## at.algebraic topology – Space with maps detected by homotopy groups in infinitely many degrees

Is there a pointed space $$(X, p)$$ such that for infinitely many integers $$ngeq 1$$ there is a map $$(X, p)to (X,p)$$ non-trivial on $$pi_n(X, p)$$?

In particular $$pi_n(X, p)$$ must be non-trivial for infinitely many $$n$$.

What if require in addition $$X$$ to be a finite-dimensional CW complex?

## combinatorics – How many graphs are there on 4 nodes with degrees 1, 1, 2, 2?

I know just the basic operations on graphs using Mathematica. But I want to know how to write a code that prints all the possible combinations of a graph with a specified number of edges.

Take for example:

How many graphs are there on 4 nodes with degrees 1, 1, 2, 2?

I know the answer mathematically is 12, but I want Mathematica to print these combinations.
Please help me and explain to me how can I write the code.

## algorithms – Random graphs with prescibed degrees and triangles

Consider two sequences of $$n$$ integers $$a_i$$ and $$b_i$$, for $$i$$ from $$1$$ to $$n$$, and build a (multi-)graph with vertices $$1$$ to $$n$$ using the following extension of the configuration model: attach to vertex $$i$$ exactly $$a_i$$ stubs, called edge stubs, and $$2cdot b_i$$ stubs, called triangle stubs; then, take random pairs of edge stubs to form edges and random triplets of pairs of triangle stubs to form triangles.

I have several questions:

• what are the conditions on the two sequences that ensure that a simple graph (no loop, no multi-edge) can be obtained?
• is there a procedure to build such a simple graph, if it exists?
• how to compute the expected number of triangles in the graph? and the number of triangles involving each vertex?
• is there a procedure that builds such a graph with the additional constraint that vertex $$i$$ belongs to exactly $$b_i$$ triangles?

An extension of the Erdös-Gallai criterion would anser the first question. An extention of Havel-Hakimi algorithm would answer the two first questions.

For the two last questions, please note that some triangle may appear due to edge stub pairings, but also that triangle stubs “pairing” may lead to additional triangles, and combinations of edge and triangle stubs may also lead to additional triangles.

Are answers to these questions known in the literature?

This model was proposed in the paper Random graphs with clustering by M. E. J. Newman, but studied only within mean field approximation, if I am correct.

## probability – χ 2 distribution with 50 degrees of freedom. Compute approximate value of P(49

`Let Y bar denote the mean of a random sample of size 100 from a χ 2 distribution with 50 degrees of freedom. Compute an approximate value of P(49 < Y < 51).`

I’ve been away from statistics for two semesters so this stuff is really difficult for me now. I don’t really remember how to approach these problems anymore and I feel like I have a lot of gaps in my knowledge. Ideally I’d like to see the step by step process on this so I can apply this to other problems that are relatively similar.

## statistics – How to handle degrees of success in roll under systems

I’m working on an RPG system that uses 2d6 roll under Skill for resolutions. On paper this system looks really good so
far, but I have one major issue: Degrees of Success, especially when it comes to Contest (Skill vs Skill) resolutions.

### Status Quo

Your character’s Attribute + Skill (e.g. Charisma + Persuasion) form a Target Number that’s between 2 and 12. You roll 2d6,
sum them, and the sum has to be equal to or lower than the Target Number. Rolling a 1 has a special positive meaning,
rolling a 6 has a special negative meaning. Additionally, 2 ones are always a success, 2 sixes are always a failure,
regardless of Skill.

### The problem

Imagine 2 parties contesting each other:

• Character A has a Target Number of 5 (pretty bad), and character B has a Target Number of 10 (pretty good).
• Character A rolls a 5 and succeeds. Character B and rolls a 6 and succeeds.
• Character B has the better Degree of Success, as the margin between the player’s roll and the character’s Skill is
bigger than for Character A.

If you say that lower is better, a character with Target Number 2 (very, very bad), who rolled a 2, will always have
a better Degree of Success over a character with a Target Number 12 (very, very good), who rolled a 3.

### Naive solution

My approach was to subtract the rolled number from the character’s Skill. You have a Target Number of 6 and rolled a 4?
`6-4=2`. You have a Target Number of 11 and rolled a 3? `11-3=8`. It works, but I’m worried that this resolution will be
too slow for actual play – we all know these sessions that last for hours and nobody is able to count straight anymore.

The best solution would allow a player to determine the Degree of Success/Failure in the same step to see if the
character succeeded or not.

### Other systems

Other systems that handle Degrees of Success for rolling under mechanics:

• Call of Cthulhu: You have certain threesholds (half your skill, 1/5 your skill) at which you score an increased Degree of Success. – very coarse when you only have 2d6 instead of a 1d100 (but could work)
• Unknown Armies: Basically like Black Jack–you roll under your Skill threshold, but as high as possible. Doubles (11, 22, 33) are criticals. – sadly doesn’t work, as ones and sixes have a special meaning. Flipping the meaning (6 is good, 1 is bad) also is iffy, as it’s flipping the understanding, that you have to roll under a threshold.
• ???

What other systems or resolution systems are there, that tackle this problem?

## statistics – How to handle degrees of success in roll under dice mechanics

I’m working on an RPG system that uses 2d6 roll under Skill for resolutions. On paper this system looks really good so
far, but I have one major issue: Degrees of Success, especially when it comes to Contest (Skill vs Skill) resolutions.

### Status Quo

Your character’s Attribute + Skill (e.g. Charisma + Persuasion) form a Target Number that’s between 2 and 12. You roll 2d6,
sum them, and the sum has to be equal to or lower than the Target Number. Rolling a 1 has a special positive meaning,
rolling a 6 has a special negative meaning. Additionally, 2 ones are always a success, 2 sixes are always a failure,
regardless of Skill.

