Figure 2
Figure 3
From these curves we can see that axial strain and shear strain both have positive linear relationships with load, while
lateral strain has a negative linear relationship with load. This is understood because if we remember from previous
6. (4 pts) The function f (x, y, z) = (x4)2 +(y4)2 +(z 12 )2 has a minimum subject to the constraint x2 +y 2 z = 0.
Find the point (a, b, c) where f attains the minimum value subject to the constraint.
(you may assume the method of Lagrange multipliers gi
Math for Econ II, Written Assignment 4 (28 points)
1. (4 pts) The figures below show the optimal point (marked with a dot) in three optimization problems with the
same constraint. Arrange the corresponding values of from smallest to largest. Assume > 0 in
Fy = 0 = P - Ry => Ry = P
Mx = 0 = -P(LTz) + Tx => Tx = P(LTz)
Mz = 0 = -P(.1125m) + Mz => Mz = P(.1125m)
The statics calculations offer us with the moment about the z axis as the product of the load P and the distance .1125m
in the x direction along the
Actually working it out leads to:
1
2
w1 rR
+ w2 + rR
+ w3 = 0
which is basically the equation for a line. The line generated for a given left image coordinate, is the epipolar line
in the right image. The corresponding point must lie on the line, or in t
We solve f = g:
h24 2x, 16 4yi = h2x, 4yi.
Thus 24 2x = 2x and 16 4y = 4y. If either x = 0 or y = 0 then wed get 24 = 0 or 16 = 0, which is
nonsense. Thus, we may divide by x and y to solve for :
=
We cross multiply the equation
242x
2x
=
164y
4y
(24 2x)(
(a) Lines plotted with right image dimenstions.
(b) Lines plotted with expanded axes.
Figure 1: Epipolar lines for the right camera given points in the left camera. The points plotted that are left pointing
triangles are the pixel points for the left came
ECE4580 Solution #6
Problem 1. (25 pts) Using the left and right camera views from the stereo problem Homework 4 (the one with the
Matlab file stereo02.mat) and the known homogeneous matrix description of the right camera frame with respect
to the left ca
The word is by origin a diminutive of "jock", the Northern English or Scots colloquial
equivalent of the first name "John," which is also used generically for "boy, or fellow"
(compare "Jack", "Dick"), at least since 1529. A familiar instance of the use o
EtO OEt
OO
EtO OEt
OH O
diethyl malonate (diethyl propanedioate)
Ethyl acetoacetate is partly enolized under normal conditions. So, you might ask, why doesnt it
immediately react with itself by the aldol reaction? There are two aspects to the answer. Firs
base
Note that aza-enolates are formed from imines, which can be made only from primary amines.
Enamines are made from aldehydes or ketones with secondary amines.
USING SPECIFIC ENOL EQUIVALENTS TO ALKYLATE ALDEHYDES AND KETONES 593
The overall sequence
R
O
Me3Si
R Ph
OO
SiMe3
R Ph
O OH
work-up
silyl ether of aldol
aldol
PhCHO
TiCl4
H2O
The immediate product is actually the silyl ether of the aldol product but this is hydrolysed during workup and the aldol is formed in good yield. The Lewis acid presumab
Now, if a second carbonyl compound is added, it too complexes with the same lithium atom. This allows
the aldol reaction to take place by a cyclic mechanism in the coordination sphere of the lithium atom.
The aldol step itself is now a very favourable int
partner is a conjugated enal that is not enolizable but that might accept the nucleophile in a conjugate
manner. In spite of these potential problems, the reaction goes in excellent yield.
O OLi
H
O
O OH
then add the electrophile 82% yield of aldol
LDA, 7
HH
compare.
+
A general solution to using formaldehyde in aldol reactions is to use the Mannich reaction. A typical
example is shown in the margin: the reaction involves an enolizable aldehyde or ketone (here we use
cyclohexanone), a secondary amine (here
HO
HH
Ofirst formed aldol
second aldol
HO
H
OH O
HO
HO
third aldol
Even this is not all. A fourth molecule of formaldehyde reacts with hydroxide ion and then reduces the
third aldol. This reduction is known as the Cannizzaro reaction, and is described in
groups altogether. Most notable among these were the nitroalkanes. Deprotonation of nitro- alkanes is
not enolization nor is the product an enolate ion, but the whole thing is so similar to enolization that it
makes sense to consider them together. You sa