A Hom-Lie algebra is a triple (L, [−,−], α), where α is a linear self-map, in which the skew-symmetric bracket satisfies an α-twisted variant of the Jacobi identity, called the Hom-Jacobi identity. When α is the identity map, the Hom-Jacobi identity reduces to the usual Jacobi identity, and L is a Lie algebra. Hom-Lie algebras and related algebras were introduced in [1] to construct deformation...

There is a fundamental dimensional mismatch between the Hong-Ou-Mandel (HOM) interferometer and two-photon (2P) states: while the latter are represented using two temporal (or spectral) dimensions, the HOM interferometer allows access to only one temporal dimension. We introduce a linear 2P interferometer containing two independent delays spanning the 2P state. By “unlocking” the fixed phase re...

let $m_r$ be a non-zero‎ ‎module and ${mathcal f}‎: ‎sigma[m_r]times sigma[m_r]‎ ‎rightarrow$ mod-$bbb{z}$ a bifunctor‎. ‎the‎ ‎$mathcal{f}$-reversibility of $m$ is defined by ${mathcal‎ ‎f}(x,y)=0 rightarrow {mathcal f}(y,x)=0$ for all non-zero $x,y$‎ ‎in $sigma[m_r]$‎. ‎hom (resp‎. ‎rej)-reversibility of $m$ is‎ ‎characterized in different ways‎. ‎among other things‎, ‎it is shown‎ ‎th...

The Drosophila HOM-C genes encode transcription factors containing the DNA-binding homeodomain. Mutations in the HOM-C genes can cause specific homeotic transformation, suggesting that the HOM-C genes determine segmental identities by acting on different target genes. However, misexpression of several HOM-C genes in the antenna disc causes similar antenna-to-leg transformations. Here we show th...

Let hom(G) denote the size of the largest clique or independent set of a graph G. In 2007, Bukh and Sudakov proved that every n-vertex graph G with hom(G) = O(log n) contains an induced subgraph with Ω(n) distinct degrees, and raised the question of deciding whether an analogous result holds for every n-vertex graph G with hom(G) = O(n), where ε > 0 is a fixed constant. Here, we answer their qu...

To any two graphs G and H one can associate a cell complex Hom (G,H) by taking all graph multihomorphisms from G to H as cells. In this paper we prove the Lovász Conjecture which states that if Hom (C2r+1, G) is k-connected, then χ(G) ≥ k + 4, where r, k ∈ Z, r ≥ 1, k ≥ −1, and C2r+1 denotes the cycle with 2r + 1 vertices. The proof requires analysis of the complexes Hom (C2r+1,Kn). For even n,...

The aim of this paper is to generalize the concept of Lie-admissible coalgebra introduced in [2] to Hom-coalgebras and to introduce Hom-Hopf algebras with some properties. These structures are based on the Hom-algebra structures introduced in [12].

Abstract The aim of this paper is to generalise the construction n -ary Hom-Lie bracket by means an $$(n-2)$$ ( n - 2 ) -cochain given algebra super case inducing -Hom-Lie superalgebras. We study notion generalized deri...