### The problem

Imagine 2 parties contesting each other:

• Character A has a Target Number of 5 (pretty bad), and character B has a Target Number of 10 (pretty good).
• Character A rolls a 5 and succeeds. Character B and rolls a 6 and succeeds.
• Character B has the better Degree of Success, as the margin between the player’s roll and the character’s Skill is
bigger than for Character A.

If you say that lower is better, a character with Target Number 2 (very, very bad), who rolled a 2, will always have
a better Degree of Success over a character with a Target Number 12 (very, very good), who rolled a 3.

### Naive solution

My approach was to subtract the rolled number from the character’s Skill. You have a Target Number of 6 and rolled a 4?
`6-4=2`. You have a Target Number of 11 and rolled a 3? `11-3=8`. It works, but I’m worried that this resolution will be
too slow for actual play – we all know these sessions that last for hours and nobody is able to count straight anymore.

The best solution would allow a player to determine the Degree of Success/Failure in the same step to see if the
character succeeded or not.

### Other systems

Other systems that handle Degrees of Success for rolling under mechanics:

• Call of Cthulhu: You have certain threesholds (half your skill, 1/5 your skill) at which you score an increased Degree of Success. – very coarse when you only have 2d6 instead of a 1d100 (but could work)
• Unknown Armies: Basically like Black Jack–you roll under your Skill threshold, but as high as possible. Doubles (11, 22, 33) are criticals. – sadly doesn’t work, as ones and sixes have a special meaning. Flipping the meaning (6 is good, 1 is bad) also is iffy, as it’s flipping the understanding, that under is better.
• ???

What other systems or resolution systems are there, that tackle this problem?

## dnd 5e – Are there guidelines for how to narrate different degrees of success/failure?

The DMG does not offer any direct guidance on how you should narrate outcomes, but it does offer a few bits of information that may be helpful here. All of these are found on page 242 of the DMG.

It specifies that most checks are straightforward, pass/fail checks, but gives a few options for “flourishes and approaches” that you can use. It offers the options of Success at a Cost (used if the roll missed the DC by a narrow margin) and Degrees of Failure (to indicate how badly you failed that attempt).

From this, we can determine a simple fact about any ability check, save, or attack roll: The numbers on the die determine how well you did beyond a simple ‘pass/fail.’

The DC is the low-end threshold to succeed in what you were trying to accomplish. If you consider the rules mentioned above, it specifies that nearly hitting the DC is cause to still allow a borderline success, that has negative consequences…and missing the DC by a large margin means a more spectacular failure.

Based on that, if you wish to fluff up your roll resolution description, you can consider the roll result’s proximity to the DC to be a measure of how well the attempt went.

To take your example of the flip…

• Barely missing the DC could be exactly as you described…you almost pull off the flip, but fail to stick the landing.

• Missing the DC by a lot could mean you slipped while trying to start the flip, and simply landed flat on your back

• Hitting the DC dead on or slightly higher means you successfully turned your flip and stuck the landing

• Surpassing the DC by a broad margin means you made that look effortless.

The same sort of thing could be applied to attack rolls, saving throws, and so on. In games I run, I do exactly this, narrating outcomes based on how close they were to the DC.

However, I will add this: once your players realize what you are doing, it’s going to start giving them hints as to what the DC is. I, personally, think this is a good thing. If a PC makes an attack roll, gets a 14 vs an AC of 20…I may describe the attack as having been ‘contemptuously swatted away’ or ‘harmlessly glancing off their armor.’ If it’s a 19 vs an AC of 20, I may describe the attack as having been ‘diverted at the last moment by a quick twist of their shield’ or ‘the creature hunkers down at the last moment and your blade scrapes across its armored shoulder, narrowly missing its mark.’

And, conversely, I may describe a roll of 22 vs an AC of 12 as the player ‘deftly slipping past their defenses’ or ‘smashing their weapons aside to land a blow.’

This gives the players an idea of what they are up against, and I think it greatly adds to the gameplay experience. After all, it’s much more flavorful to know how close you came to accomplishing something, or how far you surpassed the difficulty…than to just get a simple “You fail.” or “You succeed.”

## lo.logic – Generality of construction for \$omega\$-REA arithmetic degrees

So a common method used to construct non-zero $$omega$$-REA arithmetic degrees with various properties is to build an $$omega$$-REA operator $$J$$ satisfying the constraints that (for all $$X$$)

$$tag{1} J(X’) equiv_T J(X) oplus X’$$
$$tag{2} J(X) >_T X$$

Inductively, 1 implies that $$J(X^n) equiv_T J(emptyset) oplus X^n$$. Thus, together, these constraints ensure that $$J(emptyset)$$ isn’t arithmetic (if $$J(emptyset) leq_T 0^n$$ then $$J(0^n) equiv_T 0^n$$).

My question is whether this is fully general, i.e., if $$A$$ is $$omega$$-REA is there some $$omega$$-REA operator $$J$$ satisfying 1 such that $$A$$ and $$J(emptyset)$$ have the same arithmetic degree? And if we can also assume 2 holds if $$A$$ isn’t arithmetic?

Basically, I’m hoping someone will let me know if I’m missing some obvious elementary result or known result before I spend any time trying a hard construction.

## Textures rotated 90 degrees when importing a .glb file

I have a simple box, textured with bricks, in Blender:

I export this scene as glTF 2.0 and import into Godot. Here’s the result:

As you can see, the bricks are rotated 90 degrees for some reason. Why, and how can I make it right?

## How do I calculate a line segment that is a certain number of degrees from another line segment?

I have a line segment that is at an arbitrary angle in 3D space. I want to (in code) draw another line that shares an end point with the first line and has an angle of X degrees between them. The second line will partially overlap the first in the XZ plane.