## Difficulty to run a Wolfram Script (with FacialFeature function) on Windows Command Line

I am on Windows.

I am working on a project which consists in analysing a very high number of images. More precisely, I am using the FacialFeatures function of Mathematica to quantify the visibility of women in Hindi cinema in the last fifty years.

Here are the steps I have followed:

1. I prepare a Wolfram Script. I have borrowed the coding from this source: https://community.wolfram.com/groups/-/m/t/2288529

2. I save the Wolfram Script under the name script_test.wls in a folder which I call workplace_folder. This folder also includes another folder – film_folder – which contains the images I want to analyse.

3. I open my Windows Command Line, and I change the directory to workplace_folder

4. I run the following command: >script_test.wls film_folder output_file.json

5. Windows asks me the program I want to use to open the file; I choose wolframscript.exe

6. A new Windows Command Line window opens, some text appears (too quick for me to be able to read), and then the window closes. As if the work was done.

7. However, nothing seems to be happening. No output_file.json has been created, and no analysis seemed to have taken place.

Is there any step I got wrong?

## Guidelines for setting haggle test difficulty when using the tracking money option?

I have been running a game of WFRP 4e and my players haven’t really been enjoying tracking money much so I was thinking of switching to the optional tracking money rule on page 290 of the rule book.

Where some groups like to track every penny closely, perhaps even using chits or fantasy coins to represent in-game coins, others prefer to ignore all fiscal book-keeping. The game rules assume you are counting every coin, but if you wish to simplify money, you can do so using your Status. If an item costs less than your Status level — so if you have a Status of Silver 2, any item costing 2 silver shillings or less — you are assumed to be able to buy as much as needed of that item. Beyond that, you can buy a maximum of one item a day that costs more with a Haggle Test, with the difficulty set by the GM according to the cost of the item and the local markets.

My problem is that according to the rules only a gold tier character can reasonably afford a main gauche let alone other weapons so my players will need to make haggling tests however I couldn’t find a guideline for haggle tests when using this system.(And the regular haggle rules woul not be appropriate on the grounds that the most you can negotiate is a %50 off) So what I wish to ask is Are there any guidelines somewhere for setting haggle difficulty and if not what are the rules you use on your table?

## Is Armor Class a difficulty class (DC)?

[…] You take a status penalty equal to this value to all your checks and DCs […]

Compare it to Clumsy:

[…] You take a status penalty equal to the condition value to Dexterity-based checks and DCs, including AC, Reflex saves, ranged attack rolls, and skill checks […]

The wording of Clumsy seems to imply that AC is just one of the DCs, but is not clear. I really hope it is, otherwise Frightened is not nearly as good as I thought.

## Dice rolling mechanic where modifiers have a predictable and consistent effect on difficulty

I am looking for a dice rolling mechanic that makes it such that increasing or decreasing a modifier on the roll has a constant multiplier effect on the probability of the outcome.

Say you have to make a roll for STAT. Such a roll has a probability of success of 50%. Now say you roll with a mod of -1, this roll has a probability of success of 25%. -2 has a probability of 12.5%. -3 is 6.25% and so on, always halving. The other way around it should be the same but for the probability of failure, always being divided by the same factor.

It doesn’t have to be a multiplier of 0.5, in fact I’d much rather it was a multiplier of 0.66-0.75, not such an extreme change. The default unmodded value doesn’t need to be 50% chance of success either, it can be something else.

Is there any kind of dice rolling mechanic I can use to simulate something like this?

## Is concealed difficulty an integral part of the cliffhanger scene form?

The “Cliffhangers” section in Masters of Umdaar says that the difficulties for rolls should be concealed until rolled against:

Of course, GMs, don’t reveal a difficulty for a specific approach until a player attempts it—let them stumble around to see which methods are more effective. (MoU 28, Cliffhangers: Running the Cliffhanger)

That stands out because it’s contrary to standard practices in Fate. On the one hand, that makes it feel like an optional playstyle preference note; on the other hand, it can be read as a deliberate and noteworthy departure from Fate norms to introduce a different sort of experience to the game.

Are concealed difficulties a crucial part of the cliffhanger concept or is this just a playstyle preference of the author? What difference does concealing difficulties make to the table experience when using cliffhangers?

## Does attacking difficulty of a block cipher depend on it’s block length?

Does attacking difficulty of a block cipher depend on it’s block length compared to a substitution cipher?

