X is Compact and F is Continuous and Bijective Show That F1 is Continuous
Let $f : X \to Y$ be a bijective continuous function. Show that if $X$ is compact, then $f$ is a homeomorphism.
- Thread starter JustANoob
- Start date
- #1
JustANoob Asks: Let $f : X \to Y$ be a bijective continuous function. Show that if $X$ is compact, then $f$ is a homeomorphism.
I'm studying for a topology exam and encountered the following problem:
I struggled to solve it, so I looked up Munkres and found the following theorem (Th 26.6):
This leads me to think that maybe the first statement is false. What do you think?
- Mohamed Abduljawad
- Computer Science
- Replies: 0
Mohamed Abduljawad Asks: pd.get_dummies One-Hot Encoding but Keep the Order of Appearence
So I have this data which which has the 12 months as below: 
When I one-hot encode it using pd.get_dummies the result is this: 
You see that the new columns are ordered alphabetically losing the months order, which makes reading the data hard. Is there any way to preserve the order?
- Watson
- Mathematics
- Replies: 0
Watson Asks: Non-constant elliptic curves with everywhere good reduction over function fields
It is well-known that an elliptic curve over $\mathbb Q$ cannot have good reduction everywhere, but it is known that there are elliptic curves over number fields (like Tate's example over $\mathbb Q(\sqrt{29})$, or see here) having good reduction everywhere.
I am interested by finding such examples over global function fields:
(If the answer is yes, a follow-up question would be: can we find a non-isotrivial example, i.e., $j(E) \not\in k$?)
Thoughts:
Note that a constant elliptic curve $E_0 \times_k k(C)$ always has good reduction everywhere. For abelian varieties of dimension $g \geq 2$, it seems to be possible even in the non-isotrivial case (see here).
There is no example over $C = \mathbb P^1$. In the linked question it is claimed that "The case of elliptic curves is elementary", but I do not think so (especially if $\mathrm{char}(k) \in \{ 2, 3 \}$). This done in detail in this paper.
- Sick Nutmeg
- Mathematics
- Replies: 0
Sick Nutmeg Asks: Area enclosed between the roots of a quadratic
Let $f(x)=ax^2+bx+c$
If $f(x)$ has roots $\alpha$ and $\beta$, what is the area enclosed by $f(x)$ and the $x$-axis between $x=\alpha$ and $x=\beta$ in terms of $a,b$ and $c$?
It is also given that $\alpha>\beta.$
If $a=1$, then I thought this might be easier since you get:
$$A=\int_\beta^\alpha{(x^2-(\alpha+\beta)x+({\alpha}{\beta})x) dx}$$
But even after evaluating this, I still wasn't even able to find an answer in terms of the coefficients.
I've been at this problem for a while now, and I would love some help. Any ideas?
- ilovemath
- Mathematics
- Replies: 0
ilovemath Asks: $3$-var inequality: $\frac{bc}{\sqrt{a}+3}+\frac{ca}{\sqrt{b}+3}+\frac{ab}{\sqrt{c}+3} \leq \frac{3}{4}$ for $a+b+c=3$.
Problem: Let $a,b,c$ be positive numbers satisfied $a+b+c=3$. Prove that $$\dfrac{bc}{\sqrt{a}+3}+\dfrac{ca}{\sqrt{b}+3}+\dfrac{ab}{\sqrt{c}+3} \leq \dfrac{3}{4}$$
I've tried U.C.T method but it doesn't reach the solution. I also thought of $p,q,r$ method, but I think it will end up a ugly solution. The only useful thing I get is $\sqrt{a}+\sqrt{b}+\sqrt{c} \geq ab+bc+ca$. Anyone?
- JPRP
- Mathematics
- Replies: 0
JPRP Asks: How to construct an isomorphism?
I know the group $\mathbb{Z}_3 \times \mathbb{Z}_3 \times \mathbb{Z}_2$ is isomorphic to $\mathbb{Z}_3 \times \mathbb{Z}_6$ (since $\mathbb{Z}_3 \times \mathbb{Z}_2$ is isomorphic to $\mathbb{Z}_6$ since $2$ and $3$ are coprimes).
But how would I construct such an ismorphism?
- mark3292
- Social
- Replies: 0
mark3292 Asks: Why do people always talk about stocks that pay high dividends?
Isn't it true that on the ex-dividend date, the price of the stock goes down roughly the amount of the dividend? That is, what you gain in dividend, you lose in price drop.
(Of course, sometimes it just happens that the price does not drop as much as the dividend is worth, but then again there are times when the price actually drops more, so the average drop is equivalent to the dividend amount.)
Why is everyone making a big deal out of the amount that companies pay in dividends then? Why do some people call themselves "dividend investors"? It doesn't seem to make much sense.
Edit: Before you answer, please have a look at the most common myths debunked http://money.usnews.com/money/blogs...14/02/04/7-myths-about-dividend-paying-stocks
- Kellopeli
- Education
- Replies: 0
Kellopeli Asks: Causal inference when no obvious theory on causal model is present
A theoretical question. I am tasked to estimate the effectiveness of treatment A in comparison to treatment B. Let's say these treatments are in the field of risk management so their targets are anything but randomly chosen. However, the reasons for choosing treatment A over treatment B are opaque and differ from case to case so there are no obvious set of covariates. We do have a hunch of which attributes have a say in treatment selection so we have created a dataset containing ~ 40 covariates. Half-decent model can be built using these to predict whether observation X gets treatment A or B so propensity scores are a good and possible option and we are clearly not shooting in the wild.
I don't have a lot of data so I prefer to use methods which don't lose data (so no CEM matching for example) so have used different weighting schemes for weighted regression and different propensity score matching-set ups, also tried matching on principal components. Double ML and causal forests produced really bad results and basically were unable to differ from the raw, unbalanced difference in results between treatments so those I no longer consider.
Here comes my question: can I just opportunistically try a whole bunch of different covariate balancing schemes and just pick the one which seems to produce the best results, eg most balanced data set? Or should I first try to narrow down the set of covariates on which the balance is desired? (balancing a covariate which is in no way related to getting treatment is moot). When different schemes might produce even quite wildly different estimates, how to gauge which is the most plausible one?
I do know that treatment A produces better results than treatment B, both data and theory support it. Now just to find the most plausible estimate on exactly how much and to be able to defend my choices 
bookermorpegir1964.blogspot.com
Source: https://solveforum.com/forums/threads/let-f-x-to-y-be-a-bijective-continuous-function-show-that-if-x-is-compact-then-f-is-a-homeomorphism.1445770/
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