We establish a second anti-blocker theorem for noncommutative convex corners, show that the anti-blocking operation is continuous on bounded sets of convex corners, and define optimization parameters for a given convex corner that generalize well-known graph theoretic quantities. We define the entropy of a state with respect to a convex corner, characterize its maximum value in terms of a generalized fractional chromatic number and establish entropy splitting results that demonstrate the entropic complementarity between a convex corner and its anti-blocker. We identify two extremal tensor products of convex corners and examine the behavior of the introduced parameters with respect to tensoring. Specializing to noncommutative graphs, we obtain quantum versions of the fractional chromatic number and the clique covering number, as well as a notion of noncommutative graph entropy of a state, which we show to be continuous with respect to the state and the graph. We define the Witsenhausen rate of a noncommutative graph and compute the values of our parameters in some specific cases.

A quasiconformal tree is a doubling metric tree in which the diameter of each arc is bounded above by a fixed multiple of the distance between its endpoints. We study the geometry of these trees in two directions. First, we construct a catalog of metric trees in a purely combinatorial way, and show that every quasiconformal tree is bi-Lipschitz equivalent to one of the trees in our catalog. This is inspired by results of Herron and Meyer and of Rohde for quasi-arcs. Second, we show that a quasiconformal tree bi-Lipschitz embeds in a Euclidean space if and only if its set of leaves admits such an embedding. In particular, all quasi-arcs bi-Lipschitz embed into some Euclidean space.

The notion of epsilon multiplicity was originally defined by Ulrich and Validashti as a limsup, and they used it to detect integral dependence of modules. It is important to know if it can be realized as a limit. In this article, we show that the relative epsilon multiplicity of reduced Noetherian graded algebras over an excellent local ring exists as a limit. An important special case of Cutkosky’s result concerning epsilon multiplicity is obtained as a corollary of our main theorem. We also produce a multigraded generalization of a result due to Dao and Monataño about monomial ideals.

The trace of a Markov process is the time-changed process of the original process on the support of the Revuz measure used in the time change. In this paper, we will concentrate on the reflecting Brownian motions on certain closed strips. We will give a concrete expression of the Dirichlet forms associated with the traces of such reflecting Brownian motions on the boundary, then the limits of these traces as the distance between the upper and lower boundaries tends to 0 or ∞ will be further obtained.