2794W — Resources

Welcome to the Resources Page of 2794W: Mathematics Writing Seminar, this Spring 2020! Today is .

Sample student papers

Magical Numbers and Where to Find Them: pdf, source code: .tex, .bib
Similarities Between Integers and Polynomials: pdf, souce code: .tex, .bib
Reverse Hex: pdf, .tex, .bib, Hexgraph.png, Hexgraph2.png, Hexgraph3.png, Ygraph.png
Sample Introduction Activity

Resources for paper topics and references

Videos Talk C (for those who need to make up comment forms/library research)

TALK SUBSTITUTE Watch at least two of these graph theory videos:
Possible topics and references: Topics in the video or any graph theory topics about graphs, directed graphs, trees, binary trees, and their applications
Electronic Textbooks with chapters on Graphs and Trees:
1. Applied Discrete Structures by Al Doerr and Ken Levasseur
2. Discrete Mathematics: An Open Introduction by Oscar Levin
3. Applied Combinatorics by Mitchel Keller and William Trotter
4. Combinatorics Through Guided Discovery Free Textbook by Kenneth P. Bogart: PDF
More graph theory videos
- By PBS Infinite Series List of videos: Graph Theory

Videos Talk B (for those who need to make up comment forms/ library research)

TALK SUBSTITUTE Watch at least two these:
Possible topics and references related to the videos
- Stern-Brocot trees, numbers, see Video Infinite Fractions by Numberphile
- Continued fractions (see both Numberphile videos). More videos on continued fraction: Video Infinite fractions and the most irrational number by Mathologer
- Lucas numbers and their formulas. Follow Jim Fowler’s video (above) explanation but start with 1 and 3 instead of 1 and 1.
- Facts about the Fibonacci/Pingala numbers
- Myths about the Fibonacci/Pingala sequences which are in fact false
- Fibonacci recurrences, limit of ratios
- Other famous numbers like pi and e. See video Can we combine pi and e to make a rational number? By PBS Infinite Series
- Ptolemy’s Theorem and its connection to pentagons and the Golden Ratio. See Numberphile’s videos38-minute video A Miraculous Proof (Ptolemy’s Theorem) 7-minute video Pentagons and the Golden Ratio both by Zvezdelina Stankova.
- Wikipedia pages on these topics
- Suggestions?

Video Talk A (for those who need to make up comment forms/ library research)

TALK SUBSTITUTE Watch both videos : Frieze Patterns made by Numberphile: part 1 and part 2

Speaker: Sergei Tabachnikov at Pennsylvania State University

Possible topics related to the videos
- Conway – Coxeter frieze patterns
- Triangulations of polygons
- One of the many problems given and answered in The Mathematical Gazette Part 1 (problems), Vol. 57, no. 400, pp. 87–94 and part 2 (solutions), Vol. 57, no. 401, pp. 175–183
- Video by PBS Infinite Series Associahedra: the shapes of multiplication
- Pick two Catalan objects from this Catalan numbers page. The Wikipedia entry for “Catalan number” also has a lot of nice pictures of some Catalan objects.
- Facts about Catalan numbers from pages linked from this Catalan numbers page, by Igor Pak.
- Bijections between Catalan objects. Some are given in these slides by Richard Stanley (just do ctrl+F or Apple-F for the word ”bijection”):
Related references:
- 2015 survey article “Coxeter’s frieze patterns at the crossroads of algebra, geometry and combinatorics” by Sophie Morier-Genoud
- 2014 Slides “Weighted walks around dissected polygons — Conway-Coxeter friezes and beyond” by Christine Bessenrodt
- 2006 Slides “The combinatorics of frieze patterns and Markoff numbers” by Jim Propp
- 2018 Slides “Friezes, triangulations, continued fractions, and binary numbers” from a talk I gave
- 1973 “Triangulated Polygons and Frieze Patterns” by John Horton Conway and Harold Scott MacDonald Coxeter in The Mathematical Gazette Part 1 (problems), Vol. 57, no. 400, pp. 87–94 and part 2 (solutions), Vol. 57, no. 401, pp. 175–183
- Any Catalan number resources from below would work, since frieze patterns are Catalan objects.
- Wikipedia pages on these topics
- Suggestions?

