## New DM on House Rules, concerning Nat20 & Rule of Cool

As a DM making a house rule, am I allowed to grant "Rule of Cool" to Natural 20 rolls in exchange of giving "Rule of Uncool" effects for Natural One rolls? (but all within the boundaries of RAW as well).

• In battle, a small PC rolling Nat20 successfully maneuvers himself and climb on the back of the big bad creature (Legolas Style), only to roll Natural 1 in his attack and accidentally hit his allies.

• Failing Deception on guard with Nat1 roll additional 2 arrives and a High Ranking Knight, only to roll Nat20 and convinced the High Ranking Knight instead literally allowing you to pass the area with no consequence at all.

I just want to encourage imagination by introducing some Matrix dodges and Epic fails in the game that is somewhat lighthearted and funny especially to noobs like me.

Am I being a bad example of being a DM? Should I discontinue this approach?

## Approximation concerning Asymmetric TSP, Symmetric TSP, and Metric TSP

I always considered Symmetric TSP to be inapproximable in general, and thus by extension Asymmetric TSP as well. Once you add the condition of the triangle inequality however, you obtain Metric TSP (which can be Symmetric or Asymmetric), which is approximable (e.g. Christofides algorithm).

However, I’m having doubts after finding the following paper :

An improved approximation algorithm for ATSP
Vera Traub, Jens Vygen (https://arxiv.org/pdf/1912.00670.pdf)

In their paper, there is no mention of Metric TSP, or the triangle inequality. Does this mean that I’m misunderstanding, i.e. Asymmetric TSP is in fact approximable, even without the triangle inequality?

## Why doesn’t Mathematica provide an answer while Wolfram|Alpha does, concerning a series convergence?

Among other series I’ve been working on, I was asked to find whether $$\sum_n 1-\cos(\frac{\pi}{n})$$ converged, and Mathematica’s output to SumConvergence[1 - Cos[Pi/n], n] simply was repeating the input, without further information. Wolfram|Alpha, though, at least told me which test were or not conclusive.

I’m new to Mathematica, and even though I’ve looked both on Google and into Wolfram’s documentation, I haven’t found information that could help me figure out how to get, from Mathematica, the conditions for the convergence of a series involving something else than powers of a variable.

I would appreciate if you could give me some clues on the typical procedure to make Mathematica correctly evaluate the convergence of a series, or/and to return the conditions for convergence. Thank you in advance.

## What is the appropriate recommendation concerning making new indexes on our production database?

We are working on ERP application with a SQL server 2008 R2 database in compatibility level 80. I’m working as SQL server DBA I want to make performance tuning against our database but I’m facing many obstacles because our application may not be compatible with higher compatibility level so I cant use DMVs which may help me to find the most expensive queries which is running frequently against our production database.

I tried to run SQL server profiler to extract workload file and run this trc file on database tuning advisor to explore it’s recommendation concerning our database, including index creation and SQL server statistics. I found many opinions said that do not blindly execute DTA recommendation.

I tried to run SQL server activity monitor to discover the most expensive queries and displayed it’s execution plan and I found also recommendations to execute non-clustered indexes.

My questions are:

How can I depend on DTA or execution plan to tune performance?

If I execute these recommendations (indexes) and I face regression on performance, could I drop it easily without any threats and will it be created automatically while Index rebuild operation or rebuild indexes operation drop and create the only existed indexes?

What are the best practices to make new indexes?

## Haskell: difference behavior in ghci concerning “polymorphic recursion”

I stumbled upon some question that puzzled me, maybe it’s just a feature (or simply because I am doing first “Haskell-steps” without studying the manual too deeply, which I guess I should…

Anyway, the observation want to ask about is as follows: I did some inductive data type definition, though it’s “non-regular”, i.e., requiring polymorphic recursion. I am aware that this is problematic, so I wanted to see to what extent Haskell can handle it. The actual code is at the end of the post.

Ultimately, what I don’t understand is the difference in behavior comparing – doing a :load file.hs in the ghci as opposed to – typing in the definitions manually inside the ghci.

Actually, I am not 100% sure if it has to do with the particular kind of recursion (though for simpler things I did, no such discrepancy showed up.

For concreteness sake, the code looks like the one below: The non-regular data structure is SList (strange list). The datatype in isolation works both with loading the file or typing it manually. I am not surprized by that part. More interesting and the cause of the problem is the function slength. It’s kind of an instance of polymorphic recursion and thus potentially problematic. On the other hand, the non-regularity of the type parameter a vs. Node a, vs. Node (Node a)) etc is not really relevant in the inductive definition.

