Merge branch 'master' of git:research/least_squares/paper/topology

6 files changed, 2080 insertions, 132 deletions

diff --git a/SciPost.cls b/SciPost.cls
new file mode 100644
index 0000000..ca8b331
--- /dev/null
+++ b/SciPost.cls
@@ -0,0 +1,166 @@
+\NeedsTeXFormat{LaTeX2e}
+\ProvidesClass{SciPost} % SciPost Latex Template (2021-08)
+
+
+\LoadClass[11pt,a4paper]{article}
+
+
+% Layout
+\RequirePackage[top=12mm,bottom=12mm,left=30mm,right=30mm,head=12mm,includeheadfoot]{geometry}
+\bigskipamount 6mm
+
+% For table of contents: remove trailing dots
+\RequirePackage{tocloft}
+\renewcommand{\cftdot}{}
+% Add References to TOC
+\RequirePackage[nottoc,notlot,notlof]{tocbibind}
+
+
+% Spacings between (sub)sections:
+\RequirePackage{titlesec}
+\titlespacing*{\section}{0pt}{1.8\baselineskip}{\baselineskip}
+
+
+% Unicode characters
+\RequirePackage[utf8]{inputenc}
+
+% doi links in references
+\RequirePackage{doi}
+
+% Math formulas and symbols
+%\RequirePackage{amsmath,amssymb} % Redundant (clashes with mathdesign)
+\RequirePackage{amsmath}
+
+% Hyperrefs
+\RequirePackage{hyperref}
+
+% Include line numbers in submissions
+\RequirePackage{lineno}
+
+% SciPost BiBTeX style
+\bibliographystyle{SciPost_bibstyle}
+
+% SciPost header and footer
+\RequirePackage{fancyhdr}
+\pagestyle{fancy}
+
+\makeatletter
+  \let\ps@plain\ps@fancy
+\makeatother
+
+\RequirePackage{xcolor}
+\definecolor{scipostdeepblue}{HTML}{002B49}
+\definecolor{scipostphys}{HTML}{0019A2}
+\definecolor{scipostastro}{HTML}{946E13}
+\definecolor{scipostbio}{HTML}{2B7E13}
+\definecolor{scipostchem}{HTML}{605AAF}
+\definecolor{scipostcompsci}{HTML}{0AADAB}
+\definecolor{scipostmath}{HTML}{A10800}
+
+\RequirePackage{graphicx}
+
+\RequirePackage{cite}
+
+\RequirePackage[width=.90\textwidth]{caption}
+
+
+%% Patch lineno when used with amsmath
+\newcommand*\patchAmsMathEnvironmentForLineno[1]{%
+\expandafter\let\csname old#1\expandafter\endcsname\csname #1\endcsname
+\expandafter\let\csname oldend#1\expandafter\endcsname\csname end#1\endcsname
+\renewenvironment{#1}%
+{\linenomath\csname old#1\endcsname}%
+{\csname oldend#1\endcsname\endlinenomath}}%
+\newcommand*\patchBothAmsMathEnvironmentsForLineno[1]{%
+\patchAmsMathEnvironmentForLineno{#1}%
+\patchAmsMathEnvironmentForLineno{#1*}}%
+\AtBeginDocument{%
+\patchBothAmsMathEnvironmentsForLineno{equation}%
+\patchBothAmsMathEnvironmentsForLineno{align}%
+\patchBothAmsMathEnvironmentsForLineno{flalign}%
+\patchBothAmsMathEnvironmentsForLineno{alignat}%
+\patchBothAmsMathEnvironmentsForLineno{gather}%
+\patchBothAmsMathEnvironmentsForLineno{multline}%
+}
+%% End patch lineno
+
+
+\DeclareOption{submission}{
+  \rhead{
+    {\bf \color{scipostdeepblue} ~Submission }
+  }
+ }
+
+
+%%%%%%%%%% Physics
+\DeclareOption{Phys}{
+\lhead{
+  \colorbox{scipostphys}{\bf \color{white} ~SciPost Physics }
+}
+}
+
+\DeclareOption{PhysCore}{
+\lhead{
+  \colorbox{scipostphys}{\bf \color{white} ~SciPost Physics Core}
+}
+}
+
+\DeclareOption{PhysLectNotes}{
+  \lhead{
+    \colorbox{scipostphys}{\strut \bf \color{white} ~SciPost Physics Lecture Notes }
+  }
+}
+
+\DeclareOption{PhysProc}{
+\lhead{
+  \colorbox{scipostphys}{\strut \bf \color{white} ~SciPost Physics Proceedings }
+}
+}
+
+\DeclareOption{PhysCodeb}{
+\lhead{
+  \colorbox{scipostphys}{\strut \bf \color{white} ~SciPost Physics Codebases }
+}
+}
+
+%%%%%%%%%%% Astronomy
+\DeclareOption{Astro}{
+\lhead{
+  \colorbox{scipostastro}{\strut \bf \color{white} ~SciPost Astronomy }
+}
+}
+
+
+%%%%%%%%%%% Biology
+\DeclareOption{Bio}{
+\lhead{
+  \colorbox{scipostbio}{\strut \bf \color{white} ~SciPost Biology }
+}
+}
+
+
+%%%%%%%%%%% Chemistry
+\DeclareOption{Chem}{
+\lhead{
+  \colorbox{scipostchem}{\strut \bf \color{white} ~SciPost Chemistry }
+}
+}
+
+
+%%%%%%%%%%% Computer Science
+\DeclareOption{CompSci}{
+\lhead{
+  \colorbox{scipostcompsci}{\strut \bf \color{white} ~SciPost Computer Science }
+}
+}
+
+
+%%%%%%%%%%% Mathematics
+\DeclareOption{Math}{
+\lhead{
+  \colorbox{scipostmath}{\strut \bf \color{white} ~SciPost Mathematics }
+}
+}
+
+
+\ProcessOptions\relax
diff --git a/SciPostPhys_Template.tex b/SciPostPhys_Template.tex
new file mode 100644
index 0000000..0f47a47
--- /dev/null
+++ b/SciPostPhys_Template.tex
@@ -0,0 +1,218 @@
+% =========================================================================
+% SciPost LaTeX template
+% Version 2021-08
+%
+% Submissions to SciPost Journals should make use of this template.
+%
+% INSTRUCTIONS: simply look for the `TODO:' tokens and adapt your file.
+%
+% You can also make use of our empty "skeleton" templates for each Journal,
+% e.g. SciPostPhys_skeleton.tex
+% =========================================================================
+
+
+% TODO: uncomment ONE of the class declarations below
+
+% Class declaration format: \documentclass[submission, {DOI label of journal}]{SciPost}
+% where the DOI label of the journal should be one of:
+% Phys          (for SciPost Physics)
+% PhysCore      (for SciPost Physics Core)
+% PhysLectNotes (for SciPost Physics Lecture Notes)
+% PhysProc      (for SciPost Physics Proceedings -> !! Please use the conference-specific template which you will find on our website !!
+% PhysCodeb     (for SciPost Physics Codebases)
+% Astro         (for SciPost Astronomy)
+% Bio           (for SciPost Biology)
+% Chem          (for SciPost Chemistry)
+% CompSci       (for SciPost Computer Science)
+% Math          (for SciPost Mathematics)
+
+
+%% PHYSICS:
+% If you are submitting a paper to SciPost Physics: uncomment next line
+\documentclass[submission, Phys]{SciPost}
+% If you are submitting a paper to SciPost Physics Core: uncomment next line
+%\documentclass[submission, PhysCore]{SciPost}
+% If you are submitting a paper to SciPost Physics Lecture Notes: uncomment next line
+%\documentclass[submission, PhysLectNotes]{SciPost}
+% If you are submitting a paper to SciPost Physics Proceedings: uncomment next line
+%\documentclass[submission, PhysProc]{SciPost}
+% If you are submitting a paper to SciPost Physics Codebases: uncomment next line
+%\documentclass[submission, PhysCodeb]{SciPost}
+
+%% ASTRONOMY:
+% If you are submitting a paper to SciPost Astronomy: uncomment next line
+% \documentclass[submission, Astro]{SciPost}
+
+%% BIOLOGY:
+% If you are submitting a paper to SciPost Biology: uncomment next line
+% \documentclass[submission, Bio]{SciPost}
+
+%% CHEMISTRY:
+% If you are submitting a paper to SciPost Chemistry: uncomment next line
+% \documentclass[submission, Chem]{SciPost}
+
+%% COMPUTER SCIENCE:
+% If you are submitting a paper to SciPost Computer Science: uncomment next line
+% \documentclass[submission, CompSci]{SciPost}
+
+%% MATHEMATICS:
+% If you are submitting a paper to SciPost Mathematics: uncomment next line
+% \documentclass[submission, Math]{SciPost}
+
+
+
+% Prevent all line breaks in inline equations.
+\binoppenalty=10000
+\relpenalty=10000
+
+\hypersetup{
+    colorlinks,
+    linkcolor={red!50!black},
+    citecolor={blue!50!black},
+    urlcolor={blue!80!black}
+}
+
+\usepackage[bitstream-charter]{mathdesign}
+\urlstyle{sf}
+
+% Fix \cal and \mathcal characters look (so it's not the same as \mathscr)
+\DeclareSymbolFont{usualmathcal}{OMS}{cmsy}{m}{n}
+\DeclareSymbolFontAlphabet{\mathcal}{usualmathcal}
+
+\begin{document}
+
+% TODO: write your article's title here.
+% The article title is centered, Large boldface, and should fit in two lines
+\begin{center}{\Large \textbf{
+Article Title, as descriptive as possible, ideally fitting in two lines (approximately 150 characters) or less\\
+}}\end{center}
+
+% TODO: write the author list here. Use first name (+ other initials) + surname format.
+% Separate subsequent authors by a comma, omit comma and use "and" for the last author.
+% Mark the corresponding author with a superscript star.
+\begin{center}
+Dee E. Faa\textsuperscript{1},
+Aah B. Cee\textsuperscript{2} and
+Gee K. See\textsuperscript{3$\star$}
+\end{center}
+
+% TODO: write all affiliations here.
+% Format: institute, city, country
+\begin{center}
+{\bf 1} Affiliation1
+\\
+{\bf 2} Affiliation2
+\\
+{\bf 3} Affiliation2
+\\
+% TODO: provide email address of corresponding author
+${}^\star$ {\small \sf CorrespondingAuthor@email.address}
+\end{center}
+
+\begin{center}
+\today
+\end{center}
+
+% For convenience during refereeing (optional),
+% you can turn on line numbers by uncommenting the next line:
+%\linenumbers
+% You should run LaTeX twice in order for the line numbers to appear.
+
+\section*{Abstract}
+{\bf
+% TODO: write your abstract here.
+The abstract is in boldface, and should fit in 8 lines.
+It should be written in a clear and accessible style, emphasizing the context, the problem(s) studied, the methods used, the results obtained, the conclusions reached, and the outlook. You can add a table contents, recommended if your paper is more than 6 pages long.
+}
+
+
+% TODO: include a table of contents (optional)
+% Guideline: if your paper is longer that 6 pages, include a TOC
+% To remove the TOC, simply cut the following block
+\vspace{10pt}
+\noindent\rule{\textwidth}{1pt}
+\tableofcontents\thispagestyle{fancy}
+\noindent\rule{\textwidth}{1pt}
+\vspace{10pt}
+
+
+\section{Introduction}
+\label{sec:intro}
+% TODO: write your article here.
+The stage is yours. Write your article here.
+The bulk of the paper should be clearly divided into sections with short descriptive titles, including an introduction and a conclusion.
+
+
+
+\section{A Section}
+Use sections to structure your article's presentation.
+
+Equations should be centered; multi-line equations should be aligned.
+\begin{equation}
+H = \sum_{j=1}^N \left[J (S^x_j S^x_{j+1} + S^y_j S^y_{j+1} + \Delta S^z_j S^z_{j+1}) - h S^z_j \right].
+\end{equation}
+
+In the list of references, cited papers \cite{1931_Bethe_ZP_71} should include authors, title, journal reference (journal name, volume number (in bold), start page) and most importantly a DOI link. For a preprint \cite{arXiv:1108.2700}, please include authors, title (please ensure proper capitalization) and arXiv link. If you use BiBTeX with our style file, the right format will be automatically implemented.
+
+All equations and references should be hyperlinked to ensure ease of navigation. This also holds for [sub]sections: readers should be able to easily jump to Section \ref{sec:another}.
+
+\section{Another Section}
+\label{sec:another}
+There is no strict length limitation, but the authors are strongly encouraged to keep contents to the strict minimum necessary for peers to reproduce the research described in the paper.
+
+\subsection{A first subsection}
+You are free to use dividers as you see fit.
+\subsection{A note about figures}
+Figures should only occupy the stricly necessary space, in any case individually fitting on a single page. Each figure item should be appropriately labeled and accompanied by a descriptive caption. {\bf SciPost does not accept creative or promotional figures or artist's impressions}; on the other hand, technical drawings and scientifically accurate representations are encouraged.
+
+
+\section{Conclusion}
+You must include a conclusion.
+
+\section*{Acknowledgements}
+Acknowledgements should follow immediately after the conclusion.
+
+% TODO: include author contributions
+\paragraph{Author contributions}
+This is optional. If desired, contributions should be succinctly described in a single short paragraph, using author initials.
+
+% TODO: include funding information
+\paragraph{Funding information}
+Authors are required to provide funding information, including relevant agencies and grant numbers with linked author's initials. Correctly-provided data will be linked to funders listed in the \href{https://www.crossref.org/services/funder-registry/}{\sf Fundref registry}.
+
+
+\begin{appendix}
+
+\section{First appendix}
+Add material which is better left outside the main text in a series of Appendices labeled by capital letters.
+
+\section{About references}
+Your references should start with the comma-separated author list (initials + last name), the publication title in italics, the journal reference with volume in bold, start page number, publication year in parenthesis, completed by the DOI link (linking must be implemented before publication). If using BiBTeX, please use the style files provided  on \url{https://scipost.org/submissions/author_guidelines}. If you are using our \LaTeX template, simply add
+\begin{verbatim}
+\bibliography{your_bibtex_file}
+\end{verbatim}
+at the end of your document. If you are not using our \LaTeX template, please still use our bibstyle as
+\begin{verbatim}
+\bibliographystyle{SciPost_bibstyle}
+\end{verbatim}
+in order to simplify the production of your paper.
+\end{appendix}
+
+
+% TODO:
+% Provide your bibliography here. You have two options:
+
+% FIRST OPTION - write your entries here directly, following the example below, including Author(s), Title, Journal Ref. with year in parentheses at the end, followed by the DOI number.
+%\begin{thebibliography}{99}
+%\bibitem{1931_Bethe_ZP_71} H. A. Bethe, {\it Zur Theorie der Metalle. i. Eigenwerte und Eigenfunktionen der linearen Atomkette}, Zeit. f{\"u}r Phys. {\bf 71}, 205 (1931), \doi{10.1007\%2FBF01341708}.
+%\bibitem{arXiv:1108.2700} P. Ginsparg, {\it It was twenty years ago today... }, \url{http://arxiv.org/abs/1108.2700}.
+%\end{thebibliography}
+
+% SECOND OPTION:
+% Use your bibtex library
+% \bibliographystyle{SciPost_bibstyle} % Include this style file here only if you are not using our template
+\bibliography{SciPost_Example_BiBTeX_File.bib}
+
+\nolinenumbers
+
+\end{document}
diff --git a/SciPost_Example_BiBTeX_File.bib b/SciPost_Example_BiBTeX_File.bib
new file mode 100644
index 0000000..e68ea35
--- /dev/null
+++ b/SciPost_Example_BiBTeX_File.bib
@@ -0,0 +1,34 @@
+
+
+@Article{1931_Bethe_ZP_71,
+  author =       {H. A. Bethe},
+  title = 	 {Zur {T}heorie der {M}etalle.  I.  {E}igenwerte und {E}igenfunktionen der linearen {A}tomkette},
+  journal = 	 {Zeit. f\"ur Physik},
+  year = 	 {1931},
+  volume = 	 {71},
+  pages = 	 {205},
+  doi = {10.1007/BF01341708}
+}
+
+
+@unpublished{arXiv:1108.2700,
+  author    = {P. Ginsparg},
+  title     = {It was twenty years ago today ...},
+  eprint       = {http://arxiv.org/abs/1108.2700},
+}
+
+
+@Article{2016_Piroli_SP_1,
+  author = 	 {Lorenzo Piroli and Pasquale Calabrese and Fabian H. L. Essler},
+  title = 	 {{Quantum quenches to the attractive one-dimensional Bose gas: exact results}},
+  journal = 	 {SciPost Phys.},
+  year = 	 {2016},
+  OPTkey = 	 {},
+  volume = 	 {1},
+  number = 	 {1},
+  pages = 	 {001},
+  OPTmonth = 	 {},
+  OPTnote = 	 {},
+  OPTannote = 	 {}
+}
+
diff --git a/SciPost_bibstyle.bst b/SciPost_bibstyle.bst
new file mode 100644
index 0000000..eaee2bf
--- /dev/null
+++ b/SciPost_bibstyle.bst
@@ -0,0 +1,1367 @@
+%%
+%% This is file `SciPost_bibstyle_v1.1',
+%% generated with the docstrip utility.
+%%
+%% The original source files were:
+%%
+%% merlin.mbs  (with options: `seq-no,vonx,nm-init,ed-au,nmlm,x10,x2,m10,m2,mcite,mct-6,mct-x6,dt-end,yr-com,xmth,tit-it,vol-bf,volp-com,jpg-1,num-xser,jnm-x,isbn,doi,pp,ed,abr,xedn,jabr,and-xcom,etal-xc,etal-it,eprint,url,url-blk,nfss,')
+%% ----------------------------------------
+%% *** SciPost style version 1.1 ***
+%% 
+%% Copyright 1994-2011 Patrick W Daly
+ % ===============================================================
+ % IMPORTANT NOTICE:
+ % This bibliographic style (bst) file has been generated from one or
+ % more master bibliographic style (mbs) files, listed above.
+ %
+ % This generated file can be redistributed and/or modified under the terms
+ % of the LaTeX Project Public License Distributed from CTAN
+ % archives in directory macros/latex/base/lppl.txt; either
+ % version 1 of the License, or any later version.
+ % ===============================================================
+ % Name and version information of the main mbs file:
+ % \ProvidesFile{merlin.mbs}[2011/11/18 4.33 (PWD, AO, DPC)]
+ %   For use with BibTeX version 0.99a or later
+ %-------------------------------------------------------------------
+ % This bibliography style file is intended for texts in ENGLISH
+ % This is a numerical citation style, and as such is standard LaTeX.
+ % It requires no extra package to interface to the main text.
+ % The form of the \bibitem entries is
+ %   \bibitem{key}...
+ % Usage of \cite is as follows:
+ %   \cite{key} ==>>          [#]
+ %   \cite[chap. 2]{key} ==>> [#, chap. 2]
+ % where # is a number determined by the ordering in the reference list.
+ % The order in the reference list is that by which the works were originally
+ %   cited in the text, or that in the database.
+ %---------------------------------------------------------------------
+
+ENTRY
+  { address
+    archive
+    author
+    booktitle
+    chapter
+    doi
+    edition
+    editor
+    eid
+    eprint
+    howpublished
+    institution
+    isbn
+    journal
+    key
+    month
+    note
+    number
+    organization
+    pages
+    publisher
+    school
+    series
+    title
+    type
+    url
+    volume
+    year
+  }
+  {}
+  { label }
+INTEGERS { output.state before.all mid.sentence after.sentence after.block }
+FUNCTION {init.state.consts}
+{ #0 'before.all :=
+  #1 'mid.sentence :=
+  #2 'after.sentence :=
+  #3 'after.block :=
+}
+STRINGS { s t}
+FUNCTION {output.nonnull}
+{ 's :=
+  output.state mid.sentence =
+    { ", " * write$ }
+    { output.state after.block =
+        { ", " * write$
+          newline$
+          "\newblock " write$
+        }
+        { output.state before.all =
+            'write$
+            { add.period$ " " * write$ }
+          if$
+        }
+      if$
+      mid.sentence 'output.state :=
+    }
+  if$
+  s
+}
+FUNCTION {output}
+{ duplicate$ empty$
+    'pop$
+    'output.nonnull
+  if$
+}
+FUNCTION {output.check}
+{ 't :=
+  duplicate$ empty$
+    { pop$ "empty " t * " in " * cite$ * warning$ }
+    'output.nonnull
+  if$
+}
+FUNCTION {fin.entry}
+{ add.period$
+  write$
+  newline$
+}
+
+FUNCTION {new.block}
+{ output.state before.all =
+    'skip$
+    { after.block 'output.state := }
+  if$
+}
+FUNCTION {new.sentence}
+{ output.state after.block =
+    'skip$
+    { output.state before.all =
+        'skip$
+        { after.sentence 'output.state := }
+      if$
+    }
+  if$
+}
+FUNCTION {add.blank}
+{  " " * before.all 'output.state :=
+}
+
+FUNCTION {date.block}
+{
+  new.block
+}
+
+FUNCTION {not}
+{   { #0 }
+    { #1 }
+  if$
+}
+FUNCTION {and}
+{   'skip$
+    { pop$ #0 }
+  if$
+}
+FUNCTION {or}
+{   { pop$ #1 }
+    'skip$
+  if$
+}
+FUNCTION {new.block.checka}
+{ empty$
+    'skip$
+    'new.block
+  if$
+}
+FUNCTION {new.block.checkb}
+{ empty$
+  swap$ empty$
+  and
+    'skip$
+    'new.block
+  if$
+}
+FUNCTION {new.sentence.checka}
+{ empty$
+    'skip$
+    'new.sentence
+  if$
+}
+FUNCTION {new.sentence.checkb}
+{ empty$
+  swap$ empty$
+  and
+    'skip$
+    'new.sentence
+  if$
+}
+FUNCTION {field.or.null}
+{ duplicate$ empty$
+    { pop$ "" }
+    'skip$
+  if$
+}
+FUNCTION {emphasize}
+{ duplicate$ empty$
+    { pop$ "" }
+    { "\emph{" swap$ * "}" * }
+  if$
+}
+FUNCTION {bolden}
+{ duplicate$ empty$
+    { pop$ "" }
+    { "\textbf{" swap$ * "}" * }
+  if$
+}
+FUNCTION {tie.or.space.prefix}
+{ duplicate$ text.length$ #3 <
+    { "~" }
+    { " " }
+  if$
+  swap$
+}
+
+FUNCTION {capitalize}
+{ "u" change.case$ "t" change.case$ }
+
+FUNCTION {space.word}
+{ " " swap$ * " " * }
+ % Here are the language-specific definitions for explicit words.
+ % Each function has a name bbl.xxx where xxx is the English word.
+ % The language selected here is ENGLISH
+FUNCTION {bbl.and}
+{ "and"}
+
+FUNCTION {bbl.etal}
+{ "et~al." }
+
+FUNCTION {bbl.editors}
+{ "eds." }
+
+FUNCTION {bbl.editor}
+{ "ed." }
+
+FUNCTION {bbl.edby}
+{ "edited by" }
+
+FUNCTION {bbl.edition}
+{ "edn." }
+
+FUNCTION {bbl.volume}
+{ "vol." }
+
+FUNCTION {bbl.of}
+{ "of" }
+
+FUNCTION {bbl.number}
+{ "no." }
+
+FUNCTION {bbl.nr}
+{ "no." }
+
+FUNCTION {bbl.in}
+{ "in" }
+
+FUNCTION {bbl.pages}
+{ "pp." }
+
+FUNCTION {bbl.page}
+{ "p." }
+
+FUNCTION {bbl.chapter}
+{ "chap." }
+
+FUNCTION {bbl.techrep}
+{ "Tech. Rep." }
+
+FUNCTION {bbl.mthesis}
+{ "Master's thesis" }
+
+FUNCTION {bbl.phdthesis}
+{ "Ph.D. thesis" }
+
+MACRO {jan} {"Jan."}
+
+MACRO {feb} {"Feb."}
+
+MACRO {mar} {"Mar."}
+
+MACRO {apr} {"Apr."}
+
+MACRO {may} {"May"}
+
+MACRO {jun} {"Jun."}
+
+MACRO {jul} {"Jul."}
+
+MACRO {aug} {"Aug."}
+
+MACRO {sep} {"Sep."}
+
+MACRO {oct} {"Oct."}
+
+MACRO {nov} {"Nov."}
+
+MACRO {dec} {"Dec."}
+
+MACRO {acmcs} {"ACM Comput. Surv."}
+
+MACRO {acta} {"Acta Inf."}
+
+MACRO {cacm} {"Commun. ACM"}
+
+MACRO {ibmjrd} {"IBM J. Res. Dev."}
+
+MACRO {ibmsj} {"IBM Syst.~J."}
+
+MACRO {ieeese} {"IEEE Trans. Software Eng."}
+
+MACRO {ieeetc} {"IEEE Trans. Comput."}
+
+MACRO {ieeetcad}
+ {"IEEE Trans. Comput. Aid. Des."}
+
+MACRO {ipl} {"Inf. Process. Lett."}
+
+MACRO {jacm} {"J.~ACM"}
+
+MACRO {jcss} {"J.~Comput. Syst. Sci."}
+
+MACRO {scp} {"Sci. Comput. Program."}
+
+MACRO {sicomp} {"SIAM J. Comput."}
+
+MACRO {tocs} {"ACM Trans. Comput. Syst."}
+
+MACRO {tods} {"ACM Trans. Database Syst."}
+
+MACRO {tog} {"ACM Trans. Graphic."}
+
+MACRO {toms} {"ACM Trans. Math. Software"}
+
+MACRO {toois} {"ACM Trans. Office Inf. Syst."}
+
+MACRO {toplas} {"ACM Trans. Progr. Lang. Syst."}
+
+MACRO {tcs} {"Theor. Comput. Sci."}
+
+FUNCTION {bibinfo.check}
+{ swap$
+  duplicate$ missing$
+    {
+      pop$ pop$
+      ""
+    }
+    { duplicate$ empty$
+        {
