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...

Trigonometric functions, elliptic functions, elliptic modular forms

Fixed points for Banach and Kannan contractions in modular spaces with a graph

In this paper, we discuss the existence and uniqueness of xed points for Banach and Kannancontractions dened on modular spaces endowed with a graph. We do not impose the Δ2-conditionor the Fatou property on the modular spaces to give generalizations of some recent results. Thegiven results play as a modular version of metric xed point results.