Week 8 (Wed, March 11) to be updated after the talk

Talk Title and Speaker: Rational Parametrizations by Robert McDonald (Yale, former UConn math student)

Possible topics related to the talk

Week 7 (Wed, March 4)

Talk Title and Speaker: Fourier Analysis by Alyssa Genschaw (UConn)

Possible topics related to the talk
- Power series, definition and examples (geometric series, p-series, and power series for exponential, log, trig functions)
- Infinite series (not power series) for certain famous values, like pi, cos(pi/3), sin(pi/3), e, log(2), i.
- Various convergence tests (for example, limit comparison test, divergence test)
- You can connect power series to Week 2 talk topics, using a power series as a generating function for a sequence of positive integers.
- Fourier series, definition and examples
- Applications of Fourier series to science (for example in MRI imaging, signal processing, audio compression)
- Using power series or Fourier series to solve a differential equations (for example, during the talk you saw a Fourier series used to solve a heat equation)
- Laplace transform of a function and its application (for example, to solve a differential equation)
- Fourier transform (the speaker didn’t have time to talk about Fourier transform)
- Using power series or Fourier series to approximate a number or a function
- Suggest a topic
Some suggested references:
- Video by PBS Infinite Series The Heat Equation
- A crash course in Fourier Analysis by Leo Goldmakher use @misc when you create a bibitem for this, since this is considered an unpublished online reference
- Stewart’s Calculus textbook Chapter 11 (Sequences and Series) for facts and applications of power series
- See Week 2 talk suggested references for using a power series as a generating function of a sequence of positive integers, for example, Jim Fowler’s video for computing a formula for the Fibonacci numbers using power series
- Look for good lecture videos on power series, for example by Jim Fowler has many commercial-free lecture videos on Sequences and Series.
- A differential equations textbook (if you have taken it) for using Laplace transform to solve a differential equation
- Google!

Week 6 (Wed, Feb 26, 2020)

Talk Title and Speaker: Random Harmonic Series by Iddo Ben-Ari (UConn), slides used during the talk

Possible topics related to the talk
- Application of the Harmonic Series (from Calculus 2) for building a one-sided bridge — references: Stewart’s Calculus book)
- Application of the geometric series, example: fractals such as Sierpinski carpet, L-systems, facts and examples, how they relate to the geometric series
- Condensation test, Dirichlet’s test (for testing convergence of a series)
- Random series, definition, known facts and open questions, examples
- Suggest a topic
References suggested by the speaker:
- Probability with Martingales by David Williams, physical copy available at the UConn library
- References linked from the speaker’s slides (in pink)
Additional references:
- Harmonic series application
  - watch 10-minute video How far can you build a one-sided bridge by Jim Fowler
  - Book Stacking Problem and references
  - Stewart’s Calculus textbook Section 11.2 Series
  - Stewart’s Calculus textbook (8th ed) Problems Plus, Question 12, page 789
- A faster way to build an infinite bridge
  - Video by PBS Series: Building an infinite bridge
  - Article Overhang by Paterson, Zwick
  - Article Maximum Overhang by Paterson, Peres, Thorup, Winkler, and Zwick
- Snowflake curve, the Cantor Set, the Sierpinski carpet (and their connection to the geometric series)
  - Stewart’s Calculus textbook (8th ed) Problems Plus, Question 5, page 788
  - Stewart’s Calculus textbook (8th ed) Section 11.2, Exercise 89, page 718
  - Wikipedia entry for fractal, entry for L-system

Some more fractal topics:
- Fractals, algorithms for creating fractals-like objects like Sierpinski triangle, Sierpinski carpet, Koch curve, and L-systems
- Dragon curves
- Fractals in Computer Science
- Others (talk to me first)
Some more fractal references:
- References at the bottom of Wikipedia pages (contact me to help check out the references for free)
- Fractals and algorithms for creating fractals-like objects:
  1. Khan Academy MITK12Videos: video What is a fractal and what are they good for?
  2. Math’s Beautiful Monsters: How a destructive idea paved the way for modern math by A. Kucharski on fractals
  3. Wikipedia pages for Sierpinski Triangle, Koch Curve, and L-systems (also usually included in a intro Computer Science textbook)
- Textbook A Friendly Introduction to Mathematical Logic by Leary and Kristiansen
- Articles on fractals and L-systems
  - Article Dragon curves revisited by Tabachnikov, The Mathematical Intelligencer, 2014
- Free textbooks or website for L-systems and fractals
  - Chapter 9 (L-systems) of the textbook Python Programming in Context by Ranum and Miller, 2nd Edition (go through UConn library website for free access)
  - The Algorithmic Beauty by Prusinkiewicz and LINDENMAYER, 1990, 1996
  - A short page on L-systems with Python and Turtles, from an intro computer science textbook

Week 5 (Wed, Feb 19, 2020)

