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On Hamming-Lipschitz Type Stability of the Subdominant (Minmax) Ultrametric: Theory and Simple Proofs
arXiv:2608.04014v1 Announce Type: new Abstract: The subdominant (minmax) ultrametric is a canonical tree-structured summary of a dissimilarity matrix, arising equivalently as the ultrametric induced by single-linkage clustering. While its classical stability theory is usually formulated in $\ell_\infty$ or Gromov--Hausdorff terms, such bounds are poorly suited to sparse perturbations that alter only a few pairwise distances. We develop an $\ell_0$-type stability theory for this operator. Our analysis shows that sparse edits propagate only through the minimum spanning tree (MST): a pairwise ultrametric value can change only if its tree path crosses an edited edge or a cut newly exposed by an edited off-tree edge. This yields a sharp per-edit exposed-cut score and a tree-only global envelope, leading to Hamming--Lipschitz bounds on the number of ultrametric entries that can change. We also prove sharpness results showing that this dependence on tree geometry is unavoidable: under strict
AI Just Solved Math That Stumped Humans for Decades — And It Cost $2,000
For years, the debate over artificial intelligence has circled a single question: is it merely remixing what humans already know, or can Continue reading on Medium »
The LA Artist Who Started Selling Work to People Who Found Him Through ChatGPT
Los Angeles is one of the largest art markets in the world. Collectors, galleries, interior designers, corporate buyers, and everyday Continue reading on Medium »
Nashville uses eminent domain to block data center near zoo
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ShadowHunter D-A278: A 7-Inch FPV Drone Built for Recon
A Purpose-Built Platform for Fast Aerial ReconnaissanceContinue reading on Medium »
7 Modern Python Libraries Shaking Up AI Development in 2026
Forget heavy pipelines these lightweight imports make model orchestration feel effortlessContinue reading on Python in Plain English »
Elon Musk’s attempt at an AI Wikipedia hasn’t been updated in months
xAI's Grokipedia, an online encyclopedia with AI-generated articles that Elon Musk once promised would be a "massive improvement" over Wikipedia, apparently hasn't been updated since April 24th, according to a report from Lawfare. "As far as we can tell, no entry has changed in more than three months," Lawfare said. Grokipedia launched in v0.1 in October 2025 with an initial batch of 885,000 articles, and is now on v0.2 (released in November 2025) and has more than 6,000,000 total articles, according to grokipedia.com/live. But that live page also has a section titled "Recent changes to…" with a message under it that reads "No live edits av … Read the full story at The Verge.
Engineering Intelligent Platforms: Building an AI-Native Customer Engagement System
Building Atlas Part 15 Published by Summit Kumar Senior Software Engineer | Mobile Architect | Distributed Systems EnthusiastContinue reading on Medium »
AI Automations You Can Sell: The Complete 2026 Guide
You can build AI automations businesses pay $500 $5,000 for without writing code using n8n and free-tier AI models. This guide covers Continue reading on Medium »
X product chief Nikita Bier is leaving after one year
X head of product Nikita Bier is stepping down and says he will move into a role as an advisor, writing that "it's time to pass the torch and demote myself to my natural state: a poster." He shared the update just over a month after celebrating his one-year anniversary on the job, and just weeks after Elon Musk's combined SpaceX / X / xAI companies went public. He says that "almost every aspect of X" has been rebuilt while he was head of product, including its timeline, Android app, chat, and notifications. Going forward, the "next leaders" mentioned are Benji Taylor leading design with iOS lead Jonah Katz and Mridul Singhai leading the pr … Read the full story at The Verge.
The KOSPI Crashed 40% in a Month. Here’s the Chain Reaction Nobody Saw Coming
In June 2026, South Korea s stock market was the best story in global finance.Continue reading on Medium »
Writing Is Coming Back
Why I think we re about to witness one of the greatest renaissances of writing in human history.Continue reading on Medium »