Muellerkalkulus

Muellerkalkulus is a fictional mathematical framework presented as a generalized calculus for functionals on spaces of curves and signals. It centers on two principal operators, the Mueller derivative and the Mueller integral, which together extend classical calculus to accommodate non-smooth, impulsive, or highly irregular perturbations that arise in certain applications. In this scheme, the Mueller derivative acts on functionals by measuring their first-order response to a broad class of perturbations, while the Mueller integral provides an integration theory for functionals with respect to a designated Mueller measure. The combination is intended to supply rules analogous to the chain rule, product rule, and integration by parts in a setting that includes non-smooth data.

History and development

The concept is attributed to a fictional lineage beginning with a theoretical proposal in the late 20th or early 21st century, with subsequent elaborations by researchers in a dedicated institute. While it is not part of mainstream mathematical practice, Muellerkalkulus is described in some sources as a unifying attempt to bring together ideas from the calculus of variations, operator theory, and generalized function theory. It is presented as a framework for exploring problems where standard smooth calculus is insufficient.

Core concepts

A core idea is to replace or augment standard derivatives with Mueller derivatives, which accommodate jumps and discontinuities. The Mueller integral is designed to integrate functionals over trajectories or signals with respect to a Mueller measure. The framework proposes generalized product and chain rules and aims to yield fundamental theorems that parallel the familiar results in classical calculus, but in a broader, non-smooth context.

Applications and reception

Proposed applications include optimization problems with irregular data, analysis of dynamical systems with impulses, and numerical schemes for certain nonlinear integral equations. In the broader mathematical community, Muellerkalkulus remains a theoretical construct with limited peer-reviewed validation, and it is commonly discussed as a conceptual or historical curiosity rather than an established tool. See also: Calculus, Calculus of variations, Functional analysis, Malliavin calculus.