Following the last talk on graph homomorphisms, we continue to discuss some examples of graph homomorphisms. But this time we will focus on some models, that is, the homomorphism G → H for the graph H with fixed weights. The main reference is Section 1 in [2].

Definition. A definable orientation of a definable manifold Y of dimension m is a map s which assigns to each point y ∈ Y a generator s(y) of the local definable homology group H m (Y, Y − y) and which is locally constant in the following sense: for each proper m-ball B of Y there is a class ζB ∈ H m (Y, Y − B) such that for each p ∈ B the natural homomorphism j p : H def m (Y, Y −B)→ H m (Y, Y...

We show that if a graph H is k-colorable, then (k−1)-branching walks on H exhibit long range action, in the sense that the position of a token at time 0 constrains the configuration of its descendents arbitrarily far into the future. This long range action property is one of several investigated herein; all are similar in some respects to chromatic number but based on viewing H as the range, in...

Let A = {H1, ..., Hd} be an affine essential hyperplane arrangement in C , see [OT1], [OT2] for general facts on arrangements. We set as usual M = M(A) = C\X, X being the union of all the hyperplanes in A. One of the main problems now in hyperplane arrangement theory is to study the cohomology of the complement M with coefficients in some local system L on M , see for instance the introduction ...

We show that every Π1-elementary epimorphism between models of ZF is an isomorphism. On the other hand, nonisomorphic Σ1-elementary epimorphisms between models of ZF can be constructed, as can fully elementary epimorphisms between models of ZFC−. Elementary epimorphisms were introduced by Philipp Rothmaler in [Rot05]. A surjective homomorphism f : M → N between two model-theoretic structures is...

We prove that a Jordan homomorphism from a Banach algebra into a semisimple commutative Banach algebra is a ring homomorphism. Using a signum effectively, we can give a simple proof of the Hyers-Ulam-Rassias stability of a Jordan homomorphism between Banach algebras. As a direct corollary, we show that to each approximate Jordan homomorphism f from a Banach algebra into a semisimple commutative...