16 papers
The Mathieu Property for Compact Connected Lie Groups
Christopher D. Long
Let be a compact connected Lie group, let denote its algebra of representative functions, and let be normalized Haar…
An Explicit Counterexample to the Rank-Two Poisson Conjecture
Christopher D. Long
Let \[ {\mathcal P}_2={\mathbb C}[x,q,p,z] \] carry the canonical Poisson bracket determined by and by the vanishing of the other brackets between distinct ge…
Counterexamples to the xz-Conjecture and the Mathieu Conjecture for SU(2)
Christopher D. Long
Let \[ {\mathcal I}(h)=\int_0^1\int_{\mathbb T}h(x,z)\,\frac{dz}{2πiz}\,dx \qquad \bigl(h\in{\mathbb C}[x,z,z^{-1}]\bigr). \] We give the three-term Laurent polynomial \[ f(x,z)=(1…
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 ev…
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…
The Ballot Event for Two-Player Coupon Collection: A Renewal--Catalan Asymptotic
Christopher D. Long
We study the two-player coupon-collector competition in which two independent collectors draw one coupon each per round from a set of equally likely coupon types. Myers and Wil…
The Expiring Coupon Collector: Sliding-Window Surjection Flux and Rare-Entry Laws
Christopher D. Long
We study the coupon collector with deterministic expiration: one coupon is drawn at each time, and each coupon remains active for exactly draws. Completion occurs when all …
Terminal Defects, Growing Multiplicity, and Variance Extremality in the Double Dixie Cup Problem
Christopher D. Long
We develop a terminal-defect method for the double Dixie cup problem and use it to prove the finite-variance extremality conjecture of Doumas and Papanicolaou. For every \(m\ge1\)…
Exact Finite-Horizon Quantile Kelly for Repeated Multi-Outcome Events
Christopher D. Long
We formulate and prove an exact finite-horizon quantile theorem for repeated identical multi-outcome Kelly wagering in wealth-profile / Arrow--Debreu coordinates. For a fixed -o…
Risk-Constrained Kelly for Mutually Exclusive Outcomes: CRRA Support Invariance and Logarithmic One-Dimensional Calibration
Christopher D. Long
We study the finite mutually exclusive outcome version of risk-constrained Kelly optimization with explicit state prices. The market has outcome probabilities , state prices…
Optimal Parlay Wagering and Whitrow Asymptotics: A State-Price and Implicit-Cash Treatment
Christopher D. Long
For independent multi-outcome events under multiplicative parlay pricing, we give a short exact proof of the optimal Kelly strategy using the implicit-cash viewpoint. The proof is…
Utility-Invariant Support Selection and Eventwise Decoupling for Simultaneous Independent Multi-Outcome Bets
Christopher D. Long
For simultaneous independent events with finitely many outcomes, consider the expected-utility problem with nonnegative wagers and an endogenous cash position. We prove a short sup…
Single-Event Multinomial Full Kelly via Implicit State Positions
Christopher D. Long
For a single event with finitely many mutually exclusive outcomes, the full Kelly problem is to maximize expected log wealth over nonnegative stakes together with an optional cash…