QUESTION 12 6 points Integrate the function Fi,y) + over the rectangle 0<x<1.0 Sy<1. OA. (1 + ج - قم -5 CB. (1- د - أم - CC, (1+نم - ام - کی O D. 1 (1- م ام که E. | شی ب ام و
Suppose that a; b; c 2 R with a 6= 0 and b2 ?? 4ac < 0, so
that
r(x) = ax2 + bx + c
is an irreducible quadratic polynomial. Prove that
R[x]=r(x)R[x] =
C :
[Hint: use the Fundamental Homomorphism Theorem. You may assume
with-
out proof that an appropriate evaluation map is a ring
homomorphism.]
Suppose that a,b,cE R with a?0 and b2-4ac ? 0, so that ba c is an irreducible quadratic polynomial. Prove that Hint:...
13 please
8. b. -2 3 0 0 0 0 -1 2 0 0-4 0 3 0-2 0 3 0 0 -2 0 3 0 4 o0-1 6 0 0 1 o 2 6 0 0 -1 6 10. For any positive integer k, prove that det(4t) - de(A)*. 11. Prove that if A is invertible, then den(A-1)- I/der(A) - det(4)- 12. We know in general that A-B丰B-A for two n x n matrices. However, prove that: det(A . B)-det(B...
This Question is Numerical Analysis. Please give full proof.
2. Suppose {$0(2), 01(2),..., n(x)} is an orthogonal set of functions with respect to the L2 inner product, i.e. (, = *$3 ()bu(a)dx = 0, if j tk. Prove the Pythagorean theorem ||do + + + . . ||? = ||do|l2 + ||ói || + || 6 ||º, where || | ||2 = (f, f).
E={a,b,c,d}, L = {anbmchdm : n, m 2 0}. For example, s = aabccd e L because the symbols are in Unicode order, and #a(s) = #c(s), #(s) #c(s), #b(s) = #a(s); ac e L for the same reason; s = abcdd & L because #b(s) #a(s); and acbd & L beause the symbols are not in Unicode order. Prove that L & CFLs using the CF pumping theorem, starting by defining w such we Land |w| 2 k. Remember...
3. Let t be the co-ordinate on A (C) and let z, y be the co-ordinates on A2(C). Let f 4z? + 6xy + x-2y® E C[x, y] and let C be the curve C-V((f)) C A2(C) (You may assume without proof that f is an irreducible polynomial, therefore C is irreducible and I(C)- (f).) (a) Show that yo(t) = (2t3, 2t2 + t) defines a morphism p : A1 (C) → C. [3 marks] (b) Show that (z. У)...
a tinctlon of series y I Taylor The 6. Taylor's Remainder Theorem. fn)(0) where fw) is the n-th derivative of f, and the remainder term Ry is given by NN+1 for some point c between 0 and z. (Note. You do not need to prove Taylor's Remainder Theorem.) Problems (a) (5%) write this series for the function ez for a general N (b) (10%) Apply Taylor's Remainder Theorem to show that the Taylor series of function f = ez converges...
2. A binary string is a finite sequence v = a1a2 . . . an, where each ai is either 0 or 1. In this case n is the length of the string v. The strings a1, a1a2, . . . , a1 . . . an−1, a1 . . . an are all prefixes of v. On the set X of all binary strings consider the relations R1 and R2 defined as follows: R1 = {(w, v) | w...
(a) Suppose that f'(; y) = 0 for all x in a certain n-ball B(a) and for all vector y. Use the mean value theorem to show that it is constant in B(a). (6) Suppose that fry= 0 for a smooth vector y and all r in B(a). What can be said of I in this case?
Let y,p ~iid Exp (0), for i = 1, . . . , n. (p(y|0) for 6 to be Gamma(a, b), tha distribution of θ BeAy). Assume the prior distribution Find the posterior 2. t is, p(0) -ba/ra)ge-i exp{-be. 3. Find the posterior predictive distribution of a future observation in problem 2