+          swap$ pop$
+        }
+        { swap$
+          pop$
+        }
+      if$
+    }
+  if$
+}
+FUNCTION {bibinfo.warn}
+{ swap$
+  duplicate$ missing$
+    {
+      swap$ "missing " swap$ * " in " * cite$ * warning$ pop$
+      ""
+    }
+    { duplicate$ empty$
+        {
+          swap$ "empty " swap$ * " in " * cite$ * warning$
+        }
+        { swap$
+          pop$
+        }
+      if$
+    }
+  if$
+}
+FUNCTION {format.eprint}
+{ eprint duplicate$ empty$
+    'skip$
+    { "\eprint"
+      archive empty$
+        'skip$
+        { "[" * archive * "]" * }
+      if$
+      "{" * swap$ * "}" *
+    }
+  if$
+}
+FUNCTION {format.url}
+{
+  url
+  duplicate$ empty$
+    { pop$ "" }
+    { "\urlprefix\url{" swap$ * "}" * }
+  if$
+}
+
+INTEGERS { nameptr namesleft numnames }
+
+
+STRINGS  { bibinfo}
+
+FUNCTION {format.names}
+{ 'bibinfo :=
+  duplicate$ empty$ 'skip$ {
+  's :=
+  "" 't :=
+  #1 'nameptr :=
+  s num.names$ 'numnames :=
+  numnames 'namesleft :=
+    { namesleft #0 > }
+    { s nameptr
+      "{f.~}{vv~}{ll}{, jj}"
+      format.name$
+      bibinfo bibinfo.check
+      't :=
+      nameptr #1 >
+        {
+          nameptr #2
+          #10 +
+          #1 + =
+          numnames #2
+          #10 +
+          > and
+            { "others" 't :=
+              #1 'namesleft := }
+            'skip$
+          if$
+          namesleft #1 >
+            { ", " * t * }
+            {
+              s nameptr "{ll}" format.name$ duplicate$ "others" =
+                { 't := }
+                { pop$ }
+              if$
+              t "others" =
+                {
+                  " " * bbl.etal emphasize *
+                }
+                {
+                  bbl.and
+                  space.word * t *
+                }
+              if$
+            }
+          if$
+        }
+        't
+      if$
+      nameptr #1 + 'nameptr :=
+      namesleft #1 - 'namesleft :=
+    }
+  while$
+  } if$
+}
+FUNCTION {format.names.ed}
+{
+  format.names
+}
+FUNCTION {format.authors}
+{ author "author" format.names
+}
+FUNCTION {get.bbl.editor}
+{ editor num.names$ #1 > 'bbl.editors 'bbl.editor if$ }
+
+FUNCTION {format.editors}
+{ editor "editor" format.names duplicate$ empty$ 'skip$
+    {
+      "," *
+      " " *
+      get.bbl.editor
+      *
+    }
+  if$
+}
+FUNCTION {format.isbn}
+{ isbn "isbn" bibinfo.check
+  duplicate$ empty$ 'skip$
+    {
+      new.block
+      "ISBN " swap$ *
+    }
+  if$
+}
+
+FUNCTION {format.doi}
+{ doi empty$
+    { "" }
+    {
+      new.block
+      "\doi{" doi * "}" *
+    }
+  if$
+}
+FUNCTION {format.note}
+{
+ note empty$
+    { "" }
+    { note #1 #1 substring$
+      duplicate$ "{" =
+        'skip$
+        { output.state mid.sentence =
+          { "l" }
+          { "u" }
+        if$
+        change.case$
+        }
+      if$
+      note #2 global.max$ substring$ * "note" bibinfo.check
+    }
+  if$
+}
+
+FUNCTION {format.title}
+{ title
+  duplicate$ empty$ 'skip$
+    { "t" change.case$ }
+  if$
+  "title" bibinfo.check
+  duplicate$ empty$ 'skip$
+    {
+      emphasize
+    }
+  if$
+}
+FUNCTION {output.bibitem}
+{ newline$
+  "\bibitem{" write$
+  cite$ write$
+  "}" write$
+  newline$
+  ""
+  before.all 'output.state :=
+}
+
+FUNCTION {n.dashify}
+{
+  't :=
+  ""
+    { t empty$ not }
+    { t #1 #1 substring$ "-" =
+        { t #1 #2 substring$ "--" = not
+            { "--" *
+              t #2 global.max$ substring$ 't :=
+            }
+            {   { t #1 #1 substring$ "-" = }
+                { "-" *
+                  t #2 global.max$ substring$ 't :=
+                }
+              while$
+            }
+          if$
+        }
+        { t #1 #1 substring$ *
+          t #2 global.max$ substring$ 't :=
+        }
+      if$
+    }
+  while$
+}
+
+FUNCTION {word.in}
+{ bbl.in capitalize
+  " " * }
+
+FUNCTION {format.date}
+{
+  ""
+  duplicate$ empty$
+  year  "year"  bibinfo.check duplicate$ empty$
+    { swap$ 'skip$
+        { "there's a month but no year in " cite$ * warning$ }
+      if$
+      *
+    }
+    { swap$ 'skip$
+        {
+          swap$
+          " " * swap$
+        }
+      if$
+      *
+    }
+  if$
+  duplicate$ empty$
+    'skip$
+    {
+      before.all 'output.state :=
+    " (" swap$ * ")" *
+    }
+  if$
+}
+FUNCTION {format.btitle}
+{ title "title" bibinfo.check
+  duplicate$ empty$ 'skip$
+    {
+      emphasize
+    }
+  if$
+}
+FUNCTION {either.or.check}
+{ empty$
+    'pop$
+    { "can't use both " swap$ * " fields in " * cite$ * warning$ }
+  if$
+}
+FUNCTION {format.bvolume}
+{ volume empty$
+    { "" }
+    { bbl.volume volume tie.or.space.prefix
+      "volume" bibinfo.check * *
+      series "series" bibinfo.check
+      duplicate$ empty$ 'pop$
+        { swap$ bbl.of space.word * swap$
+          emphasize * }
+      if$
+      "volume and number" number either.or.check
+    }
+  if$
+}
+FUNCTION {format.number.series}
+{ volume empty$
+    { number empty$
+        { series field.or.null }
+        { series empty$
+            { number "number" bibinfo.check }
+            { output.state mid.sentence =
+                { bbl.number }
+                { bbl.number capitalize }
+              if$
+              number tie.or.space.prefix "number" bibinfo.check * *
+              bbl.in space.word *
+              series "series" bibinfo.check *
+            }
+          if$
+        }
+      if$
+    }
+    { "" }
+  if$
+}
+
+FUNCTION {format.edition}
+{ edition duplicate$ empty$ 'skip$
+    {
+      output.state mid.sentence =
+        { "l" }
+        { "t" }
+      if$ change.case$
+      "edition" bibinfo.check
+      " " * bbl.edition *
+    }
+  if$
+}
+INTEGERS { multiresult }
+FUNCTION {multi.page.check}
+{ 't :=
+  #0 'multiresult :=
+    { multiresult not
+      t empty$ not
+      and
+    }
+    { t #1 #1 substring$
+      duplicate$ "-" =
+      swap$ duplicate$ "," =
+      swap$ "+" =
+      or or
+        { #1 'multiresult := }
+        { t #2 global.max$ substring$ 't := }
+      if$
+    }
+  while$
+  multiresult
+}
+FUNCTION {format.pages}
+{ pages duplicate$ empty$ 'skip$
+    { duplicate$ multi.page.check
+        {
+          bbl.pages swap$
+          n.dashify
+        }
+        {
+          bbl.page swap$
+        }
+      if$
+      tie.or.space.prefix
+      "pages" bibinfo.check
+      * *
+    }
+  if$
+}
+FUNCTION {first.page}
+{ 't :=
+  ""
+    {  t empty$ not t #1 #1 substring$ "-" = not and }
+    { t #1 #1 substring$ *
+      t #2 global.max$ substring$ 't :=
+    }
+  while$
+}
+
+FUNCTION {format.journal.pages}
+{ pages duplicate$ empty$ 'pop$
+    { swap$ duplicate$ empty$
+        { pop$ pop$ format.pages }
+        {
+          ", " *
+          swap$
+          first.page
+          "pages" bibinfo.check
+          *
+        }
+      if$
+    }
+  if$
+}
+FUNCTION {format.journal.eid}
+{ eid "eid" bibinfo.check
+  duplicate$ empty$ 'pop$
+    { swap$ duplicate$ empty$ 'skip$
+      {
+          ", " *
+      }
+      if$
+      swap$ *
+    }
+  if$
+}
+FUNCTION {format.vol.num.pages}
+{ volume field.or.null
+  duplicate$ empty$ 'skip$
+    {
+      "volume" bibinfo.check
+    }
+  if$
+  bolden
+  number "number" bibinfo.check duplicate$ empty$ 'skip$
+    {
+      swap$ duplicate$ empty$
+        { "there's a number but no volume in " cite$ * warning$ }
+        'skip$
+      if$
+      swap$
+      "(" swap$ * ")" *
+    }
+  if$ *
+  eid empty$
+    { format.journal.pages }
+    { format.journal.eid }
+  if$
+}
+
+FUNCTION {format.chapter.pages}
+{ chapter empty$
+    'format.pages
+    { type empty$
+        { bbl.chapter }
+        { type "l" change.case$
+          "type" bibinfo.check
+        }
+      if$
+      chapter tie.or.space.prefix
+      "chapter" bibinfo.check
+      * *
+      pages empty$
+        'skip$
+        { ", " * format.pages * }
+      if$
+    }
+  if$
+}
+
+FUNCTION {format.booktitle}
+{
+  booktitle "booktitle" bibinfo.check
+  emphasize
+}
+FUNCTION {format.in.ed.booktitle}
+{ format.booktitle duplicate$ empty$ 'skip$
+    {
+      editor "editor" format.names.ed duplicate$ empty$ 'pop$
+        {
+          "," *
+          " " *
+          get.bbl.editor
+          ", " *
+          * swap$
+          * }
+      if$
+      word.in swap$ *
+    }
+  if$
+}
+FUNCTION {empty.misc.check}
+{ author empty$ title empty$ howpublished empty$
+  month empty$ year empty$ note empty$
+  and and and and and
+    { "all relevant fields are empty in " cite$ * warning$ }
+    'skip$
+  if$
+}
+FUNCTION {format.thesis.type}
+{ type duplicate$ empty$
+    'pop$
+    { swap$ pop$
+      "t" change.case$ "type" bibinfo.check
+    }
+  if$
+}
+FUNCTION {format.tr.number}
+{ number "number" bibinfo.check
+  type duplicate$ empty$
+    { pop$ bbl.techrep }
+    'skip$
+  if$
+  "type" bibinfo.check
+  swap$ duplicate$ empty$
+    { pop$ "t" change.case$ }
+    { tie.or.space.prefix * * }
+  if$
+}
+FUNCTION {format.article.crossref}
+{
+  key duplicate$ empty$
+    { pop$
+      journal duplicate$ empty$
+        { "need key or journal for " cite$ * " to crossref " * crossref * warning$ }
+        { "journal" bibinfo.check emphasize word.in swap$ * }
+      if$
+    }
+    { word.in swap$ * " " *}
+  if$
+  " \cite{" * crossref * "}" *
+}
+FUNCTION {format.crossref.editor}
+{ editor #1 "{vv~}{ll}" format.name$
+  "editor" bibinfo.check
+  editor num.names$ duplicate$
+  #2 >
+    { pop$
+      "editor" bibinfo.check
+      " " * bbl.etal
+      emphasize
+      *
+    }
+    { #2 <
+        'skip$
+        { editor #2 "{ff }{vv }{ll}{ jj}" format.name$ "others" =
+            {
+              "editor" bibinfo.check
+              " " * bbl.etal
+              emphasize
+              *
+            }
+            {
+             bbl.and space.word
+              * editor #2 "{vv~}{ll}" format.name$
+              "editor" bibinfo.check
+              *
+            }
+          if$
+        }
+      if$
+    }
+  if$
+}
+FUNCTION {format.book.crossref}
+{ volume duplicate$ empty$
+    { "empty volume in " cite$ * "'s crossref of " * crossref * warning$
+      pop$ word.in
+    }
+    { bbl.volume
+      capitalize
+      swap$ tie.or.space.prefix "volume" bibinfo.check * * bbl.of space.word *
+    }
+  if$
+  editor empty$
+  editor field.or.null author field.or.null =
+  or
+    { key empty$
+        { series empty$
+            { "need editor, key, or series for " cite$ * " to crossref " *
+              crossref * warning$
+              "" *
+            }
+            { series emphasize * }
+          if$
+        }
+        { key * }
+      if$
+    }
+    { format.crossref.editor * }
+  if$
+  " \cite{" * crossref * "}" *
+}
+FUNCTION {format.incoll.inproc.crossref}
+{
+  editor empty$
