined as
+yS =
+� �
+nfmax<| f|<mfmax|F[I(x)]|d2 f
+� �
+0<| f|<nfmax|F[I(x)]|d2 f
+where 1 > m > n > 0. This metric is defined as the ratio of higher to lower spatial frequency content, which is dependent upon
+aberration content, but independent of changes in overall brightness.
+12/16
+
+Microscope implementations
+Three microscopes were used to demonstrate and examine the MLAO method. The microscope implementations are briefly
+described here and fully elaborated in the supplementary document section 9A.
+In the home built 2-P system, a Newport-Spectra-Physics DeepSee femtosecond laser was used as the illumination with
+wavelength set at 850nm. Light was modulated by a Hamamatsu spatial light modulator before passing through a water
+immersion objective lens with NA equals to 1.15 and reaching the sample plane.
+A commercial Scientifica microscope system was used as the basis for our 3-P demonstration. In the 3-P system, a
+Ti:Sapphire laser passed through a pair of compressors and operated at 1300nm. Light was modulated by a Mirao 52E
+deformable mirror before reaching a water dipping objective lens with NA equals to 0.8.
+In the home built widefield 3D SIM system, two continuous wave lasers with wavelengths equal to 488 and 561nm were
+used as the illumination. Light was modulated by a ALPAO 69 deformable mirror before reaching a water dipping objective
+lens with NA of 1.1.
+Image acquisition and processing
+For 3-P imaging of live specimens, where motion was present, averaging was performed after inter-frame motion correction
+using TurboReg58. Time traces were taken from 200 raw frames captured at 4 Hz consecutively for each of the pre- and
+post-MLAO corrections.
+For the widefield/SIM results, widefield images were processed where indicated using the Fiji iterative deconvolution 3-D
+plugin59. A PSF for deconvolution was first generated using the Fiji plugin Diffraction PSF 3-D with settings the same as the
+widefield microscope. For the deconvolution, the following settings were applied: Wiener filter gamma equals to 0; both x-y
+and z direction low pass filter pixels equal to 1; maximum number of iterations equals to 100; and the iteration terminates when
+mean delta is smaller than 0.01%.
+The thresholds shown on the widefield image spectra were calculated by identifying the largest frequency in all x-y
+directions with image spectrum components higher than noise level. The noise level was identified by averaging the components
+of the highest spectral frequency, i.e. at the four corners of the image spectrum. Starting from the lowest frequency, each
+angular and radial fragment was averaged and compared to the noise level. The largest component which was still above the
+noise level was traced on the image spectra by the dashed line and identified as the threshold.
+Each 3D-SIM frame were extracted from a set of 15 image frames using the SoftWorx package (Applied Precision).60 The
+projected images were obtained by summing frames at different z depths into an extended focus xy image.
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+Acknowledgements
+This work was supported by grants from the European Research Council (to MJB: AdOMiS, No. 695140, to AMP: No. 852765),
+Wellcome Trust (to MJB: 203285/C/16/Z, to ID and MJB: 107457/Z/15/Z, to AMP: 204651/Z/16/Z, to HA: 222807/Z/21/Z),
+Engineering and Physical Sciences Research Council (to MJB: EP/W024047/1),
+Author contributions
+QH and MJB conceived the overall physics-informed approach including data pre-processing and bespoke NN architecture.
+MH, QH and MJB developed NN architectures and the training approach. QH, MH, MW, JA and DS developed the software
+packages. JW, QH, AMP set up the microscopes for the experimental demonstrations. QH performed the two-photon
+experiments, supervised by MJB. HA, JW and QH performed the three-photon experiments, supervised by AMP and MJB. JW,
+MW, DS, QH and RMP performed the widefield/SIM experiments, for which DG, TC and RMP prepared specimens, supervised
+by ID and MJB. QH performed data analysis. QH and MJB wrote the manuscript. All authors reviewed the manuscript.
+Additional information
+All experimental procedures involving animals were conducted in accordance with the UK animals in Scientific Procedures Act
+(1986).
+16/16
+
+Universal adaptive optics for
+microscopy through embedded neural
+network control: supplemental
+document
+1. MLAO PROCESS AND CNN ARCHITECTURE
+The MLAO aberration estimation process consists of two parts: image pre-processing to compute
+pseudo-PSFs from images and a CNN-based machine learning process for mode coefficient
+determination. A stack of M images over the same field of view, each with a different pre-
+determined bias phase modulation, was used to calculate pseudo-PSFs according to the procedure
+in the methods section. It was observed and understood that most of the information was
+contained within the central region of the calculated pseudo-PSFs. 1 A central patch of 32 × 32
+pixels was then cropped and used as the inputs to the CNN. Cropped pseudo-PSFs were processed
+by a sequence of convolutional layers (CL) with trainable 3 × 3 kernels, each followed by a local
+2 × 2 max-pooling and thus the x and y sizes were reduced by half but the stack size was increased
+twice going down each CL. For the input pseudo-PSFs and each of the CL outputs, a global
+max-pooling was applied and concatenated into a fully connected layer (FCL). This concatenated
+FCL was connected to the next FCL containing 32 neurons, which in turn was connected to the
+output layer, which produced the coefficients of the N chosen Zernike modes. The activation
+functions were chosen to be tanh and linear (only for the last layer connection FCL 32 and the
+output). The regularizer used was L1L2, the initializer was glorot-uniform and the optimizer
+was AdamW. The CNN architecture was built and the network training was conducted using
+TensorFlow.[1] As elaborated in the results section of the manuscript, M and N may be varied to
+suit different applications.
+The weights in the connection between the concatenated FCL and FCL32 (enclosed by a grey
+dashed square) were extracted and analysed to understand the physical significance of structures
+in the pseudo-PSFs in influencing the learning of the CNN. Further analysis of such weights is
+provided in Discussion of the main paper and section 4.
+1The process of calculating pseudo-PSFs can be interpreted as a deconvolution between two PSFs. Depending on the
+sampling size of the imaging system, most details of a deformed PSF typically occupy a central region of a few pixels. Most
+features of the pseudo-PSFs were thus captured within the central region.
+arXiv:2301.02647v1  [eess.IV]  6 Jan 2023
+
+Image stack over the 
+same field of view 
+(128×128×M)
+Cropped 
+pseudo-PSFs
+(32×32×M)
+CL
+16×16×8
+CL
+8×8×16
+CL
+4×4×32
+CL
+2×2×64
+M
+8
+16
+32
+64
+FCL
+32
+Concatenate
+FCL
+Output
+N
+Pseudo-PSF computation
+Convolution
++
+local maxpooling
+Global maxpooling
+Fully connected layers
+Calculated 
+pseudo-PSFs
+(128×128×M)
+CNN
+Fig. S1. A schematic illustration of the MLAO process and CNN architecture (enclosed by a
+black dashed square) designed for phase determination applications. CL: convolutional layer
+followed by local max-pooling; FCL: fully connected layer; M: number of input images and
+computed pseudo-PSFs; N: number of estimated output Zernike modes.
+2. ZERNIKE POLYNOMIALS AND EXAMPLE PSEUDO-PSFS
+A total of ten Zernike polynomials were used for aberration estimation and correction presented
+in the paper. A list of the polynomials, sequenced using Noll’s indices, were included in Figure
+S2 (a).
+Figure S2 (b) included some examples of pseudo-PSFs. It can be observed that when aberration
+size increases, the maximum pixel value of the Pseudo-PSF decreases; a global max-pooling of
+the pseudo-PSF extracts information related to the Strehl ratio of the PSFs. Pseudo-PSFs also have
+shapes that are related to the aberrated PSF shapes.
+3. PHYSICAL INFORMATION EMBEDDED IN THE CNN ARCHITECTURE
+As mentioned in the main paper, the bespoke CNN architecture embedded information about
+the physical effects of aberrations on images within the trainable parameters. To illustrate these
+phenomena, we designed six input patterns and two filters to calculate how values obtained
+after global max-poolings from different convolutional layers were related to the features of the
+patterns. Normally, the filters would be learned as part of the training process, but for illustrative
+purposes, we have defined them manually here.
+As shown in Figure S3, patterns 1 to 3 had the same general shape but varying sizes. They were
+all convolved with the same filter 1. Pattern 1 had the largest feature and the values obtained were
+almost constant throughout layers 1 to 5 (see Figure S3 (b)). Patterns 2 and 3 had smaller features
+and the extracted values reduced when moving further down the layers, where the embedded
+physical scales were more closely related to large scale features. Patterns 4 to 6 had the same
+general shape with four peaks positioned at the corners of a square. They were all convolved
+with filter 2, which shared a similar general shape. Pattern 4 had the smallest feature size and
+resulted a largest value in layer 2. Patterns 5 and 6 had larger feature sizes and resulted in largest
+values in layers 3 and 4, respectively. This trend confirms the expectation that layers later in the
+CNN probe larger scales in the input images. Note that all the patterns were designed in such a
+way that the maximum pixel reading (and thus the value max-pooled from layer 1) equalled to 1.
+2
+
+i = 5
+astigmatism
+8 
+coma
+9 
+trefoil
+10 
+trefoil
+11
+primary 
+spherical
+12
+secondary 
+astigmatism
+13 
+secondary 
+astigmatism
+22
+secondary 
+spherical
+6 
+astigmatism
+7 
+coma
+(a)
+(b)
+0 rad 
+0.5 rad
+i = 7 
+±1 rad
+i = 7 
+1.5 rad
+i = 7 
+2.5 rad
+i = 7 
+Aberration 
+Bias
+Pseudo-PSF
+Aberration 
+Bias
+Pseudo-PSF
+0 rad 
+0.5 rad
+i = 5 
+±1 rad
+i = 5 
+0.8 rad
+i = 5 
+1.5 rad
+i = 5
+0
+1 a.u.
+0
+π rad
+Fig. S2. (a) Zernike polynomials Noll’s index 5-13, 22. This is a whole list of the polynomi-
+als used for aberration determinations in the paper. (b) Examples of pseudo-PSFs. The first
+column is the input aberration and the second column is the bias mode used in pseudo-PSFs
+generation.
+3
+
+∗ filter1
+∗ filter2
+Local 
+max-pooling
+Convolution
+Layer
+1
+Layer
+2
+Layer
+3
+Layer
+4
+Layer
+5
+Pattern
+1
+Pattern
+4
+Pattern
+5
+Pattern
+6
+(a)
+(b)
+Pattern
+2
+Pattern
+3
+Layer 1-5
+Normalised max-pooling value
+Pattern 1
+2
+3
+4
+5
+6
+Single
+pixel
+feature
+Small
+scale
+feature
+Large
+scale
+feature
+0
+1 a.u.
+3×3
+3×3
+Fig. S3. Demonstrations of the link between feature sizes and convolutional layers. (a) Pattern
+1 to 6 each underwent a series of convolutions followed by a 2 × 2 local max-pooling. Pattern
+1 to 3 were convolved with filter 1 and pattern 4 to 6 were convolved with filter 2. For each
+layer, a global max-pooling were carried out to extract the maximum reading of each layer. The
+physical interpretations of the extracted values of the different layers were related to Strehl
+ratio (layer 1) and shapes with features ranging from small scales (layer 2) to large scales (layer
+5). The extracted readings was normalised with the readings of their respective previous layer
+and displayed in (b). The horizontal axis of each plot in (b) indicates from which layer the
+normalised maximum reading (indicated by the vertical axis) was extracted from.
+4
+
+ast2 2-P
+0
+0.4
+2N 2-P
+0
+0.4
+ast4 3-P
+0
+0.4
+4N 3-P
+0
+0.4
+ast2 widefield
+0
+0.4
+2N widefield
+0
+0.4
+Layer 1-5
+RMS of weights
+Fig. S4. Analysis of the weight distributions across convolutional layers in the CNNs trained
+for different biasing schemes and microscopes.
+4. WEIGHT ANALYSIS OF DIFFERENT TRAINED NEURAL NETWORKS
+Figure S4 shows the root-mean-square (RMS) values of the weights at the output of each section
+of the concatenated FCL following the convolutional layers of the CNN. These weights encode
+information about physical phenomena in the pseudo-PSF that is related to the spatial effects
+of aberrations on images. Higher numbered layers correspond to larger scale features. Similar
+distributions are seen for all of the ast CNNs class and all of the 2/4N class. Most notably, it can
+be seen that the 2/4N networks all carry heavier weights in layer 1, which is most similar to the
+Strehl ratio variations of the PSFs.
+5. TRAINABLE NEURAL NETWORK PARAMETERS
+The bespoke NN and data pre-processing steps were designed with knowledge of the physical
+basis of image formation. This permitted signficant reduction in NN complexity compared
+to previous methods for aberration estimation. This architecture not only allowed improved
+performances, providing insights on internal workings, but also had a structure size orders of
+magnitude smaller than common NNs used in similar applications (see the comparison in Table
+S1). This will be beneficial for future applications as NN with fewer trainable parameters would
+generally require less training data and a shorter training time. Furthermore, the simplified design
+means that there is greater potential for extending the method to more challenging applications.
+5
+
+Neural network method
+Number of trainable parameters
+ResNet[2]
+>0.27M
+Inception V3/ GoogLeNet[3, 4]
+23.6M
+Xception[5, 6]
+22.8M
+Deep Image Prior[7]
+2M
+PHASENET[8, 9]
+1M
+MLAO in this paper
+0.028M to 0.032M
+Table S1. A list of NNs used in image processing and phase determination with their number
+of trainable parameters. Inception V3[3], Xception[5] and PHASENET[8] have been directly
+demonstrated for phase determination. ResNet is a common basic NN architecture that has
+been used in many different image processing and phase determination architectures[8]. A 20
+layer ResNet is the smallest architecture proposed in the ResNet paper[2] that has ∼0.27M
+trainable parameters. Deep Image Prior employs a U-Net architecture that is a commonly
+used in many biomedical image processing applications. Deep phase decoder[10], a network
+designed for wavefront and image reconstruction, was also inspired and adapted from Deep
+Image Prior.
+6. CHOICE OF BIAS MODE
+The simplest MLAO implementation uses a pair of biased images as the input. The nature of
+the bias aberrations is a design choice. In order to investigate this, we tested individual Zernike
+modes as the bias and trained different MLAO networks with identical architecture to correct the
+same randomly generated aberrations. The loss function of the different NNs during training
+was shown in Fig. S5 (a). Results from correcting 20 randomly generated aberrations were shown
+in Fig. S5 (b).
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+0
+1k
+2k
+3k
+4k
+5k
+6k
+5
+8
+7
+6
+11
+i=
+Training epochs
+RMS loss function
+0
+0.5
+1
+1.5
+2
+2.5
+Aberration RMS / rad
+pre
+correction
+4
+5
+6
+7
+8
+11
+Bias mode i
+(a)
+(b)
+188nm/px
+5μm aberration
+free
+aberration
+1.88 rad
+Fig. S5. Testing Zernike modes as choice of bias aberration. (a) A plot of the root mean square
+(RMS) loss function against the number of epochs when training NNs of the same architec-
+ture from the same dataset but using different bias modes. (b) Statistical results of testing the
+trained NNs to correct the same sets of random aberrations over 2-P microscope images of
+beads. Twenty randomly generated aberrations consisting five Zernike modes and RMS value
+smaller than 2.2 radians were introduced for correction (dark gray bar). The remaining aberra-
+tions after correction by different networks were averaged and shown in the figure; standard
+deviations of the remaining aberrations are represented as the error bar. Insets showed an
+example of the FOV when no aberration was introduced and an example when 1.88 rad of
+aberration was introduced into the system.
+The two networks using oblique and vertical astigmatism (index i =5 and 6) converged to
+similar loss function during training (Fig. S5 (a)). The same two networks also gave similar
+6
+
+averaged remaining aberrations during experimental aberration correction on a bead sample
+(Fig. S5 (b)). The two networks using vertical and horizontal coma (index 7 and 8) also showed
+mutually similar values. This was expected as these pairs of modes (5 and 6; 7 and 8) differ only
+by rotation, which should not have an effect on how effective the networks determine aberrations.
+From these results, the NNs using astigmatism as the bias modes converged to the smallest
+loss function during training. This possibly suggested that the astigmatism modes, on average,
+allowed the network to learn more from the training data. It was also observed from the ex-
+perimental results where, in general, the NN obtained the smallest remaining aberrations. We
+therefore chose to use astigmatism as the modulation modes for the two-bias NN methods in the
+experiments conducted in this paper.
+7. TOLERANCE TO SAMPLING RATE
+As described in the paper, the networks for scanning microscopy were trained on simulated
+dataset with pixel sampling within the range of 1.0× to 1.2× of the base sampling rate (see the
+method section in the main paper for more details). However in many practical cases, there can
+be uncertainty in pixel sampling for a system or constraints on the sampling rates that may be
+used. We hence tested the tolerance of our networks to pixel sampling rates outside the range of
+the training dataset (see Fig. S6).
+0
+1
+2
+3
+4
+pre correction
+2N+1 conv
+ast2 MLAO
+2N MLAO
+219nm/px
+5μm
+188nm/px
+5μm
+156nm/px
+5μm
+5μm
+125nm/px
+Aberration RMS / rad
+Image sampling rate
+Fig. S6. Testing of robustness to pixel sampling. Statistical results of remaining aberrations
+before (red plot) and after correction using 2N+1 conv, ast2 MLAO and 2N MLAO methods.
+The results were averaged from 20 randomly generated aberrations and the SDs were shown
+as the error bars. The same algorithms were used to correct the same aberrations over images
+collected at different pixel sampling as shown by the horizontal axis. Insets show examples of
+the images collected at different sampling rates.
+In this case, 188nm per pixel was close to the sampling of the generated dataset on which the
+two NNs were trained. When images were sampled at a smaller or larger rate, ast2 MLAO and
+2N MLAO were still able to correct aberrations, but were slightly less effective.
