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' |
while read ; do
MAC=${REPLY:0:2}:${REPLY:2:2}:${REPLY:4:2}:${REPLY:6:2}:${REPLY:8:2}:${REPLY:10:2};
echo Received Address: $MAC
if ( "$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 <q <2 $, 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 $ W2.1We were done. However, in general we cannot approach even in $ W 1,1 $ Without staying anywhere. (See also this comment).

Two more comments:

  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.

Functional analysis: the positive part of the Cauchy sequence of sobolev functions is Cauchy again

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
$$
f_ {k} ^ {+} (x) = text {max} {f_ {k} (x), 0 }?
$$

Regards.