314 – Suggested reading

This is a (somewhat expanding) list of suggested additional references. Some cover topics discussed in lecture, others add new material that complements what we covered. The level varies: Some are basic, others are more advanced and portions of them may require knowledge beyond this course.

• Neil Calkin, and Herbert S. Wilf. Recounting the rationals. Amer. Math. Monthly, 107 (4), (2000), 360–363. MR1763062 (2001d:11024).
• Aimeric Malter, Dierk Schleicher, and Don Zagier. New looks at old number theory. Amer. Math. Monthly, 120 (3), (2013), 243–264. MR3030296.
• Marion Scheepers. On the Pringsheim rearrangement theorems. J. Math. Anal. Appl., 267 (2), (2002), 418–433. MR1888013 (2003j:40002).
• Zbigniew Nitecki. Subsum Sets: Intervals, Cantor Sets, and Cantorvals. ArXiv: 1106.3779.
• R. John Ferdinands. Selective sums of an infinite series. Math. Mag., 88 (3), (2015), 179–185.
• S.B. Russ. A translation of Bolzano’s paper on the intermediate value theorem. Historia Math., 7 (2), (1980), 156–185. MR0572277 (82c:01072).
• Julian F. Fleron. A note on the history of the Cantor set and Cantor function. Math. Mag., 67 (2), (1994), 136–140. MR1272828 (95c:01019).
• O. Dovgoshey, O. Martio, V. Ryazanov, and M. Vuorinen. The Cantor function. Expo. Math., 24 (1), (2006), 1–37. MR2195181 (2006k:26005).
• Keith Burns, and Boris Hasselblatt. The Sharkovsky theorem: a natural direct proof. Amer. Math. Monthly, 118 (3), (2011), 229–244. MR2800333 (2012f:37088).
• Andrew M. Bruckner, J.L. Leonard. Derivatives. Amer. Math. Monthly, 73 (4), part II, (1966), 24–56. MR0197632 (33 #5797).
• Andrew M. Bruckner. Derivatives: Why they elude classification. Math. Mag., 49 (1), (1976), no. 1, 5–11. MR0393378 (52 #14188).
• Brian R. Hunt. The prevalence of continuous nowhere differentiable functions. Proc. Amer. Math. Soc., 122 (3), (1994), 711–717. MR1260170 (95d:26009).
• Ivan Niven. Formal power series. Amer. Math. Monthly, 76, (1969), 871–889. MR0252386 (40 #5606).
• Warren P. Johnson. The curious history of Faà di Bruno’s formula. Amer. Math. Monthly, 109 (3), (2002), 217–234. MR1903577 (2003d:01019).
• E.J. McShane. A unified theory of integration. Amer. Math. Monthly, 80, (1973), 349–359. MR0318434 (47 #6981).

For the group project: Choose one of these articles. Inform me by email, to make sure it has not already been chosen. Feel free to suggest different papers or other topics, I’ll see whether we can use them.

Write (type) a note on the topic discussed in the paper you have chosen, include details of some of the results discussed there. Make sure the proofs you include contain all needed details (typically proofs in articles are more sketchy than what we are aiming for through the course), and that the write up is your own, even if modeled on the arguments in the paper. Include references as usual. Turn this in by Thursday, May 15, at 10:30 am. Feel free to turn it in earlier, of course. I encourage you, as you work through the paper, to share your progress with me during office hours, so I can give you some feedback before your final submission.

Groups:

• Booker Ahl, Dorthee Berman, and Stephanie Potter: Russ’s translation of Bolzano’s paper.
• Tim Deidrick, Justin Durflinger, and Ariel Farber: Calkin-Wilf and Malter-Schleicher-Zagier on enumerating the rationals.
• Carrie Smith, and Jordan Wilson: Fleron’s note on the history of the Cantor set and function.
• Caleb Richards, and Chris VanDerhoff: McShane’s paper on the Henstock–Kurzweil integral.
• Kenny Ballou, Sarah Devore, and Luke Warren: Nitecki’s paper on subseries.
• Farrghun Abdulrahim, and Kenneth Coiteux: Burns and Hasselblatt’s paper on Sharkovsky’s theorem.
• Tyler Clark: Niven’s paper on formal power series.
• Joe Magdaleno, and Piper Gutridge: Bruckner and Bruckner-Leonard  on derivatives.

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