+  editor field.or.null author field.or.null =
+  or
+    { key empty$
+        { format.booktitle duplicate$ empty$
+            { "need editor, key, or booktitle for " cite$ * " to crossref " *
+              crossref * warning$
+            }
+            { word.in swap$ * }
+          if$
+        }
+        { word.in key * " " *}
+      if$
+    }
+    { word.in format.crossref.editor * " " *}
+  if$
+  " \cite{" * crossref * "}" *
+}
+FUNCTION {format.org.or.pub}
+{ 't :=
+  ""
+  address empty$ t empty$ and
+    'skip$
+    {
+      t empty$
+        { address "address" bibinfo.check *
+        }
+        { t *
+          address empty$
+            'skip$
+            { ", " * address "address" bibinfo.check * }
+          if$
+        }
+      if$
+    }
+  if$
+}
+FUNCTION {format.publisher.address}
+{ publisher "publisher" bibinfo.warn format.org.or.pub
+}
+
+FUNCTION {format.organization.address}
+{ organization "organization" bibinfo.check format.org.or.pub
+}
+
+FUNCTION {article}
+{ output.bibitem
+  format.authors "author" output.check
+  new.block
+  format.title "title" output.check
+  new.block
+  crossref missing$
+    {
+      journal
+      "journal" bibinfo.check
+      "journal" output.check
+      add.blank
+      format.vol.num.pages output
+    }
+    { format.article.crossref output.nonnull
+      format.pages output
+    }
+  if$
+  format.date "year" output.check
+  format.doi output
+  new.block
+%  format.url output
+%  new.block
+  format.note output
+  new.block
+  format.eprint output
+  fin.entry
+}
+FUNCTION {book}
+{ output.bibitem
+  author empty$
+    { format.editors "author and editor" output.check
+    }
+    { format.authors output.nonnull
+      crossref missing$
+        { "author and editor" editor either.or.check }
+        'skip$
+      if$
+    }
+  if$
+  new.block
+  format.btitle "title" output.check
+  crossref missing$
+    { format.bvolume output
+      new.block
+      format.number.series output
+      new.sentence
+      format.publisher.address output
+    }
+    {
+      new.block
+      format.book.crossref output.nonnull
+    }
+  if$
+  format.edition output
+  format.isbn output
+  format.doi output
+  new.block
+%  format.url output
+%  new.block
+  format.note output
+  new.block
+  format.date "year" output.check
+  format.eprint output
+  fin.entry
+}
+FUNCTION {booklet}
+{ output.bibitem
+  format.authors output
+  new.block
+  format.title "title" output.check
+  new.block
+  howpublished "howpublished" bibinfo.check output
+  address "address" bibinfo.check output
+  format.isbn output
+  format.doi output
+  new.block
+%  format.url output
+%  new.block
+  format.note output
+  new.block
+  format.date output
+  format.eprint output
+  fin.entry
+}
+
+FUNCTION {inbook}
+{ output.bibitem
+  author empty$
+    { format.editors "author and editor" output.check
+    }
+    { format.authors output.nonnull
+      crossref missing$
+        { "author and editor" editor either.or.check }
+        'skip$
+      if$
+    }
+  if$
+  new.block
+  format.btitle "title" output.check
+  crossref missing$
+    {
+      format.bvolume output
+      format.chapter.pages "chapter and pages" output.check
+      new.block
+      format.number.series output
+      new.sentence
+      format.publisher.address output
+    }
+    {
+      format.chapter.pages "chapter and pages" output.check
+      new.block
+      format.book.crossref output.nonnull
+    }
+  if$
+  format.edition output
+  crossref missing$
+    { format.isbn output }
+    'skip$
+  if$
+  format.doi output
+  new.block
+%  format.url output
+%  new.block
+  format.note output
+  new.block
+  format.date "year" output.check
+  format.eprint output
+  fin.entry
+}
+
+FUNCTION {incollection}
+{ output.bibitem
+  format.authors "author" output.check
+  new.block
+  format.title "title" output.check
+  new.block
+  crossref missing$
+    { format.in.ed.booktitle "booktitle" output.check
+      format.bvolume output
+      format.number.series output
+      format.chapter.pages output
+      new.sentence
+      format.publisher.address output
+      format.edition output
+      format.isbn output
+    }
+    { format.incoll.inproc.crossref output.nonnull
+      format.chapter.pages output
+    }
+  if$
+  format.doi output
+  new.block
+%  format.url output
+%  new.block
+  format.note output
+  new.block
+  format.date "year" output.check
+  format.eprint output
+  fin.entry
+}
+FUNCTION {inproceedings}
+{ output.bibitem
+  format.authors "author" output.check
+  new.block
+  format.title "title" output.check
+  new.block
+  crossref missing$
+    { format.in.ed.booktitle "booktitle" output.check
+      format.bvolume output
+      format.number.series output
+      format.pages output
+      new.sentence
+      publisher empty$
+        { format.organization.address output }
+        { organization "organization" bibinfo.check output
+          format.publisher.address output
+        }
+      if$
+      format.isbn output
+    }
+    { format.incoll.inproc.crossref output.nonnull
+      format.pages output
+    }
+  if$
+  format.doi output
+  new.block
+%  format.url output
+%  new.block
+  format.note output
+  new.block
+  format.date "year" output.check
+  format.eprint output
+  fin.entry
+}
+FUNCTION {conference} { inproceedings }
+FUNCTION {manual}
+{ output.bibitem
+  author empty$
+    { organization "organization" bibinfo.check
+      duplicate$ empty$ 'pop$
+        { output
+          address "address" bibinfo.check output
+        }
+      if$
+    }
+    { format.authors output.nonnull }
+  if$
+  new.block
+  format.btitle "title" output.check
+  author empty$
+    { organization empty$
+        {
+          address new.block.checka
+          address "address" bibinfo.check output
+        }
+        'skip$
+      if$
+    }
+    {
+      organization address new.block.checkb
+      organization "organization" bibinfo.check output
+      address "address" bibinfo.check output
+    }
+  if$
+  format.edition output
+  format.doi output
+  new.block
+%  format.url output
+%  new.block
+  format.note output
+  new.block
+  format.date output
+  format.eprint output
+  fin.entry
+}
+
+FUNCTION {mastersthesis}
+{ output.bibitem
+  format.authors "author" output.check
+  new.block
+  format.btitle
+  "title" output.check
+  new.block
+  bbl.mthesis format.thesis.type output.nonnull
+  school "school" bibinfo.warn output
+  address "address" bibinfo.check output
+  format.doi output
+  new.block
+%  format.url output
+%  new.block
+  format.note output
+  new.block
+  format.date "year" output.check
+  format.eprint output
+  fin.entry
+}
+
+FUNCTION {misc}
+{ output.bibitem
+  format.authors output
+  title howpublished new.block.checkb
+  format.title output
+  howpublished new.block.checka
+  howpublished "howpublished" bibinfo.check output
+  format.doi output
+  new.block
+%  format.url output
+%  new.block
+  format.note output
+  new.block
+  format.date output
+  format.eprint output
+  fin.entry
+  empty.misc.check
+}
+FUNCTION {phdthesis}
+{ output.bibitem
+  format.authors "author" output.check
+  new.block
+  format.btitle
+  "title" output.check
+  new.block
+  bbl.phdthesis format.thesis.type output.nonnull
+  school "school" bibinfo.warn output
+  address "address" bibinfo.check output
+  format.doi output
+  new.block
+%  format.url output
+%  new.block
+  format.note output
+  new.block
+  format.date "year" output.check
+  format.eprint output
+  fin.entry
+}
+
+FUNCTION {proceedings}
+{ output.bibitem
+  editor empty$
+    { organization "organization" bibinfo.check output
+    }
+    { format.editors output.nonnull }
+  if$
+  new.block
+  format.btitle "title" output.check
+  format.bvolume output
+  format.number.series output
+  editor empty$
+    { publisher empty$
+        'skip$
+        {
+          new.sentence
+          format.publisher.address output
+        }
+      if$
+    }
+    { publisher empty$
+        {
+          new.sentence
+          format.organization.address output }
+        {
+          new.sentence
+          organization "organization" bibinfo.check output
+          format.publisher.address output
+        }
+      if$
+     }
+  if$
+  format.isbn output
+  format.doi output
+  new.block
+%  format.url output
+%  new.block
+  format.note output
+  new.block
+  format.date "year" output.check
+  format.eprint output
+  fin.entry
+}
+
+FUNCTION {techreport}
+{ output.bibitem
+  format.authors "author" output.check
+  new.block
+  format.title
+  "title" output.check
+  new.block
+  format.tr.number output.nonnull
+  institution "institution" bibinfo.warn output
+  address "address" bibinfo.check output
+  format.doi output
+  new.block
+%  format.url output
+%  new.block
+  format.note output
+  new.block
+  format.date "year" output.check
+  format.eprint output
+  fin.entry
+}
+
+FUNCTION {unpublished}
+{ output.bibitem
+  format.authors "author" output.check
+  new.block
+  format.title "title" output.check
+  format.doi output
+  new.block
+  format.url output
+  new.block
+  format.note "note" output.check
+  new.block
+  format.date output
+  format.eprint output
+  fin.entry
+}
+
+FUNCTION {default.type} { misc }
+READ
+STRINGS { longest.label }
+INTEGERS { number.label longest.label.width }
+FUNCTION {initialize.longest.label}
+{ "" 'longest.label :=
+  #1 'number.label :=
+  #0 'longest.label.width :=
+}
+FUNCTION {longest.label.pass}
+{ number.label int.to.str$ 'label :=
+  number.label #1 + 'number.label :=
+  label width$ longest.label.width >
+    { label 'longest.label :=
+      label width$ 'longest.label.width :=
+    }
+    'skip$
+  if$
+}
+EXECUTE {initialize.longest.label}
+ITERATE {longest.label.pass}
+FUNCTION {begin.bib}
+{ preamble$ empty$
+    'skip$
+    { preamble$ write$ newline$ }
+  if$
+  "\begin{thebibliography}{"  longest.label  * "}" *
+  write$ newline$
+  "\providecommand{\url}[1]{\texttt{#1}}"
+  write$ newline$
+  "\providecommand{\urlprefix}{URL }"
+  write$ newline$
+  "\expandafter\ifx\csname urlstyle\endcsname\relax"
+  write$ newline$
+  "  \providecommand{\doi}[1]{doi:\discretionary{}{}{}#1}\else"
+  write$ newline$
+  "  \providecommand{\doi}{doi:\discretionary{}{}{}\begingroup \urlstyle{rm}\Url}\fi"
+  write$ newline$
+  "\providecommand{\eprint}[2][]{\url{#2}}"
+  write$ newline$
+}
+EXECUTE {begin.bib}
+EXECUTE {init.state.consts}
+ITERATE {call.type$}
+FUNCTION {end.bib}
+{ newline$
+  "\end{thebibliography}" write$ newline$
+}
+EXECUTE {end.bib}
+%% End of customized bst file
+%%
+%% End of file `SciPost_bibstyle_v1.1'.
diff --git a/topology.bib b/topology.bib
index 293c288..316df0f 100644
--- a/topology.bib
+++ b/topology.bib
@@ -95,6 +95,35 @@
  eprinttype = {arxiv}
 }
 