+8. FURTHER THREE-PHOTON MICROSCOPE DEMONSTRATIONS
+Figure S7 showed the performance of the ast4 MLAO algorithm, for imaging neuronal activity
+at a depth of 670 µm in a mouse brain. Despite the very low SNR of the image data, the image
+quality and cell activity data were considerably improved.
+9. DETAILS OF THE EXPERIMENTAL METHODOLOGY
+Three optical systems, a 2-P, 3-P and widefield microscope, were used for demonstrations on
+different samples. Networks with different parameter settings are also adjusted for different
+applications.
+7
+
+vi
+ 
+A
+B
+C
+D
+E
+F
+G
+H
+Post MLAO it:3
+Pre MLAO
+Post MLAO it:3
+1
+2
+Post MLAO
+it:3
+Pre
+MLAO
+Post
+MLAO
+it:3
+iii
+1
+2
+Pre
+MLAO
+0
+50
+100
+0
+50
+100
+time / s
+time / s
+yS
+yI
+Images
+Images
+it:1
+it:2
+it:3
+iv
+v
+it:2
+it:1
+20μm
+A
+B
+C
+D
+E
+F
+G
+H
+GCaMP at 670 μm
+vii
+Pre MLAO
+i
+-1 rad 
+-0.5 rad
++0.5 rad
++1 rad
+ast4 MLAO
+1
+3
+2
+4
+Bias mode i=5
+ii
+Fig. S7. Three-photon microscopy imaging GCaMP neuronal activities at depth 670µm. Power
+at sample was 44 mW. Wavefronts inserted to the figures showed the phase modulations ap-
+plied by the DM at the relevant step; the common scale is indicated by the colorbar above v. i
+and iii show respectively before and after ast4 MLAO correction through three iterations (it:1
+to 3), 200 frame averages after motion correction. In iii, time traces shown to the right and
+bottom were taken from the marked lines (1) and (2) respectively. ii 1-4 shows example single-
+frame images used as inputs to the ast4 MLAO correction with the corresponding bias modes
+as insets. iv and v show the intensity metric (yI) and the sharpness metric (yS), respectively,
+calculated from single image frames, against the number of images acquired for three iterations
+ast4 MLAO. vi shows the Calcium activity of 8 cells (A-H marked on i). vii shows a histogram
+of the 200 frames collected pre MLAO (blue), post MLAO (red) and the differences between pre
+and post MLAO (yellow).
+8
+
+A
+0
+50
+100
+time / s0
+S
+time / s
+50
+1002
+1
+0
+-1
+-2
+-3based metric (y)
+Sharpness
+*ast4 MLAO
+0
+4
+8
+12
+number of
+sample exposuresFluorescence
+intensity (y,)
+*一ast4 MLAO
+0
+4
+8
+12
+number of
+sample exposures2
+1
+0
+-1
+-2
+-32
+1
+0
+-1
+-2
+-3×105
+Pre MLAO
+Post MLAO it:5
+Difference between
+post and pre MLAO
+0T
+1
+0
+2
+-T
+1
+2×105
+PreMLAO
+Post MLAO it:3
+Difference between
+post and pre MLAO0
+50
+100
+time / sA. Experimental setups
+c 
+Widefield
+3-D SIM
+microscope
+f200
+SLM
+DM
+f175
+Obj:W-D
+f60
+BX
+f175
+f125
+f200
+FS
+AP
+f175
+f75
+f400
+f50
+C1
+C2
+M
+M
+M
+PL
+M
+M
+DF
+DF
+M
+PR
+/2
+f175
+f200
+f200
+EF
+EF
+SF
+EF
+LS488
+LS561
+M
+ST
+M
+/2
+FS laser 
+HWP
+PBS
+Dump
+compressor
+M
+f50
+f200
+M
+Galvo scanners
+f206
+f30
+DF
+PZ
+EF
+PMT
+Obj:W-D
+b
+Three-photon
+microscope
+f500
+f75
+auto-
+correlator
+DM
+FS laser 
+HWP
+PBS
+Dump
+f50
+f150
+M
+M
+Galvo x
+FM
+f75
+f75
+Galvo y
+f75
+f120
+M
+M
+f150
+f75
+DF
+EF
+PMT
+PZ
+f150
+f100
+SLM
+M
+f200
+f200
+FS
+Obj:W-I
+a 
+Two-photon
+microscope
+DF
+DF
+FM
+FM
+M
+Fig. S8. Configuration of the (a) 2-P (b) 3-P (c) widefield 3-D SIM microscope. (Caption contin-
+ued on the next page.)
+9
+
+Femtosecond (FS) Laser; Continuous-wave lasers with wavelenths 488nm and 561nm (LS488 and
+LS561); half wave plate (HWP); polarisation beam splitter (PBS); laser beam dump (Dump); lens
+with focal length = x mm (fx); broadband dielectric mirror (M); flip mirror (FM); Hamamatsu
+spatial light modulator (SLM); Mirao 52E deformable mirror (DM) in the 3-P system; ALPAO
+69 deformable mirror (DM) in the widefield 3-D SIM system; aperture (AP); spatial filter (SF);
+field stopper (FS); X galvanometer (Galvo x); Y galvanometer (Galvo y); beam expansion (BX);
+half waveplate (λ/2); linear polariser (PL); polarisation rotator (PR); Olympus 40× numerical
+aperture (NA) 1.15 water immersion objective lens (Obj:W-I) used in the 2-P system; Nikon 16×
+NA 0.8 water dipping objective lens (Obj:W-D) used in the 3-P system; Olympus 60× NA 1.1
+water dipping objective lens (Obj:W-D) in the widefield 3-D SIM system; Z-piezo translation stage
+(PZ); X-Y-Z translational sample mounting stage (ST); Dichroic filter (DF) allow emission signal
+from fluorophores to be reflected through emission filter (EF) into a photo-multiplier tube (PMT)
+in a multi-photon system; cameras (C1 and C2)
+B. Sample preparation
+The 3-P results were collected from imaging male (Lhx6-eGFP)BP221Gsat; Gt(ROSA)26Sortm32(CAG-
+COP4*H134R/EYFP)Hze mice (static imaging) and female and male Tg(tetO-GCaMP6s)2Niell
+mice (calcium imaging). Mice were between 8-12 weeks of age when surgery was performed. The
+scalp was removed bilaterally from the midline to the temporalis muscles, and a metal headplate
+with a 5 mm circular imaging well was fixed to the skull with dental cement (Super-Bond C&B,
+Sun-Medical). A 4–5 mm circular craniotomy was performed during which any bleeding was
+washed away with sterile external solution or staunched with Sugi-sponges (Sugi, Kettenbach).
+Cranial windows composed of 4 or 5 mm circular glass coverslips were press-fit into the cran-
+iotomy, sealed to the skull by a thin layer of cyanoacrylate (VetBond) and fixed in place by dental
+cement.
+The widefield 3-D SIM results were collected from imaging NMJ of Drosophila larvae. For
+the immunofluorescence sample with one coloured channel, it was prepared as previously [11].
+Crawling 3rd instar larvae of wildtype Oregon-R Drosophila melanogaster were dissected on a
+Sylgard-coated Petri Dish in HL3 buffer with 0.3mM Ca2+ to prepare larval fillet [12]. Then, the
+larval fillet samples were fixed in Paraformaldehyde 4% in PBS containing 0.3% (v/v) Triton
+X-100 (PBSTX) for 30 minutes. The brains were removed post-fixation, and the fillet samples were
+transferred to a Microcentrifuge tube containing PBSTX for 45 minutes of permeabilisation. The
+samples were stained with HRP conjugated to Alexa Fluor 488 and DAPI for 1 hour at room
+temperature (21C◦). After the washes, the samples were mounted in Vectashield.
+For the 3-D SIM results collected on the Drosophila larvae sample with two coloured channels,
+it was prepared by following the protocol presented in [11]. 3rd instar Drosophila melanogaster
+larvae (Brp-GFP strain) were dissected in HL3 buffer with 0.3mM Ca2+ to prepare a so-called
+larval fillet, and the larval brains were removed. After this, larvae were stained for 15 minutes
+with HRP conjugated to Alexa Fluor 568 to visualise the neurons, washed with HL3 buffer with
+0.3mM Ca2+ and imaged in HL3 buffer without Ca2+ to prevent the larvae from moving.
+C. Network parameters
+Table S2 showed the network settings used in different imaging applications.
+10
+
+Results in
+Method label
+M
+N
+Bias
+Bias
+Corrected
+modes, i
+depths
+modes, i
+Fig. 2 (a, c, f)
+ast2 MLAO
+2
+5
+5
+±1 rad
+5–8, 11
+Fig. S3
+Fig. 2 (a, c, f)
+2N MLAO
+10
+5
+5–8, 11
+±1 rad
+5–8, 11
+Fig. 2 (b, d, e)
+ast2 MLAO
+2
+9
+5
+±1 rad
+5–13
+Fig. 2 (b, d, e)
+2N MLAO
+18
+9
+5–13
+±1 rad
+5–13
+Fig. 3 (a, b)
+ast4 MLAO
+4
+7
+5
+±0.5
+5–11
+Fig. S4
+±1 rad
+Fig. 3 (a)
+4N MLAO
+28
+7
+5–11
+±0.5
+5–11
+±1 rad
+Fig. 4
+ast2 MLAO
+2
+8
+5
+±1 rad
+5–11, 22
+Fig. 4
+2N MLAO
+2
+8
+5–11, 22
+±1 rad
+5–11, 22
+Table S2. A list of MLAO parameters chosen for different imaging applications. The Zernike
+modes were sequenced using Noll’s indices.
+REFERENCES
+1.
+M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis,
+J. Dean, M. Devin, S. Ghemawat, I. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Joze-
+fowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. Murray, C. Olah,
+M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. Tucker, V. Vanhoucke, V. Va-
+sudevan, F. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng,
+“TensorFlow: Large-scale machine learning on heterogeneous systems,” (2015). Software
+available from tensorflow.org.
+2.
+K. He, X. Zhang, S. Ren, and J. Sun, “Deep residual learning for image recognition,” in
+Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), (2016).
+3.
+T. Andersen, M. Owner-Petersen, and A. Enmark, “Neural networks for image-based wave-
+front sensing for astronomy,” Opt. Lett. 44, 4618–4621 (2019).
+4.
+C. Szegedy, W. Liu, Y. Jia, P. Sermanet, S. Reed, D. Anguelov, D. Erhan, V. Vanhoucke, and
+A. Rabinovich, “Going deeper with convolutions,” in 2015 IEEE Conference on Computer Vision
+and Pattern Recognition (CVPR), (2015), pp. 1–9.
+5.
+P. A. Khorin, A. P. Dzyuba, P. G. Serafimovich, and S. N. Khonina, “Neural networks
+application to determine the types and magnitude of aberrations from the pattern of the
+point spread function out of the focal plane,” J. Physics: Conf. Ser. 2086, 012148 (2021).
+6.
+F. Chollet, “Xception: Deep learning with depthwise separable convolutions,” (2016).
+7.
+D. Ulyanov, A. Vedaldi, and V. Lempitsky, “Deep image prior,” Int. J. Comput. Vis. 128,
+1867–1888 (2020).
+8.
+D. Saha, U. Schmidt, Q. Zhang, A. Barbotin, Q. Hu, N. Ji, M. J. Booth, M. Weigert, and E. W.
+Myers, “Practical sensorless aberration estimation for 3D microscopy with deep learning,”
+Opt. Express 28, 29044–29053 (2020).
+9.
+D. Saha and U. Schmidt, “Phasenet,” https://github.com/mpicbg-csbd/phasenet (2020).
+10.
+E. Bostan, R. Heckel, M. Chen, M. Kellman, and L. Waller, “Deep phase decoder: self-
+calibrating phase microscopy with an untrained deep neural network,” Optica 7, 559–562
+(2020).
+11.
+J. R. Brent, K. M. Werner, and B. D. McCabe, “Drosophila larval nmj dissection.” J Vis Exp
+(2009).
+12.
+R. M. Parton, A. M. Vallés, I. M. Dobbie, and I. Davis, “Drosophila Larval Fillet Preparation
+and Imaging of Neurons,” Cold Spring Harb. Protoc. 2010, pdb.prot5405 (2010).
+11
+

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+arXiv:2301.11801v1  [physics.gen-ph]  12 Jan 2023
+4D Einstein–Gauss–Bonnet gravity coupled to modified
+logarithmic nonlinear electrodynamics
+Sergey Il’ich Kruglov 1
+Department of Physics, University of Toronto,
+60 St. Georges St., Toronto, ON M5S 1A7, Canada
+Department of Chemical and Physical Sciences, University of Toronto,
+3359 Mississauga Road North, Mississauga, Ontario L5L 1C6, Canada
+Abstract
+Spherically symmetric solution in 4D Einstein–Gauss–Bonnet grav-
+ity coupled to modified logarithmic nonlinear electrodynamics (Mod-
+LogNED) is found.
+This solution at infinity possesses the charged
+black hole Reissner–Nordstr¨om behavior.
+We study the black hole
+thermodynamics, entropy, shadow, energy emission rate and quasi-
+normal modes. It was shown that black holes can possess the phase
+transitions and at some range of event horizon radii black holes are
+stable. The entropy has the logarithmic correction to the area law.
+The shadow radii were calculated for variety of parameters. We found
+that there is a peak of the black hole energy emission rate. The real
+and imaginary parts of the quasinormal modes frequencies were cal-
+culated. The energy conditions of ModLogNED are investigated.
+Keywords: Einstein−Gauss−Bonnet gravity; nonlinear electrodynamics;
+Hawking temperature; entropy; heat capacity; black hole shadow; energy
+emission rate; quasinormal modes
+1
+Introduction
+Nowadays, there are many theories of gravity that are alternatives to Ein-
+stein’s theory [1, 2]. The motivation of generalisations of Einstein’s theory of
+General Relativity (GR) is to resolve some problems in cosmology and astro-
+physics. One of important modification of GR is the Einstein–Gauss–Bonnet
+1E-mail: [email protected]
+1
+
+(EGB) theory [3, 4, 5, 6]. EGB theories do not include extra degrees of free-
+dom and field equations have second derivatives of the metric. These theories
+also prevent Ostrogradsky instability [7]. The four dimensional (4D) EGB
+theory, that includes the Einstein–Hilbert action plus GB term, is a particu-
+lar case of the Lovelock theory. It represents the generalization of Einstein’s
+GR for higher dimensions and EGB theory results covariant second-order
+field equations. The GB part of the action possesses higher order curvature
+terms.
+It is worth mentioning that at low energy the action of the het-
+erotic string theory includes higher order curvature terms [8, 9, 10, 11, 12].
+Therefore, it is of interest to study gravity action with the GB term. The
+GB term is a topological invariant in 4D and before a regularization it does
+not contribute to the equation of motion. But Glavan and Lin [13] showed
+that re-scaling the coupling constant, after the regularization, GB term con-
+tributes to the equation of motion. The consistent theory of 4D EGB gravity,
+was proposed in [14, 15, 16], is in agreement with the Lovelock theorem [5]
+and possesses two dynamical degrees of freedom breaking the temporal dif-
+feomorphism invariance. It is worth noting that the theory of [14, 15, 16],
+in the spherically-symmetric metrics, gives the solution which is a solution
+in the framework of [13] scheme (see [17]). Some aspects of 4D EGB gravity
+were considered in [18]. The black hole and wormhole type solutions in the
+effective gravity models, including higher curvature terms, were obtained in
+[19].
+Here, we study the black hole thermodynamics, the entropy, the shadow,
+the energy emission rate and quasinormal modes in the framework of the
+ModLogNED model (proposed in [20]) coupled to 4D EGB gravity. It is
+worth noting that ModLogNED model is simpler compared with logarithmic
+model [21] and generalized logarithmic model [22] because the mass and met-
+ric functions here are expressed through simple elementary functions. The
+black hole quasinormal modes, deflection angles, shadows and the Hawking
+radiation were studied in [23, 24, 25, 26, 27, 28, 29].
+The structure of the paper is as follows. In Sect. 2, we obtain the spher-
+ically symmetric solution of black holes in the 4D EGB gravity coupled to
+ModLogNED. At infinity the Reissner−Nordstr¨om behavior of the charged
+black holes takes place. The black hole thermodynamics is studied in Sect. 3.
+We calculate the Hawking temperature, the heat capacity and the entropy.
+At some parameters second order phase transitions occur. The entropy in-
+cludes the logarithmic correction to Bekenstein–Hawking entropy. In Sect. 4
+the black hole shadow is investigated. We calculate the photon sphere, the
+2
+
+event horizon, and the shadow radii. The black hole energy emission rate is
+investigate in Sect. 5. In Sect. 6 we study quasinormal modes and find com-
+plex frequencies. Section 7 is a summary. In Appendix A energy conditions
+of ModLogNED model are investigated.
+2
+4D EGB model
+The action of EGB gravity coupled to nonlinear electrodynamics (NED) in
+D-dimensions is given by
+I =
+�
+dDx√−g
+�
+1
+16πG (R + αLGB) + LNED
+�
+,
+(1)
+where G is the Newton’s constant, α has the dimension of (length)2. The
+Lagrangian of ModLogNED, proposed in [20], is
+LNED = −
+√
+2F
+8πβ ln
+�
+1 + β
+√
+2F
+�
+,
+(2)
+where we use Gaussian units. The parameter β (β ≥ 0) possesses the di-
+mension of (length)2, Fµν = ∂µAν − ∂νAµ is the field strength tensor, and
+F = (1/4)FµνF µν = (B2 − E2)/2, where B and E are the induction mag-
+netic and electric fields, correspondingly. Making use of the limit β → 0 in
+Eq. (2), we arrive at the Maxwell’s Lagrangian LM = −F/(4π). The GB
+Lagrangian has the structure
+LGB = RµναβRµναβ − 4RµνRµν + R2.
