This paper is concerned with the Abels-Garcke-Gr\"{u}n (AGG) model for two-phase flows of two viscous incompressible system with different densities. The AGG model consists of a Navier-Stokes-Cahn-Hilliard system characterized by a mixture density depending on a volume fraction and an additional flux term due to interface diffusion. We establish the global existence of weak solutions to the initial boundary value problem with the mixed boundary conditions, including the velocity, pressure and Navier-slip conditions for fluid, and the Dirichlet and Neumann conditions for volume fraction and chemical potential, together, in a cubic domain of $\mathbb{R}^3$.

The increasing complexity of artificial intelligence (AI) models has led to a rising demand for explainability in AI (XAI). Explainable Artificial Intelligence aims to make AI's decision-making processes transparent and understandable to humans. This paper examines the integral connection between XAI and mathematics, highlighting how mathematical principles can enhance the interpretability, transparency, and trustworthiness of AI models. We explore the mathematical foundations that underpin XAI techniques, examine case studies where mathematics has improved explainability, and propose future directions for integrating mathematics into XAI frameworks.

In a recent paper, we advanced a systematic procedure for generating the coefficients of the continued fraction expansion of the test function associated with a Schur stable polynomial. The procedure we used involves long and elaborate calculations and the number of required iterations equal to the degree of the polynomial plus one. In the current work, we propose new continued fraction expansions which proceed in terms of some bilinear functions. the procedure to generate the coefficients of the new expansion is extremely simple. In addition, the number of iterations required to generate the coefficients of the new proposed expansion almost equal half the degree of the polynomial, which is a significant simplification in testing the stability of such polynomials. Illustrative examples are given to highlight the advantages of the new procedure compared to the old one.

Newton's mechanics is simple. His equivalence principle is simple, as is the inverse square law of gravitational force. A simple theory should have simple solutions to simple models. A system of n particles, given their initial speed and positions along with their masses, is such a simple model. Yet, solving for n>2 is not simple. This paper discusses what could be done to overcome that problem.

Cauchy Theory is applied and extended to n-dimensional functions in Clifford algebras, showing the existence of integrals that do not exist in Euclidean spaces. It celebrates the depth of Cauchy's lecture, held on the 22nd of August, 1814, so 200 years ago, in times of bitter warfare. (I should like to recommend reading about his life, e.g. in Wikipedia.)