Let $$G=(V,E)$$ be a locally finite graph. Firstly, using calculus of variations, including direct method variation and the mountain-pass theory, we get sequences solutions to several local equations on G (the Schrödinger equation, mean field Yamabe equation). Secondly, derive uniform estimates for those solution sequences. Finally, obtain global by extracting convergent sequence solutions. Our...

We propose a notion of continuous path for locally finite metric spaces, taking inspiration from the recent development of A-theory for locally finite connected graphs. We use this notion of continuity to derive an analogue in Z of the Jordan curve theorem and to extend to a quite large class of locally finite metric spaces (containing all finite metric spaces) an inequality for the l-distortio...

In [4], Urquhart gives an example of a modal logic which has the finite model property, is recursively axiomatizable but not decidable. He poses the problem of finding such a logic containing S4. We will solve this problem by constructing a logic which in addition to the required properties also defines a locally finite variety of modal algebras. (Recall that a variety is said to be locally fin...