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1d
comment Understanding the definition of nowhere dense sets in Abbott's Understanding Analysis
For arbitrary $S$ and $B$, it may happen that all the Bosnians make fences separating themselves from the Serbs (no matter what size or shape, open or closed, just as long as there are no Serbs). More Bosnians come and stand on top of the fence. Whereas Bosnians previously only roamed inside the fences, now there are also Bosnians directly along all the borders too. (Closure) $$$$ Now if none of the Bosnians find themselves in an open region of only Bosnians, then $B$ is nowhere dense in $S$. In this case the Serbs have contained the Bosnians to isolated points.
1d
comment Understanding the definition of nowhere dense sets in Abbott's Understanding Analysis
Not sure if this helps. $S$ is the Serbs and $B$ is the Bosnians. $B$ is nowhere dense in $S$ if we can take any Bosnian (who isn't also a serb; the analogy breaks a bit here), and put an open fence around him, so that throughout the inside the fence there are only Bosnians. $$$$ On the other hand if there was a place that $B$ was dense in $S$, then there would be some Bosnian who can't fence himself off from Serbs in an open way. That Bosnian would be topologically 'inseperable' in some sense from the Serbs. $$$$ Of course, precisely what `open' means depends on the topology you have.
Apr
14
revised $\arg\max$ of an increasing function grows as the region grows.
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Apr
14
comment $\arg\max$ of an increasing function grows as the region grows.
Apologies @MichaelGrant, I misrepresented my functions $f_i$. The $f_i$ don't have inflection points.
Apr
14
revised $\arg\max$ of an increasing function grows as the region grows.
added 67 characters in body
Apr
14
revised $\arg\max$ of an increasing function grows as the region grows.
added 32 characters in body; edited tags
Apr
14
revised $\arg\max$ of an increasing function grows as the region grows.
added 26 characters in body
Apr
14
asked $\arg\max$ of an increasing function grows as the region grows.
Apr
13
comment How is this definite integral solved: $\int_{-\sqrt3}^{\sqrt3}{e^x\over(e^x+1)(x^2+1)}dx $?
What have you tried>?
Feb
13
comment Finding a special determinant, $\det(I+SD)$
Whoops you're right! I was trying to pull $D$ out instead of grouping it with $h$. Thanks
Feb
13
comment Finding a special determinant, $\det(I+SD)$
Those results are impressive, but $D$ might not be invertible.
Feb
13
asked Finding a special determinant, $\det(I+SD)$
Feb
9
revised What is the sum-capacity for a non-symmetric interference channel for information theorists?
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Feb
6
answered What is the sum-capacity for a non-symmetric interference channel for information theorists?
Feb
6
revised I want to understand uniform integrability in terms of Lebesgue integration
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Feb
6
revised I want to understand uniform integrability in terms of Lebesgue integration
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Feb
6
answered I want to understand uniform integrability in terms of Lebesgue integration
Dec
20
comment Is there any significance to complex function “monotone in norm?”
(nonconstant polynomials..)
Dec
20
comment Is there any significance to complex function “monotone in norm?”
In the converse sense, there are several interesting analytic functions for which this is true. For example, all polynomials are eventually monotone in norm. $$\left.\right.$$ This is probably pedantism, but if you didn't mean to restrict your focus to analytic functions, functions of ``monotone norm'' aren't in themselves very interesting or restrictive. They could do anything. For instance, take a natural log function that is nowhere continuous (which you can construct using two or more branches). This isn't a very nice or interesting function, but it has your property.
Dec
15
comment Integral: $\int_{-\infty}^{\infty} \frac{dx}{(e^x+x+1)^2+\pi^2}$
My guess at an approach (I have not thought about the problem yet) would be to use the standard trick with the Leibniz differentiation rule to get rid of a few of the terms in the denominator.