## real analysis: how to show that a family of open sets defined for a topology for \$ C (S) \$

Consider a subset $$xi$$ from $$C (S)$$, $$S subseteq mathbb {R}$$. let say $$f_0$$ in $$xi$$ is inside to $$xi$$ if there is a finite subset $$F$$ from $$S$$ and $$epsilon> 0$$ such that
$${f in C (S): | f (x) -f_0 (x) | < epsilon, x en F } subseteq xi$$
Set $$xi$$ is open if each function in $$xi$$ is inside to $$xi$$.

Question: How to show that the open sets defined above form a topology for $$C (S)$$

My thought:

I can't use the topology defined by the metric because they are different. I have no idea where to start.

Any help would be appreciated.

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## risk analysis – Netcat on Debian safety tips, please nc – (dknl)

This refers to a Netcat-based script that runs on a Debian-based distribution, specifically the Proxmox hypervisor (see here if https://en.wikipedia.org/wiki/Proxmox_Virtual_Environment is unknown)

You would need to run a script to start a virtual machine from a remote PC within the network. The script that runs in this Proxmox (Debian) distribution is as follows:

**nc -dknl -p 9 -u |**

stdbuf -o0 xxd -c 6 -p |
stdbuf -o0 uniq |
stdbuf -o0 grep -v 'ffffffffffff' |
if ( "\$MAC" == "0c:d2:92:48:68:9b" )
then echo STARTING VM!
qm start 101   # Proxmox Command to start Virtual machine.
fi
done

Could the Debian running over the script be exploited, as Netcat listens on port 9 UDP (of course, I could also hear another port if I change it). Naturally, anyone on the network could start a VM, but is there another risk?

## Analysis – Solve the following equations in C

1)$$(z * bar z) (z ^ 2 – 2i bar z) = 0$$

two)$$( bar z + 2i z) (z ^ 4 + 16) = 0$$

I've been trying to solve this all day. I have deep problems with complex numbers and tomorrow I have an exam about it. Please, I really need help.
You should not use the Euler form during the solution.

## real analysis: prove that \$ f (x) = x ^ {1/3} \$ is absolutely continuous.

Here is a test:

We will divide our interval into 2 intervals. $$(- 1, 0)$$ and $$(0.1)$$:

(1) for the interval $$(- 1, 0)$$, our function is increasing in $$(- 1.0)$$ then it is differentiable a.e. in $$(- 1.0)$$ for Lebesgue's theorem on p.112 and its derivative is $$f & # 39; (x) = frac {1} {3} x ^ {2/3}$$ and

our function $$f$$ it's continuous

(2) for the interval $$(0, 1)$$.

We will do this with the help of problem 37 on p. 123. Let $$epsilon> 0.$$ taking $${(c_ {i}, d_ {i}): 1 leq i leq n }$$ be a collection of non-overlapping intervals in $$(0.1)$$ such that $$sum_ {i = 1} ^ {n} (d_ {i} – c_ {i}) < epsilon ^ 3.$$ Choose $$a = frac { epsilon ^ 3} {8}.$$ Now we break the sum $$sum_ {i = 1} ^ {n} | f (d_ {i}) – f (c_ {i}) |$$ in two parts, those intervals that are in $$(0, a)$$ and those in $$(a, 1).$$ that is, we break the interval in $$to$$ Similar to $$eqn. (19)$$ on page 117. Let $$a = d_ {m}$$ for some $$m.$$

Now consider the sum over the intervals that are in $$(0, a),$$ $$sum_ {i = 1} ^ {m} | f (d_ {i}) – f (c_ {i}) | = sum_ {i = 1} ^ {m} | d_ {i} ^ {1/3} – c_ {i} ^ {1/3} | leq a ^ {1/3} = epsilon / 2.$$

This follows from the fact that $$x ^ 1/3$$ It is a growing function. Monotonicity ensures that the function does not oscillate widely.

Now we consider the sum of the intervals that are in $$(a, 1).$$

$$sum_ {i = m + 1} ^ {n} | f (d_ {i}) – f (c_ {i}) | = sum_ {i = m + 1} ^ {n} | d_ {i} ^ {1/3} – c_ {i} ^ {1/3} | = sum_ {i = m + 1} ^ {n} | d_ {i} ^ {1/3} – c_ {i} ^ {1/3} | times frac {| (d_ {i} ^ {2/3} + (d_ {i} times c_ {i}) ^ {1/3} + c_ {i} ^ {2/3}) |} {| (d_ {i} ^ {2/3} + (d_ {i} times c_ {i}) ^ {1/3} + c_ {i} ^ {2/3}) |}$$
$$= sum_ {i = m + 1} ^ {n} frac {(d_ {i} – c_ {i})} {| (d_ {i} ^ {2/3} + (d_ {i} times c_ {i}) ^ {1/3} + c_ {i} ^ {2/3}) |}$$

So,$$sum_ {i = m + 1} ^ {n} | f (d_ {i}) – f (c_ {i}) | leq sum_ {i = m + 1} ^ {n} frac {(d_ {i} – c_ {i})} {8 a ^ {2/3}} = ** frac {1} {8 a ^ 2/3} ** sum_ {i = m + 1} ^ {n} (d_ {i} – c_ {i}) < frac {1} {2 epsilon ^ 2}. epsilon ^ 3 = frac { epsilon} {2}.$$

We use $$(xy) = (x 1/3 – y 1/3) (x 2/3 + (xy) 1/3 + y 2/3).$$
Combining these two sums we see that

$$sum_ {i = 1} ^ {n} | f (d_ {i}) – f (c_ {i}) | leq ( sum_ {i = 1} ^ {m} | f (d_ {i}) – f (c_ {i}) | + sum_ {i = m + 1} ^ {n} | f (d_ { i}) – f (c_ {i}) |) < epsilon.$$

My question is:

1-It turns out that this part of my judgment $$frac {1} {8 to 2/3}$$ Is it wrong, could someone help me adjust it, please?

