## dnd 5e – How do Dimensional Shackles interact with an Arcane Archer fighter’s Banishing Arrow?

I would say that if interpreting the banishing arrow as a means of movement between the planes (in this case both forward and back), the Dimensional Shackles would prevent the secondary trip back to the plane of origin.

In addition to serving as mundane manacles, the shackles prevent a creature bound by them from using any method of extradimensional movement, including teleportation or travel to a different plane of existence.

Even though the shackles specifically state that:

They don’t prevent the creature from passing through an interdimensional portal.

Interpreting the banishing arrow to be considered an interdimensional portal would be an even bigger stretch than it being considered a type of movement between the planes.

However, the banishing arrow specifically also states that:

You use abjuration magic to try to temporarily banish your target to a harmless location in the Feywild.

So whether or not a location where some other creature puts on the shackles would be considered to be a “harmless” location is also something to consider.

## hyperbolic pde – Solving two dimensional wave equation using Fourier/Laplace transform

The Green’s function for the wave equation in two-dimensions is defined by
begin{align*} frac {partial^{2}}{partial t^{2}}G(r,t)-left(frac {partial^{2}}{partial x^{2}}+frac {partial^{2}}{partial y^{2}}right)G(r,t)=delta(t-tau)delta(x-xi)delta(y-eta). end{align*}
begin{align*} Gto 0text{ as }rtoinfty,text{ where }r^2=(x-xi)^{2}+(y-eta)^{2}. end{align*}
begin{align*} G=0text{ for }0
I was asked to derive the solution using either Fourier transform in $$x$$ and $$y$$ or Laplace transform in $$t$$. The solution is given by
begin{align*} G=frac {1}{2pi}frac {H(t-tau-r)}{sqrt{(t-tau)^{2}-r^2}},text{ where }Htext{ is the Heaviside function.} end{align*}
When I did the Fourier transform on $$x$$, I get
begin{align*} frac {partial^{2}}{partial t^{2}}tilde{G}+omega^{2}_{1}tilde{G}-frac {partial^{2}}{partial y^{2}}tilde{G}=delta(t-tau)delta(y-eta)e^{iomega_{1}xi}. end{align*}
Then I did the Fourier transform on $$y$$ to get
$$begin{equation}label{eqn:c} frac {partial^{2}}{partial t^{2}}hat{tilde{G}}+left(omega^{2}_{1}+omega^{2}_{2}right)hat{tilde{G}}=delta(t-tau)e^{i(omega_{1}xi+omega_{2}eta)}.tag{*} end{equation}$$
For $$t, $$hat{tilde{G}}=0$$. For $$t>tau$$, $$hat{tilde{G}}=c_{1}cos{left(sqrt{w^{2}_{1}+w^{2}_{2}}tright)}+c_{2}sin{left(sqrt{w^{2}_{1}+w^{2}_{2}}tright)}$$. I wonder how to match with $$t=tau$$ to determine the values of $$c_{1}$$ and $$c_{2}$$. Once I get $$hat{tilde{G}}$$, I need to do an inverse Fourier transform and use polar coordinates to get
begin{align*} G=frac {1}{2pi}int_{0}^{infty}J_{0}(kr)sin {kt}, dk. end{align*}
Then I can get the solution by using integral tables. For the Laplace transform in $$t$$, I wonder how to show that the Laplace transform of G is
begin{align*} tilde {G}=frac {1}{2pi}K_{0}(sr), end{align*}
where $$K$$ is modified Bessel function of the second kind. Any help are appreciated!

## linear algebra – What is the shape of \$S:={||Ax||_2^2=1 : ||x||_2=1, xinmathbb{C^n}}\$ on \$n\$ dimensional space?

For a given $$Ainmathbb{C}^{ntimes n}$$, we define a set $$S:={||Ax||_2^2=1 : ||x||_2=1, xinmathbb{C^n}}$$.

Here we have explicit restriction on $$x$$ given by $$||x||_2=1,$$ but I think we also have internal restrictions.

Lets do SVD: $$A=USigma V^*$$, where $$Sigma=mathrm{diag}{sigma_1,ldots,sigma_n}$$, where $$sigma_1geqcdotsgeqsigma_n.$$ Let $$x=[x_1quadcdotsquad x_n]^T$$, $$xgeqcdotsgeq x_n$$.

