@TRA_
Talks
Subword-Counting Sequences
Subword-Counting Sequences
Talk at One World Seminar on Combinatorics on words, May 20 2025.Seminar pages: http://www.i2m.univ-amu.fr/wiki/Combinatorics-on-Words-seminar/https://resear...
Hairy Ball Theorem
Hairy Ball Theorem
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Papers
The Basel Problem and Sums of Rational Functions over Intergers
The Basel Problem and Sums of Rational Functions over Intergers
We provide a general method to evaluate convergent sums of the form ∑_{k∈Z} R(k) where R is a rational function with complex coefficients. The method is entirely elementary and does not require any calculus beyond some standard limits and convergence criteria. It is inspired by a geometric solution to the famous Basel Problem given by Wästlund (2010), so we begin by demonstrating the method on the Basel Problem to serve as a pilot application. We conclude by applying our ideas to prove Euler’s factorisation for sin x which he originally used to solve the Basel Problem.
Subword-Counting Sequences
Subword-Counting Sequences
Let $w$ be a finite word over the alphabet $\{0,1\}$. For any natural number $n$, let $s_w(n)$ denote the number of occurrence of $w$ in the binary expansion of $n$ as a scattered subsequence. We study the behavior of the partial sum $\sum_{n=0}^N(-1)^{s_w(n)}$ and characterize several classes of words $w$ satisfying $\sum_{n=0}^N(-1)^{s_w(n)}= O(N^{1-ε})$ for some $ε>0$.
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