5 papers
Hitting-time mixing for the star transposition shuffle
Vanshika Jain, Evita Nestoridi
We prove a hitting-time analogue of cutoff for the star transposition shuffle on the symmetric group S_n. Let tau be the first time at which every non-top card has been selected. W…
A Note on Congruences for Weakly Holomorphic Modular Forms
Spencer Dembner, Vanshika Jain
Let be the ring of integers of a number field . Write , and suppose that $$f(z) = \sum_{n \gg - \infty}^{\infty} a_f(n) q^n \in M_{k}^{!}(\operatorname{SL}…
Hyperelliptic Curves and Newform Coefficients
Spencer Dembner, Vanshika Jain
We study which integers are admissible as Fourier coefficients of even integer weight newforms. In the specific case of the tau-function, we show that for all odd primes $\ell < 10…
Hyperbolicity of Appell Polynomials of Functions in the -Laguerre-Pólya Class
Jonas Iskander, Vanshika Jain
We present a method for proving that Jensen polynomials associated with functions in the -Laguerre-Pólya class have all real roots, and demonstrate how it can be used to constru…
Exact Formulae for the Fractional Partition Functions
Jonas Iskander, Vanshika Jain, Victoria Talvola
The partition function has been a testing ground for applications of analytic number theory to combinatorics. In particular, Hardy and Ramanujan invented the "circle method"…