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1h
comment continuity of the complex square root function
It might be easier to consider $w=e^{i \theta}$ and show that the hypotheses imply that $f(e^{i \theta}) = K e^{i { \theta \over 2}}$, where $K \in \{\pm 1\}$. Then consider $\theta \to \pm \pi$.
14h
comment Difference between measure zero and volume zero?
Well, volume zero implies measure zero and there are sets that have measure zero but no volume. So, I would just say that having volume is more restrictive rather than stronger. I'm not sure there is a nice geometric interpretation for 3). Loosely, volume requires that it behave nicely in the sense that the limits over finite partitions converge, where as measure is much more versatile, since, again loosely, it allows countable operations. The example you gave above, the rationals, is a nice example.
15h
comment Difference between measure zero and volume zero?
For 1), you have only defined measure zero, defining the (Lebesgue) measure takes more machinery. Without getting into details, a function is Riemann integrable iff the set of points of discontinuity has measure zero. Hence a set has volume iff its boundary has measure zero. Having volume is more restrictive than having measure.
15h
comment Decomposition of a measure
Well, then you will have a continuous function $F_c$ (that is piecewise linear, hence AC), and the difference $F_s = F-F_c$ which is the singular part.
17h
comment When is a series of sums the sum of the series?
If $s_n \to s, t_n \to t$ then $s_n+t_n \to s+t$.
17h
comment Show that this function is differentiable at all points
What function? All I see in a link?
17h
comment Decomposition of a measure
Draw $F$ and figure out what you have to do to remove the discontinuities.
18h
answered Find image of complex set:
20h
comment Find image of complex set:
Look at the images of $[0,\infty)$ and $i[0,\infty)$.
23h
comment about derivative of a matrix and trace
I'm not sure it helps, but I added some more detail above. In terms of the question, this is complete overkill, but the author may be leading up to something else, so I can't guess as to what her context was.
23h
revised about derivative of a matrix and trace
added 1233 characters in body
1d
comment about derivative of a matrix and trace
Note that if $s$ is a scalar, then $ s = \operatorname{tr} s$. Perhaps the author has derived the trace gradient earlier and would like to use it rather than the above expansion?
1d
comment about derivative of a matrix and trace
It is a long winded way of proving the result.
1d
comment about derivative of a matrix and trace
Taking the gradient with respect to $A^T$ is a felony in the USA.
1d
answered about derivative of a matrix and trace
1d
comment $d_1(x,y)=|x-y|$ & $d_2(x,y)=|\arctan(x)-\arctan(y)|$ equivalent on $\mathbb R$?
Note that they induce the same topology on $\mathbb{R}$, but are very different metrics. With $d_1$, the space is complete, with $d_2$ it is not.
1d
comment $d_1(x,y)=|x-y|$ & $d_2(x,y)=|\arctan(x)-\arctan(y)|$ equivalent on $\mathbb R$?
Try the sequences $x_n = n, y_n = n+1$.
1d
comment How to find in a Stability of Linear Systems. BIBO Stable, but Lyapunov Unstable System
I don't want to discourage you, but you may want to work a bit on your basic linear algebra first, as it is fairly fundamental to control theory.
1d
comment How to find in a Stability of Linear Systems. BIBO Stable, but Lyapunov Unstable System
Just solve it any way at all. Subtracting the two equations gives $3 c_2 = x_{20}-x_{10}$. Then solve for $c_1$.
2d
comment Riemann-esque sums (complex analysis)
@JohnnyBreen: A slight laziness on my part: The way I defined $\tilde{\phi}$, we have $\tilde{\phi}(t_1) = 0$, which may be quite different from $\phi(t_1)$ (since I used $[s_k,s_{k+1})$). This could be repaired easily by adding $\phi(s_n) 1_{ \{ s_{n} \}}$ to the definition of $\tilde{\phi}$, but I was lazy :-). Since we only care what happens under the integral, it doesn't matter.