## Difficulty understanding the use of arbitrary function for the worst case running time of an algorithm

In CLRS the author said

"Technically, it is an abuse to say that the running time of insertion sort is $$O(n^2)$$, since for a given $$n$$, the actual running time varies, depending on the particular input of size $$n$$. When we say “the running time is $$O(n^2)$$,” we mean that there is a function $$f(n)$$ that is $$O(n^2)$$ such that for any value of $$n$$, no matter what particular input of size $$n$$ is chosen, the running time on that input is bounded from above by the value $$f(n)$$. Equivalently, we mean that the worst-case running time is $$O(n^2)$$. "

What I have difficulties understanding is why did the author talked about an arbitrary function $$f(n)$$ instead of directly $$n^2$$.

I mean why didn’t the author wrote

"When we say “the running time is $$O(n^2)$$,” we mean that for any value of $$n$$, no matter what particular input of size $$n$$ is chosen, the running time on that input is bounded from above by the value $$cn^2$$ for some +ve $$c$$ and sufficiently large n. Equivalently, we mean that the worst-case running time is $$O(n^2)$$".

I have very limited understanding of this subject so please forgive me if my question is too basic.

## Difficulty in understanding a portion in the proof of the $\text{“white path”}$ theorem as with in CLRS text

I was going through the $$\text{DFS}$$ section of the Introduction to Algorithms by Cormen et. al. and I faced difficulty in understanding the $$\Leftarrow$$ direction of the proof of the white path theorem. Now the theorem which is the subject of this question depends on two other theorems so I present the dependence before presenting the actual theorem and the difficulty which I face in the said.

Dependencies:

Theorem 22.7 (Parenthesis theorem) In any depth-first search of a (directed or undirected) graph $$G = (V, E)$$, for any two vertices $$u$$ and $$v$$;, exactly one of the following three conditions holds:

• the intervals $$[d[u], f[u]]$$ and $$[d[v], f[v]]$$ are entirely disjoint, and neither $$u$$ nor $$v$$ is a descendant of the other in the depth-first forest,

• the interval $$[d[u], f[u]]$$ is contained entirely within the interval $$[d[v], f[v]]$$, and $$u$$ is a descendant of $$v$$; in a depth-first tree,

• the interval $$[d[v], f[v]]$$ is contained entirely within the interval $$[d[u], f[u]]$$, and $$v$$ is a descendant of $$u$$ in a depth-first tree.

Corollary 22.8 (Nesting of descendants’ intervals) Vertex $$v$$ is a proper descendant of vertex $$u$$ in the depth-first forest for a (directed or undirected) graph $$G$$ if and only if $$d[u] < d[v] < f[v] < f[u]$$.

Theorem 22.9 (White-path theorem)

In a depth-first forest of a (directed or undirected) graph $$G = (V, E)$$, vertex $$v$$ is a descendant of vertex $$u$$ if and only if at the time $$d[u]$$ that the search discovers $$u$$, vertex $$v$$ can be reached from $$u$$ along a path consisting entirely of white vertices.

Proof

$$\Rightarrow$$ : Assume that $$v$$ is a descendant of $$u$$. Let $$w$$ be any vertex on the path between $$u$$ and $$v$$ in the depth-first tree, so that $$w$$ is a descendant of $$u$$. By Corollary 22.8, $$d[u] < d[w]$$, and so $$w$$ is white at time d[u].

$$\Leftarrow$$:

1. Suppose that vertex $$v$$ is reachable from $$u$$ along a path of white vertices at time $$d[u]$$, but $$v$$ does not become a descendant of $$u$$ in the depth-first tree.
2. Without loss of generality, assume that every other vertex along the path becomes a descendant of $$u$$. (Otherwise, let $$v$$ be the closest vertex to $$u$$ along the path that doesn’t become a descendant of $$u$$.)
3. Let $$w$$ be the predecessor of $$v$$ in the path, so that $$w$$ is a descendant of $$u$$ ($$w$$ and $$u$$ may in fact be the same vertex) and, by Corollary 22.8, $$f[w] \leq f[u]$$.
4. Note that $$v$$ must be discovered after $$u$$ is discovered, but before $$w$$ is finished.$$^\dagger$$ Therefore, $$d[u] < d[v] < f[w] \leq f[u]$$.
5. Theorem 22.7 then implies that the interval $$[d[v], f[v]]$$ is contained entirely within the interval $$[d[u], f[u]]$$.$$^{\dagger\dagger}$$
6. By Corollary 22.8, $$v$$ must after all be a descendant of $$u$$. $$^\ddagger$$