+(3)
+By varying action (1) with respect to the metric we have EGB equations
+Rµν − 1
+2gµνR + αHµν = −8πGTµν,
+(4)
+Hµν = 2
+�
+RRµν − 2RµαRα
+ν − 2RµανβRαβ − RµαβγRαβγ
+ν
+�
+− 1
+2LGBgµν,
+(5)
+where Tµν is the stress (energy-momentum) tensor. To obtain the solution
+of field equations we need to use an ansatz for the interval. But the va-
+lidity of Birkhoff’s theorem [30] for our case of 4D EGB gravity coupled to
+ModLogNED model is not proven. Therefore, to simplify the problem we
+consider magnetic black holes with the static spherically symmetric metric
+3
+
+in D dimension. In addition, we assume that components of the interval are
+restricted by the relation g11 = g−1
+00 . Thus, we suppose that the metric has
+the form
+ds2 = −f(r)dt2 + dr2
+f(r) + r2dΩ2
+D−2.
+(6)
+The dΩ2
+D−2 is the line element of the unit (D − 2)-dimensional sphere. By
+following [13] we replace α by α → α/(D − 4) and taking the limit D → 4.
+We study the magnetic black holes and find F = q2/(2r4), where q is a
+magnetic charge. Then the magnetic energy density becomes [20]
+ρ = T 0
+0 = −L =
+√
+2F
+8πβ ln
+�
+1 + β
+√
+2F
+�
+=
+q
+8πβr2 ln
+�
+1 + βq
+r2
+�
+.
+(7)
+At the limit D → 4 and from Eq. (4) we obtain
+r(2αf(r) − r2 − 2α)f ′(r) − (r2 + αf(r) − 2α)f(r) + r2 − α = 2r4Gρ.
+(8)
+By virtue of Eq. (7 ) one finds
+4π
+� r
+0 r2ρdr = mM + q
+2β
+�
+r ln
+�
+1 + βq
+r2
+�
+− 2
+�
+βq arctan
+�√βq
+r
+��
+,
+(9)
+mM = 4π
+� ∞
+0
+r2ρdr = q
+2β
+� ∞
+0
+ln
+�
+1 + βq
+r2
+�
+dr = πq3/2
+2√β ,
+(10)
+where mM is the black hole magnetic mass. Making use of Eqs. (9) and (10)
+we obtain the solution to Eq. (8)
+f(r) = 1 + r2
+2α
+
+1 ±
+�
+1 + 8αG
+r3 (m + h(r)
+
+ ,
+h(r) = mM + q
+2β
+�
+r ln
+�
+1 + βq
+r2
+�
+− 2
+�
+βq arctan
+�√βq
+r
+��
+,
+(11)
+where m is the constant of integration (the Schwarzschild mass) and the total
+black hole mass is M = m+mM which is the ADM mass. At the limit β → 0
+one has
+lim
+β→0 h(r) = mM − q2/2r.
+4
+
+Then making use of Eq. (11), for the negative branch, we obtain
+lim
+β→0,α→0 f(r) = 1 − 2MG
+r
++ Gq2
+r2 ,
+that corresponds to GR coupled to Maxwell electrodynamics (the Reissner–
+Nordstr¨om solution).
+It is worth mentioning that for spherically symmetric D-dimensional line
+element (6), the Weyl tensor of the D-dimensional spatial part becomes zero
+[17]. Therefore, solution (11) corresponds to the consistent theory [14, 15, 16].
+By introducing the dimensionless variable x = r/√βq, Eq. (11) is rewritten
+in the form
+f(x) = 1 + Cx2 ± C
+�
+x4 + x(A − Bg(x)),
+(12)
+where
+A = 8αGM
+(βq)3/2, B = 4αG
+β2 , C = βq
+2α, g(x) = 2 arctan
+�1
+x
+�
+− x ln
+�
+1 + 1
+x2
+�
+.
+(13)
+We will use the negative branch in Eqs. (11) and (12) with the minus sign
+of the square root to have black holes without ghosts. As α → 0, r → ∞ the
+metric function f(r) (11), for the negative branch, becomes
+f(r) = 1 − 2MG
+r
++ Gq2
+r2 + O(r−3),
+(14)
+showing, at infinity, the Reissner−Nordstr¨om behavior of the charged black
+holes. The plot of function (12) for a particular chose of parameters, A = 15,
+C = 1 (as an example), is depicted in Fig.
+1.
+The expansion (14) was
+observed in other models (see, for example, [31]). According to Fig. 1 there
+can be two horizons or one (the extreme) horizon of black holes.
+3
+The black hole thermodynamics
+To study the black hole thermal stability we will calculate the Hawking tem-
+perature
+TH(r+) = f ′(r) |r=r+
+4π
+,
+(15)
+5
+
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+−0.5
+0
+0.5
+1
+1.5
+2
+2.5
+3
+x
+f(x)
+ 
+ 
+B=23
+B=27.5
+B=32
+Figure 1: The plot of the function f(x) for A = 15, C = 1.
+where r+ is the event horizon radius (f(r+) = 0). From Eqs. (12) and (15)
+one finds the Hawking temperature
+TH(x+) =
+1
+4π√βq
+�2Cx2
++ − 1 + BC2x2
++g′(x+)
+2x+(1 + Cx2
++)
+�
+,
+(16)
+g′(x+) = − ln
+�
+1 + 1
+x2
++
+�
+.
+Parameter A was substituted into Eq. (15) from equation f(x+) = 0. The
+plot of the dimensionless function TH(x+)√βq versus x+, for the case C = 1,
+is represented in Fig. 2. Figure 2 shows that the Hawking temperature is
+positive for some interval of event horizon radii. We will calculate the heat
+capacity to study the black hole local stability
+Cq(x+) = TH
+��� ∂S
+∂TH
+�
+q
+= ∂M(x+)
+∂TH(x+) = ∂M(x+)/∂x+
+∂TH(x+)/∂x+
+,
+(17)
+6
+
+1
+2
+3
+4
+5
+6
+7
+−0.05
+−0.04
+−0.03
+−0.02
+−0.01
+0
+0.01
+0.02
+0.03
+x+
+TH β1/2 q1/2
+ 
+ 
+B=2
+B=4
+B=6
+Figure 2: The plot of the function TH(x+)√βq at C = 1.
+where M(x+) is the black hole gravitational mass as a function of the event
+horizon radius. Making use of equation f(x+) = 0 we obtain the black hole
+mass
+M(x+) = (βq)3/2
+8αG
+�1 + 2Cx2
++
+C2x+
++ Bg(x+)
+�
+.
+(18)
+With the help of Eqs. (16) and (18) one finds
+∂M(x+)
+∂x+
+= (βq)3/2
+8αG
+�2Cx2
++ − 1
+C2x2+
++ Bg′(x+)
+�
+,
+(19)
+∂TH(x+)
+∂x+
+=
+1
+8π√βq
+�5Cx2
++ − 2C2x4
++ + 1
+x2
++(1 + Cx2
++)2
++BC2[g′(x+)(1 − Cx2
++) + x+g′′(x+)(1 + Cx2
++)]
+(1 + Cx2+)2
+�
+,
+(20)
+7
+
+g′′(x+) =
+2
+x+(x2+ + 1).
+In accordance with Eq. (17) the heat capacity has a singularity when the
+Hawking temperature possesses an extremum (∂TH(x+)/∂x+ = 0). Equa-
+tions (16) and (17) show that at one point, x+ = x1, the Hawking temper-
+ature and heat capacity become zero and the black hole remnant mass is
+formed. In another point x+ = x2 with ∂TH(x+)/∂x+ = 0, the heat capac-
+ity has a singularity where the second-order phase transition occurs. Black
+holes in the range x2 > x+ > x1 are locally stable but at x+ > x2 black holes
+are unstable. Making use of Eqs. (17), (19) and (20) the heat capacity is
+depicted in Fig. 3 at C = 1. The Hawking temperature and heat capacity
+1.5
+2
+2.5
+3
+3.5
+4
+−5000
+−4000
+−3000
+−2000
+−1000
+0
+1000
+2000
+3000
+x+
+Cqα G/(β2 q2)
+ 
+ 
+B=2
+B=4
+B=6
+Figure 3: The plot of the function Cq(x+)αG/(β2q2) at C = 1.
+are positive in the range x2 > x+ > x1 and locally stable.
+From the first law of black hole thermodynamics dM(x+) = TH(x+)dS +
+8
+
+φdq we obtain the entropy at the constant charge [32]
+S =
+� dM(x+)
+TH(x+) =
+�
+1
+TH(x+)
+∂M(x+)
+∂x+
+dx+.
+(21)
+From Eqs. (16), (19) and (21) one finds the entropy
+S = π(βq)2
+C2αG
+� 1 + Cx2
++
+x+
+dx+ = πr2
++
+G + 4πα
+G ln
+� r+
+√βq
+�
++ Const,
+(22)
+with the integration constant Const. The integration constant can be chosen
+in the form
+Const = 2πα
+G ln
+�πqβ
+G
+�
+.
+(23)
+Then making use of Eqs. (22) and (23) we obtain the black hole entropy
+S = S0 + 2πα
+G ln (S0) ,
+(24)
+with S0 = πr2
++/G being the Bekenstein–Hawking entropy and with the log-
+arithmic correction but without the coupling β. One can find same entropy
+(24) in other models [33, 34, 35].
+4
+Black holes shadows
+The light gravitational lensing leads to the formation of black hole shadow
+and a black circular disk. The Event Horizon Telescope collaboration [36] ob-
+served the image of the super-massive black hole M87*. A neutral Schwarzschild
+black hole shadow was studied in [37]. We will consider photons moving in the
+equatorial plane, ϑ = π/2. With the help of the Hamilton−Jacobi method
+one obtains the equation for the photon motion in null curves [38]
+H = 1
+2gµνpµpν = 1
+2
+�L2
+r2 − E2
+f(r) +
+˙r2
+f(r)
+�
+= 0,
+(25)
+where pµ is the photon momentum ( ˙r = ∂H/∂pr). The photon energy and
+angular momentum are constants of motion, and they are E = −pt and
+L = pφ, correspondingly. We can represent Eq. (25) as
+V + ˙r2 = 0,
+V = f(r)
+�L2
+r2 − E2
+f(r)
+�
+.
+(26)
+9
+
+Photon circular orbit radius rp can be found from equation V (rp) = V ′(r)|r=rp =
+0. Making use of Eq. (26) we find
+ξ ≡ L
+E =
+rp
+�
+f(rp)
+,
+f ′(rp)rp − 2f(rp) = 0,
+(27)
+where ξ is the impact parameter. For a distant observer as r0 → ∞, the
+shadow radius becomes rs = rp/
+�
+f(rp) (rs = ξ). By virtue of Eq. (12) and
+equation f(r+) = 0 we obtain parameters A, B and C versus x+
+A = 1 + 2Cx2
++
+C2x+
++ Bg(x+),
+B = AC2x+ − 2Cx2
++ − 1
+C2x+g(x+)
+,
+C =
+x2
++ +
+�
+x4+ + x+(A − Bg(x+))
+x+(A − Bg(x+))
+,
+(28)
+with x+ = r+/√βq. The functions (28) plots are depicted in Fig. 4. In
+accordance with Fig. 4, Subplot 1, event horizon radius x+ increases when
+parameter A increases and Subplot 2 indicates that if parameter B increases,
+the event horizon radius decreases. According to Subplot 3 of Fig. 4, when
+parameter C increases the event horizon radius x+ also increases.
+The photon sphere radii (xp), the event horizon radii (x+), and the shadow
+radii (xs) for A = 15 and C = 1 are presented in Table 1. It is worth noting
+that the null geodesics radii xp correspond to the maximum of the potential
+V (r) (V ′′ ≤ 0) and belong to unstable orbits.
+Table 1 shows that when
+Table 1: The event horizon, photon sphere and shadow dimensionless radii
+for A=15, C=1
+B
+9
+13.5
+14
+15
+16.5
+17.5
+18
+19
+x+
+6.763
+6.365
+6.317
+6.219
+6.063
+5.953
+5.896
+5.777
+xp
+10.313
+9.806
+9.746
+9.623
+9.431
+9.298
+9.229
+9.088
+xs
+18.311
+17.677
+17.603
+17.451
+17.216
+17.054
+16.971
+16.802
+parameter B increases the shadow radius xs decreases. As xs > x+ shadow
+radii are defined by rs = xs
+√βq.
+10
+
+0
+2
+4
+6
+8
+0
+10
+20
+30
+x+
+A
+Subplot 1: B = 2, 4, 6; C = 1
+ 
+ 
+0
+1
+2
+3
+4
+0
+5
+10
+15
+x+
+B
+Subplot 2: A = 8, 9, 10; C = 1
+ 
+ 
+0
+2
+4
+6
+8
+0.2
+0.4
+0.6
+0.8
+1
+x+
+C
+Subplot 3: B = 2, 6, 8; A = 20
+ 
+ 
+B=2
+B=4
+B=6
+A=8
+A=9
+A=10
+B=2
+B=6
+B=8
+Figure 4: The plots of the functions A(x+), B(x+), C(x+)
+.
+It is worth mentioning that currently there is not unique calculation of
+the shadow radius of M87* or SgrA* black holes within ModLogNED because
+our model possesses four free parameters M, α, β and q (or M, A, B and C)
+but from observations one knows only two values: the black hole mass and
+the shadow radius.
+5
+Black holes energy emission rate
+The black hole shadow, for the observer at infinity, is connected with the
+high energy absorption cross section [25, 39].
+At very high energies the
+absorption cross-section σ ≈ πr2
+s oscillates around the photon sphere. The
+11
+
+energy emission rate of black holes is given by
+d2E(ω)
+dtdω
+=
+2π3ω3r2
+s
+exp (ω/TH(r+)) − 1,
+(29)
+where ω is the emission frequency. By using dimensionless variable x+ =
+r+/√βq the black hole energy emission rate (29) becomes
+�
+βqd2E(ω)
+dtdω
+=
+2π3̟3x2
+s
+exp
+�
+̟/ ¯TH(x+)
+�
+− 1
+,
+(30)
+with ¯TH(x+) = √βqTH(x+) and ̟ = √βqω. The radiation rate versus the
+dimensionless emission frequency ¯ω for C = 1, A = 15 and B = 9, 14, 19,
+is depicted in Fig. 5. Figure 5 shows that there is a peak of the black hole
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0
+0.005
+0.01
+0.015
+0.02
+0.025
+0.03
+0.035
+ ϖ
+ 
+ 
+B=9
+B=14
+B=19
+Figure 5: The plot of the function √βq d2E(ω)
+dtdω
+vs. ̟ for B = 9, 14, 19, A = 15,
+C = 1.
+energy emission rate. When parameter B increases, the energy emission rate
+12
+
+peak becomes smaller and corresponds to the lower frequency. The black
+hole has a bigger lifetime when parameter B is bigger.
+6
+Quasinormal modes
+The stability of BHs under small perturbations are characterised by quasi-
+normal modes (QNMs) with complex frequencies ω. When Im ω < 0 modes
+are stable but if Im ω > 0 modes are unstable. Re ω, in the eikonal limit, is
+linked with the black hole radius shadow [40, 41]. Around black holes, the
+perturbations by scalar massless fields are described by the effective potential
+barrier
+V (r) = f(r)
+�f ′(r)
+r
++ l(l + 1)
+r2
+�
+,
+(31)
+with l being the multipole number l = 0, 1, 2.... Equation (31) can be rewrit-
+ten in the form
+V (x)βq = f(x)
+�f ′(x)
+x
++ l(l + 1)
+x2
+�
+.
+(32)
+Dimensionless variable V (x)βq is depicted in Fig. 6 for A = 15, B = 10,
+C = 1 (Subplot 1) and for A = 15, C = 1, l = 5 (Subplot 2). According to
+Figure 6, Subplot l, the potential barriers of effective potentials have maxima.
+For l increasing the height of the potential increases. Figure 6, Subplot 2,
+shows that when the parameter B increases the height of the potential also
+increases. The quasinormal frequencies are given by [40, 41]
+Re ω = l
+rs
+=
+l
+�
+f(rp)
+rp
+,
+Im ω = −2n + 1
+2
+√
+2rs
+�
+2f(rp) − r2
+pf ′′(rp),
+(33)
+where rs is the black hole shadow radius, rp is the black hole photon sphere
+radius, and n = 0, 1, 2, ... is the overtone number. The frequencies, at A = 15,
+C = 1, n = 5, l = 10, are given in Table 2. Because the imaginary parts
+of the frequencies in Table 2 are negative, modes are stable. The real part
+Re ω gives the oscillations frequency.
+In accordance with Table 2 when
+parameter B increasing the real part of frequency √βqRe ω increases and
+the absolute value of the frequency imaginary part | √βqIm ω | decreases.
+Therefore, when the parameter B increases the scalar perturbations oscillate
+with greater frequency and decay lower.
+13
+
+0
+10
+20
+30
+40
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+0.12
+0.14
+0.16
+0.18
+x
+V(x)qβ
+Subplot 1: l=3,5,7; A=15; B=10; C=1
+ 
+ 
+0
+10
+20
+30
+40
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+0.12
+x
+V(x)qβ
+Subplot 2: B =9,14,19; A=15; l=5; C=1
+ 
+ 
+l=3
+l=5
+l=7
+B=9
+B=14
+B=19
+Figure 6: The plot of the function V (x)βq for A = 15, C = 1.
+7
+Summary
+The exact spherically symmetric solution of magnetic black holes is obtained
+in 4D EGB gravity coupled to ModLogNED. We studied the thermodynamics
+and the thermal stability of magnetically charged black holes. The Hawking
+temperature and the heat capacity were calculated. The phase transitions
+occur when the Hawking temperature has an extremum.
