Calculate relative rotation between a vector and its negative

How do we find the rotation that rotates $v$ to –$v$, for arbitrary vector $v$.

The naive way of getting an axis using the cross product won’t work here, because the cross product of $v, -v$ is 0.

pathfinder 1e – Do negative energy attacks heal undead?

Most spells and effects that deal with positive or negative energy will state their effect on living and undead creatures. Chill Touch for example will not heal undead despite saying it deals negative energy damage. Instead, it will cause fear in the undead.

It is a case by case basis and does not have a very consistent behavior. Negative Channeled Energy to deal damage, for example, will also not affect undead. Conversely even positive energy works like this. Channeling positive energy to heal the living, will not affect undead. However when channeled to harm undead, it does not affect living.

This all seems to imply that positive and negative energy’s effects, are strongly tied to intent. Spells have a specific intent, and channeling has a specific intent. However this is just my conjecture and not strictly stated anywhere in the rules.

Is that navigation bar negative for SEO

I’m doing some SEO for a website I haven’t built for my client and it has this navigation bar:


<div align="right" id="menu">
        <table border="0" cellpadding="0" cellspacing="0">
             <td><div align="center" class="menuitem1" onmouseover="this.className='menuitem1a'" onmouseout="this.className='menuitem1'" onclick="window.location='index.php'"  >
                     <div style="margin-top:80px">Profile</div>
             <td><div align="center" class="menuitem2" onmouseover="this.className='menuitem2a'" onmouseout="this.className='menuitem2'" onclick="window.location='customers.php'">
                    <div style="margin-top:80px">Customers</div>
             <td><div align="center" class="menuitem3c">
                    <div style="margin-top:80px">Services</div>
             <td><div align="center" class="menuitem4" onmouseover="this.className='menuitem4a'" onmouseout="this.className='menuitem4'" onclick="window.location='products.php'">
                    <div style="margin-top:80px">Products</div>
             <td><div align="center" class="menuitem5" onmouseover="this.className='menuitem5a'" onmouseout="this.className='menuitem5'" onclick="window.location='contact.php'">
                     <div style="margin-top:80px">Contact</div>

First thing I noticed, it has no anchors! Second when I made a sitemap of the website only index page was there. EDIT:It also spits errors on evaluation! Does this nave a negative impact from SEO perspective? Thanks in advance!


recreational mathematics – Why dividing negative numbers gives positive outcomes

How comes that when we divide two negative numbers we get a positive answer? Consider that negative numbers are nonexistent in real life, for instance there are no -10 apples. So when we divide -10 apples among -2 people, why should we get 5 apples?? Does it mean real substances can exist from nowhere?

A closed form or a good approximation of an infinite series related to the negative binomial distribution

Does anyone know a closed form for this expression:
$$sum_{r =1}^{infty}{{alpha + 2r – 1}choose{ r – 1}}(1 – p)^{alpha + r}p^{r},$$
where $alpha geq 1$ and $0<p<1$. A good approximation of this expression works too. Thanks!

Do I need to quarantine in Colombia if I have a negative Covid test?

Colombia does require a negative Covid test for entry. If it’s impossible for you to take one they allow getting one in Colombia and quarantine until the results are in.

However, I have not found any mentioning of quarantine requirements (positive or negative) if you already have the test on entry.

The US embassy states

Are U.S. citizens required to quarantine?  Yes.

But it’s unclear whether this is a general requirement or only related to lack of testing.

Any other resource I found doesn’t mention quarantine or only in context of “no test”. However, I haven’t seen anything that specifically states: “No quarantine required if you have a negative test”.

Apparent Negative free disk space and FreeBSD 12

Just curious as to what’s happening here. This is just a non-prod VM that I have at home. Curious as to what leads to the ‘negative’ available diskspace. This is FreeBSD 12 running under VMWare ESXi 7.

FreeBSD 12.0-RELEASE-p10 FreeBSD 12.0-RELEASE-p10 GENERIC  amd64
[root@ssh1 /fixeddisk]# df -h
Filesystem     Size    Used   Avail Capacity  Mounted on
/dev/da0s1a     15G    6.5G    6.9G    48%    /
devfs          1.0K    1.0K      0B   100%    /dev
/dev/da1s1a    901G    861G    -32G   104%    /fixeddisk

ag.algebraic geometry – Negative curves on surfaces

Let $f:Xrightarrowmathbb{P}^1$ be a family of surfaces over $mathbb{C}$. Assume that for $yinmathbb{P}^1$ general $X_y = f^{-1}(y)$ is a smooth surface. Let $X_{eta}$ be the generic fiber of $f$ and $overline{X}_{eta} := X_{eta}times_{Spec(mathbb{C}(t))}Spec(overline{mathbb{C}(t)})$, where $overline{mathbb{C}(t)}$ is the algebraic closure of $mathbb{C}(t)$.

Assume that in $overline{X}_{eta}$ there is a curve of negative self-intersection $-a$. Does there is a curve of negative self-intersection $-a$ on the fiber $X_y = f^{-1}(y)$ for $yinmathbb{P}^1$ general?

dnd 5e – Do Player characters know they are under the effect of a Death Tyrant’s Negative Energy Cone?

At least, not until specific things occur as a result of being in the cone.

Targets (of spells) PHB 204

Unless a spell has a perceptible effect, a creature might not know it
was targeted by a spell at all. An effect like crackling lightning is
obvious, but a more subtle effect, such as an attempt to read a
creature’s thoughts, typically goes unnotice, unless a spell says

While the Negative Energy Cone isn’t a spell, it is a magical effect, and so it is reasonable to apply the same logic.

