Fourier analysis – standing wave formula

Let us consider $ u_ {tt} = u_ {xx} $ be the wave equation of a vibrating string $ (0, pi) $, Suppose, based on the individual variables $ u (x, t) = phi (x) psi (t) $ gives a solution. After some calculations, the following formulas are determined.

$$ phi (x) = lambda sin mx ~~~~, psi (t) = mu_1 cos mt + mu_2 sin mt $$

Why must $ m $ be an integer?

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Risk Analysis – Netcat on Debian Security Advice please nc – (dknl)

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

I would need to run a script to launch a virtual machine from a remote PC on the network. The script that runs on 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 be the above Debian script exploitedhow Netcat listens on port 9 UDP (could it of course listen to another port if I change that)? Of course, anyone on the network could start a VM, but is there another risk?

Functional Analysis – A Banach space of sequences in which the convergence towards zero depends only on its norm

Is there a Banach room? $ (C, |. |) $ With $ C subset mathbb {R} ^ { mathbb {Z}} $ satisfying:

for each $ varepsilon> 0 $is there $ k in mathbb {N} $ so if $ v = (v_n) _ {n in mathbb {Z}} in C $ Has $ | v | leq1 $, then
$$ | v_n | leq varepsilon , , , text {whenever} , , , | n | geq k? $$

Roughly speaking, this means that the rate of convergence of sequences in $ C $ basically depend on their norm.

Singularity analysis of a nonlinear differential equation

I have a nonlinear differential equation of the form

$$ y & # 39; & # 39; (x) + frac {y ((x)} {x} + Ce ^ {y (x)} = 0 $$

Most of what I have read online and in textbooks deals only with the classification of singularities of second-order linear ODEs, nothing with nonlinear problems. Does anyone have a good reference to this?

I think one way to achieve that would be to linearize the system and analyze it $ x = 0 $, something like that:

$$ begin {cases}
y ((x) = w (x) \
w ((x) = – frac1xw (x) + Ce ^ {y (x)}
end {cases} $$

and then find the Jakobian $ J $ at the point $ z $:

$$ J (z) = begin {bmatrix}
0 & 1 \
Ce ^ {y (z)} & – frac1z
end {bmatrix} $$

then analyze the equivalent equation $ y & # 39; & # 39; (x) + frac1zy ((x) + Ce ^ {y (z)} y (x) = 0 $, This obviously has a removable singularity $ z = 0 $but I do not know how that would relate to the original system.

fa.functional analysis – most general definition of differentiation

I think that depends on what you mean by "general" and what qualifies as a derivative. There are some purely syntactic definitions of differentiation that appear in category theory.

Cartesian differential categories axiomatize a differentiation operator that satisfies all higher order chain rules from the normal differential calculus (and every differentiation operator that satisfies these higher order chain rules yields a Cartesian differential category due to a free construction of Cockett and Seely).

Tangent categories axiomatize the differentiation function of mappings between manifolds. They can be described as categories with an effect according to the category of Weil algebras, which fulfills the same characteristics as the Weil extension in the category of smooth manifolds.

I'm writing this on my phone, so I'm posting links below:
http://www.math.mcgill.ca/rags/difftl/cartdiff-tac.pdf
http://www.math.mcgill.ca/rags/difftl/faaslide.pdf
https://www.mta.ca/uploadedFiles/Community/Bios/Geoff_Cruttwell/sman3.pdf
http://www.tac.mta.ca/tac/volumes/32/9/32-09.pdf

fa.functional analysis – Do all uniform representations in the infinite converge weakly towards zero?

question, To let $ G $ Be a non-compact, finally dimensional Lie group and leave $ (X, mu) $ be a radon measurement room. To let $$ rho colon G to U (L ^ 2 (X)) $$
be a consistent, highly continuous presentation. Is it true that if $ g_n to infty $, then
$$
int_X overline {h (x)} rho_ {g_n} f (x) , d mu to 0, qquad forall f, h in L ^ 2 (X)? $$

Reasonable hypotheses $ X $ can be accepted.

Here, $ g_n to infty $ means that for everyone compact $ K subset G $. $ g_n notin K $ big enough for everyone $ n in mathbb N $,


This property is true in the following cases.

  1. $ G = ( mathbb R ^ n, +) $. $ X = mathbb R ^ n $ to measure with lebesgue and $ rho_g f (x) = f (x-g) $,
  2. $ G = ( mathbb R _ {> 0}, cdot) $. $ X = mathbb R ^ n $ with measure $ d mu = frac {dx} { lvert x rvert ^ d} $, and $ rho_g f (x) = f (x / g) $,
  3. $ G = SU (1, 1) $. $ X = mathbb D $, the unit disk, with measure $ d mu = frac {4dxdy} {(1- (x ^ 2 + y ^ 2) ^ 2) ^ 2} $, and $$ rho_g f (x): = f left ( frac {az + b} { overline bz + overline a} right), qquad g = begin {bmatrix} a & b \ overline b & overline a end {bmatrix}, $$ from where $ | a | ^ 2- | b | ^ 2 = 1 $,

I've learned that the proof for example 2 is in this Math.SE. The same idea works for the same two examples and is even simpler. in both cases for all $ f in L ^ 2 (X) $And for all compacts $ A subset X $. $$ lVert rho_ {g_n} f rVert_ {L ^ 2 (A)} to 0, $$ provided that $ g_n to infty $, That's why we can argue that
$$
left lvert int_X overline {h (x)} rho_ {g_n} f (x) , d mu right rvert le lvert h rvert_ {L ^ 2 (A)} lvert rho_ {g_n} f rVert_ {L ^ 2 (A)} + lVert h rVert_ {L ^ 2 (X setminus A)} lVert rho_ {g_n} f rVert_ {L ^ 2 (X setminus ON)}. $$

The first summand tends to zero, while the second can be made arbitrarily small because of $ h in L ^ 2 (X) $, Here we use that $ rho $ is uniform.

Calculation and Analysis – Integrating a Bit by Bit Function

I try to integrate a function. But for a task, I have to do it
"Bit by bit". I have created a working example that shows the problem.

pBF := FunctionInterpolation(Piecewise({{9890/3, 500 <= z <=800}}), 
{z, 0, 1300}, InterpolationOrder->1);
dz = 1300/25;  
For(i = 1, i <= 25, i++,
Sol = Integrate(pBF, {z, (i - 1) dz, i dz})
);

I do not receive any output from For, so I'm assuming that I either made a syntax error or that Integrate can not work that way. I really have to do this "piece by piece".
Thank you in advance.

complex analysis – orders of roots of zero function

In my course, I'm not given very precise definitions, so I rely a bit on Wikipedia.

https://en.m.wikipedia.org/wiki/Zeros_and_poles

  1. Is the order of roots / poles only defined for meromorphic functions?
  2. What are the orders of the roots of the zero function, $ f (z) = 0 $?

$ f $ is holomorphic (and therefore meromorphic) but for all roots $ z_0 $ from $ f $there is not any $ n in mathbb {Z} $ so that $ (z-z_0) ^ nf (z) $ is holomorphic and not null in the neighborhood of $ z_0 $,