Banach Manifold Structure and Infinite-Dimensional Analysis for Causal Fermion Systems

A mathematical framework is developed for the analysis of causal fermion systems in the infinite-dimensional setting. It is shown that the regular spacetime point operators form a Banach manifold endowed with a canonical Fr\'echet-smooth Riemannian metric. The so-called expedient differential calculus is introduced with the purpose of treating derivatives of functions on Banach spaces which are differentiable only in certain directions. A chain rule is proven for H\"older continuous functions which are differentiable on expedient subspaces. These results are made applicable to causal fermion systems by proving that the causal Lagrangian is H\"older continuous. Moreover, H\"older continuity is analyzed for the integrated causal Lagrangian.Comment: 38 pages, LaTeX, minor improvements (published version

Fractional Calculus via Functional Calculus: Theory and Applications

This paper demonstrates the power of the functional-calculus definition of linear fractional (pseudo-)differential operators via generalised Fourier transforms. Firstly, we describe in detail how to get global causal solutions of linear fractional differential equations via this calculus. The solutions are represented as convolutions of the input functions with the related impulse responses. The suggested method via residue calculus separates an impulse response automatically into an exponentially damped (possibly oscillatory) part and a ''slow' relaxation. If an impulse response is stable it becomes automatically causal, otherwise one has to add a homogeneous solution to get causality. Secondly, we present examples and, moreover, verify the approach along experiments on viscolelastic rods. The quality of the method as an effective few-parameter model is impressively demonstrated: the chosen reference example PTFE (Teflon) shows that in contrast to standard classical models our model describes the behaviour in a wide frequency range within the accuracy of the measurement. Even dispersion effects are included. Thirdly, we conclude the paper with a survey of the required theory. There the attention is directed to the extension from the L-2-approach on the space of distributions cal D-

Gauge Invariant Fractional Electromagnetic Fields

Fractional derivatives and integrations of non-integers orders was introduced more than three centuries ago but only recently gained more attention due to its application on nonlocal phenomenas. In this context, several formulations of fractional electromagnetic fields was proposed, but all these theories suffer from the absence of an effective fractional vector calculus, and in general are non-causal or spatially asymmetric. In order to deal with these difficulties, we propose a spatially symmetric and causal gauge invariant fractional electromagnetic field from a Lagrangian formulation. From our fractional Maxwell's fields arose a definition for the fractional gradient, divergent and curl operators.Comment: accepted for publication in Physics Letters

Formalization of Transform Methods using HOL Light

Transform methods, like Laplace and Fourier, are frequently used for analyzing the dynamical behaviour of engineering and physical systems, based on their transfer function, and frequency response or the solutions of their corresponding differential equations. In this paper, we present an ongoing project, which focuses on the higher-order logic formalization of transform methods using HOL Light theorem prover. In particular, we present the motivation of the formalization, which is followed by the related work. Next, we present the task completed so far while highlighting some of the challenges faced during the formalization. Finally, we present a roadmap to achieve our objectives, the current status and the future goals for this project.Comment: 15 Pages, CICM 201