diff --git a/application/#research statement.tex# b/application/#research statement.tex#
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index 00000000..11040219
--- /dev/null
+++ b/application/#research statement.tex#	
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+\documentclass[11pt]{article}
+\usepackage[margin=1in]{geometry}
+
+\usepackage{fancyhdr}
+\pagestyle{fancy}
+
+%\usepackage{mathrsfs}
+
+\usepackage{setspace}
+
+\doublespacing
+\rhead{Anton Bobkov}
+
+\lhead{Research Statement}
+
+\begin{document}
+
+\subsection*{Abstract}
+
+In 2013, Aschenbrenner et al. investigated and developed a notion of VC-density for NIP structures, an analog of geometric dimension in an abstract setting \cite{density}. Their applications included a bound for p-adic numbers, an object of great interest and a very active area of research in mathematics. My research concentrates on improving and expanding techniques of that paper to improve the known bounds as well as computing VC-density for other NIP structures of interest. I am able to obtain new bounds for the additive reduct of p-adic numbers, Henselian valued fields, and certain families of graphs. Recent research by Chernikov and Starchenko in 2015 \cite{regularity} suggests that having good bounds on VC-density in p-adic numbers opens a path for applications to incidence combinatorics (e.g. Szemeredi-Trotter theorem).
+
+\subsection*{Introduction}
+
+The concept of VC-dimension was first introduced in 1971 by Vapnik and Chervonenkis for set systems in a probabilistic setting (see \cite{density}). The theory grew rapidly and found wide use in geometric combinatorics, computational learning theory, and machine learning. Around the same time Shelah was developing the notion of NIP ("not having the independence property"), a natural tameness property of (complete theories of) structures in model theory. In 1992 Laskowski noticed the connection between the two: theories where all uniformly definable families of sets have finite VC-dimension are exactly NIP theories. It is a wide class of theories including algebraically closed fields, differentially closed fields, modules, free groups, o-minimal structures, and ordered abelian groups. A variety of valued fields fall into this category as well, including the p-adic numbers.
+
+P-adic numbers were first introduced by Hensel in 1897, and over the following century a powerful theory was developed around them with numerous applications across a variety of disciplines, primarily in number theory, but also in physics and computer science. In 1965 Ax, Kochen and Ershov axiomatized the theory of p-adic numbers and proved a quantifier elimination result. A key insight was to connect properties of the value group and residue field to the properties of the valued field itself. In 1984 Denef proved a cell decomposition result for more general valued fields. This result was soon generalized to p-adic subanalytic and rigid analytic extensions, allowing for the later development of a more powerful technique of motivic integration. The conjunction of those model theoretic results allowed to solve a number of outstanding open problems in number theory (e.g., Artin's Conjecture on p-adic homogeneous forms).
+
+In 1997, Karpinski and Macintyre computed VC-density bounds for o-minimal structures and asked about similar bounds for p-adic numbers. VC-density is a concept closely related to VC-dimension. It comes up naturally in combinatorics with relation to packings, Hamming metric, entropic dimension and discrepancy.  VC-density is also the decisive parameter in the Epsilon-Approximation Theorem, which is one of the crucial tools for applying VC theory in computational geometry. In a model theoretic setting it is computed for families of uniformly definable sets.  In 2013, Aschenbrenner, Dolich, Haskell, Macpherson, and Starchenko computed a bound for VC-density in p-adic numbers and a number of other NIP structures \cite{density}. They observed connections to dp-rank and dp-minimality, notions first introduced by Shelah. In well behaved NIP structures families of uniformly definable sets tend to have VC-density bounded by a multiple of their dimension, a simple linear behavior. In a lot of cases including p-adic numbers this bound is not known to be optimal. My research concentrates on improving those bounds and adapting those techniques to compute VC-density in other common NIP structures of interest to mathematicians.
+
+Some of the other well behaved NIP structures are Shelah-Spencer graphs and flat graphs. Shelah-Spencer graphs are limit structures for random graphs arising naturally in a combinatorial context. Their model theory was studied by Baldwin, Shi, and Shelah in 1997, and later work of Laskowski in 2006 \cite{graph} have provided a quantifier simplification result.  Flat graphs were first studied by Podewski-Ziegler in 1978, showing that those are stable \cite{stable}, and later results gave a criterion for super stability. Flat graphs also come up naturally in combinatorics in work of Nesetril and Ossona de Mendez \cite{nowhere}. %Shelah-Spencer graphs and flat graphs are both subclasses of NIP theories, extremely well behaved model theoretically.
+
+\subsection*{Past Results and Future Work}
+
+I have studied properties of VC-densoty in Shelah-Spencer graphs. 
+I have shown that they have infinite dp-rank, so they are poorly behaved as NIP structures. I have also shown that one can obtain arbitrarily high VC-density when looking at uniformly definable families in a fixed dimension. However I'm able to bound VC-density of individual formulas in terms of edge density of the graphs they define.
+This indicates that there is potentially a notion more refined than VC-density that captures combinatorial behavior of definable sets in Shelah-Spencer graphs.
+
+I have answered an open question from \cite{density}, computing VC-density for trees viewed as a partial order. The main idea is to adapt a technique of Parigot \cite{trees} to partition trees into weakly interacting parts, with simple bounds of VC-density on each. Similar partitions come up in the Podewski-Ziegler analysis of flat graphs \cite {stable}. I am able to use that technique to show that flat graphs are dp-minimal, an important first step before establishing bounds on VC-density. I hope that I can adapt it further to compute VC-density bounds explicitly for specific families of flat graphs.
+
+The rest of my work so far deals with p-adic numbers and valued fields.
+My goal is to improve the bound $vc(n) \leq 2n - 1$ in \cite{density}.
+I was able to do so in an cerain additive reduct of p-adic numbers using a cell decomposition result from the work of Leenknegt in 2013 \cite{reduce}.
+I will explore other reducts described in that paper, to see if my techniques apply to those as well. I have also shown that a Henselian valued field of equi-characteristic zero has linear one-dimensional VC-density if its value group and its residue field have that property. This is along the lines of the results of Ax-Kochen mentioned before. The second of my remaining research goals is to adapt those techniques to higher dimensions, as well as applying them to RV sorts introduced by Flenner in 2011 \cite{value}.
+
+
+\begin{thebibliography}{9}
+
+\bibitem{density}
+	M. Aschenbrenner, A. Dolich, D. Haskell, D. Macpherson, S. Starchenko,
+	\textit{Vapnik-Chervonenkis density in some theories without the independence property}, I, preprint (2011)
+
+\bibitem{regularity}
+Artem Chernikov, Sergei Starchenko, \textit{Regularity lemma for distal structures}, 	arXiv:1507.01482
+
+\bibitem{value}
+Joseph Flenner. \textit{Relative decidability and definability in Henselian valued fields.} The
+Journal of Symbolic Logic, 76(04):1240–1260, 2011.
+	
+\bibitem{graph}
+  Michael C. Laskowski, \textit{A simpler axiomatization of the Shelah-Spencer almost sure theories},
+  Israel J. Math. \textbf{161} (2007), 157–186. MR MR2 350161	
+
+\bibitem{reduce}
+E. Leenknegt. \textit{Reducts of p-adically closed fields}, Archive for Mathematical Logic \textbf{20 pp.}
+
+\bibitem{nowhere}
+J. Nesetril and P. Ossona de Mendez. \textit{On nowhere dense graphs.} European Journal of
+Combinatorics, 32(4):600–617, 2011
+
+\bibitem{trees}
+	Michel Parigot.
+	Th\'eories d'arbres.
+	\textit{Journal of Symbolic Logic}, 47, 1982. 
+
+
+\bibitem{stable}
+	Klaus-Peter Podewski and Martin Ziegler. Stable graphs. \textit{Fund. Math.}, 100:101-107, 1978.
+
+\end{thebibliography}
+	
+\end{document}
+
+% Include edges between y as a chain minimal extension
diff --git a/application/#teaching_statement.tex# b/application/#teaching_statement.tex#
new file mode 100644
index 00000000..af2dab5e
--- /dev/null
+++ b/application/#teaching_statement.tex#
@@ -0,0 +1,109 @@
+\documentclass[11pt]{article}
+\usepackage[margin=1in]{geometry}
+
+\usepackage{fancyhdr}
+\pagestyle{fancy}
+
+%\usepackage{mathrsfs}
+
+\usepackage{setspace}
+
+\doublespacing
+\rhead{Anton Bobkov}
+
+\lhead{Teaching Statement}
+
+
+
+
+\begin{document}
+
+As UCLA student I had a wide variety of teaching experience, 
+having taught in first-year calculus classes, advanced undergraduate classes, programming courses
+as well as mentoring independent projects and reading courses for undergraduate students.
+I was a recipient of UCLA's 2016 Departmental Teaching Award.
+I have also engaged in private tutoring as well as volunteering to assist math instructors in local high schools.
+
+When I teach, my primary goal is to have students interacting with the material as soon as possible.
+Instead of waiting for them to first engage with the material only when they start their homework,
+I give out worksheets as soon as a given topic as covered.
+This way they get familiar with the basics faster which allows me to move on to harder topics and
+give more abstract explanations sooner.
+Having a class work on an assignment right away also allows me to gauge where all the students are at in their understanding of the material.
+I encourage group work when possible.
+This empahsizes a social aspect of learning and allows students to interact with their peers to explain ideas and help each other to work trough harder parts of the material.
+
+I believe that mathematics is best explained with the use of visual aid.
+I make sure to include plenty of diagrams, pictures, and graphs in my teaching.
+When possible I try to illustrate common mathematical or physical concepts using props.
+For example, as a part of explanation of sorting algorithms I use a deck of cards,
+or when explaining hyperbolic function I use a hanging rope as an illustration of catenoid.
+This helps students to visualize explained concepts as well as creating a link to the real world application of the presented abstract concepts.
+
+When teaching I strive to get as much student feedback as possible.
+For example, I always ask students attending office hours what they think of the direction the class is going and what they would like to see more.
+I also ask them to fill out an anonymous survey at several points throughout the class where they can voice criticisms and suggestions.
+I do my best to respond and adapt my teaching style according to this feedback - it has greatly helped me to form my teaching style.
+
+I have volunteered to be an independent study mentor for several students.
+They were working on a variety of projects
+
+At UCLA, I worked as a mentor for students doing independent study.
+This has encompassed a wide variety of projects.
+I have worked with students both informally and formally for university credit,
+over short periods like one quarter as well as on more serious year long projects.
+The projects' topics included pure and applied math, data analysis, computer science, and finance.
+All the projects had a strong programming component, so students always had a working program to show at the end.
+
+My main focus as a mentor was to balance encouraging students to explore on their own versus providing a structure with concrete goals and expectations.
+On one hand I would let students learn different aspects of the material, experiment with it, and see what direction they want to take their project.
+Given complexity and range of the material, however, it is easy for the students to feel overwhelmed.
+To counter that I would introduce concrete weekly goals and would often check in to talk about long term direction of the project.
+This way even if the material is dense and frustrating to get through, the student would always feel accomplished having a concrete weekly progress as well as a feel for project's overall progress.
+
+For example, one of the projects was doing image recognition of digits using neural networks.
+The student didn't have any prior experience in this area, so we started really slowly by following tutorials online.
+I would instruct to get a tutorial working, make minor changes to it, and research what other algorithms there are.
+As we progressed, the student became more familiar with the language and the code involved, gaining confidence to work on more advanced topics.
+Originally we have planned to write an algorithm recognizing chineese characters,
+but as we got through the basics, I have talked to the student and we have determined that it would be more interesting to try to adapt symbol recognition algorithm to recognizing sounds
+I value this flexibility a lot - it allows the project to evolve naturally in accord with the student's interest and their comfort with the material studied.
+
+\begin{thebibliography}{9}
+
+\bibitem{density}
+	M. Aschenbrenner, A. Dolich, D. Haskell, D. Macpherson, S. Starchenko,
+	\textit{Vapnik-Chervonenkis density in some theories without the independence property}, I, preprint (2011)
+
+\bibitem{regularity}
+Artem Chernikov, Sergei Starchenko, \textit{Regularity lemma for distal structures}, 	arXiv:1507.01482
+
+\bibitem{value}
+Joseph Flenner. \textit{Relative decidability and definability in Henselian valued fields.} The
+Journal of Symbolic Logic, 76(04):1240–1260, 2011.
+	
+\bibitem{graph}
+  Michael C. Laskowski, \textit{A simpler axiomatization of the Shelah-Spencer almost sure theories},
+  Israel J. Math. \textbf{161} (2007), 157–186. MR MR2 350161	
+
+\bibitem{reduce}
+E. Leenknegt. \textit{Reducts of p-adically closed fields}, Archive for Mathematical Logic \textbf{20 pp.}
+
+\bibitem{nowhere}
+J. Nesetril and P. Ossona de Mendez. \textit{On nowhere dense graphs.} European Journal of
+Combinatorics, 32(4):600–617, 2011
+
+\bibitem{trees}
+	Michel Parigot.
+	Th\'eories d'arbres.
+	\textit{Journal of Symbolic Logic}, 47, 1982. 
+
+
+\bibitem{stable}
+	Klaus-Peter Podewski and Martin Ziegler. Stable graphs. \textit{Fund. Math.}, 100:101-107, 1978.
+
+\end{thebibliography}
+	
+\end{document}
+
+% Include edges between y as a chain minimal extension
diff --git a/application/.teaching_statement.tex.swp b/application/.teaching_statement.tex.swp
new file mode 100644
index 00000000..9093d3f1
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diff --git a/application/DYF application.tex~ b/application/DYF application.tex~
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--- /dev/null
+++ b/application/DYF application.tex~	
@@ -0,0 +1,86 @@
+\documentclass[11pt]{article}
+\usepackage[margin=1in]{geometry}
+
+\usepackage{fancyhdr}
+\pagestyle{fancy}
+
+%\usepackage{mathrsfs}
+
+\usepackage{setspace}
+
+\doublespacing
+\rhead{Anton Bobkov}
+
+\lhead{Proposed Plan for Completing the Dissertation}
+
+
+
+
+\begin{document}
+
+\subsection*{Abstract}
+
+In 2013, Aschenbrenner et al. investigated and developed a notion of VC-density for NIP structures, an analog of geometric dimension in an abstract setting \cite{density}. Their applications included a bound for p-adic numbers, an object of great interest and a very active area of research in mathematics. My research concentrates on improving and expanding techniques of that paper to improve the known bounds as well as computing VC-density for other NIP structures of interest. I am able to obtain new bounds for the additive reduct of p-adic numbers, Henselian valued fields, and certain families of graphs. Recent research by Chernikov and Starchenko in 2015 \cite{regularity} suggests that having good bounds on VC-density in p-adic numbers opens a path for applications to incidence combinatorics (e.g. Szemeredi-Trotter theorem).
+
+\subsection*{Introduction}
+
+The concept of VC-dimension was first introduced in 1971 by Vapnik and Chervonenkis for set systems in a probabilistic setting (see \cite{density}). The theory grew rapidly and found wide use in geometric combinatorics, computational learning theory, and machine learning. Around the same time Shelah was developing the notion of NIP ("not having the independence property"), a natural tameness property of (complete theories of) structures in model theory. In 1992 Laskowski noticed the connection between the two: theories where all uniformly definable families of sets have finite VC-dimension are exactly NIP theories. It is a wide class of theories including algebraically closed fields, differentially closed fields, modules, free groups, o-minimal structures, and ordered abelian groups. A variety of valued fields fall into this category as well, including the p-adic numbers.
+
+P-adic numbers were first introduced by Hensel in 1897, and over the following century a powerful theory was developed around them with numerous applications across a variety of disciplines, primarily in number theory, but also in physics and computer science. In 1965 Ax, Kochen and Ershov axiomatized the theory of p-adic numbers and proved a quantifier elimination result. A key insight was to connect properties of the value group and residue field to the properties of the valued field itself. In 1984 Denef proved a cell decomposition result for more general valued fields. This result was soon generalized to p-adic subanalytic and rigid analytic extensions, allowing for the later development of a more powerful technique of motivic integration. The conjunction of those model theoretic results allowed to solve a number of outstanding open problems in number theory (e.g., Artin's Conjecture on p-adic homogeneous forms).
+
+In 1997, Karpinski and Macintyre computed VC-density bounds for o-minimal structures and asked about similar bounds for p-adic numbers. VC-density is a concept closely related to VC-dimension. It comes up naturally in combinatorics with relation to packings, Hamming metric, entropic dimension and discrepancy.  VC-density is also the decisive parameter in the Epsilon-Approximation Theorem, which is one of the crucial tools for applying VC theory in computational geometry. In a model theoretic setting it is computed for families of uniformly definable sets.  In 2013, Aschenbrenner, Dolich, Haskell, Macpherson, and Starchenko computed a bound for VC-density in p-adic numbers and a number of other NIP structures \cite{density}. They observed connections to dp-rank and dp-minimality, notions first introduced by Shelah. In well behaved NIP structures families of uniformly definable sets tend to have VC-density bounded by a multiple of their dimension, a simple linear behavior. In a lot of cases including p-adic numbers this bound is not known to be optimal. My research concentrates on improving those bounds and adapting those techniques to compute VC-density in other common NIP structures of interest to mathematicians.
+
+Some of the other well behaved NIP structures are Shelah-Spencer graphs and flat graphs. Shelah-Spencer graphs are limit structures for random graphs arising naturally in a combinatorial context. Their model theory was studied by Baldwin, Shi, and Shelah in 1997, and later work of Laskowski in 2006 \cite{graph} have provided a quantifier simplification result.  Flat graphs were first studied by Podewski-Ziegler in 1978, showing that those are stable \cite{stable}, and later results gave a criterion for super stability. Flat graphs also come up naturally in combinatorics in work of Nesetril and Ossona de Mendez \cite{nowhere}. %Shelah-Spencer graphs and flat graphs are both subclasses of NIP theories, extremely well behaved model theoretically.
+
+\subsection*{Past Results and Future Work}
+
+I have studied properties of VC-densoty in Shelah-Spencer graphs. 
+I have shown that they have infinite dp-rank, so they are poorly behaved as NIP structures. I have also shown that one can obtain arbitrarily high VC-density when looking at uniformly definable families in a fixed dimension. However I'm able to bound VC-density of individual formulas in terms of edge density of the graphs they define.
+This indicates that there is potentially a notion more refined than VC-density that captures combinatorial behavior of definable sets in Shelah-Spencer graphs.
+
+I have answered an open question from \cite{density}, computing VC-density for trees viewed as a partial order. The main idea is to adapt a technique of Parigot \cite{trees} to partition trees into weakly interacting parts, with simple bounds of VC-density on each. Similar partitions come up in the Podewski-Ziegler analysis of flat graphs \cite {stable}. I am able to use that technique to show that flat graphs are dp-minimal, an important first step before establishing bounds on VC-density. I hope that I can adapt it further to compute VC-density bounds explicitly for specific families of flat graphs.
+
+The rest of my work so far deals with p-adic numbers and valued fields.
+My goal is to improve the bound $vc(n) \leq 2n - 1$ in \cite{density}.
+I was able to do so in an cerain additive reduct of p-adic numbers using a cell decomposition result from the work of Leenknegt in 2013 \cite{reduce}.
+I will explore other reducts described in that paper, to see if my techniques apply to those as well. I have also shown that a Henselian valued field of equi-characteristic zero has linear one-dimensional VC-density if its value group and its residue field have that property. This is along the lines of the results of Ax-Kochen mentioned before. The second of my remaining research goals is to adapt those techniques to higher dimensions, as well as applying them to RV sorts introduced by Flenner in 2011 \cite{value}.
+
+
+\begin{thebibliography}{9}
+
+\bibitem{density}
+	M. Aschenbrenner, A. Dolich, D. Haskell, D. Macpherson, S. Starchenko,
+	\textit{Vapnik-Chervonenkis density in some theories without the independence property}, I, preprint (2011)
+
+\bibitem{regularity}
+Artem Chernikov, Sergei Starchenko, \textit{Regularity lemma for distal structures}, 	arXiv:1507.01482
+
+\bibitem{value}
+Joseph Flenner. \textit{Relative decidability and definability in Henselian valued fields.} The
+Journal of Symbolic Logic, 76(04):1240–1260, 2011.
+	
+\bibitem{graph}
+  Michael C. Laskowski, \textit{A simpler axiomatization of the Shelah-Spencer almost sure theories},
+  Israel J. Math. \textbf{161} (2007), 157–186. MR MR2 350161	
+
+\bibitem{reduce}
+E. Leenknegt. \textit{Reducts of p-adically closed fields}, Archive for Mathematical Logic \textbf{20 pp.}
+
+\bibitem{nowhere}
+J. Nesetril and P. Ossona de Mendez. \textit{On nowhere dense graphs.} European Journal of
+Combinatorics, 32(4):600–617, 2011
+
+\bibitem{trees}
+	Michel Parigot.
+	Th\'eories d'arbres.
+	\textit{Journal of Symbolic Logic}, 47, 1982. 
+
+
+\bibitem{stable}
+	Klaus-Peter Podewski and Martin Ziegler. Stable graphs. \textit{Fund. Math.}, 100:101-107, 1978.
+
+\end{thebibliography}
+	
+\end{document}
+
+% Include edges between y as a chain minimal extension
diff --git a/application/research statement.pdf b/application/research statement.pdf
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+\relax 
+\citation{density}
+\citation{regularity}
+\citation{density}
+\citation{density}
+\citation{graph}
+\citation{stable}
+\citation{nowhere}
+\citation{density}
+\citation{trees}
+\citation{stable}
+\citation{density}
+\citation{reduce}
+\citation{value}
+\bibcite{density}{1}
+\bibcite{regularity}{2}
+\bibcite{value}{3}
+\bibcite{graph}{4}
+\bibcite{reduce}{5}
+\bibcite{nowhere}{6}
+\bibcite{trees}{7}
+\bibcite{stable}{8}
diff --git a/application/research_statement.pdf b/application/research_statement.pdf
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diff --git a/application/research_statement.tex b/application/research_statement.tex
new file mode 100644
index 00000000..48cd5224
--- /dev/null
+++ b/application/research_statement.tex
@@ -0,0 +1,86 @@
+\documentclass[11pt]{article}
+\usepackage[margin=1in]{geometry}
+
+\usepackage{fancyhdr}
+\pagestyle{fancy}
+
+%\usepackage{mathrsfs}
+
+\usepackage{setspace}
+
+\doublespacing
+\rhead{Anton Bobkov}
+
+\lhead{Research Statement}
+
+
+
+
+\begin{document}
+
+\subsection*{Abstract}
+
+In 2013, Aschenbrenner et al. investigated and developed a notion of VC-density for NIP structures, an analog of geometric dimension in an abstract setting \cite{density}. Their applications included a bound for p-adic numbers, an object of great interest and a very active area of research in mathematics. My research concentrates on improving and expanding techniques of that paper to improve the known bounds as well as computing VC-density for other NIP structures of interest. I am able to obtain new bounds for the additive reduct of p-adic numbers, Henselian valued fields, and certain families of graphs. Recent research by Chernikov and Starchenko in 2015 \cite{regularity} suggests that having good bounds on VC-density in p-adic numbers opens a path for applications to incidence combinatorics (e.g. Szemeredi-Trotter theorem).
+
+\subsection*{Introduction}
+
+The concept of VC-dimension was first introduced in 1971 by Vapnik and Chervonenkis for set systems in a probabilistic setting (see \cite{density}). The theory grew rapidly and found wide use in geometric combinatorics, computational learning theory, and machine learning. Around the same time Shelah was developing the notion of NIP ("not having the independence property"), a natural tameness property of (complete theories of) structures in model theory. In 1992 Laskowski noticed the connection between the two: theories where all uniformly definable families of sets have finite VC-dimension are exactly NIP theories. It is a wide class of theories including algebraically closed fields, differentially closed fields, modules, free groups, o-minimal structures, and ordered abelian groups. A variety of valued fields fall into this category as well, including the p-adic numbers.
+
+P-adic numbers were first introduced by Hensel in 1897, and over the following century a powerful theory was developed around them with numerous applications across a variety of disciplines, primarily in number theory, but also in physics and computer science. In 1965 Ax, Kochen and Ershov axiomatized the theory of p-adic numbers and proved a quantifier elimination result. A key insight was to connect properties of the value group and residue field to the properties of the valued field itself. In 1984 Denef proved a cell decomposition result for more general valued fields. This result was soon generalized to p-adic subanalytic and rigid analytic extensions, allowing for the later development of a more powerful technique of motivic integration. The conjunction of those model theoretic results allowed to solve a number of outstanding open problems in number theory (e.g., Artin's Conjecture on p-adic homogeneous forms).
+
+In 1997, Karpinski and Macintyre computed VC-density bounds for o-minimal structures and asked about similar bounds for p-adic numbers. VC-density is a concept closely related to VC-dimension. It comes up naturally in combinatorics with relation to packings, Hamming metric, entropic dimension and discrepancy.  VC-density is also the decisive parameter in the Epsilon-Approximation Theorem, which is one of the crucial tools for applying VC theory in computational geometry. In a model theoretic setting it is computed for families of uniformly definable sets.  In 2013, Aschenbrenner, Dolich, Haskell, Macpherson, and Starchenko computed a bound for VC-density in p-adic numbers and a number of other NIP structures \cite{density}. They observed connections to dp-rank and dp-minimality, notions first introduced by Shelah. In well behaved NIP structures families of uniformly definable sets tend to have VC-density bounded by a multiple of their dimension, a simple linear behavior. In a lot of cases including p-adic numbers this bound is not known to be optimal. My research concentrates on improving those bounds and adapting those techniques to compute VC-density in other common NIP structures of interest to mathematicians.
+
+Some of the other well behaved NIP structures are Shelah-Spencer graphs and flat graphs. Shelah-Spencer graphs are limit structures for random graphs arising naturally in a combinatorial context. Their model theory was studied by Baldwin, Shi, and Shelah in 1997, and later work of Laskowski in 2006 \cite{graph} have provided a quantifier simplification result.  Flat graphs were first studied by Podewski-Ziegler in 1978, showing that those are stable \cite{stable}, and later results gave a criterion for super stability. Flat graphs also come up naturally in combinatorics in work of Nesetril and Ossona de Mendez \cite{nowhere}. %Shelah-Spencer graphs and flat graphs are both subclasses of NIP theories, extremely well behaved model theoretically.
+
+\subsection*{Past Results and Future Work}
+
+I have studied properties of VC-densoty in Shelah-Spencer graphs. 
+I have shown that they have infinite dp-rank, so they are poorly behaved as NIP structures. I have also shown that one can obtain arbitrarily high VC-density when looking at uniformly definable families in a fixed dimension. However I'm able to bound VC-density of individual formulas in terms of edge density of the graphs they define.
+This indicates that there is potentially a notion more refined than VC-density that captures combinatorial behavior of definable sets in Shelah-Spencer graphs.
+
+I have answered an open question from \cite{density}, computing VC-density for trees viewed as a partial order. The main idea is to adapt a technique of Parigot \cite{trees} to partition trees into weakly interacting parts, with simple bounds of VC-density on each. Similar partitions come up in the Podewski-Ziegler analysis of flat graphs \cite {stable}. I am able to use that technique to show that flat graphs are dp-minimal, an important first step before establishing bounds on VC-density. I hope that I can adapt it further to compute VC-density bounds explicitly for specific families of flat graphs.
+
+The rest of my work so far deals with p-adic numbers and valued fields.
+My goal is to improve the bound $vc(n) \leq 2n - 1$ in \cite{density}.
+I was able to do so in an cerain additive reduct of p-adic numbers using a cell decomposition result from the work of Leenknegt in 2013 \cite{reduce}.
+I will explore other reducts described in that paper, to see if my techniques apply to those as well. I have also shown that a Henselian valued field of equi-characteristic zero has linear one-dimensional VC-density if its value group and its residue field have that property. This is along the lines of the results of Ax-Kochen mentioned before. The second of my remaining research goals is to adapt those techniques to higher dimensions, as well as applying them to RV sorts introduced by Flenner in 2011 \cite{value}.
+
+
+\begin{thebibliography}{9}
+
+\bibitem{density}
+	M. Aschenbrenner, A. Dolich, D. Haskell, D. Macpherson, S. Starchenko,
+	\textit{Vapnik-Chervonenkis density in some theories without the independence property}, I, preprint (2011)
+
+\bibitem{regularity}
+Artem Chernikov, Sergei Starchenko, \textit{Regularity lemma for distal structures}, 	arXiv:1507.01482
+
+\bibitem{value}
+Joseph Flenner. \textit{Relative decidability and definability in Henselian valued fields.} The
+Journal of Symbolic Logic, 76(04):1240–1260, 2011.
+	
+\bibitem{graph}
+  Michael C. Laskowski, \textit{A simpler axiomatization of the Shelah-Spencer almost sure theories},
+  Israel J. Math. \textbf{161} (2007), 157–186. MR MR2 350161	
+
+\bibitem{reduce}
+E. Leenknegt. \textit{Reducts of p-adically closed fields}, Archive for Mathematical Logic \textbf{20 pp.}
+
+\bibitem{nowhere}
+J. Nesetril and P. Ossona de Mendez. \textit{On nowhere dense graphs.} European Journal of
+Combinatorics, 32(4):600–617, 2011
+
+\bibitem{trees}
+	Michel Parigot.
+	Th\'eories d'arbres.
+	\textit{Journal of Symbolic Logic}, 47, 1982. 
+
+
+\bibitem{stable}
+	Klaus-Peter Podewski and Martin Ziegler. Stable graphs. \textit{Fund. Math.}, 100:101-107, 1978.
+
+\end{thebibliography}
+	
+\end{document}
+
+% Include edges between y as a chain minimal extension
diff --git a/application/research_statement_.tex~ b/application/research_statement_.tex~
new file mode 100644
index 00000000..48cd5224
--- /dev/null
+++ b/application/research_statement_.tex~
@@ -0,0 +1,86 @@
+\documentclass[11pt]{article}
+\usepackage[margin=1in]{geometry}
+
+\usepackage{fancyhdr}
+\pagestyle{fancy}
+
+%\usepackage{mathrsfs}
+
+\usepackage{setspace}
+
+\doublespacing
+\rhead{Anton Bobkov}
+
+\lhead{Research Statement}
+
+
+
+
+\begin{document}
+
+\subsection*{Abstract}
+
+In 2013, Aschenbrenner et al. investigated and developed a notion of VC-density for NIP structures, an analog of geometric dimension in an abstract setting \cite{density}. Their applications included a bound for p-adic numbers, an object of great interest and a very active area of research in mathematics. My research concentrates on improving and expanding techniques of that paper to improve the known bounds as well as computing VC-density for other NIP structures of interest. I am able to obtain new bounds for the additive reduct of p-adic numbers, Henselian valued fields, and certain families of graphs. Recent research by Chernikov and Starchenko in 2015 \cite{regularity} suggests that having good bounds on VC-density in p-adic numbers opens a path for applications to incidence combinatorics (e.g. Szemeredi-Trotter theorem).
+
+\subsection*{Introduction}
+
+The concept of VC-dimension was first introduced in 1971 by Vapnik and Chervonenkis for set systems in a probabilistic setting (see \cite{density}). The theory grew rapidly and found wide use in geometric combinatorics, computational learning theory, and machine learning. Around the same time Shelah was developing the notion of NIP ("not having the independence property"), a natural tameness property of (complete theories of) structures in model theory. In 1992 Laskowski noticed the connection between the two: theories where all uniformly definable families of sets have finite VC-dimension are exactly NIP theories. It is a wide class of theories including algebraically closed fields, differentially closed fields, modules, free groups, o-minimal structures, and ordered abelian groups. A variety of valued fields fall into this category as well, including the p-adic numbers.
+
+P-adic numbers were first introduced by Hensel in 1897, and over the following century a powerful theory was developed around them with numerous applications across a variety of disciplines, primarily in number theory, but also in physics and computer science. In 1965 Ax, Kochen and Ershov axiomatized the theory of p-adic numbers and proved a quantifier elimination result. A key insight was to connect properties of the value group and residue field to the properties of the valued field itself. In 1984 Denef proved a cell decomposition result for more general valued fields. This result was soon generalized to p-adic subanalytic and rigid analytic extensions, allowing for the later development of a more powerful technique of motivic integration. The conjunction of those model theoretic results allowed to solve a number of outstanding open problems in number theory (e.g., Artin's Conjecture on p-adic homogeneous forms).
+
+In 1997, Karpinski and Macintyre computed VC-density bounds for o-minimal structures and asked about similar bounds for p-adic numbers. VC-density is a concept closely related to VC-dimension. It comes up naturally in combinatorics with relation to packings, Hamming metric, entropic dimension and discrepancy.  VC-density is also the decisive parameter in the Epsilon-Approximation Theorem, which is one of the crucial tools for applying VC theory in computational geometry. In a model theoretic setting it is computed for families of uniformly definable sets.  In 2013, Aschenbrenner, Dolich, Haskell, Macpherson, and Starchenko computed a bound for VC-density in p-adic numbers and a number of other NIP structures \cite{density}. They observed connections to dp-rank and dp-minimality, notions first introduced by Shelah. In well behaved NIP structures families of uniformly definable sets tend to have VC-density bounded by a multiple of their dimension, a simple linear behavior. In a lot of cases including p-adic numbers this bound is not known to be optimal. My research concentrates on improving those bounds and adapting those techniques to compute VC-density in other common NIP structures of interest to mathematicians.
+
+Some of the other well behaved NIP structures are Shelah-Spencer graphs and flat graphs. Shelah-Spencer graphs are limit structures for random graphs arising naturally in a combinatorial context. Their model theory was studied by Baldwin, Shi, and Shelah in 1997, and later work of Laskowski in 2006 \cite{graph} have provided a quantifier simplification result.  Flat graphs were first studied by Podewski-Ziegler in 1978, showing that those are stable \cite{stable}, and later results gave a criterion for super stability. Flat graphs also come up naturally in combinatorics in work of Nesetril and Ossona de Mendez \cite{nowhere}. %Shelah-Spencer graphs and flat graphs are both subclasses of NIP theories, extremely well behaved model theoretically.
+
+\subsection*{Past Results and Future Work}
+
+I have studied properties of VC-densoty in Shelah-Spencer graphs. 
+I have shown that they have infinite dp-rank, so they are poorly behaved as NIP structures. I have also shown that one can obtain arbitrarily high VC-density when looking at uniformly definable families in a fixed dimension. However I'm able to bound VC-density of individual formulas in terms of edge density of the graphs they define.
+This indicates that there is potentially a notion more refined than VC-density that captures combinatorial behavior of definable sets in Shelah-Spencer graphs.
+
+I have answered an open question from \cite{density}, computing VC-density for trees viewed as a partial order. The main idea is to adapt a technique of Parigot \cite{trees} to partition trees into weakly interacting parts, with simple bounds of VC-density on each. Similar partitions come up in the Podewski-Ziegler analysis of flat graphs \cite {stable}. I am able to use that technique to show that flat graphs are dp-minimal, an important first step before establishing bounds on VC-density. I hope that I can adapt it further to compute VC-density bounds explicitly for specific families of flat graphs.
+
+The rest of my work so far deals with p-adic numbers and valued fields.
+My goal is to improve the bound $vc(n) \leq 2n - 1$ in \cite{density}.
+I was able to do so in an cerain additive reduct of p-adic numbers using a cell decomposition result from the work of Leenknegt in 2013 \cite{reduce}.
+I will explore other reducts described in that paper, to see if my techniques apply to those as well. I have also shown that a Henselian valued field of equi-characteristic zero has linear one-dimensional VC-density if its value group and its residue field have that property. This is along the lines of the results of Ax-Kochen mentioned before. The second of my remaining research goals is to adapt those techniques to higher dimensions, as well as applying them to RV sorts introduced by Flenner in 2011 \cite{value}.
+
+
+\begin{thebibliography}{9}
+
+\bibitem{density}
+	M. Aschenbrenner, A. Dolich, D. Haskell, D. Macpherson, S. Starchenko,
+	\textit{Vapnik-Chervonenkis density in some theories without the independence property}, I, preprint (2011)
+
+\bibitem{regularity}
+Artem Chernikov, Sergei Starchenko, \textit{Regularity lemma for distal structures}, 	arXiv:1507.01482
+
+\bibitem{value}
+Joseph Flenner. \textit{Relative decidability and definability in Henselian valued fields.} The
+Journal of Symbolic Logic, 76(04):1240–1260, 2011.
+	
+\bibitem{graph}
+  Michael C. Laskowski, \textit{A simpler axiomatization of the Shelah-Spencer almost sure theories},
+  Israel J. Math. \textbf{161} (2007), 157–186. MR MR2 350161	
+
+\bibitem{reduce}
+E. Leenknegt. \textit{Reducts of p-adically closed fields}, Archive for Mathematical Logic \textbf{20 pp.}
+
+\bibitem{nowhere}
+J. Nesetril and P. Ossona de Mendez. \textit{On nowhere dense graphs.} European Journal of
+Combinatorics, 32(4):600–617, 2011
+
+\bibitem{trees}
+	Michel Parigot.
+	Th\'eories d'arbres.
+	\textit{Journal of Symbolic Logic}, 47, 1982. 
+
+
+\bibitem{stable}
+	Klaus-Peter Podewski and Martin Ziegler. Stable graphs. \textit{Fund. Math.}, 100:101-107, 1978.
+
+\end{thebibliography}
+	
+\end{document}
+
+% Include edges between y as a chain minimal extension
diff --git a/application/research_statement_IHES_IHP.pdf b/application/research_statement_IHES_IHP.pdf
new file mode 100644
index 00000000..421eecf1
Binary files /dev/null and b/application/research_statement_IHES_IHP.pdf differ
diff --git a/application/research_statement_IHES_IHP.tex b/application/research_statement_IHES_IHP.tex
new file mode 100644
index 00000000..57b845a9
--- /dev/null
+++ b/application/research_statement_IHES_IHP.tex
@@ -0,0 +1,89 @@
+\documentclass[11pt]{article}
+\usepackage[margin=1in]{geometry}
+
+\usepackage{fancyhdr}
+\pagestyle{fancy}
+
+%\usepackage{mathrsfs}
+
+\usepackage{setspace}
+
+\doublespacing
+\rhead{Anton Bobkov}
+
+\lhead{Research Statement}
+
+
+
+
+\begin{document}
+
+\subsection*{Abstract}
+
+In 2013, Aschenbrenner et al. investigated and developed a notion of VC-density for NIP structures, an analog of geometric dimension in an abstract setting \cite{density}. Their applications included a bound for p-adic numbers, an object of great interest and a very active area of research in mathematics. My research concentrates on improving and expanding techniques of that paper to improve the known bounds as well as computing VC-density for other NIP structures of interest. I am able to obtain new bounds for the additive reduct of p-adic numbers, Henselian valued fields, and certain families of graphs. Recent research by Chernikov and Starchenko in 2015 \cite{regularity} suggests that having good bounds on VC-density in p-adic numbers opens a path for applications to incidence combinatorics (e.g. Szemeredi-Trotter theorem).
+
+\subsection*{Introduction}
+
+The concept of VC-dimension was first introduced in 1971 by Vapnik and Chervonenkis for set systems in a probabilistic setting (see \cite{density}). The theory grew rapidly and found wide use in geometric combinatorics, computational learning theory, and machine learning. Around the same time Shelah was developing the notion of NIP ("not having the independence property"), a natural tameness property of (complete theories of) structures in model theory. In 1992 Laskowski noticed the connection between the two: theories where all uniformly definable families of sets have finite VC-dimension are exactly NIP theories. It is a wide class of theories including algebraically closed fields, differentially closed fields, modules, free groups, o-minimal structures, and ordered abelian groups. A variety of valued fields fall into this category as well, including the p-adic numbers.
+
+P-adic numbers were first introduced by Hensel in 1897, and over the following century a powerful theory was developed around them with numerous applications across a variety of disciplines, primarily in number theory, but also in physics and computer science. In 1965 Ax, Kochen and Ershov axiomatized the theory of p-adic numbers and proved a quantifier elimination result. A key insight was to connect properties of the value group and residue field to the properties of the valued field itself. In 1984 Denef proved a cell decomposition result for more general valued fields. This result was soon generalized to p-adic subanalytic and rigid analytic extensions, allowing for the later development of a more powerful technique of motivic integration. The conjunction of those model theoretic results allowed to solve a number of outstanding open problems in number theory (e.g., Artin's Conjecture on p-adic homogeneous forms).
+
+In 1997, Karpinski and Macintyre computed VC-density bounds for o-minimal structures and asked about similar bounds for p-adic numbers. VC-density is a concept closely related to VC-dimension. It comes up naturally in combinatorics with relation to packings, Hamming metric, entropic dimension and discrepancy.  VC-density is also the decisive parameter in the Epsilon-Approximation Theorem, which is one of the crucial tools for applying VC theory in computational geometry. In a model theoretic setting it is computed for families of uniformly definable sets.  In 2013, Aschenbrenner, Dolich, Haskell, Macpherson, and Starchenko computed a bound for VC-density in p-adic numbers and a number of other NIP structures \cite{density}. They observed connections to dp-rank and dp-minimality, notions first introduced by Shelah. In well behaved NIP structures families of uniformly definable sets tend to have VC-density bounded by a multiple of their dimension, a simple linear behavior. In a lot of cases including p-adic numbers this bound is not known to be optimal. My research concentrates on improving those bounds and adapting those techniques to compute VC-density in other common NIP structures of interest to mathematicians.
+
+Some of the other well behaved NIP structures are Shelah-Spencer graphs and flat graphs. Shelah-Spencer graphs are limit structures for random graphs arising naturally in a combinatorial context. Their model theory was studied by Baldwin, Shi, and Shelah in 1997, and later work of Laskowski in 2006 \cite{graph} have provided a quantifier simplification result.  Flat graphs were first studied by Podewski-Ziegler in 1978, showing that those are stable \cite{stable}, and later results gave a criterion for super stability. Flat graphs also come up naturally in combinatorics in work of Nesetril and Ossona de Mendez \cite{nowhere}. %Shelah-Spencer graphs and flat graphs are both subclasses of NIP theories, extremely well behaved model theoretically.
+
+\subsection*{Past Results}
+
+I have studied properties of VC-densoty in Shelah-Spencer graphs. 
+I have shown that they have infinite dp-rank, so they are poorly behaved as NIP structures. I have also shown that one can obtain arbitrarily high VC-density when looking at uniformly definable families in a fixed dimension. However I'm able to bound VC-density of individual formulas in terms of edge density of the graphs they define.
+This indicates that there is potentially a notion more refined than VC-density that captures combinatorial behavior of definable sets in Shelah-Spencer graphs.
+
+I have answered an open question from \cite{density}, computing VC-density for trees viewed as a partial order. The main idea is to adapt a technique of Parigot \cite{trees} to partition trees into weakly interacting parts, with simple bounds of VC-density on each. Similar partitions come up in the Podewski-Ziegler analysis of flat graphs \cite {stable}. I am able to use that technique to show that flat graphs are dp-minimal, an important first step before establishing bounds on VC-density. I hope that I can adapt it further to compute VC-density bounds explicitly for specific families of flat graphs.
+
+\subsection*{Short Project Description for IHES-IHP Postdoc}
+The rest of my work deals with p-adic numbers and valued fields.
+My goal is to improve the bound $vc(n) \leq 2n - 1$ in \cite{density}.
+I was able to do so in an cerain additive reduct of p-adic numbers using a cell decomposition result from the work of Leenknegt in 2013 \cite{reduce}.
+I will explore other reducts described in that paper, to see if my techniques apply to those as well.
+I have also shown that a Henselian valued field of equi-characteristic zero has linear one-dimensional VC-density if its value group and its residue field have that property. This is along the lines of the results of Ax-Kochen mentioned before.
+My research goal is to adapt those techniques to higher dimensions, as well as applying them to RV sorts introduced by Flenner in 2011 \cite{value}.
+
+
+\begin{thebibliography}{9}
+
+\bibitem{density}
+	M. Aschenbrenner, A. Dolich, D. Haskell, D. Macpherson, S. Starchenko,
+	\textit{Vapnik-Chervonenkis density in some theories without the independence property}, I, preprint (2011)
+
+\bibitem{regularity}
+Artem Chernikov, Sergei Starchenko, \textit{Regularity lemma for distal structures}, 	arXiv:1507.01482
+
+\bibitem{value}
+Joseph Flenner. \textit{Relative decidability and definability in Henselian valued fields.} The
+Journal of Symbolic Logic, 76(04):1240–1260, 2011.
+	
+\bibitem{graph}
+  Michael C. Laskowski, \textit{A simpler axiomatization of the Shelah-Spencer almost sure theories},
+  Israel J. Math. \textbf{161} (2007), 157–186. MR MR2 350161	
+
+\bibitem{reduce}
+E. Leenknegt. \textit{Reducts of p-adically closed fields}, Archive for Mathematical Logic \textbf{20 pp.}
+
+\bibitem{nowhere}
+J. Nesetril and P. Ossona de Mendez. \textit{On nowhere dense graphs.} European Journal of
+Combinatorics, 32(4):600–617, 2011
+
+\bibitem{trees}
+	Michel Parigot.
+	Th\'eories d'arbres.
+	\textit{Journal of Symbolic Logic}, 47, 1982. 
+
+
+\bibitem{stable}
+	Klaus-Peter Podewski and Martin Ziegler. Stable graphs. \textit{Fund. Math.}, 100:101-107, 1978.
+
+\end{thebibliography}
+	
+\end{document}
+
+% Include edges between y as a chain minimal extension
diff --git a/application/teaching_statement.pdf b/application/teaching_statement.pdf
new file mode 100644
index 00000000..3752037d
Binary files /dev/null and b/application/teaching_statement.pdf differ
diff --git a/application/teaching_statement.tex b/application/teaching_statement.tex
new file mode 100644
index 00000000..273566dc
--- /dev/null
+++ b/application/teaching_statement.tex
@@ -0,0 +1,102 @@
+\documentclass[11pt]{article}
+\usepackage[margin=1in]{geometry}
+
+\usepackage{fancyhdr}
+\pagestyle{fancy}
+
+%\usepackage{mathrsfs}
+
+\usepackage{setspace}
+%\doublespacing
+
+
+\rhead{Anton Bobkov}
+
+\lhead{Teaching Statement}
+
+
+
+
+\begin{document}
+
+As UCLA student I had a wide variety of teaching experience, 
+having taught in first-year calculus classes, advanced undergraduate classes, programming courses
+as well as mentoring independent projects and reading courses for undergraduate students.
+I was a recipient of UCLA's 2016 Departmental Teaching Award.
+I have also engaged in private tutoring as well as volunteering to assist math instructors in local high schools.
+
+When I teach, my primary goal is to have students interacting with the material as soon as possible.
+Instead of waiting for them to first engage with the material only when they start their homework,
+I give out worksheets as soon as a given topic as covered.
+This way they get familiar with the basics faster which allows me to move on to harder topics and
+give more abstract explanations sooner.
+Having a class work on an assignment right away also allows me to gauge where all the students are at in their understanding of the material.
+I encourage group work when possible.
+For a quicker measure of students' level of understanding of the material I prepare one or two question multiple choice quizzes which I display on a projector and ask students to text message their answer.
+This allows instanteneous feedback for me while keeping the students engaged and interacting with the material.
+If most people get the question wrong, then it is a sign for me to slow down and go over the material again.
+If majority of students answer correctly, then I can confidently move on to the next topic.
+This is a much better way to measure students' level of understanding of the material compared to soliciting answers via raised hands as that usually only covers only those students who are actively participating.
+This also provides constant feedback back to the students - if a student finds themselves struggling with quizzed or worksheet questions, they take it as an indication that they should put more time into the class.
+Normally, the students only get such feedback through the homeworks, which could be a misleading indicator for the format and difficulty of the exams.
+This empahsizes a social aspect of learning and allows students to interact with their peers to explain ideas and help each other to work trough harder parts of the material.
+
+I believe that mathematics is best explained with the use of visual aid.
+I make sure to include plenty of diagrams, pictures, and graphs in my teaching.
+Even something small - like using colors or managing board space in a good way - can have a tremendous effect in making the material clearer.
+When possible I try to demonstrate common mathematical or physical concepts using props.
+For example, as a part of explanation of sorting algorithms I use a deck of cards,
+or when explaining hyperbolic function I use a hanging rope as an illustration of catenoid.
+This helps students to visualize explained concepts as well as creating a link to the real world application of the presented abstract concepts.
+With the advance of the technology there are a lot of interactive visualization tools available online.
+I try to incorporate them into my lectures when possible.
+For example, when I was introducing students to Taylor Polynomilas I have used a Mathematica online demo for approximating trigonometric functions.
+It would display a graph of a trigonometric function overlayed with a graph of its approximation using a Taylor Polynomial.
+You can drag a slide to change the center of the approximation as well as change degree of the polynimial.
+I felt like this provided clear and intuitive demonstration of Taylor polynomials, one that is hard to replicate via usual means of drawing those graphs on blackboard.
+
+Another important component of my teaching is connecting the math explained with its real world applications.
+A nice thing about mathematics is its widespread use in all kinds of technical fields - physics and natural sciences, engineering, computer science, medicine, finance, economics as well as statistics being very useful in social sciences.
+In my experience even very abstract ideas often have useful and sometimes surprising applications.
+It is my goal to always keep students conncected with the applications of the material they study.
+I always make sure to mentinon, for example, application of exponential function to caffeine level in blood, connection of eigenvectors to facial recognition, the use of abstract groups in describing atom lattices, improper integrals for computation of terminal velocity needed to leave Earth's orbit, or how geometric series arise to describe multiplier effect in macroeconomics.
+I believe that it provides a better motivation for studying the meaterial that sometimes can get a bit abstract.
+
+When teaching I strive to get as much student feedback as possible.
+For example, I always ask students attending office hours what they think of the direction the class is going and what they would like to see more.
+I also ask them to fill out an anonymous survey at several points throughout the class where they can voice criticisms and suggestions.
+I do my best to respond and adapt my teaching style according to this feedback - it has greatly helped me to form my teaching style.
+
+When working with students one on one I try to gauge how comfortable with the material a student is and work from there.
+Instead of just explaning a given concept or a problem, I try to engage the student right away.
+This involves asking a lot of questions or ideally having the student work on a problem themselves
+and guide them through the process.
+When I was an instructor for a calculus course, I have worked with a student who was very uncomfortable with the material going into the class.
+I have spent a lot of office hours with the student.
+I would start with simple problems and once I saw that 
+
+At UCLA, I worked as a mentor for students doing independent study.
+This has encompassed a wide variety of projects.
+I have worked with students both informally and formally for university credit,
+over short periods like one quarter as well as on more serious year long projects.
+The projects' topics included pure and applied math, data analysis, computer science, and finance.
+All the projects had a strong programming component, so students always had a working program to show at the end.
+
+My main focus as a mentor was to balance encouraging students to explore on their own versus providing a structure with concrete goals and expectations.
+On one hand I would let students learn different aspects of the material, experiment with it, and see what direction they want to take their project.
+Given complexity and range of the material, however, it is easy for the students to feel overwhelmed.
+To counter that I would introduce concrete weekly goals and would often check in to talk about long term direction of the project.
+This way even if the material is dense and frustrating to get through, the student would always feel accomplished having a concrete weekly progress as well as a feel for project's overall progress.
+
+For example, one of the projects was doing image recognition of digits using neural networks.
+The student didn't have any prior experience in this area, so we started really slowly by following tutorials online.
+I would instruct to get a tutorial working, make minor changes to it, and research what other algorithms there are.
+As we progressed, the student became more familiar with the language and the code involved, gaining confidence to work on more advanced topics.
+Originally we have planned to write an algorithm recognizing chineese characters,
+but as we got through the basics, I have talked to the student and we have determined that it would be more interesting to try to adapt symbol recognition algorithm to recognizing sounds
+I value this flexibility a lot - it allows the project to evolve naturally in accord with the student's interest and their comfort with the material studied.
+
+	
+\end{document}
+
+% Include edges between y as a chain minimal extension
diff --git a/application/teaching_statement.tex~ b/application/teaching_statement.tex~
new file mode 100644
index 00000000..60b1cdd5
--- /dev/null
+++ b/application/teaching_statement.tex~
@@ -0,0 +1,102 @@
+\documentclass[11pt]{article}
+\usepackage[margin=1in]{geometry}
+
+\usepackage{fancyhdr}
+\pagestyle{fancy}
+
+%\usepackage{mathrsfs}
+
+\usepackage{setspace}
+\doublespacing
+
+
+\rhead{Anton Bobkov}
+
+\lhead{Teaching Statement}
+
+
+
+
+\begin{document}
+
+As UCLA student I had a wide variety of teaching experience, 
+having taught in first-year calculus classes, advanced undergraduate classes, programming courses
+as well as mentoring independent projects and reading courses for undergraduate students.
+I was a recipient of UCLA's 2016 Departmental Teaching Award.
+I have also engaged in private tutoring as well as volunteering to assist math instructors in local high schools.
+
+When I teach, my primary goal is to have students interacting with the material as soon as possible.
+Instead of waiting for them to first engage with the material only when they start their homework,
+I give out worksheets as soon as a given topic as covered.
+This way they get familiar with the basics faster which allows me to move on to harder topics and
+give more abstract explanations sooner.
+Having a class work on an assignment right away also allows me to gauge where all the students are at in their understanding of the material.
+I encourage group work when possible.
+For a quicker measure of students' level of understanding of the material I prepare one or two question multiple choice quizzes which I display on a projector and ask students to text message their answer.
+This allows instanteneous feedback for me while keeping the students engaged and interacting with the material.
+If most people get the question wrong, then it is a sign for me to slow down and go over the material again.
+If majority of students answer correctly, then I can confidently move on to the next topic.
+This is a much better way to measure students' level of understanding of the material compared to soliciting answers via raised hands as that usually only covers only those students who are actively participating.
+This also provides constant feedback back to the students - if a student finds themselves struggling with quizzed or worksheet questions, they take it as an indication that they should put more time into the class.
+Normally, the students only get such feedback through the homeworks, which could be a misleading indicator for the format and difficulty of the exams.
+This empahsizes a social aspect of learning and allows students to interact with their peers to explain ideas and help each other to work trough harder parts of the material.
+
+I believe that mathematics is best explained with the use of visual aid.
+I make sure to include plenty of diagrams, pictures, and graphs in my teaching.
+Even something small - like using colors or managing board space in a good way - can have a tremendous effect in making the material clearer.
+When possible I try to demonstrate common mathematical or physical concepts using props.
+For example, as a part of explanation of sorting algorithms I use a deck of cards,
+or when explaining hyperbolic function I use a hanging rope as an illustration of catenoid.
+This helps students to visualize explained concepts as well as creating a link to the real world application of the presented abstract concepts.
+With the advance of the technology there are a lot of interactive visualization tools available online.
+I try to incorporate them into my lectures when possible.
+For example, when I was introducing students to Taylor Polynomilas I have used a Mathematica online demo for approximating trigonometric functions.
+It would display a graph of a trigonometric function overlayed with a graph of its approximation using a Taylor Polynomial.
+You can drag a slide to change the center of the approximation as well as change degree of the polynimial.
+I felt like this provided clear and intuitive demonstration of Taylor polynomials, one that is hard to replicate via usual means of drawing those graphs on blackboard.
+
+Another important component of my teaching is connecting the math explained with its real world applications.
+A nice thing about mathematics is its widespread use in all kinds of technical fields - physics and natural sciences, engineering, computer science, medicine, finance, economics as well as statistics being very useful in social sciences.
+In my experience even very abstract ideas often have useful and sometimes surprising applications.
+It is my goal to always keep students conncected with the applications of the material they study.
+I always make sure to mentinon, for example, application of exponential function to caffeine level in blood, connection of eigenvectors to facial recognition, the use of abstract groups in describing atom lattices, improper integrals for computation of terminal velocity needed to leave Earth's orbit, or how geometric series arise to describe multiplier effect in macroeconomics.
+I believe that it provides a better motivation for studying the meaterial that sometimes can get a bit abstract.
+
+When teaching I strive to get as much student feedback as possible.
+For example, I always ask students attending office hours what they think of the direction the class is going and what they would like to see more.
+I also ask them to fill out an anonymous survey at several points throughout the class where they can voice criticisms and suggestions.
+I do my best to respond and adapt my teaching style according to this feedback - it has greatly helped me to form my teaching style.
+
+When working with students one on one I try to gauge how comfortable with the material a student is and work from there.
+Instead of just explaning a given concept or a problem, I try to engage the student right away.
+This involves asking a lot of questions or ideally having the student work on a problem themselves
+and guide them through the process.
+When I was an instructor for a calculus course, I have worked with a student who was very uncomfortable with the material going into the class.
+I have spent a lot of office hours with the student.
+I would start with simple problems and once I saw that 
+
+At UCLA, I worked as a mentor for students doing independent study.
+This has encompassed a wide variety of projects.
+I have worked with students both informally and formally for university credit,
+over short periods like one quarter as well as on more serious year long projects.
+The projects' topics included pure and applied math, data analysis, computer science, and finance.
+All the projects had a strong programming component, so students always had a working program to show at the end.
+
+My main focus as a mentor was to balance encouraging students to explore on their own versus providing a structure with concrete goals and expectations.
+On one hand I would let students learn different aspects of the material, experiment with it, and see what direction they want to take their project.
+Given complexity and range of the material, however, it is easy for the students to feel overwhelmed.
+To counter that I would introduce concrete weekly goals and would often check in to talk about long term direction of the project.
+This way even if the material is dense and frustrating to get through, the student would always feel accomplished having a concrete weekly progress as well as a feel for project's overall progress.
+
+For example, one of the projects was doing image recognition of digits using neural networks.
+The student didn't have any prior experience in this area, so we started really slowly by following tutorials online.
+I would instruct to get a tutorial working, make minor changes to it, and research what other algorithms there are.
+As we progressed, the student became more familiar with the language and the code involved, gaining confidence to work on more advanced topics.
+Originally we have planned to write an algorithm recognizing chineese characters,
+but as we got through the basics, I have talked to the student and we have determined that it would be more interesting to try to adapt symbol recognition algorithm to recognizing sounds
+I value this flexibility a lot - it allows the project to evolve naturally in accord with the student's interest and their comfort with the material studied.
+
+	
+\end{document}
+
+% Include edges between y as a chain minimal extension
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+\documentclass[11pt]{article}
+\usepackage[margin=1in]{geometry}
+
+\usepackage{fancyhdr}
+\pagestyle{fancy}
+
+%\usepackage{mathrsfs}
+
+\usepackage{setspace}
+
+\doublespacing
+\rhead{Anton Bobkov}
+
+\lhead{Proposed Plan for Completing the Dissertation}
+
+
+
+
+\begin{document}
+
+\subsection*{Abstract}
+
+In 2013, Aschenbrenner et al. investigated and developed a notion of VC-density for NIP structures, an analog of geometric dimension in an abstract setting \cite{density}. Their applications included a bound for p-adic numbers, an object of great interest and a very active area of research in mathematics. My research concentrates on improving and expanding techniques of that paper to improve the known bounds as well as computing VC-density for other NIP structures of interest. I am able to obtain new bounds for the additive reduct of p-adic numbers, Henselian valued fields, and certain families of graphs. Recent research by Chernikov and Starchenko in 2015 \cite{regularity} suggests that having good bounds on VC-density in p-adic numbers opens a path for applications to incidence combinatorics (e.g. Szemeredi-Trotter theorem).
+
+\subsection*{Introduction}
+
+The concept of VC-dimension was first introduced in 1971 by Vapnik and Chervonenkis for set systems in a probabilistic setting (see \cite{density}). The theory grew rapidly and found wide use in geometric combinatorics, computational learning theory, and machine learning. Around the same time Shelah was developing the notion of NIP ("not having the independence property"), a natural tameness property of (complete theories of) structures in model theory. In 1992 Laskowski noticed the connection between the two: theories where all uniformly definable families of sets have finite VC-dimension are exactly NIP theories. It is a wide class of theories including algebraically closed fields, differentially closed fields, modules, free groups, o-minimal structures, and ordered abelian groups. A variety of valued fields fall into this category as well, including the p-adic numbers.
+
+P-adic numbers were first introduced by Hensel in 1897, and over the following century a powerful theory was developed around them with numerous applications across a variety of disciplines, primarily in number theory, but also in physics and computer science. In 1965 Ax, Kochen and Ershov axiomatized the theory of p-adic numbers and proved a quantifier elimination result. A key insight was to connect properties of the value group and residue field to the properties of the valued field itself. In 1984 Denef proved a cell decomposition result for more general valued fields. This result was soon generalized to p-adic subanalytic and rigid analytic extensions, allowing for the later development of a more powerful technique of motivic integration. The conjunction of those model theoretic results allowed to solve a number of outstanding open problems in number theory (e.g., Artin's Conjecture on p-adic homogeneous forms).
+
+In 1997, Karpinski and Macintyre computed VC-density bounds for o-minimal structures and asked about similar bounds for p-adic numbers. VC-density is a concept closely related to VC-dimension. It comes up naturally in combinatorics with relation to packings, Hamming metric, entropic dimension and discrepancy.  VC-density is also the decisive parameter in the Epsilon-Approximation Theorem, which is one of the crucial tools for applying VC theory in computational geometry. In a model theoretic setting it is computed for families of uniformly definable sets.  In 2013, Aschenbrenner, Dolich, Haskell, Macpherson, and Starchenko computed a bound for VC-density in p-adic numbers and a number of other NIP structures \cite{density}. They observed connections to dp-rank and dp-minimality, notions first introduced by Shelah. In well behaved NIP structures families of uniformly definable sets tend to have VC-density bounded by a multiple of their dimension, a simple linear behavior. In a lot of cases including p-adic numbers this bound is not known to be optimal. My research concentrates on improving those bounds and adapting those techniques to compute VC-density in other common NIP structures of interest to mathematicians.
+
+Some of the other well behaved NIP structures are Shelah-Spencer graphs and flat graphs. Shelah-Spencer graphs are limit structures for random graphs arising naturally in a combinatorial context. Their model theory was studied by Baldwin, Shi, and Shelah in 1997, and later work of Laskowski in 2006 \cite{graph} have provided a quantifier simplification result.  Flat graphs were first studied by Podewski-Ziegler in 1978, showing that those are stable \cite{stable}, and later results gave a criterion for super stability. Flat graphs also come up naturally in combinatorics in work of Nesetril and Ossona de Mendez \cite{nowhere}. %Shelah-Spencer graphs and flat graphs are both subclasses of NIP theories, extremely well behaved model theoretically.
+
+\subsection*{Research Plan}
+
+
+The first chapter of my dissertation concentrates on Shelah-Spencer graphs. I have shown that they have infinite dp-rank, so they are poorly behaved as NIP structures. I have also shown that one can obtain arbitrarily high VC-density when looking at uniformly definable families in a fixed dimension. However I'm able to bound VC-density of individual formulas in terms of edge density of the graphs they define.
+
+The second chapter of my dissertation concentrates on graphs and graph-like structures. I have answered an open question from \cite{density}, computing VC-density for trees viewed as a partial order. The main idea is to adapt a technique of Parigot \cite{trees} to partition trees into weakly interacting parts, with simple bounds of VC-density on each. Similar partitions come up in the Podewski-Ziegler analysis of flat graphs \cite {stable}. I am able to use that technique to show that flat graphs are dp-minimal, an important first step before establishing bounds on VC-density. The first of my remaining research goals is to apply this partition to compute VC-densities for specific families of flat graphs.
+
+The third chapter of my dissertation deals with p-adic numbers and valued fields. I have shown that VC-density is linear for an additive reduct of p-adic numbers (using a cell decomposition result from the work of Leenknegt in 2013 \cite{reduce}). I will explore other reducts described in that paper, to see if my techniques apply to those as well. I have also shown that a Henselian valued field of equi-characteristic zero has linear one-dimensional VC-density if its value group and its residue field have that property. This is along the lines of the results of Ax-Kochen mentioned before. The second of my remaining research goals is to adapt those techniques to higher dimensions, as well as applying them to RV sorts introduced by Flenner in 2011 \cite{value}.
+
+My dissertation will therefore consist of VC-density computations for partial order trees, Shelah-Spencer graphs, flat graphs, and various valued fields, as well as any additional applications I am able to find after discussion with my advisor and my colleagues.
+
+\subsection*{Research Timeline}
+
+I propose a start date of October 2016.
+
+\begin{itemize}
+	\item March through September 2016: I will prepare and submit papers on my results for trees and Shelah-Spencer graphs. I will research families of flat graphs to see which of my techniques apply in that setting. I will generalize my result for valued fields from one dimension to multiple dimensions. I will also use this time to attend conferences to discuss my results with other mathematicians and get advice on further applications of my research.
+
+\item October 2016: I will research p-adic number reducts and RV sorts to see if my valued field techniques apply.
+
+\item November 2016: I will prepare and submit a paper containing my results for p-adic numbers and valued fields. %I will continue exploring other valued field constructions to see if my techniques would apply to those as well.
+
+\item December 2016: I will write an introduction to my thesis, defining VC-density and summarizing known results and computations.
+
+\item January 2017: I will write the first chapter of my thesis on Shelah-Spencer graphs.
+
+\item February 2017: I will write the second chapter of my thesis on trees and flat graphs.
+
+\item March 2017: I will write the third and final chapter of my thesis on p-adic numbers and valued fields. At the end of the month I will submit the thesis to my advisor.
+
+\item April 2017: I will make revisions to my thesis suggested by my advisor and submit the thesis to my committee. I will start preparing for the defense.
+
+\item May 2017: I will implement revisions to my thesis given by my committee and resubmit the final version. I will complete the defense.
+\end{itemize}
+
+\begin{thebibliography}{9}
+
+\bibitem{density}
+	M. Aschenbrenner, A. Dolich, D. Haskell, D. Macpherson, S. Starchenko,
+	\textit{Vapnik-Chervonenkis density in some theories without the independence property}, I, preprint (2011)
+
+\bibitem{regularity}
+Artem Chernikov, Sergei Starchenko, \textit{Regularity lemma for distal structures}, 	arXiv:1507.01482
+
+\bibitem{value}
+Joseph Flenner. \textit{Relative decidability and definability in Henselian valued fields.} The
+Journal of Symbolic Logic, 76(04):1240–1260, 2011.
+	
+\bibitem{graph}
+  Michael C. Laskowski, \textit{A simpler axiomatization of the Shelah-Spencer almost sure theories},
+  Israel J. Math. \textbf{161} (2007), 157–186. MR MR2 350161	
+
+\bibitem{reduce}
+E. Leenknegt. \textit{Reducts of p-adically closed fields}, Archive for Mathematical Logic \textbf{20 pp.}
+
+\bibitem{nowhere}
+J. Nesetril and P. Ossona de Mendez. \textit{On nowhere dense graphs.} European Journal of
+Combinatorics, 32(4):600–617, 2011
+
+\bibitem{trees}
+	Michel Parigot.
+	Th\'eories d'arbres.
+	\textit{Journal of Symbolic Logic}, 47, 1982. 
+
+
+\bibitem{stable}
+	Klaus-Peter Podewski and Martin Ziegler. Stable graphs. \textit{Fund. Math.}, 100:101-107, 1978.
+
+\end{thebibliography}
+	
+\end{document}
+
+% Include edges between y as a chain minimal extension
\ No newline at end of file
diff --git a/dissertation your fellowship/Shelah-Spencer VC - 2 - backup.tex b/dissertation your fellowship/Shelah-Spencer VC - 2 - backup.tex
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@@ -0,0 +1,244 @@
+\documentclass{amsart}
+
+\usepackage{../AMC_style}	
+\usepackage{../Research}
+\usepackage{../Thm}
+
+\usepackage{mathrsfs}
+
+
+
+\renewcommand{\AA}{\mathscr A}
+
+  \newcommand{\A}{\mathcal A}
+  \newcommand{\B}{\mathcal B}
+\renewcommand{\C}{\mathcal C}
+  \newcommand{\D}{\mathcal D}
+\renewcommand{\H}{\mathcal H}
+  \newcommand{\G}{\mathcal G}
+  \newcommand{\M}{\mathcal M}
+
+  \newcommand{\U}{\mathcal U}	
+
+  \newcommand{\K}{\boldface K_\alpha}
+\renewcommand{\S}{S_\alpha}
+
+\newcommand{\curly}[1]{\left\{#1\right\}}
+\newcommand{\paren}[1]{\left(#1\right)}
+\newcommand{\abs}[1]{\left|#1\right|}
+
+\providecommand{\floor}[1]{\left \lfloor #1 \right \rfloor }
+
+%\DeclareMathOperator{\dim}{dim}
+
+\title{Some vc-density computations in Shelah-Spencer graphs}
+\author{Anton Bobkov}
+\email{bobkov@math.ucla.edu}
+
+\begin{document}
+
+\maketitle
+
+Fix a formula $\phi(x, y)$ that is a minimal chain extension $\curly{M_i}_{i \in [0..k]}$ with $M_0 = \{x, y\}$ with
+\begin{itemize}
+	\item $\phi(x, y)$ determines edges and non-edges on $\curly{x, y}$.
+	\item there are no edges between $x$ and $y$.
+	\item there are no edges between $x$.
+	\item Let $\dim \paren{M_i/M_{i-1}} = -\epsilon_i$.
+	\item Let $\epsilon_L = \sum_{[1..k]} \epsilon_i$.
+	\item Let $\epsilon_U = \min_{[1..k]} \epsilon_i$.
+	\item Let $Y = \dim (y)$ considering $y$ as a graph. If $\{y\}$ are disconnected then $Y = |y|$.
+\end{itemize}
+
+We work in special family of parameter sets
+\begin{align*}
+	\AA = \curly{A \subset \U^{y} \mid \text{finite, disconnected, strongly embedded}}
+\end{align*}
+
+We estimate $\vc_\AA(\phi)$, VC-density of $\phi$ restricted to parameter sets from $\AA$.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section*{Lower bound on $\AA$}
+
+Let $n$ be the integer such that $n \epsilon_L < Y$ and $(n+1) \epsilon_L > Y$.
+
+Pick a finite $B \subset A^{|x|}$.
+
+Consider the graph $y$.
+If $y$ is not positive, then $\phi$ has no realizations over $B$.
+Otherwise, take an abstract realization of $y$, and label it by $b$.
+
+Fix $n$ arbitrary elements of $B$, label them $a_i$, with each $|a_i| = |x|$.
+Abstractly adjoin $M_i/\curly{a_i, b} = M/\curly{x,y}$ for each $i$.
+Let $\bar M = \bigcup M_i$ (disjointly).
+
+\begin{Claim}
+	$(A \cap \bar M) \leq \bar M$.
+\end{Claim}
+\begin{proof}
+	It's total dimension is $Y - n\epsilon_L > 0$ and all subextensions are positive as well.
+\end{proof}
+
+Thus a copy of $\bar M$ can be embedded over $A$ into our ambient model.
+Choice of elements of $B$ was arbitrary, thus showing that any $n$ elements can be traced out.
+Thus we have $O(|B|^n)$ many traces showing $\vc$-density of at least $n$.
+
+\begin{align*}
+	\vc_\AA(\phi) \geq \floor{\frac{Y}{\epsilon_L}}
+\end{align*}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section*{Lower bound}
+
+\begin{Claim}
+	In any random graph we can find arbitrarily large parameter set that belongs to $A$.
+\end{Claim}
+
+This shows that
+
+\begin{align*}
+	\vc(\phi) \geq \vc_\AA(\phi) \geq \floor{\frac{Y}{\epsilon_L}}
+\end{align*}
+
+\begin{Claim}
+	We can find a minimal extension $M / \{x, y\}$ with arbitrarily small dimension.
+\end{Claim}
+
+This shows that vc function is infinite in Shelah-Spencer random graphs.
+
+\begin{align*}
+	\vc(n) = \infty
+\end{align*}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section*{Upper bound on $\AA$}
+
+Let $n$ be the integer such that $n \epsilon_U < Y$ and $(n+1) \epsilon_U > Y$.
+
+Pick a trace of $\phi(x,y)$ on $A^{|x|}$ by a parameter $b$.
+
+\begin{align*}
+	B = \curly{a \in A^{|x|} \mid \phi(a, b)}
+\end{align*}
+