$$\dagger$$ : Now it is clear that since $$u$$ is the first vertex to be discovered so any other vertex (including $$v$$) is discovered after it. In point $$1$$ we assume that $$v$$ does not become the decendent of $$u$$, but by the statement that but before $$w$$ is finished I feel that this is as a result of exploring the edge $$(w,v)$$ (this exploration makes $$v$$ ultimately the descendant of $$u$$, so the proof should have ended here $$^\star$$)

$$\dagger\dagger$$ : Considering the exact statement of theorem 22.7 , I do not get which fact leads to the implication in $$5$$.

$$\ddagger$$ : The proof should have ended in the $$\star$$, but why the stretch to this line $$6$$.

Definitely I am unable to get the meaning the proof of the $$\Leftarrow$$. I hope the authors are using proof by contradiction.

(I thought of an alternate inductive prove. Let vertex $$v$$ is reachable from $$u$$ along a path of white vertices at time $$d[u]$$. We apply induction on the vertices in the white path. As a base case $$u$$ is an improper descendant of itself. Inductive hypothesis, let all vertices from $$u$$ to $$w$$ be descendants of $$u$$ , where $$w$$ is the predecessor of $$v$$ in the white path. We prove the inductive hypothesis by the exploration of the edge $$(w,v)$$. But I want to understand the proof the text.)

## What target numbers would be a certain level of difficulty under this system?

I’m writing a homebrew game system, and I found that I have an action resolution mechanic but not a good system for target numbers (I call them Success Thresholds, or STs, in this game, and from now on I’ll use that term to refer to the minimum number a player gets that can succeed).

To resolve an action, most of the time players roll 2d6 and add a modifier ranging from +0 to +3, depending on the stat. With Advantage, it is (3d6 drop lowest)+mod, and Disadvantage is (3d6 drop highest)+mod.

There are also 4 (well, 5, but one auto succeeds) levels of difficulty. The Trivial tasks are automatically successful. Easy tasks should succeed about 75-80 percent of the time, Moderate tasks should be successful 50-60% of the time, Hard tasks should be successful between 25 and 40 percent of the time, and impossible tasks shouldn’t succeed more than 25% and often more like succeeding below 10-15% or the time even with Advantage and a +3 mod.

This is an anydice program with the base probabilities for a +0 mod. I want to know what number should the ST be for each level of difficulty? I had initially considered 7 as a base difficulty for Moderate tasks, before I added modifiers to rolls.

## Difficulty in few steps in proof of “Amortized cost of $\text{Find-Set}$ operation is $\Theta(\alpha(n))$”assuming union by rank, path compression

I was reading the section of data structures for disjoint sets from the text CLRS I faced difficulty in understanding few steps in the proof of the lemma as given in the question title. Here we assume we follow union by rank and path compression heuristics. Before we move into our target lemma a few definitions and lemma is required as a prerequisites for the target lemma.

The prerequisites:

$$level(x)=\max\{k:rank[p[x]]\geq A_k(rank[x])\}$$ $$iter(x)=\max\{i:rank[p[x]]\geq A_{level(x)}^{(i)}(rank[x])\}$$ $$\phi_q(x) = \begin{cases} \alpha(n).rank[x] &\quad\text{if x is a root or rank[x]=0 }\ (\alpha(n)-level(x)).rank[x]-iter(x) &\quad\text{if x is not a root and rank[x]\geq1 }\ \end{cases}$$

Lemma 21.9: Let $$x$$ be a node that is not a root, and suppose that the $$q$$ th operation is either a $$\text{Link}$$ or $$\text{Find-Set}$$. Then after the $$q$$th operation, $$\phi_q(х) \leq \phi_{q-1}(х)$$. Moreover, if $$rank[x] \geq 1$$ and either $$level(x)$$ or $$iter(x)$$ changes due to the $$q$$ th operation, then $$\phi_q(х) < \phi_{q-1}(х) – 1$$. That is, $$x$$‘s potential cannot increase, and if it has positive rank and either $$level(x)$$ or $$iter(x)$$ changes, then $$x$$‘s potential drops by at least $$1$$.