+@article{Erba_2024_Quenches,
+ author = {Erba, Vittorio and Behrens, Freya and Krzakala, Florent and Zdeborová, Lenka},
+ title = {Quenches in the {Sherrington}–{Kirkpatrick} model},
+ journal = {Journal of Statistical Mechanics: Theory and Experiment},
+ publisher = {IOP Publishing},
+ year = {2024},
+ month = {August},
+ number = {8},
+ volume = {2024},
+ pages = {083302},
+ url = {http://dx.doi.org/10.1088/1742-5468/ad685a},
+ doi = {10.1088/1742-5468/ad685a},
+ issn = {1742-5468}
+}
+
+@article{Folena_2023_On,
+ author = {Folena, Giampaolo and Zamponi, Francesco},
+ title = {On weak ergodicity breaking in mean-field spin glasses},
+ journal = {SciPost Physics},
+ publisher = {Stichting SciPost},
+ year = {2023},
+ month = {9},
+ number = {3},
+ volume = {15},
+ pages = {109},
+ url = {https://doi.org/10.21468%2Fscipostphys.15.3.109},
+ doi = {10.21468/scipostphys.15.3.109}
+}
+
 @article{Franz_2016_The,
  author = {Franz, Silvio and Parisi, Giorgio},
  title = {The simplest model of jamming},
@@ -325,6 +354,21 @@
  issn = {1751-8121}
 }
 
