Sensitivity Analysis of Mixed Cayley Inclusion Problem with XOR-Operation

In this paper, we consider the parametric mixed Cayley inclusion problem with Exclusive or (XOR)-operation and show its equivalence with the parametric resolvent equation problem with XOR-operation. Since the sensitivity analysis, Cayley operator, inclusion problems, and XOR-operation are all applicable for solving many problems occurring in basic and applied sciences, such as financial modeling, climate models in geography, analyzing “Black Box processes”, computer programming, economics, and engineering, etc., we study the sensitivity analysis of the parametric mixed Cayley inclusion problem with XOR-operation. For this purpose, we use the equivalence of the parametric mixed Cayley inclusion problem with XOR-operation and the parametric resolvent equation problem with XOR-operation, which is an alternative approach to study the sensitivity analysis. In support of some of the concepts used in this paper, an example is provided.

Nonlinear diffusion in transparent media: The resolvent equation

We consider the partial differential equationu-f=\operatornamewithlimits{div}\biggl{(}u^{m}\frac{\nabla u}{|\nabla u|}% \biggr{)}with f nonnegative and bounded and {m\in\mathbb{R}}. We prove existence and uniqueness of solutions for both the Dirichlet problem (with bounded and nonnegative boundary datum) and the homogeneous Neumann problem. Solutions, which a priori belong to a space of truncated bounded variation functions, are shown to have zero jump part with respect to the {{\mathcal{H}}^{N-1}}-Hausdorff measure. Results and proofs extend to more general nonlinearities.

Formulations of the -functional calculus and some consequences

In this paper we introduce the two possible formulations of the -functional calculus that are based on the Fueter–Sce mapping theorem in integral form and we introduce the pseudo--resolvent equation. In the case of dimension 3 we prove the -resolvent equation and we study the analogue of the Riesz projectors associated with this calculus. The case of dimension 3 is also useful to study the quaternionic version of the -functional calculus.

Necessary and sufficient conditions for uniform stability of Volterra integro-dynamic equations using new resolvent equation

We consider the system of Volterra integro-dynamic equations and obtain necessary and sufficient conditions for the uniform stability of the zero solution employing the resolvent equation coupled with the variation of parameters formula. The resolvent equation that we use for the study of stability will have to be developed since it is unknown for time scales. At the end of the paper, we furnish an example in which we deploy an appropriate Lyapunov functional. In addition to generalization, the results of this paper provides improvements for its counterparts in integro-differential and integro-difference equations which are the most important particular cases of our equation.