We do change of a coordinates $$x=Vy,$$ then $$S:={||Sigma y||_2^2=1 : ||y||_2=1}$$. Clearly, we cannot take $$y=[1quad0quadcdotsquad 0]^T$$ if $$sigma_1^2neq1.$$ This is what I mean by internal restriction.

if we plot the set $${x:||x||_2=1}$$ on $$n$$ dimensional space we will get a sphere. What is the shape of the set $$S$$ will look like? Is there any 3D tool that can plot such sets, so that I can visualize the shape?

## integration – d’Alembert’s Solution For Two Dimensional Wave Equation

I’m having some trouble with a question. I have derived d’Alembert’s solution to the one dimensional wave equation $$u_{tt} = c^2 u_{xx}$$ with initial conditions $$u(x,0)=f(x), u_t(x,0)=g(x)$$. This is given as

$$u(x,t)= frac{1}{2}(f(x-ct) + f(x+ct)) + frac{1}{2c} int^{x+ct}_{x-ct} g(s) ds$$
The issue I’m facing is to use this result to obtain the solution to the two dimensional wave equation $$u_{tt}=c^2(u_{xx} + u_{yy})$$ with i.c $$u(x,y,0) = f(x)$$ and $$u_t(x,y,0) = g(y)$$. I did not know this was possible, any help or tips would be greatly appreciated. The question is a small thing so I don’t expect it to be the ype of solution that spans pages. Many thanks

## spells – Repercussions of creating Simulacra of Couatl, Astral Deva, or Dimensional Shambler

Suppose a character gets their hands on a wand of Simulacrum with only a single charge left. This character is wise enough to visit a library before using it so they know the full effects of the spell and the relevant errata/discussions about the design that will affect how the DM will interpret the effects (read: the player has talked with the DM). They have also gathered enough help and bonuses that they are confident they can use the wand successfully.

Rather than pursuing something grandiose such as infinite wishes from duplicating Solars or something else, this character wants reliable, free plane shift, so he is planning on creating a simulacrum of a Couatl, Astral Deva, or Dimensional Shambler. The mechanics of the plane shift spell-like-ability (such as its inaccuracy) are not important here. Rather the important question is, are there are there any consequences this character will face, either in the rules or general lore of the game, to prevent him from making such a “pet” and using its SLAs whenever he wants? Assuming the DM isn’t interested in completely shutting-down the possibility, is there anything I’m not thinking of that would make this unwise?

## dnd 5e – Does a Boggle using Dimensional Rift to attack provoke an Opportunity Attack?

### No opportunity attack happens.

I agree with the rules interpretation of this answer. An opportunity attack occurs “…when a hostile creature that you can see moves out of your reach.” The Boggle’s withdrawal after an attack does not involve moving out of the target’s reach, so no opportunity attack is made on that basis.

### Nor is this simply a technicality.

I disagree with the other answer that this is merely a technicality and that opportunity attacks could be allowed for this use of dimensional rift.

First, I observe that an opportunity attack is not just about moving out of reach. That is a necessary but not sufficient condition. The creature moving also must do so without taking the Disengage action, and without having any other feature that would limit or eliminate opportunity attacks against them (e.g. Barbarian’s third-level Eagle Spirit totem, which imposes disadvantage to opportunity attacks against the Barbarian).

There are clearly a number of specifics in the rules that offset the “moving out of reach” aspect that would normally allow an opportunity attack.

Second, it is not a logical inconsistency that an opportunity attack depends on moving out of reach but must take place before the creature is actually out of reach.

Indeed, the presence of the Disengage action makes more clear why an opportunity attack happens. It is not the movement itself, but the nature of the movement. I.e. to move away from a hostile creature in an unguarded way. The entire movement is done in an unguarded way and thus invites the opportunity attack.

The risk to the moving creature is present as soon as they try to move, thus the attack can occur before they actually move away.

The action economy is used to balance this risk, by allowing a creature to spend an action to eliminate it. Narratively, this movement is done more carefully, in a way that prevents the opportunity attack. It doesn’t mean there’s an inconsistency between the rules and the narrative movement.

