User seamus - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T08:44:14Z http://mathoverflow.net/feeds/user/4076 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/16684/when-is-the-product-of-a-set-of-numbers-greater-than-the-sum-of-them When is the product of a set of numbers greater than the sum of them? Seamus 2010-02-28T15:10:14Z 2012-03-06T16:18:22Z <p>This could well be too general a question, but I'd be interested in solutions to special cases too. Say you have some finite set of positive real numbers $x_i$, when is it the case that $\sum_i x_i > \prod_i x_i$? And when are they equal?</p> <p>The special case that prompted this was an argument about whether any number is equal to the sum of its prime factors.</p> <p>Any references or quick proofs welcome.</p> http://mathoverflow.net/questions/32368/is-monomorphism-going-in-both-directions-sufficient-for-isomorphism Is monomorphism going in both directions sufficient for isomorphism? Seamus 2010-07-18T14:52:27Z 2010-09-27T00:34:24Z <p>In category theory, it seems that a monomorphism from $A$ to $B$ and one from $B$ to $A$ should be enough to guarantee isomorphy, but it doesn't seem to be so. (If I'm right then there's something fishy with the standard definition of "subobject")</p> <p>So here's the counterexample I thought up, please explain where I went wrong.</p> <p>Consider a category consisting of 2 objects $A$ and $B$. There is a monomorphism $\phi: A \to B$ and another $\psi : B \to A$. "Close" this under composition in much the same way you do when defining a free group (that is, no non-trivial identities are allowed). I claim that this does not guarantee isomorphism. All morphisms are monic, since no identities hold, so the condition for monomorphism is trivially satisfied.</p> <p>What am I doing wrong here?</p> http://mathoverflow.net/questions/6423/set-theory-for-category-theory-beginners/23585#23585 Answer by Seamus for Set theory for category theory beginners Seamus 2010-05-05T13:47:19Z 2010-05-05T13:47:19Z <p>Lawvere and Schanuel's Conceptual Mathematics is a good introduction to category theory that assumes an absolute minimum of set theory. Peter Cameron's Sets, Logic and Categories, on the other hand, is a good introduction to sets and logic (albeit far too short), but his chapter on category theory is woefully short and would be unhelpful to anyone who doesn't already understand categories a little bit.</p> http://mathoverflow.net/questions/22851/how-would-one-extend-the-brier-score-to-an-infinite-number-of-forecasts How would one extend the Brier score to an infinite number of forecasts? Seamus 2010-04-28T13:29:26Z 2010-04-28T14:31:44Z <p>Is there a neat way to use something like the Brier score to score an infinite set of forecasts/outcomes?</p> http://mathoverflow.net/questions/18275/existence-of-convergent-subsequences-for-all-values-in-range Existence of convergent subsequences for all values in range? Seamus 2010-03-15T15:06:59Z 2010-03-16T10:46:00Z <p>Consider sequence $s(n) = \sin{nx}$. Are there values of $x$ for which the following holds: For every $y \in [-1,1]$ there is a subsequence of $s(n)$ converging to $y$? (Or perhaps just for the open interval...) Someone hypothesised that the answer is yes, and further that every $x$ that is relatively irrational with $\pi$ has this property.</p> <p>The question I am more interested in is the generalised version of this to arbitrary sequences. What are necessary and sufficient conditions for a sequence having subsequences converging to any point in the set of values the sequence visits? Does it have anything to do with properties like the function $f(n)$ being ergodic or mixing?</p> <p>(suggestions for tags welcome in comments)</p> http://mathoverflow.net/questions/10993/can-you-prove-equivalence-without-being-able-to-calculate-it/18276#18276 Answer by Seamus for Can you prove equivalence without being able to calculate it? Seamus 2010-03-15T15:15:29Z 2010-03-15T15:15:29Z <p>ZFC implies that the reals have a well ordering, but this well ordering is, in some sense, provably uncomputable.</p> <p>Given some facts about isomorphisms between well-orderings with base sets of the same cardinality, could one not prove that a well ordering of the reals is equivalent to something in an extremely non-constructive way?</p> <p>And given that you tagged this question "math-philosophy" I feel I should point out that <a href="http://en.wikipedia.org/wiki/Intuitionism" rel="nofollow">intuitionists</a> like Brouwer would have answered the original question with a resounding NO!