Problem 6, (a) Define what is meant by the weight of a word x in Z (b) Define what is meant by th...
Problem (1): Define the variable z as z -4.5; then evaluate (a) 0.4z'+3.12 -162.3z -80.7 (b) (z-23)/V2+17.5 Problem (2): The distance d from a point P(xp, yp,Zp) to the line that passes through two points 4(XA ,y,,2, ) and B(хв, Ув, zs) can be calculated by d-2S / r where r is the distance between the points A and B, given by r |(хв-%). (Yo-yA )4+ (Zg-2, )2 and S is the area of the triangle defined by the three...
Have to get an idea of how i am doing on this problem. Whould be nice to get a good explaination for each part of the problem. d1 and d2 is the two different metrics, p ,Y. Problem 2. Consider first the following definition: Definition. Let X be a set and let pand be two metrics on X. We say that p and are equivalent if the open balls in (X, p) and (x,y) are "nested". More precisely, p and...
Problem 11.21. For k є Z, we define Ak-{x є Z : x-51+ k for some 1 є z} (a) Prove that {Ak : k Z} partitions Z. (b) We denote by ~ the equivalence relation on Z that is obtained from the par- tition of part (a). Give as simple a description ofas possible; that is, given condition "C(x,y)" on x and y s x~y if and only if "C(x, y)" holds. Problem 11.21. For k є Z, we...
Problem 81 Find the point farthest from (1,3,-1) such that x2 + y2 + z2-11 and x-y+z < 3. What happens to the maximum distance if the 11 on the right side of the inequality is perturbed? 81. Suggestions (a) Take as objective the square of the distance from (x, y, z) to the point given (b) For the case of points inside the given sphere and with x-y+ z = 3, you might solve the Lagrange equations for x,...
Problem 5. Define a relation ~on R x R as (x, y) ~(a,b) if and only if either x-a or y- b. Prove or disproof, isan equivalence relation? If so, write down all the equivalence classes.
Solve the problem 6 Hint- Prob Q-[0.1] x [O, 1], A-{(z, yje Q : y z) and B-( (z, y) є Q : y2 z). Let also f be a real-valued integrable function on such that AfdV 4. lem 6. Let (i) If Jo/dV = 3 find fBfdV, and compute the value of JB(2f + 5)dV. Hint: use the Tesult of problem 5 (ii) If f > 0 on A and E c A such that Vol(A \ E) =...
a. Define what is meant by a representation of a ket state [4) and of an operator A. (10 marks) b. What is meant by coordinate space representation and momentum space representation? Define all quantities in your answer. (10 marks) c. The ground state wave function of the hydrogen atom is given by 1 *(r) = e-r/a πα3 What is the ground state wave function of the hydrogen atom in momentum space? (20 marks) Hint: Choose the z-axis along the...
12. Let g(x), h(y) and p(z) be functions and define f(x, y, z) = g(x)h(y)p(2). Let R= = {(x, y, z) E R3: a < x <b,c sy <d, eszsf} where a, b, c, d, e and f are constants. Prove the following result SS1, 5100,2)AV = L*()dx ["Mwdy ['Plzdz.
Define a relation < on Z by m <n iff |m| < |n| or (\m| = |n| 1 m <n) (a) Prove that < is a partial order on Z. (b) A partial order R on a set S is called a total order (or linear order) iff (Vx, Y ES)(x + y + ((x, y) E R V (y,x) E R)) Prove that is a total order on Z. (c) List the following elements in <-increasing order. –5, 2,...
Need only parts 5 and 6 Problem 6: 10 points Assume that X and Y are independent random variables uniformly distributed over the unit interval (0, 1) 1. Define Z = max (X, Y) as the larger of the two. Derive the CD. F. and density function for Z 2. Define W- min (X, Y) as the smaller of the two. Derive the C.D.F. and density function for W. 3. Derive the joint density of the pair (W, Z). Specify...