Abstract. Let L and M be two finite lattices. The ideal J(L,M) is a monomial ideal in a specific polynomial ring and whose minimal monomial generators correspond to lattice homomorphisms ϕ: L→M. This ideal is called the ideal of lattice homomorphism. In this paper, we study J(L,M) in the case that L is the product of two lattices L_1 and L_2 and M is the chain [2]. We first characterize the set...

In tensor networks, a geometric operation of pushing bond cut surface toward minimal corresponds to entanglement distillation. Cutting bonds defines reduced transition matrix on the and associated quantum state naturally emerges from it. We justify this picture quantitatively by evaluating trace distance between maximally entangled states surfaces in multi-scale renormalization ansatz (MERA) pr...

Let [n] = {1, 2, . . . , n} and let 2 be the collection of all subsets of [n] ordered by inclusion. C ⊆ 2 is a cutset if it meets every maximal chain in 2, and the width of C ⊆ 2 is the minimum number of chains in a chain decomposition of C. Fix 0 ≤ m ≤ l ≤ n. What is the smallest value of k such that there exists a cutset that consists only of subsets of sizes between m and l, and such that it...

Aven and Jensen (1999) proposed an approximation to the downtime distribution for arbitrary coherent systems with binary and independent repairable components, where each component follows an alternating renewal process. The approximation is based on a mixture of the duration distributions of the cut sets in the system. We extend this approach to coherent systems composed by multistate independ...

Let an edge cut partition the vertex set of an n-dimensional quadratic grid with the side length a into k subsets A1, ..., Ak with ||Ai| − |Aj || ≤ 1. We consider the problem of determining the minimal size c(n, k, a) of such a cut and present its asymptotic c(n, k, a) ∼ nan−1 n √ k as a, k →∞ and k/an → 0. The same asymptotic holds for partitioning of the n-dimensional torus. We present also s...

The embedding of finite metrics in �1 has become a fundamental tool for both combinatorial optimization and largescale data analysis. One important application is to network flow problems as there is close relation between max-flow min-cut theorems and the minimal distortion embeddings of metrics into �1. Here we show that this theory can be generalized to a larger set of combinatorial optimiza...

M. Stoer and F. Wagner and independently A. Frank have found a simple proof of the validity of Nagamochi and Ibaraki7s min-cut algorithm. This note points out some nice property of the behavior of Nagamochi and Ibaraki's rnin-cut algorithm, which also gives another simple proof of the validity of their algorithm. The proof relies only on the symmetric submodularity of the cut function. Hence, i...