+Pick $B' \subset B$, ordered $B' = \{a_i\}_{i \in I}$ such that
+\begin{align*}
+	%a_i \cap \bigcup_{j \neq i} a_j \neq \emptyset
+	a_i \cap \bigcup_{j < i} a_j \neq \emptyset
+\end{align*}
+This is always possible by starting with $B$ and taking away elements one by one.
+Call such a set a \emph{generating set} of $B$.
+
+Let $M_i / \{a_i, b\}$ be a witness of $\phi(a_i, b)$ for each $i \in I$.
+Let $\bar M = \bigcup M_i$.
+Consider $\bar M / A$.
+
+Pick $\bar M$ such that $\dim(\bar M / A)$ is maximized.
+
+$\bar M \cap A \leq \bar M$ as $A$ is strong. (Make sure $M$ is not too big!)
+Let $\bar A = A - \curly{a_i}_{i \in I}$.
+Suppose $\bar A \cap \bar M \neq \emptyset$.
+Then we can abstractly reembed $\M$ over $A$ such that $\bar A \cap \bar M = \emptyset$.
+This would increase the dimension, contradicting maximality.
+Thus we can assume $A \cap \bar M = \{a_i\}_{i \in I}$
+
+Let $\bar M_j = \bigcup_{i < j} M_i$.
+
+\begin{Lemma}
+	$\dim(\bar M_j / A) \leq j \cdot \epsilon_U$
+\end{Lemma}
+\begin{proof}
+	Proceed by induction.
+	Base case is clear.
+
+	For induction case apply lemma to $\bar M_j \cup \{a_j\}$ and $M_j / \{a_j, b\}$.
+	There are two cases
+	\begin{enumerate}
+		\item $M_j \subset \bar M' \cup \{a_j\}$.
+		In this case there are edges between $\{a_j\}$ and $M_j$ that contribute to dimension less than $-\epsilon_U$.
+		\item Otherwise $M_j$ adds extra dimension less than $-\epsilon_U$
+	\end{enumerate}
+\end{proof}
+
+Thus we have $\dim(\bar M / A) = \dim(\bar M_n / A) \leq -\epsilon_U n$.
+
+Thus as $A$ is strong we need $|B'| \epsilon_U < Y$.
+This gives us $|B'| \leq n$.
+Finally we need to relate $|B'|$ to $|B|$.
+
+Suppose we have $C \subset A^{|x|}$, finite with $|C| = N$.
+A generating set for a trace has to have size $\leq n$.
+Thus there are ${N \choose n} \leq N^n$ choices for a generating set.
+A set generated from set of size $n$ can have at most $(x|n|)^{|x|}$ elements.
+Thus a given set of size $n$ can generate at most
+\begin{align*}
+	2^{(x|n|)^{|x|}}
+\end{align*}
+sets.
+Thus the number of possible traces on $C$ is bounded above by
+\begin{align*}
+  2^{(x|n|)^{|x|}} \cdot N^n = O(N^n)
+\end{align*}
+This bounds the vc-density by $n$.
+
+\begin{align*}
+	\vc_\AA(\phi) \geq \floor{\frac{Y}{\epsilon_U}}
+\end{align*}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section*{Technical Lemmas}
+
+\begin{Lemma}
+	Suppose we have a set $B$ and a minimal pair $(M, A)$ with $A \subset B$ and $\dim(M/A) = -\epsilon$.
+Then either $M \subseteq B$ or $\dim((M \cup B)/B) < -\epsilon$.
+\end{Lemma}
+
+\begin{proof}
+	By diamond construction
+
+	\begin{align*}
+		\dim((M \cup B)/B) \leq \dim(M / (M \cap B))
+	\end{align*}
+
+	and 
+
+	\begin{align*}
+		\dim(M / (M \cap B)) &= \dim (M/A) - \dim(M / (M \cap B)) \\
+		\dim (M/A) &= -\epsilon \\
+		\dim(M / (M \cap B)) &> 0
+	\end{align*}
+\end{proof}
+
+
+
+\begin{Lemma}
+	Suppose we have a set $B$ and a minimal chain $M_n$ with $M_0 \subset B$ and dimensions $-\epsilon_i$.
+Let $\epsilon$ be the minimal of $\epsilon_i$.
+Then either $M_n \subseteq B$ or $\dim((M_n \cup B)/B) < -\epsilon$.
+\end{Lemma}
+
+
+\begin{proof}
+	Let $\bar M_i = M_i \cup B$
+
+	\begin{align*}
+		\dim(\bar M_n/B) = \dim(\bar M_n/\bar M_{n-1}) + \ldots + \dim(\bar M_2/\bar M_1) + \dim(\bar M_1/B)
+	\end{align*}
+
+	Either $M_n \subseteq B$ or one of the summands above is nonzero.
+	Apply previous lemma.
+\end{proof}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section*{Counterexamples}
+
+% Add complete graph counterexample
+% where we have a bunch of minimal extensions intersecting in a tiny way
+
+% \AA is indiscernible
+% example of indiscernible sequence that is not strong
+% example with non-strong embedding on every n-tuple of vertices?
+
+\end{document}
+
+% Include edges between y as a chain minimal extension
\ No newline at end of file
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+++ b/dissertation your fellowship/Shelah-Spencer VC - 2.tex	
@@ -0,0 +1,759 @@
+\documentclass{amsart}
+
+\usepackage{../AMC_style}	
+\usepackage{../Research}
+\usepackage{../Thm}
+
+\usepackage{mathrsfs}
+
+
+
+\renewcommand{\AA}{\mathscr A}
+  \newcommand{\II}{\mathscr I}
+  \newcommand{\MM}{\mathscr M}
+
+  \newcommand{\A}{\mathcal A}
+  \newcommand{\B}{\mathcal B}
+\renewcommand{\C}{\mathcal C}
+  \newcommand{\D}{\mathcal D}
+\renewcommand{\H}{\mathcal H}
+  \newcommand{\G}{\mathcal G}
+  \newcommand{\M}{\mathcal M}
+  \newcommand{\U}{\mathcal U}	
+  \newcommand{\X}{\mathcal X}
+  \newcommand{\Y}{\mathcal Y}
+
+  \newcommand{\K}{\boldface K_\alpha}
+\renewcommand{\S}{S_\alpha}
+
+\newcommand{\curly}[1]{\left\{#1\right\}}
+\newcommand{\paren}[1]{\left(#1\right)}
+\newcommand{\abs}[1]{\left|#1\right|}
+\newcommand{\agl}[1]{\left\langle #1 \right\rangle}
+
+\providecommand{\floor}[1]{\left \lfloor #1 \right \rfloor }
+
+%\DeclareMathOperator{\dim}{dim}
+
+\title{Some vc-density computations in Shelah-Spencer graphs}
+\author{Anton Bobkov}
+\email{bobkov@math.ucla.edu}
+
+\begin{document}
+
+\maketitle
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Preliminaries}
+
+VC density was introduced in \cite{vc_density} by Aschenbrenner, Dolich, Haskell, MacPherson, and Starchenko as a natural notion of dimension for NIP theories. In a NIP theory we can define a VC function
+
+\begin{align*}
+	\vc : \N \arr \N
+\end{align*}
+
+Where $vc(n)$ measures complexity of definable sets in an $n$-dimensional space. Simplest possible behavior is $\vc(n) = n$ for all $n$. Theories with that property are known to be dp-minimal, i.e. having the smallest possible dp-rank. In general, it is not known whether there can be a dp-minimal theory which doesn't satisfy $\vc(n)=n$.
+
+In this paper, we investigate vc-density of definable sets in Shelah-Spencer structures.
+We follow notations in \cite{laskowski}.
+In this paper we work with limit of random structure $G(n, n^{-\alpha})$ for $\alpha \in (0,1)$, irrational.
+This structure is axiomatized by $S_\alpha$.
+Our ambient model is $\MM$.
+Notations we use are $\delta(\A), \delta(\A/\B), \A \leq \B$ as well as notions of $N$-strong substructure, minimal extension, chain minimal extension, minimal pair, and $N$-strong closure.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Graph Combinatorics}
+
+We denote graph by $\A$, set of its vertices by $A$.
+When we say $\A \subseteq \B$ we mean that $A \subseteq B$ and edges of $\A$ are also edges of $\B$.
+However $\B$ may add new edges between vertices of $\A$.
+
+Fix $\alpha \in (0,1)$, irrational.
+For a finite graph $\A$ let
+\begin{align*}
+	\delta(\A) = |A| - \alpha e(\A)
+\end{align*}
+
+where $e(\A)$ is the number of edges in $\A$.
+
+For finite $\A,\B$ with $\A \subseteq \B$ define $\delta(\B/\A) = \delta(\B) - \delta(\A)$.
+We say that $\A \leq \B$ if $\A \subseteq \B$ and $\delta(\A'/\B) > 0$ for all $\A \subseteq \A' \subsetneq \B$.
+
+We say that finite $\A$ is positive if for all $\A' \subseteq \A$ we have $\delta(\A') \geq 0$.
+
+\begin{Definition}
+	We work in theory $S_\alpha$ axiomatized by
+	\begin{itemize}
+		\item Every finite substructure is positive
+		\item For a model $\MM$ given $\A \leq \B$ every embedding $f : \A \arr \MM$ extends to $g: \B \arr \MM$.
+	\end{itemize}
+\end{Definition}
+
+For $\A, \B$ positive, $(\A, \B)$ is called a minimal pair if $\A \subseteq \B$, $\delta(\B/\A) < 0$ but $\delta(\A'/\A) \geq 0$ for all proper $\A \subseteq \A' \subsetneq \B$.
+
+$\agl{\A_i}_{i \leq m}$ is called a minimal chain if $(\A_i, \A_i+1)$ is a minimal pair (for all $i < m$).
+
+For a positive $\A$ let $\delta_\A(\bar x)$ be the atomic diagram of $\A$. For positive $\A \subset \B$ let 
+
+\begin{align*}
+	\Psi_{\A,\B}(\bar x) = \delta_\A(\bar x) \wedge \exists \bar y \; \delta_\B(\bar x, \bar y)
+\end{align*}
+
+Such formula is called chain-minimal extension formula if in addition we have that there is a minimal chain starting at $\A$ and ending in $\B$. Denote such formulas as $\Psi_{\agl{\M_i}}$
+
+\begin{Theorem} [5.6 in \cite{laskowski}]
+	$S_\alpha$ admits quantifier elimination down to boolean combination of chain-minimal extension formulas.
+\end{Theorem}
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Definitions}
+
+%\begin{Definition}
+	%Let $x = (x_1, \ldots x_n), y = (y_1, \ldots, y_m)$ be variable tuples.
+	%We call a formula $\phi(x, y)$ \emph{basic} when
+	%\begin{itemize}
+		%\item $\phi(x, y)$ is a minimal chain extension, denoted by $\curly{M_i}_{i \in [0..k]}$ with $M_0 = \{x, y\}$.
+		%\item $\phi(x, y)$ determines edges and non-edges on its variables $\{x_1, \ldots x_n\} \cup \{y_1, \ldots y_m\}$.
+		%\item there is no edge between $x_i$ and $y_j$ for all $i,j$. (see note \ref{note_edges})
+		%\item Define $\mathbf x$ to be the graph on vertices $\{x_i\}$ with edges as defined by $\phi$.
+		%Similarly define $\mathbf y$.
+		%We require $\mathbf x$ and $\mathbf y$ to be positive. (see note \ref{note_positive})\
+		%%\item all elements of $y$ that are connected to $M_k - \{x,y\}$. (see note \ref{note_special})
+	%\end{itemize}
+%\end{Definition}
+
+% issue: multiple possible chain decompositions?
+
+Fix tuples $x = (x_1, \ldots x_n), y = (y_1, \ldots, y_m)$.
+We refer to chain-minimal extension formulas as basic formulas.
+Let $\phi_{\agl{\M_i}}(x, y)$ be a basic formula.
+
+\begin{Definition}
+	Define $\X$ to be the graph on vertices $\{x_i\}$ with edges as defined by $\phi_{\agl{\M_i}}$.
+	Similarly define $\Y$.
+	We define those abstractly, i.e. on a new set of vertices disjoint from $\MM$.
+\end{Definition}
+
+Note that $\X$, $\Y$ are positive as they are subgraphs of $\M_0$.
+As usual $X, Y$ will refer to vertices of those graphs.
+
+We restrict our attention to formulas that define no edges between $X$ and $Y$.
+
+\begin{Note} \label{note_edges}
+	We can handle edges between $x$ and $y$ as separate elements of the minimal chain extension.
+\end{Note}
+
+%\begin{Note} \label{note_positive}
+	%If either graph $\mathbf x$ or $\mathbf y$ is negative, then the formula would have no realizations as negative graphs cannot be embedded into our ambient model.
+%\end{Note}
+
+%\begin{Note} \label{note_special}
+	%We add the final condition to simplify our analysis. Similar techniques can be used to acquire bounds on formulas not subject to that condition.
+%\end{Note}
+
+\begin{Definition} \label{def_basic}
+	For a basic formula $\phi = \phi_{\agl{\M_i}_{i \leq k}}(x, y)$ let
+	\begin{itemize}
+		\item $\epsilon_i(\phi) = -\dim \paren{M_i/M_{i-1}}$.
+		\item $\epsilon_L(\phi) = \sum_{[1..k]} \epsilon_i(\phi)$.
+		\item $\epsilon_U(\phi) = \min_{[1..k]} \epsilon_i(\phi)$.
+		\item Let $\Y'$ be a subgraph of $\Y$ induced by vertices of $\Y$ that are connected to $M_k - (X \cup Y)$.
+		\item Let $Y(\phi) = \dim (\Y')$.
+		In particular if $\Y = \Y'$ and $\Y$ is disconnected then $Y(\phi)$ is just the arity of the tuple $y$.
+	\end{itemize}
+\end{Definition}
+
+%\begin{Definition}
+%\begin{align*}
+	%\epsilon_L(\neg \phi) &= \epsilon_L(\phi) \\
+	%\epsilon_U(\neg \phi) &= \epsilon_U(\phi) \\
+	%\epsilon_L(\phi \wedge \psi) &= \epsilon_L(\phi) + \epsilon_L(\psi) \\
+	%\epsilon_U(\phi \wedge \psi) &= \min(\epsilon_U(\phi), \epsilon_U(\psi)) \\
+	%\epsilon_L(\phi \vee \psi) &= \min(\epsilon_L(\phi), \epsilon_L(\psi)) \\
+	%\epsilon_U(\phi \vee \psi) &= \min(\epsilon_U(\phi), \epsilon_U(\psi)) 
+%\end{align*}
+%\end{Definition}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Lower bound}
+
+As a simplification for our lower bound computation we assume that all the basic formulas involved we have $\Y' = \Y$ (see Definition \ref{def_basic}).
+
+We work with formulas that are boolean combinations of basic formulas written in disjunctive-conjunctive form.
+First, we extend our definition of $\epsilon$.
+
+\begin{Definition}[Negation]
+	If $\phi$ is a basic formula, then define
+	\begin{align*}
+		\epsilon_L(\neg \phi) &= \epsilon_L(\phi)
+	\end{align*}
+\end{Definition}
+
+\begin{Definition}[Conjunction]
+	Take a collection of formulas $\phi_i(x, y)$ where each $\phi_i$ is positive or negative basic formula.
+	If both positive and negative formulas are present then $\epsilon_L(\phi) = \infty$.
+	We don't have a lower bound for that case.
+	If different formulas define $\X$ or $\Y$ differently then $\epsilon_L(\phi) = \infty$.
+	In that case of the conflicting definitions would make the formula have no realizations.
+	Otherwise
+	\begin{align*}
+		\epsilon_L(\bigwedge \phi_i) &= \sum \epsilon_L(\phi_i)
+	\end{align*}
+\end{Definition}
+
+\begin{Definition} [Disjunction]
+	Take a collection of formulas $\psi_i$ where each instance is a conjunction of positive and negative instances of basic formulas that agree on $\X$ and $\Y$. % can generalize?
+	\begin{align*}
+		\epsilon_L(\bigvee \psi_i) &= \min \epsilon_L(\psi_i)
+	\end{align*}
+\end{Definition}
+
+\begin{Theorem}
+	For a formula $\phi$ as above
+	\begin{align*}
+		\vc \phi \geq \floor{\frac{Y(\phi)}{\epsilon_L(\phi)}}
+	\end{align*}
+	where $Y(\phi)$ is $Y(\psi)$ for $\psi$ one the basic components of $\phi$ (all basic componenets agree on $\Y$).
+\end{Theorem}
+
+\begin{proof}
+	First work with a formula that is a conjunction of positive basic formulas.
+
+	\begin{align*}
+		\psi = \bigwedge_{j \leq J} \phi_j
+	\end{align*}
+	Then as we defined above
+	\begin{align*}
+		\epsilon_L(\psi) = \sum \epsilon_L(\phi_j)
+	\end{align*}
+	
+	Let $\phi$ be one of the basic formulas in $\psi$ with a chain $\agl{M_i}_{i \leq k}$.
+	Let $K_\phi = |M_k|$ i.e. the size of the extension.
+	Let $K$ be the largest such size among all $\phi_i$.
+
+	Let $n$ be the integer such that $n \epsilon_L(\psi) < Y$ and $(n+1) \epsilon_L(\psi) > Y$.
+
+	%Take an abstract realization of $y$ as dictated by $\psi$, and label it by $b$ ($b \in \MM$)
+	Label $\Y$ by an tuple $b$.
+
+	Pick parameter set $A \subset \MM$ such that 
+
+	\begin{align*}
+		A = \bigcup_{i<N} b_i
+	\end{align*}
+
+	a disjoint union where each $b_i$ is an ordered tuple of size $|x|$ connected according to $\psi$.
+	We also require $A$ to be $N \cdot I \cdot K$-strong.
+
+	Fix $n$ arbitrary elements out of $A$, label them $a_j$.
+
+	For each $\phi_i$, $a_j$ pick an abstract realization $M_{ij}$ of $\phi_i$ over $(a_j, b)$
+	(abstract meaning disjoint from $\MM$).
+	
+	Let $\bar M$ be an abstract disjoint union of all those realizations.
+
+	\begin{Claim}
+		$(A \cap \bar M) \leq \bar M$.
+	\end{Claim}
+	\begin{proof}
+		Consider some $(A \cap \bar M) \subseteq B \subseteq \bar M$.
+		Let $B_{ij} = B \cap M_{ij} \subseteq M_{ij}$.
+		Then $B_{ij}$'s are disjoint over $\bar A = A \cup b$.
+		In particular $\dim B / (\bar A \cap B) = \sum \dim B_{ij} / (\bar A \cap B_{ij})$.
+		$\dim B_{ij} / \bar A \geq -\epsilon_L(\phi_i)$ by Lemma \ref{chain_intersect}.
+		Thus $\dim B / (\bar A \cap B) \geq -n\epsilon(\psi)$.
+		Thus $\dim B / (A \cap B) \geq \dim(B) - n\epsilon(\psi)$.
+		By construction we have $Y - n\epsilon_L(\psi) > 0$ as needed.
+	\end{proof}
+
+	$|\bar M| \leq N \cdot I \cdot K$ and $A$ is $\leq N \cdot I \cdot K$-strong.
+	Thus a copy of $\bar M$ can be embedded over $A$ into our ambient model $\MM$.
+	Our choice of $b_i$'s was arbitrary, so we get ${N \choose n}$ choices out of $N|x|$ many elements.
+	Thus we have $O(|A|^n)$ many traces.
+
+	\begin{Lemma}
+		There are arbitrarily large sets with properties of $A$.
+	\end{Lemma}
+	
+	\begin{proof}
+		$A$ is positive, as each of its disjoint components is positive. Thus $0 \leq A$.
+		We apply proposition 4.4 in Laskoswki paper, embedding $A$ into our structure $\MM$ while avoiding all nonpositive extensions of size at most $N \cdot I \cdot K$.
+	\end{proof}
+
+	This shows
+
+	\begin{align*}
+		\vc \psi \geq n = \floor{\frac{Y}{\epsilon_L}}
+	\end{align*}
+
+	Now consider the formula which is a conjunction consists of negative basic formulas
+	\begin{align*}
+		\psi = \bigwedge \neg \phi_i
+	\end{align*}
+	Let
+	\begin{align*}
+		\bar \psi = \bigwedge \phi_i
+	\end{align*}
+
+	Do the construction above for $\bar \psi$ and suppose its trace is $X \subset A$ for some $b$.
+	Then over $b$ the same construction gives trace $(A - X)$ for $\psi$. Thus we get as many traces.
+	
+	Finally consider a formula which is a disjunction of formulas considered above.
+	Choose the one with the smallest $\epsilon_L$, this yields the lower bound for the entire formula.
+	%explain! non-trivial. thing of disjunction of two formulas
+	% also! what if disjunction has formulas disagreeing on x and y
+	% aslo! what if disjuction where one of the formulas is mixed positive/negative formulas
+\end{proof}
+
+\begin{Claim}
+	We can find a minimal extension $M / \{x, y\}$ with arbitrarily small dimension.
+\end{Claim}
+
+\begin{proof}
+	Put proof in here. Follow construction in Laskowski paper.
+\end{proof}
+
+This shows that vc function is infinite in Shelah-Spencer random graphs.
+
+\begin{align*}
+	\vc(n) = \infty
+\end{align*}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Upper bound}
+
+
+%\begin{Definition}
+	%Let $\phi$ be a basic formula with $M_i$ a minimal chain,
+	%$\epsilon_i$ its corresponding dimensions,
+	%and $M$ its total size.
+	%\begin{align*}
+		%U_\phi = \frac{M}{\min \epsilon_i}
+	%\end{align*}
+%\end{Definition}
+
+
+
+%\begin{Definition} [Negation]
+	%Let $\phi$ be basic
+	%\begin{align*}
+		%U_{\neg \phi} = U_{\phi}
+	%\end{align*}
+%\end{Definition}
+%
+%\begin{Definition} [Conjunction and Disjunction]
+	%Let $\phi_{ij}$ be basic or a negation of a basic formula.
+	%\begin{align*}
+		%\psi = \bigvee \bigwedge \phi_{ij}
+	%\end{align*}
+	%\begin{align*}
+		%U_\psi = \max U_{\phi_{ij}}
+	%\end{align*}
+%\end{Definition}
+
+%Let $n$ be the integer such that $n \epsilon_U < Y$ and $(n+1) \epsilon_U > Y$.
+
+%Consider a formula which is a conjunction of positive basic formulas.
+
+%Consider a case of a single basic formula $\phi(\vec x, \vec y)$.
+
+We bound the number of types of some finite collection of formulas $\Psi(\vec x, \vec y) = \curly{\phi_i(\vec x, \vec y)}_{i\in I}$ over a parameter set $B$ of size $N$,
+where $\phi_i$ is a basic formula.
+
+Fix a formula $\phi$ from our collection.
+Suppose it defines a minimal chain extension over $\{x, y\}$. 
+Record the size of that extension as $K(\phi)$ and its total dimension $\epsilon(\phi) = \epsilon_U(\phi)$.
+Define dimension of that formula $D(\phi) = |\vec y| \frac{K(\phi)}{\epsilon(\phi)}$
+Define dimension of the entire collection as $D(\Psi) = \max_{i \in I} D(\phi_i)$
+
+In general we have parameter set $B \subset \MM^{|y|}$, however without loss of generality we may work with
+a parameter set $B^{|y|}$, with $B \subset \MM$.
+
+Let $S = \floor{D(\Psi)}$.
+
+For our proof to work we also need $B$ to be $S$-strong.
+We can achieve this by taking (the unique) $S$-strong closure of $B$.
+If size of $B$ is $N$ then the size of its closure is $O(N)$.	%elaborate
+So without loss of generality we can assume that $B$ is $S$-strong.
+
+\begin{Definition}
+	A \emph{witness} of $a$ is a union of realizations of the existential formulas $\phi_i(a, b)$ for all $i, b$ so that the formula holds.
+\end{Definition}
+
+
+ % what about A \cap b???
+
+\begin{Definition}
+	For sets $C, B$ define the boundary of $C$ over $B$
+	\begin{align*}
+		\partial(C, B) = \curly{b \in B \mid \text{there is an edge between $b$ and element of $C - B$}}
+	\end{align*}
+\end{Definition}
+
+\begin{Definition}
+	For each $a$ pick some $\bar M_a$ to be its witness.
+	Define two quantities
+	\begin{itemize}
+		\item $\partial_a$ is the boundary $\partial(\bar M_a, B \cup a)$
+		\item Suppose $G_1, G_2$ are some subgraphs of our model and $a_1 \subset G_1, a_2 \subset G_2$ finite tuples of vertices.
+		Call $f \colon (G_1, a_1) \arr (G_2, a_2)$ a $\partial$-isomorphism if it is a graph isomorphism,
+		$f$ and $f^{-1}$ are constant on $B$, and
+		$f(a_1) = a_2$.
+		\item Define $\II_a$ as the $\partial$-isomorphism class of $(\bar M_a, a)$.
+	\end{itemize}
+\end{Definition}
+
+\begin{Lemma} \label {bound_trace}
+	If $\II_{a_1} = \II_{a_2}$ then $a_1, a_2$ have the same $\Psi$-type over $B$.
+\end{Lemma}
+
+\begin{proof}
+	Fix a $\partial$-isomorphism $f \colon (\bar M_{a_1}, a_1) \arr (\bar M_{a_1}, a_2)$.
+	Suppose we have $\phi(a_1, b)$ for some $b \in B$.
+	Pick witness of this existential formula $M_1 \subset \bar M_{a_1}$.
+	Then $f(M_1)$ is a witness for  $\phi(a_2, b)$.
+\end{proof}
+
+Thus to bound the number of traces it is sufficient to bound the number of possibilities for $\II_a$.
+
+\begin{Theorem} \label{main_bound}
+	\begin{align*}
+		|\partial_a| &\leq D(\Psi) \\ 
+		|\bar M_b - \bar A| &\leq D(\Psi)
+	\end{align*}
+\end{Theorem}
+
+\begin{Corollary}
+	\begin{align*}
+		\vc(\phi) \leq K(\phi) \frac{Y(\phi)}{\epsilon(\phi)}
+	\end{align*}
+\end{Corollary}
+
+\begin{proof}
+		We count possible $\partial$-isomorphism classes $\II_b$.
+		Let $W = K(\phi) \frac{Y(\phi)}{\epsilon(\phi)}$.
+		If the parameter set $A$ is of size $N$ then there are $N \choose W$ choices for boundary $\partial_b$.
+		On top of the boundary there are at most $W$ extra vertices and $(2W)^2$ extra edges.
+		Thus there are at most
+		\begin{align*}
+			W \cdot 2^{(2W)^2}
+		\end{align*}
+		configurations up to a graph isomorphism.
+		In total this gives us 
+		\begin{align*}
+			{N \choose W} \cdot W \cdot 2^{(2W)^2} = O(N^W)
+		\end{align*}
+		options for $\partial$-isomorphism classes.
+		By Lemma \ref{bound_trace} there are at most $O(N^W)$ many traces, giving the required bound.
+\end{proof}
+
+\begin{proof} \textit{(of Theorem \ref{main_bound})}
+	Fix some $b$-trace $A_b$. Enumerate $A_b = \{a_1, \ldots, a_I\}$.
+
+	Let $M_i / \{a_i, b\}$ be a witness of $\phi(a_i, b)$ for each $i \leq I$.
+	Let $\bar M_i = \bigcup_{j < i} M_j$.
+	Let $\bar M = \bigcup M_i$, a witness of $A_b$
+	
+	\begin{Claim}
+		\begin{align*}
+			&\abs{\partial(M_i M, \bar A) - \partial(M, \bar A)} \leq |M_i| = K(\phi)\\
+			&\dim(M_i M \bar A / M \bar A) > -\epsilon(\phi)
+		\end{align*}
+	\end{Claim}
+	
+	\begin{Definition}
+		$(j-1, j)$ is called a \emph{jump} if some of the following conditions happen
+		\begin{itemize}
+			\item New vertices are added outside of $\bar A$ i.e.
+				\begin{align*}
+					\bar M_j - \bar A \neq \bar M_{j-1} - \bar A
+				\end{align*}
+			\item New vertices are added to the boundary, i.e.
+				\begin{align*}
+					\partial(\bar M_j, \bar A) \neq \partial(\bar M_{j-1}, \bar A)
+				\end{align*}
+		\end{itemize}
+	\end{Definition}
+
+	\begin{Definition}
+		We now let $m_i$ count all jumps below $i$
+		%Let $d_i = \dim(\bar M_i/A)$.
+		\begin{align*}
+			m_i = \abs{\curly{j < i \mid (j-1, j) \text{ is a jump}}}
+		\end{align*}
+	\end{Definition}
+
+	\begin{Lemma} \label{ub_lemma}
+		\begin{align*}
+			\dim(\bar M_i / \bar A) &\leq -m_i \cdot \epsilon(\phi) \\
+			|\partial(\bar M_i, \bar A)| &\leq m_i \cdot K(\phi) \\
+			|\bar M_j - \bar A| &\leq m_i \cdot K(\phi)
+		\end{align*}
+	\end{Lemma}
+
+	\begin{proof} \textit{(of Lemma \ref{ub_lemma})}
+		Proceed by induction.
+		Second and third propositions are clear.
+		For the first proposition base case is clear.
+		
+		Induction step.
+		Suppose $\bar M_j \cap (A \cup b) = \bar M_{j+1}$ and $\partial(\bar M_j, A) = \partial(\bar M_{j+1}, A)$.
+		Then $m_i = m_{i+1}$ and the quantities don't change.
+		Thus assume at least one of these equalities fails.
+		
+		Apply Lemma \ref{chain_lemma} to $\bar M_j \cup (A \cup b)$ and $(M_{j+1}, a_{j+1}b)$.
+		There are two options
+		
+		\begin{itemize}
+			\item $\dim(\bar M_{j+1} \cup (A \cup b) / \bar M_i \cup (A \cup b)) \leq -\epsilon_U$.
+			This implies the proposition.
+			\item $M_{j+1} \subset \bar M_j \cup (A \cup b)$.
+			Then by our assumption it has to be $\partial(\bar M_j, A) \neq \partial(\bar M_{j+1}, A)$.
+			There are edges between $M_{j+1} \cap (\partial(\bar M_{j+1}, A) - \partial(\bar M_j, A))$ so they contribute some negative dimension $\leq \epsilon_U$.
+		\end{itemize}
+		This ends the proof for Lemma \ref{ub_lemma}.
+	\end{proof}
+	\textit{(Proof of Theorem \ref{main_bound} continued)}
+	First part of lemma \ref{ub_lemma} implies that we have $\dim(\bar M / \bar A) \leq -m_I \cdot \epsilon(\phi)$.
+	The requirement of $A$ to be $S$-strong forces
+	\begin{align*}
+		m_I \cdot \epsilon(\phi) &< Y(\phi) \\
+		m_I  &< \frac{Y(\phi)}{\epsilon(\phi)} \\
+	\end{align*}
+	%Let $W = \frac{K(\phi)Y(\phi)}{\epsilon(\phi)}$
+	Applying the rest of \ref{ub_lemma} gives us
+	\begin{align*}
+		|\partial(\bar M, A)| &\leq m_I \cdot K(\phi) \leq \frac{K(\phi)Y(\phi)}{\epsilon(\phi)} \\
+		|\bar M \cap A| &\leq m_I \cdot K(\phi) \leq \frac{K(\phi)Y(\phi)}{\epsilon(\phi)}
+	\end{align*}
+	as needed.
+	This ends the proof for Theorem \ref{main_bound}.
+\end{proof}
+
+So far we have computed an upper bound for a single basic formula $\phi$.
+
+To bound an arbitrary formula, write it as a boolean combination of basic formulas $\phi_i$ (via quantifier elimination)
+It suffices to bound vc-density for collection of formulas $\{\phi_i\}$ to obtain a bound for the original formula.
+
+In general work with a collection of basic formulas $\{\phi_i\}_{i \in I}$.
+The proof generalizes in a straightforward manner.
+Instead of $A^{|x|}$ we now work with $A^{|x|} \times I$ separating traces of different formulas.
+Formula with the largest quantity $Y(\phi)\frac{K(\phi)}{\epsilon(\phi)}$ contributes the most to the vc-density.
+Thus we have
+\begin{align*}
+	\Phi &= \{\phi_i\}_{i \in I} \\
+	\vc(\Phi) &=  \max_{i \in I} Y(\phi_i) \frac{K(\phi_i)}{\epsilon_{\phi_i}}
+\end{align*}
+
+
+%\begin{Definition}
+	%\begin{align*}
+		%d &= \dim \bar M / \bar A \\
+		%s &= |\bar M - \bar A| \\
+		%b &= |\partial(\bar M, \bar A)|
+	%\end{align*}
+%\end{Definition}
+
+
+%Thus as we consider $\bar M$ as an increasing union of witnesses to chain-minimal extensions, we see the extension with the largest ratio can contribute most to the boundary.
+%Thus is our upper bound for the boundary.
+
+
+
+
+
+
+%Now, classify every trace by the isomorphism class of $\bar M - A \cup \partial(\bar M, A)$ and by $\partial(\bar M, A)$.
+%\begin{Lemma}
+	%Suppose we have traces $b_1, b_2$ with the same components as above.
+	%Then $A_{b_1} = A_{b_2}$.
+%\end{Lemma}
+%
+%Consider $\bar M - A \cup \partial(\bar M, A)$.
+%Number of vertices is $\leq (2W)^2$.
+%Thus number of isomorphism classes $\leq 2^{(2W)^2}$.
+%
+%Consider $\partial(\bar M, A)$.
+%Let $N = |A|$.
+%Order matters, so the total number of choices for it is
+%\begin{align*}
+	%N \cdot (N-1) \cdot \ldots \cdot (N - W + 1) = \frac{N!}{(N-W)!}
+%\end{align*}
+%
+%Thus the number of possible different traces is bounded by 
+%\begin{align*}
+	%2^{(2W)^2} \cdot \frac{N!}{(N-W)!} = O(N^W)
+%\end{align*}
+%
+%Since choice of $A$ was arbitrary, this gives
+%\begin{align*}
+	%\vc{\phi} \leq W = \frac{|M|Y}{\epsilon_U}
+%\end{align*}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Technical Lemmas}
+
+\begin{Lemma}
+	Suppose we have a set $B$ and a minimal pair $(M, A)$ with $A \subset B$ and $\dim(M/A) = -\epsilon$.
+Then either $M \subseteq B$ or $\dim((M \cup B)/B) < -\epsilon$.
+\end{Lemma}
+
+\begin{proof}
+	By diamond construction
+
+	\begin{align*}
+		\dim((M \cup B)/B) \leq \dim(M / (M \cap B))
+	\end{align*}
+
+	and 
+
+	\begin{align*}
+		\dim(M / (M \cap B)) &= \dim (M/A) - \dim(M / (M \cap B)) \\
+		\dim (M/A) &= -\epsilon \\
+		\dim(M / (M \cap B)) &> 0
+	\end{align*}
+\end{proof}
+
+
+
+\begin{Lemma}	\label{chain_lemma}
+	Suppose we have a set $B$ and a minimal chain $M_n$ with $M_0 \subset B$ and dimensions $-\epsilon_i$.
+Let $\epsilon$ be the minimal of $\epsilon_i$.
+Then either $M_n \subseteq B$ or $\dim((M_n \cup B)/B) < -\epsilon$.
+\end{Lemma}
+
+
+\begin{proof}
+	Let $\bar M_i = M_i \cup B$
+
+	\begin{align*}
+		\dim(\bar M_n/B) = \dim(\bar M_n/\bar M_{n-1}) + \ldots + \dim(\bar M_2/\bar M_1) + \dim(\bar M_1/B)
+	\end{align*}
+
+	Either $M_n \subseteq B$ or one of the summands above is nonzero.
+	Apply previous lemma.
+\end{proof}
+
+\begin{Lemma} \label{chain_intersect}
+	Suppose we have a minimal chain $M_n$ with dimensions $-\epsilon_i$.
+	Let $\epsilon$ be the sum of all $\epsilon_i$.
+	Suppose we have some $B$ with $B \subseteq M_n$.
+	Then $\dim B / (M_0 \cap B) \geq -\epsilon$.
+\end{Lemma}
+
+\begin{proof}
+	Let $B_i = B \cap M_i$.
+	We have $\dim B_{i+1}/B_i \geq \dim M_{i+1}/M_i$ by minimality.
+	$\dim B / (M_0 \cap B) = \dim B_n / B_0 = \sum \dim B_{i+1}/B_i \geq -\epsilon$.
+\end{proof}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Counterexamples}
+
+% Add complete graph counterexample
+% where we have a bunch of minimal extensions intersecting in a tiny way
+% or something similar?
+
+% \AA is indiscernible
+% example of indiscernible sequence that is not strong
+% example with non-strong embedding on every n-tuple of vertices?
+
+% 2 chain minimal extension that is larger than the lower bound
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Upper bound on $\AA$}
+
+\begin{Definition}
+	\begin{align*}
+		\AA = \curly{A \subset \U^{y} \mid \text{finite, disconnected, strongly embedded}}
+	\end{align*}
+\end{Definition}
+
+Let $n$ be the integer such that $n \epsilon_U < Y$ and $(n+1) \epsilon_U > Y$.
+
+Pick a trace of $\phi(x,y)$ on $A^{|x|}$ by a parameter $b$.
+
+\begin{align*}
+	B = \curly{a \in A^{|x|} \mid \phi(a, b)}
+\end{align*}
+
+Pick $B' \subset B$, ordered $B' = \{a_i\}_{i \in I}$ such that
+\begin{align*}
+	%a_i \cap \bigcup_{j \neq i} a_j \neq \emptyset
+	a_i \cap \bigcup_{j < i} a_j \neq \emptyset
+\end{align*}
+This is always possible by starting with $B$ and taking away elements one by one.
+Call such a set a \emph{generating set} of $B$.
+
+Let $M_i / \{a_i, b\}$ be a witness of $\phi(a_i, b)$ for each $i \in I$.
+Let $\bar M = \bigcup M_i$.
+Consider $\bar M / A$.
+
+Pick $\bar M$ such that $\dim(\bar M / A)$ is maximized.
+
+$\bar M \cap A \leq \bar M$ as $A$ is strong. (Make sure $M$ is not too big!)
+Let $\bar A = A - \curly{a_i}_{i \in I}$.
+Suppose $\bar A \cap \bar M \neq \emptyset$.
+Then we can abstractly reembed $\M$ over $A$ such that $\bar A \cap \bar M = \emptyset$.
+This would increase the dimension, contradicting maximality.
+Thus we can assume $A \cap \bar M = \{a_i\}_{i \in I}$
+
+Let $\bar M_j = \bigcup_{i < j} M_i$.
+
+\begin{Lemma}
+	$\dim(\bar M_j / A) \leq j \cdot \epsilon_U$
+\end{Lemma}
+\begin{proof}
+	Proceed by induction.
+	Base case is clear.
+
+	For induction case apply lemma to $\bar M_j \cup \{a_j\}$ and $M_j / \{a_j, b\}$.
+	There are two cases
+	\begin{enumerate}
+		\item $M_j \subset \bar M' \cup \{a_j\}$.
+		In this case there are edges between $\{a_j\}$ and $M_j$ that contribute to dimension less than $-\epsilon_U$.
+		\item Otherwise $M_j$ adds extra dimension less than $-\epsilon_U$
+	\end{enumerate}
+\end{proof}
+
+Thus we have $\dim(\bar M / A) = \dim(\bar M_n / A) \leq -\epsilon_U n$.
+
+Thus as $A$ is strong we need $|B'| \epsilon_U < Y$.
+This gives us $|B'| \leq n$.
+Finally we need to relate $|B'|$ to $|B|$.
+
+Suppose we have $C \subset A^{|x|}$, finite with $|C| = N$.
+A generating set for a trace has to have size $\leq n$.
+Thus there are ${N \choose n} \leq N^n$ choices for a generating set.
+A set generated from set of size $n$ can have at most $(x|n|)^{|x|}$ elements.
+Thus a given set of size $n$ can generate at most
+\begin{align*}
+	2^{(x|n|)^{|x|}}
+\end{align*}
+sets.
+Thus the number of possible traces on $C$ is bounded above by
+\begin{align*}
+  2^{(x|n|)^{|x|}} \cdot N^n = O(N^n)
+\end{align*}
+This bounds the vc-density by $n$.
+
+\begin{align*}
+	\vc_\AA(\phi) \geq \floor{\frac{Y}{\epsilon_U}}
+\end{align*}
+
+
+\begin{thebibliography}{9}
+
+\bibitem{vc_density}
+	M. Aschenbrenner, A. Dolich, D. Haskell, D. Macpherson, S. Starchenko,
+	\textit{Vapnik-Chervonenkis density in some theories without the independence property}, I, preprint (2011)
+	
+\bibitem{laskowski}
+  Michael C. Laskowski, \textit{A simpler axiomatization of the Shelah-Spencer almost sure theories},
+  Israel J. Math. \textbf{161} (2007), 157–186. MR MR2350161	
+
+\end{thebibliography}
+	
+\end{document}
+
+% Include edges between y as a chain minimal extension
diff --git a/dissertation your fellowship/Shelah-Spencer VC.pdf b/dissertation your fellowship/Shelah-Spencer VC.pdf
new file mode 100644
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diff --git a/dissertation your fellowship/Shelah-Spencer VC.tex b/dissertation your fellowship/Shelah-Spencer VC.tex
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+\documentclass{amsart}
+
+\usepackage{../AMC_style}	
+\usepackage{../Research}
+
+\usepackage{diagrams}
+
+  \newcommand{\A}{\mathcal A}
+  \newcommand{\B}{\mathcal B}
+\renewcommand{\C}{\mathcal C}
+  \newcommand{\D}{\mathcal D}
+\renewcommand{\H}{\mathcal H}
+  \newcommand{\G}{\mathcal G}
+  \newcommand{\M}{\mathcal M}
+
+  \newcommand{\K}{\boldface K_\alpha}
+\renewcommand{\S}{S_\alpha}
+
+\begin{document}
+
+\title{Some vc-density computations in Shelah-Spencer graphs}
+\author{Anton Bobkov}
+\email{bobkov@math.ucla.edu}
+
+\begin{abstract}
+	We compute vc-densities of minimal extension formulas in Shelah-Spencer random graphs.
+\end{abstract}
+
+\maketitle
+
+We fix the density of the graph $\alpha$.
+
+\begin{Lemma}
+	For any $\A \in \K$ and $\epsilon > 0$ there exists an $\B$ such that $(\A, \B)$ is minimal and $\delta(\B/\A) < \epsilon$.
+\end{Lemma}
+
+\begin{proof}
+	Let $m$ be an integer such that $m\alpha < 1 < (m+1)\alpha$. Suppose $\A$ has less than $m+1$ vertices. Make a construction $\A_0 = \A$ and $\A_{i+1}$ is $\A_i$ with one extra vertex connected to every single vertex of $A_i$. Stop when the total number of vertices is $m+1$. Proceed as in \cite{Laskowski} 4.1. Resulting construction is still minimal.
+\end{proof}
+
+\begin{Lemma}
+	Let $\A_1 \subset \B_1$ and $\A_2 \subset \B_2$ be $\K$ structures with $(\A_2, \B_2)$ a minimal pair with $\epsilon = \delta (\B_2/\A_2)$. Let $M$ be some ambient structure. Fix embeddings of $\A_1, \B_1, \A_2$ into $M$. Assume that it is not that case that $\A_2 \subset \B_2$ and $\A_1$ is disjoint from $\A_2$ (No!). Now consider all possible embeddings $f \colon \B_2 \to M$ over $\A_1$.  Let $\A = \A_1 \cup \A_2$ and $\B_f = \B_1 \cup f(\B_2)$ with $\delta_f = \delta(\B_f/\A)$. Then $\delta_f$ is at most $\delta(\B_1 \cup \A/\A) + \epsilon$
+\end{Lemma}
+
+Fix an embedding $f$. It induces the following substructure diagram in $M$. Denote 
+\begin{align*}
+	\A &= \A_1 \cup \A_2 \\
+	\B_f^* &= \B_1 \cup f(\B_2) \\
+	\B_1^* &= \B_1 \cup \A \\
+	\B_2^* &= f(\B_2) \cup \A \\
+	\B^* &= \B_1^* \cap \B_2^*
+\end{align*}
+
+\begin{diagram}
+								&							&\B_f		\\
+								&\ruLine    	&										&\luLine	\\
+	\B_1^*      	&           	&										&					&\B_2^* \\
+								&\luLine			&										&\ruLine	\\
+								&							&\B^* \\
+								&							&\uLine \\
+								&							&\A\\
+\end{diagram}
+
+From the diagram we see that
+\begin{align*}
+	\delta(\B_f/\A) \leq \delta(\B_1^*/\A) + \delta(\B_2^*/\B^*)
+\end{align*}
+Thus all we need to do is to verify that
+\begin{align*}
+	\delta(\B_2^*/\B^*) \leq \epsilon
+\end{align*}
+Let $\B'$ denote graph induced on all the vertices in $(f(B_2) / B_1) \cup A_2$.
+Then $\B'$ is a substructure of $\B_2$ over $\A_2$. By minimality we get that $\delta(\B'/\A_2) \leq \epsilon$.
+We need to show $\delta(\B_2^*/\B^*) \leq \delta(\B'/\A_2)$.
+Do the vertex computation
+\begin{align*}
+	B_2^* - B^* &= \\
+	f(B_2) - (B_1 \cap f(B_2)) - A &= \\
+	f(B_2) - B_1 - A &= \\
+	f(B_2) - B_1 - A_2
+\end{align*}
+and
+\begin{align*}
+	B' - A_2 &=	
+	f(B_2) - B_1 - A_2
+\end{align*}
+
+So the sets of the extra vertices in the extension are the same. The base $\B_2^*/\B^*$ is larger so we can introduce some extra edges but no new vertices. This means that $\delta(\B_2^*/\B^*) \leq \delta(\B'/\A_2)$ giving us the original statement.
+
+
+Let $\phi(x,y)$ be a formula in a random graph with $|x|=|y|=1$ saying that there exists $\D$ over $\C = \{x,y\}$ such that $(\D, \C)$ is minimal with relative dimension $\epsilon$. Let $N$ be such that $N\epsilon < 1 < (N+1)\epsilon$. Then we argue that $vc(\phi) = N$.
+
+Fix a $m$-strong (for any $m > |D|$) set of non-connected vertices $A$. Fix some $a*$. We invesitgate the trace of $\phi(x, a*)$ on $A$. Suppose we have $a_1, \ldots, a_k$ satisfying $\phi(a_i, a^*)$ as witnessed by $D_i / \{a_i, a*\}$. Let $\D^* = \bigcap \D_i$ and $\C^*$
+
+Call $\M$ $n$-composite embedding if there are distinct vertices $a_1, \ldots a_n$ and $a*$ in $M$ and there are an embeddings $\D \arr \M$ with $\C$ going to $\{a_i, a^*\}$. Image of $i$-th embedding is denoted $\D_i$. Note that images of embeddings can intersect each other or $a_j$'s. Consider $\D^* = \bigcap \D_i$ and $\C^* = \{a_1, \ldots a_n, a^*\}$. Dimension of $M$ is $\delta(\D^*/\C^*)$.
+
+Lemma: Dimension of $n$-composite embedding is at most $-n\epsilon$.
+
+Note: if $\D_i$ are disjoint over $\C^*$ then the dimension is exactly $-n\epsilon$.
+
+Take $n$-composite embedding with maximal dimension. Suppose it is larger than $-n\epsilon$.
+Without loss of generality we may assume $\D_n$ intersects with $\D_1 \cup \ldots \cup \D_{n-1}$ over $\C^*$.
+Consider two cases.
+First, suppose that there is some element in $\D_n$ outside of $\D_1 \cup \ldots \cup \D_{n-1}$.
+Let $\B_1 = \D_1 \cup \ldots \cup \D_{n-1}$.
+Let $\A_1 = \{a_1, \ldots a_{n-1}\} \cup \{a*\}$.
+Let $\B_2 = \D_n$.
+Let $\A_2 = \{a_n, a*\}$.
+
+Lemma applies to the above. Above dimension is minimized when $\D_n$ is disjoint. Contradiction.
+
+Second, suppose that $\D_n \subseteq \B_1$. In particular $a_n \in \B_1$. Consider
+
+Consider sets $\B_1 \ldots \B_n$ with 
+\begin{enumerate}
+	\item $a_i \in \B_i$
+	\item $a_i \in A$
+	\item $a_i \neq a_j$
+	\item $a* \in \bigcap \B_i$
+\end{enumerate}
+ and  s.t. $\B_i / \{a*, a_i\}$ is isomorphic to $\B/\A$. We look at all the possible embeddings with those properties. We argue that a disjoint configuration minimizes total dimension of the whole construction.
+
+We argue by induction on $n$. Fix an embedding $\B_1, \ldots \B_n$ and consider possible choices for $\B_{n+1}, a_{n+1}$. We can pick $a_n$ to be an element of $A$ not used so far and embed $\B_{n+1}$ over $\{a*, a_i\}$ disjoint from the entire construction. On the other hand suppose it is embedded such that there is an intersection. We set up to apply the previous lemma. Let
+\begin{align*}
+	\B_1 &= \bigcup_{1..n} \B_i \\
+	\A_1 &= \{a_1, \ldots a_n\} \\
+	\B_2 &= \B_{n+1} \\
+	\A_2 &= \{a^*, a_{n+1}\}
+\end{align*}
+Applying the lemma say that the extra dimension cannot be larger then $\epsilon$.
+
+\begin{thebibliography}{9}
+
+\bibitem{Laskowski}
+	Michael C. Laskowski, \textsl{A simpler axiomatization of the Shelah-Spencer almost sure theories,}
+	Israel J. Math. \textbf{161} (2007), 157-186. MR MR2350161
+
+\end{thebibliography}
+
+\end{document}
diff --git a/dissertation your fellowship/diagrams.sty b/dissertation your fellowship/diagrams.sty
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+++ b/dissertation your fellowship/diagrams.sty	
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+% mangletex (24 Nov 1995) run at 14:58 BST Wednesday 11 May 2011
+\message{==================================================================}%
+\message{<Paul Taylor's commutative diagrams - version 3.94, April 2011>}%%
+%%
+%% This code runs the LaTeX \ProvidesPackage command iff it is defined.
+%% included at the request of Michael Downes <mjd@ams.org> March 2002.
+%% Put \listfiles in your LaTeX preamble to see what this is for.
+\expandafter\ifx\csname ProvidesPackage\endcsname\relax\toks0=\expandafter{%
+\fi\ProvidesPackage{diagrams}[2011/04/19 v3.94 Paul Taylor's commutative
+diagrams]%%
+\toks0=\bgroup}%%
+%%======================================================================%
+%%      TeX  macros for drawing category-theoretic diagrams             %
+%%                                                                      %
+%%                              Paul  Taylor                            %
+%%                                                                      %
+%%             		www.PaulTaylor.EU/diagrams			%
+%%	www.ctan.org/tex-archive/macros/generic/diagrams/taylor/	%
+%%			   diagrams@PaulTaylor.EU			%
+%%                                                                      %
+%%                           PLEASE READ THE MANUAL!                    %
+%%                                                                      %
+%%      Please ensure that you are registered with me as a user so that %
+%%      you can be informed of future releases.  Any electronic mail    %
+%%      message with "commutative" or "diagram" in the subject line     %
+%%      (such as your request for the package, a question about it, or  %
+%%      even an otherwise blank message) automatically registers you.   %
+%%                                                                      %
+%%                                                                      %
+%% CONTENTS:                                                            %
+%%  (O) corruption-sensitive hacks    (to approx line 331)              %
+%%      Arrow components & commands - starts approx line 1247           %
+%% (22) auxillary macros for adjustment of components                   %
+%% (23) bits of arrows  (\rhvee etc)                                    %
+%% (24) arrow commands  (\rTo etc)                                      %
+%% (25) miscellaneous                                                   %
+%% Apart from these five sections, the rest is intended to be totally   %
+%% meaningless in the undocumented version, which is approximately      %
+%% 1922  lines long. Please do not waste trees by printing it out.      %
+%%                                                                      %
+%% COPYRIGHT NOTICE:                                                    %
+%%      This package may be copied and used freely for any academic     %
+%%      (not commercial or military) purpose, on condition that it      %
+%%      is not altered in any way, and that an acknowledgement is       %
+%%      included in any published work making substantial use of it.    %
+%%                                                                      %
+%%      IT IS SUPPLIED "AS IS", WITHOUT WARRANTY, EXPRESS OR IMPLIED.   %
+%%                                                                      %
+%%      If you are doing something where mistakes cost money (or where  %
+%%      success brings financial profit) then you must use commercial   %
+%%      software, not this package. In any case, please remember to     %
+%%      keep several backup copies of all files, and check everything   %
+%%      visually before sending final copy to the publishers.           %
+%%                                                                      %
+%%      You may use this package as a (substantial) aid to writing an   %
+%%      academic research or text book on condition that                %
+%%       (i) you contact me at a suitable time to ensure that you have  %
+%%           an up-to-date version (and any infelicities can be fixed), %
+%%      (ii) you send me a copy of the book when it's published.        %
+%%                                                                      %
+%% HISTORY                                                              %
+%% 3.94 Released 11 May 2011						%
+%%	defined <= tail							%
+%% 3.93 Released 9 June 2009						%
+%%	Added support for XeTeX, with help from Apostolos Syropoulos.	%
+%% 3.92 Released 31 December 2007                                       %
+%% 3.91 Released 31 August 2006                                         %
+%%	Renamed "noPostScript" option as "UglyObsolete".		%
+%% 3.90 Released 11 April 2004                                          %
+%%      use PostScript=Rokicki not pure DVI by default                  %
+%% 3.89 Released 7 July 2002                                            %
+%%      Added support for pdftex, which is recognised automatically.    %
+%% 3.88 Released 1 September 2000                                       %
+%%      Square hook tail: \newarrow{SquareInto}{sqhook}--->             %
+%% 3.87 Released 1 September 1999                                       %
+%%      This version was used for the final 1200dpi PS copy of my book  %
+%%      ``Practical Foundations of Mathematics'' (Cambridge Univ Press) %
+%%      see   http://www.PaulTaylor.EU/Practical_Foundations            %
+%% 3.86 Released 1 September 1998                                       %
+%%      New options hug and nohug in PostScript mode: [PS,nohug] uses   %
+%%      PS for the arrows without rotating the labels, but the way of   %
+%%      calculating the actual position of these horizontal labels on   %
+%%      will remain subject to alteration for some period of time ---   %
+%%      please send me examples if you feel that adjustment is needed.  %
+%%                                                                      %
+%%      midvshaft and snake for vertical arrows                         %
+%%      New option [gap=width] (default=shortfall) to use instead of    %
+%%      ~{\;} on horizontals and PS diagonals, as this caused ^ and _   %
+%%      labels to be moved too far away from the shaft.                 %
+%%      Added >-> and <-< heads and tails, same as >> and << but the    %
+%%      shaft goes *through* the extra arrowhead.                       %
+%% 3.85 Released 20 August 1997                                         %
+%%      New option [crab=distance] shifts horizontals and PS diagonals  %
+%%      transversally by the specified distance (positive=upward).      %
+%%      New option [snake=distance] shifts midshaft horizontals and     %
+%%      PS diagonals longitudinally by the specified distance.          %
+%%      New option [leftflush], like [flushleft] but reckons alignment  %
+%%      from multiple verticals, or from text if there's no vertical.   %
+%%      Most of the history has been suppressed from the user version.  %
+%% 3.83 Released 18 May 1995                                            %
+%%      "dotted" option (set dot filler on maps)                        %
+%%      Parallel maps (\pile) outside diagrams stretch correctly.       %
+%%      Option "LaTeXeqno" uses LaTeX's equation number and style       %
+%%      for "eqno";  LaTeX's \label command picks this up.              %
+%%      Suppress warnings & 2nd pass errors with "silent" option.       %
+%% 3.81 Second alpha release 18 July 1994                               %
+%%      \overprint{text} sets text in maths and overprints it in the    %
+%%      current cell, centered in the column irrespective of other stuff%
+%%      "repositionpullbacks" option uses this for \SEpbk etc           %
+%%      \newdiagramgrid declaration, grid option and pentagon grid.     %
+%% 3.80 Alpha release for adjusted diagonals 15 July 1994.              %
+%%      Introduced landscape and portrait options.                      %
+%%      Diagonals adjusted to meet their endpoints, at last!!!!         %
+%% 3.28 Released 30 November 1993                                       %
+%%      Peter Freyd's \puncture symbol provided.                        %
+%% 3.25 Released 30 January 1993                                        %
+%%      LaTeX heads made default (unless \tenln undefined, when vee)    %
+%%      Circle, cross, little vee, little black triangle heads.         %
+%%      First-use warning when defaulted diagonal components are used.  %
+%%      AMSTEX emulation - works at least when amstex not present.      %
+%% 3.24 Release 7 Sept 1992 advertised to users.                        %
+%%      PostScript option introduced:                                   %
+%%      LaTeX, vee, curlyvee, triangle & blacktriangle heads & tails    %
+%%      TPIC option introduced as an alternative to \LaTeX@make@line.   %
+%%      Postscript arrows (basic code).                                 %
+%%      Implemented \newarrow \newarrowhead etc.                        %
+%%      Horizontal arrows outside diagram can stretch by wordspacing.   %
+%%      Added < and > for labels on left and right of arrow;            %
+%% 3.16 (20.7.90) as mass mailed; only have mangled version             %
+%% -- all following version numbers are post-facto --                   %
+%%  3   (Jan 90) stretching vertical arrows                             %
+%%  2   (Sept 89) horizontals stretch to objects; "superscript" labels  %
+%%  1   (1987) horizontal arrows stretch to edge of cell                %
+%%  0   (1986) implementation of Knuth's TeXercise 18.46 for my thesis  %
+%%======================================================================%
+
+%%======================================================================%
+%%                                                                      %
+%%      (1) CORRUPTION-SENSITIVE HACKS                                  %
+%%                                                                      %
+%%======================================================================%
+
+%%                      CORRUPTION & \catcode WARNING
+
+%% BITNET (IBM) machines may corrupt certain important characters
+%% in transmission by electronic mail:
+%%        0123456789=digits, abcdefghijklmnopqrstuvwxyz=lowers,
+%%        ABCDEFGHIJKLMNOPQRSTUVWXYZ=uppers, @=at (internal names),
+%%        {}=curly braces (grouping), \=backslash (keywords),
+%%        %=percent (comment), #=hash/sharp (argument), +=plus, -=minus,
+%%        <>=angle brackets (relations \ifnum,\ifdim), ==equals,
+%%        ,=comma, .=dot, :=colon, ;=semicolon,  =space
+%% $=dollar (maths) is only used in the "bits of maps" section
+
+%% The following characters are marked by a comment including the word "ASCII",
+%% except in comments and messages:
+%%        &=and (alignment), ~=tilde, |=vertical, []=square brackets,
+%%        ^=caret (superscript), _=underline (subscript),
+%%        "=double quote (hex), ()=round brackets,
+%%        /=slash, ?=query, !=pling/bang, 
+%% The following are no longer flagged:
+%%        `=grave/backquote (catcodes), '=acute/single quote (octal),
+
+%% The \catcode's marked * are assumed for reading this file:
+%%          \=0* {=1* }=2* $=3 &=4 return=5* #=6 ^=7 _=8 ignored=9*
+%%          space=10* letter=11* other=12 active=13 %=14* invalid=15
+%% If you want to load this package inside Stallman's "texinfo", you must do
+%%% @catcode`@\=0 \catcode`\%=14 \input diagrams \catcode`\%=12 \catcode`\\=13
+%% and then use @diagram @rTo @\ @enddiagram etc. (braces {} stay the same).
+%% Also need @catcode`@&=4.
+
+%%*** You *MUST NOT* use the internal commands (with names beginning \CD@)****
+
+%% don't load me twice!
+\ifx\diagram\isundefined\else\message{WARNING: the \string\diagram\space
+command is already defined and will not be loaded again}\expandafter\endinput
+\fi
+
+%% make @ letter, saving its old code to restore at the end of this file
+%%% look for this on the last line of the file if you think something's missing!
+%% the other \catcode assignments are to make it work with texinfo.
+\edef\cdrestoreat{%%
+\noexpand\catcode`\noexpand\@=\the\catcode`\@%%
+\noexpand\catcode`\noexpand\#=\the\catcode`\#%%
+\noexpand\catcode`\noexpand\$=\the\catcode`\$%%
+\noexpand\catcode`\noexpand\<=\the\catcode`\<%%
+\noexpand\catcode`\noexpand\>=\the\catcode`\>%%
+\noexpand\catcode`\noexpand\:=\the\catcode`\:%% Johannes L. Braams's
+\noexpand\catcode`\noexpand\;=\the\catcode`\;%% Babel languages package
+\noexpand\catcode`\noexpand\!=\the\catcode`\!%% makes these \active.
+\noexpand\catcode`\noexpand\?=\the\catcode`\?%%
+\noexpand\catcode`\noexpand\+=\the\catcode'53%% texinfo @+ is @outer@active
+}\catcode`\@=11 \catcode`\#=6 \catcode`\<=12 \catcode`\>=12 \catcode'53=12
+\catcode`\:=12 \catcode`\;=12 \catcode`\!=12 \catcode`\?=12
+
+%% Change y to n if pool_size in your implementation of TeX is small.
+%% This is reasonable if you have a small ("personal") computer, but if you
+%%% have a sun, dec, hp, ... workstation or a mainframe, complain to your local
+%% system manager and get him/her to install a version of TeX with bigger
+%% parameters. The "hash size" (number of command names) gets you next.
+\ifx\diagram@help@messages\CD@qK\let\diagram@help@messages y\fi
+
+%% The following macro is used to include literal PostScript commands in the
+%% DVI file for rotation, etc.  Since this goes beyond standard TeX, it is
+%%% dependent on the convention used by your local DVI-to-PostScript translator.
+%% Choose whichever line applies to the program used at your site, or, if
+%% yours is not listed, consult the manual, experiment with this macro and
+%% (please) tell me what is needed to make it work.
+%%
+%%
+%%% dvips (Tomas Rokicki, Radical Eye) labrea.stanford.edu /pub/dvips9999.tar.Z
+%% CTAN: dviware/dvips
+\def\cdps@Rokicki#1{\special{ps:#1}}\let\cdps@dvips\cdps@Rokicki\let
+\cdps@RadicalEye\cdps@Rokicki\let\CD@HB\cdps@Rokicki\let\CD@IK\cdps@Rokicki
+\let\CD@HB\cdps@Rokicki%%
+%% I'm not sure that the rest work.
+%%
+%% dvitps (Stephan Bechtolsheim, Integrated Computer Systems)
+%% arthur.cs.purdue.edu /pub/TeXPS-9.99.tar.Z
+\def\cdps@Bechtolsheim#1{\special{dvitps: Literal "#1"}}%
+%% ASCII two dbl quotes
+\let\cdps@dvitps\cdps@Bechtolsheim\let\cdps@IntegratedComputerSystems
+\cdps@Bechtolsheim%%
+%% dvitops (James Clark)
+%% CTAN: dviware/dvitops
+\def\cdps@Clark#1{\special{dvitops: inline #1}}%%
+\let\cdps@dvitops\cdps@Clark%%
+%% OzTeX (Andrew Trevorrow) cannot be used
+\let\cdps@OzTeX\empty\let\cdps@oztex\empty\let\cdps@Trevorrow\empty%%
+%% dvi3ps (Kevin Coombes)
+%% CTAN: dviware/dvi2ps/dvi3ps
+\def\cdps@Coombes#1{\special{ps-string #1}}%%
+%% psprint (Trevorrow) CTAN: dviware/psprint
+%% dvi2ps (Senn) CTAN: dviware/dvi2ps
+%% psdvi (Elwell) CTAN: dviware/dvi2ps/psdvi
+
+\count@=\year\multiply\count@12 \advance\count@\month%%
+\ifnum\count@>24180 %% (December 2014)
+\message{***********************************************************}%%ascii
+\message{! YOU HAVE AN OUT OF DATE VERSION OF COMMUTATIVE DIAGRAMS! *}%%
+\message{! it expired in December 2014 and is time-bombed for April *}%%
+\message{! You may get an up to date version of this package from *}%%ascii
+\message{! either www.ctan.org or www.PaulTaylor.EU/diagrams/ *}%%
+\message{***********************************************************}%%ascii
+\ifnum\count@>24183 %% (March 2015)
+\errhelp{You may press RETURN and carry on for the time being.}\message{! It
+is embarrassing to see papers in conference proceedings}\message{! and
+journals containing bugs which I had fixed years before.}\message{! It is easy
+to obtain and install a new version, which will}\errmessage{! remain
+compatible with your files. Please get it NOW.}\fi\fi
+
+\def\CD@DE{\global\let}\def\CD@RH{\outer\def}
+
+%% safe names for braces, tab (&) and maths ($), as commands and for messages
+{\escapechar\m@ne\xdef\CD@o{\string\{}\xdef\CD@yC{\string\}}%%
+%%
+%% three ASCII ampersands (ands) (&&&) appear on the next line
+\catcode`\&=4 \CD@DE\CD@Q=&\xdef\CD@S{\string\&}%%ascii three ands
+%%
+%% ASCII ^ and _ each appear twice on next line
+%% six ASCII dollars ($$$$$$) appear on the next two lines.
+\catcode`\$=3 \CD@DE\CD@k=$\CD@DE\CD@ND=$%%ascii three dollars
+\xdef\CD@nC{\string\$}\gdef\CD@LG{$$}%%ascii three dollars
+%%
+%% two ASCII underlines (__) appear on the next line.
+\catcode`\_=8 \CD@DE\CD@lJ=_%%ascii two underlines
+%%
+%% eight ASCII carets (^^^^^^^^) appear on the next three lines.
+\obeylines\catcode`\^=7 \CD@DE\@super=^%%ascii two carets
+\ifnum\newlinechar=10 \gdef\CD@uG{^^J}%%ascii two carets
+\else\ifnum\newlinechar=13 \gdef\CD@uG{^^M}%%ascii two carets
+\else\ifnum\newlinechar=-1 \gdef\CD@uG{^^J}%%ascii two carets
+\else\CD@DE\CD@uG\space\expandafter\message{! input error: \noexpand
+\newlinechar\space is ASCII \the\newlinechar, not LF=10 or CR=13.}%%
+\fi\fi\fi}%%
+
+%% avoid using <> (because I personally re-define them to mean \langle\rangle)
+\mathchardef\lessthan='30474 \mathchardef\greaterthan='30476
+
+%% LaTeX line and arrowhead font
+%% the "hit return" comments show up if the font is missing.
+\ifx\tenln\CD@qK%%
+\font\tenln=line10\relax%% Hit return - who needs diagonals?
+\fi\ifx\tenlnw\CD@qK\ifx\tenln\nullfont\let\tenlnw\nullfont\else%%
+\font\tenlnw=linew10\relax%% Hit return - who needs diagonals?
+\fi\fi%%
+
+%% report line numbers in TeX3 only
+\ifx\inputlineno\CD@qK\csname newcount\endcsname\inputlineno\inputlineno\m@ne
+\message{***************************************************}\message{!
+Obsolete TeX (version 2). You should upgrade to *}\message{! version 3, which
+has been available since 1990. *}\message{***********************************%
+****************}\fi
+
+\def\cd@shouldnt#1{\CD@KB{* THIS (#1) SHOULD NEVER HAPPEN! *}}
+
+%% turn round- and square-bracketed arguments into curly-bracketed
+\def\get@round@pair#1(#2,#3){#1{#2}{#3}}%%ascii round brackets ()
+\def\get@square@arg#1[#2]{#1{#2}}%%ascii square brackets []
+\def\CD@AE#1{\CD@PK\let\CD@DH\CD@@E\CD@@E#1,],}%%ascii sq brackets
+\def\CD@m{[}\def\CD@RD{]}\def\commdiag#1{{\let\enddiagram\relax\diagram[]#1%
+\enddiagram}}
+
+%% ASCII open square bracket occurs on next line
+\def\CD@BF{{\ifx\CD@EH[\aftergroup\get@square@arg\aftergroup\CD@YH\else
+\aftergroup\CD@JH\fi}}%%
+\def\CD@CF#1#2{\def\CD@YH{#1}\def\CD@JH{#2}\futurelet\CD@EH\CD@BF}
+
+%% ASCII vertical bar (|) occurs on the next line
+\def\CD@KK{|}
+
+\def\CD@PB{%% arguments to maps inside diagrams
+\tokcase\CD@DD:\CD@y\break@args;\catcase\@super:\upper@label;\catcase\CD@lJ:%
+\lower@label;\tokcase{~}:\middle@label;%%ascii tilde
+\tokcase<:\CD@iF;%%ascii less-than
+\tokcase>:\CD@iI;%%ascii greater-than
+\tokcase(:\CD@BC;%%)%ascii open round bracket
+\tokcase[:\optional@;%%]%ascii open square bracket
+\tokcase.:\CD@JJ;%%ascii dot 12.7.94
+\catcase\space:\eat@space;\catcase\bgroup:\positional@;\default:\CD@@A
+\break@args;\endswitch}
+
+\def\switch@arg{%% arguments to horizontal maps outside diagrams
+\catcase\@super:\upper@label;\catcase\CD@lJ:\lower@label;\tokcase[:\optional@
+;%%]%ascii open square bracket
+\tokcase.:\CD@JJ;%%ascii dot 12.7.94 % ; was : before 15.6.97
+\catcase\space:\eat@space;\catcase\bgroup:\positional@;\tokcase{~}:%
+\middle@label;%%ascii tilde (questionable!)
+\default:\CD@y\break@args;\endswitch}
+
+%% That's as much as you get to read "in clear" - the rest is private!
+
+\let\CD@tJ\relax\ifx\protect\CD@qK\let\protect\relax\fi\ifx\AtEndDocument
+\CD@qK\def\CD@PG{\CD@gB}\def\CD@GF#1#2{}\else\def\CD@PG#1{\edef\CD@CH{#1}%
+\expandafter\CD@oC\CD@CH\CD@OD}\def\CD@oC#1\CD@OD{\AtEndDocument{\typeout{%
+\CD@tA: #1}}}\def\CD@GF#1#2{\gdef#1{#2}\AtEndDocument{#1}}\fi\def\CD@ZA#1#2{%
+\def#1{\CD@PG{#2\CD@mD\CD@W}\CD@DE#1\relax}}\def\CD@uF#1\repeat{\def\CD@p{#1}%
+\CD@OF}\def\CD@OF{\CD@p\relax\expandafter\CD@OF\fi}\def\CD@sF#1\repeat{\def
+\CD@q{#1}\CD@PF}\def\CD@PF{\CD@q\relax\expandafter\CD@PF\fi}\def\CD@tF#1%
+\repeat{\def\CD@r{#1}\CD@QF}\def\CD@QF{\CD@r\relax\expandafter\CD@QF\fi}\def
+\CD@tG#1#2#3{\def#2{\let#1\iftrue}\def#3{\let#1\iffalse}#3}\if y%
+\diagram@help@messages\def\CD@rG#1#2{\csname newtoks\endcsname#1#1=%
+\expandafter{\csname#2\endcsname}}\else\csname newtoks\endcsname\no@cd@help
+\no@cd@help{See the manual}\def\CD@rG#1#2{\let#1\no@cd@help}\fi\chardef\CD@lF
+=1 \chardef\CD@lI=2 \chardef\CD@MH=5 \chardef\CD@tH=6 \chardef\CD@sH=7
+\chardef\CD@PC=9 \dimendef\CD@hI=2 \dimendef\CD@hF=3 \dimendef\CD@mF=4
+\dimendef\CD@mI=5 \dimendef\CD@wJ=6 \dimendef\CD@tI=8 \dimendef\CD@sI=9
+\skipdef\CD@uB=1 \skipdef\CD@NF=2 \skipdef\CD@tB=3 \skipdef\CD@ZE=4 \skipdef
+\CD@JK=5 \skipdef\CD@kI=6 \skipdef\CD@kF=7 \skipdef\CD@qI=8 \skipdef\CD@pI=9
+\countdef\CD@JC=9 \countdef\CD@gD=8 \countdef\CD@A=7 \def\sdef#1#2{\def#1{#2}%
+}\def\CD@L#1{\expandafter\aftergroup\csname#1\endcsname}\def\CD@RC#1{%
+\expandafter\def\csname#1\endcsname}\def\CD@sD#1{\expandafter\gdef\csname#1%
+\endcsname}\def\CD@vC#1{\expandafter\edef\csname#1\endcsname}\def\CD@nF#1#2{%
+\expandafter\let\csname#1\expandafter\endcsname\csname#2\endcsname}\def\CD@EE
+#1#2{\expandafter\CD@DE\csname#1\expandafter\endcsname\csname#2\endcsname}%
+\def\CD@AK#1{\csname#1\endcsname}\def\CD@XJ#1{\expandafter\show\csname#1%
+\endcsname}\def\CD@ZJ#1{\expandafter\showthe\csname#1\endcsname}\def\CD@WJ#1{%
+\expandafter\showbox\csname#1\endcsname}\def\CD@tA{Commutative Diagram}\edef
+\CD@kH{\string\par}\edef\CD@dC{\string\diagram}\edef\CD@HD{\string\enddiagram
+}\edef\CD@EC{\string\\}\def\CD@eF{LaTeX}\ifx\@ignoretrue\CD@qK\expandafter
+\CD@tG\csname if@ignore\endcsname\ignore@true\ignore@false\def\@ignoretrue{%
+\global\ignore@true}\def\@ignorefalse{\global\ignore@false}\fi
+
+\def\CD@g{{\ifnum0=`}\fi}\def\CD@wC{\ifnum0=`{\fi}}\def\catcase#1:{\ifcat
+\noexpand\CD@EH#1\CD@tJ\expandafter\CD@kC\else\expandafter\CD@dJ\fi}\def
+\tokcase#1:{\ifx\CD@EH#1\CD@tJ\expandafter\CD@kC\else\expandafter\CD@dJ\fi}%
+\def\CD@kC#1;#2\endswitch{#1}\def\CD@dJ#1;{}\let\endswitch\relax\def\default:%
+#1;#2\endswitch{#1}\ifx\at@\CD@qK\def\at@{@}\fi\edef\CD@P{\CD@o pt\CD@yC}%
+\CD@RC{\CD@P>}#1>#2>{\CD@z\rTo\sp{#1}\sb{#2}\CD@z}\CD@RC{\CD@P<}#1<#2<{\CD@z
+\lTo\sp{#1}\sb{#2}\CD@z}\CD@RC{\CD@P)}#1)#2){\CD@z\rTo\sp{#1}\sb{#2}\CD@z}%
+%%ascii round
+\CD@RC{\CD@P(}#1(#2({\CD@z\lTo\sp{#1}\sb{#2}\CD@z}%%ascii brack
+\def\CD@O{\def\endCD{\enddiagram}\CD@RC{\CD@P A}##1A##2A{\uTo<{##1}>{##2}%
+\CD@z\CD@z}\CD@RC{\CD@P V}##1V##2V{\dTo<{##1}>{##2}\CD@z\CD@z}\CD@RC{\CD@P=}{%
+\CD@z\hEq\CD@z}\CD@RC{\CD@P\CD@KK}{\vEq\CD@z\CD@z}\CD@RC{\CD@P\string\vert}{%
+\vEq\CD@z\CD@z}\CD@RC{\CD@P.}{\CD@z\CD@z}\let\CD@z\CD@Q}\def\CD@IE{\let\tmp
+\CD@JE\ifcat A\noexpand\CD@CH\else\ifcat=\noexpand\CD@CH\else\ifcat\relax
+\noexpand\CD@CH\else\let\tmp\at@\fi\fi\fi\tmp}\def\CD@JE#1{\CD@nF{tmp}{\CD@P
+\string#1}\ifx\tmp\relax\def\tmp{\at@#1}\fi\tmp}\def\CD@z{}\begingroup
+\aftergroup\def\aftergroup\CD@T\aftergroup{\aftergroup\def\catcode`\@\active
+\aftergroup @\endgroup{\futurelet\CD@CH\CD@IE}}\newcount\CD@uA\newcount\CD@vA
+\newcount\CD@wA\newcount\CD@xA\newdimen\CD@OA\newdimen\CD@PA\CD@tG\CD@gE
+\CD@@A\CD@y\CD@tG\CD@hE\CD@EA\CD@BA\newdimen\CD@RA\newdimen\CD@SA\newcount
+\CD@yA\newcount\CD@zA\newdimen\CD@QA\newbox\CD@DA\CD@tG\CD@lE\CD@dA\CD@bA
+\newcount\CD@LH\newcount\CD@TC\def\CD@V#1#2{\ifdim#1<#2\relax#1=#2\relax\fi}%
+\def\CD@X#1#2{\ifdim#1>#2\relax#1=#2\relax\fi}\newdimen\CD@XH\CD@XH=1sp
+\newdimen\CD@zC\CD@zC\z@\def\CD@cJ{\ifdim\CD@zC=1em\else\CD@nJ\fi}\def\CD@nJ{%
+\CD@zC1em\def\CD@NC{\fontdimen8\textfont3 }\CD@@J\CD@NJ\setbox0=\vbox{\CD@t
+\noindent\CD@k\null\penalty-9993\null\CD@ND\null\endgraf\setbox0=\lastbox
+\unskip\unpenalty\setbox1=\lastbox\global\setbox\CD@IG=\hbox{\unhbox0\unskip
+\unskip\unpenalty\setbox0=\lastbox}\global\setbox\CD@KG=\hbox{\unhbox1\unskip
+\unpenalty\setbox1=\lastbox}}}\newdimen\CD@@I\CD@@I=1true in \divide\CD@@I300
+\def\CD@zH#1{\multiply#1\tw@\advance#1\ifnum#1<\z@-\else+\fi\CD@@I\divide#1%
+\tw@\divide#1\CD@@I\multiply#1\CD@@I}\def\MapBreadth{\afterassignment\CD@gI
+\CD@LF}\newdimen\CD@LF\newdimen\CD@oI\def\CD@gI{\CD@oI\CD@LF\CD@V\CD@@I{4%
+\CD@XH}\CD@X\CD@@I\p@\CD@zH\CD@oI\ifdim\CD@LF>\z@\CD@V\CD@oI\CD@@I\fi\CD@cJ}%
+\def\CD@RJ#1{\CD@zD\count@\CD@@I#1\ifnum\count@>\z@\divide\CD@@I\count@\fi
+\CD@gI\CD@NJ}\def\CD@NJ{\dimen@\CD@QC\count@\dimen@\divide\count@5\divide
+\count@\CD@@I\edef\CD@OC{\the\count@}}\def\CD@AJ{\CD@QJ\z@}\def\CD@QJ#1{%
+\CD@tI\axisheight\advance\CD@tI#1\relax\advance\CD@tI-.5\CD@oI\CD@zH\CD@tI
+\CD@sI-\CD@tI\advance\CD@tI\CD@LF}\newdimen\CD@DC\CD@DC\z@\newdimen\CD@eJ
+\CD@eJ\z@\def\CD@CJ#1{\CD@sI#1\relax\CD@tI\CD@sI\advance\CD@tI\CD@LF\relax}%
+\def\horizhtdp{height\CD@tI depth\CD@sI}\def\axisheight{\fontdimen22\the
+\textfont\tw@}\def\script@axisheight{\fontdimen22\the\scriptfont\tw@}\def
+\ss@axisheight{\fontdimen22\the\scriptscriptfont\tw@}\def\CD@NC{0.4pt}\def
+\CD@VK{\fontdimen3\textfont\z@}\def\CD@UK{\fontdimen3\textfont\z@}\newdimen
+\PileSpacing\newdimen\CD@nA\CD@nA\z@\def\CD@RG{\ifincommdiag1.3em\else2em\fi}%
+\newdimen\CD@YB\def\CellSize{\afterassignment\CD@kB\DiagramCellHeight}%
+\newdimen\DiagramCellHeight\DiagramCellHeight-\maxdimen\newdimen
+\DiagramCellWidth\DiagramCellWidth-\maxdimen\def\CD@kB{\DiagramCellWidth
+\DiagramCellHeight}\def\CD@QC{3em}\newdimen\MapShortFall\def\MapsAbut{%
+\MapShortFall\z@\objectheight\z@\objectwidth\z@}\newdimen\CD@iA\CD@iA\z@
+\CD@tG\CD@vE\CD@aB\CD@ZB\expandafter\ifx\expandafter\iftrue\csname
+ifUglyObsoleteDiagrams\endcsname\CD@ZB\else\CD@aB\fi\CD@nF{%
+ifUglyObsoleteDiagrams}{relax}\newif\ifUglyObsoleteDiagrams\def\CD@nK{\CD@aB
+\UglyObsoleteDiagramsfalse}\def\CD@oK{\CD@ZB\UglyObsoleteDiagramstrue}\CD@vE
+\CD@nK\else\CD@oK\fi\CD@tG\CD@hK\CD@dK\CD@cK\CD@cK\def\CD@sK{\ifx\pdfoutput
+\CD@qK\else\ifx\pdfoutput\relax\else\ifnum\pdfoutput>\z@\CD@pK\fi\fi\fi} \def
+\CD@pK{\global\CD@dK\global\CD@aB\global\UglyObsoleteDiagramsfalse\global\let
+\CD@n\empty\global\let\CD@oK\relax\global\let\CD@pK\relax\global\let\CD@sK
+\relax}\def\CD@tK#1{\special{pdf: literal #1}}\ifx\pdfliteral\CD@qK\else\ifx
+\pdfliteral\relax\else\let\CD@tK\pdfliteral\fi\fi\ifx\XeTeXrevision\CD@qK
+\else\ifx\XeTeXrevision\relax\else\ifdim\XeTeXrevision pt<.996pt \expandafter
+\message{! XeTeX version \XeTeXrevision\space does not support PDF literals,
+so diagonals will not work!}\else\expandafter\message{RUNNING UNDER XETEX
+\XeTeXrevision}\CD@pK\fi\fi\fi\CD@sK\def\newarrowhead{\CD@mG h\CD@BG\CD@GG>}%
+\def\newarrowtail{\CD@mG t\CD@BG\CD@GG>}\def\newarrowmiddle{\CD@mG m\CD@BG
+\hbox@maths\empty}\def\newarrowfiller{\CD@mG f\CD@bE\CD@MK-}\def\CD@mG#1#2#3#%
+4#5#6#7#8#9{\CD@RC{r#1:#5}{#2{#6}}\CD@RC{l#1:#5}{#2{#7}}\CD@RC{d#1:#5}{#3{#8}%
+}\CD@RC{u#1:#5}{#3{#9}}\CD@vC{-#1:#5}{\expandafter\noexpand\csname-#1:#4%
+\endcsname\noexpand\CD@MC}\CD@vC{+#1:#5}{\expandafter\noexpand\csname+#1:#4%
+\endcsname\noexpand\CD@MC}}\CD@ZA\CD@MC{\CD@eF\space diagonals are used unless
+PostScript is set}\def\defaultarrowhead#1{\edef\CD@sJ{#1}\CD@@J}\def\CD@@J{%
+\CD@IJ\CD@sJ<>ht\CD@IJ\CD@sJ<>th}\def\CD@IJ#1#2#3#4#5{\CD@HJ{r#4}{#3}{l#5}{#2%
+}{r#4:#1}\CD@HJ{r#5}{#2}{l#4}{#3}{l#4:#1}\CD@HJ{d#4}{#3}{u#5}{#2}{d#4:#1}%
+\CD@HJ{d#5}{#2}{u#4}{#3}{u#4:#1}}\def\CD@HJ#1#2#3#4#5{\begingroup\aftergroup
+\CD@GJ\CD@L{#1+:#2}\CD@L{#1:#2}\CD@L{#3:#4}\CD@L{#5}\endgroup}\def\CD@GJ#1#2#%
+3#4{\csname newbox\endcsname#1\def#2{\copy#1}\def#3{\copy#1}\setbox#1=\box
+\voidb@x}\def\CD@sJ{}\CD@@J\def\CD@GJ#1#2#3#4{\setbox#1=#4}\ifx\tenln
+\nullfont\def\CD@sJ{vee}\else\let\CD@sJ\CD@eF\fi\def\CD@xF#1#2#3{\begingroup
+\aftergroup\CD@wF\CD@L{#1#2:#3#3}\CD@L{#1#2:#3}\aftergroup\CD@yF\CD@L{#1#2:#3%
+-#3}\CD@L{#1#2:#3}\endgroup}\def\CD@wF#1#2{\def#1{\hbox{\rlap{#2}\kern.4%
+\CD@zC#2}}}\def\CD@yF#1#2{\def#1{\hbox{\rlap{#2}\kern.4\CD@zC#2\kern-.4\CD@zC
+}}}\CD@xF lh>\CD@xF rt>\CD@xF rh<\CD@xF rt<\def\CD@yF#1#2{\def#1{\hbox{\kern-%
+.4\CD@zC\rlap{#2}\kern.4\CD@zC#2}}}\CD@xF rh>\CD@xF lh<\CD@xF lt>\CD@xF lt<%
+\def\CD@wF#1#2{\def#1{\vbox{\vbox to\z@{#2\vss}\nointerlineskip\kern.4\CD@zC#%
+2}}}\def\CD@yF#1#2{\def#1{\vbox{\vbox to\z@{#2\vss}\nointerlineskip\kern.4%
+\CD@zC#2\kern-.4\CD@zC}}}\CD@xF uh>\CD@xF dt>\CD@xF dh<\CD@xF dt<\def\CD@yF#1%
+#2{\def#1{\vbox{\kern-.4\CD@zC\vbox to\z@{#2\vss}\nointerlineskip\kern.4%
+\CD@zC#2}}}\CD@xF dh>\CD@xF ut>\CD@xF uh<\CD@xF ut<\def\CD@BG#1{\hbox{%
+\mathsurround\z@\offinterlineskip\CD@k\mkern-1.5mu{#1}\mkern-1.5mu\CD@ND}}%
+\def\hbox@maths#1{\hbox{\CD@k#1\CD@ND}}\def\CD@GG#1{\hbox to\CD@LF{\setbox0=%
+\hbox{\offinterlineskip\mathsurround\z@\CD@k{#1}\CD@ND}\dimen0.5\wd0\advance
+\dimen0-.5\CD@oI\CD@zH{\dimen0}\kern-\dimen0\unhbox0\hss}}\def\CD@sB#1{\hbox
+to2\CD@LF{\hss\offinterlineskip\mathsurround\z@\CD@k{#1}\CD@ND\hss}}\def
+\CD@vF#1{\hbox{\mathsurround\z@\CD@k{#1}\CD@ND}}\def\CD@bE#1{\hbox{\kern-.15%
+\CD@zC\CD@k{#1}\CD@ND\kern-.15\CD@zC}}\def\CD@MK#1{\vbox{\offinterlineskip
+\kern-.2ex\CD@GG{#1}\kern-.2ex}}\def\@fillh{\xleaders\vrule\horizhtdp}\def
+\@fillv{\xleaders\hrule width\CD@LF}\CD@nF{rf:-}{@fillh}\CD@nF{lf:-}{@fillh}%
+\CD@nF{df:-}{@fillv}\CD@nF{uf:-}{@fillv}\CD@nF{rh:}{null}\CD@nF{rm:}{null}%
+\CD@nF{rt:}{null}\CD@nF{lh:}{null}\CD@nF{lm:}{null}\CD@nF{lt:}{null}\CD@nF{dh%
+:}{null}\CD@nF{dm:}{null}\CD@nF{dt:}{null}\CD@nF{uh:}{null}\CD@nF{um:}{null}%
+\CD@nF{ut:}{null}\CD@nF{+h:}{null}\CD@nF{+m:}{null}\CD@nF{+t:}{null}\CD@nF{-h%
+:}{null}\CD@nF{-m:}{null}\CD@nF{-t:}{null}\def\CD@@D{\hbox{\vrule height 1pt
+depth-1pt width 1pt}}\CD@RC{rf:}{\CD@@D}\CD@nF{lf:}{rf:}\CD@nF{+f:}{rf:}%
+\CD@RC{df:}{\CD@@D}\CD@nF{uf:}{df:}\CD@nF{-f:}{df:}\def\CD@BD{\CD@U\null
+\CD@@D\null\CD@@D\null}\edef\CD@lG{\string\newarrow}\def\newarrow#1#2#3#4#5#6%
+{\begingroup\edef\@name{#1}\edef\CD@oJ{#2}\edef\CD@iD{#3}\edef\CD@QG{#4}\edef
+\CD@jD{#5}\edef\CD@LE{#6}\let\CD@HE\CD@sG\let\CD@FK\CD@BH\let\@x\CD@AH\ifx
+\CD@oJ\CD@iD\let\CD@oJ\empty\fi\ifx\CD@LE\CD@jD\let\CD@LE\empty\fi\def\CD@LI{%
+r}\def\CD@SF{l}\def\CD@IC{d}\def\CD@yJ{u}\def\CD@gH{+}\def\@m{-}\ifx\CD@iD
+\CD@jD\ifx\CD@QG\CD@iD\let\CD@QG\empty\fi\ifx\CD@LE\empty\ifx\CD@iD\CD@aE\let
+\@x\CD@yG\else\let\@x\CD@zG\fi\fi\else\edef\CD@a{\CD@iD\CD@oJ}\ifx\CD@a\empty
+\ifx\CD@QG\CD@jD\let\CD@QG\empty\fi\fi\fi\ifmmode\aftergroup\CD@kG\else\CD@@A
+\CD@oB rh{head\space\space}\CD@LE\CD@oB rf{filler}\CD@iD\CD@oB rm{middle}%
+\CD@QG\ifx\CD@jD\CD@iD\else\CD@oB rf{filler}\CD@jD\fi\CD@oB rt{tail\space
+\space}\CD@oJ\CD@gE\CD@HE\CD@FK\@x\CD@nG l-2+2{lu}{nw}\NorthWest\CD@nG r+2+2{%
+ru}{ne}\NorthEast\CD@nG l-2-2{ld}{sw}\SouthWest\CD@nG r+2-2{rd}{se}\SouthEast
+\else\aftergroup\CD@b\CD@L{r\@name}\fi\fi\endgroup}\def\CD@sG{\CD@vG\CD@LI
+\CD@SF rl\Horizontal@Map}\def\CD@BH{\CD@vG\CD@IC\CD@yJ du\Vertical@Map}\def
+\CD@AH{\CD@vG\CD@gH\@m+-\Vector@Map}\def\CD@yG{\CD@vG\CD@gH\@m+-\Slant@Map}%
+\def\CD@zG{\CD@vG\CD@gH\@m+-\Slope@Map}\catcode`\/=\active\def\CD@vG#1#2#3#4#%
+5{\CD@jG#1#3#5t:\CD@oJ/f:\CD@iD/m:\CD@QG/f:\CD@jD/h:\CD@LE//\CD@jG#2#4#5h:%
+\CD@LE/f:\CD@jD/m:\CD@QG/f:\CD@iD/t:\CD@oJ//}\def\CD@jG#1#2#3#4//{\edef\CD@fG
+{#2}\aftergroup\sdef\CD@L{#1\@name}\aftergroup{\aftergroup#3\CD@M#4//%
+\aftergroup}}\def\CD@M#1/{\edef\CD@EH{#1}\ifx\CD@EH\empty\else\CD@L{\CD@fG#1}%
+\expandafter\CD@M\fi}\catcode`\/=12 \def\CD@nG#1#2#3#4#5#6#7#8{\aftergroup
+\sdef\CD@L{#6\@name}\aftergroup{\CD@L{#2\@name}\if#2#4\aftergroup\CD@CI\else
+\aftergroup\CD@BI\fi\CD@L{#1\@name}%
+%% ASCII round brackets and comma (,) appear on the next line
+\aftergroup(\aftergroup#3\aftergroup,\aftergroup#5\aftergroup)\aftergroup}}%
+\def\CD@oB#1#2#3#4{\expandafter\ifx\csname#1#2:#4\endcsname\relax\CD@y\CD@gB{%
+arrow#3 "#4" undefined}\fi}\CD@rG\CD@VE{All five components must be defined
+before an arrow.}\CD@rG\CD@SE{\CD@lG, unlike \string\HorizontalMap, is a
+declaration.}\def\CD@b#1{\CD@YA{Arrows \string#1 etc could not be defined}%
+\CD@VE}\def\CD@kG{\CD@YA{misplaced \CD@lG}\CD@SE}\def\newdiagramgrid#1#2#3{%
+\CD@RC{cdgh@#1}{#2,],}%% ASCII close square bracket
+\CD@RC{cdgv@#1}{#3,],}}%% ASCII close square bracket
+\CD@tG\ifincommdiag\incommdiagtrue\incommdiagfalse\CD@tG\CD@@F\CD@IF\CD@HF
+\newcount\CD@VA\CD@VA=0 \def\CD@yH{\CD@VA6 }\def\CD@OB{\CD@VA1 \global\CD@yA1
+\CD@DE\CD@YF\empty}\def\CD@YF{}\def\CD@nB#1{\relax\CD@MD\edef\CD@vJ{#1}%
+\begingroup\CD@rE\else\ifcase\CD@VA\ifmmode\else\CD@YG\CD@E0\fi\or\CD@cE5\or
+\CD@YG\CD@F5\or\CD@YG\CD@B5\or\CD@YG\CD@B5\or\CD@YG\CD@C5\or\CD@cE7\or\CD@YG
+\CD@D7\fi\fi\endgroup\xdef\CD@YF{#1}}\def\CD@pB#1#2#3#4#5{\relax\CD@MD\xdef
+\CD@vJ{#4}\begingroup\ifnum\CD@VA<#1 \expandafter\CD@cE\ifcase\CD@VA0\or#2\or
+#3\else#2\fi\else\ifnum\CD@VA<6 \CD@tJ\CD@YG\CD@B#2\else\CD@YG\CD@G#2\fi\fi
+\endgroup\CD@DE\CD@YF\CD@vJ\ifincommdiag\let\CD@ZD#5\else\let\CD@ZD\CD@LK\fi}%
+\def\CD@yI{\global\CD@yA=\ifnum\CD@VA<5 1\else2\fi\relax}\def\CD@OI{\CD@VA
+\CD@yA}\def\CD@cE#1{\aftergroup\CD@VA\aftergroup#1\aftergroup\relax}\def
+\CD@HH{\def\CD@nB##1{\relax}\let\CD@pB\CD@FH\let\CD@yH\relax\let\CD@OB\relax
+\let\CD@yI\relax\let\CD@OI\relax}\def\CD@FH#1#2#3#4#5{\ifincommdiag\let\CD@ZD
+#5\else\xdef\CD@vJ{#4}\let\CD@ZD\CD@LK\fi}\def\CD@YG#1{\aftergroup#1%
+\aftergroup\relax\CD@cE}\def\CD@B{\CD@YE\CD@S\CD@ME\CD@Q}\def\CD@G{\CD@YE{%
+\CD@yC\CD@S}\CD@XE\CD@QD\CD@Q}\def\CD@F{\CD@YE{*\CD@S}\CD@RE\clubsuit\CD@Q}%
+\def\CD@C{\CD@YE{\CD@S*\CD@S}\CD@RE\CD@Q\clubsuit\CD@Q}\def\CD@D{\CD@YE\CD@EC
+\CD@TE\\}\def\CD@E{\CD@YE\CD@nC\CD@QE\CD@k}\def\CD@LK{\CD@YA{\CD@vJ\space
+ignored \CD@dH}\CD@WE}\def\CD@FE{}\def\CD@d{\CD@YA{maps must never be enclosed
+in braces}\CD@OE}\def\CD@dH{outside diagram}\def\CD@FC{\string\HonV, \string
+\VonH\space and \string\HmeetV}\CD@rG\CD@ME{The way that horizontal and
+vertical arrows are terminated implicitly means\CD@uG that they cannot be
+mixed with each other or with \CD@FC.}\CD@rG\CD@XE{\string\pile\space is for
+parallel horizontal arrows; verticals can just be put together in\CD@uG a cell%
+. \CD@FC\space are not meaningful in a \string\pile.}\CD@rG\CD@RE{The
+horizontal maps must point to an object, not each other (I've put in\CD@uG one
+which you're unlikely to want). Use \string\pile\space if you want them
+parallel.}\CD@rG\CD@TE{Parallel horizontal arrows must be in separate layers
+of a \string\pile.}\CD@rG\CD@QE{Horizontal arrows may be used \CD@dH s, but
+must still be in maths.}\CD@rG\CD@WE{Vertical arrows, \CD@FC\space\CD@dH s don%
+'t know where\CD@uG where to terminate.}\CD@rG\CD@OE{This prevents them from
+stretching correctly.}\def\CD@YE#1{\CD@YA{"#1" inserted \ifx\CD@YF\empty
+before \CD@vJ\else between \CD@YF\ifx\CD@YF\CD@vJ s\else\space and \CD@vJ\fi
+\fi}}\count@=\year\multiply\count@12 \advance\count@\month\ifnum\count@>24187
+\message{because this one expired in July 2015!}\expandafter\endinput\fi\def
+\Horizontal@Map{\CD@nB{horizontal map}\CD@LC\CD@TJ\CD@qD}\def\CD@TJ{\CD@GB-%
+9999 \let\CD@ZD\CD@XD\ifincommdiag\else\CD@cJ\ifinpile\else\skip2\z@ plus 1.5%
+\CD@VK minus .5\CD@UK\skip4\skip2 \fi\fi\let\CD@kD\@fillh\CD@nF{fill@dot}{rf:%
+.}}\def\Vector@Map{\CD@HK4}\def\Slant@Map{\CD@HK{\CD@EF255\else6\fi}}\def
+\Slope@Map{\CD@HK\CD@OC}\def\CD@HK#1#2#3#4#5#6{\CD@LC\def\CD@WK{2}\def\CD@aK{%
+2}\def\CD@ZK{1}\def\CD@bK{1}\let\Horizontal@Map\CD@nI\def\CD@OG{#1}\def\CD@NI
+{\CD@U#2#3#4#5#6}}\def\CD@nI{\CD@TJ\CD@JB\let\CD@ZD\CD@TD\CD@qD}\CD@tG\CD@pE
+\CD@rA\CD@qA\CD@rA\def\cds@missives{\CD@rA}\def\CD@TD{\CD@vE\let\CD@OG\CD@OC
+\CD@x\CD@zE\CD@WF\fi\setbox0\hbox{\incommdiagfalse\CD@HI}\CD@pE\CD@aD\else
+\global\CD@YC\CD@bD\fi\ifvoid6 \ifvoid7 \CD@eE\fi\fi\CD@zE\else\CD@BD\global
+\CD@YC\let\CD@CG\CD@IH\CD@YD\fi\else\CD@NI\CD@MI\global\CD@YC\CD@YD\fi}\def
+\CD@LC{\begingroup\dimen1=\MapShortFall\dimen2=\CD@RG\dimen5=\MapShortFall
+\setbox3=\box\voidb@x\setbox6=\box\voidb@x\setbox7=\box\voidb@x\CD@pD
+\mathsurround\z@\skip2\z@ plus1fill minus 1000pt\skip4\skip2 \CD@TB}\CD@tG
+\CD@tE\CD@UB\CD@TB\def\CD@U#1#2#3#4#5{\let\CD@oJ#1\let\CD@iD#2\let\CD@QG#3%
+\let\CD@jD#4\let\CD@LE#5\CD@TB\ifx\CD@iD\CD@jD\CD@UB\fi}\def\CD@qD#1#2#3#4#5{%
+\CD@U#1#2#3#4#5\CD@tD}\def\Vertical@Map{\CD@pB433{vertical map}\CD@cD\CD@LC
+\CD@GB-9995 \let\CD@kD\@fillv\CD@nF{fill@dot}{df:.}\CD@qD}\def\break@args{%
+\def\CD@tD{\CD@ZD}\CD@ZD\endgroup\aftergroup\CD@FE}\def\CD@MJ{\setbox1=\CD@oJ
+\setbox5=\CD@LE\ifvoid3 \ifx\CD@QG\null\else\setbox3=\CD@QG\fi\fi\CD@@G2%
+\CD@iD\CD@@G4\CD@jD}\def\CD@pF#1{\ifvoid1\else\CD@oF1#1\fi\ifvoid2\else\CD@oF
+2#1\fi\ifvoid3\else\CD@oF3#1\fi\ifvoid4\else\CD@oF4#1\fi\ifvoid5\else\CD@oF5#%
+1\fi} \def\CD@oF#1#2{\setbox#1\vbox{\offinterlineskip\box#1\dimen@\prevdepth
+\advance\dimen@-#2\relax\setbox0\null\dp0\dimen@\ht0-\dimen@\box0}}\def\CD@@G
+#1#2{\ifx#2\CD@kD\setbox#1=\box\voidb@x\else\setbox#1=#2\def#2{\xleaders\box#%
+1}\fi}\CD@ZA\CD@BK{\string\HorizontalMap, \string\VerticalMap\space and
+\string\DiagonalMap\CD@uG are obsolete - use \CD@lG\space to pre-define maps}%
+\def\HorizontalMap#1#2#3#4#5{\CD@BK\CD@nB{old horizontal map}\CD@LC\CD@TJ\def
+\CD@oJ{\CD@UH{#1}}\CD@SH\CD@iD{#2}\def\CD@QG{\CD@UH{#3}}\CD@SH\CD@jD{#4}\def
+\CD@LE{\CD@UH{#5}}\CD@tD}\def\VerticalMap#1#2#3#4#5{\CD@BK\CD@pB433{vertical
+map}\CD@cD\CD@LC\CD@GB-9995 \let\CD@kD\@fillv\def\CD@oJ{\CD@GG{#1}}\CD@VH
+\CD@iD{#2}\def\CD@QG{\CD@GG{#3}}\CD@VH\CD@jD{#4}\def\CD@LE{\CD@GG{#5}}\CD@tD}%
+\def\DiagonalMap#1#2#3#4#5{\CD@BK\CD@LC\def\CD@OG{4}\let\CD@kD\CD@qK\let
+\CD@ZD\CD@YD\def\CD@WK{2}\def\CD@aK{2}\def\CD@ZK{1}\def\CD@bK{1}\def\CD@QG{%
+\CD@vF{#3}}\ifPositiveGradient\let\mv\raise\def\CD@oJ{\CD@vF{#5}}\def\CD@iD{%
+\CD@vF{#4}}\def\CD@jD{\CD@vF{#2}}\def\CD@LE{\CD@vF{#1}}\else\let\mv\lower\def
+\CD@oJ{\CD@vF{#1}}\def\CD@iD{\CD@vF{#2}}\def\CD@jD{\CD@vF{#4}}\def\CD@LE{%
+\CD@vF{#5}}\fi\CD@tD}\def\CD@aE{-}\def\CD@AD{\empty}\def\CD@SH{\CD@EG\CD@bE
+\CD@aE\@fillh}\def\CD@VH{\CD@EG\CD@MK\CD@KK\@fillv}\def\CD@EG#1#2#3#4#5{\def
+\CD@CH{#5}\ifx\CD@CH#2\let#4#3\else\let#4\null\ifx\CD@CH\empty\else\ifx\CD@CH
+\CD@AD\else\let#4\CD@CH\fi\fi\fi}\def\CD@UH#1{\hbox{\mathsurround\z@
+\offinterlineskip\def\CD@CH{#1}\ifx\CD@CH\empty\else\ifx\CD@CH\CD@AD\else
+\CD@k\mkern-1.5mu{\CD@CH}\mkern-1.5mu\CD@ND\fi\fi}}\def\CD@yD#1#2{\setbox#1=%
+\hbox\bgroup\setbox0=\hbox{\CD@k\labelstyle()\CD@ND}%% ASCII round brackets
+\setbox1=\null\ht1\ht0\dp1\dp0\box1 \kern.1\CD@zC\CD@k\bgroup\labelstyle
+\aftergroup\CD@LD\CD@xD}\def\CD@LD{\CD@ND\kern.1\CD@zC\egroup\CD@tD}\def
+\CD@xD{\futurelet\CD@EH\CD@mJ}\def\CD@mJ{%% qualifiers on label arguments
+\catcase\bgroup:\CD@v;\catcase\egroup:\missing@label;\catcase\space:\CD@TF;%
+\tokcase[:\CD@XF;%%]%ascii close square bracket 
+\default:\CD@zJ;\endswitch}\def\CD@v{\let\CD@MD\CD@c\let\CD@CH}\def\CD@zJ#1{%
+\let\CD@UF\egroup{\let\actually@braces@missing@around@macro@in@label\CD@ZH
+\let\CD@MD\CD@xC\let\CD@UF\CD@VF#1%
+\actually@braces@missing@around@macro@in@label}\CD@UF}\def
+\actually@braces@missing@around@macro@in@label{\let\CD@CH=}\def\missing@label
+{\egroup\CD@YA{missing label}\CD@PE}\def\CD@xC{\egroup\missing@label}\outer
+\def\CD@ZH{}\def\CD@UF{}\def\CD@VF{\CD@wC\CD@UF}\def\CD@MD{}\def\CD@XF{\let
+\CD@N\CD@xD\get@square@arg\CD@AE}\CD@rG\CD@PE{The text which has just been
+read is not allowed within map labels.}\def\CD@c{\egroup\CD@YA{missing \CD@yC
+\space inserted after label}\CD@PE}\def\upper@label{\CD@oD\CD@yD6}\def
+\lower@label{\def\positional@{\CD@@A\break@args}\CD@yD7}\def\middle@label{%
+\CD@yD3}\CD@tG\CD@yE\CD@pD\CD@oD\def\CD@iF{\ifPositiveGradient\CD@tJ
+\expandafter\upper@label\else\expandafter\lower@label\fi}\def\CD@iI{%
+\ifPositiveGradient\CD@tJ\expandafter\lower@label\else\expandafter
+\upper@label\fi}\def\positional@{\CD@gB{labels as positional arguments are
+obsolete}\CD@yE\CD@tJ\expandafter\upper@label\else\expandafter\lower@label\fi
+-}\def\CD@tD{\futurelet\CD@EH\switch@arg}\def\eat@space{\afterassignment
+\CD@tD\let\CD@EH= }\def\CD@TF{\afterassignment\CD@xD\let\CD@EH= }\def\CD@BC{%
+\get@round@pair\CD@uD}\def\CD@uD#1#2{\def\CD@WK{#1}\def\CD@aK{#2}\CD@tD}\def
+\optional@{\let\CD@N\CD@tD\get@square@arg\CD@AE}\def\CD@JJ.{\CD@sC\CD@tD}\def
+\CD@sC{\let\CD@iD\fill@dot\let\CD@jD\fill@dot\def\CD@MI{\let\CD@iD\dfdot\let
+\CD@jD\dfdot}}\def\CD@MI{}\def\CD@@E#1,{\CD@nH#1,\begingroup\ifx\@name\CD@RD
+\CD@FF\aftergroup\CD@e\fi\aftergroup\CD@jC\else\expandafter\def\expandafter
+\CD@RF\expandafter{\csname\@name\endcsname}\expandafter\CD@vD\CD@RF\CD@KD\ifx
+\CD@RF\empty\aftergroup\CD@pC\expandafter\aftergroup\csname\CD@FB\@name
+\endcsname\expandafter\aftergroup\csname\CD@FB @\@name\endcsname\else\gdef
+\CD@GE{#1}\CD@gB{\string\relax\space inserted before `[\CD@GE'}\message{(I was
+trying to read this as a \CD@tA\ option.)}\aftergroup\CD@H\fi\fi\endgroup}%
+\def\CD@vD#1#2\CD@KD{\def\CD@RF{#2}}\def\CD@jC{\let\CD@CH\CD@N\let\CD@N\relax
+\CD@CH}\def\CD@H#1],{%% ASCII close square bracket
+\CD@jC\relax\def\CD@RF{#1}\ifx\CD@RF\empty\def\CD@RF{[\CD@GE]}%
+%% ASCII open and close square bracket
+\else\def\CD@RF{[\CD@GE,#1]}%% ASCII open and close square bracket
+\fi\CD@RF}\def\CD@pC#1#2{\ifx#2\CD@qK\ifx#1\CD@qK\CD@gB{option `\@name'
+undefined}\else#1\fi\else\CD@FF\expandafter#2\CD@GK\CD@PK\else\CD@QK\fi\fi
+\CD@DH}\CD@tG\CD@FF\CD@QK\CD@PK\def\CD@nH#1,{\CD@FF\ifx\CD@GK\CD@qK\CD@e\else
+\expandafter\CD@oH\CD@GK,#1,(,),(,)[]%
+%%ASCII 5commas two pairs round, pair square
+\fi\fi\CD@FF\else\CD@mH#1==,\fi}\def\CD@e{\CD@gB{option `\@name' needs (x,y)
+value}\CD@PK\let\@name\empty}\def\CD@mH#1=#2=#3,{\def\@name{#1}\def\CD@GK{#2}%
+\def\CD@RF{#3}\ifx\CD@RF\empty\let\CD@GK\CD@qK\fi}%
+%% ASCII 2commas 2pair round, pair square on next line
+\def\CD@oH#1(#2,#3)#4,(#5,#6)#7[]{\def\CD@GK{{#2}{#3}}\def\CD@RF{#1#4#5#6}%
+\ifx\CD@RF\empty\def\CD@RF{#7}\ifx\CD@RF\empty\CD@e\fi\else\CD@e\fi}\def
+\CD@FB{cds@}\let\CD@N\relax\def\CD@zD#1{\ifx\CD@GK\CD@qK\CD@gB{option `\@name
+' needs a value}\else#1\CD@GK\relax\fi}\def\CD@BE#1#2{\ifx\CD@GK\CD@qK#1#2%
+\relax\else#1\CD@GK\relax\fi}\def\cds@@showpair#1#2{\message{x=#1,y=#2}}\def
+\cds@@diagonalbase#1#2{\edef\CD@ZK{#1}\edef\CD@bK{#2}}\def\CD@DI#1{\def\CD@CH
+{#1}\CD@nF{@x}{cdps@#1}\ifx\CD@CH\empty\CD@f\CD@CH{cannot be used}\else\ifx
+\CD@CH\relax\CD@f\CD@CH{unknown}\else\let\CD@IK\@x\fi\fi}\def\CD@f#1#2{\CD@gB
+{PostScript translator `#1' #2}}\def\CD@PH{}\def\CD@PJ{\CD@fA\edef\CD@PH{%
+\noexpand\CD@KB{\@name\space ignored within maths}}}\def\diagramstyle{\CD@cJ
+\let\CD@N\relax\CD@CF\CD@AE\CD@AE}\let\diagramsstyle\diagramstyle\CD@tG\CD@sE
+\CD@SB\CD@RB\CD@tG\CD@qE\CD@EB\CD@DB\CD@tG\CD@oE\CD@pA\CD@oA\CD@tG\CD@iE
+\CD@HA\CD@GA\CD@HA\CD@tG\CD@jE\CD@JA\CD@IA\CD@tG\CD@kE\CD@LA\CD@KA\CD@tG
+\CD@EF\CD@DK\CD@CK\CD@tG\CD@rE\CD@JB\CD@IB\CD@tG\CD@mE\CD@gA\CD@fA\CD@tG
+\CD@nE\CD@kA\CD@jA\CD@tG\CD@AF\CD@iG\CD@hG\CD@RC{cds@ }{}\CD@RC{cds@}{}\CD@RC
+{cds@1em}{\CellSize1\CD@zC}\CD@RC{cds@1.5em}{\CellSize1.5\CD@zC}\CD@RC{cds@2%
+em}{\CellSize2\CD@zC}\CD@RC{cds@2.5em}{\CellSize2.5\CD@zC}\CD@RC{cds@3em}{%
