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).

Ap practice - performance task response the ap create performance task asks you to write about an abstraction that you developed and wrote into your code. most of the time that means identifying a function or procedure you wrote to "manage complexity" in your program. here is the actual prompt from the create performance task: 2d. capture and paste a program code segment that contains an abstraction you developed individually on your own (marked with a rectangle). this abstraction must integrate mathematical and logical concepts. explain how your abstraction manage the complexity of your program. (must not exceed 200 words) below is a segment of code from an "under the sea" program with a rectangle drawn around a portion of the code identifying an abstraction. imagine that you wrote this and are composing an ap response about how this abstraction manages complexity. (note: ignore the requirement that the abstraction integrate "mathematical and logical concepts" for this practice response. just write about managing complexity). explain how the abstraction marked with the rectangle in the code above manage complexity of this program. write your response here submit

Consider a slotted aloha system, where the time slot equals the fixed duration of each packet. assume that there are 4 stations a, b,c, d sharing the medium. (a) stations a, b,c, d receive one packet each from higher layers at times 1.3, 1.5, 2.6,5.7 respectively. show which transmissions take place when, according to the slottedaloha protocol; describe all transmissions until all four packets have been successful. when needed, each station has access to the following sequence of random number, provided by a random number generator and drawn uniformly between 0 and 1: (1) station a draws numbers: 0.31, 0.27, 0.78, 0.9, 0.9, 0.11, 0. (2) station b draws numbers: 0.45, 0.28, 0.11, 0.83, 0.37, 0.22, 0. (3)station c draws numbers: 0.1, 0.2, 0.3, 0.4, 0. (4) station d draws numbers: 0.36, 0.77, 0.9, 0.1, 0.1, 0.1, 0.1, 0. (b) in slotted aloha, a station transmits in each time slot with a given probability. what probabilities would you assign to each of the four stations so as to: (i) maximize the efficiency of the protocol? (ii) maximize fairness among the four stations? (c) will the efficiency increase or decrease if we modify slotted aloha as follows: (i) get rid of slots and allow stations to transmit immediately? (ii) implement carrier sensing? (iii) implement collision detection? (iv) implement collision avoidance?

Laire writes a letter to her grandmother, in which she describes an amusement park she visited last week. she adds pictures of that place in her letter. which feature of a word processing program will claire to remove unwanted parts of the pictures?