riemannian geometry – Critical points of the area functional restricted to CMC embeddings

For a fixed closed smooth manifolds $M^n$ and $N^{n+1}$, two $C^{k,alpha}$ embeddings $f, f’ : M to N$ are said to be equivalent if there exists $varphi in operatorname{Diff}(M)$ such that $f’ = f circ varphi$. Let $mathcal{M}$ be the set of pairs $((f), g)$, where $(f)$ is an equivalence class of CMC embeddings into the Riemannian manifold $(N,g)$ and $g$ belongs to a fixed open set $Gamma$ of $C^q$ metrics. Brian White showed that $mathcal{M}$ is a $C^{q-j}$ Banach manifold modelled on $Gamma$.

Now let $mathcal{A} : mathcal{M} to mathbb{R}$ be the area functional. I am trying to find its critical points. To this end, let $((f),g)$ be given and consider a curve $((f_t), g_t)$ in $mathcal{M}$ such that $((f_0),g_0) = ((f),g)$ and $d/dt ((f_t), g_t) = (X, h)$ at $t = 0$, where $X$ is a vector field along $f$ and $h$ is a symmetric 2-tensor. Then, by the usual variation formulae for the area and volume elements, we arrive at

$$left. frac{mathrm{d}}{mathrm{d}t} rightvert_{t=0} mathcal{A}((f_t), g_t) = – int_M H_{f} g(X, nu) , mathrm{d}A_{f^ast g} + int_M operatorname{tr}_g(h) , mathrm{d}A_{f^ast g},$$

where $H_f$ is the mean curvature of $f : M to N$, $nu$ is a unit normal for $M$ along $f$ and $operatorname{tr}_g(h)$ is just the trace of $h$ with respect to the metric $g$. How do I conclude? What are the critical points of $mathcal{A}$?

Critical incident at CTemplar, who lost all its clients’emails

CTemplar, the highly secure webmail provider, competitor of ProtonMail, has totally lost all its client directory and all their emails on July 2021 09.

CTemplar catastrophic incident with complete data loss – July 2021

As of July 11 2021, CTemplar did not provided any details regarding the cause of the incident, that was qualified by CTemplar as “catastrophic”.
SEMrush

 

dnd 5e – Spells with secondary damage effects and critical damage

Spells like like Booming Blade and Green-Flame Blade has secondary damage effects.

Booming Blade deals an extra 1d8 thunder damage if the creature hit with the spell’s initial melee attack requirement moves 5 or more feet

Green-Flame blade does a Caster’s Spellcasting Ability modifier in radiant damage to a target within 5 feet of the cantrip’s original target. This damage incerease by 1d8+ at specific levels

These extra damage stems from attack roll and does have a damage die/dice. Do these extra damage dice also double from a critical hit

Ex:
Booming Blade (at base) with a rapier would do 4d8 (2d8 weapon crit + 2d8 thunder spell crit ) on a critical and if the target moves 5+ feet it takes 1d8 thunder damage or does it take 2d8

Green-Flame Blade (at 5th Lv PC) with a rapier 4d8 (2d8 weapon crit + 2d8 radiant spell crit ) on a critical and an adjacent creature within 5 feet of the target gets an additional 1d8 + Spellcasting Ability modifier radiant damage or does it take 2d8 + SAM

What type of critical point is (0,0) in the following function?

Studying for an exam, I was given the following and had to tell what type of critical point $(0,0)$ is. The answer is "Not a saddle and not a minimum or maximum point".
I got that its supposed to be a saddle point which is why I’m confused.

$ f(x,y)in C^2 , T_2(x,y)= 3+2x+0.5x^2-xy-y^2$ Around $(0,0)$.

My efforts:
https://imgur.com/a/wDTOZRh

Emirates representative asks me whether it is critical for me to travel and presents offer of alternative arranagements

I had a weird experience with Emirates traveling from Canada to Dubai. After having checked my documents the Emirates representative asked me whether it is important for me to travel. I asked him why and he said if I do not travel then they can think up “alternate arrangements”. I am not sure what that was about, or what that meant?

