# Maxie D. Schmidt

Graduate Student and Math Enthusiast

- School of Mathematics
- Georgia Institute of Technology
- Atlanta, GA
- OrcID Author Profile
- GitHub Software Page

Graduate Student and Math Enthusiast

- School of Mathematics
- Georgia Institute of Technology
- Atlanta, GA
- OrcID Author Profile
- GitHub Software Page

My research interests are primarily in combinatorial and analytic number theory and enumerative combinatorics with an emphasis on generating function methods, continued fractions, software development, and experimental mathematics.
I am always open to exploring new interesting problems.

Georgia Institute of Technology

*Ph.D. Candidate*

University of Illinois at Urbana-Champaign

*Master of Science in Computer Science (MS)*

University of Illinois at Urbana-Champaign

*Bachelor of Science in Mathematics (BS, LAS)*

University of Illinois at Urbana-Champaign

*Bachelor of Science in Computer Science (BS, ENG)*

Northwest Missouri State University

*Associate of Science (MASMC)*

NSF GRFP National Honorable Mention

NSF GRFP National Honorable Mention

Diffenbaugh Graduate Fellowship at UIUC

Barry M. Goldwater Scholarship

RNAStructViz: Graphical base pairing analysis

*Accepted for publication in Bioinformatics*
*With Anna Kirkpatrick and Christine Heitsch*
__Full Text:__
BioRxiv (10.1101/2021.01.20.427505v1)__Software Documentation:__
RNAStructViz WIKI (on GitHub)

Exact Formulas for the Generalized Sum-of-Divisors Functions

*Accepted for publication in INTEGERS, The Electronic Journal of Combinatorial Number Theory, in
December of 2020.*
__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.

Combinatorial Sums and Identities Involving Generalized Divisor Functions with Bounded Divisors

*Published in INTEGERS*
__Full Text:__ https://arxiv.org/abs/1704.05595 (math.NT)__Full Text:__
INTEGERS (Volume 20 -- A85 -- 2020)__Keywords:__ divisor function; sum of divisors function; Lambert series.__MSC (2010):__ 30B50; 11N64; 11B37; 11B83; 11K65.

Factorization Theorems for Relatively Prime Divisor Sums, GCD Sums and Generalized Ramanujan Sums

*Published in the Ramanujan Journal*
*With Hamed Mousavi.*
__Full Text:__ https://arxiv.org/abs/1810.08373 (math.NT)__Springer Link:__ Article (10.1007/s11139-020-00323-5)__Keywords:__ divisor sum; totient function; matrix factorization; Möbius inversion; partition function.__MSC (2010):__ 11N64; 11A25; 05A17.

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

*Published in Axioms (Special Issue: Mathematical Analysis and Applications II)*
__Full Text:__ https://arxiv.org/abs/1809.03933 (math.CO)__MDPI Journal Listing:__ Axioms 2019, 8(2), 62__Keywords:__ Generating function; series transformation; gamma function; Hankel contour.__MSC (2010):__ 05A15; 30E20; 31B10; and 11B73.

Factorization Theorems for Generalized Lambert Series and Applications

*Published in the Ramanujan Journal.*
*Joint work with Mircea Merca.*
__Full Text:__ https://arxiv.org/abs/1712.00611__Keywords:__ Lambert series; factorization theorem; matrix factorization; partition function; multiplicative function.__MSC (2010):__ 11A25; 11P81; 05A17; 05A19.

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)

New Congruences and Finite Difference Equations for Generalized Factorial Functions

*Published in INTEGERS*
__Full Text:__ https://arxiv.org/abs/1701.04741 (math.CO)__Full Text:__ Volume 18 -- A78 -- 2018__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.

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.*

Generating Special Arithmetic Functions by Lambert Series Factorizations

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

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)

A Partition Identity Related to Stanley's Theorem

*Published in the American Mathematical Monthly*
*With Merca Merca*

__Journal Listing:__DOI (10.1080/00029890.2018.1521232)__PDF Preprint:__preprint link.__Keywords:__Lambert series; Euler’s totient; partition.__MSC (2010):__11A25; 11P81; 05A17; 05A19.

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.

Continued Fractions for Square Series Generating Functions

*Published in the Ramanujan Journal.*

__Full Text:__https://arxiv.org/abs/1612.02778 (math.NT)__DOI Link:__http://link.springer.com/article/10.1007/s11139-017-9971-9__Keywords:__square series; q-series; J-fraction; continued fraction; sum of squares functions; sum of divisors function; theta function; ordinary generating function.__MSC (2010):__05A15; 11Y65; 11B65; 40A15.*Temporary Link to the Author's Offprint:*available here

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

*Published in Acta Arithmetica.*

__Full Text:__https://arxiv.org/abs/1701.06257 (math.NT)

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

*Published in the Journal of Number Theory*

__Full Text:__Math.NT/1704.05200__DOI Link:__https://doi.org/10.1016/j.jnt.2017.05.023__Keywords:__divisor function; sum of divisors function; continued fraction; J-fraction.__MSC (2010):__11J70; 11Y65; 40A30; 11B65; 11A25.