+Black holes are
+thermodynamically stable at some range of event horizon radii when the
+heat capacity and the Hawking temperature are positive. The heat capacity
+has a discontinuity where the second-order phase transitions take place. The
+black hole entropy was calculated which has the logarithmic correction. We
+calculated the photon sphere radii, the event horizon radii, and the shadow
+radii. It was shown that when the model parameter B increases the black
+14
+
+Table 2: The real and the imaginary parts of the frequencies vs the parameter
+B at n = 5, l = 10, A = 15, C = 1
+B
+14
+15
+16.5
+17.5
+18
+19
+√βqRe ω
+0.568
+0.573
+0.581
+0.586
+0.589
+0.595
+−√βqIm ω
+0.2853
+0.2852
+0.2849
+0.2845
+0.2842
+0.2835
+hole energy emission rate decreases and the black hole possesses a bigger
+lifetime. We show that when the parameter B increases the scalar pertur-
+bations oscillate with greater frequency and decay lower. Other solutions in
+4D EGB gravity coupled to NED were found in [33, 34, 35].
+Appendix A
+With the spherical symmetry the energy-momentum tensor possesses the
+property T t
+t
+= T r
+r . Then, the radial pressure is pr = −T r
+r
+= −ρ. The
+tangential pressure p⊥ = −T ϑ
+ϑ = −T φ
+φ
+is given by [42]
+p⊥ = −ρ − r
+2ρ′(r),
+(A1)
+with the prime being the derivative with respect to the radius r. The Weak
+Energy Condition (WEC) is valid when ρ ≥ 0 and ρ + pk ≥ 0 (k=1,2,3) [43],
+and then the energy density is positive. According to Eq. (7) ρ ≥ 0. Making
+use of Eq. (7) we obtain
+ρ′(r) = − q
+βr3 ln
+�
+1 + qβ
+r2
+�
+−
+q2
+r3(r2 + βq) ≤ 0.
+(A2)
+Therefore WEC, ρ ≥ 0, ρ + pr ≥ 0, ρ + p⊥ ≥ 0, is satisfied. The Dominant
+Energy Condition (DEC) takes place if and only if [43] ρ ≥ 0, ρ + pk ≥ 0,
+ρ − pk ≥ 0, that includes WEC. One needs only to check the condition
+ρ − p⊥ ≥ 0. By virtue of Eqs. (7), (A1) and A(2) one finds
+ρ − p⊥ =
+q
+2βr2
+�
+ln
+�
+1 + qβ
+r2
+�
+−
+qβ
+r2 + βq
+�
+.
+(A3)
+One can verify that ρ − p⊥ ≥ 0 for any parameters. DEC is satisfied and
+therefore the sound speed is less than the speed of light. The Strong Energy
+15
+
+Condition (SEC) is valid when ρ + �3
+k=1 pk ≥ 0 [43]. From Eqs. (8)-(10) we
+obtain
+ρ +
+3
+�
+k=1
+pk = ρ + p⊥ + pr = p⊥ < 0.
+(A4)
+In accordance with Eq. (A4) SEC is not satisfied.
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+19
+

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+arXiv:2301.01500v1  [nucl-th]  4 Jan 2023
+The Langevin approach for fission of heavy and
+super-heavy nuclei∗
+F.A.Ivanyuk, S.V.Radionov
+Institute for Nuclear Research, Kyiv, Ukraine
+C.Ishizuka, S.Chiba
+Tokyo Institute of Technology, Tokyo, Japan
+In this contribution, we present the main relations of the Langevin
+approach to the description of fission or fusion-fission reactions. The results
+of Langevin calculations are shown for the mass distributions of fission
+fragments of super-heavy elements and used for the investigation of memory
+effects in nuclear fission.
+1. Introduction
+We describe the nuclear fission process by the four-dimensional set of
+the Langevin equations for the shape degrees of freedom with the shape
+given by the two-center shell model (TCSM) shape parametrization. The
+potential energy is calculated within the macroscopic-microscopic method.
+The collective mass, M, and friction, γ, tensors are defined in macroscopic
+(Werner-Wheller and wall-and-window formula) or microscopic (linear re-
+sponse theory) approaches.
+We start calculations from the ground state shape with zero collective
+velocities and solve equations until the neck radius of the nucleus turns zero
+(scission point). At the scission point, the solutions of Langevin equations
+supply complete information about the system, its shape, excitation energy,
+and collective velocities.
+This information makes it possible to calculate
+the mass distributions, the total kinetic energy, and the excitation energies
+of fission fragments. The results of numerous previous calculations are in
+reasonable agreement with the available experimental data.
+∗ Presented at the Zakopane Conference on Nuclear Physics, Zakopane, Poland, 28
+August - 4 September 2022
+(1)
+
+2
+preprint
+printed on January 5, 2023
+Below in this contribution, we present the calculated results for the mass
+distributions of super-heavy nuclei and clarify the impact of memory effects
+on the fission width of heavy nuclei.
+The physics of super-heavy elements (SHE) has a long history. The ex-
+istence of the “island of stability” was predicted at the end of the 1960s
+[1]. Nevertheless, it took almost 30 years until the alpha-decay of the ele-
+ment with Z=114 was observed experimentally at Flerov Nuclear Reactions
+Laboratory in Dubna [2].
+With the development of experimental facility, it became possible not
+only to fix the fact of formation of SHE, but examine their properties.
+One of the first property of interest – the process of fission of SHEs. For
+the successful planning and carrying out of experiments, it is crucial to
+understand what kind of fission fragments mass distribution (FFMD) one
+should expect in the result of the fission of SHEs. The two double magic
+nuclei 132Sn and 208Pb may contribute. Both have the shell correction in
+the ground state of the same magnitude.
+In order to clarify what kind of FFMD one could expect in the fission of
+SHEs, we have carried out the calculations of FFMD for a number of SHEs.
+The results are given in Section 3.
+Another problem we address in this contribution is the influence of mem-
+ory effects on the probability of the fission process. Commonly one uses the
+Markovian approximation to Langevin approach in which all quantities are
+defined at the same moment. This approximation provides reasonable re-
+sults, but its accuracy is not well established. In publications, one can find
+statements that the memory effects have a significant influence on the fusion
+or fission processes and the statements that memory effects are very small.
+To clarify this uncertainty, we have calculated the fission width using
+the Langevin approach with memory effects included in a wide range of im-
+portant parameters: the excitation energy E∗ of the system, the damping
+parameter η, the relaxation time τ. The details and results of the calcula-
+tions are given in Section 4.
+2. The Langevin approach for the fission process
+Within the Langevin approach, the fission process is described by solving
+the equations for the time evolution of the shape of nuclear surface of the fis-
+sioning system. For the shape parametrization, we use that of the two-center
+shell model (TCSM) [3] with 4 deformation parameters qµ = z0/R0, δ1, δ2, α.
+Here z0/R0 refers to the distance between the centers of left and right os-
+cillator potentials, R0 being the radius of spherical nucleus with the mass
+number A. The parameters δi describe the deformation of the right and left
+fragment tips. The fourth parameter α is the mass asymmetry and the fifth
+
+preprint
+printed on January 5, 2023
+3
+parameter of the TCSM shape parametrization ǫ was kept constant, ǫ=0.35,
+in all our calculations.
+The first-order differential equations (Langevin equations) for the time
+dependence of collective variables qµ and the conjugated momenta pµ are:
+dqµ
+dt
+=
+�
+m−1�
+µν pν,
+(1)
+dpµ
+dt
+= −∂F(q, T)
+∂qµ
+− 1
+2
+∂m−1
+νσ
+∂qµ
+pνpσ − γµνm−1
+ν�� pσ + Rµ(t).
+In Eqs. (1) the F(q, T) is the temperature-dependent free energy of the
+system, and γµν and (m−1)µν are the friction and inverse of mass tensors.
+The free energy F(q, T) is calculated within the shell correction method.
+The single particle energies are calculated with the deformed Woods-Saxon
+potential fitted to the mentioned above TCSM shapes.
+The collective inertia tensor mµν is calculated by the Werner-Wheeler
+approximation and for the friction tensor γµν we used the wall-and-window
+formula. The random force Rµ(t) is the product of the temperature-depen-
+dent strength factors gµν and the white noise ξν(t), Rµ(t) = gµνξν(t). The
+factors gµν are related to the temperature and friction tensor via the Einstein
+relation,
+gµσgσν = Tγµν
+(2)
+The temperature T is kept constant, aT 2 = E∗, or adjusted to the local
+excitation energy on each step of integration by the relation,
+aT 2 = E∗ − p2(t)/2M − [Epot(q) − Epot(qgs)].
+(3)
+Here qgs is the ground state deformation. More details are given in our
+earlier publications [4, 5, 6, 7].
+Initially, the momenta pµ are set to zero, and calculations are started
+from the ground state deformation. Such calculations are continued until the
+trajectories reach the ”scission point”, defined as the point in deformation
+space where the neck radius turns zero.
+3. Fission fragments mass distributions of super-heavy nuclei
+In order to understand what kind of mass distributions one can expect
+from the solution of Langevin equations for super-heavy nuclei, we looked
+first at the potential energy of fissioning nuclei. Fig. 1 shows the potential
+energy Edef of nuclei 296Lv and 302120 at zero temperature as a function
+of elongation (the distance R12 between the centers of mass of left and
+right parts of a nucleus) and the mass asymmetry (fragment mass number).
+
+4
+preprint
+printed on January 5, 2023
+In the top part of Fig. 1 the energy was minimized with respect to the
+deformation parameters δ1 and δ2. One sees the bottom of potential energy
+leading to almost symmetric mass splitting. There is also a hint on the mass
+asymmetric valley at AF close to AF =208.
+1.0
+1.5
+2.0
+100
+150
+200
+302120, δ1= - 0.2, δ2= 0.2 
+R12 / R0
+Fragment mass number
+-60
+-52
+-44
+-36
+-28
+-20
+-12
+-4.0
+4.0
+10
+1.0
+1.5
+2.0
+100
+150
+200
+302120,  δ1,δ2 - min.
+Fragment mass number
+-60
+-52
+-44
+-36
+-28
+-20
+-12
+-4.0
+4.0
+10
+1.0
+1.5
+2.0
+50
+100
+150
+200
+296Lv, δ1= - 0.2, δ2= 0.2 
+R12 / R0
+Fragment mass number
+-60
+-52
+-44
+-36
+-28
+-20
+-12
+-4.0
+4.0
+10
+1.0
+1.5
+2.0
+50
+100
+150
+200
+Fragment mass number
+-60
+-52
+-44
+-36
+-28
+-20
+-12
+-4.0
+4.0
+10
+296Lv, δ1,δ2 - min.
+Fig. 1. (top) The potential energy of 296Lv and 302120 at T = 0 minimized with
+respect to deformation parameters δ1 and δ2 (bottom), and at fixed values δ1 =
+−0.2 and δ2 = 0.2.
+If the trajectories followed the bottom of potential energy, the mass
+distributions would be symmetric. However, it is well known that the tra-
+jectories may deviate substantially from the bottom of the potential valley
+due to dynamic effects. We calculate the trajectories in four-dimensional
+deformation space. In this space, the local minima could lead away from
+the bottom of the potential valley. An example is shown in the bottom part
+of Fig. 1. Here we show the potential energy for fixed δ1= - 0.2 and δ2=0.2.
+One clearly sees another valley, leading to strongly mass asymmetric split-
+ting.
+In Fig. 2, we show the fission fragment mass distributions of super-heavy
+nuclei from 276Hs to 308122 as a function of fragment mass number AF . The
+FFMDs of nuclei from 276Cn to 308122 have three or four peak structures.
+The main component is the symmetric peak, split into two components in
+some isotopes. The peaks of lighter fragments are located around AF =140.
+
+preprint
+printed on January 5, 2023
+5
+5
+10
+ 
+     Fission from the ground state, ---- E
+*=10 MeV, ---- E
+*=20 MeV, ---- E
+*=30 MeV
+N =      168     170     172     174     176     178     180     182     184     186
+Z = 108    110    112    114    116    118    120    122
+5
+10
+ 
+5
+10
+ 
+5
+10
+ 
+0
+5
+10
+5
+10
+15
+ 
+140
+5
+10
+15
+ 
+5
+10
+15
+20
+ 
+40 140
+05
+10
+15
+20
+ 
+40 140
+F r a g m e n t   m a s s   n u m b e r
+Yield (%)
+40 14040 14040 14040 14040 14040 14040 14040 140
+ 
+Fig. 2. The fission fragment mass distributions of super-heavy nuclei from 276Hs to
+308122 calculated for the excitation energies E∗=10, 20 and 30 MeV as a function
+of the fragment mass number
+One can also see the strongly asymmetric peak at the mass number
+close to AF =208. The strength of the (almost) symmetric and asymmetric
+components in FFMD of SHEs depends on the proton and neutron num-
+bers of the compound nucleus. For 276Cn, the contribution of a strongly
+asymmetric peak is tiny. This contribution becomes larger for more heavy
+SHE. In some elements of SHEs with Z =116-122, the symmetric and mass-
+asymmetric peaks are of the same magnitude. More details can be found in
+[8].
+The similar strongly mass-asymmetric peaks in FFMD of SHEs were
+also found recently in [9] within the Langevin approach with the so call
+Fourier shape parametrization.
+
+6
+preprint
+printed on January 5, 2023
+4. The memory effects in nuclear fission
+In order to investigate the role of memory effects in nuclear fission, we
+exploit a simple one-dimensional model with the potential energy given by
+the two-parabolic potential (Kramers potential), see Fig. 3.
+Epot(q) = 2Vbq(q − q0)/q2
+0, 0 < q < q0; 2Vb(q − q0)(2q0 − q)q2
+0, q0 < q < 2q0.
+(4)
+The potential (4) depends on two parameters, the barrier height Vb and the
+barrier width q0. We have fixed the barrier height Vb = 6 MeV, which is
+close to the value of the fission barrier of actinide nuclei. The width of the
+barrier is somewhat uncertain. It depends on the definition of the collective
+coordinate q and the model for the potential energy. For simplicity, we have
+put here q0 = 1.0.
+For the potential (4) one can define the stiffness C = d2Epot/dq2 and
+the frequency of harmonic vibrations ω0 =
+�
+C/M. In the present work,
+we fix ¯hω0 =1.0 MeV, which is close to the frequency of collective vibra-
+tions calculated for 224Th in [10] within the microscopic linear response
+theory. Then, for the mass parameter we will have the deformation and
+temperature-independent value,
+M = C/ω2
+0 = 4Vb/(ω2
+0q2
+0).
+(5)
+For the friction coefficient ¯γ we use a slightly modified approximation of
+[10],
+¯γ/M = 0.6(T 2 + ¯h2ω2
+0/π2))/(1 + T 2/40).
+(6)
+For the temperature, we consider here two options: constant temperature
+regime and constant energy regime. In a constant temperature regime, the
+temperature is time-independent, related to the initial excitation energy E∗
+by the Fermi-gas relation, aT 2 = E∗, where a is the level density parameter
+of T¨oke and Swiatecki [11].
+The fission width calculated in a constant
+temperature regime will be denoted as Γf(T).
+At small excitations, the temperature varies with deformation and time,
+and there is no reason to consider it constant. So, it should be adjusted
+to the local excitation energy on each integration step by the relation (3).
+Correspondingly, fission width calculated in a constant energy regime is
+denoted as Γf(E).
+The fission width, Γf, is defined assuming the exponential decay of the
+number of ”particles” in the potential well,
+P(t) = e−Γf t/¯h → Γf = −¯h ln[P(t)]/t.
+(7)
+By solving the Langevin equations one will get the set of time moments tb,
+at which some trajectories would cross the barrier. From this information,
+one can find the probability P(t) and the fission width Γf, see [12].
+
+preprint
+printed on January 5, 2023
+7
+-0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+0
+5
+10
+Epot (MeV)
+q
+ A=236,  E
+*=Vb
+0.0
+0.5
+1.0
+1.5
+2.0
+0
+1000
+2000
+3000
+ 
+ 
+Γf (eV)
+η
+ Γf(T)
+ ΓLV
+ ΓHV
+T=1.5 MeV
+Fig. 3. (left) The two-parabolic potential (4) and few examples of the dynamical
+trajectories. (right) The fission width as the solution of Eqs. (1, 4, 7) calculated at
+constant temperature (open dots), and the Kramers approximations (8) for high
+and low damping limits.
+The Markovian fission width Γf(T) calculated by Eqs. (1, 4, 7) is plotted
+as function of the damping parameter η in the right part of Fig. 3.
+To
+present the results in a broader range of parameters, the damping parameter
+η ≡ ¯γ/2Mω0 in these calculations was considered as a free parameter.
+For the comparison, in Fig.3 we also show the Kramers decay width
+ΓHV , ΓLV in limits of high and low viscosity (friction) [13],
+ΓHV = ¯hω0
+2π e−Vb/T (
+�
+1 + η2 − η) ,
+ΓLV = ¯h¯γ
+M
+Vb
+T e−Vb/T .
+(8)
+As one can see, the dependence of Γf(T) on η is rather complicated. The
+fission width Γf(T) grows as function of η in low damping region (η < 0.1).
+For η > 0.2, the fission width Γf(T) decreases as function of η.
+In nuclear systems, the Markovian assumption is often too restrictive.
+We thus have to generalize the above Langevin equations to allow for finite
+memory effects. They read as [14],
+dq/dt = p(t)/M,
+(9)
+dp
+dt = −∂Epot
+∂q
+−
+� t
+0
+dt′γ(t − t′)p(t′)/M + ζ ,
+γ(t − t′) ≡ ¯γe− t−t′
+τ /τ ,
+where τ is the memory (or relaxation) time.
+The extension consists in
+allowing the friction to have a memory time, i.e., the friction reacts on past
+stages of the system, what is called a retarded friction.
+The random numbers ζ in (9) are the normally distributed random num-
+bers with the properties < ζ(t) >= 0, < ζ(t)ζ(t′) >= Tγ(t−t′). In the limit
+ω0τ << 1, one recovers the Markovian limit of nuclear fission dynamics, i.e.,
+
+8
+preprint
+printed on January 5, 2023
+when the friction force is simply given by γ ˙q(t). The random numbers ζ(t)
+in (9) satisfy the equation
+dζ(t)/dt = −ζ(t)/τ + R(t)/τ ,
+(10)
+and are used in the description of the so-called Ornstein-Uhlenbeck pro-
+cesses.
