By Sergiy Kolyada, Yuri Manin, Thomas Ward, Iu. I. Manin
This quantity features a choice of articles from the detailed software on algebraic and topological dynamics and a workshop on dynamical platforms held on the Max-Planck Institute (Bonn, Germany). It displays the intense energy of dynamical structures in its interplay with a wide variety of mathematical topics. issues lined within the e-book contain asymptotic geometric research, transformation teams, mathematics dynamics, complicated dynamics, symbolic dynamics, statistical houses of dynamical structures, and the speculation of entropy and chaos. The ebook is appropriate for graduate scholars and researchers drawn to dynamical structures.
Read or Download Algebraic And Topological Dynamics: Algebraic And Topological Dynamics, May 1-july 31, 2004, Max-planck-institut Fur Mathematik, Bonn, Germany (Contemporary Mathematics) PDF
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Additional info for Algebraic And Topological Dynamics: Algebraic And Topological Dynamics, May 1-july 31, 2004, Max-planck-institut Fur Mathematik, Bonn, Germany (Contemporary Mathematics)
These assumptions limit the number of possible, roughly spherical shell structures, each one containing twelve pentagonal units and a certain number of hexagonal units. To catalog the possible geometries, Caspar and Klug defined a number, T, which corresponds to the number of coat proteins at each corner of a triangular face of the shell. Thus, T = 1 for the shell of the satellite tobacco necrosis virus, and T = 3 for the poliovirus shell. In this virus shell model, the only T numbers allowed are 1, 3, 4, 7, 9, 12, 13, 16, 19, 21, 25, and so on.
One can also imagine the structure as being made up of five proteins gathered at each of the twelve corners, or vertices, of an icosahedron, as is shown in the figure below. In effect, its surface can be thought of as consisting of twelve protein pentagons. Larger shells have additional protein units at the corners of their triangular faces. For example, the poliovirus shell consists of 180 coat proteins, with three proteins in each corner, for a total of nine on each face. This structure can also be pictured as consisting of twelve groups of five proteins each at the twelve vertices of an icosahedron and twenty groups of six proteins each at the center of each of the faces.
In recent years, a small group of mathematicians has pioneered a novel perspective on virus self-assembly— how structural order emerges out of randomness in the microcellular realm. This research suggests that sets of simple rules, which define the way proteins stick together, automatically lead to the kinds of virus structures that biologists observe under their electron microscopes. 3 micrometers in size, viruses have highly regular structures; often they look like mineral crystals with flat faces, distinct angles, and definite edges.