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% Author, Title, etc.



%\title{Poisson Summation and Benford's Law: From values of %$L$-functions to the $3x+1$ problem to products of random %variables}
%\author{Steven J Miller}
%\date{December 2008}
%\institute{Brown University\\ %\texttt{http://www.math.brown.edu/$\sim$sjmiller}}

\Large

\title[Poisson Summation and Benford's Law]
{
  Poisson Summation and Benford's Law: From values of $L$-functions to the $3x+1$ problem to products of random variables
}

\author[Steven J Miller] %%
{
  Steven J Miller\ \ \\ \textcolor{brown!50!black}{Brown University} \\ \ \newline
  \texttt{sjmiller@math.brown.edu}\newline
  \texttt{http://www.math.brown.edu/$\sim$sjmiller}
}

\date[SF 2007]
{\alert{Joint Mathematics Meetings, January 2018\\

Special Session On Discrete Neural Networking\\ San Diego, CA}
}


% The main document

\title{Generalizations of Zeckendorf's Theorem to Two-Dimensional Sequences.}
\author[Joshua M. Siktar, Steven J. Miller]
{
  Joshua M. Siktar, Steven J. Miller\ \ \\ \textcolor{brown!50!black}{Carnegie Mellon University, Williams College} \\ \ \newline
  \texttt{jsiktar@andrew.cmu.edu, sjm1@williams.edu}
  }


\begin{document}

\frame{\titlepage}

\section[Outline]{}
%\frame{\tableofcontents}

%INTRODUCTION
\section{Introduction}
\subsection{Overview of the Presentation}
\frame
{
  \frametitle{Outline}

  \begin{itemize}
  \item<1-> How to Combinatorially Interpret Zeckendorf Decompositions \\ \
  \item<2-> Simple and Compound Jump Paths \\ \
  \item<3-> Gaussianity for Simple Jump Paths \\ \
  \item<4-> Future Research Questions \\ \
  \end{itemize}
}

%Introduction to Simple and Compound Jump Paths
\section{Motivation}
\subsection{Zeckendorf Decompositions}
\frame{
\begin{definition}[Zeckendorf Decompositions]
A \textbf{Zeckendorf Decomposition} is a way to write a natural number as the sum of non-adjacent Fibonacci Numbers.
\end{definition}
\begin{Theorem}[Zeckendorf's Theorem]
Every natural number has a unique Zeckendorf Decomposition.
\end{Theorem}
\begin{itemize}
\item<1->Example: $335 \ = \ 13 + 89 + 233$ \\ \
\item<2->Example: $1033 \ = \ 1 + 3 + 8 + 34 + 987$
\end{itemize}
}

\subsection{Construction of Zeckendorf Diagonal Sequence}
\frame{
\begin{itemize}
\item<1-> We construct a 2-dimensional sequence to motivate our combinatorial analysis. \\ \
\item<2-> Set $z_{0, 0} \ = \ 1$. \\ \
\item<3-> For each $n \in \mathbb{N}^+$, check if any downward/leftward path sums to the number. If not, add the number to the sequence so that it is added to the shortest unfilled diagonal moving from the bottom right to the top left.
\end{itemize}
}

\frame{

$\left[\begin{array}{ccccccccc}6992 & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\2200 & 6054 & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\954 & 2182 & 5328 & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\364 & 908 & 2008 & 5100 & \cdots & \cdots & \cdots & \cdots & \cdots \\138 & 342 & 862 & 1522 & 4966 & \cdots & \cdots & \cdots & \cdots \\44 & 112 & 296 & 520 & 1146 & 2952 & \cdots & \cdots & \cdots \\16 & 38 & 94 & 184 & 476 & 1102 & 2630 & \cdots & \cdots \\4 & 10 & 22 & 56 & 168 & 370 & 1052 & 2592 & \cdots \\1 & 2 & 6 & 18 & 46 & 140 & 366 & 1042 & 2270\end{array}\right]$

\begin{itemize}
\item<1-> Our goal is to enumerate how many paths are required for a linear search of a Zeckendorf decomposition.
\end{itemize}

