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1. (10 pts) Let f : (a.b) - R be bounded function and g : [a...

1. (10 pts) Let f : (a.b) - R be bounded function and g : [a, b) - IR is a function of bounded variation with majorants 91, 92 : (a.b) R, (i.e. g = 91 - 92) such that the Riemann-Stieltjes integrals and exist. Show that the Riemann-Stieltjes integral also exists and we have a 2. (10pts) Use th...

2. (a) (10 marke] (i) Provide a formal definition of a fin...

2. (a) (10 marke] (i) Provide a formal definition of a finite n-player game. (ii) In a given game, what is the difference between a mixed strategy and a pure strategy? (iii) What is a Nash equilibrium in a game? What is a mixed-strategy equilib- rium? (iv) For the following 2-player game, fin...

5. Verify the following order relations: a t2 tanht = O(t2...

5. Verify the following order relations: a t2 tanht = O(t2) as t 80. of / ( ( ( ) - €) = O(EVE) as e 0+. f) exp(e) - 1 = O(e) as E 0. i) In € III O(E-P) as E 0+ for all p &gt; 0. or (it + =Et. 8. Use the Poincaré-Lindstedt method to obtain a two-term perturb...

Equation numbers refer to items in the lecture notes More on...

Equation numbers refer to items in the lecture notes More on Monte Carlo: Variance Reduction Techniques. Q2 Find a simple example where using antithetic variates is not more efficient than using standard Monte Carlo. Q3 Show that Var(Ze) < Var(X) if and only if 0 lies between 0 and 20min, w...

G:=ni=opk(as) T: U{(0,0,0)} tER Proposition : The set T ...

G:=ni=opk(as) T: U{(0,0,0)} tER Proposition : The set T has the following properties 1) F is compact and invariant 2) There is an open set U containing r such that for each X E U, we have 0+ (x) E U for all t > 0, and Nt20 0+ t(U) = r. . 2) Show that there is a homeomorphism betwee...

4. Show that regular perturbation fails on the boundary val...

4. Show that regular perturbation fails on the boundary value problem Ey" + y' + y = 0, 0 < x < 1, , 0<<<<, y(0) = 0, y (1) = 1. Find the exact solution and sketch it for E = 0.05 and E = 0.005. Ifx= O(E), show that sy"(x) is large; if...

15. On which type of data is the Wilcoxon test used when eva...

15. On which type of data is the Wilcoxon test used when evaluating the difference between treatment conditions? a. A repeated-measures comparing two treatments b. A repeated-measures comparing more than two treatments c. Two independent treatments d. More than two independent treatments Comple...

1. (20pts) Let / : [0,1] - R be defined by f(x) = 0 sin if ...

1. (20pts) Let / : [0,1] - R be defined by f(x) = 0 sin if if a = x x E Determine whether f is of bounded variation of not. 2. (20pts) Let f, g : [a, b) R be bounded functions and g is increasing (non-decreasing). Show that for every partitions P and Q of [a, b] with P S Q. we have s(f, P, g) ...

2. Consider the initial value problem t&gt;0, u(0) ...

2. Consider the initial value problem t&gt;0, u(0) = 1, u'(0) = -1. Find a two-term perturbation approximation for 0 &lt; E " 1 and compare t graphically to a six-term Taylor series approximation (centered at t = 0) there € = 0.04. Use a numerical differential equa...

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