1. L’Hôpital’s Rule 1 1.1. The Cauchy Mean Value Theorem 1 1.2. L’Hôpital’s Rule 2 2. Newton’s Method 5 2.1. Introducing Newton’s Method 5 2.2. A Babylonian Algorithm 6 2.3. Questioning Newton’s Method 7 2.4. Introducing Infinite Sequences 7 2.5. Contractions and Fixed Points 8 2.6. Convergence of Newton’s Method 10 2.7. Quadratic Convergence of Newton’s Method 11 2.8. An example of nonconverge...

Weconsider the problemof designing optimal distributed controllerswhose impulse response has limited propagation speed. We introduce a state-space framework in which all spatially invariant systems with this property can be characterized. After establishing the closure of such systems under linear fractional transformations, we formulate the H2 optimal control problem using the model-matching f...

In this work we explore the dynamics of generalized hybrid metric-Palatini theory gravity in weak-field, slow-motion regime. We start by introducing equivalent scalar-tensor representation theory, which contains two scalar degrees freedom, and perform a conformal transformation to Einstein frame. Linear perturbations metric Minkowskian background are then studied for both fields. The effective ...

Different questions lead to the same class of functions from natural integers to integers: those which have integral difference ratios, i.e. verifying f(a)− f(b) ≡ 0 (mod (a− b)) for all a > b. We characterize this class of functions via their representations as Newton series. This class, which obviously contains all polynomials with integral coefficients, also contains unexpected functions, fo...

An efficient implementation of the high-order spectral volume (SV) method is presented for multi-dimensional conservation laws on unstructured grids. In the SV method, each simplex cell is called a spectral volume (SV), and the SV is further subdivided into polygonal (2D), or polyhedral (3D) control volumes (CVs) to support high-order data reconstructions. In the traditional implementation, Gau...

The Hermitian and skew-Hermitian splitting (HSS) method is an unconditionally convergent iteration method for solving large sparse non-Hermitian positive definite system of linear equations. By making use of the HSS iteration as the inner solver for the Newton method, we establish a class of Newton-HSS methods for solving large sparse systems of nonlinear equations with positive definite Jacobi...