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Problem 5 (Counting triominos) [20 marks/ We saw in class that every 2n x 2 board, with one squar...
Problem 5 (Counting triominos) [20 marks] We saw in class that every 2 n × 2 n board, with one sq...
Problem 5 (Counting triominos) [20 marks] We saw in class that every 2 n × 2 n board, with one square removed, could be covered with triominos. Determine a formula counting the number of triominos covering such a truncated 2 n × 2 n board. Prove this formula by induction. I have seen solutions for this question posted but they seem to use iteration rather than induction and use notation that I don't understand.
Every2x2" board, with one square removed, could be covered with triominos Determine a formula cou...
Every2x2" board, with one square removed, could be covered with triominos Determine a formula counting the number of triominos covering such a truncated 2% 2"board. Prove this formula by induction
Every2x2" board, with one square removed, could be covered with triominos Determine a formula counting the number of triominos covering such a truncated 2% 2"board. Prove this formula by induction
Question 2 Consider the differential equation We saw in class that one solution is the Bessel function (a) Suppose we h...
Question 2 Consider the differential equation We saw in class that one solution is the Bessel function (a) Suppose we have a solution to this ODE in the form y-Σχ0CnXntr where cn 0. By considering the first term of this series show that r must satisfy r2-4-0 (and hence that r = 2 or r =-2) (b) Show that any solution of the form y-ca:0G,2n-2 must satisfy C0 (c) From the theory about singular solutions we know that a linearly...
5. In class we saw that the function r(u, v) = (sin u, (2 + cos...
5. In class we saw that the function r(u, v) = (sin u, (2 + cos u) cos v, (2 + cos u) sin v), 0<u<27, 050521 parametrizes a torus T, which is depicted below. (a) Calculate ||ru x rull. (b) Show that T is smooth. (c) Find the equation of the tangent plane to T at (0,). (d) Find the surface area of T (e) Earlier in the semester, we observed that a torus can be built out of...
Question #1 (15 Marks) a) (8 Marks) Answer the following questions with True or False. 1) 2) 3) Every basic solution in the assignment problem is necessarily degenerate. The assignment problem ca...
Question #1 (15 Marks) a) (8 Marks) Answer the following questions with True or False. 1) 2) 3) Every basic solution in the assignment problem is necessarily degenerate. The assignment problem cannot be solved using the transportation technique. If the gradient vector of a function at a given point is zero, the point can only be a maximum or minimum. If a single-variable function has two local minima, it must have at least one local 4) maximum 5) The Golden...
Question 4 [35 marks in totalj An n x n matrix A is called a stochastic...
Question 4 [35 marks in totalj An n x n matrix A is called a stochastic matrix if it! satisfies two conditions: (i) all entries of A are non-negative; and (ii) the sum of entries in each column is one. If the (,) entry of A is denoted by any for ij € {1, 2,...,n}, then A is a stochastic matrix when alij 20 for all i and j and I j = 1 for all j. These matrices are...
Concepts: multi-dimension array and the Knight's Tour Problem A chess board consists of an 8 x 8 "array" of squares: int board[ROW][COL]={0}; A knight may move perpendicular to the edges o...
Concepts: multi-dimension array and the Knight's Tour Problem A chess board consists of an 8 x 8 "array" of squares: int board[ROW][COL]={0}; A knight may move perpendicular to the edges of the board, two squares in any of the 4 directions, then one square at right angles to the two-square move. The following lists all 8 possible moves a knight could make from board [3][3]: board[5][4] or board[5][2] or board[4][5] or board[4][1] or board[1][4] or board[1][2] or board[2][5] or board[2][1]...
Can someone help me solving this? B6. A study is conducted in a class of 360 students investigating the problem of screen cracking in mobile phones. We assume that a) the number of screen cracking eve...
Can someone help me solving
this?
