## dnd 5e – Limits to appearance of a phantom steed

In a recent game session, a player wanted to become invisible and then use a phantom steed to quickly and stealthfully pass through the enemy camps surrounding a besieged city. This began a discussion of whether the phantom steed summoned could be transparent, or nearly so. The spell says:

A Large quasi-real, horselike creature appears on the ground in an unoccupied space of your choice within range. You decide the creature’s appearance…The creature uses the statistics for a riding horse

So what are the limits to the appearance of the steed? While “quasi-real”, can it be made to appear so real that an observer would believe it to be a real horse? Can it be made to appear so unreal as to be seen through and receive a bonus on Stealth?

For the limit on how transparent it can be I thought the important restriction is that it has the stat block of a riding horse. Any permanent bonus to Stealth would be included within this block, and a riding horse does not have one. Thus, for me the mechanical limit to transparency would be that it could get a DM-assigned circumstantial bonus (advantage to Stealth checks depending on situation) but not a permanent bonus irrespective of circumstances (modification of stat block).

For the limit on how real it can appear, I looked to the spell Major Image, as an Illusion spell of equal level, whose illusions have the following property

A creature that uses its action to examine the image can determine that it is an illusion with a successful Intelligence (Investigation) check against your spell save DC.

Thus if the steed was summoned to appear real, it would fool any observer who was not explicitly trying to determine its nature, and even some of those depending on the roll. This approach of comparing the spell to same-school spells of equal level comes to me from 1e, with which I have more experience, and in which spells are much less clearly defined in terms of their limits. 5e, on the other hand, has the general principle that “spells do what they say they do”. Is such a comparison to another spell then valid?

1. What are the limits to the appearance of a phantom steed? How real or unreal can it appear?

2. When deciding (1), is it useful to compare the spell to Major Image?

## limits – Derivative Greater Than 0 Implies One-To-One Function In Neighborhood

Let $$f: textrm{dom}(f) rightarrow mathbb{R}.$$

Let $$x_0 in mathbb{R}.$$

Assume $$f'(x_0) > 0$$.

i.e. $$~ displaystylelim_{h rightarrow 0} frac{f(x_0 + h) – f(x_0)}{h} > 0$$

i.e. $$~ exists l > 0 textrm{ s.t. } forall varepsilon_1 > 0, exists delta_1 > 0 textrm{ s.t. } forall h in mathbb{R}, 0 < |h| < delta_1 Rightarrow Bigg| displaystylefrac{f(x_0 + h) – f(x_0)}{h} – l Bigg| < varepsilon_1$$

This implies that $$f$$ is continuous at $$x_0$$, as I have proven before.

i.e. $$~ forall varepsilon_2 > 0, exists delta_2 > 0 textrm{ s.t. } forall x in mathbb{R}, |x – x_0| < delta_2 Rightarrow |f(x) – f(x_0)| < varepsilon_2$$

Also,
$$exists delta_3 > 0 textrm{ s.t. } forall x in mathbb{R}, 0 < |x – x_0| < delta_3 Rightarrow displaystylefrac{f(x) – f(x_0)}{x – x_0} > 0$$

I would like to prove there exists an open interval containing $$x_0$$ where $$f(x)$$ does not have a value of $$f(x_0)$$ for any $$x$$ in that interval apart from $$x_0$$.
i.e. $$~ exists a, b in mathbb{R} textrm{ s.t. } a < x_0 < b wedge big( forall x in (a, b), x neq x_0 Rightarrow f(x) neq f(x_0) big)$$

Unfortunately, I cannot say anything about double differentiability of $$f$$.

Because of this, I cannot mention continuity of $$f$$ in a neighborhood of $$x_0$$ (or can I?)

Perhaps there is a counterexample and I shouldn’t try to prove this statement.

Help needed.

## limits – Proofs of \$lim_{xto 0}frac{sin(x)}{x}=lim_{xto 0}Gamma(x)sin(x)=1\$

Well it’s a little question :

How do you prove that :

$$lim_{xto 0}frac{sin(x)}{x}=lim_{xto 0}Gamma(x)sin(x)=1$$

Where we speak about the Gamma function

The most quick proof use the reflection euler formula because :

$$lim_{xto 0}Gamma(x)sin(x)=lim_{xto 0}Gamma(x)Gamma(1-x)sin(x)=1$$

Well I think there is a lot of proof so i add the tag “big-list”.

Particulary I would be happy to see a proof using the squeeze theorem .

## Mathematical limits on lossless data compression

Let’s say Bob wants to send a particular binary sequence to Alice. Imagine that Bob and Alice both have powerful machines but slow Internet connections. Bob could just send the sequence directly but the upload and the download would take a lot of time. Instead Bob could send a program that outputs the sequence. Assume Bob and Alice have agreed on the programming language they will use beforehand (be it C++ or Iota).

Two natural questions are what is the smallest possible size of the source file and how to find a program attaining the lower bound for a given sequence (e.g. can it be done algorithmically for any sequence or at least for some sequences). Have these questions been studied?

