Now that you have practiced the basic algorithm, let’s study the stable marriage problem a little bit quantitavely. here we define the following notation: on day j, let pj(m) be the rank of the woman that man m proposes to (where the first woman on his list has rank 1 and the last has rank n). also, let rj(w) be the total number of men that woman w has rejected up through day j − 1 (i. e. not including the proposals on day j). answer the following questions using the notation above.(a) prove or disprove the following claim: ∑m pj(m)−∑w rj(w) is independent of j. if it is true, also give the value of ∑m pj(m)−∑w rj(w). the notation, ∑m and ∑w , simply means that we are summing over all men and all women.(b) prove or disprove the following claim: one of the men or women must be matched to someone who is ranked in the top half of their preference list. you may assume that n is even.

this story was prepared by jim

mccullaugh in los angeles and

al stewart in new york.

new york the colossal success of "top gun" appears to have dramatically altered madison avenue's

perception of corporate sponsorship

of home video projects.

the paramount home video re- lease, featuring a 60- second diet

pepsi spot, was by no means the

first prerecorded video to carry an advertiser's plug, but it was the

first major theatrical release to do so. in the aftermath of the title's

success, paramount is widely ex- pected to ink a similar deal for

"crocodile dundee" or "star trek

iv" or both.

corporate america's new attraccds push british

recording sales

to 19% gain

london fueled by the compact

disk boom, the value of u.k. manu- facturer shipments of recordings

rose 19% in the 12 -month period ended march 1987, according to the

british phonographic industry.

total value of u.k. shipments for

the period was $755 million for all

configurations. (all value figures

are based on an exchange rate of

$1.68 to the pound sterling.)

cd sales totaled 10.7 million units

worth $121.3 million- compared

with 3.7 million units worth $39.5

million in the prior 12 months -and

accounted for two -thirds of the

overall gain in trade value. at present growth rates, says the bpi, cd

sales for calendar 1987 will reach 20

(continued on page 79)

step-by-step explanation: