real analysis – Interpretation of series solution

I have a function that is of the form

$f = a + b – cf$

I noticed that I can sove this in one of two ways. In the first way, the solution is

$f = frac{1}{1+c}(a+b)$

In the second way I can perform a recursion to get a series solution. There’s some intermediate steps to reach this realization, but in the end it has the form

$f = (a + b)*(1-c^2+c^4-c^6+ldots)$

I’ve checked and the series does in fact converge to the solution $1/(1+c)$ I presented prior. However, there is a slight difference. In the first solution, the domain for $c$ extends to infinity. In the second solution, the radius of convergence is 1, thus the domain for $c$ is bound. Here is the plot for a modest number of terms…

enter image description here

The convergence behavior is clear and everything to this point I’m good with. My question is centered around how to interpret the solution differences, specifically their domains. As shown in the figure, the domain for the first solution is infinite. Let’s assume nothing about the recursive approach was known (which lead to the series solution). Is the first solution actually wrong?

I’m trying to understand if one authors approach is superior to the other. Is there a benefit to using one approach vs the other? Within the radius of convergence they’re both the same in the limit of infinite terms, but beyond the radius of convergence, one solution goes one way while the other goes another way. The physical application does have the potential to take on values larger than $c=1$ due to what $c$ represents. The graph tells me that if $c=2$, go with the first solution approach, but I want to make sure I’m not missing something fundamental (e.g. there is an implicit assumption that $c<1$ in the first approach).

ag.algebraic geometry – Geometric interpretation of $mathbb{C}^{times}$-gerbes

Let $X$ be a (nice enough, e.g. smooth etc.) variety over the complex numbers, and let $mathcal{G}$ be a gerbe on $X$. Then $mathcal{G}$ is classified by a cohomology class in $alpha in H^2(X, mathcal{O}^{times}_X)$, and if this class happens to be torsion then the gerbe has a nice geometric interpretation. Namely, it corresponds to a Brauer-Severi variety

$$mathbb{P}(mathcal{E}) longrightarrow X,$$

where $mathcal{E}$ is an $alpha$-twisted locally free sheaf. One can recover $mathcal{G}$ as the gerbe of trivializations of $mathbb{P}(mathcal{E})$.

Now suppose we have a $mathbb{C}^{times}$-gerbe $mathcal{G}’$ on $X$. For example, $mathcal{G}’$ could be $mathcal{L}^c$, where $mathcal{L}$ is a line bundle and $c in mathbb{C}^{times}$. Can we construct from $mathcal{G}’$ a space $Y$ with some structure over $X$ in such a way that $mathcal{G}’$ can be recovered from $Y$? For example, perhaps some choices of $mathcal{G}’$ arise as gerbes of trivializations compatible with a given flat connection on a Brauer-Severi variety, or something like that.

dnd 5e – What is the source of the “spells do only what they say they do” rules interpretation principle?

Every time somebody asks a question like “can I use X spell for doing Y” the answer is usually “no” because spell descriptions are very short in 5e, and usually they don’t explicitly say a spell can do Y.

These answers are based on the “spells do only what they say they do, nothing more” principle.

What is the source of this premise?

Related question: Is there a rule for how to handle creative use of spells?

I’m asking this as a DM. My first thought was “well it is obvious, why should spells do something more”. But the more I dig into this topic, the more contradictory arguments I find. I’ve gathered all thoughts down below, if anyone is interested (upvoted comments indicate that people are).


DMG examples

The Dungeon Master’s Guide has examples of spells doing things out of their originally described scopes:

An area of desecrated ground can be any size, and a detect evil and good spell cast within range reveals its presence. (p.110)

An identify spell reveals that a creature is inside the flask (p.178)

one torch can burn a Huge tapestry, and an earthquake spell can reduce a colossus to rubble (p.247)

Other factors might help or hinder the quarry’s ability to escape, at your discretion. For example, a quarry with a faerie fire spell cast on it might have disadvantage on checks made to escape (p.253)

RPG.SE answers

There are highly upvoted answers, implying creative spell usage is a thing in 5e:

Your players are using spells creatively…
That is exactly what D&D 5 encourages.

