Regular Expression Kleene Star application – using 0s and 1s

Have a question from an online course I am trying to take for a certificate. Came across a few questions like this and need help working through the problem and the rationale to get the solution. Any help appreciated.

Consider the regular expressions R1 = (01)*(0*1*)(10) and R2 = (101)(01). Which of following is true?

10 in R1 and 101 R2

10 in R1 and 1000 R2

010 in R1 and 100 R2

010 not in R1 and 100 not in R2

Compute Intersection Of Line With Star Shaped Set

Let there be $$k$$ planes $$X_i\subseteq\mathbf{R}^3$$, all tangent to the unit sphere, in general position, represented by normal vectors $$v_i$$. Then $$\mathbf{R}^3\setminus\cup_i X_i$$ consists of bounded and unbounded segments, let $$S$$ denote the closure of the union of all bounded segments and assume that $$\mathbf{0}\in S$$. This should give a star shape w.r.t. $$\mathbf{0}$$, i.e. for all $$s\in S$$, the ‘interval’ $$[\mathbf{0},s]$$ lies within $$S$$. Given a unit vector $$w\in R^n$$, in general position to the others, I can compute all scalars $$\lambda_i$$ such that $$\lambda_iw$$ lies on $$X_i$$. What is the most efficient method to verify which (necessarily unique) positive $$\lambda_i$$ lies on the boundary of $$S$$, if I want to apply it to a lot of $$w$$s but the planes stay fixed?

The easiest way I currently see is to compute all intersection points of three planes each, then for every plane $$X_i$$, arrange a list of all triangles on that plane, that means, the three intersection points of $$X_i$$ with two out of three other planes. Now, given $$w$$, I first get the largest $$\lambda_i$$, check whether it lies in some of the triangles on $$X_i$$, if not, continue with the second largest $$\lambda_i$$, etc.

[ Drama ] Open Question : Did you like Star Trek Voyager?

It was one of my least favourite Star Trek series, only above Star Trek Discovery. I found the writing unimaginative, with few new ideas, irrational storylines (too high an opinion of the holo doctor for example) and little that made me want to watch it again.

Star Wars Beginner Box Set [closed]

I’m 42 years with Emotional and Behavioral issues(Disabled). I have all three Beginner Box Set game. How do I get my niece and her boyfriend to play one of the Star Wars Beginner Box Set games?. Backstory: But, she just read the Star Wars Core Rule book and then said: I’m not playing it. If I brought it up again. She’ll say: “Ah” to me. I have a hunch that she wants to play. But the core rule book is preventing her. She thinks that the Core Rule book for AOR, EOTH and F&D are it.

Kleene star operations

Let $$𝚺$$ be any alphabet and let $$𝑳_𝟏 \subseteq 𝚺^{∗}$$ and $$𝑳_2 \subseteq 𝚺^{∗}$$ be two non-empty languages.

a. If $$𝑳_𝟏 𝚺^{∗} \neq 𝚺^{∗}$$ than what can we say about $$L_1$$.

b.Let $$\Lambda \in L_1$$ and $$\Lambda \in L_2$$. Show using axioms and theorems of languages that $$𝑳_𝟏 𝚺^{∗}𝑳_2 = 𝚺^{∗}$$

For (a), $$\Lambda$$ should not belong to $$L_1$$ but I do not know how to prove that.

For (b),we have to prove that $$𝑳_𝟏 𝚺^{∗}𝑳_2 \subseteq 𝚺^{∗}$$ and $$𝚺^{∗} \subseteq𝑳_𝟏 𝚺^{∗}𝑳_2$$ for equality to exist. We can also distinguish two cases, when $$L = \Lambda$$, then $$\Lambda 𝚺^{∗} \Lambda = 𝚺^{∗}$$, but how can we prove that when $$L \neq \Lambda$$

Any idea

How to run a month-long chase in Star Wars D&D

I am running a D&D 5e Star Wars campaign. The characters received a prophecy that told them the only way to prevent galaxy-ending destruction was to kill the Sith Lord, and that it must be done within the month.

I would like to have it come down to the wire time-wise to help with an epic boss battle. However, the player’s are frustrated by seeming to run around in circles hunting the Sith and not getting anywhere.

How can I still make the chase take a month, but keep players invested and interested, as well as happy?

Propetries of klenne star operation

Help me proof(or give contexample) for this statements $$\quad K^* = (KK)^* \quad K^* = KK^* \quad K(LK)^* = (KL)^* \quad (K \cup L)^* = K^* \cup L^* \quad (K \cup L)^* = (K^*L^*)^* \quad (K \cup L)^* = (K^*L)^*K^* \quad$$

Where $$K^*$$ klenne star and $$K, L$$ language

[ Physics ] Open Question : Why do the Star Trek writers think that stopping engines will stop forward momentum in space?

Shouldn’t they at least know that they’d have to reverse thrusters to give the equal and opposite force to stop forward momentum? Shouldn’t someone writing for one of the premier scifi franchises know this basic fact? Verisimilitude is the hallmark of good fiction writing.

Can the motes of the Ring of Shooting Star’s “Shooting Star” effect be layered?

The Ring of Shooting Stars has the Shooting Star effect as described below:

Shooting Stars. You can expend 1 to 3 charges as an action. For every charge you expend, you launch a glowing mote of light from the ring at a point you can see within 60 feet of you. Each creature within a 15-foot cube originating from that point is showered in sparks and must make a DC15 Dexterity saving throw taking 5d4 fire damage on a failed save, or half as much damage on a successful one.

My question is can the motes be layered on top of each other to form a cube, wherein each creature must make 3 x DC15 Dexterity saving throws and be subject to 15d4 (modified by the saving throws) damage?

[ Astronomy & Space ] Open Question : In Star Trek: Voyager, they were sent to an uncharted region of space. How did they know how far from home they were or how to get home?

Wouldn’t they have needed a known star chart to plot their location and to also plot a valid route back home?