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A main drawback of classical Tikhonov regularization is that often the parameters required to apply theoretical results, e.g., the smoothness of the sought-after solution and the noise level, are unknown in practice. In this paper we investigate in new detail the residuals in Tikhonov regularization viewed as functions of the regularization parameter. We show that the residual carries, with some restrictions, the information on both the unknown solution and the noise level. By calculating approximate solutions for a large range of regularization parameters, we can extract both parameters from the residual given only one set of noisy data and the forward operator. The smoothness in the residual allows to revisit parameter choice rules and relate a-priori, a-posteriori, and heuristic rules in a novel way that blurs the lines between the classical division of the parameter choice rules. All results are accompanied by numerical experiments.

Incidents of gun violence and mass shootings continue to shock the U.S. citizenry. Each time there is a call for change, but then little happens. In this article we explore the history of gun contr...

Assembly sequence planning is always a great challenge in complex product assembly. Subassembly division is a way that can validly decrease the complexity of assembly sequence planning problem. Abo...

In order to accurately divide the coal spontaneous combustion area of multi-layer goafs in shallow coal seam group so as to take effective control measures, a mathematical model is established on t...

By examining the census records of Swedish-Caribbean colony St. Barthélemy, a GIS - Geographic Information System - can be constructed which maps the inhabitants of the harbor city of Gustavia, as well as the inhabitants' placement within the census categories of the colonial administration. This article examines these categories and how they shifted over time, as well as maps out patterns of settlement across the city. One major finding is the division between the Catholic French-Caribbean side on the east and the Anglican Anglo- and Dutch-Caribbean side of the west, the latter of which was home to a number of merchants owning high amounts of enslaved people. This spatial division was also parallel to the political differences between the merchant elite and the Swedish colonial administration. Furthermore, this article encourages additional GIS studies of predemocratic Caribbean cities, particularly regarding spatial division between whites and free people of color.

Combining insights from the new labour market segmentation approach and the labour process theory, this qualitative case study research critically examines how workplace dynamics shape the division...

In essays written between 1960 and 1963, Blanchot embarks on a new line of thought, beginning with fundamental philosophical division between language and vision. The contrast between the two domai...

In a recent Monthly note [1], Athreya and Aulicino found a closed geodesic on the regular dodecahedron at a vertex not passing through any other vertices. This work has been popularized in a Number...