Suppose a $3 \times 3$ real matrix $A$ is not similar to any upper-triangular matrix on the real field $\mathbb{R}$, that is, there is no $3 \times 3$ invertible real matrix $P$, such
that $P^{-1}AP$ is an upper-triangular matrix. Prove that $A$ is similar to a diagonal matrix on the complex...
Apologies if I am just having a mental block and missing something very obvious. Here is a conjecture that I think is obviously true, and yet I cannot prove it:
Let $X_1, X_2, \ldots, X_n, Y, Z$ be mutually independent, real-valued, non-constant random variables. (They need not be identically ...
This question, with the "random variable" and "independent" keywords in the title, was suggested with the randome-variables and independence, rejected with almost trivial edit of the post.
I need to find the following limit:
$$\lim_{(x,y) \rightarrow (0,0)} \frac{\cos(xy) -1}{x^2 y^2}$$
I did the following:
Let $f(x) = \dfrac{\cos(x)-1}{x^2}$ and $g(x,y) = xy$. Then we would have $$\lim_{(x,y) \rightarrow (0,0)} \frac{\cos(xy) -1}{x^2 y^2} = \frac{\cos(\lim_{(x,y) \rightarrow ...
And surely, limit is a tag for this question again, rejected by the same user with a rather trivial edit of using a "displaystyle" for one single formula that is alreay very clear for reading.
This is the fifth incidence of nonsensical rejection of proper taggings in a single day by one single user.
5 hours later…
user185131
5:49 AM
@T.S Yes, these cases look problematic to me, too. Do flag, and maybe also create a bookmark for this series of messages and link to it in the mod flag?
In calculus, Leibniz's rule for differentiation under the integral sign, named after Gottfried Leibniz, states that for an integral of the form
∫
a
(
x
)
b
(
x
)
f
(
x
,
t
)
d
t
,
{\displaystyle \int _{a(x)}^{b(x)}f(x,t)\,dt,}
where
−
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a
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x...
I need to find the following limit:
$$\lim_{(x,y) \rightarrow (0,0)} \frac{\cos(xy) -1}{x^2 y^2}$$
I did the following:
Let $f(x) = \dfrac{\cos(x)-1}{x^2}$ and $g(x,y) = xy$. Then we would have $$\lim_{(x,y) \rightarrow (0,0)} \frac{\cos(xy) -1}{x^2 y^2} = \frac{\cos(\lim_{(x,y) \rightarrow ...
