Existence of multiple and sign-changing solutions for a second-order nonlinear functional difference equation with periodic coefficients

In the present paper, we apply the method of invariant sets of descending flow to establish a series of criteria to ensure that a second-order nonlinear functional difference equation with periodic boundary conditions possesses at least one trivial solution and three nontrivial solutions. These nontrivial solutions consist of sign-changing solutions, positive solutions and negative solutions. Moreover, as an application of our theoretical results, an example is elaborated. Our results generalize and improve some existing ones.     

Contingent derivatives and regularization for noncoercive inverse problems

Abstract

We study the inverse problem of parameter identification in noncoercive variational problems that commonly appear in applied models. We examine the differentiability of the set-valued parameter-to-solution map using the first-order and the second-order contingent derivatives. We explore the inverse problem using the output least-squares and the modified output least-squares objectives. By regularizing the noncoercive variational problem, we obtain a single-valued regularized parameter-to-solution map and investigate its smoothness and boundedness. We also consider optimization problems using the output least-squares and the modified output least-squares objectives for the regularized variational problem. We give a complete convergence analysis showing that for the output least-squares and the modified output least-squares, the regularized minimization problems approximate the original optimization problems suitably. We also provide the first-order and the second-order adjoint method for the computation of the first-order and the second-order derivatives of the output least-squares objective. We provide discrete formulas for the gradient and the Hessian calculation and present numerical results.     

An Introduction to Influence Theory: Kinematics and Dynamics

Influence theory is a foundational theory of physics that is not based on traditional empirically defined concepts, such as positions in space and time, mass, energy, or momentum. Instead, the aim is to derive these concepts, and their empirically determined relationships, from a more primitive model. It is postulated that there exist things, which are call particles, that influence one another in a discrete and directed fashion resulting in a partially ordered set of influence events. The problem of consistent quantification of the influence events is considered. Observers are modeled as particle chains (observer chains) as if an observer were able to track a particle and quantify the influence events that the particle experiences. From these quantified influence events, consistent quantification of the universe of events based on the observer chains is studied. Herein, the kinematics and dynamics of particles from the perspective of influence theory are both reviewed and further developed.     

Computational modelling of multi-material energetic materials and systems

ABSTRACT

In this paper, we present a systematic roadmap for developing a robust and parallel multi-material reactive hydrodynamic solver that integrates historically stable algorithms with new and current modern methods to solve explosive system design problems. The Ghost Fluid Method and Riemann solvers were used to enforce appropriate interface boundary conditions. Improved performance in terms of computational work and convergence properties was achieved by modifying a local node sorting strategy that decouples ghost nodes, allowing us to set material boundary conditions via an explicit procedure, removing the need to solve a coupled system of equations numerically. The locality and explicit nature of the node sorting concept allows for greater levels of parallelism and lower computational cost when populating ghost nodes. Non-linear numerical issues endemic to the use of real Equations of State in hydro-codes were resolved by using more thermodynamically consistent forms allowing us to accurately resolve large density gradients associated with high energy detonation problems at material interfaces. Pre-computed volume tables were implemented adding to the robustness of the solver base.