No one else in the line was asked this question and the way of asking the question was bit “cringe” to think of another word.

Any one has any ideas why that line of questioning? It came out as very odd. I am thinkin to tweet Emirates customer service a complaint, unless what he had asked had some reasonable justification? Reasonable justification would not be that he was trying to limit the number of passengers to encourage social distancing because the flight was packed.

Thank you.

graphs – Find all the critical phases of the project

I’m have the following model for this problem:
“We have a project which is divided in $V = { v_1, v_2, dots, v_n}$ tasks, and each task have a dependency $N(v_i) = {v_j : j < i }$ which is a set of tasks we need to finish in order to do $v_i$. Also, each task has an associated completion time $t_i geq 0$. For simplicity $t(v_1) = 0 = t(v_f)$. We need to find all the critical tasks in the project. A critical tasks is a task such that its delay will cause a delay on the project finalization. “

My model is: create a graph such that every task will represent a vertex, and for every task $v_i$, an edge will be incoming from every task in its $N(v_i)$ set. Also, if a task is such that $v_u in N(v_i), N(v_j)$, there should be an edge $v_u,v_{uf}$ with weight $t(v_u)$ and also $v_{uf}$ will be incoming in $v_i$ and $v_j$, so that every original vertex will have $d_{out} = 1$. For every other edge $v_i,v_j$ will have weight $t(v_i)$.

My idea is to think it as a problem of maxpath, and all the vertices in this path (from $v_1$ to $v_n$) will be critical, since vertices not in this path can be delayed. I don’t know if I’m facing it the right way, but I’m open to hear suggestions / corrections.

dnd 5e – If an Echo Knight fighter makes an attack from their echo’s space and hits a paralyzed enemy, is it automatically a critical hit?

The Manifest Echo feature lets the Echo Knight fighter create an echo, and attack from the echo’s space (EGtW, p. 183):

When you take the Attack action on your turn, any attack you make with that action can originate from your space or the echo’s space. You make this choice for each attack.

Suppose an attack originating from the echo’s space hits a paralyzed creature. The Paralyzed condition states the following:

Any attack that hits the creature is a critical hit if the attacker is within 5 feet of the creature.

If the echo is within 5 feet of the paralyzed target but the Echo Knight isn’t, is it automatically a critical hit?

dnd 5e – If an attack through an Echo Knight’s echo hits a paralyzed enemy , is the attack automatically a critical hit?

“The attack originates from the echo’s space” means we treat the attacker as being in the echo’s space for rules purposes.

The Manifest Echo feature states:

any attack you make with that action can originate from your space or the echo’s space.

“Originate…from the echo’s space” indicates that for the purposes of this attack, any rules applicable for that attack treat you as though you are in the echo’s space. So for the purposes of the critical hit rule on a paralyzed creature, we treat “the attacker” as though they are in the echo’s space.

Yeah, it’s a critical hit.

Code snippet to display ID gives critical error

Unfortunately you can’t just copy code that plays a completely different role and replace users with posts and somehow expect this to do what you want it. The reason you get a fatal error is because you’re passing in too many arguments (3) to the filter, when there are only 2 supplied.

Here’s the code to achieve what you need, if I understand you correctly:


add_filter( 'manage_posts_columns', 'column_register_wpse_101322' );
add_action( 'manage_posts_custom_column', 'column_display_wpse_101322', 10, 2 );

function column_register_wpse_101322( $columns ) {
    $columns( 'uid' ) = 'ID';
    return $columns;
}

function column_display_wpse_101322( $column_name, $post_id ) {
    if ( 'uid' === $column_name ) {
        echo $post_id;
    }
}

spells – Critical hit with Acid Splash: how much damage?

Telekinetic Projectile does list that it does double damage on a critical hit.

Compare this to Acid Splash:

On a critical success, the target also takes 1 persistent acid damage.

Should this “also” mean that beside the usual damage doubling? Or a critical is just normal damage + 1 persistent?