Let M be a lattice module over the multiplicative lattice L. An L-module M is called a multiplication lattice module if for every element NÎM there exists an element aÎL such that N=a1M. Our objective is to investigate properties of prime elements of multiplication lattice modules.

Let G be a simple, oriented connected graph with n vertices and m edges. Let I (B) be the binomial ideal associated to the incidence matrix B of the graph G. Assume that IL is the lattice ideal associated to the rows of the matrix B. Also let Bi be a sub matrix of B after removing the i -th row. We introduce a graph theoretical criterion for G which is a sufficient and necessary condition for I (B) =I (Bi) and I (Bi) =IL. After that we introduce another graph theoretical criterion for G which is a sufficient and necessary condition for I (B) =IL. It is shown that the heights of I (B) and I (Bi) are equal to n - 1 and the dimensions of I (B) and I (Bi) are equal to m - n+1; then I (Bi) is a complete intersection ideal.

in this work, some of the lattice plasmon quantum features are examined. Initially, the interaction of the far-field photonic mode and the nanoparticle plasmon mode is investigated. We probe the optical properties of the array plasmon that are dramatically affected by the array geometry. It is notable to mention that the original goal of this work is to examine the quantum feature of the array plasmon. For this reason, we consider a system containing array of the plasmonic nanoparticles and quantum dots. For a complete understanding, we analyze the system with the full quantum theory. Notably, the full quantum analyzing enables us to investigate the quantum fluctuation of the array field. Here, for instance, we study the second-order correlation function and report its modeling results.

This paper addresses the need for an efficient and adaptable approach to solve linear acoustic wave equations in the lattice Boltzmann method (LBM). A novel lattice-adaptive model is introduced, derived through a Chapman–Enskog analysis, which utilizes a single relationship for the equilibrium distribution function across all lattice structures. The intended derivation begins by considering a standard equilibrium distribution function with unknown coefficients. By selecting the displacement of the acoustic wave as the zero-order microscopic moment, accurate recovery of the macroscopic wave equation is ensured. Unlike existing methods, the model simplifies the complexity associated with equilibrium distribution functions and offers greater versatility. The model is validated through extensive benchmark testing on one and two-dimensional wave propagation problems. Results demonstrate excellent agreement with analytical solutions, with maximum root mean square errors of 10-3 (<0.1% error) and minimum errors of 10-6 (<0.0001% error), indicating high predictive accuracy (>99.9%). Additionally, the model exhibits second-order spatial accuracy, with the relative error norm E_2 displaying slopes close to 2, signifying a spatial accuracy of second order. The numerical simulations show a decrease in errors as the mesh size becomes more refined.