Maxie D. Schmidt
Graduate Student and Math Enthusiast


RESEARCH INTERESTS

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.

EDUCATION

2017-Present

Georgia Institute of Technology
Ph.D. Candidate

2014

University of Illinois at Urbana-Champaign
Master of Science in Computer Science (MS)

2012

University of Illinois at Urbana-Champaign
Bachelor of Science in Mathematics (BS, LAS)

2012

University of Illinois at Urbana-Champaign
Bachelor of Science in Computer Science (BS, ENG)

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

2021

(21)

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)

2021

(20)

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.

2020

(19)

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.

2019

(18)

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.

2019

(17)

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.

2018

(16)

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.

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
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.

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
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.

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
Published in the American Mathematical Monthly With Merca Merca

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 Supervised by Professor Roy Campbell

2010

(1)

Generalized j-factorial functions, polynomials, and applications
Published in the Journal of Integer Sequences

MANUSCRIPTS



02/2021

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.

04/2020

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.

08/2017

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.

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.

TALKS AND CONFERENCES



February 2021

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]

August 2020

Ph.D. oral qualifying exam on August 14, 2020.
[Presentation slides (pdf)]   [Link to recording (video)]

November 2019

AMS Fall Southeastern Sectional Meeting Invited Speaker
Computational Aspects of Factorization Theorems for Generating Special Sums [beamer slides (pdf)]

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)]

Fall 2017

AMS Club Seminar
Sage [talk overview]

Summer 2012

Young Mathematicians Conference at OSU
Generalized j-Factorial Functions, Polynomials, and Applications

RECENT SOFTWARE

2018 - Present (2021)

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.

2020

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.

2018 - 2019

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++.

2018-2019

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.

2017-Present (2021)

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.

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

MISC DOCUMENTS

2018 - 2020

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]

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.

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