User:Mustazee

Mustazee Rahman

• Graduate student, Department of Mathematics, University of Toronto,
• 40 St. George St., Toronto, Ontario M5S 2E4, Canada
• e-mail: mustazee [dot] rahman [at] utoronto [dot] ca

My research interests are in combinatorics and probability. My advisor is Bálint Virág.

Papers (most are preprints fetched from the arxiv)

Abraham Isgur, Mustazee Rahman, Stephen Tanny

The solutions to certain nested recursions, such as Conolly's $C(n) = C(n-C(n-1))+C(n-1-C(n-2))$, with initial conditions $C(1)=1, C(2)=2$, have a well-established combinatorial interpretation in terms of counting leaves in an infinite binary tree. This tree-based interpretation, and its generalization to a similar $k$-term nested recursion, only apply to homogeneous recursions and only solve each recursion for one set of initial conditions determined by the tree. In this paper, we extend the tree-based interpretation to solve a non-homogeneous version of the $k$-term recursion that includes a constant term. To do so we introduce a tree-grafting methodology that inserts copies of a finite tree into the infinite $k$-ary tree associated with the solution of the corresponding homogeneous $k$-term recursion. This technique also solves the given non-homogeneous recursion with various sets of initial conditions.

Barnaby Dalton, Mustazee Rahman, Stephen Tanny

For many meta-Fibonacci sequences it is possible to identify a partition of the sequence into successive intervals (sometimes called blocks) with the property that the sequence behaves similarly" in each block. This partition provides insights into the sequence properties. To date, for any given sequence, only ad hoc methods have been available to identify this partition. We apply a new concept - the spot-based generation sequence - to derive a general methodology for identifying this partition for a large class of meta-Fibonacci sequences. This class includes the Conolly and Conway sequences and many of their well-behaved variants, and even some highly chaotic sequences, such as Hofstadter's famous $Q$-sequence.

Mustazee Rahman

We give a combinatorial interpretation of a classical meta-Fibonacci sequence defined by $G(n) = n - G(G(n-1))$ with the initial condition $G(1) = 1$, which appears in Hofstadter's "Gödel, Escher, Bach: An Eternal Golden Braid". The interpretation is in terms of an infinite labelled tree. We then show a couple of known properties of the sequence $G(n)$ directly from the interpretation.

Abraham Isgur and Mustazee Rahman

We study the recursions $A(n) = A(n-a-A^k(n-b)) + A(A^k(n-b))$ where $a \geq 0$, $b \geq 1$ are integers and the superscript $k$ denotes a $k$-fold composition, and also the recursion $C(n) = C(n-s-C(n-1)) + C(n-s-2-C(n-3))$ where $s \geq 0$ is an interger. We prove that under suitable initial conditions the sequences $A(n)$ and $C(n)$ will be defined for all positive integers, and be monotonic with their forward difference sequences consisting only of 0 and 1. We also show that the sequence generated by the recursion for $A(n)$ with parameters $(k,a,b) = (k,0,1)$, and initial conditions $A(1) = A(2) = 1$, satisfies $A(E_n) = E_{n-1}$ where $E_n$ is a generalized Fibonacci recursion defined by $E_n = E_{n-1} + E_{n-k}$ with $E_n = 1$ for $1 \leq n \leq k$.

Coursework on the wiki

An expository article written for Larry Guth's course in analysis: Roth's theorem on three-term arithmetic progressions.

Presentation slides and supplementary notes prepared for Pierre Milman's undergrad seminar course: An Introduction to Degree of Smooth Maps. Supplementary notes.

Notes on the restriction theorem of Stein and Tomas for Jim Colliander's PDE course in 2011: The Stein Tomas restriction theorem.

Various things from past summers

Write-ups for Jim Colliander's summer reading course:

Study group: