Consider the language L = {w |w ∈ {0, 1} ∗ , 2 × #0 (w) = #1 (w)} , here #0(w) and #1(w) means the numbers of 0s and 1s in w respectively. That is, this is the language consisting of all strings with exactly twice as many 1s as 0s: for example, 101 ∈ L, 010 ∈/ L. In this problem we will show how to construct a context-free grammar for it. (a) (2 points) For a length n word x, we define a function f (i) = (2 × number of 0s in locations [1 . . . i]) − (number of 1s in locations [1 . . . i]). Show that we have f(i) = f(j) for some i < j if and only if the substring in indices [i + 1, . . . j] has exactly twice as many 1s as 0s (i. e. xi+1,i+2,...,j ∈ L). (b) (2 points) Show that if x ∈ L starts with 0, then it can be written as x = 0w1y1z where w, y, and z are each strings with twice as many 1s as 0s (aka. w, y, z ∈ L).

Which of these options are the correct sequence of actions for content to be copied and pasted? select content, click the copy button, click the paste button, and move the insertion point to where the content needs to be inserted. click the copy button, select the content, move the insertion point to where the content needs to be inserted, and click the paste button. select the content, click the copy button, move the insertion point to where the content needs to be inserted, and click the paste button. select the content, move the insertion point to where the content needs to be inserted, click the copy button, and click the paste button.

Why is hard disk space important to an audio engineer? why are usb ports and firewire ports useful for an audio engineer? explain in 2-3 sentences. (3.0 points) here's a list of different audio software: ableton live apple inc.'s garageband apple inc.'s logic studio digidesign's pro tools propellerhead sofware's reason sony creative software's acid pro steinberg cubase steinberg nuendo choose one of the software programs listed above, and then go to that software program's web site. read about what the software program is used for, and then write 4-5 sentences about what you learned. (10.0 points) which type of software license is the most limiting? why? explain in 2-3 sentences. (3.0 points) when sending a midi channel voice message, how can you control the volume of the sound? explain in 2-3 sentences. (4.0 points)

How are roger williams, james oglethorpe, and william penn similar?