PEM Level C
21 November 2009
Topics in Geometry - Basic Concepts and Theorems
Examples: 1. (PMO 1984-85) Let the rectangle ABCD be such that AB = CD = 4 units and BC = AD = 3 units. A perpendicular is dropped to the diagonal BD from each of the vertices A
AMCS 241 Homework 5
Submitted by Mary Lai Salvaa
ID No. 156071
March 7, 2017
1. Find the MGF of a triangular distribution defined by
1
|x|
fX (x) =
1
, |x| < a
a
a
Solution:
MX (t) = E[etX ]
Z
|x|
1
dx
=
etx 1
a
a
|x|<a
Z
1
=
etx (a |x|)dx
a2 |x|<a
Z
Z
AMCS 313 Homework 5
Submitted by Mary Lai Salvaa
February 19, 2017
Deriving Exponential Variogram
V ar[Z(s1 ) Z(s2 )] = 2[C(0) C(h)]
h
h i
= 2 1 exp
a
= 2(h)
(
0 , h =n 0
o
(h) =
, h>0
c0 + c 1 exp ha
for some c > 0, c0 > 0, where c0 is the nugget effec
AMCS 313 Homework 1
Submitted by Mary Lai Salvaa
January 29, 2017
Effect of AR(1) Autocorrelation Structure on Classical Statistical Inference
Let Y1 , Y2 , , Yn N (Y , 2 ), Cov(Yi , Yj ) = 2 |ij| , i 6= j.
Y1 + Y2 , , Yn
V ar(Y ) = V ar
n
=
=
1
V ar(Y1 +
AMCS 241 Homework 4
Submitted by Mary Lai Salvaa
ID No. 156071
February 28, 2017
1. We say that the joint PDF fXY (x, y) of two random variables X and Y is circularly symmetric
if
p
fXY (x, y) = g(r) , where r = x2 + y 2
Show that if the random variables
AMCS 241 Homework 2
Submitted by Mary Lai Salvaa
ID No. 1560117
February 12, 2017
1. A sample space consists of the four points
= cfw_1 , 2 , 3 , 4
and the probabilities of the simple events are
1
1
1
1
P [cfw_1 ] = , P [cfw_2 ] = , P [cfw_3 ] = , P [cf
AMCS 241 Homework 3
Submitted by Mary Lai Salvaa
ID No. 1560117
February 21, 2017
1. Consider the Cauchy probability density function:
pY (y) =
d2
c
+ y2
a. Evaluate the ratio c/d.
Solution:
Let Y be a Cauchy random variable. The general form of the Cauch
AMCS 313 Homework 4
Submitted by Mary Lai Salvaa
February 14, 2017
mean: 0.3704
standard deviation: 1.0648
histogram
The distribution is skewed to the left and does not appear to be normally distributed.
qqplot
Spatial Statistics
Page 2
Towards the tail,
AMCS 313 Homework 7
Submitted by Mary Lai Salvaa
March 19, 2017
Optimization Problem
Quilt Plot of Data 3
data=read . csv ( " data3 . c s v " , h e a d e r=T)
x=rev ( data [ , 1 ] )
y=rev ( data [ , 2 ] )
z=rev ( data [ , 3 ] )
#1.0 QUILT PLOT OF DATA3
q
AMCS 313 Homework 6
Submitted by Mary Lai Salvaa
March 12, 2017
Variogram
#0.1 INSTALL PACKAGES
i n s t a l l . packages ( " f i e l d s " )
i n s t a l l . packages ( "geoR" )
#i n s t a l l . p a c k a g e s (" RandomFields ")
#i n s t a l l . p a c k a
AMCS 313 Homework 8
Submitted by Mary Lai Salvaa
April 18, 2017
Exercise 1
Figure 1: Left: Sea surface temperature data in degree Celsius observed on January 10, 2004 in the
southern Atlantic and Indian oceans off the coast of South Africa. As a map, the
AMCS 241: Homework 8
Assigned: Sunday Apr. 30
Due: Sunday May 7 in class (by 9:00am)
Instructions: Follow the guidelines for homework write-ups in the syllabus. If the TAs cannot understand /
read your solution, then you will not receive credit. You are r
worksheet 9:
factoring completely
Factor the following expressions completely. You may need to use more than one factoring technique.
1.
2.
3.
4.
4x2 16
t 3 + 3t2 + 4t + 12
4x3 15 + 20x2 3x
ax bx + by ay
5.