+In the top part of Fig. 4 the calculated fission width Γf(E) is shown as
+a function of the damping parameter η both for small and large excitation
+energies, E∗=10, 25 and 60 MeV, for few values of the relaxation time.
+Besides τ = 0, we choose in calculations below the two values of τ, τ =
+5 · 10−22 sec and τ = 10−21 sec.
+0
+1
+2
+0
+5
+10
+Γf (eV)
+η
+ Γf(E)
+ Γeff(T)
+E*=10 MeV, Tin=0.6 MeV
+0
+1
+2
+0
+100
+200
+300
+E*=25 MeV, Tin=1.0 MeV
+η
+0
+1
+2
+0
+1000
+2000
+3000
+E*=60 MeV. Tin=1.5 MeV
+η
+ τ=0
+ τ=5 10
+-22 sec
+ τ=10 10
+-22 sec
+0
+5
+10
+0
+5
+10
+ Γf(E)
+ Γeff(T)
+Γf (eV)
+τ  (10
+-22 sec)
+0
+5
+10
+0
+100
+200
+300
+τ  (10
+-22 sec)
+0
+5
+10
+0
+1000
+2000
+3000
+τ  (10
+-22 sec)
+ η=0.1
+ η=0.5 
+ η=1.0
+Fig. 4. (top) The dependence of the fission width Γf(E) (solid) and the approxima-
+tion (11) (dashed) on the damping parameter η for few values of the relaxation time
+τ, τ=0, τ = 5 · 10−22 sec, τ = 10−21 sec and the initial excitation energies E∗
+in=10,
+25 and 60 MeV. (bottom) The dependence of the fission width Γf(E) (solid) and
+the approximation (11) (dashed) on the relaxation time τ for a few values of the
+damping parameter η, η=0.1, 0.5 and 1.0.
+The results of Langevin calculations satisfying the energy conservation
+condition are shown in Fig. 4 by solid lines. The fission width Γf(E) grows
+
+preprint
+printed on January 5, 2023
+9
+as a function of η and decreases as a function of τ in low damping region.
+The tendency is the opposite in the high damping region; the fission width
+Γf falls as a function of η and increases as a function of τ. Such dependence
+is common both for small and large excitation energies.
+In the bottom part of Fig. 4, the fission width Γf(E) (solid lines) is shown
+as a function of the relaxation time τ for a few fixed values of the damping
+parameter η.
+The bottom part of Fig. 4 confirms the above conclusion:
+the dependence of fission width Γf on η and τ is opposite in low and high
+damping regions.
+For the comparison, we show by dashed lines in Fig. 4 the available
+analytical approximation for Γf(T, τ) [14, 15, 16],
+1
+Γeff
+=
+1
+ΓLV
++
+1
+ΓHV
+,
+ΓLV (τ) = ΓLV (0)
+1 + ω2
+0τ 2 ,
+ΓHV (τ) = ¯hλ
+2π e−Vb/T , (11)
+where λ is the largest positive solution of the secular equation,
+λ3 + λ2/τ + (¯γ/Mτ − ω2
+0)λ − ω2
+0/τ = 0 .
+(12)
+As can be seen, the results of Langevin calculations for Γf(E) are smaller
+than the analytical estimate (11) both in low and high damping limits. The
+ratio Γf(E)/Γeff is close to 1 at E∗=60 MeV and close to 0.1 at E∗=10
+MeV.
+5. Summary
+The calculated mass distributions of fission fragments of super-heavy
+nuclei from 268Hs to 308122 demonstrate a three-four peaks structure of mass
+distributions. In light super-heavies, we see the dominant mass symmetric
+peak at AF ≈ 140. With increasing mass and charge numbers of fissioning
+nuclei, the highly asymmetric peaks at AH ≈ 208 appears. In 290−296Lv
+and 290−296Og, the three peaks in FFMD are approximately of the same
+magnitude at E*=10 MeV.
+The investigation of memory effects in nuclear fission is carried out. The
+calculations presented here offer complete information on the dependence
+of fission probability on all essential parameters, the relaxation time τ, the
+damping parameter η, and the excitation energy E*.
+It turned out that the fission width Γf(E) calculated under the constant
+energy requirement is generally smaller than that calculated in the constant
+temperature regime, Γf(T), or the Bohr-Wheeler approximation.
+The dependence of the fission width Γf(E) on the relaxation time τ is
+very sensitive to the damping parameter η. In the low viscosity region, the
+fission width Γf(E) grows as a function of η and decreases as a function of τ.
+
+10
+preprint
+printed on January 5, 2023
+In the high-viscosity region, the tendency is the opposite. Such dependence
+is common both for small and large excitation energies.
+Acknowledgements. The authors are grateful to Prof. K.Pomorski
+for the valuable discussions and presentation of our results at the Zakopane
+Conference
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+[14] Y. Abe, S. Ayik, P.-G. Reinhard, and E. Suraud, Phys. Rep. 275, 49 (1996).
+[15] R.F. Grote and J.T. Hynes, Jour. Chem. Phys. 73, 2715 (1980).
+[16] D.Boilley, Y.Lallouet, Jour. Stat. Phys. 125, 477 (2006).
+

DtAzT4oBgHgl3EQfif1c/content/tmp_files/load_file.txt

@@ -0,0 +1,348 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf,len=347
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='01500v1  [nucl-th]  4 Jan 2023  The Langevin approach for fission of heavy and  super-heavy nuclei∗  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='Ivanyuk, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='Radionov  Institute for Nuclear Research, Kyiv, Ukraine  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='Ishizuka, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='Chiba  Tokyo Institute of Technology, Tokyo, Japan  In this contribution, we present the main relations of the Langevin  approach to the description of fission or fusion-fission reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' The results  of Langevin calculations are shown for the mass distributions of fission  fragments of super-heavy elements and used for the investigation of memory  effects in nuclear fission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' Introduction  We describe the nuclear fission process by the four-dimensional set of  the Langevin equations for the shape degrees of freedom with the shape  given by the two-center shell model (TCSM) shape parametrization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' The  potential energy is calculated within the macroscopic-microscopic method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  The collective mass, M, and friction, γ, tensors are defined in macroscopic  (Werner-Wheller and wall-and-window formula) or microscopic (linear re-  sponse theory) approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  We start calculations from the ground state shape with zero collective  velocities and solve equations until the neck radius of the nucleus turns zero  (scission point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' At the scission point, the solutions of Langevin equations  supply complete information about the system, its shape, excitation energy,  and collective velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  This information makes it possible to calculate  the mass distributions, the total kinetic energy, and the excitation energies  of fission fragments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' The results of numerous previous calculations are in  reasonable agreement with the available experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  ∗ Presented at the Zakopane Conference on Nuclear Physics, Zakopane, Poland, 28  August - 4 September 2022  (1)  2  preprint  printed on January 5, 2023  Below in this contribution, we present the calculated results for the mass  distributions of super-heavy nuclei and clarify the impact of memory effects  on the fission width of heavy nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  The physics of super-heavy elements (SHE) has a long history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' The ex-  istence of the “island of stability” was predicted at the end of the 1960s  [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' Nevertheless, it took almost 30 years until the alpha-decay of the ele-  ment with Z=114 was observed experimentally at Flerov Nuclear Reactions  Laboratory in Dubna [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  With the development of experimental facility, it became possible not  only to fix the fact of formation of SHE, but examine their properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  One of the first property of interest – the process of fission of SHEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' For  the successful planning and carrying out of experiments, it is crucial to  understand what kind of fission fragments mass distribution (FFMD) one  should expect in the result of the fission of SHEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' The two double magic  nuclei 132Sn and 208Pb may contribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' Both have the shell correction in  the ground state of the same magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  In order to clarify what kind of FFMD one could expect in the fission of  SHEs, we have carried out the calculations of FFMD for a number of SHEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  The results are given in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  Another problem we address in this contribution is the influence of mem-  ory effects on the probability of the fission process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' Commonly one uses the  Markovian approximation to Langevin approach in which all quantities are  defined at the same moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' This approximation provides reasonable re-  sults, but its accuracy is not well established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' In publications, one can find  statements that the memory effects have a significant influence on the fusion  or fission processes and the statements that memory effects are very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  To clarify this uncertainty, we have calculated the fission width using  the Langevin approach with memory effects included in a wide range of im-  portant parameters: the excitation energy E∗ of the system, the damping  parameter η, the relaxation time τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' The details and results of the calcula-  tions are given in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' The Langevin approach for the fission process  Within the Langevin approach, the fission process is described by solving  the equations for the time evolution of the shape of nuclear surface of the fis-  sioning system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' For the shape parametrization, we use that of the two-center  shell model (TCSM) [3] with 4 deformation parameters qµ = z0/R0, δ1, δ2, α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  Here z0/R0 refers to the distance between the centers of left and right os-  cillator potentials, R0 being the radius of spherical nucleus with the mass  number A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' The parameters δi describe the deformation of the right and left  fragment tips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' The fourth parameter α is the mass asymmetry and the fifth  preprint  printed on January 5, 2023  3  parameter of the TCSM shape parametrization ǫ was kept constant, ǫ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='35,  in all our calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  The first-order differential equations (Langevin equations) for the time  dependence of collective variables qµ and the conjugated momenta pµ are:  dqµ  dt  =  �  m−1�  µν pν,  (1)  dpµ  dt  = −∂F(q, T)  ∂qµ  − 1  2  ∂m−1  νσ  ∂qµ  pνpσ − γµνm−1  νσ pσ + Rµ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  In Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' (1) the F(q, T) is the temperature-dependent free energy of the  system, and γµν and (m−1)µν are the friction and inverse of mass tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  The free energy F(q, T) is calculated within the shell correction method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  The single particle energies are calculated with the deformed Woods-Saxon  potential fitted to the mentioned above TCSM shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  The collective inertia tensor mµν is calculated by the Werner-Wheeler  approximation and for the friction tensor γµν we used the wall-and-window  formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' The random force Rµ(t) is the product of the temperature-depen-  dent strength factors gµν and the white noise ξν(t), Rµ(t) = gµνξν(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' The  factors gµν are related to the temperature and friction tensor via the Einstein  relation,  gµσgσν = Tγµν  (2)  The temperature T is kept constant, aT 2 = E∗, or adjusted to the local  excitation energy on each step of integration by the relation,  aT 2 = E∗ − p2(t)/2M − [Epot(q) − Epot(qgs)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  (3)  Here qgs is the ground state deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' More details are given in our  earlier publications [4, 5, 6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  Initially, the momenta pµ are set to zero, and calculations are started  from the ground state deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' Such calculations are continued until the  trajectories reach the ”scission point”, defined as the point in deformation  space where the neck radius turns zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' Fission fragments mass distributions of super-heavy nuclei  In order to understand what kind of mass distributions one can expect  from the solution of Langevin equations for super-heavy nuclei, we looked  first at the potential energy of fissioning nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' 1 shows the potential  energy Edef of nuclei 296Lv and 302120 at zero temperature as a function  of elongation (the distance R12 between the centers of mass of left and  right parts of a nucleus) and the mass asymmetry (fragment mass number).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  4  preprint  printed on January 5, 2023  In the top part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' 1 the energy was minimized with respect to the  deformation parameters δ1 and δ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' One sees the bottom of potential energy  leading to almost symmetric mass splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' There is also a hint on the mass  asymmetric valley at AF close to AF =208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='0  100  150  200  302120, δ1= - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='2, δ2= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='2  R12 / R0  Fragment mass number  60  52  44  36  28  20  12  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='0  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='0  10  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='0  100  150  200  302120,  δ1,δ2 - min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  Fragment mass number  60  52  44  36  28  20  12  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='0  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='0  10  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='0  50  100  150  200  296Lv, δ1= - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='2, δ2= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='2  R12 / R0  Fragment mass number  60  52  44  36  28  20  12  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='0  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='0  10  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='0  50  100  150  200  Fragment mass number  60  52  44  36  28  20  12  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='0  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='0  10  296Lv, δ1,δ2 - min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' (top) The potential energy of 296Lv and 302120 at T = 0 minimized with  respect to deformation parameters δ1 and δ2 (bottom), and at fixed values δ1 =  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='2 and δ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  If the trajectories followed the bottom of potential energy, the mass  distributions would be symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' However, it is well known that the tra-  jectories may deviate substantially from the bottom of the potential valley  due to dynamic effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' We calculate the trajectories in four-dimensional  deformation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' In this space, the local minima could lead away from  the bottom of the potential valley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' An example is shown in the bottom part  of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' Here we show the potential energy for fixed δ1= - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='2 and δ2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  One clearly sees another valley, leading to strongly mass asymmetric split-  ting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' 2, we show the fission fragment mass distributions of super-heavy  nuclei from 276Hs to 308122 as a function of fragment mass number AF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' The  FFMDs of nuclei from 276Cn to 308122 have three or four peak structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  The main component is the symmetric peak, split into two components in  some isotopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' The peaks of lighter fragments are located around AF =140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  preprint  printed on January 5, 2023  5  5  10  Fission from the ground state, ---- E  =10 MeV, ---- E  =20 MeV, ---- E  =30 MeV  N =      168     170     172     174     176     178     180     182     184     186  Z = 108    110    112    114    116    118    120    122  5  10  5  10  5  10  0  5  10  5  10  15  140  5  10  15  5  10  15  20  40 140  05  10  15  20  40 140  F r a g m e n t   m a s s   n u m b e r  Yield (%)  40 14040 14040 14040 14040 14040 14040 14040 140  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' The fission fragment mass distributions of super-heavy nuclei from 276Hs to  308122 calculated for the excitation energies E∗=10, 20 and 30 MeV as a function  of the fragment mass number  One can also see the strongly asymmetric peak at the mass number  close to AF =208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' The strength of the (almost) symmetric and asymmetric  components in FFMD of SHEs depends on the proton and neutron num-  bers of the compound nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' For 276Cn, the contribution of a strongly  asymmetric peak is tiny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' This contribution becomes larger for more heavy  SHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' In some elements of SHEs with Z =116-122, the symmetric and mass-  asymmetric peaks are of the same magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' More details can be found in  [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  The similar strongly mass-asymmetric peaks in FFMD of SHEs were  also found recently in [9] within the Langevin approach with the so call  Fourier shape parametrization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  6  preprint  printed on January 5, 2023  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' The memory effects in nuclear fission  In order to investigate the role of memory effects in nuclear fission, we  exploit a simple one-dimensional model with the potential energy given by  the two-parabolic potential (Kramers potential), see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  Epot(q) = 2Vbq(q − q0)/q2  0, 0 < q < q0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' 2Vb(q − q0)(2q0 − q)q2  0, q0 < q < 2q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  (4)  The potential (4) depends on two parameters, the barrier height Vb and the  barrier width q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' We have fixed the barrier height Vb = 6 MeV, which is  close to the value of the fission barrier of actinide nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' The width of the  barrier is somewhat uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' It depends on the definition of the collective  coordinate q and the model for the potential energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' For simplicity, we have  put here q0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  For the potential (4) one can define the stiffness C = d2Epot/dq2 and  the frequency of harmonic vibrations ω0 =  �  C/M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' In the present work,  we fix ¯hω0 =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='0 MeV, which is close to the frequency of collective vibra-  tions calculated for 224Th in [10] within the microscopic linear response  theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' Then, for the mass parameter we will have the deformation and  temperature-independent value,  M = C/ω2  0 = 4Vb/(ω2  0q2  0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  (5)  For the friction coefficient ¯γ we use a slightly modified approximation of  [10],  ¯γ/M = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='6(T 2 + ¯h2ω2  0/π2))/(1 + T 2/40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  (6)  For the temperature, we consider here two options: constant temperature  regime and constant energy regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' In a constant temperature regime, the  temperature is time-independent, related to the initial excitation energy E∗  by the Fermi-gas relation, aT 2 = E∗, where a is the level density parameter  of T¨oke and Swiatecki [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  The fission width calculated in a constant  temperature regime will be denoted as Γf(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  At small excitations, the temperature varies with deformation and time,  and there is no reason to consider it constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' So, it should be adjusted  to the local excitation energy on each integration step by the relation (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  Correspondingly, fission width calculated in a constant energy regime is  denoted as Γf(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  The fission width, Γf, is defined assuming the exponential decay of the  number of ”particles” in the potential well,  P(t) = e−Γf t/¯h → Γf = −¯h ln[P(t)]/t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  (7)  By solving the Langevin equations one will get the set of time moments tb,  at which some trajectories would cross the barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' From this information,  one can find the probability P(t) and the fission width Γf, see [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  preprint  printed on January 5, 2023  7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='0  0  5  10  Epot (MeV)  q  A=236,  E  =Vb  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='0  0  1000  2000  3000  Γf (eV)  η  Γf(T)  ΓLV  ΓHV  T=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='5 