ag.algebraic geometry – Singular hypersurface section

Suppose $X$ is a smooth projective surface over complex numbers in $mathbb{P}^n$. Let $text{Spec}(R)$ be an affine open set of $X$. Then $R$ is a finitely generated $mathbb{C}$ algebra of dimension $2$. Let $Z:= {p_1, …p_k}$ be a set of distinct closed points in $text{Spec}(R)$. Since $X$ is smooth, there are regular parameters $x_i, y_i$ such that the maximal ideal $m_{p_i}$ is generated by $x_i$ and $y_i$. Let $I$ be ideal defined by $m^{‘}_{p_1}m^{‘}_{p_2}…m^{‘}_{p_k}$, where $m^{‘}_{p_i} = langle x_i^2, y_irangle$. Let $Z^{‘}$ be the subscheme defined by $I$.

What is the geometric interpretation of $Z^{‘}$ in terms of $Z$?

Is $Z^{‘}$ locally complete intersection?

Suppose $Y in H^0(I_{Z^{‘}}(d))$ is a degree $d$ hypersurface section. Then can we say that $Y$ is singular along $Z$ ?

reference request – Cone condition for Wave equation with Singular Speed

Consider a wave equation of the form
$$
partial_t^2u(t,x)-c(t)^2partial_x^2u(t,x)=0, quad (t,x)in (0,1)times mathbb{R}
$$

where the speed $c(t)$ is in $L^1((0,1)) cap C^1((0,1))$. This would mean that the speed may entertain logarithmic singularity.

The Cauchy problem for the above wave operator is well-posedness in $C^infty$. The well-posedness is established via an energy estimate by Colombini et. al (Colombini, F.; Del Santo, D.; Kinoshita, T.: Well-posedness of the Cauchy problem for a hyperbolic equation with non-Lipschitz coefficients. Ann. Scoula Norm Sup. Pisa Cl. Sci. 1(2002) 327-358).

Will the operator satisfy the cone condition? Especially, what will be the domain of influence when initial conditions are given at $t=0$?

Can anyone suggest some reference for determining cone condition for wave equation with singular speed?

Projective cellular varieties with singular cohomology admitting torsion different from two

Is there a simple enough example of a projective cellular variety $X$ (cellular decomposition by affine spaces) such that $H^*(X(mathbb{R}),mathbb{Z})$ has an element with torsion different from 2? (3,4,5, for example)

hardware – Singular Blade Server: A/C required in 5×5 closet?

I have a small home setup that I just play with. Mainly, I have been wanting to move my blade server elsewhere and kind of hide it away. I have several potential places, but I would have to do renovations to help protect those areas from the outside elements.

This brings me to thinking about storing it inside my closet I have. Would the server get too hot with no direct a/c, and the door will be shut, in the closet?

dg.differential geometry – Non-trivial $mathbb{R^3}rightarrowmathbb{R^3}$ maps with constant singular values

It can be proved that all $mathbb{R^2}rightarrowmathbb{R^2}$ mappings with constant singular values are affine. In three dimensions, however, there are non-trivial examples, like

$$
begin{align}
x’&=lambda_2 x -frac{1}{lambda_2}sqrt{frac{lambda_2^2-lambda_1^2}{lambda_3^2-lambda_2^2}}int_0^{z}sin f(xi),mathrm{d}xi \
y’&=lambda_2 y +frac{1}{lambda_2}sqrt{frac{lambda_2^2-lambda_1^2}{lambda_3^2-lambda_2^2}}int_0^{z}cos f(xi),mathrm{d}xi \
z’&=frac{lambda_1lambda_3}{lambda_2}z
end{align},
$$

for an arbitrary differentiable $f$ and $lambda_3>lambda_2>lambda_1$, whose differential has constant singular values $lambda_3,,lambda_2,,lambda_1$. So I wonder if one can classify or say something generic about such $mathbb{R^3}rightarrowmathbb{R^3}$ maps, as is the the case in two dimensions.

Quintic surface singular along lines

What is the maximum number of lines, along which a quintic surface in $mathbb{P}^3$ can be singular ?

Line through a singular point on a cubic surface

if P is a singular point of a cubic surface S, then there is at least one line on S through P

Hey, I come across this problem at chapter 7 in Reid’s book. Tried to think about the intersection of the tangent space with S but couldn’t go further.

dimension of quintic hypersurfaces singular at given number of points

How many quintic hypersurfaces are there which are singular at given points (need not be general) of length at least 20?Is there any upper bound of the dimension of such quintics ?

dg.differential geometry – Can we wrap a square onto itself with constant singular values?

Let $0<sigma_1<sigma_2$ satisfy $sigma_1sigma_2=1$, and let $D=(-1,1)^2$.

Does there exist a Lipschitz bijective* map $f:D to D$ such that $df$ has almost everywhere the fixed singular values $sigma_1,sigma_2$?

Is there such a diffeomorphism of $D$? (thinking of $D$ as a manifold with corners, or requiring smoothness only on the interior etc.)

*I am fine with requiring only $|f^{-1}(y)|=1$ for a.e. $y in D$; the Area formula then implies that $f$ is surjective.

Clearly, no affine map would be suitable. We somehow need a map whose singular vectors are ‘rotating’ from point to point.


Comment: If we replace $D$ with a disk then we have $ f_t:(r,theta) to (r,theta+t log r)$, which
is the flow of $log r frac{partial}{partial theta}$.

$f_t$ has constant singular values (which depend on $t$.)

ids – Processing Exceptionally High Volume Singular Flows

I wanted to know if anyone had experience sensoring single flows that generate 90 kpps or upward of traffic. My conundrum is that I use tools which I would like to be able to properly see an entire flow with (Zeek, Suricata), however both of these utilities are not capable of handling larger traffic volumes if a single connection exceeds a “worker thread’s” maximum processing rate. Because a single connection is able to generate this volume of traffic, I am not aware of a load balancing method and accompanying software that would produce similar resulting information from Zeek & Suricata(Understandably this matters a lot less with Zeek, as seeing the entire connection isn’t strictly necessary, however having a giant connection fill the rx ring results in losing subsequent connections).

For referencing I’m currently using the AF_Packet family sockets for both Suricata in Zeek, which are both in their own fanout. I get roughly 48,000pps running suricata with 4 capture threads with my current configuration and ~26,000pps per Zeekctl cluster.

I’m aware the most sensible thing to do would be to strictly analyze the flow, and then filter it out pre-sensor, however I’m doubtful policy in my org will allow me to do this, but I’ve been unlucky in my search for a satisfactory tool or reconfiguration.

I am hoping someone might have experience handling this type of situation or knows of software which has a less restrictive processing per “worker/thread” rate that would produce similar results(or honestly any results that aren’t massive amounts of packet loss).