+@article{Urbani_2024_Quantum,
+ author = {Urbani, Pierfrancesco},
+ title = {Quantum exploration of high-dimensional canyon landscapes},
+ journal = {Journal of Statistical Mechanics: Theory and Experiment},
+ publisher = {IOP Publishing},
+ year = {2024},
+ month = {August},
+ number = {8},
+ volume = {2024},
+ pages = {083301},
+ url = {http://dx.doi.org/10.1088/1742-5468/ad0635},
+ doi = {10.1088/1742-5468/ad0635},
+ issn = {1742-5468}
+}
+
 @unpublished{Urbani_2024_Statistical,
  author = {Urbani, Pierfrancesco},
  title = {Statistical physics of complex systems: glasses, spin glasses, continuous constraint satisfaction problems, high-dimensional inference and neural networks},
diff --git a/topology.tex b/topology.tex
index 3e8ce9b..4c14207 100644
--- a/topology.tex
+++ b/topology.tex
@@ -1,27 +1,25 @@
-\documentclass[a4paper,fleqn]{article}
+\documentclass[submission, Phys]{SciPost}
 
 \usepackage[utf8]{inputenc}
 \usepackage[T1]{fontenc}
-\usepackage{amsmath,amssymb,latexsym,graphicx}
-\usepackage{newtxtext,newtxmath}
-\usepackage{bbold}
+\usepackage{amsmath,latexsym,graphicx}
+\usepackage[bitstream-charter]{mathdesign}
 \usepackage[dvipsnames]{xcolor}
-\usepackage[
-  colorlinks=true,
-  urlcolor=BlueViolet,
-  citecolor=BlueViolet,
-  filecolor=BlueViolet,
-  linkcolor=BlueViolet
-]{hyperref}
-\usepackage[
-  style=phys,
-  eprint=true,
-  maxnames = 100
-]{biblatex}
 \usepackage{anyfontsize,authblk}
-\usepackage{fullpage}
 
-\addbibresource{topology.bib}
+\urlstyle{sf}
+
+% Fix \cal and \mathcal characters look (so it's not the same as \mathscr)
+\DeclareSymbolFont{usualmathcal}{OMS}{cmsy}{m}{n}
+\DeclareSymbolFontAlphabet{\mathcal}{usualmathcal}
+
+\hypersetup{
+  colorlinks=true,
+  urlcolor={blue!50!black},
+  citecolor={blue!50!black},
+  filecolor={blue!50!black},
+  linkcolor={blue!50!black}
+}
 
 \title{
   On the topology of solutions to random continuous constraint satisfaction problems
@@ -39,15 +37,17 @@
   independent Gaussian coefficients on the $(N-1)$-sphere. When solutions
   exist, they form a manifold. We compute the average Euler characteristic of
   this manifold in the limit of large $N$, and find different behavior
-  depending on the scaling of $M$ with $N$. When $\alpha=M/N$ is held constant,
-  the average characteristic is 2 whenever solutions exist. When $M$ is
-  constant, the average characteristic is also 2 up until a transition value
-  $M_\textrm{th}$, above which it is exponentially large in $N$. To better
-  interpret these results, we compute the average number of stationary points
-  of a test function on the solution manifold. In both regimes, this reveals
-  another transition between a regime with few and one with exponentially many
-  stationary points. We conjecture that this transition corresponds to a
-  geometric rather than a topological transition.
+  depending on the ratio $\alpha=M/N$. When $\alpha<\alpha_\text{onset}$, the
+  average characteristic is 2 and there is a single connected component, while
+  for $\alpha_\text{onset}<\alpha<\alpha_\text{shatter}$ a large connected
+  component coexists with many disconnected components. When
+  $\alpha>\alpha_\text{shatter}$ the large connected component vanishes, and the
+  entire manifold vanishes for $\alpha>\alpha_\text{\textsc{sat}}$. In the
+  limit $\alpha\to0$ there is a correspondence between this problem and the
+  topology of constant-energy level sets in the spherical spin glasses. We
+  conjecture that the energy $E_\text{shatter}$ associated with the vanishing of
+  the large connected component corresponds to the asymptotic limit of gradient
+  descent from a random initial condition.
 \end{abstract}
 
 \tableofcontents
@@ -71,7 +71,12 @@ to solve them \cite{Baldassi_2016_Unreasonable, Baldassi_2019_Properties,
 Beneventano_2023_On}. Here, we show how \emph{topological} information about
 the set of solutions can be calculated in a simple model of satisfying random
 nonlinear equalities. This allows us to reason about the connectivity of this
-solution set.
+solution set. The topological properties revealed by this calculation yield
+surprising results for the well-studied spherical spin glasses, where a
+topological transition thought to occur at a threshold energy $E_\text{th}$
+where marginal minima are dominant is shown to occur at a different energy
+$E_\text{shatter}$. We conjecture that this difference resolves an outstanding
+problem in gradient descent dynamics in these systems.
 
 We consider the problem of finding configurations $\mathbf x\in\mathbb R^N$
 lying on the $(N-1)$-sphere $\|\mathbf x\|^2=N$ that simultaneously satisfy $M$
@@ -80,20 +85,21 @@ constant $V_0\in\mathbb R$. The nonlinear constraints are taken to be centered
 Gaussian random functions with covariance
 \begin{equation} \label{eq:covariance}
   \overline{V_i(\mathbf x)V_j(\mathbf x')}
-  =\delta_{ij}F\left(\frac{\mathbf x\cdot\mathbf x'}N\right)
+  =\delta_{ij}f\left(\frac{\mathbf x\cdot\mathbf x'}N\right)
 \end{equation}
-for some choice of function $F$. When the covariance function $F$ is polynomial, the
-$V_k$ are also polynomial, with a term of degree $p$ in $F$ corresponding to
+for some choice of function $f$. When the covariance function $f$ is polynomial, the
+$V_k$ are also polynomial, with a term of degree $p$ in $f$ corresponding to
 all possible terms of degree $p$ in $V_k$. In particular, taking
 \begin{equation}
   V_k(\mathbf x)
-  =\sum_{p=0}^\infty\frac1{p!}\sqrt{\frac{F^{(p)}(0)}{N^p}}
+  =\sum_{p=0}^\infty\frac1{p!}\sqrt{\frac{f^{(p)}(0)}{N^p}}
   \sum_{i_1\cdots i_p}^NJ^{(k,p)}_{i_1\cdots i_p}x_{i_1}\cdots x_{i_p}
 \end{equation}
 with the elements of the tensors $J^{(k,p)}$ as independently distributed
 unit normal random variables satisfies \eqref{eq:covariance}. The size of the
-series coefficients of $F$ therefore control the variances in the coefficients
-of random polynomial constraints.
+series coefficients of $f$ therefore control the variances in the coefficients
+of random polynomial constraints. When $M=1$, this problem corresponds to the
+level set of a spherical spin glass with energy density $E=\sqrt{N}V_0$.
 
 
 This problem or small variations thereof have attracted attention recently for
@@ -104,7 +110,7 @@ tissues \cite{Fyodorov_2019_A, Fyodorov_2020_Counting,
 Montanari_2024_On, Kent-Dobias_2024_Conditioning, Kent-Dobias_2024_Algorithm-independent}. In each of these cases, the authors studied properties of
 the cost function
 \begin{equation} \label{eq:cost}
-  \mathcal C(\mathbf x)=\frac12\sum_{k=1}^MV_k(\mathbf x)^2
+  \mathscr C(\mathbf x)=\frac12\sum_{k=1}^MV_k(\mathbf x)^2
 \end{equation}
 which achieves zero only for configurations that satisfy all the constraints.
 Here we dispense with the cost function and study the set of solutions
@@ -112,21 +118,92 @@ directly.
 This set
 can be written as
 \begin{equation}
-  \Omega=\big\{\mathbf x\in\mathbb R^N\mid \|\mathbf x\|^2=N,V_k(\mathbf x)=0
+  \Omega=\big\{\mathbf x\in\mathbb R^N\mid \|\mathbf x\|^2=N,V_k(\mathbf x)=V_0
   \;\forall\;k=1,\ldots,M\big\}
 \end{equation}
-Because the constraints are all smooth functions, $\Omega$ is almost always a manifold without singular points. The conditions for a singular point are that
+Because the constraints are all smooth functions, $\Omega$ is almost always a manifold without singular points.\footnote{The conditions for a singular point are that
 $0=\frac\partial{\partial\mathbf x}V_k(\mathbf x)$ for all $k$. This is
 equivalent to asking that the constraints $V_k$ all have a stationary point at
 the same place. When the $V_k$ are independent and random, this is vanishingly
 unlikely, requiring $NM+1$ independent equations to be simultaneously satisfied.
 This means that different connected components of the set of solutions do not
-intersect, nor are there self-intersections, without extraordinary fine-tuning.
+intersect, nor are there self-intersections, without extraordinary fine-tuning.}
+We study the topology of the manifold $\Omega$ by two related means: its
+average Euler characteristic, and the average number of stationary points of a
+linear height function restricted to the manifold. These measures tell us
+complementary pieces of information, respectively the alternating sum and
+direct sum of the Betti numbers of $\Omega$. We find that for the varied cases
+we study, these two always coincide at the largest exponential order in $N$,
+putting strong constraints on the resulting topology and geometry.
+
+\section{Results}
+
+\subsection{Topology of solutions to many equations and the satisfiability transition}
+
+\begin{figure}
+  \includegraphics[width=0.245\textwidth]{figs/connected.pdf}
+  \includegraphics[width=0.245\textwidth]{figs/coexist.pdf}
+  \includegraphics[width=0.245\textwidth]{figs/shattered.pdf}
+  \includegraphics[width=0.245\textwidth]{figs/gone.pdf}
+
+  \includegraphics{figs/bar.pdf}
 
