# Bounded perturbations and flow domains: figure terms

These three figures revise the original OA-FLOW course illustrations by updating their proof locators and the explanation of the selfadjoint-sum construction. Their mathematical arrangement, labels, signs and plotted data are retained. The revised edition is dated 6 October 2026; original and revising project credit is **OA-FLOW course project contributors**.

The original and modified figure expression in the following files, and the authored annotations and arrangement in `34-derivation-domain-data.json`, are licensed under the **GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation**, with **no Invariant Sections, no Front-Cover Texts and no Back-Cover Texts**:

- `34-cocycle-and-common-domain.png` and `.svg`;
- `34-lost-second-derivative.png` and `.svg`;
- `34-summability-and-domain-window.png` and `.svg`.

A complete copy of that license is supplied in [LICENSE-GFDL-1.2.txt](LICENSE-GFDL-1.2.txt). The inherited component notices, including the earlier file identities and grants, are retained in [PRESERVED_COMPONENT_NOTICES.json](PRESERVED_COMPONENT_NOTICES.json). The lesson's dedication of newly written prose does not replace these figure and data terms. Numerical mathematical facts are not claimed as exclusive property.

The renderer `render_figures.py`, derived from the original `render-34-derivation-domain-figures.py`, retains the **MIT License** and its unchanged copyright and permission notice in [LICENSE-SOFTWARE-MIT.txt](LICENSE-SOFTWARE-MIT.txt). Reproduce all six images and the coordinate diagnostics with `python render_figures.py --output OUTPUT_DIRECTORY`. It needs Python, Matplotlib and NumPy; it launches no TeX process. SVG IDs and date metadata are fixed. The two plots use separate vertical scales, logarithmic horizontal axes and floating-point samples; their exact infinite-series bounds are proved in Section 7.

Typography and runtime libraries retain their own terms. The rendered DejaVu, STIX and Computer Modern glyph notices are supplied in [LICENSE_DEJAVU.txt](LICENSE_DEJAVU.txt), [LICENSE_STIX.txt](LICENSE_STIX.txt) and [BAKOMA_SECTION.txt](BAKOMA_SECTION.txt). Font outlines are excluded from the project-expression grant. Library and font implementations are not bundled.

The mathematical antecedent identified in the original diagrams is M. Takesaki, *Theory of Operator Algebras II*, Chapter XI, §3. The figures are original project artwork, not images taken from that source.