Death Tyrant MM 29

The death tyrant’s central eye emits an invisible, magical 150-foot
cone of negative energy. At the start of each of its turns, the tyrant
decides which way the cone faces and whether the cone is active.

creature in that area can’t regain hit points. Any humanoid that dies
there becomes a zombie
under the tyrant’s command. The dead humanoid
retrains its place in the initiative order and animates at the start
of its next turn, provided that its body hasn’t been completely

Nothing in the description of the ability describes any obvious effect, other than when trying to regain hit points or dying, so creatures won’t notice it unless either of those two things occur.

DMs may allow checks to notice where the central eye is looking

Given that the cone is emitted by the central eye, a DM may reasonably allow a character to try to determine where the cone is being directed, if said character is aware that such a cone exists. However, this is totally up to the DM as there is no specific guidance on this.

Official rules concur

The Sage Advice Compendium contains additional detailed guidance on perceiving spell effects, so again, the logic applies to our magical effect here:

Do you always know when you’re under the effect of a spell?

You’re aware that a spell is affecting you if it has a perceptible effect or if its text says you’re aware of it (see PHB , under “Targets”). Most spells are obvious. For example, fireball burns you, cure wounds heals you, and command forces you to suddenly do something you didn’t intend. Certain spells are more subtle, yet you become aware of the spell at a time specified in the spell’s description. Charm person and detect thoughts are examples of such spells.

Some spells are so subtle that you might not know you were ever under their effects. A prime example of that sort of spell is suggestion. Assuming you failed to notice the spellcaster casting the spell, you might simply remember the caster saying, “The treasure you’re looking for isn’t here. Go look for it in the room at the top of the next tower.” You failed your saving throw, and off you went to the other tower, thinking it was your idea to go there. You and your companions might deduce that you were beguiled if evidence of the spell is found. It’s ultimately up to the DM whether you discover the presence of inconspicuous spells. Discovery usually comes through the use of skills like Arcana, Investigation, Insight, and Perception or through spells like detect magic.

pr.probability – Assigning negative integer moments to random variables with Hadamard regularization

Let $Xsim F_X$ denote a continuous random variable that admits a density $f_X$ with support $mathcal S=operatorname{supp}(X)ni 0$ and assume $f_X(0)>0$. I am interested in defining a regularization $#$ of
mathsf EX^{-n}:=#int_{mathcal S} x^{-n}f_X(x)mathrm dx,quad ninBbb N,

which would assign values to negative integers moments of $X$. I am aware that uniqueness of such a regularization is unlikely and so it is important to find a regularization that suits this particular context. My thought was to use Hadamard regularization. In particular, one could define
M_{X^{-1}}(t)=mathcal Pint_{mathcal S} frac{f_X(x)}{x-t},mathrm dx,

where $mathcal P$ denotes the usual principal value. In this manner, $M_{X^{-1}}$ could serve as a sort of moment generating function so that
mathsf EX^{-n}:=frac{1}{(n-1)!}partial_t^{n-1}M_{X^{-1}}(t)|_{t=0}.

This approach does show promise however there are some undesirable aspects which I show through an example:

Suppose $Xsimmathcal N(mu,sigma^2)$. The principal value for $mathsf E(X-t)^{-1}$ can be found in the literature as
M_{X^{-1}}(t)=frac{sqrt 2}{sigma}mathcal Dleft(frac{mu-t}{sqrt 2sigma}right),

where $mathcal{D}(z)=e^{-z^{2}}int_{0}^{z}e^{t^{2}},mathrm{d}t$ is the Dawson function. If we derive the first and second “moments” we can then also find $mathsf{Var}X^{-1}=mathsf EX^{-2}-(mathsf EX^{-1})^2$. Evaluating the asymptotic expansions for $mathsf EX^{-1}$ and $mathsf{Var}X^{-1}$ as $sigmasearrow 0$ from their explicit expressions using the procedure above gives
mathsf EX^{-1}simfrac{1}{mu}+frac{sigma^2}{mu^3}+mathcal O(sigma^4)

mathsf{Var}X^{-1}simfrac{sigma^2}{mu^4}+8frac{sigma^4}{mu^6}+mathcal O(sigma^6).

Note that if we expand $g(X)=X^{-1}$ about $mu$ and compute the expected value of the first few terms w.r.t. $mathcal N(mu,sigma^2)$ we can compare the results to the above which show that our asymptotic expansions agree with what you obtain with the $delta$-method. So our regularized values for $mathsf EX^{-n}$ do encode information about the moments in the limiting sense where $mathsf{Var}X=sigma^2approx 0$ when $muneq 0$. However, let’s consider the special case $mu=0$ so that $mathsf EX^{-n}$ are interpreted as the negative central moments of $X$. If our regularization is to make any sense in this context we would hope the odd-central moments are zero while the even moments are positive. Computing the first several moments gives
n & (mathsf EX^{-n})|_{mu=0}\
1 & 0 \
2 & -frac{1}{sigma ^2} \
3 & 0 \
4 & frac{1}{3 sigma ^4} \
5 & 0 \
6 & -frac{1}{15 sigma ^6} \
7 & 0 \
8 & frac{1}{105 sigma ^8} \

which bear a remarkable resemblance to the positive central moments of the normal distribution with the addition of alternating negative signs. So it would seem that the method of regularization used here may not make total sense.

My question has to do with approaches one could take to alter this procedure to give positive even moments. Of course, I could simply force the even moments to be positive by introducing some sort of oscillating term into the definition of $M_{X^{-1}}$ that is negative when $n=2,6,10,dots$; however, without any rational for doing so my concern is taking such an ad hoc approach is simply putting a band aid on the problem. How else might I alter my definition of $M_{X^{-1}}$ above to give more appropriate results for these negative moments? Are there other regularization procedures that might be more useful for this particular application?