## Mixing colors at the intersection of tiles in Slick2D

I’m just learning Slick2D.
Can you please help and tell me how to properly mix colors on adjoining tiles? I would like to get the same effect as on the right in the picture.

Right now, I’m just using the Image.draw(x, y, Color) method. The color is applied, but it is different between tiles. (the sprite in the example is the same)

## How to perform matrix multiplication in Mixing Columns step of AES?

I am studying AES and trying to implement it. But I am having difficulty understanding the Mixing Column step. In that step we have to perform matrix multiplication between the state matrix and another fixed matrix. Here is the example given in the material I am studying from:

I am not getting the 03*2F part. How did it turn into (02*2F)xor2F? Is the material correct or does it have some mistake?

## WordPress is mixing Text/Images – Fix

Hello, you can go to our site at https://www.corykays.com and see that under alot of the sections on the home page and under the media tab, alot of the images are missing, along with some of the text and press articles we had on the home page.

Now, you can drag the mouse across them and see they are still there, but they are not showing up and we are not sure why.

We had an issue with our server yesterday and I am guessing this is what did it, but i can't figure out why.

I…

WordPress is mixing Text/Images – Fix

## How do I handle a table mixing the DM and the players’ roles too often?

So, after some time I came back to playing live (not online) RPG because a friend of mine, which is used to D&D3.5, asked me to help him with a 5e table where almost all the players are new to D&D (most have played other systems before though).

The thing is: both sides seem to be confusing what is their role in the game. For example, the very first narration from the DM forced movements on a player character (the DM moved the PC himself, instead of the player describing where she wanted the character to go). Another example is that the DM constantly narrated how the characters felt, e.g., “you are scared”, “you trust this person” (when they rolled a bad insight roll against a high deception roll), etc.

Similarly, the players frequently argue whether or not an NPC should behave like it did, which slowed the playing pace considerably. The players also seem to have some problem with suspension of disbelief – or simply trusting the DM – complaining about how something happened. For example, a maid NPC appeared “out of nowhere” and the players wasted minutes complaining on how their characters didn’t perceive the NPC passing through them before (which could easily be a teleportation magic, an illusion, or simply the NPC had a really good stealth roll, or an infinite number of other explanations).

By itself I don’t think it would be a problem if everyone was having fun, but as I mentioned, my main concern is that these arguments are slowing down the play just too much. Additionally, it’s noticeable that everyone is getting a little frustrated over everything.

The DM ended up resorting to (a bad, IMO) in-game solution (which was to knock down the PC of the player slowing down the game for a few in-game hours) so the story could continue, but besides being extremely temporary, it also (rightfully, I guess) frustrated the player more.

For clarification, the DM has no intent of screwing up the players; from what I understand, he does this in order to move the story forward. The players, on the other hand, feel that everything that does not go according to their expectations/plans is the DM trying to screw them, and waste too much time arguing about that.

# The Question

In short, how can I gently recall them that the players’ role is to describe their character actions and feelings (and the DM should avoid interfering in this part), while the DM’s role is to describe the environment, consequences of the actions and the behavior of NPCs (and the players should probably trust that if the DM said something happened, then something happened, and not waste half an hour in an argument about that being impossible, unfair or whatever).

I feel that this might be related to their previous RPG experiences which involved a little of the old “DM x Players” dynamic. That’s why sometimes the DM seems to think he should force the PCs to do something (not trusting that the player has the honesty/ability to play accordingly to what makes more sense)1 and the players feel that everything is an evil plan from the DM to kill their characters.2

If it is relevant, the adventure is a homebrew one (based on a book the DM is working on).

The players are all 20-30 years old, except for one girl who is 19 (and possibly the least problematic one – she’s very calm, with experience in 5e).

1 More details on an example already mentioned: an NPC lied to the PC. That particular PC didn’t have any kind of a priori information to know that it was a lie (although other PCs had, thus the player had) and rolled a really bad Insigh check. The DM immediately narrated it as “You trust him!”. The player agreed – and he was going to role-play as that anyway, i.e., there was no reason for the DM to force that upon the player/character.

2 The NPC that “appeared out of nowhere”, that the players wasted minutes arguing about, was actually an ally. They were worried that the NPC appeared behind them because they thought they were getting assassinated. In the end she was just an ally that was watching them closely to learn if she could trust them and help them if needed, eventually. She decided to leave the shadows when she thought there was enough evidence that the PCs were trustworthy.

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## Must this upper bound on mixing time depend on the minimum stationary probability?

It is known fact that for a finite-state, reversible and ergodic Markov chain with transition matrix $$M$$, the following control on the mixing time holds

$$\left( \frac{1}{\gamma_\star – 1}\right)\ln{2} \leq t_{mix} \leq \frac{\ln(4/\pi_\star)}{\gamma_\star}$$

where $$\pi$$ is the stationary distribution $$\pi M = \pi$$, $$\pi_\star = \min_i \pi(i),$$ the mixing time is defined by $$t_{mix} = \min \{t \geq 1 : \sup_{\mu} \|\mu M^{t-1} – \pi\|_{TV} \leq 1/4\},$$ and the absolute spectral gap $$\gamma_\star = 1 – \max \{\lambda_2(M), |\lambda_d(M)|\}.$$

Is the upper bound tight on all the parameters ? Specifically, is the dependence on $$\pi_\star$$ generally known to be necessary ?

