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

AWARDS AND HONORS



2014

NSF GRFP National Honorable Mention

2013

NSF GRFP National Honorable Mention

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

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.

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