B6. A study is conducted in a class of 360 students investigating the problem of screen cracking in mobile phones. We assume that a) the number of screen cracking events follows a Poisson distributiorn and that b) the expected rate of screen cracking is 1 in 3 per phone per year. i) Write down the formula for the probability mass function of a Poisson random variable withh 3 marks parameter X, stating also the...
please solve only part(a) and part(b) Problem 7. A 2 × n clockerboard is to be tiled using three types of tiles. The fir...
please solve only part(a) and
part(b)
Problem 7. A 2 × n clockerboard is to be tiled using three types of tiles. The first tile is a white 1 x 1 square tile. The second tile is a red 2 × 2 tile and the third one is a black 2 x 2 tile. Let t(n) denote the number of tilings of the 2 × n checkerboard using white red and black tiles. (a) Find a recursive formula for t(n)...
and Y ~ Geometric - 4 Let X ~ Geometric We assume that the random variables X and Y are statistically independent. Answer the following questions: a (3 marks) For all x E 10,1,2,...^, show that 2+1 P...
and Y ~ Geometric - 4 Let X ~ Geometric We assume that the random variables X and Y are statistically independent. Answer the following questions: a (3 marks) For all x E 10,1,2,...^, show that 2+1 P(X>x) P(x (3 = Similarly, for all y [0,1,2,...^, show that Show your working only for one of the two identities that are pre- sented above. Hint: You may use the following identity without proving it. For any non-negative integer (, we have:...
Every2x2" board, with one square removed, could be covered with triominos Determine a formula counting the number of triominos covering such a truncated 2% 2"board. Prove this formula by induction
Every2x2" board, with one square removed, could be covered with triominos Determine a formula counting the number of triominos covering such a truncated 2% 2"board. Prove this formula by induction
Question 2 Consider the differential equation We saw in class that one solution is the Bessel function (a) Suppose we have a solution to this ODE in the form y-Σχ0CnXntr where cn 0. By considering the first term of this series show that r must satisfy r2-4-0 (and hence that r = 2 or r =-2) (b) Show that any solution of the form y-ca:0G,2n-2 must satisfy C0 (c) From the theory about singular solutions we know that a linearly...
5. In class we saw that the function r(u, v) = (sin u, (2 + cos u) cos v, (2 + cos u) sin v), 0<u<27, 050521 parametrizes a torus T, which is depicted below. (a) Calculate ||ru x rull. (b) Show that T is smooth. (c) Find the equation of the tangent plane to T at (0,). (d) Find the surface area of T (e) Earlier in the semester, we observed that a torus can be built out of...
Question #1 (15 Marks) a) (8 Marks) Answer the following questions with True or False. 1) 2) 3) Every basic solution in the assignment problem is necessarily degenerate. The assignment problem cannot be solved using the transportation technique. If the gradient vector of a function at a given point is zero, the point can only be a maximum or minimum. If a single-variable function has two local minima, it must have at least one local 4) maximum 5) The Golden...
Question 4 [35 marks in totalj An n x n matrix A is called a stochastic matrix if it! satisfies two conditions: (i) all entries of A are non-negative; and (ii) the sum of entries in each column is one. If the (,) entry of A is denoted by any for ij € {1, 2,...,n}, then A is a stochastic matrix when alij 20 for all i and j and I j = 1 for all j. These matrices are...
Can someone help me solving
this?
B6. A study is conducted in a class of 360 students investigating the problem of screen cracking in mobile phones. We assume that a) the number of screen cracking events follows a Poisson distributiorn and that b) the expected rate of screen cracking is 1 in 3 per phone per year. i) Write down the formula for the probability mass function of a Poisson random variable withh 3 marks parameter X, stating also the...
please solve only part(a) and
part(b)
Problem 7. A 2 × n clockerboard is to be tiled using three types of tiles. The first tile is a white 1 x 1 square tile. The second tile is a red 2 × 2 tile and the third one is a black 2 x 2 tile. Let t(n) denote the number of tilings of the 2 × n checkerboard using white red and black tiles. (a) Find a recursive formula for t(n)...
and Y ~ Geometric - 4 Let X ~ Geometric We assume that the random variables X and Y are statistically independent. Answer the following questions: a (3 marks) For all x E 10,1,2,...^, show that 2+1 P(X>x) P(x (3 = Similarly, for all y [0,1,2,...^, show that Show your working only for one of the two identities that are pre- sented above. Hint: You may use the following identity without proving it. For any non-negative integer (, we have:...
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