## Regularity for transport equation and mixing

Consider the transport equations $$(1) \qquad \partial_t u + \operatorname{div}(bu) = 0$$ and $$(2) \qquad \partial_t u + b \cdot \nabla u= 0$$

What is the relationship between the problem of regularity of solutions for (1) or (2) and the mixing phenomenon mentioned in this note or this paper?

## two mixing rates of random dynamical system

Given random dynamical system $$(X, \mathcal{B}, (T_{\omega})_{\omega\in \Omega}, \mu)$$ where $$(\Omega, \mathbb{P})$$ is probability space with ergodic transformation $$\sigma: \Omega \to \Omega$$. Define random composition $$T_\omega^n:=T_{\sigma^{n-1}\omega} \circ \dots \circ T_\omega$$. Assume we have quasi-invariant absolutely continuous probability $$\mu_\omega:= h_\omega d\mu$$ such that $$(T_\omega)_{*} \mu_\omega=\mu_{\sigma \omega}$$.

There are two kinds of mixing rate I know so far, but could not tell the reason how it comes from.

the first type of mixing rate is : a.s. $$\omega \in \Omega$$, and any function $$\phi, \psi \in L^{\infty}(\Omega \times X)$$ (denote $$\phi_{\omega}, \psi_{\omega}: X \to \mathbb{R}$$ as function restricted on fiber $$\omega$$),

$$\left|\int \varphi_{\sigma^n \omega} \circ T^n_\omega \cdot \psi_\omega \, d\mu_\omega -\int \varphi_{\sigma^n \omega} \, d\mu_{\sigma^n \omega} \int \psi_\omega \, d\mu_\omega\right| \le C_{\omega} \cdot \|\varphi\|_\infty \cdot C_\psi \cdot e^{-n}$$

the second type of mixing rate is : a.s. $$\omega \in \Omega$$, and any function $$\phi, \psi \in L^{\infty}(\Omega \times X)$$, $$\left|\int \varphi_{\sigma^n \omega} \circ T^n_\omega \cdot \psi_\omega \, d\mu_\omega -\int \varphi_{\sigma^n \omega} \, d\mu_{\sigma^n \omega} \int \psi_\omega \, d\mu_\omega\right| \le \|\varphi\|_\infty \cdot C_\psi \cdot e^{-n}$$

The first mixing rate depends on the random environment $$\omega$$, while the second one doesn’t.

what is the reason making $$C_{\omega}$$ show up in the mixing rate? what kind of system has homogenous mixing rate, independent of $$\omega$$, i.e. does not have $$C_{\omega}$$. can we find examples in the real world to explain such difference? Thanks in advanced!

## A question on verifying the mixing time of finite groups such as the Rubik’s Cube Group

I’m interested in some questions about the computational complexity of bounding the mixing time of random walks on Cayley-graphs of finite groups like that of the Rubik’s Cube Group $$G$$. Determining that $$20$$ is the diameter (God’s number) of the Rubik’s Cube Group under the half-turn metric with Singmaster generating set $$s=\langle U, U’, U^2, D, D’, D^2,\cdots\rangle$$ was a wonderful result. I’m curious about follow-up questions, such as determining how many half-turn twists would it take to get the cube fully “mixed” to $$\epsilon$$-close to the uniform stationary distribution $$\pi$$, say in the variational distance sense.

For example, noting that there are $$18$$ moves in the half-turn metric, and calling $$n$$ the mixing time, can we say something like:

For all but a very small number of elements of $$g\in G$$, are there very close to $$\frac{18^n}{\vert G \vert}$$ ways of writing $$g$$ as words of length $$\le n$$?

My intuition is that, after the cube is fully mixed with $$n$$ moves, there should not be a large special subset $$A\subset G$$ of elements that need a lot more or a lot less than $$18^n$$ ways of writing them, starting from the solved position. On the other hand, if the cube has only been scrambled with $$m\lt n$$ twists, then there should be a large subset $$A$$ that has elements that are in some sense maybe only writable with no more than $$\frac{18^m}{2\vert G \vert}$$ different words of length $$\le m$$.

I think we can combine approximate counting techniques to parlay such gaps into an Arthur-Merlin protocol to verify the mixing time is $$\ge n$$:

• Arthur chooses a random element of $$g$$, a random hash $$h$$ mapping words of $$G$$ onto a set of size $$\frac{18^n}{\vert G \vert}$$, and a random image $$y$$ of $$h$$
• Merlin tells Arthur a word $$W$$ of length up to $$n$$ that, when applied to the starting position of the cube, equals $$g$$
• The word $$W$$ must also satisfy $$h(W)=y$$ – indicating that there are likely a lot of words of length $$\le n$$ that equal $$g$$
• Arthur repeats with Merlin to amplify as needed

Because, for groups I think, the mixing time is at least the diameter, this also may provide an Arthur-Merlin approach to bound the diameter of a large group.