## pathfinder 1e – What are the limits of the Purrodaemon’s Weapon Steep ability?

### The total effective enhancement bonus cannot exceed +4.

The effective enhancement bonus, or modified enhancement bonus, on a magic weapon includes numeric enhancement bonuses to attacks and damage (such as +1, +2, etc) plus any special ability with an enhancement bonus equivalent.

Some magic weapons have special abilities (…) A single weapon cannot have a modified bonus (enhancement bonus plus special ability bonus equivalents, including those from character abilities and spells) higher than +10. A weapon with a special ability must also have at least a +1 enhancement bonus.

The purrodaemon’s Weapon Steep ability includes an example:

The total enhancements cannot exceed a +4 effective enhancement—most purrodaemons opt to create +2 wounding weapons in this manner.

The wounding ability is equivalent to a +2 enhancement, so a +2 wounding weapon has a total effective +4 bonus. Weapon Steep has no prerequisite for the weapon itself, so these bonuses could be added to a mundane weapon, making it into a magical weapon.

As for your example, a +1 impact weapon would have a total effective bonus of +3. Weapon Steep could increase the +1 bonus to a +2. Alternatively, it could add one special ability with a +1 enhancement bonus equivalent, such as frost or shock.

The brilliant energy special ability is equivalent to a +4 enhancement bonus. A +1 impact brilliant energy weapon would have a total +7 effective enhancement bonus, which exceeds the limitations of what Weapon Steep can create. So it’s not possible.

In fact, Weapon Steep cannot add brilliant energy (or any special ability with a +4 equivalent bonus) to any weapon. This is because these special abilities can only be added to a a weapon with an existing enhancement bonus of +1 or higher. At minimum, that would be a +1 brilliant energy weapon, which has a total +5 equivalent bonus. So it’s also not possible.

Note that some special abilities have no effective enhancement bonus, such as glamered and dueling. In theory, Weapon Steep could add any and all of these bonuses to a single weapon.

## ct.category theory – Ultralimit of Metric Spaces vs. Inductive Limits of Underlying Topological Spaces

Let $${(X_n,d_n)}_{n =1}^{infty}$$ be a sequence of bounded metric spaces such that:

• $$X_n subseteq X_{n+1}$$ is a metric subspace of $$X_n$$.

Let $$omega$$ denote the Zarisky ultra-filter (i.e.: complements of finite subsets of $$mathbb{N}$$ are always in $$omega$$) and let $$X_{omega}$$ denote the corresponding ultralimit metric space.

Let $$F:Met rightarrow Top$$ be the forgetful functor. Then the inclusions of $$F(X_n)$$ into $$F(X_{n+1})$$ are continuous and thus $$(F(X_n),(mathbb{N},leq))$$ form an inductive system; hence, we can take the inductive limit in $$Top$$. Denote this inductive limit by $$varinjlim F(X_n)$$.

How are $$X_{omega}$$ and $$varinjlim F(X_n)$$ related? I expect that $$varinjlim F(X_n)$$ can be identified with a subset of $$X_{omega}$$ but (most interestingly to me) is it a dense one?

## What are the limits to repeating a move in Dungeon World?

You seem to have misunderstood one of the most fundamental rules of Dungeon World:

## You don’t make the moves

Moves just happen when their trigger is met. That means you cannot tell your GM that you wish to, for example, take the “Defend” move. You have to narrate what your character does. If that triggers a move, then the move happens.

For example, if you narrate your character slashing at the goblin with their sword, the Hack and Slash move happens, because that move is triggered by attacking enemies in melee. However, the goblin might try to grab onto your character’s back and sink their filthy fangs into your neck in retaliation. You can’t just declare you want to Hack and Slash at the goblin again (well, you can, but it’s going to require some creative narration to explain how you can use your sword against a foe in such an awkward place). Instead you might need to shake off the goblin, which might amount to Defying danger.

The circumstances may also prevent you from making certain moves: for example, if your party is fighting a dragon with scales thick enough to withstand all your blows, the GM and other players around the table might decide that your attacks don’t count as Hack and Slash. You might be required to, for example, find a weak spot first – an operation that will most likely require plenty of other moves like Spouting Lore about the dragons’ weaknesses and Defying Danger to dodge the dragon’s attacks long enough to get to the softer spot.

## The moves cut both ways

When a move calls for a roll and your result is 6 or lower, it means trouble. Sometimes the move directly tells you what the effect is, and otherwise the GM is free to make a move of their own against your character. For example, with a squirming and biting goblin on your back, failing to Defy danger could result in the GM deciding to Deal damage to your character, or Separate them from the party by stumbling down a chasm trying to shake off the goblin. Failing to Spout lore could result in Using up the party’s resources (in this case, time spent observing the dragon in hopes of a weak spot) or an Unwelcome truth being revealed – for example that the dragon’s weak spot is actually in a very dangerous place, like the inside of its mouth.

Because these moves that result from failure tend to necessarily involve changing the circumstances, you won’t be able to indefinitely repeat a move in the same circumstance — quite soon, something is going to change to push the narrative forward.

