Hi guys, I need to compute Taylor expansion of $sin(\pi z^2)$ centred in 1 with order 3.
I get:
$\pi z^2=\pi+2\pi(z-1)+\pi(z-1)^2$
$sin(y)=y-\frac{y^3}{3!}+o(y^3)$
When I substitute $y$ with $\pi+2\pi(z-1)+\pi(z-1)^2$ I have to do a lot of calculations. I need to do this expansion during an exam, so I need velocity. This way is good? Exist another faster way?
Let's say I have a collection of objects. Each object has a set of predictive attributes where the attribute may be true or false. Each attribute when true predicts "success" with a known probability, and these probabilities are not necessarily the same for different attributes. What is the pr...
Your proof is correct.
Your claim can be rephrased in the following manner:
Let $X,Y$ be topological spaces. $f:X\rightarrow Y$ is continuous if and only if $f:X\rightarrow f(X)$ is continuous.
I will provide two proofs.
Proof 1:
Forward Direction:
If $f: X\rightarrow Y$ is continuous, then...
Let $G$ be some locally compact group and $\mu$ its associated Haar measure. I am trying to adapt this proof that convolution on the locally compact group $(\Bbb{R},+)$ is associative. Here's what I have so far:
$$((f \ast g) \ast h)(u) = \int_{G} (f \ast g)(x)h(x^{-1}u) ~d \mu (x)$$
$$= \int_{...
If there are gaps in rational numbers then lets assume we have a gap between a and b, both being rational. Then we have $\frac{a+b}{2}$ which is inside the gap which essentially makes it a non-gap. What am I getting wrong?
