User:Mustazee
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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)
Solving Non-homogeneous Nested Recursions Using Trees
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.
Spot-Based Generations for Meta-Fibonacci Sequences
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.
A Combinatorial interpretation of Hofstadter's G-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.
On Variants of Conway and Conolly's Meta-Fibonacci Recursions (EJC link)
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:
