How to obtain the same result of the same system if I write the system in different ways (NDSolve)?

I did program where I solved a system when I organized in matrix form, this is the code

Clear["Global`*"]  SeedRandom[1234]  Nmax = 5; (*Number of sites*)  tini = 0; (*initial time*)  tmax = 200; (*maximal time*)  \[Sigma]2 = 0; (*Variance*)  n0 = 5; (*initial condition*)  ra = 1; (*coupling range*)  \[Psi]ini = Table[KroneckerDelta[n0 - i], {i, 1, Nmax}];  RR = RandomReal[{-Sqrt[3*\[Sigma]2], Sqrt[3*\[Sigma]2]}, Nmax];  Z = Table[     Sum[KroneckerDelta[i - j + k], {k, 1, ra}] +       Sum[KroneckerDelta[i - j - k], {k, 1, ra}], {i, 1, Nmax}, {j, 1,       Nmax}] + DiagonalMatrix[RR];  Clear[\[Psi]]  usol = NDSolveValue[{I D[\[Psi][t], t] ==      Z.\[Psi][t], \[Psi][0] == \[Psi]ini}, \[Psi], {t, tini, tmax}]  Plot[usol[t], {t, tini, tmax}] 

Now, I´m trying to solve the same system but writing the equations

Clear["Global`*"]  tini = 0;  tmax = 200;  usol = NDSolveValue[{I x1'[t] == x2[t], I x2'[t] == x1[t] + x3[t],      I x3'[t] == x2[t] + x4[t], I x4'[t] == x3[t] + x5[t],      I x5'[t] == x4[t], x1[0] == 0, x2[0] == 0, x3[0] == 0, x4[0] == 0,      x5[0] == 1}, {x1, x2, x3, x4, x5}, {t, tini, tmax}];  Plot[usol[t], {t, tini, tmax}] 

Why the second code doesn´t give me the same result if I write the same system?

What are good ways to keep the player spending in a single-player RPG?

I’m designing a game where I want the player to take the in-game money into consideration as opposed to majority where they just get rich pretty much all the time with nothing meaningful to spend it on. What are some good ways to keep them spending their money without making it feel like a tedious chore?

I’m already planning to have monsters nor quests not give gold.

And if preventing them from becoming extremely rich eventually is not possible, at least mitigate "easy" ways of generating infinite income. I want them to struggle with their money.

Are there any ways to increase the number of items I can attune to without being an Artificer?

The problem is quite simple: I want to attune to four or more magical items at once, and I don’t want to take ten levels in Artificer.

Are there any features, items, etc. that increase the number of items you can attune to without being a 10th level or higher artificer?

Ways to numerically find the monodromy group of an ODE

I’m interested in finding the monodromy group of some solutions of an ODE that I know only either approximately or numerically. For example, I might have a 3rd order ODE of which I cannot find exact solutions, but I can find them to very high accuracy as a series expansion. For the sake of the example, however, let me take an ODE of which I know the solutions, so I can compare whatever numerical method I have to the analytic solution.

Let me take the ODE $ $ f”(z)+\frac{(2-5 z) f'(z)}{4 z (1-z)}-\frac{3 f(z)}{64 (z-1)^2}=0$ $ that has the two linear independent solutions $ $ f_1=\frac{\sqrt{1-z}+\sqrt{z}+1}{2 \sqrt{\sqrt{z}+1} (1-z)^{1/8}}$ $ $ $ f_2=\frac{-\sqrt{1-z}+\sqrt{z}+1}{\sqrt{\sqrt{z}+1} (1-z)^{1/8}}$ $

Say that I start with $ 0<z<1$ , I analytically continue $ z$ to the complex plane, go around $ z=1$ , which is a branch point, and I come back to $ 0<z<1$ . From the explicit solutions, we can see that they change to $ $ \left( {\begin{array}{c} f_1 \ f_2 \ \end{array} } \right)\to \left( {\begin{array}{cc} 0 & \frac{1}{2} e^{-\frac{1}{4} i \pi }\ 2 e^{-\frac{1}{4} i \pi } & 4 \ \end{array} } \right) \left( {\begin{array}{c} f_1 \ f_2 \ \end{array} } \right) $ $

Now, let’s assume I don’t know exactly the form of $ f_1$ and $ f_2$ . Maybe I know them for example as a series expansions around z=0 with as many orders as I want. I wanna find out how the monodromy group acts on these solutions.

One way to do this would be as explained here. The idea is to discretize the path around z=1, solve the ODE as a series expansion in all of these points, and match these approximate solutions to the ones closeby. In this way we go around z=1 approximately. I’ve tried this and, assuming I’ve done this correctly, I get a sensible result, but even with 16 points and keeping 200ish orders of the series expansion, I get an error which is roughly 1-2%.