Talk Title and Speaker: Volume of Balls: from R^n to Infinite Dimensions by Marco Carfagnini (UConn, PhD candidate)
Some possible topics related to the talk
- Insert the computation details skipped by the speaker
- Using Calculus to compute the volume of a two-dimensional unit ball and a three-dimensional unit ball (Ref: Multivariable/Vector Calculus textbooks below)
- Explain the computation of volumes of higher-dimensional balls (Speaker’s notes; you can also look for printed references)
- More advanced: When the radius of the ball is 1, the volumes decreases after n=5. When the radius of the ball is not 1, determine when the volumes start to decrease as the dimension increases. (Ref: Volume and surface area of an n-dimensional sphere; Speaker’s notes; you can also look for printed references)
- Definition, examples, and facts about metric spaces (Ref: see Real Analysis textbooks below)
- For advanced students only: The space of continuous functions, an infinite-dimensional space (see pages 7 and 8 of Speaker’s notes). Write the definition, examples, and facts about this. This topic falls under an area called “functional analysis”. (Ref: Try for example Lecture notes on the space of continuous functions)
- Suggestions?
References:
- Speaker’s notes
- Any multivariable calculus textbook. If you don’t own any, see AIM open-source textbooks, under Calculus
- Introduction to Real Analysis: download one of the five AIM open-source textbooks under Real Analysis, or borrow from my office
- Look up “functional analysis” from the library database, or use free resources, for example Lecture notes on the space of continuous functions
- Wikipedia pages on these topics
- Suggestions?

Week 4 (Wed, Feb 12, 2020)

Talk Title and Speaker: Stirling’s Formula by Matt Lamoureux (former UConn math student, now Director of Data Science at the Hartford)
Some possible topics related to Stirling’s Formula (for a short paper or a section in your final paper):
- Explain the Calculus computation skipped by the speaker
- Comparing function growth, asymptotic analysis for n! and functions which grow faster or slower than n!
- Applications of the Stirling’s Formula, for example, random walks
- Definition of the Gamma function and known facts about it
- Facts and computation related to the Gaussian integral (probability)
- Facts and computation related to the Wallis product
- Facts and computation related to the Bernoulli numbers
- Asymptotic growth of popular sequences, for example, the Catalan numbers
- Suggestions?
References about Stirling’s Formula:
- E. Hairer and G. Wanner, Analysis by its History, Springer-Verlag, 1996. (Section II.10)
- R. Graham, D. Knuth, O. Patashnik, Concrete Mathematics, Addison–Wesley, 1989.(Section 2.6)
- R. Michel, On Stirling’s Formula, Amer. Math. Monthly109(2002), 388–390.
- J. M. Patin, A Very Short Proof of Stirling’s Formula, Amer. Math. Monthly 96 (1989), 41–42
- Blog post: Removing the magic from Striling’s Formula
- Wikipedia entry for Stirling’s approximation
References for related topics:
- Khan Academy: Asymptotic notation, comparing function growth, etc Faster way to build an infinite bridge:
- A Calculus textbook
- References at the bottom of the wikipedia entries for the suggested topics mentioned above

Week 3 (Wed, Feb 5, 2020) suggested references

Talk: How to Win at Hat Guessing by Nate Eldredge (U of Northern Colorado, currently visiting UConn)
Notes used by the speaker: the speaker discussed several strategies to maximize the probability of guessing hats (two colors) correctly for finitely many players (first using volunteers), then for infinitely many players; speaker skipped the Blue-eyed islanders puzzle due to lack of time.
References suggested by the speaker, start with An introduction to infinite hat problems by Hardin and Taylor

Some possible topics (for a short paper or a section in the final paper):
- Strategies for other guessing games, see below for references
- Write an explanation and solution to Blue Eyes puzzle
- Suggestions are welcome?
Recreational math references:
- By Peter Winkler MAA YouTube 12-minute lecture “Larger or Smaller”, MAA article “Games people don’t play”
Popular culture references:
- Blue Eyes puzzle explanation by the webcomic xkcd
- NY Times article “Why mathematicians now care about their hat color” by Sara Robinson

