dnd 5e – Using the optional rule on falling onto a creature, is the fall damage divided between creatures before or after damage resistance/reduction?

Lets use an example through this study: the Martial Warrior (Marty) has been knocked off of a cliff, and falls 100 feet the the bottom, into the square of a Cleric (Clarence). We will assume the DM rolls average damage. Let’s break down this fall in order from general to specific:

General Rule: PHB Fall Damage

A fall from a great height is one of the most common hazards facing an adventurer.
At the end of a fall, a creature takes 1d6 bludgeoning damage for every 10 feet it fell, to a maximum of 20d6. The creature lands prone, unless it avoids taking damage from the fall.

This is the original rule for falling, and the most general rule to follow. In this case, Marty takes the full damage of 10d6, 35 bludgeoning damage.

General Rule: TCoE Falling on another Creature

The second creature here gets to make a saving throw to avoid halving the damage. Clarence fails there save, and splits the damage with Marty: they both take 17 bludgeoning damage.

Specific rule: Monk’s Slow fall

Beginning at 4th level, you can use your reaction when you fall to reduce any falling damage you take by an amount equal to five times your monk level.

The damage is reduced “when you fall,” not when you take damage. This means the damage dealt by the fall is decreased, instead of reducing the damage once it is taken, through resistance, mitigation, division, diversion, etc.
In this specific case, Marty is now a 4th level Monk. If he falls onto Clarence’s space and Clarence fails their saving throw, then the damage of the fall is split between them. However the specific wording of Falling on a Creature states:

…and any damage resulting from the fall is divided evenly between them.

So the damage is first reduced by the Marty’s slow fall, and second split between the two. 35 damage is reduced by 5 times Marty’s Monk level to 15 damage, which is split between them: both characters take 7 bludgeoning damage.

Specific rule: Damage Resistance

If a creature or an object has resistance to a damage type, damage of that type is halved against it.

Clarence is not just a cleric, but a 17th level Forge Cleric in heavy armor, meaning their Saint of Forge and Fire feature takes effect, giving them resistance to non-magical bludgeoning damage. This means the damage taken by Clarence is halved, and only the damage taken by Clarence. So after Marty falls 100 feet, Clarence fails to get out of the way, and Marty’s Slow Fall Feature take effect, finally the damage that is taken is halved last. This means Marty would take 7 Bludgeoning damage from the fall, and Clarence would take 3 bludgeoning damage. If Marty had the resistance instead, this would still not be translated into reduced damage for the other person, as the resistances damage reduction is applied after the damage has been dealt.

Other Specific rules: Various Subclass abilities

All of the subclass features you have listed are great examples of damage reduction, and they all have a similar piece of wording that is different from the Monk’s Slow Fall.

Spirit Shield:

If you are raging and another creature you can see within 30 feet of you takes damage, you can use your reaction to reduce that damage by 2d6.

Bastion of Law:

When the warded creature takes damage, it can expend a number of those dice, roll them, and reduce the damage taken by the total rolled on those dice.

Guardian Coil:

When you or a creature you can see takes damage while within 10 feet of the tentacle, you can use your reaction to choose one of those creatures and reduce the damage to that creature by 1d8.

Song of Defense:

When you take damage, you can use your reaction to expend one spell slot and reduce that damage to you…

Unlike the Slow Fall ability, which takes effect when you fall, all of these features take effect when you take damage. These would all reduce the damage after impact, when both the falling creature and fallen upon creature have already taken the damage.
Thankfully for Clarence, his Raging Ancestral Guardian Barbarian Friend (Barb) is nearby, and she uses her Spirit Shield ability to reduce the damage of the fall by 2d6. Unfortunately for Clarence, Barb rolls garbage, and only reduces the damage by 2. According to the order of operations in the PHB, the damage would be reduced first (from 7 to 5), then halved by resistance, resulting in a final devastating total of 7 bludgeoning damage for Marty, and a brutal 2 bludgeoning damage for Clarence.

dnd 5e – When is fall damage divided between creatures using Tasha’s rule?

Tasha’s Cauldron of Everything offers the following rule:

If a creature falls into the space of a second creature and neither of them is Tiny, the second creature must succeed on a DC 15 Dexterity saving throw or be impacted by the falling creature, and any damage resulting from the fall is divided evenly between them. The impacted creature is also knocked prone, unless it is two or more sizes larger than the falling creature.

However I am unclear how this would relate to resistance, the Monk’s Slow Fall, and other damage reducing features.

The rule says “any damage resulting from the fall is divided evenly between them” but it doesn’t specify whether that damage is split before or after damage reducing features.

