## RESEARCH INTERESTS

My research interests are primarily in number theory and enumerative combinatorics with an emphasis on generating function methods, continued fractions, software development, and experimental mathematics.
Research problems I have historically been interested in include: combinatorial properties of generalized factorial functions and their symbolic polynomial expansions, generating functions and generating function transformations, generalized Stirling numbers, Jacobi-type continued fractions (J-fractions) for special series, and Lambert series generating functions. I am always open to exploring new interesting problems.

## EDUCATION

### 2014

University of Illinois at Urbana-Champaign

*Master of Science in Computer Science*

### 2012

University of Illinois at Urbana-Champaign

*Bachelor of Science in Mathematics*

### 2012

University of Illinois at Urbana-Champaign

*Bachelor of Science in Computer Science*

### 2004

Northwest Missouri State University

*Associate of Science (MASMC)*

## AWARDS AND HONORS

### 2014

NSF GRFP National Honorable Mention

### 2013

NSF GRFP National Honorable Mention

### 2012

Diffenbaugh Graduate Fellowship at UIUC

### 2010

Barry M. Goldwater Scholarship

## PUBLICATIONS

### 2018

#### (16)

Factorization Theorems for Generalized Lambert Series and Applications

*Accepted for publication in the Ramanujan Journal.*
*Joint work with Mircea Merca.*
*Submitted 07/2017*
__Keywords:__ Lambert series; factorization theorem; matrix factorization; partition function; multiplicative function.
__MSC (2010):__ 11A25; 11P81; 05A17; 05A19.

### 2018

#### (15)

Zeta Series Generating Function Transformations Related to Generalized Stirling Numbers and Partial Sums of the Hurwitz Zeta Function

*Published in the
Online Journal of Analytic Combinatorics.*
__Full Text:__ https://arxiv.org/abs/1611.00957 (math.CO)

### 2018

#### (14)

New Congruences and Finite Difference Equations for Generalized Factorial Functions

*Accepted for publication in INTEGERS.*
__Full Text:__ https://arxiv.org/abs/1701.04741 (math.CO)
__Keywords:__ continued fraction, Pochhammer symbol, multiple factorial, Pochhammer
k-symbol, prime congruence, Wilson’s theorem, Clement’s theorem, sexy prime.
__MSC (2010):__ 05A10; 11Y55; 11Y65; 11A07; 11B50.

### 2018

#### (13)

Combinatorial Identities for Generalized Stirling Numbers Expanding f-Factorial Functions and the f-Harmonic Numbers

*Published in the Journal of Integer Sequences. *

Summary of my CS499 Senior Thesis project from 2010-2011 at UIUC.
__Journal Link:__ https://cs.uwaterloo.ca/journals/JIS/VOL21/Schmidt/schmidt18.html
__Full Text:__ https://arxiv.org/abs/1611.04708 (math.CO)
__Keywords:__ factorial; multifactorial; j-factorial; Pochhammer symbol; Stirling number; generalized Stirling number; harmonic number; f-harmonic number; Stirling polynomial.
__MSC (2010):__ 11B73; 05A10; 11B75.

### 2018

#### (12)

Generating Special Arithmetic Functions by Lambert Series Factorizations

*With Mircea Merca. Accepted for publication in Contributions to Discrete Mathematics (CDM).*
__Full Text:__ https://arxiv.org/abs/1706.00393
__Keywords:__ Lambert series; factorization theorem; matrix factorization; partition function.
__MSC (2010):__ 11A25; 11P81; 05A17; 05A19.

### 2018

#### (11)

Jacobi-Type Continued Fractions and Congruences for Binomial Coefficients Modulo Integers h ≥ 2

*Published in INTEGERS.*
__Full Text:__ https://arxiv.org/abs/1702.01374 (math.CO)

### 2018

#### (10)

A Partition Identity Related to Stanley's Theorem

*With Mircea Merca. Accepted for publication in the American Mathematical Monthly, to appear in 2018.*

__Keywords:__ Lambert series; Euler’s totient; partition.
__MSC (2010):__ 11A25; 11P81; 05A17; 05A19.

### 2017

#### (9)

The partition function p(n) in terms of the classical Mobius function

*With Mircea Merca. Published in the Ramanujan Journal*

__Keywords:__ Lambert series; Mobius function; q-series; partition function.
__MSC (2010):__ 11A25; 11P81; 05A17; 05A19.

### 2017

#### (8)

Continued Fractions for Square Series Generating Functions

*Published in the Ramanujan Journal.*

### 2017

#### (7)

New Recurrence Relations and Matrix Equations for Arithmetic Functions Generated by Lambert Series

*Published in Acta Arithmetica.*

### 2017

#### (6)

Continued Fractions and q-Series Generating Functions for the Generalized Sum-of-Divisors Functions

*Published in the Journal of Number Theory*

### 2017

#### (5)

Generating Function Transformations Related to Polylogarithm Functions and the k-Order Harmonic Numbers

*Published in the Online Journal of Analytic Combinatorics*

### 2017

#### (4)

Square Series Generating Function Transformations

*Published in the Journal of Inequalities and Special Functions*

### 2017

#### (3)

Jacobi-type continued fractions for the ordinary generating functions of generalized factorial functions

*Published in the Journal of Integer Sequences*

### 2014

#### (2)

A Computer Algebra Package for Polynomial Sequence Recognition

*UIUC Master's Thesis*

### 2010

#### (1)

Generalized j-factorial functions, polynomials, and applications

*Published in the Journal of Integer Sequences*

## MANUSCRIPTS

### 09/2018

A Short Note on Integral Transformations and Conversion Formulas for Sequence Generating Functions

__Full Text:__ arXiv link to appear (math.CO)
__Keywords:__ Generating function; series transformation; gamma function; Hankel contour.
__MSC (2010):__ 05A15; 30E20; 31B10; and 11B73.

### 06/2018

Exact Formulas for the Generalized Sum-of-Divisors Functions

*Accepted for publication in the Journal of Integer Sequences. Subsequently rejected due to disagreement with editor. Currently looking for a new journal to submit this paper for review again.*
__Full Text:__ https://arxiv.org/abs/1705.03488 (math.NT)
__Keywords:__ divisor function; sum-of-divisors function; Lambert series; perfect number.
__MSC (2010):__ 30B50; 11N64; 11B83.

### 08/2017

Factorization Theorems for Hadamard Products and Higher-Order Derivatives of Lambert Series Generating Functions

__Keywords:__ Lambert series; factorization theorem; matrix factorization; partition function; Hadamard product.
__MSC (2010):__ 11A25; 11P81; 05A17; 05A19.

### 07/2017

Pair Correlation and Gap Distributions for Substitution Tilings and Generalized Ulam Sets in the Plane

*Worked closely with Jayadev Athreya on the project.*
__Full Text:__ https://arxiv.org/abs/1707.05509
__Keywords:__ substitution tiling; Ammann chair; Ulam set; directional distribution;
gap distribution; pair correlation.
__MSC (2010):__ 52C20; 06A99; 11B05; 62H11; 52C23.

### 06/2017

New Factor Pairs for Factorizations of Lambert Series Generating Functions

*With Mircea Merca. *
__Full Text:__ https://arxiv.org/abs/1706.02359
__Keywords:__ Lambert series; factorization theorem; matrix factorization; partition function.
__MSC (2010):__ 11A25; 11P81; 05A17; 05A19.

### 04/2017

Combinatorial Sums and Identities Involving Generalized Divisor Functions with Bounded Divisors

__Full Text:__ https://arxiv.org/abs/1704.05595 (math.NT)
__Keywords:__ divisor function; sum of divisors function; Lambert series.
__MSC (2010):__ 30B50; 11N64; 11B37; 11B83; 11K65.

## RECENT SOFTWARE

### 2018 -- Present

Mathematical biology group visualization code

*Continued work with Christine Heitsch's Georgia Tech
Discrete Mathematics and Molecular Biology research group.
Recent work with the group and their
growing list of software contributions
includes updating, growing, and debugging the existing code
for RNAStructViz in the current
**with-cairo* branch (see this release).
We will be submitting an application note introducing the
key features and benefits of *RNAStructViz* in an
application note publication in 2019.