}




\section{Jump Paths}

\subsection{Compound Jump Paths}

\frame{

\begin{definition}[Compound Jump Paths]
A \textbf{compound jump path} is a path on the lattice grid moving only down and to the left, where movements may be greater than one unit at a time in either direction.
\end{definition}

\begin{itemize}
\item<1->We count compound jump paths from $(a, b)$ to $(0, 0)$. \\ \
\item<2->Let the number of compound jump paths from $(a, b)$ to $(0, 0)$ be denoted as $q_{a, b}$. \\ \
\item<3->Any compound jump path must include the use of $(a, b)$ and $(0, 0)$. \\ \
\item<4->Let the number of compound jump paths from $(a, b)$ to $(0, 0)$ with $k$ steps be denoted $r_{a, b, k}$.
\end{itemize}
}

%probably need additional slide of motivation for the lattice

\subsection{Simple Jump Paths}
\frame{
\begin{definition}[Simple Jump Paths]
A \textbf{simple jump path} is a path on the lattice grid where each movement on the lattice grid consists of at least one unit movement to the left and one unit movement downward.
\end{definition}

\begin{itemize}
\item<1->We count simple jump paths from $(a, b)$ to $(0, 0)$. \\ \
\item<2->Let the number of simple jump paths from $(a, b)$ to $(0, 0)$ be denoted $s_{a, b}$. \\ \
\item<3->Any simple jump path must include the use of $(a, b)$ and $(0, 0)$. \\ \
\item<4->Let the number of simple jump paths from $(a, b)$ to $(0, 0)$ with $k$ steps be denoted $t_{a, b, k}$. 
\end{itemize}
}


%Combinatorial Identities on Simple Jump Paths
\section{Data and Identities}
\subsection{Values on Sequences} %expanded of S2016 talk, more emphasis on simple jump paths for time being

%\frame{
%\begin{itemize}
%\item<1->$\{q_{a, 0}\}^{5}_{a = 0} = \{1, 1, 2, 4, 8, 16\}$
%\item<2->$\{q_{a, 1}\}^{5}_{a = 0} = \{1, 3, 8, 20, 48, 112\}$
%\item<3->$\{q_{a, 2}\}^{5}_{a = 0} = \{2, 8, 26, 76, 208, 544\}$
%\item<4->$\{q_{a, 3}\}^{5}_{a = 0} = \{4, 20, 76, 252, 768, 2208\}$
%\item<5->$\{q_{a, 4}\}^{5}_{a = 0} = \{8, 48, 208, 768, 2568, 8016\}$
%\end{itemize}
%}

\subsection{Basic Identities}
\frame{
\begin{Lemma}[Compound Jump Path Partition Lemma]
$\forall a, b \in \mathbb{N}$, $q_{a, b} = \sum^{a + b}_{k = 1}r_{a, b, k}$.
\end{Lemma}
\vspace{0.2 cm}
\begin{Lemma}[Simple Jump Path Partition Lemma]
$\forall a, b \in \mathbb{N}$, $s_{a, b} = \sum^{\min\{a, b\}}_{k = 1}t_{a, b, k}$.
\end{Lemma}
}

\frame{
\begin{Lemma}[Enumerating Simple Jump Paths]
$\forall a, b \in \mathbb{N}, k \in \min\{a, b\}$, $t_{a, b, k} = {{a - 1}\choose{k - 1}}{{b - 1}\choose{k - 1}}$.
\end{Lemma}
\begin{itemize}
\item<1->First factor is number of ways to group $a$ objects into $k$ nonempty groups \\ \
\item<2->Second factor is number of ways to group $b$ objects into $k$ nonempty groups \\ \
\item<3->Groupings are independently determined
\end{itemize}
}


%Gaussianity
\section{Gaussianity}

\subsection{Preliminaries}
\frame{
\begin{figure}
\begin{center}
\includegraphics[width=9.5cm,height=5cm,angle=0]{simplejumppatha10b10.png}
\end{center}
\end{figure}
\begin{itemize}
\item<1->Represents $\{t_{10, 10, k}\}^{10}_{k = 1}$ \\ \
\item<2->Special case: simple jump paths over a square lattice for $n \ = \ 10$
\end{itemize}
}


\frame{
\begin{itemize}
\item<1->Want to show convergence to a normal distribution as $n \rightarrow \infty$ \\ \
\item<2->The distribution will be taken over values of $k$ that give legal jump paths for the given $n$. \\ \
\item<3-> Simple jump paths: $k \in [n]$
\end{itemize}
}

\subsection{Normal Distribution - Proof of Special Case} %the special case we studied
\frame{
\begin{itemize}
\item<1->Densities: number of simple jump paths with a fixed number of steps \\ \
\item<2->Study square lattice, i.e. where $a \ = \ b$. 
\end{itemize}
}

