Let $R\to A$ be a homomorphism of associative rings, and let $(\mathcal F,\mathcal C)$ hereditary complete cotorsion pair in $R\mathsf{-Mod}$. F_A,\mathcal C_A)$ the $A\mathsf{-Mod}$ which $\mathcal F_A$ is class all left $A$-modules whose underlying $R$-modules belong to F$. Assuming that F$-resolution dimension every $R$-module finite F$ preserved by coinduction functor $\operatorname{Hom}_R(...

The notion of ×-homotopy from [Doca] is investigated in the context of the category of pointed graphs. The main result is a long exact sequence that relates the higher homotopy groups of the space Hom∗(G,H) with the homotopy groups of Hom∗(G,H ). Here Hom∗(G,H) is a space which parameterizes pointed graph maps from G to H (a pointed version of the usual Hom complex), and H is the graph of based...

Let hom(G,H) denote the number of homomorphisms from a graph G to a graph H. In this paper we study the number of homomorphisms of trees into a path, and prove that hom(Pm, Pn) ≤ hom(Tm, Pn) ≤ hom(Sm, Pn), where Tm is any tree on m vertices, and Pm and Sm denote the path and star on m vertices, respectively. This completes the study of extremal problems concerning the number of homomorphisms be...

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...