Now in the proof below I marks the steps where I face problem

Lemma 21.12: The amortized cost of each $$\text{Find-Set}$$ operation is $$\Theta(\alpha(n))$$.

Proof: Suppose that the $$q$$ th operation is a $$\text{Find-Set}$$ and that the find path contains $$s$$ nodes. The actual cost of the $$\text{Find-Set}$$ operation is $$O(s)$$. We shall show that no node’s potential increases due to the $$\text{Find-Set}$$ and that at least $$\max\{0,s – (\alpha(n) + 2)\}$$ nodes on the find path have their potential decrease by at least $$1$$.

To see that no node’s potential increases, we first appeal to Lemma 21.9 for all nodes other than the root. If $$x$$ is the root, then its potential is $$\alpha(n) . rank[x]$$, which does not change.

Now we show that at least $$\max\{0,s – (\alpha(n) + 2)\}$$ nodes have their potential decrease by at least $$1$$. Let $$x$$ be a node on the find path such that $$rank[x] > 0$$ and $$x$$ is followed somewhere on the find path by another node $$у$$ that is not a root, where $$level(y) = level(x)$$ just before the $$\text{Find-Set}$$ operation. (Node $$у$$ need not immediately follow $$x$$ on the find path.) $$\require{color}\colorbox{yellow}{All but at most \alpha(n) + 2 nodes on the find path satisfy these constraints on x .}$$ $$\require{color}\colorbox{yellow}{Those that do not satisfy them are the firstnode on the find path (if it has rank 0 ),}$$ $$\require{color}\colorbox{yellow}{ the last node on the path (i.e., the root), and the last node w on the path for which}$$ $$\require{color}\colorbox{yellow}{ level(w) = k , for each k = 0,1,2,…, \alpha(n) – 1 .}$$

Let us fix such a node $$x$$, and we shall show that $$x$$‘s potential decreases by at least $$1$$. Let $$k = level(x) = level(y)$$. Just prior to the path compression caused by the $$\text{Find-Set}$$, we have

$$rank[p[x]] \geq A_k^{(iter(x)}(rank[x])$$ (by definition of $$iter(x)$$) ,

$$rank[p[y]] \geq A_k(rank[y])$$ (by definition of $$level(y)$$ ,

$$rank[y] > rank[p[x]]$$ (by Corollary 21.5 and because $$у$$ follows $$x$$ on the find path)

Putting these inequalities together and letting $$i$$ be the value of $$iter(x)$$ before path compression, we have

$$rank[p[y]] \geq A_k(rank[y]) \geq A_k(rank[p[x]])$$ (because $$A_k(j)$$ is strictly increasing) $$> A_k(A_k^{(iter(x)}(rank[x])) = A_k^{(i+1)}(rank[x])$$ .

Because path compression will make $$x$$ and $$у$$ have the same parent, we know that after path compression, $$rank[p[x]] = rank[p[y]]$$ and that the path compression does not decrease $$rank[p[y]]$$. Since $$rank[x]$$ does not change, after path compression we have that $$\require{color}\colorbox{pink}{ rank[p[x]]\geq A_k^{(i+1)}(rank[x]) . Thus, path compression will cause either iter(x) to }$$ $$\require{color}\colorbox{pink}{increase (to atleast i + 1 ) or level(x) to increase (which occurs if iter(x) increases}$$ $$\require{color}\colorbox{pink}{to at least rank[x] + 1 ). In either case,by Lemma 21.9, we have \phi_q(х) \leq \phi_{q-1}(х) – 1 .}$$ $$\require{color}\colorbox{pink}{Hence, x ‘s potential decreases by at least 1 .}$$

The amortized cost of the $$\text{Find-Set}$$ operation is the actual cost plus the change in potential. The actual cost is $$O(s)$$, and we have shown that the total potential decreases by at least $$\max\{0,s – (\alpha(n) + 2)\}$$. The amortized cost, therefore, is at most $$O(s) — (s — (\alpha(n) + 2)) = O(s) — s + 0(\alpha(n)) = O(\alpha(n))$$, since we can scale up the units of potential to dominate the constant hidden in $$О (s)$$. ■

In the proof above I could not get the mathematics behind the statements highlighted in yellow and pink. Can anyone help me out?