• Journal of Convergence for Information Technology
• /
• v.9 no.5
• /
• pp.7-13
• /
• 2019

Role-based access or attribute-based access control in cloud environment or big data environment need requires a suitable mathematical structure to represent a hierarchical model. This paper define the notion of multipliers and simple multipliers of lattice implication algebras that can implement a hierarchical model of role-based or attribute-based access control, and prove every multiplier is simple multiplier. Also we research the relationship between multipliers and homomorphisms of a lattice implication algebra L, and prove that the lattice [0, u] is isomorphic to a lattice $[u^{\prime},1]$ for each $u{\in}L$ and that L is isomorphic to $[u,1]{\times}[u^{\prime},1]$ as lattice implication algebras for each $u{\in}L$ satisfying $u{\vee}u^{\prime}=1$.

• Communications of Mathematical Education
• /
• v.37 no.1
• /
• pp.65-84
• /
• 2023

The method of Lagrange multipliers, one of the most fundamental algorithms for solving equality constrained optimization problems, has been widely used in basic mathematics for artificial intelligence (AI), linear algebra, optimization theory, and control theory. This method is an important tool that connects calculus and linear algebra. It is actively used in artificial intelligence algorithms including principal component analysis (PCA). Therefore, it is desired that instructors motivate students who first encounter this method in college calculus. In this paper, we provide an integrated perspective for instructors to teach the method of Lagrange multipliers effectively. First, we provide visualization materials and Python-based code, helping to understand the principle of this method. Second, we give a full explanation on the relation between Lagrange multiplier and eigenvalues of a matrix. Third, we give the proof of the first-order optimality condition, which is a fundamental of the method of Lagrange multipliers, and briefly introduce the generalized version of it in optimization. Finally, we give an example of PCA analysis on a real data. These materials can be utilized in class for teaching of the method of Lagrange multipliers.