%\frame{
%\begin{Theorem}[Li-Miller]
%Suppose $X_1, X_2, ...$ are random variables such that for all integers $m \geq 0$ we have $\lim_{n \rightarrow \infty}\mathbb{E}[X^{2m}_n] = (2m - 1)!!$ and $\lim_{n \rightarrow \infty}\E[X^{2m + 1}_n] = 0$. Then the sequence $X_1, X_2, ...$ converges weakly in distribution to the standard normal $N(0, 1)$.
%\end{Theorem}
%\begin{itemize}
%\item<1->Gives outlet to convergence to Gaussianity 
%\item<2->Only proven for ``one-dimensional" limits thus far
%\end{itemize}
%} %LI-MILLER, OMITTED FOR NOW	

\subsection{Density, Mean, Standard Deviation} %identify key parameters in proof
%\frame{
%\begin{Lemma}[Zero-Degree Binomial Coefficient Lemma]
%$\sum^{n}_{k = 0}{{n}\choose{k}}^2 \ = \ {{2n}\choose{n}}$
%\end{Lemma}
%\begin{Lemma}[First-Degree Binomial Coefficient Lemma]
%$\sum^{n}_{k = 0}k{{n}\choose{k}}^2 \ = \ \frac{n}{2}{{2n}\choose{n}} $
%\end{Lemma}
%\begin{Lemma}[Second-Degree Binomial Coefficient Lemma]
%$\sum^{n}_{k = 0}k^2{{n}\choose{k}}^2 \ = \ n^2{{2n - 2}\choose{n - 1}}$
%\end{Lemma}
%}

\frame{
\begin{Theorem}[Mean on Square Lattice]
$\forall n \in \mathbb{N}^+$, $\mu_{n + 1, n + 1} \ = \ \frac{1}{2}n + 1 \sim \frac{n}{2}$.
\end{Theorem}
\begin{itemize} 
\item<1->Derive directly from definition of first moment \\ \
\item<2->Use index shift $\sum^{n + 1}_{k = 1}k{{n}\choose{k - 1}}^2 \ = \ \sum^{n}_{k = 0}k{{n}\choose{k}}^2 + \sum^{n}_{k = 0}{{n}\choose{k}}^2$ \\ \
\item<3->Use standard techniques for evaluating binomial coefficients 
\end{itemize}
}

\frame{
\begin{Theorem}[Standard Deviation on Square Lattice]
$\forall n \in \mathbb{N}^+$, $\sigma_{n + 1, n + 1} \ = \ \frac{n}{2\sqrt{2(n - 1)}} \sim \frac{\sqrt{n}}{2\sqrt{2}}$.
\end{Theorem}
\begin{itemize} 
\item<1->Derive directly from definition of second standardized moment \\ \
\item<2->Use index shift $\sum^{n + 1}_{k = 1}\left(k - \left(\frac{1}{2}n + 1\right)\right)^2{{n}\choose{k - 1}}^2 \ = \ \sum^{n}_{k = 0}(k + 1 - \left(\frac{1}{2}n + 1\right))^2{{n}\choose{k}}^2$ \\ \
\item<3->Split into three sums via binomial expansion \\ \
\item<4->Use standard techniques for evaluating binomial coefficients 
\end{itemize}
}

\subsection{Stirling Formula Expansion} %outline use of Stirling
\frame{
\begin{itemize}
\item<1->Density function: $f_n(k + 1) \ := \ \frac{t_{n + 1, n + 1, k + 1}}{s_{n + 1, n + 1}} \ = \ \frac{{{n}\choose{k}}^2}{{{2n}\choose{n}}}$ \\ \
\item<2->Simplifying binomial coefficients gives $\frac{(n!)^4}{(k!)^2((n - k)!)^2(2n)!}$ \\ \
\item<3-> Use Stirling's Approximation on each factor: $m! \ \sim \ m^me^{-m}\sqrt{2\pi m}$ \\ \
\end{itemize}
}

\frame{ 
\begin{itemize}
\item<1-> End result of Stirling expansion is $f_n(k + 1) \ = \ \frac{n^{2n}}{k^{2k} \cdot (n - k)^{2n - 2k} \cdot 2^{2n} \cdot \frac{1}{4} \cdot \sqrt{4 \pi n}}$ \\ \
\item<2->Let $P_n(k + 1) \ :=  \ \frac{n^n}{k^k(n - k)^{n - k}2^n}$ and $S_n(k + 1) \ = \ \frac{1}{\frac{1}{2}\sqrt{\pi n}}$, then $f_n(k + 1) \ = \ P_n(k + 1)^2S_n(k + 1)$. \\ \
\item<3->Let $k \ := \ \mu_{n + 1, n + 1} + x\sigma_{n + 1, n + 1}$, then $f_n(k + 1)dk \ = \ f_n(\mu_{n} + x\sigma_{n} + 1)\sigma_{n}dx \sim f_n(\mu_n + x\sigma_n + 1)\frac{\sqrt{n}}{2}dx$
\end{itemize}
}