MeV  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' (left) The two-parabolic potential (4) and few examples of the dynamical  trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' (right) The fission width as the solution of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' (1, 4, 7) calculated at  constant temperature (open dots), and the Kramers approximations (8) for high  and low damping limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  The Markovian fission width Γf(T) calculated by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' (1, 4, 7) is plotted  as function of the damping parameter η in the right part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  To  present the results in a broader range of parameters, the damping parameter  η ≡ ¯γ/2Mω0 in these calculations was considered as a free parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  For the comparison, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='3 we also show the Kramers decay width  ΓHV , ΓLV in limits of high and low viscosity (friction) [13],  ΓHV = ¯hω0  2π e−Vb/T (  �  1 + η2 − η) ,  ΓLV = ¯h¯γ  M  Vb  T e−Vb/T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  (8)  As one can see, the dependence of Γf(T) on η is rather complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' The  fission width Γf(T) grows as function of η in low damping region (η < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  For η > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='2, the fission width Γf(T) decreases as function of η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  In nuclear systems, the Markovian assumption is often too restrictive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  We thus have to generalize the above Langevin equations to allow for finite  memory effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' They read as [14],  dq/dt = p(t)/M,  (9)  dp  dt = −∂Epot  ∂q  −  � t  0  dt′γ(t − t′)p(t′)/M + ζ ,  γ(t − t′) ≡ ¯γe− t−t′  τ /τ ,  where τ is the memory (or relaxation) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  The extension consists in  allowing the friction to have a memory time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=', the friction reacts on past  stages of the system, what is called a retarded friction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  The random numbers ζ in (9) are the normally distributed random num-  bers with the properties < ζ(t) >= 0, < ζ(t)ζ(t′) >= Tγ(t−t′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' In the limit  ω0τ << 1, one recovers the Markovian limit of nuclear fission dynamics, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=',  8  preprint  printed on January 5, 2023  when the friction force is simply given by γ ˙q(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' The random numbers ζ(t)  in (9) satisfy the equation  dζ(t)/dt = −ζ(t)/τ + R(t)/τ ,  (10)  and are used in the description of the so-called Ornstein-Uhlenbeck pro-  cesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  In the top part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' 4 the calculated fission width Γf(E) is shown as  a function of the damping parameter η both for small and large excitation  energies, E∗=10, 25 and 60 MeV, for few values of the relaxation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  Besides τ = 0, we choose in calculations below the two values of τ, τ =  5 · 10−22 sec and τ = 10−21 sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  0  1  2  0  5  10  Γf (eV)  η  Γf(E)  Γeff(T)  E*=10 MeV, Tin=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='6 MeV  0  1  2  0  100  200  300  E*=25 MeV, Tin=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='0 MeV  η  0  1  2  0  1000  2000  3000  E*=60 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' Tin=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='5 MeV  η  τ=0  τ=5 10  22 sec  τ=10 10  22 sec  0  5  10  0  5  10  Γf(E)  Γeff(T)  Γf (eV)  τ  (10  22 sec)  0  5  10  0  100  200  300  τ  (10  22 sec)  0  5  10  0  1000  2000  3000  τ  (10  22 sec)  η=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='1  η=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='5  η=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='0  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' (top) The dependence of the fission width Γf(E) (solid) and the approxima-  tion (11) (dashed) on the damping parameter η for few values of the relaxation time  τ, τ=0, τ = 5 · 10−22 sec, τ = 10−21 sec and the initial excitation energies E∗  in=10,  25 and 60 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' (bottom) The dependence of the fission width Γf(E) (solid) and  the approximation (11) (dashed) on the relaxation time τ for a few values of the  damping parameter η, η=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='5 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  The results of Langevin calculations satisfying the energy conservation  condition are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' 4 by solid lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' The fission width Γf(E) grows  preprint  printed on January 5, 2023  9  as a function of η and decreases as a function of τ in low damping region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  The tendency is the opposite in the high damping region;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' the fission width  Γf falls as a function of η and increases as a function of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' Such dependence  is common both for small and large excitation energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  In the bottom part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' 4, the fission width Γf(E) (solid lines) is shown  as a function of the relaxation time τ for a few fixed values of the damping  parameter η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  The bottom part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' 4 confirms the above conclusion:  the dependence of fission width Γf on η and τ is opposite in low and high  damping regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  For the comparison, we show by dashed lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' 4 the available  analytical approximation for Γf(T, τ) [14, 15, 16],  1  Γeff  =  1  ΓLV  +  1  ΓHV  ,  ΓLV (τ) = ΓLV (0)  1 + ω2  0τ 2 ,  ΓHV (τ) = ¯hλ  2π e−Vb/T , (11)  where λ is the largest positive solution of the secular equation,  λ3 + λ2/τ + (¯γ/Mτ − ω2  0)λ − ω2  0/τ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  (12)  As can be seen, the results of Langevin calculations for Γf(E) are smaller  than the analytical estimate (11) both in low and high damping limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' The  ratio Γf(E)/Γeff is close to 1 at E∗=60 MeV and close to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='1 at E∗=10  MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' Summary  The calculated mass distributions of fission fragments of super-heavy  nuclei from 268Hs to 308122 demonstrate a three-four peaks structure of mass  distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' In light super-heavies, we see the dominant mass symmetric  peak at AF ≈ 140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' With increasing mass and charge numbers of fissioning  nuclei, the highly asymmetric peaks at AH ≈ 208 appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' In 290−296Lv  and 290−296Og, the three peaks in FFMD are approximately of the same  magnitude at E*=10 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  The investigation of memory effects in nuclear fission is carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' The  calculations presented here offer complete information on the dependence  of fission probability on all essential parameters, the relaxation time τ, the  damping parameter η, and the excitation energy E*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  It turned out that the fission width Γf(E) calculated under the constant  energy requirement is generally smaller than that calculated in the constant  temperature regime, Γf(T), or the Bohr-Wheeler approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  The dependence of the fission width Γf(E) on the relaxation time τ is  very sensitive to the damping parameter η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' In the low viscosity region, the  fission width Γf(E) grows as a function of η and decreases as a function of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  10  preprint  printed on January 5, 2023  In the high-viscosity region, the tendency is the opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' Such dependence  is common both for small and large excitation energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content='  Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' The authors are grateful to Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'}
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+Physical Layer Security Techniques Applied to
+Vehicle-to-Everything Networks
+Leonardo B. da Silva, Evelio M. G. Fernández and Ândrei Camponogara
+Abstract— Physical Layer Security (PLS) is an emerging con-
+cept in the field of secrecy for wireless communications that can
+be used alongside cryptography to prevent unauthorized devices
+from eavesdropping a legitimate transmission. It offers low com-
+putational cost and overhead by injecting an interfering signal
+in the wiretap channels of potential eavesdroppers. This paper
+discusses the benefits of the Artificial Noise and Cooperative
+Jamming techniques in the context of Vehicle-to-everything (V2X)
+networks, which require secure data exchange with small latency.
+The simulations indicate that messages can be safely delivered
+even with devices that have low available power.
+Keywords— Wireless communication networks, Physical Layer
+Security, secrecy, Vehicle-to-everything, Artificial Noise, Cooper-
+ative Jamming.
+I. INTRODUCTION
+Urban mobility is one of the main focuses of the Internet
+of Things (IoT) when applied to smart cities, due to the
+necessity for more responsive and safe traffic control. Gener-
+ally, the solutions proposed in this scope involve the wireless
+communication between not only the vehicles themselves,
+but also with pedestrians, infrastructure, and networks. This
+paradigm is known as Vehicle-to-everything (V2X) and it can
+be standardized by protocols such as C-ITS (Cellular Intelli-
+gent Transportation System) and WAVE (Wireless Access for
+Vehicular Environment) that are based on the IEEE 802.11p
+amendment, and the Cellular-V2X (C-V2X) that implements
+the 5G standard from 3GPP (3rd Generation Partnership
+Project) [1].
+A. Problem Outline
+Due to the ever-changing location of most of the involved
+communication nodes and the time-sensitive nature of the
+data involved (brake position, vehicle speed, traffic volume,
+accident reports, etc), the transmission needs not only to occur
+at high rates, but also offer reliability through high secrecy, low
+packet loss, and small delay. Furthermore, those nodes have
+to be affordable to justify their implementation on a city-wide
+scale, thus having low power consumption and the most cost-
+efficient embedded processing unit possible [2].
+Since the main source of information security in today’s
+landscape is provided through cryptography, the secrecy con-
+straint can negatively affect most of these criteria. As a result
+L.
+B.
+da
+Silva,
+E.
+M.
+G.
+Fernandez,
+Â.
+Camponogara,
+Electri-
+cal Engineering Department, Federal University of Paraná (UFPR), Cu-
+ritiba, PR, Brazil, e-mails: [email protected], [email protected] and
+[email protected]. This study was financed in part by the Coorde-
+nação de Aperfeiçoamento de Pessoal de Nível Superior – Brasil (CAPES) –
+Finance Code 001.
+of the growth in the availability of portable and connected
+equipment with high processing capabilities, the safety mea-
+sures implemented need to match this computational power
+with proportionally longer and more complex keys to not be
+vulnerable to brute-force attacks from well-equipped malicious
+devices [2], [3]. This approach, however, is not sustainable,
+because it produces increasingly long authentication routines,
+due to the raise in computational overhead and processing cost
+as a result of the implemented security algorithms.
+B. Overview of the proposed solution
+To counterbalance this issue, this paper studies the use of
+Physical Layer Security (PLS) techniques as an additional
+protection to increase the secrecy of wireless communications
+in a V2X environment. As the name suggests, PLS is applied
+at the Physical Layer, making it an alternative that can be used
+with low processing cost when compared with cryptography,
+which is more oriented towards the computational side of the
+network stack on the Application Layer [1].
+Since cryptography techniques provide security in different
+sections of the wireless protocols, PLS is proposed as a
+complement to them, rather than a replacement [1]. Through
+the use of both approaches on the same node, it is possible to
+offer high secrecy without the necessity of infinitely growing
+key complexity.
+The PLS has its origins on the analytical proposal of
+Wyner’s wiretap channel [4], where it is described a commu-
+nication between two legitimate nodes that is spied on by an
+eavesdropper through an unauthorized channel called wiretap.
+In the modern literature, these devices are usually referred to
+as a transmitter called Alice, an authorized receiver Bob, and
+the set of K eavesdroppers named Eves.
+In the wiretap channel model shown in Fig. 1, the original
+message m is encoded and transmitted by Alice as the signal
+sa, that reaches Bob through the main channel hAB. The
+received signal yB is then decoded by Bob, obtaining the
+estimated message ˆm. Additionally, the k-th Eve can intercept
+sa through the wiretap channel hAE,k, obtaining the signal
+yE,k that when decoded produces z.
+The main focus of PLS is to guarantee that the mutual
+information between m and z is as close to zero as possible.
+When this condition is met, even if z is know, it is impossible
+for Eve to infer the contents of the original message.
+Wyner then presents a set of parameters that enable the use
+of the physical imperfections of the channel, such as noise
+and fading, to provide information secrecy by raising the level
+of confusion on undesired nodes. Rendering them unable to
+distinguish between the message and the interference.
+arXiv:2301.05123v1  [eess.SP]  12 Jan 2023
+
+Fig. 1: The wiretap channel generic model based on [4]
+Currently, plenty of techniques to provide security at the
+physical layer level have been proposed in the literature [3].
+This paper will focus on two approaches first presented in [5]:
+• Artificial Noise (AN): This approach uses a portion of
+the transmitter node’s power to inject artificially gener-
+ated noise in the eavesdropper’s channel;
+• Cooperative Jamming (CJ): This approach expands the
+AN model by proposing a connected network where
+nearby relay nodes (Charlies) send a jamming signal to
+the eavesdropper’s channel.
+To demonstrate the viability of AN and CJ applications in
+a V2X network, it is common to create stochastic geometric
+models that randomly generate streets and distribute com-
+munication nodes in a predefined area to represent an urban
+mobility scenario [6], [7]. When implementing these methods,
+metrics such as the Signal-to-Interference Ratio (SIR) are used
+to define the threshold of confusion necessary to provide se-
+crecy at the physical layer. The SIR on each eavesdropper can
+then be evaluated to determine the secrecy outage probability
+(SOP) of the data transmission with different densities of the
+involved nodes in the simulated network.
+In this paper, Section II describes the stochastic algorithms
+implemented to model a V2X network that includes streets
+and communication nodes (vehicular and planar). Section III
+presents the analytical basis of the AN and CJ techniques,
+while also introducing the SIR and SOP metrics. In Section
+IV, the results of numerical simulations are shown to illustrate
+the benefits of the considered PLS techniques on the generated
+V2X networks. Finally, Section V states some final remarks.
+Notation: IN is an identity matrix of order N, Poisson(n)
+is a Poisson distribution with mean number of arrivals n,
+CN(m, n) is a complex normal distribution with average m
+and covariance n, exp(n) is an exponential distribution with
+mean n and Gamma(m, n) is the gamma distribution with
+form m and scale n.
+II. THE V2X NETWORK MODEL
+As mentioned previously, vehicular networks are dynamic,
+with devices changing location constantly. Thus, a determin-
+istic model is not well-suited for this application. A common
+alternative is the use of stochastic geometry to represent this
+random spatial nature through a variety of different processes
+to distribute the streets and communication nodes within the
+desired coverage area [8].
+A viable option is the use of Poisson processes, as they are
+memoryless counting processes for integer arrivals [9]. In other
+words, each set of elements generated will be independent
+with a Poisson distributed integer number of uniformly spaced
+nodes. The intensity of the arrivals in these processes are
+represented by λ and the expected number of elements is
+the product of the said intensity and the Lebesgue measure,
+which in this context is essentially the spatial measurement
+associated with the object that the points will be distributed on.
+For instance, the Lebesgue measure to populate a circle is its
+area and for a line is the length. One realization of the resulting
+spatial model derived from the use of different variations of
+the Poisson processes is represented in Fig. 2.
+Fig. 2:
+Spatial simulation of the modeled V2X network.
+The color green indicates the Charlies implemented in CJ
+techniques and the Eves are in red. The planar devices are
+generated by PPPs represented by circles (◦) with intensity λ
+= 10−6/m2 for both node types. Through a PLP, the streets
+(blue lines) have been modeled with an intensity of λl = 10−3
+/m, and the vehicular devices are originated from PLP-driven
+Cox Processes indicated with triangles (△) of intensity u =
+10−3/m for both Charlies and Eves. A single Alice is indicated
+with a black × at the origin.
+In this model, the wireless devices of pedestrians and
+connected infrastructure are considered free to be positioned in
+the whole area A of the modeled network, which is a circle of
+radius r = 3 km. Thus, these “planar nodes“ are generated by
+2-D Poisson Point Processes (PPP) and the expected amount
+of elements is given by Poisson(λ · A). The set of planar
+nodes is indicated by Φ, thus the planar Eves and Charlies
+are respectively represented by ΦE and ΦC.
+The streets are represented by uniformly distributed lines
+with density µl = λl/π generated by a Poisson Line Process
+(PLP) Φl based on the second method of the Bertrand paradox
+[10], in which a set of expected Poisson(µl·2πr) midpoints are
+created [11], each with a random radius P ∈ [0, r) and angle
+θ ∈ [0, 2π). From these coordinates, a segment perpendicular
+to P is traced between two points at the edge of the circle
+of radius r. This effectively means that a pair of 1-D PPP
+points are created in the perimeter of the circular area for
+each modeled street.
+On those PLP-generated lines, a Cox process of intensity u
+
+YB
+TRANSMITTER
+MAIN CHANNEL
+RECEIVER
+m -
+(ALICE)
+hAB
+m
+(BOB)
+Ye,k
+WIRETAP CHANNEL
+k-thEAVESDROPPER
+>Z
+(EVE)3000 
+X
+Alice node
+Planar Charlies
+Q
+Planar Eves
+Vehicular Charlies
+O
+A
+Vehicular Eves
+2000
+O
+Q
+1000
+O
+F 0
+2
+X
+O
+-1000
+C
+-2000
+3000
+4000
+3000
+-2000
+-1000
+0
+1000
+2000
+3000
+4000is implemented, which is used to create the “vehicular nodes“
+on each segment [12]. These elements represent vehicles
+whose spatial distribution are constrained to a street by a 1-D
+PPP. Considering a street of length l, the number of vehicles
+in it is given by Poisson(u · l).
+The set of vehicular Eves and Charlies on each street l are
+respectively denoted by ψE and ψC. Based on these, the total
+nodes of each type can be obtained by evaluating the sets on
+the whole range of Φl [6], resulting in ΨE = {ψE(l)}l∈Φl for
+Eves and ΨC = {ψC(l)}l∈Φl for Charlies.
+Furthermore, a single deterministic transmitter (Alice) is
+included at the origin of the circle. This point is selected to
+simplify the distance calculations between a legitimate device
+and the Eve nodes, which can be planar or vehicular. This
+measurement is one of the parameters for the SIR calculations,
+that are considered to determine the effectiveness of the PLS.
+For the CJ case, auxiliary nodes (Charlies) are also modeled,
+some as planar and others as vehicular devices. Note that the
+distance between Charlies and Eves influences the power of
+the interference injected on the unauthorized channels as part
+of the jamming technique.
+III. PLS TECHNIQUES
+The PLS techniques presented in this paper are part of the
+key-less-based class [2], which implements secure information
+transmission by making the unauthorized channel’s capacity
+(CE) lower than that of the legitimate channel’s (CB). This re-
+lationship can be presented by evaluating these values through
+the Shannon-Hartley theorem, which produces the secrecy
+capacity (CS) metric as
+CS = CB − CE = log2(1 + γB) − log2(1 + γE),
+(1)
+where γB and γE are, respectively, the SIRs of Bob and Eve.
+Based on this expression, it can be inferred that in order to
+guarantee that CB is sufficiently larger than CE, the value of
+γE must be as low as possible. The approach utilized by AN
+and CJ is the injection of artificially generated interference in
+the eavesdropper channels.
+Typically, this injection is implemented with multi-antenna
+networks, as it enables the use of beamforming to selectively
+direct the transmission to legitimate receivers with minimum
+noise and high efficiency [3]. The unintended receivers on
+the other hand, intercept a signal that contains the secret
+message as well as AN. Therefore, secrecy is provided when
+the distinction between them by the Eves is improbable.
+The wireless channels in this paper are modeled with com-
+plex normal distributions (CN) which implies in a Rayleigh
+fading model. This decision provides simpler analytical equa-
+tions and also proposes a more pessimistic scenario, in which
+there is no Line-of-Sight (LoS) available. By evaluating the
+metrics in these worst-case conditions, it is possible to verify
+that even then the secrecy can be guaranteed.
+A. Artificial Noise
+In the AN scenario, the legitimate communication is es-
+tablished between a single transmitter Alice and a receiver
+Bob. Additional nodes (both planar and vehicular) that try to
+obtain Alice’s signal are then considered eavesdroppers and
+their channels will be affected by the AN.
