Daniel Asimov
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 Jul 5 revised Google question: In a country in which people only want boys corrected grammar Jul 5 answered Google question: In a country in which people only want boys Jul 5 answered Which mathematicians have influenced you the most? Jul 5 comment Which manifolds admit a diffeomorphism of order $n$? Wow, that's an interesting example. Thank you. Jul 4 comment How is it that you can guess if one of a pair of random numbers is larger with probability > 1/2? I think you mean Randall Munroe. Jul 4 answered Which manifolds admit a diffeomorphism of order $n$? Jun 17 comment Examples of common false beliefs in mathematics There are many situations where one needs to speak of a set of two numbers that may or may not be equal. E.g.: "Let x1, x2 ∈ ℝ. Then among all the open intervals containing the set {x1, x2}, none of them is contained in all the others." If one is addressing mathematicians, there is no need to specify that x1 might be equal to x2. Jun 12 awarded Commentator Jun 12 comment Examples of common false beliefs in mathematics In case that wasn't clear: F(x) = ln(x) + C_1 for x > 0, and F(x) = ln(-x) + C_2 for x < 0, where C_1 and C_2 are arbitrary real constants. Jun 12 comment Examples of common false beliefs in mathematics Really? What about the function F(x) given by ln(x) + C_1, x > 0 F(x) = ln(-x) + C_2, x < 0 for arbitrary reals C_1, C_2 ? (The appropriate technical condition is that an antiderivative be differentiable on the same domain as the function it's the antiderivative of is defined on.) Jun 12 comment What are some correct results discovered with incorrect (or no) proofs? Indeed Heawood proved the 5-color theorem. But I'm not aware that he gave an incorrect proof of the 4-color theorem. What he is known for doing is finding a flaw in an 1879 supposed proof, by Kempe, that had stood for 11 years. Perhaps at least as impressive, he determined the "Heawood number" -- an upper bound for the chromatic number -- for every compact surface, and conjectured it was the actual chromatic number. This number turned out to be the actual chromatic number of every compact surface except the Klein bottle, as shown by Ringel & Youngs (except for the sphere) in 1968. Jun 12 answered Examples of common false beliefs in mathematics Jun 12 comment What are some correct results discovered with incorrect (or no) proofs? (cont'd) This can be achieved by starting with the canonical flow on S2 x [0,1] (i.e., the one parallel to [0,1]) and introducing a "plug" -- a copy of S1 x [0,1] x[0,1] -- on which the flow is altered. See, for instance, Plugging Flows by Percell and Wilson. For those with access, at < jstor.org/stable/pdfplus/1997824.pdf >. Jun 12 comment What are some correct results discovered with incorrect (or no) proofs? Sure. The sentence starting with "Clearly" isn't. In fact there exist orientable 1-foliations (which result from C1 nonsingular vector fields as the solutions to the corresponding ODE) on even S2 x [0,1] that are entering on one boundary component and exiting on the other, without every trajectory that enters on one boundary component exiting on the other one. Jun 12 answered Who is the last mathematician that understood all of mathematics. Jun 11 answered What are some correct results discovered with incorrect (or no) proofs? Jun 1 comment Domains of homolorphy in the complex plane You probably want to assume the open set is connected. Otherwise, a locally analytic function whose domain is not connected is by convention generally not thought of a a single analytic function. (Even when it is everywhere defined by a single series, such as in Josh Shadlen's elegant answer below.) Such a function can, for instance, violate the permanence principle. There are some fascinating examples of such functions in Hille's Analytic Function Theory, vol. II. Jun 1 comment Analytic Functions over Fields other than Real or Complex Numbers This is far from satisfactory, but: Convergent power series with real coefficients make sense when interpreted as functions on the quaternions. This allows one to define a lot of old favorites like the exponential function on the quaternions. It's challenging, however, to determine the appropriate notion of "Riemann surface" for some of these, like the logarithm, since it has uncountably many branches. May 30 answered Analytic ODE with complex time May 30 revised Noninteger iterates of functions: How to get ODE from flow at a given time? FIxed partial derivative notation