We consider the Laplacian in a strip R(0,d) with the boundary condition which is Dirichlet except at the segment of a length 2a of one of the boundaries where it is switched to Neumann. This operator is known to have a non-empty and simple discrete spectrum for any a>0. There is a sequence 0<a1<a2< of critical values at which new eigenvalues emerge from the continuum when the Neumann window expands. We find the asymptotic behavior of these eigenvalues around the thresholds showing that the gap is in the leading order proportional to (aan)2 with an explicit coefficient expressed in terms of the corresponding threshold-energy resonance eigenfunction.

Suppose<i>X</i>and<i>Y</i>are locally compact Hausdorff spaces,<i>E</i>and<i>F</i>are Banach spaces, and<i>F</i>is strictly convex. We show that every linear isometry<i>T</i>from<i>C</i>₀(<i>X</i>,<i>E</i>)<i>into</i><i>C</i>₀(<i>Y</i>,<i>F</i>) is essentially a weighted composition operator<i>Tf</i>(<i>y</i>)=<i>h</i>(<i>y</i>)(<i>f</i>((<i>y</i>))).

When people in a society want to make inference about some parameter, each person may want to use data collected by other people. Information (data) exchange in social networks is usually costly, so to make reliable statistical decisions, people need to weigh the benefits and costs of information acquisition. Conflicts of interests and coordination problems will arise in the process. Classical statistics does not consider peoples incentives and interactions in the data-collection process. To address this imperfection, this work explores multi-agent Bayesian inference problems with a game theoretic social network model. Motivated by our interest in aggregate inference at the societal level, we propose a new concept, <i>finite population learning</i>, to address whether with high probability, a large fraction of people in a given finite population network can make good inference. Serving as a foundation, this concept enables

We give a topological simulation for tensor networks that we call the two-string model. In this approach we give a new way to design protocols, and we discover a new multipartite quantum communication protocol. We introduce the notion of topologically compressed transformations. Our new protocol can implement multiple, non-local compressed transformations among multi-parties using one multipartite resource state.

This is a continuation of "Mirror Principle III"(math.AG/9912038).