In this paper, we survey recent progress on the constructive theory of the Feynman operator calculus. We first develop an operator version of the Henstock-Kurzweil integral, and a new Hilbert space that allows us to construct the elementary path integral in the manner originally envisioned by Feynman. After developing our time-ordered operator theory we extend a few of the important theorems of...

Practical data analysis involves many implicit or explicit assumptions about the good behavior of the data, and excludes consideration of various potentially pathological or limit cases. In this work, we present a new general theory of data, and of data processing, to bypass some of these assumptions. The new framework presented is focused on integration, and has direct applicability to expecta...

and Applied Analysis 3 The space AC is a Banach space under the Alexiewicz norm; ‖f‖ supI⊂R| ∫ If |, where the supremum is taken over all intervals I ⊂ R. An equivalent norm is ‖f‖′ supx∈R| ∫x −∞f |. The continuous primitive integral contains the Lebesgue, Henstock-Kurzweil, and wide Denjoy integrals since their primitives are continuous functions. These three spaces of functions are not comple...

Earthquakes in subduction zones rupture the plate boundary fault in discrete segments. One factor that may control this segmentation is topography on the downgoing plate, although it is controversial whether this is by weakening or strengthening of the fault. We use multichannel seismic and gravity data to map the top of the downgoing oceanic crust offshore central Sumatra, Indonesia. Our surve...

K e y w o r d s M u l t i p l e Wiener integral, Hu-Meyer theorem, Henstock integral. 1. I N T R O D U C T I O N Classically, it is well known that the Riemann approach cannot be used to define stochastic integrals. However, it has been proved that the generalized Riemann approach (with nonuniform meshes), also called the Henstock's approach, can be used to study stochastic integrals. See detai...

See “High-fidelity drug-induced liver injury screen using human pluripotent stem cell-derived organoids,” by Shinozawa T, Kimura M, Cai Y, et al, on page 831. Drug-induced (DILI) is a leading cause of drug attrition because animals are not fully predictive DILI owing to species-specific differences in metabolism pathways and the lack genetic diversity used rodent strains. Therefore, models esse...