Question

Show that the relation ∼ deﬁned as (x_{1},y_{1})
∼ (x_{2},y_{2}) ⇔ (x_{1})
−(y_{1})^{2} = （x_{2}
）−（y_{2}）^{2} is an equivalence relation on R2.
Sketch equivalence classes of (0,1) and (1,1) Find a representative
for each distinct equivalence class.

Answer #1

Write vectors in R2 as (x,y). Define the relation on R2 by
writing (x1,y1) ∼ (x2,y2) iff y1 − sin x1 = y2 − sin x2 . Prove
that ∼ is an equivalence relation.
Find the classes [(0, 0)], [(2, π/2)] and draw them on the
plane. Describe the sets which are the equivalence classes for this
relation.

Let V=R2 with the standard scalar multiplication and nonstandard
addition given as follows: (x1, y1)⊕(x2, y2) := (x1x2, y1+y2). Show
that (V,⊕, .) is not a vector space.

(a) Show that the parametric equations
x = x1 +
(x2 −
x1)t, y
= y1 +
(y2 −
y1)t
where
0 ≤ t ≤ 1,
describe (in words) the line segment that joins the points
P1(x1,
y1)
and
P2(x2,
y2).
(b) Find parametric equations to represent the line segment
from
(−1, 6) to (1, −2).
(Enter your answer as a comma-separated list of equations. Let
x and y be in terms of t.)

Let V = W = R2. Choose the basis B = {x1, x2} of V , where x1 =
(2, 3), x2 = (4,−5) and choose the basis D = {y1,y2} of W, where y1
= (1,1), y2 = (−3,4). Find the matrix of the identity linear
mapping I : V → W with respect to these bases.

1. Assume 2 particles
L =
.5m1(x1^2+y1^2+z1^2)+.5m2(x2^2+y2^2+z2^2)-V(x1-x2)
a. Define momentum and show that it's conserved
b. Show that Newton's Third Law holds
2. Find the equation to represent the difference in ages of the
two twins in the twin paradox.
3.V,U in x direction. Galilean addition Vt=V+U. Using
Lorentz Transforms prove
Vvel=(V+U)/(1+(UV/C2)

Consider the following equations:
y1 = a1y2 +a2y3 +x1 +x2 +e1 (1) y2 = by1+2x3+x1+e2 (2) y3 =cy1+e3
(3)
Here a1, a2, b, c are unknown parameters of interest, which are all
posi- tive. x1, x2, x3 are exogenous variables (uncorrelated with
y1, y2 or y3). e1, e2, e3 are error terms.
(a) In equation (1), why y2,y3 are endogenous?
(b) what is (are) the instrumental variable(s) for y2, y3 in
equation (1)?
(no need to explain why)
(c) In...

1.suppose that Y1 and Y2 are independent random variables
2.suppose that Y1 and Y2each have a mean of A and a variance of
B
3.suppose X1 and X2 are related to Y1 and Y2 in the following
way:
X1=C/D x Y1
X2= CY1+DY2
4.suppose A, B, C, and D are constants
What is the expected value of X1?
What is the expected value of X2?
What is the variance of X1?

1.suppose that Y1 and Y2 are independent random variables
2.suppose that Y1 and Y2each have a mean of A and a variance of
B
3.suppose X1 and X2 are related to Y1 and Y2 in the following
way:
X1=C/D x Y1
X2= CY1+CY2
4.suppose A, B, C, and D are constants
What is the expected value of the expected value of X1 given
X2{E [E (X1 | X2)]}?
What is the expected value of the expected value of X2 given...

function [Ax,Ay,Az]=GravityAcc(M1,X1,Y1,Z1,X2,Y2,Z2)
%We are calculating the acceleration for object 2
G=6.67408*1e-11;
R=?
Am=G*M1/(R^2);
Ax=(X1-X2)/R*Am;
Ay=(Y1-Y2)/R*Am;
Az=(Z1-Z2)/R*Am;
end
What is R in this gravitational acceleration code?

Suppose that X1, X2, . . . , Xn are independent identically
distributed random
variables with variance σ2. Let Y1 = X2 +X3 , Y2 = X1 +X3 and
Y3 = X1 + X2. Find the following : (in terms of σ2)
(a) Var(Y1)
(b) cov(Y1 , Y2 )
(c) cov(X1 , Y1 )
(d) Var[(Y1 + Y2 + Y3)/2]

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