We show that there are universal positive constants c and C such the mixing time τmix for the fifteen puzzle in an n× n torus satisfies cn logn ≤ τmix ≤ Cn log n.

Department of Earth, Environmental and Planetary Sciences, Rice University, Houston, TX 77005, USA. Department of Earth, Planetary, and Space Sciences, University of California, Los Angeles, Los Angeles, CA 90089, USA. Department of Integrative Biology and Department of Energy Great Lakes Bioenergy Research Center, Michigan State University, East Lansing, MI 48824, USA. Department of Earth and ...

Informally, a set of abstractions of a state space S is additive if the distance between any two states in S is always greater than or equal to the sum of the corresponding distances in the abstract spaces. The first known additive abstractions, called disjoint pattern databases, were experimentally demonstrated to produce state of the art performance on certain state spaces. However, previous ...

We study a construction of the Mathieu group M12 using a game reminiscent of Loyd’s “15-puzzle”. The elements of M12 are realized as permutations on 12 of the 13 points of the finite projective plane of order 3. There is a natural extension to a “pseudogroup” M13 acting on all 13 points, which exhibits a limited form of sextuple transitivity. Another corollary of the construction is a metric, a...

A novel route is proposed for the preparation of mesopore containing zeolite ZSM-5 via in situ hydrothermal treatment of a solution containing alkali-dissolved SBA-15 containing carbonized surfactant P123 in the mesopores; it exhibited prominent stability enhancement for methanol to propylene reaction.

The successful in situ synthesis of metallophthalocyanines in mesoporous matrices is presented in this report. To overcome the high hydrophobicity of phthalocyanine (Pc) compounds in conventional wet chemical methods, a simple but effective route is developed to incorporate the metallophthalocyanines into the mesoporous matrix via an in situ reaction process between pre-loaded metal ions and 1,...

A novel type of material encapsulating phase-change materials (PCMs) is reported concerning their implication for use as thermal energy storage devices. The composites of siliceous SBA-15 or carbonaceous CMK-3 mesoporous assemblies and organic PCMs could be used to make leak-free devices that retain their capabilities over many thermal cycles for heat storage/release. A confinement effect was o...

We study a construction of the Mathieu group M12 using a game reminiscent of Loyd’s “15-puzzle”. The elements of M12 are realized as permutations on 12 of the 13 points of the finite projective plane of order 3. There is a natural extension to a “pseudogroup” M13 acting on all 13 points, which exhibits a limited form of sextuple transitivity. Another corollary of the construction is a metric, a...