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辅导代写接单- MATH 589B due May 11, 2023 Algorithms of Applied Mathematics II

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MATH 589B due May 11, 2023 Algorithms of Applied Mathematics II

Section 001, Spring 2023 (instructor: Misha Stepanov)

Final Exam

1. For a function �� : D → R, its Legendre transform is defined by ��(��) = sup��∈D ?�� − �� (��)?.

Plot the Legendre transform of ��1(��) = 12 ln?exp(��2) + exp(5 + ��)? over the interval −4 ≤ �� ≤ 4.

2. Inside a [two-dimensional] circular chamber ��2 + ��2 ≤ 1 there are three disks of radii ��1 = 12, ��2 = 13, and ��3 = 41, with the same density. Minimize the gravitational potential energy ��2(��1,��1,��2,��2,��3,��3) = ��12��1 +��2��2 +��32��3 subject to the inequality con- straints��2+��2 ≤(1−��)2forall��=1,2,3;and(��−��)2+(��−��)2 ≥ (�� + ��)2 for all �� ≠ ��.

3. Alice and Bob play the “game of the goose” or “snakes and ladders” with the following board:

Alice/Bob are at the spaces 16/3, respectively. The players are making moves alternately, a player tosses a coin and with equal probabilities moves forward by 1 or 2 spaces. It is Bob’s turn. From the spaces 4/12/15/19 a player is automatically moved to 10/16/9/6, respectively. Estimate the probabilities of Alice or Bob winning (i.e., reaching the space 20 first).

4. Each cookie box independently of others contains one of 100 trading cards with equal prob- ability. Let �� be a random variable that is the number of boxes needed to obtain a full collection. Estimate the probability P(�� ≤ 200) that a full collection is obtained from 200 boxes or less.

5. Random variables ��1, ��2, ..., ��10 are having the joint probability density function 1?10 910 ?

��5(��1,��2,...,��10) = �� exp −Õ��2 −Õ Õ ��2 ��2 ��=1 ��=1 ��=��+1

where �� is an unknown normalization constant. Depict the distribution of �� = ��1 + ��2 + ... + ��10.

Minimizing means finding the position where the minimum is achieved, also report the optimal value. You can use built-in elementary functions, linear algebra and ODE functions/solvers. Don’t use built-in functions for optimization in problems 1 and 2. You can use built-in random number generator for Unif (0, 1) only.