+\CellSize3\CD@zC}\CD@RC{cds@3.5em}{\CellSize3.5\CD@zC}\CD@RC{cds@4em}{%
+\CellSize4\CD@zC}\CD@RC{cds@4.5em}{\CellSize4.5\CD@zC}\CD@RC{cds@5em}{%
+\CellSize5\CD@zC}\CD@RC{cds@6em}{\CellSize6\CD@zC}\CD@RC{cds@7em}{\CellSize7%
+\CD@zC}\CD@RC{cds@8em}{\CellSize8\CD@zC}\def\cds@abut{\MapsAbut\dimen1\z@
+\dimen5\z@}\def\cds@alignlabels{\CD@IA\CD@KA}\def\cds@amstex{\ifincommdiag
+\CD@O\else\def\CD{\diagram[amstex]}%%ascii square brackets []
+\fi\CD@T\catcode`\@\active}\def\cds@b{\let\CD@dB\CD@bB}\def\cds@balance{\let
+\CD@hA\CD@AA}\let\cds@bottom\cds@b\def\cds@center{\cds@vcentre\cds@nobalance}%
+\let\cds@centre\cds@center\def\cds@centerdisplay{\CD@HA\CD@PJ\cds@balance}%
+\let\cds@centredisplay\cds@centerdisplay\def\cds@crab{\CD@BE\CD@DC{.5%
+\PileSpacing}}\CD@RC{cds@crab-}{\CD@DC-.5\PileSpacing}\CD@RC{cds@crab+}{%
+\CD@DC.5\PileSpacing}\CD@RC{cds@crab++}{\CD@DC1.5\PileSpacing}\CD@RC{cds@crab%
+--}{\CD@DC-1.5\PileSpacing}\def\cds@defaultsize{\CD@BE{\let\CD@QC}{3em}\CD@NJ
+}\def\cds@displayoneliner{\CD@DB}\let\cds@dotted\CD@sC\def\cds@dpi{\CD@RJ{1%
+truein}}\def\cds@dpm{\CD@RJ{100truecm}}\let\CD@XA\CD@qK\def\cds@eqno{\let
+\CD@XA\CD@GK\let\CD@EJ\empty}\def\cds@fixed{\CD@qA}\CD@tG\CD@fE\CD@J\CD@I\def
+\cds@flushleft{\CD@I\CD@GA\CD@PJ\cds@nobalance\CD@BE\CD@nA\CD@nA}\def\cds@gap
+{\CD@AJ\setbox3=\null\ht3=\CD@tI\dp3=\CD@sI\CD@BE{\wd3=}\MapShortFall} \def
+\cds@grid{\ifx\CD@GK\CD@qK\let\h@grid\relax\let\v@grid\relax\else\CD@nF{%
+h@grid}{cdgh@\CD@GK}\CD@nF{v@grid}{cdgv@\CD@GK}\ifx\h@grid\relax\CD@gB{%
+unknown grid `\CD@GK'}\else\CD@WB\fi\fi}\let\h@grid\relax\let\v@grid\relax
+\def\cds@gridx{\ifx\CD@GK\CD@qK\else\cds@grid\fi\let\CD@CH\h@grid\let\h@grid
+\v@grid\let\v@grid\CD@CH}\def\cds@h{\CD@zD\DiagramCellHeight}\def\cds@hcenter
+{\let\CD@hA\CD@aA}\let\cds@hcentre\cds@hcenter\def\cds@heads{\CD@BE{\let
+\CD@sJ}\CD@sJ\CD@@J\CD@vE\else\ifx\CD@sJ\CD@eF\else\CD@MC\fi\fi}\let
+\cds@height\cds@h\let\cds@hmiddle\cds@balance\def\cds@htriangleheight{\CD@BE
+\DiagramCellHeight\DiagramCellHeight\DiagramCellWidth1.73205%
+\DiagramCellHeight}\def\cds@htrianglewidth{\CD@BE\DiagramCellWidth
+\DiagramCellWidth\DiagramCellHeight.57735\DiagramCellWidth}\CD@tG\CD@zE\CD@eE
+\CD@dE\CD@eE\def\cds@hug{\CD@eE} \def\cds@inline{\CD@gA\let\CD@PH\empty}\def
+\cds@inlineoneliner{\CD@EB}\CD@RC{cds@l>}{\CD@zD{\let\CD@RG}\dimen2=\CD@RG}%
+\def\cds@labelstyle{\CD@zD{\let\labelstyle}}\def\cds@landscape{\CD@kA}\def
+\cds@large{\CellSize5\CD@zC}\let\CD@EJ\empty\def\CD@FJ{\refstepcounter{%
+equation}\def\CD@XA{\hbox{\@eqnnum}}}\def\cds@LaTeXeqno{\let\CD@EJ\CD@FJ}\def
+\cds@lefteqno{\CD@pA}\def\cds@leftflush{\cds@flushleft\CD@J}\def
+\cds@leftshortfall{\CD@zD{\dimen1 }}\def\cds@lowershortfall{%
+\ifPositiveGradient\cds@leftshortfall\else\cds@rightshortfall\fi}\def
+\cds@loose{\CD@VB}\def\cds@midhshaft{\CD@JA}\def\cds@midshaft{\CD@JA}\def
+\cds@midvshaft{\CD@LA}\def\cds@moreoptions{\CD@@A}\let\cds@nobalance
+\cds@hcenter\def\cds@nohcheck{\CD@HH}\def\cds@nohug{\CD@dE} \def
+\cds@nooptions{\def\CD@aC{\CD@WD}}\let\cds@noorigin\cds@nobalance\def
+\cds@nopixel{\CD@@I4\CD@XH\CD@cJ}\def\cds@UO{\CD@oK\global\let\CD@n\empty}%
+\def\cds@UglyObsolete{\cds@UO\let\cds@PS\empty}\def\CD@rK#1{\CD@gB{option `#1%
+' renamed as `UglyObsolete'}}\def\cds@noPostScript{\CD@rK{noPostScript}}\def
+\cds@noPS{\CD@rK{noPostScript}}\def\cds@notextflow{\CD@RB}\def\cds@noTPIC{%
+\CD@CK}\def\cds@objectstyle{\CD@zD{\let\objectstyle}}\def\cds@origin{\let
+\CD@hA\CD@iB}\def\cds@p{\CD@zD\PileSpacing}\let\cds@pilespacing\cds@p\def
+\cds@pixelsize{\CD@zD\CD@@I\CD@gI}\def\cds@portrait{\CD@jA}\def
+\cds@PostScript{\CD@nK\global\let\CD@n\empty\CD@BE\CD@DI\empty}\def\cds@PS{%
+\CD@nK\global\let\CD@n\empty}\CD@GF\CD@n{\typeout{\CD@tA: try the PostScript
+option for better results}}\def\cds@repositionpullbacks{\let\make@pbk\CD@fH
+\let\CD@qH\CD@pH}\def\cds@righteqno{\CD@oA}\def\cds@rightshortfall{\CD@zD{%
+\dimen5 }}\def\cds@ruleaxis{\CD@zD{\let\axisheight}}\def\cds@cmex{\let\CD@GG
+\CD@sB\let\CD@QJ\CD@CJ}\def\cds@s{\cds@height\DiagramCellWidth
+\DiagramCellHeight}\def\cds@scriptlabels{\let\labelstyle\scriptstyle}\def
+\cds@shortfall{\CD@zD\MapShortFall\dimen1\MapShortFall\dimen5\MapShortFall}%
+\def\cds@showfirstpass{\CD@BE{\let\CD@nD}\z@}\def\cds@silent{\def\CD@KB##1{}%
+\def\CD@gB##1{}}\let\cds@size\cds@s\def\cds@small{\CellSize2\CD@zC}\def
+\cds@snake{\CD@BE\CD@eJ\z@}\def\cds@t{\let\CD@dB\CD@fB}\def\cds@textflow{%
+\CD@SB\CD@PJ}\def\cds@thick{\let\CD@rF\tenlnw\CD@LF\CD@NC\CD@BE\MapBreadth{2%
+\CD@LF}\CD@@J}\def\cds@thin{\let\CD@rF\tenln\CD@BE\MapBreadth{\CD@NC}\CD@@J}%
+\def\cds@tight{\CD@WB}\let\cds@top\cds@t\def\cds@TPIC{\CD@DK}\def
+\cds@uppershortfall{\ifPositiveGradient\cds@rightshortfall\else
+\cds@leftshortfall\fi}\def\cds@vcenter{\let\CD@dB\CD@cB}\let\cds@vcentre
+\cds@vcenter\def\cds@vtriangleheight{\CD@BE\DiagramCellHeight
+\DiagramCellHeight\DiagramCellWidth.577035\DiagramCellHeight}\def
+\cds@vtrianglewidth{\CD@BE\DiagramCellWidth\DiagramCellWidth
+\DiagramCellHeight1.73205\DiagramCellWidth}\def\cds@vmiddle{\let\CD@dB\CD@eB}%
+\def\cds@w{\CD@zD\DiagramCellWidth}\let\cds@width\cds@w\def\diagram{\relax
+\protect\CD@bC}\def\enddiagram{\protect\CD@SG}\def\diagraminline{\diagram[%
+inline,moreoptions]}\def\enddiagraminline{\enddiagram}\def\CD@bC{\CD@g\CD@uI
+\incommdiagtrue\edef\CD@wI{\the\CD@NB}\global\CD@NB\z@\boxmaxdepth\maxdimen
+\everycr{}\CD@sK\everymath{}\everyhbox{}\ifx\pdfsyncstop\CD@qK\else
+\pdfsyncstop\fi\CD@aC}\def\CD@aC{\CD@y\let\CD@N\CD@ZC\CD@CF\CD@AE\CD@WD}\def
+\CD@ZC{\CD@gE\expandafter\CD@aC\else\expandafter\CD@WD\fi}\def\CD@WD{\let
+\CD@EH\relax\CD@nE\CD@vE\else\CD@hK\else\CD@KB{landscape ignored without
+PostScript}\CD@jA\fi\fi\fi\CD@EJ\setbox2=\vbox\bgroup\CD@JF\CD@VD}\def\CD@cH{%
+\CD@nE\CD@fB\else\CD@dB\fi\CD@hA\nointerlineskip\setbox0=\null\ht0-\CD@pI\dp0%
+\CD@pI\wd0\CD@kI\box0 \global\CD@QA\CD@kF\global\CD@yA\CD@XB\ifx\CD@NK\CD@qK
+\global\CD@RA\CD@kF\else\global\CD@RA\CD@NK\fi\egroup\CD@zF\CD@nE\setbox2=%
+\hbox to\dp2{\vrule height\wd2 depth\CD@QA width\z@\global\CD@QA\ht2\ht2\z@
+\dp2\z@\wd2\z@\CD@hK\CD@tK{q 0 1 -1 0 0 0 cm}\else\global\CD@iG\CD@IK{0 1
+bturn}\fi\box2\CD@gK\hss}\CD@DB\fi\ifnum\CD@yA=1 \else\CD@DB\fi\global
+\@ignorefalse\CD@mE\leavevmode\fi\ifvmode\CD@TA\else\ifmmode\CD@PH\CD@GI\else
+\CD@qE\CD@gA\fi\ifinner\CD@gA\fi\CD@mE\CD@GI\else\CD@sE\CD@QB\else\CD@TA\fi
+\fi\fi\fi\CD@dD}\def\CD@dD{\global\CD@NB\CD@wI\relax\CD@xE\global\CD@ID\else
+\aftergroup\CD@mC\fi\if@ignore\aftergroup\ignorespaces\fi\CD@wC\ignorespaces}%
+\def\CD@fB{\advance\CD@pI\dimen1\relax}\def\CD@eB{\advance\CD@pI.5\dimen1%
+\relax}\def\CD@bB{}\def\CD@cB{\CD@fB\advance\CD@pI\CD@YB\divide\CD@pI2
+\advance\CD@pI-\axisheight\relax}\def\CD@aA{}\def\CD@iB{\CD@kF\z@}\def\CD@AA{%
+\ifdim\dimen2>\CD@kF\CD@kF\dimen2 \else\dimen2\CD@kF\CD@kI\dimen0 \advance
+\CD@kI\dimen2 \fi}\def\CD@QB{\skip0\z@\relax\loop\skip1\lastskip\ifdim\skip1>%
+\z@\unskip\advance\skip0\skip1 \repeat\vadjust{\prevdepth\dp\strutbox\penalty
+\predisplaypenalty\vskip\abovedisplayskip\CD@UA\penalty\postdisplaypenalty
+\vskip\belowdisplayskip}\ifdim\skip0=\z@\else\hskip\skip0 \global\@ignoretrue
+\fi}\def\CD@TA{\CD@LG\kern-\displayindent\CD@UA\CD@LG\global\@ignoretrue}\def
+\CD@UA{\hbox to\hsize{\CD@fE\ifdim\CD@RA=\z@\else\advance\CD@QA-\CD@RA\setbox
+2=\hbox{\kern\CD@RA\box2}\fi\fi\setbox1=\hbox{\ifx\CD@XA\CD@qK\else\CD@k
+\CD@XA\CD@ND\fi}\CD@oE\CD@iE\else\advance\CD@QA\wd1 \fi\wd1\z@\box1 \fi\dimen
+0\wd2 \advance\dimen0\wd1 \advance\dimen0-\hsize\ifdim\dimen0>-\CD@nA\CD@HA
+\fi\advance\dimen0\CD@QA\ifdim\dimen0>\z@\CD@KB{wider than the page by \the
+\dimen0 }\CD@HA\fi\CD@iE\hss\else\CD@V\CD@QA\CD@nA\fi\CD@GI\hss\kern-\wd1\box
+1 }}\def\CD@GI{\CD@AF\CD@@F\else\CD@SC\global\CD@hG\fi\fi\kern\CD@QA\box2 }%
+\CD@tG\CD@wE\CD@YC\CD@XC\def\CD@JF{\CD@cJ\ifdim\DiagramCellHeight=-\maxdimen
+\DiagramCellHeight\CD@QC\fi\ifdim\DiagramCellWidth=-\maxdimen
+\DiagramCellWidth\CD@QC\fi\global\CD@XC\CD@IF\let\CD@FE\empty\let\CD@z\CD@Q
+\let\overprint\CD@eH\let\CD@s\CD@rJ\let\enddiagram\CD@ED\let\\\CD@cC\let\par
+\CD@jH\let\CD@MD\empty\let\switch@arg\CD@PB\let\shift\CD@iA\baselineskip
+\DiagramCellHeight\lineskip\z@\lineskiplimit\z@\mathsurround\z@\tabskip\z@
+\CD@OB}\def\CD@VD{\penalty-123 \begingroup\CD@jA\aftergroup\CD@K\halign
+\bgroup\global\advance\CD@NB1 \vadjust{\penalty1}\global\CD@FA\z@\CD@OB\CD@j#%
+#\CD@DD\CD@Q\CD@Q\CD@OI\CD@j##\CD@DD\cr}\def\CD@ED{\CD@MD\CD@GD\crcr\egroup
+\global\CD@JD\endgroup}\def\CD@j{\global\advance\CD@FA1 \futurelet\CD@EH\CD@i
+}\def\CD@i{\ifx\CD@EH\CD@DD\CD@tJ\hskip1sp plus 1fil \relax\let\CD@DD\relax
+\CD@vI\else\hfil\CD@k\objectstyle\let\CD@FE\CD@d\fi}\def\CD@DD{\CD@MD\relax
+\CD@yI\CD@vI\global\CD@QA\CD@iA\penalty-9993 \CD@ND\hfil\null\kern-2\CD@QA
+\null}\def\CD@cC{\cr}\def\across#1{\span\omit\mscount=#1 \global\advance
+\CD@FA\mscount\global\advance\CD@FA\m@ne\CD@sF\ifnum\mscount>2 \CD@fJ\repeat
+\ignorespaces}\def\CD@fJ{\relax\span\omit\advance\mscount\m@ne}\def\CD@qJ{%
+\ifincommdiag\ifx\CD@iD\@fillh\ifx\CD@jD\@fillh\ifdim\dimen3>\z@\else\ifdim
+\dimen2>93\CD@@I\ifdim\dimen2>18\p@\ifdim\CD@LF>\z@\count@\CD@bJ\advance
+\count@\m@ne\ifnum\count@<\z@\count@20\let\CD@aJ\CD@uJ\fi\xdef\CD@bJ{\the
+\count@}\fi\fi\fi\fi\fi\fi\fi}\def\CD@cG#1{\vrule\horizhtdp width#1\dimen@
+\kern2\dimen@}\def\CD@uJ{\rlap{\dimen@\CD@@I\CD@V\dimen@{.182\p@}\CD@zH
+\dimen@\advance\CD@tI\dimen@\CD@cG0\CD@cG0\CD@cG2\CD@cG6\CD@cG6\CD@cG2\CD@cG0%
+\CD@cG0\CD@cG2\CD@cG6\CD@cG0\CD@cG0\CD@cG2\CD@cG2\CD@cG6\CD@cG0\CD@cG0\CD@cG2%
+\CD@cG6\CD@cG2\CD@cG2\CD@cG0\CD@cG0}}\def\CD@bJ{10}\def\CD@aJ{}\def\CD@XD{%
+\CD@gE\CD@TB\fi\CD@x\CD@WF\CD@HI}\def\CD@x{\CD@QJ\CD@DC\CD@MJ\ifdim\CD@DC=\z@
+\else\CD@pF\CD@DC\fi\ifvoid3 \setbox3=\null\ht3\CD@tI\dp3\CD@sI\else\CD@V{\ht
+3}\CD@tI\CD@V{\dp3}\CD@sI\fi\dimen3=.5\wd3 \ifdim\dimen3=\z@\CD@tE\else\dimen
+3-\CD@XH\fi\else\CD@TB\fi\CD@V{\dimen2}{\wd7}\CD@V{\dimen2}{\wd6}\CD@qJ
+\advance\dimen2-2\dimen3 \dimen4.5\dimen2 \dimen2\dimen4 \advance\dimen2%
+\CD@eJ\advance\dimen4-\CD@eJ\advance\dimen2-\wd1 \advance\dimen4-\wd5 \ifvoid
+2 \else\CD@V{\ht3}{\ht2}\CD@V{\dp3}{\dp2}\CD@V{\dimen2}{\wd2}\fi\ifvoid4 \else
+\CD@V{\ht3}{\ht4}\CD@V{\dp3}{\dp4}\CD@V{\dimen4}{\wd4}\fi\advance\skip2\dimen
+2 \advance\skip4\dimen4 \CD@tE\advance\skip2\skip4 \dimen0\dimen5 \advance
+\dimen0\wd5 \skip3-\skip4 \advance\skip3-\dimen0 \let\CD@jD\empty\else\skip3%
+\z@\relax\dimen0\z@\fi}\def\CD@WF{\offinterlineskip\lineskip.2\CD@zC\ifvoid6
+\else\setbox3=\vbox{\hbox to2\dimen3{\hss\box6\hss}\box3}\fi\ifvoid7 \else
+\setbox3=\vtop{\box3 \hbox to2\dimen3{\hss\box7\hss}}\fi}\def\CD@HI{\kern
+\dimen1 \box1 \CD@aJ\CD@iD\hskip\skip2 \kern\dimen0 \ifincommdiag\CD@jE
+\penalty1\fi\kern\dimen3 \penalty\CD@GB\hskip\skip3 \null\kern-\dimen3 \else
+\hskip\skip3 \fi\box3 \CD@jD\hskip\skip4 \box5 \kern\dimen5}\def\CD@MF{\ifnum
+\CD@LH>\CD@TC\CD@V{\dimen1}\objectheight\CD@V{\dimen5}\objectheight\else\CD@V
+{\dimen1}\objectwidth\CD@V{\dimen5}\objectwidth\fi}\def\CD@Y{\begingroup
+\ifdim\dimen7=\z@\kern\dimen8 \else\ifdim\dimen6=\z@\kern\dimen9 \else\dimen5%
+\dimen6 \dimen6\dimen9 \CD@KJ\dimen4\dimen2 \CD@dG{\dimen4}\dimen6\dimen5
+\dimen7\dimen8 \CD@KJ\CD@iC{\dimen2}\ifdim\dimen2<\dimen4 \kern\dimen2 \else
+\kern\dimen4 \fi\fi\fi\endgroup}\def\CD@jJ{\CD@JI\setbox\z@\hbox{\lower
+\axisheight\hbox to\dimen2{\CD@DF\ifPositiveGradient\dimen8\ht\CD@MH\dimen9%
+\CD@mI\else\dimen8\dp3 \dimen9\dimen1 \fi\else\dimen8 \ifPositiveGradient
+\objectheight\else\z@\fi\dimen9\objectwidth\fi\advance\dimen8
+\ifPositiveGradient-\fi\axisheight\CD@Y\unhbox\z@\CD@DF\ifPositiveGradient
+\dimen8\dp3 \dimen9\dimen0 \else\dimen8\ht\CD@MH\dimen9\CD@mF\fi\else\dimen8
+\ifPositiveGradient\z@\else\objectheight\fi\dimen9\objectwidth\fi\advance
+\dimen8 \ifPositiveGradient\else-\fi\axisheight\CD@Y}}}\def\CD@bD{\dimen6
+\CD@aK\DiagramCellHeight\dimen7 \CD@WK\DiagramCellWidth\CD@jJ
+\ifPositiveGradient\advance\dimen7-\CD@ZK\DiagramCellWidth\else\dimen7 \CD@ZK
+\DiagramCellWidth\dimen6\z@\fi\advance\dimen6-\CD@bK\DiagramCellHeight\CD@mK
+\setbox0=\rlap{\kern-\dimen7 \lower\dimen6\box\z@}\ht0\z@\dp0\z@\raise
+\axisheight\box0 }\def\CD@mK{\setbox0\hbox{\ht\z@\z@\dp\z@\z@\wd\z@\z@\CD@hK
+\expandafter\CD@tK{q \CD@eK\space\CD@lK\space\CD@kK\space\CD@eK\space0 0 cm}%
+\else\global\CD@iG\CD@eD{\the\CD@TC\space\ifPositiveGradient\else-\fi\the
+\CD@LH\space bturn}\fi\box\z@\CD@gK}}\def\CD@vB{\advance\CD@hF-\CD@mI\CD@wJ
+\CD@hF\advance\CD@wJ\CD@hI\ifvoid\CD@sH\ifdim\CD@wJ<.1em\ifnum\CD@gD=\@m\else
+\CD@aG h\CD@wJ<.1em:objects overprint:\CD@FA\CD@gD\fi\fi\else\ifhbox\CD@sH
+\CD@SK\else\CD@TK\fi\advance\CD@wJ\CD@mI\CD@bH{-\CD@mI}{\box\CD@sH}{\CD@wJ}%
+\z@\fi\CD@hF-\CD@mF\CD@gD\CD@FA\CD@hI\z@}\def\CD@SK{\setbox\CD@sH=\hbox{%
+\unhbox\CD@sH\unskip\unpenalty}\setbox\CD@tH=\hbox{\unhbox\CD@tH\unskip
+\unpenalty}\setbox\CD@sH=\hbox to\CD@wJ{\CD@OA\wd\CD@sH\unhbox\CD@sH\CD@PA
+\lastkern\unkern\ifdim\CD@PA=\z@\CD@UB\advance\CD@OA-\wd\CD@tH\else\CD@TB\fi
+\ifnum\lastpenalty=\z@\else\CD@JA\unpenalty\fi\kern\CD@PA\ifdim\CD@hF<\CD@OA
+\CD@JA\fi\ifdim\CD@hI<\wd\CD@tH\CD@JA\fi\CD@jE\CD@hI\CD@wJ\advance\CD@hI-%
+\CD@OA\advance\CD@hI\wd\CD@tH\ifdim\CD@hI<2\wd\CD@tH\CD@aG h\CD@hI<2\wd\CD@tH
+:arrow too short:\CD@FA\CD@gD\fi\divide\CD@hI\tw@\CD@hF\CD@wJ\advance\CD@hF-%
+\CD@hI\fi\CD@tE\kern-\CD@hI\fi\hbox to\CD@hI{\unhbox\CD@tH}\CD@HG}}\CD@tG
+\ifinpile\inpiletrue\inpilefalse\inpilefalse\def\pile{\protect\CD@UJ\protect
+\CD@uH}\def\CD@uH#1{\CD@l#1\CD@QD}\def\CD@UJ{\CD@nB{pile}\setbox0=\vtop
+\bgroup\aftergroup\CD@lD\inpiletrue\let\CD@FE\empty\let\pile\CD@KF\let\CD@QD
+\CD@PD\let\CD@GD\CD@FD\CD@yH\baselineskip.5\PileSpacing\lineskip.1\CD@zC
+\relax\lineskiplimit\lineskip\mathsurround\z@\tabskip\z@\let\\\CD@wH}\def
+\CD@l{\CD@DE\CD@YF\empty\halign\bgroup\hfil\CD@k\let\CD@FE\CD@d\let\\\CD@vH##%
+\CD@MD\CD@ND\hfil\CD@Q\CD@R##\cr}\CD@rG\CD@NE{pile only allows one column.}%
+\CD@rG\CD@UE{you left it out!}\def\CD@R{\CD@QD\CD@Q\relax\CD@YA{missing \CD@yC
+\space inserted after \string\pile}\CD@NE}\def\CD@PD{\CD@MD\crcr\egroup
+\egroup}\def\CD@GD{\CD@MD}\def\CD@FD{\CD@MD\relax\CD@QD\CD@YA{missing \CD@yC
+\space inserted between \string\pile\space and \CD@HD}\CD@UE}\def\CD@QD{%
+\CD@MD}\def\CD@lD{\vbox{\dimen1\dp0 \unvbox0 \setbox0=\lastbox\advance\dimen1%
+\dp0 \nointerlineskip\box0 \nointerlineskip\setbox0=\null\dp0.5\dimen1\ht0-%
+\dp0 \box0}\ifincommdiag\CD@tJ\penalty-9998 \fi\xdef\CD@YF{pile}}\def\CD@vH{%
+\cr}\def\CD@wH{\noalign{\skip@\prevdepth\advance\skip@-\baselineskip
+\prevdepth\skip@}}\def\CD@KF#1{#1}\def\CD@TK{\setbox\CD@sH=\vbox{\unvbox
+\CD@sH\setbox1=\lastbox\setbox0=\box\voidb@x\CD@tF\setbox\CD@sH=\lastbox
+\ifhbox\CD@sH\CD@rC\repeat\unvbox0 \global\CD@QA\CD@ZE}\CD@ZE\CD@QA}\def
+\CD@rC{\CD@jE\setbox\CD@sH=\hbox{\unhbox\CD@sH\unskip\setbox\CD@sH=\lastbox
+\unskip\unhbox\CD@sH}\ifdim\CD@wJ<\wd\CD@sH\CD@aG h\CD@wJ<\wd\CD@sH:arrow in
+pile too short:\CD@FA\CD@gD\else\setbox\CD@sH=\hbox to\CD@wJ{\unhbox\CD@sH}%
+\fi\else\CD@gJ\fi\setbox0=\vbox{\box\CD@sH\nointerlineskip\ifvoid0 \CD@tJ\box
+1 \else\vskip\skip0 \unvbox0 \fi}\skip0=\lastskip\unskip}\def\CD@gJ{\penalty7
+\noindent\unhbox\CD@sH\unskip\setbox\CD@sH=\lastbox\unskip\unhbox\CD@sH
+\endgraf\setbox\CD@tH=\lastbox\unskip\setbox\CD@tH=\hbox{\CD@JG\unhbox\CD@tH
+\unskip\unskip\unpenalty}\ifcase\prevgraf\cd@shouldnt P\or\ifdim\CD@wJ<\wd
+\CD@tH\CD@aG h\CD@wJ<\wd\CD@sH:object in pile too wide:\CD@FA\CD@gD\setbox
+\CD@sH=\hbox to\CD@wJ{\hss\unhbox\CD@tH\hss}\else\setbox\CD@sH=\hbox to\CD@wJ
+{\hss\kern\CD@hF\unhbox\CD@tH\kern\CD@hI\hss}\fi\or\setbox\CD@sH=\lastbox
+\unskip\CD@SK\else\cd@shouldnt Q\fi\unskip\unpenalty}\def\CD@cD{\CD@MJ\ifvoid
+3 \setbox3=\null\ht3\axisheight\dp3-\ht3 \dimen3.5\CD@LF\else\dimen4\dp3
+\dimen3.5\wd3 \setbox3=\CD@GG{\box3}\dp3\dimen4 \ifdim\ht3=-\dp3 \else\CD@TB
+\fi\fi\dimen0\dimen3 \advance\dimen0-.5\CD@LF\setbox0\null\ht0\ht3\dp0\dp3\wd
+0\wd3 \ifvoid6\else\setbox6\hbox{\unhbox6\kern\dimen0\kern2pt}\dimen0\wd6 \fi
+\ifvoid7\else\setbox7\hbox{\kern2pt\kern\dimen3\unhbox7}\dimen3\wd7 \fi
+\setbox3\hbox{\ifvoid6\else\kern-\dimen0\unhbox6\fi\unhbox3 \ifvoid7\else
+\unhbox7\kern-\dimen3\fi}\ht3\ht0\dp3\dp0\wd3\wd0 \CD@tE\dimen4=\ht\CD@MH
+\advance\dimen4\dp5 \advance\dimen4\dimen1 \let\CD@jD\empty\else\dimen4\ht3
+\fi\setbox0\null\ht0\dimen4 \offinterlineskip\setbox8=\vbox spread2ex{\kern
+\dimen5 \box1 \CD@iD\vfill\CD@tE\else\kern\CD@eJ\fi\box0}\ht8=\z@\setbox9=%
+\vtop spread2ex{\kern-\ht3 \kern-\CD@eJ\box3 \CD@jD\vfill\box5 \kern\dimen1}%
+\dp9=\z@\hskip\dimen0plus.0001fil \box9 \kern-\CD@LF\box8 \CD@kE\penalty2 \fi
+\CD@tE\penalty1 \fi\kern\PileSpacing\kern-\PileSpacing\kern-.5\CD@LF\penalty
+\CD@GB\null\kern\dimen3}\def\CD@cI{\ifhbox\CD@VA\CD@KB{clashing verticals}\ht
+\CD@MH.5\dp\CD@VA\dp\CD@MH-\ht5 \CD@yB\ht\CD@MH\z@\dp\CD@MH\z@\fi\dimen1\dp
+\CD@VA\CD@xA\prevgraf\unvbox\CD@VA\CD@wA\lastpenalty\unpenalty\setbox\CD@VA=%
+\null\setbox\CD@lI=\hbox{\CD@JG\unhbox\CD@lI\unskip\unpenalty\dimen0\lastkern
+\unkern\unkern\unkern\kern\dimen0 \CD@HG}\setbox\CD@lF=\hbox{\unhbox\CD@lF
+\dimen0\lastkern\unkern\unkern\global\CD@QA\lastkern\unkern\kern\dimen0 }%
+\CD@tF\ifnum\CD@xA>4 \CD@zI\repeat\unskip\unskip\advance\CD@mF.5\wd\CD@VA
+\advance\CD@mF\wd\CD@lF\advance\CD@mI.5\wd\CD@VA\advance\CD@mI\wd\CD@lI\ifnum
+\CD@FA=\CD@lA\CD@OA.5\wd\CD@VA\edef\CD@NK{\the\CD@OA}\fi\setbox\CD@VA=\hbox{%
+\kern-\CD@mF\box\CD@lF\unhbox\CD@VA\box\CD@lI\kern-\CD@mI\penalty\CD@wA
+\penalty\CD@NB}\ht\CD@VA\dimen1 \dp\CD@VA\z@\wd\CD@VA\CD@tB\CD@vB}\def\CD@zI{%
+\ifdim\wd\CD@lF<\CD@QA\setbox\CD@lF=\hbox to\CD@QA{\CD@JG\unhbox\CD@lF}\fi
+\advance\CD@xA\m@ne\setbox\CD@VA=\hbox{\box\CD@lF\unhbox\CD@VA}\unskip\setbox
+\CD@lF=\lastbox\setbox\CD@lF=\hbox{\unhbox\CD@lF\unskip\unpenalty\dimen0%
+\lastkern\unkern\unkern\global\CD@QA\lastkern\unkern\kern\dimen0 }}\def\CD@yB
+{\dimen1\dp\CD@VA\ifhbox\CD@VA\CD@xB\else\CD@zB\fi\setbox\CD@VA=\vbox{%
+\penalty\CD@NB}\dp\CD@VA-\dp\CD@MH\wd\CD@VA\CD@tB}\def\CD@zB{\unvbox\CD@VA
+\CD@wA\lastpenalty\unpenalty\ifdim\dimen1<\ht\CD@MH\CD@aG v\dimen1<\ht\CD@MH:%
+rows overprint:\CD@NB\CD@wA\fi}\def\CD@xB{\dimen0=\ht\CD@VA\setbox\CD@VA=%
+\hbox\bgroup\advance\dimen1-\ht\CD@MH\unhbox\CD@VA\CD@xA\lastpenalty
+\unpenalty\CD@wA\lastpenalty\unpenalty\global\CD@RA-\lastkern\unkern\setbox0=%
+\lastbox\CD@tF\setbox\CD@VA=\hbox{\box0\unhbox\CD@VA}\setbox0=\lastbox\ifhbox
+0 \CD@kJ\repeat\global\CD@SA-\lastkern\unkern\global\CD@QA\CD@JK\unhbox\CD@VA
+\egroup\CD@JK\CD@QA\CD@bH{\CD@SA}{\box\CD@VA}{\CD@RA}{\dimen1}}\def\CD@kJ{%
+\setbox0=\hbox to\wd0\bgroup\unhbox0 \unskip\unpenalty\dimen7\lastkern\unkern
+\ifnum\lastpenalty=1 \unpenalty\CD@UB\else\CD@TB\fi\ifnum\lastpenalty=2
+\unpenalty\dimen2.5\dimen0\advance\dimen2-.5\dimen1\advance\dimen2-%
+\axisheight\else\dimen2\z@\fi\setbox0=\lastbox\dimen6\lastkern\unkern\setbox1%
+=\lastbox\setbox0=\vbox{\unvbox0 \CD@tE\kern-\dimen1 \else\ifdim\dimen2=\z@
+\else\kern\dimen2 \fi\fi}\ifdim\dimen0<\ht0 \CD@aG v\dimen0<\ht0:upper part of
+vertical too short:{\CD@tE\CD@NB\else\CD@wA\fi}\CD@xA\else\setbox0=\vbox to%
+\dimen0{\unvbox0}\fi\setbox1=\vtop{\unvbox1}\ifdim\dimen1<\dp1 \CD@aG v\dimen
+1<\dp1:lower part of vertical too short:\CD@NB\CD@wA\else\setbox1=\vtop to%
+\dimen1{\ifdim\dimen2=\z@\else\kern-\dimen2 \fi\unvbox1 }\fi\box1 \kern\dimen
+6 \box0 \kern\dimen7 \CD@HG\global\CD@QA\CD@JK\egroup\CD@JK\CD@QA\relax}%
+\countdef\CD@u=14 \newcount\CD@CA\newcount\CD@XB\newcount\CD@NB\let\CD@LB
+\insc@unt\newcount\CD@FA\newcount\CD@lA\let\CD@mA\CD@XB\newcount\CD@MB\CD@tG
+\CD@DF\CD@bI\CD@aI\CD@aI\def\CD@nD{-1}\def\CD@K{\ifnum\CD@nD<\z@\else
+\begingroup\scrollmode\showboxdepth\CD@nD\showboxbreadth\maxdimen\showlists
+\endgroup\fi\CD@bI\CD@zF\CD@CA=\CD@u\advance\CD@CA1 \CD@XB=\CD@CA\ifnum\CD@NB
+=1 \CD@JA\fi\advance\CD@XB\CD@NB\dimen1\z@\skip0\z@\count@=\insc@unt\advance
+\count@\CD@u\divide\count@2 \ifnum\CD@XB>\count@\CD@KB{The diagram has too
+many rows! It can't be reformatted.}\else\CD@NG\CD@WI\fi\CD@cH}\def\CD@NG{%
+\CD@NB\CD@CA\CD@uF\ifnum\CD@NB<\CD@XB\setbox\CD@NB\box\voidb@x\advance\CD@NB1%
+\relax\repeat\CD@NB\CD@CA\skip\z@\z@\CD@uF\CD@GB\lastpenalty\unpenalty\ifnum
+\CD@GB>\z@\CD@KE\repeat\ifnum\CD@GB=-123 \CD@tJ\unpenalty\else\cd@shouldnt D%
+\fi\ifx\v@grid\relax\else\CD@NB\CD@XB\advance\CD@NB\m@ne\expandafter\CD@VJ
+\v@grid\fi\CD@MB\CD@mA\CD@tB\z@\CD@XG\ifx\h@grid\relax\else\expandafter\CD@LJ
+\h@grid\fi\count@\CD@XB\advance\count@\m@ne\CD@YB\ht\count@}\def\CD@KE{%
+\ifcase\CD@GB\or\CD@MG\else\CD@uA-\lastpenalty\unpenalty\CD@vA\lastpenalty
+\unpenalty\setbox0=\lastbox\CD@WG\fi\CD@wD}\def\CD@wD{\skip1\lastskip\unskip
+\advance\skip0\skip1 \ifdim\skip1=\z@\else\expandafter\CD@wD\fi}\def\CD@MG{%
+\setbox0=\lastbox\CD@pI\dp0 \advance\CD@pI\skip\z@\skip\z@\z@\advance\CD@NF
+\CD@pI\CD@uE\ifnum\CD@NB>\CD@CA\CD@NF\DiagramCellHeight\CD@pI\CD@NF\advance
+\CD@pI-\CD@qI\fi\fi\CD@qI\ht0 \CD@NF\CD@qI\setbox\CD@NB\hbox{\unhbox\CD@NB
+\unhbox0}\dp\CD@NB\CD@pI\ht\CD@NB\CD@qI\advance\CD@NB1 }\def\CD@WG{\ifnum
+\CD@uA<\z@\advance\CD@uA\CD@XB\ifnum\CD@uA<\CD@CA\CD@UG\else\CD@OA\dp\CD@uA
+\CD@PA\ht\CD@uA\setbox\CD@uA\hbox{\box\z@\penalty\CD@vA\penalty\CD@GB\unhbox
+\CD@uA}\dp\CD@uA\CD@OA\ht\CD@uA\CD@PA\fi\else\CD@UG\fi}\def\CD@UG{\CD@KB{%
+diagonal goes outside diagram (lost)}}\def\CD@fI{\advance\CD@uA\CD@XB\ifnum
+\CD@uA<\CD@CA\CD@UG\else\ifnum\CD@uA=\CD@NB\CD@VG\else\ifnum\CD@uA>\CD@NB
+\cd@shouldnt M\else\CD@OA\dp\CD@uA\CD@PA\ht\CD@uA\setbox\CD@uA\hbox{\box\z@
+\penalty\CD@vA\penalty\CD@GB\unhbox\CD@uA}\dp\CD@uA\CD@OA\ht\CD@uA\CD@PA\fi
+\fi\fi}\def\CD@WI{\CD@t\CD@AJ\setbox\CD@PC=\hbox{\CD@k A\@super f\CD@lJ f%
+\CD@ND}\CD@ZE\z@\CD@JK\z@\CD@kI\z@\CD@kF\z@\CD@NB=\CD@XB\CD@NF\z@\CD@uB\z@
+\CD@uF\ifnum\CD@NB>\CD@CA\advance\CD@NB\m@ne\CD@qI\ht\CD@NB\CD@pI\dp\CD@NB
+\advance\CD@NF\CD@qI\CD@rI\advance\CD@uB\CD@NF\CD@KC\CD@ZI\CD@w\ht\CD@NB
+\CD@qI\dp\CD@NB\CD@pI\nointerlineskip\box\CD@NB\CD@NF\CD@pI\setbox\CD@NB\null
+\ht\CD@NB\CD@uB\repeat\CD@wB\nointerlineskip\box\CD@NB\CD@gG\CD@ZE
+\DiagramCellWidth{width}\CD@gG\CD@JK\DiagramCellHeight{height}\CD@VA\CD@LB
+\advance\CD@VA-\CD@lA\advance\CD@VA\m@ne\advance\CD@VA\CD@mA\dimen0\wd\CD@VA
+\CD@tI\axisheight\dimen1\CD@uB\advance\dimen1-\CD@YB\dimen2\CD@kI\advance
+\dimen2-\dimen0 \advance\CD@XB-\CD@CA\advance\CD@LB-\CD@lA}\count@\year
+\multiply\count@12 \advance\count@\month\ifnum\count@>24194 \loop\iftrue
+\message{gone February 2016!}\repeat\fi\def\CD@wB{\CD@qI-\CD@NF\CD@pI\CD@NF
+\setbox\CD@MH=\null\dp\CD@MH\CD@NF\ht\CD@MH-\CD@NF\CD@mF\z@\CD@mI\z@\CD@lA
+\CD@LB\advance\CD@lA-\CD@MB\advance\CD@lA\CD@mA\CD@FA\CD@LB\CD@VA\CD@MB\CD@sF
+\ifnum\CD@FA>\CD@lA\advance\CD@FA\m@ne\advance\CD@VA\m@ne\CD@tB\wd\CD@VA
+\setbox\CD@FA=\box\voidb@x\CD@yB\repeat\CD@w\ht\CD@NB\CD@qI\dp\CD@NB\CD@pI}%
+\def\CD@gG#1#2#3{\ifdim#1>.01\CD@zC\CD@PA#2\relax\advance\CD@PA#1\relax
+\advance\CD@PA.99\CD@zC\count@\CD@PA\divide\count@\CD@zC\CD@KB{increase cell #%
+3 to \the\count@ em}\fi}\def\CD@rI{\CD@FA=\CD@LB\penalty4 \noindent\unhbox
+\CD@NB\CD@sF\unskip\setbox0=\lastbox\ifhbox0 \advance\CD@FA\m@ne\setbox\CD@FA
+\hbox to\wd0{\null\penalty-9990\null\unhbox0}\repeat\CD@lA\CD@FA\advance
+\CD@FA\CD@MB\advance\CD@FA-\CD@mA\ifnum\CD@FA<\CD@LB\count@\CD@FA\advance
+\count@\m@ne\dimen0=\wd\count@\count@\CD@MB\advance\count@\m@ne\CD@tB\wd
+\count@\CD@sF\ifnum\CD@FA<\CD@LB\CD@DJ\CD@XG\dimen0\wd\CD@FA\advance\CD@FA1
+\repeat\fi\CD@sF\CD@GB\lastpenalty\unpenalty\ifnum\CD@GB>\z@\CD@vA
+\lastpenalty\unpenalty\CD@VG\repeat\endgraf\unskip\ifnum\lastpenalty=4
+\unpenalty\else\cd@shouldnt S\fi}\def\CD@VG{\advance\CD@vA\CD@lA\advance
+\CD@vA\m@ne\setbox0=\lastbox\ifnum\CD@vA<\CD@LB\setbox\CD@vA\hbox{\box0%
+\penalty\CD@GB\unhbox\CD@vA}\else\CD@UG\fi}\def\CD@bG{}\CD@tG\CD@uE\CD@WB
+\CD@VB\def\CD@DJ{\advance\dimen0\wd\CD@FA\divide\dimen0\tw@\CD@uE\dimen0%
+\DiagramCellWidth\else\CD@V{\dimen0}\DiagramCellWidth\CD@pJ\fi\advance\CD@tB
+\dimen0 }\def\CD@XG{\setbox\CD@MB=\vbox{}\dp\CD@MB=\CD@uB\wd\CD@MB\CD@tB
+\advance\CD@MB1 }\def\CD@LJ#1,{\def\CD@GK{#1}\ifx\CD@GK\CD@RD\else\advance
+\CD@tB\CD@GK\DiagramCellWidth\CD@XG\expandafter\CD@LJ\fi}\def\CD@VJ#1,{\def
+\CD@GK{#1}\ifx\CD@GK\CD@RD\else\ifnum\CD@NB>\CD@CA\CD@NF\CD@GK
+\DiagramCellHeight\advance\CD@NF-\dp\CD@NB\advance\CD@NB\m@ne\ht\CD@NB\CD@NF
+\fi\expandafter\CD@VJ\fi}\def\CD@pJ{\CD@wE\CD@OA\dimen0 \advance\CD@OA-%
+\DiagramCellWidth\ifdim\CD@OA>2\MapShortFall\CD@KB{badly drawn diagonals (see
+manual)}\let\CD@pJ\empty\fi\else\let\CD@pJ\empty\fi}\def\CD@KC{\CD@VA\CD@mA
+\CD@sF\ifnum\CD@VA<\CD@MB\dimen0\dp\CD@VA\advance\dimen0\CD@NF\dp\CD@VA\dimen
+0 \advance\CD@VA1 \repeat}\def\CD@bH#1#2#3#4{\ifnum\CD@FA<\CD@LB\CD@OA=#1%
+\relax\setbox\CD@FA=\hbox{\setbox0=#2\dimen7=#4\relax\dimen8=#3\relax\ifhbox
+\CD@FA\unhbox\CD@FA\advance\CD@OA-\lastkern\unkern\fi\ifdim\CD@OA=\z@\else
+\kern-\CD@OA\fi\raise\dimen7\box0 \kern-\dimen8 }\ifnum\CD@FA=\CD@lA\CD@V
+\CD@kF\CD@OA\fi\else\cd@shouldnt O\fi}\def\CD@w{\setbox\CD@NB=\hbox{\CD@FA
+\CD@lA\CD@VA\CD@mA\CD@PA\z@\relax\CD@sF\ifnum\CD@FA<\CD@LB\CD@tB\wd\CD@VA
+\relax\CD@eI\advance\CD@FA1 \advance\CD@VA1 \repeat}\CD@V\CD@kI{\wd\CD@NB}\wd
+\CD@NB\z@}\def\CD@eI{\ifhbox\CD@FA\CD@OA\CD@tB\relax\advance\CD@OA-\CD@PA
+\relax\ifdim\CD@OA=\z@\else\kern\CD@OA\fi\CD@PA\CD@tB\advance\CD@PA\wd\CD@FA
+\relax\unhbox\CD@FA\advance\CD@PA-\lastkern\unkern\fi}\def\CD@ZI{\setbox
+\CD@sH=\box\voidb@x\CD@VA=\CD@MB\CD@FA\CD@LB\CD@VA\CD@mA\advance\CD@VA\CD@FA
+\advance\CD@VA-\CD@lA\advance\CD@VA\m@ne\CD@tB\wd\CD@VA\count@\CD@LB\advance
+\count@\m@ne\CD@hF.5\wd\count@\advance\CD@hF\CD@tB\CD@A\m@ne\CD@gD\@m\CD@sF
+\ifnum\CD@FA>\CD@lA\advance\CD@FA\m@ne\advance\CD@hF-\CD@tB\CD@PI\wd\CD@VA
+\CD@tB\advance\CD@hF\CD@tB\advance\CD@VA\m@ne\CD@tB\wd\CD@VA\repeat\CD@mF
+\CD@kF\CD@mI-\CD@mF\CD@vB}\newcount\CD@GB\def\CD@s{}\def\CD@t{\mathsurround
+\z@\hsize\z@\rightskip\z@ plus1fil minus\maxdimen\parfillskip\z@\linepenalty
+9000 \looseness0 \hfuzz\maxdimen\hbadness10000 \clubpenalty0 \widowpenalty0
+\displaywidowpenalty0 \interlinepenalty0 \predisplaypenalty0
+\postdisplaypenalty0 \interdisplaylinepenalty0 \interfootnotelinepenalty0
+\floatingpenalty0 \brokenpenalty0 \everypar{}\leftskip\z@\parskip\z@
+\parindent\z@\pretolerance10000 \tolerance10000 \hyphenpenalty10000
+\exhyphenpenalty10000 \binoppenalty10000 \relpenalty10000 \adjdemerits0
+\doublehyphendemerits0 \finalhyphendemerits0 \CD@IA\prevdepth\z@}\newbox
+\CD@KG\newbox\CD@IG\def\CD@JG{\unhcopy\CD@KG}\def\CD@HG{\unhcopy\CD@IG}\def
+\CD@iJ{\hbox{}\penalty1\nointerlineskip}\def\CD@PI{\penalty5 \noindent\setbox
+\CD@MH=\null\CD@mF\z@\CD@mI\z@\ifnum\CD@FA<\CD@LB\ht\CD@MH\ht\CD@FA\dp\CD@MH
+\dp\CD@FA\unhbox\CD@FA\skip0=\lastskip\unskip\else\CD@OK\skip0=\z@\fi\endgraf
+\ifcase\prevgraf\cd@shouldnt Y \or\cd@shouldnt Z \or\CD@RI\or\CD@XI\else
+\CD@QI\fi\unskip\setbox0=\lastbox\unskip\unskip\unpenalty\noindent\unhbox0%
+\setbox0\lastbox\unpenalty\unskip\unskip\unpenalty\setbox0\lastbox\CD@tF
+\CD@GB\lastpenalty\unpenalty\ifnum\CD@GB>\z@\setbox\z@\lastbox\CD@lB\repeat
+\endgraf\unskip\unskip\unpenalty}\def\CD@YJ{\CD@uA\CD@XB\advance\CD@uA-\CD@NB
+\CD@vA\CD@FA\advance\CD@vA-\CD@lA\advance\CD@vA1 \expandafter\message{%
+prevgraf=\the\prevgraf at (\the\CD@uA,\the\CD@vA)}}\def\CD@XI{\CD@CE\setbox
+\CD@lI=\lastbox\setbox\CD@lI=\hbox{\unhbox\CD@lI\unskip\unpenalty}\unskip
+\ifdim\ht\CD@lI>\ht\CD@PC\setbox\CD@MH=\copy\CD@lI\else\ifdim\dp\CD@lI>\dp
+\CD@PC\setbox\CD@MH=\copy\CD@lI\else\CD@FG\CD@lI\fi\fi\advance\CD@mF.5\wd
+\CD@lI\advance\CD@mI.5\wd\CD@lI\setbox\CD@lI=\hbox{\unhbox\CD@lI\CD@HG}\CD@bH
+\CD@mF{\box\CD@lI}\CD@mI\z@\CD@yB\CD@vB}\def\CD@CE{\ifnum\CD@A>0 \advance
+\dimen0-\CD@tB\CD@iA-.5\dimen0 \CD@A-\CD@A\else\CD@A0 \CD@iA\z@\fi\setbox
+\CD@MH=\lastbox\setbox\CD@MH=\hbox{\unhbox\CD@MH\unskip\unskip\unpenalty
+\setbox0=\lastbox\global\CD@QA\lastkern\unkern}\advance\CD@iA-.5\CD@QA\unskip
+\setbox\CD@MH=\null\CD@mI\CD@iA\CD@mF-\CD@iA}\def\CD@Z{\ht\CD@MH\CD@tI\dp
+\CD@MH\CD@sI}\def\CD@FG#1{\setbox\CD@MH=\hbox{\CD@V{\ht\CD@MH}{\ht#1}\CD@V{%
+\dp\CD@MH}{\dp#1}\CD@V{\wd\CD@MH}{\wd#1}\vrule height\ht\CD@MH depth\dp\CD@MH
+width\wd\CD@MH}}\def\CD@QI{\CD@CE\CD@Z\setbox\CD@lI=\lastbox\unskip\setbox
+\CD@lF=\lastbox\unskip\setbox\CD@lF=\hbox{\unhbox\CD@lF\unskip\global\CD@yA
+\lastpenalty\unpenalty}\advance\CD@yA9999 \ifcase\CD@yA\CD@VI\or\CD@YI\or
+\CD@TI\or\CD@dI\or\CD@cI\or\CD@SI\else\cd@shouldnt9\fi}\def\CD@VI{\CD@FG
+\CD@lI\CD@UI\setbox\CD@sH=\box\CD@lF\setbox\CD@tH=\box\CD@lI}\def\CD@YI{%
+\CD@FG\CD@lF\setbox\CD@lI\hbox{\penalty8 \unhbox\CD@lI\unskip\unpenalty\ifnum
+\lastpenalty=8 \else\CD@xH\fi}\CD@UI\setbox\CD@lF=\hbox{\unhbox\CD@lF\unskip
+\unpenalty\global\setbox\CD@DA=\lastbox}\ifdim\wd\CD@lF=\z@\else\CD@xH\fi
+\setbox\CD@sH=\box\CD@DA}\def\CD@xH{\CD@KB{extra material in \string\pile
+\space cell (lost)}}\def\CD@UI{\CD@yB\ifvoid\CD@sH\else\CD@KB{Clashing
+horizontal arrows}\CD@mI.5\CD@hF\CD@mF-\CD@mI\CD@vB\CD@mI\z@\CD@mF\z@\fi
+\CD@hI\CD@hF\advance\CD@hI-\CD@mI\CD@hF-\CD@mF\CD@JC\CD@FA}\def\CD@RI{\setbox
+0\lastbox\unskip\CD@iA\z@\CD@Z\ifdim\skip0>\z@\CD@tJ\CD@A0 \else\ifnum\CD@A<1
+\CD@A0 \dimen0\CD@tB\fi\advance\CD@A1 \fi}\def\VonH{\CD@MA46\VonH{.5\CD@LF}}%
+\def\HonV{\CD@MA57\HonV{.5\CD@LF}}\def\HmeetV{\CD@MA44\HmeetV{-\MapShortFall}%
+}\def\CD@MA#1#2#3#4{\CD@pB34#1{\string#3}\CD@SD\CD@GB-999#2 \dimen0=#4\CD@tI
+\dimen0\advance\CD@tI\axisheight\CD@sI\dimen0\advance\CD@sI-\axisheight\CD@CF
+\CD@HC\CD@ZD}\def\CD@HC#1{\setbox0=\hbox{\CD@k#1\CD@ND}\dimen0.5\wd0 \CD@tI
+\ht0 \CD@sI\dp0 \CD@ZD}\def\CD@SD{\setbox0=\null\ht0=\CD@tI\dp0=\CD@sI\wd0=%
+\dimen0 \copy0\penalty\CD@GB\box0 }\def\CD@TI{\CD@GC\CD@yB}\def\CD@dI{\CD@GC
+\CD@vB}\def\CD@SI{\CD@GC\CD@yB\CD@vB}\def\CD@GC{\setbox\CD@lI=\hbox{\unhbox
+\CD@lI}\setbox\CD@lF=\hbox{\unhbox\CD@lF\global\setbox\CD@DA=\lastbox}\ht
+\CD@MH\ht\CD@DA\dp\CD@MH\dp\CD@DA\advance\CD@mF\wd\CD@DA\advance\CD@mI\wd
+\CD@lI}\CD@tG\ifPositiveGradient\CD@CI\CD@BI\CD@CI\CD@tG\ifClimbing\CD@rB
+\CD@qB\CD@rB\newcount\DiagonalChoice\DiagonalChoice\m@ne\ifx\tenln\nullfont
+\CD@tJ\def\CD@qF{\CD@KH\ifPositiveGradient/\else\CD@k\backslash\CD@ND\fi}%
+\else\def\CD@qF{\CD@rF\char\count@}\fi\let\CD@rF\tenln\def\Use@line@char#1{%
+\hbox{#1\CD@rF\ifPositiveGradient\else\advance\count@64 \fi\char\count@}}\def
+\CD@cF{\Use@line@char{\count@\CD@TC\multiply\count@8\advance\count@-9\advance
+\count@\CD@LH}}\def\CD@ZF{\Use@line@char{\ifcase\DiagonalChoice\CD@gF\or
+\CD@fF\or\CD@fF\else\CD@gF\fi}}\def\CD@gF{\ifnum\CD@TC=\z@\count@'33 \else
+\count@\CD@TC\multiply\count@\sixt@@n\advance\count@-9\advance\count@\CD@LH
+\advance\count@\CD@LH\fi}\def\CD@fF{\count@'\ifcase\CD@LH55\or\ifcase\CD@TC66%
+\or22\or52\or61\or72\fi\or\ifcase\CD@TC66\or25\or22\or63\or52\fi\or\ifcase
+\CD@TC66\or16\or36\or22\or76\fi\or\ifcase\CD@TC66\or27\or25\or67\or22\fi\fi
+\relax}\def\CD@uC#1{\hbox{#1\setbox0=\Use@line@char{#1}\ifPositiveGradient
+\else\raise.3\ht0\fi\copy0 \kern-.7\wd0 \ifPositiveGradient\raise.3\ht0\fi
+\box0}}\def\CD@jF#1{\hbox{\setbox0=#1\kern-.75\wd0 \vbox to.25\ht0{%
+\ifPositiveGradient\else\vss\fi\box0 \ifPositiveGradient\vss\fi}}}\def\CD@jI#%
+1{\hbox{\setbox0=#1\dimen0=\wd0 \vbox to.25\ht0{\ifPositiveGradient\vss\fi
+\box0 \ifPositiveGradient\else\vss\fi}\kern-.75\dimen0 }}\CD@RC{+h:>}{%
+\Use@line@char\CD@fF}\CD@RC{-h:>}{\Use@line@char\CD@gF}\CD@nF{+t:<}{-h:>}%
+\CD@nF{-t:<}{+h:>}\CD@RC{+t:>}{\CD@jF{\Use@line@char\CD@fF}}\CD@RC{-t:>}{%
+\CD@jI{\Use@line@char\CD@gF}}\CD@nF{+h:<}{-t:>}\CD@nF{-h:<}{+t:>}\CD@RC{+h:>>%
+}{\CD@uC\CD@fF}\CD@RC{-h:>>}{\CD@uC\CD@gF}\CD@nF{+t:<<}{-h:>>}\CD@nF{-t:<<}{+%
+h:>>}\CD@nF{+h:>->}{+h:>>}\CD@nF{-h:>->}{-h:>>}\CD@nF{+t:<-<}{-h:>>}\CD@nF{-t%
+:<-<}{+h:>>}\CD@RC{+t:>>}{\CD@jF{\CD@uC\CD@fF}}\CD@RC{-t:>>}{\CD@jI{\CD@uC
+\CD@gF}}\CD@nF{+h:<<}{-t:>>}\CD@nF{-h:<<}{+t:>>}\CD@nF{+t:>->}{+t:>>}\CD@nF{-%
+t:>->}{-t:>>}\CD@nF{+h:<-<}{-t:>>}\CD@nF{-h:<-<}{+t:>>}\CD@RC{+f:-}{\CD@EF
+\null\else\CD@cF\fi}\CD@nF{-f:-}{+f:-}\def\CD@tC#1#2{\vbox to#1{\vss\hbox to#%
+2{\hss.\hss}\vss}}\def\hfdot{\CD@tC{2\axisheight}{.5em}}%
+%% % .7em until 29.7.98
+\def\vfdot{\CD@tC{1ex}\z@}%% % 1.46ex until 29.7.98
+\def\CD@bF{\hbox{\dimen0=.3\CD@zC\dimen1\dimen0 \ifnum\CD@LH>\CD@TC\CD@iC{%
+\dimen1}\else\CD@dG{\dimen0}\fi\CD@tC{\dimen0}{\dimen1}}}\newarrowfiller{.}%
+\hfdot\hfdot\vfdot\vfdot\def\dfdot{\CD@bF\CD@CK}\CD@RC{+f:.}{\dfdot}\CD@RC{-f%
+:.}{\dfdot}\def\CD@@K#1{\hbox\bgroup\def\CD@CH{#1\egroup}\afterassignment
+\CD@CH%%
+\count@='}\def\lnchar{\CD@@K\CD@qF}\def\CD@dF#1{\setbox#1=\hbox{\dimen5\dimen
+#1 \setbox8=\box#1 \dimen1\wd8 \count@\dimen5 \divide\count@\dimen1 \ifnum
+\count@=0 \box8 \ifdim\dimen5<.95\dimen1 \CD@gB{diagonal line too short}\fi
+\else\dimen3=\dimen5 \advance\dimen3-\dimen1 \divide\dimen3\count@\dimen4%
+\dimen3 \CD@dG{\dimen4}\ifPositiveGradient\multiply\dimen4\m@ne\fi\dimen6%
+\dimen1 \advance\dimen6-\dimen3 \loop\raise\count@\dimen4\copy8 \ifnum\count@
+>0 \kern-\dimen6 \advance\count@\m@ne\repeat\fi}}\def\CD@CG#1{\CD@EF\CD@xJ{#1%
+}\else\CD@dF{#1}\fi}\def\CD@IH#1{}\newdimen\objectheight\objectheight1.8ex
+\newdimen\objectwidth\objectwidth1em \def\CD@YD{\dimen6=\CD@aK
+\DiagramCellHeight\dimen7=\CD@WK\DiagramCellWidth\CD@KJ\ifnum\CD@LH>0 \ifnum
+\CD@TC>0 \CD@aF\else\aftergroup\CD@VC\fi\else\aftergroup\CD@UC\fi}\def\CD@VC{%
+\CD@YA{diagonal map is nearly vertical}\CD@NA}\def\CD@UC{\CD@YA{diagonal map
+is nearly horizontal}\CD@NA}\CD@rG\CD@NA{Use an orthogonal map instead}\def
+\CD@aF{\CD@MJ\dimen3\dimen7\dimen7\dimen6\CD@iC{\dimen7}\advance\dimen3-%
+\dimen7 \CD@MF\ifnum\CD@LH>\CD@TC\advance\dimen6-\dimen1\advance\dimen6-%
+\dimen5 \CD@iC{\dimen1}\CD@iC{\dimen5}\else\dimen0\dimen1\advance\dimen0%
+\dimen5\CD@dG{\dimen0}\advance\dimen6-\dimen0 \fi\dimen2.5\dimen7\advance
+\dimen2-\dimen1 \dimen4.5\dimen7\advance\dimen4-\dimen5 \ifPositiveGradient
+\dimen0\dimen5 \advance\dimen1-\CD@WK\DiagramCellWidth\advance\dimen1 \CD@ZK
+\DiagramCellWidth\setbox6=\llap{\unhbox6\kern.1\ht2}\setbox7=\rlap{\kern.1\ht
+2\unhbox7}\else\dimen0\dimen1 \advance\dimen1-\CD@ZK\DiagramCellWidth\setbox7%
+=\llap{\unhbox7\kern.1\ht2}\setbox6=\rlap{\kern.1\ht2\unhbox6}\fi\setbox6=%
+\vbox{\box6\kern.1\wd2}\setbox7=\vtop{\kern.1\wd2\box7}\CD@dG{\dimen0}%
+\advance\dimen0-\axisheight\advance\dimen0-\CD@bK\DiagramCellHeight\dimen5-%
+\dimen0 \advance\dimen0\dimen6 \advance\dimen1.5\dimen3 \ifdim\wd3>\z@\ifdim
+\ht3>-\dp3\CD@TB\fi\fi\dimen3\dimen2 \dimen7\dimen2\advance\dimen7\dimen4
+\ifvoid3 \else\CD@tE\else\dimen8\ht3\advance\dimen8-\axisheight\CD@iC{\dimen8%
+}\CD@X{\dimen8}{.5\wd3}\dimen9\dp3\advance\dimen9\axisheight\CD@iC{\dimen9}%
+\CD@X{\dimen9}{.5\wd3}\ifPositiveGradient\advance\dimen2-\dimen9\advance
+\dimen4-\dimen8 \else\advance\dimen4-\dimen9\advance\dimen2-\dimen8 \fi\fi
+\advance\dimen3-.5\wd3 \fi\dimen9=\CD@aK\DiagramCellHeight\advance\dimen9-2%
+\DiagramCellHeight\CD@tE\advance\dimen2\dimen4 \CD@CG{2}\dimen2-\dimen0%
+\advance\dimen2\dp2 \else\CD@CG{2}\CD@CG{4}\ifPositiveGradient\dimen2-\dimen0%
+\advance\dimen2\dp2 \dimen4\dimen5\advance\dimen4-\ht4 \else\dimen4-\dimen0%
+\advance\dimen4\dp4 \dimen2\dimen5\advance\dimen2-\ht2 \fi\fi\setbox0=\hbox to%
+\z@{\kern\dimen1 \ifvoid1 \else\ifPositiveGradient\advance\dimen0-\dp1 \lower
+\dimen0 \else\advance\dimen5-\ht1 \raise\dimen5 \fi\rlap{\unhbox1}\fi\raise
+\dimen2\rlap{\unhbox2}\ifvoid3 \else\lower.5\dimen9\rlap{\kern\dimen3\unhbox3%
+}\fi\kern.5\dimen7 \lower.5\dimen9\box6 \lower.5\dimen9\box7 \kern.5\dimen7
+\CD@tE\else\raise\dimen4\llap{\unhbox4}\fi\ifvoid5 \else\ifPositiveGradient
+\advance\dimen5-\ht5 \raise\dimen5 \else\advance\dimen0-\dp5 \lower\dimen0 \fi
+\llap{\unhbox5}\fi\hss}\ht0=\axisheight\dp0=-\ht0\box0 }\def\NorthWest{\CD@BI
+\CD@rB\DiagonalChoice0 }\def\NorthEast{\CD@CI\CD@rB\DiagonalChoice1 }\def
+\SouthWest{\CD@CI\CD@qB\DiagonalChoice3 }\def\SouthEast{\CD@BI\CD@qB
+\DiagonalChoice2 }\def\CD@aD{\vadjust{\CD@uA\CD@FA\advance\CD@uA
+\ifPositiveGradient\else-\fi\CD@ZK\relax\CD@vA\CD@NB\advance\CD@vA-\CD@bK
+\relax\hbox{\advance\CD@uA\ifPositiveGradient-\fi\CD@WK\advance\CD@vA\CD@aK
+\hbox{\box6 \kern\CD@DC\kern\CD@eJ\penalty1 \box7 \box\z@}\penalty\CD@uA
+\penalty\CD@vA}\penalty\CD@uA\penalty\CD@vA\penalty104}}\def\CD@eH#1{\relax
+\vadjust{\hbox@maths{#1}\penalty\CD@FA\penalty\CD@NB\penalty\tw@}}\def\CD@lB{%
+\ifcase\CD@GB\or\or\CD@bH{.5\wd0}{\box0}{.5\wd0}\z@\or\unhbox\z@\setbox\z@
+\lastbox\CD@bH{.5\wd0}{\box0}{.5\wd0}\z@\unpenalty\unpenalty\setbox\z@
+\lastbox\or\CD@TG\else\advance\CD@GB-100 \ifnum\CD@GB<\z@\cd@shouldnt B\fi
+\setbox\z@\hbox{\kern\CD@mF\copy\CD@MH\kern\CD@mI\CD@uA\CD@XB\advance\CD@uA-%
+\CD@NB\penalty\CD@uA\CD@uA\CD@FA\advance\CD@uA-\CD@lA\penalty\CD@uA\unhbox\z@
+\global\CD@yA\lastpenalty\unpenalty\global\CD@zA\lastpenalty\unpenalty}\CD@uA
+-\CD@yA\CD@vA\CD@zA\CD@fI\fi}\def\CD@TG{\unhbox\z@\setbox\z@\lastbox\CD@uA
+\lastpenalty\unpenalty\advance\CD@uA\CD@mA\CD@vA\CD@XB\advance\CD@vA-%
+\lastpenalty\unpenalty\dimen1\lastkern\unkern\setbox3\lastbox\dimen0\lastkern
+\unkern\setbox0=\hbox to\z@{\unhbox0\setbox0\lastbox\setbox7\lastbox
+\unpenalty\CD@eJ\lastkern\unkern\CD@DC\lastkern\unkern\setbox6\lastbox\dimen7%
+\CD@tB\advance\dimen7-\wd\CD@uA\ifdim\dimen7<\z@\CD@CI\multiply\dimen7\m@ne
+\let\mv\empty\else\CD@BI\def\mv{\raise\ht1}\kern-\dimen7 \fi\ifnum\CD@vA>%
+\CD@NB\dimen6\CD@uB\advance\dimen6-\ht\CD@vA\else\dimen6\z@\fi\CD@jJ\CD@mK
+\setbox1\null\ht1\dimen6\wd1\dimen7 \dimen7\dimen2 \dimen6\wd1 \CD@KJ\CD@uA
+\CD@LH\CD@vA\CD@TC\dimen6\ht1 \CD@KJ\setbox2\null\divide\dimen2\tw@\advance
+\dimen2\CD@eJ\CD@eG{\dimen2}\wd2\dimen2 \dimen0.5\dimen7 \advance\dimen0%
+\ifPositiveGradient\else-\fi\CD@eJ\CD@dG{\dimen0}\advance\dimen0-\axisheight
+\ht2\dimen0 \dimen0\CD@DC\CD@eG{\dimen0}\advance\dimen0\ht2\ht2\dimen0 \dimen
+0\ifPositiveGradient-\fi\CD@DC\CD@dG{\dimen0}\advance\dimen0\wd2\wd2\dimen0
+\setbox4\null\dimen0 .6\CD@zC\CD@eG{\dimen0}\ht4\dimen0 \dimen0 .2\CD@zC
+\CD@dG{\dimen0}\wd4\dimen0 \dimen0\wd2 \ifvoid6\else\dimen1\ht4 \advance
+\dimen1\ht2 \CD@CC6+-\raise\dimen1\rlap{\ifPositiveGradient\advance\dimen0-%
+\wd6\advance\dimen0-\wd4 \else\advance\dimen0\wd4 \fi\kern\dimen0\box6}\fi
+\dimen0\wd2 \ifvoid7\else\dimen1\ht4 \advance\dimen1-\ht2 \CD@CC7-+\lower
+\dimen1\rlap{\ifPositiveGradient\advance\dimen0\wd4 \else\advance\dimen0-\wd7%
+\advance\dimen0-\wd4 \fi\kern\dimen0\box7}\fi\mv\box0\hss}\ht0\z@\dp0\z@
+\CD@bH{\z@}{\box\z@}{\z@}{\axisheight}}\def\CD@CC#1#2#3{\dimen4.5\wd#1 \ifdim
+\dimen4>.25\dimen7\dimen4=.25\dimen7\fi\ifdim\dimen4>\CD@zC\dimen4.4\dimen4
+\advance\dimen4.6\CD@zC\fi\CD@eG{\dimen4}\dimen5\axisheight\CD@dG{\dimen5}%
+\advance\dimen4-\dimen5 \dimen5\dimen4\CD@eG{\dimen5}\advance\dimen0%
+\ifPositiveGradient#2\else#3\fi\dimen5 \CD@dG{\dimen4}\advance\dimen1\dimen4 }
+\def\CD@eD#1{\expandafter\CD@IK{#1}}\CD@ZA\CD@EK{output is PostScript
+dependent}\def\CD@SC{\CD@IK{/bturn {gsave currentpoint currentpoint translate
+4 2 roll neg exch atan rotate neg exch neg exch translate } def /eturn {%
+currentpoint grestore moveto} def}}\def\CD@gK{\relax\CD@hK\CD@tK{Q}\else
+\CD@IK{eturn}\fi} \def\CD@OJ#1{\count@#1\relax\multiply\count@7\advance
+\count@16577\divide\count@33154 }\def\CD@fD#1{\expandafter\special{#1}} \def
+\CD@xJ#1{\setbox#1=\hbox{\dimen0\dimen#1\CD@dG{\dimen0}\CD@OJ{\dimen0}\setbox
+0=\null\ifPositiveGradient\count@-\count@\ht0\dimen0 \else\dp0\dimen0 \fi\box
+0 \CD@uA\count@\CD@OJ\CD@LF\CD@fD{pn \the\count@}\CD@fD{pa 0 0}\CD@OJ{\dimen#%
+1}\CD@fD{pa \the\count@\space\the\CD@uA}\CD@fD{fp}\kern\dimen#1}}\def\CD@JI{%
+\CD@KJ\begingroup\ifdim\dimen7<\dimen6 \dimen2=\dimen6 \dimen6=\dimen7 \dimen
+7=\dimen2 \count@\CD@LH\CD@LH\CD@TC\CD@TC\count@\else\dimen2=\dimen7 \fi
+\ifdim\dimen6>.01\p@\CD@KI\global\CD@QA\dimen0 \else\global\CD@QA\dimen7 \fi
+\endgroup\dimen2\CD@QA\CD@iK\CD@lK{\ifPositiveGradient\else-\fi\dimen6}\CD@iK
+\CD@kK{\ifPositiveGradient-\fi\dimen6}\CD@iK\CD@eK{\dimen7}}\def\CD@KI{\CD@hJ
+\ifdim\dimen7>1.73\dimen6 \divide\dimen2 4 \multiply\CD@TC2 \else\dimen2=0.%
+353553\dimen2 \advance\CD@LH-\CD@TC\multiply\CD@TC4 \fi\dimen0=4\dimen2 \CD@ZG
+4\CD@ZG{-2}\CD@ZG2\CD@ZG{-2.5}}\def\CD@AI{\begingroup\count@\dimen0 \dimen2 45%
+pt \divide\count@\dimen2 \ifdim\dimen0<\z@\advance\count@\m@ne\fi\ifodd
+\count@\advance\count@1\CD@@A\else\CD@y\fi\advance\dimen0-\count@\dimen2
+\CD@gE\multiply\dimen0\m@ne\fi\ifnum\count@<0 \multiply\count@-7 \fi\dimen3%
+\dimen1 \dimen6\dimen0 \dimen7 3754936sp \ifdim\dimen0<6\p@\def\CD@OG{4000}%
+\fi\CD@KJ\dimen2\dimen3\CD@dG{\dimen2}\CD@hJ\multiply\CD@TC-6 \dimen0\dimen2
+\CD@ZG1\CD@ZG{0.3}\dimen1\dimen0 \dimen2\dimen3 \dimen0\dimen3 \CD@ZG3\CD@ZG{%
+1.5}\CD@ZG{0.3}\divide\count@2 \CD@gE\multiply\dimen1\m@ne\fi\ifodd\count@
+\dimen2\dimen1\dimen1\dimen0\dimen0-\dimen2 \fi\divide\count@2 \ifodd\count@
+\multiply\dimen0\m@ne\multiply\dimen1\m@ne\fi\global\CD@QA\dimen0\global
+\CD@RA\dimen1\endgroup\dimen6\CD@QA\dimen7\CD@RA}\def\CD@OC{255}\let\CD@OG
+\CD@OC\def\CD@KJ{\begingroup\ifdim\dimen7<\dimen6 \dimen9\dimen7\dimen7\dimen
+6\dimen6\dimen9\CD@@A\else\CD@y\fi\dimen2\z@\dimen3\CD@XH\dimen4\CD@XH\dimen0%
+\z@\dimen8=\CD@OG\CD@XH\CD@lC\global\CD@yA\dimen\CD@gE0\else3\fi\global\CD@zA
+\dimen\CD@gE3\else0\fi\endgroup\CD@LH\CD@yA\CD@TC\CD@zA}\def\CD@lC{\count@
+\dimen6 \divide\count@\dimen7 \advance\dimen6-\count@\dimen7 \dimen9\dimen4
+\advance\dimen9\count@\dimen0 \ifdim\dimen9>\dimen8 \CD@@C\else\CD@AC\ifdim
+\dimen6>\z@\dimen9\dimen6 \dimen6\dimen7 \dimen7\dimen9 \expandafter
+\expandafter\expandafter\CD@lC\fi\fi}\def\CD@@C{\ifdim\dimen0=\z@\ifdim\dimen
+9<2\dimen8 \dimen0\dimen8 \fi\else\advance\dimen8-\dimen4 \divide\dimen8%
+\dimen0 \ifdim\count@\CD@XH<2\dimen8 \count@\dimen8 \dimen9\dimen4 \advance
+\dimen9\count@\dimen0 \CD@AC\fi\fi}\def\CD@AC{\dimen4\dimen0 \dimen0\dimen9
+\advance\dimen2\count@\dimen3 \dimen9\dimen2 \dimen2\dimen3 \dimen3\dimen9 }%
+\def\CD@ZG#1{\CD@dG{\dimen2}\advance\dimen0 #1\dimen2 }\def\CD@dG#1{\divide#1%
+\CD@TC\multiply#1\CD@LH}\def\CD@eG#1{\divide#1\CD@vA\multiply#1\CD@uA}\def
+\CD@iC#1{\divide#1\CD@LH\multiply#1\CD@TC}\def\CD@hJ{\dimen6\CD@LH\CD@XH
+\multiply\dimen6\CD@LH\dimen7\CD@TC\CD@XH\multiply\dimen7\CD@TC\CD@KJ}\def
+\CD@iK#1#2{\begingroup\dimen@#2\relax\loop\ifdim\dimen2<.4\maxdimen\multiply
+\dimen2\tw@\multiply\dimen@\tw@\repeat\divide\dimen2\@cclvi\divide\dimen@
+\dimen2\relax\multiply\dimen@\@cclvi\expandafter\CD@jK\the\dimen@\endgroup
+\let#1\CD@fK}{\catcode`p=12 \catcode`0=12 \catcode`.=12 \catcode`t=12 \gdef
+\CD@jK#1pt{\gdef\CD@fK{#1}}}\ifx\errorcontextlines\CD@qK\CD@tJ\let\CD@GH
+\relax\else\def\CD@GH{\errorcontextlines\m@ne}\fi\ifnum\inputlineno<0 \let
+\CD@CD\empty\let\CD@W\empty\let\CD@mD\relax\let\CD@uI\relax\let\CD@vI\relax
+\let\CD@zF\relax\message{! Why not upgrade to TeX version 3? (available since
+1990)}\else\def\CD@W{ at line \number\inputlineno}\def\CD@mD{ - first occurred%
+}\def\CD@uI{\edef\CD@h{\the\inputlineno}\global\let\CD@jB\CD@h}\def\CD@h{9999%
+}\def\CD@vI{\xdef\CD@jB{\the\inputlineno}}\def\CD@jB{\CD@h}\def\CD@zF{\ifnum
+\CD@h<\inputlineno\edef\CD@CD{\space at lines \CD@h--\the\inputlineno}\else
+\edef\CD@CD{\CD@W}\fi}\fi\let\CD@CD\empty\def\CD@YA#1#2{\CD@GH\errhelp=#2%
+\expandafter\errmessage{\CD@tA: #1}}\def\CD@KB#1{\begingroup\expandafter
+\message{! \CD@tA: #1\CD@CD}\ifnum\CD@XB>\CD@NB\ifnum\CD@CA>\CD@NB\else\ifnum
+\CD@lA>\CD@FA\else\ifnum\CD@LB>\CD@FA\advance\CD@XB-\CD@NB\advance\CD@FA-%
+\CD@lA\advance\CD@FA1\relax\expandafter\message{! (error detected at row \the
+\CD@XB, column \the\CD@FA, but probably caused elsewhere)}\fi\fi\fi\fi
+\endgroup}\def\CD@gB#1{{\expandafter\message{\CD@tA\space Warning: #1\CD@W}}}%
+\def\CD@CB#1#2{\CD@gB{#1 \string#2 is obsolete\CD@mD}}\def\CD@AB#1{\CD@CB{%
+Dimension}{#1}\CD@DE#1\CD@BB\CD@BB}\def\CD@BB{\CD@OA=}\def\CD@@B#1{\CD@CB{%
+Count}{#1}\CD@DE#1\CD@OH\CD@OH}\def\CD@OH{\count@=}\def\HorizontalMapLength{%
+\CD@AB\HorizontalMapLength}\def\VerticalMapHeight{\CD@AB\VerticalMapHeight}%
+\def\VerticalMapDepth{\CD@AB\VerticalMapDepth}\def\VerticalMapExtraHeight{%
+\CD@AB\VerticalMapExtraHeight}\def\VerticalMapExtraDepth{\CD@AB
+\VerticalMapExtraDepth}\def\DiagonalLineSegments{\CD@@B\DiagonalLineSegments}%
+\ifx\tenln\nullfont\CD@ZA\CD@KH{\CD@eF\space diagonal line and arrow font not
+available}\else\let\CD@KH\relax\fi\def\CD@aG#1#2<#3:#4:#5#6{\begingroup\CD@PA
+#3\relax\advance\CD@PA-#2\relax\ifdim.1em<\CD@PA\CD@uA#5\relax\CD@vA#6\relax
+\ifnum\CD@uA<\CD@vA\count@\CD@vA\advance\count@-\CD@uA\CD@KB{#4 by \the\CD@PA
+}\if#1v\let\CD@CH\CD@JK\edef\tmp{\the\CD@uA--\the\CD@vA,\the\CD@FA}\else
+\advance\count@\count@\if#1l\advance\count@-\CD@A\else\if#1r\advance\count@
+\CD@A\fi\fi\advance\CD@PA\CD@PA\let\CD@CH\CD@ZE\edef\tmp{\the\CD@NB,\the
+\CD@uA--\the\CD@vA}\fi\divide\CD@PA\count@\ifdim\CD@CH<\CD@PA\global\CD@CH
+\CD@PA\fi\fi\fi\endgroup}\CD@tG\CD@xE\CD@JD\CD@ID\CD@rG\CD@xI{See the message
+above.}\CD@rG\CD@lH{Perhaps you've forgotten to end the diagram before
+resuming the text, in\CD@uG which case some garbage may be added to the
+diagram, but we should be ok now.\CD@uG Alternatively you've left a blank line
+in the middle - TeX will now complain\CD@uG that the remaining \CD@S s are
+misplaced - so please use comments for layout.}\CD@rG\CD@hD{You have already
+closed too many brace pairs or environments; an \CD@HD\CD@uG command was (%
+over)due.}\CD@rG\CD@hH{\CD@dC\space and \CD@HD\space commands must match.}%
+\def\CD@jH{\ifnum\inputlineno=0 \else\expandafter\CD@iH\fi}\def\CD@iH{\CD@MD
+\CD@GD\crcr\CD@YA{missing \CD@HD\space inserted before \CD@kH- type "h"}%
+\CD@lH\enddiagram\CD@AG\CD@kH\par}\def\CD@AG#1{\edef\enddiagram{\noexpand
+\CD@rD{#1\CD@W}}}\def\CD@rD#1{\CD@YA{\CD@HD\space(anticipated by #1) ignored}%
+\CD@xI\let\enddiagram\CD@SG}\def\CD@SG{\CD@YA{misplaced \CD@HD\space ignored}%
+\CD@hH}\def\CD@mC{\CD@YA{missing \CD@HD\space inserted.}\CD@hD\CD@AG{closing
+group}}\ifx\DeclareOption\CD@qK\else\ifx\DeclareOption\@notprerr\else
+\DeclareOption*{\let\CD@N\relax\let\CD@DH\relax\expandafter\CD@@E
+\CurrentOption,}\fi\fi
+%%======================================================================%
+%%                                                                      %
+%%      (22) AUXILLARY MACROS FOR ADJUSTMENT OF COMPONENTS              %
+%%                                                                      %
+%%======================================================================%
+
+%% NOTE: The recommended way of defining arrow commands is now
+%%      \newarrow{Name}{tail}{filler}{middle}{filler}{head}
+%% which defines \rName, \lName, \dName and \uName using arrow parts which
+%% have themselves previously been defined using the commands
+%% \newarrowtail, \newarrowfiller, \newarrowmiddle and \newarrowhead.
+
+%% The components \rhvee etc have been retained for the time being, as an
+%% intermediate stage and to continue to support the old \HorizontalMap and
+%% \VerticalMap commands, but you should not rely on the continued existence
+%% of these macros.
+
+%% The various components usually need some correction
+%% - longitudinally, ie to prevent gaps and overprints with the shaft,
+%% - transversally,  ie to prevent "steps" in the junction with the shaft.
+%% The former can be done safely ad hoc, eg with \mkern1mu.
+%% The latter are now done with the macros \scriptaxis, \boldscriptaxis,
+%% \shifthook and \raisehook, which include pixel corrections.
+
+%% Please note that these and the other auxillary macros which follow are
+%% interim. When it becomes clear exactly what kinds of adjustments are
+%% needed for characters, this job will be done by a suitable extension
+%% to the language of \newarrowhead, etc. If you have any other ideas for
+%% transformations of general use please tell me.
+
+%% By all means experiment with other characters for arrowheads, but
+%% please, in your own interests, do not rely on macros like \rhvee,
+%% send me a copy of your definitions for distribution to other users
+%% in this file, and keep track of where your efforts get copied so
+%% that they can be replaced with the "official" version when it is
+%% incorporated.
+
+%% ***** DONT use macros with mangled names like \Cd@gH. *****
+
+\catcode`\$=3 %% make sure that $ means maths-shift
+\def\vboxtoz{\vbox to\z@}%% \z@ is in plain TeX and means 0pt
+
+%% print #1 in \scriptstyle, adjusting for the maths axis height
+\def\scriptaxis#1{\@scriptaxis{$\scriptstyle#1$}}%%
+\def\ssaxis#1{\@ssaxis{$\scriptscriptstyle#1$}}%%
+\def\@scriptaxis#1{\dimen0\axisheight\advance\dimen0-\ss@axisheight\raise
+\dimen0\hbox{#1}}\def\@ssaxis#1{\dimen0\axisheight\advance\dimen0-%
+\ss@axisheight\raise\dimen0\hbox{#1}}
+
+%% Some of the characters would look better in bold since they're
+%% taken from sub/superscript fonts; we use LaTeX's \boldmath to
+%% do this, defining this to do nothing if it doesn't exist.
+%% With the old LaTeX font selection at other than 10pt you may still
+%% get nothing happenning.  Also, PK fonts may be missing.
+%% If you have problems, DONT use boldhook or boldlittlevee.
+\ifx\boldmath\CD@qK%%
+\let\boldscriptaxis\scriptaxis%%
+\def\boldscript#1{\hbox{$\scriptstyle#1$}}%%
+\def\boldscriptscript#1{\hbox{$\scriptscriptstyle#1$}}%%
+\else\def\boldscriptaxis#1{\@scriptaxis{\boldmath$\scriptstyle#1$}}%%
+\def\boldscript#1{\hbox{\boldmath$\scriptstyle#1$}}%%
+\def\boldscriptscript#1{\hbox{\boldmath$\scriptscriptstyle#1$}}%%
+\fi
+
+%%  #1= {} or \boldmath; #2= + or -; #3=\subset or \supset
+\def\raisehook#1#2#3{\hbox{\setbox3=\hbox{#1$\scriptscriptstyle#3$}%
+%% the character to use
+\dimen0\ss@axisheight%% \scriptscriptstyle axis height
+\dimen1\axisheight\advance\dimen1-\dimen0%% difference in axis heights
+\dimen2\ht3\advance\dimen2-\dimen0%
+%%  height of char above axis (half spread)
+\advance\dimen2-0.021em\advance\dimen1 #2\dimen2%
+%% shift = axis_difference +/- half_spread
+\raise\dimen1\box3}}%% print the character
+%% Mark Dawson suggested using the width
+\def\shifthook#1#2#3{\setbox1=\hbox{#1$\scriptscriptstyle#3$}\dimen0\wd1%
+\divide\dimen0 12\CD@zH{\dimen0}%%  "u"
+\dimen1\wd1\advance\dimen1-2\dimen0 \advance\dimen1-2\CD@oI\CD@zH{\dimen1}%
+\kern#2\dimen1\box1}%% print
+
+%% use the extension font (cmex) for double vertical arrows
+\def\@cmex{\mathchar"03}%%ascii double quote
+
+%%      ************* P U L L B A C K S ************
+
+%% These will probably be replaced by something less ad hoc
+%% in a future version.
+
+\def\make@pbk#1{\setbox\tw@\hbox to\z@{#1}\ht\tw@\z@\dp\tw@\z@\box\tw@}\def
+\CD@fH#1{\overprint{\hbox to\z@{#1}}}\def\CD@qH{\kern0.11em}\def\CD@pH{\kern0%
+.35em}
+
+%% This is a hack for my book ``Practical Foundations of Mathematics''
+%% and WILL NOT BE SUPPORTED --- DO NOT USE IT!
+\def\dblvert{\def\CD@rH{\kern.5\PileSpacing}}\def\CD@rH{}
+
+\def\SEpbk{\make@pbk{\CD@qH\CD@rH\vrule depth 2.87ex height -2.75ex width 0.%
+95em \vrule height -0.66ex depth 2.87ex width 0.05em \hss}}
+
+\def\SWpbk{\make@pbk{\hss\vrule height -0.66ex depth 2.87ex width 0.05em
+\vrule depth 2.87ex height -2.75ex width 0.95em \CD@qH\CD@rH}}
+
+\def\NEpbk{\make@pbk{\CD@qH\CD@rH\vrule depth -3.81ex height 4.00ex width 0.%
+95em \vrule height 4.00ex depth -1.72ex width 0.05em \hss}}
+
+\def\NWpbk{\make@pbk{\hss\vrule height 4.00ex depth -1.72ex width 0.05em
+\vrule depth -3.81ex height 4.00ex width 0.95em \CD@qH\CD@rH}}
+
+%%  Freyd & Scedrov puncture symbol for non-commuting polygon
+\def\puncture{{\setbox0\hbox{A}\vrule height.53\ht0 depth-.47\ht0 width.35\ht
+0 \kern.12\ht0 \vrule height\ht0 depth-.65\ht0 width.06\ht0 \kern-.06\ht0
+\vrule height.35\ht0 depth0pt width.06\ht0 \kern.12\ht0 \vrule height.53\ht0
+depth-.47\ht0 width.35\ht0 }}
+
+%% 2-cells: (24.11.95)
+%%% \NEclck puts a clockwise (ie southeast) arrow to the northwest of cell etc
+\def\NEclck{\overprint{\raise2.5ex\rlap{ \CD@rH$\scriptstyle\searrow$}}}%%
+\def\NEanti{\overprint{\raise2.5ex\rlap{ \CD@rH$\scriptstyle\nwarrow$}}}%%
+\def\NWclck{\overprint{\raise2.5ex\llap{$\scriptstyle\nearrow$ \CD@rH}}}%%
+\def\NWanti{\overprint{\raise2.5ex\llap{$\scriptstyle\swarrow$ \CD@rH}}}%%
+\def\SEclck{\overprint{\lower1ex\rlap{ \CD@rH$\scriptstyle\swarrow$}}}%%
+\def\SEanti{\overprint{\lower1ex\rlap{ \CD@rH$\scriptstyle\nearrow$}}}%%
+\def\SWclck{\overprint{\lower1ex\llap{$\scriptstyle\nwarrow$ \CD@rH}}}%%
+\def\SWanti{\overprint{\lower1ex\llap{$\scriptstyle\searrow$ \CD@rH}}}
+
+%%======================================================================%
+%%                                                                      %
+%%      (23) BITS OF ARROWS                                             %
+%%                                                                      %
+%%======================================================================%
+
+%%       **********  H E A D S ***********
+
+%% \diagramstyle[heads=xxx] defines {>} as {xxx} where xxx
+%% has been defined by \newarrowhead{xxx} and \newarrowtail{xxx}
+
+%% vee head
+\def\rhvee{\mkern-10mu\greaterthan}%%
+\def\lhvee{\lessthan\mkern-10mu}%%
+\def\dhvee{\vboxtoz{\vss\hbox{$\vee$}\kern0pt}}%%
+\def\uhvee{\vboxtoz{\hbox{$\wedge$}\vss}}%%
+\newarrowhead{vee}\rhvee\lhvee\dhvee\uhvee
+
+%% little vee head
+\def\dhlvee{\vboxtoz{\vss\hbox{$\scriptstyle\vee$}\kern0pt}}%%
+\def\uhlvee{\vboxtoz{\hbox{$\scriptstyle\wedge$}\vss}}%%
+\newarrowhead{littlevee}{\mkern1mu\scriptaxis\rhvee}{\scriptaxis\lhvee}%
+\dhlvee\uhlvee\ifx\boldmath\CD@qK%%
+\newarrowhead{boldlittlevee}{\mkern1mu\scriptaxis\rhvee}{\scriptaxis\lhvee}%
+\dhlvee\uhlvee\else%%
+\def\dhblvee{\vboxtoz{\vss\boldscript\vee\kern0pt}}%%
+\def\uhblvee{\vboxtoz{\boldscript\wedge\vss}}%%
+\newarrowhead{boldlittlevee}{\mkern1mu\boldscriptaxis\rhvee}{\boldscriptaxis
+\lhvee}\dhblvee\uhblvee%%
+\fi
+
+%% curly vee head (uses AMS symbols fonts)
+\def\rhcvee{\mkern-10mu\succ}%%
+\def\lhcvee{\prec\mkern-10mu}%%
+\def\dhcvee{\vboxtoz{\vss\hbox{$\curlyvee$}\kern0pt}}%%
+\def\uhcvee{\vboxtoz{\hbox{$\curlywedge$}\vss}}%%
+\newarrowhead{curlyvee}\rhcvee\lhcvee\dhcvee\uhcvee
+
+%% double vee head %% will probably be withdrawn later
+\def\rhvvee{\mkern-13mu\gg}%% 24.8.92 changed 10mu to 13mu
+\def\lhvvee{\ll\mkern-13mu}%% to make rule go through
+\def\dhvvee{\vboxtoz{\vss\hbox{$\vee$}\kern-.6ex\hbox{$\vee$}\kern0pt}}%%
+\def\uhvvee{\vboxtoz{\hbox{$\wedge$}\kern-.6ex \hbox{$\wedge$}\vss}}%%
+\newarrowhead{doublevee}\rhvvee\lhvvee\dhvvee\uhvvee
+
+%% open and closed triangles (uses AMS symbols fonts)
+\def\triangleup{{\scriptscriptstyle\bigtriangleup}}%%
+\def\littletriangledown{{\scriptscriptstyle\triangledown}}%% AMS
+\def\rhtriangle{\triangleright\mkern1.2mu}%% 29.1.93
+\def\lhtriangle{\triangleleft\mkern.8mu}%%
+\def\uhtriangle{\vbox{\kern-.2ex \hbox{$\scriptscriptstyle\bigtriangleup$}%
+\kern-.25ex}}%%
+%% Changed \scriptstyle\triangledown to \scriptscriptstyle\bigtriangledown
+%% at the suggestion of Martin Hofmann (25.11.92) to avoid using AMS symbols
+%% and also for compatibility with upward arrow.
+\def\dhtriangle{\vbox{\kern-.28ex \hbox{$\scriptscriptstyle\bigtriangledown$}%
+\kern-.1ex}}%% 15.1.93 from -.25ex
+\def\dhblack{\vbox{\kern-.25ex\nointerlineskip\hbox{$\blacktriangledown$}}}%
+%% AMS
+\def\uhblack{\vbox{\kern-.25ex\nointerlineskip\hbox{$\blacktriangle$}}}%
+%% AMS
+\def\dhlblack{\vbox{\kern-.25ex\nointerlineskip\hbox{$\scriptstyle
+\blacktriangledown$}}}%% AMS
+\def\uhlblack{\vbox{\kern-.25ex\nointerlineskip\hbox{$\scriptstyle
+\blacktriangle$}}}%% AMS
+\newarrowhead{triangle}\rhtriangle\lhtriangle\dhtriangle\uhtriangle
+\newarrowhead{blacktriangle}{\mkern-1mu\blacktriangleright\mkern.4mu}{%
+\blacktriangleleft}\dhblack\uhblack\newarrowhead{littleblack}{\mkern-1mu%
+\scriptaxis\blacktriangleright}{\scriptaxis\blacktriangleleft\mkern-2mu}%
+\dhlblack\uhlblack
+
+%% LaTeX arrowheads
+\def\rhla{\hbox{\setbox0=\lnchar55\dimen0=\wd0\kern-.6\dimen0\ht0\z@\raise
+\axisheight\box0\kern.1\dimen0}}%%
+\def\lhla{\hbox{\setbox0=\lnchar33\dimen0=\wd0\kern.05\dimen0\ht0\z@\raise
+\axisheight\box0\kern-.5\dimen0}}%%
+\def\dhla{\vboxtoz{\vss\rlap{\lnchar77}}}%%
+\def\uhla{\vboxtoz{\setbox0=\lnchar66 \wd0\z@\kern-.15\ht0\box0\vss}}%% 1/93
+\newarrowhead{LaTeX}\rhla\lhla\dhla\uhla
+
+%% double LaTeX arrowheads %% will probably be withdrawn later
+\def\lhlala{\lhla\kern.3em\lhla}%%
+\def\rhlala{\rhla\kern.3em\rhla}%%
+\def\uhlala{\hbox{\uhla\raise-.6ex\uhla}}%%
+\def\dhlala{\hbox{\dhla\lower-.6ex\dhla}}%%
+\newarrowhead{doubleLaTeX}\rhlala\lhlala\dhlala\uhlala
+
+%% circles % \rho is a Greek letter!
+\def\hhO{\scriptaxis\bigcirc\mkern.4mu} \def\hho{{\circ}\mkern1.2mu}%
+\newarrowhead{o}\hho\hho\circ\circ%%
+\newarrowhead{O}\hhO\hhO{\scriptstyle\bigcirc}{\scriptstyle\bigcirc}%%
+
+%% crosses
+\def\rhtimes{\mkern-5mu{\times}\mkern-.8mu}\def\lhtimes{\mkern-.8mu{\times}%
+\mkern-5mu}\def\uhtimes{\setbox0=\hbox{$\times$}\ht0\axisheight\dp0-\ht0%
+\lower\ht0\box0 }\def\dhtimes{\setbox0=\hbox{$\times$}\ht0\axisheight\box0 }%
+\newarrowhead{X}\rhtimes\lhtimes\dhtimes\uhtimes\newarrowhead+++++
+
+%% empty head {} is also available
+
+%% Y from stmaryrd (vertical ones still need large adjustment)
+\newarrowhead{Y}{\mkern-3mu\Yright}{\Yleft\mkern-3mu}\Ydown\Yup
+
+%%       **********  H E A D S  with  S H A F T S  ***********
+
+%% little arrow with shaft
+\newarrowhead{->}\rightarrow\leftarrow\downarrow\uparrow
+
+%% arrow with double shaft
+%%\newarrowhead{=>}\Rightarrow\Leftarrow\Downarrow\Uparrow
+\newarrowhead{=>}\Rightarrow\Leftarrow{\@cmex7F}{\@cmex7E}
+
+%% harpoon with shaft (trailing up/left can be changed to down/right)
+\newarrowhead{harpoon}\rightharpoonup\leftharpoonup\downharpoonleft
+\upharpoonleft
+
+%% little double-headed arrow with shaft (uses AMS symbols fonts)
+\def\twoheaddownarrow{\rlap{$\downarrow$}\raise-.5ex\hbox{$\downarrow$}}%%
+\def\twoheaduparrow{\rlap{$\uparrow$}\raise.5ex\hbox{$\uparrow$}}%%
+\newarrowhead{->>}\twoheadrightarrow\twoheadleftarrow\twoheaddownarrow
+\twoheaduparrow
+
+%%       **********  T A I L S ***********
+
+%% vee tail
+\def\rtvee{\greaterthan}%%
+\def\ltvee{\mkern-1mu{\lessthan}\mkern.4mu}%% \mkern added 15.1.93
+\def\dtvee{\vee}%%
+\def\utvee{\wedge}%%
+\newarrowtail{vee}\greaterthan\ltvee\vee\wedge
+
+%% little vee tail
+\newarrowtail{littlevee}{\scriptaxis\greaterthan}{\mkern-1mu\scriptaxis
+\lessthan}{\scriptstyle\vee}{\scriptstyle\wedge}\ifx\boldmath\CD@qK
+\newarrowtail{boldlittlevee}{\scriptaxis\greaterthan}{\mkern-1mu\scriptaxis
+\lessthan}{\scriptstyle\vee}{\scriptstyle\wedge}\else\newarrowtail{%
+boldlittlevee}{\boldscriptaxis\greaterthan}{\mkern-1mu\boldscriptaxis
+\lessthan}{\boldscript\vee}{\boldscript\wedge}\fi
+
+%% curly vee tail (uses AMS symbols fonts)
+\newarrowtail{curlyvee}\succ{\mkern-1mu{\prec}\mkern.4mu}\curlyvee\curlywedge
+
+%% open and closed triangle tails (uses AMS symbols fonts)
+\def\rttriangle{\mkern1.2mu\triangleright}%% 29.1.93
+\newarrowtail{triangle}\rttriangle\lhtriangle\dhtriangle\uhtriangle
+\newarrowtail{blacktriangle}\blacktriangleright{\mkern-1mu\blacktriangleleft
+\mkern.4mu}\dhblack\uhblack\newarrowtail{littleblack}{\scriptaxis
+\blacktriangleright\mkern-2mu}{\mkern-1mu\scriptaxis\blacktriangleleft}%
+\dhlblack\uhlblack
+
+%% LaTeX tails
+\def\rtla{\hbox{\setbox0=\lnchar55\dimen0=\wd0\kern-.5\dimen0\ht0\z@\raise
+\axisheight\box0\kern-.2\dimen0}}%%
+\def\ltla{\hbox{\setbox0=\lnchar33\dimen0=\wd0\kern-.15\dimen0\ht0\z@\raise
+\axisheight\box0\kern-.5\dimen0}}%%
+\def\dtla{\vbox{\setbox0=\rlap{\lnchar77}\dimen0=\ht0\kern-.7\dimen0\box0%
+\kern-.1\dimen0}}%% 15.1.93 from -.6
+\def\utla{\vbox{\setbox0=\rlap{\lnchar66}\dimen0=\ht0\kern-.1\dimen0\box0%
+\kern-.6\dimen0}}%%
+\newarrowtail{LaTeX}\rtla\ltla\dtla\utla
+
+%% double vee tail %% will probably be withdrawn later
+\def\rtvvee{\gg\mkern-3mu}%%
+\def\ltvvee{\mkern-3mu\ll}%%
+\def\dtvvee{\vbox{\hbox{$\vee$}\kern-.6ex \hbox{$\vee$}\vss}}%%
+\def\utvvee{\vbox{\vss\hbox{$\wedge$}\kern-.6ex \hbox{$\wedge$}\kern\z@}}%%
+\newarrowtail{doublevee}\rtvvee\ltvvee\dtvvee\utvvee
+
+%% double LaTeX tails %% will probably be withdrawn later
+\def\ltlala{\ltla\kern.3em\ltla}%%
+\def\rtlala{\rtla\kern.3em\rtla}%%
+\def\utlala{\hbox{\utla\raise-.6ex\utla}}%%
+\def\dtlala{\hbox{\dtla\lower-.6ex\dtla}}%%
+\newarrowtail{doubleLaTeX}\rtlala\ltlala\dtlala\utlala
+
+%% bar (as in \mapsto)
+\def\utbar{\vrule height 0.093ex depth0pt width 0.4em}%%
+\let\dtbar\utbar%%
+\def\rtbar{\mkern1.5mu\vrule height 1.1ex depth.06ex width .04em\mkern1.5mu}%
+%%
+\let\ltbar\rtbar%%
+\newarrowtail{mapsto}\rtbar\ltbar\dtbar\utbar%%
+\newarrowtail{|}\rtbar\ltbar\dtbar\utbar%%ascii vertical bar (|)
+
+%% hooks (as in \into): choice of after/above and before/below
+
+\def\rthooka{\raisehook{}+\subset\mkern-1mu}%%
+\def\lthooka{\mkern-1mu\raisehook{}+\supset}%%
+\def\rthookb{\raisehook{}-\subset\mkern-2mu}%%
+\def\lthookb{\mkern-1mu\raisehook{}-\supset}%%
+
+\def\dthooka{\shifthook{}+\cap}%%
+\def\dthookb{\shifthook{}-\cap}%%
+\def\uthooka{\shifthook{}+\cup}%%
+\def\uthookb{\shifthook{}-\cup}%%
+
+\newarrowtail{hooka}\rthooka\lthooka\dthooka\uthooka\newarrowtail{hookb}%
+\rthookb\lthookb\dthookb\uthookb
+
+\ifx\boldmath\CD@qK\newarrowtail{boldhooka}\rthooka\lthooka\dthooka\uthooka
+\newarrowtail{boldhookb}\rthookb\lthookb\dthookb\uthookb\newarrowtail{%
+boldhook}\rthooka\lthooka\dthookb\uthooka\else\def\rtbhooka{\raisehook
+\boldmath+\subset\mkern-1mu}%%
+\def\ltbhooka{\mkern-1mu\raisehook\boldmath+\supset}%%
+\def\rtbhookb{\raisehook\boldmath-\subset\mkern-2mu}%%
+\def\ltbhookb{\mkern-1mu\raisehook\boldmath-\supset}%%
+\def\dtbhooka{\shifthook\boldmath+\cap}%%
+\def\dtbhookb{\shifthook\boldmath-\cap}%%
+\def\utbhooka{\shifthook\boldmath+\cup}%%
+\def\utbhookb{\shifthook\boldmath-\cup}%%
+\newarrowtail{boldhooka}\rtbhooka\ltbhooka\dtbhooka\utbhooka\newarrowtail{%
+boldhookb}\rtbhookb\ltbhookb\dtbhookb\utbhookb\newarrowtail{boldhook}%
+\rtbhooka\ltbhooka\dtbhooka\utbhooka\fi
+
+%% square-ended hooks (used for closed subsets in ``lifting and gluing'')
+\def\dtsqhooka{\shifthook{}+\sqcap}%%
+\def\dtsqhookb{\shifthook{}-\sqcap}%%
+\def\ltsqhooka{\mkern-1mu\raisehook{}+\sqsupset}%%
+\def\ltsqhookb{\mkern-1mu\raisehook{}-\sqsupset}%%
+\def\rtsqhooka{\raisehook{}+\sqsubset\mkern-1mu}%%
+\def\rtsqhookb{\raisehook{}-\sqsubset\mkern-2mu}%%
+\def\utsqhooka{\shifthook{}+\sqcup}%%
+\def\utsqhookb{\shifthook{}-\sqcup}%%
+\newarrowtail{sqhook}\rtsqhooka\ltsqhooka\dtsqhooka\utsqhooka
+
+%% the following seem the better choices at 10pt & 300dpi
+\newarrowtail{hook}\rthooka\lthookb\dthooka\uthooka\newarrowtail{C}\rthooka
+\lthookb\dthooka\uthooka
+
+%% circles
+\newarrowtail{o}\hho\hho\circ\circ%%
+\newarrowtail{O}\hhO\hhO{\scriptstyle\bigcirc}{\scriptstyle\bigcirc}%%
+
+%% crosses
+\newarrowtail{X}\lhtimes\rhtimes\uhtimes\dhtimes\newarrowtail+++++
+
+%% empty tail {} is also available
+
+%% Y from stmaryrd (vertical ones still need adjustment)
+\newarrowtail{Y}\Yright\Yleft\Ydown\Yup
+
+%% harpoon with shaft (trailing up/left can be changed to down/right)
+\newarrowtail{harpoon}\leftharpoondown\rightharpoondown\upharpoonright
+\downharpoonright
+
+%% arrow with double shaft
+%%\newarrowtail{<=}\Leftarrow\Rightarrow\Uparrow\Downarrow
+\newarrowtail{<=}\Leftarrow\Rightarrow{\@cmex7E}{\@cmex7F}
+
+%%      **********  F I L L E R S ***********
+
+%% shortening is up to 0.15em=2.7mu horiz and 0.35ex vertically at each end.
+
+%% dot {.}, single rule {-} and empty {} fillers are also available
+
+%% double and triple lines
+%%\newarrowfiller{=}==\Vert\Vert%%
+\newarrowfiller{=}=={\@cmex77}{\@cmex77}%% 16.1.93
+\def\vfthree{\mid\!\!\!\mid\!\!\!\mid}%%ascii
+\newarrowfiller{3}\equiv\equiv\vfthree\vfthree
+
+%% dashed line
+\def\vfdashstrut{\vrule width0pt height1.3ex depth0.7ex}%%
+\def\vfthedash{\vrule width\CD@LF height0.6ex depth 0pt}%%
+\def\hfthedash{\CD@AJ\vrule\horizhtdp width 0.26em}%%
+\def\hfdash{\mkern5.5mu\hfthedash\mkern5.5mu}%%
+\def\vfdash{\vfdashstrut\vfthedash}%%
+\newarrowfiller{dash}\hfdash\hfdash\vfdash\vfdash
+
+%%      ************* M I D D L E S ************
+
+%% plus
+\newarrowmiddle+++++
+
+%%      ************* D I A G O N A L S ************
+
+%% simple arrow heads
+%%\def\nwhTO{\nwarrow\mkern-1mu}%%
+%%\def\nehTO{\mkern-.1mu\nearrow}%%
+%%\def\sehTO{\searrow\mkern-.02mu}%%
+%%\def\swhTO{\mkern-.8mu\swarrow}%%
+
+%%======================================================================%
+%%                                                                      %
+%%      (24) ARROW COMMANDS                                             %
+%%                                                                      %
+%%======================================================================%
+
+%% change to \iftrue to get mixed heads
+\iffalse%%
+\newarrow{To}----{vee}%%
+\newarrow{Arr}----{LaTeX}%%
+\newarrow{Dotsto}....{vee}%%
+\newarrow{Dotsarr}....{LaTeX}%%
+\newarrow{Dashto}{}{dash}{}{dash}{vee}%%
+\newarrow{Dasharr}{}{dash}{}{dash}{LaTeX}%%
+\newarrow{Mapsto}{mapsto}---{vee}%%
+\newarrow{Mapsarr}{mapsto}---{LaTeX}%%
+\newarrow{IntoA}{hooka}---{vee}%%
+\newarrow{IntoB}{hookb}---{vee}%%
+\newarrow{Embed}{vee}---{vee}%%
+\newarrow{Emarr}{LaTeX}---{LaTeX}%%
+\newarrow{Onto}----{doublevee}%%
+\newarrow{Dotsonarr}....{doubleLaTeX}%%
+\newarrow{Dotsonto}....{doublevee}%%
+\newarrow{Dotsonarr}....{doubleLaTeX}%%
+\else%%
+\newarrow{To}---->%%
+\newarrow{Arr}---->%%
+\newarrow{Dotsto}....>%%
+\newarrow{Dotsarr}....>%%
+\newarrow{Dashto}{}{dash}{}{dash}>%%
+\newarrow{Dasharr}{}{dash}{}{dash}>%%
+\newarrow{Mapsto}{mapsto}--->%%
+\newarrow{Mapsarr}{mapsto}--->%%
+\newarrow{IntoA}{hooka}--->%%
+\newarrow{IntoB}{hookb}--->%%
+\newarrow{Embed}>--->%%
+\newarrow{Emarr}>--->%%
+\newarrow{Onto}----{>>}%%
+\newarrow{Dotsonarr}....{>>}%%
+\newarrow{Dotsonto}....{>>}%%
+\newarrow{Dotsonarr}....{>>}%%
+\fi%%
+
+\newarrow{Implies}===={=>}%% minimum cell height 9.5pt
+\newarrow{Project}----{triangle}%%
+\newarrow{Pto}----{harpoon}%% partial function
+\newarrow{Relto}{harpoon}---{harpoon}%% binary relation
+
+\newarrow{Eq}=====%%
+\newarrow{Line}-----%%
+\newarrow{Dots}.....%%
+\newarrow{Dashes}{}{dash}{}{dash}{}%%
+
+%% square hooked arrow (used in my ``gluing and lifting'' paper)
+\newarrow{SquareInto}{sqhook}--->
+
+%% braces and parentheses
+%% \newarrow gives inappropriate directions, so we change the names
+%% the vertical filler is too far to the right; horizontal too high
+%% the vertical middles are too low with midvshaft
+%% maybe we'll add square brackets and the integral sign one day
+\newarrowhead{cmexbra}{\@cmex7B}{\@cmex7C}{\@cmex3B}{\@cmex38}%%
+\newarrowtail{cmexbra}{\@cmex7A}{\@cmex7D}{\@cmex39}{\@cmex3A}%%
+\newarrowmiddle{cmexbra}{\braceru\bracelu}{\bracerd\braceld}{\vcenter{%
+\hbox@maths{\@cmex3D\mkern-2mu}}}%% right
+{\vcenter{\hbox@maths{\mkern2mu\@cmex3C}}}%% left
+\newarrow{@brace}{cmexbra}-{cmexbra}-{cmexbra}%% braces
+\newarrow{@parenth}{cmexbra}---{cmexbra}%% straight parentheses
+\def\rightBrace{\d@brace[thick,cmex]}%%ASCII square brackets []
+\def\leftBrace{\u@brace[thick,cmex]}%%ASCII square brackets []
+\def\upperBrace{\r@brace[thick,cmex]}%%ASCII square brackets []
+\def\lowerBrace{\l@brace[thick,cmex]}%%ASCII square brackets []
+\def\rightParenth{\d@parenth[thick,cmex]}%%ASCII square brackets []
+\def\leftParenth{\u@parenth[thick,cmex]}%%ASCII square brackets []
+\def\upperParenth{\r@parenth[thick,cmex]}%%ASCII square brackets []
+\def\lowerParenth{\l@parenth[thick,cmex]}%%ASCII square brackets []
+
+%% synonyms for reverse compatibility
+
+\let\uFrom\uTo%%
+\let\lFrom\lTo%%
+\let\uDotsfrom\uDotsto%%
+\let\lDotsfrom\lDotsto%%
+\let\uDashfrom\uDashto%%
+\let\lDashfrom\lDashto%%
+\let\uImpliedby\uImplies%%
+\let\lImpliedby\lImplies%%
+\let\uMapsfrom\uMapsto%%
+\let\lMapsfrom\lMapsto%%
+\let\lOnfrom\lOnto%%
+\let\uOnfrom\uOnto%%
+\let\lPfrom\lPto%%
+\let\uPfrom\uPto%%
+
+\let\uInfromA\uIntoA%%
+\let\uInfromB\uIntoB%%
+\let\lInfromA\lIntoA%%
+\let\lInfromB\lIntoB%%
+\let\rInto\rIntoA%%
+\let\lInto\lIntoA%%
+\let\dInto\dIntoB%%
+\let\uInto\uIntoA%%
+\let\ruInto\ruIntoA%%
+\let\luInto\luIntoA%%
+\let\rdInto\rdIntoA%%
+\let\ldInto\ldIntoA%%
+%%
+\let\hEq\rEq%%
+\let\vEq\uEq%%
+\let\hLine\rLine%%
+\let\vLine\uLine%%
+\let\hDots\rDots%%
+\let\vDots\uDots%%
+\let\hDashes\rDashes%%
+\let\vDashes\uDashes%%
+
+%%=========================================================================%
+%% The following are included for reverse compatibility only.
+%%=========================================================================%
+\let\NW\luTo\let\NE\ruTo\let\SW\ldTo\let\SE\rdTo\def\nNW{\luTo(2,3)}\def\nNE{%
+\ruTo(2,3)}%%ascii
+\def\sSW{\ldTo(2,3)}\def\sSE{\rdTo(2,3)}%%ascii
+\def\wNW{\luTo(3,2)}\def\eNE{\ruTo(3,2)}%%ascii
+\def\wSW{\ldTo(3,2)}\def\eSE{\rdTo(3,2)}%%ascii
+\def\NNW{\luTo(2,4)}\def\NNE{\ruTo(2,4)}%%ascii
+\def\SSW{\ldTo(2,4)}\def\SSE{\rdTo(2,4)}%%ascii
+\def\WNW{\luTo(4,2)}\def\ENE{\ruTo(4,2)}%%ascii
+\def\WSW{\ldTo(4,2)}\def\ESE{\rdTo(4,2)}%%ascii
+\def\NNNW{\luTo(2,6)}\def\NNNE{\ruTo(2,6)}%%ascii
+\def\SSSW{\ldTo(2,6)}\def\SSSE{\rdTo(2,6)}%%ascii
+\def\WWNW{\luTo(6,2)}\def\EENE{\ruTo(6,2)}%%ascii
+\def\WWSW{\ldTo(6,2)}\def\EESE{\rdTo(6,2)}%%ascii
+
+\let\NWd\luDotsto\let\NEd\ruDotsto\let\SWd\ldDotsto\let\SEd\rdDotsto\def\nNWd
+{\luDotsto(2,3)}\def\nNEd{\ruDotsto(2,3)}%%ascii
+\def\sSWd{\ldDotsto(2,3)}\def\sSEd{\rdDotsto(2,3)}%%ascii
+\def\wNWd{\luDotsto(3,2)}\def\eNEd{\ruDotsto(3,2)}%%ascii
+\def\wSWd{\ldDotsto(3,2)}\def\eSEd{\rdDotsto(3,2)}%%ascii
+\def\NNWd{\luDotsto(2,4)}\def\NNEd{\ruDotsto(2,4)}%%ascii
+\def\SSWd{\ldDotsto(2,4)}\def\SSEd{\rdDotsto(2,4)}%%ascii
+\def\WNWd{\luDotsto(4,2)}\def\ENEd{\ruDotsto(4,2)}%%ascii
+\def\WSWd{\ldDotsto(4,2)}\def\ESEd{\rdDotsto(4,2)}%%ascii
+\def\NNNWd{\luDotsto(2,6)}\def\NNNEd{\ruDotsto(2,6)}%%ascii
+\def\SSSWd{\ldDotsto(2,6)}\def\SSSEd{\rdDotsto(2,6)}%%ascii
+\def\WWNWd{\luDotsto(6,2)}\def\EENEd{\ruDotsto(6,2)}%%ascii
+\def\WWSWd{\ldDotsto(6,2)}\def\EESEd{\rdDotsto(6,2)}%%ascii
+
+\let\NWl\luLine\let\NEl\ruLine\let\SWl\ldLine\let\SEl\rdLine\def\nNWl{\luLine
+(2,3)}\def\nNEl{\ruLine(2,3)}%%ascii
+\def\sSWl{\ldLine(2,3)}\def\sSEl{\rdLine(2,3)}%%ascii
+\def\wNWl{\luLine(3,2)}\def\eNEl{\ruLine(3,2)}%%ascii
+\def\wSWl{\ldLine(3,2)}\def\eSEl{\rdLine(3,2)}%%ascii
+\def\NNWl{\luLine(2,4)}\def\NNEl{\ruLine(2,4)}%%ascii
+\def\SSWl{\ldLine(2,4)}\def\SSEl{\rdLine(2,4)}%%ascii
+\def\WNWl{\luLine(4,2)}\def\ENEl{\ruLine(4,2)}%%ascii
+\def\WSWl{\ldLine(4,2)}\def\ESEl{\rdLine(4,2)}%%ascii
+\def\NNNWl{\luLine(2,6)}\def\NNNEl{\ruLine(2,6)}%%ascii
+\def\SSSWl{\ldLine(2,6)}\def\SSSEl{\rdLine(2,6)}%%ascii
+\def\WWNWl{\luLine(6,2)}\def\EENEl{\ruLine(6,2)}%%ascii
+\def\WWSWl{\ldLine(6,2)}\def\EESEl{\rdLine(6,2)}%%ascii
+
+\let\NWld\luDots\let\NEld\ruDots\let\SWld\ldDots\let\SEld\rdDots\def\nNWld{%
+\luDots(2,3)}\def\nNEld{\ruDots(2,3)}%%ascii
+\def\sSWld{\ldDots(2,3)}\def\sSEld{\rdDots(2,3)}%%ascii
+\def\wNWld{\luDots(3,2)}\def\eNEld{\ruDots(3,2)}%%ascii
+\def\wSWld{\ldDots(3,2)}\def\eSEld{\rdDots(3,2)}%%ascii
+\def\NNWld{\luDots(2,4)}\def\NNEld{\ruDots(2,4)}%%ascii
+\def\SSWld{\ldDots(2,4)}\def\SSEld{\rdDots(2,4)}%%ascii
+\def\WNWld{\luDots(4,2)}\def\ENEld{\ruDots(4,2)}%%ascii
+\def\WSWld{\ldDots(4,2)}\def\ESEld{\rdDots(4,2)}%%ascii
+\def\NNNWld{\luDots(2,6)}\def\NNNEld{\ruDots(2,6)}%%ascii
+\def\SSSWld{\ldDots(2,6)}\def\SSSEld{\rdDots(2,6)}%%ascii
+\def\WWNWld{\luDots(6,2)}\def\EENEld{\ruDots(6,2)}%%ascii
+\def\WWSWld{\ldDots(6,2)}\def\EESEld{\rdDots(6,2)}%%ascii
+
+\let\NWe\luEmbed\let\NEe\ruEmbed\let\SWe\ldEmbed\let\SEe\rdEmbed\def\nNWe{%
+\luEmbed(2,3)}\def\nNEe{\ruEmbed(2,3)}%%ascii
+\def\sSWe{\ldEmbed(2,3)}\def\sSEe{\rdEmbed(2,3)}%%ascii
+\def\wNWe{\luEmbed(3,2)}\def\eNEe{\ruEmbed(3,2)}%%ascii
+\def\wSWe{\ldEmbed(3,2)}\def\eSEe{\rdEmbed(3,2)}%%ascii
+\def\NNWe{\luEmbed(2,4)}\def\NNEe{\ruEmbed(2,4)}%%ascii
+\def\SSWe{\ldEmbed(2,4)}\def\SSEe{\rdEmbed(2,4)}%%ascii
+\def\WNWe{\luEmbed(4,2)}\def\ENEe{\ruEmbed(4,2)}%%ascii
+\def\WSWe{\ldEmbed(4,2)}\def\ESEe{\rdEmbed(4,2)}%%ascii
+\def\NNNWe{\luEmbed(2,6)}\def\NNNEe{\ruEmbed(2,6)}%%ascii
+\def\SSSWe{\ldEmbed(2,6)}\def\SSSEe{\rdEmbed(2,6)}%%ascii
+\def\WWNWe{\luEmbed(6,2)}\def\EENEe{\ruEmbed(6,2)}%%ascii
+\def\WWSWe{\ldEmbed(6,2)}\def\EESEe{\rdEmbed(6,2)}%%ascii
+
+\let\NWo\luOnto\let\NEo\ruOnto\let\SWo\ldOnto\let\SEo\rdOnto\def\nNWo{\luOnto
+(2,3)}\def\nNEo{\ruOnto(2,3)}%%ascii
+\def\sSWo{\ldOnto(2,3)}\def\sSEo{\rdOnto(2,3)}%%ascii
+\def\wNWo{\luOnto(3,2)}\def\eNEo{\ruOnto(3,2)}%%ascii
+\def\wSWo{\ldOnto(3,2)}\def\eSEo{\rdOnto(3,2)}%%ascii
+\def\NNWo{\luOnto(2,4)}\def\NNEo{\ruOnto(2,4)}%%ascii
+\def\SSWo{\ldOnto(2,4)}\def\SSEo{\rdOnto(2,4)}%%ascii
+\def\WNWo{\luOnto(4,2)}\def\ENEo{\ruOnto(4,2)}%%ascii
+\def\WSWo{\ldOnto(4,2)}\def\ESEo{\rdOnto(4,2)}%%ascii
+\def\NNNWo{\luOnto(2,6)}\def\NNNEo{\ruOnto(2,6)}%%ascii
+\def\SSSWo{\ldOnto(2,6)}\def\SSSEo{\rdOnto(2,6)}%%ascii
+\def\WWNWo{\luOnto(6,2)}\def\EENEo{\ruOnto(6,2)}%%ascii
+\def\WWSWo{\ldOnto(6,2)}\def\EESEo{\rdOnto(6,2)}%%ascii
+
+\let\NWod\luDotsonto\let\NEod\ruDotsonto\let\SWod\ldDotsonto\let\SEod
+\rdDotsonto\def\nNWod{\luDotsonto(2,3)}\def\nNEod{\ruDotsonto(2,3)}%%ascii
+\def\sSWod{\ldDotsonto(2,3)}\def\sSEod{\rdDotsonto(2,3)}%%ascii
+\def\wNWod{\luDotsonto(3,2)}\def\eNEod{\ruDotsonto(3,2)}%%ascii
+\def\wSWod{\ldDotsonto(3,2)}\def\eSEod{\rdDotsonto(3,2)}%%ascii
+\def\NNWod{\luDotsonto(2,4)}\def\NNEod{\ruDotsonto(2,4)}%%ascii
+\def\SSWod{\ldDotsonto(2,4)}\def\SSEod{\rdDotsonto(2,4)}%%ascii
+\def\WNWod{\luDotsonto(4,2)}\def\ENEod{\ruDotsonto(4,2)}%%ascii
+\def\WSWod{\ldDotsonto(4,2)}\def\ESEod{\rdDotsonto(4,2)}%%ascii
+\def\NNNWod{\luDotsonto(2,6)}\def\NNNEod{\ruDotsonto(2,6)}%%ascii
+\def\SSSWod{\ldDotsonto(2,6)}\def\SSSEod{\rdDotsonto(2,6)}%%ascii
+\def\WWNWod{\luDotsonto(6,2)}\def\EENEod{\ruDotsonto(6,2)}%%ascii
+\def\WWSWod{\ldDotsonto(6,2)}\def\EESEod{\rdDotsonto(6,2)}%%ascii
+
+%%======================================================================%
+%%                                                                      %
+%%      (25) MISCELLANEOUS                                              %
+%%                                                                      %
+%%======================================================================%
+
+\def\labelstyle{%%
+\ifincommdiag%%
+\textstyle%%
+\else%%
+\scriptstyle%%
+\fi}%%
+\let\objectstyle\displaystyle
+
+\newdiagramgrid{pentagon}{0.618034,0.618034,1,1,1,1,0.618034,0.618034}{1.%
+17557,1.17557,1.902113,1.902113}
+
+\newdiagramgrid{perspective}{0.75,0.75,1.1,1.1,0.9,0.9,0.95,0.95,0.75,0.75}{0%
+.75,0.75,1.1,1.1,0.9,0.9}
+
+\diagramstyle[%%ascii open square bracket
+dpi=300,%%              office laserwriters are usually 300 dots per inch
+vmiddle,nobalance,%%    vertical and horizontal positioning
+loose,%%                allow rows and columns to stretch
+thin,%%                 line10 arrows; default rule thickness (TeXbook p447)
+pilespacing=10pt,%
+%%     parallel vertical separation (horizontals: half this)
+shortfall=4pt,%%        distance between arrowheads and their targets
+%% The following are defaulted on entry to the diagram itself.
+%% l>=2em               minimum length of horizontal arrow shafts in text
+%% l>=1em               ditto in diagrams
+%% size=3em             cell size
+%% heads=LaTeX          arrowheads
+]%%ascii close square bracket
+
+%% process options to LaTeX2e's \usepackage[options]{diagrams}
+\ifx\ProcessOptions\CD@qK\else\CD@PK\ProcessOptions\relax\CD@FF\CD@e\fi\fi
+
+%%============================== THE END ====================================
+\CD@vE\CD@hK\message{| running in pdf mode -- diagonal arrows will work
+automatically |}\else\message{| >>>>>>>> POSTSCRIPT MODE (DVIPS) IS NOW THE
+DEFAULT <<<<<<<<<<<<|}\message{|(DVI mode has not been supported since 1992
+and produces inferior|}\message{|results which are completely unsuitable for
+publication. However,|}\message{|if you really still need it, you can still
+get it by loading the |}\message{|package using ``\string\usepackage[%
+UglyObsolete]{diagrams}'' instead. ) |}\fi\else\message{| >>>>>>>> USING UGLY
+OBSOLETE DVI CODE - PLEASE STOP <<<<<<<<<<<<|}\message{|(DVI mode has not been
+supported since 1992 and produces inferior|}\message{|results which are
+completely unsuitable for publication - Please |}\message{|use the PostScript
+or PDF mode instead, for much better results.)|}\fi\cdrestoreat
+%% restore old category code for @ etc
+\message{===================================================================}%
+%% This is the end of Paul Taylor's commutative diagrams package.
+
diff --git a/dissertation your fellowship/merged_document.pdf b/dissertation your fellowship/merged_document.pdf
new file mode 100644
index 00000000..c0b5dcd0
Binary files /dev/null and b/dissertation your fellowship/merged_document.pdf differ
diff --git a/dissertation your fellowship/merged_document_2.pdf b/dissertation your fellowship/merged_document_2.pdf
new file mode 100644
index 00000000..05860ccd
Binary files /dev/null and b/dissertation your fellowship/merged_document_2.pdf differ
diff --git a/dissertation your fellowship/merged_document_3.pdf b/dissertation your fellowship/merged_document_3.pdf
new file mode 100644
index 00000000..d76df6d6
Binary files /dev/null and b/dissertation your fellowship/merged_document_3.pdf differ
diff --git a/dissertation your fellowship/personal statement.pdf b/dissertation your fellowship/personal statement.pdf
new file mode 100644
index 00000000..4d060051
Binary files /dev/null and b/dissertation your fellowship/personal statement.pdf differ
diff --git a/dissertation your fellowship/personal statement.tex b/dissertation your fellowship/personal statement.tex
new file mode 100644
index 00000000..fd3ddbc9
--- /dev/null
+++ b/dissertation your fellowship/personal statement.tex	
@@ -0,0 +1,39 @@
+\documentclass[11pt]{article}
+\usepackage[margin=1in]{geometry}
+
+\usepackage{fancyhdr}
+\pagestyle{fancy}
+
+%\usepackage{mathrsfs}
+
+\usepackage{setspace}
+
+\doublespacing
+\rhead{Anton Bobkov}
+
+\lhead{Personal Statement of Career Goals}
+
+
+
+
+\begin{document}
+\section*{Personal statement}
+
+P-adic numbers are a simple, yet a very deep construction. They were only discovered a hundred years ago, but could have been studied in classical mathematics when number theory was just forming. Their construction is simple enough to explain at the undergraduate level, yet has a very rich number theoretic structure. Normally the real numbers are constructed by first taking rational numbers in decimal form and allowing infinite decimal sequences after the decimal point. 
+Letting decimals be infinite before the decimal point yields a well behaved mathematical object as well, but with a drastically different behavior from real numbers, now depending on the base in which the decimals were written. 
+When the base is a prime number p, this constructs p-adic numbers. These were first studied exclusively within number theory, but later
+			 found applications in other areas of math, 
+			physics, and computer science. My research 
+			will allow 
+			for a finer understanding of the finite structure 
+		of polynomially definable sets in p-adic numbers. 
+			In my career as an educator I hope to increase exposure to this elegant and rich construction for students both inside and outside of mathematics.
+		
+	
+My research lies in the area of model theory, a branch of formal logic. Model theory began with G\"odel and Malcev in the 1930s, but first matured as a subject in the work of Abraham Robinson, Tarski, Vaught, and others in the 1950s. Model theory studies sets definable by first order formulas in a variety of mathematical objects. Restricting to subsets definable by simple formulas gives access to an array of powerful techniques such as indiscernible sequences and nonstandard extensions. These allow insights not otherwise accessible by classical methods. Nonstandard real numbers, for example, formalize the notion of infinitesimals. Model theory is an extremely flexible field with applications in many areas of mathematics including algebra, analysis, geometry, number theory, and combinatorics as well as some applications to computer science and quantum mechanics. In my career as a mathematician I hope to expose researchers in other fields to model theoretic methods allowing them to explore alternative approaches to classical mathematical objects.
+
+My research concentrates on the concept of VC-density, a recent notion of rank in NIP theories. The study of a structure in model theory usually starts with quantifier elimination, followed by a finer analysis of definable functions and interpretability. The study of VC-density goes one step further, looking at a structure of the asymptotic growth of finite definable families. In the most geometric examples, VC-density coincides with the natural notion of dimension. However, no geometric structure is required for the definition of VC-density, thus we can get some notion of geometric dimension for families of sets given without any geometric context! In my career as a researcher I hope to further explore this notion and introduce other model theorists to its applications.
+
+To summarize, I intend to follow a career path in academia, balancing my teaching with my research. An important part of being a mathematician is communicating and disseminating mathematical knowledge. I have enjoyed my work as a teaching assistant, and look forward to working with students at all stages of their mathematical education. Another equally important part is developing and progressing mathematical knowledge. My work in model theory has been a great motivation for me, and I plan to stay an active researcher for the rest of my mathematical career.
+\end{document}
+
diff --git a/research/10 QP reduct/QP_reduct.pdf b/research/10 QP reduct/QP_reduct.pdf
index e6df1783..ad4eb2b6 100644
Binary files a/research/10 QP reduct/QP_reduct.pdf and b/research/10 QP reduct/QP_reduct.pdf differ
diff --git a/research/10 QP reduct/QP_reduct.tex b/research/10 QP reduct/QP_reduct.tex
index 546b4728..9f613410 100644
--- a/research/10 QP reduct/QP_reduct.tex	
+++ b/research/10 QP reduct/QP_reduct.tex	
@@ -8,8 +8,8 @@
 \usepackage{pgfpages}
 \usepackage{setspace}
 
