Consider n numbers x1, x2, . . . , xn laid out on a circle and...

Consider n numbers x1, x2, . . . , xn laid out on a circle and some value α. Consider the requirement that every number equals α times the sum of its two neighbors. For example, if α were zero, this would force all the numbers to be zero. (a) Show that, no matter what α is, the system has a solution. (b) Show that if α = 1 2 , then the system has a nontrivial solution. (c) Show that if α = − 1 2 , then there is a nontrivial solution if and only if n is even.

Expert Solution

Colby Messinger answered 2 years ago

Consider n numbers x1, x2, . . . , xn laid out on a circle and some value α.

Consider n numbers x1, x2, . . . , xn laid out on a circle and some value α. Consider the requirement that every number equals α times the sum of its two neighbors. For example, if α were zero, this would force all the numbers to be zero. (a) Show that, no matter what α is, the system has a solution. (b) Show that if α = 1/2 , then the system has a nontrivial solution. (c) Show...

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Let X = {x1,x2,...,xn} a sequence of real numbers. Design an algorithm that in linear time...

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