https://www.zhumengyao.com/scidraft/tags/counterexample/Recent content in Counterexample on SciDraft | A Hugo ThemeHugoenMon, 21 Sep 2026 12:12:00 +0900https://www.zhumengyao.com/scidraft/notes/demo-note-counterexample-noncommuting/Mon, 21 Sep 2026 12:12:00 +0900https://www.zhumengyao.com/scidraft/notes/demo-note-counterexample-noncommuting/<p><strong>Demo note.</strong> This note demonstrates what the <strong>SciDraft</strong> Hugo theme can do for academic and scientific publishing — for research institutions, researchers and students. Its content is demonstration material only: readers should ignore whether the demo content is correct.</p> <p>A counterexample is the cheapest way to kill a plausible statement, and the one worth having ready concerns the order of matrix multiplication. The tempting assertion that \(AB = BA\) fails for almost every pair of matrices, and it fails on the smallest possible example.</p>https://www.zhumengyao.com/scidraft/notes/demo-note-counterexample-convex-many-minima/Mon, 21 Sep 2026 07:02:00 +0900https://www.zhumengyao.com/scidraft/notes/demo-note-counterexample-convex-many-minima/<p><strong>Demo note.</strong> This note demonstrates what the <strong>SciDraft</strong> Hugo theme can do for academic and scientific publishing — for research institutions, researchers and students. Its content is demonstration material only: readers should ignore whether the demo content is correct.</p> <p>Convexity is often described as the property that guarantees a unique optimum, and the statement is false as written: convex functions can have a whole interval of minimisers. The counterexample is a flat-bottomed function, convex everywhere and constant on a segment.</p>