Let n be a positive integer. Let S(n) = n sigma j=1 ((1/3j − 2)
− (1/3j + 1)). a) Compute the value of S(1), S(2), S(3), and S(4).
b) Make a conjecture that gives a closed form (i.e., not a
summation) formula for the value of S(n). c) Use induction to prove
your conjecture is correct.
Let J be the antipodal of A in the circumcircle of triangle ABC.
Let M be the midpoint of side BC. Let H be the orthocenter of
triangle ABC. Prove that H, M, and J are collinear.
Let J be the antipodal of A in the circumcircle of triangle ABC.
Let M be the midpoint of side BC. Let H be the orthocenter of
triangle ABC. Prove that H, M, and J are collinear.
Let F= (x2 +
y + 2 + z2) i + (exp(
x2 ) + y2)
j + (3 + x) k . Let a
> 0 and let S be part of the spherical
surface x2 + y2 +
z2 = 2az + 15a2
that is above the x-y plane and the disk formed in the
x-y plane by the circular intersection between the sphere
and the plane. Find the flux of F outward across
S.
Angular momentum: Let the Hamiltonian of our particle be
described as H = ??^2 where J is the angular momentum operator. a)
Diagonalise the Hamiltonian for j=0,1
b) Use explicit form of J+ and J- rising and lowering operators
and Jz operator to obtain the 4 lowest energy levels for j=0,1
.
Let L = {aibj | i ≠ j; i, j ≥ 0}.
Design a CFG and a PDA for this language. Provide a direct
design for both CFG and PDA (no conversions from one form to
another allowed).
. Let xj , j = 1, . . . n be n distinct values. Let yj be any n
values. Let p(x) = c1 + c2x + c3x 2 + · · · + cn x ^n−1 be the
unique polynomial that interpolates the data (xj , yj ), j = 1, . .
. , n (Vandermonde approach).
(a) Remember that (xj , yj ), j = 1, . . . , n are given. Derive
the n...
3. Let S3 act on the set A={(i,j) : 1≤i,j≤3} by σ((i, j)) =
(σ(i), σ(j)).
(a) Describe the orbits of this action.
(b) Show this is a faithful action, i.e. that the permutation
represen- tation φ:S3 →SA =S9
(c) For each σ ∈ S3, find the cycle decomposition of φ(σ) in
S9.
Let X = {1, 2, 3, 4, 5, 6} and let ∼ be given by {(1, 1),(2,
2),(3, 3),(4, 4),(5, 5),(6, 6),(1, 3),(1, 5),(2, 4),(3, 1),(3, 5),
(4, 2),(5, 1),(5, 3)}.
Is ∼ an equivalence relation? If yes, write down X/ ∼ .