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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.
Proof of: lim q →0 sin q / q = 1 This proofs of given limit uses the Squeeze Theorem. Though, getting things set up to utilize the Squeeze Theorem can be a somewha
1. A drainage system overflows into a river causing pollution when the rainfall falling in an hour exceeds 50 mm. During any given year, the maximum hourly rainfall follows an expo
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The base of an isosceles triangle and the altitude drawn from one of the congruent sides are equal to 18cm and 15cm, respectively. Find the lengths of the sides of the triangle.
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Illustration 1 In a described exam the scores for 10 students were given as: Student Mark (x) |x-x¯| A 60
Variance Consider the example of investment opportunities. The expected gains were Rs.114 and Rs.81 respectively. The fact is that an investor also looks at the dispersion befo
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