Open-ended spell descriptions

Some spells have quite open-ended wordings, like Prestidigitation:

harmless sensory effect, such as a shower of sparks, a puff of wind …

Other spells descriptions aren’t very detailed in 5e (compared to systems like Pathfinder), I guess that means it is the DM’s job to ultimately say what happens when somebody use a spell in an unusual way.

DMG supports this with its common principle, in Chapter 8: Running the Game (page 235):

Rules enable you and your players to have fun at the table. The rules serve you, not vice versa.

Trusted third-party sources

Some third-party sources encourages stretching spell limitations. See Geek&Sundry “How Watching Critical Role Made Me Better At D&D”:

A spell is typically written vague enough that you don’t have to worry about specific limitations unless you’re trying to stretch them. When in doubt, explain to the DM what you want to do and see if they’d be game

See also The Rule of Cool by Matthew Mercer.

Common sense

The strict “spells never do anything their description doesn’t mention” principle simply doesn’t work. When a player asks “Is Grease flammable?” they already challenge the frame, regardless of the answer. If the DM says “yes”, you can ignite the grease. If the DM says “no”, you can extinguish flames using the grease.

Of course, since “spells do only what they say they do”, DM might say “no” to both assumptions, but this effectively turns a tabletop role-playing game into a pen and paper computer game, boring and awkward. I don’t think this is actually RAI.

It seems a DM is supposed to resolve an unusual spell application case, using the intent behind the spell, rather its literal description. For example, it seems reasonable you should be able to use any fire-producing spell to light a torch in a non-combat situation, even when its description doesn’t explicitly say that it “ignites flammable things”. It heavily depends on the particular DM’s style, like many other things in 5e.

What is the source of the “spells do only what they say they do, nothing more” principle?

dnd 5e – What is the correct interpretation of the Gambling Results table in Xanathar’s Guide to Everything?

In Xanathar’s Guide to Everything, one of the downtime options provided in “Downtime, Revised” allows a character to gamble during their downtime to earn extra money.

Gambling

Games of chance are a way to make a fortune—and perhaps a better way to lose one.

[…]

Gambling Results
|Result | Value |
|—|—|
|0 Successes | Lose all the money you bet, and accrue a debt equal to that amount. |
|1 Success | Lose half the money you bet. |
|2 Successes | Gain the amount you bet plus half again more. |
|3 Successes | Gain double the amount you bet. |

Xanathar’s Guide to Everything, pg. 130

So if I place a bet of 100gp and make my checks against this table, how much would I have, for each category?

sql server – Can I change MSSQL interpretation of date order system wide?

I use Laravel with PHP from https://laravel.com/

MSSQL is a driver and works perfectly – with th exception of dates.

For some reason the project has decided that the follwoing is a universal date format

Y-m-d H:i:s
2021-01-21 12:00:00

However this does not seem to be universal as MSSQL interprets this as a US based date eg. The date above is invalid as 21 is considered the month not the day.

I will try to get them to allow date to be set to a custom format but this may take a while.

My MSSQL user is set to Language:British English which solves most of my issues, but not all.

Is it possible to in some way tell MSSQL to interpret a date like this

2021-01-22 12:00:00

as

YYYY-MM-DD HH:II:SS

instead of

YYYY-DD-MM HH:II:SS

which is what it is doing?

Can this be done on a system wide basis?

differential geometry – Interpretation of transport equation for a two-form

The vorticity equation arising from the Euler equation for a fluid with velocity $bf u$ reads:

$$
partial_t {bf{w}} + nabla times( {bf{w}} times {bf{u}}) =0
$$

where the vorticity is ${bf{w}} =nabla times {bf{u}}$.

