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Jan
29
revised Is “Kernel Matrix” a bad choice of wording?
added 72 characters in body
Jan
29
answered Is “Kernel Matrix” a bad choice of wording?
Jan
27
comment integrals with no analytic answer - intuition and proof
Fair enough, although there's no serious obstruction to defining the class of "elementary functions with rational constants." Is there any agreed-upon notion of which constants are "elementary," short of computability? (Obviously it would usually be nearly impossible to prove that a given number didn't belong to such a class, but we could still have an idea of what the appropriate class ought to be.)
Jan
24
revised integrals with no analytic answer - intuition and proof
added 21 characters in body
Jan
24
comment integrals with no analytic answer - intuition and proof
You're right, of course; I wasn't paying enough attention. I don't even know what the relevant notion is for a number to have a "closed form" -- maybe a value of an elementary function at a rational value of x?
Jan
24
answered integrals with no analytic answer - intuition and proof
Jan
23
comment What can go wrong if we let sigma algebra to admit the union of uncountable union of elements?
A sigma algebra on a set $X$ is, by definition, a subset $\Sigma \subseteq \mathcal{P}(X)$ which is closed under countable unions and intersections. The question is about changing that definition to "closed under arbitrary unions and countable intersections." So essentially every sigma algebra commonly used would be an example of a sigma-algebra that's not closed under uncountable unions.
Jan
23
answered “The following are equivalent” for only two statements
Jan
23
answered What can go wrong if we let sigma algebra to admit the union of uncountable union of elements?
Jan
23
awarded  Nice Question
Jan
22
comment Can you construct a field of characteristic $\neq 0, 2$ such that every one of its subrings is also a field?
Also, this is impossible in characteristic zero, since every characteristic zero field contains a copy of the integers.
Jan
22
comment Difference between pure probability and measure-theoretic probability
Did people even read the question before voting to close? It's asking for a summary of a mathematical field.
Jan
21
comment $A\subseteq B$ integral domains with surjective multiplication, then the localization by all monic polynomials evaluated at some point is nonzero
What does $v_b(S)^{-1}$ mean?
Jan
21
answered Difference between pure probability and measure-theoretic probability
Jan
15
answered probability of left maximum
Jan
14
comment Non-equivalent metrics on $PSL_2(\mathbb{R})$
Two questions: what do you mean by "metric," and what do you mean by "equivalent" here?
Jan
14
comment Geometric interpretation of point set topology
Graph theory is not quite topology in the modern sense of the word, but it's closely connected. Algebraic topology can in some sense be thought of as the study of higher-dimensional generalizations of graphs, although this study is largely informed by geometric considerations more than graph-theoretic ones. But yes, there is no especially close connection between graphs and point-set topology.
Jan
14
comment Geometric interpretation of point set topology
But if you can prove a theorem for an arbitrary topological space, why add extra assumptions just for the sake of keeping your theorem from applying to a wider range of spaces that might include some that you don't really understand?
Jan
14
comment Geometric interpretation of point set topology
The vast majority of topological spaces don't have any geometric meaning, at least any we're aware of. I mean, there are 139 distinct topological spaces with only five points. What does that mean geometrically? If something isn't at least locally compact and Hausdorff it probably doesn't have any meaning in the same sense that, say, a "nice" subset of $\mathbb{R}^n$ does. OTOH there are totally different and incompatible ways of describing geometry with topology, like the Zariski topology on a variety or complex manifold.
Jan
14
answered Geometric interpretation of point set topology