-% \doublespacing
-\usepackage[margin=.75in]{geometry}
+\doublespacing
+% \usepackage[margin=.75in]{geometry}
 % \pgfpagesuselayout{2 on 1}
 
 \renewcommand{\AA}{\mathscr A}
@@ -567,9 +567,10 @@ \section{Key Lemmas and Definitions}
   (see Lemma \ref{quantifier_elimination}).
   Fix $i,j \in I, b \in B$.
   We would like to show that 
-  \begin{align*} \label {order_equation}
+
+  \begin{equation} \label {eq:order_equation}
     \vval (p_i(d) - c_i(b)) < \vval (p_j(d) - c_j(b))  \iff \vval (p_i(d') - c_i(b)) < \vval (p_j(d') - c_j(b))
-  \end{align*}
+  \end{equation}
 
   For the following argument we will need more notation.
   Suppose $a \in \Q_p$ lies in an interval $\paren{B(t_L, \alpha_L),  B(t_U, \alpha_U)}$.
@@ -581,7 +582,7 @@ \section{Key Lemmas and Definitions}
     &\vval (p_i(d) - c_i(b)) = \vval (p_i(d') - c_i(b)) = \vval(\tbr(p_i(d)) - c_i(b)) \\
     &\vval (p_j(d) - c_j(b)) = \vval (p_j(d') - c_j(b)) = \vval(\tbr(p_j(d)) - c_j(b))  \\
   \end{align*}
-  Then it is clear that \ref{order_equation} holds.
+  Then it is clear that \eqref{eq:order_equation} holds.
 
