## pathfinder 1e – I’m building a cannibal barbarian. With a natural bite attack, will I only be able to use one attack with bite or weapons?

So I’m building an evil half-orc barbarian with a rage tree on biting and eating an opponent still alive. When I get multiple BABs, will I only be able to make a single bite attack? Even if I can’t use more than one bite around, can I still use a weapon? This might greatly alter my build if I can’t find the answer to this.

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## Ker of \$pi:G/Nto G/H\$ (natural projection)

Let $$G$$ be a group and $$N$$ and $$H$$ are normal subgroup of $$G$$ and $$N$$ is normal subgroup of $$H$$.
$$pi:G/Nto G/H$$ be natural projection, that is, $$xpmod{N}to xpmod{H}$$.

Then, I would like to formally prove $$kerpi＝H/N$$.

My attempt:
From fundamental theorem on homomorphisms,
$$(G/N)/kerpi cong G/H$$

## natural language processing – Is this problem decideable programatically?

Consider the following 3 strings:

`1tb hdd 256gb ssd`
`hdd 256gb`
`1tb hdd`

If you know, that all of these strings are part of a product title, specifically part of the title of a Laptop product, a human can easily read these out, and understand that the specific laptop for which the respective string is part of has either hdd/ssd disk in it, and if it has, he can also decide the capacity of the said disk.

Is there any way to programatically make the same assumption correctly?

I am trying to parse the attributes of some products from their titles, and this exact problem arose: some titles have the ordering of `{disktype} {capacity}`, some have the ordering of `{capacity} {disktype}`, and I can’t figure out a way to correctly identify the order, so that I can correctly parse the attribute and the data. In some cases, the titles contain `,` characters, which is very helpful in this situation, but there are a few cases where there are no other delimiters except `s` characters.

To make things worse, some product titles only contain the following structure: `1tb ssd 512gb`. Based on the human understanding, and the context of the above string, for a human, it is still decidable which type of hard disk the laptop contains, and what capacity does it have, but can a computer do this programaticaly in a safe way also?

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## natural language processing – Compute the edit distance between two words in which substitution is not allowed

This is a straighforward modification of the classical dynamic programming algorithm.

Let the first string be $$s = s_1 s_2 dots s_n$$ and the second string be $$t = t_1 t_2 dots t_m$$.
In the dynamic programming algorithm you define $$D(i)(j)$$ as the edit distance between $$s_1 s_2 dots s_i$$ and $$t_1 t_2 dots t_j$$.

The bases cases are $$D(0)(j)=j$$ and $$D(i)(0)=i$$ (for any $$i = 0, dots, n$$ and $$j=0, dots, m$$), and the recursive formula (for $$i>0$$ and $$j>0$$) is:

$$D(i)(j) = min begin{cases} 1 + D(i-1)(j) & mbox{ deletion of s_i};\ 1 + D(i)(j-1) & mbox{ insertion of t_j};\ 1 + D(i-1)(j-1) & mbox{ substitution of s_i with t_j}. end{cases}$$

You can simply modify the above formula by not contemplating the last case, i.e.:
$$D(i)(j) = min begin{cases} 1 + D(i-1)(j) & mbox{ deletion of s_i};\ 1 + D(i)(j-1) & mbox{ insertion of t_j}. end{cases}$$

By definition of $$D(cdot)(cdot)$$, the edit distance between $$s$$ and $$t$$ is $$D(n)(m)$$.

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## Can every natural number be written as \$lfloor 2^{2^s+1}/3^k rfloor\$?

Problem:

Can every natural number be written as $$lfloor frac{2^{2^s+1}}{3^k} rfloor$$ for some $$s,kin mathbb{N}$$?

Context and thoughts:

This is a generalization of these problems: Density of $$2^a 3^b$$ and Creating a number by multiplying by 2 and dividing by 3. In the second link there is a comment I did where I tried to confirm that when the exponent of the $$2$$ in the numerator can be any natural number, then the answer is yes. This procedure does not work, at least directly, in this case, because it would need that $$2^nalpha – lfloor 2^nalpha rfloor$$ to be dense in $$[0,1]$$, and in the question I linked it’s proven to be false in general. I assume this has some connections with the normality in base $$2$$ of the irrational $$alpha$$, which in our case would be $$frac{log 2}{log 3}$$ if I’m not mistaken, but I don’t know if it’s even known whether it’s normal or not. I also don’t think both problems are equivalent, density and normality seem to be way stronger than the question only for naturals.

This new version comes from this question: Integer less than 7000 achievable by …. One can notice that if the answer to the question I posted is yes, then one can produce a method to obtain all integers in the linked question because the numerators are always achievable just with the first function and $$4$$. I understand the obvious overlap between the two, but I’m actually interested in methods for questions of density like this one, because I’ve seen them in different contexts, while the linked question can have other approaches. Moreover, I think the answer to this one could end up being negative.

