bio | website | |
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location | University of California, Berkeley | |
age | 67 | |
visits | member for | 4 years, 7 months |
seen | Mar 24 at 1:18 | |
stats | profile views | 950 |
Jul 4 |
answered | Which manifolds admit a diffeomorphism of order $n$? |
Jun 17 |
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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 x<sub>1</sub>, x<sub>2</sub> ∈ ℝ. Then among all the open intervals containing the set {x<sub>1</sub>, x<sub>2</sub>}, none of them is contained in all the others." If one is addressing mathematicians, there is no need to specify that x<sub>1</sub> might be equal to x<sub>2</sub>. |
Jun 12 |
awarded | Commentator |
Jun 12 |
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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 |
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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 |
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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 |
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What are some correct results discovered with incorrect (or no) proofs?
(cont'd) This can be achieved by starting with the canonical flow on S<sup>2</sup> x [0,1] (i.e., the one parallel to [0,1]) and introducing a "plug" -- a copy of S<sup>1</sup> 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 |
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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 C<sup>1</sup> nonsingular vector fields as the solutions to the corresponding ODE) on even S<sup>2</sup> 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 |
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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 |
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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 |
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Noninteger iterates of functions: How to get ODE from flow at a given time?
FIxed partial derivative notation |
May 30 |
answered | Noninteger iterates of functions: How to get ODE from flow at a given time? |
May 30 |
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A problem of an infinite number of balls and an urn
Naturally, not everyone will agree on a philosophical issue. Which is why I said Martin Gardner's solution is "generally accepted," not "universally accepted," among mathematicians. |
May 25 |
answered | A problem of an infinite number of balls and an urn |
May 25 |
awarded | Critic |
May 25 |
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Major mathematical advances past age fifty
Just for the record, Poincaré never actually expressed his so-called conjecture as a conjecture; rather he brought it up as a question. After incorrectly making the conjecture that homology suffices to detect a 3-sphere -- and ingeniously finding a counterexample to that -- he was evidently chastened enough to refrain from phrasing what is called the Poincaré Conjecture as an actual conjecture. |