Week 2 (Wed, Jan 29, 2020) suggested references

Talk: Combinatorial Triangles by Tom Roby (UConn)
Handout and oeis.org: The speaker covered Pingala–Khayyam–YangHui–Pascal Triangle and Eulerian Numbers
Some possible topics (for a short paper or a section in your final paper):
- Various definition of the Catalan numbers; objects counted by the Catalan numbers; recurrence relation and power series for Catalan numbers
- Objects counted by the Entringer numbers; recurrence relation and power series for Entringer numbers
- Same thing for numbers on the handout but not covered during the talk (Stirling numbers for set partitions, Stirling numbers for permutations, Eulerian numbers)
- Identities involving the binomial coefficients, their combinatorial proofs, and proofs using other methods
- Applications of permutations and binomial coefficients to statistics, probability, or other topics
- Connecting this topic to Calculus 2 (power series): Use a power series as a generating function of a sequence. For example, Jim Fowler’s video for computing a formula for the Fibonacci numbers using power series
- More advanced topic: Definition of toggling (mentioned in the talk) and the current mathematical research involving dynamical algebraic combinatorics — ask me or the speaker for more information and reading materials
- More advanced topic: A generalization of the PKYP Triangle, see the article Generalized Pascal triangle for binomial coefficients of words — ask me for more information and reading materials
- Suggestions?
References suggested by the speaker (you can check out from UConn library or Interlibrary Loan):
1. See the list of references at the end of the handout
2. If you like theoretical math: List of references the speaker likes on his Spring 2017 Graduate Combinatorics course
References for specific topics:
- Alternating permutations and Entringer number: by Richard Stanley, an article A survey of alternating permutations and slides Alternating permutations
- Applications of binomial coefficients to probability: OpenIntro Statistics, try Chapter 4: Distributions, but binomial coefficients really appear everywhere
- References for Catalan numbers:
  - Catalan Numbers by Richard Stanley and Appendix B: Catalan numbers by Igor Pak called History of Catalan Numbers, borrow from the library link or from my office
  - Many free articles about the sequence and its history in Catalan Numbers Page by Igor Pak
  - A textbook on generating functions Free download, Generatingfunctionology by Wilf
  - Catalan Number Slides by Richard Stanley
References on the Internet:
- Video by PBS Infinite Series Dissecting hypercubes with Pascal’s Triangle
- Video Lecture Enumerative Combinatorics Lecture 2 by Federico Ardila: We discuss two basic principles of enumeration: the multiplication and addition principle. We use them to count permutations, subsets, and compositions.
- Wikipedia entry for Pingala–Khayyam–Yang Hui–Pascal’s Triangle, including some history.
- Using a power series as a generating function of a sequence Jim Fowler’s video for computing a formula for the Fibonacci numbers using power series
Electronic Textbook References:
1. Applied Discrete Structures by Al Doerr and Ken Levasseur, try Section 2.4 “Combinations and the Binomial Theorem”
2. Discrete Mathematics: An Open Introduction by Oscar Levin, start with Chapter 1 “Counting”
3. Applied Combinatorics by Mitchel Keller and William Trotter, start with Chapter 2 “Strings, Sets, and Binomial Coefficients”
4. Combinatorics Through Guided Discovery Free Textbook by Kenneth P. Bogart: PDF, try Sec 1.2 “Basic Counting Principles” and Sec 1.3 “Some Applications of Basic Counting Principles”
5. OpenIntro Statistics
Physical textbooks you can borrow from MONT 402 for a few days:
1. Aspects of Combinatorics by Victor Bryant
2. Combinatorics: Topics, Techniques, and Algorithms by Cameron
3. Combinatorics: Guided Tour by David Mazur
4. A Walk Through Combinatorics by Miklos Bona
5. Catalan Numbers by Richard Stanley, or borrow from the library
6. Is the Golden Ratio Really All That? by Marlena Herman
Physical textbooks you can check out from the UConn library:
1. Combinatorics and Graph Theory by Harris, Hirst, and Mossinghoff (2010)
2. Introduction to Combinatorics by Erickson (1996)
3. A Walk Through Combinatorics by Miklos Bona (available at Course Reserve)

JSTOR

To download published books and journal articles that are not freely available online, you can search on MathSciNet:https://mathscinet-ams-org.ezproxy.lib.uconn.edu/mathscinet/ JSTOR:http://rdl.lib.uconn.edu/databases/1009;go (UConn login is required for both).
For unpublished preprints, use Google Scholar and the arXiv.org

Writing convention

AMS Style Guide (PDF)

Wikipedia:Manual of Style/Mathematics

Writing proofs

Advice on Mathematical Writing by K. Conrad

Examples of proofs by induction by K. Conrad

For research papers How to write a clear math paper: some 21st century tips

Jobs requiring math skills

Math students, connect with prospective employers here: MAA Career Resource Center

Miscellaneous

Use CamScanner app or another scan app to scan your hand-written homework into PDF.