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Bill Gates spent $80 mn on French Riviera's most exclusive yacht berths
Bill Gates spent roughly $80 million securing two of the most exclusive superyacht berths on the French Riviera for a futuristic hydrogen-powered yacht project that was meant to redefine luxury sailing. The berths at Port Vauban's famed Billionaires' Quay were custom-configured to accommodate Breakthrough, the world's first hydrogen-powered superyacht, and its support vessel Wayfinder. Yet despite the enormous investment and years of planning, Gates reportedly never stepped aboard the...
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Incremental Sheaf Cohomology on Cellular Complexes: O(1)-in-n Lazy Edit Processing under Bounded Local Geometry
arXiv:2606.04227v2 Announce Type: replace Abstract: We present an algorithmic framework for incremental maintenance of first sheaf cohomology $H^1(X; \mathcal{F})$ on dynamically evolving 1-dimensional cellular complexes equipped with finite-dimensional cellular sheaves. The classical computation of $H^1$ via factorization of the coboundary matrix requires $O(n^3)$ time; when the complex evolves with a stream of $m$ edits, full recomputation after each edit costs $O(mn^3)$. Under a bounded...
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Incremental Sheaf Cohomology on Cellular Complexes: O(1)-in-n Lazy Edit Processing under Bounded Local Geometry
new Abstract: We present an algorithmic framework for incremental maintenance of first sheaf cohomology $H^1(X; \mathcal{F})$ on dynamically evolving 1-dimensional cellular complexes equipped with finite-dimensional cellular sheaves. The classical computation of $H^1$ via factorization of the coboundary matrix requires $O(n^3)$ time; when the complex evolves with a stream of $m$ edits, full recomputation after each edit costs $O(mn^3)$. Under a bounded local geometry assumption -- bounded...
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