Note that the previous test is based on Royden's fourth edition "Real Analysis."

2-In addition, the general idea of ​​the interval test $$(0.1)$$ It is not clear to me, could anyone explain it, please?

3 interval for $$(- 1.0),$$ Could someone help me complete it, please?

## pdes analysis – Solubility theory for the magnetic Laplacian

I am looking for some references on the existence of solutions to the equations of the form
$$– Delta ^ A psi + B psi = g qquad text {in} mathbb {R} ^ 2$$
where $$Delta ^ A$$ is the Laplacian magnetic operator, defined for a fixed divergence free vector field $$A = (A_1, A_2)$$ as
$$Delta ^ A = ( partial_j-iA_j) ^ 2 = Delta- | A | ^ 2-2iA cdot nabla,$$
$$psi, g: mathbb {R} ^ 2 to mathbb {C}$$ and $$B: mathbb {R} ^ 2 to mathrm {Mat} _ {2 times2} ( mathbb {R})$$. This should be equivalent to a system of two elliptical PDEs throughout the space. All the books and articles on elliptical PDEs that I have tried to verify do not address the case of the system, so any reference on this subject would also be greatly appreciated.

## Complex analysis: show that the Möbius transformation assigns each straight line to a line or circle, and assigns each circle to a line or circle

I found this statement on wiki:

These transformations preserve angles, map each straight line to a line or circle, and map each circle to a line or circle.

Speaking of the Möbius transformation, I found proof of the invariability of the angles, but nothing else. Does anyone know how we can prove it?

## complex analysis: operator integrals in the measurable Riemann mapping theorem

I'm reading Alhors's proof of the measurable Riemann mapping theorem and I don't know how to prove that if $$alpha neq 0$$what for $$1 , the next function is an element of $$L q$$:
$$f (z) = frac {1} {z (z – alpha)}$$
Someone know how to I can do?

## real analysis: approximates a unique form on the disk without disappearing anywhere the unique forms that satisfy an asymptotic disappearance of some derivatives

Leave $$mathbb {D} ^ 2$$ be the closed two-dimensional disk drive, and leave $$g: mathbb {D} ^ 2 a mathbb {R}$$ be a non constant harmonic function (soften to the limit).

Is there a sequence of smooth shapes? $$sigma_n$$ in $$mathbb {D} ^ 2$$ such that

1. $$sigma_n to dg$$ in $$L ^ 2$$.
2. $$sigma_n$$ not disappear in $$mathbb {D} ^ 2$$.
3. $$| delta d sigma_n | _ L ^ 1} a 0, | d delta sigma_n | L 1 to 0$$

That is, I want to approximate the unique way $$dg$$, which may have zeros in the domain, without disappearing forms, so that certain second derivatives become insignificant in the limit.

I know how to do that without achieving the third condition.

Let me explain a little more about that condition:

to write $$sigma = dg in Omega ^ 1 ( mathbb {D} ^ 2)$$. As $$g$$ it's harmonic $$delta sigma = 0$$. As $$sigma$$ it's accurate, we also have $$d sigma = 0$$. In fact, for a unique way $$sigma in Omega ^ 1 ( mathbb {D} ^ 2)$$, $$d sigma = delta sigma = 0$$ it is equivalent to the existence of a harmonic function $$g$$ in $$mathbb {D} ^ 2$$ such that $$sigma = dg$$.

Therefore, our desired limit form $$sigma = dg$$ satisfies $$| delta d sigma | L ^ 1 = | d delta sigma | L 1 = 0$$.

Therefore, if we could approach $$sigma$$ disappearing nowhere $$sigma_n$$ in $$W2.1$$We were done. However, in general we cannot approach even in $$W 1,1$$ Without staying anywhere. (See also this comment).

1. Trying to approach (in $$L ^ 2$$) $$g$$ by harmonic functions $$g_n$$ The differential does not disappear, nor does it work: this creates an approach that is too fast, due to the stiffness properties of the harmonic functions, hitting the topological obstruction again.
2. Even trying to approximate (again in $$L ^ 2$$) $$sigma = dg$$ with $$sigma_n$$ satisfactory $$d delta sigma_n = 0, delta d sigma_n = 0$$ can't work: yes $$sigma_n = f ^ 1_ndx + f ^ 2_ndy$$, then these conditions imply that $$f ^ 1_n$$ and $$f ^ 2_n$$ they are harmonic, which again implies a convergence too fast.
Leave $$p geq 1$$ and consider the space $$W 1, p (B)$$ where $$B subset mathbb {R} ^ {n}$$ It is the standard ball of the unit. On the other hand, let's leave $$f_ {k} in C ^ { infty} (B)$$ be a Cauchy sequence in $$W 1, p (B)$$ Soft function How can you deduce that too $$f_ {k} +$$ is a Cauchy sequence in $$W 1, p (B)$$, where $$f_ {k} +$$ are defined by
$$f_ {k} ^ {+} (x) = text {max} {f_ {k} (x), 0 }?$$