ag.algebraic geometry – Question about a proof in Berthelot’s crystalline book

Below is an excerpt from Berthelot’s book on crystalline cohomology. I don’t understand the last sentence, namely why it follows that $sigmacirc varepsilon$ is an isomorphism. For what it’s worth, $P^1$ is the sheaf of principal parts and $E$ is an $mathcal O _X$-module. I can elaborate on what $sigma,varepsilon,tau$ are, but perhaps I’m just missing some basic algebra…

We have an endomorphism of a module which becomes the identity modulo a square zero ideal. Why, in this case, is it an isomorphism?

enter image description here

probability painting rocks question – Mathematics Stack Exchange

Six rocks are sitting in a straight line. We paint them, using up to three colors (say,
R’s, W’s, and B’s). Suppose all of the 36 = 729 outcomes are equally likely.

a Find the probability that exactly 1 color of paint is used.

b Find the probability that exactly 2 colors of paint are used.

c Find the probability that all 3 colors of paint are used.

Newbie question about the security of mixing

If I am running my own node, and using a mixer to anonymize my BTC, will it not be connected anyway because the transactions are to/from my own node?

I mean I am sending the BTC to the mixer from my node, and then getting it back to another address, sure, but I am accessing that address from my own node…

Even if I am using TOR, is there not something that identifes the nodes and tells observers that these two addresses have something in common, the node on my machine?

canon – Question About Charging and Recording

canon – Question About Charging and Recording – Photography Stack Exchange

geometry – Question about a series of distance preserving transformations on points

I have a problem that asks me to

Find all length preserving transformations of the plane that send
point A to point A’ and point B to point B’ where: $A=(0,1), B=(1,1), A’=(3,2), B’=(3- frac{sqrt3}{2}, frac{3}{2})$;

and to write the transformations as a parallel transport followed by a rotation about the origin, and possibly a reflection.

I did some preliminary work and found that for the rotation, $theta = frac{pi}{6}$. I’m now left with systems of equations that involve the variables of transport. Would it just remain to solve the system for those variables? And how can I determine if a reflection is needed? Do I need to take into account the possible reflection when I write the formula for the points after translation and rotation?

algebra precalculus – Algebraic Manipulation Question When Manipulating Minimum in Epsilon Delta Proof

I am currently working my way through Spivak’s Calculus, and I cannot figure out the last basic algebra step to reduce the problem so that it reads as presented in the book’s solutions.

Specifically, I am asked to find a $delta$ such that $lvert{x^4 – a^4}rvert < epsilon ; forall x$ satisfying $lvert{x-a}rvert < delta$.

The way to tackle the problem is very straightforward. Notice that $lvert{x^2 – a^2}rvert < min{big(frac{epsilon}{2(lvert{a^2}rvert + 1)}, 1big)} implies lvert{x^4 – a^4}rvert < epsilon$. Furthermore $lvert{x – a}rvert < min{big(frac{epsilon’}{2(lvert{a}rvert + 1)}, 1big)} implies lvert{x^2 – a^2}rvert < epsilon’$. Substituting $epsilon = epsilon’$, we have:

$$
min{bigg(frac{min{big(frac{epsilon}{2(lvert{a^2}rvert + 1)}, 1big)}}{2(lvert{a}rvert + 1)}, 1bigg)}
$$
.

After I simplify, I choose:

$$
delta = min{bigg(min{big(frac{epsilon}{4(lvert{a^2}rvert + 1)(lvert{a}rvert + 1)}, frac{1}{2(lvert{a}rvert + 1)}big)}, 1bigg)}
$$

My Question:

Spivak’s solutions show that this may be reduced to

$$
delta = min{big(frac{epsilon}{4(lvert{a^2}rvert + 1)(lvert{a}rvert + 1)}, 1big)}
$$

and I cannot figure out why this is true. Why can we guarantee that

$$

begin{align}
frac{epsilon}{4(lvert{a^2}rvert + 1)(lvert{a}rvert + 1)} &< frac{1}{2(lvert{a}rvert + 1)} \
frac{epsilon}{2(lvert{a^2}rvert + 1)} &< 1
end{align}

$$

for any such $epsilon$?