My attempt at programming it for low numbers that you can try online here seems to get all decompositions from $$1$$ to $$60$$, due to the exponent the numbers get large quite fast so it starts to get slower the more you continue because the precision needs to be larger.

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## pathfinder 1e – Full attacking with natural while grappling or being grappled

My GM said that a creature with several appenges can full attack with all of their natural attacks towards me?

I know that the grapple rules specify “Instead of attempting to break or reverse the grapple, you can take any action that doesn’t require two hands to perform, such as cast a spell or make an attack or full attack with a light or one-handed weapon against any creature within your reach, including the creature that is grappling you”
I’m reading the a light weapon meaning it’s a single light weapon and not all, such as a dagger, is this incorrect?

p.s. Are hands and appendeges in any way connected or are the grappling rules completely broken while you’re using anything other than hands, say tentacles etc

My GM is a stickler for rules, so any answers with sources is greatly appreciated.

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## people – How can I get my portrait subjects to look natural and drop the cheesy smile?

Assuming this is PORTRAIT photography (where you have one on one time with no event distractions):

For very natural expressions, the thing I do is aim to make the subject feel SAFE (always). I mean: EMOTIONALLY SAFE. You must demonstrate that you are not judging them and make an effort to understand and appreciate their issues around taking photos. People are more open around people who they feel safe with. Make this a priority. People smile in an unnatural way or pose unnaturally because they haven’t created a relaxed relationship with you yet. I also coach them on being present with their body (more on this later).

I tell my subjects/clients exactly how the shoot is going to go so there’s more certainty. More certainty creates confidence and the feeling of being safe. They know how it’s going to play out. I also ask them some questions that I find super important (because I wish people who photographed me were more sensitive to these topics). As a photographer, I try my best to be a good listener.

Some questions I ask (helps build rapport and helps you know your subject):

• How do you feel about taking photos?
• How do you feel about smiling with teeth? (Some photographers assume people like to smile. This is ABSOLUTELY UNTRUE. So many people hate their teeth, hate their smile, or some other issue)
• What parts of your body do you not like? (They will tell you. And they will probably be relieved that you know this info.)

No matter how beautiful I think they are, I promise them that I will not force them to do anything they aren’t comfortable doing. And sometimes they become so relaxed during the shoot (after we agree on the boundaries of our shoot), I actually do get their beautiful smile.

Here’s an example of what I tell my clients:

“The shoot will take about 2 hours. And it usually happens in three phases. The first phase is where you and I are going to get to know each other. We’ll get some practice shots out of the way. It might be weird because there will be a lot of times where I’m mostly going to be staring at you and not saying anything. If I’m not giving you any direction, don’t worry, it’s because I’m thinking or trying to focus my lens. I’m totally going to be staring at you A LOT. I might get really close with my camera and it’s gonna feel weird. About 45 minutes in, we’re probably going to get our money shots and then if we feel like it, we can be playful and we can even try weird things.”

SO now that I’ve told them how the shoot it going to go, I tell them to pay extra attention to their body. I actually talk to them as if this were a job. “Your job is to chill out. When I’m not giving you direction, just keep doing what you’re doing because I probably really like it. Also, you might be used to smiling or sitting or standing a certain way in front of the camera, I might give you different directions. So your job is to focus: chin down because I want your eyes closer to my lens. People sometimes tilt their head so I may repeatedly tell you ‘head straight’.” Just clue your subjects in on the trappings of unnatural expressions and create the expectation that you may give them repeat instructions.

Create confident clients/subjects and make them feel emotionally safe.

I hope this is helpful.

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## ag.algebraic geometry – Explicit Natural Correspondence between Cusps of X(N) and isomorphism classes of Level N structures on Tate(q^N)

In Katz’ paper Antwerp III, section 1.4 (Ka-14) one reads (we assume $$n geq 3$$ integer):

”The scheme $$overline{M}_n – M_n$$” over $$mathbb{Z}(1/n)$$ is finite and étale, and over $$mathbb{Z}(1/n,zeta_n)$$, it is a disjoint union of sections, called the cusps of $$overline{M}_n$$,”

I would be interested to see a detailed proof of the next part of that sentence, namely:

“which in a natural way are the set of isomorphism classes of level $$n$$ structures on the Tate curve $$text{Tate}(q^n)$$ viewed over $$mathbb{Z}((q)) otimes_{mathbb{Z}} mathbb{Z}(1/n,zeta_n)$$.”

What I did: I tried to extract the relevant information in Deligne-Rapoport and Katz-Mazur but in each case, certainly for a lack of understanding on my part, I’m not able to establish this correspondence explicitly. I found the discussion of formal completion at (the divisor of) cusps well explained in both references (something which is also addressed in Katz’ paper Antwerp III on page Ka-14), but I couldn’t connect the dots for the natural correspondence above and thus my question. Feel free to ask if you need more details.