+The signal transmitted by the Alice node with NA antennas
+is composed of two terms: the first contains a message x
+intended for Bob and the second is based on a zero-forcing
+vector for the unauthorized devices [13], i.e,
+sa =
+�
+φPt
+ha
+∥ha∥x +
+�
+(1 − φ)Pt
+NA − 1 Wana,
+(2)
+where ha/∥ha∥ is the beamforming vector with the normaliza-
+tion of the Alice’s channel estimation ha ∈ CNA×1, that will
+be modeled as CN(0, INA). The AN is formed by the null-
+space orthonormal basis Wa ∈ CNA×(NA−1) and the noise
+signal na ∈ C(NA−1)×1.
+The distribution of the available power, Pt, between the
+two terms of (2) is controlled by φ ∈ {0,1}. φ = 0 means that
+all power is allocated to noise generation and no message is
+sent. Conversely, when φ = 1 the AN is not active and Pt is
+allocated entirely for data transmission.
+B. Cooperative Jamming
+The Cooperative Jamming extends the AN case, maintaining
+the single Alice-Bob authorized transmission with multiple
+Eves, however, adding auxiliary nodes in the network. These
+devices, typically called Charlies, can also be either planar or
+vehicular, just like the Eves. In contrast, they are responsible
+for providing additional security by sending jamming signals
+that further decrease the channel quality of the Eves.
+For simplicity, it is considered that only Alice will transmit
+messages in the scenarios evaluated in this paper. Hence, the
+signals sent by the Charlie nodes are made of only the AN
+(zero-forcing) portion, as follows
+sc =
+�
+Pc
+NC − 1Wcnc,
+(3)
+where NC is the number of antennas of each Charlie and PC
+is the power available for jamming. Notice that since these
+nodes are not transmitting messages, all the available power
+is directed towards CJ. Additionally, Wc ∈ CNC×(NC−1) is
+the null space orthonormal matrix and nc ∈ C(NC−1)×1 is the
+artificial noise component.
+C. Received Signals
+By considering that the channel estimation ha is precisely
+the main channel established between Alice and Bob, hAB,
+it is implied that the receiver node is not affected by the
+interference from AN or CJ. That happens because the or-
+thonormal basis Wa and Wc are null when applied to the
+authorized channels, resulting in the relationships h†
+ABWa = 0
+and h†
+ABWc = 0, respectively. Therefore, the signal received
+by Bob can be expressed as
+yB =
+�
+φPt ∥ha∥ D−α/2
+AB
+x,
+(4)
+where DAB is the distance between the devices and α > 2
+is the path loss exponent considering an NLoS scenario. The
+
+distances are obtained through simple trigonometry based on
+the coordinates randomly generated by the stochastic processes
+described in Section II.
+For the signal intercepted by the eavesdroppers, it is eval-
+uated a set of K = (ΦE + ΨE) Eves, containing both planar
+and vehicular nodes. Similar considerations are adopted for
+the Charlies in the CJ scenario, resulting in C = (ΦC + ΨC).
+As discussed when sa was presented, Alice sends a signal
+containing the secret information and AN. Since authorized
+Alice-Eves channels are not expected in the beamforming
+sense, the orthonormal basis are not null, thus the Eves
+receive interference. When the Cooperative Jamming is taken
+into consideration, Eves are also affected by the interference
+generated by the nearby Charlies through the sc signals. With
+that in mind, the signal obtained by the k-th Eve is given by
+yE,k =
+�
+φPt h†
+AE,k D−α/2
+AE,k x
++
+�
+(1 − φ)Pt
+NA − 1 h†
+AE,k Wa D−α/2
+AE,k na
++
+�
+c ∈C
+�
+Pc
+NC − 1 h†
+c,k Wc D−α/2
+c,k
+nc ,
+(5)
+which is composed of essentially three terms. The first is the
+intercepted secret message itself, the second term is the AN
+signal generated by Alice, and the third term is a sum of all
+the interference injected by the Charlie nodes. Since CJ only
+affects the last term of (5), the AN scenario can be obtained
+by simply adopting that the sum in this term is equal to zero.
+From (4) and (5), it is possible to determine the SIR of Bob
+and the K Eves. Thus, the SIR of Bob can be determined as
+γB = Ptφ ∥ha∥2 D−α
+AB,
+(6)
+and the SIR for each Eve can be obtained from (5) as follows
+γE,k =
+Pt φ
+���h†
+AE,k ha/∥ha∥
+���
+2
+D−α
+AE,k
+Pt (1−φ)
+NA−1
+���h†
+AE,k Wa
+���
+2
+D−α
+AE,k + Ic
+,
+(7)
+where Ic is the sum of the interference injected by the Charlies
+given by
+Ic =
+�
+c ∈ C
+Pc
+Nc − 1∥h†
+c,k Wc∥2 D−α
+ck ,
+(8)
+which is non-zero only in the CJ scenario. The products h†
+AE,k·
+ha/∥ha∥ and h†
+AE,k ·Wa from the Alice-Eve channel and also
+h†
+ck·Wc from Charlie-Eve produce independent identically dis-
+tributed CN random variables with unitary variance [6]. This
+enables the approximations
+���h†
+AE,k(ha/∥ha∥)
+���
+2
+∼ exp(1),
+∥h†
+AE,kWa∥2 ∼ Gamma(NA − 1, 1) and ∥h†
+c,k Wc∥2 ∼
+Gamma(NC − 1, 1).
+D. Performance metric
+Considering that Alice transmits codewords at a rate Rb with
+a secrecy rate RS ≤ CS, the redundancy rate can be defined
+as Re = Rb − RS. Then a secrecy outage event occurs when
+the channel capacity of any Eve is higher than the redundancy
+rate that Alice can provide, i.e., CE > Re.
+In a multiple passive Eves scenario, whose Channel State
+Information (CSI) are unknown, the secrecy performance is
+addressed in terms of the Secrecy Outage Probability (SOP),
+since the only available information about the Alice-Eve
+channel is its statistics. Thus, the SOP is defined as
+SOP = 1 − Pr
+�
+max
+k∈K γE,k < β
+�
+,
+(9)
+which is the complement of the probability that the highest
+SIR among all Eves is less than the threshold β = 2Re − 1.
+This means that higher values of secrecy can be obtained by
+implementing the aforementioned PLS techniques to reduce
+γE,k as much as possible.
+IV. NUMERICAL RESULTS
+Various simulations with different parameters were per-
+formed to evaluate the relationship between the SOP and the
+decrease of the SIR for the k-th Eve. Since the V2X network
+model is randomly generated, the coordinates of each node
+and street change with each run. To provide more consistent
+results, the curves presented below are the average of multiple
+realizations of each simulation configuration.
+Fig. 3 illustrates the SOP for different Pt and Pc values,
+ranging from 10 mW (10 dBm) to 1 W (30 dBm). As expected,
+when the devices have more power available for interference,
+the SOP is greatly reduced. However, for the AN scenario
+secrecy is still not guaranteed when φ grows. For CJ, the SOP
+increases in a much slower rate due to the larger amount of
+nodes jamming the signal received by the Eves.
+(a) Artificial Noise
+(b) Cooperative Jamming
+Fig. 3: SOP versus φ (25 realizations) for the AN and CJ with
+different available power {0.01, 0.1, 1} W. β = 0 dB, α = 3,
+NA = NC = 4, λE = λC = 10−6/m2 , µE = µC = 10−3/m, r
+= 3 km.
+Through the simulation results presented in Fig. 4, it can be
+easily noted that as β increases the SOP decreases, because
+
+1.0
+0.8
+0.6
+SOP
+S
+0.4
+0.2
+Pt = 0.01 W
+Pt = 0.10 W
+Pt = 1.00 W
+0.0
+0.00
+0.25
+0.50
+0.75
+1.00
+Φ1.0
+Pt = Pc = 0.01 W
+Pt = Pc = 0.10 W
+Pt = Pc = 1.00 W
+0.8
+0.6
+SOP
+0.4
+0.2
+0.0
+0.00
+0.25
+0.50
+0.75
+1.00the criteria for secrecy failure is becoming more selective.
+Furthermore, φ have an opposing effect when compared to
+β, suggesting that for higher threshold values to guarantee
+low SOP, more power needs to be allocated to interference.
+Because of that, in applications where the devices have limited
+power (such as IoT and V2X), CJ is a more economic approach
+as long as there are sufficient nearby auxiliary nodes.
+Fig. 4:
+SOP versus β (50 realizations) for the AN and CJ
+with different power allocation ratios {0.4, 0.6, 0.8}. α = 3,
+Pt = Pc = 20 dBm, NA = NC = 4, λE = λC = 10−6/m2 ,
+µE = µC = 10−3/m, r = 3 km.
+In Fig. 5, it is evaluated the influence that the proportion
+of Charlies to Eves have on the SOP. This is achieved by
+implementing different values of intensities (λ and u) for
+the Poisson processes that generate these nodes. The SOP
+grows rapidly in the AN, indicating that the available power
+is insufficient to guarantee secrecy with the given Eve density.
+For the CJ cases, however, as the number of Charlie nodes
+rises, the SOP starts to reduce, making the communication
+viable even for higher values of φ. When there are more
+Charlies than Eves it is shown that very little power needs
+to be applied in each device to provide a low SOP.
+V. CONCLUSION
+In this paper, a stochastic geometric approach was presented
+as a method to randomly generate V2X network models. The
+coordinates of these elements were then used to evaluate the
+effectiveness of PLS techniques in different realizations of
+vehicular networks subjected to path loss with NLoS.
+Both AN and CJ were introduced based on the analytical
+signals that the involved nodes transmit. Next, expressions
+were obtained for the SIR of Bob and the k-th Eve. Finally,
+the SOP was computed to evaluate the level of information
+security provided by the presented PLS techniques.
+Based on numerical results, it can be concluded that PLS
+can provide additional security for the V2X networks with
+relative low power cost, specially when both the techniques
+are combined. It is also noted that in the CJ scenario, when
+Fig. 5:
+SOP versus φ (25 realizations) for the AN and CJ
+with different λC/λE ratios {0.1, 0.5, 1, 5, 10}. β = 0 dB, α
+= 3, Pt = Pc = 10 dBm, NA = NC = 4, λE = 10−6/m2 , µE
+= 10−3/m, r = 3 km.
+there are more Charlies in the proximity, the security increases.
+Therefore, the urban networks are the most benefited by this
+technique, since it is expected a higher density of wireless
+devices in the same area in these environments.
+REFERENCES
+[1] B. M. ElHalawany, A. A. El-Banna and K. Wu, “Physical-Layer Se-
+curity and Privacy for Vehicle-to-Everything”, IEEE Communications
+Magazine, vol. 57, n. 10, pp. 84-90, 2019.
+[2] J. M. Hamamreh, H. M. Furqan and H. Arslan, “Classifications and
+Applications of Physical Layer Security Techniques for Confidentiality:
+A Comprehensive Survey”, IEEE Communications Surveys & Tutorials,
+vol. 21, n. 2, pp. 1773-1828, 2019.
+[3] A. Sanenga, G. A. Mapunda, T. M. L. Jacob, L. Marata, B. Basutli and
+J. M. Chuma, “An Overview of Key Technologies in Physical Layer
+Security”, Entropy, vol. 22, n. 11, MDPI, 2020.
+[4] A. D. Wyner, “The wire-tap channel”, The Bell System Technical
+Journal, vol. 54, n. 8, pp. 1355-1387, 1975.
+[5] R. Negi and S. Goel, “Secret communication using artificial noise”,
+VTC-2005-Fall. 2005 IEEE 62nd Vehicular Technology Conference, vol.
+3, pp. 1906-1910, 2005.
+[6] C. Wang, Z. Li, X. Xia, J. Shi, J. Si, and Y. Zou, “Physical Layer Security
+Enhancement Using Artificial Noise in Cellular Vehicle-to-Everything
+(C-V2X) Networks”, IEEE Transactions on Vehicular Technology, vol.
+69, n. 12, pp. 15253-15268, 2020.
+[7] B. Qiu and C. Jing, “Performance Analysis for Cooperative Jamming
+and Artificial Noise Aided Secure Transmission Scheme in Vehicular
+Communication Network”, Research Square Platform LLC, 2020.
+[8] M. Haenggi, Stochastic Geometry for Wireless Networks. Cambridge:
+Cambridge University Press, 2012.
+[9] R. D. Yates and D. J. Goodman, “Probability and Stochastic Processes: A
+Friendly Introduction for Electrical and Computer Engineers”, Nashville,
+TN: John Wiley & Sons, 2005.
+[10] J. Bertrand, Calcul des probabilités. Gauthier-Villars, 1889.
+[11] V. V. Chetlur and H. S. Dhillon, “Coverage Analysis of a Vehicular
+Network Modeled as Cox Process Driven by Poisson Line Process”,
+IEEE Transactions on Wireless Communications, vol. 17, n. 7, 2018.
+[12] C. Choi and F. Baccelli, “Poisson Cox Point Processes for Vehicular
+Networks”, IEEE Transactions on Vehicular Technology, vol. 67, n. 10,
+pp. 10160-10165, 2018.
+[13] L. Hu, H. Wen, B. Wu, F. Pan, R. Liao, H. Song, J. Tang, and X.
+Wang, “Cooperative Jamming for Physical Layer Security Enhancement
+in Internet of Things”, IEEE Internet of Things Journal, vol. 5, n. 1,
+2018.