6.
(a b )(x + 5) (a b )(10 x)
a 2 (x y ) + 5 (y
Practice problems for sections on September 27th and 29th.
Here are some example problems about the product, fraction and chain rules for derivatives and implicit differentiation. If you notice any errors please let me know.
1. (easy) Find the equation of
AMCS 241 Homework 6
Submitted by Mary Lai Salvaa
ID No. 156071
March 28, 2017
1. Let cfw_Xn be a sequence of RVs that converges in probability to a random variable X; that is,
P
X
Xn
Assume that the PDFs fn (x) of Xn are such that, for some N > 0,
fn (x
RC CIRCUIT
OBJECTIVE: To study the charging and discharging process for a capacitor in a simple circuit
containing an ohmic resistance, R, and a capacitance, C.
THEORY: Consider the circuit shown below in Fig. 1:
a
S
b
Vo
R
C
FIGURE 1
1. Charging the capa
PEM Level C
9 January 2010
Topics in Geometry - All about circles
Exercises: 1. Prove that a line cannot cut a circle at more than two points. 2. Two circles intersect at two points. Prove that the length of the line segment passing through one point of i
PEM Level C
28 November 2009
Topics in Geometry - Chords, Tangents and Intersecting Circles
Examples: 1. In a circle, chord AB is bisected at E by chord CD. If CE = 16 and ED = 4, nd the length of chord AB . 2. An arch is built in the form of an arc of a
PEM Level C
5 December 2009
Topics in Geometry - Cyclic Quadrilaterals
Examples: 1. Theorem: A convex quadrilateral is cyclic if and only if two opposite angles are supplementary. 2. Triangle ABC is isosceles with AB = AC . Let D and E be points on the se
PEM Level C
12 December 2009
Topics in Geometry - The Power Theorems, Power of a Point
Theorems: 1. (Power Theorem) Let P be a given point, and let be a line that passes through P and intersecting a given circle at two (not necessarily distinct) points A
PEM Level C
9 January 2010
Topics in Geometry - Power of a Point and the Radical Axis
Theorems: 1. Let P be any point, and be a given circle. Let points A and B . be a line through P that intersects at
(a) If the P is inside , then P(P, ) = P A P B . (b)
PEM Level C
16 January 2010
Topics in Geometry: More on Triangles
Theorems: 1. The perpendicular bisectors of the sides of a triangle are concurrent. 2. (Brahmaguptas Theorem) The product of the lengths of two sides of a triangle is equal to the product o
PEM Level C
6 February 2010
Topics in Geometry: Menelaus Theorem
Theorems: 1. (Menelaus Theorem) Let ABC be a triangle, and let X , Y , and Z be the feet of some cevians from A, B , and C , respectively. If X , Y , and Z are collinear, then AZ BX CY = 1.
PEM Level C
26 February 2010
Geometric Inequality
(Triangle Inequality) Let A, B , and C be points on the plane. Then AB + BC AC ; where equality holds if and only if B lies on the segment AC . (Side-Angle Inequality) In Exercises: 1. Let a, b, and c be
PEM Level C
6 March 2010
Problem Solving
1. (IMO 1968) Prove that there is one and only one triangle whose lengths of its sides are consecutive integers, and one of whose angles is twice as large as another. 2. (IMO 1976) In a convex quadrilateral of area
PEM Level C
12 February 2010
Trigonometry: Law Cosines and Sines
Theorems: 1. (Law of Cosines) In ABC , the following equations hold: a2 = b2 + c2 2bc cos A b2 = a2 + c2 2ac cos B c2 = a2 + b2 2ab cos C 2. (Extended Law of Sines) Let R be the circumradius
PEM Level C
19 February 2010
Trigonometric Identities
Exercises: 1. Let ABC be an acute-angled triangle. Prove that tan A+tan B +tan C = tan A tan B tan C. 2. Let ABC be a triangle. Prove that sin2 A B C 3 + sin2 + sin2 . 2 2 2 4
3. Let ABC be a triangle
1st Sem, A.Y. 20162017
Physics 71.1
Verification of conservation of angular momentum
Table W5: Data for verification of conservation of angular momentum
Case
1
2
Ia (kgm2 )
Ib (kgm2 )
a,i (rad/s)
b,f (rad/s)
Li (kgradm2 /s)
Lf (kgradm2 /s)
% Error
Questio