For graphs G and H , a homomorphism from G to H is a function φ : V (G) → V (H), which maps vertices adjacent in G to adjacent vertices of H . A homomorphism is locally injective if no two vertices with a common neighbor are mapped to a single vertex in H . Many cases of graph homomorphism and locally injective graph homomorphism are NPcomplete, so there is little hope to design polynomial-time...

We show how the classical notions of cohomology with local coefficients, CW-complex, covering space, homeomorphism equivalence, simple homotopy tubular neighbourhood, and spinning can be encoded on a computer used to calculate ambient isotopy invariants continuous embeddings N ↪ M one topological manifold into another. More specifically, we describe an algorithm for computing homology H n ( X ,...

We define the Homomorphism Extension (HomExt) problem: given a group G, a subgroup M ≤ G and a homomorphism φ : M → H , decide whether or not there exists a homomorphism φ̃ : G → H extending φ, i.e., φ̃|M = φ. This problem arose in the context of list-decoding homomorphism codes but is also of independent interest, both as a problem in computational group theory and as a new and natural problem i...

We study the complexity of structurally restricted homomorphism and constraint satisfaction problems. For every class of relational structures C, let LHOM(C, _) be the problem of deciding whether a structure A ∈ C has a homomorphism to a given arbitrary structure B, when each element in A is only allowed a certain subset of elements of B as its image. We prove, under a certain complexity-theore...

We discuss Matijevic–Roberts type theorem on strong F -regularity, F -purity, and Cohen–Macaulay F -injective (CMFI for short) property. Related to this problem, we also discuss the base change problem and the openness of loci of these properties. In particular, we define the notion of F -purity of homomorphisms using Radu–André homomorphisms, and prove basic properties of it. We also discuss a...

Bounds on the range of random graph homomorphism into Z, and the maximal height difference of the Gaussian random field, are presented.

Eilenberg’s variety theorem gives a bijective correspondence between varieties of languages and varieties of finite semigroups. The second author gave a similar relation between conjunctive varieties of languages and varieties of semiring homomorphisms. In this paper, we add a third component to this result by considering varieties of meet automata. We consider three significant classes of lang...

Let A be a finite-dimensional algebra over the complex numbers with nonzero identity element e. If x ∈ A, then the resolvent set associated to x is the set ρ(x) of complex numbers λ such that λ e − x is invertible, and the spectrum of x is the set σ(x) of complex numbers λ such that λ e − x is not invertible. For instance, if V is a finite-dimensional vector space over the complex numbers of po...

which are simply four points in the second quadrant on the unit circle. 2. Each couple of complex-conjugate zeros contributes a factor of the form (1−2z−1 cos θ+ z−2) to the transfer function, where θ is the angle of the complex zero. We have in the end: H(z) = (1 + z)(1 + z)(1− 2 cos( 3 4 π) z + z) 3. H(z) is a 5th degree polynomial in z−1 and therefore it has at most 6 nonzero coefficients. T...