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\title{Oral Qualifying Examination Syllabus}
\author{Eric Rowland}
\date{March 9, 2006}
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\noindent Committee:  J. Beck, R. Bumby, J. Tunnell, D. Zeilberger

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\section*{Experimental mathematics}

\noindent \textbf{Basic enumeration:}  counting arguments; inclusion-exclusion principle; generating functions; Bell numbers; Catalan numbers; Eulerian numbers; Fibonacci numbers; Stirling numbers \bigskip

\noindent \textbf{Combinatorics of subwords:}  the number of words of length $l$ on $k$ letters that do not contain a given subword; the number of occurrences of a given subword in the set of words of length $l$ on $k$ letters \bigskip

\noindent \textbf{Geometry of numbers:}  sums of two and four squares; Cauchy's polygonal number theorem \bigskip

\noindent \textbf{Calculus of finite differences:}  finite difference operator; indefinite summation; Bernoulli polynomials; Gosper's algorithm; (homogeneous and inhomogeneous) linear difference equations with constant coefficients; Petkov\v{s}ek's algorithm \bigskip

\noindent \textbf{Hypergeometric identities:}  fundamental theorem; Fasenmyer's algorithm; Zeilberger's algorithm; Wilf--Zeilberger pairs; companion identity and dual identities \bigskip

\noindent \textbf{Digit sequences:}  binary exponentiation; Gray code; Johnson--Trotter algorithm; Thue--Morse sequence \bigskip

\noindent \textbf{Binomial coefficients modulo $p^q$:}  Kummer's theorem; Lucas' theorem; Granville's theorem; fractal dimension of Pascal's triangle modulo $p$ \bigskip

\noindent \textbf{One-dimensional cellular automata:}  Jen's aperiodicity theorem; generalized linearity; positional bijectivity and reversibility; global and local nestedness \bigskip

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\section*{Diophantine approximation}

\noindent \textbf{Approximability by rationals:}  continued fractions; Dirichlet's theorem; best approximations; Hurwitz's theorem; Liouville's theorem; quadratic irrationals; Pell equation \bigskip

\noindent \textbf{Markoff spectrum:}  Markoff equation; gaps; Hall's ray \bigskip

\noindent \textbf{Transcendence theory:}  irrationality of $e$, $e^{m/n}$, $\pi$, and $\zeta(3)$; transcendence of $e$ and $\pi$; Lindemann--Weierstrass theorem; statement of Gelfond--Schneider theorem \bigskip

\noindent \textbf{Distribution modulo $1$:}  Weyl criterion; uniform distribution of $\{n \alpha\}$ for irrational $\alpha$; uniform distribution of polynomials; characterization of normal numbers \bigskip

\noindent \textbf{Integer relation algorithms:}  LLL algorithm; PSLQ algorithm \bigskip

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