+
+100
+10-1
+SOP
+10~2
+10-3
+-10
+-5
+0
+5
+10
+15
+β
+AN: Φ = 0.4
+AN: Φ = 0.6
+-
+AN: @ = 0.8
+CJ: Φ = 0.4
+CJ: Φ = 0.6
+-
+CJ: Φ = 0.81.0
+0.8
+0.6
+SOP
+0.4
+0.2
+0.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+AN
+CJ: Charlies/Eves = 0.5
+CJ: Charlies/Eves = 5.0
+CJ: Charlies/Eves = 0.1
+CJ: Charlies/Eves = 1.0
+CJ: Charlies/Eves = 10.0

ItE4T4oBgHgl3EQfhQ0t/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf,len=332
+page_content='Physical Layer Security Techniques Applied to  Vehicle-to-Everything Networks  Leonardo B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' da Silva, Evelio M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' Fernández and Ândrei Camponogara  Abstract— Physical Layer Security (PLS) is an emerging con-  cept in the field of secrecy for wireless communications that can  be used alongside cryptography to prevent unauthorized devices  from eavesdropping a legitimate transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' It offers low com-  putational cost and overhead by injecting an interfering signal  in the wiretap channels of potential eavesdroppers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' This paper  discusses the benefits of the Artificial Noise and Cooperative  Jamming techniques in the context of Vehicle-to-everything (V2X)  networks, which require secure data exchange with small latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  The simulations indicate that messages can be safely delivered  even with devices that have low available power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  Keywords— Wireless communication networks, Physical Layer  Security, secrecy, Vehicle-to-everything, Artificial Noise, Cooper-  ative Jamming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' INTRODUCTION  Urban mobility is one of the main focuses of the Internet  of Things (IoT) when applied to smart cities, due to the  necessity for more responsive and safe traffic control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' Gener-  ally, the solutions proposed in this scope involve the wireless  communication between not only the vehicles themselves,  but also with pedestrians, infrastructure, and networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' This  paradigm is known as Vehicle-to-everything (V2X) and it can  be standardized by protocols such as C-ITS (Cellular Intelli-  gent Transportation System) and WAVE (Wireless Access for  Vehicular Environment) that are based on the IEEE 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='11p  amendment, and the Cellular-V2X (C-V2X) that implements  the 5G standard from 3GPP (3rd Generation Partnership  Project) [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' Problem Outline  Due to the ever-changing location of most of the involved  communication nodes and the time-sensitive nature of the  data involved (brake position, vehicle speed, traffic volume,  accident reports, etc), the transmission needs not only to occur  at high rates, but also offer reliability through high secrecy, low  packet loss, and small delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' Furthermore, those nodes have  to be affordable to justify their implementation on a city-wide  scale, thus having low power consumption and the most cost-  efficient embedded processing unit possible [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  Since the main source of information security in today’s  landscape is provided through cryptography, the secrecy con-  straint can negatively affect most of these criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' As a result  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  da  Silva,  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  Fernandez,  Â.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  Camponogara,  Electri-  cal Engineering Department, Federal University of Paraná (UFPR), Cu-  ritiba, PR, Brazil, e-mails: leonardobarbosa@ufpr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='br, evelio@ufpr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='br and  andrei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='camponogara@ufpr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' This study was financed in part by the Coorde-  nação de Aperfeiçoamento de Pessoal de Nível Superior – Brasil (CAPES) –  Finance Code 001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  of the growth in the availability of portable and connected  equipment with high processing capabilities, the safety mea-  sures implemented need to match this computational power  with proportionally longer and more complex keys to not be  vulnerable to brute-force attacks from well-equipped malicious  devices [2], [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' This approach, however, is not sustainable,  because it produces increasingly long authentication routines,  due to the raise in computational overhead and processing cost  as a result of the implemented security algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' Overview of the proposed solution  To counterbalance this issue, this paper studies the use of  Physical Layer Security (PLS) techniques as an additional  protection to increase the secrecy of wireless communications  in a V2X environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' As the name suggests, PLS is applied  at the Physical Layer, making it an alternative that can be used  with low processing cost when compared with cryptography,  which is more oriented towards the computational side of the  network stack on the Application Layer [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  Since cryptography techniques provide security in different  sections of the wireless protocols, PLS is proposed as a  complement to them, rather than a replacement [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' Through  the use of both approaches on the same node, it is possible to  offer high secrecy without the necessity of infinitely growing  key complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  The PLS has its origins on the analytical proposal of  Wyner’s wiretap channel [4], where it is described a commu-  nication between two legitimate nodes that is spied on by an  eavesdropper through an unauthorized channel called wiretap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  In the modern literature, these devices are usually referred to  as a transmitter called Alice, an authorized receiver Bob, and  the set of K eavesdroppers named Eves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  In the wiretap channel model shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' 1, the original  message m is encoded and transmitted by Alice as the signal  sa, that reaches Bob through the main channel hAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' The  received signal yB is then decoded by Bob, obtaining the  estimated message ˆm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' Additionally, the k-th Eve can intercept  sa through the wiretap channel hAE,k, obtaining the signal  yE,k that when decoded produces z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  The main focus of PLS is to guarantee that the mutual  information between m and z is as close to zero as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  When this condition is met, even if z is know, it is impossible  for Eve to infer the contents of the original message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  Wyner then presents a set of parameters that enable the use  of the physical imperfections of the channel, such as noise  and fading, to provide information secrecy by raising the level  of confusion on undesired nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' Rendering them unable to  distinguish between the message and the interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='05123v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='SP]  12 Jan 2023  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' 1: The wiretap channel generic model based on [4]  Currently, plenty of techniques to provide security at the  physical layer level have been proposed in the literature [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  This paper will focus on two approaches first presented in [5]:  Artificial Noise (AN): This approach uses a portion of  the transmitter node’s power to inject artificially gener-  ated noise in the eavesdropper’s channel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  Cooperative Jamming (CJ): This approach expands the  AN model by proposing a connected network where  nearby relay nodes (Charlies) send a jamming signal to  the eavesdropper’s channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  To demonstrate the viability of AN and CJ applications in  a V2X network, it is common to create stochastic geometric  models that randomly generate streets and distribute com-  munication nodes in a predefined area to represent an urban  mobility scenario [6], [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' When implementing these methods,  metrics such as the Signal-to-Interference Ratio (SIR) are used  to define the threshold of confusion necessary to provide se-  crecy at the physical layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' The SIR on each eavesdropper can  then be evaluated to determine the secrecy outage probability  (SOP) of the data transmission with different densities of the  involved nodes in the simulated network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  In this paper, Section II describes the stochastic algorithms  implemented to model a V2X network that includes streets  and communication nodes (vehicular and planar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' Section III  presents the analytical basis of the AN and CJ techniques,  while also introducing the SIR and SOP metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' In Section  IV, the results of numerical simulations are shown to illustrate  the benefits of the considered PLS techniques on the generated  V2X networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' Finally, Section V states some final remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  Notation: IN is an identity matrix of order N, Poisson(n)  is a Poisson distribution with mean number of arrivals n,  CN(m, n) is a complex normal distribution with average m  and covariance n, exp(n) is an exponential distribution with  mean n and Gamma(m, n) is the gamma distribution with  form m and scale n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' THE V2X NETWORK MODEL  As mentioned previously, vehicular networks are dynamic,  with devices changing location constantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' Thus, a determin-  istic model is not well-suited for this application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' A common  alternative is the use of stochastic geometry to represent this  random spatial nature through a variety of different processes  to distribute the streets and communication nodes within the  desired coverage area [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  A viable option is the use of Poisson processes, as they are  memoryless counting processes for integer arrivals [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' In other  words, each set of elements generated will be independent  with a Poisson distributed integer number of uniformly spaced  nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' The intensity of the arrivals in these processes are  represented by λ and the expected number of elements is  the product of the said intensity and the Lebesgue measure,  which in this context is essentially the spatial measurement  associated with the object that the points will be distributed on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  For instance, the Lebesgue measure to populate a circle is its  area and for a line is the length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' One realization of the resulting  spatial model derived from the use of different variations of  the Poisson processes is represented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' 2:  Spatial simulation of the modeled V2X network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  The color green indicates the Charlies implemented in CJ  techniques and the Eves are in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' The planar devices are  generated by PPPs represented by circles (◦) with intensity λ  = 10−6/m2 for both node types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' Through a PLP, the streets  (blue lines) have been modeled with an intensity of λl = 10−3  /m, and the vehicular devices are originated from PLP-driven  Cox Processes indicated with triangles (△) of intensity u =  10−3/m for both Charlies and Eves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' A single Alice is indicated  with a black × at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  In this model, the wireless devices of pedestrians and  connected infrastructure are considered free to be positioned in  the whole area A of the modeled network, which is a circle of  radius r = 3 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' Thus, these “planar nodes“ are generated by  2-D Poisson Point Processes (PPP) and the expected amount  of elements is given by Poisson(λ · A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' The set of planar  nodes is indicated by Φ, thus the planar Eves and Charlies  are respectively represented by ΦE and ΦC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  The streets are represented by uniformly distributed lines  with density µl = λl/π generated by a Poisson Line Process  (PLP) Φl based on the second method of the Bertrand paradox  [10], in which a set of expected Poisson(µl·2πr) midpoints are  created [11], each with a random radius P ∈ [0, r) and angle  θ ∈ [0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' From these coordinates, a segment perpendicular  to P is traced between two points at the edge of the circle  of radius r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' This effectively means that a pair of 1-D PPP  points are created in the perimeter of the circular area for  each modeled street.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  On those PLP-generated lines, a Cox process of intensity u  YB  TRANSMITTER  MAIN CHANNEL  RECEIVER  m -  (ALICE)  hAB  m  (BOB)  Ye,k  WIRETAP CHANNEL  k-thEAVESDROPPER  >Z  (EVE)3000  X  Alice node  Planar Charlies  Q  Planar Eves  Vehicular Charlies  O  A  Vehicular Eves  2000  O  Q  1000  O  F 0  2  X  O  1000  C  2000  3000  4000  3000  2000  1000  0  1000  2000  3000  4000is implemented, which is used to create the “vehicular nodes“  on each segment [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' These elements represent vehicles  whose spatial distribution are constrained to a street by a 1-D  PPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' Considering a street of length l, the number of vehicles  in it is given by Poisson(u · l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  The set of vehicular Eves and Charlies on each street l are  respectively denoted by ψE and ψC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' Based on these, the total  nodes of each type can be obtained by evaluating the sets on  the whole range of Φl [6], resulting in ΨE = {ψE(l)}l∈Φl for  Eves and ΨC = {ψC(l)}l∈Φl for Charlies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  Furthermore, a single deterministic transmitter (Alice) is  included at the origin of the circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' This point is selected to  simplify the distance calculations between a legitimate device  and the Eve nodes, which can be planar or vehicular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' This  measurement is one of the parameters for the SIR calculations,  that are considered to determine the effectiveness of the PLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  For the CJ case, auxiliary nodes (Charlies) are also modeled,  some as planar and others as vehicular devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' Note that the  distance between Charlies and Eves influences the power of  the interference injected on the unauthorized channels as part  of the jamming technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' PLS TECHNIQUES  The PLS techniques presented in this paper are part of the  key-less-based class [2], which implements secure information  transmission by making the unauthorized channel’s capacity  (CE) lower than that of the legitimate channel’s (CB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' This re-  lationship can be presented by evaluating these values through  the Shannon-Hartley theorem, which produces the secrecy  capacity (CS) metric as  CS = CB − CE = log2(1 + γB) − log2(1 + γE),  (1)  where γB and γE are, respectively, the SIRs of Bob and Eve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  Based on this expression, it can be inferred that in order to  guarantee that CB is sufficiently larger than CE, the value of  γE must be as low as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' The approach utilized by AN  and CJ is the injection of artificially generated interference in  the eavesdropper channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  Typically, this injection is implemented with multi-antenna  networks, as it enables the use of beamforming to selectively  direct the transmission to legitimate receivers with minimum  noise and high efficiency [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' The unintended receivers on  the other hand, intercept a signal that contains the secret  message as well as AN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' Therefore, secrecy is provided when  the distinction between them by the Eves is improbable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  The wireless channels in this paper are modeled with com-  plex normal distributions (CN) which implies in a Rayleigh  fading model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' This decision provides simpler analytical equa-  tions and also proposes a more pessimistic scenario, in which  there is no Line-of-Sight (LoS) available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' By evaluating the  metrics in these worst-case conditions, it is possible to verify  that even then the secrecy can be guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' Artificial Noise  In the AN scenario, the legitimate communication is es-  tablished between a single transmitter Alice and a receiver  Bob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' Additional nodes (both planar and vehicular) that try to  obtain Alice’s signal are then considered eavesdroppers and  their channels will be affected by the AN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  The signal transmitted by the Alice node with NA antennas  is composed of two terms: the first contains a message x  intended for Bob and the second is based on a zero-forcing  vector for the unauthorized devices [13], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='e,  sa =  �  φPt  ha  ∥ha∥x +  �  (1 − φ)Pt  NA − 1 Wana,  (2)  where ha/∥ha∥ is the beamforming vector with the normaliza-  tion of the Alice’s channel estimation ha ∈ CNA×1, that will  be modeled as CN(0, INA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' The AN is formed by the null-  space orthonormal basis Wa ∈ CNA×(NA−1) and the noise  signal na ∈ C(NA−1)×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  The distribution of the available power, Pt, between the  two terms of (2) is controlled by φ ∈ {0,1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' φ = 0 means that  all power is allocated to noise generation and no message is  sent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' Conversely, when φ = 1 the AN is not active and Pt is  allocated entirely for data transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' Cooperative Jamming  The Cooperative Jamming extends the AN case, maintaining  the single Alice-Bob authorized transmission with multiple  Eves, however, adding auxiliary nodes in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' These  devices, typically called Charlies, can also be either planar or  vehicular, just like the Eves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' In contrast, they are responsible  for providing additional security by sending jamming signals  that further decrease the channel quality of the Eves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  For simplicity, it is considered that only Alice will transmit  messages in the scenarios evaluated in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' Hence, the  signals sent by the Charlie nodes are made of only the AN  (zero-forcing) portion, as follows  sc =  �  Pc  NC − 1Wcnc,  (3)  where NC is the number of antennas of each Charlie and PC  is the power available for jamming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' Notice that since these  nodes are not transmitting messages, all the available power  is directed towards CJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' Additionally, Wc ∈ CNC×(NC−1) is  the null space orthonormal matrix and nc ∈ C(NC−1)×1 is the  artificial noise component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' Received Signals  By considering that the channel estimation ha is precisely  the main channel established between Alice and Bob, hAB,  it is implied that the receiver node is not affected by the  interference from AN or CJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' That happens because the or-  thonormal basis Wa and Wc are null when applied to the  authorized channels, resulting in the relationships h†  ABWa = 0  and h†  ABWc = 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' Therefore, the signal received  by Bob can be expressed as  yB =  �  φPt ∥ha∥ D−α/2  AB  x,  (4)  where DAB is the distance between the devices and α > 2  is the path loss exponent considering an NLoS scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' The  distances are obtained through simple trigonometry based on  the coordinates randomly generated by the stochastic processes  described in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  For the signal intercepted by the eavesdroppers, it is eval-  uated a set of K = (ΦE + ΨE) Eves, containing both planar  and vehicular nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' Similar considerations are adopted for  the Charlies in the CJ scenario, resulting in C = (ΦC + ΨC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  As discussed when sa was presented, Alice sends a signal  containing the secret information and AN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' Since authorized  Alice-Eves channels are not expected in the beamforming  sense, the orthonormal basis are not null, thus the Eves  receive interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' When the Cooperative Jamming is taken  into consideration, Eves are also affected by the interference  generated by the nearby Charlies through the sc signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' With  that in mind, the signal obtained by the k-th Eve is given by  yE,k =  �  φPt h†  AE,k D−α/2  AE,k x  +  �  (1 − φ)Pt  NA − 1 h†  AE,k Wa D−α/2  AE,k na  +  �  c ∈C  �  Pc  NC − 1 h†  c,k Wc D−α/2  c,k  nc ,  (5)  which is composed of essentially three terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' The first is the  intercepted secret message itself, the second term is the AN  signal generated by Alice, and the third term is a sum of all  the interference injected by the Charlie nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' Since CJ only  affects the last term of (5), the AN scenario can be obtained  by simply adopting that the sum in this term is equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  From (4) and (5), it is possible to determine the SIR of Bob  and the K Eves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' Thus, the SIR of Bob can be determined as  γB = Ptφ ∥ha∥2 D−α  AB,  (6)  and the SIR for each Eve can be obtained from (5) as follows  γE,k =  Pt φ  ���h†  AE,k ha/∥ha∥  ���  2  D−α  AE,k  Pt (1−φ)  NA−1  ���h†  AE,k Wa  ���  2  D−α  AE,k + Ic  ,  (7)  where Ic is the sum of the interference injected by the Charlies  given by  Ic =  �  c ∈ C  Pc  Nc − 1∥h†  c,k Wc∥2 D−α  ck ,  (8)  which is non-zero only in the CJ scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' The products h†  AE,k·  ha/∥ha∥ and h†  AE,k ·Wa from the Alice-Eve channel and also  h†  ck·Wc from Charlie-Eve produce independent identically dis-  tributed CN random variables with unitary variance [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' This  enables the approximations  ���h†  AE,k(ha/∥ha∥)  ���  2  ∼ exp(1),  ∥h†  AE,kWa∥2 ∼ Gamma(NA − 1, 1) and ∥h†  c,k Wc∥2 ∼  Gamma(NC − 1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' Performance metric  Considering that Alice transmits codewords at a rate Rb with  a secrecy rate RS ≤ CS, the redundancy rate can be defined  as Re = Rb − RS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' Then a secrecy outage event occurs when  the channel capacity of any Eve is higher than the redundancy  rate that Alice can provide, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=', CE > Re.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  In a multiple passive Eves scenario, whose Channel State  Information (CSI) are unknown, the secrecy performance is  addressed in terms of the Secrecy Outage Probability (SOP),  since the only available information about the Alice-Eve  channel is its statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' Thus, the SOP is defined as  SOP = 1 − Pr  �  max  k∈K γE,k < β  �  ,  (9)  which is the complement of the probability that the highest  SIR among all Eves is less than the threshold β = 2Re − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  This means that higher values of secrecy can be obtained by  implementing the aforementioned PLS techniques to reduce  γE,k as much as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' NUMERICAL RESULTS  Various simulations with different parameters were per-  formed to evaluate the relationship between the SOP and the  decrease of the SIR for the k-th Eve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' Since the V2X network  model is randomly generated, the coordinates of each node  and street change with each run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' To provide more consistent  results, the curves presented below are the average of multiple  realizations of each simulation configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' 3 illustrates the SOP for different Pt and Pc values,  ranging from 10 mW (10 dBm) to 1 W (30 dBm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' As expected,  when the devices have more power available for interference,  the SOP is greatly reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' However, for the AN scenario  secrecy is still not guaranteed when φ grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' For CJ, the SOP  increases in a much slower rate due to the larger amount of  nodes jamming the signal received by the Eves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  (a) Artificial Noise  (b) Cooperative Jamming  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' 3: SOP versus φ (25 realizations) for the AN and CJ with  different available power {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='1, 1} W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' β = 0 dB, α = 3,  NA = NC = 4, λE = λC = 10−6/m2 , µE = µC = 10−3/m, r  = 3 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  Through the simulation results presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' 4, it can be  easily noted that as β increases the SOP decreases, because  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='6  SOP  S  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='2  Pt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='01 W  Pt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='10 W  Pt = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='00 W  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='00  Φ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='0  Pt = Pc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='01 W  Pt = Pc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='10 W  Pt = Pc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='00 W  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='6  SOP  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='00the criteria for secrecy failure is becoming more selective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  Furthermore, φ have an opposing effect when compared to  β, suggesting that for higher threshold values to guarantee  low SOP, more power needs to be allocated to interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  Because of that, in applications where the devices have limited  power (such as IoT and V2X), CJ is a more economic approach  as long as there are sufficient nearby auxiliary nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' 4:  SOP versus β (50 realizations) for the AN and CJ  with different power allocation ratios {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='8}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' α = 3,  Pt = Pc = 20 dBm, NA = NC = 4, λE = λC = 10−6/m2 ,  µE = µC = 10−3/m, r = 3 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' 5, it is evaluated the influence that the proportion  of Charlies to Eves have on the SOP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' This is achieved by  implementing different values of intensities (λ and u) for  the Poisson processes that generate these nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' The SOP  grows rapidly in the AN, indicating that the available power  is insufficient to guarantee secrecy with the given Eve density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  For the CJ cases, however, as the number of Charlie nodes  rises, the SOP starts to reduce, making the communication  viable even for higher values of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' When there are more  Charlies than Eves it is shown that very little power needs  to be applied in each device to provide a low SOP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' CONCLUSION  In this paper, a stochastic geometric approach was presented  as a method to randomly generate V2X network models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' The  coordinates of these elements were then used to evaluate the  effectiveness of PLS techniques in different realizations of  vehicular networks subjected to path loss with NLoS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  Both AN and CJ were introduced based on the analytical  signals that the involved nodes transmit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' Next, expressions  were obtained for the SIR of Bob and the k-th Eve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' Finally,  the SOP was computed to evaluate the level of information  security provided by the presented PLS techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  Based on numerical results, it can be concluded that PLS  can provide additional security for the V2X networks with  relative low power cost, specially when both the techniques  are combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' It is also noted that in the CJ scenario, when  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' 5:  SOP versus φ (25 realizations) for the AN and CJ  with different λC/λE ratios {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='5, 1, 5, 10}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' β = 0 dB, α  = 3, Pt = Pc = 10 dBm, NA = NC = 4, λE = 10−6/m2 , µE  = 10−3/m, r = 3 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  there are more Charlies in the proximity, the security increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  Therefore, the urban networks are the most benefited by this  technique, since it is expected a higher density of wireless  devices in the same area in these environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
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+page_content=' Wu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' Pan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' Liao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' Song, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' Tang, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content='  Wang, “Cooperative Jamming for Physical Layer Security Enhancement  in Internet of Things”, IEEE Internet of Things Journal, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
+page_content=' 5, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'}
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