Representability of a subfunctor of the functor of points of a group scheme

Let $ G$ be a group scheme over a scheme $ S$ and $ h_G:\rm{Sch}/S\longrightarrow \rm{Grp}$ the functor of points represented by $ G$ .

Let $ k$ be a subfunctor of $ h$ . Is $ k$ representable? If so, can we represent it by a (locally closed) subscheme of $ G$ over $ S$ ?

My question might be very naïve, but it is justified by a lack of familiarity with group schemes.

best points for regression on cylinder

I want to choose between 25 and 35 points in a cylinder (with diameter = height) which minimizes the worst interpolation error for a multivariate polynomial (quadratic or possibly cubic) regression, with the constraint that points are not axially aligned.

My web search has not found anything, and the only heuristic I can think of is to minimize the largest distance from the points. Is there a systematic way of selecting good points or a better figure of merit for this?

Counting the Number of Lattice Points in an $n$-Dimensional Sphere

Let $ S_n(R)$ denote the number of lattice points in an $ n$ -dimensional “sphere” with radius $ R$ . For clarification, I am interested in lattice points found both strictly inside the sphere, and on its surface. I want to count exactly how many such points there are. This count corresponds to the amount of sets of integers that are solutions to the equation $ $ a_1^2+a_2^2+a_3^2+…+a_{n-1}^2+a_n^2= R^2$ $ where the $ a_i$ ‘s are required to be integers, not necessarily positive, and sets that differ just by the order of the summands are also considered distinct, i.e for the case $ n=3, R=5$ , the following are both counted: $ 3^2+(-4)^2+0^2$ and $ (-4)^2+0^2+3^2$ . This can also be interpreted as the sum of squares function$ r_d(k)$ denotes in how many such ways I can write $ k$ as the sum of $ d$ squares. This means that $ $ S_n(R) = \sum_{k=1}^{R^2}r_n(k)$ $

I am asking this question as a general one, but I am mostly concerned about the cases of $ n=2, n=3,n=4$ .

In order not to be too lengthy and remain focused, I won’t explain why but summing this way requires to factor each $ k$ first. This is very time consuming, so I searched for better ways.

For the case of $ n=2$ , which is essentially just the count of lattice points in a circle of radius $ R$ , there is a lot of information – this is known as the Gauss Circle Problem and I have managed to find that $ $ S_2(R)= 1+\sum_{i=1}^\infty\biggl(\biggl \lfloor\frac{R^2}{4i+1}\biggl \rfloor-\biggl \lfloor\frac{R^2}{4i+3}\biggl \rfloor\biggl )$ $ For the case of $ n=4$ , I found: $ $ S_4(R)= 1+8\sum_{k=1}^{R^2}\sum_{d|k \atop {4\nmid d}}d$ $ Unfortunately, for the case of a sphere ($ n=3$ ), I have found no formula, neither for the count of lattice points inside nor on the surface of the sphere.

Although these formulas do indeed provide the count of lattice points, they are very slow in computational terms, as their running time complexity is $ O(R^2)$ . I was wondering if a more efficient way exists to count the number of such lattice points, perhaps $ O(R)$ or $ O(R^{1/2+\epsilon})$ , so that I can work with radii as big as $ 10^9$ or even more. I suspect there is a way, but I can’t find it. Not necessarily a different approach to the problem, but rather just a more clever way to reduce the order of the sum. Also, any insight about $ S_3(R)$ will be appreciated.

Convex hull partition of a set of points

Given a set $ S$ of $ n$ points in $ \mathbb R^2$ , denote by $ \mathrm{conv}(S)$ the convex hull of $ S$ . Let \begin{align*} S_1 &= \mathrm{conv}(S)\ S_{i+1} &= \mathrm{conv}\left(S \setminus \bigcup_{j=1}^i S_j\right). \end{align*}

Now $ S_1,\ldots$ forms a partition of $ S$ . Is there an $ O(n\log n)$ time algorithm for computing this partition?

Ignore all those “free” wifi access points

This has probably been asked before but I was unable to find anything relevant with searches.

My problem is, I can be using my phone with my almost unlimited 4G data, quite happily, but if I pass too close to a place offering “free” wifi, or if a bus with “free” wifi on board passes me, all of a sudden I get redirected to a “You must sign in..” page.

I don’t want to sign in, I don’t WANT to use their “free” wifi. I want to continue using my 4G data, without having to go through the rigamarole of disabling wifi every time I leave the house. I want it to IGNORE those “free” wifi things.

Can this be achieved in any meaningful way?

What is the conversion rate for Sorcery points to Spell Points?

On page 288-289 of the DMG, there is an optional rule: Variant: Spell Points.

I am frequently using this optional rule at my table. I’m interested in streamlining resource-management even further, by merging the Sorcerer’s Sorcery Points and Spell Points into a singular pool of Spell Points that replace Sorcery Points all together, which I plan to playtest in a coming one-shot.

My question is, what is the effective conversion rate for Sorcery points to Spell Points utilising the Creating Spell Slots (PHB 101) table and the Spell Point Cost (DMG 288) table?

I’m not interested in answers concerning game balance or any other ramifications on the game.

Why are Lingering Injuries triggered when a creature drops to 0 hit points “but isn’t killed outright”?

The optional rule on p.272 of the DMG lists one of the triggers for Lingering Injuries as, “When (a creature) drops to 0 hit points but isn’t killed outright.” I am curious about why “but isn’t killed outright” was included in the description. This suggests that a creature that is killed outright and then raised with, say, Raise Dead, won’t have to worry about the possibility of missing limbs, whereas one who didn’t die straight away and later died from failed death saves won’t be so lucky. This doesn’t make sense to me. Shouldn’t that bullet simply say “When it drops to 0 hit points”?

How do spell slot recovery abilities work with the Variant: Spell Points system?

I am considering using the Variant: Spell Points system from page 288-289 of the DMG. I asked a related question on the overall balance of this rule. This question is more focused on the mechanics of implementing it.

Druids and Wizards have abilities that allow them to recovery spell slots during a shot rest. For example the Arcane Recovery feature states:

[…] Once per day when you finish a short rest, you can choose expended spell slots to recover. The spell slots can have a combined level that is equal to or less than half your wizard level (rounded up), and none of the slots can be 6th level or higher.

For example, if you’re a 4th-level wizard, you can recover up to two levels worth of spell slots. You can recover either a 2nd-level slot or two 1st-level slots.

How does this feature work when using the Variant: Spell Points system?

Initially I thought you would simply regain the equivalent number of spell points to the slots you could recover. However, the conversion is somewhat ambiguous due to the flexible nature of the recovery.

To use the example from the text, a 4th level wizard can recover 2 levels worth of spells. If they recover a 2nd level slot that is equivalent to 3 spell points. If instead they choose to recover 2 1st level spell slots that would be equivalent to 4 spell points. What is the correct way to handle this?