   Case 2:
   \begin{align*}
@@ -596,12 +597,12 @@ \section{Key Lemmas and Definitions}
     &\vval (p_j(d) - c_j(b)) = \tval(p_j(d)) \text{ and } \vval (p_j(d') - c_j(b)) = \tval(p_j(d')) \\
   \end{align*}
   If $p_j(d), p_j(d')$ are close to boundary,
-  then $\tval(p_j(d)) = \tval(p_j(d'))$ and \ref{order_equation} clearly holds.
+  then $\tval(p_j(d)) = \tval(p_j(d'))$ and \eqref{eq:order_equation} clearly holds.
   Suppose then that $p_j(d), p_j(d')$ are far from boundary.
   Suppose that $p_j(d), p_j(d')$ lie in the sub-interval $\paren{B(t_L, \alpha_L),  B(t_U, \alpha_U)}$.
   Then $\tval(p_j(d)), \tval(p_j(d)') \in (\alpha_L, \alpha_U)$ (as an interval in $\Z$)
   and $\vval(\tbr(p_i(d)) - c_i(b))$ lies outside of $(\alpha_L, \alpha_U)$ by definition of sub-interval.
-  Therefore \ref{order_equation} has to hold.
+  Therefore \eqref{eq:order_equation} has to hold.
   (Note that we always have $\tval(p_j(d)), \tval(p_j(d)') \in (\alpha_L, \alpha_U]$,
   we need the far from boundary condition to avoid equality to $\alpha_U$.)
 
