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Simple proofs of open problems about the structure of involutions in the Riordan group
AbstractWe prove that if D=(g(x),f(x)) is an element of order 2 in the Riordan group then g(x)=±exp[Φ(x,xf(x)] for some antisymmetric function Φ(x,z). Also we prove that every element of order 2 in the Riordan group can be written as BMB-1 for some element B and M=(1,-1) in the Riordan group. These proofs provide solutions to two open problems presented by L. Shapiro [L.W. Shapiro, Some open questions about random walks, involutions, limiting distributions and generating functions, Adv. Appl. Math. 27 (2001) 585–596]
IST Austria Technical Report
DEC-POMDPs extend POMDPs to a multi-agent setting, where several agents operate in an uncertain environment independently to achieve a joint objective. DEC-POMDPs have been studied with finite-horizon and infinite-horizon discounted-sum objectives, and there exist solvers both for exact and approximate solutions. In this work we consider Goal-DEC-POMDPs, where given a set of target states, the objective is to ensure that the target set is reached with minimal cost. We consider the indefinite-horizon (infinite-horizon with either discounted-sum, or undiscounted-sum, where absorbing goal states have zero-cost) problem. We present a new method to solve the problem that extends methods for finite-horizon DEC- POMDPs and the RTDP-Bel approach for POMDPs. We present experimental results on several examples, and show our approach presents promising results
On Dirichlet-to-Neumann Maps, Nonlocal Interactions, and Some Applications to Fredholm Determinants
We consider Dirichlet-to-Neumann maps associated with (not necessarily
self-adjoint) Schrodinger operators describing nonlocal interactions in
, , where is an open set with a compact,
nonempty boundary satisfying certain regularity conditions. As an application
we describe a reduction of a certain ratio of Fredholm perturbation
determinants associated with operators in to Fredholm
perturbation determinants associated with operators in . This leads to an extension of a variant of a celebrated
formula due to Jost and Pais, which reduces the Fredholm perturbation
determinant associated with a Schr\"odinger operator on the half-line
, in the case of local interactions, to a Wronski determinant of
appropriate distributional solutions of the underlying Schrodinger equation.Comment: 18 page
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