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2d
comment Difference between parentheses and angle brackets in vector notation
@Zaz: My comment was meant in a light hearted way. Some folks use $]0,1[$ to indicate an open interval to avoid confusion with other notations (inner product, pairs, etc.). However, in practice there is rarely a confusion.
Jul
1
comment Are these two compositions of two functions differentiable?
It is not clear what you are asking.
Jun
29
comment Sufficient condition for $0$ to be in a closed convex hull in a Hilbert space.
@Doug: No, you just need to show that every point of $A$ is contained in the closure of $B$.
Jun
29
comment Sufficient condition for $0$ to be in a closed convex hull in a Hilbert space.
@Doug: If $0$ is not in the closure, you can find a functional $\phi$ separating the compact set $\{0\}$ from the closure of $C$. That is, there are $\alpha, \beta$ such that $\operatorname{re} \phi(x) \le \alpha < \beta \le \operatorname{re} \phi(0) = 0$ for all $x \in \overline{C}$. That is, $\sigma_C(\phi) < 0$.
Jun
28
comment derivative of a function including a vector
You notation is inconsistent, so it is difficult to guess what you are asking.
Jun
28
comment Sufficient condition for $0$ to be in a closed convex hull in a Hilbert space.
@Doug: let $\sigma_C(h) = \sup_{c \in C} \operatorname{re} \langle h, c \rangle$.
Jun
28
comment Simple Expected Value Of Continuous Variable Question
One can show it using Fubini Tonelli. Here is a similar question: math.stackexchange.com/q/1329112/27978.
Jun
28
comment Is $K =\{ S: \exists \text{ positive diagonal} D, D^TSD \;\text{diagonally dominant}\}$ convex?
$K$ is trivially convex because $0 \in K_0$ and $0^T S 0 = 0 \in K_0$ for all $S$. Presumably you have some other constraints in mind?
Jun
27
comment Map 1 to 1 and 0 to -1.
$x \mapsto -1+2x$.
Jun
27
comment Regarding linear dependence and independence for finite sequences of vectors
The authors were smoking something at the time.
Jun
27
comment Is u • v equal to |u • v|?
It might help to use $\|u\|$ rather than $|u|$ for norms, and save $|\cdot|$ for absolute values...
Jun
26
comment Prove that $I_n=\int_0^{\frac{\pi}{4}} \tan^{2n}(t)\, dt$ is convergent to $0$
Please clarify the question.
Jun
26
comment what is the geometry behind the matrix multiplication?
@gloom: On reflection I'm not so sure that the above a good approach. I was thinking of matrix multiplication as in multiplying a single vector as opposed to another matrix. Obviously matrix multiplication can be interpreted as multiplying the columns, but I don't think this will clarify anything for 11th graders.
Jun
25
comment $n! \le n^n$, $\forall$ $n \ge 1$.
Try induction, dividing, anything.
Jun
24
comment How can we prove the following limit property where n is any real number.
Does the limit apply to $x$?
Jun
24
comment How do I derive $n!$ from this series?
This seems to be an evolving question...
Jun
24
comment How do I derive $n!$ from this series?
This follows by grouping like terms. Then use $(n-k) \le (n-k-l)$ (for appropriate $k,l$, of course).
Jun
24
comment What's the name of this type of a set?
@PaddlingGhost: They are phisomorphic.
Jun
24
comment What's the name of this type of a set?
Note that $\emptyset = \{ \}$ and $\{ \emptyset \}$ are different...
Jun
24
comment $G$ open connect subset of $\mathbb{C}$ and $f: G \to \Bbb C$ analytic.
There is only a forward implication here. You just need to prove that if $f(z) \neq 0$ on $G$, then $f$ is a non zero constant. Since $f$ is analytic on $G$, it is an open map. Hence $\{|f(z)|\}_{z \in G}$ is an open interval.