diff --git a/research/10 QP reduct/QP_reduct.tex~ b/research/10 QP reduct/QP_reduct.tex~
index e6632244..546b4728 100644
--- a/research/10 QP reduct/QP_reduct.tex~	
+++ b/research/10 QP reduct/QP_reduct.tex~	
@@ -598,13 +598,23 @@ The following lemma is an adaptation of lemma 7.4 in \cite{density}.
   If $p_j(d), p_j(d')$ are close to boundary,
   then $\tval(p_j(d)) = \tval(p_j(d'))$ and \ref{order_equation} clearly holds.
   Suppose then that $p_j(d), p_j(d')$ are far from boundary.
-  Suppose that $p_j(d), p_j(d')$ lie in then sub-interval $\paren{B(t_L, \alpha_L),  B(t_U, \alpha_U)}$.
+  Suppose that $p_j(d), p_j(d')$ lie in the sub-interval $\paren{B(t_L, \alpha_L),  B(t_U, \alpha_U)}$.
   Then $\tval(p_j(d)), \tval(p_j(d)') \in (\alpha_L, \alpha_U)$ (as an interval in $\Z$)
   and $\vval(\tbr(p_i(d)) - c_i(b))$ lies outside of $(\alpha_L, \alpha_U)$ by definition of sub-interval.
   Therefore \ref{order_equation} has to hold.
+  (Note that we always have $\tval(p_j(d)), \tval(p_j(d)') \in (\alpha_L, \alpha_U]$,
+  we need the far from boundary condition to avoid equality to $\alpha_U$.)
+
+  Case 4:
+  \begin{align*}
+    &\vval (p_i(d) - c_i(b)) = \tval(p_i(d)) \text{ and } \vval (p_i(d') - c_i(b)) = \tval(p_i(d')) \\
+    &\vval (p_j(d) - c_j(b)) = \vval (p_j(d') - c_j(b)) = \vval(\tbr(p_j(d)) - c_j(b)) \\
+  \end{align*}
+  Similar to case 3.
 \end{proof}
 
