A Restless Bandit Model for Resource Allocation, Competition, and Reservation

In “A Restless Bandit Model for Resource Allocation, Competition and Reservation,” J. Fu, B. Moran, and P. G. Taylor study a resource allocation problem with varying requests and with resources of limited capacity shared by multiple requests. This problem is modeled as a set of heterogeneous restless multi-armed bandit problems (RMABPs) connected by constraints imposed by resource capacity. Following Whittle’s idea of relaxing the constraints and Weber and Weiss’s proof of asymptotic optimality, the authors propose an index policy and establish conditions for it to be asymptotically optimal in a regime where both arrival rates and capacities increase. In particular, they provide a simple sufficient condition for asymptotic optimality of the policy and, in complete generality, propose a method that generates a set of candidate policies for which asymptotic optimality can be checked. Via numerical experiments, they demonstrate the effectiveness of these results even in the pre-limit case.

DIRECTIONS SETS: A GENERALISATION OF RATIO SETS

For every integer $k\geq 2$ and every $A\subseteq \mathbb{N}$, we define the $k$-directions sets of $A$ as $D^{k}(A):=\{\boldsymbol{a}/\Vert \boldsymbol{a}\Vert :\boldsymbol{a}\in A^{k}\}$ and $D^{\text{}\underline{k}}(A):=\{\boldsymbol{a}/\Vert \boldsymbol{a}\Vert :\boldsymbol{a}\in A^{\text{}\underline{k}}\}$, where $\Vert \cdot \Vert$ is the Euclidean norm and $A^{\text{}\underline{k}}:=\{\boldsymbol{a}\in A^{k}:a_{i}\neq a_{j}\text{ for all }i\neq j\}$. Via an appropriate homeomorphism, $D^{k}(A)$ is a generalisation of the ratio set$R(A):=\{a/b:a,b\in A\}$. We study $D^{k}(A)$ and $D^{\text{}\underline{k}}(A)$ as subspaces of $S^{k-1}:=\{\boldsymbol{x}\in [0,1]^{k}:\Vert \boldsymbol{x}\Vert =1\}$. In particular, generalising a result of Bukor and Tóth, we provide a characterisation of the sets $X\subseteq S^{k-1}$ such that there exists $A\subseteq \mathbb{N}$ satisfying $D^{\text{}\underline{k}}(A)^{\prime }=X$, where $Y^{\prime }$ denotes the set of accumulation points of $Y$. Moreover, we provide a simple sufficient condition for $D^{k}(A)$ to be dense in $S^{k-1}$. We conclude with questions for further research.

Stable Polarized del Pezzo Surfaces

We give a simple sufficient condition for $K$-stability of polarized del Pezzo surfaces and for the existence of a constant scalar curvature Kähler metric in the Kähler class corresponding to the polarization.

A simple sufficient condition for triviality of obstructions in the orbifold construction for subfactors

We present a simple sufficient condition for triviality of obstructions in the orbifold construction. As an application, we can show the existence of subfactors with principal graph $D_{2n}$ without full use of Ocneanu's paragroup theory.

A Note on the Isomorphism Problem for Monomial Digraphs

Let p be a prime e be a positive integer, [Formula: see text], and let [Formula: see text] denote the finite field of q elements. Let [Formula: see text], [Formula: see text], be integers. The monomial digraph [Formula: see text] is defined as follows: the vertex set of D is [Formula: see text], and [Formula: see text] is an arc in D if [Formula: see text]. In this note we study the question of isomorphism of monomial digraphs [Formula: see text] and [Formula: see text]. Several necessary conditions and several sufficient conditions for the isomorphism are found. We conjecture that one simple sufficient condition is also a necessary one.