Now, assume that we have a 1-form $u$ and its associated “vorticity” 2-form $w=du$. Is it possible to interpret the above vorticity equation in terms of Lie derivative $L_u$ (or other kind of derivatives) and, possibly, Hodge duality? This would give a geometric meaning to the second term in the equation. Maybe something like

$$
partial_t {w} + L_u w =0 , ?
$$

probability distributions – Geometry interpretation of any continuous random variable

Given a continuous random variable $X$ with the cdf $F_X(x)$, I want to know whether there exists a random variable $mathbf{Z}$ uniformly distributed in a geometry region $mathscr{Z}_n$ in $mathbb{R}^n$ such that any one-dimensional marginal distribution $F_{mathscr{Z}_n}^1$ of $mathbf{Z}$ is close to the distribution of $X$. In addition, an asymptotic perspective is also welcome. For example, it is also meaningful to know whether $F_{mathscr{Z}_n}^1=F_X$ is possible if $n$ tends to $+infty$.


Some useful facts are given as follows:

  1. A normal distribution can be approximated by a uniform random variable distributed in a hypersphere.
  2. A negative exponential distribution can be approximated by a uniform random variable distributed in a simplex.

This question has been asked in enter link description here yesterday.

nt.number theory – Standard interpretation of permanents over finite fields

Given a $0/1$ matrix in $mathbb Z^{ntimes n}$ the standard interpretation of permanent of the matrix is the number of perfect matchings in the underlying $2n$ vertex balanced bipartite graph with biadjacency given by the matrix.

Suppose we are given a square matrix $U$ with entries in $mathbb F_{p^t}$ which is a finite field of characteristic $p>2$.

  1. What is the standard interpretation of permanent in these scenarios?

  2. Suppose the matrix is orthogonal is there a standard interpretation?

Is there reference?

intuition – Interpretation of polar derivative and apolar polynomials

If $f(x)$ is a degree $n$ polynomial and $b in mathbb{C}$ is a complex number, then the polar derivative of $f$ with respect to $b$ is defined by
$$D_b f(x) = nf(x) – (x – b)f'(x).$$
Two degree $n$ polynomials $f(x)$ and $g(x)$ are said to be apolar if
$$sum_{k=0}^n (-1)^k f^{(k)}(0)g^{(n-k)}(0) = 0,$$
and this is equivalent to
$$D_{b_1}D_{b_2}cdots D_{b_n} f(x) = 0,$$
where $b_i$ are the roots of $g(x)$ with multiplicity.

I have come across these notions in relation to Grace’s Theorem. I am trying to understand the meaning of the polar derivative and to find an interpretation of what it means for two polynomials to be apolar. Pointers to the origins of these notions would also be appreciated.

lie groups – What is the physical interpretation of each diagonal entry in the block matrix of the representations of $U(1)$

This is one of those ideas that it seems intuitively clear at first, but then it starts to blur out. I see the comment “There is no such thing as the $n$-dimensional representation of $U(1)$.” in this post and the explanation in Peter Woit’s Quantum Theory, Groups and Representations: An Introduction

enter image description here

Figure 2.2: Visualizing a representation $π : U(1) → U(n),$ along with its differential.
The spherical figure in the right-hand side of the picture is supposed to
indicate the space $U(n) ⊂ GL(n, C)$ ($GL(n, C)$ is the $n times n$ complex matrices,
$C^{n^2},$ minus the locus of matrices with zero determinant, which are those that
can’t be inverted). It has a distinguished point, the identity. The representation
$π$ takes the circle $U(1)$ to a circle inside $U(n).$ Its derivative $π’$ is a linear map
taking the tangent space $iR$ to the circle at the identity to a line in the tangent
space to $U(n)$ at the identity.


I understand how

$$R(U(1)) =begin{bmatrix}costheta & -sintheta \ sintheta & costheta end{bmatrix} in GL(2,mathbb C)$$

which can be expressed as

$$R(U(1)) =begin{bmatrix}e^{itheta} & 0 \ 0 & e^{-itheta} end{bmatrix}$$

in the basis of eigenvectors $left{ begin{bmatrix}i \ 1 end{bmatrix} , begin{bmatrix}-i \ 1 end{bmatrix}right}$

But what is the meaning of representations in higher $GL(n,mathbb C)$?