In Deligne-Rapoport?

I first looked in Deligne-Rapoport (DR), which was in Antwerp II (and so the natural place to look for the arguments):
http://smtp.math.uni-bonn.de/ag/alggeom/preprints/Lesschemas.pdf

The motivating example on pages DeRa-7 and beginning of page 8 hint to that fact. But it seems that’s not the point of view (DR) take.

“Dans le texte, nous précisons cette interprétation modulaire de l’ensemble des points à l’infini de $$mathcal{H}/Gamma(n)$$ en une interprétation modulaire de la courbe projective compactifiée $$overline{mathcal{H}/Gamma(n)}$$ de $$mathcal{H}/Gamma(n)$$.”

Here: $$mathcal{H} = { z in mathbb{C} mid Im(z) > 0 }$$ is the upper half-plane.

On page DeRa-10 they do say that $$M_n$$ can be defined as the normalization, in the field of functions of $$M_n^0(1/n)$$, of the projective $$j$$-line over $$mathbb{Z}(zeta_n)$$. (That’s what Katz and Mazur do in their book on Chapter 8.) (DR) say among other things that they prove that there exists a finite family of points $$mathbb{Z}(zeta_n)$$-points $$f_i : M_n to Spec(mathbb{Z}(zeta_n))$$ such that the sections $$f_i$$ are disjoint (incongruent modulo any prime ideal of $$mathbb{Z}(zeta_n)$$) and that $$M_n^0$$ is the complement in $$M_n$$ of the union of the ”sections at infinity” $$f_i$$.

The Tate curve is only constructed in chapter VII of (DR). But I don’t find it immediate to deduce initial assertion by Katz in Antwerp III.

In Chapter VII (sections 1 and mostly 2 seem relevant to my question), DeRa-156, (1.16.4) gives me the description of the level $$r$$-structure of the Tate curve with $$r$$ edges over $$mathbb{Z}((q^{1/r}))$$.

Moreover, $$text{Tate}(q)$$ over $$mathbb{Z}((q))$$ induces a morphism $$tau: Spec(mathbb{Z}((q))) to mathcal{M}_1$$ which identifies $$mathbb{Z}((q))$$ with the formal completion of $$mathcal{M}_1$$ along the section at infinity $$f_1$$ (Theorem 2.1).

The Néron $$n$$-gon $$C$$ over $$mathbb{Z}(zeta_n)$$ equipped with its structure of generalized elliptic curve and the natural isomorphism $$C(n) = mu_n times mathbb{Z}/nmathbb{Z}$$ defines a section at infinity $$f_n : Spec(mathbb{Z}(zeta_n)) to mathcal{M}_n$$. We also obtain an isomorphism between the $$n$$-torsion of the Tate curve with $$n$$ edges and $$mu_n times mathbb{Z}/n mathbb{Z}$$ and then we geta morphism $$Spec(mathbb{Z}(zeta_n)((q^{1/n}))) to mathcal{M}_n$$. This latter morphism identifies $$mathbb{Z}(zeta_n)((q^{1/n}))$$ with the formal completion of $$mathcal{M}_n$$ along the section at infinity $$f_n$$.

Finally, Corollary 2.5 says that the completion of $$mathcal{M}_n$$ along infinity is sum of copies of $$Spec(mathbb{Z}(zeta_n)((q^{1/n}))$$ indexed by $$SL_2(mathbb{Z}/nmathbb{Z})/pm U$$, where $$U$$ is the group of upper unipotent matrices.

It feels like the desired correspondence is there but I couldn’t extract it explicitly.

In Katz-Mazur?
I turned to the book of Katz and Mazur (see https://web.math.princeton.edu/~nmk/katz-mazur.djvu). Again, I feel I’m getting closed, but I’m not sure how to tie up the loose ends.

The point of view in (KM) doesn’t deal (explicitly?) with stacks (as in (DR)). They consider the moduli problem (contravariant functor)

$$(Gamma(N)) : textbf{Ell} to textbf{Set}$$

which classifies elliptic curves (proper smooth curves $$pi : E to S$$ with geometrically connected fibers all of genus one, given with a section $$0$$, and here $$S$$ is any scheme.) equipped with a $$Gamma(N)$$-structure (KM 3.1, page 98).

This functor is relatively representable and flat over $$textbf{Ell}$$ of constant rank $$geq 1$$, and regular of dimension $$2$$. As a functor with source $$textbf{Ell}/mathbb{Z}(1/N)$$ it is étale on the source. (First Main Theorem 5.1.1, page 129).

