14 papers
Counterexamples to the xz-Conjecture and the Mathieu Conjecture for SU(2)
Christopher D. Long
Let \[ {\mathcal I}(h)=\int_0^1\int_{\mathbb T}h(x,z)\,\frac{dz}{2Ïiz}\,dx \qquad \bigl(h\in{\mathbb C}[x,z,z^{-1}]\bigr). \] We give the three-term Laurent polynomial \[ f(x,z)=(…
Small Counterexamples to the Gaussian Moments Conjecture
Christopher D. Long
We give explicit complex polynomials in three independent standard real Gaussian variables such that \[ {\mathbb E}(P^m)=0,\qquad {\mathbb E}(QP^m)=m!\neq0 \] for every $m\ge…
Radial Transform Extremality for the Siblings of the Coupon Collector
Christopher D. Long
In the siblings version of the coupon collector, a main collector stops when every coupon type has appeared once. Duplicates are passed successively to siblings, and denote…
Extremality and Limit Laws for the Siblings of the Coupon Collector
Christopher D. Long
We study the siblings version of the coupon collector problem. A main collector stops when every coupon type has appeared at least once, duplicates are passed successively to later…
Radial Extremality for LRU Caching and the Fill--Holst Conjecture
Christopher D. Long
For the independent reference model with popularity vector , let denote the exact stationary hit rate of an LRU cache of capacity . We prove that, for e…
Clumsy and Careless: Stationary-Entry Flux in Non-monotone Coupon Collectors
Christopher D. Long
We study three nonmonotone coupon-collector models through a stationary-entry viewpoint. In such models the all-present state is not absorbing, so completion is governed not by the…