\subsection{Gaussianity Calculation}
\frame{
Apply logarithm to $P_n(k + 1) \ =  \ \frac{n^n}{k^k(n - k)^{n - k}2^n}$: 
\begin{equation}
\log P_n(k + 1) \ = \ n\log (n) - k\log (k) - (n - k)\log(n - k) - n\log(2) \nonumber
\end{equation} \\ \
Rewrite $k \ = \ \frac{n}{2} + \frac{x\sqrt{n}}{2\sqrt{2}} \ = \ \frac{n}{2}\left(1 + \frac{x}{\sqrt{2n}}\right)$ to expand $\log(k)$ and $\log(n - k)$:
\begin{equation}
\log(k) \ = \ \log\left(\frac{n}{2}\left(1 - \frac{x}{\sqrt{2n}}\right)\right) \approx \log(n) - \log(2) + \log\left(1 - \frac{x}{\sqrt{2n}}\right)  \nonumber
\end{equation}
\begin{equation}
\log(n - k)\ = \  \ \log\left(\frac{n}{2}\left(1 + \frac{x}{\sqrt{2n}}\right)\right) \approx \log(n) - \log(2) + \log\left(1 + \frac{x}{\sqrt{2n}}\right) \nonumber
\end{equation}
}


\frame{
Substitute logarithm expansions and approximate $\log\left(1 + \frac{x}{\sqrt{2n}}\right)$ and $\log\left(1 - \frac{x}{\sqrt{2n}}\right)$ to second order to conclude
\begin{equation}
\log P_n(k + 1) \sim -\frac{n}{2}\log\left(1 - \frac{x^2}{2n}\right) - \frac{x\sqrt{n}}{2}\left(\frac{x}{\sqrt{n}} + O\left(\frac{1}{n^{\frac{3}{2}}}\right)\right) \nonumber
\end{equation} \\ \
Approximate $\log\left(1 - \frac{x^2}{2n}\right)$ up to second order:
\begin{equation}
-\frac{n}{2}\left(-\frac{x^2}{2n} + O\left(\frac{1}{n^2}\right)\right) - \frac{x\sqrt{n}}{2}\left(\frac{x}{\sqrt{n}} + O\left(\frac{1}{n^{\frac{3}{2}}}\right)\right) \sim -\frac{x^2}{4}\nonumber
\end{equation}
}

\frame{
It follows that
\begin{equation}
P_n(k + 1) \sim e^{-\frac{x^2}{4}} \Rightarrow P_n(k + 1)^2 \sim e^{-\frac{x^2}{2}} \Rightarrow
\nonumber
\end{equation}
\begin{equation}
f_n(k + 1) \sim \frac{e^{-\frac{x^2}{2}}}{\sqrt{2\pi}}
\nonumber
\end{equation}
\begin{itemize}
\item<1->Normal distribution, mean $0$, standard deviation $1$.
\end{itemize}
}

%put here as a "closing" of some sorts
%
%\subsection{Supporting Data-Compound Jump Paths}
%\frame{
%\begin{itemize}
%\item<1->$\{r_{1, 1, k}\}^{3}_{k = 0} = \{0, 1, 2, 0\}$ \\ \
%\item<2->$\{r_{2, 2, k}\}^{5}_{k = 0} = \{0, 1, 7, 12, 6, 0\}$ \\ \
%\item<3->$\{r_{3, 3, k}\}^{7}_{k = 0} = \{0, 1, 14, 55, 92, 70, 20, 0\}$ \\ \
%\item<4->Data suggests Gaussianity holds in this case as well
%\end{itemize}
%}

%\frame{
%\begin{figure}
%\begin{center}
%\includegraphics[width=9.5cm,height=5cm,angle=0]{compoundjumppatha10b10.png}
%\end{center}
%\end{figure}
%\begin{itemize}
%\item<1->Represents $\{r_{10, 10, k}\}^{20}_{k = 1}$ \\ \
%\item<2->Anticipate that proof generalizes to this case
%\end{itemize}
%}

