Gauge-invariant theories and higher-degree forms

A free differential algebra is generalization of a Lie algebra in which the mathematical structure is extended by including of new Maurer-Cartan equations for higher-degree differential forms. In this article, we propose a generalization of the Chern-Weil theorem for free differential algebras containing only one p-form extension. This is achieved through a generalization of the covariant derivative, leading to an extension of the standard formula for Chern-Simons and transgression forms. We also study the possible existence of anomalies originated on this kind of structure. Some properties and particular cases are analyzed.

An Algebraic Approach to Identifiability

This paper addresses the problem of identifiability of nonlinear polynomial state-space systems. Such systems have already been studied via the input-output equations, a description that, in general, requires differential algebra. The authors use a different algebraic approach, which is based on distinguishability and observability. Employing techniques from algebraic geometry such as polynomial ideals and Gröbner bases, local as well as global results are derived. The methods are illustrated on some example systems.

Dynamic Simulation of Continuous Catalytic Reforming Process Based on Simultaneous Solution

The dynamic simulation of the continuous catalytic reforming process is of great significance to the performance prediction and optimization of the entire process. In this study, a 34-lumped mechanism model described by differential algebra was established based on the actual process conditions of the continuous catalytic reforming process in China, and an efficient dynamic simulation solution method based on simultaneous equations was proposed. First, a 34-lumped differential–algebraic mechanism model was established based on the basic principles of reforming kinetics, thermodynamics, material balance, and energy balance. Secondly, in order to solve and simulate the mechanism model composed of 144 differential equations and several algebraic equations, the method of finite-element collocation is used to discretize the differential equations and convert them into large-scale, nonlinear programming problems, and the interior point algorithm is used to estimate its parameters and verify the model. In addition, in order to avoid the problem of too long derivative solution time and too large memory in the solution process, methods such as sparse derivative and Broyden–Fletcher–Goldfarb–Shanno (BFGS) with limited storage are used to solve the problem. Finally, on the basis of model verification, dynamic simulation and sensitivity analysis of the whole process are carried out by modifying different input parameters. The results show that the mechanism model and solution method presented in this paper can quickly and accurately simulate the continuous catalytic reforming process dynamically.

Algebraic Parameter Identification of Nonlinear Vibrating Systems and Non Linearity Quantification Using the Hilbert Transformation

A novel algebraic scheme for parameters’ identification of a class of nonlinear vibrating mechanical systems is introduced. A nonlinearity index based on the Hilbert transformation is applied as an effective criterion to determine whether the system is dominantly linear or nonlinear for a specific operating condition. The online algebraic identification is then performed to compute parameters of mass and damping, as well as linear and nonlinear stiffness. The proposed algebraic parametric identification techniques are based on operational calculus of Mikusiński and differential algebra. In addition, we propose the combination of the introduced algebraic approach with signals approximation via orthogonal functions to get a suitable technique to be applied in embedded systems, as a digital signals’ processing routine based on matrix operations. A satisfactory dynamic performance of the proposed approach is proved and validated by experimental case studies to estimate significant parameters on the mechanical systems. The presented online identification approach can be extended to estimate parameters for a wide class of nonlinear oscillating electric systems that can be mathematically modelled by the Duffing equation.

Dynamics of thermoelastic half-plane by action of periodic loads and heat flows at its boundary

The problem of the dynamics of a thermoelastic half-space under periodic surface forces and heat flows is solved using the model of coupled thermoelasticity. The Green’s tensor for one boundary value problem is constructed utilizing Fourier transformation. Analytical solutions for arbitrary surface forces and heat flow using the theory of generalized functions are constructed. To solve this boundary value problem, generalized function theory, tensor and differential algebra, the operator method, and integral transformations were used. The solutions obtained make it possible to investigate the thermal stress–strain state of an array with natural and artificial thermal sources and mass power forces acting at its surface.