Problem 1

It S is a positive variable (value greater than zero) and X is also a positive variable (value greater than zero) and the function is given by:

then for all positive values of S and X, the function will be maximized for X equal to:

 a) 0
 b) 1
 c) 2
 d) -1

Problem 2

A supercomputer which is calculating a problem with infinite amount of information, breaks down the problem into steps and solves the problem. Thus there will be infinite number of steps. However, each step of the computation requires half the time of the previous step. If an infinite amount of space (memory) is available and the first step takes one unit of time then the entire problem can be solved in:

a)   one unit of time
b)   two units of time
c)   sixteen units of time
d)   A large, but finite amount of time
e)   Infinite amount of time

Explain your answer in not more than 4-5 lines.

Problem 3

A time series is given by:

where the subscript refers to time and is constant. The above series will:

a) conveage to certain value
b) enter into a period motion
c) will significantly diverge for different initial conditions
d) all the above
e) none of the above

Problem 4

is a random variable and represents the change in . Then if averaged over enough number of small steps will be:

a) effectively deterministic
b) totally random;
c) chaotic
d) partially random

Problem 5

The Dirac delta function, , has a value of infinity at and value zero everywhere else. It is a unit impulse function. The integral of the Dirac delta function:

(a) will also be a unit step function;
(b) cannot be found;
(c) will be a continuous function everywhere
(d) None of the above;

Explain your answer.

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