Characterizations of functions with higher order derivatives in $\mathcal{L}_p$

Extension of Higher Order Derivatives of Lyapunov Functions in Stability Analysis of Nonlinear Systems

The Lyapunov stability method is the most popular and applicable stability analysis tool of nonlinear dynamic systems. However, there are some bottlenecks in the Lyapunov method, such as need for negative definiteness of the Lyapunov function derivative in the direction of the system’s solutions. In this paper, we develop a new theorem to dispense the need for negative definite-ness of Lyapunov...

The period functions’ higher order derivatives

We prove a formula for the n-th derivative of the period function T in a period annulus of a planar differential system. For n = 1, we obtain Freire, Gasull and Guillamon formula for the period’s first derivative [17]. We apply such a result to hamiltonian systems with separable variables and other systems. We give some sufficient conditions for the period function of conservative second order ...

Distinguishing Properties of Higher Order Derivatives of Boolean Functions

Higher order differential cryptanalysis is based on the property of higher order derivatives of Boolean functions that the degree of a Boolean function can be reduced by at least 1 by taking a derivative on the function at any point. We define fast point as the point at which the degree can be reduced by at least 2. In this paper, we show that the fast points of a n-variable Boolean function fo...

On the higher-order derivatives of spectral functions

In this paper we are interested in the higher-order derivatives of functions of the eigenvalues of symmetric matrices with respect to the matrix argument. We describe the formula for the k-th derivative of such functions in two general cases. The first case concerns the derivatives of the composition of an arbitrary (not necessarily symmetric) k-times differentiable function with the eigenvalue...

Subordination for Higher-Order Derivatives of Multivalent Functions