In this paper, we introduce multiscale persistent functions for biomolecular structure characterization. The essential idea is to combine our multiscale rigidity functions with persistent homology analysis, so as to construct a series of multiscale persistent functions, particularly multiscale persistent entropies, for structure characterization. To clarify the fundamental idea of our method, the multiscale persistent entropy model is discussed in great detail. Mathematically, unlike the previous persistent entropy or topological entropy, a special resolution parameter is incorporated into our model. Various scales can be achieved by tuning its value. Physically, our multiscale persistent entropy can be used in conformation entropy evaluation. More specifically, it is found that our method incorporates in it a natural classification scheme. This is achieved through a density filtration of a multiscale rigidity function built from bond and/or dihedral angle distributions. To further validate our model, a systematical comparison with the traditional entropy evaluation model is done. It is found that our model is able to preserve the intrinsic topological features of biomolecular data much better than traditional approaches, particularly for resolutions in the mediate range. Moreover, our method can be successfully used in protein classification. For a test database with around nine hundred proteins, a clear separation between all-alpha and all-beta proteins can be achieved, using only the dihedral and pseudo-bond angle information. Finally, a special protein structure index (PSI) is proposed, for the first time, to describe the "regularity" of protein structures. Essentially, PSI can be used to describe the "regularity" information in any systems.

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.

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.

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.

We present an algorithm for shape reconstruction from incomplete 3D scans by fusing together two acquisition modes: 2D photographs and 3D scans. The two modes exhibit complementary characteristics: scans have depth information, but are often sparse and incomplete; photographs, on the other hand, are dense and have high resolution, but lack important depth information. In this work we fuse the two modes, taking advantage of their complementary information, to enhance 3D shape reconstruction from an incomplete scan with a 2D photograph. We compute geometrical and topological shape properties in 2D photographs and use them to reconstruct a shape from an incomplete 3D scan in a principled manner. Our key observation is that shape properties such as boundaries, smooth patches and local connectivity, can be inferred with high confidence from 2D photographs. Thus, we register the 3D scan with the 2D photograph and use scanned points as 3D depth cues for lifting 2D shape structures into 3D. Our contribution is an algorithm which significantly regularizes and enhances the problem of 3D reconstruction from partial scans by lifting 2D shape structures into 3D. We evaluate our algorithm on various shapes which are loosely scanned and photographed from different views, and compare them with state-of-the-art reconstruction methods.