Beginner question: Why are the columns on this dataframe not aligned? Date and High are pushed to the left

Why is the column of Date and High pushed to one side? How do I even out the dataframe so High, Low, etc… Adj Close gets pushed by 1 column to the right?

plugin – Question on support/deprecation status of browser plug-ins (not extensions)

plugin – Question on support/deprecation status of browser plug-ins (not extensions) – Webmasters Stack Exchange

soft question – Shapes for category theory

Most texts on category theory define a (small) diagram in a category $mathcal{A}$ as a functor $D : mathcal{I} to mathcal{A}$ on a (small) category, and $mathcal{I}$ is called the shape of the diagram. A cone from $A in mathcal{A}$ to $D$ is a morphism of functors $Delta(A) to D$, a limit is a universal cone, etc. Observe that, however, that composition in $mathcal{I}$ is never used to define the limit. One can therefore argue, and this is what I would like to discuss here, that directed multigraphs (“categories without composition”) are better suited as the shapes of diagrams:

If $Gamma$ is a directed multigraph, then a diagram of shape $Gamma$ in $mathcal{A}$ is a morphism of graphs $D : Gamma to U(mathcal{A})$, where $U$ forgets composition. A cone from $A in mathcal{A}$ to $D$ is a morphism of diagrams $Delta(A) to D$, a limit is a universal cone, etc. In my category theory textbook (published 2015) I chose this definition, which leads to an equivalent theory, but offering several advantages over the more common definition:

  1. As alreay indicated, the limit of a functor $mathcal{I} to mathcal{A}$ in $mathcal{A}$ is just the limit of the graph morphism $U(mathcal{I}) to U(mathcal{A})$ in $mathcal{A}$, so it seems awkward to have a category structure around when we do not use it all. Conversely, the limit of a graph morphism $Gamma to U(mathcal{A})$ is just the limit of the corresponding functor $mathrm{Path}(Gamma) to mathcal{A}$, so in end we end up with the same limits. (In particular, the definition cannot be wrong, and much of the discussion will be more of philosophical or pedagogical nature.)
  1. When we talk about specific types of diagrams and limits, we never really care about composition, and also never write down identities, since they are not relevant at all. For example, binary products are limits of shape$$bullet ~~ bullet$$which is just a graph with two vertices and no edges. We don’t need to write down identity morphisms in this approach. Arbitrary products are similar. An equalizer is a limit of the shape
    $$bullet rightrightarrows bullet$$
    which is just a graph with two vertices and two parallel edges between them. A fiber product is a limit of the shape
    $$bullet rightarrow bullet leftarrow bullet.$$
    Limits of shape
    $$cdots to bullet to bullet to bullet$$
    also appear very naturally. Put differently, the typical indexing categories you will find in most texts on category theory are actually already the path categories on directed multigraphs. For me this is the most convincing argument. Barr and Wells argue in their book Toposes, Triples and Theories in a similar way:

Limits were originally taken over directed index sets—partially ordered
sets in which every pair of elements has a lower bound. They were quickly generalized to arbitrary index categories. We have changed this to graphs to reflect actual mathematical practice: index categories are usually defined ad hoc and the composition of arrows is rarely made explicit. It is in fact totally irrelevant and our replacement of index categories by index graphs reflects this fact. There is no gain—or loss—in generality thereby, only an alignment of theory with practice.

  1. Let’s talk about interchanging limits. The usual formulation starts with a functor $D : mathcal{I} times mathcal{J} to mathcal{A}$. This includes, in particular all “diagonal” morphisms $D(f,g)$ for morphisms $f$ in $mathcal{I}$ and $g$ in $mathcal{J}$. However, in practice, I only want to define $D(f,j)$ and $D(i,g)$, and I don’t want to show that $D$ is a functor. For example, interchanging fiber products should be about commuting diagrams of shape
    $$begin{array}{ccccc}
    bullet & rightarrow & bullet & leftarrow & bullet \ downarrow && downarrow && downarrow \ bullet & rightarrow & bullet & leftarrow & bullet \ uparrow && uparrow && uparrow \ bullet & rightarrow & bullet & leftarrow & bulletend{array}$$

