The recent developments in the field of computational mathematics and computer graphics have shown a significant focus on enhancing the accuracy and efficiency of algorithms used in solving linear systems and modeling shapes. A notable trend is the exploration of convergence factors in iterative methods for linear systems, particularly in the context of GMRES(1), where the relationship between the system's properties and the convergence behavior is being deeply analyzed. This includes understanding how the symmetry or skew-symmetry of matrices influences the convergence rate, offering insights into optimizing these methods for better performance.nnIn the realm of computer graphics, there's a push towards improving the smoothness and continuity of shape modeling techniques. Innovations in the As-Rigid-As-Possible (ARAP) energy modification aim to eliminate artifacts like spikes and enhance the user interaction experience without compromising the method's efficiency and ease of implementation. This advancement is crucial for real-time applications, making detailed model manipulation more accessible and intuitive.nnAnother area of progress is in the development of discrete Laplacians and mean curvature approximations in three-dimensional spaces. Research is focusing on establishing the superiority of dual constructions over primal ones in terms of accuracy and sensitivity to mesh changes. This work not only provides a mathematical foundation for these claims but also suggests methods for improving approximation accuracy without the need for extensive mesh subdivision, thereby optimizing computational resources.nn### Noteworthy Papersn- The worst-case root-convergence factor of GMRES(1): This paper provides a comprehensive analysis of the convergence behavior of GMRES(1) under specific matrix conditions, offering valuable insights for optimizing linear system solvers.n- Higher Order Continuity for Smooth As-Rigid-As-Possible Shape Modeling: Introduces a significant improvement to the ARAP method, enhancing shape modeling by ensuring higher order smoothness and better user interaction.n- The Associated Discrete Laplacian in $mathbb{R}^3$ and Mean Curvature with Higher order Approximations: Presents a novel approach to discrete Laplacian and mean curvature approximation, demonstrating the advantages of dual constructions and higher order approximations in three-dimensional spaces.