A Pair of Counterexamples in Vector Calculus

It is intuitive that removing even a single point from a line disconnects it, but removing a finite set of points from a plane leaves it connected. A line disconnected by a single point. A plane remaining connected even with a few points removed. However, this basic fact leads to a non-trivial property of real and complex polynomials: not all...

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KaTeXMacros = { "\\clplus": "\\oplus", "\\clminus": "\\ominus", "\\clmul": "\\otimes", "\\cldiv": "\\oslash", "\\bclmod": "\\mathbin{\\mathrm{clmod}}", }; 1. Overview This article explains Reed-Solomon erasure codes and the problems they solve in gory detail, with the aim of providing enough background to understand how the PAR1 ...

. --> .graph { display: block; width: 300px; height: 300px; margin: 0.5em 0.2em; } .graph-container { display: inline-block; vertical-align: top; max-width: 300px; } (This was discussed on r/math and Hacker News.) 1. Overview In this article, I hope to convince you that the quintic equation is unsolvable, in the sense that I can’t...

KaTeXMacros = { "\\iroot": "\\operatorname{iroot}", "\\Bits": "\\operatorname{Bits}", "\\Err": "\\operatorname{Err}", "\\NewtonRoot": "\\mathrm{N{\\small EWTON}\\text{-}I{\\small ROOT}}", }; 1. The algorithm Today I’m going to talk about the generalization of the integer square root algorithm to higher roots. That is, given \(n\) and...