    which actually appear in practice (see also here). I don’t want to bother about all the diagonal morphisms (and the identities) in that diagram, and actually nobody does when applying “interchanging limits” in concrete examples. The theorem for directed multigraphs is as follows: Let $Gamma,Lambda$ be directed multigraphs. Consider the tensor product $Gamma otimes Lambda$ (pair the vertices, pair edges in $Gamma$ with vertices of $Lambda$, and pair edges in $Lambda$ with vertices in $Gamma$) and a diagram $D$ of shape $Gamma otimes Lambda$ in $mathcal{A}$ such that for all edges $i to j$ in $Gamma$ and edges $i’ to j’$ in $Lambda$ the diagram
    $$begin{array}{ccc} D(i,j) & rightarrow & D(i,j’) \ downarrow && downarrow \ D(i’,j) & rightarrow & D(i’,j’) end{array}$$
    commutes. Then, we have $lim_{i in Gamma} lim_{j in Lambda} D(i,j) cong lim_{(i,j) in Gamma otimes Lambda} D(i,j)$; when the left side exists, then also the right side, and they are isomorphic.
  • This is a bit vague, but for me it seems awkward and random, almost like a “type error”, that categories have two purposes in the usual theory: One purpose it to collect structured objects and their morphisms. The other purpose is to axiomatize diagram shapes. Similarly, functors have two purposes in the usual theory. I find it quite pleasant when the second purpose is fulfilled by a different thing. Also connected to that is the observation that shapes are usually small, but categories tend to be large.

Although the theory works out very well, meanwhile, I am not so confident anymore about my decision, and I am thinking about changing it in the next edition of the book. So here are some disadvantages:

  • 99% of the category theory literature (textbooks and research papers) define diagrams as functors, resp. their shapes are just small categories. It is awkward to do something which nobody else does, and this can also be irritating for the readers as well. I didn’t bother about this too much when writing the book, but I am increasingly worried about this issue.
  • Directed diagrams/colimits are indexed by directed partial orders, and here we really want a functor to ensure compatibility between the various morphisms.
  • The theory of Kan extensions has to be done with functors.

This explains hopefully enough background for the following

Questions.

  1. Can you list further mathematical advantages and disadvantages when taking directed multigraphs as the shapes of diagrams and limits / colimits?
  2. Can you name pedagogical advantages and disadvantages of this definition?
  3. Can you list other textbooks on category theory which use this definition? The book Toposes, Triples and Theories by Barr and Wells is an example, see Chapter 1, Section 7. They also define sketches in a “composition-free” way in Chapter 4. Not a book, but Grothendieck also defines diagrams this way in his famous Tohoku paper Sur quelques points d’algèbre homologique, Section 1.6.
  4. (More general side question) For those of you who already wrote a book or monograph, what other criteria did you choose to decide if a common definition should be changed? And how did you decide in the end?

spells – Question about Guards and Wards confusion effect

The 6th level Sorcerer/Wizard spell Guards and Wards features a confusion effect, quoted from the text it states:

Confusion
Where there are choices in direction—such as a corridor intersection or side passage—a minor confusion-type effect functions so as to make it 50% probable that intruders believe they are going in the opposite direction from the one they actually chose. This is an enchantment, mind-affecting effect. Saving Throw: None. Spell Resistance: Yes.
https://www.d20srd.org/srd/spells/guardsAndWards.htm

How is this implemented in the actual sense of players at the game table? Does the GM roll whenever the PCs make a choice at a corridor intersection and if the result is that they are confused, he or she states ‘you think you are going in the wrong direction.’ This seems odd as all this particular aspect of the spell seems to do is prompt a line of GM dialog whenever he is required to test and rolls 50 or under.

I would have thought it would have at least prompted an effect similar to the actual confusion spell where the test is made and if failed the player characters actually physically go in the wrong direction while still believing they are travelling in the direction they chose. But reading the spell description that isn’t what’s happening at all.

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