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Having a bit o' fun with math.


Jun
27
comment Show $\lim_{n_\rightarrow \infty}\int_0^\infty ne^{-\frac{2n^2x^2}{x + 1}}dx = \infty$
(Oops, my comment had a silly mistake.) Your answer quite decisively settles the issue.
Mar
28
comment Limit of the ratio of the logs, knowing the ratio
I'd be surprised if you could say much about the relationship between the two ratios in general.
Mar
28
comment Limit of the ratio of the logs, knowing the ratio
Not necessarily: consider $f(x)=x^a$, $g(x)=x^b$ with $a<b$. Then the first limit is zero but the second limit is again $a/b$ which gives us any positive value $<1$...
Mar
28
comment Limit of the ratio of the logs, knowing the ratio
Suppose $f(x)=x^a$ and $g(x)=x^b$, then if $a>b$ the first limit is $\infty$ but the second limit will be $a/b$ - this we can achieve any positive value for the second limit. If $f(x)=e^x$ and $g(x)=x$ then both limits are $\infty$.
Apr
18
comment On the equation $3a^2-4b^3=7^c$
BTW Twh article "Computing integral points on Mordell’s elliptic curves" by Gebel, Petho, Zimmer shows that the torsion groups are trivial (Prop.3.1) and the article may give the tools to complete the proof of the result.
Mar
31
comment On the equation $3a^2-4b^3=7^c$
There are solutions beyond the trivial $(\pm1,-1,1)$, for example $(\pm13,5,1)$, as well as families of solutions --- if $(a,b,c)$ is a solution, then clearly $(a\cdot 7^{3m}, b\cdot 7^{2m}, c+6m)$ is also a solution for $m>0$.
Mar
17
comment Using Black-Scholes Equation to “buy” stocks
Yes, and if you want to take into account the risk of bankruptcy, you can model the company as a barrier option...
Mar
17
comment Using Black-Scholes Equation to “buy” stocks
Short answer is that Black-Scholes is not likely to help you. You may want to look into models of company fundamentals. B-S is a model for pricing derivative instruments (assuming features of the stock, get value of options). [Note that nothing I say should be construed as giving financial advice. I am not a financial advisor.]
Mar
17
comment [Model Theory] Problem
No complaints with that one.
Mar
17
comment [Model Theory] Problem
Elementary equivalence is not the same thing as isomorphism, so this is not sufficient. (Two structures need not even be the same cardinality to be elementarily equivalent.) It is not clear that the assertion that a monoid is generated by a single element is expressible in a first-order sentence in the language with $0$ and $+$.
Mar
10
comment What does a hat or star means in math?
Absolutely! I guess the point is that there is no "standard" usage across all of mathematics, though there are "local" conventions. (Set theorists will have different conventions than K-theorists...)
Mar
8
comment If a basis of a topology has order no greater than the weight, is it contained in all other bases?
I would assume it is the weight of the space, the minimum cardinality of a basis.
Mar
2
comment How to compute the series $\sum_{n=0}^\infty q^{n^2}$?
The series itself is obviously very quickly converging and using a partial sum gives a good estimate. You can use the identity $\theta(-1/\tau) = (-i\tau)^{1/2}\theta(\tau)$ where $\theta(\tau) = \sum_{n=-\infty}^\infty q^{n^2}$ with $q=e^{\pi i \tau}$ to move $q$ near $1$ to near $0$ to get faster convergence for those more slowly convergent points.
Feb
14
comment What is an efficient algorithm to compute modular exponentiation of stacked exponents?
This is Fermat's little theorem: if $\gcd(b,m)=1$ then $b^{m-1}\equiv 1 \pmod{m}$.
Feb
12
comment Cardinality of all lines on $\mathbb R^{2}$ which do not contain point $(x,y)\in l$ where $x, y \in \mathbb Q$
Yep, though Ma.H did mention that the upper bound was obvious, so I didn't elaborate. (Though never hurts to give the complete argument).
Feb
12
comment Cardinal arithmetic
It's the only answer, so as bad as it is... :-) (Maybe if I remove the dependence on AC I'll get a vote?)
Feb
10
comment Cardinality of all lines on $\mathbb R^{2}$ which do not contain point $(x,y)\in l$ where $x, y \in \mathbb Q$
How many irrationals are there? (If there are $\aleph_0$ rationals and $2^{\aleph_0}$ total real numbers, then ...) There is one horizontal line for each irrational. This will give you a lower bound (since there are lots of non-horizontal lines that miss rational points) on the number of lines you are looking for.
Feb
10
comment Stone-Cech compactification problem
Yeah, and for general spaces, i guess the same argument should work (if one is okay with non-Hausdorff compactification.)
Feb
8
comment Size of the closure of a set
Yes, I wouldn't be surprised if there weren't an independence result here, something like $\text{MA}+2^{\aleph_0}=\aleph_2$ implying that every separable, sequentially-compact, (+compact?) space is $\leq\aleph_2$. For example, there's the result that if $\mathfrak{c}\leq\aleph_2$ then every compact, sequentially-compact space is pseudo-radial, etc.
Feb
8
comment Size of the closure of a set
One more comment: the claim at that "For pseudoradial Hausdorff spaces X we have |X| <= d(X)^c(X) (<= 2^d(X))" (at the at.yorku.ca link) is not correct (it is the second inequality that is a problem.) Otherwise we would have a fine contradiction coming from the fact that if $\mathfrak{c}\leq\aleph_2$ then all compact, sequentially compact spaces are pseudo-radial, but my (consistent) counterexample is a separable, compact, sequentially-compact space (hence pseudo-radial) and $\mathfrak{c}=\aleph_2$ but $|X|>2^{\aleph_0}$.