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| visits | member for | 2 years, 4 months |
| seen | 9 hours ago | |
| stats | profile views | 4,519 |
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Aug 3 |
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subset relations among Sobolev spaces and their duals @user14178: A bijection is not the same thing as equality. The set of all even numbers is in bijection with the set of all numbers, but is also a proper subset. (This is one of the possible definitions for an infinite set.) |
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Aug 3 |
revised |
Proof of non-existence of a certain metric on $S^2$ deleted 100 characters in body |
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Aug 3 |
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Proof of non-existence of a certain metric on $S^2$ @Jacob: Good point. I was thinking of a snappier / more general form of the argument the OP used when I wrote that. |
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Aug 3 |
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What are the rules for basic algebra when modulo real numbers are involved You can't do modular arithmetic with real numbers. (Actually, you can't even do modular arithmetic with rational numbers.) All you're doing is taking the remainder after some division operation. |
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Aug 3 |
answered | Proof of non-existence of a certain metric on $S^2$ |
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Aug 3 |
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Did Zariski really define the Zariski topology on the prime spectrum of a ring? @Pierre: It seems to me that you are collecting references and examples of (mis)attribution of concepts in mathematics. May I ask what it is for? If it's for an upcoming article I would certainly like to read it. |
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Aug 3 |
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Isometric actions @Chu: It might be useful if you explain why you want such an example? |
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Aug 3 |
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Sheaf cohomology: what is it and where can I learn it? @Akhil: Thanks for the reference. To think that one textbook would go from simple groups, rings, and modules all the way up to homological algebra...! I will have a look at it, and Iversen's book, sometime when I can obtain it. |
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Aug 3 |
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A question in complex analysis about analytic function It's a simple continuity argument. Can you see that $f$ cannot vanish on the domain? |
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Aug 3 |
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Why bother with Mathematics, if Gödel's Incompleteness Theorem is true? @Adrian, re your edit: One does not need to think about other universes having different mathematical axioms. Right here on Earth there are mathematicians who study the possibility of mathematics under different axiom systems. Perhaps if history turned out differently we would have the axiom of determinacy instead of the axiom of choice. |
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Aug 3 |
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Reference request: group theory I remember reading it and not learning anything about the representation theory of Lie algebras, which is a deficiency in my mind given its use in modern theoretical physics. (And I still don't know anything about it...) Could you suggest an introductory text which covers representations? |
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Aug 3 |
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countably infinite union of countably infinite sets is countable @gary: As Asaf says, in the absence of the axiom of choice, it is possible for a countable union of countable sets to be uncountable. For example, there is a model of ZF where $\mathbb{R}$ is a countable union of countable sets. |
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Aug 2 |
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Sheaf cohomology: what is it and where can I learn it? (A reference for the claim that soft sheaves are acyclic would also be much appreciated.) |
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Aug 2 |
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Sheaf cohomology: what is it and where can I learn it? @Akhil: I've just had a chance to think about your comments carefully. I can see that the fact that the cohomology of the constant sheaf agrees with de Rham cohomology follows from the fact that acyclic resolutions give the same answer as injective resolutions. (I was at first puzzled about how the de Rham complex could even be exact, let alone be an acyclic resolution, but then I realised I needed to look at stalks rather than sections.) What's still mysterious to me is why acyclic resolutions should give the same answer. Could you provide a reference or an informal explanation? |
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Aug 2 |
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Cofusing partial order “implies”, on logic and that on sets You seem to be confusing the fact that $\implies$ (or, to be more precise, $\vdash$) is a preorder on the set of propositions with the idea of reasoning about partially ordered sets. |
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Aug 2 |
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understanding of a proof of the invariance of 3-D laplacian? @Jack: It's the bit starting at ‘substituting the previous equation in’. It is literally just the chain rule. |
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Aug 1 |
answered | understanding of a proof of the invariance of 3-D laplacian? |
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Aug 1 |
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understanding of a proof of the invariance of 3-D laplacian? The first equality is the chain rule. (If this is not obvious, perhaps the notation needs to be improved.) |
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Aug 1 |
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Why bother with Mathematics, if Gödel's Incompleteness Theorem is true? A not-so-well-kept secret of mathematics is that mathematicians are not that bothered about rigour. Mathematicians have been working with things not well-defined since the dawn of time and will continue to do so. What changes is the standard of ‘well-definedness’, and from time to time there will be a crisis over the lack of foundations. People have been doing arithmetic for aeons before Peano wrote down his axioms, and for the most part, no one worried about it. Perhaps a better question to ask is, what is mathematics? |
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Aug 1 |
answered | Injective Holomorphic Functions that are not Conformal? |
