Question

Let L be a lattice. Show that a∨(b∧c)≼(a∨b)∧(a∨b) (a∧b)∨(a∧c)≼a∧(b∨c) For all a,b,c ϵ L.

Answer #1

i Show that for any Lattice L and s, t ∈ L, s ∧ (s ∨ t) = s = s
∨ (s ∧ t).

Show that the cell form factor of the FCC lattice is:
If h, k, l are partly even or odd, the summation is 0.
If h, k, l are all even or all odd, summation is 4f.
As usual, we mean hkl referred to the cubic conventional
cell.
(Please explain all steps)

Show that the reciprocal lattice of a hexagonal lattice is a
hexagonal Lattice and show its orientation w.r.t. the direct
lattice.

Let [a],[b],[c] be a subset of Zn. Show that if [a]+[b]=[a]+[c],
then [b]=[c].

Let A, B, C be three sets. Show that A ∪ B = A ∩ C ⇐⇒ B ⊆ A ⊆
C.

Let A,B and C be sets, show(Prove) that (A-B)-C =
(A-C)-(B-C).

a. Show that the reciprocal lattice for a simple cubic
lattice is also a simple cubic lattice.
b. Find the separation between closest lattice planes
that have Miller indices (110)

Let a,b,c be integers with a + b = c. Show that if w is an
integer that divides any two of a, b, and c, then w will divide the
third.

Given some constant epsilon ϵ> 0. Show how to use algorithm
ALG to get a (1- ϵ)- approximation algorithm. How we can round
number exactly? Show that the algorithm runs in polynomial time and
also give a short intuitive argument why the value is atleast (1-
ϵ)times optimal.

Let (x_n) from(n = 1 to ∞) be a sequence in R. Show that x ∈ R
is an accumulation point of (x_n) from (n=1 to ∞) if and only if,
for each ϵ > 0, there are infinitely many n ∈ N
such that |x_n − x| < ϵ

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