I would like to get these numbers with a higher precision. Is there any way that I could do this, without using the explicit form of $ f_1$ and $ f_2$ ? Maybe there’s a smart way of using NDSolve?

Are there any Feats for / ways to use spears it that reflect it’s dominance over shorter melee weapons

Spear are wildly effective weapons, in small-ish Combat Groups. In particular, when used defensively, they can keep melee attackers at bay with no ability to retaliate relatively easily. You would, for example be able to hold a 5ft-wide bridge against a mass of opponents pretty easily if they weren’t willing to rush you and sacrifice themselves.

I was disappointed to find that in DnD 5e, they really aren’t very effective or interesting weapons. They don’t even have Reach.

I’m planning to come up with some ideas for a homebrew additions to them, and propose them on here, but before I do, are there any existing official ways to make a spear behave like this?

What are some ways one might implement a more story-driven spell/skill acquisition?

What are some ways one might implement a more story-driven spell/skill acquisition? Hopefully in a way that isn’t a complete slog.

The idea of the players’ handbook being used as a shopping catalogue kinda ruins the immersion for me. It doesn’t make much sense that a character can go to sleep one day and then wake up the next suddenly knowing a bunch of new spells or skills if they happened to reach a new level. Especially that often those spells/skills have nothing to do with any in-game actions or backstory of the character. I’d love to see a game where the characters have to actively work towards acquiring their skills, and not just waiting till they’ve passed the next level threshold.

The question is largely about how a DnD5e system might get modified to fit my concerns because I don’t really have a group yet and 5e players are easiest to find. That said I’m open to hearing about other game systems.

In the context of 5e I was thinking of flipping the relationship between levels and skills/spells on its head and not have levels grant you skills but rather the achieving skills bumping your level up (at GM’s discretion). By "achieving skills" the character pursuing an RP way to acquire a skill. Depending on what’s going on in-game it might be super easy for some skills (e.g. you find that move that you’ve been trying to do all this time suddenly just works after enough practice) or verging on impossible for others (e.g. getting an ice-y spell for a character who is all about fire stuff). I imagine this could create a greater sense of anticipation and, ultimately, reward at the advancement of characters. Or I might be completely wrong and it’s only going to scare off potential players who just want to minmax.

Ways to mantain a mount

Mounted combat is really cool part of D&D, but sustaining a mount seems like it would be difficult and costly. What are the rules governing the sustainment of a mount and are there ways for low level/poorer characters to do so? Also, are there alternatives to owning and maintaining a steed that will allow you to engage in mounted combat but not have to worry about keeping a horse during the long stretches when you are not traveling or fighting on horseback? For instance, Figurines or Power and Find Steed spells can provide a temporary, "disposable" mount. Are there other methods of doing so, magical or otherwise?

In D&D 5e, what are all the ways to inflict the Paralyzed Condition on a creature?

In D&D 5e, what are all the ways to inflict the Paralyzed Condition on a creature?

I am talking about all the ways, not just spells.

  • All official actually printed-in-books sources (thus excludingg a lot of Unearthed Arcana stuff that is mostly "experimental" until actually printed in a book)

  • Even from monsters.

Personally I know only the Hold Person and Hold Monster spells.

The reason I am wondering is that I play a Barbarian and what to find ways to use my half-orc’s Savage Attack racial feature more often.


What are the fastest mundane ways to heal out of combat?

Let’s take an untemplated human who’s low on hit points. Let’s also say that, for whatever reason, they don’t have access to spellcasting, SLAs, supernatural abilities, magic items, psionics, or any class levels or feats that grant any of those things (so if there’s something that can help as a 10th level feature, but the class grants a Su ability at 9th level, that doesn’t count).

What’s the fastest way for them to heal an arbitrary number of hit points, only possessing/using entirely mundane (extraordinary) features and feats and not using any benefits tied to certain locations?

The standard method is to take a couple days of rest, regaining up to 2 hit points per level per day (4 with long-term care). Not bad, but we can probably do better.

The 6th-level Iron Heart maneuver Iron Heart Endurance takes a swift action and heals them for twice their level in hit points, but it only works until they have a bit over half their hit points. For a couple well-timed warblade levels, they can use this every other turn to get a bit above half health. That’s a start, but we need a way to fill in the rest.

There’s also the well-known Troll Blooded feat, but that’s Dragon Magazine. Let’s go for only WotC-published 3.5e (or unupdated 3e) material.

They could take Combat Vigor or Vital Recovery, but both of those require an actual encounter to take place. Maneuvers may be recovered after a minute of rest (ToB 40), but both those feats are explicitly once-per-encounter. They could carry around a bag of aggressive rats, but once they enter combat with one of them, they’re no longer meeting the "out of combat" criteria quite right.

And… That’s all I can think of. What other options does an entirely mundane character have for regaining lost hit points?