Is the damage divided between the two creatures before or after damage reducing features are calculated?

website design – What is the name of the web element that divided in columns and have illustrations on each column?

Do you know the name of this kind of element that is usually used in web design?

Searching around I have a feeling that it doesn’t really have a specific name, just “column design” or something. The reason I’m asking this is to have a deeper understanding on the illustration of this specific element. Illustration art in general, and website illustration art in specific are not something new, see for example this video about illustration art, and this article about illustration styles in web design. However, in this specific element, there is not much room for details, and both those links don’t emphasize this constrain at all. Illustration artists can still work on this element, but I think until we have a concept for it, it will be hard to solidify our skills.

Another example of this, to illustrate how I need a guideline for this specific element. It’s the missions of my project.

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hochschild cohomology – divided powers of a deformation class

Let $A$ be a (unital, associative) $k$-algebra where $k$ is a field. Given a flat deformation of $A$ one gets the deformation class $h$ in the second Hochschild cohomology $HH^2(A)$. Suppose $k$ has positive characteristic and $h$ happens to come with divided powers $h^{(p^n)}$. Does that indicate any additional structure on the deformation, something that would naturally induce such a divided power structure?

How the task is divided between multiple cores to perform in multiprocessing systems?

I am trying to find how single task is divided between two cores? what are the methods makes them work at the same time without interfering with each other?

polynomials – Give the remainder when $x^{203}-1$ is divided by $x^4-1$.

Use the roots of $x^4-1$.

Let $x^{203}-1 = P(x)(x^4-1) + Q(x)$ as polynomials. Note that we can substitute complex values and retain equality.

Now, $x^4-1$ has the roots $pm 1, pm i$. Substitute each of these in to get:
$$
Q(1) = 0 , Q(-1) =2 , Q(i) = -i-1, Q(-i) = i-1
$$

Note that $Q(x)$ is a multiple of $x-1$, so take it as $Q(x) = (ax^2+bx+c)(x-1)$. Now use the other three conditions to find $a,b,c$ and conclude.


Another way to do it, is to do :
$$
x^{203}-1 = (x^{203} – x^3) + (x^3-1)
$$

and then note that each of $pm 1, pm i$ satisfies $x^{203}-x^3 = 0$, so this polynomial is a multiple of $x^4-1$. Hence the remainder is $x^3-1$.

interpolation – Newton’s Divided Differences Iteration in Mathematica

I am trying to write a program that forms the interpolation polynomial for a given function on a given interval for any number of data points n. I wish to write a formula that will compute all of the necessary divided differences. Here is part of my code,

g(x_) := 1/(1 + x^2) 
f = Table(g(x), {x, -5.0, 5.0, 1})
x = Table(a, {a, -5, 5, 1})
 
 Table((f((i)) - f((i - 1)))/(x((i + j)) - x((i - 1))), {j, 0, 9}, {i, 2, 11 - j})

For j=0 I get a list of the correct outputs I need, {0.020362, 0.0411765, 0.1, 0.3, 0.5, -0.5, -0.3, -0.1, -0.0411765,-0.020362} but for j=1, I must find a way to have the terms in the numerator replaced with these output values over the loop, and so forth for each j. So for j=1,i=2 I wish to write a formula that will calculate (0.0411765-0.020362)/(x((3)) - x((1))) for example, if this is possible.

I am a Mathematica novice, so bear with me and I hope I have been as clear as possible. I would appreciate any help, tips, tricks or guidance if it seems my approach is not a good one.

Find the remainder when (sum n^(n!),n=1 to 17) is divided by 17?

Find the Remainder when (sum n^(n!),n=1 to 17) is divided by 17

modular arithmetic – Any easier way to get the quotient and remainder from a minus number divided by a positive one?

negative number divided by positive number, what would be remainder?

I’ve read these answers linked above, but I don’t feel I’m answered enough.
$$-27 = underbrace{-6}_qcdot underbrace{5}_d + underbrace{3}_r$$
They say this, but I don’t think I can use it later when I need to divide some large minus number with another large positive number, such as -777777 with 439. Positive number division is easy enough that no matter how young you are, even grammar school students could solve it with enough time given. I was wondering if there’s any that kind of easy method to get the quotient and remainder from the minus numbers divided by the positive numbers.

numerical methods – Proof for expanded newton’s divided differences

On the link the proof is provided by induction and everywhere else this method is used as far as I’ve seen.
But the induction here is too long and confusing, Is there any other way to prove this without induction?
Or can someone suggest a reference that explains the induction in details step by step, I have trouble to understand the induction.

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