When $$N geq 3$$, $$(Gamma(N))$$ is in fact representable by some universal elliptic curve $$E_text{univ}/Y(N)$$, where $$Y(N)$$ is a smooth affine curve (We have a rigidity.) (See (KM) Cor 2.7.2, 4.7.0 and 4.7.1)

Following (KM 8.6.3 and 8.6.8) we normalize $$Y(N)$$ near infinity to obtain $$X(N)$$ (we obtain a smooth proper curve over $$mathbb{Z}(1/N)$$ which is the normalization of the projective $$j$$-line in $$Y(N)$$).

The Tate curve $$text{Tate}(q)$$ itself represents an appropriate moduli problem $$mathcal{S}$$. Applying corollary 8.4.4 (p.235) to this and to the moduli problem $$(Gamma(N))$$ over an excellent noetherian regular ring $$R$$, we obtain an isomorphim of $$R((q))$$-schemes

$$left( (Gamma(N))_{text{Tate}(q)/R((q))} right)/ pm 1 xrightarrow{simeq} Y(N)_{R((q))}$$

where $$Aut(text{Tate}(q)/R((q))) = pm 1$$ (see Proposition 8.11.7).

Moreover, the formal completion of $$X(N)$$ along the (divisor of) cusps $$X(N) – Y(N)$$, which is a finite $$R((q))$$-scheme, is the normalization of $$R((q))$$ in the finite normal $$R((q))$$-scheme $$left( (Gamma(N))_{text{Tate}(q)/R((q))} right)/ pm 1$$.

Finally, we have

Theorem 10.8.2

There is a canonical isomorphism of $$mathbb{Z}(zeta_N)((q))$$-schemes

$$(Gamma(N))_{text{Tate}(q)/mathbb{Z}(zeta_N)((q))} simeq coprod_{text{Hom Surj }((mathbb{Z}/Nmathbb{Z})^2,mathbb{Z}/Nmathbb{Z})} Spec(mathbb{Z}(zeta_N)((q^{1/N})))$$

and

Theorem 10.9.1

(1) $$text{Cusps}(X(N))$$ is the disjoint union of $$mid text{Hom Surj }((mathbb{Z}/Nmathbb{Z})^2,mathbb{Z}/Nmathbb{Z}) mid$$ sections of $$X(N)$$ over $$mathbb{Z}(zeta_N)$$.

(2) There exists an open neighborhood $$V$$ of the cusps $$text{Cusps}((Gamma(N))) subset V subset X(N)$$ which is smooth over $$mathbb{Z}(zeta_N)$$.

(3) The formal completion of $$X(N)$$ along its cusps is the $$mathbb{Z}(zeta_N)$$-formal scheme

$$coprod_{text{Hom Surj }((mathbb{Z}/Nmathbb{Z})^2,mathbb{Z}/Nmathbb{Z})/pm 1} Spfleft( mathbb{Z}(zeta_N)((q^{1/N})) right)$$.

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## algebra precalculus – Prove that the product of four consecutive natural numbers are not the square of an integer

Prove that the product of four consecutive natural numbers are not the square of an integer

Would appreciate any thoughts and feedback on my suggested proof, which is as follows:

Let $$f(n) = n(n+1)(n+2)(n+3)$$.
Multiplying out the expression and refactoring it in a slightly different way gives
$$f(n) = n^4 + 6n^3+11n^2+6n \= n^4 + 6n^3 + 9n^2 + 2n^2 + 6n = (n^2 + 3n)^2 + 2n(n+3). tag{1}label{1}$$

We want to show that the only possible way for $$f(n)$$ to be the square of an integer is if $$f(n) = (n^2 + 3n +1 ) ^2.$$ We show this by proving that $$(n^2+3n)^2 < f(n) < (n^2+3n+2)^2$$. The left-hand side follows immediately from $$(1)$$, since $$2n(n+3) > 0$$ for all $$n geq 1$$, and the right-hand side can be verified by multiplying out both sides:
begin{align} (n^2+3n)^2 + 2n(n+3) &< (n^2+3n+2)^2 \ iff n^4 + 6n^3 + 11n^2 + 6n &< n^4 + 9n^2 + 4 + 6n^3+4n^2+12n \ iff 0 &< 2n^2 + 6n + 4 end{align}
which is true for all $$n geq 1$$. Now we note that $$n^2+3n = n(n+3)$$ is even since one of the factors $$n$$ or $$n+3$$ is even for all $$n$$. It follows that $$n^2+3n+1$$ must be odd, and so $$(n^2+3n+1)^2$$ must be odd. But $$f(n)$$ must be even, since either $$n$$ and $$(n+2)$$ is even, or $$(n+1)$$ and $$(n+3)$$ is even and an even number multiplied by an odd number is an even number. So $$f(n) neq (n^2 + 3n +1)$$ and therefore $$f(n)$$ cannot be the square of an integer for all $$n geq 1$$.

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