-When $M$ is too large, no solutions exist and $\Omega$ becomes the empty set.
-Following previous work, a replica symmetric equilibrium calculation using the
-cost function \eqref{eq:cost} predicts that solutions vanish when the ratio
-$\alpha=M/N$ is larger than $\alpha_\text{\textsc{sat}}=f'(1)/f(1)$. Based on the results of this paper, and the fact that this $\alpha_\text{\textsc{sat}}$ is consistent 
+  \caption{
+    Cartoon of the topology of the solution manifold implied by our
+    calculation. The arrow shows the vector $\mathbf x_0$ defining the height
+    function. For $V_0<V_\text{on}$, the manifold has a single connected
+    component. Above the onset with $V_\text{on}<V_0<V_\text{sh}$, the manifold
+    has a large connected component around the equator, and many disconnected
+    pieces in a certain range of latitudes. Above the shattering transition, or
+    $V_\text{sh}<V_0<V_\text{\textsc{sat}}$, the large connected
+    component vanishes and small disconnected pieces occupy the entire equatorial
+    region. Finally, above the satisfiability transition
+    $V_\text{\textsc{sat}}$ the manifold vanishes.
+  } \label{fig:cartoons}
+\end{figure}
+
+\begin{figure}
+  \includegraphics{figs/phases_1.pdf}
+  \hspace{-3em}
+  \includegraphics{figs/phases_2.pdf}
+  \hspace{-3em}
+  \includegraphics{figs/phases_3.pdf}
+
+  \caption{
+    Topological phases of the model for three different homogeneous covariance
+    functions. The onset transition $V_\text{onset}$, shattering transition
+    $V_\text{shatter}$, and satisfiability transition $V_\text{\textsc{sat}}$
+    are indicated when they exist. In the limit of $\alpha\to0$, the behavior
+    of level sets of the spherical spin glasses are recovered: the final plot
+    shows how in the pure cubic model the ground state energy $E_\text{gs}$ and threshold energy
+    $E_\text{th}$ correspond with the limits of the satisfiability and
+    shattering transitions, respectively. Note that for mixed models with
+    inhomogeneous covariance functions, $E_\text{th}$ is not the lower limit of
+    $V_\text{sh}$.
+  }
+\end{figure}
+
+\subsection{Topology of level sets of the spherical spin glasses and the dynamic threshold}
+
+\begin{figure}
+  \includegraphics{figs/dynamics_2.pdf}
+  \hspace{-0.5em}
+  \includegraphics{figs/dynamics_3.pdf}
+
+  \caption{
+    Comparison of the shattering energy $E_\text{sh}$ with the asymptotic
+    performance of gradient descent from a random initial condition in $p+s$
+    models with $p=2$ and $p=3$ and varying $s$. The values of $\lambda$ depend on $p$ and $s$ and are taken from \cite{Folena_2023_On}. The points show the asymptotic performance
+    extrapolated using two different methods and have unknown uncertainty, from \cite{Folena_2023_On}. Also
+    shown is the annealed threshold energy $E_\text{th}$, where marginal minima
+    are the most common type of stationary point. The section of $E_\text{sh}$
+    that is dashed on the left plot indicates the continuation of the annealed
+    result, whereas the solid portion gives the value calculated with a
+    {\oldstylenums 1}\textsc{frsb} ansatz.
+  }
+\end{figure}
+
+\section{The average Euler characteristic}
 
 The Euler characteristic $\chi$ of a manifold is a topological invariant \cite{Hatcher_2002_Algebraic}. It is
 perhaps most familiar in the context of connected compact orientable surfaces, where it
@@ -153,7 +230,7 @@ points. Since the sign of the determinant of the Hessian matrix of $H$ at a
 stationary point is equal to its index, if we count stationary points including
 the sign of the determinant, we arrive at the Euler characteristic, or
 \begin{equation} \label{eq:kac-rice}
-  \chi=\int_\Omega d\mathbf x\,\delta\big(\nabla H(\mathbf x)\big)\det\operatorname{Hess}H(\mathbf x)
+  \chi(\Omega)=\int_\Omega d\mathbf x\,\delta\big(\nabla H(\mathbf x)\big)\det\operatorname{Hess}H(\mathbf x)
 \end{equation}
 When the Kac--Rice formula is used to \emph{count} stationary points, the sign
 of the determinant is a nuisance that one must take pains to preserve
@@ -167,8 +244,6 @@ x)=\mathbf x_0\cdot\mathbf x$ for some $\mathbf x_0\in\mathbb R^N$ with
 $\|\mathbf x_0\|^2=N$. $H$ is a height function because when $\mathbf x_0$ is
 used as the polar axis, $H$ gives the height on the sphere.
 
-\section{The average Euler characteristic}
-
 We treat the integral over the implicitly defined manifold $\Omega$ using the
 method of Lagrange multipliers. We introduce one multiplier $\omega_0$ to
 enforce the spherical constraint and $M$ multipliers $\omega_k$ to enforce the vanishing of
@@ -176,7 +251,7 @@ each of the $V_k$, resulting in the Lagrangian
 \begin{equation}
   L(\mathbf x,\pmb\omega)
   =H(\mathbf x)+\frac12\omega_0\big(\|\mathbf x\|^2-N\big)
-  +\sum_{k=1}^M\omega_kV_k(\mathbf x)
+  +\sum_{k=1}^M\omega_k\big(V_k(\mathbf x)-V_0\big)
 \end{equation}
 The integral over $\Omega$ in \eqref{eq:kac-rice} then becomes
 \begin{equation} \label{eq:kac-rice.lagrange}
@@ -202,11 +277,11 @@ odd-dimensional manifolds. This is the signature of it in this problem.
 To evaluate the average of $\chi$ over the constraints, we first translate the $\delta$ functions and determinant to integral form, with
 \begin{align}
   \delta\big(\partial L(\mathbf x,\pmb\omega)\big)
-  =\int\frac{d\hat{\mathbf x}}{(2\pi)^N}\frac{d\hat{\pmb\omega}}{(2\pi)^{M+1}}
+  &=\int\frac{d\hat{\mathbf x}}{(2\pi)^N}\frac{d\hat{\pmb\omega}}{(2\pi)^{M+1}}
   e^{i[\hat{\mathbf x},\hat{\pmb\omega}]\cdot\partial L(\mathbf x,\pmb\omega)}
   \\
   \det\partial\partial L(\mathbf x,\pmb\omega)
-  =\int d\bar{\pmb\eta}\,d\pmb\eta\,d\bar{\pmb\gamma}\,d\pmb\gamma\,
+  &=\int d\bar{\pmb\eta}\,d\pmb\eta\,d\bar{\pmb\gamma}\,d\pmb\gamma\,
   e^{-[\bar{\pmb\eta},\bar{\pmb\gamma}]^T\partial\partial L(\mathbf x,\pmb\omega)[\pmb\eta,\pmb\gamma]}
 \end{align}
 
@@ -240,7 +315,7 @@ Since this is an exponential integrand linear in the functions $V_k$, we can ave
         +\frac{i}2\sigma_0(1)\big(\|\pmb\phi(1)\|^2-N\big)
         -iV_0\sum_{k=1}^M\sigma_k(1)
       \right] \\
-      -\frac12\int d1\,d2\,\sum_{k=1}^M\sigma_k(1)\sigma_k(2)F\left(\frac{\pmb\phi(1)\cdot\pmb\phi(2)}N\right)
+      -\frac12\int d1\,d2\,\sum_{k=1}^M\sigma_k(1)\sigma_k(2)f\left(\frac{\pmb\phi(1)\cdot\pmb\phi(2)}N\right)
     \Bigg\}
   \end{aligned}
 \end{equation}
@@ -254,8 +329,8 @@ Performing that integral yields
         H(\pmb\phi(1))
         +\frac{i}2\sigma_0(1)\big(\|\pmb\phi(1)\|^2-N\big)
       \right] \\
-    &\hspace{5em}-\frac M2V_0^2\int d1\,d2\,F\left(\frac{\pmb\phi(1)\cdot\pmb\phi(2)}N\right)^{-1}
-      -\frac M2\log\operatorname{sdet}F\left(\frac{\pmb\phi(1)\cdot\pmb\phi(2)}N\right)
+    &\hspace{5em}-\frac M2V_0^2\int d1\,d2\,f\left(\frac{\pmb\phi(1)\cdot\pmb\phi(2)}N\right)^{-1}
+      -\frac M2\log\operatorname{sdet}f\left(\frac{\pmb\phi(1)\cdot\pmb\phi(2)}N\right)
     \Bigg\}
   \end{aligned}
 \end{equation}
@@ -278,8 +353,8 @@ These new variables can replace $\pmb\phi$ in the integral using a generalized H
         \mathbb M(1)
         +\frac{i}2\sigma_0(1)\big(\mathbb Q(1,1)-1\big)
       \right] \\
-    &\hspace{5em}-\frac M2V_0^2\int d1\,d2\,F(\mathbb Q)^{-1}(1,2)
-      -\frac M2\log\operatorname{sdet}F(\mathbb Q)
+    &\hspace{5em}-\frac M2V_0^2\int d1\,d2\,f(\mathbb Q)^{-1}(1,2)
+      -\frac M2\log\operatorname{sdet}f(\mathbb Q)
       +\frac N2\log\operatorname{sdet}(\mathbb Q-\mathbb M\mathbb M^T)
     \Bigg\}
   \end{aligned}
@@ -339,8 +414,7 @@ We can treat the integral over $\sigma_0$ immediately. It gives
   =2\pi\,\delta(C-1)\,\delta(G+R)\,\bar HH
 \end{equation}
 This therefore sets $C=1$ and $G=-R$ in the remainder of the integrand, as well
-as setting everything depending on $\bar H$ and $H$ to zero.
-
+as setting everything depending on $\bar H$ and $H$ to zero. With these solutions inserted, the remaining remaining terms in the exponential give
 \begin{equation}
   \begin{aligned}
     \operatorname{sdet}(\mathbb Q-\mathbb M\mathbb M^T)
@@ -360,24 +434,59 @@ as setting everything depending on $\bar H$ and $H$ to zero.
   +\frac{2f(1)}{R^3f'(1)}\bar{\hat H}\hat H
 \end{equation}
 \begin{equation}
-  \int d1\,d2\,F(\mathbb Q)^{-1}(1,2)
+  \int d1\,d2\,f(\mathbb Q)^{-1}(1,2)
   =\left(f(1)+\frac{R^2f'(1)}{D}\right)^{-1}
   +2\frac{Rf'(1)}{(Df(1)+R^2f'(1))^2}\bar{\hat H}\hat H
 \end{equation}
+\begin{equation}
+  D=-\frac{m+R}{1-m^2} \qquad \hat m=0
+\end{equation}
+\begin{equation}
+  \mathcal S(R,D,m,\hat m\mid\alpha,V_0)
+  =\hat m-\frac\alpha2\left[
+    \log\left(1+\frac{f(1)D}{f'(1)R^2}\right)
+    +V_0^2\left(f(1)+\frac{f'(1)R^2}{D}\right)^{-1}
+  \right]
+  +\frac12\log\left(
+    1+\frac{(1-m^2)D+\hat m^2-2Rm\hat m}{R^2}
+  \right)
+\end{equation}
+When $\alpha<\alpha_\text{onset}$ this potential has maxima at $\pm m^*$ with
+$m^*=-R^*$ where its value is zero.
+\begin{equation}
+  \alpha_\text{onset}=1-\left(\frac{V_0^2}{V_0^2+f(1)}\right)^2
+\end{equation}
+\begin{equation}
+  \alpha_\text{shatter}=4V_0^2f(1)f'(1)\frac{f'(1)-f(1)}{\big((V_0^2+f(1))f'(1)-f(1)^2\big)^2}
+\end{equation}
 
