text:
\begin{document}
\pagenumbering{roman}
\begin{center}
{\Large \bf Barvinok's Rational Functions: Algorithms and Applications to Optimization, Statistics, and Algebra}
\vskip 0.2in
\centerline{By}
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\centerline{\large  Ruriko Yoshida}
\centerline{\large B.A. (University of California at Berkeley) 2000}

\vskip 0.3in
\centerline{\large  DISSERTATION}
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\centerline{\large Submitted in partial satisfaction of the requirements for the
degree of}
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\centerline{\large  DOCTOR OF PHILOSOPHY}
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\centerline{\large in}
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\centerline{\large MATHEMATICS}
\vskip 0.2in
\centerline{\large in the}
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\centerline{\large  OFFICE OF GRADUATE STUDIES}
\vskip 0.1in
\centerline{\large of the}
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\centerline{\large  UNIVERSITY OF CALIFORNIA}
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\centerline{\large  DAVIS}
\end{center}
\vskip 0.1in
\begin{center}
{\large Approved:}
\end{center}
\begin{center}
\centerline{\underbar{\hskip 2.5in}}
\vskip 0.15in
\centerline{\underbar{\hskip 2.5in}}
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\centerline{\underbar{\hskip 2.5in}}
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\centerline{\large Committee in Charge}
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\centerline{\large 2004}
\end{center}

\pagebreak

\begin{center}

  \vspace*{50pt}

To Pete Camagna,\\
for his love and support.
\end{center}

\newpage
\large
\tableofcontents
\listoftables
\listoffigures

\newpage
\large
{\Large \bf ACKNOWLEDGEMENTS} \\

\input{Aknowledge}

\newpage
\begin{center}
\underline{\bf \Large Abstract}
\end{center}

\medskip

\large
\input{ThesisAbs2}

\newpage
\pagestyle{myheadings}
\pagenumbering{arabic}
\markright{  \rm \normalsize CHAPTER 1. \hspace{0.5cm}
 Introduction}
\large
\chapter{Introduction}\label{IntroThesis}
\thispagestyle{myheadings}

\input{IntroThesis}

\newpage
\pagestyle{myheadings}
\chapter{Gr\"obner bases of toric ideals via short rational functions}\label{GBT}
\thispagestyle{myheadings}
\markright{  \rm \normalsize CHAPTER 2. \hspace{0.5cm}
 Gr\"obner bases of toric ideals via short rational functions}

\input{gbtoricfastThesis}

\newpage
\pagestyle{myheadings}
\chapter{Theoretical applications of rational functions to Mixed Integer Programming}\label{IPT}
\thispagestyle{myheadings}
\markright{  \rm \normalsize CHAPTER 3. \hspace{0.5cm}
 Theoretical applications of rational functions to MIP}

\input{MIPThesis}

\newpage
\pagestyle{myheadings}
\markright{  \rm \normalsize CHAPTER 4. \hspace{0.5cm}
Experimental results: development of {\tt LattE}}
\chapter{Experimental results: development of {\tt LattE}}\label{chapsoftware}

\thispagestyle{myheadings}

\input{softwareThesis}

\newpage
\appendix
\pagestyle{myheadings}
\markright{  \rm \normalsize APPENDIX. \hspace{0.5cm}
 User manual of {\tt LattE}}

\chapter{User manual of {\tt LattE}}\label{App}

\thispagestyle{myheadings}

\input{manualThesis}

\newpage
\pagestyle{myheadings}
\markright{  \rm \normalsize BIBLIOGRAPHY. \hspace{0.5cm}
}

config: arxiv


TITLE: A (smart) way to map out the Brillouin zone of a 2-D material
QUESTION [5 upvotes]: I am currently writing a tight-binding calculation model for various 2-D cells as part of a homework assignment. Whilst solving the problem set was quite easy I struggle at a smart way to plot the band diagram in the usual "lets visit all high symmetry points in the first BZ"-fashion.

right now I am manually 'sticking together' the vectors from point to point but that is rather tedious and doesn't seem smart to me. Unfortunately I am lacking a good idea so I am hoping there are some people here who do band calculations and can give me a hint!

REPLY [0 votes]: I think there is no general answer to that. Just try and include all important symmetry directions you can think of.
For example, in the case of the Brillouin zone shown in your left picture, how about the path K'-K-Gamma-K'-M-K-Gamma-M? (I know, the path Gamma-K now is included twice, but I couldn't think of any other possibility.)


TITLE: Flatness of Residue Field
QUESTION [2 upvotes]: My question refers to a step of in the proof of Corollary 8.5.17 in  Bosch's "Commutative Algebra and Algebraic Geometry";  see page 395
See the red tagged line below:

We consider the exact sequence
$$0 \to I/J \to R[t_1, ..., t_n]/J \to R[t_1, ..., t_n]/I \to 0$$
and we tensor it with $k(s) = \mathcal{O}_{S,s}/m_s$.
Why does it stay exact? Indeed, by assumption $\mathcal{O}_{S,s}$ is flat so it's ok to tensor it with $\mathcal{O}_{S,s}$ but what about $k(s)$? Why does it conserve the exactness?
In addition: Lemma 5.2.9:

REPLY [3 votes]: Set $T = R[t_1,\ldots, t_n]$. Our assumption that $f$ is flat at $x$ means that $(T/I)_x$ is flat over $R_s$.
We have an exact sequence:
$$ 0 \to I/J \to T/J \to T/I \to 0$$
Localization is exact so we again have the localized exact sequence:
$$0 \to (I/J)_x \to (T/J)_x \to (T/I)_x \to 0$$
Now in 5.2/9, we take $M''$ to be $(T/I)_x$ since it is flat over $R_s$, and take $N$ to be $k(s) = R_s/m_s$. We get the exact sequence we wanted:
$$0 \to (I/J)_x\otimes k(s) \to (T/J)_x\otimes k(s) \to (T/I)_x\otimes k(s) \to 0$$
This justifies the statement "[the sequence] remains exact at $x$ when tensoring over $R$ it with $k(s)$".
(Sweeping it under the rug, but we just used that Localization commutes with tensor products to identify $M\otimes_{R_s} k(s) = M\otimes_{R}k(s)$ for an $R_s$-module $M$ by the way.)
Watch out for the typo in the next line though, $J/I$ should be $I/J$.

Per the title of this question: The residue field $k(s)$ is not going to be a flat module, see e.g. is residue field ever flat over its local ring? on MathOverflow, so it's important that $N$ in 5.2/9 can be any module.


TITLE: How to prove that for $n \in \mathbb{N}$ we have $\sum_{k=2}^n \frac{1}{k}\leq \ln(n) \leq \sum_{k=1}^{n-1} \frac{1}{k}$
QUESTION [6 upvotes]: Prove that for $n \in \mathbb{N}$ we have $\sum_{k=2}^n \frac{1}{k}\leq \ln(n) \leq \sum_{k=1}^{n-1} \frac{1}{k}$ by using Riemann integral?

REPLY [1 votes]: Hint. Note that on $[k, k+1] $ we have
$$ \frac 1{k+1} \le \frac 1x \le \frac 1k $$
Now integrate over $[k, k+1] $ and sum from 1 to $ n-1$.