Likewise, it’s easy to see that the Boggle is not really moving away per se, and does not have the unguarded aspect of their movement. They are no more moving away than a fighter is moving away after they attack with a sword when they bring their sword back close to their body. Even taking into account that the Boggle ultimately winds up inaccessible by the target of their attack, there’s no inconsistency in disallowing an opportunity attack on the Boggle as they finish their attack and move back to the other side of the rift.

See also e.g. Boggle Tactics for another unofficial interpretation along these lines:

A boggle may open a Dimensional Rift (bonus action) that allows it to reach a victim and pummel him or her through the rift (Attack action), then run away (movement). It can do so without incurring an opportunity attack, because the target can’t attack back through the rift!

## matrices – Can all finite dimensional non commutative algebras be embedded into matrix rings?

Suppose I have a finite (non-)commutative ring $$R/k$$ (over a field $$k$$ of char $$0$$) with a linear “trace” function $$t: R to k$$. Can I find square matrices $$A_1,dots,A_n$$ (of some dimension $$r$$) so that I have an embedding $$f: R to M_r(k)$$ compatible with the trace functions on both sides?

One restriction I can see for the trace function on $$R$$ is that it should be invariant under cyclic permutations : $$t(a_1a_2dots a_n) = t(a_2dots a_na_1)$$. Is this the only restriction?

## matrix – Using Manipulation with two dimensional inputs

I want to have a matrix (say a 2×3 matrix) as an input of Manipulate function, where each entry takes a binary value (say, “+” or “-“). The function counts each of ++,+-,-+, and –.

For example, if the input is {{+,+},{+,-},{-,-}}, the output should be {1,1,0,1} because we have one of each of {{+,+},{+,-},{-,-}} and none of {-,+}.

I could construct a code that works for a one-dimensional case, but I have no idea how to take a matrix as an input of Manipulate function.

## adding two vectors in 3 dimensional space

This is a vector question seems not difficult but I am having issues.

In R^3 space, there is a need to add two vectors a, b (a,b ∈ R^3). a,b,c are vectors that a=(t,0,0), b=(0,t,1), so c= a+b=(t,t,1).

I plotted the chart using software below and you can see vector c is the blue line in the direction between x axis and y axis with a height of 1.

However, in terms of vector addition principles, when adding two vectors a and b, you first move vector b to the starting point of vector a (which is the origin in our case) and then add together. using this logic, a+b should equal to vector d (orange line in chart). I know that from parametric form c=(t,t,1) should be the blue line instead of orange line but what is wrong with the orange line using vector addition principles? thanks!

software plotted image of vector a,b,c

## dnd 5e – What happens when you use the knock spell on dimensional shackles?

Choose an object that you can see within range. The object can be a door, a box, a chest, a set of manacles, a padlock, or another object that contains a mundane or magical means that prevents access.
A target that is held shut by a mundane lock or that is stuck or barred becomes unlocked, unstuck, or unbarred. If the object has multiple locks, only one of them is unlocked.

If you choose a target that is held shut with arcane lock, that spell is suppressed for 10 minutes, during which time the target can be opened and shut normally.

When you cast the spell, a loud knock, audible from as far away as 300 feet, emanates from the target object.

As far as I see it, the description of the knock spell describes 4 things:

• What the knock spell can target.
• What happens if the knock spell targets something locked by a mundane lock, or is stuck or barred.
• What happens if the locking mechanism is specifically the spell ‘arcane lock.’
• The spell creates noise.

Valid spell targets include magical locks, but they aren’t a mundane lock, and we have good grounds to presume they aren’t limited to the arcane lock spell either.

Is something locked by a magical lock considered stuck for the purpose of what happens to a stuck object? This would be strange to me because why then do they specify mundane lock.

Moreover the part referring to things that are locked by a mundane lock, stuck, or barred seems to have an inclination towards the mundane (non-magical), are there magical means of making something stuck which would be included in this description? If so, where is the line between being magically locked and magically stuck?

Is the creation of noise the only result for using the knock spell on a magical lock that isn’t arcane lock?

I’m interested in the answer to this because I’m wanting to break out the dimensional shackles infusion on my artificer. Dimensional shackles seem to be magically locked for all intents and purposes. Moreover the item is designed to prevent teleportation, among any humanoid they were probably made for mages, can a mage simply open them with knock (presuming they know the spell and have it prepared)?