Abstract When solving, modeling or reasoning about complex problems, it is usually convenient to use the knowledge of a parallel physical system for representing it. This case lumped-circuit abstraction, which can be used mechanical and acoustic systems, thermal heat-diffusion problems in general partial differential equations. Integrated photonic platforms hold prospective perform signal proce...

If φ : G → G is a surjective homomorphism, we prove that the twisted Alexander polynomial of G is divisible by the twisted Alexander polynomial of G. As an application, we show non-existence of surjective homomorphism between certain knot groups.

If ϕ : G → G ′ is a surjective homomorphism, we prove that the twisted Alexander polynomial of G is divisible by the twisted Alexander polynomial of G ′. As an application, we show non-existence of surjective homomorphism between certain knot groups.

For digraphs D and H , a mapping f : V (D)→V (H) is a homomorphism of D to H if uv ∈ A(D) implies f(u)f(v) ∈ A(H). For a fixed directed or undirected graph H and an input graph D, the problem of verifying whether there exists a homomorphism of D to H has been studied in a large number of papers. We study an optimization version of this decision problem. Our optimization problem is motivated by ...

For digraphs G and H , a homomorphism of G to H is a mapping f : V (G)→V (H) such that uv ∈ A(G) implies f(u)f(v) ∈ A(H). In the minimum cost homomorphism problem we associate costs ci(u), u ∈ V (G), i ∈ V (H) with the mapping of u to i and the cost of a homomorphism f is defined ∑ u∈V (G) cf(u)(u) accordingly. Here the minimum cost homomorphism problem for a fixed digraph H , denoted by MinHOM...

With a graph G = (V,E) we associate a collection of non-negative real weights ⋃ v∈V {λi,v : 1 ≤ i ≤ m} ∪ ⋃ uv∈E{λij,uv : 1 ≤ i ≤ j ≤ m}. We consider the probability distribution on {f : V → {1, . . . ,m}} in which each f occurs with probability proportional to ∏ v∈V λf(v),v ∏ uv∈E λf(u)f(v),uv . Many well-known statistical physics models, including the Ising model with an external field and the...

We compute a twisted first cohomology group of the automorphism group of a free group with coefficients in the abelianization V of the IA-automorphism group of a free group. In particular, we show that it is generated by two crossed homomorphisms constructed with the Magnus representation and the Magnus expansion due to Morita and Kawazumi respectively. As a corollary, we see that the first Joh...