%FUTURE WORK
\section{Future Work}
\frame
{
\begin{itemize}
\item<1-> Finish generalizations of Gaussian convergence: simple jump paths where $a \neq b$ \\ \ 
\item<2->Generalize result to compound jump paths
\end{itemize}

}

%ACKNOWLEDGMENTS
\section{Acknowledgments} 
\subsection{References}
\frame{
\begin{itemize}
\item<1->H. Alpert, \emph{Differences of Multiple Fibonacci Numbers}, October 20, 2009
\item<1->I. Badinskki, C. Huffaker, N. Mccue, C. Miller, K. Miller, S. Miller, M. Stone, \emph{The M\&M Game: From Morsels to Modern Mathematics}, September 3, 2015
\item<1->O. Beckwith, A. Bower, L. Gaudet, R. Insoft, S. Li, S. Miller, P. Tosteson, \emph{The Average Gap Distribution For Generalized Zeckendorf Decompositions}, Fibonacci Quarterly, December 12, 2012.
\item<1->L. Cano, R. Diaz, \emph{Continuous Analogues for the Binomial Coefficients and the Catalan Numbers}, March 22, 2016
\end{itemize}
}

\frame{
\begin{itemize}
\item<1->R. Doward, P. Ford, E. Fourakis, P. Harris, S. Miller, E. Palsson, H. Paugh, \emph{New Behavior in Legal Decompositions Arising From Non-Positive Linear Recurrences}, September 10, 2015
\item<1->V. Guo, J. Zeng \emph{New Congruences for Sums Involving Apery Numbers or Central Delannoy Numbers}, International Journal of Number Theory, May 25, 2012
\item<1->E. Hart, \emph{The Zeckendorf Decomposition of Certain Fibonacci-Lucas Products}, Fibonacci Quarterly, November 1998
\item<1->E. Hart, L. Sanchis, \emph{On The Occurrence Of Fn in The Zeckendorf Decomposition of nFn}, February 1997

\end{itemize}
}

\frame{
\begin{itemize}
\item<1->M. Kanovich, \emph{Multiset Rewriting Over Fibonacci and Tribonacci Numbers}, Journal of Computer and System Sciences, September 2014
\item<1->M. Kologlu, G. Kopp, S. Miller, Y. Wang, \emph{On the Number of Summands in Zeckendorf Decompositons}, Journal of Number Theory, August 19, 2010.
\item<1->C. Krattenhaler, \emph{Lattice Path Enumeration}, April 17, 2015
\item<1->T. Mansour, A. Munagi, M. Shattuck \emph{Recurrence Relations and Two Dimensional Set Partitions}, Journal of Integer Sequences, March 26, 2011
\end{itemize}
}

\frame{
\begin{itemize}
\item<1->S. Miller, Y. Wang, \emph{Gaussian Behavior in Generalized Zeckendorf Decompositions}, July 14, 2011
\item<1->S. Miller, \emph{The Probability Lifesaver}, 2017
\item<1->J. Watkins, \emph{Moments and Generating Functions}, September 29, 2009
\end{itemize}
}

\subsection{Thank You}
\frame
{
  \frametitle{Thank You}

  \begin{itemize}
  \item<1-> Special thanks to... \\
  \item<2-> Carnegie Mellon University Department of Mathematical Sciences \\
  \item<3-> Williams College Department of Mathematics and Statistics \\
  \item<4-> The organizing committee for the 2018 Joint Mathematics Meetings   
  \end{itemize}
}

%Extra content for possible appendix
%\subsection{Polynomial Growth}
%\frame{
%\begin{Theorem}[Polynomial Growth Theorem]
%If $q_{n, m}$ is of the form $p_m(n)2^n \ \forall n \in \mathbb{N}^0$ where $p_m(n)$ is a $m$-degree polynomial in $n$ with positive coefficients, then $q_{n, m + 1}$ is of the form $p_{m + 1}(n)2^n$ where $p_{m + 1}(n)$ is an $m + 1$-degree polynomial in $n$ with positive coefficients.
%\end{Theorem}
%\begin{itemize}
%\item<1->One can construct closed forms for $q_{a, b}$ for a fixed $b$ and arbitrary $a$.
%\item<2->Example: $q_{a, 2} = \left(\frac{1}{4}a^2 + \frac{7}{4}a + 2\right)2^a \ \forall a \in \mathbb{N}^0$
%\item<3->Use these closed forms and induction to obtain the result.
%\end{itemize}
%}

\end{document}