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1. a) Given a digraph G = (V,E), prove that if we add a constant k to the length of every arc coming out from the root node r, the shortest path tree remains the same. Do this by using potentials:
i) Show there is a potential y* for the new costs for which the paths in the tree to each node v have cost y*v, and
ii) explain why this proves it. What is the relationship between the shortest path distances of the modified problem and those of the original problem?
b) Can adding a constant k to the length of every arc coming out from a non-root node produce a change in the shortest path tree? Justify your answer.
Assume that (xn) is a sequence of real numbers and that a, b € R with a is not eaqual to 0. (a) If (x n ) converges to x, show that (|ax n + b|) converges to |ax + b|. (b) Give
L.H.S. =cos 12+cos 60+cos 84 =cos 12+(cos 84+cos 60) =cos 12+2.cos 72 . cos 12 =(1+2sin 18)cos 12 =(1+2.(√5 -1)/4)cos 12 =(1+.(√5 -1)/2)cos 12 =(√5 +1)/2.cos 12 R.H.S =c
Q. What is the probability of getting a Royal Flush? Ans. Five cards are picked from a standard deck of 52 cards. How many different hands of five cards are possible? What
let R be a (noncommutative) ring. Given that a,b and a+b ? R are all units, prove that a^(-1)+b^(-1) is a unit
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find the series of the first twenty terms
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solve: 4ydx+xdy=0
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Consider the task of identifying a 1 cm thick breast cancer that is embedded inside a 4.2 cm thick fibroglandular breast as depicted in Fig. The cancerous tumor has a cross
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