### 2018 -- 2019

Freelance work on Chameleon Mini libraries and firmware

*Paid freelance work stemming from my independent
work on the Chameleon Mini Live Debugger
Android application which controls a Chameleon mini
(RevG or RevE) device over the phone's USB port for
more convenient NFC debugging,
pentesting, and reconaissance work in the field.
The freelance gig required me to write
this library which comes complete with
this sample application to demonstrate how to
integrate the library in custom Android applications.
Continued freelance work from this contractor has involved
writing custom cryptographic routines and customizing the
Chameleon RevG firmware source in C++ for their
private real-world commercial applications.*

### Summer 2018

Home Brew Live Streamer Application for Android

*This is an open source application written in Java for live audio and video recording of media on Android devices. The genesis of this application is that the Smashing Pumpkins are coming to Atlanta in July of 2018 and I needed a reasonable reliable custom (roll-your-own) audio recording option to take to the show with me. Therefore, I decided to write my own solution so that I can verify exactly what the application is doing with the sensitive Android camera and media recording permissions used in a background service required in the implementation of this type of app. The Google Play Store Listing for the freeware application is here.*

### 2017-Current

Chameleon Mini Live Debugger

*This application is an open source Android interface to the semi-popular Chameleon Mini boards used for emulation of NFC tags, security reconaissaince and verification, and NFC testing written in Java. It allows for live logging of NFC events with the Rev. G Chameleon Mini boards via a connection to the device over serial USB. It also supports command output and testing with the Rev. E (rebooted) firmware of these boards. The application is available on the Google Play Store in both free and paid flavors. Current as of Summer 2018, the free flavor of the app on Play Store boasts approximately 150 active users worldwide (not bad considering that users must own an ~$160 piece of hardware to use the app to start with)!*

### 2016-2017

Tiling Gap Distributions and Pair Correlation Project

*Python source code for a set of spatial statistical routines and gap distributions which can be run on a large library of custom tilings of the plane. This project was funded and supervised by Jayadev Athreya at the University of Washington while I was on sabatical from graduate school from 2016-2017. The UW website containing plots of the results of our computations with this library of tilings on the Sage Math Cloud servers is available here. I also was a graduate TA for an honors special topics course based on this research in the Spring of 2017.*

## TALKS

### October 2018

Integers Conference 2018

*Recent Work on Jacobi-Type Continued Fractions* [beamer slides (pdf)]
*J-Fractions Mathematica Package: [Package (.m file)] [examples.nb]
*

*
* *
*
### Summer 2018

George Andrews 80th Birthday Conference

*New Connections Between Partitions and Multiplicative Functions* [beamer slides (pdf)]

### Fall 2017

Undergraduate seminar at Georgia Tech (Math)

*Partition Identities Related to Stanley's Theorem*

### Fall 2017

Association of Women in Mathematics at Georgia Tech

*Partition Identities Related to Stanley's Theorem* [talk slides (pdf)]

### Summer 2012

Young Mathematicians Conference at OSU

*Generalized j-Factorial Functions, Polynomials, and Applications*

*
* *
*## MISC DOCUMENTS

### Spring 2018

Modular Forms Course Project: [pdf | Sage source tarball]

*An introduction to ECC algorithms with Sage source implementation.*

### Spring 2018

Real Analysis II Convergence Roadmap: [pdf]

*A comparison of notions of convergence including convergence in measure, almost uniform convergence, convergence in mean, and Lp space convergence. See also this link for related information. *

*
*### Summer 2018 -- Spring 2019

*Comprehensive exam review notes, prep problems, and
problem solutions.*

*Surely as anyone who has been directed to these links is
aware, the GA Tech Math Ph.D. written
qualifying exams require months of study and
otherwise careful and thoughtful preparation.
It helped me to review for my comp exams by
typing and TeX'ing up a review of the subject material
and handwritten course notes from the previous
semesters. The following links provide my typeset
review notes packets, which are a good start, but are
certainly not complete coverage of these areas.
If you would like to modify my notes to add material
for your own studies, send me an email and I can
provide the working LaTeX source files at reasonable
request since some others (*ahem*) are not as willing to make their preparation guides freely available. Unofficially, these informal guides are released under an open-source Creative Commons license for non-commercial and personal usages.*
__Analysis exam review notes__ [pdf]
__Algebra exam review notes__ [pdf]