 
+
 \begin{Note}
   This gives us an upper bound on the number of types - there are at most $|2I|!$ many choices for the order of $\tval$,
   $O(N)$ many choices for the sub-interval for each $p_i$,
@@ -793,10 +803,12 @@ For any index $i \in I$ we call it \defn{independent} if $i \in J$ and we call i
   \end{proof}	
   Additionally $a_i, a_i'$ have the same image in $\Ct$ component, so we have
   \begin{align*}
-    \val(a_i - a_i') \geq F(a_j) 
+    \val(a_i - a_i') > F(a_j) 
   \end{align*}
   As $F(a_i) \leq F(a_j)$, $a_i, a_i'$ have to lie in the same sub-interval.	
   TO DO: PROVE THE PREVIOUS SENTENCE FORMALLY
+  Suppose that $a_i$ lies in the sub-interval $\paren{B(t_L, \alpha_L),  B(t_U, \alpha_U)}$
+  and that $a_i'$ lies in the sub-interval $\paren{B(t_L', \alpha_L'),  B(t_U', \alpha_U')}$.
 \end{proof}
 
 \begin{Corollary}
diff --git a/resume/cv.pdf b/resume/cv.pdf
index a9705f00..06b60973 100644
Binary files a/resume/cv.pdf and b/resume/cv.pdf differ
diff --git a/resume/cv.tex b/resume/cv.tex
index 040139e7..04a950e7 100644
--- a/resume/cv.tex
+++ b/resume/cv.tex
@@ -74,7 +74,9 @@ \section{\textsc{Education}}
 {\sl PhD,} Mathematics (in progress)
 
 \begin{itemize}
-	\item GPA: 3.91
+        \item 2016 Excellence in Teaching Award
+        \item 2016 Girsky Fellowship
+        \item GPA: 3.91
 	\item Advanced to candidacy on June 5, 2015
 \end{itemize}
 