Indeed, the whole code loads fine in GHCi version 8.6.5, though clipping it in fails, reporting as failure that “Occurs check: cannot construct the infinite type”. Also this seems understandable, caused by the polymorphic recursive definition of slength. What I don’t understand is, why both ways of doing things in ghci behave differently (BTW also compiling with ghc works)

data Node a = Pair a a

data SList a = Nil | Cons a (SList (Node a))

slength :: SList a -> Int

slength Nil = 0

slength (Cons n r) = 1 + (slength r)

## Concerning Security Software, which programs do what and how well?

I have been using AVG Ant-virus for years, it has become bloated and intrusive. What do I need for good security? Is windows antivirus effective? Spybot Search and Destroy? Malwarebytes? How many do I use and which ones, or should I go with an all-in-one, as AVG has become, or Norton, etc.

## How a chosen prefix collision is more useful than a standard collision concerning hashing functions?

Recently a paper has been released about SHA-1 chosen prefix collision.

They present what a chosen prefix collision is, but I don’t understand how is it more interesting than a standard collision ? They briefly mention x509 rogue certificate, but I still can’t get the point.

Can anyone explain to me how interesting it is compared to standard collisions ? Hopefully with an example a bit more detailed ?

## How Concerning Is This X-CU-modified: FAKECU Text Attack?

A user clicked on the link of this email and entered their credentials, thinking the message was legitimate. However, the link didn’t redirect to the fake site, and instead their mail client sent them to the link as it was displayed (the real mail server’s web portal).

What is this (from the raw email)? X-CU-modified: FAKECU Text https: //mail.dept.example.com/ to https: //gradingzimbra.000webhostapp.com/

And which type of mail clients would actually go to the fake website?

The message appears to be duplicated in HTML, but didn’t seem to render in the user’s Apple mail, or in my Google Apps mail when the original email was forwarded to me.

I’m not sure it why it didn’t go to spam for the user, and I don’t want to send out an unnecessary warning if this attack is not actually effective. Is it?

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## Concerning certain Keller maps of $k[x,y]$

Let $$k$$ be a field of characteristic zero.

Let $$(x,y) \mapsto (p,q) \in k[x,y]$$ be a Keller map, namely, a $$k$$-algebra endomorphism of $$k[x,y]$$ with $$\operatorname{Jac}(p,q):=p_xq_y-p_yq_x \in k-\{0\}$$.

For $$A \in k[x,y]$$ denote by $$L_{a,b}(A)$$ the $$(a,b)$$-leading term of $$A$$, $$(a,b) \in \mathbb{Z}^2$$ (sometimes it is required that $$\gcd(a,b)=1$$). For example: If $$A=x^2y^2+8x^3y^3-7y^6$$, then $$l_{1,1}(A)=8x^3y^3-7y^6$$, $$l_{1,-1}(A)=x^2y^2+8x^3y^3$$, $$l_{1,0}(A)=8x^3y^3$$ and $$l_{0,1}(A)=-7y^6$$.

Is it possible to find all Keller maps satisfying the following two conditions: (i) Each of $$\{l_{1,-1}(p),l_{1,-1}(q)\}$$ is a monomial. (ii) $$l_{1,-1}(p)+l_{1,-1}(q)=0$$.

Of course, the identity map $$(x,y) \mapsto (x,y)$$ is a Keller map satisfying (i) and (ii).

What about other examples? It seems that there exist no other examples; am I missing something?

Let $$(x,y) \mapsto (p,q) \in k[x,x^{-1},y]$$ be a generalized Keller map, namely, a $$k$$-algebra homomorphism from $$k[x,y]$$ to $$k[x,x^{-1},y]$$ with $$\operatorname{Jac}(p,q):=p_xq_y-p_yq_x \in k-\{0\}$$.

Is it possible to find all generalized Keller maps satisfying the following two conditions: (i) Each of $$\{l_{1,-1}(p),l_{1,-1}(q)\}$$ is a monomial. (ii) $$l_{1,-1}(p)+l_{1,-1}(q)=0$$.

An example: $$p=x^2y+3+x^{-1}+x^{-4}+2x^{-15}$$, $$q=x^{-1}$$.

Are there other types of examples except $$p=\lambda x^{1+m}y+T$$, $$q=x^{-m}$$, where $$T \in k[x,x^{-1}]$$ has an ‘appropriate’ $$(1,-1)$$-degree?

Thank you very much!

## A conjecture concerning symmetric convex sets

Let’s suppose that $$S \subset \mathbb{R}^n$$ is convex and symmetric so:

$$$$x \in S \iff -x \in S \tag{1}$$$$

Now, if we define the radius of $$S$$ as $$R$$ such that:

$$$$R = \sup_{x \in S} \lVert x \rVert \tag{2}$$$$

and use (2) to define:

$$$$V = \{x \in S: \lVert x \rVert = R\} \tag{3}$$$$

then I conjecture that:

$$$$S = \text{conv}(V) \tag{*}$$$$

I have worked out special cases of this problem within the context of high-dimensional probability but I suspect that it’s generally true.

Might there be a theorem which guarantees this result?