Proteins perform a variety of functions in living organisms and their functions are largely determined by their shape. In this paper, we propose a novel mathematical method for designing protein-like molecules of a gi...Proteins perform a variety of functions in living organisms and their functions are largely determined by their shape. In this paper, we propose a novel mathematical method for designing protein-like molecules of a given shape. In the mathematical model, molecules are represented as loops of n-simplices (2-simplices are triangles and 3-simplices are tetrahedra). We design a new molecule of a given shape by patching together a set of smaller molecules that cover the shape. The covering set of small molecules is defined using a binary relation between sets of molecules. A new molecule is then obtained as a sum of the smaller molecules, where addition of molecules is defined using transformations acting on a set of (n + 1)-dimensional cones. Due to page limitations, only the two-dimensional case (i.e., loops of triangles) is considered. No prior knowledge of Sheaf Theory, Category Theory, or Protein Science is required. The author hopes that this paper will encourage further collaboration between Mathematics and Protein Science.展开更多
Let X be a compact complex manifold. Consider a small deformation π ： X → B of X, the dimensions of the cohomology groups of tangent sheaf Hq（xt, Txt ） may vary under this deformation. This article studies such p...Let X be a compact complex manifold. Consider a small deformation π ： X → B of X, the dimensions of the cohomology groups of tangent sheaf Hq（xt, Txt ） may vary under this deformation. This article studies such phenomena by studying the obstructions to deform a class in Hq（X, 5TX） with parameter t and gets a formula for the obstructions.展开更多
Recently, the field of differential equations has been studying in a very abstract method. Instead of considering the behaviour of one solution of a differential equation, one studies its sheaf-solutions in many kinds...Recently, the field of differential equations has been studying in a very abstract method. Instead of considering the behaviour of one solution of a differential equation, one studies its sheaf-solutions in many kinds of properties, for example, the problems of existence, comparison,... of sheaf solutions. In this paper we study some of the problems of controllability for sheaf solutions of control systems.展开更多
In this paper, we first give a 1-1 corresponds between torus C/Λ and cubic curve C in P_C^2. As complex manifold, they are isomorphic, therefore we can treat C/Λ as a variety and construction a vertex algebra sheaf ...In this paper, we first give a 1-1 corresponds between torus C/Λ and cubic curve C in P_C^2. As complex manifold, they are isomorphic, therefore we can treat C/Λ as a variety and construction a vertex algebra sheaf on it.展开更多
Antimony oxychloride Sb8O11Cl2(H2O)6 products with various morphologies including sheaf-like,rhombic-plate,oval leaf-like and quasi-wafer have been successfully synthesized via a mild and facile solution route at room...Antimony oxychloride Sb8O11Cl2(H2O)6 products with various morphologies including sheaf-like,rhombic-plate,oval leaf-like and quasi-wafer have been successfully synthesized via a mild and facile solution route at room temperature.The morphologies and structures of the as-prepared samples were characterized by X-ray powder diffraction(XRD),scanning electron microscopy(SEM) and transmission electron microscopy(TEM).A possible formation mechanism of these structures is proposed according to the experimental results and analysis.展开更多
文摘Proteins perform a variety of functions in living organisms and their functions are largely determined by their shape. In this paper, we propose a novel mathematical method for designing protein-like molecules of a given shape. In the mathematical model, molecules are represented as loops of n-simplices (2-simplices are triangles and 3-simplices are tetrahedra). We design a new molecule of a given shape by patching together a set of smaller molecules that cover the shape. The covering set of small molecules is defined using a binary relation between sets of molecules. A new molecule is then obtained as a sum of the smaller molecules, where addition of molecules is defined using transformations acting on a set of (n + 1)-dimensional cones. Due to page limitations, only the two-dimensional case (i.e., loops of triangles) is considered. No prior knowledge of Sheaf Theory, Category Theory, or Protein Science is required. The author hopes that this paper will encourage further collaboration between Mathematics and Protein Science.
基金partially supported by China-France-Russian mathematics collaboration grant,No.34000-3275100,from Sun Yat-Sen University
文摘Let X be a compact complex manifold. Consider a small deformation π ： X → B of X, the dimensions of the cohomology groups of tangent sheaf Hq（xt, Txt ） may vary under this deformation. This article studies such phenomena by studying the obstructions to deform a class in Hq（X, 5TX） with parameter t and gets a formula for the obstructions.
文摘Recently, the field of differential equations has been studying in a very abstract method. Instead of considering the behaviour of one solution of a differential equation, one studies its sheaf-solutions in many kinds of properties, for example, the problems of existence, comparison,... of sheaf solutions. In this paper we study some of the problems of controllability for sheaf solutions of control systems.
基金Supported by the National Natural Science Foundation of China(11475178,11571119)
文摘In this paper, we first give a 1-1 corresponds between torus C/Λ and cubic curve C in P_C^2. As complex manifold, they are isomorphic, therefore we can treat C/Λ as a variety and construction a vertex algebra sheaf on it.
基金supported by the National Basic Research Program of China (2010CB93470)the National Natural Science Foundation of China (50725208,20973019)
文摘Antimony oxychloride Sb8O11Cl2(H2O)6 products with various morphologies including sheaf-like,rhombic-plate,oval leaf-like and quasi-wafer have been successfully synthesized via a mild and facile solution route at room temperature.The morphologies and structures of the as-prepared samples were characterized by X-ray powder diffraction(XRD),scanning electron microscopy(SEM) and transmission electron microscopy(TEM).A possible formation mechanism of these structures is proposed according to the experimental results and analysis.