</p> http://mathoverflow.net/questions/11296/were-bourbaki-committed-to-set-theoretical-reductionism/16989#16989 Answer by Seamus for Were Bourbaki committed to set-theoretical reductionism? Seamus 2010-03-03T18:44:28Z 2010-03-03T18:44:28Z <p>Geoffrey Hellman has written something on structuralism that compares Bourbaki structuralism with category theoretic structuralism. <a href="http://www.tc.umn.edu/~hellm001/Publications/Structuralism.pdf" rel="nofollow">Here</a>. His take seems to be that they were being reductionist.</p> http://mathoverflow.net/questions/53122/mathematical-urban-legends/53905#53905 Comment by Seamus Seamus 2012-01-17T12:06:26Z 2012-01-17T12:06:26Z I was once told about a philosophy essay that started &quot;In this essay I will argue that the mind is identical to the brain, but not the other way around&quot;. http://mathoverflow.net/questions/32368/is-monomorphism-going-in-both-directions-sufficient-for-isomorphism/32379#32379 Comment by Seamus Seamus 2010-07-21T12:18:09Z 2010-07-21T12:18:09Z OK. But in general it is not true that subobjecthood determines and antisymmetric relation? But in most cases of interest, it will... http://mathoverflow.net/questions/32368/is-monomorphism-going-in-both-directions-sufficient-for-isomorphism Comment by Seamus Seamus 2010-07-18T21:38:31Z 2010-07-18T21:38:31Z Right, so the problem is more that I'm misunderstanding what sort of work the categorical subobject idea is doing. But I'm glad I was right about monomorphisms... http://mathoverflow.net/questions/32368/is-monomorphism-going-in-both-directions-sufficient-for-isomorphism/32379#32379 Comment by Seamus Seamus 2010-07-18T21:35:44Z 2010-07-18T21:35:44Z Here's the worry: in the two object category I defined above, each of A and B is a subobject of the other, but they are not isomorphic. Is this just a case where the categorical notion of &quot;subobject&quot; doesn't make sense, or have I misunderstood? http://mathoverflow.net/questions/32368/is-monomorphism-going-in-both-directions-sufficient-for-isomorphism/32379#32379 Comment by Seamus Seamus 2010-07-18T20:15:16Z 2010-07-18T20:15:16Z I don't know what you mean by &quot;the morphisms ... over X are uniquely determined&quot; http://mathoverflow.net/questions/32368/is-monomorphism-going-in-both-directions-sufficient-for-isomorphism/32379#32379 Comment by Seamus Seamus 2010-07-18T17:43:32Z 2010-07-18T17:43:32Z So to get this straight: the definition of subobject requires that the monomorphism be unique? http://mathoverflow.net/questions/32368/is-monomorphism-going-in-both-directions-sufficient-for-isomorphism Comment by Seamus Seamus 2010-07-18T17:41:25Z 2010-07-18T17:41:25Z The comment about subobjects is this. The definition of a subobject of an object A is: an object B with a monomorphism from B to A. So in my example, A is a subobject of B and B a subobject of A, but they aren't isomorphic. This seems weird, right? http://mathoverflow.net/questions/31358/can-a-mathematical-definition-be-wrong/31361#31361 Comment by Seamus Seamus 2010-07-15T14:50:12Z 2010-07-15T14:50:12Z The first appendix to Proofs and Refutations contains another example: definitions of continuity. http://mathoverflow.net/questions/4442/is-there-a-theorem-that-says-that-there-is-always-more-than-one-way-to-continue/4452#4452 Comment by Seamus Seamus 2010-03-21T17:59:39Z 2010-03-21T17:59:39Z Nice characterisation of a similar point here: <a href="http://qntm.org/1111" rel="nofollow">qntm.org/1111</a> http://mathoverflow.net/questions/18275/existence-of-convergent-subsequences-for-all-values-in-range Comment by Seamus Seamus 2010-03-15T15:37:19Z 2010-03-15T15:37:19Z Yes, I meant properties &quot;like&quot; mixing and ergodic. Sequences with the property I'm talking about seem to &quot;bounce all over the place forever&quot; in much the same way mixing functions do... http://mathoverflow.net/questions/18275/existence-of-convergent-subsequences-for-all-values-in-range/18278#18278 Comment by Seamus Seamus 2010-03-15T15:34:09Z 2010-03-15T15:34:09Z Is being dense in the interval enough to guarantee that there is a subsequence that converges to any point in the interval? I wouldn't have thought so... http://mathoverflow.net/questions/16684/when-is-the-product-of-a-set-of-numbers-greater-than-the-sum-of-them/16687#16687 Comment by Seamus Seamus 2010-03-01T14:34:57Z 2010-03-01T14:34:57Z Ah, I see the importance of set vs sequence now. Maybe I do mean sequence to allow for duplicates... http://mathoverflow.net/questions/16684/when-is-the-product-of-a-set-of-numbers-greater-than-the-sum-of-them Comment by Seamus Seamus 2010-03-01T14:32:38Z 2010-03-01T14:32:38Z I was in fact interested in both questions. I tagged number theory because I didn't know what the appropriate arXiv tag would be for the general reals question. I don't think anything hangs on my saying set rather than sequence? Surely &quot;set&quot; makes the question more general, but perhaps certain properties of sequences allow better answers in certain cases...