-\subsection{Behavior with extensively many constraints}
-
-\subsection{Behavior with finitely many constraints}
-
-The correct scaling to find a nontrivial answer with finite $M$ is to scale
-both the covariance functions and fixed constants with $N$ like
-$v_0=\frac1NV_0$, $f(q)=\frac1NF(q)$, so that $v_0$ and $f(q)$ are finite at
-large $N$. With these scalings and $M=1$, this problem reduces to examining the
-levels sets of the spherical spin glasses at energy density $E=v_0$.
+\section{Complexity of the height function}
 
-$v_0^{\chi>2}=\sqrt{2f(1)}$
+\section{Implications for the spherical spin glasses}
 
-$v_0^{m=0}=2\sqrt{f(1)-\frac{f(1)^2}{f'(1)}}$
+As indicated earlier, for $M=1$ the solution manifold corresponds to the energy
+level set of a spherical spin glass with energy density $E=\sqrt NV_0$. All the
+results from the previous sections follow, and can be translated to the spin
+glasses by taking the limit $\alpha\to0$ while scaling $V_0=\sqrt\alpha E$. With a little algebra this procedure yields
+\begin{equation}
+  E_\text{onset}=\pm\sqrt{2f(1)}
+\end{equation}
+\begin{equation}
+  E_\text{shatter}=\pm\sqrt{4f(1)\left(1-\frac{f(1)}{f'(1)}\right)}
+\end{equation}
+for the energies at which level sets of the spherical spin glasses have
+disconnected pieces appear, and that at which a large connected component
+vanishes. For the pure $p$-spin spherical spin glasses with $f(q)=\frac12q^p$,
+$E_\text{shatter}=\sqrt{2(p-1)/p}$, precisely the threshold energy in these
+models. This is expected, since threshold energy, defined as the place where
+marginal minima are dominant in the landscape, is widely understood as the
+place where level sets are broken into pieces.
+
+However, for general mixed models the threshold energy is
+\begin{equation}
+  E_\mathrm{th}=\pm\frac{f(1)[f''(1)-f'(1)]+f'(1)^2}{f'(1)\sqrt{f''(1)}}
+\end{equation}
+which generally satisfies $|E_\text{shatter}|\leq|E_\text{th}|$.
 
 \subsection{What does the average Euler characteristic tell us?}
 
@@ -426,27 +535,6 @@ Thus we find the average Euler characteristic in this simple example is 2
 despite the fact that the possible manifolds resulting from the constraints
 have characteristics of either 0 or 4.
 
-\begin{figure}
-  \includegraphics[width=0.32\columnwidth]{figs/connected.pdf}
-  \hfill
-  \includegraphics[width=0.32\columnwidth]{figs/shattered.pdf}
-  \hfill
-  \includegraphics[width=0.32\columnwidth]{figs/gone.pdf}
-
-  \includegraphics{figs/bar.pdf}
-
-  \caption{
-    Cartoon of the topology of the solution manifold implied by our
-    calculation. The arrow shows the vector $\mathbf x_0$ defining the height
-    function. The region of solutions is marked in black, and the critical points
-    of the height function restricted to this region are marked with a point.
-    For $\alpha<1$, there are few simply connected regions with most of the
-    minima and maxima contributing to the Euler characteristic concentrated at
-    the height $m^*$. For $\alpha\geq1$, there are many simply
-    connected regions and most of their minima and maxima are concentrated at
-    the equator.
-  } \label{fig:cartoons}
-\end{figure}
 
 
 \cite{Franz_2016_The, Franz_2017_Universality, Franz_2019_Critical, Annesi_2023_Star-shaped, Baldassi_2023_Typical}
@@ -471,58 +559,89 @@ have characteristics of either 0 or 4.
 
 \section{Interpretation of our results}
 
-\paragraph{Quenched average of the Euler characteristic.}
-
-\begin{equation}
-  D=\beta R
-  \qquad
-  \hat\beta=-\frac{m+\sum_aR_{1a}}{\sum_aC_{1a}}
+\[
+  E=\frac1{\hat\omega_1}\frac12\left[
+    \hat\omega_1^2f(C)+(2\omega_1\hat\omega_1R-\omega_1^2D)f'(C)+\omega_1^2(R^2-G^2)f''(C)+\log\det\frac{CD+R^2}{G^2}
+  \right]
+\]
+$D=\beta R$, $R=r_dI$, $G=-r_dI$
+\[
+  E=\frac1{\hat\omega_1}\frac12\left[
+    \hat\omega_1^2f(C)+(2\omega_1\hat\omega_1-\omega_1^2\beta)r_df'(1)+\log\det(\beta r_d^{-1}C+I)
+  \right]
+\]
+$\beta=\hat\omega_1/\omega_1$
+\[
+  E=\frac1{\hat\omega_1}\frac12\left[
+    \hat\omega_1^2f(C)+\omega_1\hat\omega_1r_df'(1)+\log\det(\hat\omega_1\omega_1^{-1}r_d^{-1}C+I)
+  \right]
+\]
+$z=r_d\omega_1$
+\[
+  E=\frac1{\hat\omega_1}\frac12\left[
+    \hat\omega_1^2f(C)+\hat \omega_1 zf'(1)+\log\det(\hat\omega_1C/z+I)
+  \right]
+\]
+\[
+  \log\chi
+  =-\hat\omega_1 E
+  +\frac12[2\omega_1\hat\omega_1r_d-\omega_1^2d_d]f'(1)
+  +\frac12\int_0^1dq\,\left[
+    \hat\omega_1^2f''(q)\chi(q)
+    +\frac1{\chi(q)+r_d^2/d_d}
+  \right]
+  -\frac12\log r_d^2
+\]
+\[
+  0=\frac12\hat\omega_1^2f''(q)-\frac12\frac1{[\chi(q)+r_d^2/d_d]^2}
+\]
+\[
+  \chi_0(q)=\frac1{\hat\omega_1}f''(q)^{-1/2}-\frac{r_d^2}{d_d}
+\]
+
+\[
+  \chi(q)=\begin{cases}
+    \chi_0(q) & q < q_0 \\
+    1-(1-m)q_1-mq & q_0 < q < q_1 \\
+    1-q & q > q_1
+  \end{cases}
+\]
+\[
+  0=\hat\omega_1r_d-\omega_1d_d
   \qquad
-  \hat m=0
-\end{equation}
-
-\begin{align}
-  &\mathcal S(m,C,R)
-  =\frac12\log\det\big[I+\hat\beta R^{-1}(C-m^2)\big] \notag \\
-  &\quad-\frac\alpha2\log\det\big[I+\hat\beta\big(R\odot f'(C)\big)^{-1}f(C)\big]
-\end{align}
-
-The quenched average of the Euler characteristic in the replica symmetric ansatz becomes for $1<\alpha<\alpha_\text{\textsc{sat}}$
-\begin{align}
-  \frac1N\overline{\log\chi}
-  =\frac12\bigg[
-    \log\left(-\frac 1{\tilde r_d}\right)
-    -\alpha\log\left(
-      1-\Delta f\frac{1+\tilde r_d}{f'(1)\tilde r_d}
-    \right) \notag \\
-    -\alpha f(0)\left(\Delta f-\frac{f'(1)\tilde r_d}{1+\tilde r_d}\right)^{-1}
-  \bigg]
-\end{align}
-where $\Delta f=f(1)-f(0)$ and $\tilde r_d$ is given by
-\begin{align}
-  \tilde r_d
-  =-\frac{f'(1)f(1)-\Delta f^2}{2(f'(1)-\Delta f)^2}
-  \bigg(
-    \alpha-2+\frac{2f'(1)f(0)}{f'(1)f(1)-\Delta f^2} \notag\\
-    +\sqrt{
-      \alpha^2
-      -4\alpha\frac{f'(1)f(0)\Delta f\big(f'(1)-\Delta f\big)}{\big(f'(1)f(1)-\Delta f^2\big)^2}
-    }
-  \bigg)
-\end{align}
-When $\alpha\to\alpha_\text{\textsc{sat}}=f'(1)/f(1)$ from below, $\tilde r_d\to -1$, which produces $N^{-1}\overline{\log\chi}\to0$.
+  \omega_1=\hat\omega_1\frac{r_d}{d_d}
+\]
+\[
+  \log\chi
+  =-\hat\omega_1 E
+  +\frac12\hat\omega_1^2r_d^2/d_df'(1)
+  +\frac12\int_0^1dq\,\left[
+    \hat\omega_1^2f''(q)\chi(q)
+    +\frac1{\chi(q)+r_d^2/d_d}
+  \right]
+  -\frac12\log r_d^2
+\]
+\[
+  0=-\frac{\hat\omega_1^2f'(1)}{d_d}+\int_0^1dq\,\frac1{(r_d^2/d_d+\chi(q))^2}
+\]
+\[
+  d_d=-\frac{1+r_d}{\int dq\,\chi(q)}r_d
+\]
+
+\paragraph{Acknowledgements}
+The authors thank Pierfrancesco Urbani for helpful conversations on these topics.
 
-\section*{Acknowledgements}
-\addcontentsline{toc}{section}{Acknowledgements}
+\paragraph{Funding information}
 JK-D is supported by a \textsc{DynSysMath} Specific Initiative of the INFN.
-The authors thank Pierfrancesco Urbani for helpful conversations on these topics.
 
 \appendix
 
 \section{Calculation of the prefactor of the average Euler characteristic}
 \label{sec:prefactor}
 
-\printbibliography
-\addcontentsline{toc}{section}{References}
+\section{The quenched shattering energy in {\oldstylenums 1}\textsc{frsb} models}
+\label{sec:1frsb}
+
+\bibliography{topology}
 
 \end{document}