The representation category of a conformal net is a unitary modular tensor category. We investigate the reconstruction program: whether all unitary modular tensor categories are representation categories of conformal nets. We give positive evidence: the fruitful theory of multi-interval Jones-Wassermann subfactors on conformal nets is also true for modular tensor categories. We construct multi-interval Jones-Wassermann subfactors for unitary modular tensor categories. We prove that these subfactors are symmetrically self-dual. It generalizes and categorifies the self-duality of finite abelian groups. We call this duality the modular self-duality, because the modularity of the modular tensor category appears in a crucial way. For each unitary modular tensor category, we obtain a sequence of unitary fusion categories. The cyclic group case gives examples of Tambara-Yamagami categories.

This paper discusses the problem that making the air flowing into the room maximized by rotating the windows. Two models are built for rotary windows with 2 and 3 sashes respectively. Based on these models and the assumption of wind blowing from different directions with equal chance, the optimal way of opening windows is calculated by using Pascal programs on computer. Finally, the optimal design for window ventilation is found via comparing maximum value of wind quantity in two situations.

In this paper we prove a new matrix Li-Yau-Hamilton (LYH) estimate for K¨ahler-Ricci flow on manifolds with nonnegative bi- sectional curvature. The form of this new LYH estimate is obtained by the interpolation consideration originated in [Ch] by Chow. This new inequality is shown to be connected with Perelman’s entropy formula through a family of differential equalities. In the rest of the paper, we show several applications of this new estimate and its corresponding estimate for linear heat equation. These include a sharp heat kernel comparison theorem, generalizing the earlier result of Li and Tian [LT], a manifold version of Stoll’s theorem [St] on the characterization of ‘algebraic divisors’, and a localized monotonicity formula for analytic subvari- eties, which sharpens the Bishop volume comparison theorem. Motivated by the connection between the heat kernel estimate and the reduced volume monotonicity of Perelman [P], we prove a sharp lower bound of the fundamental solution to the forward conjugate heat equation, which in a certain sense dual to Perelman’s monotonicity of the reduced volume. As an application of this new monotonicity formula, we show that the blow-down limit of a certain type of long-time solution is a gradient expanding soliton, generalizing an earlier result of Cao. We also illustrate the connection between the new LYH estimate and the Hessian comparison theorem of [FIN] on the forward reduced distance. Localized monotonicity formulae on entropy and forward reduced volume are also derived.

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In this paper, we study the fluid-dynamic limit for the one-dimensional Broadwell model of the nonlinear Boltzmann equation in the presence of boundaries. We consider an analogue of Maxwell's diffusive and reflective boundary conditions. The boundary layers can be classified as either compressive or expansive in terms of the associated characteristic fields. We show that both expansive and compressive boundary layers (before detachment) are nonlinearly stable and that the layer effects are localized so that the fluid dynamic approximation is valid away from the boundary. We also show that the same conclusion holds for short time without the structural conditions on the boundary layers. A rigorous estimate for the distance between the kinetic solution and the fluid-dynamic solution in terms of the mean-free path in theL∞-norm is obtained provided that the interior fluid flow is smooth. The rate of convergence is optimal.