Tetrahedral modular graph functions

Modular Functions and Modular Forms

These are the notes for Math 678, University of Michigan, Fall 1990, exactly as they were handed out during the course except for some minor revisions and corrections. Please send comments and corrections to me at [email protected].

Modular Functions

V-ATPases are rotary nanomachines that pump protons across membranes. Old as the hills, they are found in every eukaryotic cell, diligently pumping protons to acidify intracellular compartments. This enzyme can be viewed as a functional unit of two distinct subcomplexes, the ATPhydrolyzing V1 and the membrane-bound, proton-conducting V0 complex. There is good evidence that V-ATPase activity is ...

Modular Termination of Graph Transformation

We establish a machine-checkable condition which ensures that the union of two terminating hypergraph transformation systems is terminating. The condition is based on so-called sequential critical pairs which represent consecutive rule applications that are not independent. In contrast to a corresponding modularity result for term rewriting, no restrictions on the form of rules are required. Ou...

A Tetrahedral Graph of Tetrachordal Voice-leading Space

A tetrahedral graph models voice leading among the 29 T/I-type tetrachord classes. Transpositional combination plays a crucial role in the structure of the tetrahedron. A dipyramid, fusing two tetrahedra, models similar relations among the 43 T-type |4|-classes. The two graphs generalize to n-dimensional simplexes for relations among |n + 1|-classes, in modulo 12 and in other universes of even ...

Trigonometric functions, elliptic functions, elliptic modular forms