## double integration u+v substitution selection and sensitivity to limits (astronomy problem)

I’m trying to understand an integration in a paper https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1008988&tag=1 by working through it by hand

formula for perfectly focused point source from Liebe paper

The limits for both integrals are -0.5 to 0.5 representing the grid of pixels on the imager imaging a focused star and if I use Wolfram the answer comes to 0.38 as stated

I used your video as a guide to a hand solution https://www.youtube.com/watch?v=UrTGqJraxXA
I went with u = x^2+y^2 for the first substitution

My problem I’m stuck at the choice for the second substitution v as I can’t use v=1+y^2 because I don’t have a 2 from the xy term to play with (can’t

Can anybody offer any guidance as to choice of u and v

I also note that the result in the case of limits 0 to 1 being used rather than -0.5 to 0.5 the result is 2x 0.38 … again any guidance you can offer would be appreciated

The reason for this is I want to understand the effect of defocusing on centroid determination as the practice in the paper is to deliberately defocus so as to get a 3×3 pixel spot on the imager

## dnd 5e – What is a spell slot in-lore, and how does it justify the limits on casting spells?

Spell slots are something that we, as players, expend when we want our spellcasting PCs to cast a spell. It is a resource to limit how many powerful spells we can cast in a day. But for our characters in-game, they don’t exist. So what are they?

The best way I can think of to illustrate my question is via an example involving a Kenku.

I was going to ask the question “Can a (non-spellcaster) Kenku cast a Verbal-component-only spell that they have heard a spellcaster cast via Mimicry?” I knew the answer would be that they can’t, but I was wondering what the in-game justification is for this.

Out-of-game, the answer is that they do not have the Spellcasting (or Pact Magic) class feature, and therefore do not have spell slots to expend to cast the spell, but that just replaced one question with another; what is a spell slot in-game? Knowing the answer to this would justify why the Kenku perfectly mimicking the Verbal component of a spell doesn’t work in-game.

There are other terms we use: HP, AC, XP; these terms do not exist in-game. My PC won’t know what “HP” is. HP has an in-game description, as explained further in this question: What does HP represent?

In short, from the PHB, pg. 196:

Hit points represent a combination of physical and mental durability, the will to live, and luck.

So those are things that my PC might know about and understand; luck, the will to live, etc. They make sense in-game and are something my character could talk about.

## What I’ve Looked Up

A Wizard’s Spellcasting class feature (PHB, pg. 114) only describes the mechanics of what a spell slot is to the player (I didn’t check the Spellcasting class feature for all the other classes), and the Spell Slots section (PHB, pg. 201) simply says (regarding flavour):

Manipulating the fabric of magic and channeling its energy into even a simple spell is physically and mentally taxing, and higher-level spells are even more so.

But that doesn’t explain why, say, a Kenku who has learned to mimic a Wizard’s spell’s Verbal components couldn’t cast a spell (without being a spellcaster class themselves; i.e. they have no spell slots).

Sure, it might “tax” them, but surely they’d be able to pull it off at least once that day? Or is it so taxing to even say that specific word or phrase that they wouldn’t actually be able to even finish saying it “without the proper training” (e.g. being a Wizard), and thus cannot “complete” the spell? (NB: This isn’t my question, it’s just included to show my train of thought.)

So what are spell slots in-game? What in-game “thing” do they represent? Is there an in-game justification for why a character who has spell slots can cast a spell in-game, whereas a different character without spell slots could not (even if they can satisfy the spell’s components; i.e. a Kenku perfectly mimicking the Verbal component)?

If the flavour of certain spellcasting classes would influence the answer such that all classes cannot be explained by one explanation (i.e. because Warlocks have Patrons, Sorcerers have “a spark of magic within them”, divine casters have gods or ideals, etc) then this question can just focus on Wizards specifically and what their spell slots mean, since a Wizard’s relationship with magic (i.e. “learning”) is closer to how a Kenku “learns” the Verbal component via mimicry.

Also, I’m not particularly interested in a settings-specific answer, but if a specific setting would influence an answer, let’s assume the Forgotten Realms (as it is the default setting of 5e).

Just to clarify: I don’t think the Kenku should be able to do this (e.g. a level 1 Kenku overhears a high level Wizard cast wish, uh… no), my question is why not from an in-game/lore perspective.

## Is there a certain period of time we should wait before visiting Japan again (Visit limits)?

I am looking to visit Japan soon, but I am conflicted about how long I want to stay in the country (within the 90 day Tourist visit), and when I will be going. I would love to stay the full 90 days, but I would like to go for a shorter period of time, the first time I go.

I have read that people have gotten flagged for going to Japan, then another country, and back to Japan again within a few month period. I have heard you are only given entry a few times (I heard 3), as a tourist (even though I read recently someone just got back from their 12th trip). So I am not sure what is true, and if you can only go so many times.

So my questions are, are there limitations to how long you have to wait before going back to Japan, and are there limits to the amount of times one can visit the country?

I figured if I can only go a few times total, and cannot do it within x amount of time I would maximize my stay, but if I could go twice within the next 8 or so months then I would rather do that.

Note I was thinking November for a few weeks, and then February or so the second time, so I’m not sure if that is enough time between the 2.