@@ -96,44 +98,44 @@ \section{\textsc{Education}}
 \end{itemize}
 
 
-\section{\textsc{Awards and Scholarships}}
+% \section{\textsc{Awards and Scholarships}}
 
-\textbf{2011-2012:} \\
+% \textbf{2011-2012:} \\
 
-Fees: \$13,247.13 paid from unrestricted Graduate Division allocation \\
-Stipend: \$21,000 paid from departmental funds (RTG Logic) \\
-Stipend: \$4,000 paid from unrestricted Graduate Division allocation \\
-Summer'11 Stipend: \$ 3000 from departmental funds (RTG Logic) \\
-Summer'11 Stipend:\$3,000 from College funds \\
+% Fees: \$13,247.13 paid from unrestricted Graduate Division allocation \\
+% Stipend: \$21,000 paid from departmental funds (RTG Logic) \\
+% Stipend: \$4,000 paid from unrestricted Graduate Division allocation \\
+% Summer'11 Stipend: \$ 3000 from departmental funds (RTG Logic) \\
+% Summer'11 Stipend:\$3,000 from College funds \\
 
-\textbf{2012-2013:} \\
+% \textbf{2012-2013:} \\
 
-Fees: \$14,372.68 paid through TA appointments \\
-Fee remission balance: \$374.25 paid from unrestricted Graduate Division allocation  \\
-Income: \$17,655.18 TA salary \\
-Stipend: \$3,344.82 paid from unrestricted Graduate Division allocation \\				
+% Fees: \$14,372.68 paid through TA appointments \\
+% Fee remission balance: \$374.25 paid from unrestricted Graduate Division allocation  \\
+% Income: \$17,655.18 TA salary \\
+% Stipend: \$3,344.82 paid from unrestricted Graduate Division allocation \\				
   
-\textbf{2013-2014:} \\
+% \textbf{2013-2014:} \\
 
-Fees: \$2,070.75 partial fees paid from unrestricted Graduate Division allocation \\
-Fees: \$10,500 remainder of fees paid from departmental funds (RTG Logic) \\
-Stipend:  \$ 21,000 paid from departmental funds (RTG Logic) \\
-Stipend: \$4,000 paid from unrestricted Graduate Division allocation \\
-Summer'13: \$4,000 paid from departmental funds (RTG Algebra) \\
+% Fees: \$2,070.75 partial fees paid from unrestricted Graduate Division allocation \\
+% Fees: \$10,500 remainder of fees paid from departmental funds (RTG Logic) \\
+% Stipend:  \$ 21,000 paid from departmental funds (RTG Logic) \\
+% Stipend: \$4,000 paid from unrestricted Graduate Division allocation \\
+% Summer'13: \$4,000 paid from departmental funds (RTG Algebra) \\
 
-\textbf{2014-2015:} \\
+% \textbf{2014-2015:} \\
 
-Fees: \$15,203.10 paid through TA \& GSR appointment \\
-Fee remission balance: \$378.99 paid from unrestricted Graduate Division allocation  \\
-Income: \$20,621.16 TA \& GSR salary \\
-Stipend: \$378.84 paid from unrestricted Graduate Division allocation \\
+% Fees: \$15,203.10 paid through TA \& GSR appointment \\
+% Fee remission balance: \$378.99 paid from unrestricted Graduate Division allocation  \\
+% Income: \$20,621.16 TA \& GSR salary \\
+% Stipend: \$378.84 paid from unrestricted Graduate Division allocation \\
   
   
-\textbf{2015-2016:} \\
+% \textbf{2015-2016:} \\
 
-Fees: \$15,440.48 paid through TA appointments \\
-TA Income: \$11,121.43 (Spring TA salary yet to be paid during Spring'16) \\
-Summer'15: \$4,000 paid from unrestricted Graduate Division allocation \\
+% Fees: \$15,440.48 paid through TA appointments \\
+% TA Income: \$11,121.43 (Spring TA salary yet to be paid during Spring'16) \\
+% Summer'15: \$4,000 paid from unrestricted Graduate Division allocation \\
 
 
 
@@ -141,13 +143,13 @@ \section{\textsc{Awards and Scholarships}}
 %	Undergraduate Research
 %----------------------------------------------------------------------------------------
  
-\section{\textsc{Undergraduate Research}}
+% \section{\textsc{Undergraduate Research}}
 
-\textbf{Cryptography REU at Northern Kentucky University} \hfill \textbf{Summer 2009}\\
-Implemented a variant of MXL algorithm in computational algebra system MAGMA \\
+% \textbf{Cryptography REU at Northern Kentucky University} \hfill \textbf{Summer 2009}\\
+% Implemented a variant of MXL algorithm in computational algebra system MAGMA \\
 
-\textbf{Research assistant for Vladimir Vassiliev} \hfill \textbf{2008 - 2011}\\
-Numerical simulations for AGIS gamma-ray telescope. This included forward and inverse kinematics for Stewart platform, ray casting, and high precision calibration.
+% \textbf{Research assistant for Vladimir Vassiliev} \hfill \textbf{2008 - 2011}\\
+% Numerical simulations for AGIS gamma-ray telescope. This included forward and inverse kinematics for Stewart platform, ray casting, and high precision calibration.
 % I have also worked on network interfacing with Gumstix boards using CORBA as well as installing and configuring a custom linux kernel. 
 
 %----------------------------------------------------------------------------------------
@@ -158,15 +160,32 @@ \section{\textsc{Teaching}}
 
 % Intermediate C++ Programming, Linear Algebra, Calculus
 
-%\begin{itemize}
-	Math 31B: Integration and Infinite Series \hfill \textbf{2012 - 2013} \\
-	Math 33A: Linear Algebra and Applications \hfill \textbf{2012 - 2013} \\
-	PIC 10B: Intermediate Programming \hfill \textbf{Winter 2015,  Spring 2015} \\
-	PIC 20A: Principles of Java Language with Applications \hfill \textbf{Spring 2015} \\
-	PIC 40A: Introduction to Programming for Internet \hfill \textbf{Fall 2015} \\
-	Math 115B: Linear Algebra \hfill \textbf{Winter 2016} \\
-	Independent Programming Projects \hfill \textbf{Winter 2016}
-%\end{itemize}
+\begin{itemize}
+\item Teacher Assistant
+  \begin{itemize}
+  \item Math 31B: Integration and Infinite Series \hfill \textbf{2012 - 2013} \\
+  \item Math 33A: Linear Algebra and Applications \hfill \textbf{2012 - 2013} \\
+  \item PIC 10B: Intermediate Programming \hfill \textbf{Winter 2015,  Spring 2015} \\
+  \item PIC 20A: Principles of Java Language with Applications \hfill \textbf{Spring 2015} \\
+  \item PIC 40A: Introduction to Programming for Internet \hfill \textbf{Fall 2015} \\
+  \item Math 115B: Linear Algebra \hfill \textbf{Winter 2016} \\
+  \item Math 174E: Mathematics of Finance for Mathematics/Economics Students \hfill \textbf{Spring 2016, Fall 2016} \\
+  \item Independent Programming Projects \hfill \textbf{Winter 2016}
+  \end{itemize}
+\item Instructor
+  \begin{itemize}
+  \item Math 31B: Integration and Infinite Series \hfill \textbf{Summer 2016} \\
+  \item Math 32BH: Calculus of Several Variables (Honors) \hfill \textbf{Winter 2017} \\
+  \end{itemize}
+\item Undergraduate Projects Mentor
+  \begin{itemize}
+  \item Conway's game of life variations with C++/SDL graphics library \hfill \textbf{Winter 2016} \\
+  \item App development platform with Typescript 2 \hfill \textbf{Winter 2016} \\
+  \item Internet browsing data/trends visualization with Python \hfill \textbf{Winter 2016, Spring 2016} \\
+  \item Optical character recognition via neural nets with Python \hfill \textbf{Winter 2016, Spring 2016} \\
+  \item Discrete signal processing with Matlab \hfill \textbf{Fall 2017} \\
+  \end{itemize}
+\end{itemize}
 
 
 %----------------------------------------------------------------------------------------
diff --git a/resume/cv.tex~ b/resume/cv.tex~
new file mode 100644
index 00000000..040139e7
--- /dev/null
+++ b/resume/cv.tex~
@@ -0,0 +1,232 @@
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% Medium Length Graduate Curriculum Vitae
+% LaTeX Template
+% Version 1.1 (9/12/12)
+%
+% This template has been downloaded from:
+% http://www.LaTeXTemplates.com
+%
+% Original author:
+% Rensselaer Polytechnic Institute (http://www.rpi.edu/dept/arc/training/latex/resumes/)
+%
+% Important note:
+% This template requires the res.cls file to be in the same directory as the
+% .tex file. The res.cls file provides the resume style used for structuring the
+% document.
+%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%----------------------------------------------------------------------------------------
+%	PACKAGES AND OTHER DOCUMENT CONFIGURATIONS
+%----------------------------------------------------------------------------------------
+
+\documentclass[margin, 10pt]{res} % Use the res.cls style, the font size can be changed to 11pt or 12pt here
+\usepackage[T1]{fontenc}
+\usepackage{hyperref}
+\usepackage{verbatim}
+\usepackage{enumerate}
+\usepackage{enumitem}
+\usepackage{amssymb}
+\usepackage{helvet} % Default font is the helvetica postscript font
+%\usepackage{newcent} % To change the default font to the new century schoolbook postscript font uncomment this line and comment the one above
+
+%\usepackage{graphicx}
+
+%\setlength{\textwidth}{5.1in} % Text width of the document
+\setlength{\textwidth}{5.5in} % Text width of the document
+
+\hypersetup{
+     colorlinks   = true,
+     urlcolor  = blue
+}
+
+\begin{document}
+
+%----------------------------------------------------------------------------------------
+%	NAME AND ADDRESS SECTION
+%----------------------------------------------------------------------------------------
+
+\moveleft.5\hoffset\centerline{\large\bf Anton Bobkov}
+%\\ % Your name at the top{Control+Shift+F7}{Control+Shift+F5}
+\moveleft\hoffset\vbox{\hrule width\resumewidth height 1pt}\smallskip % Horizontal line after name; adjust line thickness by changing the '1pt'
+\section{\textsc{Contact Information}}
+\begin{tabular}{l|l}
+Graduate Student & {\it E-mail:}\\
+Department of Mathematics & \href{mailto:antongml@gmail.com}{antongml@gmail.com}\\
+University of California, Los Angeles & \href{mailto:bobkov@math.ucla.edu}{bobkov@math.ucla.edu}\\
+Los Angeles, CA 90095-1555 USA & {\it Website:}\\
+\phantom{lots of text that takes up a bit of space blah blah blah} & \url{www.math.ucla.edu/~bobkov/}\\
+\phantom{lots of text that takes up a bit of space blah blah blah} & {\it Phone:} (408)813-6331
+\end{tabular}
+
+
+%----------------------------------------------------------------------------------------
+
+\begin{resume}
+
+%----------------------------------------------------------------------------------------
+%	EDUCATION SECTION
+%----------------------------------------------------------------------------------------
+
+\section{\textsc{Education}}
+
+\textbf{University of California, Los Angeles} {\sl (graduate)}\hfill \textbf{Fall 2011 to present}
+{\sl PhD,} Mathematics (in progress)
+
+\begin{itemize}
+	\item GPA: 3.91
+	\item Advanced to candidacy on June 5, 2015
+\end{itemize}
+
+{\sl Advisor}: Matthias Aschenbrenner \\
+{\sl Research interests}: Mathematical logic, model theory, NIP theories, VC-density
+
+\textbf{University of California, Los Angeles} {\sl (undergraduate)}\hfill \textbf{Graduated Spring 2011}
+
+\begin{itemize}
+	\item {\sl B.S.} in Mathematics, {\sl B.A.} in Physics
+	\item Sherwood Prize
+	\item Departmental Highest Honors in Mathematics, College Honors
+	\item GPA: 3.82 (Magna Cum Laude)
+	\item William Lowell Putnam Mathematics Competition
+		\begin{itemize}
+			\item 2008 - score 30
+			\item 2009 - score 19
+		\end{itemize}
+\end{itemize}
+
+
+\section{\textsc{Awards and Scholarships}}
+
+\textbf{2011-2012:} \\
+
+Fees: \$13,247.13 paid from unrestricted Graduate Division allocation \\
+Stipend: \$21,000 paid from departmental funds (RTG Logic) \\
+Stipend: \$4,000 paid from unrestricted Graduate Division allocation \\
+Summer'11 Stipend: \$ 3000 from departmental funds (RTG Logic) \\
+Summer'11 Stipend:\$3,000 from College funds \\
+
+\textbf{2012-2013:} \\
+
+Fees: \$14,372.68 paid through TA appointments \\
+Fee remission balance: \$374.25 paid from unrestricted Graduate Division allocation  \\
+Income: \$17,655.18 TA salary \\
+Stipend: \$3,344.82 paid from unrestricted Graduate Division allocation \\				
+  
+\textbf{2013-2014:} \\
+
+Fees: \$2,070.75 partial fees paid from unrestricted Graduate Division allocation \\
+Fees: \$10,500 remainder of fees paid from departmental funds (RTG Logic) \\
+Stipend:  \$ 21,000 paid from departmental funds (RTG Logic) \\
+Stipend: \$4,000 paid from unrestricted Graduate Division allocation \\
+Summer'13: \$4,000 paid from departmental funds (RTG Algebra) \\
+
+\textbf{2014-2015:} \\
+
+Fees: \$15,203.10 paid through TA \& GSR appointment \\
+Fee remission balance: \$378.99 paid from unrestricted Graduate Division allocation  \\
+Income: \$20,621.16 TA \& GSR salary \\
+Stipend: \$378.84 paid from unrestricted Graduate Division allocation \\
+  
+  
+\textbf{2015-2016:} \\
+
+Fees: \$15,440.48 paid through TA appointments \\
+TA Income: \$11,121.43 (Spring TA salary yet to be paid during Spring'16) \\
+Summer'15: \$4,000 paid from unrestricted Graduate Division allocation \\
+
+
+
+%----------------------------------------------------------------------------------------
+%	Undergraduate Research
+%----------------------------------------------------------------------------------------
+ 
+\section{\textsc{Undergraduate Research}}
+
+\textbf{Cryptography REU at Northern Kentucky University} \hfill \textbf{Summer 2009}\\
+Implemented a variant of MXL algorithm in computational algebra system MAGMA \\
+
+\textbf{Research assistant for Vladimir Vassiliev} \hfill \textbf{2008 - 2011}\\
+Numerical simulations for AGIS gamma-ray telescope. This included forward and inverse kinematics for Stewart platform, ray casting, and high precision calibration.
+% I have also worked on network interfacing with Gumstix boards using CORBA as well as installing and configuring a custom linux kernel. 
+
+%----------------------------------------------------------------------------------------
+%	TEACHING
+%----------------------------------------------------------------------------------------
+ 
+\section{\textsc{Teaching}}
+
+% Intermediate C++ Programming, Linear Algebra, Calculus
+
+%\begin{itemize}
+	Math 31B: Integration and Infinite Series \hfill \textbf{2012 - 2013} \\
+	Math 33A: Linear Algebra and Applications \hfill \textbf{2012 - 2013} \\
+	PIC 10B: Intermediate Programming \hfill \textbf{Winter 2015,  Spring 2015} \\
+	PIC 20A: Principles of Java Language with Applications \hfill \textbf{Spring 2015} \\
+	PIC 40A: Introduction to Programming for Internet \hfill \textbf{Fall 2015} \\
+	Math 115B: Linear Algebra \hfill \textbf{Winter 2016} \\
+	Independent Programming Projects \hfill \textbf{Winter 2016}
+%\end{itemize}
+
+
+%----------------------------------------------------------------------------------------
+%	PAPERS
+%----------------------------------------------------------------------------------------
+ 
+
+
+
+\end{resume}
+
+\end{document}
+
+\section{\textsc{Papers}}
+
+Bobkov, A. {\it VC-density for trees}, in preparation  \\
+Bobkov, A. {\it Some VC-density computations for Shelah-Spencer graphs}, in preparation \\
+Bobkov, A. {\it VC-density in $\mathbb{Q}_p$-reducts}, in preparation
+
+%----------------------------------------------------------------------------------------
+%	Software
+%----------------------------------------------------------------------------------------
+ 
+\section{\textsc{Software Experience}}
+
+\textbf{Unix-like systems}\\
+I am comfortable working in command line environment, including tasks such as
+\begin{itemize}
+	\item installing and managing web-server, repository server, ssh server
+	\item code building, editing, and version control
+\end{itemize}
+
+\begin{tabular}{ll}
+\textbf{Languages} & C++, C\#, bash, Java, PHP, MAGMA \\
+\textbf{Code management} & CMake, Makefile, git, subversion, Visual Studio, Unity3D \\
+\textbf{Standards} & TCP/IP, .NET, CORBA  \\
+\end{tabular}
+
+Python, HTML, CSS, JavaScript, PHP, SQL
+
+%----------------------------------------------------------------------------------------
+%	Projects
+%----------------------------------------------------------------------------------------
+ 
+\section{\textsc{Independent Projects}}
+For more information and links visit \url{www.math.ucla.edu/~bobkov/projects.html}
+
+\textbf{Burn and Turn} \hfill \textbf{2008 - 2011}\\
+Cross-platform arcade style video game featured on \href{http://kotaku.com/5862197/burn-and-turn-combines-retro-arcade-stylings-with-tower-defense-for-combustible-fun}{Kotaku} and \href{http://indiegames.com/2011/10/trailer_burn_turn_robot_bear.html}{IndieGames}. It was coded in C++ and used OpenGL as a backend for graphics. It was created by a team of three people over a course of four years and released on iOS and Android markets.
+
+\textbf{Self Balancing Robot} \hfill \textbf{Summer 2012}\\
+A vertical self-balancing robot ran by an arduino controller coded in C++. A numerical simulation was used to determine weight distribution. Robot's position is determined by data from an accelerometer and a gyroscope combined through a Kalman filter. Balancing is done with a DC motor using PID controller.
+
+\textbf{UCLA Graduate Student Wiki} \hfill \textbf{Summer 2014}\\
+Official wiki for graduate math department at UCLA that maintains a database of qualifying exam problems. It is made on top of Semantic Media Wiki using custom extension written in PHP that allows to users to search, filter, and tag the solutions.
+
+\textbf{Decentralized Online Game} \hfill \textbf{Fall 2014 - Present}\\
+Exploration multiplayer online game that manages players and game data using peer-to-peer connections instead of relying on a central server. It is coded with C\# in Unity3D using standard TCP/IP network.
+
+\section{\textsc{Sample Code}}
+\url{https://gitorious.org/~antonbobkov}
+
+
diff --git a/resume/publication_list.pdf b/resume/publication_list.pdf
new file mode 100644
index 00000000..5194243f
Binary files /dev/null and b/resume/publication_list.pdf differ
diff --git a/resume/publication_list.tex b/resume/publication_list.tex
new file mode 100644
index 00000000..c77810dc
--- /dev/null
+++ b/resume/publication_list.tex
@@ -0,0 +1,258 @@
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% Medium Length Graduate Curriculum Vitae
+% LaTeX Template
+% Version 1.1 (9/12/12)
+%
+% This template has been downloaded from:
+% http://www.LaTeXTemplates.com
+%
+% Original author:
+% Rensselaer Polytechnic Institute (http://www.rpi.edu/dept/arc/training/latex/resumes/)
+%
+% Important note:
+% This template requires the res.cls file to be in the same directory as the
+% .tex file. The res.cls file provides the resume style used for structuring the
+% document.
+%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%----------------------------------------------------------------------------------------
+%	PACKAGES AND OTHER DOCUMENT CONFIGURATIONS
+%----------------------------------------------------------------------------------------
+
+\documentclass[margin, 10pt]{res} % Use the res.cls style, the font size can be changed to 11pt or 12pt here
+\usepackage[T1]{fontenc}
+\usepackage{hyperref}
+\usepackage{verbatim}
+\usepackage{enumerate}
+\usepackage{enumitem}
+\usepackage{amssymb}
+\usepackage{helvet} % Default font is the helvetica postscript font
+%\usepackage{newcent} % To change the default font to the new century schoolbook postscript font uncomment this line and comment the one above
+
+%\usepackage{graphicx}
+
+%\setlength{\textwidth}{5.1in} % Text width of the document
+\setlength{\textwidth}{5.5in} % Text width of the document
+
+\hypersetup{
+     colorlinks   = true,
+     urlcolor  = blue
+}
+
+\begin{document}
+
+\section{\textsc{Publication List}}
+
+Bobkov, A. {\it VC-density for trees}, in preparation  \\
+Bobkov, A. {\it Some VC-density computations for Shelah-Spencer graphs}, in preparation \\
+Bobkov, A. {\it VC-density in $\mathbb{Q}_p$-reducts}, in preparation
+
+\end{document}
+%----------------------------------------------------------------------------------------
+%	NAME AND ADDRESS SECTION
+%----------------------------------------------------------------------------------------
+
+\moveleft.5\hoffset\centerline{\large\bf Anton Bobkov}
+%\\ % Your name at the top{Control+Shift+F7}{Control+Shift+F5}
+\moveleft\hoffset\vbox{\hrule width\resumewidth height 1pt}\smallskip % Horizontal line after name; adjust line thickness by changing the '1pt'
+\section{\textsc{Contact Information}}
+\begin{tabular}{l|l}
+Graduate Student & {\it E-mail:}\\
+Department of Mathematics & \href{mailto:antongml@gmail.com}{antongml@gmail.com}\\
+University of California, Los Angeles & \href{mailto:bobkov@math.ucla.edu}{bobkov@math.ucla.edu}\\
+Los Angeles, CA 90095-1555 USA & {\it Website:}\\
+\phantom{lots of text that takes up a bit of space blah blah blah} & \url{www.math.ucla.edu/~bobkov/}\\
+\phantom{lots of text that takes up a bit of space blah blah blah} & {\it Phone:} (408)813-6331
+\end{tabular}
+
+
+%----------------------------------------------------------------------------------------
+
+\begin{resume}
+
+%----------------------------------------------------------------------------------------
+%	EDUCATION SECTION
+%----------------------------------------------------------------------------------------
+
+\section{\textsc{Education}}
+
+\textbf{University of California, Los Angeles} {\sl (graduate)}\hfill \textbf{Fall 2011 to present}
+{\sl PhD,} Mathematics (in progress)
+
+\begin{itemize}
+        \item 2016 Excellence in Teaching Award
+        \item 2016 Girsky Fellowship
+        \item GPA: 3.91
+	\item Advanced to candidacy on June 5, 2015
+\end{itemize}
+
+{\sl Advisor}: Matthias Aschenbrenner \\
+{\sl Research interests}: Mathematical logic, model theory, NIP theories, VC-density
+
+\textbf{University of California, Los Angeles} {\sl (undergraduate)}\hfill \textbf{Graduated Spring 2011}
+
+\begin{itemize}
+	\item {\sl B.S.} in Mathematics, {\sl B.A.} in Physics
+	\item Sherwood Prize
+	\item Departmental Highest Honors in Mathematics, College Honors
+	\item GPA: 3.82 (Magna Cum Laude)
+	\item William Lowell Putnam Mathematics Competition
+		\begin{itemize}
+			\item 2008 - score 30
+			\item 2009 - score 19
+		\end{itemize}
+\end{itemize}
+
+
+% \section{\textsc{Awards and Scholarships}}
+
+% \textbf{2011-2012:} \\
+
+% Fees: \$13,247.13 paid from unrestricted Graduate Division allocation \\
+% Stipend: \$21,000 paid from departmental funds (RTG Logic) \\
+% Stipend: \$4,000 paid from unrestricted Graduate Division allocation \\
+% Summer'11 Stipend: \$ 3000 from departmental funds (RTG Logic) \\
+% Summer'11 Stipend:\$3,000 from College funds \\
+
+% \textbf{2012-2013:} \\
+
+% Fees: \$14,372.68 paid through TA appointments \\
+% Fee remission balance: \$374.25 paid from unrestricted Graduate Division allocation  \\
+% Income: \$17,655.18 TA salary \\
+% Stipend: \$3,344.82 paid from unrestricted Graduate Division allocation \\				
+  
+% \textbf{2013-2014:} \\
+
+% Fees: \$2,070.75 partial fees paid from unrestricted Graduate Division allocation \\
+% Fees: \$10,500 remainder of fees paid from departmental funds (RTG Logic) \\
+% Stipend:  \$ 21,000 paid from departmental funds (RTG Logic) \\
+% Stipend: \$4,000 paid from unrestricted Graduate Division allocation \\
+% Summer'13: \$4,000 paid from departmental funds (RTG Algebra) \\
+
+% \textbf{2014-2015:} \\
+
+% Fees: \$15,203.10 paid through TA \& GSR appointment \\
+% Fee remission balance: \$378.99 paid from unrestricted Graduate Division allocation  \\
+% Income: \$20,621.16 TA \& GSR salary \\
+% Stipend: \$378.84 paid from unrestricted Graduate Division allocation \\
+  
+  
+% \textbf{2015-2016:} \\
+
+% Fees: \$15,440.48 paid through TA appointments \\
+% TA Income: \$11,121.43 (Spring TA salary yet to be paid during Spring'16) \\
+% Summer'15: \$4,000 paid from unrestricted Graduate Division allocation \\
+
+
+
+%----------------------------------------------------------------------------------------
+%	Undergraduate Research
+%----------------------------------------------------------------------------------------
+ 
+% \section{\textsc{Undergraduate Research}}
+
+% \textbf{Cryptography REU at Northern Kentucky University} \hfill \textbf{Summer 2009}\\
+% Implemented a variant of MXL algorithm in computational algebra system MAGMA \\
+
+% \textbf{Research assistant for Vladimir Vassiliev} \hfill \textbf{2008 - 2011}\\
+% Numerical simulations for AGIS gamma-ray telescope. This included forward and inverse kinematics for Stewart platform, ray casting, and high precision calibration.
+% I have also worked on network interfacing with Gumstix boards using CORBA as well as installing and configuring a custom linux kernel. 
+
+%----------------------------------------------------------------------------------------
+%	TEACHING
+%----------------------------------------------------------------------------------------
+ 
+\section{\textsc{Teaching}}
+
+% Intermediate C++ Programming, Linear Algebra, Calculus
+
+\begin{itemize}
+\item Teacher Assistant
+  \begin{itemize}
+  \item Math 31B: Integration and Infinite Series \hfill \textbf{2012 - 2013} \\
+  \item Math 33A: Linear Algebra and Applications \hfill \textbf{2012 - 2013} \\
+  \item PIC 10B: Intermediate Programming \hfill \textbf{Winter 2015,  Spring 2015} \\
+  \item PIC 20A: Principles of Java Language with Applications \hfill \textbf{Spring 2015} \\
+  \item PIC 40A: Introduction to Programming for Internet \hfill \textbf{Fall 2015} \\
+  \item Math 115B: Linear Algebra \hfill \textbf{Winter 2016} \\
+  \item Math 174E: Mathematics of Finance for Mathematics/Economics Students \hfill \textbf{Spring 2016, Fall 2016} \\
+  \item Independent Programming Projects \hfill \textbf{Winter 2016}
+  \end{itemize}
+\item Instructor
+  \begin{itemize}
+  \item Math 31B: Integration and Infinite Series \hfill \textbf{Summer 2016} \\
+  \item Math 32BH: Calculus of Several Variables (Honors) \hfill \textbf{Winter 2017} \\
+  \end{itemize}
+\item Undergraduate Projects Mentor
+  \begin{itemize}
+  \item Conway's game of life variations with C++/SDL graphics library \hfill \textbf{Winter 2016} \\
+  \item App development platform with Typescript 2 \hfill \textbf{Winter 2016} \\
+  \item Internet browsing data/trends visualization with Python \hfill \textbf{Winter 2016, Spring 2016} \\
+  \item Optical character recognition via neural nets with Python \hfill \textbf{Winter 2016, Spring 2016} \\
+  \item Discrete signal processing with Matlab \hfill \textbf{Fall 2017} \\
+  \end{itemize}
+\end{itemize}
+
+
+%----------------------------------------------------------------------------------------
+%	PAPERS
+%----------------------------------------------------------------------------------------
+ 
+
+
+
+\end{resume}
+
+\end{document}
+
+\section{\textsc{Papers}}
+
+Bobkov, A. {\it VC-density for trees}, in preparation  \\
+Bobkov, A. {\it Some VC-density computations for Shelah-Spencer graphs}, in preparation \\
+Bobkov, A. {\it VC-density in $\mathbb{Q}_p$-reducts}, in preparation
+
+%----------------------------------------------------------------------------------------
+%	Software
+%----------------------------------------------------------------------------------------
+ 
+\section{\textsc{Software Experience}}
+
+\textbf{Unix-like systems}\\
+I am comfortable working in command line environment, including tasks such as
+\begin{itemize}
+	\item installing and managing web-server, repository server, ssh server
+	\item code building, editing, and version control
+\end{itemize}
+
+\begin{tabular}{ll}
+\textbf{Languages} & C++, C\#, bash, Java, PHP, MAGMA \\
+\textbf{Code management} & CMake, Makefile, git, subversion, Visual Studio, Unity3D \\
+\textbf{Standards} & TCP/IP, .NET, CORBA  \\
+\end{tabular}
+
+Python, HTML, CSS, JavaScript, PHP, SQL
+
+%----------------------------------------------------------------------------------------
+%	Projects
+%----------------------------------------------------------------------------------------
+ 
+\section{\textsc{Independent Projects}}
+For more information and links visit \url{www.math.ucla.edu/~bobkov/projects.html}
+
+\textbf{Burn and Turn} \hfill \textbf{2008 - 2011}\\
+Cross-platform arcade style video game featured on \href{http://kotaku.com/5862197/burn-and-turn-combines-retro-arcade-stylings-with-tower-defense-for-combustible-fun}{Kotaku} and \href{http://indiegames.com/2011/10/trailer_burn_turn_robot_bear.html}{IndieGames}. It was coded in C++ and used OpenGL as a backend for graphics. It was created by a team of three people over a course of four years and released on iOS and Android markets.
+
+\textbf{Self Balancing Robot} \hfill \textbf{Summer 2012}\\
+A vertical self-balancing robot ran by an arduino controller coded in C++. A numerical simulation was used to determine weight distribution. Robot's position is determined by data from an accelerometer and a gyroscope combined through a Kalman filter. Balancing is done with a DC motor using PID controller.
+
+\textbf{UCLA Graduate Student Wiki} \hfill \textbf{Summer 2014}\\
+Official wiki for graduate math department at UCLA that maintains a database of qualifying exam problems. It is made on top of Semantic Media Wiki using custom extension written in PHP that allows to users to search, filter, and tag the solutions.
+
+\textbf{Decentralized Online Game} \hfill \textbf{Fall 2014 - Present}\\
+Exploration multiplayer online game that manages players and game data using peer-to-peer connections instead of relying on a central server. It is coded with C\# in Unity3D using standard TCP/IP network.
+
+\section{\textsc{Sample Code}}
+\url{https://gitorious.org/~antonbobkov}
+
+
diff --git a/resume/publication_list.tex~ b/resume/publication_list.tex~
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+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% Medium Length Graduate Curriculum Vitae
+% LaTeX Template
+% Version 1.1 (9/12/12)
+%
+% This template has been downloaded from:
+% http://www.LaTeXTemplates.com
+%
+% Original author:
+% Rensselaer Polytechnic Institute (http://www.rpi.edu/dept/arc/training/latex/resumes/)
+%
+% Important note:
+% This template requires the res.cls file to be in the same directory as the
+% .tex file. The res.cls file provides the resume style used for structuring the
+% document.
+%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%----------------------------------------------------------------------------------------
+%	PACKAGES AND OTHER DOCUMENT CONFIGURATIONS
+%----------------------------------------------------------------------------------------
+
+\documentclass[margin, 10pt]{res} % Use the res.cls style, the font size can be changed to 11pt or 12pt here
+\usepackage[T1]{fontenc}
+\usepackage{hyperref}
+\usepackage{verbatim}
+\usepackage{enumerate}
+\usepackage{enumitem}
+\usepackage{amssymb}
+\usepackage{helvet} % Default font is the helvetica postscript font
+%\usepackage{newcent} % To change the default font to the new century schoolbook postscript font uncomment this line and comment the one above
+
+%\usepackage{graphicx}
+
+%\setlength{\textwidth}{5.1in} % Text width of the document
+\setlength{\textwidth}{5.5in} % Text width of the document
+
+\hypersetup{
+     colorlinks   = true,
+     urlcolor  = blue
+}
+
+\begin{document}
+
+%----------------------------------------------------------------------------------------
+%	NAME AND ADDRESS SECTION
+%----------------------------------------------------------------------------------------
+
+\moveleft.5\hoffset\centerline{\large\bf Anton Bobkov}
+%\\ % Your name at the top{Control+Shift+F7}{Control+Shift+F5}
+\moveleft\hoffset\vbox{\hrule width\resumewidth height 1pt}\smallskip % Horizontal line after name; adjust line thickness by changing the '1pt'
+\section{\textsc{Contact Information}}
+\begin{tabular}{l|l}
+Graduate Student & {\it E-mail:}\\
+Department of Mathematics & \href{mailto:antongml@gmail.com}{antongml@gmail.com}\\
+University of California, Los Angeles & \href{mailto:bobkov@math.ucla.edu}{bobkov@math.ucla.edu}\\
+Los Angeles, CA 90095-1555 USA & {\it Website:}\\
+\phantom{lots of text that takes up a bit of space blah blah blah} & \url{www.math.ucla.edu/~bobkov/}\\
+\phantom{lots of text that takes up a bit of space blah blah blah} & {\it Phone:} (408)813-6331
+\end{tabular}
+
+
+%----------------------------------------------------------------------------------------
+
+\begin{resume}
+
+%----------------------------------------------------------------------------------------
+%	EDUCATION SECTION
+%----------------------------------------------------------------------------------------
+
+\section{\textsc{Education}}
+
+\textbf{University of California, Los Angeles} {\sl (graduate)}\hfill \textbf{Fall 2011 to present}
+{\sl PhD,} Mathematics (in progress)
+
+\begin{itemize}
+        \item 2016 Excellence in Teaching Award
+        \item 2016 Girsky Fellowship
+        \item GPA: 3.91
+	\item Advanced to candidacy on June 5, 2015
+\end{itemize}
+
+{\sl Advisor}: Matthias Aschenbrenner \\
+{\sl Research interests}: Mathematical logic, model theory, NIP theories, VC-density
+
+\textbf{University of California, Los Angeles} {\sl (undergraduate)}\hfill \textbf{Graduated Spring 2011}
+
+\begin{itemize}
+	\item {\sl B.S.} in Mathematics, {\sl B.A.} in Physics
+	\item Sherwood Prize
+	\item Departmental Highest Honors in Mathematics, College Honors
+	\item GPA: 3.82 (Magna Cum Laude)
+	\item William Lowell Putnam Mathematics Competition
+		\begin{itemize}
+			\item 2008 - score 30
+			\item 2009 - score 19
+		\end{itemize}
+\end{itemize}
+
+
+% \section{\textsc{Awards and Scholarships}}
+
+% \textbf{2011-2012:} \\
+
+% Fees: \$13,247.13 paid from unrestricted Graduate Division allocation \\
+% Stipend: \$21,000 paid from departmental funds (RTG Logic) \\
+% Stipend: \$4,000 paid from unrestricted Graduate Division allocation \\
+% Summer'11 Stipend: \$ 3000 from departmental funds (RTG Logic) \\
+% Summer'11 Stipend:\$3,000 from College funds \\
+
+% \textbf{2012-2013:} \\
+
+% Fees: \$14,372.68 paid through TA appointments \\
+% Fee remission balance: \$374.25 paid from unrestricted Graduate Division allocation  \\
+% Income: \$17,655.18 TA salary \\
+% Stipend: \$3,344.82 paid from unrestricted Graduate Division allocation \\				
+  
+% \textbf{2013-2014:} \\
+
+% Fees: \$2,070.75 partial fees paid from unrestricted Graduate Division allocation \\
+% Fees: \$10,500 remainder of fees paid from departmental funds (RTG Logic) \\
+% Stipend:  \$ 21,000 paid from departmental funds (RTG Logic) \\
+% Stipend: \$4,000 paid from unrestricted Graduate Division allocation \\
+% Summer'13: \$4,000 paid from departmental funds (RTG Algebra) \\
+
+% \textbf{2014-2015:} \\
+
+% Fees: \$15,203.10 paid through TA \& GSR appointment \\
+% Fee remission balance: \$378.99 paid from unrestricted Graduate Division allocation  \\
+% Income: \$20,621.16 TA \& GSR salary \\
+% Stipend: \$378.84 paid from unrestricted Graduate Division allocation \\
+  
+  
+% \textbf{2015-2016:} \\
+
+% Fees: \$15,440.48 paid through TA appointments \\
+% TA Income: \$11,121.43 (Spring TA salary yet to be paid during Spring'16) \\
+% Summer'15: \$4,000 paid from unrestricted Graduate Division allocation \\
+
+
+
+%----------------------------------------------------------------------------------------
+%	Undergraduate Research
+%----------------------------------------------------------------------------------------
+ 
+% \section{\textsc{Undergraduate Research}}
+
+% \textbf{Cryptography REU at Northern Kentucky University} \hfill \textbf{Summer 2009}\\
+% Implemented a variant of MXL algorithm in computational algebra system MAGMA \\
+
+% \textbf{Research assistant for Vladimir Vassiliev} \hfill \textbf{2008 - 2011}\\
+% Numerical simulations for AGIS gamma-ray telescope. This included forward and inverse kinematics for Stewart platform, ray casting, and high precision calibration.
+% I have also worked on network interfacing with Gumstix boards using CORBA as well as installing and configuring a custom linux kernel. 
+
+%----------------------------------------------------------------------------------------
+%	TEACHING
+%----------------------------------------------------------------------------------------
+ 
+\section{\textsc{Teaching}}
+
+% Intermediate C++ Programming, Linear Algebra, Calculus
+
+\begin{itemize}
+\item Teacher Assistant
+  \begin{itemize}
+  \item Math 31B: Integration and Infinite Series \hfill \textbf{2012 - 2013} \\
+  \item Math 33A: Linear Algebra and Applications \hfill \textbf{2012 - 2013} \\
+  \item PIC 10B: Intermediate Programming \hfill \textbf{Winter 2015,  Spring 2015} \\
+  \item PIC 20A: Principles of Java Language with Applications \hfill \textbf{Spring 2015} \\
+  \item PIC 40A: Introduction to Programming for Internet \hfill \textbf{Fall 2015} \\
+  \item Math 115B: Linear Algebra \hfill \textbf{Winter 2016} \\
+  \item Math 174E: Mathematics of Finance for Mathematics/Economics Students \hfill \textbf{Spring 2016, Fall 2016} \\
+  \item Independent Programming Projects \hfill \textbf{Winter 2016}
+  \end{itemize}
+\item Instructor
+  \begin{itemize}
+  \item Math 31B: Integration and Infinite Series \hfill \textbf{Summer 2016} \\
+  \item Math 32BH: Calculus of Several Variables (Honors) \hfill \textbf{Winter 2017} \\
+  \end{itemize}
+\item Undergraduate Projects Mentor
+  \begin{itemize}
+  \item Conway's game of life variations with C++/SDL graphics library \hfill \textbf{Winter 2016} \\
+  \item App development platform with Typescript 2 \hfill \textbf{Winter 2016} \\
+  \item Internet browsing data/trends visualization with Python \hfill \textbf{Winter 2016, Spring 2016} \\
+  \item Optical character recognition via neural nets with Python \hfill \textbf{Winter 2016, Spring 2016} \\
+  \item Discrete signal processing with Matlab \hfill \textbf{Fall 2017} \\
+  \end{itemize}
+\end{itemize}
+
+
+%----------------------------------------------------------------------------------------
+%	PAPERS
+%----------------------------------------------------------------------------------------
+ 
+
+
+
+\end{resume}
+
+\end{document}
+
+\section{\textsc{Papers}}
+
+Bobkov, A. {\it VC-density for trees}, in preparation  \\
+Bobkov, A. {\it Some VC-density computations for Shelah-Spencer graphs}, in preparation \\
+Bobkov, A. {\it VC-density in $\mathbb{Q}_p$-reducts}, in preparation
+
+%----------------------------------------------------------------------------------------
+%	Software
+%----------------------------------------------------------------------------------------
+ 
+\section{\textsc{Software Experience}}
+
+\textbf{Unix-like systems}\\
+I am comfortable working in command line environment, including tasks such as
+\begin{itemize}
+	\item installing and managing web-server, repository server, ssh server
+	\item code building, editing, and version control
+\end{itemize}
+
+\begin{tabular}{ll}
+\textbf{Languages} & C++, C\#, bash, Java, PHP, MAGMA \\
+\textbf{Code management} & CMake, Makefile, git, subversion, Visual Studio, Unity3D \\
+\textbf{Standards} & TCP/IP, .NET, CORBA  \\
+\end{tabular}
+
+Python, HTML, CSS, JavaScript, PHP, SQL
+
+%----------------------------------------------------------------------------------------
+%	Projects
+%----------------------------------------------------------------------------------------
+ 
+\section{\textsc{Independent Projects}}
+For more information and links visit \url{www.math.ucla.edu/~bobkov/projects.html}
+
+\textbf{Burn and Turn} \hfill \textbf{2008 - 2011}\\
+Cross-platform arcade style video game featured on \href{http://kotaku.com/5862197/burn-and-turn-combines-retro-arcade-stylings-with-tower-defense-for-combustible-fun}{Kotaku} and \href{http://indiegames.com/2011/10/trailer_burn_turn_robot_bear.html}{IndieGames}. It was coded in C++ and used OpenGL as a backend for graphics. It was created by a team of three people over a course of four years and released on iOS and Android markets.
+
+\textbf{Self Balancing Robot} \hfill \textbf{Summer 2012}\\
+A vertical self-balancing robot ran by an arduino controller coded in C++. A numerical simulation was used to determine weight distribution. Robot's position is determined by data from an accelerometer and a gyroscope combined through a Kalman filter. Balancing is done with a DC motor using PID controller.
+
+\textbf{UCLA Graduate Student Wiki} \hfill \textbf{Summer 2014}\\
+Official wiki for graduate math department at UCLA that maintains a database of qualifying exam problems. It is made on top of Semantic Media Wiki using custom extension written in PHP that allows to users to search, filter, and tag the solutions.
+
+\textbf{Decentralized Online Game} \hfill \textbf{Fall 2014 - Present}\\
+Exploration multiplayer online game that manages players and game data using peer-to-peer connections instead of relying on a central server. It is coded with C\# in Unity3D using standard TCP/IP network.
+
+\section{\textsc{Sample Code}}
+\url{https://gitorious.org/~antonbobkov}
+
+