1)We wish to measure a wavefront u(x,y) on a 3X3 grid (see the attached file). The sensor how ever provides us only with differences between grid points. There is sensor noise, so the difference measurements may not be consistent.

Number the grid points linearly and set up a set of linear equations (matrix form) relating the values of u to the values of du. Use least squares to get the best estimate of u(x,y). First use the exact values of du then use du with error, compare. Check by summing du's along a set of defined paths. Does least squares improve accuracy when sensor noise is included?

2)Often we are not interested in the component of u of the form L(x,y)=Ax+By+C
use least squares to find A,B,C wich provide a best fit to the u(x,y) determined from exact data.
Form u(x,y)-L(x,y) graph if possible
^u=exact value-/+error
^u is delta u

1)We wish to measure a wavefront u(x,y) on a 3X3 grid (see attached file) The sensor however provides us only with differences between grid points. There is sensor noise, so the difference measurments may not be consistent. Number the grid points linearly , and set up a set of linear equations (matrix form) relating the values of u to the values of du. Use least squares to get the best estimate of u(x,y).
First use exact value of du then use du with error, compare.
Check by summing du's along a set of defined paths.
Does least squares improve accuracy when sensor noise is included?
2)(same problem continued)Often we are not interested
in the component of u of the form L(x,y)=Ax+By+C
use least squares to find A,B,C which provide the best fit to the u(x,y) determined from the exact data.
Form u(x,y)-L(x,y), graph if possible
^u=exact value +/- error
::^u means delta u app. value of u. ...

Solution Summary

This shows how to use least squares in measuring wavefront. The number of grid points are given.

The variation attributable to factors other than the relationship between the independent variables and the explained variable in a regression analysis is represented by:
a. regression sum of squares.
b. error sum of squares.
c. total sum of squares.
d. regression mean squares.

A checkered flag used for racing is a square flag containing 64 alternating white and black squares. How many squares on the checkered flag contain an equal number of white and black squares? Be sure to describe how you arrived at your answer.

The method of leastsquares is used on time series data for
a. Eliminating the irregular movements
b. de-seasonalizing the data
c. obtaining the trend equation
d. exponentially smoothing a series

The sum of the squares of two consecutive integers is 4513. What are the integers?
English Language Mathematical Language
The two integers ?
The integers are consecutive ?
The sum of their squares is 4513

See full problem description in attached.
A company that manufactures computer chips wants to use a multiple regression model to study the effect that the variables
x1 = daily production volume
x2 = daily amount of time involved in production
have on
y = total daily production cost
If a regression model is es

Using mathematical induction, prove or disprove that all checkerboards of these shapes can be completely covered using right triominoes whenever n is a positive integer.
a) 3 x 2^n
b) 6 x 2^n
c) 3^n x 3^n
d) 6^n x 6^n

The application of the least-squares procedure to a multiple linear regression equation requires that:
a no exact linear relationships can exist among any of the independent variables
b the number of observations (n) must exceed the number of b parameters to be estimated (m)
c the number of observations (n) mus

9x4 + 4x3 - 27x2 + 12x <------ Factor by finding the GCF
x2 + 2x - 15 <--------Factor by grouping
For these problems, identify which of the methods from this lesson (GCF, grouping, difference of squares, or perfect squares) could be used to factor the polynomial.
17. x2 + 2x + 1
A This polynomial could b

Consider the following grid:
1) How many squares (of all sizes) are there in this grid?
2) How many rectangles (of all sizes) are there in this grid?
3) Can you generalize your results to an n by n grid? Can you generalize further?
Please see attachment for grid.

"The linear regression line is sometimes called the leastsquares line. Why?"
What is the idea of "leastsquares"?
What is the connection between "leastsquares" and linear regression?
Could "leastsquares" and regression be generalized to more complicated cases than lines?