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

*Published in the Online Journal of Analytic Combinatorics*

__ArXiv Link:__Math.CO/1610.09666__Journal Link:__journal website

Square Series Generating Function Transformations

*Published in the Journal of Inequalities and Special Functions*

__ArXiv Link:__Math.NT/1609.02803__Journal Link:__full-text article__MSC (2010):__05A15; 44A99; 33E99; 11B73.

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

*Published in the Journal of Integer Sequences*

__ArXiv Link:__Math.CO/1610.09691__Journal Link:__abstract__Scopus Link:__summary__MathSciNet Reference:__MR3612314__MSC (2010):__05A10; 05A15; 11A55; 11Y55; 11Y65; 11B65.

A Computer Algebra Package for Polynomial Sequence Recognition

*UIUC Master's Thesis*
*Supervised by Professor Roy Campbell*

__ArXiv Link:__Math.CO/1609.07301__Illinois Ideals:__article link__Software:__GitHub repository__In Print:__LAP Publishing site (link)

New characterizations of the summatory function of the Möbius function

__Full Text:__ arXiv/2102.05842 (math.NT)__Software Reference:__ Mertens Function Computations (GitHub)__Keywords:__ Möbius function; Mertens function; Dirichlet inverse; Liouville lambda function; prime omega function; prime counting function; Dirichlet generating function; Erdős-Kac theorem; strongly additive function.__MSC (2010):__ 11N37; 11A25; 11N60; 11N64; and 11-04.

A catalog of interesting and useful Lambert series identities

__Full Text:__ https://arxiv.org/pdf/2004.02976.pdf (math.NT, math.HO)__Keywords:__ Lambert series; Lambert series generating function; divisor sum; Anderson-Apostol sum; Dirichlet convolution; Dirichlet inverse; summatory function; generating function transformation; series identity; arithmetic function.__MSC (2010):__ 05A15, 11Y70, 11A25, and 11-00.

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

__Full Text:__ https://arxiv.org/abs/1712.00608 (math.NT)__Keywords:__ Lambert series; factorization theorem; matrix factorization; partition function; Hadamard product.__MSC (2010):__ 11A25; 11P81; 05A17; 05A19.

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.

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.

GA Tech Algebra Seminar

*Defining canonically best factorization theorems for the generating functions of
special convolution type sums*
[Slides -- no notes (pdf)]
[Slides -- with notes (pdf)]
[Talk summary]

Ph.D. oral qualifying exam on August 14, 2020.

[Presentation slides (pdf)]
[Link to recording (video)]

AMS Fall Southeastern Sectional Meeting Invited Speaker

*Computational Aspects of Factorization Theorems for Generating Special Sums*
[beamer slides (pdf)]

Integers Conference 2018

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

George Andrews 80th Birthday Conference

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

Undergraduate seminar at Georgia Tech (Math)

*Partition Identities Related to Stanley's Theorem*

Association of Women in Mathematics at Georgia Tech

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

AMS Club Seminar

*Sage* [talk overview]

Young Mathematicians Conference at OSU

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

Mathematical biology group visualization code

*Continued work with the Georgia Tech
Discrete Mathematics and Molecular Biology research group.
Recent work with the group and their
growing list of software contributions.
This includes writing major updates to the RNAStructViz
application in C/C++, re-writing the front and backend code for RNADB in PHP/JavaScript/Python, and
writing the web-based GUI for the profiling software project currently under development by gtDMMB.
*

DESFire NFC tag support for the Chameleon Mini

*
The development of this challenging software project was partially supported by
funding in the GA Tech School of Math.
It finalizes an implementation the long awaited
support for the DESFire NFC tag
specification which is now
included in the official sources
for the Chameleon Mini (RevG) device firmware.
The DESFire tag support documentation
is found here
including documentation for new Chameleon terminal commands that allow users to
clone any DESFire tag
including parameters for setting ATS bytes, UID, and other low-level factory specific
hardware settings.
*

Freelance work on Chameleon Mini Android 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.
Work includes
Java-based libraries (Mifare Classic Tool Library)
developed for the Android OS platform.
Also implemented custom cryptographic routines and customizing the
Chameleon RevG firmware
source in embedded AVR C/C++.
*

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 played a rockin three+ hour concert in Atlanta in July of 2018. I needed a reasonable reliable custom (roll-your-own) audio recording option to take to the show with me. The Google Play Store Listing for the freeware application is here. The application allows posting live feeds to both YouTube and Facebook without using the standard apps as an interface.*

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 Spring 2021, the free flavor of the app on Play Store boasts approximately 500 active users worldwide.
*

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 and supervised instructor for an honors special topics course based on this research in the Spring of 2017.*

*These informal guides are released under an open-source Creative Commons license for non-commercial and personal uses.*
__Analysis review notes__ [pdf]
__Probability review notes__ [pdf]

__Algebra review notes__ (*annotation of errors due to Y. Rahimi*) [pdf]

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

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

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.
*